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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...
pm2.01i 189 Inference associated with ...
pm2.01d 190 Deduction based on reducti...
pm2.6 191 Theorem *2.6 of [Whitehead...
pm2.61 192 Theorem *2.61 of [Whitehea...
pm2.65 193 Theorem *2.65 of [Whitehea...
pm2.65i 194 Inference for proof by con...
pm2.21dd 195 A contradiction implies an...
pm2.65d 196 Deduction for proof by con...
mto 197 The rule of modus tollens....
mtod 198 Modus tollens deduction. ...
mtoi 199 Modus tollens inference. ...
mt2 200 A rule similar to modus to...
mt3 201 A rule similar to modus to...
peirce 202 Peirce's axiom. A non-int...
looinv 203 The Inversion Axiom of the...
bijust0 204 A self-implication (see ~ ...
bijust 205 Theorem used to justify th...
impbi 208 Property of the biconditio...
impbii 209 Infer an equivalence from ...
impbidd 210 Deduce an equivalence from...
impbid21d 211 Deduce an equivalence from...
impbid 212 Deduce an equivalence from...
dfbi1 213 Relate the biconditional c...
dfbi1ALT 214 Alternate proof of ~ dfbi1...
biimp 215 Property of the biconditio...
biimpi 216 Infer an implication from ...
sylbi 217 A mixed syllogism inferenc...
sylib 218 A mixed syllogism inferenc...
sylbb 219 A mixed syllogism inferenc...
biimpr 220 Property of the biconditio...
bicom1 221 Commutative law for the bi...
bicom 222 Commutative law for the bi...
bicomd 223 Commute two sides of a bic...
bicomi 224 Inference from commutative...
impbid1 225 Infer an equivalence from ...
impbid2 226 Infer an equivalence from ...
impcon4bid 227 A variation on ~ impbid wi...
biimpri 228 Infer a converse implicati...
biimpd 229 Deduce an implication from...
mpbi 230 An inference from a bicond...
mpbir 231 An inference from a bicond...
mpbid 232 A deduction from a bicondi...
mpbii 233 An inference from a nested...
sylibr 234 A mixed syllogism inferenc...
sylbir 235 A mixed syllogism inferenc...
sylbbr 236 A mixed syllogism inferenc...
sylbb1 237 A mixed syllogism inferenc...
sylbb2 238 A mixed syllogism inferenc...
sylibd 239 A syllogism deduction. (C...
sylbid 240 A syllogism deduction. (C...
mpbidi 241 A deduction from a bicondi...
biimtrid 242 A mixed syllogism inferenc...
biimtrrid 243 A mixed syllogism inferenc...
imbitrid 244 A mixed syllogism inferenc...
syl5ibcom 245 A mixed syllogism inferenc...
imbitrrid 246 A mixed syllogism inferenc...
syl5ibrcom 247 A mixed syllogism inferenc...
biimprd 248 Deduce a converse implicat...
biimpcd 249 Deduce a commuted implicat...
biimprcd 250 Deduce a converse commuted...
imbitrdi 251 A mixed syllogism inferenc...
imbitrrdi 252 A mixed syllogism inferenc...
biimtrdi 253 A mixed syllogism inferenc...
biimtrrdi 254 A mixed syllogism inferenc...
syl7bi 255 A mixed syllogism inferenc...
syl8ib 256 A syllogism rule of infere...
mpbird 257 A deduction from a bicondi...
mpbiri 258 An inference from a nested...
sylibrd 259 A syllogism deduction. (C...
sylbird 260 A syllogism deduction. (C...
biid 261 Principle of identity for ...
biidd 262 Principle of identity with...
pm5.1im 263 Two propositions are equiv...
2th 264 Two truths are equivalent....
2thd 265 Two truths are equivalent....
monothetic 266 Two self-implications (see...
ibi 267 Inference that converts a ...
ibir 268 Inference that converts a ...
ibd 269 Deduction that converts a ...
pm5.74 270 Distribution of implicatio...
pm5.74i 271 Distribution of implicatio...
pm5.74ri 272 Distribution of implicatio...
pm5.74d 273 Distribution of implicatio...
pm5.74rd 274 Distribution of implicatio...
bitri 275 An inference from transiti...
bitr2i 276 An inference from transiti...
bitr3i 277 An inference from transiti...
bitr4i 278 An inference from transiti...
bitrd 279 Deduction form of ~ bitri ...
bitr2d 280 Deduction form of ~ bitr2i...
bitr3d 281 Deduction form of ~ bitr3i...
bitr4d 282 Deduction form of ~ bitr4i...
bitrid 283 A syllogism inference from...
bitr2id 284 A syllogism inference from...
bitr3id 285 A syllogism inference from...
bitr3di 286 A syllogism inference from...
bitrdi 287 A syllogism inference from...
bitr2di 288 A syllogism inference from...
bitr4di 289 A syllogism inference from...
bitr4id 290 A syllogism inference from...
3imtr3i 291 A mixed syllogism inferenc...
3imtr4i 292 A mixed syllogism inferenc...
3imtr3d 293 More general version of ~ ...
3imtr4d 294 More general version of ~ ...
3imtr3g 295 More general version of ~ ...
3imtr4g 296 More general version of ~ ...
3bitri 297 A chained inference from t...
3bitrri 298 A chained inference from t...
3bitr2i 299 A chained inference from t...
3bitr2ri 300 A chained inference from t...
3bitr3i 301 A chained inference from t...
3bitr3ri 302 A chained inference from t...
3bitr4i 303 A chained inference from t...
3bitr4ri 304 A chained inference from t...
3bitrd 305 Deduction from transitivit...
3bitrrd 306 Deduction from transitivit...
3bitr2d 307 Deduction from transitivit...
3bitr2rd 308 Deduction from transitivit...
3bitr3d 309 Deduction from transitivit...
3bitr3rd 310 Deduction from transitivit...
3bitr4d 311 Deduction from transitivit...
3bitr4rd 312 Deduction from transitivit...
3bitr3g 313 More general version of ~ ...
3bitr4g 314 More general version of ~ ...
notnotb 315 Double negation. Theorem ...
con34b 316 A biconditional form of co...
con4bid 317 A contraposition deduction...
notbid 318 Deduction negating both si...
notbi 319 Contraposition. Theorem *...
notbii 320 Negate both sides of a log...
con4bii 321 A contraposition inference...
mtbi 322 An inference from a bicond...
mtbir 323 An inference from a bicond...
mtbid 324 A deduction from a bicondi...
mtbird 325 A deduction from a bicondi...
mtbii 326 An inference from a bicond...
mtbiri 327 An inference from a bicond...
sylnib 328 A mixed syllogism inferenc...
sylnibr 329 A mixed syllogism inferenc...
sylnbi 330 A mixed syllogism inferenc...
sylnbir 331 A mixed syllogism inferenc...
xchnxbi 332 Replacement of a subexpres...
xchnxbir 333 Replacement of a subexpres...
xchbinx 334 Replacement of a subexpres...
xchbinxr 335 Replacement of a subexpres...
imbi2i 336 Introduce an antecedent to...
bibi2i 337 Inference adding a bicondi...
bibi1i 338 Inference adding a bicondi...
bibi12i 339 The equivalence of two equ...
imbi2d 340 Deduction adding an antece...
imbi1d 341 Deduction adding a consequ...
bibi2d 342 Deduction adding a bicondi...
bibi1d 343 Deduction adding a bicondi...
imbi12d 344 Deduction joining two equi...
bibi12d 345 Deduction joining two equi...
imbi12 346 Closed form of ~ imbi12i ....
imbi1 347 Theorem *4.84 of [Whitehea...
imbi2 348 Theorem *4.85 of [Whitehea...
imbi1i 349 Introduce a consequent to ...
imbi12i 350 Join two logical equivalen...
bibi1 351 Theorem *4.86 of [Whitehea...
bitr3 352 Closed nested implication ...
con2bi 353 Contraposition. Theorem *...
con2bid 354 A contraposition deduction...
con1bid 355 A contraposition deduction...
con1bii 356 A contraposition inference...
con2bii 357 A contraposition inference...
con1b 358 Contraposition. Bidirecti...
con2b 359 Contraposition. Bidirecti...
biimt 360 A wff is equivalent to its...
pm5.5 361 Theorem *5.5 of [Whitehead...
a1bi 362 Inference introducing a th...
mt2bi 363 A false consequent falsifi...
mtt 364 Modus-tollens-like theorem...
imnot 365 If a proposition is false,...
pm5.501 366 Theorem *5.501 of [Whitehe...
ibib 367 Implication in terms of im...
ibibr 368 Implication in terms of im...
tbt 369 A wff is equivalent to its...
nbn2 370 The negation of a wff is e...
bibif 371 Transfer negation via an e...
nbn 372 The negation of a wff is e...
nbn3 373 Transfer falsehood via equ...
pm5.21im 374 Two propositions are equiv...
2false 375 Two falsehoods are equival...
2falsed 376 Two falsehoods are equival...
pm5.21ni 377 Two propositions implying ...
pm5.21nii 378 Eliminate an antecedent im...
pm5.21ndd 379 Eliminate an antecedent im...
bija 380 Combine antecedents into a...
pm5.18 381 Theorem *5.18 of [Whitehea...
xor3 382 Two ways to express "exclu...
nbbn 383 Move negation outside of b...
biass 384 Associative law for the bi...
biluk 385 Lukasiewicz's shortest axi...
pm5.19 386 Theorem *5.19 of [Whitehea...
bi2.04 387 Logical equivalence of com...
pm5.4 388 Antecedent absorption impl...
imdi 389 Distributive law for impli...
pm5.41 390 Theorem *5.41 of [Whitehea...
imbibi 391 The antecedent of one side...
pm4.8 392 Theorem *4.8 of [Whitehead...
pm4.81 393 A formula is equivalent to...
imim21b 394 Simplify an implication be...
pm4.63 397 Theorem *4.63 of [Whitehea...
pm4.67 398 Theorem *4.67 of [Whitehea...
imnan 399 Express an implication in ...
imnani 400 Infer an implication from ...
iman 401 Implication in terms of co...
pm3.24 402 Law of noncontradiction. ...
annim 403 Express a conjunction in t...
pm4.61 404 Theorem *4.61 of [Whitehea...
pm4.65 405 Theorem *4.65 of [Whitehea...
imp 406 Importation inference. (C...
impcom 407 Importation inference with...
con3dimp 408 Variant of ~ con3d with im...
mpnanrd 409 Eliminate the right side o...
impd 410 Importation deduction. (C...
impcomd 411 Importation deduction with...
ex 412 Exportation inference. (T...
expcom 413 Exportation inference with...
expdcom 414 Commuted form of ~ expd . ...
expd 415 Exportation deduction. (C...
expcomd 416 Deduction form of ~ expcom...
imp31 417 An importation inference. ...
imp32 418 An importation inference. ...
exp31 419 An exportation inference. ...
exp32 420 An exportation inference. ...
imp4b 421 An importation inference. ...
imp4a 422 An importation inference. ...
imp4c 423 An importation inference. ...
imp4d 424 An importation inference. ...
imp41 425 An importation inference. ...
imp42 426 An importation inference. ...
imp43 427 An importation inference. ...
imp44 428 An importation inference. ...
imp45 429 An importation inference. ...
exp4b 430 An exportation inference. ...
exp4a 431 An exportation inference. ...
exp4c 432 An exportation inference. ...
exp4d 433 An exportation inference. ...
exp41 434 An exportation inference. ...
exp42 435 An exportation inference. ...
exp43 436 An exportation inference. ...
exp44 437 An exportation inference. ...
exp45 438 An exportation inference. ...
imp5d 439 An importation inference. ...
imp5a 440 An importation inference. ...
imp5g 441 An importation inference. ...
imp55 442 An importation inference. ...
imp511 443 An importation inference. ...
exp5c 444 An exportation inference. ...
exp5j 445 An exportation inference. ...
exp5l 446 An exportation inference. ...
exp53 447 An exportation inference. ...
pm3.3 448 Theorem *3.3 (Exp) of [Whi...
pm3.31 449 Theorem *3.31 (Imp) of [Wh...
impexp 450 Import-export theorem. Pa...
impancom 451 Mixed importation/commutat...
expdimp 452 A deduction version of exp...
expimpd 453 Exportation followed by a ...
impr 454 Import a wff into a right ...
impl 455 Export a wff from a left c...
expr 456 Export a wff from a right ...
expl 457 Export a wff from a left c...
ancoms 458 Inference commuting conjun...
pm3.22 459 Theorem *3.22 of [Whitehea...
ancom 460 Commutative law for conjun...
ancomd 461 Commutation of conjuncts i...
biancomi 462 Commuting conjunction in a...
biancomd 463 Commuting conjunction in a...
ancomst 464 Closed form of ~ ancoms . ...
ancomsd 465 Deduction commuting conjun...
anasss 466 Associative law for conjun...
anassrs 467 Associative law for conjun...
anass 468 Associative law for conjun...
pm3.2 469 Join antecedents with conj...
pm3.2i 470 Infer conjunction of premi...
pm3.21 471 Join antecedents with conj...
pm3.43i 472 Nested conjunction of ante...
pm3.43 473 Theorem *3.43 (Comp) of [W...
dfbi2 474 A theorem similar to the s...
dfbi 475 Definition ~ df-bi rewritt...
biimpa 476 Importation inference from...
biimpar 477 Importation inference from...
biimpac 478 Importation inference from...
biimparc 479 Importation inference from...
adantr 480 Inference adding a conjunc...
adantl 481 Inference adding a conjunc...
simpl 482 Elimination of a conjunct....
simpli 483 Inference eliminating a co...
simpr 484 Elimination of a conjunct....
simpri 485 Inference eliminating a co...
intnan 486 Introduction of conjunct i...
intnanr 487 Introduction of conjunct i...
intnand 488 Introduction of conjunct i...
intnanrd 489 Introduction of conjunct i...
adantld 490 Deduction adding a conjunc...
adantrd 491 Deduction adding a conjunc...
pm3.41 492 Theorem *3.41 of [Whitehea...
pm3.42 493 Theorem *3.42 of [Whitehea...
simpld 494 Deduction eliminating a co...
simprd 495 Deduction eliminating a co...
simprbi 496 Deduction eliminating a co...
simplbi 497 Deduction eliminating a co...
simprbda 498 Deduction eliminating a co...
simplbda 499 Deduction eliminating a co...
simplbi2 500 Deduction eliminating a co...
simplbi2comt 501 Closed form of ~ simplbi2c...
simplbi2com 502 A deduction eliminating a ...
simpl2im 503 Implication from an elimin...
simplbiim 504 Implication from an elimin...
impel 505 An inference for implicati...
mpan9 506 Modus ponens conjoining di...
sylan9 507 Nested syllogism inference...
sylan9r 508 Nested syllogism inference...
sylan9bb 509 Nested syllogism inference...
sylan9bbr 510 Nested syllogism inference...
jca 511 Deduce conjunction of the ...
jcad 512 Deduction conjoining the c...
jca2 513 Inference conjoining the c...
jca31 514 Join three consequents. (...
jca32 515 Join three consequents. (...
jcai 516 Deduction replacing implic...
jcab 517 Distributive law for impli...
pm4.76 518 Theorem *4.76 of [Whitehea...
jctil 519 Inference conjoining a the...
jctir 520 Inference conjoining a the...
jccir 521 Inference conjoining a con...
jccil 522 Inference conjoining a con...
jctl 523 Inference conjoining a the...
jctr 524 Inference conjoining a the...
jctild 525 Deduction conjoining a the...
jctird 526 Deduction conjoining a the...
iba 527 Introduction of antecedent...
ibar 528 Introduction of antecedent...
biantru 529 A wff is equivalent to its...
biantrur 530 A wff is equivalent to its...
biantrud 531 A wff is equivalent to its...
biantrurd 532 A wff is equivalent to its...
bianfi 533 A wff conjoined with false...
bianfd 534 A wff conjoined with false...
baib 535 Move conjunction outside o...
baibr 536 Move conjunction outside o...
rbaibr 537 Move conjunction outside o...
rbaib 538 Move conjunction outside o...
baibd 539 Move conjunction outside o...
rbaibd 540 Move conjunction outside o...
bianabs 541 Absorb a hypothesis into t...
pm5.44 542 Theorem *5.44 of [Whitehea...
pm5.42 543 Theorem *5.42 of [Whitehea...
ancl 544 Conjoin antecedent to left...
anclb 545 Conjoin antecedent to left...
ancr 546 Conjoin antecedent to righ...
ancrb 547 Conjoin antecedent to righ...
ancli 548 Deduction conjoining antec...
ancri 549 Deduction conjoining antec...
ancld 550 Deduction conjoining antec...
ancrd 551 Deduction conjoining antec...
impac 552 Importation with conjuncti...
anc2l 553 Conjoin antecedent to left...
anc2r 554 Conjoin antecedent to righ...
anc2li 555 Deduction conjoining antec...
anc2ri 556 Deduction conjoining antec...
pm4.71 557 Implication in terms of bi...
pm4.71r 558 Implication in terms of bi...
pm4.71i 559 Inference converting an im...
pm4.71ri 560 Inference converting an im...
pm4.71d 561 Deduction converting an im...
pm4.71rd 562 Deduction converting an im...
pm4.24 563 Theorem *4.24 of [Whitehea...
anidm 564 Idempotent law for conjunc...
anidmdbi 565 Conjunction idempotence wi...
anidms 566 Inference from idempotent ...
imdistan 567 Distribution of implicatio...
imdistani 568 Distribution of implicatio...
imdistanri 569 Distribution of implicatio...
imdistand 570 Distribution of implicatio...
imdistanda 571 Distribution of implicatio...
pm5.3 572 Theorem *5.3 of [Whitehead...
pm5.32 573 Distribution of implicatio...
pm5.32i 574 Distribution of implicatio...
pm5.32ri 575 Distribution of implicatio...
pm5.32d 576 Distribution of implicatio...
pm5.32rd 577 Distribution of implicatio...
pm5.32da 578 Distribution of implicatio...
sylan 579 A syllogism inference. (C...
sylanb 580 A syllogism inference. (C...
sylanbr 581 A syllogism inference. (C...
sylanbrc 582 Syllogism inference. (Con...
syl2anc 583 Syllogism inference combin...
syl2anc2 584 Double syllogism inference...
sylancl 585 Syllogism inference combin...
sylancr 586 Syllogism inference combin...
sylancom 587 Syllogism inference with c...
sylanblc 588 Syllogism inference combin...
sylanblrc 589 Syllogism inference combin...
syldan 590 A syllogism deduction with...
sylbida 591 A syllogism deduction. (C...
sylan2 592 A syllogism inference. (C...
sylan2b 593 A syllogism inference. (C...
sylan2br 594 A syllogism inference. (C...
syl2an 595 A double syllogism inferen...
syl2anr 596 A double syllogism inferen...
syl2anb 597 A double syllogism inferen...
syl2anbr 598 A double syllogism inferen...
sylancb 599 A syllogism inference comb...
sylancbr 600 A syllogism inference comb...
syldanl 601 A syllogism deduction with...
syland 602 A syllogism deduction. (C...
sylani 603 A syllogism inference. (C...
sylan2d 604 A syllogism deduction. (C...
sylan2i 605 A syllogism inference. (C...
syl2ani 606 A syllogism inference. (C...
syl2and 607 A syllogism deduction. (C...
anim12d 608 Conjoin antecedents and co...
anim12d1 609 Variant of ~ anim12d where...
anim1d 610 Add a conjunct to right of...
anim2d 611 Add a conjunct to left of ...
anim12i 612 Conjoin antecedents and co...
anim12ci 613 Variant of ~ anim12i with ...
anim1i 614 Introduce conjunct to both...
anim1ci 615 Introduce conjunct to both...
anim2i 616 Introduce conjunct to both...
anim12ii 617 Conjoin antecedents and co...
anim12dan 618 Conjoin antecedents and co...
im2anan9 619 Deduction joining nested i...
im2anan9r 620 Deduction joining nested i...
pm3.45 621 Theorem *3.45 (Fact) of [W...
anbi2i 622 Introduce a left conjunct ...
anbi1i 623 Introduce a right conjunct...
anbi2ci 624 Variant of ~ anbi2i with c...
anbi1ci 625 Variant of ~ anbi1i with c...
bianbi 626 Exchanging conjunction in ...
anbi12i 627 Conjoin both sides of two ...
anbi12ci 628 Variant of ~ anbi12i with ...
anbi2d 629 Deduction adding a left co...
anbi1d 630 Deduction adding a right c...
anbi12d 631 Deduction joining two equi...
anbi1 632 Introduce a right conjunct...
anbi2 633 Introduce a left conjunct ...
anbi1cd 634 Introduce a proposition as...
an2anr 635 Double commutation in conj...
pm4.38 636 Theorem *4.38 of [Whitehea...
bi2anan9 637 Deduction joining two equi...
bi2anan9r 638 Deduction joining two equi...
bi2bian9 639 Deduction joining two bico...
anbiim 640 Adding biconditional when ...
bianass 641 An inference to merge two ...
bianassc 642 An inference to merge two ...
an21 643 Swap two conjuncts. (Cont...
an12 644 Swap two conjuncts. Note ...
an32 645 A rearrangement of conjunc...
an13 646 A rearrangement of conjunc...
an31 647 A rearrangement of conjunc...
an12s 648 Swap two conjuncts in ante...
ancom2s 649 Inference commuting a nest...
an13s 650 Swap two conjuncts in ante...
an32s 651 Swap two conjuncts in ante...
ancom1s 652 Inference commuting a nest...
an31s 653 Swap two conjuncts in ante...
anass1rs 654 Commutative-associative la...
an4 655 Rearrangement of 4 conjunc...
an42 656 Rearrangement of 4 conjunc...
an43 657 Rearrangement of 4 conjunc...
an3 658 A rearrangement of conjunc...
an4s 659 Inference rearranging 4 co...
an42s 660 Inference rearranging 4 co...
anabs1 661 Absorption into embedded c...
anabs5 662 Absorption into embedded c...
anabs7 663 Absorption into embedded c...
anabsan 664 Absorption of antecedent w...
anabss1 665 Absorption of antecedent i...
anabss4 666 Absorption of antecedent i...
anabss5 667 Absorption of antecedent i...
anabsi5 668 Absorption of antecedent i...
anabsi6 669 Absorption of antecedent i...
anabsi7 670 Absorption of antecedent i...
anabsi8 671 Absorption of antecedent i...
anabss7 672 Absorption of antecedent i...
anabsan2 673 Absorption of antecedent w...
anabss3 674 Absorption of antecedent i...
anandi 675 Distribution of conjunctio...
anandir 676 Distribution of conjunctio...
anandis 677 Inference that undistribut...
anandirs 678 Inference that undistribut...
sylanl1 679 A syllogism inference. (C...
sylanl2 680 A syllogism inference. (C...
sylanr1 681 A syllogism inference. (C...
sylanr2 682 A syllogism inference. (C...
syl6an 683 A syllogism deduction comb...
syl2an2r 684 ~ syl2anr with antecedents...
syl2an2 685 ~ syl2an with antecedents ...
mpdan 686 An inference based on modu...
mpancom 687 An inference based on modu...
mpidan 688 A deduction which "stacks"...
mpan 689 An inference based on modu...
mpan2 690 An inference based on modu...
mp2an 691 An inference based on modu...
mp4an 692 An inference based on modu...
mpan2d 693 A deduction based on modus...
mpand 694 A deduction based on modus...
mpani 695 An inference based on modu...
mpan2i 696 An inference based on modu...
mp2ani 697 An inference based on modu...
mp2and 698 A deduction based on modus...
mpanl1 699 An inference based on modu...
mpanl2 700 An inference based on modu...
mpanl12 701 An inference based on modu...
mpanr1 702 An inference based on modu...
mpanr2 703 An inference based on modu...
mpanr12 704 An inference based on modu...
mpanlr1 705 An inference based on modu...
mpbirand 706 Detach truth from conjunct...
mpbiran2d 707 Detach truth from conjunct...
mpbiran 708 Detach truth from conjunct...
mpbiran2 709 Detach truth from conjunct...
mpbir2an 710 Detach a conjunction of tr...
mpbi2and 711 Detach a conjunction of tr...
mpbir2and 712 Detach a conjunction of tr...
adantll 713 Deduction adding a conjunc...
adantlr 714 Deduction adding a conjunc...
adantrl 715 Deduction adding a conjunc...
adantrr 716 Deduction adding a conjunc...
adantlll 717 Deduction adding a conjunc...
adantllr 718 Deduction adding a conjunc...
adantlrl 719 Deduction adding a conjunc...
adantlrr 720 Deduction adding a conjunc...
adantrll 721 Deduction adding a conjunc...
adantrlr 722 Deduction adding a conjunc...
adantrrl 723 Deduction adding a conjunc...
adantrrr 724 Deduction adding a conjunc...
ad2antrr 725 Deduction adding two conju...
ad2antlr 726 Deduction adding two conju...
ad2antrl 727 Deduction adding two conju...
ad2antll 728 Deduction adding conjuncts...
ad3antrrr 729 Deduction adding three con...
ad3antlr 730 Deduction adding three con...
ad4antr 731 Deduction adding 4 conjunc...
ad4antlr 732 Deduction adding 4 conjunc...
ad5antr 733 Deduction adding 5 conjunc...
ad5antlr 734 Deduction adding 5 conjunc...
ad6antr 735 Deduction adding 6 conjunc...
ad6antlr 736 Deduction adding 6 conjunc...
ad7antr 737 Deduction adding 7 conjunc...
ad7antlr 738 Deduction adding 7 conjunc...
ad8antr 739 Deduction adding 8 conjunc...
ad8antlr 740 Deduction adding 8 conjunc...
ad9antr 741 Deduction adding 9 conjunc...
ad9antlr 742 Deduction adding 9 conjunc...
ad10antr 743 Deduction adding 10 conjun...
ad10antlr 744 Deduction adding 10 conjun...
ad2ant2l 745 Deduction adding two conju...
ad2ant2r 746 Deduction adding two conju...
ad2ant2lr 747 Deduction adding two conju...
ad2ant2rl 748 Deduction adding two conju...
adantl3r 749 Deduction adding 1 conjunc...
ad4ant13 750 Deduction adding conjuncts...
ad4ant14 751 Deduction adding conjuncts...
ad4ant23 752 Deduction adding conjuncts...
ad4ant24 753 Deduction adding conjuncts...
adantl4r 754 Deduction adding 1 conjunc...
ad5ant12 755 Deduction adding conjuncts...
ad5ant13 756 Deduction adding conjuncts...
ad5ant14 757 Deduction adding conjuncts...
ad5ant15 758 Deduction adding conjuncts...
ad5ant23 759 Deduction adding conjuncts...
ad5ant24 760 Deduction adding conjuncts...
ad5ant25 761 Deduction adding conjuncts...
adantl5r 762 Deduction adding 1 conjunc...
adantl6r 763 Deduction adding 1 conjunc...
pm3.33 764 Theorem *3.33 (Syll) of [W...
pm3.34 765 Theorem *3.34 (Syll) of [W...
simpll 766 Simplification of a conjun...
simplld 767 Deduction form of ~ simpll...
simplr 768 Simplification of a conjun...
simplrd 769 Deduction eliminating a do...
simprl 770 Simplification of a conjun...
simprld 771 Deduction eliminating a do...
simprr 772 Simplification of a conjun...
simprrd 773 Deduction form of ~ simprr...
simplll 774 Simplification of a conjun...
simpllr 775 Simplification of a conjun...
simplrl 776 Simplification of a conjun...
simplrr 777 Simplification of a conjun...
simprll 778 Simplification of a conjun...
simprlr 779 Simplification of a conjun...
simprrl 780 Simplification of a conjun...
simprrr 781 Simplification of a conjun...
simp-4l 782 Simplification of a conjun...
simp-4r 783 Simplification of a conjun...
simp-5l 784 Simplification of a conjun...
simp-5r 785 Simplification of a conjun...
simp-6l 786 Simplification of a conjun...
simp-6r 787 Simplification of a conjun...
simp-7l 788 Simplification of a conjun...
simp-7r 789 Simplification of a conjun...
simp-8l 790 Simplification of a conjun...
simp-8r 791 Simplification of a conjun...
simp-9l 792 Simplification of a conjun...
simp-9r 793 Simplification of a conjun...
simp-10l 794 Simplification of a conjun...
simp-10r 795 Simplification of a conjun...
simp-11l 796 Simplification of a conjun...
simp-11r 797 Simplification of a conjun...
pm2.01da 798 Deduction based on reducti...
pm2.18da 799 Deduction based on reducti...
impbida 800 Deduce an equivalence from...
pm5.21nd 801 Eliminate an antecedent im...
pm3.35 802 Conjunctive detachment. T...
pm5.74da 803 Distribution of implicatio...
bitr 804 Theorem *4.22 of [Whitehea...
biantr 805 A transitive law of equiva...
pm4.14 806 Theorem *4.14 of [Whitehea...
pm3.37 807 Theorem *3.37 (Transp) of ...
anim12 808 Conjoin antecedents and co...
pm3.4 809 Conjunction implies implic...
exbiri 810 Inference form of ~ exbir ...
pm2.61ian 811 Elimination of an antecede...
pm2.61dan 812 Elimination of an antecede...
pm2.61ddan 813 Elimination of two anteced...
pm2.61dda 814 Elimination of two anteced...
mtand 815 A modus tollens deduction....
pm2.65da 816 Deduction for proof by con...
condan 817 Proof by contradiction. (...
biadan 818 An implication is equivale...
biadani 819 Inference associated with ...
biadaniALT 820 Alternate proof of ~ biada...
biadanii 821 Inference associated with ...
biadanid 822 Deduction associated with ...
pm5.1 823 Two propositions are equiv...
pm5.21 824 Two propositions are equiv...
pm5.35 825 Theorem *5.35 of [Whitehea...
abai 826 Introduce one conjunct as ...
pm4.45im 827 Conjunction with implicati...
impimprbi 828 An implication and its rev...
nan 829 Theorem to move a conjunct...
pm5.31 830 Theorem *5.31 of [Whitehea...
pm5.31r 831 Variant of ~ pm5.31 . (Co...
pm4.15 832 Theorem *4.15 of [Whitehea...
pm5.36 833 Theorem *5.36 of [Whitehea...
annotanannot 834 A conjunction with a negat...
pm5.33 835 Theorem *5.33 of [Whitehea...
syl12anc 836 Syllogism combined with co...
syl21anc 837 Syllogism combined with co...
syl22anc 838 Syllogism combined with co...
syl1111anc 839 Four-hypothesis eliminatio...
syldbl2 840 Stacked hypotheseis implie...
mpsyl4anc 841 An elimination deduction. ...
pm4.87 842 Theorem *4.87 of [Whitehea...
bimsc1 843 Removal of conjunct from o...
a2and 844 Deduction distributing a c...
animpimp2impd 845 Deduction deriving nested ...
pm4.64 848 Theorem *4.64 of [Whitehea...
pm4.66 849 Theorem *4.66 of [Whitehea...
pm2.53 850 Theorem *2.53 of [Whitehea...
pm2.54 851 Theorem *2.54 of [Whitehea...
imor 852 Implication in terms of di...
imori 853 Infer disjunction from imp...
imorri 854 Infer implication from dis...
pm4.62 855 Theorem *4.62 of [Whitehea...
jaoi 856 Inference disjoining the a...
jao1i 857 Add a disjunct in the ante...
jaod 858 Deduction disjoining the a...
mpjaod 859 Eliminate a disjunction in...
ori 860 Infer implication from dis...
orri 861 Infer disjunction from imp...
orrd 862 Deduce disjunction from im...
ord 863 Deduce implication from di...
orci 864 Deduction introducing a di...
olci 865 Deduction introducing a di...
orc 866 Introduction of a disjunct...
olc 867 Introduction of a disjunct...
pm1.4 868 Axiom *1.4 of [WhiteheadRu...
orcom 869 Commutative law for disjun...
orcomd 870 Commutation of disjuncts i...
orcoms 871 Commutation of disjuncts i...
orcd 872 Deduction introducing a di...
olcd 873 Deduction introducing a di...
orcs 874 Deduction eliminating disj...
olcs 875 Deduction eliminating disj...
olcnd 876 A lemma for Conjunctive No...
orcnd 877 A lemma for Conjunctive No...
mtord 878 A modus tollens deduction ...
pm3.2ni 879 Infer negated disjunction ...
pm2.45 880 Theorem *2.45 of [Whitehea...
pm2.46 881 Theorem *2.46 of [Whitehea...
pm2.47 882 Theorem *2.47 of [Whitehea...
pm2.48 883 Theorem *2.48 of [Whitehea...
pm2.49 884 Theorem *2.49 of [Whitehea...
norbi 885 If neither of two proposit...
nbior 886 If two propositions are no...
orel1 887 Elimination of disjunction...
pm2.25 888 Theorem *2.25 of [Whitehea...
orel2 889 Elimination of disjunction...
pm2.67-2 890 Slight generalization of T...
pm2.67 891 Theorem *2.67 of [Whitehea...
curryax 892 A non-intuitionistic posit...
exmid 893 Law of excluded middle, al...
exmidd 894 Law of excluded middle in ...
pm2.1 895 Theorem *2.1 of [Whitehead...
pm2.13 896 Theorem *2.13 of [Whitehea...
pm2.621 897 Theorem *2.621 of [Whitehe...
pm2.62 898 Theorem *2.62 of [Whitehea...
pm2.68 899 Theorem *2.68 of [Whitehea...
dfor2 900 Logical 'or' expressed in ...
pm2.07 901 Theorem *2.07 of [Whitehea...
pm1.2 902 Axiom *1.2 of [WhiteheadRu...
oridm 903 Idempotent law for disjunc...
pm4.25 904 Theorem *4.25 of [Whitehea...
pm2.4 905 Theorem *2.4 of [Whitehead...
pm2.41 906 Theorem *2.41 of [Whitehea...
orim12i 907 Disjoin antecedents and co...
orim1i 908 Introduce disjunct to both...
orim2i 909 Introduce disjunct to both...
orim12dALT 910 Alternate proof of ~ orim1...
orbi2i 911 Inference adding a left di...
orbi1i 912 Inference adding a right d...
orbi12i 913 Infer the disjunction of t...
orbi2d 914 Deduction adding a left di...
orbi1d 915 Deduction adding a right d...
orbi1 916 Theorem *4.37 of [Whitehea...
orbi12d 917 Deduction joining two equi...
pm1.5 918 Axiom *1.5 (Assoc) of [Whi...
or12 919 Swap two disjuncts. (Cont...
orass 920 Associative law for disjun...
pm2.31 921 Theorem *2.31 of [Whitehea...
pm2.32 922 Theorem *2.32 of [Whitehea...
pm2.3 923 Theorem *2.3 of [Whitehead...
or32 924 A rearrangement of disjunc...
or4 925 Rearrangement of 4 disjunc...
or42 926 Rearrangement of 4 disjunc...
orordi 927 Distribution of disjunctio...
orordir 928 Distribution of disjunctio...
orimdi 929 Disjunction distributes ov...
pm2.76 930 Theorem *2.76 of [Whitehea...
pm2.85 931 Theorem *2.85 of [Whitehea...
pm2.75 932 Theorem *2.75 of [Whitehea...
pm4.78 933 Implication distributes ov...
biort 934 A disjunction with a true ...
biorf 935 A wff is equivalent to its...
biortn 936 A wff is equivalent to its...
biorfi 937 The dual of ~ biorf is not...
biorfri 938 A wff is equivalent to its...
biorfriOLD 939 Obsolete proof of ~ biorfr...
pm2.26 940 Theorem *2.26 of [Whitehea...
pm2.63 941 Theorem *2.63 of [Whitehea...
pm2.64 942 Theorem *2.64 of [Whitehea...
pm2.42 943 Theorem *2.42 of [Whitehea...
pm5.11g 944 A general instance of Theo...
pm5.11 945 Theorem *5.11 of [Whitehea...
pm5.12 946 Theorem *5.12 of [Whitehea...
pm5.14 947 Theorem *5.14 of [Whitehea...
pm5.13 948 Theorem *5.13 of [Whitehea...
pm5.55 949 Theorem *5.55 of [Whitehea...
pm4.72 950 Implication in terms of bi...
imimorb 951 Simplify an implication be...
oibabs 952 Absorption of disjunction ...
orbidi 953 Disjunction distributes ov...
pm5.7 954 Disjunction distributes ov...
jaao 955 Inference conjoining and d...
jaoa 956 Inference disjoining and c...
jaoian 957 Inference disjoining the a...
jaodan 958 Deduction disjoining the a...
mpjaodan 959 Eliminate a disjunction in...
pm3.44 960 Theorem *3.44 of [Whitehea...
jao 961 Disjunction of antecedents...
jaob 962 Disjunction of antecedents...
pm4.77 963 Theorem *4.77 of [Whitehea...
pm3.48 964 Theorem *3.48 of [Whitehea...
orim12d 965 Disjoin antecedents and co...
orim1d 966 Disjoin antecedents and co...
orim2d 967 Disjoin antecedents and co...
orim2 968 Axiom *1.6 (Sum) of [White...
pm2.38 969 Theorem *2.38 of [Whitehea...
pm2.36 970 Theorem *2.36 of [Whitehea...
pm2.37 971 Theorem *2.37 of [Whitehea...
pm2.81 972 Theorem *2.81 of [Whitehea...
pm2.8 973 Theorem *2.8 of [Whitehead...
pm2.73 974 Theorem *2.73 of [Whitehea...
pm2.74 975 Theorem *2.74 of [Whitehea...
pm2.82 976 Theorem *2.82 of [Whitehea...
pm4.39 977 Theorem *4.39 of [Whitehea...
animorl 978 Conjunction implies disjun...
animorr 979 Conjunction implies disjun...
animorlr 980 Conjunction implies disjun...
animorrl 981 Conjunction implies disjun...
ianor 982 Negated conjunction in ter...
anor 983 Conjunction in terms of di...
ioran 984 Negated disjunction in ter...
pm4.52 985 Theorem *4.52 of [Whitehea...
pm4.53 986 Theorem *4.53 of [Whitehea...
pm4.54 987 Theorem *4.54 of [Whitehea...
pm4.55 988 Theorem *4.55 of [Whitehea...
pm4.56 989 Theorem *4.56 of [Whitehea...
oran 990 Disjunction in terms of co...
pm4.57 991 Theorem *4.57 of [Whitehea...
pm3.1 992 Theorem *3.1 of [Whitehead...
pm3.11 993 Theorem *3.11 of [Whitehea...
pm3.12 994 Theorem *3.12 of [Whitehea...
pm3.13 995 Theorem *3.13 of [Whitehea...
pm3.14 996 Theorem *3.14 of [Whitehea...
pm4.44 997 Theorem *4.44 of [Whitehea...
pm4.45 998 Theorem *4.45 of [Whitehea...
orabs 999 Absorption of redundant in...
oranabs 1000 Absorb a disjunct into a c...
pm5.61 1001 Theorem *5.61 of [Whitehea...
pm5.6 1002 Conjunction in antecedent ...
orcanai 1003 Change disjunction in cons...
pm4.79 1004 Theorem *4.79 of [Whitehea...
pm5.53 1005 Theorem *5.53 of [Whitehea...
ordi 1006 Distributive law for disju...
ordir 1007 Distributive law for disju...
andi 1008 Distributive law for conju...
andir 1009 Distributive law for conju...
orddi 1010 Double distributive law fo...
anddi 1011 Double distributive law fo...
pm5.17 1012 Theorem *5.17 of [Whitehea...
pm5.15 1013 Theorem *5.15 of [Whitehea...
pm5.16 1014 Theorem *5.16 of [Whitehea...
xor 1015 Two ways to express exclus...
nbi2 1016 Two ways to express "exclu...
xordi 1017 Conjunction distributes ov...
pm5.54 1018 Theorem *5.54 of [Whitehea...
pm5.62 1019 Theorem *5.62 of [Whitehea...
pm5.63 1020 Theorem *5.63 of [Whitehea...
niabn 1021 Miscellaneous inference re...
ninba 1022 Miscellaneous inference re...
pm4.43 1023 Theorem *4.43 of [Whitehea...
pm4.82 1024 Theorem *4.82 of [Whitehea...
pm4.83 1025 Theorem *4.83 of [Whitehea...
pclem6 1026 Negation inferred from emb...
bigolden 1027 Dijkstra-Scholten's Golden...
pm5.71 1028 Theorem *5.71 of [Whitehea...
pm5.75 1029 Theorem *5.75 of [Whitehea...
ecase2d 1030 Deduction for elimination ...
ecase2dOLD 1031 Obsolete version of ~ ecas...
ecase3 1032 Inference for elimination ...
ecase 1033 Inference for elimination ...
ecase3d 1034 Deduction for elimination ...
ecased 1035 Deduction for elimination ...
ecase3ad 1036 Deduction for elimination ...
ecase3adOLD 1037 Obsolete version of ~ ecas...
ccase 1038 Inference for combining ca...
ccased 1039 Deduction for combining ca...
ccase2 1040 Inference for combining ca...
4cases 1041 Inference eliminating two ...
4casesdan 1042 Deduction eliminating two ...
cases 1043 Case disjunction according...
dedlem0a 1044 Lemma for an alternate ver...
dedlem0b 1045 Lemma for an alternate ver...
dedlema 1046 Lemma for weak deduction t...
dedlemb 1047 Lemma for weak deduction t...
cases2 1048 Case disjunction according...
cases2ALT 1049 Alternate proof of ~ cases...
dfbi3 1050 An alternate definition of...
pm5.24 1051 Theorem *5.24 of [Whitehea...
4exmid 1052 The disjunction of the fou...
consensus 1053 The consensus theorem. Th...
pm4.42 1054 Theorem *4.42 of [Whitehea...
prlem1 1055 A specialized lemma for se...
prlem2 1056 A specialized lemma for se...
oplem1 1057 A specialized lemma for se...
dn1 1058 A single axiom for Boolean...
bianir 1059 A closed form of ~ mpbir ,...
jaoi2 1060 Inference removing a negat...
jaoi3 1061 Inference separating a dis...
ornld 1062 Selecting one statement fr...
dfifp2 1065 Alternate definition of th...
dfifp3 1066 Alternate definition of th...
dfifp4 1067 Alternate definition of th...
dfifp5 1068 Alternate definition of th...
dfifp6 1069 Alternate definition of th...
dfifp7 1070 Alternate definition of th...
ifpdfbi 1071 Define the biconditional a...
anifp 1072 The conditional operator i...
ifpor 1073 The conditional operator i...
ifpn 1074 Conditional operator for t...
ifptru 1075 Value of the conditional o...
ifpfal 1076 Value of the conditional o...
ifpid 1077 Value of the conditional o...
casesifp 1078 Version of ~ cases express...
ifpbi123d 1079 Equivalence deduction for ...
ifpbi23d 1080 Equivalence deduction for ...
ifpimpda 1081 Separation of the values o...
1fpid3 1082 The value of the condition...
elimh 1083 Hypothesis builder for the...
dedt 1084 The weak deduction theorem...
con3ALT 1085 Proof of ~ con3 from its a...
3orass 1090 Associative law for triple...
3orel1 1091 Partial elimination of a t...
3orrot 1092 Rotation law for triple di...
3orcoma 1093 Commutation law for triple...
3orcomb 1094 Commutation law for triple...
3anass 1095 Associative law for triple...
3anan12 1096 Convert triple conjunction...
3anan32 1097 Convert triple conjunction...
3ancoma 1098 Commutation law for triple...
3ancomb 1099 Commutation law for triple...
3anrot 1100 Rotation law for triple co...
3anrev 1101 Reversal law for triple co...
anandi3 1102 Distribution of triple con...
anandi3r 1103 Distribution of triple con...
3anidm 1104 Idempotent law for conjunc...
3an4anass 1105 Associative law for four c...
3ioran 1106 Negated triple disjunction...
3ianor 1107 Negated triple conjunction...
3anor 1108 Triple conjunction express...
3oran 1109 Triple disjunction in term...
3impa 1110 Importation from double to...
3imp 1111 Importation inference. (C...
3imp31 1112 The importation inference ...
3imp231 1113 Importation inference. (C...
3imp21 1114 The importation inference ...
3impb 1115 Importation from double to...
3impib 1116 Importation to triple conj...
3impia 1117 Importation to triple conj...
3expa 1118 Exportation from triple to...
3exp 1119 Exportation inference. (C...
3expb 1120 Exportation from triple to...
3expia 1121 Exportation from triple co...
3expib 1122 Exportation from triple co...
3com12 1123 Commutation in antecedent....
3com13 1124 Commutation in antecedent....
3comr 1125 Commutation in antecedent....
3com23 1126 Commutation in antecedent....
3coml 1127 Commutation in antecedent....
3jca 1128 Join consequents with conj...
3jcad 1129 Deduction conjoining the c...
3adant1 1130 Deduction adding a conjunc...
3adant2 1131 Deduction adding a conjunc...
3adant3 1132 Deduction adding a conjunc...
3ad2ant1 1133 Deduction adding conjuncts...
3ad2ant2 1134 Deduction adding conjuncts...
3ad2ant3 1135 Deduction adding conjuncts...
simp1 1136 Simplification of triple c...
simp2 1137 Simplification of triple c...
simp3 1138 Simplification of triple c...
simp1i 1139 Infer a conjunct from a tr...
simp2i 1140 Infer a conjunct from a tr...
simp3i 1141 Infer a conjunct from a tr...
simp1d 1142 Deduce a conjunct from a t...
simp2d 1143 Deduce a conjunct from a t...
simp3d 1144 Deduce a conjunct from a t...
simp1bi 1145 Deduce a conjunct from a t...
simp2bi 1146 Deduce a conjunct from a t...
simp3bi 1147 Deduce a conjunct from a t...
3simpa 1148 Simplification of triple c...
3simpb 1149 Simplification of triple c...
3simpc 1150 Simplification of triple c...
3anim123i 1151 Join antecedents and conse...
3anim1i 1152 Add two conjuncts to antec...
3anim2i 1153 Add two conjuncts to antec...
3anim3i 1154 Add two conjuncts to antec...
3anbi123i 1155 Join 3 biconditionals with...
3orbi123i 1156 Join 3 biconditionals with...
3anbi1i 1157 Inference adding two conju...
3anbi2i 1158 Inference adding two conju...
3anbi3i 1159 Inference adding two conju...
syl3an 1160 A triple syllogism inferen...
syl3anb 1161 A triple syllogism inferen...
syl3anbr 1162 A triple syllogism inferen...
syl3an1 1163 A syllogism inference. (C...
syl3an2 1164 A syllogism inference. (C...
syl3an3 1165 A syllogism inference. (C...
3adantl1 1166 Deduction adding a conjunc...
3adantl2 1167 Deduction adding a conjunc...
3adantl3 1168 Deduction adding a conjunc...
3adantr1 1169 Deduction adding a conjunc...
3adantr2 1170 Deduction adding a conjunc...
3adantr3 1171 Deduction adding a conjunc...
ad4ant123 1172 Deduction adding conjuncts...
ad4ant124 1173 Deduction adding conjuncts...
ad4ant134 1174 Deduction adding conjuncts...
ad4ant234 1175 Deduction adding conjuncts...
3adant1l 1176 Deduction adding a conjunc...
3adant1r 1177 Deduction adding a conjunc...
3adant2l 1178 Deduction adding a conjunc...
3adant2r 1179 Deduction adding a conjunc...
3adant3l 1180 Deduction adding a conjunc...
3adant3r 1181 Deduction adding a conjunc...
3adant3r1 1182 Deduction adding a conjunc...
3adant3r2 1183 Deduction adding a conjunc...
3adant3r3 1184 Deduction adding a conjunc...
3ad2antl1 1185 Deduction adding conjuncts...
3ad2antl2 1186 Deduction adding conjuncts...
3ad2antl3 1187 Deduction adding conjuncts...
3ad2antr1 1188 Deduction adding conjuncts...
3ad2antr2 1189 Deduction adding conjuncts...
3ad2antr3 1190 Deduction adding conjuncts...
simpl1 1191 Simplification of conjunct...
simpl2 1192 Simplification of conjunct...
simpl3 1193 Simplification of conjunct...
simpr1 1194 Simplification of conjunct...
simpr2 1195 Simplification of conjunct...
simpr3 1196 Simplification of conjunct...
simp1l 1197 Simplification of triple c...
simp1r 1198 Simplification of triple c...
simp2l 1199 Simplification of triple c...
simp2r 1200 Simplification of triple c...
simp3l 1201 Simplification of triple c...
simp3r 1202 Simplification of triple c...
simp11 1203 Simplification of doubly t...
simp12 1204 Simplification of doubly t...
simp13 1205 Simplification of doubly t...
simp21 1206 Simplification of doubly t...
simp22 1207 Simplification of doubly t...
simp23 1208 Simplification of doubly t...
simp31 1209 Simplification of doubly t...
simp32 1210 Simplification of doubly t...
simp33 1211 Simplification of doubly t...
simpll1 1212 Simplification of conjunct...
simpll2 1213 Simplification of conjunct...
simpll3 1214 Simplification of conjunct...
simplr1 1215 Simplification of conjunct...
simplr2 1216 Simplification of conjunct...
simplr3 1217 Simplification of conjunct...
simprl1 1218 Simplification of conjunct...
simprl2 1219 Simplification of conjunct...
simprl3 1220 Simplification of conjunct...
simprr1 1221 Simplification of conjunct...
simprr2 1222 Simplification of conjunct...
simprr3 1223 Simplification of conjunct...
simpl1l 1224 Simplification of conjunct...
simpl1r 1225 Simplification of conjunct...
simpl2l 1226 Simplification of conjunct...
simpl2r 1227 Simplification of conjunct...
simpl3l 1228 Simplification of conjunct...
simpl3r 1229 Simplification of conjunct...
simpr1l 1230 Simplification of conjunct...
simpr1r 1231 Simplification of conjunct...
simpr2l 1232 Simplification of conjunct...
simpr2r 1233 Simplification of conjunct...
simpr3l 1234 Simplification of conjunct...
simpr3r 1235 Simplification of conjunct...
simp1ll 1236 Simplification of conjunct...
simp1lr 1237 Simplification of conjunct...
simp1rl 1238 Simplification of conjunct...
simp1rr 1239 Simplification of conjunct...
simp2ll 1240 Simplification of conjunct...
simp2lr 1241 Simplification of conjunct...
simp2rl 1242 Simplification of conjunct...
simp2rr 1243 Simplification of conjunct...
simp3ll 1244 Simplification of conjunct...
simp3lr 1245 Simplification of conjunct...
simp3rl 1246 Simplification of conjunct...
simp3rr 1247 Simplification of conjunct...
simpl11 1248 Simplification of conjunct...
simpl12 1249 Simplification of conjunct...
simpl13 1250 Simplification of conjunct...
simpl21 1251 Simplification of conjunct...
simpl22 1252 Simplification of conjunct...
simpl23 1253 Simplification of conjunct...
simpl31 1254 Simplification of conjunct...
simpl32 1255 Simplification of conjunct...
simpl33 1256 Simplification of conjunct...
simpr11 1257 Simplification of conjunct...
simpr12 1258 Simplification of conjunct...
simpr13 1259 Simplification of conjunct...
simpr21 1260 Simplification of conjunct...
simpr22 1261 Simplification of conjunct...
simpr23 1262 Simplification of conjunct...
simpr31 1263 Simplification of conjunct...
simpr32 1264 Simplification of conjunct...
simpr33 1265 Simplification of conjunct...
simp1l1 1266 Simplification of conjunct...
simp1l2 1267 Simplification of conjunct...
simp1l3 1268 Simplification of conjunct...
simp1r1 1269 Simplification of conjunct...
simp1r2 1270 Simplification of conjunct...
simp1r3 1271 Simplification of conjunct...
simp2l1 1272 Simplification of conjunct...
simp2l2 1273 Simplification of conjunct...
simp2l3 1274 Simplification of conjunct...
simp2r1 1275 Simplification of conjunct...
simp2r2 1276 Simplification of conjunct...
simp2r3 1277 Simplification of conjunct...
simp3l1 1278 Simplification of conjunct...
simp3l2 1279 Simplification of conjunct...
simp3l3 1280 Simplification of conjunct...
simp3r1 1281 Simplification of conjunct...
simp3r2 1282 Simplification of conjunct...
simp3r3 1283 Simplification of conjunct...
simp11l 1284 Simplification of conjunct...
simp11r 1285 Simplification of conjunct...
simp12l 1286 Simplification of conjunct...
simp12r 1287 Simplification of conjunct...
simp13l 1288 Simplification of conjunct...
simp13r 1289 Simplification of conjunct...
simp21l 1290 Simplification of conjunct...
simp21r 1291 Simplification of conjunct...
simp22l 1292 Simplification of conjunct...
simp22r 1293 Simplification of conjunct...
simp23l 1294 Simplification of conjunct...
simp23r 1295 Simplification of conjunct...
simp31l 1296 Simplification of conjunct...
simp31r 1297 Simplification of conjunct...
simp32l 1298 Simplification of conjunct...
simp32r 1299 Simplification of conjunct...
simp33l 1300 Simplification of conjunct...
simp33r 1301 Simplification of conjunct...
simp111 1302 Simplification of conjunct...
simp112 1303 Simplification of conjunct...
simp113 1304 Simplification of conjunct...
simp121 1305 Simplification of conjunct...
simp122 1306 Simplification of conjunct...
simp123 1307 Simplification of conjunct...
simp131 1308 Simplification of conjunct...
simp132 1309 Simplification of conjunct...
simp133 1310 Simplification of conjunct...
simp211 1311 Simplification of conjunct...
simp212 1312 Simplification of conjunct...
simp213 1313 Simplification of conjunct...
simp221 1314 Simplification of conjunct...
simp222 1315 Simplification of conjunct...
simp223 1316 Simplification of conjunct...
simp231 1317 Simplification of conjunct...
simp232 1318 Simplification of conjunct...
simp233 1319 Simplification of conjunct...
simp311 1320 Simplification of conjunct...
simp312 1321 Simplification of conjunct...
simp313 1322 Simplification of conjunct...
simp321 1323 Simplification of conjunct...
simp322 1324 Simplification of conjunct...
simp323 1325 Simplification of conjunct...
simp331 1326 Simplification of conjunct...
simp332 1327 Simplification of conjunct...
simp333 1328 Simplification of conjunct...
3anibar 1329 Remove a hypothesis from t...
3mix1 1330 Introduction in triple dis...
3mix2 1331 Introduction in triple dis...
3mix3 1332 Introduction in triple dis...
3mix1i 1333 Introduction in triple dis...
3mix2i 1334 Introduction in triple dis...
3mix3i 1335 Introduction in triple dis...
3mix1d 1336 Deduction introducing trip...
3mix2d 1337 Deduction introducing trip...
3mix3d 1338 Deduction introducing trip...
3pm3.2i 1339 Infer conjunction of premi...
pm3.2an3 1340 Version of ~ pm3.2 for a t...
mpbir3an 1341 Detach a conjunction of tr...
mpbir3and 1342 Detach a conjunction of tr...
syl3anbrc 1343 Syllogism inference. (Con...
syl21anbrc 1344 Syllogism inference. (Con...
3imp3i2an 1345 An elimination deduction. ...
ex3 1346 Apply ~ ex to a hypothesis...
3imp1 1347 Importation to left triple...
3impd 1348 Importation deduction for ...
3imp2 1349 Importation to right tripl...
3impdi 1350 Importation inference (und...
3impdir 1351 Importation inference (und...
3exp1 1352 Exportation from left trip...
3expd 1353 Exportation deduction for ...
3exp2 1354 Exportation from right tri...
exp5o 1355 A triple exportation infer...
exp516 1356 A triple exportation infer...
exp520 1357 A triple exportation infer...
3impexp 1358 Version of ~ impexp for a ...
3an1rs 1359 Swap conjuncts. (Contribu...
3anassrs 1360 Associative law for conjun...
ad5ant245 1361 Deduction adding conjuncts...
ad5ant234 1362 Deduction adding conjuncts...
ad5ant235 1363 Deduction adding conjuncts...
ad5ant123 1364 Deduction adding conjuncts...
ad5ant124 1365 Deduction adding conjuncts...
ad5ant125 1366 Deduction adding conjuncts...
ad5ant134 1367 Deduction adding conjuncts...
ad5ant135 1368 Deduction adding conjuncts...
ad5ant145 1369 Deduction adding conjuncts...
ad5ant2345 1370 Deduction adding conjuncts...
syl3anc 1371 Syllogism combined with co...
syl13anc 1372 Syllogism combined with co...
syl31anc 1373 Syllogism combined with co...
syl112anc 1374 Syllogism combined with co...
syl121anc 1375 Syllogism combined with co...
syl211anc 1376 Syllogism combined with co...
syl23anc 1377 Syllogism combined with co...
syl32anc 1378 Syllogism combined with co...
syl122anc 1379 Syllogism combined with co...
syl212anc 1380 Syllogism combined with co...
syl221anc 1381 Syllogism combined with co...
syl113anc 1382 Syllogism combined with co...
syl131anc 1383 Syllogism combined with co...
syl311anc 1384 Syllogism combined with co...
syl33anc 1385 Syllogism combined with co...
syl222anc 1386 Syllogism combined with co...
syl123anc 1387 Syllogism combined with co...
syl132anc 1388 Syllogism combined with co...
syl213anc 1389 Syllogism combined with co...
syl231anc 1390 Syllogism combined with co...
syl312anc 1391 Syllogism combined with co...
syl321anc 1392 Syllogism combined with co...
syl133anc 1393 Syllogism combined with co...
syl313anc 1394 Syllogism combined with co...
syl331anc 1395 Syllogism combined with co...
syl223anc 1396 Syllogism combined with co...
syl232anc 1397 Syllogism combined with co...
syl322anc 1398 Syllogism combined with co...
syl233anc 1399 Syllogism combined with co...
syl323anc 1400 Syllogism combined with co...
syl332anc 1401 Syllogism combined with co...
syl333anc 1402 A syllogism inference comb...
syl3an1b 1403 A syllogism inference. (C...
syl3an2b 1404 A syllogism inference. (C...
syl3an3b 1405 A syllogism inference. (C...
syl3an1br 1406 A syllogism inference. (C...
syl3an2br 1407 A syllogism inference. (C...
syl3an3br 1408 A syllogism inference. (C...
syld3an3 1409 A syllogism inference. (C...
syld3an1 1410 A syllogism inference. (C...
syld3an2 1411 A syllogism inference. (C...
syl3anl1 1412 A syllogism inference. (C...
syl3anl2 1413 A syllogism inference. (C...
syl3anl3 1414 A syllogism inference. (C...
syl3anl 1415 A triple syllogism inferen...
syl3anr1 1416 A syllogism inference. (C...
syl3anr2 1417 A syllogism inference. (C...
syl3anr3 1418 A syllogism inference. (C...
3anidm12 1419 Inference from idempotent ...
3anidm13 1420 Inference from idempotent ...
3anidm23 1421 Inference from idempotent ...
syl2an3an 1422 ~ syl3an with antecedents ...
syl2an23an 1423 Deduction related to ~ syl...
3ori 1424 Infer implication from tri...
3jao 1425 Disjunction of three antec...
3jaob 1426 Disjunction of three antec...
3jaobOLD 1427 Obsolete version of ~ 3jao...
3jaoi 1428 Disjunction of three antec...
3jaod 1429 Disjunction of three antec...
3jaoian 1430 Disjunction of three antec...
3jaodan 1431 Disjunction of three antec...
mpjao3dan 1432 Eliminate a three-way disj...
3jaao 1433 Inference conjoining and d...
syl3an9b 1434 Nested syllogism inference...
3orbi123d 1435 Deduction joining 3 equiva...
3anbi123d 1436 Deduction joining 3 equiva...
3anbi12d 1437 Deduction conjoining and a...
3anbi13d 1438 Deduction conjoining and a...
3anbi23d 1439 Deduction conjoining and a...
3anbi1d 1440 Deduction adding conjuncts...
3anbi2d 1441 Deduction adding conjuncts...
3anbi3d 1442 Deduction adding conjuncts...
3anim123d 1443 Deduction joining 3 implic...
3orim123d 1444 Deduction joining 3 implic...
an6 1445 Rearrangement of 6 conjunc...
3an6 1446 Analogue of ~ an4 for trip...
3or6 1447 Analogue of ~ or4 for trip...
mp3an1 1448 An inference based on modu...
mp3an2 1449 An inference based on modu...
mp3an3 1450 An inference based on modu...
mp3an12 1451 An inference based on modu...
mp3an13 1452 An inference based on modu...
mp3an23 1453 An inference based on modu...
mp3an1i 1454 An inference based on modu...
mp3anl1 1455 An inference based on modu...
mp3anl2 1456 An inference based on modu...
mp3anl3 1457 An inference based on modu...
mp3anr1 1458 An inference based on modu...
mp3anr2 1459 An inference based on modu...
mp3anr3 1460 An inference based on modu...
mp3an 1461 An inference based on modu...
mpd3an3 1462 An inference based on modu...
mpd3an23 1463 An inference based on modu...
mp3and 1464 A deduction based on modus...
mp3an12i 1465 ~ mp3an with antecedents i...
mp3an2i 1466 ~ mp3an with antecedents i...
mp3an3an 1467 ~ mp3an with antecedents i...
mp3an2ani 1468 An elimination deduction. ...
biimp3a 1469 Infer implication from a l...
biimp3ar 1470 Infer implication from a l...
3anandis 1471 Inference that undistribut...
3anandirs 1472 Inference that undistribut...
ecase23d 1473 Deduction for elimination ...
3ecase 1474 Inference for elimination ...
3bior1fd 1475 A disjunction is equivalen...
3bior1fand 1476 A disjunction is equivalen...
3bior2fd 1477 A wff is equivalent to its...
3biant1d 1478 A conjunction is equivalen...
intn3an1d 1479 Introduction of a triple c...
intn3an2d 1480 Introduction of a triple c...
intn3an3d 1481 Introduction of a triple c...
an3andi 1482 Distribution of conjunctio...
an33rean 1483 Rearrange a 9-fold conjunc...
3orel2 1484 Partial elimination of a t...
3orel3 1485 Partial elimination of a t...
3orel13 1486 Elimination of two disjunc...
3pm3.2ni 1487 Triple negated disjunction...
nanan 1490 Conjunction in terms of al...
dfnan2 1491 Alternative denial in term...
nanor 1492 Alternative denial in term...
nancom 1493 Alternative denial is comm...
nannan 1494 Nested alternative denials...
nanim 1495 Implication in terms of al...
nannot 1496 Negation in terms of alter...
nanbi 1497 Biconditional in terms of ...
nanbi1 1498 Introduce a right anti-con...
nanbi2 1499 Introduce a left anti-conj...
nanbi12 1500 Join two logical equivalen...
nanbi1i 1501 Introduce a right anti-con...
nanbi2i 1502 Introduce a left anti-conj...
nanbi12i 1503 Join two logical equivalen...
nanbi1d 1504 Introduce a right anti-con...
nanbi2d 1505 Introduce a left anti-conj...
nanbi12d 1506 Join two logical equivalen...
nanass 1507 A characterization of when...
xnor 1510 Two ways to write XNOR (ex...
xorcom 1511 The connector ` \/_ ` is c...
xorass 1512 The connector ` \/_ ` is a...
excxor 1513 This tautology shows that ...
xor2 1514 Two ways to express "exclu...
xoror 1515 Exclusive disjunction impl...
xornan 1516 Exclusive disjunction impl...
xornan2 1517 XOR implies NAND (written ...
xorneg2 1518 The connector ` \/_ ` is n...
xorneg1 1519 The connector ` \/_ ` is n...
xorneg 1520 The connector ` \/_ ` is u...
xorbi12i 1521 Equality property for excl...
xorbi12d 1522 Equality property for excl...
anxordi 1523 Conjunction distributes ov...
xorexmid 1524 Exclusive-or variant of th...
norcom 1527 The connector ` -\/ ` is c...
nornot 1528 ` -. ` is expressible via ...
noran 1529 ` /\ ` is expressible via ...
noror 1530 ` \/ ` is expressible via ...
norasslem1 1531 This lemma shows the equiv...
norasslem2 1532 This lemma specializes ~ b...
norasslem3 1533 This lemma specializes ~ b...
norass 1534 A characterization of when...
trujust 1539 Soundness justification th...
tru 1541 The truth value ` T. ` is ...
dftru2 1542 An alternate definition of...
trut 1543 A proposition is equivalen...
mptru 1544 Eliminate ` T. ` as an ant...
tbtru 1545 A proposition is equivalen...
bitru 1546 A theorem is equivalent to...
trud 1547 Anything implies ` T. ` . ...
truan 1548 True can be removed from a...
fal 1551 The truth value ` F. ` is ...
nbfal 1552 The negation of a proposit...
bifal 1553 A contradiction is equival...
falim 1554 The truth value ` F. ` imp...
falimd 1555 The truth value ` F. ` imp...
dfnot 1556 Given falsum ` F. ` , we c...
inegd 1557 Negation introduction rule...
efald 1558 Deduction based on reducti...
pm2.21fal 1559 If a wff and its negation ...
truimtru 1560 A ` -> ` identity. (Contr...
truimfal 1561 A ` -> ` identity. (Contr...
falimtru 1562 A ` -> ` identity. (Contr...
falimfal 1563 A ` -> ` identity. (Contr...
nottru 1564 A ` -. ` identity. (Contr...
notfal 1565 A ` -. ` identity. (Contr...
trubitru 1566 A ` <-> ` identity. (Cont...
falbitru 1567 A ` <-> ` identity. (Cont...
trubifal 1568 A ` <-> ` identity. (Cont...
falbifal 1569 A ` <-> ` identity. (Cont...
truantru 1570 A ` /\ ` identity. (Contr...
truanfal 1571 A ` /\ ` identity. (Contr...
falantru 1572 A ` /\ ` identity. (Contr...
falanfal 1573 A ` /\ ` identity. (Contr...
truortru 1574 A ` \/ ` identity. (Contr...
truorfal 1575 A ` \/ ` identity. (Contr...
falortru 1576 A ` \/ ` identity. (Contr...
falorfal 1577 A ` \/ ` identity. (Contr...
trunantru 1578 A ` -/\ ` identity. (Cont...
trunanfal 1579 A ` -/\ ` identity. (Cont...
falnantru 1580 A ` -/\ ` identity. (Cont...
falnanfal 1581 A ` -/\ ` identity. (Cont...
truxortru 1582 A ` \/_ ` identity. (Cont...
truxorfal 1583 A ` \/_ ` identity. (Cont...
falxortru 1584 A ` \/_ ` identity. (Cont...
falxorfal 1585 A ` \/_ ` identity. (Cont...
trunortru 1586 A ` -\/ ` identity. (Cont...
trunorfal 1587 A ` -\/ ` identity. (Cont...
falnortru 1588 A ` -\/ ` identity. (Cont...
falnorfal 1589 A ` -\/ ` identity. (Cont...
hadbi123d 1592 Equality theorem for the a...
hadbi123i 1593 Equality theorem for the a...
hadass 1594 Associative law for the ad...
hadbi 1595 The adder sum is the same ...
hadcoma 1596 Commutative law for the ad...
hadcomb 1597 Commutative law for the ad...
hadrot 1598 Rotation law for the adder...
hadnot 1599 The adder sum distributes ...
had1 1600 If the first input is true...
had0 1601 If the first input is fals...
hadifp 1602 The value of the adder sum...
cador 1605 The adder carry in disjunc...
cadan 1606 The adder carry in conjunc...
cadbi123d 1607 Equality theorem for the a...
cadbi123i 1608 Equality theorem for the a...
cadcoma 1609 Commutative law for the ad...
cadcomb 1610 Commutative law for the ad...
cadrot 1611 Rotation law for the adder...
cadnot 1612 The adder carry distribute...
cad11 1613 If (at least) two inputs a...
cad1 1614 If one input is true, then...
cad0 1615 If one input is false, the...
cad0OLD 1616 Obsolete version of ~ cad0...
cadifp 1617 The value of the carry is,...
cadtru 1618 The adder carry is true as...
minimp 1619 A single axiom for minimal...
minimp-syllsimp 1620 Derivation of Syll-Simp ( ...
minimp-ax1 1621 Derivation of ~ ax-1 from ...
minimp-ax2c 1622 Derivation of a commuted f...
minimp-ax2 1623 Derivation of ~ ax-2 from ...
minimp-pm2.43 1624 Derivation of ~ pm2.43 (al...
impsingle 1625 The shortest single axiom ...
impsingle-step4 1626 Derivation of impsingle-st...
impsingle-step8 1627 Derivation of impsingle-st...
impsingle-ax1 1628 Derivation of impsingle-ax...
impsingle-step15 1629 Derivation of impsingle-st...
impsingle-step18 1630 Derivation of impsingle-st...
impsingle-step19 1631 Derivation of impsingle-st...
impsingle-step20 1632 Derivation of impsingle-st...
impsingle-step21 1633 Derivation of impsingle-st...
impsingle-step22 1634 Derivation of impsingle-st...
impsingle-step25 1635 Derivation of impsingle-st...
impsingle-imim1 1636 Derivation of impsingle-im...
impsingle-peirce 1637 Derivation of impsingle-pe...
tarski-bernays-ax2 1638 Derivation of ~ ax-2 from ...
meredith 1639 Carew Meredith's sole axio...
merlem1 1640 Step 3 of Meredith's proof...
merlem2 1641 Step 4 of Meredith's proof...
merlem3 1642 Step 7 of Meredith's proof...
merlem4 1643 Step 8 of Meredith's proof...
merlem5 1644 Step 11 of Meredith's proo...
merlem6 1645 Step 12 of Meredith's proo...
merlem7 1646 Between steps 14 and 15 of...
merlem8 1647 Step 15 of Meredith's proo...
merlem9 1648 Step 18 of Meredith's proo...
merlem10 1649 Step 19 of Meredith's proo...
merlem11 1650 Step 20 of Meredith's proo...
merlem12 1651 Step 28 of Meredith's proo...
merlem13 1652 Step 35 of Meredith's proo...
luk-1 1653 1 of 3 axioms for proposit...
luk-2 1654 2 of 3 axioms for proposit...
luk-3 1655 3 of 3 axioms for proposit...
luklem1 1656 Used to rederive standard ...
luklem2 1657 Used to rederive standard ...
luklem3 1658 Used to rederive standard ...
luklem4 1659 Used to rederive standard ...
luklem5 1660 Used to rederive standard ...
luklem6 1661 Used to rederive standard ...
luklem7 1662 Used to rederive standard ...
luklem8 1663 Used to rederive standard ...
ax1 1664 Standard propositional axi...
ax2 1665 Standard propositional axi...
ax3 1666 Standard propositional axi...
nic-dfim 1667 This theorem "defines" imp...
nic-dfneg 1668 This theorem "defines" neg...
nic-mp 1669 Derive Nicod's rule of mod...
nic-mpALT 1670 A direct proof of ~ nic-mp...
nic-ax 1671 Nicod's axiom derived from...
nic-axALT 1672 A direct proof of ~ nic-ax...
nic-imp 1673 Inference for ~ nic-mp usi...
nic-idlem1 1674 Lemma for ~ nic-id . (Con...
nic-idlem2 1675 Lemma for ~ nic-id . Infe...
nic-id 1676 Theorem ~ id expressed wit...
nic-swap 1677 The connector ` -/\ ` is s...
nic-isw1 1678 Inference version of ~ nic...
nic-isw2 1679 Inference for swapping nes...
nic-iimp1 1680 Inference version of ~ nic...
nic-iimp2 1681 Inference version of ~ nic...
nic-idel 1682 Inference to remove the tr...
nic-ich 1683 Chained inference. (Contr...
nic-idbl 1684 Double the terms. Since d...
nic-bijust 1685 Biconditional justificatio...
nic-bi1 1686 Inference to extract one s...
nic-bi2 1687 Inference to extract the o...
nic-stdmp 1688 Derive the standard modus ...
nic-luk1 1689 Proof of ~ luk-1 from ~ ni...
nic-luk2 1690 Proof of ~ luk-2 from ~ ni...
nic-luk3 1691 Proof of ~ luk-3 from ~ ni...
lukshef-ax1 1692 This alternative axiom for...
lukshefth1 1693 Lemma for ~ renicax . (Co...
lukshefth2 1694 Lemma for ~ renicax . (Co...
renicax 1695 A rederivation of ~ nic-ax...
tbw-bijust 1696 Justification for ~ tbw-ne...
tbw-negdf 1697 The definition of negation...
tbw-ax1 1698 The first of four axioms i...
tbw-ax2 1699 The second of four axioms ...
tbw-ax3 1700 The third of four axioms i...
tbw-ax4 1701 The fourth of four axioms ...
tbwsyl 1702 Used to rederive the Lukas...
tbwlem1 1703 Used to rederive the Lukas...
tbwlem2 1704 Used to rederive the Lukas...
tbwlem3 1705 Used to rederive the Lukas...
tbwlem4 1706 Used to rederive the Lukas...
tbwlem5 1707 Used to rederive the Lukas...
re1luk1 1708 ~ luk-1 derived from the T...
re1luk2 1709 ~ luk-2 derived from the T...
re1luk3 1710 ~ luk-3 derived from the T...
merco1 1711 A single axiom for proposi...
merco1lem1 1712 Used to rederive the Tarsk...
retbwax4 1713 ~ tbw-ax4 rederived from ~...
retbwax2 1714 ~ tbw-ax2 rederived from ~...
merco1lem2 1715 Used to rederive the Tarsk...
merco1lem3 1716 Used to rederive the Tarsk...
merco1lem4 1717 Used to rederive the Tarsk...
merco1lem5 1718 Used to rederive the Tarsk...
merco1lem6 1719 Used to rederive the Tarsk...
merco1lem7 1720 Used to rederive the Tarsk...
retbwax3 1721 ~ tbw-ax3 rederived from ~...
merco1lem8 1722 Used to rederive the Tarsk...
merco1lem9 1723 Used to rederive the Tarsk...
merco1lem10 1724 Used to rederive the Tarsk...
merco1lem11 1725 Used to rederive the Tarsk...
merco1lem12 1726 Used to rederive the Tarsk...
merco1lem13 1727 Used to rederive the Tarsk...
merco1lem14 1728 Used to rederive the Tarsk...
merco1lem15 1729 Used to rederive the Tarsk...
merco1lem16 1730 Used to rederive the Tarsk...
merco1lem17 1731 Used to rederive the Tarsk...
merco1lem18 1732 Used to rederive the Tarsk...
retbwax1 1733 ~ tbw-ax1 rederived from ~...
merco2 1734 A single axiom for proposi...
mercolem1 1735 Used to rederive the Tarsk...
mercolem2 1736 Used to rederive the Tarsk...
mercolem3 1737 Used to rederive the Tarsk...
mercolem4 1738 Used to rederive the Tarsk...
mercolem5 1739 Used to rederive the Tarsk...
mercolem6 1740 Used to rederive the Tarsk...
mercolem7 1741 Used to rederive the Tarsk...
mercolem8 1742 Used to rederive the Tarsk...
re1tbw1 1743 ~ tbw-ax1 rederived from ~...
re1tbw2 1744 ~ tbw-ax2 rederived from ~...
re1tbw3 1745 ~ tbw-ax3 rederived from ~...
re1tbw4 1746 ~ tbw-ax4 rederived from ~...
rb-bijust 1747 Justification for ~ rb-imd...
rb-imdf 1748 The definition of implicat...
anmp 1749 Modus ponens for ` { \/ , ...
rb-ax1 1750 The first of four axioms i...
rb-ax2 1751 The second of four axioms ...
rb-ax3 1752 The third of four axioms i...
rb-ax4 1753 The fourth of four axioms ...
rbsyl 1754 Used to rederive the Lukas...
rblem1 1755 Used to rederive the Lukas...
rblem2 1756 Used to rederive the Lukas...
rblem3 1757 Used to rederive the Lukas...
rblem4 1758 Used to rederive the Lukas...
rblem5 1759 Used to rederive the Lukas...
rblem6 1760 Used to rederive the Lukas...
rblem7 1761 Used to rederive the Lukas...
re1axmp 1762 ~ ax-mp derived from Russe...
re2luk1 1763 ~ luk-1 derived from Russe...
re2luk2 1764 ~ luk-2 derived from Russe...
re2luk3 1765 ~ luk-3 derived from Russe...
mptnan 1766 Modus ponendo tollens 1, o...
mptxor 1767 Modus ponendo tollens 2, o...
mtpor 1768 Modus tollendo ponens (inc...
mtpxor 1769 Modus tollendo ponens (ori...
stoic1a 1770 Stoic logic Thema 1 (part ...
stoic1b 1771 Stoic logic Thema 1 (part ...
stoic2a 1772 Stoic logic Thema 2 versio...
stoic2b 1773 Stoic logic Thema 2 versio...
stoic3 1774 Stoic logic Thema 3. Stat...
stoic4a 1775 Stoic logic Thema 4 versio...
stoic4b 1776 Stoic logic Thema 4 versio...
alnex 1779 Universal quantification o...
eximal 1780 An equivalence between an ...
nf2 1783 Alternate definition of no...
nf3 1784 Alternate definition of no...
nf4 1785 Alternate definition of no...
nfi 1786 Deduce that ` x ` is not f...
nfri 1787 Consequence of the definit...
nfd 1788 Deduce that ` x ` is not f...
nfrd 1789 Consequence of the definit...
nftht 1790 Closed form of ~ nfth . (...
nfntht 1791 Closed form of ~ nfnth . ...
nfntht2 1792 Closed form of ~ nfnth . ...
gen2 1794 Generalization applied twi...
mpg 1795 Modus ponens combined with...
mpgbi 1796 Modus ponens on biconditio...
mpgbir 1797 Modus ponens on biconditio...
nex 1798 Generalization rule for ne...
nfth 1799 No variable is (effectivel...
nfnth 1800 No variable is (effectivel...
hbth 1801 No variable is (effectivel...
nftru 1802 The true constant has no f...
nffal 1803 The false constant has no ...
sptruw 1804 Version of ~ sp when ` ph ...
altru 1805 For all sets, ` T. ` is tr...
alfal 1806 For all sets, ` -. F. ` is...
alim 1808 Restatement of Axiom ~ ax-...
alimi 1809 Inference quantifying both...
2alimi 1810 Inference doubly quantifyi...
ala1 1811 Add an antecedent in a uni...
al2im 1812 Closed form of ~ al2imi . ...
al2imi 1813 Inference quantifying ante...
alanimi 1814 Variant of ~ al2imi with c...
alimdh 1815 Deduction form of Theorem ...
albi 1816 Theorem 19.15 of [Margaris...
albii 1817 Inference adding universal...
2albii 1818 Inference adding two unive...
3albii 1819 Inference adding three uni...
sylgt 1820 Closed form of ~ sylg . (...
sylg 1821 A syllogism combined with ...
alrimih 1822 Inference form of Theorem ...
hbxfrbi 1823 A utility lemma to transfe...
alex 1824 Universal quantifier in te...
exnal 1825 Existential quantification...
2nalexn 1826 Part of theorem *11.5 in [...
2exnaln 1827 Theorem *11.22 in [Whitehe...
2nexaln 1828 Theorem *11.25 in [Whitehe...
alimex 1829 An equivalence between an ...
aleximi 1830 A variant of ~ al2imi : in...
alexbii 1831 Biconditional form of ~ al...
exim 1832 Theorem 19.22 of [Margaris...
eximi 1833 Inference adding existenti...
2eximi 1834 Inference adding two exist...
eximii 1835 Inference associated with ...
exa1 1836 Add an antecedent in an ex...
19.38 1837 Theorem 19.38 of [Margaris...
19.38a 1838 Under a nonfreeness hypoth...
19.38b 1839 Under a nonfreeness hypoth...
imnang 1840 Quantified implication in ...
alinexa 1841 A transformation of quanti...
exnalimn 1842 Existential quantification...
alexn 1843 A relationship between two...
2exnexn 1844 Theorem *11.51 in [Whitehe...
exbi 1845 Theorem 19.18 of [Margaris...
exbii 1846 Inference adding existenti...
2exbii 1847 Inference adding two exist...
3exbii 1848 Inference adding three exi...
nfbiit 1849 Equivalence theorem for th...
nfbii 1850 Equality theorem for the n...
nfxfr 1851 A utility lemma to transfe...
nfxfrd 1852 A utility lemma to transfe...
nfnbi 1853 A variable is nonfree in a...
nfnbiOLD 1854 Obsolete version of ~ nfnb...
nfnt 1855 If a variable is nonfree i...
nfn 1856 Inference associated with ...
nfnd 1857 Deduction associated with ...
exanali 1858 A transformation of quanti...
2exanali 1859 Theorem *11.521 in [Whiteh...
exancom 1860 Commutation of conjunction...
exan 1861 Place a conjunct in the sc...
alrimdh 1862 Deduction form of Theorem ...
eximdh 1863 Deduction from Theorem 19....
nexdh 1864 Deduction for generalizati...
albidh 1865 Formula-building rule for ...
exbidh 1866 Formula-building rule for ...
exsimpl 1867 Simplification of an exist...
exsimpr 1868 Simplification of an exist...
19.26 1869 Theorem 19.26 of [Margaris...
19.26-2 1870 Theorem ~ 19.26 with two q...
19.26-3an 1871 Theorem ~ 19.26 with tripl...
19.29 1872 Theorem 19.29 of [Margaris...
19.29r 1873 Variation of ~ 19.29 . (C...
19.29r2 1874 Variation of ~ 19.29r with...
19.29x 1875 Variation of ~ 19.29 with ...
19.35 1876 Theorem 19.35 of [Margaris...
19.35i 1877 Inference associated with ...
19.35ri 1878 Inference associated with ...
19.25 1879 Theorem 19.25 of [Margaris...
19.30 1880 Theorem 19.30 of [Margaris...
19.43 1881 Theorem 19.43 of [Margaris...
19.43OLD 1882 Obsolete proof of ~ 19.43 ...
19.33 1883 Theorem 19.33 of [Margaris...
19.33b 1884 The antecedent provides a ...
19.40 1885 Theorem 19.40 of [Margaris...
19.40-2 1886 Theorem *11.42 in [Whitehe...
19.40b 1887 The antecedent provides a ...
albiim 1888 Split a biconditional and ...
2albiim 1889 Split a biconditional and ...
exintrbi 1890 Add/remove a conjunct in t...
exintr 1891 Introduce a conjunct in th...
alsyl 1892 Universally quantified and...
nfimd 1893 If in a context ` x ` is n...
nfimt 1894 Closed form of ~ nfim and ...
nfim 1895 If ` x ` is not free in ` ...
nfand 1896 If in a context ` x ` is n...
nf3and 1897 Deduction form of bound-va...
nfan 1898 If ` x ` is not free in ` ...
nfnan 1899 If ` x ` is not free in ` ...
nf3an 1900 If ` x ` is not free in ` ...
nfbid 1901 If in a context ` x ` is n...
nfbi 1902 If ` x ` is not free in ` ...
nfor 1903 If ` x ` is not free in ` ...
nf3or 1904 If ` x ` is not free in ` ...
empty 1905 Two characterizations of t...
emptyex 1906 On the empty domain, any e...
emptyal 1907 On the empty domain, any u...
emptynf 1908 On the empty domain, any v...
ax5d 1910 Version of ~ ax-5 with ant...
ax5e 1911 A rephrasing of ~ ax-5 usi...
ax5ea 1912 If a formula holds for som...
nfv 1913 If ` x ` is not present in...
nfvd 1914 ~ nfv with antecedent. Us...
alimdv 1915 Deduction form of Theorem ...
eximdv 1916 Deduction form of Theorem ...
2alimdv 1917 Deduction form of Theorem ...
2eximdv 1918 Deduction form of Theorem ...
albidv 1919 Formula-building rule for ...
exbidv 1920 Formula-building rule for ...
nfbidv 1921 An equality theorem for no...
2albidv 1922 Formula-building rule for ...
2exbidv 1923 Formula-building rule for ...
3exbidv 1924 Formula-building rule for ...
4exbidv 1925 Formula-building rule for ...
alrimiv 1926 Inference form of Theorem ...
alrimivv 1927 Inference form of Theorem ...
alrimdv 1928 Deduction form of Theorem ...
exlimiv 1929 Inference form of Theorem ...
exlimiiv 1930 Inference (Rule C) associa...
exlimivv 1931 Inference form of Theorem ...
exlimdv 1932 Deduction form of Theorem ...
exlimdvv 1933 Deduction form of Theorem ...
exlimddv 1934 Existential elimination ru...
nexdv 1935 Deduction for generalizati...
2ax5 1936 Quantification of two vari...
stdpc5v 1937 Version of ~ stdpc5 with a...
19.21v 1938 Version of ~ 19.21 with a ...
19.32v 1939 Version of ~ 19.32 with a ...
19.31v 1940 Version of ~ 19.31 with a ...
19.23v 1941 Version of ~ 19.23 with a ...
19.23vv 1942 Theorem ~ 19.23v extended ...
pm11.53v 1943 Version of ~ pm11.53 with ...
19.36imv 1944 One direction of ~ 19.36v ...
19.36imvOLD 1945 Obsolete version of ~ 19.3...
19.36iv 1946 Inference associated with ...
19.37imv 1947 One direction of ~ 19.37v ...
19.37iv 1948 Inference associated with ...
19.41v 1949 Version of ~ 19.41 with a ...
19.41vv 1950 Version of ~ 19.41 with tw...
19.41vvv 1951 Version of ~ 19.41 with th...
19.41vvvv 1952 Version of ~ 19.41 with fo...
19.42v 1953 Version of ~ 19.42 with a ...
exdistr 1954 Distribution of existentia...
exdistrv 1955 Distribute a pair of exist...
4exdistrv 1956 Distribute two pairs of ex...
19.42vv 1957 Version of ~ 19.42 with tw...
exdistr2 1958 Distribution of existentia...
19.42vvv 1959 Version of ~ 19.42 with th...
3exdistr 1960 Distribution of existentia...
4exdistr 1961 Distribution of existentia...
weq 1962 Extend wff definition to i...
speimfw 1963 Specialization, with addit...
speimfwALT 1964 Alternate proof of ~ speim...
spimfw 1965 Specialization, with addit...
ax12i 1966 Inference that has ~ ax-12...
ax6v 1968 Axiom B7 of [Tarski] p. 75...
ax6ev 1969 At least one individual ex...
spimw 1970 Specialization. Lemma 8 o...
spimew 1971 Existential introduction, ...
speiv 1972 Inference from existential...
speivw 1973 Version of ~ spei with a d...
exgen 1974 Rule of existential genera...
extru 1975 There exists a variable su...
19.2 1976 Theorem 19.2 of [Margaris]...
19.2d 1977 Deduction associated with ...
19.8w 1978 Weak version of ~ 19.8a an...
spnfw 1979 Weak version of ~ sp . Us...
spvw 1980 Version of ~ sp when ` x `...
19.3v 1981 Version of ~ 19.3 with a d...
19.8v 1982 Version of ~ 19.8a with a ...
19.9v 1983 Version of ~ 19.9 with a d...
19.39 1984 Theorem 19.39 of [Margaris...
19.24 1985 Theorem 19.24 of [Margaris...
19.34 1986 Theorem 19.34 of [Margaris...
19.36v 1987 Version of ~ 19.36 with a ...
19.12vvv 1988 Version of ~ 19.12vv with ...
19.27v 1989 Version of ~ 19.27 with a ...
19.28v 1990 Version of ~ 19.28 with a ...
19.37v 1991 Version of ~ 19.37 with a ...
19.44v 1992 Version of ~ 19.44 with a ...
19.45v 1993 Version of ~ 19.45 with a ...
spimevw 1994 Existential introduction, ...
spimvw 1995 A weak form of specializat...
spvv 1996 Specialization, using impl...
spfalw 1997 Version of ~ sp when ` ph ...
chvarvv 1998 Implicit substitution of `...
equs4v 1999 Version of ~ equs4 with a ...
alequexv 2000 Version of ~ equs4v with i...
exsbim 2001 One direction of the equiv...
equsv 2002 If a formula does not cont...
equsalvw 2003 Version of ~ equsalv with ...
equsexvw 2004 Version of ~ equsexv with ...
cbvaliw 2005 Change bound variable. Us...
cbvalivw 2006 Change bound variable. Us...
ax7v 2008 Weakened version of ~ ax-7...
ax7v1 2009 First of two weakened vers...
ax7v2 2010 Second of two weakened ver...
equid 2011 Identity law for equality....
nfequid 2012 Bound-variable hypothesis ...
equcomiv 2013 Weaker form of ~ equcomi w...
ax6evr 2014 A commuted form of ~ ax6ev...
ax7 2015 Proof of ~ ax-7 from ~ ax7...
equcomi 2016 Commutative law for equali...
equcom 2017 Commutative law for equali...
equcomd 2018 Deduction form of ~ equcom...
equcoms 2019 An inference commuting equ...
equtr 2020 A transitive law for equal...
equtrr 2021 A transitive law for equal...
equeuclr 2022 Commuted version of ~ eque...
equeucl 2023 Equality is a left-Euclide...
equequ1 2024 An equivalence law for equ...
equequ2 2025 An equivalence law for equ...
equtr2 2026 Equality is a left-Euclide...
stdpc6 2027 One of the two equality ax...
equvinv 2028 A variable introduction la...
equvinva 2029 A modified version of the ...
equvelv 2030 A biconditional form of ~ ...
ax13b 2031 An equivalence between two...
spfw 2032 Weak version of ~ sp . Us...
spw 2033 Weak version of the specia...
cbvalw 2034 Change bound variable. Us...
cbvalvw 2035 Change bound variable. Us...
cbvexvw 2036 Change bound variable. Us...
cbvaldvaw 2037 Rule used to change the bo...
cbvexdvaw 2038 Rule used to change the bo...
cbval2vw 2039 Rule used to change bound ...
cbvex2vw 2040 Rule used to change bound ...
cbvex4vw 2041 Rule used to change bound ...
alcomimw 2042 Weak version of ~ ax-11 . ...
excomimw 2043 Weak version of ~ excomim ...
alcomw 2044 Weak version of ~ alcom an...
hbn1fw 2045 Weak version of ~ ax-10 fr...
hbn1w 2046 Weak version of ~ hbn1 . ...
hba1w 2047 Weak version of ~ hba1 . ...
hbe1w 2048 Weak version of ~ hbe1 . ...
hbalw 2049 Weak version of ~ hbal . ...
19.8aw 2050 If a formula is true, then...
exexw 2051 Existential quantification...
spaev 2052 A special instance of ~ sp...
cbvaev 2053 Change bound variable in a...
aevlem0 2054 Lemma for ~ aevlem . Inst...
aevlem 2055 Lemma for ~ aev and ~ axc1...
aeveq 2056 The antecedent ` A. x x = ...
aev 2057 A "distinctor elimination"...
aev2 2058 A version of ~ aev with tw...
hbaev 2059 All variables are effectiv...
naev 2060 If some set variables can ...
naev2 2061 Generalization of ~ hbnaev...
hbnaev 2062 Any variable is free in ` ...
sbjust 2063 Justification theorem for ...
sbt 2066 A substitution into a theo...
sbtru 2067 The result of substituting...
stdpc4 2068 The specialization axiom o...
sbtALT 2069 Alternate proof of ~ sbt ,...
2stdpc4 2070 A double specialization us...
sbi1 2071 Distribute substitution ov...
spsbim 2072 Distribute substitution ov...
spsbbi 2073 Biconditional property for...
sbimi 2074 Distribute substitution ov...
sb2imi 2075 Distribute substitution ov...
sbbii 2076 Infer substitution into bo...
2sbbii 2077 Infer double substitution ...
sbimdv 2078 Deduction substituting bot...
sbbidv 2079 Deduction substituting bot...
sban 2080 Conjunction inside and out...
sb3an 2081 Threefold conjunction insi...
spsbe 2082 Existential generalization...
sbequ 2083 Equality property for subs...
sbequi 2084 An equality theorem for su...
sb6 2085 Alternate definition of su...
2sb6 2086 Equivalence for double sub...
sb1v 2087 One direction of ~ sb5 , p...
sbv 2088 Substitution for a variabl...
sbcom4 2089 Commutativity law for subs...
pm11.07 2090 Axiom *11.07 in [Whitehead...
sbrimvw 2091 Substitution in an implica...
sbbiiev 2092 An equivalence of substitu...
sbievw 2093 Conversion of implicit sub...
sbievwOLD 2094 Obsolete version of ~ sbie...
sbiedvw 2095 Conversion of implicit sub...
2sbievw 2096 Conversion of double impli...
sbcom3vv 2097 Substituting ` y ` for ` x...
sbievw2 2098 ~ sbievw applied twice, av...
sbco2vv 2099 A composition law for subs...
cbvsbv 2100 Change the bound variable ...
sbco4lem 2101 Lemma for ~ sbco4 . It re...
sbco4 2102 Two ways of exchanging two...
equsb3 2103 Substitution in an equalit...
equsb3r 2104 Substitution applied to th...
equsb1v 2105 Substitution applied to an...
nsb 2106 Any substitution in an alw...
sbn1 2107 One direction of ~ sbn , u...
wel 2109 Extend wff definition to i...
ax8v 2111 Weakened version of ~ ax-8...
ax8v1 2112 First of two weakened vers...
ax8v2 2113 Second of two weakened ver...
ax8 2114 Proof of ~ ax-8 from ~ ax8...
elequ1 2115 An identity law for the no...
elsb1 2116 Substitution for the first...
cleljust 2117 When the class variables i...
ax9v 2119 Weakened version of ~ ax-9...
ax9v1 2120 First of two weakened vers...
ax9v2 2121 Second of two weakened ver...
ax9 2122 Proof of ~ ax-9 from ~ ax9...
elequ2 2123 An identity law for the no...
elequ2g 2124 A form of ~ elequ2 with a ...
elsb2 2125 Substitution for the secon...
elequ12 2126 An identity law for the no...
ru0 2127 The FOL statement used in ...
ax6dgen 2128 Tarski's system uses the w...
ax10w 2129 Weak version of ~ ax-10 fr...
ax11w 2130 Weak version of ~ ax-11 fr...
ax11dgen 2131 Degenerate instance of ~ a...
ax12wlem 2132 Lemma for weak version of ...
ax12w 2133 Weak version of ~ ax-12 fr...
ax12dgen 2134 Degenerate instance of ~ a...
ax12wdemo 2135 Example of an application ...
ax13w 2136 Weak version (principal in...
ax13dgen1 2137 Degenerate instance of ~ a...
ax13dgen2 2138 Degenerate instance of ~ a...
ax13dgen3 2139 Degenerate instance of ~ a...
ax13dgen4 2140 Degenerate instance of ~ a...
hbn1 2142 Alias for ~ ax-10 to be us...
hbe1 2143 The setvar ` x ` is not fr...
hbe1a 2144 Dual statement of ~ hbe1 ....
nf5-1 2145 One direction of ~ nf5 can...
nf5i 2146 Deduce that ` x ` is not f...
nf5dh 2147 Deduce that ` x ` is not f...
nf5dv 2148 Apply the definition of no...
nfnaew 2149 All variables are effectiv...
nfnaewOLD 2150 Obsolete version of ~ nfna...
nfe1 2151 The setvar ` x ` is not fr...
nfa1 2152 The setvar ` x ` is not fr...
nfna1 2153 A convenience theorem part...
nfia1 2154 Lemma 23 of [Monk2] p. 114...
nfnf1 2155 The setvar ` x ` is not fr...
modal5 2156 The analogue in our predic...
nfs1v 2157 The setvar ` x ` is not fr...
alcoms 2159 Swap quantifiers in an ant...
alcom 2160 Theorem 19.5 of [Margaris]...
alrot3 2161 Theorem *11.21 in [Whitehe...
alrot4 2162 Rotate four universal quan...
excom 2163 Theorem 19.11 of [Margaris...
excomim 2164 One direction of Theorem 1...
excom13 2165 Swap 1st and 3rd existenti...
exrot3 2166 Rotate existential quantif...
exrot4 2167 Rotate existential quantif...
hbal 2168 If ` x ` is not free in ` ...
hbald 2169 Deduction form of bound-va...
sbal 2170 Move universal quantifier ...
sbalv 2171 Quantify with new variable...
hbsbw 2172 If ` z ` is not free in ` ...
hbsbwOLD 2173 Obsolete version of ~ hbsb...
sbcom2 2174 Commutativity law for subs...
sbco4lemOLD 2175 Obsolete version of ~ sbco...
sbco4OLD 2176 Obsolete version of ~ sbco...
nfa2 2177 Lemma 24 of [Monk2] p. 114...
ax12v 2179 This is essentially Axiom ...
ax12v2 2180 It is possible to remove a...
ax12ev2 2181 Version of ~ ax12v2 rewrit...
19.8a 2182 If a wff is true, it is tr...
19.8ad 2183 If a wff is true, it is tr...
sp 2184 Specialization. A univers...
spi 2185 Inference rule of universa...
sps 2186 Generalization of antecede...
2sp 2187 A double specialization (s...
spsd 2188 Deduction generalizing ant...
19.2g 2189 Theorem 19.2 of [Margaris]...
19.21bi 2190 Inference form of ~ 19.21 ...
19.21bbi 2191 Inference removing two uni...
19.23bi 2192 Inference form of Theorem ...
nexr 2193 Inference associated with ...
qexmid 2194 Quantified excluded middle...
nf5r 2195 Consequence of the definit...
nf5ri 2196 Consequence of the definit...
nf5rd 2197 Consequence of the definit...
spimedv 2198 Deduction version of ~ spi...
spimefv 2199 Version of ~ spime with a ...
nfim1 2200 A closed form of ~ nfim . ...
nfan1 2201 A closed form of ~ nfan . ...
19.3t 2202 Closed form of ~ 19.3 and ...
19.3 2203 A wff may be quantified wi...
19.9d 2204 A deduction version of one...
19.9t 2205 Closed form of ~ 19.9 and ...
19.9 2206 A wff may be existentially...
19.21t 2207 Closed form of Theorem 19....
19.21 2208 Theorem 19.21 of [Margaris...
stdpc5 2209 An axiom scheme of standar...
19.21-2 2210 Version of ~ 19.21 with tw...
19.23t 2211 Closed form of Theorem 19....
19.23 2212 Theorem 19.23 of [Margaris...
alimd 2213 Deduction form of Theorem ...
alrimi 2214 Inference form of Theorem ...
alrimdd 2215 Deduction form of Theorem ...
alrimd 2216 Deduction form of Theorem ...
eximd 2217 Deduction form of Theorem ...
exlimi 2218 Inference associated with ...
exlimd 2219 Deduction form of Theorem ...
exlimimdd 2220 Existential elimination ru...
exlimdd 2221 Existential elimination ru...
nexd 2222 Deduction for generalizati...
albid 2223 Formula-building rule for ...
exbid 2224 Formula-building rule for ...
nfbidf 2225 An equality theorem for ef...
19.16 2226 Theorem 19.16 of [Margaris...
19.17 2227 Theorem 19.17 of [Margaris...
19.27 2228 Theorem 19.27 of [Margaris...
19.28 2229 Theorem 19.28 of [Margaris...
19.19 2230 Theorem 19.19 of [Margaris...
19.36 2231 Theorem 19.36 of [Margaris...
19.36i 2232 Inference associated with ...
19.37 2233 Theorem 19.37 of [Margaris...
19.32 2234 Theorem 19.32 of [Margaris...
19.31 2235 Theorem 19.31 of [Margaris...
19.41 2236 Theorem 19.41 of [Margaris...
19.42 2237 Theorem 19.42 of [Margaris...
19.44 2238 Theorem 19.44 of [Margaris...
19.45 2239 Theorem 19.45 of [Margaris...
spimfv 2240 Specialization, using impl...
chvarfv 2241 Implicit substitution of `...
cbv3v2 2242 Version of ~ cbv3 with two...
sbalex 2243 Equivalence of two ways to...
sbalexOLD 2244 Obsolete version of ~ sbal...
sb4av 2245 Version of ~ sb4a with a d...
sbimd 2246 Deduction substituting bot...
sbbid 2247 Deduction substituting bot...
2sbbid 2248 Deduction doubly substitut...
sbequ1 2249 An equality theorem for su...
sbequ2 2250 An equality theorem for su...
stdpc7 2251 One of the two equality ax...
sbequ12 2252 An equality theorem for su...
sbequ12r 2253 An equality theorem for su...
sbelx 2254 Elimination of substitutio...
sbequ12a 2255 An equality theorem for su...
sbid 2256 An identity theorem for su...
sbcov 2257 A composition law for subs...
sbcovOLD 2258 Obsolete version of ~ sbco...
sb6a 2259 Equivalence for substituti...
sbid2vw 2260 Reverting substitution yie...
axc16g 2261 Generalization of ~ axc16 ...
axc16 2262 Proof of older axiom ~ ax-...
axc16gb 2263 Biconditional strengthenin...
axc16nf 2264 If ~ dtru is false, then t...
axc11v 2265 Version of ~ axc11 with a ...
axc11rv 2266 Version of ~ axc11r with a...
drsb2 2267 Formula-building lemma for...
equsalv 2268 An equivalence related to ...
equsexv 2269 An equivalence related to ...
equsexvOLD 2270 Obsolete version of ~ equs...
sbft 2271 Substitution has no effect...
sbf 2272 Substitution for a variabl...
sbf2 2273 Substitution has no effect...
sbh 2274 Substitution for a variabl...
hbs1 2275 The setvar ` x ` is not fr...
nfs1f 2276 If ` x ` is not free in ` ...
sb5 2277 Alternate definition of su...
sb5OLD 2278 Obsolete version of ~ sb5 ...
sb56OLD 2279 Obsolete version of ~ sbal...
equs5av 2280 A property related to subs...
2sb5 2281 Equivalence for double sub...
sbco4lemOLDOLD 2282 Obsolete version of ~ sbco...
dfsb7 2283 An alternate definition of...
sbn 2284 Negation inside and outsid...
sbex 2285 Move existential quantifie...
nf5 2286 Alternate definition of ~ ...
nf6 2287 An alternate definition of...
nf5d 2288 Deduce that ` x ` is not f...
nf5di 2289 Since the converse holds b...
19.9h 2290 A wff may be existentially...
19.21h 2291 Theorem 19.21 of [Margaris...
19.23h 2292 Theorem 19.23 of [Margaris...
exlimih 2293 Inference associated with ...
exlimdh 2294 Deduction form of Theorem ...
equsalhw 2295 Version of ~ equsalh with ...
equsexhv 2296 An equivalence related to ...
hba1 2297 The setvar ` x ` is not fr...
hbnt 2298 Closed theorem version of ...
hbn 2299 If ` x ` is not free in ` ...
hbnd 2300 Deduction form of bound-va...
hbim1 2301 A closed form of ~ hbim . ...
hbimd 2302 Deduction form of bound-va...
hbim 2303 If ` x ` is not free in ` ...
hban 2304 If ` x ` is not free in ` ...
hb3an 2305 If ` x ` is not free in ` ...
sbi2 2306 Introduction of implicatio...
sbim 2307 Implication inside and out...
sbrim 2308 Substitution in an implica...
sbrimOLD 2309 Obsolete version of ~ sbri...
sblim 2310 Substitution in an implica...
sbor 2311 Disjunction inside and out...
sbbi 2312 Equivalence inside and out...
sblbis 2313 Introduce left bicondition...
sbrbis 2314 Introduce right biconditio...
sbrbif 2315 Introduce right biconditio...
sbnf 2316 Move nonfree predicate in ...
sbnfOLD 2317 Obsolete version of ~ sbnf...
sbiev 2318 Conversion of implicit sub...
sbievOLD 2319 Obsolete version of ~ sbie...
sbiedw 2320 Conversion of implicit sub...
axc7 2321 Show that the original axi...
axc7e 2322 Abbreviated version of ~ a...
modal-b 2323 The analogue in our predic...
19.9ht 2324 A closed version of ~ 19.9...
axc4 2325 Show that the original axi...
axc4i 2326 Inference version of ~ axc...
nfal 2327 If ` x ` is not free in ` ...
nfex 2328 If ` x ` is not free in ` ...
hbex 2329 If ` x ` is not free in ` ...
nfnf 2330 If ` x ` is not free in ` ...
19.12 2331 Theorem 19.12 of [Margaris...
nfald 2332 Deduction form of ~ nfal ....
nfexd 2333 If ` x ` is not free in ` ...
nfsbv 2334 If ` z ` is not free in ` ...
nfsbvOLD 2335 Obsolete version of ~ nfsb...
sbco2v 2336 A composition law for subs...
aaan 2337 Distribute universal quant...
aaanOLD 2338 Obsolete version of ~ aaan...
eeor 2339 Distribute existential qua...
eeorOLD 2340 Obsolete version of ~ eeor...
cbv3v 2341 Rule used to change bound ...
cbv1v 2342 Rule used to change bound ...
cbv2w 2343 Rule used to change bound ...
cbvaldw 2344 Deduction used to change b...
cbvexdw 2345 Deduction used to change b...
cbv3hv 2346 Rule used to change bound ...
cbvalv1 2347 Rule used to change bound ...
cbvexv1 2348 Rule used to change bound ...
cbval2v 2349 Rule used to change bound ...
cbvex2v 2350 Rule used to change bound ...
dvelimhw 2351 Proof of ~ dvelimh without...
pm11.53 2352 Theorem *11.53 in [Whitehe...
19.12vv 2353 Special case of ~ 19.12 wh...
eean 2354 Distribute existential qua...
eeanv 2355 Distribute a pair of exist...
eeeanv 2356 Distribute three existenti...
ee4anv 2357 Distribute two pairs of ex...
sb8v 2358 Substitution of variable i...
sb8f 2359 Substitution of variable i...
sb8fOLD 2360 Obsolete version of ~ sb8f...
sb8ef 2361 Substitution of variable i...
2sb8ef 2362 An equivalent expression f...
sb6rfv 2363 Reversed substitution. Ve...
sbnf2 2364 Two ways of expressing " `...
exsb 2365 An equivalent expression f...
2exsb 2366 An equivalent expression f...
sbbib 2367 Reversal of substitution. ...
sbbibvv 2368 Reversal of substitution. ...
cbvsbvf 2369 Change the bound variable ...
cleljustALT 2370 Alternate proof of ~ clelj...
cleljustALT2 2371 Alternate proof of ~ clelj...
equs5aALT 2372 Alternate proof of ~ equs5...
equs5eALT 2373 Alternate proof of ~ equs5...
axc11r 2374 Same as ~ axc11 but with r...
dral1v 2375 Formula-building lemma for...
dral1vOLD 2376 Obsolete version of ~ dral...
drex1v 2377 Formula-building lemma for...
drnf1v 2378 Formula-building lemma for...
drnf1vOLD 2379 Obsolete version of ~ drnf...
ax13v 2381 A weaker version of ~ ax-1...
ax13lem1 2382 A version of ~ ax13v with ...
ax13 2383 Derive ~ ax-13 from ~ ax13...
ax13lem2 2384 Lemma for ~ nfeqf2 . This...
nfeqf2 2385 An equation between setvar...
dveeq2 2386 Quantifier introduction wh...
nfeqf1 2387 An equation between setvar...
dveeq1 2388 Quantifier introduction wh...
nfeqf 2389 A variable is effectively ...
axc9 2390 Derive set.mm's original ~...
ax6e 2391 At least one individual ex...
ax6 2392 Theorem showing that ~ ax-...
axc10 2393 Show that the original axi...
spimt 2394 Closed theorem form of ~ s...
spim 2395 Specialization, using impl...
spimed 2396 Deduction version of ~ spi...
spime 2397 Existential introduction, ...
spimv 2398 A version of ~ spim with a...
spimvALT 2399 Alternate proof of ~ spimv...
spimev 2400 Distinct-variable version ...
spv 2401 Specialization, using impl...
spei 2402 Inference from existential...
chvar 2403 Implicit substitution of `...
chvarv 2404 Implicit substitution of `...
cbv3 2405 Rule used to change bound ...
cbval 2406 Rule used to change bound ...
cbvex 2407 Rule used to change bound ...
cbvalv 2408 Rule used to change bound ...
cbvexv 2409 Rule used to change bound ...
cbv1 2410 Rule used to change bound ...
cbv2 2411 Rule used to change bound ...
cbv3h 2412 Rule used to change bound ...
cbv1h 2413 Rule used to change bound ...
cbv2h 2414 Rule used to change bound ...
cbvald 2415 Deduction used to change b...
cbvexd 2416 Deduction used to change b...
cbvaldva 2417 Rule used to change the bo...
cbvexdva 2418 Rule used to change the bo...
cbval2 2419 Rule used to change bound ...
cbvex2 2420 Rule used to change bound ...
cbval2vv 2421 Rule used to change bound ...
cbvex2vv 2422 Rule used to change bound ...
cbvex4v 2423 Rule used to change bound ...
equs4 2424 Lemma used in proofs of im...
equsal 2425 An equivalence related to ...
equsex 2426 An equivalence related to ...
equsexALT 2427 Alternate proof of ~ equse...
equsalh 2428 An equivalence related to ...
equsexh 2429 An equivalence related to ...
axc15 2430 Derivation of set.mm's ori...
ax12 2431 Rederivation of Axiom ~ ax...
ax12b 2432 A bidirectional version of...
ax13ALT 2433 Alternate proof of ~ ax13 ...
axc11n 2434 Derive set.mm's original ~...
aecom 2435 Commutation law for identi...
aecoms 2436 A commutation rule for ide...
naecoms 2437 A commutation rule for dis...
axc11 2438 Show that ~ ax-c11 can be ...
hbae 2439 All variables are effectiv...
hbnae 2440 All variables are effectiv...
nfae 2441 All variables are effectiv...
nfnae 2442 All variables are effectiv...
hbnaes 2443 Rule that applies ~ hbnae ...
axc16i 2444 Inference with ~ axc16 as ...
axc16nfALT 2445 Alternate proof of ~ axc16...
dral2 2446 Formula-building lemma for...
dral1 2447 Formula-building lemma for...
dral1ALT 2448 Alternate proof of ~ dral1...
drex1 2449 Formula-building lemma for...
drex2 2450 Formula-building lemma for...
drnf1 2451 Formula-building lemma for...
drnf2 2452 Formula-building lemma for...
nfald2 2453 Variation on ~ nfald which...
nfexd2 2454 Variation on ~ nfexd which...
exdistrf 2455 Distribution of existentia...
dvelimf 2456 Version of ~ dvelimv witho...
dvelimdf 2457 Deduction form of ~ dvelim...
dvelimh 2458 Version of ~ dvelim withou...
dvelim 2459 This theorem can be used t...
dvelimv 2460 Similar to ~ dvelim with f...
dvelimnf 2461 Version of ~ dvelim using ...
dveeq2ALT 2462 Alternate proof of ~ dveeq...
equvini 2463 A variable introduction la...
equvel 2464 A variable elimination law...
equs5a 2465 A property related to subs...
equs5e 2466 A property related to subs...
equs45f 2467 Two ways of expressing sub...
equs5 2468 Lemma used in proofs of su...
dveel1 2469 Quantifier introduction wh...
dveel2 2470 Quantifier introduction wh...
axc14 2471 Axiom ~ ax-c14 is redundan...
sb6x 2472 Equivalence involving subs...
sbequ5 2473 Substitution does not chan...
sbequ6 2474 Substitution does not chan...
sb5rf 2475 Reversed substitution. Us...
sb6rf 2476 Reversed substitution. Fo...
ax12vALT 2477 Alternate proof of ~ ax12v...
2ax6elem 2478 We can always find values ...
2ax6e 2479 We can always find values ...
2sb5rf 2480 Reversed double substituti...
2sb6rf 2481 Reversed double substituti...
sbel2x 2482 Elimination of double subs...
sb4b 2483 Simplified definition of s...
sb3b 2484 Simplified definition of s...
sb3 2485 One direction of a simplif...
sb1 2486 One direction of a simplif...
sb2 2487 One direction of a simplif...
sb4a 2488 A version of one implicati...
dfsb1 2489 Alternate definition of su...
hbsb2 2490 Bound-variable hypothesis ...
nfsb2 2491 Bound-variable hypothesis ...
hbsb2a 2492 Special case of a bound-va...
sb4e 2493 One direction of a simplif...
hbsb2e 2494 Special case of a bound-va...
hbsb3 2495 If ` y ` is not free in ` ...
nfs1 2496 If ` y ` is not free in ` ...
axc16ALT 2497 Alternate proof of ~ axc16...
axc16gALT 2498 Alternate proof of ~ axc16...
equsb1 2499 Substitution applied to an...
equsb2 2500 Substitution applied to an...
dfsb2 2501 An alternate definition of...
dfsb3 2502 An alternate definition of...
drsb1 2503 Formula-building lemma for...
sb2ae 2504 In the case of two success...
sb6f 2505 Equivalence for substituti...
sb5f 2506 Equivalence for substituti...
nfsb4t 2507 A variable not free in a p...
nfsb4 2508 A variable not free in a p...
sbequ8 2509 Elimination of equality fr...
sbie 2510 Conversion of implicit sub...
sbied 2511 Conversion of implicit sub...
sbiedv 2512 Conversion of implicit sub...
2sbiev 2513 Conversion of double impli...
sbcom3 2514 Substituting ` y ` for ` x...
sbco 2515 A composition law for subs...
sbid2 2516 An identity law for substi...
sbid2v 2517 An identity law for substi...
sbidm 2518 An idempotent law for subs...
sbco2 2519 A composition law for subs...
sbco2d 2520 A composition law for subs...
sbco3 2521 A composition law for subs...
sbcom 2522 A commutativity law for su...
sbtrt 2523 Partially closed form of ~...
sbtr 2524 A partial converse to ~ sb...
sb8 2525 Substitution of variable i...
sb8e 2526 Substitution of variable i...
sb9 2527 Commutation of quantificat...
sb9i 2528 Commutation of quantificat...
sbhb 2529 Two ways of expressing " `...
nfsbd 2530 Deduction version of ~ nfs...
nfsb 2531 If ` z ` is not free in ` ...
hbsb 2532 If ` z ` is not free in ` ...
sb7f 2533 This version of ~ dfsb7 do...
sb7h 2534 This version of ~ dfsb7 do...
sb10f 2535 Hao Wang's identity axiom ...
sbal1 2536 Check out ~ sbal for a ver...
sbal2 2537 Move quantifier in and out...
2sb8e 2538 An equivalent expression f...
dfmoeu 2539 An elementary proof of ~ m...
dfeumo 2540 An elementary proof showin...
mojust 2542 Soundness justification th...
nexmo 2544 Nonexistence implies uniqu...
exmo 2545 Any proposition holds for ...
moabs 2546 Absorption of existence co...
moim 2547 The at-most-one quantifier...
moimi 2548 The at-most-one quantifier...
moimdv 2549 The at-most-one quantifier...
mobi 2550 Equivalence theorem for th...
mobii 2551 Formula-building rule for ...
mobidv 2552 Formula-building rule for ...
mobid 2553 Formula-building rule for ...
moa1 2554 If an implication holds fo...
moan 2555 "At most one" is still the...
moani 2556 "At most one" is still tru...
moor 2557 "At most one" is still the...
mooran1 2558 "At most one" imports disj...
mooran2 2559 "At most one" exports disj...
nfmo1 2560 Bound-variable hypothesis ...
nfmod2 2561 Bound-variable hypothesis ...
nfmodv 2562 Bound-variable hypothesis ...
nfmov 2563 Bound-variable hypothesis ...
nfmod 2564 Bound-variable hypothesis ...
nfmo 2565 Bound-variable hypothesis ...
mof 2566 Version of ~ df-mo with di...
mo3 2567 Alternate definition of th...
mo 2568 Equivalent definitions of ...
mo4 2569 At-most-one quantifier exp...
mo4f 2570 At-most-one quantifier exp...
eu3v 2573 An alternate way to expres...
eujust 2574 Soundness justification th...
eujustALT 2575 Alternate proof of ~ eujus...
eu6lem 2576 Lemma of ~ eu6im . A diss...
eu6 2577 Alternate definition of th...
eu6im 2578 One direction of ~ eu6 nee...
euf 2579 Version of ~ eu6 with disj...
euex 2580 Existential uniqueness imp...
eumo 2581 Existential uniqueness imp...
eumoi 2582 Uniqueness inferred from e...
exmoeub 2583 Existence implies that uni...
exmoeu 2584 Existence is equivalent to...
moeuex 2585 Uniqueness implies that ex...
moeu 2586 Uniqueness is equivalent t...
eubi 2587 Equivalence theorem for th...
eubii 2588 Introduce unique existenti...
eubidv 2589 Formula-building rule for ...
eubid 2590 Formula-building rule for ...
nfeu1 2591 Bound-variable hypothesis ...
nfeu1ALT 2592 Alternate proof of ~ nfeu1...
nfeud2 2593 Bound-variable hypothesis ...
nfeudw 2594 Bound-variable hypothesis ...
nfeud 2595 Bound-variable hypothesis ...
nfeuw 2596 Bound-variable hypothesis ...
nfeu 2597 Bound-variable hypothesis ...
dfeu 2598 Rederive ~ df-eu from the ...
dfmo 2599 Rederive ~ df-mo from the ...
euequ 2600 There exists a unique set ...
sb8eulem 2601 Lemma. Factor out the com...
sb8euv 2602 Variable substitution in u...
sb8eu 2603 Variable substitution in u...
sb8mo 2604 Variable substitution for ...
cbvmovw 2605 Change bound variable. Us...
cbvmow 2606 Rule used to change bound ...
cbvmo 2607 Rule used to change bound ...
cbveuvw 2608 Change bound variable. Us...
cbveuw 2609 Version of ~ cbveu with a ...
cbveu 2610 Rule used to change bound ...
cbveuALT 2611 Alternative proof of ~ cbv...
eu2 2612 An alternate way of defini...
eu1 2613 An alternate way to expres...
euor 2614 Introduce a disjunct into ...
euorv 2615 Introduce a disjunct into ...
euor2 2616 Introduce or eliminate a d...
sbmo 2617 Substitution into an at-mo...
eu4 2618 Uniqueness using implicit ...
euimmo 2619 Existential uniqueness imp...
euim 2620 Add unique existential qua...
moanimlem 2621 Factor out the common proo...
moanimv 2622 Introduction of a conjunct...
moanim 2623 Introduction of a conjunct...
euan 2624 Introduction of a conjunct...
moanmo 2625 Nested at-most-one quantif...
moaneu 2626 Nested at-most-one and uni...
euanv 2627 Introduction of a conjunct...
mopick 2628 "At most one" picks a vari...
moexexlem 2629 Factor out the proof skele...
2moexv 2630 Double quantification with...
moexexvw 2631 "At most one" double quant...
2moswapv 2632 A condition allowing to sw...
2euswapv 2633 A condition allowing to sw...
2euexv 2634 Double quantification with...
2exeuv 2635 Double existential uniquen...
eupick 2636 Existential uniqueness "pi...
eupicka 2637 Version of ~ eupick with c...
eupickb 2638 Existential uniqueness "pi...
eupickbi 2639 Theorem *14.26 in [Whitehe...
mopick2 2640 "At most one" can show the...
moexex 2641 "At most one" double quant...
moexexv 2642 "At most one" double quant...
2moex 2643 Double quantification with...
2euex 2644 Double quantification with...
2eumo 2645 Nested unique existential ...
2eu2ex 2646 Double existential uniquen...
2moswap 2647 A condition allowing to sw...
2euswap 2648 A condition allowing to sw...
2exeu 2649 Double existential uniquen...
2mo2 2650 Two ways of expressing "th...
2mo 2651 Two ways of expressing "th...
2mos 2652 Double "there exists at mo...
2mosOLD 2653 Obsolete version of ~ 2mos...
2eu1 2654 Double existential uniquen...
2eu1v 2655 Double existential uniquen...
2eu2 2656 Double existential uniquen...
2eu3 2657 Double existential uniquen...
2eu4 2658 This theorem provides us w...
2eu5 2659 An alternate definition of...
2eu6 2660 Two equivalent expressions...
2eu7 2661 Two equivalent expressions...
2eu8 2662 Two equivalent expressions...
euae 2663 Two ways to express "exact...
exists1 2664 Two ways to express "exact...
exists2 2665 A condition implying that ...
barbara 2666 "Barbara", one of the fund...
celarent 2667 "Celarent", one of the syl...
darii 2668 "Darii", one of the syllog...
dariiALT 2669 Alternate proof of ~ darii...
ferio 2670 "Ferio" ("Ferioque"), one ...
barbarilem 2671 Lemma for ~ barbari and th...
barbari 2672 "Barbari", one of the syll...
barbariALT 2673 Alternate proof of ~ barba...
celaront 2674 "Celaront", one of the syl...
cesare 2675 "Cesare", one of the syllo...
camestres 2676 "Camestres", one of the sy...
festino 2677 "Festino", one of the syll...
festinoALT 2678 Alternate proof of ~ festi...
baroco 2679 "Baroco", one of the syllo...
barocoALT 2680 Alternate proof of ~ festi...
cesaro 2681 "Cesaro", one of the syllo...
camestros 2682 "Camestros", one of the sy...
datisi 2683 "Datisi", one of the syllo...
disamis 2684 "Disamis", one of the syll...
ferison 2685 "Ferison", one of the syll...
bocardo 2686 "Bocardo", one of the syll...
darapti 2687 "Darapti", one of the syll...
daraptiALT 2688 Alternate proof of ~ darap...
felapton 2689 "Felapton", one of the syl...
calemes 2690 "Calemes", one of the syll...
dimatis 2691 "Dimatis", one of the syll...
fresison 2692 "Fresison", one of the syl...
calemos 2693 "Calemos", one of the syll...
fesapo 2694 "Fesapo", one of the syllo...
bamalip 2695 "Bamalip", one of the syll...
axia1 2696 Left 'and' elimination (in...
axia2 2697 Right 'and' elimination (i...
axia3 2698 'And' introduction (intuit...
axin1 2699 'Not' introduction (intuit...
axin2 2700 'Not' elimination (intuiti...
axio 2701 Definition of 'or' (intuit...
axi4 2702 Specialization (intuitioni...
axi5r 2703 Converse of ~ axc4 (intuit...
axial 2704 The setvar ` x ` is not fr...
axie1 2705 The setvar ` x ` is not fr...
axie2 2706 A key property of existent...
axi9 2707 Axiom of existence (intuit...
axi10 2708 Axiom of Quantifier Substi...
axi12 2709 Axiom of Quantifier Introd...
axbnd 2710 Axiom of Bundling (intuiti...
axexte 2712 The axiom of extensionalit...
axextg 2713 A generalization of the ax...
axextb 2714 A bidirectional version of...
axextmo 2715 There exists at most one s...
nulmo 2716 There exists at most one e...
eleq1ab 2719 Extension (in the sense of...
cleljustab 2720 Extension of ~ cleljust fr...
abid 2721 Simplification of class ab...
vexwt 2722 A standard theorem of pred...
vexw 2723 If ` ph ` is a theorem, th...
vextru 2724 Every setvar is a member o...
nfsab1 2725 Bound-variable hypothesis ...
hbab1 2726 Bound-variable hypothesis ...
hbab1OLD 2727 Obsolete version of ~ hbab...
hbab 2728 Bound-variable hypothesis ...
hbabg 2729 Bound-variable hypothesis ...
nfsab 2730 Bound-variable hypothesis ...
nfsabg 2731 Bound-variable hypothesis ...
dfcleq 2733 The defining characterizat...
cvjust 2734 Every set is a class. Pro...
ax9ALT 2735 Proof of ~ ax-9 from Tarsk...
eleq2w2 2736 A weaker version of ~ eleq...
eqriv 2737 Infer equality of classes ...
eqrdv 2738 Deduce equality of classes...
eqrdav 2739 Deduce equality of classes...
eqid 2740 Law of identity (reflexivi...
eqidd 2741 Class identity law with an...
eqeq1d 2742 Deduction from equality to...
eqeq1dALT 2743 Alternate proof of ~ eqeq1...
eqeq1 2744 Equality implies equivalen...
eqeq1i 2745 Inference from equality to...
eqcomd 2746 Deduction from commutative...
eqcom 2747 Commutative law for class ...
eqcoms 2748 Inference applying commuta...
eqcomi 2749 Inference from commutative...
neqcomd 2750 Commute an inequality. (C...
eqeq2d 2751 Deduction from equality to...
eqeq2 2752 Equality implies equivalen...
eqeq2i 2753 Inference from equality to...
eqeqan12d 2754 A useful inference for sub...
eqeqan12rd 2755 A useful inference for sub...
eqeq12d 2756 A useful inference for sub...
eqeq12 2757 Equality relationship amon...
eqeq12i 2758 A useful inference for sub...
eqeq12OLD 2759 Obsolete version of ~ eqeq...
eqeq12dOLD 2760 Obsolete version of ~ eqeq...
eqeqan12dOLD 2761 Obsolete version of ~ eqeq...
eqeqan12dALT 2762 Alternate proof of ~ eqeqa...
eqtr 2763 Transitive law for class e...
eqtr2 2764 A transitive law for class...
eqtr2OLD 2765 Obsolete version of eqtr2 ...
eqtr3 2766 A transitive law for class...
eqtr3OLD 2767 Obsolete version of ~ eqtr...
eqtri 2768 An equality transitivity i...
eqtr2i 2769 An equality transitivity i...
eqtr3i 2770 An equality transitivity i...
eqtr4i 2771 An equality transitivity i...
3eqtri 2772 An inference from three ch...
3eqtrri 2773 An inference from three ch...
3eqtr2i 2774 An inference from three ch...
3eqtr2ri 2775 An inference from three ch...
3eqtr3i 2776 An inference from three ch...
3eqtr3ri 2777 An inference from three ch...
3eqtr4i 2778 An inference from three ch...
3eqtr4ri 2779 An inference from three ch...
eqtrd 2780 An equality transitivity d...
eqtr2d 2781 An equality transitivity d...
eqtr3d 2782 An equality transitivity e...
eqtr4d 2783 An equality transitivity e...
3eqtrd 2784 A deduction from three cha...
3eqtrrd 2785 A deduction from three cha...
3eqtr2d 2786 A deduction from three cha...
3eqtr2rd 2787 A deduction from three cha...
3eqtr3d 2788 A deduction from three cha...
3eqtr3rd 2789 A deduction from three cha...
3eqtr4d 2790 A deduction from three cha...
3eqtr4rd 2791 A deduction from three cha...
eqtrid 2792 An equality transitivity d...
eqtr2id 2793 An equality transitivity d...
eqtr3id 2794 An equality transitivity d...
eqtr3di 2795 An equality transitivity d...
eqtrdi 2796 An equality transitivity d...
eqtr2di 2797 An equality transitivity d...
eqtr4di 2798 An equality transitivity d...
eqtr4id 2799 An equality transitivity d...
sylan9eq 2800 An equality transitivity d...
sylan9req 2801 An equality transitivity d...
sylan9eqr 2802 An equality transitivity d...
3eqtr3g 2803 A chained equality inferen...
3eqtr3a 2804 A chained equality inferen...
3eqtr4g 2805 A chained equality inferen...
3eqtr4a 2806 A chained equality inferen...
eq2tri 2807 A compound transitive infe...
iseqsetvlem 2808 Lemma for ~ iseqsetv-cleq ...
iseqsetv-cleq 2809 Alternate proof of ~ iseqs...
abbi 2810 Equivalent formulas yield ...
abbidv 2811 Equivalent wff's yield equ...
abbii 2812 Equivalent wff's yield equ...
abbid 2813 Equivalent wff's yield equ...
abbib 2814 Equal class abstractions r...
cbvabv 2815 Rule used to change bound ...
cbvabw 2816 Rule used to change bound ...
cbvab 2817 Rule used to change bound ...
eqabbw 2818 Version of ~ eqabb using i...
dfclel 2820 Characterization of the el...
elex2 2821 If a class contains anothe...
issettru 2822 Weak version of ~ isset . ...
iseqsetv-clel 2823 Alternate proof of ~ iseqs...
issetlem 2824 Lemma for ~ elisset and ~ ...
elissetv 2825 An element of a class exis...
elisset 2826 An element of a class exis...
eleq1w 2827 Weaker version of ~ eleq1 ...
eleq2w 2828 Weaker version of ~ eleq2 ...
eleq1d 2829 Deduction from equality to...
eleq2d 2830 Deduction from equality to...
eleq2dALT 2831 Alternate proof of ~ eleq2...
eleq1 2832 Equality implies equivalen...
eleq2 2833 Equality implies equivalen...
eleq12 2834 Equality implies equivalen...
eleq1i 2835 Inference from equality to...
eleq2i 2836 Inference from equality to...
eleq12i 2837 Inference from equality to...
eleq12d 2838 Deduction from equality to...
eleq1a 2839 A transitive-type law rela...
eqeltri 2840 Substitution of equal clas...
eqeltrri 2841 Substitution of equal clas...
eleqtri 2842 Substitution of equal clas...
eleqtrri 2843 Substitution of equal clas...
eqeltrd 2844 Substitution of equal clas...
eqeltrrd 2845 Deduction that substitutes...
eleqtrd 2846 Deduction that substitutes...
eleqtrrd 2847 Deduction that substitutes...
eqeltrid 2848 A membership and equality ...
eqeltrrid 2849 A membership and equality ...
eleqtrid 2850 A membership and equality ...
eleqtrrid 2851 A membership and equality ...
eqeltrdi 2852 A membership and equality ...
eqeltrrdi 2853 A membership and equality ...
eleqtrdi 2854 A membership and equality ...
eleqtrrdi 2855 A membership and equality ...
3eltr3i 2856 Substitution of equal clas...
3eltr4i 2857 Substitution of equal clas...
3eltr3d 2858 Substitution of equal clas...
3eltr4d 2859 Substitution of equal clas...
3eltr3g 2860 Substitution of equal clas...
3eltr4g 2861 Substitution of equal clas...
eleq2s 2862 Substitution of equal clas...
eqneltri 2863 If a class is not an eleme...
eqneltrd 2864 If a class is not an eleme...
eqneltrrd 2865 If a class is not an eleme...
neleqtrd 2866 If a class is not an eleme...
neleqtrrd 2867 If a class is not an eleme...
nelneq 2868 A way of showing two class...
nelneq2 2869 A way of showing two class...
eqsb1 2870 Substitution for the left-...
clelsb1 2871 Substitution for the first...
clelsb2 2872 Substitution for the secon...
clelsb2OLD 2873 Obsolete version of ~ clel...
cleqh 2874 Establish equality between...
hbxfreq 2875 A utility lemma to transfe...
hblem 2876 Change the free variable o...
hblemg 2877 Change the free variable o...
eqabdv 2878 Deduction from a wff to a ...
eqabcdv 2879 Deduction from a wff to a ...
eqabi 2880 Equality of a class variab...
abid1 2881 Every class is equal to a ...
abid2 2882 A simplification of class ...
eqab 2883 One direction of ~ eqabb i...
eqabb 2884 Equality of a class variab...
eqabbOLD 2885 Obsolete version of ~ eqab...
eqabcb 2886 Equality of a class variab...
eqabrd 2887 Equality of a class variab...
eqabri 2888 Equality of a class variab...
eqabcri 2889 Equality of a class variab...
clelab 2890 Membership of a class vari...
clabel 2891 Membership of a class abst...
sbab 2892 The right-hand side of the...
nfcjust 2894 Justification theorem for ...
nfci 2896 Deduce that a class ` A ` ...
nfcii 2897 Deduce that a class ` A ` ...
nfcr 2898 Consequence of the not-fre...
nfcrALT 2899 Alternate version of ~ nfc...
nfcri 2900 Consequence of the not-fre...
nfcd 2901 Deduce that a class ` A ` ...
nfcrd 2902 Consequence of the not-fre...
nfcrii 2903 Consequence of the not-fre...
nfceqdf 2904 An equality theorem for ef...
nfceqi 2905 Equality theorem for class...
nfcxfr 2906 A utility lemma to transfe...
nfcxfrd 2907 A utility lemma to transfe...
nfcv 2908 If ` x ` is disjoint from ...
nfcvd 2909 If ` x ` is disjoint from ...
nfab1 2910 Bound-variable hypothesis ...
nfnfc1 2911 The setvar ` x ` is bound ...
clelsb1fw 2912 Substitution for the first...
clelsb1f 2913 Substitution for the first...
nfab 2914 Bound-variable hypothesis ...
nfabg 2915 Bound-variable hypothesis ...
nfaba1 2916 Bound-variable hypothesis ...
nfaba1OLD 2917 Obsolete version of ~ nfab...
nfaba1g 2918 Bound-variable hypothesis ...
nfeqd 2919 Hypothesis builder for equ...
nfeld 2920 Hypothesis builder for ele...
nfnfc 2921 Hypothesis builder for ` F...
nfeq 2922 Hypothesis builder for equ...
nfel 2923 Hypothesis builder for ele...
nfeq1 2924 Hypothesis builder for equ...
nfel1 2925 Hypothesis builder for ele...
nfeq2 2926 Hypothesis builder for equ...
nfel2 2927 Hypothesis builder for ele...
drnfc1 2928 Formula-building lemma for...
drnfc1OLD 2929 Obsolete version of ~ drnf...
drnfc2 2930 Formula-building lemma for...
drnfc2OLD 2931 Obsolete version of ~ drnf...
nfabdw 2932 Bound-variable hypothesis ...
nfabdwOLD 2933 Obsolete version of ~ nfab...
nfabd 2934 Bound-variable hypothesis ...
nfabd2 2935 Bound-variable hypothesis ...
dvelimdc 2936 Deduction form of ~ dvelim...
dvelimc 2937 Version of ~ dvelim for cl...
nfcvf 2938 If ` x ` and ` y ` are dis...
nfcvf2 2939 If ` x ` and ` y ` are dis...
cleqf 2940 Establish equality between...
eqabf 2941 Equality of a class variab...
abid2f 2942 A simplification of class ...
abid2fOLD 2943 Obsolete version of ~ abid...
sbabel 2944 Theorem to move a substitu...
sbabelOLD 2945 Obsolete version of ~ sbab...
neii 2948 Inference associated with ...
neir 2949 Inference associated with ...
nne 2950 Negation of inequality. (...
neneqd 2951 Deduction eliminating ineq...
neneq 2952 From inequality to non-equ...
neqned 2953 If it is not the case that...
neqne 2954 From non-equality to inequ...
neirr 2955 No class is unequal to its...
exmidne 2956 Excluded middle with equal...
eqneqall 2957 A contradiction concerning...
nonconne 2958 Law of noncontradiction wi...
necon3ad 2959 Contrapositive law deducti...
necon3bd 2960 Contrapositive law deducti...
necon2ad 2961 Contrapositive inference f...
necon2bd 2962 Contrapositive inference f...
necon1ad 2963 Contrapositive deduction f...
necon1bd 2964 Contrapositive deduction f...
necon4ad 2965 Contrapositive inference f...
necon4bd 2966 Contrapositive inference f...
necon3d 2967 Contrapositive law deducti...
necon1d 2968 Contrapositive law deducti...
necon2d 2969 Contrapositive inference f...
necon4d 2970 Contrapositive inference f...
necon3ai 2971 Contrapositive inference f...
necon3aiOLD 2972 Obsolete version of ~ neco...
necon3bi 2973 Contrapositive inference f...
necon1ai 2974 Contrapositive inference f...
necon1bi 2975 Contrapositive inference f...
necon2ai 2976 Contrapositive inference f...
necon2bi 2977 Contrapositive inference f...
necon4ai 2978 Contrapositive inference f...
necon3i 2979 Contrapositive inference f...
necon1i 2980 Contrapositive inference f...
necon2i 2981 Contrapositive inference f...
necon4i 2982 Contrapositive inference f...
necon3abid 2983 Deduction from equality to...
necon3bbid 2984 Deduction from equality to...
necon1abid 2985 Contrapositive deduction f...
necon1bbid 2986 Contrapositive inference f...
necon4abid 2987 Contrapositive law deducti...
necon4bbid 2988 Contrapositive law deducti...
necon2abid 2989 Contrapositive deduction f...
necon2bbid 2990 Contrapositive deduction f...
necon3bid 2991 Deduction from equality to...
necon4bid 2992 Contrapositive law deducti...
necon3abii 2993 Deduction from equality to...
necon3bbii 2994 Deduction from equality to...
necon1abii 2995 Contrapositive inference f...
necon1bbii 2996 Contrapositive inference f...
necon2abii 2997 Contrapositive inference f...
necon2bbii 2998 Contrapositive inference f...
necon3bii 2999 Inference from equality to...
necom 3000 Commutation of inequality....
necomi 3001 Inference from commutative...
necomd 3002 Deduction from commutative...
nesym 3003 Characterization of inequa...
nesymi 3004 Inference associated with ...
nesymir 3005 Inference associated with ...
neeq1d 3006 Deduction for inequality. ...
neeq2d 3007 Deduction for inequality. ...
neeq12d 3008 Deduction for inequality. ...
neeq1 3009 Equality theorem for inequ...
neeq2 3010 Equality theorem for inequ...
neeq1i 3011 Inference for inequality. ...
neeq2i 3012 Inference for inequality. ...
neeq12i 3013 Inference for inequality. ...
eqnetrd 3014 Substitution of equal clas...
eqnetrrd 3015 Substitution of equal clas...
neeqtrd 3016 Substitution of equal clas...
eqnetri 3017 Substitution of equal clas...
eqnetrri 3018 Substitution of equal clas...
neeqtri 3019 Substitution of equal clas...
neeqtrri 3020 Substitution of equal clas...
neeqtrrd 3021 Substitution of equal clas...
eqnetrrid 3022 A chained equality inferen...
3netr3d 3023 Substitution of equality i...
3netr4d 3024 Substitution of equality i...
3netr3g 3025 Substitution of equality i...
3netr4g 3026 Substitution of equality i...
nebi 3027 Contraposition law for ine...
pm13.18 3028 Theorem *13.18 in [Whitehe...
pm13.181 3029 Theorem *13.181 in [Whiteh...
pm13.181OLD 3030 Obsolete version of ~ pm13...
pm2.61ine 3031 Inference eliminating an i...
pm2.21ddne 3032 A contradiction implies an...
pm2.61ne 3033 Deduction eliminating an i...
pm2.61dne 3034 Deduction eliminating an i...
pm2.61dane 3035 Deduction eliminating an i...
pm2.61da2ne 3036 Deduction eliminating two ...
pm2.61da3ne 3037 Deduction eliminating thre...
pm2.61iine 3038 Equality version of ~ pm2....
mteqand 3039 A modus tollens deduction ...
neor 3040 Logical OR with an equalit...
neanior 3041 A De Morgan's law for ineq...
ne3anior 3042 A De Morgan's law for ineq...
neorian 3043 A De Morgan's law for ineq...
nemtbir 3044 An inference from an inequ...
nelne1 3045 Two classes are different ...
nelne2 3046 Two classes are different ...
nelelne 3047 Two classes are different ...
neneor 3048 If two classes are differe...
nfne 3049 Bound-variable hypothesis ...
nfned 3050 Bound-variable hypothesis ...
nabbib 3051 Not equivalent wff's corre...
neli 3054 Inference associated with ...
nelir 3055 Inference associated with ...
nelcon3d 3056 Contrapositive law deducti...
neleq12d 3057 Equality theorem for negat...
neleq1 3058 Equality theorem for negat...
neleq2 3059 Equality theorem for negat...
nfnel 3060 Bound-variable hypothesis ...
nfneld 3061 Bound-variable hypothesis ...
nnel 3062 Negation of negated member...
elnelne1 3063 Two classes are different ...
elnelne2 3064 Two classes are different ...
pm2.24nel 3065 A contradiction concerning...
pm2.61danel 3066 Deduction eliminating an e...
rgen 3069 Generalization rule for re...
ralel 3070 All elements of a class ar...
rgenw 3071 Generalization rule for re...
rgen2w 3072 Generalization rule for re...
mprg 3073 Modus ponens combined with...
mprgbir 3074 Modus ponens on biconditio...
raln 3075 Restricted universally qua...
ralnex 3078 Relationship between restr...
dfrex2 3079 Relationship between restr...
nrex 3080 Inference adding restricte...
alral 3081 Universal quantification i...
rexex 3082 Restricted existence impli...
rextru 3083 Two ways of expressing tha...
ralimi2 3084 Inference quantifying both...
reximi2 3085 Inference quantifying both...
ralimia 3086 Inference quantifying both...
reximia 3087 Inference quantifying both...
ralimiaa 3088 Inference quantifying both...
ralimi 3089 Inference quantifying both...
reximi 3090 Inference quantifying both...
ral2imi 3091 Inference quantifying ante...
ralim 3092 Distribution of restricted...
rexim 3093 Theorem 19.22 of [Margaris...
reximiaOLD 3094 Obsolete version of ~ rexi...
ralbii2 3095 Inference adding different...
rexbii2 3096 Inference adding different...
ralbiia 3097 Inference adding restricte...
rexbiia 3098 Inference adding restricte...
ralbii 3099 Inference adding restricte...
rexbii 3100 Inference adding restricte...
ralanid 3101 Cancellation law for restr...
rexanid 3102 Cancellation law for restr...
ralcom3 3103 A commutation law for rest...
ralcom3OLD 3104 Obsolete version of ~ ralc...
dfral2 3105 Relationship between restr...
rexnal 3106 Relationship between restr...
ralinexa 3107 A transformation of restri...
rexanali 3108 A transformation of restri...
ralbi 3109 Distribute a restricted un...
rexbi 3110 Distribute restricted quan...
rexbiOLD 3111 Obsolete version of ~ rexb...
ralrexbid 3112 Formula-building rule for ...
ralrexbidOLD 3113 Obsolete version of ~ ralr...
r19.35 3114 Restricted quantifier vers...
r19.35OLD 3115 Obsolete version of ~ 19.3...
r19.26m 3116 Version of ~ 19.26 and ~ r...
r19.26 3117 Restricted quantifier vers...
r19.26-3 3118 Version of ~ r19.26 with t...
ralbiim 3119 Split a biconditional and ...
r19.29 3120 Restricted quantifier vers...
r19.29OLD 3121 Obsolete version of ~ r19....
r19.29r 3122 Restricted quantifier vers...
r19.29rOLD 3123 Obsolete version of ~ r19....
r19.29imd 3124 Theorem 19.29 of [Margaris...
r19.40 3125 Restricted quantifier vers...
r19.30 3126 Restricted quantifier vers...
r19.30OLD 3127 Obsolete version of ~ 19.3...
r19.43 3128 Restricted quantifier vers...
2ralimi 3129 Inference quantifying both...
3ralimi 3130 Inference quantifying both...
4ralimi 3131 Inference quantifying both...
5ralimi 3132 Inference quantifying both...
6ralimi 3133 Inference quantifying both...
2ralbii 3134 Inference adding two restr...
2rexbii 3135 Inference adding two restr...
3ralbii 3136 Inference adding three res...
4ralbii 3137 Inference adding four rest...
2ralbiim 3138 Split a biconditional and ...
ralnex2 3139 Relationship between two r...
ralnex3 3140 Relationship between three...
rexnal2 3141 Relationship between two r...
rexnal3 3142 Relationship between three...
nrexralim 3143 Negation of a complex pred...
r19.26-2 3144 Restricted quantifier vers...
2r19.29 3145 Theorem ~ r19.29 with two ...
r19.29d2r 3146 Theorem 19.29 of [Margaris...
r19.29d2rOLD 3147 Obsolete version of ~ r19....
r2allem 3148 Lemma factoring out common...
r2exlem 3149 Lemma factoring out common...
hbralrimi 3150 Inference from Theorem 19....
ralrimiv 3151 Inference from Theorem 19....
ralrimiva 3152 Inference from Theorem 19....
rexlimiva 3153 Inference from Theorem 19....
rexlimiv 3154 Inference from Theorem 19....
nrexdv 3155 Deduction adding restricte...
ralrimivw 3156 Inference from Theorem 19....
rexlimivw 3157 Weaker version of ~ rexlim...
ralrimdv 3158 Inference from Theorem 19....
rexlimdv 3159 Inference from Theorem 19....
ralrimdva 3160 Inference from Theorem 19....
rexlimdva 3161 Inference from Theorem 19....
rexlimdvaa 3162 Inference from Theorem 19....
rexlimdva2 3163 Inference from Theorem 19....
r19.29an 3164 A commonly used pattern in...
rexlimdv3a 3165 Inference from Theorem 19....
rexlimdvw 3166 Inference from Theorem 19....
rexlimddv 3167 Restricted existential eli...
r19.29a 3168 A commonly used pattern in...
ralimdv2 3169 Inference quantifying both...
reximdv2 3170 Deduction quantifying both...
reximdvai 3171 Deduction quantifying both...
reximdvaiOLD 3172 Obsolete version of ~ rexi...
ralimdva 3173 Deduction quantifying both...
reximdva 3174 Deduction quantifying both...
ralimdv 3175 Deduction quantifying both...
reximdv 3176 Deduction from Theorem 19....
reximddv 3177 Deduction from Theorem 19....
reximddv3 3178 Deduction from Theorem 19....
reximssdv 3179 Derivation of a restricted...
ralbidv2 3180 Formula-building rule for ...
rexbidv2 3181 Formula-building rule for ...
ralbidva 3182 Formula-building rule for ...
rexbidva 3183 Formula-building rule for ...
ralbidv 3184 Formula-building rule for ...
rexbidv 3185 Formula-building rule for ...
r19.21v 3186 Restricted quantifier vers...
r19.21vOLD 3187 Obsolete version of ~ r19....
r19.37v 3188 Restricted quantifier vers...
r19.23v 3189 Restricted quantifier vers...
r19.36v 3190 Restricted quantifier vers...
rexlimivOLD 3191 Obsolete version of ~ rexl...
rexlimivaOLD 3192 Obsolete version of ~ rexl...
rexlimivwOLD 3193 Obsolete version of ~ rexl...
r19.27v 3194 Restricted quantitifer ver...
r19.41v 3195 Restricted quantifier vers...
r19.28v 3196 Restricted quantifier vers...
r19.42v 3197 Restricted quantifier vers...
r19.32v 3198 Restricted quantifier vers...
r19.45v 3199 Restricted quantifier vers...
r19.44v 3200 One direction of a restric...
r2al 3201 Double restricted universa...
r2ex 3202 Double restricted existent...
r3al 3203 Triple restricted universa...
r3ex 3204 Triple existential quantif...
rgen2 3205 Generalization rule for re...
ralrimivv 3206 Inference from Theorem 19....
rexlimivv 3207 Inference from Theorem 19....
ralrimivva 3208 Inference from Theorem 19....
ralrimdvv 3209 Inference from Theorem 19....
rgen3 3210 Generalization rule for re...
ralrimivvva 3211 Inference from Theorem 19....
ralimdvva 3212 Deduction doubly quantifyi...
reximdvva 3213 Deduction doubly quantifyi...
ralimdvv 3214 Deduction doubly quantifyi...
ralimd4v 3215 Deduction quadrupally quan...
ralimd6v 3216 Deduction sextupally quant...
ralrimdvva 3217 Inference from Theorem 19....
rexlimdvv 3218 Inference from Theorem 19....
rexlimdvva 3219 Inference from Theorem 19....
rexlimdvvva 3220 Inference from Theorem 19....
reximddv2 3221 Double deduction from Theo...
r19.29vva 3222 A commonly used pattern ba...
r19.29vvaOLD 3223 Obsolete version of ~ r19....
2rexbiia 3224 Inference adding two restr...
2ralbidva 3225 Formula-building rule for ...
2rexbidva 3226 Formula-building rule for ...
2ralbidv 3227 Formula-building rule for ...
2rexbidv 3228 Formula-building rule for ...
rexralbidv 3229 Formula-building rule for ...
3ralbidv 3230 Formula-building rule for ...
4ralbidv 3231 Formula-building rule for ...
6ralbidv 3232 Formula-building rule for ...
r19.41vv 3233 Version of ~ r19.41v with ...
reeanlem 3234 Lemma factoring out common...
reeanv 3235 Rearrange restricted exist...
3reeanv 3236 Rearrange three restricted...
2ralor 3237 Distribute restricted univ...
2ralorOLD 3238 Obsolete version of ~ 2ral...
risset 3239 Two ways to say " ` A ` be...
nelb 3240 A definition of ` -. A e. ...
nelbOLD 3241 Obsolete version of ~ nelb...
rspw 3242 Restricted specialization....
cbvralvw 3243 Change the bound variable ...
cbvrexvw 3244 Change the bound variable ...
cbvraldva 3245 Rule used to change the bo...
cbvrexdva 3246 Rule used to change the bo...
cbvral2vw 3247 Change bound variables of ...
cbvrex2vw 3248 Change bound variables of ...
cbvral3vw 3249 Change bound variables of ...
cbvral4vw 3250 Change bound variables of ...
cbvral6vw 3251 Change bound variables of ...
cbvral8vw 3252 Change bound variables of ...
rsp 3253 Restricted specialization....
rspa 3254 Restricted specialization....
rspe 3255 Restricted specialization....
rspec 3256 Specialization rule for re...
r19.21bi 3257 Inference from Theorem 19....
r19.21be 3258 Inference from Theorem 19....
r19.21t 3259 Restricted quantifier vers...
r19.21 3260 Restricted quantifier vers...
r19.23t 3261 Closed theorem form of ~ r...
r19.23 3262 Restricted quantifier vers...
ralrimi 3263 Inference from Theorem 19....
ralrimia 3264 Inference from Theorem 19....
rexlimi 3265 Restricted quantifier vers...
ralimdaa 3266 Deduction quantifying both...
reximdai 3267 Deduction from Theorem 19....
r19.37 3268 Restricted quantifier vers...
r19.41 3269 Restricted quantifier vers...
ralrimd 3270 Inference from Theorem 19....
rexlimd2 3271 Version of ~ rexlimd with ...
rexlimd 3272 Deduction form of ~ rexlim...
r19.29af2 3273 A commonly used pattern ba...
r19.29af 3274 A commonly used pattern ba...
reximd2a 3275 Deduction quantifying both...
ralbida 3276 Formula-building rule for ...
ralbidaOLD 3277 Obsolete version of ~ ralb...
rexbida 3278 Formula-building rule for ...
ralbid 3279 Formula-building rule for ...
rexbid 3280 Formula-building rule for ...
rexbidvALT 3281 Alternate proof of ~ rexbi...
rexbidvaALT 3282 Alternate proof of ~ rexbi...
rsp2 3283 Restricted specialization,...
rsp2e 3284 Restricted specialization....
rspec2 3285 Specialization rule for re...
rspec3 3286 Specialization rule for re...
r2alf 3287 Double restricted universa...
r2exf 3288 Double restricted existent...
2ralbida 3289 Formula-building rule for ...
nfra1 3290 The setvar ` x ` is not fr...
nfre1 3291 The setvar ` x ` is not fr...
ralcom4 3292 Commutation of restricted ...
ralcom4OLD 3293 Obsolete version of ~ ralc...
rexcom4 3294 Commutation of restricted ...
ralcom 3295 Commutation of restricted ...
rexcom 3296 Commutation of restricted ...
rexcomOLD 3297 Obsolete version of ~ rexc...
rexcom4a 3298 Specialized existential co...
ralrot3 3299 Rotate three restricted un...
ralcom13 3300 Swap first and third restr...
ralcom13OLD 3301 Obsolete version of ~ ralc...
rexcom13 3302 Swap first and third restr...
rexrot4 3303 Rotate four restricted exi...
2ex2rexrot 3304 Rotate two existential qua...
nfra2w 3305 Similar to Lemma 24 of [Mo...
nfra2wOLD 3306 Obsolete version of ~ nfra...
hbra1 3307 The setvar ` x ` is not fr...
ralcomf 3308 Commutation of restricted ...
rexcomf 3309 Commutation of restricted ...
cbvralfw 3310 Rule used to change bound ...
cbvrexfw 3311 Rule used to change bound ...
cbvralw 3312 Rule used to change bound ...
cbvrexw 3313 Rule used to change bound ...
hbral 3314 Bound-variable hypothesis ...
nfraldw 3315 Deduction version of ~ nfr...
nfrexdw 3316 Deduction version of ~ nfr...
nfralw 3317 Bound-variable hypothesis ...
nfralwOLD 3318 Obsolete version of ~ nfra...
nfrexw 3319 Bound-variable hypothesis ...
r19.12 3320 Restricted quantifier vers...
r19.12OLD 3321 Obsolete version of ~ 19.1...
reean 3322 Rearrange restricted exist...
cbvralsvw 3323 Change bound variable by u...
cbvrexsvw 3324 Change bound variable by u...
cbvralsvwOLD 3325 Obsolete version of ~ cbvr...
cbvralsvwOLDOLD 3326 Obsolete version of ~ cbvr...
cbvrexsvwOLD 3327 Obsolete version of ~ cbvr...
nfraldwOLD 3328 Obsolete version of ~ nfra...
nfra2wOLDOLD 3329 Obsolete version of ~ nfra...
rexeq 3330 Equality theorem for restr...
raleq 3331 Equality theorem for restr...
raleqi 3332 Equality inference for res...
rexeqi 3333 Equality inference for res...
raleqdv 3334 Equality deduction for res...
rexeqdv 3335 Equality deduction for res...
raleqtrdv 3336 Substitution of equal clas...
rexeqtrdv 3337 Substitution of equal clas...
raleqtrrdv 3338 Substitution of equal clas...
rexeqtrrdv 3339 Substitution of equal clas...
raleqbidva 3340 Equality deduction for res...
rexeqbidva 3341 Equality deduction for res...
raleqbidvv 3342 Version of ~ raleqbidv wit...
raleqbidvvOLD 3343 Obsolete version of ~ rale...
rexeqbidvv 3344 Version of ~ rexeqbidv wit...
rexeqbidvvOLD 3345 Obsolete version of ~ rexe...
raleqbi1dv 3346 Equality deduction for res...
rexeqbi1dv 3347 Equality deduction for res...
raleqOLD 3348 Obsolete version of ~ rale...
rexeqOLD 3349 Obsolete version of ~ rale...
raleleq 3350 All elements of a class ar...
raleleqOLD 3351 Obsolete version of ~ rale...
raleqbii 3352 Equality deduction for res...
rexeqbii 3353 Equality deduction for res...
raleqbidv 3354 Equality deduction for res...
rexeqbidv 3355 Equality deduction for res...
cbvraldva2 3356 Rule used to change the bo...
cbvrexdva2 3357 Rule used to change the bo...
cbvrexdva2OLD 3358 Obsolete version of ~ cbvr...
cbvraldvaOLD 3359 Obsolete version of ~ cbvr...
cbvrexdvaOLD 3360 Obsolete version of ~ cbvr...
raleqf 3361 Equality theorem for restr...
rexeqf 3362 Equality theorem for restr...
rexeqfOLD 3363 Obsolete version of ~ rexe...
raleqbid 3364 Equality deduction for res...
rexeqbid 3365 Equality deduction for res...
sbralie 3366 Implicit to explicit subst...
sbralieALT 3367 Alternative shorter proof ...
cbvralf 3368 Rule used to change bound ...
cbvrexf 3369 Rule used to change bound ...
cbvral 3370 Rule used to change bound ...
cbvrex 3371 Rule used to change bound ...
cbvralv 3372 Change the bound variable ...
cbvrexv 3373 Change the bound variable ...
cbvralsv 3374 Change bound variable by u...
cbvrexsv 3375 Change bound variable by u...
cbvral2v 3376 Change bound variables of ...
cbvrex2v 3377 Change bound variables of ...
cbvral3v 3378 Change bound variables of ...
rgen2a 3379 Generalization rule for re...
nfrald 3380 Deduction version of ~ nfr...
nfrexd 3381 Deduction version of ~ nfr...
nfral 3382 Bound-variable hypothesis ...
nfrex 3383 Bound-variable hypothesis ...
nfra2 3384 Similar to Lemma 24 of [Mo...
ralcom2 3385 Commutation of restricted ...
reu5 3390 Restricted uniqueness in t...
reurmo 3391 Restricted existential uni...
reurex 3392 Restricted unique existenc...
mormo 3393 Unrestricted "at most one"...
rmobiia 3394 Formula-building rule for ...
reubiia 3395 Formula-building rule for ...
rmobii 3396 Formula-building rule for ...
reubii 3397 Formula-building rule for ...
rmoanid 3398 Cancellation law for restr...
reuanid 3399 Cancellation law for restr...
rmoanidOLD 3400 Obsolete version of ~ rmoa...
reuanidOLD 3401 Obsolete version of ~ reua...
2reu2rex 3402 Double restricted existent...
rmobidva 3403 Formula-building rule for ...
reubidva 3404 Formula-building rule for ...
rmobidv 3405 Formula-building rule for ...
reubidv 3406 Formula-building rule for ...
reueubd 3407 Restricted existential uni...
rmo5 3408 Restricted "at most one" i...
nrexrmo 3409 Nonexistence implies restr...
moel 3410 "At most one" element in a...
cbvrmovw 3411 Change the bound variable ...
cbvreuvw 3412 Change the bound variable ...
moelOLD 3413 Obsolete version of ~ moel...
rmobida 3414 Formula-building rule for ...
reubida 3415 Formula-building rule for ...
rmobidvaOLD 3416 Obsolete version of ~ rmob...
cbvrmow 3417 Change the bound variable ...
cbvreuw 3418 Change the bound variable ...
nfrmo1 3419 The setvar ` x ` is not fr...
nfreu1 3420 The setvar ` x ` is not fr...
nfrmow 3421 Bound-variable hypothesis ...
nfreuw 3422 Bound-variable hypothesis ...
cbvreuwOLD 3423 Obsolete version of ~ cbvr...
cbvreuvwOLD 3424 Obsolete version of ~ cbvr...
rmoeq1 3425 Equality theorem for restr...
reueq1 3426 Equality theorem for restr...
rmoeq1OLD 3427 Obsolete version of ~ rmoe...
reueq1OLD 3428 Obsolete version of ~ reue...
rmoeqd 3429 Equality deduction for res...
reueqd 3430 Equality deduction for res...
rmoeq1f 3431 Equality theorem for restr...
reueq1f 3432 Equality theorem for restr...
nfreuwOLD 3433 Obsolete version of ~ nfre...
nfrmowOLD 3434 Obsolete version of ~ nfrm...
cbvreu 3435 Change the bound variable ...
cbvrmo 3436 Change the bound variable ...
cbvrmov 3437 Change the bound variable ...
cbvreuv 3438 Change the bound variable ...
nfrmod 3439 Deduction version of ~ nfr...
nfreud 3440 Deduction version of ~ nfr...
nfrmo 3441 Bound-variable hypothesis ...
nfreu 3442 Bound-variable hypothesis ...
rabbidva2 3445 Equivalent wff's yield equ...
rabbia2 3446 Equivalent wff's yield equ...
rabbiia 3447 Equivalent formulas yield ...
rabbiiaOLD 3448 Obsolete version of ~ rabb...
rabbii 3449 Equivalent wff's correspon...
rabbidva 3450 Equivalent wff's yield equ...
rabbidv 3451 Equivalent wff's yield equ...
rabbieq 3452 Equivalent wff's correspon...
rabswap 3453 Swap with a membership rel...
cbvrabv 3454 Rule to change the bound v...
rabeqcda 3455 When ` ps ` is always true...
rabeqc 3456 A restricted class abstrac...
rabeqi 3457 Equality theorem for restr...
rabeq 3458 Equality theorem for restr...
rabeqdv 3459 Equality of restricted cla...
rabeqbidva 3460 Equality of restricted cla...
rabeqbidvaOLD 3461 Obsolete version of ~ rabe...
rabeqbidv 3462 Equality of restricted cla...
rabrabi 3463 Abstract builder restricte...
nfrab1 3464 The abstraction variable i...
rabid 3465 An "identity" law of concr...
rabidim1 3466 Membership in a restricted...
reqabi 3467 Inference from equality of...
rabrab 3468 Abstract builder restricte...
rabrabiOLD 3469 Obsolete version of ~ rabr...
rabbida4 3470 Version of ~ rabbidva2 wit...
rabbida 3471 Equivalent wff's yield equ...
rabbid 3472 Version of ~ rabbidv with ...
rabeqd 3473 Deduction form of ~ rabeq ...
rabeqbida 3474 Version of ~ rabeqbidva wi...
rabbi 3475 Equivalent wff's correspon...
rabid2f 3476 An "identity" law for rest...
rabid2im 3477 One direction of ~ rabid2 ...
rabid2 3478 An "identity" law for rest...
rabid2OLD 3479 Obsolete version of ~ rabi...
rabeqf 3480 Equality theorem for restr...
cbvrabw 3481 Rule to change the bound v...
cbvrabwOLD 3482 Obsolete version of ~ cbvr...
nfrabw 3483 A variable not free in a w...
nfrabwOLD 3484 Obsolete version of ~ nfra...
rabbidaOLD 3485 Obsolete version of ~ rabb...
nfrab 3486 A variable not free in a w...
cbvrab 3487 Rule to change the bound v...
vjust 3489 Justification theorem for ...
dfv2 3491 Alternate definition of th...
vex 3492 All setvar variables are s...
elv 3493 If a proposition is implie...
elvd 3494 If a proposition is implie...
el2v 3495 If a proposition is implie...
el3v 3496 If a proposition is implie...
el3v3 3497 If a proposition is implie...
eqv 3498 The universe contains ever...
eqvf 3499 The universe contains ever...
abv 3500 The class of sets verifyin...
abvALT 3501 Alternate proof of ~ abv ,...
isset 3502 Two ways to express that "...
cbvexeqsetf 3503 The expression ` E. x x = ...
issetft 3504 Closed theorem form of ~ i...
issetf 3505 A version of ~ isset that ...
isseti 3506 A way to say " ` A ` is a ...
issetri 3507 A way to say " ` A ` is a ...
eqvisset 3508 A class equal to a variabl...
elex 3509 If a class is a member of ...
elexOLD 3510 Obsolete version of ~ elex...
elexi 3511 If a class is a member of ...
elexd 3512 If a class is a member of ...
elex2OLD 3513 Obsolete version of ~ elex...
elex22 3514 If two classes each contai...
prcnel 3515 A proper class doesn't bel...
ralv 3516 A universal quantifier res...
rexv 3517 An existential quantifier ...
reuv 3518 A unique existential quant...
rmov 3519 An at-most-one quantifier ...
rabab 3520 A class abstraction restri...
rexcom4b 3521 Specialized existential co...
ceqsal1t 3522 One direction of ~ ceqsalt...
ceqsalt 3523 Closed theorem version of ...
ceqsralt 3524 Restricted quantifier vers...
ceqsalg 3525 A representation of explic...
ceqsalgALT 3526 Alternate proof of ~ ceqsa...
ceqsal 3527 A representation of explic...
ceqsalALT 3528 A representation of explic...
ceqsalv 3529 A representation of explic...
ceqsalvOLD 3530 Obsolete version of ~ ceqs...
ceqsralv 3531 Restricted quantifier vers...
ceqsralvOLD 3532 Obsolete version of ~ ceqs...
gencl 3533 Implicit substitution for ...
2gencl 3534 Implicit substitution for ...
3gencl 3535 Implicit substitution for ...
cgsexg 3536 Implicit substitution infe...
cgsex2g 3537 Implicit substitution infe...
cgsex4g 3538 An implicit substitution i...
cgsex4gOLD 3539 Obsolete version of ~ cgse...
ceqsex 3540 Elimination of an existent...
ceqsexOLD 3541 Obsolete version of ~ ceqs...
ceqsexv 3542 Elimination of an existent...
ceqsexvOLD 3543 Obsolete version of ~ ceqs...
ceqsexvOLDOLD 3544 Obsolete version of ~ ceqs...
ceqsexv2d 3545 Elimination of an existent...
ceqsexv2dOLD 3546 Obsolete version of ~ ceqs...
ceqsex2 3547 Elimination of two existen...
ceqsex2v 3548 Elimination of two existen...
ceqsex3v 3549 Elimination of three exist...
ceqsex4v 3550 Elimination of four existe...
ceqsex6v 3551 Elimination of six existen...
ceqsex8v 3552 Elimination of eight exist...
gencbvex 3553 Change of bound variable u...
gencbvex2 3554 Restatement of ~ gencbvex ...
gencbval 3555 Change of bound variable u...
sbhypf 3556 Introduce an explicit subs...
sbhypfOLD 3557 Obsolete version of ~ sbhy...
spcimgft 3558 Closed theorem form of ~ s...
spcimgfi1 3559 A closed version of ~ spci...
spcimgfi1OLD 3560 Obsolete version of ~ spci...
spcgft 3561 A closed version of ~ spcg...
spcimgf 3562 Rule of specialization, us...
spcimegf 3563 Existential specialization...
vtoclgft 3564 Closed theorem form of ~ v...
vtocleg 3565 Implicit substitution of a...
vtoclg 3566 Implicit substitution of a...
vtocle 3567 Implicit substitution of a...
vtocleOLD 3568 Obsolete version of ~ vtoc...
vtoclbg 3569 Implicit substitution of a...
vtocl 3570 Implicit substitution of a...
vtoclOLD 3571 Obsolete version of ~ vtoc...
vtocldf 3572 Implicit substitution of a...
vtocld 3573 Implicit substitution of a...
vtocl2d 3574 Implicit substitution of t...
vtoclef 3575 Implicit substitution of a...
vtoclf 3576 Implicit substitution of a...
vtoclfOLD 3577 Obsolete version of ~ vtoc...
vtocl2 3578 Implicit substitution of c...
vtocl3 3579 Implicit substitution of c...
vtoclb 3580 Implicit substitution of a...
vtoclgf 3581 Implicit substitution of a...
vtoclg1f 3582 Version of ~ vtoclgf with ...
vtoclgOLD 3583 Obsolete version of ~ vtoc...
vtocl2gf 3584 Implicit substitution of a...
vtocl3gf 3585 Implicit substitution of a...
vtocl2g 3586 Implicit substitution of 2...
vtocl3g 3587 Implicit substitution of a...
vtoclgaf 3588 Implicit substitution of a...
vtoclga 3589 Implicit substitution of a...
vtocl2ga 3590 Implicit substitution of 2...
vtocl2gaf 3591 Implicit substitution of 2...
vtocl2gafOLD 3592 Obsolete version of ~ vtoc...
vtocl3gaf 3593 Implicit substitution of 3...
vtocl3gafOLD 3594 Obsolete version of ~ vtoc...
vtocl3ga 3595 Implicit substitution of 3...
vtocl3gaOLD 3596 Obsolete version of ~ vtoc...
vtocl3gaOLDOLD 3597 Obsolete version of ~ vtoc...
vtocl4g 3598 Implicit substitution of 4...
vtocl4ga 3599 Implicit substitution of 4...
vtocl4gaOLD 3600 Obsolete version of ~ vtoc...
vtoclegft 3601 Implicit substitution of a...
vtoclegftOLD 3602 Obsolete version of ~ vtoc...
vtoclri 3603 Implicit substitution of a...
spcgf 3604 Rule of specialization, us...
spcegf 3605 Existential specialization...
spcimdv 3606 Restricted specialization,...
spcdv 3607 Rule of specialization, us...
spcimedv 3608 Restricted existential spe...
spcgv 3609 Rule of specialization, us...
spcegv 3610 Existential specialization...
spcedv 3611 Existential specialization...
spc2egv 3612 Existential specialization...
spc2gv 3613 Specialization with two qu...
spc2ed 3614 Existential specialization...
spc2d 3615 Specialization with 2 quan...
spc3egv 3616 Existential specialization...
spc3gv 3617 Specialization with three ...
spcv 3618 Rule of specialization, us...
spcev 3619 Existential specialization...
spc2ev 3620 Existential specialization...
rspct 3621 A closed version of ~ rspc...
rspcdf 3622 Restricted specialization,...
rspc 3623 Restricted specialization,...
rspce 3624 Restricted existential spe...
rspcimdv 3625 Restricted specialization,...
rspcimedv 3626 Restricted existential spe...
rspcdv 3627 Restricted specialization,...
rspcedv 3628 Restricted existential spe...
rspcebdv 3629 Restricted existential spe...
rspcdv2 3630 Restricted specialization,...
rspcv 3631 Restricted specialization,...
rspccv 3632 Restricted specialization,...
rspcva 3633 Restricted specialization,...
rspccva 3634 Restricted specialization,...
rspcev 3635 Restricted existential spe...
rspcdva 3636 Restricted specialization,...
rspcedvd 3637 Restricted existential spe...
rspcedvdw 3638 Version of ~ rspcedvd wher...
rspceb2dv 3639 Restricted existential spe...
rspcime 3640 Prove a restricted existen...
rspceaimv 3641 Restricted existential spe...
rspcedeq1vd 3642 Restricted existential spe...
rspcedeq2vd 3643 Restricted existential spe...
rspc2 3644 Restricted specialization ...
rspc2gv 3645 Restricted specialization ...
rspc2v 3646 2-variable restricted spec...
rspc2va 3647 2-variable restricted spec...
rspc2ev 3648 2-variable restricted exis...
2rspcedvdw 3649 Double application of ~ rs...
rspc2dv 3650 2-variable restricted spec...
rspc3v 3651 3-variable restricted spec...
rspc3ev 3652 3-variable restricted exis...
3rspcedvdw 3653 Triple application of ~ rs...
rspc3dv 3654 3-variable restricted spec...
rspc4v 3655 4-variable restricted spec...
rspc6v 3656 6-variable restricted spec...
rspc8v 3657 8-variable restricted spec...
rspceeqv 3658 Restricted existential spe...
ralxpxfr2d 3659 Transfer a universal quant...
rexraleqim 3660 Statement following from e...
eqvincg 3661 A variable introduction la...
eqvinc 3662 A variable introduction la...
eqvincf 3663 A variable introduction la...
alexeqg 3664 Two ways to express substi...
ceqex 3665 Equality implies equivalen...
ceqsexg 3666 A representation of explic...
ceqsexgv 3667 Elimination of an existent...
ceqsrexv 3668 Elimination of a restricte...
ceqsrexbv 3669 Elimination of a restricte...
ceqsralbv 3670 Elimination of a restricte...
ceqsrex2v 3671 Elimination of a restricte...
clel2g 3672 Alternate definition of me...
clel2 3673 Alternate definition of me...
clel3g 3674 Alternate definition of me...
clel3 3675 Alternate definition of me...
clel4g 3676 Alternate definition of me...
clel4 3677 Alternate definition of me...
clel5 3678 Alternate definition of cl...
pm13.183 3679 Compare theorem *13.183 in...
rr19.3v 3680 Restricted quantifier vers...
rr19.28v 3681 Restricted quantifier vers...
elab6g 3682 Membership in a class abst...
elabd2 3683 Membership in a class abst...
elabd3 3684 Membership in a class abst...
elabgt 3685 Membership in a class abst...
elabgtOLD 3686 Obsolete version of ~ elab...
elabgtOLDOLD 3687 Obsolete version of ~ elab...
elabgf 3688 Membership in a class abst...
elabf 3689 Membership in a class abst...
elabg 3690 Membership in a class abst...
elabgOLD 3691 Obsolete version of ~ elab...
elabgw 3692 Membership in a class abst...
elab2gw 3693 Membership in a class abst...
elab 3694 Membership in a class abst...
elabOLD 3695 Obsolete version of ~ elab...
elab2g 3696 Membership in a class abst...
elabd 3697 Explicit demonstration the...
elab2 3698 Membership in a class abst...
elab4g 3699 Membership in a class abst...
elab3gf 3700 Membership in a class abst...
elab3g 3701 Membership in a class abst...
elab3 3702 Membership in a class abst...
elrabi 3703 Implication for the member...
elrabf 3704 Membership in a restricted...
rabtru 3705 Abstract builder using the...
rabeqcOLD 3706 Obsolete version of ~ rabe...
elrab3t 3707 Membership in a restricted...
elrab 3708 Membership in a restricted...
elrab3 3709 Membership in a restricted...
elrabd 3710 Membership in a restricted...
elrab2 3711 Membership in a restricted...
elrab2w 3712 Membership in a restricted...
ralab 3713 Universal quantification o...
ralabOLD 3714 Obsolete version of ~ rala...
ralrab 3715 Universal quantification o...
rexab 3716 Existential quantification...
rexabOLD 3717 Obsolete version of ~ rexa...
rexrab 3718 Existential quantification...
ralab2 3719 Universal quantification o...
ralrab2 3720 Universal quantification o...
rexab2 3721 Existential quantification...
rexrab2 3722 Existential quantification...
reurab 3723 Restricted existential uni...
abidnf 3724 Identity used to create cl...
dedhb 3725 A deduction theorem for co...
class2seteq 3726 Writing a set as a class a...
nelrdva 3727 Deduce negative membership...
eqeu 3728 A condition which implies ...
moeq 3729 There exists at most one s...
eueq 3730 A class is a set if and on...
eueqi 3731 There exists a unique set ...
eueq2 3732 Equality has existential u...
eueq3 3733 Equality has existential u...
moeq3 3734 "At most one" property of ...
mosub 3735 "At most one" remains true...
mo2icl 3736 Theorem for inferring "at ...
mob2 3737 Consequence of "at most on...
moi2 3738 Consequence of "at most on...
mob 3739 Equality implied by "at mo...
moi 3740 Equality implied by "at mo...
morex 3741 Derive membership from uni...
euxfr2w 3742 Transfer existential uniqu...
euxfrw 3743 Transfer existential uniqu...
euxfr2 3744 Transfer existential uniqu...
euxfr 3745 Transfer existential uniqu...
euind 3746 Existential uniqueness via...
reu2 3747 A way to express restricte...
reu6 3748 A way to express restricte...
reu3 3749 A way to express restricte...
reu6i 3750 A condition which implies ...
eqreu 3751 A condition which implies ...
rmo4 3752 Restricted "at most one" u...
reu4 3753 Restricted uniqueness usin...
reu7 3754 Restricted uniqueness usin...
reu8 3755 Restricted uniqueness usin...
rmo3f 3756 Restricted "at most one" u...
rmo4f 3757 Restricted "at most one" u...
reu2eqd 3758 Deduce equality from restr...
reueq 3759 Equality has existential u...
rmoeq 3760 Equality's restricted exis...
rmoan 3761 Restricted "at most one" s...
rmoim 3762 Restricted "at most one" i...
rmoimia 3763 Restricted "at most one" i...
rmoimi 3764 Restricted "at most one" i...
rmoimi2 3765 Restricted "at most one" i...
2reu5a 3766 Double restricted existent...
reuimrmo 3767 Restricted uniqueness impl...
2reuswap 3768 A condition allowing swap ...
2reuswap2 3769 A condition allowing swap ...
reuxfrd 3770 Transfer existential uniqu...
reuxfr 3771 Transfer existential uniqu...
reuxfr1d 3772 Transfer existential uniqu...
reuxfr1ds 3773 Transfer existential uniqu...
reuxfr1 3774 Transfer existential uniqu...
reuind 3775 Existential uniqueness via...
2rmorex 3776 Double restricted quantifi...
2reu5lem1 3777 Lemma for ~ 2reu5 . Note ...
2reu5lem2 3778 Lemma for ~ 2reu5 . (Cont...
2reu5lem3 3779 Lemma for ~ 2reu5 . This ...
2reu5 3780 Double restricted existent...
2reurmo 3781 Double restricted quantifi...
2reurex 3782 Double restricted quantifi...
2rmoswap 3783 A condition allowing to sw...
2rexreu 3784 Double restricted existent...
cdeqi 3787 Deduce conditional equalit...
cdeqri 3788 Property of conditional eq...
cdeqth 3789 Deduce conditional equalit...
cdeqnot 3790 Distribute conditional equ...
cdeqal 3791 Distribute conditional equ...
cdeqab 3792 Distribute conditional equ...
cdeqal1 3793 Distribute conditional equ...
cdeqab1 3794 Distribute conditional equ...
cdeqim 3795 Distribute conditional equ...
cdeqcv 3796 Conditional equality for s...
cdeqeq 3797 Distribute conditional equ...
cdeqel 3798 Distribute conditional equ...
nfcdeq 3799 If we have a conditional e...
nfccdeq 3800 Variation of ~ nfcdeq for ...
rru 3801 Relative version of Russel...
ru 3802 Russell's Paradox. Propos...
ruOLD 3803 Obsolete version of ~ ru a...
dfsbcq 3806 Proper substitution of a c...
dfsbcq2 3807 This theorem, which is sim...
sbsbc 3808 Show that ~ df-sb and ~ df...
sbceq1d 3809 Equality theorem for class...
sbceq1dd 3810 Equality theorem for class...
sbceqbid 3811 Equality theorem for class...
sbc8g 3812 This is the closest we can...
sbc2or 3813 The disjunction of two equ...
sbcex 3814 By our definition of prope...
sbceq1a 3815 Equality theorem for class...
sbceq2a 3816 Equality theorem for class...
spsbc 3817 Specialization: if a formu...
spsbcd 3818 Specialization: if a formu...
sbcth 3819 A substitution into a theo...
sbcthdv 3820 Deduction version of ~ sbc...
sbcid 3821 An identity theorem for su...
nfsbc1d 3822 Deduction version of ~ nfs...
nfsbc1 3823 Bound-variable hypothesis ...
nfsbc1v 3824 Bound-variable hypothesis ...
nfsbcdw 3825 Deduction version of ~ nfs...
nfsbcw 3826 Bound-variable hypothesis ...
sbccow 3827 A composition law for clas...
nfsbcd 3828 Deduction version of ~ nfs...
nfsbc 3829 Bound-variable hypothesis ...
sbcco 3830 A composition law for clas...
sbcco2 3831 A composition law for clas...
sbc5 3832 An equivalence for class s...
sbc5ALT 3833 Alternate proof of ~ sbc5 ...
sbc6g 3834 An equivalence for class s...
sbc6gOLD 3835 Obsolete version of ~ sbc6...
sbc6 3836 An equivalence for class s...
sbc7 3837 An equivalence for class s...
cbvsbcw 3838 Change bound variables in ...
cbvsbcvw 3839 Change the bound variable ...
cbvsbc 3840 Change bound variables in ...
cbvsbcv 3841 Change the bound variable ...
sbciegft 3842 Conversion of implicit sub...
sbciegftOLD 3843 Obsolete version of ~ sbci...
sbciegf 3844 Conversion of implicit sub...
sbcieg 3845 Conversion of implicit sub...
sbciegOLD 3846 Obsolete version of ~ sbci...
sbcie2g 3847 Conversion of implicit sub...
sbcie 3848 Conversion of implicit sub...
sbciedf 3849 Conversion of implicit sub...
sbcied 3850 Conversion of implicit sub...
sbciedOLD 3851 Obsolete version of ~ sbci...
sbcied2 3852 Conversion of implicit sub...
elrabsf 3853 Membership in a restricted...
eqsbc1 3854 Substitution for the left-...
sbcng 3855 Move negation in and out o...
sbcimg 3856 Distribution of class subs...
sbcan 3857 Distribution of class subs...
sbcor 3858 Distribution of class subs...
sbcbig 3859 Distribution of class subs...
sbcn1 3860 Move negation in and out o...
sbcim1 3861 Distribution of class subs...
sbcim1OLD 3862 Obsolete version of ~ sbci...
sbcbid 3863 Formula-building deduction...
sbcbidv 3864 Formula-building deduction...
sbcbii 3865 Formula-building inference...
sbcbi1 3866 Distribution of class subs...
sbcbi2 3867 Substituting into equivale...
sbcal 3868 Move universal quantifier ...
sbcex2 3869 Move existential quantifie...
sbceqal 3870 Class version of one impli...
sbceqalOLD 3871 Obsolete version of ~ sbce...
sbeqalb 3872 Theorem *14.121 in [Whiteh...
eqsbc2 3873 Substitution for the right...
sbc3an 3874 Distribution of class subs...
sbcel1v 3875 Class substitution into a ...
sbcel2gv 3876 Class substitution into a ...
sbcel21v 3877 Class substitution into a ...
sbcimdv 3878 Substitution analogue of T...
sbcimdvOLD 3879 Obsolete version of ~ sbci...
sbctt 3880 Substitution for a variabl...
sbcgf 3881 Substitution for a variabl...
sbc19.21g 3882 Substitution for a variabl...
sbcg 3883 Substitution for a variabl...
sbcgOLD 3884 Obsolete version of ~ sbcg...
sbcgfi 3885 Substitution for a variabl...
sbc2iegf 3886 Conversion of implicit sub...
sbc2ie 3887 Conversion of implicit sub...
sbc2ieOLD 3888 Obsolete version of ~ sbc2...
sbc2iedv 3889 Conversion of implicit sub...
sbc3ie 3890 Conversion of implicit sub...
sbccomlem 3891 Lemma for ~ sbccom . (Con...
sbccomlemOLD 3892 Obsolete version of ~ sbcc...
sbccom 3893 Commutative law for double...
sbcralt 3894 Interchange class substitu...
sbcrext 3895 Interchange class substitu...
sbcralg 3896 Interchange class substitu...
sbcrex 3897 Interchange class substitu...
sbcreu 3898 Interchange class substitu...
reu8nf 3899 Restricted uniqueness usin...
sbcabel 3900 Interchange class substitu...
rspsbc 3901 Restricted quantifier vers...
rspsbca 3902 Restricted quantifier vers...
rspesbca 3903 Existence form of ~ rspsbc...
spesbc 3904 Existence form of ~ spsbc ...
spesbcd 3905 form of ~ spsbc . (Contri...
sbcth2 3906 A substitution into a theo...
ra4v 3907 Version of ~ ra4 with a di...
ra4 3908 Restricted quantifier vers...
rmo2 3909 Alternate definition of re...
rmo2i 3910 Condition implying restric...
rmo3 3911 Restricted "at most one" u...
rmob 3912 Consequence of "at most on...
rmoi 3913 Consequence of "at most on...
rmob2 3914 Consequence of "restricted...
rmoi2 3915 Consequence of "restricted...
rmoanim 3916 Introduction of a conjunct...
rmoanimALT 3917 Alternate proof of ~ rmoan...
reuan 3918 Introduction of a conjunct...
2reu1 3919 Double restricted existent...
2reu2 3920 Double restricted existent...
csb2 3923 Alternate expression for t...
csbeq1 3924 Analogue of ~ dfsbcq for p...
csbeq1d 3925 Equality deduction for pro...
csbeq2 3926 Substituting into equivale...
csbeq2d 3927 Formula-building deduction...
csbeq2dv 3928 Formula-building deduction...
csbeq2i 3929 Formula-building inference...
csbeq12dv 3930 Formula-building inference...
cbvcsbw 3931 Change bound variables in ...
cbvcsb 3932 Change bound variables in ...
cbvcsbv 3933 Change the bound variable ...
csbid 3934 Analogue of ~ sbid for pro...
csbeq1a 3935 Equality theorem for prope...
csbcow 3936 Composition law for chaine...
csbco 3937 Composition law for chaine...
csbtt 3938 Substitution doesn't affec...
csbconstgf 3939 Substitution doesn't affec...
csbconstg 3940 Substitution doesn't affec...
csbconstgOLD 3941 Obsolete version of ~ csbc...
csbgfi 3942 Substitution for a variabl...
csbconstgi 3943 The proper substitution of...
nfcsb1d 3944 Bound-variable hypothesis ...
nfcsb1 3945 Bound-variable hypothesis ...
nfcsb1v 3946 Bound-variable hypothesis ...
nfcsbd 3947 Deduction version of ~ nfc...
nfcsbw 3948 Bound-variable hypothesis ...
nfcsb 3949 Bound-variable hypothesis ...
csbhypf 3950 Introduce an explicit subs...
csbiebt 3951 Conversion of implicit sub...
csbiedf 3952 Conversion of implicit sub...
csbieb 3953 Bidirectional conversion b...
csbiebg 3954 Bidirectional conversion b...
csbiegf 3955 Conversion of implicit sub...
csbief 3956 Conversion of implicit sub...
csbie 3957 Conversion of implicit sub...
csbieOLD 3958 Obsolete version of ~ csbi...
csbied 3959 Conversion of implicit sub...
csbiedOLD 3960 Obsolete version of ~ csbi...
csbied2 3961 Conversion of implicit sub...
csbie2t 3962 Conversion of implicit sub...
csbie2 3963 Conversion of implicit sub...
csbie2g 3964 Conversion of implicit sub...
cbvrabcsfw 3965 Version of ~ cbvrabcsf wit...
cbvralcsf 3966 A more general version of ...
cbvrexcsf 3967 A more general version of ...
cbvreucsf 3968 A more general version of ...
cbvrabcsf 3969 A more general version of ...
cbvralv2 3970 Rule used to change the bo...
cbvrexv2 3971 Rule used to change the bo...
rspc2vd 3972 Deduction version of 2-var...
difjust 3978 Soundness justification th...
unjust 3980 Soundness justification th...
injust 3982 Soundness justification th...
dfin5 3984 Alternate definition for t...
dfdif2 3985 Alternate definition of cl...
eldif 3986 Expansion of membership in...
eldifd 3987 If a class is in one class...
eldifad 3988 If a class is in the diffe...
eldifbd 3989 If a class is in the diffe...
elneeldif 3990 The elements of a set diff...
velcomp 3991 Characterization of setvar...
elin 3992 Expansion of membership in...
dfss2 3994 Alternate definition of th...
dfss 3995 Variant of subclass defini...
dfss3 3997 Alternate definition of su...
dfss6 3998 Alternate definition of su...
dfssf 3999 Equivalence for subclass r...
dfss3f 4000 Equivalence for subclass r...
nfss 4001 If ` x ` is not free in ` ...
ssel 4002 Membership relationships f...
ssel2 4003 Membership relationships f...
sseli 4004 Membership implication fro...
sselii 4005 Membership inference from ...
sselid 4006 Membership inference from ...
sseld 4007 Membership deduction from ...
sselda 4008 Membership deduction from ...
sseldd 4009 Membership inference from ...
ssneld 4010 If a class is not in anoth...
ssneldd 4011 If an element is not in a ...
ssriv 4012 Inference based on subclas...
ssrd 4013 Deduction based on subclas...
ssrdv 4014 Deduction based on subclas...
sstr2 4015 Transitivity of subclass r...
sstr2OLD 4016 Obsolete version of ~ sstr...
sstr 4017 Transitivity of subclass r...
sstri 4018 Subclass transitivity infe...
sstrd 4019 Subclass transitivity dedu...
sstrid 4020 Subclass transitivity dedu...
sstrdi 4021 Subclass transitivity dedu...
sylan9ss 4022 A subclass transitivity de...
sylan9ssr 4023 A subclass transitivity de...
eqss 4024 The subclass relationship ...
eqssi 4025 Infer equality from two su...
eqssd 4026 Equality deduction from tw...
sssseq 4027 If a class is a subclass o...
eqrd 4028 Deduce equality of classes...
eqri 4029 Infer equality of classes ...
eqelssd 4030 Equality deduction from su...
ssid 4031 Any class is a subclass of...
ssidd 4032 Weakening of ~ ssid . (Co...
ssv 4033 Any class is a subclass of...
sseq1 4034 Equality theorem for subcl...
sseq2 4035 Equality theorem for the s...
sseq12 4036 Equality theorem for the s...
sseq1i 4037 An equality inference for ...
sseq2i 4038 An equality inference for ...
sseq12i 4039 An equality inference for ...
sseq1d 4040 An equality deduction for ...
sseq2d 4041 An equality deduction for ...
sseq12d 4042 An equality deduction for ...
eqsstri 4043 Substitution of equality i...
eqsstrri 4044 Substitution of equality i...
sseqtri 4045 Substitution of equality i...
sseqtrri 4046 Substitution of equality i...
eqsstrd 4047 Substitution of equality i...
eqsstrrd 4048 Substitution of equality i...
sseqtrd 4049 Substitution of equality i...
sseqtrrd 4050 Substitution of equality i...
3sstr3i 4051 Substitution of equality i...
3sstr4i 4052 Substitution of equality i...
3sstr3g 4053 Substitution of equality i...
3sstr4g 4054 Substitution of equality i...
3sstr3d 4055 Substitution of equality i...
