Metamath Proof Explorer |
This is the Unicode version. Change to GIF version |
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... |
syl5bi 241 | A mixed syllogism inferenc... |
syl5bir 242 | A mixed syllogism inferenc... |
syl5ib 243 | A mixed syllogism inferenc... |
syl5ibcom 244 | A mixed syllogism inferenc... |
syl5ibr 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... |
syl6ib 250 | A mixed syllogism inferenc... |
syl6ibr 251 | A mixed syllogism inferenc... |
syl6bi 252 | A mixed syllogism inferenc... |
syl6bir 253 | A mixed syllogism inferenc... |
syl7bi 254 | A mixed syllogism inferenc... |
syl8ib 255 | A syllogism rule of infere... |
mpbird 256 | A deduction from a bicondi... |
mpbiri 257 | An inference from a nested... |
sylibrd 258 | A syllogism deduction. (C... |
sylbird 259 | A syllogism deduction. (C... |
biid 260 | Principle of identity for ... |
biidd 261 | Principle of identity with... |
pm5.1im 262 | Two propositions are equiv... |
2th 263 | Two truths are equivalent.... |
2thd 264 | Two truths are equivalent.... |
monothetic 265 | Two self-implications (see... |
ibi 266 | Inference that converts a ... |
ibir 267 | Inference that converts a ... |
ibd 268 | Deduction that converts a ... |
pm5.74 269 | Distribution of implicatio... |
pm5.74i 270 | Distribution of implicatio... |
pm5.74ri 271 | Distribution of implicatio... |
pm5.74d 272 | Distribution of implicatio... |
pm5.74rd 273 | Distribution of implicatio... |
bitri 274 | An inference from transiti... |
bitr2i 275 | An inference from transiti... |
bitr3i 276 | An inference from transiti... |
bitr4i 277 | An inference from transiti... |
bitrd 278 | Deduction form of ~ bitri ... |
bitr2d 279 | Deduction form of ~ bitr2i... |
bitr3d 280 | Deduction form of ~ bitr3i... |
bitr4d 281 | Deduction form of ~ bitr4i... |
bitrid 282 | A syllogism inference from... |
syl5bb 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... |
jcndOLD 337 | Obsolete version of ~ jcnd... |
bibi2i 338 | Inference adding a bicondi... |
bibi1i 339 | Inference adding a bicondi... |
bibi12i 340 | The equivalence of two equ... |
imbi2d 341 | Deduction adding an antece... |
imbi1d 342 | Deduction adding a consequ... |
bibi2d 343 | Deduction adding a bicondi... |
bibi1d 344 | Deduction adding a bicondi... |
imbi12d 345 | Deduction joining two equi... |
bibi12d 346 | Deduction joining two equi... |
imbi12 347 | Closed form of ~ imbi12i .... |
imbi1 348 | Theorem *4.84 of [Whitehea... |
imbi2 349 | Theorem *4.85 of [Whitehea... |
imbi1i 350 | Introduce a consequent to ... |
imbi12i 351 | Join two logical equivalen... |
bibi1 352 | Theorem *4.86 of [Whitehea... |
bitr3 353 | Closed nested implication ... |
con2bi 354 | Contraposition. Theorem *... |
con2bid 355 | A contraposition deduction... |
con1bid 356 | A contraposition deduction... |
con1bii 357 | A contraposition inference... |
con2bii 358 | A contraposition inference... |
con1b 359 | Contraposition. Bidirecti... |
con2b 360 | Contraposition. Bidirecti... |
biimt 361 | A wff is equivalent to its... |
pm5.5 362 | Theorem *5.5 of [Whitehead... |
a1bi 363 | Inference introducing a th... |
mt2bi 364 | A false consequent falsifi... |
mtt 365 | Modus-tollens-like theorem... |
imnot 366 | If a proposition is false,... |
pm5.501 367 | Theorem *5.501 of [Whitehe... |
ibib 368 | Implication in terms of im... |
ibibr 369 | Implication in terms of im... |
tbt 370 | A wff is equivalent to its... |
nbn2 371 | The negation of a wff is e... |
bibif 372 | Transfer negation via an e... |
nbn 373 | The negation of a wff is e... |
nbn3 374 | Transfer falsehood via equ... |
pm5.21im 375 | Two propositions are equiv... |
2false 376 | Two falsehoods are equival... |
2falsed 377 | Two falsehoods are equival... |
2falsedOLD 378 | Obsolete version of ~ 2fal... |
pm5.21ni 379 | Two propositions implying ... |
pm5.21nii 380 | Eliminate an antecedent im... |
pm5.21ndd 381 | Eliminate an antecedent im... |
bija 382 | Combine antecedents into a... |
pm5.18 383 | Theorem *5.18 of [Whitehea... |
xor3 384 | Two ways to express "exclu... |
nbbn 385 | Move negation outside of b... |
biass 386 | Associative law for the bi... |
biluk 387 | Lukasiewicz's shortest axi... |
pm5.19 388 | Theorem *5.19 of [Whitehea... |
bi2.04 389 | Logical equivalence of com... |
pm5.4 390 | Antecedent absorption impl... |
imdi 391 | Distributive law for impli... |
pm5.41 392 | Theorem *5.41 of [Whitehea... |
pm4.8 393 | Theorem *4.8 of [Whitehead... |
pm4.81 394 | A formula is equivalent to... |
imim21b 395 | Simplify an implication be... |
pm4.63 398 | Theorem *4.63 of [Whitehea... |
pm4.67 399 | Theorem *4.67 of [Whitehea... |
imnan 400 | Express an implication in ... |
imnani 401 | Infer an implication from ... |
iman 402 | Implication in terms of co... |
pm3.24 403 | Law of noncontradiction. ... |
annim 404 | Express a conjunction in t... |
pm4.61 405 | Theorem *4.61 of [Whitehea... |
pm4.65 406 | Theorem *4.65 of [Whitehea... |
imp 407 | Importation inference. (C... |
impcom 408 | Importation inference with... |
con3dimp 409 | Variant of ~ con3d with im... |
mpnanrd 410 | Eliminate the right side o... |
impd 411 | Importation deduction. (C... |
impcomd 412 | Importation deduction with... |
ex 413 | Exportation inference. (T... |
expcom 414 | Exportation inference with... |
expdcom 415 | Commuted form of ~ expd . ... |
expd 416 | Exportation deduction. (C... |
expcomd 417 | Deduction form of ~ expcom... |
imp31 418 | An importation inference. ... |
imp32 419 | An importation inference. ... |
exp31 420 | An exportation inference. ... |
exp32 421 | An exportation inference. ... |
imp4b 422 | An importation inference. ... |
imp4a 423 | An importation inference. ... |
imp4c 424 | An importation inference. ... |
imp4d 425 | An importation inference. ... |
imp41 426 | An importation inference. ... |
imp42 427 | An importation inference. ... |
imp43 428 | An importation inference. ... |
imp44 429 | An importation inference. ... |
imp45 430 | An importation inference. ... |
exp4b 431 | An exportation inference. ... |
exp4a 432 | An exportation inference. ... |
exp4c 433 | An exportation inference. ... |
exp4d 434 | An exportation inference. ... |
exp41 435 | An exportation inference. ... |
exp42 436 | An exportation inference. ... |
exp43 437 | An exportation inference. ... |
exp44 438 | An exportation inference. ... |
exp45 439 | An exportation inference. ... |
imp5d 440 | An importation inference. ... |
imp5a 441 | An importation inference. ... |
imp5g 442 | An importation inference. ... |
imp55 443 | An importation inference. ... |
imp511 444 | An importation inference. ... |
exp5c 445 | An exportation inference. ... |
exp5j 446 | An exportation inference. ... |
exp5l 447 | An exportation inference. ... |
exp53 448 | An exportation inference. ... |
pm3.3 449 | Theorem *3.3 (Exp) of [Whi... |
pm3.31 450 | Theorem *3.31 (Imp) of [Wh... |
impexp 451 | Import-export theorem. Pa... |
impancom 452 | Mixed importation/commutat... |
expdimp 453 | A deduction version of exp... |
expimpd 454 | Exportation followed by a ... |
impr 455 | Import a wff into a right ... |
impl 456 | Export a wff from a left c... |
expr 457 | Export a wff from a right ... |
expl 458 | Export a wff from a left c... |
ancoms 459 | Inference commuting conjun... |
pm3.22 460 | Theorem *3.22 of [Whitehea... |
ancom 461 | Commutative law for conjun... |
ancomd 462 | Commutation of conjuncts i... |
biancomi 463 | Commuting conjunction in a... |
biancomd 464 | Commuting conjunction in a... |
ancomst 465 | Closed form of ~ ancoms . ... |
ancomsd 466 | Deduction commuting conjun... |
anasss 467 | Associative law for conjun... |
anassrs 468 | Associative law for conjun... |
anass 469 | Associative law for conjun... |
pm3.2 470 | Join antecedents with conj... |
pm3.2i 471 | Infer conjunction of premi... |
pm3.21 472 | Join antecedents with conj... |
pm3.43i 473 | Nested conjunction of ante... |
pm3.43 474 | Theorem *3.43 (Comp) of [W... |
dfbi2 475 | A theorem similar to the s... |
dfbi 476 | Definition ~ df-bi rewritt... |
biimpa 477 | Importation inference from... |
biimpar 478 | Importation inference from... |
biimpac 479 | Importation inference from... |
biimparc 480 | Importation inference from... |
adantr 481 | Inference adding a conjunc... |
adantl 482 | Inference adding a conjunc... |
simpl 483 | Elimination of a conjunct.... |
simpli 484 | Inference eliminating a co... |
simpr 485 | Elimination of a conjunct.... |
simpri 486 | Inference eliminating a co... |
intnan 487 | Introduction of conjunct i... |
intnanr 488 | Introduction of conjunct i... |
intnand 489 | Introduction of conjunct i... |
intnanrd 490 | Introduction of conjunct i... |
adantld 491 | Deduction adding a conjunc... |
adantrd 492 | Deduction adding a conjunc... |
pm3.41 493 | Theorem *3.41 of [Whitehea... |
pm3.42 494 | Theorem *3.42 of [Whitehea... |
simpld 495 | Deduction eliminating a co... |
simprd 496 | Deduction eliminating a co... |
simprbi 497 | Deduction eliminating a co... |
simplbi 498 | Deduction eliminating a co... |
simprbda 499 | Deduction eliminating a co... |
simplbda 500 | Deduction eliminating a co... |
simplbi2 501 | Deduction eliminating a co... |
simplbi2comt 502 | Closed form of ~ simplbi2c... |
simplbi2com 503 | A deduction eliminating a ... |
simpl2im 504 | Implication from an elimin... |
simplbiim 505 | Implication from an elimin... |
impel 506 | An inference for implicati... |
mpan9 507 | Modus ponens conjoining di... |
sylan9 508 | Nested syllogism inference... |
sylan9r 509 | Nested syllogism inference... |
sylan9bb 510 | Nested syllogism inference... |
sylan9bbr 511 | Nested syllogism inference... |
jca 512 | Deduce conjunction of the ... |
jcad 513 | Deduction conjoining the c... |
jca2 514 | Inference conjoining the c... |
jca31 515 | Join three consequents. (... |
jca32 516 | Join three consequents. (... |
jcai 517 | Deduction replacing implic... |
jcab 518 | Distributive law for impli... |
pm4.76 519 | Theorem *4.76 of [Whitehea... |
jctil 520 | Inference conjoining a the... |
jctir 521 | Inference conjoining a the... |
jccir 522 | Inference conjoining a con... |
jccil 523 | Inference conjoining a con... |
jctl 524 | Inference conjoining a the... |
jctr 525 | Inference conjoining a the... |
jctild 526 | Deduction conjoining a the... |
jctird 527 | Deduction conjoining a the... |
iba 528 | Introduction of antecedent... |
ibar 529 | Introduction of antecedent... |
biantru 530 | A wff is equivalent to its... |
biantrur 531 | A wff is equivalent to its... |
biantrud 532 | A wff is equivalent to its... |
biantrurd 533 | A wff is equivalent to its... |
bianfi 534 | A wff conjoined with false... |
bianfd 535 | A wff conjoined with false... |
baib 536 | Move conjunction outside o... |
baibr 537 | Move conjunction outside o... |
rbaibr 538 | Move conjunction outside o... |
rbaib 539 | Move conjunction outside o... |
baibd 540 | Move conjunction outside o... |
rbaibd 541 | Move conjunction outside o... |
bianabs 542 | Absorb a hypothesis into t... |
pm5.44 543 | Theorem *5.44 of [Whitehea... |
pm5.42 544 | Theorem *5.42 of [Whitehea... |
ancl 545 | Conjoin antecedent to left... |
anclb 546 | Conjoin antecedent to left... |
ancr 547 | Conjoin antecedent to righ... |
ancrb 548 | Conjoin antecedent to righ... |
ancli 549 | Deduction conjoining antec... |
ancri 550 | Deduction conjoining antec... |
ancld 551 | Deduction conjoining antec... |
ancrd 552 | Deduction conjoining antec... |
impac 553 | Importation with conjuncti... |
anc2l 554 | Conjoin antecedent to left... |
anc2r 555 | Conjoin antecedent to righ... |
anc2li 556 | Deduction conjoining antec... |
anc2ri 557 | Deduction conjoining antec... |
pm4.71 558 | Implication in terms of bi... |
pm4.71r 559 | Implication in terms of bi... |
pm4.71i 560 | Inference converting an im... |
pm4.71ri 561 | Inference converting an im... |
pm4.71d 562 | Deduction converting an im... |
pm4.71rd 563 | Deduction converting an im... |
pm4.24 564 | Theorem *4.24 of [Whitehea... |
anidm 565 | Idempotent law for conjunc... |
anidmdbi 566 | Conjunction idempotence wi... |
anidms 567 | Inference from idempotent ... |
imdistan 568 | Distribution of implicatio... |
imdistani 569 | Distribution of implicatio... |
imdistanri 570 | Distribution of implicatio... |
imdistand 571 | Distribution of implicatio... |
imdistanda 572 | Distribution of implicatio... |
pm5.3 573 | Theorem *5.3 of [Whitehead... |
pm5.32 574 | Distribution of implicatio... |
pm5.32i 575 | Distribution of implicatio... |
pm5.32ri 576 | Distribution of implicatio... |
pm5.32d 577 | Distribution of implicatio... |
pm5.32rd 578 | Distribution of implicatio... |
pm5.32da 579 | Distribution of implicatio... |
sylan 580 | A syllogism inference. (C... |
sylanb 581 | A syllogism inference. (C... |
sylanbr 582 | A syllogism inference. (C... |
sylanbrc 583 | Syllogism inference. (Con... |
syl2anc 584 | Syllogism inference combin... |
syl2anc2 585 | Double syllogism inference... |
sylancl 586 | Syllogism inference combin... |
sylancr 587 | Syllogism inference combin... |
sylancom 588 | Syllogism inference with c... |
sylanblc 589 | Syllogism inference combin... |
sylanblrc 590 | Syllogism inference combin... |
syldan 591 | A syllogism deduction with... |
sylbida 592 | A syllogism deduction. (C... |
sylan2 593 | A syllogism inference. (C... |
sylan2b 594 | A syllogism inference. (C... |
sylan2br 595 | A syllogism inference. (C... |
syl2an 596 | A double syllogism inferen... |
syl2anr 597 | A double syllogism inferen... |
syl2anb 598 | A double syllogism inferen... |
syl2anbr 599 | A double syllogism inferen... |
sylancb 600 | A syllogism inference comb... |
sylancbr 601 | A syllogism inference comb... |
syldanl 602 | A syllogism deduction with... |
syland 603 | A syllogism deduction. (C... |
sylani 604 | A syllogism inference. (C... |
sylan2d 605 | A syllogism deduction. (C... |
sylan2i 606 | A syllogism inference. (C... |
syl2ani 607 | A syllogism inference. (C... |
syl2and 608 | A syllogism deduction. (C... |
anim12d 609 | Conjoin antecedents and co... |
anim12d1 610 | Variant of ~ anim12d where... |
anim1d 611 | Add a conjunct to right of... |
anim2d 612 | Add a conjunct to left of ... |
anim12i 613 | Conjoin antecedents and co... |
anim12ci 614 | Variant of ~ anim12i with ... |
anim1i 615 | Introduce conjunct to both... |
anim1ci 616 | Introduce conjunct to both... |
anim2i 617 | Introduce conjunct to both... |
anim12ii 618 | Conjoin antecedents and co... |
anim12dan 619 | Conjoin antecedents and co... |
im2anan9 620 | Deduction joining nested i... |
im2anan9r 621 | Deduction joining nested i... |
pm3.45 622 | Theorem *3.45 (Fact) of [W... |
anbi2i 623 | Introduce a left conjunct ... |
anbi1i 624 | Introduce a right conjunct... |
anbi2ci 625 | Variant of ~ anbi2i with c... |
anbi1ci 626 | Variant of ~ anbi1i with c... |
anbi12i 627 | Conjoin both sides of two ... |
anbi12ci 628 | Variant of ~ anbi12i with ... |
anbi2d 629 | Deduction adding a left co... |
anbi1d 630 | Deduction adding a right c... |
anbi12d 631 | Deduction joining two equi... |
anbi1 632 | Introduce a right conjunct... |
anbi2 633 | Introduce a left conjunct ... |
anbi1cd 634 | Introduce a proposition as... |
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... |
unitreslOLD 875 | Obsolete version of ~ olcn... |
orcnd 876 | A lemma for Conjunctive No... |
mtord 877 | A modus tollens deduction ... |
pm3.2ni 878 | Infer negated disjunction ... |
pm2.45 879 | Theorem *2.45 of [Whitehea... |
pm2.46 880 | Theorem *2.46 of [Whitehea... |
pm2.47 881 | Theorem *2.47 of [Whitehea... |
pm2.48 882 | Theorem *2.48 of [Whitehea... |
pm2.49 883 | Theorem *2.49 of [Whitehea... |
norbi 884 | If neither of two proposit... |
nbior 885 | If two propositions are no... |
orel1 886 | Elimination of disjunction... |
pm2.25 887 | Theorem *2.25 of [Whitehea... |
orel2 888 | Elimination of disjunction... |
pm2.67-2 889 | Slight generalization of T... |
pm2.67 890 | Theorem *2.67 of [Whitehea... |
curryax 891 | A non-intuitionistic posit... |
exmid 892 | Law of excluded middle, al... |
exmidd 893 | Law of excluded middle in ... |
pm2.1 894 | Theorem *2.1 of [Whitehead... |
pm2.13 895 | Theorem *2.13 of [Whitehea... |
pm2.621 896 | Theorem *2.621 of [Whitehe... |
pm2.62 897 | Theorem *2.62 of [Whitehea... |
pm2.68 898 | Theorem *2.68 of [Whitehea... |
dfor2 899 | Logical 'or' expressed in ... |
pm2.07 900 | Theorem *2.07 of [Whitehea... |
pm1.2 901 | Axiom *1.2 of [WhiteheadRu... |
oridm 902 | Idempotent law for disjunc... |
pm4.25 903 | Theorem *4.25 of [Whitehea... |
pm2.4 904 | Theorem *2.4 of [Whitehead... |
pm2.41 905 | Theorem *2.41 of [Whitehea... |
orim12i 906 | Disjoin antecedents and co... |
orim1i 907 | Introduce disjunct to both... |
orim2i 908 | Introduce disjunct to both... |
orim12dALT 909 | Alternate proof of ~ orim1... |
orbi2i 910 | Inference adding a left di... |
orbi1i 911 | Inference adding a right d... |
orbi12i 912 | Infer the disjunction of t... |
orbi2d 913 | Deduction adding a left di... |
orbi1d 914 | Deduction adding a right d... |
orbi1 915 | Theorem *4.37 of [Whitehea... |
orbi12d 916 | Deduction joining two equi... |
pm1.5 917 | Axiom *1.5 (Assoc) of [Whi... |
or12 918 | Swap two disjuncts. (Cont... |
orass 919 | Associative law for disjun... |
pm2.31 920 | Theorem *2.31 of [Whitehea... |
pm2.32 921 | Theorem *2.32 of [Whitehea... |
pm2.3 922 | Theorem *2.3 of [Whitehead... |
or32 923 | A rearrangement of disjunc... |
or4 924 | Rearrangement of 4 disjunc... |
or42 925 | Rearrangement of 4 disjunc... |
orordi 926 | Distribution of disjunctio... |
orordir 927 | Distribution of disjunctio... |
orimdi 928 | Disjunction distributes ov... |
pm2.76 929 | Theorem *2.76 of [Whitehea... |
pm2.85 930 | Theorem *2.85 of [Whitehea... |
pm2.75 931 | Theorem *2.75 of [Whitehea... |
pm4.78 932 | Implication distributes ov... |
biort 933 | A disjunction with a true ... |
biorf 934 | A wff is equivalent to its... |
biortn 935 | A wff is equivalent to its... |
biorfi 936 | A wff is equivalent to its... |
pm2.26 937 | Theorem *2.26 of [Whitehea... |
pm2.63 938 | Theorem *2.63 of [Whitehea... |
pm2.64 939 | Theorem *2.64 of [Whitehea... |
pm2.42 940 | Theorem *2.42 of [Whitehea... |
pm5.11g 941 | A general instance of Theo... |
pm5.11 942 | Theorem *5.11 of [Whitehea... |
pm5.12 943 | Theorem *5.12 of [Whitehea... |
pm5.14 944 | Theorem *5.14 of [Whitehea... |
pm5.13 945 | Theorem *5.13 of [Whitehea... |
pm5.55 946 | Theorem *5.55 of [Whitehea... |
pm4.72 947 | Implication in terms of bi... |
imimorb 948 | Simplify an implication be... |
oibabs 949 | Absorption of disjunction ... |
orbidi 950 | Disjunction distributes ov... |
pm5.7 951 | Disjunction distributes ov... |
jaao 952 | Inference conjoining and d... |
jaoa 953 | Inference disjoining and c... |
jaoian 954 | Inference disjoining the a... |
jaodan 955 | Deduction disjoining the a... |
mpjaodan 956 | Eliminate a disjunction in... |
pm3.44 957 | Theorem *3.44 of [Whitehea... |
jao 958 | Disjunction of antecedents... |
jaob 959 | Disjunction of antecedents... |
pm4.77 960 | Theorem *4.77 of [Whitehea... |
pm3.48 961 | Theorem *3.48 of [Whitehea... |
orim12d 962 | Disjoin antecedents and co... |
orim1d 963 | Disjoin antecedents and co... |
orim2d 964 | Disjoin antecedents and co... |
orim2 965 | Axiom *1.6 (Sum) of [White... |
pm2.38 966 | Theorem *2.38 of [Whitehea... |
pm2.36 967 | Theorem *2.36 of [Whitehea... |
pm2.37 968 | Theorem *2.37 of [Whitehea... |
pm2.81 969 | Theorem *2.81 of [Whitehea... |
pm2.8 970 | Theorem *2.8 of [Whitehead... |
pm2.73 971 | Theorem *2.73 of [Whitehea... |
pm2.74 972 | Theorem *2.74 of [Whitehea... |
pm2.82 973 | Theorem *2.82 of [Whitehea... |
pm4.39 974 | Theorem *4.39 of [Whitehea... |
animorl 975 | Conjunction implies disjun... |
animorr 976 | Conjunction implies disjun... |
animorlr 977 | Conjunction implies disjun... |
animorrl 978 | Conjunction implies disjun... |
ianor 979 | Negated conjunction in ter... |
anor 980 | Conjunction in terms of di... |
ioran 981 | Negated disjunction in ter... |
pm4.52 982 | Theorem *4.52 of [Whitehea... |
pm4.53 983 | Theorem *4.53 of [Whitehea... |
pm4.54 984 | Theorem *4.54 of [Whitehea... |
pm4.55 985 | Theorem *4.55 of [Whitehea... |
pm4.56 986 | Theorem *4.56 of [Whitehea... |
oran 987 | Disjunction in terms of co... |
pm4.57 988 | Theorem *4.57 of [Whitehea... |
pm3.1 989 | Theorem *3.1 of [Whitehead... |
pm3.11 990 | Theorem *3.11 of [Whitehea... |
pm3.12 991 | Theorem *3.12 of [Whitehea... |
pm3.13 992 | Theorem *3.13 of [Whitehea... |
pm3.14 993 | Theorem *3.14 of [Whitehea... |
pm4.44 994 | Theorem *4.44 of [Whitehea... |
pm4.45 995 | Theorem *4.45 of [Whitehea... |
orabs 996 | Absorption of redundant in... |
oranabs 997 | Absorb a disjunct into a c... |
pm5.61 998 | Theorem *5.61 of [Whitehea... |
pm5.6 999 | Conjunction in antecedent ... |
orcanai 1000 | Change disjunction in cons... |
pm4.79 1001 | Theorem *4.79 of [Whitehea... |
pm5.53 1002 | Theorem *5.53 of [Whitehea... |
ordi 1003 | Distributive law for disju... |
ordir 1004 | Distributive law for disju... |
andi 1005 | Distributive law for conju... |
andir 1006 | Distributive law for conju... |
orddi 1007 | Double distributive law fo... |
anddi 1008 | Double distributive law fo... |
pm5.17 1009 | Theorem *5.17 of [Whitehea... |
pm5.15 1010 | Theorem *5.15 of [Whitehea... |
pm5.16 1011 | Theorem *5.16 of [Whitehea... |
xor 1012 | Two ways to express exclus... |
nbi2 1013 | Two ways to express "exclu... |
xordi 1014 | Conjunction distributes ov... |
pm5.54 1015 | Theorem *5.54 of [Whitehea... |
pm5.62 1016 | Theorem *5.62 of [Whitehea... |
pm5.63 1017 | Theorem *5.63 of [Whitehea... |
niabn 1018 | Miscellaneous inference re... |
ninba 1019 | Miscellaneous inference re... |
pm4.43 1020 | Theorem *4.43 of [Whitehea... |
pm4.82 1021 | Theorem *4.82 of [Whitehea... |
pm4.83 1022 | Theorem *4.83 of [Whitehea... |
pclem6 1023 | Negation inferred from emb... |
bigolden 1024 | Dijkstra-Scholten's Golden... |
pm5.71 1025 | Theorem *5.71 of [Whitehea... |
pm5.75 1026 | Theorem *5.75 of [Whitehea... |
ecase2d 1027 | Deduction for elimination ... |
ecase2dOLD 1028 | Obsolete version of ~ ecas... |
ecase3 1029 | Inference for elimination ... |
ecase 1030 | Inference for elimination ... |
ecase3d 1031 | Deduction for elimination ... |
ecased 1032 | Deduction for elimination ... |
ecase3ad 1033 | Deduction for elimination ... |
ecase3adOLD 1034 | Obsolete version of ~ ecas... |
ccase 1035 | Inference for combining ca... |
ccased 1036 | Deduction for combining ca... |
ccase2 1037 | Inference for combining ca... |
4cases 1038 | Inference eliminating two ... |
4casesdan 1039 | Deduction eliminating two ... |
cases 1040 | Case disjunction according... |
dedlem0a 1041 | Lemma for an alternate ver... |
dedlem0b 1042 | Lemma for an alternate ver... |
dedlema 1043 | Lemma for weak deduction t... |
dedlemb 1044 | Lemma for weak deduction t... |
cases2 1045 | Case disjunction according... |
cases2ALT 1046 | Alternate proof of ~ cases... |
dfbi3 1047 | An alternate definition of... |
pm5.24 1048 | Theorem *5.24 of [Whitehea... |
4exmid 1049 | The disjunction of the fou... |
consensus 1050 | The consensus theorem. Th... |
pm4.42 1051 | Theorem *4.42 of [Whitehea... |
prlem1 1052 | A specialized lemma for se... |
prlem2 1053 | A specialized lemma for se... |
oplem1 1054 | A specialized lemma for se... |
dn1 1055 | A single axiom for Boolean... |
bianir 1056 | A closed form of ~ mpbir ,... |
jaoi2 1057 | Inference removing a negat... |
jaoi3 1058 | Inference separating a dis... |
ornld 1059 | Selecting one statement fr... |
dfifp2 1062 | Alternate definition of th... |
dfifp3 1063 | Alternate definition of th... |
dfifp4 1064 | Alternate definition of th... |
dfifp5 1065 | Alternate definition of th... |
dfifp6 1066 | Alternate definition of th... |
dfifp7 1067 | Alternate definition of th... |
ifpdfbi 1068 | Define the biconditional a... |
anifp 1069 | The conditional operator i... |
ifpor 1070 | The conditional operator i... |
ifpn 1071 | Conditional operator for t... |
ifpnOLD 1072 | Obsolete version of ~ ifpn... |
ifptru 1073 | Value of the conditional o... |
ifpfal 1074 | Value of the conditional o... |
ifpid 1075 | Value of the conditional o... |
casesifp 1076 | Version of ~ cases express... |
ifpbi123d 1077 | Equivalence deduction for ... |
ifpbi123dOLD 1078 | Obsolete version of ~ ifpb... |
ifpbi23d 1079 | Equivalence deduction for ... |
ifpimpda 1080 | Separation of the values o... |
1fpid3 1081 | The value of the condition... |
elimh 1082 | Hypothesis builder for the... |
dedt 1083 | The weak deduction theorem... |
con3ALT 1084 | Proof of ~ con3 from its a... |
3orass 1089 | Associative law for triple... |
3orel1 1090 | Partial elimination of a t... |
3orrot 1091 | Rotation law for triple di... |
3orcoma 1092 | Commutation law for triple... |
3orcomb 1093 | Commutation law for triple... |
3anass 1094 | Associative law for triple... |
3anan12 1095 | Convert triple conjunction... |
3anan32 1096 | Convert triple conjunction... |
3ancoma 1097 | Commutation law for triple... |
3ancomb 1098 | Commutation law for triple... |
3anrot 1099 | Rotation law for triple co... |
3anrev 1100 | Reversal law for triple co... |
anandi3 1101 | Distribution of triple con... |
anandi3r 1102 | Distribution of triple con... |
3anidm 1103 | Idempotent law for conjunc... |
3an4anass 1104 | Associative law for four c... |
3ioran 1105 | Negated triple disjunction... |
3ianor 1106 | Negated triple conjunction... |
3anor 1107 | Triple conjunction express... |
3oran 1108 | Triple disjunction in term... |
3impa 1109 | Importation from double to... |
3imp 1110 | Importation inference. (C... |
3imp31 1111 | The importation inference ... |
3imp231 1112 | Importation inference. (C... |
3imp21 1113 | The importation inference ... |
3impb 1114 | Importation from double to... |
3impib 1115 | Importation to triple conj... |
3impia 1116 | Importation to triple conj... |
3expa 1117 | Exportation from triple to... |
3exp 1118 | Exportation inference. (C... |
3expb 1119 | Exportation from triple to... |
3expia 1120 | Exportation from triple co... |
3expib 1121 | Exportation from triple co... |
3com12 1122 | Commutation in antecedent.... |
3com13 1123 | Commutation in antecedent.... |
3comr 1124 | Commutation in antecedent.... |
3com23 1125 | Commutation in antecedent.... |
3coml 1126 | Commutation in antecedent.... |
3jca 1127 | Join consequents with conj... |
3jcad 1128 | Deduction conjoining the c... |
3adant1 1129 | Deduction adding a conjunc... |
3adant2 1130 | Deduction adding a conjunc... |
3adant3 1131 | Deduction adding a conjunc... |
3ad2ant1 1132 | Deduction adding conjuncts... |
3ad2ant2 1133 | Deduction adding conjuncts... |
3ad2ant3 1134 | Deduction adding conjuncts... |
simp1 1135 | Simplification of triple c... |
simp2 1136 | Simplification of triple c... |
simp3 1137 | Simplification of triple c... |
simp1i 1138 | Infer a conjunct from a tr... |
simp2i 1139 | Infer a conjunct from a tr... |
simp3i 1140 | Infer a conjunct from a tr... |
simp1d 1141 | Deduce a conjunct from a t... |
simp2d 1142 | Deduce a conjunct from a t... |
simp3d 1143 | Deduce a conjunct from a t... |
simp1bi 1144 | Deduce a conjunct from a t... |
simp2bi 1145 | Deduce a conjunct from a t... |
simp3bi 1146 | Deduce a conjunct from a t... |
3simpa 1147 | Simplification of triple c... |
3simpb 1148 | Simplification of triple c... |
3simpc 1149 | Simplification of triple c... |
3anim123i 1150 | Join antecedents and conse... |
3anim1i 1151 | Add two conjuncts to antec... |
3anim2i 1152 | Add two conjuncts to antec... |
3anim3i 1153 | Add two conjuncts to antec... |
3anbi123i 1154 | Join 3 biconditionals with... |
3orbi123i 1155 | Join 3 biconditionals with... |
3anbi1i 1156 | Inference adding two conju... |
3anbi2i 1157 | Inference adding two conju... |
3anbi3i 1158 | Inference adding two conju... |
syl3an 1159 | A triple syllogism inferen... |
syl3anb 1160 | A triple syllogism inferen... |
syl3anbr 1161 | A triple syllogism inferen... |
syl3an1 1162 | A syllogism inference. (C... |
syl3an2 1163 | A syllogism inference. (C... |
syl3an3 1164 | A syllogism inference. (C... |
3adantl1 1165 | Deduction adding a conjunc... |
3adantl2 1166 | Deduction adding a conjunc... |
3adantl3 1167 | Deduction adding a conjunc... |
3adantr1 1168 | Deduction adding a conjunc... |
3adantr2 1169 | Deduction adding a conjunc... |
3adantr3 1170 | Deduction adding a conjunc... |
ad4ant123 1171 | Deduction adding conjuncts... |
ad4ant124 1172 | Deduction adding conjuncts... |
ad4ant134 1173 | Deduction adding conjuncts... |
ad4ant234 1174 | Deduction adding conjuncts... |
3adant1l 1175 | Deduction adding a conjunc... |
3adant1r 1176 | Deduction adding a conjunc... |
3adant2l 1177 | Deduction adding a conjunc... |
3adant2r 1178 | Deduction adding a conjunc... |
3adant3l 1179 | Deduction adding a conjunc... |
3adant3r 1180 | Deduction adding a conjunc... |
3adant3r1 1181 | Deduction adding a conjunc... |
3adant3r2 1182 | Deduction adding a conjunc... |
3adant3r3 1183 | Deduction adding a conjunc... |
3ad2antl1 1184 | Deduction adding conjuncts... |
3ad2antl2 1185 | Deduction adding conjuncts... |
3ad2antl3 1186 | Deduction adding conjuncts... |
3ad2antr1 1187 | Deduction adding conjuncts... |
3ad2antr2 1188 | Deduction adding conjuncts... |
3ad2antr3 1189 | Deduction adding conjuncts... |
simpl1 1190 | Simplification of conjunct... |
simpl2 1191 | Simplification of conjunct... |
simpl3 1192 | Simplification of conjunct... |
simpr1 1193 | Simplification of conjunct... |
simpr2 1194 | Simplification of conjunct... |
simpr3 1195 | Simplification of conjunct... |
simp1l 1196 | Simplification of triple c... |
simp1r 1197 | Simplification of triple c... |
simp2l 1198 | Simplification of triple c... |
simp2r 1199 | Simplification of triple c... |
simp3l 1200 | Simplification of triple c... |
simp3r 1201 | Simplification of triple c... |
simp11 1202 | Simplification of doubly t... |
simp12 1203 | Simplification of doubly t... |
simp13 1204 | Simplification of doubly t... |
simp21 1205 | Simplification of doubly t... |
simp22 1206 | Simplification of doubly t... |
simp23 1207 | Simplification of doubly t... |
simp31 1208 | Simplification of doubly t... |
simp32 1209 | Simplification of doubly t... |
simp33 1210 | Simplification of doubly t... |
simpll1 1211 | Simplification of conjunct... |
simpll2 1212 | Simplification of conjunct... |
simpll3 1213 | Simplification of conjunct... |
simplr1 1214 | Simplification of conjunct... |
simplr2 1215 | Simplification of conjunct... |
simplr3 1216 | Simplification of conjunct... |
simprl1 1217 | Simplification of conjunct... |
simprl2 1218 | Simplification of conjunct... |
simprl3 1219 | Simplification of conjunct... |
simprr1 1220 | Simplification of conjunct... |
simprr2 1221 | Simplification of conjunct... |
simprr3 1222 | Simplification of conjunct... |
simpl1l 1223 | Simplification of conjunct... |
simpl1r 1224 | Simplification of conjunct... |
simpl2l 1225 | Simplification of conjunct... |
simpl2r 1226 | Simplification of conjunct... |
simpl3l 1227 | Simplification of conjunct... |
simpl3r 1228 | Simplification of conjunct... |
simpr1l 1229 | Simplification of conjunct... |
simpr1r 1230 | Simplification of conjunct... |
simpr2l 1231 | Simplification of conjunct... |
simpr2r 1232 | Simplification of conjunct... |
simpr3l 1233 | Simplification of conjunct... |
simpr3r 1234 | Simplification of conjunct... |
simp1ll 1235 | Simplification of conjunct... |
simp1lr 1236 | Simplification of conjunct... |
simp1rl 1237 | Simplification of conjunct... |
simp1rr 1238 | Simplification of conjunct... |
simp2ll 1239 | Simplification of conjunct... |
simp2lr 1240 | Simplification of conjunct... |
simp2rl 1241 | Simplification of conjunct... |
simp2rr 1242 | Simplification of conjunct... |
simp3ll 1243 | Simplification of conjunct... |
simp3lr 1244 | Simplification of conjunct... |
simp3rl 1245 | Simplification of conjunct... |
simp3rr 1246 | Simplification of conjunct... |
simpl11 1247 | Simplification of conjunct... |
simpl12 1248 | Simplification of conjunct... |
simpl13 1249 | Simplification of conjunct... |
simpl21 1250 | Simplification of conjunct... |
simpl22 1251 | Simplification of conjunct... |
simpl23 1252 | Simplification of conjunct... |
simpl31 1253 | Simplification of conjunct... |
simpl32 1254 | Simplification of conjunct... |
simpl33 1255 | Simplification of conjunct... |
simpr11 1256 | Simplification of conjunct... |
simpr12 1257 | Simplification of conjunct... |
simpr13 1258 | Simplification of conjunct... |
simpr21 1259 | Simplification of conjunct... |
simpr22 1260 | Simplification of conjunct... |
simpr23 1261 | Simplification of conjunct... |
simpr31 1262 | Simplification of conjunct... |
simpr32 1263 | Simplification of conjunct... |
simpr33 1264 | Simplification of conjunct... |
simp1l1 1265 | Simplification of conjunct... |
simp1l2 1266 | Simplification of conjunct... |
simp1l3 1267 | Simplification of conjunct... |
simp1r1 1268 | Simplification of conjunct... |
simp1r2 1269 | Simplification of conjunct... |
simp1r3 1270 | Simplification of conjunct... |
simp2l1 1271 | Simplification of conjunct... |
simp2l2 1272 | Simplification of conjunct... |
simp2l3 1273 | Simplification of conjunct... |
simp2r1 1274 | Simplification of conjunct... |
simp2r2 1275 | Simplification of conjunct... |
simp2r3 1276 | Simplification of conjunct... |
simp3l1 1277 | Simplification of conjunct... |
simp3l2 1278 | Simplification of conjunct... |
simp3l3 1279 | Simplification of conjunct... |
simp3r1 1280 | Simplification of conjunct... |
simp3r2 1281 | Simplification of conjunct... |
simp3r3 1282 | Simplification of conjunct... |
simp11l 1283 | Simplification of conjunct... |
simp11r 1284 | Simplification of conjunct... |
simp12l 1285 | Simplification of conjunct... |
simp12r 1286 | Simplification of conjunct... |
simp13l 1287 | Simplification of conjunct... |
simp13r 1288 | Simplification of conjunct... |
simp21l 1289 | Simplification of conjunct... |
simp21r 1290 | Simplification of conjunct... |
simp22l 1291 | Simplification of conjunct... |
simp22r 1292 | Simplification of conjunct... |
simp23l 1293 | Simplification of conjunct... |
simp23r 1294 | Simplification of conjunct... |
simp31l 1295 | Simplification of conjunct... |
simp31r 1296 | Simplification of conjunct... |
simp32l 1297 | Simplification of conjunct... |
simp32r 1298 | Simplification of conjunct... |
simp33l 1299 | Simplification of conjunct... |
simp33r 1300 | Simplification of conjunct... |
simp111 1301 | Simplification of conjunct... |
simp112 1302 | Simplification of conjunct... |
simp113 1303 | Simplification of conjunct... |
simp121 1304 | Simplification of conjunct... |
simp122 1305 | Simplification of conjunct... |
simp123 1306 | Simplification of conjunct... |
simp131 1307 | Simplification of conjunct... |
simp132 1308 | Simplification of conjunct... |
simp133 1309 | Simplification of conjunct... |
simp211 1310 | Simplification of conjunct... |
simp212 1311 | Simplification of conjunct... |
simp213 1312 | Simplification of conjunct... |
simp221 1313 | Simplification of conjunct... |
simp222 1314 | Simplification of conjunct... |
simp223 1315 | Simplification of conjunct... |
simp231 1316 | Simplification of conjunct... |
simp232 1317 | Simplification of conjunct... |
simp233 1318 | Simplification of conjunct... |
simp311 1319 | Simplification of conjunct... |
simp312 1320 | Simplification of conjunct... |
simp313 1321 | Simplification of conjunct... |
simp321 1322 | Simplification of conjunct... |
simp322 1323 | Simplification of conjunct... |
simp323 1324 | Simplification of conjunct... |
simp331 1325 | Simplification of conjunct... |
simp332 1326 | Simplification of conjunct... |
simp333 1327 | Simplification of conjunct... |
3anibar 1328 | Remove a hypothesis from t... |
3mix1 1329 | Introduction in triple dis... |
3mix2 1330 | Introduction in triple dis... |
3mix3 1331 | Introduction in triple dis... |
3mix1i 1332 | Introduction in triple dis... |
3mix2i 1333 | Introduction in triple dis... |
3mix3i 1334 | Introduction in triple dis... |
3mix1d 1335 | Deduction introducing trip... |
3mix2d 1336 | Deduction introducing trip... |
3mix3d 1337 | Deduction introducing trip... |
3pm3.2i 1338 | Infer conjunction of premi... |
pm3.2an3 1339 | Version of ~ pm3.2 for a t... |
mpbir3an 1340 | Detach a conjunction of tr... |
mpbir3and 1341 | Detach a conjunction of tr... |
syl3anbrc 1342 | Syllogism inference. (Con... |
syl21anbrc 1343 | Syllogism inference. (Con... |
3imp3i2an 1344 | An elimination deduction. ... |
ex3 1345 | Apply ~ ex to a hypothesis... |
3imp1 1346 | Importation to left triple... |
3impd 1347 | Importation deduction for ... |
3imp2 1348 | Importation to right tripl... |
3impdi 1349 | Importation inference (und... |
3impdir 1350 | Importation inference (und... |
3exp1 1351 | Exportation from left trip... |
3expd 1352 | Exportation deduction for ... |
3exp2 1353 | Exportation from right tri... |
exp5o 1354 | A triple exportation infer... |
exp516 1355 | A triple exportation infer... |
exp520 1356 | A triple exportation infer... |
3impexp 1357 | Version of ~ impexp for a ... |
3an1rs 1358 | Swap conjuncts. (Contribu... |
3anassrs 1359 | Associative law for conjun... |
ad5ant245 1360 | Deduction adding conjuncts... |
ad5ant234 1361 | Deduction adding conjuncts... |
ad5ant235 1362 | Deduction adding conjuncts... |
ad5ant123 1363 | Deduction adding conjuncts... |
ad5ant124 1364 | Deduction adding conjuncts... |
ad5ant125 1365 | Deduction adding conjuncts... |
ad5ant134 1366 | Deduction adding conjuncts... |
ad5ant135 1367 | Deduction adding conjuncts... |
ad5ant145 1368 | Deduction adding conjuncts... |
ad5ant2345 1369 | Deduction adding conjuncts... |
syl3anc 1370 | Syllogism combined with co... |
syl13anc 1371 | Syllogism combined with co... |
syl31anc 1372 | Syllogism combined with co... |
syl112anc 1373 | Syllogism combined with co... |
syl121anc 1374 | Syllogism combined with co... |
syl211anc 1375 | Syllogism combined with co... |
syl23anc 1376 | Syllogism combined with co... |
syl32anc 1377 | Syllogism combined with co... |
syl122anc 1378 | Syllogism combined with co... |
syl212anc 1379 | Syllogism combined with co... |
syl221anc 1380 | Syllogism combined with co... |
syl113anc 1381 | Syllogism combined with co... |
syl131anc 1382 | Syllogism combined with co... |
syl311anc 1383 | Syllogism combined with co... |
syl33anc 1384 | Syllogism combined with co... |
syl222anc 1385 | Syllogism combined with co... |
syl123anc 1386 | Syllogism combined with co... |
syl132anc 1387 | Syllogism combined with co... |
syl213anc 1388 | Syllogism combined with co... |
syl231anc 1389 | Syllogism combined with co... |
syl312anc 1390 | Syllogism combined with co... |
syl321anc 1391 | Syllogism combined with co... |
syl133anc 1392 | Syllogism combined with co... |
syl313anc 1393 | Syllogism combined with co... |
syl331anc 1394 | Syllogism combined with co... |
syl223anc 1395 | Syllogism combined with co... |
syl232anc 1396 | Syllogism combined with co... |
syl322anc 1397 | Syllogism combined with co... |
syl233anc 1398 | Syllogism combined with co... |
syl323anc 1399 | Syllogism combined with co... |
syl332anc 1400 | Syllogism combined with co... |
syl333anc 1401 | A syllogism inference comb... |
syl3an1b 1402 | A syllogism inference. (C... |
syl3an2b 1403 | A syllogism inference. (C... |
syl3an3b 1404 | A syllogism inference. (C... |
syl3an1br 1405 | A syllogism inference. (C... |
syl3an2br 1406 | A syllogism inference. (C... |
syl3an3br 1407 | A syllogism inference. (C... |
syld3an3 1408 | A syllogism inference. (C... |
syld3an1 1409 | A syllogism inference. (C... |
syld3an2 1410 | A syllogism inference. (C... |
syl3anl1 1411 | A syllogism inference. (C... |
syl3anl2 1412 | A syllogism inference. (C... |
syl3anl3 1413 | A syllogism inference. (C... |
syl3anl 1414 | A triple syllogism inferen... |
syl3anr1 1415 | A syllogism inference. (C... |
syl3anr2 1416 | A syllogism inference. (C... |
syl3anr3 1417 | A syllogism inference. (C... |
3anidm12 1418 | Inference from idempotent ... |
3anidm13 1419 | Inference from idempotent ... |
3anidm23 1420 | Inference from idempotent ... |
syl2an3an 1421 | ~ syl3an with antecedents ... |
syl2an23an 1422 | Deduction related to ~ syl... |
3ori 1423 | Infer implication from tri... |
3jao 1424 | Disjunction of three antec... |
3jaob 1425 | Disjunction of three antec... |
3jaoi 1426 | Disjunction of three antec... |
3jaod 1427 | Disjunction of three antec... |
3jaoian 1428 | Disjunction of three antec... |
3jaodan 1429 | Disjunction of three antec... |
mpjao3dan 1430 | Eliminate a three-way disj... |
mpjao3danOLD 1431 | Obsolete version of ~ mpja... |
3jaao 1432 | Inference conjoining and d... |
syl3an9b 1433 | Nested syllogism inference... |
3orbi123d 1434 | Deduction joining 3 equiva... |
3anbi123d 1435 | Deduction joining 3 equiva... |
3anbi12d 1436 | Deduction conjoining and a... |
3anbi13d 1437 | Deduction conjoining and a... |
3anbi23d 1438 | Deduction conjoining and a... |
3anbi1d 1439 | Deduction adding conjuncts... |
3anbi2d 1440 | Deduction adding conjuncts... |
3anbi3d 1441 | Deduction adding conjuncts... |
3anim123d 1442 | Deduction joining 3 implic... |
3orim123d 1443 | Deduction joining 3 implic... |
an6 1444 | Rearrangement of 6 conjunc... |
3an6 1445 | Analogue of ~ an4 for trip... |
3or6 1446 | Analogue of ~ or4 for trip... |
mp3an1 1447 | An inference based on modu... |
mp3an2 1448 | An inference based on modu... |
mp3an3 1449 | An inference based on modu... |
mp3an12 1450 | An inference based on modu... |
mp3an13 1451 | An inference based on modu... |
mp3an23 1452 | An inference based on modu... |
mp3an1i 1453 | An inference based on modu... |
mp3anl1 1454 | An inference based on modu... |
mp3anl2 1455 | An inference based on modu... |
mp3anl3 1456 | An inference based on modu... |
mp3anr1 1457 | An inference based on modu... |
mp3anr2 1458 | An inference based on modu... |
mp3anr3 1459 | An inference based on modu... |
mp3an 1460 | An inference based on modu... |
mpd3an3 1461 | An inference based on modu... |
mpd3an23 1462 | An inference based on modu... |
mp3and 1463 | A deduction based on modus... |
mp3an12i 1464 | ~ mp3an with antecedents i... |
mp3an2i 1465 | ~ mp3an with antecedents i... |
mp3an3an 1466 | ~ mp3an with antecedents i... |
mp3an2ani 1467 | An elimination deduction. ... |
biimp3a 1468 | Infer implication from a l... |
biimp3ar 1469 | Infer implication from a l... |
3anandis 1470 | Inference that undistribut... |
3anandirs 1471 | Inference that undistribut... |
ecase23d 1472 | Deduction for elimination ... |
3ecase 1473 | Inference for elimination ... |
3bior1fd 1474 | A disjunction is equivalen... |
3bior1fand 1475 | A disjunction is equivalen... |
3bior2fd 1476 | A wff is equivalent to its... |
3biant1d 1477 | A conjunction is equivalen... |
intn3an1d 1478 | Introduction of a triple c... |
intn3an2d 1479 | Introduction of a triple c... |
intn3an3d 1480 | Introduction of a triple c... |
an3andi 1481 | Distribution of conjunctio... |
an33rean 1482 | Rearrange a 9-fold conjunc... |
an33reanOLD 1483 | Obsolete version of ~ an33... |
3orel2 1484 | Partial elimination of a t... |
3orel3 1485 | Partial elimination of a t... |
nanan 1488 | Conjunction in terms of al... |
dfnan2 1489 | Alternative denial in term... |
nanor 1490 | Alternative denial in term... |
nancom 1491 | Alternative denial is comm... |
nannan 1492 | Nested alternative denials... |
nanim 1493 | Implication in terms of al... |
nannot 1494 | Negation in terms of alter... |
nanbi 1495 | Biconditional in terms of ... |
nanbi1 1496 | Introduce a right anti-con... |
nanbi2 1497 | Introduce a left anti-conj... |
nanbi12 1498 | Join two logical equivalen... |
nanbi1i 1499 | Introduce a right anti-con... |
nanbi2i 1500 | Introduce a left anti-conj... |
nanbi12i 1501 | Join two logical equivalen... |
nanbi1d 1502 | Introduce a right anti-con... |
nanbi2d 1503 | Introduce a left anti-conj... |
nanbi12d 1504 | Join two logical equivalen... |
nanass 1505 | A characterization of when... |
xnor 1508 | Two ways to write XNOR (ex... |
xorcom 1509 | The connector ` \/_ ` is c... |
xorcomOLD 1510 | Obsolete version of ~ xorc... |
xorass 1511 | The connector ` \/_ ` is a... |
excxor 1512 | This tautology shows that ... |
xor2 1513 | Two ways to express "exclu... |
xoror 1514 | Exclusive disjunction impl... |
xornan 1515 | Exclusive disjunction impl... |
xornan2 1516 | XOR implies NAND (written ... |
xorneg2 1517 | The connector ` \/_ ` is n... |
xorneg1 1518 | The connector ` \/_ ` is n... |
xorneg 1519 | The connector ` \/_ ` is u... |
xorbi12i 1520 | Equality property for excl... |
xorbi12iOLD 1521 | Obsolete version of ~ xorb... |
xorbi12d 1522 | Equality property for excl... |
anxordi 1523 | Conjunction distributes ov... |
xorexmid 1524 | Exclusive-or variant of th... |
norcom 1527 | The connector ` -\/ ` is c... |
norcomOLD 1528 | Obsolete version of ~ norc... |
nornot 1529 | ` -. ` is expressible via ... |
noran 1530 | ` /\ ` is expressible via ... |
noror 1531 | ` \/ ` is expressible via ... |
norasslem1 1532 | This lemma shows the equiv... |
norasslem2 1533 | This lemma specializes ~ b... |
norasslem3 1534 | This lemma specializes ~ b... |
norass 1535 | A characterization of when... |
norassOLD 1536 | Obsolete version of ~ nora... |
trujust 1541 | Soundness justification th... |
tru 1543 | The truth value ` T. ` is ... |
dftru2 1544 | An alternate definition of... |
trut 1545 | A proposition is equivalen... |
mptru 1546 | Eliminate ` T. ` as an ant... |
tbtru 1547 | A proposition is equivalen... |
bitru 1548 | A theorem is equivalent to... |
trud 1549 | Anything implies ` T. ` . ... |
truan 1550 | True can be removed from a... |
fal 1553 | The truth value ` F. ` is ... |
nbfal 1554 | The negation of a proposit... |
bifal 1555 | A contradiction is equival... |
falim 1556 | The truth value ` F. ` imp... |
falimd 1557 | The truth value ` F. ` imp... |
dfnot 1558 | Given falsum ` F. ` , we c... |
inegd 1559 | Negation introduction rule... |
efald 1560 | Deduction based on reducti... |
pm2.21fal 1561 | If a wff and its negation ... |
truimtru 1562 | A ` -> ` identity. (Contr... |
truimfal 1563 | A ` -> ` identity. (Contr... |
falimtru 1564 | A ` -> ` identity. (Contr... |
falimfal 1565 | A ` -> ` identity. (Contr... |
nottru 1566 | A ` -. ` identity. (Contr... |
notfal 1567 | A ` -. ` identity. (Contr... |
trubitru 1568 | A ` <-> ` identity. (Cont... |
falbitru 1569 | A ` <-> ` identity. (Cont... |
trubifal 1570 | A ` <-> ` identity. (Cont... |
falbifal 1571 | A ` <-> ` identity. (Cont... |
truantru 1572 | A ` /\ ` identity. (Contr... |
truanfal 1573 | A ` /\ ` identity. (Contr... |
falantru 1574 | A ` /\ ` identity. (Contr... |
falanfal 1575 | A ` /\ ` identity. (Contr... |
truortru 1576 | A ` \/ ` identity. (Contr... |
truorfal 1577 | A ` \/ ` identity. (Contr... |
falortru 1578 | A ` \/ ` identity. (Contr... |
falorfal 1579 | A ` \/ ` identity. (Contr... |
trunantru 1580 | A ` -/\ ` identity. (Cont... |
trunanfal 1581 | A ` -/\ ` identity. (Cont... |
falnantru 1582 | A ` -/\ ` identity. (Cont... |
falnanfal 1583 | A ` -/\ ` identity. (Cont... |
truxortru 1584 | A ` \/_ ` identity. (Cont... |
truxorfal 1585 | A ` \/_ ` identity. (Cont... |
falxortru 1586 | A ` \/_ ` identity. (Cont... |
falxorfal 1587 | A ` \/_ ` identity. (Cont... |
trunortru 1588 | A ` -\/ ` identity. (Cont... |
trunorfal 1589 | A ` -\/ ` identity. (Cont... |
trunorfalOLD 1590 | Obsolete version of ~ trun... |
falnortru 1591 | A ` -\/ ` identity. (Cont... |
falnorfal 1592 | A ` -\/ ` identity. (Cont... |
falnorfalOLD 1593 | Obsolete version of ~ faln... |
hadbi123d 1596 | Equality theorem for the a... |
hadbi123i 1597 | Equality theorem for the a... |
hadass 1598 | Associative law for the ad... |
hadbi 1599 | The adder sum is the same ... |
hadcoma 1600 | Commutative law for the ad... |
hadcomaOLD 1601 | Obsolete version of ~ hadc... |
hadcomb 1602 | Commutative law for the ad... |
hadrot 1603 | Rotation law for the adder... |
hadnot 1604 | The adder sum distributes ... |
had1 1605 | If the first input is true... |
had0 1606 | If the first input is fals... |
hadifp 1607 | The value of the adder sum... |
cador 1610 | The adder carry in disjunc... |
cadan 1611 | The adder carry in conjunc... |
cadbi123d 1612 | Equality theorem for the a... |
cadbi123i 1613 | Equality theorem for the a... |
cadcoma 1614 | Commutative law for the ad... |
cadcomb 1615 | Commutative law for the ad... |
cadrot 1616 | Rotation law for the adder... |
cadnot 1617 | The adder carry distribute... |
cad11 1618 | If (at least) two inputs a... |
cad1 1619 | If one input is true, then... |
cad0 1620 | If one input is false, the... |
cad0OLD 1621 | Obsolete version of ~ cad0... |
cadifp 1622 | The value of the carry is,... |
cadtru 1623 | The adder carry is true as... |
minimp 1624 | A single axiom for minimal... |
minimp-syllsimp 1625 | Derivation of Syll-Simp ( ... |
minimp-ax1 1626 | Derivation of ~ ax-1 from ... |
minimp-ax2c 1627 | Derivation of a commuted f... |
minimp-ax2 1628 | Derivation of ~ ax-2 from ... |
minimp-pm2.43 1629 | Derivation of ~ pm2.43 (al... |
impsingle 1630 | The shortest single axiom ... |
impsingle-step4 1631 | Derivation of impsingle-st... |
impsingle-step8 1632 | Derivation of impsingle-st... |
impsingle-ax1 1633 | Derivation of impsingle-ax... |
impsingle-step15 1634 | Derivation of impsingle-st... |
impsingle-step18 1635 | Derivation of impsingle-st... |
impsingle-step19 1636 | Derivation of impsingle-st... |
impsingle-step20 1637 | Derivation of impsingle-st... |
impsingle-step21 1638 | Derivation of impsingle-st... |
impsingle-step22 1639 | Derivation of impsingle-st... |
impsingle-step25 1640 | Derivation of impsingle-st... |
impsingle-imim1 1641 | Derivation of impsingle-im... |
impsingle-peirce 1642 | Derivation of impsingle-pe... |
tarski-bernays-ax2 1643 | Derivation of ~ ax-2 from ... |
meredith 1644 | Carew Meredith's sole axio... |
merlem1 1645 | Step 3 of Meredith's proof... |
merlem2 1646 | Step 4 of Meredith's proof... |
merlem3 1647 | Step 7 of Meredith's proof... |
merlem4 1648 | Step 8 of Meredith's proof... |
merlem5 1649 | Step 11 of Meredith's proo... |
merlem6 1650 | Step 12 of Meredith's proo... |
merlem7 1651 | Between steps 14 and 15 of... |
merlem8 1652 | Step 15 of Meredith's proo... |
merlem9 1653 | Step 18 of Meredith's proo... |
merlem10 1654 | Step 19 of Meredith's proo... |
merlem11 1655 | Step 20 of Meredith's proo... |
merlem12 1656 | Step 28 of Meredith's proo... |
merlem13 1657 | Step 35 of Meredith's proo... |
luk-1 1658 | 1 of 3 axioms for proposit... |
luk-2 1659 | 2 of 3 axioms for proposit... |
luk-3 1660 | 3 of 3 axioms for proposit... |
luklem1 1661 | Used to rederive standard ... |
luklem2 1662 | Used to rederive standard ... |
luklem3 1663 | Used to rederive standard ... |
luklem4 1664 | Used to rederive standard ... |
luklem5 1665 | Used to rederive standard ... |
luklem6 1666 | Used to rederive standard ... |
luklem7 1667 | Used to rederive standard ... |
luklem8 1668 | Used to rederive standard ... |
ax1 1669 | Standard propositional axi... |
ax2 1670 | Standard propositional axi... |
ax3 1671 | Standard propositional axi... |
nic-dfim 1672 | This theorem "defines" imp... |
nic-dfneg 1673 | This theorem "defines" neg... |
nic-mp 1674 | Derive Nicod's rule of mod... |
nic-mpALT 1675 | A direct proof of ~ nic-mp... |
nic-ax 1676 | Nicod's axiom derived from... |
nic-axALT 1677 | A direct proof of ~ nic-ax... |
nic-imp 1678 | Inference for ~ nic-mp usi... |
nic-idlem1 1679 | Lemma for ~ nic-id . (Con... |
nic-idlem2 1680 | Lemma for ~ nic-id . Infe... |
nic-id 1681 | Theorem ~ id expressed wit... |
nic-swap 1682 | The connector ` -/\ ` is s... |
nic-isw1 1683 | Inference version of ~ nic... |
nic-isw2 1684 | Inference for swapping nes... |
nic-iimp1 1685 | Inference version of ~ nic... |
nic-iimp2 1686 | Inference version of ~ nic... |
nic-idel 1687 | Inference to remove the tr... |
nic-ich 1688 | Chained inference. (Contr... |
nic-idbl 1689 | Double the terms. Since d... |
nic-bijust 1690 | Biconditional justificatio... |
nic-bi1 1691 | Inference to extract one s... |
nic-bi2 1692 | Inference to extract the o... |
nic-stdmp 1693 | Derive the standard modus ... |
nic-luk1 1694 | Proof of ~ luk-1 from ~ ni... |
nic-luk2 1695 | Proof of ~ luk-2 from ~ ni... |
nic-luk3 1696 | Proof of ~ luk-3 from ~ ni... |
lukshef-ax1 1697 | This alternative axiom for... |
lukshefth1 1698 | Lemma for ~ renicax . (Co... |
lukshefth2 1699 | Lemma for ~ renicax . (Co... |
renicax 1700 | A rederivation of ~ nic-ax... |
tbw-bijust 1701 | Justification for ~ tbw-ne... |
tbw-negdf 1702 | The definition of negation... |
tbw-ax1 1703 | The first of four axioms i... |
tbw-ax2 1704 | The second of four axioms ... |
tbw-ax3 1705 | The third of four axioms i... |
tbw-ax4 1706 | The fourth of four axioms ... |
tbwsyl 1707 | Used to rederive the Lukas... |
tbwlem1 1708 | Used to rederive the Lukas... |
tbwlem2 1709 | Used to rederive the Lukas... |
tbwlem3 1710 | Used to rederive the Lukas... |
tbwlem4 1711 | Used to rederive the Lukas... |
tbwlem5 1712 | Used to rederive the Lukas... |
re1luk1 1713 | ~ luk-1 derived from the T... |
re1luk2 1714 | ~ luk-2 derived from the T... |
re1luk3 1715 | ~ luk-3 derived from the T... |
merco1 1716 | A single axiom for proposi... |
merco1lem1 1717 | Used to rederive the Tarsk... |
retbwax4 1718 | ~ tbw-ax4 rederived from ~... |
retbwax2 1719 | ~ tbw-ax2 rederived from ~... |
merco1lem2 1720 | Used to rederive the Tarsk... |
merco1lem3 1721 | Used to rederive the Tarsk... |
merco1lem4 1722 | Used to rederive the Tarsk... |
merco1lem5 1723 | Used to rederive the Tarsk... |
merco1lem6 1724 | Used to rederive the Tarsk... |
merco1lem7 1725 | Used to rederive the Tarsk... |
retbwax3 1726 | ~ tbw-ax3 rederived from ~... |
merco1lem8 1727 | Used to rederive the Tarsk... |
merco1lem9 1728 | Used to rederive the Tarsk... |
merco1lem10 1729 | Used to rederive the Tarsk... |
merco1lem11 1730 | Used to rederive the Tarsk... |
merco1lem12 1731 | Used to rederive the Tarsk... |
merco1lem13 1732 | Used to rederive the Tarsk... |
merco1lem14 1733 | Used to rederive the Tarsk... |
merco1lem15 1734 | Used to rederive the Tarsk... |
merco1lem16 1735 | Used to rederive the Tarsk... |
merco1lem17 1736 | Used to rederive the Tarsk... |
merco1lem18 1737 | Used to rederive the Tarsk... |
retbwax1 1738 | ~ tbw-ax1 rederived from ~... |
merco2 1739 | A single axiom for proposi... |
mercolem1 1740 | Used to rederive the Tarsk... |
mercolem2 1741 | Used to rederive the Tarsk... |
mercolem3 1742 | Used to rederive the Tarsk... |
mercolem4 1743 | Used to rederive the Tarsk... |
mercolem5 1744 | Used to rederive the Tarsk... |
mercolem6 1745 | Used to rederive the Tarsk... |
mercolem7 1746 | Used to rederive the Tarsk... |
mercolem8 1747 | Used to rederive the Tarsk... |
re1tbw1 1748 | ~ tbw-ax1 rederived from ~... |
re1tbw2 1749 | ~ tbw-ax2 rederived from ~... |
re1tbw3 1750 | ~ tbw-ax3 rederived from ~... |
re1tbw4 1751 | ~ tbw-ax4 rederived from ~... |
rb-bijust 1752 | Justification for ~ rb-imd... |
rb-imdf 1753 | The definition of implicat... |
anmp 1754 | Modus ponens for ` { \/ , ... |
rb-ax1 1755 | The first of four axioms i... |
rb-ax2 1756 | The second of four axioms ... |
rb-ax3 1757 | The third of four axioms i... |
rb-ax4 1758 | The fourth of four axioms ... |
rbsyl 1759 | Used to rederive the Lukas... |
rblem1 1760 | Used to rederive the Lukas... |
rblem2 1761 | Used to rederive the Lukas... |
rblem3 1762 | Used to rederive the Lukas... |
rblem4 1763 | Used to rederive the Lukas... |
rblem5 1764 | Used to rederive the Lukas... |
rblem6 1765 | Used to rederive the Lukas... |
rblem7 1766 | Used to rederive the Lukas... |
re1axmp 1767 | ~ ax-mp derived from Russe... |
re2luk1 1768 | ~ luk-1 derived from Russe... |
re2luk2 1769 | ~ luk-2 derived from Russe... |
re2luk3 1770 | ~ luk-3 derived from Russe... |
mptnan 1771 | Modus ponendo tollens 1, o... |
mptxor 1772 | Modus ponendo tollens 2, o... |
mtpor 1773 | Modus tollendo ponens (inc... |
mtpxor 1774 | Modus tollendo ponens (ori... |
stoic1a 1775 | Stoic logic Thema 1 (part ... |
stoic1b 1776 | Stoic logic Thema 1 (part ... |
stoic2a 1777 | Stoic logic Thema 2 versio... |
stoic2b 1778 | Stoic logic Thema 2 versio... |
stoic3 1779 | Stoic logic Thema 3. Stat... |
stoic4a 1780 | Stoic logic Thema 4 versio... |
stoic4b 1781 | Stoic logic Thema 4 versio... |
alnex 1784 | Universal quantification o... |
eximal 1785 | An equivalence between an ... |
nf2 1788 | Alternate definition of no... |
nf3 1789 | Alternate definition of no... |
nf4 1790 | Alternate definition of no... |
nfi 1791 | Deduce that ` x ` is not f... |
nfri 1792 | Consequence of the definit... |
nfd 1793 | Deduce that ` x ` is not f... |
nfrd 1794 | Consequence of the definit... |
nftht 1795 | Closed form of ~ nfth . (... |
nfntht 1796 | Closed form of ~ nfnth . ... |
nfntht2 1797 | Closed form of ~ nfnth . ... |
gen2 1799 | Generalization applied twi... |
mpg 1800 | Modus ponens combined with... |
mpgbi 1801 | Modus ponens on biconditio... |
mpgbir 1802 | Modus ponens on biconditio... |
nex 1803 | Generalization rule for ne... |
nfth 1804 | No variable is (effectivel... |
nfnth 1805 | No variable is (effectivel... |
hbth 1806 | No variable is (effectivel... |
nftru 1807 | The true constant has no f... |
nffal 1808 | The false constant has no ... |
sptruw 1809 | Version of ~ sp when ` ph ... |
altru 1810 | For all sets, ` T. ` is tr... |
alfal 1811 | For all sets, ` -. F. ` is... |
alim 1813 | Restatement of Axiom ~ ax-... |
alimi 1814 | Inference quantifying both... |
2alimi 1815 | Inference doubly quantifyi... |
ala1 1816 | Add an antecedent in a uni... |
al2im 1817 | Closed form of ~ al2imi . ... |
al2imi 1818 | Inference quantifying ante... |
alanimi 1819 | Variant of ~ al2imi with c... |
alimdh 1820 | Deduction form of Theorem ... |
albi 1821 | Theorem 19.15 of [Margaris... |
albii 1822 | Inference adding universal... |
2albii 1823 | Inference adding two unive... |
sylgt 1824 | Closed form of ~ sylg . (... |
sylg 1825 | A syllogism combined with ... |
alrimih 1826 | Inference form of Theorem ... |
hbxfrbi 1827 | A utility lemma to transfe... |
alex 1828 | Universal quantifier in te... |
exnal 1829 | Existential quantification... |
2nalexn 1830 | Part of theorem *11.5 in [... |
2exnaln 1831 | Theorem *11.22 in [Whitehe... |
2nexaln 1832 | Theorem *11.25 in [Whitehe... |
alimex 1833 | An equivalence between an ... |
aleximi 1834 | A variant of ~ al2imi : in... |
alexbii 1835 | Biconditional form of ~ al... |
exim 1836 | Theorem 19.22 of [Margaris... |
eximi 1837 | Inference adding existenti... |
2eximi 1838 | Inference adding two exist... |
eximii 1839 | Inference associated with ... |
exa1 1840 | Add an antecedent in an ex... |
19.38 1841 | Theorem 19.38 of [Margaris... |
19.38a 1842 | Under a nonfreeness hypoth... |
19.38b 1843 | Under a nonfreeness hypoth... |
imnang 1844 | Quantified implication in ... |
alinexa 1845 | A transformation of quanti... |
exnalimn 1846 | Existential quantification... |
alexn 1847 | A relationship between two... |
2exnexn 1848 | Theorem *11.51 in [Whitehe... |
exbi 1849 | Theorem 19.18 of [Margaris... |
exbii 1850 | Inference adding existenti... |
2exbii 1851 | Inference adding two exist... |
3exbii 1852 | Inference adding three exi... |
nfbiit 1853 | Equivalence theorem for th... |
nfbii 1854 | Equality theorem for the n... |
nfxfr 1855 | A utility lemma to transfe... |
nfxfrd 1856 | A utility lemma to transfe... |
nfnbi 1857 | A variable is nonfree in a... |
nfnbiOLD 1858 | Obsolete version of ~ nfnb... |
nfnt 1859 | If a variable is nonfree i... |
nfn 1860 | Inference associated with ... |
nfnd 1861 | Deduction associated with ... |
exanali 1862 | A transformation of quanti... |
2exanali 1863 | Theorem *11.521 in [Whiteh... |
exancom 1864 | Commutation of conjunction... |
exan 1865 | Place a conjunct in the sc... |
alrimdh 1866 | Deduction form of Theorem ... |
eximdh 1867 | Deduction from Theorem 19.... |
nexdh 1868 | Deduction for generalizati... |
albidh 1869 | Formula-building rule for ... |
exbidh 1870 | Formula-building rule for ... |
exsimpl 1871 | Simplification of an exist... |
exsimpr 1872 | Simplification of an exist... |
19.26 1873 | Theorem 19.26 of [Margaris... |
19.26-2 1874 | Theorem ~ 19.26 with two q... |
19.26-3an 1875 | Theorem ~ 19.26 with tripl... |
19.29 1876 | Theorem 19.29 of [Margaris... |
19.29r 1877 | Variation of ~ 19.29 . (C... |
19.29r2 1878 | Variation of ~ 19.29r with... |
19.29x 1879 | Variation of ~ 19.29 with ... |
19.35 1880 | Theorem 19.35 of [Margaris... |
19.35i 1881 | Inference associated with ... |
19.35ri 1882 | Inference associated with ... |
19.25 1883 | Theorem 19.25 of [Margaris... |
19.30 1884 | Theorem 19.30 of [Margaris... |
19.43 1885 | Theorem 19.43 of [Margaris... |
19.43OLD 1886 | Obsolete proof of ~ 19.43 ... |
19.33 1887 | Theorem 19.33 of [Margaris... |
19.33b 1888 | The antecedent provides a ... |
19.40 1889 | Theorem 19.40 of [Margaris... |
19.40-2 1890 | Theorem *11.42 in [Whitehe... |
19.40b 1891 | The antecedent provides a ... |
albiim 1892 | Split a biconditional and ... |
2albiim 1893 | Split a biconditional and ... |
exintrbi 1894 | Add/remove a conjunct in t... |
exintr 1895 | Introduce a conjunct in th... |
alsyl 1896 | Universally quantified and... |
nfimd 1897 | If in a context ` x ` is n... |
nfimt 1898 | Closed form of ~ nfim and ... |
nfim 1899 | If ` x ` is not free in ` ... |
nfand 1900 | If in a context ` x ` is n... |
nf3and 1901 | Deduction form of bound-va... |
nfan 1902 | If ` x ` is not free in ` ... |
nfnan 1903 | If ` x ` is not free in ` ... |
nf3an 1904 | If ` x ` is not free in ` ... |
nfbid 1905 | If in a context ` x ` is n... |
nfbi 1906 | If ` x ` is not free in ` ... |
nfor 1907 | If ` x ` is not free in ` ... |
nf3or 1908 | If ` x ` is not free in ` ... |
empty 1909 | Two characterizations of t... |
emptyex 1910 | On the empty domain, any e... |
emptyal 1911 | On the empty domain, any u... |
emptynf 1912 | On the empty domain, any v... |
ax5d 1914 | Version of ~ ax-5 with ant... |
ax5e 1915 | A rephrasing of ~ ax-5 usi... |
ax5ea 1916 | If a formula holds for som... |
nfv 1917 | If ` x ` is not present in... |
nfvd 1918 | ~ nfv with antecedent. Us... |
alimdv 1919 | Deduction form of Theorem ... |
eximdv 1920 | Deduction form of Theorem ... |
2alimdv 1921 | Deduction form of Theorem ... |
2eximdv 1922 | Deduction form of Theorem ... |
albidv 1923 | Formula-building rule for ... |
exbidv 1924 | Formula-building rule for ... |
nfbidv 1925 | An equality theorem for no... |
2albidv 1926 | Formula-building rule for ... |
2exbidv 1927 | Formula-building rule for ... |
3exbidv 1928 | Formula-building rule for ... |
4exbidv 1929 | Formula-building rule for ... |
alrimiv 1930 | Inference form of Theorem ... |
alrimivv 1931 | Inference form of Theorem ... |
alrimdv 1932 | Deduction form of Theorem ... |
exlimiv 1933 | Inference form of Theorem ... |
exlimiiv 1934 | Inference (Rule C) associa... |
exlimivv 1935 | Inference form of Theorem ... |
exlimdv 1936 | Deduction form of Theorem ... |
exlimdvv 1937 | Deduction form of Theorem ... |
exlimddv 1938 | Existential elimination ru... |
nexdv 1939 | Deduction for generalizati... |
2ax5 1940 | Quantification of two vari... |
stdpc5v 1941 | Version of ~ stdpc5 with a... |
19.21v 1942 | Version of ~ 19.21 with a ... |
19.32v 1943 | Version of ~ 19.32 with a ... |
19.31v 1944 | Version of ~ 19.31 with a ... |
19.23v 1945 | Version of ~ 19.23 with a ... |
19.23vv 1946 | Theorem ~ 19.23v extended ... |
pm11.53v 1947 | Version of ~ pm11.53 with ... |
19.36imv 1948 | One direction of ~ 19.36v ... |
19.36imvOLD 1949 | Obsolete version of ~ 19.3... |
19.36iv 1950 | Inference associated with ... |
19.37imv 1951 | One direction of ~ 19.37v ... |
19.37iv 1952 | Inference associated with ... |
19.41v 1953 | Version of ~ 19.41 with a ... |
19.41vv 1954 | Version of ~ 19.41 with tw... |
19.41vvv 1955 | Version of ~ 19.41 with th... |
19.41vvvv 1956 | Version of ~ 19.41 with fo... |
19.42v 1957 | Version of ~ 19.42 with a ... |
exdistr 1958 | Distribution of existentia... |
exdistrv 1959 | Distribute a pair of exist... |
4exdistrv 1960 | Distribute two pairs of ex... |
19.42vv 1961 | Version of ~ 19.42 with tw... |
exdistr2 1962 | Distribution of existentia... |
19.42vvv 1963 | Version of ~ 19.42 with th... |
3exdistr 1964 | Distribution of existentia... |
4exdistr 1965 | Distribution of existentia... |
weq 1966 | Extend wff definition to i... |
speimfw 1967 | Specialization, with addit... |
speimfwALT 1968 | Alternate proof of ~ speim... |
spimfw 1969 | Specialization, with addit... |
ax12i 1970 | Inference that has ~ ax-12... |
ax6v 1972 | Axiom B7 of [Tarski] p. 75... |
ax6ev 1973 | At least one individual ex... |
spimw 1974 | Specialization. Lemma 8 o... |
spimew 1975 | Existential introduction, ... |
speiv 1976 | Inference from existential... |
speivw 1977 | Version of ~ spei with a d... |
exgen 1978 | Rule of existential genera... |
extru 1979 | There exists a variable su... |
19.2 1980 | Theorem 19.2 of [Margaris]... |
19.2d 1981 | Deduction associated with ... |
19.8w 1982 | Weak version of ~ 19.8a an... |
spnfw 1983 | Weak version of ~ sp . Us... |
spvw 1984 | Version of ~ sp when ` x `... |
19.3v 1985 | Version of ~ 19.3 with a d... |
19.8v 1986 | Version of ~ 19.8a with a ... |
19.9v 1987 | Version of ~ 19.9 with a d... |
19.39 1988 | Theorem 19.39 of [Margaris... |
19.24 1989 | Theorem 19.24 of [Margaris... |
19.34 1990 | Theorem 19.34 of [Margaris... |
19.36v 1991 | Version of ~ 19.36 with a ... |
19.12vvv 1992 | Version of ~ 19.12vv with ... |
19.27v 1993 | Version of ~ 19.27 with a ... |
19.28v 1994 | Version of ~ 19.28 with a ... |
19.37v 1995 | Version of ~ 19.37 with a ... |
19.44v 1996 | Version of ~ 19.44 with a ... |
19.45v 1997 | Version of ~ 19.45 with a ... |
spimevw 1998 | Existential introduction, ... |
spimvw 1999 | A weak form of specializat... |
spvv 2000 | Specialization, using impl... |
spfalw 2001 | Version of ~ sp when ` ph ... |
chvarvv 2002 | Implicit substitution of `... |
equs4v 2003 | Version of ~ equs4 with a ... |
alequexv 2004 | Version of ~ equs4v with i... |
exsbim 2005 | One direction of the equiv... |
equsv 2006 | If a formula does not cont... |
equsalvw 2007 | Version of ~ equsalv with ... |
equsexvw 2008 | Version of ~ equsexv with ... |
cbvaliw 2009 | Change bound variable. Us... |
cbvalivw 2010 | Change bound variable. Us... |
ax7v 2012 | Weakened version of ~ ax-7... |
ax7v1 2013 | First of two weakened vers... |
ax7v2 2014 | Second of two weakened ver... |
equid 2015 | Identity law for equality.... |
nfequid 2016 | Bound-variable hypothesis ... |
equcomiv 2017 | Weaker form of ~ equcomi w... |
ax6evr 2018 | A commuted form of ~ ax6ev... |
ax7 2019 | Proof of ~ ax-7 from ~ ax7... |
equcomi 2020 | Commutative law for equali... |
equcom 2021 | Commutative law for equali... |
equcomd 2022 | Deduction form of ~ equcom... |
equcoms 2023 | An inference commuting equ... |
equtr 2024 | A transitive law for equal... |
equtrr 2025 | A transitive law for equal... |
equeuclr 2026 | Commuted version of ~ eque... |
equeucl 2027 | Equality is a left-Euclide... |
equequ1 2028 | An equivalence law for equ... |
equequ2 2029 | An equivalence law for equ... |
equtr2 2030 | Equality is a left-Euclide... |
stdpc6 2031 | One of the two equality ax... |
equvinv 2032 | A variable introduction la... |
equvinva 2033 | A modified version of the ... |
equvelv 2034 | A biconditional form of ~ ... |
ax13b 2035 | An equivalence between two... |
spfw 2036 | Weak version of ~ sp . Us... |
spw 2037 | Weak version of the specia... |
cbvalw 2038 | Change bound variable. Us... |
cbvalvw 2039 | Change bound variable. Us... |
cbvexvw 2040 | Change bound variable. Us... |
cbvaldvaw 2041 | Rule used to change the bo... |
cbvexdvaw 2042 | Rule used to change the bo... |
cbval2vw 2043 | Rule used to change bound ... |
cbvex2vw 2044 | Rule used to change bound ... |
cbvex4vw 2045 | Rule used to change bound ... |
alcomiw 2046 | Weak version of ~ alcom . ... |
alcomiwOLD 2047 | Obsolete version of ~ alco... |
hbn1fw 2048 | Weak version of ~ ax-10 fr... |
hbn1w 2049 | Weak version of ~ hbn1 . ... |
hba1w 2050 | Weak version of ~ hba1 . ... |
hbe1w 2051 | Weak version of ~ hbe1 . ... |
hbalw 2052 | Weak version of ~ hbal . ... |
19.8aw 2053 | If a formula is true, then... |
exexw 2054 | Existential quantification... |
spaev 2055 | A special instance of ~ sp... |
cbvaev 2056 | Change bound variable in a... |
aevlem0 2057 | Lemma for ~ aevlem . Inst... |
aevlem 2058 | Lemma for ~ aev and ~ axc1... |
aeveq 2059 | The antecedent ` A. x x = ... |
aev 2060 | A "distinctor elimination"... |
aev2 2061 | A version of ~ aev with tw... |
hbaev 2062 | All variables are effectiv... |
naev 2063 | If some set variables can ... |
naev2 2064 | Generalization of ~ hbnaev... |
hbnaev 2065 | Any variable is free in ` ... |
sbjust 2066 | Justification theorem for ... |
sbt 2069 | A substitution into a theo... |
sbtru 2070 | The result of substituting... |
stdpc4 2071 | The specialization axiom o... |
sbtALT 2072 | Alternate proof of ~ sbt ,... |
2stdpc4 2073 | A double specialization us... |
sbi1 2074 | Distribute substitution ov... |
spsbim 2075 | Distribute substitution ov... |
spsbbi 2076 | Biconditional property for... |
sbimi 2077 | Distribute substitution ov... |
sb2imi 2078 | Distribute substitution ov... |
sbbii 2079 | Infer substitution into bo... |
2sbbii 2080 | Infer double substitution ... |
sbimdv 2081 | Deduction substituting bot... |
sbbidv 2082 | Deduction substituting bot... |
sban 2083 | Conjunction inside and out... |
sb3an 2084 | Threefold conjunction insi... |
spsbe 2085 | Existential generalization... |
sbequ 2086 | Equality property for subs... |
sbequi 2087 | An equality theorem for su... |
sb6 2088 | Alternate definition of su... |
2sb6 2089 | Equivalence for double sub... |
sb1v 2090 | One direction of ~ sb5 , p... |
sbv 2091 | Substitution for a variabl... |
sbcom4 2092 | Commutativity law for subs... |
pm11.07 2093 | Axiom *11.07 in [Whitehead... |
sbrimvw 2094 | Substitution in an implica... |
sbievw 2095 | Conversion of implicit sub... |
sbiedvw 2096 | Conversion of implicit sub... |
2sbievw 2097 | Conversion of double impli... |
sbcom3vv 2098 | Substituting ` y ` for ` x... |
sbievw2 2099 | ~ sbievw applied twice, av... |
sbco2vv 2100 | A composition law for subs... |
equsb3 2101 | Substitution in an equalit... |
equsb3r 2102 | Substitution applied to th... |
equsb1v 2103 | Substitution applied to an... |
nsb 2104 | Any substitution in an alw... |
sbn1 2105 | One direction of ~ sbn , u... |
wel 2107 | Extend wff definition to i... |
ax8v 2109 | Weakened version of ~ ax-8... |
ax8v1 2110 | First of two weakened vers... |
ax8v2 2111 | Second of two weakened ver... |
ax8 2112 | Proof of ~ ax-8 from ~ ax8... |
elequ1 2113 | An identity law for the no... |
elsb1 2114 | Substitution for the first... |
cleljust 2115 | When the class variables i... |
ax9v 2117 | Weakened version of ~ ax-9... |
ax9v1 2118 | First of two weakened vers... |
ax9v2 2119 | Second of two weakened ver... |
ax9 2120 | Proof of ~ ax-9 from ~ ax9... |
elequ2 2121 | An identity law for the no... |
elequ2g 2122 | A form of ~ elequ2 with a ... |
elsb2 2123 | Substitution for the secon... |
ax6dgen 2124 | Tarski's system uses the w... |
ax10w 2125 | Weak version of ~ ax-10 fr... |
ax11w 2126 | Weak version of ~ ax-11 fr... |
ax11dgen 2127 | Degenerate instance of ~ a... |
ax12wlem 2128 | Lemma for weak version of ... |
ax12w 2129 | Weak version of ~ ax-12 fr... |
ax12dgen 2130 | Degenerate instance of ~ a... |
ax12wdemo 2131 | Example of an application ... |
ax13w 2132 | Weak version (principal in... |
ax13dgen1 2133 | Degenerate instance of ~ a... |
ax13dgen2 2134 | Degenerate instance of ~ a... |
ax13dgen3 2135 | Degenerate instance of ~ a... |
ax13dgen4 2136 | Degenerate instance of ~ a... |
hbn1 2138 | Alias for ~ ax-10 to be us... |
hbe1 2139 | The setvar ` x ` is not fr... |
hbe1a 2140 | Dual statement of ~ hbe1 .... |
nf5-1 2141 | One direction of ~ nf5 can... |
nf5i 2142 | Deduce that ` x ` is not f... |
nf5dh 2143 | Deduce that ` x ` is not f... |
nf5dv 2144 | Apply the definition of no... |
nfnaew 2145 | All variables are effectiv... |
nfnaewOLD 2146 | Obsolete version of ~ nfna... |
nfe1 2147 | The setvar ` x ` is not fr... |
nfa1 2148 | The setvar ` x ` is not fr... |
nfna1 2149 | A convenience theorem part... |
nfia1 2150 | Lemma 23 of [Monk2] p. 114... |
nfnf1 2151 | The setvar ` x ` is not fr... |
modal5 2152 | The analogue in our predic... |
nfs1v 2153 | The setvar ` x ` is not fr... |
alcoms 2155 | Swap quantifiers in an ant... |
alcom 2156 | Theorem 19.5 of [Margaris]... |
alrot3 2157 | Theorem *11.21 in [Whitehe... |
alrot4 2158 | Rotate four universal quan... |
sbal 2159 | Move universal quantifier ... |
sbalv 2160 | Quantify with new variable... |
sbcom2 2161 | Commutativity law for subs... |
excom 2162 | Theorem 19.11 of [Margaris... |
excomim 2163 | One direction of Theorem 1... |
excom13 2164 | Swap 1st and 3rd existenti... |
exrot3 2165 | Rotate existential quantif... |
exrot4 2166 | Rotate existential quantif... |
hbal 2167 | If ` x ` is not free in ` ... |
hbald 2168 | Deduction form of bound-va... |
hbsbw 2169 | If ` z ` is not free in ` ... |
nfa2 2170 | Lemma 24 of [Monk2] p. 114... |
ax12v 2172 | This is essentially Axiom ... |
ax12v2 2173 | It is possible to remove a... |
19.8a 2174 | If a wff is true, it is tr... |
19.8ad 2175 | If a wff is true, it is tr... |
sp 2176 | Specialization. A univers... |
spi 2177 | Inference rule of universa... |
sps 2178 | Generalization of antecede... |
2sp 2179 | A double specialization (s... |
spsd 2180 | Deduction generalizing ant... |
19.2g 2181 | Theorem 19.2 of [Margaris]... |
19.21bi 2182 | Inference form of ~ 19.21 ... |
19.21bbi 2183 | Inference removing two uni... |
19.23bi 2184 | Inference form of Theorem ... |
nexr 2185 | Inference associated with ... |
qexmid 2186 | Quantified excluded middle... |
nf5r 2187 | Consequence of the definit... |
nf5ri 2188 | Consequence of the definit... |
nf5rd 2189 | Consequence of the definit... |
spimedv 2190 | Deduction version of ~ spi... |
spimefv 2191 | Version of ~ spime with a ... |
nfim1 2192 | A closed form of ~ nfim . ... |
nfan1 2193 | A closed form of ~ nfan . ... |
19.3t 2194 | Closed form of ~ 19.3 and ... |
19.3 2195 | A wff may be quantified wi... |
19.9d 2196 | A deduction version of one... |
19.9t 2197 | Closed form of ~ 19.9 and ... |
19.9 2198 | A wff may be existentially... |
19.21t 2199 | Closed form of Theorem 19.... |
19.21 2200 | Theorem 19.21 of [Margaris... |
stdpc5 2201 | An axiom scheme of standar... |
19.21-2 2202 | Version of ~ 19.21 with tw... |
19.23t 2203 | Closed form of Theorem 19.... |
19.23 2204 | Theorem 19.23 of [Margaris... |
alimd 2205 | Deduction form of Theorem ... |
alrimi 2206 | Inference form of Theorem ... |
alrimdd 2207 | Deduction form of Theorem ... |
alrimd 2208 | Deduction form of Theorem ... |
eximd 2209 | Deduction form of Theorem ... |
exlimi 2210 | Inference associated with ... |
exlimd 2211 | Deduction form of Theorem ... |
exlimimdd 2212 | Existential elimination ru... |
exlimdd 2213 | Existential elimination ru... |
nexd 2214 | Deduction for generalizati... |
albid 2215 | Formula-building rule for ... |
exbid 2216 | Formula-building rule for ... |
nfbidf 2217 | An equality theorem for ef... |
19.16 2218 | Theorem 19.16 of [Margaris... |
19.17 2219 | Theorem 19.17 of [Margaris... |
19.27 2220 | Theorem 19.27 of [Margaris... |
19.28 2221 | Theorem 19.28 of [Margaris... |
19.19 2222 | Theorem 19.19 of [Margaris... |
19.36 2223 | Theorem 19.36 of [Margaris... |
19.36i 2224 | Inference associated with ... |
19.37 2225 | Theorem 19.37 of [Margaris... |
19.32 2226 | Theorem 19.32 of [Margaris... |
19.31 2227 | Theorem 19.31 of [Margaris... |
19.41 2228 | Theorem 19.41 of [Margaris... |
19.42 2229 | Theorem 19.42 of [Margaris... |
19.44 2230 | Theorem 19.44 of [Margaris... |
19.45 2231 | Theorem 19.45 of [Margaris... |
spimfv 2232 | Specialization, using impl... |
chvarfv 2233 | Implicit substitution of `... |
cbv3v2 2234 | Version of ~ cbv3 with two... |
sbalex 2235 | Equivalence of two ways to... |
sb4av 2236 | Version of ~ sb4a with a d... |
sbimd 2237 | Deduction substituting bot... |
sbbid 2238 | Deduction substituting bot... |
2sbbid 2239 | Deduction doubly substitut... |
sbequ1 2240 | An equality theorem for su... |
sbequ2 2241 | An equality theorem for su... |
sbequ2OLD 2242 | Obsolete version of ~ sbeq... |
stdpc7 2243 | One of the two equality ax... |
sbequ12 2244 | An equality theorem for su... |
sbequ12r 2245 | An equality theorem for su... |
sbelx 2246 | Elimination of substitutio... |
sbequ12a 2247 | An equality theorem for su... |
sbid 2248 | An identity theorem for su... |
sbcov 2249 | A composition law for subs... |
sb6a 2250 | Equivalence for substituti... |
sbid2vw 2251 | Reverting substitution yie... |
axc16g 2252 | Generalization of ~ axc16 ... |
axc16 2253 | Proof of older axiom ~ ax-... |
axc16gb 2254 | Biconditional strengthenin... |
axc16nf 2255 | If ~ dtru is false, then t... |
axc11v 2256 | Version of ~ axc11 with a ... |
axc11rv 2257 | Version of ~ axc11r with a... |
drsb2 2258 | Formula-building lemma for... |
equsalv 2259 | An equivalence related to ... |
equsexv 2260 | An equivalence related to ... |
equsexvOLD 2261 | Obsolete version of ~ equs... |
sbft 2262 | Substitution has no effect... |
sbf 2263 | Substitution for a variabl... |
sbf2 2264 | Substitution has no effect... |
sbh 2265 | Substitution for a variabl... |
hbs1 2266 | The setvar ` x ` is not fr... |
nfs1f 2267 | If ` x ` is not free in ` ... |
sb5 2268 | Alternate definition of su... |
sb5OLD 2269 | Obsolete version of ~ sb5 ... |
sb56OLD 2270 | Obsolete version of ~ sbal... |
equs5av 2271 | A property related to subs... |
2sb5 2272 | Equivalence for double sub... |
sbco4lem 2273 | Lemma for ~ sbco4 . It re... |
sbco4lemOLD 2274 | Obsolete version of ~ sbco... |
sbco4 2275 | Two ways of exchanging two... |
dfsb7 2276 | An alternate definition of... |
sbn 2277 | Negation inside and outsid... |
sbex 2278 | Move existential quantifie... |
nf5 2279 | Alternate definition of ~ ... |
nf6 2280 | An alternate definition of... |
nf5d 2281 | Deduce that ` x ` is not f... |
nf5di 2282 | Since the converse holds b... |
19.9h 2283 | A wff may be existentially... |
19.21h 2284 | Theorem 19.21 of [Margaris... |
19.23h 2285 | Theorem 19.23 of [Margaris... |
exlimih 2286 | Inference associated with ... |
exlimdh 2287 | Deduction form of Theorem ... |
equsalhw 2288 | Version of ~ equsalh with ... |
equsexhv 2289 | An equivalence related to ... |
hba1 2290 | The setvar ` x ` is not fr... |
hbnt 2291 | Closed theorem version of ... |
hbn 2292 | If ` x ` is not free in ` ... |
hbnd 2293 | Deduction form of bound-va... |
hbim1 2294 | A closed form of ~ hbim . ... |
hbimd 2295 | Deduction form of bound-va... |
hbim 2296 | If ` x ` is not free in ` ... |
hban 2297 | If ` x ` is not free in ` ... |
hb3an 2298 | If ` x ` is not free in ` ... |
sbi2 2299 | Introduction of implicatio... |
sbim 2300 | Implication inside and out... |
sbrim 2301 | Substitution in an implica... |
sbrimOLD 2302 | Obsolete version of ~ sbri... |
sblim 2303 | Substitution in an implica... |
sbor 2304 | Disjunction inside and out... |
sbbi 2305 | Equivalence inside and out... |
sblbis 2306 | Introduce left bicondition... |
sbrbis 2307 | Introduce right biconditio... |
sbrbif 2308 | Introduce right biconditio... |
sbiev 2309 | Conversion of implicit sub... |
sbiedw 2310 | Conversion of implicit sub... |
axc7 2311 | Show that the original axi... |
axc7e 2312 | Abbreviated version of ~ a... |
modal-b 2313 | The analogue in our predic... |
19.9ht 2314 | A closed version of ~ 19.9... |
axc4 2315 | Show that the original axi... |
axc4i 2316 | Inference version of ~ axc... |
nfal 2317 | If ` x ` is not free in ` ... |
nfex 2318 | If ` x ` is not free in ` ... |
hbex 2319 | If ` x ` is not free in ` ... |
nfnf 2320 | If ` x ` is not free in ` ... |
19.12 2321 | Theorem 19.12 of [Margaris... |
nfald 2322 | Deduction form of ~ nfal .... |
nfexd 2323 | If ` x ` is not free in ` ... |
nfsbv 2324 | If ` z ` is not free in ` ... |
nfsbvOLD 2325 | Obsolete version of ~ nfsb... |
hbsbwOLD 2326 | Obsolete version of ~ hbsb... |
sbco2v 2327 | A composition law for subs... |
aaan 2328 | Distribute universal quant... |
aaanOLD 2329 | Obsolete version of ~ aaan... |
eeor 2330 | Distribute existential qua... |
eeorOLD 2331 | Obsolete version of ~ eeor... |
cbv3v 2332 | Rule used to change bound ... |
cbv1v 2333 | Rule used to change bound ... |
cbv2w 2334 | Rule used to change bound ... |
cbvaldw 2335 | Deduction used to change b... |
cbvexdw 2336 | Deduction used to change b... |
cbv3hv 2337 | Rule used to change bound ... |
cbvalv1 2338 | Rule used to change bound ... |
cbvexv1 2339 | Rule used to change bound ... |
cbval2v 2340 | Rule used to change bound ... |
cbval2vOLD 2341 | Obsolete version of ~ cbva... |
cbvex2v 2342 | Rule used to change bound ... |
dvelimhw 2343 | Proof of ~ dvelimh without... |
pm11.53 2344 | Theorem *11.53 in [Whitehe... |
19.12vv 2345 | Special case of ~ 19.12 wh... |
eean 2346 | Distribute existential qua... |
eeanv 2347 | Distribute a pair of exist... |
eeeanv 2348 | Distribute three existenti... |
ee4anv 2349 | Distribute two pairs of ex... |
sb8v 2350 | Substitution of variable i... |
sb8f 2351 | Substitution of variable i... |
sb8fOLD 2352 | Obsolete version of ~ sb8f... |
sb8ef 2353 | Substitution of variable i... |
2sb8ef 2354 | An equivalent expression f... |
sb6rfv 2355 | Reversed substitution. Ve... |
sbnf2 2356 | Two ways of expressing " `... |
exsb 2357 | An equivalent expression f... |
2exsb 2358 | An equivalent expression f... |
sbbib 2359 | Reversal of substitution. ... |
sbbibvv 2360 | Reversal of substitution. ... |
sbievg 2361 | Substitution applied to ex... |
cleljustALT 2362 | Alternate proof of ~ clelj... |
cleljustALT2 2363 | Alternate proof of ~ clelj... |
equs5aALT 2364 | Alternate proof of ~ equs5... |
equs5eALT 2365 | Alternate proof of ~ equs5... |
axc11r 2366 | Same as ~ axc11 but with r... |
dral1v 2367 | Formula-building lemma for... |
dral1vOLD 2368 | Obsolete version of ~ dral... |
drex1v 2369 | Formula-building lemma for... |
drnf1v 2370 | Formula-building lemma for... |
drnf1vOLD 2371 | Obsolete version of ~ drnf... |
ax13v 2373 | A weaker version of ~ ax-1... |
ax13lem1 2374 | A version of ~ ax13v with ... |
ax13 2375 | Derive ~ ax-13 from ~ ax13... |
ax13lem2 2376 | Lemma for ~ nfeqf2 . This... |
nfeqf2 2377 | An equation between setvar... |
dveeq2 2378 | Quantifier introduction wh... |
nfeqf1 2379 | An equation between setvar... |
dveeq1 2380 | Quantifier introduction wh... |
nfeqf 2381 | A variable is effectively ... |
axc9 2382 | Derive set.mm's original ~... |
ax6e 2383 | At least one individual ex... |
ax6 2384 | Theorem showing that ~ ax-... |
axc10 2385 | Show that the original axi... |
spimt 2386 | Closed theorem form of ~ s... |
spim 2387 | Specialization, using impl... |
spimed 2388 | Deduction version of ~ spi... |
spime 2389 | Existential introduction, ... |
spimv 2390 | A version of ~ spim with a... |
spimvALT 2391 | Alternate proof of ~ spimv... |
spimev 2392 | Distinct-variable version ... |
spv 2393 | Specialization, using impl... |
spei 2394 | Inference from existential... |
chvar 2395 | Implicit substitution of `... |
chvarv 2396 | Implicit substitution of `... |
cbv3 2397 | Rule used to change bound ... |
cbval 2398 | Rule used to change bound ... |
cbvex 2399 | Rule used to change bound ... |
cbvalv 2400 | Rule used to change bound ... |
cbvexv 2401 | Rule used to change bound ... |
cbv1 2402 | Rule used to change bound ... |
cbv2 2403 | Rule used to change bound ... |
cbv3h 2404 | Rule used to change bound ... |
cbv1h 2405 | Rule used to change bound ... |
cbv2h 2406 | Rule used to change bound ... |
cbvald 2407 | Deduction used to change b... |
cbvexd 2408 | Deduction used to change b... |
cbvaldva 2409 | Rule used to change the bo... |
cbvexdva 2410 | Rule used to change the bo... |
cbval2 2411 | Rule used to change bound ... |
cbvex2 2412 | Rule used to change bound ... |
cbval2vv 2413 | Rule used to change bound ... |
cbvex2vv 2414 | Rule used to change bound ... |
cbvex4v 2415 | Rule used to change bound ... |
equs4 2416 | Lemma used in proofs of im... |
equsal 2417 | An equivalence related to ... |
equsex 2418 | An equivalence related to ... |
equsexALT 2419 | Alternate proof of ~ equse... |
equsalh 2420 | An equivalence related to ... |
equsexh 2421 | An equivalence related to ... |
axc15 2422 | Derivation of set.mm's ori... |
ax12 2423 | Rederivation of Axiom ~ ax... |
ax12b 2424 | A bidirectional version of... |
ax13ALT 2425 | Alternate proof of ~ ax13 ... |
axc11n 2426 | Derive set.mm's original ~... |
aecom 2427 | Commutation law for identi... |
aecoms 2428 | A commutation rule for ide... |
naecoms 2429 | A commutation rule for dis... |
axc11 2430 | Show that ~ ax-c11 can be ... |
hbae 2431 | All variables are effectiv... |
hbnae 2432 | All variables are effectiv... |
nfae 2433 | All variables are effectiv... |
nfnae 2434 | All variables are effectiv... |
hbnaes 2435 | Rule that applies ~ hbnae ... |
axc16i 2436 | Inference with ~ axc16 as ... |
axc16nfALT 2437 | Alternate proof of ~ axc16... |
dral2 2438 | Formula-building lemma for... |
dral1 2439 | Formula-building lemma for... |
dral1ALT 2440 | Alternate proof of ~ dral1... |
drex1 2441 | Formula-building lemma for... |
drex2 2442 | Formula-building lemma for... |
drnf1 2443 | Formula-building lemma for... |
drnf2 2444 | Formula-building lemma for... |
nfald2 2445 | Variation on ~ nfald which... |
nfexd2 2446 | Variation on ~ nfexd which... |
exdistrf 2447 | Distribution of existentia... |
dvelimf 2448 | Version of ~ dvelimv witho... |
dvelimdf 2449 | Deduction form of ~ dvelim... |
dvelimh 2450 | Version of ~ dvelim withou... |
dvelim 2451 | This theorem can be used t... |
dvelimv 2452 | Similar to ~ dvelim with f... |
dvelimnf 2453 | Version of ~ dvelim using ... |
dveeq2ALT 2454 | Alternate proof of ~ dveeq... |
equvini 2455 | A variable introduction la... |
equvel 2456 | A variable elimination law... |
equs5a 2457 | A property related to subs... |
equs5e 2458 | A property related to subs... |
equs45f 2459 | Two ways of expressing sub... |
equs5 2460 | Lemma used in proofs of su... |
dveel1 2461 | Quantifier introduction wh... |
dveel2 2462 | Quantifier introduction wh... |
axc14 2463 | Axiom ~ ax-c14 is redundan... |
sb6x 2464 | Equivalence involving subs... |
sbequ5 2465 | Substitution does not chan... |
sbequ6 2466 | Substitution does not chan... |
sb5rf 2467 | Reversed substitution. Us... |
sb6rf 2468 | Reversed substitution. Fo... |
ax12vALT 2469 | Alternate proof of ~ ax12v... |
2ax6elem 2470 | We can always find values ... |
2ax6e 2471 | We can always find values ... |
2sb5rf 2472 | Reversed double substituti... |
2sb6rf 2473 | Reversed double substituti... |
sbel2x 2474 | Elimination of double subs... |
sb4b 2475 | Simplified definition of s... |
sb4bOLD 2476 | Obsolete version of ~ sb4b... |
sb3b 2477 | Simplified definition of s... |
sb3 2478 | One direction of a simplif... |
sb1 2479 | One direction of a simplif... |
sb2 2480 | One direction of a simplif... |
sb3OLD 2481 | Obsolete version of ~ sb3 ... |
sb1OLD 2482 | Obsolete version of ~ sb1 ... |
sb3bOLD 2483 | Obsolete version of ~ sb3b... |
sb4a 2484 | A version of one implicati... |
dfsb1 2485 | Alternate definition of su... |
hbsb2 2486 | Bound-variable hypothesis ... |
nfsb2 2487 | Bound-variable hypothesis ... |
hbsb2a 2488 | Special case of a bound-va... |
sb4e 2489 | One direction of a simplif... |
hbsb2e 2490 | Special case of a bound-va... |
hbsb3 2491 | If ` y ` is not free in ` ... |
nfs1 2492 | If ` y ` is not free in ` ... |
axc16ALT 2493 | Alternate proof of ~ axc16... |
axc16gALT 2494 | Alternate proof of ~ axc16... |
equsb1 2495 | Substitution applied to an... |
equsb2 2496 | Substitution applied to an... |
dfsb2 2497 | An alternate definition of... |
dfsb3 2498 | An alternate definition of... |
drsb1 2499 | Formula-building lemma for... |
sb2ae 2500 | In the case of two success... |
sb6f 2501 | Equivalence for substituti... |
sb5f 2502 | Equivalence for substituti... |
nfsb4t 2503 | A variable not free in a p... |
nfsb4 2504 | A variable not free in a p... |
sbequ8 2505 | Elimination of equality fr... |
sbie 2506 | Conversion of implicit sub... |
sbied 2507 | Conversion of implicit sub... |
sbiedv 2508 | Conversion of implicit sub... |
2sbiev 2509 | Conversion of double impli... |
sbcom3 2510 | Substituting ` y ` for ` x... |
sbco 2511 | A composition law for subs... |
sbid2 2512 | An identity law for substi... |
sbid2v 2513 | An identity law for substi... |
sbidm 2514 | An idempotent law for subs... |
sbco2 2515 | A composition law for subs... |
sbco2d 2516 | A composition law for subs... |
sbco3 2517 | A composition law for subs... |
sbcom 2518 | A commutativity law for su... |
sbtrt 2519 | Partially closed form of ~... |
sbtr 2520 | A partial converse to ~ sb... |
sb8 2521 | Substitution of variable i... |
sb8e 2522 | Substitution of variable i... |
sb9 2523 | Commutation of quantificat... |
sb9i 2524 | Commutation of quantificat... |
sbhb 2525 | Two ways of expressing " `... |
nfsbd 2526 | Deduction version of ~ nfs... |
nfsb 2527 | If ` z ` is not free in ` ... |
nfsbOLD 2528 | Obsolete version of ~ nfsb... |
hbsb 2529 | If ` z ` is not free in ` ... |
sb7f 2530 | This version of ~ dfsb7 do... |
sb7h 2531 | This version of ~ dfsb7 do... |
sb10f 2532 | Hao Wang's identity axiom ... |
sbal1 2533 | Check out ~ sbal for a ver... |
sbal2 2534 | Move quantifier in and out... |
2sb8e 2535 | An equivalent expression f... |
dfmoeu 2536 | An elementary proof of ~ m... |
dfeumo 2537 | An elementary proof showin... |
mojust 2539 | Soundness justification th... |
nexmo 2541 | Nonexistence implies uniqu... |
exmo 2542 | Any proposition holds for ... |
moabs 2543 | Absorption of existence co... |
moim 2544 | The at-most-one quantifier... |
moimi 2545 | The at-most-one quantifier... |
moimdv 2546 | The at-most-one quantifier... |
mobi 2547 | Equivalence theorem for th... |
mobii 2548 | Formula-building rule for ... |
mobidv 2549 | Formula-building rule for ... |
mobid 2550 | Formula-building rule for ... |
moa1 2551 | If an implication holds fo... |
moan 2552 | "At most one" is still the... |
moani 2553 | "At most one" is still tru... |
moor 2554 | "At most one" is still the... |
mooran1 2555 | "At most one" imports disj... |
mooran2 2556 | "At most one" exports disj... |
nfmo1 2557 | Bound-variable hypothesis ... |
nfmod2 2558 | Bound-variable hypothesis ... |
nfmodv 2559 | Bound-variable hypothesis ... |
nfmov 2560 | Bound-variable hypothesis ... |
nfmod 2561 | Bound-variable hypothesis ... |
nfmo 2562 | Bound-variable hypothesis ... |
mof 2563 | Version of ~ df-mo with di... |
mo3 2564 | Alternate definition of th... |
mo 2565 | Equivalent definitions of ... |
mo4 2566 | At-most-one quantifier exp... |
mo4f 2567 | At-most-one quantifier exp... |
eu3v 2570 | An alternate way to expres... |
eujust 2571 | Soundness justification th... |
eujustALT 2572 | Alternate proof of ~ eujus... |
eu6lem 2573 | Lemma of ~ eu6im . A diss... |
eu6 2574 | Alternate definition of th... |
eu6im 2575 | One direction of ~ eu6 nee... |
euf 2576 | Version of ~ eu6 with disj... |
euex 2577 | Existential uniqueness imp... |
eumo 2578 | Existential uniqueness imp... |
eumoi 2579 | Uniqueness inferred from e... |
exmoeub 2580 | Existence implies that uni... |
exmoeu 2581 | Existence is equivalent to... |
moeuex 2582 | Uniqueness implies that ex... |
moeu 2583 | Uniqueness is equivalent t... |
eubi 2584 | Equivalence theorem for th... |
eubii 2585 | Introduce unique existenti... |
eubidv 2586 | Formula-building rule for ... |
eubid 2587 | Formula-building rule for ... |
nfeu1 2588 | Bound-variable hypothesis ... |
nfeu1ALT 2589 | Alternate proof of ~ nfeu1... |
nfeud2 2590 | Bound-variable hypothesis ... |
nfeudw 2591 | Bound-variable hypothesis ... |
nfeud 2592 | Bound-variable hypothesis ... |
nfeuw 2593 | Bound-variable hypothesis ... |
nfeu 2594 | Bound-variable hypothesis ... |
dfeu 2595 | Rederive ~ df-eu from the ... |
dfmo 2596 | Rederive ~ df-mo from the ... |
euequ 2597 | There exists a unique set ... |
sb8eulem 2598 | Lemma. Factor out the com... |
sb8euv 2599 | Variable substitution in u... |
sb8eu 2600 | Variable substitution in u... |
sb8mo 2601 | Variable substitution for ... |
cbvmovw 2602 | Change bound variable. Us... |
cbvmow 2603 | Rule used to change bound ... |
cbvmowOLD 2604 | Obsolete version of ~ cbvm... |
cbvmo 2605 | Rule used to change bound ... |
cbveuvw 2606 | Change bound variable. Us... |
cbveuw 2607 | Version of ~ cbveu with a ... |
cbveuwOLD 2608 | Obsolete version of ~ cbve... |
cbveu 2609 | Rule used to change bound ... |
cbveuALT 2610 | Alternative proof of ~ cbv... |
eu2 2611 | An alternate way of defini... |
eu1 2612 | An alternate way to expres... |
euor 2613 | Introduce a disjunct into ... |
euorv 2614 | Introduce a disjunct into ... |
euor2 2615 | Introduce or eliminate a d... |
sbmo 2616 | Substitution into an at-mo... |
eu4 2617 | Uniqueness using implicit ... |
euimmo 2618 | Existential uniqueness imp... |
euim 2619 | Add unique existential qua... |
moanimlem 2620 | Factor out the common proo... |
moanimv 2621 | Introduction of a conjunct... |
moanim 2622 | Introduction of a conjunct... |
euan 2623 | Introduction of a conjunct... |
moanmo 2624 | Nested at-most-one quantif... |
moaneu 2625 | Nested at-most-one and uni... |
euanv 2626 | Introduction of a conjunct... |
mopick 2627 | "At most one" picks a vari... |
moexexlem 2628 | Factor out the proof skele... |
2moexv 2629 | Double quantification with... |
moexexvw 2630 | "At most one" double quant... |
2moswapv 2631 | A condition allowing to sw... |
2euswapv 2632 | A condition allowing to sw... |
2euexv 2633 | Double quantification with... |
2exeuv 2634 | Double existential uniquen... |
eupick 2635 | Existential uniqueness "pi... |
eupicka 2636 | Version of ~ eupick with c... |
eupickb 2637 | Existential uniqueness "pi... |
eupickbi 2638 | Theorem *14.26 in [Whitehe... |
mopick2 2639 | "At most one" can show the... |
moexex 2640 | "At most one" double quant... |
moexexv 2641 | "At most one" double quant... |
2moex 2642 | Double quantification with... |
2euex 2643 | Double quantification with... |
2eumo 2644 | Nested unique existential ... |
2eu2ex 2645 | Double existential uniquen... |
2moswap 2646 | A condition allowing to sw... |
2euswap 2647 | A condition allowing to sw... |
2exeu 2648 | Double existential uniquen... |
2mo2 2649 | Two ways of expressing "th... |
2mo 2650 | Two ways of expressing "th... |
2mos 2651 | Double "there exists at mo... |
2eu1 2652 | Double existential uniquen... |
2eu1v 2653 | Double existential uniquen... |
2eu2 2654 | Double existential uniquen... |
2eu3 2655 | Double existential uniquen... |
2eu4 2656 | This theorem provides us w... |
2eu5 2657 | An alternate definition of... |
2eu6 2658 | Two equivalent expressions... |
2eu7 2659 | Two equivalent expressions... |
2eu8 2660 | Two equivalent expressions... |
euae 2661 | Two ways to express "exact... |
exists1 2662 | Two ways to express "exact... |
exists2 2663 | A condition implying that ... |
barbara 2664 | "Barbara", one of the fund... |
celarent 2665 | "Celarent", one of the syl... |
darii 2666 | "Darii", one of the syllog... |
dariiALT 2667 | Alternate proof of ~ darii... |
ferio 2668 | "Ferio" ("Ferioque"), one ... |
barbarilem 2669 | Lemma for ~ barbari and th... |
barbari 2670 | "Barbari", one of the syll... |
barbariALT 2671 | Alternate proof of ~ barba... |
celaront 2672 | "Celaront", one of the syl... |
cesare 2673 | "Cesare", one of the syllo... |
camestres 2674 | "Camestres", one of the sy... |
festino 2675 | "Festino", one of the syll... |
festinoALT 2676 | Alternate proof of ~ festi... |
baroco 2677 | "Baroco", one of the syllo... |
barocoALT 2678 | Alternate proof of ~ festi... |
cesaro 2679 | "Cesaro", one of the syllo... |
camestros 2680 | "Camestros", one of the sy... |
datisi 2681 | "Datisi", one of the syllo... |
disamis 2682 | "Disamis", one of the syll... |
ferison 2683 | "Ferison", one of the syll... |
bocardo 2684 | "Bocardo", one of the syll... |
darapti 2685 | "Darapti", one of the syll... |
daraptiALT 2686 | Alternate proof of ~ darap... |
felapton 2687 | "Felapton", one of the syl... |
calemes 2688 | "Calemes", one of the syll... |
dimatis 2689 | "Dimatis", one of the syll... |
fresison 2690 | "Fresison", one of the syl... |
calemos 2691 | "Calemos", one of the syll... |
fesapo 2692 | "Fesapo", one of the syllo... |
bamalip 2693 | "Bamalip", one of the syll... |
axia1 2694 | Left 'and' elimination (in... |
axia2 2695 | Right 'and' elimination (i... |
axia3 2696 | 'And' introduction (intuit... |
axin1 2697 | 'Not' introduction (intuit... |
axin2 2698 | 'Not' elimination (intuiti... |
axio 2699 | Definition of 'or' (intuit... |
axi4 2700 | Specialization (intuitioni... |
axi5r 2701 | Converse of ~ axc4 (intuit... |
axial 2702 | The setvar ` x ` is not fr... |
axie1 2703 | The setvar ` x ` is not fr... |
axie2 2704 | A key property of existent... |
axi9 2705 | Axiom of existence (intuit... |
axi10 2706 | Axiom of Quantifier Substi... |
axi12 2707 | Axiom of Quantifier Introd... |
axbnd 2708 | Axiom of Bundling (intuiti... |
axexte 2710 | The axiom of extensionalit... |
axextg 2711 | A generalization of the ax... |
axextb 2712 | A bidirectional version of... |
axextmo 2713 | There exists at most one s... |
nulmo 2714 | There exists at most one e... |
eleq1ab 2717 | Extension (in the sense of... |
cleljustab 2718 | Extension of ~ cleljust fr... |
abid 2719 | Simplification of class ab... |
vexwt 2720 | A standard theorem of pred... |
vexw 2721 | If ` ph ` is a theorem, th... |
vextru 2722 | Every setvar is a member o... |
nfsab1 2723 | Bound-variable hypothesis ... |
hbab1 2724 | Bound-variable hypothesis ... |
hbab1OLD 2725 | Obsolete version of ~ hbab... |
hbab 2726 | Bound-variable hypothesis ... |
hbabg 2727 | Bound-variable hypothesis ... |
nfsab 2728 | Bound-variable hypothesis ... |
nfsabg 2729 | Bound-variable hypothesis ... |
dfcleq 2731 | The defining characterizat... |
cvjust 2732 | Every set is a class. Pro... |
ax9ALT 2733 | Proof of ~ ax-9 from Tarsk... |
eleq2w2 2734 | A weaker version of ~ eleq... |
eqriv 2735 | Infer equality of classes ... |
eqrdv 2736 | Deduce equality of classes... |
eqrdav 2737 | Deduce equality of classes... |
eqid 2738 | Law of identity (reflexivi... |
eqidd 2739 | Class identity law with an... |
eqeq1d 2740 | Deduction from equality to... |
eqeq1dALT 2741 | Alternate proof of ~ eqeq1... |
eqeq1 2742 | Equality implies equivalen... |
eqeq1i 2743 | Inference from equality to... |
eqcomd 2744 | Deduction from commutative... |
eqcom 2745 | Commutative law for class ... |
eqcoms 2746 | Inference applying commuta... |
eqcomi 2747 | Inference from commutative... |
neqcomd 2748 | Commute an inequality. (C... |
eqeq2d 2749 | Deduction from equality to... |
eqeq2 2750 | Equality implies equivalen... |
eqeq2i 2751 | Inference from equality to... |
eqeqan12d 2752 | A useful inference for sub... |
eqeqan12rd 2753 | A useful inference for sub... |
eqeq12d 2754 | A useful inference for sub... |
eqeq12 2755 | Equality relationship amon... |
eqeq12i 2756 | A useful inference for sub... |
eqeq12OLD 2757 | Obsolete version of ~ eqeq... |
eqeq12dOLD 2758 | Obsolete version of ~ eqeq... |
eqeqan12dOLD 2759 | Obsolete version of ~ eqeq... |
eqeqan12dALT 2760 | Alternate proof of ~ eqeqa... |
eqtr 2761 | Transitive law for class e... |
eqtr2 2762 | A transitive law for class... |
eqtr2OLD 2763 | Obsolete version of eqtr2 ... |
eqtr3 2764 | A transitive law for class... |
eqtr3OLD 2765 | Obsolete version of ~ eqtr... |
eqtri 2766 | An equality transitivity i... |
eqtr2i 2767 | An equality transitivity i... |
eqtr3i 2768 | An equality transitivity i... |
eqtr4i 2769 | An equality transitivity i... |
3eqtri 2770 | An inference from three ch... |
3eqtrri 2771 | An inference from three ch... |
3eqtr2i 2772 | An inference from three ch... |
3eqtr2ri 2773 | An inference from three ch... |
3eqtr3i 2774 | An inference from three ch... |
3eqtr3ri 2775 | An inference from three ch... |
3eqtr4i 2776 | An inference from three ch... |
3eqtr4ri 2777 | An inference from three ch... |
eqtrd 2778 | An equality transitivity d... |
eqtr2d 2779 | An equality transitivity d... |
eqtr3d 2780 | An equality transitivity e... |
eqtr4d 2781 | An equality transitivity e... |
3eqtrd 2782 | A deduction from three cha... |
3eqtrrd 2783 | A deduction from three cha... |
3eqtr2d 2784 | A deduction from three cha... |
3eqtr2rd 2785 | A deduction from three cha... |
3eqtr3d 2786 | A deduction from three cha... |
3eqtr3rd 2787 | A deduction from three cha... |
3eqtr4d 2788 | A deduction from three cha... |
3eqtr4rd 2789 | A deduction from three cha... |
eqtrid 2790 | An equality transitivity d... |
eqtr2id 2791 | An equality transitivity d... |
eqtr3id 2792 | An equality transitivity d... |
eqtr3di 2793 | An equality transitivity d... |
eqtrdi 2794 | An equality transitivity d... |
eqtr2di 2795 | An equality transitivity d... |
eqtr4di 2796 | An equality transitivity d... |
eqtr4id 2797 | An equality transitivity d... |
sylan9eq 2798 | An equality transitivity d... |
sylan9req 2799 | An equality transitivity d... |
sylan9eqr 2800 | An equality transitivity d... |
3eqtr3g 2801 | A chained equality inferen... |
3eqtr3a 2802 | A chained equality inferen... |
3eqtr4g 2803 | A chained equality inferen... |
3eqtr4a 2804 | A chained equality inferen... |
eq2tri 2805 | A compound transitive infe... |
abbi1 2806 | Equivalent formulas yield ... |
abbidv 2807 | Equivalent wff's yield equ... |
abbii 2808 | Equivalent wff's yield equ... |
abbid 2809 | Equivalent wff's yield equ... |
abbi 2810 | Equivalent formulas define... |
cbvabv 2811 | Rule used to change bound ... |
cbvabw 2812 | Rule used to change bound ... |
cbvabwOLD 2813 | Obsolete version of ~ cbva... |
cbvab 2814 | Rule used to change bound ... |
abeq2w 2815 | Version of ~ abeq2 using i... |
dfclel 2817 | Characterization of the el... |
elex2 2818 | If a class contains anothe... |
elissetv 2819 | An element of a class exis... |
elisset 2820 | An element of a class exis... |
eleq1w 2821 | Weaker version of ~ eleq1 ... |
eleq2w 2822 | Weaker version of ~ eleq2 ... |
eleq1d 2823 | Deduction from equality to... |
eleq2d 2824 | Deduction from equality to... |
eleq2dALT 2825 | Alternate proof of ~ eleq2... |
eleq1 2826 | Equality implies equivalen... |
eleq2 2827 | Equality implies equivalen... |
eleq12 2828 | Equality implies equivalen... |
eleq1i 2829 | Inference from equality to... |
eleq2i 2830 | Inference from equality to... |
eleq12i 2831 | Inference from equality to... |
eqneltri 2832 | If a class is not an eleme... |
eleq12d 2833 | Deduction from equality to... |
eleq1a 2834 | A transitive-type law rela... |
eqeltri 2835 | Substitution of equal clas... |
eqeltrri 2836 | Substitution of equal clas... |
eleqtri 2837 | Substitution of equal clas... |
eleqtrri 2838 | Substitution of equal clas... |
eqeltrd 2839 | Substitution of equal clas... |
eqeltrrd 2840 | Deduction that substitutes... |
eleqtrd 2841 | Deduction that substitutes... |
eleqtrrd 2842 | Deduction that substitutes... |
eqeltrid 2843 | A membership and equality ... |
eqeltrrid 2844 | A membership and equality ... |
eleqtrid 2845 | A membership and equality ... |
eleqtrrid 2846 | A membership and equality ... |
eqeltrdi 2847 | A membership and equality ... |
eqeltrrdi 2848 | A membership and equality ... |
eleqtrdi 2849 | A membership and equality ... |
eleqtrrdi 2850 | A membership and equality ... |
3eltr3i 2851 | Substitution of equal clas... |
3eltr4i 2852 | Substitution of equal clas... |
3eltr3d 2853 | Substitution of equal clas... |
3eltr4d 2854 | Substitution of equal clas... |
3eltr3g 2855 | Substitution of equal clas... |
3eltr4g 2856 | Substitution of equal clas... |
eleq2s 2857 | Substitution of equal clas... |
eqneltrd 2858 | If a class is not an eleme... |
eqneltrrd 2859 | If a class is not an eleme... |
neleqtrd 2860 | If a class is not an eleme... |
neleqtrrd 2861 | If a class is not an eleme... |
cleqh 2862 | Establish equality between... |
nelneq 2863 | A way of showing two class... |
nelneq2 2864 | A way of showing two class... |
eqsb1 2865 | Substitution for the left-... |
clelsb1 2866 | Substitution for the first... |
clelsb2 2867 | Substitution for the secon... |
clelsb2OLD 2868 | Obsolete version of ~ clel... |
hbxfreq 2869 | A utility lemma to transfe... |
hblem 2870 | Change the free variable o... |
hblemg 2871 | Change the free variable o... |
abeq2 2872 | Equality of a class variab... |
abeq1 2873 | Equality of a class variab... |
abeq2d 2874 | Equality of a class variab... |
abeq2i 2875 | Equality of a class variab... |
abeq1i 2876 | Equality of a class variab... |
abbi2dv 2877 | Deduction from a wff to a ... |
abbi1dv 2878 | Deduction from a wff to a ... |
abbi2i 2879 | Equality of a class variab... |
abbiOLD 2880 | Obsolete proof of ~ abbi a... |
abid1 2881 | Every class is equal to a ... |
abid2 2882 | A simplification of class ... |
clelab 2883 | Membership of a class vari... |
clelabOLD 2884 | Obsolete version of ~ clel... |
clabel 2885 | Membership of a class abst... |
sbab 2886 | The right-hand side of the... |
nfcjust 2888 | Justification theorem for ... |
nfci 2890 | Deduce that a class ` A ` ... |
nfcii 2891 | Deduce that a class ` A ` ... |
nfcr 2892 | Consequence of the not-fre... |
nfcrALT 2893 | Alternate version of ~ nfc... |
nfcri 2894 | Consequence of the not-fre... |
nfcd 2895 | Deduce that a class ` A ` ... |
nfcrd 2896 | Consequence of the not-fre... |
nfcriOLD 2897 | Obsolete version of ~ nfcr... |
nfcriOLDOLD 2898 | Obsolete version of ~ nfcr... |
nfcrii 2899 | Consequence of the not-fre... |
nfcriiOLD 2900 | Obsolete version of ~ nfcr... |
nfcriOLDOLDOLD 2901 | Obsolete version of ~ nfcr... |
nfceqdf 2902 | An equality theorem for ef... |
nfceqdfOLD 2903 | Obsolete version of ~ nfce... |
nfceqi 2904 | Equality theorem for class... |
nfcxfr 2905 | A utility lemma to transfe... |
nfcxfrd 2906 | A utility lemma to transfe... |
nfcv 2907 | If ` x ` is disjoint from ... |
nfcvd 2908 | If ` x ` is disjoint from ... |
nfab1 2909 | Bound-variable hypothesis ... |
nfnfc1 2910 | The setvar ` x ` is bound ... |
clelsb1fw 2911 | Substitution for the first... |
clelsb1f 2912 | Substitution for the first... |
nfab 2913 | Bound-variable hypothesis ... |
nfabg 2914 | Bound-variable hypothesis ... |
nfaba1 2915 | Bound-variable hypothesis ... |
nfaba1g 2916 | Bound-variable hypothesis ... |
nfeqd 2917 | Hypothesis builder for equ... |
nfeld 2918 | Hypothesis builder for ele... |
nfnfc 2919 | Hypothesis builder for ` F... |
nfeq 2920 | Hypothesis builder for equ... |
nfel 2921 | Hypothesis builder for ele... |
nfeq1 2922 | Hypothesis builder for equ... |
nfel1 2923 | Hypothesis builder for ele... |
nfeq2 2924 | Hypothesis builder for equ... |
nfel2 2925 | Hypothesis builder for ele... |
drnfc1 2926 | Formula-building lemma for... |
drnfc1OLD 2927 | Obsolete version of ~ drnf... |
drnfc2 2928 | Formula-building lemma for... |
drnfc2OLD 2929 | Obsolete version of ~ drnf... |
nfabdw 2930 | Bound-variable hypothesis ... |
nfabdwOLD 2931 | Obsolete version of ~ nfab... |
nfabd 2932 | Bound-variable hypothesis ... |
nfabd2 2933 | Bound-variable hypothesis ... |
dvelimdc 2934 | Deduction form of ~ dvelim... |
dvelimc 2935 | Version of ~ dvelim for cl... |
nfcvf 2936 | If ` x ` and ` y ` are dis... |
nfcvf2 2937 | If ` x ` and ` y ` are dis... |
cleqf 2938 | Establish equality between... |
abid2f 2939 | A simplification of class ... |
abeq2f 2940 | Equality of a class variab... |
sbabel 2941 | Theorem to move a substitu... |
sbabelOLD 2942 | Obsolete version of ~ sbab... |
neii 2945 | Inference associated with ... |
neir 2946 | Inference associated with ... |
nne 2947 | Negation of inequality. (... |
neneqd 2948 | Deduction eliminating ineq... |
neneq 2949 | From inequality to non-equ... |
neqned 2950 | If it is not the case that... |
neqne 2951 | From non-equality to inequ... |
neirr 2952 | No class is unequal to its... |
exmidne 2953 | Excluded middle with equal... |
eqneqall 2954 | A contradiction concerning... |
nonconne 2955 | Law of noncontradiction wi... |
necon3ad 2956 | Contrapositive law deducti... |
necon3bd 2957 | Contrapositive law deducti... |
necon2ad 2958 | Contrapositive inference f... |
necon2bd 2959 | Contrapositive inference f... |
necon1ad 2960 | Contrapositive deduction f... |
necon1bd 2961 | Contrapositive deduction f... |
necon4ad 2962 | Contrapositive inference f... |
necon4bd 2963 | Contrapositive inference f... |
necon3d 2964 | Contrapositive law deducti... |
necon1d 2965 | Contrapositive law deducti... |
necon2d 2966 | Contrapositive inference f... |
necon4d 2967 | Contrapositive inference f... |
necon3ai 2968 | Contrapositive inference f... |
necon3aiOLD 2969 | Obsolete version of ~ neco... |
necon3bi 2970 | Contrapositive inference f... |
necon1ai 2971 | Contrapositive inference f... |
necon1bi 2972 | Contrapositive inference f... |
necon2ai 2973 | Contrapositive inference f... |
necon2bi 2974 | Contrapositive inference f... |
necon4ai 2975 | Contrapositive inference f... |
necon3i 2976 | Contrapositive inference f... |
necon1i 2977 | Contrapositive inference f... |
necon2i 2978 | Contrapositive inference f... |
necon4i 2979 | Contrapositive inference f... |
necon3abid 2980 | Deduction from equality to... |
necon3bbid 2981 | Deduction from equality to... |
necon1abid 2982 | Contrapositive deduction f... |
necon1bbid 2983 | Contrapositive inference f... |
necon4abid 2984 | Contrapositive law deducti... |
necon4bbid 2985 | Contrapositive law deducti... |
necon2abid 2986 | Contrapositive deduction f... |
necon2bbid 2987 | Contrapositive deduction f... |
necon3bid 2988 | Deduction from equality to... |
necon4bid 2989 | Contrapositive law deducti... |
necon3abii 2990 | Deduction from equality to... |
necon3bbii 2991 | Deduction from equality to... |
necon1abii 2992 | Contrapositive inference f... |
necon1bbii 2993 | Contrapositive inference f... |
necon2abii 2994 | Contrapositive inference f... |
necon2bbii 2995 | Contrapositive inference f... |
necon3bii 2996 | Inference from equality to... |
necom 2997 | Commutation of inequality.... |
necomi 2998 | Inference from commutative... |
necomd 2999 | Deduction from commutative... |
nesym 3000 | Characterization of inequa... |
nesymi 3001 | Inference associated with ... |
nesymir 3002 | Inference associated with ... |
neeq1d 3003 | Deduction for inequality. ... |
neeq2d 3004 | Deduction for inequality. ... |
neeq12d 3005 | Deduction for inequality. ... |
neeq1 3006 | Equality theorem for inequ... |
neeq2 3007 | Equality theorem for inequ... |
neeq1i 3008 | Inference for inequality. ... |
neeq2i 3009 | Inference for inequality. ... |
neeq12i 3010 | Inference for inequality. ... |
eqnetrd 3011 | Substitution of equal clas... |
eqnetrrd 3012 | Substitution of equal clas... |
neeqtrd 3013 | Substitution of equal clas... |
eqnetri 3014 | Substitution of equal clas... |
eqnetrri 3015 | Substitution of equal clas... |
neeqtri 3016 | Substitution of equal clas... |
neeqtrri 3017 | Substitution of equal clas... |
neeqtrrd 3018 | Substitution of equal clas... |
eqnetrrid 3019 | A chained equality inferen... |
3netr3d 3020 | Substitution of equality i... |
3netr4d 3021 | Substitution of equality i... |
3netr3g 3022 | Substitution of equality i... |
3netr4g 3023 | Substitution of equality i... |
nebi 3024 | Contraposition law for ine... |
pm13.18 3025 | Theorem *13.18 in [Whitehe... |
pm13.181 3026 | Theorem *13.181 in [Whiteh... |
pm13.181OLD 3027 | Obsolete version of ~ pm13... |
pm2.61ine 3028 | Inference eliminating an i... |
pm2.21ddne 3029 | A contradiction implies an... |
pm2.61ne 3030 | Deduction eliminating an i... |
pm2.61dne 3031 | Deduction eliminating an i... |
pm2.61dane 3032 | Deduction eliminating an i... |
pm2.61da2ne 3033 | Deduction eliminating two ... |
pm2.61da3ne 3034 | Deduction eliminating thre... |
pm2.61iine 3035 | Equality version of ~ pm2.... |
neor 3036 | Logical OR with an equalit... |
neanior 3037 | A De Morgan's law for ineq... |
ne3anior 3038 | A De Morgan's law for ineq... |
neorian 3039 | A De Morgan's law for ineq... |
nemtbir 3040 | An inference from an inequ... |
nelne1 3041 | Two classes are different ... |
nelne2 3042 | Two classes are different ... |
nelelne 3043 | Two classes are different ... |
neneor 3044 | If two classes are differe... |
nfne 3045 | Bound-variable hypothesis ... |
nfned 3046 | Bound-variable hypothesis ... |
nabbi 3047 | Not equivalent wff's corre... |
mteqand 3048 | A modus tollens deduction ... |
neli 3051 | Inference associated with ... |
nelir 3052 | Inference associated with ... |
neleq12d 3053 | Equality theorem for negat... |
neleq1 3054 | Equality theorem for negat... |
neleq2 3055 | Equality theorem for negat... |
nfnel 3056 | Bound-variable hypothesis ... |
nfneld 3057 | Bound-variable hypothesis ... |
nnel 3058 | Negation of negated member... |
elnelne1 3059 | Two classes are different ... |
elnelne2 3060 | Two classes are different ... |
nelcon3d 3061 | Contrapositive law deducti... |
elnelall 3062 | A contradiction concerning... |
pm2.61danel 3063 | Deduction eliminating an e... |
rgen 3074 | Generalization rule for re... |
ralel 3075 | All elements of a class ar... |
rgenw 3076 | Generalization rule for re... |
rgen2w 3077 | Generalization rule for re... |
mprg 3078 | Modus ponens combined with... |
mprgbir 3079 | Modus ponens on biconditio... |
alral 3080 | Universal quantification i... |
raln 3081 | Restricted universally qua... |
ral2imi 3082 | Inference quantifying ante... |
ralim 3083 | Distribution of restricted... |
ralimi2 3084 | Inference quantifying both... |
ralimia 3085 | Inference quantifying both... |
ralimiaa 3086 | Inference quantifying both... |
ralimi 3087 | Inference quantifying both... |
2ralimi 3088 | Inference quantifying both... |
ralbi 3089 | Distribute a restricted un... |
ralbii2 3090 | Inference adding different... |
ralbiia 3091 | Inference adding restricte... |
ralbii 3092 | Inference adding restricte... |
2ralbii 3093 | Inference adding two restr... |
ralanid 3094 | Cancellation law for restr... |
r19.26 3095 | Restricted quantifier vers... |
r19.26-2 3096 | Restricted quantifier vers... |
r19.26-3 3097 | Version of ~ r19.26 with t... |
r19.26m 3098 | Version of ~ 19.26 and ~ r... |
ralbiim 3099 | Split a biconditional and ... |
2ralbiim 3100 | Split a biconditional and ... |
hbralrimi 3101 | Inference from Theorem 19.... |
ralrimiv 3102 | Inference from Theorem 19.... |
ralrimiva 3103 | Inference from Theorem 19.... |
ralrimivw 3104 | Inference from Theorem 19.... |
ralrimdv 3105 | Inference from Theorem 19.... |
ralrimdva 3106 | Inference from Theorem 19.... |
ralimdv2 3107 | Inference quantifying both... |
ralimdva 3108 | Deduction quantifying both... |
ralimdv 3109 | Deduction quantifying both... |
ralbidv2 3110 | Formula-building rule for ... |
ralbidva 3111 | Formula-building rule for ... |
ralbidv 3112 | Formula-building rule for ... |
r19.21v 3113 | Restricted quantifier vers... |
r19.21vOLD 3114 | Obsolete version of ~ r19.... |
r19.27v 3115 | Restricted quantitifer ver... |
r19.28v 3116 | Restricted quantifier vers... |
r2allem 3117 | Lemma factoring out common... |
r2al 3118 | Double restricted universa... |
r3al 3119 | Triple restricted universa... |
rgen2 3120 | Generalization rule for re... |
rgen3 3121 | Generalization rule for re... |
ralrimivv 3122 | Inference from Theorem 19.... |
ralrimivva 3123 | Inference from Theorem 19.... |
ralrimdvv 3124 | Inference from Theorem 19.... |
ralrimdvva 3125 | Inference from Theorem 19.... |
ralimdvva 3126 | Deduction doubly quantifyi... |
ralrimivvva 3127 | Inference from Theorem 19.... |
2ralbidva 3128 | Formula-building rule for ... |
2ralbidv 3129 | Formula-building rule for ... |
rspw 3130 | Restricted specialization.... |
rsp 3131 | Restricted specialization.... |
rspa 3132 | Restricted specialization.... |
rspec 3133 | Specialization rule for re... |
r19.21bi 3134 | Inference from Theorem 19.... |
r19.21be 3135 | Inference from Theorem 19.... |
rspec2 3136 | Specialization rule for re... |
rspec3 3137 | Specialization rule for re... |
rsp2 3138 | Restricted specialization,... |
r19.21t 3139 | Restricted quantifier vers... |
r19.21 3140 | Restricted quantifier vers... |
ralrimi 3141 | Inference from Theorem 19.... |
ralimdaa 3142 | Deduction quantifying both... |
ralrimd 3143 | Inference from Theorem 19.... |
nfra1 3144 | The setvar ` x ` is not fr... |
hbra1 3145 | The setvar ` x ` is not fr... |
hbral 3146 | Bound-variable hypothesis ... |
r2alf 3147 | Double restricted universa... |
nfraldw 3148 | Deduction version of ~ nfr... |
nfraldwOLD 3149 | Obsolete version of ~ nfra... |
nfrald 3150 | Deduction version of ~ nfr... |
nfralw 3151 | Bound-variable hypothesis ... |
nfralwOLD 3152 | Obsolete version of ~ nfra... |
nfral 3153 | Bound-variable hypothesis ... |
nfra2w 3154 | Similar to Lemma 24 of [Mo... |
nfra2wOLD 3155 | Obsolete version of ~ nfra... |
nfra2wOLDOLD 3156 | Obsolete version of ~ nfra... |
nfra2 3157 | Similar to Lemma 24 of [Mo... |
rgen2a 3158 | Generalization rule for re... |
ralbida 3159 | Formula-building rule for ... |
ralbidaOLD 3160 | Obsolete version of ~ ralb... |
ralbid 3161 | Formula-building rule for ... |
2ralbida 3162 | Formula-building rule for ... |
raleqbii 3163 | Equality deduction for res... |
ralcom4 3164 | Commutation of restricted ... |
ralcom4OLD 3165 | Obsolete version of ~ ralc... |
ralcom 3166 | Commutation of restricted ... |
ralnex 3167 | Relationship between restr... |
dfral2 3168 | Relationship between restr... |
rexnal 3169 | Relationship between restr... |
dfrex2 3170 | Relationship between restr... |
rexex 3171 | Restricted existence impli... |
rexim 3172 | Theorem 19.22 of [Margaris... |
rexbi 3173 | Distribute restricted quan... |
rexbiOLD 3174 | Obsolete version of ~ rexb... |
reximi2 3175 | Inference quantifying both... |
reximia 3176 | Inference quantifying both... |
reximiaOLD 3177 | Obsolete version of ~ rexi... |
reximi 3178 | Inference quantifying both... |
rexbii2 3179 | Inference adding different... |
rexbiia 3180 | Inference adding restricte... |
rexbii 3181 | Inference adding restricte... |
2rexbii 3182 | Inference adding two restr... |
rexanid 3183 | Cancellation law for restr... |
r19.29 3184 | Restricted quantifier vers... |
r19.29r 3185 | Restricted quantifier vers... |
r19.29imd 3186 | Theorem 19.29 of [Margaris... |
rexnal2 3187 | Relationship between two r... |
rexnal3 3188 | Relationship between three... |
ralnex2 3189 | Relationship between two r... |
ralnex3 3190 | Relationship between three... |
ralinexa 3191 | A transformation of restri... |
rexanali 3192 | A transformation of restri... |
nrexralim 3193 | Negation of a complex pred... |
risset 3194 | Two ways to say " ` A ` be... |
nelb 3195 | A definition of ` -. A e. ... |
nelbOLD 3196 | Obsolete version of ~ nelb... |
nrex 3197 | Inference adding restricte... |
nrexdv 3198 | Deduction adding restricte... |
reximdv2 3199 | Deduction quantifying both... |
reximdvai 3200 | Deduction quantifying both... |
reximdvaiOLD 3201 | Obsolete version of ~ rexi... |
reximdv 3202 | Deduction from Theorem 19.... |
reximdva 3203 | Deduction quantifying both... |
reximddv 3204 | Deduction from Theorem 19.... |
reximssdv 3205 | Derivation of a restricted... |
reximdvva 3206 | Deduction doubly quantifyi... |
reximddv2 3207 | Double deduction from Theo... |
r19.23v 3208 | Restricted quantifier vers... |
rexlimiv 3209 | Inference from Theorem 19.... |
rexlimiva 3210 | Inference from Theorem 19.... |
rexlimivw 3211 | Weaker version of ~ rexlim... |
rexlimdv 3212 | Inference from Theorem 19.... |
rexlimdva 3213 | Inference from Theorem 19.... |
rexlimdvaa 3214 | Inference from Theorem 19.... |
rexlimdv3a 3215 | Inference from Theorem 19.... |
rexlimdva2 3216 | Inference from Theorem 19.... |
r19.29an 3217 | A commonly used pattern in... |
r19.29a 3218 | A commonly used pattern in... |
rexlimdvw 3219 | Inference from Theorem 19.... |
rexlimddv 3220 | Restricted existential eli... |
rexlimivv 3221 | Inference from Theorem 19.... |
rexlimdvv 3222 | Inference from Theorem 19.... |
rexlimdvva 3223 | Inference from Theorem 19.... |
rexbidv2 3224 | Formula-building rule for ... |
rexbidva 3225 | Formula-building rule for ... |
rexbidv 3226 | Formula-building rule for ... |
2rexbiia 3227 | Inference adding two restr... |
2rexbidva 3228 | Formula-building rule for ... |
2rexbidv 3229 | Formula-building rule for ... |
rexralbidv 3230 | Formula-building rule for ... |
r2exlem 3231 | Lemma factoring out common... |
r2ex 3232 | Double restricted existent... |
rexcom4 3233 | Commutation of restricted ... |
rexcom 3234 | Commutation of restricted ... |
2ex2rexrot 3235 | Rotate two existential qua... |
rexcom4a 3236 | Specialized existential co... |
rspe 3237 | Restricted specialization.... |
rsp2e 3238 | Restricted specialization.... |
nfre1 3239 | The setvar ` x ` is not fr... |
nfrexd 3240 | Deduction version of ~ nfr... |
nfrexdg 3241 | Deduction version of ~ nfr... |
nfrex 3242 | Bound-variable hypothesis ... |
nfrexg 3243 | Bound-variable hypothesis ... |
reximdai 3244 | Deduction from Theorem 19.... |
reximd2a 3245 | Deduction quantifying both... |
r19.23t 3246 | Closed theorem form of ~ r... |
r19.23 3247 | Restricted quantifier vers... |
rexlimi 3248 | Restricted quantifier vers... |
rexlimd2 3249 | Version of ~ rexlimd with ... |
rexlimd 3250 | Deduction form of ~ rexlim... |
rexbida 3251 | Formula-building rule for ... |
rexbidvaALT 3252 | Alternate proof of ~ rexbi... |
rexbid 3253 | Formula-building rule for ... |
rexbidvALT 3254 | Alternate proof of ~ rexbi... |
ralrexbid 3255 | Formula-building rule for ... |
ralrexbidOLD 3256 | Obsolete version of ~ ralr... |
r19.12 3257 | Restricted quantifier vers... |
r19.12OLD 3258 | Obsolete version of ~ 19.1... |
r2exf 3259 | Double restricted existent... |
rexeqbii 3260 | Equality deduction for res... |
r19.29af2 3261 | A commonly used pattern ba... |
r19.29af 3262 | A commonly used pattern ba... |
2r19.29 3263 | Theorem ~ r19.29 with two ... |
r19.29d2r 3264 | Theorem 19.29 of [Margaris... |
r19.29d2rOLD 3265 | Obsolete version of ~ r19.... |
r19.29vva 3266 | A commonly used pattern ba... |
r19.29vvaOLD 3267 | Obsolete version of ~ r19.... |
r19.30 3268 | Restricted quantifier vers... |
r19.30OLD 3269 | Obsolete version of ~ 19.3... |
r19.32v 3270 | Restricted quantifier vers... |
r19.35 3271 | Restricted quantifier vers... |
r19.36v 3272 | Restricted quantifier vers... |
r19.37 3273 | Restricted quantifier vers... |
r19.37v 3274 | Restricted quantifier vers... |
r19.40 3275 | Restricted quantifier vers... |
r19.41v 3276 | Restricted quantifier vers... |
r19.41 3277 | Restricted quantifier vers... |
r19.41vv 3278 | Version of ~ r19.41v with ... |
r19.42v 3279 | Restricted quantifier vers... |
r19.43 3280 | Restricted quantifier vers... |
r19.44v 3281 | One direction of a restric... |
r19.45v 3282 | Restricted quantifier vers... |
ralcomf 3283 | Commutation of restricted ... |
rexcomf 3284 | Commutation of restricted ... |
rexcomOLD 3285 | Obsolete version of ~ rexc... |
ralcom13 3286 | Swap first and third restr... |
rexcom13 3287 | Swap first and third restr... |
ralrot3 3288 | Rotate three restricted un... |
rexrot4 3289 | Rotate four restricted exi... |
ralcom2 3290 | Commutation of restricted ... |
ralcom3 3291 | A commutation law for rest... |
reeanlem 3292 | Lemma factoring out common... |
reean 3293 | Rearrange restricted exist... |
reeanv 3294 | Rearrange restricted exist... |
3reeanv 3295 | Rearrange three restricted... |
2ralor 3296 | Distribute restricted univ... |
2ralorOLD 3297 | Obsolete version of ~ 2ral... |
reuanid 3298 | Cancellation law for restr... |
rmoanid 3299 | Cancellation law for restr... |
nfreu1 3300 | The setvar ` x ` is not fr... |
nfrmo1 3301 | The setvar ` x ` is not fr... |
nfreud 3302 | Deduction version of ~ nfr... |
nfrmod 3303 | Deduction version of ~ nfr... |
nfrmow 3304 | Bound-variable hypothesis ... |
nfreuw 3305 | Bound-variable hypothesis ... |
nfreuwOLD 3306 | Obsolete version of ~ nfre... |
nfrmowOLD 3307 | Obsolete version of ~ nfrm... |
nfreu 3308 | Bound-variable hypothesis ... |
nfrmo 3309 | Bound-variable hypothesis ... |
rabid 3310 | An "identity" law of concr... |
rabrab 3311 | Abstract builder restricte... |
rabidim1 3312 | Membership in a restricted... |
rabid2f 3313 | An "identity" law for rest... |
rabid2 3314 | An "identity" law for rest... |
rabid2OLD 3315 | Obsolete version of ~ rabi... |
rabbi 3316 | Equivalent wff's correspon... |
nfrab1 3317 | The abstraction variable i... |
nfrabw 3318 | A variable not free in a w... |
nfrabwOLD 3319 | Obsolete version of ~ nfra... |
nfrab 3320 | A variable not free in a w... |
reubida 3321 | Formula-building rule for ... |
reubidva 3322 | Formula-building rule for ... |
reubidv 3323 | Formula-building rule for ... |
reubiia 3324 | Formula-building rule for ... |
reubii 3325 | Formula-building rule for ... |
rmobida 3326 | Formula-building rule for ... |
rmobidva 3327 | Formula-building rule for ... |
rmobidvaOLD 3328 | Obsolete version of ~ rmob... |
rmobidv 3329 | Formula-building rule for ... |
rmobiia 3330 | Formula-building rule for ... |
rmobii 3331 | Formula-building rule for ... |
raleqf 3332 | Equality theorem for restr... |
rexeqf 3333 | Equality theorem for restr... |
reueq1f 3334 | Equality theorem for restr... |
rmoeq1f 3335 | Equality theorem for restr... |
raleqbidv 3336 | Equality deduction for res... |
rexeqbidv 3337 | Equality deduction for res... |
raleqbidvv 3338 | Version of ~ raleqbidv wit... |
rexeqbidvv 3339 | Version of ~ rexeqbidv wit... |
raleqbi1dv 3340 | Equality deduction for res... |
rexeqbi1dv 3341 | Equality deduction for res... |
raleq 3342 | Equality theorem for restr... |
rexeq 3343 | Equality theorem for restr... |
reueq1 3344 | Equality theorem for restr... |
rmoeq1 3345 | Equality theorem for restr... |
raleqi 3346 | Equality inference for res... |
rexeqi 3347 | Equality inference for res... |
raleqdv 3348 | Equality deduction for res... |
rexeqdv 3349 | Equality deduction for res... |
reueqd 3350 | Equality deduction for res... |
rmoeqd 3351 | Equality deduction for res... |
raleqbid 3352 | Equality deduction for res... |
rexeqbid 3353 | Equality deduction for res... |
raleqbidva 3354 | Equality deduction for res... |
rexeqbidva 3355 | Equality deduction for res... |
raleleq 3356 | All elements of a class ar... |
raleleqALT 3357 | Alternate proof of ~ ralel... |
moel 3358 | "At most one" element in a... |
moelOLD 3359 | Obsolete version of ~ moel... |
mormo 3360 | Unrestricted "at most one"... |
reu5 3361 | Restricted uniqueness in t... |
reurex 3362 | Restricted unique existenc... |
2reu2rex 3363 | Double restricted existent... |
reurmo 3364 | Restricted existential uni... |
rmo5 3365 | Restricted "at most one" i... |
nrexrmo 3366 | Nonexistence implies restr... |
reueubd 3367 | Restricted existential uni... |
cbvralfw 3368 | Rule used to change bound ... |
cbvralfwOLD 3369 | Obsolete version of ~ cbvr... |
cbvrexfw 3370 | Rule used to change bound ... |
cbvralf 3371 | Rule used to change bound ... |
cbvrexf 3372 | Rule used to change bound ... |
cbvralw 3373 | Rule used to change bound ... |
cbvrexw 3374 | Rule used to change bound ... |
cbvrmow 3375 | Change the bound variable ... |
cbvreuw 3376 | Change the bound variable ... |
cbvreuwOLD 3377 | Obsolete version of ~ cbvr... |
cbvrmowOLD 3378 | Obsolete version of ~ cbvr... |
cbvral 3379 | Rule used to change bound ... |
cbvrex 3380 | Rule used to change bound ... |
cbvreu 3381 | Change the bound variable ... |
cbvrmo 3382 | Change the bound variable ... |
cbvralvw 3383 | Change the bound variable ... |
cbvrexvw 3384 | Change the bound variable ... |
cbvrmovw 3385 | Change the bound variable ... |
cbvreuvw 3386 | Change the bound variable ... |
cbvreuvwOLD 3387 | Obsolete version of ~ cbvr... |
cbvralv 3388 | Change the bound variable ... |
cbvrexv 3389 | Change the bound variable ... |
cbvreuv 3390 | Change the bound variable ... |
cbvrmov 3391 | Change the bound variable ... |
cbvraldva2 3392 | Rule used to change the bo... |
cbvrexdva2 3393 | Rule used to change the bo... |
cbvraldva 3394 | Rule used to change the bo... |
cbvrexdva 3395 | Rule used to change the bo... |
cbvral2vw 3396 | Change bound variables of ... |
cbvrex2vw 3397 | Change bound variables of ... |
cbvral3vw 3398 | Change bound variables of ... |
cbvral2v 3399 | Change bound variables of ... |
cbvrex2v 3400 | Change bound variables of ... |
cbvral3v 3401 | Change bound variables of ... |
cbvralsvw 3402 | Change bound variable by u... |
cbvrexsvw 3403 | Change bound variable by u... |
cbvralsv 3404 | Change bound variable by u... |
cbvrexsv 3405 | Change bound variable by u... |
sbralie 3406 | Implicit to explicit subst... |
rabbiia 3407 | Equivalent formulas yield ... |
rabbii 3408 | Equivalent wff's correspon... |
rabbida 3409 | Equivalent wff's yield equ... |
rabbid 3410 | Version of ~ rabbidv with ... |
rabbidva2 3411 | Equivalent wff's yield equ... |
rabbia2 3412 | Equivalent wff's yield equ... |
rabbidva 3413 | Equivalent wff's yield equ... |
rabbidv 3414 | Equivalent wff's yield equ... |
rabeqf 3415 | Equality theorem for restr... |
rabeqi 3416 | Equality theorem for restr... |
rabeqiOLD 3417 | Obsolete version of ~ rabe... |
rabeq 3418 | Equality theorem for restr... |
rabeqdv 3419 | Equality of restricted cla... |
rabeqbidv 3420 | Equality of restricted cla... |
rabeqbidva 3421 | Equality of restricted cla... |
rabeq2i 3422 | Inference from equality of... |
rabswap 3423 | Swap with a membership rel... |
cbvrabw 3424 | Rule to change the bound v... |
cbvrab 3425 | Rule to change the bound v... |
cbvrabv 3426 | Rule to change the bound v... |
rabrabi 3427 | Abstract builder restricte... |
rabrabiOLD 3428 | Obsolete version of ~ rabr... |
rabeqcda 3429 | When ` ps ` is always true... |
ralrimia 3430 | Inference from Theorem 19.... |
ralimda 3431 | Deduction quantifying both... |
vjust 3433 | Justification theorem for ... |
dfv2 3435 | Alternate definition of th... |
vex 3436 | All setvar variables are s... |
vexOLD 3437 | Obsolete version of ~ vex ... |
elv 3438 | If a proposition is implie... |
elvd 3439 | If a proposition is implie... |
el2v 3440 | If a proposition is implie... |
eqv 3441 | The universe contains ever... |
eqvf 3442 | The universe contains ever... |
abv 3443 | The class of sets verifyin... |
abvALT 3444 | Alternate proof of ~ abv ,... |
isset 3445 | Two ways to express that "... |
issetf 3446 | A version of ~ isset that ... |
isseti 3447 | A way to say " ` A ` is a ... |
issetri 3448 | A way to say " ` A ` is a ... |
eqvisset 3449 | A class equal to a variabl... |
elex 3450 | If a class is a member of ... |
elexi 3451 | If a class is a member of ... |
elexd 3452 | If a class is a member of ... |
elex2OLD 3453 | Obsolete version of ~ elex... |
elex22 3454 | If two classes each contai... |
prcnel 3455 | A proper class doesn't bel... |
ralv 3456 | A universal quantifier res... |
rexv 3457 | An existential quantifier ... |
reuv 3458 | A unique existential quant... |
rmov 3459 | An at-most-one quantifier ... |
rabab 3460 | A class abstraction restri... |
rexcom4b 3461 | Specialized existential co... |
ceqsalt 3462 | Closed theorem version of ... |
ceqsralt 3463 | Restricted quantifier vers... |
ceqsalg 3464 | A representation of explic... |
ceqsalgALT 3465 | Alternate proof of ~ ceqsa... |
ceqsal 3466 | A representation of explic... |
ceqsalv 3467 | A representation of explic... |
ceqsalvOLD 3468 | Obsolete version of ~ ceqs... |
ceqsralv 3469 | Restricted quantifier vers... |
ceqsralvOLD 3470 | Obsolete version of ~ ceqs... |
gencl 3471 | Implicit substitution for ... |
2gencl 3472 | Implicit substitution for ... |
3gencl 3473 | Implicit substitution for ... |
cgsexg 3474 | Implicit substitution infe... |
cgsex2g 3475 | Implicit substitution infe... |
cgsex4g 3476 | An implicit substitution i... |
cgsex4gOLD 3477 | Obsolete version of ~ cgse... |
ceqsex 3478 | Elimination of an existent... |
ceqsexv 3479 | Elimination of an existent... |
ceqsexvOLD 3480 | Obsolete version of ~ ceqs... |
ceqsexv2d 3481 | Elimination of an existent... |
ceqsex2 3482 | Elimination of two existen... |
ceqsex2v 3483 | Elimination of two existen... |
ceqsex3v 3484 | Elimination of three exist... |
ceqsex4v 3485 | Elimination of four existe... |
ceqsex6v 3486 | Elimination of six existen... |
ceqsex8v 3487 | Elimination of eight exist... |
gencbvex 3488 | Change of bound variable u... |
gencbvex2 3489 | Restatement of ~ gencbvex ... |
gencbval 3490 | Change of bound variable u... |
sbhypf 3491 | Introduce an explicit subs... |
vtoclgft 3492 | Closed theorem form of ~ v... |
vtocldf 3493 | Implicit substitution of a... |
vtocld 3494 | Implicit substitution of a... |
vtocldOLD 3495 | Obsolete version of ~ vtoc... |
vtocl2d 3496 | Implicit substitution of t... |
vtoclf 3497 | Implicit substitution of a... |
vtocl 3498 | Implicit substitution of a... |
vtoclALT 3499 | Alternate proof of ~ vtocl... |
vtocl2 3500 | Implicit substitution of c... |
vtocl3 3501 | Implicit substitution of c... |
vtoclb 3502 | Implicit substitution of a... |
vtoclgf 3503 | Implicit substitution of a... |
vtoclg1f 3504 | Version of ~ vtoclgf with ... |
vtoclg 3505 | Implicit substitution of a... |
vtoclgOLD 3506 | Obsolete version of ~ vtoc... |
vtoclbg 3507 | Implicit substitution of a... |
vtocl2gf 3508 | Implicit substitution of a... |
vtocl3gf 3509 | Implicit substitution of a... |
vtocl2g 3510 | Implicit substitution of 2... |
vtocl3g 3511 | Implicit substitution of a... |
vtoclgaf 3512 | Implicit substitution of a... |
vtoclga 3513 | Implicit substitution of a... |
vtocl2ga 3514 | Implicit substitution of 2... |
vtocl2gaf 3515 | Implicit substitution of 2... |
vtocl3gaf 3516 | Implicit substitution of 3... |
vtocl3ga 3517 | Implicit substitution of 3... |
vtocl3gaOLD 3518 | Obsolete version of ~ vtoc... |
vtocl4g 3519 | Implicit substitution of 4... |
vtocl4ga 3520 | Implicit substitution of 4... |
vtocleg 3521 | Implicit substitution of a... |
vtoclegft 3522 | Implicit substitution of a... |
vtoclef 3523 | Implicit substitution of a... |
vtocle 3524 | Implicit substitution of a... |
vtoclri 3525 | Implicit substitution of a... |
spcimgft 3526 | A closed version of ~ spci... |
spcgft 3527 | A closed version of ~ spcg... |
spcimgf 3528 | Rule of specialization, us... |
spcimegf 3529 | Existential specialization... |
spcgf 3530 | Rule of specialization, us... |
spcegf 3531 | Existential specialization... |
spcimdv 3532 | Restricted specialization,... |
spcdv 3533 | Rule of specialization, us... |
spcimedv 3534 | Restricted existential spe... |
spcgv 3535 | Rule of specialization, us... |
spcegv 3536 | Existential specialization... |
spcedv 3537 | Existential specialization... |
spc2egv 3538 | Existential specialization... |
spc2gv 3539 | Specialization with two qu... |
spc2ed 3540 | Existential specialization... |
spc2d 3541 | Specialization with 2 quan... |
spc3egv 3542 | Existential specialization... |
spc3gv 3543 | Specialization with three ... |
spcv 3544 | Rule of specialization, us... |
spcev 3545 | Existential specialization... |
spc2ev 3546 | Existential specialization... |
rspct 3547 | A closed version of ~ rspc... |
rspcdf 3548 | Restricted specialization,... |
rspc 3549 | Restricted specialization,... |
rspce 3550 | Restricted existential spe... |
rspcimdv 3551 | Restricted specialization,... |
rspcimedv 3552 | Restricted existential spe... |
rspcdv 3553 | Restricted specialization,... |
rspcedv 3554 | Restricted existential spe... |
rspcebdv 3555 | Restricted existential spe... |
rspcdv2 3556 | Restricted specialization,... |
rspcv 3557 | Restricted specialization,... |
rspccv 3558 | Restricted specialization,... |
rspcva 3559 | Restricted specialization,... |
rspccva 3560 | Restricted specialization,... |
rspcev 3561 | Restricted existential spe... |
rspcdva 3562 | Restricted specialization,... |
rspcedvd 3563 | Restricted existential spe... |
rspcime 3564 | Prove a restricted existen... |
rspceaimv 3565 | Restricted existential spe... |
rspcedeq1vd 3566 | Restricted existential spe... |
rspcedeq2vd 3567 | Restricted existential spe... |
rspc2 3568 | Restricted specialization ... |
rspc2gv 3569 | Restricted specialization ... |
rspc2v 3570 | 2-variable restricted spec... |
rspc2va 3571 | 2-variable restricted spec... |
rspc2ev 3572 | 2-variable restricted exis... |
rspc3v 3573 | 3-variable restricted spec... |
rspc3ev 3574 | 3-variable restricted exis... |
rspceeqv 3575 | Restricted existential spe... |
ralxpxfr2d 3576 | Transfer a universal quant... |
rexraleqim 3577 | Statement following from e... |
eqvincg 3578 | A variable introduction la... |
eqvinc 3579 | A variable introduction la... |
eqvincf 3580 | A variable introduction la... |
alexeqg 3581 | Two ways to express substi... |
ceqex 3582 | Equality implies equivalen... |
ceqsexg 3583 | A representation of explic... |
ceqsexgv 3584 | Elimination of an existent... |
ceqsrexv 3585 | Elimination of a restricte... |
ceqsrexbv 3586 | Elimination of a restricte... |
ceqsrex2v 3587 | Elimination of a restricte... |
clel2g 3588 | Alternate definition of me... |
clel2gOLD 3589 | Obsolete version of ~ clel... |
clel2 3590 | Alternate definition of me... |
clel3g 3591 | Alternate definition of me... |
clel3 3592 | Alternate definition of me... |
clel4g 3593 | Alternate definition of me... |
clel4 3594 | Alternate definition of me... |
clel4OLD 3595 | Obsolete version of ~ clel... |
clel5 3596 | Alternate definition of cl... |
pm13.183 3597 | Compare theorem *13.183 in... |
rr19.3v 3598 | Restricted quantifier vers... |
rr19.28v 3599 | Restricted quantifier vers... |
elab6g 3600 | Membership in a class abst... |
elabd2 3601 | Membership in a class abst... |
elabd3 3602 | Membership in a class abst... |
elabgt 3603 | Membership in a class abst... |
elabgtOLD 3604 | Obsolete version of ~ elab... |
elabgf 3605 | Membership in a class abst... |
elabf 3606 | Membership in a class abst... |
elabg 3607 | Membership in a class abst... |
elabgOLD 3608 | Obsolete version of ~ elab... |
elab 3609 | Membership in a class abst... |
elabOLD 3610 | Obsolete version of ~ elab... |
elab2g 3611 | Membership in a class abst... |
elabd 3612 | Explicit demonstration the... |
elab2 3613 | Membership in a class abst... |
elab4g 3614 | Membership in a class abst... |
elab3gf 3615 | Membership in a class abst... |
elab3g 3616 | Membership in a class abst... |
elab3 3617 | Membership in a class abst... |
elrabi 3618 | Implication for the member... |
elrabiOLD 3619 | Obsolete version of ~ elra... |
elrabf 3620 | Membership in a restricted... |
rabtru 3621 | Abstract builder using the... |
rabeqc 3622 | A restricted class abstrac... |
elrab3t 3623 | Membership in a restricted... |
elrab 3624 | Membership in a restricted... |
elrab3 3625 | Membership in a restricted... |
elrabd 3626 | Membership in a restricted... |
elrab2 3627 | Membership in a restricted... |
ralab 3628 | Universal quantification o... |
ralabOLD 3629 | Obsolete version of ~ rala... |
ralrab 3630 | Universal quantification o... |
rexab 3631 | Existential quantification... |
rexabOLD 3632 | Obsolete version of ~ rexa... |
rexrab 3633 | Existential quantification... |
ralab2 3634 | Universal quantification o... |
ralrab2 3635 | Universal quantification o... |
rexab2 3636 | Existential quantification... |
rexrab2 3637 | Existential quantification... |
abidnf 3638 | Identity used to create cl... |
dedhb 3639 | A deduction theorem for co... |
nelrdva 3640 | Deduce negative membership... |
eqeu 3641 | A condition which implies ... |
moeq 3642 | There exists at most one s... |
eueq 3643 | A class is a set if and on... |
eueqi 3644 | There exists a unique set ... |
eueq2 3645 | Equality has existential u... |
eueq3 3646 | Equality has existential u... |
moeq3 3647 | "At most one" property of ... |
mosub 3648 | "At most one" remains true... |
mo2icl 3649 | Theorem for inferring "at ... |
mob2 3650 | Consequence of "at most on... |
moi2 3651 | Consequence of "at most on... |
mob 3652 | Equality implied by "at mo... |
moi 3653 | Equality implied by "at mo... |
morex 3654 | Derive membership from uni... |
euxfr2w 3655 | Transfer existential uniqu... |
euxfrw 3656 | Transfer existential uniqu... |
euxfr2 3657 | Transfer existential uniqu... |
euxfr 3658 | Transfer existential uniqu... |
euind 3659 | Existential uniqueness via... |
reu2 3660 | A way to express restricte... |
reu6 3661 | A way to express restricte... |
reu3 3662 | A way to express restricte... |
reu6i 3663 | A condition which implies ... |
eqreu 3664 | A condition which implies ... |
rmo4 3665 | Restricted "at most one" u... |
reu4 3666 | Restricted uniqueness usin... |
reu7 3667 | Restricted uniqueness usin... |
reu8 3668 | Restricted uniqueness usin... |
rmo3f 3669 | Restricted "at most one" u... |
rmo4f 3670 | Restricted "at most one" u... |
reu2eqd 3671 | Deduce equality from restr... |
reueq 3672 | Equality has existential u... |
rmoeq 3673 | Equality's restricted exis... |
rmoan 3674 | Restricted "at most one" s... |
rmoim 3675 | Restricted "at most one" i... |
rmoimia 3676 | Restricted "at most one" i... |
rmoimi 3677 | Restricted "at most one" i... |
rmoimi2 3678 | Restricted "at most one" i... |
2reu5a 3679 | Double restricted existent... |
reuimrmo 3680 | Restricted uniqueness impl... |
2reuswap 3681 | A condition allowing swap ... |
2reuswap2 3682 | A condition allowing swap ... |
reuxfrd 3683 | Transfer existential uniqu... |
reuxfr 3684 | Transfer existential uniqu... |
reuxfr1d 3685 | Transfer existential uniqu... |
reuxfr1ds 3686 | Transfer existential uniqu... |
reuxfr1 3687 | Transfer existential uniqu... |
reuind 3688 | Existential uniqueness via... |
2rmorex 3689 | Double restricted quantifi... |
2reu5lem1 3690 | Lemma for ~ 2reu5 . Note ... |
2reu5lem2 3691 | Lemma for ~ 2reu5 . (Cont... |
2reu5lem3 3692 | Lemma for ~ 2reu5 . This ... |
2reu5 3693 | Double restricted existent... |
2reurmo 3694 | Double restricted quantifi... |
2reurex 3695 | Double restricted quantifi... |
2rmoswap 3696 | A condition allowing to sw... |
2rexreu 3697 | Double restricted existent... |
cdeqi 3700 | Deduce conditional equalit... |
cdeqri 3701 | Property of conditional eq... |
cdeqth 3702 | Deduce conditional equalit... |
cdeqnot 3703 | Distribute conditional equ... |
cdeqal 3704 | Distribute conditional equ... |
cdeqab 3705 | Distribute conditional equ... |
cdeqal1 3706 | Distribute conditional equ... |
cdeqab1 3707 | Distribute conditional equ... |
cdeqim 3708 | Distribute conditional equ... |
cdeqcv 3709 | Conditional equality for s... |
cdeqeq 3710 | Distribute conditional equ... |
cdeqel 3711 | Distribute conditional equ... |
nfcdeq 3712 | If we have a conditional e... |
nfccdeq 3713 | Variation of ~ nfcdeq for ... |
rru 3714 | Relative version of Russel... |
ru 3715 | Russell's Paradox. Propos... |
dfsbcq 3718 | Proper substitution of a c... |
dfsbcq2 3719 | This theorem, which is sim... |
sbsbc 3720 | Show that ~ df-sb and ~ df... |
sbceq1d 3721 | Equality theorem for class... |
sbceq1dd 3722 | Equality theorem for class... |
sbceqbid 3723 | Equality theorem for class... |
sbc8g 3724 | This is the closest we can... |
sbc2or 3725 | The disjunction of two equ... |
sbcex 3726 | By our definition of prope... |
sbceq1a 3727 | Equality theorem for class... |
sbceq2a 3728 | Equality theorem for class... |
spsbc 3729 | Specialization: if a formu... |
spsbcd 3730 | Specialization: if a formu... |
sbcth 3731 | A substitution into a theo... |
sbcthdv 3732 | Deduction version of ~ sbc... |
sbcid 3733 | An identity theorem for su... |
nfsbc1d 3734 | Deduction version of ~ nfs... |
nfsbc1 3735 | Bound-variable hypothesis ... |
nfsbc1v 3736 | Bound-variable hypothesis ... |
nfsbcdw 3737 | Deduction version of ~ nfs... |
nfsbcw 3738 | Bound-variable hypothesis ... |
sbccow 3739 | A composition law for clas... |
nfsbcd 3740 | Deduction version of ~ nfs... |
nfsbc 3741 | Bound-variable hypothesis ... |
sbcco 3742 | A composition law for clas... |
sbcco2 3743 | A composition law for clas... |
sbc5 3744 | An equivalence for class s... |
sbc5ALT 3745 | Alternate proof of ~ sbc5 ... |
sbc6g 3746 | An equivalence for class s... |
sbc6gOLD 3747 | Obsolete version of ~ sbc6... |
sbc6 3748 | An equivalence for class s... |
sbc7 3749 | An equivalence for class s... |
cbvsbcw 3750 | Change bound variables in ... |
cbvsbcvw 3751 | Change the bound variable ... |
cbvsbc 3752 | Change bound variables in ... |
cbvsbcv 3753 | Change the bound variable ... |
sbciegft 3754 | Conversion of implicit sub... |
sbciegf 3755 | Conversion of implicit sub... |
sbcieg 3756 | Conversion of implicit sub... |
sbciegOLD 3757 | Obsolete version of ~ sbci... |
sbcie2g 3758 | Conversion of implicit sub... |
sbcie 3759 | Conversion of implicit sub... |
sbciedf 3760 | Conversion of implicit sub... |
sbcied 3761 | Conversion of implicit sub... |
sbciedOLD 3762 | Obsolete version of ~ sbci... |
sbcied2 3763 | Conversion of implicit sub... |
elrabsf 3764 | Membership in a restricted... |
eqsbc1 3765 | Substitution for the left-... |
sbcng 3766 | Move negation in and out o... |
sbcimg 3767 | Distribution of class subs... |
sbcan 3768 | Distribution of class subs... |
sbcor 3769 | Distribution of class subs... |
sbcbig 3770 | Distribution of class subs... |
sbcn1 3771 | Move negation in and out o... |
sbcim1 3772 | Distribution of class subs... |
sbcim1OLD 3773 | Obsolete version of ~ sbci... |
sbcbid 3774 | Formula-building deduction... |
sbcbidv 3775 | Formula-building deduction... |
sbcbii 3776 | Formula-building inference... |
sbcbi1 3777 | Distribution of class subs... |
sbcbi2 3778 | Substituting into equivale... |
sbcbi2OLD 3779 | Obsolete proof of ~ sbcbi2... |
sbcal 3780 | Move universal quantifier ... |
sbcex2 3781 | Move existential quantifie... |
sbceqal 3782 | Class version of one impli... |
sbceqalOLD 3783 | Obsolete version of ~ sbce... |
sbeqalb 3784 | Theorem *14.121 in [Whiteh... |
eqsbc2 3785 | Substitution for the right... |
sbc3an 3786 | Distribution of class subs... |
sbcel1v 3787 | Class substitution into a ... |
sbcel2gv 3788 | Class substitution into a ... |
sbcel21v 3789 | Class substitution into a ... |
sbcimdv 3790 | Substitution analogue of T... |
sbcimdvOLD 3791 | Obsolete version of ~ sbci... |
sbctt 3792 | Substitution for a variabl... |
sbcgf 3793 | Substitution for a variabl... |
sbc19.21g 3794 | Substitution for a variabl... |
sbcg 3795 | Substitution for a variabl... |
sbcgOLD 3796 | Obsolete version of ~ sbcg... |
sbcgfi 3797 | Substitution for a variabl... |
sbc2iegf 3798 | Conversion of implicit sub... |
sbc2ie 3799 | Conversion of implicit sub... |
sbc2ieOLD 3800 | Obsolete version of ~ sbc2... |
sbc2iedv 3801 | Conversion of implicit sub... |
sbc3ie 3802 | Conversion of implicit sub... |
sbccomlem 3803 | Lemma for ~ sbccom . (Con... |
sbccom 3804 | Commutative law for double... |
sbcralt 3805 | Interchange class substitu... |
sbcrext 3806 | Interchange class substitu... |
sbcralg 3807 | Interchange class substitu... |
sbcrex 3808 | Interchange class substitu... |
sbcreu 3809 | Interchange class substitu... |
reu8nf 3810 | Restricted uniqueness usin... |
sbcabel 3811 | Interchange class substitu... |
rspsbc 3812 | Restricted quantifier vers... |
rspsbca 3813 | Restricted quantifier vers... |
rspesbca 3814 | Existence form of ~ rspsbc... |
spesbc 3815 | Existence form of ~ spsbc ... |
spesbcd 3816 | form of ~ spsbc . (Contri... |
sbcth2 3817 | A substitution into a theo... |
ra4v 3818 | Version of ~ ra4 with a di... |
ra4 3819 | Restricted quantifier vers... |
rmo2 3820 | Alternate definition of re... |
rmo2i 3821 | Condition implying restric... |
rmo3 3822 | Restricted "at most one" u... |
rmob 3823 | Consequence of "at most on... |
rmoi 3824 | Consequence of "at most on... |
rmob2 3825 | Consequence of "restricted... |
rmoi2 3826 | Consequence of "restricted... |
rmoanim 3827 | Introduction of a conjunct... |
rmoanimALT 3828 | Alternate proof of ~ rmoan... |
reuan 3829 | Introduction of a conjunct... |
2reu1 3830 | Double restricted existent... |
2reu2 3831 | Double restricted existent... |
csb2 3834 | Alternate expression for t... |
csbeq1 3835 | Analogue of ~ dfsbcq for p... |
csbeq1d 3836 | Equality deduction for pro... |
csbeq2 3837 | Substituting into equivale... |
csbeq2d 3838 | Formula-building deduction... |
csbeq2dv 3839 | Formula-building deduction... |
csbeq2i 3840 | Formula-building inference... |
csbeq12dv 3841 | Formula-building inference... |
cbvcsbw 3842 | Change bound variables in ... |
cbvcsb 3843 | Change bound variables in ... |
cbvcsbv 3844 | Change the bound variable ... |
csbid 3845 | Analogue of ~ sbid for pro... |
csbeq1a 3846 | Equality theorem for prope... |
csbcow 3847 | Composition law for chaine... |
csbco 3848 | Composition law for chaine... |
csbtt 3849 | Substitution doesn't affec... |
csbconstgf 3850 | Substitution doesn't affec... |
csbconstg 3851 | Substitution doesn't affec... |
csbconstgOLD 3852 | Obsolete version of ~ csbc... |
csbgfi 3853 | Substitution for a variabl... |
csbconstgi 3854 | The proper substitution of... |
nfcsb1d 3855 | Bound-variable hypothesis ... |
nfcsb1 3856 | Bound-variable hypothesis ... |
nfcsb1v 3857 | Bound-variable hypothesis ... |
nfcsbd 3858 | Deduction version of ~ nfc... |
nfcsbw 3859 | Bound-variable hypothesis ... |
nfcsb 3860 | Bound-variable hypothesis ... |
csbhypf 3861 | Introduce an explicit subs... |
csbiebt 3862 | Conversion of implicit sub... |
csbiedf 3863 | Conversion of implicit sub... |
csbieb 3864 | Bidirectional conversion b... |
csbiebg 3865 | Bidirectional conversion b... |
csbiegf 3866 | Conversion of implicit sub... |
csbief 3867 | Conversion of implicit sub... |
csbie 3868 | Conversion of implicit sub... |
csbieOLD 3869 | Obsolete version of ~ csbi... |
csbied 3870 | Conversion of implicit sub... |
csbiedOLD 3871 | Obsolete version of ~ csbi... |
csbied2 3872 | Conversion of implicit sub... |
csbie2t 3873 | Conversion of implicit sub... |
csbie2 3874 | Conversion of implicit sub... |
csbie2g 3875 | Conversion of implicit sub... |
cbvrabcsfw 3876 | Version of ~ cbvrabcsf wit... |
cbvralcsf 3877 | A more general version of ... |
cbvrexcsf 3878 | A more general version of ... |
cbvreucsf 3879 | A more general version of ... |
cbvrabcsf 3880 | A more general version of ... |
cbvralv2 3881 | Rule used to change the bo... |
cbvrexv2 3882 | Rule used to change the bo... |
rspc2vd 3883 | Deduction version of 2-var... |
difjust 3889 | Soundness justification th... |
unjust 3891 | Soundness justification th... |
injust 3893 | Soundness justification th... |
dfin5 3895 | Alternate definition for t... |
dfdif2 3896 | Alternate definition of cl... |
eldif 3897 | Expansion of membership in... |
eldifd 3898 | If a class is in one class... |
eldifad 3899 | If a class is in the diffe... |
eldifbd 3900 | If a class is in the diffe... |
elneeldif 3901 | The elements of a set diff... |
velcomp 3902 | Characterization of setvar... |
elin 3903 | Expansion of membership in... |
dfss 3905 | Variant of subclass defini... |
dfss2 3907 | Alternate definition of th... |
dfss2OLD 3908 | Obsolete version of ~ dfss... |
dfss3 3909 | Alternate definition of su... |
dfss6 3910 | Alternate definition of su... |
dfss2f 3911 | Equivalence for subclass r... |
dfss3f 3912 | Equivalence for subclass r... |
nfss 3913 | If ` x ` is not free in ` ... |
ssel 3914 | Membership relationships f... |
sselOLD 3915 | Obsolete version of ~ ssel... |
ssel2 3916 | Membership relationships f... |
sseli 3917 | Membership implication fro... |
sselii 3918 | Membership inference from ... |
sselid 3919 | Membership inference from ... |
sseld 3920 | Membership deduction from ... |
sselda 3921 | Membership deduction from ... |
sseldd 3922 | Membership inference from ... |
ssneld 3923 | If a class is not in anoth... |
ssneldd 3924 | If an element is not in a ... |
ssriv 3925 | Inference based on subclas... |
ssrd 3926 | Deduction based on subclas... |
ssrdv 3927 | Deduction based on subclas... |
sstr2 3928 | Transitivity of subclass r... |
sstr 3929 | Transitivity of subclass r... |
sstri 3930 | Subclass transitivity infe... |
sstrd 3931 | Subclass transitivity dedu... |
sstrid 3932 | Subclass transitivity dedu... |
sstrdi 3933 | Subclass transitivity dedu... |
sylan9ss 3934 | A subclass transitivity de... |
sylan9ssr 3935 | A subclass transitivity de... |
eqss 3936 | The subclass relationship ... |
eqssi 3937 | Infer equality from two su... |
eqssd 3938 | Equality deduction from tw... |
sssseq 3939 | If a class is a subclass o... |
eqrd 3940 | Deduce equality of classes... |
eqri 3941 | Infer equality of classes ... |
eqelssd 3942 | Equality deduction from su... |
ssid 3943 | Any class is a subclass of... |
ssidd 3944 | Weakening of ~ ssid . (Co... |
ssv 3945 | Any class is a subclass of... |
sseq1 3946 | Equality theorem for subcl... |
sseq2 3947 | Equality theorem for the s... |
sseq12 3948 | Equality theorem for the s... |
sseq1i 3949 | An equality inference for ... |
sseq2i 3950 | An equality inference for ... |
sseq12i 3951 | An equality inference for ... |
sseq1d 3952 | An equality deduction for ... |
sseq2d 3953 | An equality deduction for ... |
sseq12d 3954 | An equality deduction for ... |
eqsstri 3955 | Substitution of equality i... |
eqsstrri 3956 | Substitution of equality i... |
sseqtri 3957 | Substitution of equality i... |
sseqtrri 3958 | Substitution of equality i... |
eqsstrd 3959 | Substitution of equality i... |
eqsstrrd 3960 | Substitution of equality i... |
sseqtrd 3961 | Substitution of equality i... |
sseqtrrd 3962 | Substitution of equality i... |
3sstr3i 3963 | Substitution of equality i... |
3sstr4i 3964 | Substitution of equality i... |
3sstr3g 3965 | Substitution of equality i... |
3sstr4g 3966 | Substitution of equality i... |
3sstr3d 3967 | Substitution of equality i... |
3sstr4d 3968 | Substitution of equality i... |
eqsstrid 3969 | A chained subclass and equ... |
eqsstrrid 3970 | A chained subclass and equ... |
sseqtrdi 3971 | A chained subclass and equ... |
sseqtrrdi 3972 | A chained subclass and equ... |
sseqtrid 3973 | Subclass transitivity dedu... |
sseqtrrid 3974 | Subclass transitivity dedu... |
eqsstrdi 3975 | A chained subclass and equ... |
eqsstrrdi 3976 | A chained subclass and equ... |
eqimss 3977 | Equality implies inclusion... |
eqimss2 3978 | Equality implies inclusion... |
eqimssi 3979 | Infer subclass relationshi... |
eqimss2i 3980 | Infer subclass relationshi... |
nssne1 3981 | Two classes are different ... |
nssne2 3982 | Two classes are different ... |
nss 3983 | Negation of subclass relat... |
nelss 3984 | Demonstrate by witnesses t... |
ssrexf 3985 | Restricted existential qua... |
ssrmof 3986 | "At most one" existential ... |
ssralv 3987 | Quantification restricted ... |
ssrexv 3988 | Existential quantification... |
ss2ralv 3989 | Two quantifications restri... |
ss2rexv 3990 | Two existential quantifica... |
ralss 3991 | Restricted universal quant... |
rexss 3992 | Restricted existential qua... |
ss2ab 3993 | Class abstractions in a su... |
abss 3994 | Class abstraction in a sub... |
ssab 3995 | Subclass of a class abstra... |
ssabral 3996 | The relation for a subclas... |
ss2abdv 3997 | Deduction of abstraction s... |
ss2abdvALT 3998 | Alternate proof of ~ ss2ab... |
ss2abdvOLD 3999 | Obsolete version of ~ ss2a... |
ss2abi 4000 | Inference of abstraction s... |
ss2abiOLD 4001 | Obsolete version of ~ ss2a... |
abssdv 4002 | Deduction of abstraction s... |
abssi 4003 | Inference of abstraction s... |
ss2rab 4004 | Restricted abstraction cla... |
rabss 4005 | Restricted class abstracti... |
ssrab 4006 | Subclass of a restricted c... |
ssrabdv 4007 | Subclass of a restricted c... |
rabssdv 4008 | Subclass of a restricted c... |
ss2rabdv 4009 | Deduction of restricted ab... |
ss2rabi 4010 | Inference of restricted ab... |
rabss2 4011 | Subclass law for restricte... |
ssab2 4012 | Subclass relation for the ... |
ssrab2 4013 | Subclass relation for a re... |
ssrab2OLD 4014 | Obsolete version of ~ ssra... |
ssrab3 4015 | Subclass relation for a re... |
rabssrabd 4016 | Subclass of a restricted c... |
ssrabeq 4017 | If the restricting class o... |
rabssab 4018 | A restricted class is a su... |
uniiunlem 4019 | A subset relationship usef... |
dfpss2 4020 | Alternate definition of pr... |
dfpss3 4021 | Alternate definition of pr... |
psseq1 4022 | Equality theorem for prope... |
psseq2 4023 | Equality theorem for prope... |
psseq1i 4024 | An equality inference for ... |
psseq2i 4025 | An equality inference for ... |
psseq12i 4026 | An equality inference for ... |
psseq1d 4027 | An equality deduction for ... |
psseq2d 4028 | An equality deduction for ... |
psseq12d 4029 | An equality deduction for ... |
pssss 4030 | A proper subclass is a sub... |
pssne 4031 | Two classes in a proper su... |
pssssd 4032 | Deduce subclass from prope... |
pssned 4033 | Proper subclasses are uneq... |
sspss 4034 | Subclass in terms of prope... |
pssirr 4035 | Proper subclass is irrefle... |
pssn2lp 4036 | Proper subclass has no 2-c... |
sspsstri 4037 | Two ways of stating tricho... |
ssnpss 4038 | Partial trichotomy law for... |
psstr 4039 | Transitive law for proper ... |
sspsstr 4040 | Transitive law for subclas... |
psssstr 4041 | Transitive law for subclas... |
psstrd 4042 | Proper subclass inclusion ... |
sspsstrd 4043 | Transitivity involving sub... |
psssstrd 4044 | Transitivity involving sub... |
npss 4045 | A class is not a proper su... |
ssnelpss 4046 | A subclass missing a membe... |
ssnelpssd 4047 | Subclass inclusion with on... |
ssexnelpss 4048 | If there is an element of ... |
dfdif3 4049 | Alternate definition of cl... |
difeq1 4050 | Equality theorem for class... |
difeq2 4051 | Equality theorem for class... |
difeq12 4052 | Equality theorem for class... |
difeq1i 4053 | Inference adding differenc... |
difeq2i 4054 | Inference adding differenc... |
difeq12i 4055 | Equality inference for cla... |
difeq1d 4056 | Deduction adding differenc... |
difeq2d 4057 | Deduction adding differenc... |
difeq12d 4058 | Equality deduction for cla... |
difeqri 4059 | Inference from membership ... |
nfdif 4060 | Bound-variable hypothesis ... |
eldifi 4061 | Implication of membership ... |
eldifn 4062 | Implication of membership ... |
elndif 4063 | A set does not belong to a... |
neldif 4064 | Implication of membership ... |
difdif 4065 | Double class difference. ... |
difss 4066 | Subclass relationship for ... |
difssd 4067 | A difference of two classe... |
difss2 4068 | If a class is contained in... |
difss2d 4069 | If a class is contained in... |
ssdifss 4070 | Preservation of a subclass... |
ddif 4071 | Double complement under un... |
ssconb 4072 | Contraposition law for sub... |
sscon 4073 | Contraposition law for sub... |
ssdif 4074 | Difference law for subsets... |
ssdifd 4075 | If ` A ` is contained in `... |
sscond 4076 | If ` A ` is contained in `... |
ssdifssd 4077 | If ` A ` is contained in `... |
ssdif2d 4078 | If ` A ` is contained in `... |
raldifb 4079 | Restricted universal quant... |
rexdifi 4080 | Restricted existential qua... |
complss 4081 | Complementation reverses i... |
compleq 4082 | Two classes are equal if a... |
elun 4083 | Expansion of membership in... |
elunnel1 4084 | A member of a union that i... |
uneqri 4085 | Inference from membership ... |
unidm 4086 | Idempotent law for union o... |
uncom 4087 | Commutative law for union ... |
equncom 4088 | If a class equals the unio... |
equncomi 4089 | Inference form of ~ equnco... |
uneq1 4090 | Equality theorem for the u... |
uneq2 4091 | Equality theorem for the u... |
uneq12 4092 | Equality theorem for the u... |
uneq1i 4093 | Inference adding union to ... |
uneq2i 4094 | Inference adding union to ... |
uneq12i 4095 | Equality inference for the... |
uneq1d 4096 | Deduction adding union to ... |
uneq2d 4097 | Deduction adding union to ... |
uneq12d 4098 | Equality deduction for the... |
nfun 4099 | Bound-variable hypothesis ... |
unass 4100 | Associative law for union ... |
un12 4101 | A rearrangement of union. ... |
un23 4102 | A rearrangement of union. ... |
un4 4103 | A rearrangement of the uni... |
unundi 4104 | Union distributes over its... |
unundir 4105 | Union distributes over its... |
ssun1 4106 | Subclass relationship for ... |
ssun2 4107 | Subclass relationship for ... |
ssun3 4108 | Subclass law for union of ... |
ssun4 4109 | Subclass law for union of ... |
elun1 4110 | Membership law for union o... |
elun2 4111 | Membership law for union o... |
elunant 4112 | A statement is true for ev... |
unss1 4113 | Subclass law for union of ... |
ssequn1 4114 | A relationship between sub... |
unss2 4115 | Subclass law for union of ... |
unss12 4116 | Subclass law for union of ... |
ssequn2 4117 | A relationship between sub... |
unss 4118 | The union of two subclasse... |
unssi 4119 | An inference showing the u... |
unssd 4120 | A deduction showing the un... |
unssad 4121 | If ` ( A u. B ) ` is conta... |
unssbd 4122 | If ` ( A u. B ) ` is conta... |
ssun 4123 | A condition that implies i... |
rexun 4124 | Restricted existential qua... |
ralunb 4125 | Restricted quantification ... |
ralun 4126 | Restricted quantification ... |
elini 4127 | Membership in an intersect... |
elind 4128 | Deduce membership in an in... |
elinel1 4129 | Membership in an intersect... |
elinel2 4130 | Membership in an intersect... |
elin2 4131 | Membership in a class defi... |
elin1d 4132 | Elementhood in the first s... |
elin2d 4133 | Elementhood in the first s... |
elin3 4134 | Membership in a class defi... |
incom 4135 | Commutative law for inters... |
ineqcom 4136 | Two ways of expressing tha... |
ineqcomi 4137 | Two ways of expressing tha... |
ineqri 4138 | Inference from membership ... |
ineq1 4139 | Equality theorem for inter... |
ineq2 4140 | Equality theorem for inter... |
ineq12 4141 | Equality theorem for inter... |
ineq1i 4142 | Equality inference for int... |
ineq2i 4143 | Equality inference for int... |
ineq12i 4144 | Equality inference for int... |
ineq1d 4145 | Equality deduction for int... |
ineq2d 4146 | Equality deduction for int... |
ineq12d 4147 | Equality deduction for int... |
ineqan12d 4148 | Equality deduction for int... |
sseqin2 4149 | A relationship between sub... |
nfin 4150 | Bound-variable hypothesis ... |
rabbi2dva 4151 | Deduction from a wff to a ... |
inidm 4152 | Idempotent law for interse... |
inass 4153 | Associative law for inters... |
in12 4154 | A rearrangement of interse... |
in32 4155 | A rearrangement of interse... |
in13 4156 | A rearrangement of interse... |
in31 4157 | A rearrangement of interse... |
inrot 4158 | Rotate the intersection of... |
in4 4159 | Rearrangement of intersect... |
inindi 4160 | Intersection distributes o... |
inindir 4161 | Intersection distributes o... |
inss1 4162 | The intersection of two cl... |
inss2 4163 | The intersection of two cl... |
ssin 4164 | Subclass of intersection. ... |
ssini 4165 | An inference showing that ... |
ssind 4166 | A deduction showing that a... |
ssrin 4167 | Add right intersection to ... |
sslin 4168 | Add left intersection to s... |
ssrind 4169 | Add right intersection to ... |
ss2in 4170 | Intersection of subclasses... |
ssinss1 4171 | Intersection preserves sub... |
inss 4172 | Inclusion of an intersecti... |
rexin 4173 | Restricted existential qua... |
dfss7 4174 | Alternate definition of su... |
symdifcom 4177 | Symmetric difference commu... |
symdifeq1 4178 | Equality theorem for symme... |
symdifeq2 4179 | Equality theorem for symme... |
nfsymdif 4180 | Hypothesis builder for sym... |
elsymdif 4181 | Membership in a symmetric ... |
dfsymdif4 4182 | Alternate definition of th... |
elsymdifxor 4183 | Membership in a symmetric ... |
dfsymdif2 4184 | Alternate definition of th... |
symdifass 4185 | Symmetric difference is as... |
difsssymdif 4186 | The symmetric difference c... |
difsymssdifssd 4187 | If the symmetric differenc... |
unabs 4188 | Absorption law for union. ... |
inabs 4189 | Absorption law for interse... |
nssinpss 4190 | Negation of subclass expre... |
nsspssun 4191 | Negation of subclass expre... |
dfss4 4192 | Subclass defined in terms ... |
dfun2 4193 | An alternate definition of... |
dfin2 4194 | An alternate definition of... |
difin 4195 | Difference with intersecti... |
ssdifim 4196 | Implication of a class dif... |
ssdifsym 4197 | Symmetric class difference... |
dfss5 4198 | Alternate definition of su... |
dfun3 4199 | Union defined in terms of ... |
dfin3 4200 | Intersection defined in te... |
dfin4 4201 | Alternate definition of th... |
invdif 4202 | Intersection with universa... |
indif 4203 | Intersection with class di... |
indif2 4204 | Bring an intersection in a... |
indif1 4205 | Bring an intersection in a... |
indifcom 4206 | Commutation law for inters... |
indi 4207 | Distributive law for inter... |
undi 4208 | Distributive law for union... |
indir 4209 | Distributive law for inter... |
undir 4210 | Distributive law for union... |
unineq 4211 | Infer equality from equali... |
uneqin 4212 | Equality of union and inte... |
difundi 4213 | Distributive law for class... |
difundir 4214 | Distributive law for class... |
difindi 4215 | Distributive law for class... |
difindir 4216 | Distributive law for class... |
indifdi 4217 | Distribute intersection ov... |
indifdir 4218 | Distribute intersection ov... |
indifdirOLD 4219 | Obsolete version of ~ indi... |
difdif2 4220 | Class difference by a clas... |
undm 4221 | De Morgan's law for union.... |
indm 4222 | De Morgan's law for inters... |
difun1 4223 | A relationship involving d... |
undif3 4224 | An equality involving clas... |
difin2 4225 | Represent a class differen... |
dif32 4226 | Swap second and third argu... |
difabs 4227 | Absorption-like law for cl... |
sscon34b 4228 | Relative complementation r... |
rcompleq 4229 | Two subclasses are equal i... |
dfsymdif3 4230 | Alternate definition of th... |
unabw 4231 | Union of two class abstrac... |
unab 4232 | Union of two class abstrac... |
inab 4233 | Intersection of two class ... |
difab 4234 | Difference of two class ab... |
abanssl 4235 | A class abstraction with a... |
abanssr 4236 | A class abstraction with a... |
notabw 4237 | A class abstraction define... |
notab 4238 | A class abstraction define... |
unrab 4239 | Union of two restricted cl... |
inrab 4240 | Intersection of two restri... |
inrab2 4241 | Intersection with a restri... |
difrab 4242 | Difference of two restrict... |
dfrab3 4243 | Alternate definition of re... |
dfrab2 4244 | Alternate definition of re... |
notrab 4245 | Complementation of restric... |
dfrab3ss 4246 | Restricted class abstracti... |
rabun2 4247 | Abstraction restricted to ... |
reuun2 4248 | Transfer uniqueness to a s... |
reuss2 4249 | Transfer uniqueness to a s... |
reuss 4250 | Transfer uniqueness to a s... |
reuun1 4251 | Transfer uniqueness to a s... |
reupick 4252 | Restricted uniqueness "pic... |
reupick3 4253 | Restricted uniqueness "pic... |
reupick2 4254 | Restricted uniqueness "pic... |
euelss 4255 | Transfer uniqueness of an ... |
dfnul4 4258 | Alternate definition of th... |
dfnul2 4259 | Alternate definition of th... |
dfnul3 4260 | Alternate definition of th... |
dfnul2OLD 4261 | Obsolete version of ~ dfnu... |
dfnul3OLD 4262 | Obsolete version of ~ dfnu... |
dfnul4OLD 4263 | Obsolete version of ~ dfnu... |
noel 4264 | The empty set has no eleme... |
noelOLD 4265 | Obsolete version of ~ noel... |
nel02 4266 | The empty set has no eleme... |
n0i 4267 | If a class has elements, t... |
ne0i 4268 | If a class has elements, t... |
ne0d 4269 | Deduction form of ~ ne0i .... |
n0ii 4270 | If a class has elements, t... |
ne0ii 4271 | If a class has elements, t... |
vn0 4272 | The universal class is not... |
vn0ALT 4273 | Alternate proof of ~ vn0 .... |
eq0f 4274 | A class is equal to the em... |
neq0f 4275 | A class is not empty if an... |
n0f 4276 | A class is nonempty if and... |
eq0 4277 | A class is equal to the em... |
eq0ALT 4278 | Alternate proof of ~ eq0 .... |
neq0 4279 | A class is not empty if an... |
n0 4280 | A class is nonempty if and... |
eq0OLDOLD 4281 | Obsolete version of ~ eq0 ... |
neq0OLD 4282 | Obsolete version of ~ neq0... |
n0OLD 4283 | Obsolete version of ~ n0 a... |
nel0 4284 | From the general negation ... |
reximdva0 4285 | Restricted existence deduc... |
rspn0 4286 | Specialization for restric... |
rspn0OLD 4287 | Obsolete version of ~ rspn... |
n0rex 4288 | There is an element in a n... |
ssn0rex 4289 | There is an element in a c... |
n0moeu 4290 | A case of equivalence of "... |
rex0 4291 | Vacuous restricted existen... |
reu0 4292 | Vacuous restricted uniquen... |
rmo0 4293 | Vacuous restricted at-most... |
0el 4294 | Membership of the empty se... |
n0el 4295 | Negated membership of the ... |
eqeuel 4296 | A condition which implies ... |
ssdif0 4297 | Subclass expressed in term... |
difn0 4298 | If the difference of two s... |
pssdifn0 4299 | A proper subclass has a no... |
pssdif 4300 | A proper subclass has a no... |
ndisj 4301 | Express that an intersecti... |
difin0ss 4302 | Difference, intersection, ... |
inssdif0 4303 | Intersection, subclass, an... |
difid 4304 | The difference between a c... |
difidALT 4305 | Alternate proof of ~ difid... |
dif0 4306 | The difference between a c... |
ab0w 4307 | The class of sets verifyin... |
ab0 4308 | The class of sets verifyin... |
ab0OLD 4309 | Obsolete version of ~ ab0 ... |
ab0ALT 4310 | Alternate proof of ~ ab0 ,... |
dfnf5 4311 | Characterization of nonfre... |
ab0orv 4312 | The class abstraction defi... |
ab0orvALT 4313 | Alternate proof of ~ ab0or... |
abn0 4314 | Nonempty class abstraction... |
abn0OLD 4315 | Obsolete version of ~ abn0... |
rab0 4316 | Any restricted class abstr... |
rabeq0w 4317 | Condition for a restricted... |
rabeq0 4318 | Condition for a restricted... |
rabn0 4319 | Nonempty restricted class ... |
rabxm 4320 | Law of excluded middle, in... |
rabnc 4321 | Law of noncontradiction, i... |
elneldisj 4322 | The set of elements ` s ` ... |
elnelun 4323 | The union of the set of el... |
un0 4324 | The union of a class with ... |
in0 4325 | The intersection of a clas... |
0un 4326 | The union of the empty set... |
0in 4327 | The intersection of the em... |
inv1 4328 | The intersection of a clas... |
unv 4329 | The union of a class with ... |
0ss 4330 | The null set is a subset o... |
ss0b 4331 | Any subset of the empty se... |
ss0 4332 | Any subset of the empty se... |
sseq0 4333 | A subclass of an empty cla... |
ssn0 4334 | A class with a nonempty su... |
0dif 4335 | The difference between the... |
abf 4336 | A class abstraction determ... |
abfOLD 4337 | Obsolete version of ~ abf ... |
eq0rdv 4338 | Deduction for equality to ... |
eq0rdvALT 4339 | Alternate proof of ~ eq0rd... |
csbprc 4340 | The proper substitution of... |
csb0 4341 | The proper substitution of... |
sbcel12 4342 | Distribute proper substitu... |
sbceqg 4343 | Distribute proper substitu... |
sbceqi 4344 | Distribution of class subs... |
sbcnel12g 4345 | Distribute proper substitu... |
sbcne12 4346 | Distribute proper substitu... |
sbcel1g 4347 | Move proper substitution i... |
sbceq1g 4348 | Move proper substitution t... |
sbcel2 4349 | Move proper substitution i... |
sbceq2g 4350 | Move proper substitution t... |
csbcom 4351 | Commutative law for double... |
sbcnestgfw 4352 | Nest the composition of tw... |
csbnestgfw 4353 | Nest the composition of tw... |
sbcnestgw 4354 | Nest the composition of tw... |
csbnestgw 4355 | Nest the composition of tw... |
sbcco3gw 4356 | Composition of two substit... |
sbcnestgf 4357 | Nest the composition of tw... |
csbnestgf 4358 | Nest the composition of tw... |
sbcnestg 4359 | Nest the composition of tw... |
csbnestg 4360 | Nest the composition of tw... |
sbcco3g 4361 | Composition of two substit... |
csbco3g 4362 | Composition of two class s... |
csbnest1g 4363 | Nest the composition of tw... |
csbidm 4364 | Idempotent law for class s... |
csbvarg 4365 | The proper substitution of... |
csbvargi 4366 | The proper substitution of... |
sbccsb 4367 | Substitution into a wff ex... |
sbccsb2 4368 | Substitution into a wff ex... |
rspcsbela 4369 | Special case related to ~ ... |
sbnfc2 4370 | Two ways of expressing " `... |
csbab 4371 | Move substitution into a c... |
csbun 4372 | Distribution of class subs... |
csbin 4373 | Distribute proper substitu... |
csbie2df 4374 | Conversion of implicit sub... |
2nreu 4375 | If there are two different... |
un00 4376 | Two classes are empty iff ... |
vss 4377 | Only the universal class h... |
0pss 4378 | The null set is a proper s... |
npss0 4379 | No set is a proper subset ... |
pssv 4380 | Any non-universal class is... |
disj 4381 | Two ways of saying that tw... |
disjOLD 4382 | Obsolete version of ~ disj... |
disjr 4383 | Two ways of saying that tw... |
disj1 4384 | Two ways of saying that tw... |
reldisj 4385 | Two ways of saying that tw... |
reldisjOLD 4386 | Obsolete version of ~ reld... |
disj3 4387 | Two ways of saying that tw... |
disjne 4388 | Members of disjoint sets a... |
disjeq0 4389 | Two disjoint sets are equa... |
disjel 4390 | A set can't belong to both... |
disj2 4391 | Two ways of saying that tw... |
disj4 4392 | Two ways of saying that tw... |
ssdisj 4393 | Intersection with a subcla... |
disjpss 4394 | A class is a proper subset... |
undisj1 4395 | The union of disjoint clas... |
undisj2 4396 | The union of disjoint clas... |
ssindif0 4397 | Subclass expressed in term... |
inelcm 4398 | The intersection of classe... |
minel 4399 | A minimum element of a cla... |
undif4 4400 | Distribute union over diff... |
disjssun 4401 | Subset relation for disjoi... |
vdif0 4402 | Universal class equality i... |
difrab0eq 4403 | If the difference between ... |
pssnel 4404 | A proper subclass has a me... |
disjdif 4405 | A class and its relative c... |
disjdifr 4406 | A class and its relative c... |
difin0 4407 | The difference of a class ... |
unvdif 4408 | The union of a class and i... |
undif1 4409 | Absorption of difference b... |
undif2 4410 | Absorption of difference b... |
undifabs 4411 | Absorption of difference b... |
inundif 4412 | The intersection and class... |
disjdif2 4413 | The difference of a class ... |
difun2 4414 | Absorption of union by dif... |
undif 4415 | Union of complementary par... |
ssdifin0 4416 | A subset of a difference d... |
ssdifeq0 4417 | A class is a subclass of i... |
ssundif 4418 | A condition equivalent to ... |
difcom 4419 | Swap the arguments of a cl... |
pssdifcom1 4420 | Two ways to express overla... |
pssdifcom2 4421 | Two ways to express non-co... |
difdifdir 4422 | Distributive law for class... |
uneqdifeq 4423 | Two ways to say that ` A `... |
raldifeq 4424 | Equality theorem for restr... |
r19.2z 4425 | Theorem 19.2 of [Margaris]... |
r19.2zb 4426 | A response to the notion t... |
r19.3rz 4427 | Restricted quantification ... |
r19.28z 4428 | Restricted quantifier vers... |
r19.3rzv 4429 | Restricted quantification ... |
r19.9rzv 4430 | Restricted quantification ... |
r19.28zv 4431 | Restricted quantifier vers... |
r19.37zv 4432 | Restricted quantifier vers... |
r19.45zv 4433 | Restricted version of Theo... |
r19.44zv 4434 | Restricted version of Theo... |
r19.27z 4435 | Restricted quantifier vers... |
r19.27zv 4436 | Restricted quantifier vers... |
r19.36zv 4437 | Restricted quantifier vers... |
ralidmw 4438 | Idempotent law for restric... |
rzal 4439 | Vacuous quantification is ... |
rzalALT 4440 | Alternate proof of ~ rzal ... |
rexn0 4441 | Restricted existential qua... |
ralidm 4442 | Idempotent law for restric... |
ral0 4443 | Vacuous universal quantifi... |
ralf0 4444 | The quantification of a fa... |
rexn0OLD 4445 | Obsolete version of ~ rexn... |
ralidmOLD 4446 | Obsolete version of ~ rali... |
ral0OLD 4447 | Obsolete version of ~ ral0... |
ralf0OLD 4448 | Obsolete version of ~ ralf... |
ralnralall 4449 | A contradiction concerning... |
falseral0 4450 | A false statement can only... |
raaan 4451 | Rearrange restricted quant... |
raaanv 4452 | Rearrange restricted quant... |
sbss 4453 | Set substitution into the ... |
sbcssg 4454 | Distribute proper substitu... |
raaan2 4455 | Rearrange restricted quant... |
2reu4lem 4456 | Lemma for ~ 2reu4 . (Cont... |
2reu4 4457 | Definition of double restr... |
csbdif 4458 | Distribution of class subs... |
dfif2 4461 | An alternate definition of... |
dfif6 4462 | An alternate definition of... |
ifeq1 4463 | Equality theorem for condi... |
ifeq2 4464 | Equality theorem for condi... |
iftrue 4465 | Value of the conditional o... |
iftruei 4466 | Inference associated with ... |
iftrued 4467 | Value of the conditional o... |
iffalse 4468 | Value of the conditional o... |
iffalsei 4469 | Inference associated with ... |
iffalsed 4470 | Value of the conditional o... |
ifnefalse 4471 | When values are unequal, b... |
ifsb 4472 | Distribute a function over... |
dfif3 4473 | Alternate definition of th... |
dfif4 4474 | Alternate definition of th... |
dfif5 4475 | Alternate definition of th... |
ifssun 4476 | A conditional class is inc... |
ifeq12 4477 | Equality theorem for condi... |
ifeq1d 4478 | Equality deduction for con... |
ifeq2d 4479 | Equality deduction for con... |
ifeq12d 4480 | Equality deduction for con... |
ifbi 4481 | Equivalence theorem for co... |
ifbid 4482 | Equivalence deduction for ... |
ifbieq1d 4483 | Equivalence/equality deduc... |
ifbieq2i 4484 | Equivalence/equality infer... |
ifbieq2d 4485 | Equivalence/equality deduc... |
ifbieq12i 4486 | Equivalence deduction for ... |
ifbieq12d 4487 | Equivalence deduction for ... |
nfifd 4488 | Deduction form of ~ nfif .... |
nfif 4489 | Bound-variable hypothesis ... |
ifeq1da 4490 | Conditional equality. (Co... |
ifeq2da 4491 | Conditional equality. (Co... |
ifeq12da 4492 | Equivalence deduction for ... |
ifbieq12d2 4493 | Equivalence deduction for ... |
ifclda 4494 | Conditional closure. (Con... |
ifeqda 4495 | Separation of the values o... |
elimif 4496 | Elimination of a condition... |
ifbothda 4497 | A wff ` th ` containing a ... |
ifboth 4498 | A wff ` th ` containing a ... |
ifid 4499 | Identical true and false a... |
eqif 4500 | Expansion of an equality w... |
ifval 4501 | Another expression of the ... |
elif 4502 | Membership in a conditiona... |
ifel 4503 | Membership of a conditiona... |
ifcl 4504 | Membership (closure) of a ... |
ifcld 4505 | Membership (closure) of a ... |
ifcli 4506 | Inference associated with ... |
ifexd 4507 | Existence of the condition... |
ifexg 4508 | Existence of the condition... |
ifex 4509 | Existence of the condition... |
ifeqor 4510 | The possible values of a c... |
ifnot 4511 | Negating the first argumen... |
ifan 4512 | Rewrite a conjunction in a... |
ifor 4513 | Rewrite a disjunction in a... |
2if2 4514 | Resolve two nested conditi... |
ifcomnan 4515 | Commute the conditions in ... |
csbif 4516 | Distribute proper substitu... |
dedth 4517 | Weak deduction theorem tha... |
dedth2h 4518 | Weak deduction theorem eli... |
dedth3h 4519 | Weak deduction theorem eli... |
dedth4h 4520 | Weak deduction theorem eli... |
dedth2v 4521 | Weak deduction theorem for... |
dedth3v 4522 | Weak deduction theorem for... |
dedth4v 4523 | Weak deduction theorem for... |
elimhyp 4524 | Eliminate a hypothesis con... |
elimhyp2v 4525 | Eliminate a hypothesis con... |
elimhyp3v 4526 | Eliminate a hypothesis con... |
elimhyp4v 4527 | Eliminate a hypothesis con... |
elimel 4528 | Eliminate a membership hyp... |
elimdhyp 4529 | Version of ~ elimhyp where... |
keephyp 4530 | Transform a hypothesis ` p... |
keephyp2v 4531 | Keep a hypothesis containi... |
keephyp3v 4532 | Keep a hypothesis containi... |
pwjust 4534 | Soundness justification th... |
elpwg 4536 | Membership in a power clas... |
elpw 4537 | Membership in a power clas... |
velpw 4538 | Setvar variable membership... |
elpwOLD 4539 | Obsolete proof of ~ elpw a... |
elpwgOLD 4540 | Obsolete proof of ~ elpwg ... |
elpwd 4541 | Membership in a power clas... |
elpwi 4542 | Subset relation implied by... |
elpwb 4543 | Characterization of the el... |
elpwid 4544 | An element of a power clas... |
elelpwi 4545 | If ` A ` belongs to a part... |
sspw 4546 | The powerclass preserves i... |
sspwi 4547 | The powerclass preserves i... |
sspwd 4548 | The powerclass preserves i... |
pweq 4549 | Equality theorem for power... |
pweqALT 4550 | Alternate proof of ~ pweq ... |
pweqi 4551 | Equality inference for pow... |
pweqd 4552 | Equality deduction for pow... |
pwunss 4553 | The power class of the uni... |
nfpw 4554 | Bound-variable hypothesis ... |
pwidg 4555 | A set is an element of its... |
pwidb 4556 | A class is an element of i... |
pwid 4557 | A set is a member of its p... |
pwss 4558 | Subclass relationship for ... |
pwundif 4559 | Break up the power class o... |
snjust 4560 | Soundness justification th... |
sneq 4571 | Equality theorem for singl... |
sneqi 4572 | Equality inference for sin... |
sneqd 4573 | Equality deduction for sin... |
dfsn2 4574 | Alternate definition of si... |
elsng 4575 | There is exactly one eleme... |
elsn 4576 | There is exactly one eleme... |
velsn 4577 | There is only one element ... |
elsni 4578 | There is at most one eleme... |
absn 4579 | Condition for a class abst... |
dfpr2 4580 | Alternate definition of a ... |
dfsn2ALT 4581 | Alternate definition of si... |
elprg 4582 | A member of a pair of clas... |
elpri 4583 | If a class is an element o... |
elpr 4584 | A member of a pair of clas... |
elpr2g 4585 | A member of a pair of sets... |
elpr2 4586 | A member of a pair of sets... |
elpr2OLD 4587 | Obsolete version of ~ elpr... |
nelpr2 4588 | If a class is not an eleme... |
nelpr1 4589 | If a class is not an eleme... |
nelpri 4590 | If an element doesn't matc... |
prneli 4591 | If an element doesn't matc... |
nelprd 4592 | If an element doesn't matc... |
eldifpr 4593 | Membership in a set with t... |
rexdifpr 4594 | Restricted existential qua... |
snidg 4595 | A set is a member of its s... |
snidb 4596 | A class is a set iff it is... |
snid 4597 | A set is a member of its s... |
vsnid 4598 | A setvar variable is a mem... |
elsn2g 4599 | There is exactly one eleme... |
elsn2 4600 | There is exactly one eleme... |
nelsn 4601 | If a class is not equal to... |
rabeqsn 4602 | Conditions for a restricte... |
rabsssn 4603 | Conditions for a restricte... |
ralsnsg 4604 | Substitution expressed in ... |
rexsns 4605 | Restricted existential qua... |
rexsngf 4606 | Restricted existential qua... |
ralsngf 4607 | Restricted universal quant... |
reusngf 4608 | Restricted existential uni... |
ralsng 4609 | Substitution expressed in ... |
rexsng 4610 | Restricted existential qua... |
reusng 4611 | Restricted existential uni... |
2ralsng 4612 | Substitution expressed in ... |
ralsngOLD 4613 | Obsolete version of ~ rals... |
rexsngOLD 4614 | Obsolete version of ~ rexs... |
rexreusng 4615 | Restricted existential uni... |
exsnrex 4616 | There is a set being the e... |
ralsn 4617 | Convert a universal quanti... |
rexsn 4618 | Convert an existential qua... |
elpwunsn 4619 | Membership in an extension... |
eqoreldif 4620 | An element of a set is eit... |
eltpg 4621 | Members of an unordered tr... |
eldiftp 4622 | Membership in a set with t... |
eltpi 4623 | A member of an unordered t... |
eltp 4624 | A member of an unordered t... |
dftp2 4625 | Alternate definition of un... |
nfpr 4626 | Bound-variable hypothesis ... |
ifpr 4627 | Membership of a conditiona... |
ralprgf 4628 | Convert a restricted unive... |
rexprgf 4629 | Convert a restricted exist... |
ralprg 4630 | Convert a restricted unive... |
ralprgOLD 4631 | Obsolete version of ~ ralp... |
rexprg 4632 | Convert a restricted exist... |
rexprgOLD 4633 | Obsolete version of ~ rexp... |
raltpg 4634 | Convert a restricted unive... |
rextpg 4635 | Convert a restricted exist... |
ralpr 4636 | Convert a restricted unive... |
rexpr 4637 | Convert a restricted exist... |
reuprg0 4638 | Convert a restricted exist... |
reuprg 4639 | Convert a restricted exist... |
reurexprg 4640 | Convert a restricted exist... |
raltp 4641 | Convert a universal quanti... |
rextp 4642 | Convert an existential qua... |
nfsn 4643 | Bound-variable hypothesis ... |
csbsng 4644 | Distribute proper substitu... |
csbprg 4645 | Distribute proper substitu... |
elinsn 4646 | If the intersection of two... |
disjsn 4647 | Intersection with the sing... |
disjsn2 4648 | Two distinct singletons ar... |
disjpr2 4649 | Two completely distinct un... |
disjprsn 4650 | The disjoint intersection ... |
disjtpsn 4651 | The disjoint intersection ... |
disjtp2 4652 | Two completely distinct un... |
snprc 4653 | The singleton of a proper ... |
snnzb 4654 | A singleton is nonempty if... |
rmosn 4655 | A restricted at-most-one q... |
r19.12sn 4656 | Special case of ~ r19.12 w... |
rabsn 4657 | Condition where a restrict... |
rabsnifsb 4658 | A restricted class abstrac... |
rabsnif 4659 | A restricted class abstrac... |
rabrsn 4660 | A restricted class abstrac... |
euabsn2 4661 | Another way to express exi... |
euabsn 4662 | Another way to express exi... |
reusn 4663 | A way to express restricte... |
absneu 4664 | Restricted existential uni... |
rabsneu 4665 | Restricted existential uni... |
eusn 4666 | Two ways to express " ` A ... |
rabsnt 4667 | Truth implied by equality ... |
prcom 4668 | Commutative law for unorde... |
preq1 4669 | Equality theorem for unord... |
preq2 4670 | Equality theorem for unord... |
preq12 4671 | Equality theorem for unord... |
preq1i 4672 | Equality inference for uno... |
preq2i 4673 | Equality inference for uno... |
preq12i 4674 | Equality inference for uno... |
preq1d 4675 | Equality deduction for uno... |
preq2d 4676 | Equality deduction for uno... |
preq12d 4677 | Equality deduction for uno... |
tpeq1 4678 | Equality theorem for unord... |
tpeq2 4679 | Equality theorem for unord... |
tpeq3 4680 | Equality theorem for unord... |
tpeq1d 4681 | Equality theorem for unord... |
tpeq2d 4682 | Equality theorem for unord... |
tpeq3d 4683 | Equality theorem for unord... |
tpeq123d 4684 | Equality theorem for unord... |
tprot 4685 | Rotation of the elements o... |
tpcoma 4686 | Swap 1st and 2nd members o... |
tpcomb 4687 | Swap 2nd and 3rd members o... |
tpass 4688 | Split off the first elemen... |
qdass 4689 | Two ways to write an unord... |
qdassr 4690 | Two ways to write an unord... |
tpidm12 4691 | Unordered triple ` { A , A... |
tpidm13 4692 | Unordered triple ` { A , B... |
tpidm23 4693 | Unordered triple ` { A , B... |
tpidm 4694 | Unordered triple ` { A , A... |
tppreq3 4695 | An unordered triple is an ... |
prid1g 4696 | An unordered pair contains... |
prid2g 4697 | An unordered pair contains... |
prid1 4698 | An unordered pair contains... |
prid2 4699 | An unordered pair contains... |
ifpprsnss 4700 | An unordered pair is a sin... |
prprc1 4701 | A proper class vanishes in... |
prprc2 4702 | A proper class vanishes in... |
prprc 4703 | An unordered pair containi... |
tpid1 4704 | One of the three elements ... |
tpid1g 4705 | Closed theorem form of ~ t... |
tpid2 4706 | One of the three elements ... |
tpid2g 4707 | Closed theorem form of ~ t... |
tpid3g 4708 | Closed theorem form of ~ t... |
tpid3 4709 | One of the three elements ... |
snnzg 4710 | The singleton of a set is ... |
snn0d 4711 | The singleton of a set is ... |
snnz 4712 | The singleton of a set is ... |
prnz 4713 | A pair containing a set is... |
prnzg 4714 | A pair containing a set is... |
tpnz 4715 | An unordered triple contai... |
tpnzd 4716 | An unordered triple contai... |
raltpd 4717 | Convert a universal quanti... |
snssg 4718 | The singleton of an elemen... |
snss 4719 | The singleton of an elemen... |
eldifsn 4720 | Membership in a set with a... |
ssdifsn 4721 | Subset of a set with an el... |
elpwdifsn 4722 | A subset of a set is an el... |
eldifsni 4723 | Membership in a set with a... |
eldifsnneq 4724 | An element of a difference... |
neldifsn 4725 | The class ` A ` is not in ... |
neldifsnd 4726 | The class ` A ` is not in ... |
rexdifsn 4727 | Restricted existential qua... |
raldifsni 4728 | Rearrangement of a propert... |
raldifsnb 4729 | Restricted universal quant... |
eldifvsn 4730 | A set is an element of the... |
difsn 4731 | An element not in a set ca... |
difprsnss 4732 | Removal of a singleton fro... |
difprsn1 4733 | Removal of a singleton fro... |
difprsn2 4734 | Removal of a singleton fro... |
diftpsn3 4735 | Removal of a singleton fro... |
difpr 4736 | Removing two elements as p... |
tpprceq3 4737 | An unordered triple is an ... |
tppreqb 4738 | An unordered triple is an ... |
difsnb 4739 | ` ( B \ { A } ) ` equals `... |
difsnpss 4740 | ` ( B \ { A } ) ` is a pro... |
snssi 4741 | The singleton of an elemen... |
snssd 4742 | The singleton of an elemen... |
difsnid 4743 | If we remove a single elem... |
eldifeldifsn 4744 | An element of a difference... |
pw0 4745 | Compute the power set of t... |
pwpw0 4746 | Compute the power set of t... |
snsspr1 4747 | A singleton is a subset of... |
snsspr2 4748 | A singleton is a subset of... |
snsstp1 4749 | A singleton is a subset of... |
snsstp2 4750 | A singleton is a subset of... |
snsstp3 4751 | A singleton is a subset of... |
prssg 4752 | A pair of elements of a cl... |
prss 4753 | A pair of elements of a cl... |
prssi 4754 | A pair of elements of a cl... |
prssd 4755 | Deduction version of ~ prs... |
prsspwg 4756 | An unordered pair belongs ... |
ssprss 4757 | A pair as subset of a pair... |
ssprsseq 4758 | A proper pair is a subset ... |
sssn 4759 | The subsets of a singleton... |
ssunsn2 4760 | The property of being sand... |
ssunsn 4761 | Possible values for a set ... |
eqsn 4762 | Two ways to express that a... |
issn 4763 | A sufficient condition for... |
n0snor2el 4764 | A nonempty set is either a... |
ssunpr 4765 | Possible values for a set ... |
sspr 4766 | The subsets of a pair. (C... |
sstp 4767 | The subsets of an unordere... |
tpss 4768 | An unordered triple of ele... |
tpssi 4769 | An unordered triple of ele... |
sneqrg 4770 | Closed form of ~ sneqr . ... |
sneqr 4771 | If the singletons of two s... |
snsssn 4772 | If a singleton is a subset... |
mosneq 4773 | There exists at most one s... |
sneqbg 4774 | Two singletons of sets are... |
snsspw 4775 | The singleton of a class i... |
prsspw 4776 | An unordered pair belongs ... |
preq1b 4777 | Biconditional equality lem... |
preq2b 4778 | Biconditional equality lem... |
preqr1 4779 | Reverse equality lemma for... |
preqr2 4780 | Reverse equality lemma for... |
preq12b 4781 | Equality relationship for ... |
opthpr 4782 | An unordered pair has the ... |
preqr1g 4783 | Reverse equality lemma for... |
preq12bg 4784 | Closed form of ~ preq12b .... |
prneimg 4785 | Two pairs are not equal if... |
prnebg 4786 | A (proper) pair is not equ... |
pr1eqbg 4787 | A (proper) pair is equal t... |
pr1nebg 4788 | A (proper) pair is not equ... |
preqsnd 4789 | Equivalence for a pair equ... |
prnesn 4790 | A proper unordered pair is... |
prneprprc 4791 | A proper unordered pair is... |
preqsn 4792 | Equivalence for a pair equ... |
preq12nebg 4793 | Equality relationship for ... |
prel12g 4794 | Equality of two unordered ... |
opthprneg 4795 | An unordered pair has the ... |
elpreqprlem 4796 | Lemma for ~ elpreqpr . (C... |
elpreqpr 4797 | Equality and membership ru... |
elpreqprb 4798 | A set is an element of an ... |
elpr2elpr 4799 | For an element ` A ` of an... |
dfopif 4800 | Rewrite ~ df-op using ` if... |
dfopifOLD 4801 | Obsolete version of ~ dfop... |
dfopg 4802 | Value of the ordered pair ... |
dfop 4803 | Value of an ordered pair w... |
opeq1 4804 | Equality theorem for order... |
opeq2 4805 | Equality theorem for order... |
opeq12 4806 | Equality theorem for order... |
opeq1i 4807 | Equality inference for ord... |
opeq2i 4808 | Equality inference for ord... |
opeq12i 4809 | Equality inference for ord... |
opeq1d 4810 | Equality deduction for ord... |
opeq2d 4811 | Equality deduction for ord... |
opeq12d 4812 | Equality deduction for ord... |
oteq1 4813 | Equality theorem for order... |
oteq2 4814 | Equality theorem for order... |
oteq3 4815 | Equality theorem for order... |
oteq1d 4816 | Equality deduction for ord... |
oteq2d 4817 | Equality deduction for ord... |
oteq3d 4818 | Equality deduction for ord... |
oteq123d 4819 | Equality deduction for ord... |
nfop 4820 | Bound-variable hypothesis ... |
nfopd 4821 | Deduction version of bound... |
csbopg 4822 | Distribution of class subs... |
opidg 4823 | The ordered pair ` <. A , ... |
opid 4824 | The ordered pair ` <. A , ... |
ralunsn 4825 | Restricted quantification ... |
2ralunsn 4826 | Double restricted quantifi... |
opprc 4827 | Expansion of an ordered pa... |
opprc1 4828 | Expansion of an ordered pa... |
opprc2 4829 | Expansion of an ordered pa... |
oprcl 4830 | If an ordered pair has an ... |
pwsn 4831 | The power set of a singlet... |
pwsnOLD 4832 | Obsolete version of ~ pwsn... |
pwpr 4833 | The power set of an unorde... |
pwtp 4834 | The power set of an unorde... |
pwpwpw0 4835 | Compute the power set of t... |
pwv 4836 | The power class of the uni... |
prproe 4837 | For an element of a proper... |
3elpr2eq 4838 | If there are three element... |
dfuni2 4841 | Alternate definition of cl... |
eluni 4842 | Membership in class union.... |
eluni2 4843 | Membership in class union.... |
elunii 4844 | Membership in class union.... |
nfunid 4845 | Deduction version of ~ nfu... |
nfuni 4846 | Bound-variable hypothesis ... |
uniss 4847 | Subclass relationship for ... |
unissi 4848 | Subclass relationship for ... |
unissd 4849 | Subclass relationship for ... |
unieq 4850 | Equality theorem for class... |
unieqOLD 4851 | Obsolete version of ~ unie... |
unieqi 4852 | Inference of equality of t... |
unieqd 4853 | Deduction of equality of t... |
eluniab 4854 | Membership in union of a c... |
elunirab 4855 | Membership in union of a c... |
uniprg 4856 | The union of a pair is the... |
unipr 4857 | The union of a pair is the... |
uniprOLD 4858 | Obsolete version of ~ unip... |
uniprgOLD 4859 | Obsolete version of ~ unip... |
unisng 4860 | A set equals the union of ... |
unisn 4861 | A set equals the union of ... |
unisn3 4862 | Union of a singleton in th... |
dfnfc2 4863 | An alternative statement o... |
uniun 4864 | The class union of the uni... |
uniin 4865 | The class union of the int... |
ssuni 4866 | Subclass relationship for ... |
uni0b 4867 | The union of a set is empt... |
uni0c 4868 | The union of a set is empt... |
uni0 4869 | The union of the empty set... |
csbuni 4870 | Distribute proper substitu... |
elssuni 4871 | An element of a class is a... |
unissel 4872 | Condition turning a subcla... |
unissb 4873 | Relationship involving mem... |
uniss2 4874 | A subclass condition on th... |
unidif 4875 | If the difference ` A \ B ... |
ssunieq 4876 | Relationship implying unio... |
unimax 4877 | Any member of a class is t... |
pwuni 4878 | A class is a subclass of t... |
dfint2 4881 | Alternate definition of cl... |
inteq 4882 | Equality law for intersect... |
inteqi 4883 | Equality inference for cla... |
inteqd 4884 | Equality deduction for cla... |
elint 4885 | Membership in class inters... |
elint2 4886 | Membership in class inters... |
elintg 4887 | Membership in class inters... |
elinti 4888 | Membership in class inters... |
nfint 4889 | Bound-variable hypothesis ... |
elintab 4890 | Membership in the intersec... |
elintrab 4891 | Membership in the intersec... |
elintrabg 4892 | Membership in the intersec... |
int0 4893 | The intersection of the em... |
intss1 4894 | An element of a class incl... |
ssint 4895 | Subclass of a class inters... |
ssintab 4896 | Subclass of the intersecti... |
ssintub 4897 | Subclass of the least uppe... |
ssmin 4898 | Subclass of the minimum va... |
intmin 4899 | Any member of a class is t... |
intss 4900 | Intersection of subclasses... |
intssuni 4901 | The intersection of a none... |
ssintrab 4902 | Subclass of the intersecti... |
unissint 4903 | If the union of a class is... |
intssuni2 4904 | Subclass relationship for ... |
intminss 4905 | Under subset ordering, the... |
intmin2 4906 | Any set is the smallest of... |
intmin3 4907 | Under subset ordering, the... |
intmin4 4908 | Elimination of a conjunct ... |
intab 4909 | The intersection of a spec... |
int0el 4910 | The intersection of a clas... |
intun 4911 | The class intersection of ... |
intprg 4912 | The intersection of a pair... |
intpr 4913 | The intersection of a pair... |
intprOLD 4914 | Obsolete version of ~ intp... |
intprgOLD 4915 | Obsolete version of ~ intp... |
intsng 4916 | Intersection of a singleto... |
intsn 4917 | The intersection of a sing... |
uniintsn 4918 | Two ways to express " ` A ... |
uniintab 4919 | The union and the intersec... |
intunsn 4920 | Theorem joining a singleto... |
rint0 4921 | Relative intersection of a... |
elrint 4922 | Membership in a restricted... |
elrint2 4923 | Membership in a restricted... |
eliun 4928 | Membership in indexed unio... |
eliin 4929 | Membership in indexed inte... |
eliuni 4930 | Membership in an indexed u... |
iuncom 4931 | Commutation of indexed uni... |
iuncom4 4932 | Commutation of union with ... |
iunconst 4933 | Indexed union of a constan... |
iinconst 4934 | Indexed intersection of a ... |
iuneqconst 4935 | Indexed union of identical... |
iuniin 4936 | Law combining indexed unio... |
iinssiun 4937 | An indexed intersection is... |
iunss1 4938 | Subclass theorem for index... |
iinss1 4939 | Subclass theorem for index... |
iuneq1 4940 | Equality theorem for index... |
iineq1 4941 | Equality theorem for index... |
ss2iun 4942 | Subclass theorem for index... |
iuneq2 4943 | Equality theorem for index... |
iineq2 4944 | Equality theorem for index... |
iuneq2i 4945 | Equality inference for ind... |
iineq2i 4946 | Equality inference for ind... |
iineq2d 4947 | Equality deduction for ind... |
iuneq2dv 4948 | Equality deduction for ind... |
iineq2dv 4949 | Equality deduction for ind... |
iuneq12df 4950 | Equality deduction for ind... |
iuneq1d 4951 | Equality theorem for index... |
iuneq12d 4952 | Equality deduction for ind... |
iuneq2d 4953 | Equality deduction for ind... |
nfiun 4954 | Bound-variable hypothesis ... |
nfiin 4955 | Bound-variable hypothesis ... |
nfiung 4956 | Bound-variable hypothesis ... |
nfiing 4957 | Bound-variable hypothesis ... |
nfiu1 4958 | Bound-variable hypothesis ... |
nfii1 4959 | Bound-variable hypothesis ... |
dfiun2g 4960 | Alternate definition of in... |
dfiun2gOLD 4961 | Obsolete version of ~ dfiu... |
dfiin2g 4962 | Alternate definition of in... |
dfiun2 4963 | Alternate definition of in... |
dfiin2 4964 | Alternate definition of in... |
dfiunv2 4965 | Define double indexed unio... |
cbviun 4966 | Rule used to change the bo... |
cbviin 4967 | Change bound variables in ... |
cbviung 4968 | Rule used to change the bo... |
cbviing 4969 | Change bound variables in ... |
cbviunv 4970 | Rule used to change the bo... |
cbviinv 4971 | Change bound variables in ... |
cbviunvg 4972 | Rule used to change the bo... |
cbviinvg 4973 | Change bound variables in ... |
iunssf 4974 | Subset theorem for an inde... |
iunss 4975 | Subset theorem for an inde... |
ssiun 4976 | Subset implication for an ... |
ssiun2 4977 | Identity law for subset of... |
ssiun2s 4978 | Subset relationship for an... |
iunss2 4979 | A subclass condition on th... |
iunssd 4980 | Subset theorem for an inde... |
iunab 4981 | The indexed union of a cla... |
iunrab 4982 | The indexed union of a res... |
iunxdif2 4983 | Indexed union with a class... |
ssiinf 4984 | Subset theorem for an inde... |
ssiin 4985 | Subset theorem for an inde... |
iinss 4986 | Subset implication for an ... |
iinss2 4987 | An indexed intersection is... |
uniiun 4988 | Class union in terms of in... |
intiin 4989 | Class intersection in term... |
iunid 4990 | An indexed union of single... |
iun0 4991 | An indexed union of the em... |
0iun 4992 | An empty indexed union is ... |
0iin 4993 | An empty indexed intersect... |
viin 4994 | Indexed intersection with ... |
iunsn 4995 | Indexed union of a singlet... |
iunn0 4996 | There is a nonempty class ... |
iinab 4997 | Indexed intersection of a ... |
iinrab 4998 | Indexed intersection of a ... |
iinrab2 4999 | Indexed intersection of a ... |
iunin2 5000 | Indexed union of intersect... |
iunin1 5001 | Indexed union of intersect... |
iinun2 5002 | Indexed intersection of un... |
iundif2 5003 | Indexed union of class dif... |
iindif1 5004 | Indexed intersection of cl... |
2iunin 5005 | Rearrange indexed unions o... |
iindif2 5006 | Indexed intersection of cl... |
iinin2 5007 | Indexed intersection of in... |
iinin1 5008 | Indexed intersection of in... |
iinvdif 5009 | The indexed intersection o... |
elriin 5010 | Elementhood in a relative ... |
riin0 5011 | Relative intersection of a... |
riinn0 5012 | Relative intersection of a... |
riinrab 5013 | Relative intersection of a... |
symdif0 5014 | Symmetric difference with ... |
symdifv 5015 | The symmetric difference w... |
symdifid 5016 | The symmetric difference o... |
iinxsng 5017 | A singleton index picks ou... |
iinxprg 5018 | Indexed intersection with ... |
iunxsng 5019 | A singleton index picks ou... |
iunxsn 5020 | A singleton index picks ou... |
iunxsngf 5021 | A singleton index picks ou... |
iunun 5022 | Separate a union in an ind... |
iunxun 5023 | Separate a union in the in... |
iunxdif3 5024 | An indexed union where som... |
iunxprg 5025 | A pair index picks out two... |
iunxiun 5026 | Separate an indexed union ... |
iinuni 5027 | A relationship involving u... |
iununi 5028 | A relationship involving u... |
sspwuni 5029 | Subclass relationship for ... |
pwssb 5030 | Two ways to express a coll... |
elpwpw 5031 | Characterization of the el... |
pwpwab 5032 | The double power class wri... |
pwpwssunieq 5033 | The class of sets whose un... |
elpwuni 5034 | Relationship for power cla... |
iinpw 5035 | The power class of an inte... |
iunpwss 5036 | Inclusion of an indexed un... |
intss2 5037 | A nonempty intersection of... |
rintn0 5038 | Relative intersection of a... |
dfdisj2 5041 | Alternate definition for d... |
disjss2 5042 | If each element of a colle... |
disjeq2 5043 | Equality theorem for disjo... |
disjeq2dv 5044 | Equality deduction for dis... |
disjss1 5045 | A subset of a disjoint col... |
disjeq1 5046 | Equality theorem for disjo... |
disjeq1d 5047 | Equality theorem for disjo... |
disjeq12d 5048 | Equality theorem for disjo... |
cbvdisj 5049 | Change bound variables in ... |
cbvdisjv 5050 | Change bound variables in ... |
nfdisjw 5051 | Bound-variable hypothesis ... |
nfdisj 5052 | Bound-variable hypothesis ... |
nfdisj1 5053 | Bound-variable hypothesis ... |
disjor 5054 | Two ways to say that a col... |
disjors 5055 | Two ways to say that a col... |
disji2 5056 | Property of a disjoint col... |
disji 5057 | Property of a disjoint col... |
invdisj 5058 | If there is a function ` C... |
invdisjrabw 5059 | Version of ~ invdisjrab wi... |
invdisjrab 5060 | The restricted class abstr... |
disjiun 5061 | A disjoint collection yiel... |
disjord 5062 | Conditions for a collectio... |
disjiunb 5063 | Two ways to say that a col... |
disjiund 5064 | Conditions for a collectio... |
sndisj 5065 | Any collection of singleto... |
0disj 5066 | Any collection of empty se... |
disjxsn 5067 | A singleton collection is ... |
disjx0 5068 | An empty collection is dis... |
disjprgw 5069 | Version of ~ disjprg with ... |
disjprg 5070 | A pair collection is disjo... |
disjxiun 5071 | An indexed union of a disj... |
disjxun 5072 | The union of two disjoint ... |
disjss3 5073 | Expand a disjoint collecti... |
breq 5076 | Equality theorem for binar... |
breq1 5077 | Equality theorem for a bin... |
breq2 5078 | Equality theorem for a bin... |
breq12 5079 | Equality theorem for a bin... |
breqi 5080 | Equality inference for bin... |
breq1i 5081 | Equality inference for a b... |
breq2i 5082 | Equality inference for a b... |
breq12i 5083 | Equality inference for a b... |
breq1d 5084 | Equality deduction for a b... |
breqd 5085 | Equality deduction for a b... |
breq2d 5086 | Equality deduction for a b... |
breq12d 5087 | Equality deduction for a b... |
breq123d 5088 | Equality deduction for a b... |
breqdi 5089 | Equality deduction for a b... |
breqan12d 5090 | Equality deduction for a b... |
breqan12rd 5091 | Equality deduction for a b... |
eqnbrtrd 5092 | Substitution of equal clas... |
nbrne1 5093 | Two classes are different ... |
nbrne2 5094 | Two classes are different ... |
eqbrtri 5095 | Substitution of equal clas... |
eqbrtrd 5096 | Substitution of equal clas... |
eqbrtrri 5097 | Substitution of equal clas... |
eqbrtrrd 5098 | Substitution of equal clas... |
breqtri 5099 | Substitution of equal clas... |
breqtrd 5100 | Substitution of equal clas... |
breqtrri 5101 | Substitution of equal clas... |
breqtrrd 5102 | Substitution of equal clas... |
3brtr3i 5103 | Substitution of equality i... |
3brtr4i 5104 | Substitution of equality i... |
3brtr3d 5105 | Substitution of equality i... |
3brtr4d 5106 | Substitution of equality i... |
3brtr3g 5107 | Substitution of equality i... |
3brtr4g 5108 | Substitution of equality i... |
eqbrtrid 5109 | A chained equality inferen... |
eqbrtrrid 5110 | A chained equality inferen... |
breqtrid 5111 | A chained equality inferen... |
breqtrrid 5112 | A chained equality inferen... |
eqbrtrdi 5113 | A chained equality inferen... |
eqbrtrrdi 5114 | A chained equality inferen... |
breqtrdi 5115 | A chained equality inferen... |
breqtrrdi 5116 | A chained equality inferen... |
ssbrd 5117 | Deduction from a subclass ... |
ssbr 5118 | Implication from a subclas... |
ssbri 5119 | Inference from a subclass ... |
nfbrd 5120 | Deduction version of bound... |
nfbr 5121 | Bound-variable hypothesis ... |
brab1 5122 | Relationship between a bin... |
br0 5123 | The empty binary relation ... |
brne0 5124 | If two sets are in a binar... |
brun 5125 | The union of two binary re... |
brin 5126 | The intersection of two re... |
brdif 5127 | The difference of two bina... |
sbcbr123 5128 | Move substitution in and o... |
sbcbr 5129 | Move substitution in and o... |
sbcbr12g 5130 | Move substitution in and o... |
sbcbr1g 5131 | Move substitution in and o... |
sbcbr2g 5132 | Move substitution in and o... |
brsymdif 5133 | Characterization of the sy... |
brralrspcev 5134 | Restricted existential spe... |
brimralrspcev 5135 | Restricted existential spe... |
opabss 5138 | The collection of ordered ... |
opabbid 5139 | Equivalent wff's yield equ... |
opabbidv 5140 | Equivalent wff's yield equ... |
opabbii 5141 | Equivalent wff's yield equ... |
nfopabd 5142 | Bound-variable hypothesis ... |
nfopab 5143 | Bound-variable hypothesis ... |
nfopab1 5144 | The first abstraction vari... |
nfopab2 5145 | The second abstraction var... |
cbvopab 5146 | Rule used to change bound ... |
cbvopabv 5147 | Rule used to change bound ... |
cbvopabvOLD 5148 | Obsolete version of ~ cbvo... |
cbvopab1 5149 | Change first bound variabl... |
cbvopab1g 5150 | Change first bound variabl... |
cbvopab2 5151 | Change second bound variab... |
cbvopab1s 5152 | Change first bound variabl... |
cbvopab1v 5153 | Rule used to change the fi... |
cbvopab1vOLD 5154 | Obsolete version of ~ cbvo... |
cbvopab2v 5155 | Rule used to change the se... |
unopab 5156 | Union of two ordered pair ... |
mpteq12da 5159 | An equality inference for ... |
mpteq12df 5160 | An equality inference for ... |
mpteq12dfOLD 5161 | Obsolete version of ~ mpte... |
mpteq12f 5162 | An equality theorem for th... |
mpteq12dva 5163 | An equality inference for ... |
mpteq12dvaOLD 5164 | Obsolete version of ~ mpte... |
mpteq12dv 5165 | An equality inference for ... |
mpteq12 5166 | An equality theorem for th... |
mpteq1 5167 | An equality theorem for th... |
mpteq1OLD 5168 | Obsolete version of ~ mpte... |
mpteq1d 5169 | An equality theorem for th... |
mpteq1i 5170 | An equality theorem for th... |
mpteq1iOLD 5171 | An equality theorem for th... |
mpteq2da 5172 | Slightly more general equa... |
mpteq2daOLD 5173 | Obsolete version of ~ mpte... |
mpteq2dva 5174 | Slightly more general equa... |
mpteq2dvaOLD 5175 | Obsolete version of ~ mpte... |
mpteq2dv 5176 | An equality inference for ... |
mpteq2ia 5177 | An equality inference for ... |
mpteq2iaOLD 5178 | Obsolete version of ~ mpte... |
mpteq2i 5179 | An equality inference for ... |
mpteq12i 5180 | An equality inference for ... |
nfmpt 5181 | Bound-variable hypothesis ... |
nfmpt1 5182 | Bound-variable hypothesis ... |
cbvmptf 5183 | Rule to change the bound v... |
cbvmptfg 5184 | Rule to change the bound v... |
cbvmpt 5185 | Rule to change the bound v... |
cbvmptg 5186 | Rule to change the bound v... |
cbvmptv 5187 | Rule to change the bound v... |
cbvmptvOLD 5188 | Obsolete version of ~ cbvm... |
cbvmptvg 5189 | Rule to change the bound v... |
mptv 5190 | Function with universal do... |
dftr2 5193 | An alternate way of defini... |
dftr5 5194 | An alternate way of defini... |
dftr3 5195 | An alternate way of defini... |
dftr4 5196 | An alternate way of defini... |
treq 5197 | Equality theorem for the t... |
trel 5198 | In a transitive class, the... |
trel3 5199 | In a transitive class, the... |
trss 5200 | An element of a transitive... |
trin 5201 | The intersection of transi... |
tr0 5202 | The empty set is transitiv... |
trv 5203 | The universe is transitive... |
triun 5204 | An indexed union of a clas... |
truni 5205 | The union of a class of tr... |
triin 5206 | An indexed intersection of... |
trint 5207 | The intersection of a clas... |
trintss 5208 | Any nonempty transitive cl... |
axrep1 5210 | The version of the Axiom o... |
axreplem 5211 | Lemma for ~ axrep2 and ~ a... |
axrep2 5212 | Axiom of Replacement expre... |
axrep3 5213 | Axiom of Replacement sligh... |
axrep4 5214 | A more traditional version... |
axrep5 5215 | Axiom of Replacement (simi... |
axrep6 5216 | A condensed form of ~ ax-r... |
axrep6g 5217 | ~ axrep6 in class notation... |
zfrepclf 5218 | An inference based on the ... |
zfrep3cl 5219 | An inference based on the ... |
zfrep4 5220 | A version of Replacement u... |
axsepgfromrep 5221 | A more general version ~ a... |
axsep 5222 | Axiom scheme of separation... |
axsepg 5224 | A more general version of ... |
zfauscl 5225 | Separation Scheme (Aussond... |
bm1.3ii 5226 | Convert implication to equ... |
ax6vsep 5227 | Derive ~ ax6v (a weakened ... |
axnulALT 5228 | Alternate proof of ~ axnul... |
axnul 5229 | The Null Set Axiom of ZF s... |
0ex 5231 | The Null Set Axiom of ZF s... |
al0ssb 5232 | The empty set is the uniqu... |
sseliALT 5233 | Alternate proof of ~ sseli... |
csbexg 5234 | The existence of proper su... |
csbex 5235 | The existence of proper su... |
unisn2 5236 | A version of ~ unisn witho... |
nalset 5237 | No set contains all sets. ... |
vnex 5238 | The universal class does n... |
vprc 5239 | The universal class is not... |
nvel 5240 | The universal class does n... |
inex1 5241 | Separation Scheme (Aussond... |
inex2 5242 | Separation Scheme (Aussond... |
inex1g 5243 | Closed-form, generalized S... |
inex2g 5244 | Sufficient condition for a... |
ssex 5245 | The subset of a set is als... |
ssexi 5246 | The subset of a set is als... |
ssexg 5247 | The subset of a set is als... |
ssexd 5248 | A subclass of a set is a s... |
prcssprc 5249 | The superclass of a proper... |
sselpwd 5250 | Elementhood to a power set... |
difexg 5251 | Existence of a difference.... |
difexi 5252 | Existence of a difference,... |
difexd 5253 | Existence of a difference.... |
zfausab 5254 | Separation Scheme (Aussond... |
rabexg 5255 | Separation Scheme in terms... |
rabex 5256 | Separation Scheme in terms... |
rabexd 5257 | Separation Scheme in terms... |
rabex2 5258 | Separation Scheme in terms... |
rab2ex 5259 | A class abstraction based ... |
elssabg 5260 | Membership in a class abst... |
intex 5261 | The intersection of a none... |
intnex 5262 | If a class intersection is... |
intexab 5263 | The intersection of a none... |
intexrab 5264 | The intersection of a none... |
iinexg 5265 | The existence of a class i... |
intabs 5266 | Absorption of a redundant ... |
inuni 5267 | The intersection of a unio... |
elpw2g 5268 | Membership in a power clas... |
elpw2 5269 | Membership in a power clas... |
elpwi2 5270 | Membership in a power clas... |
elpwi2OLD 5271 | Obsolete version of ~ elpw... |
pwnss 5272 | The power set of a set is ... |
pwne 5273 | No set equals its power se... |
difelpw 5274 | A difference is an element... |
rabelpw 5275 | A restricted class abstrac... |
class2set 5276 | Construct, from any class ... |
class2seteq 5277 | Equality theorem based on ... |
0elpw 5278 | Every power class contains... |
pwne0 5279 | A power class is never emp... |
0nep0 5280 | The empty set and its powe... |
0inp0 5281 | Something cannot be equal ... |
unidif0 5282 | The removal of the empty s... |
eqsnuniex 5283 | If a class is equal to the... |
iin0 5284 | An indexed intersection of... |
notzfaus 5285 | In the Separation Scheme ~... |
intv 5286 | The intersection of the un... |
axpweq 5287 | Two equivalent ways to exp... |
zfpow 5289 | Axiom of Power Sets expres... |
axpow2 5290 | A variant of the Axiom of ... |
axpow3 5291 | A variant of the Axiom of ... |
elALT2 5292 | Alternate proof of ~ el us... |
dtruALT2 5293 | Alternate proof of ~ dtru ... |
dtrucor 5294 | Corollary of ~ dtru . Thi... |
dtrucor2 5295 | The theorem form of the de... |
dvdemo1 5296 | Demonstration of a theorem... |
dvdemo2 5297 | Demonstration of a theorem... |
nfnid 5298 | A setvar variable is not f... |
nfcvb 5299 | The "distinctor" expressio... |
vpwex 5300 | Power set axiom: the power... |
pwexg 5301 | Power set axiom expressed ... |
pwexd 5302 | Deduction version of the p... |
pwex 5303 | Power set axiom expressed ... |
pwel 5304 | Quantitative version of ~ ... |
abssexg 5305 | Existence of a class of su... |
snexALT 5306 | Alternate proof of ~ snex ... |
p0ex 5307 | The power set of the empty... |
p0exALT 5308 | Alternate proof of ~ p0ex ... |
pp0ex 5309 | The power set of the power... |
ord3ex 5310 | The ordinal number 3 is a ... |
dtruALT 5311 | Alternate proof of ~ dtru ... |
axc16b 5312 | This theorem shows that Ax... |
eunex 5313 | Existential uniqueness imp... |
eusv1 5314 | Two ways to express single... |
eusvnf 5315 | Even if ` x ` is free in `... |
eusvnfb 5316 | Two ways to say that ` A (... |
eusv2i 5317 | Two ways to express single... |
eusv2nf 5318 | Two ways to express single... |
eusv2 5319 | Two ways to express single... |
reusv1 5320 | Two ways to express single... |
reusv2lem1 5321 | Lemma for ~ reusv2 . (Con... |
reusv2lem2 5322 | Lemma for ~ reusv2 . (Con... |
reusv2lem3 5323 | Lemma for ~ reusv2 . (Con... |
reusv2lem4 5324 | Lemma for ~ reusv2 . (Con... |
reusv2lem5 5325 | Lemma for ~ reusv2 . (Con... |
reusv2 5326 | Two ways to express single... |
reusv3i 5327 | Two ways of expressing exi... |
reusv3 5328 | Two ways to express single... |
eusv4 5329 | Two ways to express single... |
alxfr 5330 | Transfer universal quantif... |
ralxfrd 5331 | Transfer universal quantif... |
rexxfrd 5332 | Transfer universal quantif... |
ralxfr2d 5333 | Transfer universal quantif... |
rexxfr2d 5334 | Transfer universal quantif... |
ralxfrd2 5335 | Transfer universal quantif... |
rexxfrd2 5336 | Transfer existence from a ... |
ralxfr 5337 | Transfer universal quantif... |
ralxfrALT 5338 | Alternate proof of ~ ralxf... |
rexxfr 5339 | Transfer existence from a ... |
rabxfrd 5340 | Membership in a restricted... |
rabxfr 5341 | Membership in a restricted... |
reuhypd 5342 | A theorem useful for elimi... |
reuhyp 5343 | A theorem useful for elimi... |
zfpair 5344 | The Axiom of Pairing of Ze... |
axprALT 5345 | Alternate proof of ~ axpr ... |
axprlem1 5346 | Lemma for ~ axpr . There ... |
axprlem2 5347 | Lemma for ~ axpr . There ... |
axprlem3 5348 | Lemma for ~ axpr . Elimin... |
axprlem4 5349 | Lemma for ~ axpr . The fi... |
axprlem5 5350 | Lemma for ~ axpr . The se... |
axpr 5351 | Unabbreviated version of t... |
zfpair2 5353 | Derive the abbreviated ver... |
snex 5354 | A singleton is a set. The... |
prex 5355 | The Axiom of Pairing using... |
sels 5356 | If a class is a set, then ... |
el 5357 | Every set is an element of... |
elALT 5358 | Alternate proof of ~ el , ... |
dtru 5359 | At least two sets exist (o... |
snelpwi 5360 | A singleton of a set belon... |
snelpw 5361 | A singleton of a set belon... |
prelpw 5362 | A pair of two sets belongs... |
prelpwi 5363 | A pair of two sets belongs... |
rext 5364 | A theorem similar to exten... |
sspwb 5365 | The powerclass constructio... |
unipw 5366 | A class equals the union o... |
univ 5367 | The union of the universe ... |
pwtr 5368 | A class is transitive iff ... |
ssextss 5369 | An extensionality-like pri... |
ssext 5370 | An extensionality-like pri... |
nssss 5371 | Negation of subclass relat... |
pweqb 5372 | Classes are equal if and o... |
intid 5373 | The intersection of all se... |
moabex 5374 | "At most one" existence im... |
rmorabex 5375 | Restricted "at most one" e... |
euabex 5376 | The abstraction of a wff w... |
nnullss 5377 | A nonempty class (even if ... |
exss 5378 | Restricted existence in a ... |
opex 5379 | An ordered pair of classes... |
otex 5380 | An ordered triple of class... |
elopg 5381 | Characterization of the el... |
elop 5382 | Characterization of the el... |
opi1 5383 | One of the two elements in... |
opi2 5384 | One of the two elements of... |
opeluu 5385 | Each member of an ordered ... |
op1stb 5386 | Extract the first member o... |
brv 5387 | Two classes are always in ... |
opnz 5388 | An ordered pair is nonempt... |
opnzi 5389 | An ordered pair is nonempt... |
opth1 5390 | Equality of the first memb... |
opth 5391 | The ordered pair theorem. ... |
opthg 5392 | Ordered pair theorem. ` C ... |
opth1g 5393 | Equality of the first memb... |
opthg2 5394 | Ordered pair theorem. (Co... |
opth2 5395 | Ordered pair theorem. (Co... |
opthneg 5396 | Two ordered pairs are not ... |
opthne 5397 | Two ordered pairs are not ... |
otth2 5398 | Ordered triple theorem, wi... |
otth 5399 | Ordered triple theorem. (... |
otthg 5400 | Ordered triple theorem, cl... |
eqvinop 5401 | A variable introduction la... |
sbcop1 5402 | The proper substitution of... |
sbcop 5403 | The proper substitution of... |
copsexgw 5404 | Version of ~ copsexg with ... |
copsexg 5405 | Substitution of class ` A ... |
copsex2t 5406 | Closed theorem form of ~ c... |
copsex2g 5407 | Implicit substitution infe... |
copsex2gOLD 5408 | Obsolete version of ~ cops... |
copsex4g 5409 | An implicit substitution i... |
0nelop 5410 | A property of ordered pair... |
opwo0id 5411 | An ordered pair is equal t... |
opeqex 5412 | Equivalence of existence i... |
oteqex2 5413 | Equivalence of existence i... |
oteqex 5414 | Equivalence of existence i... |
opcom 5415 | An ordered pair commutes i... |
moop2 5416 | "At most one" property of ... |
opeqsng 5417 | Equivalence for an ordered... |
opeqsn 5418 | Equivalence for an ordered... |
opeqpr 5419 | Equivalence for an ordered... |
snopeqop 5420 | Equivalence for an ordered... |
propeqop 5421 | Equivalence for an ordered... |
propssopi 5422 | If a pair of ordered pairs... |
snopeqopsnid 5423 | Equivalence for an ordered... |
mosubopt 5424 | "At most one" remains true... |
mosubop 5425 | "At most one" remains true... |
euop2 5426 | Transfer existential uniqu... |
euotd 5427 | Prove existential uniquene... |
opthwiener 5428 | Justification theorem for ... |
uniop 5429 | The union of an ordered pa... |
uniopel 5430 | Ordered pair membership is... |
opthhausdorff 5431 | Justification theorem for ... |
opthhausdorff0 5432 | Justification theorem for ... |
otsndisj 5433 | The singletons consisting ... |
otiunsndisj 5434 | The union of singletons co... |
iunopeqop 5435 | Implication of an ordered ... |
brsnop 5436 | Binary relation for an ord... |
opabidw 5437 | The law of concretion. Sp... |
opabid 5438 | The law of concretion. Sp... |
elopabw 5439 | Membership in a class abst... |
elopab 5440 | Membership in a class abst... |
rexopabb 5441 | Restricted existential qua... |
vopelopabsb 5442 | The law of concretion in t... |
opelopabsb 5443 | The law of concretion in t... |
brabsb 5444 | The law of concretion in t... |
opelopabt 5445 | Closed theorem form of ~ o... |
opelopabga 5446 | The law of concretion. Th... |
brabga 5447 | The law of concretion for ... |
opelopab2a 5448 | Ordered pair membership in... |
opelopaba 5449 | The law of concretion. Th... |
braba 5450 | The law of concretion for ... |
opelopabg 5451 | The law of concretion. Th... |
brabg 5452 | The law of concretion for ... |
opelopabgf 5453 | The law of concretion. Th... |
opelopab2 5454 | Ordered pair membership in... |
opelopab 5455 | The law of concretion. Th... |
brab 5456 | The law of concretion for ... |
opelopabaf 5457 | The law of concretion. Th... |
opelopabf 5458 | The law of concretion. Th... |
ssopab2 5459 | Equivalence of ordered pai... |
ssopab2bw 5460 | Equivalence of ordered pai... |
eqopab2bw 5461 | Equivalence of ordered pai... |
ssopab2b 5462 | Equivalence of ordered pai... |
ssopab2i 5463 | Inference of ordered pair ... |
ssopab2dv 5464 | Inference of ordered pair ... |
eqopab2b 5465 | Equivalence of ordered pai... |
opabn0 5466 | Nonempty ordered pair clas... |
opab0 5467 | Empty ordered pair class a... |
csbopab 5468 | Move substitution into a c... |
csbopabgALT 5469 | Move substitution into a c... |
csbmpt12 5470 | Move substitution into a m... |
csbmpt2 5471 | Move substitution into the... |
iunopab 5472 | Move indexed union inside ... |
iunopabOLD 5473 | Obsolete version of ~ iuno... |
elopabr 5474 | Membership in an ordered-p... |
elopabran 5475 | Membership in an ordered-p... |
elopabrOLD 5476 | Obsolete version of ~ elop... |
rbropapd 5477 | Properties of a pair in an... |
rbropap 5478 | Properties of a pair in a ... |
2rbropap 5479 | Properties of a pair in a ... |
0nelopab 5480 | The empty set is never an ... |
0nelopabOLD 5481 | Obsolete version of ~ 0nel... |
brabv 5482 | If two classes are in a re... |
pwin 5483 | The power class of the int... |
pwunssOLD 5484 | Obsolete version of ~ pwun... |
pwssun 5485 | The power class of the uni... |
pwundifOLD 5486 | Obsolete proof of ~ pwundi... |
pwun 5487 | The power class of the uni... |
dfid4 5490 | The identity function expr... |
dfid2 5491 | Alternate definition of th... |
dfid3 5492 | A stronger version of ~ df... |
dfid2OLD 5493 | Obsolete version of ~ dfid... |
epelg 5496 | The membership relation an... |
epeli 5497 | The membership relation an... |
epel 5498 | The membership relation an... |
0sn0ep 5499 | An example for the members... |
epn0 5500 | The membership relation is... |
poss 5505 | Subset theorem for the par... |
poeq1 5506 | Equality theorem for parti... |
poeq2 5507 | Equality theorem for parti... |
nfpo 5508 | Bound-variable hypothesis ... |
nfso 5509 | Bound-variable hypothesis ... |
pocl 5510 | Characteristic properties ... |
poclOLD 5511 | Obsolete version of ~ pocl... |
ispod 5512 | Sufficient conditions for ... |
swopolem 5513 | Perform the substitutions ... |
swopo 5514 | A strict weak order is a p... |
poirr 5515 | A partial order is irrefle... |
potr 5516 | A partial order is a trans... |
po2nr 5517 | A partial order has no 2-c... |
po3nr 5518 | A partial order has no 3-c... |
po2ne 5519 | Two sets related by a part... |
po0 5520 | Any relation is a partial ... |
pofun 5521 | The inverse image of a par... |
sopo 5522 | A strict linear order is a... |
soss 5523 | Subset theorem for the str... |
soeq1 5524 | Equality theorem for the s... |
soeq2 5525 | Equality theorem for the s... |
sonr 5526 | A strict order relation is... |
sotr 5527 | A strict order relation is... |
solin 5528 | A strict order relation is... |
so2nr 5529 | A strict order relation ha... |
so3nr 5530 | A strict order relation ha... |
sotric 5531 | A strict order relation sa... |
sotrieq 5532 | Trichotomy law for strict ... |
sotrieq2 5533 | Trichotomy law for strict ... |
soasym 5534 | Asymmetry law for strict o... |
sotr2 5535 | A transitivity relation. ... |
issod 5536 | An irreflexive, transitive... |
issoi 5537 | An irreflexive, transitive... |
isso2i 5538 | Deduce strict ordering fro... |
so0 5539 | Any relation is a strict o... |
somo 5540 | A totally ordered set has ... |
dffr6 5547 | Alternate definition of ~ ... |
frd 5548 | A nonempty subset of an ` ... |
fri 5549 | A nonempty subset of an ` ... |
friOLD 5550 | Obsolete version of ~ fri ... |
seex 5551 | The ` R ` -preimage of an ... |
exse 5552 | Any relation on a set is s... |
dffr2 5553 | Alternate definition of we... |
dffr2ALT 5554 | Alternate proof of ~ dffr2... |
frc 5555 | Property of well-founded r... |
frss 5556 | Subset theorem for the wel... |
sess1 5557 | Subset theorem for the set... |
sess2 5558 | Subset theorem for the set... |
freq1 5559 | Equality theorem for the w... |
freq2 5560 | Equality theorem for the w... |
seeq1 5561 | Equality theorem for the s... |
seeq2 5562 | Equality theorem for the s... |
nffr 5563 | Bound-variable hypothesis ... |
nfse 5564 | Bound-variable hypothesis ... |
nfwe 5565 | Bound-variable hypothesis ... |
frirr 5566 | A well-founded relation is... |
fr2nr 5567 | A well-founded relation ha... |
fr0 5568 | Any relation is well-found... |
frminex 5569 | If an element of a well-fo... |
efrirr 5570 | A well-founded class does ... |
efrn2lp 5571 | A well-founded class conta... |
epse 5572 | The membership relation is... |
tz7.2 5573 | Similar to Theorem 7.2 of ... |
dfepfr 5574 | An alternate way of saying... |
epfrc 5575 | A subset of a well-founded... |
wess 5576 | Subset theorem for the wel... |
weeq1 5577 | Equality theorem for the w... |
weeq2 5578 | Equality theorem for the w... |
wefr 5579 | A well-ordering is well-fo... |
weso 5580 | A well-ordering is a stric... |
wecmpep 5581 | The elements of a class we... |
wetrep 5582 | On a class well-ordered by... |
wefrc 5583 | A nonempty subclass of a c... |
we0 5584 | Any relation is a well-ord... |
wereu 5585 | A nonempty subset of an ` ... |
wereu2 5586 | A nonempty subclass of an ... |
xpeq1 5603 | Equality theorem for Carte... |
xpss12 5604 | Subset theorem for Cartesi... |
xpss 5605 | A Cartesian product is inc... |
inxpssres 5606 | Intersection with a Cartes... |
relxp 5607 | A Cartesian product is a r... |
xpss1 5608 | Subset relation for Cartes... |
xpss2 5609 | Subset relation for Cartes... |
xpeq2 5610 | Equality theorem for Carte... |
elxpi 5611 | Membership in a Cartesian ... |
elxp 5612 | Membership in a Cartesian ... |
elxp2 5613 | Membership in a Cartesian ... |
xpeq12 5614 | Equality theorem for Carte... |
xpeq1i 5615 | Equality inference for Car... |
xpeq2i 5616 | Equality inference for Car... |
xpeq12i 5617 | Equality inference for Car... |
xpeq1d 5618 | Equality deduction for Car... |
xpeq2d 5619 | Equality deduction for Car... |
xpeq12d 5620 | Equality deduction for Car... |
sqxpeqd 5621 | Equality deduction for a C... |
nfxp 5622 | Bound-variable hypothesis ... |
0nelxp 5623 | The empty set is not a mem... |
0nelelxp 5624 | A member of a Cartesian pr... |
opelxp 5625 | Ordered pair membership in... |
opelxpi 5626 | Ordered pair membership in... |
opelxpd 5627 | Ordered pair membership in... |
opelvv 5628 | Ordered pair membership in... |
opelvvg 5629 | Ordered pair membership in... |
opelxp1 5630 | The first member of an ord... |
opelxp2 5631 | The second member of an or... |
otelxp1 5632 | The first member of an ord... |
otel3xp 5633 | An ordered triple is an el... |
opabssxpd 5634 | An ordered-pair class abst... |
rabxp 5635 | Class abstraction restrict... |
brxp 5636 | Binary relation on a Carte... |
pwvrel 5637 | A set is a binary relation... |
pwvabrel 5638 | The powerclass of the cart... |
brrelex12 5639 | Two classes related by a b... |
brrelex1 5640 | If two classes are related... |
brrelex2 5641 | If two classes are related... |
brrelex12i 5642 | Two classes that are relat... |
brrelex1i 5643 | The first argument of a bi... |
brrelex2i 5644 | The second argument of a b... |
nprrel12 5645 | Proper classes are not rel... |
nprrel 5646 | No proper class is related... |
0nelrel0 5647 | A binary relation does not... |
0nelrel 5648 | A binary relation does not... |
fconstmpt 5649 | Representation of a consta... |
vtoclr 5650 | Variable to class conversi... |
opthprc 5651 | Justification theorem for ... |
brel 5652 | Two things in a binary rel... |
elxp3 5653 | Membership in a Cartesian ... |
opeliunxp 5654 | Membership in a union of C... |
xpundi 5655 | Distributive law for Carte... |
xpundir 5656 | Distributive law for Carte... |
xpiundi 5657 | Distributive law for Carte... |
xpiundir 5658 | Distributive law for Carte... |
iunxpconst 5659 | Membership in a union of C... |
xpun 5660 | The Cartesian product of t... |
elvv 5661 | Membership in universal cl... |
elvvv 5662 | Membership in universal cl... |
elvvuni 5663 | An ordered pair contains i... |
brinxp2 5664 | Intersection of binary rel... |
brinxp 5665 | Intersection of binary rel... |
opelinxp 5666 | Ordered pair element in an... |
poinxp 5667 | Intersection of partial or... |
soinxp 5668 | Intersection of total orde... |
frinxp 5669 | Intersection of well-found... |
seinxp 5670 | Intersection of set-like r... |
weinxp 5671 | Intersection of well-order... |
posn 5672 | Partial ordering of a sing... |
sosn 5673 | Strict ordering on a singl... |
frsn 5674 | Founded relation on a sing... |
wesn 5675 | Well-ordering of a singlet... |
elopaelxp 5676 | Membership in an ordered-p... |
elopaelxpOLD 5677 | Obsolete version of ~ elop... |
bropaex12 5678 | Two classes related by an ... |
opabssxp 5679 | An abstraction relation is... |
brab2a 5680 | The law of concretion for ... |
optocl 5681 | Implicit substitution of c... |
2optocl 5682 | Implicit substitution of c... |
3optocl 5683 | Implicit substitution of c... |
opbrop 5684 | Ordered pair membership in... |
0xp 5685 | The Cartesian product with... |
csbxp 5686 | Distribute proper substitu... |
releq 5687 | Equality theorem for the r... |
releqi 5688 | Equality inference for the... |
releqd 5689 | Equality deduction for the... |
nfrel 5690 | Bound-variable hypothesis ... |
sbcrel 5691 | Distribute proper substitu... |
relss 5692 | Subclass theorem for relat... |
ssrel 5693 | A subclass relationship de... |
ssrelOLD 5694 | Obsolete version of ~ ssre... |
eqrel 5695 | Extensionality principle f... |
ssrel2 5696 | A subclass relationship de... |
relssi 5697 | Inference from subclass pr... |
relssdv 5698 | Deduction from subclass pr... |
eqrelriv 5699 | Inference from extensional... |
eqrelriiv 5700 | Inference from extensional... |
eqbrriv 5701 | Inference from extensional... |
eqrelrdv 5702 | Deduce equality of relatio... |
eqbrrdv 5703 | Deduction from extensional... |
eqbrrdiv 5704 | Deduction from extensional... |
eqrelrdv2 5705 | A version of ~ eqrelrdv . ... |
ssrelrel 5706 | A subclass relationship de... |
eqrelrel 5707 | Extensionality principle f... |
elrel 5708 | A member of a relation is ... |
rel0 5709 | The empty set is a relatio... |
nrelv 5710 | The universal class is not... |
relsng 5711 | A singleton is a relation ... |
relsnb 5712 | An at-most-singleton is a ... |
relsnopg 5713 | A singleton of an ordered ... |
relsn 5714 | A singleton is a relation ... |
relsnop 5715 | A singleton of an ordered ... |
copsex2gb 5716 | Implicit substitution infe... |
copsex2ga 5717 | Implicit substitution infe... |
elopaba 5718 | Membership in an ordered-p... |
xpsspw 5719 | A Cartesian product is inc... |
unixpss 5720 | The double class union of ... |
relun 5721 | The union of two relations... |
relin1 5722 | The intersection with a re... |
relin2 5723 | The intersection with a re... |
relinxp 5724 | Intersection with a Cartes... |
reldif 5725 | A difference cutting down ... |
reliun 5726 | An indexed union is a rela... |
reliin 5727 | An indexed intersection is... |
reluni 5728 | The union of a class is a ... |
relint 5729 | The intersection of a clas... |
relopabiv 5730 | A class of ordered pairs i... |
relopabv 5731 | A class of ordered pairs i... |
relopabi 5732 | A class of ordered pairs i... |
relopabiALT 5733 | Alternate proof of ~ relop... |
relopab 5734 | A class of ordered pairs i... |
mptrel 5735 | The maps-to notation alway... |
reli 5736 | The identity relation is a... |
rele 5737 | The membership relation is... |
opabid2 5738 | A relation expressed as an... |
inopab 5739 | Intersection of two ordere... |
difopab 5740 | Difference of two ordered-... |
inxp 5741 | Intersection of two Cartes... |
xpindi 5742 | Distributive law for Carte... |
xpindir 5743 | Distributive law for Carte... |
xpiindi 5744 | Distributive law for Carte... |
xpriindi 5745 | Distributive law for Carte... |
eliunxp 5746 | Membership in a union of C... |
opeliunxp2 5747 | Membership in a union of C... |
raliunxp 5748 | Write a double restricted ... |
rexiunxp 5749 | Write a double restricted ... |
ralxp 5750 | Universal quantification r... |
rexxp 5751 | Existential quantification... |
exopxfr 5752 | Transfer ordered-pair exis... |
exopxfr2 5753 | Transfer ordered-pair exis... |
djussxp 5754 | Disjoint union is a subset... |
ralxpf 5755 | Version of ~ ralxp with bo... |
rexxpf 5756 | Version of ~ rexxp with bo... |
iunxpf 5757 | Indexed union on a Cartesi... |
opabbi2dv 5758 | Deduce equality of a relat... |
relop 5759 | A necessary and sufficient... |
ideqg 5760 | For sets, the identity rel... |
ideq 5761 | For sets, the identity rel... |
ididg 5762 | A set is identical to itse... |
issetid 5763 | Two ways of expressing set... |
coss1 5764 | Subclass theorem for compo... |
coss2 5765 | Subclass theorem for compo... |
coeq1 5766 | Equality theorem for compo... |
coeq2 5767 | Equality theorem for compo... |
coeq1i 5768 | Equality inference for com... |
coeq2i 5769 | Equality inference for com... |
coeq1d 5770 | Equality deduction for com... |
coeq2d 5771 | Equality deduction for com... |
coeq12i 5772 | Equality inference for com... |
coeq12d 5773 | Equality deduction for com... |
nfco 5774 | Bound-variable hypothesis ... |
brcog 5775 | Ordered pair membership in... |
opelco2g 5776 | Ordered pair membership in... |
brcogw 5777 | Ordered pair membership in... |
eqbrrdva 5778 | Deduction from extensional... |
brco 5779 | Binary relation on a compo... |
opelco 5780 | Ordered pair membership in... |
cnvss 5781 | Subset theorem for convers... |
cnveq 5782 | Equality theorem for conve... |
cnveqi 5783 | Equality inference for con... |
cnveqd 5784 | Equality deduction for con... |
elcnv 5785 | Membership in a converse r... |
elcnv2 5786 | Membership in a converse r... |
nfcnv 5787 | Bound-variable hypothesis ... |
brcnvg 5788 | The converse of a binary r... |
opelcnvg 5789 | Ordered-pair membership in... |
opelcnv 5790 | Ordered-pair membership in... |
brcnv 5791 | The converse of a binary r... |
csbcnv 5792 | Move class substitution in... |
csbcnvgALT 5793 | Move class substitution in... |
cnvco 5794 | Distributive law of conver... |
cnvuni 5795 | The converse of a class un... |
dfdm3 5796 | Alternate definition of do... |
dfrn2 5797 | Alternate definition of ra... |
dfrn3 5798 | Alternate definition of ra... |
elrn2g 5799 | Membership in a range. (C... |
elrng 5800 | Membership in a range. (C... |
elrn2 5801 | Membership in a range. (C... |
elrn 5802 | Membership in a range. (C... |
ssrelrn 5803 | If a relation is a subset ... |
dfdm4 5804 | Alternate definition of do... |
dfdmf 5805 | Definition of domain, usin... |
csbdm 5806 | Distribute proper substitu... |
eldmg 5807 | Domain membership. Theore... |
eldm2g 5808 | Domain membership. Theore... |
eldm 5809 | Membership in a domain. T... |
eldm2 5810 | Membership in a domain. T... |
dmss 5811 | Subset theorem for domain.... |
dmeq 5812 | Equality theorem for domai... |
dmeqi 5813 | Equality inference for dom... |
dmeqd 5814 | Equality deduction for dom... |
opeldmd 5815 | Membership of first of an ... |
opeldm 5816 | Membership of first of an ... |
breldm 5817 | Membership of first of a b... |
breldmg 5818 | Membership of first of a b... |
dmun 5819 | The domain of a union is t... |
dmin 5820 | The domain of an intersect... |
breldmd 5821 | Membership of first of a b... |
dmiun 5822 | The domain of an indexed u... |
dmuni 5823 | The domain of a union. Pa... |
dmopab 5824 | The domain of a class of o... |
dmopabelb 5825 | A set is an element of the... |
dmopab2rex 5826 | The domain of an ordered p... |
dmopabss 5827 | Upper bound for the domain... |
dmopab3 5828 | The domain of a restricted... |
dm0 5829 | The domain of the empty se... |
dmi 5830 | The domain of the identity... |
dmv 5831 | The domain of the universe... |
dmep 5832 | The domain of the membersh... |
domepOLD 5833 | Obsolete proof of ~ dmep a... |
dm0rn0 5834 | An empty domain is equival... |
rn0 5835 | The range of the empty set... |
rnep 5836 | The range of the membershi... |
reldm0 5837 | A relation is empty iff it... |
dmxp 5838 | The domain of a Cartesian ... |
dmxpid 5839 | The domain of a Cartesian ... |
dmxpin 5840 | The domain of the intersec... |
xpid11 5841 | The Cartesian square is a ... |
dmcnvcnv 5842 | The domain of the double c... |
rncnvcnv 5843 | The range of the double co... |
elreldm 5844 | The first member of an ord... |
rneq 5845 | Equality theorem for range... |
rneqi 5846 | Equality inference for ran... |
rneqd 5847 | Equality deduction for ran... |
rnss 5848 | Subset theorem for range. ... |
rnssi 5849 | Subclass inference for ran... |
brelrng 5850 | The second argument of a b... |
brelrn 5851 | The second argument of a b... |
opelrn 5852 | Membership of second membe... |
releldm 5853 | The first argument of a bi... |
relelrn 5854 | The second argument of a b... |
releldmb 5855 | Membership in a domain. (... |
relelrnb 5856 | Membership in a range. (C... |
releldmi 5857 | The first argument of a bi... |
relelrni 5858 | The second argument of a b... |
dfrnf 5859 | Definition of range, using... |
nfdm 5860 | Bound-variable hypothesis ... |
nfrn 5861 | Bound-variable hypothesis ... |
dmiin 5862 | Domain of an intersection.... |
rnopab 5863 | The range of a class of or... |
rnmpt 5864 | The range of a function in... |
elrnmpt 5865 | The range of a function in... |
elrnmpt1s 5866 | Elementhood in an image se... |
elrnmpt1 5867 | Elementhood in an image se... |
elrnmptg 5868 | Membership in the range of... |
elrnmpti 5869 | Membership in the range of... |
elrnmptd 5870 | The range of a function in... |
elrnmptdv 5871 | Elementhood in the range o... |
elrnmpt2d 5872 | Elementhood in the range o... |
dfiun3g 5873 | Alternate definition of in... |
dfiin3g 5874 | Alternate definition of in... |
dfiun3 5875 | Alternate definition of in... |
dfiin3 5876 | Alternate definition of in... |
riinint 5877 | Express a relative indexed... |
relrn0 5878 | A relation is empty iff it... |
dmrnssfld 5879 | The domain and range of a ... |
dmcoss 5880 | Domain of a composition. ... |
rncoss 5881 | Range of a composition. (... |
dmcosseq 5882 | Domain of a composition. ... |
dmcoeq 5883 | Domain of a composition. ... |
rncoeq 5884 | Range of a composition. (... |
reseq1 5885 | Equality theorem for restr... |
reseq2 5886 | Equality theorem for restr... |
reseq1i 5887 | Equality inference for res... |
reseq2i 5888 | Equality inference for res... |
reseq12i 5889 | Equality inference for res... |
reseq1d 5890 | Equality deduction for res... |
reseq2d 5891 | Equality deduction for res... |
reseq12d 5892 | Equality deduction for res... |
nfres 5893 | Bound-variable hypothesis ... |
csbres 5894 | Distribute proper substitu... |
res0 5895 | A restriction to the empty... |
dfres3 5896 | Alternate definition of re... |
opelres 5897 | Ordered pair elementhood i... |
brres 5898 | Binary relation on a restr... |
opelresi 5899 | Ordered pair membership in... |
brresi 5900 | Binary relation on a restr... |
opres 5901 | Ordered pair membership in... |
resieq 5902 | A restricted identity rela... |
opelidres 5903 | ` <. A , A >. ` belongs to... |
resres 5904 | The restriction of a restr... |
resundi 5905 | Distributive law for restr... |
resundir 5906 | Distributive law for restr... |
resindi 5907 | Class restriction distribu... |
resindir 5908 | Class restriction distribu... |
inres 5909 | Move intersection into cla... |
resdifcom 5910 | Commutative law for restri... |
resiun1 5911 | Distribution of restrictio... |
resiun2 5912 | Distribution of restrictio... |
dmres 5913 | The domain of a restrictio... |
ssdmres 5914 | A domain restricted to a s... |
dmresexg 5915 | The domain of a restrictio... |
resss 5916 | A class includes its restr... |
rescom 5917 | Commutative law for restri... |
ssres 5918 | Subclass theorem for restr... |
ssres2 5919 | Subclass theorem for restr... |
relres 5920 | A restriction is a relatio... |
resabs1 5921 | Absorption law for restric... |
resabs1d 5922 | Absorption law for restric... |
resabs2 5923 | Absorption law for restric... |
residm 5924 | Idempotent law for restric... |
resima 5925 | A restriction to an image.... |
resima2 5926 | Image under a restricted c... |
rnresss 5927 | The range of a restriction... |
xpssres 5928 | Restriction of a constant ... |
elinxp 5929 | Membership in an intersect... |
elres 5930 | Membership in a restrictio... |
elsnres 5931 | Membership in restriction ... |
relssres 5932 | Simplification law for res... |
dmressnsn 5933 | The domain of a restrictio... |
eldmressnsn 5934 | The element of the domain ... |
eldmeldmressn 5935 | An element of the domain (... |
resdm 5936 | A relation restricted to i... |
resexg 5937 | The restriction of a set i... |
resexd 5938 | The restriction of a set i... |
resex 5939 | The restriction of a set i... |
resindm 5940 | When restricting a relatio... |
resdmdfsn 5941 | Restricting a relation to ... |
resopab 5942 | Restriction of a class abs... |
iss 5943 | A subclass of the identity... |
resopab2 5944 | Restriction of a class abs... |
resmpt 5945 | Restriction of the mapping... |
resmpt3 5946 | Unconditional restriction ... |
resmptf 5947 | Restriction of the mapping... |
resmptd 5948 | Restriction of the mapping... |
dfres2 5949 | Alternate definition of th... |
mptss 5950 | Sufficient condition for i... |
elidinxp 5951 | Characterization of the el... |
elidinxpid 5952 | Characterization of the el... |
elrid 5953 | Characterization of the el... |
idinxpres 5954 | The intersection of the id... |
idinxpresid 5955 | The intersection of the id... |
idssxp 5956 | A diagonal set as a subset... |
opabresid 5957 | The restricted identity re... |
mptresid 5958 | The restricted identity re... |
opabresidOLD 5959 | Obsolete version of ~ opab... |
mptresidOLD 5960 | Obsolete version of ~ mptr... |
dmresi 5961 | The domain of a restricted... |
restidsing 5962 | Restriction of the identit... |
iresn0n0 5963 | The identity function rest... |
imaeq1 5964 | Equality theorem for image... |
imaeq2 5965 | Equality theorem for image... |
imaeq1i 5966 | Equality theorem for image... |
imaeq2i 5967 | Equality theorem for image... |
imaeq1d 5968 | Equality theorem for image... |
imaeq2d 5969 | Equality theorem for image... |
imaeq12d 5970 | Equality theorem for image... |
dfima2 5971 | Alternate definition of im... |
dfima3 5972 | Alternate definition of im... |
elimag 5973 | Membership in an image. T... |
elima 5974 | Membership in an image. T... |
elima2 5975 | Membership in an image. T... |
elima3 5976 | Membership in an image. T... |
nfima 5977 | Bound-variable hypothesis ... |
nfimad 5978 | Deduction version of bound... |
imadmrn 5979 | The image of the domain of... |
imassrn 5980 | The image of a class is a ... |
mptima 5981 | Image of a function in map... |
imai 5982 | Image under the identity r... |
rnresi 5983 | The range of the restricte... |
resiima 5984 | The image of a restriction... |
ima0 5985 | Image of the empty set. T... |
0ima 5986 | Image under the empty rela... |
csbima12 5987 | Move class substitution in... |
imadisj 5988 | A class whose image under ... |
cnvimass 5989 | A preimage under any class... |
cnvimarndm 5990 | The preimage of the range ... |
imasng 5991 | The image of a singleton. ... |
relimasn 5992 | The image of a singleton. ... |
elrelimasn 5993 | Elementhood in the image o... |
elimasng1 5994 | Membership in an image of ... |
elimasn1 5995 | Membership in an image of ... |
elimasng 5996 | Membership in an image of ... |
elimasn 5997 | Membership in an image of ... |
elimasngOLD 5998 | Obsolete version of ~ elim... |
elimasni 5999 | Membership in an image of ... |
args 6000 | Two ways to express the cl... |
elinisegg 6001 | Membership in the inverse ... |
eliniseg 6002 | Membership in the inverse ... |
epin 6003 | Any set is equal to its pr... |
epini 6004 | Any set is equal to its pr... |
iniseg 6005 | An idiom that signifies an... |
inisegn0 6006 | Nonemptiness of an initial... |
dffr3 6007 | Alternate definition of we... |
dfse2 6008 | Alternate definition of se... |
imass1 6009 | Subset theorem for image. ... |
imass2 6010 | Subset theorem for image. ... |
ndmima 6011 | The image of a singleton o... |
relcnv 6012 | A converse is a relation. ... |
relbrcnvg 6013 | When ` R ` is a relation, ... |
eliniseg2 6014 | Eliminate the class existe... |
relbrcnv 6015 | When ` R ` is a relation, ... |
cotrg 6016 | Two ways of saying that th... |
cotr 6017 | Two ways of saying a relat... |
idrefALT 6018 | Alternate proof of ~ idref... |
cnvsym 6019 | Two ways of saying a relat... |
intasym 6020 | Two ways of saying a relat... |
asymref 6021 | Two ways of saying a relat... |
asymref2 6022 | Two ways of saying a relat... |
intirr 6023 | Two ways of saying a relat... |
brcodir 6024 | Two ways of saying that tw... |
codir 6025 | Two ways of saying a relat... |
qfto 6026 | A quantifier-free way of e... |
xpidtr 6027 | A Cartesian square is a tr... |
trin2 6028 | The intersection of two tr... |
poirr2 6029 | A partial order is irrefle... |
trinxp 6030 | The relation induced by a ... |
soirri 6031 | A strict order relation is... |
sotri 6032 | A strict order relation is... |
son2lpi 6033 | A strict order relation ha... |
sotri2 6034 | A transitivity relation. ... |
sotri3 6035 | A transitivity relation. ... |
poleloe 6036 | Express "less than or equa... |
poltletr 6037 | Transitive law for general... |
somin1 6038 | Property of a minimum in a... |
somincom 6039 | Commutativity of minimum i... |
somin2 6040 | Property of a minimum in a... |
soltmin 6041 | Being less than a minimum,... |
cnvopab 6042 | The converse of a class ab... |
mptcnv 6043 | The converse of a mapping ... |
cnv0 6044 | The converse of the empty ... |
cnvi 6045 | The converse of the identi... |
cnvun 6046 | The converse of a union is... |
cnvdif 6047 | Distributive law for conve... |
cnvin 6048 | Distributive law for conve... |
rnun 6049 | Distributive law for range... |
rnin 6050 | The range of an intersecti... |
rniun 6051 | The range of an indexed un... |
rnuni 6052 | The range of a union. Par... |
imaundi 6053 | Distributive law for image... |
imaundir 6054 | The image of a union. (Co... |
cnvimassrndm 6055 | The preimage of a superset... |
dminss 6056 | An upper bound for interse... |
imainss 6057 | An upper bound for interse... |
inimass 6058 | The image of an intersecti... |
inimasn 6059 | The intersection of the im... |
cnvxp 6060 | The converse of a Cartesia... |
xp0 6061 | The Cartesian product with... |
xpnz 6062 | The Cartesian product of n... |
xpeq0 6063 | At least one member of an ... |
xpdisj1 6064 | Cartesian products with di... |
xpdisj2 6065 | Cartesian products with di... |
xpsndisj 6066 | Cartesian products with tw... |
difxp 6067 | Difference of Cartesian pr... |
difxp1 6068 | Difference law for Cartesi... |
difxp2 6069 | Difference law for Cartesi... |
djudisj 6070 | Disjoint unions with disjo... |
xpdifid 6071 | The set of distinct couple... |
resdisj 6072 | A double restriction to di... |
rnxp 6073 | The range of a Cartesian p... |
dmxpss 6074 | The domain of a Cartesian ... |
rnxpss 6075 | The range of a Cartesian p... |
rnxpid 6076 | The range of a Cartesian s... |
ssxpb 6077 | A Cartesian product subcla... |
xp11 6078 | The Cartesian product of n... |
xpcan 6079 | Cancellation law for Carte... |
xpcan2 6080 | Cancellation law for Carte... |
ssrnres 6081 | Two ways to express surjec... |
rninxp 6082 | Two ways to express surjec... |
dminxp 6083 | Two ways to express totali... |
imainrect 6084 | Image by a restricted and ... |
xpima 6085 | Direct image by a Cartesia... |
xpima1 6086 | Direct image by a Cartesia... |
xpima2 6087 | Direct image by a Cartesia... |
xpimasn 6088 | Direct image of a singleto... |
sossfld 6089 | The base set of a strict o... |
sofld 6090 | The base set of a nonempty... |
cnvcnv3 6091 | The set of all ordered pai... |
dfrel2 6092 | Alternate definition of re... |
dfrel4v 6093 | A relation can be expresse... |
dfrel4 6094 | A relation can be expresse... |
cnvcnv 6095 | The double converse of a c... |
cnvcnv2 6096 | The double converse of a c... |
cnvcnvss 6097 | The double converse of a c... |
cnvrescnv 6098 | Two ways to express the co... |
cnveqb 6099 | Equality theorem for conve... |
cnveq0 6100 | A relation empty iff its c... |
dfrel3 6101 | Alternate definition of re... |
elid 6102 | Characterization of the el... |
dmresv 6103 | The domain of a universal ... |
rnresv 6104 | The range of a universal r... |
dfrn4 6105 | Range defined in terms of ... |
csbrn 6106 | Distribute proper substitu... |
rescnvcnv 6107 | The restriction of the dou... |
cnvcnvres 6108 | The double converse of the... |
imacnvcnv 6109 | The image of the double co... |
dmsnn0 6110 | The domain of a singleton ... |
rnsnn0 6111 | The range of a singleton i... |
dmsn0 6112 | The domain of the singleto... |
cnvsn0 6113 | The converse of the single... |
dmsn0el 6114 | The domain of a singleton ... |
relsn2 6115 | A singleton is a relation ... |
dmsnopg 6116 | The domain of a singleton ... |
dmsnopss 6117 | The domain of a singleton ... |
dmpropg 6118 | The domain of an unordered... |
dmsnop 6119 | The domain of a singleton ... |
dmprop 6120 | The domain of an unordered... |
dmtpop 6121 | The domain of an unordered... |
cnvcnvsn 6122 | Double converse of a singl... |
dmsnsnsn 6123 | The domain of the singleto... |
rnsnopg 6124 | The range of a singleton o... |
rnpropg 6125 | The range of a pair of ord... |
cnvsng 6126 | Converse of a singleton of... |
rnsnop 6127 | The range of a singleton o... |
op1sta 6128 | Extract the first member o... |
cnvsn 6129 | Converse of a singleton of... |
op2ndb 6130 | Extract the second member ... |
op2nda 6131 | Extract the second member ... |
opswap 6132 | Swap the members of an ord... |
cnvresima 6133 | An image under the convers... |
resdm2 6134 | A class restricted to its ... |
resdmres 6135 | Restriction to the domain ... |
resresdm 6136 | A restriction by an arbitr... |
imadmres 6137 | The image of the domain of... |
resdmss 6138 | Subset relationship for th... |
resdifdi 6139 | Distributive law for restr... |
resdifdir 6140 | Distributive law for restr... |
mptpreima 6141 | The preimage of a function... |
mptiniseg 6142 | Converse singleton image o... |
dmmpt 6143 | The domain of the mapping ... |
dmmptss 6144 | The domain of a mapping is... |
dmmptg 6145 | The domain of the mapping ... |
rnmpt0f 6146 | The range of a function in... |
rnmptn0 6147 | The range of a function in... |
relco 6148 | A composition is a relatio... |
dfco2 6149 | Alternate definition of a ... |
dfco2a 6150 | Generalization of ~ dfco2 ... |
coundi 6151 | Class composition distribu... |
coundir 6152 | Class composition distribu... |
cores 6153 | Restricted first member of... |
resco 6154 | Associative law for the re... |
imaco 6155 | Image of the composition o... |
rnco 6156 | The range of the compositi... |
rnco2 6157 | The range of the compositi... |
dmco 6158 | The domain of a compositio... |
coeq0 6159 | A composition of two relat... |
coiun 6160 | Composition with an indexe... |
cocnvcnv1 6161 | A composition is not affec... |
cocnvcnv2 6162 | A composition is not affec... |
cores2 6163 | Absorption of a reverse (p... |
co02 6164 | Composition with the empty... |
co01 6165 | Composition with the empty... |
coi1 6166 | Composition with the ident... |
coi2 6167 | Composition with the ident... |
coires1 6168 | Composition with a restric... |
coass 6169 | Associative law for class ... |
relcnvtrg 6170 | General form of ~ relcnvtr... |
relcnvtr 6171 | A relation is transitive i... |
relssdmrn 6172 | A relation is included in ... |
resssxp 6173 | If the ` R ` -image of a c... |
cnvssrndm 6174 | The converse is a subset o... |
cossxp 6175 | Composition as a subset of... |
relrelss 6176 | Two ways to describe the s... |
unielrel 6177 | The membership relation fo... |
relfld 6178 | The double union of a rela... |
relresfld 6179 | Restriction of a relation ... |
relcoi2 6180 | Composition with the ident... |
relcoi1 6181 | Composition with the ident... |
unidmrn 6182 | The double union of the co... |
relcnvfld 6183 | if ` R ` is a relation, it... |
dfdm2 6184 | Alternate definition of do... |
unixp 6185 | The double class union of ... |
unixp0 6186 | A Cartesian product is emp... |
unixpid 6187 | Field of a Cartesian squar... |
ressn 6188 | Restriction of a class to ... |
cnviin 6189 | The converse of an interse... |
cnvpo 6190 | The converse of a partial ... |
cnvso 6191 | The converse of a strict o... |
xpco 6192 | Composition of two Cartesi... |
xpcoid 6193 | Composition of two Cartesi... |
elsnxp 6194 | Membership in a Cartesian ... |
reu3op 6195 | There is a unique ordered ... |
reuop 6196 | There is a unique ordered ... |
opreu2reurex 6197 | There is a unique ordered ... |
opreu2reu 6198 | If there is a unique order... |
dfpo2 6199 | Quantifier-free definition... |
csbcog 6200 | Distribute proper substitu... |
predeq123 6203 | Equality theorem for the p... |
predeq1 6204 | Equality theorem for the p... |
predeq2 6205 | Equality theorem for the p... |
predeq3 6206 | Equality theorem for the p... |
nfpred 6207 | Bound-variable hypothesis ... |
csbpredg 6208 | Move class substitution in... |
predpredss 6209 | If ` A ` is a subset of ` ... |
predss 6210 | The predecessor class of `... |
sspred 6211 | Another subset/predecessor... |
dfpred2 6212 | An alternate definition of... |
dfpred3 6213 | An alternate definition of... |
dfpred3g 6214 | An alternate definition of... |
elpredgg 6215 | Membership in a predecesso... |
elpredg 6216 | Membership in a predecesso... |
elpredimg 6217 | Membership in a predecesso... |
elpredim 6218 | Membership in a predecesso... |
elpred 6219 | Membership in a predecesso... |
predexg 6220 | The predecessor class exis... |
predasetexOLD 6221 | Obsolete form of ~ predexg... |
dffr4 6222 | Alternate definition of we... |
predel 6223 | Membership in the predeces... |
predbrg 6224 | Closed form of ~ elpredim ... |
predtrss 6225 | If ` R ` is transitive ove... |
predpo 6226 | Property of the predecesso... |
predso 6227 | Property of the predecesso... |
setlikespec 6228 | If ` R ` is set-like in ` ... |
predidm 6229 | Idempotent law for the pre... |
predin 6230 | Intersection law for prede... |
predun 6231 | Union law for predecessor ... |
preddif 6232 | Difference law for predece... |
predep 6233 | The predecessor under the ... |
trpred 6234 | The class of predecessors ... |
preddowncl 6235 | A property of classes that... |
predpoirr 6236 | Given a partial ordering, ... |
predfrirr 6237 | Given a well-founded relat... |
pred0 6238 | The predecessor class over... |
dfse3 6239 | Alternate definition of se... |
predrelss 6240 | Subset carries from relati... |
predprc 6241 | The predecessor of a prope... |
predres 6242 | Predecessor class is unaff... |
frpomin 6243 | Every nonempty (possibly p... |
frpomin2 6244 | Every nonempty (possibly p... |
frpoind 6245 | The principle of well-foun... |
frpoinsg 6246 | Well-Founded Induction Sch... |
frpoins2fg 6247 | Well-Founded Induction sch... |
frpoins2g 6248 | Well-Founded Induction sch... |
frpoins3g 6249 | Well-Founded Induction sch... |
tz6.26 6250 | All nonempty subclasses of... |
tz6.26OLD 6251 | Obsolete proof of ~ tz6.26... |
tz6.26i 6252 | All nonempty subclasses of... |
wfi 6253 | The Principle of Well-Orde... |
wfiOLD 6254 | Obsolete proof of ~ wfi as... |
wfii 6255 | The Principle of Well-Orde... |
wfisg 6256 | Well-Ordered Induction Sch... |
wfisgOLD 6257 | Obsolete proof of ~ wfisg ... |
wfis 6258 | Well-Ordered Induction Sch... |
wfis2fg 6259 | Well-Ordered Induction Sch... |
wfis2fgOLD 6260 | Obsolete proof of ~ wfis2f... |
wfis2f 6261 | Well-Ordered Induction sch... |
wfis2g 6262 | Well-Ordered Induction Sch... |
wfis2 6263 | Well-Ordered Induction sch... |
wfis3 6264 | Well-Ordered Induction sch... |
ordeq 6273 | Equality theorem for the o... |
elong 6274 | An ordinal number is an or... |
elon 6275 | An ordinal number is an or... |
eloni 6276 | An ordinal number has the ... |
elon2 6277 | An ordinal number is an or... |
limeq 6278 | Equality theorem for the l... |
ordwe 6279 | Membership well-orders eve... |
ordtr 6280 | An ordinal class is transi... |
ordfr 6281 | Membership is well-founded... |
ordelss 6282 | An element of an ordinal c... |
trssord 6283 | A transitive subclass of a... |
ordirr 6284 | No ordinal class is a memb... |
nordeq 6285 | A member of an ordinal cla... |
ordn2lp 6286 | An ordinal class cannot be... |
tz7.5 6287 | A nonempty subclass of an ... |
ordelord 6288 | An element of an ordinal c... |
tron 6289 | The class of all ordinal n... |
ordelon 6290 | An element of an ordinal c... |
onelon 6291 | An element of an ordinal n... |
tz7.7 6292 | A transitive class belongs... |
ordelssne 6293 | For ordinal classes, membe... |
ordelpss 6294 | For ordinal classes, membe... |
ordsseleq 6295 | For ordinal classes, inclu... |
ordin 6296 | The intersection of two or... |
onin 6297 | The intersection of two or... |
ordtri3or 6298 | A trichotomy law for ordin... |
ordtri1 6299 | A trichotomy law for ordin... |
ontri1 6300 | A trichotomy law for ordin... |
ordtri2 6301 | A trichotomy law for ordin... |
ordtri3 6302 | A trichotomy law for ordin... |
ordtri4 6303 | A trichotomy law for ordin... |
orddisj 6304 | An ordinal class and its s... |
onfr 6305 | The ordinal class is well-... |
onelpss 6306 | Relationship between membe... |
onsseleq 6307 | Relationship between subse... |
onelss 6308 | An element of an ordinal n... |
ordtr1 6309 | Transitive law for ordinal... |
ordtr2 6310 | Transitive law for ordinal... |
ordtr3 6311 | Transitive law for ordinal... |
ontr1 6312 | Transitive law for ordinal... |
ontr2 6313 | Transitive law for ordinal... |
ordunidif 6314 | The union of an ordinal st... |
ordintdif 6315 | If ` B ` is smaller than `... |
onintss 6316 | If a property is true for ... |
oneqmini 6317 | A way to show that an ordi... |
ord0 6318 | The empty set is an ordina... |
0elon 6319 | The empty set is an ordina... |
ord0eln0 6320 | A nonempty ordinal contain... |
on0eln0 6321 | An ordinal number contains... |
dflim2 6322 | An alternate definition of... |
inton 6323 | The intersection of the cl... |
nlim0 6324 | The empty set is not a lim... |
limord 6325 | A limit ordinal is ordinal... |
limuni 6326 | A limit ordinal is its own... |
limuni2 6327 | The union of a limit ordin... |
0ellim 6328 | A limit ordinal contains t... |
limelon 6329 | A limit ordinal class that... |
onn0 6330 | The class of all ordinal n... |
suceq 6331 | Equality of successors. (... |
elsuci 6332 | Membership in a successor.... |
elsucg 6333 | Membership in a successor.... |
elsuc2g 6334 | Variant of membership in a... |
elsuc 6335 | Membership in a successor.... |
elsuc2 6336 | Membership in a successor.... |
nfsuc 6337 | Bound-variable hypothesis ... |
elelsuc 6338 | Membership in a successor.... |
sucel 6339 | Membership of a successor ... |
suc0 6340 | The successor of the empty... |
sucprc 6341 | A proper class is its own ... |
unisuc 6342 | A transitive class is equa... |
sssucid 6343 | A class is included in its... |
sucidg 6344 | Part of Proposition 7.23 o... |
