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List of Theorems
RefDescription
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...
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