3sstr4d 4056 Substitution of equality i...
eqsstrid 4057 A chained subclass and equ...
eqsstrrid 4058 A chained subclass and equ...
sseqtrdi 4059 A chained subclass and equ...
sseqtrrdi 4060 A chained subclass and equ...
sseqtrid 4061 Subclass transitivity dedu...
sseqtrrid 4062 Subclass transitivity dedu...
eqsstrdi 4063 A chained subclass and equ...
eqsstrrdi 4064 A chained subclass and equ...
eqimssd 4065 Equality implies inclusion...
eqimsscd 4066 Equality implies inclusion...
eqimss 4067 Equality implies inclusion...
eqimss2 4068 Equality implies inclusion...
eqimssi 4069 Infer subclass relationshi...
eqimss2i 4070 Infer subclass relationshi...
nssne1 4071 Two classes are different ...
nssne2 4072 Two classes are different ...
nss 4073 Negation of subclass relat...
nelss 4074 Demonstrate by witnesses t...
ssrexf 4075 Restricted existential qua...
ssrmof 4076 "At most one" existential ...
ssralv 4077 Quantification restricted ...
ssrexv 4078 Existential quantification...
ss2ralv 4079 Two quantifications restri...
ss2rexv 4080 Two existential quantifica...
ssralvOLD 4081 Obsolete version of ~ ssra...
ssrexvOLD 4082 Obsolete version of ~ ssre...
ralss 4083 Restricted universal quant...
rexss 4084 Restricted existential qua...
ss2ab 4085 Class abstractions in a su...
abss 4086 Class abstraction in a sub...
ssab 4087 Subclass of a class abstra...
ssabral 4088 The relation for a subclas...
ss2abdv 4089 Deduction of abstraction s...
ss2abi 4090 Inference of abstraction s...
abssdv 4091 Deduction of abstraction s...
abssdvOLD 4092 Obsolete version of ~ abss...
abssi 4093 Inference of abstraction s...
ss2rab 4094 Restricted abstraction cla...
rabss 4095 Restricted class abstracti...
ssrab 4096 Subclass of a restricted c...
ssrabdv 4097 Subclass of a restricted c...
rabssdv 4098 Subclass of a restricted c...
ss2rabdv 4099 Deduction of restricted ab...
ss2rabi 4100 Inference of restricted ab...
rabss2 4101 Subclass law for restricte...
ssab2 4102 Subclass relation for the ...
ssrab2 4103 Subclass relation for a re...
rabss3d 4104 Subclass law for restricte...
ssrab3 4105 Subclass relation for a re...
rabssrabd 4106 Subclass of a restricted c...
ssrabeq 4107 If the restricting class o...
rabssab 4108 A restricted class is a su...
eqrrabd 4109 Deduce equality with a res...
uniiunlem 4110 A subset relationship usef...
dfpss2 4111 Alternate definition of pr...
dfpss3 4112 Alternate definition of pr...
psseq1 4113 Equality theorem for prope...
psseq2 4114 Equality theorem for prope...
psseq1i 4115 An equality inference for ...
psseq2i 4116 An equality inference for ...
psseq12i 4117 An equality inference for ...
psseq1d 4118 An equality deduction for ...
psseq2d 4119 An equality deduction for ...
psseq12d 4120 An equality deduction for ...
pssss 4121 A proper subclass is a sub...
pssne 4122 Two classes in a proper su...
pssssd 4123 Deduce subclass from prope...
pssned 4124 Proper subclasses are uneq...
sspss 4125 Subclass in terms of prope...
pssirr 4126 Proper subclass is irrefle...
pssn2lp 4127 Proper subclass has no 2-c...
sspsstri 4128 Two ways of stating tricho...
ssnpss 4129 Partial trichotomy law for...
psstr 4130 Transitive law for proper ...
sspsstr 4131 Transitive law for subclas...
psssstr 4132 Transitive law for subclas...
psstrd 4133 Proper subclass inclusion ...
sspsstrd 4134 Transitivity involving sub...
psssstrd 4135 Transitivity involving sub...
npss 4136 A class is not a proper su...
ssnelpss 4137 A subclass missing a membe...
ssnelpssd 4138 Subclass inclusion with on...
ssexnelpss 4139 If there is an element of ...
dfdif3 4140 Alternate definition of cl...
dfdif3OLD 4141 Obsolete version of ~ dfdi...
difeq1 4142 Equality theorem for class...
difeq2 4143 Equality theorem for class...
difeq12 4144 Equality theorem for class...
difeq1i 4145 Inference adding differenc...
difeq2i 4146 Inference adding differenc...
difeq12i 4147 Equality inference for cla...
difeq1d 4148 Deduction adding differenc...
difeq2d 4149 Deduction adding differenc...
difeq12d 4150 Equality deduction for cla...
difeqri 4151 Inference from membership ...
nfdif 4152 Bound-variable hypothesis ...
nfdifOLD 4153 Obsolete version of ~ nfdi...
eldifi 4154 Implication of membership ...
eldifn 4155 Implication of membership ...
elndif 4156 A set does not belong to a...
neldif 4157 Implication of membership ...
difdif 4158 Double class difference. ...
difss 4159 Subclass relationship for ...
difssd 4160 A difference of two classe...
difss2 4161 If a class is contained in...
difss2d 4162 If a class is contained in...
ssdifss 4163 Preservation of a subclass...
ddif 4164 Double complement under un...
ssconb 4165 Contraposition law for sub...
sscon 4166 Contraposition law for sub...
ssdif 4167 Difference law for subsets...
ssdifd 4168 If ` A ` is contained in `...
sscond 4169 If ` A ` is contained in `...
ssdifssd 4170 If ` A ` is contained in `...
ssdif2d 4171 If ` A ` is contained in `...
raldifb 4172 Restricted universal quant...
rexdifi 4173 Restricted existential qua...
complss 4174 Complementation reverses i...
compleq 4175 Two classes are equal if a...
elun 4176 Expansion of membership in...
elunnel1 4177 A member of a union that i...
elunnel2 4178 A member of a union that i...
uneqri 4179 Inference from membership ...
unidm 4180 Idempotent law for union o...
uncom 4181 Commutative law for union ...
equncom 4182 If a class equals the unio...
equncomi 4183 Inference form of ~ equnco...
uneq1 4184 Equality theorem for the u...
uneq2 4185 Equality theorem for the u...
uneq12 4186 Equality theorem for the u...
uneq1i 4187 Inference adding union to ...
uneq2i 4188 Inference adding union to ...
uneq12i 4189 Equality inference for the...
uneq1d 4190 Deduction adding union to ...
uneq2d 4191 Deduction adding union to ...
uneq12d 4192 Equality deduction for the...
nfun 4193 Bound-variable hypothesis ...
nfunOLD 4194 Obsolete version of ~ nfun...
unass 4195 Associative law for union ...
un12 4196 A rearrangement of union. ...
un23 4197 A rearrangement of union. ...
un4 4198 A rearrangement of the uni...
unundi 4199 Union distributes over its...
unundir 4200 Union distributes over its...
ssun1 4201 Subclass relationship for ...
ssun2 4202 Subclass relationship for ...
ssun3 4203 Subclass law for union of ...
ssun4 4204 Subclass law for union of ...
elun1 4205 Membership law for union o...
elun2 4206 Membership law for union o...
elunant 4207 A statement is true for ev...
unss1 4208 Subclass law for union of ...
ssequn1 4209 A relationship between sub...
unss2 4210 Subclass law for union of ...
unss12 4211 Subclass law for union of ...
ssequn2 4212 A relationship between sub...
unss 4213 The union of two subclasse...
unssi 4214 An inference showing the u...
unssd 4215 A deduction showing the un...
unssad 4216 If ` ( A u. B ) ` is conta...
unssbd 4217 If ` ( A u. B ) ` is conta...
ssun 4218 A condition that implies i...
rexun 4219 Restricted existential qua...
ralunb 4220 Restricted quantification ...
ralun 4221 Restricted quantification ...
elini 4222 Membership in an intersect...
elind 4223 Deduce membership in an in...
elinel1 4224 Membership in an intersect...
elinel2 4225 Membership in an intersect...
elin2 4226 Membership in a class defi...
elin1d 4227 Elementhood in the first s...
elin2d 4228 Elementhood in the first s...
elin3 4229 Membership in a class defi...
incom 4230 Commutative law for inters...
ineqcom 4231 Two ways of expressing tha...
ineqcomi 4232 Two ways of expressing tha...
ineqri 4233 Inference from membership ...
ineq1 4234 Equality theorem for inter...
ineq2 4235 Equality theorem for inter...
ineq12 4236 Equality theorem for inter...
ineq1i 4237 Equality inference for int...
ineq2i 4238 Equality inference for int...
ineq12i 4239 Equality inference for int...
ineq1d 4240 Equality deduction for int...
ineq2d 4241 Equality deduction for int...
ineq12d 4242 Equality deduction for int...
ineqan12d 4243 Equality deduction for int...
sseqin2 4244 A relationship between sub...
nfin 4245 Bound-variable hypothesis ...
nfinOLD 4246 Obsolete version of ~ nfin...
rabbi2dva 4247 Deduction from a wff to a ...
inidm 4248 Idempotent law for interse...
inass 4249 Associative law for inters...
in12 4250 A rearrangement of interse...
in32 4251 A rearrangement of interse...
in13 4252 A rearrangement of interse...
in31 4253 A rearrangement of interse...
inrot 4254 Rotate the intersection of...
in4 4255 Rearrangement of intersect...
inindi 4256 Intersection distributes o...
inindir 4257 Intersection distributes o...
inss1 4258 The intersection of two cl...
inss2 4259 The intersection of two cl...
ssin 4260 Subclass of intersection. ...
ssini 4261 An inference showing that ...
ssind 4262 A deduction showing that a...
ssrin 4263 Add right intersection to ...
sslin 4264 Add left intersection to s...
ssrind 4265 Add right intersection to ...
ss2in 4266 Intersection of subclasses...
ssinss1 4267 Intersection preserves sub...
inss 4268 Inclusion of an intersecti...
rexin 4269 Restricted existential qua...
dfss7 4270 Alternate definition of su...
symdifcom 4273 Symmetric difference commu...
symdifeq1 4274 Equality theorem for symme...
symdifeq2 4275 Equality theorem for symme...
nfsymdif 4276 Hypothesis builder for sym...
elsymdif 4277 Membership in a symmetric ...
dfsymdif4 4278 Alternate definition of th...
elsymdifxor 4279 Membership in a symmetric ...
dfsymdif2 4280 Alternate definition of th...
symdifass 4281 Symmetric difference is as...
difsssymdif 4282 The symmetric difference c...
difsymssdifssd 4283 If the symmetric differenc...
unabs 4284 Absorption law for union. ...
inabs 4285 Absorption law for interse...
nssinpss 4286 Negation of subclass expre...
nsspssun 4287 Negation of subclass expre...
dfss4 4288 Subclass defined in terms ...
dfun2 4289 An alternate definition of...
dfin2 4290 An alternate definition of...
difin 4291 Difference with intersecti...
ssdifim 4292 Implication of a class dif...
ssdifsym 4293 Symmetric class difference...
dfss5 4294 Alternate definition of su...
dfun3 4295 Union defined in terms of ...
dfin3 4296 Intersection defined in te...
dfin4 4297 Alternate definition of th...
invdif 4298 Intersection with universa...
indif 4299 Intersection with class di...
indif2 4300 Bring an intersection in a...
indif1 4301 Bring an intersection in a...
indifcom 4302 Commutation law for inters...
indi 4303 Distributive law for inter...
undi 4304 Distributive law for union...
indir 4305 Distributive law for inter...
undir 4306 Distributive law for union...
unineq 4307 Infer equality from equali...
uneqin 4308 Equality of union and inte...
difundi 4309 Distributive law for class...
difundir 4310 Distributive law for class...
difindi 4311 Distributive law for class...
difindir 4312 Distributive law for class...
indifdi 4313 Distribute intersection ov...
indifdir 4314 Distribute intersection ov...
difdif2 4315 Class difference by a clas...
undm 4316 De Morgan's law for union....
indm 4317 De Morgan's law for inters...
difun1 4318 A relationship involving d...
undif3 4319 An equality involving clas...
difin2 4320 Represent a class differen...
dif32 4321 Swap second and third argu...
difabs 4322 Absorption-like law for cl...
sscon34b 4323 Relative complementation r...
rcompleq 4324 Two subclasses are equal i...
dfsymdif3 4325 Alternate definition of th...
unabw 4326 Union of two class abstrac...
unab 4327 Union of two class abstrac...
inab 4328 Intersection of two class ...
difab 4329 Difference of two class ab...
abanssl 4330 A class abstraction with a...
abanssr 4331 A class abstraction with a...
notabw 4332 A class abstraction define...
notab 4333 A class abstraction define...
unrab 4334 Union of two restricted cl...
inrab 4335 Intersection of two restri...
inrab2 4336 Intersection with a restri...
difrab 4337 Difference of two restrict...
dfrab3 4338 Alternate definition of re...
dfrab2 4339 Alternate definition of re...
rabdif 4340 Move difference in and out...
notrab 4341 Complementation of restric...
dfrab3ss 4342 Restricted class abstracti...
rabun2 4343 Abstraction restricted to ...
reuun2 4344 Transfer uniqueness to a s...
reuss2 4345 Transfer uniqueness to a s...
reuss 4346 Transfer uniqueness to a s...
reuun1 4347 Transfer uniqueness to a s...
reupick 4348 Restricted uniqueness "pic...
reupick3 4349 Restricted uniqueness "pic...
reupick2 4350 Restricted uniqueness "pic...
euelss 4351 Transfer uniqueness of an ...
dfnul4 4354 Alternate definition of th...
dfnul2 4355 Alternate definition of th...
dfnul3 4356 Alternate definition of th...
dfnul2OLD 4357 Obsolete version of ~ dfnu...
dfnul3OLD 4358 Obsolete version of ~ dfnu...
dfnul4OLD 4359 Obsolete version of ~ dfnu...
noel 4360 The empty set has no eleme...
noelOLD 4361 Obsolete version of ~ noel...
nel02 4362 The empty set has no eleme...
n0i 4363 If a class has elements, t...
ne0i 4364 If a class has elements, t...
ne0d 4365 Deduction form of ~ ne0i ....
n0ii 4366 If a class has elements, t...
ne0ii 4367 If a class has elements, t...
vn0 4368 The universal class is not...
vn0ALT 4369 Alternate proof of ~ vn0 ....
eq0f 4370 A class is equal to the em...
neq0f 4371 A class is not empty if an...
n0f 4372 A class is nonempty if and...
eq0 4373 A class is equal to the em...
eq0ALT 4374 Alternate proof of ~ eq0 ....
neq0 4375 A class is not empty if an...
n0 4376 A class is nonempty if and...
nel0 4377 From the general negation ...
reximdva0 4378 Restricted existence deduc...
rspn0 4379 Specialization for restric...
n0rex 4380 There is an element in a n...
ssn0rex 4381 There is an element in a c...
n0moeu 4382 A case of equivalence of "...
rex0 4383 Vacuous restricted existen...
reu0 4384 Vacuous restricted uniquen...
rmo0 4385 Vacuous restricted at-most...
0el 4386 Membership of the empty se...
n0el 4387 Negated membership of the ...
eqeuel 4388 A condition which implies ...
ssdif0 4389 Subclass expressed in term...
difn0 4390 If the difference of two s...
pssdifn0 4391 A proper subclass has a no...
pssdif 4392 A proper subclass has a no...
ndisj 4393 Express that an intersecti...
inn0f 4394 A nonempty intersection. ...
inn0 4395 A nonempty intersection. ...
difin0ss 4396 Difference, intersection, ...
inssdif0 4397 Intersection, subclass, an...
difid 4398 The difference between a c...
difidALT 4399 Alternate proof of ~ difid...
dif0 4400 The difference between a c...
ab0w 4401 The class of sets verifyin...
ab0 4402 The class of sets verifyin...
ab0OLD 4403 Obsolete version of ~ ab0 ...
ab0ALT 4404 Alternate proof of ~ ab0 ,...
dfnf5 4405 Characterization of nonfre...
ab0orv 4406 The class abstraction defi...
ab0orvALT 4407 Alternate proof of ~ ab0or...
abn0 4408 Nonempty class abstraction...
rab0 4409 Any restricted class abstr...
rabeq0w 4410 Condition for a restricted...
rabeq0 4411 Condition for a restricted...
rabn0 4412 Nonempty restricted class ...
rabxm 4413 Law of excluded middle, in...
rabnc 4414 Law of noncontradiction, i...
elneldisj 4415 The set of elements ` s ` ...
elnelun 4416 The union of the set of el...
un0 4417 The union of a class with ...
in0 4418 The intersection of a clas...
0un 4419 The union of the empty set...
0in 4420 The intersection of the em...
inv1 4421 The intersection of a clas...
unv 4422 The union of a class with ...
0ss 4423 The null set is a subset o...
ss0b 4424 Any subset of the empty se...
ss0 4425 Any subset of the empty se...
sseq0 4426 A subclass of an empty cla...
ssn0 4427 A class with a nonempty su...
0dif 4428 The difference between the...
abf 4429 A class abstraction determ...
eq0rdv 4430 Deduction for equality to ...
eq0rdvALT 4431 Alternate proof of ~ eq0rd...
csbprc 4432 The proper substitution of...
csb0 4433 The proper substitution of...
sbcel12 4434 Distribute proper substitu...
sbceqg 4435 Distribute proper substitu...
sbceqi 4436 Distribution of class subs...
sbcnel12g 4437 Distribute proper substitu...
sbcne12 4438 Distribute proper substitu...
sbcel1g 4439 Move proper substitution i...
sbceq1g 4440 Move proper substitution t...
sbcel2 4441 Move proper substitution i...
sbceq2g 4442 Move proper substitution t...
csbcom 4443 Commutative law for double...
sbcnestgfw 4444 Nest the composition of tw...
csbnestgfw 4445 Nest the composition of tw...
sbcnestgw 4446 Nest the composition of tw...
csbnestgw 4447 Nest the composition of tw...
sbcco3gw 4448 Composition of two substit...
sbcnestgf 4449 Nest the composition of tw...
csbnestgf 4450 Nest the composition of tw...
sbcnestg 4451 Nest the composition of tw...
csbnestg 4452 Nest the composition of tw...
sbcco3g 4453 Composition of two substit...
csbco3g 4454 Composition of two class s...
csbnest1g 4455 Nest the composition of tw...
csbidm 4456 Idempotent law for class s...
csbvarg 4457 The proper substitution of...
csbvargi 4458 The proper substitution of...
sbccsb 4459 Substitution into a wff ex...
sbccsb2 4460 Substitution into a wff ex...
rspcsbela 4461 Special case related to ~ ...
sbnfc2 4462 Two ways of expressing " `...
csbab 4463 Move substitution into a c...
csbun 4464 Distribution of class subs...
csbin 4465 Distribute proper substitu...
csbie2df 4466 Conversion of implicit sub...
2nreu 4467 If there are two different...
un00 4468 Two classes are empty iff ...
vss 4469 Only the universal class h...
0pss 4470 The null set is a proper s...
npss0 4471 No set is a proper subset ...
pssv 4472 Any non-universal class is...
disj 4473 Two ways of saying that tw...
disjr 4474 Two ways of saying that tw...
disj1 4475 Two ways of saying that tw...
reldisj 4476 Two ways of saying that tw...
disj3 4477 Two ways of saying that tw...
disjne 4478 Members of disjoint sets a...
disjeq0 4479 Two disjoint sets are equa...
disjel 4480 A set can't belong to both...
disj2 4481 Two ways of saying that tw...
disj4 4482 Two ways of saying that tw...
ssdisj 4483 Intersection with a subcla...
disjpss 4484 A class is a proper subset...
undisj1 4485 The union of disjoint clas...
undisj2 4486 The union of disjoint clas...
ssindif0 4487 Subclass expressed in term...
inelcm 4488 The intersection of classe...
minel 4489 A minimum element of a cla...
undif4 4490 Distribute union over diff...
disjssun 4491 Subset relation for disjoi...
vdif0 4492 Universal class equality i...
difrab0eq 4493 If the difference between ...
pssnel 4494 A proper subclass has a me...
disjdif 4495 A class and its relative c...
disjdifr 4496 A class and its relative c...
difin0 4497 The difference of a class ...
unvdif 4498 The union of a class and i...
undif1 4499 Absorption of difference b...
undif2 4500 Absorption of difference b...
undifabs 4501 Absorption of difference b...
inundif 4502 The intersection and class...
disjdif2 4503 The difference of a class ...
difun2 4504 Absorption of union by dif...
undif 4505 Union of complementary par...
undifr 4506 Union of complementary par...
undifrOLD 4507 Obsolete version of ~ undi...
undif5 4508 An equality involving clas...
ssdifin0 4509 A subset of a difference d...
ssdifeq0 4510 A class is a subclass of i...
ssundif 4511 A condition equivalent to ...
difcom 4512 Swap the arguments of a cl...
pssdifcom1 4513 Two ways to express overla...
pssdifcom2 4514 Two ways to express non-co...
difdifdir 4515 Distributive law for class...
uneqdifeq 4516 Two ways to say that ` A `...
raldifeq 4517 Equality theorem for restr...
r19.2z 4518 Theorem 19.2 of [Margaris]...
r19.2zb 4519 A response to the notion t...
r19.3rz 4520 Restricted quantification ...
r19.28z 4521 Restricted quantifier vers...
r19.3rzv 4522 Restricted quantification ...
r19.9rzv 4523 Restricted quantification ...
r19.28zv 4524 Restricted quantifier vers...
r19.37zv 4525 Restricted quantifier vers...
r19.45zv 4526 Restricted version of Theo...
r19.44zv 4527 Restricted version of Theo...
r19.27z 4528 Restricted quantifier vers...
r19.27zv 4529 Restricted quantifier vers...
r19.36zv 4530 Restricted quantifier vers...
ralidmw 4531 Idempotent law for restric...
rzal 4532 Vacuous quantification is ...
rzalALT 4533 Alternate proof of ~ rzal ...
rexn0 4534 Restricted existential qua...
ralidm 4535 Idempotent law for restric...
ral0 4536 Vacuous universal quantifi...
ralf0 4537 The quantification of a fa...
ralnralall 4538 A contradiction concerning...
falseral0 4539 A false statement can only...
raaan 4540 Rearrange restricted quant...
raaanv 4541 Rearrange restricted quant...
sbss 4542 Set substitution into the ...
sbcssg 4543 Distribute proper substitu...
raaan2 4544 Rearrange restricted quant...
2reu4lem 4545 Lemma for ~ 2reu4 . (Cont...
2reu4 4546 Definition of double restr...
csbdif 4547 Distribution of class subs...
dfif2 4550 An alternate definition of...
dfif6 4551 An alternate definition of...
ifeq1 4552 Equality theorem for condi...
ifeq2 4553 Equality theorem for condi...
iftrue 4554 Value of the conditional o...
iftruei 4555 Inference associated with ...
iftrued 4556 Value of the conditional o...
iffalse 4557 Value of the conditional o...
iffalsei 4558 Inference associated with ...
iffalsed 4559 Value of the conditional o...
ifnefalse 4560 When values are unequal, b...
ifsb 4561 Distribute a function over...
dfif3 4562 Alternate definition of th...
dfif4 4563 Alternate definition of th...
dfif5 4564 Alternate definition of th...
ifssun 4565 A conditional class is inc...
ifeq12 4566 Equality theorem for condi...
ifeq1d 4567 Equality deduction for con...
ifeq2d 4568 Equality deduction for con...
ifeq12d 4569 Equality deduction for con...
ifbi 4570 Equivalence theorem for co...
ifbid 4571 Equivalence deduction for ...
ifbieq1d 4572 Equivalence/equality deduc...
ifbieq2i 4573 Equivalence/equality infer...
ifbieq2d 4574 Equivalence/equality deduc...
ifbieq12i 4575 Equivalence deduction for ...
ifbieq12d 4576 Equivalence deduction for ...
nfifd 4577 Deduction form of ~ nfif ....
nfif 4578 Bound-variable hypothesis ...
ifeq1da 4579 Conditional equality. (Co...
ifeq2da 4580 Conditional equality. (Co...
ifeq12da 4581 Equivalence deduction for ...
ifbieq12d2 4582 Equivalence deduction for ...
ifclda 4583 Conditional closure. (Con...
ifeqda 4584 Separation of the values o...
elimif 4585 Elimination of a condition...
ifbothda 4586 A wff ` th ` containing a ...
ifboth 4587 A wff ` th ` containing a ...
ifid 4588 Identical true and false a...
eqif 4589 Expansion of an equality w...
ifval 4590 Another expression of the ...
elif 4591 Membership in a conditiona...
ifel 4592 Membership of a conditiona...
ifcl 4593 Membership (closure) of a ...
ifcld 4594 Membership (closure) of a ...
ifcli 4595 Inference associated with ...
ifexd 4596 Existence of the condition...
ifexg 4597 Existence of the condition...
ifex 4598 Existence of the condition...
ifeqor 4599 The possible values of a c...
ifnot 4600 Negating the first argumen...
ifan 4601 Rewrite a conjunction in a...
ifor 4602 Rewrite a disjunction in a...
2if2 4603 Resolve two nested conditi...
ifcomnan 4604 Commute the conditions in ...
csbif 4605 Distribute proper substitu...
dedth 4606 Weak deduction theorem tha...
dedth2h 4607 Weak deduction theorem eli...
dedth3h 4608 Weak deduction theorem eli...
dedth4h 4609 Weak deduction theorem eli...
dedth2v 4610 Weak deduction theorem for...
dedth3v 4611 Weak deduction theorem for...
dedth4v 4612 Weak deduction theorem for...
elimhyp 4613 Eliminate a hypothesis con...
elimhyp2v 4614 Eliminate a hypothesis con...
elimhyp3v 4615 Eliminate a hypothesis con...
elimhyp4v 4616 Eliminate a hypothesis con...
elimel 4617 Eliminate a membership hyp...
elimdhyp 4618 Version of ~ elimhyp where...
keephyp 4619 Transform a hypothesis ` p...
keephyp2v 4620 Keep a hypothesis containi...
keephyp3v 4621 Keep a hypothesis containi...
pwjust 4623 Soundness justification th...
elpwg 4625 Membership in a power clas...
elpw 4626 Membership in a power clas...
velpw 4627 Setvar variable membership...
elpwd 4628 Membership in a power clas...
elpwi 4629 Subset relation implied by...
elpwb 4630 Characterization of the el...
elpwid 4631 An element of a power clas...
elelpwi 4632 If ` A ` belongs to a part...
sspw 4633 The powerclass preserves i...
sspwi 4634 The powerclass preserves i...
sspwd 4635 The powerclass preserves i...
pweq 4636 Equality theorem for power...
pweqALT 4637 Alternate proof of ~ pweq ...
pweqi 4638 Equality inference for pow...
pweqd 4639 Equality deduction for pow...
pwunss 4640 The power class of the uni...
nfpw 4641 Bound-variable hypothesis ...
pwidg 4642 A set is an element of its...
pwidb 4643 A class is an element of i...
pwid 4644 A set is a member of its p...
pwss 4645 Subclass relationship for ...
pwundif 4646 Break up the power class o...
snjust 4647 Soundness justification th...
sneq 4658 Equality theorem for singl...
sneqi 4659 Equality inference for sin...
sneqd 4660 Equality deduction for sin...
dfsn2 4661 Alternate definition of si...
elsng 4662 There is exactly one eleme...
elsn 4663 There is exactly one eleme...
velsn 4664 There is only one element ...
elsni 4665 There is at most one eleme...
rabsneq 4666 Equality of class abstract...
absn 4667 Condition for a class abst...
dfpr2 4668 Alternate definition of a ...
dfsn2ALT 4669 Alternate definition of si...
elprg 4670 A member of a pair of clas...
elpri 4671 If a class is an element o...
elpr 4672 A member of a pair of clas...
elpr2g 4673 A member of a pair of sets...
elpr2 4674 A member of a pair of sets...
nelpr2 4675 If a class is not an eleme...
nelpr1 4676 If a class is not an eleme...
nelpri 4677 If an element doesn't matc...
prneli 4678 If an element doesn't matc...
nelprd 4679 If an element doesn't matc...
eldifpr 4680 Membership in a set with t...
rexdifpr 4681 Restricted existential qua...
snidg 4682 A set is a member of its s...
snidb 4683 A class is a set iff it is...
snid 4684 A set is a member of its s...
vsnid 4685 A setvar variable is a mem...
elsn2g 4686 There is exactly one eleme...
elsn2 4687 There is exactly one eleme...
nelsn 4688 If a class is not equal to...
rabeqsn 4689 Conditions for a restricte...
rabsssn 4690 Conditions for a restricte...
rabeqsnd 4691 Conditions for a restricte...
ralsnsg 4692 Substitution expressed in ...
rexsns 4693 Restricted existential qua...
rexsngf 4694 Restricted existential qua...
ralsngf 4695 Restricted universal quant...
reusngf 4696 Restricted existential uni...
ralsng 4697 Substitution expressed in ...
rexsng 4698 Restricted existential qua...
reusng 4699 Restricted existential uni...
2ralsng 4700 Substitution expressed in ...
ralsngOLD 4701 Obsolete version of ~ rals...
rexsngOLD 4702 Obsolete version of ~ rexs...
rexreusng 4703 Restricted existential uni...
exsnrex 4704 There is a set being the e...
ralsn 4705 Convert a universal quanti...
rexsn 4706 Convert an existential qua...
elpwunsn 4707 Membership in an extension...
eqoreldif 4708 An element of a set is eit...
eltpg 4709 Members of an unordered tr...
eldiftp 4710 Membership in a set with t...
eltpi 4711 A member of an unordered t...
eltp 4712 A member of an unordered t...
el7g 4713 Members of a set with seve...
dftp2 4714 Alternate definition of un...
nfpr 4715 Bound-variable hypothesis ...
ifpr 4716 Membership of a conditiona...
ralprgf 4717 Convert a restricted unive...
rexprgf 4718 Convert a restricted exist...
ralprg 4719 Convert a restricted unive...
ralprgOLD 4720 Obsolete version of ~ ralp...
rexprg 4721 Convert a restricted exist...
rexprgOLD 4722 Obsolete version of ~ rexp...
raltpg 4723 Convert a restricted unive...
rextpg 4724 Convert a restricted exist...
ralpr 4725 Convert a restricted unive...
rexpr 4726 Convert a restricted exist...
reuprg0 4727 Convert a restricted exist...
reuprg 4728 Convert a restricted exist...
reurexprg 4729 Convert a restricted exist...
raltp 4730 Convert a universal quanti...
rextp 4731 Convert an existential qua...
nfsn 4732 Bound-variable hypothesis ...
csbsng 4733 Distribute proper substitu...
csbprg 4734 Distribute proper substitu...
elinsn 4735 If the intersection of two...
disjsn 4736 Intersection with the sing...
disjsn2 4737 Two distinct singletons ar...
disjpr2 4738 Two completely distinct un...
disjprsn 4739 The disjoint intersection ...
disjtpsn 4740 The disjoint intersection ...
disjtp2 4741 Two completely distinct un...
snprc 4742 The singleton of a proper ...
snnzb 4743 A singleton is nonempty if...
rmosn 4744 A restricted at-most-one q...
r19.12sn 4745 Special case of ~ r19.12 w...
rabsn 4746 Condition where a restrict...
rabsnifsb 4747 A restricted class abstrac...
rabsnif 4748 A restricted class abstrac...
rabrsn 4749 A restricted class abstrac...
euabsn2 4750 Another way to express exi...
euabsn 4751 Another way to express exi...
reusn 4752 A way to express restricte...
absneu 4753 Restricted existential uni...
rabsneu 4754 Restricted existential uni...
eusn 4755 Two ways to express " ` A ...
rabsnt 4756 Truth implied by equality ...
prcom 4757 Commutative law for unorde...
preq1 4758 Equality theorem for unord...
preq2 4759 Equality theorem for unord...
preq12 4760 Equality theorem for unord...
preq1i 4761 Equality inference for uno...
preq2i 4762 Equality inference for uno...
preq12i 4763 Equality inference for uno...
preq1d 4764 Equality deduction for uno...
preq2d 4765 Equality deduction for uno...
preq12d 4766 Equality deduction for uno...
tpeq1 4767 Equality theorem for unord...
tpeq2 4768 Equality theorem for unord...
tpeq3 4769 Equality theorem for unord...
tpeq1d 4770 Equality theorem for unord...
tpeq2d 4771 Equality theorem for unord...
tpeq3d 4772 Equality theorem for unord...
tpeq123d 4773 Equality theorem for unord...
tprot 4774 Rotation of the elements o...
tpcoma 4775 Swap 1st and 2nd members o...
tpcomb 4776 Swap 2nd and 3rd members o...
tpass 4777 Split off the first elemen...
qdass 4778 Two ways to write an unord...
qdassr 4779 Two ways to write an unord...
tpidm12 4780 Unordered triple ` { A , A...
tpidm13 4781 Unordered triple ` { A , B...
tpidm23 4782 Unordered triple ` { A , B...
tpidm 4783 Unordered triple ` { A , A...
tppreq3 4784 An unordered triple is an ...
prid1g 4785 An unordered pair contains...
prid2g 4786 An unordered pair contains...
prid1 4787 An unordered pair contains...
prid2 4788 An unordered pair contains...
ifpprsnss 4789 An unordered pair is a sin...
prprc1 4790 A proper class vanishes in...
prprc2 4791 A proper class vanishes in...
prprc 4792 An unordered pair containi...
tpid1 4793 One of the three elements ...
tpid1g 4794 Closed theorem form of ~ t...
tpid2 4795 One of the three elements ...
tpid2g 4796 Closed theorem form of ~ t...
tpid3g 4797 Closed theorem form of ~ t...
tpid3 4798 One of the three elements ...
snnzg 4799 The singleton of a set is ...
snn0d 4800 The singleton of a set is ...
snnz 4801 The singleton of a set is ...
prnz 4802 A pair containing a set is...
prnzg 4803 A pair containing a set is...
tpnz 4804 An unordered triple contai...
tpnzd 4805 An unordered triple contai...
raltpd 4806 Convert a universal quanti...
snssb 4807 Characterization of the in...
snssg 4808 The singleton formed on a ...
snssgOLD 4809 Obsolete version of ~ snss...
snss 4810 The singleton of an elemen...
eldifsn 4811 Membership in a set with a...
eldifsnd 4812 Membership in a set with a...
ssdifsn 4813 Subset of a set with an el...
elpwdifsn 4814 A subset of a set is an el...
eldifsni 4815 Membership in a set with a...
eldifsnneq 4816 An element of a difference...
neldifsn 4817 The class ` A ` is not in ...
neldifsnd 4818 The class ` A ` is not in ...
rexdifsn 4819 Restricted existential qua...
raldifsni 4820 Rearrangement of a propert...
raldifsnb 4821 Restricted universal quant...
eldifvsn 4822 A set is an element of the...
difsn 4823 An element not in a set ca...
difprsnss 4824 Removal of a singleton fro...
difprsn1 4825 Removal of a singleton fro...
difprsn2 4826 Removal of a singleton fro...
diftpsn3 4827 Removal of a singleton fro...
difpr 4828 Removing two elements as p...
tpprceq3 4829 An unordered triple is an ...
tppreqb 4830 An unordered triple is an ...
difsnb 4831 ` ( B \ { A } ) ` equals `...
difsnpss 4832 ` ( B \ { A } ) ` is a pro...
snssi 4833 The singleton of an elemen...
snssd 4834 The singleton of an elemen...
difsnid 4835 If we remove a single elem...
eldifeldifsn 4836 An element of a difference...
pw0 4837 Compute the power set of t...
pwpw0 4838 Compute the power set of t...
snsspr1 4839 A singleton is a subset of...
snsspr2 4840 A singleton is a subset of...
snsstp1 4841 A singleton is a subset of...
snsstp2 4842 A singleton is a subset of...
snsstp3 4843 A singleton is a subset of...
prssg 4844 A pair of elements of a cl...
prss 4845 A pair of elements of a cl...
prssi 4846 A pair of elements of a cl...
prssd 4847 Deduction version of ~ prs...
prsspwg 4848 An unordered pair belongs ...
ssprss 4849 A pair as subset of a pair...
ssprsseq 4850 A proper pair is a subset ...
sssn 4851 The subsets of a singleton...
ssunsn2 4852 The property of being sand...
ssunsn 4853 Possible values for a set ...
eqsn 4854 Two ways to express that a...
eqsnd 4855 Deduce that a set is a sin...
eqsndOLD 4856 Obsolete version of ~ eqsn...
issn 4857 A sufficient condition for...
n0snor2el 4858 A nonempty set is either a...
ssunpr 4859 Possible values for a set ...
sspr 4860 The subsets of a pair. (C...
sstp 4861 The subsets of an unordere...
tpss 4862 An unordered triple of ele...
tpssi 4863 An unordered triple of ele...
sneqrg 4864 Closed form of ~ sneqr . ...
sneqr 4865 If the singletons of two s...
snsssn 4866 If a singleton is a subset...
mosneq 4867 There exists at most one s...
sneqbg 4868 Two singletons of sets are...
snsspw 4869 The singleton of a class i...
prsspw 4870 An unordered pair belongs ...
preq1b 4871 Biconditional equality lem...
preq2b 4872 Biconditional equality lem...
preqr1 4873 Reverse equality lemma for...
preqr2 4874 Reverse equality lemma for...
preq12b 4875 Equality relationship for ...
opthpr 4876 An unordered pair has the ...
preqr1g 4877 Reverse equality lemma for...
preq12bg 4878 Closed form of ~ preq12b ....
prneimg 4879 Two pairs are not equal if...
prnebg 4880 A (proper) pair is not equ...
pr1eqbg 4881 A (proper) pair is equal t...
pr1nebg 4882 A (proper) pair is not equ...
preqsnd 4883 Equivalence for a pair equ...
prnesn 4884 A proper unordered pair is...
prneprprc 4885 A proper unordered pair is...
preqsn 4886 Equivalence for a pair equ...
preq12nebg 4887 Equality relationship for ...
prel12g 4888 Equality of two unordered ...
opthprneg 4889 An unordered pair has the ...
elpreqprlem 4890 Lemma for ~ elpreqpr . (C...
elpreqpr 4891 Equality and membership ru...
elpreqprb 4892 A set is an element of an ...
elpr2elpr 4893 For an element ` A ` of an...
dfopif 4894 Rewrite ~ df-op using ` if...
dfopg 4895 Value of the ordered pair ...
dfop 4896 Value of an ordered pair w...
opeq1 4897 Equality theorem for order...
opeq2 4898 Equality theorem for order...
opeq12 4899 Equality theorem for order...
opeq1i 4900 Equality inference for ord...
opeq2i 4901 Equality inference for ord...
opeq12i 4902 Equality inference for ord...
opeq1d 4903 Equality deduction for ord...
opeq2d 4904 Equality deduction for ord...
opeq12d 4905 Equality deduction for ord...
oteq1 4906 Equality theorem for order...
oteq2 4907 Equality theorem for order...
oteq3 4908 Equality theorem for order...
oteq1d 4909 Equality deduction for ord...
oteq2d 4910 Equality deduction for ord...
oteq3d 4911 Equality deduction for ord...
oteq123d 4912 Equality deduction for ord...
nfop 4913 Bound-variable hypothesis ...
nfopd 4914 Deduction version of bound...
csbopg 4915 Distribution of class subs...
opidg 4916 The ordered pair ` <. A , ...
opid 4917 The ordered pair ` <. A , ...
ralunsn 4918 Restricted quantification ...
2ralunsn 4919 Double restricted quantifi...
opprc 4920 Expansion of an ordered pa...
opprc1 4921 Expansion of an ordered pa...
opprc2 4922 Expansion of an ordered pa...
oprcl 4923 If an ordered pair has an ...
pwsn 4924 The power set of a singlet...
pwpr 4925 The power set of an unorde...
pwtp 4926 The power set of an unorde...
pwpwpw0 4927 Compute the power set of t...
pwv 4928 The power class of the uni...
prproe 4929 For an element of a proper...
3elpr2eq 4930 If there are three element...
dfuni2 4933 Alternate definition of cl...
eluni 4934 Membership in class union....
eluni2 4935 Membership in class union....
elunii 4936 Membership in class union....
nfunid 4937 Deduction version of ~ nfu...
nfuni 4938 Bound-variable hypothesis ...
uniss 4939 Subclass relationship for ...
unissi 4940 Subclass relationship for ...
unissd 4941 Subclass relationship for ...
unieq 4942 Equality theorem for class...
unieqi 4943 Inference of equality of t...
unieqd 4944 Deduction of equality of t...
eluniab 4945 Membership in union of a c...
elunirab 4946 Membership in union of a c...
uniprg 4947 The union of a pair is the...
unipr 4948 The union of a pair is the...
unisng 4949 A set equals the union of ...
unisn 4950 A set equals the union of ...
unisnv 4951 A set equals the union of ...
unisn3 4952 Union of a singleton in th...
dfnfc2 4953 An alternative statement o...
uniun 4954 The class union of the uni...
uniin 4955 The class union of the int...
ssuni 4956 Subclass relationship for ...
uni0b 4957 The union of a set is empt...
uni0c 4958 The union of a set is empt...
uni0 4959 The union of the empty set...
csbuni 4960 Distribute proper substitu...
elssuni 4961 An element of a class is a...
unissel 4962 Condition turning a subcla...
unissb 4963 Relationship involving mem...
unissbOLD 4964 Obsolete version of ~ unis...
uniss2 4965 A subclass condition on th...
unidif 4966 If the difference ` A \ B ...
ssunieq 4967 Relationship implying unio...
unimax 4968 Any member of a class is t...
pwuni 4969 A class is a subclass of t...
dfint2 4972 Alternate definition of cl...
inteq 4973 Equality law for intersect...
inteqi 4974 Equality inference for cla...
inteqd 4975 Equality deduction for cla...
elint 4976 Membership in class inters...
elint2 4977 Membership in class inters...
elintg 4978 Membership in class inters...
elinti 4979 Membership in class inters...
nfint 4980 Bound-variable hypothesis ...
elintabg 4981 Two ways of saying a set i...
elintab 4982 Membership in the intersec...
elintabOLD 4983 Obsolete version of ~ elin...
elintrab 4984 Membership in the intersec...
elintrabg 4985 Membership in the intersec...
int0 4986 The intersection of the em...
intss1 4987 An element of a class incl...
ssint 4988 Subclass of a class inters...
ssintab 4989 Subclass of the intersecti...
ssintub 4990 Subclass of the least uppe...
ssmin 4991 Subclass of the minimum va...
intmin 4992 Any member of a class is t...
intss 4993 Intersection of subclasses...
intssuni 4994 The intersection of a none...
ssintrab 4995 Subclass of the intersecti...
unissint 4996 If the union of a class is...
intssuni2 4997 Subclass relationship for ...
intminss 4998 Under subset ordering, the...
intmin2 4999 Any set is the smallest of...
intmin3 5000 Under subset ordering, the...
intmin4 5001 Elimination of a conjunct ...
intab 5002 The intersection of a spec...
int0el 5003 The intersection of a clas...
intun 5004 The class intersection of ...
intprg 5005 The intersection of a pair...
intpr 5006 The intersection of a pair...
intsng 5007 Intersection of a singleto...
intsn 5008 The intersection of a sing...
uniintsn 5009 Two ways to express " ` A ...
uniintab 5010 The union and the intersec...
intunsn 5011 Theorem joining a singleto...
rint0 5012 Relative intersection of a...
elrint 5013 Membership in a restricted...
elrint2 5014 Membership in a restricted...
eliun 5019 Membership in indexed unio...
eliin 5020 Membership in indexed inte...
eliuni 5021 Membership in an indexed u...
iuncom 5022 Commutation of indexed uni...
iuncom4 5023 Commutation of union with ...
iunconst 5024 Indexed union of a constan...
iinconst 5025 Indexed intersection of a ...
iuneqconst 5026 Indexed union of identical...
iuniin 5027 Law combining indexed unio...
iinssiun 5028 An indexed intersection is...
iunss1 5029 Subclass theorem for index...
iinss1 5030 Subclass theorem for index...
iuneq1 5031 Equality theorem for index...
iineq1 5032 Equality theorem for index...
ss2iun 5033 Subclass theorem for index...
iuneq2 5034 Equality theorem for index...
iineq2 5035 Equality theorem for index...
iuneq2i 5036 Equality inference for ind...
iineq2i 5037 Equality inference for ind...
iineq2d 5038 Equality deduction for ind...
iuneq2dv 5039 Equality deduction for ind...
iineq2dv 5040 Equality deduction for ind...
iuneq12df 5041 Equality deduction for ind...
iuneq1d 5042 Equality theorem for index...
iuneq12dOLD 5043 Obsolete version of ~ iune...
iuneq12d 5044 Equality deduction for ind...
iuneq2d 5045 Equality deduction for ind...
nfiun 5046 Bound-variable hypothesis ...
nfiin 5047 Bound-variable hypothesis ...
nfiung 5048 Bound-variable hypothesis ...
nfiing 5049 Bound-variable hypothesis ...
nfiu1 5050 Bound-variable hypothesis ...
nfiu1OLD 5051 Obsolete version of ~ nfiu...
nfii1 5052 Bound-variable hypothesis ...
dfiun2g 5053 Alternate definition of in...
dfiun2gOLD 5054 Obsolete version of ~ dfiu...
dfiin2g 5055 Alternate definition of in...
dfiun2 5056 Alternate definition of in...
dfiin2 5057 Alternate definition of in...
dfiunv2 5058 Define double indexed unio...
cbviun 5059 Rule used to change the bo...
cbviin 5060 Change bound variables in ...
cbviung 5061 Rule used to change the bo...
cbviing 5062 Change bound variables in ...
cbviunv 5063 Rule used to change the bo...
cbviinv 5064 Change bound variables in ...
cbviunvg 5065 Rule used to change the bo...
cbviinvg 5066 Change bound variables in ...
iunssf 5067 Subset theorem for an inde...
iunss 5068 Subset theorem for an inde...
ssiun 5069 Subset implication for an ...
ssiun2 5070 Identity law for subset of...
ssiun2s 5071 Subset relationship for an...
iunss2 5072 A subclass condition on th...
iunssd 5073 Subset theorem for an inde...
iunab 5074 The indexed union of a cla...
iunrab 5075 The indexed union of a res...
iunxdif2 5076 Indexed union with a class...
ssiinf 5077 Subset theorem for an inde...
ssiin 5078 Subset theorem for an inde...
iinss 5079 Subset implication for an ...
iinss2 5080 An indexed intersection is...
uniiun 5081 Class union in terms of in...
intiin 5082 Class intersection in term...
iunid 5083 An indexed union of single...
iunidOLD 5084 Obsolete version of ~ iuni...
iun0 5085 An indexed union of the em...
0iun 5086 An empty indexed union is ...
0iin 5087 An empty indexed intersect...
viin 5088 Indexed intersection with ...
iunsn 5089 Indexed union of a singlet...
iunn0 5090 There is a nonempty class ...
iinab 5091 Indexed intersection of a ...
iinrab 5092 Indexed intersection of a ...
iinrab2 5093 Indexed intersection of a ...
iunin2 5094 Indexed union of intersect...
iunin1 5095 Indexed union of intersect...
iinun2 5096 Indexed intersection of un...
iundif2 5097 Indexed union of class dif...
iindif1 5098 Indexed intersection of cl...
2iunin 5099 Rearrange indexed unions o...
iindif2 5100 Indexed intersection of cl...
iinin2 5101 Indexed intersection of in...
iinin1 5102 Indexed intersection of in...
iinvdif 5103 The indexed intersection o...
elriin 5104 Elementhood in a relative ...
riin0 5105 Relative intersection of a...
riinn0 5106 Relative intersection of a...
riinrab 5107 Relative intersection of a...
symdif0 5108 Symmetric difference with ...
symdifv 5109 The symmetric difference w...
symdifid 5110 The symmetric difference o...
iinxsng 5111 A singleton index picks ou...
iinxprg 5112 Indexed intersection with ...
iunxsng 5113 A singleton index picks ou...
iunxsn 5114 A singleton index picks ou...
iunxsngf 5115 A singleton index picks ou...
iunun 5116 Separate a union in an ind...
iunxun 5117 Separate a union in the in...
iunxdif3 5118 An indexed union where som...
iunxprg 5119 A pair index picks out two...
iunxiun 5120 Separate an indexed union ...
iinuni 5121 A relationship involving u...
iununi 5122 A relationship involving u...
sspwuni 5123 Subclass relationship for ...
pwssb 5124 Two ways to express a coll...
elpwpw 5125 Characterization of the el...
pwpwab 5126 The double power class wri...
pwpwssunieq 5127 The class of sets whose un...
elpwuni 5128 Relationship for power cla...
iinpw 5129 The power class of an inte...
iunpwss 5130 Inclusion of an indexed un...
intss2 5131 A nonempty intersection of...
rintn0 5132 Relative intersection of a...
dfdisj2 5135 Alternate definition for d...
disjss2 5136 If each element of a colle...
disjeq2 5137 Equality theorem for disjo...
disjeq2dv 5138 Equality deduction for dis...
disjss1 5139 A subset of a disjoint col...
disjeq1 5140 Equality theorem for disjo...
disjeq1d 5141 Equality theorem for disjo...
disjeq12d 5142 Equality theorem for disjo...
cbvdisj 5143 Change bound variables in ...
cbvdisjv 5144 Change bound variables in ...
nfdisjw 5145 Bound-variable hypothesis ...
nfdisj 5146 Bound-variable hypothesis ...
nfdisj1 5147 Bound-variable hypothesis ...
disjor 5148 Two ways to say that a col...
disjors 5149 Two ways to say that a col...
disji2 5150 Property of a disjoint col...
disji 5151 Property of a disjoint col...
invdisj 5152 If there is a function ` C...
invdisjrab 5153 The restricted class abstr...
disjiun 5154 A disjoint collection yiel...
disjord 5155 Conditions for a collectio...
disjiunb 5156 Two ways to say that a col...
disjiund 5157 Conditions for a collectio...
sndisj 5158 Any collection of singleto...
0disj 5159 Any collection of empty se...
disjxsn 5160 A singleton collection is ...
disjx0 5161 An empty collection is dis...
disjprg 5162 A pair collection is disjo...
disjxiun 5163 An indexed union of a disj...
disjxun 5164 The union of two disjoint ...
disjss3 5165 Expand a disjoint collecti...
breq 5168 Equality theorem for binar...
breq1 5169 Equality theorem for a bin...
breq2 5170 Equality theorem for a bin...
breq12 5171 Equality theorem for a bin...
breqi 5172 Equality inference for bin...
breq1i 5173 Equality inference for a b...
breq2i 5174 Equality inference for a b...
breq12i 5175 Equality inference for a b...
breq1d 5176 Equality deduction for a b...
breqd 5177 Equality deduction for a b...
breq2d 5178 Equality deduction for a b...
breq12d 5179 Equality deduction for a b...
breq123d 5180 Equality deduction for a b...
breqdi 5181 Equality deduction for a b...
breqan12d 5182 Equality deduction for a b...
breqan12rd 5183 Equality deduction for a b...
eqnbrtrd 5184 Substitution of equal clas...
nbrne1 5185 Two classes are different ...
nbrne2 5186 Two classes are different ...
eqbrtri 5187 Substitution of equal clas...
eqbrtrd 5188 Substitution of equal clas...
eqbrtrri 5189 Substitution of equal clas...
eqbrtrrd 5190 Substitution of equal clas...
breqtri 5191 Substitution of equal clas...
breqtrd 5192 Substitution of equal clas...
breqtrri 5193 Substitution of equal clas...
breqtrrd 5194 Substitution of equal clas...
3brtr3i 5195 Substitution of equality i...
3brtr4i 5196 Substitution of equality i...
3brtr3d 5197 Substitution of equality i...
3brtr4d 5198 Substitution of equality i...
3brtr3g 5199 Substitution of equality i...
3brtr4g 5200 Substitution of equality i...
eqbrtrid 5201 A chained equality inferen...
eqbrtrrid 5202 A chained equality inferen...
breqtrid 5203 A chained equality inferen...
breqtrrid 5204 A chained equality inferen...
eqbrtrdi 5205 A chained equality inferen...
eqbrtrrdi 5206 A chained equality inferen...
breqtrdi 5207 A chained equality inferen...
breqtrrdi 5208 A chained equality inferen...
ssbrd 5209 Deduction from a subclass ...
ssbr 5210 Implication from a subclas...
ssbri 5211 Inference from a subclass ...
nfbrd 5212 Deduction version of bound...
nfbr 5213 Bound-variable hypothesis ...
brab1 5214 Relationship between a bin...
br0 5215 The empty binary relation ...
brne0 5216 If two sets are in a binar...
brun 5217 The union of two binary re...
brin 5218 The intersection of two re...
brdif 5219 The difference of two bina...
sbcbr123 5220 Move substitution in and o...
sbcbr 5221 Move substitution in and o...
sbcbr12g 5222 Move substitution in and o...
sbcbr1g 5223 Move substitution in and o...
sbcbr2g 5224 Move substitution in and o...
brsymdif 5225 Characterization of the sy...
brralrspcev 5226 Restricted existential spe...
brimralrspcev 5227 Restricted existential spe...
opabss 5230 The collection of ordered ...
opabbid 5231 Equivalent wff's yield equ...
opabbidv 5232 Equivalent wff's yield equ...
opabbii 5233 Equivalent wff's yield equ...
nfopabd 5234 Bound-variable hypothesis ...
nfopab 5235 Bound-variable hypothesis ...
nfopab1 5236 The first abstraction vari...
nfopab2 5237 The second abstraction var...
cbvopab 5238 Rule used to change bound ...
cbvopabv 5239 Rule used to change bound ...
cbvopabvOLD 5240 Obsolete version of ~ cbvo...
cbvopab1 5241 Change first bound variabl...
cbvopab1g 5242 Change first bound variabl...
cbvopab2 5243 Change second bound variab...
cbvopab1s 5244 Change first bound variabl...
cbvopab1v 5245 Rule used to change the fi...
cbvopab1vOLD 5246 Obsolete version of ~ cbvo...
cbvopab2v 5247 Rule used to change the se...
unopab 5248 Union of two ordered pair ...
mpteq12da 5251 An equality inference for ...
mpteq12df 5252 An equality inference for ...
mpteq12dfOLD 5253 Obsolete version of ~ mpte...
mpteq12f 5254 An equality theorem for th...
mpteq12dva 5255 An equality inference for ...
mpteq12dvaOLD 5256 Obsolete version of ~ mpte...
mpteq12dv 5257 An equality inference for ...
mpteq12 5258 An equality theorem for th...
mpteq1 5259 An equality theorem for th...
mpteq1OLD 5260 Obsolete version of ~ mpte...
mpteq1d 5261 An equality theorem for th...
mpteq1i 5262 An equality theorem for th...
mpteq1iOLD 5263 Obsolete version of ~ mpte...
mpteq2da 5264 Slightly more general equa...
mpteq2daOLD 5265 Obsolete version of ~ mpte...
mpteq2dva 5266 Slightly more general equa...
mpteq2dvaOLD 5267 Obsolete version of ~ mpte...
mpteq2dv 5268 An equality inference for ...
mpteq2ia 5269 An equality inference for ...
mpteq2iaOLD 5270 Obsolete version of ~ mpte...
mpteq2i 5271 An equality inference for ...
mpteq12i 5272 An equality inference for ...
nfmpt 5273 Bound-variable hypothesis ...
nfmpt1 5274 Bound-variable hypothesis ...
cbvmptf 5275 Rule to change the bound v...
cbvmptfg 5276 Rule to change the bound v...
cbvmpt 5277 Rule to change the bound v...
cbvmptg 5278 Rule to change the bound v...
cbvmptv 5279 Rule to change the bound v...
cbvmptvOLD 5280 Obsolete version of ~ cbvm...
cbvmptvg 5281 Rule to change the bound v...
mptv 5282 Function with universal do...
dftr2 5285 An alternate way of defini...
dftr2c 5286 Variant of ~ dftr2 with co...
dftr5 5287 An alternate way of defini...
dftr5OLD 5288 Obsolete version of ~ dftr...
dftr3 5289 An alternate way of defini...
dftr4 5290 An alternate way of defini...
treq 5291 Equality theorem for the t...
trel 5292 In a transitive class, the...
trel3 5293 In a transitive class, the...
trss 5294 An element of a transitive...
trin 5295 The intersection of transi...
tr0 5296 The empty set is transitiv...
trv 5297 The universe is transitive...
triun 5298 An indexed union of a clas...
truni 5299 The union of a class of tr...
triin 5300 An indexed intersection of...
trint 5301 The intersection of a clas...
trintss 5302 Any nonempty transitive cl...
axrep1 5304 The version of the Axiom o...
axreplem 5305 Lemma for ~ axrep2 and ~ a...
axrep2 5306 Axiom of Replacement expre...
axrep3 5307 Axiom of Replacement sligh...
axrep4 5308 A more traditional version...
axrep5 5309 Axiom of Replacement (simi...
axrep6 5310 A condensed form of ~ ax-r...
axrep6g 5311 ~ axrep6 in class notation...
zfrepclf 5312 An inference based on the ...
zfrep3cl 5313 An inference based on the ...
zfrep4 5314 A version of Replacement u...
axsepgfromrep 5315 A more general version ~ a...
axsep 5316 Axiom scheme of separation...
axsepg 5318 A more general version of ...
zfauscl 5319 Separation Scheme (Aussond...
bm1.3ii 5320 Convert implication to equ...
ax6vsep 5321 Derive ~ ax6v (a weakened ...
axnulALT 5322 Alternate proof of ~ axnul...
axnul 5323 The Null Set Axiom of ZF s...
0ex 5325 The Null Set Axiom of ZF s...
al0ssb 5326 The empty set is the uniqu...
sseliALT 5327 Alternate proof of ~ sseli...
csbexg 5328 The existence of proper su...
csbex 5329 The existence of proper su...
unisn2 5330 A version of ~ unisn witho...
nalset 5331 No set contains all sets. ...
vnex 5332 The universal class does n...
vprc 5333 The universal class is not...
nvel 5334 The universal class does n...
inex1 5335 Separation Scheme (Aussond...
inex2 5336 Separation Scheme (Aussond...
inex1g 5337 Closed-form, generalized S...
inex2g 5338 Sufficient condition for a...
ssex 5339 The subset of a set is als...
ssexi 5340 The subset of a set is als...
ssexg 5341 The subset of a set is als...
ssexd 5342 A subclass of a set is a s...
abexd 5343 Conditions for a class abs...
abex 5344 Conditions for a class abs...
prcssprc 5345 The superclass of a proper...
sselpwd 5346 Elementhood to a power set...
difexg 5347 Existence of a difference....
difexi 5348 Existence of a difference,...
difexd 5349 Existence of a difference....
zfausab 5350 Separation Scheme (Aussond...
elpw2g 5351 Membership in a power clas...
elpw2 5352 Membership in a power clas...
elpwi2 5353 Membership in a power clas...
rabelpw 5354 A restricted class abstrac...
rabexg 5355 Separation Scheme in terms...
rabexgOLD 5356 Obsolete proof of ~ rabexg...
rabex 5357 Separation Scheme in terms...
rabexd 5358 Separation Scheme in terms...
rabex2 5359 Separation Scheme in terms...
rab2ex 5360 A class abstraction based ...
elssabg 5361 Membership in a class abst...
intex 5362 The intersection of a none...
intnex 5363 If a class intersection is...
intexab 5364 The intersection of a none...
intexrab 5365 The intersection of a none...
iinexg 5366 The existence of a class i...
intabs 5367 Absorption of a redundant ...
inuni 5368 The intersection of a unio...
axpweq 5369 Two equivalent ways to exp...
pwnss 5370 The power set of a set is ...
pwne 5371 No set equals its power se...
difelpw 5372 A difference is an element...
class2set 5373 The class of elements of `...
0elpw 5374 Every power class contains...
pwne0 5375 A power class is never emp...
0nep0 5376 The empty set and its powe...
0inp0 5377 Something cannot be equal ...
unidif0 5378 The removal of the empty s...
eqsnuniex 5379 If a class is equal to the...
iin0 5380 An indexed intersection of...
notzfaus 5381 In the Separation Scheme ~...
intv 5382 The intersection of the un...
zfpow 5384 Axiom of Power Sets expres...
axpow2 5385 A variant of the Axiom of ...
axpow3 5386 A variant of the Axiom of ...
elALT2 5387 Alternate proof of ~ el us...
dtruALT2 5388 Alternate proof of ~ dtru ...
dtrucor 5389 Corollary of ~ dtru . Thi...
dtrucor2 5390 The theorem form of the de...
dvdemo1 5391 Demonstration of a theorem...
dvdemo2 5392 Demonstration of a theorem...
nfnid 5393 A setvar variable is not f...
nfcvb 5394 The "distinctor" expressio...
vpwex 5395 Power set axiom: the power...
pwexg 5396 Power set axiom expressed ...
pwexd 5397 Deduction version of the p...
pwex 5398 Power set axiom expressed ...
pwel 5399 Quantitative version of ~ ...
abssexg 5400 Existence of a class of su...
snexALT 5401 Alternate proof of ~ snex ...
p0ex 5402 The power set of the empty...
p0exALT 5403 Alternate proof of ~ p0ex ...
pp0ex 5404 The power set of the power...
ord3ex 5405 The ordinal number 3 is a ...
dtruALT 5406 Alternate proof of ~ dtru ...
axc16b 5407 This theorem shows that Ax...
eunex 5408 Existential uniqueness imp...
eusv1 5409 Two ways to express single...
eusvnf 5410 Even if ` x ` is free in `...
eusvnfb 5411 Two ways to say that ` A (...
eusv2i 5412 Two ways to express single...
eusv2nf 5413 Two ways to express single...
eusv2 5414 Two ways to express single...
reusv1 5415 Two ways to express single...
reusv2lem1 5416 Lemma for ~ reusv2 . (Con...
reusv2lem2 5417 Lemma for ~ reusv2 . (Con...
reusv2lem3 5418 Lemma for ~ reusv2 . (Con...
reusv2lem4 5419 Lemma for ~ reusv2 . (Con...
reusv2lem5 5420 Lemma for ~ reusv2 . (Con...
reusv2 5421 Two ways to express single...
reusv3i 5422 Two ways of expressing exi...
reusv3 5423 Two ways to express single...
eusv4 5424 Two ways to express single...
alxfr 5425 Transfer universal quantif...
ralxfrd 5426 Transfer universal quantif...
rexxfrd 5427 Transfer existential quant...
ralxfr2d 5428 Transfer universal quantif...
rexxfr2d 5429 Transfer existential quant...
ralxfrd2 5430 Transfer universal quantif...
rexxfrd2 5431 Transfer existence from a ...
ralxfr 5432 Transfer universal quantif...
ralxfrALT 5433 Alternate proof of ~ ralxf...
rexxfr 5434 Transfer existence from a ...
rabxfrd 5435 Membership in a restricted...
rabxfr 5436 Membership in a restricted...
reuhypd 5437 A theorem useful for elimi...
reuhyp 5438 A theorem useful for elimi...
zfpair 5439 The Axiom of Pairing of Ze...
axprALT 5440 Alternate proof of ~ axpr ...
axprlem1 5441 Lemma for ~ axpr . There ...
axprlem2 5442 Lemma for ~ axpr . There ...
axprlem3 5443 Lemma for ~ axpr . Elimin...
axprlem4 5444 Lemma for ~ axpr . The fi...
axprlem5 5445 Lemma for ~ axpr . The se...
axpr 5446 Unabbreviated version of t...
zfpair2 5448 Derive the abbreviated ver...
vsnex 5449 A singleton built on a set...
snexg 5450 A singleton built on a set...
snex 5451 A singleton is a set. The...
prex 5452 The Axiom of Pairing using...
exel 5453 There exist two sets, one ...
exexneq 5454 There exist two different ...
exneq 5455 Given any set (the " ` y `...
dtru 5456 Given any set (the " ` y `...
el 5457 Any set is an element of s...
sels 5458 If a class is a set, then ...
selsALT 5459 Alternate proof of ~ sels ...
elALT 5460 Alternate proof of ~ el , ...
dtruOLD 5461 Obsolete proof of ~ dtru a...
snelpwg 5462 A singleton of a set is a ...
snelpwi 5463 If a set is a member of a ...
snelpwiOLD 5464 Obsolete version of ~ snel...
snelpw 5465 A singleton of a set is a ...
prelpw 5466 An unordered pair of two s...
prelpwi 5467 If two sets are members of...
rext 5468 A theorem similar to exten...
sspwb 5469 The powerclass constructio...
unipw 5470 A class equals the union o...
univ 5471 The union of the universe ...
pwtr 5472 A class is transitive iff ...
ssextss 5473 An extensionality-like pri...
ssext 5474 An extensionality-like pri...
nssss 5475 Negation of subclass relat...
pweqb 5476 Classes are equal if and o...
intidg 5477 The intersection of all se...
intidOLD 5478 Obsolete version of ~ inti...
moabex 5479 "At most one" existence im...
rmorabex 5480 Restricted "at most one" e...
euabex 5481 The abstraction of a wff w...
nnullss 5482 A nonempty class (even if ...
exss 5483 Restricted existence in a ...
opex 5484 An ordered pair of classes...
otex 5485 An ordered triple of class...
elopg 5486 Characterization of the el...
elop 5487 Characterization of the el...
opi1 5488 One of the two elements in...
opi2 5489 One of the two elements of...
opeluu 5490 Each member of an ordered ...
op1stb 5491 Extract the first member o...
brv 5492 Two classes are always in ...
opnz 5493 An ordered pair is nonempt...
opnzi 5494 An ordered pair is nonempt...
opth1 5495 Equality of the first memb...
opth 5496 The ordered pair theorem. ...
opthg 5497 Ordered pair theorem. ` C ...
opth1g 5498 Equality of the first memb...
opthg2 5499 Ordered pair theorem. (Co...
opth2 5500 Ordered pair theorem. (Co...
opthneg 5501 Two ordered pairs are not ...
opthne 5502 Two ordered pairs are not ...
otth2 5503 Ordered triple theorem, wi...
otth 5504 Ordered triple theorem. (...
otthg 5505 Ordered triple theorem, cl...
otthne 5506 Contrapositive of the orde...
eqvinop 5507 A variable introduction la...
sbcop1 5508 The proper substitution of...
sbcop 5509 The proper substitution of...
copsexgw 5510 Version of ~ copsexg with ...
copsexg 5511 Substitution of class ` A ...
copsex2t 5512 Closed theorem form of ~ c...
copsex2g 5513 Implicit substitution infe...
copsex4g 5514 An implicit substitution i...
0nelop 5515 A property of ordered pair...
opwo0id 5516 An ordered pair is equal t...
opeqex 5517 Equivalence of existence i...
oteqex2 5518 Equivalence of existence i...
oteqex 5519 Equivalence of existence i...
opcom 5520 An ordered pair commutes i...
moop2 5521 "At most one" property of ...
opeqsng 5522 Equivalence for an ordered...
opeqsn 5523 Equivalence for an ordered...
opeqpr 5524 Equivalence for an ordered...
snopeqop 5525 Equivalence for an ordered...
propeqop 5526 Equivalence for an ordered...
propssopi 5527 If a pair of ordered pairs...
snopeqopsnid 5528 Equivalence for an ordered...
mosubopt 5529 "At most one" remains true...
mosubop 5530 "At most one" remains true...
euop2 5531 Transfer existential uniqu...
euotd 5532 Prove existential uniquene...
opthwiener 5533 Justification theorem for ...
uniop 5534 The union of an ordered pa...
uniopel 5535 Ordered pair membership is...
opthhausdorff 5536 Justification theorem for ...
opthhausdorff0 5537 Justification theorem for ...
otsndisj 5538 The singletons consisting ...
otiunsndisj 5539 The union of singletons co...
iunopeqop 5540 Implication of an ordered ...
brsnop 5541 Binary relation for an ord...
brtp 5542 A necessary and sufficient...
opabidw 5543 The law of concretion. Sp...
opabid 5544 The law of concretion. Sp...
elopabw 5545 Membership in a class abst...
elopab 5546 Membership in a class abst...
rexopabb 5547 Restricted existential qua...
vopelopabsb 5548 The law of concretion in t...
opelopabsb 5549 The law of concretion in t...
brabsb 5550 The law of concretion in t...
opelopabt 5551 Closed theorem form of ~ o...
opelopabga 5552 The law of concretion. Th...
brabga 5553 The law of concretion for ...
opelopab2a 5554 Ordered pair membership in...
opelopaba 5555 The law of concretion. Th...
braba 5556 The law of concretion for ...
opelopabg 5557 The law of concretion. Th...
brabg 5558 The law of concretion for ...
opelopabgf 5559 The law of concretion. Th...
opelopab2 5560 Ordered pair membership in...
opelopab 5561 The law of concretion. Th...
brab 5562 The law of concretion for ...
opelopabaf 5563 The law of concretion. Th...
opelopabf 5564 The law of concretion. Th...
ssopab2 5565 Equivalence of ordered pai...
ssopab2bw 5566 Equivalence of ordered pai...
eqopab2bw 5567 Equivalence of ordered pai...
ssopab2b 5568 Equivalence of ordered pai...
ssopab2i 5569 Inference of ordered pair ...
ssopab2dv 5570 Inference of ordered pair ...
eqopab2b 5571 Equivalence of ordered pai...
opabn0 5572 Nonempty ordered pair clas...
opab0 5573 Empty ordered pair class a...
csbopab 5574 Move substitution into a c...
csbopabgALT 5575 Move substitution into a c...
csbmpt12 5576 Move substitution into a m...
csbmpt2 5577 Move substitution into the...
iunopab 5578 Move indexed union inside ...
iunopabOLD 5579 Obsolete version of ~ iuno...
elopabr 5580 Membership in an ordered-p...
elopabran 5581 Membership in an ordered-p...
elopabrOLD 5582 Obsolete version of ~ elop...
rbropapd 5583 Properties of a pair in an...
rbropap 5584 Properties of a pair in a ...
2rbropap 5585 Properties of a pair in a ...
0nelopab 5586 The empty set is never an ...
0nelopabOLD 5587 Obsolete version of ~ 0nel...
brabv 5588 If two classes are in a re...
pwin 5589 The power class of the int...
pwssun 5590 The power class of the uni...
pwun 5591 The power class of the uni...
dfid4 5594 The identity function expr...
dfid2 5595 Alternate definition of th...
dfid3 5596 A stronger version of ~ df...
dfid2OLD 5597 Obsolete version of ~ dfid...
epelg 5600 The membership relation an...
epeli 5601 The membership relation an...
epel 5602 The membership relation an...
0sn0ep 5603 An example for the members...
epn0 5604 The membership relation is...
poss 5609 Subset theorem for the par...
poeq1 5610 Equality theorem for parti...
poeq2 5611 Equality theorem for parti...
poeq12d 5612 Equality deduction for par...
nfpo 5613 Bound-variable hypothesis ...
nfso 5614 Bound-variable hypothesis ...
pocl 5615 Characteristic properties ...
poclOLD 5616 Obsolete version of ~ pocl...
ispod 5617 Sufficient conditions for ...
swopolem 5618 Perform the substitutions ...
swopo 5619 A strict weak order is a p...
poirr 5620 A partial order is irrefle...
potr 5621 A partial order is a trans...
po2nr 5622 A partial order has no 2-c...
po3nr 5623 A partial order has no 3-c...
po2ne 5624 Two sets related by a part...
po0 5625 Any relation is a partial ...
pofun 5626 The inverse image of a par...
sopo 5627 A strict linear order is a...
soss 5628 Subset theorem for the str...
soeq1 5629 Equality theorem for the s...
soeq2 5630 Equality theorem for the s...
soeq12d 5631 Equality deduction for tot...
sonr 5632 A strict order relation is...
sotr 5633 A strict order relation is...
solin 5634 A strict order relation is...
so2nr 5635 A strict order relation ha...
so3nr 5636 A strict order relation ha...
sotric 5637 A strict order relation sa...
sotrieq 5638 Trichotomy law for strict ...
sotrieq2 5639 Trichotomy law for strict ...
soasym 5640 Asymmetry law for strict o...
sotr2 5641 A transitivity relation. ...
issod 5642 An irreflexive, transitive...
issoi 5643 An irreflexive, transitive...
isso2i 5644 Deduce strict ordering fro...
so0 5645 Any relation is a strict o...
somo 5646 A totally ordered set has ...
sotrine 5647 Trichotomy law for strict ...
sotr3 5648 Transitivity law for stric...
dffr6 5655 Alternate definition of ~ ...
frd 5656 A nonempty subset of an ` ...
fri 5657 A nonempty subset of an ` ...
friOLD 5658 Obsolete version of ~ fri ...
seex 5659 The ` R ` -preimage of an ...
exse 5660 Any relation on a set is s...
dffr2 5661 Alternate definition of we...
dffr2ALT 5662 Alternate proof of ~ dffr2...
frc 5663 Property of well-founded r...
frss 5664 Subset theorem for the wel...
sess1 5665 Subset theorem for the set...
sess2 5666 Subset theorem for the set...
freq1 5667 Equality theorem for the w...
freq2 5668 Equality theorem for the w...
freq12d 5669 Equality deduction for wel...
seeq1 5670 Equality theorem for the s...
seeq2 5671 Equality theorem for the s...
seeq12d 5672 Equality deduction for the...
nffr 5673 Bound-variable hypothesis ...
nfse 5674 Bound-variable hypothesis ...
nfwe 5675 Bound-variable hypothesis ...
frirr 5676 A well-founded relation is...
fr2nr 5677 A well-founded relation ha...
fr0 5678 Any relation is well-found...
frminex 5679 If an element of a well-fo...
efrirr 5680 A well-founded class does ...
efrn2lp 5681 A well-founded class conta...
epse 5682 The membership relation is...
tz7.2 5683 Similar to Theorem 7.2 of ...
dfepfr 5684 An alternate way of saying...
epfrc 5685 A subset of a well-founded...
wess 5686 Subset theorem for the wel...
weeq1 5687 Equality theorem for the w...
weeq2 5688 Equality theorem for the w...
weeq12d 5689 Equality deduction for wel...
wefr 5690 A well-ordering is well-fo...
weso 5691 A well-ordering is a stric...
wecmpep 5692 The elements of a class we...
wetrep 5693 On a class well-ordered by...
wefrc 5694 A nonempty subclass of a c...
we0 5695 Any relation is a well-ord...
wereu 5696 A nonempty subset of an ` ...
wereu2 5697 A nonempty subclass of an ...
xpeq1 5714 Equality theorem for Carte...
xpss12 5715 Subset theorem for Cartesi...
xpss 5716 A Cartesian product is inc...
inxpssres 5717 Intersection with a Cartes...
relxp 5718 A Cartesian product is a r...
xpss1 5719 Subset relation for Cartes...
xpss2 5720 Subset relation for Cartes...
xpeq2 5721 Equality theorem for Carte...
elxpi 5722 Membership in a Cartesian ...
elxp 5723 Membership in a Cartesian ...
elxp2 5724 Membership in a Cartesian ...
xpeq12 5725 Equality theorem for Carte...
xpeq1i 5726 Equality inference for Car...
xpeq2i 5727 Equality inference for Car...
xpeq12i 5728 Equality inference for Car...
xpeq1d 5729 Equality deduction for Car...
xpeq2d 5730 Equality deduction for Car...
xpeq12d 5731 Equality deduction for Car...
sqxpeqd 5732 Equality deduction for a C...
nfxp 5733 Bound-variable hypothesis ...
0nelxp 5734 The empty set is not a mem...
0nelelxp 5735 A member of a Cartesian pr...
opelxp 5736 Ordered pair membership in...
opelxpi 5737 Ordered pair membership in...
opelxpii 5738 Ordered pair membership in...
opelxpd 5739 Ordered pair membership in...
opelvv 5740 Ordered pair membership in...
opelvvg 5741 Ordered pair membership in...
opelxp1 5742 The first member of an ord...
opelxp2 5743 The second member of an or...
otelxp 5744 Ordered triple membership ...
otelxp1 5745 The first member of an ord...
otel3xp 5746 An ordered triple is an el...
opabssxpd 5747 An ordered-pair class abst...
rabxp 5748 Class abstraction restrict...
brxp 5749 Binary relation on a Carte...
pwvrel 5750 A set is a binary relation...
pwvabrel 5751 The powerclass of the cart...
brrelex12 5752 Two classes related by a b...
brrelex1 5753 If two classes are related...
brrelex2 5754 If two classes are related...
brrelex12i 5755 Two classes that are relat...
brrelex1i 5756 The first argument of a bi...
brrelex2i 5757 The second argument of a b...
nprrel12 5758 Proper classes are not rel...
nprrel 5759 No proper class is related...
0nelrel0 5760 A binary relation does not...
0nelrel 5761 A binary relation does not...
fconstmpt 5762 Representation of a consta...
vtoclr 5763 Variable to class conversi...
opthprc 5764 Justification theorem for ...
brel 5765 Two things in a binary rel...
elxp3 5766 Membership in a Cartesian ...
opeliunxp 5767 Membership in a union of C...
xpundi 5768 Distributive law for Carte...
xpundir 5769 Distributive law for Carte...
xpiundi 5770 Distributive law for Carte...
xpiundir 5771 Distributive law for Carte...
iunxpconst 5772 Membership in a union of C...
xpun 5773 The Cartesian product of t...
elvv 5774 Membership in universal cl...
elvvv 5775 Membership in universal cl...
elvvuni 5776 An ordered pair contains i...
brinxp2 5777 Intersection of binary rel...
brinxp 5778 Intersection of binary rel...
opelinxp 5779 Ordered pair element in an...
poinxp 5780 Intersection of partial or...
soinxp 5781 Intersection of total orde...
frinxp 5782 Intersection of well-found...
seinxp 5783 Intersection of set-like r...
weinxp 5784 Intersection of well-order...
posn 5785 Partial ordering of a sing...
sosn 5786 Strict ordering on a singl...
frsn 5787 Founded relation on a sing...
wesn 5788 Well-ordering of a singlet...
elopaelxp 5789 Membership in an ordered-p...
elopaelxpOLD 5790 Obsolete version of ~ elop...
bropaex12 5791 Two classes related by an ...
opabssxp 5792 An abstraction relation is...
brab2a 5793 The law of concretion for ...
optocl 5794 Implicit substitution of c...
2optocl 5795 Implicit substitution of c...
3optocl 5796 Implicit substitution of c...
opbrop 5797 Ordered pair membership in...
0xp 5798 The Cartesian product with...
csbxp 5799 Distribute proper substitu...
releq 5800 Equality theorem for the r...
releqi 5801 Equality inference for the...
releqd 5802 Equality deduction for the...
nfrel 5803 Bound-variable hypothesis ...
sbcrel 5804 Distribute proper substitu...
relss 5805 Subclass theorem for relat...
ssrel 5806 A subclass relationship de...
ssrelOLD 5807 Obsolete version of ~ ssre...
eqrel 5808 Extensionality principle f...
ssrel2 5809 A subclass relationship de...
ssrel3 5810 Subclass relation in anoth...
relssi 5811 Inference from subclass pr...
relssdv 5812 Deduction from subclass pr...
eqrelriv 5813 Inference from extensional...
eqrelriiv 5814 Inference from extensional...
eqbrriv 5815 Inference from extensional...
eqrelrdv 5816 Deduce equality of relatio...
eqbrrdv 5817 Deduction from extensional...
eqbrrdiv 5818 Deduction from extensional...
eqrelrdv2 5819 A version of ~ eqrelrdv . ...
ssrelrel 5820 A subclass relationship de...
eqrelrel 5821 Extensionality principle f...
elrel 5822 A member of a relation is ...
rel0 5823 The empty set is a relatio...
nrelv 5824 The universal class is not...
relsng 5825 A singleton is a relation ...
relsnb 5826 An at-most-singleton is a ...
relsnopg 5827 A singleton of an ordered ...
relsn 5828 A singleton is a relation ...
relsnop 5829 A singleton of an ordered ...
copsex2gb 5830 Implicit substitution infe...
copsex2ga 5831 Implicit substitution infe...
elopaba 5832 Membership in an ordered-p...
xpsspw 5833 A Cartesian product is inc...
unixpss 5834 The double class union of ...
relun 5835 The union of two relations...
relin1 5836 The intersection with a re...
relin2 5837 The intersection with a re...
relinxp 5838 Intersection with a Cartes...
reldif 5839 A difference cutting down ...
reliun 5840 An indexed union is a rela...
reliin 5841 An indexed intersection is...
reluni 5842 The union of a class is a ...
relint 5843 The intersection of a clas...
relopabiv 5844 A class of ordered pairs i...
relopabv 5845 A class of ordered pairs i...
relopabi 5846 A class of ordered pairs i...
relopabiALT 5847 Alternate proof of ~ relop...
relopab 5848 A class of ordered pairs i...
mptrel 5849 The maps-to notation alway...
reli 5850 The identity relation is a...
rele 5851 The membership relation is...
opabid2 5852 A relation expressed as an...
inopab 5853 Intersection of two ordere...
difopab 5854 Difference of two ordered-...
difopabOLD 5855 Obsolete version of ~ difo...
inxp 5856 Intersection of two Cartes...
inxpOLD 5857 Obsolete version of ~ inxp...
xpindi 5858 Distributive law for Carte...
xpindir 5859 Distributive law for Carte...
xpiindi 5860 Distributive law for Carte...
xpriindi 5861 Distributive law for Carte...
eliunxp 5862 Membership in a union of C...
opeliunxp2 5863 Membership in a union of C...
raliunxp 5864 Write a double restricted ...
rexiunxp 5865 Write a double restricted ...
ralxp 5866 Universal quantification r...
rexxp 5867 Existential quantification...
exopxfr 5868 Transfer ordered-pair exis...
exopxfr2 5869 Transfer ordered-pair exis...
djussxp 5870 Disjoint union is a subset...
ralxpf 5871 Version of ~ ralxp with bo...
rexxpf 5872 Version of ~ rexxp with bo...
iunxpf 5873 Indexed union on a Cartesi...
opabbi2dv 5874 Deduce equality of a relat...
relop 5875 A necessary and sufficient...
ideqg 5876 For sets, the identity rel...
ideq 5877 For sets, the identity rel...
ididg 5878 A set is identical to itse...
issetid 5879 Two ways of expressing set...
coss1 5880 Subclass theorem for compo...
coss2 5881 Subclass theorem for compo...
coeq1 5882 Equality theorem for compo...
coeq2 5883 Equality theorem for compo...
coeq1i 5884 Equality inference for com...
coeq2i 5885 Equality inference for com...
coeq1d 5886 Equality deduction for com...
coeq2d 5887 Equality deduction for com...
coeq12i 5888 Equality inference for com...
coeq12d 5889 Equality deduction for com...
nfco 5890 Bound-variable hypothesis ...
brcog 5891 Ordered pair membership in...
opelco2g 5892 Ordered pair membership in...
brcogw 5893 Ordered pair membership in...
eqbrrdva 5894 Deduction from extensional...
brco 5895 Binary relation on a compo...
opelco 5896 Ordered pair membership in...
cnvss 5897 Subset theorem for convers...
cnveq 5898 Equality theorem for conve...
cnveqi 5899 Equality inference for con...
cnveqd 5900 Equality deduction for con...
elcnv 5901 Membership in a converse r...
elcnv2 5902 Membership in a converse r...
nfcnv 5903 Bound-variable hypothesis ...
brcnvg 5904 The converse of a binary r...
opelcnvg 5905 Ordered-pair membership in...
opelcnv 5906 Ordered-pair membership in...
brcnv 5907 The converse of a binary r...
csbcnv 5908 Move class substitution in...
csbcnvgALT 5909 Move class substitution in...
cnvco 5910 Distributive law of conver...
cnvuni 5911 The converse of a class un...
dfdm3 5912 Alternate definition of do...
dfrn2 5913 Alternate definition of ra...
dfrn3 5914 Alternate definition of ra...
elrn2g 5915 Membership in a range. (C...
elrng 5916 Membership in a range. (C...
elrn2 5917 Membership in a range. (C...
elrn 5918 Membership in a range. (C...
ssrelrn 5919 If a relation is a subset ...
dfdm4 5920 Alternate definition of do...
dfdmf 5921 Definition of domain, usin...
csbdm 5922 Distribute proper substitu...
eldmg 5923 Domain membership. Theore...
eldm2g 5924 Domain membership. Theore...
eldm 5925 Membership in a domain. T...
eldm2 5926 Membership in a domain. T...
dmss 5927 Subset theorem for domain....
dmeq 5928 Equality theorem for domai...
dmeqi 5929 Equality inference for dom...
dmeqd 5930 Equality deduction for dom...
opeldmd 5931 Membership of first of an ...
opeldm 5932 Membership of first of an ...
breldm 5933 Membership of first of a b...
breldmg 5934 Membership of first of a b...
dmun 5935 The domain of a union is t...
dmin 5936 The domain of an intersect...
breldmd 5937 Membership of first of a b...
dmiun 5938 The domain of an indexed u...
dmuni 5939 The domain of a union. Pa...
dmopab 5940 The domain of a class of o...
dmopabelb 5941 A set is an element of the...
dmopab2rex 5942 The domain of an ordered p...
dmopabss 5943 Upper bound for the domain...
dmopab3 5944 The domain of a restricted...
dm0 5945 The domain of the empty se...
dmi 5946 The domain of the identity...
dmv 5947 The domain of the universe...
dmep 5948 The domain of the membersh...
dm0rn0 5949 An empty domain is equival...
rn0 5950 The range of the empty set...
rnep 5951 The range of the membershi...
reldm0 5952 A relation is empty iff it...
dmxp 5953 The domain of a Cartesian ...
dmxpOLD 5954 Obsolete version of ~ dmxp...
dmxpid 5955 The domain of a Cartesian ...
dmxpin 5956 The domain of the intersec...
xpid11 5957 The Cartesian square is a ...
dmcnvcnv 5958 The domain of the double c...
rncnvcnv 5959 The range of the double co...
elreldm 5960 The first member of an ord...
rneq 5961 Equality theorem for range...
rneqi 5962 Equality inference for ran...
rneqd 5963 Equality deduction for ran...
rnss 5964 Subset theorem for range. ...
rnssi 5965 Subclass inference for ran...
brelrng 5966 The second argument of a b...
brelrn 5967 The second argument of a b...
opelrn 5968 Membership of second membe...
releldm 5969 The first argument of a bi...
relelrn 5970 The second argument of a b...
releldmb 5971 Membership in a domain. (...
relelrnb 5972 Membership in a range. (C...
releldmi 5973 The first argument of a bi...
relelrni 5974 The second argument of a b...
dfrnf 5975 Definition of range, using...
nfdm 5976 Bound-variable hypothesis ...
nfrn 5977 Bound-variable hypothesis ...
dmiin 5978 Domain of an intersection....
rnopab 5979 The range of a class of or...
rnopabss 5980 Upper bound for the range ...
rnopab3 5981 The range of a restricted ...
rnmpt 5982 The range of a function in...
elrnmpt 5983 The range of a function in...
elrnmpt1s 5984 Elementhood in an image se...
elrnmpt1 5985 Elementhood in an image se...
elrnmptg 5986 Membership in the range of...
elrnmpti 5987 Membership in the range of...
elrnmptd 5988 The range of a function in...
elrnmpt1d 5989 Elementhood in an image se...
elrnmptdv 5990 Elementhood in the range o...
elrnmpt2d 5991 Elementhood in the range o...
dfiun3g 5992 Alternate definition of in...
dfiin3g 5993 Alternate definition of in...
dfiun3 5994 Alternate definition of in...
dfiin3 5995 Alternate definition of in...
riinint 5996 Express a relative indexed...
relrn0 5997 A relation is empty iff it...
dmrnssfld 5998 The domain and range of a ...
dmcoss 5999 Domain of a composition. ...
rncoss 6000 Range of a composition. (...
dmcosseq 6001 Domain of a composition. ...
dmcosseqOLD 6002 Obsolete version of ~ dmco...
dmcoeq 6003 Domain of a composition. ...
rncoeq 6004 Range of a composition. (...
reseq1 6005 Equality theorem for restr...
reseq2 6006 Equality theorem for restr...
reseq1i 6007 Equality inference for res...
reseq2i 6008 Equality inference for res...
reseq12i 6009 Equality inference for res...
reseq1d 6010 Equality deduction for res...
reseq2d 6011 Equality deduction for res...
reseq12d 6012 Equality deduction for res...
nfres 6013 Bound-variable hypothesis ...
csbres 6014 Distribute proper substitu...
res0 6015 A restriction to the empty...
dfres3 6016 Alternate definition of re...
opelres 6017 Ordered pair elementhood i...
brres 6018 Binary relation on a restr...
opelresi 6019 Ordered pair membership in...
brresi 6020 Binary relation on a restr...
opres 6021 Ordered pair membership in...
resieq 6022 A restricted identity rela...
opelidres 6023 ` <. A , A >. ` belongs to...
resres 6024 The restriction of a restr...
resundi 6025 Distributive law for restr...
resundir 6026 Distributive law for restr...
resindi 6027 Class restriction distribu...
resindir 6028 Class restriction distribu...
inres 6029 Move intersection into cla...
resdifcom 6030 Commutative law for restri...
resiun1 6031 Distribution of restrictio...
resiun2 6032 Distribution of restrictio...
resss 6033 A class includes its restr...
rescom 6034 Commutative law for restri...
ssres 6035 Subclass theorem for restr...
ssres2 6036 Subclass theorem for restr...
relres 6037 A restriction is a relatio...
resabs1 6038 Absorption law for restric...
resabs1d 6039 Absorption law for restric...
resabs2 6040 Absorption law for restric...
residm 6041 Idempotent law for restric...
dmresss 6042 The domain of a restrictio...
dmres 6043 The domain of a restrictio...
ssdmres 6044 A domain restricted to a s...
dmresexg 6045 The domain of a restrictio...
resima 6046 A restriction to an image....
resima2 6047 Image under a restricted c...
rnresss 6048 The range of a restriction...
xpssres 6049 Restriction of a constant ...
elinxp 6050 Membership in an intersect...
elres 6051 Membership in a restrictio...
elsnres 6052 Membership in restriction ...
relssres 6053 Simplification law for res...
dmressnsn 6054 The domain of a restrictio...
eldmressnsn 6055 The element of the domain ...
eldmeldmressn 6056 An element of the domain (...
resdm 6057 A relation restricted to i...
resexg 6058 The restriction of a set i...
resexd 6059 The restriction of a set i...
resex 6060 The restriction of a set i...
resindm 6061 When restricting a relatio...
resdmdfsn 6062 Restricting a relation to ...
reldisjun 6063 Split a relation into two ...
relresdm1 6064 Restriction of a disjoint ...
resopab 6065 Restriction of a class abs...
iss 6066 A subclass of the identity...
resopab2 6067 Restriction of a class abs...
resmpt 6068 Restriction of the mapping...
resmpt3 6069 Unconditional restriction ...
resmptf 6070 Restriction of the mapping...
resmptd 6071 Restriction of the mapping...
dfres2 6072 Alternate definition of th...
mptss 6073 Sufficient condition for i...
elimampt 6074 Membership in the image of...
elidinxp 6075 Characterization of the el...
elidinxpid 6076 Characterization of the el...
elrid 6077 Characterization of the el...
idinxpres 6078 The intersection of the id...
idinxpresid 6079 The intersection of the id...
idssxp 6080 A diagonal set as a subset...
opabresid 6081 The restricted identity re...
mptresid 6082 The restricted identity re...
dmresi 6083 The domain of a restricted...
restidsing 6084 Restriction of the identit...
iresn0n0 6085 The identity function rest...
imaeq1 6086 Equality theorem for image...
imaeq2 6087 Equality theorem for image...
imaeq1i 6088 Equality theorem for image...
imaeq2i 6089 Equality theorem for image...
imaeq1d 6090 Equality theorem for image...
imaeq2d 6091 Equality theorem for image...
imaeq12d 6092 Equality theorem for image...
dfima2 6093 Alternate definition of im...
dfima3 6094 Alternate definition of im...
elimag 6095 Membership in an image. T...
elima 6096 Membership in an image. T...
elima2 6097 Membership in an image. T...
elima3 6098 Membership in an image. T...
nfima 6099 Bound-variable hypothesis ...
nfimad 6100 Deduction version of bound...
imadmrn 6101 The image of the domain of...
imassrn 6102 The image of a class is a ...
mptima 6103 Image of a function in map...
mptimass 6104 Image of a function in map...
imai 6105 Image under the identity r...
rnresi 6106 The range of the restricte...
resiima 6107 The image of a restriction...
ima0 6108 Image of the empty set. T...
0ima 6109 Image under the empty rela...
csbima12 6110 Move class substitution in...
imadisj 6111 A class whose image under ...
imadisjlnd 6112 Deduction form of one nega...
cnvimass 6113 A preimage under any class...
cnvimarndm 6114 The preimage of the range ...
imasng 6115 The image of a singleton. ...
relimasn 6116 The image of a singleton. ...
elrelimasn 6117 Elementhood in the image o...
elimasng1 6118 Membership in an image of ...
elimasn1 6119 Membership in an image of ...
elimasng 6120 Membership in an image of ...
elimasn 6121 Membership in an image of ...
elimasngOLD 6122 Obsolete version of ~ elim...
elimasni 6123 Membership in an image of ...
args 6124 Two ways to express the cl...
elinisegg 6125 Membership in the inverse ...
eliniseg 6126 Membership in the inverse ...
epin 6127 Any set is equal to its pr...
epini 6128 Any set is equal to its pr...
iniseg 6129 An idiom that signifies an...
inisegn0 6130 Nonemptiness of an initial...
dffr3 6131 Alternate definition of we...
dfse2 6132 Alternate definition of se...
imass1 6133 Subset theorem for image. ...
imass2 6134 Subset theorem for image. ...
ndmima 6135 The image of a singleton o...
relcnv 6136 A converse is a relation. ...
relbrcnvg 6137 When ` R ` is a relation, ...
eliniseg2 6138 Eliminate the class existe...
relbrcnv 6139 When ` R ` is a relation, ...
relco 6140 A composition is a relatio...
cotrg 6141 Two ways of saying that th...
cotrgOLD 6142 Obsolete version of ~ cotr...
cotrgOLDOLD 6143 Obsolete version of ~ cotr...
cotr 6144 Two ways of saying a relat...
idrefALT 6145 Alternate proof of ~ idref...
cnvsym 6146 Two ways of saying a relat...
cnvsymOLD 6147 Obsolete proof of ~ cnvsym...
cnvsymOLDOLD 6148 Obsolete proof of ~ cnvsym...
intasym 6149 Two ways of saying a relat...
asymref 6150 Two ways of saying a relat...
asymref2 6151 Two ways of saying a relat...
intirr 6152 Two ways of saying a relat...
brcodir 6153 Two ways of saying that tw...
codir 6154 Two ways of saying a relat...
qfto 6155 A quantifier-free way of e...
xpidtr 6156 A Cartesian square is a tr...
trin2 6157 The intersection of two tr...
poirr2 6158 A partial order is irrefle...
trinxp 6159 The relation induced by a ...
soirri 6160 A strict order relation is...
sotri 6161 A strict order relation is...
son2lpi 6162 A strict order relation ha...
sotri2 6163 A transitivity relation. ...
sotri3 6164 A transitivity relation. ...
poleloe 6165 Express "less than or equa...
poltletr 6166 Transitive law for general...
somin1 6167 Property of a minimum in a...
somincom 6168 Commutativity of minimum i...
somin2 6169 Property of a minimum in a...
soltmin 6170 Being less than a minimum,...
cnvopab 6171 The converse of a class ab...
cnvopabOLD 6172 Obsolete version of ~ cnvo...
mptcnv 6173 The converse of a mapping ...
cnv0 6174 The converse of the empty ...
cnvi 6175 The converse of the identi...
cnvun 6176 The converse of a union is...
cnvdif 6177 Distributive law for conve...
cnvin 6178 Distributive law for conve...
rnun 6179 Distributive law for range...
rnin 6180 The range of an intersecti...
rniun 6181 The range of an indexed un...
rnuni 6182 The range of a union. Par...
imaundi 6183 Distributive law for image...
imaundir 6184 The image of a union. (Co...
cnvimassrndm 6185 The preimage of a superset...
dminss 6186 An upper bound for interse...
imainss 6187 An upper bound for interse...
inimass 6188 The image of an intersecti...
inimasn 6189 The intersection of the im...
cnvxp 6190 The converse of a Cartesia...
xp0 6191 The Cartesian product with...
xpnz 6192 The Cartesian product of n...
xpeq0 6193 At least one member of an ...
xpdisj1 6194 Cartesian products with di...
xpdisj2 6195 Cartesian products with di...
xpsndisj 6196 Cartesian products with tw...
difxp 6197 Difference of Cartesian pr...
difxp1 6198 Difference law for Cartesi...
difxp2 6199 Difference law for Cartesi...
djudisj 6200 Disjoint unions with disjo...
xpdifid 6201 The set of distinct couple...
resdisj 6202 A double restriction to di...
rnxp 6203 The range of a Cartesian p...
dmxpss 6204 The domain of a Cartesian ...
rnxpss 6205 The range of a Cartesian p...
rnxpid 6206 The range of a Cartesian s...
ssxpb 6207 A Cartesian product subcla...
xp11 6208 The Cartesian product of n...
xpcan 6209 Cancellation law for Carte...
xpcan2 6210 Cancellation law for Carte...
ssrnres 6211 Two ways to express surjec...
rninxp 6212 Two ways to express surjec...
dminxp 6213 Two ways to express totali...
imainrect 6214 Image by a restricted and ...
xpima 6215 Direct image by a Cartesia...
xpima1 6216 Direct image by a Cartesia...
xpima2 6217 Direct image by a Cartesia...
xpimasn 6218 Direct image of a singleto...
sossfld 6219 The base set of a strict o...
sofld 6220 The base set of a nonempty...
cnvcnv3 6221 The set of all ordered pai...
dfrel2 6222 Alternate definition of re...
dfrel4v 6223 A relation can be expresse...
dfrel4 6224 A relation can be expresse...
cnvcnv 6225 The double converse of a c...
cnvcnv2 6226 The double converse of a c...
cnvcnvss 6227 The double converse of a c...
cnvrescnv 6228 Two ways to express the co...
cnveqb 6229 Equality theorem for conve...
cnveq0 6230 A relation empty iff its c...
dfrel3 6231 Alternate definition of re...
elid 6232 Characterization of the el...
dmresv 6233 The domain of a universal ...
rnresv 6234 The range of a universal r...
dfrn4 6235 Range defined in terms of ...
csbrn 6236 Distribute proper substitu...
rescnvcnv 6237 The restriction of the dou...
cnvcnvres 6238 The double converse of the...
imacnvcnv 6239 The image of the double co...
dmsnn0 6240 The domain of a singleton ...
rnsnn0 6241 The range of a singleton i...
dmsn0 6242 The domain of the singleto...
cnvsn0 6243 The converse of the single...
dmsn0el 6244 The domain of a singleton ...
relsn2 6245 A singleton is a relation ...
dmsnopg 6246 The domain of a singleton ...
dmsnopss 6247 The domain of a singleton ...
dmpropg 6248 The domain of an unordered...
dmsnop 6249 The domain of a singleton ...
dmprop 6250 The domain of an unordered...
dmtpop 6251 The domain of an unordered...
cnvcnvsn 6252 Double converse of a singl...
dmsnsnsn 6253 The domain of the singleto...
rnsnopg 6254 The range of a singleton o...
rnpropg 6255 The range of a pair of ord...
cnvsng 6256 Converse of a singleton of...
rnsnop 6257 The range of a singleton o...
op1sta 6258 Extract the first member o...
cnvsn 6259 Converse of a singleton of...
op2ndb 6260 Extract the second member ...
op2nda 6261 Extract the second member ...
opswap 6262 Swap the members of an ord...
cnvresima 6263 An image under the convers...
resdm2 6264 A class restricted to its ...
resdmres 6265 Restriction to the domain ...
resresdm 6266 A restriction by an arbitr...
imadmres 6267 The image of the domain of...
resdmss 6268 Subset relationship for th...
resdifdi 6269 Distributive law for restr...
resdifdir 6270 Distributive law for restr...
mptpreima 6271 The preimage of a function...
mptiniseg 6272 Converse singleton image o...
dmmpt 6273 The domain of the mapping ...
dmmptss 6274 The domain of a mapping is...
dmmptg 6275 The domain of the mapping ...
rnmpt0f 6276 The range of a function in...
rnmptn0 6277 The range of a function in...
dfco2 6278 Alternate definition of a ...
dfco2a 6279 Generalization of ~ dfco2 ...
coundi 6280 Class composition distribu...
coundir 6281 Class composition distribu...
cores 6282 Restricted first member of...
resco 6283 Associative law for the re...
imaco 6284 Image of the composition o...
rnco 6285 The range of the compositi...
rnco2 6286 The range of the compositi...
dmco 6287 The domain of a compositio...
coeq0 6288 A composition of two relat...
coiun 6289 Composition with an indexe...
cocnvcnv1 6290 A composition is not affec...
cocnvcnv2 6291 A composition is not affec...
cores2 6292 Absorption of a reverse (p...
co02 6293 Composition with the empty...
co01 6294 Composition with the empty...
coi1 6295 Composition with the ident...
coi2 6296 Composition with the ident...
coires1 6297 Composition with a restric...
coass 6298 Associative law for class ...
relcnvtrg 6299 General form of ~ relcnvtr...
relcnvtr 6300 A relation is transitive i...
relssdmrn 6301 A relation is included in ...
relssdmrnOLD 6302 Obsolete version of ~ rels...
resssxp 6303 If the ` R ` -image of a c...
cnvssrndm 6304 The converse is a subset o...
cossxp 6305 Composition as a subset of...
relrelss 6306 Two ways to describe the s...
unielrel 6307 The membership relation fo...
relfld 6308 The double union of a rela...
relresfld 6309 Restriction of a relation ...
relcoi2 6310 Composition with the ident...
relcoi1 6311 Composition with the ident...
unidmrn 6312 The double union of the co...
relcnvfld 6313 if ` R ` is a relation, it...
dfdm2 6314 Alternate definition of do...
unixp 6315 The double class union of ...
unixp0 6316 A Cartesian product is emp...
unixpid 6317 Field of a Cartesian squar...
ressn 6318 Restriction of a class to ...
cnviin 6319 The converse of an interse...
cnvpo 6320 The converse of a partial ...
cnvso 6321 The converse of a strict o...
xpco 6322 Composition of two Cartesi...
xpcoid 6323 Composition of two Cartesi...
elsnxp 6324 Membership in a Cartesian ...
reu3op 6325 There is a unique ordered ...
reuop 6326 There is a unique ordered ...
opreu2reurex 6327 There is a unique ordered ...
opreu2reu 6328 If there is a unique order...
dfpo2 6329 Quantifier-free definition...
csbcog 6330 Distribute proper substitu...
snres0 6331 Condition for restriction ...
imaindm 6332 The image is unaffected by...
predeq123 6335 Equality theorem for the p...
predeq1 6336 Equality theorem for the p...
predeq2 6337 Equality theorem for the p...
predeq3 6338 Equality theorem for the p...
nfpred 6339 Bound-variable hypothesis ...
csbpredg 6340 Move class substitution in...
predpredss 6341 If ` A ` is a subset of ` ...
predss 6342 The predecessor class of `...
sspred 6343 Another subset/predecessor...
dfpred2 6344 An alternate definition of...
dfpred3 6345 An alternate definition of...
dfpred3g 6346 An alternate definition of...
elpredgg 6347 Membership in a predecesso...
elpredg 6348 Membership in a predecesso...
elpredimg 6349 Membership in a predecesso...
elpredim 6350 Membership in a predecesso...
elpred 6351 Membership in a predecesso...
predexg 6352 The predecessor class exis...
predasetexOLD 6353 Obsolete form of ~ predexg...
dffr4 6354 Alternate definition of we...
predel 6355 Membership in the predeces...
predtrss 6356 If ` R ` is transitive ove...
predpo 6357 Property of the predecesso...
predso 6358 Property of the predecesso...
setlikespec 6359 If ` R ` is set-like in ` ...
predidm 6360 Idempotent law for the pre...
predin 6361 Intersection law for prede...
predun 6362 Union law for predecessor ...
preddif 6363 Difference law for predece...
predep 6364 The predecessor under the ...
trpred 6365 The class of predecessors ...
preddowncl 6366 A property of classes that...
predpoirr 6367 Given a partial ordering, ...
predfrirr 6368 Given a well-founded relat...
pred0 6369 The predecessor class over...
dfse3 6370 Alternate definition of se...
predrelss 6371 Subset carries from relati...
predprc 6372 The predecessor of a prope...
predres 6373 Predecessor class is unaff...
frpomin 6374 Every nonempty (possibly p...
frpomin2 6375 Every nonempty (possibly p...
frpoind 6376 The principle of well-foun...
frpoinsg 6377 Well-Founded Induction Sch...
frpoins2fg 6378 Well-Founded Induction sch...
frpoins2g 6379 Well-Founded Induction sch...
frpoins3g 6380 Well-Founded Induction sch...
tz6.26 6381 All nonempty subclasses of...
tz6.26OLD 6382 Obsolete proof of ~ tz6.26...
tz6.26i 6383 All nonempty subclasses of...
wfi 6384 The Principle of Well-Orde...
wfiOLD 6385 Obsolete proof of ~ wfi as...
wfii 6386 The Principle of Well-Orde...
wfisg 6387 Well-Ordered Induction Sch...
wfisgOLD 6388 Obsolete version of ~ wfis...
wfis 6389 Well-Ordered Induction Sch...
wfis2fg 6390 Well-Ordered Induction Sch...
wfis2fgOLD 6391 Obsolete version of ~ wfis...
wfis2f 6392 Well-Ordered Induction sch...
wfis2g 6393 Well-Ordered Induction Sch...
wfis2 6394 Well-Ordered Induction sch...
wfis3 6395 Well-Ordered Induction sch...
ordeq 6404 Equality theorem for the o...
elong 6405 An ordinal number is an or...
elon 6406 An ordinal number is an or...
eloni 6407 An ordinal number has the ...
elon2 6408 An ordinal number is an or...
limeq 6409 Equality theorem for the l...
ordwe 6410 Membership well-orders eve...
ordtr 6411 An ordinal class is transi...
ordfr 6412 Membership is well-founded...
ordelss 6413 An element of an ordinal c...
trssord 6414 A transitive subclass of a...
ordirr 6415 No ordinal class is a memb...
nordeq 6416 A member of an ordinal cla...
ordn2lp 6417 An ordinal class cannot be...
tz7.5 6418 A nonempty subclass of an ...
ordelord 6419 An element of an ordinal c...
tron 6420 The class of all ordinal n...
ordelon 6421 An element of an ordinal c...
onelon 6422 An element of an ordinal n...
tz7.7 6423 A transitive class belongs...
ordelssne 6424 For ordinal classes, membe...
ordelpss 6425 For ordinal classes, membe...
ordsseleq 6426 For ordinal classes, inclu...
ordin 6427 The intersection of two or...
onin 6428 The intersection of two or...
ordtri3or 6429 A trichotomy law for ordin...
ordtri1 6430 A trichotomy law for ordin...
ontri1 6431 A trichotomy law for ordin...
ordtri2 6432 A trichotomy law for ordin...
ordtri3 6433 A trichotomy law for ordin...
ordtri4 6434 A trichotomy law for ordin...
orddisj 6435 An ordinal class and its s...
onfr 6436 The ordinal class is well-...
onelpss 6437 Relationship between membe...
onsseleq 6438 Relationship between subse...
onelss 6439 An element of an ordinal n...
ordtr1 6440 Transitive law for ordinal...
ordtr2 6441 Transitive law for ordinal...
ordtr3 6442 Transitive law for ordinal...
ontr1 6443 Transitive law for ordinal...
ontr2 6444 Transitive law for ordinal...
onelssex 6445 Ordinal less than is equiv...
ordunidif 6446 The union of an ordinal st...
ordintdif 6447 If ` B ` is smaller than `...
onintss 6448 If a property is true for ...
oneqmini 6449 A way to show that an ordi...
ord0 6450 The empty set is an ordina...
0elon 6451 The empty set is an ordina...
ord0eln0 6452 A nonempty ordinal contain...
on0eln0 6453 An ordinal number contains...
dflim2 6454 An alternate definition of...
inton 6455 The intersection of the cl...
nlim0 6456 The empty set is not a lim...
limord 6457 A limit ordinal is ordinal...
limuni 6458 A limit ordinal is its own...
limuni2 6459 The union of a limit ordin...
0ellim 6460 A limit ordinal contains t...
limelon 6461 A limit ordinal class that...
onn0 6462 The class of all ordinal n...
suceq 6463 Equality of successors. (...
elsuci 6464 Membership in a successor....
elsucg 6465 Membership in a successor....
elsuc2g 6466 Variant of membership in a...
elsuc 6467 Membership in a successor....
elsuc2 6468 Membership in a successor....
nfsuc 6469 Bound-variable hypothesis ...
elelsuc 6470 Membership in a successor....
sucel 6471 Membership of a successor ...
suc0 6472 The successor of the empty...
sucprc 6473 A proper class is its own ...
unisucs 6474 The union of the successor...
unisucg 6475 A transitive class is equa...
unisuc 6476 A transitive class is equa...
sssucid 6477 A class is included in its...
sucidg 6478 Part of Proposition 7.23 o...
sucid 6479 A set belongs to its succe...
nsuceq0 6480 No successor is empty. (C...
eqelsuc 6481 A set belongs to the succe...
iunsuc 6482 Inductive definition for t...
suctr 6483 The successor of a transit...
trsuc 6484 A set whose successor belo...
trsucss 6485 A member of the successor ...
ordsssuc 6486 An ordinal is a subset of ...
onsssuc 6487 A subset of an ordinal num...
ordsssuc2 6488 An ordinal subset of an or...
onmindif 6489 When its successor is subt...
ordnbtwn 6490 There is no set between an...
onnbtwn 6491 There is no set between an...
sucssel 6492 A set whose successor is a...
orddif 6493 Ordinal derived from its s...
orduniss 6494 An ordinal class includes ...
ordtri2or 6495 A trichotomy law for ordin...
ordtri2or2 6496 A trichotomy law for ordin...
ordtri2or3 6497 A consequence of total ord...
ordelinel 6498 The intersection of two or...
ordssun 6499 Property of a subclass of ...
ordequn 6500 The maximum (i.e. union) o...
ordun 6501 The maximum (i.e., union) ...
onunel 6502 The union of two ordinals ...
ordunisssuc 6503 A subclass relationship fo...
suc11 6504 The successor operation be...
onun2 6505 The union of two ordinals ...
ontr 6506 An ordinal number is a tra...
onunisuc 6507 An ordinal number is equal...
onordi 6508 An ordinal number is an or...
ontrciOLD 6509 Obsolete version of ~ ontr...
onirri 6510 An ordinal number is not a...
oneli 6511 A member of an ordinal num...
onelssi 6512 A member of an ordinal num...
onssneli 6513 An ordering law for ordina...
onssnel2i 6514 An ordering law for ordina...
onelini 6515 An element of an ordinal n...
oneluni 6516 An ordinal number equals i...
onunisuci 6517 An ordinal number is equal...
onsseli 6518 Subset is equivalent to me...
onun2i 6519 The union of two ordinal n...
unizlim 6520 An ordinal equal to its ow...
on0eqel 6521 An ordinal number either e...
snsn0non 6522 The singleton of the singl...
onxpdisj 6523 Ordinal numbers and ordere...
onnev 6524 The class of ordinal numbe...
iotajust 6526 Soundness justification th...
dfiota2 6528 Alternate definition for d...
nfiota1 6529 Bound-variable hypothesis ...
nfiotadw 6530 Deduction version of ~ nfi...
nfiotaw 6531 Bound-variable hypothesis ...
nfiotad 6532 Deduction version of ~ nfi...
nfiota 6533 Bound-variable hypothesis ...
cbviotaw 6534 Change bound variables in ...
cbviotavw 6535 Change bound variables in ...
cbviotavwOLD 6536 Obsolete version of ~ cbvi...
cbviota 6537 Change bound variables in ...
cbviotav 6538 Change bound variables in ...
sb8iota 6539 Variable substitution in d...
iotaeq 6540 Equality theorem for descr...
iotabi 6541 Equivalence theorem for de...
uniabio 6542 Part of Theorem 8.17 in [Q...
iotaval2 6543 Version of ~ iotaval using...
iotauni2 6544 Version of ~ iotauni using...
iotanul2 6545 Version of ~ iotanul using...
iotaval 6546 Theorem 8.19 in [Quine] p....
iotassuni 6547 The ` iota ` class is a su...
iotaex 6548 Theorem 8.23 in [Quine] p....
iotavalOLD 6549 Obsolete version of ~ iota...
iotauni 6550 Equivalence between two di...
iotaint 6551 Equivalence between two di...
iota1 6552 Property of iota. (Contri...
iotanul 6553 Theorem 8.22 in [Quine] p....
iotassuniOLD 6554 Obsolete version of ~ iota...
iotaexOLD 6555 Obsolete version of ~ iota...
iota4 6556 Theorem *14.22 in [Whitehe...
iota4an 6557 Theorem *14.23 in [Whitehe...
iota5 6558 A method for computing iot...
iotabidv 6559 Formula-building deduction...
iotabii 6560 Formula-building deduction...
iotacl 6561 Membership law for descrip...
iota2df 6562 A condition that allows to...
iota2d 6563 A condition that allows to...
iota2 6564 The unique element such th...
iotan0 6565 Representation of "the uni...
sniota 6566 A class abstraction with a...
dfiota4 6567 The ` iota ` operation usi...
csbiota 6568 Class substitution within ...
dffun2 6585 Alternate definition of a ...
dffun2OLD 6586 Obsolete version of ~ dffu...
dffun2OLDOLD 6587 Obsolete version of ~ dffu...
dffun6 6588 Alternate definition of a ...
dffun3 6589 Alternate definition of fu...
dffun3OLD 6590 Obsolete version of ~ dffu...
dffun4 6591 Alternate definition of a ...
dffun5 6592 Alternate definition of fu...
dffun6f 6593 Definition of function, us...
dffun6OLD 6594 Obsolete version of ~ dffu...
funmo 6595 A function has at most one...
funmoOLD 6596 Obsolete version of ~ funm...
funrel 6597 A function is a relation. ...
0nelfun 6598 A function does not contai...
funss 6599 Subclass theorem for funct...
funeq 6600 Equality theorem for funct...
funeqi 6601 Equality inference for the...
funeqd 6602 Equality deduction for the...
nffun 6603 Bound-variable hypothesis ...
sbcfung 6604 Distribute proper substitu...
funeu 6605 There is exactly one value...
funeu2 6606 There is exactly one value...
dffun7 6607 Alternate definition of a ...
dffun8 6608 Alternate definition of a ...
dffun9 6609 Alternate definition of a ...
funfn 6610 A class is a function if a...
funfnd 6611 A function is a function o...
funi 6612 The identity relation is a...
nfunv 6613 The universal class is not...
funopg 6614 A Kuratowski ordered pair ...
funopab 6615 A class of ordered pairs i...
funopabeq 6616 A class of ordered pairs o...
funopab4 6617 A class of ordered pairs o...
funmpt 6618 A function in maps-to nota...
funmpt2 6619 Functionality of a class g...
funco 6620 The composition of two fun...
funresfunco 6621 Composition of two functio...
funres 6622 A restriction of a functio...
funresd 6623 A restriction of a functio...
funssres 6624 The restriction of a funct...
fun2ssres 6625 Equality of restrictions o...
funun 6626 The union of functions wit...
fununmo 6627 If the union of classes is...
fununfun 6628 If the union of classes is...
fundif 6629 A function with removed el...
funcnvsn 6630 The converse singleton of ...
funsng 6631 A singleton of an ordered ...
fnsng 6632 Functionality and domain o...
funsn 6633 A singleton of an ordered ...
funprg 6634 A set of two pairs is a fu...
funtpg 6635 A set of three pairs is a ...
funpr 6636 A function with a domain o...
funtp 6637 A function with a domain o...
fnsn 6638 Functionality and domain o...
fnprg 6639 Function with a domain of ...
fntpg 6640 Function with a domain of ...
fntp 6641 A function with a domain o...
funcnvpr 6642 The converse pair of order...
funcnvtp 6643 The converse triple of ord...
funcnvqp 6644 The converse quadruple of ...
fun0 6645 The empty set is a functio...
funcnv0 6646 The converse of the empty ...
funcnvcnv 6647 The double converse of a f...
funcnv2 6648 A simpler equivalence for ...
funcnv 6649 The converse of a class is...
funcnv3 6650 A condition showing a clas...
fun2cnv 6651 The double converse of a c...
svrelfun 6652 A single-valued relation i...
fncnv 6653 Single-rootedness (see ~ f...
fun11 6654 Two ways of stating that `...
fununi 6655 The union of a chain (with...
funin 6656 The intersection with a fu...
funres11 6657 The restriction of a one-t...
funcnvres 6658 The converse of a restrict...
cnvresid 6659 Converse of a restricted i...
funcnvres2 6660 The converse of a restrict...
funimacnv 6661 The image of the preimage ...
funimass1 6662 A kind of contraposition l...
funimass2 6663 A kind of contraposition l...
imadif 6664 The image of a difference ...
imain 6665 The image of an intersecti...
funimaexg 6666 Axiom of Replacement using...
funimaexgOLD 6667 Obsolete version of ~ funi...
funimaex 6668 The image of a set under a...
isarep1 6669 Part of a study of the Axi...
isarep1OLD 6670 Obsolete version of ~ isar...
isarep2 6671 Part of a study of the Axi...
fneq1 6672 Equality theorem for funct...
fneq2 6673 Equality theorem for funct...
fneq1d 6674 Equality deduction for fun...
fneq2d 6675 Equality deduction for fun...
fneq12d 6676 Equality deduction for fun...
fneq12 6677 Equality theorem for funct...
fneq1i 6678 Equality inference for fun...
fneq2i 6679 Equality inference for fun...
nffn 6680 Bound-variable hypothesis ...
fnfun 6681 A function with domain is ...
fnfund 6682 A function with domain is ...
fnrel 6683 A function with domain is ...
fndm 6684 The domain of a function. ...
fndmi 6685 The domain of a function. ...
fndmd 6686 The domain of a function. ...
funfni 6687 Inference to convert a fun...
fndmu 6688 A function has a unique do...
fnbr 6689 The first argument of bina...
fnop 6690 The first argument of an o...
fneu 6691 There is exactly one value...
fneu2 6692 There is exactly one value...
fnunres1 6693 Restriction of a disjoint ...
fnunres2 6694 Restriction of a disjoint ...
fnun 6695 The union of two functions...
fnund 6696 The union of two functions...
fnunop 6697 Extension of a function wi...
fncofn 6698 Composition of a function ...
fnco 6699 Composition of two functio...
fncoOLD 6700 Obsolete version of ~ fnco...
fnresdm 6701 A function does not change...
fnresdisj 6702 A function restricted to a...
2elresin 6703 Membership in two function...
fnssresb 6704 Restriction of a function ...
fnssres 6705 Restriction of a function ...
fnssresd 6706 Restriction of a function ...
fnresin1 6707 Restriction of a function'...
fnresin2 6708 Restriction of a function'...
fnres 6709 An equivalence for functio...
idfn 6710 The identity relation is a...
fnresi 6711 The restricted identity re...
fnima 6712 The image of a function's ...
fn0 6713 A function with empty doma...
fnimadisj 6714 A class that is disjoint w...
fnimaeq0 6715 Images under a function ne...
dfmpt3 6716 Alternate definition for t...
mptfnf 6717 The maps-to notation defin...
fnmptf 6718 The maps-to notation defin...
fnopabg 6719 Functionality and domain o...
fnopab 6720 Functionality and domain o...
mptfng 6721 The maps-to notation defin...
fnmpt 6722 The maps-to notation defin...
fnmptd 6723 The maps-to notation defin...
mpt0 6724 A mapping operation with e...
fnmpti 6725 Functionality and domain o...
dmmpti 6726 Domain of the mapping oper...
dmmptd 6727 The domain of the mapping ...
mptun 6728 Union of mappings which ar...
partfun 6729 Rewrite a function defined...
feq1 6730 Equality theorem for funct...
feq2 6731 Equality theorem for funct...
feq3 6732 Equality theorem for funct...
feq23 6733 Equality theorem for funct...
feq1d 6734 Equality deduction for fun...
feq2d 6735 Equality deduction for fun...
feq3d 6736 Equality deduction for fun...
feq12d 6737 Equality deduction for fun...
feq123d 6738 Equality deduction for fun...
feq123 6739 Equality theorem for funct...
feq1i 6740 Equality inference for fun...
feq2i 6741 Equality inference for fun...
feq12i 6742 Equality inference for fun...
feq23i 6743 Equality inference for fun...
feq23d 6744 Equality deduction for fun...
nff 6745 Bound-variable hypothesis ...
sbcfng 6746 Distribute proper substitu...
sbcfg 6747 Distribute proper substitu...
elimf 6748 Eliminate a mapping hypoth...
ffn 6749 A mapping is a function wi...
ffnd 6750 A mapping is a function wi...
dffn2 6751 Any function is a mapping ...
ffun 6752 A mapping is a function. ...
ffund 6753 A mapping is a function, d...
frel 6754 A mapping is a relation. ...
freld 6755 A mapping is a relation. ...
frn 6756 The range of a mapping. (...
frnd 6757 Deduction form of ~ frn . ...
fdm 6758 The domain of a mapping. ...
fdmd 6759 Deduction form of ~ fdm . ...
fdmi 6760 Inference associated with ...
dffn3 6761 A function maps to its ran...
ffrn 6762 A function maps to its ran...
ffrnb 6763 Characterization of a func...
ffrnbd 6764 A function maps to its ran...
fss 6765 Expanding the codomain of ...
fssd 6766 Expanding the codomain of ...
fssdmd 6767 Expressing that a class is...
fssdm 6768 Expressing that a class is...
fimass 6769 The image of a class under...
fimassd 6770 The image of a class is a ...
fimacnv 6771 The preimage of the codoma...
fcof 6772 Composition of a function ...
fco 6773 Composition of two functio...
fcoOLD 6774 Obsolete version of ~ fco ...
fcod 6775 Composition of two mapping...
fco2 6776 Functionality of a composi...
fssxp 6777 A mapping is a class of or...
funssxp 6778 Two ways of specifying a p...
ffdm 6779 A mapping is a partial fun...
ffdmd 6780 The domain of a function. ...
fdmrn 6781 A different way to write `...
funcofd 6782 Composition of two functio...
fco3OLD 6783 Obsolete version of ~ func...
opelf 6784 The members of an ordered ...
fun 6785 The union of two functions...
fun2 6786 The union of two functions...
fun2d 6787 The union of functions wit...
fnfco 6788 Composition of two functio...
fssres 6789 Restriction of a function ...
fssresd 6790 Restriction of a function ...
fssres2 6791 Restriction of a restricte...
fresin 6792 An identity for the mappin...
resasplit 6793 If two functions agree on ...
fresaun 6794 The union of two functions...
fresaunres2 6795 From the union of two func...
fresaunres1 6796 From the union of two func...
fcoi1 6797 Composition of a mapping a...
fcoi2 6798 Composition of restricted ...
feu 6799 There is exactly one value...
fcnvres 6800 The converse of a restrict...
fimacnvdisj 6801 The preimage of a class di...
fint 6802 Function into an intersect...
fin 6803 Mapping into an intersecti...
f0 6804 The empty function. (Cont...
f00 6805 A class is a function with...
f0bi 6806 A function with empty doma...
f0dom0 6807 A function is empty iff it...
f0rn0 6808 If there is no element in ...
fconst 6809 A Cartesian product with a...
fconstg 6810 A Cartesian product with a...
fnconstg 6811 A Cartesian product with a...
fconst6g 6812 Constant function with loo...
fconst6 6813 A constant function as a m...
f1eq1 6814 Equality theorem for one-t...
f1eq2 6815 Equality theorem for one-t...
f1eq3 6816 Equality theorem for one-t...
nff1 6817 Bound-variable hypothesis ...
dff12 6818 Alternate definition of a ...
f1f 6819 A one-to-one mapping is a ...
f1fn 6820 A one-to-one mapping is a ...
f1fun 6821 A one-to-one mapping is a ...
f1rel 6822 A one-to-one onto mapping ...
f1dm 6823 The domain of a one-to-one...
f1ss 6824 A function that is one-to-...
f1ssr 6825 A function that is one-to-...
f1ssres 6826 A function that is one-to-...
f1resf1 6827 The restriction of an inje...
f1cnvcnv 6828 Two ways to express that a...
f1cof1 6829 Composition of two one-to-...
f1co 6830 Composition of one-to-one ...
f1coOLD 6831 Obsolete version of ~ f1co...
foeq1 6832 Equality theorem for onto ...
foeq2 6833 Equality theorem for onto ...
foeq3 6834 Equality theorem for onto ...
nffo 6835 Bound-variable hypothesis ...
fof 6836 An onto mapping is a mappi...
fofun 6837 An onto mapping is a funct...
fofn 6838 An onto mapping is a funct...
forn 6839 The codomain of an onto fu...
dffo2 6840 Alternate definition of an...
foima 6841 The image of the domain of...
dffn4 6842 A function maps onto its r...
funforn 6843 A function maps its domain...
fodmrnu 6844 An onto function has uniqu...
fimadmfo 6845 A function is a function o...
fores 6846 Restriction of an onto fun...
fimadmfoALT 6847 Alternate proof of ~ fimad...
focnvimacdmdm 6848 The preimage of the codoma...
focofo 6849 Composition of onto functi...
foco 6850 Composition of onto functi...
foconst 6851 A nonzero constant functio...
f1oeq1 6852 Equality theorem for one-t...
f1oeq2 6853 Equality theorem for one-t...
f1oeq3 6854 Equality theorem for one-t...
f1oeq23 6855 Equality theorem for one-t...
f1eq123d 6856 Equality deduction for one...
foeq123d 6857 Equality deduction for ont...
f1oeq123d 6858 Equality deduction for one...
f1oeq1d 6859 Equality deduction for one...
f1oeq2d 6860 Equality deduction for one...
f1oeq3d 6861 Equality deduction for one...
nff1o 6862 Bound-variable hypothesis ...
f1of1 6863 A one-to-one onto mapping ...
f1of 6864 A one-to-one onto mapping ...
f1ofn 6865 A one-to-one onto mapping ...
f1ofun 6866 A one-to-one onto mapping ...
f1orel 6867 A one-to-one onto mapping ...
f1odm 6868 The domain of a one-to-one...
dff1o2 6869 Alternate definition of on...
dff1o3 6870 Alternate definition of on...
f1ofo 6871 A one-to-one onto function...
dff1o4 6872 Alternate definition of on...
dff1o5 6873 Alternate definition of on...
f1orn 6874 A one-to-one function maps...
f1f1orn 6875 A one-to-one function maps...
f1ocnv 6876 The converse of a one-to-o...
f1ocnvb 6877 A relation is a one-to-one...
f1ores 6878 The restriction of a one-t...
f1orescnv 6879 The converse of a one-to-o...
f1imacnv 6880 Preimage of an image. (Co...
foimacnv 6881 A reverse version of ~ f1i...
foun 6882 The union of two onto func...
f1oun 6883 The union of two one-to-on...
f1un 6884 The union of two one-to-on...
resdif 6885 The restriction of a one-t...
resin 6886 The restriction of a one-t...
f1oco 6887 Composition of one-to-one ...
f1cnv 6888 The converse of an injecti...
funcocnv2 6889 Composition with the conve...
fococnv2 6890 The composition of an onto...
f1ococnv2 6891 The composition of a one-t...
f1cocnv2 6892 Composition of an injectiv...
f1ococnv1 6893 The composition of a one-t...
f1cocnv1 6894 Composition of an injectiv...
funcoeqres 6895 Express a constraint on a ...
f1ssf1 6896 A subset of an injective f...
f10 6897 The empty set maps one-to-...
f10d 6898 The empty set maps one-to-...
f1o00 6899 One-to-one onto mapping of...
fo00 6900 Onto mapping of the empty ...
f1o0 6901 One-to-one onto mapping of...
f1oi 6902 A restriction of the ident...
f1ovi 6903 The identity relation is a...
f1osn 6904 A singleton of an ordered ...
f1osng 6905 A singleton of an ordered ...
f1sng 6906 A singleton of an ordered ...
fsnd 6907 A singleton of an ordered ...
f1oprswap 6908 A two-element swap is a bi...
f1oprg 6909 An unordered pair of order...
tz6.12-2 6910 Function value when ` F ` ...
fveu 6911 The value of a function at...
brprcneu 6912 If ` A ` is a proper class...
brprcneuALT 6913 Alternate proof of ~ brprc...
fvprc 6914 A function's value at a pr...
fvprcALT 6915 Alternate proof of ~ fvprc...
rnfvprc 6916 The range of a function va...
fv2 6917 Alternate definition of fu...
dffv3 6918 A definition of function v...
dffv4 6919 The previous definition of...
elfv 6920 Membership in a function v...
fveq1 6921 Equality theorem for funct...
fveq2 6922 Equality theorem for funct...
fveq1i 6923 Equality inference for fun...
fveq1d 6924 Equality deduction for fun...
fveq2i 6925 Equality inference for fun...
fveq2d 6926 Equality deduction for fun...
2fveq3 6927 Equality theorem for neste...
fveq12i 6928 Equality deduction for fun...
fveq12d 6929 Equality deduction for fun...
fveqeq2d 6930 Equality deduction for fun...
fveqeq2 6931 Equality deduction for fun...
nffv 6932 Bound-variable hypothesis ...
nffvmpt1 6933 Bound-variable hypothesis ...
nffvd 6934 Deduction version of bound...
fvex 6935 The value of a class exist...
fvexi 6936 The value of a class exist...
fvexd 6937 The value of a class exist...
fvif 6938 Move a conditional outside...
iffv 6939 Move a conditional outside...
fv3 6940 Alternate definition of th...
fvres 6941 The value of a restricted ...
fvresd 6942 The value of a restricted ...
funssfv 6943 The value of a member of t...
tz6.12c 6944 Corollary of Theorem 6.12(...
tz6.12-1 6945 Function value. Theorem 6...
tz6.12-1OLD 6946 Obsolete version of ~ tz6....
tz6.12 6947 Function value. Theorem 6...
tz6.12f 6948 Function value, using boun...
tz6.12cOLD 6949 Obsolete version of ~ tz6....
tz6.12i 6950 Corollary of Theorem 6.12(...
fvbr0 6951 Two possibilities for the ...
fvrn0 6952 A function value is a memb...
fvn0fvelrn 6953 If the value of a function...
elfvunirn 6954 A function value is a subs...
fvssunirn 6955 The result of a function v...
fvssunirnOLD 6956 Obsolete version of ~ fvss...
ndmfv 6957 The value of a class outsi...
ndmfvrcl 6958 Reverse closure law for fu...
elfvdm 6959 If a function value has a ...
elfvex 6960 If a function value has a ...
elfvexd 6961 If a function value has a ...
eliman0 6962 A nonempty function value ...
nfvres 6963 The value of a non-member ...
nfunsn 6964 If the restriction of a cl...
fvfundmfvn0 6965 If the "value of a class" ...
0fv 6966 Function value of the empt...
fv2prc 6967 A function value of a func...
elfv2ex 6968 If a function value of a f...
fveqres 6969 Equal values imply equal v...
csbfv12 6970 Move class substitution in...
csbfv2g 6971 Move class substitution in...
csbfv 6972 Substitution for a functio...
funbrfv 6973 The second argument of a b...
funopfv 6974 The second element in an o...
fnbrfvb 6975 Equivalence of function va...
fnopfvb 6976 Equivalence of function va...
funbrfvb 6977 Equivalence of function va...
funopfvb 6978 Equivalence of function va...
fnbrfvb2 6979 Version of ~ fnbrfvb for f...
fdmeu 6980 There is exactly one codom...
funbrfv2b 6981 Function value in terms of...
dffn5 6982 Representation of a functi...
fnrnfv 6983 The range of a function ex...
fvelrnb 6984 A member of a function's r...
foelcdmi 6985 A member of a surjective f...
dfimafn 6986 Alternate definition of th...
dfimafn2 6987 Alternate definition of th...
funimass4 6988 Membership relation for th...
fvelima 6989 Function value in an image...
funimassd 6990 Sufficient condition for t...
fvelimad 6991 Function value in an image...
feqmptd 6992 Deduction form of ~ dffn5 ...
feqresmpt 6993 Express a restricted funct...
feqmptdf 6994 Deduction form of ~ dffn5f...
dffn5f 6995 Representation of a functi...
fvelimab 6996 Function value in an image...
fvelimabd 6997 Deduction form of ~ fvelim...
fimarab 6998 Expressing the image of a ...
unima 6999 Image of a union. (Contri...
fvi 7000 The value of the identity ...
fviss 7001 The value of the identity ...
fniinfv 7002 The indexed intersection o...
fnsnfv 7003 Singleton of function valu...
opabiotafun 7004 Define a function whose va...
opabiotadm 7005 Define a function whose va...
opabiota 7006 Define a function whose va...
fnimapr 7007 The image of a pair under ...
fnimatpd 7008 The image of an unordered ...
ssimaex 7009 The existence of a subimag...
ssimaexg 7010 The existence of a subimag...
funfv 7011 A simplified expression fo...
funfv2 7012 The value of a function. ...
funfv2f 7013 The value of a function. ...
fvun 7014 Value of the union of two ...
fvun1 7015 The value of a union when ...
fvun2 7016 The value of a union when ...
fvun1d 7017 The value of a union when ...
fvun2d 7018 The value of a union when ...
dffv2 7019 Alternate definition of fu...
dmfco 7020 Domains of a function comp...
fvco2 7021 Value of a function compos...
fvco 7022 Value of a function compos...
fvco3 7023 Value of a function compos...
fvco3d 7024 Value of a function compos...
fvco4i 7025 Conditions for a compositi...
fvopab3g 7026 Value of a function given ...
fvopab3ig 7027 Value of a function given ...
brfvopabrbr 7028 The binary relation of a f...
fvmptg 7029 Value of a function given ...
fvmpti 7030 Value of a function given ...
fvmpt 7031 Value of a function given ...
fvmpt2f 7032 Value of a function given ...
fvtresfn 7033 Functionality of a tuple-r...
fvmpts 7034 Value of a function given ...
fvmpt3 7035 Value of a function given ...
fvmpt3i 7036 Value of a function given ...
fvmptdf 7037 Deduction version of ~ fvm...
fvmptd 7038 Deduction version of ~ fvm...
fvmptd2 7039 Deduction version of ~ fvm...
mptrcl 7040 Reverse closure for a mapp...
fvmpt2i 7041 Value of a function given ...
fvmpt2 7042 Value of a function given ...
fvmptss 7043 If all the values of the m...
fvmpt2d 7044 Deduction version of ~ fvm...
fvmptex 7045 Express a function ` F ` w...
fvmptd3f 7046 Alternate deduction versio...
fvmptd2f 7047 Alternate deduction versio...
fvmptdv 7048 Alternate deduction versio...
fvmptdv2 7049 Alternate deduction versio...
mpteqb 7050 Bidirectional equality the...
fvmptt 7051 Closed theorem form of ~ f...
fvmptf 7052 Value of a function given ...
fvmptnf 7053 The value of a function gi...
fvmptd3 7054 Deduction version of ~ fvm...
fvmptd4 7055 Deduction version of ~ fvm...
fvmptn 7056 This somewhat non-intuitiv...
fvmptss2 7057 A mapping always evaluates...
elfvmptrab1w 7058 Implications for the value...
elfvmptrab1 7059 Implications for the value...
elfvmptrab 7060 Implications for the value...
fvopab4ndm 7061 Value of a function given ...
fvmptndm 7062 Value of a function given ...
fvmptrabfv 7063 Value of a function mappin...
fvopab5 7064 The value of a function th...
fvopab6 7065 Value of a function given ...
eqfnfv 7066 Equality of functions is d...
eqfnfv2 7067 Equality of functions is d...
eqfnfv3 7068 Derive equality of functio...
eqfnfvd 7069 Deduction for equality of ...
eqfnfv2f 7070 Equality of functions is d...
eqfunfv 7071 Equality of functions is d...
eqfnun 7072 Two functions on ` A u. B ...
fvreseq0 7073 Equality of restricted fun...
fvreseq1 7074 Equality of a function res...
fvreseq 7075 Equality of restricted fun...
fnmptfvd 7076 A function with a given do...
fndmdif 7077 Two ways to express the lo...
fndmdifcom 7078 The difference set between...
fndmdifeq0 7079 The difference set of two ...
fndmin 7080 Two ways to express the lo...
fneqeql 7081 Two functions are equal if...
fneqeql2 7082 Two functions are equal if...
fnreseql 7083 Two functions are equal on...
chfnrn 7084 The range of a choice func...
funfvop 7085 Ordered pair with function...
funfvbrb 7086 Two ways to say that ` A `...
fvimacnvi 7087 A member of a preimage is ...
fvimacnv 7088 The argument of a function...
funimass3 7089 A kind of contraposition l...
funimass5 7090 A subclass of a preimage i...
funconstss 7091 Two ways of specifying tha...
fvimacnvALT 7092 Alternate proof of ~ fvima...
elpreima 7093 Membership in the preimage...
elpreimad 7094 Membership in the preimage...
fniniseg 7095 Membership in the preimage...
fncnvima2 7096 Inverse images under funct...
fniniseg2 7097 Inverse point images under...
unpreima 7098 Preimage of a union. (Con...
inpreima 7099 Preimage of an intersectio...
difpreima 7100 Preimage of a difference. ...
respreima 7101 The preimage of a restrict...
cnvimainrn 7102 The preimage of the inters...
sspreima 7103 The preimage of a subset i...
iinpreima 7104 Preimage of an intersectio...
intpreima 7105 Preimage of an intersectio...
fimacnvOLD 7106 Obsolete version of ~ fima...
fimacnvinrn 7107 Taking the converse image ...
fimacnvinrn2 7108 Taking the converse image ...
rescnvimafod 7109 The restriction of a funct...
fvn0ssdmfun 7110 If a class' function value...
fnopfv 7111 Ordered pair with function...
fvelrn 7112 A function's value belongs...
nelrnfvne 7113 A function value cannot be...
fveqdmss 7114 If the empty set is not co...
fveqressseq 7115 If the empty set is not co...
fnfvelrn 7116 A function's value belongs...
ffvelcdm 7117 A function's value belongs...
fnfvelrnd 7118 A function's value belongs...
ffvelcdmi 7119 A function's value belongs...
ffvelcdmda 7120 A function's value belongs...
ffvelcdmd 7121 A function's value belongs...
feldmfvelcdm 7122 A class is an element of t...
rexrn 7123 Restricted existential qua...
ralrn 7124 Restricted universal quant...
elrnrexdm 7125 For any element in the ran...
elrnrexdmb 7126 For any element in the ran...
eldmrexrn 7127 For any element in the dom...
eldmrexrnb 7128 For any element in the dom...
fvcofneq 7129 The values of two function...
ralrnmptw 7130 A restricted quantifier ov...
rexrnmptw 7131 A restricted quantifier ov...
ralrnmpt 7132 A restricted quantifier ov...
rexrnmpt 7133 A restricted quantifier ov...
f0cli 7134 Unconditional closure of a...
dff2 7135 Alternate definition of a ...
dff3 7136 Alternate definition of a ...
dff4 7137 Alternate definition of a ...
dffo3 7138 An onto mapping expressed ...
dffo4 7139 Alternate definition of an...
dffo5 7140 Alternate definition of an...
exfo 7141 A relation equivalent to t...
dffo3f 7142 An onto mapping expressed ...
foelrn 7143 Property of a surjective f...
foelrnf 7144 Property of a surjective f...
foco2 7145 If a composition of two fu...
fmpt 7146 Functionality of the mappi...
f1ompt 7147 Express bijection for a ma...
fmpti 7148 Functionality of the mappi...
fvmptelcdm 7149 The value of a function at...
fmptd 7150 Domain and codomain of the...
fmpttd 7151 Version of ~ fmptd with in...
fmpt3d 7152 Domain and codomain of the...
fmptdf 7153 A version of ~ fmptd using...
fompt 7154 Express being onto for a m...
ffnfv 7155 A function maps to a class...
ffnfvf 7156 A function maps to a class...
fnfvrnss 7157 An upper bound for range d...
fcdmssb 7158 A function is a function i...
rnmptss 7159 The range of an operation ...
fmpt2d 7160 Domain and codomain of the...
ffvresb 7161 A necessary and sufficient...
fssrescdmd 7162 Restriction of a function ...
f1oresrab 7163 Build a bijection between ...
f1ossf1o 7164 Restricting a bijection, w...
fmptco 7165 Composition of two functio...
fmptcof 7166 Version of ~ fmptco where ...
fmptcos 7167 Composition of two functio...
cofmpt 7168 Express composition of a m...
fcompt 7169 Express composition of two...
fcoconst 7170 Composition with a constan...
fsn 7171 A function maps a singleto...
fsn2 7172 A function that maps a sin...
fsng 7173 A function maps a singleto...
fsn2g 7174 A function that maps a sin...
xpsng 7175 The Cartesian product of t...
xpprsng 7176 The Cartesian product of a...
xpsn 7177 The Cartesian product of t...
f1o2sn 7178 A singleton consisting in ...
residpr 7179 Restriction of the identit...
dfmpt 7180 Alternate definition for t...
fnasrn 7181 A function expressed as th...
idref 7182 Two ways to state that a r...
funiun 7183 A function is a union of s...
funopsn 7184 If a function is an ordere...
funop 7185 An ordered pair is a funct...
funopdmsn 7186 The domain of a function w...
funsndifnop 7187 A singleton of an ordered ...
funsneqopb 7188 A singleton of an ordered ...
ressnop0 7189 If ` A ` is not in ` C ` ,...
fpr 7190 A function with a domain o...
fprg 7191 A function with a domain o...
ftpg 7192 A function with a domain o...
ftp 7193 A function with a domain o...
fnressn 7194 A function restricted to a...
funressn 7195 A function restricted to a...
fressnfv 7196 The value of a function re...
fvrnressn 7197 If the value of a function...
fvressn 7198 The value of a function re...
fvn0fvelrnOLD 7199 Obsolete version of ~ fvn0...
fvconst 7200 The value of a constant fu...
fnsnr 7201 If a class belongs to a fu...
fnsnb 7202 A function whose domain is...
fmptsn 7203 Express a singleton functi...
fmptsng 7204 Express a singleton functi...
fmptsnd 7205 Express a singleton functi...
fmptap 7206 Append an additional value...
fmptapd 7207 Append an additional value...
fmptpr 7208 Express a pair function in...
fvresi 7209 The value of a restricted ...
fninfp 7210 Express the class of fixed...
fnelfp 7211 Property of a fixed point ...
fndifnfp 7212 Express the class of non-f...
fnelnfp 7213 Property of a non-fixed po...
fnnfpeq0 7214 A function is the identity...
fvunsn 7215 Remove an ordered pair not...
fvsng 7216 The value of a singleton o...
fvsn 7217 The value of a singleton o...
fvsnun1 7218 The value of a function wi...
fvsnun2 7219 The value of a function wi...
fnsnsplit 7220 Split a function into a si...
fsnunf 7221 Adjoining a point to a fun...
fsnunf2 7222 Adjoining a point to a pun...
fsnunfv 7223 Recover the added point fr...
fsnunres 7224 Recover the original funct...
funresdfunsn 7225 Restricting a function to ...
fvpr1g 7226 The value of a function wi...
fvpr2g 7227 The value of a function wi...
fvpr2gOLD 7228 Obsolete version of ~ fvpr...
fvpr1 7229 The value of a function wi...
fvpr1OLD 7230 Obsolete version of ~ fvpr...
fvpr2 7231 The value of a function wi...
fvpr2OLD 7232 Obsolete version of ~ fvpr...
fprb 7233 A condition for functionho...
fvtp1 7234 The first value of a funct...
fvtp2 7235 The second value of a func...
fvtp3 7236 The third value of a funct...
fvtp1g 7237 The value of a function wi...
fvtp2g 7238 The value of a function wi...
fvtp3g 7239 The value of a function wi...
tpres 7240 An unordered triple of ord...
fvconst2g 7241 The value of a constant fu...
fconst2g 7242 A constant function expres...
fvconst2 7243 The value of a constant fu...
fconst2 7244 A constant function expres...
fconst5 7245 Two ways to express that a...
rnmptc 7246 Range of a constant functi...
fnprb 7247 A function whose domain ha...
fntpb 7248 A function whose domain ha...
fnpr2g 7249 A function whose domain ha...
fpr2g 7250 A function that maps a pai...
fconstfv 7251 A constant function expres...
fconst3 7252 Two ways to express a cons...
fconst4 7253 Two ways to express a cons...
resfunexg 7254 The restriction of a funct...
resiexd 7255 The restriction of the ide...
fnex 7256 If the domain of a functio...
fnexd 7257 If the domain of a functio...
funex 7258 If the domain of a functio...
opabex 7259 Existence of a function ex...
mptexg 7260 If the domain of a functio...
mptexgf 7261 If the domain of a functio...
mptex 7262 If the domain of a functio...
mptexd 7263 If the domain of a functio...
mptrabex 7264 If the domain of a functio...
fex 7265 If the domain of a mapping...
fexd 7266 If the domain of a mapping...
mptfvmpt 7267 A function in maps-to nota...
eufnfv 7268 A function is uniquely det...
funfvima 7269 A function's value in a pr...
funfvima2 7270 A function's value in an i...
funfvima2d 7271 A function's value in a pr...
fnfvima 7272 The function value of an o...
fnfvimad 7273 A function's value belongs...
resfvresima 7274 The value of the function ...
funfvima3 7275 A class including a functi...
ralima 7276 Universal quantification u...
rexima 7277 Existential quantification...
reximaOLD 7278 Obsolete version of ~ rexi...
ralimaOLD 7279 Obsolete version of ~ rali...
fvclss 7280 Upper bound for the class ...
elabrex 7281 Elementhood in an image se...
elabrexg 7282 Elementhood in an image se...
abrexco 7283 Composition of two image m...
imaiun 7284 The image of an indexed un...
imauni 7285 The image of a union is th...
fniunfv 7286 The indexed union of a fun...
funiunfv 7287 The indexed union of a fun...
funiunfvf 7288 The indexed union of a fun...
eluniima 7289 Membership in the union of...
elunirn 7290 Membership in the union of...
elunirnALT 7291 Alternate proof of ~ eluni...
elunirn2OLD 7292 Obsolete version of ~ elfv...
fnunirn 7293 Membership in a union of s...
dff13 7294 A one-to-one function in t...
dff13f 7295 A one-to-one function in t...
f1veqaeq 7296 If the values of a one-to-...
f1cofveqaeq 7297 If the values of a composi...
f1cofveqaeqALT 7298 Alternate proof of ~ f1cof...
2f1fvneq 7299 If two one-to-one function...
f1mpt 7300 Express injection for a ma...
f1fveq 7301 Equality of function value...
f1elima 7302 Membership in the image of...
f1imass 7303 Taking images under a one-...
f1imaeq 7304 Taking images under a one-...
f1imapss 7305 Taking images under a one-...
fpropnf1 7306 A function, given by an un...
f1dom3fv3dif 7307 The function values for a ...
f1dom3el3dif 7308 The codomain of a 1-1 func...
dff14a 7309 A one-to-one function in t...
dff14b 7310 A one-to-one function in t...
f12dfv 7311 A one-to-one function with...
f13dfv 7312 A one-to-one function with...
dff1o6 7313 A one-to-one onto function...
f1ocnvfv1 7314 The converse value of the ...
f1ocnvfv2 7315 The value of the converse ...
f1ocnvfv 7316 Relationship between the v...
f1ocnvfvb 7317 Relationship between the v...
nvof1o 7318 An involution is a bijecti...
nvocnv 7319 The converse of an involut...
f1cdmsn 7320 If a one-to-one function w...
fsnex 7321 Relate a function with a s...
f1prex 7322 Relate a one-to-one functi...
f1ocnvdm 7323 The value of the converse ...
f1ocnvfvrneq 7324 If the values of a one-to-...
fcof1 7325 An application is injectiv...
fcofo 7326 An application is surjecti...
cbvfo 7327 Change bound variable betw...
cbvexfo 7328 Change bound variable betw...
cocan1 7329 An injection is left-cance...
cocan2 7330 A surjection is right-canc...
fcof1oinvd 7331 Show that a function is th...
fcof1od 7332 A function is bijective if...
2fcoidinvd 7333 Show that a function is th...
fcof1o 7334 Show that two functions ar...
2fvcoidd 7335 Show that the composition ...
2fvidf1od 7336 A function is bijective if...
2fvidinvd 7337 Show that two functions ar...
foeqcnvco 7338 Condition for function equ...
f1eqcocnv 7339 Condition for function equ...
fveqf1o 7340 Given a bijection ` F ` , ...
f1ocoima 7341 The composition of two bij...
nf1const 7342 A constant function from a...
nf1oconst 7343 A constant function from a...
f1ofvswap 7344 Swapping two values in a b...
fvf1pr 7345 Values of a one-to-one fun...
fliftrel 7346 ` F ` , a function lift, i...
fliftel 7347 Elementhood in the relatio...
fliftel1 7348 Elementhood in the relatio...
fliftcnv 7349 Converse of the relation `...
fliftfun 7350 The function ` F ` is the ...
fliftfund 7351 The function ` F ` is the ...
fliftfuns 7352 The function ` F ` is the ...
fliftf 7353 The domain and range of th...
fliftval 7354 The value of the function ...
isoeq1 7355 Equality theorem for isomo...
isoeq2 7356 Equality theorem for isomo...
isoeq3 7357 Equality theorem for isomo...
isoeq4 7358 Equality theorem for isomo...
isoeq5 7359 Equality theorem for isomo...
nfiso 7360 Bound-variable hypothesis ...
isof1o 7361 An isomorphism is a one-to...
isof1oidb 7362 A function is a bijection ...
isof1oopb 7363 A function is a bijection ...
isorel 7364 An isomorphism connects bi...
soisores 7365 Express the condition of i...
soisoi 7366 Infer isomorphism from one...
isoid 7367 Identity law for isomorphi...
isocnv 7368 Converse law for isomorphi...
isocnv2 7369 Converse law for isomorphi...
isocnv3 7370 Complementation law for is...
isores2 7371 An isomorphism from one we...
isores1 7372 An isomorphism from one we...
isores3 7373 Induced isomorphism on a s...
isotr 7374 Composition (transitive) l...
isomin 7375 Isomorphisms preserve mini...
isoini 7376 Isomorphisms preserve init...
isoini2 7377 Isomorphisms are isomorphi...
isofrlem 7378 Lemma for ~ isofr . (Cont...
isoselem 7379 Lemma for ~ isose . (Cont...
isofr 7380 An isomorphism preserves w...
isose 7381 An isomorphism preserves s...
isofr2 7382 A weak form of ~ isofr tha...
isopolem 7383 Lemma for ~ isopo . (Cont...
isopo 7384 An isomorphism preserves t...
isosolem 7385 Lemma for ~ isoso . (Cont...
isoso 7386 An isomorphism preserves t...
isowe 7387 An isomorphism preserves t...
isowe2 7388 A weak form of ~ isowe tha...
f1oiso 7389 Any one-to-one onto functi...
f1oiso2 7390 Any one-to-one onto functi...
f1owe 7391 Well-ordering of isomorphi...
weniso 7392 A set-like well-ordering h...
weisoeq 7393 Thus, there is at most one...
weisoeq2 7394 Thus, there is at most one...
knatar 7395 The Knaster-Tarski theorem...
fvresval 7396 The value of a restricted ...
funeldmb 7397 If ` (/) ` is not part of ...
eqfunresadj 7398 Law for adjoining an eleme...
eqfunressuc 7399 Law for equality of restri...
fnssintima 7400 Condition for subset of an...
imaeqsexvOLD 7401 Duplicate version of ~ ral...
imaeqsalvOLD 7402 Duplicate version of ~ ral...
canth 7403 No set ` A ` is equinumero...
ncanth 7404 Cantor's theorem fails for...
riotaeqdv 7407 Formula-building deduction...
riotabidv 7408 Formula-building deduction...
riotaeqbidv 7409 Equality deduction for res...
riotaex 7410 Restricted iota is a set. ...
riotav 7411 An iota restricted to the ...
riotauni 7412 Restricted iota in terms o...
nfriota1 7413 The abstraction variable i...
nfriotadw 7414 Deduction version of ~ nfr...
cbvriotaw 7415 Change bound variable in a...
cbvriotavw 7416 Change bound variable in a...
cbvriotavwOLD 7417 Obsolete version of ~ cbvr...
nfriotad 7418 Deduction version of ~ nfr...
nfriota 7419 A variable not free in a w...
cbvriota 7420 Change bound variable in a...
cbvriotav 7421 Change bound variable in a...
csbriota 7422 Interchange class substitu...
riotacl2 7423 Membership law for "the un...
riotacl 7424 Closure of restricted iota...
riotasbc 7425 Substitution law for descr...
riotabidva 7426 Equivalent wff's yield equ...
riotabiia 7427 Equivalent wff's yield equ...
riota1 7428 Property of restricted iot...
riota1a 7429 Property of iota. (Contri...
riota2df 7430 A deduction version of ~ r...
riota2f 7431 This theorem shows a condi...
riota2 7432 This theorem shows a condi...
riotaeqimp 7433 If two restricted iota des...
riotaprop 7434 Properties of a restricted...
riota5f 7435 A method for computing res...
riota5 7436 A method for computing res...
riotass2 7437 Restriction of a unique el...
riotass 7438 Restriction of a unique el...
moriotass 7439 Restriction of a unique el...
snriota 7440 A restricted class abstrac...
riotaxfrd 7441 Change the variable ` x ` ...
eusvobj2 7442 Specify the same property ...
eusvobj1 7443 Specify the same object in...
f1ofveu 7444 There is one domain elemen...
f1ocnvfv3 7445 Value of the converse of a...
riotaund 7446 Restricted iota equals the...
riotassuni 7447 The restricted iota class ...
riotaclb 7448 Bidirectional closure of r...
riotarab 7449 Restricted iota of a restr...
oveq 7456 Equality theorem for opera...
oveq1 7457 Equality theorem for opera...
oveq2 7458 Equality theorem for opera...
oveq12 7459 Equality theorem for opera...
oveq1i 7460 Equality inference for ope...
oveq2i 7461 Equality inference for ope...
oveq12i 7462 Equality inference for ope...
oveqi 7463 Equality inference for ope...
oveq123i 7464 Equality inference for ope...
oveq1d 7465 Equality deduction for ope...
oveq2d 7466 Equality deduction for ope...
oveqd 7467 Equality deduction for ope...
oveq12d 7468 Equality deduction for ope...
oveqan12d 7469 Equality deduction for ope...
oveqan12rd 7470 Equality deduction for ope...
oveq123d 7471 Equality deduction for ope...
fvoveq1d 7472 Equality deduction for nes...
fvoveq1 7473 Equality theorem for neste...
ovanraleqv 7474 Equality theorem for a con...
imbrov2fvoveq 7475 Equality theorem for neste...
ovrspc2v 7476 If an operation value is e...
oveqrspc2v 7477 Restricted specialization ...
oveqdr 7478 Equality of two operations...
nfovd 7479 Deduction version of bound...
nfov 7480 Bound-variable hypothesis ...
oprabidw 7481 The law of concretion. Sp...
oprabid 7482 The law of concretion. Sp...
ovex 7483 The result of an operation...
ovexi 7484 The result of an operation...
ovexd 7485 The result of an operation...
ovssunirn 7486 The result of an operation...
0ov 7487 Operation value of the emp...
ovprc 7488 The value of an operation ...
ovprc1 7489 The value of an operation ...
ovprc2 7490 The value of an operation ...
ovrcl 7491 Reverse closure for an ope...
elfvov1 7492 Utility theorem: reverse c...
elfvov2 7493 Utility theorem: reverse c...
csbov123 7494 Move class substitution in...
csbov 7495 Move class substitution in...
csbov12g 7496 Move class substitution in...
csbov1g 7497 Move class substitution in...
csbov2g 7498 Move class substitution in...
rspceov 7499 A frequently used special ...
elovimad 7500 Elementhood of the image s...
fnbrovb 7501 Value of a binary operatio...
fnotovb 7502 Equivalence of operation v...
opabbrex 7503 A collection of ordered pa...
opabresex2 7504 Restrictions of a collecti...
opabresex2d 7505 Obsolete version of ~ opab...
fvmptopab 7506 The function value of a ma...
fvmptopabOLD 7507 Obsolete version of ~ fvmp...
f1opr 7508 Condition for an operation...
brfvopab 7509 The classes involved in a ...
dfoprab2 7510 Class abstraction for oper...
reloprab 7511 An operation class abstrac...
oprabv 7512 If a pair and a class are ...
nfoprab1 7513 The abstraction variables ...
nfoprab2 7514 The abstraction variables ...
nfoprab3 7515 The abstraction variables ...
nfoprab 7516 Bound-variable hypothesis ...
oprabbid 7517 Equivalent wff's yield equ...
oprabbidv 7518 Equivalent wff's yield equ...
oprabbii 7519 Equivalent wff's yield equ...
ssoprab2 7520 Equivalence of ordered pai...
ssoprab2b 7521 Equivalence of ordered pai...
eqoprab2bw 7522 Equivalence of ordered pai...
eqoprab2b 7523 Equivalence of ordered pai...
mpoeq123 7524 An equality theorem for th...
mpoeq12 7525 An equality theorem for th...
mpoeq123dva 7526 An equality deduction for ...
mpoeq123dv 7527 An equality deduction for ...
mpoeq123i 7528 An equality inference for ...
mpoeq3dva 7529 Slightly more general equa...
mpoeq3ia 7530 An equality inference for ...
mpoeq3dv 7531 An equality deduction for ...
nfmpo1 7532 Bound-variable hypothesis ...
nfmpo2 7533 Bound-variable hypothesis ...
nfmpo 7534 Bound-variable hypothesis ...
0mpo0 7535 A mapping operation with e...
mpo0v 7536 A mapping operation with e...
mpo0 7537 A mapping operation with e...
oprab4 7538 Two ways to state the doma...
cbvoprab1 7539 Rule used to change first ...
cbvoprab2 7540 Change the second bound va...
cbvoprab12 7541 Rule used to change first ...
cbvoprab12v 7542 Rule used to change first ...
cbvoprab3 7543 Rule used to change the th...
cbvoprab3v 7544 Rule used to change the th...
cbvmpox 7545 Rule to change the bound v...
cbvmpo 7546 Rule to change the bound v...
cbvmpov 7547 Rule to change the bound v...
elimdelov 7548 Eliminate a hypothesis whi...
brif1 7549 Move a relation inside and...
ovif 7550 Move a conditional outside...
ovif2 7551 Move a conditional outside...
ovif12 7552 Move a conditional outside...
ifov 7553 Move a conditional outside...
dmoprab 7554 The domain of an operation...
dmoprabss 7555 The domain of an operation...
rnoprab 7556 The range of an operation ...
rnoprab2 7557 The range of a restricted ...
reldmoprab 7558 The domain of an operation...
oprabss 7559 Structure of an operation ...
eloprabga 7560 The law of concretion for ...
eloprabgaOLD 7561 Obsolete version of ~ elop...
eloprabg 7562 The law of concretion for ...
ssoprab2i 7563 Inference of operation cla...
mpov 7564 Operation with universal d...
mpomptx 7565 Express a two-argument fun...
mpompt 7566 Express a two-argument fun...
mpodifsnif 7567 A mapping with two argumen...
mposnif 7568 A mapping with two argumen...
fconstmpo 7569 Representation of a consta...
resoprab 7570 Restriction of an operatio...
resoprab2 7571 Restriction of an operator...
resmpo 7572 Restriction of the mapping...
funoprabg 7573 "At most one" is a suffici...
funoprab 7574 "At most one" is a suffici...
fnoprabg 7575 Functionality and domain o...
mpofun 7576 The maps-to notation for a...
fnoprab 7577 Functionality and domain o...
ffnov 7578 An operation maps to a cla...
fovcld 7579 Closure law for an operati...
fovcl 7580 Closure law for an operati...
eqfnov 7581 Equality of two operations...
eqfnov2 7582 Two operators with the sam...
fnov 7583 Representation of a functi...
mpo2eqb 7584 Bidirectional equality the...
rnmpo 7585 The range of an operation ...
reldmmpo 7586 The domain of an operation...
elrnmpog 7587 Membership in the range of...
elrnmpo 7588 Membership in the range of...
elimampo 7589 Membership in the image of...
elrnmpores 7590 Membership in the range of...
ralrnmpo 7591 A restricted quantifier ov...
rexrnmpo 7592 A restricted quantifier ov...
ovid 7593 The value of an operation ...
ovidig 7594 The value of an operation ...
ovidi 7595 The value of an operation ...
ov 7596 The value of an operation ...
ovigg 7597 The value of an operation ...
ovig 7598 The value of an operation ...
ovmpt4g 7599 Value of a function given ...
ovmpos 7600 Value of a function given ...
ov2gf 7601 The value of an operation ...
ovmpodxf 7602 Value of an operation give...
ovmpodx 7603 Value of an operation give...
ovmpod 7604 Value of an operation give...
ovmpox 7605 The value of an operation ...
ovmpoga 7606 Value of an operation give...
ovmpoa 7607 Value of an operation give...
ovmpodf 7608 Alternate deduction versio...
ovmpodv 7609 Alternate deduction versio...
ovmpodv2 7610 Alternate deduction versio...
ovmpog 7611 Value of an operation give...
ovmpo 7612 Value of an operation give...
ovmpot 7613 The value of an operation ...
fvmpopr2d 7614 Value of an operation give...
ov3 7615 The value of an operation ...
ov6g 7616 The value of an operation ...
ovg 7617 The value of an operation ...
ovres 7618 The value of a restricted ...
ovresd 7619 Lemma for converting metri...
oprres 7620 The restriction of an oper...
oprssov 7621 The value of a member of t...
fovcdm 7622 An operation's value belon...
fovcdmda 7623 An operation's value belon...
fovcdmd 7624 An operation's value belon...
fnrnov 7625 The range of an operation ...
foov 7626 An onto mapping of an oper...
fnovrn 7627 An operation's value belon...
ovelrn 7628 A member of an operation's...
funimassov 7629 Membership relation for th...
ovelimab 7630 Operation value in an imag...
ovima0 7631 An operation value is a me...
ovconst2 7632 The value of a constant op...
oprssdm 7633 Domain of closure of an op...
nssdmovg 7634 The value of an operation ...
ndmovg 7635 The value of an operation ...
ndmov 7636 The value of an operation ...
ndmovcl 7637 The closure of an operatio...
ndmovrcl 7638 Reverse closure law, when ...
ndmovcom 7639 Any operation is commutati...
ndmovass 7640 Any operation is associati...
ndmovdistr 7641 Any operation is distribut...
ndmovord 7642 Elimination of redundant a...
ndmovordi 7643 Elimination of redundant a...
caovclg 7644 Convert an operation closu...
caovcld 7645 Convert an operation closu...
caovcl 7646 Convert an operation closu...
caovcomg 7647 Convert an operation commu...
caovcomd 7648 Convert an operation commu...
caovcom 7649 Convert an operation commu...
caovassg 7650 Convert an operation assoc...
caovassd 7651 Convert an operation assoc...
caovass 7652 Convert an operation assoc...
caovcang 7653 Convert an operation cance...
caovcand 7654 Convert an operation cance...
caovcanrd 7655 Commute the arguments of a...
caovcan 7656 Convert an operation cance...
caovordig 7657 Convert an operation order...
caovordid 7658 Convert an operation order...
caovordg 7659 Convert an operation order...
caovordd 7660 Convert an operation order...
caovord2d 7661 Operation ordering law wit...
caovord3d 7662 Ordering law. (Contribute...
caovord 7663 Convert an operation order...
caovord2 7664 Operation ordering law wit...
caovord3 7665 Ordering law. (Contribute...
caovdig 7666 Convert an operation distr...
caovdid 7667 Convert an operation distr...
caovdir2d 7668 Convert an operation distr...
caovdirg 7669 Convert an operation rever...
caovdird 7670 Convert an operation distr...
caovdi 7671 Convert an operation distr...
caov32d 7672 Rearrange arguments in a c...
caov12d 7673 Rearrange arguments in a c...
caov31d 7674 Rearrange arguments in a c...
caov13d 7675 Rearrange arguments in a c...
caov4d 7676 Rearrange arguments in a c...
caov411d 7677 Rearrange arguments in a c...
caov42d 7678 Rearrange arguments in a c...
caov32 7679 Rearrange arguments in a c...
caov12 7680 Rearrange arguments in a c...
caov31 7681 Rearrange arguments in a c...
caov13 7682 Rearrange arguments in a c...
caov4 7683 Rearrange arguments in a c...
caov411 7684 Rearrange arguments in a c...
caov42 7685 Rearrange arguments in a c...
caovdir 7686 Reverse distributive law. ...
caovdilem 7687 Lemma used by real number ...
caovlem2 7688 Lemma used in real number ...
caovmo 7689 Uniqueness of inverse elem...
imaeqexov 7690 Substitute an operation va...
imaeqalov 7691 Substitute an operation va...
mpondm0 7692 The value of an operation ...
elmpocl 7693 If a two-parameter class i...
elmpocl1 7694 If a two-parameter class i...
elmpocl2 7695 If a two-parameter class i...
elovmpod 7696 Utility lemma for two-para...
elovmpo 7697 Utility lemma for two-para...
elovmporab 7698 Implications for the value...
elovmporab1w 7699 Implications for the value...
elovmporab1 7700 Implications for the value...
2mpo0 7701 If the operation value of ...
relmptopab 7702 Any function to sets of or...
f1ocnvd 7703 Describe an implicit one-t...
f1od 7704 Describe an implicit one-t...
f1ocnv2d 7705 Describe an implicit one-t...
f1o2d 7706 Describe an implicit one-t...
f1opw2 7707 A one-to-one mapping induc...
f1opw 7708 A one-to-one mapping induc...
elovmpt3imp 7709 If the value of a function...
ovmpt3rab1 7710 The value of an operation ...
ovmpt3rabdm 7711 If the value of a function...
elovmpt3rab1 7712 Implications for the value...
elovmpt3rab 7713 Implications for the value...
ofeqd 7718 Equality theorem for funct...
ofeq 7719 Equality theorem for funct...
ofreq 7720 Equality theorem for funct...
ofexg 7721 A function operation restr...
nfof 7722 Hypothesis builder for fun...
nfofr 7723 Hypothesis builder for fun...
ofrfvalg 7724 Value of a relation applie...
offval 7725 Value of an operation appl...
ofrfval 7726 Value of a relation applie...
ofval 7727 Evaluate a function operat...
ofrval 7728 Exhibit a function relatio...
offn 7729 The function operation pro...
offun 7730 The function operation pro...
offval2f 7731 The function operation exp...
ofmresval 7732 Value of a restriction of ...
fnfvof 7733 Function value of a pointw...
off 7734 The function operation pro...
ofres 7735 Restrict the operands of a...
offval2 7736 The function operation exp...
ofrfval2 7737 The function relation acti...
ofmpteq 7738 Value of a pointwise opera...
coof 7739 The composition of a _homo...
ofco 7740 The composition of a funct...
offveq 7741 Convert an identity of the...
offveqb 7742 Equivalent expressions for...
ofc1 7743 Left operation by a consta...
ofc2 7744 Right operation by a const...
ofc12 7745 Function operation on two ...
caofref 7746 Transfer a reflexive law t...
caofinvl 7747 Transfer a left inverse la...
caofid0l 7748 Transfer a left identity l...
caofid0r 7749 Transfer a right identity ...
caofid1 7750 Transfer a right absorptio...
caofid2 7751 Transfer a right absorptio...
caofcom 7752 Transfer a commutative law...
caofrss 7753 Transfer a relation subset...
caofass 7754 Transfer an associative la...
caoftrn 7755 Transfer a transitivity la...
caofdi 7756 Transfer a distributive la...
caofdir 7757 Transfer a reverse distrib...
caonncan 7758 Transfer ~ nncan -shaped l...
relrpss 7761 The proper subset relation...
brrpssg 7762 The proper subset relation...
brrpss 7763 The proper subset relation...
porpss 7764 Every class is partially o...
sorpss 7765 Express strict ordering un...
sorpssi 7766 Property of a chain of set...
sorpssun 7767 A chain of sets is closed ...
sorpssin 7768 A chain of sets is closed ...
sorpssuni 7769 In a chain of sets, a maxi...
sorpssint 7770 In a chain of sets, a mini...
sorpsscmpl 7771 The componentwise compleme...
zfun 7773 Axiom of Union expressed w...
axun2 7774 A variant of the Axiom of ...
uniex2 7775 The Axiom of Union using t...
vuniex 7776 The union of a setvar is a...
uniexg 7777 The ZF Axiom of Union in c...
uniex 7778 The Axiom of Union in clas...
uniexd 7779 Deduction version of the Z...
unexg 7780 The union of two sets is a...
unex 7781 The union of two sets is a...
unexOLD 7782 Obsolete proof of ~ unex a...
tpex 7783 An unordered triple of cla...
unexb 7784 Existence of union is equi...
unexbOLD 7785 Obsolete proof of ~ unexb ...
unexgOLD 7786 Obsolete proof of ~ unexg ...
xpexg 7787 The Cartesian product of t...
xpexd 7788 The Cartesian product of t...
3xpexg 7789 The Cartesian product of t...
xpex 7790 The Cartesian product of t...
unexd 7791 The union of two sets is a...
sqxpexg 7792 The Cartesian square of a ...
abnexg 7793 Sufficient condition for a...
abnex 7794 Sufficient condition for a...
snnex 7795 The class of all singleton...
pwnex 7796 The class of all power set...
difex2 7797 If the subtrahend of a cla...
difsnexi 7798 If the difference of a cla...
uniuni 7799 Expression for double unio...
uniexr 7800 Converse of the Axiom of U...
uniexb 7801 The Axiom of Union and its...
pwexr 7802 Converse of the Axiom of P...
pwexb 7803 The Axiom of Power Sets an...
elpwpwel 7804 A class belongs to a doubl...
eldifpw 7805 Membership in a power clas...
elpwun 7806 Membership in the power cl...
pwuncl 7807 Power classes are closed u...
iunpw 7808 An indexed union of a powe...
fr3nr 7809 A well-founded relation ha...
epne3 7810 A well-founded class conta...
dfwe2 7811 Alternate definition of we...
epweon 7812 The membership relation we...
epweonALT 7813 Alternate proof of ~ epweo...
ordon 7814 The class of all ordinal n...
onprc 7815 No set contains all ordina...
ssorduni 7816 The union of a class of or...
ssonuni 7817 The union of a set of ordi...
ssonunii 7818 The union of a set of ordi...
ordeleqon 7819 A way to express the ordin...
ordsson 7820 Any ordinal class is a sub...
dford5 7821 A class is ordinal iff it ...
onss 7822 An ordinal number is a sub...
predon 7823 The predecessor of an ordi...
predonOLD 7824 Obsolete version of ~ pred...
ssonprc 7825 Two ways of saying a class...
onuni 7826 The union of an ordinal nu...
orduni 7827 The union of an ordinal cl...
onint 7828 The intersection (infimum)...
onint0 7829 The intersection of a clas...
onssmin 7830 A nonempty class of ordina...
onminesb 7831 If a property is true for ...
onminsb 7832 If a property is true for ...
oninton 7833 The intersection of a none...
onintrab 7834 The intersection of a clas...
onintrab2 7835 An existence condition equ...
onnmin 7836 No member of a set of ordi...
onnminsb 7837 An ordinal number smaller ...
oneqmin 7838 A way to show that an ordi...
uniordint 7839 The union of a set of ordi...
onminex 7840 If a wff is true for an or...
sucon 7841 The class of all ordinal n...
sucexb 7842 A successor exists iff its...
sucexg 7843 The successor of a set is ...
sucex 7844 The successor of a set is ...
onmindif2 7845 The minimum of a class of ...
ordsuci 7846 The successor of an ordina...
sucexeloni 7847 If the successor of an ord...
sucexeloniOLD 7848 Obsolete version of ~ suce...
onsuc 7849 The successor of an ordina...
suceloniOLD 7850 Obsolete version of ~ onsu...
ordsuc 7851 A class is ordinal if and ...
ordsucOLD 7852 Obsolete version of ~ ords...
ordpwsuc 7853 The collection of ordinals...
onpwsuc 7854 The collection of ordinal ...
onsucb 7855 A class is an ordinal numb...
ordsucss 7856 The successor of an elemen...
onpsssuc 7857 An ordinal number is a pro...
ordelsuc 7858 A set belongs to an ordina...
onsucmin 7859 The successor of an ordina...
ordsucelsuc 7860 Membership is inherited by...
ordsucsssuc 7861 The subclass relationship ...
ordsucuniel 7862 Given an element ` A ` of ...
ordsucun 7863 The successor of the maxim...
ordunpr 7864 The maximum of two ordinal...
ordunel 7865 The maximum of two ordinal...
onsucuni 7866 A class of ordinal numbers...
ordsucuni 7867 An ordinal class is a subc...
orduniorsuc 7868 An ordinal class is either...
unon 7869 The class of all ordinal n...
ordunisuc 7870 An ordinal class is equal ...
orduniss2 7871 The union of the ordinal s...
onsucuni2 7872 A successor ordinal is the...
0elsuc 7873 The successor of an ordina...
limon 7874 The class of ordinal numbe...
onuniorsuc 7875 An ordinal number is eithe...
onssi 7876 An ordinal number is a sub...
onsuci 7877 The successor of an ordina...
onuniorsuciOLD 7878 Obsolete version of ~ onun...
onuninsuci 7879 An ordinal is equal to its...
onsucssi 7880 A set belongs to an ordina...
nlimsucg 7881 A successor is not a limit...
orduninsuc 7882 An ordinal class is equal ...
ordunisuc2 7883 An ordinal equal to its un...
ordzsl 7884 An ordinal is zero, a succ...
onzsl 7885 An ordinal number is zero,...
dflim3 7886 An alternate definition of...
dflim4 7887 An alternate definition of...
limsuc 7888 The successor of a member ...
limsssuc 7889 A class includes a limit o...
nlimon 7890 Two ways to express the cl...
limuni3 7891 The union of a nonempty cl...
tfi 7892 The Principle of Transfini...
tfisg 7893 A closed form of ~ tfis . ...
tfis 7894 Transfinite Induction Sche...
tfis2f 7895 Transfinite Induction Sche...
tfis2 7896 Transfinite Induction Sche...
tfis3 7897 Transfinite Induction Sche...
tfisi 7898 A transfinite induction sc...
tfinds 7899 Principle of Transfinite I...
tfindsg 7900 Transfinite Induction (inf...
tfindsg2 7901 Transfinite Induction (inf...
tfindes 7902 Transfinite Induction with...
tfinds2 7903 Transfinite Induction (inf...
tfinds3 7904 Principle of Transfinite I...
dfom2 7907 An alternate definition of...
elom 7908 Membership in omega. The ...
omsson 7909 Omega is a subset of ` On ...
limomss 7910 The class of natural numbe...
nnon 7911 A natural number is an ord...
nnoni 7912 A natural number is an ord...
nnord 7913 A natural number is ordina...
trom 7914 The class of finite ordina...
ordom 7915 The class of finite ordina...
elnn 7916 A member of a natural numb...
omon 7917 The class of natural numbe...
omelon2 7918 Omega is an ordinal number...
nnlim 7919 A natural number is not a ...
omssnlim 7920 The class of natural numbe...
limom 7921 Omega is a limit ordinal. ...
peano2b 7922 A class belongs to omega i...
nnsuc 7923 A nonzero natural number i...
omsucne 7924 A natural number is not th...
ssnlim 7925 An ordinal subclass of non...
omsinds 7926 Strong (or "total") induct...
omsindsOLD 7927 Obsolete version of ~ omsi...
omun 7928 The union of two finite or...
peano1 7929 Zero is a natural number. ...
peano1OLD 7930 Obsolete version of ~ pean...
peano2 7931 The successor of any natur...
peano3 7932 The successor of any natur...
peano4 7933 Two natural numbers are eq...
peano5 7934 The induction postulate: a...
peano5OLD 7935 Obsolete version of ~ pean...
nn0suc 7936 A natural number is either...
find 7937 The Principle of Finite In...
finds 7938 Principle of Finite Induct...
findsg 7939 Principle of Finite Induct...
finds2 7940 Principle of Finite Induct...
finds1 7941 Principle of Finite Induct...
findes 7942 Finite induction with expl...
dmexg 7943 The domain of a set is a s...
rnexg 7944 The range of a set is a se...
dmexd 7945 The domain of a set is a s...
fndmexd 7946 If a function is a set, it...
dmfex 7947 If a mapping is a set, its...
fndmexb 7948 The domain of a function i...
fdmexb 7949 The domain of a function i...
dmfexALT 7950 Alternate proof of ~ dmfex...
dmex 7951 The domain of a set is a s...
rnex 7952 The range of a set is a se...
iprc 7953 The identity function is a...
resiexg 7954 The existence of a restric...
imaexg 7955 The image of a set is a se...
imaex 7956 The image of a set is a se...
rnexd 7957 The range of a set is a se...
imaexd 7958 The image of a set is a se...
exse2 7959 Any set relation is set-li...
xpexr 7960 If a Cartesian product is ...
xpexr2 7961 If a nonempty Cartesian pr...
xpexcnv 7962 A condition where the conv...
soex 7963 If the relation in a stric...
elxp4 7964 Membership in a Cartesian ...
elxp5 7965 Membership in a Cartesian ...
cnvexg 7966 The converse of a set is a...
cnvex 7967 The converse of a set is a...
relcnvexb 7968 A relation is a set iff it...
f1oexrnex 7969 If the range of a 1-1 onto...
f1oexbi 7970 There is a one-to-one onto...
coexg 7971 The composition of two set...
coex 7972 The composition of two set...
coexd 7973 The composition of two set...
funcnvuni 7974 The union of a chain (with...
fun11uni 7975 The union of a chain (with...
fex2 7976 A function with bounded do...
fabexd 7977 Existence of a set of func...
fabexg 7978 Existence of a set of func...
fabexgOLD 7979 Obsolete version of ~ fabe...
fabex 7980 Existence of a set of func...
mapex 7981 The class of all functions...
f1oabexg 7982 The class of all 1-1-onto ...
f1oabexgOLD 7983 Obsolete version of ~ f1oa...
fiunlem 7984 Lemma for ~ fiun and ~ f1i...
fiun 7985 The union of a chain (with...
f1iun 7986 The union of a chain (with...
fviunfun 7987 The function value of an i...
ffoss 7988 Relationship between a map...
f11o 7989 Relationship between one-t...
resfunexgALT 7990 Alternate proof of ~ resfu...
cofunexg 7991 Existence of a composition...
cofunex2g 7992 Existence of a composition...
fnexALT 7993 Alternate proof of ~ fnex ...
funexw 7994 Weak version of ~ funex th...
mptexw 7995 Weak version of ~ mptex th...
funrnex 7996 If the domain of a functio...
zfrep6 7997 A version of the Axiom of ...
focdmex 7998 If the domain of an onto f...
f1dmex 7999 If the codomain of a one-t...
f1ovv 8000 The codomain/range of a 1-...
fvclex 8001 Existence of the class of ...
fvresex 8002 Existence of the class of ...
abrexexg 8003 Existence of a class abstr...
abrexexgOLD 8004 Obsolete version of ~ abre...
abrexex 8005 Existence of a class abstr...
iunexg 8006 The existence of an indexe...
abrexex2g 8007 Existence of an existentia...
opabex3d 8008 Existence of an ordered pa...
opabex3rd 8009 Existence of an ordered pa...
opabex3 8010 Existence of an ordered pa...
iunex 8011 The existence of an indexe...
abrexex2 8012 Existence of an existentia...
abexssex 8013 Existence of a class abstr...
abexex 8014 A condition where a class ...
f1oweALT 8015 Alternate proof of ~ f1owe...
wemoiso 8016 Thus, there is at most one...
wemoiso2 8017 Thus, there is at most one...
oprabexd 8018 Existence of an operator a...
oprabex 8019 Existence of an operation ...
oprabex3 8020 Existence of an operation ...
oprabrexex2 8021 Existence of an existentia...
ab2rexex 8022 Existence of a class abstr...
ab2rexex2 8023 Existence of an existentia...
xpexgALT 8024 Alternate proof of ~ xpexg...
offval3 8025 General value of ` ( F oF ...
offres 8026 Pointwise combination comm...
ofmres 8027 Equivalent expressions for...
ofmresex 8028 Existence of a restriction...
mptcnfimad 8029 The converse of a mapping ...
1stval 8034 The value of the function ...
2ndval 8035 The value of the function ...
1stnpr 8036 Value of the first-member ...
2ndnpr 8037 Value of the second-member...
1st0 8038 The value of the first-mem...
2nd0 8039 The value of the second-me...
op1st 8040 Extract the first member o...
op2nd 8041 Extract the second member ...
op1std 8042 Extract the first member o...
op2ndd 8043 Extract the second member ...
op1stg 8044 Extract the first member o...
op2ndg 8045 Extract the second member ...
ot1stg 8046 Extract the first member o...
ot2ndg 8047 Extract the second member ...
ot3rdg 8048 Extract the third member o...
1stval2 8049 Alternate value of the fun...
2ndval2 8050 Alternate value of the fun...
oteqimp 8051 The components of an order...
fo1st 8052 The ` 1st ` function maps ...
fo2nd 8053 The ` 2nd ` function maps ...
br1steqg 8054 Uniqueness condition for t...
br2ndeqg 8055 Uniqueness condition for t...
f1stres 8056 Mapping of a restriction o...
f2ndres 8057 Mapping of a restriction o...
fo1stres 8058 Onto mapping of a restrict...
fo2ndres 8059 Onto mapping of a restrict...
1st2val 8060 Value of an alternate defi...
2nd2val 8061 Value of an alternate defi...
1stcof 8062 Composition of the first m...
2ndcof 8063 Composition of the second ...
xp1st 8064 Location of the first elem...
xp2nd 8065 Location of the second ele...
elxp6 8066 Membership in a Cartesian ...
elxp7 8067 Membership in a Cartesian ...
eqopi 8068 Equality with an ordered p...
xp2 8069 Representation of Cartesia...
unielxp 8070 The membership relation fo...
1st2nd2 8071 Reconstruction of a member...
1st2ndb 8072 Reconstruction of an order...
xpopth 8073 An ordered pair theorem fo...
eqop 8074 Two ways to express equali...
eqop2 8075 Two ways to express equali...
op1steq 8076 Two ways of expressing tha...
opreuopreu 8077 There is a unique ordered ...
el2xptp 8078 A member of a nested Carte...
el2xptp0 8079 A member of a nested Carte...
el2xpss 8080 Version of ~ elrel for tri...
2nd1st 8081 Swap the members of an ord...
1st2nd 8082 Reconstruction of a member...
1stdm 8083 The first ordered pair com...
2ndrn 8084 The second ordered pair co...
1st2ndbr 8085 Express an element of a re...
releldm2 8086 Two ways of expressing mem...
reldm 8087 An expression for the doma...
releldmdifi 8088 One way of expressing memb...
funfv1st2nd 8089 The function value for the...
funelss 8090 If the first component of ...
funeldmdif 8091 Two ways of expressing mem...
sbcopeq1a 8092 Equality theorem for subst...
csbopeq1a 8093 Equality theorem for subst...
sbcoteq1a 8094 Equality theorem for subst...
dfopab2 8095 A way to define an ordered...
dfoprab3s 8096 A way to define an operati...
dfoprab3 8097 Operation class abstractio...
dfoprab4 8098 Operation class abstractio...
dfoprab4f 8099 Operation class abstractio...
opabex2 8100 Condition for an operation...
opabn1stprc 8101 An ordered-pair class abst...
opiota 8102 The property of a uniquely...
cnvoprab 8103 The converse of a class ab...
dfxp3 8104 Define the Cartesian produ...
elopabi 8105 A consequence of membershi...
eloprabi 8106 A consequence of membershi...
mpomptsx 8107 Express a two-argument fun...
mpompts 8108 Express a two-argument fun...
dmmpossx 8109 The domain of a mapping is...
fmpox 8110 Functionality, domain and ...
fmpo 8111 Functionality, domain and ...
fnmpo 8112 Functionality and domain o...
fnmpoi 8113 Functionality and domain o...
dmmpo 8114 Domain of a class given by...
ovmpoelrn 8115 An operation's value belon...
dmmpoga 8116 Domain of an operation giv...
dmmpog 8117 Domain of an operation giv...
mpoexxg 8118 Existence of an operation ...
mpoexg 8119 Existence of an operation ...
mpoexga 8120 If the domain of an operat...
mpoexw 8121 Weak version of ~ mpoex th...
mpoex 8122 If the domain of an operat...
mptmpoopabbrd 8123 The operation value of a f...
mptmpoopabbrdOLD 8124 Obsolete version of ~ mptm...
mptmpoopabovd 8125 The operation value of a f...
mptmpoopabbrdOLDOLD 8126 Obsolete version of ~ mptm...
mptmpoopabovdOLD 8127 Obsolete version of ~ mptm...
el2mpocsbcl 8128 If the operation value of ...
el2mpocl 8129 If the operation value of ...
fnmpoovd 8130 A function with a Cartesia...
offval22 8131 The function operation exp...
brovpreldm 8132 If a binary relation holds...
bropopvvv 8133 If a binary relation holds...
bropfvvvvlem 8134 Lemma for ~ bropfvvvv . (...
bropfvvvv 8135 If a binary relation holds...
ovmptss 8136 If all the values of the m...
relmpoopab 8137 Any function to sets of or...
fmpoco 8138 Composition of two functio...
oprabco 8139 Composition of a function ...
oprab2co 8140 Composition of operator ab...
df1st2 8141 An alternate possible defi...
df2nd2 8142 An alternate possible defi...
1stconst 8143 The mapping of a restricti...