sucid 6345 | A set belongs to its succe... |
nsuceq0 6346 | No successor is empty. (C... |
eqelsuc 6347 | A set belongs to the succe... |
iunsuc 6348 | Inductive definition for t... |
suctr 6349 | The successor of a transit... |
trsuc 6350 | A set whose successor belo... |
trsucss 6351 | A member of the successor ... |
ordsssuc 6352 | An ordinal is a subset of ... |
onsssuc 6353 | A subset of an ordinal num... |
ordsssuc2 6354 | An ordinal subset of an or... |
onmindif 6355 | When its successor is subt... |
ordnbtwn 6356 | There is no set between an... |
onnbtwn 6357 | There is no set between an... |
sucssel 6358 | A set whose successor is a... |
orddif 6359 | Ordinal derived from its s... |
orduniss 6360 | An ordinal class includes ... |
ordtri2or 6361 | A trichotomy law for ordin... |
ordtri2or2 6362 | A trichotomy law for ordin... |
ordtri2or3 6363 | A consequence of total ord... |
ordelinel 6364 | The intersection of two or... |
ordssun 6365 | Property of a subclass of ... |
ordequn 6366 | The maximum (i.e. union) o... |
ordun 6367 | The maximum (i.e. union) o... |
ordunisssuc 6368 | A subclass relationship fo... |
suc11 6369 | The successor operation be... |
onun2 6370 | The union of two ordinals ... |
onordi 6371 | An ordinal number is an or... |
ontrci 6372 | An ordinal number is a tra... |
onirri 6373 | An ordinal number is not a... |
oneli 6374 | A member of an ordinal num... |
onelssi 6375 | A member of an ordinal num... |
onssneli 6376 | An ordering law for ordina... |
onssnel2i 6377 | An ordering law for ordina... |
onelini 6378 | An element of an ordinal n... |
oneluni 6379 | An ordinal number equals i... |
onunisuci 6380 | An ordinal number is equal... |
onsseli 6381 | Subset is equivalent to me... |
onun2i 6382 | The union of two ordinal n... |
unizlim 6383 | An ordinal equal to its ow... |
on0eqel 6384 | An ordinal number either e... |
snsn0non 6385 | The singleton of the singl... |
onxpdisj 6386 | Ordinal numbers and ordere... |
onnev 6387 | The class of ordinal numbe... |
onnevOLD 6388 | Obsolete version of ~ onne... |
iotajust 6390 | Soundness justification th... |
dfiota2 6392 | Alternate definition for d... |
nfiota1 6393 | Bound-variable hypothesis ... |
nfiotadw 6394 | Deduction version of ~ nfi... |
nfiotaw 6395 | Bound-variable hypothesis ... |
nfiotad 6396 | Deduction version of ~ nfi... |
nfiota 6397 | Bound-variable hypothesis ... |
cbviotaw 6398 | Change bound variables in ... |
cbviotavw 6399 | Change bound variables in ... |
cbviotavwOLD 6400 | Obsolete version of ~ cbvi... |
cbviota 6401 | Change bound variables in ... |
cbviotav 6402 | Change bound variables in ... |
sb8iota 6403 | Variable substitution in d... |
iotaeq 6404 | Equality theorem for descr... |
iotabi 6405 | Equivalence theorem for de... |
uniabio 6406 | Part of Theorem 8.17 in [Q... |
iotaval 6407 | Theorem 8.19 in [Quine] p.... |
iotauni 6408 | Equivalence between two di... |
iotaint 6409 | Equivalence between two di... |
iota1 6410 | Property of iota. (Contri... |
iotanul 6411 | Theorem 8.22 in [Quine] p.... |
iotassuni 6412 | The ` iota ` class is a su... |
iotaex 6413 | Theorem 8.23 in [Quine] p.... |
iota4 6414 | Theorem *14.22 in [Whitehe... |
iota4an 6415 | Theorem *14.23 in [Whitehe... |
iota5 6416 | A method for computing iot... |
iotabidv 6417 | Formula-building deduction... |
iotabii 6418 | Formula-building deduction... |
iotacl 6419 | Membership law for descrip... |
iota2df 6420 | A condition that allows us... |
iota2d 6421 | A condition that allows us... |
iota2 6422 | The unique element such th... |
iotan0 6423 | Representation of "the uni... |
sniota 6424 | A class abstraction with a... |
dfiota4 6425 | The ` iota ` operation usi... |
csbiota 6426 | Class substitution within ... |
dffun2 6443 | Alternate definition of a ... |
dffun2OLD 6444 | Obsolete version of ~ dffu... |
dffun3 6445 | Alternate definition of fu... |
dffun4 6446 | Alternate definition of a ... |
dffun5 6447 | Alternate definition of fu... |
dffun6f 6448 | Definition of function, us... |
dffun6 6449 | Alternate definition of a ... |
funmo 6450 | A function has at most one... |
funrel 6451 | A function is a relation. ... |
0nelfun 6452 | A function does not contai... |
funss 6453 | Subclass theorem for funct... |
funeq 6454 | Equality theorem for funct... |
funeqi 6455 | Equality inference for the... |
funeqd 6456 | Equality deduction for the... |
nffun 6457 | Bound-variable hypothesis ... |
sbcfung 6458 | Distribute proper substitu... |
funeu 6459 | There is exactly one value... |
funeu2 6460 | There is exactly one value... |
dffun7 6461 | Alternate definition of a ... |
dffun8 6462 | Alternate definition of a ... |
dffun9 6463 | Alternate definition of a ... |
funfn 6464 | A class is a function if a... |
funfnd 6465 | A function is a function o... |
funi 6466 | The identity relation is a... |
nfunv 6467 | The universal class is not... |
funopg 6468 | A Kuratowski ordered pair ... |
funopab 6469 | A class of ordered pairs i... |
funopabeq 6470 | A class of ordered pairs o... |
funopab4 6471 | A class of ordered pairs o... |
funmpt 6472 | A function in maps-to nota... |
funmpt2 6473 | Functionality of a class g... |
funco 6474 | The composition of two fun... |
funresfunco 6475 | Composition of two functio... |
funres 6476 | A restriction of a functio... |
funresd 6477 | A restriction of a functio... |
funssres 6478 | The restriction of a funct... |
fun2ssres 6479 | Equality of restrictions o... |
funun 6480 | The union of functions wit... |
fununmo 6481 | If the union of classes is... |
fununfun 6482 | If the union of classes is... |
fundif 6483 | A function with removed el... |
funcnvsn 6484 | The converse singleton of ... |
funsng 6485 | A singleton of an ordered ... |
fnsng 6486 | Functionality and domain o... |
funsn 6487 | A singleton of an ordered ... |
funprg 6488 | A set of two pairs is a fu... |
funtpg 6489 | A set of three pairs is a ... |
funpr 6490 | A function with a domain o... |
funtp 6491 | A function with a domain o... |
fnsn 6492 | Functionality and domain o... |
fnprg 6493 | Function with a domain of ... |
fntpg 6494 | Function with a domain of ... |
fntp 6495 | A function with a domain o... |
funcnvpr 6496 | The converse pair of order... |
funcnvtp 6497 | The converse triple of ord... |
funcnvqp 6498 | The converse quadruple of ... |
fun0 6499 | The empty set is a functio... |
funcnv0 6500 | The converse of the empty ... |
funcnvcnv 6501 | The double converse of a f... |
funcnv2 6502 | A simpler equivalence for ... |
funcnv 6503 | The converse of a class is... |
funcnv3 6504 | A condition showing a clas... |
fun2cnv 6505 | The double converse of a c... |
svrelfun 6506 | A single-valued relation i... |
fncnv 6507 | Single-rootedness (see ~ f... |
fun11 6508 | Two ways of stating that `... |
fununi 6509 | The union of a chain (with... |
funin 6510 | The intersection with a fu... |
funres11 6511 | The restriction of a one-t... |
funcnvres 6512 | The converse of a restrict... |
cnvresid 6513 | Converse of a restricted i... |
funcnvres2 6514 | The converse of a restrict... |
funimacnv 6515 | The image of the preimage ... |
funimass1 6516 | A kind of contraposition l... |
funimass2 6517 | A kind of contraposition l... |
imadif 6518 | The image of a difference ... |
imain 6519 | The image of an intersecti... |
funimaexg 6520 | Axiom of Replacement using... |
funimaex 6521 | The image of a set under a... |
isarep1 6522 | Part of a study of the Axi... |
isarep2 6523 | Part of a study of the Axi... |
fneq1 6524 | Equality theorem for funct... |
fneq2 6525 | Equality theorem for funct... |
fneq1d 6526 | Equality deduction for fun... |
fneq2d 6527 | Equality deduction for fun... |
fneq12d 6528 | Equality deduction for fun... |
fneq12 6529 | Equality theorem for funct... |
fneq1i 6530 | Equality inference for fun... |
fneq2i 6531 | Equality inference for fun... |
nffn 6532 | Bound-variable hypothesis ... |
fnfun 6533 | A function with domain is ... |
fnfund 6534 | A function with domain is ... |
fnrel 6535 | A function with domain is ... |
fndm 6536 | The domain of a function. ... |
fndmi 6537 | The domain of a function. ... |
fndmd 6538 | The domain of a function. ... |
funfni 6539 | Inference to convert a fun... |
fndmu 6540 | A function has a unique do... |
fnbr 6541 | The first argument of bina... |
fnop 6542 | The first argument of an o... |
fneu 6543 | There is exactly one value... |
fneu2 6544 | There is exactly one value... |
fnun 6545 | The union of two functions... |
fnund 6546 | The union of two functions... |
fnunop 6547 | Extension of a function wi... |
fncofn 6548 | Composition of a function ... |
fnco 6549 | Composition of two functio... |
fncoOLD 6550 | Obsolete version of ~ fnco... |
fnresdm 6551 | A function does not change... |
fnresdisj 6552 | A function restricted to a... |
2elresin 6553 | Membership in two function... |
fnssresb 6554 | Restriction of a function ... |
fnssres 6555 | Restriction of a function ... |
fnssresd 6556 | Restriction of a function ... |
fnresin1 6557 | Restriction of a function'... |
fnresin2 6558 | Restriction of a function'... |
fnres 6559 | An equivalence for functio... |
idfn 6560 | The identity relation is a... |
fnresi 6561 | The restricted identity re... |
fnresiOLD 6562 | Obsolete proof of ~ fnresi... |
fnima 6563 | The image of a function's ... |
fn0 6564 | A function with empty doma... |
fnimadisj 6565 | A class that is disjoint w... |
fnimaeq0 6566 | Images under a function ne... |
dfmpt3 6567 | Alternate definition for t... |
mptfnf 6568 | The maps-to notation defin... |
fnmptf 6569 | The maps-to notation defin... |
fnopabg 6570 | Functionality and domain o... |
fnopab 6571 | Functionality and domain o... |
mptfng 6572 | The maps-to notation defin... |
fnmpt 6573 | The maps-to notation defin... |
fnmptd 6574 | The maps-to notation defin... |
mpt0 6575 | A mapping operation with e... |
fnmpti 6576 | Functionality and domain o... |
dmmpti 6577 | Domain of the mapping oper... |
dmmptd 6578 | The domain of the mapping ... |
mptun 6579 | Union of mappings which ar... |
partfun 6580 | Rewrite a function defined... |
feq1 6581 | Equality theorem for funct... |
feq2 6582 | Equality theorem for funct... |
feq3 6583 | Equality theorem for funct... |
feq23 6584 | Equality theorem for funct... |
feq1d 6585 | Equality deduction for fun... |
feq2d 6586 | Equality deduction for fun... |
feq3d 6587 | Equality deduction for fun... |
feq12d 6588 | Equality deduction for fun... |
feq123d 6589 | Equality deduction for fun... |
feq123 6590 | Equality theorem for funct... |
feq1i 6591 | Equality inference for fun... |
feq2i 6592 | Equality inference for fun... |
feq12i 6593 | Equality inference for fun... |
feq23i 6594 | Equality inference for fun... |
feq23d 6595 | Equality deduction for fun... |
nff 6596 | Bound-variable hypothesis ... |
sbcfng 6597 | Distribute proper substitu... |
sbcfg 6598 | Distribute proper substitu... |
elimf 6599 | Eliminate a mapping hypoth... |
ffn 6600 | A mapping is a function wi... |
ffnd 6601 | A mapping is a function wi... |
dffn2 6602 | Any function is a mapping ... |
ffun 6603 | A mapping is a function. ... |
ffund 6604 | A mapping is a function, d... |
frel 6605 | A mapping is a relation. ... |
freld 6606 | A mapping is a relation. ... |
frn 6607 | The range of a mapping. (... |
frnd 6608 | Deduction form of ~ frn . ... |
fdm 6609 | The domain of a mapping. ... |
fdmOLD 6610 | Obsolete version of ~ fdm ... |
fdmd 6611 | Deduction form of ~ fdm . ... |
fdmi 6612 | Inference associated with ... |
dffn3 6613 | A function maps to its ran... |
ffrn 6614 | A function maps to its ran... |
ffrnb 6615 | Characterization of a func... |
ffrnbd 6616 | A function maps to its ran... |
fss 6617 | Expanding the codomain of ... |
fssd 6618 | Expanding the codomain of ... |
fssdmd 6619 | Expressing that a class is... |
fssdm 6620 | Expressing that a class is... |
fimass 6621 | The image of a class under... |
fimacnv 6622 | The preimage of the codoma... |
fcof 6623 | Composition of a function ... |
fco 6624 | Composition of two functio... |
fcoOLD 6625 | Obsolete version of ~ fco ... |
fcod 6626 | Composition of two mapping... |
fco2 6627 | Functionality of a composi... |
fssxp 6628 | A mapping is a class of or... |
funssxp 6629 | Two ways of specifying a p... |
ffdm 6630 | A mapping is a partial fun... |
ffdmd 6631 | The domain of a function. ... |
fdmrn 6632 | A different way to write `... |
funcofd 6633 | Composition of two functio... |
fco3OLD 6634 | Obsolete version of ~ func... |
opelf 6635 | The members of an ordered ... |
fun 6636 | The union of two functions... |
fun2 6637 | The union of two functions... |
fun2d 6638 | The union of functions wit... |
fnfco 6639 | Composition of two functio... |
fssres 6640 | Restriction of a function ... |
fssresd 6641 | Restriction of a function ... |
fssres2 6642 | Restriction of a restricte... |
fresin 6643 | An identity for the mappin... |
resasplit 6644 | If two functions agree on ... |
fresaun 6645 | The union of two functions... |
fresaunres2 6646 | From the union of two func... |
fresaunres1 6647 | From the union of two func... |
fcoi1 6648 | Composition of a mapping a... |
fcoi2 6649 | Composition of restricted ... |
feu 6650 | There is exactly one value... |
fcnvres 6651 | The converse of a restrict... |
fimacnvdisj 6652 | The preimage of a class di... |
fint 6653 | Function into an intersect... |
fin 6654 | Mapping into an intersecti... |
f0 6655 | The empty function. (Cont... |
f00 6656 | A class is a function with... |
f0bi 6657 | A function with empty doma... |
f0dom0 6658 | A function is empty iff it... |
f0rn0 6659 | If there is no element in ... |
fconst 6660 | A Cartesian product with a... |
fconstg 6661 | A Cartesian product with a... |
fnconstg 6662 | A Cartesian product with a... |
fconst6g 6663 | Constant function with loo... |
fconst6 6664 | A constant function as a m... |
f1eq1 6665 | Equality theorem for one-t... |
f1eq2 6666 | Equality theorem for one-t... |
f1eq3 6667 | Equality theorem for one-t... |
nff1 6668 | Bound-variable hypothesis ... |
dff12 6669 | Alternate definition of a ... |
f1f 6670 | A one-to-one mapping is a ... |
f1fn 6671 | A one-to-one mapping is a ... |
f1fun 6672 | A one-to-one mapping is a ... |
f1rel 6673 | A one-to-one onto mapping ... |
f1dm 6674 | The domain of a one-to-one... |
f1dmOLD 6675 | Obsolete version of ~ f1dm... |
f1ss 6676 | A function that is one-to-... |
f1ssr 6677 | A function that is one-to-... |
f1ssres 6678 | A function that is one-to-... |
f1resf1 6679 | The restriction of an inje... |
f1cnvcnv 6680 | Two ways to express that a... |
f1cof1 6681 | Composition of two one-to-... |
f1co 6682 | Composition of one-to-one ... |
f1coOLD 6683 | Obsolete version of ~ f1co... |
foeq1 6684 | Equality theorem for onto ... |
foeq2 6685 | Equality theorem for onto ... |
foeq3 6686 | Equality theorem for onto ... |
nffo 6687 | Bound-variable hypothesis ... |
fof 6688 | An onto mapping is a mappi... |
fofun 6689 | An onto mapping is a funct... |
fofn 6690 | An onto mapping is a funct... |
forn 6691 | The codomain of an onto fu... |
dffo2 6692 | Alternate definition of an... |
foima 6693 | The image of the domain of... |
dffn4 6694 | A function maps onto its r... |
funforn 6695 | A function maps its domain... |
fodmrnu 6696 | An onto function has uniqu... |
fimadmfo 6697 | A function is a function o... |
fores 6698 | Restriction of an onto fun... |
fimadmfoALT 6699 | Alternate proof of ~ fimad... |
focnvimacdmdm 6700 | The preimage of the codoma... |
focofo 6701 | Composition of onto functi... |
foco 6702 | Composition of onto functi... |
foconst 6703 | A nonzero constant functio... |
f1oeq1 6704 | Equality theorem for one-t... |
f1oeq2 6705 | Equality theorem for one-t... |
f1oeq3 6706 | Equality theorem for one-t... |
f1oeq23 6707 | Equality theorem for one-t... |
f1eq123d 6708 | Equality deduction for one... |
foeq123d 6709 | Equality deduction for ont... |
f1oeq123d 6710 | Equality deduction for one... |
f1oeq1d 6711 | Equality deduction for one... |
f1oeq2d 6712 | Equality deduction for one... |
f1oeq3d 6713 | Equality deduction for one... |
nff1o 6714 | Bound-variable hypothesis ... |
f1of1 6715 | A one-to-one onto mapping ... |
f1of 6716 | A one-to-one onto mapping ... |
f1ofn 6717 | A one-to-one onto mapping ... |
f1ofun 6718 | A one-to-one onto mapping ... |
f1orel 6719 | A one-to-one onto mapping ... |
f1odm 6720 | The domain of a one-to-one... |
dff1o2 6721 | Alternate definition of on... |
dff1o3 6722 | Alternate definition of on... |
f1ofo 6723 | A one-to-one onto function... |
dff1o4 6724 | Alternate definition of on... |
dff1o5 6725 | Alternate definition of on... |
f1orn 6726 | A one-to-one function maps... |
f1f1orn 6727 | A one-to-one function maps... |
f1ocnv 6728 | The converse of a one-to-o... |
f1ocnvb 6729 | A relation is a one-to-one... |
f1ores 6730 | The restriction of a one-t... |
f1orescnv 6731 | The converse of a one-to-o... |
f1imacnv 6732 | Preimage of an image. (Co... |
foimacnv 6733 | A reverse version of ~ f1i... |
foun 6734 | The union of two onto func... |
f1oun 6735 | The union of two one-to-on... |
f1un 6736 | The union of two one-to-on... |
resdif 6737 | The restriction of a one-t... |
resin 6738 | The restriction of a one-t... |
f1oco 6739 | Composition of one-to-one ... |
f1cnv 6740 | The converse of an injecti... |
funcocnv2 6741 | Composition with the conve... |
fococnv2 6742 | The composition of an onto... |
f1ococnv2 6743 | The composition of a one-t... |
f1cocnv2 6744 | Composition of an injectiv... |
f1ococnv1 6745 | The composition of a one-t... |
f1cocnv1 6746 | Composition of an injectiv... |
funcoeqres 6747 | Express a constraint on a ... |
f1ssf1 6748 | A subset of an injective f... |
f10 6749 | The empty set maps one-to-... |
f10d 6750 | The empty set maps one-to-... |
f1o00 6751 | One-to-one onto mapping of... |
fo00 6752 | Onto mapping of the empty ... |
f1o0 6753 | One-to-one onto mapping of... |
f1oi 6754 | A restriction of the ident... |
f1ovi 6755 | The identity relation is a... |
f1osn 6756 | A singleton of an ordered ... |
f1osng 6757 | A singleton of an ordered ... |
f1sng 6758 | A singleton of an ordered ... |
fsnd 6759 | A singleton of an ordered ... |
f1oprswap 6760 | A two-element swap is a bi... |
f1oprg 6761 | An unordered pair of order... |
tz6.12-2 6762 | Function value when ` F ` ... |
fveu 6763 | The value of a function at... |
brprcneu 6764 | If ` A ` is a proper class... |
brprcneuALT 6765 | Alternate proof of ~ brprc... |
fvprc 6766 | A function's value at a pr... |
fvprcALT 6767 | Alternate proof of ~ fvprc... |
rnfvprc 6768 | The range of a function va... |
fv2 6769 | Alternate definition of fu... |
dffv3 6770 | A definition of function v... |
dffv4 6771 | The previous definition of... |
elfv 6772 | Membership in a function v... |
fveq1 6773 | Equality theorem for funct... |
fveq2 6774 | Equality theorem for funct... |
fveq1i 6775 | Equality inference for fun... |
fveq1d 6776 | Equality deduction for fun... |
fveq2i 6777 | Equality inference for fun... |
fveq2d 6778 | Equality deduction for fun... |
2fveq3 6779 | Equality theorem for neste... |
fveq12i 6780 | Equality deduction for fun... |
fveq12d 6781 | Equality deduction for fun... |
fveqeq2d 6782 | Equality deduction for fun... |
fveqeq2 6783 | Equality deduction for fun... |
nffv 6784 | Bound-variable hypothesis ... |
nffvmpt1 6785 | Bound-variable hypothesis ... |
nffvd 6786 | Deduction version of bound... |
fvex 6787 | The value of a class exist... |
fvexi 6788 | The value of a class exist... |
fvexd 6789 | The value of a class exist... |
fvif 6790 | Move a conditional outside... |
iffv 6791 | Move a conditional outside... |
fv3 6792 | Alternate definition of th... |
fvres 6793 | The value of a restricted ... |
fvresd 6794 | The value of a restricted ... |
funssfv 6795 | The value of a member of t... |
tz6.12-1 6796 | Function value. Theorem 6... |
tz6.12 6797 | Function value. Theorem 6... |
tz6.12f 6798 | Function value, using boun... |
tz6.12c 6799 | Corollary of Theorem 6.12(... |
tz6.12i 6800 | Corollary of Theorem 6.12(... |
fvbr0 6801 | Two possibilities for the ... |
fvrn0 6802 | A function value is a memb... |
fvssunirn 6803 | The result of a function v... |
ndmfv 6804 | The value of a class outsi... |
ndmfvrcl 6805 | Reverse closure law for fu... |
elfvdm 6806 | If a function value has a ... |
elfvex 6807 | If a function value has a ... |
elfvexd 6808 | If a function value has a ... |
eliman0 6809 | A nonempty function value ... |
nfvres 6810 | The value of a non-member ... |
nfunsn 6811 | If the restriction of a cl... |
fvfundmfvn0 6812 | If the "value of a class" ... |
0fv 6813 | Function value of the empt... |
fv2prc 6814 | A function value of a func... |
elfv2ex 6815 | If a function value of a f... |
fveqres 6816 | Equal values imply equal v... |
csbfv12 6817 | Move class substitution in... |
csbfv2g 6818 | Move class substitution in... |
csbfv 6819 | Substitution for a functio... |
funbrfv 6820 | The second argument of a b... |
funopfv 6821 | The second element in an o... |
fnbrfvb 6822 | Equivalence of function va... |
fnopfvb 6823 | Equivalence of function va... |
funbrfvb 6824 | Equivalence of function va... |
funopfvb 6825 | Equivalence of function va... |
fnbrfvb2 6826 | Version of ~ fnbrfvb for f... |
funbrfv2b 6827 | Function value in terms of... |
dffn5 6828 | Representation of a functi... |
fnrnfv 6829 | The range of a function ex... |
fvelrnb 6830 | A member of a function's r... |
foelrni 6831 | A member of a surjective f... |
dfimafn 6832 | Alternate definition of th... |
dfimafn2 6833 | Alternate definition of th... |
funimass4 6834 | Membership relation for th... |
fvelima 6835 | Function value in an image... |
fvelimad 6836 | Function value in an image... |
feqmptd 6837 | Deduction form of ~ dffn5 ... |
feqresmpt 6838 | Express a restricted funct... |
feqmptdf 6839 | Deduction form of ~ dffn5f... |
dffn5f 6840 | Representation of a functi... |
fvelimab 6841 | Function value in an image... |
fvelimabd 6842 | Deduction form of ~ fvelim... |
unima 6843 | Image of a union. (Contri... |
fvi 6844 | The value of the identity ... |
fviss 6845 | The value of the identity ... |
fniinfv 6846 | The indexed intersection o... |
fnsnfv 6847 | Singleton of function valu... |
fnsnfvOLD 6848 | Obsolete version of ~ fnsn... |
opabiotafun 6849 | Define a function whose va... |
opabiotadm 6850 | Define a function whose va... |
opabiota 6851 | Define a function whose va... |
fnimapr 6852 | The image of a pair under ... |
ssimaex 6853 | The existence of a subimag... |
ssimaexg 6854 | The existence of a subimag... |
funfv 6855 | A simplified expression fo... |
funfv2 6856 | The value of a function. ... |
funfv2f 6857 | The value of a function. ... |
fvun 6858 | Value of the union of two ... |
fvun1 6859 | The value of a union when ... |
fvun2 6860 | The value of a union when ... |
fvun1d 6861 | The value of a union when ... |
fvun2d 6862 | The value of a union when ... |
dffv2 6863 | Alternate definition of fu... |
dmfco 6864 | Domains of a function comp... |
fvco2 6865 | Value of a function compos... |
fvco 6866 | Value of a function compos... |
fvco3 6867 | Value of a function compos... |
fvco3d 6868 | Value of a function compos... |
fvco4i 6869 | Conditions for a compositi... |
fvopab3g 6870 | Value of a function given ... |
fvopab3ig 6871 | Value of a function given ... |
brfvopabrbr 6872 | The binary relation of a f... |
fvmptg 6873 | Value of a function given ... |
fvmpti 6874 | Value of a function given ... |
fvmpt 6875 | Value of a function given ... |
fvmpt2f 6876 | Value of a function given ... |
fvtresfn 6877 | Functionality of a tuple-r... |
fvmpts 6878 | Value of a function given ... |
fvmpt3 6879 | Value of a function given ... |
fvmpt3i 6880 | Value of a function given ... |
fvmptdf 6881 | Deduction version of ~ fvm... |
fvmptd 6882 | Deduction version of ~ fvm... |
fvmptd2 6883 | Deduction version of ~ fvm... |
mptrcl 6884 | Reverse closure for a mapp... |
fvmpt2i 6885 | Value of a function given ... |
fvmpt2 6886 | Value of a function given ... |
fvmptss 6887 | If all the values of the m... |
fvmpt2d 6888 | Deduction version of ~ fvm... |
fvmptex 6889 | Express a function ` F ` w... |
fvmptd3f 6890 | Alternate deduction versio... |
fvmptd2f 6891 | Alternate deduction versio... |
fvmptdv 6892 | Alternate deduction versio... |
fvmptdv2 6893 | Alternate deduction versio... |
mpteqb 6894 | Bidirectional equality the... |
fvmptt 6895 | Closed theorem form of ~ f... |
fvmptf 6896 | Value of a function given ... |
fvmptnf 6897 | The value of a function gi... |
fvmptd3 6898 | Deduction version of ~ fvm... |
fvmptn 6899 | This somewhat non-intuitiv... |
fvmptss2 6900 | A mapping always evaluates... |
elfvmptrab1w 6901 | Implications for the value... |
elfvmptrab1 6902 | Implications for the value... |
elfvmptrab 6903 | Implications for the value... |
fvopab4ndm 6904 | Value of a function given ... |
fvmptndm 6905 | Value of a function given ... |
fvmptrabfv 6906 | Value of a function mappin... |
fvopab5 6907 | The value of a function th... |
fvopab6 6908 | Value of a function given ... |
eqfnfv 6909 | Equality of functions is d... |
eqfnfv2 6910 | Equality of functions is d... |
eqfnfv3 6911 | Derive equality of functio... |
eqfnfvd 6912 | Deduction for equality of ... |
eqfnfv2f 6913 | Equality of functions is d... |
eqfunfv 6914 | Equality of functions is d... |
fvreseq0 6915 | Equality of restricted fun... |
fvreseq1 6916 | Equality of a function res... |
fvreseq 6917 | Equality of restricted fun... |
fnmptfvd 6918 | A function with a given do... |
fndmdif 6919 | Two ways to express the lo... |
fndmdifcom 6920 | The difference set between... |
fndmdifeq0 6921 | The difference set of two ... |
fndmin 6922 | Two ways to express the lo... |
fneqeql 6923 | Two functions are equal if... |
fneqeql2 6924 | Two functions are equal if... |
fnreseql 6925 | Two functions are equal on... |
chfnrn 6926 | The range of a choice func... |
funfvop 6927 | Ordered pair with function... |
funfvbrb 6928 | Two ways to say that ` A `... |
fvimacnvi 6929 | A member of a preimage is ... |
fvimacnv 6930 | The argument of a function... |
funimass3 6931 | A kind of contraposition l... |
funimass5 6932 | A subclass of a preimage i... |
funconstss 6933 | Two ways of specifying tha... |
fvimacnvALT 6934 | Alternate proof of ~ fvima... |
elpreima 6935 | Membership in the preimage... |
elpreimad 6936 | Membership in the preimage... |
fniniseg 6937 | Membership in the preimage... |
fncnvima2 6938 | Inverse images under funct... |
fniniseg2 6939 | Inverse point images under... |
unpreima 6940 | Preimage of a union. (Con... |
inpreima 6941 | Preimage of an intersectio... |
difpreima 6942 | Preimage of a difference. ... |
respreima 6943 | The preimage of a restrict... |
cnvimainrn 6944 | The preimage of the inters... |
sspreima 6945 | The preimage of a subset i... |
iinpreima 6946 | Preimage of an intersectio... |
intpreima 6947 | Preimage of an intersectio... |
fimacnvOLD 6948 | Obsolete version of ~ fima... |
fimacnvinrn 6949 | Taking the converse image ... |
fimacnvinrn2 6950 | Taking the converse image ... |
rescnvimafod 6951 | The restriction of a funct... |
fvn0ssdmfun 6952 | If a class' function value... |
fnopfv 6953 | Ordered pair with function... |
fvelrn 6954 | A function's value belongs... |
nelrnfvne 6955 | A function value cannot be... |
fveqdmss 6956 | If the empty set is not co... |
fveqressseq 6957 | If the empty set is not co... |
fnfvelrn 6958 | A function's value belongs... |
ffvelrn 6959 | A function's value belongs... |
ffvelrni 6960 | A function's value belongs... |
ffvelrnda 6961 | A function's value belongs... |
ffvelrnd 6962 | A function's value belongs... |
rexrn 6963 | Restricted existential qua... |
ralrn 6964 | Restricted universal quant... |
elrnrexdm 6965 | For any element in the ran... |
elrnrexdmb 6966 | For any element in the ran... |
eldmrexrn 6967 | For any element in the dom... |
eldmrexrnb 6968 | For any element in the dom... |
fvcofneq 6969 | The values of two function... |
ralrnmptw 6970 | A restricted quantifier ov... |
rexrnmptw 6971 | A restricted quantifier ov... |
ralrnmpt 6972 | A restricted quantifier ov... |
rexrnmpt 6973 | A restricted quantifier ov... |
f0cli 6974 | Unconditional closure of a... |
dff2 6975 | Alternate definition of a ... |
dff3 6976 | Alternate definition of a ... |
dff4 6977 | Alternate definition of a ... |
dffo3 6978 | An onto mapping expressed ... |
dffo4 6979 | Alternate definition of an... |
dffo5 6980 | Alternate definition of an... |
exfo 6981 | A relation equivalent to t... |
foelrn 6982 | Property of a surjective f... |
foco2 6983 | If a composition of two fu... |
fmpt 6984 | Functionality of the mappi... |
f1ompt 6985 | Express bijection for a ma... |
fmpti 6986 | Functionality of the mappi... |
fvmptelrn 6987 | The value of a function at... |
fmptd 6988 | Domain and codomain of the... |
fmpttd 6989 | Version of ~ fmptd with in... |
fmpt3d 6990 | Domain and codomain of the... |
fmptdf 6991 | A version of ~ fmptd using... |
ffnfv 6992 | A function maps to a class... |
ffnfvf 6993 | A function maps to a class... |
fnfvrnss 6994 | An upper bound for range d... |
frnssb 6995 | A function is a function i... |
rnmptss 6996 | The range of an operation ... |
fmpt2d 6997 | Domain and codomain of the... |
ffvresb 6998 | A necessary and sufficient... |
f1oresrab 6999 | Build a bijection between ... |
f1ossf1o 7000 | Restricting a bijection, w... |
fmptco 7001 | Composition of two functio... |
fmptcof 7002 | Version of ~ fmptco where ... |
fmptcos 7003 | Composition of two functio... |
cofmpt 7004 | Express composition of a m... |
fcompt 7005 | Express composition of two... |
fcoconst 7006 | Composition with a constan... |
fsn 7007 | A function maps a singleto... |
fsn2 7008 | A function that maps a sin... |
fsng 7009 | A function maps a singleto... |
fsn2g 7010 | A function that maps a sin... |
xpsng 7011 | The Cartesian product of t... |
xpprsng 7012 | The Cartesian product of a... |
xpsn 7013 | The Cartesian product of t... |
f1o2sn 7014 | A singleton consisting in ... |
residpr 7015 | Restriction of the identit... |
dfmpt 7016 | Alternate definition for t... |
fnasrn 7017 | A function expressed as th... |
idref 7018 | Two ways to state that a r... |
funiun 7019 | A function is a union of s... |
funopsn 7020 | If a function is an ordere... |
funop 7021 | An ordered pair is a funct... |
funopdmsn 7022 | The domain of a function w... |
funsndifnop 7023 | A singleton of an ordered ... |
funsneqopb 7024 | A singleton of an ordered ... |
ressnop0 7025 | If ` A ` is not in ` C ` ,... |
fpr 7026 | A function with a domain o... |
fprg 7027 | A function with a domain o... |
ftpg 7028 | A function with a domain o... |
ftp 7029 | A function with a domain o... |
fnressn 7030 | A function restricted to a... |
funressn 7031 | A function restricted to a... |
fressnfv 7032 | The value of a function re... |
fvrnressn 7033 | If the value of a function... |
fvressn 7034 | The value of a function re... |
fvn0fvelrn 7035 | If the value of a function... |
fvconst 7036 | The value of a constant fu... |
fnsnr 7037 | If a class belongs to a fu... |
fnsnb 7038 | A function whose domain is... |
fmptsn 7039 | Express a singleton functi... |
fmptsng 7040 | Express a singleton functi... |
fmptsnd 7041 | Express a singleton functi... |
fmptap 7042 | Append an additional value... |
fmptapd 7043 | Append an additional value... |
fmptpr 7044 | Express a pair function in... |
fvresi 7045 | The value of a restricted ... |
fninfp 7046 | Express the class of fixed... |
fnelfp 7047 | Property of a fixed point ... |
fndifnfp 7048 | Express the class of non-f... |
fnelnfp 7049 | Property of a non-fixed po... |
fnnfpeq0 7050 | A function is the identity... |
fvunsn 7051 | Remove an ordered pair not... |
fvsng 7052 | The value of a singleton o... |
fvsn 7053 | The value of a singleton o... |
fvsnun1 7054 | The value of a function wi... |
fvsnun2 7055 | The value of a function wi... |
fnsnsplit 7056 | Split a function into a si... |
fsnunf 7057 | Adjoining a point to a fun... |
fsnunf2 7058 | Adjoining a point to a pun... |
fsnunfv 7059 | Recover the added point fr... |
fsnunres 7060 | Recover the original funct... |
funresdfunsn 7061 | Restricting a function to ... |
fvpr1g 7062 | The value of a function wi... |
fvpr2g 7063 | The value of a function wi... |
fvpr2gOLD 7064 | Obsolete version of ~ fvpr... |
fvpr1 7065 | The value of a function wi... |
fvpr1OLD 7066 | Obsolete version of ~ fvpr... |
fvpr2 7067 | The value of a function wi... |
fvpr2OLD 7068 | Obsolete version of ~ fvpr... |
fprb 7069 | A condition for functionho... |
fvtp1 7070 | The first value of a funct... |
fvtp2 7071 | The second value of a func... |
fvtp3 7072 | The third value of a funct... |
fvtp1g 7073 | The value of a function wi... |
fvtp2g 7074 | The value of a function wi... |
fvtp3g 7075 | The value of a function wi... |
tpres 7076 | An unordered triple of ord... |
fvconst2g 7077 | The value of a constant fu... |
fconst2g 7078 | A constant function expres... |
fvconst2 7079 | The value of a constant fu... |
fconst2 7080 | A constant function expres... |
fconst5 7081 | Two ways to express that a... |
rnmptc 7082 | Range of a constant functi... |
rnmptcOLD 7083 | Obsolete version of ~ rnmp... |
fnprb 7084 | A function whose domain ha... |
fntpb 7085 | A function whose domain ha... |
fnpr2g 7086 | A function whose domain ha... |
fpr2g 7087 | A function that maps a pai... |
fconstfv 7088 | A constant function expres... |
fconst3 7089 | Two ways to express a cons... |
fconst4 7090 | Two ways to express a cons... |
resfunexg 7091 | The restriction of a funct... |
resiexd 7092 | The restriction of the ide... |
fnex 7093 | If the domain of a functio... |
fnexd 7094 | If the domain of a functio... |
funex 7095 | If the domain of a functio... |
opabex 7096 | Existence of a function ex... |
mptexg 7097 | If the domain of a functio... |
mptexgf 7098 | If the domain of a functio... |
mptex 7099 | If the domain of a functio... |
mptexd 7100 | If the domain of a functio... |
mptrabex 7101 | If the domain of a functio... |
fex 7102 | If the domain of a mapping... |
fexd 7103 | If the domain of a mapping... |
mptfvmpt 7104 | A function in maps-to nota... |
eufnfv 7105 | A function is uniquely det... |
funfvima 7106 | A function's value in a pr... |
funfvima2 7107 | A function's value in an i... |
funfvima2d 7108 | A function's value in a pr... |
fnfvima 7109 | The function value of an o... |
fnfvimad 7110 | A function's value belongs... |
resfvresima 7111 | The value of the function ... |
funfvima3 7112 | A class including a functi... |
rexima 7113 | Existential quantification... |
ralima 7114 | Universal quantification u... |
fvclss 7115 | Upper bound for the class ... |
elabrex 7116 | Elementhood in an image se... |
abrexco 7117 | Composition of two image m... |
imaiun 7118 | The image of an indexed un... |
imauni 7119 | The image of a union is th... |
fniunfv 7120 | The indexed union of a fun... |
funiunfv 7121 | The indexed union of a fun... |
funiunfvf 7122 | The indexed union of a fun... |
eluniima 7123 | Membership in the union of... |
elunirn 7124 | Membership in the union of... |
elunirnALT 7125 | Alternate proof of ~ eluni... |
elunirn2 7126 | Condition for the membersh... |
fnunirn 7127 | Membership in a union of s... |
dff13 7128 | A one-to-one function in t... |
dff13f 7129 | A one-to-one function in t... |
f1veqaeq 7130 | If the values of a one-to-... |
f1cofveqaeq 7131 | If the values of a composi... |
f1cofveqaeqALT 7132 | Alternate proof of ~ f1cof... |
2f1fvneq 7133 | If two one-to-one function... |
f1mpt 7134 | Express injection for a ma... |
f1fveq 7135 | Equality of function value... |
f1elima 7136 | Membership in the image of... |
f1imass 7137 | Taking images under a one-... |
f1imaeq 7138 | Taking images under a one-... |
f1imapss 7139 | Taking images under a one-... |
fpropnf1 7140 | A function, given by an un... |
f1dom3fv3dif 7141 | The function values for a ... |
f1dom3el3dif 7142 | The range of a 1-1 functio... |
dff14a 7143 | A one-to-one function in t... |
dff14b 7144 | A one-to-one function in t... |
f12dfv 7145 | A one-to-one function with... |
f13dfv 7146 | A one-to-one function with... |
dff1o6 7147 | A one-to-one onto function... |
f1ocnvfv1 7148 | The converse value of the ... |
f1ocnvfv2 7149 | The value of the converse ... |
f1ocnvfv 7150 | Relationship between the v... |
f1ocnvfvb 7151 | Relationship between the v... |
nvof1o 7152 | An involution is a bijecti... |
nvocnv 7153 | The converse of an involut... |
f1cdmsn 7154 | If a one-to-one function w... |
fsnex 7155 | Relate a function with a s... |
f1prex 7156 | Relate a one-to-one functi... |
f1ocnvdm 7157 | The value of the converse ... |
f1ocnvfvrneq 7158 | If the values of a one-to-... |
fcof1 7159 | An application is injectiv... |
fcofo 7160 | An application is surjecti... |
cbvfo 7161 | Change bound variable betw... |
cbvexfo 7162 | Change bound variable betw... |
cocan1 7163 | An injection is left-cance... |
cocan2 7164 | A surjection is right-canc... |
fcof1oinvd 7165 | Show that a function is th... |
fcof1od 7166 | A function is bijective if... |
2fcoidinvd 7167 | Show that a function is th... |
fcof1o 7168 | Show that two functions ar... |
2fvcoidd 7169 | Show that the composition ... |
2fvidf1od 7170 | A function is bijective if... |
2fvidinvd 7171 | Show that two functions ar... |
foeqcnvco 7172 | Condition for function equ... |
f1eqcocnv 7173 | Condition for function equ... |
f1eqcocnvOLD 7174 | Obsolete version of ~ f1eq... |
fveqf1o 7175 | Given a bijection ` F ` , ... |
nf1const 7176 | A constant function from a... |
nf1oconst 7177 | A constant function from a... |
f1ofvswap 7178 | Swapping two values in a b... |
fliftrel 7179 | ` F ` , a function lift, i... |
fliftel 7180 | Elementhood in the relatio... |
fliftel1 7181 | Elementhood in the relatio... |
fliftcnv 7182 | Converse of the relation `... |
fliftfun 7183 | The function ` F ` is the ... |
fliftfund 7184 | The function ` F ` is the ... |
fliftfuns 7185 | The function ` F ` is the ... |
fliftf 7186 | The domain and range of th... |
fliftval 7187 | The value of the function ... |
isoeq1 7188 | Equality theorem for isomo... |
isoeq2 7189 | Equality theorem for isomo... |
isoeq3 7190 | Equality theorem for isomo... |
isoeq4 7191 | Equality theorem for isomo... |
isoeq5 7192 | Equality theorem for isomo... |
nfiso 7193 | Bound-variable hypothesis ... |
isof1o 7194 | An isomorphism is a one-to... |
isof1oidb 7195 | A function is a bijection ... |
isof1oopb 7196 | A function is a bijection ... |
isorel 7197 | An isomorphism connects bi... |
soisores 7198 | Express the condition of i... |
soisoi 7199 | Infer isomorphism from one... |
isoid 7200 | Identity law for isomorphi... |
isocnv 7201 | Converse law for isomorphi... |
isocnv2 7202 | Converse law for isomorphi... |
isocnv3 7203 | Complementation law for is... |
isores2 7204 | An isomorphism from one we... |
isores1 7205 | An isomorphism from one we... |
isores3 7206 | Induced isomorphism on a s... |
isotr 7207 | Composition (transitive) l... |
isomin 7208 | Isomorphisms preserve mini... |
isoini 7209 | Isomorphisms preserve init... |
isoini2 7210 | Isomorphisms are isomorphi... |
isofrlem 7211 | Lemma for ~ isofr . (Cont... |
isoselem 7212 | Lemma for ~ isose . (Cont... |
isofr 7213 | An isomorphism preserves w... |
isose 7214 | An isomorphism preserves s... |
isofr2 7215 | A weak form of ~ isofr tha... |
isopolem 7216 | Lemma for ~ isopo . (Cont... |
isopo 7217 | An isomorphism preserves t... |
isosolem 7218 | Lemma for ~ isoso . (Cont... |
isoso 7219 | An isomorphism preserves t... |
isowe 7220 | An isomorphism preserves t... |
isowe2 7221 | A weak form of ~ isowe tha... |
f1oiso 7222 | Any one-to-one onto functi... |
f1oiso2 7223 | Any one-to-one onto functi... |
f1owe 7224 | Well-ordering of isomorphi... |
weniso 7225 | A set-like well-ordering h... |
weisoeq 7226 | Thus, there is at most one... |
weisoeq2 7227 | Thus, there is at most one... |
knatar 7228 | The Knaster-Tarski theorem... |
canth 7229 | No set ` A ` is equinumero... |
ncanth 7230 | Cantor's theorem fails for... |
riotaeqdv 7233 | Formula-building deduction... |
riotabidv 7234 | Formula-building deduction... |
riotaeqbidv 7235 | Equality deduction for res... |
riotaex 7236 | Restricted iota is a set. ... |
riotav 7237 | An iota restricted to the ... |
riotauni 7238 | Restricted iota in terms o... |
nfriota1 7239 | The abstraction variable i... |
nfriotadw 7240 | Deduction version of ~ nfr... |
cbvriotaw 7241 | Change bound variable in a... |
cbvriotavw 7242 | Change bound variable in a... |
cbvriotavwOLD 7243 | Obsolete version of ~ cbvr... |
nfriotad 7244 | Deduction version of ~ nfr... |
nfriota 7245 | A variable not free in a w... |
cbvriota 7246 | Change bound variable in a... |
cbvriotav 7247 | Change bound variable in a... |
csbriota 7248 | Interchange class substitu... |
riotacl2 7249 | Membership law for "the un... |
riotacl 7250 | Closure of restricted iota... |
riotasbc 7251 | Substitution law for descr... |
riotabidva 7252 | Equivalent wff's yield equ... |
riotabiia 7253 | Equivalent wff's yield equ... |
riota1 7254 | Property of restricted iot... |
riota1a 7255 | Property of iota. (Contri... |
riota2df 7256 | A deduction version of ~ r... |
riota2f 7257 | This theorem shows a condi... |
riota2 7258 | This theorem shows a condi... |
riotaeqimp 7259 | If two restricted iota des... |
riotaprop 7260 | Properties of a restricted... |
riota5f 7261 | A method for computing res... |
riota5 7262 | A method for computing res... |
riotass2 7263 | Restriction of a unique el... |
riotass 7264 | Restriction of a unique el... |
moriotass 7265 | Restriction of a unique el... |
snriota 7266 | A restricted class abstrac... |
riotaxfrd 7267 | Change the variable ` x ` ... |
eusvobj2 7268 | Specify the same property ... |
eusvobj1 7269 | Specify the same object in... |
f1ofveu 7270 | There is one domain elemen... |
f1ocnvfv3 7271 | Value of the converse of a... |
riotaund 7272 | Restricted iota equals the... |
riotassuni 7273 | The restricted iota class ... |
riotaclb 7274 | Bidirectional closure of r... |
oveq 7281 | Equality theorem for opera... |
oveq1 7282 | Equality theorem for opera... |
oveq2 7283 | Equality theorem for opera... |
oveq12 7284 | Equality theorem for opera... |
oveq1i 7285 | Equality inference for ope... |
oveq2i 7286 | Equality inference for ope... |
oveq12i 7287 | Equality inference for ope... |
oveqi 7288 | Equality inference for ope... |
oveq123i 7289 | Equality inference for ope... |
oveq1d 7290 | Equality deduction for ope... |
oveq2d 7291 | Equality deduction for ope... |
oveqd 7292 | Equality deduction for ope... |
oveq12d 7293 | Equality deduction for ope... |
oveqan12d 7294 | Equality deduction for ope... |
oveqan12rd 7295 | Equality deduction for ope... |
oveq123d 7296 | Equality deduction for ope... |
fvoveq1d 7297 | Equality deduction for nes... |
fvoveq1 7298 | Equality theorem for neste... |
ovanraleqv 7299 | Equality theorem for a con... |
imbrov2fvoveq 7300 | Equality theorem for neste... |
ovrspc2v 7301 | If an operation value is e... |
oveqrspc2v 7302 | Restricted specialization ... |
oveqdr 7303 | Equality of two operations... |
nfovd 7304 | Deduction version of bound... |
nfov 7305 | Bound-variable hypothesis ... |
oprabidw 7306 | The law of concretion. Sp... |
oprabid 7307 | The law of concretion. Sp... |
ovex 7308 | The result of an operation... |
ovexi 7309 | The result of an operation... |
ovexd 7310 | The result of an operation... |
ovssunirn 7311 | The result of an operation... |
0ov 7312 | Operation value of the emp... |
ovprc 7313 | The value of an operation ... |
ovprc1 7314 | The value of an operation ... |
ovprc2 7315 | The value of an operation ... |
ovrcl 7316 | Reverse closure for an ope... |
csbov123 7317 | Move class substitution in... |
csbov 7318 | Move class substitution in... |
csbov12g 7319 | Move class substitution in... |
csbov1g 7320 | Move class substitution in... |
csbov2g 7321 | Move class substitution in... |
rspceov 7322 | A frequently used special ... |
elovimad 7323 | Elementhood of the image s... |
fnbrovb 7324 | Value of a binary operatio... |
fnotovb 7325 | Equivalence of operation v... |
opabbrex 7326 | A collection of ordered pa... |
opabresex2 7327 | Restrictions of a collecti... |
opabresex2d 7328 | Obsolete version of ~ opab... |
fvmptopab 7329 | The function value of a ma... |
fvmptopabOLD 7330 | Obsolete version of ~ fvmp... |
f1opr 7331 | Condition for an operation... |
brfvopab 7332 | The classes involved in a ... |
dfoprab2 7333 | Class abstraction for oper... |
reloprab 7334 | An operation class abstrac... |
oprabv 7335 | If a pair and a class are ... |
nfoprab1 7336 | The abstraction variables ... |
nfoprab2 7337 | The abstraction variables ... |
nfoprab3 7338 | The abstraction variables ... |
nfoprab 7339 | Bound-variable hypothesis ... |
oprabbid 7340 | Equivalent wff's yield equ... |
oprabbidv 7341 | Equivalent wff's yield equ... |
oprabbii 7342 | Equivalent wff's yield equ... |
ssoprab2 7343 | Equivalence of ordered pai... |
ssoprab2b 7344 | Equivalence of ordered pai... |
eqoprab2bw 7345 | Equivalence of ordered pai... |
eqoprab2b 7346 | Equivalence of ordered pai... |
mpoeq123 7347 | An equality theorem for th... |
mpoeq12 7348 | An equality theorem for th... |
mpoeq123dva 7349 | An equality deduction for ... |
mpoeq123dv 7350 | An equality deduction for ... |
mpoeq123i 7351 | An equality inference for ... |
mpoeq3dva 7352 | Slightly more general equa... |
mpoeq3ia 7353 | An equality inference for ... |
mpoeq3dv 7354 | An equality deduction for ... |
nfmpo1 7355 | Bound-variable hypothesis ... |
nfmpo2 7356 | Bound-variable hypothesis ... |
nfmpo 7357 | Bound-variable hypothesis ... |
0mpo0 7358 | A mapping operation with e... |
mpo0v 7359 | A mapping operation with e... |
mpo0 7360 | A mapping operation with e... |
oprab4 7361 | Two ways to state the doma... |
cbvoprab1 7362 | Rule used to change first ... |
cbvoprab2 7363 | Change the second bound va... |
cbvoprab12 7364 | Rule used to change first ... |
cbvoprab12v 7365 | Rule used to change first ... |
cbvoprab3 7366 | Rule used to change the th... |
cbvoprab3v 7367 | Rule used to change the th... |
cbvmpox 7368 | Rule to change the bound v... |
cbvmpo 7369 | Rule to change the bound v... |
cbvmpov 7370 | Rule to change the bound v... |
elimdelov 7371 | Eliminate a hypothesis whi... |
ovif 7372 | Move a conditional outside... |
ovif2 7373 | Move a conditional outside... |
ovif12 7374 | Move a conditional outside... |
ifov 7375 | Move a conditional outside... |
dmoprab 7376 | The domain of an operation... |
dmoprabss 7377 | The domain of an operation... |
rnoprab 7378 | The range of an operation ... |
rnoprab2 7379 | The range of a restricted ... |
reldmoprab 7380 | The domain of an operation... |
oprabss 7381 | Structure of an operation ... |
eloprabga 7382 | The law of concretion for ... |
eloprabgaOLD 7383 | Obsolete version of ~ elop... |
eloprabg 7384 | The law of concretion for ... |
ssoprab2i 7385 | Inference of operation cla... |
mpov 7386 | Operation with universal d... |
mpomptx 7387 | Express a two-argument fun... |
mpompt 7388 | Express a two-argument fun... |
mpodifsnif 7389 | A mapping with two argumen... |
mposnif 7390 | A mapping with two argumen... |
fconstmpo 7391 | Representation of a consta... |
resoprab 7392 | Restriction of an operatio... |
resoprab2 7393 | Restriction of an operator... |
resmpo 7394 | Restriction of the mapping... |
funoprabg 7395 | "At most one" is a suffici... |
funoprab 7396 | "At most one" is a suffici... |
fnoprabg 7397 | Functionality and domain o... |
mpofun 7398 | The maps-to notation for a... |
mpofunOLD 7399 | Obsolete version of ~ mpof... |
fnoprab 7400 | Functionality and domain o... |
ffnov 7401 | An operation maps to a cla... |
fovcl 7402 | Closure law for an operati... |
eqfnov 7403 | Equality of two operations... |
eqfnov2 7404 | Two operators with the sam... |
fnov 7405 | Representation of a functi... |
mpo2eqb 7406 | Bidirectional equality the... |
rnmpo 7407 | The range of an operation ... |
reldmmpo 7408 | The domain of an operation... |
elrnmpog 7409 | Membership in the range of... |
elrnmpo 7410 | Membership in the range of... |
elrnmpores 7411 | Membership in the range of... |
ralrnmpo 7412 | A restricted quantifier ov... |
rexrnmpo 7413 | A restricted quantifier ov... |
ovid 7414 | The value of an operation ... |
ovidig 7415 | The value of an operation ... |
ovidi 7416 | The value of an operation ... |
ov 7417 | The value of an operation ... |
ovigg 7418 | The value of an operation ... |
ovig 7419 | The value of an operation ... |
ovmpt4g 7420 | Value of a function given ... |
ovmpos 7421 | Value of a function given ... |
ov2gf 7422 | The value of an operation ... |
ovmpodxf 7423 | Value of an operation give... |
ovmpodx 7424 | Value of an operation give... |
ovmpod 7425 | Value of an operation give... |
ovmpox 7426 | The value of an operation ... |
ovmpoga 7427 | Value of an operation give... |
ovmpoa 7428 | Value of an operation give... |
ovmpodf 7429 | Alternate deduction versio... |
ovmpodv 7430 | Alternate deduction versio... |
ovmpodv2 7431 | Alternate deduction versio... |
ovmpog 7432 | Value of an operation give... |
ovmpo 7433 | Value of an operation give... |
fvmpopr2d 7434 | Value of an operation give... |
ov3 7435 | The value of an operation ... |
ov6g 7436 | The value of an operation ... |
ovg 7437 | The value of an operation ... |
ovres 7438 | The value of a restricted ... |
ovresd 7439 | Lemma for converting metri... |
oprres 7440 | The restriction of an oper... |
oprssov 7441 | The value of a member of t... |
fovrn 7442 | An operation's value belon... |
fovrnda 7443 | An operation's value belon... |
fovrnd 7444 | An operation's value belon... |
fnrnov 7445 | The range of an operation ... |
foov 7446 | An onto mapping of an oper... |
fnovrn 7447 | An operation's value belon... |
ovelrn 7448 | A member of an operation's... |
funimassov 7449 | Membership relation for th... |
ovelimab 7450 | Operation value in an imag... |
ovima0 7451 | An operation value is a me... |
ovconst2 7452 | The value of a constant op... |
oprssdm 7453 | Domain of closure of an op... |
nssdmovg 7454 | The value of an operation ... |
ndmovg 7455 | The value of an operation ... |
ndmov 7456 | The value of an operation ... |
ndmovcl 7457 | The closure of an operatio... |
ndmovrcl 7458 | Reverse closure law, when ... |
ndmovcom 7459 | Any operation is commutati... |
ndmovass 7460 | Any operation is associati... |
ndmovdistr 7461 | Any operation is distribut... |
ndmovord 7462 | Elimination of redundant a... |
ndmovordi 7463 | Elimination of redundant a... |
caovclg 7464 | Convert an operation closu... |
caovcld 7465 | Convert an operation closu... |
caovcl 7466 | Convert an operation closu... |
caovcomg 7467 | Convert an operation commu... |
caovcomd 7468 | Convert an operation commu... |
caovcom 7469 | Convert an operation commu... |
caovassg 7470 | Convert an operation assoc... |
caovassd 7471 | Convert an operation assoc... |
caovass 7472 | Convert an operation assoc... |
caovcang 7473 | Convert an operation cance... |
caovcand 7474 | Convert an operation cance... |
caovcanrd 7475 | Commute the arguments of a... |
caovcan 7476 | Convert an operation cance... |
caovordig 7477 | Convert an operation order... |
caovordid 7478 | Convert an operation order... |
caovordg 7479 | Convert an operation order... |
caovordd 7480 | Convert an operation order... |
caovord2d 7481 | Operation ordering law wit... |
caovord3d 7482 | Ordering law. (Contribute... |
caovord 7483 | Convert an operation order... |
caovord2 7484 | Operation ordering law wit... |
caovord3 7485 | Ordering law. (Contribute... |
caovdig 7486 | Convert an operation distr... |
caovdid 7487 | Convert an operation distr... |
caovdir2d 7488 | Convert an operation distr... |
caovdirg 7489 | Convert an operation rever... |
caovdird 7490 | Convert an operation distr... |
caovdi 7491 | Convert an operation distr... |
caov32d 7492 | Rearrange arguments in a c... |
caov12d 7493 | Rearrange arguments in a c... |
caov31d 7494 | Rearrange arguments in a c... |
caov13d 7495 | Rearrange arguments in a c... |
caov4d 7496 | Rearrange arguments in a c... |
caov411d 7497 | Rearrange arguments in a c... |
caov42d 7498 | Rearrange arguments in a c... |
caov32 7499 | Rearrange arguments in a c... |
caov12 7500 | Rearrange arguments in a c... |
caov31 7501 | Rearrange arguments in a c... |
caov13 7502 | Rearrange arguments in a c... |
caov4 7503 | Rearrange arguments in a c... |
caov411 7504 | Rearrange arguments in a c... |
caov42 7505 | Rearrange arguments in a c... |
caovdir 7506 | Reverse distributive law. ... |
caovdilem 7507 | Lemma used by real number ... |
caovlem2 7508 | Lemma used in real number ... |
caovmo 7509 | Uniqueness of inverse elem... |
mpondm0 7510 | The value of an operation ... |
elmpocl 7511 | If a two-parameter class i... |
elmpocl1 7512 | If a two-parameter class i... |
elmpocl2 7513 | If a two-parameter class i... |
elovmpo 7514 | Utility lemma for two-para... |
elovmporab 7515 | Implications for the value... |
elovmporab1w 7516 | Implications for the value... |
elovmporab1 7517 | Implications for the value... |
2mpo0 7518 | If the operation value of ... |
relmptopab 7519 | Any function to sets of or... |
f1ocnvd 7520 | Describe an implicit one-t... |
f1od 7521 | Describe an implicit one-t... |
f1ocnv2d 7522 | Describe an implicit one-t... |
f1o2d 7523 | Describe an implicit one-t... |
f1opw2 7524 | A one-to-one mapping induc... |
f1opw 7525 | A one-to-one mapping induc... |
elovmpt3imp 7526 | If the value of a function... |
ovmpt3rab1 7527 | The value of an operation ... |
ovmpt3rabdm 7528 | If the value of a function... |
elovmpt3rab1 7529 | Implications for the value... |
elovmpt3rab 7530 | Implications for the value... |
ofeqd 7535 | Equality theorem for funct... |
ofeq 7536 | Equality theorem for funct... |
ofreq 7537 | Equality theorem for funct... |
ofexg 7538 | A function operation restr... |
nfof 7539 | Hypothesis builder for fun... |
nfofr 7540 | Hypothesis builder for fun... |
ofrfvalg 7541 | Value of a relation applie... |
offval 7542 | Value of an operation appl... |
ofrfval 7543 | Value of a relation applie... |
ofval 7544 | Evaluate a function operat... |
ofrval 7545 | Exhibit a function relatio... |
offn 7546 | The function operation pro... |
offun 7547 | The function operation pro... |
offval2f 7548 | The function operation exp... |
ofmresval 7549 | Value of a restriction of ... |
fnfvof 7550 | Function value of a pointw... |
off 7551 | The function operation pro... |
ofres 7552 | Restrict the operands of a... |
offval2 7553 | The function operation exp... |
ofrfval2 7554 | The function relation acti... |
ofmpteq 7555 | Value of a pointwise opera... |
ofco 7556 | The composition of a funct... |
offveq 7557 | Convert an identity of the... |
offveqb 7558 | Equivalent expressions for... |
ofc1 7559 | Left operation by a consta... |
ofc2 7560 | Right operation by a const... |
ofc12 7561 | Function operation on two ... |
caofref 7562 | Transfer a reflexive law t... |
caofinvl 7563 | Transfer a left inverse la... |
caofid0l 7564 | Transfer a left identity l... |
caofid0r 7565 | Transfer a right identity ... |
caofid1 7566 | Transfer a right absorptio... |
caofid2 7567 | Transfer a right absorptio... |
caofcom 7568 | Transfer a commutative law... |
caofrss 7569 | Transfer a relation subset... |
caofass 7570 | Transfer an associative la... |
caoftrn 7571 | Transfer a transitivity la... |
caofdi 7572 | Transfer a distributive la... |
caofdir 7573 | Transfer a reverse distrib... |
caonncan 7574 | Transfer ~ nncan -shaped l... |
relrpss 7577 | The proper subset relation... |
brrpssg 7578 | The proper subset relation... |
brrpss 7579 | The proper subset relation... |
porpss 7580 | Every class is partially o... |
sorpss 7581 | Express strict ordering un... |
sorpssi 7582 | Property of a chain of set... |
sorpssun 7583 | A chain of sets is closed ... |
sorpssin 7584 | A chain of sets is closed ... |
sorpssuni 7585 | In a chain of sets, a maxi... |
sorpssint 7586 | In a chain of sets, a mini... |
sorpsscmpl 7587 | The componentwise compleme... |
zfun 7589 | Axiom of Union expressed w... |
axun2 7590 | A variant of the Axiom of ... |
uniex2 7591 | The Axiom of Union using t... |
vuniex 7592 | The union of a setvar is a... |
uniexg 7593 | The ZF Axiom of Union in c... |
uniex 7594 | The Axiom of Union in clas... |
uniexd 7595 | Deduction version of the Z... |
unex 7596 | The union of two sets is a... |
tpex 7597 | An unordered triple of cla... |
unexb 7598 | Existence of union is equi... |
unexg 7599 | A union of two sets is a s... |
xpexg 7600 | The Cartesian product of t... |
xpexd 7601 | The Cartesian product of t... |
3xpexg 7602 | The Cartesian product of t... |
xpex 7603 | The Cartesian product of t... |
unexd 7604 | The union of two sets is a... |
sqxpexg 7605 | The Cartesian square of a ... |
abnexg 7606 | Sufficient condition for a... |
abnex 7607 | Sufficient condition for a... |
snnex 7608 | The class of all singleton... |
pwnex 7609 | The class of all power set... |
difex2 7610 | If the subtrahend of a cla... |
difsnexi 7611 | If the difference of a cla... |
uniuni 7612 | Expression for double unio... |
uniexr 7613 | Converse of the Axiom of U... |
uniexb 7614 | The Axiom of Union and its... |
pwexr 7615 | Converse of the Axiom of P... |
pwexb 7616 | The Axiom of Power Sets an... |
elpwpwel 7617 | A class belongs to a doubl... |
eldifpw 7618 | Membership in a power clas... |
elpwun 7619 | Membership in the power cl... |
pwuncl 7620 | Power classes are closed u... |
iunpw 7621 | An indexed union of a powe... |
fr3nr 7622 | A well-founded relation ha... |
epne3 7623 | A well-founded class conta... |
dfwe2 7624 | Alternate definition of we... |
epweon 7625 | The membership relation we... |
epweonOLD 7626 | Obsolete version of ~ epwe... |
ordon 7627 | The class of all ordinal n... |
onprc 7628 | No set contains all ordina... |
ssorduni 7629 | The union of a class of or... |
ssonuni 7630 | The union of a set of ordi... |
ssonunii 7631 | The union of a set of ordi... |
ordeleqon 7632 | A way to express the ordin... |
ordsson 7633 | Any ordinal class is a sub... |
onss 7634 | An ordinal number is a sub... |
predon 7635 | The predecessor of an ordi... |
predonOLD 7636 | Obsolete version of ~ pred... |
ssonprc 7637 | Two ways of saying a class... |
onuni 7638 | The union of an ordinal nu... |
orduni 7639 | The union of an ordinal cl... |
onint 7640 | The intersection (infimum)... |
onint0 7641 | The intersection of a clas... |
onssmin 7642 | A nonempty class of ordina... |
onminesb 7643 | If a property is true for ... |
onminsb 7644 | If a property is true for ... |
oninton 7645 | The intersection of a none... |
onintrab 7646 | The intersection of a clas... |
onintrab2 7647 | An existence condition equ... |
onnmin 7648 | No member of a set of ordi... |
onnminsb 7649 | An ordinal number smaller ... |
oneqmin 7650 | A way to show that an ordi... |
uniordint 7651 | The union of a set of ordi... |
onminex 7652 | If a wff is true for an or... |
sucon 7653 | The class of all ordinal n... |
sucexb 7654 | A successor exists iff its... |
sucexg 7655 | The successor of a set is ... |
sucex 7656 | The successor of a set is ... |
onmindif2 7657 | The minimum of a class of ... |
sucexeloni 7658 | If the successor of an ord... |
suceloni 7659 | The successor of an ordina... |
suceloniOLD 7660 | Obsolete version of ~ suce... |
ordsuc 7661 | The successor of an ordina... |
ordpwsuc 7662 | The collection of ordinals... |
onpwsuc 7663 | The collection of ordinal ... |
sucelon 7664 | The successor of an ordina... |
ordsucss 7665 | The successor of an elemen... |
onpsssuc 7666 | An ordinal number is a pro... |
ordelsuc 7667 | A set belongs to an ordina... |
onsucmin 7668 | The successor of an ordina... |
ordsucelsuc 7669 | Membership is inherited by... |
ordsucsssuc 7670 | The subclass relationship ... |
ordsucuniel 7671 | Given an element ` A ` of ... |
ordsucun 7672 | The successor of the maxim... |
ordunpr 7673 | The maximum of two ordinal... |
ordunel 7674 | The maximum of two ordinal... |
onsucuni 7675 | A class of ordinal numbers... |
ordsucuni 7676 | An ordinal class is a subc... |
orduniorsuc 7677 | An ordinal class is either... |
unon 7678 | The class of all ordinal n... |
ordunisuc 7679 | An ordinal class is equal ... |
orduniss2 7680 | The union of the ordinal s... |
onsucuni2 7681 | A successor ordinal is the... |
0elsuc 7682 | The successor of an ordina... |
limon 7683 | The class of ordinal numbe... |
onssi 7684 | An ordinal number is a sub... |
onsuci 7685 | The successor of an ordina... |
onuniorsuci 7686 | An ordinal number is eithe... |
onuninsuci 7687 | A limit ordinal is not a s... |
onsucssi 7688 | A set belongs to an ordina... |
nlimsucg 7689 | A successor is not a limit... |
orduninsuc 7690 | An ordinal equal to its un... |
ordunisuc2 7691 | An ordinal equal to its un... |
ordzsl 7692 | An ordinal is zero, a succ... |
onzsl 7693 | An ordinal number is zero,... |
dflim3 7694 | An alternate definition of... |
dflim4 7695 | An alternate definition of... |
limsuc 7696 | The successor of a member ... |
limsssuc 7697 | A class includes a limit o... |
nlimon 7698 | Two ways to express the cl... |
limuni3 7699 | The union of a nonempty cl... |
tfi 7700 | The Principle of Transfini... |
tfis 7701 | Transfinite Induction Sche... |
tfis2f 7702 | Transfinite Induction Sche... |
tfis2 7703 | Transfinite Induction Sche... |
tfis3 7704 | Transfinite Induction Sche... |
tfisi 7705 | A transfinite induction sc... |
tfinds 7706 | Principle of Transfinite I... |
tfindsg 7707 | Transfinite Induction (inf... |
tfindsg2 7708 | Transfinite Induction (inf... |
tfindes 7709 | Transfinite Induction with... |
tfinds2 7710 | Transfinite Induction (inf... |
tfinds3 7711 | Principle of Transfinite I... |
dfom2 7714 | An alternate definition of... |
elom 7715 | Membership in omega. The ... |
omsson 7716 | Omega is a subset of ` On ... |
limomss 7717 | The class of natural numbe... |
nnon 7718 | A natural number is an ord... |
nnoni 7719 | A natural number is an ord... |
nnord 7720 | A natural number is ordina... |
trom 7721 | The class of finite ordina... |
ordom 7722 | The class of finite ordina... |
elnn 7723 | A member of a natural numb... |
omon 7724 | The class of natural numbe... |
omelon2 7725 | Omega is an ordinal number... |
nnlim 7726 | A natural number is not a ... |
omssnlim 7727 | The class of natural numbe... |
limom 7728 | Omega is a limit ordinal. ... |
peano2b 7729 | A class belongs to omega i... |
nnsuc 7730 | A nonzero natural number i... |
omsucne 7731 | A natural number is not th... |
ssnlim 7732 | An ordinal subclass of non... |
omsinds 7733 | Strong (or "total") induct... |
omsindsOLD 7734 | Obsolete version of ~ omsi... |
peano1 7735 | Zero is a natural number. ... |
peano1OLD 7736 | Obsolete version of ~ pean... |
peano2 7737 | The successor of any natur... |
peano3 7738 | The successor of any natur... |
peano4 7739 | Two natural numbers are eq... |
peano5 7740 | The induction postulate: a... |
peano5OLD 7741 | Obsolete version of ~ pean... |
nn0suc 7742 | A natural number is either... |
find 7743 | The Principle of Finite In... |
findOLD 7744 | Obsolete version of ~ find... |
finds 7745 | Principle of Finite Induct... |
findsg 7746 | Principle of Finite Induct... |
finds2 7747 | Principle of Finite Induct... |
finds1 7748 | Principle of Finite Induct... |
findes 7749 | Finite induction with expl... |
dmexg 7750 | The domain of a set is a s... |
rnexg 7751 | The range of a set is a se... |
dmexd 7752 | The domain of a set is a s... |
fndmexd 7753 | If a function is a set, it... |
dmfex 7754 | If a mapping is a set, its... |
fndmexb 7755 | The domain of a function i... |
fdmexb 7756 | The domain of a function i... |
dmfexALT 7757 | Alternate proof of ~ dmfex... |
dmex 7758 | The domain of a set is a s... |
rnex 7759 | The range of a set is a se... |
iprc 7760 | The identity function is a... |
resiexg 7761 | The existence of a restric... |
imaexg 7762 | The image of a set is a se... |
imaex 7763 | The image of a set is a se... |
exse2 7764 | Any set relation is set-li... |
xpexr 7765 | If a Cartesian product is ... |
xpexr2 7766 | If a nonempty Cartesian pr... |
xpexcnv 7767 | A condition where the conv... |
soex 7768 | If the relation in a stric... |
elxp4 7769 | Membership in a Cartesian ... |
elxp5 7770 | Membership in a Cartesian ... |
cnvexg 7771 | The converse of a set is a... |
cnvex 7772 | The converse of a set is a... |
relcnvexb 7773 | A relation is a set iff it... |
f1oexrnex 7774 | If the range of a 1-1 onto... |
f1oexbi 7775 | There is a one-to-one onto... |
coexg 7776 | The composition of two set... |
coex 7777 | The composition of two set... |
funcnvuni 7778 | The union of a chain (with... |
fun11uni 7779 | The union of a chain (with... |
fex2 7780 | A function with bounded do... |
fabexg 7781 | Existence of a set of func... |
fabex 7782 | Existence of a set of func... |
f1oabexg 7783 | The class of all 1-1-onto ... |
fiunlem 7784 | Lemma for ~ fiun and ~ f1i... |
fiun 7785 | The union of a chain (with... |
f1iun 7786 | The union of a chain (with... |
fviunfun 7787 | The function value of an i... |
ffoss 7788 | Relationship between a map... |
f11o 7789 | Relationship between one-t... |
resfunexgALT 7790 | Alternate proof of ~ resfu... |
cofunexg 7791 | Existence of a composition... |
cofunex2g 7792 | Existence of a composition... |
fnexALT 7793 | Alternate proof of ~ fnex ... |
funexw 7794 | Weak version of ~ funex th... |
mptexw 7795 | Weak version of ~ mptex th... |
funrnex 7796 | If the domain of a functio... |
zfrep6 7797 | A version of the Axiom of ... |
fornex 7798 | If the domain of an onto f... |
f1dmex 7799 | If the codomain of a one-t... |
f1ovv 7800 | The range of a 1-1 onto fu... |
fvclex 7801 | Existence of the class of ... |
fvresex 7802 | Existence of the class of ... |
abrexexg 7803 | Existence of a class abstr... |
abrexexgOLD 7804 | Obsolete version of ~ abre... |
abrexex 7805 | Existence of a class abstr... |
iunexg 7806 | The existence of an indexe... |
abrexex2g 7807 | Existence of an existentia... |
opabex3d 7808 | Existence of an ordered pa... |
opabex3rd 7809 | Existence of an ordered pa... |
opabex3 7810 | Existence of an ordered pa... |
iunex 7811 | The existence of an indexe... |
abrexex2 7812 | Existence of an existentia... |
abexssex 7813 | Existence of a class abstr... |
abexex 7814 | A condition where a class ... |
f1oweALT 7815 | Alternate proof of ~ f1owe... |
wemoiso 7816 | Thus, there is at most one... |
wemoiso2 7817 | Thus, there is at most one... |
oprabexd 7818 | Existence of an operator a... |
oprabex 7819 | Existence of an operation ... |
oprabex3 7820 | Existence of an operation ... |
oprabrexex2 7821 | Existence of an existentia... |
ab2rexex 7822 | Existence of a class abstr... |
ab2rexex2 7823 | Existence of an existentia... |
xpexgALT 7824 | Alternate proof of ~ xpexg... |
offval3 7825 | General value of ` ( F oF ... |
offres 7826 | Pointwise combination comm... |
ofmres 7827 | Equivalent expressions for... |
ofmresex 7828 | Existence of a restriction... |
1stval 7833 | The value of the function ... |
2ndval 7834 | The value of the function ... |
1stnpr 7835 | Value of the first-member ... |
2ndnpr 7836 | Value of the second-member... |
1st0 7837 | The value of the first-mem... |
2nd0 7838 | The value of the second-me... |
op1st 7839 | Extract the first member o... |
op2nd 7840 | Extract the second member ... |
op1std 7841 | Extract the first member o... |
op2ndd 7842 | Extract the second member ... |
op1stg 7843 | Extract the first member o... |
op2ndg 7844 | Extract the second member ... |
ot1stg 7845 | Extract the first member o... |
ot2ndg 7846 | Extract the second member ... |
ot3rdg 7847 | Extract the third member o... |
1stval2 7848 | Alternate value of the fun... |
2ndval2 7849 | Alternate value of the fun... |
oteqimp 7850 | The components of an order... |
fo1st 7851 | The ` 1st ` function maps ... |
fo2nd 7852 | The ` 2nd ` function maps ... |
br1steqg 7853 | Uniqueness condition for t... |
br2ndeqg 7854 | Uniqueness condition for t... |
f1stres 7855 | Mapping of a restriction o... |
f2ndres 7856 | Mapping of a restriction o... |
fo1stres 7857 | Onto mapping of a restrict... |
fo2ndres 7858 | Onto mapping of a restrict... |
1st2val 7859 | Value of an alternate defi... |
2nd2val 7860 | Value of an alternate defi... |
1stcof 7861 | Composition of the first m... |
2ndcof 7862 | Composition of the second ... |
xp1st 7863 | Location of the first elem... |
xp2nd 7864 | Location of the second ele... |
elxp6 7865 | Membership in a Cartesian ... |
elxp7 7866 | Membership in a Cartesian ... |
eqopi 7867 | Equality with an ordered p... |
xp2 7868 | Representation of Cartesia... |
unielxp 7869 | The membership relation fo... |
1st2nd2 7870 | Reconstruction of a member... |
1st2ndb 7871 | Reconstruction of an order... |
xpopth 7872 | An ordered pair theorem fo... |
eqop 7873 | Two ways to express equali... |
eqop2 7874 | Two ways to express equali... |
op1steq 7875 | Two ways of expressing tha... |
opreuopreu 7876 | There is a unique ordered ... |
el2xptp 7877 | A member of a nested Carte... |
el2xptp0 7878 | A member of a nested Carte... |
2nd1st 7879 | Swap the members of an ord... |
1st2nd 7880 | Reconstruction of a member... |
1stdm 7881 | The first ordered pair com... |
2ndrn 7882 | The second ordered pair co... |
1st2ndbr 7883 | Express an element of a re... |
releldm2 7884 | Two ways of expressing mem... |
reldm 7885 | An expression for the doma... |
releldmdifi 7886 | One way of expressing memb... |
funfv1st2nd 7887 | The function value for the... |
funelss 7888 | If the first component of ... |
funeldmdif 7889 | Two ways of expressing mem... |
sbcopeq1a 7890 | Equality theorem for subst... |
csbopeq1a 7891 | Equality theorem for subst... |
dfopab2 7892 | A way to define an ordered... |
dfoprab3s 7893 | A way to define an operati... |
dfoprab3 7894 | Operation class abstractio... |
dfoprab4 7895 | Operation class abstractio... |
dfoprab4f 7896 | Operation class abstractio... |
opabex2 7897 | Condition for an operation... |
opabn1stprc 7898 | An ordered-pair class abst... |
opiota 7899 | The property of a uniquely... |
cnvoprab 7900 | The converse of a class ab... |
dfxp3 7901 | Define the Cartesian produ... |
elopabi 7902 | A consequence of membershi... |
eloprabi 7903 | A consequence of membershi... |
mpomptsx 7904 | Express a two-argument fun... |
mpompts 7905 | Express a two-argument fun... |
dmmpossx 7906 | The domain of a mapping is... |
fmpox 7907 | Functionality, domain and ... |
fmpo 7908 | Functionality, domain and ... |
fnmpo 7909 | Functionality and domain o... |
fnmpoi 7910 | Functionality and domain o... |
dmmpo 7911 | Domain of a class given by... |
ovmpoelrn 7912 | An operation's value belon... |
dmmpoga 7913 | Domain of an operation giv... |
dmmpogaOLD 7914 | Obsolete version of ~ dmmp... |
dmmpog 7915 | Domain of an operation giv... |
mpoexxg 7916 | Existence of an operation ... |
mpoexg 7917 | Existence of an operation ... |
mpoexga 7918 | If the domain of an operat... |
mpoexw 7919 | Weak version of ~ mpoex th... |
mpoex 7920 | If the domain of an operat... |
mptmpoopabbrd 7921 | The operation value of a f... |
mptmpoopabovd 7922 | The operation value of a f... |
mptmpoopabbrdOLD 7923 | Obsolete version of ~ mptm... |
mptmpoopabovdOLD 7924 | Obsolete version of ~ mptm... |
el2mpocsbcl 7925 | If the operation value of ... |
el2mpocl 7926 | If the operation value of ... |
fnmpoovd 7927 | A function with a Cartesia... |
offval22 7928 | The function operation exp... |
brovpreldm 7929 | If a binary relation holds... |
bropopvvv 7930 | If a binary relation holds... |
bropfvvvvlem 7931 | Lemma for ~ bropfvvvv . (... |
bropfvvvv 7932 | If a binary relation holds... |
ovmptss 7933 | If all the values of the m... |
relmpoopab 7934 | Any function to sets of or... |
fmpoco 7935 | Composition of two functio... |
oprabco 7936 | Composition of a function ... |
oprab2co 7937 | Composition of operator ab... |
df1st2 7938 | An alternate possible defi... |
df2nd2 7939 | An alternate possible defi... |
1stconst 7940 | The mapping of a restricti... |
2ndconst 7941 | The mapping of a restricti... |
dfmpo 7942 | Alternate definition for t... |
mposn 7943 | An operation (in maps-to n... |
curry1 7944 | Composition with ` ``' ( 2... |
curry1val 7945 | The value of a curried fun... |
curry1f 7946 | Functionality of a curried... |
curry2 7947 | Composition with ` ``' ( 1... |
curry2f 7948 | Functionality of a curried... |
curry2val 7949 | The value of a curried fun... |
cnvf1olem 7950 | Lemma for ~ cnvf1o . (Con... |
cnvf1o 7951 | Describe a function that m... |
fparlem1 7952 | Lemma for ~ fpar . (Contr... |
fparlem2 7953 | Lemma for ~ fpar . (Contr... |
fparlem3 7954 | Lemma for ~ fpar . (Contr... |
fparlem4 7955 | Lemma for ~ fpar . (Contr... |
fpar 7956 | Merge two functions in par... |
fsplit 7957 | A function that can be use... |
fsplitOLD 7958 | Obsolete proof of ~ fsplit... |
fsplitfpar 7959 | Merge two functions with a... |
offsplitfpar 7960 | Express the function opera... |
f2ndf 7961 | The ` 2nd ` (second compon... |
fo2ndf 7962 | The ` 2nd ` (second compon... |
f1o2ndf1 7963 | The ` 2nd ` (second compon... |
opco1 7964 | Value of an operation prec... |
opco2 7965 | Value of an operation prec... |
opco1i 7966 | Inference form of ~ opco1 ... |
frxp 7967 | A lexicographical ordering... |
xporderlem 7968 | Lemma for lexicographical ... |
poxp 7969 | A lexicographical ordering... |
soxp 7970 | A lexicographical ordering... |
wexp 7971 | A lexicographical ordering... |
fnwelem 7972 | Lemma for ~ fnwe . (Contr... |
fnwe 7973 | A variant on lexicographic... |
fnse 7974 | Condition for the well-ord... |
fvproj 7975 | Value of a function on ord... |
fimaproj 7976 | Image of a cartesian produ... |
suppval 7979 | The value of the operation... |
supp0prc 7980 | The support of a class is ... |
suppvalbr 7981 | The value of the operation... |
supp0 7982 | The support of the empty s... |
suppval1 7983 | The value of the operation... |
suppvalfng 7984 | The value of the operation... |
suppvalfn 7985 | The value of the operation... |
elsuppfng 7986 | An element of the support ... |
elsuppfn 7987 | An element of the support ... |
cnvimadfsn 7988 | The support of functions "... |
suppimacnvss 7989 | The support of functions "... |
suppimacnv 7990 | Support sets of functions ... |
frnsuppeq 7991 | Two ways of writing the su... |
frnsuppeqg 7992 | Version of ~ frnsuppeq avo... |
suppssdm 7993 | The support of a function ... |
suppsnop 7994 | The support of a singleton... |
snopsuppss 7995 | The support of a singleton... |
fvn0elsupp 7996 | If the function value for ... |
fvn0elsuppb 7997 | The function value for a g... |
rexsupp 7998 | Existential quantification... |
ressuppss 7999 | The support of the restric... |
suppun 8000 | The support of a class/fun... |
ressuppssdif 8001 | The support of the restric... |
mptsuppdifd 8002 | The support of a function ... |
mptsuppd 8003 | The support of a function ... |
extmptsuppeq 8004 | The support of an extended... |
suppfnss 8005 | The support of a function ... |
funsssuppss 8006 | The support of a function ... |
fnsuppres 8007 | Two ways to express restri... |
fnsuppeq0 8008 | The support of a function ... |
fczsupp0 8009 | The support of a constant ... |
suppss 8010 | Show that the support of a... |
suppssOLD 8011 | Obsolete version of ~ supp... |
suppssr 8012 | A function is zero outside... |
suppssrg 8013 | A function is zero outside... |
suppssov1 8014 | Formula building theorem f... |
suppssof1 8015 | Formula building theorem f... |
suppss2 8016 | Show that the support of a... |
suppsssn 8017 | Show that the support of a... |
suppssfv 8018 | Formula building theorem f... |
suppofssd 8019 | Condition for the support ... |
suppofss1d 8020 | Condition for the support ... |
suppofss2d 8021 | Condition for the support ... |
suppco 8022 | The support of the composi... |
suppcoss 8023 | The support of the composi... |
supp0cosupp0 8024 | The support of the composi... |
imacosupp 8025 | The image of the support o... |
opeliunxp2f 8026 | Membership in a union of C... |
mpoxeldm 8027 | If there is an element of ... |
mpoxneldm 8028 | If the first argument of a... |
mpoxopn0yelv 8029 | If there is an element of ... |
mpoxopynvov0g 8030 | If the second argument of ... |
mpoxopxnop0 8031 | If the first argument of a... |
mpoxopx0ov0 8032 | If the first argument of a... |
mpoxopxprcov0 8033 | If the components of the f... |
mpoxopynvov0 8034 | If the second argument of ... |
mpoxopoveq 8035 | Value of an operation give... |
mpoxopovel 8036 | Element of the value of an... |
mpoxopoveqd 8037 | Value of an operation give... |
brovex 8038 | A binary relation of the v... |
brovmpoex 8039 | A binary relation of the v... |
sprmpod 8040 | The extension of a binary ... |
tposss 8043 | Subset theorem for transpo... |
tposeq 8044 | Equality theorem for trans... |
tposeqd 8045 | Equality theorem for trans... |
tposssxp 8046 | The transposition is a sub... |
reltpos 8047 | The transposition is a rel... |
brtpos2 8048 | Value of the transposition... |
brtpos0 8049 | The behavior of ` tpos ` w... |
reldmtpos 8050 | Necessary and sufficient c... |
brtpos 8051 | The transposition swaps ar... |
ottpos 8052 | The transposition swaps th... |
relbrtpos 8053 | The transposition swaps ar... |
dmtpos 8054 | The domain of ` tpos F ` w... |
rntpos 8055 | The range of ` tpos F ` wh... |
tposexg 8056 | The transposition of a set... |
ovtpos 8057 | The transposition swaps th... |
tposfun 8058 | The transposition of a fun... |
dftpos2 8059 | Alternate definition of ` ... |
dftpos3 8060 | Alternate definition of ` ... |
dftpos4 8061 | Alternate definition of ` ... |
tpostpos 8062 | Value of the double transp... |
tpostpos2 8063 | Value of the double transp... |
tposfn2 8064 | The domain of a transposit... |
tposfo2 8065 | Condition for a surjective... |
tposf2 8066 | The domain and range of a ... |
tposf12 8067 | Condition for an injective... |
tposf1o2 8068 | Condition of a bijective t... |
tposfo 8069 | The domain and range of a ... |
tposf 8070 | The domain and range of a ... |
tposfn 8071 | Functionality of a transpo... |
tpos0 8072 | Transposition of the empty... |
tposco 8073 | Transposition of a composi... |
tpossym 8074 | Two ways to say a function... |
tposeqi 8075 | Equality theorem for trans... |
tposex 8076 | A transposition is a set. ... |
nftpos 8077 | Hypothesis builder for tra... |
tposoprab 8078 | Transposition of a class o... |
tposmpo 8079 | Transposition of a two-arg... |
tposconst 8080 | The transposition of a con... |
mpocurryd 8085 | The currying of an operati... |
mpocurryvald 8086 | The value of a curried ope... |
fvmpocurryd 8087 | The value of the value of ... |
pwuninel2 8090 | Direct proof of ~ pwuninel... |
pwuninel 8091 | The power set of the union... |
undefval 8092 | Value of the undefined val... |
undefnel2 8093 | The undefined value genera... |
undefnel 8094 | The undefined value genera... |
undefne0 8095 | The undefined value genera... |
frecseq123 8098 | Equality theorem for the w... |
nffrecs 8099 | Bound-variable hypothesis ... |
csbfrecsg 8100 | Move class substitution in... |
fpr3g 8101 | Functions defined by well-... |
frrlem1 8102 | Lemma for well-founded rec... |
frrlem2 8103 | Lemma for well-founded rec... |
frrlem3 8104 | Lemma for well-founded rec... |
frrlem4 8105 | Lemma for well-founded rec... |
frrlem5 8106 | Lemma for well-founded rec... |
frrlem6 8107 | Lemma for well-founded rec... |
frrlem7 8108 | Lemma for well-founded rec... |
frrlem8 8109 | Lemma for well-founded rec... |
frrlem9 8110 | Lemma for well-founded rec... |
frrlem10 8111 | Lemma for well-founded rec... |
frrlem11 8112 | Lemma for well-founded rec... |
frrlem12 8113 | Lemma for well-founded rec... |
frrlem13 8114 | Lemma for well-founded rec... |
frrlem14 8115 | Lemma for well-founded rec... |
fprlem1 8116 | Lemma for well-founded rec... |
fprlem2 8117 | Lemma for well-founded rec... |
fpr2a 8118 | Weak version of ~ fpr2 whi... |
fpr1 8119 | Law of well-founded recurs... |
fpr2 8120 | Law of well-founded recurs... |
fpr3 8121 | Law of well-founded recurs... |
frrrel 8122 | Show without using the axi... |
frrdmss 8123 | Show without using the axi... |
frrdmcl 8124 | Show without using the axi... |
fprfung 8125 | A "function" defined by we... |
fprresex 8126 | The restriction of a funct... |
dfwrecsOLD 8129 | Obsolete definition of the... |
wrecseq123 8130 | General equality theorem f... |
wrecseq123OLD 8131 | Obsolete proof of ~ wrecse... |
nfwrecs 8132 | Bound-variable hypothesis ... |
nfwrecsOLD 8133 | Obsolete proof of ~ nfwrec... |
wrecseq1 8134 | Equality theorem for the w... |
wrecseq2 8135 | Equality theorem for the w... |
wrecseq3 8136 | Equality theorem for the w... |
csbwrecsg 8137 | Move class substitution in... |
wfr3g 8138 | Functions defined by well-... |
wfrlem1OLD 8139 | Lemma for well-ordered rec... |
wfrlem2OLD 8140 | Lemma for well-ordered rec... |
wfrlem3OLD 8141 | Lemma for well-ordered rec... |
wfrlem3OLDa 8142 | Lemma for well-ordered rec... |
wfrlem4OLD 8143 | Lemma for well-ordered rec... |
wfrlem5OLD 8144 | Lemma for well-ordered rec... |
wfrrelOLD 8145 | Obsolete proof of ~ wfrrel... |
wfrdmssOLD 8146 | Obsolete proof of ~ wfrdms... |
wfrlem8OLD 8147 | Lemma for well-ordered rec... |
wfrdmclOLD 8148 | Obsolete proof of ~ wfrdmc... |
wfrlem10OLD 8149 | Lemma for well-ordered rec... |
wfrfunOLD 8150 | Obsolete proof of ~ wfrfun... |
wfrlem12OLD 8151 | Lemma for well-ordered rec... |
wfrlem13OLD 8152 | Lemma for well-ordered rec... |
wfrlem14OLD 8153 | Lemma for well-ordered rec... |
wfrlem15OLD 8154 | Lemma for well-ordered rec... |
wfrlem16OLD 8155 | Lemma for well-ordered rec... |
wfrlem17OLD 8156 | Without using ~ ax-rep , s... |
wfr2aOLD 8157 | Obsolete proof of ~ wfr2a ... |
wfr1OLD 8158 | Obsolete proof of ~ wfr1 a... |
wfr2OLD 8159 | Obsolete proof of ~ wfr2 a... |
wfrrel 8160 | The well-ordered recursion... |
wfrdmss 8161 | The domain of the well-ord... |
wfrdmcl 8162 | The predecessor class of a... |
wfrfun 8163 | The "function" generated b... |
wfrresex 8164 | Show without using the axi... |
wfr2a 8165 | A weak version of ~ wfr2 w... |
wfr1 8166 | The Principle of Well-Orde... |
wfr2 8167 | The Principle of Well-Orde... |
wfr3 8168 | The principle of Well-Orde... |
wfr3OLD 8169 | Obsolete form of ~ wfr3 as... |
iunon 8170 | The indexed union of a set... |
iinon 8171 | The nonempty indexed inter... |
onfununi 8172 | A property of functions on... |
onovuni 8173 | A variant of ~ onfununi fo... |
onoviun 8174 | A variant of ~ onovuni wit... |
onnseq 8175 | There are no length ` _om ... |
dfsmo2 8178 | Alternate definition of a ... |
issmo 8179 | Conditions for which ` A `... |
issmo2 8180 | Alternate definition of a ... |
smoeq 8181 | Equality theorem for stric... |
smodm 8182 | The domain of a strictly m... |
smores 8183 | A strictly monotone functi... |
smores3 8184 | A strictly monotone functi... |
smores2 8185 | A strictly monotone ordina... |
smodm2 8186 | The domain of a strictly m... |
smofvon2 8187 | The function values of a s... |
iordsmo 8188 | The identity relation rest... |
smo0 8189 | The null set is a strictly... |
smofvon 8190 | If ` B ` is a strictly mon... |
smoel 8191 | If ` x ` is less than ` y ... |
smoiun 8192 | The value of a strictly mo... |
smoiso 8193 | If ` F ` is an isomorphism... |
smoel2 8194 | A strictly monotone ordina... |
smo11 8195 | A strictly monotone ordina... |
smoord 8196 | A strictly monotone ordina... |
smoword 8197 | A strictly monotone ordina... |
smogt 8198 | A strictly monotone ordina... |
smorndom 8199 | The range of a strictly mo... |
smoiso2 8200 | The strictly monotone ordi... |
dfrecs3 8203 | The old definition of tran... |
dfrecs3OLD 8204 | Obsolete proof of ~ dfrecs... |
recseq 8205 | Equality theorem for ` rec... |
nfrecs 8206 | Bound-variable hypothesis ... |
tfrlem1 8207 | A technical lemma for tran... |
tfrlem3a 8208 | Lemma for transfinite recu... |
tfrlem3 8209 | Lemma for transfinite recu... |
tfrlem4 8210 | Lemma for transfinite recu... |
tfrlem5 8211 | Lemma for transfinite recu... |
recsfval 8212 | Lemma for transfinite recu... |
tfrlem6 8213 | Lemma for transfinite recu... |
tfrlem7 8214 | Lemma for transfinite recu... |
tfrlem8 8215 | Lemma for transfinite recu... |
tfrlem9 8216 | Lemma for transfinite recu... |
tfrlem9a 8217 | Lemma for transfinite recu... |
tfrlem10 8218 | Lemma for transfinite recu... |
tfrlem11 8219 | Lemma for transfinite recu... |
tfrlem12 8220 | Lemma for transfinite recu... |
tfrlem13 8221 | Lemma for transfinite recu... |
tfrlem14 8222 | Lemma for transfinite recu... |
tfrlem15 8223 | Lemma for transfinite recu... |
tfrlem16 8224 | Lemma for finite recursion... |
tfr1a 8225 | A weak version of ~ tfr1 w... |
tfr2a 8226 | A weak version of ~ tfr2 w... |
tfr2b 8227 | Without assuming ~ ax-rep ... |
tfr1 8228 | Principle of Transfinite R... |
tfr2 8229 | Principle of Transfinite R... |
tfr3 8230 | Principle of Transfinite R... |
tfr1ALT 8231 | Alternate proof of ~ tfr1 ... |
tfr2ALT 8232 | Alternate proof of ~ tfr2 ... |
tfr3ALT 8233 | Alternate proof of ~ tfr3 ... |
recsfnon 8234 | Strong transfinite recursi... |
recsval 8235 | Strong transfinite recursi... |
tz7.44lem1 8236 | The ordered pair abstracti... |
tz7.44-1 8237 | The value of ` F ` at ` (/... |
tz7.44-2 8238 | The value of ` F ` at a su... |
tz7.44-3 8239 | The value of ` F ` at a li... |
rdgeq1 8242 | Equality theorem for the r... |
rdgeq2 8243 | Equality theorem for the r... |
rdgeq12 8244 | Equality theorem for the r... |
nfrdg 8245 | Bound-variable hypothesis ... |
rdglem1 8246 | Lemma used with the recurs... |
rdgfun 8247 | The recursive definition g... |
rdgdmlim 8248 | The domain of the recursiv... |
rdgfnon 8249 | The recursive definition g... |
rdgvalg 8250 | Value of the recursive def... |
rdgval 8251 | Value of the recursive def... |
rdg0 8252 | The initial value of the r... |
rdgseg 8253 | The initial segments of th... |
rdgsucg 8254 | The value of the recursive... |
rdgsuc 8255 | The value of the recursive... |
rdglimg 8256 | The value of the recursive... |
rdglim 8257 | The value of the recursive... |
rdg0g 8258 | The initial value of the r... |
rdgsucmptf 8259 | The value of the recursive... |
rdgsucmptnf 8260 | The value of the recursive... |
rdgsucmpt2 8261 | This version of ~ rdgsucmp... |
rdgsucmpt 8262 | The value of the recursive... |
rdglim2 8263 | The value of the recursive... |
rdglim2a 8264 | The value of the recursive... |
rdg0n 8265 | If ` A ` is a proper class... |
frfnom 8266 | The function generated by ... |
fr0g 8267 | The initial value resultin... |
frsuc 8268 | The successor value result... |
frsucmpt 8269 | The successor value result... |
frsucmptn 8270 | The value of the finite re... |
frsucmpt2 8271 | The successor value result... |
tz7.48lem 8272 | A way of showing an ordina... |
tz7.48-2 8273 | Proposition 7.48(2) of [Ta... |
tz7.48-1 8274 | Proposition 7.48(1) of [Ta... |
tz7.48-3 8275 | Proposition 7.48(3) of [Ta... |
tz7.49 8276 | Proposition 7.49 of [Takeu... |
tz7.49c 8277 | Corollary of Proposition 7... |
seqomlem0 8280 | Lemma for ` seqom ` . Cha... |
seqomlem1 8281 | Lemma for ` seqom ` . The... |
seqomlem2 8282 | Lemma for ` seqom ` . (Co... |
seqomlem3 8283 | Lemma for ` seqom ` . (Co... |
seqomlem4 8284 | Lemma for ` seqom ` . (Co... |
seqomeq12 8285 | Equality theorem for ` seq... |
fnseqom 8286 | An index-aware recursive d... |
seqom0g 8287 | Value of an index-aware re... |
seqomsuc 8288 | Value of an index-aware re... |
omsucelsucb 8289 | Membership is inherited by... |
df1o2 8304 | Expanded value of the ordi... |
df2o3 8305 | Expanded value of the ordi... |
df2o2 8306 | Expanded value of the ordi... |
1oex 8307 | Ordinal 1 is a set. (Cont... |
2oex 8308 | ` 2o ` is a set. (Contrib... |
1on 8309 | Ordinal 1 is an ordinal nu... |
1onOLD 8310 | Obsolete version of ~ 1on ... |
2on 8311 | Ordinal 2 is an ordinal nu... |
2onOLD 8312 | Obsolete version of ~ 2on ... |
2on0 8313 | Ordinal two is not zero. ... |
3on 8314 | Ordinal 3 is an ordinal nu... |
4on 8315 | Ordinal 3 is an ordinal nu... |
1oexOLD 8316 | Obsolete version of ~ 1oex... |
2oexOLD 8317 | Obsolete version of ~ 2oex... |
1n0 8318 | Ordinal one is not equal t... |
nlim1 8319 | 1 is not a limit ordinal. ... |
nlim2 8320 | 2 is not a limit ordinal. ... |
xp01disj 8321 | Cartesian products with th... |
xp01disjl 8322 | Cartesian products with th... |
ordgt0ge1 8323 | Two ways to express that a... |
ordge1n0 8324 | An ordinal greater than or... |
el1o 8325 | Membership in ordinal one.... |
ord1eln01 8326 | An ordinal that is not 0 o... |
ord2eln012 8327 | An ordinal that is not 0, ... |
1ellim 8328 | A limit ordinal contains 1... |
2ellim 8329 | A limit ordinal contains 2... |
dif1o 8330 | Two ways to say that ` A `... |
ondif1 8331 | Two ways to say that ` A `... |
ondif2 8332 | Two ways to say that ` A `... |
2oconcl 8333 | Closure of the pair swappi... |
0lt1o 8334 | Ordinal zero is less than ... |
dif20el 8335 | An ordinal greater than on... |
0we1 8336 | The empty set is a well-or... |
brwitnlem 8337 | Lemma for relations which ... |
fnoa 8338 | Functionality and domain o... |
fnom 8339 | Functionality and domain o... |
fnoe 8340 | Functionality and domain o... |
oav 8341 | Value of ordinal addition.... |
omv 8342 | Value of ordinal multiplic... |
oe0lem 8343 | A helper lemma for ~ oe0 a... |
oev 8344 | Value of ordinal exponenti... |
oevn0 8345 | Value of ordinal exponenti... |
oa0 8346 | Addition with zero. Propo... |
om0 8347 | Ordinal multiplication wit... |
oe0m 8348 | Value of zero raised to an... |
om0x 8349 | Ordinal multiplication wit... |
oe0m0 8350 | Ordinal exponentiation wit... |
oe0m1 8351 | Ordinal exponentiation wit... |
oe0 8352 | Ordinal exponentiation wit... |
oev2 8353 | Alternate value of ordinal... |
oasuc 8354 | Addition with successor. ... |
oesuclem 8355 | Lemma for ~ oesuc . (Cont... |
omsuc 8356 | Multiplication with succes... |
oesuc 8357 | Ordinal exponentiation wit... |
onasuc 8358 | Addition with successor. ... |
onmsuc 8359 | Multiplication with succes... |
onesuc 8360 | Exponentiation with a succ... |
oa1suc 8361 | Addition with 1 is same as... |
oalim 8362 | Ordinal addition with a li... |
omlim 8363 | Ordinal multiplication wit... |
oelim 8364 | Ordinal exponentiation wit... |
oacl 8365 | Closure law for ordinal ad... |
omcl 8366 | Closure law for ordinal mu... |
oecl 8367 | Closure law for ordinal ex... |
oa0r 8368 | Ordinal addition with zero... |
om0r 8369 | Ordinal multiplication wit... |
o1p1e2 8370 | 1 + 1 = 2 for ordinal numb... |
o2p2e4 8371 | 2 + 2 = 4 for ordinal numb... |
o2p2e4OLD 8372 | Obsolete version of ~ o2p2... |
om1 8373 | Ordinal multiplication wit... |
om1r 8374 | Ordinal multiplication wit... |
oe1 8375 | Ordinal exponentiation wit... |
oe1m 8376 | Ordinal exponentiation wit... |
oaordi 8377 | Ordering property of ordin... |
oaord 8378 | Ordering property of ordin... |
oacan 8379 | Left cancellation law for ... |
oaword 8380 | Weak ordering property of ... |
oawordri 8381 | Weak ordering property of ... |
oaord1 8382 | An ordinal is less than it... |
oaword1 8383 | An ordinal is less than or... |
oaword2 8384 | An ordinal is less than or... |
oawordeulem 8385 | Lemma for ~ oawordex . (C... |
oawordeu 8386 | Existence theorem for weak... |
oawordexr 8387 | Existence theorem for weak... |
oawordex 8388 | Existence theorem for weak... |
oaordex 8389 | Existence theorem for orde... |
oa00 8390 | An ordinal sum is zero iff... |
oalimcl 8391 | The ordinal sum with a lim... |
oaass 8392 | Ordinal addition is associ... |
oarec 8393 | Recursive definition of or... |
oaf1o 8394 | Left addition by a constan... |
oacomf1olem 8395 | Lemma for ~ oacomf1o . (C... |
oacomf1o 8396 | Define a bijection from ` ... |
omordi 8397 | Ordering property of ordin... |
omord2 8398 | Ordering property of ordin... |
omord 8399 | Ordering property of ordin... |
omcan 8400 | Left cancellation law for ... |
omword 8401 | Weak ordering property of ... |
omwordi 8402 | Weak ordering property of ... |
omwordri 8403 | Weak ordering property of ... |
omword1 8404 | An ordinal is less than or... |
omword2 8405 | An ordinal is less than or... |
om00 8406 | The product of two ordinal... |
om00el 8407 | The product of two nonzero... |
omordlim 8408 | Ordering involving the pro... |
omlimcl 8409 | The product of any nonzero... |
odi 8410 | Distributive law for ordin... |
omass 8411 | Multiplication of ordinal ... |
oneo 8412 | If an ordinal number is ev... |
omeulem1 8413 | Lemma for ~ omeu : existen... |
omeulem2 8414 | Lemma for ~ omeu : uniquen... |
omopth2 8415 | An ordered pair-like theor... |
omeu 8416 | The division algorithm for... |
oen0 8417 | Ordinal exponentiation wit... |
oeordi 8418 | Ordering law for ordinal e... |
oeord 8419 | Ordering property of ordin... |
oecan 8420 | Left cancellation law for ... |
oeword 8421 | Weak ordering property of ... |
oewordi 8422 | Weak ordering property of ... |
oewordri 8423 | Weak ordering property of ... |
oeworde 8424 | Ordinal exponentiation com... |
oeordsuc 8425 | Ordering property of ordin... |
oelim2 8426 | Ordinal exponentiation wit... |
oeoalem 8427 | Lemma for ~ oeoa . (Contr... |
oeoa 8428 | Sum of exponents law for o... |
oeoelem 8429 | Lemma for ~ oeoe . (Contr... |
oeoe 8430 | Product of exponents law f... |
oelimcl 8431 | The ordinal exponential wi... |
oeeulem 8432 | Lemma for ~ oeeu . (Contr... |
oeeui 8433 | The division algorithm for... |
oeeu 8434 | The division algorithm for... |
nna0 8435 | Addition with zero. Theor... |
nnm0 8436 | Multiplication with zero. ... |
nnasuc 8437 | Addition with successor. ... |
nnmsuc 8438 | Multiplication with succes... |
nnesuc 8439 | Exponentiation with a succ... |
nna0r 8440 | Addition to zero. Remark ... |
nnm0r 8441 | Multiplication with zero. ... |
nnacl 8442 | Closure of addition of nat... |
nnmcl 8443 | Closure of multiplication ... |
nnecl 8444 | Closure of exponentiation ... |
nnacli 8445 | ` _om ` is closed under ad... |
nnmcli 8446 | ` _om ` is closed under mu... |
nnarcl 8447 | Reverse closure law for ad... |
nnacom 8448 | Addition of natural number... |
nnaordi 8449 | Ordering property of addit... |
nnaord 8450 | Ordering property of addit... |
nnaordr 8451 | Ordering property of addit... |
nnawordi 8452 | Adding to both sides of an... |
nnaass 8453 | Addition of natural number... |
nndi 8454 | Distributive law for natur... |
nnmass 8455 | Multiplication of natural ... |
nnmsucr 8456 | Multiplication with succes... |
nnmcom 8457 | Multiplication of natural ... |
nnaword 8458 | Weak ordering property of ... |
nnacan 8459 | Cancellation law for addit... |
nnaword1 8460 | Weak ordering property of ... |
nnaword2 8461 | Weak ordering property of ... |
nnmordi 8462 | Ordering property of multi... |
nnmord 8463 | Ordering property of multi... |
nnmword 8464 | Weak ordering property of ... |
nnmcan 8465 | Cancellation law for multi... |
nnmwordi 8466 | Weak ordering property of ... |
nnmwordri 8467 | Weak ordering property of ... |
nnawordex 8468 | Equivalence for weak order... |
nnaordex 8469 | Equivalence for ordering. ... |
1onn 8470 | The ordinal 1 is a natural... |
1onnALT 8471 | Shorter proof of ~ 1onn us... |
2onn 8472 | The ordinal 2 is a natural... |
2onnALT 8473 | Shorter proof of ~ 2onn us... |
3onn 8474 | The ordinal 3 is a natural... |
4onn 8475 | The ordinal 4 is a natural... |
1one2o 8476 | Ordinal one is not ordinal... |
oaabslem 8477 | Lemma for ~ oaabs . (Cont... |
oaabs 8478 | Ordinal addition absorbs a... |
oaabs2 8479 | The absorption law ~ oaabs... |
omabslem 8480 | Lemma for ~ omabs . (Cont... |
omabs 8481 | Ordinal multiplication is ... |
nnm1 8482 | Multiply an element of ` _... |
nnm2 8483 | Multiply an element of ` _... |
nn2m 8484 | Multiply an element of ` _... |
nnneo 8485 | If a natural number is eve... |
nneob 8486 | A natural number is even i... |
omsmolem 8487 | Lemma for ~ omsmo . (Cont... |
omsmo 8488 | A strictly monotonic ordin... |
omopthlem1 8489 | Lemma for ~ omopthi . (Co... |
omopthlem2 8490 | Lemma for ~ omopthi . (Co... |
omopthi 8491 | An ordered pair theorem fo... |
omopth 8492 | An ordered pair theorem fo... |
nnasmo 8493 | There is at most one left ... |
eldifsucnn 8494 | Condition for membership i... |
dfer2 8499 | Alternate definition of eq... |
dfec2 8501 | Alternate definition of ` ... |
ecexg 8502 | An equivalence class modul... |
ecexr 8503 | A nonempty equivalence cla... |
ereq1 8505 | Equality theorem for equiv... |
ereq2 8506 | Equality theorem for equiv... |
errel 8507 | An equivalence relation is... |
erdm 8508 | The domain of an equivalen... |
ercl 8509 | Elementhood in the field o... |
ersym 8510 | An equivalence relation is... |
ercl2 8511 | Elementhood in the field o... |
ersymb 8512 | An equivalence relation is... |
ertr 8513 | An equivalence relation is... |
ertrd 8514 | A transitivity relation fo... |
ertr2d 8515 | A transitivity relation fo... |
ertr3d 8516 | A transitivity relation fo... |
ertr4d 8517 | A transitivity relation fo... |
erref 8518 | An equivalence relation is... |
ercnv 8519 | The converse of an equival... |
errn 8520 | The range and domain of an... |
erssxp 8521 | An equivalence relation is... |
erex 8522 | An equivalence relation is... |
erexb 8523 | An equivalence relation is... |
iserd 8524 | A reflexive, symmetric, tr... |
iseri 8525 | A reflexive, symmetric, tr... |
iseriALT 8526 | Alternate proof of ~ iseri... |
brdifun 8527 | Evaluate the incomparabili... |
swoer 8528 | Incomparability under a st... |
swoord1 8529 | The incomparability equiva... |
swoord2 8530 | The incomparability equiva... |
swoso 8531 | If the incomparability rel... |
eqerlem 8532 | Lemma for ~ eqer . (Contr... |
eqer 8533 | Equivalence relation invol... |
ider 8534 | The identity relation is a... |
0er 8535 | The empty set is an equiva... |
eceq1 8536 | Equality theorem for equiv... |
eceq1d 8537 | Equality theorem for equiv... |
eceq2 8538 | Equality theorem for equiv... |
eceq2i 8539 | Equality theorem for the `... |
eceq2d 8540 | Equality theorem for the `... |
elecg 8541 | Membership in an equivalen... |
elec 8542 | Membership in an equivalen... |
relelec 8543 | Membership in an equivalen... |
ecss 8544 | An equivalence class is a ... |
ecdmn0 8545 | A representative of a none... |
ereldm 8546 | Equality of equivalence cl... |
erth 8547 | Basic property of equivale... |
erth2 8548 | Basic property of equivale... |
erthi 8549 | Basic property of equivale... |
erdisj 8550 | Equivalence classes do not... |
ecidsn 8551 | An equivalence class modul... |
qseq1 8552 | Equality theorem for quoti... |
qseq2 8553 | Equality theorem for quoti... |
qseq2i 8554 | Equality theorem for quoti... |
qseq2d 8555 | Equality theorem for quoti... |
qseq12 8556 | Equality theorem for quoti... |
elqsg 8557 | Closed form of ~ elqs . (... |
elqs 8558 | Membership in a quotient s... |
elqsi 8559 | Membership in a quotient s... |
elqsecl 8560 | Membership in a quotient s... |
ecelqsg 8561 | Membership of an equivalen... |
ecelqsi 8562 | Membership of an equivalen... |
ecopqsi 8563 | "Closure" law for equivale... |
qsexg 8564 | A quotient set exists. (C... |
qsex 8565 | A quotient set exists. (C... |
uniqs 8566 | The union of a quotient se... |
qsss 8567 | A quotient set is a set of... |
uniqs2 8568 | The union of a quotient se... |
snec 8569 | The singleton of an equiva... |
ecqs 8570 | Equivalence class in terms... |
ecid 8571 | A set is equal to its cose... |
qsid 8572 | A set is equal to its quot... |
ectocld 8573 | Implicit substitution of c... |
ectocl 8574 | Implicit substitution of c... |
elqsn0 8575 | A quotient set does not co... |
ecelqsdm 8576 | Membership of an equivalen... |
xpider 8577 | A Cartesian square is an e... |
iiner 8578 | The intersection of a none... |
riiner 8579 | The relative intersection ... |
erinxp 8580 | A restricted equivalence r... |
ecinxp 8581 | Restrict the relation in a... |
qsinxp 8582 | Restrict the equivalence r... |
qsdisj 8583 | Members of a quotient set ... |
qsdisj2 8584 | A quotient set is a disjoi... |
qsel 8585 | If an element of a quotien... |
uniinqs 8586 | Class union distributes ov... |
qliftlem 8587 | Lemma for theorems about a... |
qliftrel 8588 | ` F ` , a function lift, i... |
qliftel 8589 | Elementhood in the relatio... |
qliftel1 8590 | Elementhood in the relatio... |
qliftfun 8591 | The function ` F ` is the ... |
qliftfund 8592 | The function ` F ` is the ... |
qliftfuns 8593 | The function ` F ` is the ... |
qliftf 8594 | The domain and range of th... |
qliftval 8595 | The value of the function ... |
ecoptocl 8596 | Implicit substitution of c... |
2ecoptocl 8597 | Implicit substitution of c... |
3ecoptocl 8598 | Implicit substitution of c... |
brecop 8599 | Binary relation on a quoti... |
brecop2 8600 | Binary relation on a quoti... |
eroveu 8601 | Lemma for ~ erov and ~ ero... |
erovlem 8602 | Lemma for ~ erov and ~ ero... |
erov 8603 | The value of an operation ... |
eroprf 8604 | Functionality of an operat... |
erov2 8605 | The value of an operation ... |
eroprf2 8606 | Functionality of an operat... |
ecopoveq 8607 | This is the first of sever... |
ecopovsym 8608 | Assuming the operation ` F... |
ecopovtrn 8609 | Assuming that operation ` ... |
ecopover 8610 | Assuming that operation ` ... |
eceqoveq 8611 | Equality of equivalence re... |
ecovcom 8612 | Lemma used to transfer a c... |
ecovass 8613 | Lemma used to transfer an ... |
ecovdi 8614 | Lemma used to transfer a d... |
mapprc 8619 | When ` A ` is a proper cla... |
pmex 8620 | The class of all partial f... |
mapex 8621 | The class of all functions... |
fnmap 8622 | Set exponentiation has a u... |
fnpm 8623 | Partial function exponenti... |
reldmmap 8624 | Set exponentiation is a we... |
mapvalg 8625 | The value of set exponenti... |
pmvalg 8626 | The value of the partial m... |
mapval 8627 | The value of set exponenti... |
elmapg 8628 | Membership relation for se... |
elmapd 8629 | Deduction form of ~ elmapg... |
mapdm0 8630 | The empty set is the only ... |
elpmg 8631 | The predicate "is a partia... |
elpm2g 8632 | The predicate "is a partia... |
elpm2r 8633 | Sufficient condition for b... |
elpmi 8634 | A partial function is a fu... |
pmfun 8635 | A partial function is a fu... |
elmapex 8636 | Eliminate antecedent for m... |
elmapi 8637 | A mapping is a function, f... |
mapfset 8638 | If ` B ` is a set, the val... |
mapssfset 8639 | The value of the set expon... |
mapfoss 8640 | The value of the set expon... |
fsetsspwxp 8641 | The class of all functions... |
fset0 8642 | The set of functions from ... |
fsetdmprc0 8643 | The set of functions with ... |
fsetex 8644 | The set of functions betwe... |
f1setex 8645 | The set of injections betw... |
fosetex 8646 | The set of surjections bet... |
f1osetex 8647 | The set of bijections betw... |
fsetfcdm 8648 | The class of functions wit... |
fsetfocdm 8649 | The class of functions wit... |
fsetprcnex 8650 | The class of all functions... |
fsetcdmex 8651 | The class of all functions... |
fsetexb 8652 | The class of all functions... |
elmapfn 8653 | A mapping is a function wi... |
elmapfun 8654 | A mapping is always a func... |
elmapssres 8655 | A restricted mapping is a ... |
fpmg 8656 | A total function is a part... |
pmss12g 8657 | Subset relation for the se... |
pmresg 8658 | Elementhood of a restricte... |
elmap 8659 | Membership relation for se... |
mapval2 8660 | Alternate expression for t... |
elpm 8661 | The predicate "is a partia... |
elpm2 8662 | The predicate "is a partia... |
fpm 8663 | A total function is a part... |
mapsspm 8664 | Set exponentiation is a su... |
pmsspw 8665 | Partial maps are a subset ... |
mapsspw 8666 | Set exponentiation is a su... |
mapfvd 8667 | The value of a function th... |
elmapresaun 8668 | ~ fresaun transposed to ma... |
fvmptmap 8669 | Special case of ~ fvmpt fo... |
map0e 8670 | Set exponentiation with an... |
map0b 8671 | Set exponentiation with an... |
map0g 8672 | Set exponentiation is empt... |
0map0sn0 8673 | The set of mappings of the... |
mapsnd 8674 | The value of set exponenti... |
map0 8675 | Set exponentiation is empt... |
mapsn 8676 | The value of set exponenti... |
mapss 8677 | Subset inheritance for set... |
fdiagfn 8678 | Functionality of the diago... |
fvdiagfn 8679 | Functionality of the diago... |
mapsnconst 8680 | Every singleton map is a c... |
mapsncnv 8681 | Expression for the inverse... |
mapsnf1o2 8682 | Explicit bijection between... |
mapsnf1o3 8683 | Explicit bijection in the ... |
ralxpmap 8684 | Quantification over functi... |
dfixp 8687 | Eliminate the expression `... |
ixpsnval 8688 | The value of an infinite C... |
elixp2 8689 | Membership in an infinite ... |
fvixp 8690 | Projection of a factor of ... |
ixpfn 8691 | A nuple is a function. (C... |
elixp 8692 | Membership in an infinite ... |
elixpconst 8693 | Membership in an infinite ... |
ixpconstg 8694 | Infinite Cartesian product... |
ixpconst 8695 | Infinite Cartesian product... |
ixpeq1 8696 | Equality theorem for infin... |
ixpeq1d 8697 | Equality theorem for infin... |
ss2ixp 8698 | Subclass theorem for infin... |
ixpeq2 8699 | Equality theorem for infin... |
ixpeq2dva 8700 | Equality theorem for infin... |
ixpeq2dv 8701 | Equality theorem for infin... |
cbvixp 8702 | Change bound variable in a... |
cbvixpv 8703 | Change bound variable in a... |
nfixpw 8704 | Bound-variable hypothesis ... |
nfixp 8705 | Bound-variable hypothesis ... |
nfixp1 8706 | The index variable in an i... |
ixpprc 8707 | A cartesian product of pro... |
ixpf 8708 | A member of an infinite Ca... |
uniixp 8709 | The union of an infinite C... |
ixpexg 8710 | The existence of an infini... |
ixpin 8711 | The intersection of two in... |
ixpiin 8712 | The indexed intersection o... |
ixpint 8713 | The intersection of a coll... |
ixp0x 8714 | An infinite Cartesian prod... |
ixpssmap2g 8715 | An infinite Cartesian prod... |
ixpssmapg 8716 | An infinite Cartesian prod... |
0elixp 8717 | Membership of the empty se... |
ixpn0 8718 | The infinite Cartesian pro... |
ixp0 8719 | The infinite Cartesian pro... |
ixpssmap 8720 | An infinite Cartesian prod... |
resixp 8721 | Restriction of an element ... |
undifixp 8722 | Union of two projections o... |
mptelixpg 8723 | Condition for an explicit ... |
resixpfo 8724 | Restriction of elements of... |
elixpsn 8725 | Membership in a class of s... |
ixpsnf1o 8726 | A bijection between a clas... |
mapsnf1o 8727 | A bijection between a set ... |
boxriin 8728 | A rectangular subset of a ... |
boxcutc 8729 | The relative complement of... |
relen 8738 | Equinumerosity is a relati... |
reldom 8739 | Dominance is a relation. ... |
relsdom 8740 | Strict dominance is a rela... |
encv 8741 | If two classes are equinum... |
breng 8742 | Equinumerosity relation. ... |
bren 8743 | Equinumerosity relation. ... |
brenOLD 8744 | Obsolete version of ~ bren... |
brdom2g 8745 | Dominance relation. This ... |
brdomg 8746 | Dominance relation. (Cont... |
brdomgOLD 8747 | Obsolete version of ~ brdo... |
brdomi 8748 | Dominance relation. (Cont... |
brdomiOLD 8749 | Obsolete version of ~ brdo... |
brdom 8750 | Dominance relation. (Cont... |
domen 8751 | Dominance in terms of equi... |
domeng 8752 | Dominance in terms of equi... |
ctex 8753 | A countable set is a set. ... |
f1oen3g 8754 | The domain and range of a ... |
f1dom3g 8755 | The domain of a one-to-one... |
f1oen2g 8756 | The domain and range of a ... |
f1dom2g 8757 | The domain of a one-to-one... |
f1dom2gOLD 8758 | Obsolete version of ~ f1do... |
f1oeng 8759 | The domain and range of a ... |
f1domg 8760 | The domain of a one-to-one... |
f1oen 8761 | The domain and range of a ... |
f1dom 8762 | The domain of a one-to-one... |
brsdom 8763 | Strict dominance relation,... |
isfi 8764 | Express " ` A ` is finite"... |
enssdom 8765 | Equinumerosity implies dom... |
dfdom2 8766 | Alternate definition of do... |
endom 8767 | Equinumerosity implies dom... |
sdomdom 8768 | Strict dominance implies d... |
sdomnen 8769 | Strict dominance implies n... |
brdom2 8770 | Dominance in terms of stri... |
bren2 8771 | Equinumerosity expressed i... |
enrefg 8772 | Equinumerosity is reflexiv... |
enref 8773 | Equinumerosity is reflexiv... |
eqeng 8774 | Equality implies equinumer... |
domrefg 8775 | Dominance is reflexive. (... |
en2d 8776 | Equinumerosity inference f... |
en3d 8777 | Equinumerosity inference f... |
en2i 8778 | Equinumerosity inference f... |
en3i 8779 | Equinumerosity inference f... |
dom2lem 8780 | A mapping (first hypothesi... |
dom2d 8781 | A mapping (first hypothesi... |
dom3d 8782 | A mapping (first hypothesi... |
dom2 8783 | A mapping (first hypothesi... |
dom3 8784 | A mapping (first hypothesi... |
idssen 8785 | Equality implies equinumer... |
ssdomg 8786 | A set dominates its subset... |
ener 8787 | Equinumerosity is an equiv... |
ensymb 8788 | Symmetry of equinumerosity... |
ensym 8789 | Symmetry of equinumerosity... |
ensymi 8790 | Symmetry of equinumerosity... |
ensymd 8791 | Symmetry of equinumerosity... |
entr 8792 | Transitivity of equinumero... |
domtr 8793 | Transitivity of dominance ... |
entri 8794 | A chained equinumerosity i... |
entr2i 8795 | A chained equinumerosity i... |
entr3i 8796 | A chained equinumerosity i... |
entr4i 8797 | A chained equinumerosity i... |
endomtr 8798 | Transitivity of equinumero... |
domentr 8799 | Transitivity of dominance ... |
f1imaeng 8800 | If a function is one-to-on... |
f1imaen2g 8801 | If a function is one-to-on... |
f1imaen 8802 | If a function is one-to-on... |
en0 8803 | The empty set is equinumer... |
en0OLD 8804 | Obsolete version of ~ en0 ... |
en0ALT 8805 | Shorter proof of ~ en0 , d... |
en0r 8806 | The empty set is equinumer... |
ensn1 8807 | A singleton is equinumerou... |
ensn1OLD 8808 | Obsolete version of ~ ensn... |
ensn1g 8809 | A singleton is equinumerou... |
enpr1g 8810 | ` { A , A } ` has only one... |
en1 8811 | A set is equinumerous to o... |
en1OLD 8812 | Obsolete version of ~ en1 ... |
en1b 8813 | A set is equinumerous to o... |
en1bOLD 8814 | Obsolete version of ~ en1b... |
reuen1 8815 | Two ways to express "exact... |
euen1 8816 | Two ways to express "exact... |
euen1b 8817 | Two ways to express " ` A ... |
en1uniel 8818 | A singleton contains its s... |
en1unielOLD 8819 | Obsolete version of ~ en1u... |
2dom 8820 | A set that dominates ordin... |
fundmen 8821 | A function is equinumerous... |
fundmeng 8822 | A function is equinumerous... |
cnven 8823 | A relational set is equinu... |
cnvct 8824 | If a set is countable, so ... |
fndmeng 8825 | A function is equinumerate... |
mapsnend 8826 | Set exponentiation to a si... |
mapsnen 8827 | Set exponentiation to a si... |
snmapen 8828 | Set exponentiation: a sing... |
snmapen1 8829 | Set exponentiation: a sing... |
map1 8830 | Set exponentiation: ordina... |
en2sn 8831 | Two singletons are equinum... |
en2snOLD 8832 | Obsolete version of ~ en2s... |
en2snOLDOLD 8833 | Obsolete version of ~ en2s... |
snfi 8834 | A singleton is finite. (C... |
fiprc 8835 | The class of finite sets i... |
unen 8836 | Equinumerosity of union of... |
enrefnn 8837 | Equinumerosity is reflexiv... |
enpr2d 8838 | A pair with distinct eleme... |
ssct 8839 | Any subset of a countable ... |
difsnen 8840 | All decrements of a set ar... |
domdifsn 8841 | Dominance over a set with ... |
xpsnen 8842 | A set is equinumerous to i... |
xpsneng 8843 | A set is equinumerous to i... |
xp1en 8844 | One times a cardinal numbe... |
endisj 8845 | Any two sets are equinumer... |
undom 8846 | Dominance law for union. ... |
undomOLD 8847 | Obsolete version of ~ undo... |
xpcomf1o 8848 | The canonical bijection fr... |
xpcomco 8849 | Composition with the bijec... |
xpcomen 8850 | Commutative law for equinu... |
xpcomeng 8851 | Commutative law for equinu... |
xpsnen2g 8852 | A set is equinumerous to i... |
xpassen 8853 | Associative law for equinu... |
xpdom2 8854 | Dominance law for Cartesia... |
xpdom2g 8855 | Dominance law for Cartesia... |
xpdom1g 8856 | Dominance law for Cartesia... |
xpdom3 8857 | A set is dominated by its ... |
xpdom1 8858 | Dominance law for Cartesia... |
domunsncan 8859 | A singleton cancellation l... |
omxpenlem 8860 | Lemma for ~ omxpen . (Con... |
omxpen 8861 | The cardinal and ordinal p... |
omf1o 8862 | Construct an explicit bije... |
pw2f1olem 8863 | Lemma for ~ pw2f1o . (Con... |
pw2f1o 8864 | The power set of a set is ... |
pw2eng 8865 | The power set of a set is ... |
pw2en 8866 | The power set of a set is ... |
fopwdom 8867 | Covering implies injection... |
enfixsn 8868 | Given two equipollent sets... |
sucdom2OLD 8869 | Obsolete version of ~ sucd... |
sbthlem1 8870 | Lemma for ~ sbth . (Contr... |
sbthlem2 8871 | Lemma for ~ sbth . (Contr... |
sbthlem3 8872 | Lemma for ~ sbth . (Contr... |
sbthlem4 8873 | Lemma for ~ sbth . (Contr... |
sbthlem5 8874 | Lemma for ~ sbth . (Contr... |
sbthlem6 8875 | Lemma for ~ sbth . (Contr... |
sbthlem7 8876 | Lemma for ~ sbth . (Contr... |
sbthlem8 8877 | Lemma for ~ sbth . (Contr... |
sbthlem9 8878 | Lemma for ~ sbth . (Contr... |
sbthlem10 8879 | Lemma for ~ sbth . (Contr... |
sbth 8880 | Schroeder-Bernstein Theore... |
sbthb 8881 | Schroeder-Bernstein Theore... |
sbthcl 8882 | Schroeder-Bernstein Theore... |
dfsdom2 8883 | Alternate definition of st... |
brsdom2 8884 | Alternate definition of st... |
sdomnsym 8885 | Strict dominance is asymme... |
domnsym 8886 | Theorem 22(i) of [Suppes] ... |
0domg 8887 | Any set dominates the empt... |
0domgOLD 8888 | Obsolete version of ~ 0dom... |
dom0 8889 | A set dominated by the emp... |
dom0OLD 8890 | Obsolete version of ~ dom0... |
0sdomg 8891 | A set strictly dominates t... |
0sdomgOLD 8892 | Obsolete version of ~ 0sdo... |
0dom 8893 | Any set dominates the empt... |
0sdom 8894 | A set strictly dominates t... |
sdom0 8895 | The empty set does not str... |
sdom0OLD 8896 | Obsolete version of ~ sdom... |
sdomdomtr 8897 | Transitivity of strict dom... |
sdomentr 8898 | Transitivity of strict dom... |
domsdomtr 8899 | Transitivity of dominance ... |
ensdomtr 8900 | Transitivity of equinumero... |
sdomirr 8901 | Strict dominance is irrefl... |
sdomtr 8902 | Strict dominance is transi... |
sdomn2lp 8903 | Strict dominance has no 2-... |
enen1 8904 | Equality-like theorem for ... |
enen2 8905 | Equality-like theorem for ... |
domen1 8906 | Equality-like theorem for ... |
domen2 8907 | Equality-like theorem for ... |
sdomen1 8908 | Equality-like theorem for ... |
sdomen2 8909 | Equality-like theorem for ... |
domtriord 8910 | Dominance is trichotomous ... |
sdomel 8911 | For ordinals, strict domin... |
sdomdif 8912 | The difference of a set fr... |
onsdominel 8913 | An ordinal with more eleme... |
domunsn 8914 | Dominance over a set with ... |
fodomr 8915 | There exists a mapping fro... |
pwdom 8916 | Injection of sets implies ... |
canth2 8917 | Cantor's Theorem. No set ... |
canth2g 8918 | Cantor's theorem with the ... |
2pwuninel 8919 | The power set of the power... |
2pwne 8920 | No set equals the power se... |
disjen 8921 | A stronger form of ~ pwuni... |
disjenex 8922 | Existence version of ~ dis... |
domss2 8923 | A corollary of ~ disjenex ... |
domssex2 8924 | A corollary of ~ disjenex ... |
domssex 8925 | Weakening of ~ domssex2 to... |
xpf1o 8926 | Construct a bijection on a... |
xpen 8927 | Equinumerosity law for Car... |
mapen 8928 | Two set exponentiations ar... |
mapdom1 8929 | Order-preserving property ... |
mapxpen 8930 | Equinumerosity law for dou... |
xpmapenlem 8931 | Lemma for ~ xpmapen . (Co... |
xpmapen 8932 | Equinumerosity law for set... |
mapunen 8933 | Equinumerosity law for set... |
map2xp 8934 | A cardinal power with expo... |
mapdom2 8935 | Order-preserving property ... |
mapdom3 8936 | Set exponentiation dominat... |
pwen 8937 | If two sets are equinumero... |
ssenen 8938 | Equinumerosity of equinume... |
limenpsi 8939 | A limit ordinal is equinum... |
limensuci 8940 | A limit ordinal is equinum... |
limensuc 8941 | A limit ordinal is equinum... |
infensuc 8942 | Any infinite ordinal is eq... |
dif1enlem 8943 | Lemma for ~ rexdif1en and ... |
rexdif1en 8944 | If a set is equinumerous t... |
dif1en 8945 | If a set ` A ` is equinume... |
findcard 8946 | Schema for induction on th... |
findcard2 8947 | Schema for induction on th... |
findcard2s 8948 | Variation of ~ findcard2 r... |
findcard2d 8949 | Deduction version of ~ fin... |
nnfi 8950 | Natural numbers are finite... |
pssnn 8951 | A proper subset of a natur... |
ssnnfi 8952 | A subset of a natural numb... |
ssnnfiOLD 8953 | Obsolete version of ~ ssnn... |
0fin 8954 | The empty set is finite. ... |
unfi 8955 | The union of two finite se... |
ssfi 8956 | A subset of a finite set i... |
ssfiALT 8957 | Shorter proof of ~ ssfi us... |
imafi 8958 | Images of finite sets are ... |
pwfir 8959 | If the power set of a set ... |
pwfilem 8960 | Lemma for ~ pwfi . (Contr... |
pwfi 8961 | The power set of a finite ... |
diffi 8962 | If ` A ` is finite, ` ( A ... |
cnvfi 8963 | If a set is finite, its co... |
fnfi 8964 | A version of ~ fnex for fi... |
f1oenfi 8965 | If the domain of a one-to-... |
f1oenfirn 8966 | If the range of a one-to-o... |
f1domfi 8967 | If the codomain of a one-t... |
f1domfi2 8968 | If the domain of a one-to-... |
enreffi 8969 | Equinumerosity is reflexiv... |
ensymfib 8970 | Symmetry of equinumerosity... |
entrfil 8971 | Transitivity of equinumero... |
enfii 8972 | A set equinumerous to a fi... |
enfi 8973 | Equinumerous sets have the... |
enfiALT 8974 | Shorter proof of ~ enfi us... |
domfi 8975 | A set dominated by a finit... |
entrfi 8976 | Transitivity of equinumero... |
entrfir 8977 | Transitivity of equinumero... |
domtrfil 8978 | Transitivity of dominance ... |
domtrfi 8979 | Transitivity of dominance ... |
domtrfir 8980 | Transitivity of dominance ... |
f1imaenfi 8981 | If a function is one-to-on... |
ssdomfi 8982 | A finite set dominates its... |
ssdomfi2 8983 | A set dominates its finite... |
sbthfilem 8984 | Lemma for ~ sbthfi . (Con... |
sbthfi 8985 | Schroeder-Bernstein Theore... |
domnsymfi 8986 | If a set dominates a finit... |
sdomdomtrfi 8987 | Transitivity of strict dom... |
domsdomtrfi 8988 | Transitivity of dominance ... |
sucdom2 8989 | Strict dominance of a set ... |
phplem1 8990 | Lemma for Pigeonhole Princ... |
phplem2 8991 | Lemma for Pigeonhole Princ... |
nneneq 8992 | Two equinumerous natural n... |
php 8993 | Pigeonhole Principle. A n... |
php2 8994 | Corollary of Pigeonhole Pr... |
php3 8995 | Corollary of Pigeonhole Pr... |
php4 8996 | Corollary of the Pigeonhol... |
php5 8997 | Corollary of the Pigeonhol... |
phpeqd 8998 | Corollary of the Pigeonhol... |
nndomog 8999 | Cardinal ordering agrees w... |
phplem1OLD 9000 | Obsolete lemma for ~ php .... |
phplem2OLD 9001 | Obsolete lemma for ~ php .... |
phplem3OLD 9002 | Obsolete version of ~ phpl... |
phplem4OLD 9003 | Obsolete version of ~ phpl... |
nneneqOLD 9004 | Obsolete version of ~ nnen... |
phpOLD 9005 | Obsolete version of ~ php ... |
php2OLD 9006 | Obsolete version of ~ php2... |
php3OLD 9007 | Obsolete version of ~ php3... |
phpeqdOLD 9008 | Obsolete version of ~ phpe... |
nndomogOLD 9009 | Obsolete version of ~ nndo... |
snnen2oOLD 9010 | Obsolete version of ~ snne... |
onomeneq 9011 | An ordinal number equinume... |
onomeneqOLD 9012 | Obsolete version of ~ onom... |
onfin 9013 | An ordinal number is finit... |
onfin2 9014 | A set is a natural number ... |
nnfiOLD 9015 | Obsolete version of ~ nnfi... |
nndomo 9016 | Cardinal ordering agrees w... |
nnsdomo 9017 | Cardinal ordering agrees w... |
sucdom 9018 | Strict dominance of a set ... |
sucdomOLD 9019 | Obsolete version of ~ sucd... |
0sdom1dom 9020 | Strict dominance over zero... |
1sdom2 9021 | Ordinal 1 is strictly domi... |
sdom1 9022 | A set has less than one me... |
modom 9023 | Two ways to express "at mo... |
modom2 9024 | Two ways to express "at mo... |
1sdom 9025 | A set that strictly domina... |
snnen2o 9026 | A singleton ` { A } ` is n... |
unxpdomlem1 9027 | Lemma for ~ unxpdom . (Tr... |
unxpdomlem2 9028 | Lemma for ~ unxpdom . (Co... |
unxpdomlem3 9029 | Lemma for ~ unxpdom . (Co... |
unxpdom 9030 | Cartesian product dominate... |
unxpdom2 9031 | Corollary of ~ unxpdom . ... |
sucxpdom 9032 | Cartesian product dominate... |
pssinf 9033 | A set equinumerous to a pr... |
fisseneq 9034 | A finite set is equal to i... |
ominf 9035 | The set of natural numbers... |
isinf 9036 | Any set that is not finite... |
fineqvlem 9037 | Lemma for ~ fineqv . (Con... |
fineqv 9038 | If the Axiom of Infinity i... |
enfiiOLD 9039 | Obsolete version of ~ enfi... |
pssnnOLD 9040 | Obsolete version of ~ pssn... |
xpfir 9041 | The components of a nonemp... |
ssfid 9042 | A subset of a finite set i... |
infi 9043 | The intersection of two se... |
rabfi 9044 | A restricted class built f... |
finresfin 9045 | The restriction of a finit... |
f1finf1o 9046 | Any injection from one fin... |
nfielex 9047 | If a class is not finite, ... |
en1eqsn 9048 | A set with one element is ... |
en1eqsnbi 9049 | A set containing an elemen... |
dif1enALT 9050 | Alternate proof of ~ dif1e... |
enp1ilem 9051 | Lemma for uses of ~ enp1i ... |
enp1i 9052 | Proof induction for ~ en2i... |
en2 9053 | A set equinumerous to ordi... |
en3 9054 | A set equinumerous to ordi... |
en4 9055 | A set equinumerous to ordi... |
findcard2OLD 9056 | Obsolete version of ~ find... |
findcard3 9057 | Schema for strong inductio... |
ac6sfi 9058 | A version of ~ ac6s for fi... |
frfi 9059 | A partial order is well-fo... |
fimax2g 9060 | A finite set has a maximum... |
fimaxg 9061 | A finite set has a maximum... |
fisupg 9062 | Lemma showing existence an... |
wofi 9063 | A total order on a finite ... |
ordunifi 9064 | The maximum of a finite co... |
nnunifi 9065 | The union (supremum) of a ... |
unblem1 9066 | Lemma for ~ unbnn . After... |
unblem2 9067 | Lemma for ~ unbnn . The v... |
unblem3 9068 | Lemma for ~ unbnn . The v... |
unblem4 9069 | Lemma for ~ unbnn . The f... |
unbnn 9070 | Any unbounded subset of na... |
unbnn2 9071 | Version of ~ unbnn that do... |
isfinite2 9072 | Any set strictly dominated... |
nnsdomg 9073 | Omega strictly dominates a... |
isfiniteg 9074 | A set is finite iff it is ... |
infsdomnn 9075 | An infinite set strictly d... |
infn0 9076 | An infinite set is not emp... |
fin2inf 9077 | This (useless) theorem, wh... |
unfilem1 9078 | Lemma for proving that the... |
unfilem2 9079 | Lemma for proving that the... |
unfilem3 9080 | Lemma for proving that the... |
unfiOLD 9081 | Obsolete version of ~ unfi... |
unfir 9082 | If a union is finite, the ... |
unfi2 9083 | The union of two finite se... |
difinf 9084 | An infinite set ` A ` minu... |
xpfi 9085 | The Cartesian product of t... |
3xpfi 9086 | The Cartesian product of t... |
domunfican 9087 | A finite set union cancell... |
infcntss 9088 | Every infinite set has a d... |
prfi 9089 | An unordered pair is finit... |
tpfi 9090 | An unordered triple is fin... |
fiint 9091 | Equivalent ways of stating... |
fodomfi 9092 | An onto function implies d... |
fodomfib 9093 | Equivalence of an onto map... |
fofinf1o 9094 | Any surjection from one fi... |
rneqdmfinf1o 9095 | Any function from a finite... |
fidomdm 9096 | Any finite set dominates i... |
dmfi 9097 | The domain of a finite set... |
fundmfibi 9098 | A function is finite if an... |
resfnfinfin 9099 | The restriction of a funct... |
residfi 9100 | A restricted identity func... |
cnvfiALT 9101 | Shorter proof of ~ cnvfi u... |
rnfi 9102 | The range of a finite set ... |
f1dmvrnfibi 9103 | A one-to-one function whos... |
f1vrnfibi 9104 | A one-to-one function whic... |
fofi 9105 | If a function has a finite... |
f1fi 9106 | If a 1-to-1 function has a... |
iunfi 9107 | The finite union of finite... |
unifi 9108 | The finite union of finite... |
unifi2 9109 | The finite union of finite... |
infssuni 9110 | If an infinite set ` A ` i... |
unirnffid 9111 | The union of the range of ... |
imafiALT 9112 | Shorter proof of ~ imafi u... |
pwfilemOLD 9113 | Obsolete version of ~ pwfi... |
pwfiOLD 9114 | Obsolete version of ~ pwfi... |
mapfi 9115 | Set exponentiation of fini... |
ixpfi 9116 | A Cartesian product of fin... |
ixpfi2 9117 | A Cartesian product of fin... |
mptfi 9118 | A finite mapping set is fi... |
abrexfi 9119 | An image set from a finite... |
cnvimamptfin 9120 | A preimage of a mapping wi... |
elfpw 9121 | Membership in a class of f... |
unifpw 9122 | A set is the union of its ... |
f1opwfi 9123 | A one-to-one mapping induc... |
fissuni 9124 | A finite subset of a union... |
fipreima 9125 | Given a finite subset ` A ... |
finsschain 9126 | A finite subset of the uni... |
indexfi 9127 | If for every element of a ... |
relfsupp 9130 | The property of a function... |
relprcnfsupp 9131 | A proper class is never fi... |
isfsupp 9132 | The property of a class to... |
funisfsupp 9133 | The property of a function... |
fsuppimp 9134 | Implications of a class be... |
fsuppimpd 9135 | A finitely supported funct... |
fisuppfi 9136 | A function on a finite set... |
fdmfisuppfi 9137 | The support of a function ... |
fdmfifsupp 9138 | A function with a finite d... |
fsuppmptdm 9139 | A mapping with a finite do... |
fndmfisuppfi 9140 | The support of a function ... |
fndmfifsupp 9141 | A function with a finite d... |
suppeqfsuppbi 9142 | If two functions have the ... |
suppssfifsupp 9143 | If the support of a functi... |
fsuppsssupp 9144 | If the support of a functi... |
fsuppxpfi 9145 | The cartesian product of t... |
fczfsuppd 9146 | A constant function with v... |
fsuppun 9147 | The union of two finitely ... |
fsuppunfi 9148 | The union of the support o... |
fsuppunbi 9149 | If the union of two classe... |
0fsupp 9150 | The empty set is a finitel... |
snopfsupp 9151 | A singleton containing an ... |
funsnfsupp 9152 | Finite support for a funct... |
fsuppres 9153 | The restriction of a finit... |
ressuppfi 9154 | If the support of the rest... |
resfsupp 9155 | If the restriction of a fu... |
resfifsupp 9156 | The restriction of a funct... |
frnfsuppbi 9157 | Two ways of saying that a ... |
fsuppmptif 9158 | A function mapping an argu... |
sniffsupp 9159 | A function mapping all but... |
fsuppcolem 9160 | Lemma for ~ fsuppco . For... |
fsuppco 9161 | The composition of a 1-1 f... |
fsuppco2 9162 | The composition of a funct... |
fsuppcor 9163 | The composition of a funct... |
mapfienlem1 9164 | Lemma 1 for ~ mapfien . (... |
mapfienlem2 9165 | Lemma 2 for ~ mapfien . (... |
mapfienlem3 9166 | Lemma 3 for ~ mapfien . (... |
mapfien 9167 | A bijection of the base se... |
mapfien2 9168 | Equinumerousity relation f... |
fival 9171 | The set of all the finite ... |
elfi 9172 | Specific properties of an ... |
elfi2 9173 | The empty intersection nee... |
elfir 9174 | Sufficient condition for a... |
intrnfi 9175 | Sufficient condition for t... |
iinfi 9176 | An indexed intersection of... |
inelfi 9177 | The intersection of two se... |
ssfii 9178 | Any element of a set ` A `... |
fi0 9179 | The set of finite intersec... |
fieq0 9180 | A set is empty iff the cla... |
fiin 9181 | The elements of ` ( fi `` ... |
dffi2 9182 | The set of finite intersec... |
fiss 9183 | Subset relationship for fu... |
inficl 9184 | A set which is closed unde... |
fipwuni 9185 | The set of finite intersec... |
fisn 9186 | A singleton is closed unde... |
fiuni 9187 | The union of the finite in... |
fipwss 9188 | If a set is a family of su... |
elfiun 9189 | A finite intersection of e... |
dffi3 9190 | The set of finite intersec... |
fifo 9191 | Describe a surjection from... |
marypha1lem 9192 | Core induction for Philip ... |
marypha1 9193 | (Philip) Hall's marriage t... |
marypha2lem1 9194 | Lemma for ~ marypha2 . Pr... |
marypha2lem2 9195 | Lemma for ~ marypha2 . Pr... |
marypha2lem3 9196 | Lemma for ~ marypha2 . Pr... |
marypha2lem4 9197 | Lemma for ~ marypha2 . Pr... |
marypha2 9198 | Version of ~ marypha1 usin... |
dfsup2 9203 | Quantifier-free definition... |
supeq1 9204 | Equality theorem for supre... |
supeq1d 9205 | Equality deduction for sup... |
supeq1i 9206 | Equality inference for sup... |
supeq2 9207 | Equality theorem for supre... |
supeq3 9208 | Equality theorem for supre... |
supeq123d 9209 | Equality deduction for sup... |
nfsup 9210 | Hypothesis builder for sup... |
supmo 9211 | Any class ` B ` has at mos... |
supexd 9212 | A supremum is a set. (Con... |
supeu 9213 | A supremum is unique. Sim... |
supval2 9214 | Alternate expression for t... |
eqsup 9215 | Sufficient condition for a... |
eqsupd 9216 | Sufficient condition for a... |
supcl 9217 | A supremum belongs to its ... |
supub 9218 | A supremum is an upper bou... |
suplub 9219 | A supremum is the least up... |
suplub2 9220 | Bidirectional form of ~ su... |
supnub 9221 | An upper bound is not less... |
supex 9222 | A supremum is a set. (Con... |
sup00 9223 | The supremum under an empt... |
sup0riota 9224 | The supremum of an empty s... |
sup0 9225 | The supremum of an empty s... |
supmax 9226 | The greatest element of a ... |
fisup2g 9227 | A finite set satisfies the... |
fisupcl 9228 | A nonempty finite set cont... |
supgtoreq 9229 | The supremum of a finite s... |
suppr 9230 | The supremum of a pair. (... |
supsn 9231 | The supremum of a singleto... |
supisolem 9232 | Lemma for ~ supiso . (Con... |
supisoex 9233 | Lemma for ~ supiso . (Con... |
supiso 9234 | Image of a supremum under ... |
infeq1 9235 | Equality theorem for infim... |
infeq1d 9236 | Equality deduction for inf... |
infeq1i 9237 | Equality inference for inf... |
infeq2 9238 | Equality theorem for infim... |
infeq3 9239 | Equality theorem for infim... |
infeq123d 9240 | Equality deduction for inf... |
nfinf 9241 | Hypothesis builder for inf... |
infexd 9242 | An infimum is a set. (Con... |
eqinf 9243 | Sufficient condition for a... |
eqinfd 9244 | Sufficient condition for a... |
infval 9245 | Alternate expression for t... |
infcllem 9246 | Lemma for ~ infcl , ~ infl... |
infcl 9247 | An infimum belongs to its ... |
inflb 9248 | An infimum is a lower boun... |
infglb 9249 | An infimum is the greatest... |
infglbb 9250 | Bidirectional form of ~ in... |
infnlb 9251 | A lower bound is not great... |
infex 9252 | An infimum is a set. (Con... |
infmin 9253 | The smallest element of a ... |
infmo 9254 | Any class ` B ` has at mos... |
infeu 9255 | An infimum is unique. (Co... |
fimin2g 9256 | A finite set has a minimum... |
fiming 9257 | A finite set has a minimum... |
fiinfg 9258 | Lemma showing existence an... |
fiinf2g 9259 | A finite set satisfies the... |
fiinfcl 9260 | A nonempty finite set cont... |
infltoreq 9261 | The infimum of a finite se... |
infpr 9262 | The infimum of a pair. (C... |
infsupprpr 9263 | The infimum of a proper pa... |
infsn 9264 | The infimum of a singleton... |
inf00 9265 | The infimum regarding an e... |
infempty 9266 | The infimum of an empty se... |
infiso 9267 | Image of an infimum under ... |
dfoi 9270 | Rewrite ~ df-oi with abbre... |
oieq1 9271 | Equality theorem for ordin... |
oieq2 9272 | Equality theorem for ordin... |
nfoi 9273 | Hypothesis builder for ord... |
ordiso2 9274 | Generalize ~ ordiso to pro... |
ordiso 9275 | Order-isomorphic ordinal n... |
ordtypecbv 9276 | Lemma for ~ ordtype . (Co... |
ordtypelem1 9277 | Lemma for ~ ordtype . (Co... |
ordtypelem2 9278 | Lemma for ~ ordtype . (Co... |
ordtypelem3 9279 | Lemma for ~ ordtype . (Co... |
ordtypelem4 9280 | Lemma for ~ ordtype . (Co... |
ordtypelem5 9281 | Lemma for ~ ordtype . (Co... |
ordtypelem6 9282 | Lemma for ~ ordtype . (Co... |
ordtypelem7 9283 | Lemma for ~ ordtype . ` ra... |
ordtypelem8 9284 | Lemma for ~ ordtype . (Co... |
ordtypelem9 9285 | Lemma for ~ ordtype . Eit... |
ordtypelem10 9286 | Lemma for ~ ordtype . Usi... |
oi0 9287 | Definition of the ordinal ... |
oicl 9288 | The order type of the well... |
oif 9289 | The order isomorphism of t... |
oiiso2 9290 | The order isomorphism of t... |
ordtype 9291 | For any set-like well-orde... |
oiiniseg 9292 | ` ran F ` is an initial se... |
ordtype2 9293 | For any set-like well-orde... |
oiexg 9294 | The order isomorphism on a... |
oion 9295 | The order type of the well... |
oiiso 9296 | The order isomorphism of t... |
oien 9297 | The order type of a well-o... |
oieu 9298 | Uniqueness of the unique o... |
oismo 9299 | When ` A ` is a subclass o... |
oiid 9300 | The order type of an ordin... |
hartogslem1 9301 | Lemma for ~ hartogs . (Co... |
hartogslem2 9302 | Lemma for ~ hartogs . (Co... |
hartogs 9303 | The class of ordinals domi... |
wofib 9304 | The only sets which are we... |
wemaplem1 9305 | Value of the lexicographic... |
wemaplem2 9306 | Lemma for ~ wemapso . Tra... |
wemaplem3 9307 | Lemma for ~ wemapso . Tra... |
wemappo 9308 | Construct lexicographic or... |
wemapsolem 9309 | Lemma for ~ wemapso . (Co... |
wemapso 9310 | Construct lexicographic or... |
wemapso2lem 9311 | Lemma for ~ wemapso2 . (C... |
wemapso2 9312 | An alternative to having a... |
card2on 9313 | The alternate definition o... |
card2inf 9314 | The alternate definition o... |
harf 9317 | Functionality of the Harto... |
harcl 9318 | Values of the Hartogs func... |
harval 9319 | Function value of the Hart... |
elharval 9320 | The Hartogs number of a se... |
harndom 9321 | The Hartogs number of a se... |
harword 9322 | Weak ordering property of ... |
relwdom 9325 | Weak dominance is a relati... |
brwdom 9326 | Property of weak dominance... |
brwdomi 9327 | Property of weak dominance... |
brwdomn0 9328 | Weak dominance over nonemp... |
0wdom 9329 | Any set weakly dominates t... |
fowdom 9330 | An onto function implies w... |
wdomref 9331 | Reflexivity of weak domina... |
brwdom2 9332 | Alternate characterization... |
domwdom 9333 | Weak dominance is implied ... |
wdomtr 9334 | Transitivity of weak domin... |
wdomen1 9335 | Equality-like theorem for ... |
wdomen2 9336 | Equality-like theorem for ... |
wdompwdom 9337 | Weak dominance strengthens... |
canthwdom 9338 | Cantor's Theorem, stated u... |
wdom2d 9339 | Deduce weak dominance from... |
wdomd 9340 | Deduce weak dominance from... |
brwdom3 9341 | Condition for weak dominan... |
brwdom3i 9342 | Weak dominance implies exi... |
unwdomg 9343 | Weak dominance of a (disjo... |
xpwdomg 9344 | Weak dominance of a Cartes... |
wdomima2g 9345 | A set is weakly dominant o... |
wdomimag 9346 | A set is weakly dominant o... |
unxpwdom2 9347 | Lemma for ~ unxpwdom . (C... |
unxpwdom 9348 | If a Cartesian product is ... |
ixpiunwdom 9349 | Describe an onto function ... |
harwdom 9350 | The value of the Hartogs f... |
axreg2 9352 | Axiom of Regularity expres... |
zfregcl 9353 | The Axiom of Regularity wi... |
zfreg 9354 | The Axiom of Regularity us... |
elirrv 9355 | The membership relation is... |
elirr 9356 | No class is a member of it... |
elneq 9357 | A class is not equal to an... |
nelaneq 9358 | A class is not an element ... |
epinid0 9359 | The membership relation an... |
sucprcreg 9360 | A class is equal to its su... |
ruv 9361 | The Russell class is equal... |
ruALT 9362 | Alternate proof of ~ ru , ... |
zfregfr 9363 | The membership relation is... |
en2lp 9364 | No class has 2-cycle membe... |
elnanel 9365 | Two classes are not elemen... |
cnvepnep 9366 | The membership (epsilon) r... |
epnsym 9367 | The membership (epsilon) r... |
elnotel 9368 | A class cannot be an eleme... |
elnel 9369 | A class cannot be an eleme... |
en3lplem1 9370 | Lemma for ~ en3lp . (Cont... |
en3lplem2 9371 | Lemma for ~ en3lp . (Cont... |
en3lp 9372 | No class has 3-cycle membe... |
preleqg 9373 | Equality of two unordered ... |
preleq 9374 | Equality of two unordered ... |
preleqALT 9375 | Alternate proof of ~ prele... |
opthreg 9376 | Theorem for alternate repr... |
suc11reg 9377 | The successor operation be... |
dford2 9378 | Assuming ~ ax-reg , an ord... |
inf0 9379 | Existence of ` _om ` impli... |
inf1 9380 | Variation of Axiom of Infi... |
inf2 9381 | Variation of Axiom of Infi... |
inf3lema 9382 | Lemma for our Axiom of Inf... |
inf3lemb 9383 | Lemma for our Axiom of Inf... |
inf3lemc 9384 | Lemma for our Axiom of Inf... |
inf3lemd 9385 | Lemma for our Axiom of Inf... |
inf3lem1 9386 | Lemma for our Axiom of Inf... |
inf3lem2 9387 | Lemma for our Axiom of Inf... |
inf3lem3 9388 | Lemma for our Axiom of Inf... |
inf3lem4 9389 | Lemma for our Axiom of Inf... |
inf3lem5 9390 | Lemma for our Axiom of Inf... |
inf3lem6 9391 | Lemma for our Axiom of Inf... |
inf3lem7 9392 | Lemma for our Axiom of Inf... |
inf3 9393 | Our Axiom of Infinity ~ ax... |
infeq5i 9394 | Half of ~ infeq5 . (Contr... |
infeq5 9395 | The statement "there exist... |
zfinf 9397 | Axiom of Infinity expresse... |
axinf2 9398 | A standard version of Axio... |
zfinf2 9400 | A standard version of the ... |
omex 9401 | The existence of omega (th... |
axinf 9402 | The first version of the A... |
inf5 9403 | The statement "there exist... |
omelon 9404 | Omega is an ordinal number... |
dfom3 9405 | The class of natural numbe... |
elom3 9406 | A simplification of ~ elom... |
dfom4 9407 | A simplification of ~ df-o... |
dfom5 9408 | ` _om ` is the smallest li... |
oancom 9409 | Ordinal addition is not co... |
isfinite 9410 | A set is finite iff it is ... |
fict 9411 | A finite set is countable ... |
nnsdom 9412 | A natural number is strict... |
omenps 9413 | Omega is equinumerous to a... |
omensuc 9414 | The set of natural numbers... |
infdifsn 9415 | Removing a singleton from ... |
infdiffi 9416 | Removing a finite set from... |
unbnn3 9417 | Any unbounded subset of na... |
noinfep 9418 | Using the Axiom of Regular... |
cantnffval 9421 | The value of the Cantor no... |
cantnfdm 9422 | The domain of the Cantor n... |
cantnfvalf 9423 | Lemma for ~ cantnf . The ... |
cantnfs 9424 | Elementhood in the set of ... |
cantnfcl 9425 | Basic properties of the or... |
cantnfval 9426 | The value of the Cantor no... |
cantnfval2 9427 | Alternate expression for t... |
cantnfsuc 9428 | The value of the recursive... |
cantnfle 9429 | A lower bound on the ` CNF... |
cantnflt 9430 | An upper bound on the part... |
cantnflt2 9431 | An upper bound on the ` CN... |
cantnff 9432 | The ` CNF ` function is a ... |
cantnf0 9433 | The value of the zero func... |
cantnfrescl 9434 | A function is finitely sup... |
cantnfres 9435 | The ` CNF ` function respe... |
cantnfp1lem1 9436 | Lemma for ~ cantnfp1 . (C... |
cantnfp1lem2 9437 | Lemma for ~ cantnfp1 . (C... |
cantnfp1lem3 9438 | Lemma for ~ cantnfp1 . (C... |
cantnfp1 9439 | If ` F ` is created by add... |
oemapso 9440 | The relation ` T ` is a st... |
oemapval 9441 | Value of the relation ` T ... |
oemapvali 9442 | If ` F < G ` , then there ... |
cantnflem1a 9443 | Lemma for ~ cantnf . (Con... |
cantnflem1b 9444 | Lemma for ~ cantnf . (Con... |
cantnflem1c 9445 | Lemma for ~ cantnf . (Con... |
cantnflem1d 9446 | Lemma for ~ cantnf . (Con... |
cantnflem1 9447 | Lemma for ~ cantnf . This... |
cantnflem2 9448 | Lemma for ~ cantnf . (Con... |
cantnflem3 9449 | Lemma for ~ cantnf . Here... |
cantnflem4 9450 | Lemma for ~ cantnf . Comp... |
cantnf 9451 | The Cantor Normal Form the... |
oemapwe 9452 | The lexicographic order on... |
cantnffval2 9453 | An alternate definition of... |
cantnff1o 9454 | Simplify the isomorphism o... |
wemapwe 9455 | Construct lexicographic or... |
oef1o 9456 | A bijection of the base se... |
cnfcomlem 9457 | Lemma for ~ cnfcom . (Con... |
cnfcom 9458 | Any ordinal ` B ` is equin... |
cnfcom2lem 9459 | Lemma for ~ cnfcom2 . (Co... |
cnfcom2 9460 | Any nonzero ordinal ` B ` ... |
cnfcom3lem 9461 | Lemma for ~ cnfcom3 . (Co... |
cnfcom3 9462 | Any infinite ordinal ` B `... |
cnfcom3clem 9463 | Lemma for ~ cnfcom3c . (C... |
cnfcom3c 9464 | Wrap the construction of ~... |
ttrcleq 9467 | Equality theorem for trans... |
nfttrcld 9468 | Bound variable hypothesis ... |
nfttrcl 9469 | Bound variable hypothesis ... |
relttrcl 9470 | The transitive closure of ... |
brttrcl 9471 | Characterization of elemen... |
brttrcl2 9472 | Characterization of elemen... |
ssttrcl 9473 | If ` R ` is a relation, th... |
ttrcltr 9474 | The transitive closure of ... |
ttrclresv 9475 | The transitive closure of ... |
ttrclco 9476 | Composition law for the tr... |
cottrcl 9477 | Composition law for the tr... |
ttrclss 9478 | If ` R ` is a subclass of ... |
dmttrcl 9479 | The domain of a transitive... |
rnttrcl 9480 | The range of a transitive ... |
ttrclexg 9481 | If ` R ` is a set, then so... |
dfttrcl2 9482 | When ` R ` is a set and a ... |
ttrclselem1 9483 | Lemma for ~ ttrclse . Sho... |
ttrclselem2 9484 | Lemma for ~ ttrclse . Sho... |
ttrclse 9485 | If ` R ` is set-like over ... |
trcl 9486 | For any set ` A ` , show t... |
tz9.1 9487 | Every set has a transitive... |
tz9.1c 9488 | Alternate expression for t... |
epfrs 9489 | The strong form of the Axi... |
zfregs 9490 | The strong form of the Axi... |
zfregs2 9491 | Alternate strong form of t... |
setind 9492 | Set (epsilon) induction. ... |
setind2 9493 | Set (epsilon) induction, s... |
tcvalg 9496 | Value of the transitive cl... |
tcid 9497 | Defining property of the t... |
tctr 9498 | Defining property of the t... |
tcmin 9499 | Defining property of the t... |
tc2 9500 | A variant of the definitio... |
tcsni 9501 | The transitive closure of ... |
tcss 9502 | The transitive closure fun... |
tcel 9503 | The transitive closure fun... |
tcidm 9504 | The transitive closure fun... |
tc0 9505 | The transitive closure of ... |
tc00 9506 | The transitive closure is ... |
frmin 9507 | Every (possibly proper) su... |
frind 9508 | A subclass of a well-found... |
frinsg 9509 | Well-Founded Induction Sch... |
frins 9510 | Well-Founded Induction Sch... |
frins2f 9511 | Well-Founded Induction sch... |
frins2 9512 | Well-Founded Induction sch... |
frins3 9513 | Well-Founded Induction sch... |
frr3g 9514 | Functions defined by well-... |
frrlem15 9515 | Lemma for general well-fou... |
frrlem16 9516 | Lemma for general well-fou... |
frr1 9517 | Law of general well-founde... |
frr2 9518 | Law of general well-founde... |
frr3 9519 | Law of general well-founde... |
r1funlim 9524 | The cumulative hierarchy o... |
r1fnon 9525 | The cumulative hierarchy o... |
r10 9526 | Value of the cumulative hi... |
r1sucg 9527 | Value of the cumulative hi... |
r1suc 9528 | Value of the cumulative hi... |
r1limg 9529 | Value of the cumulative hi... |
r1lim 9530 | Value of the cumulative hi... |
r1fin 9531 | The first ` _om ` levels o... |
r1sdom 9532 | Each stage in the cumulati... |
r111 9533 | The cumulative hierarchy i... |
r1tr 9534 | The cumulative hierarchy o... |
r1tr2 9535 | The union of a cumulative ... |
r1ordg 9536 | Ordering relation for the ... |
r1ord3g 9537 | Ordering relation for the ... |
r1ord 9538 | Ordering relation for the ... |
r1ord2 9539 | Ordering relation for the ... |
r1ord3 9540 | Ordering relation for the ... |
r1sssuc 9541 | The value of the cumulativ... |
r1pwss 9542 | Each set of the cumulative... |
r1sscl 9543 | Each set of the cumulative... |
r1val1 9544 | The value of the cumulativ... |
tz9.12lem1 9545 | Lemma for ~ tz9.12 . (Con... |
tz9.12lem2 9546 | Lemma for ~ tz9.12 . (Con... |
tz9.12lem3 9547 | Lemma for ~ tz9.12 . (Con... |
tz9.12 9548 | A set is well-founded if a... |
tz9.13 9549 | Every set is well-founded,... |
tz9.13g 9550 | Every set is well-founded,... |
rankwflemb 9551 | Two ways of saying a set i... |
rankf 9552 | The domain and range of th... |
rankon 9553 | The rank of a set is an or... |
r1elwf 9554 | Any member of the cumulati... |
rankvalb 9555 | Value of the rank function... |
rankr1ai 9556 | One direction of ~ rankr1a... |
rankvaln 9557 | Value of the rank function... |
rankidb 9558 | Identity law for the rank ... |
rankdmr1 9559 | A rank is a member of the ... |
rankr1ag 9560 | A version of ~ rankr1a tha... |
rankr1bg 9561 | A relationship between ran... |
r1rankidb 9562 | Any set is a subset of the... |
r1elssi 9563 | The range of the ` R1 ` fu... |
r1elss 9564 | The range of the ` R1 ` fu... |
pwwf 9565 | A power set is well-founde... |
sswf 9566 | A subset of a well-founded... |
snwf 9567 | A singleton is well-founde... |
unwf 9568 | A binary union is well-fou... |
prwf 9569 | An unordered pair is well-... |
opwf 9570 | An ordered pair is well-fo... |
unir1 9571 | The cumulative hierarchy o... |
jech9.3 9572 | Every set belongs to some ... |
rankwflem 9573 | Every set is well-founded,... |
rankval 9574 | Value of the rank function... |
rankvalg 9575 | Value of the rank function... |
rankval2 9576 | Value of an alternate defi... |
uniwf 9577 | A union is well-founded if... |
rankr1clem 9578 | Lemma for ~ rankr1c . (Co... |
rankr1c 9579 | A relationship between the... |
rankidn 9580 | A relationship between the... |
rankpwi 9581 | The rank of a power set. ... |
rankelb 9582 | The membership relation is... |
wfelirr 9583 | A well-founded set is not ... |
rankval3b 9584 | The value of the rank func... |
ranksnb 9585 | The rank of a singleton. ... |
rankonidlem 9586 | Lemma for ~ rankonid . (C... |
rankonid 9587 | The rank of an ordinal num... |
onwf 9588 | The ordinals are all well-... |
onssr1 9589 | Initial segments of the or... |
rankr1g 9590 | A relationship between the... |
rankid 9591 | Identity law for the rank ... |
rankr1 9592 | A relationship between the... |
ssrankr1 9593 | A relationship between an ... |
rankr1a 9594 | A relationship between ran... |
r1val2 9595 | The value of the cumulativ... |
r1val3 9596 | The value of the cumulativ... |
rankel 9597 | The membership relation is... |
rankval3 9598 | The value of the rank func... |
bndrank 9599 | Any class whose elements h... |
unbndrank 9600 | The elements of a proper c... |
rankpw 9601 | The rank of a power set. ... |
ranklim 9602 | The rank of a set belongs ... |
r1pw 9603 | A stronger property of ` R... |
r1pwALT 9604 | Alternate shorter proof of... |
r1pwcl 9605 | The cumulative hierarchy o... |
rankssb 9606 | The subset relation is inh... |
rankss 9607 | The subset relation is inh... |
rankunb 9608 | The rank of the union of t... |
rankprb 9609 | The rank of an unordered p... |
rankopb 9610 | The rank of an ordered pai... |
rankuni2b 9611 | The value of the rank func... |
ranksn 9612 | The rank of a singleton. ... |
rankuni2 9613 | The rank of a union. Part... |
rankun 9614 | The rank of the union of t... |
rankpr 9615 | The rank of an unordered p... |
rankop 9616 | The rank of an ordered pai... |
r1rankid 9617 | Any set is a subset of the... |
rankeq0b 9618 | A set is empty iff its ran... |
rankeq0 9619 | A set is empty iff its ran... |
rankr1id 9620 | The rank of the hierarchy ... |
rankuni 9621 | The rank of a union. Part... |
rankr1b 9622 | A relationship between ran... |
ranksuc 9623 | The rank of a successor. ... |
rankuniss 9624 | Upper bound of the rank of... |
rankval4 9625 | The rank of a set is the s... |
rankbnd 9626 | The rank of a set is bound... |
rankbnd2 9627 | The rank of a set is bound... |
rankc1 9628 | A relationship that can be... |
rankc2 9629 | A relationship that can be... |
rankelun 9630 | Rank membership is inherit... |
rankelpr 9631 | Rank membership is inherit... |
rankelop 9632 | Rank membership is inherit... |
rankxpl 9633 | A lower bound on the rank ... |
rankxpu 9634 | An upper bound on the rank... |
rankfu 9635 | An upper bound on the rank... |
rankmapu 9636 | An upper bound on the rank... |
rankxplim 9637 | The rank of a Cartesian pr... |
rankxplim2 9638 | If the rank of a Cartesian... |
rankxplim3 9639 | The rank of a Cartesian pr... |
rankxpsuc 9640 | The rank of a Cartesian pr... |
tcwf 9641 | The transitive closure fun... |
tcrank 9642 | This theorem expresses two... |
scottex 9643 | Scott's trick collects all... |
scott0 9644 | Scott's trick collects all... |
scottexs 9645 | Theorem scheme version of ... |
scott0s 9646 | Theorem scheme version of ... |
cplem1 9647 | Lemma for the Collection P... |
cplem2 9648 | Lemma for the Collection P... |
cp 9649 | Collection Principle. Thi... |
bnd 9650 | A very strong generalizati... |
bnd2 9651 | A variant of the Boundedne... |
kardex 9652 | The collection of all sets... |
karden 9653 | If we allow the Axiom of R... |
htalem 9654 | Lemma for defining an emul... |
hta 9655 | A ZFC emulation of Hilbert... |
djueq12 9662 | Equality theorem for disjo... |
djueq1 9663 | Equality theorem for disjo... |
djueq2 9664 | Equality theorem for disjo... |
nfdju 9665 | Bound-variable hypothesis ... |
djuex 9666 | The disjoint union of sets... |
djuexb 9667 | The disjoint union of two ... |
djulcl 9668 | Left closure of disjoint u... |
djurcl 9669 | Right closure of disjoint ... |
djulf1o 9670 | The left injection functio... |
djurf1o 9671 | The right injection functi... |
inlresf 9672 | The left injection restric... |
inlresf1 9673 | The left injection restric... |
inrresf 9674 | The right injection restri... |
inrresf1 9675 | The right injection restri... |
djuin 9676 | The images of any classes ... |
djur 9677 | A member of a disjoint uni... |
djuss 9678 | A disjoint union is a subc... |
djuunxp 9679 | The union of a disjoint un... |
djuexALT 9680 | Alternate proof of ~ djuex... |
eldju1st 9681 | The first component of an ... |
eldju2ndl 9682 | The second component of an... |
eldju2ndr 9683 | The second component of an... |
djuun 9684 | The disjoint union of two ... |
1stinl 9685 | The first component of the... |
2ndinl 9686 | The second component of th... |
1stinr 9687 | The first component of the... |
2ndinr 9688 | The second component of th... |
updjudhf 9689 | The mapping of an element ... |
updjudhcoinlf 9690 | The composition of the map... |
updjudhcoinrg 9691 | The composition of the map... |
updjud 9692 | Universal property of the ... |
cardf2 9701 | The cardinality function i... |
cardon 9702 | The cardinal number of a s... |
isnum2 9703 | A way to express well-orde... |
isnumi 9704 | A set equinumerous to an o... |
ennum 9705 | Equinumerous sets are equi... |
finnum 9706 | Every finite set is numera... |
onenon 9707 | Every ordinal number is nu... |
tskwe 9708 | A Tarski set is well-order... |
xpnum 9709 | The cartesian product of n... |
cardval3 9710 | An alternate definition of... |
cardid2 9711 | Any numerable set is equin... |
isnum3 9712 | A set is numerable iff it ... |
oncardval 9713 | The value of the cardinal ... |
oncardid 9714 | Any ordinal number is equi... |
cardonle 9715 | The cardinal of an ordinal... |
card0 9716 | The cardinality of the emp... |
cardidm 9717 | The cardinality function i... |
oncard 9718 | A set is a cardinal number... |
ficardom 9719 | The cardinal number of a f... |
ficardid 9720 | A finite set is equinumero... |
cardnn 9721 | The cardinality of a natur... |
cardnueq0 9722 | The empty set is the only ... |
cardne 9723 | No member of a cardinal nu... |
carden2a 9724 | If two sets have equal non... |
carden2b 9725 | If two sets are equinumero... |
card1 9726 | A set has cardinality one ... |
cardsn 9727 | A singleton has cardinalit... |
carddomi2 9728 | Two sets have the dominanc... |
sdomsdomcardi 9729 | A set strictly dominates i... |
cardlim 9730 | An infinite cardinal is a ... |
cardsdomelir 9731 | A cardinal strictly domina... |
cardsdomel 9732 | A cardinal strictly domina... |
iscard 9733 | Two ways to express the pr... |
iscard2 9734 | Two ways to express the pr... |
carddom2 9735 | Two numerable sets have th... |
harcard 9736 | The class of ordinal numbe... |
cardprclem 9737 | Lemma for ~ cardprc . (Co... |
cardprc 9738 | The class of all cardinal ... |
carduni 9739 | The union of a set of card... |
cardiun 9740 | The indexed union of a set... |
cardennn 9741 | If ` A ` is equinumerous t... |
cardsucinf 9742 | The cardinality of the suc... |
cardsucnn 9743 | The cardinality of the suc... |
cardom 9744 | The set of natural numbers... |
carden2 9745 | Two numerable sets are equ... |
cardsdom2 9746 | A numerable set is strictl... |
domtri2 9747 | Trichotomy of dominance fo... |
nnsdomel 9748 | Strict dominance and eleme... |
cardval2 9749 | An alternate version of th... |
isinffi 9750 | An infinite set contains s... |
fidomtri 9751 | Trichotomy of dominance wi... |
fidomtri2 9752 | Trichotomy of dominance wi... |
harsdom 9753 | The Hartogs number of a we... |
onsdom 9754 | Any well-orderable set is ... |
harval2 9755 | An alternate expression fo... |
harsucnn 9756 | The next cardinal after a ... |
cardmin2 9757 | The smallest ordinal that ... |
pm54.43lem 9758 | In Theorem *54.43 of [Whit... |
pm54.43 9759 | Theorem *54.43 of [Whitehe... |
pr2nelem 9760 | Lemma for ~ pr2ne . (Cont... |
pr2ne 9761 | If an unordered pair has t... |
prdom2 9762 | An unordered pair has at m... |
en2eqpr 9763 | Building a set with two el... |
en2eleq 9764 | Express a set of pair card... |
en2other2 9765 | Taking the other element t... |
dif1card 9766 | The cardinality of a nonem... |
leweon 9767 | Lexicographical order is a... |
r0weon 9768 | A set-like well-ordering o... |
infxpenlem 9769 | Lemma for ~ infxpen . (Co... |
infxpen 9770 | Every infinite ordinal is ... |
xpomen 9771 | The Cartesian product of o... |
xpct 9772 | The cartesian product of t... |
infxpidm2 9773 | Every infinite well-ordera... |
infxpenc 9774 | A canonical version of ~ i... |
infxpenc2lem1 9775 | Lemma for ~ infxpenc2 . (... |
infxpenc2lem2 9776 | Lemma for ~ infxpenc2 . (... |
infxpenc2lem3 9777 | Lemma for ~ infxpenc2 . (... |
infxpenc2 9778 | Existence form of ~ infxpe... |
iunmapdisj 9779 | The union ` U_ n e. C ( A ... |
fseqenlem1 9780 | Lemma for ~ fseqen . (Con... |
fseqenlem2 9781 | Lemma for ~ fseqen . (Con... |
fseqdom 9782 | One half of ~ fseqen . (C... |
fseqen 9783 | A set that is equinumerous... |
infpwfidom 9784 | The collection of finite s... |
dfac8alem 9785 | Lemma for ~ dfac8a . If t... |
dfac8a 9786 | Numeration theorem: every ... |
dfac8b 9787 | The well-ordering theorem:... |
dfac8clem 9788 | Lemma for ~ dfac8c . (Con... |
dfac8c 9789 | If the union of a set is w... |
ac10ct 9790 | A proof of the well-orderi... |
ween 9791 | A set is numerable iff it ... |
ac5num 9792 | A version of ~ ac5b with t... |
ondomen 9793 | If a set is dominated by a... |
numdom 9794 | A set dominated by a numer... |
ssnum 9795 | A subset of a numerable se... |
onssnum 9796 | All subsets of the ordinal... |
indcardi 9797 | Indirect strong induction ... |
acnrcl 9798 | Reverse closure for the ch... |
acneq 9799 | Equality theorem for the c... |
isacn 9800 | The property of being a ch... |
acni 9801 | The property of being a ch... |
acni2 9802 | The property of being a ch... |
acni3 9803 | The property of being a ch... |
acnlem 9804 | Construct a mapping satisf... |
numacn 9805 | A well-orderable set has c... |
finacn 9806 | Every set has finite choic... |
acndom 9807 | A set with long choice seq... |
acnnum 9808 | A set ` X ` which has choi... |
acnen 9809 | The class of choice sets o... |
acndom2 9810 | A set smaller than one wit... |
acnen2 9811 | The class of sets with cho... |
fodomacn 9812 | A version of ~ fodom that ... |
fodomnum 9813 | A version of ~ fodom that ... |
fonum 9814 | A surjection maps numerabl... |
numwdom 9815 | A surjection maps numerabl... |
fodomfi2 9816 | Onto functions define domi... |
wdomfil 9817 | Weak dominance agrees with... |
infpwfien 9818 | Any infinite well-orderabl... |
inffien 9819 | The set of finite intersec... |
wdomnumr 9820 | Weak dominance agrees with... |
alephfnon 9821 | The aleph function is a fu... |
aleph0 9822 | The first infinite cardina... |
alephlim 9823 | Value of the aleph functio... |
alephsuc 9824 | Value of the aleph functio... |
alephon 9825 | An aleph is an ordinal num... |
alephcard 9826 | Every aleph is a cardinal ... |
alephnbtwn 9827 | No cardinal can be sandwic... |
alephnbtwn2 9828 | No set has equinumerosity ... |
alephordilem1 9829 | Lemma for ~ alephordi . (... |
alephordi 9830 | Strict ordering property o... |
alephord 9831 | Ordering property of the a... |
alephord2 9832 | Ordering property of the a... |
alephord2i 9833 | Ordering property of the a... |
alephord3 9834 | Ordering property of the a... |
alephsucdom 9835 | A set dominated by an alep... |
alephsuc2 9836 | An alternate representatio... |
alephdom 9837 | Relationship between inclu... |
alephgeom 9838 | Every aleph is greater tha... |
alephislim 9839 | Every aleph is a limit ord... |
aleph11 9840 | The aleph function is one-... |
alephf1 9841 | The aleph function is a on... |
alephsdom 9842 | If an ordinal is smaller t... |
alephdom2 9843 | A dominated initial ordina... |
alephle 9844 | The argument of the aleph ... |
cardaleph 9845 | Given any transfinite card... |
cardalephex 9846 | Every transfinite cardinal... |
infenaleph 9847 | An infinite numerable set ... |
isinfcard 9848 | Two ways to express the pr... |
iscard3 9849 | Two ways to express the pr... |
cardnum 9850 | Two ways to express the cl... |
alephinit 9851 | An infinite initial ordina... |
carduniima 9852 | The union of the image of ... |
cardinfima 9853 | If a mapping to cardinals ... |
alephiso 9854 | Aleph is an order isomorph... |
alephprc 9855 | The class of all transfini... |
alephsson 9856 | The class of transfinite c... |
unialeph 9857 | The union of the class of ... |
alephsmo 9858 | The aleph function is stri... |
alephf1ALT 9859 | Alternate proof of ~ aleph... |
alephfplem1 9860 | Lemma for ~ alephfp . (Co... |
alephfplem2 9861 | Lemma for ~ alephfp . (Co... |
alephfplem3 9862 | Lemma for ~ alephfp . (Co... |
alephfplem4 9863 | Lemma for ~ alephfp . (Co... |
alephfp 9864 | The aleph function has a f... |
alephfp2 9865 | The aleph function has at ... |
alephval3 9866 | An alternate way to expres... |
alephsucpw2 9867 | The power set of an aleph ... |
mappwen 9868 | Power rule for cardinal ar... |
finnisoeu 9869 | A finite totally ordered s... |
iunfictbso 9870 | Countability of a countabl... |
aceq1 9873 | Equivalence of two version... |
aceq0 9874 | Equivalence of two version... |
aceq2 9875 | Equivalence of two version... |
aceq3lem 9876 | Lemma for ~ dfac3 . (Cont... |
dfac3 9877 | Equivalence of two version... |
dfac4 9878 | Equivalence of two version... |
dfac5lem1 9879 | Lemma for ~ dfac5 . (Cont... |
dfac5lem2 9880 | Lemma for ~ dfac5 . (Cont... |
dfac5lem3 9881 | Lemma for ~ dfac5 . (Cont... |
dfac5lem4 9882 | Lemma for ~ dfac5 . (Cont... |
dfac5lem5 9883 | Lemma for ~ dfac5 . (Cont... |
dfac5 9884 | Equivalence of two version... |
dfac2a 9885 | Our Axiom of Choice (in th... |
dfac2b 9886 | Axiom of Choice (first for... |
dfac2 9887 | Axiom of Choice (first for... |
dfac7 9888 | Equivalence of the Axiom o... |
dfac0 9889 | Equivalence of two version... |
dfac1 9890 | Equivalence of two version... |
dfac8 9891 | A proof of the equivalency... |
dfac9 9892 | Equivalence of the axiom o... |
dfac10 9893 | Axiom of Choice equivalent... |
dfac10c 9894 | Axiom of Choice equivalent... |
dfac10b 9895 | Axiom of Choice equivalent... |
acacni 9896 | A choice equivalent: every... |
dfacacn 9897 | A choice equivalent: every... |
dfac13 9898 | The axiom of choice holds ... |
dfac12lem1 9899 | Lemma for ~ dfac12 . (Con... |
dfac12lem2 9900 | Lemma for ~ dfac12 . (Con... |
dfac12lem3 9901 | Lemma for ~ dfac12 . (Con... |
dfac12r 9902 | The axiom of choice holds ... |
dfac12k 9903 | Equivalence of ~ dfac12 an... |
dfac12a 9904 | The axiom of choice holds ... |
dfac12 9905 | The axiom of choice holds ... |
kmlem1 9906 | Lemma for 5-quantifier AC ... |
kmlem2 9907 | Lemma for 5-quantifier AC ... |
kmlem3 9908 | Lemma for 5-quantifier AC ... |
kmlem4 9909 | Lemma for 5-quantifier AC ... |
kmlem5 9910 | Lemma for 5-quantifier AC ... |
kmlem6 9911 | Lemma for 5-quantifier AC ... |
kmlem7 9912 | Lemma for 5-quantifier AC ... |
kmlem8 9913 | Lemma for 5-quantifier AC ... |
kmlem9 9914 | Lemma for 5-quantifier AC ... |
kmlem10 9915 | Lemma for 5-quantifier AC ... |
kmlem11 9916 | Lemma for 5-quantifier AC ... |
kmlem12 9917 | Lemma for 5-quantifier AC ... |
kmlem13 9918 | Lemma for 5-quantifier AC ... |
kmlem14 9919 | Lemma for 5-quantifier AC ... |
kmlem15 9920 | Lemma for 5-quantifier AC ... |
kmlem16 9921 | Lemma for 5-quantifier AC ... |
dfackm 9922 | Equivalence of the Axiom o... |
undjudom 9923 | Cardinal addition dominate... |
endjudisj 9924 | Equinumerosity of a disjoi... |
djuen 9925 | Disjoint unions of equinum... |
djuenun 9926 | Disjoint union is equinume... |
dju1en 9927 | Cardinal addition with car... |
dju1dif 9928 | Adding and subtracting one... |
dju1p1e2 9929 | 1+1=2 for cardinal number ... |
dju1p1e2ALT 9930 | Alternate proof of ~ dju1p... |
dju0en 9931 | Cardinal addition with car... |
xp2dju 9932 | Two times a cardinal numbe... |
djucomen 9933 | Commutative law for cardin... |
djuassen 9934 | Associative law for cardin... |
xpdjuen 9935 | Cardinal multiplication di... |
mapdjuen 9936 | Sum of exponents law for c... |
pwdjuen 9937 | Sum of exponents law for c... |
djudom1 9938 | Ordering law for cardinal ... |
djudom2 9939 | Ordering law for cardinal ... |
djudoml 9940 | A set is dominated by its ... |
djuxpdom 9941 | Cartesian product dominate... |
djufi 9942 | The disjoint union of two ... |
cdainflem 9943 | Any partition of omega int... |
djuinf 9944 | A set is infinite iff the ... |
infdju1 9945 | An infinite set is equinum... |
pwdju1 9946 | The sum of a powerset with... |
pwdjuidm 9947 | If the natural numbers inj... |
djulepw 9948 | If ` A ` is idempotent und... |
onadju 9949 | The cardinal and ordinal s... |
cardadju 9950 | The cardinal sum is equinu... |
djunum 9951 | The disjoint union of two ... |
unnum 9952 | The union of two numerable... |
nnadju 9953 | The cardinal and ordinal s... |
nnadjuALT 9954 | Shorter proof of ~ nnadju ... |
ficardadju 9955 | The disjoint union of fini... |
ficardun 9956 | The cardinality of the uni... |
ficardunOLD 9957 | Obsolete version of ~ fica... |
ficardun2 9958 | The cardinality of the uni... |
ficardun2OLD 9959 | Obsolete version of ~ fica... |
pwsdompw 9960 | Lemma for ~ domtriom . Th... |
unctb 9961 | The union of two countable... |
infdjuabs 9962 | Absorption law for additio... |
infunabs 9963 | An infinite set is equinum... |
infdju 9964 | The sum of two cardinal nu... |
infdif 9965 | The cardinality of an infi... |
infdif2 9966 | Cardinality ordering for a... |
infxpdom 9967 | Dominance law for multipli... |
infxpabs 9968 | Absorption law for multipl... |
infunsdom1 9969 | The union of two sets that... |
infunsdom 9970 | The union of two sets that... |
infxp 9971 | Absorption law for multipl... |
pwdjudom 9972 | A property of dominance ov... |
infpss 9973 | Every infinite set has an ... |
infmap2 9974 | An exponentiation law for ... |
ackbij2lem1 9975 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem1 9976 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem2 9977 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem3 9978 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem4 9979 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem5 9980 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem6 9981 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem7 9982 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem8 9983 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem9 9984 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem10 9985 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem11 9986 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem12 9987 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem13 9988 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem14 9989 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem15 9990 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem16 9991 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem17 9992 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem18 9993 | Lemma for ~ ackbij1 . (Co... |
ackbij1 9994 | The Ackermann bijection, p... |
ackbij1b 9995 | The Ackermann bijection, p... |
ackbij2lem2 9996 | Lemma for ~ ackbij2 . (Co... |
ackbij2lem3 9997 | Lemma for ~ ackbij2 . (Co... |
ackbij2lem4 9998 | Lemma for ~ ackbij2 . (Co... |
ackbij2 9999 | The Ackermann bijection, p... |
r1om 10000 | The set of hereditarily fi... |
fictb 10001 | A set is countable iff its... |
cflem 10002 | A lemma used to simplify c... |
cfval 10003 | Value of the cofinality fu... |
cff 10004 | Cofinality is a function o... |
cfub 10005 | An upper bound on cofinali... |
cflm 10006 | Value of the cofinality fu... |
cf0 10007 | Value of the cofinality fu... |
cardcf 10008 | Cofinality is a cardinal n... |
cflecard 10009 | Cofinality is bounded by t... |
cfle 10010 | Cofinality is bounded by i... |
cfon 10011 | The cofinality of any set ... |
cfeq0 10012 | Only the ordinal zero has ... |
cfsuc 10013 | Value of the cofinality fu... |
cff1 10014 | There is always a map from... |
cfflb 10015 | If there is a cofinal map ... |
cfval2 10016 | Another expression for the... |
coflim 10017 | A simpler expression for t... |
cflim3 10018 | Another expression for the... |
cflim2 10019 | The cofinality function is... |
cfom 10020 | Value of the cofinality fu... |
cfss 10021 | There is a cofinal subset ... |
cfslb 10022 | Any cofinal subset of ` A ... |
cfslbn 10023 | Any subset of ` A ` smalle... |
cfslb2n 10024 | Any small collection of sm... |
cofsmo 10025 | Any cofinal map implies th... |
cfsmolem 10026 | Lemma for ~ cfsmo . (Cont... |
cfsmo 10027 | The map in ~ cff1 can be a... |
cfcoflem 10028 | Lemma for ~ cfcof , showin... |
coftr 10029 | If there is a cofinal map ... |
cfcof 10030 | If there is a cofinal map ... |
cfidm 10031 | The cofinality function is... |
alephsing 10032 | The cofinality of a limit ... |
sornom 10033 | The range of a single-step... |
isfin1a 10048 | Definition of a Ia-finite ... |
fin1ai 10049 | Property of a Ia-finite se... |
isfin2 10050 | Definition of a II-finite ... |
fin2i 10051 | Property of a II-finite se... |
isfin3 10052 | Definition of a III-finite... |
isfin4 10053 | Definition of a IV-finite ... |
fin4i 10054 | Infer that a set is IV-inf... |
isfin5 10055 | Definition of a V-finite s... |
isfin6 10056 | Definition of a VI-finite ... |
isfin7 10057 | Definition of a VII-finite... |
sdom2en01 10058 | A set with less than two e... |
infpssrlem1 10059 | Lemma for ~ infpssr . (Co... |
infpssrlem2 10060 | Lemma for ~ infpssr . (Co... |
infpssrlem3 10061 | Lemma for ~ infpssr . (Co... |
infpssrlem4 10062 | Lemma for ~ infpssr . (Co... |
infpssrlem5 10063 | Lemma for ~ infpssr . (Co... |
infpssr 10064 | Dedekind infinity implies ... |
fin4en1 10065 | Dedekind finite is a cardi... |
ssfin4 10066 | Dedekind finite sets have ... |
domfin4 10067 | A set dominated by a Dedek... |
ominf4 10068 | ` _om ` is Dedekind infini... |
infpssALT 10069 | Alternate proof of ~ infps... |
isfin4-2 10070 | Alternate definition of IV... |
isfin4p1 10071 | Alternate definition of IV... |
fin23lem7 10072 | Lemma for ~ isfin2-2 . Th... |
fin23lem11 10073 | Lemma for ~ isfin2-2 . (C... |
fin2i2 10074 | A II-finite set contains m... |
isfin2-2 10075 | ` Fin2 ` expressed in term... |
ssfin2 10076 | A subset of a II-finite se... |
enfin2i 10077 | II-finiteness is a cardina... |
fin23lem24 10078 | Lemma for ~ fin23 . In a ... |
fincssdom 10079 | In a chain of finite sets,... |
fin23lem25 10080 | Lemma for ~ fin23 . In a ... |
fin23lem26 10081 | Lemma for ~ fin23lem22 . ... |
fin23lem23 10082 | Lemma for ~ fin23lem22 . ... |
fin23lem22 10083 | Lemma for ~ fin23 but coul... |
fin23lem27 10084 | The mapping constructed in... |
isfin3ds 10085 | Property of a III-finite s... |
ssfin3ds 10086 | A subset of a III-finite s... |
fin23lem12 10087 | The beginning of the proof... |
fin23lem13 10088 | Lemma for ~ fin23 . Each ... |
fin23lem14 10089 | Lemma for ~ fin23 . ` U ` ... |
fin23lem15 10090 | Lemma for ~ fin23 . ` U ` ... |
fin23lem16 10091 | Lemma for ~ fin23 . ` U ` ... |
fin23lem19 10092 | Lemma for ~ fin23 . The f... |
fin23lem20 10093 | Lemma for ~ fin23 . ` X ` ... |
fin23lem17 10094 | Lemma for ~ fin23 . By ? ... |
fin23lem21 10095 | Lemma for ~ fin23 . ` X ` ... |
fin23lem28 10096 | Lemma for ~ fin23 . The r... |
fin23lem29 10097 | Lemma for ~ fin23 . The r... |
fin23lem30 10098 | Lemma for ~ fin23 . The r... |
fin23lem31 10099 | Lemma for ~ fin23 . The r... |
fin23lem32 10100 | Lemma for ~ fin23 . Wrap ... |
fin23lem33 10101 | Lemma for ~ fin23 . Disch... |
fin23lem34 10102 | Lemma for ~ fin23 . Estab... |
fin23lem35 10103 | Lemma for ~ fin23 . Stric... |
fin23lem36 10104 | Lemma for ~ fin23 . Weak ... |
fin23lem38 10105 | Lemma for ~ fin23 . The c... |
fin23lem39 10106 | Lemma for ~ fin23 . Thus,... |
fin23lem40 10107 | Lemma for ~ fin23 . ` Fin2... |
fin23lem41 10108 | Lemma for ~ fin23 . A set... |
isf32lem1 10109 | Lemma for ~ isfin3-2 . De... |
isf32lem2 10110 | Lemma for ~ isfin3-2 . No... |
isf32lem3 10111 | Lemma for ~ isfin3-2 . Be... |
isf32lem4 10112 | Lemma for ~ isfin3-2 . Be... |
isf32lem5 10113 | Lemma for ~ isfin3-2 . Th... |
isf32lem6 10114 | Lemma for ~ isfin3-2 . Ea... |
isf32lem7 10115 | Lemma for ~ isfin3-2 . Di... |
isf32lem8 10116 | Lemma for ~ isfin3-2 . K ... |
isf32lem9 10117 | Lemma for ~ isfin3-2 . Co... |
isf32lem10 10118 | Lemma for isfin3-2 . Writ... |
isf32lem11 10119 | Lemma for ~ isfin3-2 . Re... |
isf32lem12 10120 | Lemma for ~ isfin3-2 . (C... |
isfin32i 10121 | One half of ~ isfin3-2 . ... |
isf33lem 10122 | Lemma for ~ isfin3-3 . (C... |
isfin3-2 10123 | Weakly Dedekind-infinite s... |
isfin3-3 10124 | Weakly Dedekind-infinite s... |
fin33i 10125 | Inference from ~ isfin3-3 ... |
compsscnvlem 10126 | Lemma for ~ compsscnv . (... |
compsscnv 10127 | Complementation on a power... |
isf34lem1 10128 | Lemma for ~ isfin3-4 . (C... |
isf34lem2 10129 | Lemma for ~ isfin3-4 . (C... |
compssiso 10130 | Complementation is an anti... |
isf34lem3 10131 | Lemma for ~ isfin3-4 . (C... |
compss 10132 | Express image under of the... |
isf34lem4 10133 | Lemma for ~ isfin3-4 . (C... |
isf34lem5 10134 | Lemma for ~ isfin3-4 . (C... |
isf34lem7 10135 | Lemma for ~ isfin3-4 . (C... |
isf34lem6 10136 | Lemma for ~ isfin3-4 . (C... |
fin34i 10137 | Inference from ~ isfin3-4 ... |
isfin3-4 10138 | Weakly Dedekind-infinite s... |
fin11a 10139 | Every I-finite set is Ia-f... |
enfin1ai 10140 | Ia-finiteness is a cardina... |
isfin1-2 10141 | A set is finite in the usu... |
isfin1-3 10142 | A set is I-finite iff ever... |
isfin1-4 10143 | A set is I-finite iff ever... |
dffin1-5 10144 | Compact quantifier-free ve... |
fin23 10145 | Every II-finite set (every... |
fin34 10146 | Every III-finite set is IV... |
isfin5-2 10147 | Alternate definition of V-... |
fin45 10148 | Every IV-finite set is V-f... |
fin56 10149 | Every V-finite set is VI-f... |
fin17 10150 | Every I-finite set is VII-... |
fin67 10151 | Every VI-finite set is VII... |
isfin7-2 10152 | A set is VII-finite iff it... |
fin71num 10153 | A well-orderable set is VI... |
dffin7-2 10154 | Class form of ~ isfin7-2 .... |
dfacfin7 10155 | Axiom of Choice equivalent... |
fin1a2lem1 10156 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem2 10157 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem3 10158 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem4 10159 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem5 10160 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem6 10161 | Lemma for ~ fin1a2 . Esta... |
fin1a2lem7 10162 | Lemma for ~ fin1a2 . Spli... |
fin1a2lem8 10163 | Lemma for ~ fin1a2 . Spli... |
fin1a2lem9 10164 | Lemma for ~ fin1a2 . In a... |
fin1a2lem10 10165 | Lemma for ~ fin1a2 . A no... |
fin1a2lem11 10166 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem12 10167 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem13 10168 | Lemma for ~ fin1a2 . (Con... |
fin12 10169 | Weak theorem which skips I... |
fin1a2s 10170 | An II-infinite set can hav... |
fin1a2 10171 | Every Ia-finite set is II-... |
itunifval 10172 | Function value of iterated... |
itunifn 10173 | Functionality of the itera... |
ituni0 10174 | A zero-fold iterated union... |
itunisuc 10175 | Successor iterated union. ... |
itunitc1 10176 | Each union iterate is a me... |
itunitc 10177 | The union of all union ite... |
ituniiun 10178 | Unwrap an iterated union f... |
hsmexlem7 10179 | Lemma for ~ hsmex . Prope... |
hsmexlem8 10180 | Lemma for ~ hsmex . Prope... |
hsmexlem9 10181 | Lemma for ~ hsmex . Prope... |
hsmexlem1 10182 | Lemma for ~ hsmex . Bound... |
hsmexlem2 10183 | Lemma for ~ hsmex . Bound... |
hsmexlem3 10184 | Lemma for ~ hsmex . Clear... |
hsmexlem4 10185 | Lemma for ~ hsmex . The c... |
hsmexlem5 10186 | Lemma for ~ hsmex . Combi... |
hsmexlem6 10187 | Lemma for ~ hsmex . (Cont... |
hsmex 10188 | The collection of heredita... |
hsmex2 10189 | The set of hereditary size... |
hsmex3 10190 | The set of hereditary size... |
axcc2lem 10192 | Lemma for ~ axcc2 . (Cont... |
axcc2 10193 | A possibly more useful ver... |
axcc3 10194 | A possibly more useful ver... |
axcc4 10195 | A version of ~ axcc3 that ... |
acncc 10196 | An ~ ax-cc equivalent: eve... |
axcc4dom 10197 | Relax the constraint on ~ ... |
domtriomlem 10198 | Lemma for ~ domtriom . (C... |
domtriom 10199 | Trichotomy of equinumerosi... |
fin41 10200 | Under countable choice, th... |
dominf 10201 | A nonempty set that is a s... |
dcomex 10203 | The Axiom of Dependent Cho... |
axdc2lem 10204 | Lemma for ~ axdc2 . We co... |
axdc2 10205 | An apparent strengthening ... |
axdc3lem 10206 | The class ` S ` of finite ... |
axdc3lem2 10207 | Lemma for ~ axdc3 . We ha... |
axdc3lem3 10208 | Simple substitution lemma ... |
axdc3lem4 10209 | Lemma for ~ axdc3 . We ha... |
axdc3 10210 | Dependent Choice. Axiom D... |
axdc4lem 10211 | Lemma for ~ axdc4 . (Cont... |
axdc4 10212 | A more general version of ... |
axcclem 10213 | Lemma for ~ axcc . (Contr... |
axcc 10214 | Although CC can be proven ... |
zfac 10216 | Axiom of Choice expressed ... |
ac2 10217 | Axiom of Choice equivalent... |
ac3 10218 | Axiom of Choice using abbr... |
axac3 10220 | This theorem asserts that ... |
ackm 10221 | A remarkable equivalent to... |
axac2 10222 | Derive ~ ax-ac2 from ~ ax-... |
axac 10223 | Derive ~ ax-ac from ~ ax-a... |
axaci 10224 | Apply a choice equivalent.... |
cardeqv 10225 | All sets are well-orderabl... |
numth3 10226 | All sets are well-orderabl... |
numth2 10227 | Numeration theorem: any se... |
numth 10228 | Numeration theorem: every ... |
ac7 10229 | An Axiom of Choice equival... |
ac7g 10230 | An Axiom of Choice equival... |
ac4 10231 | Equivalent of Axiom of Cho... |
ac4c 10232 | Equivalent of Axiom of Cho... |
ac5 10233 | An Axiom of Choice equival... |
ac5b 10234 | Equivalent of Axiom of Cho... |
ac6num 10235 | A version of ~ ac6 which t... |
ac6 10236 | Equivalent of Axiom of Cho... |
ac6c4 10237 | Equivalent of Axiom of Cho... |
ac6c5 10238 | Equivalent of Axiom of Cho... |
ac9 10239 | An Axiom of Choice equival... |
ac6s 10240 | Equivalent of Axiom of Cho... |
ac6n 10241 | Equivalent of Axiom of Cho... |
ac6s2 10242 | Generalization of the Axio... |
ac6s3 10243 | Generalization of the Axio... |
ac6sg 10244 | ~ ac6s with sethood as ant... |
ac6sf 10245 | Version of ~ ac6 with boun... |
ac6s4 10246 | Generalization of the Axio... |
ac6s5 10247 | Generalization of the Axio... |
ac8 10248 | An Axiom of Choice equival... |
ac9s 10249 | An Axiom of Choice equival... |
numthcor 10250 | Any set is strictly domina... |
weth 10251 | Well-ordering theorem: any... |
zorn2lem1 10252 | Lemma for ~ zorn2 . (Cont... |
zorn2lem2 10253 | Lemma for ~ zorn2 . (Cont... |
zorn2lem3 10254 | Lemma for ~ zorn2 . (Cont... |
zorn2lem4 10255 | Lemma for ~ zorn2 . (Cont... |
zorn2lem5 10256 | Lemma for ~ zorn2 . (Cont... |
zorn2lem6 10257 | Lemma for ~ zorn2 . (Cont... |
zorn2lem7 10258 | Lemma for ~ zorn2 . (Cont... |
zorn2g 10259 | Zorn's Lemma of [Monk1] p.... |
zorng 10260 | Zorn's Lemma. If the unio... |
zornn0g 10261 | Variant of Zorn's lemma ~ ... |
zorn2 10262 | Zorn's Lemma of [Monk1] p.... |
zorn 10263 | Zorn's Lemma. If the unio... |
zornn0 10264 | Variant of Zorn's lemma ~ ... |
ttukeylem1 10265 | Lemma for ~ ttukey . Expa... |
ttukeylem2 10266 | Lemma for ~ ttukey . A pr... |
ttukeylem3 10267 | Lemma for ~ ttukey . (Con... |
ttukeylem4 10268 | Lemma for ~ ttukey . (Con... |
ttukeylem5 10269 | Lemma for ~ ttukey . The ... |
ttukeylem6 10270 | Lemma for ~ ttukey . (Con... |
ttukeylem7 10271 | Lemma for ~ ttukey . (Con... |
ttukey2g 10272 | The Teichmüller-Tukey... |
ttukeyg 10273 | The Teichmüller-Tukey... |
ttukey 10274 | The Teichmüller-Tukey... |
axdclem 10275 | Lemma for ~ axdc . (Contr... |
axdclem2 10276 | Lemma for ~ axdc . Using ... |
axdc 10277 | This theorem derives ~ ax-... |
fodomg 10278 | An onto function implies d... |
fodom 10279 | An onto function implies d... |
dmct 10280 | The domain of a countable ... |
rnct 10281 | The range of a countable s... |
fodomb 10282 | Equivalence of an onto map... |
wdomac 10283 | When assuming AC, weak and... |
brdom3 10284 | Equivalence to a dominance... |
brdom5 10285 | An equivalence to a domina... |
brdom4 10286 | An equivalence to a domina... |
brdom7disj 10287 | An equivalence to a domina... |
brdom6disj 10288 | An equivalence to a domina... |
fin71ac 10289 | Once we allow AC, the "str... |
imadomg 10290 | An image of a function und... |
fimact 10291 | The image by a function of... |
fnrndomg 10292 | The range of a function is... |
fnct 10293 | If the domain of a functio... |
mptct 10294 | A countable mapping set is... |
iunfo 10295 | Existence of an onto funct... |
iundom2g 10296 | An upper bound for the car... |
iundomg 10297 | An upper bound for the car... |
iundom 10298 | An upper bound for the car... |
unidom 10299 | An upper bound for the car... |
uniimadom 10300 | An upper bound for the car... |
uniimadomf 10301 | An upper bound for the car... |
cardval 10302 | The value of the cardinal ... |
cardid 10303 | Any set is equinumerous to... |
cardidg 10304 | Any set is equinumerous to... |
cardidd 10305 | Any set is equinumerous to... |
cardf 10306 | The cardinality function i... |
carden 10307 | Two sets are equinumerous ... |
cardeq0 10308 | Only the empty set has car... |
unsnen 10309 | Equinumerosity of a set wi... |
carddom 10310 | Two sets have the dominanc... |
cardsdom 10311 | Two sets have the strict d... |
domtri 10312 | Trichotomy law for dominan... |
entric 10313 | Trichotomy of equinumerosi... |
entri2 10314 | Trichotomy of dominance an... |
entri3 10315 | Trichotomy of dominance. ... |
sdomsdomcard 10316 | A set strictly dominates i... |
canth3 10317 | Cantor's theorem in terms ... |
infxpidm 10318 | Every infinite class is eq... |
ondomon 10319 | The class of ordinals domi... |
cardmin 10320 | The smallest ordinal that ... |
ficard 10321 | A set is finite iff its ca... |
infinf 10322 | Equivalence between two in... |
unirnfdomd 10323 | The union of the range of ... |
konigthlem 10324 | Lemma for ~ konigth . (Co... |
konigth 10325 | Konig's Theorem. If ` m (... |
alephsucpw 10326 | The power set of an aleph ... |
aleph1 10327 | The set exponentiation of ... |
alephval2 10328 | An alternate way to expres... |
dominfac 10329 | A nonempty set that is a s... |
iunctb 10330 | The countable union of cou... |
unictb 10331 | The countable union of cou... |
infmap 10332 | An exponentiation law for ... |
alephadd 10333 | The sum of two alephs is t... |
alephmul 10334 | The product of two alephs ... |
alephexp1 10335 | An exponentiation law for ... |
alephsuc3 10336 | An alternate representatio... |
alephexp2 10337 | An expression equinumerous... |
alephreg 10338 | A successor aleph is regul... |
pwcfsdom 10339 | A corollary of Konig's The... |
cfpwsdom 10340 | A corollary of Konig's The... |
alephom 10341 | From ~ canth2 , we know th... |
smobeth 10342 | The beth function is stric... |
nd1 10343 | A lemma for proving condit... |
nd2 10344 | A lemma for proving condit... |
nd3 10345 | A lemma for proving condit... |
nd4 10346 | A lemma for proving condit... |
axextnd 10347 | A version of the Axiom of ... |
axrepndlem1 10348 | Lemma for the Axiom of Rep... |
axrepndlem2 10349 | Lemma for the Axiom of Rep... |
axrepnd 10350 | A version of the Axiom of ... |
axunndlem1 10351 | Lemma for the Axiom of Uni... |
axunnd 10352 | A version of the Axiom of ... |
axpowndlem1 10353 | Lemma for the Axiom of Pow... |
axpowndlem2 10354 | Lemma for the Axiom of Pow... |
axpowndlem3 10355 | Lemma for the Axiom of Pow... |
axpowndlem4 10356 | Lemma for the Axiom of Pow... |
axpownd 10357 | A version of the Axiom of ... |
axregndlem1 10358 | Lemma for the Axiom of Reg... |
axregndlem2 10359 | Lemma for the Axiom of Reg... |
axregnd 10360 | A version of the Axiom of ... |
axinfndlem1 10361 | Lemma for the Axiom of Inf... |
axinfnd 10362 | A version of the Axiom of ... |
axacndlem1 10363 | Lemma for the Axiom of Cho... |
axacndlem2 10364 | Lemma for the Axiom of Cho... |
axacndlem3 10365 | Lemma for the Axiom of Cho... |
axacndlem4 10366 | Lemma for the Axiom of Cho... |
axacndlem5 10367 | Lemma for the Axiom of Cho... |
axacnd 10368 | A version of the Axiom of ... |
zfcndext 10369 | Axiom of Extensionality ~ ... |
zfcndrep 10370 | Axiom of Replacement ~ ax-... |
zfcndun 10371 | Axiom of Union ~ ax-un , r... |
zfcndpow 10372 | Axiom of Power Sets ~ ax-p... |
zfcndreg 10373 | Axiom of Regularity ~ ax-r... |
zfcndinf 10374 | Axiom of Infinity ~ ax-inf... |
zfcndac 10375 | Axiom of Choice ~ ax-ac , ... |
elgch 10378 | Elementhood in the collect... |
fingch 10379 | A finite set is a GCH-set.... |
gchi 10380 | The only GCH-sets which ha... |
gchen1 10381 | If ` A <_ B < ~P A ` , and... |
gchen2 10382 | If ` A < B <_ ~P A ` , and... |
gchor 10383 | If ` A <_ B <_ ~P A ` , an... |
engch 10384 | The property of being a GC... |
gchdomtri 10385 | Under certain conditions, ... |
fpwwe2cbv 10386 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem1 10387 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem2 10388 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem3 10389 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem4 10390 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem5 10391 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem6 10392 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem7 10393 | Lemma for ~ fpwwe2 . Show... |
fpwwe2lem8 10394 | Lemma for ~ fpwwe2 . Give... |
fpwwe2lem9 10395 | Lemma for ~ fpwwe2 . Give... |
fpwwe2lem10 10396 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem11 10397 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem12 10398 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2 10399 | Given any function ` F ` f... |
fpwwecbv 10400 | Lemma for ~ fpwwe . (Cont... |
fpwwelem 10401 | Lemma for ~ fpwwe . (Cont... |
fpwwe 10402 | Given any function ` F ` f... |
canth4 10403 | An "effective" form of Can... |
canthnumlem 10404 | Lemma for ~ canthnum . (C... |
canthnum 10405 | The set of well-orderable ... |
canthwelem 10406 | Lemma for ~ canthwe . (Co... |
canthwe 10407 | The set of well-orders of ... |
canthp1lem1 10408 | Lemma for ~ canthp1 . (Co... |
canthp1lem2 10409 | Lemma for ~ canthp1 . (Co... |
canthp1 10410 | A slightly stronger form o... |
finngch 10411 | The exclusion of finite se... |
gchdju1 10412 | An infinite GCH-set is ide... |
gchinf 10413 | An infinite GCH-set is Ded... |
pwfseqlem1 10414 | Lemma for ~ pwfseq . Deri... |
pwfseqlem2 10415 | Lemma for ~ pwfseq . (Con... |
pwfseqlem3 10416 | Lemma for ~ pwfseq . Usin... |
pwfseqlem4a 10417 | Lemma for ~ pwfseqlem4 . ... |
pwfseqlem4 10418 | Lemma for ~ pwfseq . Deri... |
pwfseqlem5 10419 | Lemma for ~ pwfseq . Alth... |
pwfseq 10420 | The powerset of a Dedekind... |
pwxpndom2 10421 | The powerset of a Dedekind... |
pwxpndom 10422 | The powerset of a Dedekind... |
pwdjundom 10423 | The powerset of a Dedekind... |
gchdjuidm 10424 | An infinite GCH-set is ide... |
gchxpidm 10425 | An infinite GCH-set is ide... |
gchpwdom 10426 | A relationship between dom... |
gchaleph 10427 | If ` ( aleph `` A ) ` is a... |
gchaleph2 10428 | If ` ( aleph `` A ) ` and ... |
hargch 10429 | If ` A + ~~ ~P A ` , then ... |
alephgch 10430 | If ` ( aleph `` suc A ) ` ... |
gch2 10431 | It is sufficient to requir... |
gch3 10432 | An equivalent formulation ... |
gch-kn 10433 | The equivalence of two ver... |
gchaclem 10434 | Lemma for ~ gchac (obsolet... |
gchhar 10435 | A "local" form of ~ gchac ... |
gchacg 10436 | A "local" form of ~ gchac ... |
gchac 10437 | The Generalized Continuum ... |
elwina 10442 | Conditions of weak inacces... |
elina 10443 | Conditions of strong inacc... |
winaon 10444 | A weakly inaccessible card... |
inawinalem 10445 | Lemma for ~ inawina . (Co... |
inawina 10446 | Every strongly inaccessibl... |
omina 10447 | ` _om ` is a strongly inac... |
winacard 10448 | A weakly inaccessible card... |
winainflem 10449 | A weakly inaccessible card... |
winainf 10450 | A weakly inaccessible card... |
winalim 10451 | A weakly inaccessible card... |
winalim2 10452 | A nontrivial weakly inacce... |
winafp 10453 | A nontrivial weakly inacce... |
winafpi 10454 | This theorem, which states... |
gchina 10455 | Assuming the GCH, weakly a... |
iswun 10460 | Properties of a weak unive... |
wuntr 10461 | A weak universe is transit... |
wununi 10462 | A weak universe is closed ... |
wunpw 10463 | A weak universe is closed ... |
wunelss 10464 | The elements of a weak uni... |
wunpr 10465 | A weak universe is closed ... |
wunun 10466 | A weak universe is closed ... |
wuntp 10467 | A weak universe is closed ... |
wunss 10468 | A weak universe is closed ... |
wunin 10469 | A weak universe is closed ... |
wundif 10470 | A weak universe is closed ... |
wunint 10471 | A weak universe is closed ... |
wunsn 10472 | A weak universe is closed ... |
wunsuc 10473 | A weak universe is closed ... |
wun0 10474 | A weak universe contains t... |
wunr1om 10475 | A weak universe is infinit... |
wunom 10476 | A weak universe contains a... |
wunfi 10477 | A weak universe contains a... |
wunop 10478 | A weak universe is closed ... |
wunot 10479 | A weak universe is closed ... |
wunxp 10480 | A weak universe is closed ... |
wunpm 10481 | A weak universe is closed ... |
wunmap 10482 | A weak universe is closed ... |
wunf 10483 | A weak universe is closed ... |
wundm 10484 | A weak universe is closed ... |
wunrn 10485 | A weak universe is closed ... |
wuncnv 10486 | A weak universe is closed ... |
wunres 10487 | A weak universe is closed ... |
wunfv 10488 | A weak universe is closed ... |
wunco 10489 | A weak universe is closed ... |
wuntpos 10490 | A weak universe is closed ... |
intwun 10491 | The intersection of a coll... |
r1limwun 10492 | Each limit stage in the cu... |
r1wunlim 10493 | The weak universes in the ... |
wunex2 10494 | Construct a weak universe ... |
wunex 10495 | Construct a weak universe ... |
uniwun 10496 | Every set is contained in ... |
wunex3 10497 | Construct a weak universe ... |
wuncval 10498 | Value of the weak universe... |
wuncid 10499 | The weak universe closure ... |
wunccl 10500 | The weak universe closure ... |
wuncss 10501 | The weak universe closure ... |
wuncidm 10502 | The weak universe closure ... |
wuncval2 10503 | Our earlier expression for... |
eltskg 10506 | Properties of a Tarski cla... |
eltsk2g 10507 | Properties of a Tarski cla... |
tskpwss 10508 | First axiom of a Tarski cl... |
tskpw 10509 | Second axiom of a Tarski c... |
tsken 10510 | Third axiom of a Tarski cl... |
0tsk 10511 | The empty set is a (transi... |
tsksdom 10512 | An element of a Tarski cla... |
tskssel 10513 | A part of a Tarski class s... |
tskss 10514 | The subsets of an element ... |
tskin 10515 | The intersection of two el... |
tsksn 10516 | A singleton of an element ... |
tsktrss 10517 | A transitive element of a ... |
tsksuc 10518 | If an element of a Tarski ... |
tsk0 10519 | A nonempty Tarski class co... |
tsk1 10520 | One is an element of a non... |
tsk2 10521 | Two is an element of a non... |
2domtsk 10522 | If a Tarski class is not e... |
tskr1om 10523 | A nonempty Tarski class is... |
tskr1om2 10524 | A nonempty Tarski class co... |
tskinf 10525 | A nonempty Tarski class is... |
tskpr 10526 | If ` A ` and ` B ` are mem... |
tskop 10527 | If ` A ` and ` B ` are mem... |
tskxpss 10528 | A Cartesian product of two... |
tskwe2 10529 | A Tarski class is well-ord... |
inttsk 10530 | The intersection of a coll... |
inar1 10531 | ` ( R1 `` A ) ` for ` A ` ... |
r1omALT 10532 | Alternate proof of ~ r1om ... |
rankcf 10533 | Any set must be at least a... |
inatsk 10534 | ` ( R1 `` A ) ` for ` A ` ... |
r1omtsk 10535 | The set of hereditarily fi... |
tskord 10536 | A Tarski class contains al... |
tskcard 10537 | An even more direct relati... |
r1tskina 10538 | There is a direct relation... |
tskuni 10539 | The union of an element of... |
tskwun 10540 | A nonempty transitive Tars... |
tskint 10541 | The intersection of an ele... |
tskun 10542 | The union of two elements ... |
tskxp 10543 | The Cartesian product of t... |
tskmap 10544 | Set exponentiation is an e... |
tskurn 10545 | A transitive Tarski class ... |
elgrug 10548 | Properties of a Grothendie... |
grutr 10549 | A Grothendieck universe is... |
gruelss 10550 | A Grothendieck universe is... |
grupw 10551 | A Grothendieck universe co... |
gruss 10552 | Any subset of an element o... |
grupr 10553 | A Grothendieck universe co... |
gruurn 10554 | A Grothendieck universe co... |
gruiun 10555 | If ` B ( x ) ` is a family... |
gruuni 10556 | A Grothendieck universe co... |
grurn 10557 | A Grothendieck universe co... |
gruima 10558 | A Grothendieck universe co... |
gruel 10559 | Any element of an element ... |
grusn 10560 | A Grothendieck universe co... |
gruop 10561 | A Grothendieck universe co... |
gruun 10562 | A Grothendieck universe co... |
gruxp 10563 | A Grothendieck universe co... |
grumap 10564 | A Grothendieck universe co... |
gruixp 10565 | A Grothendieck universe co... |
gruiin 10566 | A Grothendieck universe co... |
gruf 10567 | A Grothendieck universe co... |
gruen 10568 | A Grothendieck universe co... |
gruwun 10569 | A nonempty Grothendieck un... |
intgru 10570 | The intersection of a fami... |
ingru 10571 | The intersection of a univ... |
wfgru 10572 | The wellfounded part of a ... |
grudomon 10573 | Each ordinal that is compa... |
gruina 10574 | If a Grothendieck universe... |
grur1a 10575 | A characterization of Grot... |
grur1 10576 | A characterization of Grot... |
grutsk1 10577 | Grothendieck universes are... |
grutsk 10578 | Grothendieck universes are... |
axgroth5 10580 | The Tarski-Grothendieck ax... |
axgroth2 10581 | Alternate version of the T... |
grothpw 10582 | Derive the Axiom of Power ... |
grothpwex 10583 | Derive the Axiom of Power ... |
axgroth6 10584 | The Tarski-Grothendieck ax... |
grothomex 10585 | The Tarski-Grothendieck Ax... |
grothac 10586 | The Tarski-Grothendieck Ax... |
axgroth3 10587 | Alternate version of the T... |
axgroth4 10588 | Alternate version of the T... |
grothprimlem 10589 | Lemma for ~ grothprim . E... |
grothprim 10590 | The Tarski-Grothendieck Ax... |
grothtsk 10591 | The Tarski-Grothendieck Ax... |
inaprc 10592 | An equivalent to the Tarsk... |
tskmval 10595 | Value of our tarski map. ... |
tskmid 10596 | The set ` A ` is an elemen... |
tskmcl 10597 | A Tarski class that contai... |
sstskm 10598 | Being a part of ` ( tarski... |
eltskm 10599 | Belonging to ` ( tarskiMap... |
elni 10632 | Membership in the class of... |
elni2 10633 | Membership in the class of... |
pinn 10634 | A positive integer is a na... |
pion 10635 | A positive integer is an o... |
piord 10636 | A positive integer is ordi... |
niex 10637 | The class of positive inte... |
0npi 10638 | The empty set is not a pos... |
1pi 10639 | Ordinal 'one' is a positiv... |
addpiord 10640 | Positive integer addition ... |
mulpiord 10641 | Positive integer multiplic... |
mulidpi 10642 | 1 is an identity element f... |
ltpiord 10643 | Positive integer 'less tha... |
ltsopi 10644 | Positive integer 'less tha... |
ltrelpi 10645 | Positive integer 'less tha... |
dmaddpi 10646 | Domain of addition on posi... |
dmmulpi 10647 | Domain of multiplication o... |
addclpi 10648 | Closure of addition of pos... |
mulclpi 10649 | Closure of multiplication ... |
addcompi 10650 | Addition of positive integ... |
addasspi 10651 | Addition of positive integ... |
mulcompi 10652 | Multiplication of positive... |
mulasspi 10653 | Multiplication of positive... |
distrpi 10654 | Multiplication of positive... |
addcanpi 10655 | Addition cancellation law ... |
mulcanpi 10656 | Multiplication cancellatio... |
addnidpi 10657 | There is no identity eleme... |
ltexpi 10658 | Ordering on positive integ... |
ltapi 10659 | Ordering property of addit... |
ltmpi 10660 | Ordering property of multi... |
1lt2pi 10661 | One is less than two (one ... |
nlt1pi 10662 | No positive integer is les... |
indpi 10663 | Principle of Finite Induct... |
enqbreq 10675 | Equivalence relation for p... |
enqbreq2 10676 | Equivalence relation for p... |
enqer 10677 | The equivalence relation f... |
enqex 10678 | The equivalence relation f... |
nqex 10679 | The class of positive frac... |
0nnq 10680 | The empty set is not a pos... |
elpqn 10681 | Each positive fraction is ... |
ltrelnq 10682 | Positive fraction 'less th... |
pinq 10683 | The representatives of pos... |
1nq 10684 | The positive fraction 'one... |
nqereu 10685 | There is a unique element ... |
nqerf 10686 | Corollary of ~ nqereu : th... |
nqercl 10687 | Corollary of ~ nqereu : cl... |
nqerrel 10688 | Any member of ` ( N. X. N.... |
nqerid 10689 | Corollary of ~ nqereu : th... |
enqeq 10690 | Corollary of ~ nqereu : if... |
nqereq 10691 | The function ` /Q ` acts a... |
addpipq2 10692 | Addition of positive fract... |
addpipq 10693 | Addition of positive fract... |
addpqnq 10694 | Addition of positive fract... |
mulpipq2 10695 | Multiplication of positive... |
mulpipq 10696 | Multiplication of positive... |
mulpqnq 10697 | Multiplication of positive... |
ordpipq 10698 | Ordering of positive fract... |
ordpinq 10699 | Ordering of positive fract... |
addpqf 10700 | Closure of addition on pos... |
addclnq 10701 | Closure of addition on pos... |
mulpqf 10702 | Closure of multiplication ... |
mulclnq 10703 | Closure of multiplication ... |
addnqf 10704 | Domain of addition on posi... |
mulnqf 10705 | Domain of multiplication o... |
addcompq 10706 | Addition of positive fract... |
addcomnq 10707 | Addition of positive fract... |
mulcompq 10708 | Multiplication of positive... |
mulcomnq 10709 | Multiplication of positive... |
adderpqlem 10710 | Lemma for ~ adderpq . (Co... |
mulerpqlem 10711 | Lemma for ~ mulerpq . (Co... |
adderpq 10712 | Addition is compatible wit... |
mulerpq 10713 | Multiplication is compatib... |
addassnq 10714 | Addition of positive fract... |
mulassnq 10715 | Multiplication of positive... |
mulcanenq 10716 | Lemma for distributive law... |
distrnq 10717 | Multiplication of positive... |
1nqenq 10718 | The equivalence class of r... |
mulidnq 10719 | Multiplication identity el... |
recmulnq 10720 | Relationship between recip... |
recidnq 10721 | A positive fraction times ... |
recclnq 10722 | Closure law for positive f... |
recrecnq 10723 | Reciprocal of reciprocal o... |
dmrecnq 10724 | Domain of reciprocal on po... |
ltsonq 10725 | 'Less than' is a strict or... |
lterpq 10726 | Compatibility of ordering ... |
ltanq 10727 | Ordering property of addit... |
ltmnq 10728 | Ordering property of multi... |
1lt2nq 10729 | One is less than two (one ... |
ltaddnq 10730 | The sum of two fractions i... |
ltexnq 10731 | Ordering on positive fract... |
halfnq 10732 | One-half of any positive f... |
nsmallnq 10733 | The is no smallest positiv... |
ltbtwnnq 10734 | There exists a number betw... |
ltrnq 10735 | Ordering property of recip... |
archnq 10736 | For any fraction, there is... |
npex 10742 | The class of positive real... |
elnp 10743 | Membership in positive rea... |
elnpi 10744 | Membership in positive rea... |
prn0 10745 | A positive real is not emp... |
prpssnq 10746 | A positive real is a subse... |
elprnq 10747 | A positive real is a set o... |
0npr 10748 | The empty set is not a pos... |
prcdnq 10749 | A positive real is closed ... |
prub 10750 | A positive fraction not in... |
prnmax 10751 | A positive real has no lar... |
npomex 10752 | A simplifying observation,... |
prnmadd 10753 | A positive real has no lar... |
ltrelpr 10754 | Positive real 'less than' ... |
genpv 10755 | Value of general operation... |
genpelv 10756 | Membership in value of gen... |
genpprecl 10757 | Pre-closure law for genera... |
genpdm 10758 | Domain of general operatio... |
genpn0 10759 | The result of an operation... |
genpss 10760 | The result of an operation... |
genpnnp 10761 | The result of an operation... |
genpcd 10762 | Downward closure of an ope... |
genpnmax 10763 | An operation on positive r... |
genpcl 10764 | Closure of an operation on... |
genpass 10765 | Associativity of an operat... |
plpv 10766 | Value of addition on posit... |
mpv 10767 | Value of multiplication on... |
dmplp 10768 | Domain of addition on posi... |
dmmp 10769 | Domain of multiplication o... |
nqpr 10770 | The canonical embedding of... |
1pr 10771 | The positive real number '... |
addclprlem1 10772 | Lemma to prove downward cl... |
addclprlem2 10773 | Lemma to prove downward cl... |
addclpr 10774 | Closure of addition on pos... |
mulclprlem 10775 | Lemma to prove downward cl... |
mulclpr 10776 | Closure of multiplication ... |
addcompr 10777 | Addition of positive reals... |
addasspr 10778 | Addition of positive reals... |
mulcompr 10779 | Multiplication of positive... |
mulasspr 10780 | Multiplication of positive... |
distrlem1pr 10781 | Lemma for distributive law... |
distrlem4pr 10782 | Lemma for distributive law... |
distrlem5pr 10783 | Lemma for distributive law... |
distrpr 10784 | Multiplication of positive... |
1idpr 10785 | 1 is an identity element f... |
ltprord 10786 | Positive real 'less than' ... |
psslinpr 10787 | Proper subset is a linear ... |
ltsopr 10788 | Positive real 'less than' ... |
prlem934 10789 | Lemma 9-3.4 of [Gleason] p... |
ltaddpr 10790 | The sum of two positive re... |
ltaddpr2 10791 | The sum of two positive re... |
ltexprlem1 10792 | Lemma for Proposition 9-3.... |
ltexprlem2 10793 | Lemma for Proposition 9-3.... |
ltexprlem3 10794 | Lemma for Proposition 9-3.... |
ltexprlem4 10795 | Lemma for Proposition 9-3.... |
ltexprlem5 10796 | Lemma for Proposition 9-3.... |
ltexprlem6 10797 | Lemma for Proposition 9-3.... |
ltexprlem7 10798 | Lemma for Proposition 9-3.... |
ltexpri 10799 | Proposition 9-3.5(iv) of [... |
ltaprlem 10800 | Lemma for Proposition 9-3.... |
ltapr 10801 | Ordering property of addit... |
addcanpr 10802 | Addition cancellation law ... |
prlem936 10803 | Lemma 9-3.6 of [Gleason] p... |
reclem2pr 10804 | Lemma for Proposition 9-3.... |
reclem3pr 10805 | Lemma for Proposition 9-3.... |
reclem4pr 10806 | Lemma for Proposition 9-3.... |
recexpr 10807 | The reciprocal of a positi... |
suplem1pr 10808 | The union of a nonempty, b... |
suplem2pr 10809 | The union of a set of posi... |
supexpr 10810 | The union of a nonempty, b... |
enrer 10819 | The equivalence relation f... |
nrex1 10820 | The class of signed reals ... |
enrbreq 10821 | Equivalence relation for s... |
enreceq 10822 | Equivalence class equality... |
enrex 10823 | The equivalence relation f... |
ltrelsr 10824 | Signed real 'less than' is... |
addcmpblnr 10825 | Lemma showing compatibilit... |
mulcmpblnrlem 10826 | Lemma used in lemma showin... |
mulcmpblnr 10827 | Lemma showing compatibilit... |
prsrlem1 10828 | Decomposing signed reals i... |
addsrmo 10829 | There is at most one resul... |
mulsrmo 10830 | There is at most one resul... |
addsrpr 10831 | Addition of signed reals i... |
mulsrpr 10832 | Multiplication of signed r... |
ltsrpr 10833 | Ordering of signed reals i... |
gt0srpr 10834 | Greater than zero in terms... |
0nsr 10835 | The empty set is not a sig... |
0r 10836 | The constant ` 0R ` is a s... |
1sr 10837 | The constant ` 1R ` is a s... |
m1r 10838 | The constant ` -1R ` is a ... |
addclsr 10839 | Closure of addition on sig... |
mulclsr 10840 | Closure of multiplication ... |
dmaddsr 10841 | Domain of addition on sign... |
dmmulsr 10842 | Domain of multiplication o... |
addcomsr 10843 | Addition of signed reals i... |
addasssr 10844 | Addition of signed reals i... |
mulcomsr 10845 | Multiplication of signed r... |
mulasssr 10846 | Multiplication of signed r... |
distrsr 10847 | Multiplication of signed r... |
m1p1sr 10848 | Minus one plus one is zero... |
m1m1sr 10849 | Minus one times minus one ... |
ltsosr 10850 | Signed real 'less than' is... |
0lt1sr 10851 | 0 is less than 1 for signe... |
1ne0sr 10852 | 1 and 0 are distinct for s... |
0idsr 10853 | The signed real number 0 i... |
1idsr 10854 | 1 is an identity element f... |
00sr 10855 | A signed real times 0 is 0... |
ltasr 10856 | Ordering property of addit... |
pn0sr 10857 | A signed real plus its neg... |
negexsr 10858 | Existence of negative sign... |
recexsrlem 10859 | The reciprocal of a positi... |
addgt0sr 10860 | The sum of two positive si... |
mulgt0sr 10861 | The product of two positiv... |
sqgt0sr 10862 | The square of a nonzero si... |
recexsr 10863 | The reciprocal of a nonzer... |
mappsrpr 10864 | Mapping from positive sign... |
ltpsrpr 10865 | Mapping of order from posi... |
map2psrpr 10866 | Equivalence for positive s... |
supsrlem 10867 | Lemma for supremum theorem... |
supsr 10868 | A nonempty, bounded set of... |
opelcn 10885 | Ordered pair membership in... |
opelreal 10886 | Ordered pair membership in... |
elreal 10887 | Membership in class of rea... |
elreal2 10888 | Ordered pair membership in... |
0ncn 10889 | The empty set is not a com... |
ltrelre 10890 | 'Less than' is a relation ... |
addcnsr 10891 | Addition of complex number... |
mulcnsr 10892 | Multiplication of complex ... |
eqresr 10893 | Equality of real numbers i... |
addresr 10894 | Addition of real numbers i... |
mulresr 10895 | Multiplication of real num... |
ltresr 10896 | Ordering of real subset of... |
ltresr2 10897 | Ordering of real subset of... |
dfcnqs 10898 | Technical trick to permit ... |
addcnsrec 10899 | Technical trick to permit ... |
mulcnsrec 10900 | Technical trick to permit ... |
axaddf 10901 | Addition is an operation o... |
axmulf 10902 | Multiplication is an opera... |
axcnex 10903 | The complex numbers form a... |
axresscn 10904 | The real numbers are a sub... |
ax1cn 10905 | 1 is a complex number. Ax... |
axicn 10906 | ` _i ` is a complex number... |
axaddcl 10907 | Closure law for addition o... |
axaddrcl 10908 | Closure law for addition i... |
axmulcl 10909 | Closure law for multiplica... |
axmulrcl 10910 | Closure law for multiplica... |
axmulcom 10911 | Multiplication of complex ... |
axaddass 10912 | Addition of complex number... |
axmulass 10913 | Multiplication of complex ... |
axdistr 10914 | Distributive law for compl... |
axi2m1 10915 | i-squared equals -1 (expre... |
ax1ne0 10916 | 1 and 0 are distinct. Axi... |
ax1rid 10917 | ` 1 ` is an identity eleme... |
axrnegex 10918 | Existence of negative of r... |
axrrecex 10919 | Existence of reciprocal of... |
axcnre 10920 | A complex number can be ex... |
axpre-lttri 10921 | Ordering on reals satisfie... |
axpre-lttrn 10922 | Ordering on reals is trans... |
axpre-ltadd 10923 | Ordering property of addit... |
axpre-mulgt0 10924 | The product of two positiv... |
axpre-sup 10925 | A nonempty, bounded-above ... |
wuncn 10926 | A weak universe containing... |
cnex 10952 | Alias for ~ ax-cnex . See... |
addcl 10953 | Alias for ~ ax-addcl , for... |
readdcl 10954 | Alias for ~ ax-addrcl , fo... |
mulcl 10955 | Alias for ~ ax-mulcl , for... |
remulcl 10956 | Alias for ~ ax-mulrcl , fo... |
mulcom 10957 | Alias for ~ ax-mulcom , fo... |
addass 10958 | Alias for ~ ax-addass , fo... |
mulass 10959 | Alias for ~ ax-mulass , fo... |
adddi 10960 | Alias for ~ ax-distr , for... |
recn 10961 | A real number is a complex... |
reex 10962 | The real numbers form a se... |
reelprrecn 10963 | Reals are a subset of the ... |
cnelprrecn 10964 | Complex numbers are a subs... |
elimne0 10965 | Hypothesis for weak deduct... |
adddir 10966 | Distributive law for compl... |
0cn 10967 | Zero is a complex number. ... |
0cnd 10968 | Zero is a complex number, ... |
c0ex 10969 | Zero is a set. (Contribut... |
1cnd 10970 | One is a complex number, d... |
1ex 10971 | One is a set. (Contribute... |
cnre 10972 | Alias for ~ ax-cnre , for ... |
mulid1 10973 | The number 1 is an identit... |
mulid2 10974 | Identity law for multiplic... |
1re 10975 | The number 1 is real. Thi... |
1red 10976 | The number 1 is real, dedu... |
0re 10977 | The number 0 is real. Rem... |
0red 10978 | The number 0 is real, dedu... |
mulid1i 10979 | Identity law for multiplic... |
mulid2i 10980 | Identity law for multiplic... |
addcli 10981 | Closure law for addition. ... |
mulcli 10982 | Closure law for multiplica... |
mulcomi 10983 | Commutative law for multip... |
mulcomli 10984 | Commutative law for multip... |
addassi 10985 | Associative law for additi... |
mulassi 10986 | Associative law for multip... |
adddii 10987 | Distributive law (left-dis... |
adddiri 10988 | Distributive law (right-di... |
recni 10989 | A real number is a complex... |
readdcli 10990 | Closure law for addition o... |
remulcli 10991 | Closure law for multiplica... |
mulid1d 10992 | Identity law for multiplic... |
mulid2d 10993 | Identity law for multiplic... |
addcld 10994 | Closure law for addition. ... |
mulcld 10995 | Closure law for multiplica... |
mulcomd 10996 | Commutative law for multip... |
addassd 10997 | Associative law for additi... |
mulassd 10998 | Associative law for multip... |
adddid 10999 | Distributive law (left-dis... |
adddird 11000 | Distributive law (right-di... |
adddirp1d 11001 | Distributive law, plus 1 v... |
joinlmuladdmuld 11002 | Join AB+CB into (A+C) on L... |
recnd 11003 | Deduction from real number... |
readdcld 11004 | Closure law for addition o... |
remulcld 11005 | Closure law for multiplica... |
pnfnre 11016 | Plus infinity is not a rea... |
pnfnre2 11017 | Plus infinity is not a rea... |
mnfnre 11018 | Minus infinity is not a re... |
ressxr 11019 | The standard reals are a s... |
rexpssxrxp 11020 | The Cartesian product of s... |
rexr 11021 | A standard real is an exte... |
0xr 11022 | Zero is an extended real. ... |
renepnf 11023 | No (finite) real equals pl... |
renemnf 11024 | No real equals minus infin... |
rexrd 11025 | A standard real is an exte... |
renepnfd 11026 | No (finite) real equals pl... |
renemnfd 11027 | No real equals minus infin... |
pnfex 11028 | Plus infinity exists. (Co... |
pnfxr 11029 | Plus infinity belongs to t... |
pnfnemnf 11030 | Plus and minus infinity ar... |
mnfnepnf 11031 | Minus and plus infinity ar... |
mnfxr 11032 | Minus infinity belongs to ... |
rexri 11033 | A standard real is an exte... |
1xr 11034 | ` 1 ` is an extended real ... |
renfdisj 11035 | The reals and the infiniti... |
ltrelxr 11036 | "Less than" is a relation ... |
ltrel 11037 | "Less than" is a relation.... |
lerelxr 11038 | "Less than or equal to" is... |
lerel 11039 | "Less than or equal to" is... |
xrlenlt 11040 | "Less than or equal to" ex... |
xrlenltd 11041 | "Less than or equal to" ex... |
xrltnle 11042 | "Less than" expressed in t... |
xrnltled 11043 | "Not less than" implies "l... |
ssxr 11044 | The three (non-exclusive) ... |
ltxrlt 11045 | The standard less-than ` <... |
axlttri 11046 | Ordering on reals satisfie... |
axlttrn 11047 | Ordering on reals is trans... |
axltadd 11048 | Ordering property of addit... |
axmulgt0 11049 | The product of two positiv... |
axsup 11050 | A nonempty, bounded-above ... |
lttr 11051 | Alias for ~ axlttrn , for ... |
mulgt0 11052 | The product of two positiv... |
lenlt 11053 | 'Less than or equal to' ex... |
ltnle 11054 | 'Less than' expressed in t... |
ltso 11055 | 'Less than' is a strict or... |
gtso 11056 | 'Greater than' is a strict... |
lttri2 11057 | Consequence of trichotomy.... |
lttri3 11058 | Trichotomy law for 'less t... |
lttri4 11059 | Trichotomy law for 'less t... |
letri3 11060 | Trichotomy law. (Contribu... |
leloe 11061 | 'Less than or equal to' ex... |
eqlelt 11062 | Equality in terms of 'less... |
ltle 11063 | 'Less than' implies 'less ... |
leltne 11064 | 'Less than or equal to' im... |
lelttr 11065 | Transitive law. (Contribu... |
leltletr 11066 | Transitive law, weaker for... |
ltletr 11067 | Transitive law. (Contribu... |
ltleletr 11068 | Transitive law, weaker for... |
letr 11069 | Transitive law. (Contribu... |
ltnr 11070 | 'Less than' is irreflexive... |
leid 11071 | 'Less than or equal to' is... |
ltne 11072 | 'Less than' implies not eq... |
ltnsym 11073 | 'Less than' is not symmetr... |
ltnsym2 11074 | 'Less than' is antisymmetr... |
letric 11075 | Trichotomy law. (Contribu... |
ltlen 11076 | 'Less than' expressed in t... |
eqle 11077 | Equality implies 'less tha... |
eqled 11078 | Equality implies 'less tha... |
ltadd2 11079 | Addition to both sides of ... |
ne0gt0 11080 | A nonzero nonnegative numb... |
lecasei 11081 | Ordering elimination by ca... |
lelttric 11082 | Trichotomy law. (Contribu... |
ltlecasei 11083 | Ordering elimination by ca... |
ltnri 11084 | 'Less than' is irreflexive... |
eqlei 11085 | Equality implies 'less tha... |
eqlei2 11086 | Equality implies 'less tha... |
gtneii 11087 | 'Less than' implies not eq... |
ltneii 11088 | 'Greater than' implies not... |
lttri2i 11089 | Consequence of trichotomy.... |
lttri3i 11090 | Consequence of trichotomy.... |
letri3i 11091 | Consequence of trichotomy.... |
leloei 11092 | 'Less than or equal to' in... |
ltleni 11093 | 'Less than' expressed in t... |
ltnsymi 11094 | 'Less than' is not symmetr... |
lenlti 11095 | 'Less than or equal to' in... |
ltnlei 11096 | 'Less than' in terms of 'l... |
ltlei 11097 | 'Less than' implies 'less ... |
ltleii 11098 | 'Less than' implies 'less ... |
ltnei 11099 | 'Less than' implies not eq... |
letrii 11100 | Trichotomy law for 'less t... |
lttri 11101 | 'Less than' is transitive.... |
lelttri 11102 | 'Less than or equal to', '... |
ltletri 11103 | 'Less than', 'less than or... |
letri 11104 | 'Less than or equal to' is... |
le2tri3i 11105 | Extended trichotomy law fo... |
ltadd2i 11106 | Addition to both sides of ... |
mulgt0i 11107 | The product of two positiv... |
mulgt0ii 11108 | The product of two positiv... |
ltnrd 11109 | 'Less than' is irreflexive... |
gtned 11110 | 'Less than' implies not eq... |
ltned 11111 | 'Greater than' implies not... |
ne0gt0d 11112 | A nonzero nonnegative numb... |
lttrid 11113 | Ordering on reals satisfie... |
lttri2d 11114 | Consequence of trichotomy.... |
lttri3d 11115 | Consequence of trichotomy.... |
lttri4d 11116 | Trichotomy law for 'less t... |
letri3d 11117 | Consequence of trichotomy.... |
leloed 11118 | 'Less than or equal to' in... |
eqleltd 11119 | Equality in terms of 'less... |
ltlend 11120 | 'Less than' expressed in t... |
lenltd 11121 | 'Less than or equal to' in... |
ltnled 11122 | 'Less than' in terms of 'l... |
ltled 11123 | 'Less than' implies 'less ... |
ltnsymd 11124 | 'Less than' implies 'less ... |
nltled 11125 | 'Not less than ' implies '... |
lensymd 11126 | 'Less than or equal to' im... |
letrid 11127 | Trichotomy law for 'less t... |
leltned 11128 | 'Less than or equal to' im... |
leneltd 11129 | 'Less than or equal to' an... |
mulgt0d 11130 | The product of two positiv... |
ltadd2d 11131 | Addition to both sides of ... |
letrd 11132 | Transitive law deduction f... |
lelttrd 11133 | Transitive law deduction f... |
ltadd2dd 11134 | Addition to both sides of ... |
ltletrd 11135 | Transitive law deduction f... |
lttrd 11136 | Transitive law deduction f... |
lelttrdi 11137 | If a number is less than a... |
dedekind 11138 | The Dedekind cut theorem. ... |
dedekindle 11139 | The Dedekind cut theorem, ... |
mul12 11140 | Commutative/associative la... |
mul32 11141 | Commutative/associative la... |
mul31 11142 | Commutative/associative la... |
mul4 11143 | Rearrangement of 4 factors... |
mul4r 11144 | Rearrangement of 4 factors... |
muladd11 11145 | A simple product of sums e... |
1p1times 11146 | Two times a number. (Cont... |
peano2cn 11147 | A theorem for complex numb... |
peano2re 11148 | A theorem for reals analog... |
readdcan 11149 | Cancellation law for addit... |
00id 11150 | ` 0 ` is its own additive ... |
mul02lem1 11151 | Lemma for ~ mul02 . If an... |
mul02lem2 11152 | Lemma for ~ mul02 . Zero ... |
mul02 11153 | Multiplication by ` 0 ` . ... |
mul01 11154 | Multiplication by ` 0 ` . ... |
addid1 11155 | ` 0 ` is an additive ident... |
cnegex 11156 | Existence of the negative ... |
cnegex2 11157 | Existence of a left invers... |
addid2 11158 | ` 0 ` is a left identity f... |
addcan 11159 | Cancellation law for addit... |
addcan2 11160 | Cancellation law for addit... |
addcom 11161 | Addition commutes. This u... |
addid1i 11162 | ` 0 ` is an additive ident... |
addid2i 11163 | ` 0 ` is a left identity f... |
mul02i 11164 | Multiplication by 0. Theo... |
mul01i 11165 | Multiplication by ` 0 ` . ... |
addcomi 11166 | Addition commutes. Based ... |
addcomli 11167 | Addition commutes. (Contr... |
addcani 11168 | Cancellation law for addit... |
addcan2i 11169 | Cancellation law for addit... |
mul12i 11170 | Commutative/associative la... |
mul32i 11171 | Commutative/associative la... |
mul4i 11172 | Rearrangement of 4 factors... |
mul02d 11173 | Multiplication by 0. Theo... |
mul01d 11174 | Multiplication by ` 0 ` . ... |
addid1d 11175 | ` 0 ` is an additive ident... |
addid2d 11176 | ` 0 ` is a left identity f... |
addcomd 11177 | Addition commutes. Based ... |
addcand 11178 | Cancellation law for addit... |
addcan2d 11179 | Cancellation law for addit... |
addcanad 11180 | Cancelling a term on the l... |
addcan2ad 11181 | Cancelling a term on the r... |
addneintrd 11182 | Introducing a term on the ... |
addneintr2d 11183 | Introducing a term on the ... |
mul12d 11184 | Commutative/associative la... |
mul32d 11185 | Commutative/associative la... |
mul31d 11186 | Commutative/associative la... |
mul4d 11187 | Rearrangement of 4 factors... |
muladd11r 11188 | A simple product of sums e... |
comraddd 11189 | Commute RHS addition, in d... |
ltaddneg 11190 | Adding a negative number t... |
ltaddnegr 11191 | Adding a negative number t... |
add12 11192 | Commutative/associative la... |
add32 11193 | Commutative/associative la... |
add32r 11194 | Commutative/associative la... |
add4 11195 | Rearrangement of 4 terms i... |
add42 11196 | Rearrangement of 4 terms i... |
add12i 11197 | Commutative/associative la... |
add32i 11198 | Commutative/associative la... |
add4i 11199 | Rearrangement of 4 terms i... |
add42i 11200 | Rearrangement of 4 terms i... |
add12d 11201 | Commutative/associative la... |
add32d 11202 | Commutative/associative la... |
add4d 11203 | Rearrangement of 4 terms i... |
add42d 11204 | Rearrangement of 4 terms i... |
0cnALT 11209 | Alternate proof of ~ 0cn w... |
0cnALT2 11210 | Alternate proof of ~ 0cnAL... |
negeu 11211 | Existential uniqueness of ... |
subval 11212 | Value of subtraction, whic... |
negeq 11213 | Equality theorem for negat... |
negeqi 11214 | Equality inference for neg... |
negeqd 11215 | Equality deduction for neg... |
nfnegd 11216 | Deduction version of ~ nfn... |
nfneg 11217 | Bound-variable hypothesis ... |
csbnegg 11218 | Move class substitution in... |
negex 11219 | A negative is a set. (Con... |
subcl 11220 | Closure law for subtractio... |
negcl 11221 | Closure law for negative. ... |
negicn 11222 | ` -u _i ` is a complex num... |
subf 11223 | Subtraction is an operatio... |
subadd 11224 | Relationship between subtr... |
subadd2 11225 | Relationship between subtr... |
subsub23 11226 | Swap subtrahend and result... |
pncan 11227 | Cancellation law for subtr... |
pncan2 11228 | Cancellation law for subtr... |
pncan3 11229 | Subtraction and addition o... |
npcan 11230 | Cancellation law for subtr... |
addsubass 11231 | Associative-type law for a... |
addsub 11232 | Law for addition and subtr... |
subadd23 11233 | Commutative/associative la... |
addsub12 11234 | Commutative/associative la... |
2addsub 11235 | Law for subtraction and ad... |
addsubeq4 11236 | Relation between sums and ... |
pncan3oi 11237 | Subtraction and addition o... |
mvrraddi 11238 | Move the right term in a s... |
mvlladdi 11239 | Move the left term in a su... |
subid 11240 | Subtraction of a number fr... |
subid1 11241 | Identity law for subtracti... |
npncan 11242 | Cancellation law for subtr... |
nppcan 11243 | Cancellation law for subtr... |
nnpcan 11244 | Cancellation law for subtr... |
nppcan3 11245 | Cancellation law for subtr... |
subcan2 11246 | Cancellation law for subtr... |
subeq0 11247 | If the difference between ... |
npncan2 11248 | Cancellation law for subtr... |
subsub2 11249 | Law for double subtraction... |
nncan 11250 | Cancellation law for subtr... |
subsub 11251 | Law for double subtraction... |
nppcan2 11252 | Cancellation law for subtr... |
subsub3 11253 | Law for double subtraction... |
subsub4 11254 | Law for double subtraction... |
sub32 11255 | Swap the second and third ... |
nnncan 11256 | Cancellation law for subtr... |
nnncan1 11257 | Cancellation law for subtr... |
nnncan2 11258 | Cancellation law for subtr... |
npncan3 11259 | Cancellation law for subtr... |
pnpcan 11260 | Cancellation law for mixed... |
pnpcan2 11261 | Cancellation law for mixed... |
pnncan 11262 | Cancellation law for mixed... |
ppncan 11263 | Cancellation law for mixed... |
addsub4 11264 | Rearrangement of 4 terms i... |
subadd4 11265 | Rearrangement of 4 terms i... |
sub4 11266 | Rearrangement of 4 terms i... |
neg0 11267 | Minus 0 equals 0. (Contri... |
negid 11268 | Addition of a number and i... |
negsub 11269 | Relationship between subtr... |
subneg 11270 | Relationship between subtr... |
negneg 11271 | A number is equal to the n... |
neg11 11272 | Negative is one-to-one. (... |
negcon1 11273 | Negative contraposition la... |
negcon2 11274 | Negative contraposition la... |
negeq0 11275 | A number is zero iff its n... |
subcan 11276 | Cancellation law for subtr... |
negsubdi 11277 | Distribution of negative o... |
negdi 11278 | Distribution of negative o... |
negdi2 11279 | Distribution of negative o... |
negsubdi2 11280 | Distribution of negative o... |
neg2sub 11281 | Relationship between subtr... |
renegcli 11282 | Closure law for negative o... |
resubcli 11283 | Closure law for subtractio... |
renegcl 11284 | Closure law for negative o... |
resubcl 11285 | Closure law for subtractio... |
negreb 11286 | The negative of a real is ... |
peano2cnm 11287 | "Reverse" second Peano pos... |
peano2rem 11288 | "Reverse" second Peano pos... |
negcli 11289 | Closure law for negative. ... |
negidi 11290 | Addition of a number and i... |
negnegi 11291 | A number is equal to the n... |
subidi 11292 | Subtraction of a number fr... |
subid1i 11293 | Identity law for subtracti... |
negne0bi 11294 | A number is nonzero iff it... |
negrebi 11295 | The negative of a real is ... |
negne0i 11296 | The negative of a nonzero ... |
subcli 11297 | Closure law for subtractio... |
pncan3i 11298 | Subtraction and addition o... |
negsubi 11299 | Relationship between subtr... |
subnegi 11300 | Relationship between subtr... |
subeq0i 11301 | If the difference between ... |
neg11i 11302 | Negative is one-to-one. (... |
negcon1i 11303 | Negative contraposition la... |
negcon2i 11304 | Negative contraposition la... |
negdii 11305 | Distribution of negative o... |
negsubdii 11306 | Distribution of negative o... |
negsubdi2i 11307 | Distribution of negative o... |
subaddi 11308 | Relationship between subtr... |
subadd2i 11309 | Relationship between subtr... |
subaddrii 11310 | Relationship between subtr... |
subsub23i 11311 | Swap subtrahend and result... |
addsubassi 11312 | Associative-type law for s... |
addsubi 11313 | Law for subtraction and ad... |
subcani 11314 | Cancellation law for subtr... |
subcan2i 11315 | Cancellation law for subtr... |
pnncani 11316 | Cancellation law for mixed... |
addsub4i 11317 | Rearrangement of 4 terms i... |
0reALT 11318 | Alternate proof of ~ 0re .... |
negcld 11319 | Closure law for negative. ... |
subidd 11320 | Subtraction of a number fr... |
subid1d 11321 | Identity law for subtracti... |
negidd 11322 | Addition of a number and i... |
negnegd 11323 | A number is equal to the n... |
negeq0d 11324 | A number is zero iff its n... |
negne0bd 11325 | A number is nonzero iff it... |
negcon1d 11326 | Contraposition law for una... |
negcon1ad 11327 | Contraposition law for una... |
neg11ad 11328 | The negatives of two compl... |
negned 11329 | If two complex numbers are... |
negne0d 11330 | The negative of a nonzero ... |
negrebd 11331 | The negative of a real is ... |
subcld 11332 | Closure law for subtractio... |
pncand 11333 | Cancellation law for subtr... |
pncan2d 11334 | Cancellation law for subtr... |
pncan3d 11335 | Subtraction and addition o... |
npcand 11336 | Cancellation law for subtr... |
nncand 11337 | Cancellation law for subtr... |
negsubd 11338 | Relationship between subtr... |
subnegd 11339 | Relationship between subtr... |
subeq0d 11340 | If the difference between ... |
subne0d 11341 | Two unequal numbers have n... |
subeq0ad 11342 | The difference of two comp... |
subne0ad 11343 | If the difference of two c... |
neg11d 11344 | If the difference between ... |
negdid 11345 | Distribution of negative o... |
negdi2d 11346 | Distribution of negative o... |
negsubdid 11347 | Distribution of negative o... |
negsubdi2d 11348 | Distribution of negative o... |
neg2subd 11349 | Relationship between subtr... |
subaddd 11350 | Relationship between subtr... |
subadd2d 11351 | Relationship between subtr... |
addsubassd 11352 | Associative-type law for s... |
addsubd 11353 | Law for subtraction and ad... |
subadd23d 11354 | Commutative/associative la... |
addsub12d 11355 | Commutative/associative la... |
npncand 11356 | Cancellation law for subtr... |
nppcand 11357 | Cancellation law for subtr... |
nppcan2d 11358 | Cancellation law for subtr... |
nppcan3d 11359 | Cancellation law for subtr... |
subsubd 11360 | Law for double subtraction... |
subsub2d 11361 | Law for double subtraction... |
subsub3d 11362 | Law for double subtraction... |
subsub4d 11363 | Law for double subtraction... |
sub32d 11364 | Swap the second and third ... |
nnncand 11365 | Cancellation law for subtr... |
nnncan1d 11366 | Cancellation law for subtr... |
nnncan2d 11367 | Cancellation law for subtr... |
npncan3d 11368 | Cancellation law for subtr... |
pnpcand 11369 | Cancellation law for mixed... |
pnpcan2d 11370 | Cancellation law for mixed... |
pnncand 11371 | Cancellation law for mixed... |
ppncand 11372 | Cancellation law for mixed... |
subcand 11373 | Cancellation law for subtr... |
subcan2d 11374 | Cancellation law for subtr... |
subcanad 11375 | Cancellation law for subtr... |
subneintrd 11376 | Introducing subtraction on... |
subcan2ad 11377 | Cancellation law for subtr... |
subneintr2d 11378 | Introducing subtraction on... |
addsub4d 11379 | Rearrangement of 4 terms i... |
subadd4d 11380 | Rearrangement of 4 terms i... |
sub4d 11381 | Rearrangement of 4 terms i... |
2addsubd 11382 | Law for subtraction and ad... |
addsubeq4d 11383 | Relation between sums and ... |
subeqxfrd 11384 | Transfer two terms of a su... |
mvlraddd 11385 | Move the right term in a s... |
mvlladdd 11386 | Move the left term in a su... |
mvrraddd 11387 | Move the right term in a s... |
mvrladdd 11388 | Move the left term in a su... |
assraddsubd 11389 | Associate RHS addition-sub... |
subaddeqd 11390 | Transfer two terms of a su... |
addlsub 11391 | Left-subtraction: Subtrac... |
addrsub 11392 | Right-subtraction: Subtra... |
subexsub 11393 | A subtraction law: Exchan... |
addid0 11394 | If adding a number to a an... |
addn0nid 11395 | Adding a nonzero number to... |
pnpncand 11396 | Addition/subtraction cance... |
subeqrev 11397 | Reverse the order of subtr... |
addeq0 11398 | Two complex numbers add up... |
pncan1 11399 | Cancellation law for addit... |
npcan1 11400 | Cancellation law for subtr... |
subeq0bd 11401 | If two complex numbers are... |
renegcld 11402 | Closure law for negative o... |
resubcld 11403 | Closure law for subtractio... |
negn0 11404 | The image under negation o... |
negf1o 11405 | Negation is an isomorphism... |
kcnktkm1cn 11406 | k times k minus 1 is a com... |
muladd 11407 | Product of two sums. (Con... |
subdi 11408 | Distribution of multiplica... |
subdir 11409 | Distribution of multiplica... |
ine0 11410 | The imaginary unit ` _i ` ... |
mulneg1 11411 | Product with negative is n... |
mulneg2 11412 | The product with a negativ... |
mulneg12 11413 | Swap the negative sign in ... |
mul2neg 11414 | Product of two negatives. ... |
submul2 11415 | Convert a subtraction to a... |
mulm1 11416 | Product with minus one is ... |
addneg1mul 11417 | Addition with product with... |
mulsub 11418 | Product of two differences... |
mulsub2 11419 | Swap the order of subtract... |
mulm1i 11420 | Product with minus one is ... |
mulneg1i 11421 | Product with negative is n... |
mulneg2i 11422 | Product with negative is n... |
mul2negi 11423 | Product of two negatives. ... |
subdii 11424 | Distribution of multiplica... |
subdiri 11425 | Distribution of multiplica... |
muladdi 11426 | Product of two sums. (Con... |
mulm1d 11427 | Product with minus one is ... |
mulneg1d 11428 | Product with negative is n... |
mulneg2d 11429 | Product with negative is n... |
mul2negd 11430 | Product of two negatives. ... |
subdid 11431 | Distribution of multiplica... |
subdird 11432 | Distribution of multiplica... |
muladdd 11433 | Product of two sums. (Con... |
mulsubd 11434 | Product of two differences... |
muls1d 11435 | Multiplication by one minu... |
mulsubfacd 11436 | Multiplication followed by... |
addmulsub 11437 | The product of a sum and a... |
subaddmulsub 11438 | The difference with a prod... |
mulsubaddmulsub 11439 | A special difference of a ... |
gt0ne0 11440 | Positive implies nonzero. ... |
lt0ne0 11441 | A number which is less tha... |
ltadd1 11442 | Addition to both sides of ... |
leadd1 11443 | Addition to both sides of ... |
leadd2 11444 | Addition to both sides of ... |
ltsubadd 11445 | 'Less than' relationship b... |
ltsubadd2 11446 | 'Less than' relationship b... |
lesubadd 11447 | 'Less than or equal to' re... |
lesubadd2 11448 | 'Less than or equal to' re... |
ltaddsub 11449 | 'Less than' relationship b... |
ltaddsub2 11450 | 'Less than' relationship b... |
leaddsub 11451 | 'Less than or equal to' re... |
leaddsub2 11452 | 'Less than or equal to' re... |
suble 11453 | Swap subtrahends in an ine... |
lesub 11454 | Swap subtrahends in an ine... |
ltsub23 11455 | 'Less than' relationship b... |
ltsub13 11456 | 'Less than' relationship b... |
le2add 11457 | Adding both sides of two '... |
ltleadd 11458 | Adding both sides of two o... |
leltadd 11459 | Adding both sides of two o... |
lt2add 11460 | Adding both sides of two '... |
addgt0 11461 | The sum of 2 positive numb... |
addgegt0 11462 | The sum of nonnegative and... |
addgtge0 11463 | The sum of nonnegative and... |
addge0 11464 | The sum of 2 nonnegative n... |
ltaddpos 11465 | Adding a positive number t... |
ltaddpos2 11466 | Adding a positive number t... |
ltsubpos 11467 | Subtracting a positive num... |
posdif 11468 | Comparison of two numbers ... |
lesub1 11469 | Subtraction from both side... |
lesub2 11470 | Subtraction of both sides ... |
ltsub1 11471 | Subtraction from both side... |
ltsub2 11472 | Subtraction of both sides ... |
lt2sub 11473 | Subtracting both sides of ... |
le2sub 11474 | Subtracting both sides of ... |
ltneg 11475 | Negative of both sides of ... |
ltnegcon1 11476 | Contraposition of negative... |
ltnegcon2 11477 | Contraposition of negative... |
leneg 11478 | Negative of both sides of ... |
lenegcon1 11479 | Contraposition of negative... |
lenegcon2 11480 | Contraposition of negative... |
lt0neg1 11481 | Comparison of a number and... |
lt0neg2 11482 | Comparison of a number and... |
le0neg1 11483 | Comparison of a number and... |
le0neg2 11484 | Comparison of a number and... |
addge01 11485 | A number is less than or e... |
addge02 11486 | A number is less than or e... |
add20 11487 | Two nonnegative numbers ar... |
subge0 11488 | Nonnegative subtraction. ... |
suble0 11489 | Nonpositive subtraction. ... |
leaddle0 11490 | The sum of a real number a... |
subge02 11491 | Nonnegative subtraction. ... |
lesub0 11492 | Lemma to show a nonnegativ... |
mulge0 11493 | The product of two nonnega... |
mullt0 11494 | The product of two negativ... |
msqgt0 11495 | A nonzero square is positi... |
msqge0 11496 | A square is nonnegative. ... |
0lt1 11497 | 0 is less than 1. Theorem... |
0le1 11498 | 0 is less than or equal to... |
relin01 11499 | An interval law for less t... |
ltordlem 11500 | Lemma for ~ ltord1 . (Con... |
ltord1 11501 | Infer an ordering relation... |
leord1 11502 | Infer an ordering relation... |
eqord1 11503 | A strictly increasing real... |
ltord2 11504 | Infer an ordering relation... |
leord2 11505 | Infer an ordering relation... |
eqord2 11506 | A strictly decreasing real... |
wloglei 11507 | Form of ~ wlogle where bot... |
wlogle 11508 | If the predicate ` ch ( x ... |
leidi 11509 | 'Less than or equal to' is... |
gt0ne0i 11510 | Positive means nonzero (us... |
gt0ne0ii 11511 | Positive implies nonzero. ... |
msqgt0i 11512 | A nonzero square is positi... |
msqge0i 11513 | A square is nonnegative. ... |
addgt0i 11514 | Addition of 2 positive num... |
addge0i 11515 | Addition of 2 nonnegative ... |
addgegt0i 11516 | Addition of nonnegative an... |
addgt0ii 11517 | Addition of 2 positive num... |
add20i 11518 | Two nonnegative numbers ar... |
ltnegi 11519 | Negative of both sides of ... |
lenegi 11520 | Negative of both sides of ... |
ltnegcon2i 11521 | Contraposition of negative... |
mulge0i 11522 | The product of two nonnega... |
lesub0i 11523 | Lemma to show a nonnegativ... |
ltaddposi 11524 | Adding a positive number t... |
posdifi 11525 | Comparison of two numbers ... |
ltnegcon1i 11526 | Contraposition of negative... |
lenegcon1i 11527 | Contraposition of negative... |
subge0i 11528 | Nonnegative subtraction. ... |
ltadd1i 11529 | Addition to both sides of ... |
leadd1i 11530 | Addition to both sides of ... |
leadd2i 11531 | Addition to both sides of ... |
ltsubaddi 11532 | 'Less than' relationship b... |
lesubaddi 11533 | 'Less than or equal to' re... |
ltsubadd2i 11534 | 'Less than' relationship b... |
lesubadd2i 11535 | 'Less than or equal to' re... |
ltaddsubi 11536 | 'Less than' relationship b... |
lt2addi 11537 | Adding both side of two in... |
le2addi 11538 | Adding both side of two in... |
gt0ne0d 11539 | Positive implies nonzero. ... |
lt0ne0d 11540 | Something less than zero i... |
leidd 11541 | 'Less than or equal to' is... |
msqgt0d 11542 | A nonzero square is positi... |
msqge0d 11543 | A square is nonnegative. ... |
lt0neg1d 11544 | Comparison of a number and... |
lt0neg2d 11545 | Comparison of a number and... |
le0neg1d 11546 | Comparison of a number and... |
le0neg2d 11547 | Comparison of a number and... |
addgegt0d 11548 | Addition of nonnegative an... |
addgtge0d 11549 | Addition of positive and n... |
addgt0d 11550 | Addition of 2 positive num... |
addge0d 11551 | Addition of 2 nonnegative ... |
mulge0d 11552 | The product of two nonnega... |
ltnegd 11553 | Negative of both sides of ... |
lenegd 11554 | Negative of both sides of ... |
ltnegcon1d 11555 | Contraposition of negative... |
ltnegcon2d 11556 | Contraposition of negative... |
lenegcon1d 11557 | Contraposition of negative... |
lenegcon2d 11558 | Contraposition of negative... |
ltaddposd 11559 | Adding a positive number t... |
ltaddpos2d 11560 | Adding a positive number t... |
ltsubposd 11561 | Subtracting a positive num... |
posdifd 11562 | Comparison of two numbers ... |
addge01d 11563 | A number is less than or e... |
addge02d 11564 | A number is less than or e... |
subge0d 11565 | Nonnegative subtraction. ... |
suble0d 11566 | Nonpositive subtraction. ... |
subge02d 11567 | Nonnegative subtraction. ... |
ltadd1d 11568 | Addition to both sides of ... |
leadd1d 11569 | Addition to both sides of ... |
leadd2d 11570 | Addition to both sides of ... |
ltsubaddd 11571 | 'Less than' relationship b... |
lesubaddd 11572 | 'Less than or equal to' re... |
ltsubadd2d 11573 | 'Less than' relationship b... |
lesubadd2d 11574 | 'Less than or equal to' re... |
ltaddsubd 11575 | 'Less than' relationship b... |
ltaddsub2d 11576 | 'Less than' relationship b... |
leaddsub2d 11577 | 'Less than or equal to' re... |
subled 11578 | Swap subtrahends in an ine... |
lesubd 11579 | Swap subtrahends in an ine... |
ltsub23d 11580 | 'Less than' relationship b... |
ltsub13d 11581 | 'Less than' relationship b... |
lesub1d 11582 | Subtraction from both side... |
lesub2d 11583 | Subtraction of both sides ... |
ltsub1d 11584 | Subtraction from both side... |
ltsub2d 11585 | Subtraction of both sides ... |
ltadd1dd 11586 | Addition to both sides of ... |
ltsub1dd 11587 | Subtraction from both side... |
ltsub2dd 11588 | Subtraction of both sides ... |
leadd1dd 11589 | Addition to both sides of ... |
leadd2dd 11590 | Addition to both sides of ... |
lesub1dd 11591 | Subtraction from both side... |
lesub2dd 11592 | Subtraction of both sides ... |
lesub3d 11593 | The result of subtracting ... |
le2addd 11594 | Adding both side of two in... |
le2subd 11595 | Subtracting both sides of ... |
ltleaddd 11596 | Adding both sides of two o... |
leltaddd 11597 | Adding both sides of two o... |
lt2addd 11598 | Adding both side of two in... |
lt2subd 11599 | Subtracting both sides of ... |
possumd 11600 | Condition for a positive s... |
sublt0d 11601 | When a subtraction gives a... |
ltaddsublt 11602 | Addition and subtraction o... |
1le1 11603 | One is less than or equal ... |
ixi 11604 | ` _i ` times itself is min... |
recextlem1 11605 | Lemma for ~ recex . (Cont... |
recextlem2 11606 | Lemma for ~ recex . (Cont... |
recex 11607 | Existence of reciprocal of... |
mulcand 11608 | Cancellation law for multi... |
mulcan2d 11609 | Cancellation law for multi... |
mulcanad 11610 | Cancellation of a nonzero ... |
mulcan2ad 11611 | Cancellation of a nonzero ... |
mulcan 11612 | Cancellation law for multi... |
mulcan2 11613 | Cancellation law for multi... |
mulcani 11614 | Cancellation law for multi... |
mul0or 11615 | If a product is zero, one ... |
mulne0b 11616 | The product of two nonzero... |
mulne0 11617 | The product of two nonzero... |
mulne0i 11618 | The product of two nonzero... |
muleqadd 11619 | Property of numbers whose ... |
receu 11620 | Existential uniqueness of ... |
mulnzcnopr 11621 | Multiplication maps nonzer... |
msq0i 11622 | A number is zero iff its s... |
mul0ori 11623 | If a product is zero, one ... |
msq0d 11624 | A number is zero iff its s... |
mul0ord 11625 | If a product is zero, one ... |
mulne0bd 11626 | The product of two nonzero... |
mulne0d 11627 | The product of two nonzero... |
mulcan1g 11628 | A generalized form of the ... |
mulcan2g 11629 | A generalized form of the ... |
mulne0bad 11630 | A factor of a nonzero comp... |
mulne0bbd 11631 | A factor of a nonzero comp... |
1div0 11634 | You can't divide by zero, ... |
divval 11635 | Value of division: if ` A ... |
divmul 11636 | Relationship between divis... |
divmul2 11637 | Relationship between divis... |
divmul3 11638 | Relationship between divis... |
divcl 11639 | Closure law for division. ... |
reccl 11640 | Closure law for reciprocal... |
divcan2 11641 | A cancellation law for div... |
divcan1 11642 | A cancellation law for div... |
diveq0 11643 | A ratio is zero iff the nu... |
divne0b 11644 | The ratio of nonzero numbe... |
divne0 11645 | The ratio of nonzero numbe... |
recne0 11646 | The reciprocal of a nonzer... |
recid 11647 | Multiplication of a number... |
recid2 11648 | Multiplication of a number... |
divrec 11649 | Relationship between divis... |
divrec2 11650 | Relationship between divis... |
divass 11651 | An associative law for div... |
div23 11652 | A commutative/associative ... |
div32 11653 | A commutative/associative ... |
div13 11654 | A commutative/associative ... |
div12 11655 | A commutative/associative ... |
divmulass 11656 | An associative law for div... |
divmulasscom 11657 | An associative/commutative... |
divdir 11658 | Distribution of division o... |
divcan3 11659 | A cancellation law for div... |
divcan4 11660 | A cancellation law for div... |
div11 11661 | One-to-one relationship fo... |
divid 11662 | A number divided by itself... |
div0 11663 | Division into zero is zero... |
div1 11664 | A number divided by 1 is i... |
1div1e1 11665 | 1 divided by 1 is 1. (Con... |
diveq1 11666 | Equality in terms of unit ... |
divneg 11667 | Move negative sign inside ... |
muldivdir 11668 | Distribution of division o... |
divsubdir 11669 | Distribution of division o... |
subdivcomb1 11670 | Bring a term in a subtract... |
subdivcomb2 11671 | Bring a term in a subtract... |
recrec 11672 | A number is equal to the r... |
rec11 11673 | Reciprocal is one-to-one. ... |
rec11r 11674 | Mutual reciprocals. (Cont... |
divmuldiv 11675 | Multiplication of two rati... |
divdivdiv 11676 | Division of two ratios. T... |
divcan5 11677 | Cancellation of common fac... |
divmul13 11678 | Swap the denominators in t... |
divmul24 11679 | Swap the numerators in the... |
divmuleq 11680 | Cross-multiply in an equal... |
recdiv 11681 | The reciprocal of a ratio.... |
divcan6 11682 | Cancellation of inverted f... |
divdiv32 11683 | Swap denominators in a div... |
divcan7 11684 | Cancel equal divisors in a... |
dmdcan 11685 | Cancellation law for divis... |
divdiv1 11686 | Division into a fraction. ... |
divdiv2 11687 | Division by a fraction. (... |
recdiv2 11688 | Division into a reciprocal... |
ddcan 11689 | Cancellation in a double d... |
divadddiv 11690 | Addition of two ratios. T... |
divsubdiv 11691 | Subtraction of two ratios.... |
conjmul 11692 | Two numbers whose reciproc... |
rereccl 11693 | Closure law for reciprocal... |
redivcl 11694 | Closure law for division o... |
eqneg 11695 | A number equal to its nega... |
eqnegd 11696 | A complex number equals it... |
eqnegad 11697 | If a complex number equals... |
div2neg 11698 | Quotient of two negatives.... |
divneg2 11699 | Move negative sign inside ... |
recclzi 11700 | Closure law for reciprocal... |
recne0zi 11701 | The reciprocal of a nonzer... |
recidzi 11702 | Multiplication of a number... |
div1i 11703 | A number divided by 1 is i... |
eqnegi 11704 | A number equal to its nega... |
reccli 11705 | Closure law for reciprocal... |
recidi 11706 | Multiplication of a number... |
recreci 11707 | A number is equal to the r... |
dividi 11708 | A number divided by itself... |
div0i 11709 | Division into zero is zero... |
divclzi 11710 | Closure law for division. ... |
divcan1zi 11711 | A cancellation law for div... |
divcan2zi 11712 | A cancellation law for div... |
divreczi 11713 | Relationship between divis... |
divcan3zi 11714 | A cancellation law for div... |
divcan4zi 11715 | A cancellation law for div... |
rec11i 11716 | Reciprocal is one-to-one. ... |
divcli 11717 | Closure law for division. ... |
divcan2i 11718 | A cancellation law for div... |
divcan1i 11719 | A cancellation law for div... |
divreci 11720 | Relationship between divis... |
divcan3i 11721 | A cancellation law for div... |
divcan4i 11722 | A cancellation law for div... |
divne0i 11723 | The ratio of nonzero numbe... |
rec11ii 11724 | Reciprocal is one-to-one. ... |
divasszi 11725 | An associative law for div... |
divmulzi 11726 | Relationship between divis... |
divdirzi 11727 | Distribution of division o... |
divdiv23zi 11728 | Swap denominators in a div... |
divmuli 11729 | Relationship between divis... |
divdiv32i 11730 | Swap denominators in a div... |
divassi 11731 | An associative law for div... |
divdiri 11732 | Distribution of division o... |
div23i 11733 | A commutative/associative ... |
div11i 11734 | One-to-one relationship fo... |
divmuldivi 11735 | Multiplication of two rati... |
divmul13i 11736 | Swap denominators of two r... |
divadddivi 11737 | Addition of two ratios. T... |
divdivdivi 11738 | Division of two ratios. T... |
rerecclzi 11739 | Closure law for reciprocal... |
rereccli 11740 | Closure law for reciprocal... |
redivclzi 11741 | Closure law for division o... |
redivcli 11742 | Closure law for division o... |
div1d 11743 | A number divided by 1 is i... |
reccld 11744 | Closure law for reciprocal... |
recne0d 11745 | The reciprocal of a nonzer... |
recidd 11746 | Multiplication of a number... |
recid2d 11747 | Multiplication of a number... |
recrecd 11748 | A number is equal to the r... |
dividd 11749 | A number divided by itself... |
div0d 11750 | Division into zero is zero... |
divcld 11751 | Closure law for division. ... |
divcan1d 11752 | A cancellation law for div... |
divcan2d 11753 | A cancellation law for div... |
divrecd 11754 | Relationship between divis... |
divrec2d 11755 | Relationship between divis... |
divcan3d 11756 | A cancellation law for div... |
divcan4d 11757 | A cancellation law for div... |
diveq0d 11758 | A ratio is zero iff the nu... |
diveq1d 11759 | Equality in terms of unit ... |
diveq1ad 11760 | The quotient of two comple... |
diveq0ad 11761 | A fraction of complex numb... |
divne1d 11762 | If two complex numbers are... |
divne0bd 11763 | A ratio is zero iff the nu... |
divnegd 11764 | Move negative sign inside ... |
divneg2d 11765 | Move negative sign inside ... |
div2negd 11766 | Quotient of two negatives.... |
divne0d 11767 | The ratio of nonzero numbe... |
recdivd 11768 | The reciprocal of a ratio.... |
recdiv2d 11769 | Division into a reciprocal... |
divcan6d 11770 | Cancellation of inverted f... |
ddcand 11771 | Cancellation in a double d... |
rec11d 11772 | Reciprocal is one-to-one. ... |
divmuld 11773 | Relationship between divis... |
div32d 11774 | A commutative/associative ... |
div13d 11775 | A commutative/associative ... |
divdiv32d 11776 | Swap denominators in a div... |
divcan5d 11777 | Cancellation of common fac... |
divcan5rd 11778 | Cancellation of common fac... |
divcan7d 11779 | Cancel equal divisors in a... |
dmdcand 11780 | Cancellation law for divis... |
dmdcan2d 11781 | Cancellation law for divis... |
divdiv1d 11782 | Division into a fraction. ... |
divdiv2d 11783 | Division by a fraction. (... |
divmul2d 11784 | Relationship between divis... |
divmul3d 11785 | Relationship between divis... |
divassd 11786 | An associative law for div... |
div12d 11787 | A commutative/associative ... |
div23d 11788 | A commutative/associative ... |
divdird 11789 | Distribution of division o... |
divsubdird 11790 | Distribution of division o... |
div11d 11791 | One-to-one relationship fo... |
divmuldivd 11792 | Multiplication of two rati... |
divmul13d 11793 | Swap denominators of two r... |
divmul24d 11794 | Swap the numerators in the... |
divadddivd 11795 | Addition of two ratios. T... |
divsubdivd 11796 | Subtraction of two ratios.... |
divmuleqd 11797 | Cross-multiply in an equal... |
divdivdivd 11798 | Division of two ratios. T... |
diveq1bd 11799 | If two complex numbers are... |
div2sub 11800 | Swap the order of subtract... |
div2subd 11801 | Swap subtrahend and minuen... |
rereccld 11802 | Closure law for reciprocal... |
redivcld 11803 | Closure law for division o... |
subrec 11804 | Subtraction of reciprocals... |
subreci 11805 | Subtraction of reciprocals... |
subrecd 11806 | Subtraction of reciprocals... |
mvllmuld 11807 | Move the left term in a pr... |
mvllmuli 11808 | Move the left term in a pr... |
ldiv 11809 | Left-division. (Contribut... |
rdiv 11810 | Right-division. (Contribu... |
mdiv 11811 | A division law. (Contribu... |
lineq 11812 | Solution of a (scalar) lin... |
elimgt0 11813 | Hypothesis for weak deduct... |
elimge0 11814 | Hypothesis for weak deduct... |
ltp1 11815 | A number is less than itse... |
lep1 11816 | A number is less than or e... |
ltm1 11817 | A number minus 1 is less t... |
lem1 11818 | A number minus 1 is less t... |
letrp1 11819 | A transitive property of '... |
p1le 11820 | A transitive property of p... |
recgt0 11821 | The reciprocal of a positi... |
prodgt0 11822 | Infer that a multiplicand ... |
prodgt02 11823 | Infer that a multiplier is... |
ltmul1a 11824 | Lemma for ~ ltmul1 . Mult... |
ltmul1 11825 | Multiplication of both sid... |
ltmul2 11826 | Multiplication of both sid... |
lemul1 11827 | Multiplication of both sid... |
lemul2 11828 | Multiplication of both sid... |
lemul1a 11829 | Multiplication of both sid... |
lemul2a 11830 | Multiplication of both sid... |
ltmul12a 11831 | Comparison of product of t... |
lemul12b 11832 | Comparison of product of t... |
lemul12a 11833 | Comparison of product of t... |
mulgt1 11834 | The product of two numbers... |
ltmulgt11 11835 | Multiplication by a number... |
ltmulgt12 11836 | Multiplication by a number... |
lemulge11 11837 | Multiplication by a number... |
lemulge12 11838 | Multiplication by a number... |
ltdiv1 11839 | Division of both sides of ... |
lediv1 11840 | Division of both sides of ... |
gt0div 11841 | Division of a positive num... |
ge0div 11842 | Division of a nonnegative ... |
divgt0 11843 | The ratio of two positive ... |
divge0 11844 | The ratio of nonnegative a... |
mulge0b 11845 | A condition for multiplica... |
mulle0b 11846 | A condition for multiplica... |
mulsuble0b 11847 | A condition for multiplica... |
ltmuldiv 11848 | 'Less than' relationship b... |
ltmuldiv2 11849 | 'Less than' relationship b... |
ltdivmul 11850 | 'Less than' relationship b... |
ledivmul 11851 | 'Less than or equal to' re... |
ltdivmul2 11852 | 'Less than' relationship b... |
lt2mul2div 11853 | 'Less than' relationship b... |
ledivmul2 11854 | 'Less than or equal to' re... |
lemuldiv 11855 | 'Less than or equal' relat... |
lemuldiv2 11856 | 'Less than or equal' relat... |
ltrec 11857 | The reciprocal of both sid... |
lerec 11858 | The reciprocal of both sid... |
lt2msq1 11859 | Lemma for ~ lt2msq . (Con... |
lt2msq 11860 | Two nonnegative numbers co... |
ltdiv2 11861 | Division of a positive num... |
ltrec1 11862 | Reciprocal swap in a 'less... |
lerec2 11863 | Reciprocal swap in a 'less... |
ledivdiv 11864 | Invert ratios of positive ... |
lediv2 11865 | Division of a positive num... |
ltdiv23 11866 | Swap denominator with othe... |
lediv23 11867 | Swap denominator with othe... |
lediv12a 11868 | Comparison of ratio of two... |
lediv2a 11869 | Division of both sides of ... |
reclt1 11870 | The reciprocal of a positi... |
recgt1 11871 | The reciprocal of a positi... |
recgt1i 11872 | The reciprocal of a number... |
recp1lt1 11873 | Construct a number less th... |
recreclt 11874 | Given a positive number ` ... |
le2msq 11875 | The square function on non... |
msq11 11876 | The square of a nonnegativ... |
ledivp1 11877 | "Less than or equal to" an... |
squeeze0 11878 | If a nonnegative number is... |
ltp1i 11879 | A number is less than itse... |
recgt0i 11880 | The reciprocal of a positi... |
recgt0ii 11881 | The reciprocal of a positi... |
prodgt0i 11882 | Infer that a multiplicand ... |
divgt0i 11883 | The ratio of two positive ... |
divge0i 11884 | The ratio of nonnegative a... |
ltreci 11885 | The reciprocal of both sid... |
lereci 11886 | The reciprocal of both sid... |
lt2msqi 11887 | The square function on non... |
le2msqi 11888 | The square function on non... |
msq11i 11889 | The square of a nonnegativ... |
divgt0i2i 11890 | The ratio of two positive ... |
ltrecii 11891 | The reciprocal of both sid... |
divgt0ii 11892 | The ratio of two positive ... |
ltmul1i 11893 | Multiplication of both sid... |
ltdiv1i 11894 | Division of both sides of ... |
ltmuldivi 11895 | 'Less than' relationship b... |
ltmul2i 11896 | Multiplication of both sid... |
lemul1i 11897 | Multiplication of both sid... |
lemul2i 11898 | Multiplication of both sid... |
ltdiv23i 11899 | Swap denominator with othe... |
ledivp1i 11900 | "Less than or equal to" an... |
ltdivp1i 11901 | Less-than and division rel... |
ltdiv23ii 11902 | Swap denominator with othe... |
ltmul1ii 11903 | Multiplication of both sid... |
ltdiv1ii 11904 | Division of both sides of ... |
ltp1d 11905 | A number is less than itse... |
lep1d 11906 | A number is less than or e... |
ltm1d 11907 | A number minus 1 is less t... |
lem1d 11908 | A number minus 1 is less t... |
recgt0d 11909 | The reciprocal of a positi... |
divgt0d 11910 | The ratio of two positive ... |
mulgt1d 11911 | The product of two numbers... |
lemulge11d 11912 | Multiplication by a number... |
lemulge12d 11913 | Multiplication by a number... |
lemul1ad 11914 | Multiplication of both sid... |
lemul2ad 11915 | Multiplication of both sid... |
ltmul12ad 11916 | Comparison of product of t... |
lemul12ad 11917 | Comparison of product of t... |
lemul12bd 11918 | Comparison of product of t... |
fimaxre 11919 | A finite set of real numbe... |
fimaxre2 11920 | A nonempty finite set of r... |
fimaxre3 11921 | A nonempty finite set of r... |
fiminre 11922 | A nonempty finite set of r... |
fiminre2 11923 | A nonempty finite set of r... |
negfi 11924 | The negation of a finite s... |
lbreu 11925 | If a set of reals contains... |
lbcl 11926 | If a set of reals contains... |
lble 11927 | If a set of reals contains... |
lbinf 11928 | If a set of reals contains... |
lbinfcl 11929 | If a set of reals contains... |
lbinfle 11930 | If a set of reals contains... |
sup2 11931 | A nonempty, bounded-above ... |
sup3 11932 | A version of the completen... |
infm3lem 11933 | Lemma for ~ infm3 . (Cont... |
infm3 11934 | The completeness axiom for... |
suprcl 11935 | Closure of supremum of a n... |
suprub 11936 | A member of a nonempty bou... |
suprubd 11937 | Natural deduction form of ... |
suprcld 11938 | Natural deduction form of ... |
suprlub 11939 | The supremum of a nonempty... |
suprnub 11940 | An upper bound is not less... |
suprleub 11941 | The supremum of a nonempty... |
supaddc 11942 | The supremum function dist... |
supadd 11943 | The supremum function dist... |
supmul1 11944 | The supremum function dist... |
supmullem1 11945 | Lemma for ~ supmul . (Con... |
supmullem2 11946 | Lemma for ~ supmul . (Con... |
supmul 11947 | The supremum function dist... |
sup3ii 11948 | A version of the completen... |
suprclii 11949 | Closure of supremum of a n... |
suprubii 11950 | A member of a nonempty bou... |
suprlubii 11951 | The supremum of a nonempty... |
suprnubii 11952 | An upper bound is not less... |
suprleubii 11953 | The supremum of a nonempty... |
riotaneg 11954 | The negative of the unique... |
negiso 11955 | Negation is an order anti-... |
dfinfre 11956 | The infimum of a set of re... |
infrecl 11957 | Closure of infimum of a no... |
infrenegsup 11958 | The infimum of a set of re... |
infregelb 11959 | Any lower bound of a nonem... |
infrelb 11960 | If a nonempty set of real ... |
infrefilb 11961 | The infimum of a finite se... |
supfirege 11962 | The supremum of a finite s... |
inelr 11963 | The imaginary unit ` _i ` ... |
rimul 11964 | A real number times the im... |
cru 11965 | The representation of comp... |
crne0 11966 | The real representation of... |
creur 11967 | The real part of a complex... |
creui 11968 | The imaginary part of a co... |
cju 11969 | The complex conjugate of a... |
ofsubeq0 11970 | Function analogue of ~ sub... |
ofnegsub 11971 | Function analogue of ~ neg... |
ofsubge0 11972 | Function analogue of ~ sub... |
nnexALT 11975 | Alternate proof of ~ nnex ... |
peano5nni 11976 | Peano's inductive postulat... |
nnssre 11977 | The positive integers are ... |
nnsscn 11978 | The positive integers are ... |
nnex 11979 | The set of positive intege... |
nnre 11980 | A positive integer is a re... |
nncn 11981 | A positive integer is a co... |
nnrei 11982 | A positive integer is a re... |
nncni 11983 | A positive integer is a co... |
1nn 11984 | Peano postulate: 1 is a po... |
peano2nn 11985 | Peano postulate: a success... |
dfnn2 11986 | Alternate definition of th... |
dfnn3 11987 | Alternate definition of th... |
nnred 11988 | A positive integer is a re... |
nncnd 11989 | A positive integer is a co... |
peano2nnd 11990 | Peano postulate: a success... |
nnind 11991 | Principle of Mathematical ... |
nnindALT 11992 | Principle of Mathematical ... |
nnindd 11993 | Principle of Mathematical ... |
nn1m1nn 11994 | Every positive integer is ... |
nn1suc 11995 | If a statement holds for 1... |
nnaddcl 11996 | Closure of addition of pos... |
nnmulcl 11997 | Closure of multiplication ... |
nnmulcli 11998 | Closure of multiplication ... |
nnmtmip 11999 | "Minus times minus is plus... |
nn2ge 12000 | There exists a positive in... |
nnge1 12001 | A positive integer is one ... |
nngt1ne1 12002 | A positive integer is grea... |
nnle1eq1 12003 | A positive integer is less... |
nngt0 12004 | A positive integer is posi... |
nnnlt1 12005 | A positive integer is not ... |
nnnle0 12006 | A positive integer is not ... |
nnne0 12007 | A positive integer is nonz... |
nnneneg 12008 | No positive integer is equ... |
0nnn 12009 | Zero is not a positive int... |
0nnnALT 12010 | Alternate proof of ~ 0nnn ... |
nnne0ALT 12011 | Alternate version of ~ nnn... |
nngt0i 12012 | A positive integer is posi... |
nnne0i 12013 | A positive integer is nonz... |
nndivre 12014 | The quotient of a real and... |
nnrecre 12015 | The reciprocal of a positi... |
nnrecgt0 12016 | The reciprocal of a positi... |
nnsub 12017 | Subtraction of positive in... |
nnsubi 12018 | Subtraction of positive in... |
nndiv 12019 | Two ways to express " ` A ... |
nndivtr 12020 | Transitive property of div... |
nnge1d 12021 | A positive integer is one ... |
nngt0d 12022 | A positive integer is posi... |
nnne0d 12023 | A positive integer is nonz... |
nnrecred 12024 | The reciprocal of a positi... |
nnaddcld 12025 | Closure of addition of pos... |
nnmulcld 12026 | Closure of multiplication ... |
nndivred 12027 | A positive integer is one ... |
0ne1 12044 | Zero is different from one... |
1m1e0 12045 | One minus one equals zero.... |
2nn 12046 | 2 is a positive integer. ... |
2re 12047 | The number 2 is real. (Co... |
2cn 12048 | The number 2 is a complex ... |
2cnALT 12049 | Alternate proof of ~ 2cn .... |
2ex 12050 | The number 2 is a set. (C... |
2cnd 12051 | The number 2 is a complex ... |
3nn 12052 | 3 is a positive integer. ... |
3re 12053 | The number 3 is real. (Co... |
3cn 12054 | The number 3 is a complex ... |
3ex 12055 | The number 3 is a set. (C... |
4nn 12056 | 4 is a positive integer. ... |
4re 12057 | The number 4 is real. (Co... |
4cn 12058 | The number 4 is a complex ... |
5nn 12059 | 5 is a positive integer. ... |
5re 12060 | The number 5 is real. (Co... |
5cn 12061 | The number 5 is a complex ... |
6nn 12062 | 6 is a positive integer. ... |
6re 12063 | The number 6 is real. (Co... |
6cn 12064 | The number 6 is a complex ... |
7nn 12065 | 7 is a positive integer. ... |
7re 12066 | The number 7 is real. (Co... |
7cn 12067 | The number 7 is a complex ... |
8nn 12068 | 8 is a positive integer. ... |
8re 12069 | The number 8 is real. (Co... |
8cn 12070 | The number 8 is a complex ... |
9nn 12071 | 9 is a positive integer. ... |
9re 12072 | The number 9 is real. (Co... |
9cn 12073 | The number 9 is a complex ... |
0le0 12074 | Zero is nonnegative. (Con... |
0le2 12075 | The number 0 is less than ... |
2pos 12076 | The number 2 is positive. ... |
2ne0 12077 | The number 2 is nonzero. ... |
3pos 12078 | The number 3 is positive. ... |
3ne0 12079 | The number 3 is nonzero. ... |
4pos 12080 | The number 4 is positive. ... |
4ne0 12081 | The number 4 is nonzero. ... |
5pos 12082 | The number 5 is positive. ... |
6pos 12083 | The number 6 is positive. ... |
7pos 12084 | The number 7 is positive. ... |
8pos 12085 | The number 8 is positive. ... |
9pos 12086 | The number 9 is positive. ... |
neg1cn 12087 | -1 is a complex number. (... |
neg1rr 12088 | -1 is a real number. (Con... |
neg1ne0 12089 | -1 is nonzero. (Contribut... |
neg1lt0 12090 | -1 is less than 0. (Contr... |
negneg1e1 12091 | ` -u -u 1 ` is 1. (Contri... |
1pneg1e0 12092 | ` 1 + -u 1 ` is 0. (Contr... |
0m0e0 12093 | 0 minus 0 equals 0. (Cont... |
1m0e1 12094 | 1 - 0 = 1. (Contributed b... |
0p1e1 12095 | 0 + 1 = 1. (Contributed b... |
fv0p1e1 12096 | Function value at ` N + 1 ... |
1p0e1 12097 | 1 + 0 = 1. (Contributed b... |
1p1e2 12098 | 1 + 1 = 2. (Contributed b... |
2m1e1 12099 | 2 - 1 = 1. The result is ... |
1e2m1 12100 | 1 = 2 - 1. (Contributed b... |
3m1e2 12101 | 3 - 1 = 2. (Contributed b... |
4m1e3 12102 | 4 - 1 = 3. (Contributed b... |
5m1e4 12103 | 5 - 1 = 4. (Contributed b... |
6m1e5 12104 | 6 - 1 = 5. (Contributed b... |
7m1e6 12105 | 7 - 1 = 6. (Contributed b... |
8m1e7 12106 | 8 - 1 = 7. (Contributed b... |
9m1e8 12107 | 9 - 1 = 8. (Contributed b... |
2p2e4 12108 | Two plus two equals four. ... |
2times 12109 | Two times a number. (Cont... |
times2 12110 | A number times 2. (Contri... |
2timesi 12111 | Two times a number. (Cont... |
times2i 12112 | A number times 2. (Contri... |
2txmxeqx 12113 | Two times a complex number... |
2div2e1 12114 | 2 divided by 2 is 1. (Con... |
2p1e3 12115 | 2 + 1 = 3. (Contributed b... |
1p2e3 12116 | 1 + 2 = 3. For a shorter ... |
1p2e3ALT 12117 | Alternate proof of ~ 1p2e3... |
3p1e4 12118 | 3 + 1 = 4. (Contributed b... |
4p1e5 12119 | 4 + 1 = 5. (Contributed b... |
5p1e6 12120 | 5 + 1 = 6. (Contributed b... |
6p1e7 12121 | 6 + 1 = 7. (Contributed b... |
7p1e8 12122 | 7 + 1 = 8. (Contributed b... |
8p1e9 12123 | 8 + 1 = 9. (Contributed b... |
3p2e5 12124 | 3 + 2 = 5. (Contributed b... |
3p3e6 12125 | 3 + 3 = 6. (Contributed b... |
4p2e6 12126 | 4 + 2 = 6. (Contributed b... |
4p3e7 12127 | 4 + 3 = 7. (Contributed b... |
4p4e8 12128 | 4 + 4 = 8. (Contributed b... |
5p2e7 12129 | 5 + 2 = 7. (Contributed b... |
5p3e8 12130 | 5 + 3 = 8. (Contributed b... |
5p4e9 12131 | 5 + 4 = 9. (Contributed b... |
6p2e8 12132 | 6 + 2 = 8. (Contributed b... |
6p3e9 12133 | 6 + 3 = 9. (Contributed b... |
7p2e9 12134 | 7 + 2 = 9. (Contributed b... |
1t1e1 12135 | 1 times 1 equals 1. (Cont... |
2t1e2 12136 | 2 times 1 equals 2. (Cont... |
2t2e4 12137 | 2 times 2 equals 4. (Cont... |
3t1e3 12138 | 3 times 1 equals 3. (Cont... |
3t2e6 12139 | 3 times 2 equals 6. (Cont... |
3t3e9 12140 | 3 times 3 equals 9. (Cont... |
4t2e8 12141 | 4 times 2 equals 8. (Cont... |
2t0e0 12142 | 2 times 0 equals 0. (Cont... |
4d2e2 12143 | One half of four is two. ... |
1lt2 12144 | 1 is less than 2. (Contri... |
2lt3 12145 | 2 is less than 3. (Contri... |
1lt3 12146 | 1 is less than 3. (Contri... |
3lt4 12147 | 3 is less than 4. (Contri... |
2lt4 12148 | 2 is less than 4. (Contri... |
1lt4 12149 | 1 is less than 4. (Contri... |
4lt5 12150 | 4 is less than 5. (Contri... |
3lt5 12151 | 3 is less than 5. (Contri... |
2lt5 12152 | 2 is less than 5. (Contri... |
1lt5 12153 | 1 is less than 5. (Contri... |
5lt6 12154 | 5 is less than 6. (Contri... |
4lt6 12155 | 4 is less than 6. (Contri... |
3lt6 12156 | 3 is less than 6. (Contri... |
2lt6 12157 | 2 is less than 6. (Contri... |
1lt6 12158 | 1 is less than 6. (Contri... |
6lt7 12159 | 6 is less than 7. (Contri... |
5lt7 12160 | 5 is less than 7. (Contri... |
4lt7 12161 | 4 is less than 7. (Contri... |
3lt7 12162 | 3 is less than 7. (Contri... |
2lt7 12163 | 2 is less than 7. (Contri... |
1lt7 12164 | 1 is less than 7. (Contri... |
7lt8 12165 | 7 is less than 8. (Contri... |
6lt8 12166 | 6 is less than 8. (Contri... |
5lt8 12167 | 5 is less than 8. (Contri... |
4lt8 12168 | 4 is less than 8. (Contri... |
3lt8 12169 | 3 is less than 8. (Contri... |
2lt8 12170 | 2 is less than 8. (Contri... |
1lt8 12171 | 1 is less than 8. (Contri... |
8lt9 12172 | 8 is less than 9. (Contri... |
7lt9 12173 | 7 is less than 9. (Contri... |
6lt9 12174 | 6 is less than 9. (Contri... |
5lt9 12175 | 5 is less than 9. (Contri... |
4lt9 12176 | 4 is less than 9. (Contri... |
3lt9 12177 | 3 is less than 9. (Contri... |
2lt9 12178 | 2 is less than 9. (Contri... |
1lt9 12179 | 1 is less than 9. (Contri... |
0ne2 12180 | 0 is not equal to 2. (Con... |
1ne2 12181 | 1 is not equal to 2. (Con... |
1le2 12182 | 1 is less than or equal to... |
2cnne0 12183 | 2 is a nonzero complex num... |
2rene0 12184 | 2 is a nonzero real number... |
1le3 12185 | 1 is less than or equal to... |
neg1mulneg1e1 12186 | ` -u 1 x. -u 1 ` is 1. (C... |
halfre 12187 | One-half is real. (Contri... |
halfcn 12188 | One-half is a complex numb... |
halfgt0 12189 | One-half is greater than z... |