2ndconst 8144 The mapping of a restricti...
dfmpo 8145 Alternate definition for t...
mposn 8146 An operation (in maps-to n...
curry1 8147 Composition with ` ``' ( 2...
curry1val 8148 The value of a curried fun...
curry1f 8149 Functionality of a curried...
curry2 8150 Composition with ` ``' ( 1...
curry2f 8151 Functionality of a curried...
curry2val 8152 The value of a curried fun...
cnvf1olem 8153 Lemma for ~ cnvf1o . (Con...
cnvf1o 8154 Describe a function that m...
fparlem1 8155 Lemma for ~ fpar . (Contr...
fparlem2 8156 Lemma for ~ fpar . (Contr...
fparlem3 8157 Lemma for ~ fpar . (Contr...
fparlem4 8158 Lemma for ~ fpar . (Contr...
fpar 8159 Merge two functions in par...
fsplit 8160 A function that can be use...
fsplitfpar 8161 Merge two functions with a...
offsplitfpar 8162 Express the function opera...
f2ndf 8163 The ` 2nd ` (second compon...
fo2ndf 8164 The ` 2nd ` (second compon...
f1o2ndf1 8165 The ` 2nd ` (second compon...
opco1 8166 Value of an operation prec...
opco2 8167 Value of an operation prec...
opco1i 8168 Inference form of ~ opco1 ...
frxp 8169 A lexicographical ordering...
xporderlem 8170 Lemma for lexicographical ...
poxp 8171 A lexicographical ordering...
soxp 8172 A lexicographical ordering...
wexp 8173 A lexicographical ordering...
fnwelem 8174 Lemma for ~ fnwe . (Contr...
fnwe 8175 A variant on lexicographic...
fnse 8176 Condition for the well-ord...
fvproj 8177 Value of a function on ord...
fimaproj 8178 Image of a cartesian produ...
ralxpes 8179 A version of ~ ralxp with ...
ralxp3f 8180 Restricted for all over a ...
ralxp3 8181 Restricted for all over a ...
ralxp3es 8182 Restricted for-all over a ...
frpoins3xpg 8183 Special case of founded pa...
frpoins3xp3g 8184 Special case of founded pa...
xpord2lem 8185 Lemma for Cartesian produc...
poxp2 8186 Another way of partially o...
frxp2 8187 Another way of giving a we...
xpord2pred 8188 Calculate the predecessor ...
sexp2 8189 Condition for the relation...
xpord2indlem 8190 Induction over the Cartesi...
xpord2ind 8191 Induction over the Cartesi...
xpord3lem 8192 Lemma for triple ordering....
poxp3 8193 Triple Cartesian product p...
frxp3 8194 Give well-foundedness over...
xpord3pred 8195 Calculate the predecsessor...
sexp3 8196 Show that the triple order...
xpord3inddlem 8197 Induction over the triple ...
xpord3indd 8198 Induction over the triple ...
xpord3ind 8199 Induction over the triple ...
orderseqlem 8200 Lemma for ~ poseq and ~ so...
poseq 8201 A partial ordering of ordi...
soseq 8202 A linear ordering of ordin...
suppval 8205 The value of the operation...
supp0prc 8206 The support of a class is ...
suppvalbr 8207 The value of the operation...
supp0 8208 The support of the empty s...
suppval1 8209 The value of the operation...
suppvalfng 8210 The value of the operation...
suppvalfn 8211 The value of the operation...
elsuppfng 8212 An element of the support ...
elsuppfn 8213 An element of the support ...
fvdifsupp 8214 Function value is zero out...
cnvimadfsn 8215 The support of functions "...
suppimacnvss 8216 The support of functions "...
suppimacnv 8217 Support sets of functions ...
fsuppeq 8218 Two ways of writing the su...
fsuppeqg 8219 Version of ~ fsuppeq avoid...
suppssdm 8220 The support of a function ...
suppsnop 8221 The support of a singleton...
snopsuppss 8222 The support of a singleton...
fvn0elsupp 8223 If the function value for ...
fvn0elsuppb 8224 The function value for a g...
rexsupp 8225 Existential quantification...
ressuppss 8226 The support of the restric...
suppun 8227 The support of a class/fun...
ressuppssdif 8228 The support of the restric...
mptsuppdifd 8229 The support of a function ...
mptsuppd 8230 The support of a function ...
extmptsuppeq 8231 The support of an extended...
suppfnss 8232 The support of a function ...
funsssuppss 8233 The support of a function ...
fnsuppres 8234 Two ways to express restri...
fnsuppeq0 8235 The support of a function ...
fczsupp0 8236 The support of a constant ...
suppss 8237 Show that the support of a...
suppssr 8238 A function is zero outside...
suppssrg 8239 A function is zero outside...
suppssov1 8240 Formula building theorem f...
suppssov2 8241 Formula building theorem f...
suppssof1 8242 Formula building theorem f...
suppss2 8243 Show that the support of a...
suppsssn 8244 Show that the support of a...
suppssfv 8245 Formula building theorem f...
suppofssd 8246 Condition for the support ...
suppofss1d 8247 Condition for the support ...
suppofss2d 8248 Condition for the support ...
suppco 8249 The support of the composi...
suppcoss 8250 The support of the composi...
supp0cosupp0 8251 The support of the composi...
imacosupp 8252 The image of the support o...
opeliunxp2f 8253 Membership in a union of C...
mpoxeldm 8254 If there is an element of ...
mpoxneldm 8255 If the first argument of a...
mpoxopn0yelv 8256 If there is an element of ...
mpoxopynvov0g 8257 If the second argument of ...
mpoxopxnop0 8258 If the first argument of a...
mpoxopx0ov0 8259 If the first argument of a...
mpoxopxprcov0 8260 If the components of the f...
mpoxopynvov0 8261 If the second argument of ...
mpoxopoveq 8262 Value of an operation give...
mpoxopovel 8263 Element of the value of an...
mpoxopoveqd 8264 Value of an operation give...
brovex 8265 A binary relation of the v...
brovmpoex 8266 A binary relation of the v...
sprmpod 8267 The extension of a binary ...
tposss 8270 Subset theorem for transpo...
tposeq 8271 Equality theorem for trans...
tposeqd 8272 Equality theorem for trans...
tposssxp 8273 The transposition is a sub...
reltpos 8274 The transposition is a rel...
brtpos2 8275 Value of the transposition...
brtpos0 8276 The behavior of ` tpos ` w...
reldmtpos 8277 Necessary and sufficient c...
brtpos 8278 The transposition swaps ar...
ottpos 8279 The transposition swaps th...
relbrtpos 8280 The transposition swaps ar...
dmtpos 8281 The domain of ` tpos F ` w...
rntpos 8282 The range of ` tpos F ` wh...
tposexg 8283 The transposition of a set...
ovtpos 8284 The transposition swaps th...
tposfun 8285 The transposition of a fun...
dftpos2 8286 Alternate definition of ` ...
dftpos3 8287 Alternate definition of ` ...
dftpos4 8288 Alternate definition of ` ...
tpostpos 8289 Value of the double transp...
tpostpos2 8290 Value of the double transp...
tposfn2 8291 The domain of a transposit...
tposfo2 8292 Condition for a surjective...
tposf2 8293 The domain and codomain of...
tposf12 8294 Condition for an injective...
tposf1o2 8295 Condition of a bijective t...
tposfo 8296 The domain and codomain/ra...
tposf 8297 The domain and codomain of...
tposfn 8298 Functionality of a transpo...
tpos0 8299 Transposition of the empty...
tposco 8300 Transposition of a composi...
tpossym 8301 Two ways to say a function...
tposeqi 8302 Equality theorem for trans...
tposex 8303 A transposition is a set. ...
nftpos 8304 Hypothesis builder for tra...
tposoprab 8305 Transposition of a class o...
tposmpo 8306 Transposition of a two-arg...
tposconst 8307 The transposition of a con...
mpocurryd 8312 The currying of an operati...
mpocurryvald 8313 The value of a curried ope...
fvmpocurryd 8314 The value of the value of ...
pwuninel2 8317 Proof of ~ pwuninel under ...
pwuninel 8318 The powerclass of the unio...
undefval 8319 Value of the undefined val...
undefnel2 8320 The undefined value genera...
undefnel 8321 The undefined value genera...
undefne0 8322 The undefined value genera...
frecseq123 8325 Equality theorem for the w...
nffrecs 8326 Bound-variable hypothesis ...
csbfrecsg 8327 Move class substitution in...
fpr3g 8328 Functions defined by well-...
frrlem1 8329 Lemma for well-founded rec...
frrlem2 8330 Lemma for well-founded rec...
frrlem3 8331 Lemma for well-founded rec...
frrlem4 8332 Lemma for well-founded rec...
frrlem5 8333 Lemma for well-founded rec...
frrlem6 8334 Lemma for well-founded rec...
frrlem7 8335 Lemma for well-founded rec...
frrlem8 8336 Lemma for well-founded rec...
frrlem9 8337 Lemma for well-founded rec...
frrlem10 8338 Lemma for well-founded rec...
frrlem11 8339 Lemma for well-founded rec...
frrlem12 8340 Lemma for well-founded rec...
frrlem13 8341 Lemma for well-founded rec...
frrlem14 8342 Lemma for well-founded rec...
fprlem1 8343 Lemma for well-founded rec...
fprlem2 8344 Lemma for well-founded rec...
fpr2a 8345 Weak version of ~ fpr2 whi...
fpr1 8346 Law of well-founded recurs...
fpr2 8347 Law of well-founded recurs...
fpr3 8348 Law of well-founded recurs...
frrrel 8349 Show without using the axi...
frrdmss 8350 Show without using the axi...
frrdmcl 8351 Show without using the axi...
fprfung 8352 A "function" defined by we...
fprresex 8353 The restriction of a funct...
dfwrecsOLD 8356 Obsolete definition of the...
wrecseq123 8357 General equality theorem f...
wrecseq123OLD 8358 Obsolete version of ~ wrec...
nfwrecs 8359 Bound-variable hypothesis ...
nfwrecsOLD 8360 Obsolete proof of ~ nfwrec...
wrecseq1 8361 Equality theorem for the w...
wrecseq2 8362 Equality theorem for the w...
wrecseq3 8363 Equality theorem for the w...
csbwrecsg 8364 Move class substitution in...
wfr3g 8365 Functions defined by well-...
wfrlem1OLD 8366 Lemma for well-ordered rec...
wfrlem2OLD 8367 Lemma for well-ordered rec...
wfrlem3OLD 8368 Lemma for well-ordered rec...
wfrlem3OLDa 8369 Lemma for well-ordered rec...
wfrlem4OLD 8370 Lemma for well-ordered rec...
wfrlem5OLD 8371 Lemma for well-ordered rec...
wfrrelOLD 8372 Obsolete proof of ~ wfrrel...
wfrdmssOLD 8373 Obsolete proof of ~ wfrdms...
wfrlem8OLD 8374 Lemma for well-ordered rec...
wfrdmclOLD 8375 Obsolete version of ~ wfrd...
wfrlem10OLD 8376 Lemma for well-ordered rec...
wfrfunOLD 8377 Obsolete proof of ~ wfrfun...
wfrlem12OLD 8378 Lemma for well-ordered rec...
wfrlem13OLD 8379 Lemma for well-ordered rec...
wfrlem14OLD 8380 Lemma for well-ordered rec...
wfrlem15OLD 8381 Lemma for well-ordered rec...
wfrlem16OLD 8382 Lemma for well-ordered rec...
wfrlem17OLD 8383 Without using ~ ax-rep , s...
wfr2aOLD 8384 Obsolete version of ~ wfr2...
wfr1OLD 8385 Obsolete version of ~ wfr1...
wfr2OLD 8386 Obsolete version of ~ wfr2...
wfrrel 8387 The well-ordered recursion...
wfrdmss 8388 The domain of the well-ord...
wfrdmcl 8389 The predecessor class of a...
wfrfun 8390 The "function" generated b...
wfrresex 8391 Show without using the axi...
wfr2a 8392 A weak version of ~ wfr2 w...
wfr1 8393 The Principle of Well-Orde...
wfr2 8394 The Principle of Well-Orde...
wfr3 8395 The principle of Well-Orde...
wfr3OLD 8396 Obsolete form of ~ wfr3 as...
iunon 8397 The indexed union of a set...
iinon 8398 The nonempty indexed inter...
onfununi 8399 A property of functions on...
onovuni 8400 A variant of ~ onfununi fo...
onoviun 8401 A variant of ~ onovuni wit...
onnseq 8402 There are no length ` _om ...
dfsmo2 8405 Alternate definition of a ...
issmo 8406 Conditions for which ` A `...
issmo2 8407 Alternate definition of a ...
smoeq 8408 Equality theorem for stric...
smodm 8409 The domain of a strictly m...
smores 8410 A strictly monotone functi...
smores3 8411 A strictly monotone functi...
smores2 8412 A strictly monotone ordina...
smodm2 8413 The domain of a strictly m...
smofvon2 8414 The function values of a s...
iordsmo 8415 The identity relation rest...
smo0 8416 The null set is a strictly...
smofvon 8417 If ` B ` is a strictly mon...
smoel 8418 If ` x ` is less than ` y ...
smoiun 8419 The value of a strictly mo...
smoiso 8420 If ` F ` is an isomorphism...
smoel2 8421 A strictly monotone ordina...
smo11 8422 A strictly monotone ordina...
smoord 8423 A strictly monotone ordina...
smoword 8424 A strictly monotone ordina...
smogt 8425 A strictly monotone ordina...
smocdmdom 8426 The codomain of a strictly...
smoiso2 8427 The strictly monotone ordi...
dfrecs3 8430 The old definition of tran...
dfrecs3OLD 8431 Obsolete version of ~ dfre...
recseq 8432 Equality theorem for ` rec...
nfrecs 8433 Bound-variable hypothesis ...
tfrlem1 8434 A technical lemma for tran...
tfrlem3a 8435 Lemma for transfinite recu...
tfrlem3 8436 Lemma for transfinite recu...
tfrlem4 8437 Lemma for transfinite recu...
tfrlem5 8438 Lemma for transfinite recu...
recsfval 8439 Lemma for transfinite recu...
tfrlem6 8440 Lemma for transfinite recu...
tfrlem7 8441 Lemma for transfinite recu...
tfrlem8 8442 Lemma for transfinite recu...
tfrlem9 8443 Lemma for transfinite recu...
tfrlem9a 8444 Lemma for transfinite recu...
tfrlem10 8445 Lemma for transfinite recu...
tfrlem11 8446 Lemma for transfinite recu...
tfrlem12 8447 Lemma for transfinite recu...
tfrlem13 8448 Lemma for transfinite recu...
tfrlem14 8449 Lemma for transfinite recu...
tfrlem15 8450 Lemma for transfinite recu...
tfrlem16 8451 Lemma for finite recursion...
tfr1a 8452 A weak version of ~ tfr1 w...
tfr2a 8453 A weak version of ~ tfr2 w...
tfr2b 8454 Without assuming ~ ax-rep ...
tfr1 8455 Principle of Transfinite R...
tfr2 8456 Principle of Transfinite R...
tfr3 8457 Principle of Transfinite R...
tfr1ALT 8458 Alternate proof of ~ tfr1 ...
tfr2ALT 8459 Alternate proof of ~ tfr2 ...
tfr3ALT 8460 Alternate proof of ~ tfr3 ...
recsfnon 8461 Strong transfinite recursi...
recsval 8462 Strong transfinite recursi...
tz7.44lem1 8463 The ordered pair abstracti...
tz7.44-1 8464 The value of ` F ` at ` (/...
tz7.44-2 8465 The value of ` F ` at a su...
tz7.44-3 8466 The value of ` F ` at a li...
rdgeq1 8469 Equality theorem for the r...
rdgeq2 8470 Equality theorem for the r...
rdgeq12 8471 Equality theorem for the r...
nfrdg 8472 Bound-variable hypothesis ...
rdglem1 8473 Lemma used with the recurs...
rdgfun 8474 The recursive definition g...
rdgdmlim 8475 The domain of the recursiv...
rdgfnon 8476 The recursive definition g...
rdgvalg 8477 Value of the recursive def...
rdgval 8478 Value of the recursive def...
rdg0 8479 The initial value of the r...
rdgseg 8480 The initial segments of th...
rdgsucg 8481 The value of the recursive...
rdgsuc 8482 The value of the recursive...
rdglimg 8483 The value of the recursive...
rdglim 8484 The value of the recursive...
rdg0g 8485 The initial value of the r...
rdgsucmptf 8486 The value of the recursive...
rdgsucmptnf 8487 The value of the recursive...
rdgsucmpt2 8488 This version of ~ rdgsucmp...
rdgsucmpt 8489 The value of the recursive...
rdglim2 8490 The value of the recursive...
rdglim2a 8491 The value of the recursive...
rdg0n 8492 If ` A ` is a proper class...
frfnom 8493 The function generated by ...
fr0g 8494 The initial value resultin...
frsuc 8495 The successor value result...
frsucmpt 8496 The successor value result...
frsucmptn 8497 The value of the finite re...
frsucmpt2 8498 The successor value result...
tz7.48lem 8499 A way of showing an ordina...
tz7.48-2 8500 Proposition 7.48(2) of [Ta...
tz7.48-1 8501 Proposition 7.48(1) of [Ta...
tz7.48-3 8502 Proposition 7.48(3) of [Ta...
tz7.49 8503 Proposition 7.49 of [Takeu...
tz7.49c 8504 Corollary of Proposition 7...
seqomlem0 8507 Lemma for ` seqom ` . Cha...
seqomlem1 8508 Lemma for ` seqom ` . The...
seqomlem2 8509 Lemma for ` seqom ` . (Co...
seqomlem3 8510 Lemma for ` seqom ` . (Co...
seqomlem4 8511 Lemma for ` seqom ` . (Co...
seqomeq12 8512 Equality theorem for ` seq...
fnseqom 8513 An index-aware recursive d...
seqom0g 8514 Value of an index-aware re...
seqomsuc 8515 Value of an index-aware re...
omsucelsucb 8516 Membership is inherited by...
df1o2 8531 Expanded value of the ordi...
df2o3 8532 Expanded value of the ordi...
df2o2 8533 Expanded value of the ordi...
1oex 8534 Ordinal 1 is a set. (Cont...
2oex 8535 ` 2o ` is a set. (Contrib...
1on 8536 Ordinal 1 is an ordinal nu...
1onOLD 8537 Obsolete version of ~ 1on ...
2on 8538 Ordinal 2 is an ordinal nu...
2onOLD 8539 Obsolete version of ~ 2on ...
2on0 8540 Ordinal two is not zero. ...
ord3 8541 Ordinal 3 is an ordinal cl...
3on 8542 Ordinal 3 is an ordinal nu...
4on 8543 Ordinal 4 is an ordinal nu...
1oexOLD 8544 Obsolete version of ~ 1oex...
2oexOLD 8545 Obsolete version of ~ 2oex...
1n0 8546 Ordinal one is not equal t...
nlim1 8547 1 is not a limit ordinal. ...
nlim2 8548 2 is not a limit ordinal. ...
xp01disj 8549 Cartesian products with th...
xp01disjl 8550 Cartesian products with th...
ordgt0ge1 8551 Two ways to express that a...
ordge1n0 8552 An ordinal greater than or...
el1o 8553 Membership in ordinal one....
ord1eln01 8554 An ordinal that is not 0 o...
ord2eln012 8555 An ordinal that is not 0, ...
1ellim 8556 A limit ordinal contains 1...
2ellim 8557 A limit ordinal contains 2...
dif1o 8558 Two ways to say that ` A `...
ondif1 8559 Two ways to say that ` A `...
ondif2 8560 Two ways to say that ` A `...
2oconcl 8561 Closure of the pair swappi...
0lt1o 8562 Ordinal zero is less than ...
dif20el 8563 An ordinal greater than on...
0we1 8564 The empty set is a well-or...
brwitnlem 8565 Lemma for relations which ...
fnoa 8566 Functionality and domain o...
fnom 8567 Functionality and domain o...
fnoe 8568 Functionality and domain o...
oav 8569 Value of ordinal addition....
omv 8570 Value of ordinal multiplic...
oe0lem 8571 A helper lemma for ~ oe0 a...
oev 8572 Value of ordinal exponenti...
oevn0 8573 Value of ordinal exponenti...
oa0 8574 Addition with zero. Propo...
om0 8575 Ordinal multiplication wit...
oe0m 8576 Value of zero raised to an...
om0x 8577 Ordinal multiplication wit...
oe0m0 8578 Ordinal exponentiation wit...
oe0m1 8579 Ordinal exponentiation wit...
oe0 8580 Ordinal exponentiation wit...
oev2 8581 Alternate value of ordinal...
oasuc 8582 Addition with successor. ...
oesuclem 8583 Lemma for ~ oesuc . (Cont...
omsuc 8584 Multiplication with succes...
oesuc 8585 Ordinal exponentiation wit...
onasuc 8586 Addition with successor. ...
onmsuc 8587 Multiplication with succes...
onesuc 8588 Exponentiation with a succ...
oa1suc 8589 Addition with 1 is same as...
oalim 8590 Ordinal addition with a li...
omlim 8591 Ordinal multiplication wit...
oelim 8592 Ordinal exponentiation wit...
oacl 8593 Closure law for ordinal ad...
omcl 8594 Closure law for ordinal mu...
oecl 8595 Closure law for ordinal ex...
oa0r 8596 Ordinal addition with zero...
om0r 8597 Ordinal multiplication wit...
o1p1e2 8598 1 + 1 = 2 for ordinal numb...
o2p2e4 8599 2 + 2 = 4 for ordinal numb...
om1 8600 Ordinal multiplication wit...
om1r 8601 Ordinal multiplication wit...
oe1 8602 Ordinal exponentiation wit...
oe1m 8603 Ordinal exponentiation wit...
oaordi 8604 Ordering property of ordin...
oaord 8605 Ordering property of ordin...
oacan 8606 Left cancellation law for ...
oaword 8607 Weak ordering property of ...
oawordri 8608 Weak ordering property of ...
oaord1 8609 An ordinal is less than it...
oaword1 8610 An ordinal is less than or...
oaword2 8611 An ordinal is less than or...
oawordeulem 8612 Lemma for ~ oawordex . (C...
oawordeu 8613 Existence theorem for weak...
oawordexr 8614 Existence theorem for weak...
oawordex 8615 Existence theorem for weak...
oaordex 8616 Existence theorem for orde...
oa00 8617 An ordinal sum is zero iff...
oalimcl 8618 The ordinal sum with a lim...
oaass 8619 Ordinal addition is associ...
oarec 8620 Recursive definition of or...
oaf1o 8621 Left addition by a constan...
oacomf1olem 8622 Lemma for ~ oacomf1o . (C...
oacomf1o 8623 Define a bijection from ` ...
omordi 8624 Ordering property of ordin...
omord2 8625 Ordering property of ordin...
omord 8626 Ordering property of ordin...
omcan 8627 Left cancellation law for ...
omword 8628 Weak ordering property of ...
omwordi 8629 Weak ordering property of ...
omwordri 8630 Weak ordering property of ...
omword1 8631 An ordinal is less than or...
omword2 8632 An ordinal is less than or...
om00 8633 The product of two ordinal...
om00el 8634 The product of two nonzero...
omordlim 8635 Ordering involving the pro...
omlimcl 8636 The product of any nonzero...
odi 8637 Distributive law for ordin...
omass 8638 Multiplication of ordinal ...
oneo 8639 If an ordinal number is ev...
omeulem1 8640 Lemma for ~ omeu : existen...
omeulem2 8641 Lemma for ~ omeu : uniquen...
omopth2 8642 An ordered pair-like theor...
omeu 8643 The division algorithm for...
oen0 8644 Ordinal exponentiation wit...
oeordi 8645 Ordering law for ordinal e...
oeord 8646 Ordering property of ordin...
oecan 8647 Left cancellation law for ...
oeword 8648 Weak ordering property of ...
oewordi 8649 Weak ordering property of ...
oewordri 8650 Weak ordering property of ...
oeworde 8651 Ordinal exponentiation com...
oeordsuc 8652 Ordering property of ordin...
oelim2 8653 Ordinal exponentiation wit...
oeoalem 8654 Lemma for ~ oeoa . (Contr...
oeoa 8655 Sum of exponents law for o...
oeoelem 8656 Lemma for ~ oeoe . (Contr...
oeoe 8657 Product of exponents law f...
oelimcl 8658 The ordinal exponential wi...
oeeulem 8659 Lemma for ~ oeeu . (Contr...
oeeui 8660 The division algorithm for...
oeeu 8661 The division algorithm for...
nna0 8662 Addition with zero. Theor...
nnm0 8663 Multiplication with zero. ...
nnasuc 8664 Addition with successor. ...
nnmsuc 8665 Multiplication with succes...
nnesuc 8666 Exponentiation with a succ...
nna0r 8667 Addition to zero. Remark ...
nnm0r 8668 Multiplication with zero. ...
nnacl 8669 Closure of addition of nat...
nnmcl 8670 Closure of multiplication ...
nnecl 8671 Closure of exponentiation ...
nnacli 8672 ` _om ` is closed under ad...
nnmcli 8673 ` _om ` is closed under mu...
nnarcl 8674 Reverse closure law for ad...
nnacom 8675 Addition of natural number...
nnaordi 8676 Ordering property of addit...
nnaord 8677 Ordering property of addit...
nnaordr 8678 Ordering property of addit...
nnawordi 8679 Adding to both sides of an...
nnaass 8680 Addition of natural number...
nndi 8681 Distributive law for natur...
nnmass 8682 Multiplication of natural ...
nnmsucr 8683 Multiplication with succes...
nnmcom 8684 Multiplication of natural ...
nnaword 8685 Weak ordering property of ...
nnacan 8686 Cancellation law for addit...
nnaword1 8687 Weak ordering property of ...
nnaword2 8688 Weak ordering property of ...
nnmordi 8689 Ordering property of multi...
nnmord 8690 Ordering property of multi...
nnmword 8691 Weak ordering property of ...
nnmcan 8692 Cancellation law for multi...
nnmwordi 8693 Weak ordering property of ...
nnmwordri 8694 Weak ordering property of ...
nnawordex 8695 Equivalence for weak order...
nnaordex 8696 Equivalence for ordering. ...
nnaordex2 8697 Equivalence for ordering. ...
1onn 8698 The ordinal 1 is a natural...
1onnALT 8699 Shorter proof of ~ 1onn us...
2onn 8700 The ordinal 2 is a natural...
2onnALT 8701 Shorter proof of ~ 2onn us...
3onn 8702 The ordinal 3 is a natural...
4onn 8703 The ordinal 4 is a natural...
1one2o 8704 Ordinal one is not ordinal...
oaabslem 8705 Lemma for ~ oaabs . (Cont...
oaabs 8706 Ordinal addition absorbs a...
oaabs2 8707 The absorption law ~ oaabs...
omabslem 8708 Lemma for ~ omabs . (Cont...
omabs 8709 Ordinal multiplication is ...
nnm1 8710 Multiply an element of ` _...
nnm2 8711 Multiply an element of ` _...
nn2m 8712 Multiply an element of ` _...
nnneo 8713 If a natural number is eve...
nneob 8714 A natural number is even i...
omsmolem 8715 Lemma for ~ omsmo . (Cont...
omsmo 8716 A strictly monotonic ordin...
omopthlem1 8717 Lemma for ~ omopthi . (Co...
omopthlem2 8718 Lemma for ~ omopthi . (Co...
omopthi 8719 An ordered pair theorem fo...
omopth 8720 An ordered pair theorem fo...
nnasmo 8721 There is at most one left ...
eldifsucnn 8722 Condition for membership i...
on2recsfn 8725 Show that double recursion...
on2recsov 8726 Calculate the value of the...
on2ind 8727 Double induction over ordi...
on3ind 8728 Triple induction over ordi...
coflton 8729 Cofinality theorem for ord...
cofon1 8730 Cofinality theorem for ord...
cofon2 8731 Cofinality theorem for ord...
cofonr 8732 Inverse cofinality law for...
naddfn 8733 Natural addition is a func...
naddcllem 8734 Lemma for ordinal addition...
naddcl 8735 Closure law for natural ad...
naddov 8736 The value of natural addit...
naddov2 8737 Alternate expression for n...
naddov3 8738 Alternate expression for n...
naddf 8739 Function statement for nat...
naddcom 8740 Natural addition commutes....
naddrid 8741 Ordinal zero is the additi...
naddlid 8742 Ordinal zero is the additi...
naddssim 8743 Ordinal less-than-or-equal...
naddelim 8744 Ordinal less-than is prese...
naddel1 8745 Ordinal less-than is not a...
naddel2 8746 Ordinal less-than is not a...
naddss1 8747 Ordinal less-than-or-equal...
naddss2 8748 Ordinal less-than-or-equal...
naddword1 8749 Weak-ordering principle fo...
naddword2 8750 Weak-ordering principle fo...
naddunif 8751 Uniformity theorem for nat...
naddasslem1 8752 Lemma for ~ naddass . Exp...
naddasslem2 8753 Lemma for ~ naddass . Exp...
naddass 8754 Natural ordinal addition i...
nadd32 8755 Commutative/associative la...
nadd4 8756 Rearragement of terms in a...
nadd42 8757 Rearragement of terms in a...
naddel12 8758 Natural addition to both s...
naddsuc2 8759 Natural addition with succ...
naddoa 8760 Natural addition of a natu...
omnaddcl 8761 The naturals are closed un...
dfer2 8766 Alternate definition of eq...
dfec2 8768 Alternate definition of ` ...
ecexg 8769 An equivalence class modul...
ecexr 8770 A nonempty equivalence cla...
ereq1 8772 Equality theorem for equiv...
ereq2 8773 Equality theorem for equiv...
errel 8774 An equivalence relation is...
erdm 8775 The domain of an equivalen...
ercl 8776 Elementhood in the field o...
ersym 8777 An equivalence relation is...
ercl2 8778 Elementhood in the field o...
ersymb 8779 An equivalence relation is...
ertr 8780 An equivalence relation is...
ertrd 8781 A transitivity relation fo...
ertr2d 8782 A transitivity relation fo...
ertr3d 8783 A transitivity relation fo...
ertr4d 8784 A transitivity relation fo...
erref 8785 An equivalence relation is...
ercnv 8786 The converse of an equival...
errn 8787 The range and domain of an...
erssxp 8788 An equivalence relation is...
erex 8789 An equivalence relation is...
erexb 8790 An equivalence relation is...
iserd 8791 A reflexive, symmetric, tr...
iseri 8792 A reflexive, symmetric, tr...
iseriALT 8793 Alternate proof of ~ iseri...
brinxper 8794 Conditions for a reflexive...
brdifun 8795 Evaluate the incomparabili...
swoer 8796 Incomparability under a st...
swoord1 8797 The incomparability equiva...
swoord2 8798 The incomparability equiva...
swoso 8799 If the incomparability rel...
eqerlem 8800 Lemma for ~ eqer . (Contr...
eqer 8801 Equivalence relation invol...
ider 8802 The identity relation is a...
0er 8803 The empty set is an equiva...
eceq1 8804 Equality theorem for equiv...
eceq1d 8805 Equality theorem for equiv...
eceq2 8806 Equality theorem for equiv...
eceq2i 8807 Equality theorem for the `...
eceq2d 8808 Equality theorem for the `...
elecg 8809 Membership in an equivalen...
ecref 8810 All elements are in their ...
elec 8811 Membership in an equivalen...
relelec 8812 Membership in an equivalen...
ecss 8813 An equivalence class is a ...
ecdmn0 8814 A representative of a none...
ereldm 8815 Equality of equivalence cl...
erth 8816 Basic property of equivale...
erth2 8817 Basic property of equivale...
erthi 8818 Basic property of equivale...
erdisj 8819 Equivalence classes do not...
ecidsn 8820 An equivalence class modul...
qseq1 8821 Equality theorem for quoti...
qseq2 8822 Equality theorem for quoti...
qseq2i 8823 Equality theorem for quoti...
qseq1d 8824 Equality theorem for quoti...
qseq2d 8825 Equality theorem for quoti...
qseq12 8826 Equality theorem for quoti...
0qs 8827 Quotient set with the empt...
elqsg 8828 Closed form of ~ elqs . (...
elqs 8829 Membership in a quotient s...
elqsi 8830 Membership in a quotient s...
elqsecl 8831 Membership in a quotient s...
ecelqsg 8832 Membership of an equivalen...
ecelqsi 8833 Membership of an equivalen...
ecopqsi 8834 "Closure" law for equivale...
qsexg 8835 A quotient set exists. (C...
qsex 8836 A quotient set exists. (C...
uniqs 8837 The union of a quotient se...
qsss 8838 A quotient set is a set of...
uniqs2 8839 The union of a quotient se...
snec 8840 The singleton of an equiva...
ecqs 8841 Equivalence class in terms...
ecid 8842 A set is equal to its cose...
qsid 8843 A set is equal to its quot...
ectocld 8844 Implicit substitution of c...
ectocl 8845 Implicit substitution of c...
elqsn0 8846 A quotient set does not co...
ecelqsdm 8847 Membership of an equivalen...
xpider 8848 A Cartesian square is an e...
iiner 8849 The intersection of a none...
riiner 8850 The relative intersection ...
erinxp 8851 A restricted equivalence r...
ecinxp 8852 Restrict the relation in a...
qsinxp 8853 Restrict the equivalence r...
qsdisj 8854 Members of a quotient set ...
qsdisj2 8855 A quotient set is a disjoi...
qsel 8856 If an element of a quotien...
uniinqs 8857 Class union distributes ov...
qliftlem 8858 Lemma for theorems about a...
qliftrel 8859 ` F ` , a function lift, i...
qliftel 8860 Elementhood in the relatio...
qliftel1 8861 Elementhood in the relatio...
qliftfun 8862 The function ` F ` is the ...
qliftfund 8863 The function ` F ` is the ...
qliftfuns 8864 The function ` F ` is the ...
qliftf 8865 The domain and codomain of...
qliftval 8866 The value of the function ...
ecoptocl 8867 Implicit substitution of c...
2ecoptocl 8868 Implicit substitution of c...
3ecoptocl 8869 Implicit substitution of c...
brecop 8870 Binary relation on a quoti...
brecop2 8871 Binary relation on a quoti...
eroveu 8872 Lemma for ~ erov and ~ ero...
erovlem 8873 Lemma for ~ erov and ~ ero...
erov 8874 The value of an operation ...
eroprf 8875 Functionality of an operat...
erov2 8876 The value of an operation ...
eroprf2 8877 Functionality of an operat...
ecopoveq 8878 This is the first of sever...
ecopovsym 8879 Assuming the operation ` F...
ecopovtrn 8880 Assuming that operation ` ...
ecopover 8881 Assuming that operation ` ...
eceqoveq 8882 Equality of equivalence re...
ecovcom 8883 Lemma used to transfer a c...
ecovass 8884 Lemma used to transfer an ...
ecovdi 8885 Lemma used to transfer a d...
mapprc 8890 When ` A ` is a proper cla...
pmex 8891 The class of all partial f...
mapexOLD 8892 Obsolete version of ~ mape...
fnmap 8893 Set exponentiation has a u...
fnpm 8894 Partial function exponenti...
reldmmap 8895 Set exponentiation is a we...
mapvalg 8896 The value of set exponenti...
pmvalg 8897 The value of the partial m...
mapval 8898 The value of set exponenti...
elmapg 8899 Membership relation for se...
elmapd 8900 Deduction form of ~ elmapg...
elmapdd 8901 Deduction associated with ...
mapdm0 8902 The empty set is the only ...
elpmg 8903 The predicate "is a partia...
elpm2g 8904 The predicate "is a partia...
elpm2r 8905 Sufficient condition for b...
elpmi 8906 A partial function is a fu...
pmfun 8907 A partial function is a fu...
elmapex 8908 Eliminate antecedent for m...
elmapi 8909 A mapping is a function, f...
mapfset 8910 If ` B ` is a set, the val...
mapssfset 8911 The value of the set expon...
mapfoss 8912 The value of the set expon...
fsetsspwxp 8913 The class of all functions...
fset0 8914 The set of functions from ...
fsetdmprc0 8915 The set of functions with ...
fsetex 8916 The set of functions betwe...
f1setex 8917 The set of injections betw...
fosetex 8918 The set of surjections bet...
f1osetex 8919 The set of bijections betw...
fsetfcdm 8920 The class of functions wit...
fsetfocdm 8921 The class of functions wit...
fsetprcnex 8922 The class of all functions...
fsetcdmex 8923 The class of all functions...
fsetexb 8924 The class of all functions...
elmapfn 8925 A mapping is a function wi...
elmapfun 8926 A mapping is always a func...
elmapssres 8927 A restricted mapping is a ...
fpmg 8928 A total function is a part...
pmss12g 8929 Subset relation for the se...
pmresg 8930 Elementhood of a restricte...
elmap 8931 Membership relation for se...
mapval2 8932 Alternate expression for t...
elpm 8933 The predicate "is a partia...
elpm2 8934 The predicate "is a partia...
fpm 8935 A total function is a part...
mapsspm 8936 Set exponentiation is a su...
pmsspw 8937 Partial maps are a subset ...
mapsspw 8938 Set exponentiation is a su...
mapfvd 8939 The value of a function th...
elmapresaun 8940 ~ fresaun transposed to ma...
fvmptmap 8941 Special case of ~ fvmpt fo...
map0e 8942 Set exponentiation with an...
map0b 8943 Set exponentiation with an...
map0g 8944 Set exponentiation is empt...
0map0sn0 8945 The set of mappings of the...
mapsnd 8946 The value of set exponenti...
map0 8947 Set exponentiation is empt...
mapsn 8948 The value of set exponenti...
mapss 8949 Subset inheritance for set...
fdiagfn 8950 Functionality of the diago...
fvdiagfn 8951 Functionality of the diago...
mapsnconst 8952 Every singleton map is a c...
mapsncnv 8953 Expression for the inverse...
mapsnf1o2 8954 Explicit bijection between...
mapsnf1o3 8955 Explicit bijection in the ...
ralxpmap 8956 Quantification over functi...
dfixp 8959 Eliminate the expression `...
ixpsnval 8960 The value of an infinite C...
elixp2 8961 Membership in an infinite ...
fvixp 8962 Projection of a factor of ...
ixpfn 8963 A nuple is a function. (C...
elixp 8964 Membership in an infinite ...
elixpconst 8965 Membership in an infinite ...
ixpconstg 8966 Infinite Cartesian product...
ixpconst 8967 Infinite Cartesian product...
ixpeq1 8968 Equality theorem for infin...
ixpeq1d 8969 Equality theorem for infin...
ss2ixp 8970 Subclass theorem for infin...
ixpeq2 8971 Equality theorem for infin...
ixpeq2dva 8972 Equality theorem for infin...
ixpeq2dv 8973 Equality theorem for infin...
cbvixp 8974 Change bound variable in a...
cbvixpv 8975 Change bound variable in a...
nfixpw 8976 Bound-variable hypothesis ...
nfixp 8977 Bound-variable hypothesis ...
nfixp1 8978 The index variable in an i...
ixpprc 8979 A cartesian product of pro...
ixpf 8980 A member of an infinite Ca...
uniixp 8981 The union of an infinite C...
ixpexg 8982 The existence of an infini...
ixpin 8983 The intersection of two in...
ixpiin 8984 The indexed intersection o...
ixpint 8985 The intersection of a coll...
ixp0x 8986 An infinite Cartesian prod...
ixpssmap2g 8987 An infinite Cartesian prod...
ixpssmapg 8988 An infinite Cartesian prod...
0elixp 8989 Membership of the empty se...
ixpn0 8990 The infinite Cartesian pro...
ixp0 8991 The infinite Cartesian pro...
ixpssmap 8992 An infinite Cartesian prod...
resixp 8993 Restriction of an element ...
undifixp 8994 Union of two projections o...
mptelixpg 8995 Condition for an explicit ...
resixpfo 8996 Restriction of elements of...
elixpsn 8997 Membership in a class of s...
ixpsnf1o 8998 A bijection between a clas...
mapsnf1o 8999 A bijection between a set ...
boxriin 9000 A rectangular subset of a ...
boxcutc 9001 The relative complement of...
relen 9010 Equinumerosity is a relati...
reldom 9011 Dominance is a relation. ...
relsdom 9012 Strict dominance is a rela...
encv 9013 If two classes are equinum...
breng 9014 Equinumerosity relation. ...
bren 9015 Equinumerosity relation. ...
brenOLD 9016 Obsolete version of ~ bren...
brdom2g 9017 Dominance relation. This ...
brdomg 9018 Dominance relation. (Cont...
brdomgOLD 9019 Obsolete version of ~ brdo...
brdomi 9020 Dominance relation. (Cont...
brdomiOLD 9021 Obsolete version of ~ brdo...
brdom 9022 Dominance relation. (Cont...
domen 9023 Dominance in terms of equi...
domeng 9024 Dominance in terms of equi...
ctex 9025 A countable set is a set. ...
f1oen4g 9026 The domain and range of a ...
f1dom4g 9027 The domain of a one-to-one...
f1oen3g 9028 The domain and range of a ...
f1dom3g 9029 The domain of a one-to-one...
f1oen2g 9030 The domain and range of a ...
f1dom2g 9031 The domain of a one-to-one...
f1dom2gOLD 9032 Obsolete version of ~ f1do...
f1oeng 9033 The domain and range of a ...
f1domg 9034 The domain of a one-to-one...
f1oen 9035 The domain and range of a ...
f1dom 9036 The domain of a one-to-one...
brsdom 9037 Strict dominance relation,...
isfi 9038 Express " ` A ` is finite"...
enssdom 9039 Equinumerosity implies dom...
dfdom2 9040 Alternate definition of do...
endom 9041 Equinumerosity implies dom...
sdomdom 9042 Strict dominance implies d...
sdomnen 9043 Strict dominance implies n...
brdom2 9044 Dominance in terms of stri...
bren2 9045 Equinumerosity expressed i...
enrefg 9046 Equinumerosity is reflexiv...
enref 9047 Equinumerosity is reflexiv...
eqeng 9048 Equality implies equinumer...
domrefg 9049 Dominance is reflexive. (...
en2d 9050 Equinumerosity inference f...
en3d 9051 Equinumerosity inference f...
en2i 9052 Equinumerosity inference f...
en3i 9053 Equinumerosity inference f...
dom2lem 9054 A mapping (first hypothesi...
dom2d 9055 A mapping (first hypothesi...
dom3d 9056 A mapping (first hypothesi...
dom2 9057 A mapping (first hypothesi...
dom3 9058 A mapping (first hypothesi...
idssen 9059 Equality implies equinumer...
domssl 9060 If ` A ` is a subset of ` ...
domssr 9061 If ` C ` is a superset of ...
ssdomg 9062 A set dominates its subset...
ener 9063 Equinumerosity is an equiv...
ensymb 9064 Symmetry of equinumerosity...
ensym 9065 Symmetry of equinumerosity...
ensymi 9066 Symmetry of equinumerosity...
ensymd 9067 Symmetry of equinumerosity...
entr 9068 Transitivity of equinumero...
domtr 9069 Transitivity of dominance ...
entri 9070 A chained equinumerosity i...
entr2i 9071 A chained equinumerosity i...
entr3i 9072 A chained equinumerosity i...
entr4i 9073 A chained equinumerosity i...
endomtr 9074 Transitivity of equinumero...
domentr 9075 Transitivity of dominance ...
f1imaeng 9076 If a function is one-to-on...
f1imaen2g 9077 If a function is one-to-on...
f1imaen3g 9078 If a set function is one-t...
f1imaen 9079 If a function is one-to-on...
en0 9080 The empty set is equinumer...
en0OLD 9081 Obsolete version of ~ en0 ...
en0ALT 9082 Shorter proof of ~ en0 , d...
en0r 9083 The empty set is equinumer...
ensn1 9084 A singleton is equinumerou...
ensn1OLD 9085 Obsolete version of ~ ensn...
ensn1g 9086 A singleton is equinumerou...
enpr1g 9087 ` { A , A } ` has only one...
en1 9088 A set is equinumerous to o...
en1OLD 9089 Obsolete version of ~ en1 ...
en1b 9090 A set is equinumerous to o...
en1bOLD 9091 Obsolete version of ~ en1b...
reuen1 9092 Two ways to express "exact...
euen1 9093 Two ways to express "exact...
euen1b 9094 Two ways to express " ` A ...
en1uniel 9095 A singleton contains its s...
en1unielOLD 9096 Obsolete version of ~ en1u...
2dom 9097 A set that dominates ordin...
fundmen 9098 A function is equinumerous...
fundmeng 9099 A function is equinumerous...
cnven 9100 A relational set is equinu...
cnvct 9101 If a set is countable, so ...
fndmeng 9102 A function is equinumerate...
mapsnend 9103 Set exponentiation to a si...
mapsnen 9104 Set exponentiation to a si...
snmapen 9105 Set exponentiation: a sing...
snmapen1 9106 Set exponentiation: a sing...
map1 9107 Set exponentiation: ordina...
en2sn 9108 Two singletons are equinum...
en2snOLD 9109 Obsolete version of ~ en2s...
0fi 9110 The empty set is finite. ...
snfi 9111 A singleton is finite. (C...
snfiOLD 9112 Obsolete version of ~ snfi...
fiprc 9113 The class of finite sets i...
unen 9114 Equinumerosity of union of...
enrefnn 9115 Equinumerosity is reflexiv...
en2prd 9116 Two unordered pairs are eq...
enpr2d 9117 A pair with distinct eleme...
enpr2dOLD 9118 Obsolete version of ~ enpr...
ssct 9119 Any subset of a countable ...
ssctOLD 9120 Obsolete version of ~ ssct...
difsnen 9121 All decrements of a set ar...
domdifsn 9122 Dominance over a set with ...
xpsnen 9123 A set is equinumerous to i...
xpsneng 9124 A set is equinumerous to i...
xp1en 9125 One times a cardinal numbe...
endisj 9126 Any two sets are equinumer...
undom 9127 Dominance law for union. ...
undomOLD 9128 Obsolete version of ~ undo...
xpcomf1o 9129 The canonical bijection fr...
xpcomco 9130 Composition with the bijec...
xpcomen 9131 Commutative law for equinu...
xpcomeng 9132 Commutative law for equinu...
xpsnen2g 9133 A set is equinumerous to i...
xpassen 9134 Associative law for equinu...
xpdom2 9135 Dominance law for Cartesia...
xpdom2g 9136 Dominance law for Cartesia...
xpdom1g 9137 Dominance law for Cartesia...
xpdom3 9138 A set is dominated by its ...
xpdom1 9139 Dominance law for Cartesia...
domunsncan 9140 A singleton cancellation l...
omxpenlem 9141 Lemma for ~ omxpen . (Con...
omxpen 9142 The cardinal and ordinal p...
omf1o 9143 Construct an explicit bije...
pw2f1olem 9144 Lemma for ~ pw2f1o . (Con...
pw2f1o 9145 The power set of a set is ...
pw2eng 9146 The power set of a set is ...
pw2en 9147 The power set of a set is ...
fopwdom 9148 Covering implies injection...
enfixsn 9149 Given two equipollent sets...
sucdom2OLD 9150 Obsolete version of ~ sucd...
sbthlem1 9151 Lemma for ~ sbth . (Contr...
sbthlem2 9152 Lemma for ~ sbth . (Contr...
sbthlem3 9153 Lemma for ~ sbth . (Contr...
sbthlem4 9154 Lemma for ~ sbth . (Contr...
sbthlem5 9155 Lemma for ~ sbth . (Contr...
sbthlem6 9156 Lemma for ~ sbth . (Contr...
sbthlem7 9157 Lemma for ~ sbth . (Contr...
sbthlem8 9158 Lemma for ~ sbth . (Contr...
sbthlem9 9159 Lemma for ~ sbth . (Contr...
sbthlem10 9160 Lemma for ~ sbth . (Contr...
sbth 9161 Schroeder-Bernstein Theore...
sbthb 9162 Schroeder-Bernstein Theore...
sbthcl 9163 Schroeder-Bernstein Theore...
dfsdom2 9164 Alternate definition of st...
brsdom2 9165 Alternate definition of st...
sdomnsym 9166 Strict dominance is asymme...
domnsym 9167 Theorem 22(i) of [Suppes] ...
0domg 9168 Any set dominates the empt...
0domgOLD 9169 Obsolete version of ~ 0dom...
dom0 9170 A set dominated by the emp...
dom0OLD 9171 Obsolete version of ~ dom0...
0sdomg 9172 A set strictly dominates t...
0sdomgOLD 9173 Obsolete version of ~ 0sdo...
0dom 9174 Any set dominates the empt...
0sdom 9175 A set strictly dominates t...
sdom0 9176 The empty set does not str...
sdom0OLD 9177 Obsolete version of ~ sdom...
sdomdomtr 9178 Transitivity of strict dom...
sdomentr 9179 Transitivity of strict dom...
domsdomtr 9180 Transitivity of dominance ...
ensdomtr 9181 Transitivity of equinumero...
sdomirr 9182 Strict dominance is irrefl...
sdomtr 9183 Strict dominance is transi...
sdomn2lp 9184 Strict dominance has no 2-...
enen1 9185 Equality-like theorem for ...
enen2 9186 Equality-like theorem for ...
domen1 9187 Equality-like theorem for ...
domen2 9188 Equality-like theorem for ...
sdomen1 9189 Equality-like theorem for ...
sdomen2 9190 Equality-like theorem for ...
domtriord 9191 Dominance is trichotomous ...
sdomel 9192 For ordinals, strict domin...
sdomdif 9193 The difference of a set fr...
onsdominel 9194 An ordinal with more eleme...
domunsn 9195 Dominance over a set with ...
fodomr 9196 There exists a mapping fro...
pwdom 9197 Injection of sets implies ...
canth2 9198 Cantor's Theorem. No set ...
canth2g 9199 Cantor's theorem with the ...
2pwuninel 9200 The power set of the power...
2pwne 9201 No set equals the power se...
disjen 9202 A stronger form of ~ pwuni...
disjenex 9203 Existence version of ~ dis...
domss2 9204 A corollary of ~ disjenex ...
domssex2 9205 A corollary of ~ disjenex ...
domssex 9206 Weakening of ~ domssex2 to...
xpf1o 9207 Construct a bijection on a...
xpen 9208 Equinumerosity law for Car...
mapen 9209 Two set exponentiations ar...
mapdom1 9210 Order-preserving property ...
mapxpen 9211 Equinumerosity law for dou...
xpmapenlem 9212 Lemma for ~ xpmapen . (Co...
xpmapen 9213 Equinumerosity law for set...
mapunen 9214 Equinumerosity law for set...
map2xp 9215 A cardinal power with expo...
mapdom2 9216 Order-preserving property ...
mapdom3 9217 Set exponentiation dominat...
pwen 9218 If two sets are equinumero...
ssenen 9219 Equinumerosity of equinume...
limenpsi 9220 A limit ordinal is equinum...
limensuci 9221 A limit ordinal is equinum...
limensuc 9222 A limit ordinal is equinum...
infensuc 9223 Any infinite ordinal is eq...
dif1enlem 9224 Lemma for ~ rexdif1en and ...
dif1enlemOLD 9225 Obsolete version of ~ dif1...
rexdif1en 9226 If a set is equinumerous t...
rexdif1enOLD 9227 Obsolete version of ~ rexd...
dif1en 9228 If a set ` A ` is equinume...
dif1ennn 9229 If a set ` A ` is equinume...
dif1enOLD 9230 Obsolete version of ~ dif1...
findcard 9231 Schema for induction on th...
findcard2 9232 Schema for induction on th...
findcard2s 9233 Variation of ~ findcard2 r...
findcard2d 9234 Deduction version of ~ fin...
nnfi 9235 Natural numbers are finite...
pssnn 9236 A proper subset of a natur...
ssnnfi 9237 A subset of a natural numb...
ssnnfiOLD 9238 Obsolete version of ~ ssnn...
0finOLD 9239 Obsolete version of ~ 0fi ...
unfi 9240 The union of two finite se...
unfid 9241 The union of two finite se...
ssfi 9242 A subset of a finite set i...
ssfiALT 9243 Shorter proof of ~ ssfi us...
diffi 9244 If ` A ` is finite, ` ( A ...
cnvfi 9245 If a set is finite, its co...
fnfi 9246 A version of ~ fnex for fi...
f1oenfi 9247 If the domain of a one-to-...
f1oenfirn 9248 If the range of a one-to-o...
f1domfi 9249 If the codomain of a one-t...
f1domfi2 9250 If the domain of a one-to-...
enreffi 9251 Equinumerosity is reflexiv...
ensymfib 9252 Symmetry of equinumerosity...
entrfil 9253 Transitivity of equinumero...
enfii 9254 A set equinumerous to a fi...
enfi 9255 Equinumerous sets have the...
enfiALT 9256 Shorter proof of ~ enfi us...
domfi 9257 A set dominated by a finit...
entrfi 9258 Transitivity of equinumero...
entrfir 9259 Transitivity of equinumero...
domtrfil 9260 Transitivity of dominance ...
domtrfi 9261 Transitivity of dominance ...
domtrfir 9262 Transitivity of dominance ...
f1imaenfi 9263 If a function is one-to-on...
ssdomfi 9264 A finite set dominates its...
ssdomfi2 9265 A set dominates its finite...
sbthfilem 9266 Lemma for ~ sbthfi . (Con...
sbthfi 9267 Schroeder-Bernstein Theore...
domnsymfi 9268 If a set dominates a finit...
sdomdomtrfi 9269 Transitivity of strict dom...
domsdomtrfi 9270 Transitivity of dominance ...
sucdom2 9271 Strict dominance of a set ...
phplem1 9272 Lemma for Pigeonhole Princ...
phplem2 9273 Lemma for Pigeonhole Princ...
nneneq 9274 Two equinumerous natural n...
php 9275 Pigeonhole Principle. A n...
php2 9276 Corollary of Pigeonhole Pr...
php3 9277 Corollary of Pigeonhole Pr...
php4 9278 Corollary of the Pigeonhol...
php5 9279 Corollary of the Pigeonhol...
phpeqd 9280 Corollary of the Pigeonhol...
nndomog 9281 Cardinal ordering agrees w...
phplem1OLD 9282 Obsolete lemma for ~ php a...
phplem2OLD 9283 Obsolete lemma for ~ php a...
phplem3OLD 9284 Obsolete version of ~ phpl...
phplem4OLD 9285 Obsolete version of ~ phpl...
nneneqOLD 9286 Obsolete version of ~ nnen...
phpOLD 9287 Obsolete version of ~ php ...
php2OLD 9288 Obsolete version of ~ php2...
php3OLD 9289 Obsolete version of ~ php3...
phpeqdOLD 9290 Obsolete version of ~ phpe...
nndomogOLD 9291 Obsolete version of ~ nndo...
snnen2oOLD 9292 Obsolete version of ~ snne...
onomeneq 9293 An ordinal number equinume...
onomeneqOLD 9294 Obsolete version of ~ onom...
onfin 9295 An ordinal number is finit...
onfin2 9296 A set is a natural number ...
nnfiOLD 9297 Obsolete version of ~ nnfi...
nndomo 9298 Cardinal ordering agrees w...
nnsdomo 9299 Cardinal ordering agrees w...
sucdom 9300 Strict dominance of a set ...
sucdomOLD 9301 Obsolete version of ~ sucd...
snnen2o 9302 A singleton ` { A } ` is n...
0sdom1dom 9303 Strict dominance over 0 is...
0sdom1domALT 9304 Alternate proof of ~ 0sdom...
1sdom2 9305 Ordinal 1 is strictly domi...
1sdom2ALT 9306 Alternate proof of ~ 1sdom...
sdom1 9307 A set has less than one me...
sdom1OLD 9308 Obsolete version of ~ sdom...
modom 9309 Two ways to express "at mo...
modom2 9310 Two ways to express "at mo...
rex2dom 9311 A set that has at least 2 ...
1sdom2dom 9312 Strict dominance over 1 is...
1sdom 9313 A set that strictly domina...
1sdomOLD 9314 Obsolete version of ~ 1sdo...
unxpdomlem1 9315 Lemma for ~ unxpdom . (Tr...
unxpdomlem2 9316 Lemma for ~ unxpdom . (Co...
unxpdomlem3 9317 Lemma for ~ unxpdom . (Co...
unxpdom 9318 Cartesian product dominate...
unxpdom2 9319 Corollary of ~ unxpdom . ...
sucxpdom 9320 Cartesian product dominate...
pssinf 9321 A set equinumerous to a pr...
fisseneq 9322 A finite set is equal to i...
ominf 9323 The set of natural numbers...
ominfOLD 9324 Obsolete version of ~ omin...
isinf 9325 Any set that is not finite...
isinfOLD 9326 Obsolete version of ~ isin...
fineqvlem 9327 Lemma for ~ fineqv . (Con...
fineqv 9328 If the Axiom of Infinity i...
enfiiOLD 9329 Obsolete version of ~ enfi...
xpfir 9330 The components of a nonemp...
ssfid 9331 A subset of a finite set i...
infi 9332 The intersection of two se...
rabfi 9333 A restricted class built f...
finresfin 9334 The restriction of a finit...
f1finf1o 9335 Any injection from one fin...
f1finf1oOLD 9336 Obsolete version of ~ f1fi...
nfielex 9337 If a class is not finite, ...
en1eqsn 9338 A set with one element is ...
en1eqsnOLD 9339 Obsolete version of ~ en1e...
en1eqsnbi 9340 A set containing an elemen...
dif1ennnALT 9341 Alternate proof of ~ dif1e...
enp1ilem 9342 Lemma for uses of ~ enp1i ...
enp1i 9343 Proof induction for ~ en2 ...
enp1iOLD 9344 Obsolete version of ~ enp1...
en2 9345 A set equinumerous to ordi...
en3 9346 A set equinumerous to ordi...
en4 9347 A set equinumerous to ordi...
findcard3 9348 Schema for strong inductio...
findcard3OLD 9349 Obsolete version of ~ find...
ac6sfi 9350 A version of ~ ac6s for fi...
frfi 9351 A partial order is well-fo...
fimax2g 9352 A finite set has a maximum...
fimaxg 9353 A finite set has a maximum...
fisupg 9354 Lemma showing existence an...
wofi 9355 A total order on a finite ...
ordunifi 9356 The maximum of a finite co...
nnunifi 9357 The union (supremum) of a ...
unblem1 9358 Lemma for ~ unbnn . After...
unblem2 9359 Lemma for ~ unbnn . The v...
unblem3 9360 Lemma for ~ unbnn . The v...
unblem4 9361 Lemma for ~ unbnn . The f...
unbnn 9362 Any unbounded subset of na...
unbnn2 9363 Version of ~ unbnn that do...
isfinite2 9364 Any set strictly dominated...
nnsdomg 9365 Omega strictly dominates a...
nnsdomgOLD 9366 Obsolete version of ~ nnsd...
isfiniteg 9367 A set is finite iff it is ...
infsdomnn 9368 An infinite set strictly d...
infsdomnnOLD 9369 Obsolete version of ~ infs...
infn0 9370 An infinite set is not emp...
infn0ALT 9371 Shorter proof of ~ infn0 u...
fin2inf 9372 This (useless) theorem, wh...
unfilem1 9373 Lemma for proving that the...
unfilem2 9374 Lemma for proving that the...
unfilem3 9375 Lemma for proving that the...
unfir 9376 If a union is finite, the ...
unfib 9377 A union is finite if and o...
unfi2 9378 The union of two finite se...
difinf 9379 An infinite set ` A ` minu...
fodomfi 9380 An onto function implies d...
fofi 9381 If an onto function has a ...
f1fi 9382 If a 1-to-1 function has a...
imafi 9383 Images of finite sets are ...
imafiOLD 9384 Obsolete version of ~ imaf...
pwfir 9385 If the power set of a set ...
pwfilem 9386 Lemma for ~ pwfi . (Contr...
pwfi 9387 The power set of a finite ...
xpfi 9388 The Cartesian product of t...
xpfiOLD 9389 Obsolete version of ~ xpfi...
3xpfi 9390 The Cartesian product of t...
domunfican 9391 A finite set union cancell...
infcntss 9392 Every infinite set has a d...
prfi 9393 An unordered pair is finit...
prfiALT 9394 Shorter proof of ~ prfi us...
tpfi 9395 An unordered triple is fin...
fiint 9396 Equivalent ways of stating...
fiintOLD 9397 Obsolete version of ~ fiin...
fodomfir 9398 There exists a mapping fro...
fodomfib 9399 Equivalence of an onto map...
fodomfiOLD 9400 Obsolete version of ~ fodo...
fodomfibOLD 9401 Obsolete version of ~ fodo...
fofinf1o 9402 Any surjection from one fi...
rneqdmfinf1o 9403 Any function from a finite...
fidomdm 9404 Any finite set dominates i...
dmfi 9405 The domain of a finite set...
fundmfibi 9406 A function is finite if an...
resfnfinfin 9407 The restriction of a funct...
residfi 9408 A restricted identity func...
cnvfiALT 9409 Shorter proof of ~ cnvfi u...
rnfi 9410 The range of a finite set ...
f1dmvrnfibi 9411 A one-to-one function whos...
f1vrnfibi 9412 A one-to-one function whic...
iunfi 9413 The finite union of finite...
unifi 9414 The finite union of finite...
unifi2 9415 The finite union of finite...
infssuni 9416 If an infinite set ` A ` i...
unirnffid 9417 The union of the range of ...
pwfilemOLD 9418 Obsolete version of ~ pwfi...
pwfiOLD 9419 Obsolete version of ~ pwfi...
mapfi 9420 Set exponentiation of fini...
ixpfi 9421 A Cartesian product of fin...
ixpfi2 9422 A Cartesian product of fin...
mptfi 9423 A finite mapping set is fi...
abrexfi 9424 An image set from a finite...
cnvimamptfin 9425 A preimage of a mapping wi...
elfpw 9426 Membership in a class of f...
unifpw 9427 A set is the union of its ...
f1opwfi 9428 A one-to-one mapping induc...
fissuni 9429 A finite subset of a union...
fipreima 9430 Given a finite subset ` A ...
finsschain 9431 A finite subset of the uni...
indexfi 9432 If for every element of a ...
relfsupp 9435 The property of a function...
relprcnfsupp 9436 A proper class is never fi...
isfsupp 9437 The property of a class to...
isfsuppd 9438 Deduction form of ~ isfsup...
funisfsupp 9439 The property of a function...
fsuppimp 9440 Implications of a class be...
fsuppimpd 9441 A finitely supported funct...
fsuppfund 9442 A finitely supported funct...
fisuppfi 9443 A function on a finite set...
fidmfisupp 9444 A function with a finite d...
fdmfisuppfi 9445 The support of a function ...
fdmfifsupp 9446 A function with a finite d...
fsuppmptdm 9447 A mapping with a finite do...
fndmfisuppfi 9448 The support of a function ...
fndmfifsupp 9449 A function with a finite d...
suppeqfsuppbi 9450 If two functions have the ...
suppssfifsupp 9451 If the support of a functi...
fsuppsssupp 9452 If the support of a functi...
fsuppsssuppgd 9453 If the support of a functi...
fsuppss 9454 A subset of a finitely sup...
fsuppssov1 9455 Formula building theorem f...
fsuppxpfi 9456 The cartesian product of t...
fczfsuppd 9457 A constant function with v...
fsuppun 9458 The union of two finitely ...
fsuppunfi 9459 The union of the support o...
fsuppunbi 9460 If the union of two classe...
0fsupp 9461 The empty set is a finitel...
snopfsupp 9462 A singleton containing an ...
funsnfsupp 9463 Finite support for a funct...
fsuppres 9464 The restriction of a finit...
fmptssfisupp 9465 The restriction of a mappi...
ressuppfi 9466 If the support of the rest...
resfsupp 9467 If the restriction of a fu...
resfifsupp 9468 The restriction of a funct...
ffsuppbi 9469 Two ways of saying that a ...
fsuppmptif 9470 A function mapping an argu...
sniffsupp 9471 A function mapping all but...
fsuppcolem 9472 Lemma for ~ fsuppco . For...
fsuppco 9473 The composition of a 1-1 f...
fsuppco2 9474 The composition of a funct...
fsuppcor 9475 The composition of a funct...
mapfienlem1 9476 Lemma 1 for ~ mapfien . (...
mapfienlem2 9477 Lemma 2 for ~ mapfien . (...
mapfienlem3 9478 Lemma 3 for ~ mapfien . (...
mapfien 9479 A bijection of the base se...
mapfien2 9480 Equinumerousity relation f...
fival 9483 The set of all the finite ...
elfi 9484 Specific properties of an ...
elfi2 9485 The empty intersection nee...
elfir 9486 Sufficient condition for a...
intrnfi 9487 Sufficient condition for t...
iinfi 9488 An indexed intersection of...
inelfi 9489 The intersection of two se...
ssfii 9490 Any element of a set ` A `...
fi0 9491 The set of finite intersec...
fieq0 9492 A set is empty iff the cla...
fiin 9493 The elements of ` ( fi `` ...
dffi2 9494 The set of finite intersec...
fiss 9495 Subset relationship for fu...
inficl 9496 A set which is closed unde...
fipwuni 9497 The set of finite intersec...
fisn 9498 A singleton is closed unde...
fiuni 9499 The union of the finite in...
fipwss 9500 If a set is a family of su...
elfiun 9501 A finite intersection of e...
dffi3 9502 The set of finite intersec...
fifo 9503 Describe a surjection from...
marypha1lem 9504 Core induction for Philip ...
marypha1 9505 (Philip) Hall's marriage t...
marypha2lem1 9506 Lemma for ~ marypha2 . Pr...
marypha2lem2 9507 Lemma for ~ marypha2 . Pr...
marypha2lem3 9508 Lemma for ~ marypha2 . Pr...
marypha2lem4 9509 Lemma for ~ marypha2 . Pr...
marypha2 9510 Version of ~ marypha1 usin...
dfsup2 9515 Quantifier-free definition...
supeq1 9516 Equality theorem for supre...
supeq1d 9517 Equality deduction for sup...
supeq1i 9518 Equality inference for sup...
supeq2 9519 Equality theorem for supre...
supeq3 9520 Equality theorem for supre...
supeq123d 9521 Equality deduction for sup...
nfsup 9522 Hypothesis builder for sup...
supmo 9523 Any class ` B ` has at mos...
supexd 9524 A supremum is a set. (Con...
supeu 9525 A supremum is unique. Sim...
supval2 9526 Alternate expression for t...
eqsup 9527 Sufficient condition for a...
eqsupd 9528 Sufficient condition for a...
supcl 9529 A supremum belongs to its ...
supub 9530 A supremum is an upper bou...
suplub 9531 A supremum is the least up...
suplub2 9532 Bidirectional form of ~ su...
supnub 9533 An upper bound is not less...
supex 9534 A supremum is a set. (Con...
sup00 9535 The supremum under an empt...
sup0riota 9536 The supremum of an empty s...
sup0 9537 The supremum of an empty s...
supmax 9538 The greatest element of a ...
fisup2g 9539 A finite set satisfies the...
fisupcl 9540 A nonempty finite set cont...
supgtoreq 9541 The supremum of a finite s...
suppr 9542 The supremum of a pair. (...
supsn 9543 The supremum of a singleto...
supisolem 9544 Lemma for ~ supiso . (Con...
supisoex 9545 Lemma for ~ supiso . (Con...
supiso 9546 Image of a supremum under ...
infeq1 9547 Equality theorem for infim...
infeq1d 9548 Equality deduction for inf...
infeq1i 9549 Equality inference for inf...
infeq2 9550 Equality theorem for infim...
infeq3 9551 Equality theorem for infim...
infeq123d 9552 Equality deduction for inf...
nfinf 9553 Hypothesis builder for inf...
infexd 9554 An infimum is a set. (Con...
eqinf 9555 Sufficient condition for a...
eqinfd 9556 Sufficient condition for a...
infval 9557 Alternate expression for t...
infcllem 9558 Lemma for ~ infcl , ~ infl...
infcl 9559 An infimum belongs to its ...
inflb 9560 An infimum is a lower boun...
infglb 9561 An infimum is the greatest...
infglbb 9562 Bidirectional form of ~ in...
infnlb 9563 A lower bound is not great...
infex 9564 An infimum is a set. (Con...
infmin 9565 The smallest element of a ...
infmo 9566 Any class ` B ` has at mos...
infeu 9567 An infimum is unique. (Co...
fimin2g 9568 A finite set has a minimum...
fiming 9569 A finite set has a minimum...
fiinfg 9570 Lemma showing existence an...
fiinf2g 9571 A finite set satisfies the...
fiinfcl 9572 A nonempty finite set cont...
infltoreq 9573 The infimum of a finite se...
infpr 9574 The infimum of a pair. (C...
infsupprpr 9575 The infimum of a proper pa...
infsn 9576 The infimum of a singleton...
inf00 9577 The infimum regarding an e...
infempty 9578 The infimum of an empty se...
infiso 9579 Image of an infimum under ...
dfoi 9582 Rewrite ~ df-oi with abbre...
oieq1 9583 Equality theorem for ordin...
oieq2 9584 Equality theorem for ordin...
nfoi 9585 Hypothesis builder for ord...
ordiso2 9586 Generalize ~ ordiso to pro...
ordiso 9587 Order-isomorphic ordinal n...
ordtypecbv 9588 Lemma for ~ ordtype . (Co...
ordtypelem1 9589 Lemma for ~ ordtype . (Co...
ordtypelem2 9590 Lemma for ~ ordtype . (Co...
ordtypelem3 9591 Lemma for ~ ordtype . (Co...
ordtypelem4 9592 Lemma for ~ ordtype . (Co...
ordtypelem5 9593 Lemma for ~ ordtype . (Co...
ordtypelem6 9594 Lemma for ~ ordtype . (Co...
ordtypelem7 9595 Lemma for ~ ordtype . ` ra...
ordtypelem8 9596 Lemma for ~ ordtype . (Co...
ordtypelem9 9597 Lemma for ~ ordtype . Eit...
ordtypelem10 9598 Lemma for ~ ordtype . Usi...
oi0 9599 Definition of the ordinal ...
oicl 9600 The order type of the well...
oif 9601 The order isomorphism of t...
oiiso2 9602 The order isomorphism of t...
ordtype 9603 For any set-like well-orde...
oiiniseg 9604 ` ran F ` is an initial se...
ordtype2 9605 For any set-like well-orde...
oiexg 9606 The order isomorphism on a...
oion 9607 The order type of the well...
oiiso 9608 The order isomorphism of t...
oien 9609 The order type of a well-o...
oieu 9610 Uniqueness of the unique o...
oismo 9611 When ` A ` is a subclass o...
oiid 9612 The order type of an ordin...
hartogslem1 9613 Lemma for ~ hartogs . (Co...
hartogslem2 9614 Lemma for ~ hartogs . (Co...
hartogs 9615 The class of ordinals domi...
wofib 9616 The only sets which are we...
wemaplem1 9617 Value of the lexicographic...
wemaplem2 9618 Lemma for ~ wemapso . Tra...
wemaplem3 9619 Lemma for ~ wemapso . Tra...
wemappo 9620 Construct lexicographic or...
wemapsolem 9621 Lemma for ~ wemapso . (Co...
wemapso 9622 Construct lexicographic or...
wemapso2lem 9623 Lemma for ~ wemapso2 . (C...
wemapso2 9624 An alternative to having a...
card2on 9625 The alternate definition o...
card2inf 9626 The alternate definition o...
harf 9629 Functionality of the Harto...
harcl 9630 Values of the Hartogs func...
harval 9631 Function value of the Hart...
elharval 9632 The Hartogs number of a se...
harndom 9633 The Hartogs number of a se...
harword 9634 Weak ordering property of ...
relwdom 9637 Weak dominance is a relati...
brwdom 9638 Property of weak dominance...
brwdomi 9639 Property of weak dominance...
brwdomn0 9640 Weak dominance over nonemp...
0wdom 9641 Any set weakly dominates t...
fowdom 9642 An onto function implies w...
wdomref 9643 Reflexivity of weak domina...
brwdom2 9644 Alternate characterization...
domwdom 9645 Weak dominance is implied ...
wdomtr 9646 Transitivity of weak domin...
wdomen1 9647 Equality-like theorem for ...
wdomen2 9648 Equality-like theorem for ...
wdompwdom 9649 Weak dominance strengthens...
canthwdom 9650 Cantor's Theorem, stated u...
wdom2d 9651 Deduce weak dominance from...
wdomd 9652 Deduce weak dominance from...
brwdom3 9653 Condition for weak dominan...
brwdom3i 9654 Weak dominance implies exi...
unwdomg 9655 Weak dominance of a (disjo...
xpwdomg 9656 Weak dominance of a Cartes...
wdomima2g 9657 A set is weakly dominant o...
wdomimag 9658 A set is weakly dominant o...
unxpwdom2 9659 Lemma for ~ unxpwdom . (C...
unxpwdom 9660 If a Cartesian product is ...
ixpiunwdom 9661 Describe an onto function ...
harwdom 9662 The value of the Hartogs f...
axreg2 9664 Axiom of Regularity expres...
zfregcl 9665 The Axiom of Regularity wi...
zfreg 9666 The Axiom of Regularity us...
elirrv 9667 The membership relation is...
elirr 9668 No class is a member of it...
elneq 9669 A class is not equal to an...
nelaneq 9670 A class is not an element ...
epinid0 9671 The membership relation an...
sucprcreg 9672 A class is equal to its su...
ruv 9673 The Russell class is equal...
ruALT 9674 Alternate proof of ~ ru , ...
disjcsn 9675 A class is disjoint from i...
zfregfr 9676 The membership relation is...
en2lp 9677 No class has 2-cycle membe...
elnanel 9678 Two classes are not elemen...
cnvepnep 9679 The membership (epsilon) r...
epnsym 9680 The membership (epsilon) r...
elnotel 9681 A class cannot be an eleme...
elnel 9682 A class cannot be an eleme...
en3lplem1 9683 Lemma for ~ en3lp . (Cont...
en3lplem2 9684 Lemma for ~ en3lp . (Cont...
en3lp 9685 No class has 3-cycle membe...
preleqg 9686 Equality of two unordered ...
preleq 9687 Equality of two unordered ...
preleqALT 9688 Alternate proof of ~ prele...
opthreg 9689 Theorem for alternate repr...
suc11reg 9690 The successor operation be...
dford2 9691 Assuming ~ ax-reg , an ord...
inf0 9692 Existence of ` _om ` impli...
inf1 9693 Variation of Axiom of Infi...
inf2 9694 Variation of Axiom of Infi...
inf3lema 9695 Lemma for our Axiom of Inf...
inf3lemb 9696 Lemma for our Axiom of Inf...
inf3lemc 9697 Lemma for our Axiom of Inf...
inf3lemd 9698 Lemma for our Axiom of Inf...
inf3lem1 9699 Lemma for our Axiom of Inf...
inf3lem2 9700 Lemma for our Axiom of Inf...
inf3lem3 9701 Lemma for our Axiom of Inf...
inf3lem4 9702 Lemma for our Axiom of Inf...
inf3lem5 9703 Lemma for our Axiom of Inf...
inf3lem6 9704 Lemma for our Axiom of Inf...
inf3lem7 9705 Lemma for our Axiom of Inf...
inf3 9706 Our Axiom of Infinity ~ ax...
infeq5i 9707 Half of ~ infeq5 . (Contr...
infeq5 9708 The statement "there exist...
zfinf 9710 Axiom of Infinity expresse...
axinf2 9711 A standard version of Axio...
zfinf2 9713 A standard version of the ...
omex 9714 The existence of omega (th...
axinf 9715 The first version of the A...
inf5 9716 The statement "there exist...
omelon 9717 Omega is an ordinal number...
dfom3 9718 The class of natural numbe...
elom3 9719 A simplification of ~ elom...
dfom4 9720 A simplification of ~ df-o...
dfom5 9721 ` _om ` is the smallest li...
oancom 9722 Ordinal addition is not co...
isfinite 9723 A set is finite iff it is ...
fict 9724 A finite set is countable ...
nnsdom 9725 A natural number is strict...
omenps 9726 Omega is equinumerous to a...
omensuc 9727 The set of natural numbers...
infdifsn 9728 Removing a singleton from ...
infdiffi 9729 Removing a finite set from...
unbnn3 9730 Any unbounded subset of na...
noinfep 9731 Using the Axiom of Regular...
cantnffval 9734 The value of the Cantor no...
cantnfdm 9735 The domain of the Cantor n...
cantnfvalf 9736 Lemma for ~ cantnf . The ...
cantnfs 9737 Elementhood in the set of ...
cantnfcl 9738 Basic properties of the or...
cantnfval 9739 The value of the Cantor no...
cantnfval2 9740 Alternate expression for t...
cantnfsuc 9741 The value of the recursive...
cantnfle 9742 A lower bound on the ` CNF...
cantnflt 9743 An upper bound on the part...
cantnflt2 9744 An upper bound on the ` CN...
cantnff 9745 The ` CNF ` function is a ...
cantnf0 9746 The value of the zero func...
cantnfrescl 9747 A function is finitely sup...
cantnfres 9748 The ` CNF ` function respe...
cantnfp1lem1 9749 Lemma for ~ cantnfp1 . (C...
cantnfp1lem2 9750 Lemma for ~ cantnfp1 . (C...
cantnfp1lem3 9751 Lemma for ~ cantnfp1 . (C...
cantnfp1 9752 If ` F ` is created by add...
oemapso 9753 The relation ` T ` is a st...
oemapval 9754 Value of the relation ` T ...
oemapvali 9755 If ` F < G ` , then there ...
cantnflem1a 9756 Lemma for ~ cantnf . (Con...
cantnflem1b 9757 Lemma for ~ cantnf . (Con...
cantnflem1c 9758 Lemma for ~ cantnf . (Con...
cantnflem1d 9759 Lemma for ~ cantnf . (Con...
cantnflem1 9760 Lemma for ~ cantnf . This...
cantnflem2 9761 Lemma for ~ cantnf . (Con...
cantnflem3 9762 Lemma for ~ cantnf . Here...
cantnflem4 9763 Lemma for ~ cantnf . Comp...
cantnf 9764 The Cantor Normal Form the...
oemapwe 9765 The lexicographic order on...
cantnffval2 9766 An alternate definition of...
cantnff1o 9767 Simplify the isomorphism o...
wemapwe 9768 Construct lexicographic or...
oef1o 9769 A bijection of the base se...
cnfcomlem 9770 Lemma for ~ cnfcom . (Con...
cnfcom 9771 Any ordinal ` B ` is equin...
cnfcom2lem 9772 Lemma for ~ cnfcom2 . (Co...
cnfcom2 9773 Any nonzero ordinal ` B ` ...
cnfcom3lem 9774 Lemma for ~ cnfcom3 . (Co...
cnfcom3 9775 Any infinite ordinal ` B `...
cnfcom3clem 9776 Lemma for ~ cnfcom3c . (C...
cnfcom3c 9777 Wrap the construction of ~...
ttrcleq 9780 Equality theorem for trans...
nfttrcld 9781 Bound variable hypothesis ...
nfttrcl 9782 Bound variable hypothesis ...
relttrcl 9783 The transitive closure of ...
brttrcl 9784 Characterization of elemen...
brttrcl2 9785 Characterization of elemen...
ssttrcl 9786 If ` R ` is a relation, th...
ttrcltr 9787 The transitive closure of ...
ttrclresv 9788 The transitive closure of ...
ttrclco 9789 Composition law for the tr...
cottrcl 9790 Composition law for the tr...
ttrclss 9791 If ` R ` is a subclass of ...
dmttrcl 9792 The domain of a transitive...
rnttrcl 9793 The range of a transitive ...
ttrclexg 9794 If ` R ` is a set, then so...
dfttrcl2 9795 When ` R ` is a set and a ...
ttrclselem1 9796 Lemma for ~ ttrclse . Sho...
ttrclselem2 9797 Lemma for ~ ttrclse . Sho...
ttrclse 9798 If ` R ` is set-like over ...
trcl 9799 For any set ` A ` , show t...
tz9.1 9800 Every set has a transitive...
tz9.1c 9801 Alternate expression for t...
epfrs 9802 The strong form of the Axi...
zfregs 9803 The strong form of the Axi...
zfregs2 9804 Alternate strong form of t...
setind 9805 Set (epsilon) induction. ...
setind2 9806 Set (epsilon) induction, s...
tcvalg 9809 Value of the transitive cl...
tcid 9810 Defining property of the t...
tctr 9811 Defining property of the t...
tcmin 9812 Defining property of the t...
tc2 9813 A variant of the definitio...
tcsni 9814 The transitive closure of ...
tcss 9815 The transitive closure fun...
tcel 9816 The transitive closure fun...
tcidm 9817 The transitive closure fun...
tc0 9818 The transitive closure of ...
tc00 9819 The transitive closure is ...
frmin 9820 Every (possibly proper) su...
frind 9821 A subclass of a well-found...
frinsg 9822 Well-Founded Induction Sch...
frins 9823 Well-Founded Induction Sch...
frins2f 9824 Well-Founded Induction sch...
frins2 9825 Well-Founded Induction sch...
frins3 9826 Well-Founded Induction sch...
frr3g 9827 Functions defined by well-...
frrlem15 9828 Lemma for general well-fou...
frrlem16 9829 Lemma for general well-fou...
frr1 9830 Law of general well-founde...
frr2 9831 Law of general well-founde...
frr3 9832 Law of general well-founde...
r1funlim 9837 The cumulative hierarchy o...
r1fnon 9838 The cumulative hierarchy o...
r10 9839 Value of the cumulative hi...
r1sucg 9840 Value of the cumulative hi...
r1suc 9841 Value of the cumulative hi...
r1limg 9842 Value of the cumulative hi...
r1lim 9843 Value of the cumulative hi...
r1fin 9844 The first ` _om ` levels o...
r1sdom 9845 Each stage in the cumulati...
r111 9846 The cumulative hierarchy i...
r1tr 9847 The cumulative hierarchy o...
r1tr2 9848 The union of a cumulative ...
r1ordg 9849 Ordering relation for the ...
r1ord3g 9850 Ordering relation for the ...
r1ord 9851 Ordering relation for the ...
r1ord2 9852 Ordering relation for the ...
r1ord3 9853 Ordering relation for the ...
r1sssuc 9854 The value of the cumulativ...
r1pwss 9855 Each set of the cumulative...
r1sscl 9856 Each set of the cumulative...
r1val1 9857 The value of the cumulativ...
tz9.12lem1 9858 Lemma for ~ tz9.12 . (Con...
tz9.12lem2 9859 Lemma for ~ tz9.12 . (Con...
tz9.12lem3 9860 Lemma for ~ tz9.12 . (Con...
tz9.12 9861 A set is well-founded if a...
tz9.13 9862 Every set is well-founded,...
tz9.13g 9863 Every set is well-founded,...
rankwflemb 9864 Two ways of saying a set i...
rankf 9865 The domain and codomain of...
rankon 9866 The rank of a set is an or...
r1elwf 9867 Any member of the cumulati...
rankvalb 9868 Value of the rank function...
rankr1ai 9869 One direction of ~ rankr1a...
rankvaln 9870 Value of the rank function...
rankidb 9871 Identity law for the rank ...
rankdmr1 9872 A rank is a member of the ...
rankr1ag 9873 A version of ~ rankr1a tha...
rankr1bg 9874 A relationship between ran...
r1rankidb 9875 Any set is a subset of the...
r1elssi 9876 The range of the ` R1 ` fu...
r1elss 9877 The range of the ` R1 ` fu...
pwwf 9878 A power set is well-founde...
sswf 9879 A subset of a well-founded...
snwf 9880 A singleton is well-founde...
unwf 9881 A binary union is well-fou...
prwf 9882 An unordered pair is well-...
opwf 9883 An ordered pair is well-fo...
unir1 9884 The cumulative hierarchy o...
jech9.3 9885 Every set belongs to some ...
rankwflem 9886 Every set is well-founded,...
rankval 9887 Value of the rank function...
rankvalg 9888 Value of the rank function...
rankval2 9889 Value of an alternate defi...
uniwf 9890 A union is well-founded if...
rankr1clem 9891 Lemma for ~ rankr1c . (Co...
rankr1c 9892 A relationship between the...
rankidn 9893 A relationship between the...
rankpwi 9894 The rank of a power set. ...
rankelb 9895 The membership relation is...
wfelirr 9896 A well-founded set is not ...
rankval3b 9897 The value of the rank func...
ranksnb 9898 The rank of a singleton. ...
rankonidlem 9899 Lemma for ~ rankonid . (C...
rankonid 9900 The rank of an ordinal num...
onwf 9901 The ordinals are all well-...
onssr1 9902 Initial segments of the or...
rankr1g 9903 A relationship between the...
rankid 9904 Identity law for the rank ...
rankr1 9905 A relationship between the...
ssrankr1 9906 A relationship between an ...
rankr1a 9907 A relationship between ran...
r1val2 9908 The value of the cumulativ...
r1val3 9909 The value of the cumulativ...
rankel 9910 The membership relation is...
rankval3 9911 The value of the rank func...
bndrank 9912 Any class whose elements h...
unbndrank 9913 The elements of a proper c...
rankpw 9914 The rank of a power set. ...
ranklim 9915 The rank of a set belongs ...
r1pw 9916 A stronger property of ` R...
r1pwALT 9917 Alternate shorter proof of...
r1pwcl 9918 The cumulative hierarchy o...
rankssb 9919 The subset relation is inh...
rankss 9920 The subset relation is inh...
rankunb 9921 The rank of the union of t...
rankprb 9922 The rank of an unordered p...
rankopb 9923 The rank of an ordered pai...
rankuni2b 9924 The value of the rank func...
ranksn 9925 The rank of a singleton. ...
rankuni2 9926 The rank of a union. Part...
rankun 9927 The rank of the union of t...
rankpr 9928 The rank of an unordered p...
rankop 9929 The rank of an ordered pai...
r1rankid 9930 Any set is a subset of the...
rankeq0b 9931 A set is empty iff its ran...
rankeq0 9932 A set is empty iff its ran...
rankr1id 9933 The rank of the hierarchy ...
rankuni 9934 The rank of a union. Part...
rankr1b 9935 A relationship between ran...
ranksuc 9936 The rank of a successor. ...
rankuniss 9937 Upper bound of the rank of...
rankval4 9938 The rank of a set is the s...
rankbnd 9939 The rank of a set is bound...
rankbnd2 9940 The rank of a set is bound...
rankc1 9941 A relationship that can be...
rankc2 9942 A relationship that can be...
rankelun 9943 Rank membership is inherit...
rankelpr 9944 Rank membership is inherit...
rankelop 9945 Rank membership is inherit...
rankxpl 9946 A lower bound on the rank ...
rankxpu 9947 An upper bound on the rank...
rankfu 9948 An upper bound on the rank...
rankmapu 9949 An upper bound on the rank...
rankxplim 9950 The rank of a Cartesian pr...
rankxplim2 9951 If the rank of a Cartesian...
rankxplim3 9952 The rank of a Cartesian pr...
rankxpsuc 9953 The rank of a Cartesian pr...
tcwf 9954 The transitive closure fun...
tcrank 9955 This theorem expresses two...
scottex 9956 Scott's trick collects all...
scott0 9957 Scott's trick collects all...
scottexs 9958 Theorem scheme version of ...
scott0s 9959 Theorem scheme version of ...
cplem1 9960 Lemma for the Collection P...
cplem2 9961 Lemma for the Collection P...
cp 9962 Collection Principle. Thi...
bnd 9963 A very strong generalizati...
bnd2 9964 A variant of the Boundedne...
kardex 9965 The collection of all sets...
karden 9966 If we allow the Axiom of R...
htalem 9967 Lemma for defining an emul...
hta 9968 A ZFC emulation of Hilbert...
djueq12 9975 Equality theorem for disjo...
djueq1 9976 Equality theorem for disjo...
djueq2 9977 Equality theorem for disjo...
nfdju 9978 Bound-variable hypothesis ...
djuex 9979 The disjoint union of sets...
djuexb 9980 The disjoint union of two ...
djulcl 9981 Left closure of disjoint u...
djurcl 9982 Right closure of disjoint ...
djulf1o 9983 The left injection functio...
djurf1o 9984 The right injection functi...
inlresf 9985 The left injection restric...
inlresf1 9986 The left injection restric...
inrresf 9987 The right injection restri...
inrresf1 9988 The right injection restri...
djuin 9989 The images of any classes ...
djur 9990 A member of a disjoint uni...
djuss 9991 A disjoint union is a subc...
djuunxp 9992 The union of a disjoint un...
djuexALT 9993 Alternate proof of ~ djuex...
eldju1st 9994 The first component of an ...
eldju2ndl 9995 The second component of an...
eldju2ndr 9996 The second component of an...
djuun 9997 The disjoint union of two ...
1stinl 9998 The first component of the...
2ndinl 9999 The second component of th...
1stinr 10000 The first component of the...
2ndinr 10001 The second component of th...
updjudhf 10002 The mapping of an element ...
updjudhcoinlf 10003 The composition of the map...
updjudhcoinrg 10004 The composition of the map...
updjud 10005 Universal property of the ...
cardf2 10014 The cardinality function i...
cardon 10015 The cardinal number of a s...
isnum2 10016 A way to express well-orde...
isnumi 10017 A set equinumerous to an o...
ennum 10018 Equinumerous sets are equi...
finnum 10019 Every finite set is numera...
onenon 10020 Every ordinal number is nu...
tskwe 10021 A Tarski set is well-order...
xpnum 10022 The cartesian product of n...
cardval3 10023 An alternate definition of...
cardid2 10024 Any numerable set is equin...
isnum3 10025 A set is numerable iff it ...
oncardval 10026 The value of the cardinal ...
oncardid 10027 Any ordinal number is equi...
cardonle 10028 The cardinal of an ordinal...
card0 10029 The cardinality of the emp...
cardidm 10030 The cardinality function i...
oncard 10031 A set is a cardinal number...
ficardom 10032 The cardinal number of a f...
ficardid 10033 A finite set is equinumero...
cardnn 10034 The cardinality of a natur...
cardnueq0 10035 The empty set is the only ...
cardne 10036 No member of a cardinal nu...
carden2a 10037 If two sets have equal non...
carden2b 10038 If two sets are equinumero...
card1 10039 A set has cardinality one ...
cardsn 10040 A singleton has cardinalit...
carddomi2 10041 Two sets have the dominanc...
sdomsdomcardi 10042 A set strictly dominates i...
cardlim 10043 An infinite cardinal is a ...
cardsdomelir 10044 A cardinal strictly domina...
cardsdomel 10045 A cardinal strictly domina...
iscard 10046 Two ways to express the pr...
iscard2 10047 Two ways to express the pr...
carddom2 10048 Two numerable sets have th...
harcard 10049 The class of ordinal numbe...
cardprclem 10050 Lemma for ~ cardprc . (Co...
cardprc 10051 The class of all cardinal ...
carduni 10052 The union of a set of card...
cardiun 10053 The indexed union of a set...
cardennn 10054 If ` A ` is equinumerous t...
cardsucinf 10055 The cardinality of the suc...
cardsucnn 10056 The cardinality of the suc...
cardom 10057 The set of natural numbers...
carden2 10058 Two numerable sets are equ...
cardsdom2 10059 A numerable set is strictl...
domtri2 10060 Trichotomy of dominance fo...
nnsdomel 10061 Strict dominance and eleme...
cardval2 10062 An alternate version of th...
isinffi 10063 An infinite set contains s...
fidomtri 10064 Trichotomy of dominance wi...
fidomtri2 10065 Trichotomy of dominance wi...
harsdom 10066 The Hartogs number of a we...
onsdom 10067 Any well-orderable set is ...
harval2 10068 An alternate expression fo...
harsucnn 10069 The next cardinal after a ...
cardmin2 10070 The smallest ordinal that ...
pm54.43lem 10071 In Theorem *54.43 of [Whit...
pm54.43 10072 Theorem *54.43 of [Whitehe...
enpr2 10073 An unordered pair with dis...
pr2nelemOLD 10074 Obsolete version of ~ enpr...
pr2ne 10075 If an unordered pair has t...
pr2neOLD 10076 Obsolete version of ~ pr2n...
prdom2 10077 An unordered pair has at m...
en2eqpr 10078 Building a set with two el...
en2eleq 10079 Express a set of pair card...
en2other2 10080 Taking the other element t...
dif1card 10081 The cardinality of a nonem...
leweon 10082 Lexicographical order is a...
r0weon 10083 A set-like well-ordering o...
infxpenlem 10084 Lemma for ~ infxpen . (Co...
infxpen 10085 Every infinite ordinal is ...
xpomen 10086 The Cartesian product of o...
xpct 10087 The cartesian product of t...
infxpidm2 10088 Every infinite well-ordera...
infxpenc 10089 A canonical version of ~ i...
infxpenc2lem1 10090 Lemma for ~ infxpenc2 . (...
infxpenc2lem2 10091 Lemma for ~ infxpenc2 . (...
infxpenc2lem3 10092 Lemma for ~ infxpenc2 . (...
infxpenc2 10093 Existence form of ~ infxpe...
iunmapdisj 10094 The union ` U_ n e. C ( A ...
fseqenlem1 10095 Lemma for ~ fseqen . (Con...
fseqenlem2 10096 Lemma for ~ fseqen . (Con...
fseqdom 10097 One half of ~ fseqen . (C...
fseqen 10098 A set that is equinumerous...
infpwfidom 10099 The collection of finite s...
dfac8alem 10100 Lemma for ~ dfac8a . If t...
dfac8a 10101 Numeration theorem: every ...
dfac8b 10102 The well-ordering theorem:...
dfac8clem 10103 Lemma for ~ dfac8c . (Con...
dfac8c 10104 If the union of a set is w...
ac10ct 10105 A proof of the well-orderi...
ween 10106 A set is numerable iff it ...
ac5num 10107 A version of ~ ac5b with t...
ondomen 10108 If a set is dominated by a...
numdom 10109 A set dominated by a numer...
ssnum 10110 A subset of a numerable se...
onssnum 10111 All subsets of the ordinal...
indcardi 10112 Indirect strong induction ...
acnrcl 10113 Reverse closure for the ch...
acneq 10114 Equality theorem for the c...
isacn 10115 The property of being a ch...
acni 10116 The property of being a ch...
acni2 10117 The property of being a ch...
acni3 10118 The property of being a ch...
acnlem 10119 Construct a mapping satisf...
numacn 10120 A well-orderable set has c...
finacn 10121 Every set has finite choic...
acndom 10122 A set with long choice seq...
acnnum 10123 A set ` X ` which has choi...
acnen 10124 The class of choice sets o...
acndom2 10125 A set smaller than one wit...
acnen2 10126 The class of sets with cho...
fodomacn 10127 A version of ~ fodom that ...
fodomnum 10128 A version of ~ fodom that ...
fonum 10129 A surjection maps numerabl...
numwdom 10130 A surjection maps numerabl...
fodomfi2 10131 Onto functions define domi...
wdomfil 10132 Weak dominance agrees with...
infpwfien 10133 Any infinite well-orderabl...
inffien 10134 The set of finite intersec...
wdomnumr 10135 Weak dominance agrees with...
alephfnon 10136 The aleph function is a fu...
aleph0 10137 The first infinite cardina...
alephlim 10138 Value of the aleph functio...
alephsuc 10139 Value of the aleph functio...
alephon 10140 An aleph is an ordinal num...
alephcard 10141 Every aleph is a cardinal ...
alephnbtwn 10142 No cardinal can be sandwic...
alephnbtwn2 10143 No set has equinumerosity ...
alephordilem1 10144 Lemma for ~ alephordi . (...
alephordi 10145 Strict ordering property o...
alephord 10146 Ordering property of the a...
alephord2 10147 Ordering property of the a...
alephord2i 10148 Ordering property of the a...
alephord3 10149 Ordering property of the a...
alephsucdom 10150 A set dominated by an alep...
alephsuc2 10151 An alternate representatio...
alephdom 10152 Relationship between inclu...
alephgeom 10153 Every aleph is greater tha...
alephislim 10154 Every aleph is a limit ord...
aleph11 10155 The aleph function is one-...
alephf1 10156 The aleph function is a on...
alephsdom 10157 If an ordinal is smaller t...
alephdom2 10158 A dominated initial ordina...
alephle 10159 The argument of the aleph ...
cardaleph 10160 Given any transfinite card...
cardalephex 10161 Every transfinite cardinal...
infenaleph 10162 An infinite numerable set ...
isinfcard 10163 Two ways to express the pr...
iscard3 10164 Two ways to express the pr...
cardnum 10165 Two ways to express the cl...
alephinit 10166 An infinite initial ordina...
carduniima 10167 The union of the image of ...
cardinfima 10168 If a mapping to cardinals ...
alephiso 10169 Aleph is an order isomorph...
alephprc 10170 The class of all transfini...
alephsson 10171 The class of transfinite c...
unialeph 10172 The union of the class of ...
alephsmo 10173 The aleph function is stri...
alephf1ALT 10174 Alternate proof of ~ aleph...
alephfplem1 10175 Lemma for ~ alephfp . (Co...
alephfplem2 10176 Lemma for ~ alephfp . (Co...
alephfplem3 10177 Lemma for ~ alephfp . (Co...
alephfplem4 10178 Lemma for ~ alephfp . (Co...
alephfp 10179 The aleph function has a f...
alephfp2 10180 The aleph function has at ...
alephval3 10181 An alternate way to expres...
alephsucpw2 10182 The power set of an aleph ...
mappwen 10183 Power rule for cardinal ar...
finnisoeu 10184 A finite totally ordered s...
iunfictbso 10185 Countability of a countabl...
aceq1 10188 Equivalence of two version...
aceq0 10189 Equivalence of two version...
aceq2 10190 Equivalence of two version...
aceq3lem 10191 Lemma for ~ dfac3 . (Cont...
dfac3 10192 Equivalence of two version...
dfac4 10193 Equivalence of two version...
dfac5lem1 10194 Lemma for ~ dfac5 . (Cont...
dfac5lem2 10195 Lemma for ~ dfac5 . (Cont...
dfac5lem3 10196 Lemma for ~ dfac5 . (Cont...
dfac5lem4 10197 Lemma for ~ dfac5 . (Cont...
dfac5lem5 10198 Lemma for ~ dfac5 . (Cont...
dfac5lem4OLD 10199 Obsolete version of ~ dfac...
dfac5 10200 Equivalence of two version...
dfac2a 10201 Our Axiom of Choice (in th...
dfac2b 10202 Axiom of Choice (first for...
dfac2 10203 Axiom of Choice (first for...
dfac7 10204 Equivalence of the Axiom o...
dfac0 10205 Equivalence of two version...
dfac1 10206 Equivalence of two version...
dfac8 10207 A proof of the equivalency...
dfac9 10208 Equivalence of the axiom o...
dfac10 10209 Axiom of Choice equivalent...
dfac10c 10210 Axiom of Choice equivalent...
dfac10b 10211 Axiom of Choice equivalent...
acacni 10212 A choice equivalent: every...
dfacacn 10213 A choice equivalent: every...
dfac13 10214 The axiom of choice holds ...
dfac12lem1 10215 Lemma for ~ dfac12 . (Con...
dfac12lem2 10216 Lemma for ~ dfac12 . (Con...
dfac12lem3 10217 Lemma for ~ dfac12 . (Con...
dfac12r 10218 The axiom of choice holds ...
dfac12k 10219 Equivalence of ~ dfac12 an...
dfac12a 10220 The axiom of choice holds ...
dfac12 10221 The axiom of choice holds ...
kmlem1 10222 Lemma for 5-quantifier AC ...
kmlem2 10223 Lemma for 5-quantifier AC ...
kmlem3 10224 Lemma for 5-quantifier AC ...
kmlem4 10225 Lemma for 5-quantifier AC ...
kmlem5 10226 Lemma for 5-quantifier AC ...
kmlem6 10227 Lemma for 5-quantifier AC ...
kmlem7 10228 Lemma for 5-quantifier AC ...
kmlem8 10229 Lemma for 5-quantifier AC ...
kmlem9 10230 Lemma for 5-quantifier AC ...
kmlem10 10231 Lemma for 5-quantifier AC ...
kmlem11 10232 Lemma for 5-quantifier AC ...
kmlem12 10233 Lemma for 5-quantifier AC ...
kmlem13 10234 Lemma for 5-quantifier AC ...
kmlem14 10235 Lemma for 5-quantifier AC ...
kmlem15 10236 Lemma for 5-quantifier AC ...
kmlem16 10237 Lemma for 5-quantifier AC ...
dfackm 10238 Equivalence of the Axiom o...
undjudom 10239 Cardinal addition dominate...
endjudisj 10240 Equinumerosity of a disjoi...
djuen 10241 Disjoint unions of equinum...
djuenun 10242 Disjoint union is equinume...
dju1en 10243 Cardinal addition with car...
dju1dif 10244 Adding and subtracting one...
dju1p1e2 10245 1+1=2 for cardinal number ...
dju1p1e2ALT 10246 Alternate proof of ~ dju1p...
dju0en 10247 Cardinal addition with car...
xp2dju 10248 Two times a cardinal numbe...
djucomen 10249 Commutative law for cardin...
djuassen 10250 Associative law for cardin...
xpdjuen 10251 Cardinal multiplication di...
mapdjuen 10252 Sum of exponents law for c...
pwdjuen 10253 Sum of exponents law for c...
djudom1 10254 Ordering law for cardinal ...
djudom2 10255 Ordering law for cardinal ...
djudoml 10256 A set is dominated by its ...
djuxpdom 10257 Cartesian product dominate...
djufi 10258 The disjoint union of two ...
cdainflem 10259 Any partition of omega int...
djuinf 10260 A set is infinite iff the ...
infdju1 10261 An infinite set is equinum...
pwdju1 10262 The sum of a powerset with...
pwdjuidm 10263 If the natural numbers inj...
djulepw 10264 If ` A ` is idempotent und...
onadju 10265 The cardinal and ordinal s...
cardadju 10266 The cardinal sum is equinu...
djunum 10267 The disjoint union of two ...
unnum 10268 The union of two numerable...
nnadju 10269 The cardinal and ordinal s...
nnadjuALT 10270 Shorter proof of ~ nnadju ...
ficardadju 10271 The disjoint union of fini...
ficardun 10272 The cardinality of the uni...
ficardun2 10273 The cardinality of the uni...
pwsdompw 10274 Lemma for ~ domtriom . Th...
unctb 10275 The union of two countable...
infdjuabs 10276 Absorption law for additio...
infunabs 10277 An infinite set is equinum...
infdju 10278 The sum of two cardinal nu...
infdif 10279 The cardinality of an infi...
infdif2 10280 Cardinality ordering for a...
infxpdom 10281 Dominance law for multipli...
infxpabs 10282 Absorption law for multipl...
infunsdom1 10283 The union of two sets that...
infunsdom 10284 The union of two sets that...
infxp 10285 Absorption law for multipl...
pwdjudom 10286 A property of dominance ov...
infpss 10287 Every infinite set has an ...
infmap2 10288 An exponentiation law for ...
ackbij2lem1 10289 Lemma for ~ ackbij2 . (Co...
ackbij1lem1 10290 Lemma for ~ ackbij2 . (Co...
ackbij1lem2 10291 Lemma for ~ ackbij2 . (Co...
ackbij1lem3 10292 Lemma for ~ ackbij2 . (Co...
ackbij1lem4 10293 Lemma for ~ ackbij2 . (Co...
ackbij1lem5 10294 Lemma for ~ ackbij2 . (Co...
ackbij1lem6 10295 Lemma for ~ ackbij2 . (Co...
ackbij1lem7 10296 Lemma for ~ ackbij1 . (Co...
ackbij1lem8 10297 Lemma for ~ ackbij1 . (Co...
ackbij1lem9 10298 Lemma for ~ ackbij1 . (Co...
ackbij1lem10 10299 Lemma for ~ ackbij1 . (Co...
ackbij1lem11 10300 Lemma for ~ ackbij1 . (Co...
ackbij1lem12 10301 Lemma for ~ ackbij1 . (Co...
ackbij1lem13 10302 Lemma for ~ ackbij1 . (Co...
ackbij1lem14 10303 Lemma for ~ ackbij1 . (Co...
ackbij1lem15 10304 Lemma for ~ ackbij1 . (Co...
ackbij1lem16 10305 Lemma for ~ ackbij1 . (Co...
ackbij1lem17 10306 Lemma for ~ ackbij1 . (Co...
ackbij1lem18 10307 Lemma for ~ ackbij1 . (Co...
ackbij1 10308 The Ackermann bijection, p...
ackbij1b 10309 The Ackermann bijection, p...
ackbij2lem2 10310 Lemma for ~ ackbij2 . (Co...
ackbij2lem3 10311 Lemma for ~ ackbij2 . (Co...
ackbij2lem4 10312 Lemma for ~ ackbij2 . (Co...
ackbij2 10313 The Ackermann bijection, p...
r1om 10314 The set of hereditarily fi...
fictb 10315 A set is countable iff its...
cflem 10316 A lemma used to simplify c...
cflemOLD 10317 Obsolete version of ~ cfle...
cfval 10318 Value of the cofinality fu...
cff 10319 Cofinality is a function o...
cfub 10320 An upper bound on cofinali...
cflm 10321 Value of the cofinality fu...
cf0 10322 Value of the cofinality fu...
cardcf 10323 Cofinality is a cardinal n...
cflecard 10324 Cofinality is bounded by t...
cfle 10325 Cofinality is bounded by i...
cfon 10326 The cofinality of any set ...
cfeq0 10327 Only the ordinal zero has ...
cfsuc 10328 Value of the cofinality fu...
cff1 10329 There is always a map from...
cfflb 10330 If there is a cofinal map ...
cfval2 10331 Another expression for the...
coflim 10332 A simpler expression for t...
cflim3 10333 Another expression for the...
cflim2 10334 The cofinality function is...
cfom 10335 Value of the cofinality fu...
cfss 10336 There is a cofinal subset ...
cfslb 10337 Any cofinal subset of ` A ...
cfslbn 10338 Any subset of ` A ` smalle...
cfslb2n 10339 Any small collection of sm...
cofsmo 10340 Any cofinal map implies th...
cfsmolem 10341 Lemma for ~ cfsmo . (Cont...
cfsmo 10342 The map in ~ cff1 can be a...
cfcoflem 10343 Lemma for ~ cfcof , showin...
coftr 10344 If there is a cofinal map ...
cfcof 10345 If there is a cofinal map ...
cfidm 10346 The cofinality function is...
alephsing 10347 The cofinality of a limit ...
sornom 10348 The range of a single-step...
isfin1a 10363 Definition of a Ia-finite ...
fin1ai 10364 Property of a Ia-finite se...
isfin2 10365 Definition of a II-finite ...
fin2i 10366 Property of a II-finite se...
isfin3 10367 Definition of a III-finite...
isfin4 10368 Definition of a IV-finite ...
fin4i 10369 Infer that a set is IV-inf...
isfin5 10370 Definition of a V-finite s...
isfin6 10371 Definition of a VI-finite ...
isfin7 10372 Definition of a VII-finite...
sdom2en01 10373 A set with less than two e...
infpssrlem1 10374 Lemma for ~ infpssr . (Co...
infpssrlem2 10375 Lemma for ~ infpssr . (Co...
infpssrlem3 10376 Lemma for ~ infpssr . (Co...
infpssrlem4 10377 Lemma for ~ infpssr . (Co...
infpssrlem5 10378 Lemma for ~ infpssr . (Co...
infpssr 10379 Dedekind infinity implies ...
fin4en1 10380 Dedekind finite is a cardi...
ssfin4 10381 Dedekind finite sets have ...
domfin4 10382 A set dominated by a Dedek...
ominf4 10383 ` _om ` is Dedekind infini...
infpssALT 10384 Alternate proof of ~ infps...
isfin4-2 10385 Alternate definition of IV...
isfin4p1 10386 Alternate definition of IV...
fin23lem7 10387 Lemma for ~ isfin2-2 . Th...
fin23lem11 10388 Lemma for ~ isfin2-2 . (C...
fin2i2 10389 A II-finite set contains m...
isfin2-2 10390 ` Fin2 ` expressed in term...
ssfin2 10391 A subset of a II-finite se...
enfin2i 10392 II-finiteness is a cardina...
fin23lem24 10393 Lemma for ~ fin23 . In a ...
fincssdom 10394 In a chain of finite sets,...
fin23lem25 10395 Lemma for ~ fin23 . In a ...
fin23lem26 10396 Lemma for ~ fin23lem22 . ...
fin23lem23 10397 Lemma for ~ fin23lem22 . ...
fin23lem22 10398 Lemma for ~ fin23 but coul...
fin23lem27 10399 The mapping constructed in...
isfin3ds 10400 Property of a III-finite s...
ssfin3ds 10401 A subset of a III-finite s...
fin23lem12 10402 The beginning of the proof...
fin23lem13 10403 Lemma for ~ fin23 . Each ...
fin23lem14 10404 Lemma for ~ fin23 . ` U ` ...
fin23lem15 10405 Lemma for ~ fin23 . ` U ` ...
fin23lem16 10406 Lemma for ~ fin23 . ` U ` ...
fin23lem19 10407 Lemma for ~ fin23 . The f...
fin23lem20 10408 Lemma for ~ fin23 . ` X ` ...
fin23lem17 10409 Lemma for ~ fin23 . By ? ...
fin23lem21 10410 Lemma for ~ fin23 . ` X ` ...
fin23lem28 10411 Lemma for ~ fin23 . The r...
fin23lem29 10412 Lemma for ~ fin23 . The r...
fin23lem30 10413 Lemma for ~ fin23 . The r...
fin23lem31 10414 Lemma for ~ fin23 . The r...
fin23lem32 10415 Lemma for ~ fin23 . Wrap ...
fin23lem33 10416 Lemma for ~ fin23 . Disch...
fin23lem34 10417 Lemma for ~ fin23 . Estab...
fin23lem35 10418 Lemma for ~ fin23 . Stric...
fin23lem36 10419 Lemma for ~ fin23 . Weak ...
fin23lem38 10420 Lemma for ~ fin23 . The c...
fin23lem39 10421 Lemma for ~ fin23 . Thus,...
fin23lem40 10422 Lemma for ~ fin23 . ` Fin2...
fin23lem41 10423 Lemma for ~ fin23 . A set...
isf32lem1 10424 Lemma for ~ isfin3-2 . De...
isf32lem2 10425 Lemma for ~ isfin3-2 . No...
isf32lem3 10426 Lemma for ~ isfin3-2 . Be...
isf32lem4 10427 Lemma for ~ isfin3-2 . Be...
isf32lem5 10428 Lemma for ~ isfin3-2 . Th...
isf32lem6 10429 Lemma for ~ isfin3-2 . Ea...
isf32lem7 10430 Lemma for ~ isfin3-2 . Di...
isf32lem8 10431 Lemma for ~ isfin3-2 . K ...
isf32lem9 10432 Lemma for ~ isfin3-2 . Co...
isf32lem10 10433 Lemma for isfin3-2 . Writ...
isf32lem11 10434 Lemma for ~ isfin3-2 . Re...
isf32lem12 10435 Lemma for ~ isfin3-2 . (C...
isfin32i 10436 One half of ~ isfin3-2 . ...
isf33lem 10437 Lemma for ~ isfin3-3 . (C...
isfin3-2 10438 Weakly Dedekind-infinite s...
isfin3-3 10439 Weakly Dedekind-infinite s...
fin33i 10440 Inference from ~ isfin3-3 ...
compsscnvlem 10441 Lemma for ~ compsscnv . (...
compsscnv 10442 Complementation on a power...
isf34lem1 10443 Lemma for ~ isfin3-4 . (C...
isf34lem2 10444 Lemma for ~ isfin3-4 . (C...
compssiso 10445 Complementation is an anti...
isf34lem3 10446 Lemma for ~ isfin3-4 . (C...
compss 10447 Express image under of the...
isf34lem4 10448 Lemma for ~ isfin3-4 . (C...
isf34lem5 10449 Lemma for ~ isfin3-4 . (C...
isf34lem7 10450 Lemma for ~ isfin3-4 . (C...
isf34lem6 10451 Lemma for ~ isfin3-4 . (C...
fin34i 10452 Inference from ~ isfin3-4 ...
isfin3-4 10453 Weakly Dedekind-infinite s...
fin11a 10454 Every I-finite set is Ia-f...
enfin1ai 10455 Ia-finiteness is a cardina...
isfin1-2 10456 A set is finite in the usu...
isfin1-3 10457 A set is I-finite iff ever...
isfin1-4 10458 A set is I-finite iff ever...
dffin1-5 10459 Compact quantifier-free ve...
fin23 10460 Every II-finite set (every...
fin34 10461 Every III-finite set is IV...
isfin5-2 10462 Alternate definition of V-...
fin45 10463 Every IV-finite set is V-f...
fin56 10464 Every V-finite set is VI-f...
fin17 10465 Every I-finite set is VII-...
fin67 10466 Every VI-finite set is VII...
isfin7-2 10467 A set is VII-finite iff it...
fin71num 10468 A well-orderable set is VI...
dffin7-2 10469 Class form of ~ isfin7-2 ....
dfacfin7 10470 Axiom of Choice equivalent...
fin1a2lem1 10471 Lemma for ~ fin1a2 . (Con...
fin1a2lem2 10472 Lemma for ~ fin1a2 . The ...
fin1a2lem3 10473 Lemma for ~ fin1a2 . (Con...
fin1a2lem4 10474 Lemma for ~ fin1a2 . (Con...
fin1a2lem5 10475 Lemma for ~ fin1a2 . (Con...
fin1a2lem6 10476 Lemma for ~ fin1a2 . Esta...
fin1a2lem7 10477 Lemma for ~ fin1a2 . Spli...
fin1a2lem8 10478 Lemma for ~ fin1a2 . Spli...
fin1a2lem9 10479 Lemma for ~ fin1a2 . In a...
fin1a2lem10 10480 Lemma for ~ fin1a2 . A no...
fin1a2lem11 10481 Lemma for ~ fin1a2 . (Con...
fin1a2lem12 10482 Lemma for ~ fin1a2 . (Con...
fin1a2lem13 10483 Lemma for ~ fin1a2 . (Con...
fin12 10484 Weak theorem which skips I...
fin1a2s 10485 An II-infinite set can hav...
fin1a2 10486 Every Ia-finite set is II-...
itunifval 10487 Function value of iterated...
itunifn 10488 Functionality of the itera...
ituni0 10489 A zero-fold iterated union...
itunisuc 10490 Successor iterated union. ...
itunitc1 10491 Each union iterate is a me...
itunitc 10492 The union of all union ite...
ituniiun 10493 Unwrap an iterated union f...
hsmexlem7 10494 Lemma for ~ hsmex . Prope...
hsmexlem8 10495 Lemma for ~ hsmex . Prope...
hsmexlem9 10496 Lemma for ~ hsmex . Prope...
hsmexlem1 10497 Lemma for ~ hsmex . Bound...
hsmexlem2 10498 Lemma for ~ hsmex . Bound...
hsmexlem3 10499 Lemma for ~ hsmex . Clear...
hsmexlem4 10500 Lemma for ~ hsmex . The c...
hsmexlem5 10501 Lemma for ~ hsmex . Combi...
hsmexlem6 10502 Lemma for ~ hsmex . (Cont...
hsmex 10503 The collection of heredita...
hsmex2 10504 The set of hereditary size...
hsmex3 10505 The set of hereditary size...
axcc2lem 10507 Lemma for ~ axcc2 . (Cont...
axcc2 10508 A possibly more useful ver...
axcc3 10509 A possibly more useful ver...
axcc4 10510 A version of ~ axcc3 that ...
acncc 10511 An ~ ax-cc equivalent: eve...
axcc4dom 10512 Relax the constraint on ~ ...
domtriomlem 10513 Lemma for ~ domtriom . (C...
domtriom 10514 Trichotomy of equinumerosi...
fin41 10515 Under countable choice, th...
dominf 10516 A nonempty set that is a s...
dcomex 10518 The Axiom of Dependent Cho...
axdc2lem 10519 Lemma for ~ axdc2 . We co...
axdc2 10520 An apparent strengthening ...
axdc3lem 10521 The class ` S ` of finite ...
axdc3lem2 10522 Lemma for ~ axdc3 . We ha...
axdc3lem3 10523 Simple substitution lemma ...
axdc3lem4 10524 Lemma for ~ axdc3 . We ha...
axdc3 10525 Dependent Choice. Axiom D...
axdc4lem 10526 Lemma for ~ axdc4 . (Cont...
axdc4 10527 A more general version of ...
axcclem 10528 Lemma for ~ axcc . (Contr...
axcc 10529 Although CC can be proven ...
zfac 10531 Axiom of Choice expressed ...
ac2 10532 Axiom of Choice equivalent...
ac3 10533 Axiom of Choice using abbr...
axac3 10535 This theorem asserts that ...
ackm 10536 A remarkable equivalent to...
axac2 10537 Derive ~ ax-ac2 from ~ ax-...
axac 10538 Derive ~ ax-ac from ~ ax-a...
axaci 10539 Apply a choice equivalent....
cardeqv 10540 All sets are well-orderabl...
numth3 10541 All sets are well-orderabl...
numth2 10542 Numeration theorem: any se...
numth 10543 Numeration theorem: every ...
ac7 10544 An Axiom of Choice equival...
ac7g 10545 An Axiom of Choice equival...
ac4 10546 Equivalent of Axiom of Cho...
ac4c 10547 Equivalent of Axiom of Cho...
ac5 10548 An Axiom of Choice equival...
ac5b 10549 Equivalent of Axiom of Cho...
ac6num 10550 A version of ~ ac6 which t...
ac6 10551 Equivalent of Axiom of Cho...
ac6c4 10552 Equivalent of Axiom of Cho...
ac6c5 10553 Equivalent of Axiom of Cho...
ac9 10554 An Axiom of Choice equival...
ac6s 10555 Equivalent of Axiom of Cho...
ac6n 10556 Equivalent of Axiom of Cho...
ac6s2 10557 Generalization of the Axio...
ac6s3 10558 Generalization of the Axio...
ac6sg 10559 ~ ac6s with sethood as ant...
ac6sf 10560 Version of ~ ac6 with boun...
ac6s4 10561 Generalization of the Axio...
ac6s5 10562 Generalization of the Axio...
ac8 10563 An Axiom of Choice equival...
ac9s 10564 An Axiom of Choice equival...
numthcor 10565 Any set is strictly domina...
weth 10566 Well-ordering theorem: any...
zorn2lem1 10567 Lemma for ~ zorn2 . (Cont...
zorn2lem2 10568 Lemma for ~ zorn2 . (Cont...
zorn2lem3 10569 Lemma for ~ zorn2 . (Cont...
zorn2lem4 10570 Lemma for ~ zorn2 . (Cont...
zorn2lem5 10571 Lemma for ~ zorn2 . (Cont...
zorn2lem6 10572 Lemma for ~ zorn2 . (Cont...
zorn2lem7 10573 Lemma for ~ zorn2 . (Cont...
zorn2g 10574 Zorn's Lemma of [Monk1] p....
zorng 10575 Zorn's Lemma. If the unio...
zornn0g 10576 Variant of Zorn's lemma ~ ...
zorn2 10577 Zorn's Lemma of [Monk1] p....
zorn 10578 Zorn's Lemma. If the unio...
zornn0 10579 Variant of Zorn's lemma ~ ...
ttukeylem1 10580 Lemma for ~ ttukey . Expa...
ttukeylem2 10581 Lemma for ~ ttukey . A pr...
ttukeylem3 10582 Lemma for ~ ttukey . (Con...
ttukeylem4 10583 Lemma for ~ ttukey . (Con...
ttukeylem5 10584 Lemma for ~ ttukey . The ...
ttukeylem6 10585 Lemma for ~ ttukey . (Con...
ttukeylem7 10586 Lemma for ~ ttukey . (Con...
ttukey2g 10587 The Teichmüller-Tukey...
ttukeyg 10588 The Teichmüller-Tukey...
ttukey 10589 The Teichmüller-Tukey...
axdclem 10590 Lemma for ~ axdc . (Contr...
axdclem2 10591 Lemma for ~ axdc . Using ...
axdc 10592 This theorem derives ~ ax-...
fodomg 10593 An onto function implies d...
fodom 10594 An onto function implies d...
dmct 10595 The domain of a countable ...
rnct 10596 The range of a countable s...
fodomb 10597 Equivalence of an onto map...
wdomac 10598 When assuming AC, weak and...
brdom3 10599 Equivalence to a dominance...
brdom5 10600 An equivalence to a domina...
brdom4 10601 An equivalence to a domina...
brdom7disj 10602 An equivalence to a domina...
brdom6disj 10603 An equivalence to a domina...
fin71ac 10604 Once we allow AC, the "str...
imadomg 10605 An image of a function und...
fimact 10606 The image by a function of...
fnrndomg 10607 The range of a function is...
fnct 10608 If the domain of a functio...
mptct 10609 A countable mapping set is...
iunfo 10610 Existence of an onto funct...
iundom2g 10611 An upper bound for the car...
iundomg 10612 An upper bound for the car...
iundom 10613 An upper bound for the car...
unidom 10614 An upper bound for the car...
uniimadom 10615 An upper bound for the car...
uniimadomf 10616 An upper bound for the car...
cardval 10617 The value of the cardinal ...
cardid 10618 Any set is equinumerous to...
cardidg 10619 Any set is equinumerous to...
cardidd 10620 Any set is equinumerous to...
cardf 10621 The cardinality function i...
carden 10622 Two sets are equinumerous ...
cardeq0 10623 Only the empty set has car...
unsnen 10624 Equinumerosity of a set wi...
carddom 10625 Two sets have the dominanc...
cardsdom 10626 Two sets have the strict d...
domtri 10627 Trichotomy law for dominan...
entric 10628 Trichotomy of equinumerosi...
entri2 10629 Trichotomy of dominance an...
entri3 10630 Trichotomy of dominance. ...
sdomsdomcard 10631 A set strictly dominates i...
canth3 10632 Cantor's theorem in terms ...
infxpidm 10633 Every infinite class is eq...
ondomon 10634 The class of ordinals domi...
cardmin 10635 The smallest ordinal that ...
ficard 10636 A set is finite iff its ca...
infinf 10637 Equivalence between two in...
unirnfdomd 10638 The union of the range of ...
konigthlem 10639 Lemma for ~ konigth . (Co...
konigth 10640 Konig's Theorem. If ` m (...
alephsucpw 10641 The power set of an aleph ...
aleph1 10642 The set exponentiation of ...
alephval2 10643 An alternate way to expres...
dominfac 10644 A nonempty set that is a s...
iunctb 10645 The countable union of cou...
unictb 10646 The countable union of cou...
infmap 10647 An exponentiation law for ...
alephadd 10648 The sum of two alephs is t...
alephmul 10649 The product of two alephs ...
alephexp1 10650 An exponentiation law for ...
alephsuc3 10651 An alternate representatio...
alephexp2 10652 An expression equinumerous...
alephreg 10653 A successor aleph is regul...
pwcfsdom 10654 A corollary of Konig's The...
cfpwsdom 10655 A corollary of Konig's The...
alephom 10656 From ~ canth2 , we know th...
smobeth 10657 The beth function is stric...
nd1 10658 A lemma for proving condit...
nd2 10659 A lemma for proving condit...
nd3 10660 A lemma for proving condit...
nd4 10661 A lemma for proving condit...
axextnd 10662 A version of the Axiom of ...
axrepndlem1 10663 Lemma for the Axiom of Rep...
axrepndlem2 10664 Lemma for the Axiom of Rep...
axrepnd 10665 A version of the Axiom of ...
axunndlem1 10666 Lemma for the Axiom of Uni...
axunnd 10667 A version of the Axiom of ...
axpowndlem1 10668 Lemma for the Axiom of Pow...
axpowndlem2 10669 Lemma for the Axiom of Pow...
axpowndlem3 10670 Lemma for the Axiom of Pow...
axpowndlem4 10671 Lemma for the Axiom of Pow...
axpownd 10672 A version of the Axiom of ...
axregndlem1 10673 Lemma for the Axiom of Reg...
axregndlem2 10674 Lemma for the Axiom of Reg...
axregnd 10675 A version of the Axiom of ...
axinfndlem1 10676 Lemma for the Axiom of Inf...
axinfnd 10677 A version of the Axiom of ...
axacndlem1 10678 Lemma for the Axiom of Cho...
axacndlem2 10679 Lemma for the Axiom of Cho...
axacndlem3 10680 Lemma for the Axiom of Cho...
axacndlem4 10681 Lemma for the Axiom of Cho...
axacndlem5 10682 Lemma for the Axiom of Cho...
axacnd 10683 A version of the Axiom of ...
zfcndext 10684 Axiom of Extensionality ~ ...
zfcndrep 10685 Axiom of Replacement ~ ax-...
zfcndun 10686 Axiom of Union ~ ax-un , r...
zfcndpow 10687 Axiom of Power Sets ~ ax-p...
zfcndreg 10688 Axiom of Regularity ~ ax-r...
zfcndinf 10689 Axiom of Infinity ~ ax-inf...
zfcndac 10690 Axiom of Choice ~ ax-ac , ...
elgch 10693 Elementhood in the collect...
fingch 10694 A finite set is a GCH-set....
gchi 10695 The only GCH-sets which ha...
gchen1 10696 If ` A <_ B < ~P A ` , and...
gchen2 10697 If ` A < B <_ ~P A ` , and...
gchor 10698 If ` A <_ B <_ ~P A ` , an...
engch 10699 The property of being a GC...
gchdomtri 10700 Under certain conditions, ...
fpwwe2cbv 10701 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem1 10702 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem2 10703 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem3 10704 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem4 10705 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem5 10706 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem6 10707 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem7 10708 Lemma for ~ fpwwe2 . Show...
fpwwe2lem8 10709 Lemma for ~ fpwwe2 . Give...
fpwwe2lem9 10710 Lemma for ~ fpwwe2 . Give...
fpwwe2lem10 10711 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem11 10712 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem12 10713 Lemma for ~ fpwwe2 . (Con...
fpwwe2 10714 Given any function ` F ` f...
fpwwecbv 10715 Lemma for ~ fpwwe . (Cont...
fpwwelem 10716 Lemma for ~ fpwwe . (Cont...
fpwwe 10717 Given any function ` F ` f...
canth4 10718 An "effective" form of Can...
canthnumlem 10719 Lemma for ~ canthnum . (C...
canthnum 10720 The set of well-orderable ...
canthwelem 10721 Lemma for ~ canthwe . (Co...
canthwe 10722 The set of well-orders of ...
canthp1lem1 10723 Lemma for ~ canthp1 . (Co...
canthp1lem2 10724 Lemma for ~ canthp1 . (Co...
canthp1 10725 A slightly stronger form o...
finngch 10726 The exclusion of finite se...
gchdju1 10727 An infinite GCH-set is ide...
gchinf 10728 An infinite GCH-set is Ded...
pwfseqlem1 10729 Lemma for ~ pwfseq . Deri...
pwfseqlem2 10730 Lemma for ~ pwfseq . (Con...
pwfseqlem3 10731 Lemma for ~ pwfseq . Usin...
pwfseqlem4a 10732 Lemma for ~ pwfseqlem4 . ...
pwfseqlem4 10733 Lemma for ~ pwfseq . Deri...
pwfseqlem5 10734 Lemma for ~ pwfseq . Alth...
pwfseq 10735 The powerset of a Dedekind...
pwxpndom2 10736 The powerset of a Dedekind...
pwxpndom 10737 The powerset of a Dedekind...
pwdjundom 10738 The powerset of a Dedekind...
gchdjuidm 10739 An infinite GCH-set is ide...
gchxpidm 10740 An infinite GCH-set is ide...
gchpwdom 10741 A relationship between dom...
gchaleph 10742 If ` ( aleph `` A ) ` is a...
gchaleph2 10743 If ` ( aleph `` A ) ` and ...
hargch 10744 If ` A + ~~ ~P A ` , then ...
alephgch 10745 If ` ( aleph `` suc A ) ` ...
gch2 10746 It is sufficient to requir...
gch3 10747 An equivalent formulation ...
gch-kn 10748 The equivalence of two ver...
gchaclem 10749 Lemma for ~ gchac (obsolet...
gchhar 10750 A "local" form of ~ gchac ...
gchacg 10751 A "local" form of ~ gchac ...
gchac 10752 The Generalized Continuum ...
elwina 10757 Conditions of weak inacces...
elina 10758 Conditions of strong inacc...
winaon 10759 A weakly inaccessible card...
inawinalem 10760 Lemma for ~ inawina . (Co...
inawina 10761 Every strongly inaccessibl...
omina 10762 ` _om ` is a strongly inac...
winacard 10763 A weakly inaccessible card...
winainflem 10764 A weakly inaccessible card...
winainf 10765 A weakly inaccessible card...
winalim 10766 A weakly inaccessible card...
winalim2 10767 A nontrivial weakly inacce...
winafp 10768 A nontrivial weakly inacce...
winafpi 10769 This theorem, which states...
gchina 10770 Assuming the GCH, weakly a...
iswun 10775 Properties of a weak unive...
wuntr 10776 A weak universe is transit...
wununi 10777 A weak universe is closed ...
wunpw 10778 A weak universe is closed ...
wunelss 10779 The elements of a weak uni...
wunpr 10780 A weak universe is closed ...
wunun 10781 A weak universe is closed ...
wuntp 10782 A weak universe is closed ...
wunss 10783 A weak universe is closed ...
wunin 10784 A weak universe is closed ...
wundif 10785 A weak universe is closed ...
wunint 10786 A weak universe is closed ...
wunsn 10787 A weak universe is closed ...
wunsuc 10788 A weak universe is closed ...
wun0 10789 A weak universe contains t...
wunr1om 10790 A weak universe is infinit...
wunom 10791 A weak universe contains a...
wunfi 10792 A weak universe contains a...
wunop 10793 A weak universe is closed ...
wunot 10794 A weak universe is closed ...
wunxp 10795 A weak universe is closed ...
wunpm 10796 A weak universe is closed ...
wunmap 10797 A weak universe is closed ...
wunf 10798 A weak universe is closed ...
wundm 10799 A weak universe is closed ...
wunrn 10800 A weak universe is closed ...
wuncnv 10801 A weak universe is closed ...
wunres 10802 A weak universe is closed ...
wunfv 10803 A weak universe is closed ...
wunco 10804 A weak universe is closed ...
wuntpos 10805 A weak universe is closed ...
intwun 10806 The intersection of a coll...
r1limwun 10807 Each limit stage in the cu...
r1wunlim 10808 The weak universes in the ...
wunex2 10809 Construct a weak universe ...
wunex 10810 Construct a weak universe ...
uniwun 10811 Every set is contained in ...
wunex3 10812 Construct a weak universe ...
wuncval 10813 Value of the weak universe...
wuncid 10814 The weak universe closure ...
wunccl 10815 The weak universe closure ...
wuncss 10816 The weak universe closure ...
wuncidm 10817 The weak universe closure ...
wuncval2 10818 Our earlier expression for...
eltskg 10821 Properties of a Tarski cla...
eltsk2g 10822 Properties of a Tarski cla...
tskpwss 10823 First axiom of a Tarski cl...
tskpw 10824 Second axiom of a Tarski c...
tsken 10825 Third axiom of a Tarski cl...
0tsk 10826 The empty set is a (transi...
tsksdom 10827 An element of a Tarski cla...
tskssel 10828 A part of a Tarski class s...
tskss 10829 The subsets of an element ...
tskin 10830 The intersection of two el...
tsksn 10831 A singleton of an element ...
tsktrss 10832 A transitive element of a ...
tsksuc 10833 If an element of a Tarski ...
tsk0 10834 A nonempty Tarski class co...
tsk1 10835 One is an element of a non...
tsk2 10836 Two is an element of a non...
2domtsk 10837 If a Tarski class is not e...
tskr1om 10838 A nonempty Tarski class is...
tskr1om2 10839 A nonempty Tarski class co...
tskinf 10840 A nonempty Tarski class is...
tskpr 10841 If ` A ` and ` B ` are mem...
tskop 10842 If ` A ` and ` B ` are mem...
tskxpss 10843 A Cartesian product of two...
tskwe2 10844 A Tarski class is well-ord...
inttsk 10845 The intersection of a coll...
inar1 10846 ` ( R1 `` A ) ` for ` A ` ...
r1omALT 10847 Alternate proof of ~ r1om ...
rankcf 10848 Any set must be at least a...
inatsk 10849 ` ( R1 `` A ) ` for ` A ` ...
r1omtsk 10850 The set of hereditarily fi...
tskord 10851 A Tarski class contains al...
tskcard 10852 An even more direct relati...
r1tskina 10853 There is a direct relation...
tskuni 10854 The union of an element of...
tskwun 10855 A nonempty transitive Tars...
tskint 10856 The intersection of an ele...
tskun 10857 The union of two elements ...
tskxp 10858 The Cartesian product of t...
tskmap 10859 Set exponentiation is an e...
tskurn 10860 A transitive Tarski class ...
elgrug 10863 Properties of a Grothendie...
grutr 10864 A Grothendieck universe is...
gruelss 10865 A Grothendieck universe is...
grupw 10866 A Grothendieck universe co...
gruss 10867 Any subset of an element o...
grupr 10868 A Grothendieck universe co...
gruurn 10869 A Grothendieck universe co...
gruiun 10870 If ` B ( x ) ` is a family...
gruuni 10871 A Grothendieck universe co...
grurn 10872 A Grothendieck universe co...
gruima 10873 A Grothendieck universe co...
gruel 10874 Any element of an element ...
grusn 10875 A Grothendieck universe co...
gruop 10876 A Grothendieck universe co...
gruun 10877 A Grothendieck universe co...
gruxp 10878 A Grothendieck universe co...
grumap 10879 A Grothendieck universe co...
gruixp 10880 A Grothendieck universe co...
gruiin 10881 A Grothendieck universe co...
gruf 10882 A Grothendieck universe co...
gruen 10883 A Grothendieck universe co...
gruwun 10884 A nonempty Grothendieck un...
intgru 10885 The intersection of a fami...
ingru 10886 The intersection of a univ...
wfgru 10887 The wellfounded part of a ...
grudomon 10888 Each ordinal that is compa...
gruina 10889 If a Grothendieck universe...
grur1a 10890 A characterization of Grot...
grur1 10891 A characterization of Grot...
grutsk1 10892 Grothendieck universes are...
grutsk 10893 Grothendieck universes are...
axgroth5 10895 The Tarski-Grothendieck ax...
axgroth2 10896 Alternate version of the T...
grothpw 10897 Derive the Axiom of Power ...
grothpwex 10898 Derive the Axiom of Power ...
axgroth6 10899 The Tarski-Grothendieck ax...
grothomex 10900 The Tarski-Grothendieck Ax...
grothac 10901 The Tarski-Grothendieck Ax...
axgroth3 10902 Alternate version of the T...
axgroth4 10903 Alternate version of the T...
grothprimlem 10904 Lemma for ~ grothprim . E...
grothprim 10905 The Tarski-Grothendieck Ax...
grothtsk 10906 The Tarski-Grothendieck Ax...
inaprc 10907 An equivalent to the Tarsk...
tskmval 10910 Value of our tarski map. ...
tskmid 10911 The set ` A ` is an elemen...
tskmcl 10912 A Tarski class that contai...
sstskm 10913 Being a part of ` ( tarski...
eltskm 10914 Belonging to ` ( tarskiMap...
elni 10947 Membership in the class of...
elni2 10948 Membership in the class of...
pinn 10949 A positive integer is a na...
pion 10950 A positive integer is an o...
piord 10951 A positive integer is ordi...
niex 10952 The class of positive inte...
0npi 10953 The empty set is not a pos...
1pi 10954 Ordinal 'one' is a positiv...
addpiord 10955 Positive integer addition ...
mulpiord 10956 Positive integer multiplic...
mulidpi 10957 1 is an identity element f...
ltpiord 10958 Positive integer 'less tha...
ltsopi 10959 Positive integer 'less tha...
ltrelpi 10960 Positive integer 'less tha...
dmaddpi 10961 Domain of addition on posi...
dmmulpi 10962 Domain of multiplication o...
addclpi 10963 Closure of addition of pos...
mulclpi 10964 Closure of multiplication ...
addcompi 10965 Addition of positive integ...
addasspi 10966 Addition of positive integ...
mulcompi 10967 Multiplication of positive...
mulasspi 10968 Multiplication of positive...
distrpi 10969 Multiplication of positive...
addcanpi 10970 Addition cancellation law ...
mulcanpi 10971 Multiplication cancellatio...
addnidpi 10972 There is no identity eleme...
ltexpi 10973 Ordering on positive integ...
ltapi 10974 Ordering property of addit...
ltmpi 10975 Ordering property of multi...
1lt2pi 10976 One is less than two (one ...
nlt1pi 10977 No positive integer is les...
indpi 10978 Principle of Finite Induct...
enqbreq 10990 Equivalence relation for p...
enqbreq2 10991 Equivalence relation for p...
enqer 10992 The equivalence relation f...
enqex 10993 The equivalence relation f...
nqex 10994 The class of positive frac...
0nnq 10995 The empty set is not a pos...
elpqn 10996 Each positive fraction is ...
ltrelnq 10997 Positive fraction 'less th...
pinq 10998 The representatives of pos...
1nq 10999 The positive fraction 'one...
nqereu 11000 There is a unique element ...
nqerf 11001 Corollary of ~ nqereu : th...
nqercl 11002 Corollary of ~ nqereu : cl...
nqerrel 11003 Any member of ` ( N. X. N....
nqerid 11004 Corollary of ~ nqereu : th...
enqeq 11005 Corollary of ~ nqereu : if...
nqereq 11006 The function ` /Q ` acts a...
addpipq2 11007 Addition of positive fract...
addpipq 11008 Addition of positive fract...
addpqnq 11009 Addition of positive fract...
mulpipq2 11010 Multiplication of positive...
mulpipq 11011 Multiplication of positive...
mulpqnq 11012 Multiplication of positive...
ordpipq 11013 Ordering of positive fract...
ordpinq 11014 Ordering of positive fract...
addpqf 11015 Closure of addition on pos...
addclnq 11016 Closure of addition on pos...
mulpqf 11017 Closure of multiplication ...
mulclnq 11018 Closure of multiplication ...
addnqf 11019 Domain of addition on posi...
mulnqf 11020 Domain of multiplication o...
addcompq 11021 Addition of positive fract...
addcomnq 11022 Addition of positive fract...
mulcompq 11023 Multiplication of positive...
mulcomnq 11024 Multiplication of positive...
adderpqlem 11025 Lemma for ~ adderpq . (Co...
mulerpqlem 11026 Lemma for ~ mulerpq . (Co...
adderpq 11027 Addition is compatible wit...
mulerpq 11028 Multiplication is compatib...
addassnq 11029 Addition of positive fract...
mulassnq 11030 Multiplication of positive...
mulcanenq 11031 Lemma for distributive law...
distrnq 11032 Multiplication of positive...
1nqenq 11033 The equivalence class of r...
mulidnq 11034 Multiplication identity el...
recmulnq 11035 Relationship between recip...
recidnq 11036 A positive fraction times ...
recclnq 11037 Closure law for positive f...
recrecnq 11038 Reciprocal of reciprocal o...
dmrecnq 11039 Domain of reciprocal on po...
ltsonq 11040 'Less than' is a strict or...
lterpq 11041 Compatibility of ordering ...
ltanq 11042 Ordering property of addit...
ltmnq 11043 Ordering property of multi...
1lt2nq 11044 One is less than two (one ...
ltaddnq 11045 The sum of two fractions i...
ltexnq 11046 Ordering on positive fract...
halfnq 11047 One-half of any positive f...
nsmallnq 11048 The is no smallest positiv...
ltbtwnnq 11049 There exists a number betw...
ltrnq 11050 Ordering property of recip...
archnq 11051 For any fraction, there is...
npex 11057 The class of positive real...
elnp 11058 Membership in positive rea...
elnpi 11059 Membership in positive rea...
prn0 11060 A positive real is not emp...
prpssnq 11061 A positive real is a subse...
elprnq 11062 A positive real is a set o...
0npr 11063 The empty set is not a pos...
prcdnq 11064 A positive real is closed ...
prub 11065 A positive fraction not in...
prnmax 11066 A positive real has no lar...
npomex 11067 A simplifying observation,...
prnmadd 11068 A positive real has no lar...
ltrelpr 11069 Positive real 'less than' ...
genpv 11070 Value of general operation...
genpelv 11071 Membership in value of gen...
genpprecl 11072 Pre-closure law for genera...
genpdm 11073 Domain of general operatio...
genpn0 11074 The result of an operation...
genpss 11075 The result of an operation...
genpnnp 11076 The result of an operation...
genpcd 11077 Downward closure of an ope...
genpnmax 11078 An operation on positive r...
genpcl 11079 Closure of an operation on...
genpass 11080 Associativity of an operat...
plpv 11081 Value of addition on posit...
mpv 11082 Value of multiplication on...
dmplp 11083 Domain of addition on posi...
dmmp 11084 Domain of multiplication o...
nqpr 11085 The canonical embedding of...
1pr 11086 The positive real number '...
addclprlem1 11087 Lemma to prove downward cl...
addclprlem2 11088 Lemma to prove downward cl...
addclpr 11089 Closure of addition on pos...
mulclprlem 11090 Lemma to prove downward cl...
mulclpr 11091 Closure of multiplication ...
addcompr 11092 Addition of positive reals...
addasspr 11093 Addition of positive reals...
mulcompr 11094 Multiplication of positive...
mulasspr 11095 Multiplication of positive...
distrlem1pr 11096 Lemma for distributive law...
distrlem4pr 11097 Lemma for distributive law...
distrlem5pr 11098 Lemma for distributive law...
distrpr 11099 Multiplication of positive...
1idpr 11100 1 is an identity element f...
ltprord 11101 Positive real 'less than' ...
psslinpr 11102 Proper subset is a linear ...
ltsopr 11103 Positive real 'less than' ...
prlem934 11104 Lemma 9-3.4 of [Gleason] p...
ltaddpr 11105 The sum of two positive re...
ltaddpr2 11106 The sum of two positive re...
ltexprlem1 11107 Lemma for Proposition 9-3....
ltexprlem2 11108 Lemma for Proposition 9-3....
ltexprlem3 11109 Lemma for Proposition 9-3....
ltexprlem4 11110 Lemma for Proposition 9-3....
ltexprlem5 11111 Lemma for Proposition 9-3....
ltexprlem6 11112 Lemma for Proposition 9-3....
ltexprlem7 11113 Lemma for Proposition 9-3....
ltexpri 11114 Proposition 9-3.5(iv) of [...
ltaprlem 11115 Lemma for Proposition 9-3....
ltapr 11116 Ordering property of addit...
addcanpr 11117 Addition cancellation law ...
prlem936 11118 Lemma 9-3.6 of [Gleason] p...
reclem2pr 11119 Lemma for Proposition 9-3....
reclem3pr 11120 Lemma for Proposition 9-3....
reclem4pr 11121 Lemma for Proposition 9-3....
recexpr 11122 The reciprocal of a positi...
suplem1pr 11123 The union of a nonempty, b...
suplem2pr 11124 The union of a set of posi...
supexpr 11125 The union of a nonempty, b...
enrer 11134 The equivalence relation f...
nrex1 11135 The class of signed reals ...
enrbreq 11136 Equivalence relation for s...
enreceq 11137 Equivalence class equality...
enrex 11138 The equivalence relation f...
ltrelsr 11139 Signed real 'less than' is...
addcmpblnr 11140 Lemma showing compatibilit...
mulcmpblnrlem 11141 Lemma used in lemma showin...
mulcmpblnr 11142 Lemma showing compatibilit...
prsrlem1 11143 Decomposing signed reals i...
addsrmo 11144 There is at most one resul...
mulsrmo 11145 There is at most one resul...
addsrpr 11146 Addition of signed reals i...
mulsrpr 11147 Multiplication of signed r...
ltsrpr 11148 Ordering of signed reals i...
gt0srpr 11149 Greater than zero in terms...
0nsr 11150 The empty set is not a sig...
0r 11151 The constant ` 0R ` is a s...
1sr 11152 The constant ` 1R ` is a s...
m1r 11153 The constant ` -1R ` is a ...
addclsr 11154 Closure of addition on sig...
mulclsr 11155 Closure of multiplication ...
dmaddsr 11156 Domain of addition on sign...
dmmulsr 11157 Domain of multiplication o...
addcomsr 11158 Addition of signed reals i...
addasssr 11159 Addition of signed reals i...
mulcomsr 11160 Multiplication of signed r...
mulasssr 11161 Multiplication of signed r...
distrsr 11162 Multiplication of signed r...
m1p1sr 11163 Minus one plus one is zero...
m1m1sr 11164 Minus one times minus one ...
ltsosr 11165 Signed real 'less than' is...
0lt1sr 11166 0 is less than 1 for signe...
1ne0sr 11167 1 and 0 are distinct for s...
0idsr 11168 The signed real number 0 i...
1idsr 11169 1 is an identity element f...
00sr 11170 A signed real times 0 is 0...
ltasr 11171 Ordering property of addit...
pn0sr 11172 A signed real plus its neg...
negexsr 11173 Existence of negative sign...
recexsrlem 11174 The reciprocal of a positi...
addgt0sr 11175 The sum of two positive si...
mulgt0sr 11176 The product of two positiv...
sqgt0sr 11177 The square of a nonzero si...
recexsr 11178 The reciprocal of a nonzer...
mappsrpr 11179 Mapping from positive sign...
ltpsrpr 11180 Mapping of order from posi...
map2psrpr 11181 Equivalence for positive s...
supsrlem 11182 Lemma for supremum theorem...
supsr 11183 A nonempty, bounded set of...
opelcn 11200 Ordered pair membership in...
opelreal 11201 Ordered pair membership in...
elreal 11202 Membership in class of rea...
elreal2 11203 Ordered pair membership in...
0ncn 11204 The empty set is not a com...
ltrelre 11205 'Less than' is a relation ...
addcnsr 11206 Addition of complex number...
mulcnsr 11207 Multiplication of complex ...
eqresr 11208 Equality of real numbers i...
addresr 11209 Addition of real numbers i...
mulresr 11210 Multiplication of real num...
ltresr 11211 Ordering of real subset of...
ltresr2 11212 Ordering of real subset of...
dfcnqs 11213 Technical trick to permit ...
addcnsrec 11214 Technical trick to permit ...
mulcnsrec 11215 Technical trick to permit ...
axaddf 11216 Addition is an operation o...
axmulf 11217 Multiplication is an opera...
axcnex 11218 The complex numbers form a...
axresscn 11219 The real numbers are a sub...
ax1cn 11220 1 is a complex number. Ax...
axicn 11221 ` _i ` is a complex number...
axaddcl 11222 Closure law for addition o...
axaddrcl 11223 Closure law for addition i...
axmulcl 11224 Closure law for multiplica...
axmulrcl 11225 Closure law for multiplica...
axmulcom 11226 Multiplication of complex ...
axaddass 11227 Addition of complex number...
axmulass 11228 Multiplication of complex ...
axdistr 11229 Distributive law for compl...
axi2m1 11230 i-squared equals -1 (expre...
ax1ne0 11231 1 and 0 are distinct. Axi...
ax1rid 11232 ` 1 ` is an identity eleme...
axrnegex 11233 Existence of negative of r...
axrrecex 11234 Existence of reciprocal of...
axcnre 11235 A complex number can be ex...
axpre-lttri 11236 Ordering on reals satisfie...
axpre-lttrn 11237 Ordering on reals is trans...
axpre-ltadd 11238 Ordering property of addit...
axpre-mulgt0 11239 The product of two positiv...
axpre-sup 11240 A nonempty, bounded-above ...
wuncn 11241 A weak universe containing...
cnex 11267 Alias for ~ ax-cnex . See...
addcl 11268 Alias for ~ ax-addcl , for...
readdcl 11269 Alias for ~ ax-addrcl , fo...
mulcl 11270 Alias for ~ ax-mulcl , for...
remulcl 11271 Alias for ~ ax-mulrcl , fo...
mulcom 11272 Alias for ~ ax-mulcom , fo...
addass 11273 Alias for ~ ax-addass , fo...
mulass 11274 Alias for ~ ax-mulass , fo...
adddi 11275 Alias for ~ ax-distr , for...
recn 11276 A real number is a complex...
reex 11277 The real numbers form a se...
reelprrecn 11278 Reals are a subset of the ...
cnelprrecn 11279 Complex numbers are a subs...
mpoaddf 11280 Addition is an operation o...
mpomulf 11281 Multiplication is an opera...
elimne0 11282 Hypothesis for weak deduct...
adddir 11283 Distributive law for compl...
0cn 11284 Zero is a complex number. ...
0cnd 11285 Zero is a complex number, ...
c0ex 11286 Zero is a set. (Contribut...
1cnd 11287 One is a complex number, d...
1ex 11288 One is a set. (Contribute...
cnre 11289 Alias for ~ ax-cnre , for ...
mulrid 11290 The number 1 is an identit...
mullid 11291 Identity law for multiplic...
1re 11292 The number 1 is real. Thi...
1red 11293 The number 1 is real, dedu...
0re 11294 The number 0 is real. Rem...
0red 11295 The number 0 is real, dedu...
mulridi 11296 Identity law for multiplic...
mullidi 11297 Identity law for multiplic...
addcli 11298 Closure law for addition. ...
mulcli 11299 Closure law for multiplica...
mulcomi 11300 Commutative law for multip...
mulcomli 11301 Commutative law for multip...
addassi 11302 Associative law for additi...
mulassi 11303 Associative law for multip...
adddii 11304 Distributive law (left-dis...
adddiri 11305 Distributive law (right-di...
recni 11306 A real number is a complex...
readdcli 11307 Closure law for addition o...
remulcli 11308 Closure law for multiplica...
mulridd 11309 Identity law for multiplic...
mullidd 11310 Identity law for multiplic...
addcld 11311 Closure law for addition. ...
mulcld 11312 Closure law for multiplica...
mulcomd 11313 Commutative law for multip...
addassd 11314 Associative law for additi...
mulassd 11315 Associative law for multip...
adddid 11316 Distributive law (left-dis...
adddird 11317 Distributive law (right-di...
adddirp1d 11318 Distributive law, plus 1 v...
joinlmuladdmuld 11319 Join AB+CB into (A+C) on L...
recnd 11320 Deduction from real number...
readdcld 11321 Closure law for addition o...
remulcld 11322 Closure law for multiplica...
pnfnre 11333 Plus infinity is not a rea...
pnfnre2 11334 Plus infinity is not a rea...
mnfnre 11335 Minus infinity is not a re...
ressxr 11336 The standard reals are a s...
rexpssxrxp 11337 The Cartesian product of s...
rexr 11338 A standard real is an exte...
0xr 11339 Zero is an extended real. ...
renepnf 11340 No (finite) real equals pl...
renemnf 11341 No real equals minus infin...
rexrd 11342 A standard real is an exte...
renepnfd 11343 No (finite) real equals pl...
renemnfd 11344 No real equals minus infin...
pnfex 11345 Plus infinity exists. (Co...
pnfxr 11346 Plus infinity belongs to t...
pnfnemnf 11347 Plus and minus infinity ar...
mnfnepnf 11348 Minus and plus infinity ar...
mnfxr 11349 Minus infinity belongs to ...
rexri 11350 A standard real is an exte...
1xr 11351 ` 1 ` is an extended real ...
renfdisj 11352 The reals and the infiniti...
ltrelxr 11353 "Less than" is a relation ...
ltrel 11354 "Less than" is a relation....
lerelxr 11355 "Less than or equal to" is...
lerel 11356 "Less than or equal to" is...
xrlenlt 11357 "Less than or equal to" ex...
xrlenltd 11358 "Less than or equal to" ex...
xrltnle 11359 "Less than" expressed in t...
xrnltled 11360 "Not less than" implies "l...
ssxr 11361 The three (non-exclusive) ...
ltxrlt 11362 The standard less-than ` <...
axlttri 11363 Ordering on reals satisfie...
axlttrn 11364 Ordering on reals is trans...
axltadd 11365 Ordering property of addit...
axmulgt0 11366 The product of two positiv...
axsup 11367 A nonempty, bounded-above ...
lttr 11368 Alias for ~ axlttrn , for ...
mulgt0 11369 The product of two positiv...
lenlt 11370 'Less than or equal to' ex...
ltnle 11371 'Less than' expressed in t...
ltso 11372 'Less than' is a strict or...
gtso 11373 'Greater than' is a strict...
lttri2 11374 Consequence of trichotomy....
lttri3 11375 Trichotomy law for 'less t...
lttri4 11376 Trichotomy law for 'less t...
letri3 11377 Trichotomy law. (Contribu...
leloe 11378 'Less than or equal to' ex...
eqlelt 11379 Equality in terms of 'less...
ltle 11380 'Less than' implies 'less ...
leltne 11381 'Less than or equal to' im...
lelttr 11382 Transitive law. (Contribu...
leltletr 11383 Transitive law, weaker for...
ltletr 11384 Transitive law. (Contribu...
ltleletr 11385 Transitive law, weaker for...
letr 11386 Transitive law. (Contribu...
ltnr 11387 'Less than' is irreflexive...
leid 11388 'Less than or equal to' is...
ltne 11389 'Less than' implies not eq...
ltnsym 11390 'Less than' is not symmetr...
ltnsym2 11391 'Less than' is antisymmetr...
letric 11392 Trichotomy law. (Contribu...
ltlen 11393 'Less than' expressed in t...
eqle 11394 Equality implies 'less tha...
eqled 11395 Equality implies 'less tha...
ltadd2 11396 Addition to both sides of ...
ne0gt0 11397 A nonzero nonnegative numb...
lecasei 11398 Ordering elimination by ca...
lelttric 11399 Trichotomy law. (Contribu...
ltlecasei 11400 Ordering elimination by ca...
ltnri 11401 'Less than' is irreflexive...
eqlei 11402 Equality implies 'less tha...
eqlei2 11403 Equality implies 'less tha...
gtneii 11404 'Less than' implies not eq...
ltneii 11405 'Greater than' implies not...
lttri2i 11406 Consequence of trichotomy....
lttri3i 11407 Consequence of trichotomy....
letri3i 11408 Consequence of trichotomy....
leloei 11409 'Less than or equal to' in...
ltleni 11410 'Less than' expressed in t...
ltnsymi 11411 'Less than' is not symmetr...
lenlti 11412 'Less than or equal to' in...
ltnlei 11413 'Less than' in terms of 'l...
ltlei 11414 'Less than' implies 'less ...
ltleii 11415 'Less than' implies 'less ...
ltnei 11416 'Less than' implies not eq...
letrii 11417 Trichotomy law for 'less t...
lttri 11418 'Less than' is transitive....
lelttri 11419 'Less than or equal to', '...
ltletri 11420 'Less than', 'less than or...
letri 11421 'Less than or equal to' is...
le2tri3i 11422 Extended trichotomy law fo...
ltadd2i 11423 Addition to both sides of ...
mulgt0i 11424 The product of two positiv...
mulgt0ii 11425 The product of two positiv...
ltnrd 11426 'Less than' is irreflexive...
gtned 11427 'Less than' implies not eq...
ltned 11428 'Greater than' implies not...
ne0gt0d 11429 A nonzero nonnegative numb...
lttrid 11430 Ordering on reals satisfie...
lttri2d 11431 Consequence of trichotomy....
lttri3d 11432 Consequence of trichotomy....
lttri4d 11433 Trichotomy law for 'less t...
letri3d 11434 Consequence of trichotomy....
leloed 11435 'Less than or equal to' in...
eqleltd 11436 Equality in terms of 'less...
ltlend 11437 'Less than' expressed in t...
lenltd 11438 'Less than or equal to' in...
ltnled 11439 'Less than' in terms of 'l...
ltled 11440 'Less than' implies 'less ...
ltnsymd 11441 'Less than' implies 'less ...
nltled 11442 'Not less than ' implies '...
lensymd 11443 'Less than or equal to' im...
letrid 11444 Trichotomy law for 'less t...
leltned 11445 'Less than or equal to' im...
leneltd 11446 'Less than or equal to' an...
mulgt0d 11447 The product of two positiv...
ltadd2d 11448 Addition to both sides of ...
letrd 11449 Transitive law deduction f...
lelttrd 11450 Transitive law deduction f...
ltadd2dd 11451 Addition to both sides of ...
ltletrd 11452 Transitive law deduction f...
lttrd 11453 Transitive law deduction f...
lelttrdi 11454 If a number is less than a...
dedekind 11455 The Dedekind cut theorem. ...
dedekindle 11456 The Dedekind cut theorem, ...
mul12 11457 Commutative/associative la...
mul32 11458 Commutative/associative la...
mul31 11459 Commutative/associative la...
mul4 11460 Rearrangement of 4 factors...
mul4r 11461 Rearrangement of 4 factors...
muladd11 11462 A simple product of sums e...
1p1times 11463 Two times a number. (Cont...
peano2cn 11464 A theorem for complex numb...
peano2re 11465 A theorem for reals analog...
readdcan 11466 Cancellation law for addit...
00id 11467 ` 0 ` is its own additive ...
mul02lem1 11468 Lemma for ~ mul02 . If an...
mul02lem2 11469 Lemma for ~ mul02 . Zero ...
mul02 11470 Multiplication by ` 0 ` . ...
mul01 11471 Multiplication by ` 0 ` . ...
addrid 11472 ` 0 ` is an additive ident...
cnegex 11473 Existence of the negative ...
cnegex2 11474 Existence of a left invers...
addlid 11475 ` 0 ` is a left identity f...
addcan 11476 Cancellation law for addit...
addcan2 11477 Cancellation law for addit...
addcom 11478 Addition commutes. This u...
addridi 11479 ` 0 ` is an additive ident...
addlidi 11480 ` 0 ` is a left identity f...
mul02i 11481 Multiplication by 0. Theo...
mul01i 11482 Multiplication by ` 0 ` . ...
addcomi 11483 Addition commutes. Based ...
addcomli 11484 Addition commutes. (Contr...
addcani 11485 Cancellation law for addit...
addcan2i 11486 Cancellation law for addit...
mul12i 11487 Commutative/associative la...
mul32i 11488 Commutative/associative la...
mul4i 11489 Rearrangement of 4 factors...
mul02d 11490 Multiplication by 0. Theo...
mul01d 11491 Multiplication by ` 0 ` . ...
addridd 11492 ` 0 ` is an additive ident...
addlidd 11493 ` 0 ` is a left identity f...
addcomd 11494 Addition commutes. Based ...
addcand 11495 Cancellation law for addit...
addcan2d 11496 Cancellation law for addit...
addcanad 11497 Cancelling a term on the l...
addcan2ad 11498 Cancelling a term on the r...
addneintrd 11499 Introducing a term on the ...
addneintr2d 11500 Introducing a term on the ...
mul12d 11501 Commutative/associative la...
mul32d 11502 Commutative/associative la...
mul31d 11503 Commutative/associative la...
mul4d 11504 Rearrangement of 4 factors...
muladd11r 11505 A simple product of sums e...
comraddd 11506 Commute RHS addition, in d...
ltaddneg 11507 Adding a negative number t...
ltaddnegr 11508 Adding a negative number t...
add12 11509 Commutative/associative la...
add32 11510 Commutative/associative la...
add32r 11511 Commutative/associative la...
add4 11512 Rearrangement of 4 terms i...
add42 11513 Rearrangement of 4 terms i...
add12i 11514 Commutative/associative la...
add32i 11515 Commutative/associative la...
add4i 11516 Rearrangement of 4 terms i...
add42i 11517 Rearrangement of 4 terms i...
add12d 11518 Commutative/associative la...
add32d 11519 Commutative/associative la...
add4d 11520 Rearrangement of 4 terms i...
add42d 11521 Rearrangement of 4 terms i...
0cnALT 11526 Alternate proof of ~ 0cn w...
0cnALT2 11527 Alternate proof of ~ 0cnAL...
negeu 11528 Existential uniqueness of ...
subval 11529 Value of subtraction, whic...
negeq 11530 Equality theorem for negat...
negeqi 11531 Equality inference for neg...
negeqd 11532 Equality deduction for neg...
nfnegd 11533 Deduction version of ~ nfn...
nfneg 11534 Bound-variable hypothesis ...
csbnegg 11535 Move class substitution in...
negex 11536 A negative is a set. (Con...
subcl 11537 Closure law for subtractio...
negcl 11538 Closure law for negative. ...
negicn 11539 ` -u _i ` is a complex num...
subf 11540 Subtraction is an operatio...
subadd 11541 Relationship between subtr...
subadd2 11542 Relationship between subtr...
subsub23 11543 Swap subtrahend and result...
pncan 11544 Cancellation law for subtr...
pncan2 11545 Cancellation law for subtr...
pncan3 11546 Subtraction and addition o...
npcan 11547 Cancellation law for subtr...
addsubass 11548 Associative-type law for a...
addsub 11549 Law for addition and subtr...
subadd23 11550 Commutative/associative la...
addsub12 11551 Commutative/associative la...
2addsub 11552 Law for subtraction and ad...
addsubeq4 11553 Relation between sums and ...
pncan3oi 11554 Subtraction and addition o...
mvrraddi 11555 Move the right term in a s...
mvlladdi 11556 Move the left term in a su...
subid 11557 Subtraction of a number fr...
subid1 11558 Identity law for subtracti...
npncan 11559 Cancellation law for subtr...
nppcan 11560 Cancellation law for subtr...
nnpcan 11561 Cancellation law for subtr...
nppcan3 11562 Cancellation law for subtr...
subcan2 11563 Cancellation law for subtr...
subeq0 11564 If the difference between ...
npncan2 11565 Cancellation law for subtr...
subsub2 11566 Law for double subtraction...
nncan 11567 Cancellation law for subtr...
subsub 11568 Law for double subtraction...
nppcan2 11569 Cancellation law for subtr...
subsub3 11570 Law for double subtraction...
subsub4 11571 Law for double subtraction...
sub32 11572 Swap the second and third ...
nnncan 11573 Cancellation law for subtr...
nnncan1 11574 Cancellation law for subtr...
nnncan2 11575 Cancellation law for subtr...
npncan3 11576 Cancellation law for subtr...
pnpcan 11577 Cancellation law for mixed...
pnpcan2 11578 Cancellation law for mixed...
pnncan 11579 Cancellation law for mixed...
ppncan 11580 Cancellation law for mixed...
addsub4 11581 Rearrangement of 4 terms i...
subadd4 11582 Rearrangement of 4 terms i...
sub4 11583 Rearrangement of 4 terms i...
neg0 11584 Minus 0 equals 0. (Contri...
negid 11585 Addition of a number and i...
negsub 11586 Relationship between subtr...
subneg 11587 Relationship between subtr...
negneg 11588 A number is equal to the n...
neg11 11589 Negative is one-to-one. (...
negcon1 11590 Negative contraposition la...
negcon2 11591 Negative contraposition la...
negeq0 11592 A number is zero iff its n...
subcan 11593 Cancellation law for subtr...
negsubdi 11594 Distribution of negative o...
negdi 11595 Distribution of negative o...
negdi2 11596 Distribution of negative o...
negsubdi2 11597 Distribution of negative o...
neg2sub 11598 Relationship between subtr...
renegcli 11599 Closure law for negative o...
resubcli 11600 Closure law for subtractio...
renegcl 11601 Closure law for negative o...
resubcl 11602 Closure law for subtractio...
negreb 11603 The negative of a real is ...
peano2cnm 11604 "Reverse" second Peano pos...
peano2rem 11605 "Reverse" second Peano pos...
negcli 11606 Closure law for negative. ...
negidi 11607 Addition of a number and i...
negnegi 11608 A number is equal to the n...
subidi 11609 Subtraction of a number fr...
subid1i 11610 Identity law for subtracti...
negne0bi 11611 A number is nonzero iff it...
negrebi 11612 The negative of a real is ...
negne0i 11613 The negative of a nonzero ...
subcli 11614 Closure law for subtractio...
pncan3i 11615 Subtraction and addition o...
negsubi 11616 Relationship between subtr...
subnegi 11617 Relationship between subtr...
subeq0i 11618 If the difference between ...
neg11i 11619 Negative is one-to-one. (...
negcon1i 11620 Negative contraposition la...
negcon2i 11621 Negative contraposition la...
negdii 11622 Distribution of negative o...
negsubdii 11623 Distribution of negative o...
negsubdi2i 11624 Distribution of negative o...
subaddi 11625 Relationship between subtr...
subadd2i 11626 Relationship between subtr...
subaddrii 11627 Relationship between subtr...
subsub23i 11628 Swap subtrahend and result...
addsubassi 11629 Associative-type law for s...
addsubi 11630 Law for subtraction and ad...
subcani 11631 Cancellation law for subtr...
subcan2i 11632 Cancellation law for subtr...
pnncani 11633 Cancellation law for mixed...
addsub4i 11634 Rearrangement of 4 terms i...
0reALT 11635 Alternate proof of ~ 0re ....
negcld 11636 Closure law for negative. ...
subidd 11637 Subtraction of a number fr...
subid1d 11638 Identity law for subtracti...
negidd 11639 Addition of a number and i...
negnegd 11640 A number is equal to the n...
negeq0d 11641 A number is zero iff its n...
negne0bd 11642 A number is nonzero iff it...
negcon1d 11643 Contraposition law for una...
negcon1ad 11644 Contraposition law for una...
neg11ad 11645 The negatives of two compl...
negned 11646 If two complex numbers are...
negne0d 11647 The negative of a nonzero ...
negrebd 11648 The negative of a real is ...
subcld 11649 Closure law for subtractio...
pncand 11650 Cancellation law for subtr...
pncan2d 11651 Cancellation law for subtr...
pncan3d 11652 Subtraction and addition o...
npcand 11653 Cancellation law for subtr...
nncand 11654 Cancellation law for subtr...
negsubd 11655 Relationship between subtr...
subnegd 11656 Relationship between subtr...
subeq0d 11657 If the difference between ...
subne0d 11658 Two unequal numbers have n...
subeq0ad 11659 The difference of two comp...
subne0ad 11660 If the difference of two c...
neg11d 11661 If the difference between ...
negdid 11662 Distribution of negative o...
negdi2d 11663 Distribution of negative o...
negsubdid 11664 Distribution of negative o...
negsubdi2d 11665 Distribution of negative o...
neg2subd 11666 Relationship between subtr...
subaddd 11667 Relationship between subtr...
subadd2d 11668 Relationship between subtr...
addsubassd 11669 Associative-type law for s...
addsubd 11670 Law for subtraction and ad...
subadd23d 11671 Commutative/associative la...
addsub12d 11672 Commutative/associative la...
npncand 11673 Cancellation law for subtr...
nppcand 11674 Cancellation law for subtr...
nppcan2d 11675 Cancellation law for subtr...
nppcan3d 11676 Cancellation law for subtr...
subsubd 11677 Law for double subtraction...
subsub2d 11678 Law for double subtraction...
subsub3d 11679 Law for double subtraction...
subsub4d 11680 Law for double subtraction...
sub32d 11681 Swap the second and third ...
nnncand 11682 Cancellation law for subtr...
nnncan1d 11683 Cancellation law for subtr...
nnncan2d 11684 Cancellation law for subtr...
npncan3d 11685 Cancellation law for subtr...
pnpcand 11686 Cancellation law for mixed...
pnpcan2d 11687 Cancellation law for mixed...
pnncand 11688 Cancellation law for mixed...
ppncand 11689 Cancellation law for mixed...
subcand 11690 Cancellation law for subtr...
subcan2d 11691 Cancellation law for subtr...
subcanad 11692 Cancellation law for subtr...
subneintrd 11693 Introducing subtraction on...
subcan2ad 11694 Cancellation law for subtr...
subneintr2d 11695 Introducing subtraction on...
addsub4d 11696 Rearrangement of 4 terms i...
subadd4d 11697 Rearrangement of 4 terms i...
sub4d 11698 Rearrangement of 4 terms i...
2addsubd 11699 Law for subtraction and ad...
addsubeq4d 11700 Relation between sums and ...
subeqxfrd 11701 Transfer two terms of a su...
mvlraddd 11702 Move the right term in a s...
mvlladdd 11703 Move the left term in a su...
mvrraddd 11704 Move the right term in a s...
mvrladdd 11705 Move the left term in a su...
assraddsubd 11706 Associate RHS addition-sub...
subaddeqd 11707 Transfer two terms of a su...
addlsub 11708 Left-subtraction: Subtrac...
addrsub 11709 Right-subtraction: Subtra...
subexsub 11710 A subtraction law: Exchan...
addid0 11711 If adding a number to a an...
addn0nid 11712 Adding a nonzero number to...
pnpncand 11713 Addition/subtraction cance...
subeqrev 11714 Reverse the order of subtr...
addeq0 11715 Two complex numbers add up...
pncan1 11716 Cancellation law for addit...
npcan1 11717 Cancellation law for subtr...
subeq0bd 11718 If two complex numbers are...
renegcld 11719 Closure law for negative o...
resubcld 11720 Closure law for subtractio...
negn0 11721 The image under negation o...
negf1o 11722 Negation is an isomorphism...
kcnktkm1cn 11723 k times k minus 1 is a com...
muladd 11724 Product of two sums. (Con...
subdi 11725 Distribution of multiplica...
subdir 11726 Distribution of multiplica...
ine0 11727 The imaginary unit ` _i ` ...
mulneg1 11728 Product with negative is n...
mulneg2 11729 The product with a negativ...
mulneg12 11730 Swap the negative sign in ...
mul2neg 11731 Product of two negatives. ...
submul2 11732 Convert a subtraction to a...
mulm1 11733 Product with minus one is ...
addneg1mul 11734 Addition with product with...
mulsub 11735 Product of two differences...
mulsub2 11736 Swap the order of subtract...
mulm1i 11737 Product with minus one is ...
mulneg1i 11738 Product with negative is n...
mulneg2i 11739 Product with negative is n...
mul2negi 11740 Product of two negatives. ...
subdii 11741 Distribution of multiplica...
subdiri 11742 Distribution of multiplica...
muladdi 11743 Product of two sums. (Con...
mulm1d 11744 Product with minus one is ...
mulneg1d 11745 Product with negative is n...
mulneg2d 11746 Product with negative is n...
mul2negd 11747 Product of two negatives. ...
subdid 11748 Distribution of multiplica...
subdird 11749 Distribution of multiplica...
muladdd 11750 Product of two sums. (Con...
mulsubd 11751 Product of two differences...
muls1d 11752 Multiplication by one minu...
mulsubfacd 11753 Multiplication followed by...
addmulsub 11754 The product of a sum and a...
subaddmulsub 11755 The difference with a prod...
mulsubaddmulsub 11756 A special difference of a ...
gt0ne0 11757 Positive implies nonzero. ...
lt0ne0 11758 A number which is less tha...
ltadd1 11759 Addition to both sides of ...
leadd1 11760 Addition to both sides of ...
leadd2 11761 Addition to both sides of ...
ltsubadd 11762 'Less than' relationship b...
ltsubadd2 11763 'Less than' relationship b...
lesubadd 11764 'Less than or equal to' re...
lesubadd2 11765 'Less than or equal to' re...
ltaddsub 11766 'Less than' relationship b...
ltaddsub2 11767 'Less than' relationship b...
leaddsub 11768 'Less than or equal to' re...
leaddsub2 11769 'Less than or equal to' re...
suble 11770 Swap subtrahends in an ine...
lesub 11771 Swap subtrahends in an ine...
ltsub23 11772 'Less than' relationship b...
ltsub13 11773 'Less than' relationship b...
le2add 11774 Adding both sides of two '...
ltleadd 11775 Adding both sides of two o...
leltadd 11776 Adding both sides of two o...
lt2add 11777 Adding both sides of two '...
addgt0 11778 The sum of 2 positive numb...
addgegt0 11779 The sum of nonnegative and...
addgtge0 11780 The sum of nonnegative and...
addge0 11781 The sum of 2 nonnegative n...
ltaddpos 11782 Adding a positive number t...
ltaddpos2 11783 Adding a positive number t...
ltsubpos 11784 Subtracting a positive num...
posdif 11785 Comparison of two numbers ...
lesub1 11786 Subtraction from both side...
lesub2 11787 Subtraction of both sides ...
ltsub1 11788 Subtraction from both side...
ltsub2 11789 Subtraction of both sides ...
lt2sub 11790 Subtracting both sides of ...
le2sub 11791 Subtracting both sides of ...
ltneg 11792 Negative of both sides of ...
ltnegcon1 11793 Contraposition of negative...
ltnegcon2 11794 Contraposition of negative...
leneg 11795 Negative of both sides of ...
lenegcon1 11796 Contraposition of negative...
lenegcon2 11797 Contraposition of negative...
lt0neg1 11798 Comparison of a number and...
lt0neg2 11799 Comparison of a number and...
le0neg1 11800 Comparison of a number and...
le0neg2 11801 Comparison of a number and...
addge01 11802 A number is less than or e...
addge02 11803 A number is less than or e...
add20 11804 Two nonnegative numbers ar...
subge0 11805 Nonnegative subtraction. ...
suble0 11806 Nonpositive subtraction. ...
leaddle0 11807 The sum of a real number a...
subge02 11808 Nonnegative subtraction. ...
lesub0 11809 Lemma to show a nonnegativ...
mulge0 11810 The product of two nonnega...
mullt0 11811 The product of two negativ...
msqgt0 11812 A nonzero square is positi...
msqge0 11813 A square is nonnegative. ...
0lt1 11814 0 is less than 1. Theorem...
0le1 11815 0 is less than or equal to...
relin01 11816 An interval law for less t...
ltordlem 11817 Lemma for ~ ltord1 . (Con...
ltord1 11818 Infer an ordering relation...
leord1 11819 Infer an ordering relation...
eqord1 11820 A strictly increasing real...
ltord2 11821 Infer an ordering relation...
leord2 11822 Infer an ordering relation...
eqord2 11823 A strictly decreasing real...
wloglei 11824 Form of ~ wlogle where bot...
wlogle 11825 If the predicate ` ch ( x ...
leidi 11826 'Less than or equal to' is...
gt0ne0i 11827 Positive means nonzero (us...
gt0ne0ii 11828 Positive implies nonzero. ...
msqgt0i 11829 A nonzero square is positi...
msqge0i 11830 A square is nonnegative. ...
addgt0i 11831 Addition of 2 positive num...
addge0i 11832 Addition of 2 nonnegative ...
addgegt0i 11833 Addition of nonnegative an...
addgt0ii 11834 Addition of 2 positive num...
add20i 11835 Two nonnegative numbers ar...
ltnegi 11836 Negative of both sides of ...
lenegi 11837 Negative of both sides of ...
ltnegcon2i 11838 Contraposition of negative...
mulge0i 11839 The product of two nonnega...
lesub0i 11840 Lemma to show a nonnegativ...
ltaddposi 11841 Adding a positive number t...
posdifi 11842 Comparison of two numbers ...
ltnegcon1i 11843 Contraposition of negative...
lenegcon1i 11844 Contraposition of negative...
subge0i 11845 Nonnegative subtraction. ...
ltadd1i 11846 Addition to both sides of ...
leadd1i 11847 Addition to both sides of ...
leadd2i 11848 Addition to both sides of ...
ltsubaddi 11849 'Less than' relationship b...
lesubaddi 11850 'Less than or equal to' re...
ltsubadd2i 11851 'Less than' relationship b...
lesubadd2i 11852 'Less than or equal to' re...
ltaddsubi 11853 'Less than' relationship b...
lt2addi 11854 Adding both side of two in...
le2addi 11855 Adding both side of two in...
gt0ne0d 11856 Positive implies nonzero. ...
lt0ne0d 11857 Something less than zero i...
leidd 11858 'Less than or equal to' is...
msqgt0d 11859 A nonzero square is positi...
msqge0d 11860 A square is nonnegative. ...
lt0neg1d 11861 Comparison of a number and...
lt0neg2d 11862 Comparison of a number and...
le0neg1d 11863 Comparison of a number and...
le0neg2d 11864 Comparison of a number and...
addgegt0d 11865 Addition of nonnegative an...
addgtge0d 11866 Addition of positive and n...
addgt0d 11867 Addition of 2 positive num...
addge0d 11868 Addition of 2 nonnegative ...
mulge0d 11869 The product of two nonnega...
ltnegd 11870 Negative of both sides of ...
lenegd 11871 Negative of both sides of ...
ltnegcon1d 11872 Contraposition of negative...
ltnegcon2d 11873 Contraposition of negative...
lenegcon1d 11874 Contraposition of negative...
lenegcon2d 11875 Contraposition of negative...
ltaddposd 11876 Adding a positive number t...
ltaddpos2d 11877 Adding a positive number t...
ltsubposd 11878 Subtracting a positive num...
posdifd 11879 Comparison of two numbers ...
addge01d 11880 A number is less than or e...
addge02d 11881 A number is less than or e...
subge0d 11882 Nonnegative subtraction. ...
suble0d 11883 Nonpositive subtraction. ...
subge02d 11884 Nonnegative subtraction. ...
ltadd1d 11885 Addition to both sides of ...
leadd1d 11886 Addition to both sides of ...
leadd2d 11887 Addition to both sides of ...
ltsubaddd 11888 'Less than' relationship b...
lesubaddd 11889 'Less than or equal to' re...
ltsubadd2d 11890 'Less than' relationship b...
lesubadd2d 11891 'Less than or equal to' re...
ltaddsubd 11892 'Less than' relationship b...
ltaddsub2d 11893 'Less than' relationship b...
leaddsub2d 11894 'Less than or equal to' re...
subled 11895 Swap subtrahends in an ine...
lesubd 11896 Swap subtrahends in an ine...
ltsub23d 11897 'Less than' relationship b...
ltsub13d 11898 'Less than' relationship b...
lesub1d 11899 Subtraction from both side...
lesub2d 11900 Subtraction of both sides ...
ltsub1d 11901 Subtraction from both side...
ltsub2d 11902 Subtraction of both sides ...
ltadd1dd 11903 Addition to both sides of ...
ltsub1dd 11904 Subtraction from both side...
ltsub2dd 11905 Subtraction of both sides ...
leadd1dd 11906 Addition to both sides of ...
leadd2dd 11907 Addition to both sides of ...
lesub1dd 11908 Subtraction from both side...
lesub2dd 11909 Subtraction of both sides ...
lesub3d 11910 The result of subtracting ...
le2addd 11911 Adding both side of two in...
le2subd 11912 Subtracting both sides of ...
ltleaddd 11913 Adding both sides of two o...
leltaddd 11914 Adding both sides of two o...
lt2addd 11915 Adding both side of two in...
lt2subd 11916 Subtracting both sides of ...
possumd 11917 Condition for a positive s...
sublt0d 11918 When a subtraction gives a...
ltaddsublt 11919 Addition and subtraction o...
1le1 11920 One is less than or equal ...
ixi 11921 ` _i ` times itself is min...
recextlem1 11922 Lemma for ~ recex . (Cont...
recextlem2 11923 Lemma for ~ recex . (Cont...
recex 11924 Existence of reciprocal of...
mulcand 11925 Cancellation law for multi...
mulcan2d 11926 Cancellation law for multi...
mulcanad 11927 Cancellation of a nonzero ...
mulcan2ad 11928 Cancellation of a nonzero ...
mulcan 11929 Cancellation law for multi...
mulcan2 11930 Cancellation law for multi...
mulcani 11931 Cancellation law for multi...
mul0or 11932 If a product is zero, one ...
mulne0b 11933 The product of two nonzero...
mulne0 11934 The product of two nonzero...
mulne0i 11935 The product of two nonzero...
muleqadd 11936 Property of numbers whose ...
receu 11937 Existential uniqueness of ...
mulnzcnf 11938 Multiplication maps nonzer...
msq0i 11939 A number is zero iff its s...
mul0ori 11940 If a product is zero, one ...
msq0d 11941 A number is zero iff its s...
mul0ord 11942 If a product is zero, one ...
mulne0bd 11943 The product of two nonzero...
mulne0d 11944 The product of two nonzero...
mulcan1g 11945 A generalized form of the ...
mulcan2g 11946 A generalized form of the ...
mulne0bad 11947 A factor of a nonzero comp...
mulne0bbd 11948 A factor of a nonzero comp...
1div0 11951 You can't divide by zero, ...
1div0OLD 11952 Obsolete version of ~ 1div...
divval 11953 Value of division: if ` A ...
divmul 11954 Relationship between divis...
divmul2 11955 Relationship between divis...
divmul3 11956 Relationship between divis...
divcl 11957 Closure law for division. ...
reccl 11958 Closure law for reciprocal...
divcan2 11959 A cancellation law for div...
divcan1 11960 A cancellation law for div...
diveq0 11961 A ratio is zero iff the nu...
divne0b 11962 The ratio of nonzero numbe...
divne0 11963 The ratio of nonzero numbe...
recne0 11964 The reciprocal of a nonzer...
recid 11965 Multiplication of a number...
recid2 11966 Multiplication of a number...
divrec 11967 Relationship between divis...
divrec2 11968 Relationship between divis...
divass 11969 An associative law for div...
div23 11970 A commutative/associative ...
div32 11971 A commutative/associative ...
div13 11972 A commutative/associative ...
div12 11973 A commutative/associative ...
divmulass 11974 An associative law for div...
divmulasscom 11975 An associative/commutative...
divdir 11976 Distribution of division o...
divcan3 11977 A cancellation law for div...
divcan4 11978 A cancellation law for div...
div11 11979 One-to-one relationship fo...
div11OLD 11980 Obsolete version of ~ div1...
diveq1 11981 Equality in terms of unit ...
divid 11982 A number divided by itself...
dividOLD 11983 Obsolete version of ~ divi...
div0 11984 Division into zero is zero...
div0OLD 11985 Obsolete version of ~ div0...
div1 11986 A number divided by 1 is i...
1div1e1 11987 1 divided by 1 is 1. (Con...
divneg 11988 Move negative sign inside ...
muldivdir 11989 Distribution of division o...
divsubdir 11990 Distribution of division o...
subdivcomb1 11991 Bring a term in a subtract...
subdivcomb2 11992 Bring a term in a subtract...
recrec 11993 A number is equal to the r...
rec11 11994 Reciprocal is one-to-one. ...
rec11r 11995 Mutual reciprocals. (Cont...
divmuldiv 11996 Multiplication of two rati...
divdivdiv 11997 Division of two ratios. T...
divcan5 11998 Cancellation of common fac...
divmul13 11999 Swap the denominators in t...
divmul24 12000 Swap the numerators in the...
divmuleq 12001 Cross-multiply in an equal...
recdiv 12002 The reciprocal of a ratio....
divcan6 12003 Cancellation of inverted f...
divdiv32 12004 Swap denominators in a div...
divcan7 12005 Cancel equal divisors in a...
dmdcan 12006 Cancellation law for divis...
divdiv1 12007 Division into a fraction. ...
divdiv2 12008 Division by a fraction. (...
recdiv2 12009 Division into a reciprocal...
ddcan 12010 Cancellation in a double d...
divadddiv 12011 Addition of two ratios. T...
divsubdiv 12012 Subtraction of two ratios....
conjmul 12013 Two numbers whose reciproc...
rereccl 12014 Closure law for reciprocal...
redivcl 12015 Closure law for division o...
eqneg 12016 A number equal to its nega...
eqnegd 12017 A complex number equals it...
eqnegad 12018 If a complex number equals...
div2neg 12019 Quotient of two negatives....
divneg2 12020 Move negative sign inside ...
recclzi 12021 Closure law for reciprocal...
recne0zi 12022 The reciprocal of a nonzer...
recidzi 12023 Multiplication of a number...
div1i 12024 A number divided by 1 is i...
eqnegi 12025 A number equal to its nega...
reccli 12026 Closure law for reciprocal...
recidi 12027 Multiplication of a number...
recreci 12028 A number is equal to the r...
dividi 12029 A number divided by itself...
div0i 12030 Division into zero is zero...
divclzi 12031 Closure law for division. ...
divcan1zi 12032 A cancellation law for div...
divcan2zi 12033 A cancellation law for div...
divreczi 12034 Relationship between divis...
divcan3zi 12035 A cancellation law for div...
divcan4zi 12036 A cancellation law for div...
rec11i 12037 Reciprocal is one-to-one. ...
divcli 12038 Closure law for division. ...
divcan2i 12039 A cancellation law for div...
divcan1i 12040 A cancellation law for div...
divreci 12041 Relationship between divis...
divcan3i 12042 A cancellation law for div...
divcan4i 12043 A cancellation law for div...
divne0i 12044 The ratio of nonzero numbe...
rec11ii 12045 Reciprocal is one-to-one. ...
divasszi 12046 An associative law for div...
divmulzi 12047 Relationship between divis...
divdirzi 12048 Distribution of division o...
divdiv23zi 12049 Swap denominators in a div...
divmuli 12050 Relationship between divis...
divdiv32i 12051 Swap denominators in a div...
divassi 12052 An associative law for div...
divdiri 12053 Distribution of division o...
div23i 12054 A commutative/associative ...
div11i 12055 One-to-one relationship fo...
divmuldivi 12056 Multiplication of two rati...
divmul13i 12057 Swap denominators of two r...
divadddivi 12058 Addition of two ratios. T...
divdivdivi 12059 Division of two ratios. T...
rerecclzi 12060 Closure law for reciprocal...
rereccli 12061 Closure law for reciprocal...
redivclzi 12062 Closure law for division o...
redivcli 12063 Closure law for division o...
div1d 12064 A number divided by 1 is i...
reccld 12065 Closure law for reciprocal...
recne0d 12066 The reciprocal of a nonzer...
recidd 12067 Multiplication of a number...
recid2d 12068 Multiplication of a number...
recrecd 12069 A number is equal to the r...
dividd 12070 A number divided by itself...
div0d 12071 Division into zero is zero...
divcld 12072 Closure law for division. ...
divcan1d 12073 A cancellation law for div...
divcan2d 12074 A cancellation law for div...
divrecd 12075 Relationship between divis...
divrec2d 12076 Relationship between divis...
divcan3d 12077 A cancellation law for div...
divcan4d 12078 A cancellation law for div...
diveq0d 12079 A ratio is zero iff the nu...
diveq1d 12080 Equality in terms of unit ...
diveq1ad 12081 The quotient of two comple...
diveq0ad 12082 A fraction of complex numb...
divne1d 12083 If two complex numbers are...
divne0bd 12084 A ratio is zero iff the nu...
divnegd 12085 Move negative sign inside ...
divneg2d 12086 Move negative sign inside ...
div2negd 12087 Quotient of two negatives....
divne0d 12088 The ratio of nonzero numbe...
recdivd 12089 The reciprocal of a ratio....
recdiv2d 12090 Division into a reciprocal...
divcan6d 12091 Cancellation of inverted f...
ddcand 12092 Cancellation in a double d...
rec11d 12093 Reciprocal is one-to-one. ...
divmuld 12094 Relationship between divis...
div32d 12095 A commutative/associative ...
div13d 12096 A commutative/associative ...
divdiv32d 12097 Swap denominators in a div...
divcan5d 12098 Cancellation of common fac...
divcan5rd 12099 Cancellation of common fac...
divcan7d 12100 Cancel equal divisors in a...
dmdcand 12101 Cancellation law for divis...
dmdcan2d 12102 Cancellation law for divis...
divdiv1d 12103 Division into a fraction. ...
divdiv2d 12104 Division by a fraction. (...
divmul2d 12105 Relationship between divis...
divmul3d 12106 Relationship between divis...
divassd 12107 An associative law for div...
div12d 12108 A commutative/associative ...
div23d 12109 A commutative/associative ...
divdird 12110 Distribution of division o...
divsubdird 12111 Distribution of division o...
div11d 12112 One-to-one relationship fo...
divmuldivd 12113 Multiplication of two rati...
divmul13d 12114 Swap denominators of two r...
divmul24d 12115 Swap the numerators in the...
divadddivd 12116 Addition of two ratios. T...
divsubdivd 12117 Subtraction of two ratios....
divmuleqd 12118 Cross-multiply in an equal...
divdivdivd 12119 Division of two ratios. T...
diveq1bd 12120 If two complex numbers are...
div2sub 12121 Swap the order of subtract...
div2subd 12122 Swap subtrahend and minuen...
rereccld 12123 Closure law for reciprocal...
redivcld 12124 Closure law for division o...
subrec 12125 Subtraction of reciprocals...
subreci 12126 Subtraction of reciprocals...
subrecd 12127 Subtraction of reciprocals...
mvllmuld 12128 Move the left term in a pr...
mvllmuli 12129 Move the left term in a pr...
ldiv 12130 Left-division. (Contribut...
rdiv 12131 Right-division. (Contribu...
mdiv 12132 A division law. (Contribu...
lineq 12133 Solution of a (scalar) lin...
elimgt0 12134 Hypothesis for weak deduct...
elimge0 12135 Hypothesis for weak deduct...
ltp1 12136 A number is less than itse...
lep1 12137 A number is less than or e...
ltm1 12138 A number minus 1 is less t...
lem1 12139 A number minus 1 is less t...
letrp1 12140 A transitive property of '...
p1le 12141 A transitive property of p...
recgt0 12142 The reciprocal of a positi...
prodgt0 12143 Infer that a multiplicand ...
prodgt02 12144 Infer that a multiplier is...
ltmul1a 12145 Lemma for ~ ltmul1 . Mult...
ltmul1 12146 Multiplication of both sid...
ltmul2 12147 Multiplication of both sid...
lemul1 12148 Multiplication of both sid...
lemul2 12149 Multiplication of both sid...
lemul1a 12150 Multiplication of both sid...
lemul2a 12151 Multiplication of both sid...
ltmul12a 12152 Comparison of product of t...
lemul12b 12153 Comparison of product of t...
lemul12a 12154 Comparison of product of t...
mulgt1OLD 12155 Obsolete version of ~ mulg...
ltmulgt11 12156 Multiplication by a number...
ltmulgt12 12157 Multiplication by a number...
mulgt1 12158 The product of two numbers...
lemulge11 12159 Multiplication by a number...
lemulge12 12160 Multiplication by a number...
ltdiv1 12161 Division of both sides of ...
lediv1 12162 Division of both sides of ...
gt0div 12163 Division of a positive num...
ge0div 12164 Division of a nonnegative ...
divgt0 12165 The ratio of two positive ...
divge0 12166 The ratio of nonnegative a...
mulge0b 12167 A condition for multiplica...
mulle0b 12168 A condition for multiplica...
mulsuble0b 12169 A condition for multiplica...
ltmuldiv 12170 'Less than' relationship b...
ltmuldiv2 12171 'Less than' relationship b...
ltdivmul 12172 'Less than' relationship b...
ledivmul 12173 'Less than or equal to' re...
ltdivmul2 12174 'Less than' relationship b...
lt2mul2div 12175 'Less than' relationship b...
ledivmul2 12176 'Less than or equal to' re...
lemuldiv 12177 'Less than or equal' relat...
lemuldiv2 12178 'Less than or equal' relat...
ltrec 12179 The reciprocal of both sid...
lerec 12180 The reciprocal of both sid...
lt2msq1 12181 Lemma for ~ lt2msq . (Con...
lt2msq 12182 Two nonnegative numbers co...
ltdiv2 12183 Division of a positive num...
ltrec1 12184 Reciprocal swap in a 'less...
lerec2 12185 Reciprocal swap in a 'less...
ledivdiv 12186 Invert ratios of positive ...
lediv2 12187 Division of a positive num...
ltdiv23 12188 Swap denominator with othe...
lediv23 12189 Swap denominator with othe...
lediv12a 12190 Comparison of ratio of two...
lediv2a 12191 Division of both sides of ...
reclt1 12192 The reciprocal of a positi...
recgt1 12193 The reciprocal of a positi...
recgt1i 12194 The reciprocal of a number...
recp1lt1 12195 Construct a number less th...
recreclt 12196 Given a positive number ` ...
le2msq 12197 The square function on non...
msq11 12198 The square of a nonnegativ...
ledivp1 12199 "Less than or equal to" an...
squeeze0 12200 If a nonnegative number is...
ltp1i 12201 A number is less than itse...
recgt0i 12202 The reciprocal of a positi...
recgt0ii 12203 The reciprocal of a positi...
prodgt0i 12204 Infer that a multiplicand ...
divgt0i 12205 The ratio of two positive ...
divge0i 12206 The ratio of nonnegative a...
ltreci 12207 The reciprocal of both sid...
lereci 12208 The reciprocal of both sid...
lt2msqi 12209 The square function on non...
le2msqi 12210 The square function on non...
msq11i 12211 The square of a nonnegativ...
divgt0i2i 12212 The ratio of two positive ...
ltrecii 12213 The reciprocal of both sid...
divgt0ii 12214 The ratio of two positive ...
ltmul1i 12215 Multiplication of both sid...
ltdiv1i 12216 Division of both sides of ...
ltmuldivi 12217 'Less than' relationship b...
ltmul2i 12218 Multiplication of both sid...
lemul1i 12219 Multiplication of both sid...
lemul2i 12220 Multiplication of both sid...
ltdiv23i 12221 Swap denominator with othe...
ledivp1i 12222 "Less than or equal to" an...
ltdivp1i 12223 Less-than and division rel...
ltdiv23ii 12224 Swap denominator with othe...
ltmul1ii 12225 Multiplication of both sid...
ltdiv1ii 12226 Division of both sides of ...
ltp1d 12227 A number is less than itse...
lep1d 12228 A number is less than or e...
ltm1d 12229 A number minus 1 is less t...
lem1d 12230 A number minus 1 is less t...
recgt0d 12231 The reciprocal of a positi...
divgt0d 12232 The ratio of two positive ...
mulgt1d 12233 The product of two numbers...
lemulge11d 12234 Multiplication by a number...
lemulge12d 12235 Multiplication by a number...
lemul1ad 12236 Multiplication of both sid...
lemul2ad 12237 Multiplication of both sid...
ltmul12ad 12238 Comparison of product of t...
lemul12ad 12239 Comparison of product of t...
lemul12bd 12240 Comparison of product of t...
fimaxre 12241 A finite set of real numbe...
fimaxre2 12242 A nonempty finite set of r...
fimaxre3 12243 A nonempty finite set of r...
fiminre 12244 A nonempty finite set of r...
fiminre2 12245 A nonempty finite set of r...
negfi 12246 The negation of a finite s...
lbreu 12247 If a set of reals contains...
lbcl 12248 If a set of reals contains...
lble 12249 If a set of reals contains...
lbinf 12250 If a set of reals contains...
lbinfcl 12251 If a set of reals contains...
lbinfle 12252 If a set of reals contains...
sup2 12253 A nonempty, bounded-above ...
sup3 12254 A version of the completen...
infm3lem 12255 Lemma for ~ infm3 . (Cont...
infm3 12256 The completeness axiom for...
suprcl 12257 Closure of supremum of a n...
suprub 12258 A member of a nonempty bou...
suprubd 12259 Natural deduction form of ...
suprcld 12260 Natural deduction form of ...
suprlub 12261 The supremum of a nonempty...
suprnub 12262 An upper bound is not less...
suprleub 12263 The supremum of a nonempty...
supaddc 12264 The supremum function dist...
supadd 12265 The supremum function dist...
supmul1 12266 The supremum function dist...
supmullem1 12267 Lemma for ~ supmul . (Con...
supmullem2 12268 Lemma for ~ supmul . (Con...
supmul 12269 The supremum function dist...
sup3ii 12270 A version of the completen...
suprclii 12271 Closure of supremum of a n...
suprubii 12272 A member of a nonempty bou...
suprlubii 12273 The supremum of a nonempty...
suprnubii 12274 An upper bound is not less...
suprleubii 12275 The supremum of a nonempty...
riotaneg 12276 The negative of the unique...
negiso 12277 Negation is an order anti-...
dfinfre 12278 The infimum of a set of re...
infrecl 12279 Closure of infimum of a no...
infrenegsup 12280 The infimum of a set of re...
infregelb 12281 Any lower bound of a nonem...
infrelb 12282 If a nonempty set of real ...
infrefilb 12283 The infimum of a finite se...
supfirege 12284 The supremum of a finite s...
inelr 12285 The imaginary unit ` _i ` ...
rimul 12286 A real number times the im...
cru 12287 The representation of comp...
crne0 12288 The real representation of...
creur 12289 The real part of a complex...
creui 12290 The imaginary part of a co...
cju 12291 The complex conjugate of a...
ofsubeq0 12292 Function analogue of ~ sub...
ofnegsub 12293 Function analogue of ~ neg...
ofsubge0 12294 Function analogue of ~ sub...
nnexALT 12297 Alternate proof of ~ nnex ...
peano5nni 12298 Peano's inductive postulat...
nnssre 12299 The positive integers are ...
nnsscn 12300 The positive integers are ...
nnex 12301 The set of positive intege...
nnre 12302 A positive integer is a re...
nncn 12303 A positive integer is a co...
nnrei 12304 A positive integer is a re...
nncni 12305 A positive integer is a co...
1nn 12306 Peano postulate: 1 is a po...
peano2nn 12307 Peano postulate: a success...
dfnn2 12308 Alternate definition of th...
dfnn3 12309 Alternate definition of th...
nnred 12310 A positive integer is a re...
nncnd 12311 A positive integer is a co...
peano2nnd 12312 Peano postulate: a success...
nnind 12313 Principle of Mathematical ...
nnindALT 12314 Principle of Mathematical ...
nnindd 12315 Principle of Mathematical ...
nn1m1nn 12316 Every positive integer is ...
nn1suc 12317 If a statement holds for 1...
nnaddcl 12318 Closure of addition of pos...
nnmulcl 12319 Closure of multiplication ...
nnmulcli 12320 Closure of multiplication ...
nnmtmip 12321 "Minus times minus is plus...
nn2ge 12322 There exists a positive in...
nnge1 12323 A positive integer is one ...
nngt1ne1 12324 A positive integer is grea...
nnle1eq1 12325 A positive integer is less...
nngt0 12326 A positive integer is posi...
nnnlt1 12327 A positive integer is not ...
nnnle0 12328 A positive integer is not ...
nnne0 12329 A positive integer is nonz...
nnneneg 12330 No positive integer is equ...
0nnn 12331 Zero is not a positive int...
0nnnALT 12332 Alternate proof of ~ 0nnn ...
nnne0ALT 12333 Alternate version of ~ nnn...
nngt0i 12334 A positive integer is posi...
nnne0i 12335 A positive integer is nonz...
nndivre 12336 The quotient of a real and...
nnrecre 12337 The reciprocal of a positi...
nnrecgt0 12338 The reciprocal of a positi...
nnsub 12339 Subtraction of positive in...
nnsubi 12340 Subtraction of positive in...
nndiv 12341 Two ways to express " ` A ...
nndivtr 12342 Transitive property of div...
nnge1d 12343 A positive integer is one ...
nngt0d 12344 A positive integer is posi...
nnne0d 12345 A positive integer is nonz...
nnrecred 12346 The reciprocal of a positi...
nnaddcld 12347 Closure of addition of pos...
nnmulcld 12348 Closure of multiplication ...
nndivred 12349 A positive integer is one ...
0ne1 12366 Zero is different from one...
1m1e0 12367 One minus one equals zero....
2nn 12368 2 is a positive integer. ...
2re 12369 The number 2 is real. (Co...
2cn 12370 The number 2 is a complex ...
2cnALT 12371 Alternate proof of ~ 2cn ....
2ex 12372 The number 2 is a set. (C...
2cnd 12373 The number 2 is a complex ...
3nn 12374 3 is a positive integer. ...
3re 12375 The number 3 is real. (Co...
3cn 12376 The number 3 is a complex ...
3ex 12377 The number 3 is a set. (C...
4nn 12378 4 is a positive integer. ...
4re 12379 The number 4 is real. (Co...
4cn 12380 The number 4 is a complex ...
5nn 12381 5 is a positive integer. ...
5re 12382 The number 5 is real. (Co...
5cn 12383 The number 5 is a complex ...
6nn 12384 6 is a positive integer. ...
6re 12385 The number 6 is real. (Co...
6cn 12386 The number 6 is a complex ...
7nn 12387 7 is a positive integer. ...
7re 12388 The number 7 is real. (Co...
7cn 12389 The number 7 is a complex ...
8nn 12390 8 is a positive integer. ...
8re 12391 The number 8 is real. (Co...
8cn 12392 The number 8 is a complex ...
9nn 12393 9 is a positive integer. ...
9re 12394 The number 9 is real. (Co...
9cn 12395 The number 9 is a complex ...
0le0 12396 Zero is nonnegative. (Con...
0le2 12397 The number 0 is less than ...
2pos 12398 The number 2 is positive. ...
2ne0 12399 The number 2 is nonzero. ...
3pos 12400 The number 3 is positive. ...
3ne0 12401 The number 3 is nonzero. ...
4pos 12402 The number 4 is positive. ...
4ne0 12403 The number 4 is nonzero. ...
5pos 12404 The number 5 is positive. ...
6pos 12405 The number 6 is positive. ...
7pos 12406 The number 7 is positive. ...
8pos 12407 The number 8 is positive. ...
9pos 12408 The number 9 is positive. ...
neg1cn 12409 -1 is a complex number. (...
neg1rr 12410 -1 is a real number. (Con...
neg1ne0 12411 -1 is nonzero. (Contribut...
neg1lt0 12412 -1 is less than 0. (Contr...
negneg1e1 12413 ` -u -u 1 ` is 1. (Contri...
1pneg1e0 12414 ` 1 + -u 1 ` is 0. (Contr...
0m0e0 12415 0 minus 0 equals 0. (Cont...
1m0e1 12416 1 - 0 = 1. (Contributed b...
0p1e1 12417 0 + 1 = 1. (Contributed b...
fv0p1e1 12418 Function value at ` N + 1 ...
1p0e1 12419 1 + 0 = 1. (Contributed b...
1p1e2 12420 1 + 1 = 2. (Contributed b...
2m1e1 12421 2 - 1 = 1. The result is ...
1e2m1 12422 1 = 2 - 1. (Contributed b...
3m1e2 12423 3 - 1 = 2. (Contributed b...
4m1e3 12424 4 - 1 = 3. (Contributed b...
5m1e4 12425 5 - 1 = 4. (Contributed b...
6m1e5 12426 6 - 1 = 5. (Contributed b...
7m1e6 12427 7 - 1 = 6. (Contributed b...
8m1e7 12428 8 - 1 = 7. (Contributed b...
9m1e8 12429 9 - 1 = 8. (Contributed b...
2p2e4 12430 Two plus two equals four. ...
2times 12431 Two times a number. (Cont...
times2 12432 A number times 2. (Contri...
2timesi 12433 Two times a number. (Cont...
times2i 12434 A number times 2. (Contri...
2txmxeqx 12435 Two times a complex number...
2div2e1 12436 2 divided by 2 is 1. (Con...
2p1e3 12437 2 + 1 = 3. (Contributed b...
1p2e3 12438 1 + 2 = 3. For a shorter ...
1p2e3ALT 12439 Alternate proof of ~ 1p2e3...
3p1e4 12440 3 + 1 = 4. (Contributed b...
4p1e5 12441 4 + 1 = 5. (Contributed b...
5p1e6 12442 5 + 1 = 6. (Contributed b...
6p1e7 12443 6 + 1 = 7. (Contributed b...
7p1e8 12444 7 + 1 = 8. (Contributed b...
8p1e9 12445 8 + 1 = 9. (Contributed b...
3p2e5 12446 3 + 2 = 5. (Contributed b...
3p3e6 12447 3 + 3 = 6. (Contributed b...
4p2e6 12448 4 + 2 = 6. (Contributed b...
4p3e7 12449 4 + 3 = 7. (Contributed b...
4p4e8 12450 4 + 4 = 8. (Contributed b...
5p2e7 12451 5 + 2 = 7. (Contributed b...
5p3e8 12452 5 + 3 = 8. (Contributed b...
5p4e9 12453 5 + 4 = 9. (Contributed b...
6p2e8 12454 6 + 2 = 8. (Contributed b...
6p3e9 12455 6 + 3 = 9. (Contributed b...
7p2e9 12456 7 + 2 = 9. (Contributed b...
1t1e1 12457 1 times 1 equals 1. (Cont...
2t1e2 12458 2 times 1 equals 2. (Cont...
2t2e4 12459 2 times 2 equals 4. (Cont...
3t1e3 12460 3 times 1 equals 3. (Cont...
3t2e6 12461 3 times 2 equals 6. (Cont...
3t3e9 12462 3 times 3 equals 9. (Cont...
4t2e8 12463 4 times 2 equals 8. (Cont...
2t0e0 12464 2 times 0 equals 0. (Cont...
4d2e2 12465 One half of four is two. ...
1lt2 12466 1 is less than 2. (Contri...
2lt3 12467 2 is less than 3. (Contri...
1lt3 12468 1 is less than 3. (Contri...
3lt4 12469 3 is less than 4. (Contri...
2lt4 12470 2 is less than 4. (Contri...
1lt4 12471 1 is less than 4. (Contri...
4lt5 12472 4 is less than 5. (Contri...
3lt5 12473 3 is less than 5. (Contri...
2lt5 12474 2 is less than 5. (Contri...
1lt5 12475 1 is less than 5. (Contri...
5lt6 12476 5 is less than 6. (Contri...
4lt6 12477 4 is less than 6. (Contri...
3lt6 12478 3 is less than 6. (Contri...
2lt6 12479 2 is less than 6. (Contri...
1lt6 12480 1 is less than 6. (Contri...
6lt7 12481 6 is less than 7. (Contri...
5lt7 12482 5 is less than 7. (Contri...
4lt7 12483 4 is less than 7. (Contri...
3lt7 12484 3 is less than 7. (Contri...
2lt7 12485 2 is less than 7. (Contri...
1lt7 12486 1 is less than 7. (Contri...
7lt8 12487 7 is less than 8. (Contri...
6lt8 12488 6 is less than 8. (Contri...
5lt8 12489 5 is less than 8. (Contri...
4lt8 12490 4 is less than 8. (Contri...
3lt8 12491 3 is less than 8. (Contri...
2lt8 12492 2 is less than 8. (Contri...
1lt8 12493 1 is less than 8. (Contri...
8lt9 12494 8 is less than 9. (Contri...
7lt9 12495 7 is less than 9. (Contri...
6lt9 12496 6 is less than 9. (Contri...
5lt9 12497 5 is less than 9. (Contri...
4lt9 12498 4 is less than 9. (Contri...
3lt9 12499 3 is less than 9. (Contri...
2lt9 12500 2 is less than 9. (Contri...
1lt9 12501 1 is less than 9. (Contri...
0ne2 12502 0 is not equal to 2. (Con...
1ne2 12503 1 is not equal to 2. (Con...
1le2 12504 1 is less than or equal to...
2cnne0 12505 2 is a nonzero complex num...
2rene0 12506 2 is a nonzero real number...
1le3 12507 1 is less than or equal to...
neg1mulneg1e1 12508 ` -u 1 x. -u 1 ` is 1. (C...
halfre 12509 One-half is real. (Contri...
halfcn 12510 One-half is a complex numb...
halfgt0 12511 One-half is greater than z...
halfge0 12512 One-half is not negative. ...
halflt1 12513 One-half is less than one....
1mhlfehlf 12514 Prove that 1 - 1/2 = 1/2. ...
8th4div3 12515 An eighth of four thirds i...
halfpm6th 12516 One half plus or minus one...
it0e0 12517 i times 0 equals 0. (Cont...
2mulicn 12518 ` ( 2 x. _i ) e. CC ` . (...
2muline0 12519 ` ( 2 x. _i ) =/= 0 ` . (...
halfcl 12520 Closure of half of a numbe...
rehalfcl 12521 Real closure of half. (Co...
half0 12522 Half of a number is zero i...
2halves 12523 Two halves make a whole. ...
halfpos2 12524 A number is positive iff i...
halfpos 12525 A positive number is great...
halfnneg2 12526 A number is nonnegative if...
halfaddsubcl 12527 Closure of half-sum and ha...
halfaddsub 12528 Sum and difference of half...
subhalfhalf 12529 Subtracting the half of a ...
lt2halves 12530 A sum is less than the who...
addltmul 12531 Sum is less than product f...
nominpos 12532 There is no smallest posit...
avglt1 12533 Ordering property for aver...
avglt2 12534 Ordering property for aver...
avgle1 12535 Ordering property for aver...
avgle2 12536 Ordering property for aver...
avgle 12537 The average of two numbers...
2timesd 12538 Two times a number. (Cont...
times2d 12539 A number times 2. (Contri...
halfcld 12540 Closure of half of a numbe...
2halvesd 12541 Two halves make a whole. ...
rehalfcld 12542 Real closure of half. (Co...
lt2halvesd 12543 A sum is less than the who...
rehalfcli 12544 Half a real number is real...
lt2addmuld 12545 If two real numbers are le...
add1p1 12546 Adding two times 1 to a nu...
sub1m1 12547 Subtracting two times 1 fr...
cnm2m1cnm3 12548 Subtracting 2 and afterwar...
xp1d2m1eqxm1d2 12549 A complex number increased...
div4p1lem1div2 12550 An integer greater than 5,...
nnunb 12551 The set of positive intege...
arch 12552 Archimedean property of re...
nnrecl 12553 There exists a positive in...
bndndx 12554 A bounded real sequence ` ...
elnn0 12557 Nonnegative integers expre...
nnssnn0 12558 Positive naturals are a su...
nn0ssre 12559 Nonnegative integers are a...
nn0sscn 12560 Nonnegative integers are a...
nn0ex 12561 The set of nonnegative int...
nnnn0 12562 A positive integer is a no...
nnnn0i 12563 A positive integer is a no...
nn0re 12564 A nonnegative integer is a...
nn0cn 12565 A nonnegative integer is a...
nn0rei 12566 A nonnegative integer is a...
nn0cni 12567 A nonnegative integer is a...
dfn2 12568 The set of positive intege...
elnnne0 12569 The positive integer prope...
0nn0 12570 0 is a nonnegative integer...
1nn0 12571 1 is a nonnegative integer...
2nn0 12572 2 is a nonnegative integer...
3nn0 12573 3 is a nonnegative integer...
4nn0 12574 4 is a nonnegative integer...
5nn0 12575 5 is a nonnegative integer...
6nn0 12576 6 is a nonnegative integer...
7nn0 12577 7 is a nonnegative integer...
8nn0 12578 8 is a nonnegative integer...
9nn0 12579 9 is a nonnegative integer...
nn0ge0 12580 A nonnegative integer is g...
nn0nlt0 12581 A nonnegative integer is n...
nn0ge0i 12582 Nonnegative integers are n...
nn0le0eq0 12583 A nonnegative integer is l...
nn0p1gt0 12584 A nonnegative integer incr...
nnnn0addcl 12585 A positive integer plus a ...
nn0nnaddcl 12586 A nonnegative integer plus...
0mnnnnn0 12587 The result of subtracting ...
un0addcl 12588 If ` S ` is closed under a...
un0mulcl 12589 If ` S ` is closed under m...
nn0addcl 12590 Closure of addition of non...
nn0mulcl 12591 Closure of multiplication ...
nn0addcli 12592 Closure of addition of non...
nn0mulcli 12593 Closure of multiplication ...
nn0p1nn 12594 A nonnegative integer plus...
peano2nn0 12595 Second Peano postulate for...
nnm1nn0 12596 A positive integer minus 1...
elnn0nn 12597 The nonnegative integer pr...
elnnnn0 12598 The positive integer prope...
elnnnn0b 12599 The positive integer prope...
elnnnn0c 12600 The positive integer prope...
nn0addge1 12601 A number is less than or e...
nn0addge2 12602 A number is less than or e...
nn0addge1i 12603 A number is less than or e...
nn0addge2i 12604 A number is less than or e...
nn0sub 12605 Subtraction of nonnegative...
ltsubnn0 12606 Subtracting a nonnegative ...
nn0negleid 12607 A nonnegative integer is g...
difgtsumgt 12608 If the difference of a rea...
nn0le2xi 12609 A nonnegative integer is l...
nn0lele2xi 12610 'Less than or equal to' im...
fcdmnn0supp 12611 Two ways to write the supp...
fcdmnn0fsupp 12612 A function into ` NN0 ` is...
fcdmnn0suppg 12613 Version of ~ fcdmnn0supp a...
fcdmnn0fsuppg 12614 Version of ~ fcdmnn0fsupp ...
nnnn0d 12615 A positive integer is a no...
nn0red 12616 A nonnegative integer is a...
nn0cnd 12617 A nonnegative integer is a...
nn0ge0d 12618 A nonnegative integer is g...
nn0addcld 12619 Closure of addition of non...
nn0mulcld 12620 Closure of multiplication ...
nn0readdcl 12621 Closure law for addition o...
nn0n0n1ge2 12622 A nonnegative integer whic...
nn0n0n1ge2b 12623 A nonnegative integer is n...
nn0ge2m1nn 12624 If a nonnegative integer i...
nn0ge2m1nn0 12625 If a nonnegative integer i...
nn0nndivcl 12626 Closure law for dividing o...
elxnn0 12629 An extended nonnegative in...
nn0ssxnn0 12630 The standard nonnegative i...
nn0xnn0 12631 A standard nonnegative int...
xnn0xr 12632 An extended nonnegative in...
0xnn0 12633 Zero is an extended nonneg...
pnf0xnn0 12634 Positive infinity is an ex...
nn0nepnf 12635 No standard nonnegative in...
nn0xnn0d 12636 A standard nonnegative int...
nn0nepnfd 12637 No standard nonnegative in...
xnn0nemnf 12638 No extended nonnegative in...
xnn0xrnemnf 12639 The extended nonnegative i...
xnn0nnn0pnf 12640 An extended nonnegative in...
elz 12643 Membership in the set of i...
nnnegz 12644 The negative of a positive...
zre 12645 An integer is a real. (Co...
zcn 12646 An integer is a complex nu...
zrei 12647 An integer is a real numbe...
zssre 12648 The integers are a subset ...
zsscn 12649 The integers are a subset ...
zex 12650 The set of integers exists...
elnnz 12651 Positive integer property ...
0z 12652 Zero is an integer. (Cont...
0zd 12653 Zero is an integer, deduct...
elnn0z 12654 Nonnegative integer proper...
elznn0nn 12655 Integer property expressed...
elznn0 12656 Integer property expressed...
elznn 12657 Integer property expressed...
zle0orge1 12658 There is no integer in the...
elz2 12659 Membership in the set of i...
dfz2 12660 Alternative definition of ...
zexALT 12661 Alternate proof of ~ zex ....
nnz 12662 A positive integer is an i...
nnssz 12663 Positive integers are a su...
nn0ssz 12664 Nonnegative integers are a...
nnzOLD 12665 Obsolete version of ~ nnz ...
nn0z 12666 A nonnegative integer is a...
nn0zd 12667 A nonnegative integer is a...
nnzd 12668 A positive integer is an i...
nnzi 12669 A positive integer is an i...
nn0zi 12670 A nonnegative integer is a...
elnnz1 12671 Positive integer property ...
znnnlt1 12672 An integer is not a positi...
nnzrab 12673 Positive integers expresse...
nn0zrab 12674 Nonnegative integers expre...
1z 12675 One is an integer. (Contr...
1zzd 12676 One is an integer, deducti...
2z 12677 2 is an integer. (Contrib...
3z 12678 3 is an integer. (Contrib...
4z 12679 4 is an integer. (Contrib...
znegcl 12680 Closure law for negative i...
neg1z 12681 -1 is an integer. (Contri...
znegclb 12682 A complex number is an int...
nn0negz 12683 The negative of a nonnegat...
nn0negzi 12684 The negative of a nonnegat...
zaddcl 12685 Closure of addition of int...
peano2z 12686 Second Peano postulate gen...
zsubcl 12687 Closure of subtraction of ...
peano2zm 12688 "Reverse" second Peano pos...
zletr 12689 Transitive law of ordering...
zrevaddcl 12690 Reverse closure law for ad...
znnsub 12691 The positive difference of...
znn0sub 12692 The nonnegative difference...
nzadd 12693 The sum of a real number n...
zmulcl 12694 Closure of multiplication ...
zltp1le 12695 Integer ordering relation....
zleltp1 12696 Integer ordering relation....
zlem1lt 12697 Integer ordering relation....
zltlem1 12698 Integer ordering relation....
zgt0ge1 12699 An integer greater than ` ...
nnleltp1 12700 Positive integer ordering ...
nnltp1le 12701 Positive integer ordering ...
nnaddm1cl 12702 Closure of addition of pos...
nn0ltp1le 12703 Nonnegative integer orderi...
nn0leltp1 12704 Nonnegative integer orderi...
nn0ltlem1 12705 Nonnegative integer orderi...
nn0sub2 12706 Subtraction of nonnegative...
nn0lt10b 12707 A nonnegative integer less...
nn0lt2 12708 A nonnegative integer less...
nn0le2is012 12709 A nonnegative integer whic...
nn0lem1lt 12710 Nonnegative integer orderi...
nnlem1lt 12711 Positive integer ordering ...
nnltlem1 12712 Positive integer ordering ...
nnm1ge0 12713 A positive integer decreas...
nn0ge0div 12714 Division of a nonnegative ...
zdiv 12715 Two ways to express " ` M ...
zdivadd 12716 Property of divisibility: ...
zdivmul 12717 Property of divisibility: ...
zextle 12718 An extensionality-like pro...
zextlt 12719 An extensionality-like pro...
recnz 12720 The reciprocal of a number...
btwnnz 12721 A number between an intege...
gtndiv 12722 A larger number does not d...
halfnz 12723 One-half is not an integer...
3halfnz 12724 Three halves is not an int...
suprzcl 12725 The supremum of a bounded-...
prime 12726 Two ways to express " ` A ...
msqznn 12727 The square of a nonzero in...
zneo 12728 No even integer equals an ...
nneo 12729 A positive integer is even...
nneoi 12730 A positive integer is even...
zeo 12731 An integer is even or odd....
zeo2 12732 An integer is even or odd ...
peano2uz2 12733 Second Peano postulate for...
peano5uzi 12734 Peano's inductive postulat...
peano5uzti 12735 Peano's inductive postulat...
dfuzi 12736 An expression for the uppe...
uzind 12737 Induction on the upper int...
uzind2 12738 Induction on the upper int...
uzind3 12739 Induction on the upper int...
nn0ind 12740 Principle of Mathematical ...
nn0indALT 12741 Principle of Mathematical ...
nn0indd 12742 Principle of Mathematical ...
fzind 12743 Induction on the integers ...
fnn0ind 12744 Induction on the integers ...
nn0ind-raph 12745 Principle of Mathematical ...
zindd 12746 Principle of Mathematical ...
fzindd 12747 Induction on the integers ...
btwnz 12748 Any real number can be san...
zred 12749 An integer is a real numbe...
zcnd 12750 An integer is a complex nu...
znegcld 12751 Closure law for negative i...
peano2zd 12752 Deduction from second Pean...
zaddcld 12753 Closure of addition of int...
zsubcld 12754 Closure of subtraction of ...
zmulcld 12755 Closure of multiplication ...
znnn0nn 12756 The negative of a negative...
zadd2cl 12757 Increasing an integer by 2...
zriotaneg 12758 The negative of the unique...
suprfinzcl 12759 The supremum of a nonempty...
9p1e10 12762 9 + 1 = 10. (Contributed ...
dfdec10 12763 Version of the definition ...
decex 12764 A decimal number is a set....
deceq1 12765 Equality theorem for the d...
deceq2 12766 Equality theorem for the d...
deceq1i 12767 Equality theorem for the d...
deceq2i 12768 Equality theorem for the d...
deceq12i 12769 Equality theorem for the d...
numnncl 12770 Closure for a numeral (wit...
num0u 12771 Add a zero in the units pl...
num0h 12772 Add a zero in the higher p...
numcl 12773 Closure for a decimal inte...
numsuc 12774 The successor of a decimal...
deccl 12775 Closure for a numeral. (C...
10nn 12776 10 is a positive integer. ...
10pos 12777 The number 10 is positive....
10nn0 12778 10 is a nonnegative intege...
10re 12779 The number 10 is real. (C...
decnncl 12780 Closure for a numeral. (C...
dec0u 12781 Add a zero in the units pl...
dec0h 12782 Add a zero in the higher p...
numnncl2 12783 Closure for a decimal inte...
decnncl2 12784 Closure for a decimal inte...
numlt 12785 Comparing two decimal inte...
numltc 12786 Comparing two decimal inte...
le9lt10 12787 A "decimal digit" (i.e. a ...
declt 12788 Comparing two decimal inte...
decltc 12789 Comparing two decimal inte...
declth 12790 Comparing two decimal inte...
decsuc 12791 The successor of a decimal...
3declth 12792 Comparing two decimal inte...
3decltc 12793 Comparing two decimal inte...
decle 12794 Comparing two decimal inte...
decleh 12795 Comparing two decimal inte...
declei 12796 Comparing a digit to a dec...
numlti 12797 Comparing a digit to a dec...
declti 12798 Comparing a digit to a dec...
decltdi 12799 Comparing a digit to a dec...
numsucc 12800 The successor of a decimal...
decsucc 12801 The successor of a decimal...
1e0p1 12802 The successor of zero. (C...
dec10p 12803 Ten plus an integer. (Con...
numma 12804 Perform a multiply-add of ...
nummac 12805 Perform a multiply-add of ...
numma2c 12806 Perform a multiply-add of ...
numadd 12807 Add two decimal integers `...
numaddc 12808 Add two decimal integers `...
nummul1c 12809 The product of a decimal i...
nummul2c 12810 The product of a decimal i...
decma 12811 Perform a multiply-add of ...
decmac 12812 Perform a multiply-add of ...
decma2c 12813 Perform a multiply-add of ...
decadd 12814 Add two numerals ` M ` and...
decaddc 12815 Add two numerals ` M ` and...
decaddc2 12816 Add two numerals ` M ` and...
decrmanc 12817 Perform a multiply-add of ...
decrmac 12818 Perform a multiply-add of ...
decaddm10 12819 The sum of two multiples o...
decaddi 12820 Add two numerals ` M ` and...
decaddci 12821 Add two numerals ` M ` and...
decaddci2 12822 Add two numerals ` M ` and...
decsubi 12823 Difference between a numer...
decmul1 12824 The product of a numeral w...
decmul1c 12825 The product of a numeral w...
decmul2c 12826 The product of a numeral w...
decmulnc 12827 The product of a numeral w...
11multnc 12828 The product of 11 (as nume...
decmul10add 12829 A multiplication of a numb...
6p5lem 12830 Lemma for ~ 6p5e11 and rel...
5p5e10 12831 5 + 5 = 10. (Contributed ...
6p4e10 12832 6 + 4 = 10. (Contributed ...
6p5e11 12833 6 + 5 = 11. (Contributed ...
6p6e12 12834 6 + 6 = 12. (Contributed ...
7p3e10 12835 7 + 3 = 10. (Contributed ...
7p4e11 12836 7 + 4 = 11. (Contributed ...
7p5e12 12837 7 + 5 = 12. (Contributed ...
7p6e13 12838 7 + 6 = 13. (Contributed ...
7p7e14 12839 7 + 7 = 14. (Contributed ...
8p2e10 12840 8 + 2 = 10. (Contributed ...
8p3e11 12841 8 + 3 = 11. (Contributed ...
8p4e12 12842 8 + 4 = 12. (Contributed ...
8p5e13 12843 8 + 5 = 13. (Contributed ...
8p6e14 12844 8 + 6 = 14. (Contributed ...
8p7e15 12845 8 + 7 = 15. (Contributed ...
8p8e16 12846 8 + 8 = 16. (Contributed ...
9p2e11 12847 9 + 2 = 11. (Contributed ...
9p3e12 12848 9 + 3 = 12. (Contributed ...
9p4e13 12849 9 + 4 = 13. (Contributed ...
9p5e14 12850 9 + 5 = 14. (Contributed ...
9p6e15 12851 9 + 6 = 15. (Contributed ...
9p7e16 12852 9 + 7 = 16. (Contributed ...
9p8e17 12853 9 + 8 = 17. (Contributed ...
9p9e18 12854 9 + 9 = 18. (Contributed ...
10p10e20 12855 10 + 10 = 20. (Contribute...
10m1e9 12856 10 - 1 = 9. (Contributed ...
4t3lem 12857 Lemma for ~ 4t3e12 and rel...
4t3e12 12858 4 times 3 equals 12. (Con...
4t4e16 12859 4 times 4 equals 16. (Con...
5t2e10 12860 5 times 2 equals 10. (Con...
5t3e15 12861 5 times 3 equals 15. (Con...
5t4e20 12862 5 times 4 equals 20. (Con...
5t5e25 12863 5 times 5 equals 25. (Con...
6t2e12 12864 6 times 2 equals 12. (Con...
6t3e18 12865 6 times 3 equals 18. (Con...
6t4e24 12866 6 times 4 equals 24. (Con...
6t5e30 12867 6 times 5 equals 30. (Con...
6t6e36 12868 6 times 6 equals 36. (Con...
7t2e14 12869 7 times 2 equals 14. (Con...
7t3e21 12870 7 times 3 equals 21. (Con...
7t4e28 12871 7 times 4 equals 28. (Con...
7t5e35 12872 7 times 5 equals 35. (Con...
7t6e42 12873 7 times 6 equals 42. (Con...
7t7e49 12874 7 times 7 equals 49. (Con...
8t2e16 12875 8 times 2 equals 16. (Con...
8t3e24 12876 8 times 3 equals 24. (Con...
8t4e32 12877 8 times 4 equals 32. (Con...
8t5e40 12878 8 times 5 equals 40. (Con...
8t6e48 12879 8 times 6 equals 48. (Con...
8t7e56 12880 8 times 7 equals 56. (Con...
8t8e64 12881 8 times 8 equals 64. (Con...
9t2e18 12882 9 times 2 equals 18. (Con...
9t3e27 12883 9 times 3 equals 27. (Con...
9t4e36 12884 9 times 4 equals 36. (Con...
9t5e45 12885 9 times 5 equals 45. (Con...
9t6e54 12886 9 times 6 equals 54. (Con...
9t7e63 12887 9 times 7 equals 63. (Con...
9t8e72 12888 9 times 8 equals 72. (Con...
9t9e81 12889 9 times 9 equals 81. (Con...
9t11e99 12890 9 times 11 equals 99. (Co...
9lt10 12891 9 is less than 10. (Contr...
8lt10 12892 8 is less than 10. (Contr...
7lt10 12893 7 is less than 10. (Contr...
6lt10 12894 6 is less than 10. (Contr...
5lt10 12895 5 is less than 10. (Contr...
4lt10 12896 4 is less than 10. (Contr...
3lt10 12897 3 is less than 10. (Contr...
2lt10 12898 2 is less than 10. (Contr...
1lt10 12899 1 is less than 10. (Contr...
decbin0 12900 Decompose base 4 into base...
decbin2 12901 Decompose base 4 into base...
decbin3 12902 Decompose base 4 into base...
halfthird 12903 Half minus a third. (Cont...
5recm6rec 12904 One fifth minus one sixth....
uzval 12907 The value of the upper int...
uzf 12908 The domain and codomain of...
eluz1 12909 Membership in the upper se...
eluzel2 12910 Implication of membership ...
eluz2 12911 Membership in an upper set...
eluzmn 12912 Membership in an earlier u...
eluz1i 12913 Membership in an upper set...
eluzuzle 12914 An integer in an upper set...
eluzelz 12915 A member of an upper set o...
eluzelre 12916 A member of an upper set o...
eluzelcn 12917 A member of an upper set o...
eluzle 12918 Implication of membership ...
eluz 12919 Membership in an upper set...
uzid 12920 Membership of the least me...
uzidd 12921 Membership of the least me...
uzn0 12922 The upper integers are all...
uztrn 12923 Transitive law for sets of...
uztrn2 12924 Transitive law for sets of...
uzneg 12925 Contraposition law for upp...
uzssz 12926 An upper set of integers i...
uzssre 12927 An upper set of integers i...
uzss 12928 Subset relationship for tw...
uztric 12929 Totality of the ordering r...
uz11 12930 The upper integers functio...
eluzp1m1 12931 Membership in the next upp...
eluzp1l 12932 Strict ordering implied by...
eluzp1p1 12933 Membership in the next upp...
eluzadd 12934 Membership in a later uppe...
eluzsub 12935 Membership in an earlier u...
eluzaddi 12936 Membership in a later uppe...
eluzaddiOLD 12937 Obsolete version of ~ eluz...
eluzsubi 12938 Membership in an earlier u...
eluzsubiOLD 12939 Obsolete version of ~ eluz...
eluzaddOLD 12940 Obsolete version of ~ eluz...
eluzsubOLD 12941 Obsolete version of ~ eluz...
subeluzsub 12942 Membership of a difference...
uzm1 12943 Choices for an element of ...
uznn0sub 12944 The nonnegative difference...
uzin 12945 Intersection of two upper ...
uzp1 12946 Choices for an element of ...
nn0uz 12947 Nonnegative integers expre...
nnuz 12948 Positive integers expresse...
elnnuz 12949 A positive integer express...
elnn0uz 12950 A nonnegative integer expr...
eluz2nn 12951 An integer greater than or...
eluz4eluz2 12952 An integer greater than or...
eluz4nn 12953 An integer greater than or...
eluzge2nn0 12954 If an integer is greater t...
eluz2n0 12955 An integer greater than or...
uzuzle23 12956 An integer in the upper se...
eluzge3nn 12957 If an integer is greater t...
uz3m2nn 12958 An integer greater than or...
1eluzge0 12959 1 is an integer greater th...
2eluzge0 12960 2 is an integer greater th...
2eluzge1 12961 2 is an integer greater th...
uznnssnn 12962 The upper integers startin...
raluz 12963 Restricted universal quant...
raluz2 12964 Restricted universal quant...
rexuz 12965 Restricted existential qua...
rexuz2 12966 Restricted existential qua...
2rexuz 12967 Double existential quantif...
peano2uz 12968 Second Peano postulate for...
peano2uzs 12969 Second Peano postulate for...
peano2uzr 12970 Reversed second Peano axio...
uzaddcl 12971 Addition closure law for a...
nn0pzuz 12972 The sum of a nonnegative i...
uzind4 12973 Induction on the upper set...
uzind4ALT 12974 Induction on the upper set...
uzind4s 12975 Induction on the upper set...
uzind4s2 12976 Induction on the upper set...
uzind4i 12977 Induction on the upper int...
uzwo 12978 Well-ordering principle: a...
uzwo2 12979 Well-ordering principle: a...
nnwo 12980 Well-ordering principle: a...
nnwof 12981 Well-ordering principle: a...
nnwos 12982 Well-ordering principle: a...
indstr 12983 Strong Mathematical Induct...
eluznn0 12984 Membership in a nonnegativ...
eluznn 12985 Membership in a positive u...
eluz2b1 12986 Two ways to say "an intege...
eluz2gt1 12987 An integer greater than or...
eluz2b2 12988 Two ways to say "an intege...
eluz2b3 12989 Two ways to say "an intege...
uz2m1nn 12990 One less than an integer g...
1nuz2 12991 1 is not in ` ( ZZ>= `` 2 ...
elnn1uz2 12992 A positive integer is eith...
uz2mulcl 12993 Closure of multiplication ...
indstr2 12994 Strong Mathematical Induct...
uzinfi 12995 Extract the lower bound of...
nninf 12996 The infimum of the set of ...
nn0inf 12997 The infimum of the set of ...
infssuzle 12998 The infimum of a subset of...
infssuzcl 12999 The infimum of a subset of...
ublbneg 13000 The image under negation o...
eqreznegel 13001 Two ways to express the im...
supminf 13002 The supremum of a bounded-...
lbzbi 13003 If a set of reals is bound...
zsupss 13004 Any nonempty bounded subse...
suprzcl2 13005 The supremum of a bounded-...
suprzub 13006 The supremum of a bounded-...
uzsupss 13007 Any bounded subset of an u...
nn01to3 13008 A (nonnegative) integer be...
nn0ge2m1nnALT 13009 Alternate proof of ~ nn0ge...
uzwo3 13010 Well-ordering principle: a...
zmin 13011 There is a unique smallest...
zmax 13012 There is a unique largest ...
zbtwnre 13013 There is a unique integer ...
rebtwnz 13014 There is a unique greatest...
elq 13017 Membership in the set of r...
qmulz 13018 If ` A ` is rational, then...
znq 13019 The ratio of an integer an...
qre 13020 A rational number is a rea...
zq 13021 An integer is a rational n...
qred 13022 A rational number is a rea...
zssq 13023 The integers are a subset ...
nn0ssq 13024 The nonnegative integers a...
nnssq 13025 The positive integers are ...
qssre 13026 The rationals are a subset...
qsscn 13027 The rationals are a subset...
qex 13028 The set of rational number...
nnq 13029 A positive integer is rati...
qcn 13030 A rational number is a com...
qexALT 13031 Alternate proof of ~ qex ....
qaddcl 13032 Closure of addition of rat...
qnegcl 13033 Closure law for the negati...
qmulcl 13034 Closure of multiplication ...
qsubcl 13035 Closure of subtraction of ...
qreccl 13036 Closure of reciprocal of r...
qdivcl 13037 Closure of division of rat...
qrevaddcl 13038 Reverse closure law for ad...
nnrecq 13039 The reciprocal of a positi...
irradd 13040 The sum of an irrational n...
irrmul 13041 The product of an irration...
elpq 13042 A positive rational is the...
elpqb 13043 A class is a positive rati...
rpnnen1lem2 13044 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem1 13045 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem3 13046 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem4 13047 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem5 13048 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem6 13049 Lemma for ~ rpnnen1 . (Co...
rpnnen1 13050 One half of ~ rpnnen , whe...
reexALT 13051 Alternate proof of ~ reex ...
cnref1o 13052 There is a natural one-to-...
cnexALT 13053 The set of complex numbers...
xrex 13054 The set of extended reals ...
mpoaddex 13055 The addition operation is ...
addex 13056 The addition operation is ...
mpomulex 13057 The multiplication operati...
mulex 13058 The multiplication operati...
elrp 13061 Membership in the set of p...
elrpii 13062 Membership in the set of p...
1rp 13063 1 is a positive real. (Co...
2rp 13064 2 is a positive real. (Co...
3rp 13065 3 is a positive real. (Co...
rpssre 13066 The positive reals are a s...
rpre 13067 A positive real is a real....
rpxr 13068 A positive real is an exte...
rpcn 13069 A positive real is a compl...
nnrp 13070 A positive integer is a po...
rpgt0 13071 A positive real is greater...
rpge0 13072 A positive real is greater...
rpregt0 13073 A positive real is a posit...
rprege0 13074 A positive real is a nonne...
rpne0 13075 A positive real is nonzero...
rprene0 13076 A positive real is a nonze...
rpcnne0 13077 A positive real is a nonze...
rpcndif0 13078 A positive real number is ...
ralrp 13079 Quantification over positi...
rexrp 13080 Quantification over positi...
rpaddcl 13081 Closure law for addition o...
rpmulcl 13082 Closure law for multiplica...
rpmtmip 13083 "Minus times minus is plus...
rpdivcl 13084 Closure law for division o...
rpreccl 13085 Closure law for reciprocat...
rphalfcl 13086 Closure law for half of a ...
rpgecl 13087 A number greater than or e...
rphalflt 13088 Half of a positive real is...
rerpdivcl 13089 Closure law for division o...
ge0p1rp 13090 A nonnegative number plus ...
rpneg 13091 Either a nonzero real or i...
negelrp 13092 Elementhood of a negation ...
negelrpd 13093 The negation of a negative...
0nrp 13094 Zero is not a positive rea...
ltsubrp 13095 Subtracting a positive rea...
ltaddrp 13096 Adding a positive number t...
difrp 13097 Two ways to say one number...
elrpd 13098 Membership in the set of p...
nnrpd 13099 A positive integer is a po...
zgt1rpn0n1 13100 An integer greater than 1 ...
rpred 13101 A positive real is a real....
rpxrd 13102 A positive real is an exte...
rpcnd 13103 A positive real is a compl...
rpgt0d 13104 A positive real is greater...
rpge0d 13105 A positive real is greater...
rpne0d 13106 A positive real is nonzero...
rpregt0d 13107 A positive real is real an...
rprege0d 13108 A positive real is real an...
rprene0d 13109 A positive real is a nonze...
rpcnne0d 13110 A positive real is a nonze...
rpreccld 13111 Closure law for reciprocat...
rprecred 13112 Closure law for reciprocat...
rphalfcld 13113 Closure law for half of a ...
reclt1d 13114 The reciprocal of a positi...
recgt1d 13115 The reciprocal of a positi...
rpaddcld 13116 Closure law for addition o...
rpmulcld 13117 Closure law for multiplica...
rpdivcld 13118 Closure law for division o...
ltrecd 13119 The reciprocal of both sid...
lerecd 13120 The reciprocal of both sid...
ltrec1d 13121 Reciprocal swap in a 'less...
lerec2d 13122 Reciprocal swap in a 'less...
lediv2ad 13123 Division of both sides of ...
ltdiv2d 13124 Division of a positive num...
lediv2d 13125 Division of a positive num...
ledivdivd 13126 Invert ratios of positive ...
divge1 13127 The ratio of a number over...
divlt1lt 13128 A real number divided by a...
divle1le 13129 A real number divided by a...
ledivge1le 13130 If a number is less than o...
ge0p1rpd 13131 A nonnegative number plus ...
rerpdivcld 13132 Closure law for division o...
ltsubrpd 13133 Subtracting a positive rea...
ltaddrpd 13134 Adding a positive number t...
ltaddrp2d 13135 Adding a positive number t...
ltmulgt11d 13136 Multiplication by a number...
ltmulgt12d 13137 Multiplication by a number...
gt0divd 13138 Division of a positive num...
ge0divd 13139 Division of a nonnegative ...
rpgecld 13140 A number greater than or e...
divge0d 13141 The ratio of nonnegative a...
ltmul1d 13142 The ratio of nonnegative a...
ltmul2d 13143 Multiplication of both sid...
lemul1d 13144 Multiplication of both sid...
lemul2d 13145 Multiplication of both sid...
ltdiv1d 13146 Division of both sides of ...
lediv1d 13147 Division of both sides of ...
ltmuldivd 13148 'Less than' relationship b...
ltmuldiv2d 13149 'Less than' relationship b...
lemuldivd 13150 'Less than or equal to' re...
lemuldiv2d 13151 'Less than or equal to' re...
ltdivmuld 13152 'Less than' relationship b...
ltdivmul2d 13153 'Less than' relationship b...
ledivmuld 13154 'Less than or equal to' re...
ledivmul2d 13155 'Less than or equal to' re...
ltmul1dd 13156 The ratio of nonnegative a...
ltmul2dd 13157 Multiplication of both sid...
ltdiv1dd 13158 Division of both sides of ...
lediv1dd 13159 Division of both sides of ...
lediv12ad 13160 Comparison of ratio of two...
mul2lt0rlt0 13161 If the result of a multipl...
mul2lt0rgt0 13162 If the result of a multipl...
mul2lt0llt0 13163 If the result of a multipl...
mul2lt0lgt0 13164 If the result of a multipl...
mul2lt0bi 13165 If the result of a multipl...
prodge0rd 13166 Infer that a multiplicand ...
prodge0ld 13167 Infer that a multiplier is...
ltdiv23d 13168 Swap denominator with othe...
lediv23d 13169 Swap denominator with othe...
lt2mul2divd 13170 The ratio of nonnegative a...
nnledivrp 13171 Division of a positive int...
nn0ledivnn 13172 Division of a nonnegative ...
addlelt 13173 If the sum of a real numbe...
ltxr 13180 The 'less than' binary rel...
elxr 13181 Membership in the set of e...
xrnemnf 13182 An extended real other tha...
xrnepnf 13183 An extended real other tha...
xrltnr 13184 The extended real 'less th...
ltpnf 13185 Any (finite) real is less ...
ltpnfd 13186 Any (finite) real is less ...
0ltpnf 13187 Zero is less than plus inf...
mnflt 13188 Minus infinity is less tha...
mnfltd 13189 Minus infinity is less tha...
mnflt0 13190 Minus infinity is less tha...
mnfltpnf 13191 Minus infinity is less tha...
mnfltxr 13192 Minus infinity is less tha...
pnfnlt 13193 No extended real is greate...
nltmnf 13194 No extended real is less t...
pnfge 13195 Plus infinity is an upper ...
xnn0n0n1ge2b 13196 An extended nonnegative in...
0lepnf 13197 0 less than or equal to po...
xnn0ge0 13198 An extended nonnegative in...
mnfle 13199 Minus infinity is less tha...
mnfled 13200 Minus infinity is less tha...
xrltnsym 13201 Ordering on the extended r...
xrltnsym2 13202 'Less than' is antisymmetr...
xrlttri 13203 Ordering on the extended r...
xrlttr 13204 Ordering on the extended r...
xrltso 13205 'Less than' is a strict or...
xrlttri2 13206 Trichotomy law for 'less t...
xrlttri3 13207 Trichotomy law for 'less t...
xrleloe 13208 'Less than or equal' expre...
xrleltne 13209 'Less than or equal to' im...
xrltlen 13210 'Less than' expressed in t...
dfle2 13211 Alternative definition of ...
dflt2 13212 Alternative definition of ...
xrltle 13213 'Less than' implies 'less ...
xrltled 13214 'Less than' implies 'less ...
xrleid 13215 'Less than or equal to' is...
xrleidd 13216 'Less than or equal to' is...
xrletri 13217 Trichotomy law for extende...
xrletri3 13218 Trichotomy law for extende...
xrletrid 13219 Trichotomy law for extende...
xrlelttr 13220 Transitive law for orderin...
xrltletr 13221 Transitive law for orderin...
xrletr 13222 Transitive law for orderin...
xrlttrd 13223 Transitive law for orderin...
xrlelttrd 13224 Transitive law for orderin...
xrltletrd 13225 Transitive law for orderin...
xrletrd 13226 Transitive law for orderin...
xrltne 13227 'Less than' implies not eq...
nltpnft 13228 An extended real is not le...
xgepnf 13229 An extended real which is ...
ngtmnft 13230 An extended real is not gr...
xlemnf 13231 An extended real which is ...
xrrebnd 13232 An extended real is real i...
xrre 13233 A way of proving that an e...
xrre2 13234 An extended real between t...
xrre3 13235 A way of proving that an e...
ge0gtmnf 13236 A nonnegative extended rea...
ge0nemnf 13237 A nonnegative extended rea...
xrrege0 13238 A nonnegative extended rea...
xrmax1 13239 An extended real is less t...
xrmax2 13240 An extended real is less t...
xrmin1 13241 The minimum of two extende...
xrmin2 13242 The minimum of two extende...
xrmaxeq 13243 The maximum of two extende...
xrmineq 13244 The minimum of two extende...
xrmaxlt 13245 Two ways of saying the max...
xrltmin 13246 Two ways of saying an exte...
xrmaxle 13247 Two ways of saying the max...
xrlemin 13248 Two ways of saying a numbe...
max1 13249 A number is less than or e...
max1ALT 13250 A number is less than or e...
max2 13251 A number is less than or e...
2resupmax 13252 The supremum of two real n...
min1 13253 The minimum of two numbers...
min2 13254 The minimum of two numbers...
maxle 13255 Two ways of saying the max...
lemin 13256 Two ways of saying a numbe...
maxlt 13257 Two ways of saying the max...
ltmin 13258 Two ways of saying a numbe...
lemaxle 13259 A real number which is les...
max0sub 13260 Decompose a real number in...
ifle 13261 An if statement transforms...
z2ge 13262 There exists an integer gr...
qbtwnre 13263 The rational numbers are d...
qbtwnxr 13264 The rational numbers are d...
qsqueeze 13265 If a nonnegative real is l...
qextltlem 13266 Lemma for ~ qextlt and qex...
qextlt 13267 An extensionality-like pro...
qextle 13268 An extensionality-like pro...
xralrple 13269 Show that ` A ` is less th...
alrple 13270 Show that ` A ` is less th...
xnegeq 13271 Equality of two extended n...
xnegex 13272 A negative extended real e...
xnegpnf 13273 Minus ` +oo ` . Remark of...
xnegmnf 13274 Minus ` -oo ` . Remark of...
rexneg 13275 Minus a real number. Rema...
xneg0 13276 The negative of zero. (Co...
xnegcl 13277 Closure of extended real n...
xnegneg 13278 Extended real version of ~...
xneg11 13279 Extended real version of ~...
xltnegi 13280 Forward direction of ~ xlt...
xltneg 13281 Extended real version of ~...
xleneg 13282 Extended real version of ~...
xlt0neg1 13283 Extended real version of ~...
xlt0neg2 13284 Extended real version of ~...
xle0neg1 13285 Extended real version of ~...
xle0neg2 13286 Extended real version of ~...
xaddval 13287 Value of the extended real...
xaddf 13288 The extended real addition...
xmulval 13289 Value of the extended real...
xaddpnf1 13290 Addition of positive infin...
xaddpnf2 13291 Addition of positive infin...
xaddmnf1 13292 Addition of negative infin...
xaddmnf2 13293 Addition of negative infin...
pnfaddmnf 13294 Addition of positive and n...
mnfaddpnf 13295 Addition of negative and p...
rexadd 13296 The extended real addition...
rexsub 13297 Extended real subtraction ...
rexaddd 13298 The extended real addition...
xnn0xaddcl 13299 The extended nonnegative i...
xaddnemnf 13300 Closure of extended real a...
xaddnepnf 13301 Closure of extended real a...
xnegid 13302 Extended real version of ~...
xaddcl 13303 The extended real addition...
xaddcom 13304 The extended real addition...
xaddrid 13305 Extended real version of ~...
xaddlid 13306 Extended real version of ~...
xaddridd 13307 ` 0 ` is a right identity ...
xnn0lem1lt 13308 Extended nonnegative integ...
xnn0lenn0nn0 13309 An extended nonnegative in...
xnn0le2is012 13310 An extended nonnegative in...
xnn0xadd0 13311 The sum of two extended no...
xnegdi 13312 Extended real version of ~...
xaddass 13313 Associativity of extended ...
xaddass2 13314 Associativity of extended ...
xpncan 13315 Extended real version of ~...
xnpcan 13316 Extended real version of ~...
xleadd1a 13317 Extended real version of ~...
xleadd2a 13318 Commuted form of ~ xleadd1...
xleadd1 13319 Weakened version of ~ xlea...
xltadd1 13320 Extended real version of ~...
xltadd2 13321 Extended real version of ~...
xaddge0 13322 The sum of nonnegative ext...
xle2add 13323 Extended real version of ~...
xlt2add 13324 Extended real version of ~...
xsubge0 13325 Extended real version of ~...
xposdif 13326 Extended real version of ~...
xlesubadd 13327 Under certain conditions, ...
xmullem 13328 Lemma for ~ rexmul . (Con...
xmullem2 13329 Lemma for ~ xmulneg1 . (C...
xmulcom 13330 Extended real multiplicati...
xmul01 13331 Extended real version of ~...
xmul02 13332 Extended real version of ~...
xmulneg1 13333 Extended real version of ~...
xmulneg2 13334 Extended real version of ~...
rexmul 13335 The extended real multipli...
xmulf 13336 The extended real multipli...
xmulcl 13337 Closure of extended real m...
xmulpnf1 13338 Multiplication by plus inf...
xmulpnf2 13339 Multiplication by plus inf...
xmulmnf1 13340 Multiplication by minus in...
xmulmnf2 13341 Multiplication by minus in...
xmulpnf1n 13342 Multiplication by plus inf...
xmulrid 13343 Extended real version of ~...
xmullid 13344 Extended real version of ~...
xmulm1 13345 Extended real version of ~...
xmulasslem2 13346 Lemma for ~ xmulass . (Co...
xmulgt0 13347 Extended real version of ~...
xmulge0 13348 Extended real version of ~...
xmulasslem 13349 Lemma for ~ xmulass . (Co...
xmulasslem3 13350 Lemma for ~ xmulass . (Co...
xmulass 13351 Associativity of the exten...
xlemul1a 13352 Extended real version of ~...
xlemul2a 13353 Extended real version of ~...
xlemul1 13354 Extended real version of ~...
xlemul2 13355 Extended real version of ~...
xltmul1 13356 Extended real version of ~...
xltmul2 13357 Extended real version of ~...
xadddilem 13358 Lemma for ~ xadddi . (Con...
xadddi 13359 Distributive property for ...
xadddir 13360 Commuted version of ~ xadd...
xadddi2 13361 The assumption that the mu...
xadddi2r 13362 Commuted version of ~ xadd...
x2times 13363 Extended real version of ~...
xnegcld 13364 Closure of extended real n...
xaddcld 13365 The extended real addition...
xmulcld 13366 Closure of extended real m...
xadd4d 13367 Rearrangement of 4 terms i...
xnn0add4d 13368 Rearrangement of 4 terms i...
xrsupexmnf 13369 Adding minus infinity to a...
xrinfmexpnf 13370 Adding plus infinity to a ...
xrsupsslem 13371 Lemma for ~ xrsupss . (Co...
xrinfmsslem 13372 Lemma for ~ xrinfmss . (C...
xrsupss 13373 Any subset of extended rea...
xrinfmss 13374 Any subset of extended rea...
xrinfmss2 13375 Any subset of extended rea...
xrub 13376 By quantifying only over r...
supxr 13377 The supremum of a set of e...
supxr2 13378 The supremum of a set of e...
supxrcl 13379 The supremum of an arbitra...
supxrun 13380 The supremum of the union ...
supxrmnf 13381 Adding minus infinity to a...
supxrpnf 13382 The supremum of a set of e...
supxrunb1 13383 The supremum of an unbound...
supxrunb2 13384 The supremum of an unbound...
supxrbnd1 13385 The supremum of a bounded-...
supxrbnd2 13386 The supremum of a bounded-...
xrsup0 13387 The supremum of an empty s...
supxrub 13388 A member of a set of exten...
supxrlub 13389 The supremum of a set of e...
supxrleub 13390 The supremum of a set of e...
supxrre 13391 The real and extended real...
supxrbnd 13392 The supremum of a bounded-...
supxrgtmnf 13393 The supremum of a nonempty...
supxrre1 13394 The supremum of a nonempty...
supxrre2 13395 The supremum of a nonempty...
supxrss 13396 Smaller sets of extended r...
infxrcl 13397 The infimum of an arbitrar...
infxrlb 13398 A member of a set of exten...
infxrgelb 13399 The infimum of a set of ex...
infxrre 13400 The real and extended real...
infxrmnf 13401 The infinimum of a set of ...
xrinf0 13402 The infimum of the empty s...
infxrss 13403 Larger sets of extended re...
reltre 13404 For all real numbers there...
rpltrp 13405 For all positive real numb...
reltxrnmnf 13406 For all extended real numb...
infmremnf 13407 The infimum of the reals i...
infmrp1 13408 The infimum of the positiv...
ixxval 13417 Value of the interval func...
elixx1 13418 Membership in an interval ...
ixxf 13419 The set of intervals of ex...
ixxex 13420 The set of intervals of ex...
ixxssxr 13421 The set of intervals of ex...
elixx3g 13422 Membership in a set of ope...
ixxssixx 13423 An interval is a subset of...
ixxdisj 13424 Split an interval into dis...
ixxun 13425 Split an interval into two...
ixxin 13426 Intersection of two interv...
ixxss1 13427 Subset relationship for in...
ixxss2 13428 Subset relationship for in...
ixxss12 13429 Subset relationship for in...
ixxub 13430 Extract the upper bound of...
ixxlb 13431 Extract the lower bound of...
iooex 13432 The set of open intervals ...
iooval 13433 Value of the open interval...
ioo0 13434 An empty open interval of ...
ioon0 13435 An open interval of extend...
ndmioo 13436 The open interval function...
iooid 13437 An open interval with iden...
elioo3g 13438 Membership in a set of ope...
elioore 13439 A member of an open interv...
lbioo 13440 An open interval does not ...
ubioo 13441 An open interval does not ...
iooval2 13442 Value of the open interval...
iooin 13443 Intersection of two open i...
iooss1 13444 Subset relationship for op...
iooss2 13445 Subset relationship for op...
iocval 13446 Value of the open-below, c...
icoval 13447 Value of the closed-below,...
iccval 13448 Value of the closed interv...
elioo1 13449 Membership in an open inte...
elioo2 13450 Membership in an open inte...
elioc1 13451 Membership in an open-belo...
elico1 13452 Membership in a closed-bel...
elicc1 13453 Membership in a closed int...
iccid 13454 A closed interval with ide...
ico0 13455 An empty open interval of ...
ioc0 13456 An empty open interval of ...
icc0 13457 An empty closed interval o...
dfrp2 13458 Alternate definition of th...
elicod 13459 Membership in a left-close...
icogelb 13460 An element of a left-close...
elicore 13461 A member of a left-closed ...
ubioc1 13462 The upper bound belongs to...
lbico1 13463 The lower bound belongs to...
iccleub 13464 An element of a closed int...
iccgelb 13465 An element of a closed int...
elioo5 13466 Membership in an open inte...
eliooxr 13467 A nonempty open interval s...
eliooord 13468 Ordering implied by a memb...
elioo4g 13469 Membership in an open inte...
ioossre 13470 An open interval is a set ...
ioosscn 13471 An open interval is a set ...
elioc2 13472 Membership in an open-belo...
elico2 13473 Membership in a closed-bel...
elicc2 13474 Membership in a closed rea...
elicc2i 13475 Inference for membership i...
elicc4 13476 Membership in a closed rea...
iccss 13477 Condition for a closed int...
iccssioo 13478 Condition for a closed int...
icossico 13479 Condition for a closed-bel...
iccss2 13480 Condition for a closed int...
iccssico 13481 Condition for a closed int...
iccssioo2 13482 Condition for a closed int...
iccssico2 13483 Condition for a closed int...
ioomax 13484 The open interval from min...
iccmax 13485 The closed interval from m...
ioopos 13486 The set of positive reals ...
ioorp 13487 The set of positive reals ...
iooshf 13488 Shift the arguments of the...
iocssre 13489 A closed-above interval wi...
icossre 13490 A closed-below interval wi...
iccssre 13491 A closed real interval is ...
iccssxr 13492 A closed interval is a set...
iocssxr 13493 An open-below, closed-abov...
icossxr 13494 A closed-below, open-above...
ioossicc 13495 An open interval is a subs...
iccssred 13496 A closed real interval is ...
eliccxr 13497 A member of a closed inter...
icossicc 13498 A closed-below, open-above...
iocssicc 13499 A closed-above, open-below...
ioossico 13500 An open interval is a subs...
iocssioo 13501 Condition for a closed int...
icossioo 13502 Condition for a closed int...
ioossioo 13503 Condition for an open inte...
iccsupr 13504 A nonempty subset of a clo...
elioopnf 13505 Membership in an unbounded...
elioomnf 13506 Membership in an unbounded...
elicopnf 13507 Membership in a closed unb...
repos 13508 Two ways of saying that a ...
ioof 13509 The set of open intervals ...
iccf 13510 The set of closed interval...
unirnioo 13511 The union of the range of ...
dfioo2 13512 Alternate definition of th...
ioorebas 13513 Open intervals are element...
xrge0neqmnf 13514 A nonnegative extended rea...
xrge0nre 13515 An extended real which is ...
elrege0 13516 The predicate "is a nonneg...
nn0rp0 13517 A nonnegative integer is a...
rge0ssre 13518 Nonnegative real numbers a...
elxrge0 13519 Elementhood in the set of ...
0e0icopnf 13520 0 is a member of ` ( 0 [,)...
0e0iccpnf 13521 0 is a member of ` ( 0 [,]...
ge0addcl 13522 The nonnegative reals are ...
ge0mulcl 13523 The nonnegative reals are ...
ge0xaddcl 13524 The nonnegative reals are ...
ge0xmulcl 13525 The nonnegative extended r...
lbicc2 13526 The lower bound of a close...
ubicc2 13527 The upper bound of a close...
elicc01 13528 Membership in the closed r...
elunitrn 13529 The closed unit interval i...
elunitcn 13530 The closed unit interval i...
0elunit 13531 Zero is an element of the ...
1elunit 13532 One is an element of the c...
iooneg 13533 Membership in a negated op...
iccneg 13534 Membership in a negated cl...
icoshft 13535 A shifted real is a member...
icoshftf1o 13536 Shifting a closed-below, o...
icoun 13537 The union of two adjacent ...
icodisj 13538 Adjacent left-closed right...
ioounsn 13539 The union of an open inter...
snunioo 13540 The closure of one end of ...
snunico 13541 The closure of the open en...
snunioc 13542 The closure of the open en...
prunioo 13543 The closure of an open rea...
ioodisj 13544 If the upper bound of one ...
ioojoin 13545 Join two open intervals to...
difreicc 13546 The class difference of ` ...
iccsplit 13547 Split a closed interval in...
iccshftr 13548 Membership in a shifted in...
iccshftri 13549 Membership in a shifted in...
iccshftl 13550 Membership in a shifted in...
iccshftli 13551 Membership in a shifted in...
iccdil 13552 Membership in a dilated in...
iccdili 13553 Membership in a dilated in...
icccntr 13554 Membership in a contracted...
icccntri 13555 Membership in a contracted...
divelunit 13556 A condition for a ratio to...
lincmb01cmp 13557 A linear combination of tw...
iccf1o 13558 Describe a bijection from ...
iccen 13559 Any nontrivial closed inte...
xov1plusxeqvd 13560 A complex number ` X ` is ...
unitssre 13561 ` ( 0 [,] 1 ) ` is a subse...
unitsscn 13562 The closed unit interval i...
supicc 13563 Supremum of a bounded set ...
supiccub 13564 The supremum of a bounded ...
supicclub 13565 The supremum of a bounded ...
supicclub2 13566 The supremum of a bounded ...
zltaddlt1le 13567 The sum of an integer and ...
xnn0xrge0 13568 An extended nonnegative in...
fzval 13571 The value of a finite set ...
fzval2 13572 An alternative way of expr...
fzf 13573 Establish the domain and c...
elfz1 13574 Membership in a finite set...
elfz 13575 Membership in a finite set...
elfz2 13576 Membership in a finite set...
elfzd 13577 Membership in a finite set...
elfz5 13578 Membership in a finite set...
elfz4 13579 Membership in a finite set...
elfzuzb 13580 Membership in a finite set...
eluzfz 13581 Membership in a finite set...
elfzuz 13582 A member of a finite set o...
elfzuz3 13583 Membership in a finite set...
elfzel2 13584 Membership in a finite set...
elfzel1 13585 Membership in a finite set...
elfzelz 13586 A member of a finite set o...
elfzelzd 13587 A member of a finite set o...
fzssz 13588 A finite sequence of integ...
elfzle1 13589 A member of a finite set o...
elfzle2 13590 A member of a finite set o...
elfzuz2 13591 Implication of membership ...
elfzle3 13592 Membership in a finite set...
eluzfz1 13593 Membership in a finite set...
eluzfz2 13594 Membership in a finite set...
eluzfz2b 13595 Membership in a finite set...
elfz3 13596 Membership in a finite set...
elfz1eq 13597 Membership in a finite set...
elfzubelfz 13598 If there is a member in a ...
peano2fzr 13599 A Peano-postulate-like the...
fzn0 13600 Properties of a finite int...
fz0 13601 A finite set of sequential...
fzn 13602 A finite set of sequential...
fzen 13603 A shifted finite set of se...
fz1n 13604 A 1-based finite set of se...
0nelfz1 13605 0 is not an element of a f...
0fz1 13606 Two ways to say a finite 1...
fz10 13607 There are no integers betw...
uzsubsubfz 13608 Membership of an integer g...
uzsubsubfz1 13609 Membership of an integer g...
ige3m2fz 13610 Membership of an integer g...
fzsplit2 13611 Split a finite interval of...
fzsplit 13612 Split a finite interval of...
fzdisj 13613 Condition for two finite i...
fz01en 13614 0-based and 1-based finite...
elfznn 13615 A member of a finite set o...
elfz1end 13616 A nonempty finite range of...
fz1ssnn 13617 A finite set of positive i...
fznn0sub 13618 Subtraction closure for a ...
fzmmmeqm 13619 Subtracting the difference...
fzaddel 13620 Membership of a sum in a f...
fzadd2 13621 Membership of a sum in a f...
fzsubel 13622 Membership of a difference...
fzopth 13623 A finite set of sequential...
fzass4 13624 Two ways to express a nond...
fzss1 13625 Subset relationship for fi...
fzss2 13626 Subset relationship for fi...
fzssuz 13627 A finite set of sequential...
fzsn 13628 A finite interval of integ...
fzssp1 13629 Subset relationship for fi...
fzssnn 13630 Finite sets of sequential ...
ssfzunsnext 13631 A subset of a finite seque...
ssfzunsn 13632 A subset of a finite seque...
fzsuc 13633 Join a successor to the en...
fzpred 13634 Join a predecessor to the ...
fzpreddisj 13635 A finite set of sequential...
elfzp1 13636 Append an element to a fin...
fzp1ss 13637 Subset relationship for fi...
fzelp1 13638 Membership in a set of seq...
fzp1elp1 13639 Add one to an element of a...
fznatpl1 13640 Shift membership in a fini...
fzpr 13641 A finite interval of integ...
fztp 13642 A finite interval of integ...
fz12pr 13643 An integer range between 1...
fzsuc2 13644 Join a successor to the en...
fzp1disj 13645 ` ( M ... ( N + 1 ) ) ` is...
fzdifsuc 13646 Remove a successor from th...
fzprval 13647 Two ways of defining the f...
fztpval 13648 Two ways of defining the f...
fzrev 13649 Reversal of start and end ...
fzrev2 13650 Reversal of start and end ...
fzrev2i 13651 Reversal of start and end ...
fzrev3 13652 The "complement" of a memb...
fzrev3i 13653 The "complement" of a memb...
fznn 13654 Finite set of sequential i...
elfz1b 13655 Membership in a 1-based fi...
elfz1uz 13656 Membership in a 1-based fi...
elfzm11 13657 Membership in a finite set...
uzsplit 13658 Express an upper integer s...
uzdisj 13659 The first ` N ` elements o...
fseq1p1m1 13660 Add/remove an item to/from...
fseq1m1p1 13661 Add/remove an item to/from...
fz1sbc 13662 Quantification over a one-...
elfzp1b 13663 An integer is a member of ...
elfzm1b 13664 An integer is a member of ...
elfzp12 13665 Options for membership in ...
fzm1 13666 Choices for an element of ...
fzneuz 13667 No finite set of sequentia...
fznuz 13668 Disjointness of the upper ...
uznfz 13669 Disjointness of the upper ...
fzp1nel 13670 One plus the upper bound o...
fzrevral 13671 Reversal of scanning order...
fzrevral2 13672 Reversal of scanning order...
fzrevral3 13673 Reversal of scanning order...
fzshftral 13674 Shift the scanning order i...
ige2m1fz1 13675 Membership of an integer g...
ige2m1fz 13676 Membership in a 0-based fi...
elfz2nn0 13677 Membership in a finite set...
fznn0 13678 Characterization of a fini...
elfznn0 13679 A member of a finite set o...
elfz3nn0 13680 The upper bound of a nonem...
fz0ssnn0 13681 Finite sets of sequential ...
fz1ssfz0 13682 Subset relationship for fi...
0elfz 13683 0 is an element of a finit...
nn0fz0 13684 A nonnegative integer is a...
elfz0add 13685 An element of a finite set...
fz0sn 13686 An integer range from 0 to...
fz0tp 13687 An integer range from 0 to...
fz0to3un2pr 13688 An integer range from 0 to...
fz0to4untppr 13689 An integer range from 0 to...
fz0to5un2tp 13690 An integer range from 0 to...
elfz0ubfz0 13691 An element of a finite set...
elfz0fzfz0 13692 A member of a finite set o...
fz0fzelfz0 13693 If a member of a finite se...
fznn0sub2 13694 Subtraction closure for a ...
uzsubfz0 13695 Membership of an integer g...
fz0fzdiffz0 13696 The difference of an integ...
elfzmlbm 13697 Subtracting the lower boun...
elfzmlbp 13698 Subtracting the lower boun...
fzctr 13699 Lemma for theorems about t...
difelfzle 13700 The difference of two inte...
difelfznle 13701 The difference of two inte...
nn0split 13702 Express the set of nonnega...
nn0disj 13703 The first ` N + 1 ` elemen...
fz0sn0fz1 13704 A finite set of sequential...
fvffz0 13705 The function value of a fu...
1fv 13706 A function on a singleton....
4fvwrd4 13707 The first four function va...
2ffzeq 13708 Two functions over 0-based...
preduz 13709 The value of the predecess...
prednn 13710 The value of the predecess...
prednn0 13711 The value of the predecess...
predfz 13712 Calculate the predecessor ...
fzof 13715 Functionality of the half-...
elfzoel1 13716 Reverse closure for half-o...
elfzoel2 13717 Reverse closure for half-o...
elfzoelz 13718 Reverse closure for half-o...
fzoval 13719 Value of the half-open int...
elfzo 13720 Membership in a half-open ...
elfzo2 13721 Membership in a half-open ...
elfzouz 13722 Membership in a half-open ...
nelfzo 13723 An integer not being a mem...
fzolb 13724 The left endpoint of a hal...
fzolb2 13725 The left endpoint of a hal...
elfzole1 13726 A member in a half-open in...
elfzolt2 13727 A member in a half-open in...
elfzolt3 13728 Membership in a half-open ...
elfzolt2b 13729 A member in a half-open in...
elfzolt3b 13730 Membership in a half-open ...
elfzop1le2 13731 A member in a half-open in...
fzonel 13732 A half-open range does not...
elfzouz2 13733 The upper bound of a half-...
elfzofz 13734 A half-open range is conta...
elfzo3 13735 Express membership in a ha...
fzon0 13736 A half-open integer interv...
fzossfz 13737 A half-open range is conta...
fzossz 13738 A half-open integer interv...
fzon 13739 A half-open set of sequent...
fzo0n 13740 A half-open range of nonne...
fzonlt0 13741 A half-open integer range ...
fzo0 13742 Half-open sets with equal ...
fzonnsub 13743 If ` K < N ` then ` N - K ...
fzonnsub2 13744 If ` M < N ` then ` N - M ...
fzoss1 13745 Subset relationship for ha...
fzoss2 13746 Subset relationship for ha...
fzossrbm1 13747 Subset of a half-open rang...
fzo0ss1 13748 Subset relationship for ha...
fzossnn0 13749 A half-open integer range ...
fzospliti 13750 One direction of splitting...
fzosplit 13751 Split a half-open integer ...
fzodisj 13752 Abutting half-open integer...
fzouzsplit 13753 Split an upper integer set...
fzouzdisj 13754 A half-open integer range ...
fzoun 13755 A half-open integer range ...
fzodisjsn 13756 A half-open integer range ...
prinfzo0 13757 The intersection of a half...
lbfzo0 13758 An integer is strictly gre...
elfzo0 13759 Membership in a half-open ...
elfzo0z 13760 Membership in a half-open ...
nn0p1elfzo 13761 A nonnegative integer incr...
elfzo0le 13762 A member in a half-open ra...
elfzonn0 13763 A member of a half-open ra...
fzonmapblen 13764 The result of subtracting ...
fzofzim 13765 If a nonnegative integer i...
fz1fzo0m1 13766 Translation of one between...
fzossnn 13767 Half-open integer ranges s...
elfzo1 13768 Membership in a half-open ...
fzo1fzo0n0 13769 An integer between 1 and a...
fzo0n0 13770 A half-open integer range ...
fzoaddel 13771 Translate membership in a ...
fzo0addel 13772 Translate membership in a ...
fzo0addelr 13773 Translate membership in a ...
fzoaddel2 13774 Translate membership in a ...
elfzoext 13775 Membership of an integer i...
elincfzoext 13776 Membership of an increased...
fzosubel 13777 Translate membership in a ...
fzosubel2 13778 Membership in a translated...
fzosubel3 13779 Membership in a translated...
eluzgtdifelfzo 13780 Membership of the differen...
ige2m2fzo 13781 Membership of an integer g...
fzocatel 13782 Translate membership in a ...
ubmelfzo 13783 If an integer in a 1-based...
elfzodifsumelfzo 13784 If an integer is in a half...
elfzom1elp1fzo 13785 Membership of an integer i...
elfzom1elfzo 13786 Membership in a half-open ...
fzval3 13787 Expressing a closed intege...
fz0add1fz1 13788 Translate membership in a ...
fzosn 13789 Expressing a singleton as ...
elfzomin 13790 Membership of an integer i...
zpnn0elfzo 13791 Membership of an integer i...
zpnn0elfzo1 13792 Membership of an integer i...
fzosplitsnm1 13793 Removing a singleton from ...
elfzonlteqm1 13794 If an element of a half-op...
fzonn0p1 13795 A nonnegative integer is e...
fzossfzop1 13796 A half-open range of nonne...
fzonn0p1p1 13797 If a nonnegative integer i...
elfzom1p1elfzo 13798 Increasing an element of a...
fzo0ssnn0 13799 Half-open integer ranges s...
fzo01 13800 Expressing the singleton o...
fzo12sn 13801 A 1-based half-open intege...
fzo13pr 13802 A 1-based half-open intege...
fzo0to2pr 13803 A half-open integer range ...
fzo0to3tp 13804 A half-open integer range ...
fzo0to42pr 13805 A half-open integer range ...
fzo1to4tp 13806 A half-open integer range ...
fzo0sn0fzo1 13807 A half-open range of nonne...
elfzo0l 13808 A member of a half-open ra...
fzoend 13809 The endpoint of a half-ope...
fzo0end 13810 The endpoint of a zero-bas...
ssfzo12 13811 Subset relationship for ha...
ssfzoulel 13812 If a half-open integer ran...
ssfzo12bi 13813 Subset relationship for ha...
fzoopth 13814 A half-open integer range ...
ubmelm1fzo 13815 The result of subtracting ...
fzofzp1 13816 If a point is in a half-op...
fzofzp1b 13817 If a point is in a half-op...
elfzom1b 13818 An integer is a member of ...
elfzom1elp1fzo1 13819 Membership of a nonnegativ...
elfzo1elm1fzo0 13820 Membership of a positive i...
elfzonelfzo 13821 If an element of a half-op...
fzonfzoufzol 13822 If an element of a half-op...
elfzomelpfzo 13823 An integer increased by an...
elfznelfzo 13824 A value in a finite set of...
elfznelfzob 13825 A value in a finite set of...
peano2fzor 13826 A Peano-postulate-like the...
fzosplitsn 13827 Extending a half-open rang...
fzosplitpr 13828 Extending a half-open inte...
fzosplitprm1 13829 Extending a half-open inte...
fzosplitsni 13830 Membership in a half-open ...
fzisfzounsn 13831 A finite interval of integ...
elfzr 13832 A member of a finite inter...
elfzlmr 13833 A member of a finite inter...
elfz0lmr 13834 A member of a finite inter...
fzostep1 13835 Two possibilities for a nu...
fzoshftral 13836 Shift the scanning order i...
fzind2 13837 Induction on the integers ...
fvinim0ffz 13838 The function values for th...
injresinjlem 13839 Lemma for ~ injresinj . (...
injresinj 13840 A function whose restricti...
subfzo0 13841 The difference between two...
fvf1tp 13842 Values of a one-to-one fun...
flval 13847 Value of the floor (greate...
flcl 13848 The floor (greatest intege...
reflcl 13849 The floor (greatest intege...
fllelt 13850 A basic property of the fl...
flcld 13851 The floor (greatest intege...
flle 13852 A basic property of the fl...
flltp1 13853 A basic property of the fl...
fllep1 13854 A basic property of the fl...
fraclt1 13855 The fractional part of a r...
fracle1 13856 The fractional part of a r...
fracge0 13857 The fractional part of a r...
flge 13858 The floor function value i...
fllt 13859 The floor function value i...
flflp1 13860 Move floor function betwee...
flid 13861 An integer is its own floo...
flidm 13862 The floor function is idem...
flidz 13863 A real number equals its f...
flltnz 13864 The floor of a non-integer...
flwordi 13865 Ordering relation for the ...
flword2 13866 Ordering relation for the ...
flval2 13867 An alternate way to define...
flval3 13868 An alternate way to define...
flbi 13869 A condition equivalent to ...
flbi2 13870 A condition equivalent to ...
adddivflid 13871 The floor of a sum of an i...
ico01fl0 13872 The floor of a real number...
flge0nn0 13873 The floor of a number grea...
flge1nn 13874 The floor of a number grea...
fldivnn0 13875 The floor function of a di...
refldivcl 13876 The floor function of a di...
divfl0 13877 The floor of a fraction is...
fladdz 13878 An integer can be moved in...
flzadd 13879 An integer can be moved in...
flmulnn0 13880 Move a nonnegative integer...
btwnzge0 13881 A real bounded between an ...
2tnp1ge0ge0 13882 Two times an integer plus ...
flhalf 13883 Ordering relation for the ...
fldivle 13884 The floor function of a di...
fldivnn0le 13885 The floor function of a di...
flltdivnn0lt 13886 The floor function of a di...
ltdifltdiv 13887 If the dividend of a divis...
fldiv4p1lem1div2 13888 The floor of an integer eq...
fldiv4lem1div2uz2 13889 The floor of an integer gr...
fldiv4lem1div2 13890 The floor of a positive in...
ceilval 13891 The value of the ceiling f...
dfceil2 13892 Alternative definition of ...
ceilval2 13893 The value of the ceiling f...
ceicl 13894 The ceiling function retur...
ceilcl 13895 Closure of the ceiling fun...
ceilcld 13896 Closure of the ceiling fun...
ceige 13897 The ceiling of a real numb...
ceilge 13898 The ceiling of a real numb...
ceilged 13899 The ceiling of a real numb...
ceim1l 13900 One less than the ceiling ...
ceilm1lt 13901 One less than the ceiling ...
ceile 13902 The ceiling of a real numb...
ceille 13903 The ceiling of a real numb...
ceilid 13904 An integer is its own ceil...
ceilidz 13905 A real number equals its c...
flleceil 13906 The floor of a real number...
fleqceilz 13907 A real number is an intege...
quoremz 13908 Quotient and remainder of ...
quoremnn0 13909 Quotient and remainder of ...
quoremnn0ALT 13910 Alternate proof of ~ quore...
intfrac2 13911 Decompose a real into inte...
intfracq 13912 Decompose a rational numbe...
fldiv 13913 Cancellation of the embedd...
fldiv2 13914 Cancellation of an embedde...
fznnfl 13915 Finite set of sequential i...
uzsup 13916 An upper set of integers i...
ioopnfsup 13917 An upper set of reals is u...
icopnfsup 13918 An upper set of reals is u...
rpsup 13919 The positive reals are unb...
resup 13920 The real numbers are unbou...
xrsup 13921 The extended real numbers ...
modval 13924 The value of the modulo op...
modvalr 13925 The value of the modulo op...
modcl 13926 Closure law for the modulo...
flpmodeq 13927 Partition of a division in...
modcld 13928 Closure law for the modulo...
mod0 13929 ` A mod B ` is zero iff ` ...
mulmod0 13930 The product of an integer ...
negmod0 13931 ` A ` is divisible by ` B ...
modge0 13932 The modulo operation is no...
modlt 13933 The modulo operation is le...
modelico 13934 Modular reduction produces...
moddiffl 13935 Value of the modulo operat...
moddifz 13936 The modulo operation diffe...
modfrac 13937 The fractional part of a n...
flmod 13938 The floor function express...
intfrac 13939 Break a number into its in...
zmod10 13940 An integer modulo 1 is 0. ...
zmod1congr 13941 Two arbitrary integers are...
modmulnn 13942 Move a positive integer in...
modvalp1 13943 The value of the modulo op...
zmodcl 13944 Closure law for the modulo...
zmodcld 13945 Closure law for the modulo...
zmodfz 13946 An integer mod ` B ` lies ...
zmodfzo 13947 An integer mod ` B ` lies ...
zmodfzp1 13948 An integer mod ` B ` lies ...
modid 13949 Identity law for modulo. ...
modid0 13950 A positive real number mod...
modid2 13951 Identity law for modulo. ...
zmodid2 13952 Identity law for modulo re...
zmodidfzo 13953 Identity law for modulo re...
zmodidfzoimp 13954 Identity law for modulo re...
0mod 13955 Special case: 0 modulo a p...
1mod 13956 Special case: 1 modulo a r...
modabs 13957 Absorption law for modulo....
modabs2 13958 Absorption law for modulo....
modcyc 13959 The modulo operation is pe...
modcyc2 13960 The modulo operation is pe...
modadd1 13961 Addition property of the m...
modaddabs 13962 Absorption law for modulo....
modaddmod 13963 The sum of a real number m...
muladdmodid 13964 The sum of a positive real...
mulp1mod1 13965 The product of an integer ...
modmuladd 13966 Decomposition of an intege...
modmuladdim 13967 Implication of a decomposi...
modmuladdnn0 13968 Implication of a decomposi...
negmod 13969 The negation of a number m...
m1modnnsub1 13970 Minus one modulo a positiv...
m1modge3gt1 13971 Minus one modulo an intege...
addmodid 13972 The sum of a positive inte...
addmodidr 13973 The sum of a positive inte...
modadd2mod 13974 The sum of a real number m...
modm1p1mod0 13975 If a real number modulo a ...
modltm1p1mod 13976 If a real number modulo a ...
modmul1 13977 Multiplication property of...
modmul12d 13978 Multiplication property of...
modnegd 13979 Negation property of the m...
modadd12d 13980 Additive property of the m...
modsub12d 13981 Subtraction property of th...
modsubmod 13982 The difference of a real n...
modsubmodmod 13983 The difference of a real n...
2txmodxeq0 13984 Two times a positive real ...
2submod 13985 If a real number is betwee...
modifeq2int 13986 If a nonnegative integer i...
modaddmodup 13987 The sum of an integer modu...
modaddmodlo 13988 The sum of an integer modu...
modmulmod 13989 The product of a real numb...
modmulmodr 13990 The product of an integer ...
modaddmulmod 13991 The sum of a real number a...
moddi 13992 Distribute multiplication ...
modsubdir 13993 Distribute the modulo oper...
modeqmodmin 13994 A real number equals the d...
modirr 13995 A number modulo an irratio...
modfzo0difsn 13996 For a number within a half...
modsumfzodifsn 13997 The sum of a number within...
modlteq 13998 Two nonnegative integers l...
addmodlteq 13999 Two nonnegative integers l...
om2uz0i 14000 The mapping ` G ` is a one...
om2uzsuci 14001 The value of ` G ` (see ~ ...
om2uzuzi 14002 The value ` G ` (see ~ om2...
om2uzlti 14003 Less-than relation for ` G...
om2uzlt2i 14004 The mapping ` G ` (see ~ o...
om2uzrani 14005 Range of ` G ` (see ~ om2u...
om2uzf1oi 14006 ` G ` (see ~ om2uz0i ) is ...
om2uzisoi 14007 ` G ` (see ~ om2uz0i ) is ...
om2uzoi 14008 An alternative definition ...
om2uzrdg 14009 A helper lemma for the val...
uzrdglem 14010 A helper lemma for the val...
uzrdgfni 14011 The recursive definition g...
uzrdg0i 14012 Initial value of a recursi...
uzrdgsuci 14013 Successor value of a recur...
ltweuz 14014 ` < ` is a well-founded re...
ltwenn 14015 Less than well-orders the ...
ltwefz 14016 Less than well-orders a se...
uzenom 14017 An upper integer set is de...
uzinf 14018 An upper integer set is in...
nnnfi 14019 The set of positive intege...
uzrdgxfr 14020 Transfer the value of the ...
fzennn 14021 The cardinality of a finit...
fzen2 14022 The cardinality of a finit...
cardfz 14023 The cardinality of a finit...
hashgf1o 14024 ` G ` maps ` _om ` one-to-...
fzfi 14025 A finite interval of integ...
fzfid 14026 Commonly used special case...
fzofi 14027 Half-open integer sets are...
fsequb 14028 The values of a finite rea...
fsequb2 14029 The values of a finite rea...
fseqsupcl 14030 The values of a finite rea...
fseqsupubi 14031 The values of a finite rea...
nn0ennn 14032 The nonnegative integers a...
nnenom 14033 The set of positive intege...
nnct 14034 ` NN ` is countable. (Con...
uzindi 14035 Indirect strong induction ...
axdc4uzlem 14036 Lemma for ~ axdc4uz . (Co...
axdc4uz 14037 A version of ~ axdc4 that ...
ssnn0fi 14038 A subset of the nonnegativ...
rabssnn0fi 14039 A subset of the nonnegativ...
uzsinds 14040 Strong (or "total") induct...
nnsinds 14041 Strong (or "total") induct...
nn0sinds 14042 Strong (or "total") induct...
fsuppmapnn0fiublem 14043 Lemma for ~ fsuppmapnn0fiu...
fsuppmapnn0fiub 14044 If all functions of a fini...
fsuppmapnn0fiubex 14045 If all functions of a fini...
fsuppmapnn0fiub0 14046 If all functions of a fini...
suppssfz 14047 Condition for a function o...
fsuppmapnn0ub 14048 If a function over the non...
fsuppmapnn0fz 14049 If a function over the non...
mptnn0fsupp 14050 A mapping from the nonnega...
mptnn0fsuppd 14051 A mapping from the nonnega...
mptnn0fsuppr 14052 A finitely supported mappi...
f13idfv 14053 A one-to-one function with...
seqex 14056 Existence of the sequence ...
seqeq1 14057 Equality theorem for the s...
seqeq2 14058 Equality theorem for the s...
seqeq3 14059 Equality theorem for the s...
seqeq1d 14060 Equality deduction for the...
seqeq2d 14061 Equality deduction for the...
seqeq3d 14062 Equality deduction for the...
seqeq123d 14063 Equality deduction for the...
nfseq 14064 Hypothesis builder for the...
seqval 14065 Value of the sequence buil...
seqfn 14066 The sequence builder funct...
seq1 14067 Value of the sequence buil...
seq1i 14068 Value of the sequence buil...
seqp1 14069 Value of the sequence buil...
seqexw 14070 Weak version of ~ seqex th...
seqp1d 14071 Value of the sequence buil...
seqm1 14072 Value of the sequence buil...
seqcl2 14073 Closure properties of the ...
seqf2 14074 Range of the recursive seq...
seqcl 14075 Closure properties of the ...
seqf 14076 Range of the recursive seq...
seqfveq2 14077 Equality of sequences. (C...
seqfeq2 14078 Equality of sequences. (C...
seqfveq 14079 Equality of sequences. (C...
seqfeq 14080 Equality of sequences. (C...
seqshft2 14081 Shifting the index set of ...
seqres 14082 Restricting its characteri...
serf 14083 An infinite series of comp...
serfre 14084 An infinite series of real...
monoord 14085 Ordering relation for a mo...
monoord2 14086 Ordering relation for a mo...
sermono 14087 The partial sums in an inf...
seqsplit 14088 Split a sequence into two ...
seq1p 14089 Removing the first term fr...
seqcaopr3 14090 Lemma for ~ seqcaopr2 . (...
seqcaopr2 14091 The sum of two infinite se...
seqcaopr 14092 The sum of two infinite se...
seqf1olem2a 14093 Lemma for ~ seqf1o . (Con...
seqf1olem1 14094 Lemma for ~ seqf1o . (Con...
seqf1olem2 14095 Lemma for ~ seqf1o . (Con...
seqf1o 14096 Rearrange a sum via an arb...
seradd 14097 The sum of two infinite se...
sersub 14098 The difference of two infi...
seqid3 14099 A sequence that consists e...
seqid 14100 Discarding the first few t...
seqid2 14101 The last few partial sums ...
seqhomo 14102 Apply a homomorphism to a ...
seqz 14103 If the operation ` .+ ` ha...
seqfeq4 14104 Equality of series under d...
seqfeq3 14105 Equality of series under d...
seqdistr 14106 The distributive property ...
ser0 14107 The value of the partial s...
ser0f 14108 A zero-valued infinite ser...
serge0 14109 A finite sum of nonnegativ...
serle 14110 Comparison of partial sums...
ser1const 14111 Value of the partial serie...
seqof 14112 Distribute function operat...
seqof2 14113 Distribute function operat...
expval 14116 Value of exponentiation to...
expnnval 14117 Value of exponentiation to...
exp0 14118 Value of a complex number ...
0exp0e1 14119 The zeroth power of zero e...
exp1 14120 Value of a complex number ...
expp1 14121 Value of a complex number ...
expneg 14122 Value of a complex number ...
expneg2 14123 Value of a complex number ...
expn1 14124 A complex number raised to...
expcllem 14125 Lemma for proving nonnegat...
expcl2lem 14126 Lemma for proving integer ...
nnexpcl 14127 Closure of exponentiation ...
nn0expcl 14128 Closure of exponentiation ...
zexpcl 14129 Closure of exponentiation ...
qexpcl 14130 Closure of exponentiation ...
reexpcl 14131 Closure of exponentiation ...
expcl 14132 Closure law for nonnegativ...
rpexpcl 14133 Closure law for integer ex...
qexpclz 14134 Closure of integer exponen...
reexpclz 14135 Closure of integer exponen...
expclzlem 14136 Lemma for ~ expclz . (Con...
expclz 14137 Closure law for integer ex...
m1expcl2 14138 Closure of integer exponen...
m1expcl 14139 Closure of exponentiation ...
zexpcld 14140 Closure of exponentiation ...
nn0expcli 14141 Closure of exponentiation ...
nn0sqcl 14142 The square of a nonnegativ...
expm1t 14143 Exponentiation in terms of...
1exp 14144 Value of 1 raised to an in...
expeq0 14145 A positive integer power i...
expne0 14146 A positive integer power i...
expne0i 14147 An integer power is nonzer...
expgt0 14148 A positive real raised to ...
expnegz 14149 Value of a nonzero complex...
0exp 14150 Value of zero raised to a ...
expge0 14151 A nonnegative real raised ...
expge1 14152 A real greater than or equ...
expgt1 14153 A real greater than 1 rais...
mulexp 14154 Nonnegative integer expone...
mulexpz 14155 Integer exponentiation of ...
exprec 14156 Integer exponentiation of ...
expadd 14157 Sum of exponents law for n...
expaddzlem 14158 Lemma for ~ expaddz . (Co...
expaddz 14159 Sum of exponents law for i...
expmul 14160 Product of exponents law f...
expmulz 14161 Product of exponents law f...
m1expeven 14162 Exponentiation of negative...
expsub 14163 Exponent subtraction law f...
expp1z 14164 Value of a nonzero complex...
expm1 14165 Value of a nonzero complex...
expdiv 14166 Nonnegative integer expone...
sqval 14167 Value of the square of a c...
sqneg 14168 The square of the negative...
sqsubswap 14169 Swap the order of subtract...
sqcl 14170 Closure of square. (Contr...
sqmul 14171 Distribution of squaring o...
sqeq0 14172 A complex number is zero i...
sqdiv 14173 Distribution of squaring o...
sqdivid 14174 The square of a nonzero co...
sqne0 14175 A complex number is nonzer...
resqcl 14176 Closure of squaring in rea...
resqcld 14177 Closure of squaring in rea...
sqgt0 14178 The square of a nonzero re...
sqn0rp 14179 The square of a nonzero re...
nnsqcl 14180 The positive naturals are ...
zsqcl 14181 Integers are closed under ...
qsqcl 14182 The square of a rational i...
sq11 14183 The square function is one...
nn0sq11 14184 The square function is one...
lt2sq 14185 The square function is inc...
le2sq 14186 The square function is non...
le2sq2 14187 The square function is non...
sqge0 14188 The square of a real is no...
sqge0d 14189 The square of a real is no...
zsqcl2 14190 The square of an integer i...
0expd 14191 Value of zero raised to a ...
exp0d 14192 Value of a complex number ...
exp1d 14193 Value of a complex number ...
expeq0d 14194 If a positive integer powe...
sqvald 14195 Value of square. Inferenc...
sqcld 14196 Closure of square. (Contr...
sqeq0d 14197 A number is zero iff its s...
expcld 14198 Closure law for nonnegativ...
expp1d 14199 Value of a complex number ...
expaddd 14200 Sum of exponents law for n...
expmuld 14201 Product of exponents law f...
sqrecd 14202 Square of reciprocal is re...
expclzd 14203 Closure law for integer ex...
expne0d 14204 A nonnegative integer powe...
expnegd 14205 Value of a nonzero complex...
exprecd 14206 An integer power of a reci...
expp1zd 14207 Value of a nonzero complex...
expm1d 14208 Value of a nonzero complex...
expsubd 14209 Exponent subtraction law f...
sqmuld 14210 Distribution of squaring o...
sqdivd 14211 Distribution of squaring o...
expdivd 14212 Nonnegative integer expone...
mulexpd 14213 Nonnegative integer expone...
znsqcld 14214 The square of a nonzero in...
reexpcld 14215 Closure of exponentiation ...
expge0d 14216 A nonnegative real raised ...
expge1d 14217 A real greater than or equ...
ltexp2a 14218 Exponent ordering relation...
expmordi 14219 Base ordering relationship...
rpexpmord 14220 Base ordering relationship...
expcan 14221 Cancellation law for integ...
ltexp2 14222 Strict ordering law for ex...
leexp2 14223 Ordering law for exponenti...
leexp2a 14224 Weak ordering relationship...
ltexp2r 14225 The integer powers of a fi...
leexp2r 14226 Weak ordering relationship...
leexp1a 14227 Weak base ordering relatio...
exple1 14228 A real between 0 and 1 inc...
expubnd 14229 An upper bound on ` A ^ N ...
sumsqeq0 14230 The sum of two squres of r...
sqvali 14231 Value of square. Inferenc...
sqcli 14232 Closure of square. (Contr...
sqeq0i 14233 A complex number is zero i...
sqrecii 14234 The square of a reciprocal...
sqmuli 14235 Distribution of squaring o...
sqdivi 14236 Distribution of squaring o...
resqcli 14237 Closure of square in reals...
sqgt0i 14238 The square of a nonzero re...
sqge0i 14239 The square of a real is no...
lt2sqi 14240 The square function on non...
le2sqi 14241 The square function on non...
sq11i 14242 The square function is one...
sq0 14243 The square of 0 is 0. (Co...
sq0i 14244 If a number is zero, then ...
sq0id 14245 If a number is zero, then ...
sq1 14246 The square of 1 is 1. (Co...
neg1sqe1 14247 The square of ` -u 1 ` is ...
sq2 14248 The square of 2 is 4. (Co...
sq3 14249 The square of 3 is 9. (Co...
sq4e2t8 14250 The square of 4 is 2 times...
cu2 14251 The cube of 2 is 8. (Cont...
irec 14252 The reciprocal of ` _i ` ....
i2 14253 ` _i ` squared. (Contribu...
i3 14254 ` _i ` cubed. (Contribute...
i4 14255 ` _i ` to the fourth power...
nnlesq 14256 A positive integer is less...
zzlesq 14257 An integer is less than or...
iexpcyc 14258 Taking ` _i ` to the ` K `...
expnass 14259 A counterexample showing t...
sqlecan 14260 Cancel one factor of a squ...
subsq 14261 Factor the difference of t...
subsq2 14262 Express the difference of ...
binom2i 14263 The square of a binomial. ...
subsqi 14264 Factor the difference of t...
sqeqori 14265 The squares of two complex...
subsq0i 14266 The two solutions to the d...
sqeqor 14267 The squares of two complex...
binom2 14268 The square of a binomial. ...
binom2d 14269 Deduction form of ~ binom2...
binom21 14270 Special case of ~ binom2 w...
binom2sub 14271 Expand the square of a sub...
binom2sub1 14272 Special case of ~ binom2su...
binom2subi 14273 Expand the square of a sub...
mulbinom2 14274 The square of a binomial w...
binom3 14275 The cube of a binomial. (...
sq01 14276 If a complex number equals...
zesq 14277 An integer is even iff its...
nnesq 14278 A positive integer is even...
crreczi 14279 Reciprocal of a complex nu...
bernneq 14280 Bernoulli's inequality, du...
bernneq2 14281 Variation of Bernoulli's i...
bernneq3 14282 A corollary of ~ bernneq ....
expnbnd 14283 Exponentiation with a base...
expnlbnd 14284 The reciprocal of exponent...
expnlbnd2 14285 The reciprocal of exponent...
expmulnbnd 14286 Exponentiation with a base...
digit2 14287 Two ways to express the ` ...
digit1 14288 Two ways to express the ` ...
modexp 14289 Exponentiation property of...
discr1 14290 A nonnegative quadratic fo...
discr 14291 If a quadratic polynomial ...
expnngt1 14292 If an integer power with a...
expnngt1b 14293 An integer power with an i...
sqoddm1div8 14294 A squared odd number minus...
nnsqcld 14295 The naturals are closed un...
nnexpcld 14296 Closure of exponentiation ...
nn0expcld 14297 Closure of exponentiation ...
rpexpcld 14298 Closure law for exponentia...
ltexp2rd 14299 The power of a positive nu...
reexpclzd 14300 Closure of exponentiation ...
sqgt0d 14301 The square of a nonzero re...
ltexp2d 14302 Ordering relationship for ...
leexp2d 14303 Ordering law for exponenti...
expcand 14304 Ordering relationship for ...
leexp2ad 14305 Ordering relationship for ...
leexp2rd 14306 Ordering relationship for ...
lt2sqd 14307 The square function on non...
le2sqd 14308 The square function on non...
sq11d 14309 The square function is one...
ltexp1d 14310 Elevating to a positive po...
ltexp1dd 14311 Raising both sides of 'les...
exp11nnd 14312 The function elevating non...
mulsubdivbinom2 14313 The square of a binomial w...
muldivbinom2 14314 The square of a binomial w...
sq10 14315 The square of 10 is 100. ...
sq10e99m1 14316 The square of 10 is 99 plu...
3dec 14317 A "decimal constructor" wh...
nn0le2msqi 14318 The square function on non...
nn0opthlem1 14319 A rather pretty lemma for ...
nn0opthlem2 14320 Lemma for ~ nn0opthi . (C...
nn0opthi 14321 An ordered pair theorem fo...
nn0opth2i 14322 An ordered pair theorem fo...
nn0opth2 14323 An ordered pair theorem fo...
facnn 14326 Value of the factorial fun...
fac0 14327 The factorial of 0. (Cont...
fac1 14328 The factorial of 1. (Cont...
facp1 14329 The factorial of a success...
fac2 14330 The factorial of 2. (Cont...
fac3 14331 The factorial of 3. (Cont...
fac4 14332 The factorial of 4. (Cont...
facnn2 14333 Value of the factorial fun...
faccl 14334 Closure of the factorial f...
faccld 14335 Closure of the factorial f...
facmapnn 14336 The factorial function res...
facne0 14337 The factorial function is ...
facdiv 14338 A positive integer divides...
facndiv 14339 No positive integer (great...
facwordi 14340 Ordering property of facto...
faclbnd 14341 A lower bound for the fact...
faclbnd2 14342 A lower bound for the fact...
faclbnd3 14343 A lower bound for the fact...
faclbnd4lem1 14344 Lemma for ~ faclbnd4 . Pr...
faclbnd4lem2 14345 Lemma for ~ faclbnd4 . Us...
faclbnd4lem3 14346 Lemma for ~ faclbnd4 . Th...
faclbnd4lem4 14347 Lemma for ~ faclbnd4 . Pr...
faclbnd4 14348 Variant of ~ faclbnd5 prov...
faclbnd5 14349 The factorial function gro...
faclbnd6 14350 Geometric lower bound for ...
facubnd 14351 An upper bound for the fac...
facavg 14352 The product of two factori...
bcval 14355 Value of the binomial coef...
bcval2 14356 Value of the binomial coef...
bcval3 14357 Value of the binomial coef...
bcval4 14358 Value of the binomial coef...
bcrpcl 14359 Closure of the binomial co...
bccmpl 14360 "Complementing" its second...
bcn0 14361 ` N ` choose 0 is 1. Rema...
bc0k 14362 The binomial coefficient "...
bcnn 14363 ` N ` choose ` N ` is 1. ...
bcn1 14364 Binomial coefficient: ` N ...
bcnp1n 14365 Binomial coefficient: ` N ...
bcm1k 14366 The proportion of one bino...
bcp1n 14367 The proportion of one bino...
bcp1nk 14368 The proportion of one bino...
bcval5 14369 Write out the top and bott...
bcn2 14370 Binomial coefficient: ` N ...
bcp1m1 14371 Compute the binomial coeff...
bcpasc 14372 Pascal's rule for the bino...
bccl 14373 A binomial coefficient, in...
bccl2 14374 A binomial coefficient, in...
bcn2m1 14375 Compute the binomial coeff...
bcn2p1 14376 Compute the binomial coeff...
permnn 14377 The number of permutations...
bcnm1 14378 The binomial coefficient o...
4bc3eq4 14379 The value of four choose t...
4bc2eq6 14380 The value of four choose t...
hashkf 14383 The finite part of the siz...
hashgval 14384 The value of the ` # ` fun...
hashginv 14385 The converse of ` G ` maps...
hashinf 14386 The value of the ` # ` fun...
hashbnd 14387 If ` A ` has size bounded ...
hashfxnn0 14388 The size function is a fun...
hashf 14389 The size function maps all...
hashxnn0 14390 The value of the hash func...
hashresfn 14391 Restriction of the domain ...
dmhashres 14392 Restriction of the domain ...
hashnn0pnf 14393 The value of the hash func...
hashnnn0genn0 14394 If the size of a set is no...
hashnemnf 14395 The size of a set is never...
hashv01gt1 14396 The size of a set is eithe...
hashfz1 14397 The set ` ( 1 ... N ) ` ha...
hashen 14398 Two finite sets have the s...
hasheni 14399 Equinumerous sets have the...
hasheqf1o 14400 The size of two finite set...
fiinfnf1o 14401 There is no bijection betw...
hasheqf1oi 14402 The size of two sets is eq...
hashf1rn 14403 The size of a finite set w...
hasheqf1od 14404 The size of two sets is eq...
fz1eqb 14405 Two possibly-empty 1-based...
hashcard 14406 The size function of the c...
hashcl 14407 Closure of the ` # ` funct...
hashxrcl 14408 Extended real closure of t...
hashclb 14409 Reverse closure of the ` #...
nfile 14410 The size of any infinite s...
hashvnfin 14411 A set of finite size is a ...
hashnfinnn0 14412 The size of an infinite se...
isfinite4 14413 A finite set is equinumero...
hasheq0 14414 Two ways of saying a set i...
hashneq0 14415 Two ways of saying a set i...
hashgt0n0 14416 If the size of a set is gr...
hashnncl 14417 Positive natural closure o...
hash0 14418 The empty set has size zer...
hashelne0d 14419 A set with an element has ...
hashsng 14420 The size of a singleton. ...
hashen1 14421 A set has size 1 if and on...
hash1elsn 14422 A set of size 1 with a kno...
hashrabrsn 14423 The size of a restricted c...
hashrabsn01 14424 The size of a restricted c...
hashrabsn1 14425 If the size of a restricte...
hashfn 14426 A function is equinumerous...
fseq1hash 14427 The value of the size func...
hashgadd 14428 ` G ` maps ordinal additio...
hashgval2 14429 A short expression for the...
hashdom 14430 Dominance relation for the...
hashdomi 14431 Non-strict order relation ...
hashsdom 14432 Strict dominance relation ...
hashun 14433 The size of the union of d...
hashun2 14434 The size of the union of f...
hashun3 14435 The size of the union of f...
hashinfxadd 14436 The extended real addition...
hashunx 14437 The size of the union of d...
hashge0 14438 The cardinality of a set i...
hashgt0 14439 The cardinality of a nonem...
hashge1 14440 The cardinality of a nonem...
1elfz0hash 14441 1 is an element of the fin...
hashnn0n0nn 14442 If a nonnegative integer i...
hashunsng 14443 The size of the union of a...
hashunsngx 14444 The size of the union of a...
hashunsnggt 14445 The size of a set is great...
hashprg 14446 The size of an unordered p...
elprchashprn2 14447 If one element of an unord...
hashprb 14448 The size of an unordered p...
hashprdifel 14449 The elements of an unorder...
prhash2ex 14450 There is (at least) one se...
hashle00 14451 If the size of a set is le...
hashgt0elex 14452 If the size of a set is gr...
hashgt0elexb 14453 The size of a set is great...
hashp1i 14454 Size of a finite ordinal. ...
hash1 14455 Size of a finite ordinal. ...
hash2 14456 Size of a finite ordinal. ...
hash3 14457 Size of a finite ordinal. ...
hash4 14458 Size of a finite ordinal. ...
pr0hash2ex 14459 There is (at least) one se...
hashss 14460 The size of a subset is le...
prsshashgt1 14461 The size of a superset of ...
hashin 14462 The size of the intersecti...
hashssdif 14463 The size of the difference...
hashdif 14464 The size of the difference...
hashdifsn 14465 The size of the difference...
hashdifpr 14466 The size of the difference...
hashsn01 14467 The size of a singleton is...
hashsnle1 14468 The size of a singleton is...
hashsnlei 14469 Get an upper bound on a co...
hash1snb 14470 The size of a set is 1 if ...
euhash1 14471 The size of a set is 1 in ...
hash1n0 14472 If the size of a set is 1 ...
hashgt12el 14473 In a set with more than on...
hashgt12el2 14474 In a set with more than on...
hashgt23el 14475 A set with more than two e...
hashunlei 14476 Get an upper bound on a co...
hashsslei 14477 Get an upper bound on a co...
hashfz 14478 Value of the numeric cardi...
fzsdom2 14479 Condition for finite range...
hashfzo 14480 Cardinality of a half-open...
hashfzo0 14481 Cardinality of a half-open...
hashfzp1 14482 Value of the numeric cardi...
hashfz0 14483 Value of the numeric cardi...
hashxplem 14484 Lemma for ~ hashxp . (Con...
hashxp 14485 The size of the Cartesian ...
hashmap 14486 The size of the set expone...
hashpw 14487 The size of the power set ...
hashfun 14488 A finite set is a function...
hashres 14489 The number of elements of ...
hashreshashfun 14490 The number of elements of ...
hashimarn 14491 The size of the image of a...
hashimarni 14492 If the size of the image o...
hashfundm 14493 The size of a set function...
hashf1dmrn 14494 The size of the domain of ...
hashf1dmcdm 14495 The size of the domain of ...
resunimafz0 14496 TODO-AV: Revise using ` F...
fnfz0hash 14497 The size of a function on ...
ffz0hash 14498 The size of a function on ...
fnfz0hashnn0 14499 The size of a function on ...
ffzo0hash 14500 The size of a function on ...
fnfzo0hash 14501 The size of a function on ...
fnfzo0hashnn0 14502 The value of the size func...
hashbclem 14503 Lemma for ~ hashbc : induc...
hashbc 14504 The binomial coefficient c...
hashfacen 14505 The number of bijections b...
hashf1lem1 14506 Lemma for ~ hashf1 . (Con...
hashf1lem2 14507 Lemma for ~ hashf1 . (Con...
hashf1 14508 The permutation number ` |...
hashfac 14509 A factorial counts the num...
leiso 14510 Two ways to write a strict...
leisorel 14511 Version of ~ isorel for st...
fz1isolem 14512 Lemma for ~ fz1iso . (Con...
fz1iso 14513 Any finite ordered set has...
ishashinf 14514 Any set that is not finite...
seqcoll 14515 The function ` F ` contain...
seqcoll2 14516 The function ` F ` contain...
phphashd 14517 Corollary of the Pigeonhol...
phphashrd 14518 Corollary of the Pigeonhol...
hashprlei 14519 An unordered pair has at m...
hash2pr 14520 A set of size two is an un...
hash2prde 14521 A set of size two is an un...
hash2exprb 14522 A set of size two is an un...
hash2prb 14523 A set of size two is a pro...
prprrab 14524 The set of proper pairs of...
nehash2 14525 The cardinality of a set w...
hash2prd 14526 A set of size two is an un...
hash2pwpr 14527 If the size of a subset of...
hashle2pr 14528 A nonempty set of size les...
hashle2prv 14529 A nonempty subset of a pow...
pr2pwpr 14530 The set of subsets of a pa...
hashge2el2dif 14531 A set with size at least 2...
hashge2el2difr 14532 A set with at least 2 diff...
hashge2el2difb 14533 A set has size at least 2 ...
hashdmpropge2 14534 The size of the domain of ...
hashtplei 14535 An unordered triple has at...
hashtpg 14536 The size of an unordered t...
hash7g 14537 The size of an unordered s...
hashge3el3dif 14538 A set with size at least 3...
elss2prb 14539 An element of the set of s...
hash2sspr 14540 A subset of size two is an...
exprelprel 14541 If there is an element of ...
hash3tr 14542 A set of size three is an ...
hash1to3 14543 If the size of a set is be...
hash3tpde 14544 A set of size three is an ...
hash3tpexb 14545 A set of size three is an ...
hash3tpb 14546 A set of size three is a p...
tpf1ofv0 14547 The value of a one-to-one ...
tpf1ofv1 14548 The value of a one-to-one ...
tpf1ofv2 14549 The value of a one-to-one ...
tpf 14550 A function into a (proper)...
tpfo 14551 A function onto a (proper)...
tpf1o 14552 A bijection onto a (proper...
fundmge2nop0 14553 A function with a domain c...
fundmge2nop 14554 A function with a domain c...
fun2dmnop0 14555 A function with a domain c...
fun2dmnop 14556 A function with a domain c...
hashdifsnp1 14557 If the size of a set is a ...
fi1uzind 14558 Properties of an ordered p...
brfi1uzind 14559 Properties of a binary rel...
brfi1ind 14560 Properties of a binary rel...
brfi1indALT 14561 Alternate proof of ~ brfi1...
opfi1uzind 14562 Properties of an ordered p...
opfi1ind 14563 Properties of an ordered p...
iswrd 14566 Property of being a word o...
wrdval 14567 Value of the set of words ...
iswrdi 14568 A zero-based sequence is a...
wrdf 14569 A word is a zero-based seq...
iswrdb 14570 A word over an alphabet is...
wrddm 14571 The indices of a word (i.e...
sswrd 14572 The set of words respects ...
snopiswrd 14573 A singleton of an ordered ...
wrdexg 14574 The set of words over a se...
wrdexb 14575 The set of words over a se...
wrdexi 14576 The set of words over a se...
wrdsymbcl 14577 A symbol within a word ove...
wrdfn 14578 A word is a function with ...
wrdv 14579 A word over an alphabet is...
wrdlndm 14580 The length of a word is no...
iswrdsymb 14581 An arbitrary word is a wor...
wrdfin 14582 A word is a finite set. (...
lencl 14583 The length of a word is a ...
lennncl 14584 The length of a nonempty w...
wrdffz 14585 A word is a function from ...
wrdeq 14586 Equality theorem for the s...
wrdeqi 14587 Equality theorem for the s...
iswrddm0 14588 A function with empty doma...
wrd0 14589 The empty set is a word (t...
0wrd0 14590 The empty word is the only...
ffz0iswrd 14591 A sequence with zero-based...
wrdsymb 14592 A word is a word over the ...
nfwrd 14593 Hypothesis builder for ` W...
csbwrdg 14594 Class substitution for the...
wrdnval 14595 Words of a fixed length ar...
wrdmap 14596 Words as a mapping. (Cont...
hashwrdn 14597 If there is only a finite ...
wrdnfi 14598 If there is only a finite ...
wrdsymb0 14599 A symbol at a position "ou...
wrdlenge1n0 14600 A word with length at leas...
len0nnbi 14601 The length of a word is a ...
wrdlenge2n0 14602 A word with length at leas...
wrdsymb1 14603 The first symbol of a none...
wrdlen1 14604 A word of length 1 starts ...
fstwrdne 14605 The first symbol of a none...
fstwrdne0 14606 The first symbol of a none...
eqwrd 14607 Two words are equal iff th...
elovmpowrd 14608 Implications for the value...
elovmptnn0wrd 14609 Implications for the value...
wrdred1 14610 A word truncated by a symb...
wrdred1hash 14611 The length of a word trunc...
lsw 14614 Extract the last symbol of...
lsw0 14615 The last symbol of an empt...
lsw0g 14616 The last symbol of an empt...
lsw1 14617 The last symbol of a word ...
lswcl 14618 Closure of the last symbol...
lswlgt0cl 14619 The last symbol of a nonem...
ccatfn 14622 The concatenation operator...
ccatfval 14623 Value of the concatenation...
ccatcl 14624 The concatenation of two w...
ccatlen 14625 The length of a concatenat...
ccat0 14626 The concatenation of two w...
ccatval1 14627 Value of a symbol in the l...
ccatval2 14628 Value of a symbol in the r...
ccatval3 14629 Value of a symbol in the r...
elfzelfzccat 14630 An element of a finite set...
ccatvalfn 14631 The concatenation of two w...
ccatsymb 14632 The symbol at a given posi...
ccatfv0 14633 The first symbol of a conc...
ccatval1lsw 14634 The last symbol of the lef...
ccatval21sw 14635 The first symbol of the ri...
ccatlid 14636 Concatenation of a word by...
ccatrid 14637 Concatenation of a word by...
ccatass 14638 Associative law for concat...
ccatrn 14639 The range of a concatenate...
ccatidid 14640 Concatenation of the empty...
lswccatn0lsw 14641 The last symbol of a word ...
lswccat0lsw 14642 The last symbol of a word ...
ccatalpha 14643 A concatenation of two arb...
ccatrcl1 14644 Reverse closure of a conca...
ids1 14647 Identity function protecti...
s1val 14648 Value of a singleton word....
s1rn 14649 The range of a singleton w...
s1eq 14650 Equality theorem for a sin...
s1eqd 14651 Equality theorem for a sin...
s1cl 14652 A singleton word is a word...
s1cld 14653 A singleton word is a word...
s1prc 14654 Value of a singleton word ...
s1cli 14655 A singleton word is a word...
s1len 14656 Length of a singleton word...
s1nz 14657 A singleton word is not th...
s1dm 14658 The domain of a singleton ...
s1dmALT 14659 Alternate version of ~ s1d...
s1fv 14660 Sole symbol of a singleton...
lsws1 14661 The last symbol of a singl...
eqs1 14662 A word of length 1 is a si...
wrdl1exs1 14663 A word of length 1 is a si...
wrdl1s1 14664 A word of length 1 is a si...
s111 14665 The singleton word functio...
ccatws1cl 14666 The concatenation of a wor...
ccatws1clv 14667 The concatenation of a wor...
ccat2s1cl 14668 The concatenation of two s...
ccats1alpha 14669 A concatenation of a word ...
ccatws1len 14670 The length of the concaten...
ccatws1lenp1b 14671 The length of a word is ` ...
wrdlenccats1lenm1 14672 The length of a word is th...
ccat2s1len 14673 The length of the concaten...
ccatw2s1cl 14674 The concatenation of a wor...
ccatw2s1len 14675 The length of the concaten...
ccats1val1 14676 Value of a symbol in the l...
ccats1val2 14677 Value of the symbol concat...
ccat1st1st 14678 The first symbol of a word...
ccat2s1p1 14679 Extract the first of two c...
ccat2s1p2 14680 Extract the second of two ...
ccatw2s1ass 14681 Associative law for a conc...
ccatws1n0 14682 The concatenation of a wor...
ccatws1ls 14683 The last symbol of the con...
lswccats1 14684 The last symbol of a word ...
lswccats1fst 14685 The last symbol of a nonem...
ccatw2s1p1 14686 Extract the symbol of the ...
ccatw2s1p2 14687 Extract the second of two ...
ccat2s1fvw 14688 Extract a symbol of a word...
ccat2s1fst 14689 The first symbol of the co...
swrdnznd 14692 The value of a subword ope...
swrdval 14693 Value of a subword. (Cont...
swrd00 14694 A zero length substring. ...
swrdcl 14695 Closure of the subword ext...
swrdval2 14696 Value of the subword extra...
swrdlen 14697 Length of an extracted sub...
swrdfv 14698 A symbol in an extracted s...
swrdfv0 14699 The first symbol in an ext...
swrdf 14700 A subword of a word is a f...
swrdvalfn 14701 Value of the subword extra...
swrdrn 14702 The range of a subword of ...
swrdlend 14703 The value of the subword e...
swrdnd 14704 The value of the subword e...
swrdnd2 14705 Value of the subword extra...
swrdnnn0nd 14706 The value of a subword ope...
swrdnd0 14707 The value of a subword ope...
swrd0 14708 A subword of an empty set ...
swrdrlen 14709 Length of a right-anchored...
swrdlen2 14710 Length of an extracted sub...
swrdfv2 14711 A symbol in an extracted s...
swrdwrdsymb 14712 A subword is a word over t...
swrdsb0eq 14713 Two subwords with the same...
swrdsbslen 14714 Two subwords with the same...
swrdspsleq 14715 Two words have a common su...
swrds1 14716 Extract a single symbol fr...
swrdlsw 14717 Extract the last single sy...
ccatswrd 14718 Joining two adjacent subwo...
swrdccat2 14719 Recover the right half of ...
pfxnndmnd 14722 The value of a prefix oper...
pfxval 14723 Value of a prefix operatio...
pfx00 14724 The zero length prefix is ...
pfx0 14725 A prefix of an empty set i...
pfxval0 14726 Value of a prefix operatio...
pfxcl 14727 Closure of the prefix extr...
pfxmpt 14728 Value of the prefix extrac...
pfxres 14729 Value of the subword extra...
pfxf 14730 A prefix of a word is a fu...
pfxfn 14731 Value of the prefix extrac...
pfxfv 14732 A symbol in a prefix of a ...
pfxlen 14733 Length of a prefix. (Cont...
pfxid 14734 A word is a prefix of itse...
pfxrn 14735 The range of a prefix of a...
pfxn0 14736 A prefix consisting of at ...
pfxnd 14737 The value of a prefix oper...
pfxnd0 14738 The value of a prefix oper...
pfxwrdsymb 14739 A prefix of a word is a wo...
addlenrevpfx 14740 The sum of the lengths of ...
addlenpfx 14741 The sum of the lengths of ...
pfxfv0 14742 The first symbol of a pref...
pfxtrcfv 14743 A symbol in a word truncat...
pfxtrcfv0 14744 The first symbol in a word...
pfxfvlsw 14745 The last symbol in a nonem...
pfxeq 14746 The prefixes of two words ...
pfxtrcfvl 14747 The last symbol in a word ...
pfxsuffeqwrdeq 14748 Two words are equal if and...
pfxsuff1eqwrdeq 14749 Two (nonempty) words are e...
disjwrdpfx 14750 Sets of words are disjoint...
ccatpfx 14751 Concatenating a prefix wit...
pfxccat1 14752 Recover the left half of a...
pfx1 14753 The prefix of length one o...
swrdswrdlem 14754 Lemma for ~ swrdswrd . (C...
swrdswrd 14755 A subword of a subword is ...
pfxswrd 14756 A prefix of a subword is a...
swrdpfx 14757 A subword of a prefix is a...
pfxpfx 14758 A prefix of a prefix is a ...
pfxpfxid 14759 A prefix of a prefix with ...
pfxcctswrd 14760 The concatenation of the p...
lenpfxcctswrd 14761 The length of the concaten...
lenrevpfxcctswrd 14762 The length of the concaten...
pfxlswccat 14763 Reconstruct a nonempty wor...
ccats1pfxeq 14764 The last symbol of a word ...
ccats1pfxeqrex 14765 There exists a symbol such...
ccatopth 14766 An ~ opth -like theorem fo...
ccatopth2 14767 An ~ opth -like theorem fo...
ccatlcan 14768 Concatenation of words is ...
ccatrcan 14769 Concatenation of words is ...
wrdeqs1cat 14770 Decompose a nonempty word ...
cats1un 14771 Express a word with an ext...
wrdind 14772 Perform induction over the...
wrd2ind 14773 Perform induction over the...
swrdccatfn 14774 The subword of a concatena...
swrdccatin1 14775 The subword of a concatena...
pfxccatin12lem4 14776 Lemma 4 for ~ pfxccatin12 ...
pfxccatin12lem2a 14777 Lemma for ~ pfxccatin12lem...
pfxccatin12lem1 14778 Lemma 1 for ~ pfxccatin12 ...
swrdccatin2 14779 The subword of a concatena...
pfxccatin12lem2c 14780 Lemma for ~ pfxccatin12lem...
pfxccatin12lem2 14781 Lemma 2 for ~ pfxccatin12 ...
pfxccatin12lem3 14782 Lemma 3 for ~ pfxccatin12 ...
pfxccatin12 14783 The subword of a concatena...
pfxccat3 14784 The subword of a concatena...
swrdccat 14785 The subword of a concatena...
pfxccatpfx1 14786 A prefix of a concatenatio...
pfxccatpfx2 14787 A prefix of a concatenatio...
pfxccat3a 14788 A prefix of a concatenatio...
swrdccat3blem 14789 Lemma for ~ swrdccat3b . ...
swrdccat3b 14790 A suffix of a concatenatio...
pfxccatid 14791 A prefix of a concatenatio...
ccats1pfxeqbi 14792 A word is a prefix of a wo...
swrdccatin1d 14793 The subword of a concatena...
swrdccatin2d 14794 The subword of a concatena...
pfxccatin12d 14795 The subword of a concatena...
reuccatpfxs1lem 14796 Lemma for ~ reuccatpfxs1 ....
reuccatpfxs1 14797 There is a unique word hav...
reuccatpfxs1v 14798 There is a unique word hav...
splval 14801 Value of the substring rep...
splcl 14802 Closure of the substring r...
splid 14803 Splicing a subword for the...
spllen 14804 The length of a splice. (...
splfv1 14805 Symbols to the left of a s...
splfv2a 14806 Symbols within the replace...
splval2 14807 Value of a splice, assumin...
revval 14810 Value of the word reversin...
revcl 14811 The reverse of a word is a...
revlen 14812 The reverse of a word has ...
revfv 14813 Reverse of a word at a poi...
rev0 14814 The empty word is its own ...
revs1 14815 Singleton words are their ...
revccat 14816 Antiautomorphic property o...
revrev 14817 Reversal is an involution ...
reps 14820 Construct a function mappi...
repsundef 14821 A function mapping a half-...
repsconst 14822 Construct a function mappi...
repsf 14823 The constructed function m...
repswsymb 14824 The symbols of a "repeated...
repsw 14825 A function mapping a half-...
repswlen 14826 The length of a "repeated ...
repsw0 14827 The "repeated symbol word"...
repsdf2 14828 Alternative definition of ...
repswsymball 14829 All the symbols of a "repe...
repswsymballbi 14830 A word is a "repeated symb...
repswfsts 14831 The first symbol of a none...
repswlsw 14832 The last symbol of a nonem...
repsw1 14833 The "repeated symbol word"...
repswswrd 14834 A subword of a "repeated s...
repswpfx 14835 A prefix of a repeated sym...
repswccat 14836 The concatenation of two "...
repswrevw 14837 The reverse of a "repeated...
cshfn 14840 Perform a cyclical shift f...
cshword 14841 Perform a cyclical shift f...
cshnz 14842 A cyclical shift is the em...
0csh0 14843 Cyclically shifting an emp...
cshw0 14844 A word cyclically shifted ...
cshwmodn 14845 Cyclically shifting a word...
cshwsublen 14846 Cyclically shifting a word...
cshwn 14847 A word cyclically shifted ...
cshwcl 14848 A cyclically shifted word ...
cshwlen 14849 The length of a cyclically...
cshwf 14850 A cyclically shifted word ...
cshwfn 14851 A cyclically shifted word ...
cshwrn 14852 The range of a cyclically ...
cshwidxmod 14853 The symbol at a given inde...
cshwidxmodr 14854 The symbol at a given inde...
cshwidx0mod 14855 The symbol at index 0 of a...
cshwidx0 14856 The symbol at index 0 of a...
cshwidxm1 14857 The symbol at index ((n-N)...
cshwidxm 14858 The symbol at index (n-N) ...
cshwidxn 14859 The symbol at index (n-1) ...
cshf1 14860 Cyclically shifting a word...
cshinj 14861 If a word is injectiv (reg...
repswcshw 14862 A cyclically shifted "repe...
2cshw 14863 Cyclically shifting a word...
2cshwid 14864 Cyclically shifting a word...
lswcshw 14865 The last symbol of a word ...
2cshwcom 14866 Cyclically shifting a word...
cshwleneq 14867 If the results of cyclical...
3cshw 14868 Cyclically shifting a word...
cshweqdif2 14869 If cyclically shifting two...
cshweqdifid 14870 If cyclically shifting a w...
cshweqrep 14871 If cyclically shifting a w...
cshw1 14872 If cyclically shifting a w...
cshw1repsw 14873 If cyclically shifting a w...
cshwsexa 14874 The class of (different!) ...
cshwsexaOLD 14875 Obsolete version of ~ cshw...
2cshwcshw 14876 If a word is a cyclically ...
scshwfzeqfzo 14877 For a nonempty word the se...
cshwcshid 14878 A cyclically shifted word ...
cshwcsh2id 14879 A cyclically shifted word ...
cshimadifsn 14880 The image of a cyclically ...
cshimadifsn0 14881 The image of a cyclically ...
wrdco 14882 Mapping a word by a functi...
lenco 14883 Length of a mapped word is...
s1co 14884 Mapping of a singleton wor...
revco 14885 Mapping of words (i.e., a ...
ccatco 14886 Mapping of words commutes ...
cshco 14887 Mapping of words commutes ...
swrdco 14888 Mapping of words commutes ...
pfxco 14889 Mapping of words commutes ...
lswco 14890 Mapping of (nonempty) word...
repsco 14891 Mapping of words commutes ...
cats1cld 14906 Closure of concatenation w...
cats1co 14907 Closure of concatenation w...
cats1cli 14908 Closure of concatenation w...
cats1fvn 14909 The last symbol of a conca...
cats1fv 14910 A symbol other than the la...
cats1len 14911 The length of concatenatio...
cats1cat 14912 Closure of concatenation w...
cats2cat 14913 Closure of concatenation o...
s2eqd 14914 Equality theorem for a dou...
s3eqd 14915 Equality theorem for a len...
s4eqd 14916 Equality theorem for a len...
s5eqd 14917 Equality theorem for a len...
s6eqd 14918 Equality theorem for a len...
s7eqd 14919 Equality theorem for a len...
s8eqd 14920 Equality theorem for a len...
s3eq2 14921 Equality theorem for a len...
s2cld 14922 A doubleton word is a word...
s3cld 14923 A length 3 string is a wor...
s4cld 14924 A length 4 string is a wor...
s5cld 14925 A length 5 string is a wor...
s6cld 14926 A length 6 string is a wor...
s7cld 14927 A length 7 string is a wor...
s8cld 14928 A length 7 string is a wor...
s2cl 14929 A doubleton word is a word...
s3cl 14930 A length 3 string is a wor...
s2cli 14931 A doubleton word is a word...
s3cli 14932 A length 3 string is a wor...
s4cli 14933 A length 4 string is a wor...
s5cli 14934 A length 5 string is a wor...
s6cli 14935 A length 6 string is a wor...
s7cli 14936 A length 7 string is a wor...
s8cli 14937 A length 8 string is a wor...
s2fv0 14938 Extract the first symbol f...
s2fv1 14939 Extract the second symbol ...
s2len 14940 The length of a doubleton ...
s2dm 14941 The domain of a doubleton ...
s3fv0 14942 Extract the first symbol f...
s3fv1 14943 Extract the second symbol ...
s3fv2 14944 Extract the third symbol f...
s3len 14945 The length of a length 3 s...
s4fv0 14946 Extract the first symbol f...
s4fv1 14947 Extract the second symbol ...
s4fv2 14948 Extract the third symbol f...
s4fv3 14949 Extract the fourth symbol ...
s4len 14950 The length of a length 4 s...
s5len 14951 The length of a length 5 s...
s6len 14952 The length of a length 6 s...
s7len 14953 The length of a length 7 s...
s8len 14954 The length of a length 8 s...
lsws2 14955 The last symbol of a doubl...
lsws3 14956 The last symbol of a 3 let...
lsws4 14957 The last symbol of a 4 let...
s2prop 14958 A length 2 word is an unor...
s2dmALT 14959 Alternate version of ~ s2d...
s3tpop 14960 A length 3 word is an unor...
s4prop 14961 A length 4 word is a union...
s3fn 14962 A length 3 word is a funct...
funcnvs1 14963 The converse of a singleto...
funcnvs2 14964 The converse of a length 2...
funcnvs3 14965 The converse of a length 3...
funcnvs4 14966 The converse of a length 4...
s2f1o 14967 A length 2 word with mutua...
f1oun2prg 14968 A union of unordered pairs...
s4f1o 14969 A length 4 word with mutua...
s4dom 14970 The domain of a length 4 w...
s2co 14971 Mapping a doubleton word b...
s3co 14972 Mapping a length 3 string ...
s0s1 14973 Concatenation of fixed len...
s1s2 14974 Concatenation of fixed len...
s1s3 14975 Concatenation of fixed len...
s1s4 14976 Concatenation of fixed len...
s1s5 14977 Concatenation of fixed len...
s1s6 14978 Concatenation of fixed len...
s1s7 14979 Concatenation of fixed len...
s2s2 14980 Concatenation of fixed len...
s4s2 14981 Concatenation of fixed len...
s4s3 14982 Concatenation of fixed len...
s4s4 14983 Concatenation of fixed len...
s3s4 14984 Concatenation of fixed len...
s2s5 14985 Concatenation of fixed len...
s5s2 14986 Concatenation of fixed len...
s2eq2s1eq 14987 Two length 2 words are equ...
s2eq2seq 14988 Two length 2 words are equ...
s3eqs2s1eq 14989 Two length 3 words are equ...
s3eq3seq 14990 Two length 3 words are equ...
swrds2 14991 Extract two adjacent symbo...
swrds2m 14992 Extract two adjacent symbo...
wrdlen2i 14993 Implications of a word of ...
wrd2pr2op 14994 A word of length two repre...
wrdlen2 14995 A word of length two. (Co...
wrdlen2s2 14996 A word of length two as do...
wrdl2exs2 14997 A word of length two is a ...
pfx2 14998 A prefix of length two. (...
wrd3tpop 14999 A word of length three rep...
wrdlen3s3 15000 A word of length three as ...
repsw2 15001 The "repeated symbol word"...
repsw3 15002 The "repeated symbol word"...
swrd2lsw 15003 Extract the last two symbo...
2swrd2eqwrdeq 15004 Two words of length at lea...
ccatw2s1ccatws2 15005 The concatenation of a wor...
ccat2s1fvwALT 15006 Alternate proof of ~ ccat2...
wwlktovf 15007 Lemma 1 for ~ wrd2f1tovbij...
wwlktovf1 15008 Lemma 2 for ~ wrd2f1tovbij...
wwlktovfo 15009 Lemma 3 for ~ wrd2f1tovbij...
wwlktovf1o 15010 Lemma 4 for ~ wrd2f1tovbij...
wrd2f1tovbij 15011 There is a bijection betwe...
eqwrds3 15012 A word is equal with a len...
wrdl3s3 15013 A word of length 3 is a le...
s2rn 15014 Range of a length 2 string...
s3rn 15015 Range of a length 3 string...
s7rn 15016 Range of a length 7 string...
s7f1o 15017 A length 7 word with mutua...
s3sndisj 15018 The singletons consisting ...
s3iunsndisj 15019 The union of singletons co...
ofccat 15020 Letterwise operations on w...
ofs1 15021 Letterwise operations on a...
ofs2 15022 Letterwise operations on a...
coss12d 15023 Subset deduction for compo...
trrelssd 15024 The composition of subclas...
xpcogend 15025 The most interesting case ...
xpcoidgend 15026 If two classes are not dis...
cotr2g 15027 Two ways of saying that th...
cotr2 15028 Two ways of saying a relat...
cotr3 15029 Two ways of saying a relat...
coemptyd 15030 Deduction about compositio...
xptrrel 15031 The cross product is alway...
0trrel 15032 The empty class is a trans...
cleq1lem 15033 Equality implies bijection...
cleq1 15034 Equality of relations impl...
clsslem 15035 The closure of a subclass ...
trcleq1 15040 Equality of relations impl...
trclsslem 15041 The transitive closure (as...
trcleq2lem 15042 Equality implies bijection...
cvbtrcl 15043 Change of bound variable i...
trcleq12lem 15044 Equality implies bijection...
trclexlem 15045 Existence of relation impl...
trclublem 15046 If a relation exists then ...
trclubi 15047 The Cartesian product of t...
trclubgi 15048 The union with the Cartesi...
trclub 15049 The Cartesian product of t...
trclubg 15050 The union with the Cartesi...
trclfv 15051 The transitive closure of ...
brintclab 15052 Two ways to express a bina...
brtrclfv 15053 Two ways of expressing the...
brcnvtrclfv 15054 Two ways of expressing the...
brtrclfvcnv 15055 Two ways of expressing the...
brcnvtrclfvcnv 15056 Two ways of expressing the...
trclfvss 15057 The transitive closure (as...
trclfvub 15058 The transitive closure of ...
trclfvlb 15059 The transitive closure of ...
trclfvcotr 15060 The transitive closure of ...
trclfvlb2 15061 The transitive closure of ...
trclfvlb3 15062 The transitive closure of ...
cotrtrclfv 15063 The transitive closure of ...
trclidm 15064 The transitive closure of ...
trclun 15065 Transitive closure of a un...
trclfvg 15066 The value of the transitiv...
trclfvcotrg 15067 The value of the transitiv...
reltrclfv 15068 The transitive closure of ...
dmtrclfv 15069 The domain of the transiti...
reldmrelexp 15072 The domain of the repeated...
relexp0g 15073 A relation composed zero t...
relexp0 15074 A relation composed zero t...
relexp0d 15075 A relation composed zero t...
relexpsucnnr 15076 A reduction for relation e...
relexp1g 15077 A relation composed once i...
dfid5 15078 Identity relation is equal...
dfid6 15079 Identity relation expresse...
relexp1d 15080 A relation composed once i...
relexpsucnnl 15081 A reduction for relation e...
relexpsucl 15082 A reduction for relation e...
relexpsucr 15083 A reduction for relation e...
relexpsucrd 15084 A reduction for relation e...
relexpsucld 15085 A reduction for relation e...
relexpcnv 15086 Commutation of converse an...
relexpcnvd 15087 Commutation of converse an...
relexp0rel 15088 The exponentiation of a cl...
relexprelg 15089 The exponentiation of a cl...
relexprel 15090 The exponentiation of a re...
relexpreld 15091 The exponentiation of a re...
relexpnndm 15092 The domain of an exponenti...
relexpdmg 15093 The domain of an exponenti...
relexpdm 15094 The domain of an exponenti...
relexpdmd 15095 The domain of an exponenti...
relexpnnrn 15096 The range of an exponentia...
relexprng 15097 The range of an exponentia...
relexprn 15098 The range of an exponentia...
relexprnd 15099 The range of an exponentia...
relexpfld 15100 The field of an exponentia...
relexpfldd 15101 The field of an exponentia...
relexpaddnn 15102 Relation composition becom...
relexpuzrel 15103 The exponentiation of a cl...
relexpaddg 15104 Relation composition becom...
relexpaddd 15105 Relation composition becom...
rtrclreclem1 15108 The reflexive, transitive ...
dfrtrclrec2 15109 If two elements are connec...
rtrclreclem2 15110 The reflexive, transitive ...
rtrclreclem3 15111 The reflexive, transitive ...
rtrclreclem4 15112 The reflexive, transitive ...
dfrtrcl2 15113 The two definitions ` t* `...
relexpindlem 15114 Principle of transitive in...
relexpind 15115 Principle of transitive in...
rtrclind 15116 Principle of transitive in...
shftlem 15119 Two ways to write a shifte...
shftuz 15120 A shift of the upper integ...
shftfval 15121 The value of the sequence ...
shftdm 15122 Domain of a relation shift...
shftfib 15123 Value of a fiber of the re...
shftfn 15124 Functionality and domain o...
shftval 15125 Value of a sequence shifte...
shftval2 15126 Value of a sequence shifte...
shftval3 15127 Value of a sequence shifte...
shftval4 15128 Value of a sequence shifte...
shftval5 15129 Value of a shifted sequenc...
shftf 15130 Functionality of a shifted...
2shfti 15131 Composite shift operations...
shftidt2 15132 Identity law for the shift...
shftidt 15133 Identity law for the shift...
shftcan1 15134 Cancellation law for the s...
shftcan2 15135 Cancellation law for the s...
seqshft 15136 Shifting the index set of ...
sgnval 15139 Value of the signum functi...
sgn0 15140 The signum of 0 is 0. (Co...
sgnp 15141 The signum of a positive e...
sgnrrp 15142 The signum of a positive r...
sgn1 15143 The signum of 1 is 1. (Co...
sgnpnf 15144 The signum of ` +oo ` is 1...
sgnn 15145 The signum of a negative e...
sgnmnf 15146 The signum of ` -oo ` is -...
cjval 15153 The value of the conjugate...
cjth 15154 The defining property of t...
cjf 15155 Domain and codomain of the...
cjcl 15156 The conjugate of a complex...
reval 15157 The value of the real part...
imval 15158 The value of the imaginary...
imre 15159 The imaginary part of a co...
reim 15160 The real part of a complex...
recl 15161 The real part of a complex...
imcl 15162 The imaginary part of a co...
ref 15163 Domain and codomain of the...
imf 15164 Domain and codomain of the...
crre 15165 The real part of a complex...
crim 15166 The real part of a complex...
replim 15167 Reconstruct a complex numb...
remim 15168 Value of the conjugate of ...
reim0 15169 The imaginary part of a re...
reim0b 15170 A number is real iff its i...
rereb 15171 A number is real iff it eq...
mulre 15172 A product with a nonzero r...
rere 15173 A real number equals its r...
cjreb 15174 A number is real iff it eq...
recj 15175 Real part of a complex con...
reneg 15176 Real part of negative. (C...
readd 15177 Real part distributes over...
resub 15178 Real part distributes over...
remullem 15179 Lemma for ~ remul , ~ immu...
remul 15180 Real part of a product. (...
remul2 15181 Real part of a product. (...
rediv 15182 Real part of a division. ...
imcj 15183 Imaginary part of a comple...
imneg 15184 The imaginary part of a ne...
imadd 15185 Imaginary part distributes...
imsub 15186 Imaginary part distributes...
immul 15187 Imaginary part of a produc...
immul2 15188 Imaginary part of a produc...
imdiv 15189 Imaginary part of a divisi...
cjre 15190 A real number equals its c...
cjcj 15191 The conjugate of the conju...
cjadd 15192 Complex conjugate distribu...
cjmul 15193 Complex conjugate distribu...
ipcnval 15194 Standard inner product on ...
cjmulrcl 15195 A complex number times its...
cjmulval 15196 A complex number times its...
cjmulge0 15197 A complex number times its...
cjneg 15198 Complex conjugate of negat...
addcj 15199 A number plus its conjugat...
cjsub 15200 Complex conjugate distribu...
cjexp 15201 Complex conjugate of posit...
imval2 15202 The imaginary part of a nu...
re0 15203 The real part of zero. (C...
im0 15204 The imaginary part of zero...
re1 15205 The real part of one. (Co...
im1 15206 The imaginary part of one....
rei 15207 The real part of ` _i ` . ...
imi 15208 The imaginary part of ` _i...
cj0 15209 The conjugate of zero. (C...
cji 15210 The complex conjugate of t...
cjreim 15211 The conjugate of a represe...
cjreim2 15212 The conjugate of the repre...
cj11 15213 Complex conjugate is a one...
cjne0 15214 A number is nonzero iff it...
cjdiv 15215 Complex conjugate distribu...
cnrecnv 15216 The inverse to the canonic...
sqeqd 15217 A deduction for showing tw...
recli 15218 The real part of a complex...
imcli 15219 The imaginary part of a co...
cjcli 15220 Closure law for complex co...
replimi 15221 Construct a complex number...
cjcji 15222 The conjugate of the conju...
reim0bi 15223 A number is real iff its i...
rerebi 15224 A real number equals its r...
cjrebi 15225 A number is real iff it eq...
recji 15226 Real part of a complex con...
imcji 15227 Imaginary part of a comple...
cjmulrcli 15228 A complex number times its...
cjmulvali 15229 A complex number times its...
cjmulge0i 15230 A complex number times its...
renegi 15231 Real part of negative. (C...
imnegi 15232 Imaginary part of negative...
cjnegi 15233 Complex conjugate of negat...
addcji 15234 A number plus its conjugat...
readdi 15235 Real part distributes over...
imaddi 15236 Imaginary part distributes...
remuli 15237 Real part of a product. (...
immuli 15238 Imaginary part of a produc...
cjaddi 15239 Complex conjugate distribu...
cjmuli 15240 Complex conjugate distribu...
ipcni 15241 Standard inner product on ...
cjdivi 15242 Complex conjugate distribu...
crrei 15243 The real part of a complex...
crimi 15244 The imaginary part of a co...
recld 15245 The real part of a complex...
imcld 15246 The imaginary part of a co...
cjcld 15247 Closure law for complex co...
replimd 15248 Construct a complex number...
remimd 15249 Value of the conjugate of ...
cjcjd 15250 The conjugate of the conju...
reim0bd 15251 A number is real iff its i...
rerebd 15252 A real number equals its r...
cjrebd 15253 A number is real iff it eq...
cjne0d 15254 A number is nonzero iff it...
recjd 15255 Real part of a complex con...
imcjd 15256 Imaginary part of a comple...
cjmulrcld 15257 A complex number times its...
cjmulvald 15258 A complex number times its...
cjmulge0d 15259 A complex number times its...
renegd 15260 Real part of negative. (C...
imnegd 15261 Imaginary part of negative...
cjnegd 15262 Complex conjugate of negat...
addcjd 15263 A number plus its conjugat...
cjexpd 15264 Complex conjugate of posit...
readdd 15265 Real part distributes over...
imaddd 15266 Imaginary part distributes...
resubd 15267 Real part distributes over...
imsubd 15268 Imaginary part distributes...
remuld 15269 Real part of a product. (...
immuld 15270 Imaginary part of a produc...
cjaddd 15271 Complex conjugate distribu...
cjmuld 15272 Complex conjugate distribu...
ipcnd 15273 Standard inner product on ...
cjdivd 15274 Complex conjugate distribu...
rered 15275 A real number equals its r...
reim0d 15276 The imaginary part of a re...
cjred 15277 A real number equals its c...
remul2d 15278 Real part of a product. (...
immul2d 15279 Imaginary part of a produc...
redivd 15280 Real part of a division. ...
imdivd 15281 Imaginary part of a divisi...
crred 15282 The real part of a complex...
crimd 15283 The imaginary part of a co...
sqrtval 15288 Value of square root funct...
absval 15289 The absolute value (modulu...
rennim 15290 A real number does not lie...
cnpart 15291 The specification of restr...
sqrt0 15292 The square root of zero is...
01sqrexlem1 15293 Lemma for ~ 01sqrex . (Co...
01sqrexlem2 15294 Lemma for ~ 01sqrex . (Co...
01sqrexlem3 15295 Lemma for ~ 01sqrex . (Co...
01sqrexlem4 15296 Lemma for ~ 01sqrex . (Co...
01sqrexlem5 15297 Lemma for ~ 01sqrex . (Co...
01sqrexlem6 15298 Lemma for ~ 01sqrex . (Co...
01sqrexlem7 15299 Lemma for ~ 01sqrex . (Co...
01sqrex 15300 Existence of a square root...
resqrex 15301 Existence of a square root...
sqrmo 15302 Uniqueness for the square ...
resqreu 15303 Existence and uniqueness f...
resqrtcl 15304 Closure of the square root...
resqrtthlem 15305 Lemma for ~ resqrtth . (C...
resqrtth 15306 Square root theorem over t...
remsqsqrt 15307 Square of square root. (C...
sqrtge0 15308 The square root function i...
sqrtgt0 15309 The square root function i...
sqrtmul 15310 Square root distributes ov...
sqrtle 15311 Square root is monotonic. ...
sqrtlt 15312 Square root is strictly mo...
sqrt11 15313 The square root function i...
sqrt00 15314 A square root is zero iff ...
rpsqrtcl 15315 The square root of a posit...
sqrtdiv 15316 Square root distributes ov...
sqrtneglem 15317 The square root of a negat...
sqrtneg 15318 The square root of a negat...
sqrtsq2 15319 Relationship between squar...
sqrtsq 15320 Square root of square. (C...
sqrtmsq 15321 Square root of square. (C...
sqrt1 15322 The square root of 1 is 1....
sqrt4 15323 The square root of 4 is 2....
sqrt9 15324 The square root of 9 is 3....
sqrt2gt1lt2 15325 The square root of 2 is bo...
sqrtm1 15326 The imaginary unit is the ...
nn0sqeq1 15327 A natural number with squa...
absneg 15328 Absolute value of the nega...
abscl 15329 Real closure of absolute v...
abscj 15330 The absolute value of a nu...
absvalsq 15331 Square of value of absolut...
absvalsq2 15332 Square of value of absolut...
sqabsadd 15333 Square of absolute value o...
sqabssub 15334 Square of absolute value o...
absval2 15335 Value of absolute value fu...
abs0 15336 The absolute value of 0. ...
absi 15337 The absolute value of the ...
absge0 15338 Absolute value is nonnegat...
absrpcl 15339 The absolute value of a no...
abs00 15340 The absolute value of a nu...
abs00ad 15341 A complex number is zero i...
abs00bd 15342 If a complex number is zer...
absreimsq 15343 Square of the absolute val...
absreim 15344 Absolute value of a number...
absmul 15345 Absolute value distributes...
absdiv 15346 Absolute value distributes...
absid 15347 A nonnegative number is it...
abs1 15348 The absolute value of one ...
absnid 15349 For a negative number, its...
leabs 15350 A real number is less than...
absor 15351 The absolute value of a re...
absre 15352 Absolute value of a real n...
absresq 15353 Square of the absolute val...
absmod0 15354 ` A ` is divisible by ` B ...
absexp 15355 Absolute value of positive...
absexpz 15356 Absolute value of integer ...
abssq 15357 Square can be moved in and...
sqabs 15358 The squares of two reals a...
absrele 15359 The absolute value of a co...
absimle 15360 The absolute value of a co...
max0add 15361 The sum of the positive an...
absz 15362 A real number is an intege...
nn0abscl 15363 The absolute value of an i...
zabscl 15364 The absolute value of an i...
abslt 15365 Absolute value and 'less t...
absle 15366 Absolute value and 'less t...
abssubne0 15367 If the absolute value of a...
absdiflt 15368 The absolute value of a di...
absdifle 15369 The absolute value of a di...
elicc4abs 15370 Membership in a symmetric ...
lenegsq 15371 Comparison to a nonnegativ...
releabs 15372 The real part of a number ...
recval 15373 Reciprocal expressed with ...
absidm 15374 The absolute value functio...
absgt0 15375 The absolute value of a no...
nnabscl 15376 The absolute value of a no...
abssub 15377 Swapping order of subtract...
abssubge0 15378 Absolute value of a nonneg...
abssuble0 15379 Absolute value of a nonpos...
absmax 15380 The maximum of two numbers...
abstri 15381 Triangle inequality for ab...
abs3dif 15382 Absolute value of differen...
abs2dif 15383 Difference of absolute val...
abs2dif2 15384 Difference of absolute val...
abs2difabs 15385 Absolute value of differen...
abs1m 15386 For any complex number, th...
recan 15387 Cancellation law involving...
absf 15388 Mapping domain and codomai...
abs3lem 15389 Lemma involving absolute v...
abslem2 15390 Lemma involving absolute v...
rddif 15391 The difference between a r...
absrdbnd 15392 Bound on the absolute valu...
fzomaxdiflem 15393 Lemma for ~ fzomaxdif . (...
fzomaxdif 15394 A bound on the separation ...
uzin2 15395 The upper integers are clo...
rexanuz 15396 Combine two different uppe...
rexanre 15397 Combine two different uppe...
rexfiuz 15398 Combine finitely many diff...
rexuz3 15399 Restrict the base of the u...
rexanuz2 15400 Combine two different uppe...
r19.29uz 15401 A version of ~ 19.29 for u...
r19.2uz 15402 A version of ~ r19.2z for ...
rexuzre 15403 Convert an upper real quan...
rexico 15404 Restrict the base of an up...
cau3lem 15405 Lemma for ~ cau3 . (Contr...
cau3 15406 Convert between three-quan...
cau4 15407 Change the base of a Cauch...
caubnd2 15408 A Cauchy sequence of compl...
caubnd 15409 A Cauchy sequence of compl...
sqreulem 15410 Lemma for ~ sqreu : write ...
sqreu 15411 Existence and uniqueness f...
sqrtcl 15412 Closure of the square root...
sqrtthlem 15413 Lemma for ~ sqrtth . (Con...
sqrtf 15414 Mapping domain and codomai...
sqrtth 15415 Square root theorem over t...
sqrtrege0 15416 The square root function m...
eqsqrtor 15417 Solve an equation containi...
eqsqrtd 15418 A deduction for showing th...
eqsqrt2d 15419 A deduction for showing th...
amgm2 15420 Arithmetic-geometric mean ...
sqrtthi 15421 Square root theorem. Theo...
sqrtcli 15422 The square root of a nonne...
sqrtgt0i 15423 The square root of a posit...
sqrtmsqi 15424 Square root of square. (C...
sqrtsqi 15425 Square root of square. (C...
sqsqrti 15426 Square of square root. (C...
sqrtge0i 15427 The square root of a nonne...
absidi 15428 A nonnegative number is it...
absnidi 15429 A negative number is the n...
leabsi 15430 A real number is less than...
absori 15431 The absolute value of a re...
absrei 15432 Absolute value of a real n...
sqrtpclii 15433 The square root of a posit...
sqrtgt0ii 15434 The square root of a posit...
sqrt11i 15435 The square root function i...
sqrtmuli 15436 Square root distributes ov...
sqrtmulii 15437 Square root distributes ov...
sqrtmsq2i 15438 Relationship between squar...
sqrtlei 15439 Square root is monotonic. ...
sqrtlti 15440 Square root is strictly mo...
abslti 15441 Absolute value and 'less t...
abslei 15442 Absolute value and 'less t...
cnsqrt00 15443 A square root of a complex...
absvalsqi 15444 Square of value of absolut...
absvalsq2i 15445 Square of value of absolut...
abscli 15446 Real closure of absolute v...
absge0i 15447 Absolute value is nonnegat...
absval2i 15448 Value of absolute value fu...
abs00i 15449 The absolute value of a nu...
absgt0i 15450 The absolute value of a no...
absnegi 15451 Absolute value of negative...
abscji 15452 The absolute value of a nu...
releabsi 15453 The real part of a number ...
abssubi 15454 Swapping order of subtract...
absmuli 15455 Absolute value distributes...
sqabsaddi 15456 Square of absolute value o...
sqabssubi 15457 Square of absolute value o...
absdivzi 15458 Absolute value distributes...
abstrii 15459 Triangle inequality for ab...
abs3difi 15460 Absolute value of differen...
abs3lemi 15461 Lemma involving absolute v...
rpsqrtcld 15462 The square root of a posit...
sqrtgt0d 15463 The square root of a posit...
absnidd 15464 A negative number is the n...
leabsd 15465 A real number is less than...
absord 15466 The absolute value of a re...
absred 15467 Absolute value of a real n...
resqrtcld 15468 The square root of a nonne...
sqrtmsqd 15469 Square root of square. (C...
sqrtsqd 15470 Square root of square. (C...
sqrtge0d 15471 The square root of a nonne...
sqrtnegd 15472 The square root of a negat...
absidd 15473 A nonnegative number is it...
sqrtdivd 15474 Square root distributes ov...
sqrtmuld 15475 Square root distributes ov...
sqrtsq2d 15476 Relationship between squar...
sqrtled 15477 Square root is monotonic. ...
sqrtltd 15478 Square root is strictly mo...
sqr11d 15479 The square root function i...
absltd 15480 Absolute value and 'less t...
absled 15481 Absolute value and 'less t...
abssubge0d 15482 Absolute value of a nonneg...
abssuble0d 15483 Absolute value of a nonpos...
absdifltd 15484 The absolute value of a di...
absdifled 15485 The absolute value of a di...
icodiamlt 15486 Two elements in a half-ope...
abscld 15487 Real closure of absolute v...
sqrtcld 15488 Closure of the square root...
sqrtrege0d 15489 The real part of the squar...
sqsqrtd 15490 Square root theorem. Theo...
msqsqrtd 15491 Square root theorem. Theo...
sqr00d 15492 A square root is zero iff ...
absvalsqd 15493 Square of value of absolut...
absvalsq2d 15494 Square of value of absolut...
absge0d 15495 Absolute value is nonnegat...
absval2d 15496 Value of absolute value fu...
abs00d 15497 The absolute value of a nu...
absne0d 15498 The absolute value of a nu...
absrpcld 15499 The absolute value of a no...
absnegd 15500 Absolute value of negative...
abscjd 15501 The absolute value of a nu...
releabsd 15502 The real part of a number ...
absexpd 15503 Absolute value of positive...
abssubd 15504 Swapping order of subtract...
absmuld 15505 Absolute value distributes...
absdivd 15506 Absolute value distributes...
abstrid 15507 Triangle inequality for ab...
abs2difd 15508 Difference of absolute val...
abs2dif2d 15509 Difference of absolute val...
abs2difabsd 15510 Absolute value of differen...
abs3difd 15511 Absolute value of differen...
abs3lemd 15512 Lemma involving absolute v...
reusq0 15513 A complex number is the sq...
bhmafibid1cn 15514 The Brahmagupta-Fibonacci ...
bhmafibid2cn 15515 The Brahmagupta-Fibonacci ...
bhmafibid1 15516 The Brahmagupta-Fibonacci ...
bhmafibid2 15517 The Brahmagupta-Fibonacci ...
limsupgord 15520 Ordering property of the s...
limsupcl 15521 Closure of the superior li...
limsupval 15522 The superior limit of an i...
limsupgf 15523 Closure of the superior li...
limsupgval 15524 Value of the superior limi...
limsupgle 15525 The defining property of t...
limsuple 15526 The defining property of t...
limsuplt 15527 The defining property of t...
limsupval2 15528 The superior limit, relati...
limsupgre 15529 If a sequence of real numb...
limsupbnd1 15530 If a sequence is eventuall...
limsupbnd2 15531 If a sequence is eventuall...
climrel 15540 The limit relation is a re...
rlimrel 15541 The limit relation is a re...
clim 15542 Express the predicate: Th...
rlim 15543 Express the predicate: Th...
rlim2 15544 Rewrite ~ rlim for a mappi...
rlim2lt 15545 Use strictly less-than in ...
rlim3 15546 Restrict the range of the ...
climcl 15547 Closure of the limit of a ...
rlimpm 15548 Closure of a function with...
rlimf 15549 Closure of a function with...
rlimss 15550 Domain closure of a functi...
rlimcl 15551 Closure of the limit of a ...
clim2 15552 Express the predicate: Th...
clim2c 15553 Express the predicate ` F ...
clim0 15554 Express the predicate ` F ...
clim0c 15555 Express the predicate ` F ...
rlim0 15556 Express the predicate ` B ...
rlim0lt 15557 Use strictly less-than in ...
climi 15558 Convergence of a sequence ...
climi2 15559 Convergence of a sequence ...
climi0 15560 Convergence of a sequence ...
rlimi 15561 Convergence at infinity of...
rlimi2 15562 Convergence at infinity of...
ello1 15563 Elementhood in the set of ...
ello12 15564 Elementhood in the set of ...
ello12r 15565 Sufficient condition for e...
lo1f 15566 An eventually upper bounde...
lo1dm 15567 An eventually upper bounde...
lo1bdd 15568 The defining property of a...
ello1mpt 15569 Elementhood in the set of ...
ello1mpt2 15570 Elementhood in the set of ...
ello1d 15571 Sufficient condition for e...
lo1bdd2 15572 If an eventually bounded f...
lo1bddrp 15573 Refine ~ o1bdd2 to give a ...
elo1 15574 Elementhood in the set of ...
elo12 15575 Elementhood in the set of ...
elo12r 15576 Sufficient condition for e...
o1f 15577 An eventually bounded func...
o1dm 15578 An eventually bounded func...
o1bdd 15579 The defining property of a...
lo1o1 15580 A function is eventually b...
lo1o12 15581 A function is eventually b...
elo1mpt 15582 Elementhood in the set of ...
elo1mpt2 15583 Elementhood in the set of ...
elo1d 15584 Sufficient condition for e...
o1lo1 15585 A real function is eventua...
o1lo12 15586 A lower bounded real funct...
o1lo1d 15587 A real eventually bounded ...
icco1 15588 Derive eventual boundednes...
o1bdd2 15589 If an eventually bounded f...
o1bddrp 15590 Refine ~ o1bdd2 to give a ...
climconst 15591 An (eventually) constant s...
rlimconst 15592 A constant sequence conver...
rlimclim1 15593 Forward direction of ~ rli...
rlimclim 15594 A sequence on an upper int...
climrlim2 15595 Produce a real limit from ...
climconst2 15596 A constant sequence conver...
climz 15597 The zero sequence converge...
rlimuni 15598 A real function whose doma...
rlimdm 15599 Two ways to express that a...
climuni 15600 An infinite sequence of co...
fclim 15601 The limit relation is func...
climdm 15602 Two ways to express that a...
climeu 15603 An infinite sequence of co...
climreu 15604 An infinite sequence of co...
climmo 15605 An infinite sequence of co...
rlimres 15606 The restriction of a funct...
lo1res 15607 The restriction of an even...
o1res 15608 The restriction of an even...
rlimres2 15609 The restriction of a funct...
lo1res2 15610 The restriction of a funct...
o1res2 15611 The restriction of a funct...
lo1resb 15612 The restriction of a funct...
rlimresb 15613 The restriction of a funct...
o1resb 15614 The restriction of a funct...
climeq 15615 Two functions that are eve...
lo1eq 15616 Two functions that are eve...
rlimeq 15617 Two functions that are eve...
o1eq 15618 Two functions that are eve...
climmpt 15619 Exhibit a function ` G ` w...
2clim 15620 If two sequences converge ...
climmpt2 15621 Relate an integer limit on...
climshftlem 15622 A shifted function converg...
climres 15623 A function restricted to u...
climshft 15624 A shifted function converg...
serclim0 15625 The zero series converges ...
rlimcld2 15626 If ` D ` is a closed set i...
rlimrege0 15627 The limit of a sequence of...
rlimrecl 15628 The limit of a real sequen...
rlimge0 15629 The limit of a sequence of...
climshft2 15630 A shifted function converg...
climrecl 15631 The limit of a convergent ...
climge0 15632 A nonnegative sequence con...
climabs0 15633 Convergence to zero of the...
o1co 15634 Sufficient condition for t...
o1compt 15635 Sufficient condition for t...
rlimcn1 15636 Image of a limit under a c...
rlimcn1b 15637 Image of a limit under a c...
rlimcn3 15638 Image of a limit under a c...
rlimcn2 15639 Image of a limit under a c...
climcn1 15640 Image of a limit under a c...
climcn2 15641 Image of a limit under a c...
addcn2 15642 Complex number addition is...
subcn2 15643 Complex number subtraction...
mulcn2 15644 Complex number multiplicat...
reccn2 15645 The reciprocal function is...
cn1lem 15646 A sufficient condition for...
abscn2 15647 The absolute value functio...
cjcn2 15648 The complex conjugate func...
recn2 15649 The real part function is ...
imcn2 15650 The imaginary part functio...
climcn1lem 15651 The limit of a continuous ...
climabs 15652 Limit of the absolute valu...
climcj 15653 Limit of the complex conju...
climre 15654 Limit of the real part of ...
climim 15655 Limit of the imaginary par...
rlimmptrcl 15656 Reverse closure for a real...
rlimabs 15657 Limit of the absolute valu...
rlimcj 15658 Limit of the complex conju...
rlimre 15659 Limit of the real part of ...
rlimim 15660 Limit of the imaginary par...
o1of2 15661 Show that a binary operati...
o1add 15662 The sum of two eventually ...
o1mul 15663 The product of two eventua...
o1sub 15664 The difference of two even...
rlimo1 15665 Any function with a finite...
rlimdmo1 15666 A convergent function is e...
o1rlimmul 15667 The product of an eventual...
o1const 15668 A constant function is eve...
lo1const 15669 A constant function is eve...
lo1mptrcl 15670 Reverse closure for an eve...
o1mptrcl 15671 Reverse closure for an eve...
o1add2 15672 The sum of two eventually ...
o1mul2 15673 The product of two eventua...
o1sub2 15674 The product of two eventua...
lo1add 15675 The sum of two eventually ...
lo1mul 15676 The product of an eventual...
lo1mul2 15677 The product of an eventual...
o1dif 15678 If the difference of two f...
lo1sub 15679 The difference of an event...
climadd 15680 Limit of the sum of two co...
climmul 15681 Limit of the product of tw...
climsub 15682 Limit of the difference of...
climaddc1 15683 Limit of a constant ` C ` ...
climaddc2 15684 Limit of a constant ` C ` ...
climmulc2 15685 Limit of a sequence multip...
climsubc1 15686 Limit of a constant ` C ` ...
climsubc2 15687 Limit of a constant ` C ` ...
climle 15688 Comparison of the limits o...
climsqz 15689 Convergence of a sequence ...
climsqz2 15690 Convergence of a sequence ...
rlimadd 15691 Limit of the sum of two co...
rlimaddOLD 15692 Obsolete version of ~ rlim...
rlimsub 15693 Limit of the difference of...
rlimmul 15694 Limit of the product of tw...
rlimmulOLD 15695 Obsolete version of ~ rlim...
rlimdiv 15696 Limit of the quotient of t...
rlimneg 15697 Limit of the negative of a...
rlimle 15698 Comparison of the limits o...
rlimsqzlem 15699 Lemma for ~ rlimsqz and ~ ...
rlimsqz 15700 Convergence of a sequence ...
rlimsqz2 15701 Convergence of a sequence ...
lo1le 15702 Transfer eventual upper bo...
o1le 15703 Transfer eventual boundedn...
rlimno1 15704 A function whose inverse c...
clim2ser 15705 The limit of an infinite s...
clim2ser2 15706 The limit of an infinite s...
iserex 15707 An infinite series converg...
isermulc2 15708 Multiplication of an infin...
climlec2 15709 Comparison of a constant t...
iserle 15710 Comparison of the limits o...
iserge0 15711 The limit of an infinite s...
climub 15712 The limit of a monotonic s...
climserle 15713 The partial sums of a conv...
isershft 15714 Index shift of the limit o...
isercolllem1 15715 Lemma for ~ isercoll . (C...
isercolllem2 15716 Lemma for ~ isercoll . (C...
isercolllem3 15717 Lemma for ~ isercoll . (C...
isercoll 15718 Rearrange an infinite seri...
isercoll2 15719 Generalize ~ isercoll so t...
climsup 15720 A bounded monotonic sequen...
climcau 15721 A converging sequence of c...
climbdd 15722 A converging sequence of c...
caucvgrlem 15723 Lemma for ~ caurcvgr . (C...
caurcvgr 15724 A Cauchy sequence of real ...
caucvgrlem2 15725 Lemma for ~ caucvgr . (Co...
caucvgr 15726 A Cauchy sequence of compl...
caurcvg 15727 A Cauchy sequence of real ...
caurcvg2 15728 A Cauchy sequence of real ...
caucvg 15729 A Cauchy sequence of compl...
caucvgb 15730 A function is convergent i...
serf0 15731 If an infinite series conv...
iseraltlem1 15732 Lemma for ~ iseralt . A d...
iseraltlem2 15733 Lemma for ~ iseralt . The...
iseraltlem3 15734 Lemma for ~ iseralt . Fro...
iseralt 15735 The alternating series tes...
sumex 15738 A sum is a set. (Contribu...
sumeq1 15739 Equality theorem for a sum...
nfsum1 15740 Bound-variable hypothesis ...
nfsum 15741 Bound-variable hypothesis ...
sumeq2w 15742 Equality theorem for sum, ...
sumeq2ii 15743 Equality theorem for sum, ...
sumeq2 15744 Equality theorem for sum. ...
cbvsum 15745 Change bound variable in a...
cbvsumv 15746 Change bound variable in a...
sumeq1i 15747 Equality inference for sum...
sumeq2i 15748 Equality inference for sum...
sumeq12i 15749 Equality inference for sum...
sumeq1d 15750 Equality deduction for sum...
sumeq2d 15751 Equality deduction for sum...
sumeq2dv 15752 Equality deduction for sum...
sumeq2sdv 15753 Equality deduction for sum...
sumeq2sdvOLD 15754 Obsolete version of ~ sume...
2sumeq2dv 15755 Equality deduction for dou...
sumeq12dv 15756 Equality deduction for sum...
sumeq12rdv 15757 Equality deduction for sum...
sum2id 15758 The second class argument ...
sumfc 15759 A lemma to facilitate conv...
fz1f1o 15760 A lemma for working with f...
sumrblem 15761 Lemma for ~ sumrb . (Cont...
fsumcvg 15762 The sequence of partial su...
sumrb 15763 Rebase the starting point ...
summolem3 15764 Lemma for ~ summo . (Cont...
summolem2a 15765 Lemma for ~ summo . (Cont...
summolem2 15766 Lemma for ~ summo . (Cont...
summo 15767 A sum has at most one limi...
zsum 15768 Series sum with index set ...
isum 15769 Series sum with an upper i...
fsum 15770 The value of a sum over a ...
sum0 15771 Any sum over the empty set...
sumz 15772 Any sum of zero over a sum...
fsumf1o 15773 Re-index a finite sum usin...
sumss 15774 Change the index set to a ...
fsumss 15775 Change the index set to a ...
sumss2 15776 Change the index set of a ...
fsumcvg2 15777 The sequence of partial su...
fsumsers 15778 Special case of series sum...
fsumcvg3 15779 A finite sum is convergent...
fsumser 15780 A finite sum expressed in ...
fsumcl2lem 15781 - Lemma for finite sum clo...
fsumcllem 15782 - Lemma for finite sum clo...
fsumcl 15783 Closure of a finite sum of...
fsumrecl 15784 Closure of a finite sum of...
fsumzcl 15785 Closure of a finite sum of...
fsumnn0cl 15786 Closure of a finite sum of...
fsumrpcl 15787 Closure of a finite sum of...
fsumclf 15788 Closure of a finite sum of...
fsumzcl2 15789 A finite sum with integer ...
fsumadd 15790 The sum of two finite sums...
fsumsplit 15791 Split a sum into two parts...
fsumsplitf 15792 Split a sum into two parts...
sumsnf 15793 A sum of a singleton is th...
fsumsplitsn 15794 Separate out a term in a f...
fsumsplit1 15795 Separate out a term in a f...
sumsn 15796 A sum of a singleton is th...
fsum1 15797 The finite sum of ` A ( k ...
sumpr 15798 A sum over a pair is the s...
sumtp 15799 A sum over a triple is the...
sumsns 15800 A sum of a singleton is th...
fsumm1 15801 Separate out the last term...
fzosump1 15802 Separate out the last term...
fsum1p 15803 Separate out the first ter...
fsummsnunz 15804 A finite sum all of whose ...
fsumsplitsnun 15805 Separate out a term in a f...
fsump1 15806 The addition of the next t...
isumclim 15807 An infinite sum equals the...
isumclim2 15808 A converging series conver...
isumclim3 15809 The sequence of partial fi...
sumnul 15810 The sum of a non-convergen...
isumcl 15811 The sum of a converging in...
isummulc2 15812 An infinite sum multiplied...
isummulc1 15813 An infinite sum multiplied...
isumdivc 15814 An infinite sum divided by...
isumrecl 15815 The sum of a converging in...
isumge0 15816 An infinite sum of nonnega...
isumadd 15817 Addition of infinite sums....
sumsplit 15818 Split a sum into two parts...
fsump1i 15819 Optimized version of ~ fsu...
fsum2dlem 15820 Lemma for ~ fsum2d - induc...
fsum2d 15821 Write a double sum as a su...
fsumxp 15822 Combine two sums into a si...
fsumcnv 15823 Transform a region of summ...
fsumcom2 15824 Interchange order of summa...
fsumcom 15825 Interchange order of summa...
fsum0diaglem 15826 Lemma for ~ fsum0diag . (...
fsum0diag 15827 Two ways to express "the s...
mptfzshft 15828 1-1 onto function in maps-...
fsumrev 15829 Reversal of a finite sum. ...
fsumshft 15830 Index shift of a finite su...
fsumshftm 15831 Negative index shift of a ...
fsumrev2 15832 Reversal of a finite sum. ...
fsum0diag2 15833 Two ways to express "the s...
fsummulc2 15834 A finite sum multiplied by...
fsummulc1 15835 A finite sum multiplied by...
fsumdivc 15836 A finite sum divided by a ...
fsumneg 15837 Negation of a finite sum. ...
fsumsub 15838 Split a finite sum over a ...
fsum2mul 15839 Separate the nested sum of...
fsumconst 15840 The sum of constant terms ...
fsumdifsnconst 15841 The sum of constant terms ...
modfsummodslem1 15842 Lemma 1 for ~ modfsummods ...
modfsummods 15843 Induction step for ~ modfs...
modfsummod 15844 A finite sum modulo a posi...
fsumge0 15845 If all of the terms of a f...
fsumless 15846 A shorter sum of nonnegati...
fsumge1 15847 A sum of nonnegative numbe...
fsum00 15848 A sum of nonnegative numbe...
fsumle 15849 If all of the terms of fin...
fsumlt 15850 If every term in one finit...
fsumabs 15851 Generalized triangle inequ...
telfsumo 15852 Sum of a telescoping serie...
telfsumo2 15853 Sum of a telescoping serie...
telfsum 15854 Sum of a telescoping serie...
telfsum2 15855 Sum of a telescoping serie...
fsumparts 15856 Summation by parts. (Cont...
fsumrelem 15857 Lemma for ~ fsumre , ~ fsu...
fsumre 15858 The real part of a sum. (...
fsumim 15859 The imaginary part of a su...
fsumcj 15860 The complex conjugate of a...
fsumrlim 15861 Limit of a finite sum of c...
fsumo1 15862 The finite sum of eventual...
o1fsum 15863 If ` A ( k ) ` is O(1), th...
seqabs 15864 Generalized triangle inequ...
iserabs 15865 Generalized triangle inequ...
cvgcmp 15866 A comparison test for conv...
cvgcmpub 15867 An upper bound for the lim...
cvgcmpce 15868 A comparison test for conv...
abscvgcvg 15869 An absolutely convergent s...
climfsum 15870 Limit of a finite sum of c...
fsumiun 15871 Sum over a disjoint indexe...
hashiun 15872 The cardinality of a disjo...
hash2iun 15873 The cardinality of a neste...
hash2iun1dif1 15874 The cardinality of a neste...
hashrabrex 15875 The number of elements in ...
hashuni 15876 The cardinality of a disjo...
qshash 15877 The cardinality of a set w...
ackbijnn 15878 Translate the Ackermann bi...
binomlem 15879 Lemma for ~ binom (binomia...
binom 15880 The binomial theorem: ` ( ...
binom1p 15881 Special case of the binomi...
binom11 15882 Special case of the binomi...
binom1dif 15883 A summation for the differ...
bcxmaslem1 15884 Lemma for ~ bcxmas . (Con...
bcxmas 15885 Parallel summation (Christ...
incexclem 15886 Lemma for ~ incexc . (Con...
incexc 15887 The inclusion/exclusion pr...
incexc2 15888 The inclusion/exclusion pr...
isumshft 15889 Index shift of an infinite...
isumsplit 15890 Split off the first ` N ` ...
isum1p 15891 The infinite sum of a conv...
isumnn0nn 15892 Sum from 0 to infinity in ...
isumrpcl 15893 The infinite sum of positi...
isumle 15894 Comparison of two infinite...
isumless 15895 A finite sum of nonnegativ...
isumsup2 15896 An infinite sum of nonnega...
isumsup 15897 An infinite sum of nonnega...
isumltss 15898 A partial sum of a series ...
climcndslem1 15899 Lemma for ~ climcnds : bou...
climcndslem2 15900 Lemma for ~ climcnds : bou...
climcnds 15901 The Cauchy condensation te...
divrcnv 15902 The sequence of reciprocal...
divcnv 15903 The sequence of reciprocal...
flo1 15904 The floor function satisfi...
divcnvshft 15905 Limit of a ratio function....
supcvg 15906 Extract a sequence ` f ` i...
infcvgaux1i 15907 Auxiliary theorem for appl...
infcvgaux2i 15908 Auxiliary theorem for appl...
harmonic 15909 The harmonic series ` H ` ...
arisum 15910 Arithmetic series sum of t...
arisum2 15911 Arithmetic series sum of t...
trireciplem 15912 Lemma for ~ trirecip . Sh...
trirecip 15913 The sum of the reciprocals...
expcnv 15914 A sequence of powers of a ...
explecnv 15915 A sequence of terms conver...
geoserg 15916 The value of the finite ge...
geoser 15917 The value of the finite ge...
pwdif 15918 The difference of two numb...
pwm1geoser 15919 The n-th power of a number...
geolim 15920 The partial sums in the in...
geolim2 15921 The partial sums in the ge...
georeclim 15922 The limit of a geometric s...
geo2sum 15923 The value of the finite ge...
geo2sum2 15924 The value of the finite ge...
geo2lim 15925 The value of the infinite ...
geomulcvg 15926 The geometric series conve...
geoisum 15927 The infinite sum of ` 1 + ...
geoisumr 15928 The infinite sum of recipr...
geoisum1 15929 The infinite sum of ` A ^ ...
geoisum1c 15930 The infinite sum of ` A x....
0.999... 15931 The recurring decimal 0.99...
geoihalfsum 15932 Prove that the infinite ge...
cvgrat 15933 Ratio test for convergence...
mertenslem1 15934 Lemma for ~ mertens . (Co...
mertenslem2 15935 Lemma for ~ mertens . (Co...
mertens 15936 Mertens' theorem. If ` A ...
prodf 15937 An infinite product of com...
clim2prod 15938 The limit of an infinite p...
clim2div 15939 The limit of an infinite p...
prodfmul 15940 The product of two infinit...
prodf1 15941 The value of the partial p...
prodf1f 15942 A one-valued infinite prod...
prodfclim1 15943 The constant one product c...
prodfn0 15944 No term of a nonzero infin...
prodfrec 15945 The reciprocal of an infin...
prodfdiv 15946 The quotient of two infini...
ntrivcvg 15947 A non-trivially converging...
ntrivcvgn0 15948 A product that converges t...
ntrivcvgfvn0 15949 Any value of a product seq...
ntrivcvgtail 15950 A tail of a non-trivially ...
ntrivcvgmullem 15951 Lemma for ~ ntrivcvgmul . ...
ntrivcvgmul 15952 The product of two non-tri...
prodex 15955 A product is a set. (Cont...
prodeq1f 15956 Equality theorem for a pro...
prodeq1 15957 Equality theorem for a pro...
nfcprod1 15958 Bound-variable hypothesis ...
nfcprod 15959 Bound-variable hypothesis ...
prodeq2w 15960 Equality theorem for produ...
prodeq2ii 15961 Equality theorem for produ...
prodeq2 15962 Equality theorem for produ...
cbvprod 15963 Change bound variable in a...
cbvprodv 15964 Change bound variable in a...
cbvprodi 15965 Change bound variable in a...
prodeq1i 15966 Equality inference for pro...
prodeq1iOLD 15967 Obsolete version of ~ prod...
prodeq2i 15968 Equality inference for pro...
prodeq12i 15969 Equality inference for pro...
prodeq1d 15970 Equality deduction for pro...
prodeq2d 15971 Equality deduction for pro...
prodeq2dv 15972 Equality deduction for pro...
prodeq2sdv 15973 Equality deduction for pro...
prodeq2sdvOLD 15974 Obsolete version of ~ prod...
2cprodeq2dv 15975 Equality deduction for dou...
prodeq12dv 15976 Equality deduction for pro...
prodeq12rdv 15977 Equality deduction for pro...
prod2id 15978 The second class argument ...
prodrblem 15979 Lemma for ~ prodrb . (Con...
fprodcvg 15980 The sequence of partial pr...
prodrblem2 15981 Lemma for ~ prodrb . (Con...
prodrb 15982 Rebase the starting point ...
prodmolem3 15983 Lemma for ~ prodmo . (Con...
prodmolem2a 15984 Lemma for ~ prodmo . (Con...
prodmolem2 15985 Lemma for ~ prodmo . (Con...
prodmo 15986 A product has at most one ...
zprod 15987 Series product with index ...
iprod 15988 Series product with an upp...
zprodn0 15989 Nonzero series product wit...
iprodn0 15990 Nonzero series product wit...
fprod 15991 The value of a product ove...
fprodntriv 15992 A non-triviality lemma for...
prod0 15993 A product over the empty s...
prod1 15994 Any product of one over a ...
prodfc 15995 A lemma to facilitate conv...
fprodf1o 15996 Re-index a finite product ...
prodss 15997 Change the index set to a ...
fprodss 15998 Change the index set to a ...
fprodser 15999 A finite product expressed...
fprodcl2lem 16000 Finite product closure lem...
fprodcllem 16001 Finite product closure lem...
fprodcl 16002 Closure of a finite produc...
fprodrecl 16003 Closure of a finite produc...
fprodzcl 16004 Closure of a finite produc...
fprodnncl 16005 Closure of a finite produc...
fprodrpcl 16006 Closure of a finite produc...
fprodnn0cl 16007 Closure of a finite produc...
fprodcllemf 16008 Finite product closure lem...
fprodreclf 16009 Closure of a finite produc...
fprodmul 16010 The product of two finite ...
fproddiv 16011 The quotient of two finite...
prodsn 16012 A product of a singleton i...
fprod1 16013 A finite product of only o...
prodsnf 16014 A product of a singleton i...
climprod1 16015 The limit of a product ove...
fprodsplit 16016 Split a finite product int...
fprodm1 16017 Separate out the last term...
fprod1p 16018 Separate out the first ter...
fprodp1 16019 Multiply in the last term ...
fprodm1s 16020 Separate out the last term...
fprodp1s 16021 Multiply in the last term ...
prodsns 16022 A product of the singleton...
fprodfac 16023 Factorial using product no...
fprodabs 16024 The absolute value of a fi...
fprodeq0 16025 Any finite product contain...
fprodshft 16026 Shift the index of a finit...
fprodrev 16027 Reversal of a finite produ...
fprodconst 16028 The product of constant te...
fprodn0 16029 A finite product of nonzer...
fprod2dlem 16030 Lemma for ~ fprod2d - indu...
fprod2d 16031 Write a double product as ...
fprodxp 16032 Combine two products into ...
fprodcnv 16033 Transform a product region...
fprodcom2 16034 Interchange order of multi...
fprodcom 16035 Interchange product order....
fprod0diag 16036 Two ways to express "the p...
fproddivf 16037 The quotient of two finite...
fprodsplitf 16038 Split a finite product int...
fprodsplitsn 16039 Separate out a term in a f...
fprodsplit1f 16040 Separate out a term in a f...
fprodn0f 16041 A finite product of nonzer...
fprodclf 16042 Closure of a finite produc...
fprodge0 16043 If all the terms of a fini...
fprodeq0g 16044 Any finite product contain...
fprodge1 16045 If all of the terms of a f...
fprodle 16046 If all the terms of two fi...
fprodmodd 16047 If all factors of two fini...
iprodclim 16048 An infinite product equals...
iprodclim2 16049 A converging product conve...
iprodclim3 16050 The sequence of partial fi...
iprodcl 16051 The product of a non-trivi...
iprodrecl 16052 The product of a non-trivi...
iprodmul 16053 Multiplication of infinite...
risefacval 16058 The value of the rising fa...
fallfacval 16059 The value of the falling f...
risefacval2 16060 One-based value of rising ...
fallfacval2 16061 One-based value of falling...
fallfacval3 16062 A product representation o...
risefaccllem 16063 Lemma for rising factorial...
fallfaccllem 16064 Lemma for falling factoria...
risefaccl 16065 Closure law for rising fac...
fallfaccl 16066 Closure law for falling fa...
rerisefaccl 16067 Closure law for rising fac...
refallfaccl 16068 Closure law for falling fa...
nnrisefaccl 16069 Closure law for rising fac...
zrisefaccl 16070 Closure law for rising fac...
zfallfaccl 16071 Closure law for falling fa...
nn0risefaccl 16072 Closure law for rising fac...
rprisefaccl 16073 Closure law for rising fac...
risefallfac 16074 A relationship between ris...
fallrisefac 16075 A relationship between fal...
risefall0lem 16076 Lemma for ~ risefac0 and ~...
risefac0 16077 The value of the rising fa...
fallfac0 16078 The value of the falling f...
risefacp1 16079 The value of the rising fa...
fallfacp1 16080 The value of the falling f...
risefacp1d 16081 The value of the rising fa...
fallfacp1d 16082 The value of the falling f...
risefac1 16083 The value of rising factor...
fallfac1 16084 The value of falling facto...
risefacfac 16085 Relate rising factorial to...
fallfacfwd 16086 The forward difference of ...
0fallfac 16087 The value of the zero fall...
0risefac 16088 The value of the zero risi...
binomfallfaclem1 16089 Lemma for ~ binomfallfac ....
binomfallfaclem2 16090 Lemma for ~ binomfallfac ....
binomfallfac 16091 A version of the binomial ...
binomrisefac 16092 A version of the binomial ...
fallfacval4 16093 Represent the falling fact...
bcfallfac 16094 Binomial coefficient in te...
fallfacfac 16095 Relate falling factorial t...
bpolylem 16098 Lemma for ~ bpolyval . (C...
bpolyval 16099 The value of the Bernoulli...
bpoly0 16100 The value of the Bernoulli...
bpoly1 16101 The value of the Bernoulli...
bpolycl 16102 Closure law for Bernoulli ...
bpolysum 16103 A sum for Bernoulli polyno...
bpolydiflem 16104 Lemma for ~ bpolydif . (C...
bpolydif 16105 Calculate the difference b...
fsumkthpow 16106 A closed-form expression f...
bpoly2 16107 The Bernoulli polynomials ...
bpoly3 16108 The Bernoulli polynomials ...
bpoly4 16109 The Bernoulli polynomials ...
fsumcube 16110 Express the sum of cubes i...
eftcl 16123 Closure of a term in the s...
reeftcl 16124 The terms of the series ex...
eftabs 16125 The absolute value of a te...
eftval 16126 The value of a term in the...
efcllem 16127 Lemma for ~ efcl . The se...
ef0lem 16128 The series defining the ex...
efval 16129 Value of the exponential f...
esum 16130 Value of Euler's constant ...
eff 16131 Domain and codomain of the...
efcl 16132 Closure law for the expone...
efcld 16133 Closure law for the expone...
efval2 16134 Value of the exponential f...
efcvg 16135 The series that defines th...
efcvgfsum 16136 Exponential function conve...
reefcl 16137 The exponential function i...
reefcld 16138 The exponential function i...
ere 16139 Euler's constant ` _e ` = ...
ege2le3 16140 Lemma for ~ egt2lt3 . (Co...
ef0 16141 Value of the exponential f...
efcj 16142 The exponential of a compl...
efaddlem 16143 Lemma for ~ efadd (exponen...
efadd 16144 Sum of exponents law for e...
fprodefsum 16145 Move the exponential funct...
efcan 16146 Cancellation law for expon...
efne0 16147 The exponential of a compl...
efneg 16148 The exponential of the opp...
eff2 16149 The exponential function m...
efsub 16150 Difference of exponents la...
efexp 16151 The exponential of an inte...
efzval 16152 Value of the exponential f...
efgt0 16153 The exponential of a real ...
rpefcl 16154 The exponential of a real ...
rpefcld 16155 The exponential of a real ...
eftlcvg 16156 The tail series of the exp...
eftlcl 16157 Closure of the sum of an i...
reeftlcl 16158 Closure of the sum of an i...
eftlub 16159 An upper bound on the abso...
efsep 16160 Separate out the next term...
effsumlt 16161 The partial sums of the se...
eft0val 16162 The value of the first ter...
ef4p 16163 Separate out the first fou...
efgt1p2 16164 The exponential of a posit...
efgt1p 16165 The exponential of a posit...
efgt1 16166 The exponential of a posit...
eflt 16167 The exponential function o...
efle 16168 The exponential function o...
reef11 16169 The exponential function o...
reeff1 16170 The exponential function m...
eflegeo 16171 The exponential function o...
sinval 16172 Value of the sine function...
cosval 16173 Value of the cosine functi...
sinf 16174 Domain and codomain of the...
cosf 16175 Domain and codomain of the...
sincl 16176 Closure of the sine functi...
coscl 16177 Closure of the cosine func...
tanval 16178 Value of the tangent funct...
tancl 16179 The closure of the tangent...
sincld 16180 Closure of the sine functi...
coscld 16181 Closure of the cosine func...
tancld 16182 Closure of the tangent fun...
tanval2 16183 Express the tangent functi...
tanval3 16184 Express the tangent functi...
resinval 16185 The sine of a real number ...
recosval 16186 The cosine of a real numbe...
efi4p 16187 Separate out the first fou...
resin4p 16188 Separate out the first fou...
recos4p 16189 Separate out the first fou...
resincl 16190 The sine of a real number ...
recoscl 16191 The cosine of a real numbe...
retancl 16192 The closure of the tangent...
resincld 16193 Closure of the sine functi...
recoscld 16194 Closure of the cosine func...
retancld 16195 Closure of the tangent fun...
sinneg 16196 The sine of a negative is ...
cosneg 16197 The cosines of a number an...
tanneg 16198 The tangent of a negative ...
sin0 16199 Value of the sine function...
cos0 16200 Value of the cosine functi...
tan0 16201 The value of the tangent f...
efival 16202 The exponential function i...
efmival 16203 The exponential function i...
sinhval 16204 Value of the hyperbolic si...
coshval 16205 Value of the hyperbolic co...
resinhcl 16206 The hyperbolic sine of a r...
rpcoshcl 16207 The hyperbolic cosine of a...
recoshcl 16208 The hyperbolic cosine of a...
retanhcl 16209 The hyperbolic tangent of ...
tanhlt1 16210 The hyperbolic tangent of ...
tanhbnd 16211 The hyperbolic tangent of ...
efeul 16212 Eulerian representation of...
efieq 16213 The exponentials of two im...
sinadd 16214 Addition formula for sine....
cosadd 16215 Addition formula for cosin...
tanaddlem 16216 A useful intermediate step...
tanadd 16217 Addition formula for tange...
sinsub 16218 Sine of difference. (Cont...
cossub 16219 Cosine of difference. (Co...
addsin 16220 Sum of sines. (Contribute...
subsin 16221 Difference of sines. (Con...
sinmul 16222 Product of sines can be re...
cosmul 16223 Product of cosines can be ...
addcos 16224 Sum of cosines. (Contribu...
subcos 16225 Difference of cosines. (C...
sincossq 16226 Sine squared plus cosine s...
sin2t 16227 Double-angle formula for s...
cos2t 16228 Double-angle formula for c...
cos2tsin 16229 Double-angle formula for c...
sinbnd 16230 The sine of a real number ...
cosbnd 16231 The cosine of a real numbe...
sinbnd2 16232 The sine of a real number ...
cosbnd2 16233 The cosine of a real numbe...
ef01bndlem 16234 Lemma for ~ sin01bnd and ~...
sin01bnd 16235 Bounds on the sine of a po...
cos01bnd 16236 Bounds on the cosine of a ...
cos1bnd 16237 Bounds on the cosine of 1....
cos2bnd 16238 Bounds on the cosine of 2....
sinltx 16239 The sine of a positive rea...
sin01gt0 16240 The sine of a positive rea...
cos01gt0 16241 The cosine of a positive r...
sin02gt0 16242 The sine of a positive rea...
sincos1sgn 16243 The signs of the sine and ...
sincos2sgn 16244 The signs of the sine and ...
sin4lt0 16245 The sine of 4 is negative....
absefi 16246 The absolute value of the ...
absef 16247 The absolute value of the ...
absefib 16248 A complex number is real i...
efieq1re 16249 A number whose imaginary e...
demoivre 16250 De Moivre's Formula. Proo...
demoivreALT 16251 Alternate proof of ~ demoi...
eirrlem 16254 Lemma for ~ eirr . (Contr...
eirr 16255 ` _e ` is irrational. (Co...
egt2lt3 16256 Euler's constant ` _e ` = ...
epos 16257 Euler's constant ` _e ` is...
epr 16258 Euler's constant ` _e ` is...
ene0 16259 ` _e ` is not 0. (Contrib...
ene1 16260 ` _e ` is not 1. (Contrib...
xpnnen 16261 The Cartesian product of t...
znnen 16262 The set of integers and th...
qnnen 16263 The rational numbers are c...
rpnnen2lem1 16264 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem2 16265 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem3 16266 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem4 16267 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem5 16268 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem6 16269 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem7 16270 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem8 16271 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem9 16272 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem10 16273 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem11 16274 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem12 16275 Lemma for ~ rpnnen2 . (Co...
rpnnen2 16276 The other half of ~ rpnnen...
rpnnen 16277 The cardinality of the con...
rexpen 16278 The real numbers are equin...
cpnnen 16279 The complex numbers are eq...
rucALT 16280 Alternate proof of ~ ruc ....
ruclem1 16281 Lemma for ~ ruc (the reals...
ruclem2 16282 Lemma for ~ ruc . Orderin...
ruclem3 16283 Lemma for ~ ruc . The con...
ruclem4 16284 Lemma for ~ ruc . Initial...
ruclem6 16285 Lemma for ~ ruc . Domain ...
ruclem7 16286 Lemma for ~ ruc . Success...
ruclem8 16287 Lemma for ~ ruc . The int...
ruclem9 16288 Lemma for ~ ruc . The fir...
ruclem10 16289 Lemma for ~ ruc . Every f...
ruclem11 16290 Lemma for ~ ruc . Closure...
ruclem12 16291 Lemma for ~ ruc . The sup...
ruclem13 16292 Lemma for ~ ruc . There i...
ruc 16293 The set of positive intege...
resdomq 16294 The set of rationals is st...
aleph1re 16295 There are at least aleph-o...
aleph1irr 16296 There are at least aleph-o...
cnso 16297 The complex numbers can be...
sqrt2irrlem 16298 Lemma for ~ sqrt2irr . Th...
sqrt2irr 16299 The square root of 2 is ir...
sqrt2re 16300 The square root of 2 exist...
sqrt2irr0 16301 The square root of 2 is an...
nthruc 16302 The sequence ` NN ` , ` ZZ...
nthruz 16303 The sequence ` NN ` , ` NN...
divides 16306 Define the divides relatio...
dvdsval2 16307 One nonzero integer divide...
dvdsval3 16308 One nonzero integer divide...
dvdszrcl 16309 Reverse closure for the di...
dvdsmod0 16310 If a positive integer divi...
p1modz1 16311 If a number greater than 1...
dvdsmodexp 16312 If a positive integer divi...
nndivdvds 16313 Strong form of ~ dvdsval2 ...
nndivides 16314 Definition of the divides ...
moddvds 16315 Two ways to say ` A == B `...
modm1div 16316 An integer greater than on...
dvds0lem 16317 A lemma to assist theorems...
dvds1lem 16318 A lemma to assist theorems...
dvds2lem 16319 A lemma to assist theorems...
iddvds 16320 An integer divides itself....
1dvds 16321 1 divides any integer. Th...
dvds0 16322 Any integer divides 0. Th...
negdvdsb 16323 An integer divides another...
dvdsnegb 16324 An integer divides another...
absdvdsb 16325 An integer divides another...
dvdsabsb 16326 An integer divides another...
0dvds 16327 Only 0 is divisible by 0. ...
dvdsmul1 16328 An integer divides a multi...
dvdsmul2 16329 An integer divides a multi...
iddvdsexp 16330 An integer divides a posit...
muldvds1 16331 If a product divides an in...
muldvds2 16332 If a product divides an in...
dvdscmul 16333 Multiplication by a consta...
dvdsmulc 16334 Multiplication by a consta...
dvdscmulr 16335 Cancellation law for the d...
dvdsmulcr 16336 Cancellation law for the d...
summodnegmod 16337 The sum of two integers mo...
modmulconst 16338 Constant multiplication in...
dvds2ln 16339 If an integer divides each...
dvds2add 16340 If an integer divides each...
dvds2sub 16341 If an integer divides each...
dvds2addd 16342 Deduction form of ~ dvds2a...
dvds2subd 16343 Deduction form of ~ dvds2s...
dvdstr 16344 The divides relation is tr...
dvdstrd 16345 The divides relation is tr...
dvdsmultr1 16346 If an integer divides anot...
dvdsmultr1d 16347 Deduction form of ~ dvdsmu...
dvdsmultr2 16348 If an integer divides anot...
dvdsmultr2d 16349 Deduction form of ~ dvdsmu...
ordvdsmul 16350 If an integer divides eith...
dvdssub2 16351 If an integer divides a di...
dvdsadd 16352 An integer divides another...
dvdsaddr 16353 An integer divides another...
dvdssub 16354 An integer divides another...
dvdssubr 16355 An integer divides another...
dvdsadd2b 16356 Adding a multiple of the b...
dvdsaddre2b 16357 Adding a multiple of the b...
fsumdvds 16358 If every term in a sum is ...
dvdslelem 16359 Lemma for ~ dvdsle . (Con...
dvdsle 16360 The divisors of a positive...
dvdsleabs 16361 The divisors of a nonzero ...
dvdsleabs2 16362 Transfer divisibility to a...
dvdsabseq 16363 If two integers divide eac...
dvdseq 16364 If two nonnegative integer...
divconjdvds 16365 If a nonzero integer ` M `...
dvdsdivcl 16366 The complement of a diviso...
dvdsflip 16367 An involution of the divis...
dvdsssfz1 16368 The set of divisors of a n...
dvds1 16369 The only nonnegative integ...
alzdvds 16370 Only 0 is divisible by all...
dvdsext 16371 Poset extensionality for d...
fzm1ndvds 16372 No number between ` 1 ` an...
fzo0dvdseq 16373 Zero is the only one of th...
fzocongeq 16374 Two different elements of ...
addmodlteqALT 16375 Two nonnegative integers l...
dvdsfac 16376 A positive integer divides...
dvdsexp2im 16377 If an integer divides anot...
dvdsexp 16378 A power divides a power wi...
dvdsmod 16379 Any number ` K ` whose mod...
mulmoddvds 16380 If an integer is divisible...