halfge0 12190 | One-half is not negative. ... |
halflt1 12191 | One-half is less than one.... |
1mhlfehlf 12192 | Prove that 1 - 1/2 = 1/2. ... |
8th4div3 12193 | An eighth of four thirds i... |
halfpm6th 12194 | One half plus or minus one... |
it0e0 12195 | i times 0 equals 0. (Cont... |
2mulicn 12196 | ` ( 2 x. _i ) e. CC ` . (... |
2muline0 12197 | ` ( 2 x. _i ) =/= 0 ` . (... |
halfcl 12198 | Closure of half of a numbe... |
rehalfcl 12199 | Real closure of half. (Co... |
half0 12200 | Half of a number is zero i... |
2halves 12201 | Two halves make a whole. ... |
halfpos2 12202 | A number is positive iff i... |
halfpos 12203 | A positive number is great... |
halfnneg2 12204 | A number is nonnegative if... |
halfaddsubcl 12205 | Closure of half-sum and ha... |
halfaddsub 12206 | Sum and difference of half... |
subhalfhalf 12207 | Subtracting the half of a ... |
lt2halves 12208 | A sum is less than the who... |
addltmul 12209 | Sum is less than product f... |
nominpos 12210 | There is no smallest posit... |
avglt1 12211 | Ordering property for aver... |
avglt2 12212 | Ordering property for aver... |
avgle1 12213 | Ordering property for aver... |
avgle2 12214 | Ordering property for aver... |
avgle 12215 | The average of two numbers... |
2timesd 12216 | Two times a number. (Cont... |
times2d 12217 | A number times 2. (Contri... |
halfcld 12218 | Closure of half of a numbe... |
2halvesd 12219 | Two halves make a whole. ... |
rehalfcld 12220 | Real closure of half. (Co... |
lt2halvesd 12221 | A sum is less than the who... |
rehalfcli 12222 | Half a real number is real... |
lt2addmuld 12223 | If two real numbers are le... |
add1p1 12224 | Adding two times 1 to a nu... |
sub1m1 12225 | Subtracting two times 1 fr... |
cnm2m1cnm3 12226 | Subtracting 2 and afterwar... |
xp1d2m1eqxm1d2 12227 | A complex number increased... |
div4p1lem1div2 12228 | An integer greater than 5,... |
nnunb 12229 | The set of positive intege... |
arch 12230 | Archimedean property of re... |
nnrecl 12231 | There exists a positive in... |
bndndx 12232 | A bounded real sequence ` ... |
elnn0 12235 | Nonnegative integers expre... |
nnssnn0 12236 | Positive naturals are a su... |
nn0ssre 12237 | Nonnegative integers are a... |
nn0sscn 12238 | Nonnegative integers are a... |
nn0ex 12239 | The set of nonnegative int... |
nnnn0 12240 | A positive integer is a no... |
nnnn0i 12241 | A positive integer is a no... |
nn0re 12242 | A nonnegative integer is a... |
nn0cn 12243 | A nonnegative integer is a... |
nn0rei 12244 | A nonnegative integer is a... |
nn0cni 12245 | A nonnegative integer is a... |
dfn2 12246 | The set of positive intege... |
elnnne0 12247 | The positive integer prope... |
0nn0 12248 | 0 is a nonnegative integer... |
1nn0 12249 | 1 is a nonnegative integer... |
2nn0 12250 | 2 is a nonnegative integer... |
3nn0 12251 | 3 is a nonnegative integer... |
4nn0 12252 | 4 is a nonnegative integer... |
5nn0 12253 | 5 is a nonnegative integer... |
6nn0 12254 | 6 is a nonnegative integer... |
7nn0 12255 | 7 is a nonnegative integer... |
8nn0 12256 | 8 is a nonnegative integer... |
9nn0 12257 | 9 is a nonnegative integer... |
nn0ge0 12258 | A nonnegative integer is g... |
nn0nlt0 12259 | A nonnegative integer is n... |
nn0ge0i 12260 | Nonnegative integers are n... |
nn0le0eq0 12261 | A nonnegative integer is l... |
nn0p1gt0 12262 | A nonnegative integer incr... |
nnnn0addcl 12263 | A positive integer plus a ... |
nn0nnaddcl 12264 | A nonnegative integer plus... |
0mnnnnn0 12265 | The result of subtracting ... |
un0addcl 12266 | If ` S ` is closed under a... |
un0mulcl 12267 | If ` S ` is closed under m... |
nn0addcl 12268 | Closure of addition of non... |
nn0mulcl 12269 | Closure of multiplication ... |
nn0addcli 12270 | Closure of addition of non... |
nn0mulcli 12271 | Closure of multiplication ... |
nn0p1nn 12272 | A nonnegative integer plus... |
peano2nn0 12273 | Second Peano postulate for... |
nnm1nn0 12274 | A positive integer minus 1... |
elnn0nn 12275 | The nonnegative integer pr... |
elnnnn0 12276 | The positive integer prope... |
elnnnn0b 12277 | The positive integer prope... |
elnnnn0c 12278 | The positive integer prope... |
nn0addge1 12279 | A number is less than or e... |
nn0addge2 12280 | A number is less than or e... |
nn0addge1i 12281 | A number is less than or e... |
nn0addge2i 12282 | A number is less than or e... |
nn0sub 12283 | Subtraction of nonnegative... |
ltsubnn0 12284 | Subtracting a nonnegative ... |
nn0negleid 12285 | A nonnegative integer is g... |
difgtsumgt 12286 | If the difference of a rea... |
nn0le2xi 12287 | A nonnegative integer is l... |
nn0lele2xi 12288 | 'Less than or equal to' im... |
frnnn0supp 12289 | Two ways to write the supp... |
frnnn0fsupp 12290 | A function on ` NN0 ` is f... |
frnnn0suppg 12291 | Version of ~ frnnn0supp av... |
frnnn0fsuppg 12292 | Version of ~ frnnn0fsupp a... |
nnnn0d 12293 | A positive integer is a no... |
nn0red 12294 | A nonnegative integer is a... |
nn0cnd 12295 | A nonnegative integer is a... |
nn0ge0d 12296 | A nonnegative integer is g... |
nn0addcld 12297 | Closure of addition of non... |
nn0mulcld 12298 | Closure of multiplication ... |
nn0readdcl 12299 | Closure law for addition o... |
nn0n0n1ge2 12300 | A nonnegative integer whic... |
nn0n0n1ge2b 12301 | A nonnegative integer is n... |
nn0ge2m1nn 12302 | If a nonnegative integer i... |
nn0ge2m1nn0 12303 | If a nonnegative integer i... |
nn0nndivcl 12304 | Closure law for dividing o... |
elxnn0 12307 | An extended nonnegative in... |
nn0ssxnn0 12308 | The standard nonnegative i... |
nn0xnn0 12309 | A standard nonnegative int... |
xnn0xr 12310 | An extended nonnegative in... |
0xnn0 12311 | Zero is an extended nonneg... |
pnf0xnn0 12312 | Positive infinity is an ex... |
nn0nepnf 12313 | No standard nonnegative in... |
nn0xnn0d 12314 | A standard nonnegative int... |
nn0nepnfd 12315 | No standard nonnegative in... |
xnn0nemnf 12316 | No extended nonnegative in... |
xnn0xrnemnf 12317 | The extended nonnegative i... |
xnn0nnn0pnf 12318 | An extended nonnegative in... |
elz 12321 | Membership in the set of i... |
nnnegz 12322 | The negative of a positive... |
zre 12323 | An integer is a real. (Co... |
zcn 12324 | An integer is a complex nu... |
zrei 12325 | An integer is a real numbe... |
zssre 12326 | The integers are a subset ... |
zsscn 12327 | The integers are a subset ... |
zex 12328 | The set of integers exists... |
elnnz 12329 | Positive integer property ... |
0z 12330 | Zero is an integer. (Cont... |
0zd 12331 | Zero is an integer, deduct... |
elnn0z 12332 | Nonnegative integer proper... |
elznn0nn 12333 | Integer property expressed... |
elznn0 12334 | Integer property expressed... |
elznn 12335 | Integer property expressed... |
zle0orge1 12336 | There is no integer in the... |
elz2 12337 | Membership in the set of i... |
dfz2 12338 | Alternative definition of ... |
zexALT 12339 | Alternate proof of ~ zex .... |
nnssz 12340 | Positive integers are a su... |
nn0ssz 12341 | Nonnegative integers are a... |
nnz 12342 | A positive integer is an i... |
nn0z 12343 | A nonnegative integer is a... |
nnzi 12344 | A positive integer is an i... |
nn0zi 12345 | A nonnegative integer is a... |
elnnz1 12346 | Positive integer property ... |
znnnlt1 12347 | An integer is not a positi... |
nnzrab 12348 | Positive integers expresse... |
nn0zrab 12349 | Nonnegative integers expre... |
1z 12350 | One is an integer. (Contr... |
1zzd 12351 | One is an integer, deducti... |
2z 12352 | 2 is an integer. (Contrib... |
3z 12353 | 3 is an integer. (Contrib... |
4z 12354 | 4 is an integer. (Contrib... |
znegcl 12355 | Closure law for negative i... |
neg1z 12356 | -1 is an integer. (Contri... |
znegclb 12357 | A complex number is an int... |
nn0negz 12358 | The negative of a nonnegat... |
nn0negzi 12359 | The negative of a nonnegat... |
zaddcl 12360 | Closure of addition of int... |
peano2z 12361 | Second Peano postulate gen... |
zsubcl 12362 | Closure of subtraction of ... |
peano2zm 12363 | "Reverse" second Peano pos... |
zletr 12364 | Transitive law of ordering... |
zrevaddcl 12365 | Reverse closure law for ad... |
znnsub 12366 | The positive difference of... |
znn0sub 12367 | The nonnegative difference... |
nzadd 12368 | The sum of a real number n... |
zmulcl 12369 | Closure of multiplication ... |
zltp1le 12370 | Integer ordering relation.... |
zleltp1 12371 | Integer ordering relation.... |
zlem1lt 12372 | Integer ordering relation.... |
zltlem1 12373 | Integer ordering relation.... |
zgt0ge1 12374 | An integer greater than ` ... |
nnleltp1 12375 | Positive integer ordering ... |
nnltp1le 12376 | Positive integer ordering ... |
nnaddm1cl 12377 | Closure of addition of pos... |
nn0ltp1le 12378 | Nonnegative integer orderi... |
nn0leltp1 12379 | Nonnegative integer orderi... |
nn0ltlem1 12380 | Nonnegative integer orderi... |
nn0sub2 12381 | Subtraction of nonnegative... |
nn0lt10b 12382 | A nonnegative integer less... |
nn0lt2 12383 | A nonnegative integer less... |
nn0le2is012 12384 | A nonnegative integer whic... |
nn0lem1lt 12385 | Nonnegative integer orderi... |
nnlem1lt 12386 | Positive integer ordering ... |
nnltlem1 12387 | Positive integer ordering ... |
nnm1ge0 12388 | A positive integer decreas... |
nn0ge0div 12389 | Division of a nonnegative ... |
zdiv 12390 | Two ways to express " ` M ... |
zdivadd 12391 | Property of divisibility: ... |
zdivmul 12392 | Property of divisibility: ... |
zextle 12393 | An extensionality-like pro... |
zextlt 12394 | An extensionality-like pro... |
recnz 12395 | The reciprocal of a number... |
btwnnz 12396 | A number between an intege... |
gtndiv 12397 | A larger number does not d... |
halfnz 12398 | One-half is not an integer... |
3halfnz 12399 | Three halves is not an int... |
suprzcl 12400 | The supremum of a bounded-... |
prime 12401 | Two ways to express " ` A ... |
msqznn 12402 | The square of a nonzero in... |
zneo 12403 | No even integer equals an ... |
nneo 12404 | A positive integer is even... |
nneoi 12405 | A positive integer is even... |
zeo 12406 | An integer is even or odd.... |
zeo2 12407 | An integer is even or odd ... |
peano2uz2 12408 | Second Peano postulate for... |
peano5uzi 12409 | Peano's inductive postulat... |
peano5uzti 12410 | Peano's inductive postulat... |
dfuzi 12411 | An expression for the uppe... |
uzind 12412 | Induction on the upper int... |
uzind2 12413 | Induction on the upper int... |
uzind3 12414 | Induction on the upper int... |
nn0ind 12415 | Principle of Mathematical ... |
nn0indALT 12416 | Principle of Mathematical ... |
nn0indd 12417 | Principle of Mathematical ... |
fzind 12418 | Induction on the integers ... |
fnn0ind 12419 | Induction on the integers ... |
nn0ind-raph 12420 | Principle of Mathematical ... |
zindd 12421 | Principle of Mathematical ... |
fzindd 12422 | Induction on the integers ... |
btwnz 12423 | Any real number can be san... |
nn0zd 12424 | A positive integer is an i... |
nnzd 12425 | A nonnegative integer is a... |
zred 12426 | An integer is a real numbe... |
zcnd 12427 | An integer is a complex nu... |
znegcld 12428 | Closure law for negative i... |
peano2zd 12429 | Deduction from second Pean... |
zaddcld 12430 | Closure of addition of int... |
zsubcld 12431 | Closure of subtraction of ... |
zmulcld 12432 | Closure of multiplication ... |
znnn0nn 12433 | The negative of a negative... |
zadd2cl 12434 | Increasing an integer by 2... |
zriotaneg 12435 | The negative of the unique... |
suprfinzcl 12436 | The supremum of a nonempty... |
9p1e10 12439 | 9 + 1 = 10. (Contributed ... |
dfdec10 12440 | Version of the definition ... |
decex 12441 | A decimal number is a set.... |
deceq1 12442 | Equality theorem for the d... |
deceq2 12443 | Equality theorem for the d... |
deceq1i 12444 | Equality theorem for the d... |
deceq2i 12445 | Equality theorem for the d... |
deceq12i 12446 | Equality theorem for the d... |
numnncl 12447 | Closure for a numeral (wit... |
num0u 12448 | Add a zero in the units pl... |
num0h 12449 | Add a zero in the higher p... |
numcl 12450 | Closure for a decimal inte... |
numsuc 12451 | The successor of a decimal... |
deccl 12452 | Closure for a numeral. (C... |
10nn 12453 | 10 is a positive integer. ... |
10pos 12454 | The number 10 is positive.... |
10nn0 12455 | 10 is a nonnegative intege... |
10re 12456 | The number 10 is real. (C... |
decnncl 12457 | Closure for a numeral. (C... |
dec0u 12458 | Add a zero in the units pl... |
dec0h 12459 | Add a zero in the higher p... |
numnncl2 12460 | Closure for a decimal inte... |
decnncl2 12461 | Closure for a decimal inte... |
numlt 12462 | Comparing two decimal inte... |
numltc 12463 | Comparing two decimal inte... |
le9lt10 12464 | A "decimal digit" (i.e. a ... |
declt 12465 | Comparing two decimal inte... |
decltc 12466 | Comparing two decimal inte... |
declth 12467 | Comparing two decimal inte... |
decsuc 12468 | The successor of a decimal... |
3declth 12469 | Comparing two decimal inte... |
3decltc 12470 | Comparing two decimal inte... |
decle 12471 | Comparing two decimal inte... |
decleh 12472 | Comparing two decimal inte... |
declei 12473 | Comparing a digit to a dec... |
numlti 12474 | Comparing a digit to a dec... |
declti 12475 | Comparing a digit to a dec... |
decltdi 12476 | Comparing a digit to a dec... |
numsucc 12477 | The successor of a decimal... |
decsucc 12478 | The successor of a decimal... |
1e0p1 12479 | The successor of zero. (C... |
dec10p 12480 | Ten plus an integer. (Con... |
numma 12481 | Perform a multiply-add of ... |
nummac 12482 | Perform a multiply-add of ... |
numma2c 12483 | Perform a multiply-add of ... |
numadd 12484 | Add two decimal integers `... |
numaddc 12485 | Add two decimal integers `... |
nummul1c 12486 | The product of a decimal i... |
nummul2c 12487 | The product of a decimal i... |
decma 12488 | Perform a multiply-add of ... |
decmac 12489 | Perform a multiply-add of ... |
decma2c 12490 | Perform a multiply-add of ... |
decadd 12491 | Add two numerals ` M ` and... |
decaddc 12492 | Add two numerals ` M ` and... |
decaddc2 12493 | Add two numerals ` M ` and... |
decrmanc 12494 | Perform a multiply-add of ... |
decrmac 12495 | Perform a multiply-add of ... |
decaddm10 12496 | The sum of two multiples o... |
decaddi 12497 | Add two numerals ` M ` and... |
decaddci 12498 | Add two numerals ` M ` and... |
decaddci2 12499 | Add two numerals ` M ` and... |
decsubi 12500 | Difference between a numer... |
decmul1 12501 | The product of a numeral w... |
decmul1c 12502 | The product of a numeral w... |
decmul2c 12503 | The product of a numeral w... |
decmulnc 12504 | The product of a numeral w... |
11multnc 12505 | The product of 11 (as nume... |
decmul10add 12506 | A multiplication of a numb... |
6p5lem 12507 | Lemma for ~ 6p5e11 and rel... |
5p5e10 12508 | 5 + 5 = 10. (Contributed ... |
6p4e10 12509 | 6 + 4 = 10. (Contributed ... |
6p5e11 12510 | 6 + 5 = 11. (Contributed ... |
6p6e12 12511 | 6 + 6 = 12. (Contributed ... |
7p3e10 12512 | 7 + 3 = 10. (Contributed ... |
7p4e11 12513 | 7 + 4 = 11. (Contributed ... |
7p5e12 12514 | 7 + 5 = 12. (Contributed ... |
7p6e13 12515 | 7 + 6 = 13. (Contributed ... |
7p7e14 12516 | 7 + 7 = 14. (Contributed ... |
8p2e10 12517 | 8 + 2 = 10. (Contributed ... |
8p3e11 12518 | 8 + 3 = 11. (Contributed ... |
8p4e12 12519 | 8 + 4 = 12. (Contributed ... |
8p5e13 12520 | 8 + 5 = 13. (Contributed ... |
8p6e14 12521 | 8 + 6 = 14. (Contributed ... |
8p7e15 12522 | 8 + 7 = 15. (Contributed ... |
8p8e16 12523 | 8 + 8 = 16. (Contributed ... |
9p2e11 12524 | 9 + 2 = 11. (Contributed ... |
9p3e12 12525 | 9 + 3 = 12. (Contributed ... |
9p4e13 12526 | 9 + 4 = 13. (Contributed ... |
9p5e14 12527 | 9 + 5 = 14. (Contributed ... |
9p6e15 12528 | 9 + 6 = 15. (Contributed ... |
9p7e16 12529 | 9 + 7 = 16. (Contributed ... |
9p8e17 12530 | 9 + 8 = 17. (Contributed ... |
9p9e18 12531 | 9 + 9 = 18. (Contributed ... |
10p10e20 12532 | 10 + 10 = 20. (Contribute... |
10m1e9 12533 | 10 - 1 = 9. (Contributed ... |
4t3lem 12534 | Lemma for ~ 4t3e12 and rel... |
4t3e12 12535 | 4 times 3 equals 12. (Con... |
4t4e16 12536 | 4 times 4 equals 16. (Con... |
5t2e10 12537 | 5 times 2 equals 10. (Con... |
5t3e15 12538 | 5 times 3 equals 15. (Con... |
5t4e20 12539 | 5 times 4 equals 20. (Con... |
5t5e25 12540 | 5 times 5 equals 25. (Con... |
6t2e12 12541 | 6 times 2 equals 12. (Con... |
6t3e18 12542 | 6 times 3 equals 18. (Con... |
6t4e24 12543 | 6 times 4 equals 24. (Con... |
6t5e30 12544 | 6 times 5 equals 30. (Con... |
6t6e36 12545 | 6 times 6 equals 36. (Con... |
7t2e14 12546 | 7 times 2 equals 14. (Con... |
7t3e21 12547 | 7 times 3 equals 21. (Con... |
7t4e28 12548 | 7 times 4 equals 28. (Con... |
7t5e35 12549 | 7 times 5 equals 35. (Con... |
7t6e42 12550 | 7 times 6 equals 42. (Con... |
7t7e49 12551 | 7 times 7 equals 49. (Con... |
8t2e16 12552 | 8 times 2 equals 16. (Con... |
8t3e24 12553 | 8 times 3 equals 24. (Con... |
8t4e32 12554 | 8 times 4 equals 32. (Con... |
8t5e40 12555 | 8 times 5 equals 40. (Con... |
8t6e48 12556 | 8 times 6 equals 48. (Con... |
8t7e56 12557 | 8 times 7 equals 56. (Con... |
8t8e64 12558 | 8 times 8 equals 64. (Con... |
9t2e18 12559 | 9 times 2 equals 18. (Con... |
9t3e27 12560 | 9 times 3 equals 27. (Con... |
9t4e36 12561 | 9 times 4 equals 36. (Con... |
9t5e45 12562 | 9 times 5 equals 45. (Con... |
9t6e54 12563 | 9 times 6 equals 54. (Con... |
9t7e63 12564 | 9 times 7 equals 63. (Con... |
9t8e72 12565 | 9 times 8 equals 72. (Con... |
9t9e81 12566 | 9 times 9 equals 81. (Con... |
9t11e99 12567 | 9 times 11 equals 99. (Co... |
9lt10 12568 | 9 is less than 10. (Contr... |
8lt10 12569 | 8 is less than 10. (Contr... |
7lt10 12570 | 7 is less than 10. (Contr... |
6lt10 12571 | 6 is less than 10. (Contr... |
5lt10 12572 | 5 is less than 10. (Contr... |
4lt10 12573 | 4 is less than 10. (Contr... |
3lt10 12574 | 3 is less than 10. (Contr... |
2lt10 12575 | 2 is less than 10. (Contr... |
1lt10 12576 | 1 is less than 10. (Contr... |
decbin0 12577 | Decompose base 4 into base... |
decbin2 12578 | Decompose base 4 into base... |
decbin3 12579 | Decompose base 4 into base... |
halfthird 12580 | Half minus a third. (Cont... |
5recm6rec 12581 | One fifth minus one sixth.... |
uzval 12584 | The value of the upper int... |
uzf 12585 | The domain and range of th... |
eluz1 12586 | Membership in the upper se... |
eluzel2 12587 | Implication of membership ... |
eluz2 12588 | Membership in an upper set... |
eluzmn 12589 | Membership in an earlier u... |
eluz1i 12590 | Membership in an upper set... |
eluzuzle 12591 | An integer in an upper set... |
eluzelz 12592 | A member of an upper set o... |
eluzelre 12593 | A member of an upper set o... |
eluzelcn 12594 | A member of an upper set o... |
eluzle 12595 | Implication of membership ... |
eluz 12596 | Membership in an upper set... |
uzid 12597 | Membership of the least me... |
uzidd 12598 | Membership of the least me... |
uzn0 12599 | The upper integers are all... |
uztrn 12600 | Transitive law for sets of... |
uztrn2 12601 | Transitive law for sets of... |
uzneg 12602 | Contraposition law for upp... |
uzssz 12603 | An upper set of integers i... |
uzssre 12604 | An upper set of integers i... |
uzss 12605 | Subset relationship for tw... |
uztric 12606 | Totality of the ordering r... |
uz11 12607 | The upper integers functio... |
eluzp1m1 12608 | Membership in the next upp... |
eluzp1l 12609 | Strict ordering implied by... |
eluzp1p1 12610 | Membership in the next upp... |
eluzaddi 12611 | Membership in a later uppe... |
eluzsubi 12612 | Membership in an earlier u... |
eluzadd 12613 | Membership in a later uppe... |
eluzsub 12614 | Membership in an earlier u... |
subeluzsub 12615 | Membership of a difference... |
uzm1 12616 | Choices for an element of ... |
uznn0sub 12617 | The nonnegative difference... |
uzin 12618 | Intersection of two upper ... |
uzp1 12619 | Choices for an element of ... |
nn0uz 12620 | Nonnegative integers expre... |
nnuz 12621 | Positive integers expresse... |
elnnuz 12622 | A positive integer express... |
elnn0uz 12623 | A nonnegative integer expr... |
eluz2nn 12624 | An integer greater than or... |
eluz4eluz2 12625 | An integer greater than or... |
eluz4nn 12626 | An integer greater than or... |
eluzge2nn0 12627 | If an integer is greater t... |
eluz2n0 12628 | An integer greater than or... |
uzuzle23 12629 | An integer in the upper se... |
eluzge3nn 12630 | If an integer is greater t... |
uz3m2nn 12631 | An integer greater than or... |
1eluzge0 12632 | 1 is an integer greater th... |
2eluzge0 12633 | 2 is an integer greater th... |
2eluzge1 12634 | 2 is an integer greater th... |
uznnssnn 12635 | The upper integers startin... |
raluz 12636 | Restricted universal quant... |
raluz2 12637 | Restricted universal quant... |
rexuz 12638 | Restricted existential qua... |
rexuz2 12639 | Restricted existential qua... |
2rexuz 12640 | Double existential quantif... |
peano2uz 12641 | Second Peano postulate for... |
peano2uzs 12642 | Second Peano postulate for... |
peano2uzr 12643 | Reversed second Peano axio... |
uzaddcl 12644 | Addition closure law for a... |
nn0pzuz 12645 | The sum of a nonnegative i... |
uzind4 12646 | Induction on the upper set... |
uzind4ALT 12647 | Induction on the upper set... |
uzind4s 12648 | Induction on the upper set... |
uzind4s2 12649 | Induction on the upper set... |
uzind4i 12650 | Induction on the upper int... |
uzwo 12651 | Well-ordering principle: a... |
uzwo2 12652 | Well-ordering principle: a... |
nnwo 12653 | Well-ordering principle: a... |
nnwof 12654 | Well-ordering principle: a... |
nnwos 12655 | Well-ordering principle: a... |
indstr 12656 | Strong Mathematical Induct... |
eluznn0 12657 | Membership in a nonnegativ... |
eluznn 12658 | Membership in a positive u... |
eluz2b1 12659 | Two ways to say "an intege... |
eluz2gt1 12660 | An integer greater than or... |
eluz2b2 12661 | Two ways to say "an intege... |
eluz2b3 12662 | Two ways to say "an intege... |
uz2m1nn 12663 | One less than an integer g... |
1nuz2 12664 | 1 is not in ` ( ZZ>= `` 2 ... |
elnn1uz2 12665 | A positive integer is eith... |
uz2mulcl 12666 | Closure of multiplication ... |
indstr2 12667 | Strong Mathematical Induct... |
uzinfi 12668 | Extract the lower bound of... |
nninf 12669 | The infimum of the set of ... |
nn0inf 12670 | The infimum of the set of ... |
infssuzle 12671 | The infimum of a subset of... |
infssuzcl 12672 | The infimum of a subset of... |
ublbneg 12673 | The image under negation o... |
eqreznegel 12674 | Two ways to express the im... |
supminf 12675 | The supremum of a bounded-... |
lbzbi 12676 | If a set of reals is bound... |
zsupss 12677 | Any nonempty bounded subse... |
suprzcl2 12678 | The supremum of a bounded-... |
suprzub 12679 | The supremum of a bounded-... |
uzsupss 12680 | Any bounded subset of an u... |
nn01to3 12681 | A (nonnegative) integer be... |
nn0ge2m1nnALT 12682 | Alternate proof of ~ nn0ge... |
uzwo3 12683 | Well-ordering principle: a... |
zmin 12684 | There is a unique smallest... |
zmax 12685 | There is a unique largest ... |
zbtwnre 12686 | There is a unique integer ... |
rebtwnz 12687 | There is a unique greatest... |
elq 12690 | Membership in the set of r... |
qmulz 12691 | If ` A ` is rational, then... |
znq 12692 | The ratio of an integer an... |
qre 12693 | A rational number is a rea... |
zq 12694 | An integer is a rational n... |
qred 12695 | A rational number is a rea... |
zssq 12696 | The integers are a subset ... |
nn0ssq 12697 | The nonnegative integers a... |
nnssq 12698 | The positive integers are ... |
qssre 12699 | The rationals are a subset... |
qsscn 12700 | The rationals are a subset... |
qex 12701 | The set of rational number... |
nnq 12702 | A positive integer is rati... |
qcn 12703 | A rational number is a com... |
qexALT 12704 | Alternate proof of ~ qex .... |
qaddcl 12705 | Closure of addition of rat... |
qnegcl 12706 | Closure law for the negati... |
qmulcl 12707 | Closure of multiplication ... |
qsubcl 12708 | Closure of subtraction of ... |
qreccl 12709 | Closure of reciprocal of r... |
qdivcl 12710 | Closure of division of rat... |
qrevaddcl 12711 | Reverse closure law for ad... |
nnrecq 12712 | The reciprocal of a positi... |
irradd 12713 | The sum of an irrational n... |
irrmul 12714 | The product of an irration... |
elpq 12715 | A positive rational is the... |
elpqb 12716 | A class is a positive rati... |
rpnnen1lem2 12717 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1lem1 12718 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1lem3 12719 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1lem4 12720 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1lem5 12721 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1lem6 12722 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1 12723 | One half of ~ rpnnen , whe... |
reexALT 12724 | Alternate proof of ~ reex ... |
cnref1o 12725 | There is a natural one-to-... |
cnexALT 12726 | The set of complex numbers... |
xrex 12727 | The set of extended reals ... |
addex 12728 | The addition operation is ... |
mulex 12729 | The multiplication operati... |
elrp 12732 | Membership in the set of p... |
elrpii 12733 | Membership in the set of p... |
1rp 12734 | 1 is a positive real. (Co... |
2rp 12735 | 2 is a positive real. (Co... |
3rp 12736 | 3 is a positive real. (Co... |
rpssre 12737 | The positive reals are a s... |
rpre 12738 | A positive real is a real.... |
rpxr 12739 | A positive real is an exte... |
rpcn 12740 | A positive real is a compl... |
nnrp 12741 | A positive integer is a po... |
rpgt0 12742 | A positive real is greater... |
rpge0 12743 | A positive real is greater... |
rpregt0 12744 | A positive real is a posit... |
rprege0 12745 | A positive real is a nonne... |
rpne0 12746 | A positive real is nonzero... |
rprene0 12747 | A positive real is a nonze... |
rpcnne0 12748 | A positive real is a nonze... |
rpcndif0 12749 | A positive real number is ... |
ralrp 12750 | Quantification over positi... |
rexrp 12751 | Quantification over positi... |
rpaddcl 12752 | Closure law for addition o... |
rpmulcl 12753 | Closure law for multiplica... |
rpmtmip 12754 | "Minus times minus is plus... |
rpdivcl 12755 | Closure law for division o... |
rpreccl 12756 | Closure law for reciprocat... |
rphalfcl 12757 | Closure law for half of a ... |
rpgecl 12758 | A number greater than or e... |
rphalflt 12759 | Half of a positive real is... |
rerpdivcl 12760 | Closure law for division o... |
ge0p1rp 12761 | A nonnegative number plus ... |
rpneg 12762 | Either a nonzero real or i... |
negelrp 12763 | Elementhood of a negation ... |
negelrpd 12764 | The negation of a negative... |
0nrp 12765 | Zero is not a positive rea... |
ltsubrp 12766 | Subtracting a positive rea... |
ltaddrp 12767 | Adding a positive number t... |
difrp 12768 | Two ways to say one number... |
elrpd 12769 | Membership in the set of p... |
nnrpd 12770 | A positive integer is a po... |
zgt1rpn0n1 12771 | An integer greater than 1 ... |
rpred 12772 | A positive real is a real.... |
rpxrd 12773 | A positive real is an exte... |
rpcnd 12774 | A positive real is a compl... |
rpgt0d 12775 | A positive real is greater... |
rpge0d 12776 | A positive real is greater... |
rpne0d 12777 | A positive real is nonzero... |
rpregt0d 12778 | A positive real is real an... |
rprege0d 12779 | A positive real is real an... |
rprene0d 12780 | A positive real is a nonze... |
rpcnne0d 12781 | A positive real is a nonze... |
rpreccld 12782 | Closure law for reciprocat... |
rprecred 12783 | Closure law for reciprocat... |
rphalfcld 12784 | Closure law for half of a ... |
reclt1d 12785 | The reciprocal of a positi... |
recgt1d 12786 | The reciprocal of a positi... |
rpaddcld 12787 | Closure law for addition o... |
rpmulcld 12788 | Closure law for multiplica... |
rpdivcld 12789 | Closure law for division o... |
ltrecd 12790 | The reciprocal of both sid... |
lerecd 12791 | The reciprocal of both sid... |
ltrec1d 12792 | Reciprocal swap in a 'less... |
lerec2d 12793 | Reciprocal swap in a 'less... |
lediv2ad 12794 | Division of both sides of ... |
ltdiv2d 12795 | Division of a positive num... |
lediv2d 12796 | Division of a positive num... |
ledivdivd 12797 | Invert ratios of positive ... |
divge1 12798 | The ratio of a number over... |
divlt1lt 12799 | A real number divided by a... |
divle1le 12800 | A real number divided by a... |
ledivge1le 12801 | If a number is less than o... |
ge0p1rpd 12802 | A nonnegative number plus ... |
rerpdivcld 12803 | Closure law for division o... |
ltsubrpd 12804 | Subtracting a positive rea... |
ltaddrpd 12805 | Adding a positive number t... |
ltaddrp2d 12806 | Adding a positive number t... |
ltmulgt11d 12807 | Multiplication by a number... |
ltmulgt12d 12808 | Multiplication by a number... |
gt0divd 12809 | Division of a positive num... |
ge0divd 12810 | Division of a nonnegative ... |
rpgecld 12811 | A number greater than or e... |
divge0d 12812 | The ratio of nonnegative a... |
ltmul1d 12813 | The ratio of nonnegative a... |
ltmul2d 12814 | Multiplication of both sid... |
lemul1d 12815 | Multiplication of both sid... |
lemul2d 12816 | Multiplication of both sid... |
ltdiv1d 12817 | Division of both sides of ... |
lediv1d 12818 | Division of both sides of ... |
ltmuldivd 12819 | 'Less than' relationship b... |
ltmuldiv2d 12820 | 'Less than' relationship b... |
lemuldivd 12821 | 'Less than or equal to' re... |
lemuldiv2d 12822 | 'Less than or equal to' re... |
ltdivmuld 12823 | 'Less than' relationship b... |
ltdivmul2d 12824 | 'Less than' relationship b... |
ledivmuld 12825 | 'Less than or equal to' re... |
ledivmul2d 12826 | 'Less than or equal to' re... |
ltmul1dd 12827 | The ratio of nonnegative a... |
ltmul2dd 12828 | Multiplication of both sid... |
ltdiv1dd 12829 | Division of both sides of ... |
lediv1dd 12830 | Division of both sides of ... |
lediv12ad 12831 | Comparison of ratio of two... |
mul2lt0rlt0 12832 | If the result of a multipl... |
mul2lt0rgt0 12833 | If the result of a multipl... |
mul2lt0llt0 12834 | If the result of a multipl... |
mul2lt0lgt0 12835 | If the result of a multipl... |
mul2lt0bi 12836 | If the result of a multipl... |
prodge0rd 12837 | Infer that a multiplicand ... |
prodge0ld 12838 | Infer that a multiplier is... |
ltdiv23d 12839 | Swap denominator with othe... |
lediv23d 12840 | Swap denominator with othe... |
lt2mul2divd 12841 | The ratio of nonnegative a... |
nnledivrp 12842 | Division of a positive int... |
nn0ledivnn 12843 | Division of a nonnegative ... |
addlelt 12844 | If the sum of a real numbe... |
ltxr 12851 | The 'less than' binary rel... |
elxr 12852 | Membership in the set of e... |
xrnemnf 12853 | An extended real other tha... |
xrnepnf 12854 | An extended real other tha... |
xrltnr 12855 | The extended real 'less th... |
ltpnf 12856 | Any (finite) real is less ... |
ltpnfd 12857 | Any (finite) real is less ... |
0ltpnf 12858 | Zero is less than plus inf... |
mnflt 12859 | Minus infinity is less tha... |
mnfltd 12860 | Minus infinity is less tha... |
mnflt0 12861 | Minus infinity is less tha... |
mnfltpnf 12862 | Minus infinity is less tha... |
mnfltxr 12863 | Minus infinity is less tha... |
pnfnlt 12864 | No extended real is greate... |
nltmnf 12865 | No extended real is less t... |
pnfge 12866 | Plus infinity is an upper ... |
xnn0n0n1ge2b 12867 | An extended nonnegative in... |
0lepnf 12868 | 0 less than or equal to po... |
xnn0ge0 12869 | An extended nonnegative in... |
mnfle 12870 | Minus infinity is less tha... |
xrltnsym 12871 | Ordering on the extended r... |
xrltnsym2 12872 | 'Less than' is antisymmetr... |
xrlttri 12873 | Ordering on the extended r... |
xrlttr 12874 | Ordering on the extended r... |
xrltso 12875 | 'Less than' is a strict or... |
xrlttri2 12876 | Trichotomy law for 'less t... |
xrlttri3 12877 | Trichotomy law for 'less t... |
xrleloe 12878 | 'Less than or equal' expre... |
xrleltne 12879 | 'Less than or equal to' im... |
xrltlen 12880 | 'Less than' expressed in t... |
dfle2 12881 | Alternative definition of ... |
dflt2 12882 | Alternative definition of ... |
xrltle 12883 | 'Less than' implies 'less ... |
xrltled 12884 | 'Less than' implies 'less ... |
xrleid 12885 | 'Less than or equal to' is... |
xrleidd 12886 | 'Less than or equal to' is... |
xrletri 12887 | Trichotomy law for extende... |
xrletri3 12888 | Trichotomy law for extende... |
xrletrid 12889 | Trichotomy law for extende... |
xrlelttr 12890 | Transitive law for orderin... |
xrltletr 12891 | Transitive law for orderin... |
xrletr 12892 | Transitive law for orderin... |
xrlttrd 12893 | Transitive law for orderin... |
xrlelttrd 12894 | Transitive law for orderin... |
xrltletrd 12895 | Transitive law for orderin... |
xrletrd 12896 | Transitive law for orderin... |
xrltne 12897 | 'Less than' implies not eq... |
nltpnft 12898 | An extended real is not le... |
xgepnf 12899 | An extended real which is ... |
ngtmnft 12900 | An extended real is not gr... |
xlemnf 12901 | An extended real which is ... |
xrrebnd 12902 | An extended real is real i... |
xrre 12903 | A way of proving that an e... |
xrre2 12904 | An extended real between t... |
xrre3 12905 | A way of proving that an e... |
ge0gtmnf 12906 | A nonnegative extended rea... |
ge0nemnf 12907 | A nonnegative extended rea... |
xrrege0 12908 | A nonnegative extended rea... |
xrmax1 12909 | An extended real is less t... |
xrmax2 12910 | An extended real is less t... |
xrmin1 12911 | The minimum of two extende... |
xrmin2 12912 | The minimum of two extende... |
xrmaxeq 12913 | The maximum of two extende... |
xrmineq 12914 | The minimum of two extende... |
xrmaxlt 12915 | Two ways of saying the max... |
xrltmin 12916 | Two ways of saying an exte... |
xrmaxle 12917 | Two ways of saying the max... |
xrlemin 12918 | Two ways of saying a numbe... |
max1 12919 | A number is less than or e... |
max1ALT 12920 | A number is less than or e... |
max2 12921 | A number is less than or e... |
2resupmax 12922 | The supremum of two real n... |
min1 12923 | The minimum of two numbers... |
min2 12924 | The minimum of two numbers... |
maxle 12925 | Two ways of saying the max... |
lemin 12926 | Two ways of saying a numbe... |
maxlt 12927 | Two ways of saying the max... |
ltmin 12928 | Two ways of saying a numbe... |
lemaxle 12929 | A real number which is les... |
max0sub 12930 | Decompose a real number in... |
ifle 12931 | An if statement transforms... |
z2ge 12932 | There exists an integer gr... |
qbtwnre 12933 | The rational numbers are d... |
qbtwnxr 12934 | The rational numbers are d... |
qsqueeze 12935 | If a nonnegative real is l... |
qextltlem 12936 | Lemma for ~ qextlt and qex... |
qextlt 12937 | An extensionality-like pro... |
qextle 12938 | An extensionality-like pro... |
xralrple 12939 | Show that ` A ` is less th... |
alrple 12940 | Show that ` A ` is less th... |
xnegeq 12941 | Equality of two extended n... |
xnegex 12942 | A negative extended real e... |
xnegpnf 12943 | Minus ` +oo ` . Remark of... |
xnegmnf 12944 | Minus ` -oo ` . Remark of... |
rexneg 12945 | Minus a real number. Rema... |
xneg0 12946 | The negative of zero. (Co... |
xnegcl 12947 | Closure of extended real n... |
xnegneg 12948 | Extended real version of ~... |
xneg11 12949 | Extended real version of ~... |
xltnegi 12950 | Forward direction of ~ xlt... |
xltneg 12951 | Extended real version of ~... |
xleneg 12952 | Extended real version of ~... |
xlt0neg1 12953 | Extended real version of ~... |
xlt0neg2 12954 | Extended real version of ~... |
xle0neg1 12955 | Extended real version of ~... |
xle0neg2 12956 | Extended real version of ~... |
xaddval 12957 | Value of the extended real... |
xaddf 12958 | The extended real addition... |
xmulval 12959 | Value of the extended real... |
xaddpnf1 12960 | Addition of positive infin... |
xaddpnf2 12961 | Addition of positive infin... |
xaddmnf1 12962 | Addition of negative infin... |
xaddmnf2 12963 | Addition of negative infin... |
pnfaddmnf 12964 | Addition of positive and n... |
mnfaddpnf 12965 | Addition of negative and p... |
rexadd 12966 | The extended real addition... |
rexsub 12967 | Extended real subtraction ... |
rexaddd 12968 | The extended real addition... |
xnn0xaddcl 12969 | The extended nonnegative i... |
xaddnemnf 12970 | Closure of extended real a... |
xaddnepnf 12971 | Closure of extended real a... |
xnegid 12972 | Extended real version of ~... |
xaddcl 12973 | The extended real addition... |
xaddcom 12974 | The extended real addition... |
xaddid1 12975 | Extended real version of ~... |
xaddid2 12976 | Extended real version of ~... |
xaddid1d 12977 | ` 0 ` is a right identity ... |
xnn0lem1lt 12978 | Extended nonnegative integ... |
xnn0lenn0nn0 12979 | An extended nonnegative in... |
xnn0le2is012 12980 | An extended nonnegative in... |
xnn0xadd0 12981 | The sum of two extended no... |
xnegdi 12982 | Extended real version of ~... |
xaddass 12983 | Associativity of extended ... |
xaddass2 12984 | Associativity of extended ... |
xpncan 12985 | Extended real version of ~... |
xnpcan 12986 | Extended real version of ~... |
xleadd1a 12987 | Extended real version of ~... |
xleadd2a 12988 | Commuted form of ~ xleadd1... |
xleadd1 12989 | Weakened version of ~ xlea... |
xltadd1 12990 | Extended real version of ~... |
xltadd2 12991 | Extended real version of ~... |
xaddge0 12992 | The sum of nonnegative ext... |
xle2add 12993 | Extended real version of ~... |
xlt2add 12994 | Extended real version of ~... |
xsubge0 12995 | Extended real version of ~... |
xposdif 12996 | Extended real version of ~... |
xlesubadd 12997 | Under certain conditions, ... |
xmullem 12998 | Lemma for ~ rexmul . (Con... |
xmullem2 12999 | Lemma for ~ xmulneg1 . (C... |
xmulcom 13000 | Extended real multiplicati... |
xmul01 13001 | Extended real version of ~... |
xmul02 13002 | Extended real version of ~... |
xmulneg1 13003 | Extended real version of ~... |
xmulneg2 13004 | Extended real version of ~... |
rexmul 13005 | The extended real multipli... |
xmulf 13006 | The extended real multipli... |
xmulcl 13007 | Closure of extended real m... |
xmulpnf1 13008 | Multiplication by plus inf... |
xmulpnf2 13009 | Multiplication by plus inf... |
xmulmnf1 13010 | Multiplication by minus in... |
xmulmnf2 13011 | Multiplication by minus in... |
xmulpnf1n 13012 | Multiplication by plus inf... |
xmulid1 13013 | Extended real version of ~... |
xmulid2 13014 | Extended real version of ~... |
xmulm1 13015 | Extended real version of ~... |
xmulasslem2 13016 | Lemma for ~ xmulass . (Co... |
xmulgt0 13017 | Extended real version of ~... |
xmulge0 13018 | Extended real version of ~... |
xmulasslem 13019 | Lemma for ~ xmulass . (Co... |
xmulasslem3 13020 | Lemma for ~ xmulass . (Co... |
xmulass 13021 | Associativity of the exten... |
xlemul1a 13022 | Extended real version of ~... |
xlemul2a 13023 | Extended real version of ~... |
xlemul1 13024 | Extended real version of ~... |
xlemul2 13025 | Extended real version of ~... |
xltmul1 13026 | Extended real version of ~... |
xltmul2 13027 | Extended real version of ~... |
xadddilem 13028 | Lemma for ~ xadddi . (Con... |
xadddi 13029 | Distributive property for ... |
xadddir 13030 | Commuted version of ~ xadd... |
xadddi2 13031 | The assumption that the mu... |
xadddi2r 13032 | Commuted version of ~ xadd... |
x2times 13033 | Extended real version of ~... |
xnegcld 13034 | Closure of extended real n... |
xaddcld 13035 | The extended real addition... |
xmulcld 13036 | Closure of extended real m... |
xadd4d 13037 | Rearrangement of 4 terms i... |
xnn0add4d 13038 | Rearrangement of 4 terms i... |
xrsupexmnf 13039 | Adding minus infinity to a... |
xrinfmexpnf 13040 | Adding plus infinity to a ... |
xrsupsslem 13041 | Lemma for ~ xrsupss . (Co... |
xrinfmsslem 13042 | Lemma for ~ xrinfmss . (C... |
xrsupss 13043 | Any subset of extended rea... |
xrinfmss 13044 | Any subset of extended rea... |
xrinfmss2 13045 | Any subset of extended rea... |
xrub 13046 | By quantifying only over r... |
supxr 13047 | The supremum of a set of e... |
supxr2 13048 | The supremum of a set of e... |
supxrcl 13049 | The supremum of an arbitra... |
supxrun 13050 | The supremum of the union ... |
supxrmnf 13051 | Adding minus infinity to a... |
supxrpnf 13052 | The supremum of a set of e... |
supxrunb1 13053 | The supremum of an unbound... |
supxrunb2 13054 | The supremum of an unbound... |
supxrbnd1 13055 | The supremum of a bounded-... |
supxrbnd2 13056 | The supremum of a bounded-... |
xrsup0 13057 | The supremum of an empty s... |
supxrub 13058 | A member of a set of exten... |
supxrlub 13059 | The supremum of a set of e... |
supxrleub 13060 | The supremum of a set of e... |
supxrre 13061 | The real and extended real... |
supxrbnd 13062 | The supremum of a bounded-... |
supxrgtmnf 13063 | The supremum of a nonempty... |
supxrre1 13064 | The supremum of a nonempty... |
supxrre2 13065 | The supremum of a nonempty... |
supxrss 13066 | Smaller sets of extended r... |
infxrcl 13067 | The infimum of an arbitrar... |
infxrlb 13068 | A member of a set of exten... |
infxrgelb 13069 | The infimum of a set of ex... |
infxrre 13070 | The real and extended real... |
infxrmnf 13071 | The infinimum of a set of ... |
xrinf0 13072 | The infimum of the empty s... |
infxrss 13073 | Larger sets of extended re... |
reltre 13074 | For all real numbers there... |
rpltrp 13075 | For all positive real numb... |
reltxrnmnf 13076 | For all extended real numb... |
infmremnf 13077 | The infimum of the reals i... |
infmrp1 13078 | The infimum of the positiv... |
ixxval 13087 | Value of the interval func... |
elixx1 13088 | Membership in an interval ... |
ixxf 13089 | The set of intervals of ex... |
ixxex 13090 | The set of intervals of ex... |
ixxssxr 13091 | The set of intervals of ex... |
elixx3g 13092 | Membership in a set of ope... |
ixxssixx 13093 | An interval is a subset of... |
ixxdisj 13094 | Split an interval into dis... |
ixxun 13095 | Split an interval into two... |
ixxin 13096 | Intersection of two interv... |
ixxss1 13097 | Subset relationship for in... |
ixxss2 13098 | Subset relationship for in... |
ixxss12 13099 | Subset relationship for in... |
ixxub 13100 | Extract the upper bound of... |
ixxlb 13101 | Extract the lower bound of... |
iooex 13102 | The set of open intervals ... |
iooval 13103 | Value of the open interval... |
ioo0 13104 | An empty open interval of ... |
ioon0 13105 | An open interval of extend... |
ndmioo 13106 | The open interval function... |
iooid 13107 | An open interval with iden... |
elioo3g 13108 | Membership in a set of ope... |
elioore 13109 | A member of an open interv... |
lbioo 13110 | An open interval does not ... |
ubioo 13111 | An open interval does not ... |
iooval2 13112 | Value of the open interval... |
iooin 13113 | Intersection of two open i... |
iooss1 13114 | Subset relationship for op... |
iooss2 13115 | Subset relationship for op... |
iocval 13116 | Value of the open-below, c... |
icoval 13117 | Value of the closed-below,... |
iccval 13118 | Value of the closed interv... |
elioo1 13119 | Membership in an open inte... |
elioo2 13120 | Membership in an open inte... |
elioc1 13121 | Membership in an open-belo... |
elico1 13122 | Membership in a closed-bel... |
elicc1 13123 | Membership in a closed int... |
iccid 13124 | A closed interval with ide... |
ico0 13125 | An empty open interval of ... |
ioc0 13126 | An empty open interval of ... |
icc0 13127 | An empty closed interval o... |
dfrp2 13128 | Alternate definition of th... |
elicod 13129 | Membership in a left-close... |
icogelb 13130 | An element of a left-close... |
elicore 13131 | A member of a left-closed ... |
ubioc1 13132 | The upper bound belongs to... |
lbico1 13133 | The lower bound belongs to... |
iccleub 13134 | An element of a closed int... |
iccgelb 13135 | An element of a closed int... |
elioo5 13136 | Membership in an open inte... |
eliooxr 13137 | A nonempty open interval s... |
eliooord 13138 | Ordering implied by a memb... |
elioo4g 13139 | Membership in an open inte... |
ioossre 13140 | An open interval is a set ... |
ioosscn 13141 | An open interval is a set ... |
elioc2 13142 | Membership in an open-belo... |
elico2 13143 | Membership in a closed-bel... |
elicc2 13144 | Membership in a closed rea... |
elicc2i 13145 | Inference for membership i... |
elicc4 13146 | Membership in a closed rea... |
iccss 13147 | Condition for a closed int... |
iccssioo 13148 | Condition for a closed int... |
icossico 13149 | Condition for a closed-bel... |
iccss2 13150 | Condition for a closed int... |
iccssico 13151 | Condition for a closed int... |
iccssioo2 13152 | Condition for a closed int... |
iccssico2 13153 | Condition for a closed int... |
ioomax 13154 | The open interval from min... |
iccmax 13155 | The closed interval from m... |
ioopos 13156 | The set of positive reals ... |
ioorp 13157 | The set of positive reals ... |
iooshf 13158 | Shift the arguments of the... |
iocssre 13159 | A closed-above interval wi... |
icossre 13160 | A closed-below interval wi... |
iccssre 13161 | A closed real interval is ... |
iccssxr 13162 | A closed interval is a set... |
iocssxr 13163 | An open-below, closed-abov... |
icossxr 13164 | A closed-below, open-above... |
ioossicc 13165 | An open interval is a subs... |
iccssred 13166 | A closed real interval is ... |
eliccxr 13167 | A member of a closed inter... |
icossicc 13168 | A closed-below, open-above... |
iocssicc 13169 | A closed-above, open-below... |
ioossico 13170 | An open interval is a subs... |
iocssioo 13171 | Condition for a closed int... |
icossioo 13172 | Condition for a closed int... |
ioossioo 13173 | Condition for an open inte... |
iccsupr 13174 | A nonempty subset of a clo... |
elioopnf 13175 | Membership in an unbounded... |
elioomnf 13176 | Membership in an unbounded... |
elicopnf 13177 | Membership in a closed unb... |
repos 13178 | Two ways of saying that a ... |
ioof 13179 | The set of open intervals ... |
iccf 13180 | The set of closed interval... |
unirnioo 13181 | The union of the range of ... |
dfioo2 13182 | Alternate definition of th... |
ioorebas 13183 | Open intervals are element... |
xrge0neqmnf 13184 | A nonnegative extended rea... |
xrge0nre 13185 | An extended real which is ... |
elrege0 13186 | The predicate "is a nonneg... |
nn0rp0 13187 | A nonnegative integer is a... |
rge0ssre 13188 | Nonnegative real numbers a... |
elxrge0 13189 | Elementhood in the set of ... |
0e0icopnf 13190 | 0 is a member of ` ( 0 [,)... |
0e0iccpnf 13191 | 0 is a member of ` ( 0 [,]... |
ge0addcl 13192 | The nonnegative reals are ... |
ge0mulcl 13193 | The nonnegative reals are ... |
ge0xaddcl 13194 | The nonnegative reals are ... |
ge0xmulcl 13195 | The nonnegative extended r... |
lbicc2 13196 | The lower bound of a close... |
ubicc2 13197 | The upper bound of a close... |
elicc01 13198 | Membership in the closed r... |
elunitrn 13199 | The closed unit interval i... |
elunitcn 13200 | The closed unit interval i... |
0elunit 13201 | Zero is an element of the ... |
1elunit 13202 | One is an element of the c... |
iooneg 13203 | Membership in a negated op... |
iccneg 13204 | Membership in a negated cl... |
icoshft 13205 | A shifted real is a member... |
icoshftf1o 13206 | Shifting a closed-below, o... |
icoun 13207 | The union of two adjacent ... |
icodisj 13208 | Adjacent left-closed right... |
ioounsn 13209 | The union of an open inter... |
snunioo 13210 | The closure of one end of ... |
snunico 13211 | The closure of the open en... |
snunioc 13212 | The closure of the open en... |
prunioo 13213 | The closure of an open rea... |
ioodisj 13214 | If the upper bound of one ... |
ioojoin 13215 | Join two open intervals to... |
difreicc 13216 | The class difference of ` ... |
iccsplit 13217 | Split a closed interval in... |
iccshftr 13218 | Membership in a shifted in... |
iccshftri 13219 | Membership in a shifted in... |
iccshftl 13220 | Membership in a shifted in... |
iccshftli 13221 | Membership in a shifted in... |
iccdil 13222 | Membership in a dilated in... |
iccdili 13223 | Membership in a dilated in... |
icccntr 13224 | Membership in a contracted... |
icccntri 13225 | Membership in a contracted... |
divelunit 13226 | A condition for a ratio to... |
lincmb01cmp 13227 | A linear combination of tw... |
iccf1o 13228 | Describe a bijection from ... |
iccen 13229 | Any nontrivial closed inte... |
xov1plusxeqvd 13230 | A complex number ` X ` is ... |
unitssre 13231 | ` ( 0 [,] 1 ) ` is a subse... |
unitsscn 13232 | The closed unit interval i... |
supicc 13233 | Supremum of a bounded set ... |
supiccub 13234 | The supremum of a bounded ... |
supicclub 13235 | The supremum of a bounded ... |
supicclub2 13236 | The supremum of a bounded ... |
zltaddlt1le 13237 | The sum of an integer and ... |
xnn0xrge0 13238 | An extended nonnegative in... |
fzval 13241 | The value of a finite set ... |
fzval2 13242 | An alternative way of expr... |
fzf 13243 | Establish the domain and c... |
elfz1 13244 | Membership in a finite set... |
elfz 13245 | Membership in a finite set... |
elfz2 13246 | Membership in a finite set... |
elfzd 13247 | Membership in a finite set... |
elfz5 13248 | Membership in a finite set... |
elfz4 13249 | Membership in a finite set... |
elfzuzb 13250 | Membership in a finite set... |
eluzfz 13251 | Membership in a finite set... |
elfzuz 13252 | A member of a finite set o... |
elfzuz3 13253 | Membership in a finite set... |
elfzel2 13254 | Membership in a finite set... |
elfzel1 13255 | Membership in a finite set... |
elfzelz 13256 | A member of a finite set o... |
elfzelzd 13257 | A member of a finite set o... |
fzssz 13258 | A finite sequence of integ... |
elfzle1 13259 | A member of a finite set o... |
elfzle2 13260 | A member of a finite set o... |
elfzuz2 13261 | Implication of membership ... |
elfzle3 13262 | Membership in a finite set... |
eluzfz1 13263 | Membership in a finite set... |
eluzfz2 13264 | Membership in a finite set... |
eluzfz2b 13265 | Membership in a finite set... |
elfz3 13266 | Membership in a finite set... |
elfz1eq 13267 | Membership in a finite set... |
elfzubelfz 13268 | If there is a member in a ... |
peano2fzr 13269 | A Peano-postulate-like the... |
fzn0 13270 | Properties of a finite int... |
fz0 13271 | A finite set of sequential... |
fzn 13272 | A finite set of sequential... |
fzen 13273 | A shifted finite set of se... |
fz1n 13274 | A 1-based finite set of se... |
0nelfz1 13275 | 0 is not an element of a f... |
0fz1 13276 | Two ways to say a finite 1... |
fz10 13277 | There are no integers betw... |
uzsubsubfz 13278 | Membership of an integer g... |
uzsubsubfz1 13279 | Membership of an integer g... |
ige3m2fz 13280 | Membership of an integer g... |
fzsplit2 13281 | Split a finite interval of... |
fzsplit 13282 | Split a finite interval of... |
fzdisj 13283 | Condition for two finite i... |
fz01en 13284 | 0-based and 1-based finite... |
elfznn 13285 | A member of a finite set o... |
elfz1end 13286 | A nonempty finite range of... |
fz1ssnn 13287 | A finite set of positive i... |
fznn0sub 13288 | Subtraction closure for a ... |
fzmmmeqm 13289 | Subtracting the difference... |
fzaddel 13290 | Membership of a sum in a f... |
fzadd2 13291 | Membership of a sum in a f... |
fzsubel 13292 | Membership of a difference... |
fzopth 13293 | A finite set of sequential... |
fzass4 13294 | Two ways to express a nond... |
fzss1 13295 | Subset relationship for fi... |
fzss2 13296 | Subset relationship for fi... |
fzssuz 13297 | A finite set of sequential... |
fzsn 13298 | A finite interval of integ... |
fzssp1 13299 | Subset relationship for fi... |
fzssnn 13300 | Finite sets of sequential ... |
ssfzunsnext 13301 | A subset of a finite seque... |
ssfzunsn 13302 | A subset of a finite seque... |
fzsuc 13303 | Join a successor to the en... |
fzpred 13304 | Join a predecessor to the ... |
fzpreddisj 13305 | A finite set of sequential... |
elfzp1 13306 | Append an element to a fin... |
fzp1ss 13307 | Subset relationship for fi... |
fzelp1 13308 | Membership in a set of seq... |
fzp1elp1 13309 | Add one to an element of a... |
fznatpl1 13310 | Shift membership in a fini... |
fzpr 13311 | A finite interval of integ... |
fztp 13312 | A finite interval of integ... |
fz12pr 13313 | An integer range between 1... |
fzsuc2 13314 | Join a successor to the en... |
fzp1disj 13315 | ` ( M ... ( N + 1 ) ) ` is... |
fzdifsuc 13316 | Remove a successor from th... |
fzprval 13317 | Two ways of defining the f... |
fztpval 13318 | Two ways of defining the f... |
fzrev 13319 | Reversal of start and end ... |
fzrev2 13320 | Reversal of start and end ... |
fzrev2i 13321 | Reversal of start and end ... |
fzrev3 13322 | The "complement" of a memb... |
fzrev3i 13323 | The "complement" of a memb... |
fznn 13324 | Finite set of sequential i... |
elfz1b 13325 | Membership in a 1-based fi... |
elfz1uz 13326 | Membership in a 1-based fi... |
elfzm11 13327 | Membership in a finite set... |
uzsplit 13328 | Express an upper integer s... |
uzdisj 13329 | The first ` N ` elements o... |
fseq1p1m1 13330 | Add/remove an item to/from... |
fseq1m1p1 13331 | Add/remove an item to/from... |
fz1sbc 13332 | Quantification over a one-... |
elfzp1b 13333 | An integer is a member of ... |
elfzm1b 13334 | An integer is a member of ... |
elfzp12 13335 | Options for membership in ... |
fzm1 13336 | Choices for an element of ... |
fzneuz 13337 | No finite set of sequentia... |
fznuz 13338 | Disjointness of the upper ... |
uznfz 13339 | Disjointness of the upper ... |
fzp1nel 13340 | One plus the upper bound o... |
fzrevral 13341 | Reversal of scanning order... |
fzrevral2 13342 | Reversal of scanning order... |
fzrevral3 13343 | Reversal of scanning order... |
fzshftral 13344 | Shift the scanning order i... |
ige2m1fz1 13345 | Membership of an integer g... |
ige2m1fz 13346 | Membership in a 0-based fi... |
elfz2nn0 13347 | Membership in a finite set... |
fznn0 13348 | Characterization of a fini... |
elfznn0 13349 | A member of a finite set o... |
elfz3nn0 13350 | The upper bound of a nonem... |
fz0ssnn0 13351 | Finite sets of sequential ... |
fz1ssfz0 13352 | Subset relationship for fi... |
0elfz 13353 | 0 is an element of a finit... |
nn0fz0 13354 | A nonnegative integer is a... |
elfz0add 13355 | An element of a finite set... |
fz0sn 13356 | An integer range from 0 to... |
fz0tp 13357 | An integer range from 0 to... |
fz0to3un2pr 13358 | An integer range from 0 to... |
fz0to4untppr 13359 | An integer range from 0 to... |
elfz0ubfz0 13360 | An element of a finite set... |
elfz0fzfz0 13361 | A member of a finite set o... |
fz0fzelfz0 13362 | If a member of a finite se... |
fznn0sub2 13363 | Subtraction closure for a ... |
uzsubfz0 13364 | Membership of an integer g... |
fz0fzdiffz0 13365 | The difference of an integ... |
elfzmlbm 13366 | Subtracting the lower boun... |
elfzmlbp 13367 | Subtracting the lower boun... |
fzctr 13368 | Lemma for theorems about t... |
difelfzle 13369 | The difference of two inte... |
difelfznle 13370 | The difference of two inte... |
nn0split 13371 | Express the set of nonnega... |
nn0disj 13372 | The first ` N + 1 ` elemen... |
fz0sn0fz1 13373 | A finite set of sequential... |
fvffz0 13374 | The function value of a fu... |
1fv 13375 | A function on a singleton.... |
4fvwrd4 13376 | The first four function va... |
2ffzeq 13377 | Two functions over 0-based... |
preduz 13378 | The value of the predecess... |
prednn 13379 | The value of the predecess... |
prednn0 13380 | The value of the predecess... |
predfz 13381 | Calculate the predecessor ... |
fzof 13384 | Functionality of the half-... |
elfzoel1 13385 | Reverse closure for half-o... |
elfzoel2 13386 | Reverse closure for half-o... |
elfzoelz 13387 | Reverse closure for half-o... |
fzoval 13388 | Value of the half-open int... |
elfzo 13389 | Membership in a half-open ... |
elfzo2 13390 | Membership in a half-open ... |
elfzouz 13391 | Membership in a half-open ... |
nelfzo 13392 | An integer not being a mem... |
fzolb 13393 | The left endpoint of a hal... |
fzolb2 13394 | The left endpoint of a hal... |
elfzole1 13395 | A member in a half-open in... |
elfzolt2 13396 | A member in a half-open in... |
elfzolt3 13397 | Membership in a half-open ... |
elfzolt2b 13398 | A member in a half-open in... |
elfzolt3b 13399 | Membership in a half-open ... |
elfzop1le2 13400 | A member in a half-open in... |
fzonel 13401 | A half-open range does not... |
elfzouz2 13402 | The upper bound of a half-... |
elfzofz 13403 | A half-open range is conta... |
elfzo3 13404 | Express membership in a ha... |
fzon0 13405 | A half-open integer interv... |
fzossfz 13406 | A half-open range is conta... |
fzossz 13407 | A half-open integer interv... |
fzon 13408 | A half-open set of sequent... |
fzo0n 13409 | A half-open range of nonne... |
fzonlt0 13410 | A half-open integer range ... |
fzo0 13411 | Half-open sets with equal ... |
fzonnsub 13412 | If ` K < N ` then ` N - K ... |
fzonnsub2 13413 | If ` M < N ` then ` N - M ... |
fzoss1 13414 | Subset relationship for ha... |
fzoss2 13415 | Subset relationship for ha... |
fzossrbm1 13416 | Subset of a half-open rang... |
fzo0ss1 13417 | Subset relationship for ha... |
fzossnn0 13418 | A half-open integer range ... |
fzospliti 13419 | One direction of splitting... |
fzosplit 13420 | Split a half-open integer ... |
fzodisj 13421 | Abutting half-open integer... |
fzouzsplit 13422 | Split an upper integer set... |
fzouzdisj 13423 | A half-open integer range ... |
fzoun 13424 | A half-open integer range ... |
fzodisjsn 13425 | A half-open integer range ... |
prinfzo0 13426 | The intersection of a half... |
lbfzo0 13427 | An integer is strictly gre... |
elfzo0 13428 | Membership in a half-open ... |
elfzo0z 13429 | Membership in a half-open ... |
nn0p1elfzo 13430 | A nonnegative integer incr... |
elfzo0le 13431 | A member in a half-open ra... |
elfzonn0 13432 | A member of a half-open ra... |
fzonmapblen 13433 | The result of subtracting ... |
fzofzim 13434 | If a nonnegative integer i... |
fz1fzo0m1 13435 | Translation of one between... |
fzossnn 13436 | Half-open integer ranges s... |
elfzo1 13437 | Membership in a half-open ... |
fzo1fzo0n0 13438 | An integer between 1 and a... |
fzo0n0 13439 | A half-open integer range ... |
fzoaddel 13440 | Translate membership in a ... |
fzo0addel 13441 | Translate membership in a ... |
fzo0addelr 13442 | Translate membership in a ... |
fzoaddel2 13443 | Translate membership in a ... |
elfzoext 13444 | Membership of an integer i... |
elincfzoext 13445 | Membership of an increased... |
fzosubel 13446 | Translate membership in a ... |
fzosubel2 13447 | Membership in a translated... |
fzosubel3 13448 | Membership in a translated... |
eluzgtdifelfzo 13449 | Membership of the differen... |
ige2m2fzo 13450 | Membership of an integer g... |
fzocatel 13451 | Translate membership in a ... |
ubmelfzo 13452 | If an integer in a 1-based... |
elfzodifsumelfzo 13453 | If an integer is in a half... |
elfzom1elp1fzo 13454 | Membership of an integer i... |
elfzom1elfzo 13455 | Membership in a half-open ... |
fzval3 13456 | Expressing a closed intege... |
fz0add1fz1 13457 | Translate membership in a ... |
fzosn 13458 | Expressing a singleton as ... |
elfzomin 13459 | Membership of an integer i... |
zpnn0elfzo 13460 | Membership of an integer i... |
zpnn0elfzo1 13461 | Membership of an integer i... |
fzosplitsnm1 13462 | Removing a singleton from ... |
elfzonlteqm1 13463 | If an element of a half-op... |
fzonn0p1 13464 | A nonnegative integer is e... |
fzossfzop1 13465 | A half-open range of nonne... |
fzonn0p1p1 13466 | If a nonnegative integer i... |
elfzom1p1elfzo 13467 | Increasing an element of a... |
fzo0ssnn0 13468 | Half-open integer ranges s... |
fzo01 13469 | Expressing the singleton o... |
fzo12sn 13470 | A 1-based half-open intege... |
fzo13pr 13471 | A 1-based half-open intege... |
fzo0to2pr 13472 | A half-open integer range ... |
fzo0to3tp 13473 | A half-open integer range ... |
fzo0to42pr 13474 | A half-open integer range ... |
fzo1to4tp 13475 | A half-open integer range ... |
fzo0sn0fzo1 13476 | A half-open range of nonne... |
elfzo0l 13477 | A member of a half-open ra... |
fzoend 13478 | The endpoint of a half-ope... |
fzo0end 13479 | The endpoint of a zero-bas... |
ssfzo12 13480 | Subset relationship for ha... |
ssfzoulel 13481 | If a half-open integer ran... |
ssfzo12bi 13482 | Subset relationship for ha... |
ubmelm1fzo 13483 | The result of subtracting ... |
fzofzp1 13484 | If a point is in a half-op... |
fzofzp1b 13485 | If a point is in a half-op... |
elfzom1b 13486 | An integer is a member of ... |
elfzom1elp1fzo1 13487 | Membership of a nonnegativ... |
elfzo1elm1fzo0 13488 | Membership of a positive i... |
elfzonelfzo 13489 | If an element of a half-op... |
fzonfzoufzol 13490 | If an element of a half-op... |
elfzomelpfzo 13491 | An integer increased by an... |
elfznelfzo 13492 | A value in a finite set of... |
elfznelfzob 13493 | A value in a finite set of... |
peano2fzor 13494 | A Peano-postulate-like the... |
fzosplitsn 13495 | Extending a half-open rang... |
fzosplitpr 13496 | Extending a half-open inte... |
fzosplitprm1 13497 | Extending a half-open inte... |
fzosplitsni 13498 | Membership in a half-open ... |
fzisfzounsn 13499 | A finite interval of integ... |
elfzr 13500 | A member of a finite inter... |
elfzlmr 13501 | A member of a finite inter... |
elfz0lmr 13502 | A member of a finite inter... |
fzostep1 13503 | Two possibilities for a nu... |
fzoshftral 13504 | Shift the scanning order i... |
fzind2 13505 | Induction on the integers ... |
fvinim0ffz 13506 | The function values for th... |
injresinjlem 13507 | Lemma for ~ injresinj . (... |
injresinj 13508 | A function whose restricti... |
subfzo0 13509 | The difference between two... |
flval 13514 | Value of the floor (greate... |
flcl 13515 | The floor (greatest intege... |
reflcl 13516 | The floor (greatest intege... |
fllelt 13517 | A basic property of the fl... |
flcld 13518 | The floor (greatest intege... |
flle 13519 | A basic property of the fl... |
flltp1 13520 | A basic property of the fl... |
fllep1 13521 | A basic property of the fl... |
fraclt1 13522 | The fractional part of a r... |
fracle1 13523 | The fractional part of a r... |
fracge0 13524 | The fractional part of a r... |
flge 13525 | The floor function value i... |
fllt 13526 | The floor function value i... |
flflp1 13527 | Move floor function betwee... |
flid 13528 | An integer is its own floo... |
flidm 13529 | The floor function is idem... |
flidz 13530 | A real number equals its f... |
flltnz 13531 | The floor of a non-integer... |
flwordi 13532 | Ordering relation for the ... |
flword2 13533 | Ordering relation for the ... |
flval2 13534 | An alternate way to define... |
flval3 13535 | An alternate way to define... |
flbi 13536 | A condition equivalent to ... |
flbi2 13537 | A condition equivalent to ... |
adddivflid 13538 | The floor of a sum of an i... |
ico01fl0 13539 | The floor of a real number... |
flge0nn0 13540 | The floor of a number grea... |
flge1nn 13541 | The floor of a number grea... |
fldivnn0 13542 | The floor function of a di... |
refldivcl 13543 | The floor function of a di... |
divfl0 13544 | The floor of a fraction is... |
fladdz 13545 | An integer can be moved in... |
flzadd 13546 | An integer can be moved in... |
flmulnn0 13547 | Move a nonnegative integer... |
btwnzge0 13548 | A real bounded between an ... |
2tnp1ge0ge0 13549 | Two times an integer plus ... |
flhalf 13550 | Ordering relation for the ... |
fldivle 13551 | The floor function of a di... |
fldivnn0le 13552 | The floor function of a di... |
flltdivnn0lt 13553 | The floor function of a di... |
ltdifltdiv 13554 | If the dividend of a divis... |
fldiv4p1lem1div2 13555 | The floor of an integer eq... |
fldiv4lem1div2uz2 13556 | The floor of an integer gr... |
fldiv4lem1div2 13557 | The floor of a positive in... |
ceilval 13558 | The value of the ceiling f... |
dfceil2 13559 | Alternative definition of ... |
ceilval2 13560 | The value of the ceiling f... |
ceicl 13561 | The ceiling function retur... |
ceilcl 13562 | Closure of the ceiling fun... |
ceilcld 13563 | Closure of the ceiling fun... |
ceige 13564 | The ceiling of a real numb... |
ceilge 13565 | The ceiling of a real numb... |
ceilged 13566 | The ceiling of a real numb... |
ceim1l 13567 | One less than the ceiling ... |
ceilm1lt 13568 | One less than the ceiling ... |
ceile 13569 | The ceiling of a real numb... |
ceille 13570 | The ceiling of a real numb... |
ceilid 13571 | An integer is its own ceil... |
ceilidz 13572 | A real number equals its c... |
flleceil 13573 | The floor of a real number... |
fleqceilz 13574 | A real number is an intege... |
quoremz 13575 | Quotient and remainder of ... |
quoremnn0 13576 | Quotient and remainder of ... |
quoremnn0ALT 13577 | Alternate proof of ~ quore... |
intfrac2 13578 | Decompose a real into inte... |
intfracq 13579 | Decompose a rational numbe... |
fldiv 13580 | Cancellation of the embedd... |
fldiv2 13581 | Cancellation of an embedde... |
fznnfl 13582 | Finite set of sequential i... |
uzsup 13583 | An upper set of integers i... |
ioopnfsup 13584 | An upper set of reals is u... |
icopnfsup 13585 | An upper set of reals is u... |
rpsup 13586 | The positive reals are unb... |
resup 13587 | The real numbers are unbou... |
xrsup 13588 | The extended real numbers ... |
modval 13591 | The value of the modulo op... |
modvalr 13592 | The value of the modulo op... |
modcl 13593 | Closure law for the modulo... |
flpmodeq 13594 | Partition of a division in... |
modcld 13595 | Closure law for the modulo... |
mod0 13596 | ` A mod B ` is zero iff ` ... |
mulmod0 13597 | The product of an integer ... |
negmod0 13598 | ` A ` is divisible by ` B ... |
modge0 13599 | The modulo operation is no... |
modlt 13600 | The modulo operation is le... |
modelico 13601 | Modular reduction produces... |
moddiffl 13602 | Value of the modulo operat... |
moddifz 13603 | The modulo operation diffe... |
modfrac 13604 | The fractional part of a n... |
flmod 13605 | The floor function express... |
intfrac 13606 | Break a number into its in... |
zmod10 13607 | An integer modulo 1 is 0. ... |
zmod1congr 13608 | Two arbitrary integers are... |
modmulnn 13609 | Move a positive integer in... |
modvalp1 13610 | The value of the modulo op... |
zmodcl 13611 | Closure law for the modulo... |
zmodcld 13612 | Closure law for the modulo... |
zmodfz 13613 | An integer mod ` B ` lies ... |
zmodfzo 13614 | An integer mod ` B ` lies ... |
zmodfzp1 13615 | An integer mod ` B ` lies ... |
modid 13616 | Identity law for modulo. ... |
modid0 13617 | A positive real number mod... |
modid2 13618 | Identity law for modulo. ... |
zmodid2 13619 | Identity law for modulo re... |
zmodidfzo 13620 | Identity law for modulo re... |
zmodidfzoimp 13621 | Identity law for modulo re... |
0mod 13622 | Special case: 0 modulo a p... |
1mod 13623 | Special case: 1 modulo a r... |
modabs 13624 | Absorption law for modulo.... |
modabs2 13625 | Absorption law for modulo.... |
modcyc 13626 | The modulo operation is pe... |
modcyc2 13627 | The modulo operation is pe... |
modadd1 13628 | Addition property of the m... |
modaddabs 13629 | Absorption law for modulo.... |
modaddmod 13630 | The sum of a real number m... |
muladdmodid 13631 | The sum of a positive real... |
mulp1mod1 13632 | The product of an integer ... |
modmuladd 13633 | Decomposition of an intege... |
modmuladdim 13634 | Implication of a decomposi... |
modmuladdnn0 13635 | Implication of a decomposi... |
negmod 13636 | The negation of a number m... |
m1modnnsub1 13637 | Minus one modulo a positiv... |
m1modge3gt1 13638 | Minus one modulo an intege... |
addmodid 13639 | The sum of a positive inte... |
addmodidr 13640 | The sum of a positive inte... |
modadd2mod 13641 | The sum of a real number m... |
modm1p1mod0 13642 | If a real number modulo a ... |
modltm1p1mod 13643 | If a real number modulo a ... |
modmul1 13644 | Multiplication property of... |
modmul12d 13645 | Multiplication property of... |
modnegd 13646 | Negation property of the m... |
modadd12d 13647 | Additive property of the m... |
modsub12d 13648 | Subtraction property of th... |
modsubmod 13649 | The difference of a real n... |
modsubmodmod 13650 | The difference of a real n... |
2txmodxeq0 13651 | Two times a positive real ... |
2submod 13652 | If a real number is betwee... |
modifeq2int 13653 | If a nonnegative integer i... |
modaddmodup 13654 | The sum of an integer modu... |
modaddmodlo 13655 | The sum of an integer modu... |
modmulmod 13656 | The product of a real numb... |
modmulmodr 13657 | The product of an integer ... |
modaddmulmod 13658 | The sum of a real number a... |
moddi 13659 | Distribute multiplication ... |
modsubdir 13660 | Distribute the modulo oper... |
modeqmodmin 13661 | A real number equals the d... |
modirr 13662 | A number modulo an irratio... |
modfzo0difsn 13663 | For a number within a half... |
modsumfzodifsn 13664 | The sum of a number within... |
modlteq 13665 | Two nonnegative integers l... |
addmodlteq 13666 | Two nonnegative integers l... |
om2uz0i 13667 | The mapping ` G ` is a one... |
om2uzsuci 13668 | The value of ` G ` (see ~ ... |
om2uzuzi 13669 | The value ` G ` (see ~ om2... |
om2uzlti 13670 | Less-than relation for ` G... |
om2uzlt2i 13671 | The mapping ` G ` (see ~ o... |
om2uzrani 13672 | Range of ` G ` (see ~ om2u... |
om2uzf1oi 13673 | ` G ` (see ~ om2uz0i ) is ... |
om2uzisoi 13674 | ` G ` (see ~ om2uz0i ) is ... |
om2uzoi 13675 | An alternative definition ... |
om2uzrdg 13676 | A helper lemma for the val... |
uzrdglem 13677 | A helper lemma for the val... |
uzrdgfni 13678 | The recursive definition g... |
uzrdg0i 13679 | Initial value of a recursi... |
uzrdgsuci 13680 | Successor value of a recur... |
ltweuz 13681 | ` < ` is a well-founded re... |
ltwenn 13682 | Less than well-orders the ... |
ltwefz 13683 | Less than well-orders a se... |
uzenom 13684 | An upper integer set is de... |
uzinf 13685 | An upper integer set is in... |
nnnfi 13686 | The set of positive intege... |
uzrdgxfr 13687 | Transfer the value of the ... |
fzennn 13688 | The cardinality of a finit... |
fzen2 13689 | The cardinality of a finit... |
cardfz 13690 | The cardinality of a finit... |
hashgf1o 13691 | ` G ` maps ` _om ` one-to-... |
fzfi 13692 | A finite interval of integ... |
fzfid 13693 | Commonly used special case... |
fzofi 13694 | Half-open integer sets are... |
fsequb 13695 | The values of a finite rea... |
fsequb2 13696 | The values of a finite rea... |
fseqsupcl 13697 | The values of a finite rea... |
fseqsupubi 13698 | The values of a finite rea... |
nn0ennn 13699 | The nonnegative integers a... |
nnenom 13700 | The set of positive intege... |
nnct 13701 | ` NN ` is countable. (Con... |
uzindi 13702 | Indirect strong induction ... |
axdc4uzlem 13703 | Lemma for ~ axdc4uz . (Co... |
axdc4uz 13704 | A version of ~ axdc4 that ... |
ssnn0fi 13705 | A subset of the nonnegativ... |
rabssnn0fi 13706 | A subset of the nonnegativ... |
uzsinds 13707 | Strong (or "total") induct... |
nnsinds 13708 | Strong (or "total") induct... |
nn0sinds 13709 | Strong (or "total") induct... |
fsuppmapnn0fiublem 13710 | Lemma for ~ fsuppmapnn0fiu... |
fsuppmapnn0fiub 13711 | If all functions of a fini... |
fsuppmapnn0fiubex 13712 | If all functions of a fini... |
fsuppmapnn0fiub0 13713 | If all functions of a fini... |
suppssfz 13714 | Condition for a function o... |
fsuppmapnn0ub 13715 | If a function over the non... |
fsuppmapnn0fz 13716 | If a function over the non... |
mptnn0fsupp 13717 | A mapping from the nonnega... |
mptnn0fsuppd 13718 | A mapping from the nonnega... |
mptnn0fsuppr 13719 | A finitely supported mappi... |
f13idfv 13720 | A one-to-one function with... |
seqex 13723 | Existence of the sequence ... |
seqeq1 13724 | Equality theorem for the s... |
seqeq2 13725 | Equality theorem for the s... |
seqeq3 13726 | Equality theorem for the s... |
seqeq1d 13727 | Equality deduction for the... |
seqeq2d 13728 | Equality deduction for the... |
seqeq3d 13729 | Equality deduction for the... |
seqeq123d 13730 | Equality deduction for the... |
nfseq 13731 | Hypothesis builder for the... |
seqval 13732 | Value of the sequence buil... |
seqfn 13733 | The sequence builder funct... |
seq1 13734 | Value of the sequence buil... |
seq1i 13735 | Value of the sequence buil... |
seqp1 13736 | Value of the sequence buil... |
seqexw 13737 | Weak version of ~ seqex th... |
seqp1d 13738 | Value of the sequence buil... |
seqp1iOLD 13739 | Obsolete version of ~ seqp... |
seqm1 13740 | Value of the sequence buil... |
seqcl2 13741 | Closure properties of the ... |
seqf2 13742 | Range of the recursive seq... |
seqcl 13743 | Closure properties of the ... |
seqf 13744 | Range of the recursive seq... |
seqfveq2 13745 | Equality of sequences. (C... |
seqfeq2 13746 | Equality of sequences. (C... |
seqfveq 13747 | Equality of sequences. (C... |
seqfeq 13748 | Equality of sequences. (C... |
seqshft2 13749 | Shifting the index set of ... |
seqres 13750 | Restricting its characteri... |
serf 13751 | An infinite series of comp... |
serfre 13752 | An infinite series of real... |
monoord 13753 | Ordering relation for a mo... |
monoord2 13754 | Ordering relation for a mo... |
sermono 13755 | The partial sums in an inf... |
seqsplit 13756 | Split a sequence into two ... |
seq1p 13757 | Removing the first term fr... |
seqcaopr3 13758 | Lemma for ~ seqcaopr2 . (... |
seqcaopr2 13759 | The sum of two infinite se... |
seqcaopr 13760 | The sum of two infinite se... |
seqf1olem2a 13761 | Lemma for ~ seqf1o . (Con... |
seqf1olem1 13762 | Lemma for ~ seqf1o . (Con... |
seqf1olem2 13763 | Lemma for ~ seqf1o . (Con... |
seqf1o 13764 | Rearrange a sum via an arb... |
seradd 13765 | The sum of two infinite se... |
sersub 13766 | The difference of two infi... |
seqid3 13767 | A sequence that consists e... |
seqid 13768 | Discarding the first few t... |
seqid2 13769 | The last few partial sums ... |
seqhomo 13770 | Apply a homomorphism to a ... |
seqz 13771 | If the operation ` .+ ` ha... |
seqfeq4 13772 | Equality of series under d... |
seqfeq3 13773 | Equality of series under d... |
seqdistr 13774 | The distributive property ... |
ser0 13775 | The value of the partial s... |
ser0f 13776 | A zero-valued infinite ser... |
serge0 13777 | A finite sum of nonnegativ... |
serle 13778 | Comparison of partial sums... |
ser1const 13779 | Value of the partial serie... |
seqof 13780 | Distribute function operat... |
seqof2 13781 | Distribute function operat... |
expval 13784 | Value of exponentiation to... |
expnnval 13785 | Value of exponentiation to... |
exp0 13786 | Value of a complex number ... |
0exp0e1 13787 | The zeroth power of zero e... |
exp1 13788 | Value of a complex number ... |
expp1 13789 | Value of a complex number ... |
expneg 13790 | Value of a complex number ... |
expneg2 13791 | Value of a complex number ... |
expn1 13792 | A number to the negative o... |
expcllem 13793 | Lemma for proving nonnegat... |
expcl2lem 13794 | Lemma for proving integer ... |
nnexpcl 13795 | Closure of exponentiation ... |
nn0expcl 13796 | Closure of exponentiation ... |
zexpcl 13797 | Closure of exponentiation ... |
qexpcl 13798 | Closure of exponentiation ... |
reexpcl 13799 | Closure of exponentiation ... |
expcl 13800 | Closure law for nonnegativ... |
rpexpcl 13801 | Closure law for exponentia... |
reexpclz 13802 | Closure of exponentiation ... |
qexpclz 13803 | Closure of exponentiation ... |
m1expcl2 13804 | Closure of exponentiation ... |
m1expcl 13805 | Closure of exponentiation ... |
expclzlem 13806 | Closure law for integer ex... |
expclz 13807 | Closure law for integer ex... |
zexpcld 13808 | Closure of exponentiation ... |
nn0expcli 13809 | Closure of exponentiation ... |
nn0sqcl 13810 | The square of a nonnegativ... |
expm1t 13811 | Exponentiation in terms of... |
1exp 13812 | Value of one raised to a n... |
expeq0 13813 | Positive integer exponenti... |
expne0 13814 | Positive integer exponenti... |
expne0i 13815 | Nonnegative integer expone... |
expgt0 13816 | A positive real raised to ... |
expnegz 13817 | Value of a complex number ... |
0exp 13818 | Value of zero raised to a ... |
expge0 13819 | A nonnegative real raised ... |
expge1 13820 | A real greater than or equ... |
expgt1 13821 | A real greater than 1 rais... |
mulexp 13822 | Nonnegative integer expone... |
mulexpz 13823 | Integer exponentiation of ... |
exprec 13824 | Integer exponentiation of ... |
expadd 13825 | Sum of exponents law for n... |
expaddzlem 13826 | Lemma for ~ expaddz . (Co... |
expaddz 13827 | Sum of exponents law for i... |
expmul 13828 | Product of exponents law f... |
expmulz 13829 | Product of exponents law f... |
m1expeven 13830 | Exponentiation of negative... |
expsub 13831 | Exponent subtraction law f... |
expp1z 13832 | Value of a nonzero complex... |
expm1 13833 | Value of a complex number ... |
expdiv 13834 | Nonnegative integer expone... |
sqval 13835 | Value of the square of a c... |
sqneg 13836 | The square of the negative... |
sqsubswap 13837 | Swap the order of subtract... |
sqcl 13838 | Closure of square. (Contr... |
sqmul 13839 | Distribution of square ove... |
sqeq0 13840 | A number is zero iff its s... |
sqdiv 13841 | Distribution of square ove... |
sqdivid 13842 | The square of a nonzero nu... |
sqne0 13843 | A number is nonzero iff it... |
resqcl 13844 | Closure of the square of a... |
sqgt0 13845 | The square of a nonzero re... |
sqn0rp 13846 | The square of a nonzero re... |
nnsqcl 13847 | The naturals are closed un... |
zsqcl 13848 | Integers are closed under ... |
qsqcl 13849 | The square of a rational i... |
sq11 13850 | The square function is one... |
nn0sq11 13851 | The square function is one... |
lt2sq 13852 | The square function on non... |
le2sq 13853 | The square function on non... |
le2sq2 13854 | The square of a 'less than... |
sqge0 13855 | A square of a real is nonn... |
zsqcl2 13856 | The square of an integer i... |
0expd 13857 | Value of zero raised to a ... |
exp0d 13858 | Value of a complex number ... |
exp1d 13859 | Value of a complex number ... |
expeq0d 13860 | Positive integer exponenti... |
sqvald 13861 | Value of square. Inferenc... |
sqcld 13862 | Closure of square. (Contr... |
sqeq0d 13863 | A number is zero iff its s... |
expcld 13864 | Closure law for nonnegativ... |
expp1d 13865 | Value of a complex number ... |
expaddd 13866 | Sum of exponents law for n... |
expmuld 13867 | Product of exponents law f... |
sqrecd 13868 | Square of reciprocal. (Co... |
expclzd 13869 | Closure law for integer ex... |
expne0d 13870 | Nonnegative integer expone... |
expnegd 13871 | Value of a complex number ... |
exprecd 13872 | Nonnegative integer expone... |
expp1zd 13873 | Value of a nonzero complex... |
expm1d 13874 | Value of a complex number ... |
expsubd 13875 | Exponent subtraction law f... |
sqmuld 13876 | Distribution of square ove... |
sqdivd 13877 | Distribution of square ove... |
expdivd 13878 | Nonnegative integer expone... |
mulexpd 13879 | Positive integer exponenti... |
znsqcld 13880 | The square of a nonzero in... |
reexpcld 13881 | Closure of exponentiation ... |
expge0d 13882 | A nonnegative real raised ... |
expge1d 13883 | A real greater than or equ... |
ltexp2a 13884 | Ordering relationship for ... |
expmordi 13885 | Base ordering relationship... |
rpexpmord 13886 | Base ordering relationship... |
expcan 13887 | Cancellation law for expon... |
ltexp2 13888 | Ordering law for exponenti... |
leexp2 13889 | Ordering law for exponenti... |
leexp2a 13890 | Weak ordering relationship... |
ltexp2r 13891 | The power of a positive nu... |
leexp2r 13892 | Weak ordering relationship... |
leexp1a 13893 | Weak base ordering relatio... |
exple1 13894 | A real between 0 and 1 inc... |
expubnd 13895 | An upper bound on ` A ^ N ... |
sumsqeq0 13896 | Two real numbers are equal... |
sqvali 13897 | Value of square. Inferenc... |
sqcli 13898 | Closure of square. (Contr... |
sqeq0i 13899 | A number is zero iff its s... |
sqrecii 13900 | Square of reciprocal. (Co... |
sqmuli 13901 | Distribution of square ove... |
sqdivi 13902 | Distribution of square ove... |
resqcli 13903 | Closure of square in reals... |
sqgt0i 13904 | The square of a nonzero re... |
sqge0i 13905 | A square of a real is nonn... |
lt2sqi 13906 | The square function on non... |
le2sqi 13907 | The square function on non... |
sq11i 13908 | The square function is one... |
sq0 13909 | The square of 0 is 0. (Co... |
sq0i 13910 | If a number is zero, its s... |
sq0id 13911 | If a number is zero, its s... |
sq1 13912 | The square of 1 is 1. (Co... |
neg1sqe1 13913 | ` -u 1 ` squared is 1. (C... |
sq2 13914 | The square of 2 is 4. (Co... |
sq3 13915 | The square of 3 is 9. (Co... |
sq4e2t8 13916 | The square of 4 is 2 times... |
cu2 13917 | The cube of 2 is 8. (Cont... |
irec 13918 | The reciprocal of ` _i ` .... |
i2 13919 | ` _i ` squared. (Contribu... |
i3 13920 | ` _i ` cubed. (Contribute... |
i4 13921 | ` _i ` to the fourth power... |
nnlesq 13922 | A positive integer is less... |
iexpcyc 13923 | Taking ` _i ` to the ` K `... |
expnass 13924 | A counterexample showing t... |
sqlecan 13925 | Cancel one factor of a squ... |
subsq 13926 | Factor the difference of t... |
subsq2 13927 | Express the difference of ... |
binom2i 13928 | The square of a binomial. ... |
subsqi 13929 | Factor the difference of t... |
sqeqori 13930 | The squares of two complex... |
subsq0i 13931 | The two solutions to the d... |
sqeqor 13932 | The squares of two complex... |
binom2 13933 | The square of a binomial. ... |
binom21 13934 | Special case of ~ binom2 w... |
binom2sub 13935 | Expand the square of a sub... |
binom2sub1 13936 | Special case of ~ binom2su... |
binom2subi 13937 | Expand the square of a sub... |
mulbinom2 13938 | The square of a binomial w... |
binom3 13939 | The cube of a binomial. (... |
sq01 13940 | If a complex number equals... |
zesq 13941 | An integer is even iff its... |
nnesq 13942 | A positive integer is even... |
crreczi 13943 | Reciprocal of a complex nu... |
bernneq 13944 | Bernoulli's inequality, du... |
bernneq2 13945 | Variation of Bernoulli's i... |
bernneq3 13946 | A corollary of ~ bernneq .... |
expnbnd 13947 | Exponentiation with a base... |
expnlbnd 13948 | The reciprocal of exponent... |
expnlbnd2 13949 | The reciprocal of exponent... |
expmulnbnd 13950 | Exponentiation with a base... |
digit2 13951 | Two ways to express the ` ... |
digit1 13952 | Two ways to express the ` ... |
modexp 13953 | Exponentiation property of... |
discr1 13954 | A nonnegative quadratic fo... |
discr 13955 | If a quadratic polynomial ... |
expnngt1 13956 | If an integer power with a... |
expnngt1b 13957 | An integer power with an i... |
sqoddm1div8 13958 | A squared odd number minus... |
nnsqcld 13959 | The naturals are closed un... |
nnexpcld 13960 | Closure of exponentiation ... |
nn0expcld 13961 | Closure of exponentiation ... |
rpexpcld 13962 | Closure law for exponentia... |
ltexp2rd 13963 | The power of a positive nu... |
reexpclzd 13964 | Closure of exponentiation ... |
resqcld 13965 | Closure of square in reals... |
sqge0d 13966 | A square of a real is nonn... |
sqgt0d 13967 | The square of a nonzero re... |
ltexp2d 13968 | Ordering relationship for ... |
leexp2d 13969 | Ordering law for exponenti... |
expcand 13970 | Ordering relationship for ... |
leexp2ad 13971 | Ordering relationship for ... |
leexp2rd 13972 | Ordering relationship for ... |
lt2sqd 13973 | The square function on non... |
le2sqd 13974 | The square function on non... |
sq11d 13975 | The square function is one... |
mulsubdivbinom2 13976 | The square of a binomial w... |
muldivbinom2 13977 | The square of a binomial w... |
sq10 13978 | The square of 10 is 100. ... |
sq10e99m1 13979 | The square of 10 is 99 plu... |
3dec 13980 | A "decimal constructor" wh... |
nn0le2msqi 13981 | The square function on non... |
nn0opthlem1 13982 | A rather pretty lemma for ... |
nn0opthlem2 13983 | Lemma for ~ nn0opthi . (C... |
nn0opthi 13984 | An ordered pair theorem fo... |
nn0opth2i 13985 | An ordered pair theorem fo... |
nn0opth2 13986 | An ordered pair theorem fo... |
facnn 13989 | Value of the factorial fun... |
fac0 13990 | The factorial of 0. (Cont... |
fac1 13991 | The factorial of 1. (Cont... |
facp1 13992 | The factorial of a success... |
fac2 13993 | The factorial of 2. (Cont... |
fac3 13994 | The factorial of 3. (Cont... |
fac4 13995 | The factorial of 4. (Cont... |
facnn2 13996 | Value of the factorial fun... |
faccl 13997 | Closure of the factorial f... |
faccld 13998 | Closure of the factorial f... |
facmapnn 13999 | The factorial function res... |
facne0 14000 | The factorial function is ... |
facdiv 14001 | A positive integer divides... |
facndiv 14002 | No positive integer (great... |
facwordi 14003 | Ordering property of facto... |
faclbnd 14004 | A lower bound for the fact... |
faclbnd2 14005 | A lower bound for the fact... |
faclbnd3 14006 | A lower bound for the fact... |
faclbnd4lem1 14007 | Lemma for ~ faclbnd4 . Pr... |
faclbnd4lem2 14008 | Lemma for ~ faclbnd4 . Us... |
faclbnd4lem3 14009 | Lemma for ~ faclbnd4 . Th... |
faclbnd4lem4 14010 | Lemma for ~ faclbnd4 . Pr... |
faclbnd4 14011 | Variant of ~ faclbnd5 prov... |
faclbnd5 14012 | The factorial function gro... |
faclbnd6 14013 | Geometric lower bound for ... |
facubnd 14014 | An upper bound for the fac... |
facavg 14015 | The product of two factori... |
bcval 14018 | Value of the binomial coef... |
bcval2 14019 | Value of the binomial coef... |
bcval3 14020 | Value of the binomial coef... |
bcval4 14021 | Value of the binomial coef... |
bcrpcl 14022 | Closure of the binomial co... |
bccmpl 14023 | "Complementing" its second... |
bcn0 14024 | ` N ` choose 0 is 1. Rema... |
bc0k 14025 | The binomial coefficient "... |
bcnn 14026 | ` N ` choose ` N ` is 1. ... |
bcn1 14027 | Binomial coefficient: ` N ... |
bcnp1n 14028 | Binomial coefficient: ` N ... |
bcm1k 14029 | The proportion of one bino... |
bcp1n 14030 | The proportion of one bino... |
bcp1nk 14031 | The proportion of one bino... |
bcval5 14032 | Write out the top and bott... |
bcn2 14033 | Binomial coefficient: ` N ... |
bcp1m1 14034 | Compute the binomial coeff... |
bcpasc 14035 | Pascal's rule for the bino... |
bccl 14036 | A binomial coefficient, in... |
bccl2 14037 | A binomial coefficient, in... |
bcn2m1 14038 | Compute the binomial coeff... |
bcn2p1 14039 | Compute the binomial coeff... |
permnn 14040 | The number of permutations... |
bcnm1 14041 | The binomial coefficent of... |
4bc3eq4 14042 | The value of four choose t... |
4bc2eq6 14043 | The value of four choose t... |
hashkf 14046 | The finite part of the siz... |
hashgval 14047 | The value of the ` # ` fun... |
hashginv 14048 | The converse of ` G ` maps... |
hashinf 14049 | The value of the ` # ` fun... |
hashbnd 14050 | If ` A ` has size bounded ... |
hashfxnn0 14051 | The size function is a fun... |
hashf 14052 | The size function maps all... |
hashxnn0 14053 | The value of the hash func... |
hashresfn 14054 | Restriction of the domain ... |
dmhashres 14055 | Restriction of the domain ... |
hashnn0pnf 14056 | The value of the hash func... |
hashnnn0genn0 14057 | If the size of a set is no... |
hashnemnf 14058 | The size of a set is never... |
hashv01gt1 14059 | The size of a set is eithe... |
hashfz1 14060 | The set ` ( 1 ... N ) ` ha... |
hashen 14061 | Two finite sets have the s... |
hasheni 14062 | Equinumerous sets have the... |
hasheqf1o 14063 | The size of two finite set... |
fiinfnf1o 14064 | There is no bijection betw... |
focdmex 14065 | The codomain of an onto fu... |
hasheqf1oi 14066 | The size of two sets is eq... |
hashf1rn 14067 | The size of a finite set w... |
hasheqf1od 14068 | The size of two sets is eq... |
fz1eqb 14069 | Two possibly-empty 1-based... |
hashcard 14070 | The size function of the c... |
hashcl 14071 | Closure of the ` # ` funct... |
hashxrcl 14072 | Extended real closure of t... |
hashclb 14073 | Reverse closure of the ` #... |
nfile 14074 | The size of any infinite s... |
hashvnfin 14075 | A set of finite size is a ... |
hashnfinnn0 14076 | The size of an infinite se... |
isfinite4 14077 | A finite set is equinumero... |
hasheq0 14078 | Two ways of saying a finit... |
hashneq0 14079 | Two ways of saying a set i... |
hashgt0n0 14080 | If the size of a set is gr... |
hashnncl 14081 | Positive natural closure o... |
hash0 14082 | The empty set has size zer... |
hashelne0d 14083 | A set with an element has ... |
hashsng 14084 | The size of a singleton. ... |
hashen1 14085 | A set has size 1 if and on... |
hash1elsn 14086 | A set of size 1 with a kno... |
hashrabrsn 14087 | The size of a restricted c... |
hashrabsn01 14088 | The size of a restricted c... |
hashrabsn1 14089 | If the size of a restricte... |
hashfn 14090 | A function is equinumerous... |
fseq1hash 14091 | The value of the size func... |
hashgadd 14092 | ` G ` maps ordinal additio... |
hashgval2 14093 | A short expression for the... |
hashdom 14094 | Dominance relation for the... |
hashdomi 14095 | Non-strict order relation ... |
hashsdom 14096 | Strict dominance relation ... |
hashun 14097 | The size of the union of d... |
hashun2 14098 | The size of the union of f... |
hashun3 14099 | The size of the union of f... |
hashinfxadd 14100 | The extended real addition... |
hashunx 14101 | The size of the union of d... |
hashge0 14102 | The cardinality of a set i... |
hashgt0 14103 | The cardinality of a nonem... |
hashge1 14104 | The cardinality of a nonem... |
1elfz0hash 14105 | 1 is an element of the fin... |
hashnn0n0nn 14106 | If a nonnegative integer i... |
hashunsng 14107 | The size of the union of a... |
hashunsngx 14108 | The size of the union of a... |
hashunsnggt 14109 | The size of a set is great... |
hashprg 14110 | The size of an unordered p... |
elprchashprn2 14111 | If one element of an unord... |
hashprb 14112 | The size of an unordered p... |
hashprdifel 14113 | The elements of an unorder... |
prhash2ex 14114 | There is (at least) one se... |
hashle00 14115 | If the size of a set is le... |
hashgt0elex 14116 | If the size of a set is gr... |
hashgt0elexb 14117 | The size of a set is great... |
hashp1i 14118 | Size of a finite ordinal. ... |
hash1 14119 | Size of a finite ordinal. ... |
hash2 14120 | Size of a finite ordinal. ... |
hash3 14121 | Size of a finite ordinal. ... |
hash4 14122 | Size of a finite ordinal. ... |
pr0hash2ex 14123 | There is (at least) one se... |
hashss 14124 | The size of a subset is le... |
prsshashgt1 14125 | The size of a superset of ... |
hashin 14126 | The size of the intersecti... |
hashssdif 14127 | The size of the difference... |
hashdif 14128 | The size of the difference... |
hashdifsn 14129 | The size of the difference... |
hashdifpr 14130 | The size of the difference... |
hashsn01 14131 | The size of a singleton is... |
hashsnle1 14132 | The size of a singleton is... |
hashsnlei 14133 | Get an upper bound on a co... |
hash1snb 14134 | The size of a set is 1 if ... |
euhash1 14135 | The size of a set is 1 in ... |
hash1n0 14136 | If the size of a set is 1 ... |
hashgt12el 14137 | In a set with more than on... |
hashgt12el2 14138 | In a set with more than on... |
hashgt23el 14139 | A set with more than two e... |
hashunlei 14140 | Get an upper bound on a co... |
hashsslei 14141 | Get an upper bound on a co... |
hashfz 14142 | Value of the numeric cardi... |
fzsdom2 14143 | Condition for finite range... |
hashfzo 14144 | Cardinality of a half-open... |
hashfzo0 14145 | Cardinality of a half-open... |
hashfzp1 14146 | Value of the numeric cardi... |
hashfz0 14147 | Value of the numeric cardi... |
hashxplem 14148 | Lemma for ~ hashxp . (Con... |
hashxp 14149 | The size of the Cartesian ... |
hashmap 14150 | The size of the set expone... |
hashpw 14151 | The size of the power set ... |
hashfun 14152 | A finite set is a function... |
hashres 14153 | The number of elements of ... |
hashreshashfun 14154 | The number of elements of ... |
hashimarn 14155 | The size of the image of a... |
hashimarni 14156 | If the size of the image o... |
resunimafz0 14157 | TODO-AV: Revise using ` F... |
fnfz0hash 14158 | The size of a function on ... |
ffz0hash 14159 | The size of a function on ... |
fnfz0hashnn0 14160 | The size of a function on ... |
ffzo0hash 14161 | The size of a function on ... |
fnfzo0hash 14162 | The size of a function on ... |
fnfzo0hashnn0 14163 | The value of the size func... |
hashbclem 14164 | Lemma for ~ hashbc : induc... |
hashbc 14165 | The binomial coefficient c... |
hashfacen 14166 | The number of bijections b... |
hashfacenOLD 14167 | Obsolete version of ~ hash... |
hashf1lem1 14168 | Lemma for ~ hashf1 . (Con... |
hashf1lem1OLD 14169 | Obsolete version of ~ hash... |
hashf1lem2 14170 | Lemma for ~ hashf1 . (Con... |
hashf1 14171 | The permutation number ` |... |
hashfac 14172 | A factorial counts the num... |
leiso 14173 | Two ways to write a strict... |
leisorel 14174 | Version of ~ isorel for st... |
fz1isolem 14175 | Lemma for ~ fz1iso . (Con... |
fz1iso 14176 | Any finite ordered set has... |
ishashinf 14177 | Any set that is not finite... |
seqcoll 14178 | The function ` F ` contain... |
seqcoll2 14179 | The function ` F ` contain... |
phphashd 14180 | Corollary of the Pigeonhol... |
phphashrd 14181 | Corollary of the Pigeonhol... |
hashprlei 14182 | An unordered pair has at m... |
hash2pr 14183 | A set of size two is an un... |
hash2prde 14184 | A set of size two is an un... |
hash2exprb 14185 | A set of size two is an un... |
hash2prb 14186 | A set of size two is a pro... |
prprrab 14187 | The set of proper pairs of... |
nehash2 14188 | The cardinality of a set w... |
hash2prd 14189 | A set of size two is an un... |
hash2pwpr 14190 | If the size of a subset of... |
hashle2pr 14191 | A nonempty set of size les... |
hashle2prv 14192 | A nonempty subset of a pow... |
pr2pwpr 14193 | The set of subsets of a pa... |
hashge2el2dif 14194 | A set with size at least 2... |
hashge2el2difr 14195 | A set with at least 2 diff... |
hashge2el2difb 14196 | A set has size at least 2 ... |
hashdmpropge2 14197 | The size of the domain of ... |
hashtplei 14198 | An unordered triple has at... |
hashtpg 14199 | The size of an unordered t... |
hashge3el3dif 14200 | A set with size at least 3... |
elss2prb 14201 | An element of the set of s... |
hash2sspr 14202 | A subset of size two is an... |
exprelprel 14203 | If there is an element of ... |
hash3tr 14204 | A set of size three is an ... |
hash1to3 14205 | If the size of a set is be... |
fundmge2nop0 14206 | A function with a domain c... |
fundmge2nop 14207 | A function with a domain c... |
fun2dmnop0 14208 | A function with a domain c... |
fun2dmnop 14209 | A function with a domain c... |
hashdifsnp1 14210 | If the size of a set is a ... |
fi1uzind 14211 | Properties of an ordered p... |
brfi1uzind 14212 | Properties of a binary rel... |
brfi1ind 14213 | Properties of a binary rel... |
brfi1indALT 14214 | Alternate proof of ~ brfi1... |
opfi1uzind 14215 | Properties of an ordered p... |
opfi1ind 14216 | Properties of an ordered p... |
iswrd 14219 | Property of being a word o... |
wrdval 14220 | Value of the set of words ... |
iswrdi 14221 | A zero-based sequence is a... |
wrdf 14222 | A word is a zero-based seq... |
iswrdb 14223 | A word over an alphabet is... |
wrddm 14224 | The indices of a word (i.e... |
sswrd 14225 | The set of words respects ... |
snopiswrd 14226 | A singleton of an ordered ... |
wrdexg 14227 | The set of words over a se... |
wrdexb 14228 | The set of words over a se... |
wrdexi 14229 | The set of words over a se... |
wrdsymbcl 14230 | A symbol within a word ove... |
wrdfn 14231 | A word is a function with ... |
wrdv 14232 | A word over an alphabet is... |
wrdlndm 14233 | The length of a word is no... |
iswrdsymb 14234 | An arbitrary word is a wor... |
wrdfin 14235 | A word is a finite set. (... |
lencl 14236 | The length of a word is a ... |
lennncl 14237 | The length of a nonempty w... |
wrdffz 14238 | A word is a function from ... |
wrdeq 14239 | Equality theorem for the s... |
wrdeqi 14240 | Equality theorem for the s... |
iswrddm0 14241 | A function with empty doma... |
wrd0 14242 | The empty set is a word (t... |
0wrd0 14243 | The empty word is the only... |
ffz0iswrd 14244 | A sequence with zero-based... |
wrdsymb 14245 | A word is a word over the ... |
nfwrd 14246 | Hypothesis builder for ` W... |
csbwrdg 14247 | Class substitution for the... |
wrdnval 14248 | Words of a fixed length ar... |
wrdmap 14249 | Words as a mapping. (Cont... |
hashwrdn 14250 | If there is only a finite ... |
wrdnfi 14251 | If there is only a finite ... |
wrdsymb0 14252 | A symbol at a position "ou... |
wrdlenge1n0 14253 | A word with length at leas... |
len0nnbi 14254 | The length of a word is a ... |
wrdlenge2n0 14255 | A word with length at leas... |
wrdsymb1 14256 | The first symbol of a none... |
wrdlen1 14257 | A word of length 1 starts ... |
fstwrdne 14258 | The first symbol of a none... |
fstwrdne0 14259 | The first symbol of a none... |
eqwrd 14260 | Two words are equal iff th... |
elovmpowrd 14261 | Implications for the value... |
elovmptnn0wrd 14262 | Implications for the value... |
wrdred1 14263 | A word truncated by a symb... |
wrdred1hash 14264 | The length of a word trunc... |
lsw 14267 | Extract the last symbol of... |
lsw0 14268 | The last symbol of an empt... |
lsw0g 14269 | The last symbol of an empt... |
lsw1 14270 | The last symbol of a word ... |
lswcl 14271 | Closure of the last symbol... |
lswlgt0cl 14272 | The last symbol of a nonem... |
ccatfn 14275 | The concatenation operator... |
ccatfval 14276 | Value of the concatenation... |
ccatcl 14277 | The concatenation of two w... |
ccatlen 14278 | The length of a concatenat... |
ccatlenOLD 14279 | Obsolete version of ~ ccat... |
ccat0 14280 | The concatenation of two w... |
ccatval1 14281 | Value of a symbol in the l... |
ccatval1OLD 14282 | Obsolete version of ~ ccat... |
ccatval2 14283 | Value of a symbol in the r... |
ccatval3 14284 | Value of a symbol in the r... |
elfzelfzccat 14285 | An element of a finite set... |
ccatvalfn 14286 | The concatenation of two w... |
ccatsymb 14287 | The symbol at a given posi... |
ccatfv0 14288 | The first symbol of a conc... |
ccatval1lsw 14289 | The last symbol of the lef... |
ccatval21sw 14290 | The first symbol of the ri... |
ccatlid 14291 | Concatenation of a word by... |
ccatrid 14292 | Concatenation of a word by... |
ccatass 14293 | Associative law for concat... |
ccatrn 14294 | The range of a concatenate... |
ccatidid 14295 | Concatenation of the empty... |
lswccatn0lsw 14296 | The last symbol of a word ... |
lswccat0lsw 14297 | The last symbol of a word ... |
ccatalpha 14298 | A concatenation of two arb... |
ccatrcl1 14299 | Reverse closure of a conca... |
ids1 14302 | Identity function protecti... |
s1val 14303 | Value of a singleton word.... |
s1rn 14304 | The range of a singleton w... |
s1eq 14305 | Equality theorem for a sin... |
s1eqd 14306 | Equality theorem for a sin... |
s1cl 14307 | A singleton word is a word... |
s1cld 14308 | A singleton word is a word... |
s1prc 14309 | Value of a singleton word ... |
s1cli 14310 | A singleton word is a word... |
s1len 14311 | Length of a singleton word... |
s1nz 14312 | A singleton word is not th... |
s1dm 14313 | The domain of a singleton ... |
s1dmALT 14314 | Alternate version of ~ s1d... |
s1fv 14315 | Sole symbol of a singleton... |
lsws1 14316 | The last symbol of a singl... |
eqs1 14317 | A word of length 1 is a si... |
wrdl1exs1 14318 | A word of length 1 is a si... |
wrdl1s1 14319 | A word of length 1 is a si... |
s111 14320 | The singleton word functio... |
ccatws1cl 14321 | The concatenation of a wor... |
ccatws1clv 14322 | The concatenation of a wor... |
ccat2s1cl 14323 | The concatenation of two s... |
ccats1alpha 14324 | A concatenation of a word ... |
ccatws1len 14325 | The length of the concaten... |
ccatws1lenp1b 14326 | The length of a word is ` ... |
wrdlenccats1lenm1 14327 | The length of a word is th... |
ccat2s1len 14328 | The length of the concaten... |
ccat2s1lenOLD 14329 | Obsolete version of ~ ccat... |
ccatw2s1cl 14330 | The concatenation of a wor... |
ccatw2s1len 14331 | The length of the concaten... |
ccats1val1 14332 | Value of a symbol in the l... |
ccats1val1OLD 14333 | Obsolete version of ~ ccat... |
ccats1val2 14334 | Value of the symbol concat... |
ccat1st1st 14335 | The first symbol of a word... |
ccat2s1p1 14336 | Extract the first of two c... |
ccat2s1p2 14337 | Extract the second of two ... |
ccat2s1p1OLD 14338 | Obsolete version of ~ ccat... |
ccat2s1p2OLD 14339 | Obsolete version of ~ ccat... |
ccatw2s1ass 14340 | Associative law for a conc... |
ccatw2s1assOLD 14341 | Obsolete version of ~ ccat... |
ccatws1n0 14342 | The concatenation of a wor... |
ccatws1ls 14343 | The last symbol of the con... |
lswccats1 14344 | The last symbol of a word ... |
lswccats1fst 14345 | The last symbol of a nonem... |
ccatw2s1p1 14346 | Extract the symbol of the ... |
ccatw2s1p1OLD 14347 | Obsolete version of ~ ccat... |
ccatw2s1p2 14348 | Extract the second of two ... |
ccat2s1fvw 14349 | Extract a symbol of a word... |
ccat2s1fvwOLD 14350 | Obsolete version of ~ ccat... |
ccat2s1fst 14351 | The first symbol of the co... |
ccat2s1fstOLD 14352 | Obsolete version of ~ ccat... |
swrdnznd 14355 | The value of a subword ope... |
swrdval 14356 | Value of a subword. (Cont... |
swrd00 14357 | A zero length substring. ... |
swrdcl 14358 | Closure of the subword ext... |
swrdval2 14359 | Value of the subword extra... |
swrdlen 14360 | Length of an extracted sub... |
swrdfv 14361 | A symbol in an extracted s... |
swrdfv0 14362 | The first symbol in an ext... |
swrdf 14363 | A subword of a word is a f... |
swrdvalfn 14364 | Value of the subword extra... |
swrdrn 14365 | The range of a subword of ... |
swrdlend 14366 | The value of the subword e... |
swrdnd 14367 | The value of the subword e... |
swrdnd2 14368 | Value of the subword extra... |
swrdnnn0nd 14369 | The value of a subword ope... |
swrdnd0 14370 | The value of a subword ope... |
swrd0 14371 | A subword of an empty set ... |
swrdrlen 14372 | Length of a right-anchored... |
swrdlen2 14373 | Length of an extracted sub... |
swrdfv2 14374 | A symbol in an extracted s... |
swrdwrdsymb 14375 | A subword is a word over t... |
swrdsb0eq 14376 | Two subwords with the same... |
swrdsbslen 14377 | Two subwords with the same... |
swrdspsleq 14378 | Two words have a common su... |
swrds1 14379 | Extract a single symbol fr... |
swrdlsw 14380 | Extract the last single sy... |
ccatswrd 14381 | Joining two adjacent subwo... |
swrdccat2 14382 | Recover the right half of ... |
pfxnndmnd 14385 | The value of a prefix oper... |
pfxval 14386 | Value of a prefix operatio... |
pfx00 14387 | The zero length prefix is ... |
pfx0 14388 | A prefix of an empty set i... |
pfxval0 14389 | Value of a prefix operatio... |
pfxcl 14390 | Closure of the prefix extr... |
pfxmpt 14391 | Value of the prefix extrac... |
pfxres 14392 | Value of the subword extra... |
pfxf 14393 | A prefix of a word is a fu... |
pfxfn 14394 | Value of the prefix extrac... |
pfxfv 14395 | A symbol in a prefix of a ... |
pfxlen 14396 | Length of a prefix. (Cont... |
pfxid 14397 | A word is a prefix of itse... |
pfxrn 14398 | The range of a prefix of a... |
pfxn0 14399 | A prefix consisting of at ... |
pfxnd 14400 | The value of a prefix oper... |
pfxnd0 14401 | The value of a prefix oper... |
pfxwrdsymb 14402 | A prefix of a word is a wo... |
addlenrevpfx 14403 | The sum of the lengths of ... |
addlenpfx 14404 | The sum of the lengths of ... |
pfxfv0 14405 | The first symbol of a pref... |
pfxtrcfv 14406 | A symbol in a word truncat... |
pfxtrcfv0 14407 | The first symbol in a word... |
pfxfvlsw 14408 | The last symbol in a nonem... |
pfxeq 14409 | The prefixes of two words ... |
pfxtrcfvl 14410 | The last symbol in a word ... |
pfxsuffeqwrdeq 14411 | Two words are equal if and... |
pfxsuff1eqwrdeq 14412 | Two (nonempty) words are e... |
disjwrdpfx 14413 | Sets of words are disjoint... |
ccatpfx 14414 | Concatenating a prefix wit... |
pfxccat1 14415 | Recover the left half of a... |
pfx1 14416 | The prefix of length one o... |
swrdswrdlem 14417 | Lemma for ~ swrdswrd . (C... |
swrdswrd 14418 | A subword of a subword is ... |
pfxswrd 14419 | A prefix of a subword is a... |
swrdpfx 14420 | A subword of a prefix is a... |
pfxpfx 14421 | A prefix of a prefix is a ... |
pfxpfxid 14422 | A prefix of a prefix with ... |
pfxcctswrd 14423 | The concatenation of the p... |
lenpfxcctswrd 14424 | The length of the concaten... |
lenrevpfxcctswrd 14425 | The length of the concaten... |
pfxlswccat 14426 | Reconstruct a nonempty wor... |
ccats1pfxeq 14427 | The last symbol of a word ... |
ccats1pfxeqrex 14428 | There exists a symbol such... |
ccatopth 14429 | An ~ opth -like theorem fo... |
ccatopth2 14430 | An ~ opth -like theorem fo... |
ccatlcan 14431 | Concatenation of words is ... |
ccatrcan 14432 | Concatenation of words is ... |
wrdeqs1cat 14433 | Decompose a nonempty word ... |
cats1un 14434 | Express a word with an ext... |
wrdind 14435 | Perform induction over the... |
wrd2ind 14436 | Perform induction over the... |
swrdccatfn 14437 | The subword of a concatena... |
swrdccatin1 14438 | The subword of a concatena... |
pfxccatin12lem4 14439 | Lemma 4 for ~ pfxccatin12 ... |
pfxccatin12lem2a 14440 | Lemma for ~ pfxccatin12lem... |
pfxccatin12lem1 14441 | Lemma 1 for ~ pfxccatin12 ... |
swrdccatin2 14442 | The subword of a concatena... |
pfxccatin12lem2c 14443 | Lemma for ~ pfxccatin12lem... |
pfxccatin12lem2 14444 | Lemma 2 for ~ pfxccatin12 ... |
pfxccatin12lem3 14445 | Lemma 3 for ~ pfxccatin12 ... |
pfxccatin12 14446 | The subword of a concatena... |
pfxccat3 14447 | The subword of a concatena... |
swrdccat 14448 | The subword of a concatena... |
pfxccatpfx1 14449 | A prefix of a concatenatio... |
pfxccatpfx2 14450 | A prefix of a concatenatio... |
pfxccat3a 14451 | A prefix of a concatenatio... |
swrdccat3blem 14452 | Lemma for ~ swrdccat3b . ... |
swrdccat3b 14453 | A suffix of a concatenatio... |
pfxccatid 14454 | A prefix of a concatenatio... |
ccats1pfxeqbi 14455 | A word is a prefix of a wo... |
swrdccatin1d 14456 | The subword of a concatena... |
swrdccatin2d 14457 | The subword of a concatena... |
pfxccatin12d 14458 | The subword of a concatena... |
reuccatpfxs1lem 14459 | Lemma for ~ reuccatpfxs1 .... |
reuccatpfxs1 14460 | There is a unique word hav... |
reuccatpfxs1v 14461 | There is a unique word hav... |
splval 14464 | Value of the substring rep... |
splcl 14465 | Closure of the substring r... |
splid 14466 | Splicing a subword for the... |
spllen 14467 | The length of a splice. (... |
splfv1 14468 | Symbols to the left of a s... |
splfv2a 14469 | Symbols within the replace... |
splval2 14470 | Value of a splice, assumin... |
revval 14473 | Value of the word reversin... |
revcl 14474 | The reverse of a word is a... |
revlen 14475 | The reverse of a word has ... |
revfv 14476 | Reverse of a word at a poi... |
rev0 14477 | The empty word is its own ... |
revs1 14478 | Singleton words are their ... |
revccat 14479 | Antiautomorphic property o... |
revrev 14480 | Reversal is an involution ... |
reps 14483 | Construct a function mappi... |
repsundef 14484 | A function mapping a half-... |
repsconst 14485 | Construct a function mappi... |
repsf 14486 | The constructed function m... |
repswsymb 14487 | The symbols of a "repeated... |
repsw 14488 | A function mapping a half-... |
repswlen 14489 | The length of a "repeated ... |
repsw0 14490 | The "repeated symbol word"... |
repsdf2 14491 | Alternative definition of ... |
repswsymball 14492 | All the symbols of a "repe... |
repswsymballbi 14493 | A word is a "repeated symb... |
repswfsts 14494 | The first symbol of a none... |
repswlsw 14495 | The last symbol of a nonem... |
repsw1 14496 | The "repeated symbol word"... |
repswswrd 14497 | A subword of a "repeated s... |
repswpfx 14498 | A prefix of a repeated sym... |
repswccat 14499 | The concatenation of two "... |
repswrevw 14500 | The reverse of a "repeated... |
cshfn 14503 | Perform a cyclical shift f... |
cshword 14504 | Perform a cyclical shift f... |
cshnz 14505 | A cyclical shift is the em... |
0csh0 14506 | Cyclically shifting an emp... |
cshw0 14507 | A word cyclically shifted ... |
cshwmodn 14508 | Cyclically shifting a word... |
cshwsublen 14509 | Cyclically shifting a word... |
cshwn 14510 | A word cyclically shifted ... |
cshwcl 14511 | A cyclically shifted word ... |
cshwlen 14512 | The length of a cyclically... |
cshwf 14513 | A cyclically shifted word ... |
cshwfn 14514 | A cyclically shifted word ... |
cshwrn 14515 | The range of a cyclically ... |
cshwidxmod 14516 | The symbol at a given inde... |
cshwidxmodr 14517 | The symbol at a given inde... |
cshwidx0mod 14518 | The symbol at index 0 of a... |
cshwidx0 14519 | The symbol at index 0 of a... |
cshwidxm1 14520 | The symbol at index ((n-N)... |
cshwidxm 14521 | The symbol at index (n-N) ... |
cshwidxn 14522 | The symbol at index (n-1) ... |
cshf1 14523 | Cyclically shifting a word... |
cshinj 14524 | If a word is injectiv (reg... |
repswcshw 14525 | A cyclically shifted "repe... |
2cshw 14526 | Cyclically shifting a word... |
2cshwid 14527 | Cyclically shifting a word... |
lswcshw 14528 | The last symbol of a word ... |
2cshwcom 14529 | Cyclically shifting a word... |
cshwleneq 14530 | If the results of cyclical... |
3cshw 14531 | Cyclically shifting a word... |
cshweqdif2 14532 | If cyclically shifting two... |
cshweqdifid 14533 | If cyclically shifting a w... |
cshweqrep 14534 | If cyclically shifting a w... |
cshw1 14535 | If cyclically shifting a w... |
cshw1repsw 14536 | If cyclically shifting a w... |
cshwsexa 14537 | The class of (different!) ... |
2cshwcshw 14538 | If a word is a cyclically ... |
scshwfzeqfzo 14539 | For a nonempty word the se... |
cshwcshid 14540 | A cyclically shifted word ... |
cshwcsh2id 14541 | A cyclically shifted word ... |
cshimadifsn 14542 | The image of a cyclically ... |
cshimadifsn0 14543 | The image of a cyclically ... |
wrdco 14544 | Mapping a word by a functi... |
lenco 14545 | Length of a mapped word is... |
s1co 14546 | Mapping of a singleton wor... |
revco 14547 | Mapping of words (i.e., a ... |
ccatco 14548 | Mapping of words commutes ... |
cshco 14549 | Mapping of words commutes ... |
swrdco 14550 | Mapping of words commutes ... |
pfxco 14551 | Mapping of words commutes ... |
lswco 14552 | Mapping of (nonempty) word... |
repsco 14553 | Mapping of words commutes ... |
cats1cld 14568 | Closure of concatenation w... |
cats1co 14569 | Closure of concatenation w... |
cats1cli 14570 | Closure of concatenation w... |
cats1fvn 14571 | The last symbol of a conca... |
cats1fv 14572 | A symbol other than the la... |
cats1len 14573 | The length of concatenatio... |
cats1cat 14574 | Closure of concatenation w... |
cats2cat 14575 | Closure of concatenation o... |
s2eqd 14576 | Equality theorem for a dou... |
s3eqd 14577 | Equality theorem for a len... |
s4eqd 14578 | Equality theorem for a len... |
s5eqd 14579 | Equality theorem for a len... |
s6eqd 14580 | Equality theorem for a len... |
s7eqd 14581 | Equality theorem for a len... |
s8eqd 14582 | Equality theorem for a len... |
s3eq2 14583 | Equality theorem for a len... |
s2cld 14584 | A doubleton word is a word... |
s3cld 14585 | A length 3 string is a wor... |
s4cld 14586 | A length 4 string is a wor... |
s5cld 14587 | A length 5 string is a wor... |
s6cld 14588 | A length 6 string is a wor... |
s7cld 14589 | A length 7 string is a wor... |
s8cld 14590 | A length 7 string is a wor... |
s2cl 14591 | A doubleton word is a word... |
s3cl 14592 | A length 3 string is a wor... |
s2cli 14593 | A doubleton word is a word... |
s3cli 14594 | A length 3 string is a wor... |
s4cli 14595 | A length 4 string is a wor... |
s5cli 14596 | A length 5 string is a wor... |
s6cli 14597 | A length 6 string is a wor... |
s7cli 14598 | A length 7 string is a wor... |
s8cli 14599 | A length 8 string is a wor... |
s2fv0 14600 | Extract the first symbol f... |
s2fv1 14601 | Extract the second symbol ... |
s2len 14602 | The length of a doubleton ... |
s2dm 14603 | The domain of a doubleton ... |
s3fv0 14604 | Extract the first symbol f... |
s3fv1 14605 | Extract the second symbol ... |
s3fv2 14606 | Extract the third symbol f... |
s3len 14607 | The length of a length 3 s... |
s4fv0 14608 | Extract the first symbol f... |
s4fv1 14609 | Extract the second symbol ... |
s4fv2 14610 | Extract the third symbol f... |
s4fv3 14611 | Extract the fourth symbol ... |
s4len 14612 | The length of a length 4 s... |
s5len 14613 | The length of a length 5 s... |
s6len 14614 | The length of a length 6 s... |
s7len 14615 | The length of a length 7 s... |
s8len 14616 | The length of a length 8 s... |
lsws2 14617 | The last symbol of a doubl... |
lsws3 14618 | The last symbol of a 3 let... |
lsws4 14619 | The last symbol of a 4 let... |
s2prop 14620 | A length 2 word is an unor... |
s2dmALT 14621 | Alternate version of ~ s2d... |
s3tpop 14622 | A length 3 word is an unor... |
s4prop 14623 | A length 4 word is a union... |
s3fn 14624 | A length 3 word is a funct... |
funcnvs1 14625 | The converse of a singleto... |
funcnvs2 14626 | The converse of a length 2... |
funcnvs3 14627 | The converse of a length 3... |
funcnvs4 14628 | The converse of a length 4... |
s2f1o 14629 | A length 2 word with mutua... |
f1oun2prg 14630 | A union of unordered pairs... |
s4f1o 14631 | A length 4 word with mutua... |
s4dom 14632 | The domain of a length 4 w... |
s2co 14633 | Mapping a doubleton word b... |
s3co 14634 | Mapping a length 3 string ... |
s0s1 14635 | Concatenation of fixed len... |
s1s2 14636 | Concatenation of fixed len... |
s1s3 14637 | Concatenation of fixed len... |
s1s4 14638 | Concatenation of fixed len... |
s1s5 14639 | Concatenation of fixed len... |
s1s6 14640 | Concatenation of fixed len... |
s1s7 14641 | Concatenation of fixed len... |
s2s2 14642 | Concatenation of fixed len... |
s4s2 14643 | Concatenation of fixed len... |
s4s3 14644 | Concatenation of fixed len... |
s4s4 14645 | Concatenation of fixed len... |
s3s4 14646 | Concatenation of fixed len... |
s2s5 14647 | Concatenation of fixed len... |
s5s2 14648 | Concatenation of fixed len... |
s2eq2s1eq 14649 | Two length 2 words are equ... |
s2eq2seq 14650 | Two length 2 words are equ... |
s3eqs2s1eq 14651 | Two length 3 words are equ... |
s3eq3seq 14652 | Two length 3 words are equ... |
swrds2 14653 | Extract two adjacent symbo... |
swrds2m 14654 | Extract two adjacent symbo... |
wrdlen2i 14655 | Implications of a word of ... |
wrd2pr2op 14656 | A word of length two repre... |
wrdlen2 14657 | A word of length two. (Co... |
wrdlen2s2 14658 | A word of length two as do... |
wrdl2exs2 14659 | A word of length two is a ... |
pfx2 14660 | A prefix of length two. (... |
wrd3tpop 14661 | A word of length three rep... |
wrdlen3s3 14662 | A word of length three as ... |
repsw2 14663 | The "repeated symbol word"... |
repsw3 14664 | The "repeated symbol word"... |
swrd2lsw 14665 | Extract the last two symbo... |
2swrd2eqwrdeq 14666 | Two words of length at lea... |
ccatw2s1ccatws2 14667 | The concatenation of a wor... |
ccatw2s1ccatws2OLD 14668 | Obsolete version of ~ ccat... |
ccat2s1fvwALT 14669 | Alternate proof of ~ ccat2... |
ccat2s1fvwALTOLD 14670 | Obsolete version of ~ ccat... |
wwlktovf 14671 | Lemma 1 for ~ wrd2f1tovbij... |
wwlktovf1 14672 | Lemma 2 for ~ wrd2f1tovbij... |
wwlktovfo 14673 | Lemma 3 for ~ wrd2f1tovbij... |
wwlktovf1o 14674 | Lemma 4 for ~ wrd2f1tovbij... |
wrd2f1tovbij 14675 | There is a bijection betwe... |
eqwrds3 14676 | A word is equal with a len... |
wrdl3s3 14677 | A word of length 3 is a le... |
s3sndisj 14678 | The singletons consisting ... |
s3iunsndisj 14679 | The union of singletons co... |
ofccat 14680 | Letterwise operations on w... |
ofs1 14681 | Letterwise operations on a... |
ofs2 14682 | Letterwise operations on a... |
coss12d 14683 | Subset deduction for compo... |
trrelssd 14684 | The composition of subclas... |
xpcogend 14685 | The most interesting case ... |
xpcoidgend 14686 | If two classes are not dis... |
cotr2g 14687 | Two ways of saying that th... |
cotr2 14688 | Two ways of saying a relat... |
cotr3 14689 | Two ways of saying a relat... |
coemptyd 14690 | Deduction about compositio... |
xptrrel 14691 | The cross product is alway... |
0trrel 14692 | The empty class is a trans... |
cleq1lem 14693 | Equality implies bijection... |
cleq1 14694 | Equality of relations impl... |
clsslem 14695 | The closure of a subclass ... |
trcleq1 14700 | Equality of relations impl... |
trclsslem 14701 | The transitive closure (as... |
trcleq2lem 14702 | Equality implies bijection... |
cvbtrcl 14703 | Change of bound variable i... |
trcleq12lem 14704 | Equality implies bijection... |
trclexlem 14705 | Existence of relation impl... |
trclublem 14706 | If a relation exists then ... |
trclubi 14707 | The Cartesian product of t... |
trclubgi 14708 | The union with the Cartesi... |
trclub 14709 | The Cartesian product of t... |
trclubg 14710 | The union with the Cartesi... |
trclfv 14711 | The transitive closure of ... |
brintclab 14712 | Two ways to express a bina... |
brtrclfv 14713 | Two ways of expressing the... |
brcnvtrclfv 14714 | Two ways of expressing the... |
brtrclfvcnv 14715 | Two ways of expressing the... |
brcnvtrclfvcnv 14716 | Two ways of expressing the... |
trclfvss 14717 | The transitive closure (as... |
trclfvub 14718 | The transitive closure of ... |
trclfvlb 14719 | The transitive closure of ... |
trclfvcotr 14720 | The transitive closure of ... |
trclfvlb2 14721 | The transitive closure of ... |
trclfvlb3 14722 | The transitive closure of ... |
cotrtrclfv 14723 | The transitive closure of ... |
trclidm 14724 | The transitive closure of ... |
trclun 14725 | Transitive closure of a un... |
trclfvg 14726 | The value of the transitiv... |
trclfvcotrg 14727 | The value of the transitiv... |
reltrclfv 14728 | The transitive closure of ... |
dmtrclfv 14729 | The domain of the transiti... |
reldmrelexp 14732 | The domain of the repeated... |
relexp0g 14733 | A relation composed zero t... |
relexp0 14734 | A relation composed zero t... |
relexp0d 14735 | A relation composed zero t... |
relexpsucnnr 14736 | A reduction for relation e... |
relexp1g 14737 | A relation composed once i... |
dfid5 14738 | Identity relation is equal... |
dfid6 14739 | Identity relation expresse... |
relexp1d 14740 | A relation composed once i... |
relexpsucnnl 14741 | A reduction for relation e... |
relexpsucl 14742 | A reduction for relation e... |
relexpsucr 14743 | A reduction for relation e... |
relexpsucrd 14744 | A reduction for relation e... |
relexpsucld 14745 | A reduction for relation e... |
relexpcnv 14746 | Commutation of converse an... |
relexpcnvd 14747 | Commutation of converse an... |
relexp0rel 14748 | The exponentiation of a cl... |
relexprelg 14749 | The exponentiation of a cl... |
relexprel 14750 | The exponentiation of a re... |
relexpreld 14751 | The exponentiation of a re... |
relexpnndm 14752 | The domain of an exponenti... |
relexpdmg 14753 | The domain of an exponenti... |
relexpdm 14754 | The domain of an exponenti... |
relexpdmd 14755 | The domain of an exponenti... |
relexpnnrn 14756 | The range of an exponentia... |
relexprng 14757 | The range of an exponentia... |
relexprn 14758 | The range of an exponentia... |
relexprnd 14759 | The range of an exponentia... |
relexpfld 14760 | The field of an exponentia... |
relexpfldd 14761 | The field of an exponentia... |
relexpaddnn 14762 | Relation composition becom... |
relexpuzrel 14763 | The exponentiation of a cl... |
relexpaddg 14764 | Relation composition becom... |
relexpaddd 14765 | Relation composition becom... |
rtrclreclem1 14768 | The reflexive, transitive ... |
dfrtrclrec2 14769 | If two elements are connec... |
rtrclreclem2 14770 | The reflexive, transitive ... |
rtrclreclem3 14771 | The reflexive, transitive ... |
rtrclreclem4 14772 | The reflexive, transitive ... |
dfrtrcl2 14773 | The two definitions ` t* `... |
relexpindlem 14774 | Principle of transitive in... |
relexpind 14775 | Principle of transitive in... |
rtrclind 14776 | Principle of transitive in... |
shftlem 14779 | Two ways to write a shifte... |
shftuz 14780 | A shift of the upper integ... |
shftfval 14781 | The value of the sequence ... |
shftdm 14782 | Domain of a relation shift... |
shftfib 14783 | Value of a fiber of the re... |
shftfn 14784 | Functionality and domain o... |
shftval 14785 | Value of a sequence shifte... |
shftval2 14786 | Value of a sequence shifte... |
shftval3 14787 | Value of a sequence shifte... |
shftval4 14788 | Value of a sequence shifte... |
shftval5 14789 | Value of a shifted sequenc... |
shftf 14790 | Functionality of a shifted... |
2shfti 14791 | Composite shift operations... |
shftidt2 14792 | Identity law for the shift... |
shftidt 14793 | Identity law for the shift... |
shftcan1 14794 | Cancellation law for the s... |
shftcan2 14795 | Cancellation law for the s... |
seqshft 14796 | Shifting the index set of ... |
sgnval 14799 | Value of the signum functi... |
sgn0 14800 | The signum of 0 is 0. (Co... |
sgnp 14801 | The signum of a positive e... |
sgnrrp 14802 | The signum of a positive r... |
sgn1 14803 | The signum of 1 is 1. (Co... |
sgnpnf 14804 | The signum of ` +oo ` is 1... |
sgnn 14805 | The signum of a negative e... |
sgnmnf 14806 | The signum of ` -oo ` is -... |
cjval 14813 | The value of the conjugate... |
cjth 14814 | The defining property of t... |
cjf 14815 | Domain and codomain of the... |
cjcl 14816 | The conjugate of a complex... |
reval 14817 | The value of the real part... |
imval 14818 | The value of the imaginary... |
imre 14819 | The imaginary part of a co... |
reim 14820 | The real part of a complex... |
recl 14821 | The real part of a complex... |
imcl 14822 | The imaginary part of a co... |
ref 14823 | Domain and codomain of the... |
imf 14824 | Domain and codomain of the... |
crre 14825 | The real part of a complex... |
crim 14826 | The real part of a complex... |
replim 14827 | Reconstruct a complex numb... |
remim 14828 | Value of the conjugate of ... |
reim0 14829 | The imaginary part of a re... |
reim0b 14830 | A number is real iff its i... |
rereb 14831 | A number is real iff it eq... |
mulre 14832 | A product with a nonzero r... |
rere 14833 | A real number equals its r... |
cjreb 14834 | A number is real iff it eq... |
recj 14835 | Real part of a complex con... |
reneg 14836 | Real part of negative. (C... |
readd 14837 | Real part distributes over... |
resub 14838 | Real part distributes over... |
remullem 14839 | Lemma for ~ remul , ~ immu... |
remul 14840 | Real part of a product. (... |
remul2 14841 | Real part of a product. (... |
rediv 14842 | Real part of a division. ... |
imcj 14843 | Imaginary part of a comple... |
imneg 14844 | The imaginary part of a ne... |
imadd 14845 | Imaginary part distributes... |
imsub 14846 | Imaginary part distributes... |
immul 14847 | Imaginary part of a produc... |
immul2 14848 | Imaginary part of a produc... |
imdiv 14849 | Imaginary part of a divisi... |
cjre 14850 | A real number equals its c... |
cjcj 14851 | The conjugate of the conju... |
cjadd 14852 | Complex conjugate distribu... |
cjmul 14853 | Complex conjugate distribu... |
ipcnval 14854 | Standard inner product on ... |
cjmulrcl 14855 | A complex number times its... |
cjmulval 14856 | A complex number times its... |
cjmulge0 14857 | A complex number times its... |
cjneg 14858 | Complex conjugate of negat... |
addcj 14859 | A number plus its conjugat... |
cjsub 14860 | Complex conjugate distribu... |
cjexp 14861 | Complex conjugate of posit... |
imval2 14862 | The imaginary part of a nu... |
re0 14863 | The real part of zero. (C... |
im0 14864 | The imaginary part of zero... |
re1 14865 | The real part of one. (Co... |
im1 14866 | The imaginary part of one.... |
rei 14867 | The real part of ` _i ` . ... |
imi 14868 | The imaginary part of ` _i... |
cj0 14869 | The conjugate of zero. (C... |
cji 14870 | The complex conjugate of t... |
cjreim 14871 | The conjugate of a represe... |
cjreim2 14872 | The conjugate of the repre... |
cj11 14873 | Complex conjugate is a one... |
cjne0 14874 | A number is nonzero iff it... |
cjdiv 14875 | Complex conjugate distribu... |
cnrecnv 14876 | The inverse to the canonic... |
sqeqd 14877 | A deduction for showing tw... |
recli 14878 | The real part of a complex... |
imcli 14879 | The imaginary part of a co... |
cjcli 14880 | Closure law for complex co... |
replimi 14881 | Construct a complex number... |
cjcji 14882 | The conjugate of the conju... |
reim0bi 14883 | A number is real iff its i... |
rerebi 14884 | A real number equals its r... |
cjrebi 14885 | A number is real iff it eq... |
recji 14886 | Real part of a complex con... |
imcji 14887 | Imaginary part of a comple... |
cjmulrcli 14888 | A complex number times its... |
cjmulvali 14889 | A complex number times its... |
cjmulge0i 14890 | A complex number times its... |
renegi 14891 | Real part of negative. (C... |
imnegi 14892 | Imaginary part of negative... |
cjnegi 14893 | Complex conjugate of negat... |
addcji 14894 | A number plus its conjugat... |
readdi 14895 | Real part distributes over... |
imaddi 14896 | Imaginary part distributes... |
remuli 14897 | Real part of a product. (... |
immuli 14898 | Imaginary part of a produc... |
cjaddi 14899 | Complex conjugate distribu... |
cjmuli 14900 | Complex conjugate distribu... |
ipcni 14901 | Standard inner product on ... |
cjdivi 14902 | Complex conjugate distribu... |
crrei 14903 | The real part of a complex... |
crimi 14904 | The imaginary part of a co... |
recld 14905 | The real part of a complex... |
imcld 14906 | The imaginary part of a co... |
cjcld 14907 | Closure law for complex co... |
replimd 14908 | Construct a complex number... |
remimd 14909 | Value of the conjugate of ... |
cjcjd 14910 | The conjugate of the conju... |
reim0bd 14911 | A number is real iff its i... |
rerebd 14912 | A real number equals its r... |
cjrebd 14913 | A number is real iff it eq... |
cjne0d 14914 | A number is nonzero iff it... |
recjd 14915 | Real part of a complex con... |
imcjd 14916 | Imaginary part of a comple... |
cjmulrcld 14917 | A complex number times its... |
cjmulvald 14918 | A complex number times its... |
cjmulge0d 14919 | A complex number times its... |
renegd 14920 | Real part of negative. (C... |
imnegd 14921 | Imaginary part of negative... |
cjnegd 14922 | Complex conjugate of negat... |
addcjd 14923 | A number plus its conjugat... |
cjexpd 14924 | Complex conjugate of posit... |
readdd 14925 | Real part distributes over... |
imaddd 14926 | Imaginary part distributes... |
resubd 14927 | Real part distributes over... |
imsubd 14928 | Imaginary part distributes... |
remuld 14929 | Real part of a product. (... |
immuld 14930 | Imaginary part of a produc... |
cjaddd 14931 | Complex conjugate distribu... |
cjmuld 14932 | Complex conjugate distribu... |
ipcnd 14933 | Standard inner product on ... |
cjdivd 14934 | Complex conjugate distribu... |
rered 14935 | A real number equals its r... |
reim0d 14936 | The imaginary part of a re... |
cjred 14937 | A real number equals its c... |
remul2d 14938 | Real part of a product. (... |
immul2d 14939 | Imaginary part of a produc... |
redivd 14940 | Real part of a division. ... |
imdivd 14941 | Imaginary part of a divisi... |
crred 14942 | The real part of a complex... |
crimd 14943 | The imaginary part of a co... |
sqrtval 14948 | Value of square root funct... |
absval 14949 | The absolute value (modulu... |
rennim 14950 | A real number does not lie... |
cnpart 14951 | The specification of restr... |
sqr0lem 14952 | Square root of zero. (Con... |
sqrt0 14953 | Square root of zero. (Con... |
sqrlem1 14954 | Lemma for ~ 01sqrex . (Co... |
sqrlem2 14955 | Lemma for ~ 01sqrex . (Co... |
sqrlem3 14956 | Lemma for ~ 01sqrex . (Co... |
sqrlem4 14957 | Lemma for ~ 01sqrex . (Co... |
sqrlem5 14958 | Lemma for ~ 01sqrex . (Co... |
sqrlem6 14959 | Lemma for ~ 01sqrex . (Co... |
sqrlem7 14960 | Lemma for ~ 01sqrex . (Co... |
01sqrex 14961 | Existence of a square root... |
resqrex 14962 | Existence of a square root... |
sqrmo 14963 | Uniqueness for the square ... |
resqreu 14964 | Existence and uniqueness f... |
resqrtcl 14965 | Closure of the square root... |
resqrtthlem 14966 | Lemma for ~ resqrtth . (C... |
resqrtth 14967 | Square root theorem over t... |
remsqsqrt 14968 | Square of square root. (C... |
sqrtge0 14969 | The square root function i... |
sqrtgt0 14970 | The square root function i... |
sqrtmul 14971 | Square root distributes ov... |
sqrtle 14972 | Square root is monotonic. ... |
sqrtlt 14973 | Square root is strictly mo... |
sqrt11 14974 | The square root function i... |
sqrt00 14975 | A square root is zero iff ... |
rpsqrtcl 14976 | The square root of a posit... |
sqrtdiv 14977 | Square root distributes ov... |
sqrtneglem 14978 | The square root of a negat... |
sqrtneg 14979 | The square root of a negat... |
sqrtsq2 14980 | Relationship between squar... |
sqrtsq 14981 | Square root of square. (C... |
sqrtmsq 14982 | Square root of square. (C... |
sqrt1 14983 | The square root of 1 is 1.... |
sqrt4 14984 | The square root of 4 is 2.... |
sqrt9 14985 | The square root of 9 is 3.... |
sqrt2gt1lt2 14986 | The square root of 2 is bo... |
sqrtm1 14987 | The imaginary unit is the ... |
nn0sqeq1 14988 | A natural number with squa... |
absneg 14989 | Absolute value of the oppo... |
abscl 14990 | Real closure of absolute v... |
abscj 14991 | The absolute value of a nu... |
absvalsq 14992 | Square of value of absolut... |
absvalsq2 14993 | Square of value of absolut... |
sqabsadd 14994 | Square of absolute value o... |
sqabssub 14995 | Square of absolute value o... |
absval2 14996 | Value of absolute value fu... |
abs0 14997 | The absolute value of 0. ... |
absi 14998 | The absolute value of the ... |
absge0 14999 | Absolute value is nonnegat... |
absrpcl 15000 | The absolute value of a no... |
abs00 15001 | The absolute value of a nu... |
abs00ad 15002 | A complex number is zero i... |
abs00bd 15003 | If a complex number is zer... |
absreimsq 15004 | Square of the absolute val... |
absreim 15005 | Absolute value of a number... |
absmul 15006 | Absolute value distributes... |
absdiv 15007 | Absolute value distributes... |
absid 15008 | A nonnegative number is it... |
abs1 15009 | The absolute value of one ... |
absnid 15010 | A negative number is the n... |
leabs 15011 | A real number is less than... |
absor 15012 | The absolute value of a re... |
absre 15013 | Absolute value of a real n... |
absresq 15014 | Square of the absolute val... |
absmod0 15015 | ` A ` is divisible by ` B ... |
absexp 15016 | Absolute value of positive... |
absexpz 15017 | Absolute value of integer ... |
abssq 15018 | Square can be moved in and... |
sqabs 15019 | The squares of two reals a... |
absrele 15020 | The absolute value of a co... |
absimle 15021 | The absolute value of a co... |
max0add 15022 | The sum of the positive an... |
absz 15023 | A real number is an intege... |
nn0abscl 15024 | The absolute value of an i... |
zabscl 15025 | The absolute value of an i... |
abslt 15026 | Absolute value and 'less t... |
absle 15027 | Absolute value and 'less t... |
abssubne0 15028 | If the absolute value of a... |
absdiflt 15029 | The absolute value of a di... |
absdifle 15030 | The absolute value of a di... |
elicc4abs 15031 | Membership in a symmetric ... |
lenegsq 15032 | Comparison to a nonnegativ... |
releabs 15033 | The real part of a number ... |
recval 15034 | Reciprocal expressed with ... |
absidm 15035 | The absolute value functio... |
absgt0 15036 | The absolute value of a no... |
nnabscl 15037 | The absolute value of a no... |
abssub 15038 | Swapping order of subtract... |
abssubge0 15039 | Absolute value of a nonneg... |
abssuble0 15040 | Absolute value of a nonpos... |
absmax 15041 | The maximum of two numbers... |
abstri 15042 | Triangle inequality for ab... |
abs3dif 15043 | Absolute value of differen... |
abs2dif 15044 | Difference of absolute val... |
abs2dif2 15045 | Difference of absolute val... |
abs2difabs 15046 | Absolute value of differen... |
abs1m 15047 | For any complex number, th... |
recan 15048 | Cancellation law involving... |
absf 15049 | Mapping domain and codomai... |
abs3lem 15050 | Lemma involving absolute v... |
abslem2 15051 | Lemma involving absolute v... |
rddif 15052 | The difference between a r... |
absrdbnd 15053 | Bound on the absolute valu... |
fzomaxdiflem 15054 | Lemma for ~ fzomaxdif . (... |
fzomaxdif 15055 | A bound on the separation ... |
uzin2 15056 | The upper integers are clo... |
rexanuz 15057 | Combine two different uppe... |
rexanre 15058 | Combine two different uppe... |
rexfiuz 15059 | Combine finitely many diff... |
rexuz3 15060 | Restrict the base of the u... |
rexanuz2 15061 | Combine two different uppe... |
r19.29uz 15062 | A version of ~ 19.29 for u... |
r19.2uz 15063 | A version of ~ r19.2z for ... |
rexuzre 15064 | Convert an upper real quan... |
rexico 15065 | Restrict the base of an up... |
cau3lem 15066 | Lemma for ~ cau3 . (Contr... |
cau3 15067 | Convert between three-quan... |
cau4 15068 | Change the base of a Cauch... |
caubnd2 15069 | A Cauchy sequence of compl... |
caubnd 15070 | A Cauchy sequence of compl... |
sqreulem 15071 | Lemma for ~ sqreu : write ... |
sqreu 15072 | Existence and uniqueness f... |
sqrtcl 15073 | Closure of the square root... |
sqrtthlem 15074 | Lemma for ~ sqrtth . (Con... |
sqrtf 15075 | Mapping domain and codomai... |
sqrtth 15076 | Square root theorem over t... |
sqrtrege0 15077 | The square root function m... |
eqsqrtor 15078 | Solve an equation containi... |
eqsqrtd 15079 | A deduction for showing th... |
eqsqrt2d 15080 | A deduction for showing th... |
amgm2 15081 | Arithmetic-geometric mean ... |
sqrtthi 15082 | Square root theorem. Theo... |
sqrtcli 15083 | The square root of a nonne... |
sqrtgt0i 15084 | The square root of a posit... |
sqrtmsqi 15085 | Square root of square. (C... |
sqrtsqi 15086 | Square root of square. (C... |
sqsqrti 15087 | Square of square root. (C... |
sqrtge0i 15088 | The square root of a nonne... |
absidi 15089 | A nonnegative number is it... |
absnidi 15090 | A negative number is the n... |
leabsi 15091 | A real number is less than... |
absori 15092 | The absolute value of a re... |
absrei 15093 | Absolute value of a real n... |
sqrtpclii 15094 | The square root of a posit... |
sqrtgt0ii 15095 | The square root of a posit... |
sqrt11i 15096 | The square root function i... |
sqrtmuli 15097 | Square root distributes ov... |
sqrtmulii 15098 | Square root distributes ov... |
sqrtmsq2i 15099 | Relationship between squar... |
sqrtlei 15100 | Square root is monotonic. ... |
sqrtlti 15101 | Square root is strictly mo... |
abslti 15102 | Absolute value and 'less t... |
abslei 15103 | Absolute value and 'less t... |
cnsqrt00 15104 | A square root of a complex... |
absvalsqi 15105 | Square of value of absolut... |
absvalsq2i 15106 | Square of value of absolut... |
abscli 15107 | Real closure of absolute v... |
absge0i 15108 | Absolute value is nonnegat... |
absval2i 15109 | Value of absolute value fu... |
abs00i 15110 | The absolute value of a nu... |
absgt0i 15111 | The absolute value of a no... |
absnegi 15112 | Absolute value of negative... |
abscji 15113 | The absolute value of a nu... |
releabsi 15114 | The real part of a number ... |
abssubi 15115 | Swapping order of subtract... |
absmuli 15116 | Absolute value distributes... |
sqabsaddi 15117 | Square of absolute value o... |
sqabssubi 15118 | Square of absolute value o... |
absdivzi 15119 | Absolute value distributes... |
abstrii 15120 | Triangle inequality for ab... |
abs3difi 15121 | Absolute value of differen... |
abs3lemi 15122 | Lemma involving absolute v... |
rpsqrtcld 15123 | The square root of a posit... |
sqrtgt0d 15124 | The square root of a posit... |
absnidd 15125 | A negative number is the n... |
leabsd 15126 | A real number is less than... |
absord 15127 | The absolute value of a re... |
absred 15128 | Absolute value of a real n... |
resqrtcld 15129 | The square root of a nonne... |
sqrtmsqd 15130 | Square root of square. (C... |
sqrtsqd 15131 | Square root of square. (C... |
sqrtge0d 15132 | The square root of a nonne... |
sqrtnegd 15133 | The square root of a negat... |
absidd 15134 | A nonnegative number is it... |
sqrtdivd 15135 | Square root distributes ov... |
sqrtmuld 15136 | Square root distributes ov... |
sqrtsq2d 15137 | Relationship between squar... |
sqrtled 15138 | Square root is monotonic. ... |
sqrtltd 15139 | Square root is strictly mo... |
sqr11d 15140 | The square root function i... |
absltd 15141 | Absolute value and 'less t... |
absled 15142 | Absolute value and 'less t... |
abssubge0d 15143 | Absolute value of a nonneg... |
abssuble0d 15144 | Absolute value of a nonpos... |
absdifltd 15145 | The absolute value of a di... |
absdifled 15146 | The absolute value of a di... |
icodiamlt 15147 | Two elements in a half-ope... |
abscld 15148 | Real closure of absolute v... |
sqrtcld 15149 | Closure of the square root... |
sqrtrege0d 15150 | The real part of the squar... |
sqsqrtd 15151 | Square root theorem. Theo... |
msqsqrtd 15152 | Square root theorem. Theo... |
sqr00d 15153 | A square root is zero iff ... |
absvalsqd 15154 | Square of value of absolut... |
absvalsq2d 15155 | Square of value of absolut... |
absge0d 15156 | Absolute value is nonnegat... |
absval2d 15157 | Value of absolute value fu... |
abs00d 15158 | The absolute value of a nu... |
absne0d 15159 | The absolute value of a nu... |
absrpcld 15160 | The absolute value of a no... |
absnegd 15161 | Absolute value of negative... |
abscjd 15162 | The absolute value of a nu... |
releabsd 15163 | The real part of a number ... |
absexpd 15164 | Absolute value of positive... |
abssubd 15165 | Swapping order of subtract... |
absmuld 15166 | Absolute value distributes... |
absdivd 15167 | Absolute value distributes... |
abstrid 15168 | Triangle inequality for ab... |
abs2difd 15169 | Difference of absolute val... |
abs2dif2d 15170 | Difference of absolute val... |
abs2difabsd 15171 | Absolute value of differen... |
abs3difd 15172 | Absolute value of differen... |
abs3lemd 15173 | Lemma involving absolute v... |
reusq0 15174 | A complex number is the sq... |
bhmafibid1cn 15175 | The Brahmagupta-Fibonacci ... |
bhmafibid2cn 15176 | The Brahmagupta-Fibonacci ... |
bhmafibid1 15177 | The Brahmagupta-Fibonacci ... |
bhmafibid2 15178 | The Brahmagupta-Fibonacci ... |
limsupgord 15181 | Ordering property of the s... |
limsupcl 15182 | Closure of the superior li... |
limsupval 15183 | The superior limit of an i... |
limsupgf 15184 | Closure of the superior li... |
limsupgval 15185 | Value of the superior limi... |
limsupgle 15186 | The defining property of t... |
limsuple 15187 | The defining property of t... |
limsuplt 15188 | The defining property of t... |
limsupval2 15189 | The superior limit, relati... |
limsupgre 15190 | If a sequence of real numb... |
limsupbnd1 15191 | If a sequence is eventuall... |
limsupbnd2 15192 | If a sequence is eventuall... |
climrel 15201 | The limit relation is a re... |
rlimrel 15202 | The limit relation is a re... |
clim 15203 | Express the predicate: Th... |
rlim 15204 | Express the predicate: Th... |
rlim2 15205 | Rewrite ~ rlim for a mappi... |
rlim2lt 15206 | Use strictly less-than in ... |
rlim3 15207 | Restrict the range of the ... |
climcl 15208 | Closure of the limit of a ... |
rlimpm 15209 | Closure of a function with... |
rlimf 15210 | Closure of a function with... |
rlimss 15211 | Domain closure of a functi... |
rlimcl 15212 | Closure of the limit of a ... |
clim2 15213 | Express the predicate: Th... |
clim2c 15214 | Express the predicate ` F ... |
clim0 15215 | Express the predicate ` F ... |
clim0c 15216 | Express the predicate ` F ... |
rlim0 15217 | Express the predicate ` B ... |
rlim0lt 15218 | Use strictly less-than in ... |
climi 15219 | Convergence of a sequence ... |
climi2 15220 | Convergence of a sequence ... |
climi0 15221 | Convergence of a sequence ... |
rlimi 15222 | Convergence at infinity of... |
rlimi2 15223 | Convergence at infinity of... |
ello1 15224 | Elementhood in the set of ... |
ello12 15225 | Elementhood in the set of ... |
ello12r 15226 | Sufficient condition for e... |
lo1f 15227 | An eventually upper bounde... |
lo1dm 15228 | An eventually upper bounde... |
lo1bdd 15229 | The defining property of a... |
ello1mpt 15230 | Elementhood in the set of ... |
ello1mpt2 15231 | Elementhood in the set of ... |
ello1d 15232 | Sufficient condition for e... |
lo1bdd2 15233 | If an eventually bounded f... |
lo1bddrp 15234 | Refine ~ o1bdd2 to give a ... |
elo1 15235 | Elementhood in the set of ... |
elo12 15236 | Elementhood in the set of ... |
elo12r 15237 | Sufficient condition for e... |
o1f 15238 | An eventually bounded func... |
o1dm 15239 | An eventually bounded func... |
o1bdd 15240 | The defining property of a... |
lo1o1 15241 | A function is eventually b... |
lo1o12 15242 | A function is eventually b... |
elo1mpt 15243 | Elementhood in the set of ... |
elo1mpt2 15244 | Elementhood in the set of ... |
elo1d 15245 | Sufficient condition for e... |
o1lo1 15246 | A real function is eventua... |
o1lo12 15247 | A lower bounded real funct... |
o1lo1d 15248 | A real eventually bounded ... |
icco1 15249 | Derive eventual boundednes... |
o1bdd2 15250 | If an eventually bounded f... |
o1bddrp 15251 | Refine ~ o1bdd2 to give a ... |
climconst 15252 | An (eventually) constant s... |
rlimconst 15253 | A constant sequence conver... |
rlimclim1 15254 | Forward direction of ~ rli... |
rlimclim 15255 | A sequence on an upper int... |
climrlim2 15256 | Produce a real limit from ... |
climconst2 15257 | A constant sequence conver... |
climz 15258 | The zero sequence converge... |
rlimuni 15259 | A real function whose doma... |
rlimdm 15260 | Two ways to express that a... |
climuni 15261 | An infinite sequence of co... |
fclim 15262 | The limit relation is func... |
climdm 15263 | Two ways to express that a... |
climeu 15264 | An infinite sequence of co... |
climreu 15265 | An infinite sequence of co... |
climmo 15266 | An infinite sequence of co... |
rlimres 15267 | The restriction of a funct... |
lo1res 15268 | The restriction of an even... |
o1res 15269 | The restriction of an even... |
rlimres2 15270 | The restriction of a funct... |
lo1res2 15271 | The restriction of a funct... |
o1res2 15272 | The restriction of a funct... |
lo1resb 15273 | The restriction of a funct... |
rlimresb 15274 | The restriction of a funct... |
o1resb 15275 | The restriction of a funct... |
climeq 15276 | Two functions that are eve... |
lo1eq 15277 | Two functions that are eve... |
rlimeq 15278 | Two functions that are eve... |
o1eq 15279 | Two functions that are eve... |
climmpt 15280 | Exhibit a function ` G ` w... |
2clim 15281 | If two sequences converge ... |
climmpt2 15282 | Relate an integer limit on... |
climshftlem 15283 | A shifted function converg... |
climres 15284 | A function restricted to u... |
climshft 15285 | A shifted function converg... |
serclim0 15286 | The zero series converges ... |
rlimcld2 15287 | If ` D ` is a closed set i... |
rlimrege0 15288 | The limit of a sequence of... |
rlimrecl 15289 | The limit of a real sequen... |
rlimge0 15290 | The limit of a sequence of... |
climshft2 15291 | A shifted function converg... |
climrecl 15292 | The limit of a convergent ... |
climge0 15293 | A nonnegative sequence con... |
climabs0 15294 | Convergence to zero of the... |
o1co 15295 | Sufficient condition for t... |
o1compt 15296 | Sufficient condition for t... |
rlimcn1 15297 | Image of a limit under a c... |
rlimcn1b 15298 | Image of a limit under a c... |
rlimcn3 15299 | Image of a limit under a c... |
rlimcn2 15300 | Image of a limit under a c... |
climcn1 15301 | Image of a limit under a c... |
climcn2 15302 | Image of a limit under a c... |
addcn2 15303 | Complex number addition is... |
subcn2 15304 | Complex number subtraction... |
mulcn2 15305 | Complex number multiplicat... |
reccn2 15306 | The reciprocal function is... |
cn1lem 15307 | A sufficient condition for... |
abscn2 15308 | The absolute value functio... |
cjcn2 15309 | The complex conjugate func... |
recn2 15310 | The real part function is ... |
imcn2 15311 | The imaginary part functio... |
climcn1lem 15312 | The limit of a continuous ... |
climabs 15313 | Limit of the absolute valu... |
climcj 15314 | Limit of the complex conju... |
climre 15315 | Limit of the real part of ... |
climim 15316 | Limit of the imaginary par... |
rlimmptrcl 15317 | Reverse closure for a real... |
rlimabs 15318 | Limit of the absolute valu... |
rlimcj 15319 | Limit of the complex conju... |
rlimre 15320 | Limit of the real part of ... |
rlimim 15321 | Limit of the imaginary par... |
o1of2 15322 | Show that a binary operati... |
o1add 15323 | The sum of two eventually ... |
o1mul 15324 | The product of two eventua... |
o1sub 15325 | The difference of two even... |
rlimo1 15326 | Any function with a finite... |
rlimdmo1 15327 | A convergent function is e... |
o1rlimmul 15328 | The product of an eventual... |
o1const 15329 | A constant function is eve... |
lo1const 15330 | A constant function is eve... |
lo1mptrcl 15331 | Reverse closure for an eve... |
o1mptrcl 15332 | Reverse closure for an eve... |
o1add2 15333 | The sum of two eventually ... |
o1mul2 15334 | The product of two eventua... |
o1sub2 15335 | The product of two eventua... |
lo1add 15336 | The sum of two eventually ... |
lo1mul 15337 | The product of an eventual... |
lo1mul2 15338 | The product of an eventual... |
o1dif 15339 | If the difference of two f... |
lo1sub 15340 | The difference of an event... |
climadd 15341 | Limit of the sum of two co... |
climmul 15342 | Limit of the product of tw... |
climsub 15343 | Limit of the difference of... |
climaddc1 15344 | Limit of a constant ` C ` ... |
climaddc2 15345 | Limit of a constant ` C ` ... |
climmulc2 15346 | Limit of a sequence multip... |
climsubc1 15347 | Limit of a constant ` C ` ... |
climsubc2 15348 | Limit of a constant ` C ` ... |
climle 15349 | Comparison of the limits o... |
climsqz 15350 | Convergence of a sequence ... |
climsqz2 15351 | Convergence of a sequence ... |
rlimadd 15352 | Limit of the sum of two co... |
rlimaddOLD 15353 | Obsolete version of ~ rlim... |
rlimsub 15354 | Limit of the difference of... |
rlimmul 15355 | Limit of the product of tw... |
rlimmulOLD 15356 | Obsolete version of ~ rlim... |
rlimdiv 15357 | Limit of the quotient of t... |
rlimneg 15358 | Limit of the negative of a... |
rlimle 15359 | Comparison of the limits o... |
rlimsqzlem 15360 | Lemma for ~ rlimsqz and ~ ... |
rlimsqz 15361 | Convergence of a sequence ... |
rlimsqz2 15362 | Convergence of a sequence ... |
lo1le 15363 | Transfer eventual upper bo... |
o1le 15364 | Transfer eventual boundedn... |
rlimno1 15365 | A function whose inverse c... |
clim2ser 15366 | The limit of an infinite s... |
clim2ser2 15367 | The limit of an infinite s... |
iserex 15368 | An infinite series converg... |
isermulc2 15369 | Multiplication of an infin... |
climlec2 15370 | Comparison of a constant t... |
iserle 15371 | Comparison of the limits o... |
iserge0 15372 | The limit of an infinite s... |
climub 15373 | The limit of a monotonic s... |
climserle 15374 | The partial sums of a conv... |
isershft 15375 | Index shift of the limit o... |
isercolllem1 15376 | Lemma for ~ isercoll . (C... |
isercolllem2 15377 | Lemma for ~ isercoll . (C... |
isercolllem3 15378 | Lemma for ~ isercoll . (C... |
isercoll 15379 | Rearrange an infinite seri... |
isercoll2 15380 | Generalize ~ isercoll so t... |
climsup 15381 | A bounded monotonic sequen... |
climcau 15382 | A converging sequence of c... |
climbdd 15383 | A converging sequence of c... |
caucvgrlem 15384 | Lemma for ~ caurcvgr . (C... |
caurcvgr 15385 | A Cauchy sequence of real ... |
caucvgrlem2 15386 | Lemma for ~ caucvgr . (Co... |
caucvgr 15387 | A Cauchy sequence of compl... |
caurcvg 15388 | A Cauchy sequence of real ... |
caurcvg2 15389 | A Cauchy sequence of real ... |
caucvg 15390 | A Cauchy sequence of compl... |
caucvgb 15391 | A function is convergent i... |
serf0 15392 | If an infinite series conv... |
iseraltlem1 15393 | Lemma for ~ iseralt . A d... |
iseraltlem2 15394 | Lemma for ~ iseralt . The... |
iseraltlem3 15395 | Lemma for ~ iseralt . Fro... |
iseralt 15396 | The alternating series tes... |
sumex 15399 | A sum is a set. (Contribu... |
sumeq1 15400 | Equality theorem for a sum... |
nfsum1 15401 | Bound-variable hypothesis ... |
nfsum 15402 | Bound-variable hypothesis ... |
nfsumOLD 15403 | Obsolete version of ~ nfsu... |
sumeq2w 15404 | Equality theorem for sum, ... |
sumeq2ii 15405 | Equality theorem for sum, ... |
sumeq2 15406 | Equality theorem for sum. ... |
cbvsum 15407 | Change bound variable in a... |
cbvsumv 15408 | Change bound variable in a... |
cbvsumi 15409 | Change bound variable in a... |
sumeq1i 15410 | Equality inference for sum... |
sumeq2i 15411 | Equality inference for sum... |
sumeq12i 15412 | Equality inference for sum... |
sumeq1d 15413 | Equality deduction for sum... |
sumeq2d 15414 | Equality deduction for sum... |
sumeq2dv 15415 | Equality deduction for sum... |
sumeq2sdv 15416 | Equality deduction for sum... |
2sumeq2dv 15417 | Equality deduction for dou... |
sumeq12dv 15418 | Equality deduction for sum... |
sumeq12rdv 15419 | Equality deduction for sum... |
sum2id 15420 | The second class argument ... |
sumfc 15421 | A lemma to facilitate conv... |
fz1f1o 15422 | A lemma for working with f... |
sumrblem 15423 | Lemma for ~ sumrb . (Cont... |
fsumcvg 15424 | The sequence of partial su... |
sumrb 15425 | Rebase the starting point ... |
summolem3 15426 | Lemma for ~ summo . (Cont... |
summolem2a 15427 | Lemma for ~ summo . (Cont... |
summolem2 15428 | Lemma for ~ summo . (Cont... |
summo 15429 | A sum has at most one limi... |
zsum 15430 | Series sum with index set ... |
isum 15431 | Series sum with an upper i... |
fsum 15432 | The value of a sum over a ... |
sum0 15433 | Any sum over the empty set... |
sumz 15434 | Any sum of zero over a sum... |
fsumf1o 15435 | Re-index a finite sum usin... |
sumss 15436 | Change the index set to a ... |
fsumss 15437 | Change the index set to a ... |
sumss2 15438 | Change the index set of a ... |
fsumcvg2 15439 | The sequence of partial su... |
fsumsers 15440 | Special case of series sum... |
fsumcvg3 15441 | A finite sum is convergent... |
fsumser 15442 | A finite sum expressed in ... |
fsumcl2lem 15443 | - Lemma for finite sum clo... |
fsumcllem 15444 | - Lemma for finite sum clo... |
fsumcl 15445 | Closure of a finite sum of... |
fsumrecl 15446 | Closure of a finite sum of... |
fsumzcl 15447 | Closure of a finite sum of... |
fsumnn0cl 15448 | Closure of a finite sum of... |
fsumrpcl 15449 | Closure of a finite sum of... |
fsumclf 15450 | Closure of a finite sum of... |
fsumzcl2 15451 | A finite sum with integer ... |
fsumadd 15452 | The sum of two finite sums... |
fsumsplit 15453 | Split a sum into two parts... |
fsumsplitf 15454 | Split a sum into two parts... |
sumsnf 15455 | A sum of a singleton is th... |
fsumsplitsn 15456 | Separate out a term in a f... |
fsumsplit1 15457 | Separate out a term in a f... |
sumsn 15458 | A sum of a singleton is th... |
fsum1 15459 | The finite sum of ` A ( k ... |
sumpr 15460 | A sum over a pair is the s... |
sumtp 15461 | A sum over a triple is the... |
sumsns 15462 | A sum of a singleton is th... |
fsumm1 15463 | Separate out the last term... |
fzosump1 15464 | Separate out the last term... |
fsum1p 15465 | Separate out the first ter... |
fsummsnunz 15466 | A finite sum all of whose ... |
fsumsplitsnun 15467 | Separate out a term in a f... |
fsump1 15468 | The addition of the next t... |
isumclim 15469 | An infinite sum equals the... |
isumclim2 15470 | A converging series conver... |
isumclim3 15471 | The sequence of partial fi... |
sumnul 15472 | The sum of a non-convergen... |
isumcl 15473 | The sum of a converging in... |
isummulc2 15474 | An infinite sum multiplied... |
isummulc1 15475 | An infinite sum multiplied... |
isumdivc 15476 | An infinite sum divided by... |
isumrecl 15477 | The sum of a converging in... |
isumge0 15478 | An infinite sum of nonnega... |
isumadd 15479 | Addition of infinite sums.... |
sumsplit 15480 | Split a sum into two parts... |
fsump1i 15481 | Optimized version of ~ fsu... |
fsum2dlem 15482 | Lemma for ~ fsum2d - induc... |
fsum2d 15483 | Write a double sum as a su... |
fsumxp 15484 | Combine two sums into a si... |
fsumcnv 15485 | Transform a region of summ... |
fsumcom2 15486 | Interchange order of summa... |
fsumcom 15487 | Interchange order of summa... |
fsum0diaglem 15488 | Lemma for ~ fsum0diag . (... |
fsum0diag 15489 | Two ways to express "the s... |
mptfzshft 15490 | 1-1 onto function in maps-... |
fsumrev 15491 | Reversal of a finite sum. ... |
fsumshft 15492 | Index shift of a finite su... |
fsumshftm 15493 | Negative index shift of a ... |
fsumrev2 15494 | Reversal of a finite sum. ... |
fsum0diag2 15495 | Two ways to express "the s... |
fsummulc2 15496 | A finite sum multiplied by... |
fsummulc1 15497 | A finite sum multiplied by... |
fsumdivc 15498 | A finite sum divided by a ... |
fsumneg 15499 | Negation of a finite sum. ... |
fsumsub 15500 | Split a finite sum over a ... |
fsum2mul 15501 | Separate the nested sum of... |
fsumconst 15502 | The sum of constant terms ... |
fsumdifsnconst 15503 | The sum of constant terms ... |
modfsummodslem1 15504 | Lemma 1 for ~ modfsummods ... |
modfsummods 15505 | Induction step for ~ modfs... |
modfsummod 15506 | A finite sum modulo a posi... |
fsumge0 15507 | If all of the terms of a f... |
fsumless 15508 | A shorter sum of nonnegati... |
fsumge1 15509 | A sum of nonnegative numbe... |
fsum00 15510 | A sum of nonnegative numbe... |
fsumle 15511 | If all of the terms of fin... |
fsumlt 15512 | If every term in one finit... |
fsumabs 15513 | Generalized triangle inequ... |
telfsumo 15514 | Sum of a telescoping serie... |
telfsumo2 15515 | Sum of a telescoping serie... |
telfsum 15516 | Sum of a telescoping serie... |
telfsum2 15517 | Sum of a telescoping serie... |
fsumparts 15518 | Summation by parts. (Cont... |
fsumrelem 15519 | Lemma for ~ fsumre , ~ fsu... |
fsumre 15520 | The real part of a sum. (... |
fsumim 15521 | The imaginary part of a su... |
fsumcj 15522 | The complex conjugate of a... |
fsumrlim 15523 | Limit of a finite sum of c... |
fsumo1 15524 | The finite sum of eventual... |
o1fsum 15525 | If ` A ( k ) ` is O(1), th... |
seqabs 15526 | Generalized triangle inequ... |
iserabs 15527 | Generalized triangle inequ... |
cvgcmp 15528 | A comparison test for conv... |
cvgcmpub 15529 | An upper bound for the lim... |
cvgcmpce 15530 | A comparison test for conv... |
abscvgcvg 15531 | An absolutely convergent s... |
climfsum 15532 | Limit of a finite sum of c... |
fsumiun 15533 | Sum over a disjoint indexe... |
hashiun 15534 | The cardinality of a disjo... |
hash2iun 15535 | The cardinality of a neste... |
hash2iun1dif1 15536 | The cardinality of a neste... |
hashrabrex 15537 | The number of elements in ... |
hashuni 15538 | The cardinality of a disjo... |
qshash 15539 | The cardinality of a set w... |
ackbijnn 15540 | Translate the Ackermann bi... |
binomlem 15541 | Lemma for ~ binom (binomia... |
binom 15542 | The binomial theorem: ` ( ... |
binom1p 15543 | Special case of the binomi... |
binom11 15544 | Special case of the binomi... |
binom1dif 15545 | A summation for the differ... |
bcxmaslem1 15546 | Lemma for ~ bcxmas . (Con... |
bcxmas 15547 | Parallel summation (Christ... |
incexclem 15548 | Lemma for ~ incexc . (Con... |
incexc 15549 | The inclusion/exclusion pr... |
incexc2 15550 | The inclusion/exclusion pr... |
isumshft 15551 | Index shift of an infinite... |
isumsplit 15552 | Split off the first ` N ` ... |
isum1p 15553 | The infinite sum of a conv... |
isumnn0nn 15554 | Sum from 0 to infinity in ... |
isumrpcl 15555 | The infinite sum of positi... |
isumle 15556 | Comparison of two infinite... |
isumless 15557 | A finite sum of nonnegativ... |
isumsup2 15558 | An infinite sum of nonnega... |
isumsup 15559 | An infinite sum of nonnega... |
isumltss 15560 | A partial sum of a series ... |
climcndslem1 15561 | Lemma for ~ climcnds : bou... |
climcndslem2 15562 | Lemma for ~ climcnds : bou... |
climcnds 15563 | The Cauchy condensation te... |
divrcnv 15564 | The sequence of reciprocal... |
divcnv 15565 | The sequence of reciprocal... |
flo1 15566 | The floor function satisfi... |
divcnvshft 15567 | Limit of a ratio function.... |
supcvg 15568 | Extract a sequence ` f ` i... |
infcvgaux1i 15569 | Auxiliary theorem for appl... |
infcvgaux2i 15570 | Auxiliary theorem for appl... |
harmonic 15571 | The harmonic series ` H ` ... |
arisum 15572 | Arithmetic series sum of t... |
arisum2 15573 | Arithmetic series sum of t... |
trireciplem 15574 | Lemma for ~ trirecip . Sh... |
trirecip 15575 | The sum of the reciprocals... |
expcnv 15576 | A sequence of powers of a ... |
explecnv 15577 | A sequence of terms conver... |
geoserg 15578 | The value of the finite ge... |
geoser 15579 | The value of the finite ge... |
pwdif 15580 | The difference of two numb... |
pwm1geoser 15581 | The n-th power of a number... |
geolim 15582 | The partial sums in the in... |
geolim2 15583 | The partial sums in the ge... |
georeclim 15584 | The limit of a geometric s... |
geo2sum 15585 | The value of the finite ge... |
geo2sum2 15586 | The value of the finite ge... |
geo2lim 15587 | The value of the infinite ... |
geomulcvg 15588 | The geometric series conve... |
geoisum 15589 | The infinite sum of ` 1 + ... |
geoisumr 15590 | The infinite sum of recipr... |
geoisum1 15591 | The infinite sum of ` A ^ ... |
geoisum1c 15592 | The infinite sum of ` A x.... |
0.999... 15593 | The recurring decimal 0.99... |
geoihalfsum 15594 | Prove that the infinite ge... |
cvgrat 15595 | Ratio test for convergence... |
mertenslem1 15596 | Lemma for ~ mertens . (Co... |
mertenslem2 15597 | Lemma for ~ mertens . (Co... |
mertens 15598 | Mertens' theorem. If ` A ... |
prodf 15599 | An infinite product of com... |
clim2prod 15600 | The limit of an infinite p... |
clim2div 15601 | The limit of an infinite p... |
prodfmul 15602 | The product of two infinit... |
prodf1 15603 | The value of the partial p... |
prodf1f 15604 | A one-valued infinite prod... |
prodfclim1 15605 | The constant one product c... |
prodfn0 15606 | No term of a nonzero infin... |
prodfrec 15607 | The reciprocal of an infin... |
prodfdiv 15608 | The quotient of two infini... |
ntrivcvg 15609 | A non-trivially converging... |
ntrivcvgn0 15610 | A product that converges t... |
ntrivcvgfvn0 15611 | Any value of a product seq... |
ntrivcvgtail 15612 | A tail of a non-trivially ... |
ntrivcvgmullem 15613 | Lemma for ~ ntrivcvgmul . ... |
ntrivcvgmul 15614 | The product of two non-tri... |
prodex 15617 | A product is a set. (Cont... |
prodeq1f 15618 | Equality theorem for a pro... |
prodeq1 15619 | Equality theorem for a pro... |
nfcprod1 15620 | Bound-variable hypothesis ... |
nfcprod 15621 | Bound-variable hypothesis ... |
prodeq2w 15622 | Equality theorem for produ... |
prodeq2ii 15623 | Equality theorem for produ... |
prodeq2 15624 | Equality theorem for produ... |
cbvprod 15625 | Change bound variable in a... |
cbvprodv 15626 | Change bound variable in a... |
cbvprodi 15627 | Change bound variable in a... |
prodeq1i 15628 | Equality inference for pro... |
prodeq2i 15629 | Equality inference for pro... |
prodeq12i 15630 | Equality inference for pro... |
prodeq1d 15631 | Equality deduction for pro... |
prodeq2d 15632 | Equality deduction for pro... |
prodeq2dv 15633 | Equality deduction for pro... |
prodeq2sdv 15634 | Equality deduction for pro... |
2cprodeq2dv 15635 | Equality deduction for dou... |
prodeq12dv 15636 | Equality deduction for pro... |
prodeq12rdv 15637 | Equality deduction for pro... |
prod2id 15638 | The second class argument ... |
prodrblem 15639 | Lemma for ~ prodrb . (Con... |
fprodcvg 15640 | The sequence of partial pr... |
prodrblem2 15641 | Lemma for ~ prodrb . (Con... |
prodrb 15642 | Rebase the starting point ... |
prodmolem3 15643 | Lemma for ~ prodmo . (Con... |
prodmolem2a 15644 | Lemma for ~ prodmo . (Con... |
prodmolem2 15645 | Lemma for ~ prodmo . (Con... |
prodmo 15646 | A product has at most one ... |
zprod 15647 | Series product with index ... |
iprod 15648 | Series product with an upp... |
zprodn0 15649 | Nonzero series product wit... |
iprodn0 15650 | Nonzero series product wit... |
fprod 15651 | The value of a product ove... |
fprodntriv 15652 | A non-triviality lemma for... |
prod0 15653 | A product over the empty s... |
prod1 15654 | Any product of one over a ... |
prodfc 15655 | A lemma to facilitate conv... |
fprodf1o 15656 | Re-index a finite product ... |
prodss 15657 | Change the index set to a ... |
fprodss 15658 | Change the index set to a ... |
fprodser 15659 | A finite product expressed... |
fprodcl2lem 15660 | Finite product closure lem... |
fprodcllem 15661 | Finite product closure lem... |
fprodcl 15662 | Closure of a finite produc... |
fprodrecl 15663 | Closure of a finite produc... |
fprodzcl 15664 | Closure of a finite produc... |
fprodnncl 15665 | Closure of a finite produc... |
fprodrpcl 15666 | Closure of a finite produc... |
fprodnn0cl 15667 | Closure of a finite produc... |
fprodcllemf 15668 | Finite product closure lem... |
fprodreclf 15669 | Closure of a finite produc... |
fprodmul 15670 | The product of two finite ... |
fproddiv 15671 | The quotient of two finite... |
prodsn 15672 | A product of a singleton i... |
fprod1 15673 | A finite product of only o... |
prodsnf 15674 | A product of a singleton i... |
climprod1 15675 | The limit of a product ove... |
fprodsplit 15676 | Split a finite product int... |
fprodm1 15677 | Separate out the last term... |
fprod1p 15678 | Separate out the first ter... |
fprodp1 15679 | Multiply in the last term ... |
fprodm1s 15680 | Separate out the last term... |
fprodp1s 15681 | Multiply in the last term ... |
prodsns 15682 | A product of the singleton... |
fprodfac 15683 | Factorial using product no... |
fprodabs 15684 | The absolute value of a fi... |
fprodeq0 15685 | Any finite product contain... |
fprodshft 15686 | Shift the index of a finit... |
fprodrev 15687 | Reversal of a finite produ... |
fprodconst 15688 | The product of constant te... |
fprodn0 15689 | A finite product of nonzer... |
fprod2dlem 15690 | Lemma for ~ fprod2d - indu... |
fprod2d 15691 | Write a double product as ... |
fprodxp 15692 | Combine two products into ... |
fprodcnv 15693 | Transform a product region... |
fprodcom2 15694 | Interchange order of multi... |
fprodcom 15695 | Interchange product order.... |
fprod0diag 15696 | Two ways to express "the p... |
fproddivf 15697 | The quotient of two finite... |
fprodsplitf 15698 | Split a finite product int... |
fprodsplitsn 15699 | Separate out a term in a f... |
fprodsplit1f 15700 | Separate out a term in a f... |
fprodn0f 15701 | A finite product of nonzer... |
fprodclf 15702 | Closure of a finite produc... |
fprodge0 15703 | If all the terms of a fini... |
fprodeq0g 15704 | Any finite product contain... |
fprodge1 15705 | If all of the terms of a f... |
fprodle 15706 | If all the terms of two fi... |
fprodmodd 15707 | If all factors of two fini... |
iprodclim 15708 | An infinite product equals... |
iprodclim2 15709 | A converging product conve... |
iprodclim3 15710 | The sequence of partial fi... |
iprodcl 15711 | The product of a non-trivi... |
iprodrecl 15712 | The product of a non-trivi... |
iprodmul 15713 | Multiplication of infinite... |
risefacval 15718 | The value of the rising fa... |
fallfacval 15719 | The value of the falling f... |
risefacval2 15720 | One-based value of rising ... |
fallfacval2 15721 | One-based value of falling... |
fallfacval3 15722 | A product representation o... |
risefaccllem 15723 | Lemma for rising factorial... |
fallfaccllem 15724 | Lemma for falling factoria... |
risefaccl 15725 | Closure law for rising fac... |
fallfaccl 15726 | Closure law for falling fa... |
rerisefaccl 15727 | Closure law for rising fac... |
refallfaccl 15728 | Closure law for falling fa... |
nnrisefaccl 15729 | Closure law for rising fac... |
zrisefaccl 15730 | Closure law for rising fac... |
zfallfaccl 15731 | Closure law for falling fa... |
nn0risefaccl 15732 | Closure law for rising fac... |
rprisefaccl 15733 | Closure law for rising fac... |
risefallfac 15734 | A relationship between ris... |
fallrisefac 15735 | A relationship between fal... |
risefall0lem 15736 | Lemma for ~ risefac0 and ~... |
risefac0 15737 | The value of the rising fa... |
fallfac0 15738 | The value of the falling f... |
risefacp1 15739 | The value of the rising fa... |
fallfacp1 15740 | The value of the falling f... |
risefacp1d 15741 | The value of the rising fa... |
fallfacp1d 15742 | The value of the falling f... |
risefac1 15743 | The value of rising factor... |
fallfac1 15744 | The value of falling facto... |
risefacfac 15745 | Relate rising factorial to... |
fallfacfwd 15746 | The forward difference of ... |
0fallfac 15747 | The value of the zero fall... |
0risefac 15748 | The value of the zero risi... |
binomfallfaclem1 15749 | Lemma for ~ binomfallfac .... |
binomfallfaclem2 15750 | Lemma for ~ binomfallfac .... |
binomfallfac 15751 | A version of the binomial ... |
binomrisefac 15752 | A version of the binomial ... |
fallfacval4 15753 | Represent the falling fact... |
bcfallfac 15754 | Binomial coefficient in te... |
fallfacfac 15755 | Relate falling factorial t... |
bpolylem 15758 | Lemma for ~ bpolyval . (C... |
bpolyval 15759 | The value of the Bernoulli... |
bpoly0 15760 | The value of the Bernoulli... |
bpoly1 15761 | The value of the Bernoulli... |
bpolycl 15762 | Closure law for Bernoulli ... |
bpolysum 15763 | A sum for Bernoulli polyno... |
bpolydiflem 15764 | Lemma for ~ bpolydif . (C... |
bpolydif 15765 | Calculate the difference b... |
fsumkthpow 15766 | A closed-form expression f... |
bpoly2 15767 | The Bernoulli polynomials ... |
bpoly3 15768 | The Bernoulli polynomials ... |
bpoly4 15769 | The Bernoulli polynomials ... |
fsumcube 15770 | Express the sum of cubes i... |
eftcl 15783 | Closure of a term in the s... |
reeftcl 15784 | The terms of the series ex... |
eftabs 15785 | The absolute value of a te... |
eftval 15786 | The value of a term in the... |
efcllem 15787 | Lemma for ~ efcl . The se... |
ef0lem 15788 | The series defining the ex... |
efval 15789 | Value of the exponential f... |
esum 15790 | Value of Euler's constant ... |
eff 15791 | Domain and codomain of the... |
efcl 15792 | Closure law for the expone... |
efval2 15793 | Value of the exponential f... |
efcvg 15794 | The series that defines th... |
efcvgfsum 15795 | Exponential function conve... |
reefcl 15796 | The exponential function i... |
reefcld 15797 | The exponential function i... |
ere 15798 | Euler's constant ` _e ` = ... |
ege2le3 15799 | Lemma for ~ egt2lt3 . (Co... |
ef0 15800 | Value of the exponential f... |
efcj 15801 | The exponential of a compl... |
efaddlem 15802 | Lemma for ~ efadd (exponen... |
efadd 15803 | Sum of exponents law for e... |
fprodefsum 15804 | Move the exponential funct... |
efcan 15805 | Cancellation law for expon... |
efne0 15806 | The exponential of a compl... |
efneg 15807 | The exponential of the opp... |
eff2 15808 | The exponential function m... |
efsub 15809 | Difference of exponents la... |
efexp 15810 | The exponential of an inte... |
efzval 15811 | Value of the exponential f... |
efgt0 15812 | The exponential of a real ... |
rpefcl 15813 | The exponential of a real ... |
rpefcld 15814 | The exponential of a real ... |
eftlcvg 15815 | The tail series of the exp... |
eftlcl 15816 | Closure of the sum of an i... |
reeftlcl 15817 | Closure of the sum of an i... |
eftlub 15818 | An upper bound on the abso... |
efsep 15819 | Separate out the next term... |
effsumlt 15820 | The partial sums of the se... |
eft0val 15821 | The value of the first ter... |
ef4p 15822 | Separate out the first fou... |
efgt1p2 15823 | The exponential of a posit... |
efgt1p 15824 | The exponential of a posit... |
efgt1 15825 | The exponential of a posit... |
eflt 15826 | The exponential function o... |
efle 15827 | The exponential function o... |
reef11 15828 | The exponential function o... |
reeff1 15829 | The exponential function m... |
eflegeo 15830 | The exponential function o... |
sinval 15831 | Value of the sine function... |
cosval 15832 | Value of the cosine functi... |
sinf 15833 | Domain and codomain of the... |
cosf 15834 | Domain and codomain of the... |
sincl 15835 | Closure of the sine functi... |
coscl 15836 | Closure of the cosine func... |
tanval 15837 | Value of the tangent funct... |
tancl 15838 | The closure of the tangent... |
sincld 15839 | Closure of the sine functi... |
coscld 15840 | Closure of the cosine func... |
tancld 15841 | Closure of the tangent fun... |
tanval2 15842 | Express the tangent functi... |
tanval3 15843 | Express the tangent functi... |
resinval 15844 | The sine of a real number ... |
recosval 15845 | The cosine of a real numbe... |
efi4p 15846 | Separate out the first fou... |
resin4p 15847 | Separate out the first fou... |
recos4p 15848 | Separate out the first fou... |
resincl 15849 | The sine of a real number ... |
recoscl 15850 | The cosine of a real numbe... |
retancl 15851 | The closure of the tangent... |
resincld 15852 | Closure of the sine functi... |
recoscld 15853 | Closure of the cosine func... |
retancld 15854 | Closure of the tangent fun... |
sinneg 15855 | The sine of a negative is ... |
cosneg 15856 | The cosines of a number an... |
tanneg 15857 | The tangent of a negative ... |
sin0 15858 | Value of the sine function... |
cos0 15859 | Value of the cosine functi... |
tan0 15860 | The value of the tangent f... |
efival 15861 | The exponential function i... |
efmival 15862 | The exponential function i... |
sinhval 15863 | Value of the hyperbolic si... |
coshval 15864 | Value of the hyperbolic co... |
resinhcl 15865 | The hyperbolic sine of a r... |
rpcoshcl 15866 | The hyperbolic cosine of a... |
recoshcl 15867 | The hyperbolic cosine of a... |
retanhcl 15868 | The hyperbolic tangent of ... |
tanhlt1 15869 | The hyperbolic tangent of ... |
tanhbnd 15870 | The hyperbolic tangent of ... |
efeul 15871 | Eulerian representation of... |
efieq 15872 | The exponentials of two im... |
sinadd 15873 | Addition formula for sine.... |
cosadd 15874 | Addition formula for cosin... |
tanaddlem 15875 | A useful intermediate step... |
tanadd 15876 | Addition formula for tange... |
sinsub 15877 | Sine of difference. (Cont... |
cossub 15878 | Cosine of difference. (Co... |
addsin 15879 | Sum of sines. (Contribute... |
subsin 15880 | Difference of sines. (Con... |
sinmul 15881 | Product of sines can be re... |
cosmul 15882 | Product of cosines can be ... |
addcos 15883 | Sum of cosines. (Contribu... |
subcos 15884 | Difference of cosines. (C... |
sincossq 15885 | Sine squared plus cosine s... |
sin2t 15886 | Double-angle formula for s... |
cos2t 15887 | Double-angle formula for c... |
cos2tsin 15888 | Double-angle formula for c... |
sinbnd 15889 | The sine of a real number ... |
cosbnd 15890 | The cosine of a real numbe... |
sinbnd2 15891 | The sine of a real number ... |
cosbnd2 15892 | The cosine of a real numbe... |
ef01bndlem 15893 | Lemma for ~ sin01bnd and ~... |
sin01bnd 15894 | Bounds on the sine of a po... |
cos01bnd 15895 | Bounds on the cosine of a ... |
cos1bnd 15896 | Bounds on the cosine of 1.... |
cos2bnd 15897 | Bounds on the cosine of 2.... |
sinltx 15898 | The sine of a positive rea... |
sin01gt0 15899 | The sine of a positive rea... |
cos01gt0 15900 | The cosine of a positive r... |
sin02gt0 15901 | The sine of a positive rea... |
sincos1sgn 15902 | The signs of the sine and ... |
sincos2sgn 15903 | The signs of the sine and ... |
sin4lt0 15904 | The sine of 4 is negative.... |
absefi 15905 | The absolute value of the ... |
absef 15906 | The absolute value of the ... |
absefib 15907 | A complex number is real i... |
efieq1re 15908 | A number whose imaginary e... |
demoivre 15909 | De Moivre's Formula. Proo... |
demoivreALT 15910 | Alternate proof of ~ demoi... |
eirrlem 15913 | Lemma for ~ eirr . (Contr... |
eirr 15914 | ` _e ` is irrational. (Co... |
egt2lt3 15915 | Euler's constant ` _e ` = ... |
epos 15916 | Euler's constant ` _e ` is... |
epr 15917 | Euler's constant ` _e ` is... |
ene0 15918 | ` _e ` is not 0. (Contrib... |
ene1 15919 | ` _e ` is not 1. (Contrib... |
xpnnen 15920 | The Cartesian product of t... |
znnen 15921 | The set of integers and th... |
qnnen 15922 | The rational numbers are c... |
rpnnen2lem1 15923 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem2 15924 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem3 15925 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem4 15926 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem5 15927 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem6 15928 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem7 15929 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem8 15930 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem9 15931 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem10 15932 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem11 15933 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem12 15934 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2 15935 | The other half of ~ rpnnen... |
rpnnen 15936 | The cardinality of the con... |
rexpen 15937 | The real numbers are equin... |
cpnnen 15938 | The complex numbers are eq... |
rucALT 15939 | Alternate proof of ~ ruc .... |
ruclem1 15940 | Lemma for ~ ruc (the reals... |
ruclem2 15941 | Lemma for ~ ruc . Orderin... |
ruclem3 15942 | Lemma for ~ ruc . The con... |
ruclem4 15943 | Lemma for ~ ruc . Initial... |
ruclem6 15944 | Lemma for ~ ruc . Domain ... |
ruclem7 15945 | Lemma for ~ ruc . Success... |
ruclem8 15946 | Lemma for ~ ruc . The int... |
ruclem9 15947 | Lemma for ~ ruc . The fir... |
ruclem10 15948 | Lemma for ~ ruc . Every f... |
ruclem11 15949 | Lemma for ~ ruc . Closure... |
ruclem12 15950 | Lemma for ~ ruc . The sup... |
ruclem13 15951 | Lemma for ~ ruc . There i... |
ruc 15952 | The set of positive intege... |
resdomq 15953 | The set of rationals is st... |
aleph1re 15954 | There are at least aleph-o... |
aleph1irr 15955 | There are at least aleph-o... |
cnso 15956 | The complex numbers can be... |
sqrt2irrlem 15957 | Lemma for ~ sqrt2irr . Th... |
sqrt2irr 15958 | The square root of 2 is ir... |
sqrt2re 15959 | The square root of 2 exist... |
sqrt2irr0 15960 | The square root of 2 is an... |
nthruc 15961 | The sequence ` NN ` , ` ZZ... |
nthruz 15962 | The sequence ` NN ` , ` NN... |
divides 15965 | Define the divides relatio... |
dvdsval2 15966 | One nonzero integer divide... |
dvdsval3 15967 | One nonzero integer divide... |
dvdszrcl 15968 | Reverse closure for the di... |
dvdsmod0 15969 | If a positive integer divi... |
p1modz1 15970 | If a number greater than 1... |
dvdsmodexp 15971 | If a positive integer divi... |
nndivdvds 15972 | Strong form of ~ dvdsval2 ... |
nndivides 15973 | Definition of the divides ... |
moddvds 15974 | Two ways to say ` A == B `... |
modm1div 15975 | An integer greater than on... |
dvds0lem 15976 | A lemma to assist theorems... |
dvds1lem 15977 | A lemma to assist theorems... |
dvds2lem 15978 | A lemma to assist theorems... |
iddvds 15979 | An integer divides itself.... |
1dvds 15980 | 1 divides any integer. Th... |
dvds0 15981 | Any integer divides 0. Th... |
negdvdsb 15982 | An integer divides another... |
dvdsnegb 15983 | An integer divides another... |
absdvdsb 15984 | An integer divides another... |
dvdsabsb 15985 | An integer divides another... |
0dvds 15986 | Only 0 is divisible by 0. ... |
dvdsmul1 15987 | An integer divides a multi... |
dvdsmul2 15988 | An integer divides a multi... |
iddvdsexp 15989 | An integer divides a posit... |
muldvds1 15990 | If a product divides an in... |
muldvds2 15991 | If a product divides an in... |
dvdscmul 15992 | Multiplication by a consta... |
dvdsmulc 15993 | Multiplication by a consta... |
dvdscmulr 15994 | Cancellation law for the d... |
dvdsmulcr 15995 | Cancellation law for the d... |
summodnegmod 15996 | The sum of two integers mo... |
modmulconst 15997 | Constant multiplication in... |
dvds2ln 15998 | If an integer divides each... |
dvds2add 15999 | If an integer divides each... |
dvds2sub 16000 | If an integer divides each... |
dvds2addd 16001 | Deduction form of ~ dvds2a... |
dvds2subd 16002 | Deduction form of ~ dvds2s... |
dvdstr 16003 | The divides relation is tr... |
dvdstrd 16004 | The divides relation is tr... |
dvdsmultr1 16005 | If an integer divides anot... |
dvdsmultr1d 16006 | Deduction form of ~ dvdsmu... |
dvdsmultr2 16007 | If an integer divides anot... |
dvdsmultr2d 16008 | Deduction form of ~ dvdsmu... |
ordvdsmul 16009 | If an integer divides eith... |
dvdssub2 16010 | If an integer divides a di... |
dvdsadd 16011 | An integer divides another... |
dvdsaddr 16012 | An integer divides another... |
dvdssub 16013 | An integer divides another... |
dvdssubr 16014 | An integer divides another... |
dvdsadd2b 16015 | Adding a multiple of the b... |
dvdsaddre2b 16016 | Adding a multiple of the b... |
fsumdvds 16017 | If every term in a sum is ... |
dvdslelem 16018 | Lemma for ~ dvdsle . (Con... |
dvdsle 16019 | The divisors of a positive... |
dvdsleabs 16020 | The divisors of a nonzero ... |
dvdsleabs2 16021 | Transfer divisibility to a... |
dvdsabseq 16022 | If two integers divide eac... |
dvdseq 16023 | If two nonnegative integer... |
divconjdvds 16024 | If a nonzero integer ` M `... |
dvdsdivcl 16025 | The complement of a diviso... |
dvdsflip 16026 | An involution of the divis... |
dvdsssfz1 16027 | The set of divisors of a n... |
dvds1 16028 | The only nonnegative integ... |
alzdvds 16029 | Only 0 is divisible by all... |
dvdsext 16030 | Poset extensionality for d... |
fzm1ndvds 16031 | No number between ` 1 ` an... |
fzo0dvdseq 16032 | Zero is the only one of th... |
fzocongeq 16033 | Two different elements of ... |
addmodlteqALT 16034 | Two nonnegative integers l... |
dvdsfac 16035 | A positive integer divides... |
dvdsexp2im 16036 | If an integer divides anot... |
dvdsexp 16037 | A power divides a power wi... |
dvdsmod 16038 | Any number ` K ` whose mod... |
mulmoddvds 16039 | If an integer is divisible... |
3dvds 16040 | A rule for divisibility by... |
3dvdsdec 16041 | A decimal number is divisi... |
3dvds2dec 16042 | A decimal number is divisi... |
fprodfvdvdsd 16043 | A finite product of intege... |
fproddvdsd 16044 | A finite product of intege... |
evenelz 16045 | An even number is an integ... |
zeo3 16046 | An integer is even or odd.... |
zeo4 16047 | An integer is even or odd ... |
zeneo 16048 | No even integer equals an ... |
odd2np1lem 16049 | Lemma for ~ odd2np1 . (Co... |
odd2np1 16050 | An integer is odd iff it i... |
even2n 16051 | An integer is even iff it ... |
oddm1even 16052 | An integer is odd iff its ... |
oddp1even 16053 | An integer is odd iff its ... |
oexpneg 16054 | The exponential of the neg... |
mod2eq0even 16055 | An integer is 0 modulo 2 i... |
mod2eq1n2dvds 16056 | An integer is 1 modulo 2 i... |
oddnn02np1 16057 | A nonnegative integer is o... |
oddge22np1 16058 | An integer greater than on... |
evennn02n 16059 | A nonnegative integer is e... |
evennn2n 16060 | A positive integer is even... |
2tp1odd 16061 | A number which is twice an... |
mulsucdiv2z 16062 | An integer multiplied with... |
sqoddm1div8z 16063 | A squared odd number minus... |
2teven 16064 | A number which is twice an... |
zeo5 16065 | An integer is either even ... |
evend2 16066 | An integer is even iff its... |
oddp1d2 16067 | An integer is odd iff its ... |
zob 16068 | Alternate characterization... |
oddm1d2 16069 | An integer is odd iff its ... |
ltoddhalfle 16070 | An integer is less than ha... |
halfleoddlt 16071 | An integer is greater than... |
opoe 16072 | The sum of two odds is eve... |
omoe 16073 | The difference of two odds... |
opeo 16074 | The sum of an odd and an e... |
omeo 16075 | The difference of an odd a... |
z0even 16076 | 2 divides 0. That means 0... |
n2dvds1 16077 | 2 does not divide 1. That... |
n2dvdsm1 16078 | 2 does not divide -1. Tha... |
z2even 16079 | 2 divides 2. That means 2... |
n2dvds3 16080 | 2 does not divide 3. That... |
z4even 16081 | 2 divides 4. That means 4... |
4dvdseven 16082 | An integer which is divisi... |
m1expe 16083 | Exponentiation of -1 by an... |
m1expo 16084 | Exponentiation of -1 by an... |
m1exp1 16085 | Exponentiation of negative... |
nn0enne 16086 | A positive integer is an e... |
nn0ehalf 16087 | The half of an even nonneg... |
nnehalf 16088 | The half of an even positi... |
nn0onn 16089 | An odd nonnegative integer... |
nn0o1gt2 16090 | An odd nonnegative integer... |
nno 16091 | An alternate characterizat... |
nn0o 16092 | An alternate characterizat... |
nn0ob 16093 | Alternate characterization... |
nn0oddm1d2 16094 | A positive integer is odd ... |
nnoddm1d2 16095 | A positive integer is odd ... |
sumeven 16096 | If every term in a sum is ... |
sumodd 16097 | If every term in a sum is ... |
evensumodd 16098 | If every term in a sum wit... |
oddsumodd 16099 | If every term in a sum wit... |
pwp1fsum 16100 | The n-th power of a number... |
oddpwp1fsum 16101 | An odd power of a number i... |
divalglem0 16102 | Lemma for ~ divalg . (Con... |
divalglem1 16103 | Lemma for ~ divalg . (Con... |
divalglem2 16104 | Lemma for ~ divalg . (Con... |
divalglem4 16105 | Lemma for ~ divalg . (Con... |
divalglem5 16106 | Lemma for ~ divalg . (Con... |
divalglem6 16107 | Lemma for ~ divalg . (Con... |
divalglem7 16108 | Lemma for ~ divalg . (Con... |
divalglem8 16109 | Lemma for ~ divalg . (Con... |
divalglem9 16110 | Lemma for ~ divalg . (Con... |
divalglem10 16111 | Lemma for ~ divalg . (Con... |
divalg 16112 | The division algorithm (th... |
divalgb 16113 | Express the division algor... |
divalg2 16114 | The division algorithm (th... |
divalgmod 16115 | The result of the ` mod ` ... |
divalgmodcl 16116 | The result of the ` mod ` ... |
modremain 16117 | The result of the modulo o... |
ndvdssub 16118 | Corollary of the division ... |
ndvdsadd 16119 | Corollary of the division ... |
ndvdsp1 16120 | Special case of ~ ndvdsadd... |
ndvdsi 16121 | A quick test for non-divis... |
flodddiv4 16122 | The floor of an odd intege... |
fldivndvdslt 16123 | The floor of an integer di... |
flodddiv4lt 16124 | The floor of an odd number... |
flodddiv4t2lthalf 16125 | The floor of an odd number... |
bitsfval 16130 | Expand the definition of t... |
bitsval 16131 | Expand the definition of t... |
bitsval2 16132 | Expand the definition of t... |
bitsss 16133 | The set of bits of an inte... |
bitsf 16134 | The ` bits ` function is a... |
bits0 16135 | Value of the zeroth bit. ... |
bits0e 16136 | The zeroth bit of an even ... |
bits0o 16137 | The zeroth bit of an odd n... |
bitsp1 16138 | The ` M + 1 ` -th bit of `... |
bitsp1e 16139 | The ` M + 1 ` -th bit of `... |
bitsp1o 16140 | The ` M + 1 ` -th bit of `... |
bitsfzolem 16141 | Lemma for ~ bitsfzo . (Co... |
bitsfzo 16142 | The bits of a number are a... |
bitsmod 16143 | Truncating the bit sequenc... |
bitsfi 16144 | Every number is associated... |
bitscmp 16145 | The bit complement of ` N ... |
0bits 16146 | The bits of zero. (Contri... |
m1bits 16147 | The bits of negative one. ... |
bitsinv1lem 16148 | Lemma for ~ bitsinv1 . (C... |
bitsinv1 16149 | There is an explicit inver... |
bitsinv2 16150 | There is an explicit inver... |
bitsf1ocnv 16151 | The ` bits ` function rest... |
bitsf1o 16152 | The ` bits ` function rest... |
bitsf1 16153 | The ` bits ` function is a... |
2ebits 16154 | The bits of a power of two... |
bitsinv 16155 | The inverse of the ` bits ... |
bitsinvp1 16156 | Recursive definition of th... |
sadadd2lem2 16157 | The core of the proof of ~... |
sadfval 16159 | Define the addition of two... |
sadcf 16160 | The carry sequence is a se... |
sadc0 16161 | The initial element of the... |
sadcp1 16162 | The carry sequence (which ... |
sadval 16163 | The full adder sequence is... |
sadcaddlem 16164 | Lemma for ~ sadcadd . (Co... |
sadcadd 16165 | Non-recursive definition o... |
sadadd2lem 16166 | Lemma for ~ sadadd2 . (Co... |
sadadd2 16167 | Sum of initial segments of... |
sadadd3 16168 | Sum of initial segments of... |
sadcl 16169 | The sum of two sequences i... |
sadcom 16170 | The adder sequence functio... |
saddisjlem 16171 | Lemma for ~ sadadd . (Con... |
saddisj 16172 | The sum of disjoint sequen... |
sadaddlem 16173 | Lemma for ~ sadadd . (Con... |
sadadd 16174 | For sequences that corresp... |
sadid1 16175 | The adder sequence functio... |
sadid2 16176 | The adder sequence functio... |
sadasslem 16177 | Lemma for ~ sadass . (Con... |
sadass 16178 | Sequence addition is assoc... |
sadeq 16179 | Any element of a sequence ... |
bitsres 16180 | Restrict the bits of a num... |
bitsuz 16181 | The bits of a number are a... |
bitsshft 16182 | Shifting a bit sequence to... |
smufval 16184 | The multiplication of two ... |
smupf 16185 | The sequence of partial su... |
smup0 16186 | The initial element of the... |
smupp1 16187 | The initial element of the... |
smuval 16188 | Define the addition of two... |
smuval2 16189 | The partial sum sequence s... |
smupvallem 16190 | If ` A ` only has elements... |
smucl 16191 | The product of two sequenc... |
smu01lem 16192 | Lemma for ~ smu01 and ~ sm... |
smu01 16193 | Multiplication of a sequen... |
smu02 16194 | Multiplication of a sequen... |
smupval 16195 | Rewrite the elements of th... |
smup1 16196 | Rewrite ~ smupp1 using onl... |
smueqlem 16197 | Any element of a sequence ... |
smueq 16198 | Any element of a sequence ... |
smumullem 16199 | Lemma for ~ smumul . (Con... |
smumul 16200 | For sequences that corresp... |
gcdval 16203 | The value of the ` gcd ` o... |
gcd0val 16204 | The value, by convention, ... |
gcdn0val 16205 | The value of the ` gcd ` o... |
gcdcllem1 16206 | Lemma for ~ gcdn0cl , ~ gc... |
gcdcllem2 16207 | Lemma for ~ gcdn0cl , ~ gc... |
gcdcllem3 16208 | Lemma for ~ gcdn0cl , ~ gc... |
gcdn0cl 16209 | Closure of the ` gcd ` ope... |
gcddvds 16210 | The gcd of two integers di... |
dvdslegcd 16211 | An integer which divides b... |
nndvdslegcd 16212 | A positive integer which d... |
gcdcl 16213 | Closure of the ` gcd ` ope... |
gcdnncl 16214 | Closure of the ` gcd ` ope... |
gcdcld 16215 | Closure of the ` gcd ` ope... |
gcd2n0cl 16216 | Closure of the ` gcd ` ope... |
zeqzmulgcd 16217 | An integer is the product ... |
divgcdz 16218 | An integer divided by the ... |
gcdf 16219 | Domain and codomain of the... |
gcdcom 16220 | The ` gcd ` operator is co... |
gcdcomd 16221 | The ` gcd ` operator is co... |
divgcdnn 16222 | A positive integer divided... |
divgcdnnr 16223 | A positive integer divided... |
gcdeq0 16224 | The gcd of two integers is... |
gcdn0gt0 16225 | The gcd of two integers is... |
gcd0id 16226 | The gcd of 0 and an intege... |
gcdid0 16227 | The gcd of an integer and ... |
nn0gcdid0 16228 | The gcd of a nonnegative i... |
gcdneg 16229 | Negating one operand of th... |
neggcd 16230 | Negating one operand of th... |
gcdaddmlem 16231 | Lemma for ~ gcdaddm . (Co... |
gcdaddm 16232 | Adding a multiple of one o... |
gcdadd 16233 | The GCD of two numbers is ... |
gcdid 16234 | The gcd of a number and it... |
gcd1 16235 | The gcd of a number with 1... |
gcdabs1 16236 | ` gcd ` of the absolute va... |
gcdabs2 16237 | ` gcd ` of the absolute va... |
gcdabs 16238 | The gcd of two integers is... |
gcdabsOLD 16239 | Obsolete version of ~ gcda... |
modgcd 16240 | The gcd remains unchanged ... |
1gcd 16241 | The GCD of one and an inte... |
gcdmultipled 16242 | The greatest common diviso... |
gcdmultiplez 16243 | The GCD of a multiple of a... |
gcdmultiple 16244 | The GCD of a multiple of a... |
dvdsgcdidd 16245 | The greatest common diviso... |
6gcd4e2 16246 | The greatest common diviso... |
bezoutlem1 16247 | Lemma for ~ bezout . (Con... |
bezoutlem2 16248 | Lemma for ~ bezout . (Con... |
bezoutlem3 16249 | Lemma for ~ bezout . (Con... |
bezoutlem4 16250 | Lemma for ~ bezout . (Con... |
bezout 16251 | Bézout's identity: ... |
dvdsgcd 16252 | An integer which divides e... |
dvdsgcdb 16253 | Biconditional form of ~ dv... |
dfgcd2 16254 | Alternate definition of th... |
gcdass 16255 | Associative law for ` gcd ... |
mulgcd 16256 | Distribute multiplication ... |
absmulgcd 16257 | Distribute absolute value ... |
mulgcdr 16258 | Reverse distribution law f... |
gcddiv 16259 | Division law for GCD. (Con... |
gcdmultipleOLD 16260 | Obsolete proof of ~ gcdmul... |
gcdmultiplezOLD 16261 | Obsolete proof of ~ gcdmul... |
gcdzeq 16262 | A positive integer ` A ` i... |
gcdeq 16263 | ` A ` is equal to its gcd ... |
dvdssqim 16264 | Unidirectional form of ~ d... |
dvdsmulgcd 16265 | A divisibility equivalent ... |
rpmulgcd 16266 | If ` K ` and ` M ` are rel... |
rplpwr 16267 | If ` A ` and ` B ` are rel... |
rprpwr 16268 | If ` A ` and ` B ` are rel... |
rppwr 16269 | If ` A ` and ` B ` are rel... |
sqgcd 16270 | Square distributes over gc... |
dvdssqlem 16271 | Lemma for ~ dvdssq . (Con... |
dvdssq 16272 | Two numbers are divisible ... |
bezoutr 16273 | Partial converse to ~ bezo... |
bezoutr1 16274 | Converse of ~ bezout for w... |
nn0seqcvgd 16275 | A strictly-decreasing nonn... |
seq1st 16276 | A sequence whose iteration... |
algr0 16277 | The value of the algorithm... |
algrf 16278 | An algorithm is a step fun... |
algrp1 16279 | The value of the algorithm... |
alginv 16280 | If ` I ` is an invariant o... |
algcvg 16281 | One way to prove that an a... |
algcvgblem 16282 | Lemma for ~ algcvgb . (Co... |
algcvgb 16283 | Two ways of expressing tha... |
algcvga 16284 | The countdown function ` C... |
algfx 16285 | If ` F ` reaches a fixed p... |
eucalgval2 16286 | The value of the step func... |
eucalgval 16287 | Euclid's Algorithm ~ eucal... |
eucalgf 16288 | Domain and codomain of the... |
eucalginv 16289 | The invariant of the step ... |
eucalglt 16290 | The second member of the s... |
eucalgcvga 16291 | Once Euclid's Algorithm ha... |
eucalg 16292 | Euclid's Algorithm compute... |
lcmval 16297 | Value of the ` lcm ` opera... |
lcmcom 16298 | The ` lcm ` operator is co... |
lcm0val 16299 | The value, by convention, ... |
lcmn0val 16300 | The value of the ` lcm ` o... |
lcmcllem 16301 | Lemma for ~ lcmn0cl and ~ ... |
lcmn0cl 16302 | Closure of the ` lcm ` ope... |
dvdslcm 16303 | The lcm of two integers is... |
lcmledvds 16304 | A positive integer which b... |
lcmeq0 16305 | The lcm of two integers is... |
lcmcl 16306 | Closure of the ` lcm ` ope... |
gcddvdslcm 16307 | The greatest common diviso... |
lcmneg 16308 | Negating one operand of th... |
neglcm 16309 | Negating one operand of th... |
lcmabs 16310 | The lcm of two integers is... |
lcmgcdlem 16311 | Lemma for ~ lcmgcd and ~ l... |
lcmgcd 16312 | The product of two numbers... |
lcmdvds 16313 | The lcm of two integers di... |
lcmid 16314 | The lcm of an integer and ... |
lcm1 16315 | The lcm of an integer and ... |
lcmgcdnn 16316 | The product of two positiv... |
lcmgcdeq 16317 | Two integers' absolute val... |
lcmdvdsb 16318 | Biconditional form of ~ lc... |
lcmass 16319 | Associative law for ` lcm ... |
3lcm2e6woprm 16320 | The least common multiple ... |
6lcm4e12 16321 | The least common multiple ... |
absproddvds 16322 | The absolute value of the ... |
absprodnn 16323 | The absolute value of the ... |
fissn0dvds 16324 | For each finite subset of ... |
fissn0dvdsn0 16325 | For each finite subset of ... |
lcmfval 16326 | Value of the ` _lcm ` func... |
lcmf0val 16327 | The value, by convention, ... |
lcmfn0val 16328 | The value of the ` _lcm ` ... |
lcmfnnval 16329 | The value of the ` _lcm ` ... |
lcmfcllem 16330 | Lemma for ~ lcmfn0cl and ~... |
lcmfn0cl 16331 | Closure of the ` _lcm ` fu... |
lcmfpr 16332 | The value of the ` _lcm ` ... |
lcmfcl 16333 | Closure of the ` _lcm ` fu... |
lcmfnncl 16334 | Closure of the ` _lcm ` fu... |
lcmfeq0b 16335 | The least common multiple ... |
dvdslcmf 16336 | The least common multiple ... |
lcmfledvds 16337 | A positive integer which i... |
lcmf 16338 | Characterization of the le... |
lcmf0 16339 | The least common multiple ... |
lcmfsn 16340 | The least common multiple ... |
lcmftp 16341 | The least common multiple ... |
lcmfunsnlem1 16342 | Lemma for ~ lcmfdvds and ~... |
lcmfunsnlem2lem1 16343 | Lemma 1 for ~ lcmfunsnlem2... |
lcmfunsnlem2lem2 16344 | Lemma 2 for ~ lcmfunsnlem2... |
lcmfunsnlem2 16345 | Lemma for ~ lcmfunsn and ~... |
lcmfunsnlem 16346 | Lemma for ~ lcmfdvds and ~... |
lcmfdvds 16347 | The least common multiple ... |
lcmfdvdsb 16348 | Biconditional form of ~ lc... |
lcmfunsn 16349 | The ` _lcm ` function for ... |
lcmfun 16350 | The ` _lcm ` function for ... |
lcmfass 16351 | Associative law for the ` ... |
lcmf2a3a4e12 16352 | The least common multiple ... |
lcmflefac 16353 | The least common multiple ... |
coprmgcdb 16354 | Two positive integers are ... |
ncoprmgcdne1b 16355 | Two positive integers are ... |
ncoprmgcdgt1b 16356 | Two positive integers are ... |
coprmdvds1 16357 | If two positive integers a... |
coprmdvds 16358 | Euclid's Lemma (see ProofW... |
coprmdvds2 16359 | If an integer is divisible... |
mulgcddvds 16360 | One half of ~ rpmulgcd2 , ... |
rpmulgcd2 16361 | If ` M ` is relatively pri... |
qredeq 16362 | Two equal reduced fraction... |
qredeu 16363 | Every rational number has ... |
rpmul 16364 | If ` K ` is relatively pri... |
rpdvds 16365 | If ` K ` is relatively pri... |
coprmprod 16366 | The product of the element... |
coprmproddvdslem 16367 | Lemma for ~ coprmproddvds ... |
coprmproddvds 16368 | If a positive integer is d... |
congr 16369 | Definition of congruence b... |
divgcdcoprm0 16370 | Integers divided by gcd ar... |
divgcdcoprmex 16371 | Integers divided by gcd ar... |
cncongr1 16372 | One direction of the bicon... |
cncongr2 16373 | The other direction of the... |
cncongr 16374 | Cancellability of Congruen... |
cncongrcoprm 16375 | Corollary 1 of Cancellabil... |
isprm 16378 | The predicate "is a prime ... |
prmnn 16379 | A prime number is a positi... |
prmz 16380 | A prime number is an integ... |
prmssnn 16381 | The prime numbers are a su... |
prmex 16382 | The set of prime numbers e... |
0nprm 16383 | 0 is not a prime number. ... |
1nprm 16384 | 1 is not a prime number. ... |
1idssfct 16385 | The positive divisors of a... |
isprm2lem 16386 | Lemma for ~ isprm2 . (Con... |
isprm2 16387 | The predicate "is a prime ... |
isprm3 16388 | The predicate "is a prime ... |
isprm4 16389 | The predicate "is a prime ... |
prmind2 16390 | A variation on ~ prmind as... |
prmind 16391 | Perform induction over the... |
dvdsprime 16392 | If ` M ` divides a prime, ... |
nprm 16393 | A product of two integers ... |
nprmi 16394 | An inference for composite... |
dvdsnprmd 16395 | If a number is divisible b... |
prm2orodd 16396 | A prime number is either 2... |
2prm 16397 | 2 is a prime number. (Con... |
2mulprm 16398 | A multiple of two is prime... |
3prm 16399 | 3 is a prime number. (Con... |
4nprm 16400 | 4 is not a prime number. ... |
prmuz2 16401 | A prime number is an integ... |
prmgt1 16402 | A prime number is an integ... |
prmm2nn0 16403 | Subtracting 2 from a prime... |
oddprmgt2 16404 | An odd prime is greater th... |
oddprmge3 16405 | An odd prime is greater th... |
ge2nprmge4 16406 | A composite integer greate... |
sqnprm 16407 | A square is never prime. ... |
dvdsprm 16408 | An integer greater than or... |
exprmfct 16409 | Every integer greater than... |
prmdvdsfz 16410 | Each integer greater than ... |
nprmdvds1 16411 | No prime number divides 1.... |
isprm5 16412 | One need only check prime ... |
isprm7 16413 | One need only check prime ... |
maxprmfct 16414 | The set of prime factors o... |
divgcdodd 16415 | Either ` A / ( A gcd B ) `... |
coprm 16416 | A prime number either divi... |
prmrp 16417 | Unequal prime numbers are ... |
euclemma 16418 | Euclid's lemma. A prime n... |
isprm6 16419 | A number is prime iff it s... |
prmdvdsexp 16420 | A prime divides a positive... |
prmdvdsexpb 16421 | A prime divides a positive... |
prmdvdsexpr 16422 | If a prime divides a nonne... |
prmdvdssq 16423 | Condition for a prime divi... |
prmdvdssqOLD 16424 | Obsolete version of ~ prmd... |
prmexpb 16425 | Two positive prime powers ... |
prmfac1 16426 | The factorial of a number ... |
rpexp 16427 | If two numbers ` A ` and `... |
rpexp1i 16428 | Relative primality passes ... |
rpexp12i 16429 | Relative primality passes ... |
prmndvdsfaclt 16430 | A prime number does not di... |
prmdvdsncoprmbd 16431 | Two positive integers are ... |
ncoprmlnprm 16432 | If two positive integers a... |
cncongrprm 16433 | Corollary 2 of Cancellabil... |
isevengcd2 16434 | The predicate "is an even ... |
isoddgcd1 16435 | The predicate "is an odd n... |
3lcm2e6 16436 | The least common multiple ... |
qnumval 16441 | Value of the canonical num... |
qdenval 16442 | Value of the canonical den... |
qnumdencl 16443 | Lemma for ~ qnumcl and ~ q... |
qnumcl 16444 | The canonical numerator of... |
qdencl 16445 | The canonical denominator ... |
fnum 16446 | Canonical numerator define... |
fden 16447 | Canonical denominator defi... |
qnumdenbi 16448 | Two numbers are the canoni... |
qnumdencoprm 16449 | The canonical representati... |
qeqnumdivden 16450 | Recover a rational number ... |
qmuldeneqnum 16451 | Multiplying a rational by ... |
divnumden 16452 | Calculate the reduced form... |
divdenle 16453 | Reducing a quotient never ... |
qnumgt0 16454 | A rational is positive iff... |
qgt0numnn 16455 | A rational is positive iff... |
nn0gcdsq 16456 | Squaring commutes with GCD... |
zgcdsq 16457 | ~ nn0gcdsq extended to int... |
numdensq 16458 | Squaring a rational square... |
numsq 16459 | Square commutes with canon... |
densq 16460 | Square commutes with canon... |
qden1elz 16461 | A rational is an integer i... |
zsqrtelqelz 16462 | If an integer has a ration... |
nonsq 16463 | Any integer strictly betwe... |
phival 16468 | Value of the Euler ` phi `... |
phicl2 16469 | Bounds and closure for the... |
phicl 16470 | Closure for the value of t... |
phibndlem 16471 | Lemma for ~ phibnd . (Con... |
phibnd 16472 | A slightly tighter bound o... |
phicld 16473 | Closure for the value of t... |
phi1 16474 | Value of the Euler ` phi `... |
dfphi2 16475 | Alternate definition of th... |
hashdvds 16476 | The number of numbers in a... |
phiprmpw 16477 | Value of the Euler ` phi `... |
phiprm 16478 | Value of the Euler ` phi `... |
crth 16479 | The Chinese Remainder Theo... |
phimullem 16480 | Lemma for ~ phimul . (Con... |
phimul 16481 | The Euler ` phi ` function... |
eulerthlem1 16482 | Lemma for ~ eulerth . (Co... |
eulerthlem2 16483 | Lemma for ~ eulerth . (Co... |
eulerth 16484 | Euler's theorem, a general... |
fermltl 16485 | Fermat's little theorem. ... |
prmdiv 16486 | Show an explicit expressio... |
prmdiveq 16487 | The modular inverse of ` A... |
prmdivdiv 16488 | The (modular) inverse of t... |
hashgcdlem 16489 | A correspondence between e... |
hashgcdeq 16490 | Number of initial positive... |
phisum 16491 | The divisor sum identity o... |
odzval 16492 | Value of the order functio... |
odzcllem 16493 | - Lemma for ~ odzcl , show... |
odzcl 16494 | The order of a group eleme... |
odzid 16495 | Any element raised to the ... |
odzdvds 16496 | The only powers of ` A ` t... |
odzphi 16497 | The order of any group ele... |
modprm1div 16498 | A prime number divides an ... |
m1dvdsndvds 16499 | If an integer minus 1 is d... |
modprminv 16500 | Show an explicit expressio... |
modprminveq 16501 | The modular inverse of ` A... |
vfermltl 16502 | Variant of Fermat's little... |
vfermltlALT 16503 | Alternate proof of ~ vferm... |
powm2modprm 16504 | If an integer minus 1 is d... |
reumodprminv 16505 | For any prime number and f... |
modprm0 16506 | For two positive integers ... |
nnnn0modprm0 16507 | For a positive integer and... |
modprmn0modprm0 16508 | For an integer not being 0... |
coprimeprodsq 16509 | If three numbers are copri... |
coprimeprodsq2 16510 | If three numbers are copri... |
oddprm 16511 | A prime not equal to ` 2 `... |
nnoddn2prm 16512 | A prime not equal to ` 2 `... |
oddn2prm 16513 | A prime not equal to ` 2 `... |
nnoddn2prmb 16514 | A number is a prime number... |
prm23lt5 16515 | A prime less than 5 is eit... |
prm23ge5 16516 | A prime is either 2 or 3 o... |
pythagtriplem1 16517 | Lemma for ~ pythagtrip . ... |
pythagtriplem2 16518 | Lemma for ~ pythagtrip . ... |
pythagtriplem3 16519 | Lemma for ~ pythagtrip . ... |
pythagtriplem4 16520 | Lemma for ~ pythagtrip . ... |
pythagtriplem10 16521 | Lemma for ~ pythagtrip . ... |
pythagtriplem6 16522 | Lemma for ~ pythagtrip . ... |
pythagtriplem7 16523 | Lemma for ~ pythagtrip . ... |
pythagtriplem8 16524 | Lemma for ~ pythagtrip . ... |
pythagtriplem9 16525 | Lemma for ~ pythagtrip . ... |
pythagtriplem11 16526 | Lemma for ~ pythagtrip . ... |
pythagtriplem12 16527 | Lemma for ~ pythagtrip . ... |
pythagtriplem13 16528 | Lemma for ~ pythagtrip . ... |
pythagtriplem14 16529 | Lemma for ~ pythagtrip . ... |
pythagtriplem15 16530 | Lemma for ~ pythagtrip . ... |
pythagtriplem16 16531 | Lemma for ~ pythagtrip . ... |
pythagtriplem17 16532 | Lemma for ~ pythagtrip . ... |
pythagtriplem18 16533 | Lemma for ~ pythagtrip . ... |
pythagtriplem19 16534 | Lemma for ~ pythagtrip . ... |
pythagtrip 16535 | Parameterize the Pythagore... |
iserodd 16536 | Collect the odd terms in a... |
pclem 16539 | - Lemma for the prime powe... |
pcprecl 16540 | Closure of the prime power... |
pcprendvds 16541 | Non-divisibility property ... |
pcprendvds2 16542 | Non-divisibility property ... |
pcpre1 16543 | Value of the prime power p... |
pcpremul 16544 | Multiplicative property of... |
pcval 16545 | The value of the prime pow... |
pceulem 16546 | Lemma for ~ pceu . (Contr... |
pceu 16547 | Uniqueness for the prime p... |
pczpre 16548 | Connect the prime count pr... |
pczcl 16549 | Closure of the prime power... |
pccl 16550 | Closure of the prime power... |
pccld 16551 | Closure of the prime power... |
pcmul 16552 | Multiplication property of... |
pcdiv 16553 | Division property of the p... |
pcqmul 16554 | Multiplication property of... |
pc0 16555 | The value of the prime pow... |
pc1 16556 | Value of the prime count f... |
pcqcl 16557 | Closure of the general pri... |
pcqdiv 16558 | Division property of the p... |
pcrec 16559 | Prime power of a reciproca... |
pcexp 16560 | Prime power of an exponent... |
pcxnn0cl 16561 | Extended nonnegative integ... |
pcxcl 16562 | Extended real closure of t... |
pcge0 16563 | The prime count of an inte... |
pczdvds 16564 | Defining property of the p... |
pcdvds 16565 | Defining property of the p... |
pczndvds 16566 | Defining property of the p... |
pcndvds 16567 | Defining property of the p... |
pczndvds2 16568 | The remainder after dividi... |
pcndvds2 16569 | The remainder after dividi... |
pcdvdsb 16570 | ` P ^ A ` divides ` N ` if... |
pcelnn 16571 | There are a positive numbe... |
pceq0 16572 | There are zero powers of a... |
pcidlem 16573 | The prime count of a prime... |
pcid 16574 | The prime count of a prime... |
pcneg 16575 | The prime count of a negat... |
pcabs 16576 | The prime count of an abso... |
pcdvdstr 16577 | The prime count increases ... |
pcgcd1 16578 | The prime count of a GCD i... |
pcgcd 16579 | The prime count of a GCD i... |
pc2dvds 16580 | A characterization of divi... |
pc11 16581 | The prime count function, ... |
pcz 16582 | The prime count function c... |
pcprmpw2 16583 | Self-referential expressio... |
pcprmpw 16584 | Self-referential expressio... |
dvdsprmpweq 16585 | If a positive integer divi... |
dvdsprmpweqnn 16586 | If an integer greater than... |
dvdsprmpweqle 16587 | If a positive integer divi... |
difsqpwdvds 16588 | If the difference of two s... |
pcaddlem 16589 | Lemma for ~ pcadd . The o... |
pcadd 16590 | An inequality for the prim... |
pcadd2 16591 | The inequality of ~ pcadd ... |
pcmptcl 16592 | Closure for the prime powe... |
pcmpt 16593 | Construct a function with ... |
pcmpt2 16594 | Dividing two prime count m... |
pcmptdvds 16595 | The partial products of th... |
pcprod 16596 | The product of the primes ... |
sumhash 16597 | The sum of 1 over a set is... |
fldivp1 16598 | The difference between the... |
pcfaclem 16599 | Lemma for ~ pcfac . (Cont... |
pcfac 16600 | Calculate the prime count ... |
pcbc 16601 | Calculate the prime count ... |
qexpz 16602 | If a power of a rational n... |
expnprm 16603 | A second or higher power o... |
oddprmdvds 16604 | Every positive integer whi... |
prmpwdvds 16605 | A relation involving divis... |
pockthlem 16606 | Lemma for ~ pockthg . (Co... |
pockthg 16607 | The generalized Pocklingto... |
pockthi 16608 | Pocklington's theorem, whi... |
unbenlem 16609 | Lemma for ~ unben . (Cont... |
unben 16610 | An unbounded set of positi... |
infpnlem1 16611 | Lemma for ~ infpn . The s... |
infpnlem2 16612 | Lemma for ~ infpn . For a... |
infpn 16613 | There exist infinitely man... |
infpn2 16614 | There exist infinitely man... |
prmunb 16615 | The primes are unbounded. ... |
prminf 16616 | There are an infinite numb... |
prmreclem1 16617 | Lemma for ~ prmrec . Prop... |
prmreclem2 16618 | Lemma for ~ prmrec . Ther... |
prmreclem3 16619 | Lemma for ~ prmrec . The ... |
prmreclem4 16620 | Lemma for ~ prmrec . Show... |
prmreclem5 16621 | Lemma for ~ prmrec . Here... |
prmreclem6 16622 | Lemma for ~ prmrec . If t... |
prmrec 16623 | The sum of the reciprocals... |
1arithlem1 16624 | Lemma for ~ 1arith . (Con... |
1arithlem2 16625 | Lemma for ~ 1arith . (Con... |
1arithlem3 16626 | Lemma for ~ 1arith . (Con... |
1arithlem4 16627 | Lemma for ~ 1arith . (Con... |
1arith 16628 | Fundamental theorem of ari... |
1arith2 16629 | Fundamental theorem of ari... |
elgz 16632 | Elementhood in the gaussia... |
gzcn 16633 | A gaussian integer is a co... |
zgz 16634 | An integer is a gaussian i... |
igz 16635 | ` _i ` is a gaussian integ... |
gznegcl 16636 | The gaussian integers are ... |
gzcjcl 16637 | The gaussian integers are ... |
gzaddcl 16638 | The gaussian integers are ... |
gzmulcl 16639 | The gaussian integers are ... |
gzreim 16640 | Construct a gaussian integ... |
gzsubcl 16641 | The gaussian integers are ... |
gzabssqcl 16642 | The squared norm of a gaus... |
4sqlem5 16643 | Lemma for ~ 4sq . (Contri... |
4sqlem6 16644 | Lemma for ~ 4sq . (Contri... |
4sqlem7 16645 | Lemma for ~ 4sq . (Contri... |
4sqlem8 16646 | Lemma for ~ 4sq . (Contri... |
4sqlem9 16647 | Lemma for ~ 4sq . (Contri... |
4sqlem10 16648 | Lemma for ~ 4sq . (Contri... |
4sqlem1 16649 | Lemma for ~ 4sq . The set... |
4sqlem2 16650 | Lemma for ~ 4sq . Change ... |
4sqlem3 16651 | Lemma for ~ 4sq . Suffici... |
4sqlem4a 16652 | Lemma for ~ 4sqlem4 . (Co... |
4sqlem4 16653 | Lemma for ~ 4sq . We can ... |
mul4sqlem 16654 | Lemma for ~ mul4sq : algeb... |
mul4sq 16655 | Euler's four-square identi... |
4sqlem11 16656 | Lemma for ~ 4sq . Use the... |
4sqlem12 16657 | Lemma for ~ 4sq . For any... |
4sqlem13 16658 | Lemma for ~ 4sq . (Contri... |
4sqlem14 16659 | Lemma for ~ 4sq . (Contri... |
4sqlem15 16660 | Lemma for ~ 4sq . (Contri... |
4sqlem16 16661 | Lemma for ~ 4sq . (Contri... |
4sqlem17 16662 | Lemma for ~ 4sq . (Contri... |
4sqlem18 16663 | Lemma for ~ 4sq . Inducti... |
4sqlem19 16664 | Lemma for ~ 4sq . The pro... |
4sq 16665 | Lagrange's four-square the... |
vdwapfval 16672 | Define the arithmetic prog... |
vdwapf 16673 | The arithmetic progression... |
vdwapval 16674 | Value of the arithmetic pr... |
vdwapun 16675 | Remove the first element o... |
vdwapid1 16676 | The first element of an ar... |
vdwap0 16677 | Value of a length-1 arithm... |
vdwap1 16678 | Value of a length-1 arithm... |
vdwmc 16679 | The predicate " The ` <. R... |
vdwmc2 16680 | Expand out the definition ... |
vdwpc 16681 | The predicate " The colori... |
vdwlem1 16682 | Lemma for ~ vdw . (Contri... |
vdwlem2 16683 | Lemma for ~ vdw . (Contri... |
vdwlem3 16684 | Lemma for ~ vdw . (Contri... |
vdwlem4 16685 | Lemma for ~ vdw . (Contri... |
vdwlem5 16686 | Lemma for ~ vdw . (Contri... |
vdwlem6 16687 | Lemma for ~ vdw . (Contri... |
vdwlem7 16688 | Lemma for ~ vdw . (Contri... |
vdwlem8 16689 | Lemma for ~ vdw . (Contri... |
vdwlem9 16690 | Lemma for ~ vdw . (Contri... |
vdwlem10 16691 | Lemma for ~ vdw . Set up ... |
vdwlem11 16692 | Lemma for ~ vdw . (Contri... |
vdwlem12 16693 | Lemma for ~ vdw . ` K = 2 ... |
vdwlem13 16694 | Lemma for ~ vdw . Main in... |
vdw 16695 | Van der Waerden's theorem.... |
vdwnnlem1 16696 | Corollary of ~ vdw , and l... |
vdwnnlem2 16697 | Lemma for ~ vdwnn . The s... |
vdwnnlem3 16698 | Lemma for ~ vdwnn . (Cont... |
vdwnn 16699 | Van der Waerden's theorem,... |
ramtlecl 16701 | The set ` T ` of numbers w... |
hashbcval 16703 | Value of the "binomial set... |
hashbccl 16704 | The binomial set is a fini... |
hashbcss 16705 | Subset relation for the bi... |
hashbc0 16706 | The set of subsets of size... |
hashbc2 16707 | The size of the binomial s... |
0hashbc 16708 | There are no subsets of th... |
ramval 16709 | The value of the Ramsey nu... |
ramcl2lem 16710 | Lemma for extended real cl... |
ramtcl 16711 | The Ramsey number has the ... |
ramtcl2 16712 | The Ramsey number is an in... |
ramtub 16713 | The Ramsey number is a low... |
ramub 16714 | The Ramsey number is a low... |
ramub2 16715 | It is sufficient to check ... |
rami 16716 | The defining property of a... |
ramcl2 16717 | The Ramsey number is eithe... |
ramxrcl 16718 | The Ramsey number is an ex... |
ramubcl 16719 | If the Ramsey number is up... |
ramlb 16720 | Establish a lower bound on... |
0ram 16721 | The Ramsey number when ` M... |
0ram2 16722 | The Ramsey number when ` M... |
ram0 16723 | The Ramsey number when ` R... |
0ramcl 16724 | Lemma for ~ ramcl : Exist... |
ramz2 16725 | The Ramsey number when ` F... |
ramz 16726 | The Ramsey number when ` F... |
ramub1lem1 16727 | Lemma for ~ ramub1 . (Con... |
ramub1lem2 16728 | Lemma for ~ ramub1 . (Con... |
ramub1 16729 | Inductive step for Ramsey'... |
ramcl 16730 | Ramsey's theorem: the Rams... |
ramsey 16731 | Ramsey's theorem with the ... |
prmoval 16734 | Value of the primorial fun... |
prmocl 16735 | Closure of the primorial f... |
prmone0 16736 | The primorial function is ... |
prmo0 16737 | The primorial of 0. (Cont... |
prmo1 16738 | The primorial of 1. (Cont... |
prmop1 16739 | The primorial of a success... |
prmonn2 16740 | Value of the primorial fun... |
prmo2 16741 | The primorial of 2. (Cont... |
prmo3 16742 | The primorial of 3. (Cont... |
prmdvdsprmo 16743 | The primorial of a number ... |
prmdvdsprmop 16744 | The primorial of a number ... |
fvprmselelfz 16745 | The value of the prime sel... |
fvprmselgcd1 16746 | The greatest common diviso... |
prmolefac 16747 | The primorial of a positiv... |
prmodvdslcmf 16748 | The primorial of a nonnega... |
prmolelcmf 16749 | The primorial of a positiv... |
prmgaplem1 16750 | Lemma for ~ prmgap : The ... |
prmgaplem2 16751 | Lemma for ~ prmgap : The ... |
prmgaplcmlem1 16752 | Lemma for ~ prmgaplcm : T... |
prmgaplcmlem2 16753 | Lemma for ~ prmgaplcm : T... |
prmgaplem3 16754 | Lemma for ~ prmgap . (Con... |
prmgaplem4 16755 | Lemma for ~ prmgap . (Con... |
prmgaplem5 16756 | Lemma for ~ prmgap : for e... |
prmgaplem6 16757 | Lemma for ~ prmgap : for e... |
prmgaplem7 16758 | Lemma for ~ prmgap . (Con... |
prmgaplem8 16759 | Lemma for ~ prmgap . (Con... |
prmgap 16760 | The prime gap theorem: for... |
prmgaplcm 16761 | Alternate proof of ~ prmga... |
prmgapprmolem 16762 | Lemma for ~ prmgapprmo : ... |
prmgapprmo 16763 | Alternate proof of ~ prmga... |
dec2dvds 16764 | Divisibility by two is obv... |
dec5dvds 16765 | Divisibility by five is ob... |
dec5dvds2 16766 | Divisibility by five is ob... |
dec5nprm 16767 | Divisibility by five is ob... |
dec2nprm 16768 | Divisibility by two is obv... |
modxai 16769 | Add exponents in a power m... |
mod2xi 16770 | Double exponents in a powe... |
modxp1i 16771 | Add one to an exponent in ... |
mod2xnegi 16772 | Version of ~ mod2xi with a... |
modsubi 16773 | Subtract from within a mod... |
gcdi 16774 | Calculate a GCD via Euclid... |
gcdmodi 16775 | Calculate a GCD via Euclid... |
decexp2 16776 | Calculate a power of two. ... |
numexp0 16777 | Calculate an integer power... |
numexp1 16778 | Calculate an integer power... |
numexpp1 16779 | Calculate an integer power... |
numexp2x 16780 | Double an integer power. ... |
decsplit0b 16781 | Split a decimal number int... |
decsplit0 16782 | Split a decimal number int... |
decsplit1 16783 | Split a decimal number int... |
decsplit 16784 | Split a decimal number int... |
karatsuba 16785 | The Karatsuba multiplicati... |
2exp4 16786 | Two to the fourth power is... |
2exp5 16787 | Two to the fifth power is ... |
2exp6 16788 | Two to the sixth power is ... |
2exp7 16789 | Two to the seventh power i... |
2exp8 16790 | Two to the eighth power is... |
2exp11 16791 | Two to the eleventh power ... |
2exp16 16792 | Two to the sixteenth power... |
3exp3 16793 | Three to the third power i... |
2expltfac 16794 | The factorial grows faster... |
cshwsidrepsw 16795 | If cyclically shifting a w... |
cshwsidrepswmod0 16796 | If cyclically shifting a w... |
cshwshashlem1 16797 | If cyclically shifting a w... |
cshwshashlem2 16798 | If cyclically shifting a w... |
cshwshashlem3 16799 | If cyclically shifting a w... |
cshwsdisj 16800 | The singletons resulting b... |
cshwsiun 16801 | The set of (different!) wo... |
cshwsex 16802 | The class of (different!) ... |
cshws0 16803 | The size of the set of (di... |
cshwrepswhash1 16804 | The size of the set of (di... |
cshwshashnsame 16805 | If a word (not consisting ... |
cshwshash 16806 | If a word has a length bei... |
prmlem0 16807 | Lemma for ~ prmlem1 and ~ ... |
prmlem1a 16808 | A quick proof skeleton to ... |
prmlem1 16809 | A quick proof skeleton to ... |
5prm 16810 | 5 is a prime number. (Con... |
6nprm 16811 | 6 is not a prime number. ... |
7prm 16812 | 7 is a prime number. (Con... |
8nprm 16813 | 8 is not a prime number. ... |
9nprm 16814 | 9 is not a prime number. ... |
10nprm 16815 | 10 is not a prime number. ... |
11prm 16816 | 11 is a prime number. (Co... |
13prm 16817 | 13 is a prime number. (Co... |
17prm 16818 | 17 is a prime number. (Co... |
19prm 16819 | 19 is a prime number. (Co... |
23prm 16820 | 23 is a prime number. (Co... |
prmlem2 16821 | Our last proving session g... |
37prm 16822 | 37 is a prime number. (Co... |
43prm 16823 | 43 is a prime number. (Co... |
83prm 16824 | 83 is a prime number. (Co... |
139prm 16825 | 139 is a prime number. (C... |
163prm 16826 | 163 is a prime number. (C... |
317prm 16827 | 317 is a prime number. (C... |
631prm 16828 | 631 is a prime number. (C... |
prmo4 16829 | The primorial of 4. (Cont... |
prmo5 16830 | The primorial of 5. (Cont... |
prmo6 16831 | The primorial of 6. (Cont... |
1259lem1 16832 | Lemma for ~ 1259prm . Cal... |
1259lem2 16833 | Lemma for ~ 1259prm . Cal... |
1259lem3 16834 | Lemma for ~ 1259prm . Cal... |
1259lem4 16835 | Lemma for ~ 1259prm . Cal... |
1259lem5 16836 | Lemma for ~ 1259prm . Cal... |
1259prm 16837 | 1259 is a prime number. (... |
2503lem1 16838 | Lemma for ~ 2503prm . Cal... |
2503lem2 16839 | Lemma for ~ 2503prm . Cal... |
2503lem3 16840 | Lemma for ~ 2503prm . Cal... |
2503prm 16841 | 2503 is a prime number. (... |
4001lem1 16842 | Lemma for ~ 4001prm . Cal... |
4001lem2 16843 | Lemma for ~ 4001prm . Cal... |
4001lem3 16844 | Lemma for ~ 4001prm . Cal... |
4001lem4 16845 | Lemma for ~ 4001prm . Cal... |
4001prm 16846 | 4001 is a prime number. (... |
brstruct 16849 | The structure relation is ... |
isstruct2 16850 | The property of being a st... |
structex 16851 | A structure is a set. (Co... |
structn0fun 16852 | A structure without the em... |
isstruct 16853 | The property of being a st... |
structcnvcnv 16854 | Two ways to express the re... |
structfung 16855 | The converse of the conver... |
structfun 16856 | Convert between two kinds ... |
structfn 16857 | Convert between two kinds ... |
strleun 16858 | Combine two structures int... |
strle1 16859 | Make a structure from a si... |
strle2 16860 | Make a structure from a pa... |
strle3 16861 | Make a structure from a tr... |
sbcie2s 16862 | A special version of class... |
sbcie3s 16863 | A special version of class... |
reldmsets 16866 | The structure override ope... |
setsvalg 16867 | Value of the structure rep... |
setsval 16868 | Value of the structure rep... |
fvsetsid 16869 | The value of the structure... |
fsets 16870 | The structure replacement ... |
setsdm 16871 | The domain of a structure ... |
setsfun 16872 | A structure with replaceme... |
setsfun0 16873 | A structure with replaceme... |
setsn0fun 16874 | The value of the structure... |
setsstruct2 16875 | An extensible structure wi... |
setsexstruct2 16876 | An extensible structure wi... |
setsstruct 16877 | An extensible structure wi... |
wunsets 16878 | Closure of structure repla... |
setsres 16879 | The structure replacement ... |
setsabs 16880 | Replacing the same compone... |
setscom 16881 | Component-setting is commu... |
sloteq 16884 | Equality theorem for the `... |
slotfn 16885 | A slot is a function on se... |
strfvnd 16886 | Deduction version of ~ str... |
strfvn 16887 | Value of a structure compo... |
strfvss 16888 | A structure component extr... |
wunstr 16889 | Closure of a structure ind... |
str0 16890 | All components of the empt... |
strfvi 16891 | Structure slot extractors ... |
fveqprc 16892 | Lemma for showing the equa... |
oveqprc 16893 | Lemma for showing the equa... |
wunndx 16896 | Closure of the index extra... |
ndxarg 16897 | Get the numeric argument f... |
ndxid 16898 | A structure component extr... |
strndxid 16899 | The value of a structure c... |
setsidvald 16900 | Value of the structure rep... |
setsidvaldOLD 16901 | Obsolete version of ~ sets... |
strfvd 16902 | Deduction version of ~ str... |
strfv2d 16903 | Deduction version of ~ str... |
strfv2 16904 | A variation on ~ strfv to ... |
strfv 16905 | Extract a structure compon... |
strfv3 16906 | Variant on ~ strfv for lar... |
strssd 16907 | Deduction version of ~ str... |
strss 16908 | Propagate component extrac... |
setsid 16909 | Value of the structure rep... |
setsnid 16910 | Value of the structure rep... |
setsnidOLD 16911 | Obsolete proof of ~ setsni... |
baseval 16914 | Value of the base set extr... |
baseid 16915 | Utility theorem: index-ind... |
basfn 16916 | The base set extractor is ... |
base0 16917 | The base set of the empty ... |
elbasfv 16918 | Utility theorem: reverse c... |
elbasov 16919 | Utility theorem: reverse c... |
strov2rcl 16920 | Partial reverse closure fo... |
basendx 16921 | Index value of the base se... |
basendxnn 16922 | The index value of the bas... |
basendxnnOLD 16923 | Obsolete proof of ~ basend... |
basndxelwund 16924 | The index of the base set ... |
basprssdmsets 16925 | The pair of the base index... |
opelstrbas 16926 | The base set of a structur... |
1strstr 16927 | A constructed one-slot str... |
1strstr1 16928 | A constructed one-slot str... |
1strbas 16929 | The base set of a construc... |
1strbasOLD 16930 | Obsolete proof of ~ 1strba... |
1strwunbndx 16931 | A constructed one-slot str... |
1strwun 16932 | A constructed one-slot str... |
1strwunOLD 16933 | Obsolete version of ~ 1str... |
2strstr 16934 | A constructed two-slot str... |
2strbas 16935 | The base set of a construc... |
2strop 16936 | The other slot of a constr... |
2strstr1 16937 | A constructed two-slot str... |
2strstr1OLD 16938 | Obsolete version of ~ 2str... |
2strbas1 16939 | The base set of a construc... |
2strop1 16940 | The other slot of a constr... |
reldmress 16943 | The structure restriction ... |
ressval 16944 | Value of structure restric... |
ressid2 16945 | General behavior of trivia... |
ressval2 16946 | Value of nontrivial struct... |
ressbas 16947 | Base set of a structure re... |
ressbasOLD 16948 | Obsolete proof of ~ ressba... |
ressbas2 16949 | Base set of a structure re... |
ressbasss 16950 | The base set of a restrict... |
resseqnbas 16951 | The components of an exten... |
resslemOLD 16952 | Obsolete version of ~ ress... |
ress0 16953 | All restrictions of the nu... |
ressid 16954 | Behavior of trivial restri... |
ressinbas 16955 | Restriction only cares abo... |
ressval3d 16956 | Value of structure restric... |
ressval3dOLD 16957 | Obsolete version of ~ ress... |
ressress 16958 | Restriction composition la... |
ressabs 16959 | Restriction absorption law... |
wunress 16960 | Closure of structure restr... |
wunressOLD 16961 | Obsolete proof of ~ wunres... |
plusgndx 16988 | Index value of the ~ df-pl... |
plusgid 16989 | Utility theorem: index-ind... |
plusgndxnn 16990 | The index of the slot for ... |
basendxltplusgndx 16991 | The index of the slot for ... |
basendxnplusgndx 16992 | The slot for the base set ... |
basendxnplusgndxOLD 16993 | Obsolete version of ~ base... |
grpstr 16994 | A constructed group is a s... |
grpstrndx 16995 | A constructed group is a s... |
grpbase 16996 | The base set of a construc... |
grpbaseOLD 16997 | Obsolete version of ~ grpb... |
grpplusg 16998 | The operation of a constru... |
grpplusgOLD 16999 | Obsolete version of ~ grpp... |
ressplusg 17000 | ` +g ` is unaffected by re... |
grpbasex 17001 | The base of an explicitly ... |
grpplusgx 17002 | The operation of an explic... |
mulrndx 17003 | Index value of the ~ df-mu... |
mulrid 17004 | Utility theorem: index-ind... |
basendxnmulrndx 17005 | The slot for the base set ... |
basendxnmulrndxOLD 17006 | Obsolete proof of ~ basend... |
plusgndxnmulrndx 17007 | The slot for the group (ad... |
rngstr 17008 | A constructed ring is a st... |
rngbase 17009 | The base set of a construc... |
rngplusg 17010 | The additive operation of ... |
rngmulr 17011 | The multiplicative operati... |
starvndx 17012 | Index value of the ~ df-st... |
starvid 17013 | Utility theorem: index-ind... |
starvndxnbasendx 17014 | The slot for the involutio... |
starvndxnplusgndx 17015 | The slot for the involutio... |
starvndxnmulrndx 17016 | The slot for the involutio... |
ressmulr 17017 | ` .r ` is unaffected by re... |
ressstarv 17018 | ` *r ` is unaffected by re... |
srngstr 17019 | A constructed star ring is... |
srngbase 17020 | The base set of a construc... |
srngplusg 17021 | The addition operation of ... |
srngmulr 17022 | The multiplication operati... |
srnginvl 17023 | The involution function of... |
scandx 17024 | Index value of the ~ df-sc... |
scaid 17025 | Utility theorem: index-ind... |
scandxnbasendx 17026 | The slot for the scalar is... |
scandxnplusgndx 17027 | The slot for the scalar fi... |
scandxnmulrndx 17028 | The slot for the scalar fi... |
vscandx 17029 | Index value of the ~ df-vs... |
vscaid 17030 | Utility theorem: index-ind... |
vscandxnbasendx 17031 | The slot for the scalar pr... |
vscandxnplusgndx 17032 | The slot for the scalar pr... |
vscandxnmulrndx 17033 | The slot for the scalar pr... |
vscandxnscandx 17034 | The slot for the scalar pr... |
lmodstr 17035 | A constructed left module ... |
lmodbase 17036 | The base set of a construc... |
lmodplusg 17037 | The additive operation of ... |
lmodsca 17038 | The set of scalars of a co... |
lmodvsca 17039 | The scalar product operati... |
ipndx 17040 | Index value of the ~ df-ip... |
ipid 17041 | Utility theorem: index-ind... |
ipndxnbasendx 17042 | The slot for the inner pro... |
ipndxnplusgndx 17043 | The slot for the inner pro... |
ipndxnmulrndx 17044 | The slot for the inner pro... |
slotsdifipndx 17045 | The slot for the scalar is... |
ipsstr 17046 | Lemma to shorten proofs of... |
ipsbase 17047 | The base set of a construc... |
ipsaddg 17048 | The additive operation of ... |
ipsmulr 17049 | The multiplicative operati... |
ipssca 17050 | The set of scalars of a co... |
ipsvsca 17051 | The scalar product operati... |
ipsip 17052 | The multiplicative operati... |
resssca 17053 | ` Scalar ` is unaffected b... |
ressvsca 17054 | ` .s ` is unaffected by re... |
ressip 17055 | The inner product is unaff... |
phlstr 17056 | A constructed pre-Hilbert ... |
phlbase 17057 | The base set of a construc... |
phlplusg 17058 | The additive operation of ... |
phlsca 17059 | The ring of scalars of a c... |
phlvsca 17060 | The scalar product operati... |
phlip 17061 | The inner product (Hermiti... |
tsetndx 17062 | Index value of the ~ df-ts... |
tsetid 17063 | Utility theorem: index-ind... |
tsetndxnn 17064 | The index of the slot for ... |
basendxlttsetndx 17065 | The index of the slot for ... |
tsetndxnbasendx 17066 | The slot for the topology ... |
tsetndxnplusgndx 17067 | The slot for the topology ... |
tsetndxnmulrndx 17068 | The slot for the topology ... |
tsetndxnstarvndx 17069 | The slot for the topology ... |
slotstnscsi 17070 | The slots ` Scalar ` , ` .... |
topgrpstr 17071 | A constructed topological ... |
topgrpbas 17072 | The base set of a construc... |
topgrpplusg 17073 | The additive operation of ... |
topgrptset 17074 | The topology of a construc... |
resstset 17075 | ` TopSet ` is unaffected b... |
plendx 17076 | Index value of the ~ df-pl... |
pleid 17077 | Utility theorem: self-refe... |
plendxnn 17078 | The index value of the ord... |
basendxltplendx 17079 | The index value of the ` B... |
plendxnbasendx 17080 | The slot for the order is ... |
plendxnplusgndx 17081 | The slot for the "less tha... |
plendxnmulrndx 17082 | The slot for the "less tha... |
plendxnscandx 17083 | The slot for the "less tha... |
plendxnvscandx 17084 | The slot for the "less tha... |
slotsdifplendx 17085 | The index of the slot for ... |
otpsstr 17086 | Functionality of a topolog... |
otpsbas 17087 | The base set of a topologi... |
otpstset 17088 | The open sets of a topolog... |
otpsle 17089 | The order of a topological... |
ressle 17090 | ` le ` is unaffected by re... |
ocndx 17091 | Index value of the ~ df-oc... |
ocid 17092 | Utility theorem: index-ind... |
basendxnocndx 17093 | The slot for the orthocomp... |
plendxnocndx 17094 | The slot for the orthocomp... |
dsndx 17095 | Index value of the ~ df-ds... |
dsid 17096 | Utility theorem: index-ind... |
dsndxnn 17097 | The index of the slot for ... |
basendxltdsndx 17098 | The index of the slot for ... |
dsndxnbasendx 17099 | The slot for the distance ... |
dsndxnplusgndx 17100 | The slot for the distance ... |
dsndxnmulrndx 17101 | The slot for the distance ... |
slotsdnscsi 17102 | The slots ` Scalar ` , ` .... |
dsndxntsetndx 17103 | The slot for the distance ... |
slotsdifdsndx 17104 | The index of the slot for ... |
unifndx 17105 | Index value of the ~ df-un... |
unifid 17106 | Utility theorem: index-ind... |
unifndxnn 17107 | The index of the slot for ... |
basendxltunifndx 17108 | The index of the slot for ... |
unifndxnbasendx 17109 | The slot for the uniform s... |
unifndxntsetndx 17110 | The slot for the uniform s... |
slotsdifunifndx 17111 | The index of the slot for ... |
ressunif 17112 | ` UnifSet ` is unaffected ... |
odrngstr 17113 | Functionality of an ordere... |
odrngbas 17114 | The base set of an ordered... |
odrngplusg 17115 | The addition operation of ... |
odrngmulr 17116 | The multiplication operati... |
odrngtset 17117 | The open sets of an ordere... |
odrngle 17118 | The order of an ordered me... |
odrngds 17119 | The metric of an ordered m... |
ressds 17120 | ` dist ` is unaffected by ... |
homndx 17121 | Index value of the ~ df-ho... |
homid 17122 | Utility theorem: index-ind... |
ccondx 17123 | Index value of the ~ df-cc... |
ccoid 17124 | Utility theorem: index-ind... |
slotsbhcdif 17125 | The slots ` Base ` , ` Hom... |
slotsbhcdifOLD 17126 | Obsolete proof of ~ slotsb... |
slotsdifplendx2 17127 | The index of the slot for ... |
slotsdifocndx 17128 | The index of the slot for ... |
resshom 17129 | ` Hom ` is unaffected by r... |
ressco 17130 | ` comp ` is unaffected by ... |
restfn 17135 | The subspace topology oper... |
topnfn 17136 | The topology extractor fun... |
restval 17137 | The subspace topology indu... |
elrest 17138 | The predicate "is an open ... |
elrestr 17139 | Sufficient condition for b... |
0rest 17140 | Value of the structure res... |
restid2 17141 | The subspace topology over... |
restsspw 17142 | The subspace topology is a... |
firest 17143 | The finite intersections o... |
restid 17144 | The subspace topology of t... |
topnval 17145 | Value of the topology extr... |
topnid 17146 | Value of the topology extr... |
topnpropd 17147 | The topology extractor fun... |
reldmprds 17159 | The structure product is a... |
prdsbasex 17161 | Lemma for structure produc... |
imasvalstr 17162 | An image structure value i... |
prdsvalstr 17163 | Structure product value is... |
prdsbaslem 17164 | Lemma for ~ prdsbas and si... |
prdsvallem 17165 | Lemma for ~ prdsval . (Co... |
prdsval 17166 | Value of the structure pro... |
prdssca 17167 | Scalar ring of a structure... |
prdsbas 17168 | Base set of a structure pr... |
prdsplusg 17169 | Addition in a structure pr... |
prdsmulr 17170 | Multiplication in a struct... |
prdsvsca 17171 | Scalar multiplication in a... |
prdsip 17172 | Inner product in a structu... |
prdsle 17173 | Structure product weak ord... |
prdsless 17174 | Closure of the order relat... |
prdsds 17175 | Structure product distance... |
prdsdsfn 17176 | Structure product distance... |
prdstset 17177 | Structure product topology... |
prdshom 17178 | Structure product hom-sets... |
prdsco 17179 | Structure product composit... |
prdsbas2 17180 | The base set of a structur... |
prdsbasmpt 17181 | A constructed tuple is a p... |
prdsbasfn 17182 | Points in the structure pr... |
prdsbasprj 17183 | Each point in a structure ... |
prdsplusgval 17184 | Value of a componentwise s... |
prdsplusgfval 17185 | Value of a structure produ... |
prdsmulrval 17186 | Value of a componentwise r... |
prdsmulrfval 17187 | Value of a structure produ... |
prdsleval 17188 | Value of the product order... |
prdsdsval 17189 | Value of the metric in a s... |
prdsvscaval 17190 | Scalar multiplication in a... |
prdsvscafval 17191 | Scalar multiplication of a... |
prdsbas3 17192 | The base set of an indexed... |
prdsbasmpt2 17193 | A constructed tuple is a p... |
prdsbascl 17194 | An element of the base has... |
prdsdsval2 17195 | Value of the metric in a s... |
prdsdsval3 17196 | Value of the metric in a s... |
pwsval 17197 | Value of a structure power... |
pwsbas 17198 | Base set of a structure po... |
pwselbasb 17199 | Membership in the base set... |
pwselbas 17200 | An element of a structure ... |
pwsplusgval 17201 | Value of addition in a str... |
pwsmulrval 17202 | Value of multiplication in... |
pwsle 17203 | Ordering in a structure po... |
pwsleval 17204 | Ordering in a structure po... |
pwsvscafval 17205 | Scalar multiplication in a... |
pwsvscaval 17206 | Scalar multiplication of a... |
pwssca 17207 | The ring of scalars of a s... |
pwsdiagel 17208 | Membership of diagonal ele... |
pwssnf1o 17209 | Triviality of singleton po... |
imasval 17222 | Value of an image structur... |
imasbas 17223 | The base set of an image s... |
imasds 17224 | The distance function of a... |
imasdsfn 17225 | The distance function is a... |
imasdsval 17226 | The distance function of a... |
imasdsval2 17227 | The distance function of a... |
imasplusg 17228 | The group operation in an ... |
imasmulr 17229 | The ring multiplication in... |
imassca 17230 | The scalar field of an ima... |
imasvsca 17231 | The scalar multiplication ... |
imasip 17232 | The inner product of an im... |
imastset 17233 | The topology of an image s... |
imasle 17234 | The ordering of an image s... |
f1ocpbllem 17235 | Lemma for ~ f1ocpbl . (Co... |
f1ocpbl 17236 | An injection is compatible... |
f1ovscpbl 17237 | An injection is compatible... |
f1olecpbl 17238 | An injection is compatible... |
imasaddfnlem 17239 | The image structure operat... |
imasaddvallem 17240 | The operation of an image ... |
imasaddflem 17241 | The image set operations a... |
imasaddfn 17242 | The image structure's grou... |
imasaddval 17243 | The value of an image stru... |
imasaddf 17244 | The image structure's grou... |
imasmulfn 17245 | The image structure's ring... |
imasmulval 17246 | The value of an image stru... |
imasmulf 17247 | The image structure's ring... |
imasvscafn 17248 | The image structure's scal... |
imasvscaval 17249 | The value of an image stru... |
imasvscaf 17250 | The image structure's scal... |
imasless 17251 | The order relation defined... |
imasleval 17252 | The value of the image str... |
qusval 17253 | Value of a quotient struct... |
quslem 17254 | The function in ~ qusval i... |
qusin 17255 | Restrict the equivalence r... |
qusbas 17256 | Base set of a quotient str... |
quss 17257 | The scalar field of a quot... |
divsfval 17258 | Value of the function in ~... |
ercpbllem 17259 | Lemma for ~ ercpbl . (Con... |
ercpbl 17260 | Translate the function com... |
erlecpbl 17261 | Translate the relation com... |
qusaddvallem 17262 | Value of an operation defi... |
qusaddflem 17263 | The operation of a quotien... |
qusaddval 17264 | The base set of an image s... |
qusaddf 17265 | The base set of an image s... |
qusmulval 17266 | The base set of an image s... |
qusmulf 17267 | The base set of an image s... |
fnpr2o 17268 | Function with a domain of ... |
fnpr2ob 17269 | Biconditional version of ~... |
fvpr0o 17270 | The value of a function wi... |
fvpr1o 17271 | The value of a function wi... |
fvprif 17272 | The value of the pair func... |
xpsfrnel 17273 | Elementhood in the target ... |
xpsfeq 17274 | A function on ` 2o ` is de... |
xpsfrnel2 17275 | Elementhood in the target ... |
xpscf 17276 | Equivalent condition for t... |
xpsfval 17277 | The value of the function ... |
xpsff1o 17278 | The function appearing in ... |
xpsfrn 17279 | A short expression for the... |
xpsff1o2 17280 | The function appearing in ... |
xpsval 17281 | Value of the binary struct... |
xpsrnbas 17282 | The indexed structure prod... |
xpsbas 17283 | The base set of the binary... |
xpsaddlem 17284 | Lemma for ~ xpsadd and ~ x... |
xpsadd 17285 | Value of the addition oper... |
xpsmul 17286 | Value of the multiplicatio... |
xpssca 17287 | Value of the scalar field ... |
xpsvsca 17288 | Value of the scalar multip... |
xpsless 17289 | Closure of the ordering in... |
xpsle 17290 | Value of the ordering in a... |
ismre 17299 | Property of being a Moore ... |
fnmre 17300 | The Moore collection gener... |
mresspw 17301 | A Moore collection is a su... |
mress 17302 | A Moore-closed subset is a... |
mre1cl 17303 | In any Moore collection th... |
mreintcl 17304 | A nonempty collection of c... |
mreiincl 17305 | A nonempty indexed interse... |
mrerintcl 17306 | The relative intersection ... |
mreriincl 17307 | The relative intersection ... |
mreincl 17308 | Two closed sets have a clo... |
mreuni 17309 | Since the entire base set ... |
mreunirn 17310 | Two ways to express the no... |
ismred 17311 | Properties that determine ... |
ismred2 17312 | Properties that determine ... |
mremre 17313 | The Moore collections of s... |
submre 17314 | The subcollection of a clo... |
mrcflem 17315 | The domain and range of th... |
fnmrc 17316 | Moore-closure is a well-be... |
mrcfval 17317 | Value of the function expr... |
mrcf 17318 | The Moore closure is a fun... |
mrcval 17319 | Evaluation of the Moore cl... |
mrccl 17320 | The Moore closure of a set... |
mrcsncl 17321 | The Moore closure of a sin... |
mrcid 17322 | The closure of a closed se... |
mrcssv 17323 | The closure of a set is a ... |
mrcidb 17324 | A set is closed iff it is ... |
mrcss 17325 | Closure preserves subset o... |
mrcssid 17326 | The closure of a set is a ... |
mrcidb2 17327 | A set is closed iff it con... |
mrcidm 17328 | The closure operation is i... |
mrcsscl 17329 | The closure is the minimal... |
mrcuni 17330 | Idempotence of closure und... |
mrcun 17331 | Idempotence of closure und... |
mrcssvd 17332 | The Moore closure of a set... |
mrcssd 17333 | Moore closure preserves su... |
mrcssidd 17334 | A set is contained in its ... |
mrcidmd 17335 | Moore closure is idempoten... |
mressmrcd 17336 | In a Moore system, if a se... |
submrc 17337 | In a closure system which ... |
mrieqvlemd 17338 | In a Moore system, if ` Y ... |
mrisval 17339 | Value of the set of indepe... |
ismri 17340 | Criterion for a set to be ... |
ismri2 17341 | Criterion for a subset of ... |
ismri2d 17342 | Criterion for a subset of ... |
ismri2dd 17343 | Definition of independence... |
mriss 17344 | An independent set of a Mo... |
mrissd 17345 | An independent set of a Mo... |
ismri2dad 17346 | Consequence of a set in a ... |
mrieqvd 17347 | In a Moore system, a set i... |
mrieqv2d 17348 | In a Moore system, a set i... |
mrissmrcd 17349 | In a Moore system, if an i... |
mrissmrid 17350 | In a Moore system, subsets... |
mreexd 17351 | In a Moore system, the clo... |
mreexmrid 17352 | In a Moore system whose cl... |
mreexexlemd 17353 | This lemma is used to gene... |
mreexexlem2d 17354 | Used in ~ mreexexlem4d to ... |
mreexexlem3d 17355 | Base case of the induction... |
mreexexlem4d 17356 | Induction step of the indu... |
mreexexd 17357 | Exchange-type theorem. In... |
mreexdomd 17358 | In a Moore system whose cl... |
mreexfidimd 17359 | In a Moore system whose cl... |
isacs 17360 | A set is an algebraic clos... |
acsmre 17361 | Algebraic closure systems ... |
isacs2 17362 | In the definition of an al... |
acsfiel 17363 | A set is closed in an alge... |
acsfiel2 17364 | A set is closed in an alge... |
acsmred 17365 | An algebraic closure syste... |
isacs1i 17366 | A closure system determine... |
mreacs 17367 | Algebraicity is a composab... |
acsfn 17368 | Algebraicity of a conditio... |
acsfn0 17369 | Algebraicity of a point cl... |
acsfn1 17370 | Algebraicity of a one-argu... |
acsfn1c 17371 | Algebraicity of a one-argu... |
acsfn2 17372 | Algebraicity of a two-argu... |
iscat 17381 | The predicate "is a catego... |
iscatd 17382 | Properties that determine ... |
catidex 17383 | Each object in a category ... |
catideu 17384 | Each object in a category ... |
cidfval 17385 | Each object in a category ... |
cidval 17386 | Each object in a category ... |
cidffn 17387 | The identity arrow constru... |
cidfn 17388 | The identity arrow operato... |
catidd 17389 | Deduce the identity arrow ... |
iscatd2 17390 | Version of ~ iscatd with a... |
catidcl 17391 | Each object in a category ... |
catlid 17392 | Left identity property of ... |
catrid 17393 | Right identity property of... |
catcocl 17394 | Closure of a composition a... |
catass 17395 | Associativity of compositi... |
catcone0 17396 | Composition of non-empty h... |
0catg 17397 | Any structure with an empt... |
0cat 17398 | The empty set is a categor... |
homffval 17399 | Value of the functionalize... |
fnhomeqhomf 17400 | If the Hom-set operation i... |
homfval 17401 | Value of the functionalize... |
homffn 17402 | The functionalized Hom-set... |
homfeq 17403 | Condition for two categori... |
homfeqd 17404 | If two structures have the... |
homfeqbas 17405 | Deduce equality of base se... |
homfeqval 17406 | Value of the functionalize... |
comfffval 17407 | Value of the functionalize... |
comffval 17408 | Value of the functionalize... |
comfval 17409 | Value of the functionalize... |
comfffval2 17410 | Value of the functionalize... |
comffval2 17411 | Value of the functionalize... |
comfval2 17412 | Value of the functionalize... |
comfffn 17413 | The functionalized composi... |
comffn 17414 | The functionalized composi... |
comfeq 17415 | Condition for two categori... |
comfeqd 17416 | Condition for two categori... |
comfeqval 17417 | Equality of two compositio... |
catpropd 17418 | Two structures with the sa... |
cidpropd 17419 | Two structures with the sa... |
oppcval 17422 | Value of the opposite cate... |
oppchomfval 17423 | Hom-sets of the opposite c... |
oppchomfvalOLD 17424 | Obsolete proof of ~ oppcho... |
oppchom 17425 | Hom-sets of the opposite c... |
oppccofval 17426 | Composition in the opposit... |
oppcco 17427 | Composition in the opposit... |
oppcbas 17428 | Base set of an opposite ca... |
oppcbasOLD 17429 | Obsolete version of ~ oppc... |
oppccatid 17430 | Lemma for ~ oppccat . (Co... |
oppchomf 17431 | Hom-sets of the opposite c... |
oppcid 17432 | Identity function of an op... |
oppccat 17433 | An opposite category is a ... |
2oppcbas 17434 | The double opposite catego... |
2oppchomf 17435 | The double opposite catego... |
2oppccomf 17436 | The double opposite catego... |
oppchomfpropd 17437 | If two categories have the... |
oppccomfpropd 17438 | If two categories have the... |
oppccatf 17439 | ` oppCat ` restricted to `... |
monfval 17444 | Definition of a monomorphi... |
ismon 17445 | Definition of a monomorphi... |
ismon2 17446 | Write out the monomorphism... |
monhom 17447 | A monomorphism is a morphi... |
moni 17448 | Property of a monomorphism... |
monpropd 17449 | If two categories have the... |
oppcmon 17450 | A monomorphism in the oppo... |
oppcepi 17451 | An epimorphism in the oppo... |
isepi 17452 | Definition of an epimorphi... |
isepi2 17453 | Write out the epimorphism ... |
epihom 17454 | An epimorphism is a morphi... |
epii 17455 | Property of an epimorphism... |
sectffval 17462 | Value of the section opera... |
sectfval 17463 | Value of the section relat... |
sectss 17464 | The section relation is a ... |
issect 17465 | The property " ` F ` is a ... |
issect2 17466 | Property of being a sectio... |
sectcan 17467 | If ` G ` is a section of `... |
sectco 17468 | Composition of two section... |
isofval 17469 | Function value of the func... |
invffval 17470 | Value of the inverse relat... |
invfval 17471 | Value of the inverse relat... |
isinv 17472 | Value of the inverse relat... |
invss 17473 | The inverse relation is a ... |
invsym 17474 | The inverse relation is sy... |
invsym2 17475 | The inverse relation is sy... |
invfun 17476 | The inverse relation is a ... |
isoval 17477 | The isomorphisms are the d... |
inviso1 17478 | If ` G ` is an inverse to ... |
inviso2 17479 | If ` G ` is an inverse to ... |
invf 17480 | The inverse relation is a ... |
invf1o 17481 | The inverse relation is a ... |
invinv 17482 | The inverse of the inverse... |
invco 17483 | The composition of two iso... |
dfiso2 17484 | Alternate definition of an... |
dfiso3 17485 | Alternate definition of an... |
inveq 17486 | If there are two inverses ... |
isofn 17487 | The function value of the ... |
isohom 17488 | An isomorphism is a homomo... |
isoco 17489 | The composition of two iso... |
oppcsect 17490 | A section in the opposite ... |
oppcsect2 17491 | A section in the opposite ... |
oppcinv 17492 | An inverse in the opposite... |
oppciso 17493 | An isomorphism in the oppo... |
sectmon 17494 | If ` F ` is a section of `... |
monsect 17495 | If ` F ` is a monomorphism... |
sectepi 17496 | If ` F ` is a section of `... |
episect 17497 | If ` F ` is an epimorphism... |
sectid 17498 | The identity is a section ... |
invid 17499 | The inverse of the identit... |
idiso 17500 | The identity is an isomorp... |
idinv 17501 | The inverse of the identit... |
invisoinvl 17502 | The inverse of an isomorph... |
invisoinvr 17503 | The inverse of an isomorph... |
invcoisoid 17504 | The inverse of an isomorph... |
isocoinvid 17505 | The inverse of an isomorph... |
rcaninv 17506 | Right cancellation of an i... |
cicfval 17509 | The set of isomorphic obje... |
brcic 17510 | The relation "is isomorphi... |
cic 17511 | Objects ` X ` and ` Y ` in... |
brcici 17512 | Prove that two objects are... |
cicref 17513 | Isomorphism is reflexive. ... |
ciclcl 17514 | Isomorphism implies the le... |
cicrcl 17515 | Isomorphism implies the ri... |
cicsym 17516 | Isomorphism is symmetric. ... |
cictr 17517 | Isomorphism is transitive.... |
cicer 17518 | Isomorphism is an equivale... |
sscrel 17525 | The subcategory subset rel... |
brssc 17526 | The subcategory subset rel... |
sscpwex 17527 | An analogue of ~ pwex for ... |
subcrcl 17528 | Reverse closure for the su... |
sscfn1 17529 | The subcategory subset rel... |
sscfn2 17530 | The subcategory subset rel... |
ssclem 17531 | Lemma for ~ ssc1 and simil... |
isssc 17532 | Value of the subcategory s... |
ssc1 17533 | Infer subset relation on o... |
ssc2 17534 | Infer subset relation on m... |
sscres 17535 | Any function restricted to... |
sscid 17536 | The subcategory subset rel... |
ssctr 17537 | The subcategory subset rel... |
ssceq 17538 | The subcategory subset rel... |
rescval 17539 | Value of the category rest... |
rescval2 17540 | Value of the category rest... |
rescbas 17541 | Base set of the category r... |
rescbasOLD 17542 | Obsolete version of ~ resc... |
reschom 17543 | Hom-sets of the category r... |
reschomf 17544 | Hom-sets of the category r... |
rescco 17545 | Composition in the categor... |
resccoOLD 17546 | Obsolete proof of ~ rescco... |
rescabs 17547 | Restriction absorption law... |
rescabsOLD 17548 | Obsolete proof of ~ seqp1d... |
rescabs2 17549 | Restriction absorption law... |
issubc 17550 | Elementhood in the set of ... |
issubc2 17551 | Elementhood in the set of ... |
0ssc 17552 | For any category ` C ` , t... |
0subcat 17553 | For any category ` C ` , t... |
catsubcat 17554 | For any category ` C ` , `... |
subcssc 17555 | An element in the set of s... |
subcfn 17556 | An element in the set of s... |
subcss1 17557 | The objects of a subcatego... |
subcss2 17558 | The morphisms of a subcate... |
subcidcl 17559 | The identity of the origin... |
subccocl 17560 | A subcategory is closed un... |
subccatid 17561 | A subcategory is a categor... |
subcid 17562 | The identity in a subcateg... |
subccat 17563 | A subcategory is a categor... |
issubc3 17564 | Alternate definition of a ... |
fullsubc 17565 | The full subcategory gener... |
fullresc 17566 | The category formed by str... |
resscat 17567 | A category restricted to a... |
subsubc 17568 | A subcategory of a subcate... |
relfunc 17577 | The set of functors is a r... |
funcrcl 17578 | Reverse closure for a func... |
isfunc 17579 | Value of the set of functo... |
isfuncd 17580 | Deduce that an operation i... |
funcf1 17581 | The object part of a funct... |
funcixp 17582 | The morphism part of a fun... |
funcf2 17583 | The morphism part of a fun... |
funcfn2 17584 | The morphism part of a fun... |
funcid 17585 | A functor maps each identi... |
funcco 17586 | A functor maps composition... |
funcsect 17587 | The image of a section und... |
funcinv 17588 | The image of an inverse un... |
funciso 17589 | The image of an isomorphis... |
funcoppc 17590 | A functor on categories yi... |
idfuval 17591 | Value of the identity func... |
idfu2nd 17592 | Value of the morphism part... |
idfu2 17593 | Value of the morphism part... |
idfu1st 17594 | Value of the object part o... |
idfu1 17595 | Value of the object part o... |
idfucl 17596 | The identity functor is a ... |
cofuval 17597 | Value of the composition o... |
cofu1st 17598 | Value of the object part o... |
cofu1 17599 | Value of the object part o... |
cofu2nd 17600 | Value of the morphism part... |
cofu2 17601 | Value of the morphism part... |
cofuval2 17602 | Value of the composition o... |
cofucl 17603 | The composition of two fun... |
cofuass 17604 | Functor composition is ass... |
cofulid 17605 | The identity functor is a ... |
cofurid 17606 | The identity functor is a ... |
resfval 17607 | Value of the functor restr... |
resfval2 17608 | Value of the functor restr... |
resf1st 17609 | Value of the functor restr... |
resf2nd 17610 | Value of the functor restr... |
funcres 17611 | A functor restricted to a ... |
funcres2b 17612 | Condition for a functor to... |
funcres2 17613 | A functor into a restricte... |
wunfunc 17614 | A weak universe is closed ... |
wunfuncOLD 17615 | Obsolete proof of ~ wunfun... |
funcpropd 17616 | If two categories have the... |
funcres2c 17617 | Condition for a functor to... |
fullfunc 17622 | A full functor is a functo... |
fthfunc 17623 | A faithful functor is a fu... |
relfull 17624 | The set of full functors i... |
relfth 17625 | The set of faithful functo... |
isfull 17626 | Value of the set of full f... |
isfull2 17627 | Equivalent condition for a... |
fullfo 17628 | The morphism map of a full... |
fulli 17629 | The morphism map of a full... |
isfth 17630 | Value of the set of faithf... |
isfth2 17631 | Equivalent condition for a... |
isffth2 17632 | A fully faithful functor i... |
fthf1 17633 | The morphism map of a fait... |
fthi 17634 | The morphism map of a fait... |
ffthf1o 17635 | The morphism map of a full... |
fullpropd 17636 | If two categories have the... |
fthpropd 17637 | If two categories have the... |
fulloppc 17638 | The opposite functor of a ... |
fthoppc 17639 | The opposite functor of a ... |
ffthoppc 17640 | The opposite functor of a ... |
fthsect 17641 | A faithful functor reflect... |
fthinv 17642 | A faithful functor reflect... |
fthmon 17643 | A faithful functor reflect... |
fthepi 17644 | A faithful functor reflect... |
ffthiso 17645 | A fully faithful functor r... |
fthres2b 17646 | Condition for a faithful f... |
fthres2c 17647 | Condition for a faithful f... |
fthres2 17648 | A faithful functor into a ... |
idffth 17649 | The identity functor is a ... |
cofull 17650 | The composition of two ful... |
cofth 17651 | The composition of two fai... |
coffth 17652 | The composition of two ful... |
rescfth 17653 | The inclusion functor from... |
ressffth 17654 | The inclusion functor from... |
fullres2c 17655 | Condition for a full funct... |
ffthres2c 17656 | Condition for a fully fait... |
fnfuc 17661 | The ` FuncCat ` operation ... |
natfval 17662 | Value of the function givi... |
isnat 17663 | Property of being a natura... |
isnat2 17664 | Property of being a natura... |
natffn 17665 | The natural transformation... |
natrcl 17666 | Reverse closure for a natu... |
nat1st2nd 17667 | Rewrite the natural transf... |
natixp 17668 | A natural transformation i... |
natcl 17669 | A component of a natural t... |
natfn 17670 | A natural transformation i... |
nati 17671 | Naturality property of a n... |
wunnat 17672 | A weak universe is closed ... |
wunnatOLD 17673 | Obsolete proof of ~ wunnat... |
catstr 17674 | A category structure is a ... |
fucval 17675 | Value of the functor categ... |
fuccofval 17676 | Value of the functor categ... |
fucbas 17677 | The objects of the functor... |
fuchom 17678 | The morphisms in the funct... |
fuchomOLD 17679 | Obsolete proof of ~ fuchom... |
fucco 17680 | Value of the composition o... |
fuccoval 17681 | Value of the functor categ... |
fuccocl 17682 | The composition of two nat... |
fucidcl 17683 | The identity natural trans... |
fuclid 17684 | Left identity of natural t... |
fucrid 17685 | Right identity of natural ... |
fucass 17686 | Associativity of natural t... |
fuccatid 17687 | The functor category is a ... |
fuccat 17688 | The functor category is a ... |
fucid 17689 | The identity morphism in t... |
fucsect 17690 | Two natural transformation... |
fucinv 17691 | Two natural transformation... |
invfuc 17692 | If ` V ( x ) ` is an inver... |
fuciso 17693 | A natural transformation i... |
natpropd 17694 | If two categories have the... |
fucpropd 17695 | If two categories have the... |
initofn 17702 | ` InitO ` is a function on... |
termofn 17703 | ` TermO ` is a function on... |
zeroofn 17704 | ` ZeroO ` is a function on... |
initorcl 17705 | Reverse closure for an ini... |
termorcl 17706 | Reverse closure for a term... |
zeroorcl 17707 | Reverse closure for a zero... |
initoval 17708 | The value of the initial o... |
termoval 17709 | The value of the terminal ... |
zerooval 17710 | The value of the zero obje... |
isinito 17711 | The predicate "is an initi... |
istermo 17712 | The predicate "is a termin... |
iszeroo 17713 | The predicate "is a zero o... |
isinitoi 17714 | Implication of a class bei... |
istermoi 17715 | Implication of a class bei... |
initoid 17716 | For an initial object, the... |
termoid 17717 | For a terminal object, the... |
dfinito2 17718 | An initial object is a ter... |
dftermo2 17719 | A terminal object is an in... |
dfinito3 17720 | An alternate definition of... |
dftermo3 17721 | An alternate definition of... |
initoo 17722 | An initial object is an ob... |
termoo 17723 | A terminal object is an ob... |
iszeroi 17724 | Implication of a class bei... |
2initoinv 17725 | Morphisms between two init... |
initoeu1 17726 | Initial objects are essent... |
initoeu1w 17727 | Initial objects are essent... |
initoeu2lem0 17728 | Lemma 0 for ~ initoeu2 . ... |
initoeu2lem1 17729 | Lemma 1 for ~ initoeu2 . ... |
initoeu2lem2 17730 | Lemma 2 for ~ initoeu2 . ... |
initoeu2 17731 | Initial objects are essent... |
2termoinv 17732 | Morphisms between two term... |
termoeu1 17733 | Terminal objects are essen... |
termoeu1w 17734 | Terminal objects are essen... |
homarcl 17743 | Reverse closure for an arr... |
homafval 17744 | Value of the disjointified... |
homaf 17745 | Functionality of the disjo... |
homaval 17746 | Value of the disjointified... |
elhoma 17747 | Value of the disjointified... |
elhomai 17748 | Produce an arrow from a mo... |
elhomai2 17749 | Produce an arrow from a mo... |
homarcl2 17750 | Reverse closure for the do... |
homarel 17751 | An arrow is an ordered pai... |
homa1 17752 | The first component of an ... |
homahom2 17753 | The second component of an... |
homahom 17754 | The second component of an... |
homadm 17755 | The domain of an arrow wit... |
homacd 17756 | The codomain of an arrow w... |
homadmcd 17757 | Decompose an arrow into do... |
arwval 17758 | The set of arrows is the u... |
arwrcl 17759 | The first component of an ... |
arwhoma 17760 | An arrow is contained in t... |
homarw 17761 | A hom-set is a subset of t... |
arwdm 17762 | The domain of an arrow is ... |
arwcd 17763 | The codomain of an arrow i... |
dmaf 17764 | The domain function is a f... |
cdaf 17765 | The codomain function is a... |
arwhom 17766 | The second component of an... |
arwdmcd 17767 | Decompose an arrow into do... |
idafval 17772 | Value of the identity arro... |
idaval 17773 | Value of the identity arro... |
ida2 17774 | Morphism part of the ident... |
idahom 17775 | Domain and codomain of the... |
idadm 17776 | Domain of the identity arr... |
idacd 17777 | Codomain of the identity a... |
idaf 17778 | The identity arrow functio... |
coafval 17779 | The value of the compositi... |
eldmcoa 17780 | A pair ` <. G , F >. ` is ... |
dmcoass 17781 | The domain of composition ... |
homdmcoa 17782 | If ` F : X --> Y ` and ` G... |
coaval 17783 | Value of composition for c... |
coa2 17784 | The morphism part of arrow... |
coahom 17785 | The composition of two com... |
coapm 17786 | Composition of arrows is a... |
arwlid 17787 | Left identity of a categor... |
arwrid 17788 | Right identity of a catego... |
arwass 17789 | Associativity of compositi... |
setcval 17792 | Value of the category of s... |
setcbas 17793 | Set of objects of the cate... |
setchomfval 17794 | Set of arrows of the categ... |
setchom 17795 | Set of arrows of the categ... |
elsetchom 17796 | A morphism of sets is a fu... |
setccofval 17797 | Composition in the categor... |
setcco 17798 | Composition in the categor... |
setccatid 17799 | Lemma for ~ setccat . (Co... |
setccat 17800 | The category of sets is a ... |
setcid 17801 | The identity arrow in the ... |
setcmon 17802 | A monomorphism of sets is ... |
setcepi 17803 | An epimorphism of sets is ... |
setcsect 17804 | A section in the category ... |
setcinv 17805 | An inverse in the category... |
setciso 17806 | An isomorphism in the cate... |
resssetc 17807 | The restriction of the cat... |
funcsetcres2 17808 | A functor into a smaller c... |
setc2obas 17809 | ` (/) ` and ` 1o ` are dis... |
setc2ohom 17810 | ` ( SetCat `` 2o ) ` is a ... |
cat1lem 17811 | The category of sets in a ... |
cat1 17812 | The definition of category... |
catcval 17815 | Value of the category of c... |
catcbas 17816 | Set of objects of the cate... |
catchomfval 17817 | Set of arrows of the categ... |
catchom 17818 | Set of arrows of the categ... |
catccofval 17819 | Composition in the categor... |
catcco 17820 | Composition in the categor... |
catccatid 17821 | Lemma for ~ catccat . (Co... |
catcid 17822 | The identity arrow in the ... |
catccat 17823 | The category of categories... |
resscatc 17824 | The restriction of the cat... |
catcisolem 17825 | Lemma for ~ catciso . (Co... |
catciso 17826 | A functor is an isomorphis... |
catcbascl 17827 | An element of the base set... |
catcslotelcl 17828 | A slot entry of an element... |
catcbaselcl 17829 | The base set of an element... |
catchomcl 17830 | The Hom-set of an element ... |
catcccocl 17831 | The composition operation ... |
catcoppccl 17832 | The category of categories... |
catcoppcclOLD 17833 | Obsolete proof of ~ catcop... |
catcfuccl 17834 | The category of categories... |
catcfucclOLD 17835 | Obsolete proof of ~ catcfu... |
fncnvimaeqv 17836 | The inverse images of the ... |
bascnvimaeqv 17837 | The inverse image of the u... |
estrcval 17840 | Value of the category of e... |
estrcbas 17841 | Set of objects of the cate... |
estrchomfval 17842 | Set of morphisms ("arrows"... |
estrchom 17843 | The morphisms between exte... |
elestrchom 17844 | A morphism between extensi... |
estrccofval 17845 | Composition in the categor... |
estrcco 17846 | Composition in the categor... |
estrcbasbas 17847 | An element of the base set... |
estrccatid 17848 | Lemma for ~ estrccat . (C... |
estrccat 17849 | The category of extensible... |
estrcid 17850 | The identity arrow in the ... |
estrchomfn 17851 | The Hom-set operation in t... |
estrchomfeqhom 17852 | The functionalized Hom-set... |
estrreslem1 17853 | Lemma 1 for ~ estrres . (... |
estrreslem1OLD 17854 | Obsolete version of ~ estr... |
estrreslem2 17855 | Lemma 2 for ~ estrres . (... |
estrres 17856 | Any restriction of a categ... |
funcestrcsetclem1 17857 | Lemma 1 for ~ funcestrcset... |
funcestrcsetclem2 17858 | Lemma 2 for ~ funcestrcset... |
funcestrcsetclem3 17859 | Lemma 3 for ~ funcestrcset... |
funcestrcsetclem4 17860 | Lemma 4 for ~ funcestrcset... |
funcestrcsetclem5 17861 | Lemma 5 for ~ funcestrcset... |
funcestrcsetclem6 17862 | Lemma 6 for ~ funcestrcset... |
funcestrcsetclem7 17863 | Lemma 7 for ~ funcestrcset... |
funcestrcsetclem8 17864 | Lemma 8 for ~ funcestrcset... |
funcestrcsetclem9 17865 | Lemma 9 for ~ funcestrcset... |
funcestrcsetc 17866 | The "natural forgetful fun... |
fthestrcsetc 17867 | The "natural forgetful fun... |
fullestrcsetc 17868 | The "natural forgetful fun... |
equivestrcsetc 17869 | The "natural forgetful fun... |
setc1strwun 17870 | A constructed one-slot str... |
funcsetcestrclem1 17871 | Lemma 1 for ~ funcsetcestr... |
funcsetcestrclem2 17872 | Lemma 2 for ~ funcsetcestr... |
funcsetcestrclem3 17873 | Lemma 3 for ~ funcsetcestr... |
embedsetcestrclem 17874 | Lemma for ~ embedsetcestrc... |
funcsetcestrclem4 17875 | Lemma 4 for ~ funcsetcestr... |
funcsetcestrclem5 17876 | Lemma 5 for ~ funcsetcestr... |
funcsetcestrclem6 17877 | Lemma 6 for ~ funcsetcestr... |
funcsetcestrclem7 17878 | Lemma 7 for ~ funcsetcestr... |
funcsetcestrclem8 17879 | Lemma 8 for ~ funcsetcestr... |
funcsetcestrclem9 17880 | Lemma 9 for ~ funcsetcestr... |
funcsetcestrc 17881 | The "embedding functor" fr... |
fthsetcestrc 17882 | The "embedding functor" fr... |
fullsetcestrc 17883 | The "embedding functor" fr... |
embedsetcestrc 17884 | The "embedding functor" fr... |
fnxpc 17893 | The binary product of cate... |
xpcval 17894 | Value of the binary produc... |
xpcbas 17895 | Set of objects of the bina... |
xpchomfval 17896 | Set of morphisms of the bi... |
xpchom 17897 | Set of morphisms of the bi... |
relxpchom 17898 | A hom-set in the binary pr... |
xpccofval 17899 | Value of composition in th... |
xpcco 17900 | Value of composition in th... |
xpcco1st 17901 | Value of composition in th... |
xpcco2nd 17902 | Value of composition in th... |
xpchom2 17903 | Value of the set of morphi... |
xpcco2 17904 | Value of composition in th... |
xpccatid 17905 | The product of two categor... |
xpcid 17906 | The identity morphism in t... |
xpccat 17907 | The product of two categor... |
1stfval 17908 | Value of the first project... |
1stf1 17909 | Value of the first project... |
1stf2 17910 | Value of the first project... |
2ndfval 17911 | Value of the first project... |
2ndf1 17912 | Value of the first project... |
2ndf2 17913 | Value of the first project... |
1stfcl 17914 | The first projection funct... |
2ndfcl 17915 | The second projection func... |
prfval 17916 | Value of the pairing funct... |
prf1 17917 | Value of the pairing funct... |
prf2fval 17918 | Value of the pairing funct... |