3dvds 16381 A rule for divisibility by...
3dvdsdec 16382 A decimal number is divisi...
3dvds2dec 16383 A decimal number is divisi...
fprodfvdvdsd 16384 A finite product of intege...
fproddvdsd 16385 A finite product of intege...
evenelz 16386 An even number is an integ...
zeo3 16387 An integer is even or odd....
zeo4 16388 An integer is even or odd ...
zeneo 16389 No even integer equals an ...
odd2np1lem 16390 Lemma for ~ odd2np1 . (Co...
odd2np1 16391 An integer is odd iff it i...
even2n 16392 An integer is even iff it ...
oddm1even 16393 An integer is odd iff its ...
oddp1even 16394 An integer is odd iff its ...
oexpneg 16395 The exponential of the neg...
mod2eq0even 16396 An integer is 0 modulo 2 i...
mod2eq1n2dvds 16397 An integer is 1 modulo 2 i...
oddnn02np1 16398 A nonnegative integer is o...
oddge22np1 16399 An integer greater than on...
evennn02n 16400 A nonnegative integer is e...
evennn2n 16401 A positive integer is even...
2tp1odd 16402 A number which is twice an...
mulsucdiv2z 16403 An integer multiplied with...
sqoddm1div8z 16404 A squared odd number minus...
2teven 16405 A number which is twice an...
zeo5 16406 An integer is either even ...
evend2 16407 An integer is even iff its...
oddp1d2 16408 An integer is odd iff its ...
zob 16409 Alternate characterization...
oddm1d2 16410 An integer is odd iff its ...
ltoddhalfle 16411 An integer is less than ha...
halfleoddlt 16412 An integer is greater than...
opoe 16413 The sum of two odds is eve...
omoe 16414 The difference of two odds...
opeo 16415 The sum of an odd and an e...
omeo 16416 The difference of an odd a...
z0even 16417 2 divides 0. That means 0...
n2dvds1 16418 2 does not divide 1. That...
n2dvdsm1 16419 2 does not divide -1. Tha...
z2even 16420 2 divides 2. That means 2...
n2dvds3 16421 2 does not divide 3. That...
z4even 16422 2 divides 4. That means 4...
4dvdseven 16423 An integer which is divisi...
m1expe 16424 Exponentiation of -1 by an...
m1expo 16425 Exponentiation of -1 by an...
m1exp1 16426 Exponentiation of negative...
nn0enne 16427 A positive integer is an e...
nn0ehalf 16428 The half of an even nonneg...
nnehalf 16429 The half of an even positi...
nn0onn 16430 An odd nonnegative integer...
nn0o1gt2 16431 An odd nonnegative integer...
nno 16432 An alternate characterizat...
nn0o 16433 An alternate characterizat...
nn0ob 16434 Alternate characterization...
nn0oddm1d2 16435 A positive integer is odd ...
nnoddm1d2 16436 A positive integer is odd ...
sumeven 16437 If every term in a sum is ...
sumodd 16438 If every term in a sum is ...
evensumodd 16439 If every term in a sum wit...
oddsumodd 16440 If every term in a sum wit...
pwp1fsum 16441 The n-th power of a number...
oddpwp1fsum 16442 An odd power of a number i...
divalglem0 16443 Lemma for ~ divalg . (Con...
divalglem1 16444 Lemma for ~ divalg . (Con...
divalglem2 16445 Lemma for ~ divalg . (Con...
divalglem4 16446 Lemma for ~ divalg . (Con...
divalglem5 16447 Lemma for ~ divalg . (Con...
divalglem6 16448 Lemma for ~ divalg . (Con...
divalglem7 16449 Lemma for ~ divalg . (Con...
divalglem8 16450 Lemma for ~ divalg . (Con...
divalglem9 16451 Lemma for ~ divalg . (Con...
divalglem10 16452 Lemma for ~ divalg . (Con...
divalg 16453 The division algorithm (th...
divalgb 16454 Express the division algor...
divalg2 16455 The division algorithm (th...
divalgmod 16456 The result of the ` mod ` ...
divalgmodcl 16457 The result of the ` mod ` ...
modremain 16458 The result of the modulo o...
ndvdssub 16459 Corollary of the division ...
ndvdsadd 16460 Corollary of the division ...
ndvdsp1 16461 Special case of ~ ndvdsadd...
ndvdsi 16462 A quick test for non-divis...
flodddiv4 16463 The floor of an odd intege...
fldivndvdslt 16464 The floor of an integer di...
flodddiv4lt 16465 The floor of an odd number...
flodddiv4t2lthalf 16466 The floor of an odd number...
bitsfval 16471 Expand the definition of t...
bitsval 16472 Expand the definition of t...
bitsval2 16473 Expand the definition of t...
bitsss 16474 The set of bits of an inte...
bitsf 16475 The ` bits ` function is a...
bits0 16476 Value of the zeroth bit. ...
bits0e 16477 The zeroth bit of an even ...
bits0o 16478 The zeroth bit of an odd n...
bitsp1 16479 The ` M + 1 ` -th bit of `...
bitsp1e 16480 The ` M + 1 ` -th bit of `...
bitsp1o 16481 The ` M + 1 ` -th bit of `...
bitsfzolem 16482 Lemma for ~ bitsfzo . (Co...
bitsfzo 16483 The bits of a number are a...
bitsmod 16484 Truncating the bit sequenc...
bitsfi 16485 Every number is associated...
bitscmp 16486 The bit complement of ` N ...
0bits 16487 The bits of zero. (Contri...
m1bits 16488 The bits of negative one. ...
bitsinv1lem 16489 Lemma for ~ bitsinv1 . (C...
bitsinv1 16490 There is an explicit inver...
bitsinv2 16491 There is an explicit inver...
bitsf1ocnv 16492 The ` bits ` function rest...
bitsf1o 16493 The ` bits ` function rest...
bitsf1 16494 The ` bits ` function is a...
2ebits 16495 The bits of a power of two...
bitsinv 16496 The inverse of the ` bits ...
bitsinvp1 16497 Recursive definition of th...
sadadd2lem2 16498 The core of the proof of ~...
sadfval 16500 Define the addition of two...
sadcf 16501 The carry sequence is a se...
sadc0 16502 The initial element of the...
sadcp1 16503 The carry sequence (which ...
sadval 16504 The full adder sequence is...
sadcaddlem 16505 Lemma for ~ sadcadd . (Co...
sadcadd 16506 Non-recursive definition o...
sadadd2lem 16507 Lemma for ~ sadadd2 . (Co...
sadadd2 16508 Sum of initial segments of...
sadadd3 16509 Sum of initial segments of...
sadcl 16510 The sum of two sequences i...
sadcom 16511 The adder sequence functio...
saddisjlem 16512 Lemma for ~ sadadd . (Con...
saddisj 16513 The sum of disjoint sequen...
sadaddlem 16514 Lemma for ~ sadadd . (Con...
sadadd 16515 For sequences that corresp...
sadid1 16516 The adder sequence functio...
sadid2 16517 The adder sequence functio...
sadasslem 16518 Lemma for ~ sadass . (Con...
sadass 16519 Sequence addition is assoc...
sadeq 16520 Any element of a sequence ...
bitsres 16521 Restrict the bits of a num...
bitsuz 16522 The bits of a number are a...
bitsshft 16523 Shifting a bit sequence to...
smufval 16525 The multiplication of two ...
smupf 16526 The sequence of partial su...
smup0 16527 The initial element of the...
smupp1 16528 The initial element of the...
smuval 16529 Define the addition of two...
smuval2 16530 The partial sum sequence s...
smupvallem 16531 If ` A ` only has elements...
smucl 16532 The product of two sequenc...
smu01lem 16533 Lemma for ~ smu01 and ~ sm...
smu01 16534 Multiplication of a sequen...
smu02 16535 Multiplication of a sequen...
smupval 16536 Rewrite the elements of th...
smup1 16537 Rewrite ~ smupp1 using onl...
smueqlem 16538 Any element of a sequence ...
smueq 16539 Any element of a sequence ...
smumullem 16540 Lemma for ~ smumul . (Con...
smumul 16541 For sequences that corresp...
gcdval 16544 The value of the ` gcd ` o...
gcd0val 16545 The value, by convention, ...
gcdn0val 16546 The value of the ` gcd ` o...
gcdcllem1 16547 Lemma for ~ gcdn0cl , ~ gc...
gcdcllem2 16548 Lemma for ~ gcdn0cl , ~ gc...
gcdcllem3 16549 Lemma for ~ gcdn0cl , ~ gc...
gcdn0cl 16550 Closure of the ` gcd ` ope...
gcddvds 16551 The gcd of two integers di...
dvdslegcd 16552 An integer which divides b...
nndvdslegcd 16553 A positive integer which d...
gcdcl 16554 Closure of the ` gcd ` ope...
gcdnncl 16555 Closure of the ` gcd ` ope...
gcdcld 16556 Closure of the ` gcd ` ope...
gcd2n0cl 16557 Closure of the ` gcd ` ope...
zeqzmulgcd 16558 An integer is the product ...
divgcdz 16559 An integer divided by the ...
gcdf 16560 Domain and codomain of the...
gcdcom 16561 The ` gcd ` operator is co...
gcdcomd 16562 The ` gcd ` operator is co...
divgcdnn 16563 A positive integer divided...
divgcdnnr 16564 A positive integer divided...
gcdeq0 16565 The gcd of two integers is...
gcdn0gt0 16566 The gcd of two integers is...
gcd0id 16567 The gcd of 0 and an intege...
gcdid0 16568 The gcd of an integer and ...
nn0gcdid0 16569 The gcd of a nonnegative i...
gcdneg 16570 Negating one operand of th...
neggcd 16571 Negating one operand of th...
gcdaddmlem 16572 Lemma for ~ gcdaddm . (Co...
gcdaddm 16573 Adding a multiple of one o...
gcdadd 16574 The GCD of two numbers is ...
gcdid 16575 The gcd of a number and it...
gcd1 16576 The gcd of a number with 1...
gcdabs1 16577 ` gcd ` of the absolute va...
gcdabs2 16578 ` gcd ` of the absolute va...
gcdabs 16579 The gcd of two integers is...
gcdabsOLD 16580 Obsolete version of ~ gcda...
modgcd 16581 The gcd remains unchanged ...
1gcd 16582 The GCD of one and an inte...
gcdmultipled 16583 The greatest common diviso...
gcdmultiplez 16584 The GCD of a multiple of a...
gcdmultiple 16585 The GCD of a multiple of a...
dvdsgcdidd 16586 The greatest common diviso...
6gcd4e2 16587 The greatest common diviso...
bezoutlem1 16588 Lemma for ~ bezout . (Con...
bezoutlem2 16589 Lemma for ~ bezout . (Con...
bezoutlem3 16590 Lemma for ~ bezout . (Con...
bezoutlem4 16591 Lemma for ~ bezout . (Con...
bezout 16592 Bézout's identity: ...
dvdsgcd 16593 An integer which divides e...
dvdsgcdb 16594 Biconditional form of ~ dv...
dfgcd2 16595 Alternate definition of th...
gcdass 16596 Associative law for ` gcd ...
mulgcd 16597 Distribute multiplication ...
absmulgcd 16598 Distribute absolute value ...
mulgcdr 16599 Reverse distribution law f...
gcddiv 16600 Division law for GCD. (Con...
gcdzeq 16601 A positive integer ` A ` i...
gcdeq 16602 ` A ` is equal to its gcd ...
dvdssqim 16603 Unidirectional form of ~ d...
dvdsexpim 16604 If two numbers are divisib...
dvdsmulgcd 16605 A divisibility equivalent ...
rpmulgcd 16606 If ` K ` and ` M ` are rel...
rplpwr 16607 If ` A ` and ` B ` are rel...
rprpwr 16608 If ` A ` and ` B ` are rel...
rppwr 16609 If ` A ` and ` B ` are rel...
nn0rppwr 16610 If ` A ` and ` B ` are rel...
sqgcd 16611 Square distributes over gc...
expgcd 16612 Exponentiation distributes...
nn0expgcd 16613 Exponentiation distributes...
zexpgcd 16614 Exponentiation distributes...
dvdssqlem 16615 Lemma for ~ dvdssq . (Con...
dvdssq 16616 Two numbers are divisible ...
bezoutr 16617 Partial converse to ~ bezo...
bezoutr1 16618 Converse of ~ bezout for w...
nn0seqcvgd 16619 A strictly-decreasing nonn...
seq1st 16620 A sequence whose iteration...
algr0 16621 The value of the algorithm...
algrf 16622 An algorithm is a step fun...
algrp1 16623 The value of the algorithm...
alginv 16624 If ` I ` is an invariant o...
algcvg 16625 One way to prove that an a...
algcvgblem 16626 Lemma for ~ algcvgb . (Co...
algcvgb 16627 Two ways of expressing tha...
algcvga 16628 The countdown function ` C...
algfx 16629 If ` F ` reaches a fixed p...
eucalgval2 16630 The value of the step func...
eucalgval 16631 Euclid's Algorithm ~ eucal...
eucalgf 16632 Domain and codomain of the...
eucalginv 16633 The invariant of the step ...
eucalglt 16634 The second member of the s...
eucalgcvga 16635 Once Euclid's Algorithm ha...
eucalg 16636 Euclid's Algorithm compute...
lcmval 16641 Value of the ` lcm ` opera...
lcmcom 16642 The ` lcm ` operator is co...
lcm0val 16643 The value, by convention, ...
lcmn0val 16644 The value of the ` lcm ` o...
lcmcllem 16645 Lemma for ~ lcmn0cl and ~ ...
lcmn0cl 16646 Closure of the ` lcm ` ope...
dvdslcm 16647 The lcm of two integers is...
lcmledvds 16648 A positive integer which b...
lcmeq0 16649 The lcm of two integers is...
lcmcl 16650 Closure of the ` lcm ` ope...
gcddvdslcm 16651 The greatest common diviso...
lcmneg 16652 Negating one operand of th...
neglcm 16653 Negating one operand of th...
lcmabs 16654 The lcm of two integers is...
lcmgcdlem 16655 Lemma for ~ lcmgcd and ~ l...
lcmgcd 16656 The product of two numbers...
lcmdvds 16657 The lcm of two integers di...
lcmid 16658 The lcm of an integer and ...
lcm1 16659 The lcm of an integer and ...
lcmgcdnn 16660 The product of two positiv...
lcmgcdeq 16661 Two integers' absolute val...
lcmdvdsb 16662 Biconditional form of ~ lc...
lcmass 16663 Associative law for ` lcm ...
3lcm2e6woprm 16664 The least common multiple ...
6lcm4e12 16665 The least common multiple ...
absproddvds 16666 The absolute value of the ...
absprodnn 16667 The absolute value of the ...
fissn0dvds 16668 For each finite subset of ...
fissn0dvdsn0 16669 For each finite subset of ...
lcmfval 16670 Value of the ` _lcm ` func...
lcmf0val 16671 The value, by convention, ...
lcmfn0val 16672 The value of the ` _lcm ` ...
lcmfnnval 16673 The value of the ` _lcm ` ...
lcmfcllem 16674 Lemma for ~ lcmfn0cl and ~...
lcmfn0cl 16675 Closure of the ` _lcm ` fu...
lcmfpr 16676 The value of the ` _lcm ` ...
lcmfcl 16677 Closure of the ` _lcm ` fu...
lcmfnncl 16678 Closure of the ` _lcm ` fu...
lcmfeq0b 16679 The least common multiple ...
dvdslcmf 16680 The least common multiple ...
lcmfledvds 16681 A positive integer which i...
lcmf 16682 Characterization of the le...
lcmf0 16683 The least common multiple ...
lcmfsn 16684 The least common multiple ...
lcmftp 16685 The least common multiple ...
lcmfunsnlem1 16686 Lemma for ~ lcmfdvds and ~...
lcmfunsnlem2lem1 16687 Lemma 1 for ~ lcmfunsnlem2...
lcmfunsnlem2lem2 16688 Lemma 2 for ~ lcmfunsnlem2...
lcmfunsnlem2 16689 Lemma for ~ lcmfunsn and ~...
lcmfunsnlem 16690 Lemma for ~ lcmfdvds and ~...
lcmfdvds 16691 The least common multiple ...
lcmfdvdsb 16692 Biconditional form of ~ lc...
lcmfunsn 16693 The ` _lcm ` function for ...
lcmfun 16694 The ` _lcm ` function for ...
lcmfass 16695 Associative law for the ` ...
lcmf2a3a4e12 16696 The least common multiple ...
lcmflefac 16697 The least common multiple ...
coprmgcdb 16698 Two positive integers are ...
ncoprmgcdne1b 16699 Two positive integers are ...
ncoprmgcdgt1b 16700 Two positive integers are ...
coprmdvds1 16701 If two positive integers a...
coprmdvds 16702 Euclid's Lemma (see ProofW...
coprmdvds2 16703 If an integer is divisible...
mulgcddvds 16704 One half of ~ rpmulgcd2 , ...
rpmulgcd2 16705 If ` M ` is relatively pri...
qredeq 16706 Two equal reduced fraction...
qredeu 16707 Every rational number has ...
rpmul 16708 If ` K ` is relatively pri...
rpdvds 16709 If ` K ` is relatively pri...
coprmprod 16710 The product of the element...
coprmproddvdslem 16711 Lemma for ~ coprmproddvds ...
coprmproddvds 16712 If a positive integer is d...
congr 16713 Definition of congruence b...
divgcdcoprm0 16714 Integers divided by gcd ar...
divgcdcoprmex 16715 Integers divided by gcd ar...
cncongr1 16716 One direction of the bicon...
cncongr2 16717 The other direction of the...
cncongr 16718 Cancellability of Congruen...
cncongrcoprm 16719 Corollary 1 of Cancellabil...
isprm 16722 The predicate "is a prime ...
prmnn 16723 A prime number is a positi...
prmz 16724 A prime number is an integ...
prmssnn 16725 The prime numbers are a su...
prmex 16726 The set of prime numbers e...
0nprm 16727 0 is not a prime number. ...
1nprm 16728 1 is not a prime number. ...
1idssfct 16729 The positive divisors of a...
isprm2lem 16730 Lemma for ~ isprm2 . (Con...
isprm2 16731 The predicate "is a prime ...
isprm3 16732 The predicate "is a prime ...
isprm4 16733 The predicate "is a prime ...
prmind2 16734 A variation on ~ prmind as...
prmind 16735 Perform induction over the...
dvdsprime 16736 If ` M ` divides a prime, ...
nprm 16737 A product of two integers ...
nprmi 16738 An inference for composite...
dvdsnprmd 16739 If a number is divisible b...
prm2orodd 16740 A prime number is either 2...
2prm 16741 2 is a prime number. (Con...
2mulprm 16742 A multiple of two is prime...
3prm 16743 3 is a prime number. (Con...
4nprm 16744 4 is not a prime number. ...
prmuz2 16745 A prime number is an integ...
prmgt1 16746 A prime number is an integ...
prmm2nn0 16747 Subtracting 2 from a prime...
oddprmgt2 16748 An odd prime is greater th...
oddprmge3 16749 An odd prime is greater th...
ge2nprmge4 16750 A composite integer greate...
sqnprm 16751 A square is never prime. ...
dvdsprm 16752 An integer greater than or...
exprmfct 16753 Every integer greater than...
prmdvdsfz 16754 Each integer greater than ...
nprmdvds1 16755 No prime number divides 1....
isprm5 16756 One need only check prime ...
isprm7 16757 One need only check prime ...
maxprmfct 16758 The set of prime factors o...
divgcdodd 16759 Either ` A / ( A gcd B ) `...
coprm 16760 A prime number either divi...
prmrp 16761 Unequal prime numbers are ...
euclemma 16762 Euclid's lemma. A prime n...
isprm6 16763 A number is prime iff it s...
prmdvdsexp 16764 A prime divides a positive...
prmdvdsexpb 16765 A prime divides a positive...
prmdvdsexpr 16766 If a prime divides a nonne...
prmdvdssq 16767 Condition for a prime divi...
prmexpb 16768 Two positive prime powers ...
prmfac1 16769 The factorial of a number ...
dvdszzq 16770 Divisibility for an intege...
rpexp 16771 If two numbers ` A ` and `...
rpexp1i 16772 Relative primality passes ...
rpexp12i 16773 Relative primality passes ...
prmndvdsfaclt 16774 A prime number does not di...
prmdvdsbc 16775 Condition for a prime numb...
prmdvdsncoprmbd 16776 Two positive integers are ...
ncoprmlnprm 16777 If two positive integers a...
cncongrprm 16778 Corollary 2 of Cancellabil...
isevengcd2 16779 The predicate "is an even ...
isoddgcd1 16780 The predicate "is an odd n...
3lcm2e6 16781 The least common multiple ...
qnumval 16786 Value of the canonical num...
qdenval 16787 Value of the canonical den...
qnumdencl 16788 Lemma for ~ qnumcl and ~ q...
qnumcl 16789 The canonical numerator of...
qdencl 16790 The canonical denominator ...
fnum 16791 Canonical numerator define...
fden 16792 Canonical denominator defi...
qnumdenbi 16793 Two numbers are the canoni...
qnumdencoprm 16794 The canonical representati...
qeqnumdivden 16795 Recover a rational number ...
qmuldeneqnum 16796 Multiplying a rational by ...
divnumden 16797 Calculate the reduced form...
divdenle 16798 Reducing a quotient never ...
qnumgt0 16799 A rational is positive iff...
qgt0numnn 16800 A rational is positive iff...
nn0gcdsq 16801 Squaring commutes with GCD...
zgcdsq 16802 ~ nn0gcdsq extended to int...
numdensq 16803 Squaring a rational square...
numsq 16804 Square commutes with canon...
densq 16805 Square commutes with canon...
qden1elz 16806 A rational is an integer i...
zsqrtelqelz 16807 If an integer has a ration...
nonsq 16808 Any integer strictly betwe...
numdenexp 16809 Elevating a rational numbe...
numexp 16810 Elevating to a nonnegative...
denexp 16811 Elevating to a nonnegative...
phival 16816 Value of the Euler ` phi `...
phicl2 16817 Bounds and closure for the...
phicl 16818 Closure for the value of t...
phibndlem 16819 Lemma for ~ phibnd . (Con...
phibnd 16820 A slightly tighter bound o...
phicld 16821 Closure for the value of t...
phi1 16822 Value of the Euler ` phi `...
dfphi2 16823 Alternate definition of th...
hashdvds 16824 The number of numbers in a...
phiprmpw 16825 Value of the Euler ` phi `...
phiprm 16826 Value of the Euler ` phi `...
crth 16827 The Chinese Remainder Theo...
phimullem 16828 Lemma for ~ phimul . (Con...
phimul 16829 The Euler ` phi ` function...
eulerthlem1 16830 Lemma for ~ eulerth . (Co...
eulerthlem2 16831 Lemma for ~ eulerth . (Co...
eulerth 16832 Euler's theorem, a general...
fermltl 16833 Fermat's little theorem. ...
prmdiv 16834 Show an explicit expressio...
prmdiveq 16835 The modular inverse of ` A...
prmdivdiv 16836 The (modular) inverse of t...
hashgcdlem 16837 A correspondence between e...
hashgcdeq 16838 Number of initial positive...
phisum 16839 The divisor sum identity o...
odzval 16840 Value of the order functio...
odzcllem 16841 - Lemma for ~ odzcl , show...
odzcl 16842 The order of a group eleme...
odzid 16843 Any element raised to the ...
odzdvds 16844 The only powers of ` A ` t...
odzphi 16845 The order of any group ele...
modprm1div 16846 A prime number divides an ...
m1dvdsndvds 16847 If an integer minus 1 is d...
modprminv 16848 Show an explicit expressio...
modprminveq 16849 The modular inverse of ` A...
vfermltl 16850 Variant of Fermat's little...
vfermltlALT 16851 Alternate proof of ~ vferm...
powm2modprm 16852 If an integer minus 1 is d...
reumodprminv 16853 For any prime number and f...
modprm0 16854 For two positive integers ...
nnnn0modprm0 16855 For a positive integer and...
modprmn0modprm0 16856 For an integer not being 0...
coprimeprodsq 16857 If three numbers are copri...
coprimeprodsq2 16858 If three numbers are copri...
oddprm 16859 A prime not equal to ` 2 `...
nnoddn2prm 16860 A prime not equal to ` 2 `...
oddn2prm 16861 A prime not equal to ` 2 `...
nnoddn2prmb 16862 A number is a prime number...
prm23lt5 16863 A prime less than 5 is eit...
prm23ge5 16864 A prime is either 2 or 3 o...
pythagtriplem1 16865 Lemma for ~ pythagtrip . ...
pythagtriplem2 16866 Lemma for ~ pythagtrip . ...
pythagtriplem3 16867 Lemma for ~ pythagtrip . ...
pythagtriplem4 16868 Lemma for ~ pythagtrip . ...
pythagtriplem10 16869 Lemma for ~ pythagtrip . ...
pythagtriplem6 16870 Lemma for ~ pythagtrip . ...
pythagtriplem7 16871 Lemma for ~ pythagtrip . ...
pythagtriplem8 16872 Lemma for ~ pythagtrip . ...
pythagtriplem9 16873 Lemma for ~ pythagtrip . ...
pythagtriplem11 16874 Lemma for ~ pythagtrip . ...
pythagtriplem12 16875 Lemma for ~ pythagtrip . ...
pythagtriplem13 16876 Lemma for ~ pythagtrip . ...
pythagtriplem14 16877 Lemma for ~ pythagtrip . ...
pythagtriplem15 16878 Lemma for ~ pythagtrip . ...
pythagtriplem16 16879 Lemma for ~ pythagtrip . ...
pythagtriplem17 16880 Lemma for ~ pythagtrip . ...
pythagtriplem18 16881 Lemma for ~ pythagtrip . ...
pythagtriplem19 16882 Lemma for ~ pythagtrip . ...
pythagtrip 16883 Parameterize the Pythagore...
iserodd 16884 Collect the odd terms in a...
pclem 16887 - Lemma for the prime powe...
pcprecl 16888 Closure of the prime power...
pcprendvds 16889 Non-divisibility property ...
pcprendvds2 16890 Non-divisibility property ...
pcpre1 16891 Value of the prime power p...
pcpremul 16892 Multiplicative property of...
pcval 16893 The value of the prime pow...
pceulem 16894 Lemma for ~ pceu . (Contr...
pceu 16895 Uniqueness for the prime p...
pczpre 16896 Connect the prime count pr...
pczcl 16897 Closure of the prime power...
pccl 16898 Closure of the prime power...
pccld 16899 Closure of the prime power...
pcmul 16900 Multiplication property of...
pcdiv 16901 Division property of the p...
pcqmul 16902 Multiplication property of...
pc0 16903 The value of the prime pow...
pc1 16904 Value of the prime count f...
pcqcl 16905 Closure of the general pri...
pcqdiv 16906 Division property of the p...
pcrec 16907 Prime power of a reciproca...
pcexp 16908 Prime power of an exponent...
pcxnn0cl 16909 Extended nonnegative integ...
pcxcl 16910 Extended real closure of t...
pcge0 16911 The prime count of an inte...
pczdvds 16912 Defining property of the p...
pcdvds 16913 Defining property of the p...
pczndvds 16914 Defining property of the p...
pcndvds 16915 Defining property of the p...
pczndvds2 16916 The remainder after dividi...
pcndvds2 16917 The remainder after dividi...
pcdvdsb 16918 ` P ^ A ` divides ` N ` if...
pcelnn 16919 There are a positive numbe...
pceq0 16920 There are zero powers of a...
pcidlem 16921 The prime count of a prime...
pcid 16922 The prime count of a prime...
pcneg 16923 The prime count of a negat...
pcabs 16924 The prime count of an abso...
pcdvdstr 16925 The prime count increases ...
pcgcd1 16926 The prime count of a GCD i...
pcgcd 16927 The prime count of a GCD i...
pc2dvds 16928 A characterization of divi...
pc11 16929 The prime count function, ...
pcz 16930 The prime count function c...
pcprmpw2 16931 Self-referential expressio...
pcprmpw 16932 Self-referential expressio...
dvdsprmpweq 16933 If a positive integer divi...
dvdsprmpweqnn 16934 If an integer greater than...
dvdsprmpweqle 16935 If a positive integer divi...
difsqpwdvds 16936 If the difference of two s...
pcaddlem 16937 Lemma for ~ pcadd . The o...
pcadd 16938 An inequality for the prim...
pcadd2 16939 The inequality of ~ pcadd ...
pcmptcl 16940 Closure for the prime powe...
pcmpt 16941 Construct a function with ...
pcmpt2 16942 Dividing two prime count m...
pcmptdvds 16943 The partial products of th...
pcprod 16944 The product of the primes ...
sumhash 16945 The sum of 1 over a set is...
fldivp1 16946 The difference between the...
pcfaclem 16947 Lemma for ~ pcfac . (Cont...
pcfac 16948 Calculate the prime count ...
pcbc 16949 Calculate the prime count ...
qexpz 16950 If a power of a rational n...
expnprm 16951 A second or higher power o...
oddprmdvds 16952 Every positive integer whi...
prmpwdvds 16953 A relation involving divis...
pockthlem 16954 Lemma for ~ pockthg . (Co...
pockthg 16955 The generalized Pocklingto...
pockthi 16956 Pocklington's theorem, whi...
unbenlem 16957 Lemma for ~ unben . (Cont...
unben 16958 An unbounded set of positi...
infpnlem1 16959 Lemma for ~ infpn . The s...
infpnlem2 16960 Lemma for ~ infpn . For a...
infpn 16961 There exist infinitely man...
infpn2 16962 There exist infinitely man...
prmunb 16963 The primes are unbounded. ...
prminf 16964 There are an infinite numb...
prmreclem1 16965 Lemma for ~ prmrec . Prop...
prmreclem2 16966 Lemma for ~ prmrec . Ther...
prmreclem3 16967 Lemma for ~ prmrec . The ...
prmreclem4 16968 Lemma for ~ prmrec . Show...
prmreclem5 16969 Lemma for ~ prmrec . Here...
prmreclem6 16970 Lemma for ~ prmrec . If t...
prmrec 16971 The sum of the reciprocals...
1arithlem1 16972 Lemma for ~ 1arith . (Con...
1arithlem2 16973 Lemma for ~ 1arith . (Con...
1arithlem3 16974 Lemma for ~ 1arith . (Con...
1arithlem4 16975 Lemma for ~ 1arith . (Con...
1arith 16976 Fundamental theorem of ari...
1arith2 16977 Fundamental theorem of ari...
elgz 16980 Elementhood in the gaussia...
gzcn 16981 A gaussian integer is a co...
zgz 16982 An integer is a gaussian i...
igz 16983 ` _i ` is a gaussian integ...
gznegcl 16984 The gaussian integers are ...
gzcjcl 16985 The gaussian integers are ...
gzaddcl 16986 The gaussian integers are ...
gzmulcl 16987 The gaussian integers are ...
gzreim 16988 Construct a gaussian integ...
gzsubcl 16989 The gaussian integers are ...
gzabssqcl 16990 The squared norm of a gaus...
4sqlem5 16991 Lemma for ~ 4sq . (Contri...
4sqlem6 16992 Lemma for ~ 4sq . (Contri...
4sqlem7 16993 Lemma for ~ 4sq . (Contri...
4sqlem8 16994 Lemma for ~ 4sq . (Contri...
4sqlem9 16995 Lemma for ~ 4sq . (Contri...
4sqlem10 16996 Lemma for ~ 4sq . (Contri...
4sqlem1 16997 Lemma for ~ 4sq . The set...
4sqlem2 16998 Lemma for ~ 4sq . Change ...
4sqlem3 16999 Lemma for ~ 4sq . Suffici...
4sqlem4a 17000 Lemma for ~ 4sqlem4 . (Co...
4sqlem4 17001 Lemma for ~ 4sq . We can ...
mul4sqlem 17002 Lemma for ~ mul4sq : algeb...
mul4sq 17003 Euler's four-square identi...
4sqlem11 17004 Lemma for ~ 4sq . Use the...
4sqlem12 17005 Lemma for ~ 4sq . For any...
4sqlem13 17006 Lemma for ~ 4sq . (Contri...
4sqlem14 17007 Lemma for ~ 4sq . (Contri...
4sqlem15 17008 Lemma for ~ 4sq . (Contri...
4sqlem16 17009 Lemma for ~ 4sq . (Contri...
4sqlem17 17010 Lemma for ~ 4sq . (Contri...
4sqlem18 17011 Lemma for ~ 4sq . Inducti...
4sqlem19 17012 Lemma for ~ 4sq . The pro...
4sq 17013 Lagrange's four-square the...
vdwapfval 17020 Define the arithmetic prog...
vdwapf 17021 The arithmetic progression...
vdwapval 17022 Value of the arithmetic pr...
vdwapun 17023 Remove the first element o...
vdwapid1 17024 The first element of an ar...
vdwap0 17025 Value of a length-1 arithm...
vdwap1 17026 Value of a length-1 arithm...
vdwmc 17027 The predicate " The ` <. R...
vdwmc2 17028 Expand out the definition ...
vdwpc 17029 The predicate " The colori...
vdwlem1 17030 Lemma for ~ vdw . (Contri...
vdwlem2 17031 Lemma for ~ vdw . (Contri...
vdwlem3 17032 Lemma for ~ vdw . (Contri...
vdwlem4 17033 Lemma for ~ vdw . (Contri...
vdwlem5 17034 Lemma for ~ vdw . (Contri...
vdwlem6 17035 Lemma for ~ vdw . (Contri...
vdwlem7 17036 Lemma for ~ vdw . (Contri...
vdwlem8 17037 Lemma for ~ vdw . (Contri...
vdwlem9 17038 Lemma for ~ vdw . (Contri...
vdwlem10 17039 Lemma for ~ vdw . Set up ...
vdwlem11 17040 Lemma for ~ vdw . (Contri...
vdwlem12 17041 Lemma for ~ vdw . ` K = 2 ...
vdwlem13 17042 Lemma for ~ vdw . Main in...
vdw 17043 Van der Waerden's theorem....
vdwnnlem1 17044 Corollary of ~ vdw , and l...
vdwnnlem2 17045 Lemma for ~ vdwnn . The s...
vdwnnlem3 17046 Lemma for ~ vdwnn . (Cont...
vdwnn 17047 Van der Waerden's theorem,...
ramtlecl 17049 The set ` T ` of numbers w...
hashbcval 17051 Value of the "binomial set...
hashbccl 17052 The binomial set is a fini...
hashbcss 17053 Subset relation for the bi...
hashbc0 17054 The set of subsets of size...
hashbc2 17055 The size of the binomial s...
0hashbc 17056 There are no subsets of th...
ramval 17057 The value of the Ramsey nu...
ramcl2lem 17058 Lemma for extended real cl...
ramtcl 17059 The Ramsey number has the ...
ramtcl2 17060 The Ramsey number is an in...
ramtub 17061 The Ramsey number is a low...
ramub 17062 The Ramsey number is a low...
ramub2 17063 It is sufficient to check ...
rami 17064 The defining property of a...
ramcl2 17065 The Ramsey number is eithe...
ramxrcl 17066 The Ramsey number is an ex...
ramubcl 17067 If the Ramsey number is up...
ramlb 17068 Establish a lower bound on...
0ram 17069 The Ramsey number when ` M...
0ram2 17070 The Ramsey number when ` M...
ram0 17071 The Ramsey number when ` R...
0ramcl 17072 Lemma for ~ ramcl : Exist...
ramz2 17073 The Ramsey number when ` F...
ramz 17074 The Ramsey number when ` F...
ramub1lem1 17075 Lemma for ~ ramub1 . (Con...
ramub1lem2 17076 Lemma for ~ ramub1 . (Con...
ramub1 17077 Inductive step for Ramsey'...
ramcl 17078 Ramsey's theorem: the Rams...
ramsey 17079 Ramsey's theorem with the ...
prmoval 17082 Value of the primorial fun...
prmocl 17083 Closure of the primorial f...
prmone0 17084 The primorial function is ...
prmo0 17085 The primorial of 0. (Cont...
prmo1 17086 The primorial of 1. (Cont...
prmop1 17087 The primorial of a success...
prmonn2 17088 Value of the primorial fun...
prmo2 17089 The primorial of 2. (Cont...
prmo3 17090 The primorial of 3. (Cont...
prmdvdsprmo 17091 The primorial of a number ...
prmdvdsprmop 17092 The primorial of a number ...
fvprmselelfz 17093 The value of the prime sel...
fvprmselgcd1 17094 The greatest common diviso...
prmolefac 17095 The primorial of a positiv...
prmodvdslcmf 17096 The primorial of a nonnega...
prmolelcmf 17097 The primorial of a positiv...
prmgaplem1 17098 Lemma for ~ prmgap : The ...
prmgaplem2 17099 Lemma for ~ prmgap : The ...
prmgaplcmlem1 17100 Lemma for ~ prmgaplcm : T...
prmgaplcmlem2 17101 Lemma for ~ prmgaplcm : T...
prmgaplem3 17102 Lemma for ~ prmgap . (Con...
prmgaplem4 17103 Lemma for ~ prmgap . (Con...
prmgaplem5 17104 Lemma for ~ prmgap : for e...
prmgaplem6 17105 Lemma for ~ prmgap : for e...
prmgaplem7 17106 Lemma for ~ prmgap . (Con...
prmgaplem8 17107 Lemma for ~ prmgap . (Con...
prmgap 17108 The prime gap theorem: for...
prmgaplcm 17109 Alternate proof of ~ prmga...
prmgapprmolem 17110 Lemma for ~ prmgapprmo : ...
prmgapprmo 17111 Alternate proof of ~ prmga...
dec2dvds 17112 Divisibility by two is obv...
dec5dvds 17113 Divisibility by five is ob...
dec5dvds2 17114 Divisibility by five is ob...
dec5nprm 17115 Divisibility by five is ob...
dec2nprm 17116 Divisibility by two is obv...
modxai 17117 Add exponents in a power m...
mod2xi 17118 Double exponents in a powe...
modxp1i 17119 Add one to an exponent in ...
mod2xnegi 17120 Version of ~ mod2xi with a...
modsubi 17121 Subtract from within a mod...
gcdi 17122 Calculate a GCD via Euclid...
gcdmodi 17123 Calculate a GCD via Euclid...
decexp2 17124 Calculate a power of two. ...
numexp0 17125 Calculate an integer power...
numexp1 17126 Calculate an integer power...
numexpp1 17127 Calculate an integer power...
numexp2x 17128 Double an integer power. ...
decsplit0b 17129 Split a decimal number int...
decsplit0 17130 Split a decimal number int...
decsplit1 17131 Split a decimal number int...
decsplit 17132 Split a decimal number int...
karatsuba 17133 The Karatsuba multiplicati...
2exp4 17134 Two to the fourth power is...
2exp5 17135 Two to the fifth power is ...
2exp6 17136 Two to the sixth power is ...
2exp7 17137 Two to the seventh power i...
2exp8 17138 Two to the eighth power is...
2exp11 17139 Two to the eleventh power ...
2exp16 17140 Two to the sixteenth power...
3exp3 17141 Three to the third power i...
2expltfac 17142 The factorial grows faster...
cshwsidrepsw 17143 If cyclically shifting a w...
cshwsidrepswmod0 17144 If cyclically shifting a w...
cshwshashlem1 17145 If cyclically shifting a w...
cshwshashlem2 17146 If cyclically shifting a w...
cshwshashlem3 17147 If cyclically shifting a w...
cshwsdisj 17148 The singletons resulting b...
cshwsiun 17149 The set of (different!) wo...
cshwsex 17150 The class of (different!) ...
cshws0 17151 The size of the set of (di...
cshwrepswhash1 17152 The size of the set of (di...
cshwshashnsame 17153 If a word (not consisting ...
cshwshash 17154 If a word has a length bei...
prmlem0 17155 Lemma for ~ prmlem1 and ~ ...
prmlem1a 17156 A quick proof skeleton to ...
prmlem1 17157 A quick proof skeleton to ...
5prm 17158 5 is a prime number. (Con...
6nprm 17159 6 is not a prime number. ...
7prm 17160 7 is a prime number. (Con...
8nprm 17161 8 is not a prime number. ...
9nprm 17162 9 is not a prime number. ...
10nprm 17163 10 is not a prime number. ...
11prm 17164 11 is a prime number. (Co...
13prm 17165 13 is a prime number. (Co...
17prm 17166 17 is a prime number. (Co...
19prm 17167 19 is a prime number. (Co...
23prm 17168 23 is a prime number. (Co...
prmlem2 17169 Our last proving session g...
37prm 17170 37 is a prime number. (Co...
43prm 17171 43 is a prime number. (Co...
83prm 17172 83 is a prime number. (Co...
139prm 17173 139 is a prime number. (C...
163prm 17174 163 is a prime number. (C...
317prm 17175 317 is a prime number. (C...
631prm 17176 631 is a prime number. (C...
prmo4 17177 The primorial of 4. (Cont...
prmo5 17178 The primorial of 5. (Cont...
prmo6 17179 The primorial of 6. (Cont...
1259lem1 17180 Lemma for ~ 1259prm . Cal...
1259lem2 17181 Lemma for ~ 1259prm . Cal...
1259lem3 17182 Lemma for ~ 1259prm . Cal...
1259lem4 17183 Lemma for ~ 1259prm . Cal...
1259lem5 17184 Lemma for ~ 1259prm . Cal...
1259prm 17185 1259 is a prime number. (...
2503lem1 17186 Lemma for ~ 2503prm . Cal...
2503lem2 17187 Lemma for ~ 2503prm . Cal...
2503lem3 17188 Lemma for ~ 2503prm . Cal...
2503prm 17189 2503 is a prime number. (...
4001lem1 17190 Lemma for ~ 4001prm . Cal...
4001lem2 17191 Lemma for ~ 4001prm . Cal...
4001lem3 17192 Lemma for ~ 4001prm . Cal...
4001lem4 17193 Lemma for ~ 4001prm . Cal...
4001prm 17194 4001 is a prime number. (...
brstruct 17197 The structure relation is ...
isstruct2 17198 The property of being a st...
structex 17199 A structure is a set. (Co...
structn0fun 17200 A structure without the em...
isstruct 17201 The property of being a st...
structcnvcnv 17202 Two ways to express the re...
structfung 17203 The converse of the conver...
structfun 17204 Convert between two kinds ...
structfn 17205 Convert between two kinds ...
strleun 17206 Combine two structures int...
strle1 17207 Make a structure from a si...
strle2 17208 Make a structure from a pa...
strle3 17209 Make a structure from a tr...
sbcie2s 17210 A special version of class...
sbcie3s 17211 A special version of class...
reldmsets 17214 The structure override ope...
setsvalg 17215 Value of the structure rep...
setsval 17216 Value of the structure rep...
fvsetsid 17217 The value of the structure...
fsets 17218 The structure replacement ...
setsdm 17219 The domain of a structure ...
setsfun 17220 A structure with replaceme...
setsfun0 17221 A structure with replaceme...
setsn0fun 17222 The value of the structure...
setsstruct2 17223 An extensible structure wi...
setsexstruct2 17224 An extensible structure wi...
setsstruct 17225 An extensible structure wi...
wunsets 17226 Closure of structure repla...
setsres 17227 The structure replacement ...
setsabs 17228 Replacing the same compone...
setscom 17229 Different components can b...
sloteq 17232 Equality theorem for the `...
slotfn 17233 A slot is a function on se...
strfvnd 17234 Deduction version of ~ str...
strfvn 17235 Value of a structure compo...
strfvss 17236 A structure component extr...
wunstr 17237 Closure of a structure ind...
str0 17238 All components of the empt...
strfvi 17239 Structure slot extractors ...
fveqprc 17240 Lemma for showing the equa...
oveqprc 17241 Lemma for showing the equa...
wunndx 17244 Closure of the index extra...
ndxarg 17245 Get the numeric argument f...
ndxid 17246 A structure component extr...
strndxid 17247 The value of a structure c...
setsidvald 17248 Value of the structure rep...
setsidvaldOLD 17249 Obsolete version of ~ sets...
strfvd 17250 Deduction version of ~ str...
strfv2d 17251 Deduction version of ~ str...
strfv2 17252 A variation on ~ strfv to ...
strfv 17253 Extract a structure compon...
strfv3 17254 Variant on ~ strfv for lar...
strssd 17255 Deduction version of ~ str...
strss 17256 Propagate component extrac...
setsid 17257 Value of the structure rep...
setsnid 17258 Value of the structure rep...
setsnidOLD 17259 Obsolete proof of ~ setsni...
baseval 17262 Value of the base set extr...
baseid 17263 Utility theorem: index-ind...
basfn 17264 The base set extractor is ...
base0 17265 The base set of the empty ...
elbasfv 17266 Utility theorem: reverse c...
elbasov 17267 Utility theorem: reverse c...
strov2rcl 17268 Partial reverse closure fo...
basendx 17269 Index value of the base se...
basendxnn 17270 The index value of the bas...
basendxnnOLD 17271 Obsolete proof of ~ basend...
basndxelwund 17272 The index of the base set ...
basprssdmsets 17273 The pair of the base index...
opelstrbas 17274 The base set of a structur...
1strstr 17275 A constructed one-slot str...
1strstr1 17276 A constructed one-slot str...
1strbas 17277 The base set of a construc...
1strbasOLD 17278 Obsolete proof of ~ 1strba...
1strwunbndx 17279 A constructed one-slot str...
1strwun 17280 A constructed one-slot str...
1strwunOLD 17281 Obsolete version of ~ 1str...
2strstr 17282 A constructed two-slot str...
2strbas 17283 The base set of a construc...
2strop 17284 The other slot of a constr...
2strstr1 17285 A constructed two-slot str...
2strstr1OLD 17286 Obsolete version of ~ 2str...
2strbas1 17287 The base set of a construc...
2strop1 17288 The other slot of a constr...
reldmress 17291 The structure restriction ...
ressval 17292 Value of structure restric...
ressid2 17293 General behavior of trivia...
ressval2 17294 Value of nontrivial struct...
ressbas 17295 Base set of a structure re...
ressbasOLD 17296 Obsolete proof of ~ ressba...
ressbasssg 17297 The base set of a restrict...
ressbas2 17298 Base set of a structure re...
ressbasss 17299 The base set of a restrict...
ressbasssOLD 17300 Obsolete proof of ~ ressba...
ressbasss2 17301 The base set of a restrict...
resseqnbas 17302 The components of an exten...
resslemOLD 17303 Obsolete version of ~ ress...
ress0 17304 All restrictions of the nu...
ressid 17305 Behavior of trivial restri...
ressinbas 17306 Restriction only cares abo...
ressval3d 17307 Value of structure restric...
ressval3dOLD 17308 Obsolete version of ~ ress...
ressress 17309 Restriction composition la...
ressabs 17310 Restriction absorption law...
wunress 17311 Closure of structure restr...
wunressOLD 17312 Obsolete proof of ~ wunres...
plusgndx 17339 Index value of the ~ df-pl...
plusgid 17340 Utility theorem: index-ind...
plusgndxnn 17341 The index of the slot for ...
basendxltplusgndx 17342 The index of the slot for ...
basendxnplusgndx 17343 The slot for the base set ...
basendxnplusgndxOLD 17344 Obsolete version of ~ base...
grpstr 17345 A constructed group is a s...
grpstrndx 17346 A constructed group is a s...
grpbase 17347 The base set of a construc...
grpbaseOLD 17348 Obsolete version of ~ grpb...
grpplusg 17349 The operation of a constru...
grpplusgOLD 17350 Obsolete version of ~ grpp...
ressplusg 17351 ` +g ` is unaffected by re...
grpbasex 17352 The base of an explicitly ...
grpplusgx 17353 The operation of an explic...
mulrndx 17354 Index value of the ~ df-mu...
mulridx 17355 Utility theorem: index-ind...
basendxnmulrndx 17356 The slot for the base set ...
basendxnmulrndxOLD 17357 Obsolete proof of ~ basend...
plusgndxnmulrndx 17358 The slot for the group (ad...
rngstr 17359 A constructed ring is a st...
rngbase 17360 The base set of a construc...
rngplusg 17361 The additive operation of ...
rngmulr 17362 The multiplicative operati...
starvndx 17363 Index value of the ~ df-st...
starvid 17364 Utility theorem: index-ind...
starvndxnbasendx 17365 The slot for the involutio...
starvndxnplusgndx 17366 The slot for the involutio...
starvndxnmulrndx 17367 The slot for the involutio...
ressmulr 17368 ` .r ` is unaffected by re...
ressstarv 17369 ` *r ` is unaffected by re...
srngstr 17370 A constructed star ring is...
srngbase 17371 The base set of a construc...
srngplusg 17372 The addition operation of ...
srngmulr 17373 The multiplication operati...
srnginvl 17374 The involution function of...
scandx 17375 Index value of the ~ df-sc...
scaid 17376 Utility theorem: index-ind...
scandxnbasendx 17377 The slot for the scalar is...
scandxnplusgndx 17378 The slot for the scalar fi...
scandxnmulrndx 17379 The slot for the scalar fi...
vscandx 17380 Index value of the ~ df-vs...
vscaid 17381 Utility theorem: index-ind...
vscandxnbasendx 17382 The slot for the scalar pr...
vscandxnplusgndx 17383 The slot for the scalar pr...
vscandxnmulrndx 17384 The slot for the scalar pr...
vscandxnscandx 17385 The slot for the scalar pr...
lmodstr 17386 A constructed left module ...
lmodbase 17387 The base set of a construc...
lmodplusg 17388 The additive operation of ...
lmodsca 17389 The set of scalars of a co...
lmodvsca 17390 The scalar product operati...
ipndx 17391 Index value of the ~ df-ip...
ipid 17392 Utility theorem: index-ind...
ipndxnbasendx 17393 The slot for the inner pro...
ipndxnplusgndx 17394 The slot for the inner pro...
ipndxnmulrndx 17395 The slot for the inner pro...
slotsdifipndx 17396 The slot for the scalar is...
ipsstr 17397 Lemma to shorten proofs of...
ipsbase 17398 The base set of a construc...
ipsaddg 17399 The additive operation of ...
ipsmulr 17400 The multiplicative operati...
ipssca 17401 The set of scalars of a co...
ipsvsca 17402 The scalar product operati...
ipsip 17403 The multiplicative operati...
resssca 17404 ` Scalar ` is unaffected b...
ressvsca 17405 ` .s ` is unaffected by re...
ressip 17406 The inner product is unaff...
phlstr 17407 A constructed pre-Hilbert ...
phlbase 17408 The base set of a construc...
phlplusg 17409 The additive operation of ...
phlsca 17410 The ring of scalars of a c...
phlvsca 17411 The scalar product operati...
phlip 17412 The inner product (Hermiti...
tsetndx 17413 Index value of the ~ df-ts...
tsetid 17414 Utility theorem: index-ind...
tsetndxnn 17415 The index of the slot for ...
basendxlttsetndx 17416 The index of the slot for ...
tsetndxnbasendx 17417 The slot for the topology ...
tsetndxnplusgndx 17418 The slot for the topology ...
tsetndxnmulrndx 17419 The slot for the topology ...
tsetndxnstarvndx 17420 The slot for the topology ...
slotstnscsi 17421 The slots ` Scalar ` , ` ....
topgrpstr 17422 A constructed topological ...
topgrpbas 17423 The base set of a construc...
topgrpplusg 17424 The additive operation of ...
topgrptset 17425 The topology of a construc...
resstset 17426 ` TopSet ` is unaffected b...
plendx 17427 Index value of the ~ df-pl...
pleid 17428 Utility theorem: self-refe...
plendxnn 17429 The index value of the ord...
basendxltplendx 17430 The index value of the ` B...
plendxnbasendx 17431 The slot for the order is ...
plendxnplusgndx 17432 The slot for the "less tha...
plendxnmulrndx 17433 The slot for the "less tha...
plendxnscandx 17434 The slot for the "less tha...
plendxnvscandx 17435 The slot for the "less tha...
slotsdifplendx 17436 The index of the slot for ...
otpsstr 17437 Functionality of a topolog...
otpsbas 17438 The base set of a topologi...
otpstset 17439 The open sets of a topolog...
otpsle 17440 The order of a topological...
ressle 17441 ` le ` is unaffected by re...
ocndx 17442 Index value of the ~ df-oc...
ocid 17443 Utility theorem: index-ind...
basendxnocndx 17444 The slot for the orthocomp...
plendxnocndx 17445 The slot for the orthocomp...
dsndx 17446 Index value of the ~ df-ds...
dsid 17447 Utility theorem: index-ind...
dsndxnn 17448 The index of the slot for ...
basendxltdsndx 17449 The index of the slot for ...
dsndxnbasendx 17450 The slot for the distance ...
dsndxnplusgndx 17451 The slot for the distance ...
dsndxnmulrndx 17452 The slot for the distance ...
slotsdnscsi 17453 The slots ` Scalar ` , ` ....
dsndxntsetndx 17454 The slot for the distance ...
slotsdifdsndx 17455 The index of the slot for ...
unifndx 17456 Index value of the ~ df-un...
unifid 17457 Utility theorem: index-ind...
unifndxnn 17458 The index of the slot for ...
basendxltunifndx 17459 The index of the slot for ...
unifndxnbasendx 17460 The slot for the uniform s...
unifndxntsetndx 17461 The slot for the uniform s...
slotsdifunifndx 17462 The index of the slot for ...
ressunif 17463 ` UnifSet ` is unaffected ...
odrngstr 17464 Functionality of an ordere...
odrngbas 17465 The base set of an ordered...
odrngplusg 17466 The addition operation of ...
odrngmulr 17467 The multiplication operati...
odrngtset 17468 The open sets of an ordere...
odrngle 17469 The order of an ordered me...
odrngds 17470 The metric of an ordered m...
ressds 17471 ` dist ` is unaffected by ...
homndx 17472 Index value of the ~ df-ho...
homid 17473 Utility theorem: index-ind...
ccondx 17474 Index value of the ~ df-cc...
ccoid 17475 Utility theorem: index-ind...
slotsbhcdif 17476 The slots ` Base ` , ` Hom...
slotsbhcdifOLD 17477 Obsolete proof of ~ slotsb...
slotsdifplendx2 17478 The index of the slot for ...
slotsdifocndx 17479 The index of the slot for ...
resshom 17480 ` Hom ` is unaffected by r...
ressco 17481 ` comp ` is unaffected by ...
restfn 17486 The subspace topology oper...
topnfn 17487 The topology extractor fun...
restval 17488 The subspace topology indu...
elrest 17489 The predicate "is an open ...
elrestr 17490 Sufficient condition for b...
0rest 17491 Value of the structure res...
restid2 17492 The subspace topology over...
restsspw 17493 The subspace topology is a...
firest 17494 The finite intersections o...
restid 17495 The subspace topology of t...
topnval 17496 Value of the topology extr...
topnid 17497 Value of the topology extr...
topnpropd 17498 The topology extractor fun...
reldmprds 17510 The structure product is a...
prdsbasex 17512 Lemma for structure produc...
imasvalstr 17513 An image structure value i...
prdsvalstr 17514 Structure product value is...
prdsbaslem 17515 Lemma for ~ prdsbas and si...
prdsvallem 17516 Lemma for ~ prdsval . (Co...
prdsval 17517 Value of the structure pro...
prdssca 17518 Scalar ring of a structure...
prdsbas 17519 Base set of a structure pr...
prdsplusg 17520 Addition in a structure pr...
prdsmulr 17521 Multiplication in a struct...
prdsvsca 17522 Scalar multiplication in a...
prdsip 17523 Inner product in a structu...
prdsle 17524 Structure product weak ord...
prdsless 17525 Closure of the order relat...
prdsds 17526 Structure product distance...
prdsdsfn 17527 Structure product distance...
prdstset 17528 Structure product topology...
prdshom 17529 Structure product hom-sets...
prdsco 17530 Structure product composit...
prdsbas2 17531 The base set of a structur...
prdsbasmpt 17532 A constructed tuple is a p...
prdsbasfn 17533 Points in the structure pr...
prdsbasprj 17534 Each point in a structure ...
prdsplusgval 17535 Value of a componentwise s...
prdsplusgfval 17536 Value of a structure produ...
prdsmulrval 17537 Value of a componentwise r...
prdsmulrfval 17538 Value of a structure produ...
prdsleval 17539 Value of the product order...
prdsdsval 17540 Value of the metric in a s...
prdsvscaval 17541 Scalar multiplication in a...
prdsvscafval 17542 Scalar multiplication of a...
prdsbas3 17543 The base set of an indexed...
prdsbasmpt2 17544 A constructed tuple is a p...
prdsbascl 17545 An element of the base has...
prdsdsval2 17546 Value of the metric in a s...
prdsdsval3 17547 Value of the metric in a s...
pwsval 17548 Value of a structure power...
pwsbas 17549 Base set of a structure po...
pwselbasb 17550 Membership in the base set...
pwselbas 17551 An element of a structure ...
pwsplusgval 17552 Value of addition in a str...
pwsmulrval 17553 Value of multiplication in...
pwsle 17554 Ordering in a structure po...
pwsleval 17555 Ordering in a structure po...
pwsvscafval 17556 Scalar multiplication in a...
pwsvscaval 17557 Scalar multiplication of a...
pwssca 17558 The ring of scalars of a s...
pwsdiagel 17559 Membership of diagonal ele...
pwssnf1o 17560 Triviality of singleton po...
imasval 17573 Value of an image structur...
imasbas 17574 The base set of an image s...
imasds 17575 The distance function of a...
imasdsfn 17576 The distance function is a...
imasdsval 17577 The distance function of a...
imasdsval2 17578 The distance function of a...
imasplusg 17579 The group operation in an ...
imasmulr 17580 The ring multiplication in...
imassca 17581 The scalar field of an ima...
imasvsca 17582 The scalar multiplication ...
imasip 17583 The inner product of an im...
imastset 17584 The topology of an image s...
imasle 17585 The ordering of an image s...
f1ocpbllem 17586 Lemma for ~ f1ocpbl . (Co...
f1ocpbl 17587 An injection is compatible...
f1ovscpbl 17588 An injection is compatible...
f1olecpbl 17589 An injection is compatible...
imasaddfnlem 17590 The image structure operat...
imasaddvallem 17591 The operation of an image ...
imasaddflem 17592 The image set operations a...
imasaddfn 17593 The image structure's grou...
imasaddval 17594 The value of an image stru...
imasaddf 17595 The image structure's grou...
imasmulfn 17596 The image structure's ring...
imasmulval 17597 The value of an image stru...
imasmulf 17598 The image structure's ring...
imasvscafn 17599 The image structure's scal...
imasvscaval 17600 The value of an image stru...
imasvscaf 17601 The image structure's scal...
imasless 17602 The order relation defined...
imasleval 17603 The value of the image str...
qusval 17604 Value of a quotient struct...
quslem 17605 The function in ~ qusval i...
qusin 17606 Restrict the equivalence r...
qusbas 17607 Base set of a quotient str...
quss 17608 The scalar field of a quot...
divsfval 17609 Value of the function in ~...
ercpbllem 17610 Lemma for ~ ercpbl . (Con...
ercpbl 17611 Translate the function com...
erlecpbl 17612 Translate the relation com...
qusaddvallem 17613 Value of an operation defi...
qusaddflem 17614 The operation of a quotien...
qusaddval 17615 The addition in a quotient...
qusaddf 17616 The addition in a quotient...
qusmulval 17617 The multiplication in a qu...
qusmulf 17618 The multiplication in a qu...
fnpr2o 17619 Function with a domain of ...
fnpr2ob 17620 Biconditional version of ~...
fvpr0o 17621 The value of a function wi...
fvpr1o 17622 The value of a function wi...
fvprif 17623 The value of the pair func...
xpsfrnel 17624 Elementhood in the target ...
xpsfeq 17625 A function on ` 2o ` is de...
xpsfrnel2 17626 Elementhood in the target ...
xpscf 17627 Equivalent condition for t...
xpsfval 17628 The value of the function ...
xpsff1o 17629 The function appearing in ...
xpsfrn 17630 A short expression for the...
xpsff1o2 17631 The function appearing in ...
xpsval 17632 Value of the binary struct...
xpsrnbas 17633 The indexed structure prod...
xpsbas 17634 The base set of the binary...
xpsaddlem 17635 Lemma for ~ xpsadd and ~ x...
xpsadd 17636 Value of the addition oper...
xpsmul 17637 Value of the multiplicatio...
xpssca 17638 Value of the scalar field ...
xpsvsca 17639 Value of the scalar multip...
xpsless 17640 Closure of the ordering in...
xpsle 17641 Value of the ordering in a...
ismre 17650 Property of being a Moore ...
fnmre 17651 The Moore collection gener...
mresspw 17652 A Moore collection is a su...
mress 17653 A Moore-closed subset is a...
mre1cl 17654 In any Moore collection th...
mreintcl 17655 A nonempty collection of c...
mreiincl 17656 A nonempty indexed interse...
mrerintcl 17657 The relative intersection ...
mreriincl 17658 The relative intersection ...
mreincl 17659 Two closed sets have a clo...
mreuni 17660 Since the entire base set ...
mreunirn 17661 Two ways to express the no...
ismred 17662 Properties that determine ...
ismred2 17663 Properties that determine ...
mremre 17664 The Moore collections of s...
submre 17665 The subcollection of a clo...
mrcflem 17666 The domain and codomain of...
fnmrc 17667 Moore-closure is a well-be...
mrcfval 17668 Value of the function expr...
mrcf 17669 The Moore closure is a fun...
mrcval 17670 Evaluation of the Moore cl...
mrccl 17671 The Moore closure of a set...
mrcsncl 17672 The Moore closure of a sin...
mrcid 17673 The closure of a closed se...
mrcssv 17674 The closure of a set is a ...
mrcidb 17675 A set is closed iff it is ...
mrcss 17676 Closure preserves subset o...
mrcssid 17677 The closure of a set is a ...
mrcidb2 17678 A set is closed iff it con...
mrcidm 17679 The closure operation is i...
mrcsscl 17680 The closure is the minimal...
mrcuni 17681 Idempotence of closure und...
mrcun 17682 Idempotence of closure und...
mrcssvd 17683 The Moore closure of a set...
mrcssd 17684 Moore closure preserves su...
mrcssidd 17685 A set is contained in its ...
mrcidmd 17686 Moore closure is idempoten...
mressmrcd 17687 In a Moore system, if a se...
submrc 17688 In a closure system which ...
mrieqvlemd 17689 In a Moore system, if ` Y ...
mrisval 17690 Value of the set of indepe...
ismri 17691 Criterion for a set to be ...
ismri2 17692 Criterion for a subset of ...
ismri2d 17693 Criterion for a subset of ...
ismri2dd 17694 Definition of independence...
mriss 17695 An independent set of a Mo...
mrissd 17696 An independent set of a Mo...
ismri2dad 17697 Consequence of a set in a ...
mrieqvd 17698 In a Moore system, a set i...
mrieqv2d 17699 In a Moore system, a set i...
mrissmrcd 17700 In a Moore system, if an i...
mrissmrid 17701 In a Moore system, subsets...
mreexd 17702 In a Moore system, the clo...
mreexmrid 17703 In a Moore system whose cl...
mreexexlemd 17704 This lemma is used to gene...
mreexexlem2d 17705 Used in ~ mreexexlem4d to ...
mreexexlem3d 17706 Base case of the induction...
mreexexlem4d 17707 Induction step of the indu...
mreexexd 17708 Exchange-type theorem. In...
mreexdomd 17709 In a Moore system whose cl...
mreexfidimd 17710 In a Moore system whose cl...
isacs 17711 A set is an algebraic clos...
acsmre 17712 Algebraic closure systems ...
isacs2 17713 In the definition of an al...
acsfiel 17714 A set is closed in an alge...
acsfiel2 17715 A set is closed in an alge...
acsmred 17716 An algebraic closure syste...
isacs1i 17717 A closure system determine...
mreacs 17718 Algebraicity is a composab...
acsfn 17719 Algebraicity of a conditio...
acsfn0 17720 Algebraicity of a point cl...
acsfn1 17721 Algebraicity of a one-argu...
acsfn1c 17722 Algebraicity of a one-argu...
acsfn2 17723 Algebraicity of a two-argu...
iscat 17732 The predicate "is a catego...
iscatd 17733 Properties that determine ...
catidex 17734 Each object in a category ...
catideu 17735 Each object in a category ...
cidfval 17736 Each object in a category ...
cidval 17737 Each object in a category ...
cidffn 17738 The identity arrow constru...
cidfn 17739 The identity arrow operato...
catidd 17740 Deduce the identity arrow ...
iscatd2 17741 Version of ~ iscatd with a...
catidcl 17742 Each object in a category ...
catlid 17743 Left identity property of ...
catrid 17744 Right identity property of...
catcocl 17745 Closure of a composition a...
catass 17746 Associativity of compositi...
catcone0 17747 Composition of non-empty h...
0catg 17748 Any structure with an empt...
0cat 17749 The empty set is a categor...
homffval 17750 Value of the functionalize...
fnhomeqhomf 17751 If the Hom-set operation i...
homfval 17752 Value of the functionalize...
homffn 17753 The functionalized Hom-set...
homfeq 17754 Condition for two categori...
homfeqd 17755 If two structures have the...
homfeqbas 17756 Deduce equality of base se...
homfeqval 17757 Value of the functionalize...
comfffval 17758 Value of the functionalize...
comffval 17759 Value of the functionalize...
comfval 17760 Value of the functionalize...
comfffval2 17761 Value of the functionalize...
comffval2 17762 Value of the functionalize...
comfval2 17763 Value of the functionalize...
comfffn 17764 The functionalized composi...
comffn 17765 The functionalized composi...
comfeq 17766 Condition for two categori...
comfeqd 17767 Condition for two categori...
comfeqval 17768 Equality of two compositio...
catpropd 17769 Two structures with the sa...
cidpropd 17770 Two structures with the sa...
oppcval 17773 Value of the opposite cate...
oppchomfval 17774 Hom-sets of the opposite c...
oppchomfvalOLD 17775 Obsolete proof of ~ oppcho...
oppchom 17776 Hom-sets of the opposite c...
oppccofval 17777 Composition in the opposit...
oppcco 17778 Composition in the opposit...
oppcbas 17779 Base set of an opposite ca...
oppcbasOLD 17780 Obsolete version of ~ oppc...
oppccatid 17781 Lemma for ~ oppccat . (Co...
oppchomf 17782 Hom-sets of the opposite c...
oppcid 17783 Identity function of an op...
oppccat 17784 An opposite category is a ...
2oppcbas 17785 The double opposite catego...
2oppchomf 17786 The double opposite catego...
2oppccomf 17787 The double opposite catego...
oppchomfpropd 17788 If two categories have the...
oppccomfpropd 17789 If two categories have the...
oppccatf 17790 ` oppCat ` restricted to `...
monfval 17795 Definition of a monomorphi...
ismon 17796 Definition of a monomorphi...
ismon2 17797 Write out the monomorphism...
monhom 17798 A monomorphism is a morphi...
moni 17799 Property of a monomorphism...
monpropd 17800 If two categories have the...
oppcmon 17801 A monomorphism in the oppo...
oppcepi 17802 An epimorphism in the oppo...
isepi 17803 Definition of an epimorphi...
isepi2 17804 Write out the epimorphism ...
epihom 17805 An epimorphism is a morphi...
epii 17806 Property of an epimorphism...
sectffval 17813 Value of the section opera...
sectfval 17814 Value of the section relat...
sectss 17815 The section relation is a ...
issect 17816 The property " ` F ` is a ...
issect2 17817 Property of being a sectio...
sectcan 17818 If ` G ` is a section of `...
sectco 17819 Composition of two section...
isofval 17820 Function value of the func...
invffval 17821 Value of the inverse relat...
invfval 17822 Value of the inverse relat...
isinv 17823 Value of the inverse relat...
invss 17824 The inverse relation is a ...
invsym 17825 The inverse relation is sy...
invsym2 17826 The inverse relation is sy...
invfun 17827 The inverse relation is a ...
isoval 17828 The isomorphisms are the d...
inviso1 17829 If ` G ` is an inverse to ...
inviso2 17830 If ` G ` is an inverse to ...
invf 17831 The inverse relation is a ...
invf1o 17832 The inverse relation is a ...
invinv 17833 The inverse of the inverse...
invco 17834 The composition of two iso...
dfiso2 17835 Alternate definition of an...
dfiso3 17836 Alternate definition of an...
inveq 17837 If there are two inverses ...
isofn 17838 The function value of the ...
isohom 17839 An isomorphism is a homomo...
isoco 17840 The composition of two iso...
oppcsect 17841 A section in the opposite ...
oppcsect2 17842 A section in the opposite ...
oppcinv 17843 An inverse in the opposite...
oppciso 17844 An isomorphism in the oppo...
sectmon 17845 If ` F ` is a section of `...
monsect 17846 If ` F ` is a monomorphism...
sectepi 17847 If ` F ` is a section of `...
episect 17848 If ` F ` is an epimorphism...
sectid 17849 The identity is a section ...
invid 17850 The inverse of the identit...
idiso 17851 The identity is an isomorp...
idinv 17852 The inverse of the identit...
invisoinvl 17853 The inverse of an isomorph...
invisoinvr 17854 The inverse of an isomorph...
invcoisoid 17855 The inverse of an isomorph...
isocoinvid 17856 The inverse of an isomorph...
rcaninv 17857 Right cancellation of an i...
cicfval 17860 The set of isomorphic obje...
brcic 17861 The relation "is isomorphi...
cic 17862 Objects ` X ` and ` Y ` in...
brcici 17863 Prove that two objects are...
cicref 17864 Isomorphism is reflexive. ...
ciclcl 17865 Isomorphism implies the le...
cicrcl 17866 Isomorphism implies the ri...
cicsym 17867 Isomorphism is symmetric. ...
cictr 17868 Isomorphism is transitive....
cicer 17869 Isomorphism is an equivale...
sscrel 17876 The subcategory subset rel...
brssc 17877 The subcategory subset rel...
sscpwex 17878 An analogue of ~ pwex for ...
subcrcl 17879 Reverse closure for the su...
sscfn1 17880 The subcategory subset rel...
sscfn2 17881 The subcategory subset rel...
ssclem 17882 Lemma for ~ ssc1 and simil...
isssc 17883 Value of the subcategory s...
ssc1 17884 Infer subset relation on o...
ssc2 17885 Infer subset relation on m...
sscres 17886 Any function restricted to...
sscid 17887 The subcategory subset rel...
ssctr 17888 The subcategory subset rel...
ssceq 17889 The subcategory subset rel...
rescval 17890 Value of the category rest...
rescval2 17891 Value of the category rest...
rescbas 17892 Base set of the category r...
rescbasOLD 17893 Obsolete version of ~ resc...
reschom 17894 Hom-sets of the category r...
reschomf 17895 Hom-sets of the category r...
rescco 17896 Composition in the categor...
resccoOLD 17897 Obsolete proof of ~ rescco...
rescabs 17898 Restriction absorption law...
rescabsOLD 17899 Obsolete proof of ~ seqp1d...
rescabs2 17900 Restriction absorption law...
issubc 17901 Elementhood in the set of ...
issubc2 17902 Elementhood in the set of ...
0ssc 17903 For any category ` C ` , t...
0subcat 17904 For any category ` C ` , t...
catsubcat 17905 For any category ` C ` , `...
subcssc 17906 An element in the set of s...
subcfn 17907 An element in the set of s...
subcss1 17908 The objects of a subcatego...
subcss2 17909 The morphisms of a subcate...
subcidcl 17910 The identity of the origin...
subccocl 17911 A subcategory is closed un...
subccatid 17912 A subcategory is a categor...
subcid 17913 The identity in a subcateg...
subccat 17914 A subcategory is a categor...
issubc3 17915 Alternate definition of a ...
fullsubc 17916 The full subcategory gener...
fullresc 17917 The category formed by str...
resscat 17918 A category restricted to a...
subsubc 17919 A subcategory of a subcate...
relfunc 17928 The set of functors is a r...
funcrcl 17929 Reverse closure for a func...
isfunc 17930 Value of the set of functo...
isfuncd 17931 Deduce that an operation i...
funcf1 17932 The object part of a funct...
funcixp 17933 The morphism part of a fun...
funcf2 17934 The morphism part of a fun...
funcfn2 17935 The morphism part of a fun...
funcid 17936 A functor maps each identi...
funcco 17937 A functor maps composition...
funcsect 17938 The image of a section und...
funcinv 17939 The image of an inverse un...
funciso 17940 The image of an isomorphis...
funcoppc 17941 A functor on categories yi...
idfuval 17942 Value of the identity func...
idfu2nd 17943 Value of the morphism part...
idfu2 17944 Value of the morphism part...
idfu1st 17945 Value of the object part o...
idfu1 17946 Value of the object part o...
idfucl 17947 The identity functor is a ...
cofuval 17948 Value of the composition o...
cofu1st 17949 Value of the object part o...
cofu1 17950 Value of the object part o...
cofu2nd 17951 Value of the morphism part...
cofu2 17952 Value of the morphism part...
cofuval2 17953 Value of the composition o...
cofucl 17954 The composition of two fun...
cofuass 17955 Functor composition is ass...
cofulid 17956 The identity functor is a ...
cofurid 17957 The identity functor is a ...
resfval 17958 Value of the functor restr...
resfval2 17959 Value of the functor restr...
resf1st 17960 Value of the functor restr...
resf2nd 17961 Value of the functor restr...
funcres 17962 A functor restricted to a ...
funcres2b 17963 Condition for a functor to...
funcres2 17964 A functor into a restricte...
idfusubc0 17965 The identity functor for a...
idfusubc 17966 The identity functor for a...
wunfunc 17967 A weak universe is closed ...
wunfuncOLD 17968 Obsolete proof of ~ wunfun...
funcpropd 17969 If two categories have the...
funcres2c 17970 Condition for a functor to...
fullfunc 17975 A full functor is a functo...
fthfunc 17976 A faithful functor is a fu...
relfull 17977 The set of full functors i...
relfth 17978 The set of faithful functo...
isfull 17979 Value of the set of full f...
isfull2 17980 Equivalent condition for a...
fullfo 17981 The morphism map of a full...
fulli 17982 The morphism map of a full...
isfth 17983 Value of the set of faithf...
isfth2 17984 Equivalent condition for a...
isffth2 17985 A fully faithful functor i...
fthf1 17986 The morphism map of a fait...
fthi 17987 The morphism map of a fait...
ffthf1o 17988 The morphism map of a full...
fullpropd 17989 If two categories have the...
fthpropd 17990 If two categories have the...
fulloppc 17991 The opposite functor of a ...
fthoppc 17992 The opposite functor of a ...
ffthoppc 17993 The opposite functor of a ...
fthsect 17994 A faithful functor reflect...
fthinv 17995 A faithful functor reflect...
fthmon 17996 A faithful functor reflect...
fthepi 17997 A faithful functor reflect...
ffthiso 17998 A fully faithful functor r...
fthres2b 17999 Condition for a faithful f...
fthres2c 18000 Condition for a faithful f...
fthres2 18001 A faithful functor into a ...
idffth 18002 The identity functor is a ...
cofull 18003 The composition of two ful...
cofth 18004 The composition of two fai...
coffth 18005 The composition of two ful...
rescfth 18006 The inclusion functor from...
ressffth 18007 The inclusion functor from...
fullres2c 18008 Condition for a full funct...
ffthres2c 18009 Condition for a fully fait...
inclfusubc 18010 The "inclusion functor" fr...
fnfuc 18015 The ` FuncCat ` operation ...
natfval 18016 Value of the function givi...
isnat 18017 Property of being a natura...
isnat2 18018 Property of being a natura...
natffn 18019 The natural transformation...
natrcl 18020 Reverse closure for a natu...
nat1st2nd 18021 Rewrite the natural transf...
natixp 18022 A natural transformation i...
natcl 18023 A component of a natural t...
natfn 18024 A natural transformation i...
nati 18025 Naturality property of a n...
wunnat 18026 A weak universe is closed ...
wunnatOLD 18027 Obsolete proof of ~ wunnat...
catstr 18028 A category structure is a ...
fucval 18029 Value of the functor categ...
fuccofval 18030 Value of the functor categ...
fucbas 18031 The objects of the functor...
fuchom 18032 The morphisms in the funct...
fuchomOLD 18033 Obsolete proof of ~ fuchom...
fucco 18034 Value of the composition o...
fuccoval 18035 Value of the functor categ...
fuccocl 18036 The composition of two nat...
fucidcl 18037 The identity natural trans...
fuclid 18038 Left identity of natural t...
fucrid 18039 Right identity of natural ...
fucass 18040 Associativity of natural t...
fuccatid 18041 The functor category is a ...
fuccat 18042 The functor category is a ...
fucid 18043 The identity morphism in t...
fucsect 18044 Two natural transformation...
fucinv 18045 Two natural transformation...
invfuc 18046 If ` V ( x ) ` is an inver...
fuciso 18047 A natural transformation i...
natpropd 18048 If two categories have the...
fucpropd 18049 If two categories have the...
initofn 18056 ` InitO ` is a function on...
termofn 18057 ` TermO ` is a function on...
zeroofn 18058 ` ZeroO ` is a function on...
initorcl 18059 Reverse closure for an ini...
termorcl 18060 Reverse closure for a term...
zeroorcl 18061 Reverse closure for a zero...
initoval 18062 The value of the initial o...
termoval 18063 The value of the terminal ...
zerooval 18064 The value of the zero obje...
isinito 18065 The predicate "is an initi...
istermo 18066 The predicate "is a termin...
iszeroo 18067 The predicate "is a zero o...
isinitoi 18068 Implication of a class bei...
istermoi 18069 Implication of a class bei...
initoid 18070 For an initial object, the...
termoid 18071 For a terminal object, the...
dfinito2 18072 An initial object is a ter...
dftermo2 18073 A terminal object is an in...
dfinito3 18074 An alternate definition of...
dftermo3 18075 An alternate definition of...
initoo 18076 An initial object is an ob...
termoo 18077 A terminal object is an ob...
iszeroi 18078 Implication of a class bei...
2initoinv 18079 Morphisms between two init...
initoeu1 18080 Initial objects are essent...
initoeu1w 18081 Initial objects are essent...
initoeu2lem0 18082 Lemma 0 for ~ initoeu2 . ...
initoeu2lem1 18083 Lemma 1 for ~ initoeu2 . ...
initoeu2lem2 18084 Lemma 2 for ~ initoeu2 . ...
initoeu2 18085 Initial objects are essent...
2termoinv 18086 Morphisms between two term...
termoeu1 18087 Terminal objects are essen...
termoeu1w 18088 Terminal objects are essen...
homarcl 18097 Reverse closure for an arr...
homafval 18098 Value of the disjointified...
homaf 18099 Functionality of the disjo...
homaval 18100 Value of the disjointified...
elhoma 18101 Value of the disjointified...
elhomai 18102 Produce an arrow from a mo...
elhomai2 18103 Produce an arrow from a mo...
homarcl2 18104 Reverse closure for the do...
homarel 18105 An arrow is an ordered pai...
homa1 18106 The first component of an ...
homahom2 18107 The second component of an...
homahom 18108 The second component of an...
homadm 18109 The domain of an arrow wit...
homacd 18110 The codomain of an arrow w...
homadmcd 18111 Decompose an arrow into do...
arwval 18112 The set of arrows is the u...
arwrcl 18113 The first component of an ...
arwhoma 18114 An arrow is contained in t...
homarw 18115 A hom-set is a subset of t...
arwdm 18116 The domain of an arrow is ...
arwcd 18117 The codomain of an arrow i...
dmaf 18118 The domain function is a f...
cdaf 18119 The codomain function is a...
arwhom 18120 The second component of an...
arwdmcd 18121 Decompose an arrow into do...
idafval 18126 Value of the identity arro...
idaval 18127 Value of the identity arro...
ida2 18128 Morphism part of the ident...
idahom 18129 Domain and codomain of the...
idadm 18130 Domain of the identity arr...
idacd 18131 Codomain of the identity a...
idaf 18132 The identity arrow functio...
coafval 18133 The value of the compositi...
eldmcoa 18134 A pair ` <. G , F >. ` is ...
dmcoass 18135 The domain of composition ...
homdmcoa 18136 If ` F : X --> Y ` and ` G...
coaval 18137 Value of composition for c...
coa2 18138 The morphism part of arrow...
coahom 18139 The composition of two com...
coapm 18140 Composition of arrows is a...
arwlid 18141 Left identity of a categor...
arwrid 18142 Right identity of a catego...
arwass 18143 Associativity of compositi...
setcval 18146 Value of the category of s...
setcbas 18147 Set of objects of the cate...
setchomfval 18148 Set of arrows of the categ...
setchom 18149 Set of arrows of the categ...
elsetchom 18150 A morphism of sets is a fu...
setccofval 18151 Composition in the categor...
setcco 18152 Composition in the categor...
setccatid 18153 Lemma for ~ setccat . (Co...
setccat 18154 The category of sets is a ...
setcid 18155 The identity arrow in the ...
setcmon 18156 A monomorphism of sets is ...
setcepi 18157 An epimorphism of sets is ...
setcsect 18158 A section in the category ...
setcinv 18159 An inverse in the category...
setciso 18160 An isomorphism in the cate...
resssetc 18161 The restriction of the cat...
funcsetcres2 18162 A functor into a smaller c...
setc2obas 18163 ` (/) ` and ` 1o ` are dis...
setc2ohom 18164 ` ( SetCat `` 2o ) ` is a ...
cat1lem 18165 The category of sets in a ...
cat1 18166 The definition of category...
catcval 18169 Value of the category of c...
catcbas 18170 Set of objects of the cate...
catchomfval 18171 Set of arrows of the categ...
catchom 18172 Set of arrows of the categ...
catccofval 18173 Composition in the categor...
catcco 18174 Composition in the categor...
catccatid 18175 Lemma for ~ catccat . (Co...
catcid 18176 The identity arrow in the ...
catccat 18177 The category of categories...
resscatc 18178 The restriction of the cat...
catcisolem 18179 Lemma for ~ catciso . (Co...
catciso 18180 A functor is an isomorphis...
catcbascl 18181 An element of the base set...
catcslotelcl 18182 A slot entry of an element...
catcbaselcl 18183 The base set of an element...
catchomcl 18184 The Hom-set of an element ...
catcccocl 18185 The composition operation ...
catcoppccl 18186 The category of categories...
catcoppcclOLD 18187 Obsolete proof of ~ catcop...
catcfuccl 18188 The category of categories...
catcfucclOLD 18189 Obsolete proof of ~ catcfu...
fncnvimaeqv 18190 The inverse images of the ...
bascnvimaeqv 18191 The inverse image of the u...
estrcval 18194 Value of the category of e...
estrcbas 18195 Set of objects of the cate...
estrchomfval 18196 Set of morphisms ("arrows"...
estrchom 18197 The morphisms between exte...
elestrchom 18198 A morphism between extensi...
estrccofval 18199 Composition in the categor...
estrcco 18200 Composition in the categor...
estrcbasbas 18201 An element of the base set...
estrccatid 18202 Lemma for ~ estrccat . (C...
estrccat 18203 The category of extensible...
estrcid 18204 The identity arrow in the ...
estrchomfn 18205 The Hom-set operation in t...
estrchomfeqhom 18206 The functionalized Hom-set...
estrreslem1 18207 Lemma 1 for ~ estrres . (...
estrreslem1OLD 18208 Obsolete version of ~ estr...
estrreslem2 18209 Lemma 2 for ~ estrres . (...
estrres 18210 Any restriction of a categ...
funcestrcsetclem1 18211 Lemma 1 for ~ funcestrcset...
funcestrcsetclem2 18212 Lemma 2 for ~ funcestrcset...
funcestrcsetclem3 18213 Lemma 3 for ~ funcestrcset...
funcestrcsetclem4 18214 Lemma 4 for ~ funcestrcset...
funcestrcsetclem5 18215 Lemma 5 for ~ funcestrcset...
funcestrcsetclem6 18216 Lemma 6 for ~ funcestrcset...
funcestrcsetclem7 18217 Lemma 7 for ~ funcestrcset...
funcestrcsetclem8 18218 Lemma 8 for ~ funcestrcset...
funcestrcsetclem9 18219 Lemma 9 for ~ funcestrcset...
funcestrcsetc 18220 The "natural forgetful fun...
fthestrcsetc 18221 The "natural forgetful fun...
fullestrcsetc 18222 The "natural forgetful fun...
equivestrcsetc 18223 The "natural forgetful fun...
setc1strwun 18224 A constructed one-slot str...
funcsetcestrclem1 18225 Lemma 1 for ~ funcsetcestr...
funcsetcestrclem2 18226 Lemma 2 for ~ funcsetcestr...
funcsetcestrclem3 18227 Lemma 3 for ~ funcsetcestr...
embedsetcestrclem 18228 Lemma for ~ embedsetcestrc...
funcsetcestrclem4 18229 Lemma 4 for ~ funcsetcestr...
funcsetcestrclem5 18230 Lemma 5 for ~ funcsetcestr...
funcsetcestrclem6 18231 Lemma 6 for ~ funcsetcestr...
funcsetcestrclem7 18232 Lemma 7 for ~ funcsetcestr...
funcsetcestrclem8 18233 Lemma 8 for ~ funcsetcestr...
funcsetcestrclem9 18234 Lemma 9 for ~ funcsetcestr...
funcsetcestrc 18235 The "embedding functor" fr...
fthsetcestrc 18236 The "embedding functor" fr...
fullsetcestrc 18237 The "embedding functor" fr...
embedsetcestrc 18238 The "embedding functor" fr...
fnxpc 18247 The binary product of cate...
xpcval 18248 Value of the binary produc...
xpcbas 18249 Set of objects of the bina...
xpchomfval 18250 Set of morphisms of the bi...
xpchom 18251 Set of morphisms of the bi...
relxpchom 18252 A hom-set in the binary pr...
xpccofval 18253 Value of composition in th...
xpcco 18254 Value of composition in th...
xpcco1st 18255 Value of composition in th...
xpcco2nd 18256 Value of composition in th...
xpchom2 18257 Value of the set of morphi...
xpcco2 18258 Value of composition in th...
xpccatid 18259 The product of two categor...
xpcid 18260 The identity morphism in t...
xpccat 18261 The product of two categor...
1stfval 18262 Value of the first project...
1stf1 18263 Value of the first project...
1stf2 18264 Value of the first project...
2ndfval 18265 Value of the first project...
2ndf1 18266 Value of the first project...
2ndf2 18267 Value of the first project...
1stfcl 18268 The first projection funct...
2ndfcl 18269 The second projection func...
prfval 18270 Value of the pairing funct...
prf1 18271 Value of the pairing funct...
prf2fval 18272 Value of the pairing funct...
prf2 18273 Value of the pairing funct...
prfcl 18274 The pairing of functors ` ...
prf1st 18275 Cancellation of pairing wi...
prf2nd 18276 Cancellation of pairing wi...
1st2ndprf 18277 Break a functor into a pro...
catcxpccl 18278 The category of categories...
catcxpcclOLD 18279 Obsolete proof of ~ catcxp...
xpcpropd 18280 If two categories have the...
evlfval 18289 Value of the evaluation fu...
evlf2 18290 Value of the evaluation fu...
evlf2val 18291 Value of the evaluation na...
evlf1 18292 Value of the evaluation fu...
evlfcllem 18293 Lemma for ~ evlfcl . (Con...
evlfcl 18294 The evaluation functor is ...
curfval 18295 Value of the curry functor...
curf1fval 18296 Value of the object part o...
curf1 18297 Value of the object part o...
curf11 18298 Value of the double evalua...
curf12 18299 The partially evaluated cu...
curf1cl 18300 The partially evaluated cu...
curf2 18301 Value of the curry functor...
curf2val 18302 Value of a component of th...
curf2cl 18303 The curry functor at a mor...
curfcl 18304 The curry functor of a fun...
curfpropd 18305 If two categories have the...
uncfval 18306 Value of the uncurry funct...
uncfcl 18307 The uncurry operation take...
uncf1 18308 Value of the uncurry funct...
uncf2 18309 Value of the uncurry funct...
curfuncf 18310 Cancellation of curry with...
uncfcurf 18311 Cancellation of uncurry wi...
diagval 18312 Define the diagonal functo...
diagcl 18313 The diagonal functor is a ...
diag1cl 18314 The constant functor of ` ...
diag11 18315 Value of the constant func...
diag12 18316 Value of the constant func...
diag2 18317 Value of the diagonal func...
diag2cl 18318 The diagonal functor at a ...
curf2ndf 18319 As shown in ~ diagval , th...
hofval 18324 Value of the Hom functor, ...
hof1fval 18325 The object part of the Hom...
hof1 18326 The object part of the Hom...
hof2fval 18327 The morphism part of the H...
hof2val 18328 The morphism part of the H...
hof2 18329 The morphism part of the H...
hofcllem 18330 Lemma for ~ hofcl . (Cont...
hofcl 18331 Closure of the Hom functor...
oppchofcl 18332 Closure of the opposite Ho...
yonval 18333 Value of the Yoneda embedd...
yoncl 18334 The Yoneda embedding is a ...
yon1cl 18335 The Yoneda embedding at an...
yon11 18336 Value of the Yoneda embedd...
yon12 18337 Value of the Yoneda embedd...
yon2 18338 Value of the Yoneda embedd...
hofpropd 18339 If two categories have the...
yonpropd 18340 If two categories have the...
oppcyon 18341 Value of the opposite Yone...
oyoncl 18342 The opposite Yoneda embedd...
oyon1cl 18343 The opposite Yoneda embedd...
yonedalem1 18344 Lemma for ~ yoneda . (Con...
yonedalem21 18345 Lemma for ~ yoneda . (Con...
yonedalem3a 18346 Lemma for ~ yoneda . (Con...
yonedalem4a 18347 Lemma for ~ yoneda . (Con...
yonedalem4b 18348 Lemma for ~ yoneda . (Con...
yonedalem4c 18349 Lemma for ~ yoneda . (Con...
yonedalem22 18350 Lemma for ~ yoneda . (Con...
yonedalem3b 18351 Lemma for ~ yoneda . (Con...
yonedalem3 18352 Lemma for ~ yoneda . (Con...
yonedainv 18353 The Yoneda Lemma with expl...
yonffthlem 18354 Lemma for ~ yonffth . (Co...
yoneda 18355 The Yoneda Lemma. There i...
yonffth 18356 The Yoneda Lemma. The Yon...
yoniso 18357 If the codomain is recover...
oduval 18360 Value of an order dual str...
oduleval 18361 Value of the less-equal re...
oduleg 18362 Truth of the less-equal re...
odubas 18363 Base set of an order dual ...
odubasOLD 18364 Obsolete proof of ~ odubas...
isprs 18369 Property of being a preord...
prslem 18370 Lemma for ~ prsref and ~ p...
prsref 18371 "Less than or equal to" is...
prstr 18372 "Less than or equal to" is...
isdrs 18373 Property of being a direct...
drsdir 18374 Direction of a directed se...
drsprs 18375 A directed set is a proset...
drsbn0 18376 The base of a directed set...
drsdirfi 18377 Any _finite_ number of ele...
isdrs2 18378 Directed sets may be defin...
ispos 18386 The predicate "is a poset"...
ispos2 18387 A poset is an antisymmetri...
posprs 18388 A poset is a proset. (Con...
posi 18389 Lemma for poset properties...
posref 18390 A poset ordering is reflex...
posasymb 18391 A poset ordering is asymme...
postr 18392 A poset ordering is transi...
0pos 18393 Technical lemma to simplif...
0posOLD 18394 Obsolete proof of ~ 0pos a...
isposd 18395 Properties that determine ...
isposi 18396 Properties that determine ...
isposix 18397 Properties that determine ...
isposixOLD 18398 Obsolete proof of ~ isposi...
pospropd 18399 Posethood is determined on...
odupos 18400 Being a poset is a self-du...
oduposb 18401 Being a poset is a self-du...
pltfval 18403 Value of the less-than rel...
pltval 18404 Less-than relation. ( ~ d...
pltle 18405 "Less than" implies "less ...
pltne 18406 The "less than" relation i...
pltirr 18407 The "less than" relation i...
pleval2i 18408 One direction of ~ pleval2...
pleval2 18409 "Less than or equal to" in...
pltnle 18410 "Less than" implies not co...
pltval3 18411 Alternate expression for t...
pltnlt 18412 The less-than relation imp...
pltn2lp 18413 The less-than relation has...
plttr 18414 The less-than relation is ...
pltletr 18415 Transitive law for chained...
plelttr 18416 Transitive law for chained...
pospo 18417 Write a poset structure in...
lubfval 18422 Value of the least upper b...
lubdm 18423 Domain of the least upper ...
lubfun 18424 The LUB is a function. (C...
lubeldm 18425 Member of the domain of th...
lubelss 18426 A member of the domain of ...
lubeu 18427 Unique existence proper of...
lubval 18428 Value of the least upper b...
lubcl 18429 The least upper bound func...
lubprop 18430 Properties of greatest low...
luble 18431 The greatest lower bound i...
lublecllem 18432 Lemma for ~ lublecl and ~ ...
lublecl 18433 The set of all elements le...
lubid 18434 The LUB of elements less t...
glbfval 18435 Value of the greatest lowe...
glbdm 18436 Domain of the greatest low...
glbfun 18437 The GLB is a function. (C...
glbeldm 18438 Member of the domain of th...
glbelss 18439 A member of the domain of ...
glbeu 18440 Unique existence proper of...
glbval 18441 Value of the greatest lowe...
glbcl 18442 The least upper bound func...
glbprop 18443 Properties of greatest low...
glble 18444 The greatest lower bound i...
joinfval 18445 Value of join function for...
joinfval2 18446 Value of join function for...
joindm 18447 Domain of join function fo...
joindef 18448 Two ways to say that a joi...
joinval 18449 Join value. Since both si...
joincl 18450 Closure of join of element...
joindmss 18451 Subset property of domain ...
joinval2lem 18452 Lemma for ~ joinval2 and ~...
joinval2 18453 Value of join for a poset ...
joineu 18454 Uniqueness of join of elem...
joinlem 18455 Lemma for join properties....
lejoin1 18456 A join's first argument is...
lejoin2 18457 A join's second argument i...
joinle 18458 A join is less than or equ...
meetfval 18459 Value of meet function for...
meetfval2 18460 Value of meet function for...
meetdm 18461 Domain of meet function fo...
meetdef 18462 Two ways to say that a mee...
meetval 18463 Meet value. Since both si...
meetcl 18464 Closure of meet of element...
meetdmss 18465 Subset property of domain ...
meetval2lem 18466 Lemma for ~ meetval2 and ~...
meetval2 18467 Value of meet for a poset ...
meeteu 18468 Uniqueness of meet of elem...
meetlem 18469 Lemma for meet properties....
lemeet1 18470 A meet's first argument is...
lemeet2 18471 A meet's second argument i...
meetle 18472 A meet is less than or equ...
joincomALT 18473 The join of a poset is com...
joincom 18474 The join of a poset is com...
meetcomALT 18475 The meet of a poset is com...
meetcom 18476 The meet of a poset is com...
join0 18477 Lemma for ~ odumeet . (Co...
meet0 18478 Lemma for ~ odujoin . (Co...
odulub 18479 Least upper bounds in a du...
odujoin 18480 Joins in a dual order are ...
oduglb 18481 Greatest lower bounds in a...
odumeet 18482 Meets in a dual order are ...
poslubmo 18483 Least upper bounds in a po...
posglbmo 18484 Greatest lower bounds in a...
poslubd 18485 Properties which determine...
poslubdg 18486 Properties which determine...
posglbdg 18487 Properties which determine...
istos 18490 The predicate "is a toset"...
tosso 18491 Write the totally ordered ...
tospos 18492 A Toset is a Poset. (Cont...
tleile 18493 In a Toset, any two elemen...
tltnle 18494 In a Toset, "less than" is...
p0val 18499 Value of poset zero. (Con...
p1val 18500 Value of poset zero. (Con...
p0le 18501 Any element is less than o...
ple1 18502 Any element is less than o...
islat 18505 The predicate "is a lattic...
odulatb 18506 Being a lattice is self-du...
odulat 18507 Being a lattice is self-du...
latcl2 18508 The join and meet of any t...
latlem 18509 Lemma for lattice properti...
latpos 18510 A lattice is a poset. (Co...
latjcl 18511 Closure of join operation ...
latmcl 18512 Closure of meet operation ...
latref 18513 A lattice ordering is refl...
latasymb 18514 A lattice ordering is asym...
latasym 18515 A lattice ordering is asym...
lattr 18516 A lattice ordering is tran...
latasymd 18517 Deduce equality from latti...
lattrd 18518 A lattice ordering is tran...
latjcom 18519 The join of a lattice comm...
latlej1 18520 A join's first argument is...
latlej2 18521 A join's second argument i...
latjle12 18522 A join is less than or equ...
latleeqj1 18523 "Less than or equal to" in...
latleeqj2 18524 "Less than or equal to" in...
latjlej1 18525 Add join to both sides of ...
latjlej2 18526 Add join to both sides of ...
latjlej12 18527 Add join to both sides of ...
latnlej 18528 An idiom to express that a...
latnlej1l 18529 An idiom to express that a...
latnlej1r 18530 An idiom to express that a...
latnlej2 18531 An idiom to express that a...
latnlej2l 18532 An idiom to express that a...
latnlej2r 18533 An idiom to express that a...
latjidm 18534 Lattice join is idempotent...
latmcom 18535 The join of a lattice comm...
latmle1 18536 A meet is less than or equ...
latmle2 18537 A meet is less than or equ...
latlem12 18538 An element is less than or...
latleeqm1 18539 "Less than or equal to" in...
latleeqm2 18540 "Less than or equal to" in...
latmlem1 18541 Add meet to both sides of ...
latmlem2 18542 Add meet to both sides of ...
latmlem12 18543 Add join to both sides of ...
latnlemlt 18544 Negation of "less than or ...
latnle 18545 Equivalent expressions for...
latmidm 18546 Lattice meet is idempotent...
latabs1 18547 Lattice absorption law. F...
latabs2 18548 Lattice absorption law. F...
latledi 18549 An ortholattice is distrib...
latmlej11 18550 Ordering of a meet and joi...
latmlej12 18551 Ordering of a meet and joi...
latmlej21 18552 Ordering of a meet and joi...
latmlej22 18553 Ordering of a meet and joi...
lubsn 18554 The least upper bound of a...
latjass 18555 Lattice join is associativ...
latj12 18556 Swap 1st and 2nd members o...
latj32 18557 Swap 2nd and 3rd members o...
latj13 18558 Swap 1st and 3rd members o...
latj31 18559 Swap 2nd and 3rd members o...
latjrot 18560 Rotate lattice join of 3 c...
latj4 18561 Rearrangement of lattice j...
latj4rot 18562 Rotate lattice join of 4 c...
latjjdi 18563 Lattice join distributes o...
latjjdir 18564 Lattice join distributes o...
mod1ile 18565 The weak direction of the ...
mod2ile 18566 The weak direction of the ...
latmass 18567 Lattice meet is associativ...
latdisdlem 18568 Lemma for ~ latdisd . (Co...
latdisd 18569 In a lattice, joins distri...
isclat 18572 The predicate "is a comple...
clatpos 18573 A complete lattice is a po...
clatlem 18574 Lemma for properties of a ...
clatlubcl 18575 Any subset of the base set...
clatlubcl2 18576 Any subset of the base set...
clatglbcl 18577 Any subset of the base set...
clatglbcl2 18578 Any subset of the base set...
oduclatb 18579 Being a complete lattice i...
clatl 18580 A complete lattice is a la...
isglbd 18581 Properties that determine ...
lublem 18582 Lemma for the least upper ...
lubub 18583 The LUB of a complete latt...
lubl 18584 The LUB of a complete latt...
lubss 18585 Subset law for least upper...
lubel 18586 An element of a set is les...
lubun 18587 The LUB of a union. (Cont...
clatglb 18588 Properties of greatest low...
clatglble 18589 The greatest lower bound i...
clatleglb 18590 Two ways of expressing "le...
clatglbss 18591 Subset law for greatest lo...
isdlat 18594 Property of being a distri...
dlatmjdi 18595 In a distributive lattice,...
dlatl 18596 A distributive lattice is ...
odudlatb 18597 The dual of a distributive...
dlatjmdi 18598 In a distributive lattice,...
ipostr 18601 The structure of ~ df-ipo ...
ipoval 18602 Value of the inclusion pos...
ipobas 18603 Base set of the inclusion ...
ipolerval 18604 Relation of the inclusion ...
ipotset 18605 Topology of the inclusion ...
ipole 18606 Weak order condition of th...
ipolt 18607 Strict order condition of ...
ipopos 18608 The inclusion poset on a f...
isipodrs 18609 Condition for a family of ...
ipodrscl 18610 Direction by inclusion as ...
ipodrsfi 18611 Finite upper bound propert...
fpwipodrs 18612 The finite subsets of any ...
ipodrsima 18613 The monotone image of a di...
isacs3lem 18614 An algebraic closure syste...
acsdrsel 18615 An algebraic closure syste...
isacs4lem 18616 In a closure system in whi...
isacs5lem 18617 If closure commutes with d...
acsdrscl 18618 In an algebraic closure sy...
acsficl 18619 A closure in an algebraic ...
isacs5 18620 A closure system is algebr...
isacs4 18621 A closure system is algebr...
isacs3 18622 A closure system is algebr...
acsficld 18623 In an algebraic closure sy...
acsficl2d 18624 In an algebraic closure sy...
acsfiindd 18625 In an algebraic closure sy...
acsmapd 18626 In an algebraic closure sy...
acsmap2d 18627 In an algebraic closure sy...
acsinfd 18628 In an algebraic closure sy...
acsdomd 18629 In an algebraic closure sy...
acsinfdimd 18630 In an algebraic closure sy...
acsexdimd 18631 In an algebraic closure sy...
mrelatglb 18632 Greatest lower bounds in a...
mrelatglb0 18633 The empty intersection in ...
mrelatlub 18634 Least upper bounds in a Mo...
mreclatBAD 18635 A Moore space is a complet...
isps 18640 The predicate "is a poset"...
psrel 18641 A poset is a relation. (C...
psref2 18642 A poset is antisymmetric a...
pstr2 18643 A poset is transitive. (C...
pslem 18644 Lemma for ~ psref and othe...
psdmrn 18645 The domain and range of a ...
psref 18646 A poset is reflexive. (Co...
psrn 18647 The range of a poset equal...
psasym 18648 A poset is antisymmetric. ...
pstr 18649 A poset is transitive. (C...
cnvps 18650 The converse of a poset is...
cnvpsb 18651 The converse of a poset is...
psss 18652 Any subset of a partially ...
psssdm2 18653 Field of a subposet. (Con...
psssdm 18654 Field of a subposet. (Con...
istsr 18655 The predicate is a toset. ...
istsr2 18656 The predicate is a toset. ...
tsrlin 18657 A toset is a linear order....
tsrlemax 18658 Two ways of saying a numbe...
tsrps 18659 A toset is a poset. (Cont...
cnvtsr 18660 The converse of a toset is...
tsrss 18661 Any subset of a totally or...
ledm 18662 The domain of ` <_ ` is ` ...
lern 18663 The range of ` <_ ` is ` R...
lefld 18664 The field of the 'less or ...
letsr 18665 The "less than or equal to...
isdir 18670 A condition for a relation...
reldir 18671 A direction is a relation....
dirdm 18672 A direction's domain is eq...
dirref 18673 A direction is reflexive. ...
dirtr 18674 A direction is transitive....
dirge 18675 For any two elements of a ...
tsrdir 18676 A totally ordered set is a...
ismgm 18681 The predicate "is a magma"...
ismgmn0 18682 The predicate "is a magma"...
mgmcl 18683 Closure of the operation o...
isnmgm 18684 A condition for a structur...
mgmsscl 18685 If the base set of a magma...
plusffval 18686 The group addition operati...
plusfval 18687 The group addition operati...
plusfeq 18688 If the addition operation ...
plusffn 18689 The group addition operati...
mgmplusf 18690 The group addition functio...
mgmpropd 18691 If two structures have the...
ismgmd 18692 Deduce a magma from its pr...
issstrmgm 18693 Characterize a substructur...
intopsn 18694 The internal operation for...
mgmb1mgm1 18695 The only magma with a base...
mgm0 18696 Any set with an empty base...
mgm0b 18697 The structure with an empt...
mgm1 18698 The structure with one ele...
opifismgm 18699 A structure with a group a...
mgmidmo 18700 A two-sided identity eleme...
grpidval 18701 The value of the identity ...
grpidpropd 18702 If two structures have the...
fn0g 18703 The group zero extractor i...
0g0 18704 The identity element funct...
ismgmid 18705 The identity element of a ...
mgmidcl 18706 The identity element of a ...
mgmlrid 18707 The identity element of a ...
ismgmid2 18708 Show that a given element ...
lidrideqd 18709 If there is a left and rig...
lidrididd 18710 If there is a left and rig...
grpidd 18711 Deduce the identity elemen...
mgmidsssn0 18712 Property of the set of ide...
grpinvalem 18713 Lemma for ~ grpinva . (Co...
grpinva 18714 Deduce right inverse from ...
grprida 18715 Deduce right identity from...
gsumvalx 18716 Expand out the substitutio...
gsumval 18717 Expand out the substitutio...
gsumpropd 18718 The group sum depends only...
gsumpropd2lem 18719 Lemma for ~ gsumpropd2 . ...
gsumpropd2 18720 A stronger version of ~ gs...
gsummgmpropd 18721 A stronger version of ~ gs...
gsumress 18722 The group sum in a substru...
gsumval1 18723 Value of the group sum ope...
gsum0 18724 Value of the empty group s...
gsumval2a 18725 Value of the group sum ope...
gsumval2 18726 Value of the group sum ope...
gsumsplit1r 18727 Splitting off the rightmos...
gsumprval 18728 Value of the group sum ope...
gsumpr12val 18729 Value of the group sum ope...
mgmhmrcl 18734 Reverse closure of a magma...
submgmrcl 18735 Reverse closure for submag...
ismgmhm 18736 Property of a magma homomo...
mgmhmf 18737 A magma homomorphism is a ...
mgmhmpropd 18738 Magma homomorphism depends...
mgmhmlin 18739 A magma homomorphism prese...
mgmhmf1o 18740 A magma homomorphism is bi...
idmgmhm 18741 The identity homomorphism ...
issubmgm 18742 Expand definition of a sub...
issubmgm2 18743 Submagmas are subsets that...
rabsubmgmd 18744 Deduction for proving that...
submgmss 18745 Submagmas are subsets of t...
submgmid 18746 Every magma is trivially a...
submgmcl 18747 Submagmas are closed under...
submgmmgm 18748 Submagmas are themselves m...
submgmbas 18749 The base set of a submagma...
subsubmgm 18750 A submagma of a submagma i...
resmgmhm 18751 Restriction of a magma hom...
resmgmhm2 18752 One direction of ~ resmgmh...
resmgmhm2b 18753 Restriction of the codomai...
mgmhmco 18754 The composition of magma h...
mgmhmima 18755 The homomorphic image of a...
mgmhmeql 18756 The equalizer of two magma...
submgmacs 18757 Submagmas are an algebraic...
issgrp 18760 The predicate "is a semigr...
issgrpv 18761 The predicate "is a semigr...
issgrpn0 18762 The predicate "is a semigr...
isnsgrp 18763 A condition for a structur...
sgrpmgm 18764 A semigroup is a magma. (...
sgrpass 18765 A semigroup operation is a...
sgrpcl 18766 Closure of the operation o...
sgrp0 18767 Any set with an empty base...
sgrp0b 18768 The structure with an empt...
sgrp1 18769 The structure with one ele...
issgrpd 18770 Deduce a semigroup from it...
sgrppropd 18771 If two structures are sets...
prdsplusgsgrpcl 18772 Structure product pointwis...
prdssgrpd 18773 The product of a family of...
ismnddef 18776 The predicate "is a monoid...
ismnd 18777 The predicate "is a monoid...
isnmnd 18778 A condition for a structur...
sgrpidmnd 18779 A semigroup with an identi...
mndsgrp 18780 A monoid is a semigroup. ...
mndmgm 18781 A monoid is a magma. (Con...
mndcl 18782 Closure of the operation o...
mndass 18783 A monoid operation is asso...
mndid 18784 A monoid has a two-sided i...
mndideu 18785 The two-sided identity ele...
mnd32g 18786 Commutative/associative la...
mnd12g 18787 Commutative/associative la...
mnd4g 18788 Commutative/associative la...
mndidcl 18789 The identity element of a ...
mndbn0 18790 The base set of a monoid i...
hashfinmndnn 18791 A finite monoid has positi...
mndplusf 18792 The group addition operati...
mndlrid 18793 A monoid's identity elemen...
mndlid 18794 The identity element of a ...
mndrid 18795 The identity element of a ...
ismndd 18796 Deduce a monoid from its p...
mndpfo 18797 The addition operation of ...
mndfo 18798 The addition operation of ...
mndpropd 18799 If two structures have the...
mndprop 18800 If two structures have the...
issubmnd 18801 Characterize a submonoid b...
ress0g 18802 ` 0g ` is unaffected by re...
submnd0 18803 The zero of a submonoid is...
mndinvmod 18804 Uniqueness of an inverse e...
prdsplusgcl 18805 Structure product pointwis...
prdsidlem 18806 Characterization of identi...
prdsmndd 18807 The product of a family of...
prds0g 18808 Zero in a product of monoi...
pwsmnd 18809 The structure power of a m...
pws0g 18810 Zero in a structure power ...
imasmnd2 18811 The image structure of a m...
imasmnd 18812 The image structure of a m...
imasmndf1 18813 The image of a monoid unde...
xpsmnd 18814 The binary product of mono...
xpsmnd0 18815 The identity element of a ...
mnd1 18816 The (smallest) structure r...
mnd1id 18817 The singleton element of a...
ismhm 18822 Property of a monoid homom...
ismhmd 18823 Deduction version of ~ ism...
mhmrcl1 18824 Reverse closure of a monoi...
mhmrcl2 18825 Reverse closure of a monoi...
mhmf 18826 A monoid homomorphism is a...
ismhm0 18827 Property of a monoid homom...
mhmismgmhm 18828 Each monoid homomorphism i...
mhmpropd 18829 Monoid homomorphism depend...
mhmlin 18830 A monoid homomorphism comm...
mhm0 18831 A monoid homomorphism pres...
idmhm 18832 The identity homomorphism ...
mhmf1o 18833 A monoid homomorphism is b...
mndvcl 18834 Tuple-wise additive closur...
mndvass 18835 Tuple-wise associativity i...
mndvlid 18836 Tuple-wise left identity i...
mndvrid 18837 Tuple-wise right identity ...
mhmvlin 18838 Tuple extension of monoid ...
submrcl 18839 Reverse closure for submon...
issubm 18840 Expand definition of a sub...
issubm2 18841 Submonoids are subsets tha...
issubmndb 18842 The submonoid predicate. ...
issubmd 18843 Deduction for proving a su...
mndissubm 18844 If the base set of a monoi...
resmndismnd 18845 If the base set of a monoi...
submss 18846 Submonoids are subsets of ...
submid 18847 Every monoid is trivially ...
subm0cl 18848 Submonoids contain zero. ...
submcl 18849 Submonoids are closed unde...
submmnd 18850 Submonoids are themselves ...
submbas 18851 The base set of a submonoi...
subm0 18852 Submonoids have the same i...
subsubm 18853 A submonoid of a submonoid...
0subm 18854 The zero submonoid of an a...
insubm 18855 The intersection of two su...
0mhm 18856 The constant zero linear f...
resmhm 18857 Restriction of a monoid ho...
resmhm2 18858 One direction of ~ resmhm2...
resmhm2b 18859 Restriction of the codomai...
mhmco 18860 The composition of monoid ...
mhmimalem 18861 Lemma for ~ mhmima and sim...
mhmima 18862 The homomorphic image of a...
mhmeql 18863 The equalizer of two monoi...
submacs 18864 Submonoids are an algebrai...
mndind 18865 Induction in a monoid. In...
prdspjmhm 18866 A projection from a produc...
pwspjmhm 18867 A projection from a struct...
pwsdiagmhm 18868 Diagonal monoid homomorphi...
pwsco1mhm 18869 Right composition with a f...
pwsco2mhm 18870 Left composition with a mo...
gsumvallem2 18871 Lemma for properties of th...
gsumsubm 18872 Evaluate a group sum in a ...
gsumz 18873 Value of a group sum over ...
gsumwsubmcl 18874 Closure of the composite i...
gsumws1 18875 A singleton composite reco...
gsumwcl 18876 Closure of the composite o...
gsumsgrpccat 18877 Homomorphic property of no...
gsumccat 18878 Homomorphic property of co...
gsumws2 18879 Valuation of a pair in a m...
gsumccatsn 18880 Homomorphic property of co...
gsumspl 18881 The primary purpose of the...
gsumwmhm 18882 Behavior of homomorphisms ...
gsumwspan 18883 The submonoid generated by...
frmdval 18888 Value of the free monoid c...
frmdbas 18889 The base set of a free mon...
frmdelbas 18890 An element of the base set...
frmdplusg 18891 The monoid operation of a ...
frmdadd 18892 Value of the monoid operat...
vrmdfval 18893 The canonical injection fr...
vrmdval 18894 The value of the generatin...
vrmdf 18895 The mapping from the index...
frmdmnd 18896 A free monoid is a monoid....
frmd0 18897 The identity of the free m...
frmdsssubm 18898 The set of words taking va...
frmdgsum 18899 Any word in a free monoid ...
frmdss2 18900 A subset of generators is ...
frmdup1 18901 Any assignment of the gene...
frmdup2 18902 The evaluation map has the...
frmdup3lem 18903 Lemma for ~ frmdup3 . (Co...
frmdup3 18904 Universal property of the ...
efmnd 18907 The monoid of endofunction...
efmndbas 18908 The base set of the monoid...
efmndbasabf 18909 The base set of the monoid...
elefmndbas 18910 Two ways of saying a funct...
elefmndbas2 18911 Two ways of saying a funct...
efmndbasf 18912 Elements in the monoid of ...
efmndhash 18913 The monoid of endofunction...
efmndbasfi 18914 The monoid of endofunction...
efmndfv 18915 The function value of an e...
efmndtset 18916 The topology of the monoid...
efmndplusg 18917 The group operation of a m...
efmndov 18918 The value of the group ope...
efmndcl 18919 The group operation of the...
efmndtopn 18920 The topology of the monoid...
symggrplem 18921 Lemma for ~ symggrp and ~ ...
efmndmgm 18922 The monoid of endofunction...
efmndsgrp 18923 The monoid of endofunction...
ielefmnd 18924 The identity function rest...
efmndid 18925 The identity function rest...
efmndmnd 18926 The monoid of endofunction...
efmnd0nmnd 18927 Even the monoid of endofun...
efmndbas0 18928 The base set of the monoid...
efmnd1hash 18929 The monoid of endofunction...
efmnd1bas 18930 The monoid of endofunction...
efmnd2hash 18931 The monoid of endofunction...
submefmnd 18932 If the base set of a monoi...
sursubmefmnd 18933 The set of surjective endo...
injsubmefmnd 18934 The set of injective endof...
idressubmefmnd 18935 The singleton containing o...
idresefmnd 18936 The structure with the sin...
smndex1ibas 18937 The modulo function ` I ` ...
smndex1iidm 18938 The modulo function ` I ` ...
smndex1gbas 18939 The constant functions ` (...
smndex1gid 18940 The composition of a const...
smndex1igid 18941 The composition of the mod...
smndex1basss 18942 The modulo function ` I ` ...
smndex1bas 18943 The base set of the monoid...
smndex1mgm 18944 The monoid of endofunction...
smndex1sgrp 18945 The monoid of endofunction...
smndex1mndlem 18946 Lemma for ~ smndex1mnd and...
smndex1mnd 18947 The monoid of endofunction...
smndex1id 18948 The modulo function ` I ` ...
smndex1n0mnd 18949 The identity of the monoid...
nsmndex1 18950 The base set ` B ` of the ...
smndex2dbas 18951 The doubling function ` D ...
smndex2dnrinv 18952 The doubling function ` D ...
smndex2hbas 18953 The halving functions ` H ...
smndex2dlinvh 18954 The halving functions ` H ...
mgm2nsgrplem1 18955 Lemma 1 for ~ mgm2nsgrp : ...
mgm2nsgrplem2 18956 Lemma 2 for ~ mgm2nsgrp . ...
mgm2nsgrplem3 18957 Lemma 3 for ~ mgm2nsgrp . ...
mgm2nsgrplem4 18958 Lemma 4 for ~ mgm2nsgrp : ...
mgm2nsgrp 18959 A small magma (with two el...
sgrp2nmndlem1 18960 Lemma 1 for ~ sgrp2nmnd : ...
sgrp2nmndlem2 18961 Lemma 2 for ~ sgrp2nmnd . ...
sgrp2nmndlem3 18962 Lemma 3 for ~ sgrp2nmnd . ...
sgrp2rid2 18963 A small semigroup (with tw...
sgrp2rid2ex 18964 A small semigroup (with tw...
sgrp2nmndlem4 18965 Lemma 4 for ~ sgrp2nmnd : ...
sgrp2nmndlem5 18966 Lemma 5 for ~ sgrp2nmnd : ...
sgrp2nmnd 18967 A small semigroup (with tw...
mgmnsgrpex 18968 There is a magma which is ...
sgrpnmndex 18969 There is a semigroup which...
sgrpssmgm 18970 The class of all semigroup...
mndsssgrp 18971 The class of all monoids i...
pwmndgplus 18972 The operation of the monoi...
pwmndid 18973 The identity of the monoid...
pwmnd 18974 The power set of a class `...
isgrp 18981 The predicate "is a group"...
grpmnd 18982 A group is a monoid. (Con...
grpcl 18983 Closure of the operation o...
grpass 18984 A group operation is assoc...
grpinvex 18985 Every member of a group ha...
grpideu 18986 The two-sided identity ele...
grpassd 18987 A group operation is assoc...
grpmndd 18988 A group is a monoid. (Con...
grpcld 18989 Closure of the operation o...
grpplusf 18990 The group addition operati...
grpplusfo 18991 The group addition operati...
resgrpplusfrn 18992 The underlying set of a gr...
grppropd 18993 If two structures have the...
grpprop 18994 If two structures have the...
grppropstr 18995 Generalize a specific 2-el...
grpss 18996 Show that a structure exte...
isgrpd2e 18997 Deduce a group from its pr...
isgrpd2 18998 Deduce a group from its pr...
isgrpde 18999 Deduce a group from its pr...
isgrpd 19000 Deduce a group from its pr...
isgrpi 19001 Properties that determine ...
grpsgrp 19002 A group is a semigroup. (...
grpmgmd 19003 A group is a magma, deduct...
dfgrp2 19004 Alternate definition of a ...
dfgrp2e 19005 Alternate definition of a ...
isgrpix 19006 Properties that determine ...
grpidcl 19007 The identity element of a ...
grpbn0 19008 The base set of a group is...
grplid 19009 The identity element of a ...
grprid 19010 The identity element of a ...
grplidd 19011 The identity element of a ...
grpridd 19012 The identity element of a ...
grpn0 19013 A group is not empty. (Co...
hashfingrpnn 19014 A finite group has positiv...
grprcan 19015 Right cancellation law for...
grpinveu 19016 The left inverse element o...
grpid 19017 Two ways of saying that an...
isgrpid2 19018 Properties showing that an...
grpidd2 19019 Deduce the identity elemen...
grpinvfval 19020 The inverse function of a ...
grpinvfvalALT 19021 Shorter proof of ~ grpinvf...
grpinvval 19022 The inverse of a group ele...
grpinvfn 19023 Functionality of the group...
grpinvfvi 19024 The group inverse function...
grpsubfval 19025 Group subtraction (divisio...
grpsubfvalALT 19026 Shorter proof of ~ grpsubf...
grpsubval 19027 Group subtraction (divisio...
grpinvf 19028 The group inversion operat...
grpinvcl 19029 A group element's inverse ...
grpinvcld 19030 A group element's inverse ...
grplinv 19031 The left inverse of a grou...
grprinv 19032 The right inverse of a gro...
grpinvid1 19033 The inverse of a group ele...
grpinvid2 19034 The inverse of a group ele...
isgrpinv 19035 Properties showing that a ...
grplinvd 19036 The left inverse of a grou...
grprinvd 19037 The right inverse of a gro...
grplrinv 19038 In a group, every member h...
grpidinv2 19039 A group's properties using...
grpidinv 19040 A group has a left and rig...
grpinvid 19041 The inverse of the identit...
grplcan 19042 Left cancellation law for ...
grpasscan1 19043 An associative cancellatio...
grpasscan2 19044 An associative cancellatio...
grpidrcan 19045 If right adding an element...
grpidlcan 19046 If left adding an element ...
grpinvinv 19047 Double inverse law for gro...
grpinvcnv 19048 The group inverse is its o...
grpinv11 19049 The group inverse is one-t...
grpinv11OLD 19050 Obsolete version of ~ grpi...
grpinvf1o 19051 The group inverse is a one...
grpinvnz 19052 The inverse of a nonzero g...
grpinvnzcl 19053 The inverse of a nonzero g...
grpsubinv 19054 Subtraction of an inverse....
grplmulf1o 19055 Left multiplication by a g...
grpraddf1o 19056 Right addition by a group ...
grpinvpropd 19057 If two structures have the...
grpidssd 19058 If the base set of a group...
grpinvssd 19059 If the base set of a group...
grpinvadd 19060 The inverse of the group o...
grpsubf 19061 Functionality of group sub...
grpsubcl 19062 Closure of group subtracti...
grpsubrcan 19063 Right cancellation law for...
grpinvsub 19064 Inverse of a group subtrac...
grpinvval2 19065 A ~ df-neg -like equation ...
grpsubid 19066 Subtraction of a group ele...
grpsubid1 19067 Subtraction of the identit...
grpsubeq0 19068 If the difference between ...
grpsubadd0sub 19069 Subtraction expressed as a...
grpsubadd 19070 Relationship between group...
grpsubsub 19071 Double group subtraction. ...
grpaddsubass 19072 Associative-type law for g...
grppncan 19073 Cancellation law for subtr...
grpnpcan 19074 Cancellation law for subtr...
grpsubsub4 19075 Double group subtraction (...
grppnpcan2 19076 Cancellation law for mixed...
grpnpncan 19077 Cancellation law for group...
grpnpncan0 19078 Cancellation law for group...
grpnnncan2 19079 Cancellation law for group...
dfgrp3lem 19080 Lemma for ~ dfgrp3 . (Con...
dfgrp3 19081 Alternate definition of a ...
dfgrp3e 19082 Alternate definition of a ...
grplactfval 19083 The left group action of e...
grplactval 19084 The value of the left grou...
grplactcnv 19085 The left group action of e...
grplactf1o 19086 The left group action of e...
grpsubpropd 19087 Weak property deduction fo...
grpsubpropd2 19088 Strong property deduction ...
grp1 19089 The (smallest) structure r...
grp1inv 19090 The inverse function of th...
prdsinvlem 19091 Characterization of invers...
prdsgrpd 19092 The product of a family of...
prdsinvgd 19093 Negation in a product of g...
pwsgrp 19094 A structure power of a gro...
pwsinvg 19095 Negation in a group power....
pwssub 19096 Subtraction in a group pow...
imasgrp2 19097 The image structure of a g...
imasgrp 19098 The image structure of a g...
imasgrpf1 19099 The image of a group under...
qusgrp2 19100 Prove that a quotient stru...
xpsgrp 19101 The binary product of grou...
xpsinv 19102 Value of the negation oper...
xpsgrpsub 19103 Value of the subtraction o...
mhmlem 19104 Lemma for ~ mhmmnd and ~ g...
mhmid 19105 A surjective monoid morphi...
mhmmnd 19106 The image of a monoid ` G ...
mhmfmhm 19107 The function fulfilling th...
ghmgrp 19108 The image of a group ` G `...
mulgfval 19111 Group multiple (exponentia...
mulgfvalALT 19112 Shorter proof of ~ mulgfva...
mulgval 19113 Value of the group multipl...
mulgfn 19114 Functionality of the group...
mulgfvi 19115 The group multiple operati...
mulg0 19116 Group multiple (exponentia...
mulgnn 19117 Group multiple (exponentia...
ressmulgnn 19118 Values for the group multi...
ressmulgnn0 19119 Values for the group multi...
ressmulgnnd 19120 Values for the group multi...
mulgnngsum 19121 Group multiple (exponentia...
mulgnn0gsum 19122 Group multiple (exponentia...
mulg1 19123 Group multiple (exponentia...
mulgnnp1 19124 Group multiple (exponentia...
mulg2 19125 Group multiple (exponentia...
mulgnegnn 19126 Group multiple (exponentia...
mulgnn0p1 19127 Group multiple (exponentia...
mulgnnsubcl 19128 Closure of the group multi...
mulgnn0subcl 19129 Closure of the group multi...
mulgsubcl 19130 Closure of the group multi...
mulgnncl 19131 Closure of the group multi...
mulgnn0cl 19132 Closure of the group multi...
mulgcl 19133 Closure of the group multi...
mulgneg 19134 Group multiple (exponentia...
mulgnegneg 19135 The inverse of a negative ...
mulgm1 19136 Group multiple (exponentia...
mulgnn0cld 19137 Closure of the group multi...
mulgcld 19138 Deduction associated with ...
mulgaddcomlem 19139 Lemma for ~ mulgaddcom . ...
mulgaddcom 19140 The group multiple operato...
mulginvcom 19141 The group multiple operato...
mulginvinv 19142 The group multiple operato...
mulgnn0z 19143 A group multiple of the id...
mulgz 19144 A group multiple of the id...
mulgnndir 19145 Sum of group multiples, fo...
mulgnn0dir 19146 Sum of group multiples, ge...
mulgdirlem 19147 Lemma for ~ mulgdir . (Co...
mulgdir 19148 Sum of group multiples, ge...
mulgp1 19149 Group multiple (exponentia...
mulgneg2 19150 Group multiple (exponentia...
mulgnnass 19151 Product of group multiples...
mulgnn0ass 19152 Product of group multiples...
mulgass 19153 Product of group multiples...
mulgassr 19154 Reversed product of group ...
mulgmodid 19155 Casting out multiples of t...
mulgsubdir 19156 Distribution of group mult...
mhmmulg 19157 A homomorphism of monoids ...
mulgpropd 19158 Two structures with the sa...
submmulgcl 19159 Closure of the group multi...
submmulg 19160 A group multiple is the sa...
pwsmulg 19161 Value of a group multiple ...
issubg 19168 The subgroup predicate. (...
subgss 19169 A subgroup is a subset. (...
subgid 19170 A group is a subgroup of i...
subggrp 19171 A subgroup is a group. (C...
subgbas 19172 The base of the restricted...
subgrcl 19173 Reverse closure for the su...
subg0 19174 A subgroup of a group must...
subginv 19175 The inverse of an element ...
subg0cl 19176 The group identity is an e...
subginvcl 19177 The inverse of an element ...
subgcl 19178 A subgroup is closed under...
subgsubcl 19179 A subgroup is closed under...
subgsub 19180 The subtraction of element...
subgmulgcl 19181 Closure of the group multi...
subgmulg 19182 A group multiple is the sa...
issubg2 19183 Characterize the subgroups...
issubgrpd2 19184 Prove a subgroup by closur...
issubgrpd 19185 Prove a subgroup by closur...
issubg3 19186 A subgroup is a symmetric ...
issubg4 19187 A subgroup is a nonempty s...
grpissubg 19188 If the base set of a group...
resgrpisgrp 19189 If the base set of a group...
subgsubm 19190 A subgroup is a submonoid....
subsubg 19191 A subgroup of a subgroup i...
subgint 19192 The intersection of a none...
0subg 19193 The zero subgroup of an ar...
0subgOLD 19194 Obsolete version of ~ 0sub...
trivsubgd 19195 The only subgroup of a tri...
trivsubgsnd 19196 The only subgroup of a tri...
isnsg 19197 Property of being a normal...
isnsg2 19198 Weaken the condition of ~ ...
nsgbi 19199 Defining property of a nor...
nsgsubg 19200 A normal subgroup is a sub...
nsgconj 19201 The conjugation of an elem...
isnsg3 19202 A subgroup is normal iff t...
subgacs 19203 Subgroups are an algebraic...
nsgacs 19204 Normal subgroups form an a...
elnmz 19205 Elementhood in the normali...
nmzbi 19206 Defining property of the n...
nmzsubg 19207 The normalizer N_G(S) of a...
ssnmz 19208 A subgroup is a subset of ...
isnsg4 19209 A subgroup is normal iff i...
nmznsg 19210 Any subgroup is a normal s...
0nsg 19211 The zero subgroup is norma...
nsgid 19212 The whole group is a norma...
0idnsgd 19213 The whole group and the ze...
trivnsgd 19214 The only normal subgroup o...
triv1nsgd 19215 A trivial group has exactl...
1nsgtrivd 19216 A group with exactly one n...
releqg 19217 The left coset equivalence...
eqgfval 19218 Value of the subgroup left...
eqgval 19219 Value of the subgroup left...
eqger 19220 The subgroup coset equival...
eqglact 19221 A left coset can be expres...
eqgid 19222 The left coset containing ...
eqgen 19223 Each coset is equipotent t...
eqgcpbl 19224 The subgroup coset equival...
eqg0el 19225 Equivalence class of a quo...
quselbas 19226 Membership in the base set...
quseccl0 19227 Closure of the quotient ma...
qusgrp 19228 If ` Y ` is a normal subgr...
quseccl 19229 Closure of the quotient ma...
qusadd 19230 Value of the group operati...
qus0 19231 Value of the group identit...
qusinv 19232 Value of the group inverse...
qussub 19233 Value of the group subtrac...
ecqusaddd 19234 Addition of equivalence cl...
ecqusaddcl 19235 Closure of the addition in...
lagsubg2 19236 Lagrange's theorem for fin...
lagsubg 19237 Lagrange's theorem for Gro...
eqg0subg 19238 The coset equivalence rela...
eqg0subgecsn 19239 The equivalence classes mo...
qus0subgbas 19240 The base set of a quotient...
qus0subgadd 19241 The addition in a quotient...
cycsubmel 19242 Characterization of an ele...
cycsubmcl 19243 The set of nonnegative int...
cycsubm 19244 The set of nonnegative int...
cyccom 19245 Condition for an operation...
cycsubmcom 19246 The operation of a monoid ...
cycsubggend 19247 The cyclic subgroup genera...
cycsubgcl 19248 The set of integer powers ...
cycsubgss 19249 The cyclic subgroup genera...
cycsubg 19250 The cyclic group generated...
cycsubgcld 19251 The cyclic subgroup genera...
cycsubg2 19252 The subgroup generated by ...
cycsubg2cl 19253 Any multiple of an element...
reldmghm 19256 Lemma for group homomorphi...
isghm 19257 Property of being a homomo...
isghmOLD 19258 Obsolete version of ~ isgh...
isghm3 19259 Property of a group homomo...
ghmgrp1 19260 A group homomorphism is on...
ghmgrp2 19261 A group homomorphism is on...
ghmf 19262 A group homomorphism is a ...
ghmlin 19263 A homomorphism of groups i...
ghmid 19264 A homomorphism of groups p...
ghminv 19265 A homomorphism of groups p...
ghmsub 19266 Linearity of subtraction t...
isghmd 19267 Deduction for a group homo...
ghmmhm 19268 A group homomorphism is a ...
ghmmhmb 19269 Group homomorphisms and mo...
ghmmulg 19270 A group homomorphism prese...
ghmrn 19271 The range of a homomorphis...
0ghm 19272 The constant zero linear f...
idghm 19273 The identity homomorphism ...
resghm 19274 Restriction of a homomorph...
resghm2 19275 One direction of ~ resghm2...
resghm2b 19276 Restriction of the codomai...
ghmghmrn 19277 A group homomorphism from ...
ghmco 19278 The composition of group h...
ghmima 19279 The image of a subgroup un...
ghmpreima 19280 The inverse image of a sub...
ghmeql 19281 The equalizer of two group...
ghmnsgima 19282 The image of a normal subg...
ghmnsgpreima 19283 The inverse image of a nor...
ghmker 19284 The kernel of a homomorphi...
ghmeqker 19285 Two source points map to t...
pwsdiagghm 19286 Diagonal homomorphism into...
f1ghm0to0 19287 If a group homomorphism ` ...
ghmf1 19288 Two ways of saying a group...
kerf1ghm 19289 A group homomorphism ` F `...
ghmf1o 19290 A bijective group homomorp...
conjghm 19291 Conjugation is an automorp...
conjsubg 19292 A conjugated subgroup is a...
conjsubgen 19293 A conjugated subgroup is e...
conjnmz 19294 A subgroup is unchanged un...
conjnmzb 19295 Alternative condition for ...
conjnsg 19296 A normal subgroup is uncha...
qusghm 19297 If ` Y ` is a normal subgr...
ghmpropd 19298 Group homomorphism depends...
gimfn 19303 The group isomorphism func...
isgim 19304 An isomorphism of groups i...
gimf1o 19305 An isomorphism of groups i...
gimghm 19306 An isomorphism of groups i...
isgim2 19307 A group isomorphism is a h...
subggim 19308 Behavior of subgroups unde...
gimcnv 19309 The converse of a group is...
gimco 19310 The composition of group i...
gim0to0 19311 A group isomorphism maps t...
brgic 19312 The relation "is isomorphi...
brgici 19313 Prove isomorphic by an exp...
gicref 19314 Isomorphism is reflexive. ...
giclcl 19315 Isomorphism implies the le...
gicrcl 19316 Isomorphism implies the ri...
gicsym 19317 Isomorphism is symmetric. ...
gictr 19318 Isomorphism is transitive....
gicer 19319 Isomorphism is an equivale...
gicen 19320 Isomorphic groups have equ...
gicsubgen 19321 A less trivial example of ...
ghmqusnsglem1 19322 Lemma for ~ ghmqusnsg . (...
ghmqusnsglem2 19323 Lemma for ~ ghmqusnsg . (...
ghmqusnsg 19324 The mapping ` H ` induced ...
ghmquskerlem1 19325 Lemma for ~ ghmqusker . (...
ghmquskerco 19326 In the case of theorem ~ g...
ghmquskerlem2 19327 Lemma for ~ ghmqusker . (...
ghmquskerlem3 19328 The mapping ` H ` induced ...
ghmqusker 19329 A surjective group homomor...
gicqusker 19330 The image ` H ` of a group...
isga 19333 The predicate "is a (left)...
gagrp 19334 The left argument of a gro...
gaset 19335 The right argument of a gr...
gagrpid 19336 The identity of the group ...
gaf 19337 The mapping of the group a...
gafo 19338 A group action is onto its...
gaass 19339 An "associative" property ...
ga0 19340 The action of a group on t...
gaid 19341 The trivial action of a gr...
subgga 19342 A subgroup acts on its par...
gass 19343 A subset of a group action...
gasubg 19344 The restriction of a group...
gaid2 19345 A group operation is a lef...
galcan 19346 The action of a particular...
gacan 19347 Group inverses cancel in a...
gapm 19348 The action of a particular...
gaorb 19349 The orbit equivalence rela...
gaorber 19350 The orbit equivalence rela...
gastacl 19351 The stabilizer subgroup in...
gastacos 19352 Write the coset relation f...
orbstafun 19353 Existence and uniqueness f...
orbstaval 19354 Value of the function at a...
orbsta 19355 The Orbit-Stabilizer theor...
orbsta2 19356 Relation between the size ...
cntrval 19361 Substitute definition of t...
cntzfval 19362 First level substitution f...
cntzval 19363 Definition substitution fo...
elcntz 19364 Elementhood in the central...
cntzel 19365 Membership in a centralize...
cntzsnval 19366 Special substitution for t...
elcntzsn 19367 Value of the centralizer o...
sscntz 19368 A centralizer expression f...
cntzrcl 19369 Reverse closure for elemen...
cntzssv 19370 The centralizer is uncondi...
cntzi 19371 Membership in a centralize...
elcntr 19372 Elementhood in the center ...
cntrss 19373 The center is a subset of ...
cntri 19374 Defining property of the c...
resscntz 19375 Centralizer in a substruct...
cntzsgrpcl 19376 Centralizers are closed un...
cntz2ss 19377 Centralizers reverse the s...
cntzrec 19378 Reciprocity relationship f...
cntziinsn 19379 Express any centralizer as...
cntzsubm 19380 Centralizers in a monoid a...
cntzsubg 19381 Centralizers in a group ar...
cntzidss 19382 If the elements of ` S ` c...
cntzmhm 19383 Centralizers in a monoid a...
cntzmhm2 19384 Centralizers in a monoid a...
cntrsubgnsg 19385 A central subgroup is norm...
cntrnsg 19386 The center of a group is a...
oppgval 19389 Value of the opposite grou...
oppgplusfval 19390 Value of the addition oper...
oppgplus 19391 Value of the addition oper...
setsplusg 19392 The other components of an...
oppglemOLD 19393 Obsolete version of ~ sets...
oppgbas 19394 Base set of an opposite gr...
oppgbasOLD 19395 Obsolete version of ~ oppg...
oppgtset 19396 Topology of an opposite gr...
oppgtsetOLD 19397 Obsolete version of ~ oppg...
oppgtopn 19398 Topology of an opposite gr...
oppgmnd 19399 The opposite of a monoid i...
oppgmndb 19400 Bidirectional form of ~ op...
oppgid 19401 Zero in a monoid is a symm...
oppggrp 19402 The opposite of a group is...
oppggrpb 19403 Bidirectional form of ~ op...
oppginv 19404 Inverses in a group are a ...
invoppggim 19405 The inverse is an antiauto...
oppggic 19406 Every group is (naturally)...
oppgsubm 19407 Being a submonoid is a sym...
oppgsubg 19408 Being a subgroup is a symm...
oppgcntz 19409 A centralizer in a group i...
oppgcntr 19410 The center of a group is t...
gsumwrev 19411 A sum in an opposite monoi...
symgval 19414 The value of the symmetric...
symgbas 19415 The base set of the symmet...
elsymgbas2 19416 Two ways of saying a funct...
elsymgbas 19417 Two ways of saying a funct...
symgbasf1o 19418 Elements in the symmetric ...
symgbasf 19419 A permutation (element of ...
symgbasmap 19420 A permutation (element of ...
symghash 19421 The symmetric group on ` n...
symgbasfi 19422 The symmetric group on a f...
symgfv 19423 The function value of a pe...
symgfvne 19424 The function values of a p...
symgressbas 19425 The symmetric group on ` A...
symgplusg 19426 The group operation of a s...
symgov 19427 The value of the group ope...
symgcl 19428 The group operation of the...
idresperm 19429 The identity function rest...
symgmov1 19430 For a permutation of a set...
symgmov2 19431 For a permutation of a set...
symgbas0 19432 The base set of the symmet...
symg1hash 19433 The symmetric group on a s...
symg1bas 19434 The symmetric group on a s...
symg2hash 19435 The symmetric group on a (...
symg2bas 19436 The symmetric group on a p...
0symgefmndeq 19437 The symmetric group on the...
snsymgefmndeq 19438 The symmetric group on a s...
symgpssefmnd 19439 For a set ` A ` with more ...
symgvalstruct 19440 The value of the symmetric...
symgvalstructOLD 19441 Obsolete proof of ~ symgva...
symgsubmefmnd 19442 The symmetric group on a s...
symgtset 19443 The topology of the symmet...
symggrp 19444 The symmetric group on a s...
symgid 19445 The group identity element...
symginv 19446 The group inverse in the s...
symgsubmefmndALT 19447 The symmetric group on a s...
galactghm 19448 The currying of a group ac...
lactghmga 19449 The converse of ~ galactgh...
symgtopn 19450 The topology of the symmet...
symgga 19451 The symmetric group induce...
pgrpsubgsymgbi 19452 Every permutation group is...
pgrpsubgsymg 19453 Every permutation group is...
idressubgsymg 19454 The singleton containing o...
idrespermg 19455 The structure with the sin...
cayleylem1 19456 Lemma for ~ cayley . (Con...
cayleylem2 19457 Lemma for ~ cayley . (Con...
cayley 19458 Cayley's Theorem (construc...
cayleyth 19459 Cayley's Theorem (existenc...
symgfix2 19460 If a permutation does not ...
symgextf 19461 The extension of a permuta...
symgextfv 19462 The function value of the ...
symgextfve 19463 The function value of the ...
symgextf1lem 19464 Lemma for ~ symgextf1 . (...
symgextf1 19465 The extension of a permuta...
symgextfo 19466 The extension of a permuta...
symgextf1o 19467 The extension of a permuta...
symgextsymg 19468 The extension of a permuta...
symgextres 19469 The restriction of the ext...
gsumccatsymgsn 19470 Homomorphic property of co...
gsmsymgrfixlem1 19471 Lemma 1 for ~ gsmsymgrfix ...
gsmsymgrfix 19472 The composition of permuta...
fvcosymgeq 19473 The values of two composit...
gsmsymgreqlem1 19474 Lemma 1 for ~ gsmsymgreq ....
gsmsymgreqlem2 19475 Lemma 2 for ~ gsmsymgreq ....
gsmsymgreq 19476 Two combination of permuta...
symgfixelq 19477 A permutation of a set fix...
symgfixels 19478 The restriction of a permu...
symgfixelsi 19479 The restriction of a permu...
symgfixf 19480 The mapping of a permutati...
symgfixf1 19481 The mapping of a permutati...
symgfixfolem1 19482 Lemma 1 for ~ symgfixfo . ...
symgfixfo 19483 The mapping of a permutati...
symgfixf1o 19484 The mapping of a permutati...
f1omvdmvd 19487 A permutation of any class...
f1omvdcnv 19488 A permutation and its inve...
mvdco 19489 Composing two permutations...
f1omvdconj 19490 Conjugation of a permutati...
f1otrspeq 19491 A transposition is charact...
f1omvdco2 19492 If exactly one of two perm...
f1omvdco3 19493 If a point is moved by exa...
pmtrfval 19494 The function generating tr...
pmtrval 19495 A generated transposition,...
pmtrfv 19496 General value of mapping a...
pmtrprfv 19497 In a transposition of two ...
pmtrprfv3 19498 In a transposition of two ...
pmtrf 19499 Functionality of a transpo...
pmtrmvd 19500 A transposition moves prec...
pmtrrn 19501 Transposing two points giv...
pmtrfrn 19502 A transposition (as a kind...
pmtrffv 19503 Mapping of a point under a...
pmtrrn2 19504 For any transposition ther...
pmtrfinv 19505 A transposition function i...
pmtrfmvdn0 19506 A transposition moves at l...
pmtrff1o 19507 A transposition function i...
pmtrfcnv 19508 A transposition function i...
pmtrfb 19509 An intrinsic characterizat...
pmtrfconj 19510 Any conjugate of a transpo...
symgsssg 19511 The symmetric group has su...
symgfisg 19512 The symmetric group has a ...
symgtrf 19513 Transpositions are element...
symggen 19514 The span of the transposit...
symggen2 19515 A finite permutation group...
symgtrinv 19516 To invert a permutation re...
pmtr3ncomlem1 19517 Lemma 1 for ~ pmtr3ncom . ...
pmtr3ncomlem2 19518 Lemma 2 for ~ pmtr3ncom . ...
pmtr3ncom 19519 Transpositions over sets w...
pmtrdifellem1 19520 Lemma 1 for ~ pmtrdifel . ...
pmtrdifellem2 19521 Lemma 2 for ~ pmtrdifel . ...
pmtrdifellem3 19522 Lemma 3 for ~ pmtrdifel . ...
pmtrdifellem4 19523 Lemma 4 for ~ pmtrdifel . ...
pmtrdifel 19524 A transposition of element...
pmtrdifwrdellem1 19525 Lemma 1 for ~ pmtrdifwrdel...
pmtrdifwrdellem2 19526 Lemma 2 for ~ pmtrdifwrdel...
pmtrdifwrdellem3 19527 Lemma 3 for ~ pmtrdifwrdel...
pmtrdifwrdel2lem1 19528 Lemma 1 for ~ pmtrdifwrdel...
pmtrdifwrdel 19529 A sequence of transpositio...
pmtrdifwrdel2 19530 A sequence of transpositio...
pmtrprfval 19531 The transpositions on a pa...
pmtrprfvalrn 19532 The range of the transposi...
psgnunilem1 19537 Lemma for ~ psgnuni . Giv...
psgnunilem5 19538 Lemma for ~ psgnuni . It ...
psgnunilem2 19539 Lemma for ~ psgnuni . Ind...
psgnunilem3 19540 Lemma for ~ psgnuni . Any...
psgnunilem4 19541 Lemma for ~ psgnuni . An ...
m1expaddsub 19542 Addition and subtraction o...
psgnuni 19543 If the same permutation ca...
psgnfval 19544 Function definition of the...
psgnfn 19545 Functionality and domain o...
psgndmsubg 19546 The finitary permutations ...
psgneldm 19547 Property of being a finita...
psgneldm2 19548 The finitary permutations ...
psgneldm2i 19549 A sequence of transpositio...
psgneu 19550 A finitary permutation has...
psgnval 19551 Value of the permutation s...
psgnvali 19552 A finitary permutation has...
psgnvalii 19553 Any representation of a pe...
psgnpmtr 19554 All transpositions are odd...
psgn0fv0 19555 The permutation sign funct...
sygbasnfpfi 19556 The class of non-fixed poi...
psgnfvalfi 19557 Function definition of the...
psgnvalfi 19558 Value of the permutation s...
psgnran 19559 The range of the permutati...
gsmtrcl 19560 The group sum of transposi...
psgnfitr 19561 A permutation of a finite ...
psgnfieu 19562 A permutation of a finite ...
pmtrsn 19563 The value of the transposi...
psgnsn 19564 The permutation sign funct...
psgnprfval 19565 The permutation sign funct...
psgnprfval1 19566 The permutation sign of th...
psgnprfval2 19567 The permutation sign of th...
odfval 19576 Value of the order functio...
odfvalALT 19577 Shorter proof of ~ odfval ...
odval 19578 Second substitution for th...
odlem1 19579 The group element order is...
odcl 19580 The order of a group eleme...
odf 19581 Functionality of the group...
odid 19582 Any element to the power o...
odlem2 19583 Any positive annihilator o...
odmodnn0 19584 Reduce the argument of a g...
mndodconglem 19585 Lemma for ~ mndodcong . (...
mndodcong 19586 If two multipliers are con...
mndodcongi 19587 If two multipliers are con...
oddvdsnn0 19588 The only multiples of ` A ...
odnncl 19589 If a nonzero multiple of a...
odmod 19590 Reduce the argument of a g...
oddvds 19591 The only multiples of ` A ...
oddvdsi 19592 Any group element is annih...
odcong 19593 If two multipliers are con...
odeq 19594 The ~ oddvds property uniq...
odval2 19595 A non-conditional definiti...
odcld 19596 The order of a group eleme...
odm1inv 19597 The (order-1)th multiple o...
odmulgid 19598 A relationship between the...
odmulg2 19599 The order of a multiple di...
odmulg 19600 Relationship between the o...
odmulgeq 19601 A multiple of a point of f...
odbezout 19602 If ` N ` is coprime to the...
od1 19603 The order of the group ide...
odeq1 19604 The group identity is the ...
odinv 19605 The order of the inverse o...
odf1 19606 The multiples of an elemen...
odinf 19607 The multiples of an elemen...
dfod2 19608 An alternative definition ...
odcl2 19609 The order of an element of...
oddvds2 19610 The order of an element of...
finodsubmsubg 19611 A submonoid whose elements...
0subgALT 19612 A shorter proof of ~ 0subg...
submod 19613 The order of an element is...
subgod 19614 The order of an element is...
odsubdvds 19615 The order of an element of...
odf1o1 19616 An element with zero order...
odf1o2 19617 An element with nonzero or...
odhash 19618 An element of zero order g...
odhash2 19619 If an element has nonzero ...
odhash3 19620 An element which generates...
odngen 19621 A cyclic subgroup of size ...
gexval 19622 Value of the exponent of a...
gexlem1 19623 The group element order is...
gexcl 19624 The exponent of a group is...
gexid 19625 Any element to the power o...
gexlem2 19626 Any positive annihilator o...
gexdvdsi 19627 Any group element is annih...
gexdvds 19628 The only ` N ` that annihi...
gexdvds2 19629 An integer divides the gro...
gexod 19630 Any group element is annih...
gexcl3 19631 If the order of every grou...
gexnnod 19632 Every group element has fi...
gexcl2 19633 The exponent of a finite g...
gexdvds3 19634 The exponent of a finite g...
gex1 19635 A group or monoid has expo...
ispgp 19636 A group is a ` P ` -group ...
pgpprm 19637 Reverse closure for the fi...
pgpgrp 19638 Reverse closure for the se...
pgpfi1 19639 A finite group with order ...
pgp0 19640 The identity subgroup is a...
subgpgp 19641 A subgroup of a p-group is...
sylow1lem1 19642 Lemma for ~ sylow1 . The ...
sylow1lem2 19643 Lemma for ~ sylow1 . The ...
sylow1lem3 19644 Lemma for ~ sylow1 . One ...
sylow1lem4 19645 Lemma for ~ sylow1 . The ...
sylow1lem5 19646 Lemma for ~ sylow1 . Usin...
sylow1 19647 Sylow's first theorem. If...
odcau 19648 Cauchy's theorem for the o...
pgpfi 19649 The converse to ~ pgpfi1 ....
pgpfi2 19650 Alternate version of ~ pgp...
pgphash 19651 The order of a p-group. (...
isslw 19652 The property of being a Sy...
slwprm 19653 Reverse closure for the fi...
slwsubg 19654 A Sylow ` P ` -subgroup is...
slwispgp 19655 Defining property of a Syl...
slwpss 19656 A proper superset of a Syl...
slwpgp 19657 A Sylow ` P ` -subgroup is...
pgpssslw 19658 Every ` P ` -subgroup is c...
slwn0 19659 Every finite group contain...
subgslw 19660 A Sylow subgroup that is c...
sylow2alem1 19661 Lemma for ~ sylow2a . An ...
sylow2alem2 19662 Lemma for ~ sylow2a . All...
sylow2a 19663 A named lemma of Sylow's s...
sylow2blem1 19664 Lemma for ~ sylow2b . Eva...
sylow2blem2 19665 Lemma for ~ sylow2b . Lef...
sylow2blem3 19666 Sylow's second theorem. P...
sylow2b 19667 Sylow's second theorem. A...
slwhash 19668 A sylow subgroup has cardi...
fislw 19669 The sylow subgroups of a f...
sylow2 19670 Sylow's second theorem. S...
sylow3lem1 19671 Lemma for ~ sylow3 , first...
sylow3lem2 19672 Lemma for ~ sylow3 , first...
sylow3lem3 19673 Lemma for ~ sylow3 , first...
sylow3lem4 19674 Lemma for ~ sylow3 , first...
sylow3lem5 19675 Lemma for ~ sylow3 , secon...
sylow3lem6 19676 Lemma for ~ sylow3 , secon...
sylow3 19677 Sylow's third theorem. Th...
lsmfval 19682 The subgroup sum function ...
lsmvalx 19683 Subspace sum value (for a ...
lsmelvalx 19684 Subspace sum membership (f...
lsmelvalix 19685 Subspace sum membership (f...
oppglsm 19686 The subspace sum operation...
lsmssv 19687 Subgroup sum is a subset o...
lsmless1x 19688 Subset implies subgroup su...
lsmless2x 19689 Subset implies subgroup su...
lsmub1x 19690 Subgroup sum is an upper b...
lsmub2x 19691 Subgroup sum is an upper b...
lsmval 19692 Subgroup sum value (for a ...
lsmelval 19693 Subgroup sum membership (f...
lsmelvali 19694 Subgroup sum membership (f...
lsmelvalm 19695 Subgroup sum membership an...
lsmelvalmi 19696 Membership of vector subtr...
lsmsubm 19697 The sum of two commuting s...
lsmsubg 19698 The sum of two commuting s...
lsmcom2 19699 Subgroup sum commutes. (C...
smndlsmidm 19700 The direct product is idem...
lsmub1 19701 Subgroup sum is an upper b...
lsmub2 19702 Subgroup sum is an upper b...
lsmunss 19703 Union of subgroups is a su...
lsmless1 19704 Subset implies subgroup su...
lsmless2 19705 Subset implies subgroup su...
lsmless12 19706 Subset implies subgroup su...
lsmidm 19707 Subgroup sum is idempotent...
lsmlub 19708 The least upper bound prop...
lsmss1 19709 Subgroup sum with a subset...
lsmss1b 19710 Subgroup sum with a subset...
lsmss2 19711 Subgroup sum with a subset...
lsmss2b 19712 Subgroup sum with a subset...
lsmass 19713 Subgroup sum is associativ...
mndlsmidm 19714 Subgroup sum is idempotent...
lsm01 19715 Subgroup sum with the zero...
lsm02 19716 Subgroup sum with the zero...
subglsm 19717 The subgroup sum evaluated...
lssnle 19718 Equivalent expressions for...
lsmmod 19719 The modular law holds for ...
lsmmod2 19720 Modular law dual for subgr...
lsmpropd 19721 If two structures have the...
cntzrecd 19722 Commute the "subgroups com...
lsmcntz 19723 The "subgroups commute" pr...
lsmcntzr 19724 The "subgroups commute" pr...
lsmdisj 19725 Disjointness from a subgro...
lsmdisj2 19726 Association of the disjoin...
lsmdisj3 19727 Association of the disjoin...
lsmdisjr 19728 Disjointness from a subgro...
lsmdisj2r 19729 Association of the disjoin...
lsmdisj3r 19730 Association of the disjoin...
lsmdisj2a 19731 Association of the disjoin...
lsmdisj2b 19732 Association of the disjoin...
lsmdisj3a 19733 Association of the disjoin...
lsmdisj3b 19734 Association of the disjoin...
subgdisj1 19735 Vectors belonging to disjo...
subgdisj2 19736 Vectors belonging to disjo...
subgdisjb 19737 Vectors belonging to disjo...
pj1fval 19738 The left projection functi...
pj1val 19739 The left projection functi...
pj1eu 19740 Uniqueness of a left proje...
pj1f 19741 The left projection functi...
pj2f 19742 The right projection funct...
pj1id 19743 Any element of a direct su...
pj1eq 19744 Any element of a direct su...
pj1lid 19745 The left projection functi...
pj1rid 19746 The left projection functi...
pj1ghm 19747 The left projection functi...
pj1ghm2 19748 The left projection functi...
lsmhash 19749 The order of the direct pr...
efgmval 19756 Value of the formal invers...
efgmf 19757 The formal inverse operati...
efgmnvl 19758 The inversion function on ...
efgrcl 19759 Lemma for ~ efgval . (Con...
efglem 19760 Lemma for ~ efgval . (Con...
efgval 19761 Value of the free group co...
efger 19762 Value of the free group co...
efgi 19763 Value of the free group co...
efgi0 19764 Value of the free group co...
efgi1 19765 Value of the free group co...
efgtf 19766 Value of the free group co...
efgtval 19767 Value of the extension fun...
efgval2 19768 Value of the free group co...
efgi2 19769 Value of the free group co...
efgtlen 19770 Value of the free group co...
efginvrel2 19771 The inverse of the reverse...
efginvrel1 19772 The inverse of the reverse...
efgsf 19773 Value of the auxiliary fun...
efgsdm 19774 Elementhood in the domain ...
efgsval 19775 Value of the auxiliary fun...
efgsdmi 19776 Property of the last link ...
efgsval2 19777 Value of the auxiliary fun...
efgsrel 19778 The start and end of any e...
efgs1 19779 A singleton of an irreduci...
efgs1b 19780 Every extension sequence e...
efgsp1 19781 If ` F ` is an extension s...
efgsres 19782 An initial segment of an e...
efgsfo 19783 For any word, there is a s...
efgredlema 19784 The reduced word that form...
efgredlemf 19785 Lemma for ~ efgredleme . ...
efgredlemg 19786 Lemma for ~ efgred . (Con...
efgredleme 19787 Lemma for ~ efgred . (Con...
efgredlemd 19788 The reduced word that form...
efgredlemc 19789 The reduced word that form...
efgredlemb 19790 The reduced word that form...
efgredlem 19791 The reduced word that form...
efgred 19792 The reduced word that form...
efgrelexlema 19793 If two words ` A , B ` are...
efgrelexlemb 19794 If two words ` A , B ` are...
efgrelex 19795 If two words ` A , B ` are...
efgredeu 19796 There is a unique reduced ...
efgred2 19797 Two extension sequences ha...
efgcpbllema 19798 Lemma for ~ efgrelex . De...
efgcpbllemb 19799 Lemma for ~ efgrelex . Sh...
efgcpbl 19800 Two extension sequences ha...
efgcpbl2 19801 Two extension sequences ha...
frgpval 19802 Value of the free group co...
frgpcpbl 19803 Compatibility of the group...
frgp0 19804 The free group is a group....
frgpeccl 19805 Closure of the quotient ma...
frgpgrp 19806 The free group is a group....
frgpadd 19807 Addition in the free group...
frgpinv 19808 The inverse of an element ...
frgpmhm 19809 The "natural map" from wor...
vrgpfval 19810 The canonical injection fr...
vrgpval 19811 The value of the generatin...
vrgpf 19812 The mapping from the index...
vrgpinv 19813 The inverse of a generatin...
frgpuptf 19814 Any assignment of the gene...
frgpuptinv 19815 Any assignment of the gene...
frgpuplem 19816 Any assignment of the gene...
frgpupf 19817 Any assignment of the gene...
frgpupval 19818 Any assignment of the gene...
frgpup1 19819 Any assignment of the gene...
frgpup2 19820 The evaluation map has the...
frgpup3lem 19821 The evaluation map has the...
frgpup3 19822 Universal property of the ...
0frgp 19823 The free group on zero gen...
isabl 19828 The predicate "is an Abeli...
ablgrp 19829 An Abelian group is a grou...
ablgrpd 19830 An Abelian group is a grou...
ablcmn 19831 An Abelian group is a comm...
ablcmnd 19832 An Abelian group is a comm...
iscmn 19833 The predicate "is a commut...
isabl2 19834 The predicate "is an Abeli...
cmnpropd 19835 If two structures have the...
ablpropd 19836 If two structures have the...
ablprop 19837 If two structures have the...
iscmnd 19838 Properties that determine ...
isabld 19839 Properties that determine ...
isabli 19840 Properties that determine ...
cmnmnd 19841 A commutative monoid is a ...
cmncom 19842 A commutative monoid is co...
ablcom 19843 An Abelian group operation...
cmn32 19844 Commutative/associative la...
cmn4 19845 Commutative/associative la...
cmn12 19846 Commutative/associative la...
abl32 19847 Commutative/associative la...
cmnmndd 19848 A commutative monoid is a ...
cmnbascntr 19849 The base set of a commutat...
rinvmod 19850 Uniqueness of a right inve...
ablinvadd 19851 The inverse of an Abelian ...
ablsub2inv 19852 Abelian group subtraction ...
ablsubadd 19853 Relationship between Abeli...
ablsub4 19854 Commutative/associative su...
abladdsub4 19855 Abelian group addition/sub...
abladdsub 19856 Associative-type law for g...
ablsubadd23 19857 Commutative/associative la...
ablsubaddsub 19858 Double subtraction and add...
ablpncan2 19859 Cancellation law for subtr...
ablpncan3 19860 A cancellation law for Abe...
ablsubsub 19861 Law for double subtraction...
ablsubsub4 19862 Law for double subtraction...
ablpnpcan 19863 Cancellation law for mixed...
ablnncan 19864 Cancellation law for group...
ablsub32 19865 Swap the second and third ...
ablnnncan 19866 Cancellation law for group...
ablnnncan1 19867 Cancellation law for group...
ablsubsub23 19868 Swap subtrahend and result...
mulgnn0di 19869 Group multiple of a sum, f...
mulgdi 19870 Group multiple of a sum. ...
mulgmhm 19871 The map from ` x ` to ` n ...
mulgghm 19872 The map from ` x ` to ` n ...
mulgsubdi 19873 Group multiple of a differ...
ghmfghm 19874 The function fulfilling th...
ghmcmn 19875 The image of a commutative...
ghmabl 19876 The image of an abelian gr...
invghm 19877 The inversion map is a gro...
eqgabl 19878 Value of the subgroup cose...
qusecsub 19879 Two subgroup cosets are eq...
subgabl 19880 A subgroup of an abelian g...
subcmn 19881 A submonoid of a commutati...
submcmn 19882 A submonoid of a commutati...
submcmn2 19883 A submonoid is commutative...
cntzcmn 19884 The centralizer of any sub...
cntzcmnss 19885 Any subset in a commutativ...
cntrcmnd 19886 The center of a monoid is ...
cntrabl 19887 The center of a group is a...
cntzspan 19888 If the generators commute,...
cntzcmnf 19889 Discharge the centralizer ...
ghmplusg 19890 The pointwise sum of two l...
ablnsg 19891 Every subgroup of an abeli...
odadd1 19892 The order of a product in ...
odadd2 19893 The order of a product in ...
odadd 19894 The order of a product is ...
gex2abl 19895 A group with exponent 2 (o...
gexexlem 19896 Lemma for ~ gexex . (Cont...
gexex 19897 In an abelian group with f...
torsubg 19898 The set of all elements of...
oddvdssubg 19899 The set of all elements wh...
lsmcomx 19900 Subgroup sum commutes (ext...
ablcntzd 19901 All subgroups in an abelia...
lsmcom 19902 Subgroup sum commutes. (C...
lsmsubg2 19903 The sum of two subgroups i...
lsm4 19904 Commutative/associative la...
prdscmnd 19905 The product of a family of...
prdsabld 19906 The product of a family of...
pwscmn 19907 The structure power on a c...
pwsabl 19908 The structure power on an ...
qusabl 19909 If ` Y ` is a subgroup of ...
abl1 19910 The (smallest) structure r...
abln0 19911 Abelian groups (and theref...
cnaddablx 19912 The complex numbers are an...
cnaddabl 19913 The complex numbers are an...
cnaddid 19914 The group identity element...
cnaddinv 19915 Value of the group inverse...
zaddablx 19916 The integers are an Abelia...
frgpnabllem1 19917 Lemma for ~ frgpnabl . (C...
frgpnabllem2 19918 Lemma for ~ frgpnabl . (C...
frgpnabl 19919 The free group on two or m...
imasabl 19920 The image structure of an ...
iscyg 19923 Definition of a cyclic gro...
iscyggen 19924 The property of being a cy...
iscyggen2 19925 The property of being a cy...
iscyg2 19926 A cyclic group is a group ...
cyggeninv 19927 The inverse of a cyclic ge...
cyggenod 19928 An element is the generato...
cyggenod2 19929 In an infinite cyclic grou...
iscyg3 19930 Definition of a cyclic gro...
iscygd 19931 Definition of a cyclic gro...
iscygodd 19932 Show that a group with an ...
cycsubmcmn 19933 The set of nonnegative int...
cyggrp 19934 A cyclic group is a group....
cygabl 19935 A cyclic group is abelian....
cygctb 19936 A cyclic group is countabl...
0cyg 19937 The trivial group is cycli...
prmcyg 19938 A group with prime order i...
lt6abl 19939 A group with fewer than ` ...
ghmcyg 19940 The image of a cyclic grou...
cyggex2 19941 The exponent of a cyclic g...
cyggex 19942 The exponent of a finite c...
cyggexb 19943 A finite abelian group is ...
giccyg 19944 Cyclicity is a group prope...
cycsubgcyg 19945 The cyclic subgroup genera...
cycsubgcyg2 19946 The cyclic subgroup genera...
gsumval3a 19947 Value of the group sum ope...
gsumval3eu 19948 The group sum as defined i...
gsumval3lem1 19949 Lemma 1 for ~ gsumval3 . ...
gsumval3lem2 19950 Lemma 2 for ~ gsumval3 . ...
gsumval3 19951 Value of the group sum ope...
gsumcllem 19952 Lemma for ~ gsumcl and rel...
gsumzres 19953 Extend a finite group sum ...
gsumzcl2 19954 Closure of a finite group ...
gsumzcl 19955 Closure of a finite group ...
gsumzf1o 19956 Re-index a finite group su...
gsumres 19957 Extend a finite group sum ...
gsumcl2 19958 Closure of a finite group ...
gsumcl 19959 Closure of a finite group ...
gsumf1o 19960 Re-index a finite group su...
gsumreidx 19961 Re-index a finite group su...
gsumzsubmcl 19962 Closure of a group sum in ...
gsumsubmcl 19963 Closure of a group sum in ...
gsumsubgcl 19964 Closure of a group sum in ...
gsumzaddlem 19965 The sum of two group sums....
gsumzadd 19966 The sum of two group sums....
gsumadd 19967 The sum of two group sums....
gsummptfsadd 19968 The sum of two group sums ...
gsummptfidmadd 19969 The sum of two group sums ...
gsummptfidmadd2 19970 The sum of two group sums ...
gsumzsplit 19971 Split a group sum into two...
gsumsplit 19972 Split a group sum into two...
gsumsplit2 19973 Split a group sum into two...
gsummptfidmsplit 19974 Split a group sum expresse...
gsummptfidmsplitres 19975 Split a group sum expresse...
gsummptfzsplit 19976 Split a group sum expresse...
gsummptfzsplitl 19977 Split a group sum expresse...
gsumconst 19978 Sum of a constant series. ...
gsumconstf 19979 Sum of a constant series. ...
gsummptshft 19980 Index shift of a finite gr...
gsumzmhm 19981 Apply a group homomorphism...
gsummhm 19982 Apply a group homomorphism...
gsummhm2 19983 Apply a group homomorphism...
gsummptmhm 19984 Apply a group homomorphism...
gsummulglem 19985 Lemma for ~ gsummulg and ~...
gsummulg 19986 Nonnegative multiple of a ...
gsummulgz 19987 Integer multiple of a grou...
gsumzoppg 19988 The opposite of a group su...
gsumzinv 19989 Inverse of a group sum. (...
gsuminv 19990 Inverse of a group sum. (...
gsummptfidminv 19991 Inverse of a group sum exp...
gsumsub 19992 The difference of two grou...
gsummptfssub 19993 The difference of two grou...
gsummptfidmsub 19994 The difference of two grou...
gsumsnfd 19995 Group sum of a singleton, ...
gsumsnd 19996 Group sum of a singleton, ...
gsumsnf 19997 Group sum of a singleton, ...
gsumsn 19998 Group sum of a singleton. ...
gsumpr 19999 Group sum of a pair. (Con...
gsumzunsnd 20000 Append an element to a fin...
gsumunsnfd 20001 Append an element to a fin...
gsumunsnd 20002 Append an element to a fin...
gsumunsnf 20003 Append an element to a fin...
gsumunsn 20004 Append an element to a fin...
gsumdifsnd 20005 Extract a summand from a f...
gsumpt 20006 Sum of a family that is no...
gsummptf1o 20007 Re-index a finite group su...
gsummptun 20008 Group sum of a disjoint un...
gsummpt1n0 20009 If only one summand in a f...
gsummptif1n0 20010 If only one summand in a f...
gsummptcl 20011 Closure of a finite group ...
gsummptfif1o 20012 Re-index a finite group su...
gsummptfzcl 20013 Closure of a finite group ...
gsum2dlem1 20014 Lemma 1 for ~ gsum2d . (C...
gsum2dlem2 20015 Lemma for ~ gsum2d . (Con...
gsum2d 20016 Write a sum over a two-dim...
gsum2d2lem 20017 Lemma for ~ gsum2d2 : show...
gsum2d2 20018 Write a group sum over a t...
gsumcom2 20019 Two-dimensional commutatio...
gsumxp 20020 Write a group sum over a c...
gsumcom 20021 Commute the arguments of a...
gsumcom3 20022 A commutative law for fini...
gsumcom3fi 20023 A commutative law for fini...
gsumxp2 20024 Write a group sum over a c...
prdsgsum 20025 Finite commutative sums in...
pwsgsum 20026 Finite commutative sums in...
fsfnn0gsumfsffz 20027 Replacing a finitely suppo...
nn0gsumfz 20028 Replacing a finitely suppo...
nn0gsumfz0 20029 Replacing a finitely suppo...
gsummptnn0fz 20030 A final group sum over a f...
gsummptnn0fzfv 20031 A final group sum over a f...
telgsumfzslem 20032 Lemma for ~ telgsumfzs (in...
telgsumfzs 20033 Telescoping group sum rang...
telgsumfz 20034 Telescoping group sum rang...
telgsumfz0s 20035 Telescoping finite group s...
telgsumfz0 20036 Telescoping finite group s...
telgsums 20037 Telescoping finitely suppo...
telgsum 20038 Telescoping finitely suppo...
reldmdprd 20043 The domain of the internal...
dmdprd 20044 The domain of definition o...
dmdprdd 20045 Show that a given family i...
dprddomprc 20046 A family of subgroups inde...
dprddomcld 20047 If a family of subgroups i...
dprdval0prc 20048 The internal direct produc...
dprdval 20049 The value of the internal ...
eldprd 20050 A class ` A ` is an intern...
dprdgrp 20051 Reverse closure for the in...
dprdf 20052 The function ` S ` is a fa...
dprdf2 20053 The function ` S ` is a fa...
dprdcntz 20054 The function ` S ` is a fa...
dprddisj 20055 The function ` S ` is a fa...
dprdw 20056 The property of being a fi...
dprdwd 20057 A mapping being a finitely...
dprdff 20058 A finitely supported funct...
dprdfcl 20059 A finitely supported funct...
dprdffsupp 20060 A finitely supported funct...
dprdfcntz 20061 A function on the elements...
dprdssv 20062 The internal direct produc...
dprdfid 20063 A function mapping all but...
eldprdi 20064 The domain of definition o...
dprdfinv 20065 Take the inverse of a grou...
dprdfadd 20066 Take the sum of group sums...
dprdfsub 20067 Take the difference of gro...
dprdfeq0 20068 The zero function is the o...
dprdf11 20069 Two group sums over a dire...
dprdsubg 20070 The internal direct produc...
dprdub 20071 Each factor is a subset of...
dprdlub 20072 The direct product is smal...
dprdspan 20073 The direct product is the ...
dprdres 20074 Restriction of a direct pr...
dprdss 20075 Create a direct product by...
dprdz 20076 A family consisting entire...
dprd0 20077 The empty family is an int...
dprdf1o 20078 Rearrange the index set of...
dprdf1 20079 Rearrange the index set of...
subgdmdprd 20080 A direct product in a subg...
subgdprd 20081 A direct product in a subg...
dprdsn 20082 A singleton family is an i...
dmdprdsplitlem 20083 Lemma for ~ dmdprdsplit . ...
dprdcntz2 20084 The function ` S ` is a fa...
dprddisj2 20085 The function ` S ` is a fa...
dprd2dlem2 20086 The direct product of a co...
dprd2dlem1 20087 The direct product of a co...
dprd2da 20088 The direct product of a co...
dprd2db 20089 The direct product of a co...
dprd2d2 20090 The direct product of a co...
dmdprdsplit2lem 20091 Lemma for ~ dmdprdsplit . ...
dmdprdsplit2 20092 The direct product splits ...
dmdprdsplit 20093 The direct product splits ...
dprdsplit 20094 The direct product is the ...
dmdprdpr 20095 A singleton family is an i...
dprdpr 20096 A singleton family is an i...
dpjlem 20097 Lemma for theorems about d...
dpjcntz 20098 The two subgroups that app...
dpjdisj 20099 The two subgroups that app...
dpjlsm 20100 The two subgroups that app...
dpjfval 20101 Value of the direct produc...
dpjval 20102 Value of the direct produc...
dpjf 20103 The ` X ` -th index projec...
dpjidcl 20104 The key property of projec...
dpjeq 20105 Decompose a group sum into...
dpjid 20106 The key property of projec...
dpjlid 20107 The ` X ` -th index projec...
dpjrid 20108 The ` Y ` -th index projec...
dpjghm 20109 The direct product is the ...
dpjghm2 20110 The direct product is the ...
ablfacrplem 20111 Lemma for ~ ablfacrp2 . (...
ablfacrp 20112 A finite abelian group who...
ablfacrp2 20113 The factors ` K , L ` of ~...
ablfac1lem 20114 Lemma for ~ ablfac1b . Sa...
ablfac1a 20115 The factors of ~ ablfac1b ...
ablfac1b 20116 Any abelian group is the d...
ablfac1c 20117 The factors of ~ ablfac1b ...
ablfac1eulem 20118 Lemma for ~ ablfac1eu . (...
ablfac1eu 20119 The factorization of ~ abl...
pgpfac1lem1 20120 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem2 20121 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem3a 20122 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem3 20123 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem4 20124 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem5 20125 Lemma for ~ pgpfac1 . (Co...
pgpfac1 20126 Factorization of a finite ...
pgpfaclem1 20127 Lemma for ~ pgpfac . (Con...
pgpfaclem2 20128 Lemma for ~ pgpfac . (Con...
pgpfaclem3 20129 Lemma for ~ pgpfac . (Con...
pgpfac 20130 Full factorization of a fi...
ablfaclem1 20131 Lemma for ~ ablfac . (Con...
ablfaclem2 20132 Lemma for ~ ablfac . (Con...
ablfaclem3 20133 Lemma for ~ ablfac . (Con...
ablfac 20134 The Fundamental Theorem of...
ablfac2 20135 Choose generators for each...
issimpg 20138 The predicate "is a simple...
issimpgd 20139 Deduce a simple group from...
simpggrp 20140 A simple group is a group....
simpggrpd 20141 A simple group is a group....
simpg2nsg 20142 A simple group has two nor...
trivnsimpgd 20143 Trivial groups are not sim...
simpgntrivd 20144 Simple groups are nontrivi...
simpgnideld 20145 A simple group contains a ...
simpgnsgd 20146 The only normal subgroups ...
simpgnsgeqd 20147 A normal subgroup of a sim...
2nsgsimpgd 20148 If any normal subgroup of ...
simpgnsgbid 20149 A nontrivial group is simp...
ablsimpnosubgd 20150 A subgroup of an abelian s...
ablsimpg1gend 20151 An abelian simple group is...
ablsimpgcygd 20152 An abelian simple group is...
ablsimpgfindlem1 20153 Lemma for ~ ablsimpgfind ....
ablsimpgfindlem2 20154 Lemma for ~ ablsimpgfind ....
cycsubggenodd 20155 Relationship between the o...
ablsimpgfind 20156 An abelian simple group is...
fincygsubgd 20157 The subgroup referenced in...
fincygsubgodd 20158 Calculate the order of a s...
fincygsubgodexd 20159 A finite cyclic group has ...
prmgrpsimpgd 20160 A group of prime order is ...
ablsimpgprmd 20161 An abelian simple group ha...
ablsimpgd 20162 An abelian group is simple...
fnmgp 20165 The multiplicative group o...
mgpval 20166 Value of the multiplicatio...
mgpplusg 20167 Value of the group operati...
mgplemOLD 20168 Obsolete version of ~ sets...
mgpbas 20169 Base set of the multiplica...
mgpbasOLD 20170 Obsolete version of ~ mgpb...
mgpsca 20171 The multiplication monoid ...
mgpscaOLD 20172 Obsolete version of ~ mgps...
mgptset 20173 Topology component of the ...
mgptsetOLD 20174 Obsolete version of ~ mgpt...
mgptopn 20175 Topology of the multiplica...
mgpds 20176 Distance function of the m...
mgpdsOLD 20177 Obsolete version of ~ mgpd...
mgpress 20178 Subgroup commutes with the...
mgpressOLD 20179 Obsolete version of ~ mgpr...
prdsmgp 20180 The multiplicative monoid ...
isrng 20183 The predicate "is a non-un...
rngabl 20184 A non-unital ring is an (a...
rngmgp 20185 A non-unital ring is a sem...
rngmgpf 20186 Restricted functionality o...
rnggrp 20187 A non-unital ring is a (ad...
rngass 20188 Associative law for the mu...
rngdi 20189 Distributive law for the m...
rngdir 20190 Distributive law for the m...
rngacl 20191 Closure of the addition op...
rng0cl 20192 The zero element of a non-...
rngcl 20193 Closure of the multiplicat...
rnglz 20194 The zero of a non-unital r...
rngrz 20195 The zero of a non-unital r...
rngmneg1 20196 Negation of a product in a...
rngmneg2 20197 Negation of a product in a...
rngm2neg 20198 Double negation of a produ...
rngansg 20199 Every additive subgroup of...
rngsubdi 20200 Ring multiplication distri...
rngsubdir 20201 Ring multiplication distri...
isrngd 20202 Properties that determine ...
rngpropd 20203 If two structures have the...
prdsmulrngcl 20204 Closure of the multiplicat...
prdsrngd 20205 A product of non-unital ri...
imasrng 20206 The image structure of a n...
imasrngf1 20207 The image of a non-unital ...
xpsrngd 20208 A product of two non-unita...
qusrng 20209 The quotient structure of ...
ringidval 20212 The value of the unity ele...
dfur2 20213 The multiplicative identit...
ringurd 20214 Deduce the unity element o...
issrg 20217 The predicate "is a semiri...
srgcmn 20218 A semiring is a commutativ...
srgmnd 20219 A semiring is a monoid. (...
srgmgp 20220 A semiring is a monoid und...
srgdilem 20221 Lemma for ~ srgdi and ~ sr...
srgcl 20222 Closure of the multiplicat...
srgass 20223 Associative law for the mu...
srgideu 20224 The unity element of a sem...
srgfcl 20225 Functionality of the multi...
srgdi 20226 Distributive law for the m...
srgdir 20227 Distributive law for the m...
srgidcl 20228 The unity element of a sem...
srg0cl 20229 The zero element of a semi...
srgidmlem 20230 Lemma for ~ srglidm and ~ ...
srglidm 20231 The unity element of a sem...
srgridm 20232 The unity element of a sem...
issrgid 20233 Properties showing that an...
srgacl 20234 Closure of the addition op...
srgcom 20235 Commutativity of the addit...
srgrz 20236 The zero of a semiring is ...
srglz 20237 The zero of a semiring is ...
srgisid 20238 In a semiring, the only le...
o2timesd 20239 An element of a ring-like ...
rglcom4d 20240 Restricted commutativity o...
srgo2times 20241 A semiring element plus it...
srgcom4lem 20242 Lemma for ~ srgcom4 . Thi...
srgcom4 20243 Restricted commutativity o...
srg1zr 20244 The only semiring with a b...
srgen1zr 20245 The only semiring with one...
srgmulgass 20246 An associative property be...
srgpcomp 20247 If two elements of a semir...
srgpcompp 20248 If two elements of a semir...
srgpcomppsc 20249 If two elements of a semir...
srglmhm 20250 Left-multiplication in a s...
srgrmhm 20251 Right-multiplication in a ...
srgsummulcr 20252 A finite semiring sum mult...
sgsummulcl 20253 A finite semiring sum mult...
srg1expzeq1 20254 The exponentiation (by a n...
srgbinomlem1 20255 Lemma 1 for ~ srgbinomlem ...
srgbinomlem2 20256 Lemma 2 for ~ srgbinomlem ...
srgbinomlem3 20257 Lemma 3 for ~ srgbinomlem ...
srgbinomlem4 20258 Lemma 4 for ~ srgbinomlem ...
srgbinomlem 20259 Lemma for ~ srgbinom . In...
srgbinom 20260 The binomial theorem for c...
csrgbinom 20261 The binomial theorem for c...
isring 20266 The predicate "is a (unita...
ringgrp 20267 A ring is a group. (Contr...
ringmgp 20268 A ring is a monoid under m...
iscrng 20269 A commutative ring is a ri...
crngmgp 20270 A commutative ring's multi...
ringgrpd 20271 A ring is a group. (Contr...
ringmnd 20272 A ring is a monoid under a...
ringmgm 20273 A ring is a magma. (Contr...
crngring 20274 A commutative ring is a ri...
crngringd 20275 A commutative ring is a ri...
crnggrpd 20276 A commutative ring is a gr...
mgpf 20277 Restricted functionality o...
ringdilem 20278 Properties of a unital rin...
ringcl 20279 Closure of the multiplicat...
crngcom 20280 A commutative ring's multi...
iscrng2 20281 A commutative ring is a ri...
ringass 20282 Associative law for multip...
ringideu 20283 The unity element of a rin...
crngcomd 20284 Multiplication is commutat...
crngbascntr 20285 The base set of a commutat...
ringassd 20286 Associative law for multip...
crng12d 20287 Commutative/associative la...
ringcld 20288 Closure of the multiplicat...
ringdi 20289 Distributive law for the m...
ringdir 20290 Distributive law for the m...
ringidcl 20291 The unity element of a rin...
ring0cl 20292 The zero element of a ring...
ringidmlem 20293 Lemma for ~ ringlidm and ~...
ringlidm 20294 The unity element of a rin...
ringridm 20295 The unity element of a rin...
isringid 20296 Properties showing that an...
ringlidmd 20297 The unity element of a rin...
ringridmd 20298 The unity element of a rin...
ringid 20299 The multiplication operati...
ringo2times 20300 A ring element plus itself...
ringadd2 20301 A ring element plus itself...
ringidss 20302 A subset of the multiplica...
ringacl 20303 Closure of the addition op...
ringcomlem 20304 Lemma for ~ ringcom . Thi...
ringcom 20305 Commutativity of the addit...
ringabl 20306 A ring is an Abelian group...
ringcmn 20307 A ring is a commutative mo...
ringabld 20308 A ring is an Abelian group...
ringcmnd 20309 A ring is a commutative mo...
ringrng 20310 A unital ring is a non-uni...
ringssrng 20311 The unital rings are non-u...
isringrng 20312 The predicate "is a unital...
ringpropd 20313 If two structures have the...
crngpropd 20314 If two structures have the...
ringprop 20315 If two structures have the...
isringd 20316 Properties that determine ...
iscrngd 20317 Properties that determine ...
ringlz 20318 The zero of a unital ring ...
ringrz 20319 The zero of a unital ring ...
ringlzd 20320 The zero of a unital ring ...
ringrzd 20321 The zero of a unital ring ...
ringsrg 20322 Any ring is also a semirin...
ring1eq0 20323 If one and zero are equal,...
ring1ne0 20324 If a ring has at least two...
ringinvnz1ne0 20325 In a unital ring, a left i...
ringinvnzdiv 20326 In a unital ring, a left i...
ringnegl 20327 Negation in a ring is the ...
ringnegr 20328 Negation in a ring is the ...
ringmneg1 20329 Negation of a product in a...
ringmneg2 20330 Negation of a product in a...
ringm2neg 20331 Double negation of a produ...
ringsubdi 20332 Ring multiplication distri...
ringsubdir 20333 Ring multiplication distri...
mulgass2 20334 An associative property be...
ring1 20335 The (smallest) structure r...
ringn0 20336 Rings exist. (Contributed...
ringlghm 20337 Left-multiplication in a r...
ringrghm 20338 Right-multiplication in a ...
gsummulc1OLD 20339 Obsolete version of ~ gsum...
gsummulc2OLD 20340 Obsolete version of ~ gsum...
gsummulc1 20341 A finite ring sum multipli...
gsummulc2 20342 A finite ring sum multipli...
gsummgp0 20343 If one factor in a finite ...
gsumdixp 20344 Distribute a binary produc...
prdsmulrcl 20345 A structure product of rin...
prdsringd 20346 A product of rings is a ri...
prdscrngd 20347 A product of commutative r...
prds1 20348 Value of the ring unity in...
pwsring 20349 A structure power of a rin...
pws1 20350 Value of the ring unity in...
pwscrng 20351 A structure power of a com...
pwsmgp 20352 The multiplicative group o...
pwspjmhmmgpd 20353 The projection given by ~ ...
pwsexpg 20354 Value of a group exponenti...
imasring 20355 The image structure of a r...
imasringf1 20356 The image of a ring under ...
xpsringd 20357 A product of two rings is ...
xpsring1d 20358 The multiplicative identit...
qusring2 20359 The quotient structure of ...
crngbinom 20360 The binomial theorem for c...
opprval 20363 Value of the opposite ring...
opprmulfval 20364 Value of the multiplicatio...
opprmul 20365 Value of the multiplicatio...
crngoppr 20366 In a commutative ring, the...
opprlem 20367 Lemma for ~ opprbas and ~ ...
opprlemOLD 20368 Obsolete version of ~ oppr...
opprbas 20369 Base set of an opposite ri...
opprbasOLD 20370 Obsolete proof of ~ opprba...
oppradd 20371 Addition operation of an o...
oppraddOLD 20372 Obsolete proof of ~ opprba...
opprrng 20373 An opposite non-unital rin...
opprrngb 20374 A class is a non-unital ri...
opprring 20375 An opposite ring is a ring...
opprringb 20376 Bidirectional form of ~ op...
oppr0 20377 Additive identity of an op...
oppr1 20378 Multiplicative identity of...
opprneg 20379 The negative function in a...
opprsubg 20380 Being a subgroup is a symm...
mulgass3 20381 An associative property be...
reldvdsr 20388 The divides relation is a ...
dvdsrval 20389 Value of the divides relat...
dvdsr 20390 Value of the divides relat...
dvdsr2 20391 Value of the divides relat...
dvdsrmul 20392 A left-multiple of ` X ` i...
dvdsrcl 20393 Closure of a dividing elem...
dvdsrcl2 20394 Closure of a dividing elem...
dvdsrid 20395 An element in a (unital) r...
dvdsrtr 20396 Divisibility is transitive...
dvdsrmul1 20397 The divisibility relation ...
dvdsrneg 20398 An element divides its neg...
dvdsr01 20399 In a ring, zero is divisib...
dvdsr02 20400 Only zero is divisible by ...
isunit 20401 Property of being a unit o...
1unit 20402 The multiplicative identit...
unitcl 20403 A unit is an element of th...
unitss 20404 The set of units is contai...
opprunit 20405 Being a unit is a symmetri...
crngunit 20406 Property of being a unit i...
dvdsunit 20407 A divisor of a unit is a u...
unitmulcl 20408 The product of units is a ...
unitmulclb 20409 Reversal of ~ unitmulcl in...
unitgrpbas 20410 The base set of the group ...
unitgrp 20411 The group of units is a gr...
unitabl 20412 The group of units of a co...
unitgrpid 20413 The identity of the group ...
unitsubm 20414 The group of units is a su...
invrfval 20417 Multiplicative inverse fun...
unitinvcl 20418 The inverse of a unit exis...
unitinvinv 20419 The inverse of the inverse...
ringinvcl 20420 The inverse of a unit is a...
unitlinv 20421 A unit times its inverse i...
unitrinv 20422 A unit times its inverse i...
1rinv 20423 The inverse of the ring un...
0unit 20424 The additive identity is a...
unitnegcl 20425 The negative of a unit is ...
ringunitnzdiv 20426 In a unitary ring, a unit ...
ring1nzdiv 20427 In a unitary ring, the rin...
dvrfval 20430 Division operation in a ri...
dvrval 20431 Division operation in a ri...
dvrcl 20432 Closure of division operat...
unitdvcl 20433 The units are closed under...
dvrid 20434 A ring element divided by ...
dvr1 20435 A ring element divided by ...
dvrass 20436 An associative law for div...
dvrcan1 20437 A cancellation law for div...
dvrcan3 20438 A cancellation law for div...
dvreq1 20439 Equality in terms of ratio...
dvrdir 20440 Distributive law for the d...
rdivmuldivd 20441 Multiplication of two rati...
ringinvdv 20442 Write the inverse function...
rngidpropd 20443 The ring unity depends onl...
dvdsrpropd 20444 The divisibility relation ...
unitpropd 20445 The set of units depends o...
invrpropd 20446 The ring inverse function ...
isirred 20447 An irreducible element of ...
isnirred 20448 The property of being a no...
isirred2 20449 Expand out the class diffe...
opprirred 20450 Irreducibility is symmetri...
irredn0 20451 The additive identity is n...
irredcl 20452 An irreducible element is ...
irrednu 20453 An irreducible element is ...
irredn1 20454 The multiplicative identit...
irredrmul 20455 The product of an irreduci...
irredlmul 20456 The product of a unit and ...
irredmul 20457 If product of two elements...
irredneg 20458 The negative of an irreduc...
irrednegb 20459 An element is irreducible ...
rnghmrcl 20466 Reverse closure of a non-u...
rnghmfn 20467 The mapping of two non-uni...
rnghmval 20468 The set of the non-unital ...
isrnghm 20469 A function is a non-unital...
isrnghmmul 20470 A function is a non-unital...
rnghmmgmhm 20471 A non-unital ring homomorp...
rnghmval2 20472 The non-unital ring homomo...
isrngim 20473 An isomorphism of non-unit...
rngimrcl 20474 Reverse closure for an iso...
rnghmghm 20475 A non-unital ring homomorp...
rnghmf 20476 A ring homomorphism is a f...
rnghmmul 20477 A homomorphism of non-unit...
isrnghm2d 20478 Demonstration of non-unita...
isrnghmd 20479 Demonstration of non-unita...
rnghmf1o 20480 A non-unital ring homomorp...
isrngim2 20481 An isomorphism of non-unit...
rngimf1o 20482 An isomorphism of non-unit...
rngimrnghm 20483 An isomorphism of non-unit...
rngimcnv 20484 The converse of an isomorp...
rnghmco 20485 The composition of non-uni...
idrnghm 20486 The identity homomorphism ...
c0mgm 20487 The constant mapping to ze...
c0mhm 20488 The constant mapping to ze...
c0ghm 20489 The constant mapping to ze...
c0snmgmhm 20490 The constant mapping to ze...
c0snmhm 20491 The constant mapping to ze...
c0snghm 20492 The constant mapping to ze...
rngisomfv1 20493 If there is a non-unital r...
rngisom1 20494 If there is a non-unital r...
rngisomring 20495 If there is a non-unital r...
rngisomring1 20496 If there is a non-unital r...
dfrhm2 20502 The property of a ring hom...
rhmrcl1 20504 Reverse closure of a ring ...
rhmrcl2 20505 Reverse closure of a ring ...
isrhm 20506 A function is a ring homom...
rhmmhm 20507 A ring homomorphism is a h...
rhmisrnghm 20508 Each unital ring homomorph...
isrim0OLD 20509 Obsolete version of ~ isri...
rimrcl 20510 Reverse closure for an iso...
isrim0 20511 A ring isomorphism is a ho...
rhmghm 20512 A ring homomorphism is an ...
rhmf 20513 A ring homomorphism is a f...
rhmmul 20514 A homomorphism of rings pr...
isrhm2d 20515 Demonstration of ring homo...
isrhmd 20516 Demonstration of ring homo...
rhm1 20517 Ring homomorphisms are req...
idrhm 20518 The identity homomorphism ...
rhmf1o 20519 A ring homomorphism is bij...
isrim 20520 An isomorphism of rings is...
isrimOLD 20521 Obsolete version of ~ isri...
rimf1o 20522 An isomorphism of rings is...
rimrhmOLD 20523 Obsolete version of ~ rimr...
rimrhm 20524 A ring isomorphism is a ho...
rimgim 20525 An isomorphism of rings is...
rimisrngim 20526 Each unital ring isomorphi...
rhmfn 20527 The mapping of two rings t...
rhmval 20528 The ring homomorphisms bet...
rhmco 20529 The composition of ring ho...
pwsco1rhm 20530 Right composition with a f...
pwsco2rhm 20531 Left composition with a ri...
brric 20532 The relation "is isomorphi...
brrici 20533 Prove isomorphic by an exp...
brric2 20534 The relation "is isomorphi...
ricgic 20535 If two rings are (ring) is...
rhmdvdsr 20536 A ring homomorphism preser...
rhmopp 20537 A ring homomorphism is als...
elrhmunit 20538 Ring homomorphisms preserv...
rhmunitinv 20539 Ring homomorphisms preserv...
isnzr 20542 Property of a nonzero ring...
nzrnz 20543 One and zero are different...
nzrring 20544 A nonzero ring is a ring. ...
nzrringOLD 20545 Obsolete version of ~ nzrr...
isnzr2 20546 Equivalent characterizatio...
isnzr2hash 20547 Equivalent characterizatio...
nzrpropd 20548 If two structures have the...
opprnzrb 20549 The opposite of a nonzero ...
opprnzr 20550 The opposite of a nonzero ...
ringelnzr 20551 A ring is nonzero if it ha...
nzrunit 20552 A unit is nonzero in any n...
0ringnnzr 20553 A ring is a zero ring iff ...
0ring 20554 If a ring has only one ele...
0ringdif 20555 A zero ring is a ring whic...
0ringbas 20556 The base set of a zero rin...
0ring01eq 20557 In a ring with only one el...
01eq0ring 20558 If the zero and the identi...
01eq0ringOLD 20559 Obsolete version of ~ 01eq...
0ring01eqbi 20560 In a unital ring the zero ...
0ring1eq0 20561 In a zero ring, a ring whi...
c0rhm 20562 The constant mapping to ze...
c0rnghm 20563 The constant mapping to ze...
zrrnghm 20564 The constant mapping to ze...
nrhmzr 20565 There is no ring homomorph...
islring 20568 The predicate "is a local ...
lringnzr 20569 A local ring is a nonzero ...
lringring 20570 A local ring is a ring. (...
lringnz 20571 A local ring is a nonzero ...
lringuplu 20572 If the sum of two elements...
issubrng 20575 The subring of non-unital ...
subrngss 20576 A subring is a subset. (C...
subrngid 20577 Every non-unital ring is a...
subrngrng 20578 A subring is a non-unital ...
subrngrcl 20579 Reverse closure for a subr...
subrngsubg 20580 A subring is a subgroup. ...
subrngringnsg 20581 A subring is a normal subg...
subrngbas 20582 Base set of a subring stru...
subrng0 20583 A subring always has the s...
subrngacl 20584 A subring is closed under ...
subrngmcl 20585 A subring is closed under ...
issubrng2 20586 Characterize the subrings ...
opprsubrng 20587 Being a subring is a symme...
subrngint 20588 The intersection of a none...
subrngin 20589 The intersection of two su...
subrngmre 20590 The subrings of a non-unit...
subsubrng 20591 A subring of a subring is ...
subsubrng2 20592 The set of subrings of a s...
rhmimasubrnglem 20593 Lemma for ~ rhmimasubrng :...
rhmimasubrng 20594 The homomorphic image of a...
cntzsubrng 20595 Centralizers in a non-unit...
subrngpropd 20596 If two structures have the...
issubrg 20601 The subring predicate. (C...
subrgss 20602 A subring is a subset. (C...
subrgid 20603 Every ring is a subring of...
subrgring 20604 A subring is a ring. (Con...
subrgcrng 20605 A subring of a commutative...
subrgrcl 20606 Reverse closure for a subr...
subrgsubg 20607 A subring is a subgroup. ...
subrgsubrng 20608 A subring of a unital ring...
subrg0 20609 A subring always has the s...
subrg1cl 20610 A subring contains the mul...
subrgbas 20611 Base set of a subring stru...
subrg1 20612 A subring always has the s...
subrgacl 20613 A subring is closed under ...
subrgmcl 20614 A subring is closed under ...
subrgsubm 20615 A subring is a submonoid o...
subrgdvds 20616 If an element divides anot...
subrguss 20617 A unit of a subring is a u...
subrginv 20618 A subring always has the s...
subrgdv 20619 A subring always has the s...
subrgunit 20620 An element of a ring is a ...
subrgugrp 20621 The units of a subring for...
issubrg2 20622 Characterize the subrings ...
opprsubrg 20623 Being a subring is a symme...
subrgnzr 20624 A subring of a nonzero rin...
subrgint 20625 The intersection of a none...
subrgin 20626 The intersection of two su...
subrgmre 20627 The subrings of a ring are...
subsubrg 20628 A subring of a subring is ...
subsubrg2 20629 The set of subrings of a s...
issubrg3 20630 A subring is an additive s...
resrhm 20631 Restriction of a ring homo...
resrhm2b 20632 Restriction of the codomai...
rhmeql 20633 The equalizer of two ring ...
rhmima 20634 The homomorphic image of a...
rnrhmsubrg 20635 The range of a ring homomo...
cntzsubr 20636 Centralizers in a ring are...
pwsdiagrhm 20637 Diagonal homomorphism into...
subrgpropd 20638 If two structures have the...
rhmpropd 20639 Ring homomorphism depends ...
rngcval 20642 Value of the category of n...
rnghmresfn 20643 The class of non-unital ri...
rnghmresel 20644 An element of the non-unit...
rngcbas 20645 Set of objects of the cate...
rngchomfval 20646 Set of arrows of the categ...
rngchom 20647 Set of arrows of the categ...
elrngchom 20648 A morphism of non-unital r...
rngchomfeqhom 20649 The functionalized Hom-set...
rngccofval 20650 Composition in the categor...
rngcco 20651 Composition in the categor...
dfrngc2 20652 Alternate definition of th...
rnghmsscmap2 20653 The non-unital ring homomo...
rnghmsscmap 20654 The non-unital ring homomo...
rnghmsubcsetclem1 20655 Lemma 1 for ~ rnghmsubcset...
rnghmsubcsetclem2 20656 Lemma 2 for ~ rnghmsubcset...
rnghmsubcsetc 20657 The non-unital ring homomo...
rngccat 20658 The category of non-unital...
rngcid 20659 The identity arrow in the ...
rngcsect 20660 A section in the category ...
rngcinv 20661 An inverse in the category...
rngciso 20662 An isomorphism in the cate...
rngcifuestrc 20663 The "inclusion functor" fr...
funcrngcsetc 20664 The "natural forgetful fun...
funcrngcsetcALT 20665 Alternate proof of ~ funcr...
zrinitorngc 20666 The zero ring is an initia...
zrtermorngc 20667 The zero ring is a termina...
zrzeroorngc 20668 The zero ring is a zero ob...
ringcval 20671 Value of the category of u...
rhmresfn 20672 The class of unital ring h...
rhmresel 20673 An element of the unital r...
ringcbas 20674 Set of objects of the cate...
ringchomfval 20675 Set of arrows of the categ...
ringchom 20676 Set of arrows of the categ...
elringchom 20677 A morphism of unital rings...
ringchomfeqhom 20678 The functionalized Hom-set...
ringccofval 20679 Composition in the categor...
ringcco 20680 Composition in the categor...
dfringc2 20681 Alternate definition of th...
rhmsscmap2 20682 The unital ring homomorphi...
rhmsscmap 20683 The unital ring homomorphi...
rhmsubcsetclem1 20684 Lemma 1 for ~ rhmsubcsetc ...
rhmsubcsetclem2 20685 Lemma 2 for ~ rhmsubcsetc ...
rhmsubcsetc 20686 The unital ring homomorphi...
ringccat 20687 The category of unital rin...
ringcid 20688 The identity arrow in the ...
rhmsscrnghm 20689 The unital ring homomorphi...
rhmsubcrngclem1 20690 Lemma 1 for ~ rhmsubcrngc ...
rhmsubcrngclem2 20691 Lemma 2 for ~ rhmsubcrngc ...
rhmsubcrngc 20692 The unital ring homomorphi...
rngcresringcat 20693 The restriction of the cat...
ringcsect 20694 A section in the category ...
ringcinv 20695 An inverse in the category...
ringciso 20696 An isomorphism in the cate...
ringcbasbas 20697 An element of the base set...
funcringcsetc 20698 The "natural forgetful fun...
zrtermoringc 20699 The zero ring is a termina...
zrninitoringc 20700 The zero ring is not an in...
srhmsubclem1 20701 Lemma 1 for ~ srhmsubc . ...
srhmsubclem2 20702 Lemma 2 for ~ srhmsubc . ...
srhmsubclem3 20703 Lemma 3 for ~ srhmsubc . ...
srhmsubc 20704 According to ~ df-subc , t...
sringcat 20705 The restriction of the cat...
crhmsubc 20706 According to ~ df-subc , t...
cringcat 20707 The restriction of the cat...
rngcrescrhm 20708 The category of non-unital...
rhmsubclem1 20709 Lemma 1 for ~ rhmsubc . (...
rhmsubclem2 20710 Lemma 2 for ~ rhmsubc . (...
rhmsubclem3 20711 Lemma 3 for ~ rhmsubc . (...
rhmsubclem4 20712 Lemma 4 for ~ rhmsubc . (...
rhmsubc 20713 According to ~ df-subc , t...
rhmsubccat 20714 The restriction of the cat...
rrgval 20721 Value of the set or left-r...
isrrg 20722 Membership in the set of l...
rrgeq0i 20723 Property of a left-regular...
rrgeq0 20724 Left-multiplication by a l...
rrgsupp 20725 Left multiplication by a l...
rrgss 20726 Left-regular elements are ...
unitrrg 20727 Units are regular elements...
rrgnz 20728 In a nonzero ring, the zer...
isdomn 20729 Expand definition of a dom...
domnnzr 20730 A domain is a nonzero ring...
domnring 20731 A domain is a ring. (Cont...
domneq0 20732 In a domain, a product is ...
domnmuln0 20733 In a domain, a product of ...
isdomn5 20734 The equivalence between th...
isdomn2 20735 A ring is a domain iff all...
isdomn2OLD 20736 Obsolete version of ~ isdo...
domnrrg 20737 In a domain, a nonzero ele...
isdomn6 20738 A ring is a domain iff the...
isdomn3 20739 Nonzero elements form a mu...
isdomn4 20740 A ring is a domain iff it ...
opprdomnb 20741 A class is a domain if and...
opprdomn 20742 The opposite of a domain i...
isdomn4r 20743 A ring is a domain iff it ...
domnlcanb 20744 Left-cancellation law for ...
domnlcan 20745 Left-cancellation law for ...
domnrcanb 20746 Right-cancellation law for...
domnrcan 20747 Right-cancellation law for...
domneq0r 20748 Right multiplication by a ...
isidom 20749 An integral domain is a co...
idomdomd 20750 An integral domain is a do...
idomcringd 20751 An integral domain is a co...
idomringd 20752 An integral domain is a ri...
isdrng 20757 The predicate "is a divisi...
drngunit 20758 Elementhood in the set of ...
drngui 20759 The set of units of a divi...
drngring 20760 A division ring is a ring....
drngringd 20761 A division ring is a ring....
drnggrpd 20762 A division ring is a group...
drnggrp 20763 A division ring is a group...
isfld 20764 A field is a commutative d...
flddrngd 20765 A field is a division ring...
fldcrngd 20766 A field is a commutative r...
isdrng2 20767 A division ring can equiva...
drngprop 20768 If two structures have the...
drngmgp 20769 A division ring contains a...
drngid 20770 A division ring's unity is...
drngunz 20771 A division ring's unity is...
drngnzr 20772 A division ring is a nonze...
drngdomn 20773 A division ring is a domai...
drngmcl 20774 The product of two nonzero...
drngmclOLD 20775 Obsolete version of ~ drng...
drngid2 20776 Properties showing that an...
drnginvrcl 20777 Closure of the multiplicat...
drnginvrn0 20778 The multiplicative inverse...
drnginvrcld 20779 Closure of the multiplicat...
drnginvrl 20780 Property of the multiplica...
drnginvrr 20781 Property of the multiplica...
drnginvrld 20782 Property of the multiplica...
drnginvrrd 20783 Property of the multiplica...
drngmul0or 20784 A product is zero iff one ...
drngmul0orOLD 20785 Obsolete version of ~ drng...
drngmulne0 20786 A product is nonzero iff b...
drngmuleq0 20787 An element is zero iff its...
opprdrng 20788 The opposite of a division...
isdrngd 20789 Properties that characteri...
isdrngrd 20790 Properties that characteri...
isdrngdOLD 20791 Obsolete version of ~ isdr...
isdrngrdOLD 20792 Obsolete version of ~ isdr...
drngpropd 20793 If two structures have the...
fldpropd 20794 If two structures have the...
fldidom 20795 A field is an integral dom...
fldidomOLD 20796 Obsolete version of ~ fldi...
fidomndrnglem 20797 Lemma for ~ fidomndrng . ...
fidomndrng 20798 A finite domain is a divis...
fiidomfld 20799 A finite integral domain i...
rng1nnzr 20800 The (smallest) structure r...
ring1zr 20801 The only (unital) ring wit...
rngen1zr 20802 The only (unital) ring wit...
ringen1zr 20803 The only unital ring with ...
rng1nfld 20804 The zero ring is not a fie...
issubdrg 20805 Characterize the subfields...
drhmsubc 20806 According to ~ df-subc , t...
drngcat 20807 The restriction of the cat...
fldcat 20808 The restriction of the cat...
fldc 20809 The restriction of the cat...
fldhmsubc 20810 According to ~ df-subc , t...
issdrg 20813 Property of a division sub...
sdrgrcl 20814 Reverse closure for a sub-...
sdrgdrng 20815 A sub-division-ring is a d...
sdrgsubrg 20816 A sub-division-ring is a s...
sdrgid 20817 Every division ring is a d...
sdrgss 20818 A division subring is a su...
sdrgbas 20819 Base set of a sub-division...
issdrg2 20820 Property of a division sub...
sdrgunit 20821 A unit of a sub-division-r...
imadrhmcl 20822 The image of a (nontrivial...
fldsdrgfld 20823 A sub-division-ring of a f...
acsfn1p 20824 Construction of a closure ...
subrgacs 20825 Closure property of subrin...
sdrgacs 20826 Closure property of divisi...
cntzsdrg 20827 Centralizers in division r...
subdrgint 20828 The intersection of a none...
sdrgint 20829 The intersection of a none...
primefld 20830 The smallest sub division ...
primefld0cl 20831 The prime field contains t...
primefld1cl 20832 The prime field contains t...
abvfval 20835 Value of the set of absolu...
isabv 20836 Elementhood in the set of ...
isabvd 20837 Properties that determine ...
abvrcl 20838 Reverse closure for the ab...
abvfge0 20839 An absolute value is a fun...
abvf 20840 An absolute value is a fun...
abvcl 20841 An absolute value is a fun...
abvge0 20842 The absolute value of a nu...
abveq0 20843 The value of an absolute v...
abvne0 20844 The absolute value of a no...
abvgt0 20845 The absolute value of a no...
abvmul 20846 An absolute value distribu...
abvtri 20847 An absolute value satisfie...
abv0 20848 The absolute value of zero...
abv1z 20849 The absolute value of one ...
abv1 20850 The absolute value of one ...
abvneg 20851 The absolute value of a ne...
abvsubtri 20852 An absolute value satisfie...
abvrec 20853 The absolute value distrib...
abvdiv 20854 The absolute value distrib...
abvdom 20855 Any ring with an absolute ...
abvres 20856 The restriction of an abso...
abvtrivd 20857 The trivial absolute value...
abvtrivg 20858 The trivial absolute value...
abvtriv 20859 The trivial absolute value...
abvpropd 20860 If two structures have the...
abvn0b 20861 Another characterization o...
staffval 20866 The functionalization of t...
stafval 20867 The functionalization of t...
staffn 20868 The functionalization is e...
issrng 20869 The predicate "is a star r...
srngrhm 20870 The involution function in...
srngring 20871 A star ring is a ring. (C...
srngcnv 20872 The involution function in...
srngf1o 20873 The involution function in...
srngcl 20874 The involution function in...
srngnvl 20875 The involution function in...
srngadd 20876 The involution function in...
srngmul 20877 The involution function in...
srng1 20878 The conjugate of the ring ...
srng0 20879 The conjugate of the ring ...
issrngd 20880 Properties that determine ...
idsrngd 20881 A commutative ring is a st...
islmod 20886 The predicate "is a left m...
lmodlema 20887 Lemma for properties of a ...
islmodd 20888 Properties that determine ...
lmodgrp 20889 A left module is a group. ...
lmodring 20890 The scalar component of a ...
lmodfgrp 20891 The scalar component of a ...
lmodgrpd 20892 A left module is a group. ...
lmodbn0 20893 The base set of a left mod...
lmodacl 20894 Closure of ring addition f...
lmodmcl 20895 Closure of ring multiplica...
lmodsn0 20896 The set of scalars in a le...
lmodvacl 20897 Closure of vector addition...
lmodass 20898 Left module vector sum is ...
lmodlcan 20899 Left cancellation law for ...
lmodvscl 20900 Closure of scalar product ...
lmodvscld 20901 Closure of scalar product ...
scaffval 20902 The scalar multiplication ...
scafval 20903 The scalar multiplication ...
scafeq 20904 If the scalar multiplicati...
scaffn 20905 The scalar multiplication ...
lmodscaf 20906 The scalar multiplication ...
lmodvsdi 20907 Distributive law for scala...
lmodvsdir 20908 Distributive law for scala...
lmodvsass 20909 Associative law for scalar...
lmod0cl 20910 The ring zero in a left mo...
lmod1cl 20911 The ring unity in a left m...
lmodvs1 20912 Scalar product with the ri...
lmod0vcl 20913 The zero vector is a vecto...
lmod0vlid 20914 Left identity law for the ...
lmod0vrid 20915 Right identity law for the...
lmod0vid 20916 Identity equivalent to the...
lmod0vs 20917 Zero times a vector is the...
lmodvs0 20918 Anything times the zero ve...
lmodvsmmulgdi 20919 Distributive law for a gro...
lmodfopnelem1 20920 Lemma 1 for ~ lmodfopne . ...
lmodfopnelem2 20921 Lemma 2 for ~ lmodfopne . ...
lmodfopne 20922 The (functionalized) opera...
lcomf 20923 A linear-combination sum i...
lcomfsupp 20924 A linear-combination sum i...
lmodvnegcl 20925 Closure of vector negative...
lmodvnegid 20926 Addition of a vector with ...
lmodvneg1 20927 Minus 1 times a vector is ...
lmodvsneg 20928 Multiplication of a vector...
lmodvsubcl 20929 Closure of vector subtract...
lmodcom 20930 Left module vector sum is ...
lmodabl 20931 A left module is an abelia...
lmodcmn 20932 A left module is a commuta...
lmodnegadd 20933 Distribute negation throug...
lmod4 20934 Commutative/associative la...
lmodvsubadd 20935 Relationship between vecto...
lmodvaddsub4 20936 Vector addition/subtractio...
lmodvpncan 20937 Addition/subtraction cance...
lmodvnpcan 20938 Cancellation law for vecto...
lmodvsubval2 20939 Value of vector subtractio...
lmodsubvs 20940 Subtraction of a scalar pr...
lmodsubdi 20941 Scalar multiplication dist...
lmodsubdir 20942 Scalar multiplication dist...
lmodsubeq0 20943 If the difference between ...
lmodsubid 20944 Subtraction of a vector fr...
lmodvsghm 20945 Scalar multiplication of t...
lmodprop2d 20946 If two structures have the...
lmodpropd 20947 If two structures have the...
gsumvsmul 20948 Pull a scalar multiplicati...
mptscmfsupp0 20949 A mapping to a scalar prod...
mptscmfsuppd 20950 A function mapping to a sc...
rmodislmodlem 20951 Lemma for ~ rmodislmod . ...
rmodislmod 20952 The right module ` R ` ind...
rmodislmodOLD 20953 Obsolete version of ~ rmod...
lssset 20956 The set of all (not necess...
islss 20957 The predicate "is a subspa...
islssd 20958 Properties that determine ...
lssss 20959 A subspace is a set of vec...
lssel 20960 A subspace member is a vec...
lss1 20961 The set of vectors in a le...
lssuni 20962 The union of all subspaces...
lssn0 20963 A subspace is not empty. ...
00lss 20964 The empty structure has no...
lsscl 20965 Closure property of a subs...
lssvacl 20966 Closure of vector addition...
lssvsubcl 20967 Closure of vector subtract...
lssvancl1 20968 Non-closure: if one vector...
lssvancl2 20969 Non-closure: if one vector...
lss0cl 20970 The zero vector belongs to...
lsssn0 20971 The singleton of the zero ...
lss0ss 20972 The zero subspace is inclu...
lssle0 20973 No subspace is smaller tha...
lssne0 20974 A nonzero subspace has a n...
lssvneln0 20975 A vector ` X ` which doesn...
lssneln0 20976 A vector ` X ` which doesn...
lssssr 20977 Conclude subspace ordering...
lssvscl 20978 Closure of scalar product ...
lssvnegcl 20979 Closure of negative vector...
lsssubg 20980 All subspaces are subgroup...
lsssssubg 20981 All subspaces are subgroup...
islss3 20982 A linear subspace of a mod...
lsslmod 20983 A submodule is a module. ...
lsslss 20984 The subspaces of a subspac...
islss4 20985 A linear subspace is a sub...
lss1d 20986 One-dimensional subspace (...
lssintcl 20987 The intersection of a none...
lssincl 20988 The intersection of two su...
lssmre 20989 The subspaces of a module ...
lssacs 20990 Submodules are an algebrai...
prdsvscacl 20991 Pointwise scalar multiplic...
prdslmodd 20992 The product of a family of...
pwslmod 20993 A structure power of a lef...
lspfval 20996 The span function for a le...
lspf 20997 The span function on a lef...
lspval 20998 The span of a set of vecto...
lspcl 20999 The span of a set of vecto...
lspsncl 21000 The span of a singleton is...
lspprcl 21001 The span of a pair is a su...
lsptpcl 21002 The span of an unordered t...
lspsnsubg 21003 The span of a singleton is...
00lsp 21004 ~ fvco4i lemma for linear ...
lspid 21005 The span of a subspace is ...
lspssv 21006 A span is a set of vectors...
lspss 21007 Span preserves subset orde...
lspssid 21008 A set of vectors is a subs...
lspidm 21009 The span of a set of vecto...
lspun 21010 The span of union is the s...
lspssp 21011 If a set of vectors is a s...
mrclsp 21012 Moore closure generalizes ...
lspsnss 21013 The span of the singleton ...
ellspsn3 21014 A member of the span of th...
lspprss 21015 The span of a pair of vect...
lspsnid 21016 A vector belongs to the sp...
ellspsn6 21017 Relationship between a vec...
ellspsn5b 21018 Relationship between a vec...
ellspsn5 21019 Relationship between a vec...
lspprid1 21020 A member of a pair of vect...
lspprid2 21021 A member of a pair of vect...
lspprvacl 21022 The sum of two vectors bel...
lssats2 21023 A way to express atomistic...
ellspsni 21024 A scalar product with a ve...
lspsn 21025 Span of the singleton of a...
ellspsn 21026 Member of span of the sing...
lspsnvsi 21027 Span of a scalar product o...
lspsnss2 21028 Comparable spans of single...
lspsnneg 21029 Negation does not change t...
lspsnsub 21030 Swapping subtraction order...
lspsn0 21031 Span of the singleton of t...
lsp0 21032 Span of the empty set. (C...
lspuni0 21033 Union of the span of the e...
lspun0 21034 The span of a union with t...
lspsneq0 21035 Span of the singleton is t...
lspsneq0b 21036 Equal singleton spans impl...
lmodindp1 21037 Two independent (non-colin...
lsslsp 21038 Spans in submodules corres...
lsslspOLD 21039 Obsolete version of ~ lssl...
lss0v 21040 The zero vector in a submo...
lsspropd 21041 If two structures have the...
lsppropd 21042 If two structures have the...
reldmlmhm 21049 Lemma for module homomorph...
lmimfn 21050 Lemma for module isomorphi...
islmhm 21051 Property of being a homomo...
islmhm3 21052 Property of a module homom...
lmhmlem 21053 Non-quantified consequence...
lmhmsca 21054 A homomorphism of left mod...
lmghm 21055 A homomorphism of left mod...
lmhmlmod2 21056 A homomorphism of left mod...
lmhmlmod1 21057 A homomorphism of left mod...
lmhmf 21058 A homomorphism of left mod...
lmhmlin 21059 A homomorphism of left mod...
lmodvsinv 21060 Multiplication of a vector...
lmodvsinv2 21061 Multiplying a negated vect...
islmhm2 21062 A one-equation proof of li...
islmhmd 21063 Deduction for a module hom...
0lmhm 21064 The constant zero linear f...
idlmhm 21065 The identity function on a...
invlmhm 21066 The negative function on a...
lmhmco 21067 The composition of two mod...
lmhmplusg 21068 The pointwise sum of two l...
lmhmvsca 21069 The pointwise scalar produ...
lmhmf1o 21070 A bijective module homomor...
lmhmima 21071 The image of a subspace un...
lmhmpreima 21072 The inverse image of a sub...
lmhmlsp 21073 Homomorphisms preserve spa...
lmhmrnlss 21074 The range of a homomorphis...
lmhmkerlss 21075 The kernel of a homomorphi...
reslmhm 21076 Restriction of a homomorph...
reslmhm2 21077 Expansion of the codomain ...
reslmhm2b 21078 Expansion of the codomain ...
lmhmeql 21079 The equalizer of two modul...
lspextmo 21080 A linear function is compl...
pwsdiaglmhm 21081 Diagonal homomorphism into...
pwssplit0 21082 Splitting for structure po...
pwssplit1 21083 Splitting for structure po...
pwssplit2 21084 Splitting for structure po...
pwssplit3 21085 Splitting for structure po...
islmim 21086 An isomorphism of left mod...
lmimf1o 21087 An isomorphism of left mod...
lmimlmhm 21088 An isomorphism of modules ...
lmimgim 21089 An isomorphism of modules ...
islmim2 21090 An isomorphism of left mod...
lmimcnv 21091 The converse of a bijectiv...
brlmic 21092 The relation "is isomorphi...
brlmici 21093 Prove isomorphic by an exp...
lmiclcl 21094 Isomorphism implies the le...
lmicrcl 21095 Isomorphism implies the ri...
lmicsym 21096 Module isomorphism is symm...
lmhmpropd 21097 Module homomorphism depend...
islbs 21100 The predicate " ` B ` is a...
lbsss 21101 A basis is a set of vector...
lbsel 21102 An element of a basis is a...
lbssp 21103 The span of a basis is the...
lbsind 21104 A basis is linearly indepe...
lbsind2 21105 A basis is linearly indepe...
lbspss 21106 No proper subset of a basi...
lsmcl 21107 The sum of two subspaces i...
lsmspsn 21108 Member of subspace sum of ...
lsmelval2 21109 Subspace sum membership in...
lsmsp 21110 Subspace sum in terms of s...
lsmsp2 21111 Subspace sum of spans of s...
lsmssspx 21112 Subspace sum (in its exten...
lsmpr 21113 The span of a pair of vect...
lsppreli 21114 A vector expressed as a su...
lsmelpr 21115 Two ways to say that a vec...
lsppr0 21116 The span of a vector paire...
lsppr 21117 Span of a pair of vectors....
lspprel 21118 Member of the span of a pa...
lspprabs 21119 Absorption of vector sum i...
lspvadd 21120 The span of a vector sum i...
lspsntri 21121 Triangle-type inequality f...
lspsntrim 21122 Triangle-type inequality f...
lbspropd 21123 If two structures have the...
pj1lmhm 21124 The left projection functi...
pj1lmhm2 21125 The left projection functi...
islvec 21128 The predicate "is a left v...
lvecdrng 21129 The set of scalars of a le...
lveclmod 21130 A left vector space is a l...
lveclmodd 21131 A vector space is a left m...
lvecgrpd 21132 A vector space is a group....
lsslvec 21133 A vector subspace is a vec...
lmhmlvec 21134 The property for modules t...
lvecvs0or 21135 If a scalar product is zer...
lvecvsn0 21136 A scalar product is nonzer...
lssvs0or 21137 If a scalar product belong...
lvecvscan 21138 Cancellation law for scala...
lvecvscan2 21139 Cancellation law for scala...
lvecinv 21140 Invert coefficient of scal...
lspsnvs 21141 A nonzero scalar product d...
lspsneleq 21142 Membership relation that i...
lspsncmp 21143 Comparable spans of nonzer...
lspsnne1 21144 Two ways to express that v...
lspsnne2 21145 Two ways to express that v...
lspsnnecom 21146 Swap two vectors with diff...
lspabs2 21147 Absorption law for span of...
lspabs3 21148 Absorption law for span of...
lspsneq 21149 Equal spans of singletons ...
lspsneu 21150 Nonzero vectors with equal...
ellspsn4 21151 A member of the span of th...
lspdisj 21152 The span of a vector not i...
lspdisjb 21153 A nonzero vector is not in...
lspdisj2 21154 Unequal spans are disjoint...
lspfixed 21155 Show membership in the spa...
lspexch 21156 Exchange property for span...
lspexchn1 21157 Exchange property for span...
lspexchn2 21158 Exchange property for span...
lspindpi 21159 Partial independence prope...
lspindp1 21160 Alternate way to say 3 vec...
lspindp2l 21161 Alternate way to say 3 vec...
lspindp2 21162 Alternate way to say 3 vec...
lspindp3 21163 Independence of 2 vectors ...
lspindp4 21164 (Partial) independence of ...
lvecindp 21165 Compute the ` X ` coeffici...
lvecindp2 21166 Sums of independent vector...
lspsnsubn0 21167 Unequal singleton spans im...
lsmcv 21168 Subspace sum has the cover...
lspsolvlem 21169 Lemma for ~ lspsolv . (Co...
lspsolv 21170 If ` X ` is in the span of...
lssacsex 21171 In a vector space, subspac...
lspsnat 21172 There is no subspace stric...
lspsncv0 21173 The span of a singleton co...
lsppratlem1 21174 Lemma for ~ lspprat . Let...
lsppratlem2 21175 Lemma for ~ lspprat . Sho...
lsppratlem3 21176 Lemma for ~ lspprat . In ...
lsppratlem4 21177 Lemma for ~ lspprat . In ...
lsppratlem5 21178 Lemma for ~ lspprat . Com...
lsppratlem6 21179 Lemma for ~ lspprat . Neg...
lspprat 21180 A proper subspace of the s...
islbs2 21181 An equivalent formulation ...
islbs3 21182 An equivalent formulation ...
lbsacsbs 21183 Being a basis in a vector ...
lvecdim 21184 The dimension theorem for ...
lbsextlem1 21185 Lemma for ~ lbsext . The ...
lbsextlem2 21186 Lemma for ~ lbsext . Sinc...
lbsextlem3 21187 Lemma for ~ lbsext . A ch...
lbsextlem4 21188 Lemma for ~ lbsext . ~ lbs...
lbsextg 21189 For any linearly independe...
lbsext 21190 For any linearly independe...
lbsexg 21191 Every vector space has a b...
lbsex 21192 Every vector space has a b...
lvecprop2d 21193 If two structures have the...
lvecpropd 21194 If two structures have the...
sraval 21199 Lemma for ~ srabase throug...
sralem 21200 Lemma for ~ srabase and si...
sralemOLD 21201 Obsolete version of ~ sral...
srabase 21202 Base set of a subring alge...
srabaseOLD 21203 Obsolete proof of ~ srabas...
sraaddg 21204 Additive operation of a su...
sraaddgOLD 21205 Obsolete proof of ~ sraadd...
sramulr 21206 Multiplicative operation o...
sramulrOLD 21207 Obsolete proof of ~ sramul...
srasca 21208 The set of scalars of a su...
srascaOLD 21209 Obsolete proof of ~ srasca...
sravsca 21210 The scalar product operati...
sravscaOLD 21211 Obsolete proof of ~ sravsc...
sraip 21212 The inner product operatio...
sratset 21213 Topology component of a su...
sratsetOLD 21214 Obsolete proof of ~ sratse...
sratopn 21215 Topology component of a su...
srads 21216 Distance function of a sub...
sradsOLD 21217 Obsolete proof of ~ srads ...
sraring 21218 Condition for a subring al...
sralmod 21219 The subring algebra is a l...
sralmod0 21220 The subring module inherit...
issubrgd 21221 Prove a subring by closure...
rlmfn 21222 ` ringLMod ` is a function...
rlmval 21223 Value of the ring module. ...
rlmval2 21224 Value of the ring module e...
rlmbas 21225 Base set of the ring modul...
rlmplusg 21226 Vector addition in the rin...
rlm0 21227 Zero vector in the ring mo...
rlmsub 21228 Subtraction in the ring mo...
rlmmulr 21229 Ring multiplication in the...
rlmsca 21230 Scalars in the ring module...
rlmsca2 21231 Scalars in the ring module...
rlmvsca 21232 Scalar multiplication in t...
rlmtopn 21233 Topology component of the ...
rlmds 21234 Metric component of the ri...
rlmlmod 21235 The ring module is a modul...
rlmlvec 21236 The ring module over a div...
rlmlsm 21237 Subgroup sum of the ring m...
rlmvneg 21238 Vector negation in the rin...
rlmscaf 21239 Functionalized scalar mult...
ixpsnbasval 21240 The value of an infinite C...
lidlval 21245 Value of the set of ring i...
rspval 21246 Value of the ring span fun...
lidlss 21247 An ideal is a subset of th...
lidlssbas 21248 The base set of the restri...
lidlbas 21249 A (left) ideal of a ring i...
islidl 21250 Predicate of being a (left...
rnglidlmcl 21251 A (left) ideal containing ...
rngridlmcl 21252 A right ideal (which is a ...
dflidl2rng 21253 Alternate (the usual textb...
isridlrng 21254 A right ideal is a left id...
lidl0cl 21255 An ideal contains 0. (Con...
lidlacl 21256 An ideal is closed under a...
lidlnegcl 21257 An ideal contains negative...
lidlsubg 21258 An ideal is a subgroup of ...
lidlsubcl 21259 An ideal is closed under s...
lidlmcl 21260 An ideal is closed under l...
lidl1el 21261 An ideal contains 1 iff it...
dflidl2 21262 Alternate (the usual textb...
lidl0ALT 21263 Alternate proof for ~ lidl...
rnglidl0 21264 Every non-unital ring cont...
lidl0 21265 Every ring contains a zero...
lidl1ALT 21266 Alternate proof for ~ lidl...
rnglidl1 21267 The base set of every non-...
lidl1 21268 Every ring contains a unit...
lidlacs 21269 The ideal system is an alg...
rspcl 21270 The span of a set of ring ...
rspssid 21271 The span of a set of ring ...
rsp1 21272 The span of the identity e...
rsp0 21273 The span of the zero eleme...
rspssp 21274 The ideal span of a set of...
elrspsn 21275 Membership in a principal ...
mrcrsp 21276 Moore closure generalizes ...
lidlnz 21277 A nonzero ideal contains a...
drngnidl 21278 A division ring has only t...
lidlrsppropd 21279 The left ideals and ring s...
rnglidlmmgm 21280 The multiplicative group o...
rnglidlmsgrp 21281 The multiplicative group o...
rnglidlrng 21282 A (left) ideal of a non-un...
lidlnsg 21283 An ideal is a normal subgr...
2idlval 21286 Definition of a two-sided ...
isridl 21287 A right ideal is a left id...
2idlelb 21288 Membership in a two-sided ...
2idllidld 21289 A two-sided ideal is a lef...
2idlridld 21290 A two-sided ideal is a rig...
df2idl2rng 21291 Alternate (the usual textb...
df2idl2 21292 Alternate (the usual textb...
ridl0 21293 Every ring contains a zero...
ridl1 21294 Every ring contains a unit...
2idl0 21295 Every ring contains a zero...
2idl1 21296 Every ring contains a unit...
2idlss 21297 A two-sided ideal is a sub...
2idlbas 21298 The base set of a two-side...
2idlelbas 21299 The base set of a two-side...
rng2idlsubrng 21300 A two-sided ideal of a non...
rng2idlnsg 21301 A two-sided ideal of a non...
rng2idl0 21302 The zero (additive identit...
rng2idlsubgsubrng 21303 A two-sided ideal of a non...
rng2idlsubgnsg 21304 A two-sided ideal of a non...
rng2idlsubg0 21305 The zero (additive identit...
2idlcpblrng 21306 The coset equivalence rela...
2idlcpbl 21307 The coset equivalence rela...
qus2idrng 21308 The quotient of a non-unit...
qus1 21309 The multiplicative identit...
qusring 21310 If ` S ` is a two-sided id...
qusrhm 21311 If ` S ` is a two-sided id...
rhmpreimaidl 21312 The preimage of an ideal b...
kerlidl 21313 The kernel of a ring homom...
qusmul2idl 21314 Value of the ring operatio...
crngridl 21315 In a commutative ring, the...
crng2idl 21316 In a commutative ring, a t...
qusmulrng 21317 Value of the multiplicatio...
quscrng 21318 The quotient of a commutat...
qusmulcrng 21319 Value of the ring operatio...
rhmqusnsg 21320 The mapping ` J ` induced ...
rngqiprng1elbas 21321 The ring unity of a two-si...
rngqiprngghmlem1 21322 Lemma 1 for ~ rngqiprngghm...
rngqiprngghmlem2 21323 Lemma 2 for ~ rngqiprngghm...
rngqiprngghmlem3 21324 Lemma 3 for ~ rngqiprngghm...
rngqiprngimfolem 21325 Lemma for ~ rngqiprngimfo ...
rngqiprnglinlem1 21326 Lemma 1 for ~ rngqiprnglin...
rngqiprnglinlem2 21327 Lemma 2 for ~ rngqiprnglin...
rngqiprnglinlem3 21328 Lemma 3 for ~ rngqiprnglin...
rngqiprngimf1lem 21329 Lemma for ~ rngqiprngimf1 ...
rngqipbas 21330 The base set of the produc...
rngqiprng 21331 The product of the quotien...
rngqiprngimf 21332 ` F ` is a function from (...
rngqiprngimfv 21333 The value of the function ...
rngqiprngghm 21334 ` F ` is a homomorphism of...
rngqiprngimf1 21335 ` F ` is a one-to-one func...
rngqiprngimfo 21336 ` F ` is a function from (...
rngqiprnglin 21337 ` F ` is linear with respe...
rngqiprngho 21338 ` F ` is a homomorphism of...
rngqiprngim 21339 ` F ` is an isomorphism of...
rng2idl1cntr 21340 The unity of a two-sided i...
rngringbdlem1 21341 In a unital ring, the quot...
rngringbdlem2 21342 A non-unital ring is unita...
rngringbd 21343 A non-unital ring is unita...
ring2idlqus 21344 For every unital ring ther...
ring2idlqusb 21345 A non-unital ring is unita...
rngqiprngfulem1 21346 Lemma 1 for ~ rngqiprngfu ...
rngqiprngfulem2 21347 Lemma 2 for ~ rngqiprngfu ...
rngqiprngfulem3 21348 Lemma 3 for ~ rngqiprngfu ...
rngqiprngfulem4 21349 Lemma 4 for ~ rngqiprngfu ...
rngqiprngfulem5 21350 Lemma 5 for ~ rngqiprngfu ...
rngqipring1 21351 The ring unity of the prod...
rngqiprngfu 21352 The function value of ` F ...
rngqiprngu 21353 If a non-unital ring has a...
ring2idlqus1 21354 If a non-unital ring has a...
lpival 21359 Value of the set of princi...
islpidl 21360 Property of being a princi...
lpi0 21361 The zero ideal is always p...
lpi1 21362 The unit ideal is always p...
islpir 21363 Principal ideal rings are ...
lpiss 21364 Principal ideals are a sub...
islpir2 21365 Principal ideal rings are ...
lpirring 21366 Principal ideal rings are ...
drnglpir 21367 Division rings are princip...
rspsn 21368 Membership in principal id...
lidldvgen 21369 An element generates an id...
lpigen 21370 An ideal is principal iff ...
cnfldstr 21391 The field of complex numbe...
cnfldex 21392 The field of complex numbe...
cnfldbas 21393 The base set of the field ...
mpocnfldadd 21394 The addition operation of ...
cnfldadd 21395 The addition operation of ...
mpocnfldmul 21396 The multiplication operati...
cnfldmul 21397 The multiplication operati...
cnfldcj 21398 The conjugation operation ...
cnfldtset 21399 The topology component of ...
cnfldle 21400 The ordering of the field ...
cnfldds 21401 The metric of the field of...
cnfldunif 21402 The uniform structure comp...
cnfldfun 21403 The field of complex numbe...
cnfldfunALT 21404 The field of complex numbe...
dfcnfldOLD 21405 Obsolete version of ~ df-c...
cnfldstrOLD 21406 Obsolete version of ~ cnfl...
cnfldexOLD 21407 Obsolete version of ~ cnfl...
cnfldbasOLD 21408 Obsolete version of ~ cnfl...
cnfldaddOLD 21409 Obsolete version of ~ cnfl...
cnfldmulOLD 21410 Obsolete version of ~ cnfl...
cnfldcjOLD 21411 Obsolete version of ~ cnfl...
cnfldtsetOLD 21412 Obsolete version of ~ cnfl...
cnfldleOLD 21413 Obsolete version of ~ cnfl...
cnflddsOLD 21414 Obsolete version of ~ cnfl...
cnfldunifOLD 21415 Obsolete version of ~ cnfl...
cnfldfunOLD 21416 Obsolete version of ~ cnfl...
cnfldfunALTOLD 21417 Obsolete version of ~ cnfl...
cnfldfunALTOLDOLD 21418 Obsolete proof of ~ cnfldf...
xrsstr 21419 The extended real structur...
xrsex 21420 The extended real structur...
xrsbas 21421 The base set of the extend...
xrsadd 21422 The addition operation of ...
xrsmul 21423 The multiplication operati...
xrstset 21424 The topology component of ...
xrsle 21425 The ordering of the extend...
cncrng 21426 The complex numbers form a...
cncrngOLD 21427 Obsolete version of ~ cncr...
cnring 21428 The complex numbers form a...
xrsmcmn 21429 The "multiplicative group"...
cnfld0 21430 Zero is the zero element o...
cnfld1 21431 One is the unity element o...
cnfld1OLD 21432 Obsolete version of ~ cnfl...
cnfldneg 21433 The additive inverse in th...
cnfldplusf 21434 The functionalized additio...
cnfldsub 21435 The subtraction operator i...
cndrng 21436 The complex numbers form a...
cndrngOLD 21437 Obsolete version of ~ cndr...
cnflddiv 21438 The division operation in ...
cnflddivOLD 21439 Obsolete version of ~ cnfl...
cnfldinv 21440 The multiplicative inverse...
cnfldmulg 21441 The group multiple functio...
cnfldexp 21442 The exponentiation operato...
cnsrng 21443 The complex numbers form a...
xrsmgm 21444 The "additive group" of th...
xrsnsgrp 21445 The "additive group" of th...
xrsmgmdifsgrp 21446 The "additive group" of th...
xrs1mnd 21447 The extended real numbers,...
xrs10 21448 The zero of the extended r...
xrs1cmn 21449 The extended real numbers ...
xrge0subm 21450 The nonnegative extended r...
xrge0cmn 21451 The nonnegative extended r...
xrsds 21452 The metric of the extended...
xrsdsval 21453 The metric of the extended...
xrsdsreval 21454 The metric of the extended...
xrsdsreclblem 21455 Lemma for ~ xrsdsreclb . ...
xrsdsreclb 21456 The metric of the extended...
cnsubmlem 21457 Lemma for ~ nn0subm and fr...
cnsubglem 21458 Lemma for ~ resubdrg and f...
cnsubrglem 21459 Lemma for ~ resubdrg and f...
cnsubrglemOLD 21460 Obsolete version of ~ cnsu...
cnsubdrglem 21461 Lemma for ~ resubdrg and f...
qsubdrg 21462 The rational numbers form ...
zsubrg 21463 The integers form a subrin...
gzsubrg 21464 The gaussian integers form...
nn0subm 21465 The nonnegative integers f...
rege0subm 21466 The nonnegative reals form...
absabv 21467 The regular absolute value...
zsssubrg 21468 The integers are a subset ...
qsssubdrg 21469 The rational numbers are a...
cnsubrg 21470 There are no subrings of t...
cnmgpabl 21471 The unit group of the comp...
cnmgpid 21472 The group identity element...
cnmsubglem 21473 Lemma for ~ rpmsubg and fr...
rpmsubg 21474 The positive reals form a ...
gzrngunitlem 21475 Lemma for ~ gzrngunit . (...
gzrngunit 21476 The units on ` ZZ [ _i ] `...
gsumfsum 21477 Relate a group sum on ` CC...
regsumfsum 21478 Relate a group sum on ` ( ...
expmhm 21479 Exponentiation is a monoid...
nn0srg 21480 The nonnegative integers f...
rge0srg 21481 The nonnegative real numbe...
zringcrng 21484 The ring of integers is a ...
zringring 21485 The ring of integers is a ...
zringrng 21486 The ring of integers is a ...
zringabl 21487 The ring of integers is an...
zringgrp 21488 The ring of integers is an...
zringbas 21489 The integers are the base ...
zringplusg 21490 The addition operation of ...
zringsub 21491 The subtraction of element...
zringmulg 21492 The multiplication (group ...
zringmulr 21493 The multiplication operati...
zring0 21494 The zero element of the ri...
zring1 21495 The unity element of the r...
zringnzr 21496 The ring of integers is a ...
dvdsrzring 21497 Ring divisibility in the r...
zringlpirlem1 21498 Lemma for ~ zringlpir . A...
zringlpirlem2 21499 Lemma for ~ zringlpir . A...
zringlpirlem3 21500 Lemma for ~ zringlpir . A...
zringinvg 21501 The additive inverse of an...
zringunit 21502 The units of ` ZZ ` are th...
zringlpir 21503 The integers are a princip...
zringndrg 21504 The integers are not a div...
zringcyg 21505 The integers are a cyclic ...
zringsubgval 21506 Subtraction in the ring of...
zringmpg 21507 The multiplicative group o...
prmirredlem 21508 A positive integer is irre...
dfprm2 21509 The positive irreducible e...
prmirred 21510 The irreducible elements o...
expghm 21511 Exponentiation is a group ...
mulgghm2 21512 The powers of a group elem...
mulgrhm 21513 The powers of the element ...
mulgrhm2 21514 The powers of the element ...
irinitoringc 21515 The ring of integers is an...
nzerooringczr 21516 There is no zero object in...
pzriprnglem1 21517 Lemma 1 for ~ pzriprng : `...
pzriprnglem2 21518 Lemma 2 for ~ pzriprng : ...
pzriprnglem3 21519 Lemma 3 for ~ pzriprng : ...
pzriprnglem4 21520 Lemma 4 for ~ pzriprng : `...
pzriprnglem5 21521 Lemma 5 for ~ pzriprng : `...
pzriprnglem6 21522 Lemma 6 for ~ pzriprng : `...
pzriprnglem7 21523 Lemma 7 for ~ pzriprng : `...
pzriprnglem8 21524 Lemma 8 for ~ pzriprng : `...
pzriprnglem9 21525 Lemma 9 for ~ pzriprng : ...
pzriprnglem10 21526 Lemma 10 for ~ pzriprng : ...
pzriprnglem11 21527 Lemma 11 for ~ pzriprng : ...
pzriprnglem12 21528 Lemma 12 for ~ pzriprng : ...
pzriprnglem13 21529 Lemma 13 for ~ pzriprng : ...
pzriprnglem14 21530 Lemma 14 for ~ pzriprng : ...
pzriprngALT 21531 The non-unital ring ` ( ZZ...
pzriprng1ALT 21532 The ring unity of the ring...
pzriprng 21533 The non-unital ring ` ( ZZ...
pzriprng1 21534 The ring unity of the ring...
zrhval 21543 Define the unique homomorp...
zrhval2 21544 Alternate value of the ` Z...
zrhmulg 21545 Value of the ` ZRHom ` hom...
zrhrhmb 21546 The ` ZRHom ` homomorphism...
zrhrhm 21547 The ` ZRHom ` homomorphism...
zrh1 21548 Interpretation of 1 in a r...
zrh0 21549 Interpretation of 0 in a r...
zrhpropd 21550 The ` ZZ ` ring homomorphi...
zlmval 21551 Augment an abelian group w...
zlmlem 21552 Lemma for ~ zlmbas and ~ z...
zlmlemOLD 21553 Obsolete version of ~ zlml...
zlmbas 21554 Base set of a ` ZZ ` -modu...
zlmbasOLD 21555 Obsolete version of ~ zlmb...
zlmplusg 21556 Group operation of a ` ZZ ...
zlmplusgOLD 21557 Obsolete version of ~ zlmb...
zlmmulr 21558 Ring operation of a ` ZZ `...
zlmmulrOLD 21559 Obsolete version of ~ zlmb...
zlmsca 21560 Scalar ring of a ` ZZ ` -m...
zlmvsca 21561 Scalar multiplication oper...
zlmlmod 21562 The ` ZZ ` -module operati...
chrval 21563 Definition substitution of...
chrcl 21564 Closure of the characteris...
chrid 21565 The canonical ` ZZ ` ring ...
chrdvds 21566 The ` ZZ ` ring homomorphi...
chrcong 21567 If two integers are congru...
dvdschrmulg 21568 In a ring, any multiple of...
fermltlchr 21569 A generalization of Fermat...
chrnzr 21570 Nonzero rings are precisel...
chrrhm 21571 The characteristic restric...
domnchr 21572 The characteristic of a do...
znlidl 21573 The set ` n ZZ ` is an ide...
zncrng2 21574 Making a commutative ring ...
znval 21575 The value of the ` Z/nZ ` ...
znle 21576 The value of the ` Z/nZ ` ...
znval2 21577 Self-referential expressio...
znbaslem 21578 Lemma for ~ znbas . (Cont...
znbaslemOLD 21579 Obsolete version of ~ znba...
znbas2 21580 The base set of ` Z/nZ ` i...
znbas2OLD 21581 Obsolete version of ~ znba...
znadd 21582 The additive structure of ...
znaddOLD 21583 Obsolete version of ~ znad...
znmul 21584 The multiplicative structu...
znmulOLD 21585 Obsolete version of ~ znad...
znzrh 21586 The ` ZZ ` ring homomorphi...
znbas 21587 The base set of ` Z/nZ ` s...
zncrng 21588 ` Z/nZ ` is a commutative ...
znzrh2 21589 The ` ZZ ` ring homomorphi...
znzrhval 21590 The ` ZZ ` ring homomorphi...
znzrhfo 21591 The ` ZZ ` ring homomorphi...
zncyg 21592 The group ` ZZ / n ZZ ` is...
zndvds 21593 Express equality of equiva...
zndvds0 21594 Special case of ~ zndvds w...
znf1o 21595 The function ` F ` enumera...
zzngim 21596 The ` ZZ ` ring homomorphi...
znle2 21597 The ordering of the ` Z/nZ...
znleval 21598 The ordering of the ` Z/nZ...
znleval2 21599 The ordering of the ` Z/nZ...
zntoslem 21600 Lemma for ~ zntos . (Cont...
zntos 21601 The ` Z/nZ ` structure is ...
znhash 21602 The ` Z/nZ ` structure has...
znfi 21603 The ` Z/nZ ` structure is ...
znfld 21604 The ` Z/nZ ` structure is ...
znidomb 21605 The ` Z/nZ ` structure is ...
znchr 21606 Cyclic rings are defined b...
znunit 21607 The units of ` Z/nZ ` are ...
znunithash 21608 The size of the unit group...
znrrg 21609 The regular elements of ` ...
cygznlem1 21610 Lemma for ~ cygzn . (Cont...
cygznlem2a 21611 Lemma for ~ cygzn . (Cont...
cygznlem2 21612 Lemma for ~ cygzn . (Cont...
cygznlem3 21613 A cyclic group with ` n ` ...
cygzn 21614 A cyclic group with ` n ` ...
cygth 21615 The "fundamental theorem o...
cyggic 21616 Cyclic groups are isomorph...
frgpcyg 21617 A free group is cyclic iff...
freshmansdream 21618 For a prime number ` P ` ,...
frobrhm 21619 In a commutative ring with...
cnmsgnsubg 21620 The signs form a multiplic...
cnmsgnbas 21621 The base set of the sign s...
cnmsgngrp 21622 The group of signs under m...
psgnghm 21623 The sign is a homomorphism...
psgnghm2 21624 The sign is a homomorphism...
psgninv 21625 The sign of a permutation ...
psgnco 21626 Multiplicativity of the pe...
zrhpsgnmhm 21627 Embedding of permutation s...
zrhpsgninv 21628 The embedded sign of a per...
evpmss 21629 Even permutations are perm...
psgnevpmb 21630 A class is an even permuta...
psgnodpm 21631 A permutation which is odd...
psgnevpm 21632 A permutation which is eve...
psgnodpmr 21633 If a permutation has sign ...
zrhpsgnevpm 21634 The sign of an even permut...
zrhpsgnodpm 21635 The sign of an odd permuta...
cofipsgn 21636 Composition of any class `...
zrhpsgnelbas 21637 Embedding of permutation s...
zrhcopsgnelbas 21638 Embedding of permutation s...
evpmodpmf1o 21639 The function for performin...
pmtrodpm 21640 A transposition is an odd ...
psgnfix1 21641 A permutation of a finite ...
psgnfix2 21642 A permutation of a finite ...
psgndiflemB 21643 Lemma 1 for ~ psgndif . (...
psgndiflemA 21644 Lemma 2 for ~ psgndif . (...
psgndif 21645 Embedding of permutation s...
copsgndif 21646 Embedding of permutation s...
rebase 21649 The base of the field of r...
remulg 21650 The multiplication (group ...
resubdrg 21651 The real numbers form a di...
resubgval 21652 Subtraction in the field o...
replusg 21653 The addition operation of ...
remulr 21654 The multiplication operati...
re0g 21655 The zero element of the fi...
re1r 21656 The unity element of the f...
rele2 21657 The ordering relation of t...
relt 21658 The ordering relation of t...
reds 21659 The distance of the field ...
redvr 21660 The division operation of ...
retos 21661 The real numbers are a tot...
refld 21662 The real numbers form a fi...
refldcj 21663 The conjugation operation ...
resrng 21664 The real numbers form a st...
regsumsupp 21665 The group sum over the rea...
rzgrp 21666 The quotient group ` RR / ...
isphl 21671 The predicate "is a genera...
phllvec 21672 A pre-Hilbert space is a l...
phllmod 21673 A pre-Hilbert space is a l...
phlsrng 21674 The scalar ring of a pre-H...
phllmhm 21675 The inner product of a pre...
ipcl 21676 Closure of the inner produ...
ipcj 21677 Conjugate of an inner prod...
iporthcom 21678 Orthogonality (meaning inn...
ip0l 21679 Inner product with a zero ...
ip0r 21680 Inner product with a zero ...
ipeq0 21681 The inner product of a vec...
ipdir 21682 Distributive law for inner...
ipdi 21683 Distributive law for inner...
ip2di 21684 Distributive law for inner...
ipsubdir 21685 Distributive law for inner...
ipsubdi 21686 Distributive law for inner...
ip2subdi 21687 Distributive law for inner...
ipass 21688 Associative law for inner ...
ipassr 21689 "Associative" law for seco...
ipassr2 21690 "Associative" law for inne...
ipffval 21691 The inner product operatio...
ipfval 21692 The inner product operatio...
ipfeq 21693 If the inner product opera...
ipffn 21694 The inner product operatio...
phlipf 21695 The inner product operatio...
ip2eq 21696 Two vectors are equal iff ...
isphld 21697 Properties that determine ...
phlpropd 21698 If two structures have the...
ssipeq 21699 The inner product on a sub...
phssipval 21700 The inner product on a sub...
phssip 21701 The inner product (as a fu...
phlssphl 21702 A subspace of an inner pro...
ocvfval 21709 The orthocomplement operat...
ocvval 21710 Value of the orthocompleme...
elocv 21711 Elementhood in the orthoco...
ocvi 21712 Property of a member of th...
ocvss 21713 The orthocomplement of a s...
ocvocv 21714 A set is contained in its ...
ocvlss 21715 The orthocomplement of a s...
ocv2ss 21716 Orthocomplements reverse s...
ocvin 21717 An orthocomplement has tri...
ocvsscon 21718 Two ways to say that ` S `...
ocvlsp 21719 The orthocomplement of a l...
ocv0 21720 The orthocomplement of the...
ocvz 21721 The orthocomplement of the...
ocv1 21722 The orthocomplement of the...
unocv 21723 The orthocomplement of a u...
iunocv 21724 The orthocomplement of an ...
cssval 21725 The set of closed subspace...
iscss 21726 The predicate "is a closed...
cssi 21727 Property of a closed subsp...
cssss 21728 A closed subspace is a sub...
iscss2 21729 It is sufficient to prove ...
ocvcss 21730 The orthocomplement of any...
cssincl 21731 The zero subspace is a clo...
css0 21732 The zero subspace is a clo...
css1 21733 The whole space is a close...
csslss 21734 A closed subspace of a pre...
lsmcss 21735 A subset of a pre-Hilbert ...
cssmre 21736 The closed subspaces of a ...
mrccss 21737 The Moore closure correspo...
thlval 21738 Value of the Hilbert latti...
thlbas 21739 Base set of the Hilbert la...
thlbasOLD 21740 Obsolete proof of ~ thlbas...
thlle 21741 Ordering on the Hilbert la...
thlleOLD 21742 Obsolete proof of ~ thlle ...
thlleval 21743 Ordering on the Hilbert la...
thloc 21744 Orthocomplement on the Hil...
pjfval 21751 The value of the projectio...
pjdm 21752 A subspace is in the domai...
pjpm 21753 The projection map is a pa...
pjfval2 21754 Value of the projection ma...
pjval 21755 Value of the projection ma...
pjdm2 21756 A subspace is in the domai...
pjff 21757 A projection is a linear o...
pjf 21758 A projection is a function...
pjf2 21759 A projection is a function...
pjfo 21760 A projection is a surjecti...
pjcss 21761 A projection subspace is a...
ocvpj 21762 The orthocomplement of a p...
ishil 21763 The predicate "is a Hilber...
ishil2 21764 The predicate "is a Hilber...
isobs 21765 The predicate "is an ortho...
obsip 21766 The inner product of two e...
obsipid 21767 A basis element has length...
obsrcl 21768 Reverse closure for an ort...
obsss 21769 An orthonormal basis is a ...
obsne0 21770 A basis element is nonzero...
obsocv 21771 An orthonormal basis has t...
obs2ocv 21772 The double orthocomplement...
obselocv 21773 A basis element is in the ...
obs2ss 21774 A basis has no proper subs...
obslbs 21775 An orthogonal basis is a l...
reldmdsmm 21778 The direct sum is a well-b...
dsmmval 21779 Value of the module direct...
dsmmbase 21780 Base set of the module dir...
dsmmval2 21781 Self-referential definitio...
dsmmbas2 21782 Base set of the direct sum...
dsmmfi 21783 For finite products, the d...
dsmmelbas 21784 Membership in the finitely...
dsmm0cl 21785 The all-zero vector is con...
dsmmacl 21786 The finite hull is closed ...
prdsinvgd2 21787 Negation of a single coord...
dsmmsubg 21788 The finite hull of a produ...
dsmmlss 21789 The finite hull of a produ...
dsmmlmod 21790 The direct sum of a family...
frlmval 21793 Value of the "free module"...
frlmlmod 21794 The free module is a modul...
frlmpws 21795 The free module as a restr...
frlmlss 21796 The base set of the free m...
frlmpwsfi 21797 The finite free module is ...
frlmsca 21798 The ring of scalars of a f...
frlm0 21799 Zero in a free module (rin...
frlmbas 21800 Base set of the free modul...
frlmelbas 21801 Membership in the base set...
frlmrcl 21802 If a free module is inhabi...
frlmbasfsupp 21803 Elements of the free modul...
frlmbasmap 21804 Elements of the free modul...
frlmbasf 21805 Elements of the free modul...
frlmlvec 21806 The free module over a div...
frlmfibas 21807 The base set of the finite...
elfrlmbasn0 21808 If the dimension of a free...
frlmplusgval 21809 Addition in a free module....
frlmsubgval 21810 Subtraction in a free modu...
frlmvscafval 21811 Scalar multiplication in a...
frlmvplusgvalc 21812 Coordinates of a sum with ...
frlmvscaval 21813 Coordinates of a scalar mu...
frlmplusgvalb 21814 Addition in a free module ...
frlmvscavalb 21815 Scalar multiplication in a...
frlmvplusgscavalb 21816 Addition combined with sca...
frlmgsum 21817 Finite commutative sums in...
frlmsplit2 21818 Restriction is homomorphic...
frlmsslss 21819 A subset of a free module ...
frlmsslss2 21820 A subset of a free module ...
frlmbas3 21821 An element of the base set...
mpofrlmd 21822 Elements of the free modul...
frlmip 21823 The inner product of a fre...
frlmipval 21824 The inner product of a fre...
frlmphllem 21825 Lemma for ~ frlmphl . (Co...
frlmphl 21826 Conditions for a free modu...
uvcfval 21829 Value of the unit-vector g...
uvcval 21830 Value of a single unit vec...
uvcvval 21831 Value of a unit vector coo...
uvcvvcl 21832 A coordinate of a unit vec...
uvcvvcl2 21833 A unit vector coordinate i...
uvcvv1 21834 The unit vector is one at ...
uvcvv0 21835 The unit vector is zero at...
uvcff 21836 Domain and codomain of the...
uvcf1 21837 In a nonzero ring, each un...
uvcresum 21838 Any element of a free modu...
frlmssuvc1 21839 A scalar multiple of a uni...
frlmssuvc2 21840 A nonzero scalar multiple ...
frlmsslsp 21841 A subset of a free module ...
frlmlbs 21842 The unit vectors comprise ...
frlmup1 21843 Any assignment of unit vec...
frlmup2 21844 The evaluation map has the...
frlmup3 21845 The range of such an evalu...
frlmup4 21846 Universal property of the ...
ellspd 21847 The elements of the span o...
elfilspd 21848 Simplified version of ~ el...
rellindf 21853 The independent-family pre...
islinds 21854 Property of an independent...
linds1 21855 An independent set of vect...
linds2 21856 An independent set of vect...
islindf 21857 Property of an independent...
islinds2 21858 Expanded property of an in...
islindf2 21859 Property of an independent...
lindff 21860 Functional property of a l...
lindfind 21861 A linearly independent fam...
lindsind 21862 A linearly independent set...
lindfind2 21863 In a linearly independent ...
lindsind2 21864 In a linearly independent ...
lindff1 21865 A linearly independent fam...
lindfrn 21866 The range of an independen...
f1lindf 21867 Rearranging and deleting e...
lindfres 21868 Any restriction of an inde...
lindsss 21869 Any subset of an independe...
f1linds 21870 A family constructed from ...
islindf3 21871 In a nonzero ring, indepen...
lindfmm 21872 Linear independence of a f...
lindsmm 21873 Linear independence of a s...
lindsmm2 21874 The monomorphic image of a...
lsslindf 21875 Linear independence is unc...
lsslinds 21876 Linear independence is unc...
islbs4 21877 A basis is an independent ...
lbslinds 21878 A basis is independent. (...
islinds3 21879 A subset is linearly indep...
islinds4 21880 A set is independent in a ...
lmimlbs 21881 The isomorphic image of a ...
lmiclbs 21882 Having a basis is an isomo...
islindf4 21883 A family is independent if...
islindf5 21884 A family is independent if...
indlcim 21885 An independent, spanning f...
lbslcic 21886 A module with a basis is i...
lmisfree 21887 A module has a basis iff i...
lvecisfrlm 21888 Every vector space is isom...
lmimco 21889 The composition of two iso...
lmictra 21890 Module isomorphism is tran...
uvcf1o 21891 In a nonzero ring, the map...
uvcendim 21892 In a nonzero ring, the num...
frlmisfrlm 21893 A free module is isomorphi...
frlmiscvec 21894 Every free module is isomo...
isassa 21901 The properties of an assoc...
assalem 21902 The properties of an assoc...
assaass 21903 Left-associative property ...
assaassr 21904 Right-associative property...
assalmod 21905 An associative algebra is ...
assaring 21906 An associative algebra is ...
assasca 21907 The scalars of an associat...
assa2ass 21908 Left- and right-associativ...
assa2ass2 21909 Left- and right-associativ...
isassad 21910 Sufficient condition for b...
issubassa3 21911 A subring that is also a s...
issubassa 21912 The subalgebras of an asso...
sraassab 21913 A subring algebra is an as...
sraassa 21914 The subring algebra over a...
sraassaOLD 21915 Obsolete version of ~ sraa...
rlmassa 21916 The ring module over a com...
assapropd 21917 If two structures have the...
aspval 21918 Value of the algebraic clo...
asplss 21919 The algebraic span of a se...
aspid 21920 The algebraic span of a su...
aspsubrg 21921 The algebraic span of a se...
aspss 21922 Span preserves subset orde...
aspssid 21923 A set of vectors is a subs...
asclfval 21924 Function value of the alge...
asclval 21925 Value of a mapped algebra ...
asclfn 21926 Unconditional functionalit...
asclf 21927 The algebra scalars functi...
asclghm 21928 The algebra scalars functi...
ascl0 21929 The scalar 0 embedded into...
ascl1 21930 The scalar 1 embedded into...
asclmul1 21931 Left multiplication by a l...
asclmul2 21932 Right multiplication by a ...
ascldimul 21933 The algebra scalars functi...
asclinvg 21934 The group inverse (negatio...
asclrhm 21935 The algebra scalars functi...
rnascl 21936 The set of lifted scalars ...
issubassa2 21937 A subring of a unital alge...
rnasclsubrg 21938 The scalar multiples of th...
rnasclmulcl 21939 (Vector) multiplication is...
rnasclassa 21940 The scalar multiples of th...
ressascl 21941 The lifting of scalars is ...
asclpropd 21942 If two structures have the...
aspval2 21943 The algebraic closure is t...
assamulgscmlem1 21944 Lemma 1 for ~ assamulgscm ...
assamulgscmlem2 21945 Lemma for ~ assamulgscm (i...
assamulgscm 21946 Exponentiation of a scalar...
asclmulg 21947 Apply group multiplication...
zlmassa 21948 The ` ZZ ` -module operati...
reldmpsr 21959 The multivariate power ser...
psrval 21960 Value of the multivariate ...
psrvalstr 21961 The multivariate power ser...
psrbag 21962 Elementhood in the set of ...
psrbagf 21963 A finite bag is a function...
psrbagfsupp 21964 Finite bags have finite su...
snifpsrbag 21965 A bag containing one eleme...
fczpsrbag 21966 The constant function equa...
psrbaglesupp 21967 The support of a dominated...
psrbaglecl 21968 The set of finite bags is ...
psrbagaddcl 21969 The sum of two finite bags...
psrbagcon 21970 The analogue of the statem...
psrbaglefi 21971 There are finitely many ba...
psrbagconcl 21972 The complement of a bag is...
psrbagleadd1 21973 The analogue of " ` X <_ F...
psrbagconf1o 21974 Bag complementation is a b...
gsumbagdiaglem 21975 Lemma for ~ gsumbagdiag . ...
gsumbagdiag 21976 Two-dimensional commutatio...
psrass1lem 21977 A group sum commutation us...
psrbas 21978 The base set of the multiv...
psrelbas 21979 An element of the set of p...
psrelbasfun 21980 An element of the set of p...
psrplusg 21981 The addition operation of ...
psradd 21982 The addition operation of ...
psraddcl 21983 Closure of the power serie...
psraddclOLD 21984 Obsolete version of ~ psra...
rhmpsrlem1 21985 Lemma for ~ rhmpsr et al. ...
rhmpsrlem2 21986 Lemma for ~ rhmpsr et al. ...
psrmulr 21987 The multiplication operati...
psrmulfval 21988 The multiplication operati...
psrmulval 21989 The multiplication operati...
psrmulcllem 21990 Closure of the power serie...
psrmulcl 21991 Closure of the power serie...
psrsca 21992 The scalar field of the mu...
psrvscafval 21993 The scalar multiplication ...
psrvsca 21994 The scalar multiplication ...
psrvscaval 21995 The scalar multiplication ...
psrvscacl 21996 Closure of the power serie...
psr0cl 21997 The zero element of the ri...
psr0lid 21998 The zero element of the ri...
psrnegcl 21999 The negative function in t...
psrlinv 22000 The negative function in t...
psrgrp 22001 The ring of power series i...
psrgrpOLD 22002 Obsolete proof of ~ psrgrp...
psr0 22003 The zero element of the ri...
psrneg 22004 The negative function of t...
psrlmod 22005 The ring of power series i...
psr1cl 22006 The identity element of th...
psrlidm 22007 The identity element of th...
psrridm 22008 The identity element of th...
psrass1 22009 Associative identity for t...
psrdi 22010 Distributive law for the r...
psrdir 22011 Distributive law for the r...
psrass23l 22012 Associative identity for t...
psrcom 22013 Commutative law for the ri...
psrass23 22014 Associative identities for...
psrring 22015 The ring of power series i...
psr1 22016 The identity element of th...
psrcrng 22017 The ring of power series i...
psrassa 22018 The ring of power series i...
resspsrbas 22019 A restricted power series ...
resspsradd 22020 A restricted power series ...
resspsrmul 22021 A restricted power series ...
resspsrvsca 22022 A restricted power series ...
subrgpsr 22023 A subring of the base ring...
psrascl 22024 Value of the scalar inject...
psrasclcl 22025 A scalar is lifted into a ...
mvrfval 22026 Value of the generating el...
mvrval 22027 Value of the generating el...
mvrval2 22028 Value of the generating el...
mvrid 22029 The ` X i ` -th coefficien...
mvrf 22030 The power series variable ...
mvrf1 22031 The power series variable ...
mvrcl2 22032 A power series variable is...
reldmmpl 22033 The multivariate polynomia...
mplval 22034 Value of the set of multiv...
mplbas 22035 Base set of the set of mul...
mplelbas 22036 Property of being a polyno...
mvrcl 22037 A power series variable is...
mvrf2 22038 The power series/polynomia...
mplrcl 22039 Reverse closure for the po...
mplelsfi 22040 A polynomial treated as a ...
mplval2 22041 Self-referential expressio...
mplbasss 22042 The set of polynomials is ...
mplelf 22043 A polynomial is defined as...
mplsubglem 22044 If ` A ` is an ideal of se...
mpllsslem 22045 If ` A ` is an ideal of su...
mplsubglem2 22046 Lemma for ~ mplsubg and ~ ...
mplsubg 22047 The set of polynomials is ...
mpllss 22048 The set of polynomials is ...
mplsubrglem 22049 Lemma for ~ mplsubrg . (C...
mplsubrg 22050 The set of polynomials is ...
mpl0 22051 The zero polynomial. (Con...
mplplusg 22052 Value of addition in a pol...
mplmulr 22053 Value of multiplication in...
mpladd 22054 The addition operation on ...
mplneg 22055 The negative function on m...
mplmul 22056 The multiplication operati...
mpl1 22057 The identity element of th...
mplsca 22058 The scalar field of a mult...
mplvsca2 22059 The scalar multiplication ...
mplvsca 22060 The scalar multiplication ...
mplvscaval 22061 The scalar multiplication ...
mplgrp 22062 The polynomial ring is a g...
mpllmod 22063 The polynomial ring is a l...
mplring 22064 The polynomial ring is a r...
mpllvec 22065 The polynomial ring is a v...
mplcrng 22066 The polynomial ring is a c...
mplassa 22067 The polynomial ring is an ...
mplringd 22068 The polynomial ring is a r...
mpllmodd 22069 The polynomial ring is a l...
ressmplbas2 22070 The base set of a restrict...
ressmplbas 22071 A restricted polynomial al...
ressmpladd 22072 A restricted polynomial al...
ressmplmul 22073 A restricted polynomial al...
ressmplvsca 22074 A restricted power series ...
subrgmpl 22075 A subring of the base ring...
subrgmvr 22076 The variables in a subring...
subrgmvrf 22077 The variables in a polynom...
mplmon 22078 A monomial is a polynomial...
mplmonmul 22079 The product of two monomia...
mplcoe1 22080 Decompose a polynomial int...
mplcoe3 22081 Decompose a monomial in on...
mplcoe5lem 22082 Lemma for ~ mplcoe4 . (Co...
mplcoe5 22083 Decompose a monomial into ...
mplcoe2 22084 Decompose a monomial into ...
mplbas2 22085 An alternative expression ...
ltbval 22086 Value of the well-order on...
ltbwe 22087 The finite bag order is a ...
reldmopsr 22088 Lemma for ordered power se...
opsrval 22089 The value of the "ordered ...
opsrle 22090 An alternative expression ...
opsrval2 22091 Self-referential expressio...
opsrbaslem 22092 Get a component of the ord...
opsrbaslemOLD 22093 Obsolete version of ~ opsr...
opsrbas 22094 The base set of the ordere...
opsrbasOLD 22095 Obsolete version of ~ opsr...
opsrplusg 22096 The addition operation of ...
opsrplusgOLD 22097 Obsolete version of ~ opsr...
opsrmulr 22098 The multiplication operati...
opsrmulrOLD 22099 Obsolete version of ~ opsr...
opsrvsca 22100 The scalar product operati...
opsrvscaOLD 22101 Obsolete version of ~ opsr...
opsrsca 22102 The scalar ring of the ord...
opsrscaOLD 22103 Obsolete version of ~ opsr...
opsrtoslem1 22104 Lemma for ~ opsrtos . (Co...
opsrtoslem2 22105 Lemma for ~ opsrtos . (Co...
opsrtos 22106 The ordered power series s...
opsrso 22107 The ordered power series s...
opsrcrng 22108 The ring of ordered power ...
opsrassa 22109 The ring of ordered power ...
mplmon2 22110 Express a scaled monomial....
psrbag0 22111 The empty bag is a bag. (...
psrbagsn 22112 A singleton bag is a bag. ...
mplascl 22113 Value of the scalar inject...
mplasclf 22114 The scalar injection is a ...
subrgascl 22115 The scalar injection funct...
subrgasclcl 22116 The scalars in a polynomia...
mplmon2cl 22117 A scaled monomial is a pol...
mplmon2mul 22118 Product of scaled monomial...
mplind 22119 Prove a property of polyno...
mplcoe4 22120 Decompose a polynomial int...
evlslem4 22125 The support of a tensor pr...
psrbagev1 22126 A bag of multipliers provi...
psrbagev2 22127 Closure of a sum using a b...
evlslem2 22128 A linear function on the p...
evlslem3 22129 Lemma for ~ evlseu . Poly...
evlslem6 22130 Lemma for ~ evlseu . Fini...
evlslem1 22131 Lemma for ~ evlseu , give ...
evlseu 22132 For a given interpretation...
reldmevls 22133 Well-behaved binary operat...
mpfrcl 22134 Reverse closure for the se...
evlsval 22135 Value of the polynomial ev...
evlsval2 22136 Characterizing properties ...
evlsrhm 22137 Polynomial evaluation is a...
evlssca 22138 Polynomial evaluation maps...
evlsvar 22139 Polynomial evaluation maps...
evlsgsumadd 22140 Polynomial evaluation maps...
evlsgsummul 22141 Polynomial evaluation maps...
evlspw 22142 Polynomial evaluation for ...
evlsvarpw 22143 Polynomial evaluation for ...
evlval 22144 Value of the simple/same r...
evlrhm 22145 The simple evaluation map ...
evlsscasrng 22146 The evaluation of a scalar...
evlsca 22147 Simple polynomial evaluati...
evlsvarsrng 22148 The evaluation of the vari...
evlvar 22149 Simple polynomial evaluati...
mpfconst 22150 Constants are multivariate...
mpfproj 22151 Projections are multivaria...
mpfsubrg 22152 Polynomial functions are a...
mpff 22153 Polynomial functions are f...
mpfaddcl 22154 The sum of multivariate po...
mpfmulcl 22155 The product of multivariat...
mpfind 22156 Prove a property of polyno...
selvffval 22162 Value of the "variable sel...
selvfval 22163 Value of the "variable sel...
selvval 22164 Value of the "variable sel...
reldmmhp 22166 The domain of the homogene...
mhpfval 22167 Value of the "homogeneous ...
mhpval 22168 Value of the "homogeneous ...
ismhp 22169 Property of being a homoge...
ismhp2 22170 Deduce a homogeneous polyn...
ismhp3 22171 A polynomial is homogeneou...
mhprcl 22172 Reverse closure for homoge...
mhpmpl 22173 A homogeneous polynomial i...
mhpdeg 22174 All nonzero terms of a hom...
mhp0cl 22175 The zero polynomial is hom...
mhpsclcl 22176 A scalar (or constant) pol...
mhpvarcl 22177 A power series variable is...
mhpmulcl 22178 A product of homogeneous p...
mhppwdeg 22179 Degree of a homogeneous po...
mhpaddcl 22180 Homogeneous polynomials ar...
mhpinvcl 22181 Homogeneous polynomials ar...
mhpsubg 22182 Homogeneous polynomials fo...
mhpvscacl 22183 Homogeneous polynomials ar...
mhplss 22184 Homogeneous polynomials fo...
psdffval 22186 Value of the power series ...
psdfval 22187 Give a map between power s...
psdval 22188 Evaluate the partial deriv...
psdcoef 22189 Coefficient of a term of t...
psdcl 22190 The derivative of a power ...
psdmplcl 22191 The derivative of a polyno...
psdadd 22192 The derivative of a sum is...
psdvsca 22193 The derivative of a scaled...
psdmullem 22194 Lemma for ~ psdmul . Tran...
psdmul 22195 Product rule for power ser...
psd1 22196 The derivative of one is z...
psdascl 22197 The derivative of a consta...
psr1baslem 22209 The set of finite bags on ...
psr1val 22210 Value of the ring of univa...
psr1crng 22211 The ring of univariate pow...
psr1assa 22212 The ring of univariate pow...
psr1tos 22213 The ordered power series s...
psr1bas2 22214 The base set of the ring o...
psr1bas 22215 The base set of the ring o...
vr1val 22216 The value of the generator...
vr1cl2 22217 The variable ` X ` is a me...
ply1val 22218 The value of the set of un...
ply1bas 22219 The value of the base set ...
ply1basOLD 22220 Obsolete version of ~ ply1...
ply1lss 22221 Univariate polynomials for...
ply1subrg 22222 Univariate polynomials for...
ply1crng 22223 The ring of univariate pol...
ply1assa 22224 The ring of univariate pol...
psr1bascl 22225 A univariate power series ...
psr1basf 22226 Univariate power series ba...
ply1basf 22227 Univariate polynomial base...
ply1bascl 22228 A univariate polynomial is...
ply1bascl2 22229 A univariate polynomial is...
coe1fval 22230 Value of the univariate po...
coe1fv 22231 Value of an evaluated coef...
fvcoe1 22232 Value of a multivariate co...
coe1fval3 22233 Univariate power series co...
coe1f2 22234 Functionality of univariat...
coe1fval2 22235 Univariate polynomial coef...
coe1f 22236 Functionality of univariat...
coe1fvalcl 22237 A coefficient of a univari...
coe1sfi 22238 Finite support of univaria...
coe1fsupp 22239 The coefficient vector of ...
mptcoe1fsupp 22240 A mapping involving coeffi...
coe1ae0 22241 The coefficient vector of ...
vr1cl 22242 The generator of a univari...
opsr0 22243 Zero in the ordered power ...
opsr1 22244 One in the ordered power s...
psr1plusg 22245 Value of addition in a uni...
psr1vsca 22246 Value of scalar multiplica...
psr1mulr 22247 Value of multiplication in...
ply1plusg 22248 Value of addition in a uni...
ply1vsca 22249 Value of scalar multiplica...
ply1mulr 22250 Value of multiplication in...
ply1ass23l 22251 Associative identity with ...
ressply1bas2 22252 The base set of a restrict...
ressply1bas 22253 A restricted polynomial al...
ressply1add 22254 A restricted polynomial al...
ressply1mul 22255 A restricted polynomial al...
ressply1vsca 22256 A restricted power series ...
subrgply1 22257 A subring of the base ring...
gsumply1subr 22258 Evaluate a group sum in a ...
psrbaspropd 22259 Property deduction for pow...
psrplusgpropd 22260 Property deduction for pow...
mplbaspropd 22261 Property deduction for pol...
psropprmul 22262 Reversing multiplication i...
ply1opprmul 22263 Reversing multiplication i...
00ply1bas 22264 Lemma for ~ ply1basfvi and...
ply1basfvi 22265 Protection compatibility o...
ply1plusgfvi 22266 Protection compatibility o...
ply1baspropd 22267 Property deduction for uni...
ply1plusgpropd 22268 Property deduction for uni...
opsrring 22269 Ordered power series form ...
opsrlmod 22270 Ordered power series form ...
psr1ring 22271 Univariate power series fo...
ply1ring 22272 Univariate polynomials for...
psr1lmod 22273 Univariate power series fo...
psr1sca 22274 Scalars of a univariate po...
psr1sca2 22275 Scalars of a univariate po...
ply1lmod 22276 Univariate polynomials for...
ply1sca 22277 Scalars of a univariate po...
ply1sca2 22278 Scalars of a univariate po...
ply1ascl0 22279 The zero scalar as a polyn...
ply1ascl1 22280 The multiplicative identit...
ply1mpl0 22281 The univariate polynomial ...
ply10s0 22282 Zero times a univariate po...
ply1mpl1 22283 The univariate polynomial ...
ply1ascl 22284 The univariate polynomial ...
subrg1ascl 22285 The scalar injection funct...
subrg1asclcl 22286 The scalars in a polynomia...
subrgvr1 22287 The variables in a subring...
subrgvr1cl 22288 The variables in a polynom...
coe1z 22289 The coefficient vector of ...
coe1add 22290 The coefficient vector of ...
coe1addfv 22291 A particular coefficient o...
coe1subfv 22292 A particular coefficient o...
coe1mul2lem1 22293 An equivalence for ~ coe1m...
coe1mul2lem2 22294 An equivalence for ~ coe1m...
coe1mul2 22295 The coefficient vector of ...
coe1mul 22296 The coefficient vector of ...
ply1moncl 22297 Closure of the expression ...
ply1tmcl 22298 Closure of the expression ...
coe1tm 22299 Coefficient vector of a po...
coe1tmfv1 22300 Nonzero coefficient of a p...
coe1tmfv2 22301 Zero coefficient of a poly...
coe1tmmul2 22302 Coefficient vector of a po...
coe1tmmul 22303 Coefficient vector of a po...
coe1tmmul2fv 22304 Function value of a right-...
coe1pwmul 22305 Coefficient vector of a po...
coe1pwmulfv 22306 Function value of a right-...
ply1scltm 22307 A scalar is a term with ze...
coe1sclmul 22308 Coefficient vector of a po...
coe1sclmulfv 22309 A single coefficient of a ...
coe1sclmul2 22310 Coefficient vector of a po...
ply1sclf 22311 A scalar polynomial is a p...
ply1sclcl 22312 The value of the algebra s...
coe1scl 22313 Coefficient vector of a sc...
ply1sclid 22314 Recover the base scalar fr...
ply1sclf1 22315 The polynomial scalar func...
ply1scl0 22316 The zero scalar is zero. ...
ply1scl0OLD 22317 Obsolete version of ~ ply1...
ply1scln0 22318 Nonzero scalars create non...
ply1scl1 22319 The one scalar is the unit...
ply1scl1OLD 22320 Obsolete version of ~ ply1...
ply1idvr1 22321 The identity of a polynomi...
ply1idvr1OLD 22322 Obsolete version of ~ ply1...
cply1mul 22323 The product of two constan...
ply1coefsupp 22324 The decomposition of a uni...
ply1coe 22325 Decompose a univariate pol...
eqcoe1ply1eq 22326 Two polynomials over the s...
ply1coe1eq 22327 Two polynomials over the s...
cply1coe0 22328 All but the first coeffici...
cply1coe0bi 22329 A polynomial is constant (...
coe1fzgsumdlem 22330 Lemma for ~ coe1fzgsumd (i...
coe1fzgsumd 22331 Value of an evaluated coef...
ply1scleq 22332 Equality of a constant pol...
ply1chr 22333 The characteristic of a po...
gsumsmonply1 22334 A finite group sum of scal...
gsummoncoe1 22335 A coefficient of the polyn...
gsumply1eq 22336 Two univariate polynomials...
lply1binom 22337 The binomial theorem for l...
lply1binomsc 22338 The binomial theorem for l...
ply1fermltlchr 22339 Fermat's little theorem fo...
reldmevls1 22344 Well-behaved binary operat...
ply1frcl 22345 Reverse closure for the se...
evls1fval 22346 Value of the univariate po...
evls1val 22347 Value of the univariate po...
evls1rhmlem 22348 Lemma for ~ evl1rhm and ~ ...
evls1rhm 22349 Polynomial evaluation is a...
evls1sca 22350 Univariate polynomial eval...
evls1gsumadd 22351 Univariate polynomial eval...
evls1gsummul 22352 Univariate polynomial eval...
evls1pw 22353 Univariate polynomial eval...
evls1varpw 22354 Univariate polynomial eval...
evl1fval 22355 Value of the simple/same r...
evl1val 22356 Value of the simple/same r...
evl1fval1lem 22357 Lemma for ~ evl1fval1 . (...
evl1fval1 22358 Value of the simple/same r...
evl1rhm 22359 Polynomial evaluation is a...
fveval1fvcl 22360 The function value of the ...
evl1sca 22361 Polynomial evaluation maps...
evl1scad 22362 Polynomial evaluation buil...
evl1var 22363 Polynomial evaluation maps...
evl1vard 22364 Polynomial evaluation buil...
evls1var 22365 Univariate polynomial eval...
evls1scasrng 22366 The evaluation of a scalar...
evls1varsrng 22367 The evaluation of the vari...
evl1addd 22368 Polynomial evaluation buil...
evl1subd 22369 Polynomial evaluation buil...
evl1muld 22370 Polynomial evaluation buil...
evl1vsd 22371 Polynomial evaluation buil...
evl1expd 22372 Polynomial evaluation buil...
pf1const 22373 Constants are polynomial f...
pf1id 22374 The identity is a polynomi...
pf1subrg 22375 Polynomial functions are a...
pf1rcl 22376 Reverse closure for the se...
pf1f 22377 Polynomial functions are f...
mpfpf1 22378 Convert a multivariate pol...
pf1mpf 22379 Convert a univariate polyn...
pf1addcl 22380 The sum of multivariate po...
pf1mulcl 22381 The product of multivariat...
pf1ind 22382 Prove a property of polyno...
evl1gsumdlem 22383 Lemma for ~ evl1gsumd (ind...
evl1gsumd 22384 Polynomial evaluation buil...
evl1gsumadd 22385 Univariate polynomial eval...
evl1gsumaddval 22386 Value of a univariate poly...
evl1gsummul 22387 Univariate polynomial eval...
evl1varpw 22388 Univariate polynomial eval...
evl1varpwval 22389 Value of a univariate poly...
evl1scvarpw 22390 Univariate polynomial eval...
evl1scvarpwval 22391 Value of a univariate poly...
evl1gsummon 22392 Value of a univariate poly...
evls1scafv 22393 Value of the univariate po...
evls1expd 22394 Univariate polynomial eval...
evls1varpwval 22395 Univariate polynomial eval...
evls1fpws 22396 Evaluation of a univariate...
ressply1evl 22397 Evaluation of a univariate...
evls1addd 22398 Univariate polynomial eval...
evls1muld 22399 Univariate polynomial eval...
evls1vsca 22400 Univariate polynomial eval...
asclply1subcl 22401 Closure of the algebra sca...
evls1fvcl 22402 Variant of ~ fveval1fvcl f...
evls1maprhm 22403 The function ` F ` mapping...
evls1maplmhm 22404 The function ` F ` mapping...
evls1maprnss 22405 The function ` F ` mapping...
evl1maprhm 22406 The function ` F ` mapping...
mhmcompl 22407 The composition of a monoi...
mhmcoaddmpl 22408 Show that the ring homomor...
rhmcomulmpl 22409 Show that the ring homomor...
rhmmpl 22410 Provide a ring homomorphis...
ply1vscl 22411 Closure of scalar multipli...
mhmcoply1 22412 The composition of a monoi...
rhmply1 22413 Provide a ring homomorphis...
rhmply1vr1 22414 A ring homomorphism betwee...
rhmply1vsca 22415 Apply a ring homomorphism ...
rhmply1mon 22416 Apply a ring homomorphism ...
mamufval 22419 Functional value of the ma...
mamuval 22420 Multiplication of two matr...
mamufv 22421 A cell in the multiplicati...
mamudm 22422 The domain of the matrix m...
mamufacex 22423 Every solution of the equa...
mamures 22424 Rows in a matrix product a...
grpvlinv 22425 Tuple-wise left inverse in...
grpvrinv 22426 Tuple-wise right inverse i...
ringvcl 22427 Tuple-wise multiplication ...
mamucl 22428 Operation closure of matri...
mamuass 22429 Matrix multiplication is a...
mamudi 22430 Matrix multiplication dist...
mamudir 22431 Matrix multiplication dist...
mamuvs1 22432 Matrix multiplication dist...
mamuvs2 22433 Matrix multiplication dist...
matbas0pc 22436 There is no matrix with a ...
matbas0 22437 There is no matrix for a n...
matval 22438 Value of the matrix algebr...
matrcl 22439 Reverse closure for the ma...
matbas 22440 The matrix ring has the sa...
matplusg 22441 The matrix ring has the sa...
matsca 22442 The matrix ring has the sa...
matscaOLD 22443 Obsolete proof of ~ matsca...
matvsca 22444 The matrix ring has the sa...
matvscaOLD 22445 Obsolete proof of ~ matvsc...
mat0 22446 The matrix ring has the sa...
matinvg 22447 The matrix ring has the sa...
mat0op 22448 Value of a zero matrix as ...
matsca2 22449 The scalars of the matrix ...
matbas2 22450 The base set of the matrix...
matbas2i 22451 A matrix is a function. (...
matbas2d 22452 The base set of the matrix...
eqmat 22453 Two square matrices of the...
matecl 22454 Each entry (according to W...
matecld 22455 Each entry (according to W...
matplusg2 22456 Addition in the matrix rin...
matvsca2 22457 Scalar multiplication in t...
matlmod 22458 The matrix ring is a linea...
matgrp 22459 The matrix ring is a group...
matvscl 22460 Closure of the scalar mult...
matsubg 22461 The matrix ring has the sa...
matplusgcell 22462 Addition in the matrix rin...
matsubgcell 22463 Subtraction in the matrix ...
matinvgcell 22464 Additive inversion in the ...
matvscacell 22465 Scalar multiplication in t...
matgsum 22466 Finite commutative sums in...
matmulr 22467 Multiplication in the matr...
mamumat1cl 22468 The identity matrix (as op...
mat1comp 22469 The components of the iden...
mamulid 22470 The identity matrix (as op...
mamurid 22471 The identity matrix (as op...
matring 22472 Existence of the matrix ri...
matassa 22473 Existence of the matrix al...
matmulcell 22474 Multiplication in the matr...
mpomatmul 22475 Multiplication of two N x ...
mat1 22476 Value of an identity matri...
mat1ov 22477 Entries of an identity mat...
mat1bas 22478 The identity matrix is a m...
matsc 22479 The identity matrix multip...
ofco2 22480 Distribution law for the f...
oftpos 22481 The transposition of the v...
mattposcl 22482 The transpose of a square ...
mattpostpos 22483 The transpose of the trans...
mattposvs 22484 The transposition of a mat...
mattpos1 22485 The transposition of the i...
tposmap 22486 The transposition of an I ...
mamutpos 22487 Behavior of transposes in ...
mattposm 22488 Multiplying two transposed...
matgsumcl 22489 Closure of a group sum ove...
madetsumid 22490 The identity summand in th...
matepmcl 22491 Each entry of a matrix wit...
matepm2cl 22492 Each entry of a matrix wit...
madetsmelbas 22493 A summand of the determina...
madetsmelbas2 22494 A summand of the determina...
mat0dimbas0 22495 The empty set is the one a...
mat0dim0 22496 The zero of the algebra of...
mat0dimid 22497 The identity of the algebr...
mat0dimscm 22498 The scalar multiplication ...
mat0dimcrng 22499 The algebra of matrices wi...
mat1dimelbas 22500 A matrix with dimension 1 ...
mat1dimbas 22501 A matrix with dimension 1 ...
mat1dim0 22502 The zero of the algebra of...
mat1dimid 22503 The identity of the algebr...
mat1dimscm 22504 The scalar multiplication ...
mat1dimmul 22505 The ring multiplication in...
mat1dimcrng 22506 The algebra of matrices wi...
mat1f1o 22507 There is a 1-1 function fr...
mat1rhmval 22508 The value of the ring homo...
mat1rhmelval 22509 The value of the ring homo...
mat1rhmcl 22510 The value of the ring homo...
mat1f 22511 There is a function from a...
mat1ghm 22512 There is a group homomorph...
mat1mhm 22513 There is a monoid homomorp...
mat1rhm 22514 There is a ring homomorphi...
mat1rngiso 22515 There is a ring isomorphis...
mat1ric 22516 A ring is isomorphic to th...
dmatval 22521 The set of ` N ` x ` N ` d...
dmatel 22522 A ` N ` x ` N ` diagonal m...
dmatmat 22523 An ` N ` x ` N ` diagonal ...
dmatid 22524 The identity matrix is a d...
dmatelnd 22525 An extradiagonal entry of ...
dmatmul 22526 The product of two diagona...
dmatsubcl 22527 The difference of two diag...
dmatsgrp 22528 The set of diagonal matric...
dmatmulcl 22529 The product of two diagona...
dmatsrng 22530 The set of diagonal matric...
dmatcrng 22531 The subring of diagonal ma...
dmatscmcl 22532 The multiplication of a di...
scmatval 22533 The set of ` N ` x ` N ` s...
scmatel 22534 An ` N ` x ` N ` scalar ma...
scmatscmid 22535 A scalar matrix can be exp...
scmatscmide 22536 An entry of a scalar matri...
scmatscmiddistr 22537 Distributive law for scala...
scmatmat 22538 An ` N ` x ` N ` scalar ma...
scmate 22539 An entry of an ` N ` x ` N...
scmatmats 22540 The set of an ` N ` x ` N ...
scmateALT 22541 Alternate proof of ~ scmat...
scmatscm 22542 The multiplication of a ma...
scmatid 22543 The identity matrix is a s...
scmatdmat 22544 A scalar matrix is a diago...
scmataddcl 22545 The sum of two scalar matr...
scmatsubcl 22546 The difference of two scal...
scmatmulcl 22547 The product of two scalar ...
scmatsgrp 22548 The set of scalar matrices...
scmatsrng 22549 The set of scalar matrices...
scmatcrng 22550 The subring of scalar matr...
scmatsgrp1 22551 The set of scalar matrices...
scmatsrng1 22552 The set of scalar matrices...
smatvscl 22553 Closure of the scalar mult...
scmatlss 22554 The set of scalar matrices...
scmatstrbas 22555 The set of scalar matrices...
scmatrhmval 22556 The value of the ring homo...
scmatrhmcl 22557 The value of the ring homo...
scmatf 22558 There is a function from a...
scmatfo 22559 There is a function from a...
scmatf1 22560 There is a 1-1 function fr...
scmatf1o 22561 There is a bijection betwe...
scmatghm 22562 There is a group homomorph...
scmatmhm 22563 There is a monoid homomorp...
scmatrhm 22564 There is a ring homomorphi...
scmatrngiso 22565 There is a ring isomorphis...
scmatric 22566 A ring is isomorphic to ev...
mat0scmat 22567 The empty matrix over a ri...
mat1scmat 22568 A 1-dimensional matrix ove...
mvmulfval 22571 Functional value of the ma...
mvmulval 22572 Multiplication of a vector...
mvmulfv 22573 A cell/element in the vect...
mavmulval 22574 Multiplication of a vector...
mavmulfv 22575 A cell/element in the vect...
mavmulcl 22576 Multiplication of an NxN m...
1mavmul 22577 Multiplication of the iden...
mavmulass 22578 Associativity of the multi...
mavmuldm 22579 The domain of the matrix v...
mavmulsolcl 22580 Every solution of the equa...
mavmul0 22581 Multiplication of a 0-dime...
mavmul0g 22582 The result of the 0-dimens...
mvmumamul1 22583 The multiplication of an M...
mavmumamul1 22584 The multiplication of an N...
marrepfval 22589 First substitution for the...
marrepval0 22590 Second substitution for th...
marrepval 22591 Third substitution for the...
marrepeval 22592 An entry of a matrix with ...
marrepcl 22593 Closure of the row replace...
marepvfval 22594 First substitution for the...
marepvval0 22595 Second substitution for th...
marepvval 22596 Third substitution for the...
marepveval 22597 An entry of a matrix with ...
marepvcl 22598 Closure of the column repl...
ma1repvcl 22599 Closure of the column repl...
ma1repveval 22600 An entry of an identity ma...
mulmarep1el 22601 Element by element multipl...
mulmarep1gsum1 22602 The sum of element by elem...
mulmarep1gsum2 22603 The sum of element by elem...
1marepvmarrepid 22604 Replacing the ith row by 0...
submabas 22607 Any subset of the index se...
submafval 22608 First substitution for a s...
submaval0 22609 Second substitution for a ...
submaval 22610 Third substitution for a s...
submaeval 22611 An entry of a submatrix of...
1marepvsma1 22612 The submatrix of the ident...
mdetfval 22615 First substitution for the...
mdetleib 22616 Full substitution of our d...
mdetleib2 22617 Leibniz' formula can also ...
nfimdetndef 22618 The determinant is not def...
mdetfval1 22619 First substitution of an a...
mdetleib1 22620 Full substitution of an al...
mdet0pr 22621 The determinant function f...
mdet0f1o 22622 The determinant function f...
mdet0fv0 22623 The determinant of the emp...
mdetf 22624 Functionality of the deter...
mdetcl 22625 The determinant evaluates ...
m1detdiag 22626 The determinant of a 1-dim...
mdetdiaglem 22627 Lemma for ~ mdetdiag . Pr...
mdetdiag 22628 The determinant of a diago...
mdetdiagid 22629 The determinant of a diago...
mdet1 22630 The determinant of the ide...
mdetrlin 22631 The determinant function i...
mdetrsca 22632 The determinant function i...
mdetrsca2 22633 The determinant function i...
mdetr0 22634 The determinant of a matri...
mdet0 22635 The determinant of the zer...
mdetrlin2 22636 The determinant function i...
mdetralt 22637 The determinant function i...
mdetralt2 22638 The determinant function i...
mdetero 22639 The determinant function i...
mdettpos 22640 Determinant is invariant u...
mdetunilem1 22641 Lemma for ~ mdetuni . (Co...
mdetunilem2 22642 Lemma for ~ mdetuni . (Co...
mdetunilem3 22643 Lemma for ~ mdetuni . (Co...
mdetunilem4 22644 Lemma for ~ mdetuni . (Co...
mdetunilem5 22645 Lemma for ~ mdetuni . (Co...
mdetunilem6 22646 Lemma for ~ mdetuni . (Co...
mdetunilem7 22647 Lemma for ~ mdetuni . (Co...
mdetunilem8 22648 Lemma for ~ mdetuni . (Co...
mdetunilem9 22649 Lemma for ~ mdetuni . (Co...
mdetuni0 22650 Lemma for ~ mdetuni . (Co...
mdetuni 22651 According to the definitio...
mdetmul 22652 Multiplicativity of the de...
m2detleiblem1 22653 Lemma 1 for ~ m2detleib . ...
m2detleiblem5 22654 Lemma 5 for ~ m2detleib . ...
m2detleiblem6 22655 Lemma 6 for ~ m2detleib . ...
m2detleiblem7 22656 Lemma 7 for ~ m2detleib . ...
m2detleiblem2 22657 Lemma 2 for ~ m2detleib . ...
m2detleiblem3 22658 Lemma 3 for ~ m2detleib . ...
m2detleiblem4 22659 Lemma 4 for ~ m2detleib . ...
m2detleib 22660 Leibniz' Formula for 2x2-m...
mndifsplit 22665 Lemma for ~ maducoeval2 . ...
madufval 22666 First substitution for the...
maduval 22667 Second substitution for th...
maducoeval 22668 An entry of the adjunct (c...
maducoeval2 22669 An entry of the adjunct (c...
maduf 22670 Creating the adjunct of ma...
madutpos 22671 The adjuct of a transposed...
madugsum 22672 The determinant of a matri...
madurid 22673 Multiplying a matrix with ...
madulid 22674 Multiplying the adjunct of...
minmar1fval 22675 First substitution for the...
minmar1val0 22676 Second substitution for th...
minmar1val 22677 Third substitution for the...
minmar1eval 22678 An entry of a matrix for a...
minmar1marrep 22679 The minor matrix is a spec...
minmar1cl 22680 Closure of the row replace...
maducoevalmin1 22681 The coefficients of an adj...
symgmatr01lem 22682 Lemma for ~ symgmatr01 . ...
symgmatr01 22683 Applying a permutation tha...
gsummatr01lem1 22684 Lemma A for ~ gsummatr01 ....
gsummatr01lem2 22685 Lemma B for ~ gsummatr01 ....
gsummatr01lem3 22686 Lemma 1 for ~ gsummatr01 ....
gsummatr01lem4 22687 Lemma 2 for ~ gsummatr01 ....
gsummatr01 22688 Lemma 1 for ~ smadiadetlem...
marep01ma 22689 Replacing a row of a squar...
smadiadetlem0 22690 Lemma 0 for ~ smadiadet : ...
smadiadetlem1 22691 Lemma 1 for ~ smadiadet : ...
smadiadetlem1a 22692 Lemma 1a for ~ smadiadet :...
smadiadetlem2 22693 Lemma 2 for ~ smadiadet : ...
smadiadetlem3lem0 22694 Lemma 0 for ~ smadiadetlem...
smadiadetlem3lem1 22695 Lemma 1 for ~ smadiadetlem...
smadiadetlem3lem2 22696 Lemma 2 for ~ smadiadetlem...
smadiadetlem3 22697 Lemma 3 for ~ smadiadet . ...
smadiadetlem4 22698 Lemma 4 for ~ smadiadet . ...
smadiadet 22699 The determinant of a subma...
smadiadetglem1 22700 Lemma 1 for ~ smadiadetg ....
smadiadetglem2 22701 Lemma 2 for ~ smadiadetg ....
smadiadetg 22702 The determinant of a squar...
smadiadetg0 22703 Lemma for ~ smadiadetr : v...
smadiadetr 22704 The determinant of a squar...
invrvald 22705 If a matrix multiplied wit...
matinv 22706 The inverse of a matrix is...
matunit 22707 A matrix is a unit in the ...
slesolvec 22708 Every solution of a system...
slesolinv 22709 The solution of a system o...
slesolinvbi 22710 The solution of a system o...
slesolex 22711 Every system of linear equ...
cramerimplem1 22712 Lemma 1 for ~ cramerimp : ...
cramerimplem2 22713 Lemma 2 for ~ cramerimp : ...
cramerimplem3 22714 Lemma 3 for ~ cramerimp : ...
cramerimp 22715 One direction of Cramer's ...
cramerlem1 22716 Lemma 1 for ~ cramer . (C...
cramerlem2 22717 Lemma 2 for ~ cramer . (C...
cramerlem3 22718 Lemma 3 for ~ cramer . (C...
cramer0 22719 Special case of Cramer's r...
cramer 22720 Cramer's rule. According ...
pmatring 22721 The set of polynomial matr...
pmatlmod 22722 The set of polynomial matr...
pmatassa 22723 The set of polynomial matr...
pmat0op 22724 The zero polynomial matrix...
pmat1op 22725 The identity polynomial ma...
pmat1ovd 22726 Entries of the identity po...
pmat0opsc 22727 The zero polynomial matrix...
pmat1opsc 22728 The identity polynomial ma...
pmat1ovscd 22729 Entries of the identity po...
pmatcoe1fsupp 22730 For a polynomial matrix th...
1pmatscmul 22731 The scalar product of the ...
cpmat 22738 Value of the constructor o...
cpmatpmat 22739 A constant polynomial matr...
cpmatel 22740 Property of a constant pol...
cpmatelimp 22741 Implication of a set being...
cpmatel2 22742 Another property of a cons...
cpmatelimp2 22743 Another implication of a s...
1elcpmat 22744 The identity of the ring o...
cpmatacl 22745 The set of all constant po...
cpmatinvcl 22746 The set of all constant po...
cpmatmcllem 22747 Lemma for ~ cpmatmcl . (C...
cpmatmcl 22748 The set of all constant po...
cpmatsubgpmat 22749 The set of all constant po...
cpmatsrgpmat 22750 The set of all constant po...
0elcpmat 22751 The zero of the ring of al...
mat2pmatfval 22752 Value of the matrix transf...
mat2pmatval 22753 The result of a matrix tra...
mat2pmatvalel 22754 A (matrix) element of the ...
mat2pmatbas 22755 The result of a matrix tra...
mat2pmatbas0 22756 The result of a matrix tra...
mat2pmatf 22757 The matrix transformation ...
mat2pmatf1 22758 The matrix transformation ...
mat2pmatghm 22759 The transformation of matr...
mat2pmatmul 22760 The transformation of matr...
mat2pmat1 22761 The transformation of the ...
mat2pmatmhm 22762 The transformation of matr...
mat2pmatrhm 22763 The transformation of matr...
mat2pmatlin 22764 The transformation of matr...
0mat2pmat 22765 The transformed zero matri...
idmatidpmat 22766 The transformed identity m...
d0mat2pmat 22767 The transformed empty set ...
d1mat2pmat 22768 The transformation of a ma...
mat2pmatscmxcl 22769 A transformed matrix multi...
m2cpm 22770 The result of a matrix tra...
m2cpmf 22771 The matrix transformation ...
m2cpmf1 22772 The matrix transformation ...
m2cpmghm 22773 The transformation of matr...
m2cpmmhm 22774 The transformation of matr...
m2cpmrhm 22775 The transformation of matr...
m2pmfzmap 22776 The transformed values of ...
m2pmfzgsumcl 22777 Closure of the sum of scal...
cpm2mfval 22778 Value of the inverse matri...
cpm2mval 22779 The result of an inverse m...
cpm2mvalel 22780 A (matrix) element of the ...
cpm2mf 22781 The inverse matrix transfo...
m2cpminvid 22782 The inverse transformation...
m2cpminvid2lem 22783 Lemma for ~ m2cpminvid2 . ...
m2cpminvid2 22784 The transformation applied...
m2cpmfo 22785 The matrix transformation ...
m2cpmf1o 22786 The matrix transformation ...
m2cpmrngiso 22787 The transformation of matr...
matcpmric 22788 The ring of matrices over ...
m2cpminv 22789 The inverse matrix transfo...
m2cpminv0 22790 The inverse matrix transfo...
decpmatval0 22793 The matrix consisting of t...
decpmatval 22794 The matrix consisting of t...
decpmate 22795 An entry of the matrix con...
decpmatcl 22796 Closure of the decompositi...
decpmataa0 22797 The matrix consisting of t...
decpmatfsupp 22798 The mapping to the matrice...
decpmatid 22799 The matrix consisting of t...
decpmatmullem 22800 Lemma for ~ decpmatmul . ...
decpmatmul 22801 The matrix consisting of t...
decpmatmulsumfsupp 22802 Lemma 0 for ~ pm2mpmhm . ...
pmatcollpw1lem1 22803 Lemma 1 for ~ pmatcollpw1 ...
pmatcollpw1lem2 22804 Lemma 2 for ~ pmatcollpw1 ...
pmatcollpw1 22805 Write a polynomial matrix ...
pmatcollpw2lem 22806 Lemma for ~ pmatcollpw2 . ...
pmatcollpw2 22807 Write a polynomial matrix ...
monmatcollpw 22808 The matrix consisting of t...
pmatcollpwlem 22809 Lemma for ~ pmatcollpw . ...
pmatcollpw 22810 Write a polynomial matrix ...
pmatcollpwfi 22811 Write a polynomial matrix ...
pmatcollpw3lem 22812 Lemma for ~ pmatcollpw3 an...
pmatcollpw3 22813 Write a polynomial matrix ...
pmatcollpw3fi 22814 Write a polynomial matrix ...
pmatcollpw3fi1lem1 22815 Lemma 1 for ~ pmatcollpw3f...
pmatcollpw3fi1lem2 22816 Lemma 2 for ~ pmatcollpw3f...
pmatcollpw3fi1 22817 Write a polynomial matrix ...
pmatcollpwscmatlem1 22818 Lemma 1 for ~ pmatcollpwsc...
pmatcollpwscmatlem2 22819 Lemma 2 for ~ pmatcollpwsc...
pmatcollpwscmat 22820 Write a scalar matrix over...
pm2mpf1lem 22823 Lemma for ~ pm2mpf1 . (Co...
pm2mpval 22824 Value of the transformatio...
pm2mpfval 22825 A polynomial matrix transf...
pm2mpcl 22826 The transformation of poly...
pm2mpf 22827 The transformation of poly...
pm2mpf1 22828 The transformation of poly...
pm2mpcoe1 22829 A coefficient of the polyn...
idpm2idmp 22830 The transformation of the ...
mptcoe1matfsupp 22831 The mapping extracting the...
mply1topmatcllem 22832 Lemma for ~ mply1topmatcl ...
mply1topmatval 22833 A polynomial over matrices...
mply1topmatcl 22834 A polynomial over matrices...
mp2pm2mplem1 22835 Lemma 1 for ~ mp2pm2mp . ...
mp2pm2mplem2 22836 Lemma 2 for ~ mp2pm2mp . ...
mp2pm2mplem3 22837 Lemma 3 for ~ mp2pm2mp . ...
mp2pm2mplem4 22838 Lemma 4 for ~ mp2pm2mp . ...
mp2pm2mplem5 22839 Lemma 5 for ~ mp2pm2mp . ...
mp2pm2mp 22840 A polynomial over matrices...
pm2mpghmlem2 22841 Lemma 2 for ~ pm2mpghm . ...
pm2mpghmlem1 22842 Lemma 1 for pm2mpghm . (C...
pm2mpfo 22843 The transformation of poly...
pm2mpf1o 22844 The transformation of poly...
pm2mpghm 22845 The transformation of poly...
pm2mpgrpiso 22846 The transformation of poly...
pm2mpmhmlem1 22847 Lemma 1 for ~ pm2mpmhm . ...
pm2mpmhmlem2 22848 Lemma 2 for ~ pm2mpmhm . ...
pm2mpmhm 22849 The transformation of poly...
pm2mprhm 22850 The transformation of poly...
pm2mprngiso 22851 The transformation of poly...
pmmpric 22852 The ring of polynomial mat...
monmat2matmon 22853 The transformation of a po...
pm2mp 22854 The transformation of a su...
chmatcl 22857 Closure of the characteris...
chmatval 22858 The entries of the charact...
chpmatfval 22859 Value of the characteristi...
chpmatval 22860 The characteristic polynom...
chpmatply1 22861 The characteristic polynom...
chpmatval2 22862 The characteristic polynom...
chpmat0d 22863 The characteristic polynom...
chpmat1dlem 22864 Lemma for ~ chpmat1d . (C...
chpmat1d 22865 The characteristic polynom...
chpdmatlem0 22866 Lemma 0 for ~ chpdmat . (...
chpdmatlem1 22867 Lemma 1 for ~ chpdmat . (...
chpdmatlem2 22868 Lemma 2 for ~ chpdmat . (...
chpdmatlem3 22869 Lemma 3 for ~ chpdmat . (...
chpdmat 22870 The characteristic polynom...
chpscmat 22871 The characteristic polynom...
chpscmat0 22872 The characteristic polynom...
chpscmatgsumbin 22873 The characteristic polynom...
chpscmatgsummon 22874 The characteristic polynom...
chp0mat 22875 The characteristic polynom...
chpidmat 22876 The characteristic polynom...
chmaidscmat 22877 The characteristic polynom...
fvmptnn04if 22878 The function values of a m...
fvmptnn04ifa 22879 The function value of a ma...
fvmptnn04ifb 22880 The function value of a ma...
fvmptnn04ifc 22881 The function value of a ma...
fvmptnn04ifd 22882 The function value of a ma...
chfacfisf 22883 The "characteristic factor...
chfacfisfcpmat 22884 The "characteristic factor...
chfacffsupp 22885 The "characteristic factor...
chfacfscmulcl 22886 Closure of a scaled value ...
chfacfscmul0 22887 A scaled value of the "cha...
chfacfscmulfsupp 22888 A mapping of scaled values...
chfacfscmulgsum 22889 Breaking up a sum of value...
chfacfpmmulcl 22890 Closure of the value of th...
chfacfpmmul0 22891 The value of the "characte...
chfacfpmmulfsupp 22892 A mapping of values of the...
chfacfpmmulgsum 22893 Breaking up a sum of value...
chfacfpmmulgsum2 22894 Breaking up a sum of value...
cayhamlem1 22895 Lemma 1 for ~ cayleyhamilt...
cpmadurid 22896 The right-hand fundamental...
cpmidgsum 22897 Representation of the iden...
cpmidgsumm2pm 22898 Representation of the iden...
cpmidpmatlem1 22899 Lemma 1 for ~ cpmidpmat . ...
cpmidpmatlem2 22900 Lemma 2 for ~ cpmidpmat . ...
cpmidpmatlem3 22901 Lemma 3 for ~ cpmidpmat . ...
cpmidpmat 22902 Representation of the iden...
cpmadugsumlemB 22903 Lemma B for ~ cpmadugsum ....
cpmadugsumlemC 22904 Lemma C for ~ cpmadugsum ....
cpmadugsumlemF 22905 Lemma F for ~ cpmadugsum ....
cpmadugsumfi 22906 The product of the charact...
cpmadugsum 22907 The product of the charact...
cpmidgsum2 22908 Representation of the iden...
cpmidg2sum 22909 Equality of two sums repre...
cpmadumatpolylem1 22910 Lemma 1 for ~ cpmadumatpol...
cpmadumatpolylem2 22911 Lemma 2 for ~ cpmadumatpol...
cpmadumatpoly 22912 The product of the charact...
cayhamlem2 22913 Lemma for ~ cayhamlem3 . ...
chcoeffeqlem 22914 Lemma for ~ chcoeffeq . (...
chcoeffeq 22915 The coefficients of the ch...
cayhamlem3 22916 Lemma for ~ cayhamlem4 . ...
cayhamlem4 22917 Lemma for ~ cayleyhamilton...
cayleyhamilton0 22918 The Cayley-Hamilton theore...
cayleyhamilton 22919 The Cayley-Hamilton theore...
cayleyhamiltonALT 22920 Alternate proof of ~ cayle...
cayleyhamilton1 22921 The Cayley-Hamilton theore...
istopg 22924 Express the predicate " ` ...
istop2g 22925 Express the predicate " ` ...
uniopn 22926 The union of a subset of a...
iunopn 22927 The indexed union of a sub...
inopn 22928 The intersection of two op...
fitop 22929 A topology is closed under...
fiinopn 22930 The intersection of a none...
iinopn 22931 The intersection of a none...
unopn 22932 The union of two open sets...
0opn 22933 The empty set is an open s...
0ntop 22934 The empty set is not a top...
topopn 22935 The underlying set of a to...
eltopss 22936 A member of a topology is ...
riinopn 22937 A finite indexed relative ...
rintopn 22938 A finite relative intersec...
istopon 22941 Property of being a topolo...
topontop 22942 A topology on a given base...
toponuni 22943 The base set of a topology...
topontopi 22944 A topology on a given base...
toponunii 22945 The base set of a topology...
toptopon 22946 Alternative definition of ...
toptopon2 22947 A topology is the same thi...
topontopon 22948 A topology on a set is a t...
funtopon 22949 The class ` TopOn ` is a f...
toponrestid 22950 Given a topology on a set,...
toponsspwpw 22951 The set of topologies on a...
dmtopon 22952 The domain of ` TopOn ` is...
fntopon 22953 The class ` TopOn ` is a f...
toprntopon 22954 A topology is the same thi...
toponmax 22955 The base set of a topology...
toponss 22956 A member of a topology is ...
toponcom 22957 If ` K ` is a topology on ...
toponcomb 22958 Biconditional form of ~ to...
topgele 22959 The topologies over the sa...
topsn 22960 The only topology on a sin...
istps 22963 Express the predicate "is ...
istps2 22964 Express the predicate "is ...
tpsuni 22965 The base set of a topologi...
tpstop 22966 The topology extractor on ...
tpspropd 22967 A topological space depend...
tpsprop2d 22968 A topological space depend...
topontopn 22969 Express the predicate "is ...
tsettps 22970 If the topology component ...
istpsi 22971 Properties that determine ...
eltpsg 22972 Properties that determine ...
eltpsgOLD 22973 Obsolete version of ~ eltp...
eltpsi 22974 Properties that determine ...
isbasisg 22977 Express the predicate "the...
isbasis2g 22978 Express the predicate "the...
isbasis3g 22979 Express the predicate "the...
basis1 22980 Property of a basis. (Con...
basis2 22981 Property of a basis. (Con...
fiinbas 22982 If a set is closed under f...
basdif0 22983 A basis is not affected by...
baspartn 22984 A disjoint system of sets ...
tgval 22985 The topology generated by ...
tgval2 22986 Definition of a topology g...
eltg 22987 Membership in a topology g...
eltg2 22988 Membership in a topology g...
eltg2b 22989 Membership in a topology g...
eltg4i 22990 An open set in a topology ...
eltg3i 22991 The union of a set of basi...
eltg3 22992 Membership in a topology g...
tgval3 22993 Alternate expression for t...
tg1 22994 Property of a member of a ...
tg2 22995 Property of a member of a ...
bastg 22996 A member of a basis is a s...
unitg 22997 The topology generated by ...
tgss 22998 Subset relation for genera...
tgcl 22999 Show that a basis generate...
tgclb 23000 The property ~ tgcl can be...
tgtopon 23001 A basis generates a topolo...
topbas 23002 A topology is its own basi...
tgtop 23003 A topology is its own basi...
eltop 23004 Membership in a topology, ...
eltop2 23005 Membership in a topology. ...
eltop3 23006 Membership in a topology. ...
fibas 23007 A collection of finite int...
tgdom 23008 A space has no more open s...
tgiun 23009 The indexed union of a set...
tgidm 23010 The topology generator fun...
bastop 23011 Two ways to express that a...
tgtop11 23012 The topology generation fu...
0top 23013 The singleton of the empty...
en1top 23014 ` { (/) } ` is the only to...
en2top 23015 If a topology has two elem...
tgss3 23016 A criterion for determinin...
tgss2 23017 A criterion for determinin...
basgen 23018 Given a topology ` J ` , s...
basgen2 23019 Given a topology ` J ` , s...
2basgen 23020 Conditions that determine ...
tgfiss 23021 If a subbase is included i...
tgdif0 23022 A generated topology is no...
bastop1 23023 A subset of a topology is ...
bastop2 23024 A version of ~ bastop1 tha...
distop 23025 The discrete topology on a...
topnex 23026 The class of all topologie...
distopon 23027 The discrete topology on a...
sn0topon 23028 The singleton of the empty...
sn0top 23029 The singleton of the empty...
indislem 23030 A lemma to eliminate some ...
indistopon 23031 The indiscrete topology on...
indistop 23032 The indiscrete topology on...
indisuni 23033 The base set of the indisc...
fctop 23034 The finite complement topo...
fctop2 23035 The finite complement topo...
cctop 23036 The countable complement t...
ppttop 23037 The particular point topol...
pptbas 23038 The particular point topol...
epttop 23039 The excluded point topolog...
indistpsx 23040 The indiscrete topology on...
indistps 23041 The indiscrete topology on...
indistps2 23042 The indiscrete topology on...
indistpsALT 23043 The indiscrete topology on...
indistpsALTOLD 23044 Obsolete version of ~ indi...
indistps2ALT 23045 The indiscrete topology on...
distps 23046 The discrete topology on a...
fncld 23053 The closed-set generator i...
cldval 23054 The set of closed sets of ...
ntrfval 23055 The interior function on t...
clsfval 23056 The closure function on th...
cldrcl 23057 Reverse closure of the clo...
iscld 23058 The predicate "the class `...
iscld2 23059 A subset of the underlying...
cldss 23060 A closed set is a subset o...
cldss2 23061 The set of closed sets is ...
cldopn 23062 The complement of a closed...
isopn2 23063 A subset of the underlying...
opncld 23064 The complement of an open ...
difopn 23065 The difference of a closed...
topcld 23066 The underlying set of a to...
ntrval 23067 The interior of a subset o...
clsval 23068 The closure of a subset of...
0cld 23069 The empty set is closed. ...
iincld 23070 The indexed intersection o...
intcld 23071 The intersection of a set ...
uncld 23072 The union of two closed se...
cldcls 23073 A closed subset equals its...
incld 23074 The intersection of two cl...
riincld 23075 An indexed relative inters...
iuncld 23076 A finite indexed union of ...
unicld 23077 A finite union of closed s...
clscld 23078 The closure of a subset of...
clsf 23079 The closure function is a ...
ntropn 23080 The interior of a subset o...
clsval2 23081 Express closure in terms o...
ntrval2 23082 Interior expressed in term...
ntrdif 23083 An interior of a complemen...
clsdif 23084 A closure of a complement ...
clsss 23085 Subset relationship for cl...
ntrss 23086 Subset relationship for in...
sscls 23087 A subset of a topology's u...
ntrss2 23088 A subset includes its inte...
ssntr 23089 An open subset of a set is...
clsss3 23090 The closure of a subset of...
ntrss3 23091 The interior of a subset o...
ntrin 23092 A pairwise intersection of...
cmclsopn 23093 The complement of a closur...
cmntrcld 23094 The complement of an inter...
iscld3 23095 A subset is closed iff it ...
iscld4 23096 A subset is closed iff it ...
isopn3 23097 A subset is open iff it eq...
clsidm 23098 The closure operation is i...
ntridm 23099 The interior operation is ...
clstop 23100 The closure of a topology'...
ntrtop 23101 The interior of a topology...
0ntr 23102 A subset with an empty int...
clsss2 23103 If a subset is included in...
elcls 23104 Membership in a closure. ...
elcls2 23105 Membership in a closure. ...
clsndisj 23106 Any open set containing a ...
ntrcls0 23107 A subset whose closure has...
ntreq0 23108 Two ways to say that a sub...
cldmre 23109 The closed sets of a topol...
mrccls 23110 Moore closure generalizes ...
cls0 23111 The closure of the empty s...
ntr0 23112 The interior of the empty ...
isopn3i 23113 An open subset equals its ...
elcls3 23114 Membership in a closure in...
opncldf1 23115 A bijection useful for con...
opncldf2 23116 The values of the open-clo...
opncldf3 23117 The values of the converse...
isclo 23118 A set ` A ` is clopen iff ...
isclo2 23119 A set ` A ` is clopen iff ...
discld 23120 The open sets of a discret...
sn0cld 23121 The closed sets of the top...
indiscld 23122 The closed sets of an indi...
mretopd 23123 A Moore collection which i...
toponmre 23124 The topologies over a give...
cldmreon 23125 The closed sets of a topol...
iscldtop 23126 A family is the closed set...
mreclatdemoBAD 23127 The closed subspaces of a ...
neifval 23130 Value of the neighborhood ...
neif 23131 The neighborhood function ...
neiss2 23132 A set with a neighborhood ...
neival 23133 Value of the set of neighb...
isnei 23134 The predicate "the class `...
neiint 23135 An intuitive definition of...
isneip 23136 The predicate "the class `...
neii1 23137 A neighborhood is included...
neisspw 23138 The neighborhoods of any s...
neii2 23139 Property of a neighborhood...
neiss 23140 Any neighborhood of a set ...
ssnei 23141 A set is included in any o...
elnei 23142 A point belongs to any of ...
0nnei 23143 The empty set is not a nei...
neips 23144 A neighborhood of a set is...
opnneissb 23145 An open set is a neighborh...
opnssneib 23146 Any superset of an open se...
ssnei2 23147 Any subset ` M ` of ` X ` ...
neindisj 23148 Any neighborhood of an ele...
opnneiss 23149 An open set is a neighborh...
opnneip 23150 An open set is a neighborh...
opnnei 23151 A set is open iff it is a ...
tpnei 23152 The underlying set of a to...
neiuni 23153 The union of the neighborh...
neindisj2 23154 A point ` P ` belongs to t...
topssnei 23155 A finer topology has more ...
innei 23156 The intersection of two ne...
opnneiid 23157 Only an open set is a neig...
neissex 23158 For any neighborhood ` N `...
0nei 23159 The empty set is a neighbo...
neipeltop 23160 Lemma for ~ neiptopreu . ...
neiptopuni 23161 Lemma for ~ neiptopreu . ...
neiptoptop 23162 Lemma for ~ neiptopreu . ...
neiptopnei 23163 Lemma for ~ neiptopreu . ...
neiptopreu 23164 If, to each element ` P ` ...
lpfval 23169 The limit point function o...
lpval 23170 The set of limit points of...
islp 23171 The predicate "the class `...
lpsscls 23172 The limit points of a subs...
lpss 23173 The limit points of a subs...
lpdifsn 23174 ` P ` is a limit point of ...
lpss3 23175 Subset relationship for li...
islp2 23176 The predicate " ` P ` is a...
islp3 23177 The predicate " ` P ` is a...
maxlp 23178 A point is a limit point o...
clslp 23179 The closure of a subset of...
islpi 23180 A point belonging to a set...
cldlp 23181 A subset of a topological ...
isperf 23182 Definition of a perfect sp...
isperf2 23183 Definition of a perfect sp...
isperf3 23184 A perfect space is a topol...
perflp 23185 The limit points of a perf...
perfi 23186 Property of a perfect spac...
perftop 23187 A perfect space is a topol...
restrcl 23188 Reverse closure for the su...
restbas 23189 A subspace topology basis ...
tgrest 23190 A subspace can be generate...
resttop 23191 A subspace topology is a t...
resttopon 23192 A subspace topology is a t...
restuni 23193 The underlying set of a su...
stoig 23194 The topological space buil...
restco 23195 Composition of subspaces. ...
restabs 23196 Equivalence of being a sub...
restin 23197 When the subspace region i...
restuni2 23198 The underlying set of a su...
resttopon2 23199 The underlying set of a su...
rest0 23200 The subspace topology indu...
restsn 23201 The only subspace topology...
restsn2 23202 The subspace topology indu...
restcld 23203 A closed set of a subspace...
restcldi 23204 A closed set is closed in ...
restcldr 23205 A set which is closed in t...
restopnb 23206 If ` B ` is an open subset...
ssrest 23207 If ` K ` is a finer topolo...
restopn2 23208 If ` A ` is open, then ` B...
restdis 23209 A subspace of a discrete t...
restfpw 23210 The restriction of the set...
neitr 23211 The neighborhood of a trac...
restcls 23212 A closure in a subspace to...
restntr 23213 An interior in a subspace ...
restlp 23214 The limit points of a subs...
restperf 23215 Perfection of a subspace. ...
perfopn 23216 An open subset of a perfec...
resstopn 23217 The topology of a restrict...
resstps 23218 A restricted topological s...
ordtbaslem 23219 Lemma for ~ ordtbas . In ...
ordtval 23220 Value of the order topolog...
ordtuni 23221 Value of the order topolog...
ordtbas2 23222 Lemma for ~ ordtbas . (Co...
ordtbas 23223 In a total order, the fini...
ordttopon 23224 Value of the order topolog...
ordtopn1 23225 An upward ray ` ( P , +oo ...
ordtopn2 23226 A downward ray ` ( -oo , P...
ordtopn3 23227 An open interval ` ( A , B...
ordtcld1 23228 A downward ray ` ( -oo , P...
ordtcld2 23229 An upward ray ` [ P , +oo ...
ordtcld3 23230 A closed interval ` [ A , ...
ordttop 23231 The order topology is a to...
ordtcnv 23232 The order dual generates t...
ordtrest 23233 The subspace topology of a...
ordtrest2lem 23234 Lemma for ~ ordtrest2 . (...
ordtrest2 23235 An interval-closed set ` A...
letopon 23236 The topology of the extend...
letop 23237 The topology of the extend...
letopuni 23238 The topology of the extend...
xrstopn 23239 The topology component of ...
xrstps 23240 The extended real number s...
leordtvallem1 23241 Lemma for ~ leordtval . (...
leordtvallem2 23242 Lemma for ~ leordtval . (...
leordtval2 23243 The topology of the extend...
leordtval 23244 The topology of the extend...
iccordt 23245 A closed interval is close...
iocpnfordt 23246 An unbounded above open in...
icomnfordt 23247 An unbounded above open in...
iooordt 23248 An open interval is open i...
reordt 23249 The real numbers are an op...
lecldbas 23250 The set of closed interval...
pnfnei 23251 A neighborhood of ` +oo ` ...
mnfnei 23252 A neighborhood of ` -oo ` ...
ordtrestixx 23253 The restriction of the les...
ordtresticc 23254 The restriction of the les...
lmrel 23261 The topological space conv...
lmrcl 23262 Reverse closure for the co...
lmfval 23263 The relation "sequence ` f...
cnfval 23264 The set of all continuous ...
cnpfval 23265 The function mapping the p...
iscn 23266 The predicate "the class `...
cnpval 23267 The set of all functions f...
iscnp 23268 The predicate "the class `...
iscn2 23269 The predicate "the class `...
iscnp2 23270 The predicate "the class `...
cntop1 23271 Reverse closure for a cont...
cntop2 23272 Reverse closure for a cont...
cnptop1 23273 Reverse closure for a func...
cnptop2 23274 Reverse closure for a func...
iscnp3 23275 The predicate "the class `...
cnprcl 23276 Reverse closure for a func...
cnf 23277 A continuous function is a...
cnpf 23278 A continuous function at p...
cnpcl 23279 The value of a continuous ...
cnf2 23280 A continuous function is a...
cnpf2 23281 A continuous function at p...
cnprcl2 23282 Reverse closure for a func...
tgcn 23283 The continuity predicate w...
tgcnp 23284 The "continuous at a point...
subbascn 23285 The continuity predicate w...
ssidcn 23286 The identity function is a...
cnpimaex 23287 Property of a function con...
idcn 23288 A restricted identity func...
lmbr 23289 Express the binary relatio...
lmbr2 23290 Express the binary relatio...
lmbrf 23291 Express the binary relatio...
lmconst 23292 A constant sequence conver...
lmcvg 23293 Convergence property of a ...
iscnp4 23294 The predicate "the class `...
cnpnei 23295 A condition for continuity...
cnima 23296 An open subset of the codo...
cnco 23297 The composition of two con...
cnpco 23298 The composition of a funct...
cnclima 23299 A closed subset of the cod...
iscncl 23300 A characterization of a co...
cncls2i 23301 Property of the preimage o...
cnntri 23302 Property of the preimage o...
cnclsi 23303 Property of the image of a...
cncls2 23304 Continuity in terms of clo...
cncls 23305 Continuity in terms of clo...
cnntr 23306 Continuity in terms of int...
cnss1 23307 If the topology ` K ` is f...
cnss2 23308 If the topology ` K ` is f...
cncnpi 23309 A continuous function is c...
cnsscnp 23310 The set of continuous func...
cncnp 23311 A continuous function is c...
cncnp2 23312 A continuous function is c...
cnnei 23313 Continuity in terms of nei...
cnconst2 23314 A constant function is con...
cnconst 23315 A constant function is con...
cnrest 23316 Continuity of a restrictio...
cnrest2 23317 Equivalence of continuity ...
cnrest2r 23318 Equivalence of continuity ...
cnpresti 23319 One direction of ~ cnprest...
cnprest 23320 Equivalence of continuity ...
cnprest2 23321 Equivalence of point-conti...
cndis 23322 Every function is continuo...
cnindis 23323 Every function is continuo...
cnpdis 23324 If ` A ` is an isolated po...
paste 23325 Pasting lemma. If ` A ` a...
lmfpm 23326 If ` F ` converges, then `...
lmfss 23327 Inclusion of a function ha...
lmcl 23328 Closure of a limit. (Cont...
lmss 23329 Limit on a subspace. (Con...
sslm 23330 A finer topology has fewer...
lmres 23331 A function converges iff i...
lmff 23332 If ` F ` converges, there ...
lmcls 23333 Any convergent sequence of...
lmcld 23334 Any convergent sequence of...
lmcnp 23335 The image of a convergent ...
lmcn 23336 The image of a convergent ...
ist0 23351 The predicate "is a T_0 sp...
ist1 23352 The predicate "is a T_1 sp...
ishaus 23353 The predicate "is a Hausdo...
iscnrm 23354 The property of being comp...
t0sep 23355 Any two topologically indi...
t0dist 23356 Any two distinct points in...
t1sncld 23357 In a T_1 space, singletons...
t1ficld 23358 In a T_1 space, finite set...
hausnei 23359 Neighborhood property of a...
t0top 23360 A T_0 space is a topologic...
t1top 23361 A T_1 space is a topologic...
haustop 23362 A Hausdorff space is a top...
isreg 23363 The predicate "is a regula...
regtop 23364 A regular space is a topol...
regsep 23365 In a regular space, every ...
isnrm 23366 The predicate "is a normal...
nrmtop 23367 A normal space is a topolo...
cnrmtop 23368 A completely normal space ...
iscnrm2 23369 The property of being comp...
ispnrm 23370 The property of being perf...
pnrmnrm 23371 A perfectly normal space i...
pnrmtop 23372 A perfectly normal space i...
pnrmcld 23373 A closed set in a perfectl...
pnrmopn 23374 An open set in a perfectly...
ist0-2 23375 The predicate "is a T_0 sp...
ist0-3 23376 The predicate "is a T_0 sp...
cnt0 23377 The preimage of a T_0 topo...
ist1-2 23378 An alternate characterizat...
t1t0 23379 A T_1 space is a T_0 space...
ist1-3 23380 A space is T_1 iff every p...
cnt1 23381 The preimage of a T_1 topo...
ishaus2 23382 Express the predicate " ` ...
haust1 23383 A Hausdorff space is a T_1...
hausnei2 23384 The Hausdorff condition st...
cnhaus 23385 The preimage of a Hausdorf...
nrmsep3 23386 In a normal space, given a...
nrmsep2 23387 In a normal space, any two...
nrmsep 23388 In a normal space, disjoin...
isnrm2 23389 An alternate characterizat...
isnrm3 23390 A topological space is nor...
cnrmi 23391 A subspace of a completely...
cnrmnrm 23392 A completely normal space ...
restcnrm 23393 A subspace of a completely...
resthauslem 23394 Lemma for ~ resthaus and s...
lpcls 23395 The limit points of the cl...
perfcls 23396 A subset of a perfect spac...
restt0 23397 A subspace of a T_0 topolo...
restt1 23398 A subspace of a T_1 topolo...
resthaus 23399 A subspace of a Hausdorff ...
t1sep2 23400 Any two points in a T_1 sp...
t1sep 23401 Any two distinct points in...
sncld 23402 A singleton is closed in a...
sshauslem 23403 Lemma for ~ sshaus and sim...
sst0 23404 A topology finer than a T_...
sst1 23405 A topology finer than a T_...
sshaus 23406 A topology finer than a Ha...
regsep2 23407 In a regular space, a clos...
isreg2 23408 A topological space is reg...
dnsconst 23409 If a continuous mapping to...
ordtt1 23410 The order topology is T_1 ...
lmmo 23411 A sequence in a Hausdorff ...
lmfun 23412 The convergence relation i...
dishaus 23413 A discrete topology is Hau...
ordthauslem 23414 Lemma for ~ ordthaus . (C...
ordthaus 23415 The order topology of a to...
xrhaus 23416 The topology of the extend...
iscmp 23419 The predicate "is a compac...
cmpcov 23420 An open cover of a compact...
cmpcov2 23421 Rewrite ~ cmpcov for the c...
cmpcovf 23422 Combine ~ cmpcov with ~ ac...
cncmp 23423 Compactness is respected b...
fincmp 23424 A finite topology is compa...
0cmp 23425 The singleton of the empty...
cmptop 23426 A compact topology is a to...
rncmp 23427 The image of a compact set...
imacmp 23428 The image of a compact set...
discmp 23429 A discrete topology is com...
cmpsublem 23430 Lemma for ~ cmpsub . (Con...
cmpsub 23431 Two equivalent ways of des...
tgcmp 23432 A topology generated by a ...
cmpcld 23433 A closed subset of a compa...
uncmp 23434 The union of two compact s...
fiuncmp 23435 A finite union of compact ...
sscmp 23436 A subset of a compact topo...
hauscmplem 23437 Lemma for ~ hauscmp . (Co...
hauscmp 23438 A compact subspace of a T2...
cmpfi 23439 If a topology is compact a...
cmpfii 23440 In a compact topology, a s...
bwth 23441 The glorious Bolzano-Weier...
isconn 23444 The predicate ` J ` is a c...
isconn2 23445 The predicate ` J ` is a c...
connclo 23446 The only nonempty clopen s...
conndisj 23447 If a topology is connected...
conntop 23448 A connected topology is a ...
indisconn 23449 The indiscrete topology (o...
dfconn2 23450 An alternate definition of...
connsuba 23451 Connectedness for a subspa...
connsub 23452 Two equivalent ways of say...
cnconn 23453 Connectedness is respected...
nconnsubb 23454 Disconnectedness for a sub...
connsubclo 23455 If a clopen set meets a co...
connima 23456 The image of a connected s...
conncn 23457 A continuous function from...
iunconnlem 23458 Lemma for ~ iunconn . (Co...
iunconn 23459 The indexed union of conne...
unconn 23460 The union of two connected...
clsconn 23461 The closure of a connected...
conncompid 23462 The connected component co...
conncompconn 23463 The connected component co...
conncompss 23464 The connected component co...
conncompcld 23465 The connected component co...
conncompclo 23466 The connected component co...
t1connperf 23467 A connected T_1 space is p...
is1stc 23472 The predicate "is a first-...
is1stc2 23473 An equivalent way of sayin...
1stctop 23474 A first-countable topology...
1stcclb 23475 A property of points in a ...
1stcfb 23476 For any point ` A ` in a f...
is2ndc 23477 The property of being seco...
2ndctop 23478 A second-countable topolog...
2ndci 23479 A countable basis generate...
2ndcsb 23480 Having a countable subbase...
2ndcredom 23481 A second-countable space h...
2ndc1stc 23482 A second-countable space i...
1stcrestlem 23483 Lemma for ~ 1stcrest . (C...
1stcrest 23484 A subspace of a first-coun...
2ndcrest 23485 A subspace of a second-cou...
2ndcctbss 23486 If a topology is second-co...
2ndcdisj 23487 Any disjoint family of ope...
2ndcdisj2 23488 Any disjoint collection of...
2ndcomap 23489 A surjective continuous op...
2ndcsep 23490 A second-countable topolog...
dis2ndc 23491 A discrete space is second...
1stcelcls 23492 A point belongs to the clo...
1stccnp 23493 A mapping is continuous at...
1stccn 23494 A mapping ` X --> Y ` , wh...
islly 23499 The property of being a lo...
isnlly 23500 The property of being an n...
llyeq 23501 Equality theorem for the `...
nllyeq 23502 Equality theorem for the `...
llytop 23503 A locally ` A ` space is a...
nllytop 23504 A locally ` A ` space is a...
llyi 23505 The property of a locally ...
nllyi 23506 The property of an n-local...
nlly2i 23507 Eliminate the neighborhood...
llynlly 23508 A locally ` A ` space is n...
llyssnlly 23509 A locally ` A ` space is n...
llyss 23510 The "locally" predicate re...
nllyss 23511 The "n-locally" predicate ...
subislly 23512 The property of a subspace...
restnlly 23513 If the property ` A ` pass...
restlly 23514 If the property ` A ` pass...
islly2 23515 An alternative expression ...
llyrest 23516 An open subspace of a loca...
nllyrest 23517 An open subspace of an n-l...
loclly 23518 If ` A ` is a local proper...
llyidm 23519 Idempotence of the "locall...
nllyidm 23520 Idempotence of the "n-loca...
toplly 23521 A topology is locally a to...
topnlly 23522 A topology is n-locally a ...
hauslly 23523 A Hausdorff space is local...
hausnlly 23524 A Hausdorff space is n-loc...
hausllycmp 23525 A compact Hausdorff space ...
cldllycmp 23526 A closed subspace of a loc...
lly1stc 23527 First-countability is a lo...
dislly 23528 The discrete space ` ~P X ...
disllycmp 23529 A discrete space is locall...
dis1stc 23530 A discrete space is first-...
hausmapdom 23531 If ` X ` is a first-counta...
hauspwdom 23532 Simplify the cardinal ` A ...
refrel 23539 Refinement is a relation. ...
isref 23540 The property of being a re...
refbas 23541 A refinement covers the sa...
refssex 23542 Every set in a refinement ...
ssref 23543 A subcover is a refinement...
refref 23544 Reflexivity of refinement....
reftr 23545 Refinement is transitive. ...
refun0 23546 Adding the empty set prese...
isptfin 23547 The statement "is a point-...
islocfin 23548 The statement "is a locall...
finptfin 23549 A finite cover is a point-...
ptfinfin 23550 A point covered by a point...
finlocfin 23551 A finite cover of a topolo...
locfintop 23552 A locally finite cover cov...
locfinbas 23553 A locally finite cover mus...
locfinnei 23554 A point covered by a local...
lfinpfin 23555 A locally finite cover is ...
lfinun 23556 Adding a finite set preser...
locfincmp 23557 For a compact space, the l...
unisngl 23558 Taking the union of the se...
dissnref 23559 The set of singletons is a...
dissnlocfin 23560 The set of singletons is l...
locfindis 23561 The locally finite covers ...
locfincf 23562 A locally finite cover in ...
comppfsc 23563 A space where every open c...
kgenval 23566 Value of the compact gener...
elkgen 23567 Value of the compact gener...
kgeni 23568 Property of the open sets ...
kgentopon 23569 The compact generator gene...
kgenuni 23570 The base set of the compac...
kgenftop 23571 The compact generator gene...
kgenf 23572 The compact generator is a...
kgentop 23573 A compactly generated spac...
kgenss 23574 The compact generator gene...
kgenhaus 23575 The compact generator gene...
kgencmp 23576 The compact generator topo...
kgencmp2 23577 The compact generator topo...
kgenidm 23578 The compact generator is i...
iskgen2 23579 A space is compactly gener...
iskgen3 23580 Derive the usual definitio...
llycmpkgen2 23581 A locally compact space is...
cmpkgen 23582 A compact space is compact...
llycmpkgen 23583 A locally compact space is...
1stckgenlem 23584 The one-point compactifica...
1stckgen 23585 A first-countable space is...
kgen2ss 23586 The compact generator pres...
kgencn 23587 A function from a compactl...
kgencn2 23588 A function ` F : J --> K `...
kgencn3 23589 The set of continuous func...
kgen2cn 23590 A continuous function is a...
txval 23595 Value of the binary topolo...
txuni2 23596 The underlying set of the ...
txbasex 23597 The basis for the product ...
txbas 23598 The set of Cartesian produ...
eltx 23599 A set in a product is open...
txtop 23600 The product of two topolog...
ptval 23601 The value of the product t...
ptpjpre1 23602 The preimage of a projecti...
elpt 23603 Elementhood in the bases o...
elptr 23604 A basic open set in the pr...
elptr2 23605 A basic open set in the pr...
ptbasid 23606 The base set of the produc...
ptuni2 23607 The base set for the produ...
ptbasin 23608 The basis for a product to...
ptbasin2 23609 The basis for a product to...
ptbas 23610 The basis for a product to...
ptpjpre2 23611 The basis for a product to...
ptbasfi 23612 The basis for the product ...
pttop 23613 The product topology is a ...
ptopn 23614 A basic open set in the pr...
ptopn2 23615 A sub-basic open set in th...
xkotf 23616 Functionality of function ...
xkobval 23617 Alternative expression for...
xkoval 23618 Value of the compact-open ...
xkotop 23619 The compact-open topology ...
xkoopn 23620 A basic open set of the co...
txtopi 23621 The product of two topolog...
txtopon 23622 The underlying set of the ...
txuni 23623 The underlying set of the ...
txunii 23624 The underlying set of the ...
ptuni 23625 The base set for the produ...
ptunimpt 23626 Base set of a product topo...
pttopon 23627 The base set for the produ...
pttoponconst 23628 The base set for a product...
ptuniconst 23629 The base set for a product...
xkouni 23630 The base set of the compac...
xkotopon 23631 The base set of the compac...
ptval2 23632 The value of the product t...
txopn 23633 The product of two open se...
txcld 23634 The product of two closed ...
txcls 23635 Closure of a rectangle in ...
txss12 23636 Subset property of the top...
txbasval 23637 It is sufficient to consid...
neitx 23638 The Cartesian product of t...
txcnpi 23639 Continuity of a two-argume...
tx1cn 23640 Continuity of the first pr...
tx2cn 23641 Continuity of the second p...
ptpjcn 23642 Continuity of a projection...
ptpjopn 23643 The projection map is an o...
ptcld 23644 A closed box in the produc...
ptcldmpt 23645 A closed box in the produc...
ptclsg 23646 The closure of a box in th...
ptcls 23647 The closure of a box in th...
dfac14lem 23648 Lemma for ~ dfac14 . By e...
dfac14 23649 Theorem ~ ptcls is an equi...
xkoccn 23650 The "constant function" fu...
txcnp 23651 If two functions are conti...
ptcnplem 23652 Lemma for ~ ptcnp . (Cont...
ptcnp 23653 If every projection of a f...
upxp 23654 Universal property of the ...
txcnmpt 23655 A map into the product of ...
uptx 23656 Universal property of the ...
txcn 23657 A map into the product of ...
ptcn 23658 If every projection of a f...
prdstopn 23659 Topology of a structure pr...
prdstps 23660 A structure product of top...
pwstps 23661 A structure power of a top...
txrest 23662 The subspace of a topologi...
txdis 23663 The topological product of...
txindislem 23664 Lemma for ~ txindis . (Co...
txindis 23665 The topological product of...
txdis1cn 23666 A function is jointly cont...
txlly 23667 If the property ` A ` is p...
txnlly 23668 If the property ` A ` is p...
pthaus 23669 The product of a collectio...
ptrescn 23670 Restriction is a continuou...
txtube 23671 The "tube lemma". If ` X ...
txcmplem1 23672 Lemma for ~ txcmp . (Cont...
txcmplem2 23673 Lemma for ~ txcmp . (Cont...
txcmp 23674 The topological product of...
txcmpb 23675 The topological product of...
hausdiag 23676 A topology is Hausdorff if...
hauseqlcld 23677 In a Hausdorff topology, t...
txhaus 23678 The topological product of...
txlm 23679 Two sequences converge iff...
lmcn2 23680 The image of a convergent ...
tx1stc 23681 The topological product of...
tx2ndc 23682 The topological product of...
txkgen 23683 The topological product of...
xkohaus 23684 If the codomain space is H...
xkoptsub 23685 The compact-open topology ...
xkopt 23686 The compact-open topology ...
xkopjcn 23687 Continuity of a projection...
xkoco1cn 23688 If ` F ` is a continuous f...
xkoco2cn 23689 If ` F ` is a continuous f...
xkococnlem 23690 Continuity of the composit...
xkococn 23691 Continuity of the composit...
cnmptid 23692 The identity function is c...
cnmptc 23693 A constant function is con...
cnmpt11 23694 The composition of continu...
cnmpt11f 23695 The composition of continu...
cnmpt1t 23696 The composition of continu...
cnmpt12f 23697 The composition of continu...
cnmpt12 23698 The composition of continu...
cnmpt1st 23699 The projection onto the fi...
cnmpt2nd 23700 The projection onto the se...
cnmpt2c 23701 A constant function is con...
cnmpt21 23702 The composition of continu...
cnmpt21f 23703 The composition of continu...
cnmpt2t 23704 The composition of continu...
cnmpt22 23705 The composition of continu...
cnmpt22f 23706 The composition of continu...
cnmpt1res 23707 The restriction of a conti...
cnmpt2res 23708 The restriction of a conti...
cnmptcom 23709 The argument converse of a...
cnmptkc 23710 The curried first projecti...
cnmptkp 23711 The evaluation of the inne...
cnmptk1 23712 The composition of a curri...
cnmpt1k 23713 The composition of a one-a...
cnmptkk 23714 The composition of two cur...
xkofvcn 23715 Joint continuity of the fu...
cnmptk1p 23716 The evaluation of a currie...
cnmptk2 23717 The uncurrying of a currie...
xkoinjcn 23718 Continuity of "injection",...
cnmpt2k 23719 The currying of a two-argu...
txconn 23720 The topological product of...
imasnopn 23721 If a relation graph is ope...
imasncld 23722 If a relation graph is clo...
imasncls 23723 If a relation graph is clo...
qtopval 23726 Value of the quotient topo...
qtopval2 23727 Value of the quotient topo...
elqtop 23728 Value of the quotient topo...
qtopres 23729 The quotient topology is u...
qtoptop2 23730 The quotient topology is a...
qtoptop 23731 The quotient topology is a...
elqtop2 23732 Value of the quotient topo...
qtopuni 23733 The base set of the quotie...
elqtop3 23734 Value of the quotient topo...
qtoptopon 23735 The base set of the quotie...
qtopid 23736 A quotient map is a contin...
idqtop 23737 The quotient topology indu...
qtopcmplem 23738 Lemma for ~ qtopcmp and ~ ...
qtopcmp 23739 A quotient of a compact sp...
qtopconn 23740 A quotient of a connected ...
qtopkgen 23741 A quotient of a compactly ...
basqtop 23742 An injection maps bases to...
tgqtop 23743 An injection maps generate...
qtopcld 23744 The property of being a cl...
qtopcn 23745 Universal property of a qu...
qtopss 23746 A surjective continuous fu...
qtopeu 23747 Universal property of the ...
qtoprest 23748 If ` A ` is a saturated op...
qtopomap 23749 If ` F ` is a surjective c...
qtopcmap 23750 If ` F ` is a surjective c...
imastopn 23751 The topology of an image s...
imastps 23752 The image of a topological...
qustps 23753 A quotient structure is a ...
kqfval 23754 Value of the function appe...
kqfeq 23755 Two points in the Kolmogor...
kqffn 23756 The topological indistingu...
kqval 23757 Value of the quotient topo...
kqtopon 23758 The Kolmogorov quotient is...
kqid 23759 The topological indistingu...
ist0-4 23760 The topological indistingu...
kqfvima 23761 When the image set is open...
kqsat 23762 Any open set is saturated ...
kqdisj 23763 A version of ~ imain for t...
kqcldsat 23764 Any closed set is saturate...
kqopn 23765 The topological indistingu...
kqcld 23766 The topological indistingu...
kqt0lem 23767 Lemma for ~ kqt0 . (Contr...
isr0 23768 The property " ` J ` is an...
r0cld 23769 The analogue of the T_1 ax...
regr1lem 23770 Lemma for ~ regr1 . (Cont...
regr1lem2 23771 A Kolmogorov quotient of a...
kqreglem1 23772 A Kolmogorov quotient of a...
kqreglem2 23773 If the Kolmogorov quotient...
kqnrmlem1 23774 A Kolmogorov quotient of a...
kqnrmlem2 23775 If the Kolmogorov quotient...
kqtop 23776 The Kolmogorov quotient is...
kqt0 23777 The Kolmogorov quotient is...
kqf 23778 The Kolmogorov quotient is...
r0sep 23779 The separation property of...
nrmr0reg 23780 A normal R_0 space is also...
regr1 23781 A regular space is R_1, wh...
kqreg 23782 The Kolmogorov quotient of...
kqnrm 23783 The Kolmogorov quotient of...
hmeofn 23788 The set of homeomorphisms ...
hmeofval 23789 The set of all the homeomo...
ishmeo 23790 The predicate F is a homeo...
hmeocn 23791 A homeomorphism is continu...
hmeocnvcn 23792 The converse of a homeomor...
hmeocnv 23793 The converse of a homeomor...
hmeof1o2 23794 A homeomorphism is a 1-1-o...
hmeof1o 23795 A homeomorphism is a 1-1-o...
hmeoima 23796 The image of an open set b...
hmeoopn 23797 Homeomorphisms preserve op...
hmeocld 23798 Homeomorphisms preserve cl...
hmeocls 23799 Homeomorphisms preserve cl...
hmeontr 23800 Homeomorphisms preserve in...
hmeoimaf1o 23801 The function mapping open ...
hmeores 23802 The restriction of a homeo...
hmeoco 23803 The composite of two homeo...
idhmeo 23804 The identity function is a...
hmeocnvb 23805 The converse of a homeomor...
hmeoqtop 23806 A homeomorphism is a quoti...
hmph 23807 Express the predicate ` J ...
hmphi 23808 If there is a homeomorphis...
hmphtop 23809 Reverse closure for the ho...
hmphtop1 23810 The relation "being homeom...
hmphtop2 23811 The relation "being homeom...
hmphref 23812 "Is homeomorphic to" is re...
hmphsym 23813 "Is homeomorphic to" is sy...
hmphtr 23814 "Is homeomorphic to" is tr...
hmpher 23815 "Is homeomorphic to" is an...
hmphen 23816 Homeomorphisms preserve th...
hmphsymb 23817 "Is homeomorphic to" is sy...
haushmphlem 23818 Lemma for ~ haushmph and s...
cmphmph 23819 Compactness is a topologic...
connhmph 23820 Connectedness is a topolog...
t0hmph 23821 T_0 is a topological prope...
t1hmph 23822 T_1 is a topological prope...
haushmph 23823 Hausdorff-ness is a topolo...
reghmph 23824 Regularity is a topologica...
nrmhmph 23825 Normality is a topological...
hmph0 23826 A topology homeomorphic to...
hmphdis 23827 Homeomorphisms preserve to...
hmphindis 23828 Homeomorphisms preserve to...
indishmph 23829 Equinumerous sets equipped...
hmphen2 23830 Homeomorphisms preserve th...
cmphaushmeo 23831 A continuous bijection fro...
ordthmeolem 23832 Lemma for ~ ordthmeo . (C...
ordthmeo 23833 An order isomorphism is a ...
txhmeo 23834 Lift a pair of homeomorphi...
txswaphmeolem 23835 Show inverse for the "swap...
txswaphmeo 23836 There is a homeomorphism f...
pt1hmeo 23837 The canonical homeomorphis...
ptuncnv 23838 Exhibit the converse funct...
ptunhmeo 23839 Define a homeomorphism fro...
xpstopnlem1 23840 The function ` F ` used in...
xpstps 23841 A binary product of topolo...
xpstopnlem2 23842 Lemma for ~ xpstopn . (Co...
xpstopn 23843 The topology on a binary p...
ptcmpfi 23844 A topological product of f...
xkocnv 23845 The inverse of the "curryi...
xkohmeo 23846 The Exponential Law for to...
qtopf1 23847 If a quotient map is injec...
qtophmeo 23848 If two functions on a base...
t0kq 23849 A topological space is T_0...
kqhmph 23850 A topological space is T_0...
ist1-5lem 23851 Lemma for ~ ist1-5 and sim...
t1r0 23852 A T_1 space is R_0. That ...
ist1-5 23853 A topological space is T_1...
ishaus3 23854 A topological space is Hau...
nrmreg 23855 A normal T_1 space is regu...
reghaus 23856 A regular T_0 space is Hau...
nrmhaus 23857 A T_1 normal space is Haus...
elmptrab 23858 Membership in a one-parame...
elmptrab2 23859 Membership in a one-parame...
isfbas 23860 The predicate " ` F ` is a...
fbasne0 23861 There are no empty filter ...
0nelfb 23862 No filter base contains th...
fbsspw 23863 A filter base on a set is ...
fbelss 23864 An element of the filter b...
fbdmn0 23865 The domain of a filter bas...
isfbas2 23866 The predicate " ` F ` is a...
fbasssin 23867 A filter base contains sub...
fbssfi 23868 A filter base contains sub...
fbssint 23869 A filter base contains sub...
fbncp 23870 A filter base does not con...
fbun 23871 A necessary and sufficient...
fbfinnfr 23872 No filter base containing ...
opnfbas 23873 The collection of open sup...
trfbas2 23874 Conditions for the trace o...
trfbas 23875 Conditions for the trace o...
isfil 23878 The predicate "is a filter...
filfbas 23879 A filter is a filter base....
0nelfil 23880 The empty set doesn't belo...
fileln0 23881 An element of a filter is ...
filsspw 23882 A filter is a subset of th...
filelss 23883 An element of a filter is ...
filss 23884 A filter is closed under t...
filin 23885 A filter is closed under t...
filtop 23886 The underlying set belongs...
isfil2 23887 Derive the standard axioms...
isfildlem 23888 Lemma for ~ isfild . (Con...
isfild 23889 Sufficient condition for a...
filfi 23890 A filter is closed under t...
filinn0 23891 The intersection of two el...
filintn0 23892 A filter has the finite in...
filn0 23893 The empty set is not a fil...
infil 23894 The intersection of two fi...
snfil 23895 A singleton is a filter. ...
fbasweak 23896 A filter base on any set i...
snfbas 23897 Condition for a singleton ...
fsubbas 23898 A condition for a set to g...
fbasfip 23899 A filter base has the fini...
fbunfip 23900 A helpful lemma for showin...
fgval 23901 The filter generating clas...
elfg 23902 A condition for elements o...
ssfg 23903 A filter base is a subset ...
fgss 23904 A bigger base generates a ...
fgss2 23905 A condition for a filter t...
fgfil 23906 A filter generates itself....
elfilss 23907 An element belongs to a fi...
filfinnfr 23908 No filter containing a fin...
fgcl 23909 A generated filter is a fi...
fgabs 23910 Absorption law for filter ...
neifil 23911 The neighborhoods of a non...
filunibas 23912 Recover the base set from ...
filunirn 23913 Two ways to express a filt...
filconn 23914 A filter gives rise to a c...
fbasrn 23915 Given a filter on a domain...
filuni 23916 The union of a nonempty se...
trfil1 23917 Conditions for the trace o...
trfil2 23918 Conditions for the trace o...
trfil3 23919 Conditions for the trace o...
trfilss 23920 If ` A ` is a member of th...
fgtr 23921 If ` A ` is a member of th...
trfg 23922 The trace operation and th...
trnei 23923 The trace, over a set ` A ...
cfinfil 23924 Relative complements of th...
csdfil 23925 The set of all elements wh...
supfil 23926 The supersets of a nonempt...
zfbas 23927 The set of upper sets of i...
uzrest 23928 The restriction of the set...
uzfbas 23929 The set of upper sets of i...
isufil 23934 The property of being an u...
ufilfil 23935 An ultrafilter is a filter...
ufilss 23936 For any subset of the base...
ufilb 23937 The complement is in an ul...
ufilmax 23938 Any filter finer than an u...
isufil2 23939 The maximal property of an...
ufprim 23940 An ultrafilter is a prime ...
trufil 23941 Conditions for the trace o...
filssufilg 23942 A filter is contained in s...
filssufil 23943 A filter is contained in s...
isufl 23944 Define the (strong) ultraf...
ufli 23945 Property of a set that sat...
numufl 23946 Consequence of ~ filssufil...
fiufl 23947 A finite set satisfies the...
acufl 23948 The axiom of choice implie...
ssufl 23949 If ` Y ` is a subset of ` ...
ufileu 23950 If the ultrafilter contain...
filufint 23951 A filter is equal to the i...
uffix 23952 Lemma for ~ fixufil and ~ ...
fixufil 23953 The condition describing a...
uffixfr 23954 An ultrafilter is either f...
uffix2 23955 A classification of fixed ...
uffixsn 23956 The singleton of the gener...
ufildom1 23957 An ultrafilter is generate...
uffinfix 23958 An ultrafilter containing ...
cfinufil 23959 An ultrafilter is free iff...
ufinffr 23960 An infinite subset is cont...
ufilen 23961 Any infinite set has an ul...
ufildr 23962 An ultrafilter gives rise ...
fin1aufil 23963 There are no definable fre...
fmval 23974 Introduce a function that ...
fmfil 23975 A mapping filter is a filt...
fmf 23976 Pushing-forward via a func...
fmss 23977 A finer filter produces a ...
elfm 23978 An element of a mapping fi...
elfm2 23979 An element of a mapping fi...
fmfg 23980 The image filter of a filt...
elfm3 23981 An alternate formulation o...
imaelfm 23982 An image of a filter eleme...
rnelfmlem 23983 Lemma for ~ rnelfm . (Con...
rnelfm 23984 A condition for a filter t...
fmfnfmlem1 23985 Lemma for ~ fmfnfm . (Con...
fmfnfmlem2 23986 Lemma for ~ fmfnfm . (Con...
fmfnfmlem3 23987 Lemma for ~ fmfnfm . (Con...
fmfnfmlem4 23988 Lemma for ~ fmfnfm . (Con...
fmfnfm 23989 A filter finer than an ima...
fmufil 23990 An image filter of an ultr...
fmid 23991 The filter map applied to ...
fmco 23992 Composition of image filte...
ufldom 23993 The ultrafilter lemma prop...
flimval 23994 The set of limit points of...
elflim2 23995 The predicate "is a limit ...
flimtop 23996 Reverse closure for the li...
flimneiss 23997 A filter contains the neig...
flimnei 23998 A filter contains all of t...
flimelbas 23999 A limit point of a filter ...
flimfil 24000 Reverse closure for the li...
flimtopon 24001 Reverse closure for the li...
elflim 24002 The predicate "is a limit ...
flimss2 24003 A limit point of a filter ...
flimss1 24004 A limit point of a filter ...
neiflim 24005 A point is a limit point o...
flimopn 24006 The condition for being a ...
fbflim 24007 A condition for a filter t...
fbflim2 24008 A condition for a filter b...
flimclsi 24009 The convergent points of a...
hausflimlem 24010 If ` A ` and ` B ` are bot...
hausflimi 24011 One direction of ~ hausfli...
hausflim 24012 A condition for a topology...
flimcf 24013 Fineness is properly chara...
flimrest 24014 The set of limit points in...
flimclslem 24015 Lemma for ~ flimcls . (Co...
flimcls 24016 Closure in terms of filter...
flimsncls 24017 If ` A ` is a limit point ...
hauspwpwf1 24018 Lemma for ~ hauspwpwdom . ...
hauspwpwdom 24019 If ` X ` is a Hausdorff sp...
flffval 24020 Given a topology and a fil...
flfval 24021 Given a function from a fi...
flfnei 24022 The property of being a li...
flfneii 24023 A neighborhood of a limit ...
isflf 24024 The property of being a li...
flfelbas 24025 A limit point of a functio...
flffbas 24026 Limit points of a function...
flftg 24027 Limit points of a function...
hausflf 24028 If a function has its valu...
hausflf2 24029 If a convergent function h...
cnpflfi 24030 Forward direction of ~ cnp...
cnpflf2 24031 ` F ` is continuous at poi...
cnpflf 24032 Continuity of a function a...
cnflf 24033 A function is continuous i...
cnflf2 24034 A function is continuous i...
flfcnp 24035 A continuous function pres...
lmflf 24036 The topological limit rela...
txflf 24037 Two sequences converge in ...
flfcnp2 24038 The image of a convergent ...
fclsval 24039 The set of all cluster poi...
isfcls 24040 A cluster point of a filte...
fclsfil 24041 Reverse closure for the cl...
fclstop 24042 Reverse closure for the cl...
fclstopon 24043 Reverse closure for the cl...
isfcls2 24044 A cluster point of a filte...
fclsopn 24045 Write the cluster point co...
fclsopni 24046 An open neighborhood of a ...
fclselbas 24047 A cluster point is in the ...
fclsneii 24048 A neighborhood of a cluste...
fclssscls 24049 The set of cluster points ...
fclsnei 24050 Cluster points in terms of...
supnfcls 24051 The filter of supersets of...
fclsbas 24052 Cluster points in terms of...
fclsss1 24053 A finer topology has fewer...
fclsss2 24054 A finer filter has fewer c...
fclsrest 24055 The set of cluster points ...
fclscf 24056 Characterization of finene...
flimfcls 24057 A limit point is a cluster...
fclsfnflim 24058 A filter clusters at a poi...
flimfnfcls 24059 A filter converges to a po...
fclscmpi 24060 Forward direction of ~ fcl...
fclscmp 24061 A space is compact iff eve...
uffclsflim 24062 The cluster points of an u...
ufilcmp 24063 A space is compact iff eve...
fcfval 24064 The set of cluster points ...
isfcf 24065 The property of being a cl...
fcfnei 24066 The property of being a cl...
fcfelbas 24067 A cluster point of a funct...
fcfneii 24068 A neighborhood of a cluste...
flfssfcf 24069 A limit point of a functio...
uffcfflf 24070 If the domain filter is an...
cnpfcfi 24071 Lemma for ~ cnpfcf . If a...
cnpfcf 24072 A function ` F ` is contin...
cnfcf 24073 Continuity of a function i...
flfcntr 24074 A continuous function's va...
alexsublem 24075 Lemma for ~ alexsub . (Co...
alexsub 24076 The Alexander Subbase Theo...
alexsubb 24077 Biconditional form of the ...
alexsubALTlem1 24078 Lemma for ~ alexsubALT . ...
alexsubALTlem2 24079 Lemma for ~ alexsubALT . ...
alexsubALTlem3 24080 Lemma for ~ alexsubALT . ...
alexsubALTlem4 24081 Lemma for ~ alexsubALT . ...
alexsubALT 24082 The Alexander Subbase Theo...
ptcmplem1 24083 Lemma for ~ ptcmp . (Cont...
ptcmplem2 24084 Lemma for ~ ptcmp . (Cont...
ptcmplem3 24085 Lemma for ~ ptcmp . (Cont...
ptcmplem4 24086 Lemma for ~ ptcmp . (Cont...
ptcmplem5 24087 Lemma for ~ ptcmp . (Cont...
ptcmpg 24088 Tychonoff's theorem: The ...
ptcmp 24089 Tychonoff's theorem: The ...
cnextval 24092 The function applying cont...
cnextfval 24093 The continuous extension o...
cnextrel 24094 In the general case, a con...
cnextfun 24095 If the target space is Hau...
cnextfvval 24096 The value of the continuou...
cnextf 24097 Extension by continuity. ...
cnextcn 24098 Extension by continuity. ...
cnextfres1 24099 ` F ` and its extension by...
cnextfres 24100 ` F ` and its extension by...
istmd 24105 The predicate "is a topolo...
tmdmnd 24106 A topological monoid is a ...
tmdtps 24107 A topological monoid is a ...
istgp 24108 The predicate "is a topolo...
tgpgrp 24109 A topological group is a g...
tgptmd 24110 A topological group is a t...
tgptps 24111 A topological group is a t...
tmdtopon 24112 The topology of a topologi...
tgptopon 24113 The topology of a topologi...
tmdcn 24114 In a topological monoid, t...
tgpcn 24115 In a topological group, th...
tgpinv 24116 In a topological group, th...
grpinvhmeo 24117 The inverse function in a ...
cnmpt1plusg 24118 Continuity of the group su...
cnmpt2plusg 24119 Continuity of the group su...
tmdcn2 24120 Write out the definition o...
tgpsubcn 24121 In a topological group, th...
istgp2 24122 A group with a topology is...
tmdmulg 24123 In a topological monoid, t...
tgpmulg 24124 In a topological group, th...
tgpmulg2 24125 In a topological monoid, t...
tmdgsum 24126 In a topological monoid, t...
tmdgsum2 24127 For any neighborhood ` U `...
oppgtmd 24128 The opposite of a topologi...
oppgtgp 24129 The opposite of a topologi...
distgp 24130 Any group equipped with th...
indistgp 24131 Any group equipped with th...
efmndtmd 24132 The monoid of endofunction...
tmdlactcn 24133 The left group action of e...
tgplacthmeo 24134 The left group action of e...
submtmd 24135 A submonoid of a topologic...
subgtgp 24136 A subgroup of a topologica...
symgtgp 24137 The symmetric group is a t...
subgntr 24138 A subgroup of a topologica...
opnsubg 24139 An open subgroup of a topo...
clssubg 24140 The closure of a subgroup ...
clsnsg 24141 The closure of a normal su...
cldsubg 24142 A subgroup of finite index...
tgpconncompeqg 24143 The connected component co...
tgpconncomp 24144 The identity component, th...
tgpconncompss 24145 The identity component is ...
ghmcnp 24146 A group homomorphism on to...
snclseqg 24147 The coset of the closure o...
tgphaus 24148 A topological group is Hau...
tgpt1 24149 Hausdorff and T1 are equiv...
tgpt0 24150 Hausdorff and T0 are equiv...
qustgpopn 24151 A quotient map in a topolo...
qustgplem 24152 Lemma for ~ qustgp . (Con...
qustgp 24153 The quotient of a topologi...
qustgphaus 24154 The quotient of a topologi...
prdstmdd 24155 The product of a family of...
prdstgpd 24156 The product of a family of...
tsmsfbas 24159 The collection of all sets...
tsmslem1 24160 The finite partial sums of...
tsmsval2 24161 Definition of the topologi...
tsmsval 24162 Definition of the topologi...
tsmspropd 24163 The group sum depends only...
eltsms 24164 The property of being a su...
tsmsi 24165 The property of being a su...
tsmscl 24166 A sum in a topological gro...
haustsms 24167 In a Hausdorff topological...
haustsms2 24168 In a Hausdorff topological...
tsmscls 24169 One half of ~ tgptsmscls ,...
tsmsgsum 24170 The convergent points of a...
tsmsid 24171 If a sum is finite, the us...
haustsmsid 24172 In a Hausdorff topological...
tsms0 24173 The sum of zero is zero. ...
tsmssubm 24174 Evaluate an infinite group...
tsmsres 24175 Extend an infinite group s...
tsmsf1o 24176 Re-index an infinite group...
tsmsmhm 24177 Apply a continuous group h...
tsmsadd 24178 The sum of two infinite gr...
tsmsinv 24179 Inverse of an infinite gro...
tsmssub 24180 The difference of two infi...
tgptsmscls 24181 A sum in a topological gro...
tgptsmscld 24182 The set of limit points to...
tsmssplit 24183 Split a topological group ...
tsmsxplem1 24184 Lemma for ~ tsmsxp . (Con...
tsmsxplem2 24185 Lemma for ~ tsmsxp . (Con...
tsmsxp 24186 Write a sum over a two-dim...
istrg 24195 Express the predicate " ` ...
trgtmd 24196 The multiplicative monoid ...
istdrg 24197 Express the predicate " ` ...
tdrgunit 24198 The unit group of a topolo...
trgtgp 24199 A topological ring is a to...
trgtmd2 24200 A topological ring is a to...
trgtps 24201 A topological ring is a to...
trgring 24202 A topological ring is a ri...
trggrp 24203 A topological ring is a gr...
tdrgtrg 24204 A topological division rin...
tdrgdrng 24205 A topological division rin...
tdrgring 24206 A topological division rin...
tdrgtmd 24207 A topological division rin...
tdrgtps 24208 A topological division rin...
istdrg2 24209 A topological-ring divisio...
mulrcn 24210 The functionalization of t...
invrcn2 24211 The multiplicative inverse...
invrcn 24212 The multiplicative inverse...
cnmpt1mulr 24213 Continuity of ring multipl...
cnmpt2mulr 24214 Continuity of ring multipl...
dvrcn 24215 The division function is c...
istlm 24216 The predicate " ` W ` is a...
vscacn 24217 The scalar multiplication ...
tlmtmd 24218 A topological module is a ...
tlmtps 24219 A topological module is a ...
tlmlmod 24220 A topological module is a ...
tlmtrg 24221 The scalar ring of a topol...
tlmscatps 24222 The scalar ring of a topol...
istvc 24223 A topological vector space...
tvctdrg 24224 The scalar field of a topo...
cnmpt1vsca 24225 Continuity of scalar multi...
cnmpt2vsca 24226 Continuity of scalar multi...
tlmtgp 24227 A topological vector space...
tvctlm 24228 A topological vector space...
tvclmod 24229 A topological vector space...
tvclvec 24230 A topological vector space...
ustfn 24233 The defined uniform struct...
ustval 24234 The class of all uniform s...
isust 24235 The predicate " ` U ` is a...
ustssxp 24236 Entourages are subsets of ...
ustssel 24237 A uniform structure is upw...
ustbasel 24238 The full set is always an ...
ustincl 24239 A uniform structure is clo...
ustdiag 24240 The diagonal set is includ...
ustinvel 24241 If ` V ` is an entourage, ...
ustexhalf 24242 For each entourage ` V ` t...
ustrel 24243 The elements of uniform st...
ustfilxp 24244 A uniform structure on a n...
ustne0 24245 A uniform structure cannot...
ustssco 24246 In an uniform structure, a...
ustexsym 24247 In an uniform structure, f...
ustex2sym 24248 In an uniform structure, f...
ustex3sym 24249 In an uniform structure, f...
ustref 24250 Any element of the base se...
ust0 24251 The unique uniform structu...
ustn0 24252 The empty set is not an un...
ustund 24253 If two intersecting sets `...
ustelimasn 24254 Any point ` A ` is near en...
ustneism 24255 For a point ` A ` in ` X `...
elrnustOLD 24256 Obsolete version of ~ elfv...
ustbas2 24257 Second direction for ~ ust...
ustuni 24258 The set union of a uniform...
ustbas 24259 Recover the base of an uni...
ustimasn 24260 Lemma for ~ ustuqtop . (C...
trust 24261 The trace of a uniform str...
utopval 24264 The topology induced by a ...
elutop 24265 Open sets in the topology ...
utoptop 24266 The topology induced by a ...
utopbas 24267 The base of the topology i...
utoptopon 24268 Topology induced by a unif...
restutop 24269 Restriction of a topology ...
restutopopn 24270 The restriction of the top...
ustuqtoplem 24271 Lemma for ~ ustuqtop . (C...
ustuqtop0 24272 Lemma for ~ ustuqtop . (C...
ustuqtop1 24273 Lemma for ~ ustuqtop , sim...
ustuqtop2 24274 Lemma for ~ ustuqtop . (C...
ustuqtop3 24275 Lemma for ~ ustuqtop , sim...
ustuqtop4 24276 Lemma for ~ ustuqtop . (C...
ustuqtop5 24277 Lemma for ~ ustuqtop . (C...
ustuqtop 24278 For a given uniform struct...
utopsnneiplem 24279 The neighborhoods of a poi...
utopsnneip 24280 The neighborhoods of a poi...
utopsnnei 24281 Images of singletons by en...
utop2nei 24282 For any symmetrical entour...
utop3cls 24283 Relation between a topolog...
utopreg 24284 All Hausdorff uniform spac...
ussval 24291 The uniform structure on u...
ussid 24292 In case the base of the ` ...
isusp 24293 The predicate ` W ` is a u...
ressuss 24294 Value of the uniform struc...
ressust 24295 The uniform structure of a...
ressusp 24296 The restriction of a unifo...
tusval 24297 The value of the uniform s...
tuslem 24298 Lemma for ~ tusbas , ~ tus...
tuslemOLD 24299 Obsolete proof of ~ tuslem...
tusbas 24300 The base set of a construc...
tusunif 24301 The uniform structure of a...
tususs 24302 The uniform structure of a...
tustopn 24303 The topology induced by a ...
tususp 24304 A constructed uniform spac...
tustps 24305 A constructed uniform spac...
uspreg 24306 If a uniform space is Haus...
ucnval 24309 The set of all uniformly c...
isucn 24310 The predicate " ` F ` is a...
isucn2 24311 The predicate " ` F ` is a...
ucnimalem 24312 Reformulate the ` G ` func...
ucnima 24313 An equivalent statement of...
ucnprima 24314 The preimage by a uniforml...
iducn 24315 The identity is uniformly ...
cstucnd 24316 A constant function is uni...
ucncn 24317 Uniform continuity implies...
iscfilu 24320 The predicate " ` F ` is a...
cfilufbas 24321 A Cauchy filter base is a ...
cfiluexsm 24322 For a Cauchy filter base a...
fmucndlem 24323 Lemma for ~ fmucnd . (Con...
fmucnd 24324 The image of a Cauchy filt...
cfilufg 24325 The filter generated by a ...
trcfilu 24326 Condition for the trace of...
cfiluweak 24327 A Cauchy filter base is al...
neipcfilu 24328 In an uniform space, a nei...
iscusp 24331 The predicate " ` W ` is a...
cuspusp 24332 A complete uniform space i...
cuspcvg 24333 In a complete uniform spac...
iscusp2 24334 The predicate " ` W ` is a...
cnextucn 24335 Extension by continuity. ...
ucnextcn 24336 Extension by continuity. ...
ispsmet 24337 Express the predicate " ` ...
psmetdmdm 24338 Recover the base set from ...
psmetf 24339 The distance function of a...
psmetcl 24340 Closure of the distance fu...
psmet0 24341 The distance function of a...
psmettri2 24342 Triangle inequality for th...
psmetsym 24343 The distance function of a...
psmettri 24344 Triangle inequality for th...
psmetge0 24345 The distance function of a...
psmetxrge0 24346 The distance function of a...
psmetres2 24347 Restriction of a pseudomet...
psmetlecl 24348 Real closure of an extende...
distspace 24349 A set ` X ` together with ...
ismet 24356 Express the predicate " ` ...
isxmet 24357 Express the predicate " ` ...
ismeti 24358 Properties that determine ...
isxmetd 24359 Properties that determine ...
isxmet2d 24360 It is safe to only require...
metflem 24361 Lemma for ~ metf and other...
xmetf 24362 Mapping of the distance fu...
metf 24363 Mapping of the distance fu...
xmetcl 24364 Closure of the distance fu...
metcl 24365 Closure of the distance fu...
ismet2 24366 An extended metric is a me...
metxmet 24367 A metric is an extended me...
xmetdmdm 24368 Recover the base set from ...
metdmdm 24369 Recover the base set from ...
xmetunirn 24370 Two ways to express an ext...
xmeteq0 24371 The value of an extended m...
meteq0 24372 The value of a metric is z...
xmettri2 24373 Triangle inequality for th...
mettri2 24374 Triangle inequality for th...
xmet0 24375 The distance function of a...
met0 24376 The distance function of a...
xmetge0 24377 The distance function of a...
metge0 24378 The distance function of a...
xmetlecl 24379 Real closure of an extende...
xmetsym 24380 The distance function of a...
xmetpsmet 24381 An extended metric is a ps...
xmettpos 24382 The distance function of a...
metsym 24383 The distance function of a...
xmettri 24384 Triangle inequality for th...
mettri 24385 Triangle inequality for th...
xmettri3 24386 Triangle inequality for th...
mettri3 24387 Triangle inequality for th...
xmetrtri 24388 One half of the reverse tr...
xmetrtri2 24389 The reverse triangle inequ...
metrtri 24390 Reverse triangle inequalit...
xmetgt0 24391 The distance function of a...
metgt0 24392 The distance function of a...
metn0 24393 A metric space is nonempty...
xmetres2 24394 Restriction of an extended...
metreslem 24395 Lemma for ~ metres . (Con...
metres2 24396 Lemma for ~ metres . (Con...
xmetres 24397 A restriction of an extend...
metres 24398 A restriction of a metric ...
0met 24399 The empty metric. (Contri...
prdsdsf 24400 The product metric is a fu...
prdsxmetlem 24401 The product metric is an e...
prdsxmet 24402 The product metric is an e...
prdsmet 24403 The product metric is a me...
ressprdsds 24404 Restriction of a product m...
resspwsds 24405 Restriction of a power met...
imasdsf1olem 24406 Lemma for ~ imasdsf1o . (...
imasdsf1o 24407 The distance function is t...
imasf1oxmet 24408 The image of an extended m...
imasf1omet 24409 The image of a metric is a...
xpsdsfn 24410 Closure of the metric in a...
xpsdsfn2 24411 Closure of the metric in a...
xpsxmetlem 24412 Lemma for ~ xpsxmet . (Co...
xpsxmet 24413 A product metric of extend...
xpsdsval 24414 Value of the metric in a b...
xpsmet 24415 The direct product of two ...
blfvalps 24416 The value of the ball func...
blfval 24417 The value of the ball func...
blvalps 24418 The ball around a point ` ...
blval 24419 The ball around a point ` ...
elblps 24420 Membership in a ball. (Co...
elbl 24421 Membership in a ball. (Co...
elbl2ps 24422 Membership in a ball. (Co...
elbl2 24423 Membership in a ball. (Co...
elbl3ps 24424 Membership in a ball, with...
elbl3 24425 Membership in a ball, with...
blcomps 24426 Commute the arguments to t...
blcom 24427 Commute the arguments to t...
xblpnfps 24428 The infinity ball in an ex...
xblpnf 24429 The infinity ball in an ex...
blpnf 24430 The infinity ball in a sta...
bldisj 24431 Two balls are disjoint if ...
blgt0 24432 A nonempty ball implies th...
bl2in 24433 Two balls are disjoint if ...
xblss2ps 24434 One ball is contained in a...
xblss2 24435 One ball is contained in a...
blss2ps 24436 One ball is contained in a...
blss2 24437 One ball is contained in a...
blhalf 24438 A ball of radius ` R / 2 `...
blfps 24439 Mapping of a ball. (Contr...
blf 24440 Mapping of a ball. (Contr...
blrnps 24441 Membership in the range of...
blrn 24442 Membership in the range of...
xblcntrps 24443 A ball contains its center...
xblcntr 24444 A ball contains its center...
blcntrps 24445 A ball contains its center...
blcntr 24446 A ball contains its center...
xbln0 24447 A ball is nonempty iff the...
bln0 24448 A ball is not empty. (Con...
blelrnps 24449 A ball belongs to the set ...
blelrn 24450 A ball belongs to the set ...
blssm 24451 A ball is a subset of the ...
unirnblps 24452 The union of the set of ba...
unirnbl 24453 The union of the set of ba...
blin 24454 The intersection of two ba...
ssblps 24455 The size of a ball increas...
ssbl 24456 The size of a ball increas...
blssps 24457 Any point ` P ` in a ball ...
blss 24458 Any point ` P ` in a ball ...
blssexps 24459 Two ways to express the ex...
blssex 24460 Two ways to express the ex...
ssblex 24461 A nested ball exists whose...
blin2 24462 Given any two balls and a ...
blbas 24463 The balls of a metric spac...
blres 24464 A ball in a restricted met...
xmeterval 24465 Value of the "finitely sep...
xmeter 24466 The "finitely separated" r...
xmetec 24467 The equivalence classes un...
blssec 24468 A ball centered at ` P ` i...
blpnfctr 24469 The infinity ball in an ex...
xmetresbl 24470 An extended metric restric...
mopnval 24471 An open set is a subset of...
mopntopon 24472 The set of open sets of a ...
mopntop 24473 The set of open sets of a ...
mopnuni 24474 The union of all open sets...
elmopn 24475 The defining property of a...
mopnfss 24476 The family of open sets of...
mopnm 24477 The base set of a metric s...
elmopn2 24478 A defining property of an ...
mopnss 24479 An open set of a metric sp...
isxms 24480 Express the predicate " ` ...
isxms2 24481 Express the predicate " ` ...
isms 24482 Express the predicate " ` ...
isms2 24483 Express the predicate " ` ...
xmstopn 24484 The topology component of ...
mstopn 24485 The topology component of ...
xmstps 24486 An extended metric space i...
msxms 24487 A metric space is an exten...
mstps 24488 A metric space is a topolo...
xmsxmet 24489 The distance function, sui...
msmet 24490 The distance function, sui...
msf 24491 The distance function of a...
xmsxmet2 24492 The distance function, sui...
msmet2 24493 The distance function, sui...
mscl 24494 Closure of the distance fu...
xmscl 24495 Closure of the distance fu...
xmsge0 24496 The distance function in a...
xmseq0 24497 The distance between two p...
xmssym 24498 The distance function in a...
xmstri2 24499 Triangle inequality for th...
mstri2 24500 Triangle inequality for th...
xmstri 24501 Triangle inequality for th...
mstri 24502 Triangle inequality for th...
xmstri3 24503 Triangle inequality for th...
mstri3 24504 Triangle inequality for th...
msrtri 24505 Reverse triangle inequalit...
xmspropd 24506 Property deduction for an ...
mspropd 24507 Property deduction for a m...
setsmsbas 24508 The base set of a construc...
setsmsbasOLD 24509 Obsolete proof of ~ setsms...
setsmsds 24510 The distance function of a...
setsmsdsOLD 24511 Obsolete proof of ~ setsms...
setsmstset 24512 The topology of a construc...
setsmstopn 24513 The topology of a construc...
setsxms 24514 The constructed metric spa...
setsms 24515 The constructed metric spa...
tmsval 24516 For any metric there is an...
tmslem 24517 Lemma for ~ tmsbas , ~ tms...
tmslemOLD 24518 Obsolete version of ~ tmsl...
tmsbas 24519 The base set of a construc...
tmsds 24520 The metric of a constructe...
tmstopn 24521 The topology of a construc...
tmsxms 24522 The constructed metric spa...
tmsms 24523 The constructed metric spa...
imasf1obl 24524 The image of a metric spac...
imasf1oxms 24525 The image of a metric spac...
imasf1oms 24526 The image of a metric spac...
prdsbl 24527 A ball in the product metr...
mopni 24528 An open set of a metric sp...
mopni2 24529 An open set of a metric sp...
mopni3 24530 An open set of a metric sp...
blssopn 24531 The balls of a metric spac...
unimopn 24532 The union of a collection ...
mopnin 24533 The intersection of two op...
mopn0 24534 The empty set is an open s...
rnblopn 24535 A ball of a metric space i...
blopn 24536 A ball of a metric space i...
neibl 24537 The neighborhoods around a...
blnei 24538 A ball around a point is a...
lpbl 24539 Every ball around a limit ...
blsscls2 24540 A smaller closed ball is c...
blcld 24541 A "closed ball" in a metri...
blcls 24542 The closure of an open bal...
blsscls 24543 If two concentric balls ha...
metss 24544 Two ways of saying that me...
metequiv 24545 Two ways of saying that tw...
metequiv2 24546 If there is a sequence of ...
metss2lem 24547 Lemma for ~ metss2 . (Con...
metss2 24548 If the metric ` D ` is "st...
comet 24549 The composition of an exte...
stdbdmetval 24550 Value of the standard boun...
stdbdxmet 24551 The standard bounded metri...
stdbdmet 24552 The standard bounded metri...
stdbdbl 24553 The standard bounded metri...
stdbdmopn 24554 The standard bounded metri...
mopnex 24555 The topology generated by ...
methaus 24556 The topology generated by ...
met1stc 24557 The topology generated by ...
met2ndci 24558 A separable metric space (...
met2ndc 24559 A metric space is second-c...
metrest 24560 Two alternate formulations...
ressxms 24561 The restriction of a metri...
ressms 24562 The restriction of a metri...
prdsmslem1 24563 Lemma for ~ prdsms . The ...
prdsxmslem1 24564 Lemma for ~ prdsms . The ...
prdsxmslem2 24565 Lemma for ~ prdsxms . The...
prdsxms 24566 The indexed product struct...
prdsms 24567 The indexed product struct...
pwsxms 24568 A power of an extended met...
pwsms 24569 A power of a metric space ...
xpsxms 24570 A binary product of metric...
xpsms 24571 A binary product of metric...
tmsxps 24572 Express the product of two...
tmsxpsmopn 24573 Express the product of two...
tmsxpsval 24574 Value of the product of tw...
tmsxpsval2 24575 Value of the product of tw...
metcnp3 24576 Two ways to express that `...
metcnp 24577 Two ways to say a mapping ...
metcnp2 24578 Two ways to say a mapping ...
metcn 24579 Two ways to say a mapping ...
metcnpi 24580 Epsilon-delta property of ...
metcnpi2 24581 Epsilon-delta property of ...
metcnpi3 24582 Epsilon-delta property of ...
txmetcnp 24583 Continuity of a binary ope...
txmetcn 24584 Continuity of a binary ope...
metuval 24585 Value of the uniform struc...
metustel 24586 Define a filter base ` F `...
metustss 24587 Range of the elements of t...
metustrel 24588 Elements of the filter bas...
metustto 24589 Any two elements of the fi...
metustid 24590 The identity diagonal is i...
metustsym 24591 Elements of the filter bas...
metustexhalf 24592 For any element ` A ` of t...
metustfbas 24593 The filter base generated ...
metust 24594 The uniform structure gene...
cfilucfil 24595 Given a metric ` D ` and a...
metuust 24596 The uniform structure gene...
cfilucfil2 24597 Given a metric ` D ` and a...
blval2 24598 The ball around a point ` ...
elbl4 24599 Membership in a ball, alte...
metuel 24600 Elementhood in the uniform...
metuel2 24601 Elementhood in the uniform...
metustbl 24602 The "section" image of an ...
psmetutop 24603 The topology induced by a ...
xmetutop 24604 The topology induced by a ...
xmsusp 24605 If the uniform set of a me...
restmetu 24606 The uniform structure gene...
metucn 24607 Uniform continuity in metr...
dscmet 24608 The discrete metric on any...
dscopn 24609 The discrete metric genera...
nrmmetd 24610 Show that a group norm gen...
abvmet 24611 An absolute value ` F ` ge...
nmfval 24624 The value of the norm func...
nmval 24625 The value of the norm as t...
nmfval0 24626 The value of the norm func...
nmfval2 24627 The value of the norm func...
nmval2 24628 The value of the norm on a...
nmf2 24629 The norm on a metric group...
nmpropd 24630 Weak property deduction fo...
nmpropd2 24631 Strong property deduction ...
isngp 24632 The property of being a no...
isngp2 24633 The property of being a no...
isngp3 24634 The property of being a no...
ngpgrp 24635 A normed group is a group....
ngpms 24636 A normed group is a metric...
ngpxms 24637 A normed group is an exten...
ngptps 24638 A normed group is a topolo...
ngpmet 24639 The (induced) metric of a ...
ngpds 24640 Value of the distance func...
ngpdsr 24641 Value of the distance func...
ngpds2 24642 Write the distance between...
ngpds2r 24643 Write the distance between...
ngpds3 24644 Write the distance between...
ngpds3r 24645 Write the distance between...
ngprcan 24646 Cancel right addition insi...
ngplcan 24647 Cancel left addition insid...
isngp4 24648 Express the property of be...
ngpinvds 24649 Two elements are the same ...
ngpsubcan 24650 Cancel right subtraction i...
nmf 24651 The norm on a normed group...
nmcl 24652 The norm of a normed group...
nmge0 24653 The norm of a normed group...
nmeq0 24654 The identity is the only e...
nmne0 24655 The norm of a nonzero elem...
nmrpcl 24656 The norm of a nonzero elem...
nminv 24657 The norm of a negated elem...
nmmtri 24658 The triangle inequality fo...
nmsub 24659 The norm of the difference...
nmrtri 24660 Reverse triangle inequalit...
nm2dif 24661 Inequality for the differe...
nmtri 24662 The triangle inequality fo...
nmtri2 24663 Triangle inequality for th...
ngpi 24664 The properties of a normed...
nm0 24665 Norm of the identity eleme...
nmgt0 24666 The norm of a nonzero elem...
sgrim 24667 The induced metric on a su...
sgrimval 24668 The induced metric on a su...
subgnm 24669 The norm in a subgroup. (...
subgnm2 24670 A substructure assigns the...
subgngp 24671 A normed group restricted ...
ngptgp 24672 A normed abelian group is ...
ngppropd 24673 Property deduction for a n...
reldmtng 24674 The function ` toNrmGrp ` ...
tngval 24675 Value of the function whic...
tnglem 24676 Lemma for ~ tngbas and sim...
tnglemOLD 24677 Obsolete version of ~ tngl...
tngbas 24678 The base set of a structur...
tngbasOLD 24679 Obsolete proof of ~ tngbas...
tngplusg 24680 The group addition of a st...
tngplusgOLD 24681 Obsolete proof of ~ tngplu...
tng0 24682 The group identity of a st...
tngmulr 24683 The ring multiplication of...
tngmulrOLD 24684 Obsolete proof of ~ tngmul...
tngsca 24685 The scalar ring of a struc...
tngscaOLD 24686 Obsolete proof of ~ tngsca...
tngvsca 24687 The scalar multiplication ...
tngvscaOLD 24688 Obsolete proof of ~ tngvsc...
tngip 24689 The inner product operatio...
tngipOLD 24690 Obsolete proof of ~ tngip ...
tngds 24691 The metric function of a s...
tngdsOLD 24692 Obsolete proof of ~ tngds ...
tngtset 24693 The topology generated by ...
tngtopn 24694 The topology generated by ...
tngnm 24695 The topology generated by ...
tngngp2 24696 A norm turns a group into ...
tngngpd 24697 Derive the axioms for a no...
tngngp 24698 Derive the axioms for a no...
tnggrpr 24699 If a structure equipped wi...
tngngp3 24700 Alternate definition of a ...
nrmtngdist 24701 The augmentation of a norm...
nrmtngnrm 24702 The augmentation of a norm...
tngngpim 24703 The induced metric of a no...
isnrg 24704 A normed ring is a ring wi...
nrgabv 24705 The norm of a normed ring ...
nrgngp 24706 A normed ring is a normed ...
nrgring 24707 A normed ring is a ring. ...
nmmul 24708 The norm of a product in a...
nrgdsdi 24709 Distribute a distance calc...
nrgdsdir 24710 Distribute a distance calc...
nm1 24711 The norm of one in a nonze...
unitnmn0 24712 The norm of a unit is nonz...
nminvr 24713 The norm of an inverse in ...
nmdvr 24714 The norm of a division in ...
nrgdomn 24715 A nonzero normed ring is a...
nrgtgp 24716 A normed ring is a topolog...
subrgnrg 24717 A normed ring restricted t...
tngnrg 24718 Given any absolute value o...
isnlm 24719 A normed (left) module is ...
nmvs 24720 Defining property of a nor...
nlmngp 24721 A normed module is a norme...
nlmlmod 24722 A normed module is a left ...
nlmnrg 24723 The scalar component of a ...
nlmngp2 24724 The scalar component of a ...
nlmdsdi 24725 Distribute a distance calc...
nlmdsdir 24726 Distribute a distance calc...
nlmmul0or 24727 If a scalar product is zer...
sranlm 24728 The subring algebra over a...
nlmvscnlem2 24729 Lemma for ~ nlmvscn . Com...
nlmvscnlem1 24730 Lemma for ~ nlmvscn . (Co...
nlmvscn 24731 The scalar multiplication ...
rlmnlm 24732 The ring module over a nor...
rlmnm 24733 The norm function in the r...
nrgtrg 24734 A normed ring is a topolog...
nrginvrcnlem 24735 Lemma for ~ nrginvrcn . C...
nrginvrcn 24736 The ring inverse function ...
nrgtdrg 24737 A normed division ring is ...
nlmtlm 24738 A normed module is a topol...
isnvc 24739 A normed vector space is j...
nvcnlm 24740 A normed vector space is a...
nvclvec 24741 A normed vector space is a...
nvclmod 24742 A normed vector space is a...
isnvc2 24743 A normed vector space is j...
nvctvc 24744 A normed vector space is a...
lssnlm 24745 A subspace of a normed mod...
lssnvc 24746 A subspace of a normed vec...
rlmnvc 24747 The ring module over a nor...
ngpocelbl 24748 Membership of an off-cente...
nmoffn 24755 The function producing ope...
reldmnghm 24756 Lemma for normed group hom...
reldmnmhm 24757 Lemma for module homomorph...
nmofval 24758 Value of the operator norm...
nmoval 24759 Value of the operator norm...
nmogelb 24760 Property of the operator n...
nmolb 24761 Any upper bound on the val...
nmolb2d 24762 Any upper bound on the val...
nmof 24763 The operator norm is a fun...
nmocl 24764 The operator norm of an op...
nmoge0 24765 The operator norm of an op...
nghmfval 24766 A normed group homomorphis...
isnghm 24767 A normed group homomorphis...
isnghm2 24768 A normed group homomorphis...
isnghm3 24769 A normed group homomorphis...
bddnghm 24770 A bounded group homomorphi...
nghmcl 24771 A normed group homomorphis...
nmoi 24772 The operator norm achieves...
nmoix 24773 The operator norm is a bou...
nmoi2 24774 The operator norm is a bou...
nmoleub 24775 The operator norm, defined...
nghmrcl1 24776 Reverse closure for a norm...
nghmrcl2 24777 Reverse closure for a norm...
nghmghm 24778 A normed group homomorphis...
nmo0 24779 The operator norm of the z...
nmoeq0 24780 The operator norm is zero ...
nmoco 24781 An upper bound on the oper...
nghmco 24782 The composition of normed ...
nmotri 24783 Triangle inequality for th...
nghmplusg 24784 The sum of two bounded lin...