prf2 17919 | Value of the pairing funct... |
prfcl 17920 | The pairing of functors ` ... |
prf1st 17921 | Cancellation of pairing wi... |
prf2nd 17922 | Cancellation of pairing wi... |
1st2ndprf 17923 | Break a functor into a pro... |
catcxpccl 17924 | The category of categories... |
catcxpcclOLD 17925 | Obsolete proof of ~ catcxp... |
xpcpropd 17926 | If two categories have the... |
evlfval 17935 | Value of the evaluation fu... |
evlf2 17936 | Value of the evaluation fu... |
evlf2val 17937 | Value of the evaluation na... |
evlf1 17938 | Value of the evaluation fu... |
evlfcllem 17939 | Lemma for ~ evlfcl . (Con... |
evlfcl 17940 | The evaluation functor is ... |
curfval 17941 | Value of the curry functor... |
curf1fval 17942 | Value of the object part o... |
curf1 17943 | Value of the object part o... |
curf11 17944 | Value of the double evalua... |
curf12 17945 | The partially evaluated cu... |
curf1cl 17946 | The partially evaluated cu... |
curf2 17947 | Value of the curry functor... |
curf2val 17948 | Value of a component of th... |
curf2cl 17949 | The curry functor at a mor... |
curfcl 17950 | The curry functor of a fun... |
curfpropd 17951 | If two categories have the... |
uncfval 17952 | Value of the uncurry funct... |
uncfcl 17953 | The uncurry operation take... |
uncf1 17954 | Value of the uncurry funct... |
uncf2 17955 | Value of the uncurry funct... |
curfuncf 17956 | Cancellation of curry with... |
uncfcurf 17957 | Cancellation of uncurry wi... |
diagval 17958 | Define the diagonal functo... |
diagcl 17959 | The diagonal functor is a ... |
diag1cl 17960 | The constant functor of ` ... |
diag11 17961 | Value of the constant func... |
diag12 17962 | Value of the constant func... |
diag2 17963 | Value of the diagonal func... |
diag2cl 17964 | The diagonal functor at a ... |
curf2ndf 17965 | As shown in ~ diagval , th... |
hofval 17970 | Value of the Hom functor, ... |
hof1fval 17971 | The object part of the Hom... |
hof1 17972 | The object part of the Hom... |
hof2fval 17973 | The morphism part of the H... |
hof2val 17974 | The morphism part of the H... |
hof2 17975 | The morphism part of the H... |
hofcllem 17976 | Lemma for ~ hofcl . (Cont... |
hofcl 17977 | Closure of the Hom functor... |
oppchofcl 17978 | Closure of the opposite Ho... |
yonval 17979 | Value of the Yoneda embedd... |
yoncl 17980 | The Yoneda embedding is a ... |
yon1cl 17981 | The Yoneda embedding at an... |
yon11 17982 | Value of the Yoneda embedd... |
yon12 17983 | Value of the Yoneda embedd... |
yon2 17984 | Value of the Yoneda embedd... |
hofpropd 17985 | If two categories have the... |
yonpropd 17986 | If two categories have the... |
oppcyon 17987 | Value of the opposite Yone... |
oyoncl 17988 | The opposite Yoneda embedd... |
oyon1cl 17989 | The opposite Yoneda embedd... |
yonedalem1 17990 | Lemma for ~ yoneda . (Con... |
yonedalem21 17991 | Lemma for ~ yoneda . (Con... |
yonedalem3a 17992 | Lemma for ~ yoneda . (Con... |
yonedalem4a 17993 | Lemma for ~ yoneda . (Con... |
yonedalem4b 17994 | Lemma for ~ yoneda . (Con... |
yonedalem4c 17995 | Lemma for ~ yoneda . (Con... |
yonedalem22 17996 | Lemma for ~ yoneda . (Con... |
yonedalem3b 17997 | Lemma for ~ yoneda . (Con... |
yonedalem3 17998 | Lemma for ~ yoneda . (Con... |
yonedainv 17999 | The Yoneda Lemma with expl... |
yonffthlem 18000 | Lemma for ~ yonffth . (Co... |
yoneda 18001 | The Yoneda Lemma. There i... |
yonffth 18002 | The Yoneda Lemma. The Yon... |
yoniso 18003 | If the codomain is recover... |
oduval 18006 | Value of an order dual str... |
oduleval 18007 | Value of the less-equal re... |
oduleg 18008 | Truth of the less-equal re... |
odubas 18009 | Base set of an order dual ... |
odubasOLD 18010 | Obsolete proof of ~ odubas... |
isprs 18015 | Property of being a preord... |
prslem 18016 | Lemma for ~ prsref and ~ p... |
prsref 18017 | "Less than or equal to" is... |
prstr 18018 | "Less than or equal to" is... |
isdrs 18019 | Property of being a direct... |
drsdir 18020 | Direction of a directed se... |
drsprs 18021 | A directed set is a proset... |
drsbn0 18022 | The base of a directed set... |
drsdirfi 18023 | Any _finite_ number of ele... |
isdrs2 18024 | Directed sets may be defin... |
ispos 18032 | The predicate "is a poset"... |
ispos2 18033 | A poset is an antisymmetri... |
posprs 18034 | A poset is a proset. (Con... |
posi 18035 | Lemma for poset properties... |
posref 18036 | A poset ordering is reflex... |
posasymb 18037 | A poset ordering is asymme... |
postr 18038 | A poset ordering is transi... |
0pos 18039 | Technical lemma to simplif... |
0posOLD 18040 | Obsolete proof of ~ 0pos a... |
isposd 18041 | Properties that determine ... |
isposi 18042 | Properties that determine ... |
isposix 18043 | Properties that determine ... |
isposixOLD 18044 | Obsolete proof of ~ isposi... |
pospropd 18045 | Posethood is determined on... |
odupos 18046 | Being a poset is a self-du... |
oduposb 18047 | Being a poset is a self-du... |
pltfval 18049 | Value of the less-than rel... |
pltval 18050 | Less-than relation. ( ~ d... |
pltle 18051 | "Less than" implies "less ... |
pltne 18052 | The "less than" relation i... |
pltirr 18053 | The "less than" relation i... |
pleval2i 18054 | One direction of ~ pleval2... |
pleval2 18055 | "Less than or equal to" in... |
pltnle 18056 | "Less than" implies not co... |
pltval3 18057 | Alternate expression for t... |
pltnlt 18058 | The less-than relation imp... |
pltn2lp 18059 | The less-than relation has... |
plttr 18060 | The less-than relation is ... |
pltletr 18061 | Transitive law for chained... |
plelttr 18062 | Transitive law for chained... |
pospo 18063 | Write a poset structure in... |
lubfval 18068 | Value of the least upper b... |
lubdm 18069 | Domain of the least upper ... |
lubfun 18070 | The LUB is a function. (C... |
lubeldm 18071 | Member of the domain of th... |
lubelss 18072 | A member of the domain of ... |
lubeu 18073 | Unique existence proper of... |
lubval 18074 | Value of the least upper b... |
lubcl 18075 | The least upper bound func... |
lubprop 18076 | Properties of greatest low... |
luble 18077 | The greatest lower bound i... |
lublecllem 18078 | Lemma for ~ lublecl and ~ ... |
lublecl 18079 | The set of all elements le... |
lubid 18080 | The LUB of elements less t... |
glbfval 18081 | Value of the greatest lowe... |
glbdm 18082 | Domain of the greatest low... |
glbfun 18083 | The GLB is a function. (C... |
glbeldm 18084 | Member of the domain of th... |
glbelss 18085 | A member of the domain of ... |
glbeu 18086 | Unique existence proper of... |
glbval 18087 | Value of the greatest lowe... |
glbcl 18088 | The least upper bound func... |
glbprop 18089 | Properties of greatest low... |
glble 18090 | The greatest lower bound i... |
joinfval 18091 | Value of join function for... |
joinfval2 18092 | Value of join function for... |
joindm 18093 | Domain of join function fo... |
joindef 18094 | Two ways to say that a joi... |
joinval 18095 | Join value. Since both si... |
joincl 18096 | Closure of join of element... |
joindmss 18097 | Subset property of domain ... |
joinval2lem 18098 | Lemma for ~ joinval2 and ~... |
joinval2 18099 | Value of join for a poset ... |
joineu 18100 | Uniqueness of join of elem... |
joinlem 18101 | Lemma for join properties.... |
lejoin1 18102 | A join's first argument is... |
lejoin2 18103 | A join's second argument i... |
joinle 18104 | A join is less than or equ... |
meetfval 18105 | Value of meet function for... |
meetfval2 18106 | Value of meet function for... |
meetdm 18107 | Domain of meet function fo... |
meetdef 18108 | Two ways to say that a mee... |
meetval 18109 | Meet value. Since both si... |
meetcl 18110 | Closure of meet of element... |
meetdmss 18111 | Subset property of domain ... |
meetval2lem 18112 | Lemma for ~ meetval2 and ~... |
meetval2 18113 | Value of meet for a poset ... |
meeteu 18114 | Uniqueness of meet of elem... |
meetlem 18115 | Lemma for meet properties.... |
lemeet1 18116 | A meet's first argument is... |
lemeet2 18117 | A meet's second argument i... |
meetle 18118 | A meet is less than or equ... |
joincomALT 18119 | The join of a poset is com... |
joincom 18120 | The join of a poset is com... |
meetcomALT 18121 | The meet of a poset is com... |
meetcom 18122 | The meet of a poset is com... |
join0 18123 | Lemma for ~ odumeet . (Co... |
meet0 18124 | Lemma for ~ odujoin . (Co... |
odulub 18125 | Least upper bounds in a du... |
odujoin 18126 | Joins in a dual order are ... |
oduglb 18127 | Greatest lower bounds in a... |
odumeet 18128 | Meets in a dual order are ... |
poslubmo 18129 | Least upper bounds in a po... |
posglbmo 18130 | Greatest lower bounds in a... |
poslubd 18131 | Properties which determine... |
poslubdg 18132 | Properties which determine... |
posglbdg 18133 | Properties which determine... |
istos 18136 | The predicate "is a toset"... |
tosso 18137 | Write the totally ordered ... |
tospos 18138 | A Toset is a Poset. (Cont... |
tleile 18139 | In a Toset, any two elemen... |
tltnle 18140 | In a Toset, "less than" is... |
p0val 18145 | Value of poset zero. (Con... |
p1val 18146 | Value of poset zero. (Con... |
p0le 18147 | Any element is less than o... |
ple1 18148 | Any element is less than o... |
islat 18151 | The predicate "is a lattic... |
odulatb 18152 | Being a lattice is self-du... |
odulat 18153 | Being a lattice is self-du... |
latcl2 18154 | The join and meet of any t... |
latlem 18155 | Lemma for lattice properti... |
latpos 18156 | A lattice is a poset. (Co... |
latjcl 18157 | Closure of join operation ... |
latmcl 18158 | Closure of meet operation ... |
latref 18159 | A lattice ordering is refl... |
latasymb 18160 | A lattice ordering is asym... |
latasym 18161 | A lattice ordering is asym... |
lattr 18162 | A lattice ordering is tran... |
latasymd 18163 | Deduce equality from latti... |
lattrd 18164 | A lattice ordering is tran... |
latjcom 18165 | The join of a lattice comm... |
latlej1 18166 | A join's first argument is... |
latlej2 18167 | A join's second argument i... |
latjle12 18168 | A join is less than or equ... |
latleeqj1 18169 | "Less than or equal to" in... |
latleeqj2 18170 | "Less than or equal to" in... |
latjlej1 18171 | Add join to both sides of ... |
latjlej2 18172 | Add join to both sides of ... |
latjlej12 18173 | Add join to both sides of ... |
latnlej 18174 | An idiom to express that a... |
latnlej1l 18175 | An idiom to express that a... |
latnlej1r 18176 | An idiom to express that a... |
latnlej2 18177 | An idiom to express that a... |
latnlej2l 18178 | An idiom to express that a... |
latnlej2r 18179 | An idiom to express that a... |
latjidm 18180 | Lattice join is idempotent... |
latmcom 18181 | The join of a lattice comm... |
latmle1 18182 | A meet is less than or equ... |
latmle2 18183 | A meet is less than or equ... |
latlem12 18184 | An element is less than or... |
latleeqm1 18185 | "Less than or equal to" in... |
latleeqm2 18186 | "Less than or equal to" in... |
latmlem1 18187 | Add meet to both sides of ... |
latmlem2 18188 | Add meet to both sides of ... |
latmlem12 18189 | Add join to both sides of ... |
latnlemlt 18190 | Negation of "less than or ... |
latnle 18191 | Equivalent expressions for... |
latmidm 18192 | Lattice meet is idempotent... |
latabs1 18193 | Lattice absorption law. F... |
latabs2 18194 | Lattice absorption law. F... |
latledi 18195 | An ortholattice is distrib... |
latmlej11 18196 | Ordering of a meet and joi... |
latmlej12 18197 | Ordering of a meet and joi... |
latmlej21 18198 | Ordering of a meet and joi... |
latmlej22 18199 | Ordering of a meet and joi... |
lubsn 18200 | The least upper bound of a... |
latjass 18201 | Lattice join is associativ... |
latj12 18202 | Swap 1st and 2nd members o... |
latj32 18203 | Swap 2nd and 3rd members o... |
latj13 18204 | Swap 1st and 3rd members o... |
latj31 18205 | Swap 2nd and 3rd members o... |
latjrot 18206 | Rotate lattice join of 3 c... |
latj4 18207 | Rearrangement of lattice j... |
latj4rot 18208 | Rotate lattice join of 4 c... |
latjjdi 18209 | Lattice join distributes o... |
latjjdir 18210 | Lattice join distributes o... |
mod1ile 18211 | The weak direction of the ... |
mod2ile 18212 | The weak direction of the ... |
latmass 18213 | Lattice meet is associativ... |
latdisdlem 18214 | Lemma for ~ latdisd . (Co... |
latdisd 18215 | In a lattice, joins distri... |
isclat 18218 | The predicate "is a comple... |
clatpos 18219 | A complete lattice is a po... |
clatlem 18220 | Lemma for properties of a ... |
clatlubcl 18221 | Any subset of the base set... |
clatlubcl2 18222 | Any subset of the base set... |
clatglbcl 18223 | Any subset of the base set... |
clatglbcl2 18224 | Any subset of the base set... |
oduclatb 18225 | Being a complete lattice i... |
clatl 18226 | A complete lattice is a la... |
isglbd 18227 | Properties that determine ... |
lublem 18228 | Lemma for the least upper ... |
lubub 18229 | The LUB of a complete latt... |
lubl 18230 | The LUB of a complete latt... |
lubss 18231 | Subset law for least upper... |
lubel 18232 | An element of a set is les... |
lubun 18233 | The LUB of a union. (Cont... |
clatglb 18234 | Properties of greatest low... |
clatglble 18235 | The greatest lower bound i... |
clatleglb 18236 | Two ways of expressing "le... |
clatglbss 18237 | Subset law for greatest lo... |
isdlat 18240 | Property of being a distri... |
dlatmjdi 18241 | In a distributive lattice,... |
dlatl 18242 | A distributive lattice is ... |
odudlatb 18243 | The dual of a distributive... |
dlatjmdi 18244 | In a distributive lattice,... |
ipostr 18247 | The structure of ~ df-ipo ... |
ipoval 18248 | Value of the inclusion pos... |
ipobas 18249 | Base set of the inclusion ... |
ipolerval 18250 | Relation of the inclusion ... |
ipotset 18251 | Topology of the inclusion ... |
ipole 18252 | Weak order condition of th... |
ipolt 18253 | Strict order condition of ... |
ipopos 18254 | The inclusion poset on a f... |
isipodrs 18255 | Condition for a family of ... |
ipodrscl 18256 | Direction by inclusion as ... |
ipodrsfi 18257 | Finite upper bound propert... |
fpwipodrs 18258 | The finite subsets of any ... |
ipodrsima 18259 | The monotone image of a di... |
isacs3lem 18260 | An algebraic closure syste... |
acsdrsel 18261 | An algebraic closure syste... |
isacs4lem 18262 | In a closure system in whi... |
isacs5lem 18263 | If closure commutes with d... |
acsdrscl 18264 | In an algebraic closure sy... |
acsficl 18265 | A closure in an algebraic ... |
isacs5 18266 | A closure system is algebr... |
isacs4 18267 | A closure system is algebr... |
isacs3 18268 | A closure system is algebr... |
acsficld 18269 | In an algebraic closure sy... |
acsficl2d 18270 | In an algebraic closure sy... |
acsfiindd 18271 | In an algebraic closure sy... |
acsmapd 18272 | In an algebraic closure sy... |
acsmap2d 18273 | In an algebraic closure sy... |
acsinfd 18274 | In an algebraic closure sy... |
acsdomd 18275 | In an algebraic closure sy... |
acsinfdimd 18276 | In an algebraic closure sy... |
acsexdimd 18277 | In an algebraic closure sy... |
mrelatglb 18278 | Greatest lower bounds in a... |
mrelatglb0 18279 | The empty intersection in ... |
mrelatlub 18280 | Least upper bounds in a Mo... |
mreclatBAD 18281 | A Moore space is a complet... |
isps 18286 | The predicate "is a poset"... |
psrel 18287 | A poset is a relation. (C... |
psref2 18288 | A poset is antisymmetric a... |
pstr2 18289 | A poset is transitive. (C... |
pslem 18290 | Lemma for ~ psref and othe... |
psdmrn 18291 | The domain and range of a ... |
psref 18292 | A poset is reflexive. (Co... |
psrn 18293 | The range of a poset equal... |
psasym 18294 | A poset is antisymmetric. ... |
pstr 18295 | A poset is transitive. (C... |
cnvps 18296 | The converse of a poset is... |
cnvpsb 18297 | The converse of a poset is... |
psss 18298 | Any subset of a partially ... |
psssdm2 18299 | Field of a subposet. (Con... |
psssdm 18300 | Field of a subposet. (Con... |
istsr 18301 | The predicate is a toset. ... |
istsr2 18302 | The predicate is a toset. ... |
tsrlin 18303 | A toset is a linear order.... |
tsrlemax 18304 | Two ways of saying a numbe... |
tsrps 18305 | A toset is a poset. (Cont... |
cnvtsr 18306 | The converse of a toset is... |
tsrss 18307 | Any subset of a totally or... |
ledm 18308 | The domain of ` <_ ` is ` ... |
lern 18309 | The range of ` <_ ` is ` R... |
lefld 18310 | The field of the 'less or ... |
letsr 18311 | The "less than or equal to... |
isdir 18316 | A condition for a relation... |
reldir 18317 | A direction is a relation.... |
dirdm 18318 | A direction's domain is eq... |
dirref 18319 | A direction is reflexive. ... |
dirtr 18320 | A direction is transitive.... |
dirge 18321 | For any two elements of a ... |
tsrdir 18322 | A totally ordered set is a... |
ismgm 18327 | The predicate "is a magma"... |
ismgmn0 18328 | The predicate "is a magma"... |
mgmcl 18329 | Closure of the operation o... |
isnmgm 18330 | A condition for a structur... |
mgmsscl 18331 | If the base set of a magma... |
plusffval 18332 | The group addition operati... |
plusfval 18333 | The group addition operati... |
plusfeq 18334 | If the addition operation ... |
plusffn 18335 | The group addition operati... |
mgmplusf 18336 | The group addition functio... |
issstrmgm 18337 | Characterize a substructur... |
intopsn 18338 | The internal operation for... |
mgmb1mgm1 18339 | The only magma with a base... |
mgm0 18340 | Any set with an empty base... |
mgm0b 18341 | The structure with an empt... |
mgm1 18342 | The structure with one ele... |
opifismgm 18343 | A structure with a group a... |
mgmidmo 18344 | A two-sided identity eleme... |
grpidval 18345 | The value of the identity ... |
grpidpropd 18346 | If two structures have the... |
fn0g 18347 | The group zero extractor i... |
0g0 18348 | The identity element funct... |
ismgmid 18349 | The identity element of a ... |
mgmidcl 18350 | The identity element of a ... |
mgmlrid 18351 | The identity element of a ... |
ismgmid2 18352 | Show that a given element ... |
lidrideqd 18353 | If there is a left and rig... |
lidrididd 18354 | If there is a left and rig... |
grpidd 18355 | Deduce the identity elemen... |
mgmidsssn0 18356 | Property of the set of ide... |
grprinvlem 18357 | Lemma for ~ grprinvd . (C... |
grprinvd 18358 | Deduce right inverse from ... |
grpridd 18359 | Deduce right identity from... |
gsumvalx 18360 | Expand out the substitutio... |
gsumval 18361 | Expand out the substitutio... |
gsumpropd 18362 | The group sum depends only... |
gsumpropd2lem 18363 | Lemma for ~ gsumpropd2 . ... |
gsumpropd2 18364 | A stronger version of ~ gs... |
gsummgmpropd 18365 | A stronger version of ~ gs... |
gsumress 18366 | The group sum in a substru... |
gsumval1 18367 | Value of the group sum ope... |
gsum0 18368 | Value of the empty group s... |
gsumval2a 18369 | Value of the group sum ope... |
gsumval2 18370 | Value of the group sum ope... |
gsumsplit1r 18371 | Splitting off the rightmos... |
gsumprval 18372 | Value of the group sum ope... |
gsumpr12val 18373 | Value of the group sum ope... |
issgrp 18376 | The predicate "is a semigr... |
issgrpv 18377 | The predicate "is a semigr... |
issgrpn0 18378 | The predicate "is a semigr... |
isnsgrp 18379 | A condition for a structur... |
sgrpmgm 18380 | A semigroup is a magma. (... |
sgrpass 18381 | A semigroup operation is a... |
sgrp0 18382 | Any set with an empty base... |
sgrp0b 18383 | The structure with an empt... |
sgrp1 18384 | The structure with one ele... |
ismnddef 18387 | The predicate "is a monoid... |
ismnd 18388 | The predicate "is a monoid... |
isnmnd 18389 | A condition for a structur... |
sgrpidmnd 18390 | A semigroup with an identi... |
mndsgrp 18391 | A monoid is a semigroup. ... |
mndmgm 18392 | A monoid is a magma. (Con... |
mndcl 18393 | Closure of the operation o... |
mndass 18394 | A monoid operation is asso... |
mndid 18395 | A monoid has a two-sided i... |
mndideu 18396 | The two-sided identity ele... |
mnd32g 18397 | Commutative/associative la... |
mnd12g 18398 | Commutative/associative la... |
mnd4g 18399 | Commutative/associative la... |
mndidcl 18400 | The identity element of a ... |
mndbn0 18401 | The base set of a monoid i... |
hashfinmndnn 18402 | A finite monoid has positi... |
mndplusf 18403 | The group addition operati... |
mndlrid 18404 | A monoid's identity elemen... |
mndlid 18405 | The identity element of a ... |
mndrid 18406 | The identity element of a ... |
ismndd 18407 | Deduce a monoid from its p... |
mndpfo 18408 | The addition operation of ... |
mndfo 18409 | The addition operation of ... |
mndpropd 18410 | If two structures have the... |
mndprop 18411 | If two structures have the... |
issubmnd 18412 | Characterize a submonoid b... |
ress0g 18413 | ` 0g ` is unaffected by re... |
submnd0 18414 | The zero of a submonoid is... |
mndinvmod 18415 | Uniqueness of an inverse e... |
prdsplusgcl 18416 | Structure product pointwis... |
prdsidlem 18417 | Characterization of identi... |
prdsmndd 18418 | The product of a family of... |
prds0g 18419 | Zero in a product of monoi... |
pwsmnd 18420 | The structure power of a m... |
pws0g 18421 | Zero in a structure power ... |
imasmnd2 18422 | The image structure of a m... |
imasmnd 18423 | The image structure of a m... |
imasmndf1 18424 | The image of a monoid unde... |
xpsmnd 18425 | The binary product of mono... |
mnd1 18426 | The (smallest) structure r... |
mnd1id 18427 | The singleton element of a... |
ismhm 18432 | Property of a monoid homom... |
mhmrcl1 18433 | Reverse closure of a monoi... |
mhmrcl2 18434 | Reverse closure of a monoi... |
mhmf 18435 | A monoid homomorphism is a... |
mhmpropd 18436 | Monoid homomorphism depend... |
mhmlin 18437 | A monoid homomorphism comm... |
mhm0 18438 | A monoid homomorphism pres... |
idmhm 18439 | The identity homomorphism ... |
mhmf1o 18440 | A monoid homomorphism is b... |
submrcl 18441 | Reverse closure for submon... |
issubm 18442 | Expand definition of a sub... |
issubm2 18443 | Submonoids are subsets tha... |
issubmndb 18444 | The submonoid predicate. ... |
issubmd 18445 | Deduction for proving a su... |
mndissubm 18446 | If the base set of a monoi... |
resmndismnd 18447 | If the base set of a monoi... |
submss 18448 | Submonoids are subsets of ... |
submid 18449 | Every monoid is trivially ... |
subm0cl 18450 | Submonoids contain zero. ... |
submcl 18451 | Submonoids are closed unde... |
submmnd 18452 | Submonoids are themselves ... |
submbas 18453 | The base set of a submonoi... |
subm0 18454 | Submonoids have the same i... |
subsubm 18455 | A submonoid of a submonoid... |
0subm 18456 | The zero submonoid of an a... |
insubm 18457 | The intersection of two su... |
0mhm 18458 | The constant zero linear f... |
resmhm 18459 | Restriction of a monoid ho... |
resmhm2 18460 | One direction of ~ resmhm2... |
resmhm2b 18461 | Restriction of the codomai... |
mhmco 18462 | The composition of monoid ... |
mhmima 18463 | The homomorphic image of a... |
mhmeql 18464 | The equalizer of two monoi... |
submacs 18465 | Submonoids are an algebrai... |
mndind 18466 | Induction in a monoid. In... |
prdspjmhm 18467 | A projection from a produc... |
pwspjmhm 18468 | A projection from a struct... |
pwsdiagmhm 18469 | Diagonal monoid homomorphi... |
pwsco1mhm 18470 | Right composition with a f... |
pwsco2mhm 18471 | Left composition with a mo... |
gsumvallem2 18472 | Lemma for properties of th... |
gsumsubm 18473 | Evaluate a group sum in a ... |
gsumz 18474 | Value of a group sum over ... |
gsumwsubmcl 18475 | Closure of the composite i... |
gsumws1 18476 | A singleton composite reco... |
gsumwcl 18477 | Closure of the composite o... |
gsumsgrpccat 18478 | Homomorphic property of no... |
gsumccatOLD 18479 | Obsolete version of ~ gsum... |
gsumccat 18480 | Homomorphic property of co... |
gsumws2 18481 | Valuation of a pair in a m... |
gsumccatsn 18482 | Homomorphic property of co... |
gsumspl 18483 | The primary purpose of the... |
gsumwmhm 18484 | Behavior of homomorphisms ... |
gsumwspan 18485 | The submonoid generated by... |
frmdval 18490 | Value of the free monoid c... |
frmdbas 18491 | The base set of a free mon... |
frmdelbas 18492 | An element of the base set... |
frmdplusg 18493 | The monoid operation of a ... |
frmdadd 18494 | Value of the monoid operat... |
vrmdfval 18495 | The canonical injection fr... |
vrmdval 18496 | The value of the generatin... |
vrmdf 18497 | The mapping from the index... |
frmdmnd 18498 | A free monoid is a monoid.... |
frmd0 18499 | The identity of the free m... |
frmdsssubm 18500 | The set of words taking va... |
frmdgsum 18501 | Any word in a free monoid ... |
frmdss2 18502 | A subset of generators is ... |
frmdup1 18503 | Any assignment of the gene... |
frmdup2 18504 | The evaluation map has the... |
frmdup3lem 18505 | Lemma for ~ frmdup3 . (Co... |
frmdup3 18506 | Universal property of the ... |
efmnd 18509 | The monoid of endofunction... |
efmndbas 18510 | The base set of the monoid... |
efmndbasabf 18511 | The base set of the monoid... |
elefmndbas 18512 | Two ways of saying a funct... |
elefmndbas2 18513 | Two ways of saying a funct... |
efmndbasf 18514 | Elements in the monoid of ... |
efmndhash 18515 | The monoid of endofunction... |
efmndbasfi 18516 | The monoid of endofunction... |
efmndfv 18517 | The function value of an e... |
efmndtset 18518 | The topology of the monoid... |
efmndplusg 18519 | The group operation of a m... |
efmndov 18520 | The value of the group ope... |
efmndcl 18521 | The group operation of the... |
efmndtopn 18522 | The topology of the monoid... |
symggrplem 18523 | Lemma for ~ symggrp and ~ ... |
efmndmgm 18524 | The monoid of endofunction... |
efmndsgrp 18525 | The monoid of endofunction... |
ielefmnd 18526 | The identity function rest... |
efmndid 18527 | The identity function rest... |
efmndmnd 18528 | The monoid of endofunction... |
efmnd0nmnd 18529 | Even the monoid of endofun... |
efmndbas0 18530 | The base set of the monoid... |
efmnd1hash 18531 | The monoid of endofunction... |
efmnd1bas 18532 | The monoid of endofunction... |
efmnd2hash 18533 | The monoid of endofunction... |
submefmnd 18534 | If the base set of a monoi... |
sursubmefmnd 18535 | The set of surjective endo... |
injsubmefmnd 18536 | The set of injective endof... |
idressubmefmnd 18537 | The singleton containing o... |
idresefmnd 18538 | The structure with the sin... |
smndex1ibas 18539 | The modulo function ` I ` ... |
smndex1iidm 18540 | The modulo function ` I ` ... |
smndex1gbas 18541 | The constant functions ` (... |
smndex1gid 18542 | The composition of a const... |
smndex1igid 18543 | The composition of the mod... |
smndex1basss 18544 | The modulo function ` I ` ... |
smndex1bas 18545 | The base set of the monoid... |
smndex1mgm 18546 | The monoid of endofunction... |
smndex1sgrp 18547 | The monoid of endofunction... |
smndex1mndlem 18548 | Lemma for ~ smndex1mnd and... |
smndex1mnd 18549 | The monoid of endofunction... |
smndex1id 18550 | The modulo function ` I ` ... |
smndex1n0mnd 18551 | The identity of the monoid... |
nsmndex1 18552 | The base set ` B ` of the ... |
smndex2dbas 18553 | The doubling function ` D ... |
smndex2dnrinv 18554 | The doubling function ` D ... |
smndex2hbas 18555 | The halving functions ` H ... |
smndex2dlinvh 18556 | The halving functions ` H ... |
mgm2nsgrplem1 18557 | Lemma 1 for ~ mgm2nsgrp : ... |
mgm2nsgrplem2 18558 | Lemma 2 for ~ mgm2nsgrp . ... |
mgm2nsgrplem3 18559 | Lemma 3 for ~ mgm2nsgrp . ... |
mgm2nsgrplem4 18560 | Lemma 4 for ~ mgm2nsgrp : ... |
mgm2nsgrp 18561 | A small magma (with two el... |
sgrp2nmndlem1 18562 | Lemma 1 for ~ sgrp2nmnd : ... |
sgrp2nmndlem2 18563 | Lemma 2 for ~ sgrp2nmnd . ... |
sgrp2nmndlem3 18564 | Lemma 3 for ~ sgrp2nmnd . ... |
sgrp2rid2 18565 | A small semigroup (with tw... |
sgrp2rid2ex 18566 | A small semigroup (with tw... |
sgrp2nmndlem4 18567 | Lemma 4 for ~ sgrp2nmnd : ... |
sgrp2nmndlem5 18568 | Lemma 5 for ~ sgrp2nmnd : ... |
sgrp2nmnd 18569 | A small semigroup (with tw... |
mgmnsgrpex 18570 | There is a magma which is ... |
sgrpnmndex 18571 | There is a semigroup which... |
sgrpssmgm 18572 | The class of all semigroup... |
mndsssgrp 18573 | The class of all monoids i... |
pwmndgplus 18574 | The operation of the monoi... |
pwmndid 18575 | The identity of the monoid... |
pwmnd 18576 | The power set of a class `... |
isgrp 18583 | The predicate "is a group"... |
grpmnd 18584 | A group is a monoid. (Con... |
grpcl 18585 | Closure of the operation o... |
grpass 18586 | A group operation is assoc... |
grpinvex 18587 | Every member of a group ha... |
grpideu 18588 | The two-sided identity ele... |
grpmndd 18589 | A group is a monoid. (Con... |
grpcld 18590 | Closure of the operation o... |
grpplusf 18591 | The group addition operati... |
grpplusfo 18592 | The group addition operati... |
resgrpplusfrn 18593 | The underlying set of a gr... |
grppropd 18594 | If two structures have the... |
grpprop 18595 | If two structures have the... |
grppropstr 18596 | Generalize a specific 2-el... |
grpss 18597 | Show that a structure exte... |
isgrpd2e 18598 | Deduce a group from its pr... |
isgrpd2 18599 | Deduce a group from its pr... |
isgrpde 18600 | Deduce a group from its pr... |
isgrpd 18601 | Deduce a group from its pr... |
isgrpi 18602 | Properties that determine ... |
grpsgrp 18603 | A group is a semigroup. (... |
dfgrp2 18604 | Alternate definition of a ... |
dfgrp2e 18605 | Alternate definition of a ... |
isgrpix 18606 | Properties that determine ... |
grpidcl 18607 | The identity element of a ... |
grpbn0 18608 | The base set of a group is... |
grplid 18609 | The identity element of a ... |
grprid 18610 | The identity element of a ... |
grpn0 18611 | A group is not empty. (Co... |
hashfingrpnn 18612 | A finite group has positiv... |
grprcan 18613 | Right cancellation law for... |
grpinveu 18614 | The left inverse element o... |
grpid 18615 | Two ways of saying that an... |
isgrpid2 18616 | Properties showing that an... |
grpidd2 18617 | Deduce the identity elemen... |
grpinvfval 18618 | The inverse function of a ... |
grpinvfvalALT 18619 | Shorter proof of ~ grpinvf... |
grpinvval 18620 | The inverse of a group ele... |
grpinvfn 18621 | Functionality of the group... |
grpinvfvi 18622 | The group inverse function... |
grpsubfval 18623 | Group subtraction (divisio... |
grpsubfvalALT 18624 | Shorter proof of ~ grpsubf... |
grpsubval 18625 | Group subtraction (divisio... |
grpinvf 18626 | The group inversion operat... |
grpinvcl 18627 | A group element's inverse ... |
grplinv 18628 | The left inverse of a grou... |
grprinv 18629 | The right inverse of a gro... |
grpinvid1 18630 | The inverse of a group ele... |
grpinvid2 18631 | The inverse of a group ele... |
isgrpinv 18632 | Properties showing that a ... |
grplrinv 18633 | In a group, every member h... |
grpidinv2 18634 | A group's properties using... |
grpidinv 18635 | A group has a left and rig... |
grpinvid 18636 | The inverse of the identit... |
grplcan 18637 | Left cancellation law for ... |
grpasscan1 18638 | An associative cancellatio... |
grpasscan2 18639 | An associative cancellatio... |
grpidrcan 18640 | If right adding an element... |
grpidlcan 18641 | If left adding an element ... |
grpinvinv 18642 | Double inverse law for gro... |
grpinvcnv 18643 | The group inverse is its o... |
grpinv11 18644 | The group inverse is one-t... |
grpinvf1o 18645 | The group inverse is a one... |
grpinvnz 18646 | The inverse of a nonzero g... |
grpinvnzcl 18647 | The inverse of a nonzero g... |
grpsubinv 18648 | Subtraction of an inverse.... |
grplmulf1o 18649 | Left multiplication by a g... |
grpinvpropd 18650 | If two structures have the... |
grpidssd 18651 | If the base set of a group... |
grpinvssd 18652 | If the base set of a group... |
grpinvadd 18653 | The inverse of the group o... |
grpsubf 18654 | Functionality of group sub... |
grpsubcl 18655 | Closure of group subtracti... |
grpsubrcan 18656 | Right cancellation law for... |
grpinvsub 18657 | Inverse of a group subtrac... |
grpinvval2 18658 | A ~ df-neg -like equation ... |
grpsubid 18659 | Subtraction of a group ele... |
grpsubid1 18660 | Subtraction of the identit... |
grpsubeq0 18661 | If the difference between ... |
grpsubadd0sub 18662 | Subtraction expressed as a... |
grpsubadd 18663 | Relationship between group... |
grpsubsub 18664 | Double group subtraction. ... |
grpaddsubass 18665 | Associative-type law for g... |
grppncan 18666 | Cancellation law for subtr... |
grpnpcan 18667 | Cancellation law for subtr... |
grpsubsub4 18668 | Double group subtraction (... |
grppnpcan2 18669 | Cancellation law for mixed... |
grpnpncan 18670 | Cancellation law for group... |
grpnpncan0 18671 | Cancellation law for group... |
grpnnncan2 18672 | Cancellation law for group... |
dfgrp3lem 18673 | Lemma for ~ dfgrp3 . (Con... |
dfgrp3 18674 | Alternate definition of a ... |
dfgrp3e 18675 | Alternate definition of a ... |
grplactfval 18676 | The left group action of e... |
grplactval 18677 | The value of the left grou... |
grplactcnv 18678 | The left group action of e... |
grplactf1o 18679 | The left group action of e... |
grpsubpropd 18680 | Weak property deduction fo... |
grpsubpropd2 18681 | Strong property deduction ... |
grp1 18682 | The (smallest) structure r... |
grp1inv 18683 | The inverse function of th... |
prdsinvlem 18684 | Characterization of invers... |
prdsgrpd 18685 | The product of a family of... |
prdsinvgd 18686 | Negation in a product of g... |
pwsgrp 18687 | A structure power of a gro... |
pwsinvg 18688 | Negation in a group power.... |
pwssub 18689 | Subtraction in a group pow... |
imasgrp2 18690 | The image structure of a g... |
imasgrp 18691 | The image structure of a g... |
imasgrpf1 18692 | The image of a group under... |
qusgrp2 18693 | Prove that a quotient stru... |
xpsgrp 18694 | The binary product of grou... |
mhmlem 18695 | Lemma for ~ mhmmnd and ~ g... |
mhmid 18696 | A surjective monoid morphi... |
mhmmnd 18697 | The image of a monoid ` G ... |
mhmfmhm 18698 | The function fulfilling th... |
ghmgrp 18699 | The image of a group ` G `... |
mulgfval 18702 | Group multiple (exponentia... |
mulgfvalALT 18703 | Shorter proof of ~ mulgfva... |
mulgval 18704 | Value of the group multipl... |
mulgfn 18705 | Functionality of the group... |
mulgfvi 18706 | The group multiple operati... |
mulg0 18707 | Group multiple (exponentia... |
mulgnn 18708 | Group multiple (exponentia... |
mulgnngsum 18709 | Group multiple (exponentia... |
mulgnn0gsum 18710 | Group multiple (exponentia... |
mulg1 18711 | Group multiple (exponentia... |
mulgnnp1 18712 | Group multiple (exponentia... |
mulg2 18713 | Group multiple (exponentia... |
mulgnegnn 18714 | Group multiple (exponentia... |
mulgnn0p1 18715 | Group multiple (exponentia... |
mulgnnsubcl 18716 | Closure of the group multi... |
mulgnn0subcl 18717 | Closure of the group multi... |
mulgsubcl 18718 | Closure of the group multi... |
mulgnncl 18719 | Closure of the group multi... |
mulgnn0cl 18720 | Closure of the group multi... |
mulgcl 18721 | Closure of the group multi... |
mulgneg 18722 | Group multiple (exponentia... |
mulgnegneg 18723 | The inverse of a negative ... |
mulgm1 18724 | Group multiple (exponentia... |
mulgcld 18725 | Deduction associated with ... |
mulgaddcomlem 18726 | Lemma for ~ mulgaddcom . ... |
mulgaddcom 18727 | The group multiple operato... |
mulginvcom 18728 | The group multiple operato... |
mulginvinv 18729 | The group multiple operato... |
mulgnn0z 18730 | A group multiple of the id... |
mulgz 18731 | A group multiple of the id... |
mulgnndir 18732 | Sum of group multiples, fo... |
mulgnn0dir 18733 | Sum of group multiples, ge... |
mulgdirlem 18734 | Lemma for ~ mulgdir . (Co... |
mulgdir 18735 | Sum of group multiples, ge... |
mulgp1 18736 | Group multiple (exponentia... |
mulgneg2 18737 | Group multiple (exponentia... |
mulgnnass 18738 | Product of group multiples... |
mulgnn0ass 18739 | Product of group multiples... |
mulgass 18740 | Product of group multiples... |
mulgassr 18741 | Reversed product of group ... |
mulgmodid 18742 | Casting out multiples of t... |
mulgsubdir 18743 | Subtraction of a group ele... |
mhmmulg 18744 | A homomorphism of monoids ... |
mulgpropd 18745 | Two structures with the sa... |
submmulgcl 18746 | Closure of the group multi... |
submmulg 18747 | A group multiple is the sa... |
pwsmulg 18748 | Value of a group multiple ... |
issubg 18755 | The subgroup predicate. (... |
subgss 18756 | A subgroup is a subset. (... |
subgid 18757 | A group is a subgroup of i... |
subggrp 18758 | A subgroup is a group. (C... |
subgbas 18759 | The base of the restricted... |
subgrcl 18760 | Reverse closure for the su... |
subg0 18761 | A subgroup of a group must... |
subginv 18762 | The inverse of an element ... |
subg0cl 18763 | The group identity is an e... |
subginvcl 18764 | The inverse of an element ... |
subgcl 18765 | A subgroup is closed under... |
subgsubcl 18766 | A subgroup is closed under... |
subgsub 18767 | The subtraction of element... |
subgmulgcl 18768 | Closure of the group multi... |
subgmulg 18769 | A group multiple is the sa... |
issubg2 18770 | Characterize the subgroups... |
issubgrpd2 18771 | Prove a subgroup by closur... |
issubgrpd 18772 | Prove a subgroup by closur... |
issubg3 18773 | A subgroup is a symmetric ... |
issubg4 18774 | A subgroup is a nonempty s... |
grpissubg 18775 | If the base set of a group... |
resgrpisgrp 18776 | If the base set of a group... |
subgsubm 18777 | A subgroup is a submonoid.... |
subsubg 18778 | A subgroup of a subgroup i... |
subgint 18779 | The intersection of a none... |
0subg 18780 | The zero subgroup of an ar... |
trivsubgd 18781 | The only subgroup of a tri... |
trivsubgsnd 18782 | The only subgroup of a tri... |
isnsg 18783 | Property of being a normal... |
isnsg2 18784 | Weaken the condition of ~ ... |
nsgbi 18785 | Defining property of a nor... |
nsgsubg 18786 | A normal subgroup is a sub... |
nsgconj 18787 | The conjugation of an elem... |
isnsg3 18788 | A subgroup is normal iff t... |
subgacs 18789 | Subgroups are an algebraic... |
nsgacs 18790 | Normal subgroups form an a... |
elnmz 18791 | Elementhood in the normali... |
nmzbi 18792 | Defining property of the n... |
nmzsubg 18793 | The normalizer N_G(S) of a... |
ssnmz 18794 | A subgroup is a subset of ... |
isnsg4 18795 | A subgroup is normal iff i... |
nmznsg 18796 | Any subgroup is a normal s... |
0nsg 18797 | The zero subgroup is norma... |
nsgid 18798 | The whole group is a norma... |
0idnsgd 18799 | The whole group and the ze... |
trivnsgd 18800 | The only normal subgroup o... |
triv1nsgd 18801 | A trivial group has exactl... |
1nsgtrivd 18802 | A group with exactly one n... |
releqg 18803 | The left coset equivalence... |
eqgfval 18804 | Value of the subgroup left... |
eqgval 18805 | Value of the subgroup left... |
eqger 18806 | The subgroup coset equival... |
eqglact 18807 | A left coset can be expres... |
eqgid 18808 | The left coset containing ... |
eqgen 18809 | Each coset is equipotent t... |
eqgcpbl 18810 | The subgroup coset equival... |
qusgrp 18811 | If ` Y ` is a normal subgr... |
quseccl 18812 | Closure of the quotient ma... |
qusadd 18813 | Value of the group operati... |
qus0 18814 | Value of the group identit... |
qusinv 18815 | Value of the group inverse... |
qussub 18816 | Value of the group subtrac... |
lagsubg2 18817 | Lagrange's theorem for fin... |
lagsubg 18818 | Lagrange's theorem for Gro... |
cycsubmel 18819 | Characterization of an ele... |
cycsubmcl 18820 | The set of nonnegative int... |
cycsubm 18821 | The set of nonnegative int... |
cyccom 18822 | Condition for an operation... |
cycsubmcom 18823 | The operation of a monoid ... |
cycsubggend 18824 | The cyclic subgroup genera... |
cycsubgcl 18825 | The set of integer powers ... |
cycsubgss 18826 | The cyclic subgroup genera... |
cycsubg 18827 | The cyclic group generated... |
cycsubgcld 18828 | The cyclic subgroup genera... |
cycsubg2 18829 | The subgroup generated by ... |
cycsubg2cl 18830 | Any multiple of an element... |
reldmghm 18833 | Lemma for group homomorphi... |
isghm 18834 | Property of being a homomo... |
isghm3 18835 | Property of a group homomo... |
ghmgrp1 18836 | A group homomorphism is on... |
ghmgrp2 18837 | A group homomorphism is on... |
ghmf 18838 | A group homomorphism is a ... |
ghmlin 18839 | A homomorphism of groups i... |
ghmid 18840 | A homomorphism of groups p... |
ghminv 18841 | A homomorphism of groups p... |
ghmsub 18842 | Linearity of subtraction t... |
isghmd 18843 | Deduction for a group homo... |
ghmmhm 18844 | A group homomorphism is a ... |
ghmmhmb 18845 | Group homomorphisms and mo... |
ghmmulg 18846 | A homomorphism of monoids ... |
ghmrn 18847 | The range of a homomorphis... |
0ghm 18848 | The constant zero linear f... |
idghm 18849 | The identity homomorphism ... |
resghm 18850 | Restriction of a homomorph... |
resghm2 18851 | One direction of ~ resghm2... |
resghm2b 18852 | Restriction of the codomai... |
ghmghmrn 18853 | A group homomorphism from ... |
ghmco 18854 | The composition of group h... |
ghmima 18855 | The image of a subgroup un... |
ghmpreima 18856 | The inverse image of a sub... |
ghmeql 18857 | The equalizer of two group... |
ghmnsgima 18858 | The image of a normal subg... |
ghmnsgpreima 18859 | The inverse image of a nor... |
ghmker 18860 | The kernel of a homomorphi... |
ghmeqker 18861 | Two source points map to t... |
pwsdiagghm 18862 | Diagonal homomorphism into... |
ghmf1 18863 | Two ways of saying a group... |
ghmf1o 18864 | A bijective group homomorp... |
conjghm 18865 | Conjugation is an automorp... |
conjsubg 18866 | A conjugated subgroup is a... |
conjsubgen 18867 | A conjugated subgroup is e... |
conjnmz 18868 | A subgroup is unchanged un... |
conjnmzb 18869 | Alternative condition for ... |
conjnsg 18870 | A normal subgroup is uncha... |
qusghm 18871 | If ` Y ` is a normal subgr... |
ghmpropd 18872 | Group homomorphism depends... |
gimfn 18877 | The group isomorphism func... |
isgim 18878 | An isomorphism of groups i... |
gimf1o 18879 | An isomorphism of groups i... |
gimghm 18880 | An isomorphism of groups i... |
isgim2 18881 | A group isomorphism is a h... |
subggim 18882 | Behavior of subgroups unde... |
gimcnv 18883 | The converse of a bijectiv... |
gimco 18884 | The composition of group i... |
brgic 18885 | The relation "is isomorphi... |
brgici 18886 | Prove isomorphic by an exp... |
gicref 18887 | Isomorphism is reflexive. ... |
giclcl 18888 | Isomorphism implies the le... |
gicrcl 18889 | Isomorphism implies the ri... |
gicsym 18890 | Isomorphism is symmetric. ... |
gictr 18891 | Isomorphism is transitive.... |
gicer 18892 | Isomorphism is an equivale... |
gicen 18893 | Isomorphic groups have equ... |
gicsubgen 18894 | A less trivial example of ... |
isga 18897 | The predicate "is a (left)... |
gagrp 18898 | The left argument of a gro... |
gaset 18899 | The right argument of a gr... |
gagrpid 18900 | The identity of the group ... |
gaf 18901 | The mapping of the group a... |
gafo 18902 | A group action is onto its... |
gaass 18903 | An "associative" property ... |
ga0 18904 | The action of a group on t... |
gaid 18905 | The trivial action of a gr... |
subgga 18906 | A subgroup acts on its par... |
gass 18907 | A subset of a group action... |
gasubg 18908 | The restriction of a group... |
gaid2 18909 | A group operation is a lef... |
galcan 18910 | The action of a particular... |
gacan 18911 | Group inverses cancel in a... |
gapm 18912 | The action of a particular... |
gaorb 18913 | The orbit equivalence rela... |
gaorber 18914 | The orbit equivalence rela... |
gastacl 18915 | The stabilizer subgroup in... |
gastacos 18916 | Write the coset relation f... |
orbstafun 18917 | Existence and uniqueness f... |
orbstaval 18918 | Value of the function at a... |
orbsta 18919 | The Orbit-Stabilizer theor... |
orbsta2 18920 | Relation between the size ... |
cntrval 18925 | Substitute definition of t... |
cntzfval 18926 | First level substitution f... |
cntzval 18927 | Definition substitution fo... |
elcntz 18928 | Elementhood in the central... |
cntzel 18929 | Membership in a centralize... |
cntzsnval 18930 | Special substitution for t... |
elcntzsn 18931 | Value of the centralizer o... |
sscntz 18932 | A centralizer expression f... |
cntzrcl 18933 | Reverse closure for elemen... |
cntzssv 18934 | The centralizer is uncondi... |
cntzi 18935 | Membership in a centralize... |
cntrss 18936 | The center is a subset of ... |
cntri 18937 | Defining property of the c... |
resscntz 18938 | Centralizer in a substruct... |
cntz2ss 18939 | Centralizers reverse the s... |
cntzrec 18940 | Reciprocity relationship f... |
cntziinsn 18941 | Express any centralizer as... |
cntzsubm 18942 | Centralizers in a monoid a... |
cntzsubg 18943 | Centralizers in a group ar... |
cntzidss 18944 | If the elements of ` S ` c... |
cntzmhm 18945 | Centralizers in a monoid a... |
cntzmhm2 18946 | Centralizers in a monoid a... |
cntrsubgnsg 18947 | A central subgroup is norm... |
cntrnsg 18948 | The center of a group is a... |
oppgval 18951 | Value of the opposite grou... |
oppgplusfval 18952 | Value of the addition oper... |
oppgplus 18953 | Value of the addition oper... |
setsplusg 18954 | The other components of an... |
oppglemOLD 18955 | Obsolete version of ~ sets... |
oppgbas 18956 | Base set of an opposite gr... |
oppgbasOLD 18957 | Obsolete version of ~ oppg... |
oppgtset 18958 | Topology of an opposite gr... |
oppgtsetOLD 18959 | Obsolete version of ~ oppg... |
oppgtopn 18960 | Topology of an opposite gr... |
oppgmnd 18961 | The opposite of a monoid i... |
oppgmndb 18962 | Bidirectional form of ~ op... |
oppgid 18963 | Zero in a monoid is a symm... |
oppggrp 18964 | The opposite of a group is... |
oppggrpb 18965 | Bidirectional form of ~ op... |
oppginv 18966 | Inverses in a group are a ... |
invoppggim 18967 | The inverse is an antiauto... |
oppggic 18968 | Every group is (naturally)... |
oppgsubm 18969 | Being a submonoid is a sym... |
oppgsubg 18970 | Being a subgroup is a symm... |
oppgcntz 18971 | A centralizer in a group i... |
oppgcntr 18972 | The center of a group is t... |
gsumwrev 18973 | A sum in an opposite monoi... |
symgval 18976 | The value of the symmetric... |
permsetexOLD 18977 | Obsolete version of ~ f1os... |
symgbas 18978 | The base set of the symmet... |
symgbasexOLD 18979 | Obsolete as of 8-Aug-2024.... |
elsymgbas2 18980 | Two ways of saying a funct... |
elsymgbas 18981 | Two ways of saying a funct... |
symgbasf1o 18982 | Elements in the symmetric ... |
symgbasf 18983 | A permutation (element of ... |
symgbasmap 18984 | A permutation (element of ... |
symghash 18985 | The symmetric group on ` n... |
symgbasfi 18986 | The symmetric group on a f... |
symgfv 18987 | The function value of a pe... |
symgfvne 18988 | The function values of a p... |
symgressbas 18989 | The symmetric group on ` A... |
symgplusg 18990 | The group operation of a s... |
symgov 18991 | The value of the group ope... |
symgcl 18992 | The group operation of the... |
idresperm 18993 | The identity function rest... |
symgmov1 18994 | For a permutation of a set... |
symgmov2 18995 | For a permutation of a set... |
symgbas0 18996 | The base set of the symmet... |
symg1hash 18997 | The symmetric group on a s... |
symg1bas 18998 | The symmetric group on a s... |
symg2hash 18999 | The symmetric group on a (... |
symg2bas 19000 | The symmetric group on a p... |
0symgefmndeq 19001 | The symmetric group on the... |
snsymgefmndeq 19002 | The symmetric group on a s... |
symgpssefmnd 19003 | For a set ` A ` with more ... |
symgvalstruct 19004 | The value of the symmetric... |
symgvalstructOLD 19005 | Obsolete proof of ~ symgva... |
symgsubmefmnd 19006 | The symmetric group on a s... |
symgtset 19007 | The topology of the symmet... |
symggrp 19008 | The symmetric group on a s... |
symgid 19009 | The group identity element... |
symginv 19010 | The group inverse in the s... |
symgsubmefmndALT 19011 | The symmetric group on a s... |
galactghm 19012 | The currying of a group ac... |
lactghmga 19013 | The converse of ~ galactgh... |
symgtopn 19014 | The topology of the symmet... |
symgga 19015 | The symmetric group induce... |
pgrpsubgsymgbi 19016 | Every permutation group is... |
pgrpsubgsymg 19017 | Every permutation group is... |
idressubgsymg 19018 | The singleton containing o... |
idrespermg 19019 | The structure with the sin... |
cayleylem1 19020 | Lemma for ~ cayley . (Con... |
cayleylem2 19021 | Lemma for ~ cayley . (Con... |
cayley 19022 | Cayley's Theorem (construc... |
cayleyth 19023 | Cayley's Theorem (existenc... |
symgfix2 19024 | If a permutation does not ... |
symgextf 19025 | The extension of a permuta... |
symgextfv 19026 | The function value of the ... |
symgextfve 19027 | The function value of the ... |
symgextf1lem 19028 | Lemma for ~ symgextf1 . (... |
symgextf1 19029 | The extension of a permuta... |
symgextfo 19030 | The extension of a permuta... |
symgextf1o 19031 | The extension of a permuta... |
symgextsymg 19032 | The extension of a permuta... |
symgextres 19033 | The restriction of the ext... |
gsumccatsymgsn 19034 | Homomorphic property of co... |
gsmsymgrfixlem1 19035 | Lemma 1 for ~ gsmsymgrfix ... |
gsmsymgrfix 19036 | The composition of permuta... |
fvcosymgeq 19037 | The values of two composit... |
gsmsymgreqlem1 19038 | Lemma 1 for ~ gsmsymgreq .... |
gsmsymgreqlem2 19039 | Lemma 2 for ~ gsmsymgreq .... |
gsmsymgreq 19040 | Two combination of permuta... |
symgfixelq 19041 | A permutation of a set fix... |
symgfixels 19042 | The restriction of a permu... |
symgfixelsi 19043 | The restriction of a permu... |
symgfixf 19044 | The mapping of a permutati... |
symgfixf1 19045 | The mapping of a permutati... |
symgfixfolem1 19046 | Lemma 1 for ~ symgfixfo . ... |
symgfixfo 19047 | The mapping of a permutati... |
symgfixf1o 19048 | The mapping of a permutati... |
f1omvdmvd 19051 | A permutation of any class... |
f1omvdcnv 19052 | A permutation and its inve... |
mvdco 19053 | Composing two permutations... |
f1omvdconj 19054 | Conjugation of a permutati... |
f1otrspeq 19055 | A transposition is charact... |
f1omvdco2 19056 | If exactly one of two perm... |
f1omvdco3 19057 | If a point is moved by exa... |
pmtrfval 19058 | The function generating tr... |
pmtrval 19059 | A generated transposition,... |
pmtrfv 19060 | General value of mapping a... |
pmtrprfv 19061 | In a transposition of two ... |
pmtrprfv3 19062 | In a transposition of two ... |
pmtrf 19063 | Functionality of a transpo... |
pmtrmvd 19064 | A transposition moves prec... |
pmtrrn 19065 | Transposing two points giv... |
pmtrfrn 19066 | A transposition (as a kind... |
pmtrffv 19067 | Mapping of a point under a... |
pmtrrn2 19068 | For any transposition ther... |
pmtrfinv 19069 | A transposition function i... |
pmtrfmvdn0 19070 | A transposition moves at l... |
pmtrff1o 19071 | A transposition function i... |
pmtrfcnv 19072 | A transposition function i... |
pmtrfb 19073 | An intrinsic characterizat... |
pmtrfconj 19074 | Any conjugate of a transpo... |
symgsssg 19075 | The symmetric group has su... |
symgfisg 19076 | The symmetric group has a ... |
symgtrf 19077 | Transpositions are element... |
symggen 19078 | The span of the transposit... |
symggen2 19079 | A finite permutation group... |
symgtrinv 19080 | To invert a permutation re... |
pmtr3ncomlem1 19081 | Lemma 1 for ~ pmtr3ncom . ... |
pmtr3ncomlem2 19082 | Lemma 2 for ~ pmtr3ncom . ... |
pmtr3ncom 19083 | Transpositions over sets w... |
pmtrdifellem1 19084 | Lemma 1 for ~ pmtrdifel . ... |
pmtrdifellem2 19085 | Lemma 2 for ~ pmtrdifel . ... |
pmtrdifellem3 19086 | Lemma 3 for ~ pmtrdifel . ... |
pmtrdifellem4 19087 | Lemma 4 for ~ pmtrdifel . ... |
pmtrdifel 19088 | A transposition of element... |
pmtrdifwrdellem1 19089 | Lemma 1 for ~ pmtrdifwrdel... |
pmtrdifwrdellem2 19090 | Lemma 2 for ~ pmtrdifwrdel... |
pmtrdifwrdellem3 19091 | Lemma 3 for ~ pmtrdifwrdel... |
pmtrdifwrdel2lem1 19092 | Lemma 1 for ~ pmtrdifwrdel... |
pmtrdifwrdel 19093 | A sequence of transpositio... |
pmtrdifwrdel2 19094 | A sequence of transpositio... |
pmtrprfval 19095 | The transpositions on a pa... |
pmtrprfvalrn 19096 | The range of the transposi... |
psgnunilem1 19101 | Lemma for ~ psgnuni . Giv... |
psgnunilem5 19102 | Lemma for ~ psgnuni . It ... |
psgnunilem2 19103 | Lemma for ~ psgnuni . Ind... |
psgnunilem3 19104 | Lemma for ~ psgnuni . Any... |
psgnunilem4 19105 | Lemma for ~ psgnuni . An ... |
m1expaddsub 19106 | Addition and subtraction o... |
psgnuni 19107 | If the same permutation ca... |
psgnfval 19108 | Function definition of the... |
psgnfn 19109 | Functionality and domain o... |
psgndmsubg 19110 | The finitary permutations ... |
psgneldm 19111 | Property of being a finita... |
psgneldm2 19112 | The finitary permutations ... |
psgneldm2i 19113 | A sequence of transpositio... |
psgneu 19114 | A finitary permutation has... |
psgnval 19115 | Value of the permutation s... |
psgnvali 19116 | A finitary permutation has... |
psgnvalii 19117 | Any representation of a pe... |
psgnpmtr 19118 | All transpositions are odd... |
psgn0fv0 19119 | The permutation sign funct... |
sygbasnfpfi 19120 | The class of non-fixed poi... |
psgnfvalfi 19121 | Function definition of the... |
psgnvalfi 19122 | Value of the permutation s... |
psgnran 19123 | The range of the permutati... |
gsmtrcl 19124 | The group sum of transposi... |
psgnfitr 19125 | A permutation of a finite ... |
psgnfieu 19126 | A permutation of a finite ... |
pmtrsn 19127 | The value of the transposi... |
psgnsn 19128 | The permutation sign funct... |
psgnprfval 19129 | The permutation sign funct... |
psgnprfval1 19130 | The permutation sign of th... |
psgnprfval2 19131 | The permutation sign of th... |
odfval 19140 | Value of the order functio... |
odfvalALT 19141 | Shorter proof of ~ odfval ... |
odval 19142 | Second substitution for th... |
odlem1 19143 | The group element order is... |
odcl 19144 | The order of a group eleme... |
odf 19145 | Functionality of the group... |
odid 19146 | Any element to the power o... |
odlem2 19147 | Any positive annihilator o... |
odmodnn0 19148 | Reduce the argument of a g... |
mndodconglem 19149 | Lemma for ~ mndodcong . (... |
mndodcong 19150 | If two multipliers are con... |
mndodcongi 19151 | If two multipliers are con... |
oddvdsnn0 19152 | The only multiples of ` A ... |
odnncl 19153 | If a nonzero multiple of a... |
odmod 19154 | Reduce the argument of a g... |
oddvds 19155 | The only multiples of ` A ... |
oddvdsi 19156 | Any group element is annih... |
odcong 19157 | If two multipliers are con... |
odeq 19158 | The ~ oddvds property uniq... |
odval2 19159 | A non-conditional definiti... |
odcld 19160 | The order of a group eleme... |
odmulgid 19161 | A relationship between the... |
odmulg2 19162 | The order of a multiple di... |
odmulg 19163 | Relationship between the o... |
odmulgeq 19164 | A multiple of a point of f... |
odbezout 19165 | If ` N ` is coprime to the... |
od1 19166 | The order of the group ide... |
odeq1 19167 | The group identity is the ... |
odinv 19168 | The order of the inverse o... |
odf1 19169 | The multiples of an elemen... |
odinf 19170 | The multiples of an elemen... |
dfod2 19171 | An alternative definition ... |
odcl2 19172 | The order of an element of... |
oddvds2 19173 | The order of an element of... |
submod 19174 | The order of an element is... |
subgod 19175 | The order of an element is... |
odsubdvds 19176 | The order of an element of... |
odf1o1 19177 | An element with zero order... |
odf1o2 19178 | An element with nonzero or... |
odhash 19179 | An element of zero order g... |
odhash2 19180 | If an element has nonzero ... |
odhash3 19181 | An element which generates... |
odngen 19182 | A cyclic subgroup of size ... |
gexval 19183 | Value of the exponent of a... |
gexlem1 19184 | The group element order is... |
gexcl 19185 | The exponent of a group is... |
gexid 19186 | Any element to the power o... |
gexlem2 19187 | Any positive annihilator o... |
gexdvdsi 19188 | Any group element is annih... |
gexdvds 19189 | The only ` N ` that annihi... |
gexdvds2 19190 | An integer divides the gro... |
gexod 19191 | Any group element is annih... |
gexcl3 19192 | If the order of every grou... |
gexnnod 19193 | Every group element has fi... |
gexcl2 19194 | The exponent of a finite g... |
gexdvds3 19195 | The exponent of a finite g... |
gex1 19196 | A group or monoid has expo... |
ispgp 19197 | A group is a ` P ` -group ... |
pgpprm 19198 | Reverse closure for the fi... |
pgpgrp 19199 | Reverse closure for the se... |
pgpfi1 19200 | A finite group with order ... |
pgp0 19201 | The identity subgroup is a... |
subgpgp 19202 | A subgroup of a p-group is... |
sylow1lem1 19203 | Lemma for ~ sylow1 . The ... |
sylow1lem2 19204 | Lemma for ~ sylow1 . The ... |
sylow1lem3 19205 | Lemma for ~ sylow1 . One ... |
sylow1lem4 19206 | Lemma for ~ sylow1 . The ... |
sylow1lem5 19207 | Lemma for ~ sylow1 . Usin... |
sylow1 19208 | Sylow's first theorem. If... |
odcau 19209 | Cauchy's theorem for the o... |
pgpfi 19210 | The converse to ~ pgpfi1 .... |
pgpfi2 19211 | Alternate version of ~ pgp... |
pgphash 19212 | The order of a p-group. (... |
isslw 19213 | The property of being a Sy... |
slwprm 19214 | Reverse closure for the fi... |
slwsubg 19215 | A Sylow ` P ` -subgroup is... |
slwispgp 19216 | Defining property of a Syl... |
slwpss 19217 | A proper superset of a Syl... |
slwpgp 19218 | A Sylow ` P ` -subgroup is... |
pgpssslw 19219 | Every ` P ` -subgroup is c... |
slwn0 19220 | Every finite group contain... |
subgslw 19221 | A Sylow subgroup that is c... |
sylow2alem1 19222 | Lemma for ~ sylow2a . An ... |
sylow2alem2 19223 | Lemma for ~ sylow2a . All... |
sylow2a 19224 | A named lemma of Sylow's s... |
sylow2blem1 19225 | Lemma for ~ sylow2b . Eva... |
sylow2blem2 19226 | Lemma for ~ sylow2b . Lef... |
sylow2blem3 19227 | Sylow's second theorem. P... |
sylow2b 19228 | Sylow's second theorem. A... |
slwhash 19229 | A sylow subgroup has cardi... |
fislw 19230 | The sylow subgroups of a f... |
sylow2 19231 | Sylow's second theorem. S... |
sylow3lem1 19232 | Lemma for ~ sylow3 , first... |
sylow3lem2 19233 | Lemma for ~ sylow3 , first... |
sylow3lem3 19234 | Lemma for ~ sylow3 , first... |
sylow3lem4 19235 | Lemma for ~ sylow3 , first... |
sylow3lem5 19236 | Lemma for ~ sylow3 , secon... |
sylow3lem6 19237 | Lemma for ~ sylow3 , secon... |
sylow3 19238 | Sylow's third theorem. Th... |
lsmfval 19243 | The subgroup sum function ... |
lsmvalx 19244 | Subspace sum value (for a ... |
lsmelvalx 19245 | Subspace sum membership (f... |
lsmelvalix 19246 | Subspace sum membership (f... |
oppglsm 19247 | The subspace sum operation... |
lsmssv 19248 | Subgroup sum is a subset o... |
lsmless1x 19249 | Subset implies subgroup su... |
lsmless2x 19250 | Subset implies subgroup su... |
lsmub1x 19251 | Subgroup sum is an upper b... |
lsmub2x 19252 | Subgroup sum is an upper b... |
lsmval 19253 | Subgroup sum value (for a ... |
lsmelval 19254 | Subgroup sum membership (f... |
lsmelvali 19255 | Subgroup sum membership (f... |
lsmelvalm 19256 | Subgroup sum membership an... |
lsmelvalmi 19257 | Membership of vector subtr... |
lsmsubm 19258 | The sum of two commuting s... |
lsmsubg 19259 | The sum of two commuting s... |
lsmcom2 19260 | Subgroup sum commutes. (C... |
smndlsmidm 19261 | The direct product is idem... |
lsmub1 19262 | Subgroup sum is an upper b... |
lsmub2 19263 | Subgroup sum is an upper b... |
lsmunss 19264 | Union of subgroups is a su... |
lsmless1 19265 | Subset implies subgroup su... |
lsmless2 19266 | Subset implies subgroup su... |
lsmless12 19267 | Subset implies subgroup su... |
lsmidm 19268 | Subgroup sum is idempotent... |
lsmidmOLD 19269 | Obsolete proof of ~ lsmidm... |
lsmlub 19270 | The least upper bound prop... |
lsmss1 19271 | Subgroup sum with a subset... |
lsmss1b 19272 | Subgroup sum with a subset... |
lsmss2 19273 | Subgroup sum with a subset... |
lsmss2b 19274 | Subgroup sum with a subset... |
lsmass 19275 | Subgroup sum is associativ... |
mndlsmidm 19276 | Subgroup sum is idempotent... |
lsm01 19277 | Subgroup sum with the zero... |
lsm02 19278 | Subgroup sum with the zero... |
subglsm 19279 | The subgroup sum evaluated... |
lssnle 19280 | Equivalent expressions for... |
lsmmod 19281 | The modular law holds for ... |
lsmmod2 19282 | Modular law dual for subgr... |
lsmpropd 19283 | If two structures have the... |
cntzrecd 19284 | Commute the "subgroups com... |
lsmcntz 19285 | The "subgroups commute" pr... |
lsmcntzr 19286 | The "subgroups commute" pr... |
lsmdisj 19287 | Disjointness from a subgro... |
lsmdisj2 19288 | Association of the disjoin... |
lsmdisj3 19289 | Association of the disjoin... |
lsmdisjr 19290 | Disjointness from a subgro... |
lsmdisj2r 19291 | Association of the disjoin... |
lsmdisj3r 19292 | Association of the disjoin... |
lsmdisj2a 19293 | Association of the disjoin... |
lsmdisj2b 19294 | Association of the disjoin... |
lsmdisj3a 19295 | Association of the disjoin... |
lsmdisj3b 19296 | Association of the disjoin... |
subgdisj1 19297 | Vectors belonging to disjo... |
subgdisj2 19298 | Vectors belonging to disjo... |
subgdisjb 19299 | Vectors belonging to disjo... |
pj1fval 19300 | The left projection functi... |
pj1val 19301 | The left projection functi... |
pj1eu 19302 | Uniqueness of a left proje... |
pj1f 19303 | The left projection functi... |
pj2f 19304 | The right projection funct... |
pj1id 19305 | Any element of a direct su... |
pj1eq 19306 | Any element of a direct su... |
pj1lid 19307 | The left projection functi... |
pj1rid 19308 | The left projection functi... |
pj1ghm 19309 | The left projection functi... |
pj1ghm2 19310 | The left projection functi... |
lsmhash 19311 | The order of the direct pr... |
efgmval 19318 | Value of the formal invers... |
efgmf 19319 | The formal inverse operati... |
efgmnvl 19320 | The inversion function on ... |
efgrcl 19321 | Lemma for ~ efgval . (Con... |
efglem 19322 | Lemma for ~ efgval . (Con... |
efgval 19323 | Value of the free group co... |
efger 19324 | Value of the free group co... |
efgi 19325 | Value of the free group co... |
efgi0 19326 | Value of the free group co... |
efgi1 19327 | Value of the free group co... |
efgtf 19328 | Value of the free group co... |
efgtval 19329 | Value of the extension fun... |
efgval2 19330 | Value of the free group co... |
efgi2 19331 | Value of the free group co... |
efgtlen 19332 | Value of the free group co... |
efginvrel2 19333 | The inverse of the reverse... |
efginvrel1 19334 | The inverse of the reverse... |
efgsf 19335 | Value of the auxiliary fun... |
efgsdm 19336 | Elementhood in the domain ... |
efgsval 19337 | Value of the auxiliary fun... |
efgsdmi 19338 | Property of the last link ... |
efgsval2 19339 | Value of the auxiliary fun... |
efgsrel 19340 | The start and end of any e... |
efgs1 19341 | A singleton of an irreduci... |
efgs1b 19342 | Every extension sequence e... |
efgsp1 19343 | If ` F ` is an extension s... |
efgsres 19344 | An initial segment of an e... |
efgsfo 19345 | For any word, there is a s... |
efgredlema 19346 | The reduced word that form... |
efgredlemf 19347 | Lemma for ~ efgredleme . ... |
efgredlemg 19348 | Lemma for ~ efgred . (Con... |
efgredleme 19349 | Lemma for ~ efgred . (Con... |
efgredlemd 19350 | The reduced word that form... |
efgredlemc 19351 | The reduced word that form... |
efgredlemb 19352 | The reduced word that form... |
efgredlem 19353 | The reduced word that form... |
efgred 19354 | The reduced word that form... |
efgrelexlema 19355 | If two words ` A , B ` are... |
efgrelexlemb 19356 | If two words ` A , B ` are... |
efgrelex 19357 | If two words ` A , B ` are... |
efgredeu 19358 | There is a unique reduced ... |
efgred2 19359 | Two extension sequences ha... |
efgcpbllema 19360 | Lemma for ~ efgrelex . De... |
efgcpbllemb 19361 | Lemma for ~ efgrelex . Sh... |
efgcpbl 19362 | Two extension sequences ha... |
efgcpbl2 19363 | Two extension sequences ha... |
frgpval 19364 | Value of the free group co... |
frgpcpbl 19365 | Compatibility of the group... |
frgp0 19366 | The free group is a group.... |
frgpeccl 19367 | Closure of the quotient ma... |
frgpgrp 19368 | The free group is a group.... |
frgpadd 19369 | Addition in the free group... |
frgpinv 19370 | The inverse of an element ... |
frgpmhm 19371 | The "natural map" from wor... |
vrgpfval 19372 | The canonical injection fr... |
vrgpval 19373 | The value of the generatin... |
vrgpf 19374 | The mapping from the index... |
vrgpinv 19375 | The inverse of a generatin... |
frgpuptf 19376 | Any assignment of the gene... |
frgpuptinv 19377 | Any assignment of the gene... |
frgpuplem 19378 | Any assignment of the gene... |
frgpupf 19379 | Any assignment of the gene... |
frgpupval 19380 | Any assignment of the gene... |
frgpup1 19381 | Any assignment of the gene... |
frgpup2 19382 | The evaluation map has the... |
frgpup3lem 19383 | The evaluation map has the... |
frgpup3 19384 | Universal property of the ... |
0frgp 19385 | The free group on zero gen... |
isabl 19390 | The predicate "is an Abeli... |
ablgrp 19391 | An Abelian group is a grou... |
ablgrpd 19392 | An Abelian group is a grou... |
ablcmn 19393 | An Abelian group is a comm... |
iscmn 19394 | The predicate "is a commut... |
isabl2 19395 | The predicate "is an Abeli... |
cmnpropd 19396 | If two structures have the... |
ablpropd 19397 | If two structures have the... |
ablprop 19398 | If two structures have the... |
iscmnd 19399 | Properties that determine ... |
isabld 19400 | Properties that determine ... |
isabli 19401 | Properties that determine ... |
cmnmnd 19402 | A commutative monoid is a ... |
cmncom 19403 | A commutative monoid is co... |
ablcom 19404 | An Abelian group operation... |
cmn32 19405 | Commutative/associative la... |
cmn4 19406 | Commutative/associative la... |
cmn12 19407 | Commutative/associative la... |
abl32 19408 | Commutative/associative la... |
cmnmndd 19409 | A commutative monoid is a ... |
rinvmod 19410 | Uniqueness of a right inve... |
ablinvadd 19411 | The inverse of an Abelian ... |
ablsub2inv 19412 | Abelian group subtraction ... |
ablsubadd 19413 | Relationship between Abeli... |
ablsub4 19414 | Commutative/associative su... |
abladdsub4 19415 | Abelian group addition/sub... |
abladdsub 19416 | Associative-type law for g... |
ablpncan2 19417 | Cancellation law for subtr... |
ablpncan3 19418 | A cancellation law for com... |
ablsubsub 19419 | Law for double subtraction... |
ablsubsub4 19420 | Law for double subtraction... |
ablpnpcan 19421 | Cancellation law for mixed... |
ablnncan 19422 | Cancellation law for group... |
ablsub32 19423 | Swap the second and third ... |
ablnnncan 19424 | Cancellation law for group... |
ablnnncan1 19425 | Cancellation law for group... |
ablsubsub23 19426 | Swap subtrahend and result... |
mulgnn0di 19427 | Group multiple of a sum, f... |
mulgdi 19428 | Group multiple of a sum. ... |
mulgmhm 19429 | The map from ` x ` to ` n ... |
mulgghm 19430 | The map from ` x ` to ` n ... |
mulgsubdi 19431 | Group multiple of a differ... |
ghmfghm 19432 | The function fulfilling th... |
ghmcmn 19433 | The image of a commutative... |
ghmabl 19434 | The image of an abelian gr... |
invghm 19435 | The inversion map is a gro... |
eqgabl 19436 | Value of the subgroup cose... |
subgabl 19437 | A subgroup of an abelian g... |
subcmn 19438 | A submonoid of a commutati... |
submcmn 19439 | A submonoid of a commutati... |
submcmn2 19440 | A submonoid is commutative... |
cntzcmn 19441 | The centralizer of any sub... |
cntzcmnss 19442 | Any subset in a commutativ... |
cntrcmnd 19443 | The center of a monoid is ... |
cntrabl 19444 | The center of a group is a... |
cntzspan 19445 | If the generators commute,... |
cntzcmnf 19446 | Discharge the centralizer ... |
ghmplusg 19447 | The pointwise sum of two l... |
ablnsg 19448 | Every subgroup of an abeli... |
odadd1 19449 | The order of a product in ... |
odadd2 19450 | The order of a product in ... |
odadd 19451 | The order of a product is ... |
gex2abl 19452 | A group with exponent 2 (o... |
gexexlem 19453 | Lemma for ~ gexex . (Cont... |
gexex 19454 | In an abelian group with f... |
torsubg 19455 | The set of all elements of... |
oddvdssubg 19456 | The set of all elements wh... |
lsmcomx 19457 | Subgroup sum commutes (ext... |
ablcntzd 19458 | All subgroups in an abelia... |
lsmcom 19459 | Subgroup sum commutes. (C... |
lsmsubg2 19460 | The sum of two subgroups i... |
lsm4 19461 | Commutative/associative la... |
prdscmnd 19462 | The product of a family of... |
prdsabld 19463 | The product of a family of... |
pwscmn 19464 | The structure power on a c... |
pwsabl 19465 | The structure power on an ... |
qusabl 19466 | If ` Y ` is a subgroup of ... |
abl1 19467 | The (smallest) structure r... |
abln0 19468 | Abelian groups (and theref... |
cnaddablx 19469 | The complex numbers are an... |
cnaddabl 19470 | The complex numbers are an... |
cnaddid 19471 | The group identity element... |
cnaddinv 19472 | Value of the group inverse... |
zaddablx 19473 | The integers are an Abelia... |
frgpnabllem1 19474 | Lemma for ~ frgpnabl . (C... |
frgpnabllem2 19475 | Lemma for ~ frgpnabl . (C... |
frgpnabl 19476 | The free group on two or m... |
iscyg 19479 | Definition of a cyclic gro... |
iscyggen 19480 | The property of being a cy... |
iscyggen2 19481 | The property of being a cy... |
iscyg2 19482 | A cyclic group is a group ... |
cyggeninv 19483 | The inverse of a cyclic ge... |
cyggenod 19484 | An element is the generato... |
cyggenod2 19485 | In an infinite cyclic grou... |
iscyg3 19486 | Definition of a cyclic gro... |
iscygd 19487 | Definition of a cyclic gro... |
iscygodd 19488 | Show that a group with an ... |
cycsubmcmn 19489 | The set of nonnegative int... |
cyggrp 19490 | A cyclic group is a group.... |
cygabl 19491 | A cyclic group is abelian.... |
cygablOLD 19492 | Obsolete proof of ~ cygabl... |
cygctb 19493 | A cyclic group is countabl... |
0cyg 19494 | The trivial group is cycli... |
prmcyg 19495 | A group with prime order i... |
lt6abl 19496 | A group with fewer than ` ... |
ghmcyg 19497 | The image of a cyclic grou... |
cyggex2 19498 | The exponent of a cyclic g... |
cyggex 19499 | The exponent of a finite c... |
cyggexb 19500 | A finite abelian group is ... |
giccyg 19501 | Cyclicity is a group prope... |
cycsubgcyg 19502 | The cyclic subgroup genera... |
cycsubgcyg2 19503 | The cyclic subgroup genera... |
gsumval3a 19504 | Value of the group sum ope... |
gsumval3eu 19505 | The group sum as defined i... |
gsumval3lem1 19506 | Lemma 1 for ~ gsumval3 . ... |
gsumval3lem2 19507 | Lemma 2 for ~ gsumval3 . ... |
gsumval3 19508 | Value of the group sum ope... |
gsumcllem 19509 | Lemma for ~ gsumcl and rel... |
gsumzres 19510 | Extend a finite group sum ... |
gsumzcl2 19511 | Closure of a finite group ... |
gsumzcl 19512 | Closure of a finite group ... |
gsumzf1o 19513 | Re-index a finite group su... |
gsumres 19514 | Extend a finite group sum ... |
gsumcl2 19515 | Closure of a finite group ... |
gsumcl 19516 | Closure of a finite group ... |
gsumf1o 19517 | Re-index a finite group su... |
gsumreidx 19518 | Re-index a finite group su... |
gsumzsubmcl 19519 | Closure of a group sum in ... |
gsumsubmcl 19520 | Closure of a group sum in ... |
gsumsubgcl 19521 | Closure of a group sum in ... |
gsumzaddlem 19522 | The sum of two group sums.... |
gsumzadd 19523 | The sum of two group sums.... |
gsumadd 19524 | The sum of two group sums.... |
gsummptfsadd 19525 | The sum of two group sums ... |
gsummptfidmadd 19526 | The sum of two group sums ... |
gsummptfidmadd2 19527 | The sum of two group sums ... |
gsumzsplit 19528 | Split a group sum into two... |
gsumsplit 19529 | Split a group sum into two... |
gsumsplit2 19530 | Split a group sum into two... |
gsummptfidmsplit 19531 | Split a group sum expresse... |
gsummptfidmsplitres 19532 | Split a group sum expresse... |
gsummptfzsplit 19533 | Split a group sum expresse... |
gsummptfzsplitl 19534 | Split a group sum expresse... |
gsumconst 19535 | Sum of a constant series. ... |
gsumconstf 19536 | Sum of a constant series. ... |
gsummptshft 19537 | Index shift of a finite gr... |
gsumzmhm 19538 | Apply a group homomorphism... |
gsummhm 19539 | Apply a group homomorphism... |
gsummhm2 19540 | Apply a group homomorphism... |
gsummptmhm 19541 | Apply a group homomorphism... |
gsummulglem 19542 | Lemma for ~ gsummulg and ~... |
gsummulg 19543 | Nonnegative multiple of a ... |
gsummulgz 19544 | Integer multiple of a grou... |
gsumzoppg 19545 | The opposite of a group su... |
gsumzinv 19546 | Inverse of a group sum. (... |
gsuminv 19547 | Inverse of a group sum. (... |
gsummptfidminv 19548 | Inverse of a group sum exp... |
gsumsub 19549 | The difference of two grou... |
gsummptfssub 19550 | The difference of two grou... |
gsummptfidmsub 19551 | The difference of two grou... |
gsumsnfd 19552 | Group sum of a singleton, ... |
gsumsnd 19553 | Group sum of a singleton, ... |
gsumsnf 19554 | Group sum of a singleton, ... |
gsumsn 19555 | Group sum of a singleton. ... |
gsumpr 19556 | Group sum of a pair. (Con... |
gsumzunsnd 19557 | Append an element to a fin... |
gsumunsnfd 19558 | Append an element to a fin... |
gsumunsnd 19559 | Append an element to a fin... |
gsumunsnf 19560 | Append an element to a fin... |
gsumunsn 19561 | Append an element to a fin... |
gsumdifsnd 19562 | Extract a summand from a f... |
gsumpt 19563 | Sum of a family that is no... |
gsummptf1o 19564 | Re-index a finite group su... |
gsummptun 19565 | Group sum of a disjoint un... |
gsummpt1n0 19566 | If only one summand in a f... |
gsummptif1n0 19567 | If only one summand in a f... |
gsummptcl 19568 | Closure of a finite group ... |
gsummptfif1o 19569 | Re-index a finite group su... |
gsummptfzcl 19570 | Closure of a finite group ... |
gsum2dlem1 19571 | Lemma 1 for ~ gsum2d . (C... |
gsum2dlem2 19572 | Lemma for ~ gsum2d . (Con... |
gsum2d 19573 | Write a sum over a two-dim... |
gsum2d2lem 19574 | Lemma for ~ gsum2d2 : show... |
gsum2d2 19575 | Write a group sum over a t... |
gsumcom2 19576 | Two-dimensional commutatio... |
gsumxp 19577 | Write a group sum over a c... |
gsumcom 19578 | Commute the arguments of a... |
gsumcom3 19579 | A commutative law for fini... |
gsumcom3fi 19580 | A commutative law for fini... |
gsumxp2 19581 | Write a group sum over a c... |
prdsgsum 19582 | Finite commutative sums in... |
pwsgsum 19583 | Finite commutative sums in... |
fsfnn0gsumfsffz 19584 | Replacing a finitely suppo... |
nn0gsumfz 19585 | Replacing a finitely suppo... |
nn0gsumfz0 19586 | Replacing a finitely suppo... |
gsummptnn0fz 19587 | A final group sum over a f... |
gsummptnn0fzfv 19588 | A final group sum over a f... |
telgsumfzslem 19589 | Lemma for ~ telgsumfzs (in... |
telgsumfzs 19590 | Telescoping group sum rang... |
telgsumfz 19591 | Telescoping group sum rang... |
telgsumfz0s 19592 | Telescoping finite group s... |
telgsumfz0 19593 | Telescoping finite group s... |
telgsums 19594 | Telescoping finitely suppo... |
telgsum 19595 | Telescoping finitely suppo... |
reldmdprd 19600 | The domain of the internal... |
dmdprd 19601 | The domain of definition o... |
dmdprdd 19602 | Show that a given family i... |
dprddomprc 19603 | A family of subgroups inde... |
dprddomcld 19604 | If a family of subgroups i... |
dprdval0prc 19605 | The internal direct produc... |
dprdval 19606 | The value of the internal ... |
eldprd 19607 | A class ` A ` is an intern... |
dprdgrp 19608 | Reverse closure for the in... |
dprdf 19609 | The function ` S ` is a fa... |
dprdf2 19610 | The function ` S ` is a fa... |
dprdcntz 19611 | The function ` S ` is a fa... |
dprddisj 19612 | The function ` S ` is a fa... |
dprdw 19613 | The property of being a fi... |
dprdwd 19614 | A mapping being a finitely... |
dprdff 19615 | A finitely supported funct... |
dprdfcl 19616 | A finitely supported funct... |
dprdffsupp 19617 | A finitely supported funct... |
dprdfcntz 19618 | A function on the elements... |
dprdssv 19619 | The internal direct produc... |
dprdfid 19620 | A function mapping all but... |
eldprdi 19621 | The domain of definition o... |
dprdfinv 19622 | Take the inverse of a grou... |
dprdfadd 19623 | Take the sum of group sums... |
dprdfsub 19624 | Take the difference of gro... |
dprdfeq0 19625 | The zero function is the o... |
dprdf11 19626 | Two group sums over a dire... |
dprdsubg 19627 | The internal direct produc... |
dprdub 19628 | Each factor is a subset of... |
dprdlub 19629 | The direct product is smal... |
dprdspan 19630 | The direct product is the ... |
dprdres 19631 | Restriction of a direct pr... |
dprdss 19632 | Create a direct product by... |
dprdz 19633 | A family consisting entire... |
dprd0 19634 | The empty family is an int... |
dprdf1o 19635 | Rearrange the index set of... |
dprdf1 19636 | Rearrange the index set of... |
subgdmdprd 19637 | A direct product in a subg... |
subgdprd 19638 | A direct product in a subg... |
dprdsn 19639 | A singleton family is an i... |
dmdprdsplitlem 19640 | Lemma for ~ dmdprdsplit . ... |
dprdcntz2 19641 | The function ` S ` is a fa... |
dprddisj2 19642 | The function ` S ` is a fa... |
dprd2dlem2 19643 | The direct product of a co... |
dprd2dlem1 19644 | The direct product of a co... |
dprd2da 19645 | The direct product of a co... |
dprd2db 19646 | The direct product of a co... |
dprd2d2 19647 | The direct product of a co... |
dmdprdsplit2lem 19648 | Lemma for ~ dmdprdsplit . ... |
dmdprdsplit2 19649 | The direct product splits ... |
dmdprdsplit 19650 | The direct product splits ... |
dprdsplit 19651 | The direct product is the ... |
dmdprdpr 19652 | A singleton family is an i... |
dprdpr 19653 | A singleton family is an i... |
dpjlem 19654 | Lemma for theorems about d... |
dpjcntz 19655 | The two subgroups that app... |
dpjdisj 19656 | The two subgroups that app... |
dpjlsm 19657 | The two subgroups that app... |
dpjfval 19658 | Value of the direct produc... |
dpjval 19659 | Value of the direct produc... |
dpjf 19660 | The ` X ` -th index projec... |
dpjidcl 19661 | The key property of projec... |
dpjeq 19662 | Decompose a group sum into... |
dpjid 19663 | The key property of projec... |
dpjlid 19664 | The ` X ` -th index projec... |
dpjrid 19665 | The ` Y ` -th index projec... |
dpjghm 19666 | The direct product is the ... |
dpjghm2 19667 | The direct product is the ... |
ablfacrplem 19668 | Lemma for ~ ablfacrp2 . (... |
ablfacrp 19669 | A finite abelian group who... |
ablfacrp2 19670 | The factors ` K , L ` of ~... |
ablfac1lem 19671 | Lemma for ~ ablfac1b . Sa... |
ablfac1a 19672 | The factors of ~ ablfac1b ... |
ablfac1b 19673 | Any abelian group is the d... |
ablfac1c 19674 | The factors of ~ ablfac1b ... |
ablfac1eulem 19675 | Lemma for ~ ablfac1eu . (... |
ablfac1eu 19676 | The factorization of ~ abl... |
pgpfac1lem1 19677 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1lem2 19678 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1lem3a 19679 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1lem3 19680 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1lem4 19681 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1lem5 19682 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1 19683 | Factorization of a finite ... |
pgpfaclem1 19684 | Lemma for ~ pgpfac . (Con... |
pgpfaclem2 19685 | Lemma for ~ pgpfac . (Con... |
pgpfaclem3 19686 | Lemma for ~ pgpfac . (Con... |
pgpfac 19687 | Full factorization of a fi... |
ablfaclem1 19688 | Lemma for ~ ablfac . (Con... |
ablfaclem2 19689 | Lemma for ~ ablfac . (Con... |
ablfaclem3 19690 | Lemma for ~ ablfac . (Con... |
ablfac 19691 | The Fundamental Theorem of... |
ablfac2 19692 | Choose generators for each... |
issimpg 19695 | The predicate "is a simple... |
issimpgd 19696 | Deduce a simple group from... |
simpggrp 19697 | A simple group is a group.... |
simpggrpd 19698 | A simple group is a group.... |
simpg2nsg 19699 | A simple group has two nor... |
trivnsimpgd 19700 | Trivial groups are not sim... |
simpgntrivd 19701 | Simple groups are nontrivi... |
simpgnideld 19702 | A simple group contains a ... |
simpgnsgd 19703 | The only normal subgroups ... |
simpgnsgeqd 19704 | A normal subgroup of a sim... |
2nsgsimpgd 19705 | If any normal subgroup of ... |
simpgnsgbid 19706 | A nontrivial group is simp... |
ablsimpnosubgd 19707 | A subgroup of an abelian s... |
ablsimpg1gend 19708 | An abelian simple group is... |
ablsimpgcygd 19709 | An abelian simple group is... |
ablsimpgfindlem1 19710 | Lemma for ~ ablsimpgfind .... |
ablsimpgfindlem2 19711 | Lemma for ~ ablsimpgfind .... |
cycsubggenodd 19712 | Relationship between the o... |
ablsimpgfind 19713 | An abelian simple group is... |
fincygsubgd 19714 | The subgroup referenced in... |
fincygsubgodd 19715 | Calculate the order of a s... |
fincygsubgodexd 19716 | A finite cyclic group has ... |
prmgrpsimpgd 19717 | A group of prime order is ... |
ablsimpgprmd 19718 | An abelian simple group ha... |
ablsimpgd 19719 | An abelian group is simple... |
fnmgp 19722 | The multiplicative group o... |
mgpval 19723 | Value of the multiplicatio... |
mgpplusg 19724 | Value of the group operati... |
mgplemOLD 19725 | Obsolete version of ~ sets... |
mgpbas 19726 | Base set of the multiplica... |
mgpbasOLD 19727 | Obsolete version of ~ mgpb... |
mgpsca 19728 | The multiplication monoid ... |
mgpscaOLD 19729 | Obsolete version of ~ mgps... |
mgptset 19730 | Topology component of the ... |
mgptsetOLD 19731 | Obsolete version of ~ mgpt... |
mgptopn 19732 | Topology of the multiplica... |
mgpds 19733 | Distance function of the m... |
mgpdsOLD 19734 | Obsolete version of ~ mgpd... |
mgpress 19735 | Subgroup commutes with the... |
mgpressOLD 19736 | Obsolete version of ~ mgpr... |
ringidval 19739 | The value of the unity ele... |
dfur2 19740 | The multiplicative identit... |
issrg 19743 | The predicate "is a semiri... |
srgcmn 19744 | A semiring is a commutativ... |
srgmnd 19745 | A semiring is a monoid. (... |
srgmgp 19746 | A semiring is a monoid und... |
srgi 19747 | Properties of a semiring. ... |
srgcl 19748 | Closure of the multiplicat... |
srgass 19749 | Associative law for the mu... |
srgideu 19750 | The unit element of a semi... |
srgfcl 19751 | Functionality of the multi... |
srgdi 19752 | Distributive law for the m... |
srgdir 19753 | Distributive law for the m... |
srgidcl 19754 | The unit element of a semi... |
srg0cl 19755 | The zero element of a semi... |
srgidmlem 19756 | Lemma for ~ srglidm and ~ ... |
srglidm 19757 | The unit element of a semi... |
srgridm 19758 | The unit element of a semi... |
issrgid 19759 | Properties showing that an... |
srgacl 19760 | Closure of the addition op... |
srgcom 19761 | Commutativity of the addit... |
srgrz 19762 | The zero of a semiring is ... |
srglz 19763 | The zero of a semiring is ... |
srgisid 19764 | In a semiring, the only le... |
srg1zr 19765 | The only semiring with a b... |
srgen1zr 19766 | The only semiring with one... |
srgmulgass 19767 | An associative property be... |
srgpcomp 19768 | If two elements of a semir... |
srgpcompp 19769 | If two elements of a semir... |
srgpcomppsc 19770 | If two elements of a semir... |
srglmhm 19771 | Left-multiplication in a s... |
srgrmhm 19772 | Right-multiplication in a ... |
srgsummulcr 19773 | A finite semiring sum mult... |
sgsummulcl 19774 | A finite semiring sum mult... |
srg1expzeq1 19775 | The exponentiation (by a n... |
srgbinomlem1 19776 | Lemma 1 for ~ srgbinomlem ... |
srgbinomlem2 19777 | Lemma 2 for ~ srgbinomlem ... |
srgbinomlem3 19778 | Lemma 3 for ~ srgbinomlem ... |
srgbinomlem4 19779 | Lemma 4 for ~ srgbinomlem ... |
srgbinomlem 19780 | Lemma for ~ srgbinom . In... |
srgbinom 19781 | The binomial theorem for c... |
csrgbinom 19782 | The binomial theorem for c... |
isring 19787 | The predicate "is a (unita... |
ringgrp 19788 | A ring is a group. (Contr... |
ringmgp 19789 | A ring is a monoid under m... |
iscrng 19790 | A commutative ring is a ri... |
crngmgp 19791 | A commutative ring's multi... |
ringgrpd 19792 | A ring is a group. (Contr... |
ringmnd 19793 | A ring is a monoid under a... |
ringmgm 19794 | A ring is a magma. (Contr... |
crngring 19795 | A commutative ring is a ri... |
crngringd 19796 | A commutative ring is a ri... |
crnggrpd 19797 | A commutative ring is a gr... |
mgpf 19798 | Restricted functionality o... |
ringi 19799 | Properties of a unital rin... |
ringcl 19800 | Closure of the multiplicat... |
crngcom 19801 | A commutative ring's multi... |
iscrng2 19802 | A commutative ring is a ri... |
ringass 19803 | Associative law for multip... |
ringideu 19804 | The unit element of a ring... |
ringdi 19805 | Distributive law for the m... |
ringdir 19806 | Distributive law for the m... |
ringidcl 19807 | The unit element of a ring... |
ring0cl 19808 | The zero element of a ring... |
ringidmlem 19809 | Lemma for ~ ringlidm and ~... |
ringlidm 19810 | The unit element of a ring... |
ringridm 19811 | The unit element of a ring... |
isringid 19812 | Properties showing that an... |
ringid 19813 | The multiplication operati... |
ringadd2 19814 | A ring element plus itself... |
rngo2times 19815 | A ring element plus itself... |
ringidss 19816 | A subset of the multiplica... |
ringacl 19817 | Closure of the addition op... |
ringcom 19818 | Commutativity of the addit... |
ringabl 19819 | A ring is an Abelian group... |
ringcmn 19820 | A ring is a commutative mo... |
ringpropd 19821 | If two structures have the... |
crngpropd 19822 | If two structures have the... |
ringprop 19823 | If two structures have the... |
isringd 19824 | Properties that determine ... |
iscrngd 19825 | Properties that determine ... |
ringlz 19826 | The zero of a unital ring ... |
ringrz 19827 | The zero of a unital ring ... |
ringsrg 19828 | Any ring is also a semirin... |
ring1eq0 19829 | If one and zero are equal,... |
ring1ne0 19830 | If a ring has at least two... |
ringinvnz1ne0 19831 | In a unitary ring, a left ... |
ringinvnzdiv 19832 | In a unitary ring, a left ... |
ringnegl 19833 | Negation in a ring is the ... |
rngnegr 19834 | Negation in a ring is the ... |
ringmneg1 19835 | Negation of a product in a... |
ringmneg2 19836 | Negation of a product in a... |
ringm2neg 19837 | Double negation of a produ... |
ringsubdi 19838 | Ring multiplication distri... |
rngsubdir 19839 | Ring multiplication distri... |
mulgass2 19840 | An associative property be... |
ring1 19841 | The (smallest) structure r... |
ringn0 19842 | Rings exist. (Contributed... |
ringlghm 19843 | Left-multiplication in a r... |
ringrghm 19844 | Right-multiplication in a ... |
gsummulc1 19845 | A finite ring sum multipli... |
gsummulc2 19846 | A finite ring sum multipli... |
gsummgp0 19847 | If one factor in a finite ... |
gsumdixp 19848 | Distribute a binary produc... |
prdsmgp 19849 | The multiplicative monoid ... |
prdsmulrcl 19850 | A structure product of rin... |
prdsringd 19851 | A product of rings is a ri... |
prdscrngd 19852 | A product of commutative r... |
prds1 19853 | Value of the ring unit in ... |
pwsring 19854 | A structure power of a rin... |
pws1 19855 | Value of the ring unit in ... |
pwscrng 19856 | A structure power of a com... |
pwsmgp 19857 | The multiplicative group o... |
imasring 19858 | The image structure of a r... |
qusring2 19859 | The quotient structure of ... |
crngbinom 19860 | The binomial theorem for c... |
opprval 19863 | Value of the opposite ring... |
opprmulfval 19864 | Value of the multiplicatio... |
opprmul 19865 | Value of the multiplicatio... |
crngoppr 19866 | In a commutative ring, the... |
opprlem 19867 | Lemma for ~ opprbas and ~ ... |
opprlemOLD 19868 | Obsolete version of ~ oppr... |
opprbas 19869 | Base set of an opposite ri... |
opprbasOLD 19870 | Obsolete proof of ~ opprba... |
oppradd 19871 | Addition operation of an o... |
oppraddOLD 19872 | Obsolete proof of ~ opprba... |
opprring 19873 | An opposite ring is a ring... |
opprringb 19874 | Bidirectional form of ~ op... |
oppr0 19875 | Additive identity of an op... |
oppr1 19876 | Multiplicative identity of... |
opprneg 19877 | The negative function in a... |
opprsubg 19878 | Being a subgroup is a symm... |
mulgass3 19879 | An associative property be... |
reldvdsr 19886 | The divides relation is a ... |
dvdsrval 19887 | Value of the divides relat... |
dvdsr 19888 | Value of the divides relat... |
dvdsr2 19889 | Value of the divides relat... |
dvdsrmul 19890 | A left-multiple of ` X ` i... |
dvdsrcl 19891 | Closure of a dividing elem... |
dvdsrcl2 19892 | Closure of a dividing elem... |
dvdsrid 19893 | An element in a (unital) r... |
dvdsrtr 19894 | Divisibility is transitive... |
dvdsrmul1 19895 | The divisibility relation ... |
dvdsrneg 19896 | An element divides its neg... |
dvdsr01 19897 | In a ring, zero is divisib... |
dvdsr02 19898 | Only zero is divisible by ... |
isunit 19899 | Property of being a unit o... |
1unit 19900 | The multiplicative identit... |
unitcl 19901 | A unit is an element of th... |
unitss 19902 | The set of units is contai... |
opprunit 19903 | Being a unit is a symmetri... |
crngunit 19904 | Property of being a unit i... |
dvdsunit 19905 | A divisor of a unit is a u... |
unitmulcl 19906 | The product of units is a ... |
unitmulclb 19907 | Reversal of ~ unitmulcl in... |
unitgrpbas 19908 | The base set of the group ... |
unitgrp 19909 | The group of units is a gr... |
unitabl 19910 | The group of units of a co... |
unitgrpid 19911 | The identity of the multip... |
unitsubm 19912 | The group of units is a su... |
invrfval 19915 | Multiplicative inverse fun... |
unitinvcl 19916 | The inverse of a unit exis... |
unitinvinv 19917 | The inverse of the inverse... |
ringinvcl 19918 | The inverse of a unit is a... |
unitlinv 19919 | A unit times its inverse i... |
unitrinv 19920 | A unit times its inverse i... |
1rinv 19921 | The inverse of the identit... |
0unit 19922 | The additive identity is a... |
unitnegcl 19923 | The negative of a unit is ... |
dvrfval 19926 | Division operation in a ri... |
dvrval 19927 | Division operation in a ri... |
dvrcl 19928 | Closure of division operat... |
unitdvcl 19929 | The units are closed under... |
dvrid 19930 | A cancellation law for div... |
dvr1 19931 | A cancellation law for div... |
dvrass 19932 | An associative law for div... |
dvrcan1 19933 | A cancellation law for div... |
dvrcan3 19934 | A cancellation law for div... |
dvreq1 19935 | A cancellation law for div... |
ringinvdv 19936 | Write the inverse function... |
rngidpropd 19937 | The ring identity depends ... |
dvdsrpropd 19938 | The divisibility relation ... |
unitpropd 19939 | The set of units depends o... |
invrpropd 19940 | The ring inverse function ... |
isirred 19941 | An irreducible element of ... |
isnirred 19942 | The property of being a no... |
isirred2 19943 | Expand out the class diffe... |
opprirred 19944 | Irreducibility is symmetri... |
irredn0 19945 | The additive identity is n... |
irredcl 19946 | An irreducible element is ... |
irrednu 19947 | An irreducible element is ... |
irredn1 19948 | The multiplicative identit... |
irredrmul 19949 | The product of an irreduci... |
irredlmul 19950 | The product of a unit and ... |
irredmul 19951 | If product of two elements... |
irredneg 19952 | The negative of an irreduc... |
irrednegb 19953 | An element is irreducible ... |
dfrhm2 19961 | The property of a ring hom... |
rhmrcl1 19963 | Reverse closure of a ring ... |
rhmrcl2 19964 | Reverse closure of a ring ... |
isrhm 19965 | A function is a ring homom... |
rhmmhm 19966 | A ring homomorphism is a h... |
isrim0 19967 | An isomorphism of rings is... |
rimrcl 19968 | Reverse closure for an iso... |
rhmghm 19969 | A ring homomorphism is an ... |
rhmf 19970 | A ring homomorphism is a f... |
rhmmul 19971 | A homomorphism of rings pr... |
isrhm2d 19972 | Demonstration of ring homo... |
isrhmd 19973 | Demonstration of ring homo... |
rhm1 19974 | Ring homomorphisms are req... |
idrhm 19975 | The identity homomorphism ... |
rhmf1o 19976 | A ring homomorphism is bij... |
isrim 19977 | An isomorphism of rings is... |
rimf1o 19978 | An isomorphism of rings is... |
rimrhm 19979 | An isomorphism of rings is... |
rimgim 19980 | An isomorphism of rings is... |
rhmco 19981 | The composition of ring ho... |
pwsco1rhm 19982 | Right composition with a f... |
pwsco2rhm 19983 | Left composition with a ri... |
f1ghm0to0 19984 | If a group homomorphism ` ... |
f1rhm0to0ALT 19985 | Alternate proof for ~ f1gh... |
gim0to0 19986 | A group isomorphism maps t... |
kerf1ghm 19987 | A group homomorphism ` F `... |
brric 19988 | The relation "is isomorphi... |
brric2 19989 | The relation "is isomorphi... |
ricgic 19990 | If two rings are (ring) is... |
isdrng 19995 | The predicate "is a divisi... |
drngunit 19996 | Elementhood in the set of ... |
drngui 19997 | The set of units of a divi... |
drngring 19998 | A division ring is a ring.... |
drnggrp 19999 | A division ring is a group... |
isfld 20000 | A field is a commutative d... |
isdrng2 20001 | A division ring can equiva... |
drngprop 20002 | If two structures have the... |
drngmgp 20003 | A division ring contains a... |
drngmcl 20004 | The product of two nonzero... |
drngid 20005 | A division ring's unit is ... |
drngunz 20006 | A division ring's unit is ... |
drngid2 20007 | Properties showing that an... |
drnginvrcl 20008 | Closure of the multiplicat... |
drnginvrn0 20009 | The multiplicative inverse... |
drnginvrl 20010 | Property of the multiplica... |
drnginvrr 20011 | Property of the multiplica... |
drngmul0or 20012 | A product is zero iff one ... |
drngmulne0 20013 | A product is nonzero iff b... |
drngmuleq0 20014 | An element is zero iff its... |
opprdrng 20015 | The opposite of a division... |
isdrngd 20016 | Properties that characteri... |
isdrngrd 20017 | Properties that characteri... |
drngpropd 20018 | If two structures have the... |
fldpropd 20019 | If two structures have the... |
issubrg 20024 | The subring predicate. (C... |
subrgss 20025 | A subring is a subset. (C... |
subrgid 20026 | Every ring is a subring of... |
subrgring 20027 | A subring is a ring. (Con... |
subrgcrng 20028 | A subring of a commutative... |
subrgrcl 20029 | Reverse closure for a subr... |
subrgsubg 20030 | A subring is a subgroup. ... |
subrg0 20031 | A subring always has the s... |
subrg1cl 20032 | A subring contains the mul... |
subrgbas 20033 | Base set of a subring stru... |
subrg1 20034 | A subring always has the s... |
subrgacl 20035 | A subring is closed under ... |
subrgmcl 20036 | A subgroup is closed under... |
subrgsubm 20037 | A subring is a submonoid o... |
subrgdvds 20038 | If an element divides anot... |
subrguss 20039 | A unit of a subring is a u... |
subrginv 20040 | A subring always has the s... |
subrgdv 20041 | A subring always has the s... |
subrgunit 20042 | An element of a ring is a ... |
subrgugrp 20043 | The units of a subring for... |
issubrg2 20044 | Characterize the subrings ... |
opprsubrg 20045 | Being a subring is a symme... |
subrgint 20046 | The intersection of a none... |
subrgin 20047 | The intersection of two su... |
subrgmre 20048 | The subrings of a ring are... |
issubdrg 20049 | Characterize the subfields... |
subsubrg 20050 | A subring of a subring is ... |
subsubrg2 20051 | The set of subrings of a s... |
issubrg3 20052 | A subring is an additive s... |
resrhm 20053 | Restriction of a ring homo... |
rhmeql 20054 | The equalizer of two ring ... |
rhmima 20055 | The homomorphic image of a... |
rnrhmsubrg 20056 | The range of a ring homomo... |
cntzsubr 20057 | Centralizers in a ring are... |
pwsdiagrhm 20058 | Diagonal homomorphism into... |
subrgpropd 20059 | If two structures have the... |
rhmpropd 20060 | Ring homomorphism depends ... |
issdrg 20063 | Property of a division sub... |
sdrgid 20064 | Every division ring is a d... |
sdrgss 20065 | A division subring is a su... |
issdrg2 20066 | Property of a division sub... |
acsfn1p 20067 | Construction of a closure ... |
subrgacs 20068 | Closure property of subrin... |
sdrgacs 20069 | Closure property of divisi... |
cntzsdrg 20070 | Centralizers in division r... |
subdrgint 20071 | The intersection of a none... |
sdrgint 20072 | The intersection of a none... |
primefld 20073 | The smallest sub division ... |
primefld0cl 20074 | The prime field contains t... |
primefld1cl 20075 | The prime field contains t... |
abvfval 20078 | Value of the set of absolu... |
isabv 20079 | Elementhood in the set of ... |
isabvd 20080 | Properties that determine ... |
abvrcl 20081 | Reverse closure for the ab... |
abvfge0 20082 | An absolute value is a fun... |
abvf 20083 | An absolute value is a fun... |
abvcl 20084 | An absolute value is a fun... |
abvge0 20085 | The absolute value of a nu... |
abveq0 20086 | The value of an absolute v... |
abvne0 20087 | The absolute value of a no... |
abvgt0 20088 | The absolute value of a no... |
abvmul 20089 | An absolute value distribu... |
abvtri 20090 | An absolute value satisfie... |
abv0 20091 | The absolute value of zero... |
abv1z 20092 | The absolute value of one ... |
abv1 20093 | The absolute value of one ... |
abvneg 20094 | The absolute value of a ne... |
abvsubtri 20095 | An absolute value satisfie... |
abvrec 20096 | The absolute value distrib... |
abvdiv 20097 | The absolute value distrib... |
abvdom 20098 | Any ring with an absolute ... |
abvres 20099 | The restriction of an abso... |
abvtrivd 20100 | The trivial absolute value... |
abvtriv 20101 | The trivial absolute value... |
abvpropd 20102 | If two structures have the... |
staffval 20107 | The functionalization of t... |
stafval 20108 | The functionalization of t... |
staffn 20109 | The functionalization is e... |
issrng 20110 | The predicate "is a star r... |
srngrhm 20111 | The involution function in... |
srngring 20112 | A star ring is a ring. (C... |
srngcnv 20113 | The involution function in... |
srngf1o 20114 | The involution function in... |
srngcl 20115 | The involution function in... |
srngnvl 20116 | The involution function in... |
srngadd 20117 | The involution function in... |
srngmul 20118 | The involution function in... |
srng1 20119 | The conjugate of the ring ... |
srng0 20120 | The conjugate of the ring ... |
issrngd 20121 | Properties that determine ... |
idsrngd 20122 | A commutative ring is a st... |
islmod 20127 | The predicate "is a left m... |
lmodlema 20128 | Lemma for properties of a ... |
islmodd 20129 | Properties that determine ... |
lmodgrp 20130 | A left module is a group. ... |
lmodring 20131 | The scalar component of a ... |
lmodfgrp 20132 | The scalar component of a ... |
lmodbn0 20133 | The base set of a left mod... |
lmodacl 20134 | Closure of ring addition f... |
lmodmcl 20135 | Closure of ring multiplica... |
lmodsn0 20136 | The set of scalars in a le... |
lmodvacl 20137 | Closure of vector addition... |
lmodass 20138 | Left module vector sum is ... |
lmodlcan 20139 | Left cancellation law for ... |
lmodvscl 20140 | Closure of scalar product ... |
scaffval 20141 | The scalar multiplication ... |
scafval 20142 | The scalar multiplication ... |
scafeq 20143 | If the scalar multiplicati... |
scaffn 20144 | The scalar multiplication ... |
lmodscaf 20145 | The scalar multiplication ... |
lmodvsdi 20146 | Distributive law for scala... |
lmodvsdir 20147 | Distributive law for scala... |
lmodvsass 20148 | Associative law for scalar... |
lmod0cl 20149 | The ring zero in a left mo... |
lmod1cl 20150 | The ring unit in a left mo... |
lmodvs1 20151 | Scalar product with ring u... |
lmod0vcl 20152 | The zero vector is a vecto... |
lmod0vlid 20153 | Left identity law for the ... |
lmod0vrid 20154 | Right identity law for the... |
lmod0vid 20155 | Identity equivalent to the... |
lmod0vs 20156 | Zero times a vector is the... |
lmodvs0 20157 | Anything times the zero ve... |
lmodvsmmulgdi 20158 | Distributive law for a gro... |
lmodfopnelem1 20159 | Lemma 1 for ~ lmodfopne . ... |
lmodfopnelem2 20160 | Lemma 2 for ~ lmodfopne . ... |
lmodfopne 20161 | The (functionalized) opera... |
lcomf 20162 | A linear-combination sum i... |
lcomfsupp 20163 | A linear-combination sum i... |
lmodvnegcl 20164 | Closure of vector negative... |
lmodvnegid 20165 | Addition of a vector with ... |
lmodvneg1 20166 | Minus 1 times a vector is ... |
lmodvsneg 20167 | Multiplication of a vector... |
lmodvsubcl 20168 | Closure of vector subtract... |
lmodcom 20169 | Left module vector sum is ... |
lmodabl 20170 | A left module is an abelia... |
lmodcmn 20171 | A left module is a commuta... |
lmodnegadd 20172 | Distribute negation throug... |
lmod4 20173 | Commutative/associative la... |
lmodvsubadd 20174 | Relationship between vecto... |
lmodvaddsub4 20175 | Vector addition/subtractio... |
lmodvpncan 20176 | Addition/subtraction cance... |
lmodvnpcan 20177 | Cancellation law for vecto... |
lmodvsubval2 20178 | Value of vector subtractio... |
lmodsubvs 20179 | Subtraction of a scalar pr... |
lmodsubdi 20180 | Scalar multiplication dist... |
lmodsubdir 20181 | Scalar multiplication dist... |
lmodsubeq0 20182 | If the difference between ... |
lmodsubid 20183 | Subtraction of a vector fr... |
lmodvsghm 20184 | Scalar multiplication of t... |
lmodprop2d 20185 | If two structures have the... |
lmodpropd 20186 | If two structures have the... |
gsumvsmul 20187 | Pull a scalar multiplicati... |
mptscmfsupp0 20188 | A mapping to a scalar prod... |
mptscmfsuppd 20189 | A function mapping to a sc... |
rmodislmodlem 20190 | Lemma for ~ rmodislmod . ... |
rmodislmod 20191 | The right module ` R ` ind... |
rmodislmodOLD 20192 | Obsolete version of ~ rmod... |
lssset 20195 | The set of all (not necess... |
islss 20196 | The predicate "is a subspa... |
islssd 20197 | Properties that determine ... |
lssss 20198 | A subspace is a set of vec... |
lssel 20199 | A subspace member is a vec... |
lss1 20200 | The set of vectors in a le... |
lssuni 20201 | The union of all subspaces... |
lssn0 20202 | A subspace is not empty. ... |
00lss 20203 | The empty structure has no... |
lsscl 20204 | Closure property of a subs... |
lssvsubcl 20205 | Closure of vector subtract... |
lssvancl1 20206 | Non-closure: if one vector... |
lssvancl2 20207 | Non-closure: if one vector... |
lss0cl 20208 | The zero vector belongs to... |
lsssn0 20209 | The singleton of the zero ... |
lss0ss 20210 | The zero subspace is inclu... |
lssle0 20211 | No subspace is smaller tha... |
lssne0 20212 | A nonzero subspace has a n... |
lssvneln0 20213 | A vector ` X ` which doesn... |
lssneln0 20214 | A vector ` X ` which doesn... |
lssssr 20215 | Conclude subspace ordering... |
lssvacl 20216 | Closure of vector addition... |
lssvscl 20217 | Closure of scalar product ... |
lssvnegcl 20218 | Closure of negative vector... |
lsssubg 20219 | All subspaces are subgroup... |
lsssssubg 20220 | All subspaces are subgroup... |
islss3 20221 | A linear subspace of a mod... |
lsslmod 20222 | A submodule is a module. ... |
lsslss 20223 | The subspaces of a subspac... |
islss4 20224 | A linear subspace is a sub... |
lss1d 20225 | One-dimensional subspace (... |
lssintcl 20226 | The intersection of a none... |
lssincl 20227 | The intersection of two su... |
lssmre 20228 | The subspaces of a module ... |
lssacs 20229 | Submodules are an algebrai... |
prdsvscacl 20230 | Pointwise scalar multiplic... |
prdslmodd 20231 | The product of a family of... |
pwslmod 20232 | A structure power of a lef... |
lspfval 20235 | The span function for a le... |
lspf 20236 | The span operator on a lef... |
lspval 20237 | The span of a set of vecto... |
lspcl 20238 | The span of a set of vecto... |
lspsncl 20239 | The span of a singleton is... |
lspprcl 20240 | The span of a pair is a su... |
lsptpcl 20241 | The span of an unordered t... |
lspsnsubg 20242 | The span of a singleton is... |
00lsp 20243 | ~ fvco4i lemma for linear ... |
lspid 20244 | The span of a subspace is ... |
lspssv 20245 | A span is a set of vectors... |
lspss 20246 | Span preserves subset orde... |
lspssid 20247 | A set of vectors is a subs... |
lspidm 20248 | The span of a set of vecto... |
lspun 20249 | The span of union is the s... |
lspssp 20250 | If a set of vectors is a s... |
mrclsp 20251 | Moore closure generalizes ... |
lspsnss 20252 | The span of the singleton ... |
lspsnel3 20253 | A member of the span of th... |
lspprss 20254 | The span of a pair of vect... |
lspsnid 20255 | A vector belongs to the sp... |
lspsnel6 20256 | Relationship between a vec... |
lspsnel5 20257 | Relationship between a vec... |
lspsnel5a 20258 | Relationship between a vec... |
lspprid1 20259 | A member of a pair of vect... |
lspprid2 20260 | A member of a pair of vect... |
lspprvacl 20261 | The sum of two vectors bel... |
lssats2 20262 | A way to express atomistic... |
lspsneli 20263 | A scalar product with a ve... |
lspsn 20264 | Span of the singleton of a... |
lspsnel 20265 | Member of span of the sing... |
lspsnvsi 20266 | Span of a scalar product o... |
lspsnss2 20267 | Comparable spans of single... |
lspsnneg 20268 | Negation does not change t... |
lspsnsub 20269 | Swapping subtraction order... |
lspsn0 20270 | Span of the singleton of t... |
lsp0 20271 | Span of the empty set. (C... |
lspuni0 20272 | Union of the span of the e... |
lspun0 20273 | The span of a union with t... |
lspsneq0 20274 | Span of the singleton is t... |
lspsneq0b 20275 | Equal singleton spans impl... |
lmodindp1 20276 | Two independent (non-colin... |
lsslsp 20277 | Spans in submodules corres... |
lss0v 20278 | The zero vector in a submo... |
lsspropd 20279 | If two structures have the... |
lsppropd 20280 | If two structures have the... |
reldmlmhm 20287 | Lemma for module homomorph... |
lmimfn 20288 | Lemma for module isomorphi... |
islmhm 20289 | Property of being a homomo... |
islmhm3 20290 | Property of a module homom... |
lmhmlem 20291 | Non-quantified consequence... |
lmhmsca 20292 | A homomorphism of left mod... |
lmghm 20293 | A homomorphism of left mod... |
lmhmlmod2 20294 | A homomorphism of left mod... |
lmhmlmod1 20295 | A homomorphism of left mod... |
lmhmf 20296 | A homomorphism of left mod... |
lmhmlin 20297 | A homomorphism of left mod... |
lmodvsinv 20298 | Multiplication of a vector... |
lmodvsinv2 20299 | Multiplying a negated vect... |
islmhm2 20300 | A one-equation proof of li... |
islmhmd 20301 | Deduction for a module hom... |
0lmhm 20302 | The constant zero linear f... |
idlmhm 20303 | The identity function on a... |
invlmhm 20304 | The negative function on a... |
lmhmco 20305 | The composition of two mod... |
lmhmplusg 20306 | The pointwise sum of two l... |
lmhmvsca 20307 | The pointwise scalar produ... |
lmhmf1o 20308 | A bijective module homomor... |
lmhmima 20309 | The image of a subspace un... |
lmhmpreima 20310 | The inverse image of a sub... |
lmhmlsp 20311 | Homomorphisms preserve spa... |
lmhmrnlss 20312 | The range of a homomorphis... |
lmhmkerlss 20313 | The kernel of a homomorphi... |
reslmhm 20314 | Restriction of a homomorph... |
reslmhm2 20315 | Expansion of the codomain ... |
reslmhm2b 20316 | Expansion of the codomain ... |
lmhmeql 20317 | The equalizer of two modul... |
lspextmo 20318 | A linear function is compl... |
pwsdiaglmhm 20319 | Diagonal homomorphism into... |
pwssplit0 20320 | Splitting for structure po... |
pwssplit1 20321 | Splitting for structure po... |
pwssplit2 20322 | Splitting for structure po... |
pwssplit3 20323 | Splitting for structure po... |
islmim 20324 | An isomorphism of left mod... |
lmimf1o 20325 | An isomorphism of left mod... |
lmimlmhm 20326 | An isomorphism of modules ... |
lmimgim 20327 | An isomorphism of modules ... |
islmim2 20328 | An isomorphism of left mod... |
lmimcnv 20329 | The converse of a bijectiv... |
brlmic 20330 | The relation "is isomorphi... |
brlmici 20331 | Prove isomorphic by an exp... |
lmiclcl 20332 | Isomorphism implies the le... |
lmicrcl 20333 | Isomorphism implies the ri... |
lmicsym 20334 | Module isomorphism is symm... |
lmhmpropd 20335 | Module homomorphism depend... |
islbs 20338 | The predicate " ` B ` is a... |
lbsss 20339 | A basis is a set of vector... |
lbsel 20340 | An element of a basis is a... |
lbssp 20341 | The span of a basis is the... |
lbsind 20342 | A basis is linearly indepe... |
lbsind2 20343 | A basis is linearly indepe... |
lbspss 20344 | No proper subset of a basi... |
lsmcl 20345 | The sum of two subspaces i... |
lsmspsn 20346 | Member of subspace sum of ... |
lsmelval2 20347 | Subspace sum membership in... |
lsmsp 20348 | Subspace sum in terms of s... |
lsmsp2 20349 | Subspace sum of spans of s... |
lsmssspx 20350 | Subspace sum (in its exten... |
lsmpr 20351 | The span of a pair of vect... |
lsppreli 20352 | A vector expressed as a su... |
lsmelpr 20353 | Two ways to say that a vec... |
lsppr0 20354 | The span of a vector paire... |
lsppr 20355 | Span of a pair of vectors.... |
lspprel 20356 | Member of the span of a pa... |
lspprabs 20357 | Absorption of vector sum i... |
lspvadd 20358 | The span of a vector sum i... |
lspsntri 20359 | Triangle-type inequality f... |
lspsntrim 20360 | Triangle-type inequality f... |
lbspropd 20361 | If two structures have the... |
pj1lmhm 20362 | The left projection functi... |
pj1lmhm2 20363 | The left projection functi... |
islvec 20366 | The predicate "is a left v... |
lvecdrng 20367 | The set of scalars of a le... |
lveclmod 20368 | A left vector space is a l... |
lsslvec 20369 | A vector subspace is a vec... |
lvecvs0or 20370 | If a scalar product is zer... |
lvecvsn0 20371 | A scalar product is nonzer... |
lssvs0or 20372 | If a scalar product belong... |
lvecvscan 20373 | Cancellation law for scala... |
lvecvscan2 20374 | Cancellation law for scala... |
lvecinv 20375 | Invert coefficient of scal... |
lspsnvs 20376 | A nonzero scalar product d... |
lspsneleq 20377 | Membership relation that i... |
lspsncmp 20378 | Comparable spans of nonzer... |
lspsnne1 20379 | Two ways to express that v... |
lspsnne2 20380 | Two ways to express that v... |
lspsnnecom 20381 | Swap two vectors with diff... |
lspabs2 20382 | Absorption law for span of... |
lspabs3 20383 | Absorption law for span of... |
lspsneq 20384 | Equal spans of singletons ... |
lspsneu 20385 | Nonzero vectors with equal... |
lspsnel4 20386 | A member of the span of th... |
lspdisj 20387 | The span of a vector not i... |
lspdisjb 20388 | A nonzero vector is not in... |
lspdisj2 20389 | Unequal spans are disjoint... |
lspfixed 20390 | Show membership in the spa... |
lspexch 20391 | Exchange property for span... |
lspexchn1 20392 | Exchange property for span... |
lspexchn2 20393 | Exchange property for span... |
lspindpi 20394 | Partial independence prope... |
lspindp1 20395 | Alternate way to say 3 vec... |
lspindp2l 20396 | Alternate way to say 3 vec... |
lspindp2 20397 | Alternate way to say 3 vec... |
lspindp3 20398 | Independence of 2 vectors ... |
lspindp4 20399 | (Partial) independence of ... |
lvecindp 20400 | Compute the ` X ` coeffici... |
lvecindp2 20401 | Sums of independent vector... |
lspsnsubn0 20402 | Unequal singleton spans im... |
lsmcv 20403 | Subspace sum has the cover... |
lspsolvlem 20404 | Lemma for ~ lspsolv . (Co... |
lspsolv 20405 | If ` X ` is in the span of... |
lssacsex 20406 | In a vector space, subspac... |
lspsnat 20407 | There is no subspace stric... |
lspsncv0 20408 | The span of a singleton co... |
lsppratlem1 20409 | Lemma for ~ lspprat . Let... |
lsppratlem2 20410 | Lemma for ~ lspprat . Sho... |
lsppratlem3 20411 | Lemma for ~ lspprat . In ... |
lsppratlem4 20412 | Lemma for ~ lspprat . In ... |
lsppratlem5 20413 | Lemma for ~ lspprat . Com... |
lsppratlem6 20414 | Lemma for ~ lspprat . Neg... |
lspprat 20415 | A proper subspace of the s... |
islbs2 20416 | An equivalent formulation ... |
islbs3 20417 | An equivalent formulation ... |
lbsacsbs 20418 | Being a basis in a vector ... |
lvecdim 20419 | The dimension theorem for ... |
lbsextlem1 20420 | Lemma for ~ lbsext . The ... |
lbsextlem2 20421 | Lemma for ~ lbsext . Sinc... |
lbsextlem3 20422 | Lemma for ~ lbsext . A ch... |
lbsextlem4 20423 | Lemma for ~ lbsext . ~ lbs... |
lbsextg 20424 | For any linearly independe... |
lbsext 20425 | For any linearly independe... |
lbsexg 20426 | Every vector space has a b... |
lbsex 20427 | Every vector space has a b... |
lvecprop2d 20428 | If two structures have the... |
lvecpropd 20429 | If two structures have the... |
sraval 20438 | Lemma for ~ srabase throug... |
sralem 20439 | Lemma for ~ srabase and si... |
sralemOLD 20440 | Obsolete version of ~ sral... |
srabase 20441 | Base set of a subring alge... |
srabaseOLD 20442 | Obsolete proof of ~ srabas... |
sraaddg 20443 | Additive operation of a su... |
sraaddgOLD 20444 | Obsolete proof of ~ sraadd... |
sramulr 20445 | Multiplicative operation o... |
sramulrOLD 20446 | Obsolete proof of ~ sramul... |
srasca 20447 | The set of scalars of a su... |
srascaOLD 20448 | Obsolete proof of ~ srasca... |
sravsca 20449 | The scalar product operati... |
sravscaOLD 20450 | Obsolete proof of ~ sravsc... |
sraip 20451 | The inner product operatio... |
sratset 20452 | Topology component of a su... |
sratsetOLD 20453 | Obsolete proof of ~ sratse... |
sratopn 20454 | Topology component of a su... |
srads 20455 | Distance function of a sub... |
sradsOLD 20456 | Obsolete proof of ~ srads ... |
sralmod 20457 | The subring algebra is a l... |
sralmod0 20458 | The subring module inherit... |
issubrngd2 20459 | Prove a subring by closure... |
rlmfn 20460 | ` ringLMod ` is a function... |
rlmval 20461 | Value of the ring module. ... |
lidlval 20462 | Value of the set of ring i... |
rspval 20463 | Value of the ring span fun... |
rlmval2 20464 | Value of the ring module e... |
rlmbas 20465 | Base set of the ring modul... |
rlmplusg 20466 | Vector addition in the rin... |
rlm0 20467 | Zero vector in the ring mo... |
rlmsub 20468 | Subtraction in the ring mo... |
rlmmulr 20469 | Ring multiplication in the... |
rlmsca 20470 | Scalars in the ring module... |
rlmsca2 20471 | Scalars in the ring module... |
rlmvsca 20472 | Scalar multiplication in t... |
rlmtopn 20473 | Topology component of the ... |
rlmds 20474 | Metric component of the ri... |
rlmlmod 20475 | The ring module is a modul... |
rlmlvec 20476 | The ring module over a div... |
rlmlsm 20477 | Subgroup sum of the ring m... |
rlmvneg 20478 | Vector negation in the rin... |
rlmscaf 20479 | Functionalized scalar mult... |
ixpsnbasval 20480 | The value of an infinite C... |
lidlss 20481 | An ideal is a subset of th... |
islidl 20482 | Predicate of being a (left... |
lidl0cl 20483 | An ideal contains 0. (Con... |
lidlacl 20484 | An ideal is closed under a... |
lidlnegcl 20485 | An ideal contains negative... |
lidlsubg 20486 | An ideal is a subgroup of ... |
lidlsubcl 20487 | An ideal is closed under s... |
lidlmcl 20488 | An ideal is closed under l... |
lidl1el 20489 | An ideal contains 1 iff it... |
lidl0 20490 | Every ring contains a zero... |
lidl1 20491 | Every ring contains a unit... |
lidlacs 20492 | The ideal system is an alg... |
rspcl 20493 | The span of a set of ring ... |
rspssid 20494 | The span of a set of ring ... |
rsp1 20495 | The span of the identity e... |
rsp0 20496 | The span of the zero eleme... |
rspssp 20497 | The ideal span of a set of... |
mrcrsp 20498 | Moore closure generalizes ... |
lidlnz 20499 | A nonzero ideal contains a... |
drngnidl 20500 | A division ring has only t... |
lidlrsppropd 20501 | The left ideals and ring s... |
2idlval 20504 | Definition of a two-sided ... |
2idlcpbl 20505 | The coset equivalence rela... |
qus1 20506 | The multiplicative identit... |
qusring 20507 | If ` S ` is a two-sided id... |
qusrhm 20508 | If ` S ` is a two-sided id... |
crngridl 20509 | In a commutative ring, the... |
crng2idl 20510 | In a commutative ring, a t... |
quscrng 20511 | The quotient of a commutat... |
lpival 20516 | Value of the set of princi... |
islpidl 20517 | Property of being a princi... |
lpi0 20518 | The zero ideal is always p... |
lpi1 20519 | The unit ideal is always p... |
islpir 20520 | Principal ideal rings are ... |
lpiss 20521 | Principal ideals are a sub... |
islpir2 20522 | Principal ideal rings are ... |
lpirring 20523 | Principal ideal rings are ... |
drnglpir 20524 | Division rings are princip... |
rspsn 20525 | Membership in principal id... |
lidldvgen 20526 | An element generates an id... |
lpigen 20527 | An ideal is principal iff ... |
isnzr 20530 | Property of a nonzero ring... |
nzrnz 20531 | One and zero are different... |
nzrring 20532 | A nonzero ring is a ring. ... |
drngnzr 20533 | All division rings are non... |
isnzr2 20534 | Equivalent characterizatio... |
isnzr2hash 20535 | Equivalent characterizatio... |
opprnzr 20536 | The opposite of a nonzero ... |
ringelnzr 20537 | A ring is nonzero if it ha... |
nzrunit 20538 | A unit is nonzero in any n... |
subrgnzr 20539 | A subring of a nonzero rin... |
0ringnnzr 20540 | A ring is a zero ring iff ... |
0ring 20541 | If a ring has only one ele... |
0ring01eq 20542 | In a ring with only one el... |
01eq0ring 20543 | If the zero and the identi... |
0ring01eqbi 20544 | In a unital ring the zero ... |
rng1nnzr 20545 | The (smallest) structure r... |
ring1zr 20546 | The only (unital) ring wit... |
rngen1zr 20547 | The only (unital) ring wit... |
ringen1zr 20548 | The only unital ring with ... |
rng1nfld 20549 | The zero ring is not a fie... |
rrgval 20558 | Value of the set or left-r... |
isrrg 20559 | Membership in the set of l... |
rrgeq0i 20560 | Property of a left-regular... |
rrgeq0 20561 | Left-multiplication by a l... |
rrgsupp 20562 | Left multiplication by a l... |
rrgss 20563 | Left-regular elements are ... |
unitrrg 20564 | Units are regular elements... |
isdomn 20565 | Expand definition of a dom... |
domnnzr 20566 | A domain is a nonzero ring... |
domnring 20567 | A domain is a ring. (Cont... |
domneq0 20568 | In a domain, a product is ... |
domnmuln0 20569 | In a domain, a product of ... |
isdomn2 20570 | A ring is a domain iff all... |
domnrrg 20571 | In a domain, any nonzero e... |
opprdomn 20572 | The opposite of a domain i... |
abvn0b 20573 | Another characterization o... |
drngdomn 20574 | A division ring is a domai... |
isidom 20575 | An integral domain is a co... |
fldidom 20576 | A field is an integral dom... |
fldidomOLD 20577 | Obsolete version of ~ fldi... |
fidomndrnglem 20578 | Lemma for ~ fidomndrng . ... |
fidomndrng 20579 | A finite domain is a divis... |
fiidomfld 20580 | A finite integral domain i... |
cnfldstr 20599 | The field of complex numbe... |
cnfldex 20600 | The field of complex numbe... |
cnfldbas 20601 | The base set of the field ... |
cnfldadd 20602 | The addition operation of ... |
cnfldmul 20603 | The multiplication operati... |
cnfldcj 20604 | The conjugation operation ... |
cnfldtset 20605 | The topology component of ... |
cnfldle 20606 | The ordering of the field ... |
cnfldds 20607 | The metric of the field of... |
cnfldunif 20608 | The uniform structure comp... |
cnfldfun 20609 | The field of complex numbe... |
cnfldfunALT 20610 | The field of complex numbe... |
cnfldfunALTOLD 20611 | Obsolete proof of ~ cnfldf... |
xrsstr 20612 | The extended real structur... |
xrsex 20613 | The extended real structur... |
xrsbas 20614 | The base set of the extend... |
xrsadd 20615 | The addition operation of ... |
xrsmul 20616 | The multiplication operati... |
xrstset 20617 | The topology component of ... |
xrsle 20618 | The ordering of the extend... |
cncrng 20619 | The complex numbers form a... |
cnring 20620 | The complex numbers form a... |
xrsmcmn 20621 | The "multiplicative group"... |
cnfld0 20622 | Zero is the zero element o... |
cnfld1 20623 | One is the unit element of... |
cnfldneg 20624 | The additive inverse in th... |
cnfldplusf 20625 | The functionalized additio... |
cnfldsub 20626 | The subtraction operator i... |
cndrng 20627 | The complex numbers form a... |
cnflddiv 20628 | The division operation in ... |
cnfldinv 20629 | The multiplicative inverse... |
cnfldmulg 20630 | The group multiple functio... |
cnfldexp 20631 | The exponentiation operato... |
cnsrng 20632 | The complex numbers form a... |
xrsmgm 20633 | The "additive group" of th... |
xrsnsgrp 20634 | The "additive group" of th... |
xrsmgmdifsgrp 20635 | The "additive group" of th... |
xrs1mnd 20636 | The extended real numbers,... |
xrs10 20637 | The zero of the extended r... |
xrs1cmn 20638 | The extended real numbers ... |
xrge0subm 20639 | The nonnegative extended r... |
xrge0cmn 20640 | The nonnegative extended r... |
xrsds 20641 | The metric of the extended... |
xrsdsval 20642 | The metric of the extended... |
xrsdsreval 20643 | The metric of the extended... |
xrsdsreclblem 20644 | Lemma for ~ xrsdsreclb . ... |
xrsdsreclb 20645 | The metric of the extended... |
cnsubmlem 20646 | Lemma for ~ nn0subm and fr... |
cnsubglem 20647 | Lemma for ~ resubdrg and f... |
cnsubrglem 20648 | Lemma for ~ resubdrg and f... |
cnsubdrglem 20649 | Lemma for ~ resubdrg and f... |
qsubdrg 20650 | The rational numbers form ... |
zsubrg 20651 | The integers form a subrin... |
gzsubrg 20652 | The gaussian integers form... |
nn0subm 20653 | The nonnegative integers f... |
rege0subm 20654 | The nonnegative reals form... |
absabv 20655 | The regular absolute value... |
zsssubrg 20656 | The integers are a subset ... |
qsssubdrg 20657 | The rational numbers are a... |
cnsubrg 20658 | There are no subrings of t... |
cnmgpabl 20659 | The unit group of the comp... |
cnmgpid 20660 | The group identity element... |
cnmsubglem 20661 | Lemma for ~ rpmsubg and fr... |
rpmsubg 20662 | The positive reals form a ... |
gzrngunitlem 20663 | Lemma for ~ gzrngunit . (... |
gzrngunit 20664 | The units on ` ZZ [ _i ] `... |
gsumfsum 20665 | Relate a group sum on ` CC... |
regsumfsum 20666 | Relate a group sum on ` ( ... |
expmhm 20667 | Exponentiation is a monoid... |
nn0srg 20668 | The nonnegative integers f... |
rge0srg 20669 | The nonnegative real numbe... |
zringcrng 20672 | The ring of integers is a ... |
zringring 20673 | The ring of integers is a ... |
zringabl 20674 | The ring of integers is an... |
zringgrp 20675 | The ring of integers is an... |
zringbas 20676 | The integers are the base ... |
zringplusg 20677 | The addition operation of ... |
zringmulg 20678 | The multiplication (group ... |
zringmulr 20679 | The multiplication operati... |
zring0 20680 | The neutral element of the... |
zring1 20681 | The multiplicative neutral... |
zringnzr 20682 | The ring of integers is a ... |
dvdsrzring 20683 | Ring divisibility in the r... |
zringlpirlem1 20684 | Lemma for ~ zringlpir . A... |
zringlpirlem2 20685 | Lemma for ~ zringlpir . A... |
zringlpirlem3 20686 | Lemma for ~ zringlpir . A... |
zringinvg 20687 | The additive inverse of an... |
zringunit 20688 | The units of ` ZZ ` are th... |
zringlpir 20689 | The integers are a princip... |
zringndrg 20690 | The integers are not a div... |
zringcyg 20691 | The integers are a cyclic ... |
zringsubgval 20692 | Subtraction in the ring of... |
zringmpg 20693 | The multiplication group o... |
prmirredlem 20694 | A positive integer is irre... |
dfprm2 20695 | The positive irreducible e... |
prmirred 20696 | The irreducible elements o... |
expghm 20697 | Exponentiation is a group ... |
mulgghm2 20698 | The powers of a group elem... |
mulgrhm 20699 | The powers of the element ... |
mulgrhm2 20700 | The powers of the element ... |
zrhval 20709 | Define the unique homomorp... |
zrhval2 20710 | Alternate value of the ` Z... |
zrhmulg 20711 | Value of the ` ZRHom ` hom... |
zrhrhmb 20712 | The ` ZRHom ` homomorphism... |
zrhrhm 20713 | The ` ZRHom ` homomorphism... |
zrh1 20714 | Interpretation of 1 in a r... |
zrh0 20715 | Interpretation of 0 in a r... |
zrhpropd 20716 | The ` ZZ ` ring homomorphi... |
zlmval 20717 | Augment an abelian group w... |
zlmlem 20718 | Lemma for ~ zlmbas and ~ z... |
zlmlemOLD 20719 | Obsolete version of ~ zlml... |
zlmbas 20720 | Base set of a ` ZZ ` -modu... |
zlmbasOLD 20721 | Obsolete version of ~ zlmb... |
zlmplusg 20722 | Group operation of a ` ZZ ... |
zlmplusgOLD 20723 | Obsolete version of ~ zlmb... |
zlmmulr 20724 | Ring operation of a ` ZZ `... |
zlmmulrOLD 20725 | Obsolete version of ~ zlmb... |
zlmsca 20726 | Scalar ring of a ` ZZ ` -m... |
zlmvsca 20727 | Scalar multiplication oper... |
zlmlmod 20728 | The ` ZZ ` -module operati... |
chrval 20729 | Definition substitution of... |
chrcl 20730 | Closure of the characteris... |
chrid 20731 | The canonical ` ZZ ` ring ... |
chrdvds 20732 | The ` ZZ ` ring homomorphi... |
chrcong 20733 | If two integers are congru... |
chrnzr 20734 | Nonzero rings are precisel... |
chrrhm 20735 | The characteristic restric... |
domnchr 20736 | The characteristic of a do... |
znlidl 20737 | The set ` n ZZ ` is an ide... |
zncrng2 20738 | The value of the ` Z/nZ ` ... |
znval 20739 | The value of the ` Z/nZ ` ... |
znle 20740 | The value of the ` Z/nZ ` ... |
znval2 20741 | Self-referential expressio... |
znbaslem 20742 | Lemma for ~ znbas . (Cont... |
znbaslemOLD 20743 | Obsolete version of ~ znba... |
znbas2 20744 | The base set of ` Z/nZ ` i... |
znbas2OLD 20745 | Obsolete version of ~ znba... |
znadd 20746 | The additive structure of ... |
znaddOLD 20747 | Obsolete version of ~ znad... |
znmul 20748 | The multiplicative structu... |
znmulOLD 20749 | Obsolete version of ~ znad... |
znzrh 20750 | The ` ZZ ` ring homomorphi... |
znbas 20751 | The base set of ` Z/nZ ` s... |
zncrng 20752 | ` Z/nZ ` is a commutative ... |
znzrh2 20753 | The ` ZZ ` ring homomorphi... |
znzrhval 20754 | The ` ZZ ` ring homomorphi... |
znzrhfo 20755 | The ` ZZ ` ring homomorphi... |
zncyg 20756 | The group ` ZZ / n ZZ ` is... |
zndvds 20757 | Express equality of equiva... |
zndvds0 20758 | Special case of ~ zndvds w... |
znf1o 20759 | The function ` F ` enumera... |
zzngim 20760 | The ` ZZ ` ring homomorphi... |
znle2 20761 | The ordering of the ` Z/nZ... |
znleval 20762 | The ordering of the ` Z/nZ... |
znleval2 20763 | The ordering of the ` Z/nZ... |
zntoslem 20764 | Lemma for ~ zntos . (Cont... |
zntos 20765 | The ` Z/nZ ` structure is ... |
znhash 20766 | The ` Z/nZ ` structure has... |
znfi 20767 | The ` Z/nZ ` structure is ... |
znfld 20768 | The ` Z/nZ ` structure is ... |
znidomb 20769 | The ` Z/nZ ` structure is ... |
znchr 20770 | Cyclic rings are defined b... |
znunit 20771 | The units of ` Z/nZ ` are ... |
znunithash 20772 | The size of the unit group... |
znrrg 20773 | The regular elements of ` ... |
cygznlem1 20774 | Lemma for ~ cygzn . (Cont... |
cygznlem2a 20775 | Lemma for ~ cygzn . (Cont... |
cygznlem2 20776 | Lemma for ~ cygzn . (Cont... |
cygznlem3 20777 | A cyclic group with ` n ` ... |
cygzn 20778 | A cyclic group with ` n ` ... |
cygth 20779 | The "fundamental theorem o... |
cyggic 20780 | Cyclic groups are isomorph... |
frgpcyg 20781 | A free group is cyclic iff... |
cnmsgnsubg 20782 | The signs form a multiplic... |
cnmsgnbas 20783 | The base set of the sign s... |
cnmsgngrp 20784 | The group of signs under m... |
psgnghm 20785 | The sign is a homomorphism... |
psgnghm2 20786 | The sign is a homomorphism... |
psgninv 20787 | The sign of a permutation ... |
psgnco 20788 | Multiplicativity of the pe... |
zrhpsgnmhm 20789 | Embedding of permutation s... |
zrhpsgninv 20790 | The embedded sign of a per... |
evpmss 20791 | Even permutations are perm... |
psgnevpmb 20792 | A class is an even permuta... |
psgnodpm 20793 | A permutation which is odd... |
psgnevpm 20794 | A permutation which is eve... |
psgnodpmr 20795 | If a permutation has sign ... |
zrhpsgnevpm 20796 | The sign of an even permut... |
zrhpsgnodpm 20797 | The sign of an odd permuta... |
cofipsgn 20798 | Composition of any class `... |
zrhpsgnelbas 20799 | Embedding of permutation s... |
zrhcopsgnelbas 20800 | Embedding of permutation s... |
evpmodpmf1o 20801 | The function for performin... |
pmtrodpm 20802 | A transposition is an odd ... |
psgnfix1 20803 | A permutation of a finite ... |
psgnfix2 20804 | A permutation of a finite ... |
psgndiflemB 20805 | Lemma 1 for ~ psgndif . (... |
psgndiflemA 20806 | Lemma 2 for ~ psgndif . (... |
psgndif 20807 | Embedding of permutation s... |
copsgndif 20808 | Embedding of permutation s... |
rebase 20811 | The base of the field of r... |
remulg 20812 | The multiplication (group ... |
resubdrg 20813 | The real numbers form a di... |
resubgval 20814 | Subtraction in the field o... |
replusg 20815 | The addition operation of ... |
remulr 20816 | The multiplication operati... |
re0g 20817 | The neutral element of the... |
re1r 20818 | The multiplicative neutral... |
rele2 20819 | The ordering relation of t... |
relt 20820 | The ordering relation of t... |
reds 20821 | The distance of the field ... |
redvr 20822 | The division operation of ... |
retos 20823 | The real numbers are a tot... |
refld 20824 | The real numbers form a fi... |
refldcj 20825 | The conjugation operation ... |
recrng 20826 | The real numbers form a st... |
regsumsupp 20827 | The group sum over the rea... |
rzgrp 20828 | The quotient group ` RR / ... |
isphl 20833 | The predicate "is a genera... |
phllvec 20834 | A pre-Hilbert space is a l... |
phllmod 20835 | A pre-Hilbert space is a l... |
phlsrng 20836 | The scalar ring of a pre-H... |
phllmhm 20837 | The inner product of a pre... |
ipcl 20838 | Closure of the inner produ... |
ipcj 20839 | Conjugate of an inner prod... |
iporthcom 20840 | Orthogonality (meaning inn... |
ip0l 20841 | Inner product with a zero ... |
ip0r 20842 | Inner product with a zero ... |
ipeq0 20843 | The inner product of a vec... |
ipdir 20844 | Distributive law for inner... |
ipdi 20845 | Distributive law for inner... |
ip2di 20846 | Distributive law for inner... |
ipsubdir 20847 | Distributive law for inner... |
ipsubdi 20848 | Distributive law for inner... |
ip2subdi 20849 | Distributive law for inner... |
ipass 20850 | Associative law for inner ... |
ipassr 20851 | "Associative" law for seco... |
ipassr2 20852 | "Associative" law for inne... |
ipffval 20853 | The inner product operatio... |
ipfval 20854 | The inner product operatio... |
ipfeq 20855 | If the inner product opera... |
ipffn 20856 | The inner product operatio... |
phlipf 20857 | The inner product operatio... |
ip2eq 20858 | Two vectors are equal iff ... |
isphld 20859 | Properties that determine ... |
phlpropd 20860 | If two structures have the... |
ssipeq 20861 | The inner product on a sub... |
phssipval 20862 | The inner product on a sub... |
phssip 20863 | The inner product (as a fu... |
phlssphl 20864 | A subspace of an inner pro... |
ocvfval 20871 | The orthocomplement operat... |
ocvval 20872 | Value of the orthocompleme... |
elocv 20873 | Elementhood in the orthoco... |
ocvi 20874 | Property of a member of th... |
ocvss 20875 | The orthocomplement of a s... |
ocvocv 20876 | A set is contained in its ... |
ocvlss 20877 | The orthocomplement of a s... |
ocv2ss 20878 | Orthocomplements reverse s... |
ocvin 20879 | An orthocomplement has tri... |
ocvsscon 20880 | Two ways to say that ` S `... |
ocvlsp 20881 | The orthocomplement of a l... |
ocv0 20882 | The orthocomplement of the... |
ocvz 20883 | The orthocomplement of the... |
ocv1 20884 | The orthocomplement of the... |
unocv 20885 | The orthocomplement of a u... |
iunocv 20886 | The orthocomplement of an ... |
cssval 20887 | The set of closed subspace... |
iscss 20888 | The predicate "is a closed... |
cssi 20889 | Property of a closed subsp... |
cssss 20890 | A closed subspace is a sub... |
iscss2 20891 | It is sufficient to prove ... |
ocvcss 20892 | The orthocomplement of any... |
cssincl 20893 | The zero subspace is a clo... |
css0 20894 | The zero subspace is a clo... |
css1 20895 | The whole space is a close... |
csslss 20896 | A closed subspace of a pre... |
lsmcss 20897 | A subset of a pre-Hilbert ... |
cssmre 20898 | The closed subspaces of a ... |
mrccss 20899 | The Moore closure correspo... |
thlval 20900 | Value of the Hilbert latti... |
thlbas 20901 | Base set of the Hilbert la... |
thlbasOLD 20902 | Obsolete proof of ~ thlbas... |
thlle 20903 | Ordering on the Hilbert la... |
thlleOLD 20904 | Obsolete proof of ~ thlle ... |
thlleval 20905 | Ordering on the Hilbert la... |
thloc 20906 | Orthocomplement on the Hil... |
pjfval 20913 | The value of the projectio... |
pjdm 20914 | A subspace is in the domai... |
pjpm 20915 | The projection map is a pa... |
pjfval2 20916 | Value of the projection ma... |
pjval 20917 | Value of the projection ma... |
pjdm2 20918 | A subspace is in the domai... |
pjff 20919 | A projection is a linear o... |
pjf 20920 | A projection is a function... |
pjf2 20921 | A projection is a function... |
pjfo 20922 | A projection is a surjecti... |
pjcss 20923 | A projection subspace is a... |
ocvpj 20924 | The orthocomplement of a p... |
ishil 20925 | The predicate "is a Hilber... |
ishil2 20926 | The predicate "is a Hilber... |
isobs 20927 | The predicate "is an ortho... |
obsip 20928 | The inner product of two e... |
obsipid 20929 | A basis element has unit l... |
obsrcl 20930 | Reverse closure for an ort... |
obsss 20931 | An orthonormal basis is a ... |
obsne0 20932 | A basis element is nonzero... |
obsocv 20933 | An orthonormal basis has t... |
obs2ocv 20934 | The double orthocomplement... |
obselocv 20935 | A basis element is in the ... |
obs2ss 20936 | A basis has no proper subs... |
obslbs 20937 | An orthogonal basis is a l... |
reldmdsmm 20940 | The direct sum is a well-b... |
dsmmval 20941 | Value of the module direct... |
dsmmbase 20942 | Base set of the module dir... |
dsmmval2 20943 | Self-referential definitio... |
dsmmbas2 20944 | Base set of the direct sum... |
dsmmfi 20945 | For finite products, the d... |
dsmmelbas 20946 | Membership in the finitely... |
dsmm0cl 20947 | The all-zero vector is con... |
dsmmacl 20948 | The finite hull is closed ... |
prdsinvgd2 20949 | Negation of a single coord... |
dsmmsubg 20950 | The finite hull of a produ... |
dsmmlss 20951 | The finite hull of a produ... |
dsmmlmod 20952 | The direct sum of a family... |
frlmval 20955 | Value of the "free module"... |
frlmlmod 20956 | The free module is a modul... |
frlmpws 20957 | The free module as a restr... |
frlmlss 20958 | The base set of the free m... |
frlmpwsfi 20959 | The finite free module is ... |
frlmsca 20960 | The ring of scalars of a f... |
frlm0 20961 | Zero in a free module (rin... |
frlmbas 20962 | Base set of the free modul... |
frlmelbas 20963 | Membership in the base set... |
frlmrcl 20964 | If a free module is inhabi... |
frlmbasfsupp 20965 | Elements of the free modul... |
frlmbasmap 20966 | Elements of the free modul... |
frlmbasf 20967 | Elements of the free modul... |
frlmlvec 20968 | The free module over a div... |
frlmfibas 20969 | The base set of the finite... |
elfrlmbasn0 20970 | If the dimension of a free... |
frlmplusgval 20971 | Addition in a free module.... |
frlmsubgval 20972 | Subtraction in a free modu... |
frlmvscafval 20973 | Scalar multiplication in a... |
frlmvplusgvalc 20974 | Coordinates of a sum with ... |
frlmvscaval 20975 | Coordinates of a scalar mu... |
frlmplusgvalb 20976 | Addition in a free module ... |
frlmvscavalb 20977 | Scalar multiplication in a... |
frlmvplusgscavalb 20978 | Addition combined with sca... |
frlmgsum 20979 | Finite commutative sums in... |
frlmsplit2 20980 | Restriction is homomorphic... |
frlmsslss 20981 | A subset of a free module ... |
frlmsslss2 20982 | A subset of a free module ... |
frlmbas3 20983 | An element of the base set... |
mpofrlmd 20984 | Elements of the free modul... |
frlmip 20985 | The inner product of a fre... |
frlmipval 20986 | The inner product of a fre... |
frlmphllem 20987 | Lemma for ~ frlmphl . (Co... |
frlmphl 20988 | Conditions for a free modu... |
uvcfval 20991 | Value of the unit-vector g... |
uvcval 20992 | Value of a single unit vec... |
uvcvval 20993 | Value of a unit vector coo... |
uvcvvcl 20994 | A coordinate of a unit vec... |
uvcvvcl2 20995 | A unit vector coordinate i... |
uvcvv1 20996 | The unit vector is one at ... |
uvcvv0 20997 | The unit vector is zero at... |
uvcff 20998 | Domain and range of the un... |
uvcf1 20999 | In a nonzero ring, each un... |
uvcresum 21000 | Any element of a free modu... |
frlmssuvc1 21001 | A scalar multiple of a uni... |
frlmssuvc2 21002 | A nonzero scalar multiple ... |
frlmsslsp 21003 | A subset of a free module ... |
frlmlbs 21004 | The unit vectors comprise ... |
frlmup1 21005 | Any assignment of unit vec... |
frlmup2 21006 | The evaluation map has the... |
frlmup3 21007 | The range of such an evalu... |
frlmup4 21008 | Universal property of the ... |
ellspd 21009 | The elements of the span o... |
elfilspd 21010 | Simplified version of ~ el... |
rellindf 21015 | The independent-family pre... |
islinds 21016 | Property of an independent... |
linds1 21017 | An independent set of vect... |
linds2 21018 | An independent set of vect... |
islindf 21019 | Property of an independent... |
islinds2 21020 | Expanded property of an in... |
islindf2 21021 | Property of an independent... |
lindff 21022 | Functional property of a l... |
lindfind 21023 | A linearly independent fam... |
lindsind 21024 | A linearly independent set... |
lindfind2 21025 | In a linearly independent ... |
lindsind2 21026 | In a linearly independent ... |
lindff1 21027 | A linearly independent fam... |
lindfrn 21028 | The range of an independen... |
f1lindf 21029 | Rearranging and deleting e... |
lindfres 21030 | Any restriction of an inde... |
lindsss 21031 | Any subset of an independe... |
f1linds 21032 | A family constructed from ... |
islindf3 21033 | In a nonzero ring, indepen... |
lindfmm 21034 | Linear independence of a f... |
lindsmm 21035 | Linear independence of a s... |
lindsmm2 21036 | The monomorphic image of a... |
lsslindf 21037 | Linear independence is unc... |
lsslinds 21038 | Linear independence is unc... |
islbs4 21039 | A basis is an independent ... |
lbslinds 21040 | A basis is independent. (... |
islinds3 21041 | A subset is linearly indep... |
islinds4 21042 | A set is independent in a ... |
lmimlbs 21043 | The isomorphic image of a ... |
lmiclbs 21044 | Having a basis is an isomo... |
islindf4 21045 | A family is independent if... |
islindf5 21046 | A family is independent if... |
indlcim 21047 | An independent, spanning f... |
lbslcic 21048 | A module with a basis is i... |
lmisfree 21049 | A module has a basis iff i... |
lvecisfrlm 21050 | Every vector space is isom... |
lmimco 21051 | The composition of two iso... |
lmictra 21052 | Module isomorphism is tran... |
uvcf1o 21053 | In a nonzero ring, the map... |
uvcendim 21054 | In a nonzero ring, the num... |
frlmisfrlm 21055 | A free module is isomorphi... |
frlmiscvec 21056 | Every free module is isomo... |
isassa 21063 | The properties of an assoc... |
assalem 21064 | The properties of an assoc... |
assaass 21065 | Left-associative property ... |
assaassr 21066 | Right-associative property... |
assalmod 21067 | An associative algebra is ... |
assaring 21068 | An associative algebra is ... |
assasca 21069 | An associative algebra's s... |
assa2ass 21070 | Left- and right-associativ... |
isassad 21071 | Sufficient condition for b... |
issubassa3 21072 | A subring that is also a s... |
issubassa 21073 | The subalgebras of an asso... |
sraassa 21074 | The subring algebra over a... |
rlmassa 21075 | The ring module over a com... |
assapropd 21076 | If two structures have the... |
aspval 21077 | Value of the algebraic clo... |
asplss 21078 | The algebraic span of a se... |
aspid 21079 | The algebraic span of a su... |
aspsubrg 21080 | The algebraic span of a se... |
aspss 21081 | Span preserves subset orde... |
aspssid 21082 | A set of vectors is a subs... |
asclfval 21083 | Function value of the alge... |
asclval 21084 | Value of a mapped algebra ... |
asclfn 21085 | Unconditional functionalit... |
asclf 21086 | The algebra scalars functi... |
asclghm 21087 | The algebra scalars functi... |
ascl0 21088 | The scalar 0 embedded into... |
ascl1 21089 | The scalar 1 embedded into... |
asclmul1 21090 | Left multiplication by a l... |
asclmul2 21091 | Right multiplication by a ... |
ascldimul 21092 | The algebra scalars functi... |
asclinvg 21093 | The group inverse (negatio... |
asclrhm 21094 | The scalar injection is a ... |
rnascl 21095 | The set of injected scalar... |
issubassa2 21096 | A subring of a unital alge... |
rnasclsubrg 21097 | The scalar multiples of th... |
rnasclmulcl 21098 | (Vector) multiplication is... |
rnasclassa 21099 | The scalar multiples of th... |
ressascl 21100 | The injection of scalars i... |
asclpropd 21101 | If two structures have the... |
aspval2 21102 | The algebraic closure is t... |
assamulgscmlem1 21103 | Lemma 1 for ~ assamulgscm ... |
assamulgscmlem2 21104 | Lemma for ~ assamulgscm (i... |
assamulgscm 21105 | Exponentiation of a scalar... |
zlmassa 21106 | The ` ZZ ` -module operati... |
reldmpsr 21117 | The multivariate power ser... |
psrval 21118 | Value of the multivariate ... |
psrvalstr 21119 | The multivariate power ser... |
psrbag 21120 | Elementhood in the set of ... |
psrbagf 21121 | A finite bag is a function... |
psrbagfOLD 21122 | Obsolete version of ~ psrb... |
psrbagfsupp 21123 | Finite bags have finite su... |
psrbagfsuppOLD 21124 | Obsolete version of ~ psrb... |
snifpsrbag 21125 | A bag containing one eleme... |
fczpsrbag 21126 | The constant function equa... |
psrbaglesupp 21127 | The support of a dominated... |
psrbaglesuppOLD 21128 | Obsolete version of ~ psrb... |
psrbaglecl 21129 | The set of finite bags is ... |
psrbagleclOLD 21130 | Obsolete version of ~ psrb... |
psrbagaddcl 21131 | The sum of two finite bags... |
psrbagaddclOLD 21132 | Obsolete version of ~ psrb... |
psrbagcon 21133 | The analogue of the statem... |
psrbagconOLD 21134 | Obsolete version of ~ psrb... |
psrbaglefi 21135 | There are finitely many ba... |
psrbaglefiOLD 21136 | Obsolete version of ~ psrb... |
psrbagconcl 21137 | The complement of a bag is... |
psrbagconclOLD 21138 | Obsolete version of ~ psrb... |
psrbagconf1o 21139 | Bag complementation is a b... |
psrbagconf1oOLD 21140 | Obsolete version of ~ psrb... |
gsumbagdiaglemOLD 21141 | Obsolete version of ~ gsum... |
gsumbagdiagOLD 21142 | Obsolete version of ~ gsum... |
psrass1lemOLD 21143 | Obsolete version of ~ psra... |
gsumbagdiaglem 21144 | Lemma for ~ gsumbagdiag . ... |
gsumbagdiag 21145 | Two-dimensional commutatio... |
psrass1lem 21146 | A group sum commutation us... |
psrbas 21147 | The base set of the multiv... |
psrelbas 21148 | An element of the set of p... |
psrelbasfun 21149 | An element of the set of p... |
psrplusg 21150 | The addition operation of ... |
psradd 21151 | The addition operation of ... |
psraddcl 21152 | Closure of the power serie... |
psrmulr 21153 | The multiplication operati... |
psrmulfval 21154 | The multiplication operati... |
psrmulval 21155 | The multiplication operati... |
psrmulcllem 21156 | Closure of the power serie... |
psrmulcl 21157 | Closure of the power serie... |
psrsca 21158 | The scalar field of the mu... |
psrvscafval 21159 | The scalar multiplication ... |
psrvsca 21160 | The scalar multiplication ... |
psrvscaval 21161 | The scalar multiplication ... |
psrvscacl 21162 | Closure of the power serie... |
psr0cl 21163 | The zero element of the ri... |
psr0lid 21164 | The zero element of the ri... |
psrnegcl 21165 | The negative function in t... |
psrlinv 21166 | The negative function in t... |
psrgrp 21167 | The ring of power series i... |
psr0 21168 | The zero element of the ri... |
psrneg 21169 | The negative function of t... |
psrlmod 21170 | The ring of power series i... |
psr1cl 21171 | The identity element of th... |
psrlidm 21172 | The identity element of th... |
psrridm 21173 | The identity element of th... |
psrass1 21174 | Associative identity for t... |
psrdi 21175 | Distributive law for the r... |
psrdir 21176 | Distributive law for the r... |
psrass23l 21177 | Associative identity for t... |
psrcom 21178 | Commutative law for the ri... |
psrass23 21179 | Associative identities for... |
psrring 21180 | The ring of power series i... |
psr1 21181 | The identity element of th... |
psrcrng 21182 | The ring of power series i... |
psrassa 21183 | The ring of power series i... |
resspsrbas 21184 | A restricted power series ... |
resspsradd 21185 | A restricted power series ... |
resspsrmul 21186 | A restricted power series ... |
resspsrvsca 21187 | A restricted power series ... |
subrgpsr 21188 | A subring of the base ring... |
mvrfval 21189 | Value of the generating el... |
mvrval 21190 | Value of the generating el... |
mvrval2 21191 | Value of the generating el... |
mvrid 21192 | The ` X i ` -th coefficien... |
mvrf 21193 | The power series variable ... |
mvrf1 21194 | The power series variable ... |
mvrcl2 21195 | A power series variable is... |
reldmmpl 21196 | The multivariate polynomia... |
mplval 21197 | Value of the set of multiv... |
mplbas 21198 | Base set of the set of mul... |
mplelbas 21199 | Property of being a polyno... |
mplrcl 21200 | Reverse closure for the po... |
mplelsfi 21201 | A polynomial treated as a ... |
mplval2 21202 | Self-referential expressio... |
mplbasss 21203 | The set of polynomials is ... |
mplelf 21204 | A polynomial is defined as... |
mplsubglem 21205 | If ` A ` is an ideal of se... |
mpllsslem 21206 | If ` A ` is an ideal of su... |
mplsubglem2 21207 | Lemma for ~ mplsubg and ~ ... |
mplsubg 21208 | The set of polynomials is ... |
mpllss 21209 | The set of polynomials is ... |
mplsubrglem 21210 | Lemma for ~ mplsubrg . (C... |
mplsubrg 21211 | The set of polynomials is ... |
mpl0 21212 | The zero polynomial. (Con... |
mpladd 21213 | The addition operation on ... |
mplneg 21214 | The negative function on m... |
mplmul 21215 | The multiplication operati... |
mpl1 21216 | The identity element of th... |
mplsca 21217 | The scalar field of a mult... |
mplvsca2 21218 | The scalar multiplication ... |
mplvsca 21219 | The scalar multiplication ... |
mplvscaval 21220 | The scalar multiplication ... |
mvrcl 21221 | A power series variable is... |
mplgrp 21222 | The polynomial ring is a g... |
mpllmod 21223 | The polynomial ring is a l... |
mplring 21224 | The polynomial ring is a r... |
mpllvec 21225 | The polynomial ring is a v... |
mplcrng 21226 | The polynomial ring is a c... |
mplassa 21227 | The polynomial ring is an ... |
ressmplbas2 21228 | The base set of a restrict... |
ressmplbas 21229 | A restricted polynomial al... |
ressmpladd 21230 | A restricted polynomial al... |
ressmplmul 21231 | A restricted polynomial al... |
ressmplvsca 21232 | A restricted power series ... |
subrgmpl 21233 | A subring of the base ring... |
subrgmvr 21234 | The variables in a subring... |
subrgmvrf 21235 | The variables in a polynom... |
mplmon 21236 | A monomial is a polynomial... |
mplmonmul 21237 | The product of two monomia... |
mplcoe1 21238 | Decompose a polynomial int... |
mplcoe3 21239 | Decompose a monomial in on... |
mplcoe5lem 21240 | Lemma for ~ mplcoe4 . (Co... |
mplcoe5 21241 | Decompose a monomial into ... |
mplcoe2 21242 | Decompose a monomial into ... |
mplbas2 21243 | An alternative expression ... |
ltbval 21244 | Value of the well-order on... |
ltbwe 21245 | The finite bag order is a ... |
reldmopsr 21246 | Lemma for ordered power se... |
opsrval 21247 | The value of the "ordered ... |
opsrle 21248 | An alternative expression ... |
opsrval2 21249 | Self-referential expressio... |
opsrbaslem 21250 | Get a component of the ord... |
opsrbaslemOLD 21251 | Obsolete version of ~ opsr... |
opsrbas 21252 | The base set of the ordere... |
opsrbasOLD 21253 | Obsolete version of ~ opsr... |
opsrplusg 21254 | The addition operation of ... |
opsrplusgOLD 21255 | Obsolete version of ~ opsr... |
opsrmulr 21256 | The multiplication operati... |
opsrmulrOLD 21257 | Obsolete version of ~ opsr... |
opsrvsca 21258 | The scalar product operati... |
opsrvscaOLD 21259 | Obsolete version of ~ opsr... |
opsrsca 21260 | The scalar ring of the ord... |
opsrscaOLD 21261 | Obsolete version of ~ opsr... |
opsrtoslem1 21262 | Lemma for ~ opsrtos . (Co... |
opsrtoslem2 21263 | Lemma for ~ opsrtos . (Co... |
opsrtos 21264 | The ordered power series s... |
opsrso 21265 | The ordered power series s... |
opsrcrng 21266 | The ring of ordered power ... |
opsrassa 21267 | The ring of ordered power ... |
mvrf2 21268 | The power series/polynomia... |
mplmon2 21269 | Express a scaled monomial.... |
psrbag0 21270 | The empty bag is a bag. (... |
psrbagsn 21271 | A singleton bag is a bag. ... |
mplascl 21272 | Value of the scalar inject... |
mplasclf 21273 | The scalar injection is a ... |
subrgascl 21274 | The scalar injection funct... |
subrgasclcl 21275 | The scalars in a polynomia... |
mplmon2cl 21276 | A scaled monomial is a pol... |
mplmon2mul 21277 | Product of scaled monomial... |
mplind 21278 | Prove a property of polyno... |
mplcoe4 21279 | Decompose a polynomial int... |
evlslem4 21284 | The support of a tensor pr... |
psrbagev1 21285 | A bag of multipliers provi... |
psrbagev1OLD 21286 | Obsolete version of ~ psrb... |
psrbagev2 21287 | Closure of a sum using a b... |
psrbagev2OLD 21288 | Obsolete version of ~ psrb... |
evlslem2 21289 | A linear function on the p... |
evlslem3 21290 | Lemma for ~ evlseu . Poly... |
evlslem6 21291 | Lemma for ~ evlseu . Fini... |
evlslem1 21292 | Lemma for ~ evlseu , give ... |
evlseu 21293 | For a given interpretation... |
reldmevls 21294 | Well-behaved binary operat... |
mpfrcl 21295 | Reverse closure for the se... |
evlsval 21296 | Value of the polynomial ev... |
evlsval2 21297 | Characterizing properties ... |
evlsrhm 21298 | Polynomial evaluation is a... |
evlssca 21299 | Polynomial evaluation maps... |
evlsvar 21300 | Polynomial evaluation maps... |
evlsgsumadd 21301 | Polynomial evaluation maps... |
evlsgsummul 21302 | Polynomial evaluation maps... |
evlspw 21303 | Polynomial evaluation for ... |
evlsvarpw 21304 | Polynomial evaluation for ... |
evlval 21305 | Value of the simple/same r... |
evlrhm 21306 | The simple evaluation map ... |
evlsscasrng 21307 | The evaluation of a scalar... |
evlsca 21308 | Simple polynomial evaluati... |
evlsvarsrng 21309 | The evaluation of the vari... |
evlvar 21310 | Simple polynomial evaluati... |
mpfconst 21311 | Constants are multivariate... |
mpfproj 21312 | Projections are multivaria... |
mpfsubrg 21313 | Polynomial functions are a... |
mpff 21314 | Polynomial functions are f... |
mpfaddcl 21315 | The sum of multivariate po... |
mpfmulcl 21316 | The product of multivariat... |
mpfind 21317 | Prove a property of polyno... |
selvffval 21326 | Value of the "variable sel... |
selvfval 21327 | Value of the "variable sel... |
selvval 21328 | Value of the "variable sel... |
mhpfval 21329 | Value of the "homogeneous ... |
mhpval 21330 | Value of the "homogeneous ... |
ismhp 21331 | Property of being a homoge... |
ismhp2 21332 | Deduce a homogeneous polyn... |
ismhp3 21333 | A polynomial is homogeneou... |
mhpmpl 21334 | A homogeneous polynomial i... |
mhpdeg 21335 | All nonzero terms of a hom... |
mhp0cl 21336 | The zero polynomial is hom... |
mhpsclcl 21337 | A scalar (or constant) pol... |
mhpvarcl 21338 | A power series variable is... |
mhpmulcl 21339 | A product of homogeneous p... |
mhppwdeg 21340 | Degree of a homogeneous po... |
mhpaddcl 21341 | Homogeneous polynomials ar... |
mhpinvcl 21342 | Homogeneous polynomials ar... |
mhpsubg 21343 | Homogeneous polynomials fo... |
mhpvscacl 21344 | Homogeneous polynomials ar... |
mhplss 21345 | Homogeneous polynomials fo... |
psr1baslem 21356 | The set of finite bags on ... |
psr1val 21357 | Value of the ring of univa... |
psr1crng 21358 | The ring of univariate pow... |
psr1assa 21359 | The ring of univariate pow... |
psr1tos 21360 | The ordered power series s... |
psr1bas2 21361 | The base set of the ring o... |
psr1bas 21362 | The base set of the ring o... |
vr1val 21363 | The value of the generator... |
vr1cl2 21364 | The variable ` X ` is a me... |
ply1val 21365 | The value of the set of un... |
ply1bas 21366 | The value of the base set ... |
ply1lss 21367 | Univariate polynomials for... |
ply1subrg 21368 | Univariate polynomials for... |
ply1crng 21369 | The ring of univariate pol... |
ply1assa 21370 | The ring of univariate pol... |
psr1bascl 21371 | A univariate power series ... |
psr1basf 21372 | Univariate power series ba... |
ply1basf 21373 | Univariate polynomial base... |
ply1bascl 21374 | A univariate polynomial is... |
ply1bascl2 21375 | A univariate polynomial is... |
coe1fval 21376 | Value of the univariate po... |
coe1fv 21377 | Value of an evaluated coef... |
fvcoe1 21378 | Value of a multivariate co... |
coe1fval3 21379 | Univariate power series co... |
coe1f2 21380 | Functionality of univariat... |
coe1fval2 21381 | Univariate polynomial coef... |
coe1f 21382 | Functionality of univariat... |
coe1fvalcl 21383 | A coefficient of a univari... |
coe1sfi 21384 | Finite support of univaria... |
coe1fsupp 21385 | The coefficient vector of ... |
mptcoe1fsupp 21386 | A mapping involving coeffi... |
coe1ae0 21387 | The coefficient vector of ... |
vr1cl 21388 | The generator of a univari... |
opsr0 21389 | Zero in the ordered power ... |
opsr1 21390 | One in the ordered power s... |
mplplusg 21391 | Value of addition in a pol... |
mplmulr 21392 | Value of multiplication in... |
psr1plusg 21393 | Value of addition in a uni... |
psr1vsca 21394 | Value of scalar multiplica... |
psr1mulr 21395 | Value of multiplication in... |
ply1plusg 21396 | Value of addition in a uni... |
ply1vsca 21397 | Value of scalar multiplica... |
ply1mulr 21398 | Value of multiplication in... |
ressply1bas2 21399 | The base set of a restrict... |
ressply1bas 21400 | A restricted polynomial al... |
ressply1add 21401 | A restricted polynomial al... |
ressply1mul 21402 | A restricted polynomial al... |
ressply1vsca 21403 | A restricted power series ... |
subrgply1 21404 | A subring of the base ring... |
gsumply1subr 21405 | Evaluate a group sum in a ... |
psrbaspropd 21406 | Property deduction for pow... |
psrplusgpropd 21407 | Property deduction for pow... |
mplbaspropd 21408 | Property deduction for pol... |
psropprmul 21409 | Reversing multiplication i... |
ply1opprmul 21410 | Reversing multiplication i... |
00ply1bas 21411 | Lemma for ~ ply1basfvi and... |
ply1basfvi 21412 | Protection compatibility o... |
ply1plusgfvi 21413 | Protection compatibility o... |
ply1baspropd 21414 | Property deduction for uni... |
ply1plusgpropd 21415 | Property deduction for uni... |
opsrring 21416 | Ordered power series form ... |
opsrlmod 21417 | Ordered power series form ... |
psr1ring 21418 | Univariate power series fo... |
ply1ring 21419 | Univariate polynomials for... |
psr1lmod 21420 | Univariate power series fo... |
psr1sca 21421 | Scalars of a univariate po... |
psr1sca2 21422 | Scalars of a univariate po... |
ply1lmod 21423 | Univariate polynomials for... |
ply1sca 21424 | Scalars of a univariate po... |
ply1sca2 21425 | Scalars of a univariate po... |
ply1mpl0 21426 | The univariate polynomial ... |
ply10s0 21427 | Zero times a univariate po... |
ply1mpl1 21428 | The univariate polynomial ... |
ply1ascl 21429 | The univariate polynomial ... |
subrg1ascl 21430 | The scalar injection funct... |
subrg1asclcl 21431 | The scalars in a polynomia... |
subrgvr1 21432 | The variables in a subring... |
subrgvr1cl 21433 | The variables in a polynom... |
coe1z 21434 | The coefficient vector of ... |
coe1add 21435 | The coefficient vector of ... |
coe1addfv 21436 | A particular coefficient o... |
coe1subfv 21437 | A particular coefficient o... |
coe1mul2lem1 21438 | An equivalence for ~ coe1m... |
coe1mul2lem2 21439 | An equivalence for ~ coe1m... |
coe1mul2 21440 | The coefficient vector of ... |
coe1mul 21441 | The coefficient vector of ... |
ply1moncl 21442 | Closure of the expression ... |
ply1tmcl 21443 | Closure of the expression ... |
coe1tm 21444 | Coefficient vector of a po... |
coe1tmfv1 21445 | Nonzero coefficient of a p... |
coe1tmfv2 21446 | Zero coefficient of a poly... |
coe1tmmul2 21447 | Coefficient vector of a po... |
coe1tmmul 21448 | Coefficient vector of a po... |
coe1tmmul2fv 21449 | Function value of a right-... |
coe1pwmul 21450 | Coefficient vector of a po... |
coe1pwmulfv 21451 | Function value of a right-... |
ply1scltm 21452 | A scalar is a term with ze... |
coe1sclmul 21453 | Coefficient vector of a po... |
coe1sclmulfv 21454 | A single coefficient of a ... |
coe1sclmul2 21455 | Coefficient vector of a po... |
ply1sclf 21456 | A scalar polynomial is a p... |
ply1sclcl 21457 | The value of the algebra s... |
coe1scl 21458 | Coefficient vector of a sc... |
ply1sclid 21459 | Recover the base scalar fr... |
ply1sclf1 21460 | The polynomial scalar func... |
ply1scl0 21461 | The zero scalar is zero. ... |
ply1scln0 21462 | Nonzero scalars create non... |
ply1scl1 21463 | The one scalar is the unit... |
ply1idvr1 21464 | The identity of a polynomi... |
cply1mul 21465 | The product of two constan... |
ply1coefsupp 21466 | The decomposition of a uni... |
ply1coe 21467 | Decompose a univariate pol... |
eqcoe1ply1eq 21468 | Two polynomials over the s... |
ply1coe1eq 21469 | Two polynomials over the s... |
cply1coe0 21470 | All but the first coeffici... |
cply1coe0bi 21471 | A polynomial is constant (... |
coe1fzgsumdlem 21472 | Lemma for ~ coe1fzgsumd (i... |
coe1fzgsumd 21473 | Value of an evaluated coef... |
gsumsmonply1 21474 | A finite group sum of scal... |
gsummoncoe1 21475 | A coefficient of the polyn... |
gsumply1eq 21476 | Two univariate polynomials... |
lply1binom 21477 | The binomial theorem for l... |
lply1binomsc 21478 | The binomial theorem for l... |
reldmevls1 21483 | Well-behaved binary operat... |
ply1frcl 21484 | Reverse closure for the se... |
evls1fval 21485 | Value of the univariate po... |
evls1val 21486 | Value of the univariate po... |
evls1rhmlem 21487 | Lemma for ~ evl1rhm and ~ ... |
evls1rhm 21488 | Polynomial evaluation is a... |
evls1sca 21489 | Univariate polynomial eval... |
evls1gsumadd 21490 | Univariate polynomial eval... |
evls1gsummul 21491 | Univariate polynomial eval... |
evls1pw 21492 | Univariate polynomial eval... |
evls1varpw 21493 | Univariate polynomial eval... |
evl1fval 21494 | Value of the simple/same r... |
evl1val 21495 | Value of the simple/same r... |
evl1fval1lem 21496 | Lemma for ~ evl1fval1 . (... |
evl1fval1 21497 | Value of the simple/same r... |
evl1rhm 21498 | Polynomial evaluation is a... |
fveval1fvcl 21499 | The function value of the ... |
evl1sca 21500 | Polynomial evaluation maps... |
evl1scad 21501 | Polynomial evaluation buil... |
evl1var 21502 | Polynomial evaluation maps... |
evl1vard 21503 | Polynomial evaluation buil... |
evls1var 21504 | Univariate polynomial eval... |
evls1scasrng 21505 | The evaluation of a scalar... |
evls1varsrng 21506 | The evaluation of the vari... |
evl1addd 21507 | Polynomial evaluation buil... |
evl1subd 21508 | Polynomial evaluation buil... |
evl1muld 21509 | Polynomial evaluation buil... |
evl1vsd 21510 | Polynomial evaluation buil... |
evl1expd 21511 | Polynomial evaluation buil... |
pf1const 21512 | Constants are polynomial f... |
pf1id 21513 | The identity is a polynomi... |
pf1subrg 21514 | Polynomial functions are a... |
pf1rcl 21515 | Reverse closure for the se... |
pf1f 21516 | Polynomial functions are f... |
mpfpf1 21517 | Convert a multivariate pol... |
pf1mpf 21518 | Convert a univariate polyn... |
pf1addcl 21519 | The sum of multivariate po... |
pf1mulcl 21520 | The product of multivariat... |
pf1ind 21521 | Prove a property of polyno... |
evl1gsumdlem 21522 | Lemma for ~ evl1gsumd (ind... |
evl1gsumd 21523 | Polynomial evaluation buil... |
evl1gsumadd 21524 | Univariate polynomial eval... |
evl1gsumaddval 21525 | Value of a univariate poly... |
evl1gsummul 21526 | Univariate polynomial eval... |
evl1varpw 21527 | Univariate polynomial eval... |
evl1varpwval 21528 | Value of a univariate poly... |
evl1scvarpw 21529 | Univariate polynomial eval... |
evl1scvarpwval 21530 | Value of a univariate poly... |
evl1gsummon 21531 | Value of a univariate poly... |
mamufval 21534 | Functional value of the ma... |
mamuval 21535 | Multiplication of two matr... |
mamufv 21536 | A cell in the multiplicati... |
mamudm 21537 | The domain of the matrix m... |
mamufacex 21538 | Every solution of the equa... |
mamures 21539 | Rows in a matrix product a... |
mndvcl 21540 | Tuple-wise additive closur... |
mndvass 21541 | Tuple-wise associativity i... |
mndvlid 21542 | Tuple-wise left identity i... |
mndvrid 21543 | Tuple-wise right identity ... |
grpvlinv 21544 | Tuple-wise left inverse in... |
grpvrinv 21545 | Tuple-wise right inverse i... |
mhmvlin 21546 | Tuple extension of monoid ... |
ringvcl 21547 | Tuple-wise multiplication ... |
mamucl 21548 | Operation closure of matri... |
mamuass 21549 | Matrix multiplication is a... |
mamudi 21550 | Matrix multiplication dist... |
mamudir 21551 | Matrix multiplication dist... |
mamuvs1 21552 | Matrix multiplication dist... |
mamuvs2 21553 | Matrix multiplication dist... |
matbas0pc 21556 | There is no matrix with a ... |
matbas0 21557 | There is no matrix for a n... |
matval 21558 | Value of the matrix algebr... |
matrcl 21559 | Reverse closure for the ma... |
matbas 21560 | The matrix ring has the sa... |
matplusg 21561 | The matrix ring has the sa... |
matsca 21562 | The matrix ring has the sa... |
matscaOLD 21563 | Obsolete proof of ~ matsca... |
matvsca 21564 | The matrix ring has the sa... |
matvscaOLD 21565 | Obsolete proof of ~ matvsc... |
mat0 21566 | The matrix ring has the sa... |
matinvg 21567 | The matrix ring has the sa... |
mat0op 21568 | Value of a zero matrix as ... |
matsca2 21569 | The scalars of the matrix ... |
matbas2 21570 | The base set of the matrix... |
matbas2i 21571 | A matrix is a function. (... |
matbas2d 21572 | The base set of the matrix... |
eqmat 21573 | Two square matrices of the... |
matecl 21574 | Each entry (according to W... |
matecld 21575 | Each entry (according to W... |
matplusg2 21576 | Addition in the matrix rin... |
matvsca2 21577 | Scalar multiplication in t... |
matlmod 21578 | The matrix ring is a linea... |
matgrp 21579 | The matrix ring is a group... |
matvscl 21580 | Closure of the scalar mult... |
matsubg 21581 | The matrix ring has the sa... |
matplusgcell 21582 | Addition in the matrix rin... |
matsubgcell 21583 | Subtraction in the matrix ... |
matinvgcell 21584 | Additive inversion in the ... |
matvscacell 21585 | Scalar multiplication in t... |
matgsum 21586 | Finite commutative sums in... |
matmulr 21587 | Multiplication in the matr... |
mamumat1cl 21588 | The identity matrix (as op... |
mat1comp 21589 | The components of the iden... |
mamulid 21590 | The identity matrix (as op... |
mamurid 21591 | The identity matrix (as op... |
matring 21592 | Existence of the matrix ri... |
matassa 21593 | Existence of the matrix al... |
matmulcell 21594 | Multiplication in the matr... |
mpomatmul 21595 | Multiplication of two N x ... |
mat1 21596 | Value of an identity matri... |
mat1ov 21597 | Entries of an identity mat... |
mat1bas 21598 | The identity matrix is a m... |
matsc 21599 | The identity matrix multip... |
ofco2 21600 | Distribution law for the f... |
oftpos 21601 | The transposition of the v... |
mattposcl 21602 | The transpose of a square ... |
mattpostpos 21603 | The transpose of the trans... |
mattposvs 21604 | The transposition of a mat... |
mattpos1 21605 | The transposition of the i... |
tposmap 21606 | The transposition of an I ... |
mamutpos 21607 | Behavior of transposes in ... |
mattposm 21608 | Multiplying two transposed... |
matgsumcl 21609 | Closure of a group sum ove... |
madetsumid 21610 | The identity summand in th... |
matepmcl 21611 | Each entry of a matrix wit... |
matepm2cl 21612 | Each entry of a matrix wit... |
madetsmelbas 21613 | A summand of the determina... |
madetsmelbas2 21614 | A summand of the determina... |
mat0dimbas0 21615 | The empty set is the one a... |
mat0dim0 21616 | The zero of the algebra of... |
mat0dimid 21617 | The identity of the algebr... |
mat0dimscm 21618 | The scalar multiplication ... |
mat0dimcrng 21619 | The algebra of matrices wi... |
mat1dimelbas 21620 | A matrix with dimension 1 ... |
mat1dimbas 21621 | A matrix with dimension 1 ... |
mat1dim0 21622 | The zero of the algebra of... |
mat1dimid 21623 | The identity of the algebr... |
mat1dimscm 21624 | The scalar multiplication ... |
mat1dimmul 21625 | The ring multiplication in... |
mat1dimcrng 21626 | The algebra of matrices wi... |
mat1f1o 21627 | There is a 1-1 function fr... |
mat1rhmval 21628 | The value of the ring homo... |
mat1rhmelval 21629 | The value of the ring homo... |
mat1rhmcl 21630 | The value of the ring homo... |
mat1f 21631 | There is a function from a... |
mat1ghm 21632 | There is a group homomorph... |
mat1mhm 21633 | There is a monoid homomorp... |
mat1rhm 21634 | There is a ring homomorphi... |
mat1rngiso 21635 | There is a ring isomorphis... |
mat1ric 21636 | A ring is isomorphic to th... |
dmatval 21641 | The set of ` N ` x ` N ` d... |
dmatel 21642 | A ` N ` x ` N ` diagonal m... |
dmatmat 21643 | An ` N ` x ` N ` diagonal ... |
dmatid 21644 | The identity matrix is a d... |
dmatelnd 21645 | An extradiagonal entry of ... |
dmatmul 21646 | The product of two diagona... |
dmatsubcl 21647 | The difference of two diag... |
dmatsgrp 21648 | The set of diagonal matric... |
dmatmulcl 21649 | The product of two diagona... |
dmatsrng 21650 | The set of diagonal matric... |
dmatcrng 21651 | The subring of diagonal ma... |
dmatscmcl 21652 | The multiplication of a di... |
scmatval 21653 | The set of ` N ` x ` N ` s... |
scmatel 21654 | An ` N ` x ` N ` scalar ma... |
scmatscmid 21655 | A scalar matrix can be exp... |
scmatscmide 21656 | An entry of a scalar matri... |
scmatscmiddistr 21657 | Distributive law for scala... |
scmatmat 21658 | An ` N ` x ` N ` scalar ma... |
scmate 21659 | An entry of an ` N ` x ` N... |
scmatmats 21660 | The set of an ` N ` x ` N ... |
scmateALT 21661 | Alternate proof of ~ scmat... |
scmatscm 21662 | The multiplication of a ma... |
scmatid 21663 | The identity matrix is a s... |
scmatdmat 21664 | A scalar matrix is a diago... |
scmataddcl 21665 | The sum of two scalar matr... |
scmatsubcl 21666 | The difference of two scal... |
scmatmulcl 21667 | The product of two scalar ... |
scmatsgrp 21668 | The set of scalar matrices... |
scmatsrng 21669 | The set of scalar matrices... |
scmatcrng 21670 | The subring of scalar matr... |
scmatsgrp1 21671 | The set of scalar matrices... |
scmatsrng1 21672 | The set of scalar matrices... |
smatvscl 21673 | Closure of the scalar mult... |
scmatlss 21674 | The set of scalar matrices... |
scmatstrbas 21675 | The set of scalar matrices... |
scmatrhmval 21676 | The value of the ring homo... |
scmatrhmcl 21677 | The value of the ring homo... |
scmatf 21678 | There is a function from a... |
scmatfo 21679 | There is a function from a... |
scmatf1 21680 | There is a 1-1 function fr... |
scmatf1o 21681 | There is a bijection betwe... |
scmatghm 21682 | There is a group homomorph... |
scmatmhm 21683 | There is a monoid homomorp... |
scmatrhm 21684 | There is a ring homomorphi... |
scmatrngiso 21685 | There is a ring isomorphis... |
scmatric 21686 | A ring is isomorphic to ev... |
mat0scmat 21687 | The empty matrix over a ri... |
mat1scmat 21688 | A 1-dimensional matrix ove... |
mvmulfval 21691 | Functional value of the ma... |
mvmulval 21692 | Multiplication of a vector... |
mvmulfv 21693 | A cell/element in the vect... |
mavmulval 21694 | Multiplication of a vector... |
mavmulfv 21695 | A cell/element in the vect... |
mavmulcl 21696 | Multiplication of an NxN m... |
1mavmul 21697 | Multiplication of the iden... |
mavmulass 21698 | Associativity of the multi... |
mavmuldm 21699 | The domain of the matrix v... |
mavmulsolcl 21700 | Every solution of the equa... |
mavmul0 21701 | Multiplication of a 0-dime... |
mavmul0g 21702 | The result of the 0-dimens... |
mvmumamul1 21703 | The multiplication of an M... |
mavmumamul1 21704 | The multiplication of an N... |
marrepfval 21709 | First substitution for the... |
marrepval0 21710 | Second substitution for th... |
marrepval 21711 | Third substitution for the... |
marrepeval 21712 | An entry of a matrix with ... |
marrepcl 21713 | Closure of the row replace... |
marepvfval 21714 | First substitution for the... |
marepvval0 21715 | Second substitution for th... |
marepvval 21716 | Third substitution for the... |
marepveval 21717 | An entry of a matrix with ... |
marepvcl 21718 | Closure of the column repl... |
ma1repvcl 21719 | Closure of the column repl... |
ma1repveval 21720 | An entry of an identity ma... |
mulmarep1el 21721 | Element by element multipl... |
mulmarep1gsum1 21722 | The sum of element by elem... |
mulmarep1gsum2 21723 | The sum of element by elem... |
1marepvmarrepid 21724 | Replacing the ith row by 0... |
submabas 21727 | Any subset of the index se... |
submafval 21728 | First substitution for a s... |
submaval0 21729 | Second substitution for a ... |
submaval 21730 | Third substitution for a s... |
submaeval 21731 | An entry of a submatrix of... |
1marepvsma1 21732 | The submatrix of the ident... |
mdetfval 21735 | First substitution for the... |
mdetleib 21736 | Full substitution of our d... |
mdetleib2 21737 | Leibniz' formula can also ... |
nfimdetndef 21738 | The determinant is not def... |
mdetfval1 21739 | First substitution of an a... |
mdetleib1 21740 | Full substitution of an al... |
mdet0pr 21741 | The determinant function f... |
mdet0f1o 21742 | The determinant function f... |
mdet0fv0 21743 | The determinant of the emp... |
mdetf 21744 | Functionality of the deter... |
mdetcl 21745 | The determinant evaluates ... |
m1detdiag 21746 | The determinant of a 1-dim... |
mdetdiaglem 21747 | Lemma for ~ mdetdiag . Pr... |
mdetdiag 21748 | The determinant of a diago... |
mdetdiagid 21749 | The determinant of a diago... |
mdet1 21750 | The determinant of the ide... |
mdetrlin 21751 | The determinant function i... |
mdetrsca 21752 | The determinant function i... |
mdetrsca2 21753 | The determinant function i... |
mdetr0 21754 | The determinant of a matri... |
mdet0 21755 | The determinant of the zer... |
mdetrlin2 21756 | The determinant function i... |
mdetralt 21757 | The determinant function i... |
mdetralt2 21758 | The determinant function i... |
mdetero 21759 | The determinant function i... |
mdettpos 21760 | Determinant is invariant u... |
mdetunilem1 21761 | Lemma for ~ mdetuni . (Co... |
mdetunilem2 21762 | Lemma for ~ mdetuni . (Co... |
mdetunilem3 21763 | Lemma for ~ mdetuni . (Co... |
mdetunilem4 21764 | Lemma for ~ mdetuni . (Co... |
mdetunilem5 21765 | Lemma for ~ mdetuni . (Co... |
mdetunilem6 21766 | Lemma for ~ mdetuni . (Co... |
mdetunilem7 21767 | Lemma for ~ mdetuni . (Co... |
mdetunilem8 21768 | Lemma for ~ mdetuni . (Co... |
mdetunilem9 21769 | Lemma for ~ mdetuni . (Co... |
mdetuni0 21770 | Lemma for ~ mdetuni . (Co... |
mdetuni 21771 | According to the definitio... |
mdetmul 21772 | Multiplicativity of the de... |
m2detleiblem1 21773 | Lemma 1 for ~ m2detleib . ... |
m2detleiblem5 21774 | Lemma 5 for ~ m2detleib . ... |
m2detleiblem6 21775 | Lemma 6 for ~ m2detleib . ... |
m2detleiblem7 21776 | Lemma 7 for ~ m2detleib . ... |
m2detleiblem2 21777 | Lemma 2 for ~ m2detleib . ... |
m2detleiblem3 21778 | Lemma 3 for ~ m2detleib . ... |
m2detleiblem4 21779 | Lemma 4 for ~ m2detleib . ... |
m2detleib 21780 | Leibniz' Formula for 2x2-m... |
mndifsplit 21785 | Lemma for ~ maducoeval2 . ... |
madufval 21786 | First substitution for the... |
maduval 21787 | Second substitution for th... |
maducoeval 21788 | An entry of the adjunct (c... |
maducoeval2 21789 | An entry of the adjunct (c... |
maduf 21790 | Creating the adjunct of ma... |
madutpos 21791 | The adjuct of a transposed... |
madugsum 21792 | The determinant of a matri... |
madurid 21793 | Multiplying a matrix with ... |
madulid 21794 | Multiplying the adjunct of... |
minmar1fval 21795 | First substitution for the... |
minmar1val0 21796 | Second substitution for th... |
minmar1val 21797 | Third substitution for the... |
minmar1eval 21798 | An entry of a matrix for a... |
minmar1marrep 21799 | The minor matrix is a spec... |
minmar1cl 21800 | Closure of the row replace... |
maducoevalmin1 21801 | The coefficients of an adj... |
symgmatr01lem 21802 | Lemma for ~ symgmatr01 . ... |
symgmatr01 21803 | Applying a permutation tha... |
gsummatr01lem1 21804 | Lemma A for ~ gsummatr01 .... |
gsummatr01lem2 21805 | Lemma B for ~ gsummatr01 .... |
gsummatr01lem3 21806 | Lemma 1 for ~ gsummatr01 .... |
gsummatr01lem4 21807 | Lemma 2 for ~ gsummatr01 .... |
gsummatr01 21808 | Lemma 1 for ~ smadiadetlem... |
marep01ma 21809 | Replacing a row of a squar... |
smadiadetlem0 21810 | Lemma 0 for ~ smadiadet : ... |
smadiadetlem1 21811 | Lemma 1 for ~ smadiadet : ... |
smadiadetlem1a 21812 | Lemma 1a for ~ smadiadet :... |
smadiadetlem2 21813 | Lemma 2 for ~ smadiadet : ... |
smadiadetlem3lem0 21814 | Lemma 0 for ~ smadiadetlem... |
smadiadetlem3lem1 21815 | Lemma 1 for ~ smadiadetlem... |
smadiadetlem3lem2 21816 | Lemma 2 for ~ smadiadetlem... |
smadiadetlem3 21817 | Lemma 3 for ~ smadiadet . ... |
smadiadetlem4 21818 | Lemma 4 for ~ smadiadet . ... |
smadiadet 21819 | The determinant of a subma... |
smadiadetglem1 21820 | Lemma 1 for ~ smadiadetg .... |
smadiadetglem2 21821 | Lemma 2 for ~ smadiadetg .... |
smadiadetg 21822 | The determinant of a squar... |
smadiadetg0 21823 | Lemma for ~ smadiadetr : v... |
smadiadetr 21824 | The determinant of a squar... |
invrvald 21825 | If a matrix multiplied wit... |
matinv 21826 | The inverse of a matrix is... |
matunit 21827 | A matrix is a unit in the ... |
slesolvec 21828 | Every solution of a system... |
slesolinv 21829 | The solution of a system o... |
slesolinvbi 21830 | The solution of a system o... |
slesolex 21831 | Every system of linear equ... |
cramerimplem1 21832 | Lemma 1 for ~ cramerimp : ... |
cramerimplem2 21833 | Lemma 2 for ~ cramerimp : ... |
cramerimplem3 21834 | Lemma 3 for ~ cramerimp : ... |
cramerimp 21835 | One direction of Cramer's ... |
cramerlem1 21836 | Lemma 1 for ~ cramer . (C... |
cramerlem2 21837 | Lemma 2 for ~ cramer . (C... |
cramerlem3 21838 | Lemma 3 for ~ cramer . (C... |
cramer0 21839 | Special case of Cramer's r... |
cramer 21840 | Cramer's rule. According ... |
pmatring 21841 | The set of polynomial matr... |
pmatlmod 21842 | The set of polynomial matr... |
pmatassa 21843 | The set of polynomial matr... |
pmat0op 21844 | The zero polynomial matrix... |
pmat1op 21845 | The identity polynomial ma... |
pmat1ovd 21846 | Entries of the identity po... |
pmat0opsc 21847 | The zero polynomial matrix... |
pmat1opsc 21848 | The identity polynomial ma... |
pmat1ovscd 21849 | Entries of the identity po... |
pmatcoe1fsupp 21850 | For a polynomial matrix th... |
1pmatscmul 21851 | The scalar product of the ... |
cpmat 21858 | Value of the constructor o... |
cpmatpmat 21859 | A constant polynomial matr... |
cpmatel 21860 | Property of a constant pol... |
cpmatelimp 21861 | Implication of a set being... |
cpmatel2 21862 | Another property of a cons... |
cpmatelimp2 21863 | Another implication of a s... |
1elcpmat 21864 | The identity of the ring o... |
cpmatacl 21865 | The set of all constant po... |
cpmatinvcl 21866 | The set of all constant po... |
cpmatmcllem 21867 | Lemma for ~ cpmatmcl . (C... |
cpmatmcl 21868 | The set of all constant po... |
cpmatsubgpmat 21869 | The set of all constant po... |
cpmatsrgpmat 21870 | The set of all constant po... |
0elcpmat 21871 | The zero of the ring of al... |
mat2pmatfval 21872 | Value of the matrix transf... |
mat2pmatval 21873 | The result of a matrix tra... |
mat2pmatvalel 21874 | A (matrix) element of the ... |
mat2pmatbas 21875 | The result of a matrix tra... |
mat2pmatbas0 21876 | The result of a matrix tra... |
mat2pmatf 21877 | The matrix transformation ... |
mat2pmatf1 21878 | The matrix transformation ... |
mat2pmatghm 21879 | The transformation of matr... |
mat2pmatmul 21880 | The transformation of matr... |
mat2pmat1 21881 | The transformation of the ... |
mat2pmatmhm 21882 | The transformation of matr... |
mat2pmatrhm 21883 | The transformation of matr... |
mat2pmatlin 21884 | The transformation of matr... |
0mat2pmat 21885 | The transformed zero matri... |
idmatidpmat 21886 | The transformed identity m... |
d0mat2pmat 21887 | The transformed empty set ... |
d1mat2pmat 21888 | The transformation of a ma... |
mat2pmatscmxcl 21889 | A transformed matrix multi... |
m2cpm 21890 | The result of a matrix tra... |
m2cpmf 21891 | The matrix transformation ... |
m2cpmf1 21892 | The matrix transformation ... |
m2cpmghm 21893 | The transformation of matr... |
m2cpmmhm 21894 | The transformation of matr... |
m2cpmrhm 21895 | The transformation of matr... |
m2pmfzmap 21896 | The transformed values of ... |
m2pmfzgsumcl 21897 | Closure of the sum of scal... |
cpm2mfval 21898 | Value of the inverse matri... |
cpm2mval 21899 | The result of an inverse m... |
cpm2mvalel 21900 | A (matrix) element of the ... |
cpm2mf 21901 | The inverse matrix transfo... |
m2cpminvid 21902 | The inverse transformation... |
m2cpminvid2lem 21903 | Lemma for ~ m2cpminvid2 . ... |
m2cpminvid2 21904 | The transformation applied... |
m2cpmfo 21905 | The matrix transformation ... |
m2cpmf1o 21906 | The matrix transformation ... |
m2cpmrngiso 21907 | The transformation of matr... |
matcpmric 21908 | The ring of matrices over ... |
m2cpminv 21909 | The inverse matrix transfo... |
m2cpminv0 21910 | The inverse matrix transfo... |
decpmatval0 21913 | The matrix consisting of t... |
decpmatval 21914 | The matrix consisting of t... |
decpmate 21915 | An entry of the matrix con... |
decpmatcl 21916 | Closure of the decompositi... |
decpmataa0 21917 | The matrix consisting of t... |
decpmatfsupp 21918 | The mapping to the matrice... |
decpmatid 21919 | The matrix consisting of t... |
decpmatmullem 21920 | Lemma for ~ decpmatmul . ... |
decpmatmul 21921 | The matrix consisting of t... |
decpmatmulsumfsupp 21922 | Lemma 0 for ~ pm2mpmhm . ... |
pmatcollpw1lem1 21923 | Lemma 1 for ~ pmatcollpw1 ... |
pmatcollpw1lem2 21924 | Lemma 2 for ~ pmatcollpw1 ... |
pmatcollpw1 21925 | Write a polynomial matrix ... |
pmatcollpw2lem 21926 | Lemma for ~ pmatcollpw2 . ... |
pmatcollpw2 21927 | Write a polynomial matrix ... |
monmatcollpw 21928 | The matrix consisting of t... |
pmatcollpwlem 21929 | Lemma for ~ pmatcollpw . ... |
pmatcollpw 21930 | Write a polynomial matrix ... |
pmatcollpwfi 21931 | Write a polynomial matrix ... |
pmatcollpw3lem 21932 | Lemma for ~ pmatcollpw3 an... |
pmatcollpw3 21933 | Write a polynomial matrix ... |
pmatcollpw3fi 21934 | Write a polynomial matrix ... |
pmatcollpw3fi1lem1 21935 | Lemma 1 for ~ pmatcollpw3f... |
pmatcollpw3fi1lem2 21936 | Lemma 2 for ~ pmatcollpw3f... |
pmatcollpw3fi1 21937 | Write a polynomial matrix ... |
pmatcollpwscmatlem1 21938 | Lemma 1 for ~ pmatcollpwsc... |
pmatcollpwscmatlem2 21939 | Lemma 2 for ~ pmatcollpwsc... |
pmatcollpwscmat 21940 | Write a scalar matrix over... |
pm2mpf1lem 21943 | Lemma for ~ pm2mpf1 . (Co... |
pm2mpval 21944 | Value of the transformatio... |
pm2mpfval 21945 | A polynomial matrix transf... |
pm2mpcl 21946 | The transformation of poly... |
pm2mpf 21947 | The transformation of poly... |
pm2mpf1 21948 | The transformation of poly... |
pm2mpcoe1 21949 | A coefficient of the polyn... |
idpm2idmp 21950 | The transformation of the ... |
mptcoe1matfsupp 21951 | The mapping extracting the... |
mply1topmatcllem 21952 | Lemma for ~ mply1topmatcl ... |
mply1topmatval 21953 | A polynomial over matrices... |
mply1topmatcl 21954 | A polynomial over matrices... |
mp2pm2mplem1 21955 | Lemma 1 for ~ mp2pm2mp . ... |
mp2pm2mplem2 21956 | Lemma 2 for ~ mp2pm2mp . ... |
mp2pm2mplem3 21957 | Lemma 3 for ~ mp2pm2mp . ... |
mp2pm2mplem4 21958 | Lemma 4 for ~ mp2pm2mp . ... |
mp2pm2mplem5 21959 | Lemma 5 for ~ mp2pm2mp . ... |
mp2pm2mp 21960 | A polynomial over matrices... |
pm2mpghmlem2 21961 | Lemma 2 for ~ pm2mpghm . ... |
pm2mpghmlem1 21962 | Lemma 1 for pm2mpghm . (C... |
pm2mpfo 21963 | The transformation of poly... |
pm2mpf1o 21964 | The transformation of poly... |
pm2mpghm 21965 | The transformation of poly... |
pm2mpgrpiso 21966 | The transformation of poly... |
pm2mpmhmlem1 21967 | Lemma 1 for ~ pm2mpmhm . ... |
pm2mpmhmlem2 21968 | Lemma 2 for ~ pm2mpmhm . ... |
pm2mpmhm 21969 | The transformation of poly... |
pm2mprhm 21970 | The transformation of poly... |
pm2mprngiso 21971 | The transformation of poly... |
pmmpric 21972 | The ring of polynomial mat... |
monmat2matmon 21973 | The transformation of a po... |
pm2mp 21974 | The transformation of a su... |
chmatcl 21977 | Closure of the characteris... |
chmatval 21978 | The entries of the charact... |
chpmatfval 21979 | Value of the characteristi... |
chpmatval 21980 | The characteristic polynom... |
chpmatply1 21981 | The characteristic polynom... |
chpmatval2 21982 | The characteristic polynom... |
chpmat0d 21983 | The characteristic polynom... |
chpmat1dlem 21984 | Lemma for ~ chpmat1d . (C... |
chpmat1d 21985 | The characteristic polynom... |
chpdmatlem0 21986 | Lemma 0 for ~ chpdmat . (... |
chpdmatlem1 21987 | Lemma 1 for ~ chpdmat . (... |
chpdmatlem2 21988 | Lemma 2 for ~ chpdmat . (... |
chpdmatlem3 21989 | Lemma 3 for ~ chpdmat . (... |
chpdmat 21990 | The characteristic polynom... |
chpscmat 21991 | The characteristic polynom... |
chpscmat0 21992 | The characteristic polynom... |
chpscmatgsumbin 21993 | The characteristic polynom... |
chpscmatgsummon 21994 | The characteristic polynom... |
chp0mat 21995 | The characteristic polynom... |
chpidmat 21996 | The characteristic polynom... |
chmaidscmat 21997 | The characteristic polynom... |
fvmptnn04if 21998 | The function values of a m... |
fvmptnn04ifa 21999 | The function value of a ma... |
fvmptnn04ifb 22000 | The function value of a ma... |
fvmptnn04ifc 22001 | The function value of a ma... |
fvmptnn04ifd 22002 | The function value of a ma... |
chfacfisf 22003 | The "characteristic factor... |
chfacfisfcpmat 22004 | The "characteristic factor... |
chfacffsupp 22005 | The "characteristic factor... |
chfacfscmulcl 22006 | Closure of a scaled value ... |
chfacfscmul0 22007 | A scaled value of the "cha... |
chfacfscmulfsupp 22008 | A mapping of scaled values... |
chfacfscmulgsum 22009 | Breaking up a sum of value... |
chfacfpmmulcl 22010 | Closure of the value of th... |
chfacfpmmul0 22011 | The value of the "characte... |
chfacfpmmulfsupp 22012 | A mapping of values of the... |
chfacfpmmulgsum 22013 | Breaking up a sum of value... |
chfacfpmmulgsum2 22014 | Breaking up a sum of value... |
cayhamlem1 22015 | Lemma 1 for ~ cayleyhamilt... |
cpmadurid 22016 | The right-hand fundamental... |
cpmidgsum 22017 | Representation of the iden... |
cpmidgsumm2pm 22018 | Representation of the iden... |
cpmidpmatlem1 22019 | Lemma 1 for ~ cpmidpmat . ... |
cpmidpmatlem2 22020 | Lemma 2 for ~ cpmidpmat . ... |
cpmidpmatlem3 22021 | Lemma 3 for ~ cpmidpmat . ... |
cpmidpmat 22022 | Representation of the iden... |
cpmadugsumlemB 22023 | Lemma B for ~ cpmadugsum .... |
cpmadugsumlemC 22024 | Lemma C for ~ cpmadugsum .... |
cpmadugsumlemF 22025 | Lemma F for ~ cpmadugsum .... |
cpmadugsumfi 22026 | The product of the charact... |
cpmadugsum 22027 | The product of the charact... |
cpmidgsum2 22028 | Representation of the iden... |
cpmidg2sum 22029 | Equality of two sums repre... |
cpmadumatpolylem1 22030 | Lemma 1 for ~ cpmadumatpol... |
cpmadumatpolylem2 22031 | Lemma 2 for ~ cpmadumatpol... |
cpmadumatpoly 22032 | The product of the charact... |
cayhamlem2 22033 | Lemma for ~ cayhamlem3 . ... |
chcoeffeqlem 22034 | Lemma for ~ chcoeffeq . (... |
chcoeffeq 22035 | The coefficients of the ch... |
cayhamlem3 22036 | Lemma for ~ cayhamlem4 . ... |
cayhamlem4 22037 | Lemma for ~ cayleyhamilton... |
cayleyhamilton0 22038 | The Cayley-Hamilton theore... |
cayleyhamilton 22039 | The Cayley-Hamilton theore... |
cayleyhamiltonALT 22040 | Alternate proof of ~ cayle... |
cayleyhamilton1 22041 | The Cayley-Hamilton theore... |
istopg 22044 | Express the predicate " ` ... |
istop2g 22045 | Express the predicate " ` ... |
uniopn 22046 | The union of a subset of a... |
iunopn 22047 | The indexed union of a sub... |
inopn 22048 | The intersection of two op... |
fitop 22049 | A topology is closed under... |
fiinopn 22050 | The intersection of a none... |
iinopn 22051 | The intersection of a none... |
unopn 22052 | The union of two open sets... |
0opn 22053 | The empty set is an open s... |
0ntop 22054 | The empty set is not a top... |
topopn 22055 | The underlying set of a to... |
eltopss 22056 | A member of a topology is ... |
riinopn 22057 | A finite indexed relative ... |
rintopn 22058 | A finite relative intersec... |
istopon 22061 | Property of being a topolo... |
topontop 22062 | A topology on a given base... |
toponuni 22063 | The base set of a topology... |
topontopi 22064 | A topology on a given base... |
toponunii 22065 | The base set of a topology... |
toptopon 22066 | Alternative definition of ... |
toptopon2 22067 | A topology is the same thi... |
topontopon 22068 | A topology on a set is a t... |
funtopon 22069 | The class ` TopOn ` is a f... |
toponrestid 22070 | Given a topology on a set,... |
toponsspwpw 22071 | The set of topologies on a... |
dmtopon 22072 | The domain of ` TopOn ` is... |
fntopon 22073 | The class ` TopOn ` is a f... |
toprntopon 22074 | A topology is the same thi... |
toponmax 22075 | The base set of a topology... |
toponss 22076 | A member of a topology is ... |
toponcom 22077 | If ` K ` is a topology on ... |
toponcomb 22078 | Biconditional form of ~ to... |
topgele 22079 | The topologies over the sa... |
topsn 22080 | The only topology on a sin... |
istps 22083 | Express the predicate "is ... |
istps2 22084 | Express the predicate "is ... |
tpsuni 22085 | The base set of a topologi... |
tpstop 22086 | The topology extractor on ... |
tpspropd 22087 | A topological space depend... |
tpsprop2d 22088 | A topological space depend... |
topontopn 22089 | Express the predicate "is ... |
tsettps 22090 | If the topology component ... |
istpsi 22091 | Properties that determine ... |
eltpsg 22092 | Properties that determine ... |
eltpsgOLD 22093 | Obsolete version of ~ eltp... |
eltpsi 22094 | Properties that determine ... |
isbasisg 22097 | Express the predicate "the... |
isbasis2g 22098 | Express the predicate "the... |
isbasis3g 22099 | Express the predicate "the... |
basis1 22100 | Property of a basis. (Con... |
basis2 22101 | Property of a basis. (Con... |
fiinbas 22102 | If a set is closed under f... |
basdif0 22103 | A basis is not affected by... |
baspartn 22104 | A disjoint system of sets ... |
tgval 22105 | The topology generated by ... |
tgval2 22106 | Definition of a topology g... |
eltg 22107 | Membership in a topology g... |
eltg2 22108 | Membership in a topology g... |
eltg2b 22109 | Membership in a topology g... |
eltg4i 22110 | An open set in a topology ... |
eltg3i 22111 | The union of a set of basi... |
eltg3 22112 | Membership in a topology g... |
tgval3 22113 | Alternate expression for t... |
tg1 22114 | Property of a member of a ... |
tg2 22115 | Property of a member of a ... |
bastg 22116 | A member of a basis is a s... |
unitg 22117 | The topology generated by ... |
tgss 22118 | Subset relation for genera... |
tgcl 22119 | Show that a basis generate... |
tgclb 22120 | The property ~ tgcl can be... |
tgtopon 22121 | A basis generates a topolo... |
topbas 22122 | A topology is its own basi... |
tgtop 22123 | A topology is its own basi... |
eltop 22124 | Membership in a topology, ... |
eltop2 22125 | Membership in a topology. ... |
eltop3 22126 | Membership in a topology. ... |
fibas 22127 | A collection of finite int... |
tgdom 22128 | A space has no more open s... |
tgiun 22129 | The indexed union of a set... |
tgidm 22130 | The topology generator fun... |
bastop 22131 | Two ways to express that a... |
tgtop11 22132 | The topology generation fu... |
0top 22133 | The singleton of the empty... |
en1top 22134 | ` { (/) } ` is the only to... |
en2top 22135 | If a topology has two elem... |
tgss3 22136 | A criterion for determinin... |
tgss2 22137 | A criterion for determinin... |
basgen 22138 | Given a topology ` J ` , s... |
basgen2 22139 | Given a topology ` J ` , s... |
2basgen 22140 | Conditions that determine ... |
tgfiss 22141 | If a subbase is included i... |
tgdif0 22142 | A generated topology is no... |
bastop1 22143 | A subset of a topology is ... |
bastop2 22144 | A version of ~ bastop1 tha... |
distop 22145 | The discrete topology on a... |
topnex 22146 | The class of all topologie... |
distopon 22147 | The discrete topology on a... |
sn0topon 22148 | The singleton of the empty... |
sn0top 22149 | The singleton of the empty... |
indislem 22150 | A lemma to eliminate some ... |
indistopon 22151 | The indiscrete topology on... |
indistop 22152 | The indiscrete topology on... |
indisuni 22153 | The base set of the indisc... |
fctop 22154 | The finite complement topo... |
fctop2 22155 | The finite complement topo... |
cctop 22156 | The countable complement t... |
ppttop 22157 | The particular point topol... |
pptbas 22158 | The particular point topol... |
epttop 22159 | The excluded point topolog... |
indistpsx 22160 | The indiscrete topology on... |
indistps 22161 | The indiscrete topology on... |
indistps2 22162 | The indiscrete topology on... |
indistpsALT 22163 | The indiscrete topology on... |
indistpsALTOLD 22164 | Obsolete proof of ~ indist... |
indistps2ALT 22165 | The indiscrete topology on... |
distps 22166 | The discrete topology on a... |
fncld 22173 | The closed-set generator i... |
cldval 22174 | The set of closed sets of ... |
ntrfval 22175 | The interior function on t... |
clsfval 22176 | The closure function on th... |
cldrcl 22177 | Reverse closure of the clo... |
iscld 22178 | The predicate "the class `... |
iscld2 22179 | A subset of the underlying... |
cldss 22180 | A closed set is a subset o... |
cldss2 22181 | The set of closed sets is ... |
cldopn 22182 | The complement of a closed... |
isopn2 22183 | A subset of the underlying... |
opncld 22184 | The complement of an open ... |
difopn 22185 | The difference of a closed... |
topcld 22186 | The underlying set of a to... |
ntrval 22187 | The interior of a subset o... |
clsval 22188 | The closure of a subset of... |
0cld 22189 | The empty set is closed. ... |
iincld 22190 | The indexed intersection o... |
intcld 22191 | The intersection of a set ... |
uncld 22192 | The union of two closed se... |
cldcls 22193 | A closed subset equals its... |
incld 22194 | The intersection of two cl... |
riincld 22195 | An indexed relative inters... |
iuncld 22196 | A finite indexed union of ... |
unicld 22197 | A finite union of closed s... |
clscld 22198 | The closure of a subset of... |
clsf 22199 | The closure function is a ... |
ntropn 22200 | The interior of a subset o... |
clsval2 22201 | Express closure in terms o... |
ntrval2 22202 | Interior expressed in term... |
ntrdif 22203 | An interior of a complemen... |
clsdif 22204 | A closure of a complement ... |
clsss 22205 | Subset relationship for cl... |
ntrss 22206 | Subset relationship for in... |
sscls 22207 | A subset of a topology's u... |
ntrss2 22208 | A subset includes its inte... |
ssntr 22209 | An open subset of a set is... |
clsss3 22210 | The closure of a subset of... |
ntrss3 22211 | The interior of a subset o... |
ntrin 22212 | A pairwise intersection of... |
cmclsopn 22213 | The complement of a closur... |
cmntrcld 22214 | The complement of an inter... |
iscld3 22215 | A subset is closed iff it ... |
iscld4 22216 | A subset is closed iff it ... |
isopn3 22217 | A subset is open iff it eq... |
clsidm 22218 | The closure operation is i... |
ntridm 22219 | The interior operation is ... |
clstop 22220 | The closure of a topology'... |
ntrtop 22221 | The interior of a topology... |
0ntr 22222 | A subset with an empty int... |
clsss2 22223 | If a subset is included in... |
elcls 22224 | Membership in a closure. ... |
elcls2 22225 | Membership in a closure. ... |
clsndisj 22226 | Any open set containing a ... |
ntrcls0 22227 | A subset whose closure has... |
ntreq0 22228 | Two ways to say that a sub... |
cldmre 22229 | The closed sets of a topol... |
mrccls 22230 | Moore closure generalizes ... |
cls0 22231 | The closure of the empty s... |
ntr0 22232 | The interior of the empty ... |
isopn3i 22233 | An open subset equals its ... |
elcls3 22234 | Membership in a closure in... |
opncldf1 22235 | A bijection useful for con... |
opncldf2 22236 | The values of the open-clo... |
opncldf3 22237 | The values of the converse... |
isclo 22238 | A set ` A ` is clopen iff ... |
isclo2 22239 | A set ` A ` is clopen iff ... |
discld 22240 | The open sets of a discret... |
sn0cld 22241 | The closed sets of the top... |
indiscld 22242 | The closed sets of an indi... |
mretopd 22243 | A Moore collection which i... |
toponmre 22244 | The topologies over a give... |
cldmreon 22245 | The closed sets of a topol... |
iscldtop 22246 | A family is the closed set... |
mreclatdemoBAD 22247 | The closed subspaces of a ... |
neifval 22250 | Value of the neighborhood ... |
neif 22251 | The neighborhood function ... |
neiss2 22252 | A set with a neighborhood ... |
neival 22253 | Value of the set of neighb... |
isnei 22254 | The predicate "the class `... |
neiint 22255 | An intuitive definition of... |
isneip 22256 | The predicate "the class `... |
neii1 22257 | A neighborhood is included... |
neisspw 22258 | The neighborhoods of any s... |
neii2 22259 | Property of a neighborhood... |
neiss 22260 | Any neighborhood of a set ... |
ssnei 22261 | A set is included in any o... |
elnei 22262 | A point belongs to any of ... |
0nnei 22263 | The empty set is not a nei... |
neips 22264 | A neighborhood of a set is... |
opnneissb 22265 | An open set is a neighborh... |
opnssneib 22266 | Any superset of an open se... |
ssnei2 22267 | Any subset ` M ` of ` X ` ... |
neindisj 22268 | Any neighborhood of an ele... |
opnneiss 22269 | An open set is a neighborh... |
opnneip 22270 | An open set is a neighborh... |
opnnei 22271 | A set is open iff it is a ... |
tpnei 22272 | The underlying set of a to... |
neiuni 22273 | The union of the neighborh... |
neindisj2 22274 | A point ` P ` belongs to t... |
topssnei 22275 | A finer topology has more ... |
innei 22276 | The intersection of two ne... |
opnneiid 22277 | Only an open set is a neig... |
neissex 22278 | For any neighborhood ` N `... |
0nei 22279 | The empty set is a neighbo... |
neipeltop 22280 | Lemma for ~ neiptopreu . ... |
neiptopuni 22281 | Lemma for ~ neiptopreu . ... |
neiptoptop 22282 | Lemma for ~ neiptopreu . ... |
neiptopnei 22283 | Lemma for ~ neiptopreu . ... |
neiptopreu 22284 | If, to each element ` P ` ... |
lpfval 22289 | The limit point function o... |
lpval 22290 | The set of limit points of... |
islp 22291 | The predicate "the class `... |
lpsscls 22292 | The limit points of a subs... |
lpss 22293 | The limit points of a subs... |
lpdifsn 22294 | ` P ` is a limit point of ... |
lpss3 22295 | Subset relationship for li... |
islp2 22296 | The predicate " ` P ` is a... |
islp3 22297 | The predicate " ` P ` is a... |
maxlp 22298 | A point is a limit point o... |
clslp 22299 | The closure of a subset of... |
islpi 22300 | A point belonging to a set... |
cldlp 22301 | A subset of a topological ... |
isperf 22302 | Definition of a perfect sp... |
isperf2 22303 | Definition of a perfect sp... |
isperf3 22304 | A perfect space is a topol... |
perflp 22305 | The limit points of a perf... |
perfi 22306 | Property of a perfect spac... |
perftop 22307 | A perfect space is a topol... |
restrcl 22308 | Reverse closure for the su... |
restbas 22309 | A subspace topology basis ... |
tgrest 22310 | A subspace can be generate... |
resttop 22311 | A subspace topology is a t... |
resttopon 22312 | A subspace topology is a t... |
restuni 22313 | The underlying set of a su... |
stoig 22314 | The topological space buil... |
restco 22315 | Composition of subspaces. ... |
restabs 22316 | Equivalence of being a sub... |
restin 22317 | When the subspace region i... |
restuni2 22318 | The underlying set of a su... |
resttopon2 22319 | The underlying set of a su... |
rest0 22320 | The subspace topology indu... |
restsn 22321 | The only subspace topology... |
restsn2 22322 | The subspace topology indu... |
restcld 22323 | A closed set of a subspace... |
restcldi 22324 | A closed set is closed in ... |
restcldr 22325 | A set which is closed in t... |
restopnb 22326 | If ` B ` is an open subset... |
ssrest 22327 | If ` K ` is a finer topolo... |
restopn2 22328 | If ` A ` is open, then ` B... |
restdis 22329 | A subspace of a discrete t... |
restfpw 22330 | The restriction of the set... |
neitr 22331 | The neighborhood of a trac... |
restcls 22332 | A closure in a subspace to... |
restntr 22333 | An interior in a subspace ... |
restlp 22334 | The limit points of a subs... |
restperf 22335 | Perfection of a subspace. ... |
perfopn 22336 | An open subset of a perfec... |
resstopn 22337 | The topology of a restrict... |
resstps 22338 | A restricted topological s... |
ordtbaslem 22339 | Lemma for ~ ordtbas . In ... |
ordtval 22340 | Value of the order topolog... |
ordtuni 22341 | Value of the order topolog... |
ordtbas2 22342 | Lemma for ~ ordtbas . (Co... |
ordtbas 22343 | In a total order, the fini... |
ordttopon 22344 | Value of the order topolog... |
ordtopn1 22345 | An upward ray ` ( P , +oo ... |
ordtopn2 22346 | A downward ray ` ( -oo , P... |
ordtopn3 22347 | An open interval ` ( A , B... |
ordtcld1 22348 | A downward ray ` ( -oo , P... |
ordtcld2 22349 | An upward ray ` [ P , +oo ... |
ordtcld3 22350 | A closed interval ` [ A , ... |
ordttop 22351 | The order topology is a to... |
ordtcnv 22352 | The order dual generates t... |
ordtrest 22353 | The subspace topology of a... |
ordtrest2lem 22354 | Lemma for ~ ordtrest2 . (... |
ordtrest2 22355 | An interval-closed set ` A... |
letopon 22356 | The topology of the extend... |
letop 22357 | The topology of the extend... |
letopuni 22358 | The topology of the extend... |
xrstopn 22359 | The topology component of ... |
xrstps 22360 | The extended real number s... |
leordtvallem1 22361 | Lemma for ~ leordtval . (... |
leordtvallem2 22362 | Lemma for ~ leordtval . (... |
leordtval2 22363 | The topology of the extend... |
leordtval 22364 | The topology of the extend... |
iccordt 22365 | A closed interval is close... |
iocpnfordt 22366 | An unbounded above open in... |
icomnfordt 22367 | An unbounded above open in... |
iooordt 22368 | An open interval is open i... |
reordt 22369 | The real numbers are an op... |
lecldbas 22370 | The set of closed interval... |
pnfnei 22371 | A neighborhood of ` +oo ` ... |
mnfnei 22372 | A neighborhood of ` -oo ` ... |
ordtrestixx 22373 | The restriction of the les... |
ordtresticc 22374 | The restriction of the les... |
lmrel 22381 | The topological space conv... |
lmrcl 22382 | Reverse closure for the co... |
lmfval 22383 | The relation "sequence ` f... |
cnfval 22384 | The set of all continuous ... |
cnpfval 22385 | The function mapping the p... |
iscn 22386 | The predicate "the class `... |
cnpval 22387 | The set of all functions f... |
iscnp 22388 | The predicate "the class `... |
iscn2 22389 | The predicate "the class `... |
iscnp2 22390 | The predicate "the class `... |
cntop1 22391 | Reverse closure for a cont... |
cntop2 22392 | Reverse closure for a cont... |
cnptop1 22393 | Reverse closure for a func... |
cnptop2 22394 | Reverse closure for a func... |
iscnp3 22395 | The predicate "the class `... |
cnprcl 22396 | Reverse closure for a func... |
cnf 22397 | A continuous function is a... |
cnpf 22398 | A continuous function at p... |
cnpcl 22399 | The value of a continuous ... |
cnf2 22400 | A continuous function is a... |
cnpf2 22401 | A continuous function at p... |
cnprcl2 22402 | Reverse closure for a func... |
tgcn 22403 | The continuity predicate w... |
tgcnp 22404 | The "continuous at a point... |
subbascn 22405 | The continuity predicate w... |
ssidcn 22406 | The identity function is a... |
cnpimaex 22407 | Property of a function con... |
idcn 22408 | A restricted identity func... |
lmbr 22409 | Express the binary relatio... |
lmbr2 22410 | Express the binary relatio... |
lmbrf 22411 | Express the binary relatio... |
lmconst 22412 | A constant sequence conver... |
lmcvg 22413 | Convergence property of a ... |
iscnp4 22414 | The predicate "the class `... |
cnpnei 22415 | A condition for continuity... |
cnima 22416 | An open subset of the codo... |
cnco 22417 | The composition of two con... |
cnpco 22418 | The composition of a funct... |
cnclima 22419 | A closed subset of the cod... |
iscncl 22420 | A characterization of a co... |
cncls2i 22421 | Property of the preimage o... |
cnntri 22422 | Property of the preimage o... |
cnclsi 22423 | Property of the image of a... |
cncls2 22424 | Continuity in terms of clo... |
cncls 22425 | Continuity in terms of clo... |
cnntr 22426 | Continuity in terms of int... |
cnss1 22427 | If the topology ` K ` is f... |
cnss2 22428 | If the topology ` K ` is f... |
cncnpi 22429 | A continuous function is c... |
cnsscnp 22430 | The set of continuous func... |
cncnp 22431 | A continuous function is c... |
cncnp2 22432 | A continuous function is c... |
cnnei 22433 | Continuity in terms of nei... |
cnconst2 22434 | A constant function is con... |
cnconst 22435 | A constant function is con... |
cnrest 22436 | Continuity of a restrictio... |
cnrest2 22437 | Equivalence of continuity ... |
cnrest2r 22438 | Equivalence of continuity ... |
cnpresti 22439 | One direction of ~ cnprest... |
cnprest 22440 | Equivalence of continuity ... |
cnprest2 22441 | Equivalence of point-conti... |
cndis 22442 | Every function is continuo... |
cnindis 22443 | Every function is continuo... |
cnpdis 22444 | If ` A ` is an isolated po... |
paste 22445 | Pasting lemma. If ` A ` a... |
lmfpm 22446 | If ` F ` converges, then `... |
lmfss 22447 | Inclusion of a function ha... |
lmcl 22448 | Closure of a limit. (Cont... |
lmss 22449 | Limit on a subspace. (Con... |
sslm 22450 | A finer topology has fewer... |
lmres 22451 | A function converges iff i... |
lmff 22452 | If ` F ` converges, there ... |
lmcls 22453 | Any convergent sequence of... |
lmcld 22454 | Any convergent sequence of... |
lmcnp 22455 | The image of a convergent ... |
lmcn 22456 | The image of a convergent ... |
ist0 22471 | The predicate "is a T_0 sp... |
ist1 22472 | The predicate "is a T_1 sp... |
ishaus 22473 | The predicate "is a Hausdo... |
iscnrm 22474 | The property of being comp... |
t0sep 22475 | Any two topologically indi... |
t0dist 22476 | Any two distinct points in... |
t1sncld 22477 | In a T_1 space, singletons... |
t1ficld 22478 | In a T_1 space, finite set... |
hausnei 22479 | Neighborhood property of a... |
t0top 22480 | A T_0 space is a topologic... |
t1top 22481 | A T_1 space is a topologic... |
haustop 22482 | A Hausdorff space is a top... |
isreg 22483 | The predicate "is a regula... |
regtop 22484 | A regular space is a topol... |
regsep 22485 | In a regular space, every ... |
isnrm 22486 | The predicate "is a normal... |
nrmtop 22487 | A normal space is a topolo... |
cnrmtop 22488 | A completely normal space ... |
iscnrm2 22489 | The property of being comp... |
ispnrm 22490 | The property of being perf... |
pnrmnrm 22491 | A perfectly normal space i... |
pnrmtop 22492 | A perfectly normal space i... |
pnrmcld 22493 | A closed set in a perfectl... |
pnrmopn 22494 | An open set in a perfectly... |
ist0-2 22495 | The predicate "is a T_0 sp... |
ist0-3 22496 | The predicate "is a T_0 sp... |
cnt0 22497 | The preimage of a T_0 topo... |
ist1-2 22498 | An alternate characterizat... |
t1t0 22499 | A T_1 space is a T_0 space... |
ist1-3 22500 | A space is T_1 iff every p... |
cnt1 22501 | The preimage of a T_1 topo... |
ishaus2 22502 | Express the predicate " ` ... |
haust1 22503 | A Hausdorff space is a T_1... |
hausnei2 22504 | The Hausdorff condition st... |
cnhaus 22505 | The preimage of a Hausdorf... |
nrmsep3 22506 | In a normal space, given a... |
nrmsep2 22507 | In a normal space, any two... |
nrmsep 22508 | In a normal space, disjoin... |
isnrm2 22509 | An alternate characterizat... |
isnrm3 22510 | A topological space is nor... |
cnrmi 22511 | A subspace of a completely... |
cnrmnrm 22512 | A completely normal space ... |
restcnrm 22513 | A subspace of a completely... |
resthauslem 22514 | Lemma for ~ resthaus and s... |
lpcls 22515 | The limit points of the cl... |
perfcls 22516 | A subset of a perfect spac... |
restt0 22517 | A subspace of a T_0 topolo... |
restt1 22518 | A subspace of a T_1 topolo... |
resthaus 22519 | A subspace of a Hausdorff ... |
t1sep2 22520 | Any two points in a T_1 sp... |
t1sep 22521 | Any two distinct points in... |
sncld 22522 | A singleton is closed in a... |
sshauslem 22523 | Lemma for ~ sshaus and sim... |
sst0 22524 | A topology finer than a T_... |
sst1 22525 | A topology finer than a T_... |
sshaus 22526 | A topology finer than a Ha... |
regsep2 22527 | In a regular space, a clos... |
isreg2 22528 | A topological space is reg... |
dnsconst 22529 | If a continuous mapping to... |
ordtt1 22530 | The order topology is T_1 ... |
lmmo 22531 | A sequence in a Hausdorff ... |
lmfun 22532 | The convergence relation i... |
dishaus 22533 | A discrete topology is Hau... |
ordthauslem 22534 | Lemma for ~ ordthaus . (C... |
ordthaus 22535 | The order topology of a to... |
xrhaus 22536 | The topology of the extend... |
iscmp 22539 | The predicate "is a compac... |
cmpcov 22540 | An open cover of a compact... |
cmpcov2 22541 | Rewrite ~ cmpcov for the c... |
cmpcovf 22542 | Combine ~ cmpcov with ~ ac... |
cncmp 22543 | Compactness is respected b... |
fincmp 22544 | A finite topology is compa... |
0cmp 22545 | The singleton of the empty... |
cmptop 22546 | A compact topology is a to... |
rncmp 22547 | The image of a compact set... |
imacmp 22548 | The image of a compact set... |
discmp 22549 | A discrete topology is com... |
cmpsublem 22550 | Lemma for ~ cmpsub . (Con... |
cmpsub 22551 | Two equivalent ways of des... |
tgcmp 22552 | A topology generated by a ... |
cmpcld 22553 | A closed subset of a compa... |
uncmp 22554 | The union of two compact s... |
fiuncmp 22555 | A finite union of compact ... |
sscmp 22556 | A subset of a compact topo... |
hauscmplem 22557 | Lemma for ~ hauscmp . (Co... |
hauscmp 22558 | A compact subspace of a T2... |
cmpfi 22559 | If a topology is compact a... |
cmpfii 22560 | In a compact topology, a s... |
bwth 22561 | The glorious Bolzano-Weier... |
isconn 22564 | The predicate ` J ` is a c... |
isconn2 22565 | The predicate ` J ` is a c... |
connclo 22566 | The only nonempty clopen s... |
conndisj 22567 | If a topology is connected... |
conntop 22568 | A connected topology is a ... |
indisconn 22569 | The indiscrete topology (o... |
dfconn2 22570 | An alternate definition of... |
connsuba 22571 | Connectedness for a subspa... |
connsub 22572 | Two equivalent ways of say... |
cnconn 22573 | Connectedness is respected... |
nconnsubb 22574 | Disconnectedness for a sub... |
connsubclo 22575 | If a clopen set meets a co... |
connima 22576 | The image of a connected s... |
conncn 22577 | A continuous function from... |
iunconnlem 22578 | Lemma for ~ iunconn . (Co... |
iunconn 22579 | The indexed union of conne... |
unconn 22580 | The union of two connected... |
clsconn 22581 | The closure of a connected... |
conncompid 22582 | The connected component co... |
conncompconn 22583 | The connected component co... |
conncompss 22584 | The connected component co... |
conncompcld 22585 | The connected component co... |
conncompclo 22586 | The connected component co... |
t1connperf 22587 | A connected T_1 space is p... |
is1stc 22592 | The predicate "is a first-... |
is1stc2 22593 | An equivalent way of sayin... |
1stctop 22594 | A first-countable topology... |
1stcclb 22595 | A property of points in a ... |
1stcfb 22596 | For any point ` A ` in a f... |
is2ndc 22597 | The property of being seco... |
2ndctop 22598 | A second-countable topolog... |
2ndci 22599 | A countable basis generate... |
2ndcsb 22600 | Having a countable subbase... |
2ndcredom 22601 | A second-countable space h... |
2ndc1stc 22602 | A second-countable space i... |
1stcrestlem 22603 | Lemma for ~ 1stcrest . (C... |
1stcrest 22604 | A subspace of a first-coun... |
2ndcrest 22605 | A subspace of a second-cou... |
2ndcctbss 22606 | If a topology is second-co... |
2ndcdisj 22607 | Any disjoint family of ope... |
2ndcdisj2 22608 | Any disjoint collection of... |
2ndcomap 22609 | A surjective continuous op... |
2ndcsep 22610 | A second-countable topolog... |
dis2ndc 22611 | A discrete space is second... |
1stcelcls 22612 | A point belongs to the clo... |
1stccnp 22613 | A mapping is continuous at... |
1stccn 22614 | A mapping ` X --> Y ` , wh... |
islly 22619 | The property of being a lo... |
isnlly 22620 | The property of being an n... |
llyeq 22621 | Equality theorem for the `... |
nllyeq 22622 | Equality theorem for the `... |
llytop 22623 | A locally ` A ` space is a... |
nllytop 22624 | A locally ` A ` space is a... |
llyi 22625 | The property of a locally ... |
nllyi 22626 | The property of an n-local... |
nlly2i 22627 | Eliminate the neighborhood... |
llynlly 22628 | A locally ` A ` space is n... |
llyssnlly 22629 | A locally ` A ` space is n... |
llyss 22630 | The "locally" predicate re... |
nllyss 22631 | The "n-locally" predicate ... |
subislly 22632 | The property of a subspace... |
restnlly 22633 | If the property ` A ` pass... |
restlly 22634 | If the property ` A ` pass... |
islly2 22635 | An alternative expression ... |
llyrest 22636 | An open subspace of a loca... |
nllyrest 22637 | An open subspace of an n-l... |
loclly 22638 | If ` A ` is a local proper... |
llyidm 22639 | Idempotence of the "locall... |
nllyidm 22640 | Idempotence of the "n-loca... |
toplly 22641 | A topology is locally a to... |
topnlly 22642 | A topology is n-locally a ... |
hauslly 22643 | A Hausdorff space is local... |
hausnlly 22644 | A Hausdorff space is n-loc... |
hausllycmp 22645 | A compact Hausdorff space ... |
cldllycmp 22646 | A closed subspace of a loc... |
lly1stc 22647 | First-countability is a lo... |
dislly 22648 | The discrete space ` ~P X ... |
disllycmp 22649 | A discrete space is locall... |
dis1stc 22650 | A discrete space is first-... |
hausmapdom 22651 | If ` X ` is a first-counta... |
hauspwdom 22652 | Simplify the cardinal ` A ... |
refrel 22659 | Refinement is a relation. ... |
isref 22660 | The property of being a re... |
refbas 22661 | A refinement covers the sa... |
refssex 22662 | Every set in a refinement ... |
ssref 22663 | A subcover is a refinement... |
refref 22664 | Reflexivity of refinement.... |
reftr 22665 | Refinement is transitive. ... |
refun0 22666 | Adding the empty set prese... |
isptfin 22667 | The statement "is a point-... |
islocfin 22668 | The statement "is a locall... |
finptfin 22669 | A finite cover is a point-... |
ptfinfin 22670 | A point covered by a point... |
finlocfin 22671 | A finite cover of a topolo... |
locfintop 22672 | A locally finite cover cov... |
locfinbas 22673 | A locally finite cover mus... |
locfinnei 22674 | A point covered by a local... |
lfinpfin 22675 | A locally finite cover is ... |
lfinun 22676 | Adding a finite set preser... |
locfincmp 22677 | For a compact space, the l... |
unisngl 22678 | Taking the union of the se... |
dissnref 22679 | The set of singletons is a... |
dissnlocfin 22680 | The set of singletons is l... |
locfindis 22681 | The locally finite covers ... |
locfincf 22682 | A locally finite cover in ... |
comppfsc 22683 | A space where every open c... |
kgenval 22686 | Value of the compact gener... |
elkgen 22687 | Value of the compact gener... |
kgeni 22688 | Property of the open sets ... |
kgentopon 22689 | The compact generator gene... |
kgenuni 22690 | The base set of the compac... |
kgenftop 22691 | The compact generator gene... |
kgenf 22692 | The compact generator is a... |
kgentop 22693 | A compactly generated spac... |
kgenss 22694 | The compact generator gene... |
kgenhaus 22695 | The compact generator gene... |
kgencmp 22696 | The compact generator topo... |
kgencmp2 22697 | The compact generator topo... |
kgenidm 22698 | The compact generator is i... |
iskgen2 22699 | A space is compactly gener... |
iskgen3 22700 | Derive the usual definitio... |
llycmpkgen2 22701 | A locally compact space is... |
cmpkgen 22702 | A compact space is compact... |
llycmpkgen 22703 | A locally compact space is... |
1stckgenlem 22704 | The one-point compactifica... |
1stckgen 22705 | A first-countable space is... |
kgen2ss 22706 | The compact generator pres... |
kgencn 22707 | A function from a compactl... |
kgencn2 22708 | A function ` F : J --> K `... |
kgencn3 22709 | The set of continuous func... |
kgen2cn 22710 | A continuous function is a... |
txval 22715 | Value of the binary topolo... |
txuni2 22716 | The underlying set of the ... |
txbasex 22717 | The basis for the product ... |
txbas 22718 | The set of Cartesian produ... |
eltx 22719 | A set in a product is open... |
txtop 22720 | The product of two topolog... |
ptval 22721 | The value of the product t... |
ptpjpre1 22722 | The preimage of a projecti... |
elpt 22723 | Elementhood in the bases o... |
elptr 22724 | A basic open set in the pr... |
elptr2 22725 | A basic open set in the pr... |
ptbasid 22726 | The base set of the produc... |
ptuni2 22727 | The base set for the produ... |
ptbasin 22728 | The basis for a product to... |
ptbasin2 22729 | The basis for a product to... |
ptbas 22730 | The basis for a product to... |
ptpjpre2 22731 | The basis for a product to... |
ptbasfi 22732 | The basis for the product ... |
pttop 22733 | The product topology is a ... |
ptopn 22734 | A basic open set in the pr... |
ptopn2 22735 | A sub-basic open set in th... |
xkotf 22736 | Functionality of function ... |
xkobval 22737 | Alternative expression for... |
xkoval 22738 | Value of the compact-open ... |
xkotop 22739 | The compact-open topology ... |
xkoopn 22740 | A basic open set of the co... |
txtopi 22741 | The product of two topolog... |
txtopon 22742 | The underlying set of the ... |
txuni 22743 | The underlying set of the ... |
txunii 22744 | The underlying set of the ... |
ptuni 22745 | The base set for the produ... |
ptunimpt 22746 | Base set of a product topo... |
pttopon 22747 | The base set for the produ... |
pttoponconst 22748 | The base set for a product... |
ptuniconst 22749 | The base set for a product... |
xkouni 22750 | The base set of the compac... |
xkotopon 22751 | The base set of the compac... |
ptval2 22752 | The value of the product t... |
txopn 22753 | The product of two open se... |
txcld 22754 | The product of two closed ... |
txcls 22755 | Closure of a rectangle in ... |
txss12 22756 | Subset property of the top... |
txbasval 22757 | It is sufficient to consid... |
neitx 22758 | The Cartesian product of t... |
txcnpi 22759 | Continuity of a two-argume... |
tx1cn 22760 | Continuity of the first pr... |
tx2cn 22761 | Continuity of the second p... |
ptpjcn 22762 | Continuity of a projection... |
ptpjopn 22763 | The projection map is an o... |
ptcld 22764 | A closed box in the produc... |
ptcldmpt 22765 | A closed box in the produc... |
ptclsg 22766 | The closure of a box in th... |
ptcls 22767 | The closure of a box in th... |
dfac14lem 22768 | Lemma for ~ dfac14 . By e... |
dfac14 22769 | Theorem ~ ptcls is an equi... |
xkoccn 22770 | The "constant function" fu... |
txcnp 22771 | If two functions are conti... |
ptcnplem 22772 | Lemma for ~ ptcnp . (Cont... |
ptcnp 22773 | If every projection of a f... |
upxp 22774 | Universal property of the ... |
txcnmpt 22775 | A map into the product of ... |
uptx 22776 | Universal property of the ... |
txcn 22777 | A map into the product of ... |
ptcn 22778 | If every projection of a f... |
prdstopn 22779 | Topology of a structure pr... |
prdstps 22780 | A structure product of top... |
pwstps 22781 | A structure power of a top... |
txrest 22782 | The subspace of a topologi... |
txdis 22783 | The topological product of... |
txindislem 22784 | Lemma for ~ txindis . (Co... |
txindis 22785 | The topological product of... |
txdis1cn 22786 | A function is jointly cont... |
txlly 22787 | If the property ` A ` is p... |
txnlly 22788 | If the property ` A ` is p... |
pthaus 22789 | The product of a collectio... |
ptrescn 22790 | Restriction is a continuou... |
txtube 22791 | The "tube lemma". If ` X ... |
txcmplem1 22792 | Lemma for ~ txcmp . (Cont... |
txcmplem2 22793 | Lemma for ~ txcmp . (Cont... |
txcmp 22794 | The topological product of... |
txcmpb 22795 | The topological product of... |
hausdiag 22796 | A topology is Hausdorff if... |
hauseqlcld 22797 | In a Hausdorff topology, t... |
txhaus 22798 | The topological product of... |
txlm 22799 | Two sequences converge iff... |
lmcn2 22800 | The image of a convergent ... |
tx1stc 22801 | The topological product of... |
tx2ndc 22802 | The topological product of... |
txkgen 22803 | The topological product of... |
xkohaus 22804 | If the codomain space is H... |
xkoptsub 22805 | The compact-open topology ... |
xkopt 22806 | The compact-open topology ... |
xkopjcn 22807 | Continuity of a projection... |
xkoco1cn 22808 | If ` F ` is a continuous f... |
xkoco2cn 22809 | If ` F ` is a continuous f... |
xkococnlem 22810 | Continuity of the composit... |
xkococn 22811 | Continuity of the composit... |
cnmptid 22812 | The identity function is c... |
cnmptc 22813 | A constant function is con... |
cnmpt11 22814 | The composition of continu... |
cnmpt11f 22815 | The composition of continu... |
cnmpt1t 22816 | The composition of continu... |
cnmpt12f 22817 | The composition of continu... |
cnmpt12 22818 | The composition of continu... |
cnmpt1st 22819 | The projection onto the fi... |
cnmpt2nd 22820 | The projection onto the se... |
cnmpt2c 22821 | A constant function is con... |
cnmpt21 22822 | The composition of continu... |
cnmpt21f 22823 | The composition of continu... |
cnmpt2t 22824 | The composition of continu... |
cnmpt22 22825 | The composition of continu... |
cnmpt22f 22826 | The composition of continu... |
cnmpt1res 22827 | The restriction of a conti... |
cnmpt2res 22828 | The restriction of a conti... |
cnmptcom 22829 | The argument converse of a... |
cnmptkc 22830 | The curried first projecti... |
cnmptkp 22831 | The evaluation of the inne... |
cnmptk1 22832 | The composition of a curri... |
cnmpt1k 22833 | The composition of a one-a... |
cnmptkk 22834 | The composition of two cur... |
xkofvcn 22835 | Joint continuity of the fu... |
cnmptk1p 22836 | The evaluation of a currie... |
cnmptk2 22837 | The uncurrying of a currie... |
xkoinjcn 22838 | Continuity of "injection",... |
cnmpt2k 22839 | The currying of a two-argu... |
txconn 22840 | The topological product of... |
imasnopn 22841 | If a relation graph is ope... |
imasncld 22842 | If a relation graph is clo... |
imasncls 22843 | If a relation graph is clo... |
qtopval 22846 | Value of the quotient topo... |
qtopval2 22847 | Value of the quotient topo... |
elqtop 22848 | Value of the quotient topo... |
qtopres 22849 | The quotient topology is u... |
qtoptop2 22850 | The quotient topology is a... |
qtoptop 22851 | The quotient topology is a... |
elqtop2 22852 | Value of the quotient topo... |
qtopuni 22853 | The base set of the quotie... |
elqtop3 22854 | Value of the quotient topo... |
qtoptopon 22855 | The base set of the quotie... |
qtopid 22856 | A quotient map is a contin... |
idqtop 22857 | The quotient topology indu... |
qtopcmplem 22858 | Lemma for ~ qtopcmp and ~ ... |
qtopcmp 22859 | A quotient of a compact sp... |
qtopconn 22860 | A quotient of a connected ... |
qtopkgen 22861 | A quotient of a compactly ... |
basqtop 22862 | An injection maps bases to... |
tgqtop 22863 | An injection maps generate... |
qtopcld 22864 | The property of being a cl... |
qtopcn 22865 | Universal property of a qu... |
qtopss 22866 | A surjective continuous fu... |
qtopeu 22867 | Universal property of the ... |
qtoprest 22868 | If ` A ` is a saturated op... |
qtopomap 22869 | If ` F ` is a surjective c... |
qtopcmap 22870 | If ` F ` is a surjective c... |
imastopn 22871 | The topology of an image s... |
imastps 22872 | The image of a topological... |
qustps 22873 | A quotient structure is a ... |
kqfval 22874 | Value of the function appe... |
kqfeq 22875 | Two points in the Kolmogor... |
kqffn 22876 | The topological indistingu... |
kqval 22877 | Value of the quotient topo... |
kqtopon 22878 | The Kolmogorov quotient is... |
kqid 22879 | The topological indistingu... |
ist0-4 22880 | The topological indistingu... |
kqfvima 22881 | When the image set is open... |
kqsat 22882 | Any open set is saturated ... |
kqdisj 22883 | A version of ~ imain for t... |
kqcldsat 22884 | Any closed set is saturate... |
kqopn 22885 | The topological indistingu... |
kqcld 22886 | The topological indistingu... |
kqt0lem 22887 | Lemma for ~ kqt0 . (Contr... |
isr0 22888 | The property " ` J ` is an... |
r0cld 22889 | The analogue of the T_1 ax... |
regr1lem 22890 | Lemma for ~ regr1 . (Cont... |
regr1lem2 22891 | A Kolmogorov quotient of a... |
kqreglem1 22892 | A Kolmogorov quotient of a... |
kqreglem2 22893 | If the Kolmogorov quotient... |
kqnrmlem1 22894 | A Kolmogorov quotient of a... |
kqnrmlem2 22895 | If the Kolmogorov quotient... |
kqtop 22896 | The Kolmogorov quotient is... |
kqt0 22897 | The Kolmogorov quotient is... |
kqf 22898 | The Kolmogorov quotient is... |
r0sep 22899 | The separation property of... |
nrmr0reg 22900 | A normal R_0 space is also... |
regr1 22901 | A regular space is R_1, wh... |
kqreg 22902 | The Kolmogorov quotient of... |
kqnrm 22903 | The Kolmogorov quotient of... |
hmeofn 22908 | The set of homeomorphisms ... |
hmeofval 22909 | The set of all the homeomo... |
ishmeo 22910 | The predicate F is a homeo... |
hmeocn 22911 | A homeomorphism is continu... |
hmeocnvcn 22912 | The converse of a homeomor... |
hmeocnv 22913 | The converse of a homeomor... |
hmeof1o2 22914 | A homeomorphism is a 1-1-o... |
hmeof1o 22915 | A homeomorphism is a 1-1-o... |
hmeoima 22916 | The image of an open set b... |
hmeoopn 22917 | Homeomorphisms preserve op... |
hmeocld 22918 | Homeomorphisms preserve cl... |
hmeocls 22919 | Homeomorphisms preserve cl... |
hmeontr 22920 | Homeomorphisms preserve in... |
hmeoimaf1o 22921 | The function mapping open ... |
hmeores 22922 | The restriction of a homeo... |
hmeoco 22923 | The composite of two homeo... |
idhmeo 22924 | The identity function is a... |
hmeocnvb 22925 | The converse of a homeomor... |
hmeoqtop 22926 | A homeomorphism is a quoti... |
hmph 22927 | Express the predicate ` J ... |
hmphi 22928 | If there is a homeomorphis... |
hmphtop 22929 | Reverse closure for the ho... |
hmphtop1 22930 | The relation "being homeom... |
hmphtop2 22931 | The relation "being homeom... |
hmphref 22932 | "Is homeomorphic to" is re... |
hmphsym 22933 | "Is homeomorphic to" is sy... |
hmphtr 22934 | "Is homeomorphic to" is tr... |
hmpher 22935 | "Is homeomorphic to" is an... |
hmphen 22936 | Homeomorphisms preserve th... |
hmphsymb 22937 | "Is homeomorphic to" is sy... |
haushmphlem 22938 | Lemma for ~ haushmph and s... |
cmphmph 22939 | Compactness is a topologic... |
connhmph 22940 | Connectedness is a topolog... |
t0hmph 22941 | T_0 is a topological prope... |
t1hmph 22942 | T_1 is a topological prope... |
haushmph 22943 | Hausdorff-ness is a topolo... |
reghmph 22944 | Regularity is a topologica... |
nrmhmph 22945 | Normality is a topological... |
hmph0 22946 | A topology homeomorphic to... |
hmphdis 22947 | Homeomorphisms preserve to... |
hmphindis 22948 | Homeomorphisms preserve to... |
indishmph 22949 | Equinumerous sets equipped... |
hmphen2 22950 | Homeomorphisms preserve th... |
cmphaushmeo 22951 | A continuous bijection fro... |
ordthmeolem 22952 | Lemma for ~ ordthmeo . (C... |
ordthmeo 22953 | An order isomorphism is a ... |
txhmeo 22954 | Lift a pair of homeomorphi... |
txswaphmeolem 22955 | Show inverse for the "swap... |
txswaphmeo 22956 | There is a homeomorphism f... |
pt1hmeo 22957 | The canonical homeomorphis... |
ptuncnv 22958 | Exhibit the converse funct... |
ptunhmeo 22959 | Define a homeomorphism fro... |
xpstopnlem1 22960 | The function ` F ` used in... |
xpstps 22961 | A binary product of topolo... |
xpstopnlem2 22962 | Lemma for ~ xpstopn . (Co... |
xpstopn 22963 | The topology on a binary p... |
ptcmpfi 22964 | A topological product of f... |
xkocnv 22965 | The inverse of the "curryi... |
xkohmeo 22966 | The Exponential Law for to... |
qtopf1 22967 | If a quotient map is injec... |
qtophmeo 22968 | If two functions on a base... |
t0kq 22969 | A topological space is T_0... |
kqhmph 22970 | A topological space is T_0... |
ist1-5lem 22971 | Lemma for ~ ist1-5 and sim... |
t1r0 22972 | A T_1 space is R_0. That ... |
ist1-5 22973 | A topological space is T_1... |
ishaus3 22974 | A topological space is Hau... |
nrmreg 22975 | A normal T_1 space is regu... |
reghaus 22976 | A regular T_0 space is Hau... |
nrmhaus 22977 | A T_1 normal space is Haus... |
elmptrab 22978 | Membership in a one-parame... |
elmptrab2 22979 | Membership in a one-parame... |
isfbas 22980 | The predicate " ` F ` is a... |
fbasne0 22981 | There are no empty filter ... |
0nelfb 22982 | No filter base contains th... |
fbsspw 22983 | A filter base on a set is ... |
fbelss 22984 | An element of the filter b... |
fbdmn0 22985 | The domain of a filter bas... |
isfbas2 22986 | The predicate " ` F ` is a... |
fbasssin 22987 | A filter base contains sub... |
fbssfi 22988 | A filter base contains sub... |
fbssint 22989 | A filter base contains sub... |
fbncp 22990 | A filter base does not con... |
fbun 22991 | A necessary and sufficient... |
fbfinnfr 22992 | No filter base containing ... |
opnfbas 22993 | The collection of open sup... |
trfbas2 22994 | Conditions for the trace o... |
trfbas 22995 | Conditions for the trace o... |
isfil 22998 | The predicate "is a filter... |
filfbas 22999 | A filter is a filter base.... |
0nelfil 23000 | The empty set doesn't belo... |
fileln0 23001 | An element of a filter is ... |
filsspw 23002 | A filter is a subset of th... |
filelss 23003 | An element of a filter is ... |
filss 23004 | A filter is closed under t... |
filin 23005 | A filter is closed under t... |
filtop 23006 | The underlying set belongs... |
isfil2 23007 | Derive the standard axioms... |
isfildlem 23008 | Lemma for ~ isfild . (Con... |
isfild 23009 | Sufficient condition for a... |
filfi 23010 | A filter is closed under t... |
filinn0 23011 | The intersection of two el... |
filintn0 23012 | A filter has the finite in... |
filn0 23013 | The empty set is not a fil... |
infil 23014 | The intersection of two fi... |
snfil 23015 | A singleton is a filter. ... |
fbasweak 23016 | A filter base on any set i... |
snfbas 23017 | Condition for a singleton ... |
fsubbas 23018 | A condition for a set to g... |
fbasfip 23019 | A filter base has the fini... |
fbunfip 23020 | A helpful lemma for showin... |
fgval 23021 | The filter generating clas... |
elfg 23022 | A condition for elements o... |
ssfg 23023 | A filter base is a subset ... |
fgss 23024 | A bigger base generates a ... |
fgss2 23025 | A condition for a filter t... |
fgfil 23026 | A filter generates itself.... |
elfilss 23027 | An element belongs to a fi... |
filfinnfr 23028 | No filter containing a fin... |
fgcl 23029 | A generated filter is a fi... |
fgabs 23030 | Absorption law for filter ... |
neifil 23031 | The neighborhoods of a non... |
filunibas 23032 | Recover the base set from ... |
filunirn 23033 | Two ways to express a filt... |
filconn 23034 | A filter gives rise to a c... |
fbasrn 23035 | Given a filter on a domain... |
filuni 23036 | The union of a nonempty se... |
trfil1 23037 | Conditions for the trace o... |
trfil2 23038 | Conditions for the trace o... |
trfil3 23039 | Conditions for the trace o... |
trfilss 23040 | If ` A ` is a member of th... |
fgtr 23041 | If ` A ` is a member of th... |
trfg 23042 | The trace operation and th... |
trnei 23043 | The trace, over a set ` A ... |
cfinfil 23044 | Relative complements of th... |
csdfil 23045 | The set of all elements wh... |
supfil 23046 | The supersets of a nonempt... |
zfbas 23047 | The set of upper sets of i... |
uzrest 23048 | The restriction of the set... |
uzfbas 23049 | The set of upper sets of i... |
isufil 23054 | The property of being an u... |
ufilfil 23055 | An ultrafilter is a filter... |
ufilss 23056 | For any subset of the base... |
ufilb 23057 | The complement is in an ul... |
ufilmax 23058 | Any filter finer than an u... |
isufil2 23059 | The maximal property of an... |
ufprim 23060 | An ultrafilter is a prime ... |
trufil 23061 | Conditions for the trace o... |
filssufilg 23062 | A filter is contained in s... |
filssufil 23063 | A filter is contained in s... |
isufl 23064 | Define the (strong) ultraf... |
ufli 23065 | Property of a set that sat... |
numufl 23066 | Consequence of ~ filssufil... |
fiufl 23067 | A finite set satisfies the... |
acufl 23068 | The axiom of choice implie... |
ssufl 23069 | If ` Y ` is a subset of ` ... |
ufileu 23070 | If the ultrafilter contain... |
filufint 23071 | A filter is equal to the i... |
uffix 23072 | Lemma for ~ fixufil and ~ ... |
fixufil 23073 | The condition describing a... |
uffixfr 23074 | An ultrafilter is either f... |
uffix2 23075 | A classification of fixed ... |
uffixsn 23076 | The singleton of the gener... |
ufildom1 23077 | An ultrafilter is generate... |
uffinfix 23078 | An ultrafilter containing ... |
cfinufil 23079 | An ultrafilter is free iff... |
ufinffr 23080 | An infinite subset is cont... |
ufilen 23081 | Any infinite set has an ul... |
ufildr 23082 | An ultrafilter gives rise ... |
fin1aufil 23083 | There are no definable fre... |
fmval 23094 | Introduce a function that ... |
fmfil 23095 | A mapping filter is a filt... |
fmf 23096 | Pushing-forward via a func... |
fmss 23097 | A finer filter produces a ... |
elfm 23098 | An element of a mapping fi... |
elfm2 23099 | An element of a mapping fi... |
fmfg 23100 | The image filter of a filt... |
elfm3 23101 | An alternate formulation o... |
imaelfm 23102 | An image of a filter eleme... |
rnelfmlem 23103 | Lemma for ~ rnelfm . (Con... |
rnelfm 23104 | A condition for a filter t... |
fmfnfmlem1 23105 | Lemma for ~ fmfnfm . (Con... |
fmfnfmlem2 23106 | Lemma for ~ fmfnfm . (Con... |
fmfnfmlem3 23107 | Lemma for ~ fmfnfm . (Con... |
fmfnfmlem4 23108 | Lemma for ~ fmfnfm . (Con... |
fmfnfm 23109 | A filter finer than an ima... |
fmufil 23110 | An image filter of an ultr... |
fmid 23111 | The filter map applied to ... |
fmco 23112 | Composition of image filte... |
ufldom 23113 | The ultrafilter lemma prop... |
flimval 23114 | The set of limit points of... |
elflim2 23115 | The predicate "is a limit ... |
flimtop 23116 | Reverse closure for the li... |
flimneiss 23117 | A filter contains the neig... |
flimnei 23118 | A filter contains all of t... |
flimelbas 23119 | A limit point of a filter ... |
flimfil 23120 | Reverse closure for the li... |
flimtopon 23121 | Reverse closure for the li... |
elflim 23122 | The predicate "is a limit ... |
flimss2 23123 | A limit point of a filter ... |
flimss1 23124 | A limit point of a filter ... |
neiflim 23125 | A point is a limit point o... |
flimopn 23126 | The condition for being a ... |
fbflim 23127 | A condition for a filter t... |
fbflim2 23128 | A condition for a filter b... |
flimclsi 23129 | The convergent points of a... |
hausflimlem 23130 | If ` A ` and ` B ` are bot... |
hausflimi 23131 | One direction of ~ hausfli... |
hausflim 23132 | A condition for a topology... |
flimcf 23133 | Fineness is properly chara... |
flimrest 23134 | The set of limit points in... |
flimclslem 23135 | Lemma for ~ flimcls . (Co... |
flimcls 23136 | Closure in terms of filter... |
flimsncls 23137 | If ` A ` is a limit point ... |
hauspwpwf1 23138 | Lemma for ~ hauspwpwdom . ... |
hauspwpwdom 23139 | If ` X ` is a Hausdorff sp... |
flffval 23140 | Given a topology and a fil... |
flfval 23141 | Given a function from a fi... |
flfnei 23142 | The property of being a li... |
flfneii 23143 | A neighborhood of a limit ... |
isflf 23144 | The property of being a li... |
flfelbas 23145 | A limit point of a functio... |
flffbas 23146 | Limit points of a function... |
flftg 23147 | Limit points of a function... |
hausflf 23148 | If a function has its valu... |
hausflf2 23149 | If a convergent function h... |
cnpflfi 23150 | Forward direction of ~ cnp... |
cnpflf2 23151 | ` F ` is continuous at poi... |
cnpflf 23152 | Continuity of a function a... |
cnflf 23153 | A function is continuous i... |
cnflf2 23154 | A function is continuous i... |
flfcnp 23155 | A continuous function pres... |
lmflf 23156 | The topological limit rela... |
txflf 23157 | Two sequences converge in ... |
flfcnp2 23158 | The image of a convergent ... |
fclsval 23159 | The set of all cluster poi... |
isfcls 23160 | A cluster point of a filte... |
fclsfil 23161 | Reverse closure for the cl... |
fclstop 23162 | Reverse closure for the cl... |
fclstopon 23163 | Reverse closure for the cl... |
isfcls2 23164 | A cluster point of a filte... |
fclsopn 23165 | Write the cluster point co... |
fclsopni 23166 | An open neighborhood of a ... |
fclselbas 23167 | A cluster point is in the ... |
fclsneii 23168 | A neighborhood of a cluste... |
fclssscls 23169 | The set of cluster points ... |
fclsnei 23170 | Cluster points in terms of... |
supnfcls 23171 | The filter of supersets of... |
fclsbas 23172 | Cluster points in terms of... |
fclsss1 23173 | A finer topology has fewer... |
fclsss2 23174 | A finer filter has fewer c... |
fclsrest 23175 | The set of cluster points ... |
fclscf 23176 | Characterization of finene... |
flimfcls 23177 | A limit point is a cluster... |
fclsfnflim 23178 | A filter clusters at a poi... |
flimfnfcls 23179 | A filter converges to a po... |
fclscmpi 23180 | Forward direction of ~ fcl... |
fclscmp 23181 | A space is compact iff eve... |
uffclsflim 23182 | The cluster points of an u... |
ufilcmp 23183 | A space is compact iff eve... |
fcfval 23184 | The set of cluster points ... |
isfcf 23185 | The property of being a cl... |
fcfnei 23186 | The property of being a cl... |
fcfelbas 23187 | A cluster point of a funct... |
fcfneii 23188 | A neighborhood of a cluste... |
flfssfcf 23189 | A limit point of a functio... |
uffcfflf 23190 | If the domain filter is an... |
cnpfcfi 23191 | Lemma for ~ cnpfcf . If a... |
cnpfcf 23192 | A function ` F ` is contin... |
cnfcf 23193 | Continuity of a function i... |
flfcntr 23194 | A continuous function's va... |
alexsublem 23195 | Lemma for ~ alexsub . (Co... |
alexsub 23196 | The Alexander Subbase Theo... |
alexsubb 23197 | Biconditional form of the ... |
alexsubALTlem1 23198 | Lemma for ~ alexsubALT . ... |
alexsubALTlem2 23199 | Lemma for ~ alexsubALT . ... |
alexsubALTlem3 23200 | Lemma for ~ alexsubALT . ... |
alexsubALTlem4 23201 | Lemma for ~ alexsubALT . ... |
alexsubALT 23202 | The Alexander Subbase Theo... |
ptcmplem1 23203 | Lemma for ~ ptcmp . (Cont... |
ptcmplem2 23204 | Lemma for ~ ptcmp . (Cont... |
ptcmplem3 23205 | Lemma for ~ ptcmp . (Cont... |
ptcmplem4 23206 | Lemma for ~ ptcmp . (Cont... |
ptcmplem5 23207 | Lemma for ~ ptcmp . (Cont... |
ptcmpg 23208 | Tychonoff's theorem: The ... |
ptcmp 23209 | Tychonoff's theorem: The ... |
cnextval 23212 | The function applying cont... |
cnextfval 23213 | The continuous extension o... |
cnextrel 23214 | In the general case, a con... |
cnextfun 23215 | If the target space is Hau... |
cnextfvval 23216 | The value of the continuou... |
cnextf 23217 | Extension by continuity. ... |
cnextcn 23218 | Extension by continuity. ... |
cnextfres1 23219 | ` F ` and its extension by... |
cnextfres 23220 | ` F ` and its extension by... |
istmd 23225 | The predicate "is a topolo... |
tmdmnd 23226 | A topological monoid is a ... |
tmdtps 23227 | A topological monoid is a ... |
istgp 23228 | The predicate "is a topolo... |
tgpgrp 23229 | A topological group is a g... |
tgptmd 23230 | A topological group is a t... |
tgptps 23231 | A topological group is a t... |
tmdtopon 23232 | The topology of a topologi... |
tgptopon 23233 | The topology of a topologi... |
tmdcn 23234 | In a topological monoid, t... |
tgpcn 23235 | In a topological group, th... |
tgpinv 23236 | In a topological group, th... |
grpinvhmeo 23237 | The inverse function in a ... |
cnmpt1plusg 23238 | Continuity of the group su... |
cnmpt2plusg 23239 | Continuity of the group su... |
tmdcn2 23240 | Write out the definition o... |
tgpsubcn 23241 | In a topological group, th... |
istgp2 23242 | A group with a topology is... |
tmdmulg 23243 | In a topological monoid, t... |
tgpmulg 23244 | In a topological group, th... |
tgpmulg2 23245 | In a topological monoid, t... |
tmdgsum 23246 | In a topological monoid, t... |
tmdgsum2 23247 | For any neighborhood ` U `... |
oppgtmd 23248 | The opposite of a topologi... |
oppgtgp 23249 | The opposite of a topologi... |
distgp 23250 | Any group equipped with th... |
indistgp 23251 | Any group equipped with th... |
efmndtmd 23252 | The monoid of endofunction... |
tmdlactcn 23253 | The left group action of e... |
tgplacthmeo 23254 | The left group action of e... |
submtmd 23255 | A submonoid of a topologic... |
subgtgp 23256 | A subgroup of a topologica... |
symgtgp 23257 | The symmetric group is a t... |
subgntr 23258 | A subgroup of a topologica... |
opnsubg 23259 | An open subgroup of a topo... |
clssubg 23260 | The closure of a subgroup ... |
clsnsg 23261 | The closure of a normal su... |
cldsubg 23262 | A subgroup of finite index... |
tgpconncompeqg 23263 | The connected component co... |
tgpconncomp 23264 | The identity component, th... |
tgpconncompss 23265 | The identity component is ... |
ghmcnp 23266 | A group homomorphism on to... |
snclseqg 23267 | The coset of the closure o... |
tgphaus 23268 | A topological group is Hau... |
tgpt1 23269 | Hausdorff and T1 are equiv... |
tgpt0 23270 | Hausdorff and T0 are equiv... |
qustgpopn 23271 | A quotient map in a topolo... |
qustgplem 23272 | Lemma for ~ qustgp . (Con... |
qustgp 23273 | The quotient of a topologi... |
qustgphaus 23274 | The quotient of a topologi... |
prdstmdd 23275 | The product of a family of... |
prdstgpd 23276 | The product of a family of... |
tsmsfbas 23279 | The collection of all sets... |
tsmslem1 23280 | The finite partial sums of... |
tsmsval2 23281 | Definition of the topologi... |
tsmsval 23282 | Definition of the topologi... |
tsmspropd 23283 | The group sum depends only... |
eltsms 23284 | The property of being a su... |
tsmsi 23285 | The property of being a su... |
tsmscl 23286 | A sum in a topological gro... |
haustsms 23287 | In a Hausdorff topological... |
haustsms2 23288 | In a Hausdorff topological... |
tsmscls 23289 | One half of ~ tgptsmscls ,... |
tsmsgsum 23290 | The convergent points of a... |
tsmsid 23291 | If a sum is finite, the us... |
haustsmsid 23292 | In a Hausdorff topological... |
tsms0 23293 | The sum of zero is zero. ... |
tsmssubm 23294 | Evaluate an infinite group... |
tsmsres 23295 | Extend an infinite group s... |
tsmsf1o 23296 | Re-index an infinite group... |
tsmsmhm 23297 | Apply a continuous group h... |
tsmsadd 23298 | The sum of two infinite gr... |
tsmsinv 23299 | Inverse of an infinite gro... |
tsmssub 23300 | The difference of two infi... |
tgptsmscls 23301 | A sum in a topological gro... |
tgptsmscld 23302 | The set of limit points to... |
tsmssplit 23303 | Split a topological group ... |
tsmsxplem1 23304 | Lemma for ~ tsmsxp . (Con... |
tsmsxplem2 23305 | Lemma for ~ tsmsxp . (Con... |
tsmsxp 23306 | Write a sum over a two-dim... |
istrg 23315 | Express the predicate " ` ... |
trgtmd 23316 | The multiplicative monoid ... |
istdrg 23317 | Express the predicate " ` ... |
tdrgunit 23318 | The unit group of a topolo... |
trgtgp 23319 | A topological ring is a to... |
trgtmd2 23320 | A topological ring is a to... |
trgtps 23321 | A topological ring is a to... |
trgring 23322 | A topological ring is a ri... |
trggrp 23323 | A topological ring is a gr... |
tdrgtrg 23324 | A topological division rin... |
tdrgdrng 23325 | A topological division rin... |
tdrgring 23326 | A topological division rin... |
tdrgtmd 23327 | A topological division rin... |
tdrgtps 23328 | A topological division rin... |
istdrg2 23329 | A topological-ring divisio... |
mulrcn 23330 | The functionalization of t... |
invrcn2 23331 | The multiplicative inverse... |
invrcn 23332 | The multiplicative inverse... |
cnmpt1mulr 23333 | Continuity of ring multipl... |
cnmpt2mulr 23334 | Continuity of ring multipl... |
dvrcn 23335 | The division function is c... |
istlm 23336 | The predicate " ` W ` is a... |
vscacn 23337 | The scalar multiplication ... |
tlmtmd 23338 | A topological module is a ... |
tlmtps 23339 | A topological module is a ... |
tlmlmod 23340 | A topological module is a ... |
tlmtrg 23341 | The scalar ring of a topol... |
tlmscatps 23342 | The scalar ring of a topol... |
istvc 23343 | A topological vector space... |
tvctdrg 23344 | The scalar field of a topo... |
cnmpt1vsca 23345 | Continuity of scalar multi... |
cnmpt2vsca 23346 | Continuity of scalar multi... |
tlmtgp 23347 | A topological vector space... |
tvctlm 23348 | A topological vector space... |
tvclmod 23349 | A topological vector space... |
tvclvec 23350 | A topological vector space... |
ustfn 23353 | The defined uniform struct... |
ustval 23354 | The class of all uniform s... |
isust 23355 | The predicate " ` U ` is a... |
ustssxp 23356 | Entourages are subsets of ... |
ustssel 23357 | A uniform structure is upw... |
ustbasel 23358 | The full set is always an ... |
ustincl 23359 | A uniform structure is clo... |
ustdiag 23360 | The diagonal set is includ... |
ustinvel 23361 | If ` V ` is an entourage, ... |
ustexhalf 23362 | For each entourage ` V ` t... |
ustrel 23363 | The elements of uniform st... |
ustfilxp 23364 | A uniform structure on a n... |
ustne0 23365 | A uniform structure cannot... |
ustssco 23366 | In an uniform structure, a... |
ustexsym 23367 | In an uniform structure, f... |
ustex2sym 23368 | In an uniform structure, f... |
ustex3sym 23369 | In an uniform structure, f... |
ustref 23370 | Any element of the base se... |
ust0 23371 | The unique uniform structu... |
ustn0 23372 | The empty set is not an un... |
ustund 23373 | If two intersecting sets `... |
ustelimasn 23374 | Any point ` A ` is near en... |
ustneism 23375 | For a point ` A ` in ` X `... |
elrnust 23376 | First direction for ~ ustb... |
ustbas2 23377 | Second direction for ~ ust... |
ustuni 23378 | The set union of a uniform... |
ustbas 23379 | Recover the base of an uni... |
ustimasn 23380 | Lemma for ~ ustuqtop . (C... |
trust 23381 | The trace of a uniform str... |
utopval 23384 | The topology induced by a ... |
elutop 23385 | Open sets in the topology ... |
utoptop 23386 | The topology induced by a ... |
utopbas 23387 | The base of the topology i... |
utoptopon 23388 | Topology induced by a unif... |
restutop 23389 | Restriction of a topology ... |
restutopopn 23390 | The restriction of the top... |
ustuqtoplem 23391 | Lemma for ~ ustuqtop . (C... |
ustuqtop0 23392 | Lemma for ~ ustuqtop . (C... |
ustuqtop1 23393 | Lemma for ~ ustuqtop , sim... |
ustuqtop2 23394 | Lemma for ~ ustuqtop . (C... |
ustuqtop3 23395 | Lemma for ~ ustuqtop , sim... |
ustuqtop4 23396 | Lemma for ~ ustuqtop . (C... |
ustuqtop5 23397 | Lemma for ~ ustuqtop . (C... |
ustuqtop 23398 | For a given uniform struct... |
utopsnneiplem 23399 | The neighborhoods of a poi... |
utopsnneip 23400 | The neighborhoods of a poi... |
utopsnnei 23401 | Images of singletons by en... |
utop2nei 23402 | For any symmetrical entour... |
utop3cls 23403 | Relation between a topolog... |
utopreg 23404 | All Hausdorff uniform spac... |
ussval 23411 | The uniform structure on u... |
ussid 23412 | In case the base of the ` ... |
isusp 23413 | The predicate ` W ` is a u... |
ressuss 23414 | Value of the uniform struc... |
ressust 23415 | The uniform structure of a... |
ressusp 23416 | The restriction of a unifo... |
tusval 23417 | The value of the uniform s... |
tuslem 23418 | Lemma for ~ tusbas , ~ tus... |
tuslemOLD 23419 | Obsolete proof of ~ tuslem... |
tusbas 23420 | The base set of a construc... |
tusunif 23421 | The uniform structure of a... |
tususs 23422 | The uniform structure of a... |
tustopn 23423 | The topology induced by a ... |
tususp 23424 | A constructed uniform spac... |
tustps 23425 | A constructed uniform spac... |
uspreg 23426 | If a uniform space is Haus... |
ucnval 23429 | The set of all uniformly c... |
isucn 23430 | The predicate " ` F ` is a... |
isucn2 23431 | The predicate " ` F ` is a... |
ucnimalem 23432 | Reformulate the ` G ` func... |
ucnima 23433 | An equivalent statement of... |
ucnprima 23434 | The preimage by a uniforml... |
iducn 23435 | The identity is uniformly ... |
cstucnd 23436 | A constant function is uni... |
ucncn 23437 | Uniform continuity implies... |
iscfilu 23440 | The predicate " ` F ` is a... |
cfilufbas 23441 | A Cauchy filter base is a ... |
cfiluexsm 23442 | For a Cauchy filter base a... |
fmucndlem 23443 | Lemma for ~ fmucnd . (Con... |
fmucnd 23444 | The image of a Cauchy filt... |
cfilufg 23445 | The filter generated by a ... |
trcfilu 23446 | Condition for the trace of... |
cfiluweak 23447 | A Cauchy filter base is al... |
neipcfilu 23448 | In an uniform space, a nei... |
iscusp 23451 | The predicate " ` W ` is a... |
cuspusp 23452 | A complete uniform space i... |
cuspcvg 23453 | In a complete uniform spac... |
iscusp2 23454 | The predicate " ` W ` is a... |
cnextucn 23455 | Extension by continuity. ... |
ucnextcn 23456 | Extension by continuity. ... |
ispsmet 23457 | Express the predicate " ` ... |
psmetdmdm 23458 | Recover the base set from ... |
psmetf 23459 | The distance function of a... |
psmetcl 23460 | Closure of the distance fu... |
psmet0 23461 | The distance function of a... |
psmettri2 23462 | Triangle inequality for th... |
psmetsym 23463 | The distance function of a... |
psmettri 23464 | Triangle inequality for th... |
psmetge0 23465 | The distance function of a... |
psmetxrge0 23466 | The distance function of a... |
psmetres2 23467 | Restriction of a pseudomet... |
psmetlecl 23468 | Real closure of an extende... |
distspace 23469 | A set ` X ` together with ... |
ismet 23476 | Express the predicate " ` ... |
isxmet 23477 | Express the predicate " ` ... |
ismeti 23478 | Properties that determine ... |
isxmetd 23479 | Properties that determine ... |
isxmet2d 23480 | It is safe to only require... |
metflem 23481 | Lemma for ~ metf and other... |
xmetf 23482 | Mapping of the distance fu... |
metf 23483 | Mapping of the distance fu... |
xmetcl 23484 | Closure of the distance fu... |
metcl 23485 | Closure of the distance fu... |
ismet2 23486 | An extended metric is a me... |
metxmet 23487 | A metric is an extended me... |
xmetdmdm 23488 | Recover the base set from ... |
metdmdm 23489 | Recover the base set from ... |
xmetunirn 23490 | Two ways to express an ext... |
xmeteq0 23491 | The value of an extended m... |
meteq0 23492 | The value of a metric is z... |
xmettri2 23493 | Triangle inequality for th... |
mettri2 23494 | Triangle inequality for th... |
xmet0 23495 | The distance function of a... |
met0 23496 | The distance function of a... |
xmetge0 23497 | The distance function of a... |
metge0 23498 | The distance function of a... |
xmetlecl 23499 | Real closure of an extende... |
xmetsym 23500 | The distance function of a... |
xmetpsmet 23501 | An extended metric is a ps... |
xmettpos 23502 | The distance function of a... |
metsym 23503 | The distance function of a... |
xmettri 23504 | Triangle inequality for th... |
mettri 23505 | Triangle inequality for th... |
xmettri3 23506 | Triangle inequality for th... |
mettri3 23507 | Triangle inequality for th... |
xmetrtri 23508 | One half of the reverse tr... |
xmetrtri2 23509 | The reverse triangle inequ... |
metrtri 23510 | Reverse triangle inequalit... |
xmetgt0 23511 | The distance function of a... |
metgt0 23512 | The distance function of a... |
metn0 23513 | A metric space is nonempty... |
xmetres2 23514 | Restriction of an extended... |
metreslem 23515 | Lemma for ~ metres . (Con... |
metres2 23516 | Lemma for ~ metres . (Con... |
xmetres 23517 | A restriction of an extend... |
metres 23518 | A restriction of a metric ... |
0met 23519 | The empty metric. (Contri... |
prdsdsf 23520 | The product metric is a fu... |
prdsxmetlem 23521 | The product metric is an e... |
prdsxmet 23522 | The product metric is an e... |
prdsmet 23523 | The product metric is a me... |
ressprdsds 23524 | Restriction of a product m... |
resspwsds 23525 | Restriction of a power met... |
imasdsf1olem 23526 | Lemma for ~ imasdsf1o . (... |
imasdsf1o 23527 | The distance function is t... |
imasf1oxmet 23528 | The image of an extended m... |
imasf1omet 23529 | The image of a metric is a... |
xpsdsfn 23530 | Closure of the metric in a... |
xpsdsfn2 23531 | Closure of the metric in a... |
xpsxmetlem 23532 | Lemma for ~ xpsxmet . (Co... |
xpsxmet 23533 | A product metric of extend... |
xpsdsval 23534 | Value of the metric in a b... |
xpsmet 23535 | The direct product of two ... |
blfvalps 23536 | The value of the ball func... |
blfval 23537 | The value of the ball func... |
blvalps 23538 | The ball around a point ` ... |
blval 23539 | The ball around a point ` ... |
elblps 23540 | Membership in a ball. (Co... |
elbl 23541 | Membership in a ball. (Co... |
elbl2ps 23542 | Membership in a ball. (Co... |
elbl2 23543 | Membership in a ball. (Co... |
elbl3ps 23544 | Membership in a ball, with... |
elbl3 23545 | Membership in a ball, with... |
blcomps 23546 | Commute the arguments to t... |
blcom 23547 | Commute the arguments to t... |
xblpnfps 23548 | The infinity ball in an ex... |
xblpnf 23549 | The infinity ball in an ex... |
blpnf 23550 | The infinity ball in a sta... |
bldisj 23551 | Two balls are disjoint if ... |
blgt0 23552 | A nonempty ball implies th... |
bl2in 23553 | Two balls are disjoint if ... |
xblss2ps 23554 | One ball is contained in a... |
xblss2 23555 | One ball is contained in a... |
blss2ps 23556 | One ball is contained in a... |
blss2 23557 | One ball is contained in a... |
blhalf 23558 | A ball of radius ` R / 2 `... |
blfps 23559 | Mapping of a ball. (Contr... |
blf 23560 | Mapping of a ball. (Contr... |
blrnps 23561 | Membership in the range of... |
blrn 23562 | Membership in the range of... |
xblcntrps 23563 | A ball contains its center... |
xblcntr 23564 | A ball contains its center... |
blcntrps 23565 | A ball contains its center... |
blcntr 23566 | A ball contains its center... |
xbln0 23567 | A ball is nonempty iff the... |
bln0 23568 | A ball is not empty. (Con... |
blelrnps 23569 | A ball belongs to the set ... |
blelrn 23570 | A ball belongs to the set ... |
blssm 23571 | A ball is a subset of the ... |
unirnblps 23572 | The union of the set of ba... |
unirnbl 23573 | The union of the set of ba... |
blin 23574 | The intersection of two ba... |
ssblps 23575 | The size of a ball increas... |
ssbl 23576 | The size of a ball increas... |
blssps 23577 | Any point ` P ` in a ball ... |
blss 23578 | Any point ` P ` in a ball ... |
blssexps 23579 | Two ways to express the ex... |
blssex 23580 | Two ways to express the ex... |
ssblex 23581 | A nested ball exists whose... |
blin2 23582 | Given any two balls and a ... |
blbas 23583 | The balls of a metric spac... |
blres 23584 | A ball in a restricted met... |
xmeterval 23585 | Value of the "finitely sep... |
xmeter 23586 | The "finitely separated" r... |
xmetec 23587 | The equivalence classes un... |
blssec 23588 | A ball centered at ` P ` i... |
blpnfctr 23589 | The infinity ball in an ex... |
xmetresbl 23590 | An extended metric restric... |
mopnval 23591 | An open set is a subset of... |
mopntopon 23592 | The set of open sets of a ... |
mopntop 23593 | The set of open sets of a ... |
mopnuni 23594 | The union of all open sets... |
elmopn 23595 | The defining property of a... |
mopnfss 23596 | The family of open sets of... |
mopnm 23597 | The base set of a metric s... |
elmopn2 23598 | A defining property of an ... |
mopnss 23599 | An open set of a metric sp... |
isxms 23600 | Express the predicate " ` ... |
isxms2 23601 | Express the predicate " ` ... |
isms 23602 | Express the predicate " ` ... |
isms2 23603 | Express the predicate " ` ... |
xmstopn 23604 | The topology component of ... |
mstopn 23605 | The topology component of ... |
xmstps 23606 | An extended metric space i... |
msxms 23607 | A metric space is an exten... |
mstps 23608 | A metric space is a topolo... |
xmsxmet 23609 | The distance function, sui... |
msmet 23610 | The distance function, sui... |
msf 23611 | The distance function of a... |
xmsxmet2 23612 | The distance function, sui... |
msmet2 23613 | The distance function, sui... |
mscl 23614 | Closure of the distance fu... |
xmscl 23615 | Closure of the distance fu... |
xmsge0 23616 | The distance function in a... |
xmseq0 23617 | The distance between two p... |
xmssym 23618 | The distance function in a... |
xmstri2 23619 | Triangle inequality for th... |
mstri2 23620 | Triangle inequality for th... |
xmstri 23621 | Triangle inequality for th... |
mstri 23622 | Triangle inequality for th... |
xmstri3 23623 | Triangle inequality for th... |
mstri3 23624 | Triangle inequality for th... |
msrtri 23625 | Reverse triangle inequalit... |
xmspropd 23626 | Property deduction for an ... |
mspropd 23627 | Property deduction for a m... |
setsmsbas 23628 | The base set of a construc... |
setsmsbasOLD 23629 | Obsolete proof of ~ setsms... |
setsmsds 23630 | The distance function of a... |
setsmsdsOLD 23631 | Obsolete proof of ~ setsms... |
setsmstset 23632 | The topology of a construc... |
setsmstopn 23633 | The topology of a construc... |
setsxms 23634 | The constructed metric spa... |
setsms 23635 | The constructed metric spa... |
tmsval 23636 | For any metric there is an... |
tmslem 23637 | Lemma for ~ tmsbas , ~ tms... |
tmslemOLD 23638 | Obsolete version of ~ tmsl... |
tmsbas 23639 | The base set of a construc... |
tmsds 23640 | The metric of a constructe... |
tmstopn 23641 | The topology of a construc... |
tmsxms 23642 | The constructed metric spa... |
tmsms 23643 | The constructed metric spa... |
imasf1obl 23644 | The image of a metric spac... |
imasf1oxms 23645 | The image of a metric spac... |
imasf1oms 23646 | The image of a metric spac... |
prdsbl 23647 | A ball in the product metr... |
mopni 23648 | An open set of a metric sp... |
mopni2 23649 | An open set of a metric sp... |
mopni3 23650 | An open set of a metric sp... |
blssopn 23651 | The balls of a metric spac... |
unimopn 23652 | The union of a collection ... |
mopnin 23653 | The intersection of two op... |
mopn0 23654 | The empty set is an open s... |
rnblopn 23655 | A ball of a metric space i... |
blopn 23656 | A ball of a metric space i... |
neibl 23657 | The neighborhoods around a... |
blnei 23658 | A ball around a point is a... |
lpbl 23659 | Every ball around a limit ... |
blsscls2 23660 | A smaller closed ball is c... |
blcld 23661 | A "closed ball" in a metri... |
blcls 23662 | The closure of an open bal... |
blsscls 23663 | If two concentric balls ha... |
metss 23664 | Two ways of saying that me... |
metequiv 23665 | Two ways of saying that tw... |
metequiv2 23666 | If there is a sequence of ... |
metss2lem 23667 | Lemma for ~ metss2 . (Con... |
metss2 23668 | If the metric ` D ` is "st... |
comet 23669 | The composition of an exte... |
stdbdmetval 23670 | Value of the standard boun... |
stdbdxmet 23671 | The standard bounded metri... |
stdbdmet 23672 | The standard bounded metri... |
stdbdbl 23673 | The standard bounded metri... |
stdbdmopn 23674 | The standard bounded metri... |
mopnex 23675 | The topology generated by ... |
methaus 23676 | The topology generated by ... |
met1stc 23677 | The topology generated by ... |
met2ndci 23678 | A separable metric space (... |
met2ndc 23679 | A metric space is second-c... |
metrest 23680 | Two alternate formulations... |
ressxms 23681 | The restriction of a metri... |
ressms 23682 | The restriction of a metri... |
prdsmslem1 23683 | Lemma for ~ prdsms . The ... |
prdsxmslem1 23684 | Lemma for ~ prdsms . The ... |
prdsxmslem2 23685 | Lemma for ~ prdsxms . The... |
prdsxms 23686 | The indexed product struct... |
prdsms 23687 | The indexed product struct... |
pwsxms 23688 | A power of an extended met... |
pwsms 23689 | A power of a metric space ... |
xpsxms 23690 | A binary product of metric... |
xpsms 23691 | A binary product of metric... |
tmsxps 23692 | Express the product of two... |
tmsxpsmopn 23693 | Express the product of two... |
tmsxpsval 23694 | Value of the product of tw... |
tmsxpsval2 23695 | Value of the product of tw... |
metcnp3 23696 | Two ways to express that `... |
metcnp 23697 | Two ways to say a mapping ... |
metcnp2 23698 | Two ways to say a mapping ... |
metcn 23699 | Two ways to say a mapping ... |
metcnpi 23700 | Epsilon-delta property of ... |
metcnpi2 23701 | Epsilon-delta property of ... |
metcnpi3 23702 | Epsilon-delta property of ... |
txmetcnp 23703 | Continuity of a binary ope... |
txmetcn 23704 | Continuity of a binary ope... |
metuval 23705 | Value of the uniform struc... |
metustel 23706 | Define a filter base ` F `... |
metustss 23707 | Range of the elements of t... |
metustrel 23708 | Elements of the filter bas... |
metustto 23709 | Any two elements of the fi... |
metustid 23710 | The identity diagonal is i... |
metustsym 23711 | Elements of the filter bas... |
metustexhalf 23712 | For any element ` A ` of t... |
metustfbas 23713 | The filter base generated ... |
metust 23714 | The uniform structure gene... |
cfilucfil 23715 | Given a metric ` D ` and a... |
metuust 23716 | The uniform structure gene... |
cfilucfil2 23717 | Given a metric ` D ` and a... |
blval2 23718 | The ball around a point ` ... |
elbl4 23719 | Membership in a ball, alte... |
metuel 23720 | Elementhood in the uniform... |
metuel2 23721 | Elementhood in the uniform... |
metustbl 23722 | The "section" image of an ... |
psmetutop 23723 | The topology induced by a ... |
xmetutop 23724 | The topology induced by a ... |
xmsusp 23725 | If the uniform set of a me... |
restmetu 23726 | The uniform structure gene... |
metucn 23727 | Uniform continuity in metr... |
dscmet 23728 | The discrete metric on any... |
dscopn 23729 | The discrete metric genera... |
nrmmetd 23730 | Show that a group norm gen... |
abvmet 23731 | An absolute value ` F ` ge... |
nmfval 23744 | The value of the norm func... |
nmval 23745 | The value of the norm as t... |
nmfval0 23746 | The value of the norm func... |
nmfval2 23747 | The value of the norm func... |
nmval2 23748 | The value of the norm on a... |
nmf2 23749 | The norm on a metric group... |
nmpropd 23750 | Weak property deduction fo... |
nmpropd2 23751 | Strong property deduction ... |
isngp 23752 | The property of being a no... |
isngp2 23753 | The property of being a no... |
isngp3 23754 | The property of being a no... |
ngpgrp 23755 | A normed group is a group.... |
ngpms 23756 | A normed group is a metric... |
ngpxms 23757 | A normed group is an exten... |
ngptps 23758 | A normed group is a topolo... |
ngpmet 23759 | The (induced) metric of a ... |
ngpds 23760 | Value of the distance func... |
ngpdsr 23761 | Value of the distance func... |
ngpds2 23762 | Write the distance between... |
ngpds2r 23763 | Write the distance between... |
ngpds3 23764 | Write the distance between... |
ngpds3r 23765 | Write the distance between... |
ngprcan 23766 | Cancel right addition insi... |
ngplcan 23767 | Cancel left addition insid... |
isngp4 23768 | Express the property of be... |
ngpinvds 23769 | Two elements are the same ... |
ngpsubcan 23770 | Cancel right subtraction i... |
nmf 23771 | The norm on a normed group... |
nmcl 23772 | The norm of a normed group... |
nmge0 23773 | The norm of a normed group... |
nmeq0 23774 | The identity is the only e... |
nmne0 23775 | The norm of a nonzero elem... |
nmrpcl 23776 | The norm of a nonzero elem... |
nminv 23777 | The norm of a negated elem... |
nmmtri 23778 | The triangle inequality fo... |
nmsub 23779 | The norm of the difference... |
nmrtri 23780 | Reverse triangle inequalit... |
nm2dif 23781 | Inequality for the differe... |
nmtri 23782 | The triangle inequality fo... |
nmtri2 23783 | Triangle inequality for th... |
ngpi 23784 | The properties of a normed... |
nm0 23785 | Norm of the identity eleme... |
nmgt0 23786 | The norm of a nonzero elem... |
sgrim 23787 | The induced metric on a su... |
sgrimval 23788 | The induced metric on a su... |
subgnm 23789 | The norm in a subgroup. (... |
subgnm2 23790 | A substructure assigns the... |
subgngp 23791 | A normed group restricted ... |