0nghm 24785 The zero operator is a nor...
nmoid 24786 The operator norm of the i...
idnghm 24787 The identity operator is a...
nmods 24788 Upper bound for the distan...
nghmcn 24789 A normed group homomorphis...
isnmhm 24790 A normed module homomorphi...
nmhmrcl1 24791 Reverse closure for a norm...
nmhmrcl2 24792 Reverse closure for a norm...
nmhmlmhm 24793 A normed module homomorphi...
nmhmnghm 24794 A normed module homomorphi...
nmhmghm 24795 A normed module homomorphi...
isnmhm2 24796 A normed module homomorphi...
nmhmcl 24797 A normed module homomorphi...
idnmhm 24798 The identity operator is a...
0nmhm 24799 The zero operator is a bou...
nmhmco 24800 The composition of bounded...
nmhmplusg 24801 The sum of two bounded lin...
qtopbaslem 24802 The set of open intervals ...
qtopbas 24803 The set of open intervals ...
retopbas 24804 A basis for the standard t...
retop 24805 The standard topology on t...
uniretop 24806 The underlying set of the ...
retopon 24807 The standard topology on t...
retps 24808 The standard topological s...
iooretop 24809 Open intervals are open se...
icccld 24810 Closed intervals are close...
icopnfcld 24811 Right-unbounded closed int...
iocmnfcld 24812 Left-unbounded closed inte...
qdensere 24813 ` QQ ` is dense in the sta...
cnmetdval 24814 Value of the distance func...
cnmet 24815 The absolute value metric ...
cnxmet 24816 The absolute value metric ...
cnbl0 24817 Two ways to write the open...
cnblcld 24818 Two ways to write the clos...
cnfldms 24819 The complex number field i...
cnfldxms 24820 The complex number field i...
cnfldtps 24821 The complex number field i...
cnfldnm 24822 The norm of the field of c...
cnngp 24823 The complex numbers form a...
cnnrg 24824 The complex numbers form a...
cnfldtopn 24825 The topology of the comple...
cnfldtopon 24826 The topology of the comple...
cnfldtop 24827 The topology of the comple...
cnfldhaus 24828 The topology of the comple...
unicntop 24829 The underlying set of the ...
cnopn 24830 The set of complex numbers...
zringnrg 24831 The ring of integers is a ...
remetdval 24832 Value of the distance func...
remet 24833 The absolute value metric ...
rexmet 24834 The absolute value metric ...
bl2ioo 24835 A ball in terms of an open...
ioo2bl 24836 An open interval of reals ...
ioo2blex 24837 An open interval of reals ...
blssioo 24838 The balls of the standard ...
tgioo 24839 The topology generated by ...
qdensere2 24840 ` QQ ` is dense in ` RR ` ...
blcvx 24841 An open ball in the comple...
rehaus 24842 The standard topology on t...
tgqioo 24843 The topology generated by ...
re2ndc 24844 The standard topology on t...
resubmet 24845 The subspace topology indu...
tgioo2 24846 The standard topology on t...
rerest 24847 The subspace topology indu...
tgioo3 24848 The standard topology on t...
xrtgioo 24849 The topology on the extend...
xrrest 24850 The subspace topology indu...
xrrest2 24851 The subspace topology indu...
xrsxmet 24852 The metric on the extended...
xrsdsre 24853 The metric on the extended...
xrsblre 24854 Any ball of the metric of ...
xrsmopn 24855 The metric on the extended...
zcld 24856 The integers are a closed ...
recld2 24857 The real numbers are a clo...
zcld2 24858 The integers are a closed ...
zdis 24859 The integers are a discret...
sszcld 24860 Every subset of the intege...
reperflem 24861 A subset of the real numbe...
reperf 24862 The real numbers are a per...
cnperf 24863 The complex numbers are a ...
iccntr 24864 The interior of a closed i...
icccmplem1 24865 Lemma for ~ icccmp . (Con...
icccmplem2 24866 Lemma for ~ icccmp . (Con...
icccmplem3 24867 Lemma for ~ icccmp . (Con...
icccmp 24868 A closed interval in ` RR ...
reconnlem1 24869 Lemma for ~ reconn . Conn...
reconnlem2 24870 Lemma for ~ reconn . (Con...
reconn 24871 A subset of the reals is c...
retopconn 24872 Corollary of ~ reconn . T...
iccconn 24873 A closed interval is conne...
opnreen 24874 Every nonempty open set is...
rectbntr0 24875 A countable subset of the ...
xrge0gsumle 24876 A finite sum in the nonneg...
xrge0tsms 24877 Any finite or infinite sum...
xrge0tsms2 24878 Any finite or infinite sum...
metdcnlem 24879 The metric function of a m...
xmetdcn2 24880 The metric function of an ...
xmetdcn 24881 The metric function of an ...
metdcn2 24882 The metric function of a m...
metdcn 24883 The metric function of a m...
msdcn 24884 The metric function of a m...
cnmpt1ds 24885 Continuity of the metric f...
cnmpt2ds 24886 Continuity of the metric f...
nmcn 24887 The norm of a normed group...
ngnmcncn 24888 The norm of a normed group...
abscn 24889 The absolute value functio...
metdsval 24890 Value of the "distance to ...
metdsf 24891 The distance from a point ...
metdsge 24892 The distance from the poin...
metds0 24893 If a point is in a set, it...
metdstri 24894 A generalization of the tr...
metdsle 24895 The distance from a point ...
metdsre 24896 The distance from a point ...
metdseq0 24897 The distance from a point ...
metdscnlem 24898 Lemma for ~ metdscn . (Co...
metdscn 24899 The function ` F ` which g...
metdscn2 24900 The function ` F ` which g...
metnrmlem1a 24901 Lemma for ~ metnrm . (Con...
metnrmlem1 24902 Lemma for ~ metnrm . (Con...
metnrmlem2 24903 Lemma for ~ metnrm . (Con...
metnrmlem3 24904 Lemma for ~ metnrm . (Con...
metnrm 24905 A metric space is normal. ...
metreg 24906 A metric space is regular....
addcnlem 24907 Lemma for ~ addcn , ~ subc...
addcn 24908 Complex number addition is...
subcn 24909 Complex number subtraction...
mulcn 24910 Complex number multiplicat...
divcnOLD 24911 Obsolete version of ~ divc...
mpomulcn 24912 Complex number multiplicat...
divcn 24913 Complex number division is...
cnfldtgp 24914 The complex numbers form a...
fsumcn 24915 A finite sum of functions ...
fsum2cn 24916 Version of ~ fsumcn for tw...
expcn 24917 The power function on comp...
divccn 24918 Division by a nonzero cons...
expcnOLD 24919 Obsolete version of ~ expc...
divccnOLD 24920 Obsolete version of ~ divc...
sqcn 24921 The square function on com...
iitopon 24926 The unit interval is a top...
iitop 24927 The unit interval is a top...
iiuni 24928 The base set of the unit i...
dfii2 24929 Alternate definition of th...
dfii3 24930 Alternate definition of th...
dfii4 24931 Alternate definition of th...
dfii5 24932 The unit interval expresse...
iicmp 24933 The unit interval is compa...
iiconn 24934 The unit interval is conne...
cncfval 24935 The value of the continuou...
elcncf 24936 Membership in the set of c...
elcncf2 24937 Version of ~ elcncf with a...
cncfrss 24938 Reverse closure of the con...
cncfrss2 24939 Reverse closure of the con...
cncff 24940 A continuous complex funct...
cncfi 24941 Defining property of a con...
elcncf1di 24942 Membership in the set of c...
elcncf1ii 24943 Membership in the set of c...
rescncf 24944 A continuous complex funct...
cncfcdm 24945 Change the codomain of a c...
cncfss 24946 The set of continuous func...
climcncf 24947 Image of a limit under a c...
abscncf 24948 Absolute value is continuo...
recncf 24949 Real part is continuous. ...
imcncf 24950 Imaginary part is continuo...
cjcncf 24951 Complex conjugate is conti...
mulc1cncf 24952 Multiplication by a consta...
divccncf 24953 Division by a constant is ...
cncfco 24954 The composition of two con...
cncfcompt2 24955 Composition of continuous ...
cncfmet 24956 Relate complex function co...
cncfcn 24957 Relate complex function co...
cncfcn1 24958 Relate complex function co...
cncfmptc 24959 A constant function is a c...
cncfmptid 24960 The identity function is a...
cncfmpt1f 24961 Composition of continuous ...
cncfmpt2f 24962 Composition of continuous ...
cncfmpt2ss 24963 Composition of continuous ...
addccncf 24964 Adding a constant is a con...
idcncf 24965 The identity function is a...
sub1cncf 24966 Subtracting a constant is ...
sub2cncf 24967 Subtraction from a constan...
cdivcncf 24968 Division with a constant n...
negcncf 24969 The negative function is c...
negcncfOLD 24970 Obsolete version of ~ negc...
negfcncf 24971 The negative of a continuo...
abscncfALT 24972 Absolute value is continuo...
cncfcnvcn 24973 Rewrite ~ cmphaushmeo for ...
expcncf 24974 The power function on comp...
cnmptre 24975 Lemma for ~ iirevcn and re...
cnmpopc 24976 Piecewise definition of a ...
iirev 24977 Reverse the unit interval....
iirevcn 24978 The reversion function is ...
iihalf1 24979 Map the first half of ` II...
iihalf1cn 24980 The first half function is...
iihalf1cnOLD 24981 Obsolete version of ~ iiha...
iihalf2 24982 Map the second half of ` I...
iihalf2cn 24983 The second half function i...
iihalf2cnOLD 24984 Obsolete version of ~ iiha...
elii1 24985 Divide the unit interval i...
elii2 24986 Divide the unit interval i...
iimulcl 24987 The unit interval is close...
iimulcn 24988 Multiplication is a contin...
iimulcnOLD 24989 Obsolete version of ~ iimu...
icoopnst 24990 A half-open interval start...
iocopnst 24991 A half-open interval endin...
icchmeo 24992 The natural bijection from...
icchmeoOLD 24993 Obsolete version of ~ icch...
icopnfcnv 24994 Define a bijection from ` ...
icopnfhmeo 24995 The defined bijection from...
iccpnfcnv 24996 Define a bijection from ` ...
iccpnfhmeo 24997 The defined bijection from...
xrhmeo 24998 The bijection from ` [ -u ...
xrhmph 24999 The extended reals are hom...
xrcmp 25000 The topology of the extend...
xrconn 25001 The topology of the extend...
icccvx 25002 A linear combination of tw...
oprpiece1res1 25003 Restriction to the first p...
oprpiece1res2 25004 Restriction to the second ...
cnrehmeo 25005 The canonical bijection fr...
cnrehmeoOLD 25006 Obsolete version of ~ cnre...
cnheiborlem 25007 Lemma for ~ cnheibor . (C...
cnheibor 25008 Heine-Borel theorem for co...
cnllycmp 25009 The topology on the comple...
rellycmp 25010 The topology on the reals ...
bndth 25011 The Boundedness Theorem. ...
evth 25012 The Extreme Value Theorem....
evth2 25013 The Extreme Value Theorem,...
lebnumlem1 25014 Lemma for ~ lebnum . The ...
lebnumlem2 25015 Lemma for ~ lebnum . As a...
lebnumlem3 25016 Lemma for ~ lebnum . By t...
lebnum 25017 The Lebesgue number lemma,...
xlebnum 25018 Generalize ~ lebnum to ext...
lebnumii 25019 Specialize the Lebesgue nu...
ishtpy 25025 Membership in the class of...
htpycn 25026 A homotopy is a continuous...
htpyi 25027 A homotopy evaluated at it...
ishtpyd 25028 Deduction for membership i...
htpycom 25029 Given a homotopy from ` F ...
htpyid 25030 A homotopy from a function...
htpyco1 25031 Compose a homotopy with a ...
htpyco2 25032 Compose a homotopy with a ...
htpycc 25033 Concatenate two homotopies...
isphtpy 25034 Membership in the class of...
phtpyhtpy 25035 A path homotopy is a homot...
phtpycn 25036 A path homotopy is a conti...
phtpyi 25037 Membership in the class of...
phtpy01 25038 Two path-homotopic paths h...
isphtpyd 25039 Deduction for membership i...
isphtpy2d 25040 Deduction for membership i...
phtpycom 25041 Given a homotopy from ` F ...
phtpyid 25042 A homotopy from a path to ...
phtpyco2 25043 Compose a path homotopy wi...
phtpycc 25044 Concatenate two path homot...
phtpcrel 25046 The path homotopy relation...
isphtpc 25047 The relation "is path homo...
phtpcer 25048 Path homotopy is an equiva...
phtpc01 25049 Path homotopic paths have ...
reparphti 25050 Lemma for ~ reparpht . (C...
reparphtiOLD 25051 Obsolete version of ~ repa...
reparpht 25052 Reparametrization lemma. ...
phtpcco2 25053 Compose a path homotopy wi...
pcofval 25064 The value of the path conc...
pcoval 25065 The concatenation of two p...
pcovalg 25066 Evaluate the concatenation...
pcoval1 25067 Evaluate the concatenation...
pco0 25068 The starting point of a pa...
pco1 25069 The ending point of a path...
pcoval2 25070 Evaluate the concatenation...
pcocn 25071 The concatenation of two p...
copco 25072 The composition of a conca...
pcohtpylem 25073 Lemma for ~ pcohtpy . (Co...
pcohtpy 25074 Homotopy invariance of pat...
pcoptcl 25075 A constant function is a p...
pcopt 25076 Concatenation with a point...
pcopt2 25077 Concatenation with a point...
pcoass 25078 Order of concatenation doe...
pcorevcl 25079 Closure for a reversed pat...
pcorevlem 25080 Lemma for ~ pcorev . Prov...
pcorev 25081 Concatenation with the rev...
pcorev2 25082 Concatenation with the rev...
pcophtb 25083 The path homotopy equivale...
om1val 25084 The definition of the loop...
om1bas 25085 The base set of the loop s...
om1elbas 25086 Elementhood in the base se...
om1addcl 25087 Closure of the group opera...
om1plusg 25088 The group operation (which...
om1tset 25089 The topology of the loop s...
om1opn 25090 The topology of the loop s...
pi1val 25091 The definition of the fund...
pi1bas 25092 The base set of the fundam...
pi1blem 25093 Lemma for ~ pi1buni . (Co...
pi1buni 25094 Another way to write the l...
pi1bas2 25095 The base set of the fundam...
pi1eluni 25096 Elementhood in the base se...
pi1bas3 25097 The base set of the fundam...
pi1cpbl 25098 The group operation, loop ...
elpi1 25099 The elements of the fundam...
elpi1i 25100 The elements of the fundam...
pi1addf 25101 The group operation of ` p...
pi1addval 25102 The concatenation of two p...
pi1grplem 25103 Lemma for ~ pi1grp . (Con...
pi1grp 25104 The fundamental group is a...
pi1id 25105 The identity element of th...
pi1inv 25106 An inverse in the fundamen...
pi1xfrf 25107 Functionality of the loop ...
pi1xfrval 25108 The value of the loop tran...
pi1xfr 25109 Given a path ` F ` and its...
pi1xfrcnvlem 25110 Given a path ` F ` between...
pi1xfrcnv 25111 Given a path ` F ` between...
pi1xfrgim 25112 The mapping ` G ` between ...
pi1cof 25113 Functionality of the loop ...
pi1coval 25114 The value of the loop tran...
pi1coghm 25115 The mapping ` G ` between ...
isclm 25118 A subcomplex module is a l...
clmsca 25119 The ring of scalars ` F ` ...
clmsubrg 25120 The base set of the ring o...
clmlmod 25121 A subcomplex module is a l...
clmgrp 25122 A subcomplex module is an ...
clmabl 25123 A subcomplex module is an ...
clmring 25124 The scalar ring of a subco...
clmfgrp 25125 The scalar ring of a subco...
clm0 25126 The zero of the scalar rin...
clm1 25127 The identity of the scalar...
clmadd 25128 The addition of the scalar...
clmmul 25129 The multiplication of the ...
clmcj 25130 The conjugation of the sca...
isclmi 25131 Reverse direction of ~ isc...
clmzss 25132 The scalar ring of a subco...
clmsscn 25133 The scalar ring of a subco...
clmsub 25134 Subtraction in the scalar ...
clmneg 25135 Negation in the scalar rin...
clmneg1 25136 Minus one is in the scalar...
clmabs 25137 Norm in the scalar ring of...
clmacl 25138 Closure of ring addition f...
clmmcl 25139 Closure of ring multiplica...
clmsubcl 25140 Closure of ring subtractio...
lmhmclm 25141 The domain of a linear ope...
clmvscl 25142 Closure of scalar product ...
clmvsass 25143 Associative law for scalar...
clmvscom 25144 Commutative law for the sc...
clmvsdir 25145 Distributive law for scala...
clmvsdi 25146 Distributive law for scala...
clmvs1 25147 Scalar product with ring u...
clmvs2 25148 A vector plus itself is tw...
clm0vs 25149 Zero times a vector is the...
clmopfne 25150 The (functionalized) opera...
isclmp 25151 The predicate "is a subcom...
isclmi0 25152 Properties that determine ...
clmvneg1 25153 Minus 1 times a vector is ...
clmvsneg 25154 Multiplication of a vector...
clmmulg 25155 The group multiple functio...
clmsubdir 25156 Scalar multiplication dist...
clmpm1dir 25157 Subtractive distributive l...
clmnegneg 25158 Double negative of a vecto...
clmnegsubdi2 25159 Distribution of negative o...
clmsub4 25160 Rearrangement of 4 terms i...
clmvsrinv 25161 A vector minus itself. (C...
clmvslinv 25162 Minus a vector plus itself...
clmvsubval 25163 Value of vector subtractio...
clmvsubval2 25164 Value of vector subtractio...
clmvz 25165 Two ways to express the ne...
zlmclm 25166 The ` ZZ ` -module operati...
clmzlmvsca 25167 The scalar product of a su...
nmoleub2lem 25168 Lemma for ~ nmoleub2a and ...
nmoleub2lem3 25169 Lemma for ~ nmoleub2a and ...
nmoleub2lem2 25170 Lemma for ~ nmoleub2a and ...
nmoleub2a 25171 The operator norm is the s...
nmoleub2b 25172 The operator norm is the s...
nmoleub3 25173 The operator norm is the s...
nmhmcn 25174 A linear operator over a n...
cmodscexp 25175 The powers of ` _i ` belon...
cmodscmulexp 25176 The scalar product of a ve...
cvslvec 25179 A subcomplex vector space ...
cvsclm 25180 A subcomplex vector space ...
iscvs 25181 A subcomplex vector space ...
iscvsp 25182 The predicate "is a subcom...
iscvsi 25183 Properties that determine ...
cvsi 25184 The properties of a subcom...
cvsunit 25185 Unit group of the scalar r...
cvsdiv 25186 Division of the scalar rin...
cvsdivcl 25187 The scalar field of a subc...
cvsmuleqdivd 25188 An equality involving rati...
cvsdiveqd 25189 An equality involving rati...
cnlmodlem1 25190 Lemma 1 for ~ cnlmod . (C...
cnlmodlem2 25191 Lemma 2 for ~ cnlmod . (C...
cnlmodlem3 25192 Lemma 3 for ~ cnlmod . (C...
cnlmod4 25193 Lemma 4 for ~ cnlmod . (C...
cnlmod 25194 The set of complex numbers...
cnstrcvs 25195 The set of complex numbers...
cnrbas 25196 The set of complex numbers...
cnrlmod 25197 The complex left module of...
cnrlvec 25198 The complex left module of...
cncvs 25199 The complex left module of...
recvs 25200 The field of the real numb...
recvsOLD 25201 Obsolete version of ~ recv...
qcvs 25202 The field of rational numb...
zclmncvs 25203 The ring of integers as le...
isncvsngp 25204 A normed subcomplex vector...
isncvsngpd 25205 Properties that determine ...
ncvsi 25206 The properties of a normed...
ncvsprp 25207 Proportionality property o...
ncvsge0 25208 The norm of a scalar produ...
ncvsm1 25209 The norm of the opposite o...
ncvsdif 25210 The norm of the difference...
ncvspi 25211 The norm of a vector plus ...
ncvs1 25212 From any nonzero vector of...
cnrnvc 25213 The module of complex numb...
cnncvs 25214 The module of complex numb...
cnnm 25215 The norm of the normed sub...
ncvspds 25216 Value of the distance func...
cnindmet 25217 The metric induced on the ...
cnncvsaddassdemo 25218 Derive the associative law...
cnncvsmulassdemo 25219 Derive the associative law...
cnncvsabsnegdemo 25220 Derive the absolute value ...
iscph 25225 A subcomplex pre-Hilbert s...
cphphl 25226 A subcomplex pre-Hilbert s...
cphnlm 25227 A subcomplex pre-Hilbert s...
cphngp 25228 A subcomplex pre-Hilbert s...
cphlmod 25229 A subcomplex pre-Hilbert s...
cphlvec 25230 A subcomplex pre-Hilbert s...
cphnvc 25231 A subcomplex pre-Hilbert s...
cphsubrglem 25232 Lemma for ~ cphsubrg . (C...
cphreccllem 25233 Lemma for ~ cphreccl . (C...
cphsca 25234 A subcomplex pre-Hilbert s...
cphsubrg 25235 The scalar field of a subc...
cphreccl 25236 The scalar field of a subc...
cphdivcl 25237 The scalar field of a subc...
cphcjcl 25238 The scalar field of a subc...
cphsqrtcl 25239 The scalar field of a subc...
cphabscl 25240 The scalar field of a subc...
cphsqrtcl2 25241 The scalar field of a subc...
cphsqrtcl3 25242 If the scalar field of a s...
cphqss 25243 The scalar field of a subc...
cphclm 25244 A subcomplex pre-Hilbert s...
cphnmvs 25245 Norm of a scalar product. ...
cphipcl 25246 An inner product is a memb...
cphnmfval 25247 The value of the norm in a...
cphnm 25248 The square of the norm is ...
nmsq 25249 The square of the norm is ...
cphnmf 25250 The norm of a vector is a ...
cphnmcl 25251 The norm of a vector is a ...
reipcl 25252 An inner product of an ele...
ipge0 25253 The inner product in a sub...
cphipcj 25254 Conjugate of an inner prod...
cphipipcj 25255 An inner product times its...
cphorthcom 25256 Orthogonality (meaning inn...
cphip0l 25257 Inner product with a zero ...
cphip0r 25258 Inner product with a zero ...
cphipeq0 25259 The inner product of a vec...
cphdir 25260 Distributive law for inner...
cphdi 25261 Distributive law for inner...
cph2di 25262 Distributive law for inner...
cphsubdir 25263 Distributive law for inner...
cphsubdi 25264 Distributive law for inner...
cph2subdi 25265 Distributive law for inner...
cphass 25266 Associative law for inner ...
cphassr 25267 "Associative" law for seco...
cph2ass 25268 Move scalar multiplication...
cphassi 25269 Associative law for the fi...
cphassir 25270 "Associative" law for the ...
cphpyth 25271 The pythagorean theorem fo...
tcphex 25272 Lemma for ~ tcphbas and si...
tcphval 25273 Define a function to augme...
tcphbas 25274 The base set of a subcompl...
tchplusg 25275 The addition operation of ...
tcphsub 25276 The subtraction operation ...
tcphmulr 25277 The ring operation of a su...
tcphsca 25278 The scalar field of a subc...
tcphvsca 25279 The scalar multiplication ...
tcphip 25280 The inner product of a sub...
tcphtopn 25281 The topology of a subcompl...
tcphphl 25282 Augmentation of a subcompl...
tchnmfval 25283 The norm of a subcomplex p...
tcphnmval 25284 The norm of a subcomplex p...
cphtcphnm 25285 The norm of a norm-augment...
tcphds 25286 The distance of a pre-Hilb...
phclm 25287 A pre-Hilbert space whose ...
tcphcphlem3 25288 Lemma for ~ tcphcph : real...
ipcau2 25289 The Cauchy-Schwarz inequal...
tcphcphlem1 25290 Lemma for ~ tcphcph : the ...
tcphcphlem2 25291 Lemma for ~ tcphcph : homo...
tcphcph 25292 The standard definition of...
ipcau 25293 The Cauchy-Schwarz inequal...
nmparlem 25294 Lemma for ~ nmpar . (Cont...
nmpar 25295 A subcomplex pre-Hilbert s...
cphipval2 25296 Value of the inner product...
4cphipval2 25297 Four times the inner produ...
cphipval 25298 Value of the inner product...
ipcnlem2 25299 The inner product operatio...
ipcnlem1 25300 The inner product operatio...
ipcn 25301 The inner product operatio...
cnmpt1ip 25302 Continuity of inner produc...
cnmpt2ip 25303 Continuity of inner produc...
csscld 25304 A "closed subspace" in a s...
clsocv 25305 The orthogonal complement ...
cphsscph 25306 A subspace of a subcomplex...
lmmbr 25313 Express the binary relatio...
lmmbr2 25314 Express the binary relatio...
lmmbr3 25315 Express the binary relatio...
lmmcvg 25316 Convergence property of a ...
lmmbrf 25317 Express the binary relatio...
lmnn 25318 A condition that implies c...
cfilfval 25319 The set of Cauchy filters ...
iscfil 25320 The property of being a Ca...
iscfil2 25321 The property of being a Ca...
cfilfil 25322 A Cauchy filter is a filte...
cfili 25323 Property of a Cauchy filte...
cfil3i 25324 A Cauchy filter contains b...
cfilss 25325 A filter finer than a Cauc...
fgcfil 25326 The Cauchy filter conditio...
fmcfil 25327 The Cauchy filter conditio...
iscfil3 25328 A filter is Cauchy iff it ...
cfilfcls 25329 Similar to ultrafilters ( ...
caufval 25330 The set of Cauchy sequence...
iscau 25331 Express the property " ` F...
iscau2 25332 Express the property " ` F...
iscau3 25333 Express the Cauchy sequenc...
iscau4 25334 Express the property " ` F...
iscauf 25335 Express the property " ` F...
caun0 25336 A metric with a Cauchy seq...
caufpm 25337 Inclusion of a Cauchy sequ...
caucfil 25338 A Cauchy sequence predicat...
iscmet 25339 The property " ` D ` is a ...
cmetcvg 25340 The convergence of a Cauch...
cmetmet 25341 A complete metric space is...
cmetmeti 25342 A complete metric space is...
cmetcaulem 25343 Lemma for ~ cmetcau . (Co...
cmetcau 25344 The convergence of a Cauch...
iscmet3lem3 25345 Lemma for ~ iscmet3 . (Co...
iscmet3lem1 25346 Lemma for ~ iscmet3 . (Co...
iscmet3lem2 25347 Lemma for ~ iscmet3 . (Co...
iscmet3 25348 The property " ` D ` is a ...
iscmet2 25349 A metric ` D ` is complete...
cfilresi 25350 A Cauchy filter on a metri...
cfilres 25351 Cauchy filter on a metric ...
caussi 25352 Cauchy sequence on a metri...
causs 25353 Cauchy sequence on a metri...
equivcfil 25354 If the metric ` D ` is "st...
equivcau 25355 If the metric ` D ` is "st...
lmle 25356 If the distance from each ...
nglmle 25357 If the norm of each member...
lmclim 25358 Relate a limit on the metr...
lmclimf 25359 Relate a limit on the metr...
metelcls 25360 A point belongs to the clo...
metcld 25361 A subset of a metric space...
metcld2 25362 A subset of a metric space...
caubl 25363 Sufficient condition to en...
caublcls 25364 The convergent point of a ...
metcnp4 25365 Two ways to say a mapping ...
metcn4 25366 Two ways to say a mapping ...
iscmet3i 25367 Properties that determine ...
lmcau 25368 Every convergent sequence ...
flimcfil 25369 Every convergent filter in...
metsscmetcld 25370 A complete subspace of a m...
cmetss 25371 A subspace of a complete m...
equivcmet 25372 If two metrics are strongl...
relcmpcmet 25373 If ` D ` is a metric space...
cmpcmet 25374 A compact metric space is ...
cfilucfil3 25375 Given a metric ` D ` and a...
cfilucfil4 25376 Given a metric ` D ` and a...
cncmet 25377 The set of complex numbers...
recmet 25378 The real numbers are a com...
bcthlem1 25379 Lemma for ~ bcth . Substi...
bcthlem2 25380 Lemma for ~ bcth . The ba...
bcthlem3 25381 Lemma for ~ bcth . The li...
bcthlem4 25382 Lemma for ~ bcth . Given ...
bcthlem5 25383 Lemma for ~ bcth . The pr...
bcth 25384 Baire's Category Theorem. ...
bcth2 25385 Baire's Category Theorem, ...
bcth3 25386 Baire's Category Theorem, ...
isbn 25393 A Banach space is a normed...
bnsca 25394 The scalar field of a Bana...
bnnvc 25395 A Banach space is a normed...
bnnlm 25396 A Banach space is a normed...
bnngp 25397 A Banach space is a normed...
bnlmod 25398 A Banach space is a left m...
bncms 25399 A Banach space is a comple...
iscms 25400 A complete metric space is...
cmscmet 25401 The induced metric on a co...
bncmet 25402 The induced metric on Bana...
cmsms 25403 A complete metric space is...
cmspropd 25404 Property deduction for a c...
cmssmscld 25405 The restriction of a metri...
cmsss 25406 The restriction of a compl...
lssbn 25407 A subspace of a Banach spa...
cmetcusp1 25408 If the uniform set of a co...
cmetcusp 25409 The uniform space generate...
cncms 25410 The field of complex numbe...
cnflduss 25411 The uniform structure of t...
cnfldcusp 25412 The field of complex numbe...
resscdrg 25413 The real numbers are a sub...
cncdrg 25414 The only complete subfield...
srabn 25415 The subring algebra over a...
rlmbn 25416 The ring module over a com...
ishl 25417 The predicate "is a subcom...
hlbn 25418 Every subcomplex Hilbert s...
hlcph 25419 Every subcomplex Hilbert s...
hlphl 25420 Every subcomplex Hilbert s...
hlcms 25421 Every subcomplex Hilbert s...
hlprlem 25422 Lemma for ~ hlpr . (Contr...
hlress 25423 The scalar field of a subc...
hlpr 25424 The scalar field of a subc...
ishl2 25425 A Hilbert space is a compl...
cphssphl 25426 A Banach subspace of a sub...
cmslssbn 25427 A complete linear subspace...
cmscsscms 25428 A closed subspace of a com...
bncssbn 25429 A closed subspace of a Ban...
cssbn 25430 A complete subspace of a n...
csschl 25431 A complete subspace of a c...
cmslsschl 25432 A complete linear subspace...
chlcsschl 25433 A closed subspace of a sub...
retopn 25434 The topology of the real n...
recms 25435 The real numbers form a co...
reust 25436 The Uniform structure of t...
recusp 25437 The real numbers form a co...
rrxval 25442 Value of the generalized E...
rrxbase 25443 The base of the generalize...
rrxprds 25444 Expand the definition of t...
rrxip 25445 The inner product of the g...
rrxnm 25446 The norm of the generalize...
rrxcph 25447 Generalized Euclidean real...
rrxds 25448 The distance over generali...
rrxvsca 25449 The scalar product over ge...
rrxplusgvscavalb 25450 The result of the addition...
rrxsca 25451 The field of real numbers ...
rrx0 25452 The zero ("origin") in a g...
rrx0el 25453 The zero ("origin") in a g...
csbren 25454 Cauchy-Schwarz-Bunjakovsky...
trirn 25455 Triangle inequality in R^n...
rrxf 25456 Euclidean vectors as funct...
rrxfsupp 25457 Euclidean vectors are of f...
rrxsuppss 25458 Support of Euclidean vecto...
rrxmvallem 25459 Support of the function us...
rrxmval 25460 The value of the Euclidean...
rrxmfval 25461 The value of the Euclidean...
rrxmetlem 25462 Lemma for ~ rrxmet . (Con...
rrxmet 25463 Euclidean space is a metri...
rrxdstprj1 25464 The distance between two p...
rrxbasefi 25465 The base of the generalize...
rrxdsfi 25466 The distance over generali...
rrxmetfi 25467 Euclidean space is a metri...
rrxdsfival 25468 The value of the Euclidean...
ehlval 25469 Value of the Euclidean spa...
ehlbase 25470 The base of the Euclidean ...
ehl0base 25471 The base of the Euclidean ...
ehl0 25472 The Euclidean space of dim...
ehleudis 25473 The Euclidean distance fun...
ehleudisval 25474 The value of the Euclidean...
ehl1eudis 25475 The Euclidean distance fun...
ehl1eudisval 25476 The value of the Euclidean...
ehl2eudis 25477 The Euclidean distance fun...
ehl2eudisval 25478 The value of the Euclidean...
minveclem1 25479 Lemma for ~ minvec . The ...
minveclem4c 25480 Lemma for ~ minvec . The ...
minveclem2 25481 Lemma for ~ minvec . Any ...
minveclem3a 25482 Lemma for ~ minvec . ` D `...
minveclem3b 25483 Lemma for ~ minvec . The ...
minveclem3 25484 Lemma for ~ minvec . The ...
minveclem4a 25485 Lemma for ~ minvec . ` F `...
minveclem4b 25486 Lemma for ~ minvec . The ...
minveclem4 25487 Lemma for ~ minvec . The ...
minveclem5 25488 Lemma for ~ minvec . Disc...
minveclem6 25489 Lemma for ~ minvec . Any ...
minveclem7 25490 Lemma for ~ minvec . Sinc...
minvec 25491 Minimizing vector theorem,...
pjthlem1 25492 Lemma for ~ pjth . (Contr...
pjthlem2 25493 Lemma for ~ pjth . (Contr...
pjth 25494 Projection Theorem: Any H...
pjth2 25495 Projection Theorem with ab...
cldcss 25496 Corollary of the Projectio...
cldcss2 25497 Corollary of the Projectio...
hlhil 25498 Corollary of the Projectio...
addcncf 25499 The addition of two contin...
subcncf 25500 The subtraction of two con...
mulcncf 25501 The multiplication of two ...
mulcncfOLD 25502 Obsolete version of ~ mulc...
divcncf 25503 The quotient of two contin...
pmltpclem1 25504 Lemma for ~ pmltpc . (Con...
pmltpclem2 25505 Lemma for ~ pmltpc . (Con...
pmltpc 25506 Any function on the reals ...
ivthlem1 25507 Lemma for ~ ivth . The se...
ivthlem2 25508 Lemma for ~ ivth . Show t...
ivthlem3 25509 Lemma for ~ ivth , the int...
ivth 25510 The intermediate value the...
ivth2 25511 The intermediate value the...
ivthle 25512 The intermediate value the...
ivthle2 25513 The intermediate value the...
ivthicc 25514 The interval between any t...
evthicc 25515 Specialization of the Extr...
evthicc2 25516 Combine ~ ivthicc with ~ e...
cniccbdd 25517 A continuous function on a...
ovolfcl 25522 Closure for the interval e...
ovolfioo 25523 Unpack the interval coveri...
ovolficc 25524 Unpack the interval coveri...
ovolficcss 25525 Any (closed) interval cove...
ovolfsval 25526 The value of the interval ...
ovolfsf 25527 Closure for the interval l...
ovolsf 25528 Closure for the partial su...
ovolval 25529 The value of the outer mea...
elovolmlem 25530 Lemma for ~ elovolm and re...
elovolm 25531 Elementhood in the set ` M...
elovolmr 25532 Sufficient condition for e...
ovolmge0 25533 The set ` M ` is composed ...
ovolcl 25534 The volume of a set is an ...
ovollb 25535 The outer volume is a lowe...
ovolgelb 25536 The outer volume is the gr...
ovolge0 25537 The volume of a set is alw...
ovolf 25538 The domain and codomain of...
ovollecl 25539 If an outer volume is boun...
ovolsslem 25540 Lemma for ~ ovolss . (Con...
ovolss 25541 The volume of a set is mon...
ovolsscl 25542 If a set is contained in a...
ovolssnul 25543 A subset of a nullset is n...
ovollb2lem 25544 Lemma for ~ ovollb2 . (Co...
ovollb2 25545 It is often more convenien...
ovolctb 25546 The volume of a denumerabl...
ovolq 25547 The rational numbers have ...
ovolctb2 25548 The volume of a countable ...
ovol0 25549 The empty set has 0 outer ...
ovolfi 25550 A finite set has 0 outer L...
ovolsn 25551 A singleton has 0 outer Le...
ovolunlem1a 25552 Lemma for ~ ovolun . (Con...
ovolunlem1 25553 Lemma for ~ ovolun . (Con...
ovolunlem2 25554 Lemma for ~ ovolun . (Con...
ovolun 25555 The Lebesgue outer measure...
ovolunnul 25556 Adding a nullset does not ...
ovolfiniun 25557 The Lebesgue outer measure...
ovoliunlem1 25558 Lemma for ~ ovoliun . (Co...
ovoliunlem2 25559 Lemma for ~ ovoliun . (Co...
ovoliunlem3 25560 Lemma for ~ ovoliun . (Co...
ovoliun 25561 The Lebesgue outer measure...
ovoliun2 25562 The Lebesgue outer measure...
ovoliunnul 25563 A countable union of nulls...
shft2rab 25564 If ` B ` is a shift of ` A...
ovolshftlem1 25565 Lemma for ~ ovolshft . (C...
ovolshftlem2 25566 Lemma for ~ ovolshft . (C...
ovolshft 25567 The Lebesgue outer measure...
sca2rab 25568 If ` B ` is a scale of ` A...
ovolscalem1 25569 Lemma for ~ ovolsca . (Co...
ovolscalem2 25570 Lemma for ~ ovolshft . (C...
ovolsca 25571 The Lebesgue outer measure...
ovolicc1 25572 The measure of a closed in...
ovolicc2lem1 25573 Lemma for ~ ovolicc2 . (C...
ovolicc2lem2 25574 Lemma for ~ ovolicc2 . (C...
ovolicc2lem3 25575 Lemma for ~ ovolicc2 . (C...
ovolicc2lem4 25576 Lemma for ~ ovolicc2 . (C...
ovolicc2lem5 25577 Lemma for ~ ovolicc2 . (C...
ovolicc2 25578 The measure of a closed in...
ovolicc 25579 The measure of a closed in...
ovolicopnf 25580 The measure of a right-unb...
ovolre 25581 The measure of the real nu...
ismbl 25582 The predicate " ` A ` is L...
ismbl2 25583 From ~ ovolun , it suffice...
volres 25584 A self-referencing abbrevi...
volf 25585 The domain and codomain of...
mblvol 25586 The volume of a measurable...
mblss 25587 A measurable set is a subs...
mblsplit 25588 The defining property of m...
volss 25589 The Lebesgue measure is mo...
cmmbl 25590 The complement of a measur...
nulmbl 25591 A nullset is measurable. ...
nulmbl2 25592 A set of outer measure zer...
unmbl 25593 A union of measurable sets...
shftmbl 25594 A shift of a measurable se...
0mbl 25595 The empty set is measurabl...
rembl 25596 The set of all real number...
unidmvol 25597 The union of the Lebesgue ...
inmbl 25598 An intersection of measura...
difmbl 25599 A difference of measurable...
finiunmbl 25600 A finite union of measurab...
volun 25601 The Lebesgue measure funct...
volinun 25602 Addition of non-disjoint s...
volfiniun 25603 The volume of a disjoint f...
iundisj 25604 Rewrite a countable union ...
iundisj2 25605 A disjoint union is disjoi...
voliunlem1 25606 Lemma for ~ voliun . (Con...
voliunlem2 25607 Lemma for ~ voliun . (Con...
voliunlem3 25608 Lemma for ~ voliun . (Con...
iunmbl 25609 The measurable sets are cl...
voliun 25610 The Lebesgue measure funct...
volsuplem 25611 Lemma for ~ volsup . (Con...
volsup 25612 The volume of the limit of...
iunmbl2 25613 The measurable sets are cl...
ioombl1lem1 25614 Lemma for ~ ioombl1 . (Co...
ioombl1lem2 25615 Lemma for ~ ioombl1 . (Co...
ioombl1lem3 25616 Lemma for ~ ioombl1 . (Co...
ioombl1lem4 25617 Lemma for ~ ioombl1 . (Co...
ioombl1 25618 An open right-unbounded in...
icombl1 25619 A closed unbounded-above i...
icombl 25620 A closed-below, open-above...
ioombl 25621 An open real interval is m...
iccmbl 25622 A closed real interval is ...
iccvolcl 25623 A closed real interval has...
ovolioo 25624 The measure of an open int...
volioo 25625 The measure of an open int...
ioovolcl 25626 An open real interval has ...
ovolfs2 25627 Alternative expression for...
ioorcl2 25628 An open interval with fini...
ioorf 25629 Define a function from ope...
ioorval 25630 Define a function from ope...
ioorinv2 25631 The function ` F ` is an "...
ioorinv 25632 The function ` F ` is an "...
ioorcl 25633 The function ` F ` does no...
uniiccdif 25634 A union of closed interval...
uniioovol 25635 A disjoint union of open i...
uniiccvol 25636 An almost-disjoint union o...
uniioombllem1 25637 Lemma for ~ uniioombl . (...
uniioombllem2a 25638 Lemma for ~ uniioombl . (...
uniioombllem2 25639 Lemma for ~ uniioombl . (...
uniioombllem3a 25640 Lemma for ~ uniioombl . (...
uniioombllem3 25641 Lemma for ~ uniioombl . (...
uniioombllem4 25642 Lemma for ~ uniioombl . (...
uniioombllem5 25643 Lemma for ~ uniioombl . (...
uniioombllem6 25644 Lemma for ~ uniioombl . (...
uniioombl 25645 A disjoint union of open i...
uniiccmbl 25646 An almost-disjoint union o...
dyadf 25647 The function ` F ` returns...
dyadval 25648 Value of the dyadic ration...
dyadovol 25649 Volume of a dyadic rationa...
dyadss 25650 Two closed dyadic rational...
dyaddisjlem 25651 Lemma for ~ dyaddisj . (C...
dyaddisj 25652 Two closed dyadic rational...
dyadmaxlem 25653 Lemma for ~ dyadmax . (Co...
dyadmax 25654 Any nonempty set of dyadic...
dyadmbllem 25655 Lemma for ~ dyadmbl . (Co...
dyadmbl 25656 Any union of dyadic ration...
opnmbllem 25657 Lemma for ~ opnmbl . (Con...
opnmbl 25658 All open sets are measurab...
opnmblALT 25659 All open sets are measurab...
subopnmbl 25660 Sets which are open in a m...
volsup2 25661 The volume of ` A ` is the...
volcn 25662 The function formed by res...
volivth 25663 The Intermediate Value The...
vitalilem1 25664 Lemma for ~ vitali . (Con...
vitalilem2 25665 Lemma for ~ vitali . (Con...
vitalilem3 25666 Lemma for ~ vitali . (Con...
vitalilem4 25667 Lemma for ~ vitali . (Con...
vitalilem5 25668 Lemma for ~ vitali . (Con...
vitali 25669 If the reals can be well-o...
ismbf1 25680 The predicate " ` F ` is a...
mbff 25681 A measurable function is a...
mbfdm 25682 The domain of a measurable...
mbfconstlem 25683 Lemma for ~ mbfconst and r...
ismbf 25684 The predicate " ` F ` is a...
ismbfcn 25685 A complex function is meas...
mbfima 25686 Definitional property of a...
mbfimaicc 25687 The preimage of any closed...
mbfimasn 25688 The preimage of a point un...
mbfconst 25689 A constant function is mea...
mbf0 25690 The empty function is meas...
mbfid 25691 The identity function is m...
mbfmptcl 25692 Lemma for the ` MblFn ` pr...
mbfdm2 25693 The domain of a measurable...
ismbfcn2 25694 A complex function is meas...
ismbfd 25695 Deduction to prove measura...
ismbf2d 25696 Deduction to prove measura...
mbfeqalem1 25697 Lemma for ~ mbfeqalem2 . ...
mbfeqalem2 25698 Lemma for ~ mbfeqa . (Con...
mbfeqa 25699 If two functions are equal...
mbfres 25700 The restriction of a measu...
mbfres2 25701 Measurability of a piecewi...
mbfss 25702 Change the domain of a mea...
mbfmulc2lem 25703 Multiplication by a consta...
mbfmulc2re 25704 Multiplication by a consta...
mbfmax 25705 The maximum of two functio...
mbfneg 25706 The negative of a measurab...
mbfpos 25707 The positive part of a mea...
mbfposr 25708 Converse to ~ mbfpos . (C...
mbfposb 25709 A function is measurable i...
ismbf3d 25710 Simplified form of ~ ismbf...
mbfimaopnlem 25711 Lemma for ~ mbfimaopn . (...
mbfimaopn 25712 The preimage of any open s...
mbfimaopn2 25713 The preimage of any set op...
cncombf 25714 The composition of a conti...
cnmbf 25715 A continuous function is m...
mbfaddlem 25716 The sum of two measurable ...
mbfadd 25717 The sum of two measurable ...
mbfsub 25718 The difference of two meas...
mbfmulc2 25719 A complex constant times a...
mbfsup 25720 The supremum of a sequence...
mbfinf 25721 The infimum of a sequence ...
mbflimsup 25722 The limit supremum of a se...
mbflimlem 25723 The pointwise limit of a s...
mbflim 25724 The pointwise limit of a s...
0pval 25727 The zero function evaluate...
0plef 25728 Two ways to say that the f...
0pledm 25729 Adjust the domain of the l...
isi1f 25730 The predicate " ` F ` is a...
i1fmbf 25731 Simple functions are measu...
i1ff 25732 A simple function is a fun...
i1frn 25733 A simple function has fini...
i1fima 25734 Any preimage of a simple f...
i1fima2 25735 Any preimage of a simple f...
i1fima2sn 25736 Preimage of a singleton. ...
i1fd 25737 A simplified set of assump...
i1f0rn 25738 Any simple function takes ...
itg1val 25739 The value of the integral ...
itg1val2 25740 The value of the integral ...
itg1cl 25741 Closure of the integral on...
itg1ge0 25742 Closure of the integral on...
i1f0 25743 The zero function is simpl...
itg10 25744 The zero function has zero...
i1f1lem 25745 Lemma for ~ i1f1 and ~ itg...
i1f1 25746 Base case simple functions...
itg11 25747 The integral of an indicat...
itg1addlem1 25748 Decompose a preimage, whic...
i1faddlem 25749 Decompose the preimage of ...
i1fmullem 25750 Decompose the preimage of ...
i1fadd 25751 The sum of two simple func...
i1fmul 25752 The pointwise product of t...
itg1addlem2 25753 Lemma for ~ itg1add . The...
itg1addlem3 25754 Lemma for ~ itg1add . (Co...
itg1addlem4 25755 Lemma for ~ itg1add . (Co...
itg1addlem4OLD 25756 Obsolete version of ~ itg1...
itg1addlem5 25757 Lemma for ~ itg1add . (Co...
itg1add 25758 The integral of a sum of s...
i1fmulclem 25759 Decompose the preimage of ...
i1fmulc 25760 A nonnegative constant tim...
itg1mulc 25761 The integral of a constant...
i1fres 25762 The "restriction" of a sim...
i1fpos 25763 The positive part of a sim...
i1fposd 25764 Deduction form of ~ i1fpos...
i1fsub 25765 The difference of two simp...
itg1sub 25766 The integral of a differen...
itg10a 25767 The integral of a simple f...
itg1ge0a 25768 The integral of an almost ...
itg1lea 25769 Approximate version of ~ i...
itg1le 25770 If one simple function dom...
itg1climres 25771 Restricting the simple fun...
mbfi1fseqlem1 25772 Lemma for ~ mbfi1fseq . (...
mbfi1fseqlem2 25773 Lemma for ~ mbfi1fseq . (...
mbfi1fseqlem3 25774 Lemma for ~ mbfi1fseq . (...
mbfi1fseqlem4 25775 Lemma for ~ mbfi1fseq . T...
mbfi1fseqlem5 25776 Lemma for ~ mbfi1fseq . V...
mbfi1fseqlem6 25777 Lemma for ~ mbfi1fseq . V...
mbfi1fseq 25778 A characterization of meas...
mbfi1flimlem 25779 Lemma for ~ mbfi1flim . (...
mbfi1flim 25780 Any real measurable functi...
mbfmullem2 25781 Lemma for ~ mbfmul . (Con...
mbfmullem 25782 Lemma for ~ mbfmul . (Con...
mbfmul 25783 The product of two measura...
itg2lcl 25784 The set of lower sums is a...
itg2val 25785 Value of the integral on n...
itg2l 25786 Elementhood in the set ` L...
itg2lr 25787 Sufficient condition for e...
xrge0f 25788 A real function is a nonne...
itg2cl 25789 The integral of a nonnegat...
itg2ub 25790 The integral of a nonnegat...
itg2leub 25791 Any upper bound on the int...
itg2ge0 25792 The integral of a nonnegat...
itg2itg1 25793 The integral of a nonnegat...
itg20 25794 The integral of the zero f...
itg2lecl 25795 If an ` S.2 ` integral is ...
itg2le 25796 If one function dominates ...
itg2const 25797 Integral of a constant fun...
itg2const2 25798 When the base set of a con...
itg2seq 25799 Definitional property of t...
itg2uba 25800 Approximate version of ~ i...
itg2lea 25801 Approximate version of ~ i...
itg2eqa 25802 Approximate equality of in...
itg2mulclem 25803 Lemma for ~ itg2mulc . (C...
itg2mulc 25804 The integral of a nonnegat...
itg2splitlem 25805 Lemma for ~ itg2split . (...
itg2split 25806 The ` S.2 ` integral split...
itg2monolem1 25807 Lemma for ~ itg2mono . We...
itg2monolem2 25808 Lemma for ~ itg2mono . (C...
itg2monolem3 25809 Lemma for ~ itg2mono . (C...
itg2mono 25810 The Monotone Convergence T...
itg2i1fseqle 25811 Subject to the conditions ...
itg2i1fseq 25812 Subject to the conditions ...
itg2i1fseq2 25813 In an extension to the res...
itg2i1fseq3 25814 Special case of ~ itg2i1fs...
itg2addlem 25815 Lemma for ~ itg2add . (Co...
itg2add 25816 The ` S.2 ` integral is li...
itg2gt0 25817 If the function ` F ` is s...
itg2cnlem1 25818 Lemma for ~ itgcn . (Cont...
itg2cnlem2 25819 Lemma for ~ itgcn . (Cont...
itg2cn 25820 A sort of absolute continu...
ibllem 25821 Conditioned equality theor...
isibl 25822 The predicate " ` F ` is i...
isibl2 25823 The predicate " ` F ` is i...
iblmbf 25824 An integrable function is ...
iblitg 25825 If a function is integrabl...
dfitg 25826 Evaluate the class substit...
itgex 25827 An integral is a set. (Co...
itgeq1f 25828 Equality theorem for an in...
itgeq1fOLD 25829 Obsolete version of ~ itge...
itgeq1 25830 Equality theorem for an in...
nfitg1 25831 Bound-variable hypothesis ...
nfitg 25832 Bound-variable hypothesis ...
cbvitg 25833 Change bound variable in a...
cbvitgv 25834 Change bound variable in a...
itgeq2 25835 Equality theorem for an in...
itgresr 25836 The domain of an integral ...
itg0 25837 The integral of anything o...
itgz 25838 The integral of zero on an...
itgeq2dv 25839 Equality theorem for an in...
itgmpt 25840 Change bound variable in a...
itgcl 25841 The integral of an integra...
itgvallem 25842 Substitution lemma. (Cont...
itgvallem3 25843 Lemma for ~ itgposval and ...
ibl0 25844 The zero function is integ...
iblcnlem1 25845 Lemma for ~ iblcnlem . (C...
iblcnlem 25846 Expand out the universal q...
itgcnlem 25847 Expand out the sum in ~ df...
iblrelem 25848 Integrability of a real fu...
iblposlem 25849 Lemma for ~ iblpos . (Con...
iblpos 25850 Integrability of a nonnega...
iblre 25851 Integrability of a real fu...
itgrevallem1 25852 Lemma for ~ itgposval and ...
itgposval 25853 The integral of a nonnegat...
itgreval 25854 Decompose the integral of ...
itgrecl 25855 Real closure of an integra...
iblcn 25856 Integrability of a complex...
itgcnval 25857 Decompose the integral of ...
itgre 25858 Real part of an integral. ...
itgim 25859 Imaginary part of an integ...
iblneg 25860 The negative of an integra...
itgneg 25861 Negation of an integral. ...
iblss 25862 A subset of an integrable ...
iblss2 25863 Change the domain of an in...
itgitg2 25864 Transfer an integral using...
i1fibl 25865 A simple function is integ...
itgitg1 25866 Transfer an integral using...
itgle 25867 Monotonicity of an integra...
itgge0 25868 The integral of a positive...
itgss 25869 Expand the set of an integ...
itgss2 25870 Expand the set of an integ...
itgeqa 25871 Approximate equality of in...
itgss3 25872 Expand the set of an integ...
itgioo 25873 Equality of integrals on o...
itgless 25874 Expand the integral of a n...
iblconst 25875 A constant function is int...
itgconst 25876 Integral of a constant fun...
ibladdlem 25877 Lemma for ~ ibladd . (Con...
ibladd 25878 Add two integrals over the...
iblsub 25879 Subtract two integrals ove...
itgaddlem1 25880 Lemma for ~ itgadd . (Con...
itgaddlem2 25881 Lemma for ~ itgadd . (Con...
itgadd 25882 Add two integrals over the...
itgsub 25883 Subtract two integrals ove...
itgfsum 25884 Take a finite sum of integ...
iblabslem 25885 Lemma for ~ iblabs . (Con...
iblabs 25886 The absolute value of an i...
iblabsr 25887 A measurable function is i...
iblmulc2 25888 Multiply an integral by a ...
itgmulc2lem1 25889 Lemma for ~ itgmulc2 : pos...
itgmulc2lem2 25890 Lemma for ~ itgmulc2 : rea...
itgmulc2 25891 Multiply an integral by a ...
itgabs 25892 The triangle inequality fo...
itgsplit 25893 The ` S. ` integral splits...
itgspliticc 25894 The ` S. ` integral splits...
itgsplitioo 25895 The ` S. ` integral splits...
bddmulibl 25896 A bounded function times a...
bddibl 25897 A bounded function is inte...
cniccibl 25898 A continuous function on a...
bddiblnc 25899 Choice-free proof of ~ bdd...
cnicciblnc 25900 Choice-free proof of ~ cni...
itggt0 25901 The integral of a strictly...
itgcn 25902 Transfer ~ itg2cn to the f...
ditgeq1 25905 Equality theorem for the d...
ditgeq2 25906 Equality theorem for the d...
ditgeq3 25907 Equality theorem for the d...
ditgeq3dv 25908 Equality theorem for the d...
ditgex 25909 A directed integral is a s...
ditg0 25910 Value of the directed inte...
cbvditg 25911 Change bound variable in a...
cbvditgv 25912 Change bound variable in a...
ditgpos 25913 Value of the directed inte...
ditgneg 25914 Value of the directed inte...
ditgcl 25915 Closure of a directed inte...
ditgswap 25916 Reverse a directed integra...
ditgsplitlem 25917 Lemma for ~ ditgsplit . (...
ditgsplit 25918 This theorem is the raison...
reldv 25927 The derivative function is...
limcvallem 25928 Lemma for ~ ellimc . (Con...
limcfval 25929 Value and set bounds on th...
ellimc 25930 Value of the limit predica...
limcrcl 25931 Reverse closure for the li...
limccl 25932 Closure of the limit opera...
limcdif 25933 It suffices to consider fu...
ellimc2 25934 Write the definition of a ...
limcnlp 25935 If ` B ` is not a limit po...
ellimc3 25936 Write the epsilon-delta de...
limcflflem 25937 Lemma for ~ limcflf . (Co...
limcflf 25938 The limit operator can be ...
limcmo 25939 If ` B ` is a limit point ...
limcmpt 25940 Express the limit operator...
limcmpt2 25941 Express the limit operator...
limcresi 25942 Any limit of ` F ` is also...
limcres 25943 If ` B ` is an interior po...
cnplimc 25944 A function is continuous a...
cnlimc 25945 ` F ` is a continuous func...
cnlimci 25946 If ` F ` is a continuous f...
cnmptlimc 25947 If ` F ` is a continuous f...
limccnp 25948 If the limit of ` F ` at `...
limccnp2 25949 The image of a convergent ...
limcco 25950 Composition of two limits....
limciun 25951 A point is a limit of ` F ...
limcun 25952 A point is a limit of ` F ...
dvlem 25953 Closure for a difference q...
dvfval 25954 Value and set bounds on th...
eldv 25955 The differentiable predica...
dvcl 25956 The derivative function ta...
dvbssntr 25957 The set of differentiable ...
dvbss 25958 The set of differentiable ...
dvbsss 25959 The set of differentiable ...
perfdvf 25960 The derivative is a functi...
recnprss 25961 Both ` RR ` and ` CC ` are...
recnperf 25962 Both ` RR ` and ` CC ` are...
dvfg 25963 Explicitly write out the f...
dvf 25964 The derivative is a functi...
dvfcn 25965 The derivative is a functi...
dvreslem 25966 Lemma for ~ dvres . (Cont...
dvres2lem 25967 Lemma for ~ dvres2 . (Con...
dvres 25968 Restriction of a derivativ...
dvres2 25969 Restriction of the base se...
dvres3 25970 Restriction of a complex d...
dvres3a 25971 Restriction of a complex d...
dvidlem 25972 Lemma for ~ dvid and ~ dvc...
dvmptresicc 25973 Derivative of a function r...
dvconst 25974 Derivative of a constant f...
dvid 25975 Derivative of the identity...
dvcnp 25976 The difference quotient is...
dvcnp2 25977 A function is continuous a...
dvcnp2OLD 25978 Obsolete version of ~ dvcn...
dvcn 25979 A differentiable function ...
dvnfval 25980 Value of the iterated deri...
dvnff 25981 The iterated derivative is...
dvn0 25982 Zero times iterated deriva...
dvnp1 25983 Successor iterated derivat...
dvn1 25984 One times iterated derivat...
dvnf 25985 The N-times derivative is ...
dvnbss 25986 The set of N-times differe...
dvnadd 25987 The ` N ` -th derivative o...
dvn2bss 25988 An N-times differentiable ...
dvnres 25989 Multiple derivative versio...
cpnfval 25990 Condition for n-times cont...
fncpn 25991 The ` C^n ` object is a fu...
elcpn 25992 Condition for n-times cont...
cpnord 25993 ` C^n ` conditions are ord...
cpncn 25994 A ` C^n ` function is cont...
cpnres 25995 The restriction of a ` C^n...
dvaddbr 25996 The sum rule for derivativ...
dvmulbr 25997 The product rule for deriv...
dvmulbrOLD 25998 Obsolete version of ~ dvmu...
dvadd 25999 The sum rule for derivativ...
dvmul 26000 The product rule for deriv...
dvaddf 26001 The sum rule for everywher...
dvmulf 26002 The product rule for every...
dvcmul 26003 The product rule when one ...
dvcmulf 26004 The product rule when one ...
dvcobr 26005 The chain rule for derivat...
dvcobrOLD 26006 Obsolete version of ~ dvco...
dvco 26007 The chain rule for derivat...
dvcof 26008 The chain rule for everywh...
dvcjbr 26009 The derivative of the conj...
dvcj 26010 The derivative of the conj...
dvfre 26011 The derivative of a real f...
dvnfre 26012 The ` N ` -th derivative o...
dvexp 26013 Derivative of a power func...
dvexp2 26014 Derivative of an exponenti...
dvrec 26015 Derivative of the reciproc...
dvmptres3 26016 Function-builder for deriv...
dvmptid 26017 Function-builder for deriv...
dvmptc 26018 Function-builder for deriv...
dvmptcl 26019 Closure lemma for ~ dvmptc...
dvmptadd 26020 Function-builder for deriv...
dvmptmul 26021 Function-builder for deriv...
dvmptres2 26022 Function-builder for deriv...
dvmptres 26023 Function-builder for deriv...
dvmptcmul 26024 Function-builder for deriv...
dvmptdivc 26025 Function-builder for deriv...
dvmptneg 26026 Function-builder for deriv...
dvmptsub 26027 Function-builder for deriv...
dvmptcj 26028 Function-builder for deriv...
dvmptre 26029 Function-builder for deriv...
dvmptim 26030 Function-builder for deriv...
dvmptntr 26031 Function-builder for deriv...
dvmptco 26032 Function-builder for deriv...
dvrecg 26033 Derivative of the reciproc...
dvmptdiv 26034 Function-builder for deriv...
dvmptfsum 26035 Function-builder for deriv...
dvcnvlem 26036 Lemma for ~ dvcnvre . (Co...
dvcnv 26037 A weak version of ~ dvcnvr...
dvexp3 26038 Derivative of an exponenti...
dveflem 26039 Derivative of the exponent...
dvef 26040 Derivative of the exponent...
dvsincos 26041 Derivative of the sine and...
dvsin 26042 Derivative of the sine fun...
dvcos 26043 Derivative of the cosine f...
dvferm1lem 26044 Lemma for ~ dvferm . (Con...
dvferm1 26045 One-sided version of ~ dvf...
dvferm2lem 26046 Lemma for ~ dvferm . (Con...
dvferm2 26047 One-sided version of ~ dvf...
dvferm 26048 Fermat's theorem on statio...
rollelem 26049 Lemma for ~ rolle . (Cont...
rolle 26050 Rolle's theorem. If ` F `...
cmvth 26051 Cauchy's Mean Value Theore...
cmvthOLD 26052 Obsolete version of ~ cmvt...
mvth 26053 The Mean Value Theorem. I...
dvlip 26054 A function with derivative...
dvlipcn 26055 A complex function with de...
dvlip2 26056 Combine the results of ~ d...
c1liplem1 26057 Lemma for ~ c1lip1 . (Con...
c1lip1 26058 C^1 functions are Lipschit...
c1lip2 26059 C^1 functions are Lipschit...
c1lip3 26060 C^1 functions are Lipschit...
dveq0 26061 If a continuous function h...
dv11cn 26062 Two functions defined on a...
dvgt0lem1 26063 Lemma for ~ dvgt0 and ~ dv...
dvgt0lem2 26064 Lemma for ~ dvgt0 and ~ dv...
dvgt0 26065 A function on a closed int...
dvlt0 26066 A function on a closed int...
dvge0 26067 A function on a closed int...
dvle 26068 If ` A ( x ) , C ( x ) ` a...
dvivthlem1 26069 Lemma for ~ dvivth . (Con...
dvivthlem2 26070 Lemma for ~ dvivth . (Con...
dvivth 26071 Darboux' theorem, or the i...
dvne0 26072 A function on a closed int...
dvne0f1 26073 A function on a closed int...
lhop1lem 26074 Lemma for ~ lhop1 . (Cont...
lhop1 26075 L'Hôpital's Rule for...
lhop2 26076 L'Hôpital's Rule for...
lhop 26077 L'Hôpital's Rule. I...
dvcnvrelem1 26078 Lemma for ~ dvcnvre . (Co...
dvcnvrelem2 26079 Lemma for ~ dvcnvre . (Co...
dvcnvre 26080 The derivative rule for in...
dvcvx 26081 A real function with stric...
dvfsumle 26082 Compare a finite sum to an...
dvfsumleOLD 26083 Obsolete version of ~ dvfs...
dvfsumge 26084 Compare a finite sum to an...
dvfsumabs 26085 Compare a finite sum to an...
dvmptrecl 26086 Real closure of a derivati...
dvfsumrlimf 26087 Lemma for ~ dvfsumrlim . ...
dvfsumlem1 26088 Lemma for ~ dvfsumrlim . ...
dvfsumlem2 26089 Lemma for ~ dvfsumrlim . ...
dvfsumlem2OLD 26090 Obsolete version of ~ dvfs...
dvfsumlem3 26091 Lemma for ~ dvfsumrlim . ...
dvfsumlem4 26092 Lemma for ~ dvfsumrlim . ...
dvfsumrlimge0 26093 Lemma for ~ dvfsumrlim . ...
dvfsumrlim 26094 Compare a finite sum to an...
dvfsumrlim2 26095 Compare a finite sum to an...
dvfsumrlim3 26096 Conjoin the statements of ...
dvfsum2 26097 The reverse of ~ dvfsumrli...
ftc1lem1 26098 Lemma for ~ ftc1a and ~ ft...
ftc1lem2 26099 Lemma for ~ ftc1 . (Contr...
ftc1a 26100 The Fundamental Theorem of...
ftc1lem3 26101 Lemma for ~ ftc1 . (Contr...
ftc1lem4 26102 Lemma for ~ ftc1 . (Contr...
ftc1lem5 26103 Lemma for ~ ftc1 . (Contr...
ftc1lem6 26104 Lemma for ~ ftc1 . (Contr...
ftc1 26105 The Fundamental Theorem of...
ftc1cn 26106 Strengthen the assumptions...
ftc2 26107 The Fundamental Theorem of...
ftc2ditglem 26108 Lemma for ~ ftc2ditg . (C...
ftc2ditg 26109 Directed integral analogue...
itgparts 26110 Integration by parts. If ...
itgsubstlem 26111 Lemma for ~ itgsubst . (C...
itgsubst 26112 Integration by ` u ` -subs...
itgpowd 26113 The integral of a monomial...
reldmmdeg 26118 Multivariate degree is a b...
tdeglem1 26119 Functionality of the total...
tdeglem3 26120 Additivity of the total de...
tdeglem4 26121 There is only one multi-in...
tdeglem2 26122 Simplification of total de...
mdegfval 26123 Value of the multivariate ...
mdegval 26124 Value of the multivariate ...
mdegleb 26125 Property of being of limit...
mdeglt 26126 If there is an upper limit...
mdegldg 26127 A nonzero polynomial has s...
mdegxrcl 26128 Closure of polynomial degr...
mdegxrf 26129 Functionality of polynomia...
mdegcl 26130 Sharp closure for multivar...
mdeg0 26131 Degree of the zero polynom...
mdegnn0cl 26132 Degree of a nonzero polyno...
degltlem1 26133 Theorem on arithmetic of e...
degltp1le 26134 Theorem on arithmetic of e...
mdegaddle 26135 The degree of a sum is at ...
mdegvscale 26136 The degree of a scalar mul...
mdegvsca 26137 The degree of a scalar mul...
mdegle0 26138 A polynomial has nonpositi...
mdegmullem 26139 Lemma for ~ mdegmulle2 . ...
mdegmulle2 26140 The multivariate degree of...
deg1fval 26141 Relate univariate polynomi...
deg1xrf 26142 Functionality of univariat...
deg1xrcl 26143 Closure of univariate poly...
deg1cl 26144 Sharp closure of univariat...
mdegpropd 26145 Property deduction for pol...
deg1fvi 26146 Univariate polynomial degr...
deg1propd 26147 Property deduction for pol...
deg1z 26148 Degree of the zero univari...
deg1nn0cl 26149 Degree of a nonzero univar...
deg1n0ima 26150 Degree image of a set of p...
deg1nn0clb 26151 A polynomial is nonzero if...
deg1lt0 26152 A polynomial is zero iff i...
deg1ldg 26153 A nonzero univariate polyn...
deg1ldgn 26154 An index at which a polyno...
deg1ldgdomn 26155 A nonzero univariate polyn...
deg1leb 26156 Property of being of limit...
deg1val 26157 Value of the univariate de...
deg1lt 26158 If the degree of a univari...
deg1ge 26159 Conversely, a nonzero coef...
coe1mul3 26160 The coefficient vector of ...
coe1mul4 26161 Value of the "leading" coe...
deg1addle 26162 The degree of a sum is at ...
deg1addle2 26163 If both factors have degre...
deg1add 26164 Exact degree of a sum of t...
deg1vscale 26165 The degree of a scalar tim...
deg1vsca 26166 The degree of a scalar tim...
deg1invg 26167 The degree of the negated ...
deg1suble 26168 The degree of a difference...
deg1sub 26169 Exact degree of a differen...
deg1mulle2 26170 Produce a bound on the pro...
deg1sublt 26171 Subtraction of two polynom...
deg1le0 26172 A polynomial has nonpositi...
deg1sclle 26173 A scalar polynomial has no...
deg1scl 26174 A nonzero scalar polynomia...
deg1mul2 26175 Degree of multiplication o...
deg1mul 26176 Degree of multiplication o...
deg1mul3 26177 Degree of multiplication o...
deg1mul3le 26178 Degree of multiplication o...
deg1tmle 26179 Limiting degree of a polyn...
deg1tm 26180 Exact degree of a polynomi...
deg1pwle 26181 Limiting degree of a varia...
deg1pw 26182 Exact degree of a variable...
ply1nz 26183 Univariate polynomials ove...
ply1nzb 26184 Univariate polynomials are...
ply1domn 26185 Corollary of ~ deg1mul2 : ...
ply1idom 26186 The ring of univariate pol...
ply1divmo 26197 Uniqueness of a quotient i...
ply1divex 26198 Lemma for ~ ply1divalg : e...
ply1divalg 26199 The division algorithm for...
ply1divalg2 26200 Reverse the order of multi...
uc1pval 26201 Value of the set of unitic...
isuc1p 26202 Being a unitic polynomial....
mon1pval 26203 Value of the set of monic ...
ismon1p 26204 Being a monic polynomial. ...
uc1pcl 26205 Unitic polynomials are pol...
mon1pcl 26206 Monic polynomials are poly...
uc1pn0 26207 Unitic polynomials are not...
mon1pn0 26208 Monic polynomials are not ...
uc1pdeg 26209 Unitic polynomials have no...
uc1pldg 26210 Unitic polynomials have un...
mon1pldg 26211 Unitic polynomials have on...
mon1puc1p 26212 Monic polynomials are unit...
uc1pmon1p 26213 Make a unitic polynomial m...
deg1submon1p 26214 The difference of two moni...
mon1pid 26215 Monicity and degree of the...
q1pval 26216 Value of the univariate po...
q1peqb 26217 Characterizing property of...
q1pcl 26218 Closure of the quotient by...
r1pval 26219 Value of the polynomial re...
r1pcl 26220 Closure of remainder follo...
r1pdeglt 26221 The remainder has a degree...
r1pid 26222 Express the original polyn...
r1pid2 26223 Identity law for polynomia...
dvdsq1p 26224 Divisibility in a polynomi...
dvdsr1p 26225 Divisibility in a polynomi...
ply1remlem 26226 A term of the form ` x - N...
ply1rem 26227 The polynomial remainder t...
facth1 26228 The factor theorem and its...
fta1glem1 26229 Lemma for ~ fta1g . (Cont...
fta1glem2 26230 Lemma for ~ fta1g . (Cont...
fta1g 26231 The one-sided fundamental ...
fta1blem 26232 Lemma for ~ fta1b . (Cont...
fta1b 26233 The assumption that ` R ` ...
idomrootle 26234 No element of an integral ...
drnguc1p 26235 Over a division ring, all ...
ig1peu 26236 There is a unique monic po...
ig1pval 26237 Substitutions for the poly...
ig1pval2 26238 Generator of the zero idea...
ig1pval3 26239 Characterizing properties ...
ig1pcl 26240 The monic generator of an ...
ig1pdvds 26241 The monic generator of an ...
ig1prsp 26242 Any ideal of polynomials o...
ply1lpir 26243 The ring of polynomials ov...
ply1pid 26244 The polynomials over a fie...
plyco0 26253 Two ways to say that a fun...
plyval 26254 Value of the polynomial se...
plybss 26255 Reverse closure of the par...
elply 26256 Definition of a polynomial...
elply2 26257 The coefficient function c...
plyun0 26258 The set of polynomials is ...
plyf 26259 A polynomial is a function...
plyss 26260 The polynomial set functio...
plyssc 26261 Every polynomial ring is c...
elplyr 26262 Sufficient condition for e...
elplyd 26263 Sufficient condition for e...
ply1termlem 26264 Lemma for ~ ply1term . (C...
ply1term 26265 A one-term polynomial. (C...
plypow 26266 A power is a polynomial. ...
plyconst 26267 A constant function is a p...
ne0p 26268 A test to show that a poly...
ply0 26269 The zero function is a pol...
plyid 26270 The identity function is a...
plyeq0lem 26271 Lemma for ~ plyeq0 . If `...
plyeq0 26272 If a polynomial is zero at...
plypf1 26273 Write the set of complex p...
plyaddlem1 26274 Derive the coefficient fun...
plymullem1 26275 Derive the coefficient fun...
plyaddlem 26276 Lemma for ~ plyadd . (Con...
plymullem 26277 Lemma for ~ plymul . (Con...
plyadd 26278 The sum of two polynomials...
plymul 26279 The product of two polynom...
plysub 26280 The difference of two poly...
plyaddcl 26281 The sum of two polynomials...
plymulcl 26282 The product of two polynom...
plysubcl 26283 The difference of two poly...
coeval 26284 Value of the coefficient f...
coeeulem 26285 Lemma for ~ coeeu . (Cont...
coeeu 26286 Uniqueness of the coeffici...
coelem 26287 Lemma for properties of th...
coeeq 26288 If ` A ` satisfies the pro...
dgrval 26289 Value of the degree functi...
dgrlem 26290 Lemma for ~ dgrcl and simi...
coef 26291 The domain and codomain of...
coef2 26292 The domain and codomain of...
coef3 26293 The domain and codomain of...
dgrcl 26294 The degree of any polynomi...
dgrub 26295 If the ` M ` -th coefficie...
dgrub2 26296 All the coefficients above...
dgrlb 26297 If all the coefficients ab...
coeidlem 26298 Lemma for ~ coeid . (Cont...
coeid 26299 Reconstruct a polynomial a...
coeid2 26300 Reconstruct a polynomial a...
coeid3 26301 Reconstruct a polynomial a...
plyco 26302 The composition of two pol...
coeeq2 26303 Compute the coefficient fu...
dgrle 26304 Given an explicit expressi...
dgreq 26305 If the highest term in a p...
0dgr 26306 A constant function has de...
0dgrb 26307 A function has degree zero...
dgrnznn 26308 A nonzero polynomial with ...
coefv0 26309 The result of evaluating a...
coeaddlem 26310 Lemma for ~ coeadd and ~ d...
coemullem 26311 Lemma for ~ coemul and ~ d...
coeadd 26312 The coefficient function o...
coemul 26313 A coefficient of a product...
coe11 26314 The coefficient function i...
coemulhi 26315 The leading coefficient of...
coemulc 26316 The coefficient function i...
coe0 26317 The coefficients of the ze...
coesub 26318 The coefficient function o...
coe1termlem 26319 The coefficient function o...
coe1term 26320 The coefficient function o...
dgr1term 26321 The degree of a monomial. ...
plycn 26322 A polynomial is a continuo...
plycnOLD 26323 Obsolete version of ~ plyc...
dgr0 26324 The degree of the zero pol...
coeidp 26325 The coefficients of the id...
dgrid 26326 The degree of the identity...
dgreq0 26327 The leading coefficient of...
dgrlt 26328 Two ways to say that the d...
dgradd 26329 The degree of a sum of pol...
dgradd2 26330 The degree of a sum of pol...
dgrmul2 26331 The degree of a product of...
dgrmul 26332 The degree of a product of...
dgrmulc 26333 Scalar multiplication by a...
dgrsub 26334 The degree of a difference...
dgrcolem1 26335 The degree of a compositio...
dgrcolem2 26336 Lemma for ~ dgrco . (Cont...
dgrco 26337 The degree of a compositio...
plycjlem 26338 Lemma for ~ plycj and ~ co...
plycj 26339 The double conjugation of ...
coecj 26340 Double conjugation of a po...
plyrecj 26341 A polynomial with real coe...
plymul0or 26342 Polynomial multiplication ...
ofmulrt 26343 The set of roots of a prod...
plyreres 26344 Real-coefficient polynomia...
dvply1 26345 Derivative of a polynomial...
dvply2g 26346 The derivative of a polyno...
dvply2gOLD 26347 Obsolete version of ~ dvpl...
dvply2 26348 The derivative of a polyno...
dvnply2 26349 Polynomials have polynomia...
dvnply 26350 Polynomials have polynomia...
plycpn 26351 Polynomials are smooth. (...
quotval 26354 Value of the quotient func...
plydivlem1 26355 Lemma for ~ plydivalg . (...
plydivlem2 26356 Lemma for ~ plydivalg . (...
plydivlem3 26357 Lemma for ~ plydivex . Ba...
plydivlem4 26358 Lemma for ~ plydivex . In...
plydivex 26359 Lemma for ~ plydivalg . (...
plydiveu 26360 Lemma for ~ plydivalg . (...
plydivalg 26361 The division algorithm on ...
quotlem 26362 Lemma for properties of th...
quotcl 26363 The quotient of two polyno...
quotcl2 26364 Closure of the quotient fu...
quotdgr 26365 Remainder property of the ...
plyremlem 26366 Closure of a linear factor...
plyrem 26367 The polynomial remainder t...
facth 26368 The factor theorem. If a ...
fta1lem 26369 Lemma for ~ fta1 . (Contr...
fta1 26370 The easy direction of the ...
quotcan 26371 Exact division with a mult...
vieta1lem1 26372 Lemma for ~ vieta1 . (Con...
vieta1lem2 26373 Lemma for ~ vieta1 : induc...
vieta1 26374 The first-order Vieta's fo...
plyexmo 26375 An infinite set of values ...
elaa 26378 Elementhood in the set of ...
aacn 26379 An algebraic number is a c...
aasscn 26380 The algebraic numbers are ...
elqaalem1 26381 Lemma for ~ elqaa . The f...
elqaalem2 26382 Lemma for ~ elqaa . (Cont...
elqaalem3 26383 Lemma for ~ elqaa . (Cont...
elqaa 26384 The set of numbers generat...
qaa 26385 Every rational number is a...
qssaa 26386 The rational numbers are c...
iaa 26387 The imaginary unit is alge...
aareccl 26388 The reciprocal of an algeb...
aacjcl 26389 The conjugate of an algebr...
aannenlem1 26390 Lemma for ~ aannen . (Con...
aannenlem2 26391 Lemma for ~ aannen . (Con...
aannenlem3 26392 The algebraic numbers are ...
aannen 26393 The algebraic numbers are ...
aalioulem1 26394 Lemma for ~ aaliou . An i...
aalioulem2 26395 Lemma for ~ aaliou . (Con...
aalioulem3 26396 Lemma for ~ aaliou . (Con...
aalioulem4 26397 Lemma for ~ aaliou . (Con...
aalioulem5 26398 Lemma for ~ aaliou . (Con...
aalioulem6 26399 Lemma for ~ aaliou . (Con...
aaliou 26400 Liouville's theorem on dio...
geolim3 26401 Geometric series convergen...
aaliou2 26402 Liouville's approximation ...
aaliou2b 26403 Liouville's approximation ...
aaliou3lem1 26404 Lemma for ~ aaliou3 . (Co...
aaliou3lem2 26405 Lemma for ~ aaliou3 . (Co...
aaliou3lem3 26406 Lemma for ~ aaliou3 . (Co...
aaliou3lem8 26407 Lemma for ~ aaliou3 . (Co...
aaliou3lem4 26408 Lemma for ~ aaliou3 . (Co...
aaliou3lem5 26409 Lemma for ~ aaliou3 . (Co...
aaliou3lem6 26410 Lemma for ~ aaliou3 . (Co...
aaliou3lem7 26411 Lemma for ~ aaliou3 . (Co...
aaliou3lem9 26412 Example of a "Liouville nu...
aaliou3 26413 Example of a "Liouville nu...
taylfvallem1 26418 Lemma for ~ taylfval . (C...
taylfvallem 26419 Lemma for ~ taylfval . (C...
taylfval 26420 Define the Taylor polynomi...
eltayl 26421 Value of the Taylor series...
taylf 26422 The Taylor series defines ...
tayl0 26423 The Taylor series is alway...
taylplem1 26424 Lemma for ~ taylpfval and ...
taylplem2 26425 Lemma for ~ taylpfval and ...
taylpfval 26426 Define the Taylor polynomi...
taylpf 26427 The Taylor polynomial is a...
taylpval 26428 Value of the Taylor polyno...
taylply2 26429 The Taylor polynomial is a...
taylply2OLD 26430 Obsolete version of ~ tayl...
taylply 26431 The Taylor polynomial is a...
dvtaylp 26432 The derivative of the Tayl...
dvntaylp 26433 The ` M ` -th derivative o...
dvntaylp0 26434 The first ` N ` derivative...
taylthlem1 26435 Lemma for ~ taylth . This...
taylthlem2 26436 Lemma for ~ taylth . (Con...
taylthlem2OLD 26437 Obsolete version of ~ tayl...
taylth 26438 Taylor's theorem. The Tay...
ulmrel 26441 The uniform limit relation...
ulmscl 26442 Closure of the base set in...
ulmval 26443 Express the predicate: Th...
ulmcl 26444 Closure of a uniform limit...
ulmf 26445 Closure of a uniform limit...
ulmpm 26446 Closure of a uniform limit...
ulmf2 26447 Closure of a uniform limit...
ulm2 26448 Simplify ~ ulmval when ` F...
ulmi 26449 The uniform limit property...
ulmclm 26450 A uniform limit of functio...
ulmres 26451 A sequence of functions co...
ulmshftlem 26452 Lemma for ~ ulmshft . (Co...
ulmshft 26453 A sequence of functions co...
ulm0 26454 Every function converges u...
ulmuni 26455 A sequence of functions un...
ulmdm 26456 Two ways to express that a...
ulmcaulem 26457 Lemma for ~ ulmcau and ~ u...
ulmcau 26458 A sequence of functions co...
ulmcau2 26459 A sequence of functions co...
ulmss 26460 A uniform limit of functio...
ulmbdd 26461 A uniform limit of bounded...
ulmcn 26462 A uniform limit of continu...
ulmdvlem1 26463 Lemma for ~ ulmdv . (Cont...
ulmdvlem2 26464 Lemma for ~ ulmdv . (Cont...
ulmdvlem3 26465 Lemma for ~ ulmdv . (Cont...
ulmdv 26466 If ` F ` is a sequence of ...
mtest 26467 The Weierstrass M-test. I...
mtestbdd 26468 Given the hypotheses of th...
mbfulm 26469 A uniform limit of measura...
iblulm 26470 A uniform limit of integra...
itgulm 26471 A uniform limit of integra...
itgulm2 26472 A uniform limit of integra...
pserval 26473 Value of the function ` G ...
pserval2 26474 Value of the function ` G ...
psergf 26475 The sequence of terms in t...
radcnvlem1 26476 Lemma for ~ radcnvlt1 , ~ ...
radcnvlem2 26477 Lemma for ~ radcnvlt1 , ~ ...
radcnvlem3 26478 Lemma for ~ radcnvlt1 , ~ ...
radcnv0 26479 Zero is always a convergen...
radcnvcl 26480 The radius of convergence ...
radcnvlt1 26481 If ` X ` is within the ope...
radcnvlt2 26482 If ` X ` is within the ope...
radcnvle 26483 If ` X ` is a convergent p...
dvradcnv 26484 The radius of convergence ...
pserulm 26485 If ` S ` is a region conta...
psercn2 26486 Since by ~ pserulm the ser...
psercn2OLD 26487 Obsolete version of ~ pser...
psercnlem2 26488 Lemma for ~ psercn . (Con...
psercnlem1 26489 Lemma for ~ psercn . (Con...
psercn 26490 An infinite series converg...
pserdvlem1 26491 Lemma for ~ pserdv . (Con...
pserdvlem2 26492 Lemma for ~ pserdv . (Con...
pserdv 26493 The derivative of a power ...
pserdv2 26494 The derivative of a power ...
abelthlem1 26495 Lemma for ~ abelth . (Con...
abelthlem2 26496 Lemma for ~ abelth . The ...
abelthlem3 26497 Lemma for ~ abelth . (Con...
abelthlem4 26498 Lemma for ~ abelth . (Con...
abelthlem5 26499 Lemma for ~ abelth . (Con...
abelthlem6 26500 Lemma for ~ abelth . (Con...
abelthlem7a 26501 Lemma for ~ abelth . (Con...
abelthlem7 26502 Lemma for ~ abelth . (Con...
abelthlem8 26503 Lemma for ~ abelth . (Con...
abelthlem9 26504 Lemma for ~ abelth . By a...
abelth 26505 Abel's theorem. If the po...
abelth2 26506 Abel's theorem, restricted...
efcn 26507 The exponential function i...
sincn 26508 Sine is continuous. (Cont...
coscn 26509 Cosine is continuous. (Co...
reeff1olem 26510 Lemma for ~ reeff1o . (Co...
reeff1o 26511 The real exponential funct...
reefiso 26512 The exponential function o...
efcvx 26513 The exponential function o...
reefgim 26514 The exponential function i...
pilem1 26515 Lemma for ~ pire , ~ pigt2...
pilem2 26516 Lemma for ~ pire , ~ pigt2...
pilem3 26517 Lemma for ~ pire , ~ pigt2...
pigt2lt4 26518 ` _pi ` is between 2 and 4...
sinpi 26519 The sine of ` _pi ` is 0. ...
pire 26520 ` _pi ` is a real number. ...
picn 26521 ` _pi ` is a complex numbe...
pipos 26522 ` _pi ` is positive. (Con...
pirp 26523 ` _pi ` is a positive real...
negpicn 26524 ` -u _pi ` is a real numbe...
sinhalfpilem 26525 Lemma for ~ sinhalfpi and ...
halfpire 26526 ` _pi / 2 ` is real. (Con...
neghalfpire 26527 ` -u _pi / 2 ` is real. (...
neghalfpirx 26528 ` -u _pi / 2 ` is an exten...
pidiv2halves 26529 Adding ` _pi / 2 ` to itse...
sinhalfpi 26530 The sine of ` _pi / 2 ` is...
coshalfpi 26531 The cosine of ` _pi / 2 ` ...
cosneghalfpi 26532 The cosine of ` -u _pi / 2...
efhalfpi 26533 The exponential of ` _i _p...
cospi 26534 The cosine of ` _pi ` is `...
efipi 26535 The exponential of ` _i x....
eulerid 26536 Euler's identity. (Contri...
sin2pi 26537 The sine of ` 2 _pi ` is 0...
cos2pi 26538 The cosine of ` 2 _pi ` is...
ef2pi 26539 The exponential of ` 2 _pi...
ef2kpi 26540 If ` K ` is an integer, th...
efper 26541 The exponential function i...
sinperlem 26542 Lemma for ~ sinper and ~ c...
sinper 26543 The sine function is perio...
cosper 26544 The cosine function is per...
sin2kpi 26545 If ` K ` is an integer, th...
cos2kpi 26546 If ` K ` is an integer, th...
sin2pim 26547 Sine of a number subtracte...
cos2pim 26548 Cosine of a number subtrac...
sinmpi 26549 Sine of a number less ` _p...
cosmpi 26550 Cosine of a number less ` ...
sinppi 26551 Sine of a number plus ` _p...
cosppi 26552 Cosine of a number plus ` ...
efimpi 26553 The exponential function a...
sinhalfpip 26554 The sine of ` _pi / 2 ` pl...
sinhalfpim 26555 The sine of ` _pi / 2 ` mi...
coshalfpip 26556 The cosine of ` _pi / 2 ` ...
coshalfpim 26557 The cosine of ` _pi / 2 ` ...
ptolemy 26558 Ptolemy's Theorem. This t...
sincosq1lem 26559 Lemma for ~ sincosq1sgn . ...
sincosq1sgn 26560 The signs of the sine and ...
sincosq2sgn 26561 The signs of the sine and ...
sincosq3sgn 26562 The signs of the sine and ...
sincosq4sgn 26563 The signs of the sine and ...
coseq00topi 26564 Location of the zeroes of ...
coseq0negpitopi 26565 Location of the zeroes of ...
tanrpcl 26566 Positive real closure of t...
tangtx 26567 The tangent function is gr...
tanabsge 26568 The tangent function is gr...
sinq12gt0 26569 The sine of a number stric...
sinq12ge0 26570 The sine of a number betwe...
sinq34lt0t 26571 The sine of a number stric...
cosq14gt0 26572 The cosine of a number str...
cosq14ge0 26573 The cosine of a number bet...
sincosq1eq 26574 Complementarity of the sin...
sincos4thpi 26575 The sine and cosine of ` _...
tan4thpi 26576 The tangent of ` _pi / 4 `...
tan4thpiOLD 26577 Obsolete version of ~ tan4...
sincos6thpi 26578 The sine and cosine of ` _...
sincos3rdpi 26579 The sine and cosine of ` _...
pigt3 26580 ` _pi ` is greater than 3....
pige3 26581 ` _pi ` is greater than or...
pige3ALT 26582 Alternate proof of ~ pige3...
abssinper 26583 The absolute value of sine...
sinkpi 26584 The sine of an integer mul...
coskpi 26585 The absolute value of the ...
sineq0 26586 A complex number whose sin...
coseq1 26587 A complex number whose cos...
cos02pilt1 26588 Cosine is less than one be...
cosq34lt1 26589 Cosine is less than one in...
efeq1 26590 A complex number whose exp...
cosne0 26591 The cosine function has no...
cosordlem 26592 Lemma for ~ cosord . (Con...
cosord 26593 Cosine is decreasing over ...
cos0pilt1 26594 Cosine is between minus on...
cos11 26595 Cosine is one-to-one over ...
sinord 26596 Sine is increasing over th...
recosf1o 26597 The cosine function is a b...
resinf1o 26598 The sine function is a bij...
tanord1 26599 The tangent function is st...
tanord 26600 The tangent function is st...
tanregt0 26601 The real part of the tange...
negpitopissre 26602 The interval ` ( -u _pi (,...
efgh 26603 The exponential function o...
efif1olem1 26604 Lemma for ~ efif1o . (Con...
efif1olem2 26605 Lemma for ~ efif1o . (Con...
efif1olem3 26606 Lemma for ~ efif1o . (Con...
efif1olem4 26607 The exponential function o...
efif1o 26608 The exponential function o...
efifo 26609 The exponential function o...
eff1olem 26610 The exponential function m...
eff1o 26611 The exponential function m...
efabl 26612 The image of a subgroup of...
efsubm 26613 The image of a subgroup of...
circgrp 26614 The circle group ` T ` is ...
circsubm 26615 The circle group ` T ` is ...
logrn 26620 The range of the natural l...
ellogrn 26621 Write out the property ` A...
dflog2 26622 The natural logarithm func...
relogrn 26623 The range of the natural l...
logrncn 26624 The range of the natural l...
eff1o2 26625 The exponential function r...
logf1o 26626 The natural logarithm func...
dfrelog 26627 The natural logarithm func...
relogf1o 26628 The natural logarithm func...
logrncl 26629 Closure of the natural log...
logcl 26630 Closure of the natural log...
logimcl 26631 Closure of the imaginary p...
logcld 26632 The logarithm of a nonzero...
logimcld 26633 The imaginary part of the ...
logimclad 26634 The imaginary part of the ...
abslogimle 26635 The imaginary part of the ...
logrnaddcl 26636 The range of the natural l...
relogcl 26637 Closure of the natural log...
eflog 26638 Relationship between the n...
logeq0im1 26639 If the logarithm of a numb...
logccne0 26640 The logarithm isn't 0 if i...
logne0 26641 Logarithm of a non-1 posit...
reeflog 26642 Relationship between the n...
logef 26643 Relationship between the n...
relogef 26644 Relationship between the n...
logeftb 26645 Relationship between the n...
relogeftb 26646 Relationship between the n...
log1 26647 The natural logarithm of `...
loge 26648 The natural logarithm of `...
logi 26649 The natural logarithm of `...
logneg 26650 The natural logarithm of a...
logm1 26651 The natural logarithm of n...
lognegb 26652 If a number has imaginary ...
relogoprlem 26653 Lemma for ~ relogmul and ~...
relogmul 26654 The natural logarithm of t...
relogdiv 26655 The natural logarithm of t...
explog 26656 Exponentiation of a nonzer...
reexplog 26657 Exponentiation of a positi...
relogexp 26658 The natural logarithm of p...
relog 26659 Real part of a logarithm. ...
relogiso 26660 The natural logarithm func...
reloggim 26661 The natural logarithm is a...
logltb 26662 The natural logarithm func...
logfac 26663 The logarithm of a factori...
eflogeq 26664 Solve an equation involvin...
logleb 26665 Natural logarithm preserve...
rplogcl 26666 Closure of the logarithm f...
logge0 26667 The logarithm of a number ...
logcj 26668 The natural logarithm dist...
efiarg 26669 The exponential of the "ar...
cosargd 26670 The cosine of the argument...
cosarg0d 26671 The cosine of the argument...
argregt0 26672 Closure of the argument of...
argrege0 26673 Closure of the argument of...
argimgt0 26674 Closure of the argument of...
argimlt0 26675 Closure of the argument of...
logimul 26676 Multiplying a number by ` ...
logneg2 26677 The logarithm of the negat...
logmul2 26678 Generalization of ~ relogm...
logdiv2 26679 Generalization of ~ relogd...
abslogle 26680 Bound on the magnitude of ...
tanarg 26681 The basic relation between...
logdivlti 26682 The ` log x / x ` function...
logdivlt 26683 The ` log x / x ` function...
logdivle 26684 The ` log x / x ` function...
relogcld 26685 Closure of the natural log...
reeflogd 26686 Relationship between the n...
relogmuld 26687 The natural logarithm of t...
relogdivd 26688 The natural logarithm of t...
logled 26689 Natural logarithm preserve...
relogefd 26690 Relationship between the n...
rplogcld 26691 Closure of the logarithm f...
logge0d 26692 The logarithm of a number ...
logge0b 26693 The logarithm of a number ...
loggt0b 26694 The logarithm of a number ...
logle1b 26695 The logarithm of a number ...
loglt1b 26696 The logarithm of a number ...
divlogrlim 26697 The inverse logarithm func...
logno1 26698 The logarithm function is ...
dvrelog 26699 The derivative of the real...
relogcn 26700 The real logarithm functio...
ellogdm 26701 Elementhood in the "contin...
logdmn0 26702 A number in the continuous...
logdmnrp 26703 A number in the continuous...
logdmss 26704 The continuity domain of `...
logcnlem2 26705 Lemma for ~ logcn . (Cont...
logcnlem3 26706 Lemma for ~ logcn . (Cont...
logcnlem4 26707 Lemma for ~ logcn . (Cont...
logcnlem5 26708 Lemma for ~ logcn . (Cont...
logcn 26709 The logarithm function is ...
dvloglem 26710 Lemma for ~ dvlog . (Cont...
logdmopn 26711 The "continuous domain" of...
logf1o2 26712 The logarithm maps its con...
dvlog 26713 The derivative of the comp...
dvlog2lem 26714 Lemma for ~ dvlog2 . (Con...
dvlog2 26715 The derivative of the comp...
advlog 26716 The antiderivative of the ...
advlogexp 26717 The antiderivative of a po...
efopnlem1 26718 Lemma for ~ efopn . (Cont...
efopnlem2 26719 Lemma for ~ efopn . (Cont...
efopn 26720 The exponential map is an ...
logtayllem 26721 Lemma for ~ logtayl . (Co...
logtayl 26722 The Taylor series for ` -u...
logtaylsum 26723 The Taylor series for ` -u...
logtayl2 26724 Power series expression fo...
logccv 26725 The natural logarithm func...
cxpval 26726 Value of the complex power...
cxpef 26727 Value of the complex power...
0cxp 26728 Value of the complex power...
cxpexpz 26729 Relate the complex power f...
cxpexp 26730 Relate the complex power f...
logcxp 26731 Logarithm of a complex pow...
cxp0 26732 Value of the complex power...
cxp1 26733 Value of the complex power...
1cxp 26734 Value of the complex power...
ecxp 26735 Write the exponential func...
cxpcl 26736 Closure of the complex pow...
recxpcl 26737 Real closure of the comple...
rpcxpcl 26738 Positive real closure of t...
cxpne0 26739 Complex exponentiation is ...
cxpeq0 26740 Complex exponentiation is ...
cxpadd 26741 Sum of exponents law for c...
cxpp1 26742 Value of a nonzero complex...
cxpneg 26743 Value of a complex number ...
cxpsub 26744 Exponent subtraction law f...
cxpge0 26745 Nonnegative exponentiation...
mulcxplem 26746 Lemma for ~ mulcxp . (Con...
mulcxp 26747 Complex exponentiation of ...
cxprec 26748 Complex exponentiation of ...
divcxp 26749 Complex exponentiation of ...
cxpmul 26750 Product of exponents law f...
cxpmul2 26751 Product of exponents law f...
cxproot 26752 The complex power function...
cxpmul2z 26753 Generalize ~ cxpmul2 to ne...
abscxp 26754 Absolute value of a power,...
abscxp2 26755 Absolute value of a power,...
cxplt 26756 Ordering property for comp...
cxple 26757 Ordering property for comp...
cxplea 26758 Ordering property for comp...
cxple2 26759 Ordering property for comp...
cxplt2 26760 Ordering property for comp...
cxple2a 26761 Ordering property for comp...
cxplt3 26762 Ordering property for comp...
cxple3 26763 Ordering property for comp...
cxpsqrtlem 26764 Lemma for ~ cxpsqrt . (Co...
cxpsqrt 26765 The complex exponential fu...
logsqrt 26766 Logarithm of a square root...
cxp0d 26767 Value of the complex power...
cxp1d 26768 Value of the complex power...
1cxpd 26769 Value of the complex power...
cxpcld 26770 Closure of the complex pow...
cxpmul2d 26771 Product of exponents law f...
0cxpd 26772 Value of the complex power...
cxpexpzd 26773 Relate the complex power f...
cxpefd 26774 Value of the complex power...
cxpne0d 26775 Complex exponentiation is ...
cxpp1d 26776 Value of a nonzero complex...
cxpnegd 26777 Value of a complex number ...
cxpmul2zd 26778 Generalize ~ cxpmul2 to ne...
cxpaddd 26779 Sum of exponents law for c...
cxpsubd 26780 Exponent subtraction law f...
cxpltd 26781 Ordering property for comp...
cxpled 26782 Ordering property for comp...
cxplead 26783 Ordering property for comp...
divcxpd 26784 Complex exponentiation of ...
recxpcld 26785 Positive real closure of t...
cxpge0d 26786 Nonnegative exponentiation...
cxple2ad 26787 Ordering property for comp...
cxplt2d 26788 Ordering property for comp...
cxple2d 26789 Ordering property for comp...
mulcxpd 26790 Complex exponentiation of ...
recxpf1lem 26791 Complex exponentiation on ...
cxpsqrtth 26792 Square root theorem over t...
2irrexpq 26793 There exist irrational num...
cxprecd 26794 Complex exponentiation of ...
rpcxpcld 26795 Positive real closure of t...
logcxpd 26796 Logarithm of a complex pow...
cxplt3d 26797 Ordering property for comp...
cxple3d 26798 Ordering property for comp...
cxpmuld 26799 Product of exponents law f...
cxpgt0d 26800 A positive real raised to ...
cxpcom 26801 Commutative law for real e...
dvcxp1 26802 The derivative of a comple...
dvcxp2 26803 The derivative of a comple...
dvsqrt 26804 The derivative of the real...
dvcncxp1 26805 Derivative of complex powe...
dvcnsqrt 26806 Derivative of square root ...
cxpcn 26807 Domain of continuity of th...
cxpcnOLD 26808 Obsolete version of ~ cxpc...
cxpcn2 26809 Continuity of the complex ...
cxpcn3lem 26810 Lemma for ~ cxpcn3 . (Con...
cxpcn3 26811 Extend continuity of the c...
resqrtcn 26812 Continuity of the real squ...
sqrtcn 26813 Continuity of the square r...
cxpaddlelem 26814 Lemma for ~ cxpaddle . (C...
cxpaddle 26815 Ordering property for comp...
abscxpbnd 26816 Bound on the absolute valu...
root1id 26817 Property of an ` N ` -th r...
root1eq1 26818 The only powers of an ` N ...
root1cj 26819 Within the ` N ` -th roots...
cxpeq 26820 Solve an equation involvin...
zrtelqelz 26821 If the ` N ` -th root of a...
zrtdvds 26822 A positive integer root di...
rtprmirr 26823 The root of a prime number...
loglesqrt 26824 An upper bound on the loga...
logreclem 26825 Symmetry of the natural lo...
logrec 26826 Logarithm of a reciprocal ...
logbval 26829 Define the value of the ` ...
logbcl 26830 General logarithm closure....
logbid1 26831 General logarithm is 1 whe...
logb1 26832 The logarithm of ` 1 ` to ...
elogb 26833 The general logarithm of a...
logbchbase 26834 Change of base for logarit...
relogbval 26835 Value of the general logar...
relogbcl 26836 Closure of the general log...
relogbzcl 26837 Closure of the general log...
relogbreexp 26838 Power law for the general ...
relogbzexp 26839 Power law for the general ...
relogbmul 26840 The logarithm of the produ...
relogbmulexp 26841 The logarithm of the produ...
relogbdiv 26842 The logarithm of the quoti...
relogbexp 26843 Identity law for general l...
nnlogbexp 26844 Identity law for general l...
logbrec 26845 Logarithm of a reciprocal ...
logbleb 26846 The general logarithm func...
logblt 26847 The general logarithm func...
relogbcxp 26848 Identity law for the gener...
cxplogb 26849 Identity law for the gener...
relogbcxpb 26850 The logarithm is the inver...
logbmpt 26851 The general logarithm to a...
logbf 26852 The general logarithm to a...
logbfval 26853 The general logarithm of a...
relogbf 26854 The general logarithm to a...
logblog 26855 The general logarithm to t...
logbgt0b 26856 The logarithm of a positiv...
logbgcd1irr 26857 The logarithm of an intege...
2logb9irr 26858 Example for ~ logbgcd1irr ...
logbprmirr 26859 The logarithm of a prime t...
2logb3irr 26860 Example for ~ logbprmirr ....
2logb9irrALT 26861 Alternate proof of ~ 2logb...
sqrt2cxp2logb9e3 26862 The square root of two to ...
2irrexpqALT 26863 Alternate proof of ~ 2irre...
angval 26864 Define the angle function,...
angcan 26865 Cancel a constant multipli...
angneg 26866 Cancel a negative sign in ...
angvald 26867 The (signed) angle between...
angcld 26868 The (signed) angle between...
angrteqvd 26869 Two vectors are at a right...
cosangneg2d 26870 The cosine of the angle be...
angrtmuld 26871 Perpendicularity of two ve...
ang180lem1 26872 Lemma for ~ ang180 . Show...
ang180lem2 26873 Lemma for ~ ang180 . Show...
ang180lem3 26874 Lemma for ~ ang180 . Sinc...
ang180lem4 26875 Lemma for ~ ang180 . Redu...
ang180lem5 26876 Lemma for ~ ang180 : Redu...
ang180 26877 The sum of angles ` m A B ...
lawcoslem1 26878 Lemma for ~ lawcos . Here...
lawcos 26879 Law of cosines (also known...
pythag 26880 Pythagorean theorem. Give...
isosctrlem1 26881 Lemma for ~ isosctr . (Co...
isosctrlem2 26882 Lemma for ~ isosctr . Cor...
isosctrlem3 26883 Lemma for ~ isosctr . Cor...
isosctr 26884 Isosceles triangle theorem...
ssscongptld 26885 If two triangles have equa...
affineequiv 26886 Equivalence between two wa...
affineequiv2 26887 Equivalence between two wa...
affineequiv3 26888 Equivalence between two wa...
affineequiv4 26889 Equivalence between two wa...
affineequivne 26890 Equivalence between two wa...
angpieqvdlem 26891 Equivalence used in the pr...
angpieqvdlem2 26892 Equivalence used in ~ angp...
angpined 26893 If the angle at ABC is ` _...
angpieqvd 26894 The angle ABC is ` _pi ` i...
chordthmlem 26895 If ` M ` is the midpoint o...
chordthmlem2 26896 If M is the midpoint of AB...
chordthmlem3 26897 If M is the midpoint of AB...
chordthmlem4 26898 If P is on the segment AB ...
chordthmlem5 26899 If P is on the segment AB ...
chordthm 26900 The intersecting chords th...
heron 26901 Heron's formula gives the ...
quad2 26902 The quadratic equation, wi...
quad 26903 The quadratic equation. (...
1cubrlem 26904 The cube roots of unity. ...
1cubr 26905 The cube roots of unity. ...
dcubic1lem 26906 Lemma for ~ dcubic1 and ~ ...
dcubic2 26907 Reverse direction of ~ dcu...
dcubic1 26908 Forward direction of ~ dcu...
dcubic 26909 Solutions to the depressed...
mcubic 26910 Solutions to a monic cubic...
cubic2 26911 The solution to the genera...
cubic 26912 The cubic equation, which ...
binom4 26913 Work out a quartic binomia...
dquartlem1 26914 Lemma for ~ dquart . (Con...
dquartlem2 26915 Lemma for ~ dquart . (Con...
dquart 26916 Solve a depressed quartic ...
quart1cl 26917 Closure lemmas for ~ quart...
quart1lem 26918 Lemma for ~ quart1 . (Con...
quart1 26919 Depress a quartic equation...
quartlem1 26920 Lemma for ~ quart . (Cont...
quartlem2 26921 Closure lemmas for ~ quart...
quartlem3 26922 Closure lemmas for ~ quart...
quartlem4 26923 Closure lemmas for ~ quart...
quart 26924 The quartic equation, writ...
asinlem 26931 The argument to the logari...
asinlem2 26932 The argument to the logari...
asinlem3a 26933 Lemma for ~ asinlem3 . (C...
asinlem3 26934 The argument to the logari...
asinf 26935 Domain and codomain of the...
asincl 26936 Closure for the arcsin fun...
acosf 26937 Domain and codoamin of the...
acoscl 26938 Closure for the arccos fun...
atandm 26939 Since the property is a li...
atandm2 26940 This form of ~ atandm is a...
atandm3 26941 A compact form of ~ atandm...
atandm4 26942 A compact form of ~ atandm...
atanf 26943 Domain and codoamin of the...
atancl 26944 Closure for the arctan fun...
asinval 26945 Value of the arcsin functi...
acosval 26946 Value of the arccos functi...
atanval 26947 Value of the arctan functi...
atanre 26948 A real number is in the do...
asinneg 26949 The arcsine function is od...
acosneg 26950 The negative symmetry rela...
efiasin 26951 The exponential of the arc...
sinasin 26952 The arcsine function is an...
cosacos 26953 The arccosine function is ...
asinsinlem 26954 Lemma for ~ asinsin . (Co...
asinsin 26955 The arcsine function compo...
acoscos 26956 The arccosine function is ...
asin1 26957 The arcsine of ` 1 ` is ` ...
acos1 26958 The arccosine of ` 1 ` is ...
reasinsin 26959 The arcsine function compo...
asinsinb 26960 Relationship between sine ...
acoscosb 26961 Relationship between cosin...
asinbnd 26962 The arcsine function has r...
acosbnd 26963 The arccosine function has...
asinrebnd 26964 Bounds on the arcsine func...
asinrecl 26965 The arcsine function is re...
acosrecl 26966 The arccosine function is ...
cosasin 26967 The cosine of the arcsine ...
sinacos 26968 The sine of the arccosine ...
atandmneg 26969 The domain of the arctange...
atanneg 26970 The arctangent function is...
atan0 26971 The arctangent of zero is ...
atandmcj 26972 The arctangent function di...
atancj 26973 The arctangent function di...
atanrecl 26974 The arctangent function is...
efiatan 26975 Value of the exponential o...
atanlogaddlem 26976 Lemma for ~ atanlogadd . ...
atanlogadd 26977 The rule ` sqrt ( z w ) = ...
atanlogsublem 26978 Lemma for ~ atanlogsub . ...
atanlogsub 26979 A variation on ~ atanlogad...
efiatan2 26980 Value of the exponential o...
2efiatan 26981 Value of the exponential o...
tanatan 26982 The arctangent function is...
atandmtan 26983 The tangent function has r...
cosatan 26984 The cosine of an arctangen...
cosatanne0 26985 The arctangent function ha...
atantan 26986 The arctangent function is...
atantanb 26987 Relationship between tange...
atanbndlem 26988 Lemma for ~ atanbnd . (Co...
atanbnd 26989 The arctangent function is...
atanord 26990 The arctangent function is...
atan1 26991 The arctangent of ` 1 ` is...
bndatandm 26992 A point in the open unit d...
atans 26993 The "domain of continuity"...
atans2 26994 It suffices to show that `...
atansopn 26995 The domain of continuity o...
atansssdm 26996 The domain of continuity o...
ressatans 26997 The real number line is a ...
dvatan 26998 The derivative of the arct...
atancn 26999 The arctangent is a contin...
atantayl 27000 The Taylor series for ` ar...
atantayl2 27001 The Taylor series for ` ar...
atantayl3 27002 The Taylor series for ` ar...
leibpilem1 27003 Lemma for ~ leibpi . (Con...
leibpilem2 27004 The Leibniz formula for ` ...
leibpi 27005 The Leibniz formula for ` ...
leibpisum 27006 The Leibniz formula for ` ...
log2cnv 27007 Using the Taylor series fo...
log2tlbnd 27008 Bound the error term in th...
log2ublem1 27009 Lemma for ~ log2ub . The ...
log2ublem2 27010 Lemma for ~ log2ub . (Con...
log2ublem3 27011 Lemma for ~ log2ub . In d...
log2ub 27012 ` log 2 ` is less than ` 2...
log2le1 27013 ` log 2 ` is less than ` 1...
birthdaylem1 27014 Lemma for ~ birthday . (C...
birthdaylem2 27015 For general ` N ` and ` K ...
birthdaylem3 27016 For general ` N ` and ` K ...
birthday 27017 The Birthday Problem. The...
dmarea 27020 The domain of the area fun...
areambl 27021 The fibers of a measurable...
areass 27022 A measurable region is a s...
dfarea 27023 Rewrite ~ df-area self-ref...
areaf 27024 Area measurement is a func...
areacl 27025 The area of a measurable r...
areage0 27026 The area of a measurable r...
areaval 27027 The area of a measurable r...
rlimcnp 27028 Relate a limit of a real-v...
rlimcnp2 27029 Relate a limit of a real-v...
rlimcnp3 27030 Relate a limit of a real-v...
xrlimcnp 27031 Relate a limit of a real-v...
efrlim 27032 The limit of the sequence ...
efrlimOLD 27033 Obsolete version of ~ efrl...
dfef2 27034 The limit of the sequence ...
cxplim 27035 A power to a negative expo...
sqrtlim 27036 The inverse square root fu...
rlimcxp 27037 Any power to a positive ex...
o1cxp 27038 An eventually bounded func...
cxp2limlem 27039 A linear factor grows slow...
cxp2lim 27040 Any power grows slower tha...
cxploglim 27041 The logarithm grows slower...
cxploglim2 27042 Every power of the logarit...
divsqrtsumlem 27043 Lemma for ~ divsqrsum and ...
divsqrsumf 27044 The function ` F ` used in...
divsqrsum 27045 The sum ` sum_ n <_ x ( 1 ...
divsqrtsum2 27046 A bound on the distance of...
divsqrtsumo1 27047 The sum ` sum_ n <_ x ( 1 ...
cvxcl 27048 Closure of a 0-1 linear co...
scvxcvx 27049 A strictly convex function...
jensenlem1 27050 Lemma for ~ jensen . (Con...
jensenlem2 27051 Lemma for ~ jensen . (Con...
jensen 27052 Jensen's inequality, a fin...
amgmlem 27053 Lemma for ~ amgm . (Contr...
amgm 27054 Inequality of arithmetic a...
logdifbnd 27057 Bound on the difference of...
logdiflbnd 27058 Lower bound on the differe...
emcllem1 27059 Lemma for ~ emcl . The se...
emcllem2 27060 Lemma for ~ emcl . ` F ` i...
emcllem3 27061 Lemma for ~ emcl . The fu...
emcllem4 27062 Lemma for ~ emcl . The di...
emcllem5 27063 Lemma for ~ emcl . The pa...
emcllem6 27064 Lemma for ~ emcl . By the...
emcllem7 27065 Lemma for ~ emcl and ~ har...
emcl 27066 Closure and bounds for the...
harmonicbnd 27067 A bound on the harmonic se...
harmonicbnd2 27068 A bound on the harmonic se...
emre 27069 The Euler-Mascheroni const...
emgt0 27070 The Euler-Mascheroni const...
harmonicbnd3 27071 A bound on the harmonic se...
harmoniclbnd 27072 A bound on the harmonic se...
harmonicubnd 27073 A bound on the harmonic se...
harmonicbnd4 27074 The asymptotic behavior of...
fsumharmonic 27075 Bound a finite sum based o...
zetacvg 27078 The zeta series is converg...
eldmgm 27085 Elementhood in the set of ...
dmgmaddn0 27086 If ` A ` is not a nonposit...
dmlogdmgm 27087 If ` A ` is in the continu...
rpdmgm 27088 A positive real number is ...
dmgmn0 27089 If ` A ` is not a nonposit...
dmgmaddnn0 27090 If ` A ` is not a nonposit...
dmgmdivn0 27091 Lemma for ~ lgamf . (Cont...
lgamgulmlem1 27092 Lemma for ~ lgamgulm . (C...
lgamgulmlem2 27093 Lemma for ~ lgamgulm . (C...
lgamgulmlem3 27094 Lemma for ~ lgamgulm . (C...
lgamgulmlem4 27095 Lemma for ~ lgamgulm . (C...
lgamgulmlem5 27096 Lemma for ~ lgamgulm . (C...
lgamgulmlem6 27097 The series ` G ` is unifor...
lgamgulm 27098 The series ` G ` is unifor...
lgamgulm2 27099 Rewrite the limit of the s...
lgambdd 27100 The log-Gamma function is ...
lgamucov 27101 The ` U ` regions used in ...
lgamucov2 27102 The ` U ` regions used in ...
lgamcvglem 27103 Lemma for ~ lgamf and ~ lg...
lgamcl 27104 The log-Gamma function is ...
lgamf 27105 The log-Gamma function is ...
gamf 27106 The Gamma function is a co...
gamcl 27107 The exponential of the log...
eflgam 27108 The exponential of the log...
gamne0 27109 The Gamma function is neve...
igamval 27110 Value of the inverse Gamma...
igamz 27111 Value of the inverse Gamma...
igamgam 27112 Value of the inverse Gamma...
igamlgam 27113 Value of the inverse Gamma...
igamf 27114 Closure of the inverse Gam...
igamcl 27115 Closure of the inverse Gam...
gamigam 27116 The Gamma function is the ...
lgamcvg 27117 The series ` G ` converges...
lgamcvg2 27118 The series ` G ` converges...
gamcvg 27119 The pointwise exponential ...
lgamp1 27120 The functional equation of...
gamp1 27121 The functional equation of...
gamcvg2lem 27122 Lemma for ~ gamcvg2 . (Co...
gamcvg2 27123 An infinite product expres...
regamcl 27124 The Gamma function is real...
relgamcl 27125 The log-Gamma function is ...
rpgamcl 27126 The log-Gamma function is ...
lgam1 27127 The log-Gamma function at ...
gam1 27128 The log-Gamma function at ...
facgam 27129 The Gamma function general...
gamfac 27130 The Gamma function general...
wilthlem1 27131 The only elements that are...
wilthlem2 27132 Lemma for ~ wilth : induct...
wilthlem3 27133 Lemma for ~ wilth . Here ...
wilth 27134 Wilson's theorem. A numbe...
wilthimp 27135 The forward implication of...
ftalem1 27136 Lemma for ~ fta : "growth...
ftalem2 27137 Lemma for ~ fta . There e...
ftalem3 27138 Lemma for ~ fta . There e...
ftalem4 27139 Lemma for ~ fta : Closure...
ftalem5 27140 Lemma for ~ fta : Main pr...
ftalem6 27141 Lemma for ~ fta : Dischar...
ftalem7 27142 Lemma for ~ fta . Shift t...
fta 27143 The Fundamental Theorem of...
basellem1 27144 Lemma for ~ basel . Closu...
basellem2 27145 Lemma for ~ basel . Show ...
basellem3 27146 Lemma for ~ basel . Using...
basellem4 27147 Lemma for ~ basel . By ~ ...
basellem5 27148 Lemma for ~ basel . Using...
basellem6 27149 Lemma for ~ basel . The f...
basellem7 27150 Lemma for ~ basel . The f...
basellem8 27151 Lemma for ~ basel . The f...
basellem9 27152 Lemma for ~ basel . Since...
basel 27153 The sum of the inverse squ...
efnnfsumcl 27166 Finite sum closure in the ...
ppisval 27167 The set of primes less tha...
ppisval2 27168 The set of primes less tha...
ppifi 27169 The set of primes less tha...
prmdvdsfi 27170 The set of prime divisors ...
chtf 27171 Domain and codoamin of the...
chtcl 27172 Real closure of the Chebys...
chtval 27173 Value of the Chebyshev fun...
efchtcl 27174 The Chebyshev function is ...
chtge0 27175 The Chebyshev function is ...
vmaval 27176 Value of the von Mangoldt ...
isppw 27177 Two ways to say that ` A `...
isppw2 27178 Two ways to say that ` A `...
vmappw 27179 Value of the von Mangoldt ...
vmaprm 27180 Value of the von Mangoldt ...
vmacl 27181 Closure for the von Mangol...
vmaf 27182 Functionality of the von M...
efvmacl 27183 The von Mangoldt is closed...
vmage0 27184 The von Mangoldt function ...
chpval 27185 Value of the second Chebys...
chpf 27186 Functionality of the secon...
chpcl 27187 Closure for the second Che...
efchpcl 27188 The second Chebyshev funct...
chpge0 27189 The second Chebyshev funct...
ppival 27190 Value of the prime-countin...
ppival2 27191 Value of the prime-countin...
ppival2g 27192 Value of the prime-countin...
ppif 27193 Domain and codomain of the...
ppicl 27194 Real closure of the prime-...
muval 27195 The value of the Möbi...
muval1 27196 The value of the Möbi...
muval2 27197 The value of the Möbi...
isnsqf 27198 Two ways to say that a num...
issqf 27199 Two ways to say that a num...
sqfpc 27200 The prime count of a squar...
dvdssqf 27201 A divisor of a squarefree ...
sqf11 27202 A squarefree number is com...
muf 27203 The Möbius function i...
mucl 27204 Closure of the Möbius...
sgmval 27205 The value of the divisor f...
sgmval2 27206 The value of the divisor f...
0sgm 27207 The value of the sum-of-di...
sgmf 27208 The divisor function is a ...
sgmcl 27209 Closure of the divisor fun...
sgmnncl 27210 Closure of the divisor fun...
mule1 27211 The Möbius function t...
chtfl 27212 The Chebyshev function doe...
chpfl 27213 The second Chebyshev funct...
ppiprm 27214 The prime-counting functio...
ppinprm 27215 The prime-counting functio...
chtprm 27216 The Chebyshev function at ...
chtnprm 27217 The Chebyshev function at ...
chpp1 27218 The second Chebyshev funct...
chtwordi 27219 The Chebyshev function is ...
chpwordi 27220 The second Chebyshev funct...
chtdif 27221 The difference of the Cheb...
efchtdvds 27222 The exponentiated Chebyshe...
ppifl 27223 The prime-counting functio...
ppip1le 27224 The prime-counting functio...
ppiwordi 27225 The prime-counting functio...
ppidif 27226 The difference of the prim...
ppi1 27227 The prime-counting functio...
cht1 27228 The Chebyshev function at ...
vma1 27229 The von Mangoldt function ...
chp1 27230 The second Chebyshev funct...
ppi1i 27231 Inference form of ~ ppiprm...
ppi2i 27232 Inference form of ~ ppinpr...
ppi2 27233 The prime-counting functio...
ppi3 27234 The prime-counting functio...
cht2 27235 The Chebyshev function at ...
cht3 27236 The Chebyshev function at ...
ppinncl 27237 Closure of the prime-count...
chtrpcl 27238 Closure of the Chebyshev f...
ppieq0 27239 The prime-counting functio...
ppiltx 27240 The prime-counting functio...
prmorcht 27241 Relate the primorial (prod...
mumullem1 27242 Lemma for ~ mumul . A mul...
mumullem2 27243 Lemma for ~ mumul . The p...
mumul 27244 The Möbius function i...
sqff1o 27245 There is a bijection from ...
fsumdvdsdiaglem 27246 A "diagonal commutation" o...
fsumdvdsdiag 27247 A "diagonal commutation" o...
fsumdvdscom 27248 A double commutation of di...
dvdsppwf1o 27249 A bijection from the divis...
dvdsflf1o 27250 A bijection from the numbe...
dvdsflsumcom 27251 A sum commutation from ` s...
fsumfldivdiaglem 27252 Lemma for ~ fsumfldivdiag ...
fsumfldivdiag 27253 The right-hand side of ~ d...
musum 27254 The sum of the Möbius...
musumsum 27255 Evaluate a collapsing sum ...
muinv 27256 The Möbius inversion ...
mpodvdsmulf1o 27257 If ` M ` and ` N ` are two...
fsumdvdsmul 27258 Product of two divisor sum...
dvdsmulf1o 27259 If ` M ` and ` N ` are two...
fsumdvdsmulOLD 27260 Obsolete version of ~ fsum...
sgmppw 27261 The value of the divisor f...
0sgmppw 27262 A prime power ` P ^ K ` ha...
1sgmprm 27263 The sum of divisors for a ...
1sgm2ppw 27264 The sum of the divisors of...
sgmmul 27265 The divisor function for f...
ppiublem1 27266 Lemma for ~ ppiub . (Cont...
ppiublem2 27267 A prime greater than ` 3 `...
ppiub 27268 An upper bound on the prim...
vmalelog 27269 The von Mangoldt function ...
chtlepsi 27270 The first Chebyshev functi...
chprpcl 27271 Closure of the second Cheb...
chpeq0 27272 The second Chebyshev funct...
chteq0 27273 The first Chebyshev functi...
chtleppi 27274 Upper bound on the ` theta...
chtublem 27275 Lemma for ~ chtub . (Cont...
chtub 27276 An upper bound on the Cheb...
fsumvma 27277 Rewrite a sum over the von...
fsumvma2 27278 Apply ~ fsumvma for the co...
pclogsum 27279 The logarithmic analogue o...
vmasum 27280 The sum of the von Mangold...
logfac2 27281 Another expression for the...
chpval2 27282 Express the second Chebysh...
chpchtsum 27283 The second Chebyshev funct...
chpub 27284 An upper bound on the seco...
logfacubnd 27285 A simple upper bound on th...
logfaclbnd 27286 A lower bound on the logar...
logfacbnd3 27287 Show the stronger statemen...
logfacrlim 27288 Combine the estimates ~ lo...
logexprlim 27289 The sum ` sum_ n <_ x , lo...
logfacrlim2 27290 Write out ~ logfacrlim as ...
mersenne 27291 A Mersenne prime is a prim...
perfect1 27292 Euclid's contribution to t...
perfectlem1 27293 Lemma for ~ perfect . (Co...
perfectlem2 27294 Lemma for ~ perfect . (Co...
perfect 27295 The Euclid-Euler theorem, ...
dchrval 27298 Value of the group of Diri...
dchrbas 27299 Base set of the group of D...
dchrelbas 27300 A Dirichlet character is a...
dchrelbas2 27301 A Dirichlet character is a...
dchrelbas3 27302 A Dirichlet character is a...
dchrelbasd 27303 A Dirichlet character is a...
dchrrcl 27304 Reverse closure for a Diri...
dchrmhm 27305 A Dirichlet character is a...
dchrf 27306 A Dirichlet character is a...
dchrelbas4 27307 A Dirichlet character is a...
dchrzrh1 27308 Value of a Dirichlet chara...
dchrzrhcl 27309 A Dirichlet character take...
dchrzrhmul 27310 A Dirichlet character is c...
dchrplusg 27311 Group operation on the gro...
dchrmul 27312 Group operation on the gro...
dchrmulcl 27313 Closure of the group opera...
dchrn0 27314 A Dirichlet character is n...
dchr1cl 27315 Closure of the principal D...
dchrmullid 27316 Left identity for the prin...
dchrinvcl 27317 Closure of the group inver...
dchrabl 27318 The set of Dirichlet chara...
dchrfi 27319 The group of Dirichlet cha...
dchrghm 27320 A Dirichlet character rest...
dchr1 27321 Value of the principal Dir...
dchreq 27322 A Dirichlet character is d...
dchrresb 27323 A Dirichlet character is d...
dchrabs 27324 A Dirichlet character take...
dchrinv 27325 The inverse of a Dirichlet...
dchrabs2 27326 A Dirichlet character take...
dchr1re 27327 The principal Dirichlet ch...
dchrptlem1 27328 Lemma for ~ dchrpt . (Con...
dchrptlem2 27329 Lemma for ~ dchrpt . (Con...
dchrptlem3 27330 Lemma for ~ dchrpt . (Con...
dchrpt 27331 For any element other than...
dchrsum2 27332 An orthogonality relation ...
dchrsum 27333 An orthogonality relation ...
sumdchr2 27334 Lemma for ~ sumdchr . (Co...
dchrhash 27335 There are exactly ` phi ( ...
sumdchr 27336 An orthogonality relation ...
dchr2sum 27337 An orthogonality relation ...
sum2dchr 27338 An orthogonality relation ...
bcctr 27339 Value of the central binom...
pcbcctr 27340 Prime count of a central b...
bcmono 27341 The binomial coefficient i...
bcmax 27342 The binomial coefficient t...
bcp1ctr 27343 Ratio of two central binom...
bclbnd 27344 A bound on the binomial co...
efexple 27345 Convert a bound on a power...
bpos1lem 27346 Lemma for ~ bpos1 . (Cont...
bpos1 27347 Bertrand's postulate, chec...
bposlem1 27348 An upper bound on the prim...
bposlem2 27349 There are no odd primes in...
bposlem3 27350 Lemma for ~ bpos . Since ...
bposlem4 27351 Lemma for ~ bpos . (Contr...
bposlem5 27352 Lemma for ~ bpos . Bound ...
bposlem6 27353 Lemma for ~ bpos . By usi...
bposlem7 27354 Lemma for ~ bpos . The fu...
bposlem8 27355 Lemma for ~ bpos . Evalua...
bposlem9 27356 Lemma for ~ bpos . Derive...
bpos 27357 Bertrand's postulate: ther...
zabsle1 27360 ` { -u 1 , 0 , 1 } ` is th...
lgslem1 27361 When ` a ` is coprime to t...
lgslem2 27362 The set ` Z ` of all integ...
lgslem3 27363 The set ` Z ` of all integ...
lgslem4 27364 Lemma for ~ lgsfcl2 . (Co...
lgsval 27365 Value of the Legendre symb...
lgsfval 27366 Value of the function ` F ...
lgsfcl2 27367 The function ` F ` is clos...
lgscllem 27368 The Legendre symbol is an ...
lgsfcl 27369 Closure of the function ` ...
lgsfle1 27370 The function ` F ` has mag...
lgsval2lem 27371 Lemma for ~ lgsval2 . (Co...
lgsval4lem 27372 Lemma for ~ lgsval4 . (Co...
lgscl2 27373 The Legendre symbol is an ...
lgs0 27374 The Legendre symbol when t...
lgscl 27375 The Legendre symbol is an ...
lgsle1 27376 The Legendre symbol has ab...
lgsval2 27377 The Legendre symbol at a p...
lgs2 27378 The Legendre symbol at ` 2...
lgsval3 27379 The Legendre symbol at an ...
lgsvalmod 27380 The Legendre symbol is equ...
lgsval4 27381 Restate ~ lgsval for nonze...
lgsfcl3 27382 Closure of the function ` ...
lgsval4a 27383 Same as ~ lgsval4 for posi...
lgscl1 27384 The value of the Legendre ...
lgsneg 27385 The Legendre symbol is eit...
lgsneg1 27386 The Legendre symbol for no...
lgsmod 27387 The Legendre (Jacobi) symb...
lgsdilem 27388 Lemma for ~ lgsdi and ~ lg...
lgsdir2lem1 27389 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem2 27390 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem3 27391 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem4 27392 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem5 27393 Lemma for ~ lgsdir2 . (Co...
lgsdir2 27394 The Legendre symbol is com...
lgsdirprm 27395 The Legendre symbol is com...
lgsdir 27396 The Legendre symbol is com...
lgsdilem2 27397 Lemma for ~ lgsdi . (Cont...
lgsdi 27398 The Legendre symbol is com...
lgsne0 27399 The Legendre symbol is non...
lgsabs1 27400 The Legendre symbol is non...
lgssq 27401 The Legendre symbol at a s...
lgssq2 27402 The Legendre symbol at a s...
lgsprme0 27403 The Legendre symbol at any...
1lgs 27404 The Legendre symbol at ` 1...
lgs1 27405 The Legendre symbol at ` 1...
lgsmodeq 27406 The Legendre (Jacobi) symb...
lgsmulsqcoprm 27407 The Legendre (Jacobi) symb...
lgsdirnn0 27408 Variation on ~ lgsdir vali...
lgsdinn0 27409 Variation on ~ lgsdi valid...
lgsqrlem1 27410 Lemma for ~ lgsqr . (Cont...
lgsqrlem2 27411 Lemma for ~ lgsqr . (Cont...
lgsqrlem3 27412 Lemma for ~ lgsqr . (Cont...
lgsqrlem4 27413 Lemma for ~ lgsqr . (Cont...
lgsqrlem5 27414 Lemma for ~ lgsqr . (Cont...
lgsqr 27415 The Legendre symbol for od...
lgsqrmod 27416 If the Legendre symbol of ...
lgsqrmodndvds 27417 If the Legendre symbol of ...
lgsdchrval 27418 The Legendre symbol functi...
lgsdchr 27419 The Legendre symbol functi...
gausslemma2dlem0a 27420 Auxiliary lemma 1 for ~ ga...
gausslemma2dlem0b 27421 Auxiliary lemma 2 for ~ ga...
gausslemma2dlem0c 27422 Auxiliary lemma 3 for ~ ga...
gausslemma2dlem0d 27423 Auxiliary lemma 4 for ~ ga...
gausslemma2dlem0e 27424 Auxiliary lemma 5 for ~ ga...
gausslemma2dlem0f 27425 Auxiliary lemma 6 for ~ ga...
gausslemma2dlem0g 27426 Auxiliary lemma 7 for ~ ga...
gausslemma2dlem0h 27427 Auxiliary lemma 8 for ~ ga...
gausslemma2dlem0i 27428 Auxiliary lemma 9 for ~ ga...
gausslemma2dlem1a 27429 Lemma for ~ gausslemma2dle...
gausslemma2dlem1 27430 Lemma 1 for ~ gausslemma2d...
gausslemma2dlem2 27431 Lemma 2 for ~ gausslemma2d...
gausslemma2dlem3 27432 Lemma 3 for ~ gausslemma2d...
gausslemma2dlem4 27433 Lemma 4 for ~ gausslemma2d...
gausslemma2dlem5a 27434 Lemma for ~ gausslemma2dle...
gausslemma2dlem5 27435 Lemma 5 for ~ gausslemma2d...
gausslemma2dlem6 27436 Lemma 6 for ~ gausslemma2d...
gausslemma2dlem7 27437 Lemma 7 for ~ gausslemma2d...
gausslemma2d 27438 Gauss' Lemma (see also the...
lgseisenlem1 27439 Lemma for ~ lgseisen . If...
lgseisenlem2 27440 Lemma for ~ lgseisen . Th...
lgseisenlem3 27441 Lemma for ~ lgseisen . (C...
lgseisenlem4 27442 Lemma for ~ lgseisen . (C...
lgseisen 27443 Eisenstein's lemma, an exp...
lgsquadlem1 27444 Lemma for ~ lgsquad . Cou...
lgsquadlem2 27445 Lemma for ~ lgsquad . Cou...
lgsquadlem3 27446 Lemma for ~ lgsquad . (Co...
lgsquad 27447 The Law of Quadratic Recip...
lgsquad2lem1 27448 Lemma for ~ lgsquad2 . (C...
lgsquad2lem2 27449 Lemma for ~ lgsquad2 . (C...
lgsquad2 27450 Extend ~ lgsquad to coprim...
lgsquad3 27451 Extend ~ lgsquad2 to integ...
m1lgs 27452 The first supplement to th...
2lgslem1a1 27453 Lemma 1 for ~ 2lgslem1a . ...
2lgslem1a2 27454 Lemma 2 for ~ 2lgslem1a . ...
2lgslem1a 27455 Lemma 1 for ~ 2lgslem1 . ...
2lgslem1b 27456 Lemma 2 for ~ 2lgslem1 . ...
2lgslem1c 27457 Lemma 3 for ~ 2lgslem1 . ...
2lgslem1 27458 Lemma 1 for ~ 2lgs . (Con...
2lgslem2 27459 Lemma 2 for ~ 2lgs . (Con...
2lgslem3a 27460 Lemma for ~ 2lgslem3a1 . ...
2lgslem3b 27461 Lemma for ~ 2lgslem3b1 . ...
2lgslem3c 27462 Lemma for ~ 2lgslem3c1 . ...
2lgslem3d 27463 Lemma for ~ 2lgslem3d1 . ...
2lgslem3a1 27464 Lemma 1 for ~ 2lgslem3 . ...
2lgslem3b1 27465 Lemma 2 for ~ 2lgslem3 . ...
2lgslem3c1 27466 Lemma 3 for ~ 2lgslem3 . ...
2lgslem3d1 27467 Lemma 4 for ~ 2lgslem3 . ...
2lgslem3 27468 Lemma 3 for ~ 2lgs . (Con...
2lgs2 27469 The Legendre symbol for ` ...
2lgslem4 27470 Lemma 4 for ~ 2lgs : speci...
2lgs 27471 The second supplement to t...
2lgsoddprmlem1 27472 Lemma 1 for ~ 2lgsoddprm ....
2lgsoddprmlem2 27473 Lemma 2 for ~ 2lgsoddprm ....
2lgsoddprmlem3a 27474 Lemma 1 for ~ 2lgsoddprmle...
2lgsoddprmlem3b 27475 Lemma 2 for ~ 2lgsoddprmle...
2lgsoddprmlem3c 27476 Lemma 3 for ~ 2lgsoddprmle...
2lgsoddprmlem3d 27477 Lemma 4 for ~ 2lgsoddprmle...
2lgsoddprmlem3 27478 Lemma 3 for ~ 2lgsoddprm ....
2lgsoddprmlem4 27479 Lemma 4 for ~ 2lgsoddprm ....
2lgsoddprm 27480 The second supplement to t...
2sqlem1 27481 Lemma for ~ 2sq . (Contri...
2sqlem2 27482 Lemma for ~ 2sq . (Contri...
mul2sq 27483 Fibonacci's identity (actu...
2sqlem3 27484 Lemma for ~ 2sqlem5 . (Co...
2sqlem4 27485 Lemma for ~ 2sqlem5 . (Co...
2sqlem5 27486 Lemma for ~ 2sq . If a nu...
2sqlem6 27487 Lemma for ~ 2sq . If a nu...
2sqlem7 27488 Lemma for ~ 2sq . (Contri...
2sqlem8a 27489 Lemma for ~ 2sqlem8 . (Co...
2sqlem8 27490 Lemma for ~ 2sq . (Contri...
2sqlem9 27491 Lemma for ~ 2sq . (Contri...
2sqlem10 27492 Lemma for ~ 2sq . Every f...
2sqlem11 27493 Lemma for ~ 2sq . (Contri...
2sq 27494 All primes of the form ` 4...
2sqblem 27495 Lemma for ~ 2sqb . (Contr...
2sqb 27496 The converse to ~ 2sq . (...
2sq2 27497 ` 2 ` is the sum of square...
2sqn0 27498 If the sum of two squares ...
2sqcoprm 27499 If the sum of two squares ...
2sqmod 27500 Given two decompositions o...
2sqmo 27501 There exists at most one d...
2sqnn0 27502 All primes of the form ` 4...
2sqnn 27503 All primes of the form ` 4...
addsq2reu 27504 For each complex number ` ...
addsqn2reu 27505 For each complex number ` ...
addsqrexnreu 27506 For each complex number, t...
addsqnreup 27507 There is no unique decompo...
addsq2nreurex 27508 For each complex number ` ...
addsqn2reurex2 27509 For each complex number ` ...
2sqreulem1 27510 Lemma 1 for ~ 2sqreu . (C...
2sqreultlem 27511 Lemma for ~ 2sqreult . (C...
2sqreultblem 27512 Lemma for ~ 2sqreultb . (...
2sqreunnlem1 27513 Lemma 1 for ~ 2sqreunn . ...
2sqreunnltlem 27514 Lemma for ~ 2sqreunnlt . ...
2sqreunnltblem 27515 Lemma for ~ 2sqreunnltb . ...
2sqreulem2 27516 Lemma 2 for ~ 2sqreu etc. ...
2sqreulem3 27517 Lemma 3 for ~ 2sqreu etc. ...
2sqreulem4 27518 Lemma 4 for ~ 2sqreu et. ...
2sqreunnlem2 27519 Lemma 2 for ~ 2sqreunn . ...
2sqreu 27520 There exists a unique deco...
2sqreunn 27521 There exists a unique deco...
2sqreult 27522 There exists a unique deco...
2sqreultb 27523 There exists a unique deco...
2sqreunnlt 27524 There exists a unique deco...
2sqreunnltb 27525 There exists a unique deco...
2sqreuop 27526 There exists a unique deco...
2sqreuopnn 27527 There exists a unique deco...
2sqreuoplt 27528 There exists a unique deco...
2sqreuopltb 27529 There exists a unique deco...
2sqreuopnnlt 27530 There exists a unique deco...
2sqreuopnnltb 27531 There exists a unique deco...
2sqreuopb 27532 There exists a unique deco...
chebbnd1lem1 27533 Lemma for ~ chebbnd1 : sho...
chebbnd1lem2 27534 Lemma for ~ chebbnd1 : Sh...
chebbnd1lem3 27535 Lemma for ~ chebbnd1 : get...
chebbnd1 27536 The Chebyshev bound: The ...
chtppilimlem1 27537 Lemma for ~ chtppilim . (...
chtppilimlem2 27538 Lemma for ~ chtppilim . (...
chtppilim 27539 The ` theta ` function is ...
chto1ub 27540 The ` theta ` function is ...
chebbnd2 27541 The Chebyshev bound, part ...
chto1lb 27542 The ` theta ` function is ...
chpchtlim 27543 The ` psi ` and ` theta ` ...
chpo1ub 27544 The ` psi ` function is up...
chpo1ubb 27545 The ` psi ` function is up...
vmadivsum 27546 The sum of the von Mangold...
vmadivsumb 27547 Give a total bound on the ...
rplogsumlem1 27548 Lemma for ~ rplogsum . (C...
rplogsumlem2 27549 Lemma for ~ rplogsum . Eq...
dchrisum0lem1a 27550 Lemma for ~ dchrisum0lem1 ...
rpvmasumlem 27551 Lemma for ~ rpvmasum . Ca...
dchrisumlema 27552 Lemma for ~ dchrisum . Le...
dchrisumlem1 27553 Lemma for ~ dchrisum . Le...
dchrisumlem2 27554 Lemma for ~ dchrisum . Le...
dchrisumlem3 27555 Lemma for ~ dchrisum . Le...
dchrisum 27556 If ` n e. [ M , +oo ) |-> ...
dchrmusumlema 27557 Lemma for ~ dchrmusum and ...
dchrmusum2 27558 The sum of the Möbius...
dchrvmasumlem1 27559 An alternative expression ...
dchrvmasum2lem 27560 Give an expression for ` l...
dchrvmasum2if 27561 Combine the results of ~ d...
dchrvmasumlem2 27562 Lemma for ~ dchrvmasum . ...
dchrvmasumlem3 27563 Lemma for ~ dchrvmasum . ...
dchrvmasumlema 27564 Lemma for ~ dchrvmasum and...
dchrvmasumiflem1 27565 Lemma for ~ dchrvmasumif ....
dchrvmasumiflem2 27566 Lemma for ~ dchrvmasum . ...
dchrvmasumif 27567 An asymptotic approximatio...
dchrvmaeq0 27568 The set ` W ` is the colle...
dchrisum0fval 27569 Value of the function ` F ...
dchrisum0fmul 27570 The function ` F ` , the d...
dchrisum0ff 27571 The function ` F ` is a re...
dchrisum0flblem1 27572 Lemma for ~ dchrisum0flb ....
dchrisum0flblem2 27573 Lemma for ~ dchrisum0flb ....
dchrisum0flb 27574 The divisor sum of a real ...
dchrisum0fno1 27575 The sum ` sum_ k <_ x , F ...
rpvmasum2 27576 A partial result along the...
dchrisum0re 27577 Suppose ` X ` is a non-pri...
dchrisum0lema 27578 Lemma for ~ dchrisum0 . A...
dchrisum0lem1b 27579 Lemma for ~ dchrisum0lem1 ...
dchrisum0lem1 27580 Lemma for ~ dchrisum0 . (...
dchrisum0lem2a 27581 Lemma for ~ dchrisum0 . (...
dchrisum0lem2 27582 Lemma for ~ dchrisum0 . (...
dchrisum0lem3 27583 Lemma for ~ dchrisum0 . (...
dchrisum0 27584 The sum ` sum_ n e. NN , X...
dchrisumn0 27585 The sum ` sum_ n e. NN , X...
dchrmusumlem 27586 The sum of the Möbius...
dchrvmasumlem 27587 The sum of the Möbius...
dchrmusum 27588 The sum of the Möbius...
dchrvmasum 27589 The sum of the von Mangold...
rpvmasum 27590 The sum of the von Mangold...
rplogsum 27591 The sum of ` log p / p ` o...
dirith2 27592 Dirichlet's theorem: there...
dirith 27593 Dirichlet's theorem: there...
mudivsum 27594 Asymptotic formula for ` s...
mulogsumlem 27595 Lemma for ~ mulogsum . (C...
mulogsum 27596 Asymptotic formula for ...
logdivsum 27597 Asymptotic analysis of ...
mulog2sumlem1 27598 Asymptotic formula for ...
mulog2sumlem2 27599 Lemma for ~ mulog2sum . (...
mulog2sumlem3 27600 Lemma for ~ mulog2sum . (...
mulog2sum 27601 Asymptotic formula for ...
vmalogdivsum2 27602 The sum ` sum_ n <_ x , La...
vmalogdivsum 27603 The sum ` sum_ n <_ x , La...
2vmadivsumlem 27604 Lemma for ~ 2vmadivsum . ...
2vmadivsum 27605 The sum ` sum_ m n <_ x , ...
logsqvma 27606 A formula for ` log ^ 2 ( ...
logsqvma2 27607 The Möbius inverse of...
log2sumbnd 27608 Bound on the difference be...
selberglem1 27609 Lemma for ~ selberg . Est...
selberglem2 27610 Lemma for ~ selberg . (Co...
selberglem3 27611 Lemma for ~ selberg . Est...
selberg 27612 Selberg's symmetry formula...
selbergb 27613 Convert eventual boundedne...
selberg2lem 27614 Lemma for ~ selberg2 . Eq...
selberg2 27615 Selberg's symmetry formula...
selberg2b 27616 Convert eventual boundedne...
chpdifbndlem1 27617 Lemma for ~ chpdifbnd . (...
chpdifbndlem2 27618 Lemma for ~ chpdifbnd . (...
chpdifbnd 27619 A bound on the difference ...
logdivbnd 27620 A bound on a sum of logs, ...
selberg3lem1 27621 Introduce a log weighting ...
selberg3lem2 27622 Lemma for ~ selberg3 . Eq...
selberg3 27623 Introduce a log weighting ...
selberg4lem1 27624 Lemma for ~ selberg4 . Eq...
selberg4 27625 The Selberg symmetry formu...
pntrval 27626 Define the residual of the...
pntrf 27627 Functionality of the resid...
pntrmax 27628 There is a bound on the re...
pntrsumo1 27629 A bound on a sum over ` R ...
pntrsumbnd 27630 A bound on a sum over ` R ...
pntrsumbnd2 27631 A bound on a sum over ` R ...
selbergr 27632 Selberg's symmetry formula...
selberg3r 27633 Selberg's symmetry formula...
selberg4r 27634 Selberg's symmetry formula...
selberg34r 27635 The sum of ~ selberg3r and...
pntsval 27636 Define the "Selberg functi...
pntsf 27637 Functionality of the Selbe...
selbergs 27638 Selberg's symmetry formula...
selbergsb 27639 Selberg's symmetry formula...
pntsval2 27640 The Selberg function can b...
pntrlog2bndlem1 27641 The sum of ~ selberg3r and...
pntrlog2bndlem2 27642 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem3 27643 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem4 27644 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem5 27645 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem6a 27646 Lemma for ~ pntrlog2bndlem...
pntrlog2bndlem6 27647 Lemma for ~ pntrlog2bnd . ...
pntrlog2bnd 27648 A bound on ` R ( x ) log ^...
pntpbnd1a 27649 Lemma for ~ pntpbnd . (Co...
pntpbnd1 27650 Lemma for ~ pntpbnd . (Co...
pntpbnd2 27651 Lemma for ~ pntpbnd . (Co...
pntpbnd 27652 Lemma for ~ pnt . Establi...
pntibndlem1 27653 Lemma for ~ pntibnd . (Co...
pntibndlem2a 27654 Lemma for ~ pntibndlem2 . ...
pntibndlem2 27655 Lemma for ~ pntibnd . The...
pntibndlem3 27656 Lemma for ~ pntibnd . Pac...
pntibnd 27657 Lemma for ~ pnt . Establi...
pntlemd 27658 Lemma for ~ pnt . Closure...
pntlemc 27659 Lemma for ~ pnt . Closure...
pntlema 27660 Lemma for ~ pnt . Closure...
pntlemb 27661 Lemma for ~ pnt . Unpack ...
pntlemg 27662 Lemma for ~ pnt . Closure...
pntlemh 27663 Lemma for ~ pnt . Bounds ...
pntlemn 27664 Lemma for ~ pnt . The "na...
pntlemq 27665 Lemma for ~ pntlemj . (Co...
pntlemr 27666 Lemma for ~ pntlemj . (Co...
pntlemj 27667 Lemma for ~ pnt . The ind...
pntlemi 27668 Lemma for ~ pnt . Elimina...
pntlemf 27669 Lemma for ~ pnt . Add up ...
pntlemk 27670 Lemma for ~ pnt . Evaluat...
pntlemo 27671 Lemma for ~ pnt . Combine...
pntleme 27672 Lemma for ~ pnt . Package...
pntlem3 27673 Lemma for ~ pnt . Equatio...
pntlemp 27674 Lemma for ~ pnt . Wrappin...
pntleml 27675 Lemma for ~ pnt . Equatio...
pnt3 27676 The Prime Number Theorem, ...
pnt2 27677 The Prime Number Theorem, ...
pnt 27678 The Prime Number Theorem: ...
abvcxp 27679 Raising an absolute value ...
padicfval 27680 Value of the p-adic absolu...
padicval 27681 Value of the p-adic absolu...
ostth2lem1 27682 Lemma for ~ ostth2 , altho...
qrngbas 27683 The base set of the field ...
qdrng 27684 The rationals form a divis...
qrng0 27685 The zero element of the fi...
qrng1 27686 The unity element of the f...
qrngneg 27687 The additive inverse in th...
qrngdiv 27688 The division operation in ...
qabvle 27689 By using induction on ` N ...
qabvexp 27690 Induct the product rule ~ ...
ostthlem1 27691 Lemma for ~ ostth . If tw...
ostthlem2 27692 Lemma for ~ ostth . Refin...
qabsabv 27693 The regular absolute value...
padicabv 27694 The p-adic absolute value ...
padicabvf 27695 The p-adic absolute value ...
padicabvcxp 27696 All positive powers of the...
ostth1 27697 - Lemma for ~ ostth : triv...
ostth2lem2 27698 Lemma for ~ ostth2 . (Con...
ostth2lem3 27699 Lemma for ~ ostth2 . (Con...
ostth2lem4 27700 Lemma for ~ ostth2 . (Con...
ostth2 27701 - Lemma for ~ ostth : regu...
ostth3 27702 - Lemma for ~ ostth : p-ad...
ostth 27703 Ostrowski's theorem, which...
elno 27710 Membership in the surreals...
elnoOLD 27711 Obsolete version of ~ elno...
sltval 27712 The value of the surreal l...
bdayval 27713 The value of the birthday ...
nofun 27714 A surreal is a function. ...
nodmon 27715 The domain of a surreal is...
norn 27716 The range of a surreal is ...
nofnbday 27717 A surreal is a function ov...
nodmord 27718 The domain of a surreal ha...
elno2 27719 An alternative condition f...
elno3 27720 Another condition for memb...
sltval2 27721 Alternate expression for s...
nofv 27722 The function value of a su...
nosgnn0 27723 ` (/) ` is not a surreal s...
nosgnn0i 27724 If ` X ` is a surreal sign...
noreson 27725 The restriction of a surre...
sltintdifex 27726 If ` A
sltres 27727 If the restrictions of two...
noxp1o 27728 The Cartesian product of a...
noseponlem 27729 Lemma for ~ nosepon . Con...
nosepon 27730 Given two unequal surreals...
noextend 27731 Extending a surreal by one...
noextendseq 27732 Extend a surreal by a sequ...
noextenddif 27733 Calculate the place where ...
noextendlt 27734 Extending a surreal with a...
noextendgt 27735 Extending a surreal with a...
nolesgn2o 27736 Given ` A ` less-than or e...
nolesgn2ores 27737 Given ` A ` less-than or e...
nogesgn1o 27738 Given ` A ` greater than o...
nogesgn1ores 27739 Given ` A ` greater than o...
sltsolem1 27740 Lemma for ~ sltso . The "...
sltso 27741 Less-than totally orders t...
bdayfo 27742 The birthday function maps...
fvnobday 27743 The value of a surreal at ...
nosepnelem 27744 Lemma for ~ nosepne . (Co...
nosepne 27745 The value of two non-equal...
nosep1o 27746 If the value of a surreal ...
nosep2o 27747 If the value of a surreal ...
nosepdmlem 27748 Lemma for ~ nosepdm . (Co...
nosepdm 27749 The first place two surrea...
nosepeq 27750 The values of two surreals...
nosepssdm 27751 Given two non-equal surrea...
nodenselem4 27752 Lemma for ~ nodense . Sho...
nodenselem5 27753 Lemma for ~ nodense . If ...
nodenselem6 27754 The restriction of a surre...
nodenselem7 27755 Lemma for ~ nodense . ` A ...
nodenselem8 27756 Lemma for ~ nodense . Giv...
nodense 27757 Given two distinct surreal...
bdayimaon 27758 Lemma for full-eta propert...
nolt02olem 27759 Lemma for ~ nolt02o . If ...
nolt02o 27760 Given ` A ` less-than ` B ...
nogt01o 27761 Given ` A ` greater than `...
noresle 27762 Restriction law for surrea...
nomaxmo 27763 A class of surreals has at...
nominmo 27764 A class of surreals has at...
nosupprefixmo 27765 In any class of surreals, ...
noinfprefixmo 27766 In any class of surreals, ...
nosupcbv 27767 Lemma to change bound vari...
nosupno 27768 The next several theorems ...
nosupdm 27769 The domain of the surreal ...
nosupbday 27770 Birthday bounding law for ...
nosupfv 27771 The value of surreal supre...
nosupres 27772 A restriction law for surr...
nosupbnd1lem1 27773 Lemma for ~ nosupbnd1 . E...
nosupbnd1lem2 27774 Lemma for ~ nosupbnd1 . W...
nosupbnd1lem3 27775 Lemma for ~ nosupbnd1 . I...
nosupbnd1lem4 27776 Lemma for ~ nosupbnd1 . I...
nosupbnd1lem5 27777 Lemma for ~ nosupbnd1 . I...
nosupbnd1lem6 27778 Lemma for ~ nosupbnd1 . E...
nosupbnd1 27779 Bounding law from below fo...
nosupbnd2lem1 27780 Bounding law from above wh...
nosupbnd2 27781 Bounding law from above fo...
noinfcbv 27782 Change bound variables for...
noinfno 27783 The next several theorems ...
noinfdm 27784 Next, we calculate the dom...
noinfbday 27785 Birthday bounding law for ...
noinffv 27786 The value of surreal infim...
noinfres 27787 The restriction of surreal...
noinfbnd1lem1 27788 Lemma for ~ noinfbnd1 . E...
noinfbnd1lem2 27789 Lemma for ~ noinfbnd1 . W...
noinfbnd1lem3 27790 Lemma for ~ noinfbnd1 . I...
noinfbnd1lem4 27791 Lemma for ~ noinfbnd1 . I...
noinfbnd1lem5 27792 Lemma for ~ noinfbnd1 . I...
noinfbnd1lem6 27793 Lemma for ~ noinfbnd1 . E...
noinfbnd1 27794 Bounding law from above fo...
noinfbnd2lem1 27795 Bounding law from below wh...
noinfbnd2 27796 Bounding law from below fo...
nosupinfsep 27797 Given two sets of surreals...
noetasuplem1 27798 Lemma for ~ noeta . Estab...
noetasuplem2 27799 Lemma for ~ noeta . The r...
noetasuplem3 27800 Lemma for ~ noeta . ` Z ` ...
noetasuplem4 27801 Lemma for ~ noeta . When ...
noetainflem1 27802 Lemma for ~ noeta . Estab...
noetainflem2 27803 Lemma for ~ noeta . The r...
noetainflem3 27804 Lemma for ~ noeta . ` W ` ...
noetainflem4 27805 Lemma for ~ noeta . If ` ...
noetalem1 27806 Lemma for ~ noeta . Eithe...
noetalem2 27807 Lemma for ~ noeta . The f...
noeta 27808 The full-eta axiom for the...
sltirr 27811 Surreal less-than is irref...
slttr 27812 Surreal less-than is trans...
sltasym 27813 Surreal less-than is asymm...
sltlin 27814 Surreal less-than obeys tr...
slttrieq2 27815 Trichotomy law for surreal...
slttrine 27816 Trichotomy law for surreal...
slenlt 27817 Surreal less-than or equal...
sltnle 27818 Surreal less-than in terms...
sleloe 27819 Surreal less-than or equal...
sletri3 27820 Trichotomy law for surreal...
sltletr 27821 Surreal transitive law. (...
slelttr 27822 Surreal transitive law. (...
sletr 27823 Surreal transitive law. (...
slttrd 27824 Surreal less-than is trans...
sltletrd 27825 Surreal less-than is trans...
slelttrd 27826 Surreal less-than is trans...
sletrd 27827 Surreal less-than or equal...
slerflex 27828 Surreal less-than or equal...
sletric 27829 Surreal trichotomy law. (...
maxs1 27830 A surreal is less than or ...
maxs2 27831 A surreal is less than or ...
mins1 27832 The minimum of two surreal...
mins2 27833 The minimum of two surreal...
sltled 27834 Surreal less-than implies ...
sltne 27835 Surreal less-than implies ...
sltlend 27836 Surreal less-than in terms...
bdayfun 27837 The birthday function is a...
bdayfn 27838 The birthday function is a...
bdaydm 27839 The birthday function's do...
bdayrn 27840 The birthday function's ra...
bdayelon 27841 The value of the birthday ...
nocvxminlem 27842 Lemma for ~ nocvxmin . Gi...
nocvxmin 27843 Given a nonempty convex cl...
noprc 27844 The surreal numbers are a ...
noeta2 27849 A version of ~ noeta with ...
brsslt 27850 Binary relation form of th...
ssltex1 27851 The first argument of surr...
ssltex2 27852 The second argument of sur...
ssltss1 27853 The first argument of surr...
ssltss2 27854 The second argument of sur...
ssltsep 27855 The separation property of...
ssltd 27856 Deduce surreal set less-th...
ssltsn 27857 Surreal set less-than of t...
ssltsepc 27858 Two elements of separated ...
ssltsepcd 27859 Two elements of separated ...
sssslt1 27860 Relation between surreal s...
sssslt2 27861 Relation between surreal s...
nulsslt 27862 The empty set is less-than...
nulssgt 27863 The empty set is greater t...
conway 27864 Conway's Simplicity Theore...
scutval 27865 The value of the surreal c...
scutcut 27866 Cut properties of the surr...
scutcl 27867 Closure law for surreal cu...
scutcld 27868 Closure law for surreal cu...
scutbday 27869 The birthday of the surrea...
eqscut 27870 Condition for equality to ...
eqscut2 27871 Condition for equality to ...
sslttr 27872 Transitive law for surreal...
ssltun1 27873 Union law for surreal set ...
ssltun2 27874 Union law for surreal set ...
scutun12 27875 Union law for surreal cuts...
dmscut 27876 The domain of the surreal ...
scutf 27877 Functionality statement fo...
etasslt 27878 A restatement of ~ noeta u...
etasslt2 27879 A version of ~ etasslt wit...
scutbdaybnd 27880 An upper bound on the birt...
scutbdaybnd2 27881 An upper bound on the birt...
scutbdaybnd2lim 27882 An upper bound on the birt...
scutbdaylt 27883 If a surreal lies in a gap...
slerec 27884 A comparison law for surre...
sltrec 27885 A comparison law for surre...
ssltdisj 27886 If ` A ` preceeds ` B ` , ...
0sno 27891 Surreal zero is a surreal....
1sno 27892 Surreal one is a surreal. ...
bday0s 27893 Calculate the birthday of ...
0slt1s 27894 Surreal zero is less than ...
bday0b 27895 The only surreal with birt...
bday1s 27896 The birthday of surreal on...
cuteq0 27897 Condition for a surreal cu...
cuteq1 27898 Condition for a surreal cu...
sgt0ne0 27899 A positive surreal is not ...
sgt0ne0d 27900 A positive surreal is not ...
madeval 27911 The value of the made by f...
madeval2 27912 Alternative characterizati...
oldval 27913 The value of the old optio...
newval 27914 The value of the new optio...
madef 27915 The made function is a fun...
oldf 27916 The older function is a fu...
newf 27917 The new function is a func...
old0 27918 No surreal is older than `...
madessno 27919 Made sets are surreals. (...
oldssno 27920 Old sets are surreals. (C...
newssno 27921 New sets are surreals. (C...
leftval 27922 The value of the left opti...
rightval 27923 The value of the right opt...
leftf 27924 The functionality of the l...
rightf 27925 The functionality of the r...
elmade 27926 Membership in the made fun...
elmade2 27927 Membership in the made fun...
elold 27928 Membership in an old set. ...
ssltleft 27929 A surreal is greater than ...
ssltright 27930 A surreal is less than its...
lltropt 27931 The left options of a surr...
made0 27932 The only surreal made on d...
new0 27933 The only surreal new on da...
old1 27934 The only surreal older tha...
madess 27935 If ` A ` is less than or e...
oldssmade 27936 The older-than set is a su...
leftssold 27937 The left options are a sub...
rightssold 27938 The right options are a su...
leftssno 27939 The left set of a surreal ...
rightssno 27940 The right set of a surreal...
madecut 27941 Given a section that is a ...
madeun 27942 The made set is the union ...
madeoldsuc 27943 The made set is the old se...
oldsuc 27944 The value of the old set a...
oldlim 27945 The value of the old set a...
madebdayim 27946 If a surreal is a member o...
oldbdayim 27947 If ` X ` is in the old set...
oldirr 27948 No surreal is a member of ...
leftirr 27949 No surreal is a member of ...
rightirr 27950 No surreal is a member of ...
left0s 27951 The left set of ` 0s ` is ...
right0s 27952 The right set of ` 0s ` is...
left1s 27953 The left set of ` 1s ` is ...
right1s 27954 The right set of ` 1s ` is...
lrold 27955 The union of the left and ...
madebdaylemold 27956 Lemma for ~ madebday . If...
madebdaylemlrcut 27957 Lemma for ~ madebday . If...
madebday 27958 A surreal is part of the s...
oldbday 27959 A surreal is part of the s...
newbday 27960 A surreal is an element of...
lrcut 27961 A surreal is equal to the ...
scutfo 27962 The surreal cut function i...
sltn0 27963 If ` X ` is less than ` Y ...
lruneq 27964 If two surreals share a bi...
sltlpss 27965 If two surreals share a bi...
slelss 27966 If two surreals ` A ` and ...
0elold 27967 Zero is in the old set of ...
0elleft 27968 Zero is in the left set of...
0elright 27969 Zero is in the right set o...
madefi 27970 The made set of an ordinal...
oldfi 27971 The old set of an ordinal ...
cofsslt 27972 If every element of ` A ` ...
coinitsslt 27973 If ` B ` is coinitial with...
cofcut1 27974 If ` C ` is cofinal with `...
cofcut1d 27975 If ` C ` is cofinal with `...
cofcut2 27976 If ` A ` and ` C ` are mut...
cofcut2d 27977 If ` A ` and ` C ` are mut...
cofcutr 27978 If ` X ` is the cut of ` A...
cofcutr1d 27979 If ` X ` is the cut of ` A...
cofcutr2d 27980 If ` X ` is the cut of ` A...
cofcutrtime 27981 If ` X ` is the cut of ` A...
cofcutrtime1d 27982 If ` X ` is a timely cut o...
cofcutrtime2d 27983 If ` X ` is a timely cut o...
cofss 27984 Cofinality for a subset. ...
coiniss 27985 Coinitiality for a subset....
cutlt 27986 Eliminating all elements b...
cutpos 27987 Reduce the elements of a c...
cutmax 27988 If ` A ` has a maximum, th...
cutmin 27989 If ` B ` has a minimum, th...
lrrecval 27992 The next step in the devel...
lrrecval2 27993 Next, we establish an alte...
lrrecpo 27994 Now, we establish that ` R...
lrrecse 27995 Next, we show that ` R ` i...
lrrecfr 27996 Now we show that ` R ` is ...
lrrecpred 27997 Finally, we calculate the ...
noinds 27998 Induction principle for a ...
norecfn 27999 Surreal recursion over one...
norecov 28000 Calculate the value of the...
noxpordpo 28003 To get through most of the...
noxpordfr 28004 Next we establish the foun...
noxpordse 28005 Next we establish the set-...
noxpordpred 28006 Next we calculate the pred...
no2indslem 28007 Double induction on surrea...
no2inds 28008 Double induction on surrea...
norec2fn 28009 The double-recursion opera...
norec2ov 28010 The value of the double-re...
no3inds 28011 Triple induction over surr...
addsfn 28014 Surreal addition is a func...
addsval 28015 The value of surreal addit...
addsval2 28016 The value of surreal addit...
addsrid 28017 Surreal addition to zero i...
addsridd 28018 Surreal addition to zero i...
addscom 28019 Surreal addition commutes....
addscomd 28020 Surreal addition commutes....
addslid 28021 Surreal addition to zero i...
addsproplem1 28022 Lemma for surreal addition...
addsproplem2 28023 Lemma for surreal addition...
addsproplem3 28024 Lemma for surreal addition...
addsproplem4 28025 Lemma for surreal addition...
addsproplem5 28026 Lemma for surreal addition...
addsproplem6 28027 Lemma for surreal addition...
addsproplem7 28028 Lemma for surreal addition...
addsprop 28029 Inductively show that surr...
addscutlem 28030 Lemma for ~ addscut . Sho...
addscut 28031 Demonstrate the cut proper...
addscut2 28032 Show that the cut involved...
addscld 28033 Surreal numbers are closed...
addscl 28034 Surreal numbers are closed...
addsf 28035 Function statement for sur...
addsfo 28036 Surreal addition is onto. ...
peano2no 28037 A theorem for surreals tha...
sltadd1im 28038 Surreal less-than is prese...
sltadd2im 28039 Surreal less-than is prese...
sleadd1im 28040 Surreal less-than or equal...
sleadd2im 28041 Surreal less-than or equal...
sleadd1 28042 Addition to both sides of ...
sleadd2 28043 Addition to both sides of ...
sltadd2 28044 Addition to both sides of ...
sltadd1 28045 Addition to both sides of ...
addscan2 28046 Cancellation law for surre...
addscan1 28047 Cancellation law for surre...
sleadd1d 28048 Addition to both sides of ...
sleadd2d 28049 Addition to both sides of ...
sltadd2d 28050 Addition to both sides of ...
sltadd1d 28051 Addition to both sides of ...
addscan2d 28052 Cancellation law for surre...
addscan1d 28053 Cancellation law for surre...
addsuniflem 28054 Lemma for ~ addsunif . St...
addsunif 28055 Uniformity theorem for sur...
addsasslem1 28056 Lemma for addition associa...
addsasslem2 28057 Lemma for addition associa...
addsass 28058 Surreal addition is associ...
addsassd 28059 Surreal addition is associ...
adds32d 28060 Commutative/associative la...
adds12d 28061 Commutative/associative la...
adds4d 28062 Rearrangement of four term...
adds42d 28063 Rearrangement of four term...
sltaddpos1d 28064 Addition of a positive num...
sltaddpos2d 28065 Addition of a positive num...
slt2addd 28066 Adding both sides of two s...
addsgt0d 28067 The sum of two positive su...
sltp1d 28068 A surreal is less than its...
addsbdaylem 28069 Lemma for ~ addsbday . (C...
addsbday 28070 The birthday of the sum of...
negsfn 28075 Surreal negation is a func...
subsfn 28076 Surreal subtraction is a f...
negsval 28077 The value of the surreal n...
negs0s 28078 Negative surreal zero is s...
negs1s 28079 An expression for negative...
negsproplem1 28080 Lemma for surreal negation...
negsproplem2 28081 Lemma for surreal negation...
negsproplem3 28082 Lemma for surreal negation...
negsproplem4 28083 Lemma for surreal negation...
negsproplem5 28084 Lemma for surreal negation...
negsproplem6 28085 Lemma for surreal negation...
negsproplem7 28086 Lemma for surreal negation...
negsprop 28087 Show closure and ordering ...
negscl 28088 The surreals are closed un...
negscld 28089 The surreals are closed un...
sltnegim 28090 The forward direction of t...
negscut 28091 The cut properties of surr...
negscut2 28092 The cut that defines surre...
negsid 28093 Surreal addition of a numb...
negsidd 28094 Surreal addition of a numb...
negsex 28095 Every surreal has a negati...
negnegs 28096 A surreal is equal to the ...
sltneg 28097 Negative of both sides of ...
sleneg 28098 Negative of both sides of ...
sltnegd 28099 Negative of both sides of ...
slenegd 28100 Negative of both sides of ...
negs11 28101 Surreal negation is one-to...
negsdi 28102 Distribution of surreal ne...
slt0neg2d 28103 Comparison of a surreal an...
negsf 28104 Function statement for sur...
negsfo 28105 Function statement for sur...
negsf1o 28106 Surreal negation is a bije...
negsunif 28107 Uniformity property for su...
negsbdaylem 28108 Lemma for ~ negsbday . Bo...
negsbday 28109 Negation of a surreal numb...
subsval 28110 The value of surreal subtr...
subsvald 28111 The value of surreal subtr...
subscl 28112 Closure law for surreal su...
subscld 28113 Closure law for surreal su...
subsf 28114 Function statement for sur...
subsfo 28115 Surreal subtraction is an ...
negsval2 28116 Surreal negation in terms ...
negsval2d 28117 Surreal negation in terms ...
subsid1 28118 Identity law for subtracti...
subsid 28119 Subtraction of a surreal f...
subadds 28120 Relationship between addit...
subaddsd 28121 Relationship between addit...
pncans 28122 Cancellation law for surre...
pncan3s 28123 Subtraction and addition o...
pncan2s 28124 Cancellation law for surre...
npcans 28125 Cancellation law for surre...
sltsub1 28126 Subtraction from both side...
sltsub2 28127 Subtraction from both side...
sltsub1d 28128 Subtraction from both side...
sltsub2d 28129 Subtraction from both side...
negsubsdi2d 28130 Distribution of negative o...
addsubsassd 28131 Associative-type law for s...
addsubsd 28132 Law for surreal addition a...
sltsubsubbd 28133 Equivalence for the surrea...
sltsubsub2bd 28134 Equivalence for the surrea...
sltsubsub3bd 28135 Equivalence for the surrea...
slesubsubbd 28136 Equivalence for the surrea...
slesubsub2bd 28137 Equivalence for the surrea...
slesubsub3bd 28138 Equivalence for the surrea...
sltsubaddd 28139 Surreal less-than relation...
sltsubadd2d 28140 Surreal less-than relation...
sltaddsubd 28141 Surreal less-than relation...
sltaddsub2d 28142 Surreal less-than relation...
slesubaddd 28143 Surreal less-than or equal...
subsubs4d 28144 Law for double surreal sub...
subsubs2d 28145 Law for double surreal sub...
nncansd 28146 Cancellation law for surre...
posdifsd 28147 Comparison of two surreals...
sltsubposd 28148 Subtraction of a positive ...
subsge0d 28149 Non-negative subtraction. ...
addsubs4d 28150 Rearrangement of four term...
sltm1d 28151 A surreal is greater than ...
mulsfn 28154 Surreal multiplication is ...
mulsval 28155 The value of surreal multi...
mulsval2lem 28156 Lemma for ~ mulsval2 . Ch...
mulsval2 28157 The value of surreal multi...
muls01 28158 Surreal multiplication by ...
mulsrid 28159 Surreal one is a right ide...
mulsridd 28160 Surreal one is a right ide...
mulsproplemcbv 28161 Lemma for surreal multipli...
mulsproplem1 28162 Lemma for surreal multipli...
mulsproplem2 28163 Lemma for surreal multipli...
mulsproplem3 28164 Lemma for surreal multipli...
mulsproplem4 28165 Lemma for surreal multipli...
mulsproplem5 28166 Lemma for surreal multipli...
mulsproplem6 28167 Lemma for surreal multipli...
mulsproplem7 28168 Lemma for surreal multipli...
mulsproplem8 28169 Lemma for surreal multipli...
mulsproplem9 28170 Lemma for surreal multipli...
mulsproplem10 28171 Lemma for surreal multipli...
mulsproplem11 28172 Lemma for surreal multipli...
mulsproplem12 28173 Lemma for surreal multipli...
mulsproplem13 28174 Lemma for surreal multipli...
mulsproplem14 28175 Lemma for surreal multipli...
mulsprop 28176 Surreals are closed under ...
mulscutlem 28177 Lemma for ~ mulscut . Sta...
mulscut 28178 Show the cut properties of...
mulscut2 28179 Show that the cut involved...
mulscl 28180 The surreals are closed un...
mulscld 28181 The surreals are closed un...
sltmul 28182 An ordering relationship f...
sltmuld 28183 An ordering relationship f...
slemuld 28184 An ordering relationship f...
mulscom 28185 Surreal multiplication com...
mulscomd 28186 Surreal multiplication com...
muls02 28187 Surreal multiplication by ...
mulslid 28188 Surreal one is a left iden...
mulslidd 28189 Surreal one is a left iden...
mulsgt0 28190 The product of two positiv...
mulsgt0d 28191 The product of two positiv...
mulsge0d 28192 The product of two non-neg...
ssltmul1 28193 One surreal set less-than ...
ssltmul2 28194 One surreal set less-than ...
mulsuniflem 28195 Lemma for ~ mulsunif . St...
mulsunif 28196 Surreal multiplication has...
addsdilem1 28197 Lemma for surreal distribu...
addsdilem2 28198 Lemma for surreal distribu...
addsdilem3 28199 Lemma for ~ addsdi . Show...
addsdilem4 28200 Lemma for ~ addsdi . Show...
addsdi 28201 Distributive law for surre...
addsdid 28202 Distributive law for surre...
addsdird 28203 Distributive law for surre...
subsdid 28204 Distribution of surreal mu...
subsdird 28205 Distribution of surreal mu...
mulnegs1d 28206 Product with negative is n...
mulnegs2d 28207 Product with negative is n...
mul2negsd 28208 Surreal product of two neg...
mulsasslem1 28209 Lemma for ~ mulsass . Exp...
mulsasslem2 28210 Lemma for ~ mulsass . Exp...
mulsasslem3 28211 Lemma for ~ mulsass . Dem...
mulsass 28212 Associative law for surrea...
mulsassd 28213 Associative law for surrea...
muls4d 28214 Rearrangement of four surr...
mulsunif2lem 28215 Lemma for ~ mulsunif2 . S...
mulsunif2 28216 Alternate expression for s...
sltmul2 28217 Multiplication of both sid...
sltmul2d 28218 Multiplication of both sid...
sltmul1d 28219 Multiplication of both sid...
slemul2d 28220 Multiplication of both sid...
slemul1d 28221 Multiplication of both sid...
sltmulneg1d 28222 Multiplication of both sid...
sltmulneg2d 28223 Multiplication of both sid...
mulscan2dlem 28224 Lemma for ~ mulscan2d . C...
mulscan2d 28225 Cancellation of surreal mu...
mulscan1d 28226 Cancellation of surreal mu...
muls12d 28227 Commutative/associative la...
slemul1ad 28228 Multiplication of both sid...
sltmul12ad 28229 Comparison of the product ...
divsmo 28230 Uniqueness of surreal inve...
muls0ord 28231 If a surreal product is ze...
mulsne0bd 28232 The product of two non-zer...
divsval 28235 The value of surreal divis...
norecdiv 28236 If a surreal has a recipro...
noreceuw 28237 If a surreal has a recipro...
divsmulw 28238 Relationship between surre...
divsmulwd 28239 Relationship between surre...
divsclw 28240 Weak division closure law....
divsclwd 28241 Weak division closure law....
divscan2wd 28242 A weak cancellation law fo...
divscan1wd 28243 A weak cancellation law fo...
sltdivmulwd 28244 Surreal less-than relation...
sltdivmul2wd 28245 Surreal less-than relation...
sltmuldivwd 28246 Surreal less-than relation...
sltmuldiv2wd 28247 Surreal less-than relation...
divsasswd 28248 An associative law for sur...
divs1 28249 A surreal divided by one i...
precsexlemcbv 28250 Lemma for surreal reciproc...
precsexlem1 28251 Lemma for surreal reciproc...
precsexlem2 28252 Lemma for surreal reciproc...
precsexlem3 28253 Lemma for surreal reciproc...
precsexlem4 28254 Lemma for surreal reciproc...
precsexlem5 28255 Lemma for surreal reciproc...
precsexlem6 28256 Lemma for surreal reciproc...
precsexlem7 28257 Lemma for surreal reciproc...
precsexlem8 28258 Lemma for surreal reciproc...
precsexlem9 28259 Lemma for surreal reciproc...
precsexlem10 28260 Lemma for surreal reciproc...
precsexlem11 28261 Lemma for surreal reciproc...
precsex 28262 Every positive surreal has...
recsex 28263 A non-zero surreal has a r...
recsexd 28264 A non-zero surreal has a r...
divsmul 28265 Relationship between surre...
divsmuld 28266 Relationship between surre...
divscl 28267 Surreal division closure l...
divscld 28268 Surreal division closure l...
divscan2d 28269 A cancellation law for sur...
divscan1d 28270 A cancellation law for sur...
sltdivmuld 28271 Surreal less-than relation...
sltdivmul2d 28272 Surreal less-than relation...
sltmuldivd 28273 Surreal less-than relation...
sltmuldiv2d 28274 Surreal less-than relation...
divsassd 28275 An associative law for sur...
divmuldivsd 28276 Multiplication of two surr...
divdivs1d 28277 Surreal division into a fr...
divsrecd 28278 Relationship between surre...
divsdird 28279 Distribution of surreal di...
divscan3d 28280 A cancellation law for sur...
abssval 28283 The value of surreal absol...
absscl 28284 Closure law for surreal ab...
abssid 28285 The absolute value of a no...
abs0s 28286 The absolute value of surr...
abssnid 28287 For a negative surreal, it...
absmuls 28288 Surreal absolute value dis...
abssge0 28289 The absolute value of a su...
abssor 28290 The absolute value of a su...
abssneg 28291 Surreal absolute value of ...
sleabs 28292 A surreal is less than or ...
absslt 28293 Surreal absolute value and...
elons 28296 Membership in the class of...
onssno 28297 The surreal ordinals are a...
onsno 28298 A surreal ordinal is a sur...
0ons 28299 Surreal zero is a surreal ...
1ons 28300 Surreal one is a surreal o...
elons2 28301 A surreal is ordinal iff i...
elons2d 28302 The cut of any set of surr...
sltonold 28303 The class of ordinals less...
sltonex 28304 The class of ordinals less...
onscutleft 28305 A surreal ordinal is equal...
onaddscl 28306 The surreal ordinals are c...
onmulscl 28307 The surreal ordinals are c...
peano2ons 28308 The successor of a surreal...
seqsex 28311 Existence of the surreal s...
seqseq123d 28312 Equality deduction for the...
nfseqs 28313 Hypothesis builder for the...
seqsval 28314 The value of the surreal s...
noseqex 28315 The next several theorems ...
noseq0 28316 The surreal ` A ` is a mem...
noseqp1 28317 One plus an element of ` Z...
noseqind 28318 Peano's inductive postulat...
noseqinds 28319 Induction schema for surre...
noseqssno 28320 A surreal sequence is a su...
noseqno 28321 An element of a surreal se...
om2noseq0 28322 The mapping ` G ` is a one...
om2noseqsuc 28323 The value of ` G ` at a su...
om2noseqfo 28324 Function statement for ` G...
om2noseqlt 28325 Surreal less-than relation...
om2noseqlt2 28326 The mapping ` G ` preserve...
om2noseqf1o 28327 ` G ` is a bijection. (Co...
om2noseqiso 28328 ` G ` is an isomorphism fr...
om2noseqoi 28329 An alternative definition ...
om2noseqrdg 28330 A helper lemma for the val...
noseqrdglem 28331 A helper lemma for the val...
noseqrdgfn 28332 The recursive definition g...
noseqrdg0 28333 Initial value of a recursi...
noseqrdgsuc 28334 Successor value of a recur...
seqsfn 28335 The surreal sequence build...
seqs1 28336 The value of the surreal s...
seqsp1 28337 The value of the surreal s...
n0sex 28342 The set of all non-negativ...
nnsex 28343 The set of all positive su...
peano5n0s 28344 Peano's inductive postulat...
n0ssno 28345 The non-negative surreal i...
nnssn0s 28346 The positive surreal integ...
nnssno 28347 The positive surreal integ...
n0sno 28348 A non-negative surreal int...
nnsno 28349 A positive surreal integer...
n0snod 28350 A non-negative surreal int...
nnsnod 28351 A positive surreal integer...
nnn0s 28352 A positive surreal integer...
nnn0sd 28353 A positive surreal integer...
0n0s 28354 Peano postulate: ` 0s ` is...
peano2n0s 28355 Peano postulate: the succe...
dfn0s2 28356 Alternate definition of th...
n0sind 28357 Principle of Mathematical ...
n0scut 28358 A cut form for surreal nat...
n0ons 28359 A surreal natural is a sur...
nnne0s 28360 A surreal positive integer...
n0sge0 28361 A non-negative integer is ...
nnsgt0 28362 A positive integer is grea...
elnns 28363 Membership in the positive...
elnns2 28364 A positive surreal integer...
n0s0suc 28365 A non-negative surreal int...
nnsge1 28366 A positive surreal integer...
n0addscl 28367 The non-negative surreal i...
n0mulscl 28368 The non-negative surreal i...
nnaddscl 28369 The positive surreal integ...
nnmulscl 28370 The positive surreal integ...
1n0s 28371 Surreal one is a non-negat...
1nns 28372 Surreal one is a positive ...
peano2nns 28373 Peano postulate for positi...
n0sbday 28374 A non-negative surreal int...
n0ssold 28375 The non-negative surreal i...
nnsrecgt0d 28376 The reciprocal of a positi...
seqn0sfn 28377 The surreal sequence build...
eln0s 28378 A non-negative surreal int...
n0s0m1 28379 Every non-negative surreal...
n0subs 28380 Subtraction of non-negativ...
n0p1nns 28381 One plus a non-negative su...
dfnns2 28382 Alternate definition of th...
nnsind 28383 Principle of Mathematical ...
zsex 28386 The surreal integers form ...
zssno 28387 The surreal integers are a...
zno 28388 A surreal integer is a sur...
znod 28389 A surreal integer is a sur...
elzs 28390 Membership in the set of s...
nnzsubs 28391 The difference of two surr...
nnzs 28392 A positive surreal integer...
nnzsd 28393 A positive surreal integer...
0zs 28394 Zero is a surreal integer....
n0zs 28395 A non-negative surreal int...
n0zsd 28396 A non-negative surreal int...
1zs 28397 One is a surreal integer. ...
znegscl 28398 The surreal integers are c...
znegscld 28399 The surreal integers are c...
zaddscl 28400 The surreal integers are c...
zaddscld 28401 The surreal integers are c...
zsubscld 28402 The surreal integers are c...
zmulscld 28403 The surreal integers are c...
elzn0s 28404 A surreal integer is a sur...
elzs2 28405 A surreal integer is eithe...
eln0zs 28406 Non-negative surreal integ...
elnnzs 28407 Positive surreal integer p...
elznns 28408 Surreal integer property e...
zn0subs 28409 The non-negative differenc...
peano5uzs 28410 Peano's inductive postulat...
uzsind 28411 Induction on the upper sur...
zsbday 28412 A surreal integer has a fi...
zscut 28413 A cut expression for surre...
1p1e2s 28420 One plus one is two. Surr...
no2times 28421 Version of ~ 2times for su...
2nns 28422 Surreal two is a surreal n...
2sno 28423 Surreal two is a surreal n...
2ne0s 28424 Surreal two is non-zero. ...
n0seo 28425 A non-negative surreal int...
zseo 28426 A surreal integer is eithe...
nohalf 28427 An explicit expression for...
expsval 28428 The value of surreal expon...
expsnnval 28429 Value of surreal exponenti...
exps0 28430 Surreal exponentiation to ...
exps1 28431 Surreal exponentiation to ...
expsp1 28432 Value of a surreal number ...
expscl 28433 Closure law for surreal ex...
expsne0 28434 A non-negative surreal int...
expsgt0 28435 A non-negative surreal int...
halfcut 28436 Relate the cut of twice of...
cutpw2 28437 A cut expression for inver...
pw2bday 28438 The inverses of powers of ...
addhalfcut 28439 The cut of a surreal non-n...
pw2cut 28440 Extend ~ halfcut to arbitr...
elzs12 28441 Membership in the dyadic f...
zs12ex 28442 The class of dyadic fracti...
zzs12 28443 A surreal integer is a dya...
zs12bday 28444 A dyadic fraction has a fi...
elreno 28447 Membership in the set of s...
recut 28448 The cut involved in defini...