ngptgp 23792 | A normed abelian group is ... |
ngppropd 23793 | Property deduction for a n... |
reldmtng 23794 | The function ` toNrmGrp ` ... |
tngval 23795 | Value of the function whic... |
tnglem 23796 | Lemma for ~ tngbas and sim... |
tnglemOLD 23797 | Obsolete version of ~ tngl... |
tngbas 23798 | The base set of a structur... |
tngbasOLD 23799 | Obsolete proof of ~ tngbas... |
tngplusg 23800 | The group addition of a st... |
tngplusgOLD 23801 | Obsolete proof of ~ tngplu... |
tng0 23802 | The group identity of a st... |
tngmulr 23803 | The ring multiplication of... |
tngmulrOLD 23804 | Obsolete proof of ~ tngmul... |
tngsca 23805 | The scalar ring of a struc... |
tngscaOLD 23806 | Obsolete proof of ~ tngsca... |
tngvsca 23807 | The scalar multiplication ... |
tngvscaOLD 23808 | Obsolete proof of ~ tngvsc... |
tngip 23809 | The inner product operatio... |
tngipOLD 23810 | Obsolete proof of ~ tngip ... |
tngds 23811 | The metric function of a s... |
tngdsOLD 23812 | Obsolete proof of ~ tngds ... |
tngtset 23813 | The topology generated by ... |
tngtopn 23814 | The topology generated by ... |
tngnm 23815 | The topology generated by ... |
tngngp2 23816 | A norm turns a group into ... |
tngngpd 23817 | Derive the axioms for a no... |
tngngp 23818 | Derive the axioms for a no... |
tnggrpr 23819 | If a structure equipped wi... |
tngngp3 23820 | Alternate definition of a ... |
nrmtngdist 23821 | The augmentation of a norm... |
nrmtngnrm 23822 | The augmentation of a norm... |
tngngpim 23823 | The induced metric of a no... |
isnrg 23824 | A normed ring is a ring wi... |
nrgabv 23825 | The norm of a normed ring ... |
nrgngp 23826 | A normed ring is a normed ... |
nrgring 23827 | A normed ring is a ring. ... |
nmmul 23828 | The norm of a product in a... |
nrgdsdi 23829 | Distribute a distance calc... |
nrgdsdir 23830 | Distribute a distance calc... |
nm1 23831 | The norm of one in a nonze... |
unitnmn0 23832 | The norm of a unit is nonz... |
nminvr 23833 | The norm of an inverse in ... |
nmdvr 23834 | The norm of a division in ... |
nrgdomn 23835 | A nonzero normed ring is a... |
nrgtgp 23836 | A normed ring is a topolog... |
subrgnrg 23837 | A normed ring restricted t... |
tngnrg 23838 | Given any absolute value o... |
isnlm 23839 | A normed (left) module is ... |
nmvs 23840 | Defining property of a nor... |
nlmngp 23841 | A normed module is a norme... |
nlmlmod 23842 | A normed module is a left ... |
nlmnrg 23843 | The scalar component of a ... |
nlmngp2 23844 | The scalar component of a ... |
nlmdsdi 23845 | Distribute a distance calc... |
nlmdsdir 23846 | Distribute a distance calc... |
nlmmul0or 23847 | If a scalar product is zer... |
sranlm 23848 | The subring algebra over a... |
nlmvscnlem2 23849 | Lemma for ~ nlmvscn . Com... |
nlmvscnlem1 23850 | Lemma for ~ nlmvscn . (Co... |
nlmvscn 23851 | The scalar multiplication ... |
rlmnlm 23852 | The ring module over a nor... |
rlmnm 23853 | The norm function in the r... |
nrgtrg 23854 | A normed ring is a topolog... |
nrginvrcnlem 23855 | Lemma for ~ nrginvrcn . C... |
nrginvrcn 23856 | The ring inverse function ... |
nrgtdrg 23857 | A normed division ring is ... |
nlmtlm 23858 | A normed module is a topol... |
isnvc 23859 | A normed vector space is j... |
nvcnlm 23860 | A normed vector space is a... |
nvclvec 23861 | A normed vector space is a... |
nvclmod 23862 | A normed vector space is a... |
isnvc2 23863 | A normed vector space is j... |
nvctvc 23864 | A normed vector space is a... |
lssnlm 23865 | A subspace of a normed mod... |
lssnvc 23866 | A subspace of a normed vec... |
rlmnvc 23867 | The ring module over a nor... |
ngpocelbl 23868 | Membership of an off-cente... |
nmoffn 23875 | The function producing ope... |
reldmnghm 23876 | Lemma for normed group hom... |
reldmnmhm 23877 | Lemma for module homomorph... |
nmofval 23878 | Value of the operator norm... |
nmoval 23879 | Value of the operator norm... |
nmogelb 23880 | Property of the operator n... |
nmolb 23881 | Any upper bound on the val... |
nmolb2d 23882 | Any upper bound on the val... |
nmof 23883 | The operator norm is a fun... |
nmocl 23884 | The operator norm of an op... |
nmoge0 23885 | The operator norm of an op... |
nghmfval 23886 | A normed group homomorphis... |
isnghm 23887 | A normed group homomorphis... |
isnghm2 23888 | A normed group homomorphis... |
isnghm3 23889 | A normed group homomorphis... |
bddnghm 23890 | A bounded group homomorphi... |
nghmcl 23891 | A normed group homomorphis... |
nmoi 23892 | The operator norm achieves... |
nmoix 23893 | The operator norm is a bou... |
nmoi2 23894 | The operator norm is a bou... |
nmoleub 23895 | The operator norm, defined... |
nghmrcl1 23896 | Reverse closure for a norm... |
nghmrcl2 23897 | Reverse closure for a norm... |
nghmghm 23898 | A normed group homomorphis... |
nmo0 23899 | The operator norm of the z... |
nmoeq0 23900 | The operator norm is zero ... |
nmoco 23901 | An upper bound on the oper... |
nghmco 23902 | The composition of normed ... |
nmotri 23903 | Triangle inequality for th... |
nghmplusg 23904 | The sum of two bounded lin... |
0nghm 23905 | The zero operator is a nor... |
nmoid 23906 | The operator norm of the i... |
idnghm 23907 | The identity operator is a... |
nmods 23908 | Upper bound for the distan... |
nghmcn 23909 | A normed group homomorphis... |
isnmhm 23910 | A normed module homomorphi... |
nmhmrcl1 23911 | Reverse closure for a norm... |
nmhmrcl2 23912 | Reverse closure for a norm... |
nmhmlmhm 23913 | A normed module homomorphi... |
nmhmnghm 23914 | A normed module homomorphi... |
nmhmghm 23915 | A normed module homomorphi... |
isnmhm2 23916 | A normed module homomorphi... |
nmhmcl 23917 | A normed module homomorphi... |
idnmhm 23918 | The identity operator is a... |
0nmhm 23919 | The zero operator is a bou... |
nmhmco 23920 | The composition of bounded... |
nmhmplusg 23921 | The sum of two bounded lin... |
qtopbaslem 23922 | The set of open intervals ... |
qtopbas 23923 | The set of open intervals ... |
retopbas 23924 | A basis for the standard t... |
retop 23925 | The standard topology on t... |
uniretop 23926 | The underlying set of the ... |
retopon 23927 | The standard topology on t... |
retps 23928 | The standard topological s... |
iooretop 23929 | Open intervals are open se... |
icccld 23930 | Closed intervals are close... |
icopnfcld 23931 | Right-unbounded closed int... |
iocmnfcld 23932 | Left-unbounded closed inte... |
qdensere 23933 | ` QQ ` is dense in the sta... |
cnmetdval 23934 | Value of the distance func... |
cnmet 23935 | The absolute value metric ... |
cnxmet 23936 | The absolute value metric ... |
cnbl0 23937 | Two ways to write the open... |
cnblcld 23938 | Two ways to write the clos... |
cnfldms 23939 | The complex number field i... |
cnfldxms 23940 | The complex number field i... |
cnfldtps 23941 | The complex number field i... |
cnfldnm 23942 | The norm of the field of c... |
cnngp 23943 | The complex numbers form a... |
cnnrg 23944 | The complex numbers form a... |
cnfldtopn 23945 | The topology of the comple... |
cnfldtopon 23946 | The topology of the comple... |
cnfldtop 23947 | The topology of the comple... |
cnfldhaus 23948 | The topology of the comple... |
unicntop 23949 | The underlying set of the ... |
cnopn 23950 | The set of complex numbers... |
zringnrg 23951 | The ring of integers is a ... |
remetdval 23952 | Value of the distance func... |
remet 23953 | The absolute value metric ... |
rexmet 23954 | The absolute value metric ... |
bl2ioo 23955 | A ball in terms of an open... |
ioo2bl 23956 | An open interval of reals ... |
ioo2blex 23957 | An open interval of reals ... |
blssioo 23958 | The balls of the standard ... |
tgioo 23959 | The topology generated by ... |
qdensere2 23960 | ` QQ ` is dense in ` RR ` ... |
blcvx 23961 | An open ball in the comple... |
rehaus 23962 | The standard topology on t... |
tgqioo 23963 | The topology generated by ... |
re2ndc 23964 | The standard topology on t... |
resubmet 23965 | The subspace topology indu... |
tgioo2 23966 | The standard topology on t... |
rerest 23967 | The subspace topology indu... |
tgioo3 23968 | The standard topology on t... |
xrtgioo 23969 | The topology on the extend... |
xrrest 23970 | The subspace topology indu... |
xrrest2 23971 | The subspace topology indu... |
xrsxmet 23972 | The metric on the extended... |
xrsdsre 23973 | The metric on the extended... |
xrsblre 23974 | Any ball of the metric of ... |
xrsmopn 23975 | The metric on the extended... |
zcld 23976 | The integers are a closed ... |
recld2 23977 | The real numbers are a clo... |
zcld2 23978 | The integers are a closed ... |
zdis 23979 | The integers are a discret... |
sszcld 23980 | Every subset of the intege... |
reperflem 23981 | A subset of the real numbe... |
reperf 23982 | The real numbers are a per... |
cnperf 23983 | The complex numbers are a ... |
iccntr 23984 | The interior of a closed i... |
icccmplem1 23985 | Lemma for ~ icccmp . (Con... |
icccmplem2 23986 | Lemma for ~ icccmp . (Con... |
icccmplem3 23987 | Lemma for ~ icccmp . (Con... |
icccmp 23988 | A closed interval in ` RR ... |
reconnlem1 23989 | Lemma for ~ reconn . Conn... |
reconnlem2 23990 | Lemma for ~ reconn . (Con... |
reconn 23991 | A subset of the reals is c... |
retopconn 23992 | Corollary of ~ reconn . T... |
iccconn 23993 | A closed interval is conne... |
opnreen 23994 | Every nonempty open set is... |
rectbntr0 23995 | A countable subset of the ... |
xrge0gsumle 23996 | A finite sum in the nonneg... |
xrge0tsms 23997 | Any finite or infinite sum... |
xrge0tsms2 23998 | Any finite or infinite sum... |
metdcnlem 23999 | The metric function of a m... |
xmetdcn2 24000 | The metric function of an ... |
xmetdcn 24001 | The metric function of an ... |
metdcn2 24002 | The metric function of a m... |
metdcn 24003 | The metric function of a m... |
msdcn 24004 | The metric function of a m... |
cnmpt1ds 24005 | Continuity of the metric f... |
cnmpt2ds 24006 | Continuity of the metric f... |
nmcn 24007 | The norm of a normed group... |
ngnmcncn 24008 | The norm of a normed group... |
abscn 24009 | The absolute value functio... |
metdsval 24010 | Value of the "distance to ... |
metdsf 24011 | The distance from a point ... |
metdsge 24012 | The distance from the poin... |
metds0 24013 | If a point is in a set, it... |
metdstri 24014 | A generalization of the tr... |
metdsle 24015 | The distance from a point ... |
metdsre 24016 | The distance from a point ... |
metdseq0 24017 | The distance from a point ... |
metdscnlem 24018 | Lemma for ~ metdscn . (Co... |
metdscn 24019 | The function ` F ` which g... |
metdscn2 24020 | The function ` F ` which g... |
metnrmlem1a 24021 | Lemma for ~ metnrm . (Con... |
metnrmlem1 24022 | Lemma for ~ metnrm . (Con... |
metnrmlem2 24023 | Lemma for ~ metnrm . (Con... |
metnrmlem3 24024 | Lemma for ~ metnrm . (Con... |
metnrm 24025 | A metric space is normal. ... |
metreg 24026 | A metric space is regular.... |
addcnlem 24027 | Lemma for ~ addcn , ~ subc... |
addcn 24028 | Complex number addition is... |
subcn 24029 | Complex number subtraction... |
mulcn 24030 | Complex number multiplicat... |
divcn 24031 | Complex number division is... |
cnfldtgp 24032 | The complex numbers form a... |
fsumcn 24033 | A finite sum of functions ... |
fsum2cn 24034 | Version of ~ fsumcn for tw... |
expcn 24035 | The power function on comp... |
divccn 24036 | Division by a nonzero cons... |
sqcn 24037 | The square function on com... |
iitopon 24042 | The unit interval is a top... |
iitop 24043 | The unit interval is a top... |
iiuni 24044 | The base set of the unit i... |
dfii2 24045 | Alternate definition of th... |
dfii3 24046 | Alternate definition of th... |
dfii4 24047 | Alternate definition of th... |
dfii5 24048 | The unit interval expresse... |
iicmp 24049 | The unit interval is compa... |
iiconn 24050 | The unit interval is conne... |
cncfval 24051 | The value of the continuou... |
elcncf 24052 | Membership in the set of c... |
elcncf2 24053 | Version of ~ elcncf with a... |
cncfrss 24054 | Reverse closure of the con... |
cncfrss2 24055 | Reverse closure of the con... |
cncff 24056 | A continuous complex funct... |
cncfi 24057 | Defining property of a con... |
elcncf1di 24058 | Membership in the set of c... |
elcncf1ii 24059 | Membership in the set of c... |
rescncf 24060 | A continuous complex funct... |
cncffvrn 24061 | Change the codomain of a c... |
cncfss 24062 | The set of continuous func... |
climcncf 24063 | Image of a limit under a c... |
abscncf 24064 | Absolute value is continuo... |
recncf 24065 | Real part is continuous. ... |
imcncf 24066 | Imaginary part is continuo... |
cjcncf 24067 | Complex conjugate is conti... |
mulc1cncf 24068 | Multiplication by a consta... |
divccncf 24069 | Division by a constant is ... |
cncfco 24070 | The composition of two con... |
cncfcompt2 24071 | Composition of continuous ... |
cncfmet 24072 | Relate complex function co... |
cncfcn 24073 | Relate complex function co... |
cncfcn1 24074 | Relate complex function co... |
cncfmptc 24075 | A constant function is a c... |
cncfmptid 24076 | The identity function is a... |
cncfmpt1f 24077 | Composition of continuous ... |
cncfmpt2f 24078 | Composition of continuous ... |
cncfmpt2ss 24079 | Composition of continuous ... |
addccncf 24080 | Adding a constant is a con... |
idcncf 24081 | The identity function is a... |
sub1cncf 24082 | Subtracting a constant is ... |
sub2cncf 24083 | Subtraction from a constan... |
cdivcncf 24084 | Division with a constant n... |
negcncf 24085 | The negative function is c... |
negfcncf 24086 | The negative of a continuo... |
abscncfALT 24087 | Absolute value is continuo... |
cncfcnvcn 24088 | Rewrite ~ cmphaushmeo for ... |
expcncf 24089 | The power function on comp... |
cnmptre 24090 | Lemma for ~ iirevcn and re... |
cnmpopc 24091 | Piecewise definition of a ... |
iirev 24092 | Reverse the unit interval.... |
iirevcn 24093 | The reversion function is ... |
iihalf1 24094 | Map the first half of ` II... |
iihalf1cn 24095 | The first half function is... |
iihalf2 24096 | Map the second half of ` I... |
iihalf2cn 24097 | The second half function i... |
elii1 24098 | Divide the unit interval i... |
elii2 24099 | Divide the unit interval i... |
iimulcl 24100 | The unit interval is close... |
iimulcn 24101 | Multiplication is a contin... |
icoopnst 24102 | A half-open interval start... |
iocopnst 24103 | A half-open interval endin... |
icchmeo 24104 | The natural bijection from... |
icopnfcnv 24105 | Define a bijection from ` ... |
icopnfhmeo 24106 | The defined bijection from... |
iccpnfcnv 24107 | Define a bijection from ` ... |
iccpnfhmeo 24108 | The defined bijection from... |
xrhmeo 24109 | The bijection from ` [ -u ... |
xrhmph 24110 | The extended reals are hom... |
xrcmp 24111 | The topology of the extend... |
xrconn 24112 | The topology of the extend... |
icccvx 24113 | A linear combination of tw... |
oprpiece1res1 24114 | Restriction to the first p... |
oprpiece1res2 24115 | Restriction to the second ... |
cnrehmeo 24116 | The canonical bijection fr... |
cnheiborlem 24117 | Lemma for ~ cnheibor . (C... |
cnheibor 24118 | Heine-Borel theorem for co... |
cnllycmp 24119 | The topology on the comple... |
rellycmp 24120 | The topology on the reals ... |
bndth 24121 | The Boundedness Theorem. ... |
evth 24122 | The Extreme Value Theorem.... |
evth2 24123 | The Extreme Value Theorem,... |
lebnumlem1 24124 | Lemma for ~ lebnum . The ... |
lebnumlem2 24125 | Lemma for ~ lebnum . As a... |
lebnumlem3 24126 | Lemma for ~ lebnum . By t... |
lebnum 24127 | The Lebesgue number lemma,... |
xlebnum 24128 | Generalize ~ lebnum to ext... |
lebnumii 24129 | Specialize the Lebesgue nu... |
ishtpy 24135 | Membership in the class of... |
htpycn 24136 | A homotopy is a continuous... |
htpyi 24137 | A homotopy evaluated at it... |
ishtpyd 24138 | Deduction for membership i... |
htpycom 24139 | Given a homotopy from ` F ... |
htpyid 24140 | A homotopy from a function... |
htpyco1 24141 | Compose a homotopy with a ... |
htpyco2 24142 | Compose a homotopy with a ... |
htpycc 24143 | Concatenate two homotopies... |
isphtpy 24144 | Membership in the class of... |
phtpyhtpy 24145 | A path homotopy is a homot... |
phtpycn 24146 | A path homotopy is a conti... |
phtpyi 24147 | Membership in the class of... |
phtpy01 24148 | Two path-homotopic paths h... |
isphtpyd 24149 | Deduction for membership i... |
isphtpy2d 24150 | Deduction for membership i... |
phtpycom 24151 | Given a homotopy from ` F ... |
phtpyid 24152 | A homotopy from a path to ... |
phtpyco2 24153 | Compose a path homotopy wi... |
phtpycc 24154 | Concatenate two path homot... |
phtpcrel 24156 | The path homotopy relation... |
isphtpc 24157 | The relation "is path homo... |
phtpcer 24158 | Path homotopy is an equiva... |
phtpc01 24159 | Path homotopic paths have ... |
reparphti 24160 | Lemma for ~ reparpht . (C... |
reparpht 24161 | Reparametrization lemma. ... |
phtpcco2 24162 | Compose a path homotopy wi... |
pcofval 24173 | The value of the path conc... |
pcoval 24174 | The concatenation of two p... |
pcovalg 24175 | Evaluate the concatenation... |
pcoval1 24176 | Evaluate the concatenation... |
pco0 24177 | The starting point of a pa... |
pco1 24178 | The ending point of a path... |
pcoval2 24179 | Evaluate the concatenation... |
pcocn 24180 | The concatenation of two p... |
copco 24181 | The composition of a conca... |
pcohtpylem 24182 | Lemma for ~ pcohtpy . (Co... |
pcohtpy 24183 | Homotopy invariance of pat... |
pcoptcl 24184 | A constant function is a p... |
pcopt 24185 | Concatenation with a point... |
pcopt2 24186 | Concatenation with a point... |
pcoass 24187 | Order of concatenation doe... |
pcorevcl 24188 | Closure for a reversed pat... |
pcorevlem 24189 | Lemma for ~ pcorev . Prov... |
pcorev 24190 | Concatenation with the rev... |
pcorev2 24191 | Concatenation with the rev... |
pcophtb 24192 | The path homotopy equivale... |
om1val 24193 | The definition of the loop... |
om1bas 24194 | The base set of the loop s... |
om1elbas 24195 | Elementhood in the base se... |
om1addcl 24196 | Closure of the group opera... |
om1plusg 24197 | The group operation (which... |
om1tset 24198 | The topology of the loop s... |
om1opn 24199 | The topology of the loop s... |
pi1val 24200 | The definition of the fund... |
pi1bas 24201 | The base set of the fundam... |
pi1blem 24202 | Lemma for ~ pi1buni . (Co... |
pi1buni 24203 | Another way to write the l... |
pi1bas2 24204 | The base set of the fundam... |
pi1eluni 24205 | Elementhood in the base se... |
pi1bas3 24206 | The base set of the fundam... |
pi1cpbl 24207 | The group operation, loop ... |
elpi1 24208 | The elements of the fundam... |
elpi1i 24209 | The elements of the fundam... |
pi1addf 24210 | The group operation of ` p... |
pi1addval 24211 | The concatenation of two p... |
pi1grplem 24212 | Lemma for ~ pi1grp . (Con... |
pi1grp 24213 | The fundamental group is a... |
pi1id 24214 | The identity element of th... |
pi1inv 24215 | An inverse in the fundamen... |
pi1xfrf 24216 | Functionality of the loop ... |
pi1xfrval 24217 | The value of the loop tran... |
pi1xfr 24218 | Given a path ` F ` and its... |
pi1xfrcnvlem 24219 | Given a path ` F ` between... |
pi1xfrcnv 24220 | Given a path ` F ` between... |
pi1xfrgim 24221 | The mapping ` G ` between ... |
pi1cof 24222 | Functionality of the loop ... |
pi1coval 24223 | The value of the loop tran... |
pi1coghm 24224 | The mapping ` G ` between ... |
isclm 24227 | A subcomplex module is a l... |
clmsca 24228 | The ring of scalars ` F ` ... |
clmsubrg 24229 | The base set of the ring o... |
clmlmod 24230 | A subcomplex module is a l... |
clmgrp 24231 | A subcomplex module is an ... |
clmabl 24232 | A subcomplex module is an ... |
clmring 24233 | The scalar ring of a subco... |
clmfgrp 24234 | The scalar ring of a subco... |
clm0 24235 | The zero of the scalar rin... |
clm1 24236 | The identity of the scalar... |
clmadd 24237 | The addition of the scalar... |
clmmul 24238 | The multiplication of the ... |
clmcj 24239 | The conjugation of the sca... |
isclmi 24240 | Reverse direction of ~ isc... |
clmzss 24241 | The scalar ring of a subco... |
clmsscn 24242 | The scalar ring of a subco... |
clmsub 24243 | Subtraction in the scalar ... |
clmneg 24244 | Negation in the scalar rin... |
clmneg1 24245 | Minus one is in the scalar... |
clmabs 24246 | Norm in the scalar ring of... |
clmacl 24247 | Closure of ring addition f... |
clmmcl 24248 | Closure of ring multiplica... |
clmsubcl 24249 | Closure of ring subtractio... |
lmhmclm 24250 | The domain of a linear ope... |
clmvscl 24251 | Closure of scalar product ... |
clmvsass 24252 | Associative law for scalar... |
clmvscom 24253 | Commutative law for the sc... |
clmvsdir 24254 | Distributive law for scala... |
clmvsdi 24255 | Distributive law for scala... |
clmvs1 24256 | Scalar product with ring u... |
clmvs2 24257 | A vector plus itself is tw... |
clm0vs 24258 | Zero times a vector is the... |
clmopfne 24259 | The (functionalized) opera... |
isclmp 24260 | The predicate "is a subcom... |
isclmi0 24261 | Properties that determine ... |
clmvneg1 24262 | Minus 1 times a vector is ... |
clmvsneg 24263 | Multiplication of a vector... |
clmmulg 24264 | The group multiple functio... |
clmsubdir 24265 | Scalar multiplication dist... |
clmpm1dir 24266 | Subtractive distributive l... |
clmnegneg 24267 | Double negative of a vecto... |
clmnegsubdi2 24268 | Distribution of negative o... |
clmsub4 24269 | Rearrangement of 4 terms i... |
clmvsrinv 24270 | A vector minus itself. (C... |
clmvslinv 24271 | Minus a vector plus itself... |
clmvsubval 24272 | Value of vector subtractio... |
clmvsubval2 24273 | Value of vector subtractio... |
clmvz 24274 | Two ways to express the ne... |
zlmclm 24275 | The ` ZZ ` -module operati... |
clmzlmvsca 24276 | The scalar product of a su... |
nmoleub2lem 24277 | Lemma for ~ nmoleub2a and ... |
nmoleub2lem3 24278 | Lemma for ~ nmoleub2a and ... |
nmoleub2lem2 24279 | Lemma for ~ nmoleub2a and ... |
nmoleub2a 24280 | The operator norm is the s... |
nmoleub2b 24281 | The operator norm is the s... |
nmoleub3 24282 | The operator norm is the s... |
nmhmcn 24283 | A linear operator over a n... |
cmodscexp 24284 | The powers of ` _i ` belon... |
cmodscmulexp 24285 | The scalar product of a ve... |
cvslvec 24288 | A subcomplex vector space ... |
cvsclm 24289 | A subcomplex vector space ... |
iscvs 24290 | A subcomplex vector space ... |
iscvsp 24291 | The predicate "is a subcom... |
iscvsi 24292 | Properties that determine ... |
cvsi 24293 | The properties of a subcom... |
cvsunit 24294 | Unit group of the scalar r... |
cvsdiv 24295 | Division of the scalar rin... |
cvsdivcl 24296 | The scalar field of a subc... |
cvsmuleqdivd 24297 | An equality involving rati... |
cvsdiveqd 24298 | An equality involving rati... |
cnlmodlem1 24299 | Lemma 1 for ~ cnlmod . (C... |
cnlmodlem2 24300 | Lemma 2 for ~ cnlmod . (C... |
cnlmodlem3 24301 | Lemma 3 for ~ cnlmod . (C... |
cnlmod4 24302 | Lemma 4 for ~ cnlmod . (C... |
cnlmod 24303 | The set of complex numbers... |
cnstrcvs 24304 | The set of complex numbers... |
cnrbas 24305 | The set of complex numbers... |
cnrlmod 24306 | The complex left module of... |
cnrlvec 24307 | The complex left module of... |
cncvs 24308 | The complex left module of... |
recvs 24309 | The field of the real numb... |
recvsOLD 24310 | Obsolete version of ~ recv... |
qcvs 24311 | The field of rational numb... |
zclmncvs 24312 | The ring of integers as le... |
isncvsngp 24313 | A normed subcomplex vector... |
isncvsngpd 24314 | Properties that determine ... |
ncvsi 24315 | The properties of a normed... |
ncvsprp 24316 | Proportionality property o... |
ncvsge0 24317 | The norm of a scalar produ... |
ncvsm1 24318 | The norm of the opposite o... |
ncvsdif 24319 | The norm of the difference... |
ncvspi 24320 | The norm of a vector plus ... |
ncvs1 24321 | From any nonzero vector of... |
cnrnvc 24322 | The module of complex numb... |
cnncvs 24323 | The module of complex numb... |
cnnm 24324 | The norm of the normed sub... |
ncvspds 24325 | Value of the distance func... |
cnindmet 24326 | The metric induced on the ... |
cnncvsaddassdemo 24327 | Derive the associative law... |
cnncvsmulassdemo 24328 | Derive the associative law... |
cnncvsabsnegdemo 24329 | Derive the absolute value ... |
iscph 24334 | A subcomplex pre-Hilbert s... |
cphphl 24335 | A subcomplex pre-Hilbert s... |
cphnlm 24336 | A subcomplex pre-Hilbert s... |
cphngp 24337 | A subcomplex pre-Hilbert s... |
cphlmod 24338 | A subcomplex pre-Hilbert s... |
cphlvec 24339 | A subcomplex pre-Hilbert s... |
cphnvc 24340 | A subcomplex pre-Hilbert s... |
cphsubrglem 24341 | Lemma for ~ cphsubrg . (C... |
cphreccllem 24342 | Lemma for ~ cphreccl . (C... |
cphsca 24343 | A subcomplex pre-Hilbert s... |
cphsubrg 24344 | The scalar field of a subc... |
cphreccl 24345 | The scalar field of a subc... |
cphdivcl 24346 | The scalar field of a subc... |
cphcjcl 24347 | The scalar field of a subc... |
cphsqrtcl 24348 | The scalar field of a subc... |
cphabscl 24349 | The scalar field of a subc... |
cphsqrtcl2 24350 | The scalar field of a subc... |
cphsqrtcl3 24351 | If the scalar field of a s... |
cphqss 24352 | The scalar field of a subc... |
cphclm 24353 | A subcomplex pre-Hilbert s... |
cphnmvs 24354 | Norm of a scalar product. ... |
cphipcl 24355 | An inner product is a memb... |
cphnmfval 24356 | The value of the norm in a... |
cphnm 24357 | The square of the norm is ... |
nmsq 24358 | The square of the norm is ... |
cphnmf 24359 | The norm of a vector is a ... |
cphnmcl 24360 | The norm of a vector is a ... |
reipcl 24361 | An inner product of an ele... |
ipge0 24362 | The inner product in a sub... |
cphipcj 24363 | Conjugate of an inner prod... |
cphipipcj 24364 | An inner product times its... |
cphorthcom 24365 | Orthogonality (meaning inn... |
cphip0l 24366 | Inner product with a zero ... |
cphip0r 24367 | Inner product with a zero ... |
cphipeq0 24368 | The inner product of a vec... |
cphdir 24369 | Distributive law for inner... |
cphdi 24370 | Distributive law for inner... |
cph2di 24371 | Distributive law for inner... |
cphsubdir 24372 | Distributive law for inner... |
cphsubdi 24373 | Distributive law for inner... |
cph2subdi 24374 | Distributive law for inner... |
cphass 24375 | Associative law for inner ... |
cphassr 24376 | "Associative" law for seco... |
cph2ass 24377 | Move scalar multiplication... |
cphassi 24378 | Associative law for the fi... |
cphassir 24379 | "Associative" law for the ... |
cphpyth 24380 | The pythagorean theorem fo... |
tcphex 24381 | Lemma for ~ tcphbas and si... |
tcphval 24382 | Define a function to augme... |
tcphbas 24383 | The base set of a subcompl... |
tchplusg 24384 | The addition operation of ... |
tcphsub 24385 | The subtraction operation ... |
tcphmulr 24386 | The ring operation of a su... |
tcphsca 24387 | The scalar field of a subc... |
tcphvsca 24388 | The scalar multiplication ... |
tcphip 24389 | The inner product of a sub... |
tcphtopn 24390 | The topology of a subcompl... |
tcphphl 24391 | Augmentation of a subcompl... |
tchnmfval 24392 | The norm of a subcomplex p... |
tcphnmval 24393 | The norm of a subcomplex p... |
cphtcphnm 24394 | The norm of a norm-augment... |
tcphds 24395 | The distance of a pre-Hilb... |
phclm 24396 | A pre-Hilbert space whose ... |
tcphcphlem3 24397 | Lemma for ~ tcphcph : real... |
ipcau2 24398 | The Cauchy-Schwarz inequal... |
tcphcphlem1 24399 | Lemma for ~ tcphcph : the ... |
tcphcphlem2 24400 | Lemma for ~ tcphcph : homo... |
tcphcph 24401 | The standard definition of... |
ipcau 24402 | The Cauchy-Schwarz inequal... |
nmparlem 24403 | Lemma for ~ nmpar . (Cont... |
nmpar 24404 | A subcomplex pre-Hilbert s... |
cphipval2 24405 | Value of the inner product... |
4cphipval2 24406 | Four times the inner produ... |
cphipval 24407 | Value of the inner product... |
ipcnlem2 24408 | The inner product operatio... |
ipcnlem1 24409 | The inner product operatio... |
ipcn 24410 | The inner product operatio... |
cnmpt1ip 24411 | Continuity of inner produc... |
cnmpt2ip 24412 | Continuity of inner produc... |
csscld 24413 | A "closed subspace" in a s... |
clsocv 24414 | The orthogonal complement ... |
cphsscph 24415 | A subspace of a subcomplex... |
lmmbr 24422 | Express the binary relatio... |
lmmbr2 24423 | Express the binary relatio... |
lmmbr3 24424 | Express the binary relatio... |
lmmcvg 24425 | Convergence property of a ... |
lmmbrf 24426 | Express the binary relatio... |
lmnn 24427 | A condition that implies c... |
cfilfval 24428 | The set of Cauchy filters ... |
iscfil 24429 | The property of being a Ca... |
iscfil2 24430 | The property of being a Ca... |
cfilfil 24431 | A Cauchy filter is a filte... |
cfili 24432 | Property of a Cauchy filte... |
cfil3i 24433 | A Cauchy filter contains b... |
cfilss 24434 | A filter finer than a Cauc... |
fgcfil 24435 | The Cauchy filter conditio... |
fmcfil 24436 | The Cauchy filter conditio... |
iscfil3 24437 | A filter is Cauchy iff it ... |
cfilfcls 24438 | Similar to ultrafilters ( ... |
caufval 24439 | The set of Cauchy sequence... |
iscau 24440 | Express the property " ` F... |
iscau2 24441 | Express the property " ` F... |
iscau3 24442 | Express the Cauchy sequenc... |
iscau4 24443 | Express the property " ` F... |
iscauf 24444 | Express the property " ` F... |
caun0 24445 | A metric with a Cauchy seq... |
caufpm 24446 | Inclusion of a Cauchy sequ... |
caucfil 24447 | A Cauchy sequence predicat... |
iscmet 24448 | The property " ` D ` is a ... |
cmetcvg 24449 | The convergence of a Cauch... |
cmetmet 24450 | A complete metric space is... |
cmetmeti 24451 | A complete metric space is... |
cmetcaulem 24452 | Lemma for ~ cmetcau . (Co... |
cmetcau 24453 | The convergence of a Cauch... |
iscmet3lem3 24454 | Lemma for ~ iscmet3 . (Co... |
iscmet3lem1 24455 | Lemma for ~ iscmet3 . (Co... |
iscmet3lem2 24456 | Lemma for ~ iscmet3 . (Co... |
iscmet3 24457 | The property " ` D ` is a ... |
iscmet2 24458 | A metric ` D ` is complete... |
cfilresi 24459 | A Cauchy filter on a metri... |
cfilres 24460 | Cauchy filter on a metric ... |
caussi 24461 | Cauchy sequence on a metri... |
causs 24462 | Cauchy sequence on a metri... |
equivcfil 24463 | If the metric ` D ` is "st... |
equivcau 24464 | If the metric ` D ` is "st... |
lmle 24465 | If the distance from each ... |
nglmle 24466 | If the norm of each member... |
lmclim 24467 | Relate a limit on the metr... |
lmclimf 24468 | Relate a limit on the metr... |
metelcls 24469 | A point belongs to the clo... |
metcld 24470 | A subset of a metric space... |
metcld2 24471 | A subset of a metric space... |
caubl 24472 | Sufficient condition to en... |
caublcls 24473 | The convergent point of a ... |
metcnp4 24474 | Two ways to say a mapping ... |
metcn4 24475 | Two ways to say a mapping ... |
iscmet3i 24476 | Properties that determine ... |
lmcau 24477 | Every convergent sequence ... |
flimcfil 24478 | Every convergent filter in... |
metsscmetcld 24479 | A complete subspace of a m... |
cmetss 24480 | A subspace of a complete m... |
equivcmet 24481 | If two metrics are strongl... |
relcmpcmet 24482 | If ` D ` is a metric space... |
cmpcmet 24483 | A compact metric space is ... |
cfilucfil3 24484 | Given a metric ` D ` and a... |
cfilucfil4 24485 | Given a metric ` D ` and a... |
cncmet 24486 | The set of complex numbers... |
recmet 24487 | The real numbers are a com... |
bcthlem1 24488 | Lemma for ~ bcth . Substi... |
bcthlem2 24489 | Lemma for ~ bcth . The ba... |
bcthlem3 24490 | Lemma for ~ bcth . The li... |
bcthlem4 24491 | Lemma for ~ bcth . Given ... |
bcthlem5 24492 | Lemma for ~ bcth . The pr... |
bcth 24493 | Baire's Category Theorem. ... |
bcth2 24494 | Baire's Category Theorem, ... |
bcth3 24495 | Baire's Category Theorem, ... |
isbn 24502 | A Banach space is a normed... |
bnsca 24503 | The scalar field of a Bana... |
bnnvc 24504 | A Banach space is a normed... |
bnnlm 24505 | A Banach space is a normed... |
bnngp 24506 | A Banach space is a normed... |
bnlmod 24507 | A Banach space is a left m... |
bncms 24508 | A Banach space is a comple... |
iscms 24509 | A complete metric space is... |
cmscmet 24510 | The induced metric on a co... |
bncmet 24511 | The induced metric on Bana... |
cmsms 24512 | A complete metric space is... |
cmspropd 24513 | Property deduction for a c... |
cmssmscld 24514 | The restriction of a metri... |
cmsss 24515 | The restriction of a compl... |
lssbn 24516 | A subspace of a Banach spa... |
cmetcusp1 24517 | If the uniform set of a co... |
cmetcusp 24518 | The uniform space generate... |
cncms 24519 | The field of complex numbe... |
cnflduss 24520 | The uniform structure of t... |
cnfldcusp 24521 | The field of complex numbe... |
resscdrg 24522 | The real numbers are a sub... |
cncdrg 24523 | The only complete subfield... |
srabn 24524 | The subring algebra over a... |
rlmbn 24525 | The ring module over a com... |
ishl 24526 | The predicate "is a subcom... |
hlbn 24527 | Every subcomplex Hilbert s... |
hlcph 24528 | Every subcomplex Hilbert s... |
hlphl 24529 | Every subcomplex Hilbert s... |
hlcms 24530 | Every subcomplex Hilbert s... |
hlprlem 24531 | Lemma for ~ hlpr . (Contr... |
hlress 24532 | The scalar field of a subc... |
hlpr 24533 | The scalar field of a subc... |
ishl2 24534 | A Hilbert space is a compl... |
cphssphl 24535 | A Banach subspace of a sub... |
cmslssbn 24536 | A complete linear subspace... |
cmscsscms 24537 | A closed subspace of a com... |
bncssbn 24538 | A closed subspace of a Ban... |
cssbn 24539 | A complete subspace of a n... |
csschl 24540 | A complete subspace of a c... |
cmslsschl 24541 | A complete linear subspace... |
chlcsschl 24542 | A closed subspace of a sub... |
retopn 24543 | The topology of the real n... |
recms 24544 | The real numbers form a co... |
reust 24545 | The Uniform structure of t... |
recusp 24546 | The real numbers form a co... |
rrxval 24551 | Value of the generalized E... |
rrxbase 24552 | The base of the generalize... |
rrxprds 24553 | Expand the definition of t... |
rrxip 24554 | The inner product of the g... |
rrxnm 24555 | The norm of the generalize... |
rrxcph 24556 | Generalized Euclidean real... |
rrxds 24557 | The distance over generali... |
rrxvsca 24558 | The scalar product over ge... |
rrxplusgvscavalb 24559 | The result of the addition... |
rrxsca 24560 | The field of real numbers ... |
rrx0 24561 | The zero ("origin") in a g... |
rrx0el 24562 | The zero ("origin") in a g... |
csbren 24563 | Cauchy-Schwarz-Bunjakovsky... |
trirn 24564 | Triangle inequality in R^n... |
rrxf 24565 | Euclidean vectors as funct... |
rrxfsupp 24566 | Euclidean vectors are of f... |
rrxsuppss 24567 | Support of Euclidean vecto... |
rrxmvallem 24568 | Support of the function us... |
rrxmval 24569 | The value of the Euclidean... |
rrxmfval 24570 | The value of the Euclidean... |
rrxmetlem 24571 | Lemma for ~ rrxmet . (Con... |
rrxmet 24572 | Euclidean space is a metri... |
rrxdstprj1 24573 | The distance between two p... |
rrxbasefi 24574 | The base of the generalize... |
rrxdsfi 24575 | The distance over generali... |
rrxmetfi 24576 | Euclidean space is a metri... |
rrxdsfival 24577 | The value of the Euclidean... |
ehlval 24578 | Value of the Euclidean spa... |
ehlbase 24579 | The base of the Euclidean ... |
ehl0base 24580 | The base of the Euclidean ... |
ehl0 24581 | The Euclidean space of dim... |
ehleudis 24582 | The Euclidean distance fun... |
ehleudisval 24583 | The value of the Euclidean... |
ehl1eudis 24584 | The Euclidean distance fun... |
ehl1eudisval 24585 | The value of the Euclidean... |
ehl2eudis 24586 | The Euclidean distance fun... |
ehl2eudisval 24587 | The value of the Euclidean... |
minveclem1 24588 | Lemma for ~ minvec . The ... |
minveclem4c 24589 | Lemma for ~ minvec . The ... |
minveclem2 24590 | Lemma for ~ minvec . Any ... |
minveclem3a 24591 | Lemma for ~ minvec . ` D `... |
minveclem3b 24592 | Lemma for ~ minvec . The ... |
minveclem3 24593 | Lemma for ~ minvec . The ... |
minveclem4a 24594 | Lemma for ~ minvec . ` F `... |
minveclem4b 24595 | Lemma for ~ minvec . The ... |
minveclem4 24596 | Lemma for ~ minvec . The ... |
minveclem5 24597 | Lemma for ~ minvec . Disc... |
minveclem6 24598 | Lemma for ~ minvec . Any ... |
minveclem7 24599 | Lemma for ~ minvec . Sinc... |
minvec 24600 | Minimizing vector theorem,... |
pjthlem1 24601 | Lemma for ~ pjth . (Contr... |
pjthlem2 24602 | Lemma for ~ pjth . (Contr... |
pjth 24603 | Projection Theorem: Any H... |
pjth2 24604 | Projection Theorem with ab... |
cldcss 24605 | Corollary of the Projectio... |
cldcss2 24606 | Corollary of the Projectio... |
hlhil 24607 | Corollary of the Projectio... |
addcncf 24608 | The addition of two contin... |
subcncf 24609 | The addition of two contin... |
mulcncf 24610 | The multiplication of two ... |
divcncf 24611 | The quotient of two contin... |
pmltpclem1 24612 | Lemma for ~ pmltpc . (Con... |
pmltpclem2 24613 | Lemma for ~ pmltpc . (Con... |
pmltpc 24614 | Any function on the reals ... |
ivthlem1 24615 | Lemma for ~ ivth . The se... |
ivthlem2 24616 | Lemma for ~ ivth . Show t... |
ivthlem3 24617 | Lemma for ~ ivth , the int... |
ivth 24618 | The intermediate value the... |
ivth2 24619 | The intermediate value the... |
ivthle 24620 | The intermediate value the... |
ivthle2 24621 | The intermediate value the... |
ivthicc 24622 | The interval between any t... |
evthicc 24623 | Specialization of the Extr... |
evthicc2 24624 | Combine ~ ivthicc with ~ e... |
cniccbdd 24625 | A continuous function on a... |
ovolfcl 24630 | Closure for the interval e... |
ovolfioo 24631 | Unpack the interval coveri... |
ovolficc 24632 | Unpack the interval coveri... |
ovolficcss 24633 | Any (closed) interval cove... |
ovolfsval 24634 | The value of the interval ... |
ovolfsf 24635 | Closure for the interval l... |
ovolsf 24636 | Closure for the partial su... |
ovolval 24637 | The value of the outer mea... |
elovolmlem 24638 | Lemma for ~ elovolm and re... |
elovolm 24639 | Elementhood in the set ` M... |
elovolmr 24640 | Sufficient condition for e... |
ovolmge0 24641 | The set ` M ` is composed ... |
ovolcl 24642 | The volume of a set is an ... |
ovollb 24643 | The outer volume is a lowe... |
ovolgelb 24644 | The outer volume is the gr... |
ovolge0 24645 | The volume of a set is alw... |
ovolf 24646 | The domain and range of th... |
ovollecl 24647 | If an outer volume is boun... |
ovolsslem 24648 | Lemma for ~ ovolss . (Con... |
ovolss 24649 | The volume of a set is mon... |
ovolsscl 24650 | If a set is contained in a... |
ovolssnul 24651 | A subset of a nullset is n... |
ovollb2lem 24652 | Lemma for ~ ovollb2 . (Co... |
ovollb2 24653 | It is often more convenien... |
ovolctb 24654 | The volume of a denumerabl... |
ovolq 24655 | The rational numbers have ... |
ovolctb2 24656 | The volume of a countable ... |
ovol0 24657 | The empty set has 0 outer ... |
ovolfi 24658 | A finite set has 0 outer L... |
ovolsn 24659 | A singleton has 0 outer Le... |
ovolunlem1a 24660 | Lemma for ~ ovolun . (Con... |
ovolunlem1 24661 | Lemma for ~ ovolun . (Con... |
ovolunlem2 24662 | Lemma for ~ ovolun . (Con... |
ovolun 24663 | The Lebesgue outer measure... |
ovolunnul 24664 | Adding a nullset does not ... |
ovolfiniun 24665 | The Lebesgue outer measure... |
ovoliunlem1 24666 | Lemma for ~ ovoliun . (Co... |
ovoliunlem2 24667 | Lemma for ~ ovoliun . (Co... |
ovoliunlem3 24668 | Lemma for ~ ovoliun . (Co... |
ovoliun 24669 | The Lebesgue outer measure... |
ovoliun2 24670 | The Lebesgue outer measure... |
ovoliunnul 24671 | A countable union of nulls... |
shft2rab 24672 | If ` B ` is a shift of ` A... |
ovolshftlem1 24673 | Lemma for ~ ovolshft . (C... |
ovolshftlem2 24674 | Lemma for ~ ovolshft . (C... |
ovolshft 24675 | The Lebesgue outer measure... |
sca2rab 24676 | If ` B ` is a scale of ` A... |
ovolscalem1 24677 | Lemma for ~ ovolsca . (Co... |
ovolscalem2 24678 | Lemma for ~ ovolshft . (C... |
ovolsca 24679 | The Lebesgue outer measure... |
ovolicc1 24680 | The measure of a closed in... |
ovolicc2lem1 24681 | Lemma for ~ ovolicc2 . (C... |
ovolicc2lem2 24682 | Lemma for ~ ovolicc2 . (C... |
ovolicc2lem3 24683 | Lemma for ~ ovolicc2 . (C... |
ovolicc2lem4 24684 | Lemma for ~ ovolicc2 . (C... |
ovolicc2lem5 24685 | Lemma for ~ ovolicc2 . (C... |
ovolicc2 24686 | The measure of a closed in... |
ovolicc 24687 | The measure of a closed in... |
ovolicopnf 24688 | The measure of a right-unb... |
ovolre 24689 | The measure of the real nu... |
ismbl 24690 | The predicate " ` A ` is L... |
ismbl2 24691 | From ~ ovolun , it suffice... |
volres 24692 | A self-referencing abbrevi... |
volf 24693 | The domain and range of th... |
mblvol 24694 | The volume of a measurable... |
mblss 24695 | A measurable set is a subs... |
mblsplit 24696 | The defining property of m... |
volss 24697 | The Lebesgue measure is mo... |
cmmbl 24698 | The complement of a measur... |
nulmbl 24699 | A nullset is measurable. ... |
nulmbl2 24700 | A set of outer measure zer... |
unmbl 24701 | A union of measurable sets... |
shftmbl 24702 | A shift of a measurable se... |
0mbl 24703 | The empty set is measurabl... |
rembl 24704 | The set of all real number... |
unidmvol 24705 | The union of the Lebesgue ... |
inmbl 24706 | An intersection of measura... |
difmbl 24707 | A difference of measurable... |
finiunmbl 24708 | A finite union of measurab... |
volun 24709 | The Lebesgue measure funct... |
volinun 24710 | Addition of non-disjoint s... |
volfiniun 24711 | The volume of a disjoint f... |
iundisj 24712 | Rewrite a countable union ... |
iundisj2 24713 | A disjoint union is disjoi... |
voliunlem1 24714 | Lemma for ~ voliun . (Con... |
voliunlem2 24715 | Lemma for ~ voliun . (Con... |
voliunlem3 24716 | Lemma for ~ voliun . (Con... |
iunmbl 24717 | The measurable sets are cl... |
voliun 24718 | The Lebesgue measure funct... |
volsuplem 24719 | Lemma for ~ volsup . (Con... |
volsup 24720 | The volume of the limit of... |
iunmbl2 24721 | The measurable sets are cl... |
ioombl1lem1 24722 | Lemma for ~ ioombl1 . (Co... |
ioombl1lem2 24723 | Lemma for ~ ioombl1 . (Co... |
ioombl1lem3 24724 | Lemma for ~ ioombl1 . (Co... |
ioombl1lem4 24725 | Lemma for ~ ioombl1 . (Co... |
ioombl1 24726 | An open right-unbounded in... |
icombl1 24727 | A closed unbounded-above i... |
icombl 24728 | A closed-below, open-above... |
ioombl 24729 | An open real interval is m... |
iccmbl 24730 | A closed real interval is ... |
iccvolcl 24731 | A closed real interval has... |
ovolioo 24732 | The measure of an open int... |
volioo 24733 | The measure of an open int... |
ioovolcl 24734 | An open real interval has ... |
ovolfs2 24735 | Alternative expression for... |
ioorcl2 24736 | An open interval with fini... |
ioorf 24737 | Define a function from ope... |
ioorval 24738 | Define a function from ope... |
ioorinv2 24739 | The function ` F ` is an "... |
ioorinv 24740 | The function ` F ` is an "... |
ioorcl 24741 | The function ` F ` does no... |
uniiccdif 24742 | A union of closed interval... |
uniioovol 24743 | A disjoint union of open i... |
uniiccvol 24744 | An almost-disjoint union o... |
uniioombllem1 24745 | Lemma for ~ uniioombl . (... |
uniioombllem2a 24746 | Lemma for ~ uniioombl . (... |
uniioombllem2 24747 | Lemma for ~ uniioombl . (... |
uniioombllem3a 24748 | Lemma for ~ uniioombl . (... |
uniioombllem3 24749 | Lemma for ~ uniioombl . (... |
uniioombllem4 24750 | Lemma for ~ uniioombl . (... |
uniioombllem5 24751 | Lemma for ~ uniioombl . (... |
uniioombllem6 24752 | Lemma for ~ uniioombl . (... |
uniioombl 24753 | A disjoint union of open i... |
uniiccmbl 24754 | An almost-disjoint union o... |
dyadf 24755 | The function ` F ` returns... |
dyadval 24756 | Value of the dyadic ration... |
dyadovol 24757 | Volume of a dyadic rationa... |
dyadss 24758 | Two closed dyadic rational... |
dyaddisjlem 24759 | Lemma for ~ dyaddisj . (C... |
dyaddisj 24760 | Two closed dyadic rational... |
dyadmaxlem 24761 | Lemma for ~ dyadmax . (Co... |
dyadmax 24762 | Any nonempty set of dyadic... |
dyadmbllem 24763 | Lemma for ~ dyadmbl . (Co... |
dyadmbl 24764 | Any union of dyadic ration... |
opnmbllem 24765 | Lemma for ~ opnmbl . (Con... |
opnmbl 24766 | All open sets are measurab... |
opnmblALT 24767 | All open sets are measurab... |
subopnmbl 24768 | Sets which are open in a m... |
volsup2 24769 | The volume of ` A ` is the... |
volcn 24770 | The function formed by res... |
volivth 24771 | The Intermediate Value The... |
vitalilem1 24772 | Lemma for ~ vitali . (Con... |
vitalilem2 24773 | Lemma for ~ vitali . (Con... |
vitalilem3 24774 | Lemma for ~ vitali . (Con... |
vitalilem4 24775 | Lemma for ~ vitali . (Con... |
vitalilem5 24776 | Lemma for ~ vitali . (Con... |
vitali 24777 | If the reals can be well-o... |
ismbf1 24788 | The predicate " ` F ` is a... |
mbff 24789 | A measurable function is a... |
mbfdm 24790 | The domain of a measurable... |
mbfconstlem 24791 | Lemma for ~ mbfconst and r... |
ismbf 24792 | The predicate " ` F ` is a... |
ismbfcn 24793 | A complex function is meas... |
mbfima 24794 | Definitional property of a... |
mbfimaicc 24795 | The preimage of any closed... |
mbfimasn 24796 | The preimage of a point un... |
mbfconst 24797 | A constant function is mea... |
mbf0 24798 | The empty function is meas... |
mbfid 24799 | The identity function is m... |
mbfmptcl 24800 | Lemma for the ` MblFn ` pr... |
mbfdm2 24801 | The domain of a measurable... |
ismbfcn2 24802 | A complex function is meas... |
ismbfd 24803 | Deduction to prove measura... |
ismbf2d 24804 | Deduction to prove measura... |
mbfeqalem1 24805 | Lemma for ~ mbfeqalem2 . ... |
mbfeqalem2 24806 | Lemma for ~ mbfeqa . (Con... |
mbfeqa 24807 | If two functions are equal... |
mbfres 24808 | The restriction of a measu... |
mbfres2 24809 | Measurability of a piecewi... |
mbfss 24810 | Change the domain of a mea... |
mbfmulc2lem 24811 | Multiplication by a consta... |
mbfmulc2re 24812 | Multiplication by a consta... |
mbfmax 24813 | The maximum of two functio... |
mbfneg 24814 | The negative of a measurab... |
mbfpos 24815 | The positive part of a mea... |
mbfposr 24816 | Converse to ~ mbfpos . (C... |
mbfposb 24817 | A function is measurable i... |
ismbf3d 24818 | Simplified form of ~ ismbf... |
mbfimaopnlem 24819 | Lemma for ~ mbfimaopn . (... |
mbfimaopn 24820 | The preimage of any open s... |
mbfimaopn2 24821 | The preimage of any set op... |
cncombf 24822 | The composition of a conti... |
cnmbf 24823 | A continuous function is m... |
mbfaddlem 24824 | The sum of two measurable ... |
mbfadd 24825 | The sum of two measurable ... |
mbfsub 24826 | The difference of two meas... |
mbfmulc2 24827 | A complex constant times a... |
mbfsup 24828 | The supremum of a sequence... |
mbfinf 24829 | The infimum of a sequence ... |
mbflimsup 24830 | The limit supremum of a se... |
mbflimlem 24831 | The pointwise limit of a s... |
mbflim 24832 | The pointwise limit of a s... |
0pval 24835 | The zero function evaluate... |
0plef 24836 | Two ways to say that the f... |
0pledm 24837 | Adjust the domain of the l... |
isi1f 24838 | The predicate " ` F ` is a... |
i1fmbf 24839 | Simple functions are measu... |
i1ff 24840 | A simple function is a fun... |
i1frn 24841 | A simple function has fini... |
i1fima 24842 | Any preimage of a simple f... |
i1fima2 24843 | Any preimage of a simple f... |
i1fima2sn 24844 | Preimage of a singleton. ... |
i1fd 24845 | A simplified set of assump... |
i1f0rn 24846 | Any simple function takes ... |
itg1val 24847 | The value of the integral ... |
itg1val2 24848 | The value of the integral ... |
itg1cl 24849 | Closure of the integral on... |
itg1ge0 24850 | Closure of the integral on... |
i1f0 24851 | The zero function is simpl... |
itg10 24852 | The zero function has zero... |
i1f1lem 24853 | Lemma for ~ i1f1 and ~ itg... |
i1f1 24854 | Base case simple functions... |
itg11 24855 | The integral of an indicat... |
itg1addlem1 24856 | Decompose a preimage, whic... |
i1faddlem 24857 | Decompose the preimage of ... |
i1fmullem 24858 | Decompose the preimage of ... |
i1fadd 24859 | The sum of two simple func... |
i1fmul 24860 | The pointwise product of t... |
itg1addlem2 24861 | Lemma for ~ itg1add . The... |
itg1addlem3 24862 | Lemma for ~ itg1add . (Co... |
itg1addlem4 24863 | Lemma for ~ itg1add . (Co... |
itg1addlem4OLD 24864 | Obsolete version of ~ itg1... |
itg1addlem5 24865 | Lemma for ~ itg1add . (Co... |
itg1add 24866 | The integral of a sum of s... |
i1fmulclem 24867 | Decompose the preimage of ... |
i1fmulc 24868 | A nonnegative constant tim... |
itg1mulc 24869 | The integral of a constant... |
i1fres 24870 | The "restriction" of a sim... |
i1fpos 24871 | The positive part of a sim... |
i1fposd 24872 | Deduction form of ~ i1fpos... |
i1fsub 24873 | The difference of two simp... |
itg1sub 24874 | The integral of a differen... |
itg10a 24875 | The integral of a simple f... |
itg1ge0a 24876 | The integral of an almost ... |
itg1lea 24877 | Approximate version of ~ i... |
itg1le 24878 | If one simple function dom... |
itg1climres 24879 | Restricting the simple fun... |
mbfi1fseqlem1 24880 | Lemma for ~ mbfi1fseq . (... |
mbfi1fseqlem2 24881 | Lemma for ~ mbfi1fseq . (... |
mbfi1fseqlem3 24882 | Lemma for ~ mbfi1fseq . (... |
mbfi1fseqlem4 24883 | Lemma for ~ mbfi1fseq . T... |
mbfi1fseqlem5 24884 | Lemma for ~ mbfi1fseq . V... |
mbfi1fseqlem6 24885 | Lemma for ~ mbfi1fseq . V... |
mbfi1fseq 24886 | A characterization of meas... |
mbfi1flimlem 24887 | Lemma for ~ mbfi1flim . (... |
mbfi1flim 24888 | Any real measurable functi... |
mbfmullem2 24889 | Lemma for ~ mbfmul . (Con... |
mbfmullem 24890 | Lemma for ~ mbfmul . (Con... |
mbfmul 24891 | The product of two measura... |
itg2lcl 24892 | The set of lower sums is a... |
itg2val 24893 | Value of the integral on n... |
itg2l 24894 | Elementhood in the set ` L... |
itg2lr 24895 | Sufficient condition for e... |
xrge0f 24896 | A real function is a nonne... |
itg2cl 24897 | The integral of a nonnegat... |
itg2ub 24898 | The integral of a nonnegat... |
itg2leub 24899 | Any upper bound on the int... |
itg2ge0 24900 | The integral of a nonnegat... |
itg2itg1 24901 | The integral of a nonnegat... |
itg20 24902 | The integral of the zero f... |
itg2lecl 24903 | If an ` S.2 ` integral is ... |
itg2le 24904 | If one function dominates ... |
itg2const 24905 | Integral of a constant fun... |
itg2const2 24906 | When the base set of a con... |
itg2seq 24907 | Definitional property of t... |
itg2uba 24908 | Approximate version of ~ i... |
itg2lea 24909 | Approximate version of ~ i... |
itg2eqa 24910 | Approximate equality of in... |
itg2mulclem 24911 | Lemma for ~ itg2mulc . (C... |
itg2mulc 24912 | The integral of a nonnegat... |
itg2splitlem 24913 | Lemma for ~ itg2split . (... |
itg2split 24914 | The ` S.2 ` integral split... |
itg2monolem1 24915 | Lemma for ~ itg2mono . We... |
itg2monolem2 24916 | Lemma for ~ itg2mono . (C... |
itg2monolem3 24917 | Lemma for ~ itg2mono . (C... |
itg2mono 24918 | The Monotone Convergence T... |
itg2i1fseqle 24919 | Subject to the conditions ... |
itg2i1fseq 24920 | Subject to the conditions ... |
itg2i1fseq2 24921 | In an extension to the res... |
itg2i1fseq3 24922 | Special case of ~ itg2i1fs... |
itg2addlem 24923 | Lemma for ~ itg2add . (Co... |
itg2add 24924 | The ` S.2 ` integral is li... |
itg2gt0 24925 | If the function ` F ` is s... |
itg2cnlem1 24926 | Lemma for ~ itgcn . (Cont... |
itg2cnlem2 24927 | Lemma for ~ itgcn . (Cont... |
itg2cn 24928 | A sort of absolute continu... |
ibllem 24929 | Conditioned equality theor... |
isibl 24930 | The predicate " ` F ` is i... |
isibl2 24931 | The predicate " ` F ` is i... |
iblmbf 24932 | An integrable function is ... |
iblitg 24933 | If a function is integrabl... |
dfitg 24934 | Evaluate the class substit... |
itgex 24935 | An integral is a set. (Co... |
itgeq1f 24936 | Equality theorem for an in... |
itgeq1 24937 | Equality theorem for an in... |
nfitg1 24938 | Bound-variable hypothesis ... |
nfitg 24939 | Bound-variable hypothesis ... |
cbvitg 24940 | Change bound variable in a... |
cbvitgv 24941 | Change bound variable in a... |
itgeq2 24942 | Equality theorem for an in... |
itgresr 24943 | The domain of an integral ... |
itg0 24944 | The integral of anything o... |
itgz 24945 | The integral of zero on an... |
itgeq2dv 24946 | Equality theorem for an in... |
itgmpt 24947 | Change bound variable in a... |
itgcl 24948 | The integral of an integra... |
itgvallem 24949 | Substitution lemma. (Cont... |
itgvallem3 24950 | Lemma for ~ itgposval and ... |
ibl0 24951 | The zero function is integ... |
iblcnlem1 24952 | Lemma for ~ iblcnlem . (C... |
iblcnlem 24953 | Expand out the universal q... |
itgcnlem 24954 | Expand out the sum in ~ df... |
iblrelem 24955 | Integrability of a real fu... |
iblposlem 24956 | Lemma for ~ iblpos . (Con... |
iblpos 24957 | Integrability of a nonnega... |
iblre 24958 | Integrability of a real fu... |
itgrevallem1 24959 | Lemma for ~ itgposval and ... |
itgposval 24960 | The integral of a nonnegat... |
itgreval 24961 | Decompose the integral of ... |
itgrecl 24962 | Real closure of an integra... |
iblcn 24963 | Integrability of a complex... |
itgcnval 24964 | Decompose the integral of ... |
itgre 24965 | Real part of an integral. ... |
itgim 24966 | Imaginary part of an integ... |
iblneg 24967 | The negative of an integra... |
itgneg 24968 | Negation of an integral. ... |
iblss 24969 | A subset of an integrable ... |
iblss2 24970 | Change the domain of an in... |
itgitg2 24971 | Transfer an integral using... |
i1fibl 24972 | A simple function is integ... |
itgitg1 24973 | Transfer an integral using... |
itgle 24974 | Monotonicity of an integra... |
itgge0 24975 | The integral of a positive... |
itgss 24976 | Expand the set of an integ... |
itgss2 24977 | Expand the set of an integ... |
itgeqa 24978 | Approximate equality of in... |
itgss3 24979 | Expand the set of an integ... |
itgioo 24980 | Equality of integrals on o... |
itgless 24981 | Expand the integral of a n... |
iblconst 24982 | A constant function is int... |
itgconst 24983 | Integral of a constant fun... |
ibladdlem 24984 | Lemma for ~ ibladd . (Con... |
ibladd 24985 | Add two integrals over the... |
iblsub 24986 | Subtract two integrals ove... |
itgaddlem1 24987 | Lemma for ~ itgadd . (Con... |
itgaddlem2 24988 | Lemma for ~ itgadd . (Con... |
itgadd 24989 | Add two integrals over the... |
itgsub 24990 | Subtract two integrals ove... |
itgfsum 24991 | Take a finite sum of integ... |
iblabslem 24992 | Lemma for ~ iblabs . (Con... |
iblabs 24993 | The absolute value of an i... |
iblabsr 24994 | A measurable function is i... |
iblmulc2 24995 | Multiply an integral by a ... |
itgmulc2lem1 24996 | Lemma for ~ itgmulc2 : pos... |
itgmulc2lem2 24997 | Lemma for ~ itgmulc2 : rea... |
itgmulc2 24998 | Multiply an integral by a ... |
itgabs 24999 | The triangle inequality fo... |
itgsplit 25000 | The ` S. ` integral splits... |
itgspliticc 25001 | The ` S. ` integral splits... |
itgsplitioo 25002 | The ` S. ` integral splits... |
bddmulibl 25003 | A bounded function times a... |
bddibl 25004 | A bounded function is inte... |
cniccibl 25005 | A continuous function on a... |
bddiblnc 25006 | Choice-free proof of ~ bdd... |
cnicciblnc 25007 | Choice-free proof of ~ cni... |
itggt0 25008 | The integral of a strictly... |
itgcn 25009 | Transfer ~ itg2cn to the f... |
ditgeq1 25012 | Equality theorem for the d... |
ditgeq2 25013 | Equality theorem for the d... |
ditgeq3 25014 | Equality theorem for the d... |
ditgeq3dv 25015 | Equality theorem for the d... |
ditgex 25016 | A directed integral is a s... |
ditg0 25017 | Value of the directed inte... |
cbvditg 25018 | Change bound variable in a... |
cbvditgv 25019 | Change bound variable in a... |
ditgpos 25020 | Value of the directed inte... |
ditgneg 25021 | Value of the directed inte... |
ditgcl 25022 | Closure of a directed inte... |
ditgswap 25023 | Reverse a directed integra... |
ditgsplitlem 25024 | Lemma for ~ ditgsplit . (... |
ditgsplit 25025 | This theorem is the raison... |
reldv 25034 | The derivative function is... |
limcvallem 25035 | Lemma for ~ ellimc . (Con... |
limcfval 25036 | Value and set bounds on th... |
ellimc 25037 | Value of the limit predica... |
limcrcl 25038 | Reverse closure for the li... |
limccl 25039 | Closure of the limit opera... |
limcdif 25040 | It suffices to consider fu... |
ellimc2 25041 | Write the definition of a ... |
limcnlp 25042 | If ` B ` is not a limit po... |
ellimc3 25043 | Write the epsilon-delta de... |
limcflflem 25044 | Lemma for ~ limcflf . (Co... |
limcflf 25045 | The limit operator can be ... |
limcmo 25046 | If ` B ` is a limit point ... |
limcmpt 25047 | Express the limit operator... |
limcmpt2 25048 | Express the limit operator... |
limcresi 25049 | Any limit of ` F ` is also... |
limcres 25050 | If ` B ` is an interior po... |
cnplimc 25051 | A function is continuous a... |
cnlimc 25052 | ` F ` is a continuous func... |
cnlimci 25053 | If ` F ` is a continuous f... |
cnmptlimc 25054 | If ` F ` is a continuous f... |
limccnp 25055 | If the limit of ` F ` at `... |
limccnp2 25056 | The image of a convergent ... |
limcco 25057 | Composition of two limits.... |
limciun 25058 | A point is a limit of ` F ... |
limcun 25059 | A point is a limit of ` F ... |
dvlem 25060 | Closure for a difference q... |
dvfval 25061 | Value and set bounds on th... |
eldv 25062 | The differentiable predica... |
dvcl 25063 | The derivative function ta... |
dvbssntr 25064 | The set of differentiable ... |
dvbss 25065 | The set of differentiable ... |
dvbsss 25066 | The set of differentiable ... |
perfdvf 25067 | The derivative is a functi... |
recnprss 25068 | Both ` RR ` and ` CC ` are... |
recnperf 25069 | Both ` RR ` and ` CC ` are... |
dvfg 25070 | Explicitly write out the f... |
dvf 25071 | The derivative is a functi... |
dvfcn 25072 | The derivative is a functi... |
dvreslem 25073 | Lemma for ~ dvres . (Cont... |
dvres2lem 25074 | Lemma for ~ dvres2 . (Con... |
dvres 25075 | Restriction of a derivativ... |
dvres2 25076 | Restriction of the base se... |
dvres3 25077 | Restriction of a complex d... |
dvres3a 25078 | Restriction of a complex d... |
dvidlem 25079 | Lemma for ~ dvid and ~ dvc... |
dvmptresicc 25080 | Derivative of a function r... |
dvconst 25081 | Derivative of a constant f... |
dvid 25082 | Derivative of the identity... |
dvcnp 25083 | The difference quotient is... |
dvcnp2 25084 | A function is continuous a... |
dvcn 25085 | A differentiable function ... |
dvnfval 25086 | Value of the iterated deri... |
dvnff 25087 | The iterated derivative is... |
dvn0 25088 | Zero times iterated deriva... |
dvnp1 25089 | Successor iterated derivat... |
dvn1 25090 | One times iterated derivat... |
dvnf 25091 | The N-times derivative is ... |
dvnbss 25092 | The set of N-times differe... |
dvnadd 25093 | The ` N ` -th derivative o... |
dvn2bss 25094 | An N-times differentiable ... |
dvnres 25095 | Multiple derivative versio... |
cpnfval 25096 | Condition for n-times cont... |
fncpn 25097 | The ` C^n ` object is a fu... |
elcpn 25098 | Condition for n-times cont... |
cpnord 25099 | ` C^n ` conditions are ord... |
cpncn 25100 | A ` C^n ` function is cont... |
cpnres 25101 | The restriction of a ` C^n... |
dvaddbr 25102 | The sum rule for derivativ... |
dvmulbr 25103 | The product rule for deriv... |
dvadd 25104 | The sum rule for derivativ... |
dvmul 25105 | The product rule for deriv... |
dvaddf 25106 | The sum rule for everywher... |
dvmulf 25107 | The product rule for every... |
dvcmul 25108 | The product rule when one ... |
dvcmulf 25109 | The product rule when one ... |
dvcobr 25110 | The chain rule for derivat... |
dvco 25111 | The chain rule for derivat... |
dvcof 25112 | The chain rule for everywh... |
dvcjbr 25113 | The derivative of the conj... |
dvcj 25114 | The derivative of the conj... |
dvfre 25115 | The derivative of a real f... |
dvnfre 25116 | The ` N ` -th derivative o... |
dvexp 25117 | Derivative of a power func... |
dvexp2 25118 | Derivative of an exponenti... |
dvrec 25119 | Derivative of the reciproc... |
dvmptres3 25120 | Function-builder for deriv... |
dvmptid 25121 | Function-builder for deriv... |
dvmptc 25122 | Function-builder for deriv... |
dvmptcl 25123 | Closure lemma for ~ dvmptc... |
dvmptadd 25124 | Function-builder for deriv... |
dvmptmul 25125 | Function-builder for deriv... |
dvmptres2 25126 | Function-builder for deriv... |
dvmptres 25127 | Function-builder for deriv... |
dvmptcmul 25128 | Function-builder for deriv... |
dvmptdivc 25129 | Function-builder for deriv... |
dvmptneg 25130 | Function-builder for deriv... |
dvmptsub 25131 | Function-builder for deriv... |
dvmptcj 25132 | Function-builder for deriv... |
dvmptre 25133 | Function-builder for deriv... |
dvmptim 25134 | Function-builder for deriv... |
dvmptntr 25135 | Function-builder for deriv... |
dvmptco 25136 | Function-builder for deriv... |
dvrecg 25137 | Derivative of the reciproc... |
dvmptdiv 25138 | Function-builder for deriv... |
dvmptfsum 25139 | Function-builder for deriv... |
dvcnvlem 25140 | Lemma for ~ dvcnvre . (Co... |
dvcnv 25141 | A weak version of ~ dvcnvr... |
dvexp3 25142 | Derivative of an exponenti... |
dveflem 25143 | Derivative of the exponent... |
dvef 25144 | Derivative of the exponent... |
dvsincos 25145 | Derivative of the sine and... |
dvsin 25146 | Derivative of the sine fun... |
dvcos 25147 | Derivative of the cosine f... |
dvferm1lem 25148 | Lemma for ~ dvferm . (Con... |
dvferm1 25149 | One-sided version of ~ dvf... |
dvferm2lem 25150 | Lemma for ~ dvferm . (Con... |
dvferm2 25151 | One-sided version of ~ dvf... |
dvferm 25152 | Fermat's theorem on statio... |
rollelem 25153 | Lemma for ~ rolle . (Cont... |
rolle 25154 | Rolle's theorem. If ` F `... |
cmvth 25155 | Cauchy's Mean Value Theore... |
mvth 25156 | The Mean Value Theorem. I... |
dvlip 25157 | A function with derivative... |
dvlipcn 25158 | A complex function with de... |
dvlip2 25159 | Combine the results of ~ d... |
c1liplem1 25160 | Lemma for ~ c1lip1 . (Con... |
c1lip1 25161 | C^1 functions are Lipschit... |
c1lip2 25162 | C^1 functions are Lipschit... |
c1lip3 25163 | C^1 functions are Lipschit... |
dveq0 25164 | If a continuous function h... |
dv11cn 25165 | Two functions defined on a... |
dvgt0lem1 25166 | Lemma for ~ dvgt0 and ~ dv... |
dvgt0lem2 25167 | Lemma for ~ dvgt0 and ~ dv... |
dvgt0 25168 | A function on a closed int... |
dvlt0 25169 | A function on a closed int... |
dvge0 25170 | A function on a closed int... |
dvle 25171 | If ` A ( x ) , C ( x ) ` a... |
dvivthlem1 25172 | Lemma for ~ dvivth . (Con... |
dvivthlem2 25173 | Lemma for ~ dvivth . (Con... |
dvivth 25174 | Darboux' theorem, or the i... |
dvne0 25175 | A function on a closed int... |
dvne0f1 25176 | A function on a closed int... |
lhop1lem 25177 | Lemma for ~ lhop1 . (Cont... |
lhop1 25178 | L'Hôpital's Rule for... |
lhop2 25179 | L'Hôpital's Rule for... |
lhop 25180 | L'Hôpital's Rule. I... |
dvcnvrelem1 25181 | Lemma for ~ dvcnvre . (Co... |
dvcnvrelem2 25182 | Lemma for ~ dvcnvre . (Co... |
dvcnvre 25183 | The derivative rule for in... |
dvcvx 25184 | A real function with stric... |
dvfsumle 25185 | Compare a finite sum to an... |
dvfsumge 25186 | Compare a finite sum to an... |
dvfsumabs 25187 | Compare a finite sum to an... |
dvmptrecl 25188 | Real closure of a derivati... |
dvfsumrlimf 25189 | Lemma for ~ dvfsumrlim . ... |
dvfsumlem1 25190 | Lemma for ~ dvfsumrlim . ... |
dvfsumlem2 25191 | Lemma for ~ dvfsumrlim . ... |
dvfsumlem3 25192 | Lemma for ~ dvfsumrlim . ... |
dvfsumlem4 25193 | Lemma for ~ dvfsumrlim . ... |
dvfsumrlimge0 25194 | Lemma for ~ dvfsumrlim . ... |
dvfsumrlim 25195 | Compare a finite sum to an... |
dvfsumrlim2 25196 | Compare a finite sum to an... |
dvfsumrlim3 25197 | Conjoin the statements of ... |
dvfsum2 25198 | The reverse of ~ dvfsumrli... |
ftc1lem1 25199 | Lemma for ~ ftc1a and ~ ft... |
ftc1lem2 25200 | Lemma for ~ ftc1 . (Contr... |
ftc1a 25201 | The Fundamental Theorem of... |
ftc1lem3 25202 | Lemma for ~ ftc1 . (Contr... |
ftc1lem4 25203 | Lemma for ~ ftc1 . (Contr... |
ftc1lem5 25204 | Lemma for ~ ftc1 . (Contr... |
ftc1lem6 25205 | Lemma for ~ ftc1 . (Contr... |
ftc1 25206 | The Fundamental Theorem of... |
ftc1cn 25207 | Strengthen the assumptions... |
ftc2 25208 | The Fundamental Theorem of... |
ftc2ditglem 25209 | Lemma for ~ ftc2ditg . (C... |
ftc2ditg 25210 | Directed integral analogue... |
itgparts 25211 | Integration by parts. If ... |
itgsubstlem 25212 | Lemma for ~ itgsubst . (C... |
itgsubst 25213 | Integration by ` u ` -subs... |
itgpowd 25214 | The integral of a monomial... |
reldmmdeg 25219 | Multivariate degree is a b... |
tdeglem1 25220 | Functionality of the total... |
tdeglem1OLD 25221 | Obsolete version of ~ tdeg... |
tdeglem3 25222 | Additivity of the total de... |
tdeglem3OLD 25223 | Obsolete version of ~ tdeg... |
tdeglem4 25224 | There is only one multi-in... |
tdeglem4OLD 25225 | Obsolete version of ~ tdeg... |
tdeglem2 25226 | Simplification of total de... |
mdegfval 25227 | Value of the multivariate ... |
mdegval 25228 | Value of the multivariate ... |
mdegleb 25229 | Property of being of limit... |
mdeglt 25230 | If there is an upper limit... |
mdegldg 25231 | A nonzero polynomial has s... |
mdegxrcl 25232 | Closure of polynomial degr... |
mdegxrf 25233 | Functionality of polynomia... |
mdegcl 25234 | Sharp closure for multivar... |
mdeg0 25235 | Degree of the zero polynom... |
mdegnn0cl 25236 | Degree of a nonzero polyno... |
degltlem1 25237 | Theorem on arithmetic of e... |
degltp1le 25238 | Theorem on arithmetic of e... |
mdegaddle 25239 | The degree of a sum is at ... |
mdegvscale 25240 | The degree of a scalar mul... |
mdegvsca 25241 | The degree of a scalar mul... |
mdegle0 25242 | A polynomial has nonpositi... |
mdegmullem 25243 | Lemma for ~ mdegmulle2 . ... |
mdegmulle2 25244 | The multivariate degree of... |
deg1fval 25245 | Relate univariate polynomi... |
deg1xrf 25246 | Functionality of univariat... |
deg1xrcl 25247 | Closure of univariate poly... |
deg1cl 25248 | Sharp closure of univariat... |
mdegpropd 25249 | Property deduction for pol... |
deg1fvi 25250 | Univariate polynomial degr... |
deg1propd 25251 | Property deduction for pol... |
deg1z 25252 | Degree of the zero univari... |
deg1nn0cl 25253 | Degree of a nonzero univar... |
deg1n0ima 25254 | Degree image of a set of p... |
deg1nn0clb 25255 | A polynomial is nonzero if... |
deg1lt0 25256 | A polynomial is zero iff i... |
deg1ldg 25257 | A nonzero univariate polyn... |
deg1ldgn 25258 | An index at which a polyno... |
deg1ldgdomn 25259 | A nonzero univariate polyn... |
deg1leb 25260 | Property of being of limit... |
deg1val 25261 | Value of the univariate de... |
deg1lt 25262 | If the degree of a univari... |
deg1ge 25263 | Conversely, a nonzero coef... |
coe1mul3 25264 | The coefficient vector of ... |
coe1mul4 25265 | Value of the "leading" coe... |
deg1addle 25266 | The degree of a sum is at ... |
deg1addle2 25267 | If both factors have degre... |
deg1add 25268 | Exact degree of a sum of t... |
deg1vscale 25269 | The degree of a scalar tim... |
deg1vsca 25270 | The degree of a scalar tim... |
deg1invg 25271 | The degree of the negated ... |
deg1suble 25272 | The degree of a difference... |
deg1sub 25273 | Exact degree of a differen... |
deg1mulle2 25274 | Produce a bound on the pro... |
deg1sublt 25275 | Subtraction of two polynom... |
deg1le0 25276 | A polynomial has nonpositi... |
deg1sclle 25277 | A scalar polynomial has no... |
deg1scl 25278 | A nonzero scalar polynomia... |
deg1mul2 25279 | Degree of multiplication o... |
deg1mul3 25280 | Degree of multiplication o... |
deg1mul3le 25281 | Degree of multiplication o... |
deg1tmle 25282 | Limiting degree of a polyn... |
deg1tm 25283 | Exact degree of a polynomi... |
deg1pwle 25284 | Limiting degree of a varia... |
deg1pw 25285 | Exact degree of a variable... |
ply1nz 25286 | Univariate polynomials ove... |
ply1nzb 25287 | Univariate polynomials are... |
ply1domn 25288 | Corollary of ~ deg1mul2 : ... |
ply1idom 25289 | The ring of univariate pol... |
ply1divmo 25300 | Uniqueness of a quotient i... |
ply1divex 25301 | Lemma for ~ ply1divalg : e... |
ply1divalg 25302 | The division algorithm for... |
ply1divalg2 25303 | Reverse the order of multi... |
uc1pval 25304 | Value of the set of unitic... |
isuc1p 25305 | Being a unitic polynomial.... |
mon1pval 25306 | Value of the set of monic ... |
ismon1p 25307 | Being a monic polynomial. ... |
uc1pcl 25308 | Unitic polynomials are pol... |
mon1pcl 25309 | Monic polynomials are poly... |
uc1pn0 25310 | Unitic polynomials are not... |
mon1pn0 25311 | Monic polynomials are not ... |
uc1pdeg 25312 | Unitic polynomials have no... |
uc1pldg 25313 | Unitic polynomials have un... |
mon1pldg 25314 | Unitic polynomials have on... |
mon1puc1p 25315 | Monic polynomials are unit... |
uc1pmon1p 25316 | Make a unitic polynomial m... |
deg1submon1p 25317 | The difference of two moni... |
q1pval 25318 | Value of the univariate po... |
q1peqb 25319 | Characterizing property of... |
q1pcl 25320 | Closure of the quotient by... |
r1pval 25321 | Value of the polynomial re... |
r1pcl 25322 | Closure of remainder follo... |
r1pdeglt 25323 | The remainder has a degree... |
r1pid 25324 | Express the original polyn... |
dvdsq1p 25325 | Divisibility in a polynomi... |
dvdsr1p 25326 | Divisibility in a polynomi... |
ply1remlem 25327 | A term of the form ` x - N... |
ply1rem 25328 | The polynomial remainder t... |
facth1 25329 | The factor theorem and its... |
fta1glem1 25330 | Lemma for ~ fta1g . (Cont... |
fta1glem2 25331 | Lemma for ~ fta1g . (Cont... |
fta1g 25332 | The one-sided fundamental ... |
fta1blem 25333 | Lemma for ~ fta1b . (Cont... |
fta1b 25334 | The assumption that ` R ` ... |
drnguc1p 25335 | Over a division ring, all ... |
ig1peu 25336 | There is a unique monic po... |
ig1pval 25337 | Substitutions for the poly... |
ig1pval2 25338 | Generator of the zero idea... |
ig1pval3 25339 | Characterizing properties ... |
ig1pcl 25340 | The monic generator of an ... |
ig1pdvds 25341 | The monic generator of an ... |
ig1prsp 25342 | Any ideal of polynomials o... |
ply1lpir 25343 | The ring of polynomials ov... |
ply1pid 25344 | The polynomials over a fie... |
plyco0 25353 | Two ways to say that a fun... |
plyval 25354 | Value of the polynomial se... |
plybss 25355 | Reverse closure of the par... |
elply 25356 | Definition of a polynomial... |
elply2 25357 | The coefficient function c... |
plyun0 25358 | The set of polynomials is ... |
plyf 25359 | The polynomial is a functi... |
plyss 25360 | The polynomial set functio... |
plyssc 25361 | Every polynomial ring is c... |
elplyr 25362 | Sufficient condition for e... |
elplyd 25363 | Sufficient condition for e... |
ply1termlem 25364 | Lemma for ~ ply1term . (C... |
ply1term 25365 | A one-term polynomial. (C... |
plypow 25366 | A power is a polynomial. ... |
plyconst 25367 | A constant function is a p... |
ne0p 25368 | A test to show that a poly... |
ply0 25369 | The zero function is a pol... |
plyid 25370 | The identity function is a... |
plyeq0lem 25371 | Lemma for ~ plyeq0 . If `... |
plyeq0 25372 | If a polynomial is zero at... |
plypf1 25373 | Write the set of complex p... |
plyaddlem1 25374 | Derive the coefficient fun... |
plymullem1 25375 | Derive the coefficient fun... |
plyaddlem 25376 | Lemma for ~ plyadd . (Con... |
plymullem 25377 | Lemma for ~ plymul . (Con... |
plyadd 25378 | The sum of two polynomials... |
plymul 25379 | The product of two polynom... |
plysub 25380 | The difference of two poly... |
plyaddcl 25381 | The sum of two polynomials... |
plymulcl 25382 | The product of two polynom... |
plysubcl 25383 | The difference of two poly... |
coeval 25384 | Value of the coefficient f... |
coeeulem 25385 | Lemma for ~ coeeu . (Cont... |
coeeu 25386 | Uniqueness of the coeffici... |
coelem 25387 | Lemma for properties of th... |
coeeq 25388 | If ` A ` satisfies the pro... |
dgrval 25389 | Value of the degree functi... |
dgrlem 25390 | Lemma for ~ dgrcl and simi... |
coef 25391 | The domain and range of th... |
coef2 25392 | The domain and range of th... |
coef3 25393 | The domain and range of th... |
dgrcl 25394 | The degree of any polynomi... |
dgrub 25395 | If the ` M ` -th coefficie... |
dgrub2 25396 | All the coefficients above... |
dgrlb 25397 | If all the coefficients ab... |
coeidlem 25398 | Lemma for ~ coeid . (Cont... |
coeid 25399 | Reconstruct a polynomial a... |
coeid2 25400 | Reconstruct a polynomial a... |
coeid3 25401 | Reconstruct a polynomial a... |
plyco 25402 | The composition of two pol... |
coeeq2 25403 | Compute the coefficient fu... |
dgrle 25404 | Given an explicit expressi... |
dgreq 25405 | If the highest term in a p... |
0dgr 25406 | A constant function has de... |
0dgrb 25407 | A function has degree zero... |
dgrnznn 25408 | A nonzero polynomial with ... |
coefv0 25409 | The result of evaluating a... |
coeaddlem 25410 | Lemma for ~ coeadd and ~ d... |
coemullem 25411 | Lemma for ~ coemul and ~ d... |
coeadd 25412 | The coefficient function o... |
coemul 25413 | A coefficient of a product... |
coe11 25414 | The coefficient function i... |
coemulhi 25415 | The leading coefficient of... |
coemulc 25416 | The coefficient function i... |
coe0 25417 | The coefficients of the ze... |
coesub 25418 | The coefficient function o... |
coe1termlem 25419 | The coefficient function o... |
coe1term 25420 | The coefficient function o... |
dgr1term 25421 | The degree of a monomial. ... |
plycn 25422 | A polynomial is a continuo... |
dgr0 25423 | The degree of the zero pol... |
coeidp 25424 | The coefficients of the id... |
dgrid 25425 | The degree of the identity... |
dgreq0 25426 | The leading coefficient of... |
dgrlt 25427 | Two ways to say that the d... |
dgradd 25428 | The degree of a sum of pol... |
dgradd2 25429 | The degree of a sum of pol... |
dgrmul2 25430 | The degree of a product of... |
dgrmul 25431 | The degree of a product of... |
dgrmulc 25432 | Scalar multiplication by a... |
dgrsub 25433 | The degree of a difference... |
dgrcolem1 25434 | The degree of a compositio... |
dgrcolem2 25435 | Lemma for ~ dgrco . (Cont... |
dgrco 25436 | The degree of a compositio... |
plycjlem 25437 | Lemma for ~ plycj and ~ co... |
plycj 25438 | The double conjugation of ... |
coecj 25439 | Double conjugation of a po... |
plyrecj 25440 | A polynomial with real coe... |
plymul0or 25441 | Polynomial multiplication ... |
ofmulrt 25442 | The set of roots of a prod... |
plyreres 25443 | Real-coefficient polynomia... |
dvply1 25444 | Derivative of a polynomial... |
dvply2g 25445 | The derivative of a polyno... |
dvply2 25446 | The derivative of a polyno... |
dvnply2 25447 | Polynomials have polynomia... |
dvnply 25448 | Polynomials have polynomia... |
plycpn 25449 | Polynomials are smooth. (... |
quotval 25452 | Value of the quotient func... |
plydivlem1 25453 | Lemma for ~ plydivalg . (... |
plydivlem2 25454 | Lemma for ~ plydivalg . (... |
plydivlem3 25455 | Lemma for ~ plydivex . Ba... |
plydivlem4 25456 | Lemma for ~ plydivex . In... |
plydivex 25457 | Lemma for ~ plydivalg . (... |
plydiveu 25458 | Lemma for ~ plydivalg . (... |
plydivalg 25459 | The division algorithm on ... |
quotlem 25460 | Lemma for properties of th... |
quotcl 25461 | The quotient of two polyno... |
quotcl2 25462 | Closure of the quotient fu... |
quotdgr 25463 | Remainder property of the ... |
plyremlem 25464 | Closure of a linear factor... |
plyrem 25465 | The polynomial remainder t... |
facth 25466 | The factor theorem. If a ... |
fta1lem 25467 | Lemma for ~ fta1 . (Contr... |
fta1 25468 | The easy direction of the ... |
quotcan 25469 | Exact division with a mult... |
vieta1lem1 25470 | Lemma for ~ vieta1 . (Con... |
vieta1lem2 25471 | Lemma for ~ vieta1 : induc... |
vieta1 25472 | The first-order Vieta's fo... |
plyexmo 25473 | An infinite set of values ... |
elaa 25476 | Elementhood in the set of ... |
aacn 25477 | An algebraic number is a c... |
aasscn 25478 | The algebraic numbers are ... |
elqaalem1 25479 | Lemma for ~ elqaa . The f... |
elqaalem2 25480 | Lemma for ~ elqaa . (Cont... |
elqaalem3 25481 | Lemma for ~ elqaa . (Cont... |
elqaa 25482 | The set of numbers generat... |
qaa 25483 | Every rational number is a... |
qssaa 25484 | The rational numbers are c... |
iaa 25485 | The imaginary unit is alge... |
aareccl 25486 | The reciprocal of an algeb... |
aacjcl 25487 | The conjugate of an algebr... |
aannenlem1 25488 | Lemma for ~ aannen . (Con... |
aannenlem2 25489 | Lemma for ~ aannen . (Con... |
aannenlem3 25490 | The algebraic numbers are ... |
aannen 25491 | The algebraic numbers are ... |
aalioulem1 25492 | Lemma for ~ aaliou . An i... |
aalioulem2 25493 | Lemma for ~ aaliou . (Con... |
aalioulem3 25494 | Lemma for ~ aaliou . (Con... |
aalioulem4 25495 | Lemma for ~ aaliou . (Con... |
aalioulem5 25496 | Lemma for ~ aaliou . (Con... |
aalioulem6 25497 | Lemma for ~ aaliou . (Con... |
aaliou 25498 | Liouville's theorem on dio... |
geolim3 25499 | Geometric series convergen... |
aaliou2 25500 | Liouville's approximation ... |
aaliou2b 25501 | Liouville's approximation ... |
aaliou3lem1 25502 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem2 25503 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem3 25504 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem8 25505 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem4 25506 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem5 25507 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem6 25508 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem7 25509 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem9 25510 | Example of a "Liouville nu... |
aaliou3 25511 | Example of a "Liouville nu... |
taylfvallem1 25516 | Lemma for ~ taylfval . (C... |
taylfvallem 25517 | Lemma for ~ taylfval . (C... |
taylfval 25518 | Define the Taylor polynomi... |
eltayl 25519 | Value of the Taylor series... |
taylf 25520 | The Taylor series defines ... |
tayl0 25521 | The Taylor series is alway... |
taylplem1 25522 | Lemma for ~ taylpfval and ... |
taylplem2 25523 | Lemma for ~ taylpfval and ... |
taylpfval 25524 | Define the Taylor polynomi... |
taylpf 25525 | The Taylor polynomial is a... |
taylpval 25526 | Value of the Taylor polyno... |
taylply2 25527 | The Taylor polynomial is a... |
taylply 25528 | The Taylor polynomial is a... |
dvtaylp 25529 | The derivative of the Tayl... |
dvntaylp 25530 | The ` M ` -th derivative o... |
dvntaylp0 25531 | The first ` N ` derivative... |
taylthlem1 25532 | Lemma for ~ taylth . This... |
taylthlem2 25533 | Lemma for ~ taylth . (Con... |
taylth 25534 | Taylor's theorem. The Tay... |
ulmrel 25537 | The uniform limit relation... |
ulmscl 25538 | Closure of the base set in... |
ulmval 25539 | Express the predicate: Th... |
ulmcl 25540 | Closure of a uniform limit... |
ulmf 25541 | Closure of a uniform limit... |
ulmpm 25542 | Closure of a uniform limit... |
ulmf2 25543 | Closure of a uniform limit... |
ulm2 25544 | Simplify ~ ulmval when ` F... |
ulmi 25545 | The uniform limit property... |
ulmclm 25546 | A uniform limit of functio... |
ulmres 25547 | A sequence of functions co... |
ulmshftlem 25548 | Lemma for ~ ulmshft . (Co... |
ulmshft 25549 | A sequence of functions co... |
ulm0 25550 | Every function converges u... |
ulmuni 25551 | A sequence of functions un... |
ulmdm 25552 | Two ways to express that a... |
ulmcaulem 25553 | Lemma for ~ ulmcau and ~ u... |
ulmcau 25554 | A sequence of functions co... |
ulmcau2 25555 | A sequence of functions co... |
ulmss 25556 | A uniform limit of functio... |
ulmbdd 25557 | A uniform limit of bounded... |
ulmcn 25558 | A uniform limit of continu... |
ulmdvlem1 25559 | Lemma for ~ ulmdv . (Cont... |
ulmdvlem2 25560 | Lemma for ~ ulmdv . (Cont... |
ulmdvlem3 25561 | Lemma for ~ ulmdv . (Cont... |
ulmdv 25562 | If ` F ` is a sequence of ... |
mtest 25563 | The Weierstrass M-test. I... |
mtestbdd 25564 | Given the hypotheses of th... |
mbfulm 25565 | A uniform limit of measura... |
iblulm 25566 | A uniform limit of integra... |
itgulm 25567 | A uniform limit of integra... |
itgulm2 25568 | A uniform limit of integra... |
pserval 25569 | Value of the function ` G ... |
pserval2 25570 | Value of the function ` G ... |
psergf 25571 | The sequence of terms in t... |
radcnvlem1 25572 | Lemma for ~ radcnvlt1 , ~ ... |
radcnvlem2 25573 | Lemma for ~ radcnvlt1 , ~ ... |
radcnvlem3 25574 | Lemma for ~ radcnvlt1 , ~ ... |
radcnv0 25575 | Zero is always a convergen... |
radcnvcl 25576 | The radius of convergence ... |
radcnvlt1 25577 | If ` X ` is within the ope... |
radcnvlt2 25578 | If ` X ` is within the ope... |
radcnvle 25579 | If ` X ` is a convergent p... |
dvradcnv 25580 | The radius of convergence ... |
pserulm 25581 | If ` S ` is a region conta... |
psercn2 25582 | Since by ~ pserulm the ser... |
psercnlem2 25583 | Lemma for ~ psercn . (Con... |
psercnlem1 25584 | Lemma for ~ psercn . (Con... |
psercn 25585 | An infinite series converg... |
pserdvlem1 25586 | Lemma for ~ pserdv . (Con... |
pserdvlem2 25587 | Lemma for ~ pserdv . (Con... |
pserdv 25588 | The derivative of a power ... |
pserdv2 25589 | The derivative of a power ... |
abelthlem1 25590 | Lemma for ~ abelth . (Con... |
abelthlem2 25591 | Lemma for ~ abelth . The ... |
abelthlem3 25592 | Lemma for ~ abelth . (Con... |
abelthlem4 25593 | Lemma for ~ abelth . (Con... |
abelthlem5 25594 | Lemma for ~ abelth . (Con... |
abelthlem6 25595 | Lemma for ~ abelth . (Con... |
abelthlem7a 25596 | Lemma for ~ abelth . (Con... |
abelthlem7 25597 | Lemma for ~ abelth . (Con... |
abelthlem8 25598 | Lemma for ~ abelth . (Con... |
abelthlem9 25599 | Lemma for ~ abelth . By a... |
abelth 25600 | Abel's theorem. If the po... |
abelth2 25601 | Abel's theorem, restricted... |
efcn 25602 | The exponential function i... |
sincn 25603 | Sine is continuous. (Cont... |
coscn 25604 | Cosine is continuous. (Co... |
reeff1olem 25605 | Lemma for ~ reeff1o . (Co... |
reeff1o 25606 | The real exponential funct... |
reefiso 25607 | The exponential function o... |
efcvx 25608 | The exponential function o... |
reefgim 25609 | The exponential function i... |
pilem1 25610 | Lemma for ~ pire , ~ pigt2... |
pilem2 25611 | Lemma for ~ pire , ~ pigt2... |
pilem3 25612 | Lemma for ~ pire , ~ pigt2... |
pigt2lt4 25613 | ` _pi ` is between 2 and 4... |
sinpi 25614 | The sine of ` _pi ` is 0. ... |
pire 25615 | ` _pi ` is a real number. ... |
picn 25616 | ` _pi ` is a complex numbe... |
pipos 25617 | ` _pi ` is positive. (Con... |
pirp 25618 | ` _pi ` is a positive real... |
negpicn 25619 | ` -u _pi ` is a real numbe... |
sinhalfpilem 25620 | Lemma for ~ sinhalfpi and ... |
halfpire 25621 | ` _pi / 2 ` is real. (Con... |
neghalfpire 25622 | ` -u _pi / 2 ` is real. (... |
neghalfpirx 25623 | ` -u _pi / 2 ` is an exten... |
pidiv2halves 25624 | Adding ` _pi / 2 ` to itse... |
sinhalfpi 25625 | The sine of ` _pi / 2 ` is... |
coshalfpi 25626 | The cosine of ` _pi / 2 ` ... |
cosneghalfpi 25627 | The cosine of ` -u _pi / 2... |
efhalfpi 25628 | The exponential of ` _i _p... |
cospi 25629 | The cosine of ` _pi ` is `... |
efipi 25630 | The exponential of ` _i x.... |
eulerid 25631 | Euler's identity. (Contri... |
sin2pi 25632 | The sine of ` 2 _pi ` is 0... |
cos2pi 25633 | The cosine of ` 2 _pi ` is... |
ef2pi 25634 | The exponential of ` 2 _pi... |
ef2kpi 25635 | If ` K ` is an integer, th... |
efper 25636 | The exponential function i... |
sinperlem 25637 | Lemma for ~ sinper and ~ c... |
sinper 25638 | The sine function is perio... |
cosper 25639 | The cosine function is per... |
sin2kpi 25640 | If ` K ` is an integer, th... |
cos2kpi 25641 | If ` K ` is an integer, th... |
sin2pim 25642 | Sine of a number subtracte... |
cos2pim 25643 | Cosine of a number subtrac... |
sinmpi 25644 | Sine of a number less ` _p... |
cosmpi 25645 | Cosine of a number less ` ... |
sinppi 25646 | Sine of a number plus ` _p... |
cosppi 25647 | Cosine of a number plus ` ... |
efimpi 25648 | The exponential function a... |
sinhalfpip 25649 | The sine of ` _pi / 2 ` pl... |
sinhalfpim 25650 | The sine of ` _pi / 2 ` mi... |
coshalfpip 25651 | The cosine of ` _pi / 2 ` ... |
coshalfpim 25652 | The cosine of ` _pi / 2 ` ... |
ptolemy 25653 | Ptolemy's Theorem. This t... |
sincosq1lem 25654 | Lemma for ~ sincosq1sgn . ... |
sincosq1sgn 25655 | The signs of the sine and ... |
sincosq2sgn 25656 | The signs of the sine and ... |
sincosq3sgn 25657 | The signs of the sine and ... |
sincosq4sgn 25658 | The signs of the sine and ... |
coseq00topi 25659 | Location of the zeroes of ... |
coseq0negpitopi 25660 | Location of the zeroes of ... |
tanrpcl 25661 | Positive real closure of t... |
tangtx 25662 | The tangent function is gr... |
tanabsge 25663 | The tangent function is gr... |
sinq12gt0 25664 | The sine of a number stric... |
sinq12ge0 25665 | The sine of a number betwe... |
sinq34lt0t 25666 | The sine of a number stric... |
cosq14gt0 25667 | The cosine of a number str... |
cosq14ge0 25668 | The cosine of a number bet... |
sincosq1eq 25669 | Complementarity of the sin... |
sincos4thpi 25670 | The sine and cosine of ` _... |
tan4thpi 25671 | The tangent of ` _pi / 4 `... |
sincos6thpi 25672 | The sine and cosine of ` _... |
sincos3rdpi 25673 | The sine and cosine of ` _... |
pigt3 25674 | ` _pi ` is greater than 3.... |
pige3 25675 | ` _pi ` is greater than or... |
pige3ALT 25676 | Alternate proof of ~ pige3... |
abssinper 25677 | The absolute value of sine... |
sinkpi 25678 | The sine of an integer mul... |
coskpi 25679 | The absolute value of the ... |
sineq0 25680 | A complex number whose sin... |
coseq1 25681 | A complex number whose cos... |
cos02pilt1 25682 | Cosine is less than one be... |
cosq34lt1 25683 | Cosine is less than one in... |
efeq1 25684 | A complex number whose exp... |
cosne0 25685 | The cosine function has no... |
cosordlem 25686 | Lemma for ~ cosord . (Con... |
cosord 25687 | Cosine is decreasing over ... |
cos0pilt1 25688 | Cosine is between minus on... |
cos11 25689 | Cosine is one-to-one over ... |
sinord 25690 | Sine is increasing over th... |
recosf1o 25691 | The cosine function is a b... |
resinf1o 25692 | The sine function is a bij... |
tanord1 25693 | The tangent function is st... |
tanord 25694 | The tangent function is st... |
tanregt0 25695 | The real part of the tange... |
negpitopissre 25696 | The interval ` ( -u _pi (,... |
efgh 25697 | The exponential function o... |
efif1olem1 25698 | Lemma for ~ efif1o . (Con... |
efif1olem2 25699 | Lemma for ~ efif1o . (Con... |
efif1olem3 25700 | Lemma for ~ efif1o . (Con... |
efif1olem4 25701 | The exponential function o... |
efif1o 25702 | The exponential function o... |
efifo 25703 | The exponential function o... |
eff1olem 25704 | The exponential function m... |
eff1o 25705 | The exponential function m... |
efabl 25706 | The image of a subgroup of... |
efsubm 25707 | The image of a subgroup of... |
circgrp 25708 | The circle group ` T ` is ... |
circsubm 25709 | The circle group ` T ` is ... |
logrn 25714 | The range of the natural l... |
ellogrn 25715 | Write out the property ` A... |
dflog2 25716 | The natural logarithm func... |
relogrn 25717 | The range of the natural l... |
logrncn 25718 | The range of the natural l... |
eff1o2 25719 | The exponential function r... |
logf1o 25720 | The natural logarithm func... |
dfrelog 25721 | The natural logarithm func... |
relogf1o 25722 | The natural logarithm func... |
logrncl 25723 | Closure of the natural log... |
logcl 25724 | Closure of the natural log... |
logimcl 25725 | Closure of the imaginary p... |
logcld 25726 | The logarithm of a nonzero... |
logimcld 25727 | The imaginary part of the ... |
logimclad 25728 | The imaginary part of the ... |
abslogimle 25729 | The imaginary part of the ... |
logrnaddcl 25730 | The range of the natural l... |
relogcl 25731 | Closure of the natural log... |
eflog 25732 | Relationship between the n... |
logeq0im1 25733 | If the logarithm of a numb... |
logccne0 25734 | The logarithm isn't 0 if i... |
logne0 25735 | Logarithm of a non-1 posit... |
reeflog 25736 | Relationship between the n... |
logef 25737 | Relationship between the n... |
relogef 25738 | Relationship between the n... |
logeftb 25739 | Relationship between the n... |
relogeftb 25740 | Relationship between the n... |
log1 25741 | The natural logarithm of `... |
loge 25742 | The natural logarithm of `... |
logneg 25743 | The natural logarithm of a... |
logm1 25744 | The natural logarithm of n... |
lognegb 25745 | If a number has imaginary ... |
relogoprlem 25746 | Lemma for ~ relogmul and ~... |
relogmul 25747 | The natural logarithm of t... |
relogdiv 25748 | The natural logarithm of t... |
explog 25749 | Exponentiation of a nonzer... |
reexplog 25750 | Exponentiation of a positi... |
relogexp 25751 | The natural logarithm of p... |
relog 25752 | Real part of a logarithm. ... |
relogiso 25753 | The natural logarithm func... |
reloggim 25754 | The natural logarithm is a... |
logltb 25755 | The natural logarithm func... |
logfac 25756 | The logarithm of a factori... |
eflogeq 25757 | Solve an equation involvin... |
logleb 25758 | Natural logarithm preserve... |
rplogcl 25759 | Closure of the logarithm f... |
logge0 25760 | The logarithm of a number ... |
logcj 25761 | The natural logarithm dist... |
efiarg 25762 | The exponential of the "ar... |
cosargd 25763 | The cosine of the argument... |
cosarg0d 25764 | The cosine of the argument... |
argregt0 25765 | Closure of the argument of... |
argrege0 25766 | Closure of the argument of... |
argimgt0 25767 | Closure of the argument of... |
argimlt0 25768 | Closure of the argument of... |
logimul 25769 | Multiplying a number by ` ... |
logneg2 25770 | The logarithm of the negat... |
logmul2 25771 | Generalization of ~ relogm... |
logdiv2 25772 | Generalization of ~ relogd... |
abslogle 25773 | Bound on the magnitude of ... |
tanarg 25774 | The basic relation between... |
logdivlti 25775 | The ` log x / x ` function... |
logdivlt 25776 | The ` log x / x ` function... |
logdivle 25777 | The ` log x / x ` function... |
relogcld 25778 | Closure of the natural log... |
reeflogd 25779 | Relationship between the n... |
relogmuld 25780 | The natural logarithm of t... |
relogdivd 25781 | The natural logarithm of t... |
logled 25782 | Natural logarithm preserve... |
relogefd 25783 | Relationship between the n... |
rplogcld 25784 | Closure of the logarithm f... |
logge0d 25785 | The logarithm of a number ... |
logge0b 25786 | The logarithm of a number ... |
loggt0b 25787 | The logarithm of a number ... |
logle1b 25788 | The logarithm of a number ... |
loglt1b 25789 | The logarithm of a number ... |
divlogrlim 25790 | The inverse logarithm func... |
logno1 25791 | The logarithm function is ... |
dvrelog 25792 | The derivative of the real... |
relogcn 25793 | The real logarithm functio... |
ellogdm 25794 | Elementhood in the "contin... |
logdmn0 25795 | A number in the continuous... |
logdmnrp 25796 | A number in the continuous... |
logdmss 25797 | The continuity domain of `... |
logcnlem2 25798 | Lemma for ~ logcn . (Cont... |
logcnlem3 25799 | Lemma for ~ logcn . (Cont... |
logcnlem4 25800 | Lemma for ~ logcn . (Cont... |
logcnlem5 25801 | Lemma for ~ logcn . (Cont... |
logcn 25802 | The logarithm function is ... |
dvloglem 25803 | Lemma for ~ dvlog . (Cont... |
logdmopn 25804 | The "continuous domain" of... |
logf1o2 25805 | The logarithm maps its con... |
dvlog 25806 | The derivative of the comp... |
dvlog2lem 25807 | Lemma for ~ dvlog2 . (Con... |
dvlog2 25808 | The derivative of the comp... |
advlog 25809 | The antiderivative of the ... |
advlogexp 25810 | The antiderivative of a po... |
efopnlem1 25811 | Lemma for ~ efopn . (Cont... |
efopnlem2 25812 | Lemma for ~ efopn . (Cont... |
efopn 25813 | The exponential map is an ... |
logtayllem 25814 | Lemma for ~ logtayl . (Co... |
logtayl 25815 | The Taylor series for ` -u... |
logtaylsum 25816 | The Taylor series for ` -u... |
logtayl2 25817 | Power series expression fo... |
logccv 25818 | The natural logarithm func... |
cxpval 25819 | Value of the complex power... |
cxpef 25820 | Value of the complex power... |
0cxp 25821 | Value of the complex power... |
cxpexpz 25822 | Relate the complex power f... |
cxpexp 25823 | Relate the complex power f... |
logcxp 25824 | Logarithm of a complex pow... |
cxp0 25825 | Value of the complex power... |
cxp1 25826 | Value of the complex power... |
1cxp 25827 | Value of the complex power... |
ecxp 25828 | Write the exponential func... |
cxpcl 25829 | Closure of the complex pow... |
recxpcl 25830 | Real closure of the comple... |
rpcxpcl 25831 | Positive real closure of t... |
cxpne0 25832 | Complex exponentiation is ... |
cxpeq0 25833 | Complex exponentiation is ... |
cxpadd 25834 | Sum of exponents law for c... |
cxpp1 25835 | Value of a nonzero complex... |
cxpneg 25836 | Value of a complex number ... |
cxpsub 25837 | Exponent subtraction law f... |
cxpge0 25838 | Nonnegative exponentiation... |
mulcxplem 25839 | Lemma for ~ mulcxp . (Con... |
mulcxp 25840 | Complex exponentiation of ... |
cxprec 25841 | Complex exponentiation of ... |
divcxp 25842 | Complex exponentiation of ... |
cxpmul 25843 | Product of exponents law f... |
cxpmul2 25844 | Product of exponents law f... |
cxproot 25845 | The complex power function... |
cxpmul2z 25846 | Generalize ~ cxpmul2 to ne... |
abscxp 25847 | Absolute value of a power,... |
abscxp2 25848 | Absolute value of a power,... |
cxplt 25849 | Ordering property for comp... |
cxple 25850 | Ordering property for comp... |
cxplea 25851 | Ordering property for comp... |
cxple2 25852 | Ordering property for comp... |
cxplt2 25853 | Ordering property for comp... |
cxple2a 25854 | Ordering property for comp... |
cxplt3 25855 | Ordering property for comp... |
cxple3 25856 | Ordering property for comp... |
cxpsqrtlem 25857 | Lemma for ~ cxpsqrt . (Co... |
cxpsqrt 25858 | The complex exponential fu... |
logsqrt 25859 | Logarithm of a square root... |
cxp0d 25860 | Value of the complex power... |
cxp1d 25861 | Value of the complex power... |
1cxpd 25862 | Value of the complex power... |
cxpcld 25863 | Closure of the complex pow... |
cxpmul2d 25864 | Product of exponents law f... |
0cxpd 25865 | Value of the complex power... |
cxpexpzd 25866 | Relate the complex power f... |
cxpefd 25867 | Value of the complex power... |
cxpne0d 25868 | Complex exponentiation is ... |
cxpp1d 25869 | Value of a nonzero complex... |
cxpnegd 25870 | Value of a complex number ... |
cxpmul2zd 25871 | Generalize ~ cxpmul2 to ne... |
cxpaddd 25872 | Sum of exponents law for c... |
cxpsubd 25873 | Exponent subtraction law f... |
cxpltd 25874 | Ordering property for comp... |
cxpled 25875 | Ordering property for comp... |
cxplead 25876 | Ordering property for comp... |
divcxpd 25877 | Complex exponentiation of ... |
recxpcld 25878 | Positive real closure of t... |
cxpge0d 25879 | Nonnegative exponentiation... |
cxple2ad 25880 | Ordering property for comp... |
cxplt2d 25881 | Ordering property for comp... |
cxple2d 25882 | Ordering property for comp... |
mulcxpd 25883 | Complex exponentiation of ... |
cxpsqrtth 25884 | Square root theorem over t... |
2irrexpq 25885 | There exist irrational num... |
cxprecd 25886 | Complex exponentiation of ... |
rpcxpcld 25887 | Positive real closure of t... |
logcxpd 25888 | Logarithm of a complex pow... |
cxplt3d 25889 | Ordering property for comp... |
cxple3d 25890 | Ordering property for comp... |
cxpmuld 25891 | Product of exponents law f... |
cxpcom 25892 | Commutative law for real e... |
dvcxp1 25893 | The derivative of a comple... |
dvcxp2 25894 | The derivative of a comple... |
dvsqrt 25895 | The derivative of the real... |
dvcncxp1 25896 | Derivative of complex powe... |
dvcnsqrt 25897 | Derivative of square root ... |
cxpcn 25898 | Domain of continuity of th... |
cxpcn2 25899 | Continuity of the complex ... |
cxpcn3lem 25900 | Lemma for ~ cxpcn3 . (Con... |
cxpcn3 25901 | Extend continuity of the c... |
resqrtcn 25902 | Continuity of the real squ... |
sqrtcn 25903 | Continuity of the square r... |
cxpaddlelem 25904 | Lemma for ~ cxpaddle . (C... |
cxpaddle 25905 | Ordering property for comp... |
abscxpbnd 25906 | Bound on the absolute valu... |
root1id 25907 | Property of an ` N ` -th r... |
root1eq1 25908 | The only powers of an ` N ... |
root1cj 25909 | Within the ` N ` -th roots... |
cxpeq 25910 | Solve an equation involvin... |
loglesqrt 25911 | An upper bound on the loga... |
logreclem 25912 | Symmetry of the natural lo... |
logrec 25913 | Logarithm of a reciprocal ... |
logbval 25916 | Define the value of the ` ... |
logbcl 25917 | General logarithm closure.... |
logbid1 25918 | General logarithm is 1 whe... |
logb1 25919 | The logarithm of ` 1 ` to ... |
elogb 25920 | The general logarithm of a... |
logbchbase 25921 | Change of base for logarit... |
relogbval 25922 | Value of the general logar... |
relogbcl 25923 | Closure of the general log... |
relogbzcl 25924 | Closure of the general log... |
relogbreexp 25925 | Power law for the general ... |
relogbzexp 25926 | Power law for the general ... |
relogbmul 25927 | The logarithm of the produ... |
relogbmulexp 25928 | The logarithm of the produ... |
relogbdiv 25929 | The logarithm of the quoti... |
relogbexp 25930 | Identity law for general l... |
nnlogbexp 25931 | Identity law for general l... |
logbrec 25932 | Logarithm of a reciprocal ... |
logbleb 25933 | The general logarithm func... |
logblt 25934 | The general logarithm func... |
relogbcxp 25935 | Identity law for the gener... |
cxplogb 25936 | Identity law for the gener... |
relogbcxpb 25937 | The logarithm is the inver... |
logbmpt 25938 | The general logarithm to a... |
logbf 25939 | The general logarithm to a... |
logbfval 25940 | The general logarithm of a... |
relogbf 25941 | The general logarithm to a... |
logblog 25942 | The general logarithm to t... |
logbgt0b 25943 | The logarithm of a positiv... |
logbgcd1irr 25944 | The logarithm of an intege... |
2logb9irr 25945 | Example for ~ logbgcd1irr ... |
logbprmirr 25946 | The logarithm of a prime t... |
2logb3irr 25947 | Example for ~ logbprmirr .... |
2logb9irrALT 25948 | Alternate proof of ~ 2logb... |
sqrt2cxp2logb9e3 25949 | The square root of two to ... |
2irrexpqALT 25950 | Alternate proof of ~ 2irre... |
angval 25951 | Define the angle function,... |
angcan 25952 | Cancel a constant multipli... |
angneg 25953 | Cancel a negative sign in ... |
angvald 25954 | The (signed) angle between... |
angcld 25955 | The (signed) angle between... |
angrteqvd 25956 | Two vectors are at a right... |
cosangneg2d 25957 | The cosine of the angle be... |
angrtmuld 25958 | Perpendicularity of two ve... |
ang180lem1 25959 | Lemma for ~ ang180 . Show... |
ang180lem2 25960 | Lemma for ~ ang180 . Show... |
ang180lem3 25961 | Lemma for ~ ang180 . Sinc... |
ang180lem4 25962 | Lemma for ~ ang180 . Redu... |
ang180lem5 25963 | Lemma for ~ ang180 : Redu... |
ang180 25964 | The sum of angles ` m A B ... |
lawcoslem1 25965 | Lemma for ~ lawcos . Here... |
lawcos 25966 | Law of cosines (also known... |
pythag 25967 | Pythagorean theorem. Give... |
isosctrlem1 25968 | Lemma for ~ isosctr . (Co... |
isosctrlem2 25969 | Lemma for ~ isosctr . Cor... |
isosctrlem3 25970 | Lemma for ~ isosctr . Cor... |
isosctr 25971 | Isosceles triangle theorem... |
ssscongptld 25972 | If two triangles have equa... |
affineequiv 25973 | Equivalence between two wa... |
affineequiv2 25974 | Equivalence between two wa... |
affineequiv3 25975 | Equivalence between two wa... |
affineequiv4 25976 | Equivalence between two wa... |
affineequivne 25977 | Equivalence between two wa... |
angpieqvdlem 25978 | Equivalence used in the pr... |
angpieqvdlem2 25979 | Equivalence used in ~ angp... |
angpined 25980 | If the angle at ABC is ` _... |
angpieqvd 25981 | The angle ABC is ` _pi ` i... |
chordthmlem 25982 | If ` M ` is the midpoint o... |
chordthmlem2 25983 | If M is the midpoint of AB... |
chordthmlem3 25984 | If M is the midpoint of AB... |
chordthmlem4 25985 | If P is on the segment AB ... |
chordthmlem5 25986 | If P is on the segment AB ... |
chordthm 25987 | The intersecting chords th... |
heron 25988 | Heron's formula gives the ... |
quad2 25989 | The quadratic equation, wi... |
quad 25990 | The quadratic equation. (... |
1cubrlem 25991 | The cube roots of unity. ... |
1cubr 25992 | The cube roots of unity. ... |
dcubic1lem 25993 | Lemma for ~ dcubic1 and ~ ... |
dcubic2 25994 | Reverse direction of ~ dcu... |
dcubic1 25995 | Forward direction of ~ dcu... |
dcubic 25996 | Solutions to the depressed... |
mcubic 25997 | Solutions to a monic cubic... |
cubic2 25998 | The solution to the genera... |
cubic 25999 | The cubic equation, which ... |
binom4 26000 | Work out a quartic binomia... |
dquartlem1 26001 | Lemma for ~ dquart . (Con... |
dquartlem2 26002 | Lemma for ~ dquart . (Con... |
dquart 26003 | Solve a depressed quartic ... |
quart1cl 26004 | Closure lemmas for ~ quart... |
quart1lem 26005 | Lemma for ~ quart1 . (Con... |
quart1 26006 | Depress a quartic equation... |
quartlem1 26007 | Lemma for ~ quart . (Cont... |
quartlem2 26008 | Closure lemmas for ~ quart... |
quartlem3 26009 | Closure lemmas for ~ quart... |
quartlem4 26010 | Closure lemmas for ~ quart... |
quart 26011 | The quartic equation, writ... |
asinlem 26018 | The argument to the logari... |
asinlem2 26019 | The argument to the logari... |
asinlem3a 26020 | Lemma for ~ asinlem3 . (C... |
asinlem3 26021 | The argument to the logari... |
asinf 26022 | Domain and range of the ar... |
asincl 26023 | Closure for the arcsin fun... |
acosf 26024 | Domain and range of the ar... |
acoscl 26025 | Closure for the arccos fun... |
atandm 26026 | Since the property is a li... |
atandm2 26027 | This form of ~ atandm is a... |
atandm3 26028 | A compact form of ~ atandm... |
atandm4 26029 | A compact form of ~ atandm... |
atanf 26030 | Domain and range of the ar... |
atancl 26031 | Closure for the arctan fun... |
asinval 26032 | Value of the arcsin functi... |
acosval 26033 | Value of the arccos functi... |
atanval 26034 | Value of the arctan functi... |
atanre 26035 | A real number is in the do... |
asinneg 26036 | The arcsine function is od... |
acosneg 26037 | The negative symmetry rela... |
efiasin 26038 | The exponential of the arc... |
sinasin 26039 | The arcsine function is an... |
cosacos 26040 | The arccosine function is ... |
asinsinlem 26041 | Lemma for ~ asinsin . (Co... |
asinsin 26042 | The arcsine function compo... |
acoscos 26043 | The arccosine function is ... |
asin1 26044 | The arcsine of ` 1 ` is ` ... |
acos1 26045 | The arccosine of ` 1 ` is ... |
reasinsin 26046 | The arcsine function compo... |
asinsinb 26047 | Relationship between sine ... |
acoscosb 26048 | Relationship between cosin... |
asinbnd 26049 | The arcsine function has r... |
acosbnd 26050 | The arccosine function has... |
asinrebnd 26051 | Bounds on the arcsine func... |
asinrecl 26052 | The arcsine function is re... |
acosrecl 26053 | The arccosine function is ... |
cosasin 26054 | The cosine of the arcsine ... |
sinacos 26055 | The sine of the arccosine ... |
atandmneg 26056 | The domain of the arctange... |
atanneg 26057 | The arctangent function is... |
atan0 26058 | The arctangent of zero is ... |
atandmcj 26059 | The arctangent function di... |
atancj 26060 | The arctangent function di... |
atanrecl 26061 | The arctangent function is... |
efiatan 26062 | Value of the exponential o... |
atanlogaddlem 26063 | Lemma for ~ atanlogadd . ... |
atanlogadd 26064 | The rule ` sqrt ( z w ) = ... |
atanlogsublem 26065 | Lemma for ~ atanlogsub . ... |
atanlogsub 26066 | A variation on ~ atanlogad... |
efiatan2 26067 | Value of the exponential o... |
2efiatan 26068 | Value of the exponential o... |
tanatan 26069 | The arctangent function is... |
atandmtan 26070 | The tangent function has r... |
cosatan 26071 | The cosine of an arctangen... |
cosatanne0 26072 | The arctangent function ha... |
atantan 26073 | The arctangent function is... |
atantanb 26074 | Relationship between tange... |
atanbndlem 26075 | Lemma for ~ atanbnd . (Co... |
atanbnd 26076 | The arctangent function is... |
atanord 26077 | The arctangent function is... |
atan1 26078 | The arctangent of ` 1 ` is... |
bndatandm 26079 | A point in the open unit d... |
atans 26080 | The "domain of continuity"... |
atans2 26081 | It suffices to show that `... |
atansopn 26082 | The domain of continuity o... |
atansssdm 26083 | The domain of continuity o... |
ressatans 26084 | The real number line is a ... |
dvatan 26085 | The derivative of the arct... |
atancn 26086 | The arctangent is a contin... |
atantayl 26087 | The Taylor series for ` ar... |
atantayl2 26088 | The Taylor series for ` ar... |
atantayl3 26089 | The Taylor series for ` ar... |
leibpilem1 26090 | Lemma for ~ leibpi . (Con... |
leibpilem2 26091 | The Leibniz formula for ` ... |
leibpi 26092 | The Leibniz formula for ` ... |
leibpisum 26093 | The Leibniz formula for ` ... |
log2cnv 26094 | Using the Taylor series fo... |
log2tlbnd 26095 | Bound the error term in th... |
log2ublem1 26096 | Lemma for ~ log2ub . The ... |
log2ublem2 26097 | Lemma for ~ log2ub . (Con... |
log2ublem3 26098 | Lemma for ~ log2ub . In d... |
log2ub 26099 | ` log 2 ` is less than ` 2... |
log2le1 26100 | ` log 2 ` is less than ` 1... |
birthdaylem1 26101 | Lemma for ~ birthday . (C... |
birthdaylem2 26102 | For general ` N ` and ` K ... |
birthdaylem3 26103 | For general ` N ` and ` K ... |
birthday 26104 | The Birthday Problem. The... |
dmarea 26107 | The domain of the area fun... |
areambl 26108 | The fibers of a measurable... |
areass 26109 | A measurable region is a s... |
dfarea 26110 | Rewrite ~ df-area self-ref... |
areaf 26111 | Area measurement is a func... |
areacl 26112 | The area of a measurable r... |
areage0 26113 | The area of a measurable r... |
areaval 26114 | The area of a measurable r... |
rlimcnp 26115 | Relate a limit of a real-v... |
rlimcnp2 26116 | Relate a limit of a real-v... |
rlimcnp3 26117 | Relate a limit of a real-v... |
xrlimcnp 26118 | Relate a limit of a real-v... |
efrlim 26119 | The limit of the sequence ... |
dfef2 26120 | The limit of the sequence ... |
cxplim 26121 | A power to a negative expo... |
sqrtlim 26122 | The inverse square root fu... |
rlimcxp 26123 | Any power to a positive ex... |
o1cxp 26124 | An eventually bounded func... |
cxp2limlem 26125 | A linear factor grows slow... |
cxp2lim 26126 | Any power grows slower tha... |
cxploglim 26127 | The logarithm grows slower... |
cxploglim2 26128 | Every power of the logarit... |
divsqrtsumlem 26129 | Lemma for ~ divsqrsum and ... |
divsqrsumf 26130 | The function ` F ` used in... |
divsqrsum 26131 | The sum ` sum_ n <_ x ( 1 ... |
divsqrtsum2 26132 | A bound on the distance of... |
divsqrtsumo1 26133 | The sum ` sum_ n <_ x ( 1 ... |
cvxcl 26134 | Closure of a 0-1 linear co... |
scvxcvx 26135 | A strictly convex function... |
jensenlem1 26136 | Lemma for ~ jensen . (Con... |
jensenlem2 26137 | Lemma for ~ jensen . (Con... |
jensen 26138 | Jensen's inequality, a fin... |
amgmlem 26139 | Lemma for ~ amgm . (Contr... |
amgm 26140 | Inequality of arithmetic a... |
logdifbnd 26143 | Bound on the difference of... |
logdiflbnd 26144 | Lower bound on the differe... |
emcllem1 26145 | Lemma for ~ emcl . The se... |
emcllem2 26146 | Lemma for ~ emcl . ` F ` i... |
emcllem3 26147 | Lemma for ~ emcl . The fu... |
emcllem4 26148 | Lemma for ~ emcl . The di... |
emcllem5 26149 | Lemma for ~ emcl . The pa... |
emcllem6 26150 | Lemma for ~ emcl . By the... |
emcllem7 26151 | Lemma for ~ emcl and ~ har... |
emcl 26152 | Closure and bounds for the... |
harmonicbnd 26153 | A bound on the harmonic se... |
harmonicbnd2 26154 | A bound on the harmonic se... |
emre 26155 | The Euler-Mascheroni const... |
emgt0 26156 | The Euler-Mascheroni const... |
harmonicbnd3 26157 | A bound on the harmonic se... |
harmoniclbnd 26158 | A bound on the harmonic se... |
harmonicubnd 26159 | A bound on the harmonic se... |
harmonicbnd4 26160 | The asymptotic behavior of... |
fsumharmonic 26161 | Bound a finite sum based o... |
zetacvg 26164 | The zeta series is converg... |
eldmgm 26171 | Elementhood in the set of ... |
dmgmaddn0 26172 | If ` A ` is not a nonposit... |
dmlogdmgm 26173 | If ` A ` is in the continu... |
rpdmgm 26174 | A positive real number is ... |
dmgmn0 26175 | If ` A ` is not a nonposit... |
dmgmaddnn0 26176 | If ` A ` is not a nonposit... |
dmgmdivn0 26177 | Lemma for ~ lgamf . (Cont... |
lgamgulmlem1 26178 | Lemma for ~ lgamgulm . (C... |
lgamgulmlem2 26179 | Lemma for ~ lgamgulm . (C... |
lgamgulmlem3 26180 | Lemma for ~ lgamgulm . (C... |
lgamgulmlem4 26181 | Lemma for ~ lgamgulm . (C... |
lgamgulmlem5 26182 | Lemma for ~ lgamgulm . (C... |
lgamgulmlem6 26183 | The series ` G ` is unifor... |
lgamgulm 26184 | The series ` G ` is unifor... |
lgamgulm2 26185 | Rewrite the limit of the s... |
lgambdd 26186 | The log-Gamma function is ... |
lgamucov 26187 | The ` U ` regions used in ... |
lgamucov2 26188 | The ` U ` regions used in ... |
lgamcvglem 26189 | Lemma for ~ lgamf and ~ lg... |
lgamcl 26190 | The log-Gamma function is ... |
lgamf 26191 | The log-Gamma function is ... |
gamf 26192 | The Gamma function is a co... |
gamcl 26193 | The exponential of the log... |
eflgam 26194 | The exponential of the log... |
gamne0 26195 | The Gamma function is neve... |
igamval 26196 | Value of the inverse Gamma... |
igamz 26197 | Value of the inverse Gamma... |
igamgam 26198 | Value of the inverse Gamma... |
igamlgam 26199 | Value of the inverse Gamma... |
igamf 26200 | Closure of the inverse Gam... |
igamcl 26201 | Closure of the inverse Gam... |
gamigam 26202 | The Gamma function is the ... |
lgamcvg 26203 | The series ` G ` converges... |
lgamcvg2 26204 | The series ` G ` converges... |
gamcvg 26205 | The pointwise exponential ... |
lgamp1 26206 | The functional equation of... |
gamp1 26207 | The functional equation of... |
gamcvg2lem 26208 | Lemma for ~ gamcvg2 . (Co... |
gamcvg2 26209 | An infinite product expres... |
regamcl 26210 | The Gamma function is real... |
relgamcl 26211 | The log-Gamma function is ... |
rpgamcl 26212 | The log-Gamma function is ... |
lgam1 26213 | The log-Gamma function at ... |
gam1 26214 | The log-Gamma function at ... |
facgam 26215 | The Gamma function general... |
gamfac 26216 | The Gamma function general... |
wilthlem1 26217 | The only elements that are... |
wilthlem2 26218 | Lemma for ~ wilth : induct... |
wilthlem3 26219 | Lemma for ~ wilth . Here ... |
wilth 26220 | Wilson's theorem. A numbe... |
wilthimp 26221 | The forward implication of... |
ftalem1 26222 | Lemma for ~ fta : "growth... |
ftalem2 26223 | Lemma for ~ fta . There e... |
ftalem3 26224 | Lemma for ~ fta . There e... |
ftalem4 26225 | Lemma for ~ fta : Closure... |
ftalem5 26226 | Lemma for ~ fta : Main pr... |
ftalem6 26227 | Lemma for ~ fta : Dischar... |
ftalem7 26228 | Lemma for ~ fta . Shift t... |
fta 26229 | The Fundamental Theorem of... |
basellem1 26230 | Lemma for ~ basel . Closu... |
basellem2 26231 | Lemma for ~ basel . Show ... |
basellem3 26232 | Lemma for ~ basel . Using... |
basellem4 26233 | Lemma for ~ basel . By ~ ... |
basellem5 26234 | Lemma for ~ basel . Using... |
basellem6 26235 | Lemma for ~ basel . The f... |
basellem7 26236 | Lemma for ~ basel . The f... |
basellem8 26237 | Lemma for ~ basel . The f... |
basellem9 26238 | Lemma for ~ basel . Since... |
basel 26239 | The sum of the inverse squ... |
efnnfsumcl 26252 | Finite sum closure in the ... |
ppisval 26253 | The set of primes less tha... |
ppisval2 26254 | The set of primes less tha... |
ppifi 26255 | The set of primes less tha... |
prmdvdsfi 26256 | The set of prime divisors ... |
chtf 26257 | Domain and range of the Ch... |
chtcl 26258 | Real closure of the Chebys... |
chtval 26259 | Value of the Chebyshev fun... |
efchtcl 26260 | The Chebyshev function is ... |
chtge0 26261 | The Chebyshev function is ... |
vmaval 26262 | Value of the von Mangoldt ... |
isppw 26263 | Two ways to say that ` A `... |
isppw2 26264 | Two ways to say that ` A `... |
vmappw 26265 | Value of the von Mangoldt ... |
vmaprm 26266 | Value of the von Mangoldt ... |
vmacl 26267 | Closure for the von Mangol... |
vmaf 26268 | Functionality of the von M... |
efvmacl 26269 | The von Mangoldt is closed... |
vmage0 26270 | The von Mangoldt function ... |
chpval 26271 | Value of the second Chebys... |
chpf 26272 | Functionality of the secon... |
chpcl 26273 | Closure for the second Che... |
efchpcl 26274 | The second Chebyshev funct... |
chpge0 26275 | The second Chebyshev funct... |
ppival 26276 | Value of the prime-countin... |
ppival2 26277 | Value of the prime-countin... |
ppival2g 26278 | Value of the prime-countin... |
ppif 26279 | Domain and range of the pr... |
ppicl 26280 | Real closure of the prime-... |
muval 26281 | The value of the Möbi... |
muval1 26282 | The value of the Möbi... |
muval2 26283 | The value of the Möbi... |
isnsqf 26284 | Two ways to say that a num... |
issqf 26285 | Two ways to say that a num... |
sqfpc 26286 | The prime count of a squar... |
dvdssqf 26287 | A divisor of a squarefree ... |
sqf11 26288 | A squarefree number is com... |
muf 26289 | The Möbius function i... |
mucl 26290 | Closure of the Möbius... |
sgmval 26291 | The value of the divisor f... |
sgmval2 26292 | The value of the divisor f... |
0sgm 26293 | The value of the sum-of-di... |
sgmf 26294 | The divisor function is a ... |
sgmcl 26295 | Closure of the divisor fun... |
sgmnncl 26296 | Closure of the divisor fun... |
mule1 26297 | The Möbius function t... |
chtfl 26298 | The Chebyshev function doe... |
chpfl 26299 | The second Chebyshev funct... |
ppiprm 26300 | The prime-counting functio... |
ppinprm 26301 | The prime-counting functio... |
chtprm 26302 | The Chebyshev function at ... |
chtnprm 26303 | The Chebyshev function at ... |
chpp1 26304 | The second Chebyshev funct... |
chtwordi 26305 | The Chebyshev function is ... |
chpwordi 26306 | The second Chebyshev funct... |
chtdif 26307 | The difference of the Cheb... |
efchtdvds 26308 | The exponentiated Chebyshe... |
ppifl 26309 | The prime-counting functio... |
ppip1le 26310 | The prime-counting functio... |
ppiwordi 26311 | The prime-counting functio... |
ppidif 26312 | The difference of the prim... |
ppi1 26313 | The prime-counting functio... |
cht1 26314 | The Chebyshev function at ... |
vma1 26315 | The von Mangoldt function ... |
chp1 26316 | The second Chebyshev funct... |
ppi1i 26317 | Inference form of ~ ppiprm... |
ppi2i 26318 | Inference form of ~ ppinpr... |
ppi2 26319 | The prime-counting functio... |
ppi3 26320 | The prime-counting functio... |
cht2 26321 | The Chebyshev function at ... |
cht3 26322 | The Chebyshev function at ... |
ppinncl 26323 | Closure of the prime-count... |
chtrpcl 26324 | Closure of the Chebyshev f... |
ppieq0 26325 | The prime-counting functio... |
ppiltx 26326 | The prime-counting functio... |
prmorcht 26327 | Relate the primorial (prod... |
mumullem1 26328 | Lemma for ~ mumul . A mul... |
mumullem2 26329 | Lemma for ~ mumul . The p... |
mumul 26330 | The Möbius function i... |
sqff1o 26331 | There is a bijection from ... |
fsumdvdsdiaglem 26332 | A "diagonal commutation" o... |
fsumdvdsdiag 26333 | A "diagonal commutation" o... |
fsumdvdscom 26334 | A double commutation of di... |
dvdsppwf1o 26335 | A bijection from the divis... |
dvdsflf1o 26336 | A bijection from the numbe... |
dvdsflsumcom 26337 | A sum commutation from ` s... |
fsumfldivdiaglem 26338 | Lemma for ~ fsumfldivdiag ... |
fsumfldivdiag 26339 | The right-hand side of ~ d... |
musum 26340 | The sum of the Möbius... |
musumsum 26341 | Evaluate a collapsing sum ... |
muinv 26342 | The Möbius inversion ... |
dvdsmulf1o 26343 | If ` M ` and ` N ` are two... |
fsumdvdsmul 26344 | Product of two divisor sum... |
sgmppw 26345 | The value of the divisor f... |
0sgmppw 26346 | A prime power ` P ^ K ` ha... |
1sgmprm 26347 | The sum of divisors for a ... |
1sgm2ppw 26348 | The sum of the divisors of... |
sgmmul 26349 | The divisor function for f... |
ppiublem1 26350 | Lemma for ~ ppiub . (Cont... |
ppiublem2 26351 | A prime greater than ` 3 `... |
ppiub 26352 | An upper bound on the prim... |
vmalelog 26353 | The von Mangoldt function ... |
chtlepsi 26354 | The first Chebyshev functi... |
chprpcl 26355 | Closure of the second Cheb... |
chpeq0 26356 | The second Chebyshev funct... |
chteq0 26357 | The first Chebyshev functi... |
chtleppi 26358 | Upper bound on the ` theta... |
chtublem 26359 | Lemma for ~ chtub . (Cont... |
chtub 26360 | An upper bound on the Cheb... |
fsumvma 26361 | Rewrite a sum over the von... |
fsumvma2 26362 | Apply ~ fsumvma for the co... |
pclogsum 26363 | The logarithmic analogue o... |
vmasum 26364 | The sum of the von Mangold... |
logfac2 26365 | Another expression for the... |
chpval2 26366 | Express the second Chebysh... |
chpchtsum 26367 | The second Chebyshev funct... |
chpub 26368 | An upper bound on the seco... |
logfacubnd 26369 | A simple upper bound on th... |
logfaclbnd 26370 | A lower bound on the logar... |
logfacbnd3 26371 | Show the stronger statemen... |
logfacrlim 26372 | Combine the estimates ~ lo... |
logexprlim 26373 | The sum ` sum_ n <_ x , lo... |
logfacrlim2 26374 | Write out ~ logfacrlim as ... |
mersenne 26375 | A Mersenne prime is a prim... |
perfect1 26376 | Euclid's contribution to t... |
perfectlem1 26377 | Lemma for ~ perfect . (Co... |
perfectlem2 26378 | Lemma for ~ perfect . (Co... |
perfect 26379 | The Euclid-Euler theorem, ... |
dchrval 26382 | Value of the group of Diri... |
dchrbas 26383 | Base set of the group of D... |
dchrelbas 26384 | A Dirichlet character is a... |
dchrelbas2 26385 | A Dirichlet character is a... |
dchrelbas3 26386 | A Dirichlet character is a... |
dchrelbasd 26387 | A Dirichlet character is a... |
dchrrcl 26388 | Reverse closure for a Diri... |
dchrmhm 26389 | A Dirichlet character is a... |
dchrf 26390 | A Dirichlet character is a... |
dchrelbas4 26391 | A Dirichlet character is a... |
dchrzrh1 26392 | Value of a Dirichlet chara... |
dchrzrhcl 26393 | A Dirichlet character take... |
dchrzrhmul 26394 | A Dirichlet character is c... |
dchrplusg 26395 | Group operation on the gro... |
dchrmul 26396 | Group operation on the gro... |
dchrmulcl 26397 | Closure of the group opera... |
dchrn0 26398 | A Dirichlet character is n... |
dchr1cl 26399 | Closure of the principal D... |
dchrmulid2 26400 | Left identity for the prin... |
dchrinvcl 26401 | Closure of the group inver... |
dchrabl 26402 | The set of Dirichlet chara... |
dchrfi 26403 | The group of Dirichlet cha... |
dchrghm 26404 | A Dirichlet character rest... |
dchr1 26405 | Value of the principal Dir... |
dchreq 26406 | A Dirichlet character is d... |
dchrresb 26407 | A Dirichlet character is d... |
dchrabs 26408 | A Dirichlet character take... |
dchrinv 26409 | The inverse of a Dirichlet... |
dchrabs2 26410 | A Dirichlet character take... |
dchr1re 26411 | The principal Dirichlet ch... |
dchrptlem1 26412 | Lemma for ~ dchrpt . (Con... |
dchrptlem2 26413 | Lemma for ~ dchrpt . (Con... |
dchrptlem3 26414 | Lemma for ~ dchrpt . (Con... |
dchrpt 26415 | For any element other than... |
dchrsum2 26416 | An orthogonality relation ... |
dchrsum 26417 | An orthogonality relation ... |
sumdchr2 26418 | Lemma for ~ sumdchr . (Co... |
dchrhash 26419 | There are exactly ` phi ( ... |
sumdchr 26420 | An orthogonality relation ... |
dchr2sum 26421 | An orthogonality relation ... |
sum2dchr 26422 | An orthogonality relation ... |
bcctr 26423 | Value of the central binom... |
pcbcctr 26424 | Prime count of a central b... |
bcmono 26425 | The binomial coefficient i... |
bcmax 26426 | The binomial coefficient t... |
bcp1ctr 26427 | Ratio of two central binom... |
bclbnd 26428 | A bound on the binomial co... |
efexple 26429 | Convert a bound on a power... |
bpos1lem 26430 | Lemma for ~ bpos1 . (Cont... |
bpos1 26431 | Bertrand's postulate, chec... |
bposlem1 26432 | An upper bound on the prim... |
bposlem2 26433 | There are no odd primes in... |
bposlem3 26434 | Lemma for ~ bpos . Since ... |
bposlem4 26435 | Lemma for ~ bpos . (Contr... |
bposlem5 26436 | Lemma for ~ bpos . Bound ... |
bposlem6 26437 | Lemma for ~ bpos . By usi... |
bposlem7 26438 | Lemma for ~ bpos . The fu... |
bposlem8 26439 | Lemma for ~ bpos . Evalua... |
bposlem9 26440 | Lemma for ~ bpos . Derive... |
bpos 26441 | Bertrand's postulate: ther... |
zabsle1 26444 | ` { -u 1 , 0 , 1 } ` is th... |
lgslem1 26445 | When ` a ` is coprime to t... |
lgslem2 26446 | The set ` Z ` of all integ... |
lgslem3 26447 | The set ` Z ` of all integ... |
lgslem4 26448 | Lemma for ~ lgsfcl2 . (Co... |
lgsval 26449 | Value of the Legendre symb... |
lgsfval 26450 | Value of the function ` F ... |
lgsfcl2 26451 | The function ` F ` is clos... |
lgscllem 26452 | The Legendre symbol is an ... |
lgsfcl 26453 | Closure of the function ` ... |
lgsfle1 26454 | The function ` F ` has mag... |
lgsval2lem 26455 | Lemma for ~ lgsval2 . (Co... |
lgsval4lem 26456 | Lemma for ~ lgsval4 . (Co... |
lgscl2 26457 | The Legendre symbol is an ... |
lgs0 26458 | The Legendre symbol when t... |
lgscl 26459 | The Legendre symbol is an ... |
lgsle1 26460 | The Legendre symbol has ab... |
lgsval2 26461 | The Legendre symbol at a p... |
lgs2 26462 | The Legendre symbol at ` 2... |
lgsval3 26463 | The Legendre symbol at an ... |
lgsvalmod 26464 | The Legendre symbol is equ... |
lgsval4 26465 | Restate ~ lgsval for nonze... |
lgsfcl3 26466 | Closure of the function ` ... |
lgsval4a 26467 | Same as ~ lgsval4 for posi... |
lgscl1 26468 | The value of the Legendre ... |
lgsneg 26469 | The Legendre symbol is eit... |
lgsneg1 26470 | The Legendre symbol for no... |
lgsmod 26471 | The Legendre (Jacobi) symb... |
lgsdilem 26472 | Lemma for ~ lgsdi and ~ lg... |
lgsdir2lem1 26473 | Lemma for ~ lgsdir2 . (Co... |
lgsdir2lem2 26474 | Lemma for ~ lgsdir2 . (Co... |
lgsdir2lem3 26475 | Lemma for ~ lgsdir2 . (Co... |
lgsdir2lem4 26476 | Lemma for ~ lgsdir2 . (Co... |
lgsdir2lem5 26477 | Lemma for ~ lgsdir2 . (Co... |
lgsdir2 26478 | The Legendre symbol is com... |
lgsdirprm 26479 | The Legendre symbol is com... |
lgsdir 26480 | The Legendre symbol is com... |
lgsdilem2 26481 | Lemma for ~ lgsdi . (Cont... |
lgsdi 26482 | The Legendre symbol is com... |
lgsne0 26483 | The Legendre symbol is non... |
lgsabs1 26484 | The Legendre symbol is non... |
lgssq 26485 | The Legendre symbol at a s... |
lgssq2 26486 | The Legendre symbol at a s... |
lgsprme0 26487 | The Legendre symbol at any... |
1lgs 26488 | The Legendre symbol at ` 1... |
lgs1 26489 | The Legendre symbol at ` 1... |
lgsmodeq 26490 | The Legendre (Jacobi) symb... |
lgsmulsqcoprm 26491 | The Legendre (Jacobi) symb... |
lgsdirnn0 26492 | Variation on ~ lgsdir vali... |
lgsdinn0 26493 | Variation on ~ lgsdi valid... |
lgsqrlem1 26494 | Lemma for ~ lgsqr . (Cont... |
lgsqrlem2 26495 | Lemma for ~ lgsqr . (Cont... |
lgsqrlem3 26496 | Lemma for ~ lgsqr . (Cont... |
lgsqrlem4 26497 | Lemma for ~ lgsqr . (Cont... |
lgsqrlem5 26498 | Lemma for ~ lgsqr . (Cont... |
lgsqr 26499 | The Legendre symbol for od... |
lgsqrmod 26500 | If the Legendre symbol of ... |
lgsqrmodndvds 26501 | If the Legendre symbol of ... |
lgsdchrval 26502 | The Legendre symbol functi... |
lgsdchr 26503 | The Legendre symbol functi... |
gausslemma2dlem0a 26504 | Auxiliary lemma 1 for ~ ga... |
gausslemma2dlem0b 26505 | Auxiliary lemma 2 for ~ ga... |
gausslemma2dlem0c 26506 | Auxiliary lemma 3 for ~ ga... |
gausslemma2dlem0d 26507 | Auxiliary lemma 4 for ~ ga... |
gausslemma2dlem0e 26508 | Auxiliary lemma 5 for ~ ga... |
gausslemma2dlem0f 26509 | Auxiliary lemma 6 for ~ ga... |
gausslemma2dlem0g 26510 | Auxiliary lemma 7 for ~ ga... |
gausslemma2dlem0h 26511 | Auxiliary lemma 8 for ~ ga... |
gausslemma2dlem0i 26512 | Auxiliary lemma 9 for ~ ga... |
gausslemma2dlem1a 26513 | Lemma for ~ gausslemma2dle... |
gausslemma2dlem1 26514 | Lemma 1 for ~ gausslemma2d... |
gausslemma2dlem2 26515 | Lemma 2 for ~ gausslemma2d... |
gausslemma2dlem3 26516 | Lemma 3 for ~ gausslemma2d... |
gausslemma2dlem4 26517 | Lemma 4 for ~ gausslemma2d... |
gausslemma2dlem5a 26518 | Lemma for ~ gausslemma2dle... |
gausslemma2dlem5 26519 | Lemma 5 for ~ gausslemma2d... |
gausslemma2dlem6 26520 | Lemma 6 for ~ gausslemma2d... |
gausslemma2dlem7 26521 | Lemma 7 for ~ gausslemma2d... |
gausslemma2d 26522 | Gauss' Lemma (see also the... |
lgseisenlem1 26523 | Lemma for ~ lgseisen . If... |
lgseisenlem2 26524 | Lemma for ~ lgseisen . Th... |
lgseisenlem3 26525 | Lemma for ~ lgseisen . (C... |
lgseisenlem4 26526 | Lemma for ~ lgseisen . Th... |
lgseisen 26527 | Eisenstein's lemma, an exp... |
lgsquadlem1 26528 | Lemma for ~ lgsquad . Cou... |
lgsquadlem2 26529 | Lemma for ~ lgsquad . Cou... |
lgsquadlem3 26530 | Lemma for ~ lgsquad . (Co... |
lgsquad 26531 | The Law of Quadratic Recip... |
lgsquad2lem1 26532 | Lemma for ~ lgsquad2 . (C... |
lgsquad2lem2 26533 | Lemma for ~ lgsquad2 . (C... |
lgsquad2 26534 | Extend ~ lgsquad to coprim... |
lgsquad3 26535 | Extend ~ lgsquad2 to integ... |
m1lgs 26536 | The first supplement to th... |
2lgslem1a1 26537 | Lemma 1 for ~ 2lgslem1a . ... |
2lgslem1a2 26538 | Lemma 2 for ~ 2lgslem1a . ... |
2lgslem1a 26539 | Lemma 1 for ~ 2lgslem1 . ... |
2lgslem1b 26540 | Lemma 2 for ~ 2lgslem1 . ... |
2lgslem1c 26541 | Lemma 3 for ~ 2lgslem1 . ... |
2lgslem1 26542 | Lemma 1 for ~ 2lgs . (Con... |
2lgslem2 26543 | Lemma 2 for ~ 2lgs . (Con... |
2lgslem3a 26544 | Lemma for ~ 2lgslem3a1 . ... |
2lgslem3b 26545 | Lemma for ~ 2lgslem3b1 . ... |
2lgslem3c 26546 | Lemma for ~ 2lgslem3c1 . ... |
2lgslem3d 26547 | Lemma for ~ 2lgslem3d1 . ... |
2lgslem3a1 26548 | Lemma 1 for ~ 2lgslem3 . ... |
2lgslem3b1 26549 | Lemma 2 for ~ 2lgslem3 . ... |
2lgslem3c1 26550 | Lemma 3 for ~ 2lgslem3 . ... |
2lgslem3d1 26551 | Lemma 4 for ~ 2lgslem3 . ... |
2lgslem3 26552 | Lemma 3 for ~ 2lgs . (Con... |
2lgs2 26553 | The Legendre symbol for ` ... |
2lgslem4 26554 | Lemma 4 for ~ 2lgs : speci... |
2lgs 26555 | The second supplement to t... |
2lgsoddprmlem1 26556 | Lemma 1 for ~ 2lgsoddprm .... |
2lgsoddprmlem2 26557 | Lemma 2 for ~ 2lgsoddprm .... |
2lgsoddprmlem3a 26558 | Lemma 1 for ~ 2lgsoddprmle... |
2lgsoddprmlem3b 26559 | Lemma 2 for ~ 2lgsoddprmle... |
2lgsoddprmlem3c 26560 | Lemma 3 for ~ 2lgsoddprmle... |
2lgsoddprmlem3d 26561 | Lemma 4 for ~ 2lgsoddprmle... |
2lgsoddprmlem3 26562 | Lemma 3 for ~ 2lgsoddprm .... |
2lgsoddprmlem4 26563 | Lemma 4 for ~ 2lgsoddprm .... |
2lgsoddprm 26564 | The second supplement to t... |
2sqlem1 26565 | Lemma for ~ 2sq . (Contri... |
2sqlem2 26566 | Lemma for ~ 2sq . (Contri... |
mul2sq 26567 | Fibonacci's identity (actu... |
2sqlem3 26568 | Lemma for ~ 2sqlem5 . (Co... |
2sqlem4 26569 | Lemma for ~ 2sqlem5 . (Co... |
2sqlem5 26570 | Lemma for ~ 2sq . If a nu... |
2sqlem6 26571 | Lemma for ~ 2sq . If a nu... |
2sqlem7 26572 | Lemma for ~ 2sq . (Contri... |
2sqlem8a 26573 | Lemma for ~ 2sqlem8 . (Co... |
2sqlem8 26574 | Lemma for ~ 2sq . (Contri... |
2sqlem9 26575 | Lemma for ~ 2sq . (Contri... |
2sqlem10 26576 | Lemma for ~ 2sq . Every f... |
2sqlem11 26577 | Lemma for ~ 2sq . (Contri... |
2sq 26578 | All primes of the form ` 4... |
2sqblem 26579 | Lemma for ~ 2sqb . (Contr... |
2sqb 26580 | The converse to ~ 2sq . (... |
2sq2 26581 | ` 2 ` is the sum of square... |
2sqn0 26582 | If the sum of two squares ... |
2sqcoprm 26583 | If the sum of two squares ... |
2sqmod 26584 | Given two decompositions o... |
2sqmo 26585 | There exists at most one d... |
2sqnn0 26586 | All primes of the form ` 4... |
2sqnn 26587 | All primes of the form ` 4... |
addsq2reu 26588 | For each complex number ` ... |
addsqn2reu 26589 | For each complex number ` ... |
addsqrexnreu 26590 | For each complex number, t... |
addsqnreup 26591 | There is no unique decompo... |
addsq2nreurex 26592 | For each complex number ` ... |
addsqn2reurex2 26593 | For each complex number ` ... |
2sqreulem1 26594 | Lemma 1 for ~ 2sqreu . (C... |
2sqreultlem 26595 | Lemma for ~ 2sqreult . (C... |
2sqreultblem 26596 | Lemma for ~ 2sqreultb . (... |
2sqreunnlem1 26597 | Lemma 1 for ~ 2sqreunn . ... |
2sqreunnltlem 26598 | Lemma for ~ 2sqreunnlt . ... |
2sqreunnltblem 26599 | Lemma for ~ 2sqreunnltb . ... |
2sqreulem2 26600 | Lemma 2 for ~ 2sqreu etc. ... |
2sqreulem3 26601 | Lemma 3 for ~ 2sqreu etc. ... |
2sqreulem4 26602 | Lemma 4 for ~ 2sqreu et. ... |
2sqreunnlem2 26603 | Lemma 2 for ~ 2sqreunn . ... |
2sqreu 26604 | There exists a unique deco... |
2sqreunn 26605 | There exists a unique deco... |
2sqreult 26606 | There exists a unique deco... |
2sqreultb 26607 | There exists a unique deco... |
2sqreunnlt 26608 | There exists a unique deco... |
2sqreunnltb 26609 | There exists a unique deco... |
2sqreuop 26610 | There exists a unique deco... |
2sqreuopnn 26611 | There exists a unique deco... |
2sqreuoplt 26612 | There exists a unique deco... |
2sqreuopltb 26613 | There exists a unique deco... |
2sqreuopnnlt 26614 | There exists a unique deco... |
2sqreuopnnltb 26615 | There exists a unique deco... |
2sqreuopb 26616 | There exists a unique deco... |
chebbnd1lem1 26617 | Lemma for ~ chebbnd1 : sho... |
chebbnd1lem2 26618 | Lemma for ~ chebbnd1 : Sh... |
chebbnd1lem3 26619 | Lemma for ~ chebbnd1 : get... |
chebbnd1 26620 | The Chebyshev bound: The ... |
chtppilimlem1 26621 | Lemma for ~ chtppilim . (... |
chtppilimlem2 26622 | Lemma for ~ chtppilim . (... |
chtppilim 26623 | The ` theta ` function is ... |
chto1ub 26624 | The ` theta ` function is ... |
chebbnd2 26625 | The Chebyshev bound, part ... |
chto1lb 26626 | The ` theta ` function is ... |
chpchtlim 26627 | The ` psi ` and ` theta ` ... |
chpo1ub 26628 | The ` psi ` function is up... |
chpo1ubb 26629 | The ` psi ` function is up... |
vmadivsum 26630 | The sum of the von Mangold... |
vmadivsumb 26631 | Give a total bound on the ... |
rplogsumlem1 26632 | Lemma for ~ rplogsum . (C... |
rplogsumlem2 26633 | Lemma for ~ rplogsum . Eq... |
dchrisum0lem1a 26634 | Lemma for ~ dchrisum0lem1 ... |
rpvmasumlem 26635 | Lemma for ~ rpvmasum . Ca... |
dchrisumlema 26636 | Lemma for ~ dchrisum . Le... |
dchrisumlem1 26637 | Lemma for ~ dchrisum . Le... |
dchrisumlem2 26638 | Lemma for ~ dchrisum . Le... |
dchrisumlem3 26639 | Lemma for ~ dchrisum . Le... |
dchrisum 26640 | If ` n e. [ M , +oo ) |-> ... |
dchrmusumlema 26641 | Lemma for ~ dchrmusum and ... |
dchrmusum2 26642 | The sum of the Möbius... |
dchrvmasumlem1 26643 | An alternative expression ... |
dchrvmasum2lem 26644 | Give an expression for ` l... |
dchrvmasum2if 26645 | Combine the results of ~ d... |
dchrvmasumlem2 26646 | Lemma for ~ dchrvmasum . ... |
dchrvmasumlem3 26647 | Lemma for ~ dchrvmasum . ... |
dchrvmasumlema 26648 | Lemma for ~ dchrvmasum and... |
dchrvmasumiflem1 26649 | Lemma for ~ dchrvmasumif .... |
dchrvmasumiflem2 26650 | Lemma for ~ dchrvmasum . ... |
dchrvmasumif 26651 | An asymptotic approximatio... |
dchrvmaeq0 26652 | The set ` W ` is the colle... |
dchrisum0fval 26653 | Value of the function ` F ... |
dchrisum0fmul 26654 | The function ` F ` , the d... |
dchrisum0ff 26655 | The function ` F ` is a re... |
dchrisum0flblem1 26656 | Lemma for ~ dchrisum0flb .... |
dchrisum0flblem2 26657 | Lemma for ~ dchrisum0flb .... |
dchrisum0flb 26658 | The divisor sum of a real ... |
dchrisum0fno1 26659 | The sum ` sum_ k <_ x , F ... |
rpvmasum2 26660 | A partial result along the... |
dchrisum0re 26661 | Suppose ` X ` is a non-pri... |
dchrisum0lema 26662 | Lemma for ~ dchrisum0 . A... |
dchrisum0lem1b 26663 | Lemma for ~ dchrisum0lem1 ... |
dchrisum0lem1 26664 | Lemma for ~ dchrisum0 . (... |
dchrisum0lem2a 26665 | Lemma for ~ dchrisum0 . (... |
dchrisum0lem2 26666 | Lemma for ~ dchrisum0 . (... |
dchrisum0lem3 26667 | Lemma for ~ dchrisum0 . (... |
dchrisum0 26668 | The sum ` sum_ n e. NN , X... |
dchrisumn0 26669 | The sum ` sum_ n e. NN , X... |
dchrmusumlem 26670 | The sum of the Möbius... |
dchrvmasumlem 26671 | The sum of the Möbius... |
dchrmusum 26672 | The sum of the Möbius... |
dchrvmasum 26673 | The sum of the von Mangold... |
rpvmasum 26674 | The sum of the von Mangold... |
rplogsum 26675 | The sum of ` log p / p ` o... |
dirith2 26676 | Dirichlet's theorem: there... |
dirith 26677 | Dirichlet's theorem: there... |
mudivsum 26678 | Asymptotic formula for ` s... |
mulogsumlem 26679 | Lemma for ~ mulogsum . (C... |
mulogsum 26680 | Asymptotic formula for ... |
logdivsum 26681 | Asymptotic analysis of ... |
mulog2sumlem1 26682 | Asymptotic formula for ... |
mulog2sumlem2 26683 | Lemma for ~ mulog2sum . (... |
mulog2sumlem3 26684 | Lemma for ~ mulog2sum . (... |
mulog2sum 26685 | Asymptotic formula for ... |
vmalogdivsum2 26686 | The sum ` sum_ n <_ x , La... |
vmalogdivsum 26687 | The sum ` sum_ n <_ x , La... |
2vmadivsumlem 26688 | Lemma for ~ 2vmadivsum . ... |
2vmadivsum 26689 | The sum ` sum_ m n <_ x , ... |
logsqvma 26690 | A formula for ` log ^ 2 ( ... |
logsqvma2 26691 | The Möbius inverse of... |
log2sumbnd 26692 | Bound on the difference be... |
selberglem1 26693 | Lemma for ~ selberg . Est... |
selberglem2 26694 | Lemma for ~ selberg . (Co... |
selberglem3 26695 | Lemma for ~ selberg . Est... |
selberg 26696 | Selberg's symmetry formula... |
selbergb 26697 | Convert eventual boundedne... |
selberg2lem 26698 | Lemma for ~ selberg2 . Eq... |
selberg2 26699 | Selberg's symmetry formula... |
selberg2b 26700 | Convert eventual boundedne... |
chpdifbndlem1 26701 | Lemma for ~ chpdifbnd . (... |
chpdifbndlem2 26702 | Lemma for ~ chpdifbnd . (... |
chpdifbnd 26703 | A bound on the difference ... |
logdivbnd 26704 | A bound on a sum of logs, ... |
selberg3lem1 26705 | Introduce a log weighting ... |
selberg3lem2 26706 | Lemma for ~ selberg3 . Eq... |
selberg3 26707 | Introduce a log weighting ... |
selberg4lem1 26708 | Lemma for ~ selberg4 . Eq... |
selberg4 26709 | The Selberg symmetry formu... |
pntrval 26710 | Define the residual of the... |
pntrf 26711 | Functionality of the resid... |
pntrmax 26712 | There is a bound on the re... |
pntrsumo1 26713 | A bound on a sum over ` R ... |
pntrsumbnd 26714 | A bound on a sum over ` R ... |
pntrsumbnd2 26715 | A bound on a sum over ` R ... |
selbergr 26716 | Selberg's symmetry formula... |
selberg3r 26717 | Selberg's symmetry formula... |
selberg4r 26718 | Selberg's symmetry formula... |
selberg34r 26719 | The sum of ~ selberg3r and... |
pntsval 26720 | Define the "Selberg functi... |
pntsf 26721 | Functionality of the Selbe... |
selbergs 26722 | Selberg's symmetry formula... |
selbergsb 26723 | Selberg's symmetry formula... |
pntsval2 26724 | The Selberg function can b... |
pntrlog2bndlem1 26725 | The sum of ~ selberg3r and... |
pntrlog2bndlem2 26726 | Lemma for ~ pntrlog2bnd . ... |
pntrlog2bndlem3 26727 | Lemma for ~ pntrlog2bnd . ... |
pntrlog2bndlem4 26728 | Lemma for ~ pntrlog2bnd . ... |
pntrlog2bndlem5 26729 | Lemma for ~ pntrlog2bnd . ... |
pntrlog2bndlem6a 26730 | Lemma for ~ pntrlog2bndlem... |
pntrlog2bndlem6 26731 | Lemma for ~ pntrlog2bnd . ... |
pntrlog2bnd 26732 | A bound on ` R ( x ) log ^... |
pntpbnd1a 26733 | Lemma for ~ pntpbnd . (Co... |
pntpbnd1 26734 | Lemma for ~ pntpbnd . (Co... |
pntpbnd2 26735 | Lemma for ~ pntpbnd . (Co... |
pntpbnd 26736 | Lemma for ~ pnt . Establi... |
pntibndlem1 26737 | Lemma for ~ pntibnd . (Co... |
pntibndlem2a 26738 | Lemma for ~ pntibndlem2 . ... |
pntibndlem2 26739 | Lemma for ~ pntibnd . The... |
pntibndlem3 26740 | Lemma for ~ pntibnd . Pac... |
pntibnd 26741 | Lemma for ~ pnt . Establi... |
pntlemd 26742 | Lemma for ~ pnt . Closure... |
pntlemc 26743 | Lemma for ~ pnt . Closure... |
pntlema 26744 | Lemma for ~ pnt . Closure... |
pntlemb 26745 | Lemma for ~ pnt . Unpack ... |
pntlemg 26746 | Lemma for ~ pnt . Closure... |
pntlemh 26747 | Lemma for ~ pnt . Bounds ... |
pntlemn 26748 | Lemma for ~ pnt . The "na... |
pntlemq 26749 | Lemma for ~ pntlemj . (Co... |
pntlemr 26750 | Lemma for ~ pntlemj . (Co... |
pntlemj 26751 | Lemma for ~ pnt . The ind... |
pntlemi 26752 | Lemma for ~ pnt . Elimina... |
pntlemf 26753 | Lemma for ~ pnt . Add up ... |
pntlemk 26754 | Lemma for ~ pnt . Evaluat... |
pntlemo 26755 | Lemma for ~ pnt . Combine... |
pntleme 26756 | Lemma for ~ pnt . Package... |
pntlem3 26757 | Lemma for ~ pnt . Equatio... |
pntlemp 26758 | Lemma for ~ pnt . Wrappin... |
pntleml 26759 | Lemma for ~ pnt . Equatio... |
pnt3 26760 | The Prime Number Theorem, ... |
pnt2 26761 | The Prime Number Theorem, ... |
pnt 26762 | The Prime Number Theorem: ... |
abvcxp 26763 | Raising an absolute value ... |
padicfval 26764 | Value of the p-adic absolu... |
padicval 26765 | Value of the p-adic absolu... |
ostth2lem1 26766 | Lemma for ~ ostth2 , altho... |
qrngbas 26767 | The base set of the field ... |
qdrng 26768 | The rationals form a divis... |
qrng0 26769 | The zero element of the fi... |
qrng1 26770 | The unit element of the fi... |
qrngneg 26771 | The additive inverse in th... |
qrngdiv 26772 | The division operation in ... |
qabvle 26773 | By using induction on ` N ... |
qabvexp 26774 | Induct the product rule ~ ... |
ostthlem1 26775 | Lemma for ~ ostth . If tw... |
ostthlem2 26776 | Lemma for ~ ostth . Refin... |
qabsabv 26777 | The regular absolute value... |
padicabv 26778 | The p-adic absolute value ... |
padicabvf 26779 | The p-adic absolute value ... |
padicabvcxp 26780 | All positive powers of the... |
ostth1 26781 | - Lemma for ~ ostth : triv... |
ostth2lem2 26782 | Lemma for ~ ostth2 . (Con... |
ostth2lem3 26783 | Lemma for ~ ostth2 . (Con... |
ostth2lem4 26784 | Lemma for ~ ostth2 . (Con... |
ostth2 26785 | - Lemma for ~ ostth : regu... |
ostth3 26786 | - Lemma for ~ ostth : p-ad... |
ostth 26787 | Ostrowski's theorem, which... |
itvndx 26798 | Index value of the Interva... |
lngndx 26799 | Index value of the "line" ... |
itvid 26800 | Utility theorem: index-ind... |
lngid 26801 | Utility theorem: index-ind... |
slotsinbpsd 26802 | The slots ` Base ` , ` +g ... |
slotslnbpsd 26803 | The slots ` Base ` , ` +g ... |
lngndxnitvndx 26804 | The slot for the line is n... |
trkgstr 26805 | Functionality of a Tarski ... |
trkgbas 26806 | The base set of a Tarski g... |
trkgdist 26807 | The measure of a distance ... |
trkgitv 26808 | The congruence relation in... |
istrkgc 26815 | Property of being a Tarski... |
istrkgb 26816 | Property of being a Tarski... |
istrkgcb 26817 | Property of being a Tarski... |
istrkge 26818 | Property of fulfilling Euc... |
istrkgl 26819 | Building lines from the se... |
istrkgld 26820 | Property of fulfilling the... |
istrkg2ld 26821 | Property of fulfilling the... |
istrkg3ld 26822 | Property of fulfilling the... |
axtgcgrrflx 26823 | Axiom of reflexivity of co... |
axtgcgrid 26824 | Axiom of identity of congr... |
axtgsegcon 26825 | Axiom of segment construct... |
axtg5seg 26826 | Five segments axiom, Axiom... |
axtgbtwnid 26827 | Identity of Betweenness. ... |
axtgpasch 26828 | Axiom of (Inner) Pasch, Ax... |
axtgcont1 26829 | Axiom of Continuity. Axio... |
axtgcont 26830 | Axiom of Continuity. Axio... |
axtglowdim2 26831 | Lower dimension axiom for ... |
axtgupdim2 26832 | Upper dimension axiom for ... |
axtgeucl 26833 | Euclid's Axiom. Axiom A10... |
tgjustf 26834 | Given any function ` F ` ,... |
tgjustr 26835 | Given any equivalence rela... |
tgjustc1 26836 | A justification for using ... |
tgjustc2 26837 | A justification for using ... |
tgcgrcomimp 26838 | Congruence commutes on the... |
tgcgrcomr 26839 | Congruence commutes on the... |
tgcgrcoml 26840 | Congruence commutes on the... |
tgcgrcomlr 26841 | Congruence commutes on bot... |
tgcgreqb 26842 | Congruence and equality. ... |
tgcgreq 26843 | Congruence and equality. ... |
tgcgrneq 26844 | Congruence and equality. ... |
tgcgrtriv 26845 | Degenerate segments are co... |
tgcgrextend 26846 | Link congruence over a pai... |
tgsegconeq 26847 | Two points that satisfy th... |
tgbtwntriv2 26848 | Betweenness always holds f... |
tgbtwncom 26849 | Betweenness commutes. The... |
tgbtwncomb 26850 | Betweenness commutes, bico... |
tgbtwnne 26851 | Betweenness and inequality... |
tgbtwntriv1 26852 | Betweenness always holds f... |
tgbtwnswapid 26853 | If you can swap the first ... |
tgbtwnintr 26854 | Inner transitivity law for... |
tgbtwnexch3 26855 | Exchange the first endpoin... |
tgbtwnouttr2 26856 | Outer transitivity law for... |
tgbtwnexch2 26857 | Exchange the outer point o... |
tgbtwnouttr 26858 | Outer transitivity law for... |
tgbtwnexch 26859 | Outer transitivity law for... |
tgtrisegint 26860 | A line segment between two... |
tglowdim1 26861 | Lower dimension axiom for ... |
tglowdim1i 26862 | Lower dimension axiom for ... |
tgldimor 26863 | Excluded-middle like state... |
tgldim0eq 26864 | In dimension zero, any two... |
tgldim0itv 26865 | In dimension zero, any two... |
tgldim0cgr 26866 | In dimension zero, any two... |
tgbtwndiff 26867 | There is always a ` c ` di... |
tgdim01 26868 | In geometries of dimension... |
tgifscgr 26869 | Inner five segment congrue... |
tgcgrsub 26870 | Removing identical parts f... |
iscgrg 26873 | The congruence property fo... |
iscgrgd 26874 | The property for two seque... |
iscgrglt 26875 | The property for two seque... |
trgcgrg 26876 | The property for two trian... |
trgcgr 26877 | Triangle congruence. (Con... |
ercgrg 26878 | The shape congruence relat... |
tgcgrxfr 26879 | A line segment can be divi... |
cgr3id 26880 | Reflexivity law for three-... |
cgr3simp1 26881 | Deduce segment congruence ... |
cgr3simp2 26882 | Deduce segment congruence ... |
cgr3simp3 26883 | Deduce segment congruence ... |
cgr3swap12 26884 | Permutation law for three-... |
cgr3swap23 26885 | Permutation law for three-... |
cgr3swap13 26886 | Permutation law for three-... |
cgr3rotr 26887 | Permutation law for three-... |
cgr3rotl 26888 | Permutation law for three-... |
trgcgrcom 26889 | Commutative law for three-... |
cgr3tr 26890 | Transitivity law for three... |
tgbtwnxfr 26891 | A condition for extending ... |
tgcgr4 26892 | Two quadrilaterals to be c... |
isismt 26895 | Property of being an isome... |
ismot 26896 | Property of being an isome... |
motcgr 26897 | Property of a motion: dist... |
idmot 26898 | The identity is a motion. ... |
motf1o 26899 | Motions are bijections. (... |
motcl 26900 | Closure of motions. (Cont... |
motco 26901 | The composition of two mot... |
cnvmot 26902 | The converse of a motion i... |
motplusg 26903 | The operation for motions ... |
motgrp 26904 | The motions of a geometry ... |
motcgrg 26905 | Property of a motion: dist... |
motcgr3 26906 | Property of a motion: dist... |
tglng 26907 | Lines of a Tarski Geometry... |
tglnfn 26908 | Lines as functions. (Cont... |
tglnunirn 26909 | Lines are sets of points. ... |
tglnpt 26910 | Lines are sets of points. ... |
tglngne 26911 | It takes two different poi... |
tglngval 26912 | The line going through poi... |
tglnssp 26913 | Lines are subset of the ge... |
tgellng 26914 | Property of lying on the l... |
tgcolg 26915 | We choose the notation ` (... |
btwncolg1 26916 | Betweenness implies coline... |
btwncolg2 26917 | Betweenness implies coline... |
btwncolg3 26918 | Betweenness implies coline... |
colcom 26919 | Swapping the points defini... |
colrot1 26920 | Rotating the points defini... |
colrot2 26921 | Rotating the points defini... |
ncolcom 26922 | Swapping non-colinear poin... |
ncolrot1 26923 | Rotating non-colinear poin... |
ncolrot2 26924 | Rotating non-colinear poin... |
tgdim01ln 26925 | In geometries of dimension... |
ncoltgdim2 26926 | If there are three non-col... |
lnxfr 26927 | Transfer law for colineari... |
lnext 26928 | Extend a line with a missi... |
tgfscgr 26929 | Congruence law for the gen... |
lncgr 26930 | Congruence rule for lines.... |
lnid 26931 | Identity law for points on... |
tgidinside 26932 | Law for finding a point in... |
tgbtwnconn1lem1 26933 | Lemma for ~ tgbtwnconn1 . ... |
tgbtwnconn1lem2 26934 | Lemma for ~ tgbtwnconn1 . ... |
tgbtwnconn1lem3 26935 | Lemma for ~ tgbtwnconn1 . ... |
tgbtwnconn1 26936 | Connectivity law for betwe... |
tgbtwnconn2 26937 | Another connectivity law f... |
tgbtwnconn3 26938 | Inner connectivity law for... |
tgbtwnconnln3 26939 | Derive colinearity from be... |
tgbtwnconn22 26940 | Double connectivity law fo... |
tgbtwnconnln1 26941 | Derive colinearity from be... |
tgbtwnconnln2 26942 | Derive colinearity from be... |
legval 26945 | Value of the less-than rel... |
legov 26946 | Value of the less-than rel... |
legov2 26947 | An equivalent definition o... |
legid 26948 | Reflexivity of the less-th... |
btwnleg 26949 | Betweenness implies less-t... |
legtrd 26950 | Transitivity of the less-t... |
legtri3 26951 | Equality from the less-tha... |
legtrid 26952 | Trichotomy law for the les... |
leg0 26953 | Degenerated (zero-length) ... |
legeq 26954 | Deduce equality from "less... |
legbtwn 26955 | Deduce betweenness from "l... |
tgcgrsub2 26956 | Removing identical parts f... |
ltgseg 26957 | The set ` E ` denotes the ... |
ltgov 26958 | Strict "shorter than" geom... |
legov3 26959 | An equivalent definition o... |
legso 26960 | The "shorter than" relatio... |
ishlg 26963 | Rays : Definition 6.1 of ... |
hlcomb 26964 | The half-line relation com... |
hlcomd 26965 | The half-line relation com... |
hlne1 26966 | The half-line relation imp... |
hlne2 26967 | The half-line relation imp... |
hlln 26968 | The half-line relation imp... |
hleqnid 26969 | The endpoint does not belo... |
hlid 26970 | The half-line relation is ... |
hltr 26971 | The half-line relation is ... |
hlbtwn 26972 | Betweenness is a sufficien... |
btwnhl1 26973 | Deduce half-line from betw... |
btwnhl2 26974 | Deduce half-line from betw... |
btwnhl 26975 | Swap betweenness for a hal... |
lnhl 26976 | Either a point ` C ` on th... |
hlcgrex 26977 | Construct a point on a hal... |
hlcgreulem 26978 | Lemma for ~ hlcgreu . (Co... |
hlcgreu 26979 | The point constructed in ~... |
btwnlng1 26980 | Betweenness implies coline... |
btwnlng2 26981 | Betweenness implies coline... |
btwnlng3 26982 | Betweenness implies coline... |
lncom 26983 | Swapping the points defini... |
lnrot1 26984 | Rotating the points defini... |
lnrot2 26985 | Rotating the points defini... |
ncolne1 26986 | Non-colinear points are di... |
ncolne2 26987 | Non-colinear points are di... |
tgisline 26988 | The property of being a pr... |
tglnne 26989 | It takes two different poi... |
tglndim0 26990 | There are no lines in dime... |
tgelrnln 26991 | The property of being a pr... |
tglineeltr 26992 | Transitivity law for lines... |
tglineelsb2 26993 | If ` S ` lies on PQ , then... |
tglinerflx1 26994 | Reflexivity law for line m... |
tglinerflx2 26995 | Reflexivity law for line m... |
tglinecom 26996 | Commutativity law for line... |
tglinethru 26997 | If ` A ` is a line contain... |
tghilberti1 26998 | There is a line through an... |
tghilberti2 26999 | There is at most one line ... |
tglinethrueu 27000 | There is a unique line goi... |
tglnne0 27001 | A line ` A ` has at least ... |
tglnpt2 27002 | Find a second point on a l... |
tglineintmo 27003 | Two distinct lines interse... |
tglineineq 27004 | Two distinct lines interse... |
tglineneq 27005 | Given three non-colinear p... |
tglineinteq 27006 | Two distinct lines interse... |
ncolncol 27007 | Deduce non-colinearity fro... |
coltr 27008 | A transitivity law for col... |
coltr3 27009 | A transitivity law for col... |
colline 27010 | Three points are colinear ... |
tglowdim2l 27011 | Reformulation of the lower... |
tglowdim2ln 27012 | There is always one point ... |
mirreu3 27015 | Existential uniqueness of ... |
mirval 27016 | Value of the point inversi... |
mirfv 27017 | Value of the point inversi... |
mircgr 27018 | Property of the image by t... |
mirbtwn 27019 | Property of the image by t... |
ismir 27020 | Property of the image by t... |
mirf 27021 | Point inversion as functio... |
mircl 27022 | Closure of the point inver... |
mirmir 27023 | The point inversion functi... |
mircom 27024 | Variation on ~ mirmir . (... |
mirreu 27025 | Any point has a unique ant... |
mireq 27026 | Equality deduction for poi... |
mirinv 27027 | The only invariant point o... |
mirne 27028 | Mirror of non-center point... |
mircinv 27029 | The center point is invari... |
mirf1o 27030 | The point inversion functi... |
miriso 27031 | The point inversion functi... |
mirbtwni 27032 | Point inversion preserves ... |
mirbtwnb 27033 | Point inversion preserves ... |
mircgrs 27034 | Point inversion preserves ... |
mirmir2 27035 | Point inversion of a point... |
mirmot 27036 | Point investion is a motio... |
mirln 27037 | If two points are on the s... |
mirln2 27038 | If a point and its mirror ... |
mirconn 27039 | Point inversion of connect... |
mirhl 27040 | If two points ` X ` and ` ... |
mirbtwnhl 27041 | If the center of the point... |
mirhl2 27042 | Deduce half-line relation ... |
mircgrextend 27043 | Link congruence over a pai... |
mirtrcgr 27044 | Point inversion of one poi... |
mirauto 27045 | Point inversion preserves ... |
miduniq 27046 | Uniqueness of the middle p... |
miduniq1 27047 | Uniqueness of the middle p... |
miduniq2 27048 | If two point inversions co... |
colmid 27049 | Colinearity and equidistan... |
symquadlem 27050 | Lemma of the symetrial qua... |
krippenlem 27051 | Lemma for ~ krippen . We ... |
krippen 27052 | Krippenlemma (German for c... |
midexlem 27053 | Lemma for the existence of... |
israg 27058 | Property for 3 points A, B... |
ragcom 27059 | Commutative rule for right... |
ragcol 27060 | The right angle property i... |
ragmir 27061 | Right angle property is pr... |
mirrag 27062 | Right angle is conserved b... |
ragtrivb 27063 | Trivial right angle. Theo... |
ragflat2 27064 | Deduce equality from two r... |
ragflat 27065 | Deduce equality from two r... |
ragtriva 27066 | Trivial right angle. Theo... |
ragflat3 27067 | Right angle and colinearit... |
ragcgr 27068 | Right angle and colinearit... |
motrag 27069 | Right angles are preserved... |
ragncol 27070 | Right angle implies non-co... |
perpln1 27071 | Derive a line from perpend... |
perpln2 27072 | Derive a line from perpend... |
isperp 27073 | Property for 2 lines A, B ... |
perpcom 27074 | The "perpendicular" relati... |
perpneq 27075 | Two perpendicular lines ar... |
isperp2 27076 | Property for 2 lines A, B,... |
isperp2d 27077 | One direction of ~ isperp2... |
ragperp 27078 | Deduce that two lines are ... |
footexALT 27079 | Alternative version of ~ f... |
footexlem1 27080 | Lemma for ~ footex . (Con... |
footexlem2 27081 | Lemma for ~ footex . (Con... |
footex 27082 | From a point ` C ` outside... |
foot 27083 | From a point ` C ` outside... |
footne 27084 | Uniqueness of the foot poi... |
footeq 27085 | Uniqueness of the foot poi... |
hlperpnel 27086 | A point on a half-line whi... |
perprag 27087 | Deduce a right angle from ... |
perpdragALT 27088 | Deduce a right angle from ... |
perpdrag 27089 | Deduce a right angle from ... |
colperp 27090 | Deduce a perpendicularity ... |
colperpexlem1 27091 | Lemma for ~ colperp . Fir... |
colperpexlem2 27092 | Lemma for ~ colperpex . S... |
colperpexlem3 27093 | Lemma for ~ colperpex . C... |
colperpex 27094 | In dimension 2 and above, ... |
mideulem2 27095 | Lemma for ~ opphllem , whi... |
opphllem 27096 | Lemma 8.24 of [Schwabhause... |
mideulem 27097 | Lemma for ~ mideu . We ca... |
midex 27098 | Existence of the midpoint,... |
mideu 27099 | Existence and uniqueness o... |
islnopp 27100 | The property for two point... |
islnoppd 27101 | Deduce that ` A ` and ` B ... |
oppne1 27102 | Points lying on opposite s... |
oppne2 27103 | Points lying on opposite s... |
oppne3 27104 | Points lying on opposite s... |
oppcom 27105 | Commutativity rule for "op... |
opptgdim2 27106 | If two points opposite to ... |
oppnid 27107 | The "opposite to a line" r... |
opphllem1 27108 | Lemma for ~ opphl . (Cont... |
opphllem2 27109 | Lemma for ~ opphl . Lemma... |
opphllem3 27110 | Lemma for ~ opphl : We as... |
opphllem4 27111 | Lemma for ~ opphl . (Cont... |
opphllem5 27112 | Second part of Lemma 9.4 o... |
opphllem6 27113 | First part of Lemma 9.4 of... |
oppperpex 27114 | Restating ~ colperpex usin... |
opphl 27115 | If two points ` A ` and ` ... |
outpasch 27116 | Axiom of Pasch, outer form... |
hlpasch 27117 | An application of the axio... |
ishpg 27120 | Value of the half-plane re... |
hpgbr 27121 | Half-planes : property for... |
hpgne1 27122 | Points on the open half pl... |
hpgne2 27123 | Points on the open half pl... |
lnopp2hpgb 27124 | Theorem 9.8 of [Schwabhaus... |
lnoppnhpg 27125 | If two points lie on the o... |
hpgerlem 27126 | Lemma for the proof that t... |
hpgid 27127 | The half-plane relation is... |
hpgcom 27128 | The half-plane relation co... |
hpgtr 27129 | The half-plane relation is... |
colopp 27130 | Opposite sides of a line f... |
colhp 27131 | Half-plane relation for co... |
hphl 27132 | If two points are on the s... |
midf 27137 | Midpoint as a function. (... |
midcl 27138 | Closure of the midpoint. ... |
ismidb 27139 | Property of the midpoint. ... |
midbtwn 27140 | Betweenness of midpoint. ... |
midcgr 27141 | Congruence of midpoint. (... |
midid 27142 | Midpoint of a null segment... |
midcom 27143 | Commutativity rule for the... |
mirmid 27144 | Point inversion preserves ... |
lmieu 27145 | Uniqueness of the line mir... |
lmif 27146 | Line mirror as a function.... |
lmicl 27147 | Closure of the line mirror... |
islmib 27148 | Property of the line mirro... |
lmicom 27149 | The line mirroring functio... |
lmilmi 27150 | Line mirroring is an invol... |
lmireu 27151 | Any point has a unique ant... |
lmieq 27152 | Equality deduction for lin... |
lmiinv 27153 | The invariants of the line... |
lmicinv 27154 | The mirroring line is an i... |
lmimid 27155 | If we have a right angle, ... |
lmif1o 27156 | The line mirroring functio... |
lmiisolem 27157 | Lemma for ~ lmiiso . (Con... |
lmiiso 27158 | The line mirroring functio... |
lmimot 27159 | Line mirroring is a motion... |
hypcgrlem1 27160 | Lemma for ~ hypcgr , case ... |
hypcgrlem2 27161 | Lemma for ~ hypcgr , case ... |
hypcgr 27162 | If the catheti of two righ... |
lmiopp 27163 | Line mirroring produces po... |
lnperpex 27164 | Existence of a perpendicul... |
trgcopy 27165 | Triangle construction: a c... |
trgcopyeulem 27166 | Lemma for ~ trgcopyeu . (... |
trgcopyeu 27167 | Triangle construction: a c... |
iscgra 27170 | Property for two angles AB... |
iscgra1 27171 | A special version of ~ isc... |
iscgrad 27172 | Sufficient conditions for ... |
cgrane1 27173 | Angles imply inequality. ... |
cgrane2 27174 | Angles imply inequality. ... |
cgrane3 27175 | Angles imply inequality. ... |
cgrane4 27176 | Angles imply inequality. ... |
cgrahl1 27177 | Angle congruence is indepe... |
cgrahl2 27178 | Angle congruence is indepe... |
cgracgr 27179 | First direction of proposi... |
cgraid 27180 | Angle congruence is reflex... |
cgraswap 27181 | Swap rays in a congruence ... |
cgrcgra 27182 | Triangle congruence implie... |
cgracom 27183 | Angle congruence commutes.... |
cgratr 27184 | Angle congruence is transi... |
flatcgra 27185 | Flat angles are congruent.... |
cgraswaplr 27186 | Swap both side of angle co... |
cgrabtwn 27187 | Angle congruence preserves... |
cgrahl 27188 | Angle congruence preserves... |
cgracol 27189 | Angle congruence preserves... |
cgrancol 27190 | Angle congruence preserves... |
dfcgra2 27191 | This is the full statement... |
sacgr 27192 | Supplementary angles of co... |
oacgr 27193 | Vertical angle theorem. V... |
acopy 27194 | Angle construction. Theor... |
acopyeu 27195 | Angle construction. Theor... |
isinag 27199 | Property for point ` X ` t... |
isinagd 27200 | Sufficient conditions for ... |
inagflat 27201 | Any point lies in a flat a... |
inagswap 27202 | Swap the order of the half... |
inagne1 27203 | Deduce inequality from the... |
inagne2 27204 | Deduce inequality from the... |
inagne3 27205 | Deduce inequality from the... |
inaghl 27206 | The "point lie in angle" r... |
isleag 27208 | Geometrical "less than" pr... |
isleagd 27209 | Sufficient condition for "... |
leagne1 27210 | Deduce inequality from the... |
leagne2 27211 | Deduce inequality from the... |
leagne3 27212 | Deduce inequality from the... |
leagne4 27213 | Deduce inequality from the... |
cgrg3col4 27214 | Lemma 11.28 of [Schwabhaus... |
tgsas1 27215 | First congruence theorem: ... |
tgsas 27216 | First congruence theorem: ... |
tgsas2 27217 | First congruence theorem: ... |
tgsas3 27218 | First congruence theorem: ... |
tgasa1 27219 | Second congruence theorem:... |
tgasa 27220 | Second congruence theorem:... |
tgsss1 27221 | Third congruence theorem: ... |
tgsss2 27222 | Third congruence theorem: ... |
tgsss3 27223 | Third congruence theorem: ... |
dfcgrg2 27224 | Congruence for two triangl... |
isoas 27225 | Congruence theorem for iso... |
iseqlg 27228 | Property of a triangle bei... |
iseqlgd 27229 | Condition for a triangle t... |
f1otrgds 27230 | Convenient lemma for ~ f1o... |
f1otrgitv 27231 | Convenient lemma for ~ f1o... |
f1otrg 27232 | A bijection between bases ... |
f1otrge 27233 | A bijection between bases ... |
ttgval 27236 | Define a function to augme... |
ttgvalOLD 27237 | Obsolete proof of ~ ttgval... |
ttglem 27238 | Lemma for ~ ttgbas , ~ ttg... |
ttglemOLD 27239 | Obsolete version of ~ ttgl... |
ttgbas 27240 | The base set of a subcompl... |
ttgbasOLD 27241 | Obsolete proof of ~ ttgbas... |
ttgplusg 27242 | The addition operation of ... |
ttgplusgOLD 27243 | Obsolete proof of ~ ttgplu... |
ttgsub 27244 | The subtraction operation ... |
ttgvsca 27245 | The scalar product of a su... |
ttgvscaOLD 27246 | Obsolete proof of ~ ttgvsc... |
ttgds 27247 | The metric of a subcomplex... |
ttgdsOLD 27248 | Obsolete proof of ~ ttgds ... |
ttgitvval 27249 | Betweenness for a subcompl... |
ttgelitv 27250 | Betweenness for a subcompl... |
ttgbtwnid 27251 | Any subcomplex module equi... |
ttgcontlem1 27252 | Lemma for % ttgcont . (Co... |
xmstrkgc 27253 | Any metric space fulfills ... |
cchhllem 27254 | Lemma for chlbas and chlvs... |
cchhllemOLD 27255 | Obsolete version of ~ cchh... |
elee 27262 | Membership in a Euclidean ... |
mptelee 27263 | A condition for a mapping ... |
eleenn 27264 | If ` A ` is in ` ( EE `` N... |
eleei 27265 | The forward direction of ~... |
eedimeq 27266 | A point belongs to at most... |
brbtwn 27267 | The binary relation form o... |
brcgr 27268 | The binary relation form o... |
fveere 27269 | The function value of a po... |
fveecn 27270 | The function value of a po... |
eqeefv 27271 | Two points are equal iff t... |
eqeelen 27272 | Two points are equal iff t... |
brbtwn2 27273 | Alternate characterization... |
colinearalglem1 27274 | Lemma for ~ colinearalg . ... |
colinearalglem2 27275 | Lemma for ~ colinearalg . ... |
colinearalglem3 27276 | Lemma for ~ colinearalg . ... |
colinearalglem4 27277 | Lemma for ~ colinearalg . ... |
colinearalg 27278 | An algebraic characterizat... |
eleesub 27279 | Membership of a subtractio... |
eleesubd 27280 | Membership of a subtractio... |
axdimuniq 27281 | The unique dimension axiom... |
axcgrrflx 27282 | ` A ` is as far from ` B `... |
axcgrtr 27283 | Congruence is transitive. ... |
axcgrid 27284 | If there is no distance be... |
axsegconlem1 27285 | Lemma for ~ axsegcon . Ha... |
axsegconlem2 27286 | Lemma for ~ axsegcon . Sh... |
axsegconlem3 27287 | Lemma for ~ axsegcon . Sh... |
axsegconlem4 27288 | Lemma for ~ axsegcon . Sh... |
axsegconlem5 27289 | Lemma for ~ axsegcon . Sh... |
axsegconlem6 27290 | Lemma for ~ axsegcon . Sh... |
axsegconlem7 27291 | Lemma for ~ axsegcon . Sh... |
axsegconlem8 27292 | Lemma for ~ axsegcon . Sh... |
axsegconlem9 27293 | Lemma for ~ axsegcon . Sh... |
axsegconlem10 27294 | Lemma for ~ axsegcon . Sh... |
axsegcon 27295 | Any segment ` A B ` can be... |
ax5seglem1 27296 | Lemma for ~ ax5seg . Rexp... |
ax5seglem2 27297 | Lemma for ~ ax5seg . Rexp... |
ax5seglem3a 27298 | Lemma for ~ ax5seg . (Con... |
ax5seglem3 27299 | Lemma for ~ ax5seg . Comb... |
ax5seglem4 27300 | Lemma for ~ ax5seg . Give... |
ax5seglem5 27301 | Lemma for ~ ax5seg . If `... |
ax5seglem6 27302 | Lemma for ~ ax5seg . Give... |
ax5seglem7 27303 | Lemma for ~ ax5seg . An a... |
ax5seglem8 27304 | Lemma for ~ ax5seg . Use ... |
ax5seglem9 27305 | Lemma for ~ ax5seg . Take... |
ax5seg 27306 | The five segment axiom. T... |
axbtwnid 27307 | Points are indivisible. T... |
axpaschlem 27308 | Lemma for ~ axpasch . Set... |
axpasch 27309 | The inner Pasch axiom. Ta... |
axlowdimlem1 27310 | Lemma for ~ axlowdim . Es... |
axlowdimlem2 27311 | Lemma for ~ axlowdim . Sh... |
axlowdimlem3 27312 | Lemma for ~ axlowdim . Se... |
axlowdimlem4 27313 | Lemma for ~ axlowdim . Se... |
axlowdimlem5 27314 | Lemma for ~ axlowdim . Sh... |
axlowdimlem6 27315 | Lemma for ~ axlowdim . Sh... |
axlowdimlem7 27316 | Lemma for ~ axlowdim . Se... |
axlowdimlem8 27317 | Lemma for ~ axlowdim . Ca... |
axlowdimlem9 27318 | Lemma for ~ axlowdim . Ca... |
axlowdimlem10 27319 | Lemma for ~ axlowdim . Se... |
axlowdimlem11 27320 | Lemma for ~ axlowdim . Ca... |
axlowdimlem12 27321 | Lemma for ~ axlowdim . Ca... |
axlowdimlem13 27322 | Lemma for ~ axlowdim . Es... |
axlowdimlem14 27323 | Lemma for ~ axlowdim . Ta... |
axlowdimlem15 27324 | Lemma for ~ axlowdim . Se... |
axlowdimlem16 27325 | Lemma for ~ axlowdim . Se... |
axlowdimlem17 27326 | Lemma for ~ axlowdim . Es... |
axlowdim1 27327 | The lower dimension axiom ... |
axlowdim2 27328 | The lower two-dimensional ... |
axlowdim 27329 | The general lower dimensio... |
axeuclidlem 27330 | Lemma for ~ axeuclid . Ha... |
axeuclid 27331 | Euclid's axiom. Take an a... |
axcontlem1 27332 | Lemma for ~ axcont . Chan... |
axcontlem2 27333 | Lemma for ~ axcont . The ... |
axcontlem3 27334 | Lemma for ~ axcont . Give... |
axcontlem4 27335 | Lemma for ~ axcont . Give... |
axcontlem5 27336 | Lemma for ~ axcont . Comp... |
axcontlem6 27337 | Lemma for ~ axcont . Stat... |
axcontlem7 27338 | Lemma for ~ axcont . Give... |
axcontlem8 27339 | Lemma for ~ axcont . A po... |
axcontlem9 27340 | Lemma for ~ axcont . Give... |
axcontlem10 27341 | Lemma for ~ axcont . Give... |
axcontlem11 27342 | Lemma for ~ axcont . Elim... |
axcontlem12 27343 | Lemma for ~ axcont . Elim... |
axcont 27344 | The axiom of continuity. ... |
eengv 27347 | The value of the Euclidean... |
eengstr 27348 | The Euclidean geometry as ... |
eengbas 27349 | The Base of the Euclidean ... |
ebtwntg 27350 | The betweenness relation u... |
ecgrtg 27351 | The congruence relation us... |
elntg 27352 | The line definition in the... |
elntg2 27353 | The line definition in the... |
eengtrkg 27354 | The geometry structure for... |
eengtrkge 27355 | The geometry structure for... |
edgfid 27358 | Utility theorem: index-ind... |
edgfndx 27359 | Index value of the ~ df-ed... |
edgfndxnn 27360 | The index value of the edg... |
edgfndxid 27361 | The value of the edge func... |
edgfndxidOLD 27362 | Obsolete version of ~ edgf... |
basendxltedgfndx 27363 | The index value of the ` B... |
baseltedgfOLD 27364 | Obsolete proof of ~ basend... |
basendxnedgfndx 27365 | The slots ` Base ` and ` .... |
vtxval 27370 | The set of vertices of a g... |
iedgval 27371 | The set of indexed edges o... |
1vgrex 27372 | A graph with at least one ... |
opvtxval 27373 | The set of vertices of a g... |
opvtxfv 27374 | The set of vertices of a g... |
opvtxov 27375 | The set of vertices of a g... |
opiedgval 27376 | The set of indexed edges o... |
opiedgfv 27377 | The set of indexed edges o... |
opiedgov 27378 | The set of indexed edges o... |
opvtxfvi 27379 | The set of vertices of a g... |
opiedgfvi 27380 | The set of indexed edges o... |
funvtxdmge2val 27381 | The set of vertices of an ... |
funiedgdmge2val 27382 | The set of indexed edges o... |
funvtxdm2val 27383 | The set of vertices of an ... |
funiedgdm2val 27384 | The set of indexed edges o... |
funvtxval0 27385 | The set of vertices of an ... |
basvtxval 27386 | The set of vertices of a g... |
edgfiedgval 27387 | The set of indexed edges o... |
funvtxval 27388 | The set of vertices of a g... |
funiedgval 27389 | The set of indexed edges o... |
structvtxvallem 27390 | Lemma for ~ structvtxval a... |
structvtxval 27391 | The set of vertices of an ... |
structiedg0val 27392 | The set of indexed edges o... |
structgrssvtxlem 27393 | Lemma for ~ structgrssvtx ... |
structgrssvtx 27394 | The set of vertices of a g... |
structgrssiedg 27395 | The set of indexed edges o... |
struct2grstr 27396 | A graph represented as an ... |
struct2grvtx 27397 | The set of vertices of a g... |
struct2griedg 27398 | The set of indexed edges o... |
graop 27399 | Any representation of a gr... |
grastruct 27400 | Any representation of a gr... |
gropd 27401 | If any representation of a... |
grstructd 27402 | If any representation of a... |
gropeld 27403 | If any representation of a... |
grstructeld 27404 | If any representation of a... |
setsvtx 27405 | The vertices of a structur... |
setsiedg 27406 | The (indexed) edges of a s... |
snstrvtxval 27407 | The set of vertices of a g... |
snstriedgval 27408 | The set of indexed edges o... |
vtxval0 27409 | Degenerated case 1 for ver... |
iedgval0 27410 | Degenerated case 1 for edg... |
vtxvalsnop 27411 | Degenerated case 2 for ver... |
iedgvalsnop 27412 | Degenerated case 2 for edg... |
vtxval3sn 27413 | Degenerated case 3 for ver... |
iedgval3sn 27414 | Degenerated case 3 for edg... |
vtxvalprc 27415 | Degenerated case 4 for ver... |
iedgvalprc 27416 | Degenerated case 4 for edg... |
edgval 27419 | The edges of a graph. (Co... |
iedgedg 27420 | An indexed edge is an edge... |
edgopval 27421 | The edges of a graph repre... |
edgov 27422 | The edges of a graph repre... |
edgstruct 27423 | The edges of a graph repre... |
edgiedgb 27424 | A set is an edge iff it is... |
edg0iedg0 27425 | There is no edge in a grap... |
isuhgr 27430 | The predicate "is an undir... |
isushgr 27431 | The predicate "is an undir... |
uhgrf 27432 | The edge function of an un... |
ushgrf 27433 | The edge function of an un... |
uhgrss 27434 | An edge is a subset of ver... |
uhgreq12g 27435 | If two sets have the same ... |
uhgrfun 27436 | The edge function of an un... |
uhgrn0 27437 | An edge is a nonempty subs... |
lpvtx 27438 | The endpoints of a loop (w... |
ushgruhgr 27439 | An undirected simple hyper... |
isuhgrop 27440 | The property of being an u... |
uhgr0e 27441 | The empty graph, with vert... |
uhgr0vb 27442 | The null graph, with no ve... |
uhgr0 27443 | The null graph represented... |
uhgrun 27444 | The union ` U ` of two (un... |
uhgrunop 27445 | The union of two (undirect... |
ushgrun 27446 | The union ` U ` of two (un... |
ushgrunop 27447 | The union of two (undirect... |
uhgrstrrepe 27448 | Replacing (or adding) the ... |
incistruhgr 27449 | An _incidence structure_ `... |
isupgr 27454 | The property of being an u... |
wrdupgr 27455 | The property of being an u... |
upgrf 27456 | The edge function of an un... |
upgrfn 27457 | The edge function of an un... |
upgrss 27458 | An edge is a subset of ver... |
upgrn0 27459 | An edge is a nonempty subs... |
upgrle 27460 | An edge of an undirected p... |
upgrfi 27461 | An edge is a finite subset... |
upgrex 27462 | An edge is an unordered pa... |
upgrbi 27463 | Show that an unordered pai... |
upgrop 27464 | A pseudograph represented ... |
isumgr 27465 | The property of being an u... |
isumgrs 27466 | The simplified property of... |
wrdumgr 27467 | The property of being an u... |
umgrf 27468 | The edge function of an un... |
umgrfn 27469 | The edge function of an un... |
umgredg2 27470 | An edge of a multigraph ha... |
umgrbi 27471 | Show that an unordered pai... |
upgruhgr 27472 | An undirected pseudograph ... |
umgrupgr 27473 | An undirected multigraph i... |
umgruhgr 27474 | An undirected multigraph i... |
upgrle2 27475 | An edge of an undirected p... |
umgrnloopv 27476 | In a multigraph, there is ... |
umgredgprv 27477 | In a multigraph, an edge i... |
umgrnloop 27478 | In a multigraph, there is ... |
umgrnloop0 27479 | A multigraph has no loops.... |
umgr0e 27480 | The empty graph, with vert... |
upgr0e 27481 | The empty graph, with vert... |
upgr1elem 27482 | Lemma for ~ upgr1e and ~ u... |
upgr1e 27483 | A pseudograph with one edg... |
upgr0eop 27484 | The empty graph, with vert... |
upgr1eop 27485 | A pseudograph with one edg... |
upgr0eopALT 27486 | Alternate proof of ~ upgr0... |
upgr1eopALT 27487 | Alternate proof of ~ upgr1... |
upgrun 27488 | The union ` U ` of two pse... |
upgrunop 27489 | The union of two pseudogra... |
umgrun 27490 | The union ` U ` of two mul... |
umgrunop 27491 | The union of two multigrap... |
umgrislfupgrlem 27492 | Lemma for ~ umgrislfupgr a... |
umgrislfupgr 27493 | A multigraph is a loop-fre... |
lfgredgge2 27494 | An edge of a loop-free gra... |
lfgrnloop 27495 | A loop-free graph has no l... |
uhgredgiedgb 27496 | In a hypergraph, a set is ... |
uhgriedg0edg0 27497 | A hypergraph has no edges ... |
uhgredgn0 27498 | An edge of a hypergraph is... |
edguhgr 27499 | An edge of a hypergraph is... |
uhgredgrnv 27500 | An edge of a hypergraph co... |
uhgredgss 27501 | The set of edges of a hype... |
upgredgss 27502 | The set of edges of a pseu... |
umgredgss 27503 | The set of edges of a mult... |
edgupgr 27504 | Properties of an edge of a... |
edgumgr 27505 | Properties of an edge of a... |
uhgrvtxedgiedgb 27506 | In a hypergraph, a vertex ... |
upgredg 27507 | For each edge in a pseudog... |
umgredg 27508 | For each edge in a multigr... |
upgrpredgv 27509 | An edge of a pseudograph a... |
umgrpredgv 27510 | An edge of a multigraph al... |
upgredg2vtx 27511 | For a vertex incident to a... |
upgredgpr 27512 | If a proper pair (of verti... |
edglnl 27513 | The edges incident with a ... |
numedglnl 27514 | The number of edges incide... |
umgredgne 27515 | An edge of a multigraph al... |
umgrnloop2 27516 | A multigraph has no loops.... |
umgredgnlp 27517 | An edge of a multigraph is... |
isuspgr 27522 | The property of being a si... |
isusgr 27523 | The property of being a si... |
uspgrf 27524 | The edge function of a sim... |
usgrf 27525 | The edge function of a sim... |
isusgrs 27526 | The property of being a si... |
usgrfs 27527 | The edge function of a sim... |
usgrfun 27528 | The edge function of a sim... |
usgredgss 27529 | The set of edges of a simp... |
edgusgr 27530 | An edge of a simple graph ... |
isuspgrop 27531 | The property of being an u... |
isusgrop 27532 | The property of being an u... |
usgrop 27533 | A simple graph represented... |
isausgr 27534 | The property of an unorder... |
ausgrusgrb 27535 | The equivalence of the def... |
usgrausgri 27536 | A simple graph represented... |
ausgrumgri 27537 | If an alternatively define... |
ausgrusgri 27538 | The equivalence of the def... |
usgrausgrb 27539 | The equivalence of the def... |
usgredgop 27540 | An edge of a simple graph ... |
usgrf1o 27541 | The edge function of a sim... |
usgrf1 27542 | The edge function of a sim... |
uspgrf1oedg 27543 | The edge function of a sim... |
usgrss 27544 | An edge is a subset of ver... |
uspgrushgr 27545 | A simple pseudograph is an... |
uspgrupgr 27546 | A simple pseudograph is an... |
uspgrupgrushgr 27547 | A graph is a simple pseudo... |
usgruspgr 27548 | A simple graph is a simple... |
usgrumgr 27549 | A simple graph is an undir... |
usgrumgruspgr 27550 | A graph is a simple graph ... |
usgruspgrb 27551 | A class is a simple graph ... |
usgrupgr 27552 | A simple graph is an undir... |
usgruhgr 27553 | A simple graph is an undir... |
usgrislfuspgr 27554 | A simple graph is a loop-f... |
uspgrun 27555 | The union ` U ` of two sim... |
uspgrunop 27556 | The union of two simple ps... |
usgrun 27557 | The union ` U ` of two sim... |
usgrunop 27558 | The union of two simple gr... |
usgredg2 27559 | The value of the "edge fun... |
usgredg2ALT 27560 | Alternate proof of ~ usgre... |
usgredgprv 27561 | In a simple graph, an edge... |
usgredgprvALT 27562 | Alternate proof of ~ usgre... |
usgredgppr 27563 | An edge of a simple graph ... |
usgrpredgv 27564 | An edge of a simple graph ... |
edgssv2 27565 | An edge of a simple graph ... |
usgredg 27566 | For each edge in a simple ... |
usgrnloopv 27567 | In a simple graph, there i... |
usgrnloopvALT 27568 | Alternate proof of ~ usgrn... |
usgrnloop 27569 | In a simple graph, there i... |
usgrnloopALT 27570 | Alternate proof of ~ usgrn... |
usgrnloop0 27571 | A simple graph has no loop... |
usgrnloop0ALT 27572 | Alternate proof of ~ usgrn... |
usgredgne 27573 | An edge of a simple graph ... |
usgrf1oedg 27574 | The edge function of a sim... |
uhgr2edg 27575 | If a vertex is adjacent to... |
umgr2edg 27576 | If a vertex is adjacent to... |
usgr2edg 27577 | If a vertex is adjacent to... |
umgr2edg1 27578 | If a vertex is adjacent to... |
usgr2edg1 27579 | If a vertex is adjacent to... |
umgrvad2edg 27580 | If a vertex is adjacent to... |
umgr2edgneu 27581 | If a vertex is adjacent to... |
usgrsizedg 27582 | In a simple graph, the siz... |
usgredg3 27583 | The value of the "edge fun... |
usgredg4 27584 | For a vertex incident to a... |
usgredgreu 27585 | For a vertex incident to a... |
usgredg2vtx 27586 | For a vertex incident to a... |
uspgredg2vtxeu 27587 | For a vertex incident to a... |
usgredg2vtxeu 27588 | For a vertex incident to a... |
usgredg2vtxeuALT 27589 | Alternate proof of ~ usgre... |
uspgredg2vlem 27590 | Lemma for ~ uspgredg2v . ... |
uspgredg2v 27591 | In a simple pseudograph, t... |
usgredg2vlem1 27592 | Lemma 1 for ~ usgredg2v . ... |
usgredg2vlem2 27593 | Lemma 2 for ~ usgredg2v . ... |
usgredg2v 27594 | In a simple graph, the map... |
usgriedgleord 27595 | Alternate version of ~ usg... |
ushgredgedg 27596 | In a simple hypergraph the... |
usgredgedg 27597 | In a simple graph there is... |
ushgredgedgloop 27598 | In a simple hypergraph the... |
uspgredgleord 27599 | In a simple pseudograph th... |
usgredgleord 27600 | In a simple graph the numb... |
usgredgleordALT 27601 | Alternate proof for ~ usgr... |
usgrstrrepe 27602 | Replacing (or adding) the ... |
usgr0e 27603 | The empty graph, with vert... |
usgr0vb 27604 | The null graph, with no ve... |
uhgr0v0e 27605 | The null graph, with no ve... |
uhgr0vsize0 27606 | The size of a hypergraph w... |
uhgr0edgfi 27607 | A graph of order 0 (i.e. w... |
usgr0v 27608 | The null graph, with no ve... |
uhgr0vusgr 27609 | The null graph, with no ve... |
usgr0 27610 | The null graph represented... |
uspgr1e 27611 | A simple pseudograph with ... |
usgr1e 27612 | A simple graph with one ed... |
usgr0eop 27613 | The empty graph, with vert... |
uspgr1eop 27614 | A simple pseudograph with ... |
uspgr1ewop 27615 | A simple pseudograph with ... |
uspgr1v1eop 27616 | A simple pseudograph with ... |
usgr1eop 27617 | A simple graph with (at le... |
uspgr2v1e2w 27618 | A simple pseudograph with ... |
usgr2v1e2w 27619 | A simple graph with two ve... |
edg0usgr 27620 | A class without edges is a... |
lfuhgr1v0e 27621 | A loop-free hypergraph wit... |
usgr1vr 27622 | A simple graph with one ve... |
usgr1v 27623 | A class with one (or no) v... |
usgr1v0edg 27624 | A class with one (or no) v... |
usgrexmpldifpr 27625 | Lemma for ~ usgrexmpledg :... |
usgrexmplef 27626 | Lemma for ~ usgrexmpl . (... |
usgrexmpllem 27627 | Lemma for ~ usgrexmpl . (... |
usgrexmplvtx 27628 | The vertices ` 0 , 1 , 2 ,... |
usgrexmpledg 27629 | The edges ` { 0 , 1 } , { ... |
usgrexmpl 27630 | ` G ` is a simple graph of... |
griedg0prc 27631 | The class of empty graphs ... |
griedg0ssusgr 27632 | The class of all simple gr... |
usgrprc 27633 | The class of simple graphs... |
relsubgr 27636 | The class of the subgraph ... |
subgrv 27637 | If a class is a subgraph o... |
issubgr 27638 | The property of a set to b... |
issubgr2 27639 | The property of a set to b... |
subgrprop 27640 | The properties of a subgra... |
subgrprop2 27641 | The properties of a subgra... |
uhgrissubgr 27642 | The property of a hypergra... |
subgrprop3 27643 | The properties of a subgra... |
egrsubgr 27644 | An empty graph consisting ... |
0grsubgr 27645 | The null graph (represente... |
0uhgrsubgr 27646 | The null graph (as hypergr... |
uhgrsubgrself 27647 | A hypergraph is a subgraph... |
subgrfun 27648 | The edge function of a sub... |
subgruhgrfun 27649 | The edge function of a sub... |
subgreldmiedg 27650 | An element of the domain o... |
subgruhgredgd 27651 | An edge of a subgraph of a... |
subumgredg2 27652 | An edge of a subgraph of a... |
subuhgr 27653 | A subgraph of a hypergraph... |
subupgr 27654 | A subgraph of a pseudograp... |
subumgr 27655 | A subgraph of a multigraph... |
subusgr 27656 | A subgraph of a simple gra... |
uhgrspansubgrlem 27657 | Lemma for ~ uhgrspansubgr ... |
uhgrspansubgr 27658 | A spanning subgraph ` S ` ... |
uhgrspan 27659 | A spanning subgraph ` S ` ... |
upgrspan 27660 | A spanning subgraph ` S ` ... |
umgrspan 27661 | A spanning subgraph ` S ` ... |
usgrspan 27662 | A spanning subgraph ` S ` ... |
uhgrspanop 27663 | A spanning subgraph of a h... |
upgrspanop 27664 | A spanning subgraph of a p... |
umgrspanop 27665 | A spanning subgraph of a m... |
usgrspanop 27666 | A spanning subgraph of a s... |
uhgrspan1lem1 27667 | Lemma 1 for ~ uhgrspan1 . ... |
uhgrspan1lem2 27668 | Lemma 2 for ~ uhgrspan1 . ... |
uhgrspan1lem3 27669 | Lemma 3 for ~ uhgrspan1 . ... |
uhgrspan1 27670 | The induced subgraph ` S `... |
upgrreslem 27671 | Lemma for ~ upgrres . (Co... |
umgrreslem 27672 | Lemma for ~ umgrres and ~ ... |
upgrres 27673 | A subgraph obtained by rem... |
umgrres 27674 | A subgraph obtained by rem... |
usgrres 27675 | A subgraph obtained by rem... |
upgrres1lem1 27676 | Lemma 1 for ~ upgrres1 . ... |
umgrres1lem 27677 | Lemma for ~ umgrres1 . (C... |
upgrres1lem2 27678 | Lemma 2 for ~ upgrres1 . ... |
upgrres1lem3 27679 | Lemma 3 for ~ upgrres1 . ... |
upgrres1 27680 | A pseudograph obtained by ... |
umgrres1 27681 | A multigraph obtained by r... |
usgrres1 27682 | Restricting a simple graph... |
isfusgr 27685 | The property of being a fi... |
fusgrvtxfi 27686 | A finite simple graph has ... |
isfusgrf1 27687 | The property of being a fi... |
isfusgrcl 27688 | The property of being a fi... |
fusgrusgr 27689 | A finite simple graph is a... |
opfusgr 27690 | A finite simple graph repr... |
usgredgffibi 27691 | The number of edges in a s... |
fusgredgfi 27692 | In a finite simple graph t... |
usgr1v0e 27693 | The size of a (finite) sim... |
usgrfilem 27694 | In a finite simple graph, ... |
fusgrfisbase 27695 | Induction base for ~ fusgr... |
fusgrfisstep 27696 | Induction step in ~ fusgrf... |
fusgrfis 27697 | A finite simple graph is o... |
fusgrfupgrfs 27698 | A finite simple graph is a... |
nbgrprc0 27701 | The set of neighbors is em... |
nbgrcl 27702 | If a class ` X ` has at le... |
nbgrval 27703 | The set of neighbors of a ... |
dfnbgr2 27704 | Alternate definition of th... |
dfnbgr3 27705 | Alternate definition of th... |
nbgrnvtx0 27706 | If a class ` X ` is not a ... |
nbgrel 27707 | Characterization of a neig... |
nbgrisvtx 27708 | Every neighbor ` N ` of a ... |
nbgrssvtx 27709 | The neighbors of a vertex ... |
nbuhgr 27710 | The set of neighbors of a ... |
nbupgr 27711 | The set of neighbors of a ... |
nbupgrel 27712 | A neighbor of a vertex in ... |
nbumgrvtx 27713 | The set of neighbors of a ... |
nbumgr 27714 | The set of neighbors of an... |
nbusgrvtx 27715 | The set of neighbors of a ... |
nbusgr 27716 | The set of neighbors of an... |
nbgr2vtx1edg 27717 | If a graph has two vertice... |
nbuhgr2vtx1edgblem 27718 | Lemma for ~ nbuhgr2vtx1edg... |
nbuhgr2vtx1edgb 27719 | If a hypergraph has two ve... |
nbusgreledg 27720 | A class/vertex is a neighb... |
uhgrnbgr0nb 27721 | A vertex which is not endp... |
nbgr0vtxlem 27722 | Lemma for ~ nbgr0vtx and ~... |
nbgr0vtx 27723 | In a null graph (with no v... |
nbgr0edg 27724 | In an empty graph (with no... |
nbgr1vtx 27725 | In a graph with one vertex... |
nbgrnself 27726 | A vertex in a graph is not... |
nbgrnself2 27727 | A class ` X ` is not a nei... |
nbgrssovtx 27728 | The neighbors of a vertex ... |
nbgrssvwo2 27729 | The neighbors of a vertex ... |
nbgrsym 27730 | In a graph, the neighborho... |
nbupgrres 27731 | The neighborhood of a vert... |
usgrnbcnvfv 27732 | Applying the edge function... |
nbusgredgeu 27733 | For each neighbor of a ver... |
edgnbusgreu 27734 | For each edge incident to ... |
nbusgredgeu0 27735 | For each neighbor of a ver... |
nbusgrf1o0 27736 | The mapping of neighbors o... |
nbusgrf1o1 27737 | The set of neighbors of a ... |
nbusgrf1o 27738 | The set of neighbors of a ... |
nbedgusgr 27739 | The number of neighbors of... |
edgusgrnbfin 27740 | The number of neighbors of... |
nbusgrfi 27741 | The class of neighbors of ... |
nbfiusgrfi 27742 | The class of neighbors of ... |
hashnbusgrnn0 27743 | The number of neighbors of... |
nbfusgrlevtxm1 27744 | The number of neighbors of... |
nbfusgrlevtxm2 27745 | If there is a vertex which... |
nbusgrvtxm1 27746 | If the number of neighbors... |
nb3grprlem1 27747 | Lemma 1 for ~ nb3grpr . (... |
nb3grprlem2 27748 | Lemma 2 for ~ nb3grpr . (... |
nb3grpr 27749 | The neighbors of a vertex ... |
nb3grpr2 27750 | The neighbors of a vertex ... |
nb3gr2nb 27751 | If the neighbors of two ve... |
uvtxval 27754 | The set of all universal v... |
uvtxel 27755 | A universal vertex, i.e. a... |
uvtxisvtx 27756 | A universal vertex is a ve... |
uvtxssvtx 27757 | The set of the universal v... |
vtxnbuvtx 27758 | A universal vertex has all... |
uvtxnbgrss 27759 | A universal vertex has all... |
uvtxnbgrvtx 27760 | A universal vertex is neig... |
uvtx0 27761 | There is no universal vert... |
isuvtx 27762 | The set of all universal v... |
uvtxel1 27763 | Characterization of a univ... |
uvtx01vtx 27764 | If a graph/class has no ed... |
uvtx2vtx1edg 27765 | If a graph has two vertice... |
uvtx2vtx1edgb 27766 | If a hypergraph has two ve... |
uvtxnbgr 27767 | A universal vertex has all... |
uvtxnbgrb 27768 | A vertex is universal iff ... |
uvtxusgr 27769 | The set of all universal v... |
uvtxusgrel 27770 | A universal vertex, i.e. a... |
uvtxnm1nbgr 27771 | A universal vertex has ` n... |
nbusgrvtxm1uvtx 27772 | If the number of neighbors... |
uvtxnbvtxm1 27773 | A universal vertex has ` n... |
nbupgruvtxres 27774 | The neighborhood of a univ... |
uvtxupgrres 27775 | A universal vertex is univ... |
cplgruvtxb 27780 | A graph ` G ` is complete ... |
prcliscplgr 27781 | A proper class (representi... |
iscplgr 27782 | The property of being a co... |
iscplgrnb 27783 | A graph is complete iff al... |
iscplgredg 27784 | A graph ` G ` is complete ... |
iscusgr 27785 | The property of being a co... |
cusgrusgr 27786 | A complete simple graph is... |
cusgrcplgr 27787 | A complete simple graph is... |
iscusgrvtx 27788 | A simple graph is complete... |
cusgruvtxb 27789 | A simple graph is complete... |
iscusgredg 27790 | A simple graph is complete... |
cusgredg 27791 | In a complete simple graph... |
cplgr0 27792 | The null graph (with no ve... |
cusgr0 27793 | The null graph (with no ve... |
cplgr0v 27794 | A null graph (with no vert... |
cusgr0v 27795 | A graph with no vertices a... |
cplgr1vlem 27796 | Lemma for ~ cplgr1v and ~ ... |
cplgr1v 27797 | A graph with one vertex is... |
cusgr1v 27798 | A graph with one vertex an... |
cplgr2v 27799 | An undirected hypergraph w... |
cplgr2vpr 27800 | An undirected hypergraph w... |
nbcplgr 27801 | In a complete graph, each ... |
cplgr3v 27802 | A pseudograph with three (... |
cusgr3vnbpr 27803 | The neighbors of a vertex ... |
cplgrop 27804 | A complete graph represent... |
cusgrop 27805 | A complete simple graph re... |
cusgrexilem1 27806 | Lemma 1 for ~ cusgrexi . ... |
usgrexilem 27807 | Lemma for ~ usgrexi . (Co... |
usgrexi 27808 | An arbitrary set regarded ... |
cusgrexilem2 27809 | Lemma 2 for ~ cusgrexi . ... |
cusgrexi 27810 | An arbitrary set ` V ` reg... |
cusgrexg 27811 | For each set there is a se... |
structtousgr 27812 | Any (extensible) structure... |
structtocusgr 27813 | Any (extensible) structure... |
cffldtocusgr 27814 | The field of complex numbe... |
cusgrres 27815 | Restricting a complete sim... |
cusgrsizeindb0 27816 | Base case of the induction... |
cusgrsizeindb1 27817 | Base case of the induction... |
cusgrsizeindslem 27818 | Lemma for ~ cusgrsizeinds ... |
cusgrsizeinds 27819 | Part 1 of induction step i... |
cusgrsize2inds 27820 | Induction step in ~ cusgrs... |
cusgrsize 27821 | The size of a finite compl... |
cusgrfilem1 27822 | Lemma 1 for ~ cusgrfi . (... |
cusgrfilem2 27823 | Lemma 2 for ~ cusgrfi . (... |
cusgrfilem3 27824 | Lemma 3 for ~ cusgrfi . (... |
cusgrfi 27825 | If the size of a complete ... |
usgredgsscusgredg 27826 | A simple graph is a subgra... |
usgrsscusgr 27827 | A simple graph is a subgra... |
sizusglecusglem1 27828 | Lemma 1 for ~ sizusglecusg... |
sizusglecusglem2 27829 | Lemma 2 for ~ sizusglecusg... |
sizusglecusg 27830 | The size of a simple graph... |
fusgrmaxsize 27831 | The maximum size of a fini... |
vtxdgfval 27834 | The value of the vertex de... |
vtxdgval 27835 | The degree of a vertex. (... |
vtxdgfival 27836 | The degree of a vertex for... |
vtxdgop 27837 | The vertex degree expresse... |
vtxdgf 27838 | The vertex degree function... |
vtxdgelxnn0 27839 | The degree of a vertex is ... |
vtxdg0v 27840 | The degree of a vertex in ... |
vtxdg0e 27841 | The degree of a vertex in ... |
vtxdgfisnn0 27842 | The degree of a vertex in ... |
vtxdgfisf 27843 | The vertex degree function... |
vtxdeqd 27844 | Equality theorem for the v... |
vtxduhgr0e 27845 | The degree of a vertex in ... |
vtxdlfuhgr1v 27846 | The degree of the vertex i... |
vdumgr0 27847 | A vertex in a multigraph h... |
vtxdun 27848 | The degree of a vertex in ... |
vtxdfiun 27849 | The degree of a vertex in ... |
vtxduhgrun 27850 | The degree of a vertex in ... |
vtxduhgrfiun 27851 | The degree of a vertex in ... |
vtxdlfgrval 27852 | The value of the vertex de... |
vtxdumgrval 27853 | The value of the vertex de... |
vtxdusgrval 27854 | The value of the vertex de... |
vtxd0nedgb 27855 | A vertex has degree 0 iff ... |
vtxdushgrfvedglem 27856 | Lemma for ~ vtxdushgrfvedg... |
vtxdushgrfvedg 27857 | The value of the vertex de... |
vtxdusgrfvedg 27858 | The value of the vertex de... |
vtxduhgr0nedg 27859 | If a vertex in a hypergrap... |
vtxdumgr0nedg 27860 | If a vertex in a multigrap... |
vtxduhgr0edgnel 27861 | A vertex in a hypergraph h... |
vtxdusgr0edgnel 27862 | A vertex in a simple graph... |
vtxdusgr0edgnelALT 27863 | Alternate proof of ~ vtxdu... |
vtxdgfusgrf 27864 | The vertex degree function... |
vtxdgfusgr 27865 | In a finite simple graph, ... |
fusgrn0degnn0 27866 | In a nonempty, finite grap... |
1loopgruspgr 27867 | A graph with one edge whic... |
1loopgredg 27868 | The set of edges in a grap... |
1loopgrnb0 27869 | In a graph (simple pseudog... |
1loopgrvd2 27870 | The vertex degree of a one... |
1loopgrvd0 27871 | The vertex degree of a one... |
1hevtxdg0 27872 | The vertex degree of verte... |
1hevtxdg1 27873 | The vertex degree of verte... |
1hegrvtxdg1 27874 | The vertex degree of a gra... |
1hegrvtxdg1r 27875 | The vertex degree of a gra... |
1egrvtxdg1 27876 | The vertex degree of a one... |
1egrvtxdg1r 27877 | The vertex degree of a one... |
1egrvtxdg0 27878 | The vertex degree of a one... |
p1evtxdeqlem 27879 | Lemma for ~ p1evtxdeq and ... |
p1evtxdeq 27880 | If an edge ` E ` which doe... |
p1evtxdp1 27881 | If an edge ` E ` (not bein... |
uspgrloopvtx 27882 | The set of vertices in a g... |
uspgrloopvtxel 27883 | A vertex in a graph (simpl... |
uspgrloopiedg 27884 | The set of edges in a grap... |
uspgrloopedg 27885 | The set of edges in a grap... |
uspgrloopnb0 27886 | In a graph (simple pseudog... |
uspgrloopvd2 27887 | The vertex degree of a one... |
umgr2v2evtx 27888 | The set of vertices in a m... |
umgr2v2evtxel 27889 | A vertex in a multigraph w... |
umgr2v2eiedg 27890 | The edge function in a mul... |
umgr2v2eedg 27891 | The set of edges in a mult... |
umgr2v2e 27892 | A multigraph with two edge... |
umgr2v2enb1 27893 | In a multigraph with two e... |
umgr2v2evd2 27894 | In a multigraph with two e... |
hashnbusgrvd 27895 | In a simple graph, the num... |
usgruvtxvdb 27896 | In a finite simple graph w... |
vdiscusgrb 27897 | A finite simple graph with... |
vdiscusgr 27898 | In a finite complete simpl... |
vtxdusgradjvtx 27899 | The degree of a vertex in ... |
usgrvd0nedg 27900 | If a vertex in a simple gr... |
uhgrvd00 27901 | If every vertex in a hyper... |
usgrvd00 27902 | If every vertex in a simpl... |
vdegp1ai 27903 | The induction step for a v... |
vdegp1bi 27904 | The induction step for a v... |
vdegp1ci 27905 | The induction step for a v... |
vtxdginducedm1lem1 27906 | Lemma 1 for ~ vtxdginduced... |
vtxdginducedm1lem2 27907 | Lemma 2 for ~ vtxdginduced... |
vtxdginducedm1lem3 27908 | Lemma 3 for ~ vtxdginduced... |
vtxdginducedm1lem4 27909 | Lemma 4 for ~ vtxdginduced... |
vtxdginducedm1 27910 | The degree of a vertex ` v... |
vtxdginducedm1fi 27911 | The degree of a vertex ` v... |
finsumvtxdg2ssteplem1 27912 | Lemma for ~ finsumvtxdg2ss... |
finsumvtxdg2ssteplem2 27913 | Lemma for ~ finsumvtxdg2ss... |
finsumvtxdg2ssteplem3 27914 | Lemma for ~ finsumvtxdg2ss... |
finsumvtxdg2ssteplem4 27915 | Lemma for ~ finsumvtxdg2ss... |
finsumvtxdg2sstep 27916 | Induction step of ~ finsum... |
finsumvtxdg2size 27917 | The sum of the degrees of ... |
fusgr1th 27918 | The sum of the degrees of ... |
finsumvtxdgeven 27919 | The sum of the degrees of ... |
vtxdgoddnumeven 27920 | The number of vertices of ... |
fusgrvtxdgonume 27921 | The number of vertices of ... |
isrgr 27926 | The property of a class be... |
rgrprop 27927 | The properties of a k-regu... |
isrusgr 27928 | The property of being a k-... |
rusgrprop 27929 | The properties of a k-regu... |
rusgrrgr 27930 | A k-regular simple graph i... |
rusgrusgr 27931 | A k-regular simple graph i... |
finrusgrfusgr 27932 | A finite regular simple gr... |
isrusgr0 27933 | The property of being a k-... |
rusgrprop0 27934 | The properties of a k-regu... |
usgreqdrusgr 27935 | If all vertices in a simpl... |
fusgrregdegfi 27936 | In a nonempty finite simpl... |
fusgrn0eqdrusgr 27937 | If all vertices in a nonem... |
frusgrnn0 27938 | In a nonempty finite k-reg... |
0edg0rgr 27939 | A graph is 0-regular if it... |
uhgr0edg0rgr 27940 | A hypergraph is 0-regular ... |
uhgr0edg0rgrb 27941 | A hypergraph is 0-regular ... |
usgr0edg0rusgr 27942 | A simple graph is 0-regula... |
0vtxrgr 27943 | A null graph (with no vert... |
0vtxrusgr 27944 | A graph with no vertices a... |
0uhgrrusgr 27945 | The null graph as hypergra... |
0grrusgr 27946 | The null graph represented... |
0grrgr 27947 | The null graph represented... |
cusgrrusgr 27948 | A complete simple graph wi... |
cusgrm1rusgr 27949 | A finite simple graph with... |
rusgrpropnb 27950 | The properties of a k-regu... |
rusgrpropedg 27951 | The properties of a k-regu... |
rusgrpropadjvtx 27952 | The properties of a k-regu... |
rusgrnumwrdl2 27953 | In a k-regular simple grap... |
rusgr1vtxlem 27954 | Lemma for ~ rusgr1vtx . (... |
rusgr1vtx 27955 | If a k-regular simple grap... |
rgrusgrprc 27956 | The class of 0-regular sim... |
rusgrprc 27957 | The class of 0-regular sim... |
rgrprc 27958 | The class of 0-regular gra... |
rgrprcx 27959 | The class of 0-regular gra... |
rgrx0ndm 27960 | 0 is not in the domain of ... |
rgrx0nd 27961 | The potentially alternativ... |
ewlksfval 27968 | The set of s-walks of edge... |
isewlk 27969 | Conditions for a function ... |
ewlkprop 27970 | Properties of an s-walk of... |
ewlkinedg 27971 | The intersection (common v... |
ewlkle 27972 | An s-walk of edges is also... |
upgrewlkle2 27973 | In a pseudograph, there is... |
wkslem1 27974 | Lemma 1 for walks to subst... |
wkslem2 27975 | Lemma 2 for walks to subst... |
wksfval 27976 | The set of walks (in an un... |
iswlk 27977 | Properties of a pair of fu... |
wlkprop 27978 | Properties of a walk. (Co... |
wlkv 27979 | The classes involved in a ... |
iswlkg 27980 | Generalization of ~ iswlk ... |
wlkf 27981 | The mapping enumerating th... |
wlkcl 27982 | A walk has length ` # ( F ... |
wlkp 27983 | The mapping enumerating th... |
wlkpwrd 27984 | The sequence of vertices o... |
wlklenvp1 27985 | The number of vertices of ... |
wksv 27986 | The class of walks is a se... |
wksvOLD 27987 | Obsolete version of ~ wksv... |
wlkn0 27988 | The sequence of vertices o... |
wlklenvm1 27989 | The number of edges of a w... |
ifpsnprss 27990 | Lemma for ~ wlkvtxeledg : ... |
wlkvtxeledg 27991 | Each pair of adjacent vert... |
wlkvtxiedg 27992 | The vertices of a walk are... |
relwlk 27993 | The set ` ( Walks `` G ) `... |
wlkvv 27994 | If there is at least one w... |
wlkop 27995 | A walk is an ordered pair.... |
wlkcpr 27996 | A walk as class with two c... |
wlk2f 27997 | If there is a walk ` W ` t... |
wlkcomp 27998 | A walk expressed by proper... |
wlkcompim 27999 | Implications for the prope... |
wlkelwrd 28000 | The components of a walk a... |
wlkeq 28001 | Conditions for two walks (... |
edginwlk 28002 | The value of the edge func... |
upgredginwlk 28003 | The value of the edge func... |
iedginwlk 28004 | The value of the edge func... |
wlkl1loop 28005 | A walk of length 1 from a ... |
wlk1walk 28006 | A walk is a 1-walk "on the... |
wlk1ewlk 28007 | A walk is an s-walk "on th... |
upgriswlk 28008 | Properties of a pair of fu... |
upgrwlkedg 28009 | The edges of a walk in a p... |
upgrwlkcompim 28010 | Implications for the prope... |
wlkvtxedg 28011 | The vertices of a walk are... |
upgrwlkvtxedg 28012 | The pairs of connected ver... |
uspgr2wlkeq 28013 | Conditions for two walks w... |
uspgr2wlkeq2 28014 | Conditions for two walks w... |
uspgr2wlkeqi 28015 | Conditions for two walks w... |
umgrwlknloop 28016 | In a multigraph, each walk... |
wlkResOLD 28017 | Obsolete version of ~ opab... |
wlkv0 28018 | If there is a walk in the ... |
g0wlk0 28019 | There is no walk in a null... |
0wlk0 28020 | There is no walk for the e... |
wlk0prc 28021 | There is no walk in a null... |
wlklenvclwlk 28022 | The number of vertices in ... |
wlklenvclwlkOLD 28023 | Obsolete version of ~ wlkl... |
wlkson 28024 | The set of walks between t... |
iswlkon 28025 | Properties of a pair of fu... |
wlkonprop 28026 | Properties of a walk betwe... |
wlkpvtx 28027 | A walk connects vertices. ... |
wlkepvtx 28028 | The endpoints of a walk ar... |
wlkoniswlk 28029 | A walk between two vertice... |
wlkonwlk 28030 | A walk is a walk between i... |
wlkonwlk1l 28031 | A walk is a walk from its ... |
wlksoneq1eq2 28032 | Two walks with identical s... |
wlkonl1iedg 28033 | If there is a walk between... |
wlkon2n0 28034 | The length of a walk betwe... |
2wlklem 28035 | Lemma for theorems for wal... |
upgr2wlk 28036 | Properties of a pair of fu... |
wlkreslem 28037 | Lemma for ~ wlkres . (Con... |
wlkres 28038 | The restriction ` <. H , Q... |
redwlklem 28039 | Lemma for ~ redwlk . (Con... |
redwlk 28040 | A walk ending at the last ... |
wlkp1lem1 28041 | Lemma for ~ wlkp1 . (Cont... |
wlkp1lem2 28042 | Lemma for ~ wlkp1 . (Cont... |
wlkp1lem3 28043 | Lemma for ~ wlkp1 . (Cont... |
wlkp1lem4 28044 | Lemma for ~ wlkp1 . (Cont... |
wlkp1lem5 28045 | Lemma for ~ wlkp1 . (Cont... |
wlkp1lem6 28046 | Lemma for ~ wlkp1 . (Cont... |
wlkp1lem7 28047 | Lemma for ~ wlkp1 . (Cont... |
wlkp1lem8 28048 | Lemma for ~ wlkp1 . (Cont... |
wlkp1 28049 | Append one path segment (e... |
wlkdlem1 28050 | Lemma 1 for ~ wlkd . (Con... |
wlkdlem2 28051 | Lemma 2 for ~ wlkd . (Con... |
wlkdlem3 28052 | Lemma 3 for ~ wlkd . (Con... |
wlkdlem4 28053 | Lemma 4 for ~ wlkd . (Con... |
wlkd 28054 | Two words representing a w... |
lfgrwlkprop 28055 | Two adjacent vertices in a... |
lfgriswlk 28056 | Conditions for a pair of f... |
lfgrwlknloop 28057 | In a loop-free graph, each... |
reltrls 28062 | The set ` ( Trails `` G ) ... |
trlsfval 28063 | The set of trails (in an u... |
istrl 28064 | Conditions for a pair of c... |
trliswlk 28065 | A trail is a walk. (Contr... |
trlf1 28066 | The enumeration ` F ` of a... |
trlreslem 28067 | Lemma for ~ trlres . Form... |
trlres 28068 | The restriction ` <. H , Q... |
upgrtrls 28069 | The set of trails in a pse... |
upgristrl 28070 | Properties of a pair of fu... |
upgrf1istrl 28071 | Properties of a pair of a ... |
wksonproplem 28072 | Lemma for theorems for pro... |
wksonproplemOLD 28073 | Obsolete version of ~ wkso... |
trlsonfval 28074 | The set of trails between ... |
istrlson 28075 | Properties of a pair of fu... |
trlsonprop 28076 | Properties of a trail betw... |
trlsonistrl 28077 | A trail between two vertic... |
trlsonwlkon 28078 | A trail between two vertic... |
trlontrl 28079 | A trail is a trail between... |
relpths 28088 | The set ` ( Paths `` G ) `... |
pthsfval 28089 | The set of paths (in an un... |
spthsfval 28090 | The set of simple paths (i... |
ispth 28091 | Conditions for a pair of c... |
isspth 28092 | Conditions for a pair of c... |
pthistrl 28093 | A path is a trail (in an u... |
spthispth 28094 | A simple path is a path (i... |
pthiswlk 28095 | A path is a walk (in an un... |
spthiswlk 28096 | A simple path is a walk (i... |
pthdivtx 28097 | The inner vertices of a pa... |
pthdadjvtx 28098 | The adjacent vertices of a... |
2pthnloop 28099 | A path of length at least ... |
upgr2pthnlp 28100 | A path of length at least ... |
spthdifv 28101 | The vertices of a simple p... |
spthdep 28102 | A simple path (at least of... |
pthdepisspth 28103 | A path with different star... |
upgrwlkdvdelem 28104 | Lemma for ~ upgrwlkdvde . ... |
upgrwlkdvde 28105 | In a pseudograph, all edge... |
upgrspthswlk 28106 | The set of simple paths in... |
upgrwlkdvspth 28107 | A walk consisting of diffe... |
pthsonfval 28108 | The set of paths between t... |
spthson 28109 | The set of simple paths be... |
ispthson 28110 | Properties of a pair of fu... |
isspthson 28111 | Properties of a pair of fu... |
pthsonprop 28112 | Properties of a path betwe... |
spthonprop 28113 | Properties of a simple pat... |
pthonispth 28114 | A path between two vertice... |
pthontrlon 28115 | A path between two vertice... |
pthonpth 28116 | A path is a path between i... |
isspthonpth 28117 | A pair of functions is a s... |
spthonisspth 28118 | A simple path between to v... |
spthonpthon 28119 | A simple path between two ... |
spthonepeq 28120 | The endpoints of a simple ... |
uhgrwkspthlem1 28121 | Lemma 1 for ~ uhgrwkspth .... |
uhgrwkspthlem2 28122 | Lemma 2 for ~ uhgrwkspth .... |
uhgrwkspth 28123 | Any walk of length 1 betwe... |
usgr2wlkneq 28124 | The vertices and edges are... |
usgr2wlkspthlem1 28125 | Lemma 1 for ~ usgr2wlkspth... |
usgr2wlkspthlem2 28126 | Lemma 2 for ~ usgr2wlkspth... |
usgr2wlkspth 28127 | In a simple graph, any wal... |
usgr2trlncl 28128 | In a simple graph, any tra... |
usgr2trlspth 28129 | In a simple graph, any tra... |
usgr2pthspth 28130 | In a simple graph, any pat... |
usgr2pthlem 28131 | Lemma for ~ usgr2pth . (C... |
usgr2pth 28132 | In a simple graph, there i... |
usgr2pth0 28133 | In a simply graph, there i... |
pthdlem1 28134 | Lemma 1 for ~ pthd . (Con... |
pthdlem2lem 28135 | Lemma for ~ pthdlem2 . (C... |
pthdlem2 28136 | Lemma 2 for ~ pthd . (Con... |
pthd 28137 | Two words representing a t... |
clwlks 28140 | The set of closed walks (i... |
isclwlk 28141 | A pair of functions repres... |
clwlkiswlk 28142 | A closed walk is a walk (i... |
clwlkwlk 28143 | Closed walks are walks (in... |
clwlkswks 28144 | Closed walks are walks (in... |
isclwlke 28145 | Properties of a pair of fu... |
isclwlkupgr 28146 | Properties of a pair of fu... |
clwlkcomp 28147 | A closed walk expressed by... |
clwlkcompim 28148 | Implications for the prope... |
upgrclwlkcompim 28149 | Implications for the prope... |
clwlkcompbp 28150 | Basic properties of the co... |
clwlkl1loop 28151 | A closed walk of length 1 ... |
crcts 28156 | The set of circuits (in an... |
cycls 28157 | The set of cycles (in an u... |
iscrct 28158 | Sufficient and necessary c... |
iscycl 28159 | Sufficient and necessary c... |
crctprop 28160 | The properties of a circui... |
cyclprop 28161 | The properties of a cycle:... |
crctisclwlk 28162 | A circuit is a closed walk... |
crctistrl 28163 | A circuit is a trail. (Co... |
crctiswlk 28164 | A circuit is a walk. (Con... |
cyclispth 28165 | A cycle is a path. (Contr... |
cycliswlk 28166 | A cycle is a walk. (Contr... |
cycliscrct 28167 | A cycle is a circuit. (Co... |
cyclnspth 28168 | A (non-trivial) cycle is n... |
cyclispthon 28169 | A cycle is a path starting... |
lfgrn1cycl 28170 | In a loop-free graph there... |
usgr2trlncrct 28171 | In a simple graph, any tra... |
umgrn1cycl 28172 | In a multigraph graph (wit... |
uspgrn2crct 28173 | In a simple pseudograph th... |
usgrn2cycl 28174 | In a simple graph there ar... |
crctcshwlkn0lem1 28175 | Lemma for ~ crctcshwlkn0 .... |
crctcshwlkn0lem2 28176 | Lemma for ~ crctcshwlkn0 .... |
crctcshwlkn0lem3 28177 | Lemma for ~ crctcshwlkn0 .... |
crctcshwlkn0lem4 28178 | Lemma for ~ crctcshwlkn0 .... |
crctcshwlkn0lem5 28179 | Lemma for ~ crctcshwlkn0 .... |
crctcshwlkn0lem6 28180 | Lemma for ~ crctcshwlkn0 .... |
crctcshwlkn0lem7 28181 | Lemma for ~ crctcshwlkn0 .... |
crctcshlem1 28182 | Lemma for ~ crctcsh . (Co... |
crctcshlem2 28183 | Lemma for ~ crctcsh . (Co... |
crctcshlem3 28184 | Lemma for ~ crctcsh . (Co... |
crctcshlem4 28185 | Lemma for ~ crctcsh . (Co... |
crctcshwlkn0 28186 | Cyclically shifting the in... |
crctcshwlk 28187 | Cyclically shifting the in... |
crctcshtrl 28188 | Cyclically shifting the in... |
crctcsh 28189 | Cyclically shifting the in... |
wwlks 28200 | The set of walks (in an un... |
iswwlks 28201 | A word over the set of ver... |
wwlksn 28202 | The set of walks (in an un... |
iswwlksn 28203 | A word over the set of ver... |
wwlksnprcl 28204 | Derivation of the length o... |
iswwlksnx 28205 | Properties of a word to re... |
wwlkbp 28206 | Basic properties of a walk... |
wwlknbp 28207 | Basic properties of a walk... |
wwlknp 28208 | Properties of a set being ... |
wwlknbp1 28209 | Other basic properties of ... |
wwlknvtx 28210 | The symbols of a word ` W ... |
wwlknllvtx 28211 | If a word ` W ` represents... |
wwlknlsw 28212 | If a word represents a wal... |
wspthsn 28213 | The set of simple paths of... |
iswspthn 28214 | An element of the set of s... |
wspthnp 28215 | Properties of a set being ... |
wwlksnon 28216 | The set of walks of a fixe... |
wspthsnon 28217 | The set of simple paths of... |
iswwlksnon 28218 | The set of walks of a fixe... |
wwlksnon0 28219 | Sufficient conditions for ... |
wwlksonvtx 28220 | If a word ` W ` represents... |
iswspthsnon 28221 | The set of simple paths of... |
wwlknon 28222 | An element of the set of w... |
wspthnon 28223 | An element of the set of s... |
wspthnonp 28224 | Properties of a set being ... |
wspthneq1eq2 28225 | Two simple paths with iden... |
wwlksn0s 28226 | The set of all walks as wo... |
wwlkssswrd 28227 | Walks (represented by word... |
wwlksn0 28228 | A walk of length 0 is repr... |
0enwwlksnge1 28229 | In graphs without edges, t... |
wwlkswwlksn 28230 | A walk of a fixed length a... |
wwlkssswwlksn 28231 | The walks of a fixed lengt... |
wlkiswwlks1 28232 | The sequence of vertices i... |
wlklnwwlkln1 28233 | The sequence of vertices i... |
wlkiswwlks2lem1 28234 | Lemma 1 for ~ wlkiswwlks2 ... |
wlkiswwlks2lem2 28235 | Lemma 2 for ~ wlkiswwlks2 ... |
wlkiswwlks2lem3 28236 | Lemma 3 for ~ wlkiswwlks2 ... |
wlkiswwlks2lem4 28237 | Lemma 4 for ~ wlkiswwlks2 ... |
wlkiswwlks2lem5 28238 | Lemma 5 for ~ wlkiswwlks2 ... |
wlkiswwlks2lem6 28239 | Lemma 6 for ~ wlkiswwlks2 ... |
wlkiswwlks2 28240 | A walk as word corresponds... |
wlkiswwlks 28241 | A walk as word corresponds... |
wlkiswwlksupgr2 28242 | A walk as word corresponds... |
wlkiswwlkupgr 28243 | A walk as word corresponds... |
wlkswwlksf1o 28244 | The mapping of (ordinary) ... |
wlkswwlksen 28245 | The set of walks as words ... |
wwlksm1edg 28246 | Removing the trailing edge... |
wlklnwwlkln2lem 28247 | Lemma for ~ wlklnwwlkln2 a... |
wlklnwwlkln2 28248 | A walk of length ` N ` as ... |
wlklnwwlkn 28249 | A walk of length ` N ` as ... |
wlklnwwlklnupgr2 28250 | A walk of length ` N ` as ... |
wlklnwwlknupgr 28251 | A walk of length ` N ` as ... |
wlknewwlksn 28252 | If a walk in a pseudograph... |
wlknwwlksnbij 28253 | The mapping ` ( t e. T |->... |
wlknwwlksnen 28254 | In a simple pseudograph, t... |
wlknwwlksneqs 28255 | The set of walks of a fixe... |
wwlkseq 28256 | Equality of two walks (as ... |
wwlksnred 28257 | Reduction of a walk (as wo... |
wwlksnext 28258 | Extension of a walk (as wo... |
wwlksnextbi 28259 | Extension of a walk (as wo... |
wwlksnredwwlkn 28260 | For each walk (as word) of... |
wwlksnredwwlkn0 28261 | For each walk (as word) of... |
wwlksnextwrd 28262 | Lemma for ~ wwlksnextbij .... |
wwlksnextfun 28263 | Lemma for ~ wwlksnextbij .... |
wwlksnextinj 28264 | Lemma for ~ wwlksnextbij .... |
wwlksnextsurj 28265 | Lemma for ~ wwlksnextbij .... |
wwlksnextbij0 28266 | Lemma for ~ wwlksnextbij .... |
wwlksnextbij 28267 | There is a bijection betwe... |
wwlksnexthasheq 28268 | The number of the extensio... |
disjxwwlksn 28269 | Sets of walks (as words) e... |
wwlksnndef 28270 | Conditions for ` WWalksN `... |
wwlksnfi 28271 | The number of walks repres... |
wlksnfi 28272 | The number of walks of fix... |
wlksnwwlknvbij 28273 | There is a bijection betwe... |
wwlksnextproplem1 28274 | Lemma 1 for ~ wwlksnextpro... |
wwlksnextproplem2 28275 | Lemma 2 for ~ wwlksnextpro... |
wwlksnextproplem3 28276 | Lemma 3 for ~ wwlksnextpro... |
wwlksnextprop 28277 | Adding additional properti... |
disjxwwlkn 28278 | Sets of walks (as words) e... |
hashwwlksnext 28279 | Number of walks (as words)... |
wwlksnwwlksnon 28280 | A walk of fixed length is ... |
wspthsnwspthsnon 28281 | A simple path of fixed len... |
wspthsnonn0vne 28282 | If the set of simple paths... |
wspthsswwlkn 28283 | The set of simple paths of... |
wspthnfi 28284 | In a finite graph, the set... |
wwlksnonfi 28285 | In a finite graph, the set... |
wspthsswwlknon 28286 | The set of simple paths of... |
wspthnonfi 28287 | In a finite graph, the set... |
wspniunwspnon 28288 | The set of nonempty simple... |
wspn0 28289 | If there are no vertices, ... |
2wlkdlem1 28290 | Lemma 1 for ~ 2wlkd . (Co... |
2wlkdlem2 28291 | Lemma 2 for ~ 2wlkd . (Co... |
2wlkdlem3 28292 | Lemma 3 for ~ 2wlkd . (Co... |
2wlkdlem4 28293 | Lemma 4 for ~ 2wlkd . (Co... |
2wlkdlem5 28294 | Lemma 5 for ~ 2wlkd . (Co... |
2pthdlem1 28295 | Lemma 1 for ~ 2pthd . (Co... |
2wlkdlem6 28296 | Lemma 6 for ~ 2wlkd . (Co... |
2wlkdlem7 28297 | Lemma 7 for ~ 2wlkd . (Co... |
2wlkdlem8 28298 | Lemma 8 for ~ 2wlkd . (Co... |
2wlkdlem9 28299 | Lemma 9 for ~ 2wlkd . (Co... |
2wlkdlem10 28300 | Lemma 10 for ~ 3wlkd . (C... |
2wlkd 28301 | Construction of a walk fro... |
2wlkond 28302 | A walk of length 2 from on... |
2trld 28303 | Construction of a trail fr... |
2trlond 28304 | A trail of length 2 from o... |
2pthd 28305 | A path of length 2 from on... |
2spthd 28306 | A simple path of length 2 ... |
2pthond 28307 | A simple path of length 2 ... |
2pthon3v 28308 | For a vertex adjacent to t... |
umgr2adedgwlklem 28309 | Lemma for ~ umgr2adedgwlk ... |
umgr2adedgwlk 28310 | In a multigraph, two adjac... |
umgr2adedgwlkon 28311 | In a multigraph, two adjac... |
umgr2adedgwlkonALT 28312 | Alternate proof for ~ umgr... |
umgr2adedgspth 28313 | In a multigraph, two adjac... |
umgr2wlk 28314 | In a multigraph, there is ... |
umgr2wlkon 28315 | For each pair of adjacent ... |
elwwlks2s3 28316 | A walk of length 2 as word... |
midwwlks2s3 28317 | There is a vertex between ... |
wwlks2onv 28318 | If a length 3 string repre... |
elwwlks2ons3im 28319 | A walk as word of length 2... |
elwwlks2ons3 28320 | For each walk of length 2 ... |
s3wwlks2on 28321 | A length 3 string which re... |
umgrwwlks2on 28322 | A walk of length 2 between... |
wwlks2onsym 28323 | There is a walk of length ... |
elwwlks2on 28324 | A walk of length 2 between... |
elwspths2on 28325 | A simple path of length 2 ... |
wpthswwlks2on 28326 | For two different vertices... |
2wspdisj 28327 | All simple paths of length... |
2wspiundisj 28328 | All simple paths of length... |
usgr2wspthons3 28329 | A simple path of length 2 ... |
usgr2wspthon 28330 | A simple path of length 2 ... |
elwwlks2 28331 | A walk of length 2 between... |
elwspths2spth 28332 | A simple path of length 2 ... |
rusgrnumwwlkl1 28333 | In a k-regular graph, ther... |
rusgrnumwwlkslem 28334 | Lemma for ~ rusgrnumwwlks ... |
rusgrnumwwlklem 28335 | Lemma for ~ rusgrnumwwlk e... |
rusgrnumwwlkb0 28336 | Induction base 0 for ~ rus... |
rusgrnumwwlkb1 28337 | Induction base 1 for ~ rus... |
rusgr0edg 28338 | Special case for graphs wi... |
rusgrnumwwlks 28339 | Induction step for ~ rusgr... |
rusgrnumwwlk 28340 | In a ` K `-regular graph, ... |
rusgrnumwwlkg 28341 | In a ` K `-regular graph, ... |
rusgrnumwlkg 28342 | In a k-regular graph, the ... |
clwwlknclwwlkdif 28343 | The set ` A ` of walks of ... |
clwwlknclwwlkdifnum 28344 | In a ` K `-regular graph, ... |
clwwlk 28347 | The set of closed walks (i... |
isclwwlk 28348 | Properties of a word to re... |
clwwlkbp 28349 | Basic properties of a clos... |
clwwlkgt0 28350 | There is no empty closed w... |
clwwlksswrd 28351 | Closed walks (represented ... |
clwwlk1loop 28352 | A closed walk of length 1 ... |
clwwlkccatlem 28353 | Lemma for ~ clwwlkccat : i... |
clwwlkccat 28354 | The concatenation of two w... |
umgrclwwlkge2 28355 | A closed walk in a multigr... |
clwlkclwwlklem2a1 28356 | Lemma 1 for ~ clwlkclwwlkl... |
clwlkclwwlklem2a2 28357 | Lemma 2 for ~ clwlkclwwlkl... |
clwlkclwwlklem2a3 28358 | Lemma 3 for ~ clwlkclwwlkl... |
clwlkclwwlklem2fv1 28359 | Lemma 4a for ~ clwlkclwwlk... |
clwlkclwwlklem2fv2 28360 | Lemma 4b for ~ clwlkclwwlk... |
clwlkclwwlklem2a4 28361 | Lemma 4 for ~ clwlkclwwlkl... |
clwlkclwwlklem2a 28362 | Lemma for ~ clwlkclwwlklem... |
clwlkclwwlklem1 28363 | Lemma 1 for ~ clwlkclwwlk ... |
clwlkclwwlklem2 28364 | Lemma 2 for ~ clwlkclwwlk ... |
clwlkclwwlklem3 28365 | Lemma 3 for ~ clwlkclwwlk ... |
clwlkclwwlk 28366 | A closed walk as word of l... |
clwlkclwwlk2 28367 | A closed walk corresponds ... |
clwlkclwwlkflem 28368 | Lemma for ~ clwlkclwwlkf .... |
clwlkclwwlkf1lem2 28369 | Lemma 2 for ~ clwlkclwwlkf... |
clwlkclwwlkf1lem3 28370 | Lemma 3 for ~ clwlkclwwlkf... |
clwlkclwwlkfolem 28371 | Lemma for ~ clwlkclwwlkfo ... |
clwlkclwwlkf 28372 | ` F ` is a function from t... |
clwlkclwwlkfo 28373 | ` F ` is a function from t... |
clwlkclwwlkf1 28374 | ` F ` is a one-to-one func... |
clwlkclwwlkf1o 28375 | ` F ` is a bijection betwe... |
clwlkclwwlken 28376 | The set of the nonempty cl... |
clwwisshclwwslemlem 28377 | Lemma for ~ clwwisshclwwsl... |
clwwisshclwwslem 28378 | Lemma for ~ clwwisshclwws ... |
clwwisshclwws 28379 | Cyclically shifting a clos... |
clwwisshclwwsn 28380 | Cyclically shifting a clos... |
erclwwlkrel 28381 | ` .~ ` is a relation. (Co... |
erclwwlkeq 28382 | Two classes are equivalent... |
erclwwlkeqlen 28383 | If two classes are equival... |
erclwwlkref 28384 | ` .~ ` is a reflexive rela... |
erclwwlksym 28385 | ` .~ ` is a symmetric rela... |
erclwwlktr 28386 | ` .~ ` is a transitive rel... |
erclwwlk 28387 | ` .~ ` is an equivalence r... |
clwwlkn 28390 | The set of closed walks of... |
isclwwlkn 28391 | A word over the set of ver... |
clwwlkn0 28392 | There is no closed walk of... |
clwwlkneq0 28393 | Sufficient conditions for ... |
clwwlkclwwlkn 28394 | A closed walk of a fixed l... |
clwwlksclwwlkn 28395 | The closed walks of a fixe... |
clwwlknlen 28396 | The length of a word repre... |
clwwlknnn 28397 | The length of a closed wal... |
clwwlknwrd 28398 | A closed walk of a fixed l... |
clwwlknbp 28399 | Basic properties of a clos... |
isclwwlknx 28400 | Characterization of a word... |
clwwlknp 28401 | Properties of a set being ... |
clwwlknwwlksn 28402 | A word representing a clos... |
clwwlknlbonbgr1 28403 | The last but one vertex in... |
clwwlkinwwlk 28404 | If the initial vertex of a... |
clwwlkn1 28405 | A closed walk of length 1 ... |
loopclwwlkn1b 28406 | The singleton word consist... |
clwwlkn1loopb 28407 | A word represents a closed... |
clwwlkn2 28408 | A closed walk of length 2 ... |
clwwlknfi 28409 | If there is only a finite ... |
clwwlkel 28410 | Obtaining a closed walk (a... |
clwwlkf 28411 | Lemma 1 for ~ clwwlkf1o : ... |
clwwlkfv 28412 | Lemma 2 for ~ clwwlkf1o : ... |
clwwlkf1 28413 | Lemma 3 for ~ clwwlkf1o : ... |
clwwlkfo 28414 | Lemma 4 for ~ clwwlkf1o : ... |
clwwlkf1o 28415 | F is a 1-1 onto function, ... |
clwwlken 28416 | The set of closed walks of... |
clwwlknwwlkncl 28417 | Obtaining a closed walk (a... |
clwwlkwwlksb 28418 | A nonempty word over verti... |
clwwlknwwlksnb 28419 | A word over vertices repre... |
clwwlkext2edg 28420 | If a word concatenated wit... |
wwlksext2clwwlk 28421 | If a word represents a wal... |
wwlksubclwwlk 28422 | Any prefix of a word repre... |
clwwnisshclwwsn 28423 | Cyclically shifting a clos... |
eleclclwwlknlem1 28424 | Lemma 1 for ~ eleclclwwlkn... |
eleclclwwlknlem2 28425 | Lemma 2 for ~ eleclclwwlkn... |
clwwlknscsh 28426 | The set of cyclical shifts... |
clwwlknccat 28427 | The concatenation of two w... |
umgr2cwwk2dif 28428 | If a word represents a clo... |
umgr2cwwkdifex 28429 | If a word represents a clo... |
erclwwlknrel 28430 | ` .~ ` is a relation. (Co... |
erclwwlkneq 28431 | Two classes are equivalent... |
erclwwlkneqlen 28432 | If two classes are equival... |
erclwwlknref 28433 | ` .~ ` is a reflexive rela... |
erclwwlknsym 28434 | ` .~ ` is a symmetric rela... |
erclwwlkntr 28435 | ` .~ ` is a transitive rel... |
erclwwlkn 28436 | ` .~ ` is an equivalence r... |
qerclwwlknfi 28437 | The quotient set of the se... |
hashclwwlkn0 28438 | The number of closed walks... |
eclclwwlkn1 28439 | An equivalence class accor... |
eleclclwwlkn 28440 | A member of an equivalence... |
hashecclwwlkn1 28441 | The size of every equivale... |
umgrhashecclwwlk 28442 | The size of every equivale... |
fusgrhashclwwlkn 28443 | The size of the set of clo... |
clwwlkndivn 28444 | The size of the set of clo... |
clwlknf1oclwwlknlem1 28445 | Lemma 1 for ~ clwlknf1oclw... |
clwlknf1oclwwlknlem2 28446 | Lemma 2 for ~ clwlknf1oclw... |
clwlknf1oclwwlknlem3 28447 | Lemma 3 for ~ clwlknf1oclw... |
clwlknf1oclwwlkn 28448 | There is a one-to-one onto... |
clwlkssizeeq 28449 | The size of the set of clo... |
clwlksndivn 28450 | The size of the set of clo... |
clwwlknonmpo 28453 | ` ( ClWWalksNOn `` G ) ` i... |
clwwlknon 28454 | The set of closed walks on... |
isclwwlknon 28455 | A word over the set of ver... |
clwwlk0on0 28456 | There is no word over the ... |
clwwlknon0 28457 | Sufficient conditions for ... |
clwwlknonfin 28458 | In a finite graph ` G ` , ... |
clwwlknonel 28459 | Characterization of a word... |
clwwlknonccat 28460 | The concatenation of two w... |
clwwlknon1 28461 | The set of closed walks on... |
clwwlknon1loop 28462 | If there is a loop at vert... |
clwwlknon1nloop 28463 | If there is no loop at ver... |
clwwlknon1sn 28464 | The set of (closed) walks ... |
clwwlknon1le1 28465 | There is at most one (clos... |
clwwlknon2 28466 | The set of closed walks on... |
clwwlknon2x 28467 | The set of closed walks on... |
s2elclwwlknon2 28468 | Sufficient conditions of a... |
clwwlknon2num 28469 | In a ` K `-regular graph `... |
clwwlknonwwlknonb 28470 | A word over vertices repre... |
clwwlknonex2lem1 28471 | Lemma 1 for ~ clwwlknonex2... |
clwwlknonex2lem2 28472 | Lemma 2 for ~ clwwlknonex2... |
clwwlknonex2 28473 | Extending a closed walk ` ... |
clwwlknonex2e 28474 | Extending a closed walk ` ... |
clwwlknondisj 28475 | The sets of closed walks o... |
clwwlknun 28476 | The set of closed walks of... |
clwwlkvbij 28477 | There is a bijection betwe... |
0ewlk 28478 | The empty set (empty seque... |
1ewlk 28479 | A sequence of 1 edge is an... |
0wlk 28480 | A pair of an empty set (of... |
is0wlk 28481 | A pair of an empty set (of... |
0wlkonlem1 28482 | Lemma 1 for ~ 0wlkon and ~... |
0wlkonlem2 28483 | Lemma 2 for ~ 0wlkon and ~... |
0wlkon 28484 | A walk of length 0 from a ... |
0wlkons1 28485 | A walk of length 0 from a ... |
0trl 28486 | A pair of an empty set (of... |
is0trl 28487 | A pair of an empty set (of... |
0trlon 28488 | A trail of length 0 from a... |
0pth 28489 | A pair of an empty set (of... |
0spth 28490 | A pair of an empty set (of... |
0pthon 28491 | A path of length 0 from a ... |
0pthon1 28492 | A path of length 0 from a ... |
0pthonv 28493 | For each vertex there is a... |
0clwlk 28494 | A pair of an empty set (of... |
0clwlkv 28495 | Any vertex (more precisely... |
0clwlk0 28496 | There is no closed walk in... |
0crct 28497 | A pair of an empty set (of... |
0cycl 28498 | A pair of an empty set (of... |
1pthdlem1 28499 | Lemma 1 for ~ 1pthd . (Co... |
1pthdlem2 28500 | Lemma 2 for ~ 1pthd . (Co... |
1wlkdlem1 28501 | Lemma 1 for ~ 1wlkd . (Co... |
1wlkdlem2 28502 | Lemma 2 for ~ 1wlkd . (Co... |
1wlkdlem3 28503 | Lemma 3 for ~ 1wlkd . (Co... |
1wlkdlem4 28504 | Lemma 4 for ~ 1wlkd . (Co... |
1wlkd 28505 | In a graph with two vertic... |
1trld 28506 | In a graph with two vertic... |
1pthd 28507 | In a graph with two vertic... |
1pthond 28508 | In a graph with two vertic... |
upgr1wlkdlem1 28509 | Lemma 1 for ~ upgr1wlkd . ... |
upgr1wlkdlem2 28510 | Lemma 2 for ~ upgr1wlkd . ... |
upgr1wlkd 28511 | In a pseudograph with two ... |
upgr1trld 28512 | In a pseudograph with two ... |
upgr1pthd 28513 | In a pseudograph with two ... |
upgr1pthond 28514 | In a pseudograph with two ... |
lppthon 28515 | A loop (which is an edge a... |
lp1cycl 28516 | A loop (which is an edge a... |
1pthon2v 28517 | For each pair of adjacent ... |
1pthon2ve 28518 | For each pair of adjacent ... |
wlk2v2elem1 28519 | Lemma 1 for ~ wlk2v2e : ` ... |
wlk2v2elem2 28520 | Lemma 2 for ~ wlk2v2e : T... |
wlk2v2e 28521 | In a graph with two vertic... |
ntrl2v2e 28522 | A walk which is not a trai... |
3wlkdlem1 28523 | Lemma 1 for ~ 3wlkd . (Co... |
3wlkdlem2 28524 | Lemma 2 for ~ 3wlkd . (Co... |
3wlkdlem3 28525 | Lemma 3 for ~ 3wlkd . (Co... |
3wlkdlem4 28526 | Lemma 4 for ~ 3wlkd . (Co... |
3wlkdlem5 28527 | Lemma 5 for ~ 3wlkd . (Co... |
3pthdlem1 28528 | Lemma 1 for ~ 3pthd . (Co... |
3wlkdlem6 28529 | Lemma 6 for ~ 3wlkd . (Co... |
3wlkdlem7 28530 | Lemma 7 for ~ 3wlkd . (Co... |
3wlkdlem8 28531 | Lemma 8 for ~ 3wlkd . (Co... |
3wlkdlem9 28532 | Lemma 9 for ~ 3wlkd . (Co... |
3wlkdlem10 28533 | Lemma 10 for ~ 3wlkd . (C... |
3wlkd 28534 | Construction of a walk fro... |
3wlkond 28535 | A walk of length 3 from on... |
3trld 28536 | Construction of a trail fr... |
3trlond 28537 | A trail of length 3 from o... |
3pthd 28538 | A path of length 3 from on... |
3pthond 28539 | A path of length 3 from on... |
3spthd 28540 | A simple path of length 3 ... |
3spthond 28541 | A simple path of length 3 ... |
3cycld 28542 | Construction of a 3-cycle ... |
3cyclpd 28543 | Construction of a 3-cycle ... |
upgr3v3e3cycl 28544 | If there is a cycle of len... |
uhgr3cyclexlem 28545 | Lemma for ~ uhgr3cyclex . ... |
uhgr3cyclex 28546 | If there are three differe... |
umgr3cyclex 28547 | If there are three (differ... |
umgr3v3e3cycl 28548 | If and only if there is a ... |
upgr4cycl4dv4e 28549 | If there is a cycle of len... |
dfconngr1 28552 | Alternative definition of ... |
isconngr 28553 | The property of being a co... |
isconngr1 28554 | The property of being a co... |
cusconngr 28555 | A complete hypergraph is c... |
0conngr 28556 | A graph without vertices i... |
0vconngr 28557 | A graph without vertices i... |
1conngr 28558 | A graph with (at most) one... |
conngrv2edg 28559 | A vertex in a connected gr... |
vdn0conngrumgrv2 28560 | A vertex in a connected mu... |
releupth 28563 | The set ` ( EulerPaths `` ... |
eupths 28564 | The Eulerian paths on the ... |
iseupth 28565 | The property " ` <. F , P ... |
iseupthf1o 28566 | The property " ` <. F , P ... |
eupthi 28567 | Properties of an Eulerian ... |
eupthf1o 28568 | The ` F ` function in an E... |
eupthfi 28569 | Any graph with an Eulerian... |
eupthseg 28570 | The ` N ` -th edge in an e... |
upgriseupth 28571 | The property " ` <. F , P ... |
upgreupthi 28572 | Properties of an Eulerian ... |
upgreupthseg 28573 | The ` N ` -th edge in an e... |
eupthcl 28574 | An Eulerian path has lengt... |
eupthistrl 28575 | An Eulerian path is a trai... |
eupthiswlk 28576 | An Eulerian path is a walk... |
eupthpf 28577 | The ` P ` function in an E... |
eupth0 28578 | There is an Eulerian path ... |
eupthres 28579 | The restriction ` <. H , Q... |
eupthp1 28580 | Append one path segment to... |
eupth2eucrct 28581 | Append one path segment to... |
eupth2lem1 28582 | Lemma for ~ eupth2 . (Con... |
eupth2lem2 28583 | Lemma for ~ eupth2 . (Con... |
trlsegvdeglem1 28584 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem2 28585 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem3 28586 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem4 28587 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem5 28588 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem6 28589 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem7 28590 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeg 28591 | Formerly part of proof of ... |
eupth2lem3lem1 28592 | Lemma for ~ eupth2lem3 . ... |
eupth2lem3lem2 28593 | Lemma for ~ eupth2lem3 . ... |
eupth2lem3lem3 28594 | Lemma for ~ eupth2lem3 , f... |
eupth2lem3lem4 28595 | Lemma for ~ eupth2lem3 , f... |
eupth2lem3lem5 28596 | Lemma for ~ eupth2 . (Con... |
eupth2lem3lem6 28597 | Formerly part of proof of ... |
eupth2lem3lem7 28598 | Lemma for ~ eupth2lem3 : ... |
eupthvdres 28599 | Formerly part of proof of ... |
eupth2lem3 28600 | Lemma for ~ eupth2 . (Con... |
eupth2lemb 28601 | Lemma for ~ eupth2 (induct... |
eupth2lems 28602 | Lemma for ~ eupth2 (induct... |
eupth2 28603 | The only vertices of odd d... |
eulerpathpr 28604 | A graph with an Eulerian p... |
eulerpath 28605 | A pseudograph with an Eule... |
eulercrct 28606 | A pseudograph with an Eule... |
eucrctshift 28607 | Cyclically shifting the in... |
eucrct2eupth1 28608 | Removing one edge ` ( I ``... |
eucrct2eupth 28609 | Removing one edge ` ( I ``... |
konigsbergvtx 28610 | The set of vertices of the... |
konigsbergiedg 28611 | The indexed edges of the K... |
konigsbergiedgw 28612 | The indexed edges of the K... |
konigsbergssiedgwpr 28613 | Each subset of the indexed... |
konigsbergssiedgw 28614 | Each subset of the indexed... |
konigsbergumgr 28615 | The Königsberg graph ... |
konigsberglem1 28616 | Lemma 1 for ~ konigsberg :... |
konigsberglem2 28617 | Lemma 2 for ~ konigsberg :... |
konigsberglem3 28618 | Lemma 3 for ~ konigsberg :... |
konigsberglem4 28619 | Lemma 4 for ~ konigsberg :... |
konigsberglem5 28620 | Lemma 5 for ~ konigsberg :... |
konigsberg 28621 | The Königsberg Bridge... |
isfrgr 28624 | The property of being a fr... |
frgrusgr 28625 | A friendship graph is a si... |
frgr0v 28626 | Any null graph (set with n... |
frgr0vb 28627 | Any null graph (without ve... |
frgruhgr0v 28628 | Any null graph (without ve... |
frgr0 28629 | The null graph (graph with... |
frcond1 28630 | The friendship condition: ... |
frcond2 28631 | The friendship condition: ... |
frgreu 28632 | Variant of ~ frcond2 : An... |
frcond3 28633 | The friendship condition, ... |
frcond4 28634 | The friendship condition, ... |
frgr1v 28635 | Any graph with (at most) o... |
nfrgr2v 28636 | Any graph with two (differ... |
frgr3vlem1 28637 | Lemma 1 for ~ frgr3v . (C... |
frgr3vlem2 28638 | Lemma 2 for ~ frgr3v . (C... |
frgr3v 28639 | Any graph with three verti... |
1vwmgr 28640 | Every graph with one verte... |
3vfriswmgrlem 28641 | Lemma for ~ 3vfriswmgr . ... |
3vfriswmgr 28642 | Every friendship graph wit... |
1to2vfriswmgr 28643 | Every friendship graph wit... |
1to3vfriswmgr 28644 | Every friendship graph wit... |
1to3vfriendship 28645 | The friendship theorem for... |
2pthfrgrrn 28646 | Between any two (different... |
2pthfrgrrn2 28647 | Between any two (different... |
2pthfrgr 28648 | Between any two (different... |
3cyclfrgrrn1 28649 | Every vertex in a friendsh... |
3cyclfrgrrn 28650 | Every vertex in a friendsh... |
3cyclfrgrrn2 28651 | Every vertex in a friendsh... |
3cyclfrgr 28652 | Every vertex in a friendsh... |
4cycl2v2nb 28653 | In a (maybe degenerate) 4-... |
4cycl2vnunb 28654 | In a 4-cycle, two distinct... |
n4cyclfrgr 28655 | There is no 4-cycle in a f... |
4cyclusnfrgr 28656 | A graph with a 4-cycle is ... |
frgrnbnb 28657 | If two neighbors ` U ` and... |
frgrconngr 28658 | A friendship graph is conn... |
vdgn0frgrv2 28659 | A vertex in a friendship g... |
vdgn1frgrv2 28660 | Any vertex in a friendship... |
vdgn1frgrv3 28661 | Any vertex in a friendship... |
vdgfrgrgt2 28662 | Any vertex in a friendship... |
frgrncvvdeqlem1 28663 | Lemma 1 for ~ frgrncvvdeq ... |
frgrncvvdeqlem2 28664 | Lemma 2 for ~ frgrncvvdeq ... |
frgrncvvdeqlem3 28665 | Lemma 3 for ~ frgrncvvdeq ... |
frgrncvvdeqlem4 28666 | Lemma 4 for ~ frgrncvvdeq ... |
frgrncvvdeqlem5 28667 | Lemma 5 for ~ frgrncvvdeq ... |
frgrncvvdeqlem6 28668 | Lemma 6 for ~ frgrncvvdeq ... |
frgrncvvdeqlem7 28669 | Lemma 7 for ~ frgrncvvdeq ... |
frgrncvvdeqlem8 28670 | Lemma 8 for ~ frgrncvvdeq ... |
frgrncvvdeqlem9 28671 | Lemma 9 for ~ frgrncvvdeq ... |
frgrncvvdeqlem10 28672 | Lemma 10 for ~ frgrncvvdeq... |
frgrncvvdeq 28673 | In a friendship graph, two... |
frgrwopreglem4a 28674 | In a friendship graph any ... |
frgrwopreglem5a 28675 | If a friendship graph has ... |
frgrwopreglem1 28676 | Lemma 1 for ~ frgrwopreg :... |
frgrwopreglem2 28677 | Lemma 2 for ~ frgrwopreg .... |
frgrwopreglem3 28678 | Lemma 3 for ~ frgrwopreg .... |
frgrwopreglem4 28679 | Lemma 4 for ~ frgrwopreg .... |
frgrwopregasn 28680 | According to statement 5 i... |
frgrwopregbsn 28681 | According to statement 5 i... |
frgrwopreg1 28682 | According to statement 5 i... |
frgrwopreg2 28683 | According to statement 5 i... |
frgrwopreglem5lem 28684 | Lemma for ~ frgrwopreglem5... |
frgrwopreglem5 28685 | Lemma 5 for ~ frgrwopreg .... |
frgrwopreglem5ALT 28686 | Alternate direct proof of ... |
frgrwopreg 28687 | In a friendship graph ther... |
frgrregorufr0 28688 | In a friendship graph ther... |
frgrregorufr 28689 | If there is a vertex havin... |
frgrregorufrg 28690 | If there is a vertex havin... |
frgr2wwlkeu 28691 | For two different vertices... |
frgr2wwlkn0 28692 | In a friendship graph, the... |
frgr2wwlk1 28693 | In a friendship graph, the... |
frgr2wsp1 28694 | In a friendship graph, the... |
frgr2wwlkeqm 28695 | If there is a (simple) pat... |
frgrhash2wsp 28696 | The number of simple paths... |
fusgreg2wsplem 28697 | Lemma for ~ fusgreg2wsp an... |
fusgr2wsp2nb 28698 | The set of paths of length... |
fusgreghash2wspv 28699 | According to statement 7 i... |
fusgreg2wsp 28700 | In a finite simple graph, ... |
2wspmdisj 28701 | The sets of paths of lengt... |
fusgreghash2wsp 28702 | In a finite k-regular grap... |
frrusgrord0lem 28703 | Lemma for ~ frrusgrord0 . ... |
frrusgrord0 28704 | If a nonempty finite frien... |
frrusgrord 28705 | If a nonempty finite frien... |
numclwwlk2lem1lem 28706 | Lemma for ~ numclwwlk2lem1... |
2clwwlklem 28707 | Lemma for ~ clwwnonrepclww... |
clwwnrepclwwn 28708 | If the initial vertex of a... |
clwwnonrepclwwnon 28709 | If the initial vertex of a... |
2clwwlk2clwwlklem 28710 | Lemma for ~ 2clwwlk2clwwlk... |
2clwwlk 28711 | Value of operation ` C ` ,... |
2clwwlk2 28712 | The set ` ( X C 2 ) ` of d... |
2clwwlkel 28713 | Characterization of an ele... |
2clwwlk2clwwlk 28714 | An element of the value of... |
numclwwlk1lem2foalem 28715 | Lemma for ~ numclwwlk1lem2... |
extwwlkfab 28716 | The set ` ( X C N ) ` of d... |
extwwlkfabel 28717 | Characterization of an ele... |
numclwwlk1lem2foa 28718 | Going forth and back from ... |
numclwwlk1lem2f 28719 | ` T ` is a function, mappi... |
numclwwlk1lem2fv 28720 | Value of the function ` T ... |
numclwwlk1lem2f1 28721 | ` T ` is a 1-1 function. ... |
numclwwlk1lem2fo 28722 | ` T ` is an onto function.... |
numclwwlk1lem2f1o 28723 | ` T ` is a 1-1 onto functi... |
numclwwlk1lem2 28724 | The set of double loops of... |
numclwwlk1 28725 | Statement 9 in [Huneke] p.... |
clwwlknonclwlknonf1o 28726 | ` F ` is a bijection betwe... |
clwwlknonclwlknonen 28727 | The sets of the two repres... |
dlwwlknondlwlknonf1olem1 28728 | Lemma 1 for ~ dlwwlknondlw... |
dlwwlknondlwlknonf1o 28729 | ` F ` is a bijection betwe... |
dlwwlknondlwlknonen 28730 | The sets of the two repres... |
wlkl0 28731 | There is exactly one walk ... |
clwlknon2num 28732 | There are k walks of lengt... |
numclwlk1lem1 28733 | Lemma 1 for ~ numclwlk1 (S... |
numclwlk1lem2 28734 | Lemma 2 for ~ numclwlk1 (S... |
numclwlk1 28735 | Statement 9 in [Huneke] p.... |
numclwwlkovh0 28736 | Value of operation ` H ` ,... |
numclwwlkovh 28737 | Value of operation ` H ` ,... |
numclwwlkovq 28738 | Value of operation ` Q ` ,... |
numclwwlkqhash 28739 | In a ` K `-regular graph, ... |
numclwwlk2lem1 28740 | In a friendship graph, for... |
numclwlk2lem2f 28741 | ` R ` is a function mappin... |
numclwlk2lem2fv 28742 | Value of the function ` R ... |
numclwlk2lem2f1o 28743 | ` R ` is a 1-1 onto functi... |
numclwwlk2lem3 28744 | In a friendship graph, the... |
numclwwlk2 28745 | Statement 10 in [Huneke] p... |
numclwwlk3lem1 28746 | Lemma 2 for ~ numclwwlk3 .... |
numclwwlk3lem2lem 28747 | Lemma for ~ numclwwlk3lem2... |
numclwwlk3lem2 28748 | Lemma 1 for ~ numclwwlk3 :... |
numclwwlk3 28749 | Statement 12 in [Huneke] p... |
numclwwlk4 28750 | The total number of closed... |
numclwwlk5lem 28751 | Lemma for ~ numclwwlk5 . ... |
numclwwlk5 28752 | Statement 13 in [Huneke] p... |
numclwwlk7lem 28753 | Lemma for ~ numclwwlk7 , ~... |
numclwwlk6 28754 | For a prime divisor ` P ` ... |
numclwwlk7 28755 | Statement 14 in [Huneke] p... |
numclwwlk8 28756 | The size of the set of clo... |
frgrreggt1 28757 | If a finite nonempty frien... |
frgrreg 28758 | If a finite nonempty frien... |
frgrregord013 28759 | If a finite friendship gra... |
frgrregord13 28760 | If a nonempty finite frien... |
frgrogt3nreg 28761 | If a finite friendship gra... |
friendshipgt3 28762 | The friendship theorem for... |
friendship 28763 | The friendship theorem: I... |
conventions 28764 |
H... |
conventions-labels 28765 |
... |
conventions-comments 28766 |
... |
natded 28767 | Here are typical n... |
ex-natded5.2 28768 | Theorem 5.2 of [Clemente] ... |
ex-natded5.2-2 28769 | A more efficient proof of ... |
ex-natded5.2i 28770 | The same as ~ ex-natded5.2... |
ex-natded5.3 28771 | Theorem 5.3 of [Clemente] ... |
ex-natded5.3-2 28772 | A more efficient proof of ... |
ex-natded5.3i 28773 | The same as ~ ex-natded5.3... |
ex-natded5.5 28774 | Theorem 5.5 of [Clemente] ... |
ex-natded5.7 28775 | Theorem 5.7 of [Clemente] ... |
ex-natded5.7-2 28776 | A more efficient proof of ... |
ex-natded5.8 28777 | Theorem 5.8 of [Clemente] ... |
ex-natded5.8-2 28778 | A more efficient proof of ... |
ex-natded5.13 28779 | Theorem 5.13 of [Clemente]... |
ex-natded5.13-2 28780 | A more efficient proof of ... |
ex-natded9.20 28781 | Theorem 9.20 of [Clemente]... |
ex-natded9.20-2 28782 | A more efficient proof of ... |
ex-natded9.26 28783 | Theorem 9.26 of [Clemente]... |
ex-natded9.26-2 28784 | A more efficient proof of ... |
ex-or 28785 | Example for ~ df-or . Exa... |
ex-an 28786 | Example for ~ df-an . Exa... |
ex-dif 28787 | Example for ~ df-dif . Ex... |
ex-un 28788 | Example for ~ df-un . Exa... |
ex-in 28789 | Example for ~ df-in . Exa... |
ex-uni 28790 | Example for ~ df-uni . Ex... |
ex-ss 28791 | Example for ~ df-ss . Exa... |
ex-pss 28792 | Example for ~ df-pss . Ex... |
ex-pw 28793 | Example for ~ df-pw . Exa... |
ex-pr 28794 | Example for ~ df-pr . (Co... |
ex-br 28795 | Example for ~ df-br . Exa... |
ex-opab 28796 | Example for ~ df-opab . E... |
ex-eprel 28797 | Example for ~ df-eprel . ... |
ex-id 28798 | Example for ~ df-id . Exa... |
ex-po 28799 | Example for ~ df-po . Exa... |
ex-xp 28800 | Example for ~ df-xp . Exa... |
ex-cnv 28801 | Example for ~ df-cnv . Ex... |
ex-co 28802 | Example for ~ df-co . Exa... |
ex-dm 28803 | Example for ~ df-dm . Exa... |
ex-rn 28804 | Example for ~ df-rn . Exa... |
ex-res 28805 | Example for ~ df-res . Ex... |
ex-ima 28806 | Example for ~ df-ima . Ex... |
ex-fv 28807 | Example for ~ df-fv . Exa... |
ex-1st 28808 | Example for ~ df-1st . Ex... |
ex-2nd 28809 | Example for ~ df-2nd . Ex... |
1kp2ke3k 28810 | Example for ~ df-dec , 100... |
ex-fl 28811 | Example for ~ df-fl . Exa... |
ex-ceil 28812 | Example for ~ df-ceil . (... |
ex-mod 28813 | Example for ~ df-mod . (C... |
ex-exp 28814 | Example for ~ df-exp . (C... |
ex-fac 28815 | Example for ~ df-fac . (C... |
ex-bc 28816 | Example for ~ df-bc . (Co... |
ex-hash 28817 | Example for ~ df-hash . (... |
ex-sqrt 28818 | Example for ~ df-sqrt . (... |
ex-abs 28819 | Example for ~ df-abs . (C... |
ex-dvds 28820 | Example for ~ df-dvds : 3 ... |
ex-gcd 28821 | Example for ~ df-gcd . (C... |
ex-lcm 28822 | Example for ~ df-lcm . (C... |
ex-prmo 28823 | Example for ~ df-prmo : ` ... |
aevdemo 28824 | Proof illustrating the com... |
ex-ind-dvds 28825 | Example of a proof by indu... |
ex-fpar 28826 | Formalized example provide... |
avril1 28827 | Poisson d'Avril's Theorem.... |
2bornot2b 28828 | The law of excluded middle... |
helloworld 28829 | The classic "Hello world" ... |
1p1e2apr1 28830 | One plus one equals two. ... |
eqid1 28831 | Law of identity (reflexivi... |
1div0apr 28832 | Division by zero is forbid... |
topnfbey 28833 | Nothing seems to be imposs... |
9p10ne21 28834 | 9 + 10 is not equal to 21.... |
9p10ne21fool 28835 | 9 + 10 equals 21. This as... |
isplig 28838 | The predicate "is a planar... |
ispligb 28839 | The predicate "is a planar... |
tncp 28840 | In any planar incidence ge... |
l2p 28841 | For any line in a planar i... |
lpni 28842 | For any line in a planar i... |
nsnlplig 28843 | There is no "one-point lin... |
nsnlpligALT 28844 | Alternate version of ~ nsn... |
n0lplig 28845 | There is no "empty line" i... |
n0lpligALT 28846 | Alternate version of ~ n0l... |
eulplig 28847 | Through two distinct point... |
pliguhgr 28848 | Any planar incidence geome... |
dummylink 28849 | Alias for ~ a1ii that may ... |
id1 28850 | Alias for ~ idALT that may... |
isgrpo 28859 | The predicate "is a group ... |
isgrpoi 28860 | Properties that determine ... |
grpofo 28861 | A group operation maps ont... |
grpocl 28862 | Closure law for a group op... |
grpolidinv 28863 | A group has a left identit... |
grpon0 28864 | The base set of a group is... |
grpoass 28865 | A group operation is assoc... |
grpoidinvlem1 28866 | Lemma for ~ grpoidinv . (... |
grpoidinvlem2 28867 | Lemma for ~ grpoidinv . (... |
grpoidinvlem3 28868 | Lemma for ~ grpoidinv . (... |
grpoidinvlem4 28869 | Lemma for ~ grpoidinv . (... |
grpoidinv 28870 | A group has a left and rig... |
grpoideu 28871 | The left identity element ... |
grporndm 28872 | A group's range in terms o... |
0ngrp 28873 | The empty set is not a gro... |
gidval 28874 | The value of the identity ... |
grpoidval 28875 | Lemma for ~ grpoidcl and o... |
grpoidcl 28876 | The identity element of a ... |
grpoidinv2 28877 | A group's properties using... |
grpolid 28878 | The identity element of a ... |
grporid 28879 | The identity element of a ... |
grporcan 28880 | Right cancellation law for... |
grpoinveu 28881 | The left inverse element o... |
grpoid 28882 | Two ways of saying that an... |
grporn 28883 | The range of a group opera... |
grpoinvfval 28884 | The inverse function of a ... |
grpoinvval 28885 | The inverse of a group ele... |
grpoinvcl 28886 | A group element's inverse ... |
grpoinv 28887 | The properties of a group ... |
grpolinv 28888 | The left inverse of a grou... |
grporinv 28889 | The right inverse of a gro... |
grpoinvid1 28890 | The inverse of a group ele... |
grpoinvid2 28891 | The inverse of a group ele... |
grpolcan 28892 | Left cancellation law for ... |
grpo2inv 28893 | Double inverse law for gro... |
grpoinvf 28894 | Mapping of the inverse fun... |
grpoinvop 28895 | The inverse of the group o... |
grpodivfval 28896 | Group division (or subtrac... |
grpodivval 28897 | Group division (or subtrac... |
grpodivinv 28898 | Group division by an inver... |
grpoinvdiv 28899 | Inverse of a group divisio... |
grpodivf 28900 | Mapping for group division... |
grpodivcl 28901 | Closure of group division ... |
grpodivdiv 28902 | Double group division. (C... |
grpomuldivass 28903 | Associative-type law for m... |
grpodivid 28904 | Division of a group member... |
grponpcan 28905 | Cancellation law for group... |
isablo 28908 | The predicate "is an Abeli... |
ablogrpo 28909 | An Abelian group operation... |
ablocom 28910 | An Abelian group operation... |
ablo32 28911 | Commutative/associative la... |
ablo4 28912 | Commutative/associative la... |
isabloi 28913 | Properties that determine ... |
ablomuldiv 28914 | Law for group multiplicati... |
ablodivdiv 28915 | Law for double group divis... |
ablodivdiv4 28916 | Law for double group divis... |
ablodiv32 28917 | Swap the second and third ... |
ablonncan 28918 | Cancellation law for group... |
ablonnncan1 28919 | Cancellation law for group... |
vcrel 28922 | The class of all complex v... |
vciOLD 28923 | Obsolete version of ~ cvsi... |
vcsm 28924 | Functionality of th scalar... |
vccl 28925 | Closure of the scalar prod... |
vcidOLD 28926 | Identity element for the s... |
vcdi 28927 | Distributive law for the s... |
vcdir 28928 | Distributive law for the s... |
vcass 28929 | Associative law for the sc... |
vc2OLD 28930 | A vector plus itself is tw... |
vcablo 28931 | Vector addition is an Abel... |
vcgrp 28932 | Vector addition is a group... |
vclcan 28933 | Left cancellation law for ... |
vczcl 28934 | The zero vector is a vecto... |
vc0rid 28935 | The zero vector is a right... |
vc0 28936 | Zero times a vector is the... |
vcz 28937 | Anything times the zero ve... |
vcm 28938 | Minus 1 times a vector is ... |
isvclem 28939 | Lemma for ~ isvcOLD . (Co... |
vcex 28940 | The components of a comple... |
isvcOLD 28941 | The predicate "is a comple... |
isvciOLD 28942 | Properties that determine ... |
cnaddabloOLD 28943 | Obsolete version of ~ cnad... |
cnidOLD 28944 | Obsolete version of ~ cnad... |
cncvcOLD 28945 | Obsolete version of ~ cncv... |
nvss 28955 | Structure of the class of ... |
nvvcop 28956 | A normed complex vector sp... |
nvrel 28964 | The class of all normed co... |
vafval 28965 | Value of the function for ... |
bafval 28966 | Value of the function for ... |
smfval 28967 | Value of the function for ... |
0vfval 28968 | Value of the function for ... |
nmcvfval 28969 | Value of the norm function... |
nvop2 28970 | A normed complex vector sp... |
nvvop 28971 | The vector space component... |
isnvlem 28972 | Lemma for ~ isnv . (Contr... |
nvex 28973 | The components of a normed... |
isnv 28974 | The predicate "is a normed... |
isnvi 28975 | Properties that determine ... |
nvi 28976 | The properties of a normed... |
nvvc 28977 | The vector space component... |
nvablo 28978 | The vector addition operat... |
nvgrp 28979 | The vector addition operat... |
nvgf 28980 | Mapping for the vector add... |
nvsf 28981 | Mapping for the scalar mul... |
nvgcl 28982 | Closure law for the vector... |
nvcom 28983 | The vector addition (group... |
nvass 28984 | The vector addition (group... |
nvadd32 28985 | Commutative/associative la... |
nvrcan 28986 | Right cancellation law for... |
nvadd4 28987 | Rearrangement of 4 terms i... |
nvscl 28988 | Closure law for the scalar... |
nvsid 28989 | Identity element for the s... |
nvsass 28990 | Associative law for the sc... |
nvscom 28991 | Commutative law for the sc... |
nvdi 28992 | Distributive law for the s... |
nvdir 28993 | Distributive law for the s... |
nv2 28994 | A vector plus itself is tw... |
vsfval 28995 | Value of the function for ... |
nvzcl 28996 | Closure law for the zero v... |
nv0rid 28997 | The zero vector is a right... |
nv0lid 28998 | The zero vector is a left ... |
nv0 28999 | Zero times a vector is the... |
nvsz 29000 | Anything times the zero ve... |
nvinv 29001 | Minus 1 times a vector is ... |
nvinvfval 29002 | Function for the negative ... |
nvm 29003 | Vector subtraction in term... |
nvmval 29004 | Value of vector subtractio... |
nvmval2 29005 | Value of vector subtractio... |
nvmfval 29006 | Value of the function for ... |
nvmf 29007 | Mapping for the vector sub... |
nvmcl 29008 | Closure law for the vector... |
nvnnncan1 29009 | Cancellation law for vecto... |
nvmdi 29010 | Distributive law for scala... |
nvnegneg 29011 | Double negative of a vecto... |
nvmul0or 29012 | If a scalar product is zer... |
nvrinv 29013 | A vector minus itself. (C... |
nvlinv 29014 | Minus a vector plus itself... |
nvpncan2 29015 | Cancellation law for vecto... |
nvpncan 29016 | Cancellation law for vecto... |
nvaddsub 29017 | Commutative/associative la... |
nvnpcan 29018 | Cancellation law for a nor... |
nvaddsub4 29019 | Rearrangement of 4 terms i... |
nvmeq0 29020 | The difference between two... |
nvmid 29021 | A vector minus itself is t... |
nvf 29022 | Mapping for the norm funct... |
nvcl 29023 | The norm of a normed compl... |
nvcli 29024 | The norm of a normed compl... |
nvs 29025 | Proportionality property o... |
nvsge0 29026 | The norm of a scalar produ... |
nvm1 29027 | The norm of the negative o... |
nvdif 29028 | The norm of the difference... |
nvpi 29029 | The norm of a vector plus ... |
nvz0 29030 | The norm of a zero vector ... |
nvz 29031 | The norm of a vector is ze... |
nvtri 29032 | Triangle inequality for th... |
nvmtri 29033 | Triangle inequality for th... |
nvabs 29034 | Norm difference property o... |
nvge0 29035 | The norm of a normed compl... |
nvgt0 29036 | A nonzero norm is positive... |
nv1 29037 | From any nonzero vector, c... |
nvop 29038 | A complex inner product sp... |
cnnv 29039 | The set of complex numbers... |
cnnvg 29040 | The vector addition (group... |
cnnvba 29041 | The base set of the normed... |
cnnvs 29042 | The scalar product operati... |
cnnvnm 29043 | The norm operation of the ... |
cnnvm 29044 | The vector subtraction ope... |
elimnv 29045 | Hypothesis elimination lem... |
elimnvu 29046 | Hypothesis elimination lem... |
imsval 29047 | Value of the induced metri... |
imsdval 29048 | Value of the induced metri... |
imsdval2 29049 | Value of the distance func... |
nvnd 29050 | The norm of a normed compl... |
imsdf 29051 | Mapping for the induced me... |
imsmetlem 29052 | Lemma for ~ imsmet . (Con... |
imsmet 29053 | The induced metric of a no... |
imsxmet 29054 | The induced metric of a no... |
cnims 29055 | The metric induced on the ... |
vacn 29056 | Vector addition is jointly... |
nmcvcn 29057 | The norm of a normed compl... |
nmcnc 29058 | The norm of a normed compl... |
smcnlem 29059 | Lemma for ~ smcn . (Contr... |
smcn 29060 | Scalar multiplication is j... |
vmcn 29061 | Vector subtraction is join... |
dipfval 29064 | The inner product function... |
ipval 29065 | Value of the inner product... |
ipval2lem2 29066 | Lemma for ~ ipval3 . (Con... |
ipval2lem3 29067 | Lemma for ~ ipval3 . (Con... |
ipval2lem4 29068 | Lemma for ~ ipval3 . (Con... |
ipval2 29069 | Expansion of the inner pro... |
4ipval2 29070 | Four times the inner produ... |
ipval3 29071 | Expansion of the inner pro... |
ipidsq 29072 | The inner product of a vec... |
ipnm 29073 | Norm expressed in terms of... |
dipcl 29074 | An inner product is a comp... |
ipf 29075 | Mapping for the inner prod... |
dipcj 29076 | The complex conjugate of a... |
ipipcj 29077 | An inner product times its... |
diporthcom 29078 | Orthogonality (meaning inn... |
dip0r 29079 | Inner product with a zero ... |
dip0l 29080 | Inner product with a zero ... |
ipz 29081 | The inner product of a vec... |
dipcn 29082 | Inner product is jointly c... |
sspval 29085 | The set of all subspaces o... |
isssp 29086 | The predicate "is a subspa... |
sspid 29087 | A normed complex vector sp... |
sspnv 29088 | A subspace is a normed com... |
sspba 29089 | The base set of a subspace... |
sspg 29090 | Vector addition on a subsp... |
sspgval 29091 | Vector addition on a subsp... |
ssps 29092 | Scalar multiplication on a... |
sspsval 29093 | Scalar multiplication on a... |
sspmlem 29094 | Lemma for ~ sspm and other... |
sspmval 29095 | Vector addition on a subsp... |
sspm 29096 | Vector subtraction on a su... |
sspz 29097 | The zero vector of a subsp... |
sspn 29098 | The norm on a subspace is ... |
sspnval 29099 | The norm on a subspace in ... |
sspimsval 29100 | The induced metric on a su... |
sspims 29101 | The induced metric on a su... |
lnoval 29114 | The set of linear operator... |
islno 29115 | The predicate "is a linear... |
lnolin 29116 | Basic linearity property o... |
lnof 29117 | A linear operator is a map... |
lno0 29118 | The value of a linear oper... |
lnocoi 29119 | The composition of two lin... |
lnoadd 29120 | Addition property of a lin... |
lnosub 29121 | Subtraction property of a ... |
lnomul 29122 | Scalar multiplication prop... |
nvo00 29123 | Two ways to express a zero... |
nmoofval 29124 | The operator norm function... |
nmooval 29125 | The operator norm function... |
nmosetre 29126 | The set in the supremum of... |
nmosetn0 29127 | The set in the supremum of... |
nmoxr 29128 | The norm of an operator is... |
nmooge0 29129 | The norm of an operator is... |
nmorepnf 29130 | The norm of an operator is... |
nmoreltpnf 29131 | The norm of any operator i... |
nmogtmnf 29132 | The norm of an operator is... |
nmoolb 29133 | A lower bound for an opera... |
nmoubi 29134 | An upper bound for an oper... |
nmoub3i 29135 | An upper bound for an oper... |
nmoub2i 29136 | An upper bound for an oper... |
nmobndi 29137 | Two ways to express that a... |
nmounbi 29138 | Two ways two express that ... |
nmounbseqi 29139 | An unbounded operator dete... |
nmounbseqiALT 29140 | Alternate shorter proof of... |
nmobndseqi 29141 | A bounded sequence determi... |
nmobndseqiALT 29142 | Alternate shorter proof of... |
bloval 29143 | The class of bounded linea... |
isblo 29144 | The predicate "is a bounde... |
isblo2 29145 | The predicate "is a bounde... |
bloln 29146 | A bounded operator is a li... |
blof 29147 | A bounded operator is an o... |
nmblore 29148 | The norm of a bounded oper... |
0ofval 29149 | The zero operator between ... |
0oval 29150 | Value of the zero operator... |
0oo 29151 | The zero operator is an op... |
0lno 29152 | The zero operator is linea... |
nmoo0 29153 | The operator norm of the z... |
0blo 29154 | The zero operator is a bou... |
nmlno0lem 29155 | Lemma for ~ nmlno0i . (Co... |
nmlno0i 29156 | The norm of a linear opera... |
nmlno0 29157 | The norm of a linear opera... |
nmlnoubi 29158 | An upper bound for the ope... |
nmlnogt0 29159 | The norm of a nonzero line... |
lnon0 29160 | The domain of a nonzero li... |
nmblolbii 29161 | A lower bound for the norm... |
nmblolbi 29162 | A lower bound for the norm... |
isblo3i 29163 | The predicate "is a bounde... |
blo3i 29164 | Properties that determine ... |
blometi 29165 | Upper bound for the distan... |
blocnilem 29166 | Lemma for ~ blocni and ~ l... |
blocni 29167 | A linear operator is conti... |
lnocni 29168 | If a linear operator is co... |
blocn 29169 | A linear operator is conti... |
blocn2 29170 | A bounded linear operator ... |
ajfval 29171 | The adjoint function. (Co... |
hmoval 29172 | The set of Hermitian (self... |
ishmo 29173 | The predicate "is a hermit... |
phnv 29176 | Every complex inner produc... |
phrel 29177 | The class of all complex i... |
phnvi 29178 | Every complex inner produc... |
isphg 29179 | The predicate "is a comple... |
phop 29180 | A complex inner product sp... |
cncph 29181 | The set of complex numbers... |
elimph 29182 | Hypothesis elimination lem... |
elimphu 29183 | Hypothesis elimination lem... |
isph 29184 | The predicate "is an inner... |
phpar2 29185 | The parallelogram law for ... |
phpar 29186 | The parallelogram law for ... |
ip0i 29187 | A slight variant of Equati... |
ip1ilem 29188 | Lemma for ~ ip1i . (Contr... |
ip1i 29189 | Equation 6.47 of [Ponnusam... |
ip2i 29190 | Equation 6.48 of [Ponnusam... |
ipdirilem 29191 | Lemma for ~ ipdiri . (Con... |
ipdiri 29192 | Distributive law for inner... |
ipasslem1 29193 | Lemma for ~ ipassi . Show... |
ipasslem2 29194 | Lemma for ~ ipassi . Show... |
ipasslem3 29195 | Lemma for ~ ipassi . Show... |
ipasslem4 29196 | Lemma for ~ ipassi . Show... |
ipasslem5 29197 | Lemma for ~ ipassi . Show... |
ipasslem7 29198 | Lemma for ~ ipassi . Show... |
ipasslem8 29199 | Lemma for ~ ipassi . By ~... |
ipasslem9 29200 | Lemma for ~ ipassi . Conc... |
ipasslem10 29201 | Lemma for ~ ipassi . Show... |
ipasslem11 29202 | Lemma for ~ ipassi . Show... |
ipassi 29203 | Associative law for inner ... |
dipdir 29204 | Distributive law for inner... |
dipdi 29205 | Distributive law for inner... |
ip2dii 29206 | Inner product of two sums.... |
dipass 29207 | Associative law for inner ... |
dipassr 29208 | "Associative" law for seco... |
dipassr2 29209 | "Associative" law for inne... |
dipsubdir 29210 | Distributive law for inner... |
dipsubdi 29211 | Distributive law for inner... |
pythi 29212 | The Pythagorean theorem fo... |
siilem1 29213 | Lemma for ~ sii . (Contri... |
siilem2 29214 | Lemma for ~ sii . (Contri... |
siii 29215 | Inference from ~ sii . (C... |
sii 29216 | Obsolete version of ~ ipca... |
ipblnfi 29217 | A function ` F ` generated... |
ip2eqi 29218 | Two vectors are equal iff ... |
phoeqi 29219 | A condition implying that ... |
ajmoi 29220 | Every operator has at most... |
ajfuni 29221 | The adjoint function is a ... |
ajfun 29222 | The adjoint function is a ... |
ajval 29223 | Value of the adjoint funct... |
iscbn 29226 | A complex Banach space is ... |
cbncms 29227 | The induced metric on comp... |
bnnv 29228 | Every complex Banach space... |
bnrel 29229 | The class of all complex B... |
bnsscmcl 29230 | A subspace of a Banach spa... |
cnbn 29231 | The set of complex numbers... |
ubthlem1 29232 | Lemma for ~ ubth . The fu... |
ubthlem2 29233 | Lemma for ~ ubth . Given ... |
ubthlem3 29234 | Lemma for ~ ubth . Prove ... |
ubth 29235 | Uniform Boundedness Theore... |
minvecolem1 29236 | Lemma for ~ minveco . The... |
minvecolem2 29237 | Lemma for ~ minveco . Any... |
minvecolem3 29238 | Lemma for ~ minveco . The... |
minvecolem4a 29239 | Lemma for ~ minveco . ` F ... |
minvecolem4b 29240 | Lemma for ~ minveco . The... |
minvecolem4c 29241 | Lemma for ~ minveco . The... |
minvecolem4 29242 | Lemma for ~ minveco . The... |
minvecolem5 29243 | Lemma for ~ minveco . Dis... |
minvecolem6 29244 | Lemma for ~ minveco . Any... |
minvecolem7 29245 | Lemma for ~ minveco . Sin... |
minveco 29246 | Minimizing vector theorem,... |
ishlo 29249 | The predicate "is a comple... |
hlobn 29250 | Every complex Hilbert spac... |
hlph 29251 | Every complex Hilbert spac... |
hlrel 29252 | The class of all complex H... |
hlnv 29253 | Every complex Hilbert spac... |
hlnvi 29254 | Every complex Hilbert spac... |
hlvc 29255 | Every complex Hilbert spac... |
hlcmet 29256 | The induced metric on a co... |
hlmet 29257 | The induced metric on a co... |
hlpar2 29258 | The parallelogram law sati... |
hlpar 29259 | The parallelogram law sati... |
hlex 29260 | The base set of a Hilbert ... |
hladdf 29261 | Mapping for Hilbert space ... |
hlcom 29262 | Hilbert space vector addit... |
hlass 29263 | Hilbert space vector addit... |
hl0cl 29264 | The Hilbert space zero vec... |
hladdid 29265 | Hilbert space addition wit... |
hlmulf 29266 | Mapping for Hilbert space ... |
hlmulid 29267 | Hilbert space scalar multi... |
hlmulass 29268 | Hilbert space scalar multi... |
hldi 29269 | Hilbert space scalar multi... |
hldir 29270 | Hilbert space scalar multi... |
hlmul0 29271 | Hilbert space scalar multi... |
hlipf 29272 | Mapping for Hilbert space ... |
hlipcj 29273 | Conjugate law for Hilbert ... |
hlipdir 29274 | Distributive law for Hilbe... |
hlipass 29275 | Associative law for Hilber... |
hlipgt0 29276 | The inner product of a Hil... |
hlcompl 29277 | Completeness of a Hilbert ... |
cnchl 29278 | The set of complex numbers... |
htthlem 29279 | Lemma for ~ htth . The co... |
htth 29280 | Hellinger-Toeplitz Theorem... |
The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
h2hva 29336 | The group (addition) opera... |
h2hsm 29337 | The scalar product operati... |
h2hnm 29338 | The norm function of Hilbe... |
h2hvs 29339 | The vector subtraction ope... |
h2hmetdval 29340 | Value of the distance func... |
h2hcau 29341 | The Cauchy sequences of Hi... |
h2hlm 29342 | The limit sequences of Hil... |
axhilex-zf 29343 | Derive Axiom ~ ax-hilex fr... |
axhfvadd-zf 29344 | Derive Axiom ~ ax-hfvadd f... |
axhvcom-zf 29345 | Derive Axiom ~ ax-hvcom fr... |
axhvass-zf 29346 | Derive Axiom ~ ax-hvass fr... |
axhv0cl-zf 29347 | Derive Axiom ~ ax-hv0cl fr... |
axhvaddid-zf 29348 | Derive Axiom ~ ax-hvaddid ... |
axhfvmul-zf 29349 | Derive Axiom ~ ax-hfvmul f... |
axhvmulid-zf 29350 | Derive Axiom ~ ax-hvmulid ... |
axhvmulass-zf 29351 | Derive Axiom ~ ax-hvmulass... |
axhvdistr1-zf 29352 | Derive Axiom ~ ax-hvdistr1... |
axhvdistr2-zf 29353 | Derive Axiom ~ ax-hvdistr2... |
axhvmul0-zf 29354 | Derive Axiom ~ ax-hvmul0 f... |
axhfi-zf 29355 | Derive Axiom ~ ax-hfi from... |
axhis1-zf 29356 | Derive Axiom ~ ax-his1 fro... |
axhis2-zf 29357 | Derive Axiom ~ ax-his2 fro... |
axhis3-zf 29358 | Derive Axiom ~ ax-his3 fro... |
axhis4-zf 29359 | Derive Axiom ~ ax-his4 fro... |
axhcompl-zf 29360 | Derive Axiom ~ ax-hcompl f... |
hvmulex 29373 | The Hilbert space scalar p... |
hvaddcl 29374 | Closure of vector addition... |
hvmulcl 29375 | Closure of scalar multipli... |
hvmulcli 29376 | Closure inference for scal... |
hvsubf 29377 | Mapping domain and codomai... |
hvsubval 29378 | Value of vector subtractio... |
hvsubcl 29379 | Closure of vector subtract... |
hvaddcli 29380 | Closure of vector addition... |
hvcomi 29381 | Commutation of vector addi... |
hvsubvali 29382 | Value of vector subtractio... |
hvsubcli 29383 | Closure of vector subtract... |
ifhvhv0 29384 | Prove ` if ( A e. ~H , A ,... |
hvaddid2 29385 | Addition with the zero vec... |
hvmul0 29386 | Scalar multiplication with... |
hvmul0or 29387 | If a scalar product is zer... |
hvsubid 29388 | Subtraction of a vector fr... |
hvnegid 29389 | Addition of negative of a ... |
hv2neg 29390 | Two ways to express the ne... |
hvaddid2i 29391 | Addition with the zero vec... |
hvnegidi 29392 | Addition of negative of a ... |
hv2negi 29393 | Two ways to express the ne... |
hvm1neg 29394 | Convert minus one times a ... |
hvaddsubval 29395 | Value of vector addition i... |
hvadd32 29396 | Commutative/associative la... |
hvadd12 29397 | Commutative/associative la... |
hvadd4 29398 | Hilbert vector space addit... |
hvsub4 29399 | Hilbert vector space addit... |
hvaddsub12 29400 | Commutative/associative la... |
hvpncan 29401 | Addition/subtraction cance... |
hvpncan2 29402 | Addition/subtraction cance... |
hvaddsubass 29403 | Associativity of sum and d... |
hvpncan3 29404 | Subtraction and addition o... |
hvmulcom 29405 | Scalar multiplication comm... |
hvsubass 29406 | Hilbert vector space assoc... |
hvsub32 29407 | Hilbert vector space commu... |
hvmulassi 29408 | Scalar multiplication asso... |
hvmulcomi 29409 | Scalar multiplication comm... |
hvmul2negi 29410 | Double negative in scalar ... |
hvsubdistr1 29411 | Scalar multiplication dist... |
hvsubdistr2 29412 | Scalar multiplication dist... |
hvdistr1i 29413 | Scalar multiplication dist... |
hvsubdistr1i 29414 | Scalar multiplication dist... |
hvassi 29415 | Hilbert vector space assoc... |
hvadd32i 29416 | Hilbert vector space commu... |
hvsubassi 29417 | Hilbert vector space assoc... |
hvsub32i 29418 | Hilbert vector space commu... |
hvadd12i 29419 | Hilbert vector space commu... |
hvadd4i 29420 | Hilbert vector space addit... |
hvsubsub4i 29421 | Hilbert vector space addit... |
hvsubsub4 29422 | Hilbert vector space addit... |
hv2times 29423 | Two times a vector. (Cont... |
hvnegdii 29424 | Distribution of negative o... |
hvsubeq0i 29425 | If the difference between ... |
hvsubcan2i 29426 | Vector cancellation law. ... |
hvaddcani 29427 | Cancellation law for vecto... |
hvsubaddi 29428 | Relationship between vecto... |
hvnegdi 29429 | Distribution of negative o... |
hvsubeq0 29430 | If the difference between ... |
hvaddeq0 29431 | If the sum of two vectors ... |
hvaddcan 29432 | Cancellation law for vecto... |
hvaddcan2 29433 | Cancellation law for vecto... |
hvmulcan 29434 | Cancellation law for scala... |
hvmulcan2 29435 | Cancellation law for scala... |
hvsubcan 29436 | Cancellation law for vecto... |
hvsubcan2 29437 | Cancellation law for vecto... |
hvsub0 29438 | Subtraction of a zero vect... |
hvsubadd 29439 | Relationship between vecto... |
hvaddsub4 29440 | Hilbert vector space addit... |
hicl 29442 | Closure of inner product. ... |
hicli 29443 | Closure inference for inne... |
his5 29448 | Associative law for inner ... |
his52 29449 | Associative law for inner ... |
his35 29450 | Move scalar multiplication... |
his35i 29451 | Move scalar multiplication... |
his7 29452 | Distributive law for inner... |
hiassdi 29453 | Distributive/associative l... |
his2sub 29454 | Distributive law for inner... |
his2sub2 29455 | Distributive law for inner... |
hire 29456 | A necessary and sufficient... |
hiidrcl 29457 | Real closure of inner prod... |
hi01 29458 | Inner product with the 0 v... |
hi02 29459 | Inner product with the 0 v... |
hiidge0 29460 | Inner product with self is... |
his6 29461 | Zero inner product with se... |
his1i 29462 | Conjugate law for inner pr... |
abshicom 29463 | Commuted inner products ha... |
hial0 29464 | A vector whose inner produ... |
hial02 29465 | A vector whose inner produ... |
hisubcomi 29466 | Two vector subtractions si... |
hi2eq 29467 | Lemma used to prove equali... |
hial2eq 29468 | Two vectors whose inner pr... |
hial2eq2 29469 | Two vectors whose inner pr... |
orthcom 29470 | Orthogonality commutes. (... |
normlem0 29471 | Lemma used to derive prope... |
normlem1 29472 | Lemma used to derive prope... |
normlem2 29473 | Lemma used to derive prope... |
normlem3 29474 | Lemma used to derive prope... |
normlem4 29475 | Lemma used to derive prope... |
normlem5 29476 | Lemma used to derive prope... |
normlem6 29477 | Lemma used to derive prope... |
normlem7 29478 | Lemma used to derive prope... |
normlem8 29479 | Lemma used to derive prope... |
normlem9 29480 | Lemma used to derive prope... |
normlem7tALT 29481 | Lemma used to derive prope... |
bcseqi 29482 | Equality case of Bunjakova... |
normlem9at 29483 | Lemma used to derive prope... |
dfhnorm2 29484 | Alternate definition of th... |
normf 29485 | The norm function maps fro... |
normval 29486 | The value of the norm of a... |
normcl 29487 | Real closure of the norm o... |
normge0 29488 | The norm of a vector is no... |
normgt0 29489 | The norm of nonzero vector... |
norm0 29490 | The norm of a zero vector.... |
norm-i 29491 | Theorem 3.3(i) of [Beran] ... |
normne0 29492 | A norm is nonzero iff its ... |
normcli 29493 | Real closure of the norm o... |
normsqi 29494 | The square of a norm. (Co... |
norm-i-i 29495 | Theorem 3.3(i) of [Beran] ... |
normsq 29496 | The square of a norm. (Co... |
normsub0i 29497 | Two vectors are equal iff ... |
normsub0 29498 | Two vectors are equal iff ... |
norm-ii-i 29499 | Triangle inequality for no... |
norm-ii 29500 | Triangle inequality for no... |
norm-iii-i 29501 | Theorem 3.3(iii) of [Beran... |
norm-iii 29502 | Theorem 3.3(iii) of [Beran... |
normsubi 29503 | Negative doesn't change th... |
normpythi 29504 | Analogy to Pythagorean the... |
normsub 29505 | Swapping order of subtract... |
normneg 29506 | The norm of a vector equal... |
normpyth 29507 | Analogy to Pythagorean the... |
normpyc 29508 | Corollary to Pythagorean t... |
norm3difi 29509 | Norm of differences around... |
norm3adifii 29510 | Norm of differences around... |
norm3lem 29511 | Lemma involving norm of di... |
norm3dif 29512 | Norm of differences around... |
norm3dif2 29513 | Norm of differences around... |
norm3lemt 29514 | Lemma involving norm of di... |
norm3adifi 29515 | Norm of differences around... |
normpari 29516 | Parallelogram law for norm... |
normpar 29517 | Parallelogram law for norm... |
normpar2i 29518 | Corollary of parallelogram... |
polid2i 29519 | Generalized polarization i... |
polidi 29520 | Polarization identity. Re... |
polid 29521 | Polarization identity. Re... |
hilablo 29522 | Hilbert space vector addit... |
hilid 29523 | The group identity element... |
hilvc 29524 | Hilbert space is a complex... |
hilnormi 29525 | Hilbert space norm in term... |
hilhhi 29526 | Deduce the structure of Hi... |
hhnv 29527 | Hilbert space is a normed ... |
hhva 29528 | The group (addition) opera... |
hhba 29529 | The base set of Hilbert sp... |
hh0v 29530 | The zero vector of Hilbert... |
hhsm 29531 | The scalar product operati... |
hhvs 29532 | The vector subtraction ope... |
hhnm 29533 | The norm function of Hilbe... |
hhims 29534 | The induced metric of Hilb... |
hhims2 29535 | Hilbert space distance met... |
hhmet 29536 | The induced metric of Hilb... |
hhxmet 29537 | The induced metric of Hilb... |
hhmetdval 29538 | Value of the distance func... |
hhip 29539 | The inner product operatio... |
hhph 29540 | The Hilbert space of the H... |
bcsiALT 29541 | Bunjakovaskij-Cauchy-Schwa... |
bcsiHIL 29542 | Bunjakovaskij-Cauchy-Schwa... |
bcs 29543 | Bunjakovaskij-Cauchy-Schwa... |
bcs2 29544 | Corollary of the Bunjakova... |
bcs3 29545 | Corollary of the Bunjakova... |
hcau 29546 | Member of the set of Cauch... |
hcauseq 29547 | A Cauchy sequences on a Hi... |
hcaucvg 29548 | A Cauchy sequence on a Hil... |
seq1hcau 29549 | A sequence on a Hilbert sp... |
hlimi 29550 | Express the predicate: Th... |
hlimseqi 29551 | A sequence with a limit on... |
hlimveci 29552 | Closure of the limit of a ... |
hlimconvi 29553 | Convergence of a sequence ... |
hlim2 29554 | The limit of a sequence on... |
hlimadd 29555 | Limit of the sum of two se... |
hilmet 29556 | The Hilbert space norm det... |
hilxmet 29557 | The Hilbert space norm det... |
hilmetdval 29558 | Value of the distance func... |
hilims 29559 | Hilbert space distance met... |
hhcau 29560 | The Cauchy sequences of Hi... |
hhlm 29561 | The limit sequences of Hil... |
hhcmpl 29562 | Lemma used for derivation ... |
hilcompl 29563 | Lemma used for derivation ... |
hhcms 29565 | The Hilbert space induced ... |
hhhl 29566 | The Hilbert space structur... |
hilcms 29567 | The Hilbert space norm det... |
hilhl 29568 | The Hilbert space of the H... |
issh 29570 | Subspace ` H ` of a Hilber... |
issh2 29571 | Subspace ` H ` of a Hilber... |
shss 29572 | A subspace is a subset of ... |
shel 29573 | A member of a subspace of ... |
shex 29574 | The set of subspaces of a ... |
shssii 29575 | A closed subspace of a Hil... |
sheli 29576 | A member of a subspace of ... |
shelii 29577 | A member of a subspace of ... |
sh0 29578 | The zero vector belongs to... |
shaddcl 29579 | Closure of vector addition... |
shmulcl 29580 | Closure of vector scalar m... |
issh3 29581 | Subspace ` H ` of a Hilber... |
shsubcl 29582 | Closure of vector subtract... |
isch 29584 | Closed subspace ` H ` of a... |
isch2 29585 | Closed subspace ` H ` of a... |
chsh 29586 | A closed subspace is a sub... |
chsssh 29587 | Closed subspaces are subsp... |
chex 29588 | The set of closed subspace... |
chshii 29589 | A closed subspace is a sub... |
ch0 29590 | The zero vector belongs to... |
chss 29591 | A closed subspace of a Hil... |
chel 29592 | A member of a closed subsp... |
chssii 29593 | A closed subspace of a Hil... |
cheli 29594 | A member of a closed subsp... |
chelii 29595 | A member of a closed subsp... |
chlimi 29596 | The limit property of a cl... |
hlim0 29597 | The zero sequence in Hilbe... |
hlimcaui 29598 | If a sequence in Hilbert s... |
hlimf 29599 | Function-like behavior of ... |
hlimuni 29600 | A Hilbert space sequence c... |
hlimreui 29601 | The limit of a Hilbert spa... |
hlimeui 29602 | The limit of a Hilbert spa... |
isch3 29603 | A Hilbert subspace is clos... |
chcompl 29604 | Completeness of a closed s... |
helch 29605 | The unit Hilbert lattice e... |
ifchhv 29606 | Prove ` if ( A e. CH , A ,... |
helsh 29607 | Hilbert space is a subspac... |
shsspwh 29608 | Subspaces are subsets of H... |
chsspwh 29609 | Closed subspaces are subse... |
hsn0elch 29610 | The zero subspace belongs ... |
norm1 29611 | From any nonzero Hilbert s... |
norm1exi 29612 | A normalized vector exists... |
norm1hex 29613 | A normalized vector can ex... |
elch0 29616 | Membership in zero for clo... |
h0elch 29617 | The zero subspace is a clo... |
h0elsh 29618 | The zero subspace is a sub... |
hhssva 29619 | The vector addition operat... |
hhsssm 29620 | The scalar multiplication ... |
hhssnm 29621 | The norm operation on a su... |
issubgoilem 29622 | Lemma for ~ hhssabloilem .... |
hhssabloilem 29623 | Lemma for ~ hhssabloi . F... |
hhssabloi 29624 | Abelian group property of ... |
hhssablo 29625 | Abelian group property of ... |
hhssnv 29626 | Normed complex vector spac... |
hhssnvt 29627 | Normed complex vector spac... |
hhsst 29628 | A member of ` SH ` is a su... |
hhshsslem1 29629 | Lemma for ~ hhsssh . (Con... |
hhshsslem2 29630 | Lemma for ~ hhsssh . (Con... |
hhsssh 29631 | The predicate " ` H ` is a... |
hhsssh2 29632 | The predicate " ` H ` is a... |
hhssba 29633 | The base set of a subspace... |
hhssvs 29634 | The vector subtraction ope... |
hhssvsf 29635 | Mapping of the vector subt... |
hhssims 29636 | Induced metric of a subspa... |
hhssims2 29637 | Induced metric of a subspa... |
hhssmet 29638 | Induced metric of a subspa... |
hhssmetdval 29639 | Value of the distance func... |
hhsscms 29640 | The induced metric of a cl... |
hhssbnOLD 29641 | Obsolete version of ~ cssb... |
ocval 29642 | Value of orthogonal comple... |
ocel 29643 | Membership in orthogonal c... |
shocel 29644 | Membership in orthogonal c... |
ocsh 29645 | The orthogonal complement ... |
shocsh 29646 | The orthogonal complement ... |
ocss 29647 | An orthogonal complement i... |
shocss 29648 | An orthogonal complement i... |
occon 29649 | Contraposition law for ort... |
occon2 29650 | Double contraposition for ... |
occon2i 29651 | Double contraposition for ... |
oc0 29652 | The zero vector belongs to... |
ocorth 29653 | Members of a subset and it... |
shocorth 29654 | Members of a subspace and ... |
ococss 29655 | Inclusion in complement of... |
shococss 29656 | Inclusion in complement of... |
shorth 29657 | Members of orthogonal subs... |
ocin 29658 | Intersection of a Hilbert ... |
occon3 29659 | Hilbert lattice contraposi... |
ocnel 29660 | A nonzero vector in the co... |
chocvali 29661 | Value of the orthogonal co... |
shuni 29662 | Two subspaces with trivial... |
chocunii 29663 | Lemma for uniqueness part ... |
pjhthmo 29664 | Projection Theorem, unique... |
occllem 29665 | Lemma for ~ occl . (Contr... |
occl 29666 | Closure of complement of H... |
shoccl 29667 | Closure of complement of H... |
choccl 29668 | Closure of complement of H... |
choccli 29669 | Closure of ` CH ` orthocom... |
shsval 29674 | Value of subspace sum of t... |
shsss 29675 | The subspace sum is a subs... |
shsel 29676 | Membership in the subspace... |
shsel3 29677 | Membership in the subspace... |
shseli 29678 | Membership in subspace sum... |
shscli 29679 | Closure of subspace sum. ... |
shscl 29680 | Closure of subspace sum. ... |
shscom 29681 | Commutative law for subspa... |
shsva 29682 | Vector sum belongs to subs... |
shsel1 29683 | A subspace sum contains a ... |
shsel2 29684 | A subspace sum contains a ... |
shsvs 29685 | Vector subtraction belongs... |
shsub1 29686 | Subspace sum is an upper b... |
shsub2 29687 | Subspace sum is an upper b... |
choc0 29688 | The orthocomplement of the... |
choc1 29689 | The orthocomplement of the... |
chocnul 29690 | Orthogonal complement of t... |
shintcli 29691 | Closure of intersection of... |
shintcl 29692 | The intersection of a none... |
chintcli 29693 | The intersection of a none... |
chintcl 29694 | The intersection (infimum)... |
spanval 29695 | Value of the linear span o... |
hsupval 29696 | Value of supremum of set o... |
chsupval 29697 | The value of the supremum ... |
spancl 29698 | The span of a subset of Hi... |
elspancl 29699 | A member of a span is a ve... |
shsupcl 29700 | Closure of the subspace su... |
hsupcl 29701 | Closure of supremum of set... |
chsupcl 29702 | Closure of supremum of sub... |
hsupss 29703 | Subset relation for suprem... |
chsupss 29704 | Subset relation for suprem... |
hsupunss 29705 | The union of a set of Hilb... |
chsupunss 29706 | The union of a set of clos... |
spanss2 29707 | A subset of Hilbert space ... |
shsupunss 29708 | The union of a set of subs... |
spanid 29709 | A subspace of Hilbert spac... |
spanss 29710 | Ordering relationship for ... |
spanssoc 29711 | The span of a subset of Hi... |
sshjval 29712 | Value of join for subsets ... |
shjval 29713 | Value of join in ` SH ` . ... |
chjval 29714 | Value of join in ` CH ` . ... |
chjvali 29715 | Value of join in ` CH ` . ... |
sshjval3 29716 | Value of join for subsets ... |
sshjcl 29717 | Closure of join for subset... |
shjcl 29718 | Closure of join in ` SH ` ... |
chjcl 29719 | Closure of join in ` CH ` ... |
shjcom 29720 | Commutative law for Hilber... |
shless 29721 | Subset implies subset of s... |
shlej1 29722 | Add disjunct to both sides... |
shlej2 29723 | Add disjunct to both sides... |
shincli 29724 | Closure of intersection of... |
shscomi 29725 | Commutative law for subspa... |
shsvai 29726 | Vector sum belongs to subs... |
shsel1i 29727 | A subspace sum contains a ... |
shsel2i 29728 | A subspace sum contains a ... |
shsvsi 29729 | Vector subtraction belongs... |
shunssi 29730 | Union is smaller than subs... |
shunssji 29731 | Union is smaller than Hilb... |
shsleji 29732 | Subspace sum is smaller th... |
shjcomi 29733 | Commutative law for join i... |
shsub1i 29734 | Subspace sum is an upper b... |
shsub2i 29735 | Subspace sum is an upper b... |
shub1i 29736 | Hilbert lattice join is an... |
shjcli 29737 | Closure of ` CH ` join. (... |
shjshcli 29738 | ` SH ` closure of join. (... |
shlessi 29739 | Subset implies subset of s... |
shlej1i 29740 | Add disjunct to both sides... |
shlej2i 29741 | Add disjunct to both sides... |
shslej 29742 | Subspace sum is smaller th... |
shincl 29743 | Closure of intersection of... |
shub1 29744 | Hilbert lattice join is an... |
shub2 29745 | A subspace is a subset of ... |
shsidmi 29746 | Idempotent law for Hilbert... |
shslubi 29747 | The least upper bound law ... |
shlesb1i 29748 | Hilbert lattice ordering i... |
shsval2i 29749 | An alternate way to expres... |
shsval3i 29750 | An alternate way to expres... |
shmodsi 29751 | The modular law holds for ... |
shmodi 29752 | The modular law is implied... |
pjhthlem1 29753 | Lemma for ~ pjhth . (Cont... |
pjhthlem2 29754 | Lemma for ~ pjhth . (Cont... |
pjhth 29755 | Projection Theorem: Any H... |
pjhtheu 29756 | Projection Theorem: Any H... |
pjhfval 29758 | The value of the projectio... |
pjhval 29759 | Value of a projection. (C... |
pjpreeq 29760 | Equality with a projection... |
pjeq 29761 | Equality with a projection... |
axpjcl 29762 | Closure of a projection in... |
pjhcl 29763 | Closure of a projection in... |
omlsilem 29764 | Lemma for orthomodular law... |
omlsii 29765 | Subspace inference form of... |
omlsi 29766 | Subspace form of orthomodu... |
ococi 29767 | Complement of complement o... |
ococ 29768 | Complement of complement o... |
dfch2 29769 | Alternate definition of th... |
ococin 29770 | The double complement is t... |
hsupval2 29771 | Alternate definition of su... |
chsupval2 29772 | The value of the supremum ... |
sshjval2 29773 | Value of join in the set o... |
chsupid 29774 | A subspace is the supremum... |
chsupsn 29775 | Value of supremum of subse... |
shlub 29776 | Hilbert lattice join is th... |
shlubi 29777 | Hilbert lattice join is th... |
pjhtheu2 29778 | Uniqueness of ` y ` for th... |
pjcli 29779 | Closure of a projection in... |
pjhcli 29780 | Closure of a projection in... |
pjpjpre 29781 | Decomposition of a vector ... |
axpjpj 29782 | Decomposition of a vector ... |
pjclii 29783 | Closure of a projection in... |
pjhclii 29784 | Closure of a projection in... |
pjpj0i 29785 | Decomposition of a vector ... |
pjpji 29786 | Decomposition of a vector ... |
pjpjhth 29787 | Projection Theorem: Any H... |
pjpjhthi 29788 | Projection Theorem: Any H... |
pjop 29789 | Orthocomplement projection... |
pjpo 29790 | Projection in terms of ort... |
pjopi 29791 | Orthocomplement projection... |
pjpoi 29792 | Projection in terms of ort... |
pjoc1i 29793 | Projection of a vector in ... |
pjchi 29794 | Projection of a vector in ... |
pjoccl 29795 | The part of a vector that ... |
pjoc1 29796 | Projection of a vector in ... |
pjomli 29797 | Subspace form of orthomodu... |
pjoml 29798 | Subspace form of orthomodu... |
pjococi 29799 | Proof of orthocomplement t... |
pjoc2i 29800 | Projection of a vector in ... |
pjoc2 29801 | Projection of a vector in ... |
sh0le 29802 | The zero subspace is the s... |
ch0le 29803 | The zero subspace is the s... |
shle0 29804 | No subspace is smaller tha... |
chle0 29805 | No Hilbert lattice element... |
chnlen0 29806 | A Hilbert lattice element ... |
ch0pss 29807 | The zero subspace is a pro... |
orthin 29808 | The intersection of orthog... |
ssjo 29809 | The lattice join of a subs... |
shne0i 29810 | A nonzero subspace has a n... |
shs0i 29811 | Hilbert subspace sum with ... |
shs00i 29812 | Two subspaces are zero iff... |
ch0lei 29813 | The closed subspace zero i... |
chle0i 29814 | No Hilbert closed subspace... |
chne0i 29815 | A nonzero closed subspace ... |
chocini 29816 | Intersection of a closed s... |
chj0i 29817 | Join with lattice zero in ... |
chm1i 29818 | Meet with lattice one in `... |
chjcli 29819 | Closure of ` CH ` join. (... |
chsleji 29820 | Subspace sum is smaller th... |
chseli 29821 | Membership in subspace sum... |
chincli 29822 | Closure of Hilbert lattice... |
chsscon3i 29823 | Hilbert lattice contraposi... |
chsscon1i 29824 | Hilbert lattice contraposi... |
chsscon2i 29825 | Hilbert lattice contraposi... |
chcon2i 29826 | Hilbert lattice contraposi... |
chcon1i 29827 | Hilbert lattice contraposi... |
chcon3i 29828 | Hilbert lattice contraposi... |
chunssji 29829 | Union is smaller than ` CH... |
chjcomi 29830 | Commutative law for join i... |
chub1i 29831 | ` CH ` join is an upper bo... |
chub2i 29832 | ` CH ` join is an upper bo... |
chlubi 29833 | Hilbert lattice join is th... |
chlubii 29834 | Hilbert lattice join is th... |
chlej1i 29835 | Add join to both sides of ... |
chlej2i 29836 | Add join to both sides of ... |
chlej12i 29837 | Add join to both sides of ... |
chlejb1i 29838 | Hilbert lattice ordering i... |
chdmm1i 29839 | De Morgan's law for meet i... |
chdmm2i 29840 | De Morgan's law for meet i... |
chdmm3i 29841 | De Morgan's law for meet i... |
chdmm4i 29842 | De Morgan's law for meet i... |
chdmj1i 29843 | De Morgan's law for join i... |
chdmj2i 29844 | De Morgan's law for join i... |
chdmj3i 29845 | De Morgan's law for join i... |
chdmj4i 29846 | De Morgan's law for join i... |
chnlei 29847 | Equivalent expressions for... |
chjassi 29848 | Associative law for Hilber... |
chj00i 29849 | Two Hilbert lattice elemen... |
chjoi 29850 | The join of a closed subsp... |
chj1i 29851 | Join with Hilbert lattice ... |
chm0i 29852 | Meet with Hilbert lattice ... |
chm0 29853 | Meet with Hilbert lattice ... |
shjshsi 29854 | Hilbert lattice join equal... |
shjshseli 29855 | A closed subspace sum equa... |
chne0 29856 | A nonzero closed subspace ... |
chocin 29857 | Intersection of a closed s... |
chssoc 29858 | A closed subspace less tha... |
chj0 29859 | Join with Hilbert lattice ... |
chslej 29860 | Subspace sum is smaller th... |
chincl 29861 | Closure of Hilbert lattice... |
chsscon3 29862 | Hilbert lattice contraposi... |
chsscon1 29863 | Hilbert lattice contraposi... |
chsscon2 29864 | Hilbert lattice contraposi... |
chpsscon3 29865 | Hilbert lattice contraposi... |
chpsscon1 29866 | Hilbert lattice contraposi... |
chpsscon2 29867 | Hilbert lattice contraposi... |
chjcom 29868 | Commutative law for Hilber... |
chub1 29869 | Hilbert lattice join is gr... |
chub2 29870 | Hilbert lattice join is gr... |
chlub 29871 | Hilbert lattice join is th... |
chlej1 29872 | Add join to both sides of ... |
chlej2 29873 | Add join to both sides of ... |
chlejb1 29874 | Hilbert lattice ordering i... |
chlejb2 29875 | Hilbert lattice ordering i... |
chnle 29876 | Equivalent expressions for... |
chjo 29877 | The join of a closed subsp... |
chabs1 29878 | Hilbert lattice absorption... |
chabs2 29879 | Hilbert lattice absorption... |
chabs1i 29880 | Hilbert lattice absorption... |
chabs2i 29881 | Hilbert lattice absorption... |
chjidm 29882 | Idempotent law for Hilbert... |
chjidmi 29883 | Idempotent law for Hilbert... |
chj12i 29884 | A rearrangement of Hilbert... |
chj4i 29885 | Rearrangement of the join ... |
chjjdiri 29886 | Hilbert lattice join distr... |
chdmm1 29887 | De Morgan's law for meet i... |
chdmm2 29888 | De Morgan's law for meet i... |
chdmm3 29889 | De Morgan's law for meet i... |
chdmm4 29890 | De Morgan's law for meet i... |
chdmj1 29891 | De Morgan's law for join i... |
chdmj2 29892 | De Morgan's law for join i... |
chdmj3 29893 | De Morgan's law for join i... |
chdmj4 29894 | De Morgan's law for join i... |
chjass 29895 | Associative law for Hilber... |
chj12 29896 | A rearrangement of Hilbert... |
chj4 29897 | Rearrangement of the join ... |
ledii 29898 | An ortholattice is distrib... |
lediri 29899 | An ortholattice is distrib... |
lejdii 29900 | An ortholattice is distrib... |
lejdiri 29901 | An ortholattice is distrib... |
ledi 29902 | An ortholattice is distrib... |
spansn0 29903 | The span of the singleton ... |
span0 29904 | The span of the empty set ... |
elspani 29905 | Membership in the span of ... |
spanuni 29906 | The span of a union is the... |
spanun 29907 | The span of a union is the... |
sshhococi 29908 | The join of two Hilbert sp... |
hne0 29909 | Hilbert space has a nonzer... |
chsup0 29910 | The supremum of the empty ... |
h1deoi 29911 | Membership in orthocomplem... |
h1dei 29912 | Membership in 1-dimensiona... |
h1did 29913 | A generating vector belong... |
h1dn0 29914 | A nonzero vector generates... |
h1de2i 29915 | Membership in 1-dimensiona... |
h1de2bi 29916 | Membership in 1-dimensiona... |
h1de2ctlem 29917 | Lemma for ~ h1de2ci . (Co... |
h1de2ci 29918 | Membership in 1-dimensiona... |
spansni 29919 | The span of a singleton in... |
elspansni 29920 | Membership in the span of ... |
spansn 29921 | The span of a singleton in... |
spansnch 29922 | The span of a Hilbert spac... |
spansnsh 29923 | The span of a Hilbert spac... |
spansnchi 29924 | The span of a singleton in... |
spansnid 29925 | A vector belongs to the sp... |
spansnmul 29926 | A scalar product with a ve... |
elspansncl 29927 | A member of a span of a si... |
elspansn 29928 | Membership in the span of ... |
elspansn2 29929 | Membership in the span of ... |
spansncol 29930 | The singletons of collinea... |
spansneleqi 29931 | Membership relation implie... |
spansneleq 29932 | Membership relation that i... |
spansnss 29933 | The span of the singleton ... |
elspansn3 29934 | A member of the span of th... |
elspansn4 29935 | A span membership conditio... |
elspansn5 29936 | A vector belonging to both... |
spansnss2 29937 | The span of the singleton ... |
normcan 29938 | Cancellation-type law that... |
pjspansn 29939 | A projection on the span o... |
spansnpji 29940 | A subset of Hilbert space ... |
spanunsni 29941 | The span of the union of a... |
spanpr 29942 | The span of a pair of vect... |
h1datomi 29943 | A 1-dimensional subspace i... |
h1datom 29944 | A 1-dimensional subspace i... |
cmbr 29946 | Binary relation expressing... |
pjoml2i 29947 | Variation of orthomodular ... |
pjoml3i 29948 | Variation of orthomodular ... |
pjoml4i 29949 | Variation of orthomodular ... |
pjoml5i 29950 | The orthomodular law. Rem... |
pjoml6i 29951 | An equivalent of the ortho... |
cmbri 29952 | Binary relation expressing... |
cmcmlem 29953 | Commutation is symmetric. ... |
cmcmi 29954 | Commutation is symmetric. ... |
cmcm2i 29955 | Commutation with orthocomp... |
cmcm3i 29956 | Commutation with orthocomp... |
cmcm4i 29957 | Commutation with orthocomp... |
cmbr2i 29958 | Alternate definition of th... |
cmcmii 29959 | Commutation is symmetric. ... |
cmcm2ii 29960 | Commutation with orthocomp... |
cmcm3ii 29961 | Commutation with orthocomp... |
cmbr3i 29962 | Alternate definition for t... |
cmbr4i 29963 | Alternate definition for t... |
lecmi 29964 | Comparable Hilbert lattice... |
lecmii 29965 | Comparable Hilbert lattice... |
cmj1i 29966 | A Hilbert lattice element ... |
cmj2i 29967 | A Hilbert lattice element ... |
cmm1i 29968 | A Hilbert lattice element ... |
cmm2i 29969 | A Hilbert lattice element ... |
cmbr3 29970 | Alternate definition for t... |
cm0 29971 | The zero Hilbert lattice e... |
cmidi 29972 | The commutes relation is r... |
pjoml2 29973 | Variation of orthomodular ... |
pjoml3 29974 | Variation of orthomodular ... |
pjoml5 29975 | The orthomodular law. Rem... |
cmcm 29976 | Commutation is symmetric. ... |
cmcm3 29977 | Commutation with orthocomp... |
cmcm2 29978 | Commutation with orthocomp... |
lecm 29979 | Comparable Hilbert lattice... |
fh1 29980 | Foulis-Holland Theorem. I... |
fh2 29981 | Foulis-Holland Theorem. I... |
cm2j 29982 | A lattice element that com... |
fh1i 29983 | Foulis-Holland Theorem. I... |
fh2i 29984 | Foulis-Holland Theorem. I... |
fh3i 29985 | Variation of the Foulis-Ho... |
fh4i 29986 | Variation of the Foulis-Ho... |
cm2ji 29987 | A lattice element that com... |
cm2mi 29988 | A lattice element that com... |
qlax1i 29989 | One of the equations showi... |
qlax2i 29990 | One of the equations showi... |
qlax3i 29991 | One of the equations showi... |
qlax4i 29992 | One of the equations showi... |
qlax5i 29993 | One of the equations showi... |
qlaxr1i 29994 | One of the conditions show... |
qlaxr2i 29995 | One of the conditions show... |
qlaxr4i 29996 | One of the conditions show... |
qlaxr5i 29997 | One of the conditions show... |
qlaxr3i 29998 | A variation of the orthomo... |
chscllem1 29999 | Lemma for ~ chscl . (Cont... |
chscllem2 30000 | Lemma for ~ chscl . (Cont... |
chscllem3 30001 | Lemma for ~ chscl . (Cont... |
chscllem4 30002 | Lemma for ~ chscl . (Cont... |
chscl 30003 | The subspace sum of two cl... |
osumi 30004 | If two closed subspaces of... |
osumcori 30005 | Corollary of ~ osumi . (C... |
osumcor2i 30006 | Corollary of ~ osumi , sho... |
osum 30007 | If two closed subspaces of... |
spansnji 30008 | The subspace sum of a clos... |
spansnj 30009 | The subspace sum of a clos... |
spansnscl 30010 | The subspace sum of a clos... |
sumspansn 30011 | The sum of two vectors bel... |
spansnm0i 30012 | The meet of different one-... |
nonbooli 30013 | A Hilbert lattice with two... |
spansncvi 30014 | Hilbert space has the cove... |
spansncv 30015 | Hilbert space has the cove... |
5oalem1 30016 | Lemma for orthoarguesian l... |
5oalem2 30017 | Lemma for orthoarguesian l... |
5oalem3 30018 | Lemma for orthoarguesian l... |
5oalem4 30019 | Lemma for orthoarguesian l... |
5oalem5 30020 | Lemma for orthoarguesian l... |
5oalem6 30021 | Lemma for orthoarguesian l... |
5oalem7 30022 | Lemma for orthoarguesian l... |
5oai 30023 | Orthoarguesian law 5OA. Th... |
3oalem1 30024 | Lemma for 3OA (weak) ortho... |
3oalem2 30025 | Lemma for 3OA (weak) ortho... |
3oalem3 30026 | Lemma for 3OA (weak) ortho... |
3oalem4 30027 | Lemma for 3OA (weak) ortho... |
3oalem5 30028 | Lemma for 3OA (weak) ortho... |
3oalem6 30029 | Lemma for 3OA (weak) ortho... |
3oai 30030 | 3OA (weak) orthoarguesian ... |
pjorthi 30031 | Projection components on o... |
pjch1 30032 | Property of identity proje... |
pjo 30033 | The orthogonal projection.... |
pjcompi 30034 | Component of a projection.... |
pjidmi 30035 | A projection is idempotent... |
pjadjii 30036 | A projection is self-adjoi... |
pjaddii 30037 | Projection of vector sum i... |
pjinormii 30038 | The inner product of a pro... |
pjmulii 30039 | Projection of (scalar) pro... |
pjsubii 30040 | Projection of vector diffe... |
pjsslem 30041 | Lemma for subset relations... |
pjss2i 30042 | Subset relationship for pr... |
pjssmii 30043 | Projection meet property. ... |
pjssge0ii 30044 | Theorem 4.5(iv)->(v) of [B... |
pjdifnormii 30045 | Theorem 4.5(v)<->(vi) of [... |
pjcji 30046 | The projection on a subspa... |
pjadji 30047 | A projection is self-adjoi... |
pjaddi 30048 | Projection of vector sum i... |
pjinormi 30049 | The inner product of a pro... |
pjsubi 30050 | Projection of vector diffe... |
pjmuli 30051 | Projection of scalar produ... |
pjige0i 30052 | The inner product of a pro... |
pjige0 30053 | The inner product of a pro... |
pjcjt2 30054 | The projection on a subspa... |
pj0i 30055 | The projection of the zero... |
pjch 30056 | Projection of a vector in ... |
pjid 30057 | The projection of a vector... |
pjvec 30058 | The set of vectors belongi... |
pjocvec 30059 | The set of vectors belongi... |
pjocini 30060 | Membership of projection i... |
pjini 30061 | Membership of projection i... |
pjjsi 30062 | A sufficient condition for... |
pjfni 30063 | Functionality of a project... |
pjrni 30064 | The range of a projection.... |
pjfoi 30065 | A projection maps onto its... |
pjfi 30066 | The mapping of a projectio... |
pjvi 30067 | The value of a projection ... |
pjhfo 30068 | A projection maps onto its... |
pjrn 30069 | The range of a projection.... |
pjhf 30070 | The mapping of a projectio... |
pjfn 30071 | Functionality of a project... |
pjsumi 30072 | The projection on a subspa... |
pj11i 30073 | One-to-one correspondence ... |
pjdsi 30074 | Vector decomposition into ... |
pjds3i 30075 | Vector decomposition into ... |
pj11 30076 | One-to-one correspondence ... |
pjmfn 30077 | Functionality of the proje... |
pjmf1 30078 | The projector function map... |
pjoi0 30079 | The inner product of proje... |
pjoi0i 30080 | The inner product of proje... |
pjopythi 30081 | Pythagorean theorem for pr... |
pjopyth 30082 | Pythagorean theorem for pr... |
pjnormi 30083 | The norm of the projection... |
pjpythi 30084 | Pythagorean theorem for pr... |
pjneli 30085 | If a vector does not belon... |
pjnorm 30086 | The norm of the projection... |
pjpyth 30087 | Pythagorean theorem for pr... |
pjnel 30088 | If a vector does not belon... |
pjnorm2 30089 | A vector belongs to the su... |
mayete3i 30090 | Mayet's equation E_3. Par... |
mayetes3i 30091 | Mayet's equation E^*_3, de... |
hosmval 30097 | Value of the sum of two Hi... |
hommval 30098 | Value of the scalar produc... |
hodmval 30099 | Value of the difference of... |
hfsmval 30100 | Value of the sum of two Hi... |
hfmmval 30101 | Value of the scalar produc... |
hosval 30102 | Value of the sum of two Hi... |
homval 30103 | Value of the scalar produc... |
hodval 30104 | Value of the difference of... |
hfsval 30105 | Value of the sum of two Hi... |
hfmval 30106 | Value of the scalar produc... |
hoscl 30107 | Closure of the sum of two ... |
homcl 30108 | Closure of the scalar prod... |
hodcl 30109 | Closure of the difference ... |
ho0val 30112 | Value of the zero Hilbert ... |
ho0f 30113 | Functionality of the zero ... |
df0op2 30114 | Alternate definition of Hi... |
dfiop2 30115 | Alternate definition of Hi... |
hoif 30116 | Functionality of the Hilbe... |
hoival 30117 | The value of the Hilbert s... |
hoico1 30118 | Composition with the Hilbe... |
hoico2 30119 | Composition with the Hilbe... |
hoaddcl 30120 | The sum of Hilbert space o... |
homulcl 30121 | The scalar product of a Hi... |
hoeq 30122 | Equality of Hilbert space ... |
hoeqi 30123 | Equality of Hilbert space ... |
hoscli 30124 | Closure of Hilbert space o... |
hodcli 30125 | Closure of Hilbert space o... |
hocoi 30126 | Composition of Hilbert spa... |
hococli 30127 | Closure of composition of ... |
hocofi 30128 | Mapping of composition of ... |
hocofni 30129 | Functionality of compositi... |
hoaddcli 30130 | Mapping of sum of Hilbert ... |
hosubcli 30131 | Mapping of difference of H... |
hoaddfni 30132 | Functionality of sum of Hi... |
hosubfni 30133 | Functionality of differenc... |
hoaddcomi 30134 | Commutativity of sum of Hi... |
hosubcl 30135 | Mapping of difference of H... |
hoaddcom 30136 | Commutativity of sum of Hi... |
hodsi 30137 | Relationship between Hilbe... |
hoaddassi 30138 | Associativity of sum of Hi... |
hoadd12i 30139 | Commutative/associative la... |
hoadd32i 30140 | Commutative/associative la... |
hocadddiri 30141 | Distributive law for Hilbe... |
hocsubdiri 30142 | Distributive law for Hilbe... |
ho2coi 30143 | Double composition of Hilb... |
hoaddass 30144 | Associativity of sum of Hi... |
hoadd32 30145 | Commutative/associative la... |
hoadd4 30146 | Rearrangement of 4 terms i... |
hocsubdir 30147 | Distributive law for Hilbe... |
hoaddid1i 30148 | Sum of a Hilbert space ope... |
hodidi 30149 | Difference of a Hilbert sp... |
ho0coi 30150 | Composition of the zero op... |
hoid1i 30151 | Composition of Hilbert spa... |
hoid1ri 30152 | Composition of Hilbert spa... |
hoaddid1 30153 | Sum of a Hilbert space ope... |
hodid 30154 | Difference of a Hilbert sp... |
hon0 30155 | A Hilbert space operator i... |
hodseqi 30156 | Subtraction and addition o... |
ho0subi 30157 | Subtraction of Hilbert spa... |
honegsubi 30158 | Relationship between Hilbe... |
ho0sub 30159 | Subtraction of Hilbert spa... |
hosubid1 30160 | The zero operator subtract... |
honegsub 30161 | Relationship between Hilbe... |
homulid2 30162 | An operator equals its sca... |
homco1 30163 | Associative law for scalar... |
homulass 30164 | Scalar product associative... |
hoadddi 30165 | Scalar product distributiv... |
hoadddir 30166 | Scalar product reverse dis... |
homul12 30167 | Swap first and second fact... |
honegneg 30168 | Double negative of a Hilbe... |
hosubneg 30169 | Relationship between opera... |
hosubdi 30170 | Scalar product distributiv... |
honegdi 30171 | Distribution of negative o... |
honegsubdi 30172 | Distribution of negative o... |
honegsubdi2 30173 | Distribution of negative o... |
hosubsub2 30174 | Law for double subtraction... |
hosub4 30175 | Rearrangement of 4 terms i... |
hosubadd4 30176 | Rearrangement of 4 terms i... |
hoaddsubass 30177 | Associative-type law for a... |
hoaddsub 30178 | Law for operator addition ... |
hosubsub 30179 | Law for double subtraction... |
hosubsub4 30180 | Law for double subtraction... |
ho2times 30181 | Two times a Hilbert space ... |
hoaddsubassi 30182 | Associativity of sum and d... |
hoaddsubi 30183 | Law for sum and difference... |
hosd1i 30184 | Hilbert space operator sum... |
hosd2i 30185 | Hilbert space operator sum... |
hopncani 30186 | Hilbert space operator can... |
honpcani 30187 | Hilbert space operator can... |
hosubeq0i 30188 | If the difference between ... |
honpncani 30189 | Hilbert space operator can... |
ho01i 30190 | A condition implying that ... |
ho02i 30191 | A condition implying that ... |
hoeq1 30192 | A condition implying that ... |
hoeq2 30193 | A condition implying that ... |
adjmo 30194 | Every Hilbert space operat... |
adjsym 30195 | Symmetry property of an ad... |
eigrei 30196 | A necessary and sufficient... |
eigre 30197 | A necessary and sufficient... |
eigposi 30198 | A sufficient condition (fi... |
eigorthi 30199 | A necessary and sufficient... |
eigorth 30200 | A necessary and sufficient... |
nmopval 30218 | Value of the norm of a Hil... |
elcnop 30219 | Property defining a contin... |
ellnop 30220 | Property defining a linear... |
lnopf 30221 | A linear Hilbert space ope... |
elbdop 30222 | Property defining a bounde... |
bdopln 30223 | A bounded linear Hilbert s... |
bdopf 30224 | A bounded linear Hilbert s... |
nmopsetretALT 30225 | The set in the supremum of... |
nmopsetretHIL 30226 | The set in the supremum of... |
nmopsetn0 30227 | The set in the supremum of... |
nmopxr 30228 | The norm of a Hilbert spac... |
nmoprepnf 30229 | The norm of a Hilbert spac... |
nmopgtmnf 30230 | The norm of a Hilbert spac... |
nmopreltpnf 30231 | The norm of a Hilbert spac... |
nmopre 30232 | The norm of a bounded oper... |
elbdop2 30233 | Property defining a bounde... |
elunop 30234 | Property defining a unitar... |
elhmop 30235 | Property defining a Hermit... |
hmopf 30236 | A Hermitian operator is a ... |
hmopex 30237 | The class of Hermitian ope... |
nmfnval 30238 | Value of the norm of a Hil... |
nmfnsetre 30239 | The set in the supremum of... |
nmfnsetn0 30240 | The set in the supremum of... |
nmfnxr 30241 | The norm of any Hilbert sp... |
nmfnrepnf 30242 | The norm of a Hilbert spac... |
nlfnval 30243 | Value of the null space of... |
elcnfn 30244 | Property defining a contin... |
ellnfn 30245 | Property defining a linear... |
lnfnf 30246 | A linear Hilbert space fun... |
dfadj2 30247 | Alternate definition of th... |
funadj 30248 | Functionality of the adjoi... |
dmadjss 30249 | The domain of the adjoint ... |
dmadjop 30250 | A member of the domain of ... |
adjeu 30251 | Elementhood in the domain ... |
adjval 30252 | Value of the adjoint funct... |
adjval2 30253 | Value of the adjoint funct... |
cnvadj 30254 | The adjoint function equal... |
funcnvadj 30255 | The converse of the adjoin... |
adj1o 30256 | The adjoint function maps ... |
dmadjrn 30257 | The adjoint of an operator... |
eigvecval 30258 | The set of eigenvectors of... |
eigvalfval 30259 | The eigenvalues of eigenve... |
specval 30260 | The value of the spectrum ... |
speccl 30261 | The spectrum of an operato... |
hhlnoi 30262 | The linear operators of Hi... |
hhnmoi 30263 | The norm of an operator in... |
hhbloi 30264 | A bounded linear operator ... |
hh0oi 30265 | The zero operator in Hilbe... |
hhcno 30266 | The continuous operators o... |
hhcnf 30267 | The continuous functionals... |
dmadjrnb 30268 | The adjoint of an operator... |
nmoplb 30269 | A lower bound for an opera... |
nmopub 30270 | An upper bound for an oper... |
nmopub2tALT 30271 | An upper bound for an oper... |
nmopub2tHIL 30272 | An upper bound for an oper... |
nmopge0 30273 | The norm of any Hilbert sp... |
nmopgt0 30274 | A linear Hilbert space ope... |
cnopc 30275 | Basic continuity property ... |
lnopl 30276 | Basic linearity property o... |
unop 30277 | Basic inner product proper... |
unopf1o 30278 | A unitary operator in Hilb... |
unopnorm 30279 | A unitary operator is idem... |
cnvunop 30280 | The inverse (converse) of ... |
unopadj 30281 | The inverse (converse) of ... |
unoplin 30282 | A unitary operator is line... |
counop 30283 | The composition of two uni... |
hmop 30284 | Basic inner product proper... |
hmopre 30285 | The inner product of the v... |
nmfnlb 30286 | A lower bound for a functi... |
nmfnleub 30287 | An upper bound for the nor... |
nmfnleub2 30288 | An upper bound for the nor... |
nmfnge0 30289 | The norm of any Hilbert sp... |
elnlfn 30290 | Membership in the null spa... |
elnlfn2 30291 | Membership in the null spa... |
cnfnc 30292 | Basic continuity property ... |
lnfnl 30293 | Basic linearity property o... |
adjcl 30294 | Closure of the adjoint of ... |
adj1 30295 | Property of an adjoint Hil... |
adj2 30296 | Property of an adjoint Hil... |
adjeq 30297 | A property that determines... |
adjadj 30298 | Double adjoint. Theorem 3... |
adjvalval 30299 | Value of the value of the ... |
unopadj2 30300 | The adjoint of a unitary o... |
hmopadj 30301 | A Hermitian operator is se... |
hmdmadj 30302 | Every Hermitian operator h... |
hmopadj2 30303 | An operator is Hermitian i... |
hmoplin 30304 | A Hermitian operator is li... |
brafval 30305 | The bra of a vector, expre... |
braval 30306 | A bra-ket juxtaposition, e... |
braadd 30307 | Linearity property of bra ... |
bramul 30308 | Linearity property of bra ... |
brafn 30309 | The bra function is a func... |
bralnfn 30310 | The Dirac bra function is ... |
bracl 30311 | Closure of the bra functio... |
bra0 30312 | The Dirac bra of the zero ... |
brafnmul 30313 | Anti-linearity property of... |
kbfval 30314 | The outer product of two v... |
kbop 30315 | The outer product of two v... |
kbval 30316 | The value of the operator ... |
kbmul 30317 | Multiplication property of... |
kbpj 30318 | If a vector ` A ` has norm... |
eleigvec 30319 | Membership in the set of e... |
eleigvec2 30320 | Membership in the set of e... |
eleigveccl 30321 | Closure of an eigenvector ... |
eigvalval 30322 | The eigenvalue of an eigen... |
eigvalcl 30323 | An eigenvalue is a complex... |
eigvec1 30324 | Property of an eigenvector... |
eighmre 30325 | The eigenvalues of a Hermi... |
eighmorth 30326 | Eigenvectors of a Hermitia... |
nmopnegi 30327 | Value of the norm of the n... |
lnop0 30328 | The value of a linear Hilb... |
lnopmul 30329 | Multiplicative property of... |
lnopli 30330 | Basic scalar product prope... |
lnopfi 30331 | A linear Hilbert space ope... |
lnop0i 30332 | The value of a linear Hilb... |
lnopaddi 30333 | Additive property of a lin... |
lnopmuli 30334 | Multiplicative property of... |
lnopaddmuli 30335 | Sum/product property of a ... |
lnopsubi 30336 | Subtraction property for a... |
lnopsubmuli 30337 | Subtraction/product proper... |
lnopmulsubi 30338 | Product/subtraction proper... |
homco2 30339 | Move a scalar product out ... |
idunop 30340 | The identity function (res... |
0cnop 30341 | The identically zero funct... |
0cnfn 30342 | The identically zero funct... |
idcnop 30343 | The identity function (res... |
idhmop 30344 | The Hilbert space identity... |
0hmop 30345 | The identically zero funct... |
0lnop 30346 | The identically zero funct... |
0lnfn 30347 | The identically zero funct... |
nmop0 30348 | The norm of the zero opera... |
nmfn0 30349 | The norm of the identicall... |
hmopbdoptHIL 30350 | A Hermitian operator is a ... |
hoddii 30351 | Distributive law for Hilbe... |
hoddi 30352 | Distributive law for Hilbe... |
nmop0h 30353 | The norm of any operator o... |
idlnop 30354 | The identity function (res... |
0bdop 30355 | The identically zero opera... |
adj0 30356 | Adjoint of the zero operat... |
nmlnop0iALT 30357 | A linear operator with a z... |
nmlnop0iHIL 30358 | A linear operator with a z... |
nmlnopgt0i 30359 | A linear Hilbert space ope... |
nmlnop0 30360 | A linear operator with a z... |
nmlnopne0 30361 | A linear operator with a n... |
lnopmi 30362 | The scalar product of a li... |
lnophsi 30363 | The sum of two linear oper... |
lnophdi 30364 | The difference of two line... |
lnopcoi 30365 | The composition of two lin... |
lnopco0i 30366 | The composition of a linea... |
lnopeq0lem1 30367 | Lemma for ~ lnopeq0i . Ap... |
lnopeq0lem2 30368 | Lemma for ~ lnopeq0i . (C... |
lnopeq0i 30369 | A condition implying that ... |
lnopeqi 30370 | Two linear Hilbert space o... |
lnopeq 30371 | Two linear Hilbert space o... |
lnopunilem1 30372 | Lemma for ~ lnopunii . (C... |
lnopunilem2 30373 | Lemma for ~ lnopunii . (C... |
lnopunii 30374 | If a linear operator (whos... |
elunop2 30375 | An operator is unitary iff... |
nmopun 30376 | Norm of a unitary Hilbert ... |
unopbd 30377 | A unitary operator is a bo... |
lnophmlem1 30378 | Lemma for ~ lnophmi . (Co... |
lnophmlem2 30379 | Lemma for ~ lnophmi . (Co... |
lnophmi 30380 | A linear operator is Hermi... |
lnophm 30381 | A linear operator is Hermi... |
hmops 30382 | The sum of two Hermitian o... |
hmopm 30383 | The scalar product of a He... |
hmopd 30384 | The difference of two Herm... |
hmopco 30385 | The composition of two com... |
nmbdoplbi 30386 | A lower bound for the norm... |
nmbdoplb 30387 | A lower bound for the norm... |
nmcexi 30388 | Lemma for ~ nmcopexi and ~... |
nmcopexi 30389 | The norm of a continuous l... |
nmcoplbi 30390 | A lower bound for the norm... |
nmcopex 30391 | The norm of a continuous l... |
nmcoplb 30392 | A lower bound for the norm... |
nmophmi 30393 | The norm of the scalar pro... |
bdophmi 30394 | The scalar product of a bo... |
lnconi 30395 | Lemma for ~ lnopconi and ~... |
lnopconi 30396 | A condition equivalent to ... |
lnopcon 30397 | A condition equivalent to ... |
lnopcnbd 30398 | A linear operator is conti... |
lncnopbd 30399 | A continuous linear operat... |
lncnbd 30400 | A continuous linear operat... |
lnopcnre 30401 | A linear operator is conti... |
lnfnli 30402 | Basic property of a linear... |
lnfnfi 30403 | A linear Hilbert space fun... |
lnfn0i 30404 | The value of a linear Hilb... |
lnfnaddi 30405 | Additive property of a lin... |
lnfnmuli 30406 | Multiplicative property of... |
lnfnaddmuli 30407 | Sum/product property of a ... |
lnfnsubi 30408 | Subtraction property for a... |
lnfn0 30409 | The value of a linear Hilb... |
lnfnmul 30410 | Multiplicative property of... |
nmbdfnlbi 30411 | A lower bound for the norm... |
nmbdfnlb 30412 | A lower bound for the norm... |
nmcfnexi 30413 | The norm of a continuous l... |
nmcfnlbi 30414 | A lower bound for the norm... |
nmcfnex 30415 | The norm of a continuous l... |
nmcfnlb 30416 | A lower bound of the norm ... |
lnfnconi 30417 | A condition equivalent to ... |
lnfncon 30418 | A condition equivalent to ... |
lnfncnbd 30419 | A linear functional is con... |
imaelshi 30420 | The image of a subspace un... |
rnelshi 30421 | The range of a linear oper... |
nlelshi 30422 | The null space of a linear... |
nlelchi 30423 | The null space of a contin... |
riesz3i 30424 | A continuous linear functi... |
riesz4i 30425 | A continuous linear functi... |
riesz4 30426 | A continuous linear functi... |
riesz1 30427 | Part 1 of the Riesz repres... |
riesz2 30428 | Part 2 of the Riesz repres... |
cnlnadjlem1 30429 | Lemma for ~ cnlnadji (Theo... |
cnlnadjlem2 30430 | Lemma for ~ cnlnadji . ` G... |
cnlnadjlem3 30431 | Lemma for ~ cnlnadji . By... |
cnlnadjlem4 30432 | Lemma for ~ cnlnadji . Th... |
cnlnadjlem5 30433 | Lemma for ~ cnlnadji . ` F... |
cnlnadjlem6 30434 | Lemma for ~ cnlnadji . ` F... |
cnlnadjlem7 30435 | Lemma for ~ cnlnadji . He... |
cnlnadjlem8 30436 | Lemma for ~ cnlnadji . ` F... |
cnlnadjlem9 30437 | Lemma for ~ cnlnadji . ` F... |
cnlnadji 30438 | Every continuous linear op... |
cnlnadjeui 30439 | Every continuous linear op... |
cnlnadjeu 30440 | Every continuous linear op... |
cnlnadj 30441 | Every continuous linear op... |
cnlnssadj 30442 | Every continuous linear Hi... |
bdopssadj 30443 | Every bounded linear Hilbe... |
bdopadj 30444 | Every bounded linear Hilbe... |
adjbdln 30445 | The adjoint of a bounded l... |
adjbdlnb 30446 | An operator is bounded and... |
adjbd1o 30447 | The mapping of adjoints of... |
adjlnop 30448 | The adjoint of an operator... |
adjsslnop 30449 | Every operator with an adj... |
nmopadjlei 30450 | Property of the norm of an... |
nmopadjlem 30451 | Lemma for ~ nmopadji . (C... |
nmopadji 30452 | Property of the norm of an... |
adjeq0 30453 | An operator is zero iff it... |
adjmul 30454 | The adjoint of the scalar ... |
adjadd 30455 | The adjoint of the sum of ... |
nmoptrii 30456 | Triangle inequality for th... |
nmopcoi 30457 | Upper bound for the norm o... |
bdophsi 30458 | The sum of two bounded lin... |
bdophdi 30459 | The difference between two... |
bdopcoi 30460 | The composition of two bou... |
nmoptri2i 30461 | Triangle-type inequality f... |
adjcoi 30462 | The adjoint of a compositi... |
nmopcoadji 30463 | The norm of an operator co... |
nmopcoadj2i 30464 | The norm of an operator co... |
nmopcoadj0i 30465 | An operator composed with ... |
unierri 30466 | If we approximate a chain ... |
branmfn 30467 | The norm of the bra functi... |
brabn 30468 | The bra of a vector is a b... |
rnbra 30469 | The set of bras equals the... |
bra11 30470 | The bra function maps vect... |
bracnln 30471 | A bra is a continuous line... |
cnvbraval 30472 | Value of the converse of t... |
cnvbracl 30473 | Closure of the converse of... |
cnvbrabra 30474 | The converse bra of the br... |
bracnvbra 30475 | The bra of the converse br... |
bracnlnval 30476 | The vector that a continuo... |
cnvbramul 30477 | Multiplication property of... |
kbass1 30478 | Dirac bra-ket associative ... |
kbass2 30479 | Dirac bra-ket associative ... |
kbass3 30480 | Dirac bra-ket associative ... |
kbass4 30481 | Dirac bra-ket associative ... |
kbass5 30482 | Dirac bra-ket associative ... |
kbass6 30483 | Dirac bra-ket associative ... |
leopg 30484 | Ordering relation for posi... |
leop 30485 | Ordering relation for oper... |
leop2 30486 | Ordering relation for oper... |
leop3 30487 | Operator ordering in terms... |
leoppos 30488 | Binary relation defining a... |
leoprf2 30489 | The ordering relation for ... |
leoprf 30490 | The ordering relation for ... |
leopsq 30491 | The square of a Hermitian ... |
0leop 30492 | The zero operator is a pos... |
idleop 30493 | The identity operator is a... |
leopadd 30494 | The sum of two positive op... |
leopmuli 30495 | The scalar product of a no... |
leopmul 30496 | The scalar product of a po... |
leopmul2i 30497 | Scalar product applied to ... |
leoptri 30498 | The positive operator orde... |
leoptr 30499 | The positive operator orde... |
leopnmid 30500 | A bounded Hermitian operat... |
nmopleid 30501 | A nonzero, bounded Hermiti... |
opsqrlem1 30502 | Lemma for opsqri . (Contr... |
opsqrlem2 30503 | Lemma for opsqri . ` F `` ... |
opsqrlem3 30504 | Lemma for opsqri . (Contr... |
opsqrlem4 30505 | Lemma for opsqri . (Contr... |
opsqrlem5 30506 | Lemma for opsqri . (Contr... |
opsqrlem6 30507 | Lemma for opsqri . (Contr... |
pjhmopi 30508 | A projector is a Hermitian... |
pjlnopi 30509 | A projector is a linear op... |
pjnmopi 30510 | The operator norm of a pro... |
pjbdlni 30511 | A projector is a bounded l... |
pjhmop 30512 | A projection is a Hermitia... |
hmopidmchi 30513 | An idempotent Hermitian op... |
hmopidmpji 30514 | An idempotent Hermitian op... |
hmopidmch 30515 | An idempotent Hermitian op... |
hmopidmpj 30516 | An idempotent Hermitian op... |
pjsdii 30517 | Distributive law for Hilbe... |
pjddii 30518 | Distributive law for Hilbe... |
pjsdi2i 30519 | Chained distributive law f... |
pjcoi 30520 | Composition of projections... |
pjcocli 30521 | Closure of composition of ... |
pjcohcli 30522 | Closure of composition of ... |
pjadjcoi 30523 | Adjoint of composition of ... |
pjcofni 30524 | Functionality of compositi... |
pjss1coi 30525 | Subset relationship for pr... |
pjss2coi 30526 | Subset relationship for pr... |
pjssmi 30527 | Projection meet property. ... |
pjssge0i 30528 | Theorem 4.5(iv)->(v) of [B... |
pjdifnormi 30529 | Theorem 4.5(v)<->(vi) of [... |
pjnormssi 30530 | Theorem 4.5(i)<->(vi) of [... |
pjorthcoi 30531 | Composition of projections... |
pjscji 30532 | The projection of orthogon... |
pjssumi 30533 | The projection on a subspa... |
pjssposi 30534 | Projector ordering can be ... |
pjordi 30535 | The definition of projecto... |
pjssdif2i 30536 | The projection subspace of... |
pjssdif1i 30537 | A necessary and sufficient... |
pjimai 30538 | The image of a projection.... |
pjidmcoi 30539 | A projection is idempotent... |
pjoccoi 30540 | Composition of projections... |
pjtoi 30541 | Subspace sum of projection... |
pjoci 30542 | Projection of orthocomplem... |
pjidmco 30543 | A projection operator is i... |
dfpjop 30544 | Definition of projection o... |
pjhmopidm 30545 | Two ways to express the se... |
elpjidm 30546 | A projection operator is i... |
elpjhmop 30547 | A projection operator is H... |
0leopj 30548 | A projector is a positive ... |
pjadj2 30549 | A projector is self-adjoin... |
pjadj3 30550 | A projector is self-adjoin... |
elpjch 30551 | Reconstruction of the subs... |
elpjrn 30552 | Reconstruction of the subs... |
pjinvari 30553 | A closed subspace ` H ` wi... |
pjin1i 30554 | Lemma for Theorem 1.22 of ... |
pjin2i 30555 | Lemma for Theorem 1.22 of ... |
pjin3i 30556 | Lemma for Theorem 1.22 of ... |
pjclem1 30557 | Lemma for projection commu... |
pjclem2 30558 | Lemma for projection commu... |
pjclem3 30559 | Lemma for projection commu... |
pjclem4a 30560 | Lemma for projection commu... |
pjclem4 30561 | Lemma for projection commu... |
pjci 30562 | Two subspaces commute iff ... |
pjcmul1i 30563 | A necessary and sufficient... |
pjcmul2i 30564 | The projection subspace of... |
pjcohocli 30565 | Closure of composition of ... |
pjadj2coi 30566 | Adjoint of double composit... |
pj2cocli 30567 | Closure of double composit... |
pj3lem1 30568 | Lemma for projection tripl... |
pj3si 30569 | Stronger projection triple... |
pj3i 30570 | Projection triplet theorem... |
pj3cor1i 30571 | Projection triplet corolla... |
pjs14i 30572 | Theorem S-14 of Watanabe, ... |
isst 30575 | Property of a state. (Con... |
ishst 30576 | Property of a complex Hilb... |
sticl 30577 | ` [ 0 , 1 ] ` closure of t... |
stcl 30578 | Real closure of the value ... |
hstcl 30579 | Closure of the value of a ... |
hst1a 30580 | Unit value of a Hilbert-sp... |
hstel2 30581 | Properties of a Hilbert-sp... |
hstorth 30582 | Orthogonality property of ... |
hstosum 30583 | Orthogonal sum property of... |
hstoc 30584 | Sum of a Hilbert-space-val... |
hstnmoc 30585 | Sum of norms of a Hilbert-... |
stge0 30586 | The value of a state is no... |
stle1 30587 | The value of a state is le... |
hstle1 30588 | The norm of the value of a... |
hst1h 30589 | The norm of a Hilbert-spac... |
hst0h 30590 | The norm of a Hilbert-spac... |
hstpyth 30591 | Pythagorean property of a ... |
hstle 30592 | Ordering property of a Hil... |
hstles 30593 | Ordering property of a Hil... |
hstoh 30594 | A Hilbert-space-valued sta... |
hst0 30595 | A Hilbert-space-valued sta... |
sthil 30596 | The value of a state at th... |
stj 30597 | The value of a state on a ... |
sto1i 30598 | The state of a subspace pl... |
sto2i 30599 | The state of the orthocomp... |
stge1i 30600 | If a state is greater than... |
stle0i 30601 | If a state is less than or... |
stlei 30602 | Ordering law for states. ... |
stlesi 30603 | Ordering law for states. ... |
stji1i 30604 | Join of components of Sasa... |
stm1i 30605 | State of component of unit... |
stm1ri 30606 | State of component of unit... |
stm1addi 30607 | Sum of states whose meet i... |
staddi 30608 | If the sum of 2 states is ... |
stm1add3i 30609 | Sum of states whose meet i... |
stadd3i 30610 | If the sum of 3 states is ... |
st0 30611 | The state of the zero subs... |
strlem1 30612 | Lemma for strong state the... |
strlem2 30613 | Lemma for strong state the... |
strlem3a 30614 | Lemma for strong state the... |
strlem3 30615 | Lemma for strong state the... |
strlem4 30616 | Lemma for strong state the... |
strlem5 30617 | Lemma for strong state the... |
strlem6 30618 | Lemma for strong state the... |
stri 30619 | Strong state theorem. The... |
strb 30620 | Strong state theorem (bidi... |
hstrlem2 30621 | Lemma for strong set of CH... |
hstrlem3a 30622 | Lemma for strong set of CH... |
hstrlem3 30623 | Lemma for strong set of CH... |
hstrlem4 30624 | Lemma for strong set of CH... |
hstrlem5 30625 | Lemma for strong set of CH... |
hstrlem6 30626 | Lemma for strong set of CH... |
hstri 30627 | Hilbert space admits a str... |
hstrbi 30628 | Strong CH-state theorem (b... |
largei 30629 | A Hilbert lattice admits a... |
jplem1 30630 | Lemma for Jauch-Piron theo... |
jplem2 30631 | Lemma for Jauch-Piron theo... |
jpi 30632 | The function ` S ` , that ... |
golem1 30633 | Lemma for Godowski's equat... |
golem2 30634 | Lemma for Godowski's equat... |
goeqi 30635 | Godowski's equation, shown... |
stcltr1i 30636 | Property of a strong class... |
stcltr2i 30637 | Property of a strong class... |
stcltrlem1 30638 | Lemma for strong classical... |
stcltrlem2 30639 | Lemma for strong classical... |
stcltrthi 30640 | Theorem for classically st... |
cvbr 30644 | Binary relation expressing... |
cvbr2 30645 | Binary relation expressing... |
cvcon3 30646 | Contraposition law for the... |
cvpss 30647 | The covers relation implie... |
cvnbtwn 30648 | The covers relation implie... |
cvnbtwn2 30649 | The covers relation implie... |
cvnbtwn3 30650 | The covers relation implie... |
cvnbtwn4 30651 | The covers relation implie... |
cvnsym 30652 | The covers relation is not... |
cvnref 30653 | The covers relation is not... |
cvntr 30654 | The covers relation is not... |
spansncv2 30655 | Hilbert space has the cove... |
mdbr 30656 | Binary relation expressing... |
mdi 30657 | Consequence of the modular... |
mdbr2 30658 | Binary relation expressing... |
mdbr3 30659 | Binary relation expressing... |
mdbr4 30660 | Binary relation expressing... |
dmdbr 30661 | Binary relation expressing... |
dmdmd 30662 | The dual modular pair prop... |
mddmd 30663 | The modular pair property ... |
dmdi 30664 | Consequence of the dual mo... |
dmdbr2 30665 | Binary relation expressing... |
dmdi2 30666 | Consequence of the dual mo... |
dmdbr3 30667 | Binary relation expressing... |
dmdbr4 30668 | Binary relation expressing... |
dmdi4 30669 | Consequence of the dual mo... |
dmdbr5 30670 | Binary relation expressing... |
mddmd2 30671 | Relationship between modul... |
mdsl0 30672 | A sublattice condition tha... |
ssmd1 30673 | Ordering implies the modul... |
ssmd2 30674 | Ordering implies the modul... |
ssdmd1 30675 | Ordering implies the dual ... |
ssdmd2 30676 | Ordering implies the dual ... |
dmdsl3 30677 | Sublattice mapping for a d... |
mdsl3 30678 | Sublattice mapping for a m... |
mdslle1i 30679 | Order preservation of the ... |
mdslle2i 30680 | Order preservation of the ... |
mdslj1i 30681 | Join preservation of the o... |
mdslj2i 30682 | Meet preservation of the r... |
mdsl1i 30683 | If the modular pair proper... |
mdsl2i 30684 | If the modular pair proper... |
mdsl2bi 30685 | If the modular pair proper... |
cvmdi 30686 | The covering property impl... |
mdslmd1lem1 30687 | Lemma for ~ mdslmd1i . (C... |
mdslmd1lem2 30688 | Lemma for ~ mdslmd1i . (C... |
mdslmd1lem3 30689 | Lemma for ~ mdslmd1i . (C... |
mdslmd1lem4 30690 | Lemma for ~ mdslmd1i . (C... |
mdslmd1i 30691 | Preservation of the modula... |
mdslmd2i 30692 | Preservation of the modula... |
mdsldmd1i 30693 | Preservation of the dual m... |
mdslmd3i 30694 | Modular pair conditions th... |
mdslmd4i 30695 | Modular pair condition tha... |
csmdsymi 30696 | Cross-symmetry implies M-s... |
mdexchi 30697 | An exchange lemma for modu... |
cvmd 30698 | The covering property impl... |
cvdmd 30699 | The covering property impl... |
ela 30701 | Atoms in a Hilbert lattice... |
elat2 30702 | Expanded membership relati... |
elatcv0 30703 | A Hilbert lattice element ... |
atcv0 30704 | An atom covers the zero su... |
atssch 30705 | Atoms are a subset of the ... |
atelch 30706 | An atom is a Hilbert latti... |
atne0 30707 | An atom is not the Hilbert... |
atss 30708 | A lattice element smaller ... |
atsseq 30709 | Two atoms in a subset rela... |
atcveq0 30710 | A Hilbert lattice element ... |
h1da 30711 | A 1-dimensional subspace i... |
spansna 30712 | The span of the singleton ... |
sh1dle 30713 | A 1-dimensional subspace i... |
ch1dle 30714 | A 1-dimensional subspace i... |
atom1d 30715 | The 1-dimensional subspace... |
superpos 30716 | Superposition Principle. ... |
chcv1 30717 | The Hilbert lattice has th... |
chcv2 30718 | The Hilbert lattice has th... |
chjatom 30719 | The join of a closed subsp... |
shatomici 30720 | The lattice of Hilbert sub... |
hatomici 30721 | The Hilbert lattice is ato... |
hatomic 30722 | A Hilbert lattice is atomi... |
shatomistici 30723 | The lattice of Hilbert sub... |
hatomistici 30724 | ` CH ` is atomistic, i.e. ... |
chpssati 30725 | Two Hilbert lattice elemen... |
chrelati 30726 | The Hilbert lattice is rel... |
chrelat2i 30727 | A consequence of relative ... |
cvati 30728 | If a Hilbert lattice eleme... |
cvbr4i 30729 | An alternate way to expres... |
cvexchlem 30730 | Lemma for ~ cvexchi . (Co... |
cvexchi 30731 | The Hilbert lattice satisf... |
chrelat2 30732 | A consequence of relative ... |
chrelat3 30733 | A consequence of relative ... |
chrelat3i 30734 | A consequence of the relat... |
chrelat4i 30735 | A consequence of relative ... |
cvexch 30736 | The Hilbert lattice satisf... |
cvp 30737 | The Hilbert lattice satisf... |
atnssm0 30738 | The meet of a Hilbert latt... |
atnemeq0 30739 | The meet of distinct atoms... |
atssma 30740 | The meet with an atom's su... |
atcv0eq 30741 | Two atoms covering the zer... |
atcv1 30742 | Two atoms covering the zer... |
atexch 30743 | The Hilbert lattice satisf... |
atomli 30744 | An assertion holding in at... |
atoml2i 30745 | An assertion holding in at... |
atordi 30746 | An ordering law for a Hilb... |
atcvatlem 30747 | Lemma for ~ atcvati . (Co... |
atcvati 30748 | A nonzero Hilbert lattice ... |
atcvat2i 30749 | A Hilbert lattice element ... |
atord 30750 | An ordering law for a Hilb... |
atcvat2 30751 | A Hilbert lattice element ... |
chirredlem1 30752 | Lemma for ~ chirredi . (C... |
chirredlem2 30753 | Lemma for ~ chirredi . (C... |
chirredlem3 30754 | Lemma for ~ chirredi . (C... |
chirredlem4 30755 | Lemma for ~ chirredi . (C... |
chirredi 30756 | The Hilbert lattice is irr... |
chirred 30757 | The Hilbert lattice is irr... |
atcvat3i 30758 | A condition implying that ... |
atcvat4i 30759 | A condition implying exist... |
atdmd 30760 | Two Hilbert lattice elemen... |
atmd 30761 | Two Hilbert lattice elemen... |
atmd2 30762 | Two Hilbert lattice elemen... |
atabsi 30763 | Absorption of an incompara... |
atabs2i 30764 | Absorption of an incompara... |
mdsymlem1 30765 | Lemma for ~ mdsymi . (Con... |
mdsymlem2 30766 | Lemma for ~ mdsymi . (Con... |
mdsymlem3 30767 | Lemma for ~ mdsymi . (Con... |
mdsymlem4 30768 | Lemma for ~ mdsymi . This... |
mdsymlem5 30769 | Lemma for ~ mdsymi . (Con... |
mdsymlem6 30770 | Lemma for ~ mdsymi . This... |
mdsymlem7 30771 | Lemma for ~ mdsymi . Lemm... |
mdsymlem8 30772 | Lemma for ~ mdsymi . Lemm... |
mdsymi 30773 | M-symmetry of the Hilbert ... |
mdsym 30774 | M-symmetry of the Hilbert ... |
dmdsym 30775 | Dual M-symmetry of the Hil... |
atdmd2 30776 | Two Hilbert lattice elemen... |
sumdmdii 30777 | If the subspace sum of two... |
cmmdi 30778 | Commuting subspaces form a... |
cmdmdi 30779 | Commuting subspaces form a... |
sumdmdlem 30780 | Lemma for ~ sumdmdi . The... |
sumdmdlem2 30781 | Lemma for ~ sumdmdi . (Co... |
sumdmdi 30782 | The subspace sum of two Hi... |
dmdbr4ati 30783 | Dual modular pair property... |
dmdbr5ati 30784 | Dual modular pair property... |
dmdbr6ati 30785 | Dual modular pair property... |
dmdbr7ati 30786 | Dual modular pair property... |
mdoc1i 30787 | Orthocomplements form a mo... |
mdoc2i 30788 | Orthocomplements form a mo... |
dmdoc1i 30789 | Orthocomplements form a du... |
dmdoc2i 30790 | Orthocomplements form a du... |
mdcompli 30791 | A condition equivalent to ... |
dmdcompli 30792 | A condition equivalent to ... |
mddmdin0i 30793 | If dual modular implies mo... |
cdjreui 30794 | A member of the sum of dis... |
cdj1i 30795 | Two ways to express " ` A ... |
cdj3lem1 30796 | A property of " ` A ` and ... |
cdj3lem2 30797 | Lemma for ~ cdj3i . Value... |
cdj3lem2a 30798 | Lemma for ~ cdj3i . Closu... |
cdj3lem2b 30799 | Lemma for ~ cdj3i . The f... |
cdj3lem3 30800 | Lemma for ~ cdj3i . Value... |
cdj3lem3a 30801 | Lemma for ~ cdj3i . Closu... |
cdj3lem3b 30802 | Lemma for ~ cdj3i . The s... |
cdj3i 30803 | Two ways to express " ` A ... |
The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
mathbox 30804 | (_This theorem is a dummy ... |
sa-abvi 30805 | A theorem about the univer... |
xfree 30806 | A partial converse to ~ 19... |
xfree2 30807 | A partial converse to ~ 19... |
addltmulALT 30808 | A proof readability experi... |
bian1d 30809 | Adding a superfluous conju... |
or3di 30810 | Distributive law for disju... |
or3dir 30811 | Distributive law for disju... |
3o1cs 30812 | Deduction eliminating disj... |
3o2cs 30813 | Deduction eliminating disj... |
3o3cs 30814 | Deduction eliminating disj... |
sbc2iedf 30815 | Conversion of implicit sub... |
rspc2daf 30816 | Double restricted speciali... |
nelbOLDOLD 30817 | Obsolete version of ~ nelb... |
ralcom4f 30818 | Commutation of restricted ... |
rexcom4f 30819 | Commutation of restricted ... |
19.9d2rf 30820 | A deduction version of one... |
19.9d2r 30821 | A deduction version of one... |
r19.29ffa 30822 | A commonly used pattern ba... |
eqtrb 30823 | A transposition of equalit... |
opsbc2ie 30824 | Conversion of implicit sub... |
opreu2reuALT 30825 | Correspondence between uni... |
2reucom 30828 | Double restricted existent... |
2reu2rex1 30829 | Double restricted existent... |
2reureurex 30830 | Double restricted existent... |
2reu2reu2 30831 | Double restricted existent... |
opreu2reu1 30832 | Equivalent definition of t... |
sq2reunnltb 30833 | There exists a unique deco... |
addsqnot2reu 30834 | For each complex number ` ... |
sbceqbidf 30835 | Equality theorem for class... |
sbcies 30836 | A special version of class... |
mo5f 30837 | Alternate definition of "a... |
nmo 30838 | Negation of "at most one".... |
reuxfrdf 30839 | Transfer existential uniqu... |
rexunirn 30840 | Restricted existential qua... |
rmoxfrd 30841 | Transfer "at most one" res... |
rmoun 30842 | "At most one" restricted e... |
rmounid 30843 | A case where an "at most o... |
dmrab 30844 | Domain of a restricted cla... |
difrab2 30845 | Difference of two restrict... |
rabexgfGS 30846 | Separation Scheme in terms... |
rabsnel 30847 | Truth implied by equality ... |
rabeqsnd 30848 | Conditions for a restricte... |
eqrrabd 30849 | Deduce equality with a res... |
foresf1o 30850 | From a surjective function... |
rabfodom 30851 | Domination relation for re... |
abrexdomjm 30852 | An indexed set is dominate... |
abrexdom2jm 30853 | An indexed set is dominate... |
abrexexd 30854 | Existence of a class abstr... |
elabreximd 30855 | Class substitution in an i... |
elabreximdv 30856 | Class substitution in an i... |
abrexss 30857 | A necessary condition for ... |
elunsn 30858 | Elementhood to a union wit... |
nelun 30859 | Negated membership for a u... |
snsssng 30860 | If a singleton is a subset... |
rabss3d 30861 | Subclass law for restricte... |
inin 30862 | Intersection with an inter... |
inindif 30863 | See ~ inundif . (Contribu... |
difininv 30864 | Condition for the intersec... |
difeq 30865 | Rewriting an equation with... |
eqdif 30866 | If both set differences of... |
undif5 30867 | An equality involving clas... |
indifbi 30868 | Two ways to express equali... |
diffib 30869 | Case where ~ diffi is a bi... |
difxp1ss 30870 | Difference law for Cartesi... |
difxp2ss 30871 | Difference law for Cartesi... |
undifr 30872 | Union of complementary par... |
indifundif 30873 | A remarkable equation with... |
elpwincl1 30874 | Closure of intersection wi... |
elpwdifcl 30875 | Closure of class differenc... |
elpwiuncl 30876 | Closure of indexed union w... |
eqsnd 30877 | Deduce that a set is a sin... |
elpreq 30878 | Equality wihin a pair. (C... |
nelpr 30879 | A set ` A ` not in a pair ... |
inpr0 30880 | Rewrite an empty intersect... |
neldifpr1 30881 | The first element of a pai... |
neldifpr2 30882 | The second element of a pa... |
unidifsnel 30883 | The other element of a pai... |
unidifsnne 30884 | The other element of a pai... |
ifeqeqx 30885 | An equality theorem tailor... |
elimifd 30886 | Elimination of a condition... |
elim2if 30887 | Elimination of two conditi... |
elim2ifim 30888 | Elimination of two conditi... |
ifeq3da 30889 | Given an expression ` C ` ... |
uniinn0 30890 | Sufficient and necessary c... |
uniin1 30891 | Union of intersection. Ge... |
uniin2 30892 | Union of intersection. Ge... |
difuncomp 30893 | Express a class difference... |
elpwunicl 30894 | Closure of a set union wit... |
cbviunf 30895 | Rule used to change the bo... |
iuneq12daf 30896 | Equality deduction for ind... |
iunin1f 30897 | Indexed union of intersect... |
ssiun3 30898 | Subset equivalence for an ... |
ssiun2sf 30899 | Subset relationship for an... |
iuninc 30900 | The union of an increasing... |
iundifdifd 30901 | The intersection of a set ... |
iundifdif 30902 | The intersection of a set ... |
iunrdx 30903 | Re-index an indexed union.... |
iunpreima 30904 | Preimage of an indexed uni... |
iunrnmptss 30905 | A subset relation for an i... |
iunxunsn 30906 | Appending a set to an inde... |
iunxunpr 30907 | Appending two sets to an i... |
iinabrex 30908 | Rewriting an indexed inter... |
disjnf 30909 | In case ` x ` is not free ... |
cbvdisjf 30910 | Change bound variables in ... |
disjss1f 30911 | A subset of a disjoint col... |
disjeq1f 30912 | Equality theorem for disjo... |
disjxun0 30913 | Simplify a disjoint union.... |
disjdifprg 30914 | A trivial partition into a... |
disjdifprg2 30915 | A trivial partition of a s... |
disji2f 30916 | Property of a disjoint col... |
disjif 30917 | Property of a disjoint col... |
disjorf 30918 | Two ways to say that a col... |
disjorsf 30919 | Two ways to say that a col... |
disjif2 30920 | Property of a disjoint col... |
disjabrex 30921 | Rewriting a disjoint colle... |
disjabrexf 30922 | Rewriting a disjoint colle... |
disjpreima 30923 | A preimage of a disjoint s... |
disjrnmpt 30924 | Rewriting a disjoint colle... |
disjin 30925 | If a collection is disjoin... |
disjin2 30926 | If a collection is disjoin... |
disjxpin 30927 | Derive a disjunction over ... |
iundisjf 30928 | Rewrite a countable union ... |
iundisj2f 30929 | A disjoint union is disjoi... |
disjrdx 30930 | Re-index a disjunct collec... |
disjex 30931 | Two ways to say that two c... |
disjexc 30932 | A variant of ~ disjex , ap... |
disjunsn 30933 | Append an element to a dis... |
disjun0 30934 | Adding the empty element p... |
disjiunel 30935 | A set of elements B of a d... |
disjuniel 30936 | A set of elements B of a d... |
xpdisjres 30937 | Restriction of a constant ... |
opeldifid 30938 | Ordered pair elementhood o... |
difres 30939 | Case when class difference... |
imadifxp 30940 | Image of the difference wi... |
relfi 30941 | A relation (set) is finite... |
reldisjun 30942 | Split a relation into two ... |
0res 30943 | Restriction of the empty f... |
funresdm1 30944 | Restriction of a disjoint ... |
fnunres1 30945 | Restriction of a disjoint ... |
fcoinver 30946 | Build an equivalence relat... |
fcoinvbr 30947 | Binary relation for the eq... |
brabgaf 30948 | The law of concretion for ... |
brelg 30949 | Two things in a binary rel... |
br8d 30950 | Substitution for an eight-... |
opabdm 30951 | Domain of an ordered-pair ... |
opabrn 30952 | Range of an ordered-pair c... |
opabssi 30953 | Sufficient condition for a... |
opabid2ss 30954 | One direction of ~ opabid2... |
ssrelf 30955 | A subclass relationship de... |
eqrelrd2 30956 | A version of ~ eqrelrdv2 w... |
erbr3b 30957 | Biconditional for equivale... |
iunsnima 30958 | Image of a singleton by an... |
iunsnima2 30959 | Version of ~ iunsnima with... |
ac6sf2 30960 | Alternate version of ~ ac6... |
fnresin 30961 | Restriction of a function ... |
f1o3d 30962 | Describe an implicit one-t... |
eldmne0 30963 | A function of nonempty dom... |
f1rnen 30964 | Equinumerosity of the rang... |
rinvf1o 30965 | Sufficient conditions for ... |
fresf1o 30966 | Conditions for a restricti... |
nfpconfp 30967 | The set of fixed points of... |
fmptco1f1o 30968 | The action of composing (t... |
cofmpt2 30969 | Express composition of a m... |
f1mptrn 30970 | Express injection for a ma... |
dfimafnf 30971 | Alternate definition of th... |
funimass4f 30972 | Membership relation for th... |
elimampt 30973 | Membership in the image of... |
suppss2f 30974 | Show that the support of a... |
fovcld 30975 | Closure law for an operati... |
ofrn 30976 | The range of the function ... |
ofrn2 30977 | The range of the function ... |
off2 30978 | The function operation pro... |
ofresid 30979 | Applying an operation rest... |
fimarab 30980 | Expressing the image of a ... |
unipreima 30981 | Preimage of a class union.... |
opfv 30982 | Value of a function produc... |
xppreima 30983 | The preimage of a Cartesia... |
2ndimaxp 30984 | Image of a cartesian produ... |
djussxp2 30985 | Stronger version of ~ djus... |
2ndresdju 30986 | The ` 2nd ` function restr... |
2ndresdjuf1o 30987 | The ` 2nd ` function restr... |
xppreima2 30988 | The preimage of a Cartesia... |
abfmpunirn 30989 | Membership in a union of a... |
rabfmpunirn 30990 | Membership in a union of a... |
abfmpeld 30991 | Membership in an element o... |
abfmpel 30992 | Membership in an element o... |
fmptdF 30993 | Domain and codomain of the... |
fmptcof2 30994 | Composition of two functio... |
fcomptf 30995 | Express composition of two... |
acunirnmpt 30996 | Axiom of choice for the un... |
acunirnmpt2 30997 | Axiom of choice for the un... |
acunirnmpt2f 30998 | Axiom of choice for the un... |
aciunf1lem 30999 | Choice in an index union. ... |
aciunf1 31000 | Choice in an index union. ... |
ofoprabco 31001 | Function operation as a co... |
ofpreima 31002 | Express the preimage of a ... |
ofpreima2 31003 | Express the preimage of a ... |
funcnvmpt 31004 | Condition for a function i... |
funcnv5mpt 31005 | Two ways to say that a fun... |
funcnv4mpt 31006 | Two ways to say that a fun... |
preimane 31007 | Different elements have di... |
fnpreimac 31008 | Choose a set ` x ` contain... |
fgreu 31009 | Exactly one point of a fun... |
fcnvgreu 31010 | If the converse of a relat... |
rnmposs 31011 | The range of an operation ... |
mptssALT 31012 | Deduce subset relation of ... |
dfcnv2 31013 | Alternative definition of ... |
fnimatp 31014 | The image of an unordered ... |
fnunres2 31015 | Restriction of a disjoint ... |
mpomptxf 31016 | Express a two-argument fun... |
suppovss 31017 | A bound for the support of... |
fvdifsupp 31018 | Function value is zero out... |
fmptssfisupp 31019 | The restriction of a mappi... |
suppiniseg 31020 | Relation between the suppo... |
fsuppinisegfi 31021 | The initial segment ` ( ``... |
fressupp 31022 | The restriction of a funct... |
fdifsuppconst 31023 | A function is a zero const... |
ressupprn 31024 | The range of a function re... |
supppreima 31025 | Express the support of a f... |
fsupprnfi 31026 | Finite support implies fin... |
cosnopne 31027 | Composition of two ordered... |
cosnop 31028 | Composition of two ordered... |
cnvprop 31029 | Converse of a pair of orde... |
brprop 31030 | Binary relation for a pair... |
mptprop 31031 | Rewrite pairs of ordered p... |
coprprop 31032 | Composition of two pairs o... |
gtiso 31033 | Two ways to write a strict... |
isoun 31034 | Infer an isomorphism from ... |
disjdsct 31035 | A disjoint collection is d... |
df1stres 31036 | Definition for a restricti... |
df2ndres 31037 | Definition for a restricti... |
1stpreimas 31038 | The preimage of a singleto... |
1stpreima 31039 | The preimage by ` 1st ` is... |
2ndpreima 31040 | The preimage by ` 2nd ` is... |
curry2ima 31041 | The image of a curried fun... |
preiman0 31042 | The preimage of a nonempty... |
intimafv 31043 | The intersection of an ima... |
supssd 31044 | Inequality deduction for s... |
infssd 31045 | Inequality deduction for i... |
imafi2 31046 | The image by a finite set ... |
unifi3 31047 | If a union is finite, then... |
snct 31048 | A singleton is countable. ... |
prct 31049 | An unordered pair is count... |
mpocti 31050 | An operation is countable ... |
abrexct 31051 | An image set of a countabl... |
mptctf 31052 | A countable mapping set is... |
abrexctf 31053 | An image set of a countabl... |
padct 31054 | Index a countable set with... |
cnvoprabOLD 31055 | The converse of a class ab... |
f1od2 31056 | Sufficient condition for a... |
fcobij 31057 | Composing functions with a... |
fcobijfs 31058 | Composing finitely support... |
suppss3 31059 | Deduce a function's suppor... |
fsuppcurry1 31060 | Finite support of a currie... |
fsuppcurry2 31061 | Finite support of a currie... |
offinsupp1 31062 | Finite support for a funct... |
ffs2 31063 | Rewrite a function's suppo... |
ffsrn 31064 | The range of a finitely su... |
resf1o 31065 | Restriction of functions t... |
maprnin 31066 | Restricting the range of t... |
fpwrelmapffslem 31067 | Lemma for ~ fpwrelmapffs .... |
fpwrelmap 31068 | Define a canonical mapping... |
fpwrelmapffs 31069 | Define a canonical mapping... |
creq0 31070 | The real representation of... |
1nei 31071 | The imaginary unit ` _i ` ... |
1neg1t1neg1 31072 | An integer unit times itse... |
nnmulge 31073 | Multiplying by a positive ... |
lt2addrd 31074 | If the right-hand side of ... |
xrlelttric 31075 | Trichotomy law for extende... |
xaddeq0 31076 | Two extended reals which a... |
xrinfm 31077 | The extended real numbers ... |
le2halvesd 31078 | A sum is less than the who... |
xraddge02 31079 | A number is less than or e... |
xrge0addge 31080 | A number is less than or e... |
xlt2addrd 31081 | If the right-hand side of ... |
xrsupssd 31082 | Inequality deduction for s... |
xrge0infss 31083 | Any subset of nonnegative ... |
xrge0infssd 31084 | Inequality deduction for i... |
xrge0addcld 31085 | Nonnegative extended reals... |
xrge0subcld 31086 | Condition for closure of n... |
infxrge0lb 31087 | A member of a set of nonne... |
infxrge0glb 31088 | The infimum of a set of no... |
infxrge0gelb 31089 | The infimum of a set of no... |
xrofsup 31090 | The supremum is preserved ... |
supxrnemnf 31091 | The supremum of a nonempty... |
xnn0gt0 31092 | Nonzero extended nonnegati... |
xnn01gt 31093 | An extended nonnegative in... |
nn0xmulclb 31094 | Finite multiplication in t... |
joiniooico 31095 | Disjoint joining an open i... |
ubico 31096 | A right-open interval does... |
xeqlelt 31097 | Equality in terms of 'less... |
eliccelico 31098 | Relate elementhood to a cl... |
elicoelioo 31099 | Relate elementhood to a cl... |
iocinioc2 31100 | Intersection between two o... |
xrdifh 31101 | Class difference of a half... |
iocinif 31102 | Relate intersection of two... |
difioo 31103 | The difference between two... |
difico 31104 | The difference between two... |
uzssico 31105 | Upper integer sets are a s... |
fz2ssnn0 31106 | A finite set of sequential... |
nndiffz1 31107 | Upper set of the positive ... |
ssnnssfz 31108 | For any finite subset of `... |
fzne1 31109 | Elementhood in a finite se... |
fzm1ne1 31110 | Elementhood of an integer ... |
fzspl 31111 | Split the last element of ... |
fzdif2 31112 | Split the last element of ... |
fzodif2 31113 | Split the last element of ... |
fzodif1 31114 | Set difference of two half... |
fzsplit3 31115 | Split a finite interval of... |
bcm1n 31116 | The proportion of one bino... |
iundisjfi 31117 | Rewrite a countable union ... |
iundisj2fi 31118 | A disjoint union is disjoi... |
iundisjcnt 31119 | Rewrite a countable union ... |
iundisj2cnt 31120 | A countable disjoint union... |
fzone1 31121 | Elementhood in a half-open... |
fzom1ne1 31122 | Elementhood in a half-open... |
f1ocnt 31123 | Given a countable set ` A ... |
fz1nnct 31124 | NN and integer ranges star... |
fz1nntr 31125 | NN and integer ranges star... |
hashunif 31126 | The cardinality of a disjo... |
hashxpe 31127 | The size of the Cartesian ... |
hashgt1 31128 | Restate "set contains at l... |
dvdszzq 31129 | Divisibility for an intege... |
prmdvdsbc 31130 | Condition for a prime numb... |
numdenneg 31131 | Numerator and denominator ... |
divnumden2 31132 | Calculate the reduced form... |
nnindf 31133 | Principle of Mathematical ... |
nn0min 31134 | Extracting the minimum pos... |
subne0nn 31135 | A nonnegative difference i... |
ltesubnnd 31136 | Subtracting an integer num... |
fprodeq02 31137 | If one of the factors is z... |
pr01ssre 31138 | The range of the indicator... |
fprodex01 31139 | A product of factors equal... |
prodpr 31140 | A product over a pair is t... |
prodtp 31141 | A product over a triple is... |
fsumub 31142 | An upper bound for a term ... |
fsumiunle 31143 | Upper bound for a sum of n... |
dfdec100 31144 | Split the hundreds from a ... |
dp2eq1 31147 | Equality theorem for the d... |
dp2eq2 31148 | Equality theorem for the d... |
dp2eq1i 31149 | Equality theorem for the d... |
dp2eq2i 31150 | Equality theorem for the d... |
dp2eq12i 31151 | Equality theorem for the d... |
dp20u 31152 | Add a zero in the tenths (... |
dp20h 31153 | Add a zero in the unit pla... |
dp2cl 31154 | Closure for the decimal fr... |
dp2clq 31155 | Closure for a decimal frac... |
rpdp2cl 31156 | Closure for a decimal frac... |
rpdp2cl2 31157 | Closure for a decimal frac... |
dp2lt10 31158 | Decimal fraction builds re... |
dp2lt 31159 | Comparing two decimal frac... |
dp2ltsuc 31160 | Comparing a decimal fracti... |
dp2ltc 31161 | Comparing two decimal expa... |
dpval 31164 | Define the value of the de... |
dpcl 31165 | Prove that the closure of ... |
dpfrac1 31166 | Prove a simple equivalence... |
dpval2 31167 | Value of the decimal point... |
dpval3 31168 | Value of the decimal point... |
dpmul10 31169 | Multiply by 10 a decimal e... |
decdiv10 31170 | Divide a decimal number by... |
dpmul100 31171 | Multiply by 100 a decimal ... |
dp3mul10 31172 | Multiply by 10 a decimal e... |
dpmul1000 31173 | Multiply by 1000 a decimal... |
dpval3rp 31174 | Value of the decimal point... |
dp0u 31175 | Add a zero in the tenths p... |
dp0h 31176 | Remove a zero in the units... |
rpdpcl 31177 | Closure of the decimal poi... |
dplt 31178 | Comparing two decimal expa... |
dplti 31179 | Comparing a decimal expans... |
dpgti 31180 | Comparing a decimal expans... |
dpltc 31181 | Comparing two decimal inte... |
dpexpp1 31182 | Add one zero to the mantis... |
0dp2dp 31183 | Multiply by 10 a decimal e... |
dpadd2 31184 | Addition with one decimal,... |
dpadd 31185 | Addition with one decimal.... |
dpadd3 31186 | Addition with two decimals... |
dpmul 31187 | Multiplication with one de... |
dpmul4 31188 | An upper bound to multipli... |
threehalves 31189 | Example theorem demonstrat... |
1mhdrd 31190 | Example theorem demonstrat... |
xdivval 31193 | Value of division: the (un... |
xrecex 31194 | Existence of reciprocal of... |
xmulcand 31195 | Cancellation law for exten... |
xreceu 31196 | Existential uniqueness of ... |
xdivcld 31197 | Closure law for the extend... |
xdivcl 31198 | Closure law for the extend... |
xdivmul 31199 | Relationship between divis... |
rexdiv 31200 | The extended real division... |
xdivrec 31201 | Relationship between divis... |
xdivid 31202 | A number divided by itself... |
xdiv0 31203 | Division into zero is zero... |
xdiv0rp 31204 | Division into zero is zero... |
eliccioo 31205 | Membership in a closed int... |
elxrge02 31206 | Elementhood in the set of ... |
xdivpnfrp 31207 | Plus infinity divided by a... |
rpxdivcld 31208 | Closure law for extended d... |
xrpxdivcld 31209 | Closure law for extended d... |
wrdfd 31210 | A word is a zero-based seq... |
wrdres 31211 | Condition for the restrict... |
wrdsplex 31212 | Existence of a split of a ... |
pfx1s2 31213 | The prefix of length 1 of ... |
pfxrn2 31214 | The range of a prefix of a... |
pfxrn3 31215 | Express the range of a pre... |
pfxf1 31216 | Condition for a prefix to ... |
s1f1 31217 | Conditions for a length 1 ... |
s2rn 31218 | Range of a length 2 string... |
s2f1 31219 | Conditions for a length 2 ... |
s3rn 31220 | Range of a length 3 string... |
s3f1 31221 | Conditions for a length 3 ... |
s3clhash 31222 | Closure of the words of le... |
ccatf1 31223 | Conditions for a concatena... |
pfxlsw2ccat 31224 | Reconstruct a word from it... |
wrdt2ind 31225 | Perform an induction over ... |
swrdrn2 31226 | The range of a subword is ... |
swrdrn3 31227 | Express the range of a sub... |
swrdf1 31228 | Condition for a subword to... |
swrdrndisj 31229 | Condition for the range of... |
splfv3 31230 | Symbols to the right of a ... |
1cshid 31231 | Cyclically shifting a sing... |
cshw1s2 31232 | Cyclically shifting a leng... |
cshwrnid 31233 | Cyclically shifting a word... |
cshf1o 31234 | Condition for the cyclic s... |
ressplusf 31235 | The group operation functi... |
ressnm 31236 | The norm in a restricted s... |
abvpropd2 31237 | Weaker version of ~ abvpro... |
oppgle 31238 | less-than relation of an o... |
oppgleOLD 31239 | Obsolete version of ~ oppg... |
oppglt 31240 | less-than relation of an o... |
ressprs 31241 | The restriction of a prose... |
oduprs 31242 | Being a proset is a self-d... |
posrasymb 31243 | A poset ordering is asymet... |
resspos 31244 | The restriction of a Poset... |
resstos 31245 | The restriction of a Toset... |
odutos 31246 | Being a toset is a self-du... |
tlt2 31247 | In a Toset, two elements m... |
tlt3 31248 | In a Toset, two elements m... |
trleile 31249 | In a Toset, two elements m... |
toslublem 31250 | Lemma for ~ toslub and ~ x... |
toslub 31251 | In a toset, the lowest upp... |
tosglblem 31252 | Lemma for ~ tosglb and ~ x... |
tosglb 31253 | Same theorem as ~ toslub ,... |
clatp0cl 31254 | The poset zero of a comple... |
clatp1cl 31255 | The poset one of a complet... |
mntoval 31260 | Operation value of the mon... |
ismnt 31261 | Express the statement " ` ... |
ismntd 31262 | Property of being a monoto... |
mntf 31263 | A monotone function is a f... |
mgcoval 31264 | Operation value of the mon... |
mgcval 31265 | Monotone Galois connection... |
mgcf1 31266 | The lower adjoint ` F ` of... |
mgcf2 31267 | The upper adjoint ` G ` of... |
mgccole1 31268 | An inequality for the kern... |
mgccole2 31269 | Inequality for the closure... |
mgcmnt1 31270 | The lower adjoint ` F ` of... |
mgcmnt2 31271 | The upper adjoint ` G ` of... |
mgcmntco 31272 | A Galois connection like s... |
dfmgc2lem 31273 | Lemma for dfmgc2, backward... |
dfmgc2 31274 | Alternate definition of th... |
mgcmnt1d 31275 | Galois connection implies ... |
mgcmnt2d 31276 | Galois connection implies ... |
mgccnv 31277 | The inverse Galois connect... |
pwrssmgc 31278 | Given a function ` F ` , e... |
mgcf1olem1 31279 | Property of a Galois conne... |
mgcf1olem2 31280 | Property of a Galois conne... |
mgcf1o 31281 | Given a Galois connection,... |
xrs0 31284 | The zero of the extended r... |
xrslt 31285 | The "strictly less than" r... |
xrsinvgval 31286 | The inversion operation in... |
xrsmulgzz 31287 | The "multiple" function in... |
xrstos 31288 | The extended real numbers ... |
xrsclat 31289 | The extended real numbers ... |
xrsp0 31290 | The poset 0 of the extende... |
xrsp1 31291 | The poset 1 of the extende... |
ressmulgnn 31292 | Values for the group multi... |
ressmulgnn0 31293 | Values for the group multi... |
xrge0base 31294 | The base of the extended n... |
xrge00 31295 | The zero of the extended n... |
xrge0plusg 31296 | The additive law of the ex... |
xrge0le 31297 | The "less than or equal to... |
xrge0mulgnn0 31298 | The group multiple functio... |
xrge0addass 31299 | Associativity of extended ... |
xrge0addgt0 31300 | The sum of nonnegative and... |
xrge0adddir 31301 | Right-distributivity of ex... |
xrge0adddi 31302 | Left-distributivity of ext... |
xrge0npcan 31303 | Extended nonnegative real ... |
fsumrp0cl 31304 | Closure of a finite sum of... |
abliso 31305 | The image of an Abelian gr... |
gsumsubg 31306 | The group sum in a subgrou... |
gsumsra 31307 | The group sum in a subring... |
gsummpt2co 31308 | Split a finite sum into a ... |
gsummpt2d 31309 | Express a finite sum over ... |
lmodvslmhm 31310 | Scalar multiplication in a... |
gsumvsmul1 31311 | Pull a scalar multiplicati... |
gsummptres 31312 | Extend a finite group sum ... |
gsummptres2 31313 | Extend a finite group sum ... |
gsumzresunsn 31314 | Append an element to a fin... |
gsumpart 31315 | Express a group sum as a d... |
gsumhashmul 31316 | Express a group sum by gro... |
xrge0tsmsd 31317 | Any finite or infinite sum... |
xrge0tsmsbi 31318 | Any limit of a finite or i... |
xrge0tsmseq 31319 | Any limit of a finite or i... |
cntzun 31320 | The centralizer of a union... |
cntzsnid 31321 | The centralizer of the ide... |
cntrcrng 31322 | The center of a ring is a ... |
isomnd 31327 | A (left) ordered monoid is... |
isogrp 31328 | A (left-)ordered group is ... |
ogrpgrp 31329 | A left-ordered group is a ... |
omndmnd 31330 | A left-ordered monoid is a... |
omndtos 31331 | A left-ordered monoid is a... |
omndadd 31332 | In an ordered monoid, the ... |
omndaddr 31333 | In a right ordered monoid,... |
omndadd2d 31334 | In a commutative left orde... |
omndadd2rd 31335 | In a left- and right- orde... |
submomnd 31336 | A submonoid of an ordered ... |
xrge0omnd 31337 | The nonnegative extended r... |
omndmul2 31338 | In an ordered monoid, the ... |
omndmul3 31339 | In an ordered monoid, the ... |
omndmul 31340 | In a commutative ordered m... |
ogrpinv0le 31341 | In an ordered group, the o... |
ogrpsub 31342 | In an ordered group, the o... |
ogrpaddlt 31343 | In an ordered group, stric... |
ogrpaddltbi 31344 | In a right ordered group, ... |
ogrpaddltrd 31345 | In a right ordered group, ... |
ogrpaddltrbid 31346 | In a right ordered group, ... |
ogrpsublt 31347 | In an ordered group, stric... |
ogrpinv0lt 31348 | In an ordered group, the o... |
ogrpinvlt 31349 | In an ordered group, the o... |
gsumle 31350 | A finite sum in an ordered... |
symgfcoeu 31351 | Uniqueness property of per... |
symgcom 31352 | Two permutations ` X ` and... |
symgcom2 31353 | Two permutations ` X ` and... |
symgcntz 31354 | All elements of a (finite)... |
odpmco 31355 | The composition of two odd... |
symgsubg 31356 | The value of the group sub... |
pmtrprfv2 31357 | In a transposition of two ... |
pmtrcnel 31358 | Composing a permutation ` ... |
pmtrcnel2 31359 | Variation on ~ pmtrcnel . ... |
pmtrcnelor 31360 | Composing a permutation ` ... |
pmtridf1o 31361 | Transpositions of ` X ` an... |
pmtridfv1 31362 | Value at X of the transpos... |
pmtridfv2 31363 | Value at Y of the transpos... |
psgnid 31364 | Permutation sign of the id... |
psgndmfi 31365 | For a finite base set, the... |
pmtrto1cl 31366 | Useful lemma for the follo... |
psgnfzto1stlem 31367 | Lemma for ~ psgnfzto1st . ... |
fzto1stfv1 31368 | Value of our permutation `... |
fzto1st1 31369 | Special case where the per... |
fzto1st 31370 | The function moving one el... |
fzto1stinvn 31371 | Value of the inverse of ou... |
psgnfzto1st 31372 | The permutation sign for m... |
tocycval 31375 | Value of the cycle builder... |
tocycfv 31376 | Function value of a permut... |
tocycfvres1 31377 | A cyclic permutation is a ... |
tocycfvres2 31378 | A cyclic permutation is th... |
cycpmfvlem 31379 | Lemma for ~ cycpmfv1 and ~... |
cycpmfv1 31380 | Value of a cycle function ... |
cycpmfv2 31381 | Value of a cycle function ... |
cycpmfv3 31382 | Values outside of the orbi... |
cycpmcl 31383 | Cyclic permutations are pe... |
tocycf 31384 | The permutation cycle buil... |
tocyc01 31385 | Permutation cycles built f... |
cycpm2tr 31386 | A cyclic permutation of 2 ... |
cycpm2cl 31387 | Closure for the 2-cycles. ... |
cyc2fv1 31388 | Function value of a 2-cycl... |
cyc2fv2 31389 | Function value of a 2-cycl... |
trsp2cyc 31390 | Exhibit the word a transpo... |
cycpmco2f1 31391 | The word U used in ~ cycpm... |
cycpmco2rn 31392 | The orbit of the compositi... |
cycpmco2lem1 31393 | Lemma for ~ cycpmco2 . (C... |
cycpmco2lem2 31394 | Lemma for ~ cycpmco2 . (C... |
cycpmco2lem3 31395 | Lemma for ~ cycpmco2 . (C... |
cycpmco2lem4 31396 | Lemma for ~ cycpmco2 . (C... |
cycpmco2lem5 31397 | Lemma for ~ cycpmco2 . (C... |
cycpmco2lem6 31398 | Lemma for ~ cycpmco2 . (C... |
cycpmco2lem7 31399 | Lemma for ~ cycpmco2 . (C... |
cycpmco2 31400 | The composition of a cycli... |
cyc2fvx 31401 | Function value of a 2-cycl... |
cycpm3cl 31402 | Closure of the 3-cycles in... |
cycpm3cl2 31403 | Closure of the 3-cycles in... |
cyc3fv1 31404 | Function value of a 3-cycl... |
cyc3fv2 31405 | Function value of a 3-cycl... |
cyc3fv3 31406 | Function value of a 3-cycl... |
cyc3co2 31407 | Represent a 3-cycle as a c... |
cycpmconjvlem 31408 | Lemma for ~ cycpmconjv . ... |
cycpmconjv 31409 | A formula for computing co... |
cycpmrn 31410 | The range of the word used... |
tocyccntz 31411 | All elements of a (finite)... |
evpmval 31412 | Value of the set of even p... |
cnmsgn0g 31413 | The neutral element of the... |
evpmsubg 31414 | The alternating group is a... |
evpmid 31415 | The identity is an even pe... |
altgnsg 31416 | The alternating group ` ( ... |
cyc3evpm 31417 | 3-Cycles are even permutat... |
cyc3genpmlem 31418 | Lemma for ~ cyc3genpm . (... |
cyc3genpm 31419 | The alternating group ` A ... |
cycpmgcl 31420 | Cyclic permutations are pe... |
cycpmconjslem1 31421 | Lemma for ~ cycpmconjs . ... |
cycpmconjslem2 31422 | Lemma for ~ cycpmconjs . ... |
cycpmconjs 31423 | All cycles of the same len... |
cyc3conja 31424 | All 3-cycles are conjugate... |
sgnsv 31427 | The sign mapping. (Contri... |
sgnsval 31428 | The sign value. (Contribu... |
sgnsf 31429 | The sign function. (Contr... |
inftmrel 31434 | The infinitesimal relation... |
isinftm 31435 | Express ` x ` is infinites... |
isarchi 31436 | Express the predicate " ` ... |
pnfinf 31437 | Plus infinity is an infini... |
xrnarchi 31438 | The completed real line is... |
isarchi2 31439 | Alternative way to express... |
submarchi 31440 | A submonoid is archimedean... |
isarchi3 31441 | This is the usual definiti... |
archirng 31442 | Property of Archimedean or... |
archirngz 31443 | Property of Archimedean le... |
archiexdiv 31444 | In an Archimedean group, g... |
archiabllem1a 31445 | Lemma for ~ archiabl : In... |
archiabllem1b 31446 | Lemma for ~ archiabl . (C... |
archiabllem1 31447 | Archimedean ordered groups... |
archiabllem2a 31448 | Lemma for ~ archiabl , whi... |
archiabllem2c 31449 | Lemma for ~ archiabl . (C... |
archiabllem2b 31450 | Lemma for ~ archiabl . (C... |
archiabllem2 31451 | Archimedean ordered groups... |
archiabl 31452 | Archimedean left- and righ... |
isslmd 31455 | The predicate "is a semimo... |
slmdlema 31456 | Lemma for properties of a ... |
lmodslmd 31457 | Left semimodules generaliz... |
slmdcmn 31458 | A semimodule is a commutat... |
slmdmnd 31459 | A semimodule is a monoid. ... |
slmdsrg 31460 | The scalar component of a ... |
slmdbn0 31461 | The base set of a semimodu... |
slmdacl 31462 | Closure of ring addition f... |
slmdmcl 31463 | Closure of ring multiplica... |
slmdsn0 31464 | The set of scalars in a se... |
slmdvacl 31465 | Closure of vector addition... |
slmdass 31466 | Semiring left module vecto... |
slmdvscl 31467 | Closure of scalar product ... |
slmdvsdi 31468 | Distributive law for scala... |
slmdvsdir 31469 | Distributive law for scala... |
slmdvsass 31470 | Associative law for scalar... |
slmd0cl 31471 | The ring zero in a semimod... |
slmd1cl 31472 | The ring unit in a semirin... |
slmdvs1 31473 | Scalar product with ring u... |
slmd0vcl 31474 | The zero vector is a vecto... |
slmd0vlid 31475 | Left identity law for the ... |
slmd0vrid 31476 | Right identity law for the... |
slmd0vs 31477 | Zero times a vector is the... |
slmdvs0 31478 | Anything times the zero ve... |
gsumvsca1 31479 | Scalar product of a finite... |
gsumvsca2 31480 | Scalar product of a finite... |
prmsimpcyc 31481 | A group of prime order is ... |
rngurd 31482 | Deduce the unit of a ring ... |
dvdschrmulg 31483 | In a ring, any multiple of... |
freshmansdream 31484 | For a prime number ` P ` ,... |
frobrhm 31485 | In a commutative ring with... |
ress1r 31486 | ` 1r ` is unaffected by re... |
dvrdir 31487 | Distributive law for the d... |
rdivmuldivd 31488 | Multiplication of two rati... |
ringinvval 31489 | The ring inverse expressed... |
dvrcan5 31490 | Cancellation law for commo... |
subrgchr 31491 | If ` A ` is a subring of `... |
rmfsupp2 31492 | A mapping of a multiplicat... |
primefldchr 31493 | The characteristic of a pr... |
isorng 31498 | An ordered ring is a ring ... |
orngring 31499 | An ordered ring is a ring.... |
orngogrp 31500 | An ordered ring is an orde... |
isofld 31501 | An ordered field is a fiel... |
orngmul 31502 | In an ordered ring, the or... |
orngsqr 31503 | In an ordered ring, all sq... |
ornglmulle 31504 | In an ordered ring, multip... |
orngrmulle 31505 | In an ordered ring, multip... |
ornglmullt 31506 | In an ordered ring, multip... |
orngrmullt 31507 | In an ordered ring, multip... |
orngmullt 31508 | In an ordered ring, the st... |
ofldfld 31509 | An ordered field is a fiel... |
ofldtos 31510 | An ordered field is a tota... |
orng0le1 31511 | In an ordered ring, the ri... |
ofldlt1 31512 | In an ordered field, the r... |
ofldchr 31513 | The characteristic of an o... |
suborng 31514 | Every subring of an ordere... |
subofld 31515 | Every subfield of an order... |
isarchiofld 31516 | Axiom of Archimedes : a ch... |
rhmdvdsr 31517 | A ring homomorphism preser... |
rhmopp 31518 | A ring homomorphism is als... |
elrhmunit 31519 | Ring homomorphisms preserv... |
rhmdvd 31520 | A ring homomorphism preser... |
rhmunitinv 31521 | Ring homomorphisms preserv... |
kerunit 31522 | If a unit element lies in ... |
reldmresv 31525 | The scalar restriction is ... |
resvval 31526 | Value of structure restric... |
resvid2 31527 | General behavior of trivia... |
resvval2 31528 | Value of nontrivial struct... |
resvsca 31529 | Base set of a structure re... |
resvlem 31530 | Other elements of a scalar... |
resvlemOLD 31531 | Obsolete version of ~ resv... |
resvbas 31532 | ` Base ` is unaffected by ... |
resvbasOLD 31533 | Obsolete proof of ~ resvba... |
resvplusg 31534 | ` +g ` is unaffected by sc... |
resvplusgOLD 31535 | Obsolete proof of ~ resvpl... |
resvvsca 31536 | ` .s ` is unaffected by sc... |
resvvscaOLD 31537 | Obsolete proof of ~ resvvs... |
resvmulr 31538 | ` .r ` is unaffected by sc... |
resvmulrOLD 31539 | Obsolete proof of ~ resvmu... |
resv0g 31540 | ` 0g ` is unaffected by sc... |
resv1r 31541 | ` 1r ` is unaffected by sc... |
resvcmn 31542 | Scalar restriction preserv... |
gzcrng 31543 | The gaussian integers form... |
reofld 31544 | The real numbers form an o... |
nn0omnd 31545 | The nonnegative integers f... |
rearchi 31546 | The field of the real numb... |
nn0archi 31547 | The monoid of the nonnegat... |
xrge0slmod 31548 | The extended nonnegative r... |
qusker 31549 | The kernel of a quotient m... |
eqgvscpbl 31550 | The left coset equivalence... |
qusvscpbl 31551 | The quotient map distribut... |
qusscaval 31552 | Value of the scalar multip... |
imaslmod 31553 | The image structure of a l... |
quslmod 31554 | If ` G ` is a submodule in... |
quslmhm 31555 | If ` G ` is a submodule of... |
ecxpid 31556 | The equivalence class of a... |
eqg0el 31557 | Equivalence class of a quo... |
qsxpid 31558 | The quotient set of a cart... |
qusxpid 31559 | The Group quotient equival... |
qustriv 31560 | The quotient of a group ` ... |
qustrivr 31561 | Converse of ~ qustriv . (... |
znfermltl 31562 | Fermat's little theorem in... |
islinds5 31563 | A set is linearly independ... |
ellspds 31564 | Variation on ~ ellspd . (... |
0ellsp 31565 | Zero is in all spans. (Co... |
0nellinds 31566 | The group identity cannot ... |
rspsnel 31567 | Membership in a principal ... |
rspsnid 31568 | A principal ideal contains... |
elrsp 31569 | Write the elements of a ri... |
rspidlid 31570 | The ideal span of an ideal... |
pidlnz 31571 | A principal ideal generate... |
lbslsp 31572 | Any element of a left modu... |
lindssn 31573 | Any singleton of a nonzero... |
lindflbs 31574 | Conditions for an independ... |
linds2eq 31575 | Deduce equality of element... |
lindfpropd 31576 | Property deduction for lin... |
lindspropd 31577 | Property deduction for lin... |
elgrplsmsn 31578 | Membership in a sumset wit... |
lsmsnorb 31579 | The sumset of a group with... |
lsmsnorb2 31580 | The sumset of a single ele... |
elringlsm 31581 | Membership in a product of... |
elringlsmd 31582 | Membership in a product of... |
ringlsmss 31583 | Closure of the product of ... |
ringlsmss1 31584 | The product of an ideal ` ... |
ringlsmss2 31585 | The product with an ideal ... |
lsmsnpridl 31586 | The product of the ring wi... |
lsmsnidl 31587 | The product of the ring wi... |
lsmidllsp 31588 | The sum of two ideals is t... |
lsmidl 31589 | The sum of two ideals is a... |
lsmssass 31590 | Group sum is associative, ... |
grplsm0l 31591 | Sumset with the identity s... |
grplsmid 31592 | The direct sum of an eleme... |
quslsm 31593 | Express the image by the q... |
qusima 31594 | The image of a subgroup by... |
nsgqus0 31595 | A normal subgroup ` N ` is... |
nsgmgclem 31596 | Lemma for ~ nsgmgc . (Con... |
nsgmgc 31597 | There is a monotone Galois... |
nsgqusf1olem1 31598 | Lemma for ~ nsgqusf1o . (... |
nsgqusf1olem2 31599 | Lemma for ~ nsgqusf1o . (... |
nsgqusf1olem3 31600 | Lemma for ~ nsgqusf1o . (... |
nsgqusf1o 31601 | The canonical projection h... |
intlidl 31602 | The intersection of a none... |
rhmpreimaidl 31603 | The preimage of an ideal b... |
kerlidl 31604 | The kernel of a ring homom... |
0ringidl 31605 | The zero ideal is the only... |
elrspunidl 31606 | Elementhood to the span of... |
lidlincl 31607 | Ideals are closed under in... |
idlinsubrg 31608 | The intersection between a... |
rhmimaidl 31609 | The image of an ideal ` I ... |
prmidlval 31612 | The class of prime ideals ... |
isprmidl 31613 | The predicate "is a prime ... |
prmidlnr 31614 | A prime ideal is a proper ... |
prmidl 31615 | The main property of a pri... |
prmidl2 31616 | A condition that shows an ... |
idlmulssprm 31617 | Let ` P ` be a prime ideal... |
pridln1 31618 | A proper ideal cannot cont... |
prmidlidl 31619 | A prime ideal is an ideal.... |
prmidlssidl 31620 | Prime ideals as a subset o... |
lidlnsg 31621 | An ideal is a normal subgr... |
cringm4 31622 | Commutative/associative la... |
isprmidlc 31623 | The predicate "is prime id... |
prmidlc 31624 | Property of a prime ideal ... |
0ringprmidl 31625 | The trivial ring does not ... |
prmidl0 31626 | The zero ideal of a commut... |
rhmpreimaprmidl 31627 | The preimage of a prime id... |
qsidomlem1 31628 | If the quotient ring of a ... |
qsidomlem2 31629 | A quotient by a prime idea... |
qsidom 31630 | An ideal ` I ` in the comm... |
mxidlval 31633 | The set of maximal ideals ... |
ismxidl 31634 | The predicate "is a maxima... |
mxidlidl 31635 | A maximal ideal is an idea... |
mxidlnr 31636 | A maximal ideal is proper.... |
mxidlmax 31637 | A maximal ideal is a maxim... |
mxidln1 31638 | One is not contained in an... |
mxidlnzr 31639 | A ring with a maximal idea... |
mxidlprm 31640 | Every maximal ideal is pri... |
ssmxidllem 31641 | The set ` P ` used in the ... |
ssmxidl 31642 | Let ` R ` be a ring, and l... |
krull 31643 | Krull's theorem: Any nonz... |
mxidlnzrb 31644 | A ring is nonzero if and o... |
idlsrgstr 31647 | A constructed semiring of ... |
idlsrgval 31648 | Lemma for ~ idlsrgbas thro... |
idlsrgbas 31649 | Baae of the ideals of a ri... |
idlsrgplusg 31650 | Additive operation of the ... |
idlsrg0g 31651 | The zero ideal is the addi... |
idlsrgmulr 31652 | Multiplicative operation o... |
idlsrgtset 31653 | Topology component of the ... |
idlsrgmulrval 31654 | Value of the ring multipli... |
idlsrgmulrcl 31655 | Ideals of a ring ` R ` are... |
idlsrgmulrss1 31656 | In a commutative ring, the... |
idlsrgmulrss2 31657 | The product of two ideals ... |
idlsrgmulrssin 31658 | In a commutative ring, the... |
idlsrgmnd 31659 | The ideals of a ring form ... |
idlsrgcmnd 31660 | The ideals of a ring form ... |
isufd 31663 | The property of being a Un... |
rprmval 31664 | The prime elements of a ri... |
isrprm 31665 | Property for ` P ` to be a... |
asclmulg 31666 | Apply group multiplication... |
fply1 31667 | Conditions for a function ... |
ply1scleq 31668 | Equality of a constant pol... |
ply1chr 31669 | The characteristic of a po... |
ply1fermltl 31670 | Fermat's little theorem fo... |
sra1r 31671 | The multiplicative neutral... |
sraring 31672 | Condition for a subring al... |
sradrng 31673 | Condition for a subring al... |
srasubrg 31674 | A subring of the original ... |
sralvec 31675 | Given a sub division ring ... |
srafldlvec 31676 | Given a subfield ` F ` of ... |
drgext0g 31677 | The additive neutral eleme... |
drgextvsca 31678 | The scalar multiplication ... |
drgext0gsca 31679 | The additive neutral eleme... |
drgextsubrg 31680 | The scalar field is a subr... |
drgextlsp 31681 | The scalar field is a subs... |
drgextgsum 31682 | Group sum in a division ri... |
lvecdimfi 31683 | Finite version of ~ lvecdi... |
dimval 31686 | The dimension of a vector ... |
dimvalfi 31687 | The dimension of a vector ... |
dimcl 31688 | Closure of the vector spac... |
lvecdim0i 31689 | A vector space of dimensio... |
lvecdim0 31690 | A vector space of dimensio... |
lssdimle 31691 | The dimension of a linear ... |
dimpropd 31692 | If two structures have the... |
rgmoddim 31693 | The left vector space indu... |
frlmdim 31694 | Dimension of a free left m... |
tnglvec 31695 | Augmenting a structure wit... |
tngdim 31696 | Dimension of a left vector... |
rrxdim 31697 | Dimension of the generaliz... |
matdim 31698 | Dimension of the space of ... |
lbslsat 31699 | A nonzero vector ` X ` is ... |
lsatdim 31700 | A line, spanned by a nonze... |
drngdimgt0 31701 | The dimension of a vector ... |
lmhmlvec2 31702 | A homomorphism of left vec... |
kerlmhm 31703 | The kernel of a vector spa... |
imlmhm 31704 | The image of a vector spac... |
lindsunlem 31705 | Lemma for ~ lindsun . (Co... |
lindsun 31706 | Condition for the union of... |
lbsdiflsp0 31707 | The linear spans of two di... |
dimkerim 31708 | Given a linear map ` F ` b... |
qusdimsum 31709 | Let ` W ` be a vector spac... |
fedgmullem1 31710 | Lemma for ~ fedgmul . (Co... |
fedgmullem2 31711 | Lemma for ~ fedgmul . (Co... |
fedgmul 31712 | The multiplicativity formu... |
relfldext 31721 | The field extension is a r... |
brfldext 31722 | The field extension relati... |
ccfldextrr 31723 | The field of the complex n... |
fldextfld1 31724 | A field extension is only ... |
fldextfld2 31725 | A field extension is only ... |
fldextsubrg 31726 | Field extension implies a ... |
fldextress 31727 | Field extension implies a ... |
brfinext 31728 | The finite field extension... |
extdgval 31729 | Value of the field extensi... |
fldextsralvec 31730 | The subring algebra associ... |
extdgcl 31731 | Closure of the field exten... |
extdggt0 31732 | Degrees of field extension... |
fldexttr 31733 | Field extension is a trans... |
fldextid 31734 | The field extension relati... |
extdgid 31735 | A trivial field extension ... |
extdgmul 31736 | The multiplicativity formu... |
finexttrb 31737 | The extension ` E ` of ` K... |
extdg1id 31738 | If the degree of the exten... |
extdg1b 31739 | The degree of the extensio... |
fldextchr 31740 | The characteristic of a su... |
ccfldsrarelvec 31741 | The subring algebra of the... |
ccfldextdgrr 31742 | The degree of the field ex... |
smatfval 31745 | Value of the submatrix. (... |
smatrcl 31746 | Closure of the rectangular... |
smatlem 31747 | Lemma for the next theorem... |
smattl 31748 | Entries of a submatrix, to... |
smattr 31749 | Entries of a submatrix, to... |
smatbl 31750 | Entries of a submatrix, bo... |
smatbr 31751 | Entries of a submatrix, bo... |
smatcl 31752 | Closure of the square subm... |
matmpo 31753 | Write a square matrix as a... |
1smat1 31754 | The submatrix of the ident... |
submat1n 31755 | One case where the submatr... |
submatres 31756 | Special case where the sub... |
submateqlem1 31757 | Lemma for ~ submateq . (C... |
submateqlem2 31758 | Lemma for ~ submateq . (C... |
submateq 31759 | Sufficient condition for t... |
submatminr1 31760 | If we take a submatrix by ... |
lmatval 31763 | Value of the literal matri... |
lmatfval 31764 | Entries of a literal matri... |
lmatfvlem 31765 | Useful lemma to extract li... |
lmatcl 31766 | Closure of the literal mat... |
lmat22lem 31767 | Lemma for ~ lmat22e11 and ... |
lmat22e11 31768 | Entry of a 2x2 literal mat... |
lmat22e12 31769 | Entry of a 2x2 literal mat... |
lmat22e21 31770 | Entry of a 2x2 literal mat... |
lmat22e22 31771 | Entry of a 2x2 literal mat... |
lmat22det 31772 | The determinant of a liter... |
mdetpmtr1 31773 | The determinant of a matri... |
mdetpmtr2 31774 | The determinant of a matri... |
mdetpmtr12 31775 | The determinant of a matri... |
mdetlap1 31776 | A Laplace expansion of the... |
madjusmdetlem1 31777 | Lemma for ~ madjusmdet . ... |
madjusmdetlem2 31778 | Lemma for ~ madjusmdet . ... |
madjusmdetlem3 31779 | Lemma for ~ madjusmdet . ... |
madjusmdetlem4 31780 | Lemma for ~ madjusmdet . ... |
madjusmdet 31781 | Express the cofactor of th... |
mdetlap 31782 | Laplace expansion of the d... |
ist0cld 31783 | The predicate "is a T_0 sp... |
txomap 31784 | Given two open maps ` F ` ... |
qtopt1 31785 | If every equivalence class... |
qtophaus 31786 | If an open map's graph in ... |
circtopn 31787 | The topology of the unit c... |
circcn 31788 | The function gluing the re... |
reff 31789 | For any cover refinement, ... |
locfinreflem 31790 | A locally finite refinemen... |
locfinref 31791 | A locally finite refinemen... |
iscref 31794 | The property that every op... |
crefeq 31795 | Equality theorem for the "... |
creftop 31796 | A space where every open c... |
crefi 31797 | The property that every op... |
crefdf 31798 | A formulation of ~ crefi e... |
crefss 31799 | The "every open cover has ... |
cmpcref 31800 | Equivalent definition of c... |
cmpfiref 31801 | Every open cover of a Comp... |
ldlfcntref 31804 | Every open cover of a Lind... |
ispcmp 31807 | The predicate "is a paraco... |
cmppcmp 31808 | Every compact space is par... |
dispcmp 31809 | Every discrete space is pa... |
pcmplfin 31810 | Given a paracompact topolo... |
pcmplfinf 31811 | Given a paracompact topolo... |
rspecval 31814 | Value of the spectrum of t... |
rspecbas 31815 | The prime ideals form the ... |
rspectset 31816 | Topology component of the ... |
rspectopn 31817 | The topology component of ... |
zarcls0 31818 | The closure of the identit... |
zarcls1 31819 | The unit ideal ` B ` is th... |
zarclsun 31820 | The union of two closed se... |
zarclsiin 31821 | In a Zariski topology, the... |
zarclsint 31822 | The intersection of a fami... |
zarclssn 31823 | The closed points of Zaris... |
zarcls 31824 | The open sets of the Zaris... |
zartopn 31825 | The Zariski topology is a ... |
zartop 31826 | The Zariski topology is a ... |
zartopon 31827 | The points of the Zariski ... |
zar0ring 31828 | The Zariski Topology of th... |
zart0 31829 | The Zariski topology is T_... |
zarmxt1 31830 | The Zariski topology restr... |
zarcmplem 31831 | Lemma for ~ zarcmp . (Con... |
zarcmp 31832 | The Zariski topology is co... |
rspectps 31833 | The spectrum of a ring ` R... |
rhmpreimacnlem 31834 | Lemma for ~ rhmpreimacn . ... |
rhmpreimacn 31835 | The function mapping a pri... |
metidval 31840 | Value of the metric identi... |
metidss 31841 | As a relation, the metric ... |
metidv 31842 | ` A ` and ` B ` identify b... |
metideq 31843 | Basic property of the metr... |
metider 31844 | The metric identification ... |
pstmval 31845 | Value of the metric induce... |
pstmfval 31846 | Function value of the metr... |
pstmxmet 31847 | The metric induced by a ps... |
hauseqcn 31848 | In a Hausdorff topology, t... |
elunitge0 31849 | An element of the closed u... |
unitssxrge0 31850 | The closed unit interval i... |
unitdivcld 31851 | Necessary conditions for a... |
iistmd 31852 | The closed unit interval f... |
unicls 31853 | The union of the closed se... |
tpr2tp 31854 | The usual topology on ` ( ... |
tpr2uni 31855 | The usual topology on ` ( ... |
xpinpreima 31856 | Rewrite the cartesian prod... |
xpinpreima2 31857 | Rewrite the cartesian prod... |
sqsscirc1 31858 | The complex square of side... |
sqsscirc2 31859 | The complex square of side... |
cnre2csqlem 31860 | Lemma for ~ cnre2csqima . ... |
cnre2csqima 31861 | Image of a centered square... |
tpr2rico 31862 | For any point of an open s... |
cnvordtrestixx 31863 | The restriction of the 'gr... |
prsdm 31864 | Domain of the relation of ... |
prsrn 31865 | Range of the relation of a... |
prsss 31866 | Relation of a subproset. ... |
prsssdm 31867 | Domain of a subproset rela... |
ordtprsval 31868 | Value of the order topolog... |
ordtprsuni 31869 | Value of the order topolog... |
ordtcnvNEW 31870 | The order dual generates t... |
ordtrestNEW 31871 | The subspace topology of a... |
ordtrest2NEWlem 31872 | Lemma for ~ ordtrest2NEW .... |
ordtrest2NEW 31873 | An interval-closed set ` A... |
ordtconnlem1 31874 | Connectedness in the order... |
ordtconn 31875 | Connectedness in the order... |
mndpluscn 31876 | A mapping that is both a h... |
mhmhmeotmd 31877 | Deduce a Topological Monoi... |
rmulccn 31878 | Multiplication by a real c... |
raddcn 31879 | Addition in the real numbe... |
xrmulc1cn 31880 | The operation multiplying ... |
fmcncfil 31881 | The image of a Cauchy filt... |
xrge0hmph 31882 | The extended nonnegative r... |
xrge0iifcnv 31883 | Define a bijection from ` ... |
xrge0iifcv 31884 | The defined function's val... |
xrge0iifiso 31885 | The defined bijection from... |
xrge0iifhmeo 31886 | Expose a homeomorphism fro... |
xrge0iifhom 31887 | The defined function from ... |
xrge0iif1 31888 | Condition for the defined ... |
xrge0iifmhm 31889 | The defined function from ... |
xrge0pluscn 31890 | The addition operation of ... |
xrge0mulc1cn 31891 | The operation multiplying ... |
xrge0tps 31892 | The extended nonnegative r... |
xrge0topn 31893 | The topology of the extend... |
xrge0haus 31894 | The topology of the extend... |
xrge0tmd 31895 | The extended nonnegative r... |
xrge0tmdALT 31896 | Alternate proof of ~ xrge0... |
lmlim 31897 | Relate a limit in a given ... |
lmlimxrge0 31898 | Relate a limit in the nonn... |
rge0scvg 31899 | Implication of convergence... |
fsumcvg4 31900 | A serie with finite suppor... |
pnfneige0 31901 | A neighborhood of ` +oo ` ... |
lmxrge0 31902 | Express "sequence ` F ` co... |
lmdvg 31903 | If a monotonic sequence of... |
lmdvglim 31904 | If a monotonic real number... |
pl1cn 31905 | A univariate polynomial is... |
zringnm 31908 | The norm (function) for a ... |
zzsnm 31909 | The norm of the ring of th... |
zlm0 31910 | Zero of a ` ZZ ` -module. ... |
zlm1 31911 | Unit of a ` ZZ ` -module (... |
zlmds 31912 | Distance in a ` ZZ ` -modu... |
zlmdsOLD 31913 | Obsolete proof of ~ zlmds ... |
zlmtset 31914 | Topology in a ` ZZ ` -modu... |
zlmtsetOLD 31915 | Obsolete proof of ~ zlmtse... |
zlmnm 31916 | Norm of a ` ZZ ` -module (... |
zhmnrg 31917 | The ` ZZ ` -module built f... |
nmmulg 31918 | The norm of a group produc... |
zrhnm 31919 | The norm of the image by `... |
cnzh 31920 | The ` ZZ ` -module of ` CC... |
rezh 31921 | The ` ZZ ` -module of ` RR... |
qqhval 31924 | Value of the canonical hom... |
zrhf1ker 31925 | The kernel of the homomorp... |
zrhchr 31926 | The kernel of the homomorp... |
zrhker 31927 | The kernel of the homomorp... |
zrhunitpreima 31928 | The preimage by ` ZRHom ` ... |
elzrhunit 31929 | Condition for the image by... |
elzdif0 31930 | Lemma for ~ qqhval2 . (Co... |
qqhval2lem 31931 | Lemma for ~ qqhval2 . (Co... |
qqhval2 31932 | Value of the canonical hom... |
qqhvval 31933 | Value of the canonical hom... |
qqh0 31934 | The image of ` 0 ` by the ... |
qqh1 31935 | The image of ` 1 ` by the ... |
qqhf 31936 | ` QQHom ` as a function. ... |
qqhvq 31937 | The image of a quotient by... |
qqhghm 31938 | The ` QQHom ` homomorphism... |
qqhrhm 31939 | The ` QQHom ` homomorphism... |
qqhnm 31940 | The norm of the image by `... |
qqhcn 31941 | The ` QQHom ` homomorphism... |
qqhucn 31942 | The ` QQHom ` homomorphism... |
rrhval 31946 | Value of the canonical hom... |
rrhcn 31947 | If the topology of ` R ` i... |
rrhf 31948 | If the topology of ` R ` i... |
isrrext 31950 | Express the property " ` R... |
rrextnrg 31951 | An extension of ` RR ` is ... |
rrextdrg 31952 | An extension of ` RR ` is ... |
rrextnlm 31953 | The norm of an extension o... |
rrextchr 31954 | The ring characteristic of... |
rrextcusp 31955 | An extension of ` RR ` is ... |
rrexttps 31956 | An extension of ` RR ` is ... |
rrexthaus 31957 | The topology of an extensi... |
rrextust 31958 | The uniformity of an exten... |
rerrext 31959 | The field of the real numb... |
cnrrext 31960 | The field of the complex n... |
qqtopn 31961 | The topology of the field ... |
rrhfe 31962 | If ` R ` is an extension o... |
rrhcne 31963 | If ` R ` is an extension o... |
rrhqima 31964 | The ` RRHom ` homomorphism... |
rrh0 31965 | The image of ` 0 ` by the ... |
xrhval 31968 | The value of the embedding... |
zrhre 31969 | The ` ZRHom ` homomorphism... |
qqhre 31970 | The ` QQHom ` homomorphism... |
rrhre 31971 | The ` RRHom ` homomorphism... |
relmntop 31974 | Manifold is a relation. (... |
ismntoplly 31975 | Property of being a manifo... |
ismntop 31976 | Property of being a manifo... |
nexple 31977 | A lower bound for an expon... |
indv 31980 | Value of the indicator fun... |
indval 31981 | Value of the indicator fun... |
indval2 31982 | Alternate value of the ind... |
indf 31983 | An indicator function as a... |
indfval 31984 | Value of the indicator fun... |
ind1 31985 | Value of the indicator fun... |
ind0 31986 | Value of the indicator fun... |
ind1a 31987 | Value of the indicator fun... |
indpi1 31988 | Preimage of the singleton ... |
indsum 31989 | Finite sum of a product wi... |
indsumin 31990 | Finite sum of a product wi... |
prodindf 31991 | The product of indicators ... |
indf1o 31992 | The bijection between a po... |
indpreima 31993 | A function with range ` { ... |
indf1ofs 31994 | The bijection between fini... |
esumex 31997 | An extended sum is a set b... |
esumcl 31998 | Closure for extended sum i... |
esumeq12dvaf 31999 | Equality deduction for ext... |
esumeq12dva 32000 | Equality deduction for ext... |
esumeq12d 32001 | Equality deduction for ext... |
esumeq1 32002 | Equality theorem for an ex... |
esumeq1d 32003 | Equality theorem for an ex... |
esumeq2 32004 | Equality theorem for exten... |
esumeq2d 32005 | Equality deduction for ext... |
esumeq2dv 32006 | Equality deduction for ext... |
esumeq2sdv 32007 | Equality deduction for ext... |
nfesum1 32008 | Bound-variable hypothesis ... |
nfesum2 32009 | Bound-variable hypothesis ... |
cbvesum 32010 | Change bound variable in a... |
cbvesumv 32011 | Change bound variable in a... |
esumid 32012 | Identify the extended sum ... |
esumgsum 32013 | A finite extended sum is t... |
esumval 32014 | Develop the value of the e... |
esumel 32015 | The extended sum is a limi... |
esumnul 32016 | Extended sum over the empt... |
esum0 32017 | Extended sum of zero. (Co... |
esumf1o 32018 | Re-index an extended sum u... |
esumc 32019 | Convert from the collectio... |
esumrnmpt 32020 | Rewrite an extended sum in... |
esumsplit 32021 | Split an extended sum into... |
esummono 32022 | Extended sum is monotonic.... |
esumpad 32023 | Extend an extended sum by ... |
esumpad2 32024 | Remove zeroes from an exte... |
esumadd 32025 | Addition of infinite sums.... |
esumle 32026 | If all of the terms of an ... |
gsumesum 32027 | Relate a group sum on ` ( ... |
esumlub 32028 | The extended sum is the lo... |
esumaddf 32029 | Addition of infinite sums.... |
esumlef 32030 | If all of the terms of an ... |
esumcst 32031 | The extended sum of a cons... |
esumsnf 32032 | The extended sum of a sing... |
esumsn 32033 | The extended sum of a sing... |
esumpr 32034 | Extended sum over a pair. ... |
esumpr2 32035 | Extended sum over a pair, ... |
esumrnmpt2 32036 | Rewrite an extended sum in... |
esumfzf 32037 | Formulating a partial exte... |
esumfsup 32038 | Formulating an extended su... |
esumfsupre 32039 | Formulating an extended su... |
esumss 32040 | Change the index set to a ... |
esumpinfval 32041 | The value of the extended ... |
esumpfinvallem 32042 | Lemma for ~ esumpfinval . ... |
esumpfinval 32043 | The value of the extended ... |
esumpfinvalf 32044 | Same as ~ esumpfinval , mi... |
esumpinfsum 32045 | The value of the extended ... |
esumpcvgval 32046 | The value of the extended ... |
esumpmono 32047 | The partial sums in an ext... |
esumcocn 32048 | Lemma for ~ esummulc2 and ... |
esummulc1 32049 | An extended sum multiplied... |
esummulc2 32050 | An extended sum multiplied... |
esumdivc 32051 | An extended sum divided by... |
hashf2 32052 | Lemma for ~ hasheuni . (C... |
hasheuni 32053 | The cardinality of a disjo... |
esumcvg 32054 | The sequence of partial su... |
esumcvg2 32055 | Simpler version of ~ esumc... |
esumcvgsum 32056 | The value of the extended ... |
esumsup 32057 | Express an extended sum as... |
esumgect 32058 | "Send ` n ` to ` +oo ` " i... |
esumcvgre 32059 | All terms of a converging ... |
esum2dlem 32060 | Lemma for ~ esum2d (finite... |
esum2d 32061 | Write a double extended su... |
esumiun 32062 | Sum over a nonnecessarily ... |
ofceq 32065 | Equality theorem for funct... |
ofcfval 32066 | Value of an operation appl... |
ofcval 32067 | Evaluate a function/consta... |
ofcfn 32068 | The function operation pro... |
ofcfeqd2 32069 | Equality theorem for funct... |
ofcfval3 32070 | General value of ` ( F oFC... |
ofcf 32071 | The function/constant oper... |
ofcfval2 32072 | The function operation exp... |
ofcfval4 32073 | The function/constant oper... |
ofcc 32074 | Left operation by a consta... |
ofcof 32075 | Relate function operation ... |
sigaex 32078 | Lemma for ~ issiga and ~ i... |
sigaval 32079 | The set of sigma-algebra w... |
issiga 32080 | An alternative definition ... |
isrnsiga 32081 | The property of being a si... |
0elsiga 32082 | A sigma-algebra contains t... |
baselsiga 32083 | A sigma-algebra contains i... |
sigasspw 32084 | A sigma-algebra is a set o... |
sigaclcu 32085 | A sigma-algebra is closed ... |
sigaclcuni 32086 | A sigma-algebra is closed ... |
sigaclfu 32087 | A sigma-algebra is closed ... |
sigaclcu2 32088 | A sigma-algebra is closed ... |
sigaclfu2 32089 | A sigma-algebra is closed ... |
sigaclcu3 32090 | A sigma-algebra is closed ... |
issgon 32091 | Property of being a sigma-... |
sgon 32092 | A sigma-algebra is a sigma... |
elsigass 32093 | An element of a sigma-alge... |
elrnsiga 32094 | Dropping the base informat... |
isrnsigau 32095 | The property of being a si... |
unielsiga 32096 | A sigma-algebra contains i... |
dmvlsiga 32097 | Lebesgue-measurable subset... |
pwsiga 32098 | Any power set forms a sigm... |
prsiga 32099 | The smallest possible sigm... |
sigaclci 32100 | A sigma-algebra is closed ... |
difelsiga 32101 | A sigma-algebra is closed ... |
unelsiga 32102 | A sigma-algebra is closed ... |
inelsiga 32103 | A sigma-algebra is closed ... |
sigainb 32104 | Building a sigma-algebra f... |
insiga 32105 | The intersection of a coll... |
sigagenval 32108 | Value of the generated sig... |
sigagensiga 32109 | A generated sigma-algebra ... |
sgsiga 32110 | A generated sigma-algebra ... |
unisg 32111 | The sigma-algebra generate... |
dmsigagen 32112 | A sigma-algebra can be gen... |
sssigagen 32113 | A set is a subset of the s... |
sssigagen2 32114 | A subset of the generating... |
elsigagen 32115 | Any element of a set is al... |
elsigagen2 32116 | Any countable union of ele... |
sigagenss 32117 | The generated sigma-algebr... |
sigagenss2 32118 | Sufficient condition for i... |
sigagenid 32119 | The sigma-algebra generate... |
ispisys 32120 | The property of being a pi... |
ispisys2 32121 | The property of being a pi... |
inelpisys 32122 | Pi-systems are closed unde... |
sigapisys 32123 | All sigma-algebras are pi-... |
isldsys 32124 | The property of being a la... |
pwldsys 32125 | The power set of the unive... |
unelldsys 32126 | Lambda-systems are closed ... |
sigaldsys 32127 | All sigma-algebras are lam... |
ldsysgenld 32128 | The intersection of all la... |
sigapildsyslem 32129 | Lemma for ~ sigapildsys . ... |
sigapildsys 32130 | Sigma-algebra are exactly ... |
ldgenpisyslem1 32131 | Lemma for ~ ldgenpisys . ... |
ldgenpisyslem2 32132 | Lemma for ~ ldgenpisys . ... |
ldgenpisyslem3 32133 | Lemma for ~ ldgenpisys . ... |
ldgenpisys 32134 | The lambda system ` E ` ge... |
dynkin 32135 | Dynkin's lambda-pi theorem... |
isros 32136 | The property of being a ri... |
rossspw 32137 | A ring of sets is a collec... |
0elros 32138 | A ring of sets contains th... |
unelros 32139 | A ring of sets is closed u... |
difelros 32140 | A ring of sets is closed u... |
inelros 32141 | A ring of sets is closed u... |
fiunelros 32142 | A ring of sets is closed u... |
issros 32143 | The property of being a se... |
srossspw 32144 | A semiring of sets is a co... |
0elsros 32145 | A semiring of sets contain... |
inelsros 32146 | A semiring of sets is clos... |
diffiunisros 32147 | In semiring of sets, compl... |
rossros 32148 | Rings of sets are semiring... |
brsiga 32151 | The Borel Algebra on real ... |
brsigarn 32152 | The Borel Algebra is a sig... |
brsigasspwrn 32153 | The Borel Algebra is a set... |
unibrsiga 32154 | The union of the Borel Alg... |
cldssbrsiga 32155 | A Borel Algebra contains a... |
sxval 32158 | Value of the product sigma... |
sxsiga 32159 | A product sigma-algebra is... |
sxsigon 32160 | A product sigma-algebra is... |
sxuni 32161 | The base set of a product ... |
elsx 32162 | The cartesian product of t... |
measbase 32165 | The base set of a measure ... |
measval 32166 | The value of the ` measure... |
ismeas 32167 | The property of being a me... |
isrnmeas 32168 | The property of being a me... |
dmmeas 32169 | The domain of a measure is... |
measbasedom 32170 | The base set of a measure ... |
measfrge0 32171 | A measure is a function ov... |
measfn 32172 | A measure is a function on... |
measvxrge0 32173 | The values of a measure ar... |
measvnul 32174 | The measure of the empty s... |
measge0 32175 | A measure is nonnegative. ... |
measle0 32176 | If the measure of a given ... |
measvun 32177 | The measure of a countable... |
measxun2 32178 | The measure the union of t... |
measun 32179 | The measure the union of t... |
measvunilem 32180 | Lemma for ~ measvuni . (C... |
measvunilem0 32181 | Lemma for ~ measvuni . (C... |
measvuni 32182 | The measure of a countable... |
measssd 32183 | A measure is monotone with... |
measunl 32184 | A measure is sub-additive ... |
measiuns 32185 | The measure of the union o... |
measiun 32186 | A measure is sub-additive.... |
meascnbl 32187 | A measure is continuous fr... |
measinblem 32188 | Lemma for ~ measinb . (Co... |
measinb 32189 | Building a measure restric... |
measres 32190 | Building a measure restric... |
measinb2 32191 | Building a measure restric... |
measdivcst 32192 | Division of a measure by a... |
measdivcstALTV 32193 | Alternate version of ~ mea... |
cntmeas 32194 | The Counting measure is a ... |
pwcntmeas 32195 | The counting measure is a ... |
cntnevol 32196 | Counting and Lebesgue meas... |
voliune 32197 | The Lebesgue measure funct... |
volfiniune 32198 | The Lebesgue measure funct... |
volmeas 32199 | The Lebesgue measure is a ... |
ddeval1 32202 | Value of the delta measure... |
ddeval0 32203 | Value of the delta measure... |
ddemeas 32204 | The Dirac delta measure is... |
relae 32208 | 'almost everywhere' is a r... |
brae 32209 | 'almost everywhere' relati... |
braew 32210 | 'almost everywhere' relati... |
truae 32211 | A truth holds almost every... |
aean 32212 | A conjunction holds almost... |
faeval 32214 | Value of the 'almost every... |
relfae 32215 | The 'almost everywhere' bu... |
brfae 32216 | 'almost everywhere' relati... |
ismbfm 32219 | The predicate " ` F ` is a... |
elunirnmbfm 32220 | The property of being a me... |
mbfmfun 32221 | A measurable function is a... |
mbfmf 32222 | A measurable function as a... |
isanmbfm 32223 | The predicate to be a meas... |
mbfmcnvima 32224 | The preimage by a measurab... |
mbfmbfm 32225 | A measurable function to a... |
mbfmcst 32226 | A constant function is mea... |
1stmbfm 32227 | The first projection map i... |
2ndmbfm 32228 | The second projection map ... |
imambfm 32229 | If the sigma-algebra in th... |
cnmbfm 32230 | A continuous function is m... |
mbfmco 32231 | The composition of two mea... |
mbfmco2 32232 | The pair building of two m... |
mbfmvolf 32233 | Measurable functions with ... |
elmbfmvol2 32234 | Measurable functions with ... |
mbfmcnt 32235 | All functions are measurab... |
br2base 32236 | The base set for the gener... |
dya2ub 32237 | An upper bound for a dyadi... |
sxbrsigalem0 32238 | The closed half-spaces of ... |
sxbrsigalem3 32239 | The sigma-algebra generate... |
dya2iocival 32240 | The function ` I ` returns... |
dya2iocress 32241 | Dyadic intervals are subse... |
dya2iocbrsiga 32242 | Dyadic intervals are Borel... |
dya2icobrsiga 32243 | Dyadic intervals are Borel... |
dya2icoseg 32244 | For any point and any clos... |
dya2icoseg2 32245 | For any point and any open... |
dya2iocrfn 32246 | The function returning dya... |
dya2iocct 32247 | The dyadic rectangle set i... |
dya2iocnrect 32248 | For any point of an open r... |
dya2iocnei 32249 | For any point of an open s... |
dya2iocuni 32250 | Every open set of ` ( RR X... |
dya2iocucvr 32251 | The dyadic rectangular set... |
sxbrsigalem1 32252 | The Borel algebra on ` ( R... |
sxbrsigalem2 32253 | The sigma-algebra generate... |
sxbrsigalem4 32254 | The Borel algebra on ` ( R... |
sxbrsigalem5 32255 | First direction for ~ sxbr... |
sxbrsigalem6 32256 | First direction for ~ sxbr... |
sxbrsiga 32257 | The product sigma-algebra ... |
omsval 32260 | Value of the function mapp... |
omsfval 32261 | Value of the outer measure... |
omscl 32262 | A closure lemma for the co... |
omsf 32263 | A constructed outer measur... |
oms0 32264 | A constructed outer measur... |
omsmon 32265 | A constructed outer measur... |
omssubaddlem 32266 | For any small margin ` E `... |
omssubadd 32267 | A constructed outer measur... |
carsgval 32270 | Value of the Caratheodory ... |
carsgcl 32271 | Closure of the Caratheodor... |
elcarsg 32272 | Property of being a Carath... |
baselcarsg 32273 | The universe set, ` O ` , ... |
0elcarsg 32274 | The empty set is Caratheod... |
carsguni 32275 | The union of all Caratheod... |
elcarsgss 32276 | Caratheodory measurable se... |
difelcarsg 32277 | The Caratheodory measurabl... |
inelcarsg 32278 | The Caratheodory measurabl... |
unelcarsg 32279 | The Caratheodory-measurabl... |
difelcarsg2 32280 | The Caratheodory-measurabl... |
carsgmon 32281 | Utility lemma: Apply mono... |
carsgsigalem 32282 | Lemma for the following th... |
fiunelcarsg 32283 | The Caratheodory measurabl... |
carsgclctunlem1 32284 | Lemma for ~ carsgclctun . ... |
carsggect 32285 | The outer measure is count... |
carsgclctunlem2 32286 | Lemma for ~ carsgclctun . ... |
carsgclctunlem3 32287 | Lemma for ~ carsgclctun . ... |
carsgclctun 32288 | The Caratheodory measurabl... |
carsgsiga 32289 | The Caratheodory measurabl... |
omsmeas 32290 | The restriction of a const... |
pmeasmono 32291 | This theorem's hypotheses ... |
pmeasadd 32292 | A premeasure on a ring of ... |
itgeq12dv 32293 | Equality theorem for an in... |
sitgval 32299 | Value of the simple functi... |
issibf 32300 | The predicate " ` F ` is a... |
sibf0 32301 | The constant zero function... |
sibfmbl 32302 | A simple function is measu... |
sibff 32303 | A simple function is a fun... |
sibfrn 32304 | A simple function has fini... |
sibfima 32305 | Any preimage of a singleto... |
sibfinima 32306 | The measure of the interse... |
sibfof 32307 | Applying function operatio... |
sitgfval 32308 | Value of the Bochner integ... |
sitgclg 32309 | Closure of the Bochner int... |
sitgclbn 32310 | Closure of the Bochner int... |
sitgclcn 32311 | Closure of the Bochner int... |
sitgclre 32312 | Closure of the Bochner int... |
sitg0 32313 | The integral of the consta... |
sitgf 32314 | The integral for simple fu... |
sitgaddlemb 32315 | Lemma for * sitgadd . (Co... |
sitmval 32316 | Value of the simple functi... |
sitmfval 32317 | Value of the integral dist... |
sitmcl 32318 | Closure of the integral di... |
sitmf 32319 | The integral metric as a f... |
oddpwdc 32321 | Lemma for ~ eulerpart . T... |
oddpwdcv 32322 | Lemma for ~ eulerpart : va... |
eulerpartlemsv1 32323 | Lemma for ~ eulerpart . V... |
eulerpartlemelr 32324 | Lemma for ~ eulerpart . (... |
eulerpartlemsv2 32325 | Lemma for ~ eulerpart . V... |
eulerpartlemsf 32326 | Lemma for ~ eulerpart . (... |
eulerpartlems 32327 | Lemma for ~ eulerpart . (... |
eulerpartlemsv3 32328 | Lemma for ~ eulerpart . V... |
eulerpartlemgc 32329 | Lemma for ~ eulerpart . (... |
eulerpartleme 32330 | Lemma for ~ eulerpart . (... |
eulerpartlemv 32331 | Lemma for ~ eulerpart . (... |
eulerpartlemo 32332 | Lemma for ~ eulerpart : ` ... |
eulerpartlemd 32333 | Lemma for ~ eulerpart : ` ... |
eulerpartlem1 32334 | Lemma for ~ eulerpart . (... |
eulerpartlemb 32335 | Lemma for ~ eulerpart . T... |
eulerpartlemt0 32336 | Lemma for ~ eulerpart . (... |
eulerpartlemf 32337 | Lemma for ~ eulerpart : O... |
eulerpartlemt 32338 | Lemma for ~ eulerpart . (... |
eulerpartgbij 32339 | Lemma for ~ eulerpart : T... |
eulerpartlemgv 32340 | Lemma for ~ eulerpart : va... |
eulerpartlemr 32341 | Lemma for ~ eulerpart . (... |
eulerpartlemmf 32342 | Lemma for ~ eulerpart . (... |
eulerpartlemgvv 32343 | Lemma for ~ eulerpart : va... |
eulerpartlemgu 32344 | Lemma for ~ eulerpart : R... |
eulerpartlemgh 32345 | Lemma for ~ eulerpart : T... |
eulerpartlemgf 32346 | Lemma for ~ eulerpart : I... |
eulerpartlemgs2 32347 | Lemma for ~ eulerpart : T... |
eulerpartlemn 32348 | Lemma for ~ eulerpart . (... |
eulerpart 32349 | Euler's theorem on partiti... |
subiwrd 32352 | Lemma for ~ sseqp1 . (Con... |
subiwrdlen 32353 | Length of a subword of an ... |
iwrdsplit 32354 | Lemma for ~ sseqp1 . (Con... |
sseqval 32355 | Value of the strong sequen... |
sseqfv1 32356 | Value of the strong sequen... |
sseqfn 32357 | A strong recursive sequenc... |
sseqmw 32358 | Lemma for ~ sseqf amd ~ ss... |
sseqf 32359 | A strong recursive sequenc... |
sseqfres 32360 | The first elements in the ... |
sseqfv2 32361 | Value of the strong sequen... |
sseqp1 32362 | Value of the strong sequen... |
fiblem 32365 | Lemma for ~ fib0 , ~ fib1 ... |
fib0 32366 | Value of the Fibonacci seq... |
fib1 32367 | Value of the Fibonacci seq... |
fibp1 32368 | Value of the Fibonacci seq... |
fib2 32369 | Value of the Fibonacci seq... |
fib3 32370 | Value of the Fibonacci seq... |
fib4 32371 | Value of the Fibonacci seq... |
fib5 32372 | Value of the Fibonacci seq... |
fib6 32373 | Value of the Fibonacci seq... |
elprob 32376 | The property of being a pr... |
domprobmeas 32377 | A probability measure is a... |
domprobsiga 32378 | The domain of a probabilit... |
probtot 32379 | The probability of the uni... |
prob01 32380 | A probability is an elemen... |
probnul 32381 | The probability of the emp... |
unveldomd 32382 | The universe is an element... |
unveldom 32383 | The universe is an element... |
nuleldmp 32384 | The empty set is an elemen... |
probcun 32385 | The probability of the uni... |
probun 32386 | The probability of the uni... |
probdif 32387 | The probability of the dif... |
probinc 32388 | A probability law is incre... |
probdsb 32389 | The probability of the com... |
probmeasd 32390 | A probability measure is a... |
probvalrnd 32391 | The value of a probability... |
probtotrnd 32392 | The probability of the uni... |
totprobd 32393 | Law of total probability, ... |
totprob 32394 | Law of total probability. ... |
probfinmeasb 32395 | Build a probability measur... |
probfinmeasbALTV 32396 | Alternate version of ~ pro... |
probmeasb 32397 | Build a probability from a... |
cndprobval 32400 | The value of the condition... |
cndprobin 32401 | An identity linking condit... |
cndprob01 32402 | The conditional probabilit... |
cndprobtot 32403 | The conditional probabilit... |
cndprobnul 32404 | The conditional probabilit... |
cndprobprob 32405 | The conditional probabilit... |
bayesth 32406 | Bayes Theorem. (Contribut... |
rrvmbfm 32409 | A real-valued random varia... |
isrrvv 32410 | Elementhood to the set of ... |
rrvvf 32411 | A real-valued random varia... |
rrvfn 32412 | A real-valued random varia... |
rrvdm 32413 | The domain of a random var... |
rrvrnss 32414 | The range of a random vari... |
rrvf2 32415 | A real-valued random varia... |
rrvdmss 32416 | The domain of a random var... |
rrvfinvima 32417 | For a real-value random va... |
0rrv 32418 | The constant function equa... |
rrvadd 32419 | The sum of two random vari... |
rrvmulc 32420 | A random variable multipli... |
rrvsum 32421 | An indexed sum of random v... |
orvcval 32424 | Value of the preimage mapp... |
orvcval2 32425 | Another way to express the... |
elorvc 32426 | Elementhood of a preimage.... |
orvcval4 32427 | The value of the preimage ... |
orvcoel 32428 | If the relation produces o... |
orvccel 32429 | If the relation produces c... |
elorrvc 32430 | Elementhood of a preimage ... |
orrvcval4 32431 | The value of the preimage ... |
orrvcoel 32432 | If the relation produces o... |
orrvccel 32433 | If the relation produces c... |
orvcgteel 32434 | Preimage maps produced by ... |
orvcelval 32435 | Preimage maps produced by ... |
orvcelel 32436 | Preimage maps produced by ... |
dstrvval 32437 | The value of the distribut... |
dstrvprob 32438 | The distribution of a rand... |
orvclteel 32439 | Preimage maps produced by ... |
dstfrvel 32440 | Elementhood of preimage ma... |
dstfrvunirn 32441 | The limit of all preimage ... |
orvclteinc 32442 | Preimage maps produced by ... |
dstfrvinc 32443 | A cumulative distribution ... |
dstfrvclim1 32444 | The limit of the cumulativ... |
coinfliplem 32445 | Division in the extended r... |
coinflipprob 32446 | The ` P ` we defined for c... |
coinflipspace 32447 | The space of our coin-flip... |
coinflipuniv 32448 | The universe of our coin-f... |
coinfliprv 32449 | The ` X ` we defined for c... |
coinflippv 32450 | The probability of heads i... |
coinflippvt 32451 | The probability of tails i... |
ballotlemoex 32452 | ` O ` is a set. (Contribu... |
ballotlem1 32453 | The size of the universe i... |
ballotlemelo 32454 | Elementhood in ` O ` . (C... |
ballotlem2 32455 | The probability that the f... |
ballotlemfval 32456 | The value of ` F ` . (Con... |
ballotlemfelz 32457 | ` ( F `` C ) ` has values ... |
ballotlemfp1 32458 | If the ` J ` th ballot is ... |
ballotlemfc0 32459 | ` F ` takes value 0 betwee... |
ballotlemfcc 32460 | ` F ` takes value 0 betwee... |
ballotlemfmpn 32461 | ` ( F `` C ) ` finishes co... |
ballotlemfval0 32462 | ` ( F `` C ) ` always star... |
ballotleme 32463 | Elements of ` E ` . (Cont... |
ballotlemodife 32464 | Elements of ` ( O \ E ) ` ... |
ballotlem4 32465 | If the first pick is a vot... |
ballotlem5 32466 | If A is not ahead througho... |
ballotlemi 32467 | Value of ` I ` for a given... |
ballotlemiex 32468 | Properties of ` ( I `` C )... |
ballotlemi1 32469 | The first tie cannot be re... |
ballotlemii 32470 | The first tie cannot be re... |
ballotlemsup 32471 | The set of zeroes of ` F `... |
ballotlemimin 32472 | ` ( I `` C ) ` is the firs... |
ballotlemic 32473 | If the first vote is for B... |
ballotlem1c 32474 | If the first vote is for A... |
ballotlemsval 32475 | Value of ` S ` . (Contrib... |
ballotlemsv 32476 | Value of ` S ` evaluated a... |
ballotlemsgt1 32477 | ` S ` maps values less tha... |
ballotlemsdom 32478 | Domain of ` S ` for a give... |
ballotlemsel1i 32479 | The range ` ( 1 ... ( I ``... |
ballotlemsf1o 32480 | The defined ` S ` is a bij... |
ballotlemsi 32481 | The image by ` S ` of the ... |
ballotlemsima 32482 | The image by ` S ` of an i... |
ballotlemieq 32483 | If two countings share the... |
ballotlemrval 32484 | Value of ` R ` . (Contrib... |
ballotlemscr 32485 | The image of ` ( R `` C ) ... |
ballotlemrv 32486 | Value of ` R ` evaluated a... |
ballotlemrv1 32487 | Value of ` R ` before the ... |
ballotlemrv2 32488 | Value of ` R ` after the t... |
ballotlemro 32489 | Range of ` R ` is included... |
ballotlemgval 32490 | Expand the value of ` .^ `... |
ballotlemgun 32491 | A property of the defined ... |
ballotlemfg 32492 | Express the value of ` ( F... |
ballotlemfrc 32493 | Express the value of ` ( F... |
ballotlemfrci 32494 | Reverse counting preserves... |
ballotlemfrceq 32495 | Value of ` F ` for a rever... |
ballotlemfrcn0 32496 | Value of ` F ` for a rever... |
ballotlemrc 32497 | Range of ` R ` . (Contrib... |
ballotlemirc 32498 | Applying ` R ` does not ch... |
ballotlemrinv0 32499 | Lemma for ~ ballotlemrinv ... |
ballotlemrinv 32500 | ` R ` is its own inverse :... |
ballotlem1ri 32501 | When the vote on the first... |
ballotlem7 32502 | ` R ` is a bijection betwe... |
ballotlem8 32503 | There are as many counting... |
ballotth 32504 | Bertrand's ballot problem ... |
sgncl 32505 | Closure of the signum. (C... |
sgnclre 32506 | Closure of the signum. (C... |
sgnneg 32507 | Negation of the signum. (... |
sgn3da 32508 | A conditional containing a... |
sgnmul 32509 | Signum of a product. (Con... |
sgnmulrp2 32510 | Multiplication by a positi... |
sgnsub 32511 | Subtraction of a number of... |
sgnnbi 32512 | Negative signum. (Contrib... |
sgnpbi 32513 | Positive signum. (Contrib... |
sgn0bi 32514 | Zero signum. (Contributed... |
sgnsgn 32515 | Signum is idempotent. (Co... |
sgnmulsgn 32516 | If two real numbers are of... |
sgnmulsgp 32517 | If two real numbers are of... |
fzssfzo 32518 | Condition for an integer i... |
gsumncl 32519 | Closure of a group sum in ... |
gsumnunsn 32520 | Closure of a group sum in ... |
ccatmulgnn0dir 32521 | Concatenation of words fol... |
ofcccat 32522 | Letterwise operations on w... |
ofcs1 32523 | Letterwise operations on a... |
ofcs2 32524 | Letterwise operations on a... |
plymul02 32525 | Product of a polynomial wi... |
plymulx0 32526 | Coefficients of a polynomi... |
plymulx 32527 | Coefficients of a polynomi... |
plyrecld 32528 | Closure of a polynomial wi... |
signsplypnf 32529 | The quotient of a polynomi... |
signsply0 32530 | Lemma for the rule of sign... |
signspval 32531 | The value of the skipping ... |
signsw0glem 32532 | Neutral element property o... |
signswbase 32533 | The base of ` W ` is the u... |
signswplusg 32534 | The operation of ` W ` . ... |
signsw0g 32535 | The neutral element of ` W... |
signswmnd 32536 | ` W ` is a monoid structur... |
signswrid 32537 | The zero-skipping operatio... |
signswlid 32538 | The zero-skipping operatio... |
signswn0 32539 | The zero-skipping operatio... |
signswch 32540 | The zero-skipping operatio... |
signslema 32541 | Computational part of ~~? ... |
signstfv 32542 | Value of the zero-skipping... |
signstfval 32543 | Value of the zero-skipping... |
signstcl 32544 | Closure of the zero skippi... |
signstf 32545 | The zero skipping sign wor... |
signstlen 32546 | Length of the zero skippin... |
signstf0 32547 | Sign of a single letter wo... |
signstfvn 32548 | Zero-skipping sign in a wo... |
signsvtn0 32549 | If the last letter is nonz... |
signstfvp 32550 | Zero-skipping sign in a wo... |
signstfvneq0 32551 | In case the first letter i... |
signstfvcl 32552 | Closure of the zero skippi... |
signstfvc 32553 | Zero-skipping sign in a wo... |
signstres 32554 | Restriction of a zero skip... |
signstfveq0a 32555 | Lemma for ~ signstfveq0 . ... |
signstfveq0 32556 | In case the last letter is... |
signsvvfval 32557 | The value of ` V ` , which... |
signsvvf 32558 | ` V ` is a function. (Con... |
signsvf0 32559 | There is no change of sign... |
signsvf1 32560 | In a single-letter word, w... |
signsvfn 32561 | Number of changes in a wor... |
signsvtp 32562 | Adding a letter of the sam... |
signsvtn 32563 | Adding a letter of a diffe... |
signsvfpn 32564 | Adding a letter of the sam... |
signsvfnn 32565 | Adding a letter of a diffe... |
signlem0 32566 | Adding a zero as the highe... |
signshf 32567 | ` H ` , corresponding to t... |
signshwrd 32568 | ` H ` , corresponding to t... |
signshlen 32569 | Length of ` H ` , correspo... |
signshnz 32570 | ` H ` is not the empty wor... |
efcld 32571 | Closure law for the expone... |
iblidicc 32572 | The identity function is i... |
rpsqrtcn 32573 | Continuity of the real pos... |
divsqrtid 32574 | A real number divided by i... |
cxpcncf1 32575 | The power function on comp... |
efmul2picn 32576 | Multiplying by ` ( _i x. (... |
fct2relem 32577 | Lemma for ~ ftc2re . (Con... |
ftc2re 32578 | The Fundamental Theorem of... |
fdvposlt 32579 | Functions with a positive ... |
fdvneggt 32580 | Functions with a negative ... |
fdvposle 32581 | Functions with a nonnegati... |
fdvnegge 32582 | Functions with a nonpositi... |
prodfzo03 32583 | A product of three factors... |
actfunsnf1o 32584 | The action ` F ` of extend... |
actfunsnrndisj 32585 | The action ` F ` of extend... |
itgexpif 32586 | The basis for the circle m... |
fsum2dsub 32587 | Lemma for ~ breprexp - Re-... |
reprval 32590 | Value of the representatio... |
repr0 32591 | There is exactly one repre... |
reprf 32592 | Members of the representat... |
reprsum 32593 | Sums of values of the memb... |
reprle 32594 | Upper bound to the terms i... |
reprsuc 32595 | Express the representation... |
reprfi 32596 | Bounded representations ar... |
reprss 32597 | Representations with terms... |
reprinrn 32598 | Representations with term ... |
reprlt 32599 | There are no representatio... |
hashreprin 32600 | Express a sum of represent... |
reprgt 32601 | There are no representatio... |
reprinfz1 32602 | For the representation of ... |
reprfi2 32603 | Corollary of ~ reprinfz1 .... |
reprfz1 32604 | Corollary of ~ reprinfz1 .... |
hashrepr 32605 | Develop the number of repr... |
reprpmtf1o 32606 | Transposing ` 0 ` and ` X ... |
reprdifc 32607 | Express the representation... |
chpvalz 32608 | Value of the second Chebys... |
chtvalz 32609 | Value of the Chebyshev fun... |
breprexplema 32610 | Lemma for ~ breprexp (indu... |
breprexplemb 32611 | Lemma for ~ breprexp (clos... |
breprexplemc 32612 | Lemma for ~ breprexp (indu... |
breprexp 32613 | Express the ` S ` th power... |
breprexpnat 32614 | Express the ` S ` th power... |
vtsval 32617 | Value of the Vinogradov tr... |
vtscl 32618 | Closure of the Vinogradov ... |
vtsprod 32619 | Express the Vinogradov tri... |
circlemeth 32620 | The Hardy, Littlewood and ... |
circlemethnat 32621 | The Hardy, Littlewood and ... |
circlevma 32622 | The Circle Method, where t... |
circlemethhgt 32623 | The circle method, where t... |
hgt750lemc 32627 | An upper bound to the summ... |
hgt750lemd 32628 | An upper bound to the summ... |
hgt749d 32629 | A deduction version of ~ a... |
logdivsqrle 32630 | Conditions for ` ( ( log `... |
hgt750lem 32631 | Lemma for ~ tgoldbachgtd .... |
hgt750lem2 32632 | Decimal multiplication gal... |
hgt750lemf 32633 | Lemma for the statement 7.... |
hgt750lemg 32634 | Lemma for the statement 7.... |
oddprm2 32635 | Two ways to write the set ... |
hgt750lemb 32636 | An upper bound on the cont... |
hgt750lema 32637 | An upper bound on the cont... |
hgt750leme 32638 | An upper bound on the cont... |
tgoldbachgnn 32639 | Lemma for ~ tgoldbachgtd .... |
tgoldbachgtde 32640 | Lemma for ~ tgoldbachgtd .... |
tgoldbachgtda 32641 | Lemma for ~ tgoldbachgtd .... |
tgoldbachgtd 32642 | Odd integers greater than ... |
tgoldbachgt 32643 | Odd integers greater than ... |
istrkg2d 32646 | Property of fulfilling dim... |
axtglowdim2ALTV 32647 | Alternate version of ~ axt... |
axtgupdim2ALTV 32648 | Alternate version of ~ axt... |
afsval 32651 | Value of the AFS relation ... |
brafs 32652 | Binary relation form of th... |
tg5segofs 32653 | Rephrase ~ axtg5seg using ... |
lpadval 32656 | Value of the ` leftpad ` f... |
lpadlem1 32657 | Lemma for the ` leftpad ` ... |
lpadlem3 32658 | Lemma for ~ lpadlen1 . (C... |
lpadlen1 32659 | Length of a left-padded wo... |
lpadlem2 32660 | Lemma for the ` leftpad ` ... |
lpadlen2 32661 | Length of a left-padded wo... |
lpadmax 32662 | Length of a left-padded wo... |
lpadleft 32663 | The contents of prefix of ... |
lpadright 32664 | The suffix of a left-padde... |
bnj170 32677 | ` /\ ` -manipulation. (Co... |
bnj240 32678 | ` /\ ` -manipulation. (Co... |
bnj248 32679 | ` /\ ` -manipulation. (Co... |
bnj250 32680 | ` /\ ` -manipulation. (Co... |
bnj251 32681 | ` /\ ` -manipulation. (Co... |
bnj252 32682 | ` /\ ` -manipulation. (Co... |
bnj253 32683 | ` /\ ` -manipulation. (Co... |
bnj255 32684 | ` /\ ` -manipulation. (Co... |
bnj256 32685 | ` /\ ` -manipulation. (Co... |
bnj257 32686 | ` /\ ` -manipulation. (Co... |
bnj258 32687 | ` /\ ` -manipulation. (Co... |
bnj268 32688 | ` /\ ` -manipulation. (Co... |
bnj290 32689 | ` /\ ` -manipulation. (Co... |
bnj291 32690 | ` /\ ` -manipulation. (Co... |
bnj312 32691 | ` /\ ` -manipulation. (Co... |
bnj334 32692 | ` /\ ` -manipulation. (Co... |
bnj345 32693 | ` /\ ` -manipulation. (Co... |
bnj422 32694 | ` /\ ` -manipulation. (Co... |
bnj432 32695 | ` /\ ` -manipulation. (Co... |
bnj446 32696 | ` /\ ` -manipulation. (Co... |
bnj23 32697 | First-order logic and set ... |
bnj31 32698 | First-order logic and set ... |
bnj62 32699 | First-order logic and set ... |
bnj89 32700 | First-order logic and set ... |
bnj90 32701 | First-order logic and set ... |
bnj101 32702 | First-order logic and set ... |
bnj105 32703 | First-order logic and set ... |
bnj115 32704 | First-order logic and set ... |
bnj132 32705 | First-order logic and set ... |
bnj133 32706 | First-order logic and set ... |
bnj156 32707 | First-order logic and set ... |
bnj158 32708 | First-order logic and set ... |
bnj168 32709 | First-order logic and set ... |
bnj206 32710 | First-order logic and set ... |
bnj216 32711 | First-order logic and set ... |
bnj219 32712 | First-order logic and set ... |
bnj226 32713 | First-order logic and set ... |
bnj228 32714 | First-order logic and set ... |
bnj519 32715 | First-order logic and set ... |
bnj521 32716 | First-order logic and set ... |
bnj524 32717 | First-order logic and set ... |
bnj525 32718 | First-order logic and set ... |
bnj534 32719 | First-order logic and set ... |
bnj538 32720 | First-order logic and set ... |
bnj529 32721 | First-order logic and set ... |
bnj551 32722 | First-order logic and set ... |
bnj563 32723 | First-order logic and set ... |
bnj564 32724 | First-order logic and set ... |
bnj593 32725 | First-order logic and set ... |
bnj596 32726 | First-order logic and set ... |
bnj610 32727 | Pass from equality ( ` x =... |
bnj642 32728 | ` /\ ` -manipulation. (Co... |
bnj643 32729 | ` /\ ` -manipulation. (Co... |
bnj645 32730 | ` /\ ` -manipulation. (Co... |
bnj658 32731 | ` /\ ` -manipulation. (Co... |
bnj667 32732 | ` /\ ` -manipulation. (Co... |
bnj705 32733 | ` /\ ` -manipulation. (Co... |
bnj706 32734 | ` /\ ` -manipulation. (Co... |
bnj707 32735 | ` /\ ` -manipulation. (Co... |
bnj708 32736 | ` /\ ` -manipulation. (Co... |
bnj721 32737 | ` /\ ` -manipulation. (Co... |
bnj832 32738 | ` /\ ` -manipulation. (Co... |
bnj835 32739 | ` /\ ` -manipulation. (Co... |
bnj836 32740 | ` /\ ` -manipulation. (Co... |
bnj837 32741 | ` /\ ` -manipulation. (Co... |
bnj769 32742 | ` /\ ` -manipulation. (Co... |
bnj770 32743 | ` /\ ` -manipulation. (Co... |
bnj771 32744 | ` /\ ` -manipulation. (Co... |
bnj887 32745 | ` /\ ` -manipulation. (Co... |
bnj918 32746 | First-order logic and set ... |
bnj919 32747 | First-order logic and set ... |
bnj923 32748 | First-order logic and set ... |
bnj927 32749 | First-order logic and set ... |
bnj931 32750 | First-order logic and set ... |
bnj937 32751 | First-order logic and set ... |
bnj941 32752 | First-order logic and set ... |
bnj945 32753 | Technical lemma for ~ bnj6... |
bnj946 32754 | First-order logic and set ... |
bnj951 32755 | ` /\ ` -manipulation. (Co... |
bnj956 32756 | First-order logic and set ... |
bnj976 32757 | First-order logic and set ... |
bnj982 32758 | First-order logic and set ... |
bnj1019 32759 | First-order logic and set ... |
bnj1023 32760 | First-order logic and set ... |
bnj1095 32761 | First-order logic and set ... |
bnj1096 32762 | First-order logic and set ... |
bnj1098 32763 | First-order logic and set ... |
bnj1101 32764 | First-order logic and set ... |
bnj1113 32765 | First-order logic and set ... |
bnj1109 32766 | First-order logic and set ... |
bnj1131 32767 | First-order logic and set ... |
bnj1138 32768 | First-order logic and set ... |
bnj1142 32769 | First-order logic and set ... |
bnj1143 32770 | First-order logic and set ... |
bnj1146 32771 | First-order logic and set ... |
bnj1149 32772 | First-order logic and set ... |
bnj1185 32773 | First-order logic and set ... |
bnj1196 32774 | First-order logic and set ... |
bnj1198 32775 | First-order logic and set ... |
bnj1209 32776 | First-order logic and set ... |
bnj1211 32777 | First-order logic and set ... |
bnj1213 32778 | First-order logic and set ... |
bnj1212 32779 | First-order logic and set ... |
bnj1219 32780 | First-order logic and set ... |
bnj1224 32781 | First-order logic and set ... |
bnj1230 32782 | First-order logic and set ... |
bnj1232 32783 | First-order logic and set ... |
bnj1235 32784 | First-order logic and set ... |
bnj1239 32785 | First-order logic and set ... |
bnj1238 32786 | First-order logic and set ... |
bnj1241 32787 | First-order logic and set ... |
bnj1247 32788 | First-order logic and set ... |
bnj1254 32789 | First-order logic and set ... |
bnj1262 32790 | First-order logic and set ... |
bnj1266 32791 | First-order logic and set ... |
bnj1265 32792 | First-order logic and set ... |
bnj1275 32793 | First-order logic and set ... |
bnj1276 32794 | First-order logic and set ... |
bnj1292 32795 | First-order logic and set ... |
bnj1293 32796 | First-order logic and set ... |
bnj1294 32797 | First-order logic and set ... |
bnj1299 32798 | First-order logic and set ... |
bnj1304 32799 | First-order logic and set ... |
bnj1316 32800 | First-order logic and set ... |
bnj1317 32801 | First-order logic and set ... |
bnj1322 32802 | First-order logic and set ... |
bnj1340 32803 | First-order logic and set ... |
bnj1345 32804 | First-order logic and set ... |
bnj1350 32805 | First-order logic and set ... |
bnj1351 32806 | First-order logic and set ... |
bnj1352 32807 | First-order logic and set ... |
bnj1361 32808 | First-order logic and set ... |
bnj1366 32809 | First-order logic and set ... |
bnj1379 32810 | First-order logic and set ... |
bnj1383 32811 | First-order logic and set ... |
bnj1385 32812 | First-order logic and set ... |
bnj1386 32813 | First-order logic and set ... |
bnj1397 32814 | First-order logic and set ... |
bnj1400 32815 | First-order logic and set ... |
bnj1405 32816 | First-order logic and set ... |
bnj1422 32817 | First-order logic and set ... |
bnj1424 32818 | First-order logic and set ... |
bnj1436 32819 | First-order logic and set ... |
bnj1441 32820 | First-order logic and set ... |
bnj1441g 32821 | First-order logic and set ... |
bnj1454 32822 | First-order logic and set ... |
bnj1459 32823 | First-order logic and set ... |
bnj1464 32824 | Conversion of implicit sub... |
bnj1465 32825 | First-order logic and set ... |
bnj1468 32826 | Conversion of implicit sub... |
bnj1476 32827 | First-order logic and set ... |
bnj1502 32828 | First-order logic and set ... |
bnj1503 32829 | First-order logic and set ... |
bnj1517 32830 | First-order logic and set ... |
bnj1521 32831 | First-order logic and set ... |
bnj1533 32832 | First-order logic and set ... |
bnj1534 32833 | First-order logic and set ... |
bnj1536 32834 | First-order logic and set ... |
bnj1538 32835 | First-order logic and set ... |
bnj1541 32836 | First-order logic and set ... |
bnj1542 32837 | First-order logic and set ... |
bnj110 32838 | Well-founded induction res... |
bnj157 32839 | Well-founded induction res... |
bnj66 32840 | Technical lemma for ~ bnj6... |
bnj91 32841 | First-order logic and set ... |
bnj92 32842 | First-order logic and set ... |
bnj93 32843 | Technical lemma for ~ bnj9... |
bnj95 32844 | Technical lemma for ~ bnj1... |
bnj96 32845 | Technical lemma for ~ bnj1... |
bnj97 32846 | Technical lemma for ~ bnj1... |
bnj98 32847 | Technical lemma for ~ bnj1... |
bnj106 32848 | First-order logic and set ... |
bnj118 32849 | First-order logic and set ... |
bnj121 32850 | First-order logic and set ... |
bnj124 32851 | Technical lemma for ~ bnj1... |
bnj125 32852 | Technical lemma for ~ bnj1... |
bnj126 32853 | Technical lemma for ~ bnj1... |
bnj130 32854 | Technical lemma for ~ bnj1... |
bnj149 32855 | Technical lemma for ~ bnj1... |
bnj150 32856 | Technical lemma for ~ bnj1... |
bnj151 32857 | Technical lemma for ~ bnj1... |
bnj154 32858 | Technical lemma for ~ bnj1... |
bnj155 32859 | Technical lemma for ~ bnj1... |
bnj153 32860 | Technical lemma for ~ bnj8... |
bnj207 32861 | Technical lemma for ~ bnj8... |
bnj213 32862 | First-order logic and set ... |
bnj222 32863 | Technical lemma for ~ bnj2... |
bnj229 32864 | Technical lemma for ~ bnj5... |
bnj517 32865 | Technical lemma for ~ bnj5... |
bnj518 32866 | Technical lemma for ~ bnj8... |
bnj523 32867 | Technical lemma for ~ bnj8... |
bnj526 32868 | Technical lemma for ~ bnj8... |
bnj528 32869 | Technical lemma for ~ bnj8... |
bnj535 32870 | Technical lemma for ~ bnj8... |
bnj539 32871 | Technical lemma for ~ bnj8... |
bnj540 32872 | Technical lemma for ~ bnj8... |
bnj543 32873 | Technical lemma for ~ bnj8... |
bnj544 32874 | Technical lemma for ~ bnj8... |
bnj545 32875 | Technical lemma for ~ bnj8... |
bnj546 32876 | Technical lemma for ~ bnj8... |
bnj548 32877 | Technical lemma for ~ bnj8... |
bnj553 32878 | Technical lemma for ~ bnj8... |
bnj554 32879 | Technical lemma for ~ bnj8... |
bnj556 32880 | Technical lemma for ~ bnj8... |
bnj557 32881 | Technical lemma for ~ bnj8... |
bnj558 32882 | Technical lemma for ~ bnj8... |
bnj561 32883 | Technical lemma for ~ bnj8... |
bnj562 32884 | Technical lemma for ~ bnj8... |
bnj570 32885 | Technical lemma for ~ bnj8... |
bnj571 32886 | Technical lemma for ~ bnj8... |
bnj605 32887 | Technical lemma. This lem... |
bnj581 32888 | Technical lemma for ~ bnj5... |
bnj589 32889 | Technical lemma for ~ bnj8... |
bnj590 32890 | Technical lemma for ~ bnj8... |
bnj591 32891 | Technical lemma for ~ bnj8... |
bnj594 32892 | Technical lemma for ~ bnj8... |
bnj580 32893 | Technical lemma for ~ bnj5... |
bnj579 32894 | Technical lemma for ~ bnj8... |
bnj602 32895 | Equality theorem for the `... |
bnj607 32896 | Technical lemma for ~ bnj8... |
bnj609 32897 | Technical lemma for ~ bnj8... |
bnj611 32898 | Technical lemma for ~ bnj8... |
bnj600 32899 | Technical lemma for ~ bnj8... |
bnj601 32900 | Technical lemma for ~ bnj8... |
bnj852 32901 | Technical lemma for ~ bnj6... |
bnj864 32902 | Technical lemma for ~ bnj6... |
bnj865 32903 | Technical lemma for ~ bnj6... |
bnj873 32904 | Technical lemma for ~ bnj6... |
bnj849 32905 | Technical lemma for ~ bnj6... |
bnj882 32906 | Definition (using hypothes... |
bnj18eq1 32907 | Equality theorem for trans... |
bnj893 32908 | Property of ` _trCl ` . U... |
bnj900 32909 | Technical lemma for ~ bnj6... |
bnj906 32910 | Property of ` _trCl ` . (... |
bnj908 32911 | Technical lemma for ~ bnj6... |
bnj911 32912 | Technical lemma for ~ bnj6... |
bnj916 32913 | Technical lemma for ~ bnj6... |
bnj917 32914 | Technical lemma for ~ bnj6... |
bnj934 32915 | Technical lemma for ~ bnj6... |
bnj929 32916 | Technical lemma for ~ bnj6... |
bnj938 32917 | Technical lemma for ~ bnj6... |
bnj944 32918 | Technical lemma for ~ bnj6... |
bnj953 32919 | Technical lemma for ~ bnj6... |
bnj958 32920 | Technical lemma for ~ bnj6... |
bnj1000 32921 | Technical lemma for ~ bnj8... |
bnj965 32922 | Technical lemma for ~ bnj8... |
bnj964 32923 | Technical lemma for ~ bnj6... |
bnj966 32924 | Technical lemma for ~ bnj6... |
bnj967 32925 | Technical lemma for ~ bnj6... |
bnj969 32926 | Technical lemma for ~ bnj6... |
bnj970 32927 | Technical lemma for ~ bnj6... |
bnj910 32928 | Technical lemma for ~ bnj6... |
bnj978 32929 | Technical lemma for ~ bnj6... |
bnj981 32930 | Technical lemma for ~ bnj6... |
bnj983 32931 | Technical lemma for ~ bnj6... |
bnj984 32932 | Technical lemma for ~ bnj6... |
bnj985v 32933 | Version of ~ bnj985 with a... |
bnj985 32934 | Technical lemma for ~ bnj6... |
bnj986 32935 | Technical lemma for ~ bnj6... |
bnj996 32936 | Technical lemma for ~ bnj6... |
bnj998 32937 | Technical lemma for ~ bnj6... |
bnj999 32938 | Technical lemma for ~ bnj6... |
bnj1001 32939 | Technical lemma for ~ bnj6... |
bnj1006 32940 | Technical lemma for ~ bnj6... |
bnj1014 32941 | Technical lemma for ~ bnj6... |
bnj1015 32942 | Technical lemma for ~ bnj6... |
bnj1018g 32943 | Version of ~ bnj1018 with ... |
bnj1018 32944 | Technical lemma for ~ bnj6... |
bnj1020 32945 | Technical lemma for ~ bnj6... |
bnj1021 32946 | Technical lemma for ~ bnj6... |
bnj907 32947 | Technical lemma for ~ bnj6... |
bnj1029 32948 | Property of ` _trCl ` . (... |
bnj1033 32949 | Technical lemma for ~ bnj6... |
bnj1034 32950 | Technical lemma for ~ bnj6... |
bnj1039 32951 | Technical lemma for ~ bnj6... |
bnj1040 32952 | Technical lemma for ~ bnj6... |
bnj1047 32953 | Technical lemma for ~ bnj6... |
bnj1049 32954 | Technical lemma for ~ bnj6... |
bnj1052 32955 | Technical lemma for ~ bnj6... |
bnj1053 32956 | Technical lemma for ~ bnj6... |
bnj1071 32957 | Technical lemma for ~ bnj6... |
bnj1083 32958 | Technical lemma for ~ bnj6... |
bnj1090 32959 | Technical lemma for ~ bnj6... |
bnj1093 32960 | Technical lemma for ~ bnj6... |
bnj1097 32961 | Technical lemma for ~ bnj6... |
bnj1110 32962 | Technical lemma for ~ bnj6... |
bnj1112 32963 | Technical lemma for ~ bnj6... |
bnj1118 32964 | Technical lemma for ~ bnj6... |
bnj1121 32965 | Technical lemma for ~ bnj6... |
bnj1123 32966 | Technical lemma for ~ bnj6... |
bnj1030 32967 | Technical lemma for ~ bnj6... |
bnj1124 32968 | Property of ` _trCl ` . (... |
bnj1133 32969 | Technical lemma for ~ bnj6... |
bnj1128 32970 | Technical lemma for ~ bnj6... |
bnj1127 32971 | Property of ` _trCl ` . (... |
bnj1125 32972 | Property of ` _trCl ` . (... |
bnj1145 32973 | Technical lemma for ~ bnj6... |
bnj1147 32974 | Property of ` _trCl ` . (... |
bnj1137 32975 | Property of ` _trCl ` . (... |
bnj1148 32976 | Property of ` _pred ` . (... |
bnj1136 32977 | Technical lemma for ~ bnj6... |
bnj1152 32978 | Technical lemma for ~ bnj6... |
bnj1154 32979 | Property of ` Fr ` . (Con... |
bnj1171 32980 | Technical lemma for ~ bnj6... |
bnj1172 32981 | Technical lemma for ~ bnj6... |
bnj1173 32982 | Technical lemma for ~ bnj6... |
bnj1174 32983 | Technical lemma for ~ bnj6... |
bnj1175 32984 | Technical lemma for ~ bnj6... |
bnj1176 32985 | Technical lemma for ~ bnj6... |
bnj1177 32986 | Technical lemma for ~ bnj6... |
bnj1186 32987 | Technical lemma for ~ bnj6... |
bnj1190 32988 | Technical lemma for ~ bnj6... |
bnj1189 32989 | Technical lemma for ~ bnj6... |
bnj69 32990 | Existence of a minimal ele... |
bnj1228 32991 | Existence of a minimal ele... |
bnj1204 32992 | Well-founded induction. T... |
bnj1234 32993 | Technical lemma for ~ bnj6... |
bnj1245 32994 | Technical lemma for ~ bnj6... |
bnj1256 32995 | Technical lemma for ~ bnj6... |
bnj1259 32996 | Technical lemma for ~ bnj6... |
bnj1253 32997 | Technical lemma for ~ bnj6... |
bnj1279 32998 | Technical lemma for ~ bnj6... |
bnj1286 32999 | Technical lemma for ~ bnj6... |
bnj1280 33000 | Technical lemma for ~ bnj6... |
bnj1296 33001 | Technical lemma for ~ bnj6... |
bnj1309 33002 | Technical lemma for ~ bnj6... |
bnj1307 33003 | Technical lemma for ~ bnj6... |
bnj1311 33004 | Technical lemma for ~ bnj6... |
bnj1318 33005 | Technical lemma for ~ bnj6... |
bnj1326 33006 | Technical lemma for ~ bnj6... |
bnj1321 33007 | Technical lemma for ~ bnj6... |
bnj1364 33008 | Property of ` _FrSe ` . (... |
bnj1371 33009 | Technical lemma for ~ bnj6... |
bnj1373 33010 | Technical lemma for ~ bnj6... |
bnj1374 33011 | Technical lemma for ~ bnj6... |
bnj1384 33012 | Technical lemma for ~ bnj6... |
bnj1388 33013 | Technical lemma for ~ bnj6... |
bnj1398 33014 | Technical lemma for ~ bnj6... |
bnj1413 33015 | Property of ` _trCl ` . (... |
bnj1408 33016 | Technical lemma for ~ bnj1... |
bnj1414 33017 | Property of ` _trCl ` . (... |
bnj1415 33018 | Technical lemma for ~ bnj6... |
bnj1416 33019 | Technical lemma for ~ bnj6... |
bnj1418 33020 | Property of ` _pred ` . (... |
bnj1417 33021 | Technical lemma for ~ bnj6... |
bnj1421 33022 | Technical lemma for ~ bnj6... |
bnj1444 33023 | Technical lemma for ~ bnj6... |
bnj1445 33024 | Technical lemma for ~ bnj6... |
bnj1446 33025 | Technical lemma for ~ bnj6... |
bnj1447 33026 | Technical lemma for ~ bnj6... |
bnj1448 33027 | Technical lemma for ~ bnj6... |
bnj1449 33028 | Technical lemma for ~ bnj6... |
bnj1442 33029 | Technical lemma for ~ bnj6... |
bnj1450 33030 | Technical lemma for ~ bnj6... |
bnj1423 33031 | Technical lemma for ~ bnj6... |
bnj1452 33032 | Technical lemma for ~ bnj6... |
bnj1466 33033 | Technical lemma for ~ bnj6... |
bnj1467 33034 | Technical lemma for ~ bnj6... |
bnj1463 33035 | Technical lemma for ~ bnj6... |
bnj1489 33036 | Technical lemma for ~ bnj6... |
bnj1491 33037 | Technical lemma for ~ bnj6... |
bnj1312 33038 | Technical lemma for ~ bnj6... |
bnj1493 33039 | Technical lemma for ~ bnj6... |
bnj1497 33040 | Technical lemma for ~ bnj6... |
bnj1498 33041 | Technical lemma for ~ bnj6... |
bnj60 33042 | Well-founded recursion, pa... |
bnj1514 33043 | Technical lemma for ~ bnj1... |
bnj1518 33044 | Technical lemma for ~ bnj1... |
bnj1519 33045 | Technical lemma for ~ bnj1... |
bnj1520 33046 | Technical lemma for ~ bnj1... |
bnj1501 33047 | Technical lemma for ~ bnj1... |
bnj1500 33048 | Well-founded recursion, pa... |
bnj1525 33049 | Technical lemma for ~ bnj1... |
bnj1529 33050 | Technical lemma for ~ bnj1... |
bnj1523 33051 | Technical lemma for ~ bnj1... |
bnj1522 33052 | Well-founded recursion, pa... |
exdifsn 33053 | There exists an element in... |
srcmpltd 33054 | If a statement is true for... |
prsrcmpltd 33055 | If a statement is true for... |
dff15 33056 | A one-to-one function in t... |
f1resveqaeq 33057 | If a function restricted t... |
f1resrcmplf1dlem 33058 | Lemma for ~ f1resrcmplf1d ... |
f1resrcmplf1d 33059 | If a function's restrictio... |
funen1cnv 33060 | If a function is equinumer... |
fnrelpredd 33061 | A function that preserves ... |
cardpred 33062 | The cardinality function p... |
nummin 33063 | Every nonempty class of nu... |
fineqvrep 33064 | If the Axiom of Infinity i... |
fineqvpow 33065 | If the Axiom of Infinity i... |
fineqvac 33066 | If the Axiom of Infinity i... |
fineqvacALT 33067 | Shorter proof of ~ fineqva... |
zltp1ne 33068 | Integer ordering relation.... |
nnltp1ne 33069 | Positive integer ordering ... |
nn0ltp1ne 33070 | Nonnegative integer orderi... |
0nn0m1nnn0 33071 | A number is zero if and on... |
f1resfz0f1d 33072 | If a function with a seque... |
fisshasheq 33073 | A finite set is equal to i... |
hashfundm 33074 | The size of a set function... |
hashf1dmrn 33075 | The size of the domain of ... |
hashf1dmcdm 33076 | The size of the domain of ... |
revpfxsfxrev 33077 | The reverse of a prefix of... |
swrdrevpfx 33078 | A subword expressed in ter... |
lfuhgr 33079 | A hypergraph is loop-free ... |
lfuhgr2 33080 | A hypergraph is loop-free ... |
lfuhgr3 33081 | A hypergraph is loop-free ... |
cplgredgex 33082 | Any two (distinct) vertice... |
cusgredgex 33083 | Any two (distinct) vertice... |
cusgredgex2 33084 | Any two distinct vertices ... |
pfxwlk 33085 | A prefix of a walk is a wa... |
revwlk 33086 | The reverse of a walk is a... |
revwlkb 33087 | Two words represent a walk... |
swrdwlk 33088 | Two matching subwords of a... |
pthhashvtx 33089 | A graph containing a path ... |
pthisspthorcycl 33090 | A path is either a simple ... |
spthcycl 33091 | A walk is a trivial path i... |
usgrgt2cycl 33092 | A non-trivial cycle in a s... |
usgrcyclgt2v 33093 | A simple graph with a non-... |
subgrwlk 33094 | If a walk exists in a subg... |
subgrtrl 33095 | If a trail exists in a sub... |
subgrpth 33096 | If a path exists in a subg... |
subgrcycl 33097 | If a cycle exists in a sub... |
cusgr3cyclex 33098 | Every complete simple grap... |
loop1cycl 33099 | A hypergraph has a cycle o... |
2cycld 33100 | Construction of a 2-cycle ... |
2cycl2d 33101 | Construction of a 2-cycle ... |
umgr2cycllem 33102 | Lemma for ~ umgr2cycl . (... |
umgr2cycl 33103 | A multigraph with two dist... |
dfacycgr1 33106 | An alternate definition of... |
isacycgr 33107 | The property of being an a... |
isacycgr1 33108 | The property of being an a... |
acycgrcycl 33109 | Any cycle in an acyclic gr... |
acycgr0v 33110 | A null graph (with no vert... |
acycgr1v 33111 | A multigraph with one vert... |
acycgr2v 33112 | A simple graph with two ve... |
prclisacycgr 33113 | A proper class (representi... |
acycgrislfgr 33114 | An acyclic hypergraph is a... |
upgracycumgr 33115 | An acyclic pseudograph is ... |
umgracycusgr 33116 | An acyclic multigraph is a... |
upgracycusgr 33117 | An acyclic pseudograph is ... |
cusgracyclt3v 33118 | A complete simple graph is... |
pthacycspth 33119 | A path in an acyclic graph... |
acycgrsubgr 33120 | The subgraph of an acyclic... |
quartfull 33127 | The quartic equation, writ... |
deranglem 33128 | Lemma for derangements. (... |
derangval 33129 | Define the derangement fun... |
derangf 33130 | The derangement number is ... |
derang0 33131 | The derangement number of ... |
derangsn 33132 | The derangement number of ... |
derangenlem 33133 | One half of ~ derangen . ... |
derangen 33134 | The derangement number is ... |
subfacval 33135 | The subfactorial is define... |
derangen2 33136 | Write the derangement numb... |
subfacf 33137 | The subfactorial is a func... |
subfaclefac 33138 | The subfactorial is less t... |
subfac0 33139 | The subfactorial at zero. ... |
subfac1 33140 | The subfactorial at one. ... |
subfacp1lem1 33141 | Lemma for ~ subfacp1 . Th... |
subfacp1lem2a 33142 | Lemma for ~ subfacp1 . Pr... |
subfacp1lem2b 33143 | Lemma for ~ subfacp1 . Pr... |
subfacp1lem3 33144 | Lemma for ~ subfacp1 . In... |
subfacp1lem4 33145 | Lemma for ~ subfacp1 . Th... |
subfacp1lem5 33146 | Lemma for ~ subfacp1 . In... |
subfacp1lem6 33147 | Lemma for ~ subfacp1 . By... |
subfacp1 33148 | A two-term recurrence for ... |
subfacval2 33149 | A closed-form expression f... |
subfaclim 33150 | The subfactorial converges... |
subfacval3 33151 | Another closed form expres... |
derangfmla 33152 | The derangements formula, ... |
erdszelem1 33153 | Lemma for ~ erdsze . (Con... |
erdszelem2 33154 | Lemma for ~ erdsze . (Con... |
erdszelem3 33155 | Lemma for ~ erdsze . (Con... |
erdszelem4 33156 | Lemma for ~ erdsze . (Con... |
erdszelem5 33157 | Lemma for ~ erdsze . (Con... |
erdszelem6 33158 | Lemma for ~ erdsze . (Con... |
erdszelem7 33159 | Lemma for ~ erdsze . (Con... |
erdszelem8 33160 | Lemma for ~ erdsze . (Con... |
erdszelem9 33161 | Lemma for ~ erdsze . (Con... |
erdszelem10 33162 | Lemma for ~ erdsze . (Con... |
erdszelem11 33163 | Lemma for ~ erdsze . (Con... |
erdsze 33164 | The Erdős-Szekeres th... |
erdsze2lem1 33165 | Lemma for ~ erdsze2 . (Co... |
erdsze2lem2 33166 | Lemma for ~ erdsze2 . (Co... |
erdsze2 33167 | Generalize the statement o... |
kur14lem1 33168 | Lemma for ~ kur14 . (Cont... |
kur14lem2 33169 | Lemma for ~ kur14 . Write... |
kur14lem3 33170 | Lemma for ~ kur14 . A clo... |
kur14lem4 33171 | Lemma for ~ kur14 . Compl... |
kur14lem5 33172 | Lemma for ~ kur14 . Closu... |
kur14lem6 33173 | Lemma for ~ kur14 . If ` ... |
kur14lem7 33174 | Lemma for ~ kur14 : main p... |
kur14lem8 33175 | Lemma for ~ kur14 . Show ... |
kur14lem9 33176 | Lemma for ~ kur14 . Since... |
kur14lem10 33177 | Lemma for ~ kur14 . Disch... |
kur14 33178 | Kuratowski's closure-compl... |
ispconn 33185 | The property of being a pa... |
pconncn 33186 | The property of being a pa... |
pconntop 33187 | A simply connected space i... |
issconn 33188 | The property of being a si... |
sconnpconn 33189 | A simply connected space i... |
sconntop 33190 | A simply connected space i... |
sconnpht 33191 | A closed path in a simply ... |
cnpconn 33192 | An image of a path-connect... |
pconnconn 33193 | A path-connected space is ... |
txpconn 33194 | The topological product of... |
ptpconn 33195 | The topological product of... |
indispconn 33196 | The indiscrete topology (o... |
connpconn 33197 | A connected and locally pa... |
qtoppconn 33198 | A quotient of a path-conne... |
pconnpi1 33199 | All fundamental groups in ... |
sconnpht2 33200 | Any two paths in a simply ... |
sconnpi1 33201 | A path-connected topologic... |
txsconnlem 33202 | Lemma for ~ txsconn . (Co... |
txsconn 33203 | The topological product of... |
cvxpconn 33204 | A convex subset of the com... |
cvxsconn 33205 | A convex subset of the com... |
blsconn 33206 | An open ball in the comple... |
cnllysconn 33207 | The topology of the comple... |
resconn 33208 | A subset of ` RR ` is simp... |
ioosconn 33209 | An open interval is simply... |
iccsconn 33210 | A closed interval is simpl... |
retopsconn 33211 | The real numbers are simpl... |
iccllysconn 33212 | A closed interval is local... |
rellysconn 33213 | The real numbers are local... |
iisconn 33214 | The unit interval is simpl... |
iillysconn 33215 | The unit interval is local... |
iinllyconn 33216 | The unit interval is local... |
fncvm 33219 | Lemma for covering maps. ... |
cvmscbv 33220 | Change bound variables in ... |
iscvm 33221 | The property of being a co... |
cvmtop1 33222 | Reverse closure for a cove... |
cvmtop2 33223 | Reverse closure for a cove... |
cvmcn 33224 | A covering map is a contin... |
cvmcov 33225 | Property of a covering map... |
cvmsrcl 33226 | Reverse closure for an eve... |
cvmsi 33227 | One direction of ~ cvmsval... |
cvmsval 33228 | Elementhood in the set ` S... |
cvmsss 33229 | An even covering is a subs... |
cvmsn0 33230 | An even covering is nonemp... |
cvmsuni 33231 | An even covering of ` U ` ... |
cvmsdisj 33232 | An even covering of ` U ` ... |
cvmshmeo 33233 | Every element of an even c... |
cvmsf1o 33234 | ` F ` , localized to an el... |
cvmscld 33235 | The sets of an even coveri... |
cvmsss2 33236 | An open subset of an evenl... |
cvmcov2 33237 | The covering map property ... |
cvmseu 33238 | Every element in ` U. T ` ... |
cvmsiota 33239 | Identify the unique elemen... |
cvmopnlem 33240 | Lemma for ~ cvmopn . (Con... |
cvmfolem 33241 | Lemma for ~ cvmfo . (Cont... |
cvmopn 33242 | A covering map is an open ... |
cvmliftmolem1 33243 | Lemma for ~ cvmliftmo . (... |
cvmliftmolem2 33244 | Lemma for ~ cvmliftmo . (... |
cvmliftmoi 33245 | A lift of a continuous fun... |
cvmliftmo 33246 | A lift of a continuous fun... |
cvmliftlem1 33247 | Lemma for ~ cvmlift . In ... |
cvmliftlem2 33248 | Lemma for ~ cvmlift . ` W ... |
cvmliftlem3 33249 | Lemma for ~ cvmlift . Sin... |
cvmliftlem4 33250 | Lemma for ~ cvmlift . The... |
cvmliftlem5 33251 | Lemma for ~ cvmlift . Def... |
cvmliftlem6 33252 | Lemma for ~ cvmlift . Ind... |
cvmliftlem7 33253 | Lemma for ~ cvmlift . Pro... |
cvmliftlem8 33254 | Lemma for ~ cvmlift . The... |
cvmliftlem9 33255 | Lemma for ~ cvmlift . The... |
cvmliftlem10 33256 | Lemma for ~ cvmlift . The... |
cvmliftlem11 33257 | Lemma for ~ cvmlift . (Co... |
cvmliftlem13 33258 | Lemma for ~ cvmlift . The... |
cvmliftlem14 33259 | Lemma for ~ cvmlift . Put... |
cvmliftlem15 33260 | Lemma for ~ cvmlift . Dis... |
cvmlift 33261 | One of the important prope... |
cvmfo 33262 | A covering map is an onto ... |
cvmliftiota 33263 | Write out a function ` H `... |
cvmlift2lem1 33264 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem9a 33265 | Lemma for ~ cvmlift2 and ~... |
cvmlift2lem2 33266 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem3 33267 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem4 33268 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem5 33269 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem6 33270 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem7 33271 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem8 33272 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem9 33273 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem10 33274 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem11 33275 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem12 33276 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem13 33277 | Lemma for ~ cvmlift2 . (C... |
cvmlift2 33278 | A two-dimensional version ... |
cvmliftphtlem 33279 | Lemma for ~ cvmliftpht . ... |
cvmliftpht 33280 | If ` G ` and ` H ` are pat... |
cvmlift3lem1 33281 | Lemma for ~ cvmlift3 . (C... |
cvmlift3lem2 33282 | Lemma for ~ cvmlift2 . (C... |
cvmlift3lem3 33283 | Lemma for ~ cvmlift2 . (C... |
cvmlift3lem4 33284 | Lemma for ~ cvmlift2 . (C... |
cvmlift3lem5 33285 | Lemma for ~ cvmlift2 . (C... |
cvmlift3lem6 33286 | Lemma for ~ cvmlift3 . (C... |
cvmlift3lem7 33287 | Lemma for ~ cvmlift3 . (C... |
cvmlift3lem8 33288 | Lemma for ~ cvmlift2 . (C... |
cvmlift3lem9 33289 | Lemma for ~ cvmlift2 . (C... |
cvmlift3 33290 | A general version of ~ cvm... |
snmlff 33291 | The function ` F ` from ~ ... |
snmlfval 33292 | The function ` F ` from ~ ... |
snmlval 33293 | The property " ` A ` is si... |
snmlflim 33294 | If ` A ` is simply normal,... |
goel 33309 | A "Godel-set of membership... |
goelel3xp 33310 | A "Godel-set of membership... |
goeleq12bg 33311 | Two "Godel-set of membersh... |
gonafv 33312 | The "Godel-set for the She... |
goaleq12d 33313 | Equality of the "Godel-set... |
gonanegoal 33314 | The Godel-set for the Shef... |
satf 33315 | The satisfaction predicate... |
satfsucom 33316 | The satisfaction predicate... |
satfn 33317 | The satisfaction predicate... |
satom 33318 | The satisfaction predicate... |
satfvsucom 33319 | The satisfaction predicate... |
satfv0 33320 | The value of the satisfact... |
satfvsuclem1 33321 | Lemma 1 for ~ satfvsuc . ... |
satfvsuclem2 33322 | Lemma 2 for ~ satfvsuc . ... |
satfvsuc 33323 | The value of the satisfact... |
satfv1lem 33324 | Lemma for ~ satfv1 . (Con... |
satfv1 33325 | The value of the satisfact... |
satfsschain 33326 | The binary relation of a s... |
satfvsucsuc 33327 | The satisfaction predicate... |
satfbrsuc 33328 | The binary relation of a s... |
satfrel 33329 | The value of the satisfact... |
satfdmlem 33330 | Lemma for ~ satfdm . (Con... |
satfdm 33331 | The domain of the satisfac... |
satfrnmapom 33332 | The range of the satisfact... |
satfv0fun 33333 | The value of the satisfact... |
satf0 33334 | The satisfaction predicate... |
satf0sucom 33335 | The satisfaction predicate... |
satf00 33336 | The value of the satisfact... |
satf0suclem 33337 | Lemma for ~ satf0suc , ~ s... |
satf0suc 33338 | The value of the satisfact... |
satf0op 33339 | An element of a value of t... |
satf0n0 33340 | The value of the satisfact... |
sat1el2xp 33341 | The first component of an ... |
fmlafv 33342 | The valid Godel formulas o... |
fmla 33343 | The set of all valid Godel... |
fmla0 33344 | The valid Godel formulas o... |
fmla0xp 33345 | The valid Godel formulas o... |
fmlasuc0 33346 | The valid Godel formulas o... |
fmlafvel 33347 | A class is a valid Godel f... |
fmlasuc 33348 | The valid Godel formulas o... |
fmla1 33349 | The valid Godel formulas o... |
isfmlasuc 33350 | The characterization of a ... |
fmlasssuc 33351 | The Godel formulas of heig... |
fmlaomn0 33352 | The empty set is not a God... |
fmlan0 33353 | The empty set is not a God... |
gonan0 33354 | The "Godel-set of NAND" is... |
goaln0 33355 | The "Godel-set of universa... |
gonarlem 33356 | Lemma for ~ gonar (inducti... |
gonar 33357 | If the "Godel-set of NAND"... |
goalrlem 33358 | Lemma for ~ goalr (inducti... |
goalr 33359 | If the "Godel-set of unive... |
fmla0disjsuc 33360 | The set of valid Godel for... |
fmlasucdisj 33361 | The valid Godel formulas o... |
satfdmfmla 33362 | The domain of the satisfac... |
satffunlem 33363 | Lemma for ~ satffunlem1lem... |
satffunlem1lem1 33364 | Lemma for ~ satffunlem1 . ... |
satffunlem1lem2 33365 | Lemma 2 for ~ satffunlem1 ... |
satffunlem2lem1 33366 | Lemma 1 for ~ satffunlem2 ... |
dmopab3rexdif 33367 | The domain of an ordered p... |
satffunlem2lem2 33368 | Lemma 2 for ~ satffunlem2 ... |
satffunlem1 33369 | Lemma 1 for ~ satffun : in... |
satffunlem2 33370 | Lemma 2 for ~ satffun : in... |
satffun 33371 | The value of the satisfact... |
satff 33372 | The satisfaction predicate... |
satfun 33373 | The satisfaction predicate... |
satfvel 33374 | An element of the value of... |
satfv0fvfmla0 33375 | The value of the satisfact... |
satefv 33376 | The simplified satisfactio... |
sate0 33377 | The simplified satisfactio... |
satef 33378 | The simplified satisfactio... |
sate0fv0 33379 | A simplified satisfaction ... |
satefvfmla0 33380 | The simplified satisfactio... |
sategoelfvb 33381 | Characterization of a valu... |
sategoelfv 33382 | Condition of a valuation `... |
ex-sategoelel 33383 | Example of a valuation of ... |
ex-sategoel 33384 | Instance of ~ sategoelfv f... |
satfv1fvfmla1 33385 | The value of the satisfact... |
2goelgoanfmla1 33386 | Two Godel-sets of membersh... |
satefvfmla1 33387 | The simplified satisfactio... |
ex-sategoelelomsuc 33388 | Example of a valuation of ... |
ex-sategoelel12 33389 | Example of a valuation of ... |
prv 33390 | The "proves" relation on a... |
elnanelprv 33391 | The wff ` ( A e. B -/\ B e... |
prv0 33392 | Every wff encoded as ` U `... |
prv1n 33393 | No wff encoded as a Godel-... |
mvtval 33462 | The set of variable typeco... |
mrexval 33463 | The set of "raw expression... |
mexval 33464 | The set of expressions, wh... |
mexval2 33465 | The set of expressions, wh... |
mdvval 33466 | The set of disjoint variab... |
mvrsval 33467 | The set of variables in an... |
mvrsfpw 33468 | The set of variables in an... |
mrsubffval 33469 | The substitution of some v... |
mrsubfval 33470 | The substitution of some v... |
mrsubval 33471 | The substitution of some v... |
mrsubcv 33472 | The value of a substituted... |
mrsubvr 33473 | The value of a substituted... |
mrsubff 33474 | A substitution is a functi... |
mrsubrn 33475 | Although it is defined for... |
mrsubff1 33476 | When restricted to complet... |
mrsubff1o 33477 | When restricted to complet... |
mrsub0 33478 | The value of the substitut... |
mrsubf 33479 | A substitution is a functi... |
mrsubccat 33480 | Substitution distributes o... |
mrsubcn 33481 | A substitution does not ch... |
elmrsubrn 33482 | Characterization of the su... |
mrsubco 33483 | The composition of two sub... |
mrsubvrs 33484 | The set of variables in a ... |
msubffval 33485 | A substitution applied to ... |
msubfval 33486 | A substitution applied to ... |
msubval 33487 | A substitution applied to ... |
msubrsub 33488 | A substitution applied to ... |
msubty 33489 | The type of a substituted ... |
elmsubrn 33490 | Characterization of substi... |
msubrn 33491 | Although it is defined for... |
msubff 33492 | A substitution is a functi... |
msubco 33493 | The composition of two sub... |
msubf 33494 | A substitution is a functi... |
mvhfval 33495 | Value of the function mapp... |
mvhval 33496 | Value of the function mapp... |
mpstval 33497 | A pre-statement is an orde... |
elmpst 33498 | Property of being a pre-st... |
msrfval 33499 | Value of the reduct of a p... |
msrval 33500 | Value of the reduct of a p... |
mpstssv 33501 | A pre-statement is an orde... |
mpst123 33502 | Decompose a pre-statement ... |
mpstrcl 33503 | The elements of a pre-stat... |
msrf 33504 | The reduct of a pre-statem... |
msrrcl 33505 | If ` X ` and ` Y ` have th... |
mstaval 33506 | Value of the set of statem... |
msrid 33507 | The reduct of a statement ... |
msrfo 33508 | The reduct of a pre-statem... |
mstapst 33509 | A statement is a pre-state... |
elmsta 33510 | Property of being a statem... |
ismfs 33511 | A formal system is a tuple... |
mfsdisj 33512 | The constants and variable... |
mtyf2 33513 | The type function maps var... |
mtyf 33514 | The type function maps var... |
mvtss 33515 | The set of variable typeco... |
maxsta 33516 | An axiom is a statement. ... |
mvtinf 33517 | Each variable typecode has... |
msubff1 33518 | When restricted to complet... |
msubff1o 33519 | When restricted to complet... |
mvhf 33520 | The function mapping varia... |
mvhf1 33521 | The function mapping varia... |
msubvrs 33522 | The set of variables in a ... |
mclsrcl 33523 | Reverse closure for the cl... |
mclsssvlem 33524 | Lemma for ~ mclsssv . (Co... |
mclsval 33525 | The function mapping varia... |
mclsssv 33526 | The closure of a set of ex... |
ssmclslem 33527 | Lemma for ~ ssmcls . (Con... |
vhmcls 33528 | All variable hypotheses ar... |
ssmcls 33529 | The original expressions a... |
ss2mcls 33530 | The closure is monotonic u... |
mclsax 33531 | The closure is closed unde... |
mclsind 33532 | Induction theorem for clos... |
mppspstlem 33533 | Lemma for ~ mppspst . (Co... |
mppsval 33534 | Definition of a provable p... |
elmpps 33535 | Definition of a provable p... |
mppspst 33536 | A provable pre-statement i... |
mthmval 33537 | A theorem is a pre-stateme... |
elmthm 33538 | A theorem is a pre-stateme... |
mthmi 33539 | A statement whose reduct i... |
mthmsta 33540 | A theorem is a pre-stateme... |
mppsthm 33541 | A provable pre-statement i... |
mthmblem 33542 | Lemma for ~ mthmb . (Cont... |
mthmb 33543 | If two statements have the... |
mthmpps 33544 | Given a theorem, there is ... |
mclsppslem 33545 | The closure is closed unde... |
mclspps 33546 | The closure is closed unde... |
problem1 33623 | Practice problem 1. Clues... |
problem2 33624 | Practice problem 2. Clues... |
problem3 33625 | Practice problem 3. Clues... |
problem4 33626 | Practice problem 4. Clues... |
problem5 33627 | Practice problem 5. Clues... |
quad3 33628 | Variant of quadratic equat... |
climuzcnv 33629 | Utility lemma to convert b... |
sinccvglem 33630 | ` ( ( sin `` x ) / x ) ~~>... |
sinccvg 33631 | ` ( ( sin `` x ) / x ) ~~>... |
circum 33632 | The circumference of a cir... |
elfzm12 33633 | Membership in a curtailed ... |
nn0seqcvg 33634 | A strictly-decreasing nonn... |
lediv2aALT 33635 | Division of both sides of ... |
abs2sqlei 33636 | The absolute values of two... |
abs2sqlti 33637 | The absolute values of two... |
abs2sqle 33638 | The absolute values of two... |
abs2sqlt 33639 | The absolute values of two... |
abs2difi 33640 | Difference of absolute val... |
abs2difabsi 33641 | Absolute value of differen... |
axextprim 33642 | ~ ax-ext without distinct ... |
axrepprim 33643 | ~ ax-rep without distinct ... |
axunprim 33644 | ~ ax-un without distinct v... |
axpowprim 33645 | ~ ax-pow without distinct ... |
axregprim 33646 | ~ ax-reg without distinct ... |
axinfprim 33647 | ~ ax-inf without distinct ... |
axacprim 33648 | ~ ax-ac without distinct v... |
untelirr 33649 | We call a class "untanged"... |
untuni 33650 | The union of a class is un... |
untsucf 33651 | If a class is untangled, t... |
unt0 33652 | The null set is untangled.... |
untint 33653 | If there is an untangled e... |
efrunt 33654 | If ` A ` is well-founded b... |
untangtr 33655 | A transitive class is unta... |
3pm3.2ni 33656 | Triple negated disjunction... |
3jaodd 33657 | Double deduction form of ~... |
3orit 33658 | Closed form of ~ 3ori . (... |
biimpexp 33659 | A biconditional in the ant... |
3orel13 33660 | Elimination of two disjunc... |
onelssex 33661 | Ordinal less than is equiv... |
nepss 33662 | Two classes are unequal if... |
3ccased 33663 | Triple disjunction form of... |
dfso3 33664 | Expansion of the definitio... |
brtpid1 33665 | A binary relation involvin... |
brtpid2 33666 | A binary relation involvin... |
brtpid3 33667 | A binary relation involvin... |
ceqsrexv2 33668 | Alternate elimitation of a... |
iota5f 33669 | A method for computing iot... |
ceqsralv2 33670 | Alternate elimination of a... |
dford5 33671 | A class is ordinal iff it ... |
jath 33672 | Closed form of ~ ja . Pro... |
riotarab 33673 | Restricted iota of a restr... |
reurab 33674 | Restricted existential uni... |
snres0 33675 | Condition for restriction ... |
fnssintima 33676 | Condition for subset of an... |
xpab 33677 | Cross product of two class... |
ralxpes 33678 | A version of ~ ralxp with ... |
ot2elxp 33679 | Ordered triple membership ... |
ot21std 33680 | Extract the first member o... |
ot22ndd 33681 | Extract the second member ... |
otthne 33682 | Contrapositive of the orde... |
elxpxp 33683 | Membership in a triple cro... |
elxpxpss 33684 | Version of ~ elrel for tri... |
ralxp3f 33685 | Restricted for all over a ... |
ralxp3 33686 | Restricted for-all over a ... |
sbcoteq1a 33687 | Equality theorem for subst... |
ralxp3es 33688 | Restricted for-all over a ... |
onunel 33689 | The union of two ordinals ... |
imaeqsexv 33690 | Substitute a function valu... |
imaeqsalv 33691 | Substitute a function valu... |
nnuni 33692 | The union of a finite ordi... |
sqdivzi 33693 | Distribution of square ove... |
supfz 33694 | The supremum of a finite s... |
inffz 33695 | The infimum of a finite se... |
fz0n 33696 | The sequence ` ( 0 ... ( N... |
shftvalg 33697 | Value of a sequence shifte... |
divcnvlin 33698 | Limit of the ratio of two ... |
climlec3 33699 | Comparison of a constant t... |
logi 33700 | Calculate the logarithm of... |
iexpire 33701 | ` _i ` raised to itself is... |
bcneg1 33702 | The binomial coefficent ov... |
bcm1nt 33703 | The proportion of one bion... |
bcprod 33704 | A product identity for bin... |
bccolsum 33705 | A column-sum rule for bino... |
iprodefisumlem 33706 | Lemma for ~ iprodefisum . ... |
iprodefisum 33707 | Applying the exponential f... |
iprodgam 33708 | An infinite product versio... |
faclimlem1 33709 | Lemma for ~ faclim . Clos... |
faclimlem2 33710 | Lemma for ~ faclim . Show... |
faclimlem3 33711 | Lemma for ~ faclim . Alge... |
faclim 33712 | An infinite product expres... |
iprodfac 33713 | An infinite product expres... |
faclim2 33714 | Another factorial limit du... |
gcd32 33715 | Swap the second and third ... |
gcdabsorb 33716 | Absorption law for gcd. (... |
brtp 33717 | A condition for a binary r... |
dftr6 33718 | A potential definition of ... |
coep 33719 | Composition with the membe... |
coepr 33720 | Composition with the conve... |
dffr5 33721 | A quantifier-free definiti... |
dfso2 33722 | Quantifier-free definition... |
br8 33723 | Substitution for an eight-... |
br6 33724 | Substitution for a six-pla... |
br4 33725 | Substitution for a four-pl... |
cnvco1 33726 | Another distributive law o... |
cnvco2 33727 | Another distributive law o... |
eldm3 33728 | Quantifier-free definition... |
elrn3 33729 | Quantifier-free definition... |
pocnv 33730 | The converse of a partial ... |
socnv 33731 | The converse of a strict o... |
sotrd 33732 | Transitivity law for stric... |
sotr3 33733 | Transitivity law for stric... |
sotrine 33734 | Trichotomy law for strict ... |
eqfunresadj 33735 | Law for adjoining an eleme... |
eqfunressuc 33736 | Law for equality of restri... |
funeldmb 33737 | If ` (/) ` is not part of ... |
elintfv 33738 | Membership in an intersect... |
funpsstri 33739 | A condition for subset tri... |
fundmpss 33740 | If a class ` F ` is a prop... |
fvresval 33741 | The value of a function at... |
funsseq 33742 | Given two functions with e... |
fununiq 33743 | The uniqueness condition o... |
funbreq 33744 | An equality condition for ... |
br1steq 33745 | Uniqueness condition for t... |
br2ndeq 33746 | Uniqueness condition for t... |
dfdm5 33747 | Definition of domain in te... |
dfrn5 33748 | Definition of range in ter... |
opelco3 33749 | Alternate way of saying th... |
elima4 33750 | Quantifier-free expression... |
fv1stcnv 33751 | The value of the converse ... |
fv2ndcnv 33752 | The value of the converse ... |
imaindm 33753 | The image is unaffected by... |
setinds 33754 | Principle of set induction... |
setinds2f 33755 | ` _E ` induction schema, u... |
setinds2 33756 | ` _E ` induction schema, u... |
elpotr 33757 | A class of transitive sets... |
dford5reg 33758 | Given ~ ax-reg , an ordina... |
dfon2lem1 33759 | Lemma for ~ dfon2 . (Cont... |
dfon2lem2 33760 | Lemma for ~ dfon2 . (Cont... |
dfon2lem3 33761 | Lemma for ~ dfon2 . All s... |
dfon2lem4 33762 | Lemma for ~ dfon2 . If tw... |
dfon2lem5 33763 | Lemma for ~ dfon2 . Two s... |
dfon2lem6 33764 | Lemma for ~ dfon2 . A tra... |
dfon2lem7 33765 | Lemma for ~ dfon2 . All e... |
dfon2lem8 33766 | Lemma for ~ dfon2 . The i... |
dfon2lem9 33767 | Lemma for ~ dfon2 . A cla... |
dfon2 33768 | ` On ` consists of all set... |
rdgprc0 33769 | The value of the recursive... |
rdgprc 33770 | The value of the recursive... |
dfrdg2 33771 | Alternate definition of th... |
dfrdg3 33772 | Generalization of ~ dfrdg2... |
axextdfeq 33773 | A version of ~ ax-ext for ... |
ax8dfeq 33774 | A version of ~ ax-8 for us... |
axextdist 33775 | ~ ax-ext with distinctors ... |
axextbdist 33776 | ~ axextb with distinctors ... |
19.12b 33777 | Version of ~ 19.12vv with ... |
exnel 33778 | There is always a set not ... |
distel 33779 | Distinctors in terms of me... |
axextndbi 33780 | ~ axextnd as a bicondition... |
hbntg 33781 | A more general form of ~ h... |
hbimtg 33782 | A more general and closed ... |
hbaltg 33783 | A more general and closed ... |
hbng 33784 | A more general form of ~ h... |
hbimg 33785 | A more general form of ~ h... |
tfisg 33786 | A closed form of ~ tfis . ... |
frpoins3xpg 33787 | Special case of founded pa... |
frpoins3xp3g 33788 | Special case of founded pa... |
xpord2lem 33789 | Lemma for cross product or... |
poxp2 33790 | Another way of partially o... |
frxp2 33791 | Another way of giving a fo... |
xpord2pred 33792 | Calculate the predecessor ... |
sexp2 33793 | Condition for the relation... |
xpord2ind 33794 | Induction over the cross p... |
xpord3lem 33795 | Lemma for triple ordering.... |
poxp3 33796 | Triple cross product parti... |
frxp3 33797 | Give foundedness over a tr... |
xpord3pred 33798 | Calculate the predecsessor... |
sexp3 33799 | Show that the triple order... |
xpord3ind 33800 | Induction over the triple ... |
orderseqlem 33801 | Lemma for ~ poseq and ~ so... |
poseq 33802 | A partial ordering of sequ... |
soseq 33803 | A linear ordering of seque... |
wsuceq123 33808 | Equality theorem for well-... |
wsuceq1 33809 | Equality theorem for well-... |
wsuceq2 33810 | Equality theorem for well-... |
wsuceq3 33811 | Equality theorem for well-... |
nfwsuc 33812 | Bound-variable hypothesis ... |
wlimeq12 33813 | Equality theorem for the l... |
wlimeq1 33814 | Equality theorem for the l... |
wlimeq2 33815 | Equality theorem for the l... |
nfwlim 33816 | Bound-variable hypothesis ... |
elwlim 33817 | Membership in the limit cl... |
wzel 33818 | The zero of a well-founded... |
wsuclem 33819 | Lemma for the supremum pro... |
wsucex 33820 | Existence theorem for well... |
wsuccl 33821 | If ` X ` is a set with an ... |
wsuclb 33822 | A well-founded successor i... |
wlimss 33823 | The class of limit points ... |
on2recsfn 33826 | Show that double recursion... |
on2recsov 33827 | Calculate the value of the... |
on2ind 33828 | Double induction over ordi... |
on3ind 33829 | Triple induction over ordi... |
naddfn 33830 | Natural addition is a func... |
naddcllem 33831 | Lemma for ordinal addition... |
naddcl 33832 | Closure law for natural ad... |
naddov 33833 | The value of natural addit... |
naddov2 33834 | Alternate expression for n... |
naddcom 33835 | Natural addition commutes.... |
naddid1 33836 | Ordinal zero is the additi... |
naddssim 33837 | Ordinal less-than-or-equal... |
naddelim 33838 | Ordinal less-than is prese... |
naddel1 33839 | Ordinal less-than is not a... |
naddel2 33840 | Ordinal less-than is not a... |
naddss1 33841 | Ordinal less-than-or-equal... |
naddss2 33842 | Ordinal less-than-or-equal... |
elno 33849 | Membership in the surreals... |
sltval 33850 | The value of the surreal l... |
bdayval 33851 | The value of the birthday ... |
nofun 33852 | A surreal is a function. ... |
nodmon 33853 | The domain of a surreal is... |
norn 33854 | The range of a surreal is ... |
nofnbday 33855 | A surreal is a function ov... |
nodmord 33856 | The domain of a surreal ha... |
elno2 33857 | An alternative condition f... |
elno3 33858 | Another condition for memb... |
sltval2 33859 | Alternate expression for s... |
nofv 33860 | The function value of a su... |
nosgnn0 33861 | ` (/) ` is not a surreal s... |
nosgnn0i 33862 | If ` X ` is a surreal sign... |
noreson 33863 | The restriction of a surre... |
sltintdifex 33864 |
If ` A |
sltres 33865 | If the restrictions of two... |
noxp1o 33866 | The Cartesian product of a... |
noseponlem 33867 | Lemma for ~ nosepon . Con... |
nosepon 33868 | Given two unequal surreals... |
noextend 33869 | Extending a surreal by one... |
noextendseq 33870 | Extend a surreal by a sequ... |
noextenddif 33871 | Calculate the place where ... |
noextendlt 33872 | Extending a surreal with a... |
noextendgt 33873 | Extending a surreal with a... |
nolesgn2o 33874 | Given ` A ` less than or e... |
nolesgn2ores 33875 | Given ` A ` less than or e... |
nogesgn1o 33876 | Given ` A ` greater than o... |
nogesgn1ores 33877 | Given ` A ` greater than o... |
sltsolem1 33878 | Lemma for ~ sltso . The s... |
sltso 33879 | Surreal less than totally ... |
bdayfo 33880 | The birthday function maps... |
fvnobday 33881 | The value of a surreal at ... |
nosepnelem 33882 | Lemma for ~ nosepne . (Co... |
nosepne 33883 | The value of two non-equal... |
nosep1o 33884 | If the value of a surreal ... |
nosep2o 33885 | If the value of a surreal ... |
nosepdmlem 33886 | Lemma for ~ nosepdm . (Co... |
nosepdm 33887 | The first place two surrea... |
nosepeq 33888 | The values of two surreals... |
nosepssdm 33889 | Given two non-equal surrea... |
nodenselem4 33890 | Lemma for ~ nodense . Sho... |
nodenselem5 33891 | Lemma for ~ nodense . If ... |
nodenselem6 33892 | The restriction of a surre... |
nodenselem7 33893 | Lemma for ~ nodense . ` A ... |
nodenselem8 33894 | Lemma for ~ nodense . Giv... |
nodense 33895 | Given two distinct surreal... |
bdayimaon 33896 | Lemma for full-eta propert... |
nolt02olem 33897 | Lemma for ~ nolt02o . If ... |
nolt02o 33898 | Given ` A ` less than ` B ... |
nogt01o 33899 | Given ` A ` greater than `... |
noresle 33900 | Restriction law for surrea... |
nomaxmo 33901 | A class of surreals has at... |
nominmo 33902 | A class of surreals has at... |
nosupprefixmo 33903 | In any class of surreals, ... |
noinfprefixmo 33904 | In any class of surreals, ... |
nosupcbv 33905 | Lemma to change bound vari... |
nosupno 33906 | The next several theorems ... |
nosupdm 33907 | The domain of the surreal ... |
nosupbday 33908 | Birthday bounding law for ... |
nosupfv 33909 | The value of surreal supre... |
nosupres 33910 | A restriction law for surr... |
nosupbnd1lem1 33911 | Lemma for ~ nosupbnd1 . E... |
nosupbnd1lem2 33912 | Lemma for ~ nosupbnd1 . W... |
nosupbnd1lem3 33913 | Lemma for ~ nosupbnd1 . I... |
nosupbnd1lem4 33914 | Lemma for ~ nosupbnd1 . I... |
nosupbnd1lem5 33915 | Lemma for ~ nosupbnd1 . I... |
nosupbnd1lem6 33916 | Lemma for ~ nosupbnd1 . E... |
nosupbnd1 33917 | Bounding law from below fo... |
nosupbnd2lem1 33918 | Bounding law from above wh... |
nosupbnd2 33919 | Bounding law from above fo... |
noinfcbv 33920 | Change bound variables for... |
noinfno 33921 | The next several theorems ... |
noinfdm 33922 | Next, we calculate the dom... |
noinfbday 33923 | Birthday bounding law for ... |
noinffv 33924 | The value of surreal infim... |
noinfres 33925 | The restriction of surreal... |
noinfbnd1lem1 33926 | Lemma for ~ noinfbnd1 . E... |
noinfbnd1lem2 33927 | Lemma for ~ noinfbnd1 . W... |
noinfbnd1lem3 33928 | Lemma for ~ noinfbnd1 . I... |
noinfbnd1lem4 33929 | Lemma for ~ noinfbnd1 . I... |
noinfbnd1lem5 33930 | Lemma for ~ noinfbnd1 . I... |
noinfbnd1lem6 33931 | Lemma for ~ noinfbnd1 . E... |
noinfbnd1 33932 | Bounding law from above fo... |
noinfbnd2lem1 33933 | Bounding law from below wh... |
noinfbnd2 33934 | Bounding law from below fo... |
nosupinfsep 33935 | Given two sets of surreals... |
noetasuplem1 33936 | Lemma for ~ noeta . Estab... |
noetasuplem2 33937 | Lemma for ~ noeta . The r... |
noetasuplem3 33938 | Lemma for ~ noeta . ` Z ` ... |
noetasuplem4 33939 | Lemma for ~ noeta . When ... |
noetainflem1 33940 | Lemma for ~ noeta . Estab... |
noetainflem2 33941 | Lemma for ~ noeta . The r... |
noetainflem3 33942 | Lemma for ~ noeta . ` W ` ... |
noetainflem4 33943 | Lemma for ~ noeta . If ` ... |
noetalem1 33944 | Lemma for ~ noeta . Eithe... |
noetalem2 33945 | Lemma for ~ noeta . The f... |
noeta 33946 | The full-eta axiom for the... |
sltirr 33949 | Surreal less than is irref... |
slttr 33950 | Surreal less than is trans... |
sltasym 33951 | Surreal less than is asymm... |
sltlin 33952 | Surreal less than obeys tr... |
slttrieq2 33953 | Trichotomy law for surreal... |
slttrine 33954 | Trichotomy law for surreal... |
slenlt 33955 | Surreal less than or equal... |
sltnle 33956 | Surreal less than in terms... |
sleloe 33957 | Surreal less than or equal... |
sletri3 33958 | Trichotomy law for surreal... |
sltletr 33959 | Surreal transitive law. (... |
slelttr 33960 | Surreal transitive law. (... |
sletr 33961 | Surreal transitive law. (... |
slttrd 33962 | Surreal less than is trans... |
sltletrd 33963 | Surreal less than is trans... |
slelttrd 33964 | Surreal less than is trans... |
sletrd 33965 | Surreal less than or equal... |
slerflex 33966 | Surreal less than or equal... |
bdayfun 33967 | The birthday function is a... |
bdayfn 33968 | The birthday function is a... |
bdaydm 33969 | The birthday function's do... |
bdayrn 33970 | The birthday function's ra... |
bdayelon 33971 | The value of the birthday ... |
nocvxminlem 33972 | Lemma for ~ nocvxmin . Gi... |
nocvxmin 33973 | Given a nonempty convex cl... |
noprc 33974 | The surreal numbers are a ... |
noeta2 33979 | A version of ~ noeta with ... |
brsslt 33980 | Binary relation form of th... |
ssltex1 33981 | The first argument of surr... |
ssltex2 33982 | The second argument of sur... |
ssltss1 33983 | The first argument of surr... |
ssltss2 33984 | The second argument of sur... |
ssltsep 33985 | The separation property of... |
ssltd 33986 | Deduce surreal set less th... |
ssltsepc 33987 | Two elements of separated ... |
ssltsepcd 33988 | Two elements of separated ... |
sssslt1 33989 | Relationship between surre... |
sssslt2 33990 | Relationship between surre... |
nulsslt 33991 | The empty set is less than... |
nulssgt 33992 | The empty set is greater t... |
conway 33993 | Conway's Simplicity Theore... |
scutval 33994 | The value of the surreal c... |
scutcut 33995 | Cut properties of the surr... |
scutcl 33996 | Closure law for surreal cu... |
scutcld 33997 | Closure law for surreal cu... |
scutbday 33998 | The birthday of the surrea... |
eqscut 33999 | Condition for equality to ... |
eqscut2 34000 | Condition for equality to ... |
sslttr 34001 | Transitive law for surreal... |
ssltun1 34002 | Union law for surreal set ... |
ssltun2 34003 | Union law for surreal set ... |
scutun12 34004 | Union law for surreal cuts... |
dmscut 34005 | The domain of the surreal ... |
scutf 34006 | Functionality statement fo... |
etasslt 34007 | A restatement of ~ noeta u... |
etasslt2 34008 | A version of ~ etasslt wit... |
scutbdaybnd 34009 | An upper bound on the birt... |
scutbdaybnd2 34010 | An upper bound on the birt... |
scutbdaybnd2lim 34011 | An upper bound on the birt... |
scutbdaylt 34012 | If a surreal lies in a gap... |
slerec 34013 | A comparison law for surre... |
sltrec 34014 | A comparison law for surre... |
ssltdisj 34015 | If ` A ` preceeds ` B ` , ... |
0sno 34020 | Surreal zero is a surreal.... |
1sno 34021 | Surreal one is a surreal. ... |
bday0s 34022 | Calculate the birthday of ... |
0slt1s 34023 | Surreal zero is less than ... |
bday0b 34024 | The only surreal with birt... |
bday1s 34025 | The birthday of surreal on... |
madeval 34036 | The value of the made by f... |
madeval2 34037 | Alternative characterizati... |
oldval 34038 | The value of the old optio... |
newval 34039 | The value of the new optio... |
madef 34040 | The made function is a fun... |
oldf 34041 | The older function is a fu... |
newf 34042 | The new function is a func... |
old0 34043 | No surreal is older than `... |
madessno 34044 | Made sets are surreals. (... |
oldssno 34045 | Old sets are surreals. (C... |
newssno 34046 | New sets are surreals. (C... |
leftval 34047 | The value of the left opti... |
rightval 34048 | The value of the right opt... |
leftf 34049 | The functionality of the l... |
rightf 34050 | The functionality of the r... |
elmade 34051 | Membership in the made fun... |
elmade2 34052 | Membership in the made fun... |
elold 34053 | Membership in an old set. ... |
ssltleft 34054 | A surreal is greater than ... |
ssltright 34055 | A surreal is less than its... |
lltropt 34056 | The left options of a surr... |
made0 34057 | The only surreal made on d... |
new0 34058 | The only surreal new on da... |
madess 34059 | If ` A ` is less than or e... |
oldssmade 34060 | The older-than set is a su... |
leftssold 34061 | The left options are a sub... |
rightssold 34062 | The right options are a su... |
leftssno 34063 | The left set of a surreal ... |
rightssno 34064 | The right set of a surreal... |
madecut 34065 | Given a section that is a ... |
madeun 34066 | The made set is the union ... |
madeoldsuc 34067 | The made set is the old se... |
oldsuc 34068 | The value of the old set a... |
oldlim 34069 | The value of the old set a... |
madebdayim 34070 | If a surreal is a member o... |
oldbdayim 34071 | If ` X ` is in the old set... |
oldirr 34072 | No surreal is a member of ... |
leftirr 34073 | No surreal is a member of ... |
rightirr 34074 | No surreal is a member of ... |
left0s 34075 | The left set of ` 0s ` is ... |
right0s 34076 | The right set of ` 0s ` is... |
lrold 34077 | The union of the left and ... |
madebdaylemold 34078 | Lemma for ~ madebday . If... |
madebdaylemlrcut 34079 | Lemma for ~ madebday . If... |
madebday 34080 | A surreal is part of the s... |
oldbday 34081 | A surreal is part of the s... |
newbday 34082 | A surreal is an element of... |
lrcut 34083 | A surreal is equal to the ... |
scutfo 34084 | The surreal cut function i... |
sltn0 34085 | If ` X ` is less than ` Y ... |
lruneq 34086 | If two surreals share a bi... |
sltlpss 34087 | If two surreals share a bi... |
cofsslt 34088 | If every element of ` A ` ... |
coinitsslt 34089 | If ` B ` is coinitial with... |
cofcut1 34090 | If ` C ` is cofinal with `... |
cofcut2 34091 | If ` A ` and ` C ` are mut... |
cofcutr 34092 | If ` X ` is the cut of ` A... |
cofcutrtime 34093 | If ` X ` is the cut of ` A... |
lrrecval 34096 | The next step in the devel... |
lrrecval2 34097 | Next, we establish an alte... |
lrrecpo 34098 | Now, we establish that ` R... |
lrrecse 34099 | Next, we show that ` R ` i... |
lrrecfr 34100 | Now we show that ` R ` is ... |
lrrecpred 34101 | Finally, we calculate the ... |
noinds 34102 | Induction principle for a ... |
norecfn 34103 | Surreal recursion over one... |
norecov 34104 | Calculate the value of the... |
noxpordpo 34107 | To get through most of the... |
noxpordfr 34108 | Next we establish the foun... |
noxpordse 34109 | Next we establish the set-... |
noxpordpred 34110 | Next we calculate the pred... |
no2indslem 34111 | Double induction on surrea... |
no2inds 34112 | Double induction on surrea... |
norec2fn 34113 | The double-recursion opera... |
norec2ov 34114 | The value of the double-re... |
no3inds 34115 | Triple induction over surr... |
negsfn 34122 | Surreal negation is a func... |
negsval 34123 | The value of the surreal n... |
negs0s 34124 | Negative surreal zero is s... |
addsfn 34125 | Surreal addition is a func... |
addsval 34126 | The value of surreal addit... |
addsid1 34127 | Surreal addition to zero i... |
addsid1d 34128 | Surreal addition to zero i... |
addscom 34129 | Surreal addition commutes.... |
addscomd 34130 | Surreal addition commutes.... |
addscllem1 34131 | Lemma for addscl (future) ... |
txpss3v 34180 | A tail Cartesian product i... |
txprel 34181 | A tail Cartesian product i... |
brtxp 34182 | Characterize a ternary rel... |
brtxp2 34183 | The binary relation over a... |
dfpprod2 34184 | Expanded definition of par... |
pprodcnveq 34185 | A converse law for paralle... |
pprodss4v 34186 | The parallel product is a ... |
brpprod 34187 | Characterize a quaternary ... |
brpprod3a 34188 | Condition for parallel pro... |
brpprod3b 34189 | Condition for parallel pro... |
relsset 34190 | The subset class is a bina... |
brsset 34191 | For sets, the ` SSet ` bin... |
idsset 34192 | ` _I ` is equal to the int... |
eltrans 34193 | Membership in the class of... |
dfon3 34194 | A quantifier-free definiti... |
dfon4 34195 | Another quantifier-free de... |
brtxpsd 34196 | Expansion of a common form... |
brtxpsd2 34197 | Another common abbreviatio... |
brtxpsd3 34198 | A third common abbreviatio... |
relbigcup 34199 | The ` Bigcup ` relationshi... |
brbigcup 34200 | Binary relation over ` Big... |
dfbigcup2 34201 | ` Bigcup ` using maps-to n... |
fobigcup 34202 | ` Bigcup ` maps the univer... |
fnbigcup 34203 | ` Bigcup ` is a function o... |
fvbigcup 34204 | For sets, ` Bigcup ` yield... |
elfix 34205 | Membership in the fixpoint... |
elfix2 34206 | Alternative membership in ... |
dffix2 34207 | The fixpoints of a class i... |
fixssdm 34208 | The fixpoints of a class a... |
fixssrn 34209 | The fixpoints of a class a... |
fixcnv 34210 | The fixpoints of a class a... |
fixun 34211 | The fixpoint operator dist... |
ellimits 34212 | Membership in the class of... |
limitssson 34213 | The class of all limit ord... |
dfom5b 34214 | A quantifier-free definiti... |
sscoid 34215 | A condition for subset and... |
dffun10 34216 | Another potential definiti... |
elfuns 34217 | Membership in the class of... |
elfunsg 34218 | Closed form of ~ elfuns . ... |
brsingle 34219 | The binary relation form o... |
elsingles 34220 | Membership in the class of... |
fnsingle 34221 | The singleton relationship... |
fvsingle 34222 | The value of the singleton... |
dfsingles2 34223 | Alternate definition of th... |
snelsingles 34224 | A singleton is a member of... |
dfiota3 34225 | A definition of iota using... |
dffv5 34226 | Another quantifier-free de... |
unisnif 34227 | Express union of singleton... |
brimage 34228 | Binary relation form of th... |
brimageg 34229 | Closed form of ~ brimage .... |
funimage 34230 | ` Image A ` is a function.... |
fnimage 34231 | ` Image R ` is a function ... |
imageval 34232 | The image functor in maps-... |
fvimage 34233 | Value of the image functor... |
brcart 34234 | Binary relation form of th... |
brdomain 34235 | Binary relation form of th... |
brrange 34236 | Binary relation form of th... |
brdomaing 34237 | Closed form of ~ brdomain ... |
brrangeg 34238 | Closed form of ~ brrange .... |
brimg 34239 | Binary relation form of th... |
brapply 34240 | Binary relation form of th... |
brcup 34241 | Binary relation form of th... |
brcap 34242 | Binary relation form of th... |
brsuccf 34243 | Binary relation form of th... |
funpartlem 34244 | Lemma for ~ funpartfun . ... |
funpartfun 34245 | The functional part of ` F... |
funpartss 34246 | The functional part of ` F... |
funpartfv 34247 | The function value of the ... |
fullfunfnv 34248 | The full functional part o... |
fullfunfv 34249 | The function value of the ... |
brfullfun 34250 | A binary relation form con... |
brrestrict 34251 | Binary relation form of th... |
dfrecs2 34252 | A quantifier-free definiti... |
dfrdg4 34253 | A quantifier-free definiti... |
dfint3 34254 | Quantifier-free definition... |
imagesset 34255 | The Image functor applied ... |
brub 34256 | Binary relation form of th... |
brlb 34257 | Binary relation form of th... |
altopex 34262 | Alternative ordered pairs ... |
altopthsn 34263 | Two alternate ordered pair... |
altopeq12 34264 | Equality for alternate ord... |
altopeq1 34265 | Equality for alternate ord... |
altopeq2 34266 | Equality for alternate ord... |
altopth1 34267 | Equality of the first memb... |
altopth2 34268 | Equality of the second mem... |
altopthg 34269 | Alternate ordered pair the... |
altopthbg 34270 | Alternate ordered pair the... |
altopth 34271 | The alternate ordered pair... |
altopthb 34272 | Alternate ordered pair the... |
altopthc 34273 | Alternate ordered pair the... |
altopthd 34274 | Alternate ordered pair the... |
altxpeq1 34275 | Equality for alternate Car... |
altxpeq2 34276 | Equality for alternate Car... |
elaltxp 34277 | Membership in alternate Ca... |
altopelaltxp 34278 | Alternate ordered pair mem... |
altxpsspw 34279 | An inclusion rule for alte... |
altxpexg 34280 | The alternate Cartesian pr... |
rankaltopb 34281 | Compute the rank of an alt... |
nfaltop 34282 | Bound-variable hypothesis ... |
sbcaltop 34283 | Distribution of class subs... |
cgrrflx2d 34286 | Deduction form of ~ axcgrr... |
cgrtr4d 34287 | Deduction form of ~ axcgrt... |
cgrtr4and 34288 | Deduction form of ~ axcgrt... |
cgrrflx 34289 | Reflexivity law for congru... |
cgrrflxd 34290 | Deduction form of ~ cgrrfl... |
cgrcomim 34291 | Congruence commutes on the... |
cgrcom 34292 | Congruence commutes betwee... |
cgrcomand 34293 | Deduction form of ~ cgrcom... |
cgrtr 34294 | Transitivity law for congr... |
cgrtrand 34295 | Deduction form of ~ cgrtr ... |
cgrtr3 34296 | Transitivity law for congr... |
cgrtr3and 34297 | Deduction form of ~ cgrtr3... |
cgrcoml 34298 | Congruence commutes on the... |
cgrcomr 34299 | Congruence commutes on the... |
cgrcomlr 34300 | Congruence commutes on bot... |
cgrcomland 34301 | Deduction form of ~ cgrcom... |
cgrcomrand 34302 | Deduction form of ~ cgrcom... |
cgrcomlrand 34303 | Deduction form of ~ cgrcom... |
cgrtriv 34304 | Degenerate segments are co... |
cgrid2 34305 | Identity law for congruenc... |
cgrdegen 34306 | Two congruent segments are... |
brofs 34307 | Binary relation form of th... |
5segofs 34308 | Rephrase ~ ax5seg using th... |
ofscom 34309 | The outer five segment pre... |
cgrextend 34310 | Link congruence over a pai... |
cgrextendand 34311 | Deduction form of ~ cgrext... |
segconeq 34312 | Two points that satisfy th... |
segconeu 34313 | Existential uniqueness ver... |
btwntriv2 34314 | Betweenness always holds f... |
btwncomim 34315 | Betweenness commutes. Imp... |
btwncom 34316 | Betweenness commutes. (Co... |
btwncomand 34317 | Deduction form of ~ btwnco... |
btwntriv1 34318 | Betweenness always holds f... |
btwnswapid 34319 | If you can swap the first ... |
btwnswapid2 34320 | If you can swap arguments ... |
btwnintr 34321 | Inner transitivity law for... |
btwnexch3 34322 | Exchange the first endpoin... |
btwnexch3and 34323 | Deduction form of ~ btwnex... |
btwnouttr2 34324 | Outer transitivity law for... |
btwnexch2 34325 | Exchange the outer point o... |
btwnouttr 34326 | Outer transitivity law for... |
btwnexch 34327 | Outer transitivity law for... |
btwnexchand 34328 | Deduction form of ~ btwnex... |
btwndiff 34329 | There is always a ` c ` di... |
trisegint 34330 | A line segment between two... |
funtransport 34333 | The ` TransportTo ` relati... |
fvtransport 34334 | Calculate the value of the... |
transportcl 34335 | Closure law for segment tr... |
transportprops 34336 | Calculate the defining pro... |
brifs 34345 | Binary relation form of th... |
ifscgr 34346 | Inner five segment congrue... |
cgrsub 34347 | Removing identical parts f... |
brcgr3 34348 | Binary relation form of th... |
cgr3permute3 34349 | Permutation law for three-... |
cgr3permute1 34350 | Permutation law for three-... |
cgr3permute2 34351 | Permutation law for three-... |
cgr3permute4 34352 | Permutation law for three-... |
cgr3permute5 34353 | Permutation law for three-... |
cgr3tr4 34354 | Transitivity law for three... |
cgr3com 34355 | Commutativity law for thre... |
cgr3rflx 34356 | Identity law for three-pla... |
cgrxfr 34357 | A line segment can be divi... |
btwnxfr 34358 | A condition for extending ... |
colinrel 34359 | Colinearity is a relations... |
brcolinear2 34360 | Alternate colinearity bina... |
brcolinear 34361 | The binary relation form o... |
colinearex 34362 | The colinear predicate exi... |
colineardim1 34363 | If ` A ` is colinear with ... |
colinearperm1 34364 | Permutation law for coline... |
colinearperm3 34365 | Permutation law for coline... |
colinearperm2 34366 | Permutation law for coline... |
colinearperm4 34367 | Permutation law for coline... |
colinearperm5 34368 | Permutation law for coline... |
colineartriv1 34369 | Trivial case of colinearit... |
colineartriv2 34370 | Trivial case of colinearit... |
btwncolinear1 34371 | Betweenness implies coline... |
btwncolinear2 34372 | Betweenness implies coline... |
btwncolinear3 34373 | Betweenness implies coline... |
btwncolinear4 34374 | Betweenness implies coline... |
btwncolinear5 34375 | Betweenness implies coline... |
btwncolinear6 34376 | Betweenness implies coline... |
colinearxfr 34377 | Transfer law for colineari... |
lineext 34378 | Extend a line with a missi... |
brofs2 34379 | Change some conditions for... |
brifs2 34380 | Change some conditions for... |
brfs 34381 | Binary relation form of th... |
fscgr 34382 | Congruence law for the gen... |
linecgr 34383 | Congruence rule for lines.... |
linecgrand 34384 | Deduction form of ~ linecg... |
lineid 34385 | Identity law for points on... |
idinside 34386 | Law for finding a point in... |
endofsegid 34387 | If ` A ` , ` B ` , and ` C... |
endofsegidand 34388 | Deduction form of ~ endofs... |
btwnconn1lem1 34389 | Lemma for ~ btwnconn1 . T... |
btwnconn1lem2 34390 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem3 34391 | Lemma for ~ btwnconn1 . E... |
btwnconn1lem4 34392 | Lemma for ~ btwnconn1 . A... |
btwnconn1lem5 34393 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem6 34394 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem7 34395 | Lemma for ~ btwnconn1 . U... |
btwnconn1lem8 34396 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem9 34397 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem10 34398 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem11 34399 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem12 34400 | Lemma for ~ btwnconn1 . U... |
btwnconn1lem13 34401 | Lemma for ~ btwnconn1 . B... |
btwnconn1lem14 34402 | Lemma for ~ btwnconn1 . F... |
btwnconn1 34403 | Connectitivy law for betwe... |
btwnconn2 34404 | Another connectivity law f... |
btwnconn3 34405 | Inner connectivity law for... |
midofsegid 34406 | If two points fall in the ... |
segcon2 34407 | Generalization of ~ axsegc... |
brsegle 34410 | Binary relation form of th... |
brsegle2 34411 | Alternate characterization... |
seglecgr12im 34412 | Substitution law for segme... |
seglecgr12 34413 | Substitution law for segme... |
seglerflx 34414 | Segment comparison is refl... |
seglemin 34415 | Any segment is at least as... |
segletr 34416 | Segment less than is trans... |
segleantisym 34417 | Antisymmetry law for segme... |
seglelin 34418 | Linearity law for segment ... |
btwnsegle 34419 | If ` B ` falls between ` A... |
colinbtwnle 34420 | Given three colinear point... |
broutsideof 34423 | Binary relation form of ` ... |
broutsideof2 34424 | Alternate form of ` Outsid... |
outsidene1 34425 | Outsideness implies inequa... |
outsidene2 34426 | Outsideness implies inequa... |
btwnoutside 34427 | A principle linking outsid... |
broutsideof3 34428 | Characterization of outsid... |
outsideofrflx 34429 | Reflexivity of outsideness... |
outsideofcom 34430 | Commutativity law for outs... |
outsideoftr 34431 | Transitivity law for outsi... |
outsideofeq 34432 | Uniqueness law for ` Outsi... |
outsideofeu 34433 | Given a nondegenerate ray,... |
outsidele 34434 | Relate ` OutsideOf ` to ` ... |
outsideofcol 34435 | Outside of implies colinea... |
funray 34442 | Show that the ` Ray ` rela... |
fvray 34443 | Calculate the value of the... |
funline 34444 | Show that the ` Line ` rel... |
linedegen 34445 | When ` Line ` is applied w... |
fvline 34446 | Calculate the value of the... |
liness 34447 | A line is a subset of the ... |
fvline2 34448 | Alternate definition of a ... |
lineunray 34449 | A line is composed of a po... |
lineelsb2 34450 | If ` S ` lies on ` P Q ` ,... |
linerflx1 34451 | Reflexivity law for line m... |
linecom 34452 | Commutativity law for line... |
linerflx2 34453 | Reflexivity law for line m... |
ellines 34454 | Membership in the set of a... |
linethru 34455 | If ` A ` is a line contain... |
hilbert1.1 34456 | There is a line through an... |
hilbert1.2 34457 | There is at most one line ... |
linethrueu 34458 | There is a unique line goi... |
lineintmo 34459 | Two distinct lines interse... |
fwddifval 34464 | Calculate the value of the... |
fwddifnval 34465 | The value of the forward d... |
fwddifn0 34466 | The value of the n-iterate... |
fwddifnp1 34467 | The value of the n-iterate... |
rankung 34468 | The rank of the union of t... |
ranksng 34469 | The rank of a singleton. ... |
rankelg 34470 | The membership relation is... |
rankpwg 34471 | The rank of a power set. ... |
rank0 34472 | The rank of the empty set ... |
rankeq1o 34473 | The only set with rank ` 1... |
elhf 34476 | Membership in the heredita... |
elhf2 34477 | Alternate form of membersh... |
elhf2g 34478 | Hereditarily finiteness vi... |
0hf 34479 | The empty set is a heredit... |
hfun 34480 | The union of two HF sets i... |
hfsn 34481 | The singleton of an HF set... |
hfadj 34482 | Adjoining one HF element t... |
hfelhf 34483 | Any member of an HF set is... |
hftr 34484 | The class of all hereditar... |
hfext 34485 | Extensionality for HF sets... |
hfuni 34486 | The union of an HF set is ... |
hfpw 34487 | The power class of an HF s... |
hfninf 34488 | ` _om ` is not hereditaril... |
a1i14 34489 | Add two antecedents to a w... |
a1i24 34490 | Add two antecedents to a w... |
exp5d 34491 | An exportation inference. ... |
exp5g 34492 | An exportation inference. ... |
exp5k 34493 | An exportation inference. ... |
exp56 34494 | An exportation inference. ... |
exp58 34495 | An exportation inference. ... |
exp510 34496 | An exportation inference. ... |
exp511 34497 | An exportation inference. ... |
exp512 34498 | An exportation inference. ... |
3com12d 34499 | Commutation in consequent.... |
imp5p 34500 | A triple importation infer... |
imp5q 34501 | A triple importation infer... |
ecase13d 34502 | Deduction for elimination ... |
subtr 34503 | Transitivity of implicit s... |
subtr2 34504 | Transitivity of implicit s... |
trer 34505 | A relation intersected wit... |
elicc3 34506 | An equivalent membership c... |
finminlem 34507 | A useful lemma about finit... |
gtinf 34508 | Any number greater than an... |
opnrebl 34509 | A set is open in the stand... |
opnrebl2 34510 | A set is open in the stand... |
nn0prpwlem 34511 | Lemma for ~ nn0prpw . Use... |
nn0prpw 34512 | Two nonnegative integers a... |
topbnd 34513 | Two equivalent expressions... |
opnbnd 34514 | A set is open iff it is di... |
cldbnd 34515 | A set is closed iff it con... |
ntruni 34516 | A union of interiors is a ... |
clsun 34517 | A pairwise union of closur... |
clsint2 34518 | The closure of an intersec... |
opnregcld 34519 | A set is regularly closed ... |
cldregopn 34520 | A set if regularly open if... |
neiin 34521 | Two neighborhoods intersec... |
hmeoclda 34522 | Homeomorphisms preserve cl... |
hmeocldb 34523 | Homeomorphisms preserve cl... |
ivthALT 34524 | An alternate proof of the ... |
fnerel 34527 | Fineness is a relation. (... |
isfne 34528 | The predicate " ` B ` is f... |
isfne4 34529 | The predicate " ` B ` is f... |
isfne4b 34530 | A condition for a topology... |
isfne2 34531 | The predicate " ` B ` is f... |
isfne3 34532 | The predicate " ` B ` is f... |
fnebas 34533 | A finer cover covers the s... |
fnetg 34534 | A finer cover generates a ... |
fnessex 34535 | If ` B ` is finer than ` A... |
fneuni 34536 | If ` B ` is finer than ` A... |
fneint 34537 | If a cover is finer than a... |
fness 34538 | A cover is finer than its ... |
fneref 34539 | Reflexivity of the finenes... |
fnetr 34540 | Transitivity of the finene... |
fneval 34541 | Two covers are finer than ... |
fneer 34542 | Fineness intersected with ... |
topfne 34543 | Fineness for covers corres... |
topfneec 34544 | A cover is equivalent to a... |
topfneec2 34545 | A topology is precisely id... |
fnessref 34546 | A cover is finer iff it ha... |
refssfne 34547 | A cover is a refinement if... |
neibastop1 34548 | A collection of neighborho... |
neibastop2lem 34549 | Lemma for ~ neibastop2 . ... |
neibastop2 34550 | In the topology generated ... |
neibastop3 34551 | The topology generated by ... |
topmtcl 34552 | The meet of a collection o... |
topmeet 34553 | Two equivalent formulation... |
topjoin 34554 | Two equivalent formulation... |
fnemeet1 34555 | The meet of a collection o... |
fnemeet2 34556 | The meet of equivalence cl... |
fnejoin1 34557 | Join of equivalence classe... |
fnejoin2 34558 | Join of equivalence classe... |
fgmin 34559 | Minimality property of a g... |
neifg 34560 | The neighborhood filter of... |
tailfval 34561 | The tail function for a di... |
tailval 34562 | The tail of an element in ... |
eltail 34563 | An element of a tail. (Co... |
tailf 34564 | The tail function of a dir... |
tailini 34565 | A tail contains its initia... |
tailfb 34566 | The collection of tails of... |
filnetlem1 34567 | Lemma for ~ filnet . Chan... |
filnetlem2 34568 | Lemma for ~ filnet . The ... |
filnetlem3 34569 | Lemma for ~ filnet . (Con... |
filnetlem4 34570 | Lemma for ~ filnet . (Con... |
filnet 34571 | A filter has the same conv... |
tb-ax1 34572 | The first of three axioms ... |
tb-ax2 34573 | The second of three axioms... |
tb-ax3 34574 | The third of three axioms ... |
tbsyl 34575 | The weak syllogism from Ta... |
re1ax2lem 34576 | Lemma for ~ re1ax2 . (Con... |
re1ax2 34577 | ~ ax-2 rederived from the ... |
naim1 34578 | Constructor theorem for ` ... |
naim2 34579 | Constructor theorem for ` ... |
naim1i 34580 | Constructor rule for ` -/\... |
naim2i 34581 | Constructor rule for ` -/\... |
naim12i 34582 | Constructor rule for ` -/\... |
nabi1i 34583 | Constructor rule for ` -/\... |
nabi2i 34584 | Constructor rule for ` -/\... |
nabi12i 34585 | Constructor rule for ` -/\... |
df3nandALT1 34588 | The double nand expressed ... |
df3nandALT2 34589 | The double nand expressed ... |
andnand1 34590 | Double and in terms of dou... |
imnand2 34591 | An ` -> ` nand relation. ... |
nalfal 34592 | Not all sets hold ` F. ` a... |
nexntru 34593 | There does not exist a set... |
nexfal 34594 | There does not exist a set... |
neufal 34595 | There does not exist exact... |
neutru 34596 | There does not exist exact... |
nmotru 34597 | There does not exist at mo... |
mofal 34598 | There exist at most one se... |
nrmo 34599 | "At most one" restricted e... |
meran1 34600 | A single axiom for proposi... |
meran2 34601 | A single axiom for proposi... |
meran3 34602 | A single axiom for proposi... |
waj-ax 34603 | A single axiom for proposi... |
lukshef-ax2 34604 | A single axiom for proposi... |
arg-ax 34605 | A single axiom for proposi... |
negsym1 34606 | In the paper "On Variable ... |
imsym1 34607 | A symmetry with ` -> ` . ... |
bisym1 34608 | A symmetry with ` <-> ` . ... |
consym1 34609 | A symmetry with ` /\ ` . ... |
dissym1 34610 | A symmetry with ` \/ ` . ... |
nandsym1 34611 | A symmetry with ` -/\ ` . ... |
unisym1 34612 | A symmetry with ` A. ` . ... |
exisym1 34613 | A symmetry with ` E. ` . ... |
unqsym1 34614 | A symmetry with ` E! ` . ... |
amosym1 34615 | A symmetry with ` E* ` . ... |
subsym1 34616 | A symmetry with ` [ x / y ... |
ontopbas 34617 | An ordinal number is a top... |
onsstopbas 34618 | The class of ordinal numbe... |
onpsstopbas 34619 | The class of ordinal numbe... |
ontgval 34620 | The topology generated fro... |
ontgsucval 34621 | The topology generated fro... |
onsuctop 34622 | A successor ordinal number... |
onsuctopon 34623 | One of the topologies on a... |
ordtoplem 34624 | Membership of the class of... |
ordtop 34625 | An ordinal is a topology i... |
onsucconni 34626 | A successor ordinal number... |
onsucconn 34627 | A successor ordinal number... |
ordtopconn 34628 | An ordinal topology is con... |
onintopssconn 34629 | An ordinal topology is con... |
onsuct0 34630 | A successor ordinal number... |
ordtopt0 34631 | An ordinal topology is T_0... |
onsucsuccmpi 34632 | The successor of a success... |
onsucsuccmp 34633 | The successor of a success... |
limsucncmpi 34634 | The successor of a limit o... |
limsucncmp 34635 | The successor of a limit o... |
ordcmp 34636 | An ordinal topology is com... |
ssoninhaus 34637 | The ordinal topologies ` 1... |
onint1 34638 | The ordinal T_1 spaces are... |
oninhaus 34639 | The ordinal Hausdorff spac... |
fveleq 34640 | Please add description her... |
findfvcl 34641 | Please add description her... |
findreccl 34642 | Please add description her... |
findabrcl 34643 | Please add description her... |
nnssi2 34644 | Convert a theorem for real... |
nnssi3 34645 | Convert a theorem for real... |
nndivsub 34646 | Please add description her... |
nndivlub 34647 | A factor of a positive int... |
ee7.2aOLD 34650 | Lemma for Euclid's Element... |
dnival 34651 | Value of the "distance to ... |
dnicld1 34652 | Closure theorem for the "d... |
dnicld2 34653 | Closure theorem for the "d... |
dnif 34654 | The "distance to nearest i... |
dnizeq0 34655 | The distance to nearest in... |
dnizphlfeqhlf 34656 | The distance to nearest in... |
rddif2 34657 | Variant of ~ rddif . (Con... |
dnibndlem1 34658 | Lemma for ~ dnibnd . (Con... |
dnibndlem2 34659 | Lemma for ~ dnibnd . (Con... |
dnibndlem3 34660 | Lemma for ~ dnibnd . (Con... |
dnibndlem4 34661 | Lemma for ~ dnibnd . (Con... |
dnibndlem5 34662 | Lemma for ~ dnibnd . (Con... |
dnibndlem6 34663 | Lemma for ~ dnibnd . (Con... |
dnibndlem7 34664 | Lemma for ~ dnibnd . (Con... |
dnibndlem8 34665 | Lemma for ~ dnibnd . (Con... |
dnibndlem9 34666 | Lemma for ~ dnibnd . (Con... |
dnibndlem10 34667 | Lemma for ~ dnibnd . (Con... |
dnibndlem11 34668 | Lemma for ~ dnibnd . (Con... |
dnibndlem12 34669 | Lemma for ~ dnibnd . (Con... |
dnibndlem13 34670 | Lemma for ~ dnibnd . (Con... |
dnibnd 34671 | The "distance to nearest i... |
dnicn 34672 | The "distance to nearest i... |
knoppcnlem1 34673 | Lemma for ~ knoppcn . (Co... |
knoppcnlem2 34674 | Lemma for ~ knoppcn . (Co... |
knoppcnlem3 34675 | Lemma for ~ knoppcn . (Co... |
knoppcnlem4 34676 | Lemma for ~ knoppcn . (Co... |
knoppcnlem5 34677 | Lemma for ~ knoppcn . (Co... |
knoppcnlem6 34678 | Lemma for ~ knoppcn . (Co... |
knoppcnlem7 34679 | Lemma for ~ knoppcn . (Co... |
knoppcnlem8 34680 | Lemma for ~ knoppcn . (Co... |
knoppcnlem9 34681 | Lemma for ~ knoppcn . (Co... |
knoppcnlem10 34682 | Lemma for ~ knoppcn . (Co... |
knoppcnlem11 34683 | Lemma for ~ knoppcn . (Co... |
knoppcn 34684 | The continuous nowhere dif... |
knoppcld 34685 | Closure theorem for Knopp'... |
unblimceq0lem 34686 | Lemma for ~ unblimceq0 . ... |
unblimceq0 34687 | If ` F ` is unbounded near... |
unbdqndv1 34688 | If the difference quotient... |
unbdqndv2lem1 34689 | Lemma for ~ unbdqndv2 . (... |
unbdqndv2lem2 34690 | Lemma for ~ unbdqndv2 . (... |
unbdqndv2 34691 | Variant of ~ unbdqndv1 wit... |
knoppndvlem1 34692 | Lemma for ~ knoppndv . (C... |
knoppndvlem2 34693 | Lemma for ~ knoppndv . (C... |
knoppndvlem3 34694 | Lemma for ~ knoppndv . (C... |
knoppndvlem4 34695 | Lemma for ~ knoppndv . (C... |
knoppndvlem5 34696 | Lemma for ~ knoppndv . (C... |
knoppndvlem6 34697 | Lemma for ~ knoppndv . (C... |
knoppndvlem7 34698 | Lemma for ~ knoppndv . (C... |
knoppndvlem8 34699 | Lemma for ~ knoppndv . (C... |
knoppndvlem9 34700 | Lemma for ~ knoppndv . (C... |
knoppndvlem10 34701 | Lemma for ~ knoppndv . (C... |
knoppndvlem11 34702 | Lemma for ~ knoppndv . (C... |
knoppndvlem12 34703 | Lemma for ~ knoppndv . (C... |
knoppndvlem13 34704 | Lemma for ~ knoppndv . (C... |
knoppndvlem14 34705 | Lemma for ~ knoppndv . (C... |
knoppndvlem15 34706 | Lemma for ~ knoppndv . (C... |
knoppndvlem16 34707 | Lemma for ~ knoppndv . (C... |
knoppndvlem17 34708 | Lemma for ~ knoppndv . (C... |
knoppndvlem18 34709 | Lemma for ~ knoppndv . (C... |
knoppndvlem19 34710 | Lemma for ~ knoppndv . (C... |
knoppndvlem20 34711 | Lemma for ~ knoppndv . (C... |
knoppndvlem21 34712 | Lemma for ~ knoppndv . (C... |
knoppndvlem22 34713 | Lemma for ~ knoppndv . (C... |
knoppndv 34714 | The continuous nowhere dif... |
knoppf 34715 | Knopp's function is a func... |
knoppcn2 34716 | Variant of ~ knoppcn with ... |
cnndvlem1 34717 | Lemma for ~ cnndv . (Cont... |
cnndvlem2 34718 | Lemma for ~ cnndv . (Cont... |
cnndv 34719 | There exists a continuous ... |
bj-mp2c 34720 | A double modus ponens infe... |
bj-mp2d 34721 | A double modus ponens infe... |
bj-0 34722 | A syntactic theorem. See ... |
bj-1 34723 | In this proof, the use of ... |
bj-a1k 34724 | Weakening of ~ ax-1 . As ... |
bj-poni 34725 | Inference associated with ... |
bj-nnclav 34726 | When ` F. ` is substituted... |
bj-nnclavi 34727 | Inference associated with ... |
bj-nnclavc 34728 | Commuted form of ~ bj-nncl... |
bj-nnclavci 34729 | Inference associated with ... |
bj-jarrii 34730 | Inference associated with ... |
bj-imim21 34731 | The propositional function... |
bj-imim21i 34732 | Inference associated with ... |
bj-peircestab 34733 | Over minimal implicational... |
bj-stabpeirce 34734 | This minimal implicational... |
bj-syl66ib 34735 | A mixed syllogism inferenc... |
bj-orim2 34736 | Proof of ~ orim2 from the ... |
bj-currypeirce 34737 | Curry's axiom ~ curryax (a... |
bj-peircecurry 34738 | Peirce's axiom ~ peirce im... |
bj-animbi 34739 | Conjunction in terms of im... |
bj-currypara 34740 | Curry's paradox. Note tha... |
bj-con2com 34741 | A commuted form of the con... |
bj-con2comi 34742 | Inference associated with ... |
bj-pm2.01i 34743 | Inference associated with ... |
bj-nimn 34744 | If a formula is true, then... |
bj-nimni 34745 | Inference associated with ... |
bj-peircei 34746 | Inference associated with ... |
bj-looinvi 34747 | Inference associated with ... |
bj-looinvii 34748 | Inference associated with ... |
bj-mt2bi 34749 | Version of ~ mt2 where the... |
bj-ntrufal 34750 | The negation of a theorem ... |
bj-fal 34751 | Shortening of ~ fal using ... |
bj-jaoi1 34752 | Shortens ~ orfa2 (58>53), ... |
bj-jaoi2 34753 | Shortens ~ consensus (110>... |
bj-dfbi4 34754 | Alternate definition of th... |
bj-dfbi5 34755 | Alternate definition of th... |
bj-dfbi6 34756 | Alternate definition of th... |
bj-bijust0ALT 34757 | Alternate proof of ~ bijus... |
bj-bijust00 34758 | A self-implication does no... |
bj-consensus 34759 | Version of ~ consensus exp... |
bj-consensusALT 34760 | Alternate proof of ~ bj-co... |
bj-df-ifc 34761 | Candidate definition for t... |
bj-dfif 34762 | Alternate definition of th... |
bj-ififc 34763 | A biconditional connecting... |
bj-imbi12 34764 | Uncurried (imported) form ... |
bj-biorfi 34765 | This should be labeled "bi... |
bj-falor 34766 | Dual of ~ truan (which has... |
bj-falor2 34767 | Dual of ~ truan . (Contri... |
bj-bibibi 34768 | A property of the bicondit... |
bj-imn3ani 34769 | Duplication of ~ bnj1224 .... |
bj-andnotim 34770 | Two ways of expressing a c... |
bj-bi3ant 34771 | This used to be in the mai... |
bj-bisym 34772 | This used to be in the mai... |
bj-bixor 34773 | Equivalence of two ternary... |
bj-axdd2 34774 | This implication, proved u... |
bj-axd2d 34775 | This implication, proved u... |
bj-axtd 34776 | This implication, proved f... |
bj-gl4 34777 | In a normal modal logic, t... |
bj-axc4 34778 | Over minimal calculus, the... |
prvlem1 34783 | An elementary property of ... |
prvlem2 34784 | An elementary property of ... |
bj-babygodel 34785 | See the section header com... |
bj-babylob 34786 | See the section header com... |
bj-godellob 34787 | Proof of Gödel's theo... |
bj-genr 34788 | Generalization rule on the... |
bj-genl 34789 | Generalization rule on the... |
bj-genan 34790 | Generalization rule on a c... |
bj-mpgs 34791 | From a closed form theorem... |
bj-2alim 34792 | Closed form of ~ 2alimi . ... |
bj-2exim 34793 | Closed form of ~ 2eximi . ... |
bj-alanim 34794 | Closed form of ~ alanimi .... |
bj-2albi 34795 | Closed form of ~ 2albii . ... |
bj-notalbii 34796 | Equivalence of universal q... |
bj-2exbi 34797 | Closed form of ~ 2exbii . ... |
bj-3exbi 34798 | Closed form of ~ 3exbii . ... |
bj-sylgt2 34799 | Uncurried (imported) form ... |
bj-alrimg 34800 | The general form of the *a... |
bj-alrimd 34801 | A slightly more general ~ ... |
bj-sylget 34802 | Dual statement of ~ sylgt ... |
bj-sylget2 34803 | Uncurried (imported) form ... |
bj-exlimg 34804 | The general form of the *e... |
bj-sylge 34805 | Dual statement of ~ sylg (... |
bj-exlimd 34806 | A slightly more general ~ ... |
bj-nfimexal 34807 | A weak from of nonfreeness... |
bj-alexim 34808 | Closed form of ~ aleximi .... |
bj-nexdh 34809 | Closed form of ~ nexdh (ac... |
bj-nexdh2 34810 | Uncurried (imported) form ... |
bj-hbxfrbi 34811 | Closed form of ~ hbxfrbi .... |
bj-hbyfrbi 34812 | Version of ~ bj-hbxfrbi wi... |
bj-exalim 34813 | Distribute quantifiers ove... |
bj-exalimi 34814 | An inference for distribut... |
bj-exalims 34815 | Distributing quantifiers o... |
bj-exalimsi 34816 | An inference for distribut... |
bj-ax12ig 34817 | A lemma used to prove a we... |
bj-ax12i 34818 | A weakening of ~ bj-ax12ig... |
bj-nfimt 34819 | Closed form of ~ nfim and ... |
bj-cbvalimt 34820 | A lemma in closed form use... |
bj-cbveximt 34821 | A lemma in closed form use... |
bj-eximALT 34822 | Alternate proof of ~ exim ... |
bj-aleximiALT 34823 | Alternate proof of ~ alexi... |
bj-eximcom 34824 | A commuted form of ~ exim ... |
bj-ax12wlem 34825 | A lemma used to prove a we... |
bj-cbvalim 34826 | A lemma used to prove ~ bj... |
bj-cbvexim 34827 | A lemma used to prove ~ bj... |
bj-cbvalimi 34828 | An equality-free general i... |
bj-cbveximi 34829 | An equality-free general i... |
bj-cbval 34830 | Changing a bound variable ... |
bj-cbvex 34831 | Changing a bound variable ... |
bj-ssbeq 34834 | Substitution in an equalit... |
bj-ssblem1 34835 | A lemma for the definiens ... |
bj-ssblem2 34836 | An instance of ~ ax-11 pro... |
bj-ax12v 34837 | A weaker form of ~ ax-12 a... |
bj-ax12 34838 | Remove a DV condition from... |
bj-ax12ssb 34839 | Axiom ~ bj-ax12 expressed ... |
bj-19.41al 34840 | Special case of ~ 19.41 pr... |
bj-equsexval 34841 | Special case of ~ equsexv ... |
bj-subst 34842 | Proof of ~ sbalex from cor... |
bj-ssbid2 34843 | A special case of ~ sbequ2... |
bj-ssbid2ALT 34844 | Alternate proof of ~ bj-ss... |
bj-ssbid1 34845 | A special case of ~ sbequ1... |
bj-ssbid1ALT 34846 | Alternate proof of ~ bj-ss... |
bj-ax6elem1 34847 | Lemma for ~ bj-ax6e . (Co... |
bj-ax6elem2 34848 | Lemma for ~ bj-ax6e . (Co... |
bj-ax6e 34849 | Proof of ~ ax6e (hence ~ a... |
bj-spimvwt 34850 | Closed form of ~ spimvw . ... |
bj-spnfw 34851 | Theorem close to a closed ... |
bj-cbvexiw 34852 | Change bound variable. Th... |
bj-cbvexivw 34853 | Change bound variable. Th... |
bj-modald 34854 | A short form of the axiom ... |
bj-denot 34855 | A weakening of ~ ax-6 and ... |
bj-eqs 34856 | A lemma for substitutions,... |
bj-cbvexw 34857 | Change bound variable. Th... |
bj-ax12w 34858 | The general statement that... |
bj-ax89 34859 | A theorem which could be u... |
bj-elequ12 34860 | An identity law for the no... |
bj-cleljusti 34861 | One direction of ~ cleljus... |
bj-alcomexcom 34862 | Commutation of universal q... |
bj-hbalt 34863 | Closed form of ~ hbal . W... |
axc11n11 34864 | Proof of ~ axc11n from { ~... |
axc11n11r 34865 | Proof of ~ axc11n from { ~... |
bj-axc16g16 34866 | Proof of ~ axc16g from { ~... |
bj-ax12v3 34867 | A weak version of ~ ax-12 ... |
bj-ax12v3ALT 34868 | Alternate proof of ~ bj-ax... |
bj-sb 34869 | A weak variant of ~ sbid2 ... |
bj-modalbe 34870 | The predicate-calculus ver... |
bj-spst 34871 | Closed form of ~ sps . On... |
bj-19.21bit 34872 | Closed form of ~ 19.21bi .... |
bj-19.23bit 34873 | Closed form of ~ 19.23bi .... |
bj-nexrt 34874 | Closed form of ~ nexr . C... |
bj-alrim 34875 | Closed form of ~ alrimi . ... |
bj-alrim2 34876 | Uncurried (imported) form ... |
bj-nfdt0 34877 | A theorem close to a close... |
bj-nfdt 34878 | Closed form of ~ nf5d and ... |
bj-nexdt 34879 | Closed form of ~ nexd . (... |
bj-nexdvt 34880 | Closed form of ~ nexdv . ... |
bj-alexbiex 34881 | Adding a second quantifier... |
bj-exexbiex 34882 | Adding a second quantifier... |
bj-alalbial 34883 | Adding a second quantifier... |
bj-exalbial 34884 | Adding a second quantifier... |
bj-19.9htbi 34885 | Strengthening ~ 19.9ht by ... |
bj-hbntbi 34886 | Strengthening ~ hbnt by re... |
bj-biexal1 34887 | A general FOL biconditiona... |
bj-biexal2 34888 | When ` ph ` is substituted... |
bj-biexal3 34889 | When ` ph ` is substituted... |
bj-bialal 34890 | When ` ph ` is substituted... |
bj-biexex 34891 | When ` ph ` is substituted... |
bj-hbext 34892 | Closed form of ~ hbex . (... |
bj-nfalt 34893 | Closed form of ~ nfal . (... |
bj-nfext 34894 | Closed form of ~ nfex . (... |
bj-eeanvw 34895 | Version of ~ exdistrv with... |
bj-modal4 34896 | First-order logic form of ... |
bj-modal4e 34897 | First-order logic form of ... |
bj-modalb 34898 | A short form of the axiom ... |
bj-wnf1 34899 | When ` ph ` is substituted... |
bj-wnf2 34900 | When ` ph ` is substituted... |
bj-wnfanf 34901 | When ` ph ` is substituted... |
bj-wnfenf 34902 | When ` ph ` is substituted... |
bj-substax12 34903 | Equivalent form of the axi... |
bj-substw 34904 | Weak form of the LHS of ~ ... |
bj-nnfbi 34907 | If two formulas are equiva... |
bj-nnfbd 34908 | If two formulas are equiva... |
bj-nnfbii 34909 | If two formulas are equiva... |
bj-nnfa 34910 | Nonfreeness implies the eq... |
bj-nnfad 34911 | Nonfreeness implies the eq... |
bj-nnfai 34912 | Nonfreeness implies the eq... |
bj-nnfe 34913 | Nonfreeness implies the eq... |
bj-nnfed 34914 | Nonfreeness implies the eq... |
bj-nnfei 34915 | Nonfreeness implies the eq... |
bj-nnfea 34916 | Nonfreeness implies the eq... |
bj-nnfead 34917 | Nonfreeness implies the eq... |
bj-nnfeai 34918 | Nonfreeness implies the eq... |
bj-dfnnf2 34919 | Alternate definition of ~ ... |
bj-nnfnfTEMP 34920 | New nonfreeness implies ol... |
bj-wnfnf 34921 | When ` ph ` is substituted... |
bj-nnfnt 34922 | A variable is nonfree in a... |
bj-nnftht 34923 | A variable is nonfree in a... |
bj-nnfth 34924 | A variable is nonfree in a... |
bj-nnfnth 34925 | A variable is nonfree in t... |
bj-nnfim1 34926 | A consequence of nonfreene... |
bj-nnfim2 34927 | A consequence of nonfreene... |
bj-nnfim 34928 | Nonfreeness in the anteced... |
bj-nnfimd 34929 | Nonfreeness in the anteced... |
bj-nnfan 34930 | Nonfreeness in both conjun... |
bj-nnfand 34931 | Nonfreeness in both conjun... |
bj-nnfor 34932 | Nonfreeness in both disjun... |
bj-nnford 34933 | Nonfreeness in both disjun... |
bj-nnfbit 34934 | Nonfreeness in both sides ... |
bj-nnfbid 34935 | Nonfreeness in both sides ... |
bj-nnfv 34936 | A non-occurring variable i... |
bj-nnf-alrim 34937 | Proof of the closed form o... |
bj-nnf-exlim 34938 | Proof of the closed form o... |
bj-dfnnf3 34939 | Alternate definition of no... |
bj-nfnnfTEMP 34940 | New nonfreeness is equival... |
bj-nnfa1 34941 | See ~ nfa1 . (Contributed... |
bj-nnfe1 34942 | See ~ nfe1 . (Contributed... |
bj-19.12 34943 | See ~ 19.12 . Could be la... |
bj-nnflemaa 34944 | One of four lemmas for non... |
bj-nnflemee 34945 | One of four lemmas for non... |
bj-nnflemae 34946 | One of four lemmas for non... |
bj-nnflemea 34947 | One of four lemmas for non... |
bj-nnfalt 34948 | See ~ nfal and ~ bj-nfalt ... |
bj-nnfext 34949 | See ~ nfex and ~ bj-nfext ... |
bj-stdpc5t 34950 | Alias of ~ bj-nnf-alrim fo... |
bj-19.21t 34951 | Statement ~ 19.21t proved ... |
bj-19.23t 34952 | Statement ~ 19.23t proved ... |
bj-19.36im 34953 | One direction of ~ 19.36 f... |
bj-19.37im 34954 | One direction of ~ 19.37 f... |
bj-19.42t 34955 | Closed form of ~ 19.42 fro... |
bj-19.41t 34956 | Closed form of ~ 19.41 fro... |
bj-sbft 34957 | Version of ~ sbft using ` ... |
bj-pm11.53vw 34958 | Version of ~ pm11.53v with... |
bj-pm11.53v 34959 | Version of ~ pm11.53v with... |
bj-pm11.53a 34960 | A variant of ~ pm11.53v . ... |
bj-equsvt 34961 | A variant of ~ equsv . (C... |
bj-equsalvwd 34962 | Variant of ~ equsalvw . (... |
bj-equsexvwd 34963 | Variant of ~ equsexvw . (... |
bj-sbievwd 34964 | Variant of ~ sbievw . (Co... |
bj-axc10 34965 | Alternate proof of ~ axc10... |
bj-alequex 34966 | A fol lemma. See ~ aleque... |
bj-spimt2 34967 | A step in the proof of ~ s... |
bj-cbv3ta 34968 | Closed form of ~ cbv3 . (... |
bj-cbv3tb 34969 | Closed form of ~ cbv3 . (... |
bj-hbsb3t 34970 | A theorem close to a close... |
bj-hbsb3 34971 | Shorter proof of ~ hbsb3 .... |
bj-nfs1t 34972 | A theorem close to a close... |
bj-nfs1t2 34973 | A theorem close to a close... |
bj-nfs1 34974 | Shorter proof of ~ nfs1 (t... |
bj-axc10v 34975 | Version of ~ axc10 with a ... |
bj-spimtv 34976 | Version of ~ spimt with a ... |
bj-cbv3hv2 34977 | Version of ~ cbv3h with tw... |
bj-cbv1hv 34978 | Version of ~ cbv1h with a ... |
bj-cbv2hv 34979 | Version of ~ cbv2h with a ... |
bj-cbv2v 34980 | Version of ~ cbv2 with a d... |
bj-cbvaldv 34981 | Version of ~ cbvald with a... |
bj-cbvexdv 34982 | Version of ~ cbvexd with a... |
bj-cbval2vv 34983 | Version of ~ cbval2vv with... |
bj-cbvex2vv 34984 | Version of ~ cbvex2vv with... |
bj-cbvaldvav 34985 | Version of ~ cbvaldva with... |
bj-cbvexdvav 34986 | Version of ~ cbvexdva with... |
bj-cbvex4vv 34987 | Version of ~ cbvex4v with ... |
bj-equsalhv 34988 | Version of ~ equsalh with ... |
bj-axc11nv 34989 | Version of ~ axc11n with a... |
bj-aecomsv 34990 | Version of ~ aecoms with a... |
bj-axc11v 34991 | Version of ~ axc11 with a ... |
bj-drnf2v 34992 | Version of ~ drnf2 with a ... |
bj-equs45fv 34993 | Version of ~ equs45f with ... |
bj-hbs1 34994 | Version of ~ hbsb2 with a ... |
bj-nfs1v 34995 | Version of ~ nfsb2 with a ... |
bj-hbsb2av 34996 | Version of ~ hbsb2a with a... |
bj-hbsb3v 34997 | Version of ~ hbsb3 with a ... |
bj-nfsab1 34998 | Remove dependency on ~ ax-... |
bj-dtru 34999 | Remove dependency on ~ ax-... |
bj-dtrucor2v 35000 | Version of ~ dtrucor2 with... |
bj-hbaeb2 35001 | Biconditional version of a... |
bj-hbaeb 35002 | Biconditional version of ~... |
bj-hbnaeb 35003 | Biconditional version of ~... |
bj-dvv 35004 | A special instance of ~ bj... |
bj-equsal1t 35005 | Duplication of ~ wl-equsal... |
bj-equsal1ti 35006 | Inference associated with ... |
bj-equsal1 35007 | One direction of ~ equsal ... |
bj-equsal2 35008 | One direction of ~ equsal ... |
bj-equsal 35009 | Shorter proof of ~ equsal ... |
stdpc5t 35010 | Closed form of ~ stdpc5 . ... |
bj-stdpc5 35011 | More direct proof of ~ std... |
2stdpc5 35012 | A double ~ stdpc5 (one dir... |
bj-19.21t0 35013 | Proof of ~ 19.21t from ~ s... |
exlimii 35014 | Inference associated with ... |
ax11-pm 35015 | Proof of ~ ax-11 similar t... |
ax6er 35016 | Commuted form of ~ ax6e . ... |
exlimiieq1 35017 | Inferring a theorem when i... |
exlimiieq2 35018 | Inferring a theorem when i... |
ax11-pm2 35019 | Proof of ~ ax-11 from the ... |
bj-sbsb 35020 | Biconditional showing two ... |
bj-dfsb2 35021 | Alternate (dual) definitio... |
bj-sbf3 35022 | Substitution has no effect... |
bj-sbf4 35023 | Substitution has no effect... |
bj-sbnf 35024 | Move nonfree predicate in ... |
bj-eu3f 35025 | Version of ~ eu3v where th... |
bj-sblem1 35026 | Lemma for substitution. (... |
bj-sblem2 35027 | Lemma for substitution. (... |
bj-sblem 35028 | Lemma for substitution. (... |
bj-sbievw1 35029 | Lemma for substitution. (... |
bj-sbievw2 35030 | Lemma for substitution. (... |
bj-sbievw 35031 | Lemma for substitution. C... |
bj-sbievv 35032 | Version of ~ sbie with a s... |
bj-moeub 35033 | Uniqueness is equivalent t... |
bj-sbidmOLD 35034 | Obsolete proof of ~ sbidm ... |
bj-dvelimdv 35035 | Deduction form of ~ dvelim... |
bj-dvelimdv1 35036 | Curried (exported) form of... |
bj-dvelimv 35037 | A version of ~ dvelim usin... |
bj-nfeel2 35038 | Nonfreeness in a membershi... |
bj-axc14nf 35039 | Proof of a version of ~ ax... |
bj-axc14 35040 | Alternate proof of ~ axc14... |
mobidvALT 35041 | Alternate proof of ~ mobid... |
sbn1ALT 35042 | Alternate proof of ~ sbn1 ... |
eliminable1 35043 | A theorem used to prove th... |
eliminable2a 35044 | A theorem used to prove th... |
eliminable2b 35045 | A theorem used to prove th... |
eliminable2c 35046 | A theorem used to prove th... |
eliminable3a 35047 | A theorem used to prove th... |
eliminable3b 35048 | A theorem used to prove th... |
eliminable-velab 35049 | A theorem used to prove th... |
eliminable-veqab 35050 | A theorem used to prove th... |
eliminable-abeqv 35051 | A theorem used to prove th... |
eliminable-abeqab 35052 | A theorem used to prove th... |
eliminable-abelv 35053 | A theorem used to prove th... |
eliminable-abelab 35054 | A theorem used to prove th... |
bj-denoteslem 35055 | Lemma for ~ bj-denotes . ... |
bj-denotes 35056 | This would be the justific... |
bj-issettru 35057 | Weak version of ~ isset wi... |
bj-elabtru 35058 | This is as close as we can... |
bj-issetwt 35059 | Closed form of ~ bj-issetw... |
bj-issetw 35060 | The closest one can get to... |
bj-elissetALT 35061 | Alternate proof of ~ eliss... |
bj-issetiv 35062 | Version of ~ bj-isseti wit... |
bj-isseti 35063 | Version of ~ isseti with a... |
bj-ralvw 35064 | A weak version of ~ ralv n... |
bj-rexvw 35065 | A weak version of ~ rexv n... |
bj-rababw 35066 | A weak version of ~ rabab ... |
bj-rexcom4bv 35067 | Version of ~ rexcom4b and ... |
bj-rexcom4b 35068 | Remove from ~ rexcom4b dep... |
bj-ceqsalt0 35069 | The FOL content of ~ ceqsa... |
bj-ceqsalt1 35070 | The FOL content of ~ ceqsa... |
bj-ceqsalt 35071 | Remove from ~ ceqsalt depe... |
bj-ceqsaltv 35072 | Version of ~ bj-ceqsalt wi... |
bj-ceqsalg0 35073 | The FOL content of ~ ceqsa... |
bj-ceqsalg 35074 | Remove from ~ ceqsalg depe... |
bj-ceqsalgALT 35075 | Alternate proof of ~ bj-ce... |
bj-ceqsalgv 35076 | Version of ~ bj-ceqsalg wi... |
bj-ceqsalgvALT 35077 | Alternate proof of ~ bj-ce... |
bj-ceqsal 35078 | Remove from ~ ceqsal depen... |
bj-ceqsalv 35079 | Remove from ~ ceqsalv depe... |
bj-spcimdv 35080 | Remove from ~ spcimdv depe... |
bj-spcimdvv 35081 | Remove from ~ spcimdv depe... |
elelb 35082 | Equivalence between two co... |
bj-pwvrelb 35083 | Characterization of the el... |
bj-nfcsym 35084 | The nonfreeness quantifier... |
bj-sbeqALT 35085 | Substitution in an equalit... |
bj-sbeq 35086 | Distribute proper substitu... |
bj-sbceqgALT 35087 | Distribute proper substitu... |
bj-csbsnlem 35088 | Lemma for ~ bj-csbsn (in t... |
bj-csbsn 35089 | Substitution in a singleto... |
bj-sbel1 35090 | Version of ~ sbcel1g when ... |
bj-abv 35091 | The class of sets verifyin... |
bj-abvALT 35092 | Alternate version of ~ bj-... |
bj-ab0 35093 | The class of sets verifyin... |
bj-abf 35094 | Shorter proof of ~ abf (wh... |
bj-csbprc 35095 | More direct proof of ~ csb... |
bj-exlimvmpi 35096 | A Fol lemma ( ~ exlimiv fo... |
bj-exlimmpi 35097 | Lemma for ~ bj-vtoclg1f1 (... |
bj-exlimmpbi 35098 | Lemma for theorems of the ... |
bj-exlimmpbir 35099 | Lemma for theorems of the ... |
bj-vtoclf 35100 | Remove dependency on ~ ax-... |
bj-vtocl 35101 | Remove dependency on ~ ax-... |
bj-vtoclg1f1 35102 | The FOL content of ~ vtocl... |
bj-vtoclg1f 35103 | Reprove ~ vtoclg1f from ~ ... |
bj-vtoclg1fv 35104 | Version of ~ bj-vtoclg1f w... |
bj-vtoclg 35105 | A version of ~ vtoclg with... |
bj-rabbida2 35106 | Version of ~ rabbidva2 wit... |
bj-rabeqd 35107 | Deduction form of ~ rabeq ... |
bj-rabeqbid 35108 | Version of ~ rabeqbidv wit... |
bj-rabeqbida 35109 | Version of ~ rabeqbidva wi... |
bj-seex 35110 | Version of ~ seex with a d... |
bj-nfcf 35111 | Version of ~ df-nfc with a... |
bj-zfauscl 35112 | General version of ~ zfaus... |
bj-elabd2ALT 35113 | Alternate proof of ~ elabd... |
bj-unrab 35114 | Generalization of ~ unrab ... |
bj-inrab 35115 | Generalization of ~ inrab ... |
bj-inrab2 35116 | Shorter proof of ~ inrab .... |
bj-inrab3 35117 | Generalization of ~ dfrab3... |
bj-rabtr 35118 | Restricted class abstracti... |
bj-rabtrALT 35119 | Alternate proof of ~ bj-ra... |
bj-rabtrAUTO 35120 | Proof of ~ bj-rabtr found ... |
bj-gabss 35123 | Inclusion of generalized c... |
bj-gabssd 35124 | Inclusion of generalized c... |
bj-gabeqd 35125 | Equality of generalized cl... |
bj-gabeqis 35126 | Equality of generalized cl... |
bj-elgab 35127 | Elements of a generalized ... |
bj-gabima 35128 | Generalized class abstract... |
bj-ru0 35131 | The FOL part of Russell's ... |
bj-ru1 35132 | A version of Russell's par... |
bj-ru 35133 | Remove dependency on ~ ax-... |
currysetlem 35134 | Lemma for ~ currysetlem , ... |
curryset 35135 | Curry's paradox in set the... |
currysetlem1 35136 | Lemma for ~ currysetALT . ... |
currysetlem2 35137 | Lemma for ~ currysetALT . ... |
currysetlem3 35138 | Lemma for ~ currysetALT . ... |
currysetALT 35139 | Alternate proof of ~ curry... |
bj-n0i 35140 | Inference associated with ... |
bj-disjcsn 35141 | A class is disjoint from i... |
bj-disjsn01 35142 | Disjointness of the single... |
bj-0nel1 35143 | The empty set does not bel... |
bj-1nel0 35144 | ` 1o ` does not belong to ... |
bj-xpimasn 35145 | The image of a singleton, ... |
bj-xpima1sn 35146 | The image of a singleton b... |
bj-xpima1snALT 35147 | Alternate proof of ~ bj-xp... |
bj-xpima2sn 35148 | The image of a singleton b... |
bj-xpnzex 35149 | If the first factor of a p... |
bj-xpexg2 35150 | Curried (exported) form of... |
bj-xpnzexb 35151 | If the first factor of a p... |
bj-cleq 35152 | Substitution property for ... |
bj-snsetex 35153 | The class of sets "whose s... |
bj-clex 35154 | Sethood of certain classes... |
bj-sngleq 35157 | Substitution property for ... |
bj-elsngl 35158 | Characterization of the el... |
bj-snglc 35159 | Characterization of the el... |
bj-snglss 35160 | The singletonization of a ... |
bj-0nelsngl 35161 | The empty set is not a mem... |
bj-snglinv 35162 | Inverse of singletonizatio... |
bj-snglex 35163 | A class is a set if and on... |
bj-tageq 35166 | Substitution property for ... |
bj-eltag 35167 | Characterization of the el... |
bj-0eltag 35168 | The empty set belongs to t... |
bj-tagn0 35169 | The tagging of a class is ... |
bj-tagss 35170 | The tagging of a class is ... |
bj-snglsstag 35171 | The singletonization is in... |
bj-sngltagi 35172 | The singletonization is in... |
bj-sngltag 35173 | The singletonization and t... |
bj-tagci 35174 | Characterization of the el... |
bj-tagcg 35175 | Characterization of the el... |
bj-taginv 35176 | Inverse of tagging. (Cont... |
bj-tagex 35177 | A class is a set if and on... |
bj-xtageq 35178 | The products of a given cl... |
bj-xtagex 35179 | The product of a set and t... |
bj-projeq 35182 | Substitution property for ... |
bj-projeq2 35183 | Substitution property for ... |
bj-projun 35184 | The class projection on a ... |
bj-projex 35185 | Sethood of the class proje... |
bj-projval 35186 | Value of the class project... |
bj-1upleq 35189 | Substitution property for ... |
bj-pr1eq 35192 | Substitution property for ... |
bj-pr1un 35193 | The first projection prese... |
bj-pr1val 35194 | Value of the first project... |
bj-pr11val 35195 | Value of the first project... |
bj-pr1ex 35196 | Sethood of the first proje... |
bj-1uplth 35197 | The characteristic propert... |
bj-1uplex 35198 | A monuple is a set if and ... |
bj-1upln0 35199 | A monuple is nonempty. (C... |
bj-2upleq 35202 | Substitution property for ... |
bj-pr21val 35203 | Value of the first project... |
bj-pr2eq 35206 | Substitution property for ... |
bj-pr2un 35207 | The second projection pres... |
bj-pr2val 35208 | Value of the second projec... |
bj-pr22val 35209 | Value of the second projec... |
bj-pr2ex 35210 | Sethood of the second proj... |
bj-2uplth 35211 | The characteristic propert... |
bj-2uplex 35212 | A couple is a set if and o... |
bj-2upln0 35213 | A couple is nonempty. (Co... |
bj-2upln1upl 35214 | A couple is never equal to... |
bj-rcleqf 35215 | Relative version of ~ cleq... |
bj-rcleq 35216 | Relative version of ~ dfcl... |
bj-reabeq 35217 | Relative form of ~ abeq2 .... |
bj-disj2r 35218 | Relative version of ~ ssdi... |
bj-sscon 35219 | Contraposition law for rel... |
eleq2w2ALT 35220 | Alternate proof of ~ eleq2... |
bj-clel3gALT 35221 | Alternate proof of ~ clel3... |
bj-pw0ALT 35222 | Alternate proof of ~ pw0 .... |
bj-sselpwuni 35223 | Quantitative version of ~ ... |
bj-unirel 35224 | Quantitative version of ~ ... |
bj-elpwg 35225 | If the intersection of two... |
bj-vjust 35226 | Justification theorem for ... |
bj-nul 35227 | Two formulations of the ax... |
bj-nuliota 35228 | Definition of the empty se... |
bj-nuliotaALT 35229 | Alternate proof of ~ bj-nu... |
bj-vtoclgfALT 35230 | Alternate proof of ~ vtocl... |
bj-elsn12g 35231 | Join of ~ elsng and ~ elsn... |
bj-elsnb 35232 | Biconditional version of ~... |
bj-pwcfsdom 35233 | Remove hypothesis from ~ p... |
bj-grur1 35234 | Remove hypothesis from ~ g... |
bj-bm1.3ii 35235 | The extension of a predica... |
bj-dfid2ALT 35236 | Alternate version of ~ dfi... |
bj-0nelopab 35237 | The empty set is never an ... |
bj-brrelex12ALT 35238 | Two classes related by a b... |
bj-epelg 35239 | The membership relation an... |
bj-epelb 35240 | Two classes are related by... |
bj-nsnid 35241 | A set does not contain the... |
bj-rdg0gALT 35242 | Alternate proof of ~ rdg0g... |
bj-evaleq 35243 | Equality theorem for the `... |
bj-evalfun 35244 | The evaluation at a class ... |
bj-evalfn 35245 | The evaluation at a class ... |
bj-evalval 35246 | Value of the evaluation at... |
bj-evalid 35247 | The evaluation at a set of... |
bj-ndxarg 35248 | Proof of ~ ndxarg from ~ b... |
bj-evalidval 35249 | Closed general form of ~ s... |
bj-rest00 35252 | An elementwise intersectio... |
bj-restsn 35253 | An elementwise intersectio... |
bj-restsnss 35254 | Special case of ~ bj-rests... |
bj-restsnss2 35255 | Special case of ~ bj-rests... |
bj-restsn0 35256 | An elementwise intersectio... |
bj-restsn10 35257 | Special case of ~ bj-rests... |
bj-restsnid 35258 | The elementwise intersecti... |
bj-rest10 35259 | An elementwise intersectio... |
bj-rest10b 35260 | Alternate version of ~ bj-... |
bj-restn0 35261 | An elementwise intersectio... |
bj-restn0b 35262 | Alternate version of ~ bj-... |
bj-restpw 35263 | The elementwise intersecti... |
bj-rest0 35264 | An elementwise intersectio... |
bj-restb 35265 | An elementwise intersectio... |
bj-restv 35266 | An elementwise intersectio... |
bj-resta 35267 | An elementwise intersectio... |
bj-restuni 35268 | The union of an elementwis... |
bj-restuni2 35269 | The union of an elementwis... |
bj-restreg 35270 | A reformulation of the axi... |
bj-raldifsn 35271 | All elements in a set sati... |
bj-0int 35272 | If ` A ` is a collection o... |
bj-mooreset 35273 | A Moore collection is a se... |
bj-ismoore 35276 | Characterization of Moore ... |
bj-ismoored0 35277 | Necessary condition to be ... |
bj-ismoored 35278 | Necessary condition to be ... |
bj-ismoored2 35279 | Necessary condition to be ... |
bj-ismooredr 35280 | Sufficient condition to be... |
bj-ismooredr2 35281 | Sufficient condition to be... |
bj-discrmoore 35282 | The powerclass ` ~P A ` is... |
bj-0nmoore 35283 | The empty set is not a Moo... |
bj-snmoore 35284 | A singleton is a Moore col... |
bj-snmooreb 35285 | A singleton is a Moore col... |
bj-prmoore 35286 | A pair formed of two neste... |
bj-0nelmpt 35287 | The empty set is not an el... |
bj-mptval 35288 | Value of a function given ... |
bj-dfmpoa 35289 | An equivalent definition o... |
bj-mpomptALT 35290 | Alternate proof of ~ mpomp... |
setsstrset 35307 | Relation between ~ df-sets... |
bj-nfald 35308 | Variant of ~ nfald . (Con... |
bj-nfexd 35309 | Variant of ~ nfexd . (Con... |
copsex2d 35310 | Implicit substitution dedu... |
copsex2b 35311 | Biconditional form of ~ co... |
opelopabd 35312 | Membership of an ordere pa... |
opelopabb 35313 | Membership of an ordered p... |
opelopabbv 35314 | Membership of an ordered p... |
bj-opelrelex 35315 | The coordinates of an orde... |
bj-opelresdm 35316 | If an ordered pair is in a... |
bj-brresdm 35317 | If two classes are related... |
brabd0 35318 | Expressing that two sets a... |
brabd 35319 | Expressing that two sets a... |
bj-brab2a1 35320 | "Unbounded" version of ~ b... |
bj-opabssvv 35321 | A variant of ~ relopabiv (... |
bj-funidres 35322 | The restricted identity re... |
bj-opelidb 35323 | Characterization of the or... |
bj-opelidb1 35324 | Characterization of the or... |
bj-inexeqex 35325 | Lemma for ~ bj-opelid (but... |
bj-elsn0 35326 | If the intersection of two... |
bj-opelid 35327 | Characterization of the or... |
bj-ideqg 35328 | Characterization of the cl... |
bj-ideqgALT 35329 | Alternate proof of ~ bj-id... |
bj-ideqb 35330 | Characterization of classe... |
bj-idres 35331 | Alternate expression for t... |
bj-opelidres 35332 | Characterization of the or... |
bj-idreseq 35333 | Sufficient condition for t... |
bj-idreseqb 35334 | Characterization for two c... |
bj-ideqg1 35335 | For sets, the identity rel... |
bj-ideqg1ALT 35336 | Alternate proof of bj-ideq... |
bj-opelidb1ALT 35337 | Characterization of the co... |
bj-elid3 35338 | Characterization of the co... |
bj-elid4 35339 | Characterization of the el... |
bj-elid5 35340 | Characterization of the el... |
bj-elid6 35341 | Characterization of the el... |
bj-elid7 35342 | Characterization of the el... |
bj-diagval 35345 | Value of the functionalize... |
bj-diagval2 35346 | Value of the functionalize... |
bj-eldiag 35347 | Characterization of the el... |
bj-eldiag2 35348 | Characterization of the el... |
bj-imdirvallem 35351 | Lemma for ~ bj-imdirval an... |
bj-imdirval 35352 | Value of the functionalize... |
bj-imdirval2lem 35353 | Lemma for ~ bj-imdirval2 a... |
bj-imdirval2 35354 | Value of the functionalize... |
bj-imdirval3 35355 | Value of the functionalize... |
bj-imdiridlem 35356 | Lemma for ~ bj-imdirid and... |
bj-imdirid 35357 | Functorial property of the... |
bj-opelopabid 35358 | Membership in an ordered-p... |
bj-opabco 35359 | Composition of ordered-pai... |
bj-xpcossxp 35360 | The composition of two Car... |
bj-imdirco 35361 | Functorial property of the... |
bj-iminvval 35364 | Value of the functionalize... |
bj-iminvval2 35365 | Value of the functionalize... |
bj-iminvid 35366 | Functorial property of the... |
bj-inftyexpitaufo 35373 | The function ` inftyexpita... |
bj-inftyexpitaudisj 35376 | An element of the circle a... |
bj-inftyexpiinv 35379 | Utility theorem for the in... |
bj-inftyexpiinj 35380 | Injectivity of the paramet... |
bj-inftyexpidisj 35381 | An element of the circle a... |
bj-ccinftydisj 35384 | The circle at infinity is ... |
bj-elccinfty 35385 | A lemma for infinite exten... |
bj-ccssccbar 35388 | Complex numbers are extend... |
bj-ccinftyssccbar 35389 | Infinite extended complex ... |
bj-pinftyccb 35392 | The class ` pinfty ` is an... |
bj-pinftynrr 35393 | The extended complex numbe... |
bj-minftyccb 35396 | The class ` minfty ` is an... |
bj-minftynrr 35397 | The extended complex numbe... |
bj-pinftynminfty 35398 | The extended complex numbe... |
bj-rrhatsscchat 35407 | The real projective line i... |
bj-imafv 35422 | If the direct image of a s... |
bj-funun 35423 | Value of a function expres... |
bj-fununsn1 35424 | Value of a function expres... |
bj-fununsn2 35425 | Value of a function expres... |
bj-fvsnun1 35426 | The value of a function wi... |
bj-fvsnun2 35427 | The value of a function wi... |
bj-fvmptunsn1 35428 | Value of a function expres... |
bj-fvmptunsn2 35429 | Value of a function expres... |
bj-iomnnom 35430 | The canonical bijection fr... |
bj-smgrpssmgm 35439 | Semigroups are magmas. (C... |
bj-smgrpssmgmel 35440 | Semigroups are magmas (ele... |
bj-mndsssmgrp 35441 | Monoids are semigroups. (... |
bj-mndsssmgrpel 35442 | Monoids are semigroups (el... |
bj-cmnssmnd 35443 | Commutative monoids are mo... |
bj-cmnssmndel 35444 | Commutative monoids are mo... |
bj-grpssmnd 35445 | Groups are monoids. (Cont... |
bj-grpssmndel 35446 | Groups are monoids (elemen... |
bj-ablssgrp 35447 | Abelian groups are groups.... |
bj-ablssgrpel 35448 | Abelian groups are groups ... |
bj-ablsscmn 35449 | Abelian groups are commuta... |
bj-ablsscmnel 35450 | Abelian groups are commuta... |
bj-modssabl 35451 | (The additive groups of) m... |
bj-vecssmod 35452 | Vector spaces are modules.... |
bj-vecssmodel 35453 | Vector spaces are modules ... |
bj-finsumval0 35456 | Value of a finite sum. (C... |
bj-fvimacnv0 35457 | Variant of ~ fvimacnv wher... |
bj-isvec 35458 | The predicate "is a vector... |
bj-fldssdrng 35459 | Fields are division rings.... |
bj-flddrng 35460 | Fields are division rings ... |
bj-rrdrg 35461 | The field of real numbers ... |
bj-isclm 35462 | The predicate "is a subcom... |
bj-isrvec 35465 | The predicate "is a real v... |
bj-rvecmod 35466 | Real vector spaces are mod... |
bj-rvecssmod 35467 | Real vector spaces are mod... |
bj-rvecrr 35468 | The field of scalars of a ... |
bj-isrvecd 35469 | The predicate "is a real v... |
bj-rvecvec 35470 | Real vector spaces are vec... |
bj-isrvec2 35471 | The predicate "is a real v... |
bj-rvecssvec 35472 | Real vector spaces are vec... |
bj-rveccmod 35473 | Real vector spaces are sub... |
bj-rvecsscmod 35474 | Real vector spaces are sub... |
bj-rvecsscvec 35475 | Real vector spaces are sub... |
bj-rveccvec 35476 | Real vector spaces are sub... |
bj-rvecssabl 35477 | (The additive groups of) r... |
bj-rvecabl 35478 | (The additive groups of) r... |
bj-subcom 35479 | A consequence of commutati... |
bj-lineqi 35480 | Solution of a (scalar) lin... |
bj-bary1lem 35481 | Lemma for ~ bj-bary1 : exp... |
bj-bary1lem1 35482 | Lemma for bj-bary1: comput... |
bj-bary1 35483 | Barycentric coordinates in... |
bj-endval 35486 | Value of the monoid of end... |
bj-endbase 35487 | Base set of the monoid of ... |
bj-endcomp 35488 | Composition law of the mon... |
bj-endmnd 35489 | The monoid of endomorphism... |
taupilem3 35490 | Lemma for tau-related theo... |
taupilemrplb 35491 | A set of positive reals ha... |
taupilem1 35492 | Lemma for ~ taupi . A pos... |
taupilem2 35493 | Lemma for ~ taupi . The s... |
taupi 35494 | Relationship between ` _ta... |
dfgcd3 35495 | Alternate definition of th... |
irrdifflemf 35496 | Lemma for ~ irrdiff . The... |
irrdiff 35497 | The irrationals are exactl... |
iccioo01 35498 | The closed unit interval i... |
csbrecsg 35499 | Move class substitution in... |
csbrdgg 35500 | Move class substitution in... |
csboprabg 35501 | Move class substitution in... |
csbmpo123 35502 | Move class substitution in... |
con1bii2 35503 | A contraposition inference... |
con2bii2 35504 | A contraposition inference... |
vtoclefex 35505 | Implicit substitution of a... |
rnmptsn 35506 | The range of a function ma... |
f1omptsnlem 35507 | This is the core of the pr... |
f1omptsn 35508 | A function mapping to sing... |
mptsnunlem 35509 | This is the core of the pr... |
mptsnun 35510 | A class ` B ` is equal to ... |
dissneqlem 35511 | This is the core of the pr... |
dissneq 35512 | Any topology that contains... |
exlimim 35513 | Closed form of ~ exlimimd ... |
exlimimd 35514 | Existential elimination ru... |
exellim 35515 | Closed form of ~ exellimdd... |
exellimddv 35516 | Eliminate an antecedent wh... |
topdifinfindis 35517 | Part of Exercise 3 of [Mun... |
topdifinffinlem 35518 | This is the core of the pr... |
topdifinffin 35519 | Part of Exercise 3 of [Mun... |
topdifinf 35520 | Part of Exercise 3 of [Mun... |
topdifinfeq 35521 | Two different ways of defi... |
icorempo 35522 | Closed-below, open-above i... |
icoreresf 35523 | Closed-below, open-above i... |
icoreval 35524 | Value of the closed-below,... |
icoreelrnab 35525 | Elementhood in the set of ... |
isbasisrelowllem1 35526 | Lemma for ~ isbasisrelowl ... |
isbasisrelowllem2 35527 | Lemma for ~ isbasisrelowl ... |
icoreclin 35528 | The set of closed-below, o... |
isbasisrelowl 35529 | The set of all closed-belo... |
icoreunrn 35530 | The union of all closed-be... |
istoprelowl 35531 | The set of all closed-belo... |
icoreelrn 35532 | A class abstraction which ... |
iooelexlt 35533 | An element of an open inte... |
relowlssretop 35534 | The lower limit topology o... |
relowlpssretop 35535 | The lower limit topology o... |
sucneqond 35536 | Inequality of an ordinal s... |
sucneqoni 35537 | Inequality of an ordinal s... |
onsucuni3 35538 | If an ordinal number has a... |
1oequni2o 35539 | The ordinal number ` 1o ` ... |
rdgsucuni 35540 | If an ordinal number has a... |
rdgeqoa 35541 | If a recursive function wi... |
elxp8 35542 | Membership in a Cartesian ... |
cbveud 35543 | Deduction used to change b... |
cbvreud 35544 | Deduction used to change b... |
difunieq 35545 | The difference of unions i... |
inunissunidif 35546 | Theorem about subsets of t... |
rdgellim 35547 | Elementhood in a recursive... |
rdglimss 35548 | A recursive definition at ... |
rdgssun 35549 | In a recursive definition ... |
exrecfnlem 35550 | Lemma for ~ exrecfn . (Co... |
exrecfn 35551 | Theorem about the existenc... |
exrecfnpw 35552 | For any base set, a set wh... |
finorwe 35553 | If the Axiom of Infinity i... |
dffinxpf 35556 | This theorem is the same a... |
finxpeq1 35557 | Equality theorem for Carte... |
finxpeq2 35558 | Equality theorem for Carte... |
csbfinxpg 35559 | Distribute proper substitu... |
finxpreclem1 35560 | Lemma for ` ^^ ` recursion... |
finxpreclem2 35561 | Lemma for ` ^^ ` recursion... |
finxp0 35562 | The value of Cartesian exp... |
finxp1o 35563 | The value of Cartesian exp... |
finxpreclem3 35564 | Lemma for ` ^^ ` recursion... |
finxpreclem4 35565 | Lemma for ` ^^ ` recursion... |
finxpreclem5 35566 | Lemma for ` ^^ ` recursion... |
finxpreclem6 35567 | Lemma for ` ^^ ` recursion... |
finxpsuclem 35568 | Lemma for ~ finxpsuc . (C... |
finxpsuc 35569 | The value of Cartesian exp... |
finxp2o 35570 | The value of Cartesian exp... |
finxp3o 35571 | The value of Cartesian exp... |
finxpnom 35572 | Cartesian exponentiation w... |
finxp00 35573 | Cartesian exponentiation o... |
iunctb2 35574 | Using the axiom of countab... |
domalom 35575 | A class which dominates ev... |
isinf2 35576 | The converse of ~ isinf . ... |
ctbssinf 35577 | Using the axiom of choice,... |
ralssiun 35578 | The index set of an indexe... |
nlpineqsn 35579 | For every point ` p ` of a... |
nlpfvineqsn 35580 | Given a subset ` A ` of ` ... |
fvineqsnf1 35581 | A theorem about functions ... |
fvineqsneu 35582 | A theorem about functions ... |
fvineqsneq 35583 | A theorem about functions ... |
pibp16 35584 | Property P000016 of pi-bas... |
pibp19 35585 | Property P000019 of pi-bas... |
pibp21 35586 | Property P000021 of pi-bas... |
pibt1 35587 | Theorem T000001 of pi-base... |
pibt2 35588 | Theorem T000002 of pi-base... |
wl-section-prop 35589 | Intuitionistic logic is no... |
wl-section-boot 35593 | In this section, I provide... |
wl-luk-imim1i 35594 | Inference adding common co... |
wl-luk-syl 35595 | An inference version of th... |
wl-luk-imtrid 35596 | A syllogism rule of infere... |
wl-luk-pm2.18d 35597 | Deduction based on reducti... |
wl-luk-con4i 35598 | Inference rule. Copy of ~... |
wl-luk-pm2.24i 35599 | Inference rule. Copy of ~... |
wl-luk-a1i 35600 | Inference rule. Copy of ~... |
wl-luk-mpi 35601 | A nested modus ponens infe... |
wl-luk-imim2i 35602 | Inference adding common an... |
wl-luk-imtrdi 35603 | A syllogism rule of infere... |
wl-luk-ax3 35604 | ~ ax-3 proved from Lukasie... |
wl-luk-ax1 35605 | ~ ax-1 proved from Lukasie... |
wl-luk-pm2.27 35606 | This theorem, called "Asse... |
wl-luk-com12 35607 | Inference that swaps (comm... |
wl-luk-pm2.21 35608 | From a wff and its negatio... |
wl-luk-con1i 35609 | A contraposition inference... |
wl-luk-ja 35610 | Inference joining the ante... |
wl-luk-imim2 35611 | A closed form of syllogism... |
wl-luk-a1d 35612 | Deduction introducing an e... |
wl-luk-ax2 35613 | ~ ax-2 proved from Lukasie... |
wl-luk-id 35614 | Principle of identity. Th... |
wl-luk-notnotr 35615 | Converse of double negatio... |
wl-luk-pm2.04 35616 | Swap antecedents. Theorem... |
wl-section-impchain 35617 | An implication like ` ( ps... |
wl-impchain-mp-x 35618 | This series of theorems pr... |
wl-impchain-mp-0 35619 | This theorem is the start ... |
wl-impchain-mp-1 35620 | This theorem is in fact a ... |
wl-impchain-mp-2 35621 | This theorem is in fact a ... |
wl-impchain-com-1.x 35622 | It is often convenient to ... |
wl-impchain-com-1.1 35623 | A degenerate form of antec... |
wl-impchain-com-1.2 35624 | This theorem is in fact a ... |
wl-impchain-com-1.3 35625 | This theorem is in fact a ... |
wl-impchain-com-1.4 35626 | This theorem is in fact a ... |
wl-impchain-com-n.m 35627 | This series of theorems al... |
wl-impchain-com-2.3 35628 | This theorem is in fact a ... |
wl-impchain-com-2.4 35629 | This theorem is in fact a ... |
wl-impchain-com-3.2.1 35630 | This theorem is in fact a ... |
wl-impchain-a1-x 35631 | If an implication chain is... |
wl-impchain-a1-1 35632 | Inference rule, a copy of ... |
wl-impchain-a1-2 35633 | Inference rule, a copy of ... |
wl-impchain-a1-3 35634 | Inference rule, a copy of ... |
wl-ifp-ncond1 35635 | If one case of an ` if- ` ... |
wl-ifp-ncond2 35636 | If one case of an ` if- ` ... |
wl-ifpimpr 35637 | If one case of an ` if- ` ... |
wl-ifp4impr 35638 | If one case of an ` if- ` ... |
wl-df-3xor 35639 | Alternative definition of ... |
wl-df3xor2 35640 | Alternative definition of ... |
wl-df3xor3 35641 | Alternative form of ~ wl-d... |
wl-3xortru 35642 | If the first input is true... |
wl-3xorfal 35643 | If the first input is fals... |
wl-3xorbi 35644 | Triple xor can be replaced... |
wl-3xorbi2 35645 | Alternative form of ~ wl-3... |
wl-3xorbi123d 35646 | Equivalence theorem for tr... |
wl-3xorbi123i 35647 | Equivalence theorem for tr... |
wl-3xorrot 35648 | Rotation law for triple xo... |
wl-3xorcoma 35649 | Commutative law for triple... |
wl-3xorcomb 35650 | Commutative law for triple... |
wl-3xornot1 35651 | Flipping the first input f... |
wl-3xornot 35652 | Triple xor distributes ove... |
wl-1xor 35653 | In the recursive scheme ... |
wl-2xor 35654 | In the recursive scheme ... |
wl-df-3mintru2 35655 | Alternative definition of ... |
wl-df2-3mintru2 35656 | The adder carry in disjunc... |
wl-df3-3mintru2 35657 | The adder carry in conjunc... |
wl-df4-3mintru2 35658 | An alternative definition ... |
wl-1mintru1 35659 | Using the recursion formul... |
wl-1mintru2 35660 | Using the recursion formul... |
wl-2mintru1 35661 | Using the recursion formul... |
wl-2mintru2 35662 | Using the recursion formul... |
wl-df3maxtru1 35663 | Assuming "(n+1)-maxtru1" `... |
wl-ax13lem1 35665 | A version of ~ ax-wl-13v w... |
wl-mps 35666 | Replacing a nested consequ... |
wl-syls1 35667 | Replacing a nested consequ... |
wl-syls2 35668 | Replacing a nested anteced... |
wl-embant 35669 | A true wff can always be a... |
wl-orel12 35670 | In a conjunctive normal fo... |
wl-cases2-dnf 35671 | A particular instance of ~... |
wl-cbvmotv 35672 | Change bound variable. Us... |
wl-moteq 35673 | Change bound variable. Us... |
wl-motae 35674 | Change bound variable. Us... |
wl-moae 35675 | Two ways to express "at mo... |
wl-euae 35676 | Two ways to express "exact... |
wl-nax6im 35677 | The following series of th... |
wl-hbae1 35678 | This specialization of ~ h... |
wl-naevhba1v 35679 | An instance of ~ hbn1w app... |
wl-spae 35680 | Prove an instance of ~ sp ... |
wl-speqv 35681 | Under the assumption ` -. ... |
wl-19.8eqv 35682 | Under the assumption ` -. ... |
wl-19.2reqv 35683 | Under the assumption ` -. ... |
wl-nfalv 35684 | If ` x ` is not present in... |
wl-nfimf1 35685 | An antecedent is irrelevan... |
wl-nfae1 35686 | Unlike ~ nfae , this speci... |
wl-nfnae1 35687 | Unlike ~ nfnae , this spec... |
wl-aetr 35688 | A transitive law for varia... |
wl-axc11r 35689 | Same as ~ axc11r , but usi... |
wl-dral1d 35690 | A version of ~ dral1 with ... |
wl-cbvalnaed 35691 | ~ wl-cbvalnae with a conte... |
wl-cbvalnae 35692 | A more general version of ... |
wl-exeq 35693 | The semantics of ` E. x y ... |
wl-aleq 35694 | The semantics of ` A. x y ... |
wl-nfeqfb 35695 | Extend ~ nfeqf to an equiv... |
wl-nfs1t 35696 | If ` y ` is not free in ` ... |
wl-equsalvw 35697 | Version of ~ equsalv with ... |
wl-equsald 35698 | Deduction version of ~ equ... |
wl-equsal 35699 | A useful equivalence relat... |
wl-equsal1t 35700 | The expression ` x = y ` i... |
wl-equsalcom 35701 | This simple equivalence ea... |
wl-equsal1i 35702 | The antecedent ` x = y ` i... |
wl-sb6rft 35703 | A specialization of ~ wl-e... |
wl-cbvalsbi 35704 | Change bounded variables i... |
wl-sbrimt 35705 | Substitution with a variab... |
wl-sblimt 35706 | Substitution with a variab... |
wl-sb8t 35707 | Substitution of variable i... |
wl-sb8et 35708 | Substitution of variable i... |
wl-sbhbt 35709 | Closed form of ~ sbhb . C... |
wl-sbnf1 35710 | Two ways expressing that `... |
wl-equsb3 35711 | ~ equsb3 with a distinctor... |
wl-equsb4 35712 | Substitution applied to an... |
wl-2sb6d 35713 | Version of ~ 2sb6 with a c... |
wl-sbcom2d-lem1 35714 | Lemma used to prove ~ wl-s... |
wl-sbcom2d-lem2 35715 | Lemma used to prove ~ wl-s... |
wl-sbcom2d 35716 | Version of ~ sbcom2 with a... |
wl-sbalnae 35717 | A theorem used in eliminat... |
wl-sbal1 35718 | A theorem used in eliminat... |
wl-sbal2 35719 | Move quantifier in and out... |
wl-2spsbbi 35720 | ~ spsbbi applied twice. (... |
wl-lem-exsb 35721 | This theorem provides a ba... |
wl-lem-nexmo 35722 | This theorem provides a ba... |
wl-lem-moexsb 35723 | The antecedent ` A. x ( ph... |
wl-alanbii 35724 | This theorem extends ~ ala... |
wl-mo2df 35725 | Version of ~ mof with a co... |
wl-mo2tf 35726 | Closed form of ~ mof with ... |
wl-eudf 35727 | Version of ~ eu6 with a co... |
wl-eutf 35728 | Closed form of ~ eu6 with ... |
wl-euequf 35729 | ~ euequ proved with a dist... |
wl-mo2t 35730 | Closed form of ~ mof . (C... |
wl-mo3t 35731 | Closed form of ~ mo3 . (C... |
wl-sb8eut 35732 | Substitution of variable i... |
wl-sb8mot 35733 | Substitution of variable i... |
wl-axc11rc11 35734 | Proving ~ axc11r from ~ ax... |
wl-ax11-lem1 35736 | A transitive law for varia... |
wl-ax11-lem2 35737 | Lemma. (Contributed by Wo... |
wl-ax11-lem3 35738 | Lemma. (Contributed by Wo... |
wl-ax11-lem4 35739 | Lemma. (Contributed by Wo... |
wl-ax11-lem5 35740 | Lemma. (Contributed by Wo... |
wl-ax11-lem6 35741 | Lemma. (Contributed by Wo... |
wl-ax11-lem7 35742 | Lemma. (Contributed by Wo... |
wl-ax11-lem8 35743 | Lemma. (Contributed by Wo... |
wl-ax11-lem9 35744 | The easy part when ` x ` c... |
wl-ax11-lem10 35745 | We now have prepared every... |
wl-clabv 35746 | Variant of ~ df-clab , whe... |
wl-dfclab 35747 | Rederive ~ df-clab from ~ ... |
wl-clabtv 35748 | Using class abstraction in... |
wl-clabt 35749 | Using class abstraction in... |
rabiun 35750 | Abstraction restricted to ... |
iundif1 35751 | Indexed union of class dif... |
imadifss 35752 | The difference of images i... |
cureq 35753 | Equality theorem for curry... |
unceq 35754 | Equality theorem for uncur... |
curf 35755 | Functional property of cur... |
uncf 35756 | Functional property of unc... |
curfv 35757 | Value of currying. (Contr... |
uncov 35758 | Value of uncurrying. (Con... |
curunc 35759 | Currying of uncurrying. (... |
unccur 35760 | Uncurrying of currying. (... |
phpreu 35761 | Theorem related to pigeonh... |
finixpnum 35762 | A finite Cartesian product... |
fin2solem 35763 | Lemma for ~ fin2so . (Con... |
fin2so 35764 | Any totally ordered Tarski... |
ltflcei 35765 | Theorem to move the floor ... |
leceifl 35766 | Theorem to move the floor ... |
sin2h 35767 | Half-angle rule for sine. ... |
cos2h 35768 | Half-angle rule for cosine... |
tan2h 35769 | Half-angle rule for tangen... |
lindsadd 35770 | In a vector space, the uni... |
lindsdom 35771 | A linearly independent set... |
lindsenlbs 35772 | A maximal linearly indepen... |
matunitlindflem1 35773 | One direction of ~ matunit... |
matunitlindflem2 35774 | One direction of ~ matunit... |
matunitlindf 35775 | A matrix over a field is i... |
ptrest 35776 | Expressing a restriction o... |
ptrecube 35777 | Any point in an open set o... |
poimirlem1 35778 | Lemma for ~ poimir - the v... |
poimirlem2 35779 | Lemma for ~ poimir - conse... |
poimirlem3 35780 | Lemma for ~ poimir to add ... |
poimirlem4 35781 | Lemma for ~ poimir connect... |
poimirlem5 35782 | Lemma for ~ poimir to esta... |
poimirlem6 35783 | Lemma for ~ poimir establi... |
poimirlem7 35784 | Lemma for ~ poimir , simil... |
poimirlem8 35785 | Lemma for ~ poimir , estab... |
poimirlem9 35786 | Lemma for ~ poimir , estab... |
poimirlem10 35787 | Lemma for ~ poimir establi... |
poimirlem11 35788 | Lemma for ~ poimir connect... |
poimirlem12 35789 | Lemma for ~ poimir connect... |
poimirlem13 35790 | Lemma for ~ poimir - for a... |
poimirlem14 35791 | Lemma for ~ poimir - for a... |
poimirlem15 35792 | Lemma for ~ poimir , that ... |
poimirlem16 35793 | Lemma for ~ poimir establi... |
poimirlem17 35794 | Lemma for ~ poimir establi... |
poimirlem18 35795 | Lemma for ~ poimir stating... |
poimirlem19 35796 | Lemma for ~ poimir establi... |
poimirlem20 35797 | Lemma for ~ poimir establi... |
poimirlem21 35798 | Lemma for ~ poimir stating... |
poimirlem22 35799 | Lemma for ~ poimir , that ... |
poimirlem23 35800 | Lemma for ~ poimir , two w... |
poimirlem24 35801 | Lemma for ~ poimir , two w... |
poimirlem25 35802 | Lemma for ~ poimir stating... |
poimirlem26 35803 | Lemma for ~ poimir showing... |
poimirlem27 35804 | Lemma for ~ poimir showing... |
poimirlem28 35805 | Lemma for ~ poimir , a var... |
poimirlem29 35806 | Lemma for ~ poimir connect... |
poimirlem30 35807 | Lemma for ~ poimir combini... |
poimirlem31 35808 | Lemma for ~ poimir , assig... |
poimirlem32 35809 | Lemma for ~ poimir , combi... |
poimir 35810 | Poincare-Miranda theorem. ... |
broucube 35811 | Brouwer - or as Kulpa call... |
heicant 35812 | Heine-Cantor theorem: a co... |
opnmbllem0 35813 | Lemma for ~ ismblfin ; cou... |
mblfinlem1 35814 | Lemma for ~ ismblfin , ord... |
mblfinlem2 35815 | Lemma for ~ ismblfin , eff... |
mblfinlem3 35816 | The difference between two... |
mblfinlem4 35817 | Backward direction of ~ is... |
ismblfin 35818 | Measurability in terms of ... |
ovoliunnfl 35819 | ~ ovoliun is incompatible ... |
ex-ovoliunnfl 35820 | Demonstration of ~ ovoliun... |
voliunnfl 35821 | ~ voliun is incompatible w... |
volsupnfl 35822 | ~ volsup is incompatible w... |
mbfresfi 35823 | Measurability of a piecewi... |
mbfposadd 35824 | If the sum of two measurab... |
cnambfre 35825 | A real-valued, a.e. contin... |
dvtanlem 35826 | Lemma for ~ dvtan - the do... |
dvtan 35827 | Derivative of tangent. (C... |
itg2addnclem 35828 | An alternate expression fo... |
itg2addnclem2 35829 | Lemma for ~ itg2addnc . T... |
itg2addnclem3 35830 | Lemma incomprehensible in ... |
itg2addnc 35831 | Alternate proof of ~ itg2a... |
itg2gt0cn 35832 | ~ itg2gt0 holds on functio... |
ibladdnclem 35833 | Lemma for ~ ibladdnc ; cf ... |
ibladdnc 35834 | Choice-free analogue of ~ ... |
itgaddnclem1 35835 | Lemma for ~ itgaddnc ; cf.... |
itgaddnclem2 35836 | Lemma for ~ itgaddnc ; cf.... |
itgaddnc 35837 | Choice-free analogue of ~ ... |
iblsubnc 35838 | Choice-free analogue of ~ ... |
itgsubnc 35839 | Choice-free analogue of ~ ... |
iblabsnclem 35840 | Lemma for ~ iblabsnc ; cf.... |
iblabsnc 35841 | Choice-free analogue of ~ ... |
iblmulc2nc 35842 | Choice-free analogue of ~ ... |
itgmulc2nclem1 35843 | Lemma for ~ itgmulc2nc ; c... |
itgmulc2nclem2 35844 | Lemma for ~ itgmulc2nc ; c... |
itgmulc2nc 35845 | Choice-free analogue of ~ ... |
itgabsnc 35846 | Choice-free analogue of ~ ... |
itggt0cn 35847 | ~ itggt0 holds for continu... |
ftc1cnnclem 35848 | Lemma for ~ ftc1cnnc ; cf.... |
ftc1cnnc 35849 | Choice-free proof of ~ ftc... |
ftc1anclem1 35850 | Lemma for ~ ftc1anc - the ... |
ftc1anclem2 35851 | Lemma for ~ ftc1anc - rest... |
ftc1anclem3 35852 | Lemma for ~ ftc1anc - the ... |
ftc1anclem4 35853 | Lemma for ~ ftc1anc . (Co... |
ftc1anclem5 35854 | Lemma for ~ ftc1anc , the ... |
ftc1anclem6 35855 | Lemma for ~ ftc1anc - cons... |
ftc1anclem7 35856 | Lemma for ~ ftc1anc . (Co... |
ftc1anclem8 35857 | Lemma for ~ ftc1anc . (Co... |
ftc1anc 35858 | ~ ftc1a holds for function... |
ftc2nc 35859 | Choice-free proof of ~ ftc... |
asindmre 35860 | Real part of domain of dif... |
dvasin 35861 | Derivative of arcsine. (C... |
dvacos 35862 | Derivative of arccosine. ... |
dvreasin 35863 | Real derivative of arcsine... |
dvreacos 35864 | Real derivative of arccosi... |
areacirclem1 35865 | Antiderivative of cross-se... |
areacirclem2 35866 | Endpoint-inclusive continu... |
areacirclem3 35867 | Integrability of cross-sec... |
areacirclem4 35868 | Endpoint-inclusive continu... |
areacirclem5 35869 | Finding the cross-section ... |
areacirc 35870 | The area of a circle of ra... |
unirep 35871 | Define a quantity whose de... |
cover2 35872 | Two ways of expressing the... |
cover2g 35873 | Two ways of expressing the... |
brabg2 35874 | Relation by a binary relat... |
opelopab3 35875 | Ordered pair membership in... |
cocanfo 35876 | Cancellation of a surjecti... |
brresi2 35877 | Restriction of a binary re... |
fnopabeqd 35878 | Equality deduction for fun... |
fvopabf4g 35879 | Function value of an opera... |
eqfnun 35880 | Two functions on ` A u. B ... |
fnopabco 35881 | Composition of a function ... |
opropabco 35882 | Composition of an operator... |
cocnv 35883 | Composition with a functio... |
f1ocan1fv 35884 | Cancel a composition by a ... |
f1ocan2fv 35885 | Cancel a composition by th... |
inixp 35886 | Intersection of Cartesian ... |
upixp 35887 | Universal property of the ... |
abrexdom 35888 | An indexed set is dominate... |
abrexdom2 35889 | An indexed set is dominate... |
ac6gf 35890 | Axiom of Choice. (Contrib... |
indexa 35891 | If for every element of an... |
indexdom 35892 | If for every element of an... |
frinfm 35893 | A subset of a well-founded... |
welb 35894 | A nonempty subset of a wel... |
supex2g 35895 | Existence of supremum. (C... |
supclt 35896 | Closure of supremum. (Con... |
supubt 35897 | Upper bound property of su... |
filbcmb 35898 | Combine a finite set of lo... |
fzmul 35899 | Membership of a product in... |
sdclem2 35900 | Lemma for ~ sdc . (Contri... |
sdclem1 35901 | Lemma for ~ sdc . (Contri... |
sdc 35902 | Strong dependent choice. ... |
fdc 35903 | Finite version of dependen... |
fdc1 35904 | Variant of ~ fdc with no s... |
seqpo 35905 | Two ways to say that a seq... |
incsequz 35906 | An increasing sequence of ... |
incsequz2 35907 | An increasing sequence of ... |
nnubfi 35908 | A bounded above set of pos... |
nninfnub 35909 | An infinite set of positiv... |
subspopn 35910 | An open set is open in the... |
neificl 35911 | Neighborhoods are closed u... |
lpss2 35912 | Limit points of a subset a... |
metf1o 35913 | Use a bijection with a met... |
blssp 35914 | A ball in the subspace met... |
mettrifi 35915 | Generalized triangle inequ... |
lmclim2 35916 | A sequence in a metric spa... |
geomcau 35917 | If the distance between co... |
caures 35918 | The restriction of a Cauch... |
caushft 35919 | A shifted Cauchy sequence ... |
constcncf 35920 | A constant function is a c... |
cnres2 35921 | The restriction of a conti... |
cnresima 35922 | A continuous function is c... |
cncfres 35923 | A continuous function on c... |
istotbnd 35927 | The predicate "is a totall... |
istotbnd2 35928 | The predicate "is a totall... |
istotbnd3 35929 | A metric space is totally ... |
totbndmet 35930 | The predicate "totally bou... |
0totbnd 35931 | The metric (there is only ... |
sstotbnd2 35932 | Condition for a subset of ... |
sstotbnd 35933 | Condition for a subset of ... |
sstotbnd3 35934 | Use a net that is not nece... |
totbndss 35935 | A subset of a totally boun... |
equivtotbnd 35936 | If the metric ` M ` is "st... |
isbnd 35938 | The predicate "is a bounde... |
bndmet 35939 | A bounded metric space is ... |
isbndx 35940 | A "bounded extended metric... |
isbnd2 35941 | The predicate "is a bounde... |
isbnd3 35942 | A metric space is bounded ... |
isbnd3b 35943 | A metric space is bounded ... |
bndss 35944 | A subset of a bounded metr... |
blbnd 35945 | A ball is bounded. (Contr... |
ssbnd 35946 | A subset of a metric space... |
totbndbnd 35947 | A totally bounded metric s... |
equivbnd 35948 | If the metric ` M ` is "st... |
bnd2lem 35949 | Lemma for ~ equivbnd2 and ... |
equivbnd2 35950 | If balls are totally bound... |
prdsbnd 35951 | The product metric over fi... |
prdstotbnd 35952 | The product metric over fi... |
prdsbnd2 35953 | If balls are totally bound... |
cntotbnd 35954 | A subset of the complex nu... |
cnpwstotbnd 35955 | A subset of ` A ^ I ` , wh... |
ismtyval 35958 | The set of isometries betw... |
isismty 35959 | The condition "is an isome... |
ismtycnv 35960 | The inverse of an isometry... |
ismtyima 35961 | The image of a ball under ... |
ismtyhmeolem 35962 | Lemma for ~ ismtyhmeo . (... |
ismtyhmeo 35963 | An isometry is a homeomorp... |
ismtybndlem 35964 | Lemma for ~ ismtybnd . (C... |
ismtybnd 35965 | Isometries preserve bounde... |
ismtyres 35966 | A restriction of an isomet... |
heibor1lem 35967 | Lemma for ~ heibor1 . A c... |
heibor1 35968 | One half of ~ heibor , tha... |
heiborlem1 35969 | Lemma for ~ heibor . We w... |
heiborlem2 35970 | Lemma for ~ heibor . Subs... |
heiborlem3 35971 | Lemma for ~ heibor . Usin... |
heiborlem4 35972 | Lemma for ~ heibor . Usin... |
heiborlem5 35973 | Lemma for ~ heibor . The ... |
heiborlem6 35974 | Lemma for ~ heibor . Sinc... |
heiborlem7 35975 | Lemma for ~ heibor . Sinc... |
heiborlem8 35976 | Lemma for ~ heibor . The ... |
heiborlem9 35977 | Lemma for ~ heibor . Disc... |
heiborlem10 35978 | Lemma for ~ heibor . The ... |
heibor 35979 | Generalized Heine-Borel Th... |
bfplem1 35980 | Lemma for ~ bfp . The seq... |
bfplem2 35981 | Lemma for ~ bfp . Using t... |
bfp 35982 | Banach fixed point theorem... |
rrnval 35985 | The n-dimensional Euclidea... |
rrnmval 35986 | The value of the Euclidean... |
rrnmet 35987 | Euclidean space is a metri... |
rrndstprj1 35988 | The distance between two p... |
rrndstprj2 35989 | Bound on the distance betw... |
rrncmslem 35990 | Lemma for ~ rrncms . (Con... |
rrncms 35991 | Euclidean space is complet... |
repwsmet 35992 | The supremum metric on ` R... |
rrnequiv 35993 | The supremum metric on ` R... |
rrntotbnd 35994 | A set in Euclidean space i... |
rrnheibor 35995 | Heine-Borel theorem for Eu... |
ismrer1 35996 | An isometry between ` RR `... |
reheibor 35997 | Heine-Borel theorem for re... |
iccbnd 35998 | A closed interval in ` RR ... |
icccmpALT 35999 | A closed interval in ` RR ... |
isass 36004 | The predicate "is an assoc... |
isexid 36005 | The predicate ` G ` has a ... |
ismgmOLD 36008 | Obsolete version of ~ ismg... |
clmgmOLD 36009 | Obsolete version of ~ mgmc... |
opidonOLD 36010 | Obsolete version of ~ mndp... |
rngopidOLD 36011 | Obsolete version of ~ mndp... |
opidon2OLD 36012 | Obsolete version of ~ mndp... |
isexid2 36013 | If ` G e. ( Magma i^i ExId... |
exidu1 36014 | Uniqueness of the left and... |
idrval 36015 | The value of the identity ... |
iorlid 36016 | A magma right and left ide... |
cmpidelt 36017 | A magma right and left ide... |
smgrpismgmOLD 36020 | Obsolete version of ~ sgrp... |
issmgrpOLD 36021 | Obsolete version of ~ issg... |
smgrpmgm 36022 | A semigroup is a magma. (... |
smgrpassOLD 36023 | Obsolete version of ~ sgrp... |
mndoissmgrpOLD 36026 | Obsolete version of ~ mnds... |
mndoisexid 36027 | A monoid has an identity e... |
mndoismgmOLD 36028 | Obsolete version of ~ mndm... |
mndomgmid 36029 | A monoid is a magma with a... |
ismndo 36030 | The predicate "is a monoid... |
ismndo1 36031 | The predicate "is a monoid... |
ismndo2 36032 | The predicate "is a monoid... |
grpomndo 36033 | A group is a monoid. (Con... |
exidcl 36034 | Closure of the binary oper... |
exidreslem 36035 | Lemma for ~ exidres and ~ ... |
exidres 36036 | The restriction of a binar... |
exidresid 36037 | The restriction of a binar... |
ablo4pnp 36038 | A commutative/associative ... |
grpoeqdivid 36039 | Two group elements are equ... |
grposnOLD 36040 | The group operation for th... |
elghomlem1OLD 36043 | Obsolete as of 15-Mar-2020... |
elghomlem2OLD 36044 | Obsolete as of 15-Mar-2020... |
elghomOLD 36045 | Obsolete version of ~ isgh... |
ghomlinOLD 36046 | Obsolete version of ~ ghml... |
ghomidOLD 36047 | Obsolete version of ~ ghmi... |
ghomf 36048 | Mapping property of a grou... |
ghomco 36049 | The composition of two gro... |
ghomdiv 36050 | Group homomorphisms preser... |
grpokerinj 36051 | A group homomorphism is in... |
relrngo 36054 | The class of all unital ri... |
isrngo 36055 | The predicate "is a (unita... |
isrngod 36056 | Conditions that determine ... |
rngoi 36057 | The properties of a unital... |
rngosm 36058 | Functionality of the multi... |
rngocl 36059 | Closure of the multiplicat... |
rngoid 36060 | The multiplication operati... |
rngoideu 36061 | The unit element of a ring... |
rngodi 36062 | Distributive law for the m... |
rngodir 36063 | Distributive law for the m... |
rngoass 36064 | Associative law for the mu... |
rngo2 36065 | A ring element plus itself... |
rngoablo 36066 | A ring's addition operatio... |
rngoablo2 36067 | In a unital ring the addit... |
rngogrpo 36068 | A ring's addition operatio... |
rngone0 36069 | The base set of a ring is ... |
rngogcl 36070 | Closure law for the additi... |
rngocom 36071 | The addition operation of ... |
rngoaass 36072 | The addition operation of ... |
rngoa32 36073 | The addition operation of ... |
rngoa4 36074 | Rearrangement of 4 terms i... |
rngorcan 36075 | Right cancellation law for... |
rngolcan 36076 | Left cancellation law for ... |
rngo0cl 36077 | A ring has an additive ide... |
rngo0rid 36078 | The additive identity of a... |
rngo0lid 36079 | The additive identity of a... |
rngolz 36080 | The zero of a unital ring ... |
rngorz 36081 | The zero of a unital ring ... |
rngosn3 36082 | Obsolete as of 25-Jan-2020... |
rngosn4 36083 | Obsolete as of 25-Jan-2020... |
rngosn6 36084 | Obsolete as of 25-Jan-2020... |
rngonegcl 36085 | A ring is closed under neg... |
rngoaddneg1 36086 | Adding the negative in a r... |
rngoaddneg2 36087 | Adding the negative in a r... |
rngosub 36088 | Subtraction in a ring, in ... |
rngmgmbs4 36089 | The range of an internal o... |
rngodm1dm2 36090 | In a unital ring the domai... |
rngorn1 36091 | In a unital ring the range... |
rngorn1eq 36092 | In a unital ring the range... |
rngomndo 36093 | In a unital ring the multi... |
rngoidmlem 36094 | The unit of a ring is an i... |
rngolidm 36095 | The unit of a ring is an i... |
rngoridm 36096 | The unit of a ring is an i... |
rngo1cl 36097 | The unit of a ring belongs... |
rngoueqz 36098 | Obsolete as of 23-Jan-2020... |
rngonegmn1l 36099 | Negation in a ring is the ... |
rngonegmn1r 36100 | Negation in a ring is the ... |
rngoneglmul 36101 | Negation of a product in a... |
rngonegrmul 36102 | Negation of a product in a... |
rngosubdi 36103 | Ring multiplication distri... |
rngosubdir 36104 | Ring multiplication distri... |
zerdivemp1x 36105 | In a unitary ring a left i... |
isdivrngo 36108 | The predicate "is a divisi... |
drngoi 36109 | The properties of a divisi... |
gidsn 36110 | Obsolete as of 23-Jan-2020... |
zrdivrng 36111 | The zero ring is not a div... |
dvrunz 36112 | In a division ring the uni... |
isgrpda 36113 | Properties that determine ... |
isdrngo1 36114 | The predicate "is a divisi... |
divrngcl 36115 | The product of two nonzero... |
isdrngo2 36116 | A division ring is a ring ... |
isdrngo3 36117 | A division ring is a ring ... |
rngohomval 36122 | The set of ring homomorphi... |
isrngohom 36123 | The predicate "is a ring h... |
rngohomf 36124 | A ring homomorphism is a f... |
rngohomcl 36125 | Closure law for a ring hom... |
rngohom1 36126 | A ring homomorphism preser... |
rngohomadd 36127 | Ring homomorphisms preserv... |
rngohommul 36128 | Ring homomorphisms preserv... |
rngogrphom 36129 | A ring homomorphism is a g... |
rngohom0 36130 | A ring homomorphism preser... |
rngohomsub 36131 | Ring homomorphisms preserv... |
rngohomco 36132 | The composition of two rin... |
rngokerinj 36133 | A ring homomorphism is inj... |
rngoisoval 36135 | The set of ring isomorphis... |
isrngoiso 36136 | The predicate "is a ring i... |
rngoiso1o 36137 | A ring isomorphism is a bi... |
rngoisohom 36138 | A ring isomorphism is a ri... |
rngoisocnv 36139 | The inverse of a ring isom... |
rngoisoco 36140 | The composition of two rin... |
isriscg 36142 | The ring isomorphism relat... |
isrisc 36143 | The ring isomorphism relat... |
risc 36144 | The ring isomorphism relat... |
risci 36145 | Determine that two rings a... |
riscer 36146 | Ring isomorphism is an equ... |
iscom2 36153 | A device to add commutativ... |
iscrngo 36154 | The predicate "is a commut... |
iscrngo2 36155 | The predicate "is a commut... |
iscringd 36156 | Conditions that determine ... |
flddivrng 36157 | A field is a division ring... |
crngorngo 36158 | A commutative ring is a ri... |
crngocom 36159 | The multiplication operati... |
crngm23 36160 | Commutative/associative la... |
crngm4 36161 | Commutative/associative la... |
fldcrng 36162 | A field is a commutative r... |
isfld2 36163 | The predicate "is a field"... |
crngohomfo 36164 | The image of a homomorphis... |
idlval 36171 | The class of ideals of a r... |
isidl 36172 | The predicate "is an ideal... |
isidlc 36173 | The predicate "is an ideal... |
idlss 36174 | An ideal of ` R ` is a sub... |
idlcl 36175 | An element of an ideal is ... |
idl0cl 36176 | An ideal contains ` 0 ` . ... |
idladdcl 36177 | An ideal is closed under a... |
idllmulcl 36178 | An ideal is closed under m... |
idlrmulcl 36179 | An ideal is closed under m... |
idlnegcl 36180 | An ideal is closed under n... |
idlsubcl 36181 | An ideal is closed under s... |
rngoidl 36182 | A ring ` R ` is an ` R ` i... |
0idl 36183 | The set containing only ` ... |
1idl 36184 | Two ways of expressing the... |
0rngo 36185 | In a ring, ` 0 = 1 ` iff t... |
divrngidl 36186 | The only ideals in a divis... |
intidl 36187 | The intersection of a none... |
inidl 36188 | The intersection of two id... |
unichnidl 36189 | The union of a nonempty ch... |
keridl 36190 | The kernel of a ring homom... |
pridlval 36191 | The class of prime ideals ... |
ispridl 36192 | The predicate "is a prime ... |
pridlidl 36193 | A prime ideal is an ideal.... |
pridlnr 36194 | A prime ideal is a proper ... |
pridl 36195 | The main property of a pri... |
ispridl2 36196 | A condition that shows an ... |
maxidlval 36197 | The set of maximal ideals ... |
ismaxidl 36198 | The predicate "is a maxima... |
maxidlidl 36199 | A maximal ideal is an idea... |
maxidlnr 36200 | A maximal ideal is proper.... |
maxidlmax 36201 | A maximal ideal is a maxim... |
maxidln1 36202 | One is not contained in an... |
maxidln0 36203 | A ring with a maximal idea... |
isprrngo 36208 | The predicate "is a prime ... |
prrngorngo 36209 | A prime ring is a ring. (... |
smprngopr 36210 | A simple ring (one whose o... |
divrngpr 36211 | A division ring is a prime... |
isdmn 36212 | The predicate "is a domain... |
isdmn2 36213 | The predicate "is a domain... |
dmncrng 36214 | A domain is a commutative ... |
dmnrngo 36215 | A domain is a ring. (Cont... |
flddmn 36216 | A field is a domain. (Con... |
igenval 36219 | The ideal generated by a s... |
igenss 36220 | A set is a subset of the i... |
igenidl 36221 | The ideal generated by a s... |
igenmin 36222 | The ideal generated by a s... |
igenidl2 36223 | The ideal generated by an ... |
igenval2 36224 | The ideal generated by a s... |
prnc 36225 | A principal ideal (an idea... |
isfldidl 36226 | Determine if a ring is a f... |
isfldidl2 36227 | Determine if a ring is a f... |
ispridlc 36228 | The predicate "is a prime ... |
pridlc 36229 | Property of a prime ideal ... |
pridlc2 36230 | Property of a prime ideal ... |
pridlc3 36231 | Property of a prime ideal ... |
isdmn3 36232 | The predicate "is a domain... |
dmnnzd 36233 | A domain has no zero-divis... |
dmncan1 36234 | Cancellation law for domai... |
dmncan2 36235 | Cancellation law for domai... |
efald2 36236 | A proof by contradiction. ... |
notbinot1 36237 | Simplification rule of neg... |
bicontr 36238 | Biconditional of its own n... |
impor 36239 | An equivalent formula for ... |
orfa 36240 | The falsum ` F. ` can be r... |
notbinot2 36241 | Commutation rule between n... |
biimpor 36242 | A rewriting rule for bicon... |
orfa1 36243 | Add a contradicting disjun... |
orfa2 36244 | Remove a contradicting dis... |
bifald 36245 | Infer the equivalence to a... |
orsild 36246 | A lemma for not-or-not eli... |
orsird 36247 | A lemma for not-or-not eli... |
cnf1dd 36248 | A lemma for Conjunctive No... |
cnf2dd 36249 | A lemma for Conjunctive No... |
cnfn1dd 36250 | A lemma for Conjunctive No... |
cnfn2dd 36251 | A lemma for Conjunctive No... |
or32dd 36252 | A rearrangement of disjunc... |
notornotel1 36253 | A lemma for not-or-not eli... |
notornotel2 36254 | A lemma for not-or-not eli... |
contrd 36255 | A proof by contradiction, ... |
an12i 36256 | An inference from commutin... |
exmid2 36257 | An excluded middle law. (... |
selconj 36258 | An inference for selecting... |
truconj 36259 | Add true as a conjunct. (... |
orel 36260 | An inference for disjuncti... |
negel 36261 | An inference for negation ... |
botel 36262 | An inference for bottom el... |
tradd 36263 | Add top ad a conjunct. (C... |
gm-sbtru 36264 | Substitution does not chan... |
sbfal 36265 | Substitution does not chan... |
sbcani 36266 | Distribution of class subs... |
sbcori 36267 | Distribution of class subs... |
sbcimi 36268 | Distribution of class subs... |
sbcni 36269 | Move class substitution in... |
sbali 36270 | Discard class substitution... |
sbexi 36271 | Discard class substitution... |
sbcalf 36272 | Move universal quantifier ... |
sbcexf 36273 | Move existential quantifie... |
sbcalfi 36274 | Move universal quantifier ... |
sbcexfi 36275 | Move existential quantifie... |
spsbcdi 36276 | A lemma for eliminating a ... |
alrimii 36277 | A lemma for introducing a ... |
spesbcdi 36278 | A lemma for introducing an... |
exlimddvf 36279 | A lemma for eliminating an... |
exlimddvfi 36280 | A lemma for eliminating an... |
sbceq1ddi 36281 | A lemma for eliminating in... |
sbccom2lem 36282 | Lemma for ~ sbccom2 . (Co... |
sbccom2 36283 | Commutative law for double... |
sbccom2f 36284 | Commutative law for double... |
sbccom2fi 36285 | Commutative law for double... |
csbcom2fi 36286 | Commutative law for double... |
fald 36287 | Refutation of falsity, in ... |
tsim1 36288 | A Tseitin axiom for logica... |
tsim2 36289 | A Tseitin axiom for logica... |
tsim3 36290 | A Tseitin axiom for logica... |
tsbi1 36291 | A Tseitin axiom for logica... |
tsbi2 36292 | A Tseitin axiom for logica... |
tsbi3 36293 | A Tseitin axiom for logica... |
tsbi4 36294 | A Tseitin axiom for logica... |
tsxo1 36295 | A Tseitin axiom for logica... |
tsxo2 36296 | A Tseitin axiom for logica... |
tsxo3 36297 | A Tseitin axiom for logica... |
tsxo4 36298 | A Tseitin axiom for logica... |
tsan1 36299 | A Tseitin axiom for logica... |
tsan2 36300 | A Tseitin axiom for logica... |
tsan3 36301 | A Tseitin axiom for logica... |
tsna1 36302 | A Tseitin axiom for logica... |
tsna2 36303 | A Tseitin axiom for logica... |
tsna3 36304 | A Tseitin axiom for logica... |
tsor1 36305 | A Tseitin axiom for logica... |
tsor2 36306 | A Tseitin axiom for logica... |
tsor3 36307 | A Tseitin axiom for logica... |
ts3an1 36308 | A Tseitin axiom for triple... |
ts3an2 36309 | A Tseitin axiom for triple... |
ts3an3 36310 | A Tseitin axiom for triple... |
ts3or1 36311 | A Tseitin axiom for triple... |
ts3or2 36312 | A Tseitin axiom for triple... |
ts3or3 36313 | A Tseitin axiom for triple... |
iuneq2f 36314 | Equality deduction for ind... |
rabeq12f 36315 | Equality deduction for res... |
csbeq12 36316 | Equality deduction for sub... |
sbeqi 36317 | Equality deduction for sub... |
ralbi12f 36318 | Equality deduction for res... |
oprabbi 36319 | Equality deduction for cla... |
mpobi123f 36320 | Equality deduction for map... |
iuneq12f 36321 | Equality deduction for ind... |
iineq12f 36322 | Equality deduction for ind... |
opabbi 36323 | Equality deduction for cla... |
mptbi12f 36324 | Equality deduction for map... |
orcomdd 36325 | Commutativity of logic dis... |
scottexf 36326 | A version of ~ scottex wit... |
scott0f 36327 | A version of ~ scott0 with... |
scottn0f 36328 | A version of ~ scott0f wit... |
ac6s3f 36329 | Generalization of the Axio... |
ac6s6 36330 | Generalization of the Axio... |
ac6s6f 36331 | Generalization of the Axio... |
el2v1 36370 | New way ( ~ elv , and the ... |
el3v 36371 | New way ( ~ elv , and the ... |
el3v1 36372 | New way ( ~ elv , and the ... |
el3v2 36373 | New way ( ~ elv , and the ... |
el3v3 36374 | New way ( ~ elv , and the ... |
el3v12 36375 | New way ( ~ elv , and the ... |
el3v13 36376 | New way ( ~ elv , and the ... |
el3v23 36377 | New way ( ~ elv , and the ... |
an2anr 36378 | Double commutation in conj... |
anan 36379 | Multiple commutations in c... |
triantru3 36380 | A wff is equivalent to its... |
eqeltr 36381 | Substitution of equal clas... |
eqelb 36382 | Substitution of equal clas... |
eqeqan2d 36383 | Implication of introducing... |
inres2 36384 | Two ways of expressing the... |
coideq 36385 | Equality theorem for compo... |
nexmo1 36386 | If there is no case where ... |
3albii 36387 | Inference adding three uni... |
3ralbii 36388 | Inference adding three res... |
ssrabi 36389 | Inference of restricted ab... |
rabbieq 36390 | Equivalent wff's correspon... |
rabimbieq 36391 | Restricted equivalent wff'... |
abeqin 36392 | Intersection with class ab... |
abeqinbi 36393 | Intersection with class ab... |
rabeqel 36394 | Class element of a restric... |
eqrelf 36395 | The equality connective be... |
releleccnv 36396 | Elementhood in a converse ... |
releccnveq 36397 | Equality of converse ` R `... |
opelvvdif 36398 | Negated elementhood of ord... |
vvdifopab 36399 | Ordered-pair class abstrac... |
brvdif 36400 | Binary relation with unive... |
brvdif2 36401 | Binary relation with unive... |
brvvdif 36402 | Binary relation with the c... |
brvbrvvdif 36403 | Binary relation with the c... |
brcnvep 36404 | The converse of the binary... |
elecALTV 36405 | Elementhood in the ` R ` -... |
brcnvepres 36406 | Restricted converse epsilo... |
brres2 36407 | Binary relation on a restr... |
eldmres 36408 | Elementhood in the domain ... |
eldm4 36409 | Elementhood in a domain. ... |
eldmres2 36410 | Elementhood in the domain ... |
eceq1i 36411 | Equality theorem for ` C `... |
elecres 36412 | Elementhood in the restric... |
ecres 36413 | Restricted coset of ` B ` ... |
ecres2 36414 | The restricted coset of ` ... |
eccnvepres 36415 | Restricted converse epsilo... |
eleccnvep 36416 | Elementhood in the convers... |
eccnvep 36417 | The converse epsilon coset... |
extep 36418 | Property of epsilon relati... |
eccnvepres2 36419 | The restricted converse ep... |
eccnvepres3 36420 | Condition for a restricted... |
eldmqsres 36421 | Elementhood in a restricte... |
eldmqsres2 36422 | Elementhood in a restricte... |
qsss1 36423 | Subclass theorem for quoti... |
qseq1i 36424 | Equality theorem for quoti... |
qseq1d 36425 | Equality theorem for quoti... |
brinxprnres 36426 | Binary relation on a restr... |
inxprnres 36427 | Restriction of a class as ... |
dfres4 36428 | Alternate definition of th... |
exan3 36429 | Equivalent expressions wit... |
exanres 36430 | Equivalent expressions wit... |
exanres3 36431 | Equivalent expressions wit... |
exanres2 36432 | Equivalent expressions wit... |
cnvepres 36433 | Restricted converse epsilo... |
ssrel3 36434 | Subclass relation in anoth... |
eqrel2 36435 | Equality of relations. (C... |
rncnv 36436 | Range of converse is the d... |
dfdm6 36437 | Alternate definition of do... |
dfrn6 36438 | Alternate definition of ra... |
rncnvepres 36439 | The range of the restricte... |
dmecd 36440 | Equality of the coset of `... |
dmec2d 36441 | Equality of the coset of `... |
brid 36442 | Property of the identity b... |
ideq2 36443 | For sets, the identity bin... |
idresssidinxp 36444 | Condition for the identity... |
idreseqidinxp 36445 | Condition for the identity... |
extid 36446 | Property of identity relat... |
inxpss 36447 | Two ways to say that an in... |
idinxpss 36448 | Two ways to say that an in... |
inxpss3 36449 | Two ways to say that an in... |
inxpss2 36450 | Two ways to say that inter... |
inxpssidinxp 36451 | Two ways to say that inter... |
idinxpssinxp 36452 | Two ways to say that inter... |
idinxpssinxp2 36453 | Identity intersection with... |
idinxpssinxp3 36454 | Identity intersection with... |
idinxpssinxp4 36455 | Identity intersection with... |
relcnveq3 36456 | Two ways of saying a relat... |
relcnveq 36457 | Two ways of saying a relat... |
relcnveq2 36458 | Two ways of saying a relat... |
relcnveq4 36459 | Two ways of saying a relat... |
qsresid 36460 | Simplification of a specia... |
n0elqs 36461 | Two ways of expressing tha... |
n0elqs2 36462 | Two ways of expressing tha... |
ecex2 36463 | Condition for a coset to b... |
uniqsALTV 36464 | The union of a quotient se... |
imaexALTV 36465 | Existence of an image of a... |
ecexALTV 36466 | Existence of a coset, like... |
rnresequniqs 36467 | The range of a restriction... |
n0el2 36468 | Two ways of expressing tha... |
cnvepresex 36469 | Sethood condition for the ... |
eccnvepex 36470 | The converse epsilon coset... |
cnvepimaex 36471 | The image of converse epsi... |
cnvepima 36472 | The image of converse epsi... |
inex3 36473 | Sufficient condition for t... |
inxpex 36474 | Sufficient condition for a... |
eqres 36475 | Converting a class constan... |
brrabga 36476 | The law of concretion for ... |
brcnvrabga 36477 | The law of concretion for ... |
opideq 36478 | Equality conditions for or... |
iss2 36479 | A subclass of the identity... |
eldmcnv 36480 | Elementhood in a domain of... |
dfrel5 36481 | Alternate definition of th... |
dfrel6 36482 | Alternate definition of th... |
cnvresrn 36483 | Converse restricted to ran... |
ecin0 36484 | Two ways of saying that th... |
ecinn0 36485 | Two ways of saying that th... |
ineleq 36486 | Equivalence of restricted ... |
inecmo 36487 | Equivalence of a double re... |
inecmo2 36488 | Equivalence of a double re... |
ineccnvmo 36489 | Equivalence of a double re... |
alrmomorn 36490 | Equivalence of an "at most... |
alrmomodm 36491 | Equivalence of an "at most... |
ineccnvmo2 36492 | Equivalence of a double un... |
inecmo3 36493 | Equivalence of a double un... |
moantr 36494 | Sufficient condition for t... |
brabidgaw 36495 | The law of concretion for ... |
brabidga 36496 | The law of concretion for ... |
inxp2 36497 | Intersection with a Cartes... |
opabf 36498 | A class abstraction of a c... |
ec0 36499 | The empty-coset of a class... |
0qs 36500 | Quotient set with the empt... |
xrnss3v 36502 | A range Cartesian product ... |
xrnrel 36503 | A range Cartesian product ... |
brxrn 36504 | Characterize a ternary rel... |
brxrn2 36505 | A characterization of the ... |
dfxrn2 36506 | Alternate definition of th... |
xrneq1 36507 | Equality theorem for the r... |
xrneq1i 36508 | Equality theorem for the r... |
xrneq1d 36509 | Equality theorem for the r... |
xrneq2 36510 | Equality theorem for the r... |
xrneq2i 36511 | Equality theorem for the r... |
xrneq2d 36512 | Equality theorem for the r... |
xrneq12 36513 | Equality theorem for the r... |
xrneq12i 36514 | Equality theorem for the r... |
xrneq12d 36515 | Equality theorem for the r... |
elecxrn 36516 | Elementhood in the ` ( R |... |
ecxrn 36517 | The ` ( R |X. S ) ` -coset... |
xrninxp 36518 | Intersection of a range Ca... |
xrninxp2 36519 | Intersection of a range Ca... |
xrninxpex 36520 | Sufficient condition for t... |
inxpxrn 36521 | Two ways to express the in... |
br1cnvxrn2 36522 | The converse of a binary r... |
elec1cnvxrn2 36523 | Elementhood in the convers... |
rnxrn 36524 | Range of the range Cartesi... |
rnxrnres 36525 | Range of a range Cartesian... |
rnxrncnvepres 36526 | Range of a range Cartesian... |
rnxrnidres 36527 | Range of a range Cartesian... |
xrnres 36528 | Two ways to express restri... |
xrnres2 36529 | Two ways to express restri... |
xrnres3 36530 | Two ways to express restri... |
xrnres4 36531 | Two ways to express restri... |
xrnresex 36532 | Sufficient condition for a... |
xrnidresex 36533 | Sufficient condition for a... |
xrncnvepresex 36534 | Sufficient condition for a... |
brin2 36535 | Binary relation on an inte... |
brin3 36536 | Binary relation on an inte... |
dfcoss2 36539 | Alternate definition of th... |
dfcoss3 36540 | Alternate definition of th... |
dfcoss4 36541 | Alternate definition of th... |
cossex 36542 | If ` A ` is a set then the... |
cosscnvex 36543 | If ` A ` is a set then the... |
1cosscnvepresex 36544 | Sufficient condition for a... |
1cossxrncnvepresex 36545 | Sufficient condition for a... |
relcoss 36546 | Cosets by ` R ` is a relat... |
relcoels 36547 | Coelements on ` A ` is a r... |
cossss 36548 | Subclass theorem for the c... |
cosseq 36549 | Equality theorem for the c... |
cosseqi 36550 | Equality theorem for the c... |
cosseqd 36551 | Equality theorem for the c... |
1cossres 36552 | The class of cosets by a r... |
dfcoels 36553 | Alternate definition of th... |
brcoss 36554 | ` A ` and ` B ` are cosets... |
brcoss2 36555 | Alternate form of the ` A ... |
brcoss3 36556 | Alternate form of the ` A ... |
brcosscnvcoss 36557 | For sets, the ` A ` and ` ... |
brcoels 36558 | ` B ` and ` C ` are coelem... |
cocossss 36559 | Two ways of saying that co... |
cnvcosseq 36560 | The converse of cosets by ... |
br2coss 36561 | Cosets by ` ,~ R ` binary ... |
br1cossres 36562 | ` B ` and ` C ` are cosets... |
br1cossres2 36563 | ` B ` and ` C ` are cosets... |
relbrcoss 36564 | ` A ` and ` B ` are cosets... |
br1cossinres 36565 | ` B ` and ` C ` are cosets... |
br1cossxrnres 36566 | ` <. B , C >. ` and ` <. D... |
br1cossinidres 36567 | ` B ` and ` C ` are cosets... |
br1cossincnvepres 36568 | ` B ` and ` C ` are cosets... |
br1cossxrnidres 36569 | ` <. B , C >. ` and ` <. D... |
br1cossxrncnvepres 36570 | ` <. B , C >. ` and ` <. D... |
dmcoss3 36571 | The domain of cosets is th... |
dmcoss2 36572 | The domain of cosets is th... |
rncossdmcoss 36573 | The range of cosets is the... |
dm1cosscnvepres 36574 | The domain of cosets of th... |
dmcoels 36575 | The domain of coelements i... |
eldmcoss 36576 | Elementhood in the domain ... |
eldmcoss2 36577 | Elementhood in the domain ... |
eldm1cossres 36578 | Elementhood in the domain ... |
eldm1cossres2 36579 | Elementhood in the domain ... |
refrelcosslem 36580 | Lemma for the left side of... |
refrelcoss3 36581 | The class of cosets by ` R... |
refrelcoss2 36582 | The class of cosets by ` R... |
symrelcoss3 36583 | The class of cosets by ` R... |
symrelcoss2 36584 | The class of cosets by ` R... |
cossssid 36585 | Equivalent expressions for... |
cossssid2 36586 | Equivalent expressions for... |
cossssid3 36587 | Equivalent expressions for... |
cossssid4 36588 | Equivalent expressions for... |
cossssid5 36589 | Equivalent expressions for... |
brcosscnv 36590 | ` A ` and ` B ` are cosets... |
brcosscnv2 36591 | ` A ` and ` B ` are cosets... |
br1cosscnvxrn 36592 | ` A ` and ` B ` are cosets... |
1cosscnvxrn 36593 | Cosets by the converse ran... |
cosscnvssid3 36594 | Equivalent expressions for... |
cosscnvssid4 36595 | Equivalent expressions for... |
cosscnvssid5 36596 | Equivalent expressions for... |
coss0 36597 | Cosets by the empty set ar... |
cossid 36598 | Cosets by the identity rel... |
cosscnvid 36599 | Cosets by the converse ide... |
trcoss 36600 | Sufficient condition for t... |
eleccossin 36601 | Two ways of saying that th... |
trcoss2 36602 | Equivalent expressions for... |
elrels2 36604 | The element of the relatio... |
elrelsrel 36605 | The element of the relatio... |
elrelsrelim 36606 | The element of the relatio... |
elrels5 36607 | Equivalent expressions for... |
elrels6 36608 | Equivalent expressions for... |
elrelscnveq3 36609 | Two ways of saying a relat... |
elrelscnveq 36610 | Two ways of saying a relat... |
elrelscnveq2 36611 | Two ways of saying a relat... |
elrelscnveq4 36612 | Two ways of saying a relat... |
cnvelrels 36613 | The converse of a set is a... |
cosselrels 36614 | Cosets of sets are element... |
cosscnvelrels 36615 | Cosets of converse sets ar... |
dfssr2 36617 | Alternate definition of th... |
relssr 36618 | The subset relation is a r... |
brssr 36619 | The subset relation and su... |
brssrid 36620 | Any set is a subset of its... |
issetssr 36621 | Two ways of expressing set... |
brssrres 36622 | Restricted subset binary r... |
br1cnvssrres 36623 | Restricted converse subset... |
brcnvssr 36624 | The converse of a subset r... |
brcnvssrid 36625 | Any set is a converse subs... |
br1cossxrncnvssrres 36626 | ` <. B , C >. ` and ` <. D... |
extssr 36627 | Property of subset relatio... |
dfrefrels2 36631 | Alternate definition of th... |
dfrefrels3 36632 | Alternate definition of th... |
dfrefrel2 36633 | Alternate definition of th... |
dfrefrel3 36634 | Alternate definition of th... |
elrefrels2 36635 | Element of the class of re... |
elrefrels3 36636 | Element of the class of re... |
elrefrelsrel 36637 | For sets, being an element... |
refreleq 36638 | Equality theorem for refle... |
refrelid 36639 | Identity relation is refle... |
refrelcoss 36640 | The class of cosets by ` R... |
dfcnvrefrels2 36644 | Alternate definition of th... |
dfcnvrefrels3 36645 | Alternate definition of th... |
dfcnvrefrel2 36646 | Alternate definition of th... |
dfcnvrefrel3 36647 | Alternate definition of th... |
elcnvrefrels2 36648 | Element of the class of co... |
elcnvrefrels3 36649 | Element of the class of co... |
elcnvrefrelsrel 36650 | For sets, being an element... |
cnvrefrelcoss2 36651 | Necessary and sufficient c... |
cosselcnvrefrels2 36652 | Necessary and sufficient c... |
cosselcnvrefrels3 36653 | Necessary and sufficient c... |
cosselcnvrefrels4 36654 | Necessary and sufficient c... |
cosselcnvrefrels5 36655 | Necessary and sufficient c... |
dfsymrels2 36659 | Alternate definition of th... |
dfsymrels3 36660 | Alternate definition of th... |
dfsymrels4 36661 | Alternate definition of th... |
dfsymrels5 36662 | Alternate definition of th... |
dfsymrel2 36663 | Alternate definition of th... |
dfsymrel3 36664 | Alternate definition of th... |
dfsymrel4 36665 | Alternate definition of th... |
dfsymrel5 36666 | Alternate definition of th... |
elsymrels2 36667 | Element of the class of sy... |
elsymrels3 36668 | Element of the class of sy... |
elsymrels4 36669 | Element of the class of sy... |
elsymrels5 36670 | Element of the class of sy... |
elsymrelsrel 36671 | For sets, being an element... |
symreleq 36672 | Equality theorem for symme... |
symrelim 36673 | Symmetric relation implies... |
symrelcoss 36674 | The class of cosets by ` R... |
idsymrel 36675 | The identity relation is s... |
epnsymrel 36676 | The membership (epsilon) r... |
symrefref2 36677 | Symmetry is a sufficient c... |
symrefref3 36678 | Symmetry is a sufficient c... |
refsymrels2 36679 | Elements of the class of r... |
refsymrels3 36680 | Elements of the class of r... |
refsymrel2 36681 | A relation which is reflex... |
refsymrel3 36682 | A relation which is reflex... |
elrefsymrels2 36683 | Elements of the class of r... |
elrefsymrels3 36684 | Elements of the class of r... |
elrefsymrelsrel 36685 | For sets, being an element... |
dftrrels2 36689 | Alternate definition of th... |
dftrrels3 36690 | Alternate definition of th... |
dftrrel2 36691 | Alternate definition of th... |
dftrrel3 36692 | Alternate definition of th... |
eltrrels2 36693 | Element of the class of tr... |
eltrrels3 36694 | Element of the class of tr... |
eltrrelsrel 36695 | For sets, being an element... |
trreleq 36696 | Equality theorem for the t... |
dfeqvrels2 36701 | Alternate definition of th... |
dfeqvrels3 36702 | Alternate definition of th... |
dfeqvrel2 36703 | Alternate definition of th... |
dfeqvrel3 36704 | Alternate definition of th... |
eleqvrels2 36705 | Element of the class of eq... |
eleqvrels3 36706 | Element of the class of eq... |
eleqvrelsrel 36707 | For sets, being an element... |
elcoeleqvrels 36708 | Elementhood in the coeleme... |
elcoeleqvrelsrel 36709 | For sets, being an element... |
eqvrelrel 36710 | An equivalence relation is... |
eqvrelrefrel 36711 | An equivalence relation is... |
eqvrelsymrel 36712 | An equivalence relation is... |
eqvreltrrel 36713 | An equivalence relation is... |
eqvrelim 36714 | Equivalence relation impli... |
eqvreleq 36715 | Equality theorem for equiv... |
eqvreleqi 36716 | Equality theorem for equiv... |
eqvreleqd 36717 | Equality theorem for equiv... |
eqvrelsym 36718 | An equivalence relation is... |
eqvrelsymb 36719 | An equivalence relation is... |
eqvreltr 36720 | An equivalence relation is... |
eqvreltrd 36721 | A transitivity relation fo... |
eqvreltr4d 36722 | A transitivity relation fo... |
eqvrelref 36723 | An equivalence relation is... |
eqvrelth 36724 | Basic property of equivale... |
eqvrelcl 36725 | Elementhood in the field o... |
eqvrelthi 36726 | Basic property of equivale... |
eqvreldisj 36727 | Equivalence classes do not... |
qsdisjALTV 36728 | Elements of a quotient set... |
eqvrelqsel 36729 | If an element of a quotien... |
eqvrelcoss 36730 | Two ways to express equiva... |
eqvrelcoss3 36731 | Two ways to express equiva... |
eqvrelcoss2 36732 | Two ways to express equiva... |
eqvrelcoss4 36733 | Two ways to express equiva... |
dfcoeleqvrels 36734 | Alternate definition of th... |
dfcoeleqvrel 36735 | Alternate definition of th... |
brredunds 36739 | Binary relation on the cla... |
brredundsredund 36740 | For sets, binary relation ... |
redundss3 36741 | Implication of redundancy ... |
redundeq1 36742 | Equivalence of redundancy ... |
redundpim3 36743 | Implication of redundancy ... |
redundpbi1 36744 | Equivalence of redundancy ... |
refrelsredund4 36745 | The naive version of the c... |
refrelsredund2 36746 | The naive version of the c... |
refrelsredund3 36747 | The naive version of the c... |
refrelredund4 36748 | The naive version of the d... |
refrelredund2 36749 | The naive version of the d... |
refrelredund3 36750 | The naive version of the d... |
dmqseq 36753 | Equality theorem for domai... |
dmqseqi 36754 | Equality theorem for domai... |
dmqseqd 36755 | Equality theorem for domai... |
dmqseqeq1 36756 | Equality theorem for domai... |
dmqseqeq1i 36757 | Equality theorem for domai... |
dmqseqeq1d 36758 | Equality theorem for domai... |
brdmqss 36759 | The domain quotient binary... |
brdmqssqs 36760 | If ` A ` and ` R ` are set... |
n0eldmqs 36761 | The empty set is not an el... |
n0eldmqseq 36762 | The empty set is not an el... |
n0el3 36763 | Two ways of expressing tha... |
cnvepresdmqss 36764 | The domain quotient binary... |
cnvepresdmqs 36765 | The domain quotient predic... |
unidmqs 36766 | The range of a relation is... |
unidmqseq 36767 | The union of the domain qu... |
dmqseqim 36768 | If the domain quotient of ... |
dmqseqim2 36769 | Lemma for ~ erim2 . (Cont... |
releldmqs 36770 | Elementhood in the domain ... |
eldmqs1cossres 36771 | Elementhood in the domain ... |
releldmqscoss 36772 | Elementhood in the domain ... |
dmqscoelseq 36773 | Two ways to express the eq... |
dmqs1cosscnvepreseq 36774 | Two ways to express the eq... |
brers 36779 | Binary equivalence relatio... |
dferALTV2 36780 | Equivalence relation with ... |
erALTVeq1 36781 | Equality theorem for equiv... |
erALTVeq1i 36782 | Equality theorem for equiv... |
erALTVeq1d 36783 | Equality theorem for equiv... |
dfmember 36784 | Alternate definition of th... |
dfmember2 36785 | Alternate definition of th... |
dfmember3 36786 | Alternate definition of th... |
eqvreldmqs 36787 | Two ways to express member... |
brerser 36788 | Binary equivalence relatio... |
erim2 36789 | Equivalence relation on it... |
erim 36790 | Equivalence relation on it... |
dffunsALTV 36794 | Alternate definition of th... |
dffunsALTV2 36795 | Alternate definition of th... |
dffunsALTV3 36796 | Alternate definition of th... |
dffunsALTV4 36797 | Alternate definition of th... |
dffunsALTV5 36798 | Alternate definition of th... |
dffunALTV2 36799 | Alternate definition of th... |
dffunALTV3 36800 | Alternate definition of th... |
dffunALTV4 36801 | Alternate definition of th... |
dffunALTV5 36802 | Alternate definition of th... |
elfunsALTV 36803 | Elementhood in the class o... |
elfunsALTV2 36804 | Elementhood in the class o... |
elfunsALTV3 36805 | Elementhood in the class o... |
elfunsALTV4 36806 | Elementhood in the class o... |
elfunsALTV5 36807 | Elementhood in the class o... |
elfunsALTVfunALTV 36808 | The element of the class o... |
funALTVfun 36809 | Our definition of the func... |
funALTVss 36810 | Subclass theorem for funct... |
funALTVeq 36811 | Equality theorem for funct... |
funALTVeqi 36812 | Equality inference for the... |
funALTVeqd 36813 | Equality deduction for the... |
dfdisjs 36819 | Alternate definition of th... |
dfdisjs2 36820 | Alternate definition of th... |
dfdisjs3 36821 | Alternate definition of th... |
dfdisjs4 36822 | Alternate definition of th... |
dfdisjs5 36823 | Alternate definition of th... |
dfdisjALTV 36824 | Alternate definition of th... |
dfdisjALTV2 36825 | Alternate definition of th... |
dfdisjALTV3 36826 | Alternate definition of th... |
dfdisjALTV4 36827 | Alternate definition of th... |
dfdisjALTV5 36828 | Alternate definition of th... |
dfeldisj2 36829 | Alternate definition of th... |
dfeldisj3 36830 | Alternate definition of th... |
dfeldisj4 36831 | Alternate definition of th... |
dfeldisj5 36832 | Alternate definition of th... |
eldisjs 36833 | Elementhood in the class o... |
eldisjs2 36834 | Elementhood in the class o... |
eldisjs3 36835 | Elementhood in the class o... |
eldisjs4 36836 | Elementhood in the class o... |
eldisjs5 36837 | Elementhood in the class o... |
eldisjsdisj 36838 | The element of the class o... |
eleldisjs 36839 | Elementhood in the disjoin... |
eleldisjseldisj 36840 | The element of the disjoin... |
disjrel 36841 | Disjoint relation is a rel... |
disjss 36842 | Subclass theorem for disjo... |
disjssi 36843 | Subclass theorem for disjo... |
disjssd 36844 | Subclass theorem for disjo... |
disjeq 36845 | Equality theorem for disjo... |
disjeqi 36846 | Equality theorem for disjo... |
disjeqd 36847 | Equality theorem for disjo... |
disjdmqseqeq1 36848 | Lemma for the equality the... |
eldisjss 36849 | Subclass theorem for disjo... |
eldisjssi 36850 | Subclass theorem for disjo... |
eldisjssd 36851 | Subclass theorem for disjo... |
eldisjeq 36852 | Equality theorem for disjo... |
eldisjeqi 36853 | Equality theorem for disjo... |
eldisjeqd 36854 | Equality theorem for disjo... |
disjxrn 36855 | Two ways of saying that a ... |
disjorimxrn 36856 | Disjointness condition for... |
disjimxrn 36857 | Disjointness condition for... |
disjimres 36858 | Disjointness condition for... |
disjimin 36859 | Disjointness condition for... |
disjiminres 36860 | Disjointness condition for... |
disjimxrnres 36861 | Disjointness condition for... |
disjALTV0 36862 | The null class is disjoint... |
disjALTVid 36863 | The class of identity rela... |
disjALTVidres 36864 | The class of identity rela... |
disjALTVinidres 36865 | The intersection with rest... |
disjALTVxrnidres 36866 | The class of range Cartesi... |
prtlem60 36867 | Lemma for ~ prter3 . (Con... |
bicomdd 36868 | Commute two sides of a bic... |
jca2r 36869 | Inference conjoining the c... |
jca3 36870 | Inference conjoining the c... |
prtlem70 36871 | Lemma for ~ prter3 : a rea... |
ibdr 36872 | Reverse of ~ ibd . (Contr... |
prtlem100 36873 | Lemma for ~ prter3 . (Con... |
prtlem5 36874 | Lemma for ~ prter1 , ~ prt... |
prtlem80 36875 | Lemma for ~ prter2 . (Con... |
brabsb2 36876 | A closed form of ~ brabsb ... |
eqbrrdv2 36877 | Other version of ~ eqbrrdi... |
prtlem9 36878 | Lemma for ~ prter3 . (Con... |
prtlem10 36879 | Lemma for ~ prter3 . (Con... |
prtlem11 36880 | Lemma for ~ prter2 . (Con... |
prtlem12 36881 | Lemma for ~ prtex and ~ pr... |
prtlem13 36882 | Lemma for ~ prter1 , ~ prt... |
prtlem16 36883 | Lemma for ~ prtex , ~ prte... |
prtlem400 36884 | Lemma for ~ prter2 and als... |
erprt 36887 | The quotient set of an equ... |
prtlem14 36888 | Lemma for ~ prter1 , ~ prt... |
prtlem15 36889 | Lemma for ~ prter1 and ~ p... |
prtlem17 36890 | Lemma for ~ prter2 . (Con... |
prtlem18 36891 | Lemma for ~ prter2 . (Con... |
prtlem19 36892 | Lemma for ~ prter2 . (Con... |
prter1 36893 | Every partition generates ... |
prtex 36894 | The equivalence relation g... |
prter2 36895 | The quotient set of the eq... |
prter3 36896 | For every partition there ... |
axc5 36907 | This theorem repeats ~ sp ... |
ax4fromc4 36908 | Rederivation of Axiom ~ ax... |
ax10fromc7 36909 | Rederivation of Axiom ~ ax... |
ax6fromc10 36910 | Rederivation of Axiom ~ ax... |
hba1-o 36911 | The setvar ` x ` is not fr... |
axc4i-o 36912 | Inference version of ~ ax-... |
equid1 36913 | Proof of ~ equid from our ... |
equcomi1 36914 | Proof of ~ equcomi from ~ ... |
aecom-o 36915 | Commutation law for identi... |
aecoms-o 36916 | A commutation rule for ide... |
hbae-o 36917 | All variables are effectiv... |
dral1-o 36918 | Formula-building lemma for... |
ax12fromc15 36919 | Rederivation of Axiom ~ ax... |
ax13fromc9 36920 | Derive ~ ax-13 from ~ ax-c... |
ax5ALT 36921 | Axiom to quantify a variab... |
sps-o 36922 | Generalization of antecede... |
hbequid 36923 | Bound-variable hypothesis ... |
nfequid-o 36924 | Bound-variable hypothesis ... |
axc5c7 36925 | Proof of a single axiom th... |
axc5c7toc5 36926 | Rederivation of ~ ax-c5 fr... |
axc5c7toc7 36927 | Rederivation of ~ ax-c7 fr... |
axc711 36928 | Proof of a single axiom th... |
nfa1-o 36929 | ` x ` is not free in ` A. ... |
axc711toc7 36930 | Rederivation of ~ ax-c7 fr... |
axc711to11 36931 | Rederivation of ~ ax-11 fr... |
axc5c711 36932 | Proof of a single axiom th... |
axc5c711toc5 36933 | Rederivation of ~ ax-c5 fr... |
axc5c711toc7 36934 | Rederivation of ~ ax-c7 fr... |
axc5c711to11 36935 | Rederivation of ~ ax-11 fr... |
equidqe 36936 | ~ equid with existential q... |
axc5sp1 36937 | A special case of ~ ax-c5 ... |
equidq 36938 | ~ equid with universal qua... |
equid1ALT 36939 | Alternate proof of ~ equid... |
axc11nfromc11 36940 | Rederivation of ~ ax-c11n ... |
naecoms-o 36941 | A commutation rule for dis... |
hbnae-o 36942 | All variables are effectiv... |
dvelimf-o 36943 | Proof of ~ dvelimh that us... |
dral2-o 36944 | Formula-building lemma for... |
aev-o 36945 | A "distinctor elimination"... |
ax5eq 36946 | Theorem to add distinct qu... |
dveeq2-o 36947 | Quantifier introduction wh... |
axc16g-o 36948 | A generalization of Axiom ... |
dveeq1-o 36949 | Quantifier introduction wh... |
dveeq1-o16 36950 | Version of ~ dveeq1 using ... |
ax5el 36951 | Theorem to add distinct qu... |
axc11n-16 36952 | This theorem shows that, g... |
dveel2ALT 36953 | Alternate proof of ~ dveel... |
ax12f 36954 | Basis step for constructin... |
ax12eq 36955 | Basis step for constructin... |
ax12el 36956 | Basis step for constructin... |
ax12indn 36957 | Induction step for constru... |
ax12indi 36958 | Induction step for constru... |
ax12indalem 36959 | Lemma for ~ ax12inda2 and ... |
ax12inda2ALT 36960 | Alternate proof of ~ ax12i... |
ax12inda2 36961 | Induction step for constru... |
ax12inda 36962 | Induction step for constru... |
ax12v2-o 36963 | Rederivation of ~ ax-c15 f... |
ax12a2-o 36964 | Derive ~ ax-c15 from a hyp... |
axc11-o 36965 | Show that ~ ax-c11 can be ... |
fsumshftd 36966 | Index shift of a finite su... |
riotaclbgBAD 36968 | Closure of restricted iota... |
riotaclbBAD 36969 | Closure of restricted iota... |
riotasvd 36970 | Deduction version of ~ rio... |
riotasv2d 36971 | Value of description binde... |
riotasv2s 36972 | The value of description b... |
riotasv 36973 | Value of description binde... |
riotasv3d 36974 | A property ` ch ` holding ... |
elimhyps 36975 | A version of ~ elimhyp usi... |
dedths 36976 | A version of weak deductio... |
renegclALT 36977 | Closure law for negative o... |
elimhyps2 36978 | Generalization of ~ elimhy... |
dedths2 36979 | Generalization of ~ dedths... |
nfcxfrdf 36980 | A utility lemma to transfe... |
nfded 36981 | A deduction theorem that c... |
nfded2 36982 | A deduction theorem that c... |
nfunidALT2 36983 | Deduction version of ~ nfu... |
nfunidALT 36984 | Deduction version of ~ nfu... |
nfopdALT 36985 | Deduction version of bound... |
cnaddcom 36986 | Recover the commutative la... |
toycom 36987 | Show the commutative law f... |
lshpset 36992 | The set of all hyperplanes... |
islshp 36993 | The predicate "is a hyperp... |
islshpsm 36994 | Hyperplane properties expr... |
lshplss 36995 | A hyperplane is a subspace... |
lshpne 36996 | A hyperplane is not equal ... |
lshpnel 36997 | A hyperplane's generating ... |
lshpnelb 36998 | The subspace sum of a hype... |
lshpnel2N 36999 | Condition that determines ... |
lshpne0 37000 | The member of the span in ... |
lshpdisj 37001 | A hyperplane and the span ... |
lshpcmp 37002 | If two hyperplanes are com... |
lshpinN 37003 | The intersection of two di... |
lsatset 37004 | The set of all 1-dim subsp... |
islsat 37005 | The predicate "is a 1-dim ... |
lsatlspsn2 37006 | The span of a nonzero sing... |
lsatlspsn 37007 | The span of a nonzero sing... |
islsati 37008 | A 1-dim subspace (atom) (o... |
lsateln0 37009 | A 1-dim subspace (atom) (o... |
lsatlss 37010 | The set of 1-dim subspaces... |
lsatlssel 37011 | An atom is a subspace. (C... |
lsatssv 37012 | An atom is a set of vector... |
lsatn0 37013 | A 1-dim subspace (atom) of... |
lsatspn0 37014 | The span of a vector is an... |
lsator0sp 37015 | The span of a vector is ei... |
lsatssn0 37016 | A subspace (or any class) ... |
lsatcmp 37017 | If two atoms are comparabl... |
lsatcmp2 37018 | If an atom is included in ... |
lsatel 37019 | A nonzero vector in an ato... |
lsatelbN 37020 | A nonzero vector in an ato... |
lsat2el 37021 | Two atoms sharing a nonzer... |
lsmsat 37022 | Convert comparison of atom... |
lsatfixedN 37023 | Show equality with the spa... |
lsmsatcv 37024 | Subspace sum has the cover... |
lssatomic 37025 | The lattice of subspaces i... |
lssats 37026 | The lattice of subspaces i... |
lpssat 37027 | Two subspaces in a proper ... |
lrelat 37028 | Subspaces are relatively a... |
lssatle 37029 | The ordering of two subspa... |
lssat 37030 | Two subspaces in a proper ... |
islshpat 37031 | Hyperplane properties expr... |
lcvfbr 37034 | The covers relation for a ... |
lcvbr 37035 | The covers relation for a ... |
lcvbr2 37036 | The covers relation for a ... |
lcvbr3 37037 | The covers relation for a ... |
lcvpss 37038 | The covers relation implie... |
lcvnbtwn 37039 | The covers relation implie... |
lcvntr 37040 | The covers relation is not... |
lcvnbtwn2 37041 | The covers relation implie... |
lcvnbtwn3 37042 | The covers relation implie... |
lsmcv2 37043 | Subspace sum has the cover... |
lcvat 37044 | If a subspace covers anoth... |
lsatcv0 37045 | An atom covers the zero su... |
lsatcveq0 37046 | A subspace covered by an a... |
lsat0cv 37047 | A subspace is an atom iff ... |
lcvexchlem1 37048 | Lemma for ~ lcvexch . (Co... |
lcvexchlem2 37049 | Lemma for ~ lcvexch . (Co... |
lcvexchlem3 37050 | Lemma for ~ lcvexch . (Co... |
lcvexchlem4 37051 | Lemma for ~ lcvexch . (Co... |
lcvexchlem5 37052 | Lemma for ~ lcvexch . (Co... |
lcvexch 37053 | Subspaces satisfy the exch... |
lcvp 37054 | Covering property of Defin... |
lcv1 37055 | Covering property of a sub... |
lcv2 37056 | Covering property of a sub... |
lsatexch 37057 | The atom exchange property... |
lsatnle 37058 | The meet of a subspace and... |
lsatnem0 37059 | The meet of distinct atoms... |
lsatexch1 37060 | The atom exch1ange propert... |
lsatcv0eq 37061 | If the sum of two atoms co... |
lsatcv1 37062 | Two atoms covering the zer... |
lsatcvatlem 37063 | Lemma for ~ lsatcvat . (C... |
lsatcvat 37064 | A nonzero subspace less th... |
lsatcvat2 37065 | A subspace covered by the ... |
lsatcvat3 37066 | A condition implying that ... |
islshpcv 37067 | Hyperplane properties expr... |
l1cvpat 37068 | A subspace covered by the ... |
l1cvat 37069 | Create an atom under an el... |
lshpat 37070 | Create an atom under a hyp... |
lflset 37073 | The set of linear function... |
islfl 37074 | The predicate "is a linear... |
lfli 37075 | Property of a linear funct... |
islfld 37076 | Properties that determine ... |
lflf 37077 | A linear functional is a f... |
lflcl 37078 | A linear functional value ... |
lfl0 37079 | A linear functional is zer... |
lfladd 37080 | Property of a linear funct... |
lflsub 37081 | Property of a linear funct... |
lflmul 37082 | Property of a linear funct... |
lfl0f 37083 | The zero function is a fun... |
lfl1 37084 | A nonzero functional has a... |
lfladdcl 37085 | Closure of addition of two... |
lfladdcom 37086 | Commutativity of functiona... |
lfladdass 37087 | Associativity of functiona... |
lfladd0l 37088 | Functional addition with t... |
lflnegcl 37089 | Closure of the negative of... |
lflnegl 37090 | A functional plus its nega... |
lflvscl 37091 | Closure of a scalar produc... |
lflvsdi1 37092 | Distributive law for (righ... |
lflvsdi2 37093 | Reverse distributive law f... |
lflvsdi2a 37094 | Reverse distributive law f... |
lflvsass 37095 | Associative law for (right... |
lfl0sc 37096 | The (right vector space) s... |
lflsc0N 37097 | The scalar product with th... |
lfl1sc 37098 | The (right vector space) s... |
lkrfval 37101 | The kernel of a functional... |
lkrval 37102 | Value of the kernel of a f... |
ellkr 37103 | Membership in the kernel o... |
lkrval2 37104 | Value of the kernel of a f... |
ellkr2 37105 | Membership in the kernel o... |
lkrcl 37106 | A member of the kernel of ... |
lkrf0 37107 | The value of a functional ... |
lkr0f 37108 | The kernel of the zero fun... |
lkrlss 37109 | The kernel of a linear fun... |
lkrssv 37110 | The kernel of a linear fun... |
lkrsc 37111 | The kernel of a nonzero sc... |
lkrscss 37112 | The kernel of a scalar pro... |
eqlkr 37113 | Two functionals with the s... |
eqlkr2 37114 | Two functionals with the s... |
eqlkr3 37115 | Two functionals with the s... |
lkrlsp 37116 | The subspace sum of a kern... |
lkrlsp2 37117 | The subspace sum of a kern... |
lkrlsp3 37118 | The subspace sum of a kern... |
lkrshp 37119 | The kernel of a nonzero fu... |
lkrshp3 37120 | The kernels of nonzero fun... |
lkrshpor 37121 | The kernel of a functional... |
lkrshp4 37122 | A kernel is a hyperplane i... |
lshpsmreu 37123 | Lemma for ~ lshpkrex . Sh... |
lshpkrlem1 37124 | Lemma for ~ lshpkrex . Th... |
lshpkrlem2 37125 | Lemma for ~ lshpkrex . Th... |
lshpkrlem3 37126 | Lemma for ~ lshpkrex . De... |
lshpkrlem4 37127 | Lemma for ~ lshpkrex . Pa... |
lshpkrlem5 37128 | Lemma for ~ lshpkrex . Pa... |
lshpkrlem6 37129 | Lemma for ~ lshpkrex . Sh... |
lshpkrcl 37130 | The set ` G ` defined by h... |
lshpkr 37131 | The kernel of functional `... |
lshpkrex 37132 | There exists a functional ... |
lshpset2N 37133 | The set of all hyperplanes... |
islshpkrN 37134 | The predicate "is a hyperp... |
lfl1dim 37135 | Equivalent expressions for... |
lfl1dim2N 37136 | Equivalent expressions for... |
ldualset 37139 | Define the (left) dual of ... |
ldualvbase 37140 | The vectors of a dual spac... |
ldualelvbase 37141 | Utility theorem for conver... |
ldualfvadd 37142 | Vector addition in the dua... |
ldualvadd 37143 | Vector addition in the dua... |
ldualvaddcl 37144 | The value of vector additi... |
ldualvaddval 37145 | The value of the value of ... |
ldualsca 37146 | The ring of scalars of the... |
ldualsbase 37147 | Base set of scalar ring fo... |
ldualsaddN 37148 | Scalar addition for the du... |
ldualsmul 37149 | Scalar multiplication for ... |
ldualfvs 37150 | Scalar product operation f... |
ldualvs 37151 | Scalar product operation v... |
ldualvsval 37152 | Value of scalar product op... |
ldualvscl 37153 | The scalar product operati... |
ldualvaddcom 37154 | Commutative law for vector... |
ldualvsass 37155 | Associative law for scalar... |
ldualvsass2 37156 | Associative law for scalar... |
ldualvsdi1 37157 | Distributive law for scala... |
ldualvsdi2 37158 | Reverse distributive law f... |
ldualgrplem 37159 | Lemma for ~ ldualgrp . (C... |
ldualgrp 37160 | The dual of a vector space... |
ldual0 37161 | The zero scalar of the dua... |
ldual1 37162 | The unit scalar of the dua... |
ldualneg 37163 | The negative of a scalar o... |
ldual0v 37164 | The zero vector of the dua... |
ldual0vcl 37165 | The dual zero vector is a ... |
lduallmodlem 37166 | Lemma for ~ lduallmod . (... |
lduallmod 37167 | The dual of a left module ... |
lduallvec 37168 | The dual of a left vector ... |
ldualvsub 37169 | The value of vector subtra... |
ldualvsubcl 37170 | Closure of vector subtract... |
ldualvsubval 37171 | The value of the value of ... |
ldualssvscl 37172 | Closure of scalar product ... |
ldualssvsubcl 37173 | Closure of vector subtract... |
ldual0vs 37174 | Scalar zero times a functi... |
lkr0f2 37175 | The kernel of the zero fun... |
lduallkr3 37176 | The kernels of nonzero fun... |
lkrpssN 37177 | Proper subset relation bet... |
lkrin 37178 | Intersection of the kernel... |
eqlkr4 37179 | Two functionals with the s... |
ldual1dim 37180 | Equivalent expressions for... |
ldualkrsc 37181 | The kernel of a nonzero sc... |
lkrss 37182 | The kernel of a scalar pro... |
lkrss2N 37183 | Two functionals with kerne... |
lkreqN 37184 | Proportional functionals h... |
lkrlspeqN 37185 | Condition for colinear fun... |
isopos 37194 | The predicate "is an ortho... |
opposet 37195 | Every orthoposet is a pose... |
oposlem 37196 | Lemma for orthoposet prope... |
op01dm 37197 | Conditions necessary for z... |
op0cl 37198 | An orthoposet has a zero e... |
op1cl 37199 | An orthoposet has a unit e... |
op0le 37200 | Orthoposet zero is less th... |
ople0 37201 | An element less than or eq... |
opnlen0 37202 | An element not less than a... |
lub0N 37203 | The least upper bound of t... |
opltn0 37204 | A lattice element greater ... |
ople1 37205 | Any element is less than t... |
op1le 37206 | If the orthoposet unit is ... |
glb0N 37207 | The greatest lower bound o... |
opoccl 37208 | Closure of orthocomplement... |
opococ 37209 | Double negative law for or... |
opcon3b 37210 | Contraposition law for ort... |
opcon2b 37211 | Orthocomplement contraposi... |
opcon1b 37212 | Orthocomplement contraposi... |
oplecon3 37213 | Contraposition law for ort... |
oplecon3b 37214 | Contraposition law for ort... |
oplecon1b 37215 | Contraposition law for str... |
opoc1 37216 | Orthocomplement of orthopo... |
opoc0 37217 | Orthocomplement of orthopo... |
opltcon3b 37218 | Contraposition law for str... |
opltcon1b 37219 | Contraposition law for str... |
opltcon2b 37220 | Contraposition law for str... |
opexmid 37221 | Law of excluded middle for... |
opnoncon 37222 | Law of contradiction for o... |
riotaocN 37223 | The orthocomplement of the... |
cmtfvalN 37224 | Value of commutes relation... |
cmtvalN 37225 | Equivalence for commutes r... |
isolat 37226 | The predicate "is an ortho... |
ollat 37227 | An ortholattice is a latti... |
olop 37228 | An ortholattice is an orth... |
olposN 37229 | An ortholattice is a poset... |
isolatiN 37230 | Properties that determine ... |
oldmm1 37231 | De Morgan's law for meet i... |
oldmm2 37232 | De Morgan's law for meet i... |
oldmm3N 37233 | De Morgan's law for meet i... |
oldmm4 37234 | De Morgan's law for meet i... |
oldmj1 37235 | De Morgan's law for join i... |
oldmj2 37236 | De Morgan's law for join i... |
oldmj3 37237 | De Morgan's law for join i... |
oldmj4 37238 | De Morgan's law for join i... |
olj01 37239 | An ortholattice element jo... |
olj02 37240 | An ortholattice element jo... |
olm11 37241 | The meet of an ortholattic... |
olm12 37242 | The meet of an ortholattic... |
latmassOLD 37243 | Ortholattice meet is assoc... |
latm12 37244 | A rearrangement of lattice... |
latm32 37245 | A rearrangement of lattice... |
latmrot 37246 | Rotate lattice meet of 3 c... |
latm4 37247 | Rearrangement of lattice m... |
latmmdiN 37248 | Lattice meet distributes o... |
latmmdir 37249 | Lattice meet distributes o... |
olm01 37250 | Meet with lattice zero is ... |
olm02 37251 | Meet with lattice zero is ... |
isoml 37252 | The predicate "is an ortho... |
isomliN 37253 | Properties that determine ... |
omlol 37254 | An orthomodular lattice is... |
omlop 37255 | An orthomodular lattice is... |
omllat 37256 | An orthomodular lattice is... |
omllaw 37257 | The orthomodular law. (Co... |
omllaw2N 37258 | Variation of orthomodular ... |
omllaw3 37259 | Orthomodular law equivalen... |
omllaw4 37260 | Orthomodular law equivalen... |
omllaw5N 37261 | The orthomodular law. Rem... |
cmtcomlemN 37262 | Lemma for ~ cmtcomN . ( ~... |
cmtcomN 37263 | Commutation is symmetric. ... |
cmt2N 37264 | Commutation with orthocomp... |
cmt3N 37265 | Commutation with orthocomp... |
cmt4N 37266 | Commutation with orthocomp... |
cmtbr2N 37267 | Alternate definition of th... |
cmtbr3N 37268 | Alternate definition for t... |
cmtbr4N 37269 | Alternate definition for t... |
lecmtN 37270 | Ordered elements commute. ... |
cmtidN 37271 | Any element commutes with ... |
omlfh1N 37272 | Foulis-Holland Theorem, pa... |
omlfh3N 37273 | Foulis-Holland Theorem, pa... |
omlmod1i2N 37274 | Analogue of modular law ~ ... |
omlspjN 37275 | Contraction of a Sasaki pr... |
cvrfval 37282 | Value of covers relation "... |
cvrval 37283 | Binary relation expressing... |
cvrlt 37284 | The covers relation implie... |
cvrnbtwn 37285 | There is no element betwee... |
ncvr1 37286 | No element covers the latt... |
cvrletrN 37287 | Property of an element abo... |
cvrval2 37288 | Binary relation expressing... |
cvrnbtwn2 37289 | The covers relation implie... |
cvrnbtwn3 37290 | The covers relation implie... |
cvrcon3b 37291 | Contraposition law for the... |
cvrle 37292 | The covers relation implie... |
cvrnbtwn4 37293 | The covers relation implie... |
cvrnle 37294 | The covers relation implie... |
cvrne 37295 | The covers relation implie... |
cvrnrefN 37296 | The covers relation is not... |
cvrcmp 37297 | If two lattice elements th... |
cvrcmp2 37298 | If two lattice elements co... |
pats 37299 | The set of atoms in a pose... |
isat 37300 | The predicate "is an atom"... |
isat2 37301 | The predicate "is an atom"... |
atcvr0 37302 | An atom covers zero. ( ~ ... |
atbase 37303 | An atom is a member of the... |
atssbase 37304 | The set of atoms is a subs... |
0ltat 37305 | An atom is greater than ze... |
leatb 37306 | A poset element less than ... |
leat 37307 | A poset element less than ... |
leat2 37308 | A nonzero poset element le... |
leat3 37309 | A poset element less than ... |
meetat 37310 | The meet of any element wi... |
meetat2 37311 | The meet of any element wi... |
isatl 37313 | The predicate "is an atomi... |
atllat 37314 | An atomic lattice is a lat... |
atlpos 37315 | An atomic lattice is a pos... |
atl0dm 37316 | Condition necessary for ze... |
atl0cl 37317 | An atomic lattice has a ze... |
atl0le 37318 | Orthoposet zero is less th... |
atlle0 37319 | An element less than or eq... |
atlltn0 37320 | A lattice element greater ... |
isat3 37321 | The predicate "is an atom"... |
atn0 37322 | An atom is not zero. ( ~ ... |
atnle0 37323 | An atom is not less than o... |
atlen0 37324 | A lattice element is nonze... |
atcmp 37325 | If two atoms are comparabl... |
atncmp 37326 | Frequently-used variation ... |
atnlt 37327 | Two atoms cannot satisfy t... |
atcvreq0 37328 | An element covered by an a... |
atncvrN 37329 | Two atoms cannot satisfy t... |
atlex 37330 | Every nonzero element of a... |
atnle 37331 | Two ways of expressing "an... |
atnem0 37332 | The meet of distinct atoms... |
atlatmstc 37333 | An atomic, complete, ortho... |
atlatle 37334 | The ordering of two Hilber... |
atlrelat1 37335 | An atomistic lattice with ... |
iscvlat 37337 | The predicate "is an atomi... |
iscvlat2N 37338 | The predicate "is an atomi... |
cvlatl 37339 | An atomic lattice with the... |
cvllat 37340 | An atomic lattice with the... |
cvlposN 37341 | An atomic lattice with the... |
cvlexch1 37342 | An atomic covering lattice... |
cvlexch2 37343 | An atomic covering lattice... |
cvlexchb1 37344 | An atomic covering lattice... |
cvlexchb2 37345 | An atomic covering lattice... |
cvlexch3 37346 | An atomic covering lattice... |
cvlexch4N 37347 | An atomic covering lattice... |
cvlatexchb1 37348 | A version of ~ cvlexchb1 f... |
cvlatexchb2 37349 | A version of ~ cvlexchb2 f... |
cvlatexch1 37350 | Atom exchange property. (... |
cvlatexch2 37351 | Atom exchange property. (... |
cvlatexch3 37352 | Atom exchange property. (... |
cvlcvr1 37353 | The covering property. Pr... |
cvlcvrp 37354 | A Hilbert lattice satisfie... |
cvlatcvr1 37355 | An atom is covered by its ... |
cvlatcvr2 37356 | An atom is covered by its ... |
cvlsupr2 37357 | Two equivalent ways of exp... |
cvlsupr3 37358 | Two equivalent ways of exp... |
cvlsupr4 37359 | Consequence of superpositi... |
cvlsupr5 37360 | Consequence of superpositi... |
cvlsupr6 37361 | Consequence of superpositi... |
cvlsupr7 37362 | Consequence of superpositi... |
cvlsupr8 37363 | Consequence of superpositi... |
ishlat1 37366 | The predicate "is a Hilber... |
ishlat2 37367 | The predicate "is a Hilber... |
ishlat3N 37368 | The predicate "is a Hilber... |
ishlatiN 37369 | Properties that determine ... |
hlomcmcv 37370 | A Hilbert lattice is ortho... |
hloml 37371 | A Hilbert lattice is ortho... |
hlclat 37372 | A Hilbert lattice is compl... |
hlcvl 37373 | A Hilbert lattice is an at... |
hlatl 37374 | A Hilbert lattice is atomi... |
hlol 37375 | A Hilbert lattice is an or... |
hlop 37376 | A Hilbert lattice is an or... |
hllat 37377 | A Hilbert lattice is a lat... |
hllatd 37378 | Deduction form of ~ hllat ... |
hlomcmat 37379 | A Hilbert lattice is ortho... |
hlpos 37380 | A Hilbert lattice is a pos... |
hlatjcl 37381 | Closure of join operation.... |
hlatjcom 37382 | Commutatitivity of join op... |
hlatjidm 37383 | Idempotence of join operat... |
hlatjass 37384 | Lattice join is associativ... |
hlatj12 37385 | Swap 1st and 2nd members o... |
hlatj32 37386 | Swap 2nd and 3rd members o... |
hlatjrot 37387 | Rotate lattice join of 3 c... |
hlatj4 37388 | Rearrangement of lattice j... |
hlatlej1 37389 | A join's first argument is... |
hlatlej2 37390 | A join's second argument i... |
glbconN 37391 | De Morgan's law for GLB an... |
glbconxN 37392 | De Morgan's law for GLB an... |
atnlej1 37393 | If an atom is not less tha... |
atnlej2 37394 | If an atom is not less tha... |
hlsuprexch 37395 | A Hilbert lattice has the ... |
hlexch1 37396 | A Hilbert lattice has the ... |
hlexch2 37397 | A Hilbert lattice has the ... |
hlexchb1 37398 | A Hilbert lattice has the ... |
hlexchb2 37399 | A Hilbert lattice has the ... |
hlsupr 37400 | A Hilbert lattice has the ... |
hlsupr2 37401 | A Hilbert lattice has the ... |
hlhgt4 37402 | A Hilbert lattice has a he... |
hlhgt2 37403 | A Hilbert lattice has a he... |
hl0lt1N 37404 | Lattice 0 is less than lat... |
hlexch3 37405 | A Hilbert lattice has the ... |
hlexch4N 37406 | A Hilbert lattice has the ... |
hlatexchb1 37407 | A version of ~ hlexchb1 fo... |
hlatexchb2 37408 | A version of ~ hlexchb2 fo... |
hlatexch1 37409 | Atom exchange property. (... |
hlatexch2 37410 | Atom exchange property. (... |
hlatmstcOLDN 37411 | An atomic, complete, ortho... |
hlatle 37412 | The ordering of two Hilber... |
hlateq 37413 | The equality of two Hilber... |
hlrelat1 37414 | An atomistic lattice with ... |
hlrelat5N 37415 | An atomistic lattice with ... |
hlrelat 37416 | A Hilbert lattice is relat... |
hlrelat2 37417 | A consequence of relative ... |
exatleN 37418 | A condition for an atom to... |
hl2at 37419 | A Hilbert lattice has at l... |
atex 37420 | At least one atom exists. ... |
intnatN 37421 | If the intersection with a... |
2llnne2N 37422 | Condition implying that tw... |
2llnneN 37423 | Condition implying that tw... |
cvr1 37424 | A Hilbert lattice has the ... |
cvr2N 37425 | Less-than and covers equiv... |
hlrelat3 37426 | The Hilbert lattice is rel... |
cvrval3 37427 | Binary relation expressing... |
cvrval4N 37428 | Binary relation expressing... |
cvrval5 37429 | Binary relation expressing... |
cvrp 37430 | A Hilbert lattice satisfie... |
atcvr1 37431 | An atom is covered by its ... |
atcvr2 37432 | An atom is covered by its ... |
cvrexchlem 37433 | Lemma for ~ cvrexch . ( ~... |
cvrexch 37434 | A Hilbert lattice satisfie... |
cvratlem 37435 | Lemma for ~ cvrat . ( ~ a... |
cvrat 37436 | A nonzero Hilbert lattice ... |
ltltncvr 37437 | A chained strong ordering ... |
ltcvrntr 37438 | Non-transitive condition f... |
cvrntr 37439 | The covers relation is not... |
atcvr0eq 37440 | The covers relation is not... |
lnnat 37441 | A line (the join of two di... |
atcvrj0 37442 | Two atoms covering the zer... |
cvrat2 37443 | A Hilbert lattice element ... |
atcvrneN 37444 | Inequality derived from at... |
atcvrj1 37445 | Condition for an atom to b... |
atcvrj2b 37446 | Condition for an atom to b... |
atcvrj2 37447 | Condition for an atom to b... |
atleneN 37448 | Inequality derived from at... |
atltcvr 37449 | An equivalence of less-tha... |
atle 37450 | Any nonzero element has an... |
atlt 37451 | Two atoms are unequal iff ... |
atlelt 37452 | Transfer less-than relatio... |
2atlt 37453 | Given an atom less than an... |
atexchcvrN 37454 | Atom exchange property. V... |
atexchltN 37455 | Atom exchange property. V... |
cvrat3 37456 | A condition implying that ... |
cvrat4 37457 | A condition implying exist... |
cvrat42 37458 | Commuted version of ~ cvra... |
2atjm 37459 | The meet of a line (expres... |
atbtwn 37460 | Property of a 3rd atom ` R... |
atbtwnexOLDN 37461 | There exists a 3rd atom ` ... |
atbtwnex 37462 | Given atoms ` P ` in ` X `... |
3noncolr2 37463 | Two ways to express 3 non-... |
3noncolr1N 37464 | Two ways to express 3 non-... |
hlatcon3 37465 | Atom exchange combined wit... |
hlatcon2 37466 | Atom exchange combined wit... |
4noncolr3 37467 | A way to express 4 non-col... |
4noncolr2 37468 | A way to express 4 non-col... |
4noncolr1 37469 | A way to express 4 non-col... |
athgt 37470 | A Hilbert lattice, whose h... |
3dim0 37471 | There exists a 3-dimension... |
3dimlem1 37472 | Lemma for ~ 3dim1 . (Cont... |
3dimlem2 37473 | Lemma for ~ 3dim1 . (Cont... |
3dimlem3a 37474 | Lemma for ~ 3dim3 . (Cont... |
3dimlem3 37475 | Lemma for ~ 3dim1 . (Cont... |
3dimlem3OLDN 37476 | Lemma for ~ 3dim1 . (Cont... |
3dimlem4a 37477 | Lemma for ~ 3dim3 . (Cont... |
3dimlem4 37478 | Lemma for ~ 3dim1 . (Cont... |
3dimlem4OLDN 37479 | Lemma for ~ 3dim1 . (Cont... |
3dim1lem5 37480 | Lemma for ~ 3dim1 . (Cont... |
3dim1 37481 | Construct a 3-dimensional ... |
3dim2 37482 | Construct 2 new layers on ... |
3dim3 37483 | Construct a new layer on t... |
2dim 37484 | Generate a height-3 elemen... |
1dimN 37485 | An atom is covered by a he... |
1cvrco 37486 | The orthocomplement of an ... |
1cvratex 37487 | There exists an atom less ... |
1cvratlt 37488 | An atom less than or equal... |
1cvrjat 37489 | An element covered by the ... |
1cvrat 37490 | Create an atom under an el... |
ps-1 37491 | The join of two atoms ` R ... |
ps-2 37492 | Lattice analogue for the p... |
2atjlej 37493 | Two atoms are different if... |
hlatexch3N 37494 | Rearrange join of atoms in... |
hlatexch4 37495 | Exchange 2 atoms. (Contri... |
ps-2b 37496 | Variation of projective ge... |
3atlem1 37497 | Lemma for ~ 3at . (Contri... |
3atlem2 37498 | Lemma for ~ 3at . (Contri... |
3atlem3 37499 | Lemma for ~ 3at . (Contri... |
3atlem4 37500 | Lemma for ~ 3at . (Contri... |
3atlem5 37501 | Lemma for ~ 3at . (Contri... |
3atlem6 37502 | Lemma for ~ 3at . (Contri... |
3atlem7 37503 | Lemma for ~ 3at . (Contri... |
3at 37504 | Any three non-colinear ato... |
llnset 37519 | The set of lattice lines i... |
islln 37520 | The predicate "is a lattic... |
islln4 37521 | The predicate "is a lattic... |
llni 37522 | Condition implying a latti... |
llnbase 37523 | A lattice line is a lattic... |
islln3 37524 | The predicate "is a lattic... |
islln2 37525 | The predicate "is a lattic... |
llni2 37526 | The join of two different ... |
llnnleat 37527 | An atom cannot majorize a ... |
llnneat 37528 | A lattice line is not an a... |
2atneat 37529 | The join of two distinct a... |
llnn0 37530 | A lattice line is nonzero.... |
islln2a 37531 | The predicate "is a lattic... |
llnle 37532 | Any element greater than 0... |
atcvrlln2 37533 | An atom under a line is co... |
atcvrlln 37534 | An element covering an ato... |
llnexatN 37535 | Given an atom on a line, t... |
llncmp 37536 | If two lattice lines are c... |
llnnlt 37537 | Two lattice lines cannot s... |
2llnmat 37538 | Two intersecting lines int... |
2at0mat0 37539 | Special case of ~ 2atmat0 ... |
2atmat0 37540 | The meet of two unequal li... |
2atm 37541 | An atom majorized by two d... |
ps-2c 37542 | Variation of projective ge... |
lplnset 37543 | The set of lattice planes ... |
islpln 37544 | The predicate "is a lattic... |
islpln4 37545 | The predicate "is a lattic... |
lplni 37546 | Condition implying a latti... |
islpln3 37547 | The predicate "is a lattic... |
lplnbase 37548 | A lattice plane is a latti... |
islpln5 37549 | The predicate "is a lattic... |
islpln2 37550 | The predicate "is a lattic... |
lplni2 37551 | The join of 3 different at... |
lvolex3N 37552 | There is an atom outside o... |
llnmlplnN 37553 | The intersection of a line... |
lplnle 37554 | Any element greater than 0... |
lplnnle2at 37555 | A lattice line (or atom) c... |
lplnnleat 37556 | A lattice plane cannot maj... |
lplnnlelln 37557 | A lattice plane is not les... |
2atnelpln 37558 | The join of two atoms is n... |
lplnneat 37559 | No lattice plane is an ato... |
lplnnelln 37560 | No lattice plane is a latt... |
lplnn0N 37561 | A lattice plane is nonzero... |
islpln2a 37562 | The predicate "is a lattic... |
islpln2ah 37563 | The predicate "is a lattic... |
lplnriaN 37564 | Property of a lattice plan... |
lplnribN 37565 | Property of a lattice plan... |
lplnric 37566 | Property of a lattice plan... |
lplnri1 37567 | Property of a lattice plan... |
lplnri2N 37568 | Property of a lattice plan... |
lplnri3N 37569 | Property of a lattice plan... |
lplnllnneN 37570 | Two lattice lines defined ... |
llncvrlpln2 37571 | A lattice line under a lat... |
llncvrlpln 37572 | An element covering a latt... |
2lplnmN 37573 | If the join of two lattice... |
2llnmj 37574 | The meet of two lattice li... |
2atmat 37575 | The meet of two intersecti... |
lplncmp 37576 | If two lattice planes are ... |
lplnexatN 37577 | Given a lattice line on a ... |
lplnexllnN 37578 | Given an atom on a lattice... |
lplnnlt 37579 | Two lattice planes cannot ... |
2llnjaN 37580 | The join of two different ... |
2llnjN 37581 | The join of two different ... |
2llnm2N 37582 | The meet of two different ... |
2llnm3N 37583 | Two lattice lines in a lat... |
2llnm4 37584 | Two lattice lines that maj... |
2llnmeqat 37585 | An atom equals the interse... |
lvolset 37586 | The set of 3-dim lattice v... |
islvol 37587 | The predicate "is a 3-dim ... |
islvol4 37588 | The predicate "is a 3-dim ... |
lvoli 37589 | Condition implying a 3-dim... |
islvol3 37590 | The predicate "is a 3-dim ... |
lvoli3 37591 | Condition implying a 3-dim... |
lvolbase 37592 | A 3-dim lattice volume is ... |
islvol5 37593 | The predicate "is a 3-dim ... |
islvol2 37594 | The predicate "is a 3-dim ... |
lvoli2 37595 | The join of 4 different at... |
lvolnle3at 37596 | A lattice plane (or lattic... |
lvolnleat 37597 | An atom cannot majorize a ... |
lvolnlelln 37598 | A lattice line cannot majo... |
lvolnlelpln 37599 | A lattice plane cannot maj... |
3atnelvolN 37600 | The join of 3 atoms is not... |
2atnelvolN 37601 | The join of two atoms is n... |
lvolneatN 37602 | No lattice volume is an at... |
lvolnelln 37603 | No lattice volume is a lat... |
lvolnelpln 37604 | No lattice volume is a lat... |
lvoln0N 37605 | A lattice volume is nonzer... |
islvol2aN 37606 | The predicate "is a lattic... |
4atlem0a 37607 | Lemma for ~ 4at . (Contri... |
4atlem0ae 37608 | Lemma for ~ 4at . (Contri... |
4atlem0be 37609 | Lemma for ~ 4at . (Contri... |
4atlem3 37610 | Lemma for ~ 4at . Break i... |
4atlem3a 37611 | Lemma for ~ 4at . Break i... |
4atlem3b 37612 | Lemma for ~ 4at . Break i... |
4atlem4a 37613 | Lemma for ~ 4at . Frequen... |
4atlem4b 37614 | Lemma for ~ 4at . Frequen... |
4atlem4c 37615 | Lemma for ~ 4at . Frequen... |
4atlem4d 37616 | Lemma for ~ 4at . Frequen... |
4atlem9 37617 | Lemma for ~ 4at . Substit... |
4atlem10a 37618 | Lemma for ~ 4at . Substit... |
4atlem10b 37619 | Lemma for ~ 4at . Substit... |
4atlem10 37620 | Lemma for ~ 4at . Combine... |
4atlem11a 37621 | Lemma for ~ 4at . Substit... |
4atlem11b 37622 | Lemma for ~ 4at . Substit... |
4atlem11 37623 | Lemma for ~ 4at . Combine... |
4atlem12a 37624 | Lemma for ~ 4at . Substit... |
4atlem12b 37625 | Lemma for ~ 4at . Substit... |
4atlem12 37626 | Lemma for ~ 4at . Combine... |
4at 37627 | Four atoms determine a lat... |
4at2 37628 | Four atoms determine a lat... |
lplncvrlvol2 37629 | A lattice line under a lat... |
lplncvrlvol 37630 | An element covering a latt... |
lvolcmp 37631 | If two lattice planes are ... |
lvolnltN 37632 | Two lattice volumes cannot... |
2lplnja 37633 | The join of two different ... |
2lplnj 37634 | The join of two different ... |
2lplnm2N 37635 | The meet of two different ... |
2lplnmj 37636 | The meet of two lattice pl... |
dalemkehl 37637 | Lemma for ~ dath . Freque... |
dalemkelat 37638 | Lemma for ~ dath . Freque... |
dalemkeop 37639 | Lemma for ~ dath . Freque... |
dalempea 37640 | Lemma for ~ dath . Freque... |
dalemqea 37641 | Lemma for ~ dath . Freque... |
dalemrea 37642 | Lemma for ~ dath . Freque... |
dalemsea 37643 | Lemma for ~ dath . Freque... |
dalemtea 37644 | Lemma for ~ dath . Freque... |
dalemuea 37645 | Lemma for ~ dath . Freque... |
dalemyeo 37646 | Lemma for ~ dath . Freque... |
dalemzeo 37647 | Lemma for ~ dath . Freque... |
dalemclpjs 37648 | Lemma for ~ dath . Freque... |
dalemclqjt 37649 | Lemma for ~ dath . Freque... |
dalemclrju 37650 | Lemma for ~ dath . Freque... |
dalem-clpjq 37651 | Lemma for ~ dath . Freque... |
dalemceb 37652 | Lemma for ~ dath . Freque... |
dalempeb 37653 | Lemma for ~ dath . Freque... |
dalemqeb 37654 | Lemma for ~ dath . Freque... |
dalemreb 37655 | Lemma for ~ dath . Freque... |
dalemseb 37656 | Lemma for ~ dath . Freque... |
dalemteb 37657 | Lemma for ~ dath . Freque... |
dalemueb 37658 | Lemma for ~ dath . Freque... |
dalempjqeb 37659 | Lemma for ~ dath . Freque... |
dalemsjteb 37660 | Lemma for ~ dath . Freque... |
dalemtjueb 37661 | Lemma for ~ dath . Freque... |
dalemqrprot 37662 | Lemma for ~ dath . Freque... |
dalemyeb 37663 | Lemma for ~ dath . Freque... |
dalemcnes 37664 | Lemma for ~ dath . Freque... |
dalempnes 37665 | Lemma for ~ dath . Freque... |
dalemqnet 37666 | Lemma for ~ dath . Freque... |
dalempjsen 37667 | Lemma for ~ dath . Freque... |
dalemply 37668 | Lemma for ~ dath . Freque... |
dalemsly 37669 | Lemma for ~ dath . Freque... |
dalemswapyz 37670 | Lemma for ~ dath . Swap t... |
dalemrot 37671 | Lemma for ~ dath . Rotate... |
dalemrotyz 37672 | Lemma for ~ dath . Rotate... |
dalem1 37673 | Lemma for ~ dath . Show t... |
dalemcea 37674 | Lemma for ~ dath . Freque... |
dalem2 37675 | Lemma for ~ dath . Show t... |
dalemdea 37676 | Lemma for ~ dath . Freque... |
dalemeea 37677 | Lemma for ~ dath . Freque... |
dalem3 37678 | Lemma for ~ dalemdnee . (... |
dalem4 37679 | Lemma for ~ dalemdnee . (... |
dalemdnee 37680 | Lemma for ~ dath . Axis o... |
dalem5 37681 | Lemma for ~ dath . Atom `... |
dalem6 37682 | Lemma for ~ dath . Analog... |
dalem7 37683 | Lemma for ~ dath . Analog... |
dalem8 37684 | Lemma for ~ dath . Plane ... |
dalem-cly 37685 | Lemma for ~ dalem9 . Cent... |
dalem9 37686 | Lemma for ~ dath . Since ... |
dalem10 37687 | Lemma for ~ dath . Atom `... |
dalem11 37688 | Lemma for ~ dath . Analog... |
dalem12 37689 | Lemma for ~ dath . Analog... |
dalem13 37690 | Lemma for ~ dalem14 . (Co... |
dalem14 37691 | Lemma for ~ dath . Planes... |
dalem15 37692 | Lemma for ~ dath . The ax... |
dalem16 37693 | Lemma for ~ dath . The at... |
dalem17 37694 | Lemma for ~ dath . When p... |
dalem18 37695 | Lemma for ~ dath . Show t... |
dalem19 37696 | Lemma for ~ dath . Show t... |
dalemccea 37697 | Lemma for ~ dath . Freque... |
dalemddea 37698 | Lemma for ~ dath . Freque... |
dalem-ccly 37699 | Lemma for ~ dath . Freque... |
dalem-ddly 37700 | Lemma for ~ dath . Freque... |
dalemccnedd 37701 | Lemma for ~ dath . Freque... |
dalemclccjdd 37702 | Lemma for ~ dath . Freque... |
dalemcceb 37703 | Lemma for ~ dath . Freque... |
dalemswapyzps 37704 | Lemma for ~ dath . Swap t... |
dalemrotps 37705 | Lemma for ~ dath . Rotate... |
dalemcjden 37706 | Lemma for ~ dath . Show t... |
dalem20 37707 | Lemma for ~ dath . Show t... |
dalem21 37708 | Lemma for ~ dath . Show t... |
dalem22 37709 | Lemma for ~ dath . Show t... |
dalem23 37710 | Lemma for ~ dath . Show t... |
dalem24 37711 | Lemma for ~ dath . Show t... |
dalem25 37712 | Lemma for ~ dath . Show t... |
dalem27 37713 | Lemma for ~ dath . Show t... |
dalem28 37714 | Lemma for ~ dath . Lemma ... |
dalem29 37715 | Lemma for ~ dath . Analog... |
dalem30 37716 | Lemma for ~ dath . Analog... |
dalem31N 37717 | Lemma for ~ dath . Analog... |
dalem32 37718 | Lemma for ~ dath . Analog... |
dalem33 37719 | Lemma for ~ dath . Analog... |
dalem34 37720 | Lemma for ~ dath . Analog... |
dalem35 37721 | Lemma for ~ dath . Analog... |
dalem36 37722 | Lemma for ~ dath . Analog... |
dalem37 37723 | Lemma for ~ dath . Analog... |
dalem38 37724 | Lemma for ~ dath . Plane ... |
dalem39 37725 | Lemma for ~ dath . Auxili... |
dalem40 37726 | Lemma for ~ dath . Analog... |
dalem41 37727 | Lemma for ~ dath . (Contr... |
dalem42 37728 | Lemma for ~ dath . Auxili... |
dalem43 37729 | Lemma for ~ dath . Planes... |
dalem44 37730 | Lemma for ~ dath . Dummy ... |
dalem45 37731 | Lemma for ~ dath . Dummy ... |
dalem46 37732 | Lemma for ~ dath . Analog... |
dalem47 37733 | Lemma for ~ dath . Analog... |
dalem48 37734 | Lemma for ~ dath . Analog... |
dalem49 37735 | Lemma for ~ dath . Analog... |
dalem50 37736 | Lemma for ~ dath . Analog... |
dalem51 37737 | Lemma for ~ dath . Constr... |
dalem52 37738 | Lemma for ~ dath . Lines ... |
dalem53 37739 | Lemma for ~ dath . The au... |
dalem54 37740 | Lemma for ~ dath . Line `... |
dalem55 37741 | Lemma for ~ dath . Lines ... |
dalem56 37742 | Lemma for ~ dath . Analog... |
dalem57 37743 | Lemma for ~ dath . Axis o... |
dalem58 37744 | Lemma for ~ dath . Analog... |
dalem59 37745 | Lemma for ~ dath . Analog... |
dalem60 37746 | Lemma for ~ dath . ` B ` i... |
dalem61 37747 | Lemma for ~ dath . Show t... |
dalem62 37748 | Lemma for ~ dath . Elimin... |
dalem63 37749 | Lemma for ~ dath . Combin... |
dath 37750 | Desargues's theorem of pro... |
dath2 37751 | Version of Desargues's the... |
lineset 37752 | The set of lines in a Hilb... |
isline 37753 | The predicate "is a line".... |
islinei 37754 | Condition implying "is a l... |
pointsetN 37755 | The set of points in a Hil... |
ispointN 37756 | The predicate "is a point"... |
atpointN 37757 | The singleton of an atom i... |
psubspset 37758 | The set of projective subs... |
ispsubsp 37759 | The predicate "is a projec... |
ispsubsp2 37760 | The predicate "is a projec... |
psubspi 37761 | Property of a projective s... |
psubspi2N 37762 | Property of a projective s... |
0psubN 37763 | The empty set is a project... |
snatpsubN 37764 | The singleton of an atom i... |
pointpsubN 37765 | A point (singleton of an a... |
linepsubN 37766 | A line is a projective sub... |
atpsubN 37767 | The set of all atoms is a ... |
psubssat 37768 | A projective subspace cons... |
psubatN 37769 | A member of a projective s... |
pmapfval 37770 | The projective map of a Hi... |
pmapval 37771 | Value of the projective ma... |
elpmap 37772 | Member of a projective map... |
pmapssat 37773 | The projective map of a Hi... |
pmapssbaN 37774 | A weakening of ~ pmapssat ... |
pmaple 37775 | The projective map of a Hi... |
pmap11 37776 | The projective map of a Hi... |
pmapat 37777 | The projective map of an a... |
elpmapat 37778 | Member of the projective m... |
pmap0 37779 | Value of the projective ma... |
pmapeq0 37780 | A projective map value is ... |
pmap1N 37781 | Value of the projective ma... |
pmapsub 37782 | The projective map of a Hi... |
pmapglbx 37783 | The projective map of the ... |
pmapglb 37784 | The projective map of the ... |
pmapglb2N 37785 | The projective map of the ... |
pmapglb2xN 37786 | The projective map of the ... |
pmapmeet 37787 | The projective map of a me... |
isline2 37788 | Definition of line in term... |
linepmap 37789 | A line described with a pr... |
isline3 37790 | Definition of line in term... |
isline4N 37791 | Definition of line in term... |
lneq2at 37792 | A line equals the join of ... |
lnatexN 37793 | There is an atom in a line... |
lnjatN 37794 | Given an atom in a line, t... |
lncvrelatN 37795 | A lattice element covered ... |
lncvrat 37796 | A line covers the atoms it... |
lncmp 37797 | If two lines are comparabl... |
2lnat 37798 | Two intersecting lines int... |
2atm2atN 37799 | Two joins with a common at... |
2llnma1b 37800 | Generalization of ~ 2llnma... |
2llnma1 37801 | Two different intersecting... |
2llnma3r 37802 | Two different intersecting... |
2llnma2 37803 | Two different intersecting... |
2llnma2rN 37804 | Two different intersecting... |
cdlema1N 37805 | A condition for required f... |
cdlema2N 37806 | A condition for required f... |
cdlemblem 37807 | Lemma for ~ cdlemb . (Con... |
cdlemb 37808 | Given two atoms not less t... |
paddfval 37811 | Projective subspace sum op... |
paddval 37812 | Projective subspace sum op... |
elpadd 37813 | Member of a projective sub... |
elpaddn0 37814 | Member of projective subsp... |
paddvaln0N 37815 | Projective subspace sum op... |
elpaddri 37816 | Condition implying members... |
elpaddatriN 37817 | Condition implying members... |
elpaddat 37818 | Membership in a projective... |
elpaddatiN 37819 | Consequence of membership ... |
elpadd2at 37820 | Membership in a projective... |
elpadd2at2 37821 | Membership in a projective... |
paddunssN 37822 | Projective subspace sum in... |
elpadd0 37823 | Member of projective subsp... |
paddval0 37824 | Projective subspace sum wi... |
padd01 37825 | Projective subspace sum wi... |
padd02 37826 | Projective subspace sum wi... |
paddcom 37827 | Projective subspace sum co... |
paddssat 37828 | A projective subspace sum ... |
sspadd1 37829 | A projective subspace sum ... |
sspadd2 37830 | A projective subspace sum ... |
paddss1 37831 | Subset law for projective ... |
paddss2 37832 | Subset law for projective ... |
paddss12 37833 | Subset law for projective ... |
paddasslem1 37834 | Lemma for ~ paddass . (Co... |
paddasslem2 37835 | Lemma for ~ paddass . (Co... |
paddasslem3 37836 | Lemma for ~ paddass . Res... |
paddasslem4 37837 | Lemma for ~ paddass . Com... |
paddasslem5 37838 | Lemma for ~ paddass . Sho... |
paddasslem6 37839 | Lemma for ~ paddass . (Co... |
paddasslem7 37840 | Lemma for ~ paddass . Com... |
paddasslem8 37841 | Lemma for ~ paddass . (Co... |
paddasslem9 37842 | Lemma for ~ paddass . Com... |
paddasslem10 37843 | Lemma for ~ paddass . Use... |
paddasslem11 37844 | Lemma for ~ paddass . The... |
paddasslem12 37845 | Lemma for ~ paddass . The... |
paddasslem13 37846 | Lemma for ~ paddass . The... |
paddasslem14 37847 | Lemma for ~ paddass . Rem... |
paddasslem15 37848 | Lemma for ~ paddass . Use... |
paddasslem16 37849 | Lemma for ~ paddass . Use... |
paddasslem17 37850 | Lemma for ~ paddass . The... |
paddasslem18 37851 | Lemma for ~ paddass . Com... |
paddass 37852 | Projective subspace sum is... |
padd12N 37853 | Commutative/associative la... |
padd4N 37854 | Rearrangement of 4 terms i... |
paddidm 37855 | Projective subspace sum is... |
paddclN 37856 | The projective sum of two ... |
paddssw1 37857 | Subset law for projective ... |
paddssw2 37858 | Subset law for projective ... |
paddss 37859 | Subset law for projective ... |
pmodlem1 37860 | Lemma for ~ pmod1i . (Con... |
pmodlem2 37861 | Lemma for ~ pmod1i . (Con... |
pmod1i 37862 | The modular law holds in a... |
pmod2iN 37863 | Dual of the modular law. ... |
pmodN 37864 | The modular law for projec... |
pmodl42N 37865 | Lemma derived from modular... |
pmapjoin 37866 | The projective map of the ... |
pmapjat1 37867 | The projective map of the ... |
pmapjat2 37868 | The projective map of the ... |
pmapjlln1 37869 | The projective map of the ... |
hlmod1i 37870 | A version of the modular l... |
atmod1i1 37871 | Version of modular law ~ p... |
atmod1i1m 37872 | Version of modular law ~ p... |
atmod1i2 37873 | Version of modular law ~ p... |
llnmod1i2 37874 | Version of modular law ~ p... |
atmod2i1 37875 | Version of modular law ~ p... |
atmod2i2 37876 | Version of modular law ~ p... |
llnmod2i2 37877 | Version of modular law ~ p... |
atmod3i1 37878 | Version of modular law tha... |
atmod3i2 37879 | Version of modular law tha... |
atmod4i1 37880 | Version of modular law tha... |
atmod4i2 37881 | Version of modular law tha... |
llnexchb2lem 37882 | Lemma for ~ llnexchb2 . (... |
llnexchb2 37883 | Line exchange property (co... |
llnexch2N 37884 | Line exchange property (co... |
dalawlem1 37885 | Lemma for ~ dalaw . Speci... |
dalawlem2 37886 | Lemma for ~ dalaw . Utili... |
dalawlem3 37887 | Lemma for ~ dalaw . First... |
dalawlem4 37888 | Lemma for ~ dalaw . Secon... |
dalawlem5 37889 | Lemma for ~ dalaw . Speci... |
dalawlem6 37890 | Lemma for ~ dalaw . First... |
dalawlem7 37891 | Lemma for ~ dalaw . Secon... |
dalawlem8 37892 | Lemma for ~ dalaw . Speci... |
dalawlem9 37893 | Lemma for ~ dalaw . Speci... |
dalawlem10 37894 | Lemma for ~ dalaw . Combi... |
dalawlem11 37895 | Lemma for ~ dalaw . First... |
dalawlem12 37896 | Lemma for ~ dalaw . Secon... |
dalawlem13 37897 | Lemma for ~ dalaw . Speci... |
dalawlem14 37898 | Lemma for ~ dalaw . Combi... |
dalawlem15 37899 | Lemma for ~ dalaw . Swap ... |
dalaw 37900 | Desargues's law, derived f... |
pclfvalN 37903 | The projective subspace cl... |
pclvalN 37904 | Value of the projective su... |
pclclN 37905 | Closure of the projective ... |
elpclN 37906 | Membership in the projecti... |
elpcliN 37907 | Implication of membership ... |
pclssN 37908 | Ordering is preserved by s... |
pclssidN 37909 | A set of atoms is included... |
pclidN 37910 | The projective subspace cl... |
pclbtwnN 37911 | A projective subspace sand... |
pclunN 37912 | The projective subspace cl... |
pclun2N 37913 | The projective subspace cl... |
pclfinN 37914 | The projective subspace cl... |
pclcmpatN 37915 | The set of projective subs... |
polfvalN 37918 | The projective subspace po... |
polvalN 37919 | Value of the projective su... |
polval2N 37920 | Alternate expression for v... |
polsubN 37921 | The polarity of a set of a... |
polssatN 37922 | The polarity of a set of a... |
pol0N 37923 | The polarity of the empty ... |
pol1N 37924 | The polarity of the whole ... |
2pol0N 37925 | The closed subspace closur... |
polpmapN 37926 | The polarity of a projecti... |
2polpmapN 37927 | Double polarity of a proje... |
2polvalN 37928 | Value of double polarity. ... |
2polssN 37929 | A set of atoms is a subset... |
3polN 37930 | Triple polarity cancels to... |
polcon3N 37931 | Contraposition law for pol... |
2polcon4bN 37932 | Contraposition law for pol... |
polcon2N 37933 | Contraposition law for pol... |
polcon2bN 37934 | Contraposition law for pol... |
pclss2polN 37935 | The projective subspace cl... |
pcl0N 37936 | The projective subspace cl... |
pcl0bN 37937 | The projective subspace cl... |
pmaplubN 37938 | The LUB of a projective ma... |
sspmaplubN 37939 | A set of atoms is a subset... |
2pmaplubN 37940 | Double projective map of a... |
paddunN 37941 | The closure of the project... |
poldmj1N 37942 | De Morgan's law for polari... |
pmapj2N 37943 | The projective map of the ... |
pmapocjN 37944 | The projective map of the ... |
polatN 37945 | The polarity of the single... |
2polatN 37946 | Double polarity of the sin... |
pnonsingN 37947 | The intersection of a set ... |
psubclsetN 37950 | The set of closed projecti... |
ispsubclN 37951 | The predicate "is a closed... |
psubcliN 37952 | Property of a closed proje... |
psubcli2N 37953 | Property of a closed proje... |
psubclsubN 37954 | A closed projective subspa... |
psubclssatN 37955 | A closed projective subspa... |
pmapidclN 37956 | Projective map of the LUB ... |
0psubclN 37957 | The empty set is a closed ... |
1psubclN 37958 | The set of all atoms is a ... |
atpsubclN 37959 | A point (singleton of an a... |
pmapsubclN 37960 | A projective map value is ... |
ispsubcl2N 37961 | Alternate predicate for "i... |
psubclinN 37962 | The intersection of two cl... |
paddatclN 37963 | The projective sum of a cl... |
pclfinclN 37964 | The projective subspace cl... |
linepsubclN 37965 | A line is a closed project... |
polsubclN 37966 | A polarity is a closed pro... |
poml4N 37967 | Orthomodular law for proje... |
poml5N 37968 | Orthomodular law for proje... |
poml6N 37969 | Orthomodular law for proje... |
osumcllem1N 37970 | Lemma for ~ osumclN . (Co... |
osumcllem2N 37971 | Lemma for ~ osumclN . (Co... |
osumcllem3N 37972 | Lemma for ~ osumclN . (Co... |
osumcllem4N 37973 | Lemma for ~ osumclN . (Co... |
osumcllem5N 37974 | Lemma for ~ osumclN . (Co... |
osumcllem6N 37975 | Lemma for ~ osumclN . Use... |
osumcllem7N 37976 | Lemma for ~ osumclN . (Co... |
osumcllem8N 37977 | Lemma for ~ osumclN . (Co... |
osumcllem9N 37978 | Lemma for ~ osumclN . (Co... |
osumcllem10N 37979 | Lemma for ~ osumclN . Con... |
osumcllem11N 37980 | Lemma for ~ osumclN . (Co... |
osumclN 37981 | Closure of orthogonal sum.... |
pmapojoinN 37982 | For orthogonal elements, p... |
pexmidN 37983 | Excluded middle law for cl... |
pexmidlem1N 37984 | Lemma for ~ pexmidN . Hol... |
pexmidlem2N 37985 | Lemma for ~ pexmidN . (Co... |
pexmidlem3N 37986 | Lemma for ~ pexmidN . Use... |
pexmidlem4N 37987 | Lemma for ~ pexmidN . (Co... |
pexmidlem5N 37988 | Lemma for ~ pexmidN . (Co... |
pexmidlem6N 37989 | Lemma for ~ pexmidN . (Co... |
pexmidlem7N 37990 | Lemma for ~ pexmidN . Con... |
pexmidlem8N 37991 | Lemma for ~ pexmidN . The... |
pexmidALTN 37992 | Excluded middle law for cl... |
pl42lem1N 37993 | Lemma for ~ pl42N . (Cont... |
pl42lem2N 37994 | Lemma for ~ pl42N . (Cont... |
pl42lem3N 37995 | Lemma for ~ pl42N . (Cont... |
pl42lem4N 37996 | Lemma for ~ pl42N . (Cont... |
pl42N 37997 | Law holding in a Hilbert l... |
watfvalN 38006 | The W atoms function. (Co... |
watvalN 38007 | Value of the W atoms funct... |
iswatN 38008 | The predicate "is a W atom... |
lhpset 38009 | The set of co-atoms (latti... |
islhp 38010 | The predicate "is a co-ato... |
islhp2 38011 | The predicate "is a co-ato... |
lhpbase 38012 | A co-atom is a member of t... |
lhp1cvr 38013 | The lattice unit covers a ... |
lhplt 38014 | An atom under a co-atom is... |
lhp2lt 38015 | The join of two atoms unde... |
lhpexlt 38016 | There exists an atom less ... |
lhp0lt 38017 | A co-atom is greater than ... |
lhpn0 38018 | A co-atom is nonzero. TOD... |
lhpexle 38019 | There exists an atom under... |
lhpexnle 38020 | There exists an atom not u... |
lhpexle1lem 38021 | Lemma for ~ lhpexle1 and o... |
lhpexle1 38022 | There exists an atom under... |
lhpexle2lem 38023 | Lemma for ~ lhpexle2 . (C... |
lhpexle2 38024 | There exists atom under a ... |
lhpexle3lem 38025 | There exists atom under a ... |
lhpexle3 38026 | There exists atom under a ... |
lhpex2leN 38027 | There exist at least two d... |
lhpoc 38028 | The orthocomplement of a c... |
lhpoc2N 38029 | The orthocomplement of an ... |
lhpocnle 38030 | The orthocomplement of a c... |
lhpocat 38031 | The orthocomplement of a c... |
lhpocnel 38032 | The orthocomplement of a c... |
lhpocnel2 38033 | The orthocomplement of a c... |
lhpjat1 38034 | The join of a co-atom (hyp... |
lhpjat2 38035 | The join of a co-atom (hyp... |
lhpj1 38036 | The join of a co-atom (hyp... |
lhpmcvr 38037 | The meet of a lattice hype... |
lhpmcvr2 38038 | Alternate way to express t... |
lhpmcvr3 38039 | Specialization of ~ lhpmcv... |
lhpmcvr4N 38040 | Specialization of ~ lhpmcv... |
lhpmcvr5N 38041 | Specialization of ~ lhpmcv... |
lhpmcvr6N 38042 | Specialization of ~ lhpmcv... |
lhpm0atN 38043 | If the meet of a lattice h... |
lhpmat 38044 | An element covered by the ... |
lhpmatb 38045 | An element covered by the ... |
lhp2at0 38046 | Join and meet with differe... |
lhp2atnle 38047 | Inequality for 2 different... |
lhp2atne 38048 | Inequality for joins with ... |
lhp2at0nle 38049 | Inequality for 2 different... |
lhp2at0ne 38050 | Inequality for joins with ... |
lhpelim 38051 | Eliminate an atom not unde... |
lhpmod2i2 38052 | Modular law for hyperplane... |
lhpmod6i1 38053 | Modular law for hyperplane... |
lhprelat3N 38054 | The Hilbert lattice is rel... |
cdlemb2 38055 | Given two atoms not under ... |
lhple 38056 | Property of a lattice elem... |
lhpat 38057 | Create an atom under a co-... |
lhpat4N 38058 | Property of an atom under ... |
lhpat2 38059 | Create an atom under a co-... |
lhpat3 38060 | There is only one atom und... |
4atexlemk 38061 | Lemma for ~ 4atexlem7 . (... |
4atexlemw 38062 | Lemma for ~ 4atexlem7 . (... |
4atexlempw 38063 | Lemma for ~ 4atexlem7 . (... |
4atexlemp 38064 | Lemma for ~ 4atexlem7 . (... |
4atexlemq 38065 | Lemma for ~ 4atexlem7 . (... |
4atexlems 38066 | Lemma for ~ 4atexlem7 . (... |
4atexlemt 38067 | Lemma for ~ 4atexlem7 . (... |
4atexlemutvt 38068 | Lemma for ~ 4atexlem7 . (... |
4atexlempnq 38069 | Lemma for ~ 4atexlem7 . (... |
4atexlemnslpq 38070 | Lemma for ~ 4atexlem7 . (... |
4atexlemkl 38071 | Lemma for ~ 4atexlem7 . (... |
4atexlemkc 38072 | Lemma for ~ 4atexlem7 . (... |
4atexlemwb 38073 | Lemma for ~ 4atexlem7 . (... |
4atexlempsb 38074 | Lemma for ~ 4atexlem7 . (... |
4atexlemqtb 38075 | Lemma for ~ 4atexlem7 . (... |
4atexlempns 38076 | Lemma for ~ 4atexlem7 . (... |
4atexlemswapqr 38077 | Lemma for ~ 4atexlem7 . S... |
4atexlemu 38078 | Lemma for ~ 4atexlem7 . (... |
4atexlemv 38079 | Lemma for ~ 4atexlem7 . (... |
4atexlemunv 38080 | Lemma for ~ 4atexlem7 . (... |
4atexlemtlw 38081 | Lemma for ~ 4atexlem7 . (... |
4atexlemntlpq 38082 | Lemma for ~ 4atexlem7 . (... |
4atexlemc 38083 | Lemma for ~ 4atexlem7 . (... |
4atexlemnclw 38084 | Lemma for ~ 4atexlem7 . (... |
4atexlemex2 38085 | Lemma for ~ 4atexlem7 . S... |
4atexlemcnd 38086 | Lemma for ~ 4atexlem7 . (... |
4atexlemex4 38087 | Lemma for ~ 4atexlem7 . S... |
4atexlemex6 38088 | Lemma for ~ 4atexlem7 . (... |
4atexlem7 38089 | Whenever there are at leas... |
4atex 38090 | Whenever there are at leas... |
4atex2 38091 | More general version of ~ ... |
4atex2-0aOLDN 38092 | Same as ~ 4atex2 except th... |
4atex2-0bOLDN 38093 | Same as ~ 4atex2 except th... |
4atex2-0cOLDN 38094 | Same as ~ 4atex2 except th... |
4atex3 38095 | More general version of ~ ... |
lautset 38096 | The set of lattice automor... |
islaut 38097 | The predicate "is a lattic... |
lautle 38098 | Less-than or equal propert... |
laut1o 38099 | A lattice automorphism is ... |
laut11 38100 | One-to-one property of a l... |
lautcl 38101 | A lattice automorphism val... |
lautcnvclN 38102 | Reverse closure of a latti... |
lautcnvle 38103 | Less-than or equal propert... |
lautcnv 38104 | The converse of a lattice ... |
lautlt 38105 | Less-than property of a la... |
lautcvr 38106 | Covering property of a lat... |
lautj 38107 | Meet property of a lattice... |
lautm 38108 | Meet property of a lattice... |
lauteq 38109 | A lattice automorphism arg... |
idlaut 38110 | The identity function is a... |
lautco 38111 | The composition of two lat... |
pautsetN 38112 | The set of projective auto... |
ispautN 38113 | The predicate "is a projec... |
ldilfset 38122 | The mapping from fiducial ... |
ldilset 38123 | The set of lattice dilatio... |
isldil 38124 | The predicate "is a lattic... |
ldillaut 38125 | A lattice dilation is an a... |
ldil1o 38126 | A lattice dilation is a on... |
ldilval 38127 | Value of a lattice dilatio... |
idldil 38128 | The identity function is a... |
ldilcnv 38129 | The converse of a lattice ... |
ldilco 38130 | The composition of two lat... |
ltrnfset 38131 | The set of all lattice tra... |
ltrnset 38132 | The set of lattice transla... |
isltrn 38133 | The predicate "is a lattic... |
isltrn2N 38134 | The predicate "is a lattic... |
ltrnu 38135 | Uniqueness property of a l... |
ltrnldil 38136 | A lattice translation is a... |
ltrnlaut 38137 | A lattice translation is a... |
ltrn1o 38138 | A lattice translation is a... |
ltrncl 38139 | Closure of a lattice trans... |
ltrn11 38140 | One-to-one property of a l... |
ltrncnvnid 38141 | If a translation is differ... |
ltrncoidN 38142 | Two translations are equal... |
ltrnle 38143 | Less-than or equal propert... |
ltrncnvleN 38144 | Less-than or equal propert... |
ltrnm 38145 | Lattice translation of a m... |
ltrnj 38146 | Lattice translation of a m... |
ltrncvr 38147 | Covering property of a lat... |
ltrnval1 38148 | Value of a lattice transla... |
ltrnid 38149 | A lattice translation is t... |
ltrnnid 38150 | If a lattice translation i... |
ltrnatb 38151 | The lattice translation of... |
ltrncnvatb 38152 | The converse of the lattic... |
ltrnel 38153 | The lattice translation of... |
ltrnat 38154 | The lattice translation of... |
ltrncnvat 38155 | The converse of the lattic... |
ltrncnvel 38156 | The converse of the lattic... |
ltrncoelN 38157 | Composition of lattice tra... |
ltrncoat 38158 | Composition of lattice tra... |
ltrncoval 38159 | Two ways to express value ... |
ltrncnv 38160 | The converse of a lattice ... |
ltrn11at 38161 | Frequently used one-to-one... |
ltrneq2 38162 | The equality of two transl... |
ltrneq 38163 | The equality of two transl... |
idltrn 38164 | The identity function is a... |
ltrnmw 38165 | Property of lattice transl... |
dilfsetN 38166 | The mapping from fiducial ... |
dilsetN 38167 | The set of dilations for a... |
isdilN 38168 | The predicate "is a dilati... |
trnfsetN 38169 | The mapping from fiducial ... |
trnsetN 38170 | The set of translations fo... |
istrnN 38171 | The predicate "is a transl... |
trlfset 38174 | The set of all traces of l... |
trlset 38175 | The set of traces of latti... |
trlval 38176 | The value of the trace of ... |
trlval2 38177 | The value of the trace of ... |
trlcl 38178 | Closure of the trace of a ... |
trlcnv 38179 | The trace of the converse ... |
trljat1 38180 | The value of a translation... |
trljat2 38181 | The value of a translation... |
trljat3 38182 | The value of a translation... |
trlat 38183 | If an atom differs from it... |
trl0 38184 | If an atom not under the f... |
trlator0 38185 | The trace of a lattice tra... |
trlatn0 38186 | The trace of a lattice tra... |
trlnidat 38187 | The trace of a lattice tra... |
ltrnnidn 38188 | If a lattice translation i... |
ltrnideq 38189 | Property of the identity l... |
trlid0 38190 | The trace of the identity ... |
trlnidatb 38191 | A lattice translation is n... |
trlid0b 38192 | A lattice translation is t... |
trlnid 38193 | Different translations wit... |
ltrn2ateq 38194 | Property of the equality o... |
ltrnateq 38195 | If any atom (under ` W ` )... |
ltrnatneq 38196 | If any atom (under ` W ` )... |
ltrnatlw 38197 | If the value of an atom eq... |
trlle 38198 | The trace of a lattice tra... |
trlne 38199 | The trace of a lattice tra... |
trlnle 38200 | The atom not under the fid... |
trlval3 38201 | The value of the trace of ... |
trlval4 38202 | The value of the trace of ... |
trlval5 38203 | The value of the trace of ... |
arglem1N 38204 | Lemma for Desargues's law.... |
cdlemc1 38205 | Part of proof of Lemma C i... |
cdlemc2 38206 | Part of proof of Lemma C i... |
cdlemc3 38207 | Part of proof of Lemma C i... |
cdlemc4 38208 | Part of proof of Lemma C i... |
cdlemc5 38209 | Lemma for ~ cdlemc . (Con... |
cdlemc6 38210 | Lemma for ~ cdlemc . (Con... |
cdlemc 38211 | Lemma C in [Crawley] p. 11... |
cdlemd1 38212 | Part of proof of Lemma D i... |
cdlemd2 38213 | Part of proof of Lemma D i... |
cdlemd3 38214 | Part of proof of Lemma D i... |
cdlemd4 38215 | Part of proof of Lemma D i... |
cdlemd5 38216 | Part of proof of Lemma D i... |
cdlemd6 38217 | Part of proof of Lemma D i... |
cdlemd7 38218 | Part of proof of Lemma D i... |
cdlemd8 38219 | Part of proof of Lemma D i... |
cdlemd9 38220 | Part of proof of Lemma D i... |
cdlemd 38221 | If two translations agree ... |
ltrneq3 38222 | Two translations agree at ... |
cdleme00a 38223 | Part of proof of Lemma E i... |
cdleme0aa 38224 | Part of proof of Lemma E i... |
cdleme0a 38225 | Part of proof of Lemma E i... |
cdleme0b 38226 | Part of proof of Lemma E i... |
cdleme0c 38227 | Part of proof of Lemma E i... |
cdleme0cp 38228 | Part of proof of Lemma E i... |
cdleme0cq 38229 | Part of proof of Lemma E i... |
cdleme0dN 38230 | Part of proof of Lemma E i... |
cdleme0e 38231 | Part of proof of Lemma E i... |
cdleme0fN 38232 | Part of proof of Lemma E i... |
cdleme0gN 38233 | Part of proof of Lemma E i... |
cdlemeulpq 38234 | Part of proof of Lemma E i... |
cdleme01N 38235 | Part of proof of Lemma E i... |
cdleme02N 38236 | Part of proof of Lemma E i... |
cdleme0ex1N 38237 | Part of proof of Lemma E i... |
cdleme0ex2N 38238 | Part of proof of Lemma E i... |
cdleme0moN 38239 | Part of proof of Lemma E i... |
cdleme1b 38240 | Part of proof of Lemma E i... |
cdleme1 38241 | Part of proof of Lemma E i... |
cdleme2 38242 | Part of proof of Lemma E i... |
cdleme3b 38243 | Part of proof of Lemma E i... |
cdleme3c 38244 | Part of proof of Lemma E i... |
cdleme3d 38245 | Part of proof of Lemma E i... |
cdleme3e 38246 | Part of proof of Lemma E i... |
cdleme3fN 38247 | Part of proof of Lemma E i... |
cdleme3g 38248 | Part of proof of Lemma E i... |
cdleme3h 38249 | Part of proof of Lemma E i... |
cdleme3fa 38250 | Part of proof of Lemma E i... |
cdleme3 38251 | Part of proof of Lemma E i... |
cdleme4 38252 | Part of proof of Lemma E i... |
cdleme4a 38253 | Part of proof of Lemma E i... |
cdleme5 38254 | Part of proof of Lemma E i... |
cdleme6 38255 | Part of proof of Lemma E i... |
cdleme7aa 38256 | Part of proof of Lemma E i... |
cdleme7a 38257 | Part of proof of Lemma E i... |
cdleme7b 38258 | Part of proof of Lemma E i... |
cdleme7c 38259 | Part of proof of Lemma E i... |
cdleme7d 38260 | Part of proof of Lemma E i... |
cdleme7e 38261 | Part of proof of Lemma E i... |
cdleme7ga 38262 | Part of proof of Lemma E i... |
cdleme7 38263 | Part of proof of Lemma E i... |
cdleme8 38264 | Part of proof of Lemma E i... |
cdleme9a 38265 | Part of proof of Lemma E i... |
cdleme9b 38266 | Utility lemma for Lemma E ... |
cdleme9 38267 | Part of proof of Lemma E i... |
cdleme10 38268 | Part of proof of Lemma E i... |
cdleme8tN 38269 | Part of proof of Lemma E i... |
cdleme9taN 38270 | Part of proof of Lemma E i... |
cdleme9tN 38271 | Part of proof of Lemma E i... |
cdleme10tN 38272 | Part of proof of Lemma E i... |
cdleme16aN 38273 | Part of proof of Lemma E i... |
cdleme11a 38274 | Part of proof of Lemma E i... |
cdleme11c 38275 | Part of proof of Lemma E i... |
cdleme11dN 38276 | Part of proof of Lemma E i... |
cdleme11e 38277 | Part of proof of Lemma E i... |
cdleme11fN 38278 | Part of proof of Lemma E i... |
cdleme11g 38279 | Part of proof of Lemma E i... |
cdleme11h 38280 | Part of proof of Lemma E i... |
cdleme11j 38281 | Part of proof of Lemma E i... |
cdleme11k 38282 | Part of proof of Lemma E i... |
cdleme11l 38283 | Part of proof of Lemma E i... |
cdleme11 38284 | Part of proof of Lemma E i... |
cdleme12 38285 | Part of proof of Lemma E i... |
cdleme13 38286 | Part of proof of Lemma E i... |
cdleme14 38287 | Part of proof of Lemma E i... |
cdleme15a 38288 | Part of proof of Lemma E i... |
cdleme15b 38289 | Part of proof of Lemma E i... |
cdleme15c 38290 | Part of proof of Lemma E i... |
cdleme15d 38291 | Part of proof of Lemma E i... |
cdleme15 38292 | Part of proof of Lemma E i... |
cdleme16b 38293 | Part of proof of Lemma E i... |
cdleme16c 38294 | Part of proof of Lemma E i... |
cdleme16d 38295 | Part of proof of Lemma E i... |
cdleme16e 38296 | Part of proof of Lemma E i... |
cdleme16f 38297 | Part of proof of Lemma E i... |
cdleme16g 38298 | Part of proof of Lemma E i... |
cdleme16 38299 | Part of proof of Lemma E i... |
cdleme17a 38300 | Part of proof of Lemma E i... |
cdleme17b 38301 | Lemma leading to ~ cdleme1... |
cdleme17c 38302 | Part of proof of Lemma E i... |
cdleme17d1 38303 | Part of proof of Lemma E i... |
cdleme0nex 38304 | Part of proof of Lemma E i... |
cdleme18a 38305 | Part of proof of Lemma E i... |
cdleme18b 38306 | Part of proof of Lemma E i... |
cdleme18c 38307 | Part of proof of Lemma E i... |
cdleme22gb 38308 | Utility lemma for Lemma E ... |
cdleme18d 38309 | Part of proof of Lemma E i... |
cdlemesner 38310 | Part of proof of Lemma E i... |
cdlemedb 38311 | Part of proof of Lemma E i... |
cdlemeda 38312 | Part of proof of Lemma E i... |
cdlemednpq 38313 | Part of proof of Lemma E i... |
cdlemednuN 38314 | Part of proof of Lemma E i... |
cdleme20zN 38315 | Part of proof of Lemma E i... |
cdleme20y 38316 | Part of proof of Lemma E i... |
cdleme19a 38317 | Part of proof of Lemma E i... |
cdleme19b 38318 | Part of proof of Lemma E i... |
cdleme19c 38319 | Part of proof of Lemma E i... |
cdleme19d 38320 | Part of proof of Lemma E i... |
cdleme19e 38321 | Part of proof of Lemma E i... |
cdleme19f 38322 | Part of proof of Lemma E i... |
cdleme20aN 38323 | Part of proof of Lemma E i... |
cdleme20bN 38324 | Part of proof of Lemma E i... |
cdleme20c 38325 | Part of proof of Lemma E i... |
cdleme20d 38326 | Part of proof of Lemma E i... |
cdleme20e 38327 | Part of proof of Lemma E i... |
cdleme20f 38328 | Part of proof of Lemma E i... |
cdleme20g 38329 | Part of proof of Lemma E i... |
cdleme20h 38330 | Part of proof of Lemma E i... |
cdleme20i 38331 | Part of proof of Lemma E i... |
cdleme20j 38332 | Part of proof of Lemma E i... |
cdleme20k 38333 | Part of proof of Lemma E i... |
cdleme20l1 38334 | Part of proof of Lemma E i... |
cdleme20l2 38335 | Part of proof of Lemma E i... |
cdleme20l 38336 | Part of proof of Lemma E i... |
cdleme20m 38337 | Part of proof of Lemma E i... |
cdleme20 38338 | Combine ~ cdleme19f and ~ ... |
cdleme21a 38339 | Part of proof of Lemma E i... |
cdleme21b 38340 | Part of proof of Lemma E i... |
cdleme21c 38341 | Part of proof of Lemma E i... |
cdleme21at 38342 | Part of proof of Lemma E i... |
cdleme21ct 38343 | Part of proof of Lemma E i... |
cdleme21d 38344 | Part of proof of Lemma E i... |
cdleme21e 38345 | Part of proof of Lemma E i... |
cdleme21f 38346 | Part of proof of Lemma E i... |
cdleme21g 38347 | Part of proof of Lemma E i... |
cdleme21h 38348 | Part of proof of Lemma E i... |
cdleme21i 38349 | Part of proof of Lemma E i... |
cdleme21j 38350 | Combine ~ cdleme20 and ~ c... |
cdleme21 38351 | Part of proof of Lemma E i... |
cdleme21k 38352 | Eliminate ` S =/= T ` cond... |
cdleme22aa 38353 | Part of proof of Lemma E i... |
cdleme22a 38354 | Part of proof of Lemma E i... |
cdleme22b 38355 | Part of proof of Lemma E i... |
cdleme22cN 38356 | Part of proof of Lemma E i... |
cdleme22d 38357 | Part of proof of Lemma E i... |
cdleme22e 38358 | Part of proof of Lemma E i... |
cdleme22eALTN 38359 | Part of proof of Lemma E i... |
cdleme22f 38360 | Part of proof of Lemma E i... |
cdleme22f2 38361 | Part of proof of Lemma E i... |
cdleme22g 38362 | Part of proof of Lemma E i... |
cdleme23a 38363 | Part of proof of Lemma E i... |
cdleme23b 38364 | Part of proof of Lemma E i... |
cdleme23c 38365 | Part of proof of Lemma E i... |
cdleme24 38366 | Quantified version of ~ cd... |
cdleme25a 38367 | Lemma for ~ cdleme25b . (... |
cdleme25b 38368 | Transform ~ cdleme24 . TO... |
cdleme25c 38369 | Transform ~ cdleme25b . (... |
cdleme25dN 38370 | Transform ~ cdleme25c . (... |
cdleme25cl 38371 | Show closure of the unique... |
cdleme25cv 38372 | Change bound variables in ... |
cdleme26e 38373 | Part of proof of Lemma E i... |
cdleme26ee 38374 | Part of proof of Lemma E i... |
cdleme26eALTN 38375 | Part of proof of Lemma E i... |
cdleme26fALTN 38376 | Part of proof of Lemma E i... |
cdleme26f 38377 | Part of proof of Lemma E i... |
cdleme26f2ALTN 38378 | Part of proof of Lemma E i... |
cdleme26f2 38379 | Part of proof of Lemma E i... |
cdleme27cl 38380 | Part of proof of Lemma E i... |
cdleme27a 38381 | Part of proof of Lemma E i... |
cdleme27b 38382 | Lemma for ~ cdleme27N . (... |
cdleme27N 38383 | Part of proof of Lemma E i... |
cdleme28a 38384 | Lemma for ~ cdleme25b . T... |
cdleme28b 38385 | Lemma for ~ cdleme25b . T... |
cdleme28c 38386 | Part of proof of Lemma E i... |
cdleme28 38387 | Quantified version of ~ cd... |
cdleme29ex 38388 | Lemma for ~ cdleme29b . (... |
cdleme29b 38389 | Transform ~ cdleme28 . (C... |
cdleme29c 38390 | Transform ~ cdleme28b . (... |
cdleme29cl 38391 | Show closure of the unique... |
cdleme30a 38392 | Part of proof of Lemma E i... |
cdleme31so 38393 | Part of proof of Lemma E i... |
cdleme31sn 38394 | Part of proof of Lemma E i... |
cdleme31sn1 38395 | Part of proof of Lemma E i... |
cdleme31se 38396 | Part of proof of Lemma D i... |
cdleme31se2 38397 | Part of proof of Lemma D i... |
cdleme31sc 38398 | Part of proof of Lemma E i... |
cdleme31sde 38399 | Part of proof of Lemma D i... |
cdleme31snd 38400 | Part of proof of Lemma D i... |
cdleme31sdnN 38401 | Part of proof of Lemma E i... |
cdleme31sn1c 38402 | Part of proof of Lemma E i... |
cdleme31sn2 38403 | Part of proof of Lemma E i... |
cdleme31fv 38404 | Part of proof of Lemma E i... |
cdleme31fv1 38405 | Part of proof of Lemma E i... |
cdleme31fv1s 38406 | Part of proof of Lemma E i... |
cdleme31fv2 38407 | Part of proof of Lemma E i... |
cdleme31id 38408 | Part of proof of Lemma E i... |
cdlemefrs29pre00 38409 | ***START OF VALUE AT ATOM ... |
cdlemefrs29bpre0 38410 | TODO fix comment. (Contri... |
cdlemefrs29bpre1 38411 | TODO: FIX COMMENT. (Contr... |
cdlemefrs29cpre1 38412 | TODO: FIX COMMENT. (Contr... |
cdlemefrs29clN 38413 | TODO: NOT USED? Show clo... |
cdlemefrs32fva 38414 | Part of proof of Lemma E i... |
cdlemefrs32fva1 38415 | Part of proof of Lemma E i... |
cdlemefr29exN 38416 | Lemma for ~ cdlemefs29bpre... |
cdlemefr27cl 38417 | Part of proof of Lemma E i... |
cdlemefr32sn2aw 38418 | Show that ` [_ R / s ]_ N ... |
cdlemefr32snb 38419 | Show closure of ` [_ R / s... |
cdlemefr29bpre0N 38420 | TODO fix comment. (Contri... |
cdlemefr29clN 38421 | Show closure of the unique... |
cdleme43frv1snN 38422 | Value of ` [_ R / s ]_ N `... |
cdlemefr32fvaN 38423 | Part of proof of Lemma E i... |
cdlemefr32fva1 38424 | Part of proof of Lemma E i... |
cdlemefr31fv1 38425 | Value of ` ( F `` R ) ` wh... |
cdlemefs29pre00N 38426 | FIX COMMENT. TODO: see if ... |
cdlemefs27cl 38427 | Part of proof of Lemma E i... |
cdlemefs32sn1aw 38428 | Show that ` [_ R / s ]_ N ... |
cdlemefs32snb 38429 | Show closure of ` [_ R / s... |
cdlemefs29bpre0N 38430 | TODO: FIX COMMENT. (Contr... |
cdlemefs29bpre1N 38431 | TODO: FIX COMMENT. (Contr... |
cdlemefs29cpre1N 38432 | TODO: FIX COMMENT. (Contr... |
cdlemefs29clN 38433 | Show closure of the unique... |
cdleme43fsv1snlem 38434 | Value of ` [_ R / s ]_ N `... |
cdleme43fsv1sn 38435 | Value of ` [_ R / s ]_ N `... |
cdlemefs32fvaN 38436 | Part of proof of Lemma E i... |
cdlemefs32fva1 38437 | Part of proof of Lemma E i... |
cdlemefs31fv1 38438 | Value of ` ( F `` R ) ` wh... |
cdlemefr44 38439 | Value of f(r) when r is an... |
cdlemefs44 38440 | Value of f_s(r) when r is ... |
cdlemefr45 38441 | Value of f(r) when r is an... |
cdlemefr45e 38442 | Explicit expansion of ~ cd... |
cdlemefs45 38443 | Value of f_s(r) when r is ... |
cdlemefs45ee 38444 | Explicit expansion of ~ cd... |
cdlemefs45eN 38445 | Explicit expansion of ~ cd... |
cdleme32sn1awN 38446 | Show that ` [_ R / s ]_ N ... |
cdleme41sn3a 38447 | Show that ` [_ R / s ]_ N ... |
cdleme32sn2awN 38448 | Show that ` [_ R / s ]_ N ... |
cdleme32snaw 38449 | Show that ` [_ R / s ]_ N ... |
cdleme32snb 38450 | Show closure of ` [_ R / s... |
cdleme32fva 38451 | Part of proof of Lemma D i... |
cdleme32fva1 38452 | Part of proof of Lemma D i... |
cdleme32fvaw 38453 | Show that ` ( F `` R ) ` i... |
cdleme32fvcl 38454 | Part of proof of Lemma D i... |
cdleme32a 38455 | Part of proof of Lemma D i... |
cdleme32b 38456 | Part of proof of Lemma D i... |
cdleme32c 38457 | Part of proof of Lemma D i... |
cdleme32d 38458 | Part of proof of Lemma D i... |
cdleme32e 38459 | Part of proof of Lemma D i... |
cdleme32f 38460 | Part of proof of Lemma D i... |
cdleme32le 38461 | Part of proof of Lemma D i... |
cdleme35a 38462 | Part of proof of Lemma E i... |
cdleme35fnpq 38463 | Part of proof of Lemma E i... |
cdleme35b 38464 | Part of proof of Lemma E i... |
cdleme35c 38465 | Part of proof of Lemma E i... |
cdleme35d 38466 | Part of proof of Lemma E i... |
cdleme35e 38467 | Part of proof of Lemma E i... |
cdleme35f 38468 | Part of proof of Lemma E i... |
cdleme35g 38469 | Part of proof of Lemma E i... |
cdleme35h 38470 | Part of proof of Lemma E i... |
cdleme35h2 38471 | Part of proof of Lemma E i... |
cdleme35sn2aw 38472 | Part of proof of Lemma E i... |
cdleme35sn3a 38473 | Part of proof of Lemma E i... |
cdleme36a 38474 | Part of proof of Lemma E i... |
cdleme36m 38475 | Part of proof of Lemma E i... |
cdleme37m 38476 | Part of proof of Lemma E i... |
cdleme38m 38477 | Part of proof of Lemma E i... |
cdleme38n 38478 | Part of proof of Lemma E i... |
cdleme39a 38479 | Part of proof of Lemma E i... |
cdleme39n 38480 | Part of proof of Lemma E i... |
cdleme40m 38481 | Part of proof of Lemma E i... |
cdleme40n 38482 | Part of proof of Lemma E i... |
cdleme40v 38483 | Part of proof of Lemma E i... |
cdleme40w 38484 | Part of proof of Lemma E i... |
cdleme42a 38485 | Part of proof of Lemma E i... |
cdleme42c 38486 | Part of proof of Lemma E i... |
cdleme42d 38487 | Part of proof of Lemma E i... |
cdleme41sn3aw 38488 | Part of proof of Lemma E i... |
cdleme41sn4aw 38489 | Part of proof of Lemma E i... |
cdleme41snaw 38490 | Part of proof of Lemma E i... |
cdleme41fva11 38491 | Part of proof of Lemma E i... |
cdleme42b 38492 | Part of proof of Lemma E i... |
cdleme42e 38493 | Part of proof of Lemma E i... |
cdleme42f 38494 | Part of proof of Lemma E i... |
cdleme42g 38495 | Part of proof of Lemma E i... |
cdleme42h 38496 | Part of proof of Lemma E i... |
cdleme42i 38497 | Part of proof of Lemma E i... |
cdleme42k 38498 | Part of proof of Lemma E i... |
cdleme42ke 38499 | Part of proof of Lemma E i... |
cdleme42keg 38500 | Part of proof of Lemma E i... |
cdleme42mN 38501 | Part of proof of Lemma E i... |
cdleme42mgN 38502 | Part of proof of Lemma E i... |
cdleme43aN 38503 | Part of proof of Lemma E i... |
cdleme43bN 38504 | Lemma for Lemma E in [Craw... |
cdleme43cN 38505 | Part of proof of Lemma E i... |
cdleme43dN 38506 | Part of proof of Lemma E i... |
cdleme46f2g2 38507 | Conversion for ` G ` to re... |
cdleme46f2g1 38508 | Conversion for ` G ` to re... |
cdleme17d2 38509 | Part of proof of Lemma E i... |
cdleme17d3 38510 | TODO: FIX COMMENT. (Contr... |
cdleme17d4 38511 | TODO: FIX COMMENT. (Contr... |
cdleme17d 38512 | Part of proof of Lemma E i... |
cdleme48fv 38513 | Part of proof of Lemma D i... |
cdleme48fvg 38514 | Remove ` P =/= Q ` conditi... |
cdleme46fvaw 38515 | Show that ` ( F `` R ) ` i... |
cdleme48bw 38516 | TODO: fix comment. TODO: ... |
cdleme48b 38517 | TODO: fix comment. (Contr... |
cdleme46frvlpq 38518 | Show that ` ( F `` S ) ` i... |
cdleme46fsvlpq 38519 | Show that ` ( F `` R ) ` i... |
cdlemeg46fvcl 38520 | TODO: fix comment. (Contr... |
cdleme4gfv 38521 | Part of proof of Lemma D i... |
cdlemeg47b 38522 | TODO: FIX COMMENT. (Contr... |
cdlemeg47rv 38523 | Value of g_s(r) when r is ... |
cdlemeg47rv2 38524 | Value of g_s(r) when r is ... |
cdlemeg49le 38525 | Part of proof of Lemma D i... |
cdlemeg46bOLDN 38526 | TODO FIX COMMENT. (Contrib... |
cdlemeg46c 38527 | TODO FIX COMMENT. (Contrib... |
cdlemeg46rvOLDN 38528 | Value of g_s(r) when r is ... |
cdlemeg46rv2OLDN 38529 | Value of g_s(r) when r is ... |
cdlemeg46fvaw 38530 | Show that ` ( F `` R ) ` i... |
cdlemeg46nlpq 38531 | Show that ` ( G `` S ) ` i... |
cdlemeg46ngfr 38532 | TODO FIX COMMENT g(f(s))=s... |
cdlemeg46nfgr 38533 | TODO FIX COMMENT f(g(s))=s... |
cdlemeg46sfg 38534 | TODO FIX COMMENT f(r) ` \/... |
cdlemeg46fjgN 38535 | NOT NEEDED? TODO FIX COMM... |
cdlemeg46rjgN 38536 | NOT NEEDED? TODO FIX COMM... |
cdlemeg46fjv 38537 | TODO FIX COMMENT f(r) ` \/... |
cdlemeg46fsfv 38538 | TODO FIX COMMENT f(r) ` \/... |
cdlemeg46frv 38539 | TODO FIX COMMENT. (f(r) ` ... |
cdlemeg46v1v2 38540 | TODO FIX COMMENT v_1 = v_2... |
cdlemeg46vrg 38541 | TODO FIX COMMENT v_1 ` <_ ... |
cdlemeg46rgv 38542 | TODO FIX COMMENT r ` <_ ` ... |
cdlemeg46req 38543 | TODO FIX COMMENT r = (v_1 ... |
cdlemeg46gfv 38544 | TODO FIX COMMENT p. 115 pe... |
cdlemeg46gfr 38545 | TODO FIX COMMENT p. 116 pe... |
cdlemeg46gfre 38546 | TODO FIX COMMENT p. 116 pe... |
cdlemeg46gf 38547 | TODO FIX COMMENT Eliminate... |
cdlemeg46fgN 38548 | TODO FIX COMMENT p. 116 pe... |
cdleme48d 38549 | TODO: fix comment. (Contr... |
cdleme48gfv1 38550 | TODO: fix comment. (Contr... |
cdleme48gfv 38551 | TODO: fix comment. (Contr... |
cdleme48fgv 38552 | TODO: fix comment. (Contr... |
cdlemeg49lebilem 38553 | Part of proof of Lemma D i... |
cdleme50lebi 38554 | Part of proof of Lemma D i... |
cdleme50eq 38555 | Part of proof of Lemma D i... |
cdleme50f 38556 | Part of proof of Lemma D i... |
cdleme50f1 38557 | Part of proof of Lemma D i... |
cdleme50rnlem 38558 | Part of proof of Lemma D i... |
cdleme50rn 38559 | Part of proof of Lemma D i... |
cdleme50f1o 38560 | Part of proof of Lemma D i... |
cdleme50laut 38561 | Part of proof of Lemma D i... |
cdleme50ldil 38562 | Part of proof of Lemma D i... |
cdleme50trn1 38563 | Part of proof that ` F ` i... |
cdleme50trn2a 38564 | Part of proof that ` F ` i... |
cdleme50trn2 38565 | Part of proof that ` F ` i... |
cdleme50trn12 38566 | Part of proof that ` F ` i... |
cdleme50trn3 38567 | Part of proof that ` F ` i... |
cdleme50trn123 38568 | Part of proof that ` F ` i... |
cdleme51finvfvN 38569 | Part of proof of Lemma E i... |
cdleme51finvN 38570 | Part of proof of Lemma E i... |
cdleme50ltrn 38571 | Part of proof of Lemma E i... |
cdleme51finvtrN 38572 | Part of proof of Lemma E i... |
cdleme50ex 38573 | Part of Lemma E in [Crawle... |
cdleme 38574 | Lemma E in [Crawley] p. 11... |
cdlemf1 38575 | Part of Lemma F in [Crawle... |
cdlemf2 38576 | Part of Lemma F in [Crawle... |
cdlemf 38577 | Lemma F in [Crawley] p. 11... |
cdlemfnid 38578 | ~ cdlemf with additional c... |
cdlemftr3 38579 | Special case of ~ cdlemf s... |
cdlemftr2 38580 | Special case of ~ cdlemf s... |
cdlemftr1 38581 | Part of proof of Lemma G o... |
cdlemftr0 38582 | Special case of ~ cdlemf s... |
trlord 38583 | The ordering of two Hilber... |
cdlemg1a 38584 | Shorter expression for ` G... |
cdlemg1b2 38585 | This theorem can be used t... |
cdlemg1idlemN 38586 | Lemma for ~ cdlemg1idN . ... |
cdlemg1fvawlemN 38587 | Lemma for ~ ltrniotafvawN ... |
cdlemg1ltrnlem 38588 | Lemma for ~ ltrniotacl . ... |
cdlemg1finvtrlemN 38589 | Lemma for ~ ltrniotacnvN .... |
cdlemg1bOLDN 38590 | This theorem can be used t... |
cdlemg1idN 38591 | Version of ~ cdleme31id wi... |
ltrniotafvawN 38592 | Version of ~ cdleme46fvaw ... |
ltrniotacl 38593 | Version of ~ cdleme50ltrn ... |
ltrniotacnvN 38594 | Version of ~ cdleme51finvt... |
ltrniotaval 38595 | Value of the unique transl... |
ltrniotacnvval 38596 | Converse value of the uniq... |
ltrniotaidvalN 38597 | Value of the unique transl... |
ltrniotavalbN 38598 | Value of the unique transl... |
cdlemeiota 38599 | A translation is uniquely ... |
cdlemg1ci2 38600 | Any function of the form o... |
cdlemg1cN 38601 | Any translation belongs to... |
cdlemg1cex 38602 | Any translation is one of ... |
cdlemg2cN 38603 | Any translation belongs to... |
cdlemg2dN 38604 | This theorem can be used t... |
cdlemg2cex 38605 | Any translation is one of ... |
cdlemg2ce 38606 | Utility theorem to elimina... |
cdlemg2jlemOLDN 38607 | Part of proof of Lemma E i... |
cdlemg2fvlem 38608 | Lemma for ~ cdlemg2fv . (... |
cdlemg2klem 38609 | ~ cdleme42keg with simpler... |
cdlemg2idN 38610 | Version of ~ cdleme31id wi... |
cdlemg3a 38611 | Part of proof of Lemma G i... |
cdlemg2jOLDN 38612 | TODO: Replace this with ~... |
cdlemg2fv 38613 | Value of a translation in ... |
cdlemg2fv2 38614 | Value of a translation in ... |
cdlemg2k 38615 | ~ cdleme42keg with simpler... |
cdlemg2kq 38616 | ~ cdlemg2k with ` P ` and ... |
cdlemg2l 38617 | TODO: FIX COMMENT. (Contr... |
cdlemg2m 38618 | TODO: FIX COMMENT. (Contr... |
cdlemg5 38619 | TODO: Is there a simpler ... |
cdlemb3 38620 | Given two atoms not under ... |
cdlemg7fvbwN 38621 | Properties of a translatio... |
cdlemg4a 38622 | TODO: FIX COMMENT If fg(p... |
cdlemg4b1 38623 | TODO: FIX COMMENT. (Contr... |
cdlemg4b2 38624 | TODO: FIX COMMENT. (Contr... |
cdlemg4b12 38625 | TODO: FIX COMMENT. (Contr... |
cdlemg4c 38626 | TODO: FIX COMMENT. (Contr... |
cdlemg4d 38627 | TODO: FIX COMMENT. (Contr... |
cdlemg4e 38628 | TODO: FIX COMMENT. (Contr... |
cdlemg4f 38629 | TODO: FIX COMMENT. (Contr... |
cdlemg4g 38630 | TODO: FIX COMMENT. (Contr... |
cdlemg4 38631 | TODO: FIX COMMENT. (Contr... |
cdlemg6a 38632 | TODO: FIX COMMENT. TODO: ... |
cdlemg6b 38633 | TODO: FIX COMMENT. TODO: ... |
cdlemg6c 38634 | TODO: FIX COMMENT. (Contr... |
cdlemg6d 38635 | TODO: FIX COMMENT. (Contr... |
cdlemg6e 38636 | TODO: FIX COMMENT. (Contr... |
cdlemg6 38637 | TODO: FIX COMMENT. (Contr... |
cdlemg7fvN 38638 | Value of a translation com... |
cdlemg7aN 38639 | TODO: FIX COMMENT. (Contr... |
cdlemg7N 38640 | TODO: FIX COMMENT. (Contr... |
cdlemg8a 38641 | TODO: FIX COMMENT. (Contr... |
cdlemg8b 38642 | TODO: FIX COMMENT. (Contr... |
cdlemg8c 38643 | TODO: FIX COMMENT. (Contr... |
cdlemg8d 38644 | TODO: FIX COMMENT. (Contr... |
cdlemg8 38645 | TODO: FIX COMMENT. (Contr... |
cdlemg9a 38646 | TODO: FIX COMMENT. (Contr... |
cdlemg9b 38647 | The triples ` <. P , ( F `... |
cdlemg9 38648 | The triples ` <. P , ( F `... |
cdlemg10b 38649 | TODO: FIX COMMENT. TODO: ... |
cdlemg10bALTN 38650 | TODO: FIX COMMENT. TODO: ... |
cdlemg11a 38651 | TODO: FIX COMMENT. (Contr... |
cdlemg11aq 38652 | TODO: FIX COMMENT. TODO: ... |
cdlemg10c 38653 | TODO: FIX COMMENT. TODO: ... |
cdlemg10a 38654 | TODO: FIX COMMENT. (Contr... |
cdlemg10 38655 | TODO: FIX COMMENT. (Contr... |
cdlemg11b 38656 | TODO: FIX COMMENT. (Contr... |
cdlemg12a 38657 | TODO: FIX COMMENT. (Contr... |
cdlemg12b 38658 | The triples ` <. P , ( F `... |
cdlemg12c 38659 | The triples ` <. P , ( F `... |
cdlemg12d 38660 | TODO: FIX COMMENT. (Contr... |
cdlemg12e 38661 | TODO: FIX COMMENT. (Contr... |
cdlemg12f 38662 | TODO: FIX COMMENT. (Contr... |
cdlemg12g 38663 | TODO: FIX COMMENT. TODO: ... |
cdlemg12 38664 | TODO: FIX COMMENT. (Contr... |
cdlemg13a 38665 | TODO: FIX COMMENT. (Contr... |
cdlemg13 38666 | TODO: FIX COMMENT. (Contr... |
cdlemg14f 38667 | TODO: FIX COMMENT. (Contr... |
cdlemg14g 38668 | TODO: FIX COMMENT. (Contr... |
cdlemg15a 38669 | Eliminate the ` ( F `` P )... |
cdlemg15 38670 | Eliminate the ` ( (... |
cdlemg16 38671 | Part of proof of Lemma G o... |
cdlemg16ALTN 38672 | This version of ~ cdlemg16... |
cdlemg16z 38673 | Eliminate ` ( ( F `... |
cdlemg16zz 38674 | Eliminate ` P =/= Q ` from... |
cdlemg17a 38675 | TODO: FIX COMMENT. (Contr... |
cdlemg17b 38676 | Part of proof of Lemma G i... |
cdlemg17dN 38677 | TODO: fix comment. (Contr... |
cdlemg17dALTN 38678 | Same as ~ cdlemg17dN with ... |
cdlemg17e 38679 | TODO: fix comment. (Contr... |
cdlemg17f 38680 | TODO: fix comment. (Contr... |
cdlemg17g 38681 | TODO: fix comment. (Contr... |
cdlemg17h 38682 | TODO: fix comment. (Contr... |
cdlemg17i 38683 | TODO: fix comment. (Contr... |
cdlemg17ir 38684 | TODO: fix comment. (Contr... |
cdlemg17j 38685 | TODO: fix comment. (Contr... |
cdlemg17pq 38686 | Utility theorem for swappi... |
cdlemg17bq 38687 | ~ cdlemg17b with ` P ` and... |
cdlemg17iqN 38688 | ~ cdlemg17i with ` P ` and... |
cdlemg17irq 38689 | ~ cdlemg17ir with ` P ` an... |
cdlemg17jq 38690 | ~ cdlemg17j with ` P ` and... |
cdlemg17 38691 | Part of Lemma G of [Crawle... |
cdlemg18a 38692 | Show two lines are differe... |
cdlemg18b 38693 | Lemma for ~ cdlemg18c . T... |
cdlemg18c 38694 | Show two lines intersect a... |
cdlemg18d 38695 | Show two lines intersect a... |
cdlemg18 38696 | Show two lines intersect a... |
cdlemg19a 38697 | Show two lines intersect a... |
cdlemg19 38698 | Show two lines intersect a... |
cdlemg20 38699 | Show two lines intersect a... |
cdlemg21 38700 | Version of cdlemg19 with `... |
cdlemg22 38701 | ~ cdlemg21 with ` ( F `` P... |
cdlemg24 38702 | Combine ~ cdlemg16z and ~ ... |
cdlemg37 38703 | Use ~ cdlemg8 to eliminate... |
cdlemg25zz 38704 | ~ cdlemg16zz restated for ... |
cdlemg26zz 38705 | ~ cdlemg16zz restated for ... |
cdlemg27a 38706 | For use with case when ` (... |
cdlemg28a 38707 | Part of proof of Lemma G o... |
cdlemg31b0N 38708 | TODO: Fix comment. (Cont... |
cdlemg31b0a 38709 | TODO: Fix comment. (Cont... |
cdlemg27b 38710 | TODO: Fix comment. (Cont... |
cdlemg31a 38711 | TODO: fix comment. (Contr... |
cdlemg31b 38712 | TODO: fix comment. (Contr... |
cdlemg31c 38713 | Show that when ` N ` is an... |
cdlemg31d 38714 | Eliminate ` ( F `` P ) =/=... |
cdlemg33b0 38715 | TODO: Fix comment. (Cont... |
cdlemg33c0 38716 | TODO: Fix comment. (Cont... |
cdlemg28b 38717 | Part of proof of Lemma G o... |
cdlemg28 38718 | Part of proof of Lemma G o... |
cdlemg29 38719 | Eliminate ` ( F `` P ) =/=... |
cdlemg33a 38720 | TODO: Fix comment. (Cont... |
cdlemg33b 38721 | TODO: Fix comment. (Cont... |
cdlemg33c 38722 | TODO: Fix comment. (Cont... |
cdlemg33d 38723 | TODO: Fix comment. (Cont... |
cdlemg33e 38724 | TODO: Fix comment. (Cont... |
cdlemg33 38725 | Combine ~ cdlemg33b , ~ cd... |
cdlemg34 38726 | Use cdlemg33 to eliminate ... |
cdlemg35 38727 | TODO: Fix comment. TODO:... |
cdlemg36 38728 | Use cdlemg35 to eliminate ... |
cdlemg38 38729 | Use ~ cdlemg37 to eliminat... |
cdlemg39 38730 | Eliminate ` =/= ` conditio... |
cdlemg40 38731 | Eliminate ` P =/= Q ` cond... |
cdlemg41 38732 | Convert ~ cdlemg40 to func... |
ltrnco 38733 | The composition of two tra... |
trlcocnv 38734 | Swap the arguments of the ... |
trlcoabs 38735 | Absorption into a composit... |
trlcoabs2N 38736 | Absorption of the trace of... |
trlcoat 38737 | The trace of a composition... |
trlcocnvat 38738 | Commonly used special case... |
trlconid 38739 | The composition of two dif... |
trlcolem 38740 | Lemma for ~ trlco . (Cont... |
trlco 38741 | The trace of a composition... |
trlcone 38742 | If two translations have d... |
cdlemg42 38743 | Part of proof of Lemma G o... |
cdlemg43 38744 | Part of proof of Lemma G o... |
cdlemg44a 38745 | Part of proof of Lemma G o... |
cdlemg44b 38746 | Eliminate ` ( F `` P ) =/=... |
cdlemg44 38747 | Part of proof of Lemma G o... |
cdlemg47a 38748 | TODO: fix comment. TODO: ... |
cdlemg46 38749 | Part of proof of Lemma G o... |
cdlemg47 38750 | Part of proof of Lemma G o... |
cdlemg48 38751 | Eliminate ` h ` from ~ cdl... |
ltrncom 38752 | Composition is commutative... |
ltrnco4 38753 | Rearrange a composition of... |
trljco 38754 | Trace joined with trace of... |
trljco2 38755 | Trace joined with trace of... |
tgrpfset 38758 | The translation group maps... |
tgrpset 38759 | The translation group for ... |
tgrpbase 38760 | The base set of the transl... |
tgrpopr 38761 | The group operation of the... |
tgrpov 38762 | The group operation value ... |
tgrpgrplem 38763 | Lemma for ~ tgrpgrp . (Co... |
tgrpgrp 38764 | The translation group is a... |
tgrpabl 38765 | The translation group is a... |
tendofset 38772 | The set of all trace-prese... |
tendoset 38773 | The set of trace-preservin... |
istendo 38774 | The predicate "is a trace-... |
tendotp 38775 | Trace-preserving property ... |
istendod 38776 | Deduce the predicate "is a... |
tendof 38777 | Functionality of a trace-p... |
tendoeq1 38778 | Condition determining equa... |
tendovalco 38779 | Value of composition of tr... |
tendocoval 38780 | Value of composition of en... |
tendocl 38781 | Closure of a trace-preserv... |
tendoco2 38782 | Distribution of compositio... |
tendoidcl 38783 | The identity is a trace-pr... |
tendo1mul 38784 | Multiplicative identity mu... |
tendo1mulr 38785 | Multiplicative identity mu... |
tendococl 38786 | The composition of two tra... |
tendoid 38787 | The identity value of a tr... |
tendoeq2 38788 | Condition determining equa... |
tendoplcbv 38789 | Define sum operation for t... |
tendopl 38790 | Value of endomorphism sum ... |
tendopl2 38791 | Value of result of endomor... |
tendoplcl2 38792 | Value of result of endomor... |
tendoplco2 38793 | Value of result of endomor... |
tendopltp 38794 | Trace-preserving property ... |
tendoplcl 38795 | Endomorphism sum is a trac... |
tendoplcom 38796 | The endomorphism sum opera... |
tendoplass 38797 | The endomorphism sum opera... |
tendodi1 38798 | Endomorphism composition d... |
tendodi2 38799 | Endomorphism composition d... |
tendo0cbv 38800 | Define additive identity f... |
tendo02 38801 | Value of additive identity... |
tendo0co2 38802 | The additive identity trac... |
tendo0tp 38803 | Trace-preserving property ... |
tendo0cl 38804 | The additive identity is a... |
tendo0pl 38805 | Property of the additive i... |
tendo0plr 38806 | Property of the additive i... |
tendoicbv 38807 | Define inverse function fo... |
tendoi 38808 | Value of inverse endomorph... |
tendoi2 38809 | Value of additive inverse ... |
tendoicl 38810 | Closure of the additive in... |
tendoipl 38811 | Property of the additive i... |
tendoipl2 38812 | Property of the additive i... |
erngfset 38813 | The division rings on trac... |
erngset 38814 | The division ring on trace... |
erngbase 38815 | The base set of the divisi... |
erngfplus 38816 | Ring addition operation. ... |
erngplus 38817 | Ring addition operation. ... |
erngplus2 38818 | Ring addition operation. ... |
erngfmul 38819 | Ring multiplication operat... |
erngmul 38820 | Ring addition operation. ... |
erngfset-rN 38821 | The division rings on trac... |
erngset-rN 38822 | The division ring on trace... |
erngbase-rN 38823 | The base set of the divisi... |
erngfplus-rN 38824 | Ring addition operation. ... |
erngplus-rN 38825 | Ring addition operation. ... |
erngplus2-rN 38826 | Ring addition operation. ... |
erngfmul-rN 38827 | Ring multiplication operat... |
erngmul-rN 38828 | Ring addition operation. ... |
cdlemh1 38829 | Part of proof of Lemma H o... |
cdlemh2 38830 | Part of proof of Lemma H o... |
cdlemh 38831 | Lemma H of [Crawley] p. 11... |
cdlemi1 38832 | Part of proof of Lemma I o... |
cdlemi2 38833 | Part of proof of Lemma I o... |
cdlemi 38834 | Lemma I of [Crawley] p. 11... |
cdlemj1 38835 | Part of proof of Lemma J o... |
cdlemj2 38836 | Part of proof of Lemma J o... |
cdlemj3 38837 | Part of proof of Lemma J o... |
tendocan 38838 | Cancellation law: if the v... |
tendoid0 38839 | A trace-preserving endomor... |
tendo0mul 38840 | Additive identity multipli... |
tendo0mulr 38841 | Additive identity multipli... |
tendo1ne0 38842 | The identity (unity) is no... |
tendoconid 38843 | The composition (product) ... |
tendotr 38844 | The trace of the value of ... |
cdlemk1 38845 | Part of proof of Lemma K o... |
cdlemk2 38846 | Part of proof of Lemma K o... |
cdlemk3 38847 | Part of proof of Lemma K o... |
cdlemk4 38848 | Part of proof of Lemma K o... |
cdlemk5a 38849 | Part of proof of Lemma K o... |
cdlemk5 38850 | Part of proof of Lemma K o... |
cdlemk6 38851 | Part of proof of Lemma K o... |
cdlemk8 38852 | Part of proof of Lemma K o... |
cdlemk9 38853 | Part of proof of Lemma K o... |
cdlemk9bN 38854 | Part of proof of Lemma K o... |
cdlemki 38855 | Part of proof of Lemma K o... |
cdlemkvcl 38856 | Part of proof of Lemma K o... |
cdlemk10 38857 | Part of proof of Lemma K o... |
cdlemksv 38858 | Part of proof of Lemma K o... |
cdlemksel 38859 | Part of proof of Lemma K o... |
cdlemksat 38860 | Part of proof of Lemma K o... |
cdlemksv2 38861 | Part of proof of Lemma K o... |
cdlemk7 38862 | Part of proof of Lemma K o... |
cdlemk11 38863 | Part of proof of Lemma K o... |
cdlemk12 38864 | Part of proof of Lemma K o... |
cdlemkoatnle 38865 | Utility lemma. (Contribut... |
cdlemk13 38866 | Part of proof of Lemma K o... |
cdlemkole 38867 | Utility lemma. (Contribut... |
cdlemk14 38868 | Part of proof of Lemma K o... |
cdlemk15 38869 | Part of proof of Lemma K o... |
cdlemk16a 38870 | Part of proof of Lemma K o... |
cdlemk16 38871 | Part of proof of Lemma K o... |
cdlemk17 38872 | Part of proof of Lemma K o... |
cdlemk1u 38873 | Part of proof of Lemma K o... |
cdlemk5auN 38874 | Part of proof of Lemma K o... |
cdlemk5u 38875 | Part of proof of Lemma K o... |
cdlemk6u 38876 | Part of proof of Lemma K o... |
cdlemkj 38877 | Part of proof of Lemma K o... |
cdlemkuvN 38878 | Part of proof of Lemma K o... |
cdlemkuel 38879 | Part of proof of Lemma K o... |
cdlemkuat 38880 | Part of proof of Lemma K o... |
cdlemkuv2 38881 | Part of proof of Lemma K o... |
cdlemk18 38882 | Part of proof of Lemma K o... |
cdlemk19 38883 | Part of proof of Lemma K o... |
cdlemk7u 38884 | Part of proof of Lemma K o... |
cdlemk11u 38885 | Part of proof of Lemma K o... |
cdlemk12u 38886 | Part of proof of Lemma K o... |
cdlemk21N 38887 | Part of proof of Lemma K o... |
cdlemk20 38888 | Part of proof of Lemma K o... |
cdlemkoatnle-2N 38889 | Utility lemma. (Contribut... |
cdlemk13-2N 38890 | Part of proof of Lemma K o... |
cdlemkole-2N 38891 | Utility lemma. (Contribut... |
cdlemk14-2N 38892 | Part of proof of Lemma K o... |
cdlemk15-2N 38893 | Part of proof of Lemma K o... |
cdlemk16-2N 38894 | Part of proof of Lemma K o... |
cdlemk17-2N 38895 | Part of proof of Lemma K o... |
cdlemkj-2N 38896 | Part of proof of Lemma K o... |
cdlemkuv-2N 38897 | Part of proof of Lemma K o... |
cdlemkuel-2N 38898 | Part of proof of Lemma K o... |
cdlemkuv2-2 38899 | Part of proof of Lemma K o... |
cdlemk18-2N 38900 | Part of proof of Lemma K o... |
cdlemk19-2N 38901 | Part of proof of Lemma K o... |
cdlemk7u-2N 38902 | Part of proof of Lemma K o... |
cdlemk11u-2N 38903 | Part of proof of Lemma K o... |
cdlemk12u-2N 38904 | Part of proof of Lemma K o... |
cdlemk21-2N 38905 | Part of proof of Lemma K o... |
cdlemk20-2N 38906 | Part of proof of Lemma K o... |
cdlemk22 38907 | Part of proof of Lemma K o... |
cdlemk30 38908 | Part of proof of Lemma K o... |
cdlemkuu 38909 | Convert between function a... |
cdlemk31 38910 | Part of proof of Lemma K o... |
cdlemk32 38911 | Part of proof of Lemma K o... |
cdlemkuel-3 38912 | Part of proof of Lemma K o... |
cdlemkuv2-3N 38913 | Part of proof of Lemma K o... |
cdlemk18-3N 38914 | Part of proof of Lemma K o... |
cdlemk22-3 38915 | Part of proof of Lemma K o... |
cdlemk23-3 38916 | Part of proof of Lemma K o... |
cdlemk24-3 38917 | Part of proof of Lemma K o... |
cdlemk25-3 38918 | Part of proof of Lemma K o... |
cdlemk26b-3 38919 | Part of proof of Lemma K o... |
cdlemk26-3 38920 | Part of proof of Lemma K o... |
cdlemk27-3 38921 | Part of proof of Lemma K o... |
cdlemk28-3 38922 | Part of proof of Lemma K o... |
cdlemk33N 38923 | Part of proof of Lemma K o... |
cdlemk34 38924 | Part of proof of Lemma K o... |
cdlemk29-3 38925 | Part of proof of Lemma K o... |
cdlemk35 38926 | Part of proof of Lemma K o... |
cdlemk36 38927 | Part of proof of Lemma K o... |
cdlemk37 38928 | Part of proof of Lemma K o... |
cdlemk38 38929 | Part of proof of Lemma K o... |
cdlemk39 38930 | Part of proof of Lemma K o... |
cdlemk40 38931 | TODO: fix comment. (Contr... |
cdlemk40t 38932 | TODO: fix comment. (Contr... |
cdlemk40f 38933 | TODO: fix comment. (Contr... |
cdlemk41 38934 | Part of proof of Lemma K o... |
cdlemkfid1N 38935 | Lemma for ~ cdlemkfid3N . ... |
cdlemkid1 38936 | Lemma for ~ cdlemkid . (C... |
cdlemkfid2N 38937 | Lemma for ~ cdlemkfid3N . ... |
cdlemkid2 38938 | Lemma for ~ cdlemkid . (C... |
cdlemkfid3N 38939 | TODO: is this useful or sh... |
cdlemky 38940 | Part of proof of Lemma K o... |
cdlemkyu 38941 | Convert between function a... |
cdlemkyuu 38942 | ~ cdlemkyu with some hypot... |
cdlemk11ta 38943 | Part of proof of Lemma K o... |
cdlemk19ylem 38944 | Lemma for ~ cdlemk19y . (... |
cdlemk11tb 38945 | Part of proof of Lemma K o... |
cdlemk19y 38946 | ~ cdlemk19 with simpler hy... |
cdlemkid3N 38947 | Lemma for ~ cdlemkid . (C... |
cdlemkid4 38948 | Lemma for ~ cdlemkid . (C... |
cdlemkid5 38949 | Lemma for ~ cdlemkid . (C... |
cdlemkid 38950 | The value of the tau funct... |
cdlemk35s 38951 | Substitution version of ~ ... |
cdlemk35s-id 38952 | Substitution version of ~ ... |
cdlemk39s 38953 | Substitution version of ~ ... |
cdlemk39s-id 38954 | Substitution version of ~ ... |
cdlemk42 38955 | Part of proof of Lemma K o... |
cdlemk19xlem 38956 | Lemma for ~ cdlemk19x . (... |
cdlemk19x 38957 | ~ cdlemk19 with simpler hy... |
cdlemk42yN 38958 | Part of proof of Lemma K o... |
cdlemk11tc 38959 | Part of proof of Lemma K o... |
cdlemk11t 38960 | Part of proof of Lemma K o... |
cdlemk45 38961 | Part of proof of Lemma K o... |
cdlemk46 38962 | Part of proof of Lemma K o... |
cdlemk47 38963 | Part of proof of Lemma K o... |
cdlemk48 38964 | Part of proof of Lemma K o... |
cdlemk49 38965 | Part of proof of Lemma K o... |
cdlemk50 38966 | Part of proof of Lemma K o... |
cdlemk51 38967 | Part of proof of Lemma K o... |
cdlemk52 38968 | Part of proof of Lemma K o... |
cdlemk53a 38969 | Lemma for ~ cdlemk53 . (C... |
cdlemk53b 38970 | Lemma for ~ cdlemk53 . (C... |
cdlemk53 38971 | Part of proof of Lemma K o... |
cdlemk54 38972 | Part of proof of Lemma K o... |
cdlemk55a 38973 | Lemma for ~ cdlemk55 . (C... |
cdlemk55b 38974 | Lemma for ~ cdlemk55 . (C... |
cdlemk55 38975 | Part of proof of Lemma K o... |
cdlemkyyN 38976 | Part of proof of Lemma K o... |
cdlemk43N 38977 | Part of proof of Lemma K o... |
cdlemk35u 38978 | Substitution version of ~ ... |
cdlemk55u1 38979 | Lemma for ~ cdlemk55u . (... |
cdlemk55u 38980 | Part of proof of Lemma K o... |
cdlemk39u1 38981 | Lemma for ~ cdlemk39u . (... |
cdlemk39u 38982 | Part of proof of Lemma K o... |
cdlemk19u1 38983 | ~ cdlemk19 with simpler hy... |
cdlemk19u 38984 | Part of Lemma K of [Crawle... |
cdlemk56 38985 | Part of Lemma K of [Crawle... |
cdlemk19w 38986 | Use a fixed element to eli... |
cdlemk56w 38987 | Use a fixed element to eli... |
cdlemk 38988 | Lemma K of [Crawley] p. 11... |
tendoex 38989 | Generalization of Lemma K ... |
cdleml1N 38990 | Part of proof of Lemma L o... |
cdleml2N 38991 | Part of proof of Lemma L o... |
cdleml3N 38992 | Part of proof of Lemma L o... |
cdleml4N 38993 | Part of proof of Lemma L o... |
cdleml5N 38994 | Part of proof of Lemma L o... |
cdleml6 38995 | Part of proof of Lemma L o... |
cdleml7 38996 | Part of proof of Lemma L o... |
cdleml8 38997 | Part of proof of Lemma L o... |
cdleml9 38998 | Part of proof of Lemma L o... |
dva1dim 38999 | Two expressions for the 1-... |
dvhb1dimN 39000 | Two expressions for the 1-... |
erng1lem 39001 | Value of the endomorphism ... |
erngdvlem1 39002 | Lemma for ~ eringring . (... |
erngdvlem2N 39003 | Lemma for ~ eringring . (... |
erngdvlem3 39004 | Lemma for ~ eringring . (... |
erngdvlem4 39005 | Lemma for ~ erngdv . (Con... |
eringring 39006 | An endomorphism ring is a ... |
erngdv 39007 | An endomorphism ring is a ... |
erng0g 39008 | The division ring zero of ... |
erng1r 39009 | The division ring unit of ... |
erngdvlem1-rN 39010 | Lemma for ~ eringring . (... |
erngdvlem2-rN 39011 | Lemma for ~ eringring . (... |
erngdvlem3-rN 39012 | Lemma for ~ eringring . (... |
erngdvlem4-rN 39013 | Lemma for ~ erngdv . (Con... |
erngring-rN 39014 | An endomorphism ring is a ... |
erngdv-rN 39015 | An endomorphism ring is a ... |
dvafset 39018 | The constructed partial ve... |
dvaset 39019 | The constructed partial ve... |
dvasca 39020 | The ring base set of the c... |
dvabase 39021 | The ring base set of the c... |
dvafplusg 39022 | Ring addition operation fo... |
dvaplusg 39023 | Ring addition operation fo... |
dvaplusgv 39024 | Ring addition operation fo... |
dvafmulr 39025 | Ring multiplication operat... |
dvamulr 39026 | Ring multiplication operat... |
dvavbase 39027 | The vectors (vector base s... |
dvafvadd 39028 | The vector sum operation f... |
dvavadd 39029 | Ring addition operation fo... |
dvafvsca 39030 | Ring addition operation fo... |
dvavsca 39031 | Ring addition operation fo... |
tendospcl 39032 | Closure of endomorphism sc... |
tendospass 39033 | Associative law for endomo... |
tendospdi1 39034 | Forward distributive law f... |
tendocnv 39035 | Converse of a trace-preser... |
tendospdi2 39036 | Reverse distributive law f... |
tendospcanN 39037 | Cancellation law for trace... |
dvaabl 39038 | The constructed partial ve... |
dvalveclem 39039 | Lemma for ~ dvalvec . (Co... |
dvalvec 39040 | The constructed partial ve... |
dva0g 39041 | The zero vector of partial... |
diaffval 39044 | The partial isomorphism A ... |
diafval 39045 | The partial isomorphism A ... |
diaval 39046 | The partial isomorphism A ... |
diaelval 39047 | Member of the partial isom... |
diafn 39048 | Functionality and domain o... |
diadm 39049 | Domain of the partial isom... |
diaeldm 39050 | Member of domain of the pa... |
diadmclN 39051 | A member of domain of the ... |
diadmleN 39052 | A member of domain of the ... |
dian0 39053 | The value of the partial i... |
dia0eldmN 39054 | The lattice zero belongs t... |
dia1eldmN 39055 | The fiducial hyperplane (t... |
diass 39056 | The value of the partial i... |
diael 39057 | A member of the value of t... |
diatrl 39058 | Trace of a member of the p... |
diaelrnN 39059 | Any value of the partial i... |
dialss 39060 | The value of partial isomo... |
diaord 39061 | The partial isomorphism A ... |
dia11N 39062 | The partial isomorphism A ... |
diaf11N 39063 | The partial isomorphism A ... |
diaclN 39064 | Closure of partial isomorp... |
diacnvclN 39065 | Closure of partial isomorp... |
dia0 39066 | The value of the partial i... |
dia1N 39067 | The value of the partial i... |
dia1elN 39068 | The largest subspace in th... |
diaglbN 39069 | Partial isomorphism A of a... |
diameetN 39070 | Partial isomorphism A of a... |
diainN 39071 | Inverse partial isomorphis... |
diaintclN 39072 | The intersection of partia... |
diasslssN 39073 | The partial isomorphism A ... |
diassdvaN 39074 | The partial isomorphism A ... |
dia1dim 39075 | Two expressions for the 1-... |
dia1dim2 39076 | Two expressions for a 1-di... |
dia1dimid 39077 | A vector (translation) bel... |
dia2dimlem1 39078 | Lemma for ~ dia2dim . Sho... |
dia2dimlem2 39079 | Lemma for ~ dia2dim . Def... |
dia2dimlem3 39080 | Lemma for ~ dia2dim . Def... |
dia2dimlem4 39081 | Lemma for ~ dia2dim . Sho... |
dia2dimlem5 39082 | Lemma for ~ dia2dim . The... |
dia2dimlem6 39083 | Lemma for ~ dia2dim . Eli... |
dia2dimlem7 39084 | Lemma for ~ dia2dim . Eli... |
dia2dimlem8 39085 | Lemma for ~ dia2dim . Eli... |
dia2dimlem9 39086 | Lemma for ~ dia2dim . Eli... |
dia2dimlem10 39087 | Lemma for ~ dia2dim . Con... |
dia2dimlem11 39088 | Lemma for ~ dia2dim . Con... |
dia2dimlem12 39089 | Lemma for ~ dia2dim . Obt... |
dia2dimlem13 39090 | Lemma for ~ dia2dim . Eli... |
dia2dim 39091 | A two-dimensional subspace... |
dvhfset 39094 | The constructed full vecto... |
dvhset 39095 | The constructed full vecto... |
dvhsca 39096 | The ring of scalars of the... |
dvhbase 39097 | The ring base set of the c... |
dvhfplusr 39098 | Ring addition operation fo... |
dvhfmulr 39099 | Ring multiplication operat... |
dvhmulr 39100 | Ring multiplication operat... |
dvhvbase 39101 | The vectors (vector base s... |
dvhelvbasei 39102 | Vector membership in the c... |
dvhvaddcbv 39103 | Change bound variables to ... |
dvhvaddval 39104 | The vector sum operation f... |
dvhfvadd 39105 | The vector sum operation f... |
dvhvadd 39106 | The vector sum operation f... |
dvhopvadd 39107 | The vector sum operation f... |
dvhopvadd2 39108 | The vector sum operation f... |
dvhvaddcl 39109 | Closure of the vector sum ... |
dvhvaddcomN 39110 | Commutativity of vector su... |
dvhvaddass 39111 | Associativity of vector su... |
dvhvscacbv 39112 | Change bound variables to ... |
dvhvscaval 39113 | The scalar product operati... |
dvhfvsca 39114 | Scalar product operation f... |
dvhvsca 39115 | Scalar product operation f... |
dvhopvsca 39116 | Scalar product operation f... |
dvhvscacl 39117 | Closure of the scalar prod... |
tendoinvcl 39118 | Closure of multiplicative ... |
tendolinv 39119 | Left multiplicative invers... |
tendorinv 39120 | Right multiplicative inver... |
dvhgrp 39121 | The full vector space ` U ... |
dvhlveclem 39122 | Lemma for ~ dvhlvec . TOD... |
dvhlvec 39123 | The full vector space ` U ... |
dvhlmod 39124 | The full vector space ` U ... |
dvh0g 39125 | The zero vector of vector ... |
dvheveccl 39126 | Properties of a unit vecto... |
dvhopclN 39127 | Closure of a ` DVecH ` vec... |
dvhopaddN 39128 | Sum of ` DVecH ` vectors e... |
dvhopspN 39129 | Scalar product of ` DVecH ... |
dvhopN 39130 | Decompose a ` DVecH ` vect... |
dvhopellsm 39131 | Ordered pair membership in... |
cdlemm10N 39132 | The image of the map ` G `... |
docaffvalN 39135 | Subspace orthocomplement f... |
docafvalN 39136 | Subspace orthocomplement f... |
docavalN 39137 | Subspace orthocomplement f... |
docaclN 39138 | Closure of subspace orthoc... |
diaocN 39139 | Value of partial isomorphi... |
doca2N 39140 | Double orthocomplement of ... |
doca3N 39141 | Double orthocomplement of ... |
dvadiaN 39142 | Any closed subspace is a m... |
diarnN 39143 | Partial isomorphism A maps... |
diaf1oN 39144 | The partial isomorphism A ... |
djaffvalN 39147 | Subspace join for ` DVecA ... |
djafvalN 39148 | Subspace join for ` DVecA ... |
djavalN 39149 | Subspace join for ` DVecA ... |
djaclN 39150 | Closure of subspace join f... |
djajN 39151 | Transfer lattice join to `... |
dibffval 39154 | The partial isomorphism B ... |
dibfval 39155 | The partial isomorphism B ... |
dibval 39156 | The partial isomorphism B ... |
dibopelvalN 39157 | Member of the partial isom... |
dibval2 39158 | Value of the partial isomo... |
dibopelval2 39159 | Member of the partial isom... |
dibval3N 39160 | Value of the partial isomo... |
dibelval3 39161 | Member of the partial isom... |
dibopelval3 39162 | Member of the partial isom... |
dibelval1st 39163 | Membership in value of the... |
dibelval1st1 39164 | Membership in value of the... |
dibelval1st2N 39165 | Membership in value of the... |
dibelval2nd 39166 | Membership in value of the... |
dibn0 39167 | The value of the partial i... |
dibfna 39168 | Functionality and domain o... |
dibdiadm 39169 | Domain of the partial isom... |
dibfnN 39170 | Functionality and domain o... |
dibdmN 39171 | Domain of the partial isom... |
dibeldmN 39172 | Member of domain of the pa... |
dibord 39173 | The isomorphism B for a la... |
dib11N 39174 | The isomorphism B for a la... |
dibf11N 39175 | The partial isomorphism A ... |
dibclN 39176 | Closure of partial isomorp... |
dibvalrel 39177 | The value of partial isomo... |
dib0 39178 | The value of partial isomo... |
dib1dim 39179 | Two expressions for the 1-... |
dibglbN 39180 | Partial isomorphism B of a... |
dibintclN 39181 | The intersection of partia... |
dib1dim2 39182 | Two expressions for a 1-di... |
dibss 39183 | The partial isomorphism B ... |
diblss 39184 | The value of partial isomo... |
diblsmopel 39185 | Membership in subspace sum... |
dicffval 39188 | The partial isomorphism C ... |
dicfval 39189 | The partial isomorphism C ... |
dicval 39190 | The partial isomorphism C ... |
dicopelval 39191 | Membership in value of the... |
dicelvalN 39192 | Membership in value of the... |
dicval2 39193 | The partial isomorphism C ... |
dicelval3 39194 | Member of the partial isom... |
dicopelval2 39195 | Membership in value of the... |
dicelval2N 39196 | Membership in value of the... |
dicfnN 39197 | Functionality and domain o... |
dicdmN 39198 | Domain of the partial isom... |
dicvalrelN 39199 | The value of partial isomo... |
dicssdvh 39200 | The partial isomorphism C ... |
dicelval1sta 39201 | Membership in value of the... |
dicelval1stN 39202 | Membership in value of the... |
dicelval2nd 39203 | Membership in value of the... |
dicvaddcl 39204 | Membership in value of the... |
dicvscacl 39205 | Membership in value of the... |
dicn0 39206 | The value of the partial i... |
diclss 39207 | The value of partial isomo... |
diclspsn 39208 | The value of isomorphism C... |
cdlemn2 39209 | Part of proof of Lemma N o... |
cdlemn2a 39210 | Part of proof of Lemma N o... |
cdlemn3 39211 | Part of proof of Lemma N o... |
cdlemn4 39212 | Part of proof of Lemma N o... |
cdlemn4a 39213 | Part of proof of Lemma N o... |
cdlemn5pre 39214 | Part of proof of Lemma N o... |
cdlemn5 39215 | Part of proof of Lemma N o... |
cdlemn6 39216 | Part of proof of Lemma N o... |
cdlemn7 39217 | Part of proof of Lemma N o... |
cdlemn8 39218 | Part of proof of Lemma N o... |
cdlemn9 39219 | Part of proof of Lemma N o... |
cdlemn10 39220 | Part of proof of Lemma N o... |
cdlemn11a 39221 | Part of proof of Lemma N o... |
cdlemn11b 39222 | Part of proof of Lemma N o... |
cdlemn11c 39223 | Part of proof of Lemma N o... |
cdlemn11pre 39224 | Part of proof of Lemma N o... |
cdlemn11 39225 | Part of proof of Lemma N o... |
cdlemn 39226 | Lemma N of [Crawley] p. 12... |
dihordlem6 39227 | Part of proof of Lemma N o... |
dihordlem7 39228 | Part of proof of Lemma N o... |
dihordlem7b 39229 | Part of proof of Lemma N o... |
dihjustlem 39230 | Part of proof after Lemma ... |
dihjust 39231 | Part of proof after Lemma ... |
dihord1 39232 | Part of proof after Lemma ... |
dihord2a 39233 | Part of proof after Lemma ... |
dihord2b 39234 | Part of proof after Lemma ... |
dihord2cN 39235 | Part of proof after Lemma ... |
dihord11b 39236 | Part of proof after Lemma ... |
dihord10 39237 | Part of proof after Lemma ... |
dihord11c 39238 | Part of proof after Lemma ... |
dihord2pre 39239 | Part of proof after Lemma ... |
dihord2pre2 39240 | Part of proof after Lemma ... |
dihord2 39241 | Part of proof after Lemma ... |
dihffval 39244 | The isomorphism H for a la... |
dihfval 39245 | Isomorphism H for a lattic... |
dihval 39246 | Value of isomorphism H for... |
dihvalc 39247 | Value of isomorphism H for... |
dihlsscpre 39248 | Closure of isomorphism H f... |
dihvalcqpre 39249 | Value of isomorphism H for... |
dihvalcq 39250 | Value of isomorphism H for... |
dihvalb 39251 | Value of isomorphism H for... |
dihopelvalbN 39252 | Ordered pair member of the... |
dihvalcqat 39253 | Value of isomorphism H for... |
dih1dimb 39254 | Two expressions for a 1-di... |
dih1dimb2 39255 | Isomorphism H at an atom u... |
dih1dimc 39256 | Isomorphism H at an atom n... |
dib2dim 39257 | Extend ~ dia2dim to partia... |
dih2dimb 39258 | Extend ~ dib2dim to isomor... |
dih2dimbALTN 39259 | Extend ~ dia2dim to isomor... |
dihopelvalcqat 39260 | Ordered pair member of the... |
dihvalcq2 39261 | Value of isomorphism H for... |
dihopelvalcpre 39262 | Member of value of isomorp... |
dihopelvalc 39263 | Member of value of isomorp... |
dihlss 39264 | The value of isomorphism H... |
dihss 39265 | The value of isomorphism H... |
dihssxp 39266 | An isomorphism H value is ... |
dihopcl 39267 | Closure of an ordered pair... |
xihopellsmN 39268 | Ordered pair membership in... |
dihopellsm 39269 | Ordered pair membership in... |
dihord6apre 39270 | Part of proof that isomorp... |
dihord3 39271 | The isomorphism H for a la... |
dihord4 39272 | The isomorphism H for a la... |
dihord5b 39273 | Part of proof that isomorp... |
dihord6b 39274 | Part of proof that isomorp... |
dihord6a 39275 | Part of proof that isomorp... |
dihord5apre 39276 | Part of proof that isomorp... |
dihord5a 39277 | Part of proof that isomorp... |
dihord 39278 | The isomorphism H is order... |
dih11 39279 | The isomorphism H is one-t... |
dihf11lem 39280 | Functionality of the isomo... |
dihf11 39281 | The isomorphism H for a la... |
dihfn 39282 | Functionality and domain o... |
dihdm 39283 | Domain of isomorphism H. (... |
dihcl 39284 | Closure of isomorphism H. ... |
dihcnvcl 39285 | Closure of isomorphism H c... |
dihcnvid1 39286 | The converse isomorphism o... |
dihcnvid2 39287 | The isomorphism of a conve... |
dihcnvord 39288 | Ordering property for conv... |
dihcnv11 39289 | The converse of isomorphis... |
dihsslss 39290 | The isomorphism H maps to ... |
dihrnlss 39291 | The isomorphism H maps to ... |
dihrnss 39292 | The isomorphism H maps to ... |
dihvalrel 39293 | The value of isomorphism H... |
dih0 39294 | The value of isomorphism H... |
dih0bN 39295 | A lattice element is zero ... |
dih0vbN 39296 | A vector is zero iff its s... |
dih0cnv 39297 | The isomorphism H converse... |
dih0rn 39298 | The zero subspace belongs ... |
dih0sb 39299 | A subspace is zero iff the... |
dih1 39300 | The value of isomorphism H... |
dih1rn 39301 | The full vector space belo... |
dih1cnv 39302 | The isomorphism H converse... |
dihwN 39303 | Value of isomorphism H at ... |
dihmeetlem1N 39304 | Isomorphism H of a conjunc... |
dihglblem5apreN 39305 | A conjunction property of ... |
dihglblem5aN 39306 | A conjunction property of ... |
dihglblem2aN 39307 | Lemma for isomorphism H of... |
dihglblem2N 39308 | The GLB of a set of lattic... |
dihglblem3N 39309 | Isomorphism H of a lattice... |
dihglblem3aN 39310 | Isomorphism H of a lattice... |
dihglblem4 39311 | Isomorphism H of a lattice... |
dihglblem5 39312 | Isomorphism H of a lattice... |
dihmeetlem2N 39313 | Isomorphism H of a conjunc... |
dihglbcpreN 39314 | Isomorphism H of a lattice... |
dihglbcN 39315 | Isomorphism H of a lattice... |
dihmeetcN 39316 | Isomorphism H of a lattice... |
dihmeetbN 39317 | Isomorphism H of a lattice... |
dihmeetbclemN 39318 | Lemma for isomorphism H of... |
dihmeetlem3N 39319 | Lemma for isomorphism H of... |
dihmeetlem4preN 39320 | Lemma for isomorphism H of... |
dihmeetlem4N 39321 | Lemma for isomorphism H of... |
dihmeetlem5 39322 | Part of proof that isomorp... |
dihmeetlem6 39323 | Lemma for isomorphism H of... |
dihmeetlem7N 39324 | Lemma for isomorphism H of... |
dihjatc1 39325 | Lemma for isomorphism H of... |
dihjatc2N 39326 | Isomorphism H of join with... |
dihjatc3 39327 | Isomorphism H of join with... |
dihmeetlem8N 39328 | Lemma for isomorphism H of... |
dihmeetlem9N 39329 | Lemma for isomorphism H of... |
dihmeetlem10N 39330 | Lemma for isomorphism H of... |
dihmeetlem11N 39331 | Lemma for isomorphism H of... |
dihmeetlem12N 39332 | Lemma for isomorphism H of... |
dihmeetlem13N 39333 | Lemma for isomorphism H of... |
dihmeetlem14N 39334 | Lemma for isomorphism H of... |
dihmeetlem15N 39335 | Lemma for isomorphism H of... |
dihmeetlem16N 39336 | Lemma for isomorphism H of... |
dihmeetlem17N 39337 | Lemma for isomorphism H of... |
dihmeetlem18N 39338 | Lemma for isomorphism H of... |
dihmeetlem19N 39339 | Lemma for isomorphism H of... |
dihmeetlem20N 39340 | Lemma for isomorphism H of... |
dihmeetALTN 39341 | Isomorphism H of a lattice... |
dih1dimatlem0 39342 | Lemma for ~ dih1dimat . (... |
dih1dimatlem 39343 | Lemma for ~ dih1dimat . (... |
dih1dimat 39344 | Any 1-dimensional subspace... |
dihlsprn 39345 | The span of a vector belon... |
dihlspsnssN 39346 | A subspace included in a 1... |
dihlspsnat 39347 | The inverse isomorphism H ... |
dihatlat 39348 | The isomorphism H of an at... |
dihat 39349 | There exists at least one ... |
dihpN 39350 | The value of isomorphism H... |
dihlatat 39351 | The reverse isomorphism H ... |
dihatexv 39352 | There is a nonzero vector ... |
dihatexv2 39353 | There is a nonzero vector ... |
dihglblem6 39354 | Isomorphism H of a lattice... |
dihglb 39355 | Isomorphism H of a lattice... |
dihglb2 39356 | Isomorphism H of a lattice... |
dihmeet 39357 | Isomorphism H of a lattice... |
dihintcl 39358 | The intersection of closed... |
dihmeetcl 39359 | Closure of closed subspace... |
dihmeet2 39360 | Reverse isomorphism H of a... |
dochffval 39363 | Subspace orthocomplement f... |
dochfval 39364 | Subspace orthocomplement f... |
dochval 39365 | Subspace orthocomplement f... |
dochval2 39366 | Subspace orthocomplement f... |
dochcl 39367 | Closure of subspace orthoc... |
dochlss 39368 | A subspace orthocomplement... |
dochssv 39369 | A subspace orthocomplement... |
dochfN 39370 | Domain and codomain of the... |
dochvalr 39371 | Orthocomplement of a close... |
doch0 39372 | Orthocomplement of the zer... |
doch1 39373 | Orthocomplement of the uni... |
dochoc0 39374 | The zero subspace is close... |
dochoc1 39375 | The unit subspace (all vec... |
dochvalr2 39376 | Orthocomplement of a close... |
dochvalr3 39377 | Orthocomplement of a close... |
doch2val2 39378 | Double orthocomplement for... |
dochss 39379 | Subset law for orthocomple... |
dochocss 39380 | Double negative law for or... |
dochoc 39381 | Double negative law for or... |
dochsscl 39382 | If a set of vectors is inc... |
dochoccl 39383 | A set of vectors is closed... |
dochord 39384 | Ordering law for orthocomp... |
dochord2N 39385 | Ordering law for orthocomp... |
dochord3 39386 | Ordering law for orthocomp... |
doch11 39387 | Orthocomplement is one-to-... |
dochsordN 39388 | Strict ordering law for or... |
dochn0nv 39389 | An orthocomplement is nonz... |
dihoml4c 39390 | Version of ~ dihoml4 with ... |
dihoml4 39391 | Orthomodular law for const... |
dochspss 39392 | The span of a set of vecto... |
dochocsp 39393 | The span of an orthocomple... |
dochspocN 39394 | The span of an orthocomple... |
dochocsn 39395 | The double orthocomplement... |
dochsncom 39396 | Swap vectors in an orthoco... |
dochsat 39397 | The double orthocomplement... |
dochshpncl 39398 | If a hyperplane is not clo... |
dochlkr 39399 | Equivalent conditions for ... |
dochkrshp 39400 | The closure of a kernel is... |
dochkrshp2 39401 | Properties of the closure ... |
dochkrshp3 39402 | Properties of the closure ... |
dochkrshp4 39403 | Properties of the closure ... |
dochdmj1 39404 | De Morgan-like law for sub... |
dochnoncon 39405 | Law of noncontradiction. ... |
dochnel2 39406 | A nonzero member of a subs... |
dochnel 39407 | A nonzero vector doesn't b... |
djhffval 39410 | Subspace join for ` DVecH ... |
djhfval 39411 | Subspace join for ` DVecH ... |
djhval 39412 | Subspace join for ` DVecH ... |
djhval2 39413 | Value of subspace join for... |
djhcl 39414 | Closure of subspace join f... |
djhlj 39415 | Transfer lattice join to `... |
djhljjN 39416 | Lattice join in terms of `... |
djhjlj 39417 | ` DVecH ` vector space clo... |
djhj 39418 | ` DVecH ` vector space clo... |
djhcom 39419 | Subspace join commutes. (... |
djhspss 39420 | Subspace span of union is ... |
djhsumss 39421 | Subspace sum is a subset o... |
dihsumssj 39422 | The subspace sum of two is... |
djhunssN 39423 | Subspace union is a subset... |
dochdmm1 39424 | De Morgan-like law for clo... |
djhexmid 39425 | Excluded middle property o... |
djh01 39426 | Closed subspace join with ... |
djh02 39427 | Closed subspace join with ... |
djhlsmcl 39428 | A closed subspace sum equa... |
djhcvat42 39429 | A covering property. ( ~ ... |
dihjatb 39430 | Isomorphism H of lattice j... |
dihjatc 39431 | Isomorphism H of lattice j... |
dihjatcclem1 39432 | Lemma for isomorphism H of... |
dihjatcclem2 39433 | Lemma for isomorphism H of... |
dihjatcclem3 39434 | Lemma for ~ dihjatcc . (C... |
dihjatcclem4 39435 | Lemma for isomorphism H of... |
dihjatcc 39436 | Isomorphism H of lattice j... |
dihjat 39437 | Isomorphism H of lattice j... |
dihprrnlem1N 39438 | Lemma for ~ dihprrn , show... |
dihprrnlem2 39439 | Lemma for ~ dihprrn . (Co... |
dihprrn 39440 | The span of a vector pair ... |
djhlsmat 39441 | The sum of two subspace at... |
dihjat1lem 39442 | Subspace sum of a closed s... |
dihjat1 39443 | Subspace sum of a closed s... |
dihsmsprn 39444 | Subspace sum of a closed s... |
dihjat2 39445 | The subspace sum of a clos... |
dihjat3 39446 | Isomorphism H of lattice j... |
dihjat4 39447 | Transfer the subspace sum ... |
dihjat6 39448 | Transfer the subspace sum ... |
dihsmsnrn 39449 | The subspace sum of two si... |
dihsmatrn 39450 | The subspace sum of a clos... |
dihjat5N 39451 | Transfer lattice join with... |
dvh4dimat 39452 | There is an atom that is o... |
dvh3dimatN 39453 | There is an atom that is o... |
dvh2dimatN 39454 | Given an atom, there exist... |
dvh1dimat 39455 | There exists an atom. (Co... |
dvh1dim 39456 | There exists a nonzero vec... |
dvh4dimlem 39457 | Lemma for ~ dvh4dimN . (C... |
dvhdimlem 39458 | Lemma for ~ dvh2dim and ~ ... |
dvh2dim 39459 | There is a vector that is ... |
dvh3dim 39460 | There is a vector that is ... |
dvh4dimN 39461 | There is a vector that is ... |
dvh3dim2 39462 | There is a vector that is ... |
dvh3dim3N 39463 | There is a vector that is ... |
dochsnnz 39464 | The orthocomplement of a s... |
dochsatshp 39465 | The orthocomplement of a s... |
dochsatshpb 39466 | The orthocomplement of a s... |
dochsnshp 39467 | The orthocomplement of a n... |
dochshpsat 39468 | A hyperplane is closed iff... |
dochkrsat 39469 | The orthocomplement of a k... |
dochkrsat2 39470 | The orthocomplement of a k... |
dochsat0 39471 | The orthocomplement of a k... |
dochkrsm 39472 | The subspace sum of a clos... |
dochexmidat 39473 | Special case of excluded m... |
dochexmidlem1 39474 | Lemma for ~ dochexmid . H... |
dochexmidlem2 39475 | Lemma for ~ dochexmid . (... |
dochexmidlem3 39476 | Lemma for ~ dochexmid . U... |
dochexmidlem4 39477 | Lemma for ~ dochexmid . (... |
dochexmidlem5 39478 | Lemma for ~ dochexmid . (... |
dochexmidlem6 39479 | Lemma for ~ dochexmid . (... |
dochexmidlem7 39480 | Lemma for ~ dochexmid . C... |
dochexmidlem8 39481 | Lemma for ~ dochexmid . T... |
dochexmid 39482 | Excluded middle law for cl... |
dochsnkrlem1 39483 | Lemma for ~ dochsnkr . (C... |
dochsnkrlem2 39484 | Lemma for ~ dochsnkr . (C... |
dochsnkrlem3 39485 | Lemma for ~ dochsnkr . (C... |
dochsnkr 39486 | A (closed) kernel expresse... |
dochsnkr2 39487 | Kernel of the explicit fun... |
dochsnkr2cl 39488 | The ` X ` determining func... |
dochflcl 39489 | Closure of the explicit fu... |
dochfl1 39490 | The value of the explicit ... |
dochfln0 39491 | The value of a functional ... |
dochkr1 39492 | A nonzero functional has a... |
dochkr1OLDN 39493 | A nonzero functional has a... |
lpolsetN 39496 | The set of polarities of a... |
islpolN 39497 | The predicate "is a polari... |
islpoldN 39498 | Properties that determine ... |
lpolfN 39499 | Functionality of a polarit... |
lpolvN 39500 | The polarity of the whole ... |
lpolconN 39501 | Contraposition property of... |
lpolsatN 39502 | The polarity of an atomic ... |
lpolpolsatN 39503 | Property of a polarity. (... |
dochpolN 39504 | The subspace orthocompleme... |
lcfl1lem 39505 | Property of a functional w... |
lcfl1 39506 | Property of a functional w... |
lcfl2 39507 | Property of a functional w... |
lcfl3 39508 | Property of a functional w... |
lcfl4N 39509 | Property of a functional w... |
lcfl5 39510 | Property of a functional w... |
lcfl5a 39511 | Property of a functional w... |
lcfl6lem 39512 | Lemma for ~ lcfl6 . A fun... |
lcfl7lem 39513 | Lemma for ~ lcfl7N . If t... |
lcfl6 39514 | Property of a functional w... |
lcfl7N 39515 | Property of a functional w... |
lcfl8 39516 | Property of a functional w... |
lcfl8a 39517 | Property of a functional w... |
lcfl8b 39518 | Property of a nonzero func... |
lcfl9a 39519 | Property implying that a f... |
lclkrlem1 39520 | The set of functionals hav... |
lclkrlem2a 39521 | Lemma for ~ lclkr . Use ~... |
lclkrlem2b 39522 | Lemma for ~ lclkr . (Cont... |
lclkrlem2c 39523 | Lemma for ~ lclkr . (Cont... |
lclkrlem2d 39524 | Lemma for ~ lclkr . (Cont... |
lclkrlem2e 39525 | Lemma for ~ lclkr . The k... |
lclkrlem2f 39526 | Lemma for ~ lclkr . Const... |
lclkrlem2g 39527 | Lemma for ~ lclkr . Compa... |
lclkrlem2h 39528 | Lemma for ~ lclkr . Elimi... |
lclkrlem2i 39529 | Lemma for ~ lclkr . Elimi... |
lclkrlem2j 39530 | Lemma for ~ lclkr . Kerne... |
lclkrlem2k 39531 | Lemma for ~ lclkr . Kerne... |
lclkrlem2l 39532 | Lemma for ~ lclkr . Elimi... |
lclkrlem2m 39533 | Lemma for ~ lclkr . Const... |
lclkrlem2n 39534 | Lemma for ~ lclkr . (Cont... |
lclkrlem2o 39535 | Lemma for ~ lclkr . When ... |
lclkrlem2p 39536 | Lemma for ~ lclkr . When ... |
lclkrlem2q 39537 | Lemma for ~ lclkr . The s... |
lclkrlem2r 39538 | Lemma for ~ lclkr . When ... |
lclkrlem2s 39539 | Lemma for ~ lclkr . Thus,... |
lclkrlem2t 39540 | Lemma for ~ lclkr . We el... |
lclkrlem2u 39541 | Lemma for ~ lclkr . ~ lclk... |
lclkrlem2v 39542 | Lemma for ~ lclkr . When ... |
lclkrlem2w 39543 | Lemma for ~ lclkr . This ... |
lclkrlem2x 39544 | Lemma for ~ lclkr . Elimi... |
lclkrlem2y 39545 | Lemma for ~ lclkr . Resta... |
lclkrlem2 39546 | The set of functionals hav... |
lclkr 39547 | The set of functionals wit... |
lcfls1lem 39548 | Property of a functional w... |
lcfls1N 39549 | Property of a functional w... |
lcfls1c 39550 | Property of a functional w... |
lclkrslem1 39551 | The set of functionals hav... |
lclkrslem2 39552 | The set of functionals hav... |
lclkrs 39553 | The set of functionals hav... |
lclkrs2 39554 | The set of functionals wit... |
lcfrvalsnN 39555 | Reconstruction from the du... |
lcfrlem1 39556 | Lemma for ~ lcfr . Note t... |
lcfrlem2 39557 | Lemma for ~ lcfr . (Contr... |
lcfrlem3 39558 | Lemma for ~ lcfr . (Contr... |
lcfrlem4 39559 | Lemma for ~ lcfr . (Contr... |
lcfrlem5 39560 | Lemma for ~ lcfr . The se... |
lcfrlem6 39561 | Lemma for ~ lcfr . Closur... |
lcfrlem7 39562 | Lemma for ~ lcfr . Closur... |
lcfrlem8 39563 | Lemma for ~ lcf1o and ~ lc... |
lcfrlem9 39564 | Lemma for ~ lcf1o . (This... |
lcf1o 39565 | Define a function ` J ` th... |
lcfrlem10 39566 | Lemma for ~ lcfr . (Contr... |
lcfrlem11 39567 | Lemma for ~ lcfr . (Contr... |
lcfrlem12N 39568 | Lemma for ~ lcfr . (Contr... |
lcfrlem13 39569 | Lemma for ~ lcfr . (Contr... |
lcfrlem14 39570 | Lemma for ~ lcfr . (Contr... |
lcfrlem15 39571 | Lemma for ~ lcfr . (Contr... |
lcfrlem16 39572 | Lemma for ~ lcfr . (Contr... |
lcfrlem17 39573 | Lemma for ~ lcfr . Condit... |
lcfrlem18 39574 | Lemma for ~ lcfr . (Contr... |
lcfrlem19 39575 | Lemma for ~ lcfr . (Contr... |
lcfrlem20 39576 | Lemma for ~ lcfr . (Contr... |
lcfrlem21 39577 | Lemma for ~ lcfr . (Contr... |
lcfrlem22 39578 | Lemma for ~ lcfr . (Contr... |
lcfrlem23 39579 | Lemma for ~ lcfr . TODO: ... |
lcfrlem24 39580 | Lemma for ~ lcfr . (Contr... |
lcfrlem25 39581 | Lemma for ~ lcfr . Specia... |
lcfrlem26 39582 | Lemma for ~ lcfr . Specia... |
lcfrlem27 39583 | Lemma for ~ lcfr . Specia... |
lcfrlem28 39584 | Lemma for ~ lcfr . TODO: ... |
lcfrlem29 39585 | Lemma for ~ lcfr . (Contr... |
lcfrlem30 39586 | Lemma for ~ lcfr . (Contr... |
lcfrlem31 39587 | Lemma for ~ lcfr . (Contr... |
lcfrlem32 39588 | Lemma for ~ lcfr . (Contr... |
lcfrlem33 39589 | Lemma for ~ lcfr . (Contr... |
lcfrlem34 39590 | Lemma for ~ lcfr . (Contr... |
lcfrlem35 39591 | Lemma for ~ lcfr . (Contr... |
lcfrlem36 39592 | Lemma for ~ lcfr . (Contr... |
lcfrlem37 39593 | Lemma for ~ lcfr . (Contr... |
lcfrlem38 39594 | Lemma for ~ lcfr . Combin... |
lcfrlem39 39595 | Lemma for ~ lcfr . Elimin... |
lcfrlem40 39596 | Lemma for ~ lcfr . Elimin... |
lcfrlem41 39597 | Lemma for ~ lcfr . Elimin... |
lcfrlem42 39598 | Lemma for ~ lcfr . Elimin... |
lcfr 39599 | Reconstruction of a subspa... |
lcdfval 39602 | Dual vector space of funct... |
lcdval 39603 | Dual vector space of funct... |
lcdval2 39604 | Dual vector space of funct... |
lcdlvec 39605 | The dual vector space of f... |
lcdlmod 39606 | The dual vector space of f... |
lcdvbase 39607 | Vector base set of a dual ... |
lcdvbasess 39608 | The vector base set of the... |
lcdvbaselfl 39609 | A vector in the base set o... |
lcdvbasecl 39610 | Closure of the value of a ... |
lcdvadd 39611 | Vector addition for the cl... |
lcdvaddval 39612 | The value of the value of ... |
lcdsca 39613 | The ring of scalars of the... |
lcdsbase 39614 | Base set of scalar ring fo... |
lcdsadd 39615 | Scalar addition for the cl... |
lcdsmul 39616 | Scalar multiplication for ... |
lcdvs 39617 | Scalar product for the clo... |
lcdvsval 39618 | Value of scalar product op... |
lcdvscl 39619 | The scalar product operati... |
lcdlssvscl 39620 | Closure of scalar product ... |
lcdvsass 39621 | Associative law for scalar... |
lcd0 39622 | The zero scalar of the clo... |
lcd1 39623 | The unit scalar of the clo... |
lcdneg 39624 | The unit scalar of the clo... |
lcd0v 39625 | The zero functional in the... |
lcd0v2 39626 | The zero functional in the... |
lcd0vvalN 39627 | Value of the zero function... |
lcd0vcl 39628 | Closure of the zero functi... |
lcd0vs 39629 | A scalar zero times a func... |
lcdvs0N 39630 | A scalar times the zero fu... |
lcdvsub 39631 | The value of vector subtra... |
lcdvsubval 39632 | The value of the value of ... |
lcdlss 39633 | Subspaces of a dual vector... |
lcdlss2N 39634 | Subspaces of a dual vector... |
lcdlsp 39635 | Span in the set of functio... |
lcdlkreqN 39636 | Colinear functionals have ... |
lcdlkreq2N 39637 | Colinear functionals have ... |
mapdffval 39640 | Projectivity from vector s... |
mapdfval 39641 | Projectivity from vector s... |
mapdval 39642 | Value of projectivity from... |
mapdvalc 39643 | Value of projectivity from... |
mapdval2N 39644 | Value of projectivity from... |
mapdval3N 39645 | Value of projectivity from... |
mapdval4N 39646 | Value of projectivity from... |
mapdval5N 39647 | Value of projectivity from... |
mapdordlem1a 39648 | Lemma for ~ mapdord . (Co... |
mapdordlem1bN 39649 | Lemma for ~ mapdord . (Co... |
mapdordlem1 39650 | Lemma for ~ mapdord . (Co... |
mapdordlem2 39651 | Lemma for ~ mapdord . Ord... |
mapdord 39652 | Ordering property of the m... |
mapd11 39653 | The map defined by ~ df-ma... |
mapddlssN 39654 | The mapping of a subspace ... |
mapdsn 39655 | Value of the map defined b... |
mapdsn2 39656 | Value of the map defined b... |
mapdsn3 39657 | Value of the map defined b... |
mapd1dim2lem1N 39658 | Value of the map defined b... |
mapdrvallem2 39659 | Lemma for ~ mapdrval . TO... |
mapdrvallem3 39660 | Lemma for ~ mapdrval . (C... |
mapdrval 39661 | Given a dual subspace ` R ... |
mapd1o 39662 | The map defined by ~ df-ma... |
mapdrn 39663 | Range of the map defined b... |
mapdunirnN 39664 | Union of the range of the ... |
mapdrn2 39665 | Range of the map defined b... |
mapdcnvcl 39666 | Closure of the converse of... |
mapdcl 39667 | Closure the value of the m... |
mapdcnvid1N 39668 | Converse of the value of t... |
mapdsord 39669 | Strong ordering property o... |
mapdcl2 39670 | The mapping of a subspace ... |
mapdcnvid2 39671 | Value of the converse of t... |
mapdcnvordN 39672 | Ordering property of the c... |
mapdcnv11N 39673 | The converse of the map de... |
mapdcv 39674 | Covering property of the c... |
mapdincl 39675 | Closure of dual subspace i... |
mapdin 39676 | Subspace intersection is p... |
mapdlsmcl 39677 | Closure of dual subspace s... |
mapdlsm 39678 | Subspace sum is preserved ... |
mapd0 39679 | Projectivity map of the ze... |
mapdcnvatN 39680 | Atoms are preserved by the... |
mapdat 39681 | Atoms are preserved by the... |
mapdspex 39682 | The map of a span equals t... |
mapdn0 39683 | Transfer nonzero property ... |
mapdncol 39684 | Transfer non-colinearity f... |
mapdindp 39685 | Transfer (part of) vector ... |
mapdpglem1 39686 | Lemma for ~ mapdpg . Baer... |
mapdpglem2 39687 | Lemma for ~ mapdpg . Baer... |
mapdpglem2a 39688 | Lemma for ~ mapdpg . (Con... |
mapdpglem3 39689 | Lemma for ~ mapdpg . Baer... |
mapdpglem4N 39690 | Lemma for ~ mapdpg . (Con... |
mapdpglem5N 39691 | Lemma for ~ mapdpg . (Con... |
mapdpglem6 39692 | Lemma for ~ mapdpg . Baer... |
mapdpglem8 39693 | Lemma for ~ mapdpg . Baer... |
mapdpglem9 39694 | Lemma for ~ mapdpg . Baer... |
mapdpglem10 39695 | Lemma for ~ mapdpg . Baer... |
mapdpglem11 39696 | Lemma for ~ mapdpg . (Con... |
mapdpglem12 39697 | Lemma for ~ mapdpg . TODO... |
mapdpglem13 39698 | Lemma for ~ mapdpg . (Con... |
mapdpglem14 39699 | Lemma for ~ mapdpg . (Con... |
mapdpglem15 39700 | Lemma for ~ mapdpg . (Con... |
mapdpglem16 39701 | Lemma for ~ mapdpg . Baer... |
mapdpglem17N 39702 | Lemma for ~ mapdpg . Baer... |
mapdpglem18 39703 | Lemma for ~ mapdpg . Baer... |
mapdpglem19 39704 | Lemma for ~ mapdpg . Baer... |
mapdpglem20 39705 | Lemma for ~ mapdpg . Baer... |
mapdpglem21 39706 | Lemma for ~ mapdpg . (Con... |
mapdpglem22 39707 | Lemma for ~ mapdpg . Baer... |
mapdpglem23 39708 | Lemma for ~ mapdpg . Baer... |
mapdpglem30a 39709 | Lemma for ~ mapdpg . (Con... |
mapdpglem30b 39710 | Lemma for ~ mapdpg . (Con... |
mapdpglem25 39711 | Lemma for ~ mapdpg . Baer... |
mapdpglem26 39712 | Lemma for ~ mapdpg . Baer... |
mapdpglem27 39713 | Lemma for ~ mapdpg . Baer... |
mapdpglem29 39714 | Lemma for ~ mapdpg . Baer... |
mapdpglem28 39715 | Lemma for ~ mapdpg . Baer... |
mapdpglem30 39716 | Lemma for ~ mapdpg . Baer... |
mapdpglem31 39717 | Lemma for ~ mapdpg . Baer... |
mapdpglem24 39718 | Lemma for ~ mapdpg . Exis... |
mapdpglem32 39719 | Lemma for ~ mapdpg . Uniq... |
mapdpg 39720 | Part 1 of proof of the fir... |
baerlem3lem1 39721 | Lemma for ~ baerlem3 . (C... |
baerlem5alem1 39722 | Lemma for ~ baerlem5a . (... |
baerlem5blem1 39723 | Lemma for ~ baerlem5b . (... |
baerlem3lem2 39724 | Lemma for ~ baerlem3 . (C... |
baerlem5alem2 39725 | Lemma for ~ baerlem5a . (... |
baerlem5blem2 39726 | Lemma for ~ baerlem5b . (... |
baerlem3 39727 | An equality that holds whe... |
baerlem5a 39728 | An equality that holds whe... |
baerlem5b 39729 | An equality that holds whe... |
baerlem5amN 39730 | An equality that holds whe... |
baerlem5bmN 39731 | An equality that holds whe... |
baerlem5abmN 39732 | An equality that holds whe... |
mapdindp0 39733 | Vector independence lemma.... |
mapdindp1 39734 | Vector independence lemma.... |
mapdindp2 39735 | Vector independence lemma.... |
mapdindp3 39736 | Vector independence lemma.... |
mapdindp4 39737 | Vector independence lemma.... |
mapdhval 39738 | Lemmma for ~~? mapdh . (C... |
mapdhval0 39739 | Lemmma for ~~? mapdh . (C... |
mapdhval2 39740 | Lemmma for ~~? mapdh . (C... |
mapdhcl 39741 | Lemmma for ~~? mapdh . (C... |
mapdheq 39742 | Lemmma for ~~? mapdh . Th... |
mapdheq2 39743 | Lemmma for ~~? mapdh . On... |
mapdheq2biN 39744 | Lemmma for ~~? mapdh . Pa... |
mapdheq4lem 39745 | Lemma for ~ mapdheq4 . Pa... |
mapdheq4 39746 | Lemma for ~~? mapdh . Par... |
mapdh6lem1N 39747 | Lemma for ~ mapdh6N . Par... |
mapdh6lem2N 39748 | Lemma for ~ mapdh6N . Par... |
mapdh6aN 39749 | Lemma for ~ mapdh6N . Par... |
mapdh6b0N 39750 | Lemmma for ~ mapdh6N . (C... |
mapdh6bN 39751 | Lemmma for ~ mapdh6N . (C... |
mapdh6cN 39752 | Lemmma for ~ mapdh6N . (C... |
mapdh6dN 39753 | Lemmma for ~ mapdh6N . (C... |
mapdh6eN 39754 | Lemmma for ~ mapdh6N . Pa... |
mapdh6fN 39755 | Lemmma for ~ mapdh6N . Pa... |
mapdh6gN 39756 | Lemmma for ~ mapdh6N . Pa... |
mapdh6hN 39757 | Lemmma for ~ mapdh6N . Pa... |
mapdh6iN 39758 | Lemmma for ~ mapdh6N . El... |
mapdh6jN 39759 | Lemmma for ~ mapdh6N . El... |
mapdh6kN 39760 | Lemmma for ~ mapdh6N . El... |
mapdh6N 39761 | Part (6) of [Baer] p. 47 l... |
mapdh7eN 39762 | Part (7) of [Baer] p. 48 l... |
mapdh7cN 39763 | Part (7) of [Baer] p. 48 l... |
mapdh7dN 39764 | Part (7) of [Baer] p. 48 l... |
mapdh7fN 39765 | Part (7) of [Baer] p. 48 l... |
mapdh75e 39766 | Part (7) of [Baer] p. 48 l... |
mapdh75cN 39767 | Part (7) of [Baer] p. 48 l... |
mapdh75d 39768 | Part (7) of [Baer] p. 48 l... |
mapdh75fN 39769 | Part (7) of [Baer] p. 48 l... |
hvmapffval 39772 | Map from nonzero vectors t... |
hvmapfval 39773 | Map from nonzero vectors t... |
hvmapval 39774 | Value of map from nonzero ... |
hvmapvalvalN 39775 | Value of value of map (i.e... |
hvmapidN 39776 | The value of the vector to... |
hvmap1o 39777 | The vector to functional m... |
hvmapclN 39778 | Closure of the vector to f... |
hvmap1o2 39779 | The vector to functional m... |
hvmapcl2 39780 | Closure of the vector to f... |
hvmaplfl 39781 | The vector to functional m... |
hvmaplkr 39782 | Kernel of the vector to fu... |
mapdhvmap 39783 | Relationship between ` map... |
lspindp5 39784 | Obtain an independent vect... |
hdmaplem1 39785 | Lemma to convert a frequen... |
hdmaplem2N 39786 | Lemma to convert a frequen... |
hdmaplem3 39787 | Lemma to convert a frequen... |
hdmaplem4 39788 | Lemma to convert a frequen... |
mapdh8a 39789 | Part of Part (8) in [Baer]... |
mapdh8aa 39790 | Part of Part (8) in [Baer]... |
mapdh8ab 39791 | Part of Part (8) in [Baer]... |
mapdh8ac 39792 | Part of Part (8) in [Baer]... |
mapdh8ad 39793 | Part of Part (8) in [Baer]... |
mapdh8b 39794 | Part of Part (8) in [Baer]... |
mapdh8c 39795 | Part of Part (8) in [Baer]... |
mapdh8d0N 39796 | Part of Part (8) in [Baer]... |
mapdh8d 39797 | Part of Part (8) in [Baer]... |
mapdh8e 39798 | Part of Part (8) in [Baer]... |
mapdh8g 39799 | Part of Part (8) in [Baer]... |
mapdh8i 39800 | Part of Part (8) in [Baer]... |
mapdh8j 39801 | Part of Part (8) in [Baer]... |
mapdh8 39802 | Part (8) in [Baer] p. 48. ... |
mapdh9a 39803 | Lemma for part (9) in [Bae... |
mapdh9aOLDN 39804 | Lemma for part (9) in [Bae... |
hdmap1ffval 39809 | Preliminary map from vecto... |
hdmap1fval 39810 | Preliminary map from vecto... |
hdmap1vallem 39811 | Value of preliminary map f... |
hdmap1val 39812 | Value of preliminary map f... |
hdmap1val0 39813 | Value of preliminary map f... |
hdmap1val2 39814 | Value of preliminary map f... |
hdmap1eq 39815 | The defining equation for ... |
hdmap1cbv 39816 | Frequently used lemma to c... |
hdmap1valc 39817 | Connect the value of the p... |
hdmap1cl 39818 | Convert closure theorem ~ ... |
hdmap1eq2 39819 | Convert ~ mapdheq2 to use ... |
hdmap1eq4N 39820 | Convert ~ mapdheq4 to use ... |
hdmap1l6lem1 39821 | Lemma for ~ hdmap1l6 . Pa... |
hdmap1l6lem2 39822 | Lemma for ~ hdmap1l6 . Pa... |
hdmap1l6a 39823 | Lemma for ~ hdmap1l6 . Pa... |
hdmap1l6b0N 39824 | Lemmma for ~ hdmap1l6 . (... |
hdmap1l6b 39825 | Lemmma for ~ hdmap1l6 . (... |
hdmap1l6c 39826 | Lemmma for ~ hdmap1l6 . (... |
hdmap1l6d 39827 | Lemmma for ~ hdmap1l6 . (... |
hdmap1l6e 39828 | Lemmma for ~ hdmap1l6 . P... |
hdmap1l6f 39829 | Lemmma for ~ hdmap1l6 . P... |
hdmap1l6g 39830 | Lemmma for ~ hdmap1l6 . P... |
hdmap1l6h 39831 | Lemmma for ~ hdmap1l6 . P... |
hdmap1l6i 39832 | Lemmma for ~ hdmap1l6 . E... |
hdmap1l6j 39833 | Lemmma for ~ hdmap1l6 . E... |
hdmap1l6k 39834 | Lemmma for ~ hdmap1l6 . E... |
hdmap1l6 39835 | Part (6) of [Baer] p. 47 l... |
hdmap1eulem 39836 | Lemma for ~ hdmap1eu . TO... |
hdmap1eulemOLDN 39837 | Lemma for ~ hdmap1euOLDN .... |
hdmap1eu 39838 | Convert ~ mapdh9a to use t... |
hdmap1euOLDN 39839 | Convert ~ mapdh9aOLDN to u... |
hdmapffval 39840 | Map from vectors to functi... |
hdmapfval 39841 | Map from vectors to functi... |
hdmapval 39842 | Value of map from vectors ... |
hdmapfnN 39843 | Functionality of map from ... |
hdmapcl 39844 | Closure of map from vector... |
hdmapval2lem 39845 | Lemma for ~ hdmapval2 . (... |
hdmapval2 39846 | Value of map from vectors ... |
hdmapval0 39847 | Value of map from vectors ... |
hdmapeveclem 39848 | Lemma for ~ hdmapevec . T... |
hdmapevec 39849 | Value of map from vectors ... |
hdmapevec2 39850 | The inner product of the r... |
hdmapval3lemN 39851 | Value of map from vectors ... |
hdmapval3N 39852 | Value of map from vectors ... |
hdmap10lem 39853 | Lemma for ~ hdmap10 . (Co... |
hdmap10 39854 | Part 10 in [Baer] p. 48 li... |
hdmap11lem1 39855 | Lemma for ~ hdmapadd . (C... |
hdmap11lem2 39856 | Lemma for ~ hdmapadd . (C... |
hdmapadd 39857 | Part 11 in [Baer] p. 48 li... |
hdmapeq0 39858 | Part of proof of part 12 i... |
hdmapnzcl 39859 | Nonzero vector closure of ... |
hdmapneg 39860 | Part of proof of part 12 i... |
hdmapsub 39861 | Part of proof of part 12 i... |
hdmap11 39862 | Part of proof of part 12 i... |
hdmaprnlem1N 39863 | Part of proof of part 12 i... |
hdmaprnlem3N 39864 | Part of proof of part 12 i... |
hdmaprnlem3uN 39865 | Part of proof of part 12 i... |
hdmaprnlem4tN 39866 | Lemma for ~ hdmaprnN . TO... |
hdmaprnlem4N 39867 | Part of proof of part 12 i... |
hdmaprnlem6N 39868 | Part of proof of part 12 i... |
hdmaprnlem7N 39869 | Part of proof of part 12 i... |
hdmaprnlem8N 39870 | Part of proof of part 12 i... |
hdmaprnlem9N 39871 | Part of proof of part 12 i... |
hdmaprnlem3eN 39872 | Lemma for ~ hdmaprnN . (C... |
hdmaprnlem10N 39873 | Lemma for ~ hdmaprnN . Sh... |
hdmaprnlem11N 39874 | Lemma for ~ hdmaprnN . Sh... |
hdmaprnlem15N 39875 | Lemma for ~ hdmaprnN . El... |
hdmaprnlem16N 39876 | Lemma for ~ hdmaprnN . El... |
hdmaprnlem17N 39877 | Lemma for ~ hdmaprnN . In... |
hdmaprnN 39878 | Part of proof of part 12 i... |
hdmapf1oN 39879 | Part 12 in [Baer] p. 49. ... |
hdmap14lem1a 39880 | Prior to part 14 in [Baer]... |
hdmap14lem2a 39881 | Prior to part 14 in [Baer]... |
hdmap14lem1 39882 | Prior to part 14 in [Baer]... |
hdmap14lem2N 39883 | Prior to part 14 in [Baer]... |
hdmap14lem3 39884 | Prior to part 14 in [Baer]... |
hdmap14lem4a 39885 | Simplify ` ( A \ { Q } ) `... |
hdmap14lem4 39886 | Simplify ` ( A \ { Q } ) `... |
hdmap14lem6 39887 | Case where ` F ` is zero. ... |
hdmap14lem7 39888 | Combine cases of ` F ` . ... |
hdmap14lem8 39889 | Part of proof of part 14 i... |
hdmap14lem9 39890 | Part of proof of part 14 i... |
hdmap14lem10 39891 | Part of proof of part 14 i... |
hdmap14lem11 39892 | Part of proof of part 14 i... |
hdmap14lem12 39893 | Lemma for proof of part 14... |
hdmap14lem13 39894 | Lemma for proof of part 14... |
hdmap14lem14 39895 | Part of proof of part 14 i... |
hdmap14lem15 39896 | Part of proof of part 14 i... |
hgmapffval 39899 | Map from the scalar divisi... |
hgmapfval 39900 | Map from the scalar divisi... |
hgmapval 39901 | Value of map from the scal... |
hgmapfnN 39902 | Functionality of scalar si... |
hgmapcl 39903 | Closure of scalar sigma ma... |
hgmapdcl 39904 | Closure of the vector spac... |
hgmapvs 39905 | Part 15 of [Baer] p. 50 li... |
hgmapval0 39906 | Value of the scalar sigma ... |
hgmapval1 39907 | Value of the scalar sigma ... |
hgmapadd 39908 | Part 15 of [Baer] p. 50 li... |
hgmapmul 39909 | Part 15 of [Baer] p. 50 li... |
hgmaprnlem1N 39910 | Lemma for ~ hgmaprnN . (C... |
hgmaprnlem2N 39911 | Lemma for ~ hgmaprnN . Pa... |
hgmaprnlem3N 39912 | Lemma for ~ hgmaprnN . El... |
hgmaprnlem4N 39913 | Lemma for ~ hgmaprnN . El... |
hgmaprnlem5N 39914 | Lemma for ~ hgmaprnN . El... |
hgmaprnN 39915 | Part of proof of part 16 i... |
hgmap11 39916 | The scalar sigma map is on... |
hgmapf1oN 39917 | The scalar sigma map is a ... |
hgmapeq0 39918 | The scalar sigma map is ze... |
hdmapipcl 39919 | The inner product (Hermiti... |
hdmapln1 39920 | Linearity property that wi... |
hdmaplna1 39921 | Additive property of first... |
hdmaplns1 39922 | Subtraction property of fi... |
hdmaplnm1 39923 | Multiplicative property of... |
hdmaplna2 39924 | Additive property of secon... |
hdmapglnm2 39925 | g-linear property of secon... |
hdmapgln2 39926 | g-linear property that wil... |
hdmaplkr 39927 | Kernel of the vector to du... |
hdmapellkr 39928 | Membership in the kernel (... |
hdmapip0 39929 | Zero property that will be... |
hdmapip1 39930 | Construct a proportional v... |
hdmapip0com 39931 | Commutation property of Ba... |
hdmapinvlem1 39932 | Line 27 in [Baer] p. 110. ... |
hdmapinvlem2 39933 | Line 28 in [Baer] p. 110, ... |
hdmapinvlem3 39934 | Line 30 in [Baer] p. 110, ... |
hdmapinvlem4 39935 | Part 1.1 of Proposition 1 ... |
hdmapglem5 39936 | Part 1.2 in [Baer] p. 110 ... |
hgmapvvlem1 39937 | Involution property of sca... |
hgmapvvlem2 39938 | Lemma for ~ hgmapvv . Eli... |
hgmapvvlem3 39939 | Lemma for ~ hgmapvv . Eli... |
hgmapvv 39940 | Value of a double involuti... |
hdmapglem7a 39941 | Lemma for ~ hdmapg . (Con... |
hdmapglem7b 39942 | Lemma for ~ hdmapg . (Con... |
hdmapglem7 39943 | Lemma for ~ hdmapg . Line... |
hdmapg 39944 | Apply the scalar sigma fun... |
hdmapoc 39945 | Express our constructed or... |
hlhilset 39948 | The final Hilbert space co... |
hlhilsca 39949 | The scalar of the final co... |
hlhilbase 39950 | The base set of the final ... |
hlhilplus 39951 | The vector addition for th... |
hlhilslem 39952 | Lemma for ~ hlhilsbase etc... |
hlhilslemOLD 39953 | Obsolete version of ~ hlhi... |
hlhilsbase 39954 | The scalar base set of the... |
hlhilsbaseOLD 39955 | Obsolete version of ~ hlhi... |
hlhilsplus 39956 | Scalar addition for the fi... |
hlhilsplusOLD 39957 | Obsolete version of ~ hlhi... |
hlhilsmul 39958 | Scalar multiplication for ... |
hlhilsmulOLD 39959 | Obsolete version of ~ hlhi... |
hlhilsbase2 39960 | The scalar base set of the... |
hlhilsplus2 39961 | Scalar addition for the fi... |
hlhilsmul2 39962 | Scalar multiplication for ... |
hlhils0 39963 | The scalar ring zero for t... |
hlhils1N 39964 | The scalar ring unity for ... |
hlhilvsca 39965 | The scalar product for the... |
hlhilip 39966 | Inner product operation fo... |
hlhilipval 39967 | Value of inner product ope... |
hlhilnvl 39968 | The involution operation o... |
hlhillvec 39969 | The final constructed Hilb... |
hlhildrng 39970 | The star division ring for... |
hlhilsrnglem 39971 | Lemma for ~ hlhilsrng . (... |
hlhilsrng 39972 | The star division ring for... |
hlhil0 39973 | The zero vector for the fi... |
hlhillsm 39974 | The vector sum operation f... |
hlhilocv 39975 | The orthocomplement for th... |
hlhillcs 39976 | The closed subspaces of th... |
hlhilphllem 39977 | Lemma for ~ hlhil . (Cont... |
hlhilhillem 39978 | Lemma for ~ hlhil . (Cont... |
hlathil 39979 | Construction of a Hilbert ... |
leexp1ad 39980 | Weak base ordering relatio... |
relogbcld 39981 | Closure of the general log... |
relogbexpd 39982 | Identity law for general l... |
relogbzexpd 39983 | Power law for the general ... |
logblebd 39984 | The general logarithm is m... |
uzindd 39985 | Induction on the upper int... |
fzadd2d 39986 | Membership of a sum in a f... |
zltlem1d 39987 | Integer ordering relation,... |
zltp1led 39988 | Integer ordering relation,... |
fzne2d 39989 | Elementhood in a finite se... |
eqfnfv2d2 39990 | Equality of functions is d... |
fzsplitnd 39991 | Split a finite interval of... |
fzsplitnr 39992 | Split a finite interval of... |
addassnni 39993 | Associative law for additi... |
addcomnni 39994 | Commutative law for additi... |
mulassnni 39995 | Associative law for multip... |
mulcomnni 39996 | Commutative law for multip... |
gcdcomnni 39997 | Commutative law for gcd. ... |
gcdnegnni 39998 | Negation invariance for gc... |
neggcdnni 39999 | Negation invariance for gc... |
bccl2d 40000 | Closure of the binomial co... |
recbothd 40001 | Take reciprocal on both si... |
gcdmultiplei 40002 | The GCD of a multiple of a... |
gcdaddmzz2nni 40003 | Adding a multiple of one o... |
gcdaddmzz2nncomi 40004 | Adding a multiple of one o... |
gcdnncli 40005 | Closure of the gcd operato... |
muldvds1d 40006 | If a product divides an in... |
muldvds2d 40007 | If a product divides an in... |
nndivdvdsd 40008 | A positive integer divides... |
nnproddivdvdsd 40009 | A product of natural numbe... |
coprmdvds2d 40010 | If an integer is divisible... |
12gcd5e1 40011 | The gcd of 12 and 5 is 1. ... |
60gcd6e6 40012 | The gcd of 60 and 6 is 6. ... |
60gcd7e1 40013 | The gcd of 60 and 7 is 1. ... |
420gcd8e4 40014 | The gcd of 420 and 8 is 4.... |
lcmeprodgcdi 40015 | Calculate the least common... |
12lcm5e60 40016 | The lcm of 12 and 5 is 60.... |
60lcm6e60 40017 | The lcm of 60 and 6 is 60.... |
60lcm7e420 40018 | The lcm of 60 and 7 is 420... |
420lcm8e840 40019 | The lcm of 420 and 8 is 84... |
lcmfunnnd 40020 | Useful equation to calcula... |
lcm1un 40021 | Least common multiple of n... |
lcm2un 40022 | Least common multiple of n... |
lcm3un 40023 | Least common multiple of n... |
lcm4un 40024 | Least common multiple of n... |
lcm5un 40025 | Least common multiple of n... |
lcm6un 40026 | Least common multiple of n... |
lcm7un 40027 | Least common multiple of n... |
lcm8un 40028 | Least common multiple of n... |
3factsumint1 40029 | Move constants out of inte... |
3factsumint2 40030 | Move constants out of inte... |
3factsumint3 40031 | Move constants out of inte... |
3factsumint4 40032 | Move constants out of inte... |
3factsumint 40033 | Helpful equation for lcm i... |
resopunitintvd 40034 | Restrict continuous functi... |
resclunitintvd 40035 | Restrict continuous functi... |
resdvopclptsd 40036 | Restrict derivative on uni... |
lcmineqlem1 40037 | Part of lcm inequality lem... |
lcmineqlem2 40038 | Part of lcm inequality lem... |
lcmineqlem3 40039 | Part of lcm inequality lem... |
lcmineqlem4 40040 | Part of lcm inequality lem... |
lcmineqlem5 40041 | Technical lemma for recipr... |
lcmineqlem6 40042 | Part of lcm inequality lem... |
lcmineqlem7 40043 | Derivative of 1-x for chai... |
lcmineqlem8 40044 | Derivative of (1-x)^(N-M).... |
lcmineqlem9 40045 | (1-x)^(N-M) is continuous.... |
lcmineqlem10 40046 | Induction step of ~ lcmine... |
lcmineqlem11 40047 | Induction step, continuati... |
lcmineqlem12 40048 | Base case for induction. ... |
lcmineqlem13 40049 | Induction proof for lcm in... |
lcmineqlem14 40050 | Technical lemma for inequa... |
lcmineqlem15 40051 | F times the least common m... |
lcmineqlem16 40052 | Technical divisibility lem... |
lcmineqlem17 40053 | Inequality of 2^{2n}. (Co... |
lcmineqlem18 40054 | Technical lemma to shift f... |
lcmineqlem19 40055 | Dividing implies inequalit... |
lcmineqlem20 40056 | Inequality for lcm lemma. ... |
lcmineqlem21 40057 | The lcm inequality lemma w... |
lcmineqlem22 40058 | The lcm inequality lemma w... |
lcmineqlem23 40059 | Penultimate step to the lc... |
lcmineqlem 40060 | The least common multiple ... |
3exp7 40061 | 3 to the power of 7 equals... |
3lexlogpow5ineq1 40062 | First inequality in inequa... |
3lexlogpow5ineq2 40063 | Second inequality in inequ... |
3lexlogpow5ineq4 40064 | Sharper logarithm inequali... |
3lexlogpow5ineq3 40065 | Combined inequality chain ... |
3lexlogpow2ineq1 40066 | Result for bound in AKS in... |
3lexlogpow2ineq2 40067 | Result for bound in AKS in... |
3lexlogpow5ineq5 40068 | Result for bound in AKS in... |
intlewftc 40069 | Inequality inference by in... |
aks4d1lem1 40070 | Technical lemma to reduce ... |
aks4d1p1p1 40071 | Exponential law for finite... |
dvrelog2 40072 | The derivative of the loga... |
dvrelog3 40073 | The derivative of the loga... |
dvrelog2b 40074 | Derivative of the binary l... |
0nonelalab 40075 | Technical lemma for open i... |
dvrelogpow2b 40076 | Derivative of the power of... |
aks4d1p1p3 40077 | Bound of a ceiling of the ... |
aks4d1p1p2 40078 | Rewrite ` A ` in more suit... |
aks4d1p1p4 40079 | Technical step for inequal... |
dvle2 40080 | Collapsed ~ dvle . (Contr... |
aks4d1p1p6 40081 | Inequality lift to differe... |
aks4d1p1p7 40082 | Bound of intermediary of i... |
aks4d1p1p5 40083 | Show inequality for existe... |
aks4d1p1 40084 | Show inequality for existe... |
aks4d1p2 40085 | Technical lemma for existe... |
aks4d1p3 40086 | There exists a small enoug... |
aks4d1p4 40087 | There exists a small enoug... |
aks4d1p5 40088 | Show that ` N ` and ` R ` ... |
aks4d1p6 40089 | The maximal prime power ex... |
aks4d1p7d1 40090 | Technical step in AKS lemm... |
aks4d1p7 40091 | Technical step in AKS lemm... |
aks4d1p8d1 40092 | If a prime divides one num... |
aks4d1p8d2 40093 | Any prime power dividing a... |
aks4d1p8d3 40094 | The remainder of a divisio... |
aks4d1p8 40095 | Show that ` N ` and ` R ` ... |
aks4d1p9 40096 | Show that the order is bou... |
aks4d1 40097 | Lemma 4.1 from ~ https://w... |
5bc2eq10 40098 | The value of 5 choose 2. ... |
facp2 40099 | The factorial of a success... |
2np3bcnp1 40100 | Part of induction step for... |
2ap1caineq 40101 | Inequality for Theorem 6.6... |
sticksstones1 40102 | Different strictly monoton... |
sticksstones2 40103 | The range function on stri... |
sticksstones3 40104 | The range function on stri... |
sticksstones4 40105 | Equinumerosity lemma for s... |
sticksstones5 40106 | Count the number of strict... |
sticksstones6 40107 | Function induces an order ... |
sticksstones7 40108 | Closure property of sticks... |
sticksstones8 40109 | Establish mapping between ... |
sticksstones9 40110 | Establish mapping between ... |
sticksstones10 40111 | Establish mapping between ... |
sticksstones11 40112 | Establish bijective mappin... |
sticksstones12a 40113 | Establish bijective mappin... |
sticksstones12 40114 | Establish bijective mappin... |
sticksstones13 40115 | Establish bijective mappin... |
sticksstones14 40116 | Sticks and stones with def... |
sticksstones15 40117 | Sticks and stones with alm... |
sticksstones16 40118 | Sticks and stones with col... |
sticksstones17 40119 | Extend sticks and stones t... |
sticksstones18 40120 | Extend sticks and stones t... |
sticksstones19 40121 | Extend sticks and stones t... |
sticksstones20 40122 | Lift sticks and stones to ... |
sticksstones21 40123 | Lift sticks and stones to ... |
sticksstones22 40124 | Non-exhaustive sticks and ... |
metakunt1 40125 | A is an endomapping. (Con... |
metakunt2 40126 | A is an endomapping. (Con... |
metakunt3 40127 | Value of A. (Contributed b... |
metakunt4 40128 | Value of A. (Contributed b... |
metakunt5 40129 | C is the left inverse for ... |
metakunt6 40130 | C is the left inverse for ... |
metakunt7 40131 | C is the left inverse for ... |
metakunt8 40132 | C is the left inverse for ... |
metakunt9 40133 | C is the left inverse for ... |
metakunt10 40134 | C is the right inverse for... |
metakunt11 40135 | C is the right inverse for... |
metakunt12 40136 | C is the right inverse for... |
metakunt13 40137 | C is the right inverse for... |
metakunt14 40138 | A is a primitive permutati... |
metakunt15 40139 | Construction of another pe... |
metakunt16 40140 | Construction of another pe... |
metakunt17 40141 | The union of three disjoin... |
metakunt18 40142 | Disjoint domains and codom... |
metakunt19 40143 | Domains on restrictions of... |
metakunt20 40144 | Show that B coincides on t... |
metakunt21 40145 | Show that B coincides on t... |
metakunt22 40146 | Show that B coincides on t... |
metakunt23 40147 | B coincides on the union o... |
metakunt24 40148 | Technical condition such t... |
metakunt25 40149 | B is a permutation. (Cont... |
metakunt26 40150 | Construction of one soluti... |
metakunt27 40151 | Construction of one soluti... |
metakunt28 40152 | Construction of one soluti... |
metakunt29 40153 | Construction of one soluti... |
metakunt30 40154 | Construction of one soluti... |
metakunt31 40155 | Construction of one soluti... |
metakunt32 40156 | Construction of one soluti... |
metakunt33 40157 | Construction of one soluti... |
metakunt34 40158 | ` D ` is a permutation. (... |
andiff 40159 | Adding biconditional when ... |
fac2xp3 40160 | Factorial of 2x+3, sublemm... |
prodsplit 40161 | Product split into two fac... |
2xp3dxp2ge1d 40162 | 2x+3 is greater than or eq... |
factwoffsmonot 40163 | A factorial with offset is... |
bicomdALT 40164 | Alternate proof of ~ bicom... |
elabgw 40165 | Membership in a class abst... |
elab2gw 40166 | Membership in a class abst... |
elrab2w 40167 | Membership in a restricted... |
ruvALT 40168 | Alternate proof of ~ ruv w... |
sn-wcdeq 40169 | Alternative to ~ wcdeq and... |
acos1half 40170 | The arccosine of ` 1 / 2 `... |
isdomn5 40171 | The right conjunct in the ... |
isdomn4 40172 | A ring is a domain iff it ... |
ioin9i8 40173 | Miscellaneous inference cr... |
jaodd 40174 | Double deduction form of ~... |
syl3an12 40175 | A double syllogism inferen... |
sbtd 40176 | A true statement is true u... |
sbor2 40177 | One direction of ~ sbor , ... |
19.9dev 40178 | ~ 19.9d in the case of an ... |
rspcedvdw 40179 | Version of ~ rspcedvd wher... |
2rspcedvdw 40180 | Double application of ~ rs... |
3rspcedvdw 40181 | Triple application of ~ rs... |
3rspcedvd 40182 | Triple application of ~ rs... |
eqimssd 40183 | Equality implies inclusion... |
rabdif 40184 | Move difference in and out... |
sn-axrep5v 40185 | A condensed form of ~ axre... |
sn-axprlem3 40186 | ~ axprlem3 using only Tars... |
sn-el 40187 | A version of ~ el with an ... |
sn-dtru 40188 | ~ dtru without ~ ax-8 or ~... |
sn-iotalem 40189 | An unused lemma showing th... |
sn-iotalemcor 40190 | Corollary of ~ sn-iotalem ... |
abbi1sn 40191 | Originally part of ~ uniab... |
iotavallem 40192 | Version of ~ iotaval using... |
sn-iotauni 40193 | Version of ~ iotauni using... |
sn-iotanul 40194 | Version of ~ iotanul using... |
sn-iotaval 40195 | ~ iotaval without ~ ax-10 ... |
sn-iotassuni 40196 | ~ iotassuni without ~ ax-1... |
sn-iotaex 40197 | ~ iotaex without ~ ax-10 ,... |
brif1 40198 | Move a relation inside and... |
brif2 40199 | Move a relation inside and... |
brif12 40200 | Move a relation inside and... |
pssexg 40201 | The proper subset of a set... |
pssn0 40202 | A proper superset is nonem... |
psspwb 40203 | Classes are proper subclas... |
xppss12 40204 | Proper subset theorem for ... |
elpwbi 40205 | Membership in a power set,... |
opelxpii 40206 | Ordered pair membership in... |
imaopab 40207 | The image of a class of or... |
fnsnbt 40208 | A function's domain is a s... |
fnimasnd 40209 | The image of a function by... |
fvmptd4 40210 | Deduction version of ~ fvm... |
ofun 40211 | A function operation of un... |
dfqs2 40212 | Alternate definition of qu... |
dfqs3 40213 | Alternate definition of qu... |
qseq12d 40214 | Equality theorem for quoti... |
qsalrel 40215 | The quotient set is equal ... |
elmapdd 40216 | Deduction associated with ... |
isfsuppd 40217 | Deduction form of ~ isfsup... |
fzosumm1 40218 | Separate out the last term... |
ccatcan2d 40219 | Cancellation law for conca... |
nelsubginvcld 40220 | The inverse of a non-subgr... |
nelsubgcld 40221 | A non-subgroup-member plus... |
nelsubgsubcld 40222 | A non-subgroup-member minu... |
rnasclg 40223 | The set of injected scalar... |
selvval2lem1 40224 | ` T ` is an associative al... |
selvval2lem2 40225 | ` D ` is a ring homomorphi... |
selvval2lem3 40226 | The third argument passed ... |
selvval2lemn 40227 | A lemma to illustrate the ... |
selvval2lem4 40228 | The fourth argument passed... |
selvval2lem5 40229 | The fifth argument passed ... |
selvcl 40230 | Closure of the "variable s... |
frlmfielbas 40231 | The vectors of a finite fr... |
frlmfzwrd 40232 | A vector of a module with ... |
frlmfzowrd 40233 | A vector of a module with ... |
frlmfzolen 40234 | The dimension of a vector ... |
frlmfzowrdb 40235 | The vectors of a module wi... |
frlmfzoccat 40236 | The concatenation of two v... |
frlmvscadiccat 40237 | Scalar multiplication dist... |
ismhmd 40238 | Deduction version of ~ ism... |
ablcmnd 40239 | An Abelian group is a comm... |
ringcld 40240 | Closure of the multiplicat... |
ringassd 40241 | Associative law for multip... |
ringlidmd 40242 | The unit element of a ring... |
ringridmd 40243 | The unit element of a ring... |
ringabld 40244 | A ring is an Abelian group... |
ringcmnd 40245 | A ring is a commutative mo... |
drngringd 40246 | A division ring is a ring.... |
drnggrpd 40247 | A division ring is a group... |
drnginvrcld 40248 | Closure of the multiplicat... |
drnginvrn0d 40249 | A multiplicative inverse i... |
drnginvrld 40250 | Property of the multiplica... |
drnginvrrd 40251 | Property of the multiplica... |
drngmulcanad 40252 | Cancellation of a nonzero ... |
drngmulcan2ad 40253 | Cancellation of a nonzero ... |
drnginvmuld 40254 | Inverse of a nonzero produ... |
fldcrngd 40255 | A field is a commutative r... |
lmodgrpd 40256 | A left module is a group. ... |
lvecgrp 40257 | A vector space is a group.... |
lveclmodd 40258 | A vector space is a left m... |
lvecgrpd 40259 | A vector space is a group.... |
lvecring 40260 | The scalar component of a ... |
lmhmlvec 40261 | The property for modules t... |
frlm0vald 40262 | All coordinates of the zer... |
frlmsnic 40263 | Given a free module with a... |
uvccl 40264 | A unit vector is a vector.... |
uvcn0 40265 | A unit vector is nonzero. ... |
pwselbasr 40266 | The reverse direction of ~... |
pwspjmhmmgpd 40267 | The projection given by ~ ... |
pwsexpg 40268 | Value of a group exponenti... |
pwsgprod 40269 | Finite products in a power... |
mplascl0 40270 | The zero scalar as a polyn... |
evl0 40271 | The zero polynomial evalua... |
evlsval3 40272 | Give a formula for the pol... |
evlsscaval 40273 | Polynomial evaluation buil... |
evlsvarval 40274 | Polynomial evaluation buil... |
evlsbagval 40275 | Polynomial evaluation buil... |
evlsexpval 40276 | Polynomial evaluation buil... |
evlsaddval 40277 | Polynomial evaluation buil... |
evlsmulval 40278 | Polynomial evaluation buil... |
fsuppind 40279 | Induction on functions ` F... |
fsuppssindlem1 40280 | Lemma for ~ fsuppssind . ... |
fsuppssindlem2 40281 | Lemma for ~ fsuppssind . ... |
fsuppssind 40282 | Induction on functions ` F... |
mhpind 40283 | The homogeneous polynomial... |
mhphflem 40284 | Lemma for ~ mhphf . Add s... |
mhphf 40285 | A homogeneous polynomial d... |
mhphf2 40286 | A homogeneous polynomial d... |
mhphf3 40287 | A homogeneous polynomial d... |
mhphf4 40288 | A homogeneous polynomial d... |
c0exALT 40289 | Alternate proof of ~ c0ex ... |
0cnALT3 40290 | Alternate proof of ~ 0cn u... |
elre0re 40291 | Specialized version of ~ 0... |
1t1e1ALT 40292 | Alternate proof of ~ 1t1e1... |
remulcan2d 40293 | ~ mulcan2d for real number... |
readdid1addid2d 40294 | Given some real number ` B... |
sn-1ne2 40295 | A proof of ~ 1ne2 without ... |
nnn1suc 40296 | A positive integer that is... |
nnadd1com 40297 | Addition with 1 is commuta... |
nnaddcom 40298 | Addition is commutative fo... |
nnaddcomli 40299 | Version of ~ addcomli for ... |
nnadddir 40300 | Right-distributivity for n... |
nnmul1com 40301 | Multiplication with 1 is c... |
nnmulcom 40302 | Multiplication is commutat... |
mvrrsubd 40303 | Move a subtraction in the ... |
laddrotrd 40304 | Rotate the variables right... |
raddcom12d 40305 | Swap the first two variabl... |
lsubrotld 40306 | Rotate the variables left ... |
lsubcom23d 40307 | Swap the second and third ... |
addsubeq4com 40308 | Relation between sums and ... |
sqsumi 40309 | A sum squared. (Contribut... |
negn0nposznnd 40310 | Lemma for ~ dffltz . (Con... |
sqmid3api 40311 | Value of the square of the... |
decaddcom 40312 | Commute ones place in addi... |
sqn5i 40313 | The square of a number end... |
sqn5ii 40314 | The square of a number end... |
decpmulnc 40315 | Partial products algorithm... |
decpmul 40316 | Partial products algorithm... |
sqdeccom12 40317 | The square of a number in ... |
sq3deccom12 40318 | Variant of ~ sqdeccom12 wi... |
235t711 40319 | Calculate a product by lon... |
ex-decpmul 40320 | Example usage of ~ decpmul... |
oexpreposd 40321 | Lemma for ~ dffltz . TODO... |
ltexp1d 40322 | ~ ltmul1d for exponentiati... |
ltexp1dd 40323 | Raising both sides of 'les... |
exp11nnd 40324 | ~ sq11d for positive real ... |
exp11d 40325 | ~ exp11nnd for nonzero int... |
0dvds0 40326 | 0 divides 0. (Contributed... |
absdvdsabsb 40327 | Divisibility is invariant ... |
dvdsexpim 40328 | ~ dvdssqim generalized to ... |
gcdnn0id 40329 | The ` gcd ` of a nonnegati... |
gcdle1d 40330 | The greatest common diviso... |
gcdle2d 40331 | The greatest common diviso... |
dvdsexpad 40332 | Deduction associated with ... |
nn0rppwr 40333 | If ` A ` and ` B ` are rel... |
expgcd 40334 | Exponentiation distributes... |
nn0expgcd 40335 | Exponentiation distributes... |
zexpgcd 40336 | Exponentiation distributes... |
numdenexp 40337 | ~ numdensq extended to non... |
numexp 40338 | ~ numsq extended to nonneg... |
denexp 40339 | ~ densq extended to nonneg... |
dvdsexpnn 40340 | ~ dvdssqlem generalized to... |
dvdsexpnn0 40341 | ~ dvdsexpnn generalized to... |
dvdsexpb 40342 | ~ dvdssq generalized to po... |
posqsqznn 40343 | When a positive rational s... |
cxpgt0d 40344 | A positive real raised to ... |
zrtelqelz 40345 | ~ zsqrtelqelz generalized ... |
zrtdvds 40346 | A positive integer root di... |
rtprmirr 40347 | The root of a prime number... |
resubval 40350 | Value of real subtraction,... |
renegeulemv 40351 | Lemma for ~ renegeu and si... |
renegeulem 40352 | Lemma for ~ renegeu and si... |
renegeu 40353 | Existential uniqueness of ... |
rernegcl 40354 | Closure law for negative r... |
renegadd 40355 | Relationship between real ... |
renegid 40356 | Addition of a real number ... |
reneg0addid2 40357 | Negative zero is a left ad... |
resubeulem1 40358 | Lemma for ~ resubeu . A v... |
resubeulem2 40359 | Lemma for ~ resubeu . A v... |
resubeu 40360 | Existential uniqueness of ... |
rersubcl 40361 | Closure for real subtracti... |
resubadd 40362 | Relation between real subt... |
resubaddd 40363 | Relationship between subtr... |
resubf 40364 | Real subtraction is an ope... |
repncan2 40365 | Addition and subtraction o... |
repncan3 40366 | Addition and subtraction o... |
readdsub 40367 | Law for addition and subtr... |
reladdrsub 40368 | Move LHS of a sum into RHS... |
reltsub1 40369 | Subtraction from both side... |
reltsubadd2 40370 | 'Less than' relationship b... |
resubcan2 40371 | Cancellation law for real ... |
resubsub4 40372 | Law for double subtraction... |
rennncan2 40373 | Cancellation law for real ... |
renpncan3 40374 | Cancellation law for real ... |
repnpcan 40375 | Cancellation law for addit... |
reppncan 40376 | Cancellation law for mixed... |
resubidaddid1lem 40377 | Lemma for ~ resubidaddid1 ... |
resubidaddid1 40378 | Any real number subtracted... |
resubdi 40379 | Distribution of multiplica... |
re1m1e0m0 40380 | Equality of two left-addit... |
sn-00idlem1 40381 | Lemma for ~ sn-00id . (Co... |
sn-00idlem2 40382 | Lemma for ~ sn-00id . (Co... |
sn-00idlem3 40383 | Lemma for ~ sn-00id . (Co... |
sn-00id 40384 | ~ 00id proven without ~ ax... |
re0m0e0 40385 | Real number version of ~ 0... |
readdid2 40386 | Real number version of ~ a... |
sn-addid2 40387 | ~ addid2 without ~ ax-mulc... |
remul02 40388 | Real number version of ~ m... |
sn-0ne2 40389 | ~ 0ne2 without ~ ax-mulcom... |
remul01 40390 | Real number version of ~ m... |
resubid 40391 | Subtraction of a real numb... |
readdid1 40392 | Real number version of ~ a... |
resubid1 40393 | Real number version of ~ s... |
renegneg 40394 | A real number is equal to ... |
readdcan2 40395 | Commuted version of ~ read... |
renegid2 40396 | Commuted version of ~ rene... |
sn-it0e0 40397 | Proof of ~ it0e0 without ~... |
sn-negex12 40398 | A combination of ~ cnegex ... |
sn-negex 40399 | Proof of ~ cnegex without ... |
sn-negex2 40400 | Proof of ~ cnegex2 without... |
sn-addcand 40401 | ~ addcand without ~ ax-mul... |
sn-addid1 40402 | ~ addid1 without ~ ax-mulc... |
sn-addcan2d 40403 | ~ addcan2d without ~ ax-mu... |
reixi 40404 | ~ ixi without ~ ax-mulcom ... |
rei4 40405 | ~ i4 without ~ ax-mulcom .... |
sn-addid0 40406 | A number that sums to itse... |
sn-mul01 40407 | ~ mul01 without ~ ax-mulco... |
sn-subeu 40408 | ~ negeu without ~ ax-mulco... |
sn-subcl 40409 | ~ subcl without ~ ax-mulco... |
sn-subf 40410 | ~ subf without ~ ax-mulcom... |
resubeqsub 40411 | Equivalence between real s... |
subresre 40412 | Subtraction restricted to ... |
addinvcom 40413 | A number commutes with its... |
remulinvcom 40414 | A left multiplicative inve... |
remulid2 40415 | Commuted version of ~ ax-1... |
sn-1ticom 40416 | Lemma for ~ sn-mulid2 and ... |
sn-mulid2 40417 | ~ mulid2 without ~ ax-mulc... |
it1ei 40418 | ` 1 ` is a multiplicative ... |
ipiiie0 40419 | The multiplicative inverse... |
remulcand 40420 | Commuted version of ~ remu... |
sn-0tie0 40421 | Lemma for ~ sn-mul02 . Co... |
sn-mul02 40422 | ~ mul02 without ~ ax-mulco... |
sn-ltaddpos 40423 | ~ ltaddpos without ~ ax-mu... |
reposdif 40424 | Comparison of two numbers ... |
relt0neg1 40425 | Comparison of a real and i... |
relt0neg2 40426 | Comparison of a real and i... |
mulgt0con1dlem 40427 | Lemma for ~ mulgt0con1d . ... |
mulgt0con1d 40428 | Counterpart to ~ mulgt0con... |
mulgt0con2d 40429 | Lemma for ~ mulgt0b2d and ... |
mulgt0b2d 40430 | Biconditional, deductive f... |
sn-ltmul2d 40431 | ~ ltmul2d without ~ ax-mul... |
sn-0lt1 40432 | ~ 0lt1 without ~ ax-mulcom... |
sn-ltp1 40433 | ~ ltp1 without ~ ax-mulcom... |
reneg1lt0 40434 | Lemma for ~ sn-inelr . (C... |
sn-inelr 40435 | ~ inelr without ~ ax-mulco... |
itrere 40436 | ` _i ` times a real is rea... |
retire 40437 | Commuted version of ~ itre... |
cnreeu 40438 | The reals in the expressio... |
sn-sup2 40439 | ~ sup2 with exactly the sa... |
prjspval 40442 | Value of the projective sp... |
prjsprel 40443 | Utility theorem regarding ... |
prjspertr 40444 | The relation in ` PrjSp ` ... |
prjsperref 40445 | The relation in ` PrjSp ` ... |
prjspersym 40446 | The relation in ` PrjSp ` ... |
prjsper 40447 | The relation used to defin... |
prjspreln0 40448 | Two nonzero vectors are eq... |
prjspvs 40449 | A nonzero multiple of a ve... |
prjsprellsp 40450 | Two vectors are equivalent... |
prjspeclsp 40451 | The vectors equivalent to ... |
prjspval2 40452 | Alternate definition of pr... |
prjspnval 40455 | Value of the n-dimensional... |
prjspnerlem 40456 | A lemma showing that the e... |
prjspnval2 40457 | Value of the n-dimensional... |
prjspner 40458 | The relation used to defin... |
prjspnvs 40459 | A nonzero multiple of a ve... |
0prjspnlem 40460 | Lemma for ~ 0prjspn . The... |
prjspnfv01 40461 | Any vector is equivalent t... |
prjspner01 40462 | Any vector is equivalent t... |
prjspner1 40463 | Two vectors whose zeroth c... |
0prjspnrel 40464 | In the zero-dimensional pr... |
0prjspn 40465 | A zero-dimensional project... |
prjcrvfval 40468 | Value of the projective cu... |
prjcrvval 40469 | Value of the projective cu... |
prjcrv0 40470 | The "curve" (zero set) cor... |
dffltz 40471 | Fermat's Last Theorem (FLT... |
fltmul 40472 | A counterexample to FLT st... |
fltdiv 40473 | A counterexample to FLT st... |
flt0 40474 | A counterexample for FLT d... |
fltdvdsabdvdsc 40475 | Any factor of both ` A ` a... |
fltabcoprmex 40476 | A counterexample to FLT im... |
fltaccoprm 40477 | A counterexample to FLT wi... |
fltbccoprm 40478 | A counterexample to FLT wi... |
fltabcoprm 40479 | A counterexample to FLT wi... |
infdesc 40480 | Infinite descent. The hyp... |
fltne 40481 | If a counterexample to FLT... |
flt4lem 40482 | Raising a number to the fo... |
flt4lem1 40483 | Satisfy the antecedent use... |
flt4lem2 40484 | If ` A ` is even, ` B ` is... |
flt4lem3 40485 | Equivalent to ~ pythagtrip... |
flt4lem4 40486 | If the product of two copr... |
flt4lem5 40487 | In the context of the lemm... |
flt4lem5elem 40488 | Version of ~ fltaccoprm an... |
flt4lem5a 40489 | Part 1 of Equation 1 of ... |
flt4lem5b 40490 | Part 2 of Equation 1 of ... |
flt4lem5c 40491 | Part 2 of Equation 2 of ... |
flt4lem5d 40492 | Part 3 of Equation 2 of ... |
flt4lem5e 40493 | Satisfy the hypotheses of ... |
flt4lem5f 40494 | Final equation of ~... |
flt4lem6 40495 | Remove shared factors in a... |
flt4lem7 40496 | Convert ~ flt4lem5f into a... |
nna4b4nsq 40497 | Strengthening of Fermat's ... |
fltltc 40498 | ` ( C ^ N ) ` is the large... |
fltnltalem 40499 | Lemma for ~ fltnlta . A l... |
fltnlta 40500 | In a Fermat counterexample... |
binom2d 40501 | Deduction form of binom2. ... |
cu3addd 40502 | Cube of sum of three numbe... |
sqnegd 40503 | The square of the negative... |
negexpidd 40504 | The sum of a real number t... |
rexlimdv3d 40505 | An extended version of ~ r... |
3cubeslem1 40506 | Lemma for ~ 3cubes . (Con... |
3cubeslem2 40507 | Lemma for ~ 3cubes . Used... |
3cubeslem3l 40508 | Lemma for ~ 3cubes . (Con... |
3cubeslem3r 40509 | Lemma for ~ 3cubes . (Con... |
3cubeslem3 40510 | Lemma for ~ 3cubes . (Con... |
3cubeslem4 40511 | Lemma for ~ 3cubes . This... |
3cubes 40512 | Every rational number is a... |
rntrclfvOAI 40513 | The range of the transitiv... |
moxfr 40514 | Transfer at-most-one betwe... |
imaiinfv 40515 | Indexed intersection of an... |
elrfi 40516 | Elementhood in a set of re... |
elrfirn 40517 | Elementhood in a set of re... |
elrfirn2 40518 | Elementhood in a set of re... |
cmpfiiin 40519 | In a compact topology, a s... |
ismrcd1 40520 | Any function from the subs... |
ismrcd2 40521 | Second half of ~ ismrcd1 .... |
istopclsd 40522 | A closure function which s... |
ismrc 40523 | A function is a Moore clos... |
isnacs 40526 | Expand definition of Noeth... |
nacsfg 40527 | In a Noetherian-type closu... |
isnacs2 40528 | Express Noetherian-type cl... |
mrefg2 40529 | Slight variation on finite... |
mrefg3 40530 | Slight variation on finite... |
nacsacs 40531 | A closure system of Noethe... |
isnacs3 40532 | A choice-free order equiva... |
incssnn0 40533 | Transitivity induction of ... |
nacsfix 40534 | An increasing sequence of ... |
constmap 40535 | A constant (represented wi... |
mapco2g 40536 | Renaming indices in a tupl... |
mapco2 40537 | Post-composition (renaming... |
mapfzcons 40538 | Extending a one-based mapp... |
mapfzcons1 40539 | Recover prefix mapping fro... |
mapfzcons1cl 40540 | A nonempty mapping has a p... |
mapfzcons2 40541 | Recover added element from... |
mptfcl 40542 | Interpret range of a maps-... |
mzpclval 40547 | Substitution lemma for ` m... |
elmzpcl 40548 | Double substitution lemma ... |
mzpclall 40549 | The set of all functions w... |
mzpcln0 40550 | Corollary of ~ mzpclall : ... |
mzpcl1 40551 | Defining property 1 of a p... |
mzpcl2 40552 | Defining property 2 of a p... |
mzpcl34 40553 | Defining properties 3 and ... |
mzpval 40554 | Value of the ` mzPoly ` fu... |
dmmzp 40555 | ` mzPoly ` is defined for ... |
mzpincl 40556 | Polynomial closedness is a... |
mzpconst 40557 | Constant functions are pol... |
mzpf 40558 | A polynomial function is a... |
mzpproj 40559 | A projection function is p... |
mzpadd 40560 | The pointwise sum of two p... |
mzpmul 40561 | The pointwise product of t... |
mzpconstmpt 40562 | A constant function expres... |
mzpaddmpt 40563 | Sum of polynomial function... |
mzpmulmpt 40564 | Product of polynomial func... |
mzpsubmpt 40565 | The difference of two poly... |
mzpnegmpt 40566 | Negation of a polynomial f... |
mzpexpmpt 40567 | Raise a polynomial functio... |
mzpindd 40568 | "Structural" induction to ... |
mzpmfp 40569 | Relationship between multi... |
mzpsubst 40570 | Substituting polynomials f... |
mzprename 40571 | Simplified version of ~ mz... |
mzpresrename 40572 | A polynomial is a polynomi... |
mzpcompact2lem 40573 | Lemma for ~ mzpcompact2 . ... |
mzpcompact2 40574 | Polynomials are finitary o... |
coeq0i 40575 | ~ coeq0 but without explic... |
fzsplit1nn0 40576 | Split a finite 1-based set... |
eldiophb 40579 | Initial expression of Diop... |
eldioph 40580 | Condition for a set to be ... |
diophrw 40581 | Renaming and adding unused... |
eldioph2lem1 40582 | Lemma for ~ eldioph2 . Co... |
eldioph2lem2 40583 | Lemma for ~ eldioph2 . Co... |
eldioph2 40584 | Construct a Diophantine se... |
eldioph2b 40585 | While Diophantine sets wer... |
eldiophelnn0 40586 | Remove antecedent on ` B `... |
eldioph3b 40587 | Define Diophantine sets in... |
eldioph3 40588 | Inference version of ~ eld... |
ellz1 40589 | Membership in a lower set ... |
lzunuz 40590 | The union of a lower set o... |
fz1eqin 40591 | Express a one-based finite... |
lzenom 40592 | Lower integers are countab... |
elmapresaunres2 40593 | ~ fresaunres2 transposed t... |
diophin 40594 | If two sets are Diophantin... |
diophun 40595 | If two sets are Diophantin... |
eldiophss 40596 | Diophantine sets are sets ... |
diophrex 40597 | Projecting a Diophantine s... |
eq0rabdioph 40598 | This is the first of a num... |
eqrabdioph 40599 | Diophantine set builder fo... |
0dioph 40600 | The null set is Diophantin... |
vdioph 40601 | The "universal" set (as la... |
anrabdioph 40602 | Diophantine set builder fo... |
orrabdioph 40603 | Diophantine set builder fo... |
3anrabdioph 40604 | Diophantine set builder fo... |
3orrabdioph 40605 | Diophantine set builder fo... |
2sbcrex 40606 | Exchange an existential qu... |
sbcrexgOLD 40607 | Interchange class substitu... |
2sbcrexOLD 40608 | Exchange an existential qu... |
sbc2rex 40609 | Exchange a substitution wi... |
sbc2rexgOLD 40610 | Exchange a substitution wi... |
sbc4rex 40611 | Exchange a substitution wi... |
sbc4rexgOLD 40612 | Exchange a substitution wi... |
sbcrot3 40613 | Rotate a sequence of three... |
sbcrot5 40614 | Rotate a sequence of five ... |
sbccomieg 40615 | Commute two explicit subst... |
rexrabdioph 40616 | Diophantine set builder fo... |
rexfrabdioph 40617 | Diophantine set builder fo... |
2rexfrabdioph 40618 | Diophantine set builder fo... |
3rexfrabdioph 40619 | Diophantine set builder fo... |
4rexfrabdioph 40620 | Diophantine set builder fo... |
6rexfrabdioph 40621 | Diophantine set builder fo... |
7rexfrabdioph 40622 | Diophantine set builder fo... |
rabdiophlem1 40623 | Lemma for arithmetic dioph... |
rabdiophlem2 40624 | Lemma for arithmetic dioph... |
elnn0rabdioph 40625 | Diophantine set builder fo... |
rexzrexnn0 40626 | Rewrite an existential qua... |
lerabdioph 40627 | Diophantine set builder fo... |
eluzrabdioph 40628 | Diophantine set builder fo... |
elnnrabdioph 40629 | Diophantine set builder fo... |
ltrabdioph 40630 | Diophantine set builder fo... |
nerabdioph 40631 | Diophantine set builder fo... |
dvdsrabdioph 40632 | Divisibility is a Diophant... |
eldioph4b 40633 | Membership in ` Dioph ` ex... |
eldioph4i 40634 | Forward-only version of ~ ... |
diophren 40635 | Change variables in a Diop... |
rabrenfdioph 40636 | Change variable numbers in... |
rabren3dioph 40637 | Change variable numbers in... |
fphpd 40638 | Pigeonhole principle expre... |
fphpdo 40639 | Pigeonhole principle for s... |
ctbnfien 40640 | An infinite subset of a co... |
fiphp3d 40641 | Infinite pigeonhole princi... |
rencldnfilem 40642 | Lemma for ~ rencldnfi . (... |
rencldnfi 40643 | A set of real numbers whic... |
irrapxlem1 40644 | Lemma for ~ irrapx1 . Div... |
irrapxlem2 40645 | Lemma for ~ irrapx1 . Two... |
irrapxlem3 40646 | Lemma for ~ irrapx1 . By ... |
irrapxlem4 40647 | Lemma for ~ irrapx1 . Eli... |
irrapxlem5 40648 | Lemma for ~ irrapx1 . Swi... |
irrapxlem6 40649 | Lemma for ~ irrapx1 . Exp... |
irrapx1 40650 | Dirichlet's approximation ... |
pellexlem1 40651 | Lemma for ~ pellex . Arit... |
pellexlem2 40652 | Lemma for ~ pellex . Arit... |
pellexlem3 40653 | Lemma for ~ pellex . To e... |
pellexlem4 40654 | Lemma for ~ pellex . Invo... |
pellexlem5 40655 | Lemma for ~ pellex . Invo... |
pellexlem6 40656 | Lemma for ~ pellex . Doin... |
pellex 40657 | Every Pell equation has a ... |
pell1qrval 40668 | Value of the set of first-... |
elpell1qr 40669 | Membership in a first-quad... |
pell14qrval 40670 | Value of the set of positi... |
elpell14qr 40671 | Membership in the set of p... |
pell1234qrval 40672 | Value of the set of genera... |
elpell1234qr 40673 | Membership in the set of g... |
pell1234qrre 40674 | General Pell solutions are... |
pell1234qrne0 40675 | No solution to a Pell equa... |
pell1234qrreccl 40676 | General solutions of the P... |
pell1234qrmulcl 40677 | General solutions of the P... |
pell14qrss1234 40678 | A positive Pell solution i... |
pell14qrre 40679 | A positive Pell solution i... |
pell14qrne0 40680 | A positive Pell solution i... |
pell14qrgt0 40681 | A positive Pell solution i... |
pell14qrrp 40682 | A positive Pell solution i... |
pell1234qrdich 40683 | A general Pell solution is... |
elpell14qr2 40684 | A number is a positive Pel... |
pell14qrmulcl 40685 | Positive Pell solutions ar... |
pell14qrreccl 40686 | Positive Pell solutions ar... |
pell14qrdivcl 40687 | Positive Pell solutions ar... |
pell14qrexpclnn0 40688 | Lemma for ~ pell14qrexpcl ... |
pell14qrexpcl 40689 | Positive Pell solutions ar... |
pell1qrss14 40690 | First-quadrant Pell soluti... |
pell14qrdich 40691 | A positive Pell solution i... |
pell1qrge1 40692 | A Pell solution in the fir... |
pell1qr1 40693 | 1 is a Pell solution and i... |
elpell1qr2 40694 | The first quadrant solutio... |
pell1qrgaplem 40695 | Lemma for ~ pell1qrgap . ... |
pell1qrgap 40696 | First-quadrant Pell soluti... |
pell14qrgap 40697 | Positive Pell solutions ar... |
pell14qrgapw 40698 | Positive Pell solutions ar... |
pellqrexplicit 40699 | Condition for a calculated... |
infmrgelbi 40700 | Any lower bound of a nonem... |
pellqrex 40701 | There is a nontrivial solu... |
pellfundval 40702 | Value of the fundamental s... |
pellfundre 40703 | The fundamental solution o... |
pellfundge 40704 | Lower bound on the fundame... |
pellfundgt1 40705 | Weak lower bound on the Pe... |
pellfundlb 40706 | A nontrivial first quadran... |
pellfundglb 40707 | If a real is larger than t... |
pellfundex 40708 | The fundamental solution a... |
pellfund14gap 40709 | There are no solutions bet... |
pellfundrp 40710 | The fundamental Pell solut... |
pellfundne1 40711 | The fundamental Pell solut... |
reglogcl 40712 | General logarithm is a rea... |
reglogltb 40713 | General logarithm preserve... |
reglogleb 40714 | General logarithm preserve... |
reglogmul 40715 | Multiplication law for gen... |
reglogexp 40716 | Power law for general log.... |
reglogbas 40717 | General log of the base is... |
reglog1 40718 | General log of 1 is 0. (C... |
reglogexpbas 40719 | General log of a power of ... |
pellfund14 40720 | Every positive Pell soluti... |
pellfund14b 40721 | The positive Pell solution... |
rmxfval 40726 | Value of the X sequence. ... |
rmyfval 40727 | Value of the Y sequence. ... |
rmspecsqrtnq 40728 | The discriminant used to d... |
rmspecnonsq 40729 | The discriminant used to d... |
qirropth 40730 | This lemma implements the ... |
rmspecfund 40731 | The base of exponent used ... |
rmxyelqirr 40732 | The solutions used to cons... |
rmxypairf1o 40733 | The function used to extra... |
rmxyelxp 40734 | Lemma for ~ frmx and ~ frm... |
frmx 40735 | The X sequence is a nonneg... |
frmy 40736 | The Y sequence is an integ... |
rmxyval 40737 | Main definition of the X a... |
rmspecpos 40738 | The discriminant used to d... |
rmxycomplete 40739 | The X and Y sequences take... |
rmxynorm 40740 | The X and Y sequences defi... |
rmbaserp 40741 | The base of exponentiation... |
rmxyneg 40742 | Negation law for X and Y s... |
rmxyadd 40743 | Addition formula for X and... |
rmxy1 40744 | Value of the X and Y seque... |
rmxy0 40745 | Value of the X and Y seque... |
rmxneg 40746 | Negation law (even functio... |
rmx0 40747 | Value of X sequence at 0. ... |
rmx1 40748 | Value of X sequence at 1. ... |
rmxadd 40749 | Addition formula for X seq... |
rmyneg 40750 | Negation formula for Y seq... |
rmy0 40751 | Value of Y sequence at 0. ... |
rmy1 40752 | Value of Y sequence at 1. ... |
rmyadd 40753 | Addition formula for Y seq... |
rmxp1 40754 | Special addition-of-1 form... |
rmyp1 40755 | Special addition of 1 form... |
rmxm1 40756 | Subtraction of 1 formula f... |
rmym1 40757 | Subtraction of 1 formula f... |
rmxluc 40758 | The X sequence is a Lucas ... |
rmyluc 40759 | The Y sequence is a Lucas ... |
rmyluc2 40760 | Lucas sequence property of... |
rmxdbl 40761 | "Double-angle formula" for... |
rmydbl 40762 | "Double-angle formula" for... |
monotuz 40763 | A function defined on an u... |
monotoddzzfi 40764 | A function which is odd an... |
monotoddzz 40765 | A function (given implicit... |
oddcomabszz 40766 | An odd function which take... |
2nn0ind 40767 | Induction on nonnegative i... |
zindbi 40768 | Inductively transfer a pro... |
rmxypos 40769 | For all nonnegative indice... |
ltrmynn0 40770 | The Y-sequence is strictly... |
ltrmxnn0 40771 | The X-sequence is strictly... |
lermxnn0 40772 | The X-sequence is monotoni... |
rmxnn 40773 | The X-sequence is defined ... |
ltrmy 40774 | The Y-sequence is strictly... |
rmyeq0 40775 | Y is zero only at zero. (... |
rmyeq 40776 | Y is one-to-one. (Contrib... |
lermy 40777 | Y is monotonic (non-strict... |
rmynn 40778 | ` rmY ` is positive for po... |
rmynn0 40779 | ` rmY ` is nonnegative for... |
rmyabs 40780 | ` rmY ` commutes with ` ab... |
jm2.24nn 40781 | X(n) is strictly greater t... |
jm2.17a 40782 | First half of lemma 2.17 o... |
jm2.17b 40783 | Weak form of the second ha... |
jm2.17c 40784 | Second half of lemma 2.17 ... |
jm2.24 40785 | Lemma 2.24 of [JonesMatija... |
rmygeid 40786 | Y(n) increases faster than... |
congtr 40787 | A wff of the form ` A || (... |
congadd 40788 | If two pairs of numbers ar... |
congmul 40789 | If two pairs of numbers ar... |
congsym 40790 | Congruence mod ` A ` is a ... |
congneg 40791 | If two integers are congru... |
congsub 40792 | If two pairs of numbers ar... |
congid 40793 | Every integer is congruent... |
mzpcong 40794 | Polynomials commute with c... |
congrep 40795 | Every integer is congruent... |
congabseq 40796 | If two integers are congru... |
acongid 40797 | A wff like that in this th... |
acongsym 40798 | Symmetry of alternating co... |
acongneg2 40799 | Negate right side of alter... |
acongtr 40800 | Transitivity of alternatin... |
acongeq12d 40801 | Substitution deduction for... |
acongrep 40802 | Every integer is alternati... |
fzmaxdif 40803 | Bound on the difference be... |
fzneg 40804 | Reflection of a finite ran... |
acongeq 40805 | Two numbers in the fundame... |
dvdsacongtr 40806 | Alternating congruence pas... |
coprmdvdsb 40807 | Multiplication by a coprim... |
modabsdifz 40808 | Divisibility in terms of m... |
dvdsabsmod0 40809 | Divisibility in terms of m... |
jm2.18 40810 | Theorem 2.18 of [JonesMati... |
jm2.19lem1 40811 | Lemma for ~ jm2.19 . X an... |
jm2.19lem2 40812 | Lemma for ~ jm2.19 . (Con... |
jm2.19lem3 40813 | Lemma for ~ jm2.19 . (Con... |
jm2.19lem4 40814 | Lemma for ~ jm2.19 . Exte... |
jm2.19 40815 | Lemma 2.19 of [JonesMatija... |
jm2.21 40816 | Lemma for ~ jm2.20nn . Ex... |
jm2.22 40817 | Lemma for ~ jm2.20nn . Ap... |
jm2.23 40818 | Lemma for ~ jm2.20nn . Tr... |
jm2.20nn 40819 | Lemma 2.20 of [JonesMatija... |
jm2.25lem1 40820 | Lemma for ~ jm2.26 . (Con... |
jm2.25 40821 | Lemma for ~ jm2.26 . Rema... |
jm2.26a 40822 | Lemma for ~ jm2.26 . Reve... |
jm2.26lem3 40823 | Lemma for ~ jm2.26 . Use ... |
jm2.26 40824 | Lemma 2.26 of [JonesMatija... |
jm2.15nn0 40825 | Lemma 2.15 of [JonesMatija... |
jm2.16nn0 40826 | Lemma 2.16 of [JonesMatija... |
jm2.27a 40827 | Lemma for ~ jm2.27 . Reve... |
jm2.27b 40828 | Lemma for ~ jm2.27 . Expa... |
jm2.27c 40829 | Lemma for ~ jm2.27 . Forw... |
jm2.27 40830 | Lemma 2.27 of [JonesMatija... |
jm2.27dlem1 40831 | Lemma for ~ rmydioph . Su... |
jm2.27dlem2 40832 | Lemma for ~ rmydioph . Th... |
jm2.27dlem3 40833 | Lemma for ~ rmydioph . In... |
jm2.27dlem4 40834 | Lemma for ~ rmydioph . In... |
jm2.27dlem5 40835 | Lemma for ~ rmydioph . Us... |
rmydioph 40836 | ~ jm2.27 restated in terms... |
rmxdiophlem 40837 | X can be expressed in term... |
rmxdioph 40838 | X is a Diophantine functio... |
jm3.1lem1 40839 | Lemma for ~ jm3.1 . (Cont... |
jm3.1lem2 40840 | Lemma for ~ jm3.1 . (Cont... |
jm3.1lem3 40841 | Lemma for ~ jm3.1 . (Cont... |
jm3.1 40842 | Diophantine expression for... |
expdiophlem1 40843 | Lemma for ~ expdioph . Fu... |
expdiophlem2 40844 | Lemma for ~ expdioph . Ex... |
expdioph 40845 | The exponential function i... |
setindtr 40846 | Set induction for sets con... |
setindtrs 40847 | Set induction scheme witho... |
dford3lem1 40848 | Lemma for ~ dford3 . (Con... |
dford3lem2 40849 | Lemma for ~ dford3 . (Con... |
dford3 40850 | Ordinals are precisely the... |
dford4 40851 | ~ dford3 expressed in prim... |
wopprc 40852 | Unrelated: Wiener pairs t... |
rpnnen3lem 40853 | Lemma for ~ rpnnen3 . (Co... |
rpnnen3 40854 | Dedekind cut injection of ... |
axac10 40855 | Characterization of choice... |
harinf 40856 | The Hartogs number of an i... |
wdom2d2 40857 | Deduction for weak dominan... |
ttac 40858 | Tarski's theorem about cho... |
pw2f1ocnv 40859 | Define a bijection between... |
pw2f1o2 40860 | Define a bijection between... |
pw2f1o2val 40861 | Function value of the ~ pw... |
pw2f1o2val2 40862 | Membership in a mapped set... |
soeq12d 40863 | Equality deduction for tot... |
freq12d 40864 | Equality deduction for fou... |
weeq12d 40865 | Equality deduction for wel... |
limsuc2 40866 | Limit ordinals in the sens... |
wepwsolem 40867 | Transfer an ordering on ch... |
wepwso 40868 | A well-ordering induces a ... |
dnnumch1 40869 | Define an enumeration of a... |
dnnumch2 40870 | Define an enumeration (wea... |
dnnumch3lem 40871 | Value of the ordinal injec... |
dnnumch3 40872 | Define an injection from a... |
dnwech 40873 | Define a well-ordering fro... |
fnwe2val 40874 | Lemma for ~ fnwe2 . Subst... |
fnwe2lem1 40875 | Lemma for ~ fnwe2 . Subst... |
fnwe2lem2 40876 | Lemma for ~ fnwe2 . An el... |
fnwe2lem3 40877 | Lemma for ~ fnwe2 . Trich... |
fnwe2 40878 | A well-ordering can be con... |
aomclem1 40879 | Lemma for ~ dfac11 . This... |
aomclem2 40880 | Lemma for ~ dfac11 . Succ... |
aomclem3 40881 | Lemma for ~ dfac11 . Succ... |
aomclem4 40882 | Lemma for ~ dfac11 . Limi... |
aomclem5 40883 | Lemma for ~ dfac11 . Comb... |
aomclem6 40884 | Lemma for ~ dfac11 . Tran... |
aomclem7 40885 | Lemma for ~ dfac11 . ` ( R... |
aomclem8 40886 | Lemma for ~ dfac11 . Perf... |
dfac11 40887 | The right-hand side of thi... |
kelac1 40888 | Kelley's choice, basic for... |
kelac2lem 40889 | Lemma for ~ kelac2 and ~ d... |
kelac2 40890 | Kelley's choice, most comm... |
dfac21 40891 | Tychonoff's theorem is a c... |
islmodfg 40894 | Property of a finitely gen... |
islssfg 40895 | Property of a finitely gen... |
islssfg2 40896 | Property of a finitely gen... |
islssfgi 40897 | Finitely spanned subspaces... |
fglmod 40898 | Finitely generated left mo... |
lsmfgcl 40899 | The sum of two finitely ge... |
islnm 40902 | Property of being a Noethe... |
islnm2 40903 | Property of being a Noethe... |
lnmlmod 40904 | A Noetherian left module i... |
lnmlssfg 40905 | A submodule of Noetherian ... |
lnmlsslnm 40906 | All submodules of a Noethe... |
lnmfg 40907 | A Noetherian left module i... |
kercvrlsm 40908 | The domain of a linear fun... |
lmhmfgima 40909 | A homomorphism maps finite... |
lnmepi 40910 | Epimorphic images of Noeth... |
lmhmfgsplit 40911 | If the kernel and range of... |
lmhmlnmsplit 40912 | If the kernel and range of... |
lnmlmic 40913 | Noetherian is an invariant... |
pwssplit4 40914 | Splitting for structure po... |
filnm 40915 | Finite left modules are No... |
pwslnmlem0 40916 | Zeroeth powers are Noether... |
pwslnmlem1 40917 | First powers are Noetheria... |
pwslnmlem2 40918 | A sum of powers is Noether... |
pwslnm 40919 | Finite powers of Noetheria... |
unxpwdom3 40920 | Weaker version of ~ unxpwd... |
pwfi2f1o 40921 | The ~ pw2f1o bijection rel... |
pwfi2en 40922 | Finitely supported indicat... |
frlmpwfi 40923 | Formal linear combinations... |
gicabl 40924 | Being Abelian is a group i... |
imasgim 40925 | A relabeling of the elemen... |
isnumbasgrplem1 40926 | A set which is equipollent... |
harn0 40927 | The Hartogs number of a se... |
numinfctb 40928 | A numerable infinite set c... |
isnumbasgrplem2 40929 | If the (to be thought of a... |
isnumbasgrplem3 40930 | Every nonempty numerable s... |
isnumbasabl 40931 | A set is numerable iff it ... |
isnumbasgrp 40932 | A set is numerable iff it ... |
dfacbasgrp 40933 | A choice equivalent in abs... |
islnr 40936 | Property of a left-Noether... |
lnrring 40937 | Left-Noetherian rings are ... |
lnrlnm 40938 | Left-Noetherian rings have... |
islnr2 40939 | Property of being a left-N... |
islnr3 40940 | Relate left-Noetherian rin... |
lnr2i 40941 | Given an ideal in a left-N... |
lpirlnr 40942 | Left principal ideal rings... |
lnrfrlm 40943 | Finite-dimensional free mo... |
lnrfg 40944 | Finitely-generated modules... |
lnrfgtr 40945 | A submodule of a finitely ... |
hbtlem1 40948 | Value of the leading coeff... |
hbtlem2 40949 | Leading coefficient ideals... |
hbtlem7 40950 | Functionality of leading c... |
hbtlem4 40951 | The leading ideal function... |
hbtlem3 40952 | The leading ideal function... |
hbtlem5 40953 | The leading ideal function... |
hbtlem6 40954 | There is a finite set of p... |
hbt 40955 | The Hilbert Basis Theorem ... |
dgrsub2 40960 | Subtracting two polynomial... |
elmnc 40961 | Property of a monic polyno... |
mncply 40962 | A monic polynomial is a po... |
mnccoe 40963 | A monic polynomial has lea... |
mncn0 40964 | A monic polynomial is not ... |
dgraaval 40969 | Value of the degree functi... |
dgraalem 40970 | Properties of the degree o... |
dgraacl 40971 | Closure of the degree func... |
dgraaf 40972 | Degree function on algebra... |
dgraaub 40973 | Upper bound on degree of a... |
dgraa0p 40974 | A rational polynomial of d... |
mpaaeu 40975 | An algebraic number has ex... |
mpaaval 40976 | Value of the minimal polyn... |
mpaalem 40977 | Properties of the minimal ... |
mpaacl 40978 | Minimal polynomial is a po... |
mpaadgr 40979 | Minimal polynomial has deg... |
mpaaroot 40980 | The minimal polynomial of ... |
mpaamn 40981 | Minimal polynomial is moni... |
itgoval 40986 | Value of the integral-over... |
aaitgo 40987 | The standard algebraic num... |
itgoss 40988 | An integral element is int... |
itgocn 40989 | All integral elements are ... |
cnsrexpcl 40990 | Exponentiation is closed i... |
fsumcnsrcl 40991 | Finite sums are closed in ... |
cnsrplycl 40992 | Polynomials are closed in ... |
rgspnval 40993 | Value of the ring-span of ... |
rgspncl 40994 | The ring-span of a set is ... |
rgspnssid 40995 | The ring-span of a set con... |
rgspnmin 40996 | The ring-span is contained... |
rgspnid 40997 | The span of a subring is i... |
rngunsnply 40998 | Adjoining one element to a... |
flcidc 40999 | Finite linear combinations... |
algstr 41002 | Lemma to shorten proofs of... |
algbase 41003 | The base set of a construc... |
algaddg 41004 | The additive operation of ... |
algmulr 41005 | The multiplicative operati... |
algsca 41006 | The set of scalars of a co... |
algvsca 41007 | The scalar product operati... |
mendval 41008 | Value of the module endomo... |
mendbas 41009 | Base set of the module end... |
mendplusgfval 41010 | Addition in the module end... |
mendplusg 41011 | A specific addition in the... |
mendmulrfval 41012 | Multiplication in the modu... |
mendmulr 41013 | A specific multiplication ... |
mendsca 41014 | The module endomorphism al... |
mendvscafval 41015 | Scalar multiplication in t... |
mendvsca 41016 | A specific scalar multipli... |
mendring 41017 | The module endomorphism al... |
mendlmod 41018 | The module endomorphism al... |
mendassa 41019 | The module endomorphism al... |
idomrootle 41020 | No element of an integral ... |
idomodle 41021 | Limit on the number of ` N... |
fiuneneq 41022 | Two finite sets of equal s... |
idomsubgmo 41023 | The units of an integral d... |
proot1mul 41024 | Any primitive ` N ` -th ro... |
proot1hash 41025 | If an integral domain has ... |
proot1ex 41026 | The complex field has prim... |
isdomn3 41029 | Nonzero elements form a mu... |
mon1pid 41030 | Monicity and degree of the... |
mon1psubm 41031 | Monic polynomials are a mu... |
deg1mhm 41032 | Homomorphic property of th... |
cytpfn 41033 | Functionality of the cyclo... |
cytpval 41034 | Substitutions for the Nth ... |
fgraphopab 41035 | Express a function as a su... |
fgraphxp 41036 | Express a function as a su... |
hausgraph 41037 | The graph of a continuous ... |
iocunico 41042 | Split an open interval int... |
iocinico 41043 | The intersection of two se... |
iocmbl 41044 | An open-below, closed-abov... |
cnioobibld 41045 | A bounded, continuous func... |
arearect 41046 | The area of a rectangle wh... |
areaquad 41047 | The area of a quadrilatera... |
nlimsuc 41048 | A successor is not a limit... |
nlim1NEW 41049 | 1 is not a limit ordinal. ... |
nlim2NEW 41050 | 2 is not a limit ordinal. ... |
nlim3 41051 | 3 is not a limit ordinal. ... |
nlim4 41052 | 4 is not a limit ordinal. ... |
oa1un 41053 | Given ` A e. On ` , let ` ... |
oa1cl 41054 | ` A +o 1o ` is in ` On ` .... |
0finon 41055 | 0 is a finite ordinal. Se... |
1finon 41056 | 1 is a finite ordinal. Se... |
2finon 41057 | 2 is a finite ordinal. Se... |
3finon 41058 | 3 is a finite ordinal. Se... |
4finon 41059 | 4 is a finite ordinal. Se... |
finona1cl 41060 | The finite ordinals are cl... |
finonex 41061 | The finite ordinals are a ... |
fzunt 41062 | Union of two adjacent fini... |
fzuntd 41063 | Union of two adjacent fini... |
fzunt1d 41064 | Union of two overlapping f... |
fzuntgd 41065 | Union of two adjacent or o... |
ifpan123g 41066 | Conjunction of conditional... |
ifpan23 41067 | Conjunction of conditional... |
ifpdfor2 41068 | Define or in terms of cond... |
ifporcor 41069 | Corollary of commutation o... |
ifpdfan2 41070 | Define and with conditiona... |
ifpancor 41071 | Corollary of commutation o... |
ifpdfor 41072 | Define or in terms of cond... |
ifpdfan 41073 | Define and with conditiona... |
ifpbi2 41074 | Equivalence theorem for co... |
ifpbi3 41075 | Equivalence theorem for co... |
ifpim1 41076 | Restate implication as con... |
ifpnot 41077 | Restate negated wff as con... |
ifpid2 41078 | Restate wff as conditional... |
ifpim2 41079 | Restate implication as con... |
ifpbi23 41080 | Equivalence theorem for co... |
ifpbiidcor 41081 | Restatement of ~ biid . (... |
ifpbicor 41082 | Corollary of commutation o... |
ifpxorcor 41083 | Corollary of commutation o... |
ifpbi1 41084 | Equivalence theorem for co... |
ifpnot23 41085 | Negation of conditional lo... |
ifpnotnotb 41086 | Factor conditional logic o... |
ifpnorcor 41087 | Corollary of commutation o... |
ifpnancor 41088 | Corollary of commutation o... |
ifpnot23b 41089 | Negation of conditional lo... |
ifpbiidcor2 41090 | Restatement of ~ biid . (... |
ifpnot23c 41091 | Negation of conditional lo... |
ifpnot23d 41092 | Negation of conditional lo... |
ifpdfnan 41093 | Define nand as conditional... |
ifpdfxor 41094 | Define xor as conditional ... |
ifpbi12 41095 | Equivalence theorem for co... |
ifpbi13 41096 | Equivalence theorem for co... |
ifpbi123 41097 | Equivalence theorem for co... |
ifpidg 41098 | Restate wff as conditional... |
ifpid3g 41099 | Restate wff as conditional... |
ifpid2g 41100 | Restate wff as conditional... |
ifpid1g 41101 | Restate wff as conditional... |
ifpim23g 41102 | Restate implication as con... |
ifpim3 41103 | Restate implication as con... |
ifpnim1 41104 | Restate negated implicatio... |
ifpim4 41105 | Restate implication as con... |
ifpnim2 41106 | Restate negated implicatio... |
ifpim123g 41107 | Implication of conditional... |
ifpim1g 41108 | Implication of conditional... |
ifp1bi 41109 | Substitute the first eleme... |
ifpbi1b 41110 | When the first variable is... |
ifpimimb 41111 | Factor conditional logic o... |
ifpororb 41112 | Factor conditional logic o... |
ifpananb 41113 | Factor conditional logic o... |
ifpnannanb 41114 | Factor conditional logic o... |
ifpor123g 41115 | Disjunction of conditional... |
ifpimim 41116 | Consequnce of implication.... |
ifpbibib 41117 | Factor conditional logic o... |
ifpxorxorb 41118 | Factor conditional logic o... |
rp-fakeimass 41119 | A special case where impli... |
rp-fakeanorass 41120 | A special case where a mix... |
rp-fakeoranass 41121 | A special case where a mix... |
rp-fakeinunass 41122 | A special case where a mix... |
rp-fakeuninass 41123 | A special case where a mix... |
rp-isfinite5 41124 | A set is said to be finite... |
rp-isfinite6 41125 | A set is said to be finite... |
intabssd 41126 | When for each element ` y ... |
eu0 41127 | There is only one empty se... |
epelon2 41128 | Over the ordinal numbers, ... |
ontric3g 41129 | For all ` x , y e. On ` , ... |
dfsucon 41130 | ` A ` is called a successo... |
snen1g 41131 | A singleton is equinumerou... |
snen1el 41132 | A singleton is equinumerou... |
sn1dom 41133 | A singleton is dominated b... |
pr2dom 41134 | An unordered pair is domin... |
tr3dom 41135 | An unordered triple is dom... |
ensucne0 41136 | A class equinumerous to a ... |
ensucne0OLD 41137 | A class equinumerous to a ... |
dfom6 41138 | Let ` _om ` be defined to ... |
infordmin 41139 | ` _om ` is the smallest in... |
iscard4 41140 | Two ways to express the pr... |
minregex 41141 | Given any cardinal number ... |
minregex2 41142 | Given any cardinal number ... |
iscard5 41143 | Two ways to express the pr... |
elrncard 41144 | Let us define a cardinal n... |
harval3 41145 | ` ( har `` A ) ` is the le... |
harval3on 41146 | For any ordinal number ` A... |
omssrncard 41147 | All natural numbers are ca... |
0iscard 41148 | 0 is a cardinal number. (... |
1iscard 41149 | 1 is a cardinal number. (... |
omiscard 41150 | ` _om ` is a cardinal numb... |
sucomisnotcard 41151 | ` _om +o 1o ` is not a car... |
nna1iscard 41152 | For any natural number, th... |
har2o 41153 | The least cardinal greater... |
en2pr 41154 | A class is equinumerous to... |
pr2cv 41155 | If an unordered pair is eq... |
pr2el1 41156 | If an unordered pair is eq... |
pr2cv1 41157 | If an unordered pair is eq... |
pr2el2 41158 | If an unordered pair is eq... |
pr2cv2 41159 | If an unordered pair is eq... |
pren2 41160 | An unordered pair is equin... |
pr2eldif1 41161 | If an unordered pair is eq... |
pr2eldif2 41162 | If an unordered pair is eq... |
pren2d 41163 | A pair of two distinct set... |
aleph1min 41164 | ` ( aleph `` 1o ) ` is the... |
alephiso2 41165 | ` aleph ` is a strictly or... |
alephiso3 41166 | ` aleph ` is a strictly or... |
pwelg 41167 | The powerclass is an eleme... |
pwinfig 41168 | The powerclass of an infin... |
pwinfi2 41169 | The powerclass of an infin... |
pwinfi3 41170 | The powerclass of an infin... |
pwinfi 41171 | The powerclass of an infin... |
fipjust 41172 | A definition of the finite... |
cllem0 41173 | The class of all sets with... |
superficl 41174 | The class of all supersets... |
superuncl 41175 | The class of all supersets... |
ssficl 41176 | The class of all subsets o... |
ssuncl 41177 | The class of all subsets o... |
ssdifcl 41178 | The class of all subsets o... |
sssymdifcl 41179 | The class of all subsets o... |
fiinfi 41180 | If two classes have the fi... |
rababg 41181 | Condition when restricted ... |
elintabg 41182 | Two ways of saying a set i... |
elinintab 41183 | Two ways of saying a set i... |
elmapintrab 41184 | Two ways to say a set is a... |
elinintrab 41185 | Two ways of saying a set i... |
inintabss 41186 | Upper bound on intersectio... |
inintabd 41187 | Value of the intersection ... |
xpinintabd 41188 | Value of the intersection ... |
relintabex 41189 | If the intersection of a c... |
elcnvcnvintab 41190 | Two ways of saying a set i... |
relintab 41191 | Value of the intersection ... |
nonrel 41192 | A non-relation is equal to... |
elnonrel 41193 | Only an ordered pair where... |
cnvssb 41194 | Subclass theorem for conve... |
relnonrel 41195 | The non-relation part of a... |
cnvnonrel 41196 | The converse of the non-re... |
brnonrel 41197 | A non-relation cannot rela... |
dmnonrel 41198 | The domain of the non-rela... |
rnnonrel 41199 | The range of the non-relat... |
resnonrel 41200 | A restriction of the non-r... |
imanonrel 41201 | An image under the non-rel... |
cononrel1 41202 | Composition with the non-r... |
cononrel2 41203 | Composition with the non-r... |
elmapintab 41204 | Two ways to say a set is a... |
fvnonrel 41205 | The function value of any ... |
elinlem 41206 | Two ways to say a set is a... |
elcnvcnvlem 41207 | Two ways to say a set is a... |
cnvcnvintabd 41208 | Value of the relationship ... |
elcnvlem 41209 | Two ways to say a set is a... |
elcnvintab 41210 | Two ways of saying a set i... |
cnvintabd 41211 | Value of the converse of t... |
undmrnresiss 41212 | Two ways of saying the ide... |
reflexg 41213 | Two ways of saying a relat... |
cnvssco 41214 | A condition weaker than re... |
refimssco 41215 | Reflexive relations are su... |
cleq2lem 41216 | Equality implies bijection... |
cbvcllem 41217 | Change of bound variable i... |
clublem 41218 | If a superset ` Y ` of ` X... |
clss2lem 41219 | The closure of a property ... |
dfid7 41220 | Definition of identity rel... |
mptrcllem 41221 | Show two versions of a clo... |
cotrintab 41222 | The intersection of a clas... |
rclexi 41223 | The reflexive closure of a... |
rtrclexlem 41224 | Existence of relation impl... |
rtrclex 41225 | The reflexive-transitive c... |
trclubgNEW 41226 | If a relation exists then ... |
trclubNEW 41227 | If a relation exists then ... |
trclexi 41228 | The transitive closure of ... |
rtrclexi 41229 | The reflexive-transitive c... |
clrellem 41230 | When the property ` ps ` h... |
clcnvlem 41231 | When ` A ` , an upper boun... |
cnvtrucl0 41232 | The converse of the trivia... |
cnvrcl0 41233 | The converse of the reflex... |
cnvtrcl0 41234 | The converse of the transi... |
dmtrcl 41235 | The domain of the transiti... |
rntrcl 41236 | The range of the transitiv... |
dfrtrcl5 41237 | Definition of reflexive-tr... |
trcleq2lemRP 41238 | Equality implies bijection... |
sqrtcvallem1 41239 | Two ways of saying a compl... |
reabsifneg 41240 | Alternate expression for t... |
reabsifnpos 41241 | Alternate expression for t... |
reabsifpos 41242 | Alternate expression for t... |
reabsifnneg 41243 | Alternate expression for t... |
reabssgn 41244 | Alternate expression for t... |
sqrtcvallem2 41245 | Equivalent to saying that ... |
sqrtcvallem3 41246 | Equivalent to saying that ... |
sqrtcvallem4 41247 | Equivalent to saying that ... |
sqrtcvallem5 41248 | Equivalent to saying that ... |
sqrtcval 41249 | Explicit formula for the c... |
sqrtcval2 41250 | Explicit formula for the c... |
resqrtval 41251 | Real part of the complex s... |
imsqrtval 41252 | Imaginary part of the comp... |
resqrtvalex 41253 | Example for ~ resqrtval . ... |
imsqrtvalex 41254 | Example for ~ imsqrtval . ... |
al3im 41255 | Version of ~ ax-4 for a ne... |
intima0 41256 | Two ways of expressing the... |
elimaint 41257 | Element of image of inters... |
cnviun 41258 | Converse of indexed union.... |
imaiun1 41259 | The image of an indexed un... |
coiun1 41260 | Composition with an indexe... |
elintima 41261 | Element of intersection of... |
intimass 41262 | The image under the inters... |
intimass2 41263 | The image under the inters... |
intimag 41264 | Requirement for the image ... |
intimasn 41265 | Two ways to express the im... |
intimasn2 41266 | Two ways to express the im... |
ss2iundf 41267 | Subclass theorem for index... |
ss2iundv 41268 | Subclass theorem for index... |
cbviuneq12df 41269 | Rule used to change the bo... |
cbviuneq12dv 41270 | Rule used to change the bo... |
conrel1d 41271 | Deduction about compositio... |
conrel2d 41272 | Deduction about compositio... |
trrelind 41273 | The intersection of transi... |
xpintrreld 41274 | The intersection of a tran... |
restrreld 41275 | The restriction of a trans... |
trrelsuperreldg 41276 | Concrete construction of a... |
trficl 41277 | The class of all transitiv... |
cnvtrrel 41278 | The converse of a transiti... |
trrelsuperrel2dg 41279 | Concrete construction of a... |
dfrcl2 41282 | Reflexive closure of a rel... |
dfrcl3 41283 | Reflexive closure of a rel... |
dfrcl4 41284 | Reflexive closure of a rel... |
relexp2 41285 | A set operated on by the r... |
relexpnul 41286 | If the domain and range of... |
eliunov2 41287 | Membership in the indexed ... |
eltrclrec 41288 | Membership in the indexed ... |
elrtrclrec 41289 | Membership in the indexed ... |
briunov2 41290 | Two classes related by the... |
brmptiunrelexpd 41291 | If two elements are connec... |
fvmptiunrelexplb0d 41292 | If the indexed union range... |
fvmptiunrelexplb0da 41293 | If the indexed union range... |
fvmptiunrelexplb1d 41294 | If the indexed union range... |
brfvid 41295 | If two elements are connec... |
brfvidRP 41296 | If two elements are connec... |
fvilbd 41297 | A set is a subset of its i... |
fvilbdRP 41298 | A set is a subset of its i... |
brfvrcld 41299 | If two elements are connec... |
brfvrcld2 41300 | If two elements are connec... |
fvrcllb0d 41301 | A restriction of the ident... |
fvrcllb0da 41302 | A restriction of the ident... |
fvrcllb1d 41303 | A set is a subset of its i... |
brtrclrec 41304 | Two classes related by the... |
brrtrclrec 41305 | Two classes related by the... |
briunov2uz 41306 | Two classes related by the... |
eliunov2uz 41307 | Membership in the indexed ... |
ov2ssiunov2 41308 | Any particular operator va... |
relexp0eq 41309 | The zeroth power of relati... |
iunrelexp0 41310 | Simplification of zeroth p... |
relexpxpnnidm 41311 | Any positive power of a Ca... |
relexpiidm 41312 | Any power of any restricti... |
relexpss1d 41313 | The relational power of a ... |
comptiunov2i 41314 | The composition two indexe... |
corclrcl 41315 | The reflexive closure is i... |
iunrelexpmin1 41316 | The indexed union of relat... |
relexpmulnn 41317 | With exponents limited to ... |
relexpmulg 41318 | With ordered exponents, th... |
trclrelexplem 41319 | The union of relational po... |
iunrelexpmin2 41320 | The indexed union of relat... |
relexp01min 41321 | With exponents limited to ... |
relexp1idm 41322 | Repeated raising a relatio... |
relexp0idm 41323 | Repeated raising a relatio... |
relexp0a 41324 | Absorbtion law for zeroth ... |
relexpxpmin 41325 | The composition of powers ... |
relexpaddss 41326 | The composition of two pow... |
iunrelexpuztr 41327 | The indexed union of relat... |
dftrcl3 41328 | Transitive closure of a re... |
brfvtrcld 41329 | If two elements are connec... |
fvtrcllb1d 41330 | A set is a subset of its i... |
trclfvcom 41331 | The transitive closure of ... |
cnvtrclfv 41332 | The converse of the transi... |
cotrcltrcl 41333 | The transitive closure is ... |
trclimalb2 41334 | Lower bound for image unde... |
brtrclfv2 41335 | Two ways to indicate two e... |
trclfvdecomr 41336 | The transitive closure of ... |
trclfvdecoml 41337 | The transitive closure of ... |
dmtrclfvRP 41338 | The domain of the transiti... |
rntrclfvRP 41339 | The range of the transitiv... |
rntrclfv 41340 | The range of the transitiv... |
dfrtrcl3 41341 | Reflexive-transitive closu... |
brfvrtrcld 41342 | If two elements are connec... |
fvrtrcllb0d 41343 | A restriction of the ident... |
fvrtrcllb0da 41344 | A restriction of the ident... |
fvrtrcllb1d 41345 | A set is a subset of its i... |
dfrtrcl4 41346 | Reflexive-transitive closu... |
corcltrcl 41347 | The composition of the ref... |
cortrcltrcl 41348 | Composition with the refle... |
corclrtrcl 41349 | Composition with the refle... |
cotrclrcl 41350 | The composition of the ref... |
cortrclrcl 41351 | Composition with the refle... |
cotrclrtrcl 41352 | Composition with the refle... |
cortrclrtrcl 41353 | The reflexive-transitive c... |
frege77d 41354 | If the images of both ` { ... |
frege81d 41355 | If the image of ` U ` is a... |
frege83d 41356 | If the image of the union ... |
frege96d 41357 | If ` C ` follows ` A ` in ... |
frege87d 41358 | If the images of both ` { ... |
frege91d 41359 | If ` B ` follows ` A ` in ... |
frege97d 41360 | If ` A ` contains all elem... |
frege98d 41361 | If ` C ` follows ` A ` and... |
frege102d 41362 | If either ` A ` and ` C ` ... |
frege106d 41363 | If ` B ` follows ` A ` in ... |
frege108d 41364 | If either ` A ` and ` C ` ... |
frege109d 41365 | If ` A ` contains all elem... |
frege114d 41366 | If either ` R ` relates ` ... |
frege111d 41367 | If either ` A ` and ` C ` ... |
frege122d 41368 | If ` F ` is a function, ` ... |
frege124d 41369 | If ` F ` is a function, ` ... |
frege126d 41370 | If ` F ` is a function, ` ... |
frege129d 41371 | If ` F ` is a function and... |
frege131d 41372 | If ` F ` is a function and... |
frege133d 41373 | If ` F ` is a function and... |
dfxor4 41374 | Express exclusive-or in te... |
dfxor5 41375 | Express exclusive-or in te... |
df3or2 41376 | Express triple-or in terms... |
df3an2 41377 | Express triple-and in term... |
nev 41378 | Express that not every set... |
0pssin 41379 | Express that an intersecti... |
dfhe2 41382 | The property of relation `... |
dfhe3 41383 | The property of relation `... |
heeq12 41384 | Equality law for relations... |
heeq1 41385 | Equality law for relations... |
heeq2 41386 | Equality law for relations... |
sbcheg 41387 | Distribute proper substitu... |
hess 41388 | Subclass law for relations... |
xphe 41389 | Any Cartesian product is h... |
0he 41390 | The empty relation is here... |
0heALT 41391 | The empty relation is here... |
he0 41392 | Any relation is hereditary... |
unhe1 41393 | The union of two relations... |
snhesn 41394 | Any singleton is hereditar... |
idhe 41395 | The identity relation is h... |
psshepw 41396 | The relation between sets ... |
sshepw 41397 | The relation between sets ... |
rp-simp2-frege 41400 | Simplification of triple c... |
rp-simp2 41401 | Simplification of triple c... |
rp-frege3g 41402 | Add antecedent to ~ ax-fre... |
frege3 41403 | Add antecedent to ~ ax-fre... |
rp-misc1-frege 41404 | Double-use of ~ ax-frege2 ... |
rp-frege24 41405 | Introducing an embedded an... |
rp-frege4g 41406 | Deduction related to distr... |
frege4 41407 | Special case of closed for... |
frege5 41408 | A closed form of ~ syl . ... |
rp-7frege 41409 | Distribute antecedent and ... |
rp-4frege 41410 | Elimination of a nested an... |
rp-6frege 41411 | Elimination of a nested an... |
rp-8frege 41412 | Eliminate antecedent when ... |
rp-frege25 41413 | Closed form for ~ a1dd . ... |
frege6 41414 | A closed form of ~ imim2d ... |
axfrege8 41415 | Swap antecedents. Identic... |
frege7 41416 | A closed form of ~ syl6 . ... |
frege26 41418 | Identical to ~ idd . Prop... |
frege27 41419 | We cannot (at the same tim... |
frege9 41420 | Closed form of ~ syl with ... |
frege12 41421 | A closed form of ~ com23 .... |
frege11 41422 | Elimination of a nested an... |
frege24 41423 | Closed form for ~ a1d . D... |
frege16 41424 | A closed form of ~ com34 .... |
frege25 41425 | Closed form for ~ a1dd . ... |
frege18 41426 | Closed form of a syllogism... |
frege22 41427 | A closed form of ~ com45 .... |
frege10 41428 | Result commuting anteceden... |
frege17 41429 | A closed form of ~ com3l .... |
frege13 41430 | A closed form of ~ com3r .... |
frege14 41431 | Closed form of a deduction... |
frege19 41432 | A closed form of ~ syl6 . ... |
frege23 41433 | Syllogism followed by rota... |
frege15 41434 | A closed form of ~ com4r .... |
frege21 41435 | Replace antecedent in ante... |
frege20 41436 | A closed form of ~ syl8 . ... |
axfrege28 41437 | Contraposition. Identical... |
frege29 41439 | Closed form of ~ con3d . ... |
frege30 41440 | Commuted, closed form of ~... |
axfrege31 41441 | Identical to ~ notnotr . ... |
frege32 41443 | Deduce ~ con1 from ~ con3 ... |
frege33 41444 | If ` ph ` or ` ps ` takes ... |
frege34 41445 | If as a conseqence of the ... |
frege35 41446 | Commuted, closed form of ~... |
frege36 41447 | The case in which ` ps ` i... |
frege37 41448 | If ` ch ` is a necessary c... |
frege38 41449 | Identical to ~ pm2.21 . P... |
frege39 41450 | Syllogism between ~ pm2.18... |
frege40 41451 | Anything implies ~ pm2.18 ... |
axfrege41 41452 | Identical to ~ notnot . A... |
frege42 41454 | Not not ~ id . Propositio... |
frege43 41455 | If there is a choice only ... |
frege44 41456 | Similar to a commuted ~ pm... |
frege45 41457 | Deduce ~ pm2.6 from ~ con1... |
frege46 41458 | If ` ps ` holds when ` ph ... |
frege47 41459 | Deduce consequence follows... |
frege48 41460 | Closed form of syllogism w... |
frege49 41461 | Closed form of deduction w... |
frege50 41462 | Closed form of ~ jaoi . P... |
frege51 41463 | Compare with ~ jaod . Pro... |
axfrege52a 41464 | Justification for ~ ax-fre... |
frege52aid 41466 | The case when the content ... |
frege53aid 41467 | Specialization of ~ frege5... |
frege53a 41468 | Lemma for ~ frege55a . Pr... |
axfrege54a 41469 | Justification for ~ ax-fre... |
frege54cor0a 41471 | Synonym for logical equiva... |
frege54cor1a 41472 | Reflexive equality. (Cont... |
frege55aid 41473 | Lemma for ~ frege57aid . ... |
frege55lem1a 41474 | Necessary deduction regard... |
frege55lem2a 41475 | Core proof of Proposition ... |
frege55a 41476 | Proposition 55 of [Frege18... |
frege55cor1a 41477 | Proposition 55 of [Frege18... |
frege56aid 41478 | Lemma for ~ frege57aid . ... |
frege56a 41479 | Proposition 56 of [Frege18... |
frege57aid 41480 | This is the all imporant f... |
frege57a 41481 | Analogue of ~ frege57aid .... |
axfrege58a 41482 | Identical to ~ anifp . Ju... |
frege58acor 41484 | Lemma for ~ frege59a . (C... |
frege59a 41485 | A kind of Aristotelian inf... |
frege60a 41486 | Swap antecedents of ~ ax-f... |
frege61a 41487 | Lemma for ~ frege65a . Pr... |
frege62a 41488 | A kind of Aristotelian inf... |
frege63a 41489 | Proposition 63 of [Frege18... |
frege64a 41490 | Lemma for ~ frege65a . Pr... |
frege65a 41491 | A kind of Aristotelian inf... |
frege66a 41492 | Swap antecedents of ~ freg... |
frege67a 41493 | Lemma for ~ frege68a . Pr... |
frege68a 41494 | Combination of applying a ... |
axfrege52c 41495 | Justification for ~ ax-fre... |
frege52b 41497 | The case when the content ... |
frege53b 41498 | Lemma for frege102 (via ~ ... |
axfrege54c 41499 | Reflexive equality of clas... |
frege54b 41501 | Reflexive equality of sets... |
frege54cor1b 41502 | Reflexive equality. (Cont... |
frege55lem1b 41503 | Necessary deduction regard... |
frege55lem2b 41504 | Lemma for ~ frege55b . Co... |
frege55b 41505 | Lemma for ~ frege57b . Pr... |
frege56b 41506 | Lemma for ~ frege57b . Pr... |
frege57b 41507 | Analogue of ~ frege57aid .... |
axfrege58b 41508 | If ` A. x ph ` is affirmed... |
frege58bid 41510 | If ` A. x ph ` is affirmed... |
frege58bcor 41511 | Lemma for ~ frege59b . (C... |
frege59b 41512 | A kind of Aristotelian inf... |
frege60b 41513 | Swap antecedents of ~ ax-f... |
frege61b 41514 | Lemma for ~ frege65b . Pr... |
frege62b 41515 | A kind of Aristotelian inf... |
frege63b 41516 | Lemma for ~ frege91 . Pro... |
frege64b 41517 | Lemma for ~ frege65b . Pr... |
frege65b 41518 | A kind of Aristotelian inf... |
frege66b 41519 | Swap antecedents of ~ freg... |
frege67b 41520 | Lemma for ~ frege68b . Pr... |
frege68b 41521 | Combination of applying a ... |
frege53c 41522 | Proposition 53 of [Frege18... |
frege54cor1c 41523 | Reflexive equality. (Cont... |
frege55lem1c 41524 | Necessary deduction regard... |
frege55lem2c 41525 | Core proof of Proposition ... |
frege55c 41526 | Proposition 55 of [Frege18... |
frege56c 41527 | Lemma for ~ frege57c . Pr... |
frege57c 41528 | Swap order of implication ... |
frege58c 41529 | Principle related to ~ sp ... |
frege59c 41530 | A kind of Aristotelian inf... |
frege60c 41531 | Swap antecedents of ~ freg... |
frege61c 41532 | Lemma for ~ frege65c . Pr... |
frege62c 41533 | A kind of Aristotelian inf... |
frege63c 41534 | Analogue of ~ frege63b . ... |
frege64c 41535 | Lemma for ~ frege65c . Pr... |
frege65c 41536 | A kind of Aristotelian inf... |
frege66c 41537 | Swap antecedents of ~ freg... |
frege67c 41538 | Lemma for ~ frege68c . Pr... |
frege68c 41539 | Combination of applying a ... |
dffrege69 41540 | If from the proposition th... |
frege70 41541 | Lemma for ~ frege72 . Pro... |
frege71 41542 | Lemma for ~ frege72 . Pro... |
frege72 41543 | If property ` A ` is hered... |
frege73 41544 | Lemma for ~ frege87 . Pro... |
frege74 41545 | If ` X ` has a property ` ... |
frege75 41546 | If from the proposition th... |
dffrege76 41547 | If from the two propositio... |
frege77 41548 | If ` Y ` follows ` X ` in ... |
frege78 41549 | Commuted form of of ~ freg... |
frege79 41550 | Distributed form of ~ freg... |
frege80 41551 | Add additional condition t... |
frege81 41552 | If ` X ` has a property ` ... |
frege82 41553 | Closed-form deduction base... |
frege83 41554 | Apply commuted form of ~ f... |
frege84 41555 | Commuted form of ~ frege81... |
frege85 41556 | Commuted form of ~ frege77... |
frege86 41557 | Conclusion about element o... |
frege87 41558 | If ` Z ` is a result of an... |
frege88 41559 | Commuted form of ~ frege87... |
frege89 41560 | One direction of ~ dffrege... |
frege90 41561 | Add antecedent to ~ frege8... |
frege91 41562 | Every result of an applica... |
frege92 41563 | Inference from ~ frege91 .... |
frege93 41564 | Necessary condition for tw... |
frege94 41565 | Looking one past a pair re... |
frege95 41566 | Looking one past a pair re... |
frege96 41567 | Every result of an applica... |
frege97 41568 | The property of following ... |
frege98 41569 | If ` Y ` follows ` X ` and... |
dffrege99 41570 | If ` Z ` is identical with... |
frege100 41571 | One direction of ~ dffrege... |
frege101 41572 | Lemma for ~ frege102 . Pr... |
frege102 41573 | If ` Z ` belongs to the ` ... |
frege103 41574 | Proposition 103 of [Frege1... |
frege104 41575 | Proposition 104 of [Frege1... |
frege105 41576 | Proposition 105 of [Frege1... |
frege106 41577 | Whatever follows ` X ` in ... |
frege107 41578 | Proposition 107 of [Frege1... |
frege108 41579 | If ` Y ` belongs to the ` ... |
frege109 41580 | The property of belonging ... |
frege110 41581 | Proposition 110 of [Frege1... |
frege111 41582 | If ` Y ` belongs to the ` ... |
frege112 41583 | Identity implies belonging... |
frege113 41584 | Proposition 113 of [Frege1... |
frege114 41585 | If ` X ` belongs to the ` ... |
dffrege115 41586 | If from the circumstance t... |
frege116 41587 | One direction of ~ dffrege... |
frege117 41588 | Lemma for ~ frege118 . Pr... |
frege118 41589 | Simplified application of ... |
frege119 41590 | Lemma for ~ frege120 . Pr... |
frege120 41591 | Simplified application of ... |
frege121 41592 | Lemma for ~ frege122 . Pr... |
frege122 41593 | If ` X ` is a result of an... |
frege123 41594 | Lemma for ~ frege124 . Pr... |
frege124 41595 | If ` X ` is a result of an... |
frege125 41596 | Lemma for ~ frege126 . Pr... |
frege126 41597 | If ` M ` follows ` Y ` in ... |
frege127 41598 | Communte antecedents of ~ ... |
frege128 41599 | Lemma for ~ frege129 . Pr... |
frege129 41600 | If the procedure ` R ` is ... |
frege130 41601 | Lemma for ~ frege131 . Pr... |
frege131 41602 | If the procedure ` R ` is ... |
frege132 41603 | Lemma for ~ frege133 . Pr... |
frege133 41604 | If the procedure ` R ` is ... |
enrelmap 41605 | The set of all possible re... |
enrelmapr 41606 | The set of all possible re... |
enmappw 41607 | The set of all mappings fr... |
enmappwid 41608 | The set of all mappings fr... |
rfovd 41609 | Value of the operator, ` (... |
rfovfvd 41610 | Value of the operator, ` (... |
rfovfvfvd 41611 | Value of the operator, ` (... |
rfovcnvf1od 41612 | Properties of the operator... |
rfovcnvd 41613 | Value of the converse of t... |
rfovf1od 41614 | The value of the operator,... |
rfovcnvfvd 41615 | Value of the converse of t... |
fsovd 41616 | Value of the operator, ` (... |
fsovrfovd 41617 | The operator which gives a... |
fsovfvd 41618 | Value of the operator, ` (... |
fsovfvfvd 41619 | Value of the operator, ` (... |
fsovfd 41620 | The operator, ` ( A O B ) ... |
fsovcnvlem 41621 | The ` O ` operator, which ... |
fsovcnvd 41622 | The value of the converse ... |
fsovcnvfvd 41623 | The value of the converse ... |
fsovf1od 41624 | The value of ` ( A O B ) `... |
dssmapfvd 41625 | Value of the duality opera... |
dssmapfv2d 41626 | Value of the duality opera... |
dssmapfv3d 41627 | Value of the duality opera... |
dssmapnvod 41628 | For any base set ` B ` the... |
dssmapf1od 41629 | For any base set ` B ` the... |
dssmap2d 41630 | For any base set ` B ` the... |
or3or 41631 | Decompose disjunction into... |
andi3or 41632 | Distribute over triple dis... |
uneqsn 41633 | If a union of classes is e... |
df3o2 41634 | Ordinal 3 is the unordered... |
df3o3 41635 | Ordinal 3, fully expanded.... |
brfvimex 41636 | If a binary relation holds... |
brovmptimex 41637 | If a binary relation holds... |
brovmptimex1 41638 | If a binary relation holds... |
brovmptimex2 41639 | If a binary relation holds... |
brcoffn 41640 | Conditions allowing the de... |
brcofffn 41641 | Conditions allowing the de... |
brco2f1o 41642 | Conditions allowing the de... |
brco3f1o 41643 | Conditions allowing the de... |
ntrclsbex 41644 | If (pseudo-)interior and (... |
ntrclsrcomplex 41645 | The relative complement of... |
neik0imk0p 41646 | Kuratowski's K0 axiom impl... |
ntrk2imkb 41647 | If an interior function is... |
ntrkbimka 41648 | If the interiors of disjoi... |
ntrk0kbimka 41649 | If the interiors of disjoi... |
clsk3nimkb 41650 | If the base set is not emp... |
clsk1indlem0 41651 | The ansatz closure functio... |
clsk1indlem2 41652 | The ansatz closure functio... |
clsk1indlem3 41653 | The ansatz closure functio... |
clsk1indlem4 41654 | The ansatz closure functio... |
clsk1indlem1 41655 | The ansatz closure functio... |
clsk1independent 41656 | For generalized closure fu... |
neik0pk1imk0 41657 | Kuratowski's K0' and K1 ax... |
isotone1 41658 | Two different ways to say ... |
isotone2 41659 | Two different ways to say ... |
ntrk1k3eqk13 41660 | An interior function is bo... |
ntrclsf1o 41661 | If (pseudo-)interior and (... |
ntrclsnvobr 41662 | If (pseudo-)interior and (... |
ntrclsiex 41663 | If (pseudo-)interior and (... |
ntrclskex 41664 | If (pseudo-)interior and (... |
ntrclsfv1 41665 | If (pseudo-)interior and (... |
ntrclsfv2 41666 | If (pseudo-)interior and (... |
ntrclselnel1 41667 | If (pseudo-)interior and (... |
ntrclselnel2 41668 | If (pseudo-)interior and (... |
ntrclsfv 41669 | The value of the interior ... |
ntrclsfveq1 41670 | If interior and closure fu... |
ntrclsfveq2 41671 | If interior and closure fu... |
ntrclsfveq 41672 | If interior and closure fu... |
ntrclsss 41673 | If interior and closure fu... |
ntrclsneine0lem 41674 | If (pseudo-)interior and (... |
ntrclsneine0 41675 | If (pseudo-)interior and (... |
ntrclscls00 41676 | If (pseudo-)interior and (... |
ntrclsiso 41677 | If (pseudo-)interior and (... |
ntrclsk2 41678 | An interior function is co... |
ntrclskb 41679 | The interiors of disjoint ... |
ntrclsk3 41680 | The intersection of interi... |
ntrclsk13 41681 | The interior of the inters... |
ntrclsk4 41682 | Idempotence of the interio... |
ntrneibex 41683 | If (pseudo-)interior and (... |
ntrneircomplex 41684 | The relative complement of... |
ntrneif1o 41685 | If (pseudo-)interior and (... |
ntrneiiex 41686 | If (pseudo-)interior and (... |
ntrneinex 41687 | If (pseudo-)interior and (... |
ntrneicnv 41688 | If (pseudo-)interior and (... |
ntrneifv1 41689 | If (pseudo-)interior and (... |
ntrneifv2 41690 | If (pseudo-)interior and (... |
ntrneiel 41691 | If (pseudo-)interior and (... |
ntrneifv3 41692 | The value of the neighbors... |
ntrneineine0lem 41693 | If (pseudo-)interior and (... |
ntrneineine1lem 41694 | If (pseudo-)interior and (... |
ntrneifv4 41695 | The value of the interior ... |
ntrneiel2 41696 | Membership in iterated int... |
ntrneineine0 41697 | If (pseudo-)interior and (... |
ntrneineine1 41698 | If (pseudo-)interior and (... |
ntrneicls00 41699 | If (pseudo-)interior and (... |
ntrneicls11 41700 | If (pseudo-)interior and (... |
ntrneiiso 41701 | If (pseudo-)interior and (... |
ntrneik2 41702 | An interior function is co... |
ntrneix2 41703 | An interior (closure) func... |
ntrneikb 41704 | The interiors of disjoint ... |
ntrneixb 41705 | The interiors (closures) o... |
ntrneik3 41706 | The intersection of interi... |
ntrneix3 41707 | The closure of the union o... |
ntrneik13 41708 | The interior of the inters... |
ntrneix13 41709 | The closure of the union o... |
ntrneik4w 41710 | Idempotence of the interio... |
ntrneik4 41711 | Idempotence of the interio... |
clsneibex 41712 | If (pseudo-)closure and (p... |
clsneircomplex 41713 | The relative complement of... |
clsneif1o 41714 | If a (pseudo-)closure func... |
clsneicnv 41715 | If a (pseudo-)closure func... |
clsneikex 41716 | If closure and neighborhoo... |
clsneinex 41717 | If closure and neighborhoo... |
clsneiel1 41718 | If a (pseudo-)closure func... |
clsneiel2 41719 | If a (pseudo-)closure func... |
clsneifv3 41720 | Value of the neighborhoods... |
clsneifv4 41721 | Value of the closure (inte... |
neicvgbex 41722 | If (pseudo-)neighborhood a... |
neicvgrcomplex 41723 | The relative complement of... |
neicvgf1o 41724 | If neighborhood and conver... |
neicvgnvo 41725 | If neighborhood and conver... |
neicvgnvor 41726 | If neighborhood and conver... |
neicvgmex 41727 | If the neighborhoods and c... |
neicvgnex 41728 | If the neighborhoods and c... |
neicvgel1 41729 | A subset being an element ... |
neicvgel2 41730 | The complement of a subset... |
neicvgfv 41731 | The value of the neighborh... |
ntrrn 41732 | The range of the interior ... |
ntrf 41733 | The interior function of a... |
ntrf2 41734 | The interior function is a... |
ntrelmap 41735 | The interior function is a... |
clsf2 41736 | The closure function is a ... |
clselmap 41737 | The closure function is a ... |
dssmapntrcls 41738 | The interior and closure o... |
dssmapclsntr 41739 | The closure and interior o... |
gneispa 41740 | Each point ` p ` of the ne... |
gneispb 41741 | Given a neighborhood ` N `... |
gneispace2 41742 | The predicate that ` F ` i... |
gneispace3 41743 | The predicate that ` F ` i... |
gneispace 41744 | The predicate that ` F ` i... |
gneispacef 41745 | A generic neighborhood spa... |
gneispacef2 41746 | A generic neighborhood spa... |
gneispacefun 41747 | A generic neighborhood spa... |
gneispacern 41748 | A generic neighborhood spa... |
gneispacern2 41749 | A generic neighborhood spa... |
gneispace0nelrn 41750 | A generic neighborhood spa... |
gneispace0nelrn2 41751 | A generic neighborhood spa... |
gneispace0nelrn3 41752 | A generic neighborhood spa... |
gneispaceel 41753 | Every neighborhood of a po... |
gneispaceel2 41754 | Every neighborhood of a po... |
gneispacess 41755 | All supersets of a neighbo... |
gneispacess2 41756 | All supersets of a neighbo... |
k0004lem1 41757 | Application of ~ ssin to r... |
k0004lem2 41758 | A mapping with a particula... |
k0004lem3 41759 | When the value of a mappin... |
k0004val 41760 | The topological simplex of... |
k0004ss1 41761 | The topological simplex of... |
k0004ss2 41762 | The topological simplex of... |
k0004ss3 41763 | The topological simplex of... |
k0004val0 41764 | The topological simplex of... |
inductionexd 41765 | Simple induction example. ... |
wwlemuld 41766 | Natural deduction form of ... |
leeq1d 41767 | Specialization of ~ breq1d... |
leeq2d 41768 | Specialization of ~ breq2d... |
absmulrposd 41769 | Specialization of absmuld ... |
imadisjld 41770 | Natural dduction form of o... |
imadisjlnd 41771 | Natural deduction form of ... |
wnefimgd 41772 | The image of a mapping fro... |
fco2d 41773 | Natural deduction form of ... |
wfximgfd 41774 | The value of a function on... |
extoimad 41775 | If |f(x)| <= C for all x t... |
imo72b2lem0 41776 | Lemma for ~ imo72b2 . (Co... |
suprleubrd 41777 | Natural deduction form of ... |
imo72b2lem2 41778 | Lemma for ~ imo72b2 . (Co... |
suprlubrd 41779 | Natural deduction form of ... |
imo72b2lem1 41780 | Lemma for ~ imo72b2 . (Co... |
lemuldiv3d 41781 | 'Less than or equal to' re... |
lemuldiv4d 41782 | 'Less than or equal to' re... |
imo72b2 41783 | IMO 1972 B2. (14th Intern... |
int-addcomd 41784 | AdditionCommutativity gene... |
int-addassocd 41785 | AdditionAssociativity gene... |
int-addsimpd 41786 | AdditionSimplification gen... |
int-mulcomd 41787 | MultiplicationCommutativit... |
int-mulassocd 41788 | MultiplicationAssociativit... |
int-mulsimpd 41789 | MultiplicationSimplificati... |
int-leftdistd 41790 | AdditionMultiplicationLeft... |
int-rightdistd 41791 | AdditionMultiplicationRigh... |
int-sqdefd 41792 | SquareDefinition generator... |
int-mul11d 41793 | First MultiplicationOne ge... |
int-mul12d 41794 | Second MultiplicationOne g... |
int-add01d 41795 | First AdditionZero generat... |
int-add02d 41796 | Second AdditionZero genera... |
int-sqgeq0d 41797 | SquareGEQZero generator ru... |
int-eqprincd 41798 | PrincipleOfEquality genera... |
int-eqtransd 41799 | EqualityTransitivity gener... |
int-eqmvtd 41800 | EquMoveTerm generator rule... |
int-eqineqd 41801 | EquivalenceImpliesDoubleIn... |
int-ineqmvtd 41802 | IneqMoveTerm generator rul... |
int-ineq1stprincd 41803 | FirstPrincipleOfInequality... |
int-ineq2ndprincd 41804 | SecondPrincipleOfInequalit... |
int-ineqtransd 41805 | InequalityTransitivity gen... |
unitadd 41806 | Theorem used in conjunctio... |
gsumws3 41807 | Valuation of a length 3 wo... |
gsumws4 41808 | Valuation of a length 4 wo... |
amgm2d 41809 | Arithmetic-geometric mean ... |
amgm3d 41810 | Arithmetic-geometric mean ... |
amgm4d 41811 | Arithmetic-geometric mean ... |
spALT 41812 | ~ sp can be proven from th... |
elnelneqd 41813 | Two classes are not equal ... |
elnelneq2d 41814 | Two classes are not equal ... |
rr-spce 41815 | Prove an existential. (Co... |
rexlimdvaacbv 41816 | Unpack a restricted existe... |
rexlimddvcbvw 41817 | Unpack a restricted existe... |
rexlimddvcbv 41818 | Unpack a restricted existe... |
rr-elrnmpt3d 41819 | Elementhood in an image se... |
finnzfsuppd 41820 | If a function is zero outs... |
rr-phpd 41821 | Equivalent of ~ php withou... |
suceqd 41822 | Deduction associated with ... |
tfindsd 41823 | Deduction associated with ... |
mnringvald 41826 | Value of the monoid ring f... |
mnringnmulrd 41827 | Components of a monoid rin... |
mnringnmulrdOLD 41828 | Obsolete version of ~ mnri... |
mnringbased 41829 | The base set of a monoid r... |
mnringbasedOLD 41830 | Obsolete version of ~ mnri... |
mnringbaserd 41831 | The base set of a monoid r... |
mnringelbased 41832 | Membership in the base set... |
mnringbasefd 41833 | Elements of a monoid ring ... |
mnringbasefsuppd 41834 | Elements of a monoid ring ... |
mnringaddgd 41835 | The additive operation of ... |
mnringaddgdOLD 41836 | Obsolete version of ~ mnri... |
mnring0gd 41837 | The additive identity of a... |
mnring0g2d 41838 | The additive identity of a... |
mnringmulrd 41839 | The ring product of a mono... |
mnringscad 41840 | The scalar ring of a monoi... |
mnringscadOLD 41841 | Obsolete version of ~ mnri... |
mnringvscad 41842 | The scalar product of a mo... |
mnringvscadOLD 41843 | Obsolete version of ~ mnri... |
mnringlmodd 41844 | Monoid rings are left modu... |
mnringmulrvald 41845 | Value of multiplication in... |
mnringmulrcld 41846 | Monoid rings are closed un... |
gru0eld 41847 | A nonempty Grothendieck un... |
grusucd 41848 | Grothendieck universes are... |
r1rankcld 41849 | Any rank of the cumulative... |
grur1cld 41850 | Grothendieck universes are... |
grurankcld 41851 | Grothendieck universes are... |
grurankrcld 41852 | If a Grothendieck universe... |
scotteqd 41855 | Equality theorem for the S... |
scotteq 41856 | Closed form of ~ scotteqd ... |
nfscott 41857 | Bound-variable hypothesis ... |
scottabf 41858 | Value of the Scott operati... |
scottab 41859 | Value of the Scott operati... |
scottabes 41860 | Value of the Scott operati... |
scottss 41861 | Scott's trick produces a s... |
elscottab 41862 | An element of the output o... |
scottex2 41863 | ~ scottex expressed using ... |
scotteld 41864 | The Scott operation sends ... |
scottelrankd 41865 | Property of a Scott's tric... |
scottrankd 41866 | Rank of a nonempty Scott's... |
gruscottcld 41867 | If a Grothendieck universe... |
dfcoll2 41870 | Alternate definition of th... |
colleq12d 41871 | Equality theorem for the c... |
colleq1 41872 | Equality theorem for the c... |
colleq2 41873 | Equality theorem for the c... |
nfcoll 41874 | Bound-variable hypothesis ... |
collexd 41875 | The output of the collecti... |
cpcolld 41876 | Property of the collection... |
cpcoll2d 41877 | ~ cpcolld with an extra ex... |
grucollcld 41878 | A Grothendieck universe co... |
ismnu 41879 | The hypothesis of this the... |
mnuop123d 41880 | Operations of a minimal un... |
mnussd 41881 | Minimal universes are clos... |
mnuss2d 41882 | ~ mnussd with arguments pr... |
mnu0eld 41883 | A nonempty minimal univers... |
mnuop23d 41884 | Second and third operation... |
mnupwd 41885 | Minimal universes are clos... |
mnusnd 41886 | Minimal universes are clos... |
mnuprssd 41887 | A minimal universe contain... |
mnuprss2d 41888 | Special case of ~ mnuprssd... |
mnuop3d 41889 | Third operation of a minim... |
mnuprdlem1 41890 | Lemma for ~ mnuprd . (Con... |
mnuprdlem2 41891 | Lemma for ~ mnuprd . (Con... |
mnuprdlem3 41892 | Lemma for ~ mnuprd . (Con... |
mnuprdlem4 41893 | Lemma for ~ mnuprd . Gene... |
mnuprd 41894 | Minimal universes are clos... |
mnuunid 41895 | Minimal universes are clos... |
mnuund 41896 | Minimal universes are clos... |
mnutrcld 41897 | Minimal universes contain ... |
mnutrd 41898 | Minimal universes are tran... |
mnurndlem1 41899 | Lemma for ~ mnurnd . (Con... |
mnurndlem2 41900 | Lemma for ~ mnurnd . Dedu... |
mnurnd 41901 | Minimal universes contain ... |
mnugrud 41902 | Minimal universes are Grot... |
grumnudlem 41903 | Lemma for ~ grumnud . (Co... |
grumnud 41904 | Grothendieck universes are... |
grumnueq 41905 | The class of Grothendieck ... |
expandan 41906 | Expand conjunction to prim... |
expandexn 41907 | Expand an existential quan... |
expandral 41908 | Expand a restricted univer... |
expandrexn 41909 | Expand a restricted existe... |
expandrex 41910 | Expand a restricted existe... |
expanduniss 41911 | Expand ` U. A C_ B ` to pr... |
ismnuprim 41912 | Express the predicate on `... |
rr-grothprimbi 41913 | Express "every set is cont... |
inagrud 41914 | Inaccessible levels of the... |
inaex 41915 | Assuming the Tarski-Grothe... |
gruex 41916 | Assuming the Tarski-Grothe... |
rr-groth 41917 | An equivalent of ~ ax-grot... |
rr-grothprim 41918 | An equivalent of ~ ax-grot... |
ismnushort 41919 | Express the predicate on `... |
dfuniv2 41920 | Alternative definition of ... |
rr-grothshortbi 41921 | Express "every set is cont... |
rr-grothshort 41922 | A shorter equivalent of ~ ... |
nanorxor 41923 | 'nand' is equivalent to th... |
undisjrab 41924 | Union of two disjoint rest... |
iso0 41925 | The empty set is an ` R , ... |
ssrecnpr 41926 | ` RR ` is a subset of both... |
seff 41927 | Let set ` S ` be the real ... |
sblpnf 41928 | The infinity ball in the a... |
prmunb2 41929 | The primes are unbounded. ... |
dvgrat 41930 | Ratio test for divergence ... |
cvgdvgrat 41931 | Ratio test for convergence... |
radcnvrat 41932 | Let ` L ` be the limit, if... |
reldvds 41933 | The divides relation is in... |
nznngen 41934 | All positive integers in t... |
nzss 41935 | The set of multiples of _m... |
nzin 41936 | The intersection of the se... |
nzprmdif 41937 | Subtract one prime's multi... |
hashnzfz 41938 | Special case of ~ hashdvds... |
hashnzfz2 41939 | Special case of ~ hashnzfz... |
hashnzfzclim 41940 | As the upper bound ` K ` o... |
caofcan 41941 | Transfer a cancellation la... |
ofsubid 41942 | Function analogue of ~ sub... |
ofmul12 41943 | Function analogue of ~ mul... |
ofdivrec 41944 | Function analogue of ~ div... |
ofdivcan4 41945 | Function analogue of ~ div... |
ofdivdiv2 41946 | Function analogue of ~ div... |
lhe4.4ex1a 41947 | Example of the Fundamental... |
dvsconst 41948 | Derivative of a constant f... |
dvsid 41949 | Derivative of the identity... |
dvsef 41950 | Derivative of the exponent... |
expgrowthi 41951 | Exponential growth and dec... |
dvconstbi 41952 | The derivative of a functi... |
expgrowth 41953 | Exponential growth and dec... |
bccval 41956 | Value of the generalized b... |
bcccl 41957 | Closure of the generalized... |
bcc0 41958 | The generalized binomial c... |
bccp1k 41959 | Generalized binomial coeff... |
bccm1k 41960 | Generalized binomial coeff... |
bccn0 41961 | Generalized binomial coeff... |
bccn1 41962 | Generalized binomial coeff... |
bccbc 41963 | The binomial coefficient a... |
uzmptshftfval 41964 | When ` F ` is a maps-to fu... |
dvradcnv2 41965 | The radius of convergence ... |
binomcxplemwb 41966 | Lemma for ~ binomcxp . Th... |
binomcxplemnn0 41967 | Lemma for ~ binomcxp . Wh... |
binomcxplemrat 41968 | Lemma for ~ binomcxp . As... |
binomcxplemfrat 41969 | Lemma for ~ binomcxp . ~ b... |
binomcxplemradcnv 41970 | Lemma for ~ binomcxp . By... |
binomcxplemdvbinom 41971 | Lemma for ~ binomcxp . By... |
binomcxplemcvg 41972 | Lemma for ~ binomcxp . Th... |
binomcxplemdvsum 41973 | Lemma for ~ binomcxp . Th... |
binomcxplemnotnn0 41974 | Lemma for ~ binomcxp . Wh... |
binomcxp 41975 | Generalize the binomial th... |
pm10.12 41976 | Theorem *10.12 in [Whitehe... |
pm10.14 41977 | Theorem *10.14 in [Whitehe... |
pm10.251 41978 | Theorem *10.251 in [Whiteh... |
pm10.252 41979 | Theorem *10.252 in [Whiteh... |
pm10.253 41980 | Theorem *10.253 in [Whiteh... |
albitr 41981 | Theorem *10.301 in [Whiteh... |
pm10.42 41982 | Theorem *10.42 in [Whitehe... |
pm10.52 41983 | Theorem *10.52 in [Whitehe... |
pm10.53 41984 | Theorem *10.53 in [Whitehe... |
pm10.541 41985 | Theorem *10.541 in [Whiteh... |
pm10.542 41986 | Theorem *10.542 in [Whiteh... |
pm10.55 41987 | Theorem *10.55 in [Whitehe... |
pm10.56 41988 | Theorem *10.56 in [Whitehe... |
pm10.57 41989 | Theorem *10.57 in [Whitehe... |
2alanimi 41990 | Removes two universal quan... |
2al2imi 41991 | Removes two universal quan... |
pm11.11 41992 | Theorem *11.11 in [Whitehe... |
pm11.12 41993 | Theorem *11.12 in [Whitehe... |
19.21vv 41994 | Compare Theorem *11.3 in [... |
2alim 41995 | Theorem *11.32 in [Whitehe... |
2albi 41996 | Theorem *11.33 in [Whitehe... |
2exim 41997 | Theorem *11.34 in [Whitehe... |
2exbi 41998 | Theorem *11.341 in [Whiteh... |
spsbce-2 41999 | Theorem *11.36 in [Whitehe... |
19.33-2 42000 | Theorem *11.421 in [Whiteh... |
19.36vv 42001 | Theorem *11.43 in [Whitehe... |
19.31vv 42002 | Theorem *11.44 in [Whitehe... |
19.37vv 42003 | Theorem *11.46 in [Whitehe... |
19.28vv 42004 | Theorem *11.47 in [Whitehe... |
pm11.52 42005 | Theorem *11.52 in [Whitehe... |
aaanv 42006 | Theorem *11.56 in [Whitehe... |
pm11.57 42007 | Theorem *11.57 in [Whitehe... |
pm11.58 42008 | Theorem *11.58 in [Whitehe... |
pm11.59 42009 | Theorem *11.59 in [Whitehe... |
pm11.6 42010 | Theorem *11.6 in [Whitehea... |
pm11.61 42011 | Theorem *11.61 in [Whitehe... |
pm11.62 42012 | Theorem *11.62 in [Whitehe... |
pm11.63 42013 | Theorem *11.63 in [Whitehe... |
pm11.7 42014 | Theorem *11.7 in [Whitehea... |
pm11.71 42015 | Theorem *11.71 in [Whitehe... |
sbeqal1 42016 | If ` x = y ` always implie... |
sbeqal1i 42017 | Suppose you know ` x = y `... |
sbeqal2i 42018 | If ` x = y ` implies ` x =... |
axc5c4c711 42019 | Proof of a theorem that ca... |
axc5c4c711toc5 42020 | Rederivation of ~ sp from ... |
axc5c4c711toc4 42021 | Rederivation of ~ axc4 fro... |
axc5c4c711toc7 42022 | Rederivation of ~ axc7 fro... |
axc5c4c711to11 42023 | Rederivation of ~ ax-11 fr... |
axc11next 42024 | This theorem shows that, g... |
pm13.13a 42025 | One result of theorem *13.... |
pm13.13b 42026 | Theorem *13.13 in [Whitehe... |
pm13.14 42027 | Theorem *13.14 in [Whitehe... |
pm13.192 42028 | Theorem *13.192 in [Whiteh... |
pm13.193 42029 | Theorem *13.193 in [Whiteh... |
pm13.194 42030 | Theorem *13.194 in [Whiteh... |
pm13.195 42031 | Theorem *13.195 in [Whiteh... |
pm13.196a 42032 | Theorem *13.196 in [Whiteh... |
2sbc6g 42033 | Theorem *13.21 in [Whitehe... |
2sbc5g 42034 | Theorem *13.22 in [Whitehe... |
iotain 42035 | Equivalence between two di... |
iotaexeu 42036 | The iota class exists. Th... |
iotasbc 42037 | Definition *14.01 in [Whit... |
iotasbc2 42038 | Theorem *14.111 in [Whiteh... |
pm14.12 42039 | Theorem *14.12 in [Whitehe... |
pm14.122a 42040 | Theorem *14.122 in [Whiteh... |
pm14.122b 42041 | Theorem *14.122 in [Whiteh... |
pm14.122c 42042 | Theorem *14.122 in [Whiteh... |
pm14.123a 42043 | Theorem *14.123 in [Whiteh... |
pm14.123b 42044 | Theorem *14.123 in [Whiteh... |
pm14.123c 42045 | Theorem *14.123 in [Whiteh... |
pm14.18 42046 | Theorem *14.18 in [Whitehe... |
iotaequ 42047 | Theorem *14.2 in [Whitehea... |
iotavalb 42048 | Theorem *14.202 in [Whiteh... |
iotasbc5 42049 | Theorem *14.205 in [Whiteh... |
pm14.24 42050 | Theorem *14.24 in [Whitehe... |
iotavalsb 42051 | Theorem *14.242 in [Whiteh... |
sbiota1 42052 | Theorem *14.25 in [Whitehe... |
sbaniota 42053 | Theorem *14.26 in [Whitehe... |
eubiOLD 42054 | Obsolete proof of ~ eubi a... |
iotasbcq 42055 | Theorem *14.272 in [Whiteh... |
elnev 42056 | Any set that contains one ... |
rusbcALT 42057 | A version of Russell's par... |
compeq 42058 | Equality between two ways ... |
compne 42059 | The complement of ` A ` is... |
compab 42060 | Two ways of saying "the co... |
conss2 42061 | Contrapositive law for sub... |
conss1 42062 | Contrapositive law for sub... |
ralbidar 42063 | More general form of ~ ral... |
rexbidar 42064 | More general form of ~ rex... |
dropab1 42065 | Theorem to aid use of the ... |
dropab2 42066 | Theorem to aid use of the ... |
ipo0 42067 | If the identity relation p... |
ifr0 42068 | A class that is founded by... |
ordpss 42069 | ~ ordelpss with an anteced... |
fvsb 42070 | Explicit substitution of a... |
fveqsb 42071 | Implicit substitution of a... |
xpexb 42072 | A Cartesian product exists... |
trelpss 42073 | An element of a transitive... |
addcomgi 42074 | Generalization of commutat... |
addrval 42084 | Value of the operation of ... |
subrval 42085 | Value of the operation of ... |
mulvval 42086 | Value of the operation of ... |
addrfv 42087 | Vector addition at a value... |
subrfv 42088 | Vector subtraction at a va... |
mulvfv 42089 | Scalar multiplication at a... |
addrfn 42090 | Vector addition produces a... |
subrfn 42091 | Vector subtraction produce... |
mulvfn 42092 | Scalar multiplication prod... |
addrcom 42093 | Vector addition is commuta... |
idiALT 42097 | Placeholder for ~ idi . T... |
exbir 42098 | Exportation implication al... |
3impexpbicom 42099 | Version of ~ 3impexp where... |
3impexpbicomi 42100 | Inference associated with ... |
bi1imp 42101 | Importation inference simi... |
bi2imp 42102 | Importation inference simi... |
bi3impb 42103 | Similar to ~ 3impb with im... |
bi3impa 42104 | Similar to ~ 3impa with im... |
bi23impib 42105 | ~ 3impib with the inner im... |
bi13impib 42106 | ~ 3impib with the outer im... |
bi123impib 42107 | ~ 3impib with the implicat... |
bi13impia 42108 | ~ 3impia with the outer im... |
bi123impia 42109 | ~ 3impia with the implicat... |
bi33imp12 42110 | ~ 3imp with innermost impl... |
bi23imp13 42111 | ~ 3imp with middle implica... |
bi13imp23 42112 | ~ 3imp with outermost impl... |
bi13imp2 42113 | Similar to ~ 3imp except t... |
bi12imp3 42114 | Similar to ~ 3imp except a... |
bi23imp1 42115 | Similar to ~ 3imp except a... |
bi123imp0 42116 | Similar to ~ 3imp except a... |
4animp1 42117 | A single hypothesis unific... |
4an31 42118 | A rearrangement of conjunc... |
4an4132 42119 | A rearrangement of conjunc... |
expcomdg 42120 | Biconditional form of ~ ex... |
iidn3 42121 | ~ idn3 without virtual ded... |
ee222 42122 | ~ e222 without virtual ded... |
ee3bir 42123 | Right-biconditional form o... |
ee13 42124 | ~ e13 without virtual dedu... |
ee121 42125 | ~ e121 without virtual ded... |
ee122 42126 | ~ e122 without virtual ded... |
ee333 42127 | ~ e333 without virtual ded... |
ee323 42128 | ~ e323 without virtual ded... |
3ornot23 42129 | If the second and third di... |
orbi1r 42130 | ~ orbi1 with order of disj... |
3orbi123 42131 | ~ pm4.39 with a 3-conjunct... |
syl5imp 42132 | Closed form of ~ syl5 . D... |
impexpd 42133 | The following User's Proof... |
com3rgbi 42134 | The following User's Proof... |
impexpdcom 42135 | The following User's Proof... |
ee1111 42136 | Non-virtual deduction form... |
pm2.43bgbi 42137 | Logical equivalence of a 2... |
pm2.43cbi 42138 | Logical equivalence of a 3... |
ee233 42139 | Non-virtual deduction form... |
imbi13 42140 | Join three logical equival... |
ee33 42141 | Non-virtual deduction form... |
con5 42142 | Biconditional contrapositi... |
con5i 42143 | Inference form of ~ con5 .... |
exlimexi 42144 | Inference similar to Theor... |
sb5ALT 42145 | Equivalence for substituti... |
eexinst01 42146 | ~ exinst01 without virtual... |
eexinst11 42147 | ~ exinst11 without virtual... |
vk15.4j 42148 | Excercise 4j of Unit 15 of... |
notnotrALT 42149 | Converse of double negatio... |
con3ALT2 42150 | Contraposition. Alternate... |
ssralv2 42151 | Quantification restricted ... |
sbc3or 42152 | ~ sbcor with a 3-disjuncts... |
alrim3con13v 42153 | Closed form of ~ alrimi wi... |
rspsbc2 42154 | ~ rspsbc with two quantify... |
sbcoreleleq 42155 | Substitution of a setvar v... |
tratrb 42156 | If a class is transitive a... |
ordelordALT 42157 | An element of an ordinal c... |
sbcim2g 42158 | Distribution of class subs... |
sbcbi 42159 | Implication form of ~ sbcb... |
trsbc 42160 | Formula-building inference... |
truniALT 42161 | The union of a class of tr... |
onfrALTlem5 42162 | Lemma for ~ onfrALT . (Co... |
onfrALTlem4 42163 | Lemma for ~ onfrALT . (Co... |
onfrALTlem3 42164 | Lemma for ~ onfrALT . (Co... |
ggen31 42165 | ~ gen31 without virtual de... |
onfrALTlem2 42166 | Lemma for ~ onfrALT . (Co... |
cbvexsv 42167 | A theorem pertaining to th... |
onfrALTlem1 42168 | Lemma for ~ onfrALT . (Co... |
onfrALT 42169 | The membership relation is... |
19.41rg 42170 | Closed form of right-to-le... |
opelopab4 42171 | Ordered pair membership in... |
2pm13.193 42172 | ~ pm13.193 for two variabl... |
hbntal 42173 | A closed form of ~ hbn . ~... |
hbimpg 42174 | A closed form of ~ hbim . ... |
hbalg 42175 | Closed form of ~ hbal . D... |
hbexg 42176 | Closed form of ~ nfex . D... |
ax6e2eq 42177 | Alternate form of ~ ax6e f... |
ax6e2nd 42178 | If at least two sets exist... |
ax6e2ndeq 42179 | "At least two sets exist" ... |
2sb5nd 42180 | Equivalence for double sub... |
2uasbanh 42181 | Distribute the unabbreviat... |
2uasban 42182 | Distribute the unabbreviat... |
e2ebind 42183 | Absorption of an existenti... |
elpwgded 42184 | ~ elpwgdedVD in convention... |
trelded 42185 | Deduction form of ~ trel .... |
jaoded 42186 | Deduction form of ~ jao . ... |
sbtT 42187 | A substitution into a theo... |
not12an2impnot1 42188 | If a double conjunction is... |
in1 42191 | Inference form of ~ df-vd1... |
iin1 42192 | ~ in1 without virtual dedu... |
dfvd1ir 42193 | Inference form of ~ df-vd1... |
idn1 42194 | Virtual deduction identity... |
dfvd1imp 42195 | Left-to-right part of defi... |
dfvd1impr 42196 | Right-to-left part of defi... |
dfvd2 42199 | Definition of a 2-hypothes... |
dfvd2an 42202 | Definition of a 2-hypothes... |
dfvd2ani 42203 | Inference form of ~ dfvd2a... |
dfvd2anir 42204 | Right-to-left inference fo... |
dfvd2i 42205 | Inference form of ~ dfvd2 ... |
dfvd2ir 42206 | Right-to-left inference fo... |
dfvd3 42211 | Definition of a 3-hypothes... |
dfvd3i 42212 | Inference form of ~ dfvd3 ... |
dfvd3ir 42213 | Right-to-left inference fo... |
dfvd3an 42214 | Definition of a 3-hypothes... |
dfvd3ani 42215 | Inference form of ~ dfvd3a... |
dfvd3anir 42216 | Right-to-left inference fo... |
vd01 42217 | A virtual hypothesis virtu... |
vd02 42218 | Two virtual hypotheses vir... |
vd03 42219 | A theorem is virtually inf... |
vd12 42220 | A virtual deduction with 1... |
vd13 42221 | A virtual deduction with 1... |
vd23 42222 | A virtual deduction with 2... |
dfvd2imp 42223 | The virtual deduction form... |
dfvd2impr 42224 | A 2-antecedent nested impl... |
in2 42225 | The virtual deduction intr... |
int2 42226 | The virtual deduction intr... |
iin2 42227 | ~ in2 without virtual dedu... |
in2an 42228 | The virtual deduction intr... |
in3 42229 | The virtual deduction intr... |
iin3 42230 | ~ in3 without virtual dedu... |
in3an 42231 | The virtual deduction intr... |
int3 42232 | The virtual deduction intr... |
idn2 42233 | Virtual deduction identity... |
iden2 42234 | Virtual deduction identity... |
idn3 42235 | Virtual deduction identity... |
gen11 42236 | Virtual deduction generali... |
gen11nv 42237 | Virtual deduction generali... |
gen12 42238 | Virtual deduction generali... |
gen21 42239 | Virtual deduction generali... |
gen21nv 42240 | Virtual deduction form of ... |
gen31 42241 | Virtual deduction generali... |
gen22 42242 | Virtual deduction generali... |
ggen22 42243 | ~ gen22 without virtual de... |
exinst 42244 | Existential Instantiation.... |
exinst01 42245 | Existential Instantiation.... |
exinst11 42246 | Existential Instantiation.... |
e1a 42247 | A Virtual deduction elimin... |
el1 42248 | A Virtual deduction elimin... |
e1bi 42249 | Biconditional form of ~ e1... |
e1bir 42250 | Right biconditional form o... |
e2 42251 | A virtual deduction elimin... |
e2bi 42252 | Biconditional form of ~ e2... |
e2bir 42253 | Right biconditional form o... |
ee223 42254 | ~ e223 without virtual ded... |
e223 42255 | A virtual deduction elimin... |
e222 42256 | A virtual deduction elimin... |
e220 42257 | A virtual deduction elimin... |
ee220 42258 | ~ e220 without virtual ded... |
e202 42259 | A virtual deduction elimin... |
ee202 42260 | ~ e202 without virtual ded... |
e022 42261 | A virtual deduction elimin... |
ee022 42262 | ~ e022 without virtual ded... |
e002 42263 | A virtual deduction elimin... |
ee002 42264 | ~ e002 without virtual ded... |
e020 42265 | A virtual deduction elimin... |
ee020 42266 | ~ e020 without virtual ded... |
e200 42267 | A virtual deduction elimin... |
ee200 42268 | ~ e200 without virtual ded... |
e221 42269 | A virtual deduction elimin... |
ee221 42270 | ~ e221 without virtual ded... |
e212 42271 | A virtual deduction elimin... |
ee212 42272 | ~ e212 without virtual ded... |
e122 42273 | A virtual deduction elimin... |
e112 42274 | A virtual deduction elimin... |
ee112 42275 | ~ e112 without virtual ded... |
e121 42276 | A virtual deduction elimin... |
e211 42277 | A virtual deduction elimin... |
ee211 42278 | ~ e211 without virtual ded... |
e210 42279 | A virtual deduction elimin... |
ee210 42280 | ~ e210 without virtual ded... |
e201 42281 | A virtual deduction elimin... |
ee201 42282 | ~ e201 without virtual ded... |
e120 42283 | A virtual deduction elimin... |
ee120 42284 | Virtual deduction rule ~ e... |
e021 42285 | A virtual deduction elimin... |
ee021 42286 | ~ e021 without virtual ded... |
e012 42287 | A virtual deduction elimin... |
ee012 42288 | ~ e012 without virtual ded... |
e102 42289 | A virtual deduction elimin... |
ee102 42290 | ~ e102 without virtual ded... |
e22 42291 | A virtual deduction elimin... |
e22an 42292 | Conjunction form of ~ e22 ... |
ee22an 42293 | ~ e22an without virtual de... |
e111 42294 | A virtual deduction elimin... |
e1111 42295 | A virtual deduction elimin... |
e110 42296 | A virtual deduction elimin... |
ee110 42297 | ~ e110 without virtual ded... |
e101 42298 | A virtual deduction elimin... |
ee101 42299 | ~ e101 without virtual ded... |
e011 42300 | A virtual deduction elimin... |
ee011 42301 | ~ e011 without virtual ded... |
e100 42302 | A virtual deduction elimin... |
ee100 42303 | ~ e100 without virtual ded... |
e010 42304 | A virtual deduction elimin... |
ee010 42305 | ~ e010 without virtual ded... |
e001 42306 | A virtual deduction elimin... |
ee001 42307 | ~ e001 without virtual ded... |
e11 42308 | A virtual deduction elimin... |
e11an 42309 | Conjunction form of ~ e11 ... |
ee11an 42310 | ~ e11an without virtual de... |
e01 42311 | A virtual deduction elimin... |
e01an 42312 | Conjunction form of ~ e01 ... |
ee01an 42313 | ~ e01an without virtual de... |
e10 42314 | A virtual deduction elimin... |
e10an 42315 | Conjunction form of ~ e10 ... |
ee10an 42316 | ~ e10an without virtual de... |
e02 42317 | A virtual deduction elimin... |
e02an 42318 | Conjunction form of ~ e02 ... |
ee02an 42319 | ~ e02an without virtual de... |
eel021old 42320 | ~ el021old without virtual... |
el021old 42321 | A virtual deduction elimin... |
eel132 42322 | ~ syl2an with antecedents ... |
eel000cT 42323 | An elimination deduction. ... |
eel0TT 42324 | An elimination deduction. ... |
eelT00 42325 | An elimination deduction. ... |
eelTTT 42326 | An elimination deduction. ... |
eelT11 42327 | An elimination deduction. ... |
eelT1 42328 | Syllogism inference combin... |
eelT12 42329 | An elimination deduction. ... |
eelTT1 42330 | An elimination deduction. ... |
eelT01 42331 | An elimination deduction. ... |
eel0T1 42332 | An elimination deduction. ... |
eel12131 42333 | An elimination deduction. ... |
eel2131 42334 | ~ syl2an with antecedents ... |
eel3132 42335 | ~ syl2an with antecedents ... |
eel0321old 42336 | ~ el0321old without virtua... |
el0321old 42337 | A virtual deduction elimin... |
eel2122old 42338 | ~ el2122old without virtua... |
el2122old 42339 | A virtual deduction elimin... |
eel0000 42340 | Elimination rule similar t... |
eel00001 42341 | An elimination deduction. ... |
eel00000 42342 | Elimination rule similar ~... |
eel11111 42343 | Five-hypothesis eliminatio... |
e12 42344 | A virtual deduction elimin... |
e12an 42345 | Conjunction form of ~ e12 ... |
el12 42346 | Virtual deduction form of ... |
e20 42347 | A virtual deduction elimin... |
e20an 42348 | Conjunction form of ~ e20 ... |
ee20an 42349 | ~ e20an without virtual de... |
e21 42350 | A virtual deduction elimin... |
e21an 42351 | Conjunction form of ~ e21 ... |
ee21an 42352 | ~ e21an without virtual de... |
e333 42353 | A virtual deduction elimin... |
e33 42354 | A virtual deduction elimin... |
e33an 42355 | Conjunction form of ~ e33 ... |
ee33an 42356 | ~ e33an without virtual de... |
e3 42357 | Meta-connective form of ~ ... |
e3bi 42358 | Biconditional form of ~ e3... |
e3bir 42359 | Right biconditional form o... |
e03 42360 | A virtual deduction elimin... |
ee03 42361 | ~ e03 without virtual dedu... |
e03an 42362 | Conjunction form of ~ e03 ... |
ee03an 42363 | Conjunction form of ~ ee03... |
e30 42364 | A virtual deduction elimin... |
ee30 42365 | ~ e30 without virtual dedu... |
e30an 42366 | A virtual deduction elimin... |
ee30an 42367 | Conjunction form of ~ ee30... |
e13 42368 | A virtual deduction elimin... |
e13an 42369 | A virtual deduction elimin... |
ee13an 42370 | ~ e13an without virtual de... |
e31 42371 | A virtual deduction elimin... |
ee31 42372 | ~ e31 without virtual dedu... |
e31an 42373 | A virtual deduction elimin... |
ee31an 42374 | ~ e31an without virtual de... |
e23 42375 | A virtual deduction elimin... |
e23an 42376 | A virtual deduction elimin... |
ee23an 42377 | ~ e23an without virtual de... |
e32 42378 | A virtual deduction elimin... |
ee32 42379 | ~ e32 without virtual dedu... |
e32an 42380 | A virtual deduction elimin... |
ee32an 42381 | ~ e33an without virtual de... |
e123 42382 | A virtual deduction elimin... |
ee123 42383 | ~ e123 without virtual ded... |
el123 42384 | A virtual deduction elimin... |
e233 42385 | A virtual deduction elimin... |
e323 42386 | A virtual deduction elimin... |
e000 42387 | A virtual deduction elimin... |
e00 42388 | Elimination rule identical... |
e00an 42389 | Elimination rule identical... |
eel00cT 42390 | An elimination deduction. ... |
eelTT 42391 | An elimination deduction. ... |
e0a 42392 | Elimination rule identical... |
eelT 42393 | An elimination deduction. ... |
eel0cT 42394 | An elimination deduction. ... |
eelT0 42395 | An elimination deduction. ... |
e0bi 42396 | Elimination rule identical... |
e0bir 42397 | Elimination rule identical... |
uun0.1 42398 | Convention notation form o... |
un0.1 42399 | ` T. ` is the constant tru... |
uunT1 42400 | A deduction unionizing a n... |
uunT1p1 42401 | A deduction unionizing a n... |
uunT21 42402 | A deduction unionizing a n... |
uun121 42403 | A deduction unionizing a n... |
uun121p1 42404 | A deduction unionizing a n... |
uun132 42405 | A deduction unionizing a n... |
uun132p1 42406 | A deduction unionizing a n... |
anabss7p1 42407 | A deduction unionizing a n... |
un10 42408 | A unionizing deduction. (... |
un01 42409 | A unionizing deduction. (... |
un2122 42410 | A deduction unionizing a n... |
uun2131 42411 | A deduction unionizing a n... |
uun2131p1 42412 | A deduction unionizing a n... |
uunTT1 42413 | A deduction unionizing a n... |
uunTT1p1 42414 | A deduction unionizing a n... |
uunTT1p2 42415 | A deduction unionizing a n... |
uunT11 42416 | A deduction unionizing a n... |
uunT11p1 42417 | A deduction unionizing a n... |
uunT11p2 42418 | A deduction unionizing a n... |
uunT12 42419 | A deduction unionizing a n... |
uunT12p1 42420 | A deduction unionizing a n... |
uunT12p2 42421 | A deduction unionizing a n... |
uunT12p3 42422 | A deduction unionizing a n... |
uunT12p4 42423 | A deduction unionizing a n... |
uunT12p5 42424 | A deduction unionizing a n... |
uun111 42425 | A deduction unionizing a n... |
3anidm12p1 42426 | A deduction unionizing a n... |
3anidm12p2 42427 | A deduction unionizing a n... |
uun123 42428 | A deduction unionizing a n... |
uun123p1 42429 | A deduction unionizing a n... |
uun123p2 42430 | A deduction unionizing a n... |
uun123p3 42431 | A deduction unionizing a n... |
uun123p4 42432 | A deduction unionizing a n... |
uun2221 42433 | A deduction unionizing a n... |
uun2221p1 42434 | A deduction unionizing a n... |
uun2221p2 42435 | A deduction unionizing a n... |
3impdirp1 42436 | A deduction unionizing a n... |
3impcombi 42437 | A 1-hypothesis proposition... |
trsspwALT 42438 | Virtual deduction proof of... |
trsspwALT2 42439 | Virtual deduction proof of... |
trsspwALT3 42440 | Short predicate calculus p... |
sspwtr 42441 | Virtual deduction proof of... |
sspwtrALT 42442 | Virtual deduction proof of... |
sspwtrALT2 42443 | Short predicate calculus p... |
pwtrVD 42444 | Virtual deduction proof of... |
pwtrrVD 42445 | Virtual deduction proof of... |
suctrALT 42446 | The successor of a transit... |
snssiALTVD 42447 | Virtual deduction proof of... |
snssiALT 42448 | If a class is an element o... |
snsslVD 42449 | Virtual deduction proof of... |
snssl 42450 | If a singleton is a subcla... |
snelpwrVD 42451 | Virtual deduction proof of... |
unipwrVD 42452 | Virtual deduction proof of... |
unipwr 42453 | A class is a subclass of t... |
sstrALT2VD 42454 | Virtual deduction proof of... |
sstrALT2 42455 | Virtual deduction proof of... |
suctrALT2VD 42456 | Virtual deduction proof of... |
suctrALT2 42457 | Virtual deduction proof of... |
elex2VD 42458 | Virtual deduction proof of... |
elex22VD 42459 | Virtual deduction proof of... |
eqsbc2VD 42460 | Virtual deduction proof of... |
zfregs2VD 42461 | Virtual deduction proof of... |
tpid3gVD 42462 | Virtual deduction proof of... |
en3lplem1VD 42463 | Virtual deduction proof of... |
en3lplem2VD 42464 | Virtual deduction proof of... |
en3lpVD 42465 | Virtual deduction proof of... |
simplbi2VD 42466 | Virtual deduction proof of... |
3ornot23VD 42467 | Virtual deduction proof of... |
orbi1rVD 42468 | Virtual deduction proof of... |
bitr3VD 42469 | Virtual deduction proof of... |
3orbi123VD 42470 | Virtual deduction proof of... |
sbc3orgVD 42471 | Virtual deduction proof of... |
19.21a3con13vVD 42472 | Virtual deduction proof of... |
exbirVD 42473 | Virtual deduction proof of... |
exbiriVD 42474 | Virtual deduction proof of... |
rspsbc2VD 42475 | Virtual deduction proof of... |
3impexpVD 42476 | Virtual deduction proof of... |
3impexpbicomVD 42477 | Virtual deduction proof of... |
3impexpbicomiVD 42478 | Virtual deduction proof of... |
sbcoreleleqVD 42479 | Virtual deduction proof of... |
hbra2VD 42480 | Virtual deduction proof of... |
tratrbVD 42481 | Virtual deduction proof of... |
al2imVD 42482 | Virtual deduction proof of... |
syl5impVD 42483 | Virtual deduction proof of... |
idiVD 42484 | Virtual deduction proof of... |
ancomstVD 42485 | Closed form of ~ ancoms . ... |
ssralv2VD 42486 | Quantification restricted ... |
ordelordALTVD 42487 | An element of an ordinal c... |
equncomVD 42488 | If a class equals the unio... |
equncomiVD 42489 | Inference form of ~ equnco... |
sucidALTVD 42490 | A set belongs to its succe... |
sucidALT 42491 | A set belongs to its succe... |
sucidVD 42492 | A set belongs to its succe... |
imbi12VD 42493 | Implication form of ~ imbi... |
imbi13VD 42494 | Join three logical equival... |
sbcim2gVD 42495 | Distribution of class subs... |
sbcbiVD 42496 | Implication form of ~ sbcb... |
trsbcVD 42497 | Formula-building inference... |
truniALTVD 42498 | The union of a class of tr... |
ee33VD 42499 | Non-virtual deduction form... |
trintALTVD 42500 | The intersection of a clas... |
trintALT 42501 | The intersection of a clas... |
undif3VD 42502 | The first equality of Exer... |
sbcssgVD 42503 | Virtual deduction proof of... |
csbingVD 42504 | Virtual deduction proof of... |
onfrALTlem5VD 42505 | Virtual deduction proof of... |
onfrALTlem4VD 42506 | Virtual deduction proof of... |
onfrALTlem3VD 42507 | Virtual deduction proof of... |
simplbi2comtVD 42508 | Virtual deduction proof of... |
onfrALTlem2VD 42509 | Virtual deduction proof of... |
onfrALTlem1VD 42510 | Virtual deduction proof of... |
onfrALTVD 42511 | Virtual deduction proof of... |
csbeq2gVD 42512 | Virtual deduction proof of... |
csbsngVD 42513 | Virtual deduction proof of... |
csbxpgVD 42514 | Virtual deduction proof of... |
csbresgVD 42515 | Virtual deduction proof of... |
csbrngVD 42516 | Virtual deduction proof of... |
csbima12gALTVD 42517 | Virtual deduction proof of... |
csbunigVD 42518 | Virtual deduction proof of... |
csbfv12gALTVD 42519 | Virtual deduction proof of... |
con5VD 42520 | Virtual deduction proof of... |
relopabVD 42521 | Virtual deduction proof of... |
19.41rgVD 42522 | Virtual deduction proof of... |
2pm13.193VD 42523 | Virtual deduction proof of... |
hbimpgVD 42524 | Virtual deduction proof of... |
hbalgVD 42525 | Virtual deduction proof of... |
hbexgVD 42526 | Virtual deduction proof of... |
ax6e2eqVD 42527 | The following User's Proof... |
ax6e2ndVD 42528 | The following User's Proof... |
ax6e2ndeqVD 42529 | The following User's Proof... |
2sb5ndVD 42530 | The following User's Proof... |
2uasbanhVD 42531 | The following User's Proof... |
e2ebindVD 42532 | The following User's Proof... |
sb5ALTVD 42533 | The following User's Proof... |
vk15.4jVD 42534 | The following User's Proof... |
notnotrALTVD 42535 | The following User's Proof... |
con3ALTVD 42536 | The following User's Proof... |
elpwgdedVD 42537 | Membership in a power clas... |
sspwimp 42538 | If a class is a subclass o... |
sspwimpVD 42539 | The following User's Proof... |
sspwimpcf 42540 | If a class is a subclass o... |
sspwimpcfVD 42541 | The following User's Proof... |
suctrALTcf 42542 | The sucessor of a transiti... |
suctrALTcfVD 42543 | The following User's Proof... |
suctrALT3 42544 | The successor of a transit... |
sspwimpALT 42545 | If a class is a subclass o... |
unisnALT 42546 | A set equals the union of ... |
notnotrALT2 42547 | Converse of double negatio... |
sspwimpALT2 42548 | If a class is a subclass o... |
e2ebindALT 42549 | Absorption of an existenti... |
ax6e2ndALT 42550 | If at least two sets exist... |
ax6e2ndeqALT 42551 | "At least two sets exist" ... |
2sb5ndALT 42552 | Equivalence for double sub... |
chordthmALT 42553 | The intersecting chords th... |
isosctrlem1ALT 42554 | Lemma for ~ isosctr . Thi... |
iunconnlem2 42555 | The indexed union of conne... |
iunconnALT 42556 | The indexed union of conne... |
sineq0ALT 42557 | A complex number whose sin... |
evth2f 42558 | A version of ~ evth2 using... |
elunif 42559 | A version of ~ eluni using... |
rzalf 42560 | A version of ~ rzal using ... |
fvelrnbf 42561 | A version of ~ fvelrnb usi... |
rfcnpre1 42562 | If F is a continuous funct... |
ubelsupr 42563 | If U belongs to A and U is... |
fsumcnf 42564 | A finite sum of functions ... |
mulltgt0 42565 | The product of a negative ... |
rspcegf 42566 | A version of ~ rspcev usin... |
rabexgf 42567 | A version of ~ rabexg usin... |
fcnre 42568 | A function continuous with... |
sumsnd 42569 | A sum of a singleton is th... |
evthf 42570 | A version of ~ evth using ... |
cnfex 42571 | The class of continuous fu... |
fnchoice 42572 | For a finite set, a choice... |
refsumcn 42573 | A finite sum of continuous... |
rfcnpre2 42574 | If ` F ` is a continuous f... |
cncmpmax 42575 | When the hypothesis for th... |
rfcnpre3 42576 | If F is a continuous funct... |
rfcnpre4 42577 | If F is a continuous funct... |
sumpair 42578 | Sum of two distinct comple... |
rfcnnnub 42579 | Given a real continuous fu... |
refsum2cnlem1 42580 | This is the core Lemma for... |
refsum2cn 42581 | The sum of two continuus r... |
elunnel2 42582 | A member of a union that i... |
adantlllr 42583 | Deduction adding a conjunc... |
3adantlr3 42584 | Deduction adding a conjunc... |
nnxrd 42585 | A natural number is an ext... |
3adantll2 42586 | Deduction adding a conjunc... |
3adantll3 42587 | Deduction adding a conjunc... |
ssnel 42588 | If not element of a set, t... |
elabrexg 42589 | Elementhood in an image se... |
sncldre 42590 | A singleton is closed w.r.... |
n0p 42591 | A polynomial with a nonzer... |
pm2.65ni 42592 | Inference rule for proof b... |
pwssfi 42593 | Every element of the power... |
iuneq2df 42594 | Equality deduction for ind... |
nnfoctb 42595 | There exists a mapping fro... |
ssinss1d 42596 | Intersection preserves sub... |
elpwinss 42597 | An element of the powerset... |
unidmex 42598 | If ` F ` is a set, then ` ... |
ndisj2 42599 | A non-disjointness conditi... |
zenom 42600 | The set of integer numbers... |
uzwo4 42601 | Well-ordering principle: a... |
unisn0 42602 | The union of the singleton... |
ssin0 42603 | If two classes are disjoin... |
inabs3 42604 | Absorption law for interse... |
pwpwuni 42605 | Relationship between power... |
disjiun2 42606 | In a disjoint collection, ... |
0pwfi 42607 | The empty set is in any po... |
ssinss2d 42608 | Intersection preserves sub... |
zct 42609 | The set of integer numbers... |
pwfin0 42610 | A finite set always belong... |
uzct 42611 | An upper integer set is co... |
iunxsnf 42612 | A singleton index picks ou... |
fiiuncl 42613 | If a set is closed under t... |
iunp1 42614 | The addition of the next s... |
fiunicl 42615 | If a set is closed under t... |
ixpeq2d 42616 | Equality theorem for infin... |
disjxp1 42617 | The sets of a cartesian pr... |
disjsnxp 42618 | The sets in the cartesian ... |
eliind 42619 | Membership in indexed inte... |
rspcef 42620 | Restricted existential spe... |
inn0f 42621 | A nonempty intersection. ... |
ixpssmapc 42622 | An infinite Cartesian prod... |
inn0 42623 | A nonempty intersection. ... |
elintd 42624 | Membership in class inters... |
ssdf 42625 | A sufficient condition for... |
brneqtrd 42626 | Substitution of equal clas... |
ssnct 42627 | A set containing an uncoun... |
ssuniint 42628 | Sufficient condition for b... |
elintdv 42629 | Membership in class inters... |
ssd 42630 | A sufficient condition for... |
ralimralim 42631 | Introducing any antecedent... |
snelmap 42632 | Membership of the element ... |
xrnmnfpnf 42633 | An extended real that is n... |
nelrnmpt 42634 | Non-membership in the rang... |
iuneq1i 42635 | Equality theorem for index... |
nssrex 42636 | Negation of subclass relat... |
ssinc 42637 | Inclusion relation for a m... |
ssdec 42638 | Inclusion relation for a m... |
elixpconstg 42639 | Membership in an infinite ... |
iineq1d 42640 | Equality theorem for index... |
metpsmet 42641 | A metric is a pseudometric... |
ixpssixp 42642 | Subclass theorem for infin... |
ballss3 42643 | A sufficient condition for... |
iunincfi 42644 | Given a sequence of increa... |
nsstr 42645 | If it's not a subclass, it... |
rexanuz3 42646 | Combine two different uppe... |
cbvmpo2 42647 | Rule to change the second ... |
cbvmpo1 42648 | Rule to change the first b... |
eliuniin 42649 | Indexed union of indexed i... |
ssabf 42650 | Subclass of a class abstra... |
pssnssi 42651 | A proper subclass does not... |
rabidim2 42652 | Membership in a restricted... |
eluni2f 42653 | Membership in class union.... |
eliin2f 42654 | Membership in indexed inte... |
nssd 42655 | Negation of subclass relat... |
iineq12dv 42656 | Equality deduction for ind... |
supxrcld 42657 | The supremum of an arbitra... |
elrestd 42658 | A sufficient condition for... |
eliuniincex 42659 | Counterexample to show tha... |
eliincex 42660 | Counterexample to show tha... |
eliinid 42661 | Membership in an indexed i... |
abssf 42662 | Class abstraction in a sub... |
supxrubd 42663 | A member of a set of exten... |
ssrabf 42664 | Subclass of a restricted c... |
eliin2 42665 | Membership in indexed inte... |
ssrab2f 42666 | Subclass relation for a re... |
restuni3 42667 | The underlying set of a su... |
rabssf 42668 | Restricted class abstracti... |
eliuniin2 42669 | Indexed union of indexed i... |
restuni4 42670 | The underlying set of a su... |
restuni6 42671 | The underlying set of a su... |
restuni5 42672 | The underlying set of a su... |
unirestss 42673 | The union of an elementwis... |
iniin1 42674 | Indexed intersection of in... |
iniin2 42675 | Indexed intersection of in... |
cbvrabv2 42676 | A more general version of ... |
cbvrabv2w 42677 | A more general version of ... |
iinssiin 42678 | Subset implication for an ... |
eliind2 42679 | Membership in indexed inte... |
iinssd 42680 | Subset implication for an ... |
rabbida2 42681 | Equivalent wff's yield equ... |
iinexd 42682 | The existence of an indexe... |
rabexf 42683 | Separation Scheme in terms... |
rabbida3 42684 | Equivalent wff's yield equ... |
r19.36vf 42685 | Restricted quantifier vers... |
raleqd 42686 | Equality deduction for res... |
iinssf 42687 | Subset implication for an ... |
iinssdf 42688 | Subset implication for an ... |
resabs2i 42689 | Absorption law for restric... |
ssdf2 42690 | A sufficient condition for... |
rabssd 42691 | Restricted class abstracti... |
rexnegd 42692 | Minus a real number. (Con... |
rexlimd3 42693 | * Inference from Theorem 1... |
resabs1i 42694 | Absorption law for restric... |
nel1nelin 42695 | Membership in an intersect... |
nel2nelin 42696 | Membership in an intersect... |
nel1nelini 42697 | Membership in an intersect... |
nel2nelini 42698 | Membership in an intersect... |
eliunid 42699 | Membership in indexed unio... |
reximddv3 42700 | Deduction from Theorem 19.... |
reximdd 42701 | Deduction from Theorem 19.... |
unfid 42702 | The union of two finite se... |
feq1dd 42703 | Equality deduction for fun... |
fnresdmss 42704 | A function does not change... |
fmptsnxp 42705 | Maps-to notation and Carte... |
fvmpt2bd 42706 | Value of a function given ... |
rnmptfi 42707 | The range of a function wi... |
fresin2 42708 | Restriction of a function ... |
ffi 42709 | A function with finite dom... |
suprnmpt 42710 | An explicit bound for the ... |
rnffi 42711 | The range of a function wi... |
mptelpm 42712 | A function in maps-to nota... |
rnmptpr 42713 | Range of a function define... |
resmpti 42714 | Restriction of the mapping... |
founiiun 42715 | Union expressed as an inde... |
rnresun 42716 | Distribution law for range... |
dffo3f 42717 | An onto mapping expressed ... |
elrnmptf 42718 | The range of a function in... |
rnmptssrn 42719 | Inclusion relation for two... |
disjf1 42720 | A 1 to 1 mapping built fro... |
rnsnf 42721 | The range of a function wh... |
wessf1ornlem 42722 | Given a function ` F ` on ... |
wessf1orn 42723 | Given a function ` F ` on ... |
foelrnf 42724 | Property of a surjective f... |
nelrnres 42725 | If ` A ` is not in the ran... |
disjrnmpt2 42726 | Disjointness of the range ... |
elrnmpt1sf 42727 | Elementhood in an image se... |
founiiun0 42728 | Union expressed as an inde... |
disjf1o 42729 | A bijection built from dis... |
fompt 42730 | Express being onto for a m... |
disjinfi 42731 | Only a finite number of di... |
fvovco 42732 | Value of the composition o... |
ssnnf1octb 42733 | There exists a bijection b... |
nnf1oxpnn 42734 | There is a bijection betwe... |
rnmptssd 42735 | The range of an operation ... |
projf1o 42736 | A biijection from a set to... |
fvmap 42737 | Function value for a membe... |
fvixp2 42738 | Projection of a factor of ... |
fidmfisupp 42739 | A function with a finite d... |
choicefi 42740 | For a finite set, a choice... |
mpct 42741 | The exponentiation of a co... |
cnmetcoval 42742 | Value of the distance func... |
fcomptss 42743 | Express composition of two... |
elmapsnd 42744 | Membership in a set expone... |
mapss2 42745 | Subset inheritance for set... |
fsneq 42746 | Equality condition for two... |
difmap 42747 | Difference of two sets exp... |
unirnmap 42748 | Given a subset of a set ex... |
inmap 42749 | Intersection of two sets e... |
fcoss 42750 | Composition of two mapping... |
fsneqrn 42751 | Equality condition for two... |
difmapsn 42752 | Difference of two sets exp... |
mapssbi 42753 | Subset inheritance for set... |
unirnmapsn 42754 | Equality theorem for a sub... |
iunmapss 42755 | The indexed union of set e... |
ssmapsn 42756 | A subset ` C ` of a set ex... |
iunmapsn 42757 | The indexed union of set e... |
absfico 42758 | Mapping domain and codomai... |
icof 42759 | The set of left-closed rig... |
elpmrn 42760 | The range of a partial fun... |
imaexi 42761 | The image of a set is a se... |
axccdom 42762 | Relax the constraint on ax... |
dmmptdf 42763 | The domain of the mapping ... |
elpmi2 42764 | The domain of a partial fu... |
dmrelrnrel 42765 | A relation preserving func... |
fvcod 42766 | Value of a function compos... |
elrnmpoid 42767 | Membership in the range of... |
axccd 42768 | An alternative version of ... |
axccd2 42769 | An alternative version of ... |
funimassd 42770 | Sufficient condition for t... |
fimassd 42771 | The image of a class is a ... |
feqresmptf 42772 | Express a restricted funct... |
elrnmpt1d 42773 | Elementhood in an image se... |
dmresss 42774 | The domain of a restrictio... |
dmmptssf 42775 | The domain of a mapping is... |
dmmptdf2 42776 | The domain of the mapping ... |
dmuz 42777 | Domain of the upper intege... |
fmptd2f 42778 | Domain and codomain of the... |
mpteq1df 42779 | An equality theorem for th... |
mpteq1dfOLD 42780 | Obsolete version of ~ mpte... |
mptexf 42781 | If the domain of a functio... |
fvmpt4 42782 | Value of a function given ... |
fmptf 42783 | Functionality of the mappi... |
resimass 42784 | The image of a restriction... |
mptssid 42785 | The mapping operation expr... |
mptfnd 42786 | The maps-to notation defin... |
mpteq12daOLD 42787 | Obsolete version of ~ mpte... |
rnmptlb 42788 | Boundness below of the ran... |
rnmptbddlem 42789 | Boundness of the range of ... |
rnmptbdd 42790 | Boundness of the range of ... |
mptima2 42791 | Image of a function in map... |
funimaeq 42792 | Membership relation for th... |
rnmptssf 42793 | The range of an operation ... |
rnmptbd2lem 42794 | Boundness below of the ran... |
rnmptbd2 42795 | Boundness below of the ran... |
infnsuprnmpt 42796 | The indexed infimum of rea... |
suprclrnmpt 42797 | Closure of the indexed sup... |
suprubrnmpt2 42798 | A member of a nonempty ind... |
suprubrnmpt 42799 | A member of a nonempty ind... |
rnmptssdf 42800 | The range of an operation ... |
rnmptbdlem 42801 | Boundness above of the ran... |
rnmptbd 42802 | Boundness above of the ran... |
rnmptss2 42803 | The range of an operation ... |
elmptima 42804 | The image of a function in... |
ralrnmpt3 42805 | A restricted quantifier ov... |
fvelima2 42806 | Function value in an image... |
rnmptssbi 42807 | The range of an operation ... |
fnfvelrnd 42808 | A function's value belongs... |
imass2d 42809 | Subset theorem for image. ... |
imassmpt 42810 | Membership relation for th... |
fpmd 42811 | A total function is a part... |
fconst7 42812 | An alternative way to expr... |
sub2times 42813 | Subtracting from a number,... |
abssubrp 42814 | The distance of two distin... |
elfzfzo 42815 | Relationship between membe... |
oddfl 42816 | Odd number representation ... |
abscosbd 42817 | Bound for the absolute val... |
mul13d 42818 | Commutative/associative la... |
negpilt0 42819 | Negative ` _pi ` is negati... |
dstregt0 42820 | A complex number ` A ` tha... |
subadd4b 42821 | Rearrangement of 4 terms i... |
xrlttri5d 42822 | Not equal and not larger i... |
neglt 42823 | The negative of a positive... |
zltlesub 42824 | If an integer ` N ` is les... |
divlt0gt0d 42825 | The ratio of a negative nu... |
subsub23d 42826 | Swap subtrahend and result... |
2timesgt 42827 | Double of a positive real ... |
reopn 42828 | The reals are open with re... |
sub31 42829 | Swap the first and third t... |
nnne1ge2 42830 | A positive integer which i... |
lefldiveq 42831 | A closed enough, smaller r... |
negsubdi3d 42832 | Distribution of negative o... |
ltdiv2dd 42833 | Division of a positive num... |
abssinbd 42834 | Bound for the absolute val... |
halffl 42835 | Floor of ` ( 1 / 2 ) ` . ... |
monoords 42836 | Ordering relation for a st... |
hashssle 42837 | The size of a subset of a ... |
lttri5d 42838 | Not equal and not larger i... |
fzisoeu 42839 | A finite ordered set has a... |
lt3addmuld 42840 | If three real numbers are ... |
absnpncan2d 42841 | Triangular inequality, com... |
fperiodmullem 42842 | A function with period ` T... |
fperiodmul 42843 | A function with period T i... |
upbdrech 42844 | Choice of an upper bound f... |
lt4addmuld 42845 | If four real numbers are l... |
absnpncan3d 42846 | Triangular inequality, com... |
upbdrech2 42847 | Choice of an upper bound f... |
ssfiunibd 42848 | A finite union of bounded ... |
fzdifsuc2 42849 | Remove a successor from th... |
fzsscn 42850 | A finite sequence of integ... |
divcan8d 42851 | A cancellation law for div... |
dmmcand 42852 | Cancellation law for divis... |
fzssre 42853 | A finite sequence of integ... |
bccld 42854 | A binomial coefficient, in... |
leadd12dd 42855 | Addition to both sides of ... |
fzssnn0 42856 | A finite set of sequential... |
xreqle 42857 | Equality implies 'less tha... |
xaddid2d 42858 | ` 0 ` is a left identity f... |
xadd0ge 42859 | A number is less than or e... |
elfzolem1 42860 | A member in a half-open in... |
xrgtned 42861 | 'Greater than' implies not... |
xrleneltd 42862 | 'Less than or equal to' an... |
xaddcomd 42863 | The extended real addition... |
supxrre3 42864 | The supremum of a nonempty... |
uzfissfz 42865 | For any finite subset of t... |
xleadd2d 42866 | Addition of extended reals... |
suprltrp 42867 | The supremum of a nonempty... |
xleadd1d 42868 | Addition of extended reals... |
xreqled 42869 | Equality implies 'less tha... |
xrgepnfd 42870 | An extended real greater t... |
xrge0nemnfd 42871 | A nonnegative extended rea... |
supxrgere 42872 | If a real number can be ap... |
iuneqfzuzlem 42873 | Lemma for ~ iuneqfzuz : he... |
iuneqfzuz 42874 | If two unions indexed by u... |
xle2addd 42875 | Adding both side of two in... |
supxrgelem 42876 | If an extended real number... |
supxrge 42877 | If an extended real number... |
suplesup 42878 | If any element of ` A ` ca... |
infxrglb 42879 | The infimum of a set of ex... |
xadd0ge2 42880 | A number is less than or e... |
nepnfltpnf 42881 | An extended real that is n... |
ltadd12dd 42882 | Addition to both sides of ... |
nemnftgtmnft 42883 | An extended real that is n... |
xrgtso 42884 | 'Greater than' is a strict... |
rpex 42885 | The positive reals form a ... |
xrge0ge0 42886 | A nonnegative extended rea... |
xrssre 42887 | A subset of extended reals... |
ssuzfz 42888 | A finite subset of the upp... |
absfun 42889 | The absolute value is a fu... |
infrpge 42890 | The infimum of a nonempty,... |
xrlexaddrp 42891 | If an extended real number... |
supsubc 42892 | The supremum function dist... |
xralrple2 42893 | Show that ` A ` is less th... |
nnuzdisj 42894 | The first ` N ` elements o... |
ltdivgt1 42895 | Divsion by a number greate... |
xrltned 42896 | 'Less than' implies not eq... |
nnsplit 42897 | Express the set of positiv... |
divdiv3d 42898 | Division into a fraction. ... |
abslt2sqd 42899 | Comparison of the square o... |
qenom 42900 | The set of rational number... |
qct 42901 | The set of rational number... |
xrltnled 42902 | 'Less than' in terms of 'l... |
lenlteq 42903 | 'less than or equal to' bu... |
xrred 42904 | An extended real that is n... |
rr2sscn2 42905 | The cartesian square of ` ... |
infxr 42906 | The infimum of a set of ex... |
infxrunb2 42907 | The infimum of an unbounde... |
infxrbnd2 42908 | The infimum of a bounded-b... |
infleinflem1 42909 | Lemma for ~ infleinf , cas... |
infleinflem2 42910 | Lemma for ~ infleinf , whe... |
infleinf 42911 | If any element of ` B ` ca... |
xralrple4 42912 | Show that ` A ` is less th... |
xralrple3 42913 | Show that ` A ` is less th... |
eluzelzd 42914 | A member of an upper set o... |
suplesup2 42915 | If any element of ` A ` is... |
recnnltrp 42916 | ` N ` is a natural number ... |
nnn0 42917 | The set of positive intege... |
fzct 42918 | A finite set of sequential... |
rpgtrecnn 42919 | Any positive real number i... |
fzossuz 42920 | A half-open integer interv... |
infxrrefi 42921 | The real and extended real... |
xrralrecnnle 42922 | Show that ` A ` is less th... |
fzoct 42923 | A finite set of sequential... |
frexr 42924 | A function taking real val... |
nnrecrp 42925 | The reciprocal of a positi... |
reclt0d 42926 | The reciprocal of a negati... |
lt0neg1dd 42927 | If a number is negative, i... |
mnfled 42928 | Minus infinity is less tha... |
infxrcld 42929 | The infimum of an arbitrar... |
xrralrecnnge 42930 | Show that ` A ` is less th... |
reclt0 42931 | The reciprocal of a negati... |
ltmulneg 42932 | Multiplying by a negative ... |
allbutfi 42933 | For all but finitely many.... |
ltdiv23neg 42934 | Swap denominator with othe... |
xreqnltd 42935 | A consequence of trichotom... |
mnfnre2 42936 | Minus infinity is not a re... |
zssxr 42937 | The integers are a subset ... |
fisupclrnmpt 42938 | A nonempty finite indexed ... |
supxrunb3 42939 | The supremum of an unbound... |
elfzod 42940 | Membership in a half-open ... |
fimaxre4 42941 | A nonempty finite set of r... |
ren0 42942 | The set of reals is nonemp... |
eluzelz2 42943 | A member of an upper set o... |
resabs2d 42944 | Absorption law for restric... |
uzid2 42945 | Membership of the least me... |
supxrleubrnmpt 42946 | The supremum of a nonempty... |
uzssre2 42947 | An upper set of integers i... |
uzssd 42948 | Subset relationship for tw... |
eluzd 42949 | Membership in an upper set... |
infxrlbrnmpt2 42950 | A member of a nonempty ind... |
xrre4 42951 | An extended real is real i... |
uz0 42952 | The upper integers functio... |
eluzelz2d 42953 | A member of an upper set o... |
infleinf2 42954 | If any element in ` B ` is... |
unb2ltle 42955 | "Unbounded below" expresse... |
uzidd2 42956 | Membership of the least me... |
uzssd2 42957 | Subset relationship for tw... |
rexabslelem 42958 | An indexed set of absolute... |
rexabsle 42959 | An indexed set of absolute... |
allbutfiinf 42960 | Given a "for all but finit... |
supxrrernmpt 42961 | The real and extended real... |
suprleubrnmpt 42962 | The supremum of a nonempty... |
infrnmptle 42963 | An indexed infimum of exte... |
infxrunb3 42964 | The infimum of an unbounde... |
uzn0d 42965 | The upper integers are all... |
uzssd3 42966 | Subset relationship for tw... |
rexabsle2 42967 | An indexed set of absolute... |
infxrunb3rnmpt 42968 | The infimum of an unbounde... |
supxrre3rnmpt 42969 | The indexed supremum of a ... |
uzublem 42970 | A set of reals, indexed by... |
uzub 42971 | A set of reals, indexed by... |
ssrexr 42972 | A subset of the reals is a... |
supxrmnf2 42973 | Removing minus infinity fr... |
supxrcli 42974 | The supremum of an arbitra... |
uzid3 42975 | Membership of the least me... |
infxrlesupxr 42976 | The supremum of a nonempty... |
xnegeqd 42977 | Equality of two extended n... |
xnegrecl 42978 | The extended real negative... |
xnegnegi 42979 | Extended real version of ~... |
xnegeqi 42980 | Equality of two extended n... |
nfxnegd 42981 | Deduction version of ~ nfx... |
xnegnegd 42982 | Extended real version of ~... |
uzred 42983 | An upper integer is a real... |
xnegcli 42984 | Closure of extended real n... |
supminfrnmpt 42985 | The indexed supremum of a ... |
infxrpnf 42986 | Adding plus infinity to a ... |
infxrrnmptcl 42987 | The infimum of an arbitrar... |
leneg2d 42988 | Negative of one side of 'l... |
supxrltinfxr 42989 | The supremum of the empty ... |
max1d 42990 | A number is less than or e... |
supxrleubrnmptf 42991 | The supremum of a nonempty... |
nleltd 42992 | 'Not less than or equal to... |
zxrd 42993 | An integer is an extended ... |
infxrgelbrnmpt 42994 | The infimum of an indexed ... |
rphalfltd 42995 | Half of a positive real is... |
uzssz2 42996 | An upper set of integers i... |
leneg3d 42997 | Negative of one side of 'l... |
max2d 42998 | A number is less than or e... |
uzn0bi 42999 | The upper integers functio... |
xnegrecl2 43000 | If the extended real negat... |
nfxneg 43001 | Bound-variable hypothesis ... |
uzxrd 43002 | An upper integer is an ext... |
infxrpnf2 43003 | Removing plus infinity fro... |
supminfxr 43004 | The extended real suprema ... |
infrpgernmpt 43005 | The infimum of a nonempty,... |
xnegre 43006 | An extended real is real i... |
xnegrecl2d 43007 | If the extended real negat... |
uzxr 43008 | An upper integer is an ext... |
supminfxr2 43009 | The extended real suprema ... |
xnegred 43010 | An extended real is real i... |
supminfxrrnmpt 43011 | The indexed supremum of a ... |
min1d 43012 | The minimum of two numbers... |
min2d 43013 | The minimum of two numbers... |
pnfged 43014 | Plus infinity is an upper ... |
xrnpnfmnf 43015 | An extended real that is n... |
uzsscn 43016 | An upper set of integers i... |
absimnre 43017 | The absolute value of the ... |
uzsscn2 43018 | An upper set of integers i... |
xrtgcntopre 43019 | The standard topologies on... |
absimlere 43020 | The absolute value of the ... |
rpssxr 43021 | The positive reals are a s... |
monoordxrv 43022 | Ordering relation for a mo... |
monoordxr 43023 | Ordering relation for a mo... |
monoord2xrv 43024 | Ordering relation for a mo... |
monoord2xr 43025 | Ordering relation for a mo... |
xrpnf 43026 | An extended real is plus i... |
xlenegcon1 43027 | Extended real version of ~... |
xlenegcon2 43028 | Extended real version of ~... |
gtnelioc 43029 | A real number larger than ... |
ioossioc 43030 | An open interval is a subs... |
ioondisj2 43031 | A condition for two open i... |
ioondisj1 43032 | A condition for two open i... |
ioogtlb 43033 | An element of a closed int... |
evthiccabs 43034 | Extreme Value Theorem on y... |
ltnelicc 43035 | A real number smaller than... |
eliood 43036 | Membership in an open real... |
iooabslt 43037 | An upper bound for the dis... |
gtnelicc 43038 | A real number greater than... |
iooinlbub 43039 | An open interval has empty... |
iocgtlb 43040 | An element of a left-open ... |
iocleub 43041 | An element of a left-open ... |
eliccd 43042 | Membership in a closed rea... |
eliccre 43043 | A member of a closed inter... |
eliooshift 43044 | Element of an open interva... |
eliocd 43045 | Membership in a left-open ... |
icoltub 43046 | An element of a left-close... |
eliocre 43047 | A member of a left-open ri... |
iooltub 43048 | An element of an open inte... |
ioontr 43049 | The interior of an interva... |
snunioo1 43050 | The closure of one end of ... |
lbioc 43051 | A left-open right-closed i... |
ioomidp 43052 | The midpoint is an element... |
iccdifioo 43053 | If the open inverval is re... |
iccdifprioo 43054 | An open interval is the cl... |
ioossioobi 43055 | Biconditional form of ~ io... |
iccshift 43056 | A closed interval shifted ... |
iccsuble 43057 | An upper bound to the dist... |
iocopn 43058 | A left-open right-closed i... |
eliccelioc 43059 | Membership in a closed int... |
iooshift 43060 | An open interval shifted b... |
iccintsng 43061 | Intersection of two adiace... |
icoiccdif 43062 | Left-closed right-open int... |
icoopn 43063 | A left-closed right-open i... |
icoub 43064 | A left-closed, right-open ... |
eliccxrd 43065 | Membership in a closed rea... |
pnfel0pnf 43066 | ` +oo ` is a nonnegative e... |
eliccnelico 43067 | An element of a closed int... |
eliccelicod 43068 | A member of a closed inter... |
ge0xrre 43069 | A nonnegative extended rea... |
ge0lere 43070 | A nonnegative extended Rea... |
elicores 43071 | Membership in a left-close... |
inficc 43072 | The infimum of a nonempty ... |
qinioo 43073 | The rational numbers are d... |
lenelioc 43074 | A real number smaller than... |
ioonct 43075 | A nonempty open interval i... |
xrgtnelicc 43076 | A real number greater than... |
iccdificc 43077 | The difference of two clos... |
iocnct 43078 | A nonempty left-open, righ... |
iccnct 43079 | A closed interval, with mo... |
iooiinicc 43080 | A closed interval expresse... |
iccgelbd 43081 | An element of a closed int... |
iooltubd 43082 | An element of an open inte... |
icoltubd 43083 | An element of a left-close... |
qelioo 43084 | The rational numbers are d... |
tgqioo2 43085 | Every open set of reals is... |
iccleubd 43086 | An element of a closed int... |
elioored 43087 | A member of an open interv... |
ioogtlbd 43088 | An element of a closed int... |
ioofun 43089 | ` (,) ` is a function. (C... |
icomnfinre 43090 | A left-closed, right-open,... |
sqrlearg 43091 | The square compared with i... |
ressiocsup 43092 | If the supremum belongs to... |
ressioosup 43093 | If the supremum does not b... |
iooiinioc 43094 | A left-open, right-closed ... |
ressiooinf 43095 | If the infimum does not be... |
icogelbd 43096 | An element of a left-close... |
iocleubd 43097 | An element of a left-open ... |
uzinico 43098 | An upper interval of integ... |
preimaiocmnf 43099 | Preimage of a right-closed... |
uzinico2 43100 | An upper interval of integ... |
uzinico3 43101 | An upper interval of integ... |
icossico2 43102 | Condition for a closed-bel... |
dmico 43103 | The domain of the closed-b... |
ndmico 43104 | The closed-below, open-abo... |
uzubioo 43105 | The upper integers are unb... |
uzubico 43106 | The upper integers are unb... |
uzubioo2 43107 | The upper integers are unb... |
uzubico2 43108 | The upper integers are unb... |
iocgtlbd 43109 | An element of a left-open ... |
xrtgioo2 43110 | The topology on the extend... |
tgioo4 43111 | The standard topology on t... |
fsummulc1f 43112 | Closure of a finite sum of... |
fsumnncl 43113 | Closure of a nonempty, fin... |
fsumge0cl 43114 | The finite sum of nonnegat... |
fsumf1of 43115 | Re-index a finite sum usin... |
fsumiunss 43116 | Sum over a disjoint indexe... |
fsumreclf 43117 | Closure of a finite sum of... |
fsumlessf 43118 | A shorter sum of nonnegati... |
fsumsupp0 43119 | Finite sum of function val... |
fsumsermpt 43120 | A finite sum expressed in ... |
fmul01 43121 | Multiplying a finite numbe... |
fmulcl 43122 | If ' Y ' is closed under t... |
fmuldfeqlem1 43123 | induction step for the pro... |
fmuldfeq 43124 | X and Z are two equivalent... |
fmul01lt1lem1 43125 | Given a finite multiplicat... |
fmul01lt1lem2 43126 | Given a finite multiplicat... |
fmul01lt1 43127 | Given a finite multiplicat... |
cncfmptss 43128 | A continuous complex funct... |
rrpsscn 43129 | The positive reals are a s... |
mulc1cncfg 43130 | A version of ~ mulc1cncf u... |
infrglb 43131 | The infimum of a nonempty ... |
expcnfg 43132 | If ` F ` is a complex cont... |
prodeq2ad 43133 | Equality deduction for pro... |
fprodsplit1 43134 | Separate out a term in a f... |
fprodexp 43135 | Positive integer exponenti... |
fprodabs2 43136 | The absolute value of a fi... |
fprod0 43137 | A finite product with a ze... |
mccllem 43138 | * Induction step for ~ mcc... |
mccl 43139 | A multinomial coefficient,... |
fprodcnlem 43140 | A finite product of functi... |
fprodcn 43141 | A finite product of functi... |
clim1fr1 43142 | A class of sequences of fr... |
isumneg 43143 | Negation of a converging s... |
climrec 43144 | Limit of the reciprocal of... |
climmulf 43145 | A version of ~ climmul usi... |
climexp 43146 | The limit of natural power... |
climinf 43147 | A bounded monotonic noninc... |
climsuselem1 43148 | The subsequence index ` I ... |
climsuse 43149 | A subsequence ` G ` of a c... |
climrecf 43150 | A version of ~ climrec usi... |
climneg 43151 | Complex limit of the negat... |
climinff 43152 | A version of ~ climinf usi... |
climdivf 43153 | Limit of the ratio of two ... |
climreeq 43154 | If ` F ` is a real functio... |
ellimciota 43155 | An explicit value for the ... |
climaddf 43156 | A version of ~ climadd usi... |
mullimc 43157 | Limit of the product of tw... |
ellimcabssub0 43158 | An equivalent condition fo... |
limcdm0 43159 | If a function has empty do... |
islptre 43160 | An equivalence condition f... |
limccog 43161 | Limit of the composition o... |
limciccioolb 43162 | The limit of a function at... |
climf 43163 | Express the predicate: Th... |
mullimcf 43164 | Limit of the multiplicatio... |
constlimc 43165 | Limit of constant function... |
rexlim2d 43166 | Inference removing two res... |
idlimc 43167 | Limit of the identity func... |
divcnvg 43168 | The sequence of reciprocal... |
limcperiod 43169 | If ` F ` is a periodic fun... |
limcrecl 43170 | If ` F ` is a real-valued ... |
sumnnodd 43171 | A series indexed by ` NN `... |
lptioo2 43172 | The upper bound of an open... |
lptioo1 43173 | The lower bound of an open... |
elprn1 43174 | A member of an unordered p... |
elprn2 43175 | A member of an unordered p... |
limcmptdm 43176 | The domain of a maps-to fu... |
clim2f 43177 | Express the predicate: Th... |
limcicciooub 43178 | The limit of a function at... |
ltmod 43179 | A sufficient condition for... |
islpcn 43180 | A characterization for a l... |
lptre2pt 43181 | If a set in the real line ... |
limsupre 43182 | If a sequence is bounded, ... |
limcresiooub 43183 | The left limit doesn't cha... |
limcresioolb 43184 | The right limit doesn't ch... |
limcleqr 43185 | If the left and the right ... |
lptioo2cn 43186 | The upper bound of an open... |
lptioo1cn 43187 | The lower bound of an open... |
neglimc 43188 | Limit of the negative func... |
addlimc 43189 | Sum of two limits. (Contr... |
0ellimcdiv 43190 | If the numerator converges... |
clim2cf 43191 | Express the predicate ` F ... |
limclner 43192 | For a limit point, both fr... |
sublimc 43193 | Subtraction of two limits.... |
reclimc 43194 | Limit of the reciprocal of... |
clim0cf 43195 | Express the predicate ` F ... |
limclr 43196 | For a limit point, both fr... |
divlimc 43197 | Limit of the quotient of t... |
expfac 43198 | Factorial grows faster tha... |
climconstmpt 43199 | A constant sequence conver... |
climresmpt 43200 | A function restricted to u... |
climsubmpt 43201 | Limit of the difference of... |
climsubc2mpt 43202 | Limit of the difference of... |
climsubc1mpt 43203 | Limit of the difference of... |
fnlimfv 43204 | The value of the limit fun... |
climreclf 43205 | The limit of a convergent ... |
climeldmeq 43206 | Two functions that are eve... |
climf2 43207 | Express the predicate: Th... |
fnlimcnv 43208 | The sequence of function v... |
climeldmeqmpt 43209 | Two functions that are eve... |
climfveq 43210 | Two functions that are eve... |
clim2f2 43211 | Express the predicate: Th... |
climfveqmpt 43212 | Two functions that are eve... |
climd 43213 | Express the predicate: Th... |
clim2d 43214 | The limit of complex numbe... |
fnlimfvre 43215 | The limit function of real... |
allbutfifvre 43216 | Given a sequence of real-v... |
climleltrp 43217 | The limit of complex numbe... |
fnlimfvre2 43218 | The limit function of real... |
fnlimf 43219 | The limit function of real... |
fnlimabslt 43220 | A sequence of function val... |
climfveqf 43221 | Two functions that are eve... |
climmptf 43222 | Exhibit a function ` G ` w... |
climfveqmpt3 43223 | Two functions that are eve... |
climeldmeqf 43224 | Two functions that are eve... |
climreclmpt 43225 | The limit of B convergent ... |
limsupref 43226 | If a sequence is bounded, ... |
limsupbnd1f 43227 | If a sequence is eventuall... |
climbddf 43228 | A converging sequence of c... |
climeqf 43229 | Two functions that are eve... |
climeldmeqmpt3 43230 | Two functions that are eve... |
limsupcld 43231 | Closure of the superior li... |
climfv 43232 | The limit of a convergent ... |
limsupval3 43233 | The superior limit of an i... |
climfveqmpt2 43234 | Two functions that are eve... |
limsup0 43235 | The superior limit of the ... |
climeldmeqmpt2 43236 | Two functions that are eve... |
limsupresre 43237 | The supremum limit of a fu... |
climeqmpt 43238 | Two functions that are eve... |
climfvd 43239 | The limit of a convergent ... |
limsuplesup 43240 | An upper bound for the sup... |
limsupresico 43241 | The superior limit doesn't... |
limsuppnfdlem 43242 | If the restriction of a fu... |
limsuppnfd 43243 | If the restriction of a fu... |
limsupresuz 43244 | If the real part of the do... |
limsupub 43245 | If the limsup is not ` +oo... |
limsupres 43246 | The superior limit of a re... |
climinf2lem 43247 | A convergent, nonincreasin... |
climinf2 43248 | A convergent, nonincreasin... |
limsupvaluz 43249 | The superior limit, when t... |
limsupresuz2 43250 | If the domain of a functio... |
limsuppnflem 43251 | If the restriction of a fu... |
limsuppnf 43252 | If the restriction of a fu... |
limsupubuzlem 43253 | If the limsup is not ` +oo... |
limsupubuz 43254 | For a real-valued function... |
climinf2mpt 43255 | A bounded below, monotonic... |
climinfmpt 43256 | A bounded below, monotonic... |
climinf3 43257 | A convergent, nonincreasin... |
limsupvaluzmpt 43258 | The superior limit, when t... |
limsupequzmpt2 43259 | Two functions that are eve... |
limsupubuzmpt 43260 | If the limsup is not ` +oo... |
limsupmnflem 43261 | The superior limit of a fu... |
limsupmnf 43262 | The superior limit of a fu... |
limsupequzlem 43263 | Two functions that are eve... |
limsupequz 43264 | Two functions that are eve... |
limsupre2lem 43265 | Given a function on the ex... |
limsupre2 43266 | Given a function on the ex... |
limsupmnfuzlem 43267 | The superior limit of a fu... |
limsupmnfuz 43268 | The superior limit of a fu... |
limsupequzmptlem 43269 | Two functions that are eve... |
limsupequzmpt 43270 | Two functions that are eve... |
limsupre2mpt 43271 | Given a function on the ex... |
limsupequzmptf 43272 | Two functions that are eve... |
limsupre3lem 43273 | Given a function on the ex... |
limsupre3 43274 | Given a function on the ex... |
limsupre3mpt 43275 | Given a function on the ex... |
limsupre3uzlem 43276 | Given a function on the ex... |
limsupre3uz 43277 | Given a function on the ex... |
limsupreuz 43278 | Given a function on the re... |
limsupvaluz2 43279 | The superior limit, when t... |
limsupreuzmpt 43280 | Given a function on the re... |
supcnvlimsup 43281 | If a function on a set of ... |
supcnvlimsupmpt 43282 | If a function on a set of ... |
0cnv 43283 | If ` (/) ` is a complex nu... |
climuzlem 43284 | Express the predicate: Th... |
climuz 43285 | Express the predicate: Th... |
lmbr3v 43286 | Express the binary relatio... |
climisp 43287 | If a sequence converges to... |
lmbr3 43288 | Express the binary relatio... |
climrescn 43289 | A sequence converging w.r.... |
climxrrelem 43290 | If a seqence ranging over ... |
climxrre 43291 | If a sequence ranging over... |
limsuplt2 43294 | The defining property of t... |
liminfgord 43295 | Ordering property of the i... |
limsupvald 43296 | The superior limit of a se... |
limsupresicompt 43297 | The superior limit doesn't... |
limsupcli 43298 | Closure of the superior li... |
liminfgf 43299 | Closure of the inferior li... |
liminfval 43300 | The inferior limit of a se... |
climlimsup 43301 | A sequence of real numbers... |
limsupge 43302 | The defining property of t... |
liminfgval 43303 | Value of the inferior limi... |
liminfcl 43304 | Closure of the inferior li... |
liminfvald 43305 | The inferior limit of a se... |
liminfval5 43306 | The inferior limit of an i... |
limsupresxr 43307 | The superior limit of a fu... |
liminfresxr 43308 | The inferior limit of a fu... |
liminfval2 43309 | The superior limit, relati... |
climlimsupcex 43310 | Counterexample for ~ climl... |
liminfcld 43311 | Closure of the inferior li... |
liminfresico 43312 | The inferior limit doesn't... |
limsup10exlem 43313 | The range of the given fun... |
limsup10ex 43314 | The superior limit of a fu... |
liminf10ex 43315 | The inferior limit of a fu... |
liminflelimsuplem 43316 | The superior limit is grea... |
liminflelimsup 43317 | The superior limit is grea... |
limsupgtlem 43318 | For any positive real, the... |
limsupgt 43319 | Given a sequence of real n... |
liminfresre 43320 | The inferior limit of a fu... |
liminfresicompt 43321 | The inferior limit doesn't... |
liminfltlimsupex 43322 | An example where the ` lim... |
liminfgelimsup 43323 | The inferior limit is grea... |
liminfvalxr 43324 | Alternate definition of ` ... |
liminfresuz 43325 | If the real part of the do... |
liminflelimsupuz 43326 | The superior limit is grea... |
liminfvalxrmpt 43327 | Alternate definition of ` ... |
liminfresuz2 43328 | If the domain of a functio... |
liminfgelimsupuz 43329 | The inferior limit is grea... |
liminfval4 43330 | Alternate definition of ` ... |
liminfval3 43331 | Alternate definition of ` ... |
liminfequzmpt2 43332 | Two functions that are eve... |
liminfvaluz 43333 | Alternate definition of ` ... |
liminf0 43334 | The inferior limit of the ... |
limsupval4 43335 | Alternate definition of ` ... |
liminfvaluz2 43336 | Alternate definition of ` ... |
liminfvaluz3 43337 | Alternate definition of ` ... |
liminflelimsupcex 43338 | A counterexample for ~ lim... |
limsupvaluz3 43339 | Alternate definition of ` ... |
liminfvaluz4 43340 | Alternate definition of ` ... |
limsupvaluz4 43341 | Alternate definition of ` ... |
climliminflimsupd 43342 | If a sequence of real numb... |
liminfreuzlem 43343 | Given a function on the re... |
liminfreuz 43344 | Given a function on the re... |
liminfltlem 43345 | Given a sequence of real n... |
liminflt 43346 | Given a sequence of real n... |
climliminf 43347 | A sequence of real numbers... |
liminflimsupclim 43348 | A sequence of real numbers... |
climliminflimsup 43349 | A sequence of real numbers... |
climliminflimsup2 43350 | A sequence of real numbers... |
climliminflimsup3 43351 | A sequence of real numbers... |
climliminflimsup4 43352 | A sequence of real numbers... |
limsupub2 43353 | A extended real valued fun... |
limsupubuz2 43354 | A sequence with values in ... |
xlimpnfxnegmnf 43355 | A sequence converges to ` ... |
liminflbuz2 43356 | A sequence with values in ... |
liminfpnfuz 43357 | The inferior limit of a fu... |
liminflimsupxrre 43358 | A sequence with values in ... |
xlimrel 43361 | The limit on extended real... |
xlimres 43362 | A function converges iff i... |
xlimcl 43363 | The limit of a sequence of... |
rexlimddv2 43364 | Restricted existential eli... |
xlimclim 43365 | Given a sequence of reals,... |
xlimconst 43366 | A constant sequence conver... |
climxlim 43367 | A converging sequence in t... |
xlimbr 43368 | Express the binary relatio... |
fuzxrpmcn 43369 | A function mapping from an... |
cnrefiisplem 43370 | Lemma for ~ cnrefiisp (som... |
cnrefiisp 43371 | A non-real, complex number... |
xlimxrre 43372 | If a sequence ranging over... |
xlimmnfvlem1 43373 | Lemma for ~ xlimmnfv : the... |
xlimmnfvlem2 43374 | Lemma for ~ xlimmnf : the ... |
xlimmnfv 43375 | A function converges to mi... |
xlimconst2 43376 | A sequence that eventually... |
xlimpnfvlem1 43377 | Lemma for ~ xlimpnfv : the... |
xlimpnfvlem2 43378 | Lemma for ~ xlimpnfv : the... |
xlimpnfv 43379 | A function converges to pl... |
xlimclim2lem 43380 | Lemma for ~ xlimclim2 . H... |
xlimclim2 43381 | Given a sequence of extend... |
xlimmnf 43382 | A function converges to mi... |
xlimpnf 43383 | A function converges to pl... |
xlimmnfmpt 43384 | A function converges to pl... |
xlimpnfmpt 43385 | A function converges to pl... |
climxlim2lem 43386 | In this lemma for ~ climxl... |
climxlim2 43387 | A sequence of extended rea... |
dfxlim2v 43388 | An alternative definition ... |
dfxlim2 43389 | An alternative definition ... |
climresd 43390 | A function restricted to u... |
climresdm 43391 | A real function converges ... |
dmclimxlim 43392 | A real valued sequence tha... |
xlimmnflimsup2 43393 | A sequence of extended rea... |
xlimuni 43394 | An infinite sequence conve... |
xlimclimdm 43395 | A sequence of extended rea... |
xlimfun 43396 | The convergence relation o... |
xlimmnflimsup 43397 | If a sequence of extended ... |
xlimdm 43398 | Two ways to express that a... |
xlimpnfxnegmnf2 43399 | A sequence converges to ` ... |
xlimresdm 43400 | A function converges in th... |
xlimpnfliminf 43401 | If a sequence of extended ... |
xlimpnfliminf2 43402 | A sequence of extended rea... |
xlimliminflimsup 43403 | A sequence of extended rea... |
xlimlimsupleliminf 43404 | A sequence of extended rea... |
coseq0 43405 | A complex number whose cos... |
sinmulcos 43406 | Multiplication formula for... |
coskpi2 43407 | The cosine of an integer m... |
cosnegpi 43408 | The cosine of negative ` _... |
sinaover2ne0 43409 | If ` A ` in ` ( 0 , 2 _pi ... |
cosknegpi 43410 | The cosine of an integer m... |
mulcncff 43411 | The multiplication of two ... |
cncfmptssg 43412 | A continuous complex funct... |
constcncfg 43413 | A constant function is a c... |
idcncfg 43414 | The identity function is a... |
cncfshift 43415 | A periodic continuous func... |
resincncf 43416 | ` sin ` restricted to real... |
addccncf2 43417 | Adding a constant is a con... |
0cnf 43418 | The empty set is a continu... |
fsumcncf 43419 | The finite sum of continuo... |
cncfperiod 43420 | A periodic continuous func... |
subcncff 43421 | The subtraction of two con... |
negcncfg 43422 | The opposite of a continuo... |
cnfdmsn 43423 | A function with a singleto... |
cncfcompt 43424 | Composition of continuous ... |
addcncff 43425 | The sum of two continuous ... |
ioccncflimc 43426 | Limit at the upper bound o... |
cncfuni 43427 | A complex function on a su... |
icccncfext 43428 | A continuous function on a... |
cncficcgt0 43429 | A the absolute value of a ... |
icocncflimc 43430 | Limit at the lower bound, ... |
cncfdmsn 43431 | A complex function with a ... |
divcncff 43432 | The quotient of two contin... |
cncfshiftioo 43433 | A periodic continuous func... |
cncfiooicclem1 43434 | A continuous function ` F ... |
cncfiooicc 43435 | A continuous function ` F ... |
cncfiooiccre 43436 | A continuous function ` F ... |
cncfioobdlem 43437 | ` G ` actually extends ` F... |
cncfioobd 43438 | A continuous function ` F ... |
jumpncnp 43439 | Jump discontinuity or disc... |
cxpcncf2 43440 | The complex power function... |
fprodcncf 43441 | The finite product of cont... |
add1cncf 43442 | Addition to a constant is ... |
add2cncf 43443 | Addition to a constant is ... |
sub1cncfd 43444 | Subtracting a constant is ... |
sub2cncfd 43445 | Subtraction from a constan... |
fprodsub2cncf 43446 | ` F ` is continuous. (Con... |
fprodadd2cncf 43447 | ` F ` is continuous. (Con... |
fprodsubrecnncnvlem 43448 | The sequence ` S ` of fini... |
fprodsubrecnncnv 43449 | The sequence ` S ` of fini... |
fprodaddrecnncnvlem 43450 | The sequence ` S ` of fini... |
fprodaddrecnncnv 43451 | The sequence ` S ` of fini... |
dvsinexp 43452 | The derivative of sin^N . ... |
dvcosre 43453 | The real derivative of the... |
dvsinax 43454 | Derivative exercise: the d... |
dvsubf 43455 | The subtraction rule for e... |
dvmptconst 43456 | Function-builder for deriv... |
dvcnre 43457 | From compex differentiatio... |
dvmptidg 43458 | Function-builder for deriv... |
dvresntr 43459 | Function-builder for deriv... |
fperdvper 43460 | The derivative of a period... |
dvasinbx 43461 | Derivative exercise: the d... |
dvresioo 43462 | Restriction of a derivativ... |
dvdivf 43463 | The quotient rule for ever... |
dvdivbd 43464 | A sufficient condition for... |
dvsubcncf 43465 | A sufficient condition for... |
dvmulcncf 43466 | A sufficient condition for... |
dvcosax 43467 | Derivative exercise: the d... |
dvdivcncf 43468 | A sufficient condition for... |
dvbdfbdioolem1 43469 | Given a function with boun... |
dvbdfbdioolem2 43470 | A function on an open inte... |
dvbdfbdioo 43471 | A function on an open inte... |
ioodvbdlimc1lem1 43472 | If ` F ` has bounded deriv... |
ioodvbdlimc1lem2 43473 | Limit at the lower bound o... |
ioodvbdlimc1 43474 | A real function with bound... |
ioodvbdlimc2lem 43475 | Limit at the upper bound o... |
ioodvbdlimc2 43476 | A real function with bound... |
dvdmsscn 43477 | ` X ` is a subset of ` CC ... |
dvmptmulf 43478 | Function-builder for deriv... |
dvnmptdivc 43479 | Function-builder for itera... |
dvdsn1add 43480 | If ` K ` divides ` N ` but... |
dvxpaek 43481 | Derivative of the polynomi... |
dvnmptconst 43482 | The ` N ` -th derivative o... |
dvnxpaek 43483 | The ` n ` -th derivative o... |
dvnmul 43484 | Function-builder for the `... |
dvmptfprodlem 43485 | Induction step for ~ dvmpt... |
dvmptfprod 43486 | Function-builder for deriv... |
dvnprodlem1 43487 | ` D ` is bijective. (Cont... |
dvnprodlem2 43488 | Induction step for ~ dvnpr... |
dvnprodlem3 43489 | The multinomial formula fo... |
dvnprod 43490 | The multinomial formula fo... |
itgsin0pilem1 43491 | Calculation of the integra... |
ibliccsinexp 43492 | sin^n on a closed interval... |
itgsin0pi 43493 | Calculation of the integra... |
iblioosinexp 43494 | sin^n on an open integral ... |
itgsinexplem1 43495 | Integration by parts is ap... |
itgsinexp 43496 | A recursive formula for th... |
iblconstmpt 43497 | A constant function is int... |
itgeq1d 43498 | Equality theorem for an in... |
mbfres2cn 43499 | Measurability of a piecewi... |
vol0 43500 | The measure of the empty s... |
ditgeqiooicc 43501 | A function ` F ` on an ope... |
volge0 43502 | The volume of a set is alw... |
cnbdibl 43503 | A continuous bounded funct... |
snmbl 43504 | A singleton is measurable.... |
ditgeq3d 43505 | Equality theorem for the d... |
iblempty 43506 | The empty function is inte... |
iblsplit 43507 | The union of two integrabl... |
volsn 43508 | A singleton has 0 Lebesgue... |
itgvol0 43509 | If the domani is negligibl... |
itgcoscmulx 43510 | Exercise: the integral of ... |
iblsplitf 43511 | A version of ~ iblsplit us... |
ibliooicc 43512 | If a function is integrabl... |
volioc 43513 | The measure of a left-open... |
iblspltprt 43514 | If a function is integrabl... |
itgsincmulx 43515 | Exercise: the integral of ... |
itgsubsticclem 43516 | lemma for ~ itgsubsticc . ... |
itgsubsticc 43517 | Integration by u-substitut... |
itgioocnicc 43518 | The integral of a piecewis... |
iblcncfioo 43519 | A continuous function ` F ... |
itgspltprt 43520 | The ` S. ` integral splits... |
itgiccshift 43521 | The integral of a function... |
itgperiod 43522 | The integral of a periodic... |
itgsbtaddcnst 43523 | Integral substitution, add... |
volico 43524 | The measure of left-closed... |
sublevolico 43525 | The Lebesgue measure of a ... |
dmvolss 43526 | Lebesgue measurable sets a... |
ismbl3 43527 | The predicate " ` A ` is L... |
volioof 43528 | The function that assigns ... |
ovolsplit 43529 | The Lebesgue outer measure... |
fvvolioof 43530 | The function value of the ... |
volioore 43531 | The measure of an open int... |
fvvolicof 43532 | The function value of the ... |
voliooico 43533 | An open interval and a lef... |
ismbl4 43534 | The predicate " ` A ` is L... |
volioofmpt 43535 | ` ( ( vol o. (,) ) o. F ) ... |
volicoff 43536 | ` ( ( vol o. [,) ) o. F ) ... |
voliooicof 43537 | The Lebesgue measure of op... |
volicofmpt 43538 | ` ( ( vol o. [,) ) o. F ) ... |
volicc 43539 | The Lebesgue measure of a ... |
voliccico 43540 | A closed interval and a le... |
mbfdmssre 43541 | The domain of a measurable... |
stoweidlem1 43542 | Lemma for ~ stoweid . Thi... |
stoweidlem2 43543 | lemma for ~ stoweid : here... |
stoweidlem3 43544 | Lemma for ~ stoweid : if `... |
stoweidlem4 43545 | Lemma for ~ stoweid : a cl... |
stoweidlem5 43546 | There exists a δ as ... |
stoweidlem6 43547 | Lemma for ~ stoweid : two ... |
stoweidlem7 43548 | This lemma is used to prov... |
stoweidlem8 43549 | Lemma for ~ stoweid : two ... |
stoweidlem9 43550 | Lemma for ~ stoweid : here... |
stoweidlem10 43551 | Lemma for ~ stoweid . Thi... |
stoweidlem11 43552 | This lemma is used to prov... |
stoweidlem12 43553 | Lemma for ~ stoweid . Thi... |
stoweidlem13 43554 | Lemma for ~ stoweid . Thi... |
stoweidlem14 43555 | There exists a ` k ` as in... |
stoweidlem15 43556 | This lemma is used to prov... |
stoweidlem16 43557 | Lemma for ~ stoweid . The... |
stoweidlem17 43558 | This lemma proves that the... |
stoweidlem18 43559 | This theorem proves Lemma ... |
stoweidlem19 43560 | If a set of real functions... |
stoweidlem20 43561 | If a set A of real functio... |
stoweidlem21 43562 | Once the Stone Weierstrass... |
stoweidlem22 43563 | If a set of real functions... |
stoweidlem23 43564 | This lemma is used to prov... |
stoweidlem24 43565 | This lemma proves that for... |
stoweidlem25 43566 | This lemma proves that for... |
stoweidlem26 43567 | This lemma is used to prov... |
stoweidlem27 43568 | This lemma is used to prov... |
stoweidlem28 43569 | There exists a δ as ... |
stoweidlem29 43570 | When the hypothesis for th... |
stoweidlem30 43571 | This lemma is used to prov... |
stoweidlem31 43572 | This lemma is used to prov... |
stoweidlem32 43573 | If a set A of real functio... |
stoweidlem33 43574 | If a set of real functions... |
stoweidlem34 43575 | This lemma proves that for... |
stoweidlem35 43576 | This lemma is used to prov... |
stoweidlem36 43577 | This lemma is used to prov... |
stoweidlem37 43578 | This lemma is used to prov... |
stoweidlem38 43579 | This lemma is used to prov... |
stoweidlem39 43580 | This lemma is used to prov... |
stoweidlem40 43581 | This lemma proves that q_n... |
stoweidlem41 43582 | This lemma is used to prov... |
stoweidlem42 43583 | This lemma is used to prov... |
stoweidlem43 43584 | This lemma is used to prov... |
stoweidlem44 43585 | This lemma is used to prov... |
stoweidlem45 43586 | This lemma proves that, gi... |
stoweidlem46 43587 | This lemma proves that set... |
stoweidlem47 43588 | Subtracting a constant fro... |
stoweidlem48 43589 | This lemma is used to prov... |
stoweidlem49 43590 | There exists a function q_... |
stoweidlem50 43591 | This lemma proves that set... |
stoweidlem51 43592 | There exists a function x ... |
stoweidlem52 43593 | There exists a neighborhoo... |
stoweidlem53 43594 | This lemma is used to prov... |
stoweidlem54 43595 | There exists a function ` ... |
stoweidlem55 43596 | This lemma proves the exis... |
stoweidlem56 43597 | This theorem proves Lemma ... |
stoweidlem57 43598 | There exists a function x ... |
stoweidlem58 43599 | This theorem proves Lemma ... |
stoweidlem59 43600 | This lemma proves that the... |
stoweidlem60 43601 | This lemma proves that the... |
stoweidlem61 43602 | This lemma proves that the... |
stoweidlem62 43603 | This theorem proves the St... |
stoweid 43604 | This theorem proves the St... |
stowei 43605 | This theorem proves the St... |
wallispilem1 43606 | ` I ` is monotone: increas... |
wallispilem2 43607 | A first set of properties ... |
wallispilem3 43608 | I maps to real values. (C... |
wallispilem4 43609 | ` F ` maps to explicit exp... |
wallispilem5 43610 | The sequence ` H ` converg... |
wallispi 43611 | Wallis' formula for π :... |
wallispi2lem1 43612 | An intermediate step betwe... |
wallispi2lem2 43613 | Two expressions are proven... |
wallispi2 43614 | An alternative version of ... |
stirlinglem1 43615 | A simple limit of fraction... |
stirlinglem2 43616 | ` A ` maps to positive rea... |
stirlinglem3 43617 | Long but simple algebraic ... |
stirlinglem4 43618 | Algebraic manipulation of ... |
stirlinglem5 43619 | If ` T ` is between ` 0 ` ... |
stirlinglem6 43620 | A series that converges to... |
stirlinglem7 43621 | Algebraic manipulation of ... |
stirlinglem8 43622 | If ` A ` converges to ` C ... |
stirlinglem9 43623 | ` ( ( B `` N ) - ( B `` ( ... |
stirlinglem10 43624 | A bound for any B(N)-B(N +... |
stirlinglem11 43625 | ` B ` is decreasing. (Con... |
stirlinglem12 43626 | The sequence ` B ` is boun... |
stirlinglem13 43627 | ` B ` is decreasing and ha... |
stirlinglem14 43628 | The sequence ` A ` converg... |
stirlinglem15 43629 | The Stirling's formula is ... |
stirling 43630 | Stirling's approximation f... |
stirlingr 43631 | Stirling's approximation f... |
dirkerval 43632 | The N_th Dirichlet Kernel.... |
dirker2re 43633 | The Dirichlet Kernel value... |
dirkerdenne0 43634 | The Dirichlet Kernel denom... |
dirkerval2 43635 | The N_th Dirichlet Kernel ... |
dirkerre 43636 | The Dirichlet Kernel at an... |
dirkerper 43637 | the Dirichlet Kernel has p... |
dirkerf 43638 | For any natural number ` N... |
dirkertrigeqlem1 43639 | Sum of an even number of a... |
dirkertrigeqlem2 43640 | Trigonomic equality lemma ... |
dirkertrigeqlem3 43641 | Trigonometric equality lem... |
dirkertrigeq 43642 | Trigonometric equality for... |
dirkeritg 43643 | The definite integral of t... |
dirkercncflem1 43644 | If ` Y ` is a multiple of ... |
dirkercncflem2 43645 | Lemma used to prove that t... |
dirkercncflem3 43646 | The Dirichlet Kernel is co... |
dirkercncflem4 43647 | The Dirichlet Kernel is co... |
dirkercncf 43648 | For any natural number ` N... |
fourierdlem1 43649 | A partition interval is a ... |
fourierdlem2 43650 | Membership in a partition.... |
fourierdlem3 43651 | Membership in a partition.... |
fourierdlem4 43652 | ` E ` is a function that m... |
fourierdlem5 43653 | ` S ` is a function. (Con... |
fourierdlem6 43654 | ` X ` is in the periodic p... |
fourierdlem7 43655 | The difference between the... |
fourierdlem8 43656 | A partition interval is a ... |
fourierdlem9 43657 | ` H ` is a complex functio... |
fourierdlem10 43658 | Condition on the bounds of... |
fourierdlem11 43659 | If there is a partition, t... |
fourierdlem12 43660 | A point of a partition is ... |
fourierdlem13 43661 | Value of ` V ` in terms of... |
fourierdlem14 43662 | Given the partition ` V ` ... |
fourierdlem15 43663 | The range of the partition... |
fourierdlem16 43664 | The coefficients of the fo... |
fourierdlem17 43665 | The defined ` L ` is actua... |
fourierdlem18 43666 | The function ` S ` is cont... |
fourierdlem19 43667 | If two elements of ` D ` h... |
fourierdlem20 43668 | Every interval in the part... |
fourierdlem21 43669 | The coefficients of the fo... |
fourierdlem22 43670 | The coefficients of the fo... |
fourierdlem23 43671 | If ` F ` is continuous and... |
fourierdlem24 43672 | A sufficient condition for... |
fourierdlem25 43673 | If ` C ` is not in the ran... |
fourierdlem26 43674 | Periodic image of a point ... |
fourierdlem27 43675 | A partition open interval ... |
fourierdlem28 43676 | Derivative of ` ( F `` ( X... |
fourierdlem29 43677 | Explicit function value fo... |
fourierdlem30 43678 | Sum of three small pieces ... |
fourierdlem31 43679 | If ` A ` is finite and for... |
fourierdlem32 43680 | Limit of a continuous func... |
fourierdlem33 43681 | Limit of a continuous func... |
fourierdlem34 43682 | A partition is one to one.... |
fourierdlem35 43683 | There is a single point in... |
fourierdlem36 43684 | ` F ` is an isomorphism. ... |
fourierdlem37 43685 | ` I ` is a function that m... |
fourierdlem38 43686 | The function ` F ` is cont... |
fourierdlem39 43687 | Integration by parts of ... |
fourierdlem40 43688 | ` H ` is a continuous func... |
fourierdlem41 43689 | Lemma used to prove that e... |
fourierdlem42 43690 | The set of points in a mov... |
fourierdlem43 43691 | ` K ` is a real function. ... |
fourierdlem44 43692 | A condition for having ` (... |
fourierdlem46 43693 | The function ` F ` has a l... |
fourierdlem47 43694 | For ` r ` large enough, th... |
fourierdlem48 43695 | The given periodic functio... |
fourierdlem49 43696 | The given periodic functio... |
fourierdlem50 43697 | Continuity of ` O ` and it... |
fourierdlem51 43698 | ` X ` is in the periodic p... |
fourierdlem52 43699 | d16:d17,d18:jca |- ( ph ->... |
fourierdlem53 43700 | The limit of ` F ( s ) ` a... |
fourierdlem54 43701 | Given a partition ` Q ` an... |
fourierdlem55 43702 | ` U ` is a real function. ... |
fourierdlem56 43703 | Derivative of the ` K ` fu... |
fourierdlem57 43704 | The derivative of ` O ` . ... |
fourierdlem58 43705 | The derivative of ` K ` is... |
fourierdlem59 43706 | The derivative of ` H ` is... |
fourierdlem60 43707 | Given a differentiable fun... |
fourierdlem61 43708 | Given a differentiable fun... |
fourierdlem62 43709 | The function ` K ` is cont... |
fourierdlem63 43710 | The upper bound of interva... |
fourierdlem64 43711 | The partition ` V ` is fin... |
fourierdlem65 43712 | The distance of two adjace... |
fourierdlem66 43713 | Value of the ` G ` functio... |
fourierdlem67 43714 | ` G ` is a function. (Con... |
fourierdlem68 43715 | The derivative of ` O ` is... |
fourierdlem69 43716 | A piecewise continuous fun... |
fourierdlem70 43717 | A piecewise continuous fun... |
fourierdlem71 43718 | A periodic piecewise conti... |
fourierdlem72 43719 | The derivative of ` O ` is... |
fourierdlem73 43720 | A version of the Riemann L... |
fourierdlem74 43721 | Given a piecewise smooth f... |
fourierdlem75 43722 | Given a piecewise smooth f... |
fourierdlem76 43723 | Continuity of ` O ` and it... |
fourierdlem77 43724 | If ` H ` is bounded, then ... |
fourierdlem78 43725 | ` G ` is continuous when r... |
fourierdlem79 43726 | ` E ` projects every inter... |
fourierdlem80 43727 | The derivative of ` O ` is... |
fourierdlem81 43728 | The integral of a piecewis... |
fourierdlem82 43729 | Integral by substitution, ... |
fourierdlem83 43730 | The fourier partial sum fo... |
fourierdlem84 43731 | If ` F ` is piecewise coni... |
fourierdlem85 43732 | Limit of the function ` G ... |
fourierdlem86 43733 | Continuity of ` O ` and it... |
fourierdlem87 43734 | The integral of ` G ` goes... |
fourierdlem88 43735 | Given a piecewise continuo... |
fourierdlem89 43736 | Given a piecewise continuo... |
fourierdlem90 43737 | Given a piecewise continuo... |
fourierdlem91 43738 | Given a piecewise continuo... |
fourierdlem92 43739 | The integral of a piecewis... |
fourierdlem93 43740 | Integral by substitution (... |
fourierdlem94 43741 | For a piecewise smooth fun... |
fourierdlem95 43742 | Algebraic manipulation of ... |
fourierdlem96 43743 | limit for ` F ` at the low... |
fourierdlem97 43744 | ` F ` is continuous on the... |
fourierdlem98 43745 | ` F ` is continuous on the... |
fourierdlem99 43746 | limit for ` F ` at the upp... |
fourierdlem100 43747 | A piecewise continuous fun... |
fourierdlem101 43748 | Integral by substitution f... |
fourierdlem102 43749 | For a piecewise smooth fun... |
fourierdlem103 43750 | The half lower part of the... |
fourierdlem104 43751 | The half upper part of the... |
fourierdlem105 43752 | A piecewise continuous fun... |
fourierdlem106 43753 | For a piecewise smooth fun... |
fourierdlem107 43754 | The integral of a piecewis... |
fourierdlem108 43755 | The integral of a piecewis... |
fourierdlem109 43756 | The integral of a piecewis... |
fourierdlem110 43757 | The integral of a piecewis... |
fourierdlem111 43758 | The fourier partial sum fo... |
fourierdlem112 43759 | Here abbreviations (local ... |
fourierdlem113 43760 | Fourier series convergence... |
fourierdlem114 43761 | Fourier series convergence... |
fourierdlem115 43762 | Fourier serier convergence... |
fourierd 43763 | Fourier series convergence... |
fourierclimd 43764 | Fourier series convergence... |
fourierclim 43765 | Fourier series convergence... |
fourier 43766 | Fourier series convergence... |
fouriercnp 43767 | If ` F ` is continuous at ... |
fourier2 43768 | Fourier series convergence... |
sqwvfoura 43769 | Fourier coefficients for t... |
sqwvfourb 43770 | Fourier series ` B ` coeff... |
fourierswlem 43771 | The Fourier series for the... |
fouriersw 43772 | Fourier series convergence... |
fouriercn 43773 | If the derivative of ` F `... |
elaa2lem 43774 | Elementhood in the set of ... |
elaa2 43775 | Elementhood in the set of ... |
etransclem1 43776 | ` H ` is a function. (Con... |
etransclem2 43777 | Derivative of ` G ` . (Co... |
etransclem3 43778 | The given ` if ` term is a... |
etransclem4 43779 | ` F ` expressed as a finit... |
etransclem5 43780 | A change of bound variable... |
etransclem6 43781 | A change of bound variable... |
etransclem7 43782 | The given product is an in... |
etransclem8 43783 | ` F ` is a function. (Con... |
etransclem9 43784 | If ` K ` divides ` N ` but... |
etransclem10 43785 | The given ` if ` term is a... |
etransclem11 43786 | A change of bound variable... |
etransclem12 43787 | ` C ` applied to ` N ` . ... |
etransclem13 43788 | ` F ` applied to ` Y ` . ... |
etransclem14 43789 | Value of the term ` T ` , ... |
etransclem15 43790 | Value of the term ` T ` , ... |
etransclem16 43791 | Every element in the range... |
etransclem17 43792 | The ` N ` -th derivative o... |
etransclem18 43793 | The given function is inte... |
etransclem19 43794 | The ` N ` -th derivative o... |
etransclem20 43795 | ` H ` is smooth. (Contrib... |
etransclem21 43796 | The ` N ` -th derivative o... |
etransclem22 43797 | The ` N ` -th derivative o... |
etransclem23 43798 | This is the claim proof in... |
etransclem24 43799 | ` P ` divides the I -th de... |
etransclem25 43800 | ` P ` factorial divides th... |
etransclem26 43801 | Every term in the sum of t... |
etransclem27 43802 | The ` N ` -th derivative o... |
etransclem28 43803 | ` ( P - 1 ) ` factorial di... |
etransclem29 43804 | The ` N ` -th derivative o... |
etransclem30 43805 | The ` N ` -th derivative o... |
etransclem31 43806 | The ` N ` -th derivative o... |
etransclem32 43807 | This is the proof for the ... |
etransclem33 43808 | ` F ` is smooth. (Contrib... |
etransclem34 43809 | The ` N ` -th derivative o... |
etransclem35 43810 | ` P ` does not divide the ... |
etransclem36 43811 | The ` N ` -th derivative o... |
etransclem37 43812 | ` ( P - 1 ) ` factorial di... |
etransclem38 43813 | ` P ` divides the I -th de... |
etransclem39 43814 | ` G ` is a function. (Con... |
etransclem40 43815 | The ` N ` -th derivative o... |
etransclem41 43816 | ` P ` does not divide the ... |
etransclem42 43817 | The ` N ` -th derivative o... |
etransclem43 43818 | ` G ` is a continuous func... |
etransclem44 43819 | The given finite sum is no... |
etransclem45 43820 | ` K ` is an integer. (Con... |
etransclem46 43821 | This is the proof for equa... |
etransclem47 43822 | ` _e ` is transcendental. ... |
etransclem48 43823 | ` _e ` is transcendental. ... |
etransc 43824 | ` _e ` is transcendental. ... |
rrxtopn 43825 | The topology of the genera... |
rrxngp 43826 | Generalized Euclidean real... |
rrxtps 43827 | Generalized Euclidean real... |
rrxtopnfi 43828 | The topology of the n-dime... |
rrxtopon 43829 | The topology on generalize... |
rrxtop 43830 | The topology on generalize... |
rrndistlt 43831 | Given two points in the sp... |
rrxtoponfi 43832 | The topology on n-dimensio... |
rrxunitopnfi 43833 | The base set of the standa... |
rrxtopn0 43834 | The topology of the zero-d... |
qndenserrnbllem 43835 | n-dimensional rational num... |
qndenserrnbl 43836 | n-dimensional rational num... |
rrxtopn0b 43837 | The topology of the zero-d... |
qndenserrnopnlem 43838 | n-dimensional rational num... |
qndenserrnopn 43839 | n-dimensional rational num... |
qndenserrn 43840 | n-dimensional rational num... |
rrxsnicc 43841 | A multidimensional singlet... |
rrnprjdstle 43842 | The distance between two p... |
rrndsmet 43843 | ` D ` is a metric for the ... |
rrndsxmet 43844 | ` D ` is an extended metri... |
ioorrnopnlem 43845 | The a point in an indexed ... |
ioorrnopn 43846 | The indexed product of ope... |
ioorrnopnxrlem 43847 | Given a point ` F ` that b... |
ioorrnopnxr 43848 | The indexed product of ope... |
issal 43855 | Express the predicate " ` ... |
pwsal 43856 | The power set of a given s... |
salunicl 43857 | SAlg sigma-algebra is clos... |
saluncl 43858 | The union of two sets in a... |
prsal 43859 | The pair of the empty set ... |
saldifcl 43860 | The complement of an eleme... |
0sal 43861 | The empty set belongs to e... |
salgenval 43862 | The sigma-algebra generate... |
saliuncl 43863 | SAlg sigma-algebra is clos... |
salincl 43864 | The intersection of two se... |
saluni 43865 | A set is an element of any... |
saliincl 43866 | SAlg sigma-algebra is clos... |
saldifcl2 43867 | The difference of two elem... |
intsaluni 43868 | The union of an arbitrary ... |
intsal 43869 | The arbitrary intersection... |
salgenn0 43870 | The set used in the defini... |
salgencl 43871 | ` SalGen ` actually genera... |
issald 43872 | Sufficient condition to pr... |
salexct 43873 | An example of nontrivial s... |
sssalgen 43874 | A set is a subset of the s... |
salgenss 43875 | The sigma-algebra generate... |
salgenuni 43876 | The base set of the sigma-... |
issalgend 43877 | One side of ~ dfsalgen2 . ... |
salexct2 43878 | An example of a subset tha... |
unisalgen 43879 | The union of a set belongs... |
dfsalgen2 43880 | Alternate characterization... |
salexct3 43881 | An example of a sigma-alge... |
salgencntex 43882 | This counterexample shows ... |
salgensscntex 43883 | This counterexample shows ... |
issalnnd 43884 | Sufficient condition to pr... |
dmvolsal 43885 | Lebesgue measurable sets f... |
saldifcld 43886 | The complement of an eleme... |
saluncld 43887 | The union of two sets in a... |
salgencld 43888 | ` SalGen ` actually genera... |
0sald 43889 | The empty set belongs to e... |
iooborel 43890 | An open interval is a Bore... |
salincld 43891 | The intersection of two se... |
salunid 43892 | A set is an element of any... |
unisalgen2 43893 | The union of a set belongs... |
bor1sal 43894 | The Borel sigma-algebra on... |
iocborel 43895 | A left-open, right-closed ... |
subsaliuncllem 43896 | A subspace sigma-algebra i... |
subsaliuncl 43897 | A subspace sigma-algebra i... |
subsalsal 43898 | A subspace sigma-algebra i... |
subsaluni 43899 | A set belongs to the subsp... |
sge0rnre 43902 | When ` sum^ ` is applied t... |
fge0icoicc 43903 | If ` F ` maps to nonnegati... |
sge0val 43904 | The value of the sum of no... |
fge0npnf 43905 | If ` F ` maps to nonnegati... |
sge0rnn0 43906 | The range used in the defi... |
sge0vald 43907 | The value of the sum of no... |
fge0iccico 43908 | A range of nonnegative ext... |
gsumge0cl 43909 | Closure of group sum, for ... |
sge0reval 43910 | Value of the sum of nonneg... |
sge0pnfval 43911 | If a term in the sum of no... |
fge0iccre 43912 | A range of nonnegative ext... |
sge0z 43913 | Any nonnegative extended s... |
sge00 43914 | The sum of nonnegative ext... |
fsumlesge0 43915 | Every finite subsum of non... |
sge0revalmpt 43916 | Value of the sum of nonneg... |
sge0sn 43917 | A sum of a nonnegative ext... |
sge0tsms 43918 | ` sum^ ` applied to a nonn... |
sge0cl 43919 | The arbitrary sum of nonne... |
sge0f1o 43920 | Re-index a nonnegative ext... |
sge0snmpt 43921 | A sum of a nonnegative ext... |
sge0ge0 43922 | The sum of nonnegative ext... |
sge0xrcl 43923 | The arbitrary sum of nonne... |
sge0repnf 43924 | The of nonnegative extende... |
sge0fsum 43925 | The arbitrary sum of a fin... |
sge0rern 43926 | If the sum of nonnegative ... |
sge0supre 43927 | If the arbitrary sum of no... |
sge0fsummpt 43928 | The arbitrary sum of a fin... |
sge0sup 43929 | The arbitrary sum of nonne... |
sge0less 43930 | A shorter sum of nonnegati... |
sge0rnbnd 43931 | The range used in the defi... |
sge0pr 43932 | Sum of a pair of nonnegati... |
sge0gerp 43933 | The arbitrary sum of nonne... |
sge0pnffigt 43934 | If the sum of nonnegative ... |
sge0ssre 43935 | If a sum of nonnegative ex... |
sge0lefi 43936 | A sum of nonnegative exten... |
sge0lessmpt 43937 | A shorter sum of nonnegati... |
sge0ltfirp 43938 | If the sum of nonnegative ... |
sge0prle 43939 | The sum of a pair of nonne... |
sge0gerpmpt 43940 | The arbitrary sum of nonne... |
sge0resrnlem 43941 | The sum of nonnegative ext... |
sge0resrn 43942 | The sum of nonnegative ext... |
sge0ssrempt 43943 | If a sum of nonnegative ex... |
sge0resplit 43944 | ` sum^ ` splits into two p... |
sge0le 43945 | If all of the terms of sum... |
sge0ltfirpmpt 43946 | If the extended sum of non... |
sge0split 43947 | Split a sum of nonnegative... |
sge0lempt 43948 | If all of the terms of sum... |
sge0splitmpt 43949 | Split a sum of nonnegative... |
sge0ss 43950 | Change the index set to a ... |
sge0iunmptlemfi 43951 | Sum of nonnegative extende... |
sge0p1 43952 | The addition of the next t... |
sge0iunmptlemre 43953 | Sum of nonnegative extende... |
sge0fodjrnlem 43954 | Re-index a nonnegative ext... |
sge0fodjrn 43955 | Re-index a nonnegative ext... |
sge0iunmpt 43956 | Sum of nonnegative extende... |
sge0iun 43957 | Sum of nonnegative extende... |
sge0nemnf 43958 | The generalized sum of non... |
sge0rpcpnf 43959 | The sum of an infinite num... |
sge0rernmpt 43960 | If the sum of nonnegative ... |
sge0lefimpt 43961 | A sum of nonnegative exten... |
nn0ssge0 43962 | Nonnegative integers are n... |
sge0clmpt 43963 | The generalized sum of non... |
sge0ltfirpmpt2 43964 | If the extended sum of non... |
sge0isum 43965 | If a series of nonnegative... |
sge0xrclmpt 43966 | The generalized sum of non... |
sge0xp 43967 | Combine two generalized su... |
sge0isummpt 43968 | If a series of nonnegative... |
sge0ad2en 43969 | The value of the infinite ... |
sge0isummpt2 43970 | If a series of nonnegative... |
sge0xaddlem1 43971 | The extended addition of t... |
sge0xaddlem2 43972 | The extended addition of t... |
sge0xadd 43973 | The extended addition of t... |
sge0fsummptf 43974 | The generalized sum of a f... |
sge0snmptf 43975 | A sum of a nonnegative ext... |
sge0ge0mpt 43976 | The sum of nonnegative ext... |
sge0repnfmpt 43977 | The of nonnegative extende... |
sge0pnffigtmpt 43978 | If the generalized sum of ... |
sge0splitsn 43979 | Separate out a term in a g... |
sge0pnffsumgt 43980 | If the sum of nonnegative ... |
sge0gtfsumgt 43981 | If the generalized sum of ... |
sge0uzfsumgt 43982 | If a real number is smalle... |
sge0pnfmpt 43983 | If a term in the sum of no... |
sge0seq 43984 | A series of nonnegative re... |
sge0reuz 43985 | Value of the generalized s... |
sge0reuzb 43986 | Value of the generalized s... |
ismea 43989 | Express the predicate " ` ... |
dmmeasal 43990 | The domain of a measure is... |
meaf 43991 | A measure is a function th... |
mea0 43992 | The measure of the empty s... |
nnfoctbdjlem 43993 | There exists a mapping fro... |
nnfoctbdj 43994 | There exists a mapping fro... |
meadjuni 43995 | The measure of the disjoin... |
meacl 43996 | The measure of a set is a ... |
iundjiunlem 43997 | The sets in the sequence `... |
iundjiun 43998 | Given a sequence ` E ` of ... |
meaxrcl 43999 | The measure of a set is an... |
meadjun 44000 | The measure of the union o... |
meassle 44001 | The measure of a set is gr... |
meaunle 44002 | The measure of the union o... |
meadjiunlem 44003 | The sum of nonnegative ext... |
meadjiun 44004 | The measure of the disjoin... |
ismeannd 44005 | Sufficient condition to pr... |
meaiunlelem 44006 | The measure of the union o... |
meaiunle 44007 | The measure of the union o... |
psmeasurelem 44008 | ` M ` applied to a disjoin... |
psmeasure 44009 | Point supported measure, R... |
voliunsge0lem 44010 | The Lebesgue measure funct... |
voliunsge0 44011 | The Lebesgue measure funct... |
volmea 44012 | The Lebeasgue measure on t... |
meage0 44013 | If the measure of a measur... |
meadjunre 44014 | The measure of the union o... |
meassre 44015 | If the measure of a measur... |
meale0eq0 44016 | A measure that is less tha... |
meadif 44017 | The measure of the differe... |
meaiuninclem 44018 | Measures are continuous fr... |
meaiuninc 44019 | Measures are continuous fr... |
meaiuninc2 44020 | Measures are continuous fr... |
meaiunincf 44021 | Measures are continuous fr... |
meaiuninc3v 44022 | Measures are continuous fr... |
meaiuninc3 44023 | Measures are continuous fr... |
meaiininclem 44024 | Measures are continuous fr... |
meaiininc 44025 | Measures are continuous fr... |
meaiininc2 44026 | Measures are continuous fr... |
caragenval 44031 | The sigma-algebra generate... |
isome 44032 | Express the predicate " ` ... |
caragenel 44033 | Membership in the Caratheo... |
omef 44034 | An outer measure is a func... |
ome0 44035 | The outer measure of the e... |
omessle 44036 | The outer measure of a set... |
omedm 44037 | The domain of an outer mea... |
caragensplit 44038 | If ` E ` is in the set gen... |
caragenelss 44039 | An element of the Caratheo... |
carageneld 44040 | Membership in the Caratheo... |
omecl 44041 | The outer measure of a set... |
caragenss 44042 | The sigma-algebra generate... |
omeunile 44043 | The outer measure of the u... |
caragen0 44044 | The empty set belongs to a... |
omexrcl 44045 | The outer measure of a set... |
caragenunidm 44046 | The base set of an outer m... |
caragensspw 44047 | The sigma-algebra generate... |
omessre 44048 | If the outer measure of a ... |
caragenuni 44049 | The base set of the sigma-... |
caragenuncllem 44050 | The Caratheodory's constru... |
caragenuncl 44051 | The Caratheodory's constru... |
caragendifcl 44052 | The Caratheodory's constru... |
caragenfiiuncl 44053 | The Caratheodory's constru... |
omeunle 44054 | The outer measure of the u... |
omeiunle 44055 | The outer measure of the i... |
omelesplit 44056 | The outer measure of a set... |
omeiunltfirp 44057 | If the outer measure of a ... |
omeiunlempt 44058 | The outer measure of the i... |
carageniuncllem1 44059 | The outer measure of ` A i... |
carageniuncllem2 44060 | The Caratheodory's constru... |
carageniuncl 44061 | The Caratheodory's constru... |
caragenunicl 44062 | The Caratheodory's constru... |
caragensal 44063 | Caratheodory's method gene... |
caratheodorylem1 44064 | Lemma used to prove that C... |
caratheodorylem2 44065 | Caratheodory's constructio... |
caratheodory 44066 | Caratheodory's constructio... |
0ome 44067 | The map that assigns 0 to ... |
isomenndlem 44068 | ` O ` is sub-additive w.r.... |
isomennd 44069 | Sufficient condition to pr... |
caragenel2d 44070 | Membership in the Caratheo... |
omege0 44071 | If the outer measure of a ... |
omess0 44072 | If the outer measure of a ... |
caragencmpl 44073 | A measure built with the C... |
vonval 44078 | Value of the Lebesgue meas... |
ovnval 44079 | Value of the Lebesgue oute... |
elhoi 44080 | Membership in a multidimen... |
icoresmbl 44081 | A closed-below, open-above... |
hoissre 44082 | The projection of a half-o... |
ovnval2 44083 | Value of the Lebesgue oute... |
volicorecl 44084 | The Lebesgue measure of a ... |
hoiprodcl 44085 | The pre-measure of half-op... |
hoicvr 44086 | ` I ` is a countable set o... |
hoissrrn 44087 | A half-open interval is a ... |
ovn0val 44088 | The Lebesgue outer measure... |
ovnn0val 44089 | The value of a (multidimen... |
ovnval2b 44090 | Value of the Lebesgue oute... |
volicorescl 44091 | The Lebesgue measure of a ... |
ovnprodcl 44092 | The product used in the de... |
hoiprodcl2 44093 | The pre-measure of half-op... |
hoicvrrex 44094 | Any subset of the multidim... |
ovnsupge0 44095 | The set used in the defini... |
ovnlecvr 44096 | Given a subset of multidim... |
ovnpnfelsup 44097 | ` +oo ` is an element of t... |
ovnsslelem 44098 | The (multidimensional, non... |
ovnssle 44099 | The (multidimensional) Leb... |
ovnlerp 44100 | The Lebesgue outer measure... |
ovnf 44101 | The Lebesgue outer measure... |
ovncvrrp 44102 | The Lebesgue outer measure... |
ovn0lem 44103 | For any finite dimension, ... |
ovn0 44104 | For any finite dimension, ... |
ovncl 44105 | The Lebesgue outer measure... |
ovn02 44106 | For the zero-dimensional s... |
ovnxrcl 44107 | The Lebesgue outer measure... |
ovnsubaddlem1 44108 | The Lebesgue outer measure... |
ovnsubaddlem2 44109 | ` ( voln* `` X ) ` is suba... |
ovnsubadd 44110 | ` ( voln* `` X ) ` is suba... |
ovnome 44111 | ` ( voln* `` X ) ` is an o... |
vonmea 44112 | ` ( voln `` X ) ` is a mea... |
volicon0 44113 | The measure of a nonempty ... |
hsphoif 44114 | ` H ` is a function (that ... |
hoidmvval 44115 | The dimensional volume of ... |
hoissrrn2 44116 | A half-open interval is a ... |
hsphoival 44117 | ` H ` is a function (that ... |
hoiprodcl3 44118 | The pre-measure of half-op... |
volicore 44119 | The Lebesgue measure of a ... |
hoidmvcl 44120 | The dimensional volume of ... |
hoidmv0val 44121 | The dimensional volume of ... |
hoidmvn0val 44122 | The dimensional volume of ... |
hsphoidmvle2 44123 | The dimensional volume of ... |
hsphoidmvle 44124 | The dimensional volume of ... |
hoidmvval0 44125 | The dimensional volume of ... |
hoiprodp1 44126 | The dimensional volume of ... |
sge0hsphoire 44127 | If the generalized sum of ... |
hoidmvval0b 44128 | The dimensional volume of ... |
hoidmv1lelem1 44129 | The supremum of ` U ` belo... |
hoidmv1lelem2 44130 | This is the contradiction ... |
hoidmv1lelem3 44131 | The dimensional volume of ... |
hoidmv1le 44132 | The dimensional volume of ... |
hoidmvlelem1 44133 | The supremum of ` U ` belo... |
hoidmvlelem2 44134 | This is the contradiction ... |
hoidmvlelem3 44135 | This is the contradiction ... |
hoidmvlelem4 44136 | The dimensional volume of ... |
hoidmvlelem5 44137 | The dimensional volume of ... |
hoidmvle 44138 | The dimensional volume of ... |
ovnhoilem1 44139 | The Lebesgue outer measure... |
ovnhoilem2 44140 | The Lebesgue outer measure... |
ovnhoi 44141 | The Lebesgue outer measure... |
dmovn 44142 | The domain of the Lebesgue... |
hoicoto2 44143 | The half-open interval exp... |
dmvon 44144 | Lebesgue measurable n-dime... |
hoi2toco 44145 | The half-open interval exp... |
hoidifhspval 44146 | ` D ` is a function that r... |
hspval 44147 | The value of the half-spac... |
ovnlecvr2 44148 | Given a subset of multidim... |
ovncvr2 44149 | ` B ` and ` T ` are the le... |
dmovnsal 44150 | The domain of the Lebesgue... |
unidmovn 44151 | Base set of the n-dimensio... |
rrnmbl 44152 | The set of n-dimensional R... |
hoidifhspval2 44153 | ` D ` is a function that r... |
hspdifhsp 44154 | A n-dimensional half-open ... |
unidmvon 44155 | Base set of the n-dimensio... |
hoidifhspf 44156 | ` D ` is a function that r... |
hoidifhspval3 44157 | ` D ` is a function that r... |
hoidifhspdmvle 44158 | The dimensional volume of ... |
voncmpl 44159 | The Lebesgue measure is co... |
hoiqssbllem1 44160 | The center of the n-dimens... |
hoiqssbllem2 44161 | The center of the n-dimens... |
hoiqssbllem3 44162 | A n-dimensional ball conta... |
hoiqssbl 44163 | A n-dimensional ball conta... |
hspmbllem1 44164 | Any half-space of the n-di... |
hspmbllem2 44165 | Any half-space of the n-di... |
hspmbllem3 44166 | Any half-space of the n-di... |
hspmbl 44167 | Any half-space of the n-di... |
hoimbllem 44168 | Any n-dimensional half-ope... |
hoimbl 44169 | Any n-dimensional half-ope... |
opnvonmbllem1 44170 | The half-open interval exp... |
opnvonmbllem2 44171 | An open subset of the n-di... |
opnvonmbl 44172 | An open subset of the n-di... |
opnssborel 44173 | Open sets of a generalized... |
borelmbl 44174 | All Borel subsets of the n... |
volicorege0 44175 | The Lebesgue measure of a ... |
isvonmbl 44176 | The predicate " ` A ` is m... |
mblvon 44177 | The n-dimensional Lebesgue... |
vonmblss 44178 | n-dimensional Lebesgue mea... |
volico2 44179 | The measure of left-closed... |
vonmblss2 44180 | n-dimensional Lebesgue mea... |
ovolval2lem 44181 | The value of the Lebesgue ... |
ovolval2 44182 | The value of the Lebesgue ... |
ovnsubadd2lem 44183 | ` ( voln* `` X ) ` is suba... |
ovnsubadd2 44184 | ` ( voln* `` X ) ` is suba... |
ovolval3 44185 | The value of the Lebesgue ... |
ovnsplit 44186 | The n-dimensional Lebesgue... |
ovolval4lem1 44187 | |- ( ( ph /\ n e. A ) -> ... |
ovolval4lem2 44188 | The value of the Lebesgue ... |
ovolval4 44189 | The value of the Lebesgue ... |
ovolval5lem1 44190 | ` |- ( ph -> ( sum^ `` ( n... |
ovolval5lem2 44191 | ` |- ( ( ph /\ n e. NN ) -... |
ovolval5lem3 44192 | The value of the Lebesgue ... |
ovolval5 44193 | The value of the Lebesgue ... |
ovnovollem1 44194 | if ` F ` is a cover of ` B... |
ovnovollem2 44195 | if ` I ` is a cover of ` (... |
ovnovollem3 44196 | The 1-dimensional Lebesgue... |
ovnovol 44197 | The 1-dimensional Lebesgue... |
vonvolmbllem 44198 | If a subset ` B ` of real ... |
vonvolmbl 44199 | A subset of Real numbers i... |
vonvol 44200 | The 1-dimensional Lebesgue... |
vonvolmbl2 44201 | A subset ` X ` of the spac... |
vonvol2 44202 | The 1-dimensional Lebesgue... |
hoimbl2 44203 | Any n-dimensional half-ope... |
voncl 44204 | The Lebesgue measure of a ... |
vonhoi 44205 | The Lebesgue outer measure... |
vonxrcl 44206 | The Lebesgue measure of a ... |
ioosshoi 44207 | A n-dimensional open inter... |
vonn0hoi 44208 | The Lebesgue outer measure... |
von0val 44209 | The Lebesgue measure (for ... |
vonhoire 44210 | The Lebesgue measure of a ... |
iinhoiicclem 44211 | A n-dimensional closed int... |
iinhoiicc 44212 | A n-dimensional closed int... |
iunhoiioolem 44213 | A n-dimensional open inter... |
iunhoiioo 44214 | A n-dimensional open inter... |
ioovonmbl 44215 | Any n-dimensional open int... |
iccvonmbllem 44216 | Any n-dimensional closed i... |
iccvonmbl 44217 | Any n-dimensional closed i... |
vonioolem1 44218 | The sequence of the measur... |
vonioolem2 44219 | The n-dimensional Lebesgue... |
vonioo 44220 | The n-dimensional Lebesgue... |
vonicclem1 44221 | The sequence of the measur... |
vonicclem2 44222 | The n-dimensional Lebesgue... |
vonicc 44223 | The n-dimensional Lebesgue... |
snvonmbl 44224 | A n-dimensional singleton ... |
vonn0ioo 44225 | The n-dimensional Lebesgue... |
vonn0icc 44226 | The n-dimensional Lebesgue... |
ctvonmbl 44227 | Any n-dimensional countabl... |
vonn0ioo2 44228 | The n-dimensional Lebesgue... |
vonsn 44229 | The n-dimensional Lebesgue... |
vonn0icc2 44230 | The n-dimensional Lebesgue... |
vonct 44231 | The n-dimensional Lebesgue... |
vitali2 44232 | There are non-measurable s... |
pimltmnf2f 44235 | Given a real-valued functi... |
pimltmnf2 44236 | Given a real-valued functi... |
preimagelt 44237 | The preimage of a right-op... |
preimalegt 44238 | The preimage of a left-ope... |
pimconstlt0 44239 | Given a constant function,... |
pimconstlt1 44240 | Given a constant function,... |
pimltpnf 44241 | Given a real-valued functi... |
pimgtpnf2f 44242 | Given a real-valued functi... |
pimgtpnf2 44243 | Given a real-valued functi... |
salpreimagelt 44244 | If all the preimages of le... |
pimrecltpos 44245 | The preimage of an unbound... |
salpreimalegt 44246 | If all the preimages of ri... |
pimiooltgt 44247 | The preimage of an open in... |
preimaicomnf 44248 | Preimage of an open interv... |
pimltpnf2f 44249 | Given a real-valued functi... |
pimltpnf2 44250 | Given a real-valued functi... |
pimgtmnf2 44251 | Given a real-valued functi... |
pimdecfgtioc 44252 | Given a nonincreasing func... |
pimincfltioc 44253 | Given a nondecreasing func... |
pimdecfgtioo 44254 | Given a nondecreasing func... |
pimincfltioo 44255 | Given a nondecreasing func... |
preimaioomnf 44256 | Preimage of an open interv... |
preimageiingt 44257 | A preimage of a left-close... |
preimaleiinlt 44258 | A preimage of a left-open,... |
pimgtmnf 44259 | Given a real-valued functi... |
pimrecltneg 44260 | The preimage of an unbound... |
salpreimagtge 44261 | If all the preimages of le... |
salpreimaltle 44262 | If all the preimages of ri... |
issmflem 44263 | The predicate " ` F ` is a... |
issmf 44264 | The predicate " ` F ` is a... |
salpreimalelt 44265 | If all the preimages of ri... |
salpreimagtlt 44266 | If all the preimages of le... |
smfpreimalt 44267 | Given a function measurabl... |
smff 44268 | A function measurable w.r.... |
smfdmss 44269 | The domain of a function m... |
issmff 44270 | The predicate " ` F ` is a... |
issmfd 44271 | A sufficient condition for... |
smfpreimaltf 44272 | Given a function measurabl... |
issmfdf 44273 | A sufficient condition for... |
sssmf 44274 | The restriction of a sigma... |
mbfresmf 44275 | A real-valued measurable f... |
cnfsmf 44276 | A continuous function is m... |
incsmflem 44277 | A nondecreasing function i... |
incsmf 44278 | A real-valued, nondecreasi... |
smfsssmf 44279 | If a function is measurabl... |
issmflelem 44280 | The predicate " ` F ` is a... |
issmfle 44281 | The predicate " ` F ` is a... |
smfpimltmpt 44282 | Given a function measurabl... |
smfpimltxr 44283 | Given a function measurabl... |
issmfdmpt 44284 | A sufficient condition for... |
smfconst 44285 | Given a sigma-algebra over... |
sssmfmpt 44286 | The restriction of a sigma... |
cnfrrnsmf 44287 | A function, continuous fro... |
smfid 44288 | The identity function is B... |
bormflebmf 44289 | A Borel measurable functio... |
smfpreimale 44290 | Given a function measurabl... |
issmfgtlem 44291 | The predicate " ` F ` is a... |
issmfgt 44292 | The predicate " ` F ` is a... |
issmfled 44293 | A sufficient condition for... |
smfpimltxrmpt 44294 | Given a function measurabl... |
smfmbfcex 44295 | A constant function, with ... |
issmfgtd 44296 | A sufficient condition for... |
smfpreimagt 44297 | Given a function measurabl... |
smfaddlem1 44298 | Given the sum of two funct... |
smfaddlem2 44299 | The sum of two sigma-measu... |
smfadd 44300 | The sum of two sigma-measu... |
decsmflem 44301 | A nonincreasing function i... |
decsmf 44302 | A real-valued, nonincreasi... |
smfpreimagtf 44303 | Given a function measurabl... |
issmfgelem 44304 | The predicate " ` F ` is a... |
issmfge 44305 | The predicate " ` F ` is a... |
smflimlem1 44306 | Lemma for the proof that t... |
smflimlem2 44307 | Lemma for the proof that t... |
smflimlem3 44308 | The limit of sigma-measura... |
smflimlem4 44309 | Lemma for the proof that t... |
smflimlem5 44310 | Lemma for the proof that t... |
smflimlem6 44311 | Lemma for the proof that t... |
smflim 44312 | The limit of sigma-measura... |
nsssmfmbflem 44313 | The sigma-measurable funct... |
nsssmfmbf 44314 | The sigma-measurable funct... |
smfpimgtxr 44315 | Given a function measurabl... |
smfpimgtmpt 44316 | Given a function measurabl... |
smfpreimage 44317 | Given a function measurabl... |
mbfpsssmf 44318 | Real-valued measurable fun... |
smfpimgtxrmpt 44319 | Given a function measurabl... |
smfpimioompt 44320 | Given a function measurabl... |
smfpimioo 44321 | Given a function measurabl... |
smfresal 44322 | Given a sigma-measurable f... |
smfrec 44323 | The reciprocal of a sigma-... |
smfres 44324 | The restriction of sigma-m... |
smfmullem1 44325 | The multiplication of two ... |
smfmullem2 44326 | The multiplication of two ... |
smfmullem3 44327 | The multiplication of two ... |
smfmullem4 44328 | The multiplication of two ... |
smfmul 44329 | The multiplication of two ... |
smfmulc1 44330 | A sigma-measurable functio... |
smfdiv 44331 | The fraction of two sigma-... |
smfpimbor1lem1 44332 | Every open set belongs to ... |
smfpimbor1lem2 44333 | Given a sigma-measurable f... |
smfpimbor1 44334 | Given a sigma-measurable f... |
smf2id 44335 | Twice the identity functio... |
smfco 44336 | The composition of a Borel... |
smfneg 44337 | The negative of a sigma-me... |
smffmpt 44338 | A function measurable w.r.... |
smflim2 44339 | The limit of a sequence of... |
smfpimcclem 44340 | Lemma for ~ smfpimcc given... |
smfpimcc 44341 | Given a countable set of s... |
issmfle2d 44342 | A sufficient condition for... |
smflimmpt 44343 | The limit of a sequence of... |
smfsuplem1 44344 | The supremum of a countabl... |
smfsuplem2 44345 | The supremum of a countabl... |
smfsuplem3 44346 | The supremum of a countabl... |
smfsup 44347 | The supremum of a countabl... |
smfsupmpt 44348 | The supremum of a countabl... |
smfsupxr 44349 | The supremum of a countabl... |
smfinflem 44350 | The infimum of a countable... |
smfinf 44351 | The infimum of a countable... |
smfinfmpt 44352 | The infimum of a countable... |
smflimsuplem1 44353 | If ` H ` converges, the ` ... |
smflimsuplem2 44354 | The superior limit of a se... |
smflimsuplem3 44355 | The limit of the ` ( H `` ... |
smflimsuplem4 44356 | If ` H ` converges, the ` ... |
smflimsuplem5 44357 | ` H ` converges to the sup... |
smflimsuplem6 44358 | The superior limit of a se... |
smflimsuplem7 44359 | The superior limit of a se... |
smflimsuplem8 44360 | The superior limit of a se... |
smflimsup 44361 | The superior limit of a se... |
smflimsupmpt 44362 | The superior limit of a se... |
smfliminflem 44363 | The inferior limit of a co... |
smfliminf 44364 | The inferior limit of a co... |
smfliminfmpt 44365 | The inferior limit of a co... |
sigarval 44366 | Define the signed area by ... |
sigarim 44367 | Signed area takes value in... |
sigarac 44368 | Signed area is anticommuta... |
sigaraf 44369 | Signed area is additive by... |
sigarmf 44370 | Signed area is additive (w... |
sigaras 44371 | Signed area is additive by... |
sigarms 44372 | Signed area is additive (w... |
sigarls 44373 | Signed area is linear by t... |
sigarid 44374 | Signed area of a flat para... |
sigarexp 44375 | Expand the signed area for... |
sigarperm 44376 | Signed area ` ( A - C ) G ... |
sigardiv 44377 | If signed area between vec... |
sigarimcd 44378 | Signed area takes value in... |
sigariz 44379 | If signed area is zero, th... |
sigarcol 44380 | Given three points ` A ` ,... |
sharhght 44381 | Let ` A B C ` be a triangl... |
sigaradd 44382 | Subtracting (double) area ... |
cevathlem1 44383 | Ceva's theorem first lemma... |
cevathlem2 44384 | Ceva's theorem second lemm... |
cevath 44385 | Ceva's theorem. Let ` A B... |
simpcntrab 44386 | The center of a simple gro... |
hirstL-ax3 44387 | The third axiom of a syste... |
ax3h 44388 | Recover ~ ax-3 from ~ hirs... |
aibandbiaiffaiffb 44389 | A closed form showing (a i... |
aibandbiaiaiffb 44390 | A closed form showing (a i... |
notatnand 44391 | Do not use. Use intnanr i... |
aistia 44392 | Given a is equivalent to `... |
aisfina 44393 | Given a is equivalent to `... |
bothtbothsame 44394 | Given both a, b are equiva... |
bothfbothsame 44395 | Given both a, b are equiva... |
aiffbbtat 44396 | Given a is equivalent to b... |
aisbbisfaisf 44397 | Given a is equivalent to b... |
axorbtnotaiffb 44398 | Given a is exclusive to b,... |
aiffnbandciffatnotciffb 44399 | Given a is equivalent to (... |
axorbciffatcxorb 44400 | Given a is equivalent to (... |
aibnbna 44401 | Given a implies b, (not b)... |
aibnbaif 44402 | Given a implies b, not b, ... |
aiffbtbat 44403 | Given a is equivalent to b... |
astbstanbst 44404 | Given a is equivalent to T... |
aistbistaandb 44405 | Given a is equivalent to T... |
aisbnaxb 44406 | Given a is equivalent to b... |
atbiffatnnb 44407 | If a implies b, then a imp... |
bisaiaisb 44408 | Application of bicom1 with... |
atbiffatnnbalt 44409 | If a implies b, then a imp... |
abnotbtaxb 44410 | Assuming a, not b, there e... |
abnotataxb 44411 | Assuming not a, b, there e... |
conimpf 44412 | Assuming a, not b, and a i... |
conimpfalt 44413 | Assuming a, not b, and a i... |
aistbisfiaxb 44414 | Given a is equivalent to T... |
aisfbistiaxb 44415 | Given a is equivalent to F... |
aifftbifffaibif 44416 | Given a is equivalent to T... |
aifftbifffaibifff 44417 | Given a is equivalent to T... |
atnaiana 44418 | Given a, it is not the cas... |
ainaiaandna 44419 | Given a, a implies it is n... |
abcdta 44420 | Given (((a and b) and c) a... |
abcdtb 44421 | Given (((a and b) and c) a... |
abcdtc 44422 | Given (((a and b) and c) a... |
abcdtd 44423 | Given (((a and b) and c) a... |
abciffcbatnabciffncba 44424 | Operands in a biconditiona... |
abciffcbatnabciffncbai 44425 | Operands in a biconditiona... |
nabctnabc 44426 | not ( a -> ( b /\ c ) ) we... |
jabtaib 44427 | For when pm3.4 lacks a pm3... |
onenotinotbothi 44428 | From one negated implicati... |
twonotinotbothi 44429 | From these two negated imp... |
clifte 44430 | show d is the same as an i... |
cliftet 44431 | show d is the same as an i... |
clifteta 44432 | show d is the same as an i... |
cliftetb 44433 | show d is the same as an i... |
confun 44434 | Given the hypotheses there... |
confun2 44435 | Confun simplified to two p... |
confun3 44436 | Confun's more complex form... |
confun4 44437 | An attempt at derivative. ... |
confun5 44438 | An attempt at derivative. ... |
plcofph 44439 | Given, a,b and a "definiti... |
pldofph 44440 | Given, a,b c, d, "definiti... |
plvcofph 44441 | Given, a,b,d, and "definit... |
plvcofphax 44442 | Given, a,b,d, and "definit... |
plvofpos 44443 | rh is derivable because ON... |
mdandyv0 44444 | Given the equivalences set... |
mdandyv1 44445 | Given the equivalences set... |
mdandyv2 44446 | Given the equivalences set... |
mdandyv3 44447 | Given the equivalences set... |
mdandyv4 44448 | Given the equivalences set... |
mdandyv5 44449 | Given the equivalences set... |
mdandyv6 44450 | Given the equivalences set... |
mdandyv7 44451 | Given the equivalences set... |
mdandyv8 44452 | Given the equivalences set... |
mdandyv9 44453 | Given the equivalences set... |
mdandyv10 44454 | Given the equivalences set... |
mdandyv11 44455 | Given the equivalences set... |
mdandyv12 44456 | Given the equivalences set... |
mdandyv13 44457 | Given the equivalences set... |
mdandyv14 44458 | Given the equivalences set... |
mdandyv15 44459 | Given the equivalences set... |
mdandyvr0 44460 | Given the equivalences set... |
mdandyvr1 44461 | Given the equivalences set... |
mdandyvr2 44462 | Given the equivalences set... |
mdandyvr3 44463 | Given the equivalences set... |
mdandyvr4 44464 | Given the equivalences set... |
mdandyvr5 44465 | Given the equivalences set... |
mdandyvr6 44466 | Given the equivalences set... |
mdandyvr7 44467 | Given the equivalences set... |
mdandyvr8 44468 | Given the equivalences set... |
mdandyvr9 44469 | Given the equivalences set... |
mdandyvr10 44470 | Given the equivalences set... |
mdandyvr11 44471 | Given the equivalences set... |
mdandyvr12 44472 | Given the equivalences set... |
mdandyvr13 44473 | Given the equivalences set... |
mdandyvr14 44474 | Given the equivalences set... |
mdandyvr15 44475 | Given the equivalences set... |
mdandyvrx0 44476 | Given the exclusivities se... |
mdandyvrx1 44477 | Given the exclusivities se... |
mdandyvrx2 44478 | Given the exclusivities se... |
mdandyvrx3 44479 | Given the exclusivities se... |
mdandyvrx4 44480 | Given the exclusivities se... |
mdandyvrx5 44481 | Given the exclusivities se... |
mdandyvrx6 44482 | Given the exclusivities se... |
mdandyvrx7 44483 | Given the exclusivities se... |
mdandyvrx8 44484 | Given the exclusivities se... |
mdandyvrx9 44485 | Given the exclusivities se... |
mdandyvrx10 44486 | Given the exclusivities se... |
mdandyvrx11 44487 | Given the exclusivities se... |
mdandyvrx12 44488 | Given the exclusivities se... |
mdandyvrx13 44489 | Given the exclusivities se... |
mdandyvrx14 44490 | Given the exclusivities se... |
mdandyvrx15 44491 | Given the exclusivities se... |
H15NH16TH15IH16 44492 | Given 15 hypotheses and a ... |
dandysum2p2e4 44493 | CONTRADICTION PROVED AT 1 ... |
mdandysum2p2e4 44494 | CONTRADICTION PROVED AT 1 ... |
adh-jarrsc 44495 | Replacement of a nested an... |
adh-minim 44496 | A single axiom for minimal... |
adh-minim-ax1-ax2-lem1 44497 | First lemma for the deriva... |
adh-minim-ax1-ax2-lem2 44498 | Second lemma for the deriv... |
adh-minim-ax1-ax2-lem3 44499 | Third lemma for the deriva... |
adh-minim-ax1-ax2-lem4 44500 | Fourth lemma for the deriv... |
adh-minim-ax1 44501 | Derivation of ~ ax-1 from ... |
adh-minim-ax2-lem5 44502 | Fifth lemma for the deriva... |
adh-minim-ax2-lem6 44503 | Sixth lemma for the deriva... |
adh-minim-ax2c 44504 | Derivation of a commuted f... |
adh-minim-ax2 44505 | Derivation of ~ ax-2 from ... |
adh-minim-idALT 44506 | Derivation of ~ id (reflex... |
adh-minim-pm2.43 44507 | Derivation of ~ pm2.43 Whi... |
adh-minimp 44508 | Another single axiom for m... |
adh-minimp-jarr-imim1-ax2c-lem1 44509 | First lemma for the deriva... |
adh-minimp-jarr-lem2 44510 | Second lemma for the deriv... |
adh-minimp-jarr-ax2c-lem3 44511 | Third lemma for the deriva... |
adh-minimp-sylsimp 44512 | Derivation of ~ jarr (also... |
adh-minimp-ax1 44513 | Derivation of ~ ax-1 from ... |
adh-minimp-imim1 44514 | Derivation of ~ imim1 ("le... |
adh-minimp-ax2c 44515 | Derivation of a commuted f... |
adh-minimp-ax2-lem4 44516 | Fourth lemma for the deriv... |
adh-minimp-ax2 44517 | Derivation of ~ ax-2 from ... |
adh-minimp-idALT 44518 | Derivation of ~ id (reflex... |
adh-minimp-pm2.43 44519 | Derivation of ~ pm2.43 Whi... |
eusnsn 44520 | There is a unique element ... |
absnsb 44521 | If the class abstraction `... |
euabsneu 44522 | Another way to express exi... |
elprneb 44523 | An element of a proper uno... |
oppr 44524 | Equality for ordered pairs... |
opprb 44525 | Equality for unordered pai... |
or2expropbilem1 44526 | Lemma 1 for ~ or2expropbi ... |
or2expropbilem2 44527 | Lemma 2 for ~ or2expropbi ... |
or2expropbi 44528 | If two classes are strictl... |
eubrv 44529 | If there is a unique set w... |
eubrdm 44530 | If there is a unique set w... |
eldmressn 44531 | Element of the domain of a... |
iota0def 44532 | Example for a defined iota... |
iota0ndef 44533 | Example for an undefined i... |
fveqvfvv 44534 | If a function's value at a... |
fnresfnco 44535 | Composition of two functio... |
funcoressn 44536 | A composition restricted t... |
funressnfv 44537 | A restriction to a singlet... |
funressndmfvrn 44538 | The value of a function ` ... |
funressnvmo 44539 | A function restricted to a... |
funressnmo 44540 | A function restricted to a... |
funressneu 44541 | There is exactly one value... |
fresfo 44542 | Conditions for a restricti... |
fsetsniunop 44543 | The class of all functions... |
fsetabsnop 44544 | The class of all functions... |
fsetsnf 44545 | The mapping of an element ... |
fsetsnf1 44546 | The mapping of an element ... |
fsetsnfo 44547 | The mapping of an element ... |
fsetsnf1o 44548 | The mapping of an element ... |
fsetsnprcnex 44549 | The class of all functions... |
cfsetssfset 44550 | The class of constant func... |
cfsetsnfsetfv 44551 | The function value of the ... |
cfsetsnfsetf 44552 | The mapping of the class o... |
cfsetsnfsetf1 44553 | The mapping of the class o... |
cfsetsnfsetfo 44554 | The mapping of the class o... |
cfsetsnfsetf1o 44555 | The mapping of the class o... |
fsetprcnexALT 44556 | First version of proof for... |
fcoreslem1 44557 | Lemma 1 for ~ fcores . (C... |
fcoreslem2 44558 | Lemma 2 for ~ fcores . (C... |
fcoreslem3 44559 | Lemma 3 for ~ fcores . (C... |
fcoreslem4 44560 | Lemma 4 for ~ fcores . (C... |
fcores 44561 | Every composite function `... |
fcoresf1lem 44562 | Lemma for ~ fcoresf1 . (C... |
fcoresf1 44563 | If a composition is inject... |
fcoresf1b 44564 | A composition is injective... |
fcoresfo 44565 | If a composition is surjec... |
fcoresfob 44566 | A composition is surjectiv... |
fcoresf1ob 44567 | A composition is bijective... |
f1cof1blem 44568 | Lemma for ~ f1cof1b and ~ ... |
f1cof1b 44569 | If the range of ` F ` equa... |
funfocofob 44570 | If the domain of a functio... |
fnfocofob 44571 | If the domain of a functio... |
focofob 44572 | If the domain of a functio... |
f1ocof1ob 44573 | If the range of ` F ` equa... |
f1ocof1ob2 44574 | If the range of ` F ` equa... |
aiotajust 44576 | Soundness justification th... |
dfaiota2 44578 | Alternate definition of th... |
reuabaiotaiota 44579 | The iota and the alternate... |
reuaiotaiota 44580 | The iota and the alternate... |
aiotaexb 44581 | The alternate iota over a ... |
aiotavb 44582 | The alternate iota over a ... |
aiotaint 44583 | This is to ~ df-aiota what... |
dfaiota3 44584 | Alternate definition of ` ... |
iotan0aiotaex 44585 | If the iota over a wff ` p... |
aiotaexaiotaiota 44586 | The alternate iota over a ... |
aiotaval 44587 | Theorem 8.19 in [Quine] p.... |
aiota0def 44588 | Example for a defined alte... |
aiota0ndef 44589 | Example for an undefined a... |
r19.32 44590 | Theorem 19.32 of [Margaris... |
rexsb 44591 | An equivalent expression f... |
rexrsb 44592 | An equivalent expression f... |
2rexsb 44593 | An equivalent expression f... |
2rexrsb 44594 | An equivalent expression f... |
cbvral2 44595 | Change bound variables of ... |
cbvrex2 44596 | Change bound variables of ... |
ralndv1 44597 | Example for a theorem abou... |
ralndv2 44598 | Second example for a theor... |
reuf1odnf 44599 | There is exactly one eleme... |
reuf1od 44600 | There is exactly one eleme... |
euoreqb 44601 | There is a set which is eq... |
2reu3 44602 | Double restricted existent... |
2reu7 44603 | Two equivalent expressions... |
2reu8 44604 | Two equivalent expressions... |
2reu8i 44605 | Implication of a double re... |
2reuimp0 44606 | Implication of a double re... |
2reuimp 44607 | Implication of a double re... |
ralbinrald 44614 | Elemination of a restricte... |
nvelim 44615 | If a class is the universa... |
alneu 44616 | If a statement holds for a... |
eu2ndop1stv 44617 | If there is a unique secon... |
dfateq12d 44618 | Equality deduction for "de... |
nfdfat 44619 | Bound-variable hypothesis ... |
dfdfat2 44620 | Alternate definition of th... |
fundmdfat 44621 | A function is defined at a... |
dfatprc 44622 | A function is not defined ... |
dfatelrn 44623 | The value of a function ` ... |
dfafv2 44624 | Alternative definition of ... |
afveq12d 44625 | Equality deduction for fun... |
afveq1 44626 | Equality theorem for funct... |
afveq2 44627 | Equality theorem for funct... |
nfafv 44628 | Bound-variable hypothesis ... |
csbafv12g 44629 | Move class substitution in... |
afvfundmfveq 44630 | If a class is a function r... |
afvnfundmuv 44631 | If a set is not in the dom... |
ndmafv 44632 | The value of a class outsi... |
afvvdm 44633 | If the function value of a... |
nfunsnafv 44634 | If the restriction of a cl... |
afvvfunressn 44635 | If the function value of a... |
afvprc 44636 | A function's value at a pr... |
afvvv 44637 | If a function's value at a... |
afvpcfv0 44638 | If the value of the altern... |
afvnufveq 44639 | The value of the alternati... |
afvvfveq 44640 | The value of the alternati... |
afv0fv0 44641 | If the value of the altern... |
afvfvn0fveq 44642 | If the function's value at... |
afv0nbfvbi 44643 | The function's value at an... |
afvfv0bi 44644 | The function's value at an... |
afveu 44645 | The value of a function at... |
fnbrafvb 44646 | Equivalence of function va... |
fnopafvb 44647 | Equivalence of function va... |
funbrafvb 44648 | Equivalence of function va... |
funopafvb 44649 | Equivalence of function va... |
funbrafv 44650 | The second argument of a b... |
funbrafv2b 44651 | Function value in terms of... |
dfafn5a 44652 | Representation of a functi... |
dfafn5b 44653 | Representation of a functi... |
fnrnafv 44654 | The range of a function ex... |
afvelrnb 44655 | A member of a function's r... |
afvelrnb0 44656 | A member of a function's r... |
dfaimafn 44657 | Alternate definition of th... |
dfaimafn2 44658 | Alternate definition of th... |
afvelima 44659 | Function value in an image... |
afvelrn 44660 | A function's value belongs... |
fnafvelrn 44661 | A function's value belongs... |
fafvelrn 44662 | A function's value belongs... |
ffnafv 44663 | A function maps to a class... |
afvres 44664 | The value of a restricted ... |
tz6.12-afv 44665 | Function value. Theorem 6... |
tz6.12-1-afv 44666 | Function value (Theorem 6.... |
dmfcoafv 44667 | Domains of a function comp... |
afvco2 44668 | Value of a function compos... |
rlimdmafv 44669 | Two ways to express that a... |
aoveq123d 44670 | Equality deduction for ope... |
nfaov 44671 | Bound-variable hypothesis ... |
csbaovg 44672 | Move class substitution in... |
aovfundmoveq 44673 | If a class is a function r... |
aovnfundmuv 44674 | If an ordered pair is not ... |
ndmaov 44675 | The value of an operation ... |
ndmaovg 44676 | The value of an operation ... |
aovvdm 44677 | If the operation value of ... |
nfunsnaov 44678 | If the restriction of a cl... |
aovvfunressn 44679 | If the operation value of ... |
aovprc 44680 | The value of an operation ... |
aovrcl 44681 | Reverse closure for an ope... |
aovpcov0 44682 | If the alternative value o... |
aovnuoveq 44683 | The alternative value of t... |
aovvoveq 44684 | The alternative value of t... |
aov0ov0 44685 | If the alternative value o... |
aovovn0oveq 44686 | If the operation's value a... |
aov0nbovbi 44687 | The operation's value on a... |
aovov0bi 44688 | The operation's value on a... |
rspceaov 44689 | A frequently used special ... |
fnotaovb 44690 | Equivalence of operation v... |
ffnaov 44691 | An operation maps to a cla... |
faovcl 44692 | Closure law for an operati... |
aovmpt4g 44693 | Value of a function given ... |
aoprssdm 44694 | Domain of closure of an op... |
ndmaovcl 44695 | The "closure" of an operat... |
ndmaovrcl 44696 | Reverse closure law, in co... |
ndmaovcom 44697 | Any operation is commutati... |
ndmaovass 44698 | Any operation is associati... |
ndmaovdistr 44699 | Any operation is distribut... |
dfatafv2iota 44702 | If a function is defined a... |
ndfatafv2 44703 | The alternate function val... |
ndfatafv2undef 44704 | The alternate function val... |
dfatafv2ex 44705 | The alternate function val... |
afv2ex 44706 | The alternate function val... |
afv2eq12d 44707 | Equality deduction for fun... |
afv2eq1 44708 | Equality theorem for funct... |
afv2eq2 44709 | Equality theorem for funct... |
nfafv2 44710 | Bound-variable hypothesis ... |
csbafv212g 44711 | Move class substitution in... |
fexafv2ex 44712 | The alternate function val... |
ndfatafv2nrn 44713 | The alternate function val... |
ndmafv2nrn 44714 | The value of a class outsi... |
funressndmafv2rn 44715 | The alternate function val... |
afv2ndefb 44716 | Two ways to say that an al... |
nfunsnafv2 44717 | If the restriction of a cl... |
afv2prc 44718 | A function's value at a pr... |
dfatafv2rnb 44719 | The alternate function val... |
afv2orxorb 44720 | If a set is in the range o... |
dmafv2rnb 44721 | The alternate function val... |
fundmafv2rnb 44722 | The alternate function val... |
afv2elrn 44723 | An alternate function valu... |
afv20defat 44724 | If the alternate function ... |
fnafv2elrn 44725 | An alternate function valu... |
fafv2elrn 44726 | An alternate function valu... |
fafv2elrnb 44727 | An alternate function valu... |
frnvafv2v 44728 | If the codomain of a funct... |
tz6.12-2-afv2 44729 | Function value when ` F ` ... |
afv2eu 44730 | The value of a function at... |
afv2res 44731 | The value of a restricted ... |
tz6.12-afv2 44732 | Function value (Theorem 6.... |
tz6.12-1-afv2 44733 | Function value (Theorem 6.... |
tz6.12c-afv2 44734 | Corollary of Theorem 6.12(... |
tz6.12i-afv2 44735 | Corollary of Theorem 6.12(... |
funressnbrafv2 44736 | The second argument of a b... |
dfatbrafv2b 44737 | Equivalence of function va... |
dfatopafv2b 44738 | Equivalence of function va... |
funbrafv2 44739 | The second argument of a b... |
fnbrafv2b 44740 | Equivalence of function va... |
fnopafv2b 44741 | Equivalence of function va... |
funbrafv22b 44742 | Equivalence of function va... |
funopafv2b 44743 | Equivalence of function va... |
dfatsnafv2 44744 | Singleton of function valu... |
dfafv23 44745 | A definition of function v... |
dfatdmfcoafv2 44746 | Domain of a function compo... |
dfatcolem 44747 | Lemma for ~ dfatco . (Con... |
dfatco 44748 | The predicate "defined at"... |
afv2co2 44749 | Value of a function compos... |
rlimdmafv2 44750 | Two ways to express that a... |
dfafv22 44751 | Alternate definition of ` ... |
afv2ndeffv0 44752 | If the alternate function ... |
dfatafv2eqfv 44753 | If a function is defined a... |
afv2rnfveq 44754 | If the alternate function ... |
afv20fv0 44755 | If the alternate function ... |
afv2fvn0fveq 44756 | If the function's value at... |
afv2fv0 44757 | If the function's value at... |
afv2fv0b 44758 | The function's value at an... |
afv2fv0xorb 44759 | If a set is in the range o... |
an4com24 44760 | Rearrangement of 4 conjunc... |
3an4ancom24 44761 | Commutative law for a conj... |
4an21 44762 | Rearrangement of 4 conjunc... |
dfnelbr2 44765 | Alternate definition of th... |
nelbr 44766 | The binary relation of a s... |
nelbrim 44767 | If a set is related to ano... |
nelbrnel 44768 | A set is related to anothe... |
nelbrnelim 44769 | If a set is related to ano... |
ralralimp 44770 | Selecting one of two alter... |
otiunsndisjX 44771 | The union of singletons co... |
fvifeq 44772 | Equality of function value... |
rnfdmpr 44773 | The range of a one-to-one ... |
imarnf1pr 44774 | The image of the range of ... |
funop1 44775 | A function is an ordered p... |
fun2dmnopgexmpl 44776 | A function with a domain c... |
opabresex0d 44777 | A collection of ordered pa... |
opabbrfex0d 44778 | A collection of ordered pa... |
opabresexd 44779 | A collection of ordered pa... |
opabbrfexd 44780 | A collection of ordered pa... |
f1oresf1orab 44781 | Build a bijection by restr... |
f1oresf1o 44782 | Build a bijection by restr... |
f1oresf1o2 44783 | Build a bijection by restr... |
fvmptrab 44784 | Value of a function mappin... |
fvmptrabdm 44785 | Value of a function mappin... |
cnambpcma 44786 | ((a-b)+c)-a = c-a holds fo... |
cnapbmcpd 44787 | ((a+b)-c)+d = ((a+d)+b)-c ... |
addsubeq0 44788 | The sum of two complex num... |
leaddsuble 44789 | Addition and subtraction o... |
2leaddle2 44790 | If two real numbers are le... |
ltnltne 44791 | Variant of trichotomy law ... |
p1lep2 44792 | A real number increasd by ... |
ltsubsubaddltsub 44793 | If the result of subtracti... |
zm1nn 44794 | An integer minus 1 is posi... |
readdcnnred 44795 | The sum of a real number a... |
resubcnnred 44796 | The difference of a real n... |
recnmulnred 44797 | The product of a real numb... |
cndivrenred 44798 | The quotient of an imagina... |
sqrtnegnre 44799 | The square root of a negat... |
nn0resubcl 44800 | Closure law for subtractio... |
zgeltp1eq 44801 | If an integer is between a... |
1t10e1p1e11 44802 | 11 is 1 times 10 to the po... |
deccarry 44803 | Add 1 to a 2 digit number ... |
eluzge0nn0 44804 | If an integer is greater t... |
nltle2tri 44805 | Negated extended trichotom... |
ssfz12 44806 | Subset relationship for fi... |
elfz2z 44807 | Membership of an integer i... |
2elfz3nn0 44808 | If there are two elements ... |
fz0addcom 44809 | The addition of two member... |
2elfz2melfz 44810 | If the sum of two integers... |
fz0addge0 44811 | The sum of two integers in... |
elfzlble 44812 | Membership of an integer i... |
elfzelfzlble 44813 | Membership of an element o... |
fzopred 44814 | Join a predecessor to the ... |
fzopredsuc 44815 | Join a predecessor and a s... |
1fzopredsuc 44816 | Join 0 and a successor to ... |
el1fzopredsuc 44817 | An element of an open inte... |
subsubelfzo0 44818 | Subtracting a difference f... |
fzoopth 44819 | A half-open integer range ... |
2ffzoeq 44820 | Two functions over a half-... |
m1mod0mod1 44821 | An integer decreased by 1 ... |
elmod2 44822 | An integer modulo 2 is eit... |
smonoord 44823 | Ordering relation for a st... |
fsummsndifre 44824 | A finite sum with one of i... |
fsumsplitsndif 44825 | Separate out a term in a f... |
fsummmodsndifre 44826 | A finite sum of summands m... |
fsummmodsnunz 44827 | A finite sum of summands m... |
setsidel 44828 | The injected slot is an el... |
setsnidel 44829 | The injected slot is an el... |
setsv 44830 | The value of the structure... |
preimafvsnel 44831 | The preimage of a function... |
preimafvn0 44832 | The preimage of a function... |
uniimafveqt 44833 | The union of the image of ... |
uniimaprimaeqfv 44834 | The union of the image of ... |
setpreimafvex 44835 | The class ` P ` of all pre... |
elsetpreimafvb 44836 | The characterization of an... |
elsetpreimafv 44837 | An element of the class ` ... |
elsetpreimafvssdm 44838 | An element of the class ` ... |
fvelsetpreimafv 44839 | There is an element in a p... |
preimafvelsetpreimafv 44840 | The preimage of a function... |
preimafvsspwdm 44841 | The class ` P ` of all pre... |
0nelsetpreimafv 44842 | The empty set is not an el... |
elsetpreimafvbi 44843 | An element of the preimage... |
elsetpreimafveqfv 44844 | The elements of the preima... |
eqfvelsetpreimafv 44845 | If an element of the domai... |
elsetpreimafvrab 44846 | An element of the preimage... |
imaelsetpreimafv 44847 | The image of an element of... |
uniimaelsetpreimafv 44848 | The union of the image of ... |
elsetpreimafveq 44849 | If two preimages of functi... |
fundcmpsurinjlem1 44850 | Lemma 1 for ~ fundcmpsurin... |
fundcmpsurinjlem2 44851 | Lemma 2 for ~ fundcmpsurin... |
fundcmpsurinjlem3 44852 | Lemma 3 for ~ fundcmpsurin... |
imasetpreimafvbijlemf 44853 | Lemma for ~ imasetpreimafv... |
imasetpreimafvbijlemfv 44854 | Lemma for ~ imasetpreimafv... |
imasetpreimafvbijlemfv1 44855 | Lemma for ~ imasetpreimafv... |
imasetpreimafvbijlemf1 44856 | Lemma for ~ imasetpreimafv... |
imasetpreimafvbijlemfo 44857 | Lemma for ~ imasetpreimafv... |
imasetpreimafvbij 44858 | The mapping ` H ` is a bij... |
fundcmpsurbijinjpreimafv 44859 | Every function ` F : A -->... |
fundcmpsurinjpreimafv 44860 | Every function ` F : A -->... |
fundcmpsurinj 44861 | Every function ` F : A -->... |
fundcmpsurbijinj 44862 | Every function ` F : A -->... |
fundcmpsurinjimaid 44863 | Every function ` F : A -->... |
fundcmpsurinjALT 44864 | Alternate proof of ~ fundc... |
iccpval 44867 | Partition consisting of a ... |
iccpart 44868 | A special partition. Corr... |
iccpartimp 44869 | Implications for a class b... |
iccpartres 44870 | The restriction of a parti... |
iccpartxr 44871 | If there is a partition, t... |
iccpartgtprec 44872 | If there is a partition, t... |
iccpartipre 44873 | If there is a partition, t... |
iccpartiltu 44874 | If there is a partition, t... |
iccpartigtl 44875 | If there is a partition, t... |
iccpartlt 44876 | If there is a partition, t... |
iccpartltu 44877 | If there is a partition, t... |
iccpartgtl 44878 | If there is a partition, t... |
iccpartgt 44879 | If there is a partition, t... |
iccpartleu 44880 | If there is a partition, t... |
iccpartgel 44881 | If there is a partition, t... |
iccpartrn 44882 | If there is a partition, t... |
iccpartf 44883 | The range of the partition... |
iccpartel 44884 | If there is a partition, t... |
iccelpart 44885 | An element of any partitio... |
iccpartiun 44886 | A half-open interval of ex... |
icceuelpartlem 44887 | Lemma for ~ icceuelpart . ... |
icceuelpart 44888 | An element of a partitione... |
iccpartdisj 44889 | The segments of a partitio... |
iccpartnel 44890 | A point of a partition is ... |
fargshiftfv 44891 | If a class is a function, ... |
fargshiftf 44892 | If a class is a function, ... |
fargshiftf1 44893 | If a function is 1-1, then... |
fargshiftfo 44894 | If a function is onto, the... |
fargshiftfva 44895 | The values of a shifted fu... |
lswn0 44896 | The last symbol of a not e... |
nfich1 44899 | The first interchangeable ... |
nfich2 44900 | The second interchangeable... |
ichv 44901 | Setvar variables are inter... |
ichf 44902 | Setvar variables are inter... |
ichid 44903 | A setvar variable is alway... |
icht 44904 | A theorem is interchangeab... |
ichbidv 44905 | Formula building rule for ... |
ichcircshi 44906 | The setvar variables are i... |
ichan 44907 | If two setvar variables ar... |
ichn 44908 | Negation does not affect i... |
ichim 44909 | Formula building rule for ... |
dfich2 44910 | Alternate definition of th... |
ichcom 44911 | The interchangeability of ... |
ichbi12i 44912 | Equivalence for interchang... |
icheqid 44913 | In an equality for the sam... |
icheq 44914 | In an equality of setvar v... |
ichnfimlem 44915 | Lemma for ~ ichnfim : A s... |
ichnfim 44916 | If in an interchangeabilit... |
ichnfb 44917 | If ` x ` and ` y ` are int... |
ichal 44918 | Move a universal quantifie... |
ich2al 44919 | Two setvar variables are a... |
ich2ex 44920 | Two setvar variables are a... |
ichexmpl1 44921 | Example for interchangeabl... |
ichexmpl2 44922 | Example for interchangeabl... |
ich2exprop 44923 | If the setvar variables ar... |
ichnreuop 44924 | If the setvar variables ar... |
ichreuopeq 44925 | If the setvar variables ar... |
sprid 44926 | Two identical representati... |
elsprel 44927 | An unordered pair is an el... |
spr0nelg 44928 | The empty set is not an el... |
sprval 44931 | The set of all unordered p... |
sprvalpw 44932 | The set of all unordered p... |
sprssspr 44933 | The set of all unordered p... |
spr0el 44934 | The empty set is not an un... |
sprvalpwn0 44935 | The set of all unordered p... |
sprel 44936 | An element of the set of a... |
prssspr 44937 | An element of a subset of ... |
prelspr 44938 | An unordered pair of eleme... |
prsprel 44939 | The elements of a pair fro... |
prsssprel 44940 | The elements of a pair fro... |
sprvalpwle2 44941 | The set of all unordered p... |
sprsymrelfvlem 44942 | Lemma for ~ sprsymrelf and... |
sprsymrelf1lem 44943 | Lemma for ~ sprsymrelf1 . ... |
sprsymrelfolem1 44944 | Lemma 1 for ~ sprsymrelfo ... |
sprsymrelfolem2 44945 | Lemma 2 for ~ sprsymrelfo ... |
sprsymrelfv 44946 | The value of the function ... |
sprsymrelf 44947 | The mapping ` F ` is a fun... |
sprsymrelf1 44948 | The mapping ` F ` is a one... |
sprsymrelfo 44949 | The mapping ` F ` is a fun... |
sprsymrelf1o 44950 | The mapping ` F ` is a bij... |
sprbisymrel 44951 | There is a bijection betwe... |
sprsymrelen 44952 | The class ` P ` of subsets... |
prpair 44953 | Characterization of a prop... |
prproropf1olem0 44954 | Lemma 0 for ~ prproropf1o ... |
prproropf1olem1 44955 | Lemma 1 for ~ prproropf1o ... |
prproropf1olem2 44956 | Lemma 2 for ~ prproropf1o ... |
prproropf1olem3 44957 | Lemma 3 for ~ prproropf1o ... |
prproropf1olem4 44958 | Lemma 4 for ~ prproropf1o ... |
prproropf1o 44959 | There is a bijection betwe... |
prproropen 44960 | The set of proper pairs an... |
prproropreud 44961 | There is exactly one order... |
pairreueq 44962 | Two equivalent representat... |
paireqne 44963 | Two sets are not equal iff... |
prprval 44966 | The set of all proper unor... |
prprvalpw 44967 | The set of all proper unor... |
prprelb 44968 | An element of the set of a... |
prprelprb 44969 | A set is an element of the... |
prprspr2 44970 | The set of all proper unor... |
prprsprreu 44971 | There is a unique proper u... |
prprreueq 44972 | There is a unique proper u... |
sbcpr 44973 | The proper substitution of... |
reupr 44974 | There is a unique unordere... |
reuprpr 44975 | There is a unique proper u... |
poprelb 44976 | Equality for unordered pai... |
2exopprim 44977 | The existence of an ordere... |
reuopreuprim 44978 | There is a unique unordere... |
fmtno 44981 | The ` N ` th Fermat number... |
fmtnoge3 44982 | Each Fermat number is grea... |
fmtnonn 44983 | Each Fermat number is a po... |
fmtnom1nn 44984 | A Fermat number minus one ... |
fmtnoodd 44985 | Each Fermat number is odd.... |
fmtnorn 44986 | A Fermat number is a funct... |
fmtnof1 44987 | The enumeration of the Fer... |
fmtnoinf 44988 | The set of Fermat numbers ... |
fmtnorec1 44989 | The first recurrence relat... |
sqrtpwpw2p 44990 | The floor of the square ro... |
fmtnosqrt 44991 | The floor of the square ro... |
fmtno0 44992 | The ` 0 ` th Fermat number... |
fmtno1 44993 | The ` 1 ` st Fermat number... |
fmtnorec2lem 44994 | Lemma for ~ fmtnorec2 (ind... |
fmtnorec2 44995 | The second recurrence rela... |
fmtnodvds 44996 | Any Fermat number divides ... |
goldbachthlem1 44997 | Lemma 1 for ~ goldbachth .... |
goldbachthlem2 44998 | Lemma 2 for ~ goldbachth .... |
goldbachth 44999 | Goldbach's theorem: Two d... |
fmtnorec3 45000 | The third recurrence relat... |
fmtnorec4 45001 | The fourth recurrence rela... |
fmtno2 45002 | The ` 2 ` nd Fermat number... |
fmtno3 45003 | The ` 3 ` rd Fermat number... |
fmtno4 45004 | The ` 4 ` th Fermat number... |
fmtno5lem1 45005 | Lemma 1 for ~ fmtno5 . (C... |
fmtno5lem2 45006 | Lemma 2 for ~ fmtno5 . (C... |
fmtno5lem3 45007 | Lemma 3 for ~ fmtno5 . (C... |
fmtno5lem4 45008 | Lemma 4 for ~ fmtno5 . (C... |
fmtno5 45009 | The ` 5 ` th Fermat number... |
fmtno0prm 45010 | The ` 0 ` th Fermat number... |
fmtno1prm 45011 | The ` 1 ` st Fermat number... |
fmtno2prm 45012 | The ` 2 ` nd Fermat number... |
257prm 45013 | 257 is a prime number (the... |
fmtno3prm 45014 | The ` 3 ` rd Fermat number... |
odz2prm2pw 45015 | Any power of two is coprim... |
fmtnoprmfac1lem 45016 | Lemma for ~ fmtnoprmfac1 :... |
fmtnoprmfac1 45017 | Divisor of Fermat number (... |
fmtnoprmfac2lem1 45018 | Lemma for ~ fmtnoprmfac2 .... |
fmtnoprmfac2 45019 | Divisor of Fermat number (... |
fmtnofac2lem 45020 | Lemma for ~ fmtnofac2 (Ind... |
fmtnofac2 45021 | Divisor of Fermat number (... |
fmtnofac1 45022 | Divisor of Fermat number (... |
fmtno4sqrt 45023 | The floor of the square ro... |
fmtno4prmfac 45024 | If P was a (prime) factor ... |
fmtno4prmfac193 45025 | If P was a (prime) factor ... |
fmtno4nprmfac193 45026 | 193 is not a (prime) facto... |
fmtno4prm 45027 | The ` 4 `-th Fermat number... |
65537prm 45028 | 65537 is a prime number (t... |
fmtnofz04prm 45029 | The first five Fermat numb... |
fmtnole4prm 45030 | The first five Fermat numb... |
fmtno5faclem1 45031 | Lemma 1 for ~ fmtno5fac . ... |
fmtno5faclem2 45032 | Lemma 2 for ~ fmtno5fac . ... |
fmtno5faclem3 45033 | Lemma 3 for ~ fmtno5fac . ... |
fmtno5fac 45034 | The factorisation of the `... |
fmtno5nprm 45035 | The ` 5 ` th Fermat number... |
prmdvdsfmtnof1lem1 45036 | Lemma 1 for ~ prmdvdsfmtno... |
prmdvdsfmtnof1lem2 45037 | Lemma 2 for ~ prmdvdsfmtno... |
prmdvdsfmtnof 45038 | The mapping of a Fermat nu... |
prmdvdsfmtnof1 45039 | The mapping of a Fermat nu... |
prminf2 45040 | The set of prime numbers i... |
2pwp1prm 45041 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
2pwp1prmfmtno 45042 | Every prime number of the ... |
m2prm 45043 | The second Mersenne number... |
m3prm 45044 | The third Mersenne number ... |
flsqrt 45045 | A condition equivalent to ... |
flsqrt5 45046 | The floor of the square ro... |
3ndvds4 45047 | 3 does not divide 4. (Con... |
139prmALT 45048 | 139 is a prime number. In... |
31prm 45049 | 31 is a prime number. In ... |
m5prm 45050 | The fifth Mersenne number ... |
127prm 45051 | 127 is a prime number. (C... |
m7prm 45052 | The seventh Mersenne numbe... |
m11nprm 45053 | The eleventh Mersenne numb... |
mod42tp1mod8 45054 | If a number is ` 3 ` modul... |
sfprmdvdsmersenne 45055 | If ` Q ` is a safe prime (... |
sgprmdvdsmersenne 45056 | If ` P ` is a Sophie Germa... |
lighneallem1 45057 | Lemma 1 for ~ lighneal . ... |
lighneallem2 45058 | Lemma 2 for ~ lighneal . ... |
lighneallem3 45059 | Lemma 3 for ~ lighneal . ... |
lighneallem4a 45060 | Lemma 1 for ~ lighneallem4... |
lighneallem4b 45061 | Lemma 2 for ~ lighneallem4... |
lighneallem4 45062 | Lemma 3 for ~ lighneal . ... |
lighneal 45063 | If a power of a prime ` P ... |
modexp2m1d 45064 | The square of an integer w... |
proththdlem 45065 | Lemma for ~ proththd . (C... |
proththd 45066 | Proth's theorem (1878). I... |
5tcu2e40 45067 | 5 times the cube of 2 is 4... |
3exp4mod41 45068 | 3 to the fourth power is -... |
41prothprmlem1 45069 | Lemma 1 for ~ 41prothprm .... |
41prothprmlem2 45070 | Lemma 2 for ~ 41prothprm .... |
41prothprm 45071 | 41 is a _Proth prime_. (C... |
quad1 45072 | A condition for a quadrati... |
requad01 45073 | A condition for a quadrati... |
requad1 45074 | A condition for a quadrati... |
requad2 45075 | A condition for a quadrati... |
iseven 45080 | The predicate "is an even ... |
isodd 45081 | The predicate "is an odd n... |
evenz 45082 | An even number is an integ... |
oddz 45083 | An odd number is an intege... |
evendiv2z 45084 | The result of dividing an ... |
oddp1div2z 45085 | The result of dividing an ... |
oddm1div2z 45086 | The result of dividing an ... |
isodd2 45087 | The predicate "is an odd n... |
dfodd2 45088 | Alternate definition for o... |
dfodd6 45089 | Alternate definition for o... |
dfeven4 45090 | Alternate definition for e... |
evenm1odd 45091 | The predecessor of an even... |
evenp1odd 45092 | The successor of an even n... |
oddp1eveni 45093 | The successor of an odd nu... |
oddm1eveni 45094 | The predecessor of an odd ... |
evennodd 45095 | An even number is not an o... |
oddneven 45096 | An odd number is not an ev... |
enege 45097 | The negative of an even nu... |
onego 45098 | The negative of an odd num... |
m1expevenALTV 45099 | Exponentiation of -1 by an... |
m1expoddALTV 45100 | Exponentiation of -1 by an... |
dfeven2 45101 | Alternate definition for e... |
dfodd3 45102 | Alternate definition for o... |
iseven2 45103 | The predicate "is an even ... |
isodd3 45104 | The predicate "is an odd n... |
2dvdseven 45105 | 2 divides an even number. ... |
m2even 45106 | A multiple of 2 is an even... |
2ndvdsodd 45107 | 2 does not divide an odd n... |
2dvdsoddp1 45108 | 2 divides an odd number in... |
2dvdsoddm1 45109 | 2 divides an odd number de... |
dfeven3 45110 | Alternate definition for e... |
dfodd4 45111 | Alternate definition for o... |
dfodd5 45112 | Alternate definition for o... |
zefldiv2ALTV 45113 | The floor of an even numbe... |
zofldiv2ALTV 45114 | The floor of an odd numer ... |
oddflALTV 45115 | Odd number representation ... |
iseven5 45116 | The predicate "is an even ... |
isodd7 45117 | The predicate "is an odd n... |
dfeven5 45118 | Alternate definition for e... |
dfodd7 45119 | Alternate definition for o... |
gcd2odd1 45120 | The greatest common diviso... |
zneoALTV 45121 | No even integer equals an ... |
zeoALTV 45122 | An integer is even or odd.... |
zeo2ALTV 45123 | An integer is even or odd ... |
nneoALTV 45124 | A positive integer is even... |
nneoiALTV 45125 | A positive integer is even... |
odd2np1ALTV 45126 | An integer is odd iff it i... |
oddm1evenALTV 45127 | An integer is odd iff its ... |
oddp1evenALTV 45128 | An integer is odd iff its ... |
oexpnegALTV 45129 | The exponential of the neg... |
oexpnegnz 45130 | The exponential of the neg... |
bits0ALTV 45131 | Value of the zeroth bit. ... |
bits0eALTV 45132 | The zeroth bit of an even ... |
bits0oALTV 45133 | The zeroth bit of an odd n... |
divgcdoddALTV 45134 | Either ` A / ( A gcd B ) `... |
opoeALTV 45135 | The sum of two odds is eve... |
opeoALTV 45136 | The sum of an odd and an e... |
omoeALTV 45137 | The difference of two odds... |
omeoALTV 45138 | The difference of an odd a... |
oddprmALTV 45139 | A prime not equal to ` 2 `... |
0evenALTV 45140 | 0 is an even number. (Con... |
0noddALTV 45141 | 0 is not an odd number. (... |
1oddALTV 45142 | 1 is an odd number. (Cont... |
1nevenALTV 45143 | 1 is not an even number. ... |
2evenALTV 45144 | 2 is an even number. (Con... |
2noddALTV 45145 | 2 is not an odd number. (... |
nn0o1gt2ALTV 45146 | An odd nonnegative integer... |
nnoALTV 45147 | An alternate characterizat... |
nn0oALTV 45148 | An alternate characterizat... |
nn0e 45149 | An alternate characterizat... |
nneven 45150 | An alternate characterizat... |
nn0onn0exALTV 45151 | For each odd nonnegative i... |
nn0enn0exALTV 45152 | For each even nonnegative ... |
nnennexALTV 45153 | For each even positive int... |
nnpw2evenALTV 45154 | 2 to the power of a positi... |
epoo 45155 | The sum of an even and an ... |
emoo 45156 | The difference of an even ... |
epee 45157 | The sum of two even number... |
emee 45158 | The difference of two even... |
evensumeven 45159 | If a summand is even, the ... |
3odd 45160 | 3 is an odd number. (Cont... |
4even 45161 | 4 is an even number. (Con... |
5odd 45162 | 5 is an odd number. (Cont... |
6even 45163 | 6 is an even number. (Con... |
7odd 45164 | 7 is an odd number. (Cont... |
8even 45165 | 8 is an even number. (Con... |
evenprm2 45166 | A prime number is even iff... |
oddprmne2 45167 | Every prime number not bei... |
oddprmuzge3 45168 | A prime number which is od... |
evenltle 45169 | If an even number is great... |
odd2prm2 45170 | If an odd number is the su... |
even3prm2 45171 | If an even number is the s... |
mogoldbblem 45172 | Lemma for ~ mogoldbb . (C... |
perfectALTVlem1 45173 | Lemma for ~ perfectALTV . ... |
perfectALTVlem2 45174 | Lemma for ~ perfectALTV . ... |
perfectALTV 45175 | The Euclid-Euler theorem, ... |
fppr 45178 | The set of Fermat pseudopr... |
fpprmod 45179 | The set of Fermat pseudopr... |
fpprel 45180 | A Fermat pseudoprime to th... |
fpprbasnn 45181 | The base of a Fermat pseud... |
fpprnn 45182 | A Fermat pseudoprime to th... |
fppr2odd 45183 | A Fermat pseudoprime to th... |
11t31e341 45184 | 341 is the product of 11 a... |
2exp340mod341 45185 | Eight to the eighth power ... |
341fppr2 45186 | 341 is the (smallest) _Pou... |
4fppr1 45187 | 4 is the (smallest) Fermat... |
8exp8mod9 45188 | Eight to the eighth power ... |
9fppr8 45189 | 9 is the (smallest) Fermat... |
dfwppr 45190 | Alternate definition of a ... |
fpprwppr 45191 | A Fermat pseudoprime to th... |
fpprwpprb 45192 | An integer ` X ` which is ... |
fpprel2 45193 | An alternate definition fo... |
nfermltl8rev 45194 | Fermat's little theorem wi... |
nfermltl2rev 45195 | Fermat's little theorem wi... |
nfermltlrev 45196 | Fermat's little theorem re... |
isgbe 45203 | The predicate "is an even ... |
isgbow 45204 | The predicate "is a weak o... |
isgbo 45205 | The predicate "is an odd G... |
gbeeven 45206 | An even Goldbach number is... |
gbowodd 45207 | A weak odd Goldbach number... |
gbogbow 45208 | A (strong) odd Goldbach nu... |
gboodd 45209 | An odd Goldbach number is ... |
gbepos 45210 | Any even Goldbach number i... |
gbowpos 45211 | Any weak odd Goldbach numb... |
gbopos 45212 | Any odd Goldbach number is... |
gbegt5 45213 | Any even Goldbach number i... |
gbowgt5 45214 | Any weak odd Goldbach numb... |
gbowge7 45215 | Any weak odd Goldbach numb... |
gboge9 45216 | Any odd Goldbach number is... |
gbege6 45217 | Any even Goldbach number i... |
gbpart6 45218 | The Goldbach partition of ... |
gbpart7 45219 | The (weak) Goldbach partit... |
gbpart8 45220 | The Goldbach partition of ... |
gbpart9 45221 | The (strong) Goldbach part... |
gbpart11 45222 | The (strong) Goldbach part... |
6gbe 45223 | 6 is an even Goldbach numb... |
7gbow 45224 | 7 is a weak odd Goldbach n... |
8gbe 45225 | 8 is an even Goldbach numb... |
9gbo 45226 | 9 is an odd Goldbach numbe... |
11gbo 45227 | 11 is an odd Goldbach numb... |
stgoldbwt 45228 | If the strong ternary Gold... |
sbgoldbwt 45229 | If the strong binary Goldb... |
sbgoldbst 45230 | If the strong binary Goldb... |
sbgoldbaltlem1 45231 | Lemma 1 for ~ sbgoldbalt :... |
sbgoldbaltlem2 45232 | Lemma 2 for ~ sbgoldbalt :... |
sbgoldbalt 45233 | An alternate (related to t... |
sbgoldbb 45234 | If the strong binary Goldb... |
sgoldbeven3prm 45235 | If the binary Goldbach con... |
sbgoldbm 45236 | If the strong binary Goldb... |
mogoldbb 45237 | If the modern version of t... |
sbgoldbmb 45238 | The strong binary Goldbach... |
sbgoldbo 45239 | If the strong binary Goldb... |
nnsum3primes4 45240 | 4 is the sum of at most 3 ... |
nnsum4primes4 45241 | 4 is the sum of at most 4 ... |
nnsum3primesprm 45242 | Every prime is "the sum of... |
nnsum4primesprm 45243 | Every prime is "the sum of... |
nnsum3primesgbe 45244 | Any even Goldbach number i... |
nnsum4primesgbe 45245 | Any even Goldbach number i... |
nnsum3primesle9 45246 | Every integer greater than... |
nnsum4primesle9 45247 | Every integer greater than... |
nnsum4primesodd 45248 | If the (weak) ternary Gold... |
nnsum4primesoddALTV 45249 | If the (strong) ternary Go... |
evengpop3 45250 | If the (weak) ternary Gold... |
evengpoap3 45251 | If the (strong) ternary Go... |
nnsum4primeseven 45252 | If the (weak) ternary Gold... |
nnsum4primesevenALTV 45253 | If the (strong) ternary Go... |
wtgoldbnnsum4prm 45254 | If the (weak) ternary Gold... |
stgoldbnnsum4prm 45255 | If the (strong) ternary Go... |
bgoldbnnsum3prm 45256 | If the binary Goldbach con... |
bgoldbtbndlem1 45257 | Lemma 1 for ~ bgoldbtbnd :... |
bgoldbtbndlem2 45258 | Lemma 2 for ~ bgoldbtbnd .... |
bgoldbtbndlem3 45259 | Lemma 3 for ~ bgoldbtbnd .... |
bgoldbtbndlem4 45260 | Lemma 4 for ~ bgoldbtbnd .... |
bgoldbtbnd 45261 | If the binary Goldbach con... |
tgoldbachgtALTV 45264 | Variant of Thierry Arnoux'... |
bgoldbachlt 45265 | The binary Goldbach conjec... |
tgblthelfgott 45267 | The ternary Goldbach conje... |
tgoldbachlt 45268 | The ternary Goldbach conje... |
tgoldbach 45269 | The ternary Goldbach conje... |
isomgrrel 45274 | The isomorphy relation for... |
isomgr 45275 | The isomorphy relation for... |
isisomgr 45276 | Implications of two graphs... |
isomgreqve 45277 | A set is isomorphic to a h... |
isomushgr 45278 | The isomorphy relation for... |
isomuspgrlem1 45279 | Lemma 1 for ~ isomuspgr . ... |
isomuspgrlem2a 45280 | Lemma 1 for ~ isomuspgrlem... |
isomuspgrlem2b 45281 | Lemma 2 for ~ isomuspgrlem... |
isomuspgrlem2c 45282 | Lemma 3 for ~ isomuspgrlem... |
isomuspgrlem2d 45283 | Lemma 4 for ~ isomuspgrlem... |
isomuspgrlem2e 45284 | Lemma 5 for ~ isomuspgrlem... |
isomuspgrlem2 45285 | Lemma 2 for ~ isomuspgr . ... |
isomuspgr 45286 | The isomorphy relation for... |
isomgrref 45287 | The isomorphy relation is ... |
isomgrsym 45288 | The isomorphy relation is ... |
isomgrsymb 45289 | The isomorphy relation is ... |
isomgrtrlem 45290 | Lemma for ~ isomgrtr . (C... |
isomgrtr 45291 | The isomorphy relation is ... |
strisomgrop 45292 | A graph represented as an ... |
ushrisomgr 45293 | A simple hypergraph (with ... |
1hegrlfgr 45294 | A graph ` G ` with one hyp... |
upwlksfval 45297 | The set of simple walks (i... |
isupwlk 45298 | Properties of a pair of fu... |
isupwlkg 45299 | Generalization of ~ isupwl... |
upwlkbprop 45300 | Basic properties of a simp... |
upwlkwlk 45301 | A simple walk is a walk. ... |
upgrwlkupwlk 45302 | In a pseudograph, a walk i... |
upgrwlkupwlkb 45303 | In a pseudograph, the defi... |
upgrisupwlkALT 45304 | Alternate proof of ~ upgri... |
upgredgssspr 45305 | The set of edges of a pseu... |
uspgropssxp 45306 | The set ` G ` of "simple p... |
uspgrsprfv 45307 | The value of the function ... |
uspgrsprf 45308 | The mapping ` F ` is a fun... |
uspgrsprf1 45309 | The mapping ` F ` is a one... |
uspgrsprfo 45310 | The mapping ` F ` is a fun... |
uspgrsprf1o 45311 | The mapping ` F ` is a bij... |
uspgrex 45312 | The class ` G ` of all "si... |
uspgrbispr 45313 | There is a bijection betwe... |
uspgrspren 45314 | The set ` G ` of the "simp... |
uspgrymrelen 45315 | The set ` G ` of the "simp... |
uspgrbisymrel 45316 | There is a bijection betwe... |
uspgrbisymrelALT 45317 | Alternate proof of ~ uspgr... |
ovn0dmfun 45318 | If a class operation value... |
xpsnopab 45319 | A Cartesian product with a... |
xpiun 45320 | A Cartesian product expres... |
ovn0ssdmfun 45321 | If a class' operation valu... |
fnxpdmdm 45322 | The domain of the domain o... |
cnfldsrngbas 45323 | The base set of a subring ... |
cnfldsrngadd 45324 | The group addition operati... |
cnfldsrngmul 45325 | The ring multiplication op... |
plusfreseq 45326 | If the empty set is not co... |
mgmplusfreseq 45327 | If the empty set is not co... |
0mgm 45328 | A set with an empty base s... |
mgmpropd 45329 | If two structures have the... |
ismgmd 45330 | Deduce a magma from its pr... |
mgmhmrcl 45335 | Reverse closure of a magma... |
submgmrcl 45336 | Reverse closure for submag... |
ismgmhm 45337 | Property of a magma homomo... |
mgmhmf 45338 | A magma homomorphism is a ... |
mgmhmpropd 45339 | Magma homomorphism depends... |
mgmhmlin 45340 | A magma homomorphism prese... |
mgmhmf1o 45341 | A magma homomorphism is bi... |
idmgmhm 45342 | The identity homomorphism ... |
issubmgm 45343 | Expand definition of a sub... |
issubmgm2 45344 | Submagmas are subsets that... |
rabsubmgmd 45345 | Deduction for proving that... |
submgmss 45346 | Submagmas are subsets of t... |
submgmid 45347 | Every magma is trivially a... |
submgmcl 45348 | Submagmas are closed under... |
submgmmgm 45349 | Submagmas are themselves m... |
submgmbas 45350 | The base set of a submagma... |
subsubmgm 45351 | A submagma of a submagma i... |
resmgmhm 45352 | Restriction of a magma hom... |
resmgmhm2 45353 | One direction of ~ resmgmh... |
resmgmhm2b 45354 | Restriction of the codomai... |
mgmhmco 45355 | The composition of magma h... |
mgmhmima 45356 | The homomorphic image of a... |
mgmhmeql 45357 | The equalizer of two magma... |
submgmacs 45358 | Submagmas are an algebraic... |
ismhm0 45359 | Property of a monoid homom... |
mhmismgmhm 45360 | Each monoid homomorphism i... |
opmpoismgm 45361 | A structure with a group a... |
copissgrp 45362 | A structure with a constan... |
copisnmnd 45363 | A structure with a constan... |
0nodd 45364 | 0 is not an odd integer. ... |
1odd 45365 | 1 is an odd integer. (Con... |
2nodd 45366 | 2 is not an odd integer. ... |
oddibas 45367 | Lemma 1 for ~ oddinmgm : ... |
oddiadd 45368 | Lemma 2 for ~ oddinmgm : ... |
oddinmgm 45369 | The structure of all odd i... |
nnsgrpmgm 45370 | The structure of positive ... |
nnsgrp 45371 | The structure of positive ... |
nnsgrpnmnd 45372 | The structure of positive ... |
nn0mnd 45373 | The set of nonnegative int... |
gsumsplit2f 45374 | Split a group sum into two... |
gsumdifsndf 45375 | Extract a summand from a f... |
gsumfsupp 45376 | A group sum of a family ca... |
iscllaw 45383 | The predicate "is a closed... |
iscomlaw 45384 | The predicate "is a commut... |
clcllaw 45385 | Closure of a closed operat... |
isasslaw 45386 | The predicate "is an assoc... |
asslawass 45387 | Associativity of an associ... |
mgmplusgiopALT 45388 | Slot 2 (group operation) o... |
sgrpplusgaopALT 45389 | Slot 2 (group operation) o... |
intopval 45396 | The internal (binary) oper... |
intop 45397 | An internal (binary) opera... |
clintopval 45398 | The closed (internal binar... |
assintopval 45399 | The associative (closed in... |
assintopmap 45400 | The associative (closed in... |
isclintop 45401 | The predicate "is a closed... |
clintop 45402 | A closed (internal binary)... |
assintop 45403 | An associative (closed int... |
isassintop 45404 | The predicate "is an assoc... |
clintopcllaw 45405 | The closure law holds for ... |
assintopcllaw 45406 | The closure low holds for ... |
assintopasslaw 45407 | The associative low holds ... |
assintopass 45408 | An associative (closed int... |
ismgmALT 45417 | The predicate "is a magma"... |
iscmgmALT 45418 | The predicate "is a commut... |
issgrpALT 45419 | The predicate "is a semigr... |
iscsgrpALT 45420 | The predicate "is a commut... |
mgm2mgm 45421 | Equivalence of the two def... |
sgrp2sgrp 45422 | Equivalence of the two def... |
idfusubc0 45423 | The identity functor for a... |
idfusubc 45424 | The identity functor for a... |
inclfusubc 45425 | The "inclusion functor" fr... |
lmod0rng 45426 | If the scalar ring of a mo... |
nzrneg1ne0 45427 | The additive inverse of th... |
0ringdif 45428 | A zero ring is a ring whic... |
0ringbas 45429 | The base set of a zero rin... |
0ring1eq0 45430 | In a zero ring, a ring whi... |
nrhmzr 45431 | There is no ring homomorph... |
isrng 45434 | The predicate "is a non-un... |
rngabl 45435 | A non-unital ring is an (a... |
rngmgp 45436 | A non-unital ring is a sem... |
ringrng 45437 | A unital ring is a (non-un... |
ringssrng 45438 | The unital rings are (non-... |
isringrng 45439 | The predicate "is a unital... |
rngdir 45440 | Distributive law for the m... |
rngcl 45441 | Closure of the multiplicat... |
rnglz 45442 | The zero of a nonunital ri... |
rnghmrcl 45447 | Reverse closure of a non-u... |
rnghmfn 45448 | The mapping of two non-uni... |
rnghmval 45449 | The set of the non-unital ... |
isrnghm 45450 | A function is a non-unital... |
isrnghmmul 45451 | A function is a non-unital... |
rnghmmgmhm 45452 | A non-unital ring homomorp... |
rnghmval2 45453 | The non-unital ring homomo... |
isrngisom 45454 | An isomorphism of non-unit... |
rngimrcl 45455 | Reverse closure for an iso... |
rnghmghm 45456 | A non-unital ring homomorp... |
rnghmf 45457 | A ring homomorphism is a f... |
rnghmmul 45458 | A homomorphism of non-unit... |
isrnghm2d 45459 | Demonstration of non-unita... |
isrnghmd 45460 | Demonstration of non-unita... |
rnghmf1o 45461 | A non-unital ring homomorp... |
isrngim 45462 | An isomorphism of non-unit... |
rngimf1o 45463 | An isomorphism of non-unit... |
rngimrnghm 45464 | An isomorphism of non-unit... |
rnghmco 45465 | The composition of non-uni... |
idrnghm 45466 | The identity homomorphism ... |
c0mgm 45467 | The constant mapping to ze... |
c0mhm 45468 | The constant mapping to ze... |
c0ghm 45469 | The constant mapping to ze... |
c0rhm 45470 | The constant mapping to ze... |
c0rnghm 45471 | The constant mapping to ze... |
c0snmgmhm 45472 | The constant mapping to ze... |
c0snmhm 45473 | The constant mapping to ze... |
c0snghm 45474 | The constant mapping to ze... |
zrrnghm 45475 | The constant mapping to ze... |
rhmfn 45476 | The mapping of two rings t... |
rhmval 45477 | The ring homomorphisms bet... |
rhmisrnghm 45478 | Each unital ring homomorph... |
lidldomn1 45479 | If a (left) ideal (which i... |
lidlssbas 45480 | The base set of the restri... |
lidlbas 45481 | A (left) ideal of a ring i... |
lidlabl 45482 | A (left) ideal of a ring i... |
lidlmmgm 45483 | The multiplicative group o... |
lidlmsgrp 45484 | The multiplicative group o... |
lidlrng 45485 | A (left) ideal of a ring i... |
zlidlring 45486 | The zero (left) ideal of a... |
uzlidlring 45487 | Only the zero (left) ideal... |
lidldomnnring 45488 | A (left) ideal of a domain... |
0even 45489 | 0 is an even integer. (Co... |
1neven 45490 | 1 is not an even integer. ... |
2even 45491 | 2 is an even integer. (Co... |
2zlidl 45492 | The even integers are a (l... |
2zrng 45493 | The ring of integers restr... |
2zrngbas 45494 | The base set of R is the s... |
2zrngadd 45495 | The group addition operati... |
2zrng0 45496 | The additive identity of R... |
2zrngamgm 45497 | R is an (additive) magma. ... |
2zrngasgrp 45498 | R is an (additive) semigro... |
2zrngamnd 45499 | R is an (additive) monoid.... |
2zrngacmnd 45500 | R is a commutative (additi... |
2zrngagrp 45501 | R is an (additive) group. ... |
2zrngaabl 45502 | R is an (additive) abelian... |
2zrngmul 45503 | The ring multiplication op... |
2zrngmmgm 45504 | R is a (multiplicative) ma... |
2zrngmsgrp 45505 | R is a (multiplicative) se... |
2zrngALT 45506 | The ring of integers restr... |
2zrngnmlid 45507 | R has no multiplicative (l... |
2zrngnmrid 45508 | R has no multiplicative (r... |
2zrngnmlid2 45509 | R has no multiplicative (l... |
2zrngnring 45510 | R is not a unital ring. (... |
cznrnglem 45511 | Lemma for ~ cznrng : The ... |
cznabel 45512 | The ring constructed from ... |
cznrng 45513 | The ring constructed from ... |
cznnring 45514 | The ring constructed from ... |
rngcvalALTV 45519 | Value of the category of n... |
rngcval 45520 | Value of the category of n... |
rnghmresfn 45521 | The class of non-unital ri... |
rnghmresel 45522 | An element of the non-unit... |
rngcbas 45523 | Set of objects of the cate... |
rngchomfval 45524 | Set of arrows of the categ... |
rngchom 45525 | Set of arrows of the categ... |
elrngchom 45526 | A morphism of non-unital r... |
rngchomfeqhom 45527 | The functionalized Hom-set... |
rngccofval 45528 | Composition in the categor... |
rngcco 45529 | Composition in the categor... |
dfrngc2 45530 | Alternate definition of th... |
rnghmsscmap2 45531 | The non-unital ring homomo... |
rnghmsscmap 45532 | The non-unital ring homomo... |
rnghmsubcsetclem1 45533 | Lemma 1 for ~ rnghmsubcset... |
rnghmsubcsetclem2 45534 | Lemma 2 for ~ rnghmsubcset... |
rnghmsubcsetc 45535 | The non-unital ring homomo... |
rngccat 45536 | The category of non-unital... |
rngcid 45537 | The identity arrow in the ... |
rngcsect 45538 | A section in the category ... |
rngcinv 45539 | An inverse in the category... |
rngciso 45540 | An isomorphism in the cate... |
rngcbasALTV 45541 | Set of objects of the cate... |
rngchomfvalALTV 45542 | Set of arrows of the categ... |
rngchomALTV 45543 | Set of arrows of the categ... |
elrngchomALTV 45544 | A morphism of non-unital r... |
rngccofvalALTV 45545 | Composition in the categor... |
rngccoALTV 45546 | Composition in the categor... |
rngccatidALTV 45547 | Lemma for ~ rngccatALTV . ... |
rngccatALTV 45548 | The category of non-unital... |
rngcidALTV 45549 | The identity arrow in the ... |
rngcsectALTV 45550 | A section in the category ... |
rngcinvALTV 45551 | An inverse in the category... |
rngcisoALTV 45552 | An isomorphism in the cate... |
rngchomffvalALTV 45553 | The value of the functiona... |
rngchomrnghmresALTV 45554 | The value of the functiona... |
rngcifuestrc 45555 | The "inclusion functor" fr... |
funcrngcsetc 45556 | The "natural forgetful fun... |
funcrngcsetcALT 45557 | Alternate proof of ~ funcr... |
zrinitorngc 45558 | The zero ring is an initia... |
zrtermorngc 45559 | The zero ring is a termina... |
zrzeroorngc 45560 | The zero ring is a zero ob... |
ringcvalALTV 45565 | Value of the category of r... |
ringcval 45566 | Value of the category of u... |
rhmresfn 45567 | The class of unital ring h... |
rhmresel 45568 | An element of the unital r... |
ringcbas 45569 | Set of objects of the cate... |
ringchomfval 45570 | Set of arrows of the categ... |
ringchom 45571 | Set of arrows of the categ... |
elringchom 45572 | A morphism of unital rings... |
ringchomfeqhom 45573 | The functionalized Hom-set... |
ringccofval 45574 | Composition in the categor... |
ringcco 45575 | Composition in the categor... |
dfringc2 45576 | Alternate definition of th... |
rhmsscmap2 45577 | The unital ring homomorphi... |
rhmsscmap 45578 | The unital ring homomorphi... |
rhmsubcsetclem1 45579 | Lemma 1 for ~ rhmsubcsetc ... |
rhmsubcsetclem2 45580 | Lemma 2 for ~ rhmsubcsetc ... |
rhmsubcsetc 45581 | The unital ring homomorphi... |
ringccat 45582 | The category of unital rin... |
ringcid 45583 | The identity arrow in the ... |
rhmsscrnghm 45584 | The unital ring homomorphi... |
rhmsubcrngclem1 45585 | Lemma 1 for ~ rhmsubcrngc ... |
rhmsubcrngclem2 45586 | Lemma 2 for ~ rhmsubcrngc ... |
rhmsubcrngc 45587 | The unital ring homomorphi... |
rngcresringcat 45588 | The restriction of the cat... |
ringcsect 45589 | A section in the category ... |
ringcinv 45590 | An inverse in the category... |
ringciso 45591 | An isomorphism in the cate... |
ringcbasbas 45592 | An element of the base set... |
funcringcsetc 45593 | The "natural forgetful fun... |
funcringcsetcALTV2lem1 45594 | Lemma 1 for ~ funcringcset... |
funcringcsetcALTV2lem2 45595 | Lemma 2 for ~ funcringcset... |
funcringcsetcALTV2lem3 45596 | Lemma 3 for ~ funcringcset... |
funcringcsetcALTV2lem4 45597 | Lemma 4 for ~ funcringcset... |
funcringcsetcALTV2lem5 45598 | Lemma 5 for ~ funcringcset... |
funcringcsetcALTV2lem6 45599 | Lemma 6 for ~ funcringcset... |
funcringcsetcALTV2lem7 45600 | Lemma 7 for ~ funcringcset... |
funcringcsetcALTV2lem8 45601 | Lemma 8 for ~ funcringcset... |
funcringcsetcALTV2lem9 45602 | Lemma 9 for ~ funcringcset... |
funcringcsetcALTV2 45603 | The "natural forgetful fun... |
ringcbasALTV 45604 | Set of objects of the cate... |
ringchomfvalALTV 45605 | Set of arrows of the categ... |
ringchomALTV 45606 | Set of arrows of the categ... |
elringchomALTV 45607 | A morphism of rings is a f... |
ringccofvalALTV 45608 | Composition in the categor... |
ringccoALTV 45609 | Composition in the categor... |
ringccatidALTV 45610 | Lemma for ~ ringccatALTV .... |
ringccatALTV 45611 | The category of rings is a... |
ringcidALTV 45612 | The identity arrow in the ... |
ringcsectALTV 45613 | A section in the category ... |
ringcinvALTV 45614 | An inverse in the category... |
ringcisoALTV 45615 | An isomorphism in the cate... |
ringcbasbasALTV 45616 | An element of the base set... |
funcringcsetclem1ALTV 45617 | Lemma 1 for ~ funcringcset... |
funcringcsetclem2ALTV 45618 | Lemma 2 for ~ funcringcset... |
funcringcsetclem3ALTV 45619 | Lemma 3 for ~ funcringcset... |
funcringcsetclem4ALTV 45620 | Lemma 4 for ~ funcringcset... |
funcringcsetclem5ALTV 45621 | Lemma 5 for ~ funcringcset... |
funcringcsetclem6ALTV 45622 | Lemma 6 for ~ funcringcset... |
funcringcsetclem7ALTV 45623 | Lemma 7 for ~ funcringcset... |
funcringcsetclem8ALTV 45624 | Lemma 8 for ~ funcringcset... |
funcringcsetclem9ALTV 45625 | Lemma 9 for ~ funcringcset... |
funcringcsetcALTV 45626 | The "natural forgetful fun... |
irinitoringc 45627 | The ring of integers is an... |
zrtermoringc 45628 | The zero ring is a termina... |
zrninitoringc 45629 | The zero ring is not an in... |
nzerooringczr 45630 | There is no zero object in... |
srhmsubclem1 45631 | Lemma 1 for ~ srhmsubc . ... |
srhmsubclem2 45632 | Lemma 2 for ~ srhmsubc . ... |
srhmsubclem3 45633 | Lemma 3 for ~ srhmsubc . ... |
srhmsubc 45634 | According to ~ df-subc , t... |
sringcat 45635 | The restriction of the cat... |
crhmsubc 45636 | According to ~ df-subc , t... |
cringcat 45637 | The restriction of the cat... |
drhmsubc 45638 | According to ~ df-subc , t... |
drngcat 45639 | The restriction of the cat... |
fldcat 45640 | The restriction of the cat... |
fldc 45641 | The restriction of the cat... |
fldhmsubc 45642 | According to ~ df-subc , t... |
rngcrescrhm 45643 | The category of non-unital... |
rhmsubclem1 45644 | Lemma 1 for ~ rhmsubc . (... |
rhmsubclem2 45645 | Lemma 2 for ~ rhmsubc . (... |
rhmsubclem3 45646 | Lemma 3 for ~ rhmsubc . (... |
rhmsubclem4 45647 | Lemma 4 for ~ rhmsubc . (... |
rhmsubc 45648 | According to ~ df-subc , t... |
rhmsubccat 45649 | The restriction of the cat... |
srhmsubcALTVlem1 45650 | Lemma 1 for ~ srhmsubcALTV... |
srhmsubcALTVlem2 45651 | Lemma 2 for ~ srhmsubcALTV... |
srhmsubcALTV 45652 | According to ~ df-subc , t... |
sringcatALTV 45653 | The restriction of the cat... |
crhmsubcALTV 45654 | According to ~ df-subc , t... |
cringcatALTV 45655 | The restriction of the cat... |
drhmsubcALTV 45656 | According to ~ df-subc , t... |
drngcatALTV 45657 | The restriction of the cat... |
fldcatALTV 45658 | The restriction of the cat... |
fldcALTV 45659 | The restriction of the cat... |
fldhmsubcALTV 45660 | According to ~ df-subc , t... |
rngcrescrhmALTV 45661 | The category of non-unital... |
rhmsubcALTVlem1 45662 | Lemma 1 for ~ rhmsubcALTV ... |
rhmsubcALTVlem2 45663 | Lemma 2 for ~ rhmsubcALTV ... |
rhmsubcALTVlem3 45664 | Lemma 3 for ~ rhmsubcALTV ... |
rhmsubcALTVlem4 45665 | Lemma 4 for ~ rhmsubcALTV ... |
rhmsubcALTV 45666 | According to ~ df-subc , t... |
rhmsubcALTVcat 45667 | The restriction of the cat... |
opeliun2xp 45668 | Membership of an ordered p... |
eliunxp2 45669 | Membership in a union of C... |
mpomptx2 45670 | Express a two-argument fun... |
cbvmpox2 45671 | Rule to change the bound v... |
dmmpossx2 45672 | The domain of a mapping is... |
mpoexxg2 45673 | Existence of an operation ... |
ovmpordxf 45674 | Value of an operation give... |
ovmpordx 45675 | Value of an operation give... |
ovmpox2 45676 | The value of an operation ... |
fdmdifeqresdif 45677 | The restriction of a condi... |
offvalfv 45678 | The function operation exp... |
ofaddmndmap 45679 | The function operation app... |
mapsnop 45680 | A singleton of an ordered ... |
fprmappr 45681 | A function with a domain o... |
mapprop 45682 | An unordered pair containi... |
ztprmneprm 45683 | A prime is not an integer ... |
2t6m3t4e0 45684 | 2 times 6 minus 3 times 4 ... |
ssnn0ssfz 45685 | For any finite subset of `... |
nn0sumltlt 45686 | If the sum of two nonnegat... |
bcpascm1 45687 | Pascal's rule for the bino... |
altgsumbc 45688 | The sum of binomial coeffi... |
altgsumbcALT 45689 | Alternate proof of ~ altgs... |
zlmodzxzlmod 45690 | The ` ZZ `-module ` ZZ X. ... |
zlmodzxzel 45691 | An element of the (base se... |
zlmodzxz0 45692 | The ` 0 ` of the ` ZZ `-mo... |
zlmodzxzscm 45693 | The scalar multiplication ... |
zlmodzxzadd 45694 | The addition of the ` ZZ `... |
zlmodzxzsubm 45695 | The subtraction of the ` Z... |
zlmodzxzsub 45696 | The subtraction of the ` Z... |
mgpsumunsn 45697 | Extract a summand/factor f... |
mgpsumz 45698 | If the group sum for the m... |
mgpsumn 45699 | If the group sum for the m... |
exple2lt6 45700 | A nonnegative integer to t... |
pgrple2abl 45701 | Every symmetric group on a... |
pgrpgt2nabl 45702 | Every symmetric group on a... |
invginvrid 45703 | Identity for a multiplicat... |
rmsupp0 45704 | The support of a mapping o... |
domnmsuppn0 45705 | The support of a mapping o... |
rmsuppss 45706 | The support of a mapping o... |
mndpsuppss 45707 | The support of a mapping o... |
scmsuppss 45708 | The support of a mapping o... |
rmsuppfi 45709 | The support of a mapping o... |
rmfsupp 45710 | A mapping of a multiplicat... |
mndpsuppfi 45711 | The support of a mapping o... |
mndpfsupp 45712 | A mapping of a scalar mult... |
scmsuppfi 45713 | The support of a mapping o... |
scmfsupp 45714 | A mapping of a scalar mult... |
suppmptcfin 45715 | The support of a mapping w... |
mptcfsupp 45716 | A mapping with value 0 exc... |
fsuppmptdmf 45717 | A mapping with a finite do... |
lmodvsmdi 45718 | Multiple distributive law ... |
gsumlsscl 45719 | Closure of a group sum in ... |
assaascl0 45720 | The scalar 0 embedded into... |
assaascl1 45721 | The scalar 1 embedded into... |
ply1vr1smo 45722 | The variable in a polynomi... |
ply1ass23l 45723 | Associative identity with ... |
ply1sclrmsm 45724 | The ring multiplication of... |
coe1id 45725 | Coefficient vector of the ... |
coe1sclmulval 45726 | The value of the coefficie... |
ply1mulgsumlem1 45727 | Lemma 1 for ~ ply1mulgsum ... |
ply1mulgsumlem2 45728 | Lemma 2 for ~ ply1mulgsum ... |
ply1mulgsumlem3 45729 | Lemma 3 for ~ ply1mulgsum ... |
ply1mulgsumlem4 45730 | Lemma 4 for ~ ply1mulgsum ... |
ply1mulgsum 45731 | The product of two polynom... |
evl1at0 45732 | Polynomial evaluation for ... |
evl1at1 45733 | Polynomial evaluation for ... |
linply1 45734 | A term of the form ` x - C... |
lineval 45735 | A term of the form ` x - C... |
linevalexample 45736 | The polynomial ` x - 3 ` o... |
dmatALTval 45741 | The algebra of ` N ` x ` N... |
dmatALTbas 45742 | The base set of the algebr... |
dmatALTbasel 45743 | An element of the base set... |
dmatbas 45744 | The set of all ` N ` x ` N... |
lincop 45749 | A linear combination as op... |
lincval 45750 | The value of a linear comb... |
dflinc2 45751 | Alternative definition of ... |
lcoop 45752 | A linear combination as op... |
lcoval 45753 | The value of a linear comb... |
lincfsuppcl 45754 | A linear combination of ve... |
linccl 45755 | A linear combination of ve... |
lincval0 45756 | The value of an empty line... |
lincvalsng 45757 | The linear combination ove... |
lincvalsn 45758 | The linear combination ove... |
lincvalpr 45759 | The linear combination ove... |
lincval1 45760 | The linear combination ove... |
lcosn0 45761 | Properties of a linear com... |
lincvalsc0 45762 | The linear combination whe... |
lcoc0 45763 | Properties of a linear com... |
linc0scn0 45764 | If a set contains the zero... |
lincdifsn 45765 | A vector is a linear combi... |
linc1 45766 | A vector is a linear combi... |
lincellss 45767 | A linear combination of a ... |
lco0 45768 | The set of empty linear co... |
lcoel0 45769 | The zero vector is always ... |
lincsum 45770 | The sum of two linear comb... |
lincscm 45771 | A linear combinations mult... |
lincsumcl 45772 | The sum of two linear comb... |
lincscmcl 45773 | The multiplication of a li... |
lincsumscmcl 45774 | The sum of a linear combin... |
lincolss 45775 | According to the statement... |
ellcoellss 45776 | Every linear combination o... |
lcoss 45777 | A set of vectors of a modu... |
lspsslco 45778 | Lemma for ~ lspeqlco . (C... |
lcosslsp 45779 | Lemma for ~ lspeqlco . (C... |
lspeqlco 45780 | Equivalence of a _span_ of... |
rellininds 45784 | The class defining the rel... |
linindsv 45786 | The classes of the module ... |
islininds 45787 | The property of being a li... |
linindsi 45788 | The implications of being ... |
linindslinci 45789 | The implications of being ... |
islinindfis 45790 | The property of being a li... |
islinindfiss 45791 | The property of being a li... |
linindscl 45792 | A linearly independent set... |
lindepsnlininds 45793 | A linearly dependent subse... |
islindeps 45794 | The property of being a li... |
lincext1 45795 | Property 1 of an extension... |
lincext2 45796 | Property 2 of an extension... |
lincext3 45797 | Property 3 of an extension... |
lindslinindsimp1 45798 | Implication 1 for ~ lindsl... |
lindslinindimp2lem1 45799 | Lemma 1 for ~ lindslininds... |
lindslinindimp2lem2 45800 | Lemma 2 for ~ lindslininds... |
lindslinindimp2lem3 45801 | Lemma 3 for ~ lindslininds... |
lindslinindimp2lem4 45802 | Lemma 4 for ~ lindslininds... |
lindslinindsimp2lem5 45803 | Lemma 5 for ~ lindslininds... |
lindslinindsimp2 45804 | Implication 2 for ~ lindsl... |
lindslininds 45805 | Equivalence of definitions... |
linds0 45806 | The empty set is always a ... |
el0ldep 45807 | A set containing the zero ... |
el0ldepsnzr 45808 | A set containing the zero ... |
lindsrng01 45809 | Any subset of a module is ... |
lindszr 45810 | Any subset of a module ove... |
snlindsntorlem 45811 | Lemma for ~ snlindsntor . ... |
snlindsntor 45812 | A singleton is linearly in... |
ldepsprlem 45813 | Lemma for ~ ldepspr . (Co... |
ldepspr 45814 | If a vector is a scalar mu... |
lincresunit3lem3 45815 | Lemma 3 for ~ lincresunit3... |
lincresunitlem1 45816 | Lemma 1 for properties of ... |
lincresunitlem2 45817 | Lemma for properties of a ... |
lincresunit1 45818 | Property 1 of a specially ... |
lincresunit2 45819 | Property 2 of a specially ... |
lincresunit3lem1 45820 | Lemma 1 for ~ lincresunit3... |
lincresunit3lem2 45821 | Lemma 2 for ~ lincresunit3... |
lincresunit3 45822 | Property 3 of a specially ... |
lincreslvec3 45823 | Property 3 of a specially ... |
islindeps2 45824 | Conditions for being a lin... |
islininds2 45825 | Implication of being a lin... |
isldepslvec2 45826 | Alternative definition of ... |
lindssnlvec 45827 | A singleton not containing... |
lmod1lem1 45828 | Lemma 1 for ~ lmod1 . (Co... |
lmod1lem2 45829 | Lemma 2 for ~ lmod1 . (Co... |
lmod1lem3 45830 | Lemma 3 for ~ lmod1 . (Co... |
lmod1lem4 45831 | Lemma 4 for ~ lmod1 . (Co... |
lmod1lem5 45832 | Lemma 5 for ~ lmod1 . (Co... |
lmod1 45833 | The (smallest) structure r... |
lmod1zr 45834 | The (smallest) structure r... |
lmod1zrnlvec 45835 | There is a (left) module (... |
lmodn0 45836 | Left modules exist. (Cont... |
zlmodzxzequa 45837 | Example of an equation wit... |
zlmodzxznm 45838 | Example of a linearly depe... |
zlmodzxzldeplem 45839 | A and B are not equal. (C... |
zlmodzxzequap 45840 | Example of an equation wit... |
zlmodzxzldeplem1 45841 | Lemma 1 for ~ zlmodzxzldep... |
zlmodzxzldeplem2 45842 | Lemma 2 for ~ zlmodzxzldep... |
zlmodzxzldeplem3 45843 | Lemma 3 for ~ zlmodzxzldep... |
zlmodzxzldeplem4 45844 | Lemma 4 for ~ zlmodzxzldep... |
zlmodzxzldep 45845 | { A , B } is a linearly de... |
ldepsnlinclem1 45846 | Lemma 1 for ~ ldepsnlinc .... |
ldepsnlinclem2 45847 | Lemma 2 for ~ ldepsnlinc .... |
lvecpsslmod 45848 | The class of all (left) ve... |
ldepsnlinc 45849 | The reverse implication of... |
ldepslinc 45850 | For (left) vector spaces, ... |
suppdm 45851 | If the range of a function... |
eluz2cnn0n1 45852 | An integer greater than 1 ... |
divge1b 45853 | The ratio of a real number... |
divgt1b 45854 | The ratio of a real number... |
ltsubaddb 45855 | Equivalence for the "less ... |
ltsubsubb 45856 | Equivalence for the "less ... |
ltsubadd2b 45857 | Equivalence for the "less ... |
divsub1dir 45858 | Distribution of division o... |
expnegico01 45859 | An integer greater than 1 ... |
elfzolborelfzop1 45860 | An element of a half-open ... |
pw2m1lepw2m1 45861 | 2 to the power of a positi... |
zgtp1leeq 45862 | If an integer is between a... |
flsubz 45863 | An integer can be moved in... |
fldivmod 45864 | Expressing the floor of a ... |
mod0mul 45865 | If an integer is 0 modulo ... |
modn0mul 45866 | If an integer is not 0 mod... |
m1modmmod 45867 | An integer decreased by 1 ... |
difmodm1lt 45868 | The difference between an ... |
nn0onn0ex 45869 | For each odd nonnegative i... |
nn0enn0ex 45870 | For each even nonnegative ... |
nnennex 45871 | For each even positive int... |
nneop 45872 | A positive integer is even... |
nneom 45873 | A positive integer is even... |
nn0eo 45874 | A nonnegative integer is e... |
nnpw2even 45875 | 2 to the power of a positi... |
zefldiv2 45876 | The floor of an even integ... |
zofldiv2 45877 | The floor of an odd intege... |
nn0ofldiv2 45878 | The floor of an odd nonneg... |
flnn0div2ge 45879 | The floor of a positive in... |
flnn0ohalf 45880 | The floor of the half of a... |
logcxp0 45881 | Logarithm of a complex pow... |
regt1loggt0 45882 | The natural logarithm for ... |
fdivval 45885 | The quotient of two functi... |
fdivmpt 45886 | The quotient of two functi... |
fdivmptf 45887 | The quotient of two functi... |
refdivmptf 45888 | The quotient of two functi... |
fdivpm 45889 | The quotient of two functi... |
refdivpm 45890 | The quotient of two functi... |
fdivmptfv 45891 | The function value of a qu... |
refdivmptfv 45892 | The function value of a qu... |
bigoval 45895 | Set of functions of order ... |
elbigofrcl 45896 | Reverse closure of the "bi... |
elbigo 45897 | Properties of a function o... |
elbigo2 45898 | Properties of a function o... |
elbigo2r 45899 | Sufficient condition for a... |
elbigof 45900 | A function of order G(x) i... |
elbigodm 45901 | The domain of a function o... |
elbigoimp 45902 | The defining property of a... |
elbigolo1 45903 | A function (into the posit... |
rege1logbrege0 45904 | The general logarithm, wit... |
rege1logbzge0 45905 | The general logarithm, wit... |
fllogbd 45906 | A real number is between t... |
relogbmulbexp 45907 | The logarithm of the produ... |
relogbdivb 45908 | The logarithm of the quoti... |
logbge0b 45909 | The logarithm of a number ... |
logblt1b 45910 | The logarithm of a number ... |
fldivexpfllog2 45911 | The floor of a positive re... |
nnlog2ge0lt1 45912 | A positive integer is 1 if... |
logbpw2m1 45913 | The floor of the binary lo... |
fllog2 45914 | The floor of the binary lo... |
blenval 45917 | The binary length of an in... |
blen0 45918 | The binary length of 0. (... |
blenn0 45919 | The binary length of a "nu... |
blenre 45920 | The binary length of a pos... |
blennn 45921 | The binary length of a pos... |
blennnelnn 45922 | The binary length of a pos... |
blennn0elnn 45923 | The binary length of a non... |
blenpw2 45924 | The binary length of a pow... |
blenpw2m1 45925 | The binary length of a pow... |
nnpw2blen 45926 | A positive integer is betw... |
nnpw2blenfzo 45927 | A positive integer is betw... |
nnpw2blenfzo2 45928 | A positive integer is eith... |
nnpw2pmod 45929 | Every positive integer can... |
blen1 45930 | The binary length of 1. (... |
blen2 45931 | The binary length of 2. (... |
nnpw2p 45932 | Every positive integer can... |
nnpw2pb 45933 | A number is a positive int... |
blen1b 45934 | The binary length of a non... |
blennnt2 45935 | The binary length of a pos... |
nnolog2flm1 45936 | The floor of the binary lo... |
blennn0em1 45937 | The binary length of the h... |
blennngt2o2 45938 | The binary length of an od... |
blengt1fldiv2p1 45939 | The binary length of an in... |
blennn0e2 45940 | The binary length of an ev... |
digfval 45943 | Operation to obtain the ` ... |
digval 45944 | The ` K ` th digit of a no... |
digvalnn0 45945 | The ` K ` th digit of a no... |
nn0digval 45946 | The ` K ` th digit of a no... |
dignn0fr 45947 | The digits of the fraction... |
dignn0ldlem 45948 | Lemma for ~ dignnld . (Co... |
dignnld 45949 | The leading digits of a po... |
dig2nn0ld 45950 | The leading digits of a po... |
dig2nn1st 45951 | The first (relevant) digit... |
dig0 45952 | All digits of 0 are 0. (C... |
digexp 45953 | The ` K ` th digit of a po... |
dig1 45954 | All but one digits of 1 ar... |
0dig1 45955 | The ` 0 ` th digit of 1 is... |
0dig2pr01 45956 | The integers 0 and 1 corre... |
dig2nn0 45957 | A digit of a nonnegative i... |
0dig2nn0e 45958 | The last bit of an even in... |
0dig2nn0o 45959 | The last bit of an odd int... |
dig2bits 45960 | The ` K ` th digit of a no... |
dignn0flhalflem1 45961 | Lemma 1 for ~ dignn0flhalf... |
dignn0flhalflem2 45962 | Lemma 2 for ~ dignn0flhalf... |
dignn0ehalf 45963 | The digits of the half of ... |
dignn0flhalf 45964 | The digits of the rounded ... |
nn0sumshdiglemA 45965 | Lemma for ~ nn0sumshdig (i... |
nn0sumshdiglemB 45966 | Lemma for ~ nn0sumshdig (i... |
nn0sumshdiglem1 45967 | Lemma 1 for ~ nn0sumshdig ... |
nn0sumshdiglem2 45968 | Lemma 2 for ~ nn0sumshdig ... |
nn0sumshdig 45969 | A nonnegative integer can ... |
nn0mulfsum 45970 | Trivial algorithm to calcu... |
nn0mullong 45971 | Standard algorithm (also k... |
naryfval 45974 | The set of the n-ary (endo... |
naryfvalixp 45975 | The set of the n-ary (endo... |
naryfvalel 45976 | An n-ary (endo)function on... |
naryrcl 45977 | Reverse closure for n-ary ... |
naryfvalelfv 45978 | The value of an n-ary (end... |
naryfvalelwrdf 45979 | An n-ary (endo)function on... |
0aryfvalel 45980 | A nullary (endo)function o... |
0aryfvalelfv 45981 | The value of a nullary (en... |
1aryfvalel 45982 | A unary (endo)function on ... |
fv1arycl 45983 | Closure of a unary (endo)f... |
1arympt1 45984 | A unary (endo)function in ... |
1arympt1fv 45985 | The value of a unary (endo... |
1arymaptfv 45986 | The value of the mapping o... |
1arymaptf 45987 | The mapping of unary (endo... |
1arymaptf1 45988 | The mapping of unary (endo... |
1arymaptfo 45989 | The mapping of unary (endo... |
1arymaptf1o 45990 | The mapping of unary (endo... |
1aryenef 45991 | The set of unary (endo)fun... |
1aryenefmnd 45992 | The set of unary (endo)fun... |
2aryfvalel 45993 | A binary (endo)function on... |
fv2arycl 45994 | Closure of a binary (endo)... |
2arympt 45995 | A binary (endo)function in... |
2arymptfv 45996 | The value of a binary (end... |
2arymaptfv 45997 | The value of the mapping o... |
2arymaptf 45998 | The mapping of binary (end... |
2arymaptf1 45999 | The mapping of binary (end... |
2arymaptfo 46000 | The mapping of binary (end... |
2arymaptf1o 46001 | The mapping of binary (end... |
2aryenef 46002 | The set of binary (endo)fu... |
itcoval 46007 | The value of the function ... |
itcoval0 46008 | A function iterated zero t... |
itcoval1 46009 | A function iterated once. ... |
itcoval2 46010 | A function iterated twice.... |
itcoval3 46011 | A function iterated three ... |
itcoval0mpt 46012 | A mapping iterated zero ti... |
itcovalsuc 46013 | The value of the function ... |
itcovalsucov 46014 | The value of the function ... |
itcovalendof 46015 | The n-th iterate of an end... |
itcovalpclem1 46016 | Lemma 1 for ~ itcovalpc : ... |
itcovalpclem2 46017 | Lemma 2 for ~ itcovalpc : ... |
itcovalpc 46018 | The value of the function ... |
itcovalt2lem2lem1 46019 | Lemma 1 for ~ itcovalt2lem... |
itcovalt2lem2lem2 46020 | Lemma 2 for ~ itcovalt2lem... |
itcovalt2lem1 46021 | Lemma 1 for ~ itcovalt2 : ... |
itcovalt2lem2 46022 | Lemma 2 for ~ itcovalt2 : ... |
itcovalt2 46023 | The value of the function ... |
ackvalsuc1mpt 46024 | The Ackermann function at ... |
ackvalsuc1 46025 | The Ackermann function at ... |
ackval0 46026 | The Ackermann function at ... |
ackval1 46027 | The Ackermann function at ... |
ackval2 46028 | The Ackermann function at ... |
ackval3 46029 | The Ackermann function at ... |
ackendofnn0 46030 | The Ackermann function at ... |
ackfnnn0 46031 | The Ackermann function at ... |
ackval0val 46032 | The Ackermann function at ... |
ackvalsuc0val 46033 | The Ackermann function at ... |
ackvalsucsucval 46034 | The Ackermann function at ... |
ackval0012 46035 | The Ackermann function at ... |
ackval1012 46036 | The Ackermann function at ... |
ackval2012 46037 | The Ackermann function at ... |
ackval3012 46038 | The Ackermann function at ... |
ackval40 46039 | The Ackermann function at ... |
ackval41a 46040 | The Ackermann function at ... |
ackval41 46041 | The Ackermann function at ... |
ackval42 46042 | The Ackermann function at ... |
ackval42a 46043 | The Ackermann function at ... |
ackval50 46044 | The Ackermann function at ... |
fv1prop 46045 | The function value of unor... |
fv2prop 46046 | The function value of unor... |
submuladdmuld 46047 | Transformation of a sum of... |
affinecomb1 46048 | Combination of two real af... |
affinecomb2 46049 | Combination of two real af... |
affineid 46050 | Identity of an affine comb... |
1subrec1sub 46051 | Subtract the reciprocal of... |
resum2sqcl 46052 | The sum of two squares of ... |
resum2sqgt0 46053 | The sum of the square of a... |
resum2sqrp 46054 | The sum of the square of a... |
resum2sqorgt0 46055 | The sum of the square of t... |
reorelicc 46056 | Membership in and outside ... |
rrx2pxel 46057 | The x-coordinate of a poin... |
rrx2pyel 46058 | The y-coordinate of a poin... |
prelrrx2 46059 | An unordered pair of order... |
prelrrx2b 46060 | An unordered pair of order... |
rrx2pnecoorneor 46061 | If two different points ` ... |
rrx2pnedifcoorneor 46062 | If two different points ` ... |
rrx2pnedifcoorneorr 46063 | If two different points ` ... |
rrx2xpref1o 46064 | There is a bijection betwe... |
rrx2xpreen 46065 | The set of points in the t... |
rrx2plord 46066 | The lexicographical orderi... |
rrx2plord1 46067 | The lexicographical orderi... |
rrx2plord2 46068 | The lexicographical orderi... |
rrx2plordisom 46069 | The set of points in the t... |
rrx2plordso 46070 | The lexicographical orderi... |
ehl2eudisval0 46071 | The Euclidean distance of ... |
ehl2eudis0lt 46072 | An upper bound of the Eucl... |
lines 46077 | The lines passing through ... |
line 46078 | The line passing through t... |
rrxlines 46079 | Definition of lines passin... |
rrxline 46080 | The line passing through t... |
rrxlinesc 46081 | Definition of lines passin... |
rrxlinec 46082 | The line passing through t... |
eenglngeehlnmlem1 46083 | Lemma 1 for ~ eenglngeehln... |
eenglngeehlnmlem2 46084 | Lemma 2 for ~ eenglngeehln... |
eenglngeehlnm 46085 | The line definition in the... |
rrx2line 46086 | The line passing through t... |
rrx2vlinest 46087 | The vertical line passing ... |
rrx2linest 46088 | The line passing through t... |
rrx2linesl 46089 | The line passing through t... |
rrx2linest2 46090 | The line passing through t... |
elrrx2linest2 46091 | The line passing through t... |
spheres 46092 | The spheres for given cent... |
sphere 46093 | A sphere with center ` X `... |
rrxsphere 46094 | The sphere with center ` M... |
2sphere 46095 | The sphere with center ` M... |
2sphere0 46096 | The sphere around the orig... |
line2ylem 46097 | Lemma for ~ line2y . This... |
line2 46098 | Example for a line ` G ` p... |
line2xlem 46099 | Lemma for ~ line2x . This... |
line2x 46100 | Example for a horizontal l... |
line2y 46101 | Example for a vertical lin... |
itsclc0lem1 46102 | Lemma for theorems about i... |
itsclc0lem2 46103 | Lemma for theorems about i... |
itsclc0lem3 46104 | Lemma for theorems about i... |
itscnhlc0yqe 46105 | Lemma for ~ itsclc0 . Qua... |
itschlc0yqe 46106 | Lemma for ~ itsclc0 . Qua... |
itsclc0yqe 46107 | Lemma for ~ itsclc0 . Qua... |
itsclc0yqsollem1 46108 | Lemma 1 for ~ itsclc0yqsol... |
itsclc0yqsollem2 46109 | Lemma 2 for ~ itsclc0yqsol... |
itsclc0yqsol 46110 | Lemma for ~ itsclc0 . Sol... |
itscnhlc0xyqsol 46111 | Lemma for ~ itsclc0 . Sol... |
itschlc0xyqsol1 46112 | Lemma for ~ itsclc0 . Sol... |
itschlc0xyqsol 46113 | Lemma for ~ itsclc0 . Sol... |
itsclc0xyqsol 46114 | Lemma for ~ itsclc0 . Sol... |
itsclc0xyqsolr 46115 | Lemma for ~ itsclc0 . Sol... |
itsclc0xyqsolb 46116 | Lemma for ~ itsclc0 . Sol... |
itsclc0 46117 | The intersection points of... |
itsclc0b 46118 | The intersection points of... |
itsclinecirc0 46119 | The intersection points of... |
itsclinecirc0b 46120 | The intersection points of... |
itsclinecirc0in 46121 | The intersection points of... |
itsclquadb 46122 | Quadratic equation for the... |
itsclquadeu 46123 | Quadratic equation for the... |
2itscplem1 46124 | Lemma 1 for ~ 2itscp . (C... |
2itscplem2 46125 | Lemma 2 for ~ 2itscp . (C... |
2itscplem3 46126 | Lemma D for ~ 2itscp . (C... |
2itscp 46127 | A condition for a quadrati... |
itscnhlinecirc02plem1 46128 | Lemma 1 for ~ itscnhlineci... |
itscnhlinecirc02plem2 46129 | Lemma 2 for ~ itscnhlineci... |
itscnhlinecirc02plem3 46130 | Lemma 3 for ~ itscnhlineci... |
itscnhlinecirc02p 46131 | Intersection of a nonhoriz... |
inlinecirc02plem 46132 | Lemma for ~ inlinecirc02p ... |
inlinecirc02p 46133 | Intersection of a line wit... |
inlinecirc02preu 46134 | Intersection of a line wit... |
pm4.71da 46135 | Deduction converting a bic... |
logic1 46136 | Distribution of implicatio... |
logic1a 46137 | Variant of ~ logic1 . (Co... |
logic2 46138 | Variant of ~ logic1 . (Co... |
pm5.32dav 46139 | Distribution of implicatio... |
pm5.32dra 46140 | Reverse distribution of im... |
exp12bd 46141 | The import-export theorem ... |
mpbiran3d 46142 | Equivalence with a conjunc... |
mpbiran4d 46143 | Equivalence with a conjunc... |
dtrucor3 46144 | An example of how ~ ax-5 w... |
ralbidb 46145 | Formula-building rule for ... |
ralbidc 46146 | Formula-building rule for ... |
r19.41dv 46147 | A complex deduction form o... |
rspceb2dv 46148 | Restricted existential spe... |
rextru 46149 | Two ways of expressing "at... |
rmotru 46150 | Two ways of expressing "at... |
reutru 46151 | Two ways of expressing "ex... |
reutruALT 46152 | Alternate proof for ~ reut... |
ssdisjd 46153 | Subset preserves disjointn... |
ssdisjdr 46154 | Subset preserves disjointn... |
disjdifb 46155 | Relative complement is ant... |
predisj 46156 | Preimages of disjoint sets... |
vsn 46157 | The singleton of the unive... |
mosn 46158 | "At most one" element in a... |
mo0 46159 | "At most one" element in a... |
mosssn 46160 | "At most one" element in a... |
mo0sn 46161 | Two ways of expressing "at... |
mosssn2 46162 | Two ways of expressing "at... |
unilbss 46163 | Superclass of the greatest... |
inpw 46164 | Two ways of expressing a c... |
mof0 46165 | There is at most one funct... |
mof02 46166 | A variant of ~ mof0 . (Co... |
mof0ALT 46167 | Alternate proof for ~ mof0... |
eufsnlem 46168 | There is exactly one funct... |
eufsn 46169 | There is exactly one funct... |
eufsn2 46170 | There is exactly one funct... |
mofsn 46171 | There is at most one funct... |
mofsn2 46172 | There is at most one funct... |
mofsssn 46173 | There is at most one funct... |
mofmo 46174 | There is at most one funct... |
mofeu 46175 | The uniqueness of a functi... |
elfvne0 46176 | If a function value has a ... |
fdomne0 46177 | A function with non-empty ... |
f1sn2g 46178 | A function that maps a sin... |
f102g 46179 | A function that maps the e... |
f1mo 46180 | A function that maps a set... |
f002 46181 | A function with an empty c... |
map0cor 46182 | A function exists iff an e... |
fvconstr 46183 | Two ways of expressing ` A... |
fvconstrn0 46184 | Two ways of expressing ` A... |
fvconstr2 46185 | Two ways of expressing ` A... |
fvconst0ci 46186 | A constant function's valu... |
fvconstdomi 46187 | A constant function's valu... |
f1omo 46188 | There is at most one eleme... |
f1omoALT 46189 | There is at most one eleme... |
iccin 46190 | Intersection of two closed... |
iccdisj2 46191 | If the upper bound of one ... |
iccdisj 46192 | If the upper bound of one ... |
mreuniss 46193 | The union of a collection ... |
clduni 46194 | The union of closed sets i... |
opncldeqv 46195 | Conditions on open sets ar... |
opndisj 46196 | Two ways of saying that tw... |
clddisj 46197 | Two ways of saying that tw... |
neircl 46198 | Reverse closure of the nei... |
opnneilem 46199 | Lemma factoring out common... |
opnneir 46200 | If something is true for a... |
opnneirv 46201 | A variant of ~ opnneir wit... |
opnneilv 46202 | The converse of ~ opnneir ... |
opnneil 46203 | A variant of ~ opnneilv . ... |
opnneieqv 46204 | The equivalence between ne... |
opnneieqvv 46205 | The equivalence between ne... |
restcls2lem 46206 | A closed set in a subspace... |
restcls2 46207 | A closed set in a subspace... |
restclsseplem 46208 | Lemma for ~ restclssep . ... |
restclssep 46209 | Two disjoint closed sets i... |
cnneiima 46210 | Given a continuous functio... |
iooii 46211 | Open intervals are open se... |
icccldii 46212 | Closed intervals are close... |
i0oii 46213 | ` ( 0 [,) A ) ` is open in... |
io1ii 46214 | ` ( A (,] 1 ) ` is open in... |
sepnsepolem1 46215 | Lemma for ~ sepnsepo . (C... |
sepnsepolem2 46216 | Open neighborhood and neig... |
sepnsepo 46217 | Open neighborhood and neig... |
sepdisj 46218 | Separated sets are disjoin... |
seposep 46219 | If two sets are separated ... |
sepcsepo 46220 | If two sets are separated ... |
sepfsepc 46221 | If two sets are separated ... |
seppsepf 46222 | If two sets are precisely ... |
seppcld 46223 | If two sets are precisely ... |
isnrm4 46224 | A topological space is nor... |
dfnrm2 46225 | A topological space is nor... |
dfnrm3 46226 | A topological space is nor... |
iscnrm3lem1 46227 | Lemma for ~ iscnrm3 . Sub... |
iscnrm3lem2 46228 | Lemma for ~ iscnrm3 provin... |
iscnrm3lem3 46229 | Lemma for ~ iscnrm3lem4 . ... |
iscnrm3lem4 46230 | Lemma for ~ iscnrm3lem5 an... |
iscnrm3lem5 46231 | Lemma for ~ iscnrm3l . (C... |
iscnrm3lem6 46232 | Lemma for ~ iscnrm3lem7 . ... |
iscnrm3lem7 46233 | Lemma for ~ iscnrm3rlem8 a... |
iscnrm3rlem1 46234 | Lemma for ~ iscnrm3rlem2 .... |
iscnrm3rlem2 46235 | Lemma for ~ iscnrm3rlem3 .... |
iscnrm3rlem3 46236 | Lemma for ~ iscnrm3r . Th... |
iscnrm3rlem4 46237 | Lemma for ~ iscnrm3rlem8 .... |
iscnrm3rlem5 46238 | Lemma for ~ iscnrm3rlem6 .... |
iscnrm3rlem6 46239 | Lemma for ~ iscnrm3rlem7 .... |
iscnrm3rlem7 46240 | Lemma for ~ iscnrm3rlem8 .... |
iscnrm3rlem8 46241 | Lemma for ~ iscnrm3r . Di... |
iscnrm3r 46242 | Lemma for ~ iscnrm3 . If ... |
iscnrm3llem1 46243 | Lemma for ~ iscnrm3l . Cl... |
iscnrm3llem2 46244 | Lemma for ~ iscnrm3l . If... |
iscnrm3l 46245 | Lemma for ~ iscnrm3 . Giv... |
iscnrm3 46246 | A completely normal topolo... |
iscnrm3v 46247 | A topology is completely n... |
iscnrm4 46248 | A completely normal topolo... |
isprsd 46249 | Property of being a preord... |
lubeldm2 46250 | Member of the domain of th... |
glbeldm2 46251 | Member of the domain of th... |
lubeldm2d 46252 | Member of the domain of th... |
glbeldm2d 46253 | Member of the domain of th... |
lubsscl 46254 | If a subset of ` S ` conta... |
glbsscl 46255 | If a subset of ` S ` conta... |
lubprlem 46256 | Lemma for ~ lubprdm and ~ ... |
lubprdm 46257 | The set of two comparable ... |
lubpr 46258 | The LUB of the set of two ... |
glbprlem 46259 | Lemma for ~ glbprdm and ~ ... |
glbprdm 46260 | The set of two comparable ... |
glbpr 46261 | The GLB of the set of two ... |
joindm2 46262 | The join of any two elemen... |
joindm3 46263 | The join of any two elemen... |
meetdm2 46264 | The meet of any two elemen... |
meetdm3 46265 | The meet of any two elemen... |
posjidm 46266 | Poset join is idempotent. ... |
posmidm 46267 | Poset meet is idempotent. ... |
toslat 46268 | A toset is a lattice. (Co... |
isclatd 46269 | The predicate "is a comple... |
intubeu 46270 | Existential uniqueness of ... |
unilbeu 46271 | Existential uniqueness of ... |
ipolublem 46272 | Lemma for ~ ipolubdm and ~... |
ipolubdm 46273 | The domain of the LUB of t... |
ipolub 46274 | The LUB of the inclusion p... |
ipoglblem 46275 | Lemma for ~ ipoglbdm and ~... |
ipoglbdm 46276 | The domain of the GLB of t... |
ipoglb 46277 | The GLB of the inclusion p... |
ipolub0 46278 | The LUB of the empty set i... |
ipolub00 46279 | The LUB of the empty set i... |
ipoglb0 46280 | The GLB of the empty set i... |
mrelatlubALT 46281 | Least upper bounds in a Mo... |
mrelatglbALT 46282 | Greatest lower bounds in a... |
mreclat 46283 | A Moore space is a complet... |
topclat 46284 | A topology is a complete l... |
toplatglb0 46285 | The empty intersection in ... |
toplatlub 46286 | Least upper bounds in a to... |
toplatglb 46287 | Greatest lower bounds in a... |
toplatjoin 46288 | Joins in a topology are re... |
toplatmeet 46289 | Meets in a topology are re... |
topdlat 46290 | A topology is a distributi... |
catprslem 46291 | Lemma for ~ catprs . (Con... |
catprs 46292 | A preorder can be extracte... |
catprs2 46293 | A category equipped with t... |
catprsc 46294 | A construction of the preo... |
catprsc2 46295 | An alternate construction ... |
endmndlem 46296 | A diagonal hom-set in a ca... |
idmon 46297 | An identity arrow, or an i... |
idepi 46298 | An identity arrow, or an i... |
funcf2lem 46299 | A utility theorem for prov... |
isthinc 46302 | The predicate "is a thin c... |
isthinc2 46303 | A thin category is a categ... |
isthinc3 46304 | A thin category is a categ... |
thincc 46305 | A thin category is a categ... |
thinccd 46306 | A thin category is a categ... |
thincssc 46307 | A thin category is a categ... |
isthincd2lem1 46308 | Lemma for ~ isthincd2 and ... |
thincmo2 46309 | Morphisms in the same hom-... |
thincmo 46310 | There is at most one morph... |
thincmoALT 46311 | Alternate proof for ~ thin... |
thincmod 46312 | At most one morphism in ea... |
thincn0eu 46313 | In a thin category, a hom-... |
thincid 46314 | In a thin category, a morp... |
thincmon 46315 | In a thin category, all mo... |
thincepi 46316 | In a thin category, all mo... |
isthincd2lem2 46317 | Lemma for ~ isthincd2 . (... |
isthincd 46318 | The predicate "is a thin c... |
isthincd2 46319 | The predicate " ` C ` is a... |
oppcthin 46320 | The opposite category of a... |
subthinc 46321 | A subcategory of a thin ca... |
functhinclem1 46322 | Lemma for ~ functhinc . G... |
functhinclem2 46323 | Lemma for ~ functhinc . (... |
functhinclem3 46324 | Lemma for ~ functhinc . T... |
functhinclem4 46325 | Lemma for ~ functhinc . O... |
functhinc 46326 | A functor to a thin catego... |
fullthinc 46327 | A functor to a thin catego... |
fullthinc2 46328 | A full functor to a thin c... |
thincfth 46329 | A functor from a thin cate... |
thincciso 46330 | Two thin categories are is... |
0thincg 46331 | Any structure with an empt... |
0thinc 46332 | The empty category (see ~ ... |
indthinc 46333 | An indiscrete category in ... |
indthincALT 46334 | An alternate proof for ~ i... |
prsthinc 46335 | Preordered sets as categor... |
setcthin 46336 | A category of sets all of ... |
setc2othin 46337 | The category ` ( SetCat ``... |
thincsect 46338 | In a thin category, one mo... |
thincsect2 46339 | In a thin category, ` F ` ... |
thincinv 46340 | In a thin category, ` F ` ... |
thinciso 46341 | In a thin category, ` F : ... |
thinccic 46342 | In a thin category, two ob... |
prstcval 46345 | Lemma for ~ prstcnidlem an... |
prstcnidlem 46346 | Lemma for ~ prstcnid and ~... |
prstcnid 46347 | Components other than ` Ho... |
prstcbas 46348 | The base set is unchanged.... |
prstcleval 46349 | Value of the less-than-or-... |
prstclevalOLD 46350 | Obsolete proof of ~ prstcl... |
prstcle 46351 | Value of the less-than-or-... |
prstcocval 46352 | Orthocomplementation is un... |
prstcocvalOLD 46353 | Obsolete proof of ~ prstco... |
prstcoc 46354 | Orthocomplementation is un... |
prstchomval 46355 | Hom-sets of the constructe... |
prstcprs 46356 | The category is a preorder... |
prstcthin 46357 | The preordered set is equi... |
prstchom 46358 | Hom-sets of the constructe... |
prstchom2 46359 | Hom-sets of the constructe... |
prstchom2ALT 46360 | Hom-sets of the constructe... |
postcpos 46361 | The converted category is ... |
postcposALT 46362 | Alternate proof for ~ post... |
postc 46363 | The converted category is ... |
mndtcval 46366 | Value of the category buil... |
mndtcbasval 46367 | The base set of the catego... |
mndtcbas 46368 | The category built from a ... |
mndtcob 46369 | Lemma for ~ mndtchom and ~... |
mndtcbas2 46370 | Two objects in a category ... |
mndtchom 46371 | The only hom-set of the ca... |
mndtcco 46372 | The composition of the cat... |
mndtcco2 46373 | The composition of the cat... |
mndtccatid 46374 | Lemma for ~ mndtccat and ~... |
mndtccat 46375 | The function value is a ca... |
mndtcid 46376 | The identity morphism, or ... |
grptcmon 46377 | All morphisms in a categor... |
grptcepi 46378 | All morphisms in a categor... |
nfintd 46379 | Bound-variable hypothesis ... |
nfiund 46380 | Bound-variable hypothesis ... |
nfiundg 46381 | Bound-variable hypothesis ... |
iunord 46382 | The indexed union of a col... |
iunordi 46383 | The indexed union of a col... |
spd 46384 | Specialization deduction, ... |
spcdvw 46385 | A version of ~ spcdv where... |
tfis2d 46386 | Transfinite Induction Sche... |
bnd2d 46387 | Deduction form of ~ bnd2 .... |
dffun3f 46388 | Alternate definition of fu... |
setrecseq 46391 | Equality theorem for set r... |
nfsetrecs 46392 | Bound-variable hypothesis ... |
setrec1lem1 46393 | Lemma for ~ setrec1 . Thi... |
setrec1lem2 46394 | Lemma for ~ setrec1 . If ... |
setrec1lem3 46395 | Lemma for ~ setrec1 . If ... |
setrec1lem4 46396 | Lemma for ~ setrec1 . If ... |
setrec1 46397 | This is the first of two f... |
setrec2fun 46398 | This is the second of two ... |
setrec2lem1 46399 | Lemma for ~ setrec2 . The... |
setrec2lem2 46400 | Lemma for ~ setrec2 . The... |
setrec2 46401 | This is the second of two ... |
setrec2v 46402 | Version of ~ setrec2 with ... |
setis 46403 | Version of ~ setrec2 expre... |
elsetrecslem 46404 | Lemma for ~ elsetrecs . A... |
elsetrecs 46405 | A set ` A ` is an element ... |
setrecsss 46406 | The ` setrecs ` operator r... |
setrecsres 46407 | A recursively generated cl... |
vsetrec 46408 | Construct ` _V ` using set... |
0setrec 46409 | If a function sends the em... |
onsetreclem1 46410 | Lemma for ~ onsetrec . (C... |
onsetreclem2 46411 | Lemma for ~ onsetrec . (C... |
onsetreclem3 46412 | Lemma for ~ onsetrec . (C... |
onsetrec 46413 | Construct ` On ` using set... |
elpglem1 46416 | Lemma for ~ elpg . (Contr... |
elpglem2 46417 | Lemma for ~ elpg . (Contr... |
elpglem3 46418 | Lemma for ~ elpg . (Contr... |
elpg 46419 | Membership in the class of... |
sbidd 46420 | An identity theorem for su... |
sbidd-misc 46421 | An identity theorem for su... |
gte-lte 46426 | Simple relationship betwee... |
gt-lt 46427 | Simple relationship betwee... |
gte-lteh 46428 | Relationship between ` <_ ... |
gt-lth 46429 | Relationship between ` < `... |
ex-gt 46430 | Simple example of ` > ` , ... |
ex-gte 46431 | Simple example of ` >_ ` ,... |
sinhval-named 46438 | Value of the named sinh fu... |
coshval-named 46439 | Value of the named cosh fu... |
tanhval-named 46440 | Value of the named tanh fu... |
sinh-conventional 46441 | Conventional definition of... |
sinhpcosh 46442 | Prove that ` ( sinh `` A )... |
secval 46449 | Value of the secant functi... |
cscval 46450 | Value of the cosecant func... |
cotval 46451 | Value of the cotangent fun... |
seccl 46452 | The closure of the secant ... |
csccl 46453 | The closure of the cosecan... |
cotcl 46454 | The closure of the cotange... |
reseccl 46455 | The closure of the secant ... |
recsccl 46456 | The closure of the cosecan... |
recotcl 46457 | The closure of the cotange... |
recsec 46458 | The reciprocal of secant i... |
reccsc 46459 | The reciprocal of cosecant... |
reccot 46460 | The reciprocal of cotangen... |
rectan 46461 | The reciprocal of tangent ... |
sec0 46462 | The value of the secant fu... |
onetansqsecsq 46463 | Prove the tangent squared ... |
cotsqcscsq 46464 | Prove the tangent squared ... |
ifnmfalse 46465 | If A is not a member of B,... |
logb2aval 46466 | Define the value of the ` ... |
comraddi 46473 | Commute RHS addition. See... |
mvlraddi 46474 | Move the right term in a s... |
mvrladdi 46475 | Move the left term in a su... |
assraddsubi 46476 | Associate RHS addition-sub... |
joinlmuladdmuli 46477 | Join AB+CB into (A+C) on L... |
joinlmulsubmuld 46478 | Join AB-CB into (A-C) on L... |
joinlmulsubmuli 46479 | Join AB-CB into (A-C) on L... |
mvlrmuld 46480 | Move the right term in a p... |
mvlrmuli 46481 | Move the right term in a p... |
i2linesi 46482 | Solve for the intersection... |
i2linesd 46483 | Solve for the intersection... |
alimp-surprise 46484 | Demonstrate that when usin... |
alimp-no-surprise 46485 | There is no "surprise" in ... |
empty-surprise 46486 | Demonstrate that when usin... |
empty-surprise2 46487 | "Prove" that false is true... |
eximp-surprise 46488 | Show what implication insi... |
eximp-surprise2 46489 | Show that "there exists" w... |
alsconv 46494 | There is an equivalence be... |
alsi1d 46495 | Deduction rule: Given "al... |
alsi2d 46496 | Deduction rule: Given "al... |
alsc1d 46497 | Deduction rule: Given "al... |
alsc2d 46498 | Deduction rule: Given "al... |
alscn0d 46499 | Deduction rule: Given "al... |
alsi-no-surprise 46500 | Demonstrate that there is ... |
5m4e1 46501 | Prove that 5 - 4 = 1. (Co... |
2p2ne5 46502 | Prove that ` 2 + 2 =/= 5 `... |
resolution 46503 | Resolution rule. This is ... |
testable 46504 | In classical logic all wff... |
aacllem 46505 | Lemma for other theorems a... |
amgmwlem 46506 | Weighted version of ~ amgm... |
amgmlemALT 46507 | Alternate proof of ~ amgml... |
amgmw2d 46508 | Weighted arithmetic-geomet... |
young2d 46509 | Young's inequality for ` n... |
et-ltneverrefl 46510 | Less-than class is never r... |
natlocalincr 46511 | Global monotonicity on hal... |
natglobalincr 46512 | Local monotonicity on half... |
upwordnul 46515 | Empty set is an increasing... |
upwordisword 46516 | Any increasing sequence is... |
singoutnword 46517 | Singleton with character o... |
singoutnupword 46518 | Singleton with character o... |
upwordsing 46519 | Singleton is an increasing... |
upwordsseti 46520 | Strictly increasing sequen... |
tworepnotupword 46521 | Word of two matching chara... |
Copyright terms: Public domain | W3C validator |