0reno 28449 Surreal zero is a surreal ...
renegscl 28450 The surreal reals are clos...
readdscl 28451 The surreal reals are clos...
remulscllem1 28452 Lemma for ~ remulscl . Sp...
remulscllem2 28453 Lemma for ~ remulscl . Bo...
remulscl 28454 The surreal reals are clos...
itvndx 28465 Index value of the Interva...
lngndx 28466 Index value of the "line" ...
itvid 28467 Utility theorem: index-ind...
lngid 28468 Utility theorem: index-ind...
slotsinbpsd 28469 The slots ` Base ` , ` +g ...
slotslnbpsd 28470 The slots ` Base ` , ` +g ...
lngndxnitvndx 28471 The slot for the line is n...
trkgstr 28472 Functionality of a Tarski ...
trkgbas 28473 The base set of a Tarski g...
trkgdist 28474 The measure of a distance ...
trkgitv 28475 The congruence relation in...
istrkgc 28482 Property of being a Tarski...
istrkgb 28483 Property of being a Tarski...
istrkgcb 28484 Property of being a Tarski...
istrkge 28485 Property of fulfilling Euc...
istrkgl 28486 Building lines from the se...
istrkgld 28487 Property of fulfilling the...
istrkg2ld 28488 Property of fulfilling the...
istrkg3ld 28489 Property of fulfilling the...
axtgcgrrflx 28490 Axiom of reflexivity of co...
axtgcgrid 28491 Axiom of identity of congr...
axtgsegcon 28492 Axiom of segment construct...
axtg5seg 28493 Five segments axiom, Axiom...
axtgbtwnid 28494 Identity of Betweenness. ...
axtgpasch 28495 Axiom of (Inner) Pasch, Ax...
axtgcont1 28496 Axiom of Continuity. Axio...
axtgcont 28497 Axiom of Continuity. Axio...
axtglowdim2 28498 Lower dimension axiom for ...
axtgupdim2 28499 Upper dimension axiom for ...
axtgeucl 28500 Euclid's Axiom. Axiom A10...
tgjustf 28501 Given any function ` F ` ,...
tgjustr 28502 Given any equivalence rela...
tgjustc1 28503 A justification for using ...
tgjustc2 28504 A justification for using ...
tgcgrcomimp 28505 Congruence commutes on the...
tgcgrcomr 28506 Congruence commutes on the...
tgcgrcoml 28507 Congruence commutes on the...
tgcgrcomlr 28508 Congruence commutes on bot...
tgcgreqb 28509 Congruence and equality. ...
tgcgreq 28510 Congruence and equality. ...
tgcgrneq 28511 Congruence and equality. ...
tgcgrtriv 28512 Degenerate segments are co...
tgcgrextend 28513 Link congruence over a pai...
tgsegconeq 28514 Two points that satisfy th...
tgbtwntriv2 28515 Betweenness always holds f...
tgbtwncom 28516 Betweenness commutes. The...
tgbtwncomb 28517 Betweenness commutes, bico...
tgbtwnne 28518 Betweenness and inequality...
tgbtwntriv1 28519 Betweenness always holds f...
tgbtwnswapid 28520 If you can swap the first ...
tgbtwnintr 28521 Inner transitivity law for...
tgbtwnexch3 28522 Exchange the first endpoin...
tgbtwnouttr2 28523 Outer transitivity law for...
tgbtwnexch2 28524 Exchange the outer point o...
tgbtwnouttr 28525 Outer transitivity law for...
tgbtwnexch 28526 Outer transitivity law for...
tgtrisegint 28527 A line segment between two...
tglowdim1 28528 Lower dimension axiom for ...
tglowdim1i 28529 Lower dimension axiom for ...
tgldimor 28530 Excluded-middle like state...
tgldim0eq 28531 In dimension zero, any two...
tgldim0itv 28532 In dimension zero, any two...
tgldim0cgr 28533 In dimension zero, any two...
tgbtwndiff 28534 There is always a ` c ` di...
tgdim01 28535 In geometries of dimension...
tgifscgr 28536 Inner five segment congrue...
tgcgrsub 28537 Removing identical parts f...
iscgrg 28540 The congruence property fo...
iscgrgd 28541 The property for two seque...
iscgrglt 28542 The property for two seque...
trgcgrg 28543 The property for two trian...
trgcgr 28544 Triangle congruence. (Con...
ercgrg 28545 The shape congruence relat...
tgcgrxfr 28546 A line segment can be divi...
cgr3id 28547 Reflexivity law for three-...
cgr3simp1 28548 Deduce segment congruence ...
cgr3simp2 28549 Deduce segment congruence ...
cgr3simp3 28550 Deduce segment congruence ...
cgr3swap12 28551 Permutation law for three-...
cgr3swap23 28552 Permutation law for three-...
cgr3swap13 28553 Permutation law for three-...
cgr3rotr 28554 Permutation law for three-...
cgr3rotl 28555 Permutation law for three-...
trgcgrcom 28556 Commutative law for three-...
cgr3tr 28557 Transitivity law for three...
tgbtwnxfr 28558 A condition for extending ...
tgcgr4 28559 Two quadrilaterals to be c...
isismt 28562 Property of being an isome...
ismot 28563 Property of being an isome...
motcgr 28564 Property of a motion: dist...
idmot 28565 The identity is a motion. ...
motf1o 28566 Motions are bijections. (...
motcl 28567 Closure of motions. (Cont...
motco 28568 The composition of two mot...
cnvmot 28569 The converse of a motion i...
motplusg 28570 The operation for motions ...
motgrp 28571 The motions of a geometry ...
motcgrg 28572 Property of a motion: dist...
motcgr3 28573 Property of a motion: dist...
tglng 28574 Lines of a Tarski Geometry...
tglnfn 28575 Lines as functions. (Cont...
tglnunirn 28576 Lines are sets of points. ...
tglnpt 28577 Lines are sets of points. ...
tglngne 28578 It takes two different poi...
tglngval 28579 The line going through poi...
tglnssp 28580 Lines are subset of the ge...
tgellng 28581 Property of lying on the l...
tgcolg 28582 We choose the notation ` (...
btwncolg1 28583 Betweenness implies coline...
btwncolg2 28584 Betweenness implies coline...
btwncolg3 28585 Betweenness implies coline...
colcom 28586 Swapping the points defini...
colrot1 28587 Rotating the points defini...
colrot2 28588 Rotating the points defini...
ncolcom 28589 Swapping non-colinear poin...
ncolrot1 28590 Rotating non-colinear poin...
ncolrot2 28591 Rotating non-colinear poin...
tgdim01ln 28592 In geometries of dimension...
ncoltgdim2 28593 If there are three non-col...
lnxfr 28594 Transfer law for colineari...
lnext 28595 Extend a line with a missi...
tgfscgr 28596 Congruence law for the gen...
lncgr 28597 Congruence rule for lines....
lnid 28598 Identity law for points on...
tgidinside 28599 Law for finding a point in...
tgbtwnconn1lem1 28600 Lemma for ~ tgbtwnconn1 . ...
tgbtwnconn1lem2 28601 Lemma for ~ tgbtwnconn1 . ...
tgbtwnconn1lem3 28602 Lemma for ~ tgbtwnconn1 . ...
tgbtwnconn1 28603 Connectivity law for betwe...
tgbtwnconn2 28604 Another connectivity law f...
tgbtwnconn3 28605 Inner connectivity law for...
tgbtwnconnln3 28606 Derive colinearity from be...
tgbtwnconn22 28607 Double connectivity law fo...
tgbtwnconnln1 28608 Derive colinearity from be...
tgbtwnconnln2 28609 Derive colinearity from be...
legval 28612 Value of the less-than rel...
legov 28613 Value of the less-than rel...
legov2 28614 An equivalent definition o...
legid 28615 Reflexivity of the less-th...
btwnleg 28616 Betweenness implies less-t...
legtrd 28617 Transitivity of the less-t...
legtri3 28618 Equality from the less-tha...
legtrid 28619 Trichotomy law for the les...
leg0 28620 Degenerated (zero-length) ...
legeq 28621 Deduce equality from "less...
legbtwn 28622 Deduce betweenness from "l...
tgcgrsub2 28623 Removing identical parts f...
ltgseg 28624 The set ` E ` denotes the ...
ltgov 28625 Strict "shorter than" geom...
legov3 28626 An equivalent definition o...
legso 28627 The "shorter than" relatio...
ishlg 28630 Rays : Definition 6.1 of ...
hlcomb 28631 The half-line relation com...
hlcomd 28632 The half-line relation com...
hlne1 28633 The half-line relation imp...
hlne2 28634 The half-line relation imp...
hlln 28635 The half-line relation imp...
hleqnid 28636 The endpoint does not belo...
hlid 28637 The half-line relation is ...
hltr 28638 The half-line relation is ...
hlbtwn 28639 Betweenness is a sufficien...
btwnhl1 28640 Deduce half-line from betw...
btwnhl2 28641 Deduce half-line from betw...
btwnhl 28642 Swap betweenness for a hal...
lnhl 28643 Either a point ` C ` on th...
hlcgrex 28644 Construct a point on a hal...
hlcgreulem 28645 Lemma for ~ hlcgreu . (Co...
hlcgreu 28646 The point constructed in ~...
btwnlng1 28647 Betweenness implies coline...
btwnlng2 28648 Betweenness implies coline...
btwnlng3 28649 Betweenness implies coline...
lncom 28650 Swapping the points defini...
lnrot1 28651 Rotating the points defini...
lnrot2 28652 Rotating the points defini...
ncolne1 28653 Non-colinear points are di...
ncolne2 28654 Non-colinear points are di...
tgisline 28655 The property of being a pr...
tglnne 28656 It takes two different poi...
tglndim0 28657 There are no lines in dime...
tgelrnln 28658 The property of being a pr...
tglineeltr 28659 Transitivity law for lines...
tglineelsb2 28660 If ` S ` lies on PQ , then...
tglinerflx1 28661 Reflexivity law for line m...
tglinerflx2 28662 Reflexivity law for line m...
tglinecom 28663 Commutativity law for line...
tglinethru 28664 If ` A ` is a line contain...
tghilberti1 28665 There is a line through an...
tghilberti2 28666 There is at most one line ...
tglinethrueu 28667 There is a unique line goi...
tglnne0 28668 A line ` A ` has at least ...
tglnpt2 28669 Find a second point on a l...
tglineintmo 28670 Two distinct lines interse...
tglineineq 28671 Two distinct lines interse...
tglineneq 28672 Given three non-colinear p...
tglineinteq 28673 Two distinct lines interse...
ncolncol 28674 Deduce non-colinearity fro...
coltr 28675 A transitivity law for col...
coltr3 28676 A transitivity law for col...
colline 28677 Three points are colinear ...
tglowdim2l 28678 Reformulation of the lower...
tglowdim2ln 28679 There is always one point ...
mirreu3 28682 Existential uniqueness of ...
mirval 28683 Value of the point inversi...
mirfv 28684 Value of the point inversi...
mircgr 28685 Property of the image by t...
mirbtwn 28686 Property of the image by t...
ismir 28687 Property of the image by t...
mirf 28688 Point inversion as functio...
mircl 28689 Closure of the point inver...
mirmir 28690 The point inversion functi...
mircom 28691 Variation on ~ mirmir . (...
mirreu 28692 Any point has a unique ant...
mireq 28693 Equality deduction for poi...
mirinv 28694 The only invariant point o...
mirne 28695 Mirror of non-center point...
mircinv 28696 The center point is invari...
mirf1o 28697 The point inversion functi...
miriso 28698 The point inversion functi...
mirbtwni 28699 Point inversion preserves ...
mirbtwnb 28700 Point inversion preserves ...
mircgrs 28701 Point inversion preserves ...
mirmir2 28702 Point inversion of a point...
mirmot 28703 Point investion is a motio...
mirln 28704 If two points are on the s...
mirln2 28705 If a point and its mirror ...
mirconn 28706 Point inversion of connect...
mirhl 28707 If two points ` X ` and ` ...
mirbtwnhl 28708 If the center of the point...
mirhl2 28709 Deduce half-line relation ...
mircgrextend 28710 Link congruence over a pai...
mirtrcgr 28711 Point inversion of one poi...
mirauto 28712 Point inversion preserves ...
miduniq 28713 Uniqueness of the middle p...
miduniq1 28714 Uniqueness of the middle p...
miduniq2 28715 If two point inversions co...
colmid 28716 Colinearity and equidistan...
symquadlem 28717 Lemma of the symetrial qua...
krippenlem 28718 Lemma for ~ krippen . We ...
krippen 28719 Krippenlemma (German for c...
midexlem 28720 Lemma for the existence of...
israg 28725 Property for 3 points A, B...
ragcom 28726 Commutative rule for right...
ragcol 28727 The right angle property i...
ragmir 28728 Right angle property is pr...
mirrag 28729 Right angle is conserved b...
ragtrivb 28730 Trivial right angle. Theo...
ragflat2 28731 Deduce equality from two r...
ragflat 28732 Deduce equality from two r...
ragtriva 28733 Trivial right angle. Theo...
ragflat3 28734 Right angle and colinearit...
ragcgr 28735 Right angle and colinearit...
motrag 28736 Right angles are preserved...
ragncol 28737 Right angle implies non-co...
perpln1 28738 Derive a line from perpend...
perpln2 28739 Derive a line from perpend...
isperp 28740 Property for 2 lines A, B ...
perpcom 28741 The "perpendicular" relati...
perpneq 28742 Two perpendicular lines ar...
isperp2 28743 Property for 2 lines A, B,...
isperp2d 28744 One direction of ~ isperp2...
ragperp 28745 Deduce that two lines are ...
footexALT 28746 Alternative version of ~ f...
footexlem1 28747 Lemma for ~ footex . (Con...
footexlem2 28748 Lemma for ~ footex . (Con...
footex 28749 From a point ` C ` outside...
foot 28750 From a point ` C ` outside...
footne 28751 Uniqueness of the foot poi...
footeq 28752 Uniqueness of the foot poi...
hlperpnel 28753 A point on a half-line whi...
perprag 28754 Deduce a right angle from ...
perpdragALT 28755 Deduce a right angle from ...
perpdrag 28756 Deduce a right angle from ...
colperp 28757 Deduce a perpendicularity ...
colperpexlem1 28758 Lemma for ~ colperp . Fir...
colperpexlem2 28759 Lemma for ~ colperpex . S...
colperpexlem3 28760 Lemma for ~ colperpex . C...
colperpex 28761 In dimension 2 and above, ...
mideulem2 28762 Lemma for ~ opphllem , whi...
opphllem 28763 Lemma 8.24 of [Schwabhause...
mideulem 28764 Lemma for ~ mideu . We ca...
midex 28765 Existence of the midpoint,...
mideu 28766 Existence and uniqueness o...
islnopp 28767 The property for two point...
islnoppd 28768 Deduce that ` A ` and ` B ...
oppne1 28769 Points lying on opposite s...
oppne2 28770 Points lying on opposite s...
oppne3 28771 Points lying on opposite s...
oppcom 28772 Commutativity rule for "op...
opptgdim2 28773 If two points opposite to ...
oppnid 28774 The "opposite to a line" r...
opphllem1 28775 Lemma for ~ opphl . (Cont...
opphllem2 28776 Lemma for ~ opphl . Lemma...
opphllem3 28777 Lemma for ~ opphl : We as...
opphllem4 28778 Lemma for ~ opphl . (Cont...
opphllem5 28779 Second part of Lemma 9.4 o...
opphllem6 28780 First part of Lemma 9.4 of...
oppperpex 28781 Restating ~ colperpex usin...
opphl 28782 If two points ` A ` and ` ...
outpasch 28783 Axiom of Pasch, outer form...
hlpasch 28784 An application of the axio...
ishpg 28787 Value of the half-plane re...
hpgbr 28788 Half-planes : property for...
hpgne1 28789 Points on the open half pl...
hpgne2 28790 Points on the open half pl...
lnopp2hpgb 28791 Theorem 9.8 of [Schwabhaus...
lnoppnhpg 28792 If two points lie on the o...
hpgerlem 28793 Lemma for the proof that t...
hpgid 28794 The half-plane relation is...
hpgcom 28795 The half-plane relation co...
hpgtr 28796 The half-plane relation is...
colopp 28797 Opposite sides of a line f...
colhp 28798 Half-plane relation for co...
hphl 28799 If two points are on the s...
midf 28804 Midpoint as a function. (...
midcl 28805 Closure of the midpoint. ...
ismidb 28806 Property of the midpoint. ...
midbtwn 28807 Betweenness of midpoint. ...
midcgr 28808 Congruence of midpoint. (...
midid 28809 Midpoint of a null segment...
midcom 28810 Commutativity rule for the...
mirmid 28811 Point inversion preserves ...
lmieu 28812 Uniqueness of the line mir...
lmif 28813 Line mirror as a function....
lmicl 28814 Closure of the line mirror...
islmib 28815 Property of the line mirro...
lmicom 28816 The line mirroring functio...
lmilmi 28817 Line mirroring is an invol...
lmireu 28818 Any point has a unique ant...
lmieq 28819 Equality deduction for lin...
lmiinv 28820 The invariants of the line...
lmicinv 28821 The mirroring line is an i...
lmimid 28822 If we have a right angle, ...
lmif1o 28823 The line mirroring functio...
lmiisolem 28824 Lemma for ~ lmiiso . (Con...
lmiiso 28825 The line mirroring functio...
lmimot 28826 Line mirroring is a motion...
hypcgrlem1 28827 Lemma for ~ hypcgr , case ...
hypcgrlem2 28828 Lemma for ~ hypcgr , case ...
hypcgr 28829 If the catheti of two righ...
lmiopp 28830 Line mirroring produces po...
lnperpex 28831 Existence of a perpendicul...
trgcopy 28832 Triangle construction: a c...
trgcopyeulem 28833 Lemma for ~ trgcopyeu . (...
trgcopyeu 28834 Triangle construction: a c...
iscgra 28837 Property for two angles AB...
iscgra1 28838 A special version of ~ isc...
iscgrad 28839 Sufficient conditions for ...
cgrane1 28840 Angles imply inequality. ...
cgrane2 28841 Angles imply inequality. ...
cgrane3 28842 Angles imply inequality. ...
cgrane4 28843 Angles imply inequality. ...
cgrahl1 28844 Angle congruence is indepe...
cgrahl2 28845 Angle congruence is indepe...
cgracgr 28846 First direction of proposi...
cgraid 28847 Angle congruence is reflex...
cgraswap 28848 Swap rays in a congruence ...
cgrcgra 28849 Triangle congruence implie...
cgracom 28850 Angle congruence commutes....
cgratr 28851 Angle congruence is transi...
flatcgra 28852 Flat angles are congruent....
cgraswaplr 28853 Swap both side of angle co...
cgrabtwn 28854 Angle congruence preserves...
cgrahl 28855 Angle congruence preserves...
cgracol 28856 Angle congruence preserves...
cgrancol 28857 Angle congruence preserves...
dfcgra2 28858 This is the full statement...
sacgr 28859 Supplementary angles of co...
oacgr 28860 Vertical angle theorem. V...
acopy 28861 Angle construction. Theor...
acopyeu 28862 Angle construction. Theor...
isinag 28866 Property for point ` X ` t...
isinagd 28867 Sufficient conditions for ...
inagflat 28868 Any point lies in a flat a...
inagswap 28869 Swap the order of the half...
inagne1 28870 Deduce inequality from the...
inagne2 28871 Deduce inequality from the...
inagne3 28872 Deduce inequality from the...
inaghl 28873 The "point lie in angle" r...
isleag 28875 Geometrical "less than" pr...
isleagd 28876 Sufficient condition for "...
leagne1 28877 Deduce inequality from the...
leagne2 28878 Deduce inequality from the...
leagne3 28879 Deduce inequality from the...
leagne4 28880 Deduce inequality from the...
cgrg3col4 28881 Lemma 11.28 of [Schwabhaus...
tgsas1 28882 First congruence theorem: ...
tgsas 28883 First congruence theorem: ...
tgsas2 28884 First congruence theorem: ...
tgsas3 28885 First congruence theorem: ...
tgasa1 28886 Second congruence theorem:...
tgasa 28887 Second congruence theorem:...
tgsss1 28888 Third congruence theorem: ...
tgsss2 28889 Third congruence theorem: ...
tgsss3 28890 Third congruence theorem: ...
dfcgrg2 28891 Congruence for two triangl...
isoas 28892 Congruence theorem for iso...
iseqlg 28895 Property of a triangle bei...
iseqlgd 28896 Condition for a triangle t...
f1otrgds 28897 Convenient lemma for ~ f1o...
f1otrgitv 28898 Convenient lemma for ~ f1o...
f1otrg 28899 A bijection between bases ...
f1otrge 28900 A bijection between bases ...
ttgval 28903 Define a function to augme...
ttgvalOLD 28904 Obsolete proof of ~ ttgval...
ttglem 28905 Lemma for ~ ttgbas , ~ ttg...
ttglemOLD 28906 Obsolete version of ~ ttgl...
ttgbas 28907 The base set of a subcompl...
ttgbasOLD 28908 Obsolete proof of ~ ttgbas...
ttgplusg 28909 The addition operation of ...
ttgplusgOLD 28910 Obsolete proof of ~ ttgplu...
ttgsub 28911 The subtraction operation ...
ttgvsca 28912 The scalar product of a su...
ttgvscaOLD 28913 Obsolete proof of ~ ttgvsc...
ttgds 28914 The metric of a subcomplex...
ttgdsOLD 28915 Obsolete proof of ~ ttgds ...
ttgitvval 28916 Betweenness for a subcompl...
ttgelitv 28917 Betweenness for a subcompl...
ttgbtwnid 28918 Any subcomplex module equi...
ttgcontlem1 28919 Lemma for % ttgcont . (Co...
xmstrkgc 28920 Any metric space fulfills ...
cchhllem 28921 Lemma for chlbas and chlvs...
cchhllemOLD 28922 Obsolete version of ~ cchh...
elee 28929 Membership in a Euclidean ...
mptelee 28930 A condition for a mapping ...
eleenn 28931 If ` A ` is in ` ( EE `` N...
eleei 28932 The forward direction of ~...
eedimeq 28933 A point belongs to at most...
brbtwn 28934 The binary relation form o...
brcgr 28935 The binary relation form o...
fveere 28936 The function value of a po...
fveecn 28937 The function value of a po...
eqeefv 28938 Two points are equal iff t...
eqeelen 28939 Two points are equal iff t...
brbtwn2 28940 Alternate characterization...
colinearalglem1 28941 Lemma for ~ colinearalg . ...
colinearalglem2 28942 Lemma for ~ colinearalg . ...
colinearalglem3 28943 Lemma for ~ colinearalg . ...
colinearalglem4 28944 Lemma for ~ colinearalg . ...
colinearalg 28945 An algebraic characterizat...
eleesub 28946 Membership of a subtractio...
eleesubd 28947 Membership of a subtractio...
axdimuniq 28948 The unique dimension axiom...
axcgrrflx 28949 ` A ` is as far from ` B `...
axcgrtr 28950 Congruence is transitive. ...
axcgrid 28951 If there is no distance be...
axsegconlem1 28952 Lemma for ~ axsegcon . Ha...
axsegconlem2 28953 Lemma for ~ axsegcon . Sh...
axsegconlem3 28954 Lemma for ~ axsegcon . Sh...
axsegconlem4 28955 Lemma for ~ axsegcon . Sh...
axsegconlem5 28956 Lemma for ~ axsegcon . Sh...
axsegconlem6 28957 Lemma for ~ axsegcon . Sh...
axsegconlem7 28958 Lemma for ~ axsegcon . Sh...
axsegconlem8 28959 Lemma for ~ axsegcon . Sh...
axsegconlem9 28960 Lemma for ~ axsegcon . Sh...
axsegconlem10 28961 Lemma for ~ axsegcon . Sh...
axsegcon 28962 Any segment ` A B ` can be...
ax5seglem1 28963 Lemma for ~ ax5seg . Rexp...
ax5seglem2 28964 Lemma for ~ ax5seg . Rexp...
ax5seglem3a 28965 Lemma for ~ ax5seg . (Con...
ax5seglem3 28966 Lemma for ~ ax5seg . Comb...
ax5seglem4 28967 Lemma for ~ ax5seg . Give...
ax5seglem5 28968 Lemma for ~ ax5seg . If `...
ax5seglem6 28969 Lemma for ~ ax5seg . Give...
ax5seglem7 28970 Lemma for ~ ax5seg . An a...
ax5seglem8 28971 Lemma for ~ ax5seg . Use ...
ax5seglem9 28972 Lemma for ~ ax5seg . Take...
ax5seg 28973 The five segment axiom. T...
axbtwnid 28974 Points are indivisible. T...
axpaschlem 28975 Lemma for ~ axpasch . Set...
axpasch 28976 The inner Pasch axiom. Ta...
axlowdimlem1 28977 Lemma for ~ axlowdim . Es...
axlowdimlem2 28978 Lemma for ~ axlowdim . Sh...
axlowdimlem3 28979 Lemma for ~ axlowdim . Se...
axlowdimlem4 28980 Lemma for ~ axlowdim . Se...
axlowdimlem5 28981 Lemma for ~ axlowdim . Sh...
axlowdimlem6 28982 Lemma for ~ axlowdim . Sh...
axlowdimlem7 28983 Lemma for ~ axlowdim . Se...
axlowdimlem8 28984 Lemma for ~ axlowdim . Ca...
axlowdimlem9 28985 Lemma for ~ axlowdim . Ca...
axlowdimlem10 28986 Lemma for ~ axlowdim . Se...
axlowdimlem11 28987 Lemma for ~ axlowdim . Ca...
axlowdimlem12 28988 Lemma for ~ axlowdim . Ca...
axlowdimlem13 28989 Lemma for ~ axlowdim . Es...
axlowdimlem14 28990 Lemma for ~ axlowdim . Ta...
axlowdimlem15 28991 Lemma for ~ axlowdim . Se...
axlowdimlem16 28992 Lemma for ~ axlowdim . Se...
axlowdimlem17 28993 Lemma for ~ axlowdim . Es...
axlowdim1 28994 The lower dimension axiom ...
axlowdim2 28995 The lower two-dimensional ...
axlowdim 28996 The general lower dimensio...
axeuclidlem 28997 Lemma for ~ axeuclid . Ha...
axeuclid 28998 Euclid's axiom. Take an a...
axcontlem1 28999 Lemma for ~ axcont . Chan...
axcontlem2 29000 Lemma for ~ axcont . The ...
axcontlem3 29001 Lemma for ~ axcont . Give...
axcontlem4 29002 Lemma for ~ axcont . Give...
axcontlem5 29003 Lemma for ~ axcont . Comp...
axcontlem6 29004 Lemma for ~ axcont . Stat...
axcontlem7 29005 Lemma for ~ axcont . Give...
axcontlem8 29006 Lemma for ~ axcont . A po...
axcontlem9 29007 Lemma for ~ axcont . Give...
axcontlem10 29008 Lemma for ~ axcont . Give...
axcontlem11 29009 Lemma for ~ axcont . Elim...
axcontlem12 29010 Lemma for ~ axcont . Elim...
axcont 29011 The axiom of continuity. ...
eengv 29014 The value of the Euclidean...
eengstr 29015 The Euclidean geometry as ...
eengbas 29016 The Base of the Euclidean ...
ebtwntg 29017 The betweenness relation u...
ecgrtg 29018 The congruence relation us...
elntg 29019 The line definition in the...
elntg2 29020 The line definition in the...
eengtrkg 29021 The geometry structure for...
eengtrkge 29022 The geometry structure for...
edgfid 29025 Utility theorem: index-ind...
edgfndx 29026 Index value of the ~ df-ed...
edgfndxnn 29027 The index value of the edg...
edgfndxid 29028 The value of the edge func...
edgfndxidOLD 29029 Obsolete version of ~ edgf...
basendxltedgfndx 29030 The index value of the ` B...
baseltedgfOLD 29031 Obsolete proof of ~ basend...
basendxnedgfndx 29032 The slots ` Base ` and ` ....
vtxval 29037 The set of vertices of a g...
iedgval 29038 The set of indexed edges o...
1vgrex 29039 A graph with at least one ...
opvtxval 29040 The set of vertices of a g...
opvtxfv 29041 The set of vertices of a g...
opvtxov 29042 The set of vertices of a g...
opiedgval 29043 The set of indexed edges o...
opiedgfv 29044 The set of indexed edges o...
opiedgov 29045 The set of indexed edges o...
opvtxfvi 29046 The set of vertices of a g...
opiedgfvi 29047 The set of indexed edges o...
funvtxdmge2val 29048 The set of vertices of an ...
funiedgdmge2val 29049 The set of indexed edges o...
funvtxdm2val 29050 The set of vertices of an ...
funiedgdm2val 29051 The set of indexed edges o...
funvtxval0 29052 The set of vertices of an ...
basvtxval 29053 The set of vertices of a g...
edgfiedgval 29054 The set of indexed edges o...
funvtxval 29055 The set of vertices of a g...
funiedgval 29056 The set of indexed edges o...
structvtxvallem 29057 Lemma for ~ structvtxval a...
structvtxval 29058 The set of vertices of an ...
structiedg0val 29059 The set of indexed edges o...
structgrssvtxlem 29060 Lemma for ~ structgrssvtx ...
structgrssvtx 29061 The set of vertices of a g...
structgrssiedg 29062 The set of indexed edges o...
struct2grstr 29063 A graph represented as an ...
struct2grvtx 29064 The set of vertices of a g...
struct2griedg 29065 The set of indexed edges o...
graop 29066 Any representation of a gr...
grastruct 29067 Any representation of a gr...
gropd 29068 If any representation of a...
grstructd 29069 If any representation of a...
gropeld 29070 If any representation of a...
grstructeld 29071 If any representation of a...
setsvtx 29072 The vertices of a structur...
setsiedg 29073 The (indexed) edges of a s...
snstrvtxval 29074 The set of vertices of a g...
snstriedgval 29075 The set of indexed edges o...
vtxval0 29076 Degenerated case 1 for ver...
iedgval0 29077 Degenerated case 1 for edg...
vtxvalsnop 29078 Degenerated case 2 for ver...
iedgvalsnop 29079 Degenerated case 2 for edg...
vtxval3sn 29080 Degenerated case 3 for ver...
iedgval3sn 29081 Degenerated case 3 for edg...
vtxvalprc 29082 Degenerated case 4 for ver...
iedgvalprc 29083 Degenerated case 4 for edg...
edgval 29086 The edges of a graph. (Co...
iedgedg 29087 An indexed edge is an edge...
edgopval 29088 The edges of a graph repre...
edgov 29089 The edges of a graph repre...
edgstruct 29090 The edges of a graph repre...
edgiedgb 29091 A set is an edge iff it is...
edg0iedg0 29092 There is no edge in a grap...
isuhgr 29097 The predicate "is an undir...
isushgr 29098 The predicate "is an undir...
uhgrf 29099 The edge function of an un...
ushgrf 29100 The edge function of an un...
uhgrss 29101 An edge is a subset of ver...
uhgreq12g 29102 If two sets have the same ...
uhgrfun 29103 The edge function of an un...
uhgrn0 29104 An edge is a nonempty subs...
lpvtx 29105 The endpoints of a loop (w...
ushgruhgr 29106 An undirected simple hyper...
isuhgrop 29107 The property of being an u...
uhgr0e 29108 The empty graph, with vert...
uhgr0vb 29109 The null graph, with no ve...
uhgr0 29110 The null graph represented...
uhgrun 29111 The union ` U ` of two (un...
uhgrunop 29112 The union of two (undirect...
ushgrun 29113 The union ` U ` of two (un...
ushgrunop 29114 The union of two (undirect...
uhgrstrrepe 29115 Replacing (or adding) the ...
incistruhgr 29116 An _incidence structure_ `...
isupgr 29121 The property of being an u...
wrdupgr 29122 The property of being an u...
upgrf 29123 The edge function of an un...
upgrfn 29124 The edge function of an un...
upgrss 29125 An edge is a subset of ver...
upgrn0 29126 An edge is a nonempty subs...
upgrle 29127 An edge of an undirected p...
upgrfi 29128 An edge is a finite subset...
upgrex 29129 An edge is an unordered pa...
upgrbi 29130 Show that an unordered pai...
upgrop 29131 A pseudograph represented ...
isumgr 29132 The property of being an u...
isumgrs 29133 The simplified property of...
wrdumgr 29134 The property of being an u...
umgrf 29135 The edge function of an un...
umgrfn 29136 The edge function of an un...
umgredg2 29137 An edge of a multigraph ha...
umgrbi 29138 Show that an unordered pai...
upgruhgr 29139 An undirected pseudograph ...
umgrupgr 29140 An undirected multigraph i...
umgruhgr 29141 An undirected multigraph i...
upgrle2 29142 An edge of an undirected p...
umgrnloopv 29143 In a multigraph, there is ...
umgredgprv 29144 In a multigraph, an edge i...
umgrnloop 29145 In a multigraph, there is ...
umgrnloop0 29146 A multigraph has no loops....
umgr0e 29147 The empty graph, with vert...
upgr0e 29148 The empty graph, with vert...
upgr1elem 29149 Lemma for ~ upgr1e and ~ u...
upgr1e 29150 A pseudograph with one edg...
upgr0eop 29151 The empty graph, with vert...
upgr1eop 29152 A pseudograph with one edg...
upgr0eopALT 29153 Alternate proof of ~ upgr0...
upgr1eopALT 29154 Alternate proof of ~ upgr1...
upgrun 29155 The union ` U ` of two pse...
upgrunop 29156 The union of two pseudogra...
umgrun 29157 The union ` U ` of two mul...
umgrunop 29158 The union of two multigrap...
umgrislfupgrlem 29159 Lemma for ~ umgrislfupgr a...
umgrislfupgr 29160 A multigraph is a loop-fre...
lfgredgge2 29161 An edge of a loop-free gra...
lfgrnloop 29162 A loop-free graph has no l...
uhgredgiedgb 29163 In a hypergraph, a set is ...
uhgriedg0edg0 29164 A hypergraph has no edges ...
uhgredgn0 29165 An edge of a hypergraph is...
edguhgr 29166 An edge of a hypergraph is...
uhgredgrnv 29167 An edge of a hypergraph co...
uhgredgss 29168 The set of edges of a hype...
upgredgss 29169 The set of edges of a pseu...
umgredgss 29170 The set of edges of a mult...
edgupgr 29171 Properties of an edge of a...
edgumgr 29172 Properties of an edge of a...
uhgrvtxedgiedgb 29173 In a hypergraph, a vertex ...
upgredg 29174 For each edge in a pseudog...
umgredg 29175 For each edge in a multigr...
upgrpredgv 29176 An edge of a pseudograph a...
umgrpredgv 29177 An edge of a multigraph al...
upgredg2vtx 29178 For a vertex incident to a...
upgredgpr 29179 If a proper pair (of verti...
edglnl 29180 The edges incident with a ...
numedglnl 29181 The number of edges incide...
umgredgne 29182 An edge of a multigraph al...
umgrnloop2 29183 A multigraph has no loops....
umgredgnlp 29184 An edge of a multigraph is...
isuspgr 29189 The property of being a si...
isusgr 29190 The property of being a si...
uspgrf 29191 The edge function of a sim...
usgrf 29192 The edge function of a sim...
isusgrs 29193 The property of being a si...
usgrfs 29194 The edge function of a sim...
usgrfun 29195 The edge function of a sim...
usgredgss 29196 The set of edges of a simp...
edgusgr 29197 An edge of a simple graph ...
isuspgrop 29198 The property of being an u...
isusgrop 29199 The property of being an u...
usgrop 29200 A simple graph represented...
isausgr 29201 The property of an unorder...
ausgrusgrb 29202 The equivalence of the def...
usgrausgri 29203 A simple graph represented...
ausgrumgri 29204 If an alternatively define...
ausgrusgri 29205 The equivalence of the def...
usgrausgrb 29206 The equivalence of the def...
usgredgop 29207 An edge of a simple graph ...
usgrf1o 29208 The edge function of a sim...
usgrf1 29209 The edge function of a sim...
uspgrf1oedg 29210 The edge function of a sim...
usgrss 29211 An edge is a subset of ver...
uspgredgiedg 29212 In a simple pseudograph, f...
uspgriedgedg 29213 In a simple pseudograph, f...
uspgrushgr 29214 A simple pseudograph is an...
uspgrupgr 29215 A simple pseudograph is an...
uspgrupgrushgr 29216 A graph is a simple pseudo...
usgruspgr 29217 A simple graph is a simple...
usgrumgr 29218 A simple graph is an undir...
usgrumgruspgr 29219 A graph is a simple graph ...
usgruspgrb 29220 A class is a simple graph ...
uspgruhgr 29221 An undirected simple pseud...
usgrupgr 29222 A simple graph is an undir...
usgruhgr 29223 A simple graph is an undir...
usgrislfuspgr 29224 A simple graph is a loop-f...
uspgrun 29225 The union ` U ` of two sim...
uspgrunop 29226 The union of two simple ps...
usgrun 29227 The union ` U ` of two sim...
usgrunop 29228 The union of two simple gr...
usgredg2 29229 The value of the "edge fun...
usgredg2ALT 29230 Alternate proof of ~ usgre...
usgredgprv 29231 In a simple graph, an edge...
usgredgprvALT 29232 Alternate proof of ~ usgre...
usgredgppr 29233 An edge of a simple graph ...
usgrpredgv 29234 An edge of a simple graph ...
edgssv2 29235 An edge of a simple graph ...
usgredg 29236 For each edge in a simple ...
usgrnloopv 29237 In a simple graph, there i...
usgrnloopvALT 29238 Alternate proof of ~ usgrn...
usgrnloop 29239 In a simple graph, there i...
usgrnloopALT 29240 Alternate proof of ~ usgrn...
usgrnloop0 29241 A simple graph has no loop...
usgrnloop0ALT 29242 Alternate proof of ~ usgrn...
usgredgne 29243 An edge of a simple graph ...
usgrf1oedg 29244 The edge function of a sim...
uhgr2edg 29245 If a vertex is adjacent to...
umgr2edg 29246 If a vertex is adjacent to...
usgr2edg 29247 If a vertex is adjacent to...
umgr2edg1 29248 If a vertex is adjacent to...
usgr2edg1 29249 If a vertex is adjacent to...
umgrvad2edg 29250 If a vertex is adjacent to...
umgr2edgneu 29251 If a vertex is adjacent to...
usgrsizedg 29252 In a simple graph, the siz...
usgredg3 29253 The value of the "edge fun...
usgredg4 29254 For a vertex incident to a...
usgredgreu 29255 For a vertex incident to a...
usgredg2vtx 29256 For a vertex incident to a...
uspgredg2vtxeu 29257 For a vertex incident to a...
usgredg2vtxeu 29258 For a vertex incident to a...
usgredg2vtxeuALT 29259 Alternate proof of ~ usgre...
uspgredg2vlem 29260 Lemma for ~ uspgredg2v . ...
uspgredg2v 29261 In a simple pseudograph, t...
usgredg2vlem1 29262 Lemma 1 for ~ usgredg2v . ...
usgredg2vlem2 29263 Lemma 2 for ~ usgredg2v . ...
usgredg2v 29264 In a simple graph, the map...
usgriedgleord 29265 Alternate version of ~ usg...
ushgredgedg 29266 In a simple hypergraph the...
usgredgedg 29267 In a simple graph there is...
ushgredgedgloop 29268 In a simple hypergraph the...
uspgredgleord 29269 In a simple pseudograph th...
usgredgleord 29270 In a simple graph the numb...
usgredgleordALT 29271 Alternate proof for ~ usgr...
usgrstrrepe 29272 Replacing (or adding) the ...
usgr0e 29273 The empty graph, with vert...
usgr0vb 29274 The null graph, with no ve...
uhgr0v0e 29275 The null graph, with no ve...
uhgr0vsize0 29276 The size of a hypergraph w...
uhgr0edgfi 29277 A graph of order 0 (i.e. w...
usgr0v 29278 The null graph, with no ve...
uhgr0vusgr 29279 The null graph, with no ve...
usgr0 29280 The null graph represented...
uspgr1e 29281 A simple pseudograph with ...
usgr1e 29282 A simple graph with one ed...
usgr0eop 29283 The empty graph, with vert...
uspgr1eop 29284 A simple pseudograph with ...
uspgr1ewop 29285 A simple pseudograph with ...
uspgr1v1eop 29286 A simple pseudograph with ...
usgr1eop 29287 A simple graph with (at le...
uspgr2v1e2w 29288 A simple pseudograph with ...
usgr2v1e2w 29289 A simple graph with two ve...
edg0usgr 29290 A class without edges is a...
lfuhgr1v0e 29291 A loop-free hypergraph wit...
usgr1vr 29292 A simple graph with one ve...
usgr1v 29293 A class with one (or no) v...
usgr1v0edg 29294 A class with one (or no) v...
usgrexmpldifpr 29295 Lemma for ~ usgrexmpledg :...
usgrexmplef 29296 Lemma for ~ usgrexmpl . (...
usgrexmpllem 29297 Lemma for ~ usgrexmpl . (...
usgrexmplvtx 29298 The vertices ` 0 , 1 , 2 ,...
usgrexmpledg 29299 The edges ` { 0 , 1 } , { ...
usgrexmpl 29300 ` G ` is a simple graph of...
griedg0prc 29301 The class of empty graphs ...
griedg0ssusgr 29302 The class of all simple gr...
usgrprc 29303 The class of simple graphs...
relsubgr 29306 The class of the subgraph ...
subgrv 29307 If a class is a subgraph o...
issubgr 29308 The property of a set to b...
issubgr2 29309 The property of a set to b...
subgrprop 29310 The properties of a subgra...
subgrprop2 29311 The properties of a subgra...
uhgrissubgr 29312 The property of a hypergra...
subgrprop3 29313 The properties of a subgra...
egrsubgr 29314 An empty graph consisting ...
0grsubgr 29315 The null graph (represente...
0uhgrsubgr 29316 The null graph (as hypergr...
uhgrsubgrself 29317 A hypergraph is a subgraph...
subgrfun 29318 The edge function of a sub...
subgruhgrfun 29319 The edge function of a sub...
subgreldmiedg 29320 An element of the domain o...
subgruhgredgd 29321 An edge of a subgraph of a...
subumgredg2 29322 An edge of a subgraph of a...
subuhgr 29323 A subgraph of a hypergraph...
subupgr 29324 A subgraph of a pseudograp...
subumgr 29325 A subgraph of a multigraph...
subusgr 29326 A subgraph of a simple gra...
uhgrspansubgrlem 29327 Lemma for ~ uhgrspansubgr ...
uhgrspansubgr 29328 A spanning subgraph ` S ` ...
uhgrspan 29329 A spanning subgraph ` S ` ...
upgrspan 29330 A spanning subgraph ` S ` ...
umgrspan 29331 A spanning subgraph ` S ` ...
usgrspan 29332 A spanning subgraph ` S ` ...
uhgrspanop 29333 A spanning subgraph of a h...
upgrspanop 29334 A spanning subgraph of a p...
umgrspanop 29335 A spanning subgraph of a m...
usgrspanop 29336 A spanning subgraph of a s...
uhgrspan1lem1 29337 Lemma 1 for ~ uhgrspan1 . ...
uhgrspan1lem2 29338 Lemma 2 for ~ uhgrspan1 . ...
uhgrspan1lem3 29339 Lemma 3 for ~ uhgrspan1 . ...
uhgrspan1 29340 The induced subgraph ` S `...
upgrreslem 29341 Lemma for ~ upgrres . (Co...
umgrreslem 29342 Lemma for ~ umgrres and ~ ...
upgrres 29343 A subgraph obtained by rem...
umgrres 29344 A subgraph obtained by rem...
usgrres 29345 A subgraph obtained by rem...
upgrres1lem1 29346 Lemma 1 for ~ upgrres1 . ...
umgrres1lem 29347 Lemma for ~ umgrres1 . (C...
upgrres1lem2 29348 Lemma 2 for ~ upgrres1 . ...
upgrres1lem3 29349 Lemma 3 for ~ upgrres1 . ...
upgrres1 29350 A pseudograph obtained by ...
umgrres1 29351 A multigraph obtained by r...
usgrres1 29352 Restricting a simple graph...
isfusgr 29355 The property of being a fi...
fusgrvtxfi 29356 A finite simple graph has ...
isfusgrf1 29357 The property of being a fi...
isfusgrcl 29358 The property of being a fi...
fusgrusgr 29359 A finite simple graph is a...
opfusgr 29360 A finite simple graph repr...
usgredgffibi 29361 The number of edges in a s...
fusgredgfi 29362 In a finite simple graph t...
usgr1v0e 29363 The size of a (finite) sim...
usgrfilem 29364 In a finite simple graph, ...
fusgrfisbase 29365 Induction base for ~ fusgr...
fusgrfisstep 29366 Induction step in ~ fusgrf...
fusgrfis 29367 A finite simple graph is o...
fusgrfupgrfs 29368 A finite simple graph is a...
nbgrprc0 29371 The set of neighbors is em...
nbgrcl 29372 If a class ` X ` has at le...
nbgrval 29373 The set of neighbors of a ...
dfnbgr2 29374 Alternate definition of th...
dfnbgr3 29375 Alternate definition of th...
nbgrnvtx0 29376 If a class ` X ` is not a ...
nbgrel 29377 Characterization of a neig...
nbgrisvtx 29378 Every neighbor ` N ` of a ...
nbgrssvtx 29379 The neighbors of a vertex ...
nbuhgr 29380 The set of neighbors of a ...
nbupgr 29381 The set of neighbors of a ...
nbupgrel 29382 A neighbor of a vertex in ...
nbumgrvtx 29383 The set of neighbors of a ...
nbumgr 29384 The set of neighbors of an...
nbusgrvtx 29385 The set of neighbors of a ...
nbusgr 29386 The set of neighbors of an...
nbgr2vtx1edg 29387 If a graph has two vertice...
nbuhgr2vtx1edgblem 29388 Lemma for ~ nbuhgr2vtx1edg...
nbuhgr2vtx1edgb 29389 If a hypergraph has two ve...
nbusgreledg 29390 A class/vertex is a neighb...
uhgrnbgr0nb 29391 A vertex which is not endp...
nbgr0vtx 29392 In a null graph (with no v...
nbgr0edglem 29393 Lemma for ~ nbgr0edg and ~...
nbgr0edg 29394 In an empty graph (with no...
nbgr1vtx 29395 In a graph with one vertex...
nbgrnself 29396 A vertex in a graph is not...
nbgrnself2 29397 A class ` X ` is not a nei...
nbgrssovtx 29398 The neighbors of a vertex ...
nbgrssvwo2 29399 The neighbors of a vertex ...
nbgrsym 29400 In a graph, the neighborho...
nbupgrres 29401 The neighborhood of a vert...
usgrnbcnvfv 29402 Applying the edge function...
nbusgredgeu 29403 For each neighbor of a ver...
edgnbusgreu 29404 For each edge incident to ...
nbusgredgeu0 29405 For each neighbor of a ver...
nbusgrf1o0 29406 The mapping of neighbors o...
nbusgrf1o1 29407 The set of neighbors of a ...
nbusgrf1o 29408 The set of neighbors of a ...
nbedgusgr 29409 The number of neighbors of...
edgusgrnbfin 29410 The number of neighbors of...
nbusgrfi 29411 The class of neighbors of ...
nbfiusgrfi 29412 The class of neighbors of ...
hashnbusgrnn0 29413 The number of neighbors of...
nbfusgrlevtxm1 29414 The number of neighbors of...
nbfusgrlevtxm2 29415 If there is a vertex which...
nbusgrvtxm1 29416 If the number of neighbors...
nb3grprlem1 29417 Lemma 1 for ~ nb3grpr . (...
nb3grprlem2 29418 Lemma 2 for ~ nb3grpr . (...
nb3grpr 29419 The neighbors of a vertex ...
nb3grpr2 29420 The neighbors of a vertex ...
nb3gr2nb 29421 If the neighbors of two ve...
uvtxval 29424 The set of all universal v...
uvtxel 29425 A universal vertex, i.e. a...
uvtxisvtx 29426 A universal vertex is a ve...
uvtxssvtx 29427 The set of the universal v...
vtxnbuvtx 29428 A universal vertex has all...
uvtxnbgrss 29429 A universal vertex has all...
uvtxnbgrvtx 29430 A universal vertex is neig...
uvtx0 29431 There is no universal vert...
isuvtx 29432 The set of all universal v...
uvtxel1 29433 Characterization of a univ...
uvtx01vtx 29434 If a graph/class has no ed...
uvtx2vtx1edg 29435 If a graph has two vertice...
uvtx2vtx1edgb 29436 If a hypergraph has two ve...
uvtxnbgr 29437 A universal vertex has all...
uvtxnbgrb 29438 A vertex is universal iff ...
uvtxusgr 29439 The set of all universal v...
uvtxusgrel 29440 A universal vertex, i.e. a...
uvtxnm1nbgr 29441 A universal vertex has ` n...
nbusgrvtxm1uvtx 29442 If the number of neighbors...
uvtxnbvtxm1 29443 A universal vertex has ` n...
nbupgruvtxres 29444 The neighborhood of a univ...
uvtxupgrres 29445 A universal vertex is univ...
cplgruvtxb 29450 A graph ` G ` is complete ...
prcliscplgr 29451 A proper class (representi...
iscplgr 29452 The property of being a co...
iscplgrnb 29453 A graph is complete iff al...
iscplgredg 29454 A graph ` G ` is complete ...
iscusgr 29455 The property of being a co...
cusgrusgr 29456 A complete simple graph is...
cusgrcplgr 29457 A complete simple graph is...
iscusgrvtx 29458 A simple graph is complete...
cusgruvtxb 29459 A simple graph is complete...
iscusgredg 29460 A simple graph is complete...
cusgredg 29461 In a complete simple graph...
cplgr0 29462 The null graph (with no ve...
cusgr0 29463 The null graph (with no ve...
cplgr0v 29464 A null graph (with no vert...
cusgr0v 29465 A graph with no vertices a...
cplgr1vlem 29466 Lemma for ~ cplgr1v and ~ ...
cplgr1v 29467 A graph with one vertex is...
cusgr1v 29468 A graph with one vertex an...
cplgr2v 29469 An undirected hypergraph w...
cplgr2vpr 29470 An undirected hypergraph w...
nbcplgr 29471 In a complete graph, each ...
cplgr3v 29472 A pseudograph with three (...
cusgr3vnbpr 29473 The neighbors of a vertex ...
cplgrop 29474 A complete graph represent...
cusgrop 29475 A complete simple graph re...
cusgrexilem1 29476 Lemma 1 for ~ cusgrexi . ...
usgrexilem 29477 Lemma for ~ usgrexi . (Co...
usgrexi 29478 An arbitrary set regarded ...
cusgrexilem2 29479 Lemma 2 for ~ cusgrexi . ...
cusgrexi 29480 An arbitrary set ` V ` reg...
cusgrexg 29481 For each set there is a se...
structtousgr 29482 Any (extensible) structure...
structtocusgr 29483 Any (extensible) structure...
cffldtocusgr 29484 The field of complex numbe...
cffldtocusgrOLD 29485 Obsolete version of ~ cffl...
cusgrres 29486 Restricting a complete sim...
cusgrsizeindb0 29487 Base case of the induction...
cusgrsizeindb1 29488 Base case of the induction...
cusgrsizeindslem 29489 Lemma for ~ cusgrsizeinds ...
cusgrsizeinds 29490 Part 1 of induction step i...
cusgrsize2inds 29491 Induction step in ~ cusgrs...
cusgrsize 29492 The size of a finite compl...
cusgrfilem1 29493 Lemma 1 for ~ cusgrfi . (...
cusgrfilem2 29494 Lemma 2 for ~ cusgrfi . (...
cusgrfilem3 29495 Lemma 3 for ~ cusgrfi . (...
cusgrfi 29496 If the size of a complete ...
usgredgsscusgredg 29497 A simple graph is a subgra...
usgrsscusgr 29498 A simple graph is a subgra...
sizusglecusglem1 29499 Lemma 1 for ~ sizusglecusg...
sizusglecusglem2 29500 Lemma 2 for ~ sizusglecusg...
sizusglecusg 29501 The size of a simple graph...
fusgrmaxsize 29502 The maximum size of a fini...
vtxdgfval 29505 The value of the vertex de...
vtxdgval 29506 The degree of a vertex. (...
vtxdgfival 29507 The degree of a vertex for...
vtxdgop 29508 The vertex degree expresse...
vtxdgf 29509 The vertex degree function...
vtxdgelxnn0 29510 The degree of a vertex is ...
vtxdg0v 29511 The degree of a vertex in ...
vtxdg0e 29512 The degree of a vertex in ...
vtxdgfisnn0 29513 The degree of a vertex in ...
vtxdgfisf 29514 The vertex degree function...
vtxdeqd 29515 Equality theorem for the v...
vtxduhgr0e 29516 The degree of a vertex in ...
vtxdlfuhgr1v 29517 The degree of the vertex i...
vdumgr0 29518 A vertex in a multigraph h...
vtxdun 29519 The degree of a vertex in ...
vtxdfiun 29520 The degree of a vertex in ...
vtxduhgrun 29521 The degree of a vertex in ...
vtxduhgrfiun 29522 The degree of a vertex in ...
vtxdlfgrval 29523 The value of the vertex de...
vtxdumgrval 29524 The value of the vertex de...
vtxdusgrval 29525 The value of the vertex de...
vtxd0nedgb 29526 A vertex has degree 0 iff ...
vtxdushgrfvedglem 29527 Lemma for ~ vtxdushgrfvedg...
vtxdushgrfvedg 29528 The value of the vertex de...
vtxdusgrfvedg 29529 The value of the vertex de...
vtxduhgr0nedg 29530 If a vertex in a hypergrap...
vtxdumgr0nedg 29531 If a vertex in a multigrap...
vtxduhgr0edgnel 29532 A vertex in a hypergraph h...
vtxdusgr0edgnel 29533 A vertex in a simple graph...
vtxdusgr0edgnelALT 29534 Alternate proof of ~ vtxdu...
vtxdgfusgrf 29535 The vertex degree function...
vtxdgfusgr 29536 In a finite simple graph, ...
fusgrn0degnn0 29537 In a nonempty, finite grap...
1loopgruspgr 29538 A graph with one edge whic...
1loopgredg 29539 The set of edges in a grap...
1loopgrnb0 29540 In a graph (simple pseudog...
1loopgrvd2 29541 The vertex degree of a one...
1loopgrvd0 29542 The vertex degree of a one...
1hevtxdg0 29543 The vertex degree of verte...
1hevtxdg1 29544 The vertex degree of verte...
1hegrvtxdg1 29545 The vertex degree of a gra...
1hegrvtxdg1r 29546 The vertex degree of a gra...
1egrvtxdg1 29547 The vertex degree of a one...
1egrvtxdg1r 29548 The vertex degree of a one...
1egrvtxdg0 29549 The vertex degree of a one...
p1evtxdeqlem 29550 Lemma for ~ p1evtxdeq and ...
p1evtxdeq 29551 If an edge ` E ` which doe...
p1evtxdp1 29552 If an edge ` E ` (not bein...
uspgrloopvtx 29553 The set of vertices in a g...
uspgrloopvtxel 29554 A vertex in a graph (simpl...
uspgrloopiedg 29555 The set of edges in a grap...
uspgrloopedg 29556 The set of edges in a grap...
uspgrloopnb0 29557 In a graph (simple pseudog...
uspgrloopvd2 29558 The vertex degree of a one...
umgr2v2evtx 29559 The set of vertices in a m...
umgr2v2evtxel 29560 A vertex in a multigraph w...
umgr2v2eiedg 29561 The edge function in a mul...
umgr2v2eedg 29562 The set of edges in a mult...
umgr2v2e 29563 A multigraph with two edge...
umgr2v2enb1 29564 In a multigraph with two e...
umgr2v2evd2 29565 In a multigraph with two e...
hashnbusgrvd 29566 In a simple graph, the num...
usgruvtxvdb 29567 In a finite simple graph w...
vdiscusgrb 29568 A finite simple graph with...
vdiscusgr 29569 In a finite complete simpl...
vtxdusgradjvtx 29570 The degree of a vertex in ...
usgrvd0nedg 29571 If a vertex in a simple gr...
uhgrvd00 29572 If every vertex in a hyper...
usgrvd00 29573 If every vertex in a simpl...
vdegp1ai 29574 The induction step for a v...
vdegp1bi 29575 The induction step for a v...
vdegp1ci 29576 The induction step for a v...
vtxdginducedm1lem1 29577 Lemma 1 for ~ vtxdginduced...
vtxdginducedm1lem2 29578 Lemma 2 for ~ vtxdginduced...
vtxdginducedm1lem3 29579 Lemma 3 for ~ vtxdginduced...
vtxdginducedm1lem4 29580 Lemma 4 for ~ vtxdginduced...
vtxdginducedm1 29581 The degree of a vertex ` v...
vtxdginducedm1fi 29582 The degree of a vertex ` v...
finsumvtxdg2ssteplem1 29583 Lemma for ~ finsumvtxdg2ss...
finsumvtxdg2ssteplem2 29584 Lemma for ~ finsumvtxdg2ss...
finsumvtxdg2ssteplem3 29585 Lemma for ~ finsumvtxdg2ss...
finsumvtxdg2ssteplem4 29586 Lemma for ~ finsumvtxdg2ss...
finsumvtxdg2sstep 29587 Induction step of ~ finsum...
finsumvtxdg2size 29588 The sum of the degrees of ...
fusgr1th 29589 The sum of the degrees of ...
finsumvtxdgeven 29590 The sum of the degrees of ...
vtxdgoddnumeven 29591 The number of vertices of ...
fusgrvtxdgonume 29592 The number of vertices of ...
isrgr 29597 The property of a class be...
rgrprop 29598 The properties of a k-regu...
isrusgr 29599 The property of being a k-...
rusgrprop 29600 The properties of a k-regu...
rusgrrgr 29601 A k-regular simple graph i...
rusgrusgr 29602 A k-regular simple graph i...
finrusgrfusgr 29603 A finite regular simple gr...
isrusgr0 29604 The property of being a k-...
rusgrprop0 29605 The properties of a k-regu...
usgreqdrusgr 29606 If all vertices in a simpl...
fusgrregdegfi 29607 In a nonempty finite simpl...
fusgrn0eqdrusgr 29608 If all vertices in a nonem...
frusgrnn0 29609 In a nonempty finite k-reg...
0edg0rgr 29610 A graph is 0-regular if it...
uhgr0edg0rgr 29611 A hypergraph is 0-regular ...
uhgr0edg0rgrb 29612 A hypergraph is 0-regular ...
usgr0edg0rusgr 29613 A simple graph is 0-regula...
0vtxrgr 29614 A null graph (with no vert...
0vtxrusgr 29615 A graph with no vertices a...
0uhgrrusgr 29616 The null graph as hypergra...
0grrusgr 29617 The null graph represented...
0grrgr 29618 The null graph represented...
cusgrrusgr 29619 A complete simple graph wi...
cusgrm1rusgr 29620 A finite simple graph with...
rusgrpropnb 29621 The properties of a k-regu...
rusgrpropedg 29622 The properties of a k-regu...
rusgrpropadjvtx 29623 The properties of a k-regu...
rusgrnumwrdl2 29624 In a k-regular simple grap...
rusgr1vtxlem 29625 Lemma for ~ rusgr1vtx . (...
rusgr1vtx 29626 If a k-regular simple grap...
rgrusgrprc 29627 The class of 0-regular sim...
rusgrprc 29628 The class of 0-regular sim...
rgrprc 29629 The class of 0-regular gra...
rgrprcx 29630 The class of 0-regular gra...
rgrx0ndm 29631 0 is not in the domain of ...
rgrx0nd 29632 The potentially alternativ...
ewlksfval 29639 The set of s-walks of edge...
isewlk 29640 Conditions for a function ...
ewlkprop 29641 Properties of an s-walk of...
ewlkinedg 29642 The intersection (common v...
ewlkle 29643 An s-walk of edges is also...
upgrewlkle2 29644 In a pseudograph, there is...
wkslem1 29645 Lemma 1 for walks to subst...
wkslem2 29646 Lemma 2 for walks to subst...
wksfval 29647 The set of walks (in an un...
iswlk 29648 Properties of a pair of fu...
wlkprop 29649 Properties of a walk. (Co...
wlkv 29650 The classes involved in a ...
iswlkg 29651 Generalization of ~ iswlk ...
wlkf 29652 The mapping enumerating th...
wlkcl 29653 A walk has length ` # ( F ...
wlkp 29654 The mapping enumerating th...
wlkpwrd 29655 The sequence of vertices o...
wlklenvp1 29656 The number of vertices of ...
wksv 29657 The class of walks is a se...
wksvOLD 29658 Obsolete version of ~ wksv...
wlkn0 29659 The sequence of vertices o...
wlklenvm1 29660 The number of edges of a w...
ifpsnprss 29661 Lemma for ~ wlkvtxeledg : ...
wlkvtxeledg 29662 Each pair of adjacent vert...
wlkvtxiedg 29663 The vertices of a walk are...
relwlk 29664 The set ` ( Walks `` G ) `...
wlkvv 29665 If there is at least one w...
wlkop 29666 A walk is an ordered pair....
wlkcpr 29667 A walk as class with two c...
wlk2f 29668 If there is a walk ` W ` t...
wlkcomp 29669 A walk expressed by proper...
wlkcompim 29670 Implications for the prope...
wlkelwrd 29671 The components of a walk a...
wlkeq 29672 Conditions for two walks (...
edginwlk 29673 The value of the edge func...
upgredginwlk 29674 The value of the edge func...
iedginwlk 29675 The value of the edge func...
wlkl1loop 29676 A walk of length 1 from a ...
wlk1walk 29677 A walk is a 1-walk "on the...
wlk1ewlk 29678 A walk is an s-walk "on th...
upgriswlk 29679 Properties of a pair of fu...
upgrwlkedg 29680 The edges of a walk in a p...
upgrwlkcompim 29681 Implications for the prope...
wlkvtxedg 29682 The vertices of a walk are...
upgrwlkvtxedg 29683 The pairs of connected ver...
uspgr2wlkeq 29684 Conditions for two walks w...
uspgr2wlkeq2 29685 Conditions for two walks w...
uspgr2wlkeqi 29686 Conditions for two walks w...
umgrwlknloop 29687 In a multigraph, each walk...
wlkResOLD 29688 Obsolete version of ~ opab...
wlkv0 29689 If there is a walk in the ...
g0wlk0 29690 There is no walk in a null...
0wlk0 29691 There is no walk for the e...
wlk0prc 29692 There is no walk in a null...
wlklenvclwlk 29693 The number of vertices in ...
wlkson 29694 The set of walks between t...
iswlkon 29695 Properties of a pair of fu...
wlkonprop 29696 Properties of a walk betwe...
wlkpvtx 29697 A walk connects vertices. ...
wlkepvtx 29698 The endpoints of a walk ar...
wlkoniswlk 29699 A walk between two vertice...
wlkonwlk 29700 A walk is a walk between i...
wlkonwlk1l 29701 A walk is a walk from its ...
wlksoneq1eq2 29702 Two walks with identical s...
wlkonl1iedg 29703 If there is a walk between...
wlkon2n0 29704 The length of a walk betwe...
2wlklem 29705 Lemma for theorems for wal...
upgr2wlk 29706 Properties of a pair of fu...
wlkreslem 29707 Lemma for ~ wlkres . (Con...
wlkres 29708 The restriction ` <. H , Q...
redwlklem 29709 Lemma for ~ redwlk . (Con...
redwlk 29710 A walk ending at the last ...
wlkp1lem1 29711 Lemma for ~ wlkp1 . (Cont...
wlkp1lem2 29712 Lemma for ~ wlkp1 . (Cont...
wlkp1lem3 29713 Lemma for ~ wlkp1 . (Cont...
wlkp1lem4 29714 Lemma for ~ wlkp1 . (Cont...
wlkp1lem5 29715 Lemma for ~ wlkp1 . (Cont...
wlkp1lem6 29716 Lemma for ~ wlkp1 . (Cont...
wlkp1lem7 29717 Lemma for ~ wlkp1 . (Cont...
wlkp1lem8 29718 Lemma for ~ wlkp1 . (Cont...
wlkp1 29719 Append one path segment (e...
wlkdlem1 29720 Lemma 1 for ~ wlkd . (Con...
wlkdlem2 29721 Lemma 2 for ~ wlkd . (Con...
wlkdlem3 29722 Lemma 3 for ~ wlkd . (Con...
wlkdlem4 29723 Lemma 4 for ~ wlkd . (Con...
wlkd 29724 Two words representing a w...
lfgrwlkprop 29725 Two adjacent vertices in a...
lfgriswlk 29726 Conditions for a pair of f...
lfgrwlknloop 29727 In a loop-free graph, each...
reltrls 29732 The set ` ( Trails `` G ) ...
trlsfval 29733 The set of trails (in an u...
istrl 29734 Conditions for a pair of c...
trliswlk 29735 A trail is a walk. (Contr...
trlf1 29736 The enumeration ` F ` of a...
trlreslem 29737 Lemma for ~ trlres . Form...
trlres 29738 The restriction ` <. H , Q...
upgrtrls 29739 The set of trails in a pse...
upgristrl 29740 Properties of a pair of fu...
upgrf1istrl 29741 Properties of a pair of a ...
wksonproplem 29742 Lemma for theorems for pro...
wksonproplemOLD 29743 Obsolete version of ~ wkso...
trlsonfval 29744 The set of trails between ...
istrlson 29745 Properties of a pair of fu...
trlsonprop 29746 Properties of a trail betw...
trlsonistrl 29747 A trail between two vertic...
trlsonwlkon 29748 A trail between two vertic...
trlontrl 29749 A trail is a trail between...
relpths 29758 The set ` ( Paths `` G ) `...
pthsfval 29759 The set of paths (in an un...
spthsfval 29760 The set of simple paths (i...
ispth 29761 Conditions for a pair of c...
isspth 29762 Conditions for a pair of c...
pthistrl 29763 A path is a trail (in an u...
spthispth 29764 A simple path is a path (i...
pthiswlk 29765 A path is a walk (in an un...
spthiswlk 29766 A simple path is a walk (i...
pthdivtx 29767 The inner vertices of a pa...
pthdadjvtx 29768 The adjacent vertices of a...
2pthnloop 29769 A path of length at least ...
upgr2pthnlp 29770 A path of length at least ...
spthdifv 29771 The vertices of a simple p...
spthdep 29772 A simple path (at least of...
pthdepisspth 29773 A path with different star...
upgrwlkdvdelem 29774 Lemma for ~ upgrwlkdvde . ...
upgrwlkdvde 29775 In a pseudograph, all edge...
upgrspthswlk 29776 The set of simple paths in...
upgrwlkdvspth 29777 A walk consisting of diffe...
pthsonfval 29778 The set of paths between t...
spthson 29779 The set of simple paths be...
ispthson 29780 Properties of a pair of fu...
isspthson 29781 Properties of a pair of fu...
pthsonprop 29782 Properties of a path betwe...
spthonprop 29783 Properties of a simple pat...
pthonispth 29784 A path between two vertice...
pthontrlon 29785 A path between two vertice...
pthonpth 29786 A path is a path between i...
isspthonpth 29787 A pair of functions is a s...
spthonisspth 29788 A simple path between to v...
spthonpthon 29789 A simple path between two ...
spthonepeq 29790 The endpoints of a simple ...
uhgrwkspthlem1 29791 Lemma 1 for ~ uhgrwkspth ....
uhgrwkspthlem2 29792 Lemma 2 for ~ uhgrwkspth ....
uhgrwkspth 29793 Any walk of length 1 betwe...
usgr2wlkneq 29794 The vertices and edges are...
usgr2wlkspthlem1 29795 Lemma 1 for ~ usgr2wlkspth...
usgr2wlkspthlem2 29796 Lemma 2 for ~ usgr2wlkspth...
usgr2wlkspth 29797 In a simple graph, any wal...
usgr2trlncl 29798 In a simple graph, any tra...
usgr2trlspth 29799 In a simple graph, any tra...
usgr2pthspth 29800 In a simple graph, any pat...
usgr2pthlem 29801 Lemma for ~ usgr2pth . (C...
usgr2pth 29802 In a simple graph, there i...
usgr2pth0 29803 In a simply graph, there i...
pthdlem1 29804 Lemma 1 for ~ pthd . (Con...
pthdlem2lem 29805 Lemma for ~ pthdlem2 . (C...
pthdlem2 29806 Lemma 2 for ~ pthd . (Con...
pthd 29807 Two words representing a t...
clwlks 29810 The set of closed walks (i...
isclwlk 29811 A pair of functions repres...
clwlkiswlk 29812 A closed walk is a walk (i...
clwlkwlk 29813 Closed walks are walks (in...
clwlkswks 29814 Closed walks are walks (in...
isclwlke 29815 Properties of a pair of fu...
isclwlkupgr 29816 Properties of a pair of fu...
clwlkcomp 29817 A closed walk expressed by...
clwlkcompim 29818 Implications for the prope...
upgrclwlkcompim 29819 Implications for the prope...
clwlkcompbp 29820 Basic properties of the co...
clwlkl1loop 29821 A closed walk of length 1 ...
crcts 29826 The set of circuits (in an...
cycls 29827 The set of cycles (in an u...
iscrct 29828 Sufficient and necessary c...
iscycl 29829 Sufficient and necessary c...
crctprop 29830 The properties of a circui...
cyclprop 29831 The properties of a cycle:...
crctisclwlk 29832 A circuit is a closed walk...
crctistrl 29833 A circuit is a trail. (Co...
crctiswlk 29834 A circuit is a walk. (Con...
cyclispth 29835 A cycle is a path. (Contr...
cycliswlk 29836 A cycle is a walk. (Contr...
cycliscrct 29837 A cycle is a circuit. (Co...
cyclnspth 29838 A (non-trivial) cycle is n...
cyclispthon 29839 A cycle is a path starting...
lfgrn1cycl 29840 In a loop-free graph there...
usgr2trlncrct 29841 In a simple graph, any tra...
umgrn1cycl 29842 In a multigraph graph (wit...
uspgrn2crct 29843 In a simple pseudograph th...
usgrn2cycl 29844 In a simple graph there ar...
crctcshwlkn0lem1 29845 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem2 29846 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem3 29847 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem4 29848 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem5 29849 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem6 29850 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem7 29851 Lemma for ~ crctcshwlkn0 ....
crctcshlem1 29852 Lemma for ~ crctcsh . (Co...
crctcshlem2 29853 Lemma for ~ crctcsh . (Co...
crctcshlem3 29854 Lemma for ~ crctcsh . (Co...
crctcshlem4 29855 Lemma for ~ crctcsh . (Co...
crctcshwlkn0 29856 Cyclically shifting the in...
crctcshwlk 29857 Cyclically shifting the in...
crctcshtrl 29858 Cyclically shifting the in...
crctcsh 29859 Cyclically shifting the in...
wwlks 29870 The set of walks (in an un...
iswwlks 29871 A word over the set of ver...
wwlksn 29872 The set of walks (in an un...
iswwlksn 29873 A word over the set of ver...
wwlksnprcl 29874 Derivation of the length o...
iswwlksnx 29875 Properties of a word to re...
wwlkbp 29876 Basic properties of a walk...
wwlknbp 29877 Basic properties of a walk...
wwlknp 29878 Properties of a set being ...
wwlknbp1 29879 Other basic properties of ...
wwlknvtx 29880 The symbols of a word ` W ...
wwlknllvtx 29881 If a word ` W ` represents...
wwlknlsw 29882 If a word represents a wal...
wspthsn 29883 The set of simple paths of...
iswspthn 29884 An element of the set of s...
wspthnp 29885 Properties of a set being ...
wwlksnon 29886 The set of walks of a fixe...
wspthsnon 29887 The set of simple paths of...
iswwlksnon 29888 The set of walks of a fixe...
wwlksnon0 29889 Sufficient conditions for ...
wwlksonvtx 29890 If a word ` W ` represents...
iswspthsnon 29891 The set of simple paths of...
wwlknon 29892 An element of the set of w...
wspthnon 29893 An element of the set of s...
wspthnonp 29894 Properties of a set being ...
wspthneq1eq2 29895 Two simple paths with iden...
wwlksn0s 29896 The set of all walks as wo...
wwlkssswrd 29897 Walks (represented by word...
wwlksn0 29898 A walk of length 0 is repr...
0enwwlksnge1 29899 In graphs without edges, t...
wwlkswwlksn 29900 A walk of a fixed length a...
wwlkssswwlksn 29901 The walks of a fixed lengt...
wlkiswwlks1 29902 The sequence of vertices i...
wlklnwwlkln1 29903 The sequence of vertices i...
wlkiswwlks2lem1 29904 Lemma 1 for ~ wlkiswwlks2 ...
wlkiswwlks2lem2 29905 Lemma 2 for ~ wlkiswwlks2 ...
wlkiswwlks2lem3 29906 Lemma 3 for ~ wlkiswwlks2 ...
wlkiswwlks2lem4 29907 Lemma 4 for ~ wlkiswwlks2 ...
wlkiswwlks2lem5 29908 Lemma 5 for ~ wlkiswwlks2 ...
wlkiswwlks2lem6 29909 Lemma 6 for ~ wlkiswwlks2 ...
wlkiswwlks2 29910 A walk as word corresponds...
wlkiswwlks 29911 A walk as word corresponds...
wlkiswwlksupgr2 29912 A walk as word corresponds...
wlkiswwlkupgr 29913 A walk as word corresponds...
wlkswwlksf1o 29914 The mapping of (ordinary) ...
wlkswwlksen 29915 The set of walks as words ...
wwlksm1edg 29916 Removing the trailing edge...
wlklnwwlkln2lem 29917 Lemma for ~ wlklnwwlkln2 a...
wlklnwwlkln2 29918 A walk of length ` N ` as ...
wlklnwwlkn 29919 A walk of length ` N ` as ...
wlklnwwlklnupgr2 29920 A walk of length ` N ` as ...
wlklnwwlknupgr 29921 A walk of length ` N ` as ...
wlknewwlksn 29922 If a walk in a pseudograph...
wlknwwlksnbij 29923 The mapping ` ( t e. T |->...
wlknwwlksnen 29924 In a simple pseudograph, t...
wlknwwlksneqs 29925 The set of walks of a fixe...
wwlkseq 29926 Equality of two walks (as ...
wwlksnred 29927 Reduction of a walk (as wo...
wwlksnext 29928 Extension of a walk (as wo...
wwlksnextbi 29929 Extension of a walk (as wo...
wwlksnredwwlkn 29930 For each walk (as word) of...
wwlksnredwwlkn0 29931 For each walk (as word) of...
wwlksnextwrd 29932 Lemma for ~ wwlksnextbij ....
wwlksnextfun 29933 Lemma for ~ wwlksnextbij ....
wwlksnextinj 29934 Lemma for ~ wwlksnextbij ....
wwlksnextsurj 29935 Lemma for ~ wwlksnextbij ....
wwlksnextbij0 29936 Lemma for ~ wwlksnextbij ....
wwlksnextbij 29937 There is a bijection betwe...
wwlksnexthasheq 29938 The number of the extensio...
disjxwwlksn 29939 Sets of walks (as words) e...
wwlksnndef 29940 Conditions for ` WWalksN `...
wwlksnfi 29941 The number of walks repres...
wlksnfi 29942 The number of walks of fix...
wlksnwwlknvbij 29943 There is a bijection betwe...
wwlksnextproplem1 29944 Lemma 1 for ~ wwlksnextpro...
wwlksnextproplem2 29945 Lemma 2 for ~ wwlksnextpro...
wwlksnextproplem3 29946 Lemma 3 for ~ wwlksnextpro...
wwlksnextprop 29947 Adding additional properti...
disjxwwlkn 29948 Sets of walks (as words) e...
hashwwlksnext 29949 Number of walks (as words)...
wwlksnwwlksnon 29950 A walk of fixed length is ...
wspthsnwspthsnon 29951 A simple path of fixed len...
wspthsnonn0vne 29952 If the set of simple paths...
wspthsswwlkn 29953 The set of simple paths of...
wspthnfi 29954 In a finite graph, the set...
wwlksnonfi 29955 In a finite graph, the set...
wspthsswwlknon 29956 The set of simple paths of...
wspthnonfi 29957 In a finite graph, the set...
wspniunwspnon 29958 The set of nonempty simple...
wspn0 29959 If there are no vertices, ...
2wlkdlem1 29960 Lemma 1 for ~ 2wlkd . (Co...
2wlkdlem2 29961 Lemma 2 for ~ 2wlkd . (Co...
2wlkdlem3 29962 Lemma 3 for ~ 2wlkd . (Co...
2wlkdlem4 29963 Lemma 4 for ~ 2wlkd . (Co...
2wlkdlem5 29964 Lemma 5 for ~ 2wlkd . (Co...
2pthdlem1 29965 Lemma 1 for ~ 2pthd . (Co...
2wlkdlem6 29966 Lemma 6 for ~ 2wlkd . (Co...
2wlkdlem7 29967 Lemma 7 for ~ 2wlkd . (Co...
2wlkdlem8 29968 Lemma 8 for ~ 2wlkd . (Co...
2wlkdlem9 29969 Lemma 9 for ~ 2wlkd . (Co...
2wlkdlem10 29970 Lemma 10 for ~ 3wlkd . (C...
2wlkd 29971 Construction of a walk fro...
2wlkond 29972 A walk of length 2 from on...
2trld 29973 Construction of a trail fr...
2trlond 29974 A trail of length 2 from o...
2pthd 29975 A path of length 2 from on...
2spthd 29976 A simple path of length 2 ...
2pthond 29977 A simple path of length 2 ...
2pthon3v 29978 For a vertex adjacent to t...
umgr2adedgwlklem 29979 Lemma for ~ umgr2adedgwlk ...
umgr2adedgwlk 29980 In a multigraph, two adjac...
umgr2adedgwlkon 29981 In a multigraph, two adjac...
umgr2adedgwlkonALT 29982 Alternate proof for ~ umgr...
umgr2adedgspth 29983 In a multigraph, two adjac...
umgr2wlk 29984 In a multigraph, there is ...
umgr2wlkon 29985 For each pair of adjacent ...
elwwlks2s3 29986 A walk of length 2 as word...
midwwlks2s3 29987 There is a vertex between ...
wwlks2onv 29988 If a length 3 string repre...
elwwlks2ons3im 29989 A walk as word of length 2...
elwwlks2ons3 29990 For each walk of length 2 ...
s3wwlks2on 29991 A length 3 string which re...
umgrwwlks2on 29992 A walk of length 2 between...
wwlks2onsym 29993 There is a walk of length ...
elwwlks2on 29994 A walk of length 2 between...
elwspths2on 29995 A simple path of length 2 ...
wpthswwlks2on 29996 For two different vertices...
2wspdisj 29997 All simple paths of length...
2wspiundisj 29998 All simple paths of length...
usgr2wspthons3 29999 A simple path of length 2 ...
usgr2wspthon 30000 A simple path of length 2 ...
elwwlks2 30001 A walk of length 2 between...
elwspths2spth 30002 A simple path of length 2 ...
rusgrnumwwlkl1 30003 In a k-regular graph, ther...
rusgrnumwwlkslem 30004 Lemma for ~ rusgrnumwwlks ...
rusgrnumwwlklem 30005 Lemma for ~ rusgrnumwwlk e...
rusgrnumwwlkb0 30006 Induction base 0 for ~ rus...
rusgrnumwwlkb1 30007 Induction base 1 for ~ rus...
rusgr0edg 30008 Special case for graphs wi...
rusgrnumwwlks 30009 Induction step for ~ rusgr...
rusgrnumwwlk 30010 In a ` K `-regular graph, ...
rusgrnumwwlkg 30011 In a ` K `-regular graph, ...
rusgrnumwlkg 30012 In a k-regular graph, the ...
clwwlknclwwlkdif 30013 The set ` A ` of walks of ...
clwwlknclwwlkdifnum 30014 In a ` K `-regular graph, ...
clwwlk 30017 The set of closed walks (i...
isclwwlk 30018 Properties of a word to re...
clwwlkbp 30019 Basic properties of a clos...
clwwlkgt0 30020 There is no empty closed w...
clwwlksswrd 30021 Closed walks (represented ...
clwwlk1loop 30022 A closed walk of length 1 ...
clwwlkccatlem 30023 Lemma for ~ clwwlkccat : i...
clwwlkccat 30024 The concatenation of two w...
umgrclwwlkge2 30025 A closed walk in a multigr...
clwlkclwwlklem2a1 30026 Lemma 1 for ~ clwlkclwwlkl...
clwlkclwwlklem2a2 30027 Lemma 2 for ~ clwlkclwwlkl...
clwlkclwwlklem2a3 30028 Lemma 3 for ~ clwlkclwwlkl...
clwlkclwwlklem2fv1 30029 Lemma 4a for ~ clwlkclwwlk...
clwlkclwwlklem2fv2 30030 Lemma 4b for ~ clwlkclwwlk...
clwlkclwwlklem2a4 30031 Lemma 4 for ~ clwlkclwwlkl...
clwlkclwwlklem2a 30032 Lemma for ~ clwlkclwwlklem...
clwlkclwwlklem1 30033 Lemma 1 for ~ clwlkclwwlk ...
clwlkclwwlklem2 30034 Lemma 2 for ~ clwlkclwwlk ...
clwlkclwwlklem3 30035 Lemma 3 for ~ clwlkclwwlk ...
clwlkclwwlk 30036 A closed walk as word of l...
clwlkclwwlk2 30037 A closed walk corresponds ...
clwlkclwwlkflem 30038 Lemma for ~ clwlkclwwlkf ....
clwlkclwwlkf1lem2 30039 Lemma 2 for ~ clwlkclwwlkf...
clwlkclwwlkf1lem3 30040 Lemma 3 for ~ clwlkclwwlkf...
clwlkclwwlkfolem 30041 Lemma for ~ clwlkclwwlkfo ...
clwlkclwwlkf 30042 ` F ` is a function from t...
clwlkclwwlkfo 30043 ` F ` is a function from t...
clwlkclwwlkf1 30044 ` F ` is a one-to-one func...
clwlkclwwlkf1o 30045 ` F ` is a bijection betwe...
clwlkclwwlken 30046 The set of the nonempty cl...
clwwisshclwwslemlem 30047 Lemma for ~ clwwisshclwwsl...
clwwisshclwwslem 30048 Lemma for ~ clwwisshclwws ...
clwwisshclwws 30049 Cyclically shifting a clos...
clwwisshclwwsn 30050 Cyclically shifting a clos...
erclwwlkrel 30051 ` .~ ` is a relation. (Co...
erclwwlkeq 30052 Two classes are equivalent...
erclwwlkeqlen 30053 If two classes are equival...
erclwwlkref 30054 ` .~ ` is a reflexive rela...
erclwwlksym 30055 ` .~ ` is a symmetric rela...
erclwwlktr 30056 ` .~ ` is a transitive rel...
erclwwlk 30057 ` .~ ` is an equivalence r...
clwwlkn 30060 The set of closed walks of...
isclwwlkn 30061 A word over the set of ver...
clwwlkn0 30062 There is no closed walk of...
clwwlkneq0 30063 Sufficient conditions for ...
clwwlkclwwlkn 30064 A closed walk of a fixed l...
clwwlksclwwlkn 30065 The closed walks of a fixe...
clwwlknlen 30066 The length of a word repre...
clwwlknnn 30067 The length of a closed wal...
clwwlknwrd 30068 A closed walk of a fixed l...
clwwlknbp 30069 Basic properties of a clos...
isclwwlknx 30070 Characterization of a word...
clwwlknp 30071 Properties of a set being ...
clwwlknwwlksn 30072 A word representing a clos...
clwwlknlbonbgr1 30073 The last but one vertex in...
clwwlkinwwlk 30074 If the initial vertex of a...
clwwlkn1 30075 A closed walk of length 1 ...
loopclwwlkn1b 30076 The singleton word consist...
clwwlkn1loopb 30077 A word represents a closed...
clwwlkn2 30078 A closed walk of length 2 ...
clwwlknfi 30079 If there is only a finite ...
clwwlkel 30080 Obtaining a closed walk (a...
clwwlkf 30081 Lemma 1 for ~ clwwlkf1o : ...
clwwlkfv 30082 Lemma 2 for ~ clwwlkf1o : ...
clwwlkf1 30083 Lemma 3 for ~ clwwlkf1o : ...
clwwlkfo 30084 Lemma 4 for ~ clwwlkf1o : ...
clwwlkf1o 30085 F is a 1-1 onto function, ...
clwwlken 30086 The set of closed walks of...
clwwlknwwlkncl 30087 Obtaining a closed walk (a...
clwwlkwwlksb 30088 A nonempty word over verti...
clwwlknwwlksnb 30089 A word over vertices repre...
clwwlkext2edg 30090 If a word concatenated wit...
wwlksext2clwwlk 30091 If a word represents a wal...
wwlksubclwwlk 30092 Any prefix of a word repre...
clwwnisshclwwsn 30093 Cyclically shifting a clos...
eleclclwwlknlem1 30094 Lemma 1 for ~ eleclclwwlkn...
eleclclwwlknlem2 30095 Lemma 2 for ~ eleclclwwlkn...
clwwlknscsh 30096 The set of cyclical shifts...
clwwlknccat 30097 The concatenation of two w...
umgr2cwwk2dif 30098 If a word represents a clo...
umgr2cwwkdifex 30099 If a word represents a clo...
erclwwlknrel 30100 ` .~ ` is a relation. (Co...
erclwwlkneq 30101 Two classes are equivalent...
erclwwlkneqlen 30102 If two classes are equival...
erclwwlknref 30103 ` .~ ` is a reflexive rela...
erclwwlknsym 30104 ` .~ ` is a symmetric rela...
erclwwlkntr 30105 ` .~ ` is a transitive rel...
erclwwlkn 30106 ` .~ ` is an equivalence r...
qerclwwlknfi 30107 The quotient set of the se...
hashclwwlkn0 30108 The number of closed walks...
eclclwwlkn1 30109 An equivalence class accor...
eleclclwwlkn 30110 A member of an equivalence...
hashecclwwlkn1 30111 The size of every equivale...
umgrhashecclwwlk 30112 The size of every equivale...
fusgrhashclwwlkn 30113 The size of the set of clo...
clwwlkndivn 30114 The size of the set of clo...
clwlknf1oclwwlknlem1 30115 Lemma 1 for ~ clwlknf1oclw...
clwlknf1oclwwlknlem2 30116 Lemma 2 for ~ clwlknf1oclw...
clwlknf1oclwwlknlem3 30117 Lemma 3 for ~ clwlknf1oclw...
clwlknf1oclwwlkn 30118 There is a one-to-one onto...
clwlkssizeeq 30119 The size of the set of clo...
clwlksndivn 30120 The size of the set of clo...
clwwlknonmpo 30123 ` ( ClWWalksNOn `` G ) ` i...
clwwlknon 30124 The set of closed walks on...
isclwwlknon 30125 A word over the set of ver...
clwwlk0on0 30126 There is no word over the ...
clwwlknon0 30127 Sufficient conditions for ...
clwwlknonfin 30128 In a finite graph ` G ` , ...
clwwlknonel 30129 Characterization of a word...
clwwlknonccat 30130 The concatenation of two w...
clwwlknon1 30131 The set of closed walks on...
clwwlknon1loop 30132 If there is a loop at vert...
clwwlknon1nloop 30133 If there is no loop at ver...
clwwlknon1sn 30134 The set of (closed) walks ...
clwwlknon1le1 30135 There is at most one (clos...
clwwlknon2 30136 The set of closed walks on...
clwwlknon2x 30137 The set of closed walks on...
s2elclwwlknon2 30138 Sufficient conditions of a...
clwwlknon2num 30139 In a ` K `-regular graph `...
clwwlknonwwlknonb 30140 A word over vertices repre...
clwwlknonex2lem1 30141 Lemma 1 for ~ clwwlknonex2...
clwwlknonex2lem2 30142 Lemma 2 for ~ clwwlknonex2...
clwwlknonex2 30143 Extending a closed walk ` ...
clwwlknonex2e 30144 Extending a closed walk ` ...
clwwlknondisj 30145 The sets of closed walks o...
clwwlknun 30146 The set of closed walks of...
clwwlkvbij 30147 There is a bijection betwe...
0ewlk 30148 The empty set (empty seque...
1ewlk 30149 A sequence of 1 edge is an...
0wlk 30150 A pair of an empty set (of...
is0wlk 30151 A pair of an empty set (of...
0wlkonlem1 30152 Lemma 1 for ~ 0wlkon and ~...
0wlkonlem2 30153 Lemma 2 for ~ 0wlkon and ~...
0wlkon 30154 A walk of length 0 from a ...
0wlkons1 30155 A walk of length 0 from a ...
0trl 30156 A pair of an empty set (of...
is0trl 30157 A pair of an empty set (of...
0trlon 30158 A trail of length 0 from a...
0pth 30159 A pair of an empty set (of...
0spth 30160 A pair of an empty set (of...
0pthon 30161 A path of length 0 from a ...
0pthon1 30162 A path of length 0 from a ...
0pthonv 30163 For each vertex there is a...
0clwlk 30164 A pair of an empty set (of...
0clwlkv 30165 Any vertex (more precisely...
0clwlk0 30166 There is no closed walk in...
0crct 30167 A pair of an empty set (of...
0cycl 30168 A pair of an empty set (of...
1pthdlem1 30169 Lemma 1 for ~ 1pthd . (Co...
1pthdlem2 30170 Lemma 2 for ~ 1pthd . (Co...
1wlkdlem1 30171 Lemma 1 for ~ 1wlkd . (Co...
1wlkdlem2 30172 Lemma 2 for ~ 1wlkd . (Co...
1wlkdlem3 30173 Lemma 3 for ~ 1wlkd . (Co...
1wlkdlem4 30174 Lemma 4 for ~ 1wlkd . (Co...
1wlkd 30175 In a graph with two vertic...
1trld 30176 In a graph with two vertic...
1pthd 30177 In a graph with two vertic...
1pthond 30178 In a graph with two vertic...
upgr1wlkdlem1 30179 Lemma 1 for ~ upgr1wlkd . ...
upgr1wlkdlem2 30180 Lemma 2 for ~ upgr1wlkd . ...
upgr1wlkd 30181 In a pseudograph with two ...
upgr1trld 30182 In a pseudograph with two ...
upgr1pthd 30183 In a pseudograph with two ...
upgr1pthond 30184 In a pseudograph with two ...
lppthon 30185 A loop (which is an edge a...
lp1cycl 30186 A loop (which is an edge a...
1pthon2v 30187 For each pair of adjacent ...
1pthon2ve 30188 For each pair of adjacent ...
wlk2v2elem1 30189 Lemma 1 for ~ wlk2v2e : ` ...
wlk2v2elem2 30190 Lemma 2 for ~ wlk2v2e : T...
wlk2v2e 30191 In a graph with two vertic...
ntrl2v2e 30192 A walk which is not a trai...
3wlkdlem1 30193 Lemma 1 for ~ 3wlkd . (Co...
3wlkdlem2 30194 Lemma 2 for ~ 3wlkd . (Co...
3wlkdlem3 30195 Lemma 3 for ~ 3wlkd . (Co...
3wlkdlem4 30196 Lemma 4 for ~ 3wlkd . (Co...
3wlkdlem5 30197 Lemma 5 for ~ 3wlkd . (Co...
3pthdlem1 30198 Lemma 1 for ~ 3pthd . (Co...
3wlkdlem6 30199 Lemma 6 for ~ 3wlkd . (Co...
3wlkdlem7 30200 Lemma 7 for ~ 3wlkd . (Co...
3wlkdlem8 30201 Lemma 8 for ~ 3wlkd . (Co...
3wlkdlem9 30202 Lemma 9 for ~ 3wlkd . (Co...
3wlkdlem10 30203 Lemma 10 for ~ 3wlkd . (C...
3wlkd 30204 Construction of a walk fro...
3wlkond 30205 A walk of length 3 from on...
3trld 30206 Construction of a trail fr...
3trlond 30207 A trail of length 3 from o...
3pthd 30208 A path of length 3 from on...
3pthond 30209 A path of length 3 from on...
3spthd 30210 A simple path of length 3 ...
3spthond 30211 A simple path of length 3 ...
3cycld 30212 Construction of a 3-cycle ...
3cyclpd 30213 Construction of a 3-cycle ...
upgr3v3e3cycl 30214 If there is a cycle of len...
uhgr3cyclexlem 30215 Lemma for ~ uhgr3cyclex . ...
uhgr3cyclex 30216 If there are three differe...
umgr3cyclex 30217 If there are three (differ...
umgr3v3e3cycl 30218 If and only if there is a ...
upgr4cycl4dv4e 30219 If there is a cycle of len...
dfconngr1 30222 Alternative definition of ...
isconngr 30223 The property of being a co...
isconngr1 30224 The property of being a co...
cusconngr 30225 A complete hypergraph is c...
0conngr 30226 A graph without vertices i...
0vconngr 30227 A graph without vertices i...
1conngr 30228 A graph with (at most) one...
conngrv2edg 30229 A vertex in a connected gr...
vdn0conngrumgrv2 30230 A vertex in a connected mu...
releupth 30233 The set ` ( EulerPaths `` ...
eupths 30234 The Eulerian paths on the ...
iseupth 30235 The property " ` <. F , P ...
iseupthf1o 30236 The property " ` <. F , P ...
eupthi 30237 Properties of an Eulerian ...
eupthf1o 30238 The ` F ` function in an E...
eupthfi 30239 Any graph with an Eulerian...
eupthseg 30240 The ` N ` -th edge in an e...
upgriseupth 30241 The property " ` <. F , P ...
upgreupthi 30242 Properties of an Eulerian ...
upgreupthseg 30243 The ` N ` -th edge in an e...
eupthcl 30244 An Eulerian path has lengt...
eupthistrl 30245 An Eulerian path is a trai...
eupthiswlk 30246 An Eulerian path is a walk...
eupthpf 30247 The ` P ` function in an E...
eupth0 30248 There is an Eulerian path ...
eupthres 30249 The restriction ` <. H , Q...
eupthp1 30250 Append one path segment to...
eupth2eucrct 30251 Append one path segment to...
eupth2lem1 30252 Lemma for ~ eupth2 . (Con...
eupth2lem2 30253 Lemma for ~ eupth2 . (Con...
trlsegvdeglem1 30254 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem2 30255 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem3 30256 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem4 30257 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem5 30258 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem6 30259 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem7 30260 Lemma for ~ trlsegvdeg . ...
trlsegvdeg 30261 Formerly part of proof of ...
eupth2lem3lem1 30262 Lemma for ~ eupth2lem3 . ...
eupth2lem3lem2 30263 Lemma for ~ eupth2lem3 . ...
eupth2lem3lem3 30264 Lemma for ~ eupth2lem3 , f...
eupth2lem3lem4 30265 Lemma for ~ eupth2lem3 , f...
eupth2lem3lem5 30266 Lemma for ~ eupth2 . (Con...
eupth2lem3lem6 30267 Formerly part of proof of ...
eupth2lem3lem7 30268 Lemma for ~ eupth2lem3 : ...
eupthvdres 30269 Formerly part of proof of ...
eupth2lem3 30270 Lemma for ~ eupth2 . (Con...
eupth2lemb 30271 Lemma for ~ eupth2 (induct...
eupth2lems 30272 Lemma for ~ eupth2 (induct...
eupth2 30273 The only vertices of odd d...
eulerpathpr 30274 A graph with an Eulerian p...
eulerpath 30275 A pseudograph with an Eule...
eulercrct 30276 A pseudograph with an Eule...
eucrctshift 30277 Cyclically shifting the in...
eucrct2eupth1 30278 Removing one edge ` ( I ``...
eucrct2eupth 30279 Removing one edge ` ( I ``...
konigsbergvtx 30280 The set of vertices of the...
konigsbergiedg 30281 The indexed edges of the K...
konigsbergiedgw 30282 The indexed edges of the K...
konigsbergssiedgwpr 30283 Each subset of the indexed...
konigsbergssiedgw 30284 Each subset of the indexed...
konigsbergumgr 30285 The Königsberg graph ...
konigsberglem1 30286 Lemma 1 for ~ konigsberg :...
konigsberglem2 30287 Lemma 2 for ~ konigsberg :...
konigsberglem3 30288 Lemma 3 for ~ konigsberg :...
konigsberglem4 30289 Lemma 4 for ~ konigsberg :...
konigsberglem5 30290 Lemma 5 for ~ konigsberg :...
konigsberg 30291 The Königsberg Bridge...
isfrgr 30294 The property of being a fr...
frgrusgr 30295 A friendship graph is a si...
frgr0v 30296 Any null graph (set with n...
frgr0vb 30297 Any null graph (without ve...
frgruhgr0v 30298 Any null graph (without ve...
frgr0 30299 The null graph (graph with...
frcond1 30300 The friendship condition: ...
frcond2 30301 The friendship condition: ...
frgreu 30302 Variant of ~ frcond2 : An...
frcond3 30303 The friendship condition, ...
frcond4 30304 The friendship condition, ...
frgr1v 30305 Any graph with (at most) o...
nfrgr2v 30306 Any graph with two (differ...
frgr3vlem1 30307 Lemma 1 for ~ frgr3v . (C...
frgr3vlem2 30308 Lemma 2 for ~ frgr3v . (C...
frgr3v 30309 Any graph with three verti...
1vwmgr 30310 Every graph with one verte...
3vfriswmgrlem 30311 Lemma for ~ 3vfriswmgr . ...
3vfriswmgr 30312 Every friendship graph wit...
1to2vfriswmgr 30313 Every friendship graph wit...
1to3vfriswmgr 30314 Every friendship graph wit...
1to3vfriendship 30315 The friendship theorem for...
2pthfrgrrn 30316 Between any two (different...
2pthfrgrrn2 30317 Between any two (different...
2pthfrgr 30318 Between any two (different...
3cyclfrgrrn1 30319 Every vertex in a friendsh...
3cyclfrgrrn 30320 Every vertex in a friendsh...
3cyclfrgrrn2 30321 Every vertex in a friendsh...
3cyclfrgr 30322 Every vertex in a friendsh...
4cycl2v2nb 30323 In a (maybe degenerate) 4-...
4cycl2vnunb 30324 In a 4-cycle, two distinct...
n4cyclfrgr 30325 There is no 4-cycle in a f...
4cyclusnfrgr 30326 A graph with a 4-cycle is ...
frgrnbnb 30327 If two neighbors ` U ` and...
frgrconngr 30328 A friendship graph is conn...
vdgn0frgrv2 30329 A vertex in a friendship g...
vdgn1frgrv2 30330 Any vertex in a friendship...
vdgn1frgrv3 30331 Any vertex in a friendship...
vdgfrgrgt2 30332 Any vertex in a friendship...
frgrncvvdeqlem1 30333 Lemma 1 for ~ frgrncvvdeq ...
frgrncvvdeqlem2 30334 Lemma 2 for ~ frgrncvvdeq ...
frgrncvvdeqlem3 30335 Lemma 3 for ~ frgrncvvdeq ...
frgrncvvdeqlem4 30336 Lemma 4 for ~ frgrncvvdeq ...
frgrncvvdeqlem5 30337 Lemma 5 for ~ frgrncvvdeq ...
frgrncvvdeqlem6 30338 Lemma 6 for ~ frgrncvvdeq ...
frgrncvvdeqlem7 30339 Lemma 7 for ~ frgrncvvdeq ...
frgrncvvdeqlem8 30340 Lemma 8 for ~ frgrncvvdeq ...
frgrncvvdeqlem9 30341 Lemma 9 for ~ frgrncvvdeq ...
frgrncvvdeqlem10 30342 Lemma 10 for ~ frgrncvvdeq...
frgrncvvdeq 30343 In a friendship graph, two...
frgrwopreglem4a 30344 In a friendship graph any ...
frgrwopreglem5a 30345 If a friendship graph has ...
frgrwopreglem1 30346 Lemma 1 for ~ frgrwopreg :...
frgrwopreglem2 30347 Lemma 2 for ~ frgrwopreg ....
frgrwopreglem3 30348 Lemma 3 for ~ frgrwopreg ....
frgrwopreglem4 30349 Lemma 4 for ~ frgrwopreg ....
frgrwopregasn 30350 According to statement 5 i...
frgrwopregbsn 30351 According to statement 5 i...
frgrwopreg1 30352 According to statement 5 i...
frgrwopreg2 30353 According to statement 5 i...
frgrwopreglem5lem 30354 Lemma for ~ frgrwopreglem5...
frgrwopreglem5 30355 Lemma 5 for ~ frgrwopreg ....
frgrwopreglem5ALT 30356 Alternate direct proof of ...
frgrwopreg 30357 In a friendship graph ther...
frgrregorufr0 30358 In a friendship graph ther...
frgrregorufr 30359 If there is a vertex havin...
frgrregorufrg 30360 If there is a vertex havin...
frgr2wwlkeu 30361 For two different vertices...
frgr2wwlkn0 30362 In a friendship graph, the...
frgr2wwlk1 30363 In a friendship graph, the...
frgr2wsp1 30364 In a friendship graph, the...
frgr2wwlkeqm 30365 If there is a (simple) pat...
frgrhash2wsp 30366 The number of simple paths...
fusgreg2wsplem 30367 Lemma for ~ fusgreg2wsp an...
fusgr2wsp2nb 30368 The set of paths of length...
fusgreghash2wspv 30369 According to statement 7 i...
fusgreg2wsp 30370 In a finite simple graph, ...
2wspmdisj 30371 The sets of paths of lengt...
fusgreghash2wsp 30372 In a finite k-regular grap...
frrusgrord0lem 30373 Lemma for ~ frrusgrord0 . ...
frrusgrord0 30374 If a nonempty finite frien...
frrusgrord 30375 If a nonempty finite frien...
numclwwlk2lem1lem 30376 Lemma for ~ numclwwlk2lem1...
2clwwlklem 30377 Lemma for ~ clwwnonrepclww...
clwwnrepclwwn 30378 If the initial vertex of a...
clwwnonrepclwwnon 30379 If the initial vertex of a...
2clwwlk2clwwlklem 30380 Lemma for ~ 2clwwlk2clwwlk...
2clwwlk 30381 Value of operation ` C ` ,...
2clwwlk2 30382 The set ` ( X C 2 ) ` of d...
2clwwlkel 30383 Characterization of an ele...
2clwwlk2clwwlk 30384 An element of the value of...
numclwwlk1lem2foalem 30385 Lemma for ~ numclwwlk1lem2...
extwwlkfab 30386 The set ` ( X C N ) ` of d...
extwwlkfabel 30387 Characterization of an ele...
numclwwlk1lem2foa 30388 Going forth and back from ...
numclwwlk1lem2f 30389 ` T ` is a function, mappi...
numclwwlk1lem2fv 30390 Value of the function ` T ...
numclwwlk1lem2f1 30391 ` T ` is a 1-1 function. ...
numclwwlk1lem2fo 30392 ` T ` is an onto function....
numclwwlk1lem2f1o 30393 ` T ` is a 1-1 onto functi...
numclwwlk1lem2 30394 The set of double loops of...
numclwwlk1 30395 Statement 9 in [Huneke] p....
clwwlknonclwlknonf1o 30396 ` F ` is a bijection betwe...
clwwlknonclwlknonen 30397 The sets of the two repres...
dlwwlknondlwlknonf1olem1 30398 Lemma 1 for ~ dlwwlknondlw...
dlwwlknondlwlknonf1o 30399 ` F ` is a bijection betwe...
dlwwlknondlwlknonen 30400 The sets of the two repres...
wlkl0 30401 There is exactly one walk ...
clwlknon2num 30402 There are k walks of lengt...
numclwlk1lem1 30403 Lemma 1 for ~ numclwlk1 (S...
numclwlk1lem2 30404 Lemma 2 for ~ numclwlk1 (S...
numclwlk1 30405 Statement 9 in [Huneke] p....
numclwwlkovh0 30406 Value of operation ` H ` ,...
numclwwlkovh 30407 Value of operation ` H ` ,...
numclwwlkovq 30408 Value of operation ` Q ` ,...
numclwwlkqhash 30409 In a ` K `-regular graph, ...
numclwwlk2lem1 30410 In a friendship graph, for...
numclwlk2lem2f 30411 ` R ` is a function mappin...
numclwlk2lem2fv 30412 Value of the function ` R ...
numclwlk2lem2f1o 30413 ` R ` is a 1-1 onto functi...
numclwwlk2lem3 30414 In a friendship graph, the...
numclwwlk2 30415 Statement 10 in [Huneke] p...
numclwwlk3lem1 30416 Lemma 2 for ~ numclwwlk3 ....
numclwwlk3lem2lem 30417 Lemma for ~ numclwwlk3lem2...
numclwwlk3lem2 30418 Lemma 1 for ~ numclwwlk3 :...
numclwwlk3 30419 Statement 12 in [Huneke] p...
numclwwlk4 30420 The total number of closed...
numclwwlk5lem 30421 Lemma for ~ numclwwlk5 . ...
numclwwlk5 30422 Statement 13 in [Huneke] p...
numclwwlk7lem 30423 Lemma for ~ numclwwlk7 , ~...
numclwwlk6 30424 For a prime divisor ` P ` ...
numclwwlk7 30425 Statement 14 in [Huneke] p...
numclwwlk8 30426 The size of the set of clo...
frgrreggt1 30427 If a finite nonempty frien...
frgrreg 30428 If a finite nonempty frien...
frgrregord013 30429 If a finite friendship gra...
frgrregord13 30430 If a nonempty finite frien...
frgrogt3nreg 30431 If a finite friendship gra...
friendshipgt3 30432 The friendship theorem for...
friendship 30433 The friendship theorem: I...
conventions 30434

H...

conventions-labels 30435

...

conventions-comments 30436

...

natded 30437 Here are typical n...
ex-natded5.2 30438 Theorem 5.2 of [Clemente] ...
ex-natded5.2-2 30439 A more efficient proof of ...
ex-natded5.2i 30440 The same as ~ ex-natded5.2...
ex-natded5.3 30441 Theorem 5.3 of [Clemente] ...
ex-natded5.3-2 30442 A more efficient proof of ...
ex-natded5.3i 30443 The same as ~ ex-natded5.3...
ex-natded5.5 30444 Theorem 5.5 of [Clemente] ...
ex-natded5.7 30445 Theorem 5.7 of [Clemente] ...
ex-natded5.7-2 30446 A more efficient proof of ...
ex-natded5.8 30447 Theorem 5.8 of [Clemente] ...
ex-natded5.8-2 30448 A more efficient proof of ...
ex-natded5.13 30449 Theorem 5.13 of [Clemente]...
ex-natded5.13-2 30450 A more efficient proof of ...
ex-natded9.20 30451 Theorem 9.20 of [Clemente]...
ex-natded9.20-2 30452 A more efficient proof of ...
ex-natded9.26 30453 Theorem 9.26 of [Clemente]...
ex-natded9.26-2 30454 A more efficient proof of ...
ex-or 30455 Example for ~ df-or . Exa...
ex-an 30456 Example for ~ df-an . Exa...
ex-dif 30457 Example for ~ df-dif . Ex...
ex-un 30458 Example for ~ df-un . Exa...
ex-in 30459 Example for ~ df-in . Exa...
ex-uni 30460 Example for ~ df-uni . Ex...
ex-ss 30461 Example for ~ df-ss . Exa...
ex-pss 30462 Example for ~ df-pss . Ex...
ex-pw 30463 Example for ~ df-pw . Exa...
ex-pr 30464 Example for ~ df-pr . (Co...
ex-br 30465 Example for ~ df-br . Exa...
ex-opab 30466 Example for ~ df-opab . E...
ex-eprel 30467 Example for ~ df-eprel . ...
ex-id 30468 Example for ~ df-id . Exa...
ex-po 30469 Example for ~ df-po . Exa...
ex-xp 30470 Example for ~ df-xp . Exa...
ex-cnv 30471 Example for ~ df-cnv . Ex...
ex-co 30472 Example for ~ df-co . Exa...
ex-dm 30473 Example for ~ df-dm . Exa...
ex-rn 30474 Example for ~ df-rn . Exa...
ex-res 30475 Example for ~ df-res . Ex...
ex-ima 30476 Example for ~ df-ima . Ex...
ex-fv 30477 Example for ~ df-fv . Exa...
ex-1st 30478 Example for ~ df-1st . Ex...
ex-2nd 30479 Example for ~ df-2nd . Ex...
1kp2ke3k 30480 Example for ~ df-dec , 100...
ex-fl 30481 Example for ~ df-fl . Exa...
ex-ceil 30482 Example for ~ df-ceil . (...
ex-mod 30483 Example for ~ df-mod . (C...
ex-exp 30484 Example for ~ df-exp . (C...
ex-fac 30485 Example for ~ df-fac . (C...
ex-bc 30486 Example for ~ df-bc . (Co...
ex-hash 30487 Example for ~ df-hash . (...
ex-sqrt 30488 Example for ~ df-sqrt . (...
ex-abs 30489 Example for ~ df-abs . (C...
ex-dvds 30490 Example for ~ df-dvds : 3 ...
ex-gcd 30491 Example for ~ df-gcd . (C...
ex-lcm 30492 Example for ~ df-lcm . (C...
ex-prmo 30493 Example for ~ df-prmo : ` ...
aevdemo 30494 Proof illustrating the com...
ex-ind-dvds 30495 Example of a proof by indu...
ex-fpar 30496 Formalized example provide...
avril1 30497 Poisson d'Avril's Theorem....
2bornot2b 30498 The law of excluded middle...
helloworld 30499 The classic "Hello world" ...
1p1e2apr1 30500 One plus one equals two. ...
eqid1 30501 Law of identity (reflexivi...
1div0apr 30502 Division by zero is forbid...
topnfbey 30503 Nothing seems to be imposs...
9p10ne21 30504 9 + 10 is not equal to 21....
9p10ne21fool 30505 9 + 10 equals 21. This as...
nrt2irr 30507 The ` N ` -th root of 2 is...
isplig 30510 The predicate "is a planar...
ispligb 30511 The predicate "is a planar...
tncp 30512 In any planar incidence ge...
l2p 30513 For any line in a planar i...
lpni 30514 For any line in a planar i...
nsnlplig 30515 There is no "one-point lin...
nsnlpligALT 30516 Alternate version of ~ nsn...
n0lplig 30517 There is no "empty line" i...
n0lpligALT 30518 Alternate version of ~ n0l...
eulplig 30519 Through two distinct point...
pliguhgr 30520 Any planar incidence geome...
dummylink 30521 Alias for ~ a1ii that may ...
id1 30522 Alias for ~ idALT that may...
isgrpo 30531 The predicate "is a group ...
isgrpoi 30532 Properties that determine ...
grpofo 30533 A group operation maps ont...
grpocl 30534 Closure law for a group op...
grpolidinv 30535 A group has a left identit...
grpon0 30536 The base set of a group is...
grpoass 30537 A group operation is assoc...
grpoidinvlem1 30538 Lemma for ~ grpoidinv . (...
grpoidinvlem2 30539 Lemma for ~ grpoidinv . (...
grpoidinvlem3 30540 Lemma for ~ grpoidinv . (...
grpoidinvlem4 30541 Lemma for ~ grpoidinv . (...
grpoidinv 30542 A group has a left and rig...
grpoideu 30543 The left identity element ...
grporndm 30544 A group's range in terms o...
0ngrp 30545 The empty set is not a gro...
gidval 30546 The value of the identity ...
grpoidval 30547 Lemma for ~ grpoidcl and o...
grpoidcl 30548 The identity element of a ...
grpoidinv2 30549 A group's properties using...
grpolid 30550 The identity element of a ...
grporid 30551 The identity element of a ...
grporcan 30552 Right cancellation law for...
grpoinveu 30553 The left inverse element o...
grpoid 30554 Two ways of saying that an...
grporn 30555 The range of a group opera...
grpoinvfval 30556 The inverse function of a ...
grpoinvval 30557 The inverse of a group ele...
grpoinvcl 30558 A group element's inverse ...
grpoinv 30559 The properties of a group ...
grpolinv 30560 The left inverse of a grou...
grporinv 30561 The right inverse of a gro...
grpoinvid1 30562 The inverse of a group ele...
grpoinvid2 30563 The inverse of a group ele...
grpolcan 30564 Left cancellation law for ...
grpo2inv 30565 Double inverse law for gro...
grpoinvf 30566 Mapping of the inverse fun...
grpoinvop 30567 The inverse of the group o...
grpodivfval 30568 Group division (or subtrac...
grpodivval 30569 Group division (or subtrac...
grpodivinv 30570 Group division by an inver...
grpoinvdiv 30571 Inverse of a group divisio...
grpodivf 30572 Mapping for group division...
grpodivcl 30573 Closure of group division ...
grpodivdiv 30574 Double group division. (C...
grpomuldivass 30575 Associative-type law for m...
grpodivid 30576 Division of a group member...
grponpcan 30577 Cancellation law for group...
isablo 30580 The predicate "is an Abeli...
ablogrpo 30581 An Abelian group operation...
ablocom 30582 An Abelian group operation...
ablo32 30583 Commutative/associative la...
ablo4 30584 Commutative/associative la...
isabloi 30585 Properties that determine ...
ablomuldiv 30586 Law for group multiplicati...
ablodivdiv 30587 Law for double group divis...
ablodivdiv4 30588 Law for double group divis...
ablodiv32 30589 Swap the second and third ...
ablonncan 30590 Cancellation law for group...
ablonnncan1 30591 Cancellation law for group...
vcrel 30594 The class of all complex v...
vciOLD 30595 Obsolete version of ~ cvsi...
vcsm 30596 Functionality of th scalar...
vccl 30597 Closure of the scalar prod...
vcidOLD 30598 Identity element for the s...
vcdi 30599 Distributive law for the s...
vcdir 30600 Distributive law for the s...
vcass 30601 Associative law for the sc...
vc2OLD 30602 A vector plus itself is tw...
vcablo 30603 Vector addition is an Abel...
vcgrp 30604 Vector addition is a group...
vclcan 30605 Left cancellation law for ...
vczcl 30606 The zero vector is a vecto...
vc0rid 30607 The zero vector is a right...
vc0 30608 Zero times a vector is the...
vcz 30609 Anything times the zero ve...
vcm 30610 Minus 1 times a vector is ...
isvclem 30611 Lemma for ~ isvcOLD . (Co...
vcex 30612 The components of a comple...
isvcOLD 30613 The predicate "is a comple...
isvciOLD 30614 Properties that determine ...
cnaddabloOLD 30615 Obsolete version of ~ cnad...
cnidOLD 30616 Obsolete version of ~ cnad...
cncvcOLD 30617 Obsolete version of ~ cncv...
nvss 30627 Structure of the class of ...
nvvcop 30628 A normed complex vector sp...
nvrel 30636 The class of all normed co...
vafval 30637 Value of the function for ...
bafval 30638 Value of the function for ...
smfval 30639 Value of the function for ...
0vfval 30640 Value of the function for ...
nmcvfval 30641 Value of the norm function...
nvop2 30642 A normed complex vector sp...
nvvop 30643 The vector space component...
isnvlem 30644 Lemma for ~ isnv . (Contr...
nvex 30645 The components of a normed...
isnv 30646 The predicate "is a normed...
isnvi 30647 Properties that determine ...
nvi 30648 The properties of a normed...
nvvc 30649 The vector space component...
nvablo 30650 The vector addition operat...
nvgrp 30651 The vector addition operat...
nvgf 30652 Mapping for the vector add...
nvsf 30653 Mapping for the scalar mul...
nvgcl 30654 Closure law for the vector...
nvcom 30655 The vector addition (group...
nvass 30656 The vector addition (group...
nvadd32 30657 Commutative/associative la...
nvrcan 30658 Right cancellation law for...
nvadd4 30659 Rearrangement of 4 terms i...
nvscl 30660 Closure law for the scalar...
nvsid 30661 Identity element for the s...
nvsass 30662 Associative law for the sc...
nvscom 30663 Commutative law for the sc...
nvdi 30664 Distributive law for the s...
nvdir 30665 Distributive law for the s...
nv2 30666 A vector plus itself is tw...
vsfval 30667 Value of the function for ...
nvzcl 30668 Closure law for the zero v...
nv0rid 30669 The zero vector is a right...
nv0lid 30670 The zero vector is a left ...
nv0 30671 Zero times a vector is the...
nvsz 30672 Anything times the zero ve...
nvinv 30673 Minus 1 times a vector is ...
nvinvfval 30674 Function for the negative ...
nvm 30675 Vector subtraction in term...
nvmval 30676 Value of vector subtractio...
nvmval2 30677 Value of vector subtractio...
nvmfval 30678 Value of the function for ...
nvmf 30679 Mapping for the vector sub...
nvmcl 30680 Closure law for the vector...
nvnnncan1 30681 Cancellation law for vecto...
nvmdi 30682 Distributive law for scala...
nvnegneg 30683 Double negative of a vecto...
nvmul0or 30684 If a scalar product is zer...
nvrinv 30685 A vector minus itself. (C...
nvlinv 30686 Minus a vector plus itself...
nvpncan2 30687 Cancellation law for vecto...
nvpncan 30688 Cancellation law for vecto...
nvaddsub 30689 Commutative/associative la...
nvnpcan 30690 Cancellation law for a nor...
nvaddsub4 30691 Rearrangement of 4 terms i...
nvmeq0 30692 The difference between two...
nvmid 30693 A vector minus itself is t...
nvf 30694 Mapping for the norm funct...
nvcl 30695 The norm of a normed compl...
nvcli 30696 The norm of a normed compl...
nvs 30697 Proportionality property o...
nvsge0 30698 The norm of a scalar produ...
nvm1 30699 The norm of the negative o...
nvdif 30700 The norm of the difference...
nvpi 30701 The norm of a vector plus ...
nvz0 30702 The norm of a zero vector ...
nvz 30703 The norm of a vector is ze...
nvtri 30704 Triangle inequality for th...
nvmtri 30705 Triangle inequality for th...
nvabs 30706 Norm difference property o...
nvge0 30707 The norm of a normed compl...
nvgt0 30708 A nonzero norm is positive...
nv1 30709 From any nonzero vector, c...
nvop 30710 A complex inner product sp...
cnnv 30711 The set of complex numbers...
cnnvg 30712 The vector addition (group...
cnnvba 30713 The base set of the normed...
cnnvs 30714 The scalar product operati...
cnnvnm 30715 The norm operation of the ...
cnnvm 30716 The vector subtraction ope...
elimnv 30717 Hypothesis elimination lem...
elimnvu 30718 Hypothesis elimination lem...
imsval 30719 Value of the induced metri...
imsdval 30720 Value of the induced metri...
imsdval2 30721 Value of the distance func...
nvnd 30722 The norm of a normed compl...
imsdf 30723 Mapping for the induced me...
imsmetlem 30724 Lemma for ~ imsmet . (Con...
imsmet 30725 The induced metric of a no...
imsxmet 30726 The induced metric of a no...
cnims 30727 The metric induced on the ...
vacn 30728 Vector addition is jointly...
nmcvcn 30729 The norm of a normed compl...
nmcnc 30730 The norm of a normed compl...
smcnlem 30731 Lemma for ~ smcn . (Contr...
smcn 30732 Scalar multiplication is j...
vmcn 30733 Vector subtraction is join...
dipfval 30736 The inner product function...
ipval 30737 Value of the inner product...
ipval2lem2 30738 Lemma for ~ ipval3 . (Con...
ipval2lem3 30739 Lemma for ~ ipval3 . (Con...
ipval2lem4 30740 Lemma for ~ ipval3 . (Con...
ipval2 30741 Expansion of the inner pro...
4ipval2 30742 Four times the inner produ...
ipval3 30743 Expansion of the inner pro...
ipidsq 30744 The inner product of a vec...
ipnm 30745 Norm expressed in terms of...
dipcl 30746 An inner product is a comp...
ipf 30747 Mapping for the inner prod...
dipcj 30748 The complex conjugate of a...
ipipcj 30749 An inner product times its...
diporthcom 30750 Orthogonality (meaning inn...
dip0r 30751 Inner product with a zero ...
dip0l 30752 Inner product with a zero ...
ipz 30753 The inner product of a vec...
dipcn 30754 Inner product is jointly c...
sspval 30757 The set of all subspaces o...
isssp 30758 The predicate "is a subspa...
sspid 30759 A normed complex vector sp...
sspnv 30760 A subspace is a normed com...
sspba 30761 The base set of a subspace...
sspg 30762 Vector addition on a subsp...
sspgval 30763 Vector addition on a subsp...
ssps 30764 Scalar multiplication on a...
sspsval 30765 Scalar multiplication on a...
sspmlem 30766 Lemma for ~ sspm and other...
sspmval 30767 Vector addition on a subsp...
sspm 30768 Vector subtraction on a su...
sspz 30769 The zero vector of a subsp...
sspn 30770 The norm on a subspace is ...
sspnval 30771 The norm on a subspace in ...
sspimsval 30772 The induced metric on a su...
sspims 30773 The induced metric on a su...
lnoval 30786 The set of linear operator...
islno 30787 The predicate "is a linear...
lnolin 30788 Basic linearity property o...
lnof 30789 A linear operator is a map...
lno0 30790 The value of a linear oper...
lnocoi 30791 The composition of two lin...
lnoadd 30792 Addition property of a lin...
lnosub 30793 Subtraction property of a ...
lnomul 30794 Scalar multiplication prop...
nvo00 30795 Two ways to express a zero...
nmoofval 30796 The operator norm function...
nmooval 30797 The operator norm function...
nmosetre 30798 The set in the supremum of...
nmosetn0 30799 The set in the supremum of...
nmoxr 30800 The norm of an operator is...
nmooge0 30801 The norm of an operator is...
nmorepnf 30802 The norm of an operator is...
nmoreltpnf 30803 The norm of any operator i...
nmogtmnf 30804 The norm of an operator is...
nmoolb 30805 A lower bound for an opera...
nmoubi 30806 An upper bound for an oper...
nmoub3i 30807 An upper bound for an oper...
nmoub2i 30808 An upper bound for an oper...
nmobndi 30809 Two ways to express that a...
nmounbi 30810 Two ways two express that ...
nmounbseqi 30811 An unbounded operator dete...
nmounbseqiALT 30812 Alternate shorter proof of...
nmobndseqi 30813 A bounded sequence determi...
nmobndseqiALT 30814 Alternate shorter proof of...
bloval 30815 The class of bounded linea...
isblo 30816 The predicate "is a bounde...
isblo2 30817 The predicate "is a bounde...
bloln 30818 A bounded operator is a li...
blof 30819 A bounded operator is an o...
nmblore 30820 The norm of a bounded oper...
0ofval 30821 The zero operator between ...
0oval 30822 Value of the zero operator...
0oo 30823 The zero operator is an op...
0lno 30824 The zero operator is linea...
nmoo0 30825 The operator norm of the z...
0blo 30826 The zero operator is a bou...
nmlno0lem 30827 Lemma for ~ nmlno0i . (Co...
nmlno0i 30828 The norm of a linear opera...
nmlno0 30829 The norm of a linear opera...
nmlnoubi 30830 An upper bound for the ope...
nmlnogt0 30831 The norm of a nonzero line...
lnon0 30832 The domain of a nonzero li...
nmblolbii 30833 A lower bound for the norm...
nmblolbi 30834 A lower bound for the norm...
isblo3i 30835 The predicate "is a bounde...
blo3i 30836 Properties that determine ...
blometi 30837 Upper bound for the distan...
blocnilem 30838 Lemma for ~ blocni and ~ l...
blocni 30839 A linear operator is conti...
lnocni 30840 If a linear operator is co...
blocn 30841 A linear operator is conti...
blocn2 30842 A bounded linear operator ...
ajfval 30843 The adjoint function. (Co...
hmoval 30844 The set of Hermitian (self...
ishmo 30845 The predicate "is a hermit...
phnv 30848 Every complex inner produc...
phrel 30849 The class of all complex i...
phnvi 30850 Every complex inner produc...
isphg 30851 The predicate "is a comple...
phop 30852 A complex inner product sp...
cncph 30853 The set of complex numbers...
elimph 30854 Hypothesis elimination lem...
elimphu 30855 Hypothesis elimination lem...
isph 30856 The predicate "is an inner...
phpar2 30857 The parallelogram law for ...
phpar 30858 The parallelogram law for ...
ip0i 30859 A slight variant of Equati...
ip1ilem 30860 Lemma for ~ ip1i . (Contr...
ip1i 30861 Equation 6.47 of [Ponnusam...
ip2i 30862 Equation 6.48 of [Ponnusam...
ipdirilem 30863 Lemma for ~ ipdiri . (Con...
ipdiri 30864 Distributive law for inner...
ipasslem1 30865 Lemma for ~ ipassi . Show...
ipasslem2 30866 Lemma for ~ ipassi . Show...
ipasslem3 30867 Lemma for ~ ipassi . Show...
ipasslem4 30868 Lemma for ~ ipassi . Show...
ipasslem5 30869 Lemma for ~ ipassi . Show...
ipasslem7 30870 Lemma for ~ ipassi . Show...
ipasslem8 30871 Lemma for ~ ipassi . By ~...
ipasslem9 30872 Lemma for ~ ipassi . Conc...
ipasslem10 30873 Lemma for ~ ipassi . Show...
ipasslem11 30874 Lemma for ~ ipassi . Show...
ipassi 30875 Associative law for inner ...
dipdir 30876 Distributive law for inner...
dipdi 30877 Distributive law for inner...
ip2dii 30878 Inner product of two sums....
dipass 30879 Associative law for inner ...
dipassr 30880 "Associative" law for seco...
dipassr2 30881 "Associative" law for inne...
dipsubdir 30882 Distributive law for inner...
dipsubdi 30883 Distributive law for inner...
pythi 30884 The Pythagorean theorem fo...
siilem1 30885 Lemma for ~ sii . (Contri...
siilem2 30886 Lemma for ~ sii . (Contri...
siii 30887 Inference from ~ sii . (C...
sii 30888 Obsolete version of ~ ipca...
ipblnfi 30889 A function ` F ` generated...
ip2eqi 30890 Two vectors are equal iff ...
phoeqi 30891 A condition implying that ...
ajmoi 30892 Every operator has at most...
ajfuni 30893 The adjoint function is a ...
ajfun 30894 The adjoint function is a ...
ajval 30895 Value of the adjoint funct...
iscbn 30898 A complex Banach space is ...
cbncms 30899 The induced metric on comp...
bnnv 30900 Every complex Banach space...
bnrel 30901 The class of all complex B...
bnsscmcl 30902 A subspace of a Banach spa...
cnbn 30903 The set of complex numbers...
ubthlem1 30904 Lemma for ~ ubth . The fu...
ubthlem2 30905 Lemma for ~ ubth . Given ...
ubthlem3 30906 Lemma for ~ ubth . Prove ...
ubth 30907 Uniform Boundedness Theore...
minvecolem1 30908 Lemma for ~ minveco . The...
minvecolem2 30909 Lemma for ~ minveco . Any...
minvecolem3 30910 Lemma for ~ minveco . The...
minvecolem4a 30911 Lemma for ~ minveco . ` F ...
minvecolem4b 30912 Lemma for ~ minveco . The...
minvecolem4c 30913 Lemma for ~ minveco . The...
minvecolem4 30914 Lemma for ~ minveco . The...
minvecolem5 30915 Lemma for ~ minveco . Dis...
minvecolem6 30916 Lemma for ~ minveco . Any...
minvecolem7 30917 Lemma for ~ minveco . Sin...
minveco 30918 Minimizing vector theorem,...
ishlo 30921 The predicate "is a comple...
hlobn 30922 Every complex Hilbert spac...
hlph 30923 Every complex Hilbert spac...
hlrel 30924 The class of all complex H...
hlnv 30925 Every complex Hilbert spac...
hlnvi 30926 Every complex Hilbert spac...
hlvc 30927 Every complex Hilbert spac...
hlcmet 30928 The induced metric on a co...
hlmet 30929 The induced metric on a co...
hlpar2 30930 The parallelogram law sati...
hlpar 30931 The parallelogram law sati...
hlex 30932 The base set of a Hilbert ...
hladdf 30933 Mapping for Hilbert space ...
hlcom 30934 Hilbert space vector addit...
hlass 30935 Hilbert space vector addit...
hl0cl 30936 The Hilbert space zero vec...
hladdid 30937 Hilbert space addition wit...
hlmulf 30938 Mapping for Hilbert space ...
hlmulid 30939 Hilbert space scalar multi...
hlmulass 30940 Hilbert space scalar multi...
hldi 30941 Hilbert space scalar multi...
hldir 30942 Hilbert space scalar multi...
hlmul0 30943 Hilbert space scalar multi...
hlipf 30944 Mapping for Hilbert space ...
hlipcj 30945 Conjugate law for Hilbert ...
hlipdir 30946 Distributive law for Hilbe...
hlipass 30947 Associative law for Hilber...
hlipgt0 30948 The inner product of a Hil...
hlcompl 30949 Completeness of a Hilbert ...
cnchl 30950 The set of complex numbers...
htthlem 30951 Lemma for ~ htth . The co...
htth 30952 Hellinger-Toeplitz Theorem...
The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here
h2hva 31008 The group (addition) opera...
h2hsm 31009 The scalar product operati...
h2hnm 31010 The norm function of Hilbe...
h2hvs 31011 The vector subtraction ope...
h2hmetdval 31012 Value of the distance func...
h2hcau 31013 The Cauchy sequences of Hi...
h2hlm 31014 The limit sequences of Hil...
axhilex-zf 31015 Derive Axiom ~ ax-hilex fr...
axhfvadd-zf 31016 Derive Axiom ~ ax-hfvadd f...
axhvcom-zf 31017 Derive Axiom ~ ax-hvcom fr...
axhvass-zf 31018 Derive Axiom ~ ax-hvass fr...
axhv0cl-zf 31019 Derive Axiom ~ ax-hv0cl fr...
axhvaddid-zf 31020 Derive Axiom ~ ax-hvaddid ...
axhfvmul-zf 31021 Derive Axiom ~ ax-hfvmul f...
axhvmulid-zf 31022 Derive Axiom ~ ax-hvmulid ...
axhvmulass-zf 31023 Derive Axiom ~ ax-hvmulass...
axhvdistr1-zf 31024 Derive Axiom ~ ax-hvdistr1...
axhvdistr2-zf 31025 Derive Axiom ~ ax-hvdistr2...
axhvmul0-zf 31026 Derive Axiom ~ ax-hvmul0 f...
axhfi-zf 31027 Derive Axiom ~ ax-hfi from...
axhis1-zf 31028 Derive Axiom ~ ax-his1 fro...
axhis2-zf 31029 Derive Axiom ~ ax-his2 fro...
axhis3-zf 31030 Derive Axiom ~ ax-his3 fro...
axhis4-zf 31031 Derive Axiom ~ ax-his4 fro...
axhcompl-zf 31032 Derive Axiom ~ ax-hcompl f...
hvmulex 31045 The Hilbert space scalar p...
hvaddcl 31046 Closure of vector addition...
hvmulcl 31047 Closure of scalar multipli...
hvmulcli 31048 Closure inference for scal...
hvsubf 31049 Mapping domain and codomai...
hvsubval 31050 Value of vector subtractio...
hvsubcl 31051 Closure of vector subtract...
hvaddcli 31052 Closure of vector addition...
hvcomi 31053 Commutation of vector addi...
hvsubvali 31054 Value of vector subtractio...
hvsubcli 31055 Closure of vector subtract...
ifhvhv0 31056 Prove ` if ( A e. ~H , A ,...
hvaddlid 31057 Addition with the zero vec...
hvmul0 31058 Scalar multiplication with...
hvmul0or 31059 If a scalar product is zer...
hvsubid 31060 Subtraction of a vector fr...
hvnegid 31061 Addition of negative of a ...
hv2neg 31062 Two ways to express the ne...
hvaddlidi 31063 Addition with the zero vec...
hvnegidi 31064 Addition of negative of a ...
hv2negi 31065 Two ways to express the ne...
hvm1neg 31066 Convert minus one times a ...
hvaddsubval 31067 Value of vector addition i...
hvadd32 31068 Commutative/associative la...
hvadd12 31069 Commutative/associative la...
hvadd4 31070 Hilbert vector space addit...
hvsub4 31071 Hilbert vector space addit...
hvaddsub12 31072 Commutative/associative la...
hvpncan 31073 Addition/subtraction cance...
hvpncan2 31074 Addition/subtraction cance...
hvaddsubass 31075 Associativity of sum and d...
hvpncan3 31076 Subtraction and addition o...
hvmulcom 31077 Scalar multiplication comm...
hvsubass 31078 Hilbert vector space assoc...
hvsub32 31079 Hilbert vector space commu...
hvmulassi 31080 Scalar multiplication asso...
hvmulcomi 31081 Scalar multiplication comm...
hvmul2negi 31082 Double negative in scalar ...
hvsubdistr1 31083 Scalar multiplication dist...
hvsubdistr2 31084 Scalar multiplication dist...
hvdistr1i 31085 Scalar multiplication dist...
hvsubdistr1i 31086 Scalar multiplication dist...
hvassi 31087 Hilbert vector space assoc...
hvadd32i 31088 Hilbert vector space commu...
hvsubassi 31089 Hilbert vector space assoc...
hvsub32i 31090 Hilbert vector space commu...
hvadd12i 31091 Hilbert vector space commu...
hvadd4i 31092 Hilbert vector space addit...
hvsubsub4i 31093 Hilbert vector space addit...
hvsubsub4 31094 Hilbert vector space addit...
hv2times 31095 Two times a vector. (Cont...
hvnegdii 31096 Distribution of negative o...
hvsubeq0i 31097 If the difference between ...
hvsubcan2i 31098 Vector cancellation law. ...
hvaddcani 31099 Cancellation law for vecto...
hvsubaddi 31100 Relationship between vecto...
hvnegdi 31101 Distribution of negative o...
hvsubeq0 31102 If the difference between ...
hvaddeq0 31103 If the sum of two vectors ...
hvaddcan 31104 Cancellation law for vecto...
hvaddcan2 31105 Cancellation law for vecto...
hvmulcan 31106 Cancellation law for scala...
hvmulcan2 31107 Cancellation law for scala...
hvsubcan 31108 Cancellation law for vecto...
hvsubcan2 31109 Cancellation law for vecto...
hvsub0 31110 Subtraction of a zero vect...
hvsubadd 31111 Relationship between vecto...
hvaddsub4 31112 Hilbert vector space addit...
hicl 31114 Closure of inner product. ...
hicli 31115 Closure inference for inne...
his5 31120 Associative law for inner ...
his52 31121 Associative law for inner ...
his35 31122 Move scalar multiplication...
his35i 31123 Move scalar multiplication...
his7 31124 Distributive law for inner...
hiassdi 31125 Distributive/associative l...
his2sub 31126 Distributive law for inner...
his2sub2 31127 Distributive law for inner...
hire 31128 A necessary and sufficient...
hiidrcl 31129 Real closure of inner prod...
hi01 31130 Inner product with the 0 v...
hi02 31131 Inner product with the 0 v...
hiidge0 31132 Inner product with self is...
his6 31133 Zero inner product with se...
his1i 31134 Conjugate law for inner pr...
abshicom 31135 Commuted inner products ha...
hial0 31136 A vector whose inner produ...
hial02 31137 A vector whose inner produ...
hisubcomi 31138 Two vector subtractions si...
hi2eq 31139 Lemma used to prove equali...
hial2eq 31140 Two vectors whose inner pr...
hial2eq2 31141 Two vectors whose inner pr...
orthcom 31142 Orthogonality commutes. (...
normlem0 31143 Lemma used to derive prope...
normlem1 31144 Lemma used to derive prope...
normlem2 31145 Lemma used to derive prope...
normlem3 31146 Lemma used to derive prope...
normlem4 31147 Lemma used to derive prope...
normlem5 31148 Lemma used to derive prope...
normlem6 31149 Lemma used to derive prope...
normlem7 31150 Lemma used to derive prope...
normlem8 31151 Lemma used to derive prope...
normlem9 31152 Lemma used to derive prope...
normlem7tALT 31153 Lemma used to derive prope...
bcseqi 31154 Equality case of Bunjakova...
normlem9at 31155 Lemma used to derive prope...
dfhnorm2 31156 Alternate definition of th...
normf 31157 The norm function maps fro...
normval 31158 The value of the norm of a...
normcl 31159 Real closure of the norm o...
normge0 31160 The norm of a vector is no...
normgt0 31161 The norm of nonzero vector...
norm0 31162 The norm of a zero vector....
norm-i 31163 Theorem 3.3(i) of [Beran] ...
normne0 31164 A norm is nonzero iff its ...
normcli 31165 Real closure of the norm o...
normsqi 31166 The square of a norm. (Co...
norm-i-i 31167 Theorem 3.3(i) of [Beran] ...
normsq 31168 The square of a norm. (Co...
normsub0i 31169 Two vectors are equal iff ...
normsub0 31170 Two vectors are equal iff ...
norm-ii-i 31171 Triangle inequality for no...
norm-ii 31172 Triangle inequality for no...
norm-iii-i 31173 Theorem 3.3(iii) of [Beran...
norm-iii 31174 Theorem 3.3(iii) of [Beran...
normsubi 31175 Negative doesn't change th...
normpythi 31176 Analogy to Pythagorean the...
normsub 31177 Swapping order of subtract...
normneg 31178 The norm of a vector equal...
normpyth 31179 Analogy to Pythagorean the...
normpyc 31180 Corollary to Pythagorean t...
norm3difi 31181 Norm of differences around...
norm3adifii 31182 Norm of differences around...
norm3lem 31183 Lemma involving norm of di...
norm3dif 31184 Norm of differences around...
norm3dif2 31185 Norm of differences around...
norm3lemt 31186 Lemma involving norm of di...
norm3adifi 31187 Norm of differences around...
normpari 31188 Parallelogram law for norm...
normpar 31189 Parallelogram law for norm...
normpar2i 31190 Corollary of parallelogram...
polid2i 31191 Generalized polarization i...
polidi 31192 Polarization identity. Re...
polid 31193 Polarization identity. Re...
hilablo 31194 Hilbert space vector addit...
hilid 31195 The group identity element...
hilvc 31196 Hilbert space is a complex...
hilnormi 31197 Hilbert space norm in term...
hilhhi 31198 Deduce the structure of Hi...
hhnv 31199 Hilbert space is a normed ...
hhva 31200 The group (addition) opera...
hhba 31201 The base set of Hilbert sp...
hh0v 31202 The zero vector of Hilbert...
hhsm 31203 The scalar product operati...
hhvs 31204 The vector subtraction ope...
hhnm 31205 The norm function of Hilbe...
hhims 31206 The induced metric of Hilb...
hhims2 31207 Hilbert space distance met...
hhmet 31208 The induced metric of Hilb...
hhxmet 31209 The induced metric of Hilb...
hhmetdval 31210 Value of the distance func...
hhip 31211 The inner product operatio...
hhph 31212 The Hilbert space of the H...
bcsiALT 31213 Bunjakovaskij-Cauchy-Schwa...
bcsiHIL 31214 Bunjakovaskij-Cauchy-Schwa...
bcs 31215 Bunjakovaskij-Cauchy-Schwa...
bcs2 31216 Corollary of the Bunjakova...
bcs3 31217 Corollary of the Bunjakova...
hcau 31218 Member of the set of Cauch...
hcauseq 31219 A Cauchy sequences on a Hi...
hcaucvg 31220 A Cauchy sequence on a Hil...
seq1hcau 31221 A sequence on a Hilbert sp...
hlimi 31222 Express the predicate: Th...
hlimseqi 31223 A sequence with a limit on...
hlimveci 31224 Closure of the limit of a ...
hlimconvi 31225 Convergence of a sequence ...
hlim2 31226 The limit of a sequence on...
hlimadd 31227 Limit of the sum of two se...
hilmet 31228 The Hilbert space norm det...
hilxmet 31229 The Hilbert space norm det...
hilmetdval 31230 Value of the distance func...
hilims 31231 Hilbert space distance met...
hhcau 31232 The Cauchy sequences of Hi...
hhlm 31233 The limit sequences of Hil...
hhcmpl 31234 Lemma used for derivation ...
hilcompl 31235 Lemma used for derivation ...
hhcms 31237 The Hilbert space induced ...
hhhl 31238 The Hilbert space structur...
hilcms 31239 The Hilbert space norm det...
hilhl 31240 The Hilbert space of the H...
issh 31242 Subspace ` H ` of a Hilber...
issh2 31243 Subspace ` H ` of a Hilber...
shss 31244 A subspace is a subset of ...
shel 31245 A member of a subspace of ...
shex 31246 The set of subspaces of a ...
shssii 31247 A closed subspace of a Hil...
sheli 31248 A member of a subspace of ...
shelii 31249 A member of a subspace of ...
sh0 31250 The zero vector belongs to...
shaddcl 31251 Closure of vector addition...
shmulcl 31252 Closure of vector scalar m...
issh3 31253 Subspace ` H ` of a Hilber...
shsubcl 31254 Closure of vector subtract...
isch 31256 Closed subspace ` H ` of a...
isch2 31257 Closed subspace ` H ` of a...
chsh 31258 A closed subspace is a sub...
chsssh 31259 Closed subspaces are subsp...
chex 31260 The set of closed subspace...
chshii 31261 A closed subspace is a sub...
ch0 31262 The zero vector belongs to...
chss 31263 A closed subspace of a Hil...
chel 31264 A member of a closed subsp...
chssii 31265 A closed subspace of a Hil...
cheli 31266 A member of a closed subsp...
chelii 31267 A member of a closed subsp...
chlimi 31268 The limit property of a cl...
hlim0 31269 The zero sequence in Hilbe...
hlimcaui 31270 If a sequence in Hilbert s...
hlimf 31271 Function-like behavior of ...
hlimuni 31272 A Hilbert space sequence c...
hlimreui 31273 The limit of a Hilbert spa...
hlimeui 31274 The limit of a Hilbert spa...
isch3 31275 A Hilbert subspace is clos...
chcompl 31276 Completeness of a closed s...
helch 31277 The Hilbert lattice one (w...
ifchhv 31278 Prove ` if ( A e. CH , A ,...
helsh 31279 Hilbert space is a subspac...
shsspwh 31280 Subspaces are subsets of H...
chsspwh 31281 Closed subspaces are subse...
hsn0elch 31282 The zero subspace belongs ...
norm1 31283 From any nonzero Hilbert s...
norm1exi 31284 A normalized vector exists...
norm1hex 31285 A normalized vector can ex...
elch0 31288 Membership in zero for clo...
h0elch 31289 The zero subspace is a clo...
h0elsh 31290 The zero subspace is a sub...
hhssva 31291 The vector addition operat...
hhsssm 31292 The scalar multiplication ...
hhssnm 31293 The norm operation on a su...
issubgoilem 31294 Lemma for ~ hhssabloilem ....
hhssabloilem 31295 Lemma for ~ hhssabloi . F...
hhssabloi 31296 Abelian group property of ...
hhssablo 31297 Abelian group property of ...
hhssnv 31298 Normed complex vector spac...
hhssnvt 31299 Normed complex vector spac...
hhsst 31300 A member of ` SH ` is a su...
hhshsslem1 31301 Lemma for ~ hhsssh . (Con...
hhshsslem2 31302 Lemma for ~ hhsssh . (Con...
hhsssh 31303 The predicate " ` H ` is a...
hhsssh2 31304 The predicate " ` H ` is a...
hhssba 31305 The base set of a subspace...
hhssvs 31306 The vector subtraction ope...
hhssvsf 31307 Mapping of the vector subt...
hhssims 31308 Induced metric of a subspa...
hhssims2 31309 Induced metric of a subspa...
hhssmet 31310 Induced metric of a subspa...
hhssmetdval 31311 Value of the distance func...
hhsscms 31312 The induced metric of a cl...
hhssbnOLD 31313 Obsolete version of ~ cssb...
ocval 31314 Value of orthogonal comple...
ocel 31315 Membership in orthogonal c...
shocel 31316 Membership in orthogonal c...
ocsh 31317 The orthogonal complement ...
shocsh 31318 The orthogonal complement ...
ocss 31319 An orthogonal complement i...
shocss 31320 An orthogonal complement i...
occon 31321 Contraposition law for ort...
occon2 31322 Double contraposition for ...
occon2i 31323 Double contraposition for ...
oc0 31324 The zero vector belongs to...
ocorth 31325 Members of a subset and it...
shocorth 31326 Members of a subspace and ...
ococss 31327 Inclusion in complement of...
shococss 31328 Inclusion in complement of...
shorth 31329 Members of orthogonal subs...
ocin 31330 Intersection of a Hilbert ...
occon3 31331 Hilbert lattice contraposi...
ocnel 31332 A nonzero vector in the co...
chocvali 31333 Value of the orthogonal co...
shuni 31334 Two subspaces with trivial...
chocunii 31335 Lemma for uniqueness part ...
pjhthmo 31336 Projection Theorem, unique...
occllem 31337 Lemma for ~ occl . (Contr...
occl 31338 Closure of complement of H...
shoccl 31339 Closure of complement of H...
choccl 31340 Closure of complement of H...
choccli 31341 Closure of ` CH ` orthocom...
shsval 31346 Value of subspace sum of t...
shsss 31347 The subspace sum is a subs...
shsel 31348 Membership in the subspace...
shsel3 31349 Membership in the subspace...
shseli 31350 Membership in subspace sum...
shscli 31351 Closure of subspace sum. ...
shscl 31352 Closure of subspace sum. ...
shscom 31353 Commutative law for subspa...
shsva 31354 Vector sum belongs to subs...
shsel1 31355 A subspace sum contains a ...
shsel2 31356 A subspace sum contains a ...
shsvs 31357 Vector subtraction belongs...
shsub1 31358 Subspace sum is an upper b...
shsub2 31359 Subspace sum is an upper b...
choc0 31360 The orthocomplement of the...
choc1 31361 The orthocomplement of the...
chocnul 31362 Orthogonal complement of t...
shintcli 31363 Closure of intersection of...
shintcl 31364 The intersection of a none...
chintcli 31365 The intersection of a none...
chintcl 31366 The intersection (infimum)...
spanval 31367 Value of the linear span o...
hsupval 31368 Value of supremum of set o...
chsupval 31369 The value of the supremum ...
spancl 31370 The span of a subset of Hi...
elspancl 31371 A member of a span is a ve...
shsupcl 31372 Closure of the subspace su...
hsupcl 31373 Closure of supremum of set...
chsupcl 31374 Closure of supremum of sub...
hsupss 31375 Subset relation for suprem...
chsupss 31376 Subset relation for suprem...
hsupunss 31377 The union of a set of Hilb...
chsupunss 31378 The union of a set of clos...
spanss2 31379 A subset of Hilbert space ...
shsupunss 31380 The union of a set of subs...
spanid 31381 A subspace of Hilbert spac...
spanss 31382 Ordering relationship for ...
spanssoc 31383 The span of a subset of Hi...
sshjval 31384 Value of join for subsets ...
shjval 31385 Value of join in ` SH ` . ...
chjval 31386 Value of join in ` CH ` . ...
chjvali 31387 Value of join in ` CH ` . ...
sshjval3 31388 Value of join for subsets ...
sshjcl 31389 Closure of join for subset...
shjcl 31390 Closure of join in ` SH ` ...
chjcl 31391 Closure of join in ` CH ` ...
shjcom 31392 Commutative law for Hilber...
shless 31393 Subset implies subset of s...
shlej1 31394 Add disjunct to both sides...
shlej2 31395 Add disjunct to both sides...
shincli 31396 Closure of intersection of...
shscomi 31397 Commutative law for subspa...
shsvai 31398 Vector sum belongs to subs...
shsel1i 31399 A subspace sum contains a ...
shsel2i 31400 A subspace sum contains a ...
shsvsi 31401 Vector subtraction belongs...
shunssi 31402 Union is smaller than subs...
shunssji 31403 Union is smaller than Hilb...
shsleji 31404 Subspace sum is smaller th...
shjcomi 31405 Commutative law for join i...
shsub1i 31406 Subspace sum is an upper b...
shsub2i 31407 Subspace sum is an upper b...
shub1i 31408 Hilbert lattice join is an...
shjcli 31409 Closure of ` CH ` join. (...
shjshcli 31410 ` SH ` closure of join. (...
shlessi 31411 Subset implies subset of s...
shlej1i 31412 Add disjunct to both sides...
shlej2i 31413 Add disjunct to both sides...
shslej 31414 Subspace sum is smaller th...
shincl 31415 Closure of intersection of...
shub1 31416 Hilbert lattice join is an...
shub2 31417 A subspace is a subset of ...
shsidmi 31418 Idempotent law for Hilbert...
shslubi 31419 The least upper bound law ...
shlesb1i 31420 Hilbert lattice ordering i...
shsval2i 31421 An alternate way to expres...
shsval3i 31422 An alternate way to expres...
shmodsi 31423 The modular law holds for ...
shmodi 31424 The modular law is implied...
pjhthlem1 31425 Lemma for ~ pjhth . (Cont...
pjhthlem2 31426 Lemma for ~ pjhth . (Cont...
pjhth 31427 Projection Theorem: Any H...
pjhtheu 31428 Projection Theorem: Any H...
pjhfval 31430 The value of the projectio...
pjhval 31431 Value of a projection. (C...
pjpreeq 31432 Equality with a projection...
pjeq 31433 Equality with a projection...
axpjcl 31434 Closure of a projection in...
pjhcl 31435 Closure of a projection in...
omlsilem 31436 Lemma for orthomodular law...
omlsii 31437 Subspace inference form of...
omlsi 31438 Subspace form of orthomodu...
ococi 31439 Complement of complement o...
ococ 31440 Complement of complement o...
dfch2 31441 Alternate definition of th...
ococin 31442 The double complement is t...
hsupval2 31443 Alternate definition of su...
chsupval2 31444 The value of the supremum ...
sshjval2 31445 Value of join in the set o...
chsupid 31446 A subspace is the supremum...
chsupsn 31447 Value of supremum of subse...
shlub 31448 Hilbert lattice join is th...
shlubi 31449 Hilbert lattice join is th...
pjhtheu2 31450 Uniqueness of ` y ` for th...
pjcli 31451 Closure of a projection in...
pjhcli 31452 Closure of a projection in...
pjpjpre 31453 Decomposition of a vector ...
axpjpj 31454 Decomposition of a vector ...
pjclii 31455 Closure of a projection in...
pjhclii 31456 Closure of a projection in...
pjpj0i 31457 Decomposition of a vector ...
pjpji 31458 Decomposition of a vector ...
pjpjhth 31459 Projection Theorem: Any H...
pjpjhthi 31460 Projection Theorem: Any H...
pjop 31461 Orthocomplement projection...
pjpo 31462 Projection in terms of ort...
pjopi 31463 Orthocomplement projection...
pjpoi 31464 Projection in terms of ort...
pjoc1i 31465 Projection of a vector in ...
pjchi 31466 Projection of a vector in ...
pjoccl 31467 The part of a vector that ...
pjoc1 31468 Projection of a vector in ...
pjomli 31469 Subspace form of orthomodu...
pjoml 31470 Subspace form of orthomodu...
pjococi 31471 Proof of orthocomplement t...
pjoc2i 31472 Projection of a vector in ...
pjoc2 31473 Projection of a vector in ...
sh0le 31474 The zero subspace is the s...
ch0le 31475 The zero subspace is the s...
shle0 31476 No subspace is smaller tha...
chle0 31477 No Hilbert lattice element...
chnlen0 31478 A Hilbert lattice element ...
ch0pss 31479 The zero subspace is a pro...
orthin 31480 The intersection of orthog...
ssjo 31481 The lattice join of a subs...
shne0i 31482 A nonzero subspace has a n...
shs0i 31483 Hilbert subspace sum with ...
shs00i 31484 Two subspaces are zero iff...
ch0lei 31485 The closed subspace zero i...
chle0i 31486 No Hilbert closed subspace...
chne0i 31487 A nonzero closed subspace ...
chocini 31488 Intersection of a closed s...
chj0i 31489 Join with lattice zero in ...
chm1i 31490 Meet with lattice one in `...
chjcli 31491 Closure of ` CH ` join. (...
chsleji 31492 Subspace sum is smaller th...
chseli 31493 Membership in subspace sum...
chincli 31494 Closure of Hilbert lattice...
chsscon3i 31495 Hilbert lattice contraposi...
chsscon1i 31496 Hilbert lattice contraposi...
chsscon2i 31497 Hilbert lattice contraposi...
chcon2i 31498 Hilbert lattice contraposi...
chcon1i 31499 Hilbert lattice contraposi...
chcon3i 31500 Hilbert lattice contraposi...
chunssji 31501 Union is smaller than ` CH...
chjcomi 31502 Commutative law for join i...
chub1i 31503 ` CH ` join is an upper bo...
chub2i 31504 ` CH ` join is an upper bo...
chlubi 31505 Hilbert lattice join is th...
chlubii 31506 Hilbert lattice join is th...
chlej1i 31507 Add join to both sides of ...
chlej2i 31508 Add join to both sides of ...
chlej12i 31509 Add join to both sides of ...
chlejb1i 31510 Hilbert lattice ordering i...
chdmm1i 31511 De Morgan's law for meet i...
chdmm2i 31512 De Morgan's law for meet i...
chdmm3i 31513 De Morgan's law for meet i...
chdmm4i 31514 De Morgan's law for meet i...
chdmj1i 31515 De Morgan's law for join i...
chdmj2i 31516 De Morgan's law for join i...
chdmj3i 31517 De Morgan's law for join i...
chdmj4i 31518 De Morgan's law for join i...
chnlei 31519 Equivalent expressions for...
chjassi 31520 Associative law for Hilber...
chj00i 31521 Two Hilbert lattice elemen...
chjoi 31522 The join of a closed subsp...
chj1i 31523 Join with Hilbert lattice ...
chm0i 31524 Meet with Hilbert lattice ...
chm0 31525 Meet with Hilbert lattice ...
shjshsi 31526 Hilbert lattice join equal...
shjshseli 31527 A closed subspace sum equa...
chne0 31528 A nonzero closed subspace ...
chocin 31529 Intersection of a closed s...
chssoc 31530 A closed subspace less tha...
chj0 31531 Join with Hilbert lattice ...
chslej 31532 Subspace sum is smaller th...
chincl 31533 Closure of Hilbert lattice...
chsscon3 31534 Hilbert lattice contraposi...
chsscon1 31535 Hilbert lattice contraposi...
chsscon2 31536 Hilbert lattice contraposi...
chpsscon3 31537 Hilbert lattice contraposi...
chpsscon1 31538 Hilbert lattice contraposi...
chpsscon2 31539 Hilbert lattice contraposi...
chjcom 31540 Commutative law for Hilber...
chub1 31541 Hilbert lattice join is gr...
chub2 31542 Hilbert lattice join is gr...
chlub 31543 Hilbert lattice join is th...
chlej1 31544 Add join to both sides of ...
chlej2 31545 Add join to both sides of ...
chlejb1 31546 Hilbert lattice ordering i...
chlejb2 31547 Hilbert lattice ordering i...
chnle 31548 Equivalent expressions for...
chjo 31549 The join of a closed subsp...
chabs1 31550 Hilbert lattice absorption...
chabs2 31551 Hilbert lattice absorption...
chabs1i 31552 Hilbert lattice absorption...
chabs2i 31553 Hilbert lattice absorption...
chjidm 31554 Idempotent law for Hilbert...
chjidmi 31555 Idempotent law for Hilbert...
chj12i 31556 A rearrangement of Hilbert...
chj4i 31557 Rearrangement of the join ...
chjjdiri 31558 Hilbert lattice join distr...
chdmm1 31559 De Morgan's law for meet i...
chdmm2 31560 De Morgan's law for meet i...
chdmm3 31561 De Morgan's law for meet i...
chdmm4 31562 De Morgan's law for meet i...
chdmj1 31563 De Morgan's law for join i...
chdmj2 31564 De Morgan's law for join i...
chdmj3 31565 De Morgan's law for join i...
chdmj4 31566 De Morgan's law for join i...
chjass 31567 Associative law for Hilber...
chj12 31568 A rearrangement of Hilbert...
chj4 31569 Rearrangement of the join ...
ledii 31570 An ortholattice is distrib...
lediri 31571 An ortholattice is distrib...
lejdii 31572 An ortholattice is distrib...
lejdiri 31573 An ortholattice is distrib...
ledi 31574 An ortholattice is distrib...
spansn0 31575 The span of the singleton ...
span0 31576 The span of the empty set ...
elspani 31577 Membership in the span of ...
spanuni 31578 The span of a union is the...
spanun 31579 The span of a union is the...
sshhococi 31580 The join of two Hilbert sp...
hne0 31581 Hilbert space has a nonzer...
chsup0 31582 The supremum of the empty ...
h1deoi 31583 Membership in orthocomplem...
h1dei 31584 Membership in 1-dimensiona...
h1did 31585 A generating vector belong...
h1dn0 31586 A nonzero vector generates...
h1de2i 31587 Membership in 1-dimensiona...
h1de2bi 31588 Membership in 1-dimensiona...
h1de2ctlem 31589 Lemma for ~ h1de2ci . (Co...
h1de2ci 31590 Membership in 1-dimensiona...
spansni 31591 The span of a singleton in...
elspansni 31592 Membership in the span of ...
spansn 31593 The span of a singleton in...
spansnch 31594 The span of a Hilbert spac...
spansnsh 31595 The span of a Hilbert spac...
spansnchi 31596 The span of a singleton in...
spansnid 31597 A vector belongs to the sp...
spansnmul 31598 A scalar product with a ve...
elspansncl 31599 A member of a span of a si...
elspansn 31600 Membership in the span of ...
elspansn2 31601 Membership in the span of ...
spansncol 31602 The singletons of collinea...
spansneleqi 31603 Membership relation implie...
spansneleq 31604 Membership relation that i...
spansnss 31605 The span of the singleton ...
elspansn3 31606 A member of the span of th...
elspansn4 31607 A span membership conditio...
elspansn5 31608 A vector belonging to both...
spansnss2 31609 The span of the singleton ...
normcan 31610 Cancellation-type law that...
pjspansn 31611 A projection on the span o...
spansnpji 31612 A subset of Hilbert space ...
spanunsni 31613 The span of the union of a...
spanpr 31614 The span of a pair of vect...
h1datomi 31615 A 1-dimensional subspace i...
h1datom 31616 A 1-dimensional subspace i...
cmbr 31618 Binary relation expressing...
pjoml2i 31619 Variation of orthomodular ...
pjoml3i 31620 Variation of orthomodular ...
pjoml4i 31621 Variation of orthomodular ...
pjoml5i 31622 The orthomodular law. Rem...
pjoml6i 31623 An equivalent of the ortho...
cmbri 31624 Binary relation expressing...
cmcmlem 31625 Commutation is symmetric. ...
cmcmi 31626 Commutation is symmetric. ...
cmcm2i 31627 Commutation with orthocomp...
cmcm3i 31628 Commutation with orthocomp...
cmcm4i 31629 Commutation with orthocomp...
cmbr2i 31630 Alternate definition of th...
cmcmii 31631 Commutation is symmetric. ...
cmcm2ii 31632 Commutation with orthocomp...
cmcm3ii 31633 Commutation with orthocomp...
cmbr3i 31634 Alternate definition for t...
cmbr4i 31635 Alternate definition for t...
lecmi 31636 Comparable Hilbert lattice...
lecmii 31637 Comparable Hilbert lattice...
cmj1i 31638 A Hilbert lattice element ...
cmj2i 31639 A Hilbert lattice element ...
cmm1i 31640 A Hilbert lattice element ...
cmm2i 31641 A Hilbert lattice element ...
cmbr3 31642 Alternate definition for t...
cm0 31643 The zero Hilbert lattice e...
cmidi 31644 The commutes relation is r...
pjoml2 31645 Variation of orthomodular ...
pjoml3 31646 Variation of orthomodular ...
pjoml5 31647 The orthomodular law. Rem...
cmcm 31648 Commutation is symmetric. ...
cmcm3 31649 Commutation with orthocomp...
cmcm2 31650 Commutation with orthocomp...
lecm 31651 Comparable Hilbert lattice...
fh1 31652 Foulis-Holland Theorem. I...
fh2 31653 Foulis-Holland Theorem. I...
cm2j 31654 A lattice element that com...
fh1i 31655 Foulis-Holland Theorem. I...
fh2i 31656 Foulis-Holland Theorem. I...
fh3i 31657 Variation of the Foulis-Ho...
fh4i 31658 Variation of the Foulis-Ho...
cm2ji 31659 A lattice element that com...
cm2mi 31660 A lattice element that com...
qlax1i 31661 One of the equations showi...
qlax2i 31662 One of the equations showi...
qlax3i 31663 One of the equations showi...
qlax4i 31664 One of the equations showi...
qlax5i 31665 One of the equations showi...
qlaxr1i 31666 One of the conditions show...
qlaxr2i 31667 One of the conditions show...
qlaxr4i 31668 One of the conditions show...
qlaxr5i 31669 One of the conditions show...
qlaxr3i 31670 A variation of the orthomo...
chscllem1 31671 Lemma for ~ chscl . (Cont...
chscllem2 31672 Lemma for ~ chscl . (Cont...
chscllem3 31673 Lemma for ~ chscl . (Cont...
chscllem4 31674 Lemma for ~ chscl . (Cont...
chscl 31675 The subspace sum of two cl...
osumi 31676 If two closed subspaces of...
osumcori 31677 Corollary of ~ osumi . (C...
osumcor2i 31678 Corollary of ~ osumi , sho...
osum 31679 If two closed subspaces of...
spansnji 31680 The subspace sum of a clos...
spansnj 31681 The subspace sum of a clos...
spansnscl 31682 The subspace sum of a clos...
sumspansn 31683 The sum of two vectors bel...
spansnm0i 31684 The meet of different one-...
nonbooli 31685 A Hilbert lattice with two...
spansncvi 31686 Hilbert space has the cove...
spansncv 31687 Hilbert space has the cove...
5oalem1 31688 Lemma for orthoarguesian l...
5oalem2 31689 Lemma for orthoarguesian l...
5oalem3 31690 Lemma for orthoarguesian l...
5oalem4 31691 Lemma for orthoarguesian l...
5oalem5 31692 Lemma for orthoarguesian l...
5oalem6 31693 Lemma for orthoarguesian l...
5oalem7 31694 Lemma for orthoarguesian l...
5oai 31695 Orthoarguesian law 5OA. Th...
3oalem1 31696 Lemma for 3OA (weak) ortho...
3oalem2 31697 Lemma for 3OA (weak) ortho...
3oalem3 31698 Lemma for 3OA (weak) ortho...
3oalem4 31699 Lemma for 3OA (weak) ortho...
3oalem5 31700 Lemma for 3OA (weak) ortho...
3oalem6 31701 Lemma for 3OA (weak) ortho...
3oai 31702 3OA (weak) orthoarguesian ...
pjorthi 31703 Projection components on o...
pjch1 31704 Property of identity proje...
pjo 31705 The orthogonal projection....
pjcompi 31706 Component of a projection....
pjidmi 31707 A projection is idempotent...
pjadjii 31708 A projection is self-adjoi...
pjaddii 31709 Projection of vector sum i...
pjinormii 31710 The inner product of a pro...
pjmulii 31711 Projection of (scalar) pro...
pjsubii 31712 Projection of vector diffe...
pjsslem 31713 Lemma for subset relations...
pjss2i 31714 Subset relationship for pr...
pjssmii 31715 Projection meet property. ...
pjssge0ii 31716 Theorem 4.5(iv)->(v) of [B...
pjdifnormii 31717 Theorem 4.5(v)<->(vi) of [...
pjcji 31718 The projection on a subspa...
pjadji 31719 A projection is self-adjoi...
pjaddi 31720 Projection of vector sum i...
pjinormi 31721 The inner product of a pro...
pjsubi 31722 Projection of vector diffe...
pjmuli 31723 Projection of scalar produ...
pjige0i 31724 The inner product of a pro...
pjige0 31725 The inner product of a pro...
pjcjt2 31726 The projection on a subspa...
pj0i 31727 The projection of the zero...
pjch 31728 Projection of a vector in ...
pjid 31729 The projection of a vector...
pjvec 31730 The set of vectors belongi...
pjocvec 31731 The set of vectors belongi...
pjocini 31732 Membership of projection i...
pjini 31733 Membership of projection i...
pjjsi 31734 A sufficient condition for...
pjfni 31735 Functionality of a project...
pjrni 31736 The range of a projection....
pjfoi 31737 A projection maps onto its...
pjfi 31738 The mapping of a projectio...
pjvi 31739 The value of a projection ...
pjhfo 31740 A projection maps onto its...
pjrn 31741 The range of a projection....
pjhf 31742 The mapping of a projectio...
pjfn 31743 Functionality of a project...
pjsumi 31744 The projection on a subspa...
pj11i 31745 One-to-one correspondence ...
pjdsi 31746 Vector decomposition into ...
pjds3i 31747 Vector decomposition into ...
pj11 31748 One-to-one correspondence ...
pjmfn 31749 Functionality of the proje...
pjmf1 31750 The projector function map...
pjoi0 31751 The inner product of proje...
pjoi0i 31752 The inner product of proje...
pjopythi 31753 Pythagorean theorem for pr...
pjopyth 31754 Pythagorean theorem for pr...
pjnormi 31755 The norm of the projection...
pjpythi 31756 Pythagorean theorem for pr...
pjneli 31757 If a vector does not belon...
pjnorm 31758 The norm of the projection...
pjpyth 31759 Pythagorean theorem for pr...
pjnel 31760 If a vector does not belon...
pjnorm2 31761 A vector belongs to the su...
mayete3i 31762 Mayet's equation E_3. Par...
mayetes3i 31763 Mayet's equation E^*_3, de...
hosmval 31769 Value of the sum of two Hi...
hommval 31770 Value of the scalar produc...
hodmval 31771 Value of the difference of...
hfsmval 31772 Value of the sum of two Hi...
hfmmval 31773 Value of the scalar produc...
hosval 31774 Value of the sum of two Hi...
homval 31775 Value of the scalar produc...
hodval 31776 Value of the difference of...
hfsval 31777 Value of the sum of two Hi...
hfmval 31778 Value of the scalar produc...
hoscl 31779 Closure of the sum of two ...
homcl 31780 Closure of the scalar prod...
hodcl 31781 Closure of the difference ...
ho0val 31784 Value of the zero Hilbert ...
ho0f 31785 Functionality of the zero ...
df0op2 31786 Alternate definition of Hi...
dfiop2 31787 Alternate definition of Hi...
hoif 31788 Functionality of the Hilbe...
hoival 31789 The value of the Hilbert s...
hoico1 31790 Composition with the Hilbe...
hoico2 31791 Composition with the Hilbe...
hoaddcl 31792 The sum of Hilbert space o...
homulcl 31793 The scalar product of a Hi...
hoeq 31794 Equality of Hilbert space ...
hoeqi 31795 Equality of Hilbert space ...
hoscli 31796 Closure of Hilbert space o...
hodcli 31797 Closure of Hilbert space o...
hocoi 31798 Composition of Hilbert spa...
hococli 31799 Closure of composition of ...
hocofi 31800 Mapping of composition of ...
hocofni 31801 Functionality of compositi...
hoaddcli 31802 Mapping of sum of Hilbert ...
hosubcli 31803 Mapping of difference of H...
hoaddfni 31804 Functionality of sum of Hi...
hosubfni 31805 Functionality of differenc...
hoaddcomi 31806 Commutativity of sum of Hi...
hosubcl 31807 Mapping of difference of H...
hoaddcom 31808 Commutativity of sum of Hi...
hodsi 31809 Relationship between Hilbe...
hoaddassi 31810 Associativity of sum of Hi...
hoadd12i 31811 Commutative/associative la...
hoadd32i 31812 Commutative/associative la...
hocadddiri 31813 Distributive law for Hilbe...
hocsubdiri 31814 Distributive law for Hilbe...
ho2coi 31815 Double composition of Hilb...
hoaddass 31816 Associativity of sum of Hi...
hoadd32 31817 Commutative/associative la...
hoadd4 31818 Rearrangement of 4 terms i...
hocsubdir 31819 Distributive law for Hilbe...
hoaddridi 31820 Sum of a Hilbert space ope...
hodidi 31821 Difference of a Hilbert sp...
ho0coi 31822 Composition of the zero op...
hoid1i 31823 Composition of Hilbert spa...
hoid1ri 31824 Composition of Hilbert spa...
hoaddrid 31825 Sum of a Hilbert space ope...
hodid 31826 Difference of a Hilbert sp...
hon0 31827 A Hilbert space operator i...
hodseqi 31828 Subtraction and addition o...
ho0subi 31829 Subtraction of Hilbert spa...
honegsubi 31830 Relationship between Hilbe...
ho0sub 31831 Subtraction of Hilbert spa...
hosubid1 31832 The zero operator subtract...
honegsub 31833 Relationship between Hilbe...
homullid 31834 An operator equals its sca...
homco1 31835 Associative law for scalar...
homulass 31836 Scalar product associative...
hoadddi 31837 Scalar product distributiv...
hoadddir 31838 Scalar product reverse dis...
homul12 31839 Swap first and second fact...
honegneg 31840 Double negative of a Hilbe...
hosubneg 31841 Relationship between opera...
hosubdi 31842 Scalar product distributiv...
honegdi 31843 Distribution of negative o...
honegsubdi 31844 Distribution of negative o...
honegsubdi2 31845 Distribution of negative o...
hosubsub2 31846 Law for double subtraction...
hosub4 31847 Rearrangement of 4 terms i...
hosubadd4 31848 Rearrangement of 4 terms i...
hoaddsubass 31849 Associative-type law for a...
hoaddsub 31850 Law for operator addition ...
hosubsub 31851 Law for double subtraction...
hosubsub4 31852 Law for double subtraction...
ho2times 31853 Two times a Hilbert space ...
hoaddsubassi 31854 Associativity of sum and d...
hoaddsubi 31855 Law for sum and difference...
hosd1i 31856 Hilbert space operator sum...
hosd2i 31857 Hilbert space operator sum...
hopncani 31858 Hilbert space operator can...
honpcani 31859 Hilbert space operator can...
hosubeq0i 31860 If the difference between ...
honpncani 31861 Hilbert space operator can...
ho01i 31862 A condition implying that ...
ho02i 31863 A condition implying that ...
hoeq1 31864 A condition implying that ...
hoeq2 31865 A condition implying that ...
adjmo 31866 Every Hilbert space operat...
adjsym 31867 Symmetry property of an ad...
eigrei 31868 A necessary and sufficient...
eigre 31869 A necessary and sufficient...
eigposi 31870 A sufficient condition (fi...
eigorthi 31871 A necessary and sufficient...
eigorth 31872 A necessary and sufficient...
nmopval 31890 Value of the norm of a Hil...
elcnop 31891 Property defining a contin...
ellnop 31892 Property defining a linear...
lnopf 31893 A linear Hilbert space ope...
elbdop 31894 Property defining a bounde...
bdopln 31895 A bounded linear Hilbert s...
bdopf 31896 A bounded linear Hilbert s...
nmopsetretALT 31897 The set in the supremum of...
nmopsetretHIL 31898 The set in the supremum of...
nmopsetn0 31899 The set in the supremum of...
nmopxr 31900 The norm of a Hilbert spac...
nmoprepnf 31901 The norm of a Hilbert spac...
nmopgtmnf 31902 The norm of a Hilbert spac...
nmopreltpnf 31903 The norm of a Hilbert spac...
nmopre 31904 The norm of a bounded oper...
elbdop2 31905 Property defining a bounde...
elunop 31906 Property defining a unitar...
elhmop 31907 Property defining a Hermit...
hmopf 31908 A Hermitian operator is a ...
hmopex 31909 The class of Hermitian ope...
nmfnval 31910 Value of the norm of a Hil...
nmfnsetre 31911 The set in the supremum of...
nmfnsetn0 31912 The set in the supremum of...
nmfnxr 31913 The norm of any Hilbert sp...
nmfnrepnf 31914 The norm of a Hilbert spac...
nlfnval 31915 Value of the null space of...
elcnfn 31916 Property defining a contin...
ellnfn 31917 Property defining a linear...
lnfnf 31918 A linear Hilbert space fun...
dfadj2 31919 Alternate definition of th...
funadj 31920 Functionality of the adjoi...
dmadjss 31921 The domain of the adjoint ...
dmadjop 31922 A member of the domain of ...
adjeu 31923 Elementhood in the domain ...
adjval 31924 Value of the adjoint funct...
adjval2 31925 Value of the adjoint funct...
cnvadj 31926 The adjoint function equal...
funcnvadj 31927 The converse of the adjoin...
adj1o 31928 The adjoint function maps ...
dmadjrn 31929 The adjoint of an operator...
eigvecval 31930 The set of eigenvectors of...
eigvalfval 31931 The eigenvalues of eigenve...
specval 31932 The value of the spectrum ...
speccl 31933 The spectrum of an operato...
hhlnoi 31934 The linear operators of Hi...
hhnmoi 31935 The norm of an operator in...
hhbloi 31936 A bounded linear operator ...
hh0oi 31937 The zero operator in Hilbe...
hhcno 31938 The continuous operators o...
hhcnf 31939 The continuous functionals...
dmadjrnb 31940 The adjoint of an operator...
nmoplb 31941 A lower bound for an opera...
nmopub 31942 An upper bound for an oper...
nmopub2tALT 31943 An upper bound for an oper...
nmopub2tHIL 31944 An upper bound for an oper...
nmopge0 31945 The norm of any Hilbert sp...
nmopgt0 31946 A linear Hilbert space ope...
cnopc 31947 Basic continuity property ...
lnopl 31948 Basic linearity property o...
unop 31949 Basic inner product proper...
unopf1o 31950 A unitary operator in Hilb...
unopnorm 31951 A unitary operator is idem...
cnvunop 31952 The inverse (converse) of ...
unopadj 31953 The inverse (converse) of ...
unoplin 31954 A unitary operator is line...
counop 31955 The composition of two uni...
hmop 31956 Basic inner product proper...
hmopre 31957 The inner product of the v...
nmfnlb 31958 A lower bound for a functi...
nmfnleub 31959 An upper bound for the nor...
nmfnleub2 31960 An upper bound for the nor...
nmfnge0 31961 The norm of any Hilbert sp...
elnlfn 31962 Membership in the null spa...
elnlfn2 31963 Membership in the null spa...
cnfnc 31964 Basic continuity property ...
lnfnl 31965 Basic linearity property o...
adjcl 31966 Closure of the adjoint of ...
adj1 31967 Property of an adjoint Hil...
adj2 31968 Property of an adjoint Hil...
adjeq 31969 A property that determines...
adjadj 31970 Double adjoint. Theorem 3...
adjvalval 31971 Value of the value of the ...
unopadj2 31972 The adjoint of a unitary o...
hmopadj 31973 A Hermitian operator is se...
hmdmadj 31974 Every Hermitian operator h...
hmopadj2 31975 An operator is Hermitian i...
hmoplin 31976 A Hermitian operator is li...
brafval 31977 The bra of a vector, expre...
braval 31978 A bra-ket juxtaposition, e...
braadd 31979 Linearity property of bra ...
bramul 31980 Linearity property of bra ...
brafn 31981 The bra function is a func...
bralnfn 31982 The Dirac bra function is ...
bracl 31983 Closure of the bra functio...
bra0 31984 The Dirac bra of the zero ...
brafnmul 31985 Anti-linearity property of...
kbfval 31986 The outer product of two v...
kbop 31987 The outer product of two v...
kbval 31988 The value of the operator ...
kbmul 31989 Multiplication property of...
kbpj 31990 If a vector ` A ` has norm...
eleigvec 31991 Membership in the set of e...
eleigvec2 31992 Membership in the set of e...
eleigveccl 31993 Closure of an eigenvector ...
eigvalval 31994 The eigenvalue of an eigen...
eigvalcl 31995 An eigenvalue is a complex...
eigvec1 31996 Property of an eigenvector...
eighmre 31997 The eigenvalues of a Hermi...
eighmorth 31998 Eigenvectors of a Hermitia...
nmopnegi 31999 Value of the norm of the n...
lnop0 32000 The value of a linear Hilb...
lnopmul 32001 Multiplicative property of...
lnopli 32002 Basic scalar product prope...
lnopfi 32003 A linear Hilbert space ope...
lnop0i 32004 The value of a linear Hilb...
lnopaddi 32005 Additive property of a lin...
lnopmuli 32006 Multiplicative property of...
lnopaddmuli 32007 Sum/product property of a ...
lnopsubi 32008 Subtraction property for a...
lnopsubmuli 32009 Subtraction/product proper...
lnopmulsubi 32010 Product/subtraction proper...
homco2 32011 Move a scalar product out ...
idunop 32012 The identity function (res...
0cnop 32013 The identically zero funct...
0cnfn 32014 The identically zero funct...
idcnop 32015 The identity function (res...
idhmop 32016 The Hilbert space identity...
0hmop 32017 The identically zero funct...
0lnop 32018 The identically zero funct...
0lnfn 32019 The identically zero funct...
nmop0 32020 The norm of the zero opera...
nmfn0 32021 The norm of the identicall...
hmopbdoptHIL 32022 A Hermitian operator is a ...
hoddii 32023 Distributive law for Hilbe...
hoddi 32024 Distributive law for Hilbe...
nmop0h 32025 The norm of any operator o...
idlnop 32026 The identity function (res...
0bdop 32027 The identically zero opera...
adj0 32028 Adjoint of the zero operat...
nmlnop0iALT 32029 A linear operator with a z...
nmlnop0iHIL 32030 A linear operator with a z...
nmlnopgt0i 32031 A linear Hilbert space ope...
nmlnop0 32032 A linear operator with a z...
nmlnopne0 32033 A linear operator with a n...
lnopmi 32034 The scalar product of a li...
lnophsi 32035 The sum of two linear oper...
lnophdi 32036 The difference of two line...
lnopcoi 32037 The composition of two lin...
lnopco0i 32038 The composition of a linea...
lnopeq0lem1 32039 Lemma for ~ lnopeq0i . Ap...
lnopeq0lem2 32040 Lemma for ~ lnopeq0i . (C...
lnopeq0i 32041 A condition implying that ...
lnopeqi 32042 Two linear Hilbert space o...
lnopeq 32043 Two linear Hilbert space o...
lnopunilem1 32044 Lemma for ~ lnopunii . (C...
lnopunilem2 32045 Lemma for ~ lnopunii . (C...
lnopunii 32046 If a linear operator (whos...
elunop2 32047 An operator is unitary iff...
nmopun 32048 Norm of a unitary Hilbert ...
unopbd 32049 A unitary operator is a bo...
lnophmlem1 32050 Lemma for ~ lnophmi . (Co...
lnophmlem2 32051 Lemma for ~ lnophmi . (Co...
lnophmi 32052 A linear operator is Hermi...
lnophm 32053 A linear operator is Hermi...
hmops 32054 The sum of two Hermitian o...
hmopm 32055 The scalar product of a He...
hmopd 32056 The difference of two Herm...
hmopco 32057 The composition of two com...
nmbdoplbi 32058 A lower bound for the norm...
nmbdoplb 32059 A lower bound for the norm...
nmcexi 32060 Lemma for ~ nmcopexi and ~...
nmcopexi 32061 The norm of a continuous l...
nmcoplbi 32062 A lower bound for the norm...
nmcopex 32063 The norm of a continuous l...
nmcoplb 32064 A lower bound for the norm...
nmophmi 32065 The norm of the scalar pro...
bdophmi 32066 The scalar product of a bo...
lnconi 32067 Lemma for ~ lnopconi and ~...
lnopconi 32068 A condition equivalent to ...
lnopcon 32069 A condition equivalent to ...
lnopcnbd 32070 A linear operator is conti...
lncnopbd 32071 A continuous linear operat...
lncnbd 32072 A continuous linear operat...
lnopcnre 32073 A linear operator is conti...
lnfnli 32074 Basic property of a linear...
lnfnfi 32075 A linear Hilbert space fun...
lnfn0i 32076 The value of a linear Hilb...
lnfnaddi 32077 Additive property of a lin...
lnfnmuli 32078 Multiplicative property of...
lnfnaddmuli 32079 Sum/product property of a ...
lnfnsubi 32080 Subtraction property for a...
lnfn0 32081 The value of a linear Hilb...
lnfnmul 32082 Multiplicative property of...
nmbdfnlbi 32083 A lower bound for the norm...
nmbdfnlb 32084 A lower bound for the norm...
nmcfnexi 32085 The norm of a continuous l...
nmcfnlbi 32086 A lower bound for the norm...
nmcfnex 32087 The norm of a continuous l...
nmcfnlb 32088 A lower bound of the norm ...
lnfnconi 32089 A condition equivalent to ...
lnfncon 32090 A condition equivalent to ...
lnfncnbd 32091 A linear functional is con...
imaelshi 32092 The image of a subspace un...
rnelshi 32093 The range of a linear oper...
nlelshi 32094 The null space of a linear...
nlelchi 32095 The null space of a contin...
riesz3i 32096 A continuous linear functi...
riesz4i 32097 A continuous linear functi...
riesz4 32098 A continuous linear functi...
riesz1 32099 Part 1 of the Riesz repres...
riesz2 32100 Part 2 of the Riesz repres...
cnlnadjlem1 32101 Lemma for ~ cnlnadji (Theo...
cnlnadjlem2 32102 Lemma for ~ cnlnadji . ` G...
cnlnadjlem3 32103 Lemma for ~ cnlnadji . By...
cnlnadjlem4 32104 Lemma for ~ cnlnadji . Th...
cnlnadjlem5 32105 Lemma for ~ cnlnadji . ` F...
cnlnadjlem6 32106 Lemma for ~ cnlnadji . ` F...
cnlnadjlem7 32107 Lemma for ~ cnlnadji . He...
cnlnadjlem8 32108 Lemma for ~ cnlnadji . ` F...
cnlnadjlem9 32109 Lemma for ~ cnlnadji . ` F...
cnlnadji 32110 Every continuous linear op...
cnlnadjeui 32111 Every continuous linear op...
cnlnadjeu 32112 Every continuous linear op...
cnlnadj 32113 Every continuous linear op...
cnlnssadj 32114 Every continuous linear Hi...
bdopssadj 32115 Every bounded linear Hilbe...
bdopadj 32116 Every bounded linear Hilbe...
adjbdln 32117 The adjoint of a bounded l...
adjbdlnb 32118 An operator is bounded and...
adjbd1o 32119 The mapping of adjoints of...
adjlnop 32120 The adjoint of an operator...
adjsslnop 32121 Every operator with an adj...
nmopadjlei 32122 Property of the norm of an...
nmopadjlem 32123 Lemma for ~ nmopadji . (C...
nmopadji 32124 Property of the norm of an...
adjeq0 32125 An operator is zero iff it...
adjmul 32126 The adjoint of the scalar ...
adjadd 32127 The adjoint of the sum of ...
nmoptrii 32128 Triangle inequality for th...
nmopcoi 32129 Upper bound for the norm o...
bdophsi 32130 The sum of two bounded lin...
bdophdi 32131 The difference between two...
bdopcoi 32132 The composition of two bou...
nmoptri2i 32133 Triangle-type inequality f...
adjcoi 32134 The adjoint of a compositi...
nmopcoadji 32135 The norm of an operator co...
nmopcoadj2i 32136 The norm of an operator co...
nmopcoadj0i 32137 An operator composed with ...
unierri 32138 If we approximate a chain ...
branmfn 32139 The norm of the bra functi...
brabn 32140 The bra of a vector is a b...
rnbra 32141 The set of bras equals the...
bra11 32142 The bra function maps vect...
bracnln 32143 A bra is a continuous line...
cnvbraval 32144 Value of the converse of t...
cnvbracl 32145 Closure of the converse of...
cnvbrabra 32146 The converse bra of the br...
bracnvbra 32147 The bra of the converse br...
bracnlnval 32148 The vector that a continuo...
cnvbramul 32149 Multiplication property of...
kbass1 32150 Dirac bra-ket associative ...
kbass2 32151 Dirac bra-ket associative ...
kbass3 32152 Dirac bra-ket associative ...
kbass4 32153 Dirac bra-ket associative ...
kbass5 32154 Dirac bra-ket associative ...
kbass6 32155 Dirac bra-ket associative ...
leopg 32156 Ordering relation for posi...
leop 32157 Ordering relation for oper...
leop2 32158 Ordering relation for oper...
leop3 32159 Operator ordering in terms...
leoppos 32160 Binary relation defining a...
leoprf2 32161 The ordering relation for ...
leoprf 32162 The ordering relation for ...
leopsq 32163 The square of a Hermitian ...
0leop 32164 The zero operator is a pos...
idleop 32165 The identity operator is a...
leopadd 32166 The sum of two positive op...
leopmuli 32167 The scalar product of a no...
leopmul 32168 The scalar product of a po...
leopmul2i 32169 Scalar product applied to ...
leoptri 32170 The positive operator orde...
leoptr 32171 The positive operator orde...
leopnmid 32172 A bounded Hermitian operat...
nmopleid 32173 A nonzero, bounded Hermiti...
opsqrlem1 32174 Lemma for opsqri . (Contr...
opsqrlem2 32175 Lemma for opsqri . ` F `` ...
opsqrlem3 32176 Lemma for opsqri . (Contr...
opsqrlem4 32177 Lemma for opsqri . (Contr...
opsqrlem5 32178 Lemma for opsqri . (Contr...
opsqrlem6 32179 Lemma for opsqri . (Contr...
pjhmopi 32180 A projector is a Hermitian...
pjlnopi 32181 A projector is a linear op...
pjnmopi 32182 The operator norm of a pro...
pjbdlni 32183 A projector is a bounded l...
pjhmop 32184 A projection is a Hermitia...
hmopidmchi 32185 An idempotent Hermitian op...
hmopidmpji 32186 An idempotent Hermitian op...
hmopidmch 32187 An idempotent Hermitian op...
hmopidmpj 32188 An idempotent Hermitian op...
pjsdii 32189 Distributive law for Hilbe...
pjddii 32190 Distributive law for Hilbe...
pjsdi2i 32191 Chained distributive law f...
pjcoi 32192 Composition of projections...
pjcocli 32193 Closure of composition of ...
pjcohcli 32194 Closure of composition of ...
pjadjcoi 32195 Adjoint of composition of ...
pjcofni 32196 Functionality of compositi...
pjss1coi 32197 Subset relationship for pr...
pjss2coi 32198 Subset relationship for pr...
pjssmi 32199 Projection meet property. ...
pjssge0i 32200 Theorem 4.5(iv)->(v) of [B...
pjdifnormi 32201 Theorem 4.5(v)<->(vi) of [...
pjnormssi 32202 Theorem 4.5(i)<->(vi) of [...
pjorthcoi 32203 Composition of projections...
pjscji 32204 The projection of orthogon...
pjssumi 32205 The projection on a subspa...
pjssposi 32206 Projector ordering can be ...
pjordi 32207 The definition of projecto...
pjssdif2i 32208 The projection subspace of...
pjssdif1i 32209 A necessary and sufficient...
pjimai 32210 The image of a projection....
pjidmcoi 32211 A projection is idempotent...
pjoccoi 32212 Composition of projections...
pjtoi 32213 Subspace sum of projection...
pjoci 32214 Projection of orthocomplem...
pjidmco 32215 A projection operator is i...
dfpjop 32216 Definition of projection o...
pjhmopidm 32217 Two ways to express the se...
elpjidm 32218 A projection operator is i...
elpjhmop 32219 A projection operator is H...
0leopj 32220 A projector is a positive ...
pjadj2 32221 A projector is self-adjoin...
pjadj3 32222 A projector is self-adjoin...
elpjch 32223 Reconstruction of the subs...
elpjrn 32224 Reconstruction of the subs...
pjinvari 32225 A closed subspace ` H ` wi...
pjin1i 32226 Lemma for Theorem 1.22 of ...
pjin2i 32227 Lemma for Theorem 1.22 of ...
pjin3i 32228 Lemma for Theorem 1.22 of ...
pjclem1 32229 Lemma for projection commu...
pjclem2 32230 Lemma for projection commu...
pjclem3 32231 Lemma for projection commu...
pjclem4a 32232 Lemma for projection commu...
pjclem4 32233 Lemma for projection commu...
pjci 32234 Two subspaces commute iff ...
pjcmul1i 32235 A necessary and sufficient...
pjcmul2i 32236 The projection subspace of...
pjcohocli 32237 Closure of composition of ...
pjadj2coi 32238 Adjoint of double composit...
pj2cocli 32239 Closure of double composit...
pj3lem1 32240 Lemma for projection tripl...
pj3si 32241 Stronger projection triple...
pj3i 32242 Projection triplet theorem...
pj3cor1i 32243 Projection triplet corolla...
pjs14i 32244 Theorem S-14 of Watanabe, ...
isst 32247 Property of a state. (Con...
ishst 32248 Property of a complex Hilb...
sticl 32249 ` [ 0 , 1 ] ` closure of t...
stcl 32250 Real closure of the value ...
hstcl 32251 Closure of the value of a ...
hst1a 32252 Unit value of a Hilbert-sp...
hstel2 32253 Properties of a Hilbert-sp...
hstorth 32254 Orthogonality property of ...
hstosum 32255 Orthogonal sum property of...
hstoc 32256 Sum of a Hilbert-space-val...
hstnmoc 32257 Sum of norms of a Hilbert-...
stge0 32258 The value of a state is no...
stle1 32259 The value of a state is le...
hstle1 32260 The norm of the value of a...
hst1h 32261 The norm of a Hilbert-spac...
hst0h 32262 The norm of a Hilbert-spac...
hstpyth 32263 Pythagorean property of a ...
hstle 32264 Ordering property of a Hil...
hstles 32265 Ordering property of a Hil...
hstoh 32266 A Hilbert-space-valued sta...
hst0 32267 A Hilbert-space-valued sta...
sthil 32268 The value of a state at th...
stj 32269 The value of a state on a ...
sto1i 32270 The state of a subspace pl...
sto2i 32271 The state of the orthocomp...
stge1i 32272 If a state is greater than...
stle0i 32273 If a state is less than or...
stlei 32274 Ordering law for states. ...
stlesi 32275 Ordering law for states. ...
stji1i 32276 Join of components of Sasa...
stm1i 32277 State of component of unit...
stm1ri 32278 State of component of unit...
stm1addi 32279 Sum of states whose meet i...
staddi 32280 If the sum of 2 states is ...
stm1add3i 32281 Sum of states whose meet i...
stadd3i 32282 If the sum of 3 states is ...
st0 32283 The state of the zero subs...
strlem1 32284 Lemma for strong state the...
strlem2 32285 Lemma for strong state the...
strlem3a 32286 Lemma for strong state the...
strlem3 32287 Lemma for strong state the...
strlem4 32288 Lemma for strong state the...
strlem5 32289 Lemma for strong state the...
strlem6 32290 Lemma for strong state the...
stri 32291 Strong state theorem. The...
strb 32292 Strong state theorem (bidi...
hstrlem2 32293 Lemma for strong set of CH...
hstrlem3a 32294 Lemma for strong set of CH...
hstrlem3 32295 Lemma for strong set of CH...
hstrlem4 32296 Lemma for strong set of CH...
hstrlem5 32297 Lemma for strong set of CH...
hstrlem6 32298 Lemma for strong set of CH...
hstri 32299 Hilbert space admits a str...
hstrbi 32300 Strong CH-state theorem (b...
largei 32301 A Hilbert lattice admits a...
jplem1 32302 Lemma for Jauch-Piron theo...
jplem2 32303 Lemma for Jauch-Piron theo...
jpi 32304 The function ` S ` , that ...
golem1 32305 Lemma for Godowski's equat...
golem2 32306 Lemma for Godowski's equat...
goeqi 32307 Godowski's equation, shown...
stcltr1i 32308 Property of a strong class...
stcltr2i 32309 Property of a strong class...
stcltrlem1 32310 Lemma for strong classical...
stcltrlem2 32311 Lemma for strong classical...
stcltrthi 32312 Theorem for classically st...
cvbr 32316 Binary relation expressing...
cvbr2 32317 Binary relation expressing...
cvcon3 32318 Contraposition law for the...
cvpss 32319 The covers relation implie...
cvnbtwn 32320 The covers relation implie...
cvnbtwn2 32321 The covers relation implie...
cvnbtwn3 32322 The covers relation implie...
cvnbtwn4 32323 The covers relation implie...
cvnsym 32324 The covers relation is not...
cvnref 32325 The covers relation is not...
cvntr 32326 The covers relation is not...
spansncv2 32327 Hilbert space has the cove...
mdbr 32328 Binary relation expressing...
mdi 32329 Consequence of the modular...
mdbr2 32330 Binary relation expressing...
mdbr3 32331 Binary relation expressing...
mdbr4 32332 Binary relation expressing...
dmdbr 32333 Binary relation expressing...
dmdmd 32334 The dual modular pair prop...
mddmd 32335 The modular pair property ...
dmdi 32336 Consequence of the dual mo...
dmdbr2 32337 Binary relation expressing...
dmdi2 32338 Consequence of the dual mo...
dmdbr3 32339 Binary relation expressing...
dmdbr4 32340 Binary relation expressing...
dmdi4 32341 Consequence of the dual mo...
dmdbr5 32342 Binary relation expressing...
mddmd2 32343 Relationship between modul...
mdsl0 32344 A sublattice condition tha...
ssmd1 32345 Ordering implies the modul...
ssmd2 32346 Ordering implies the modul...
ssdmd1 32347 Ordering implies the dual ...
ssdmd2 32348 Ordering implies the dual ...
dmdsl3 32349 Sublattice mapping for a d...
mdsl3 32350 Sublattice mapping for a m...
mdslle1i 32351 Order preservation of the ...
mdslle2i 32352 Order preservation of the ...
mdslj1i 32353 Join preservation of the o...
mdslj2i 32354 Meet preservation of the r...
mdsl1i 32355 If the modular pair proper...
mdsl2i 32356 If the modular pair proper...
mdsl2bi 32357 If the modular pair proper...
cvmdi 32358 The covering property impl...
mdslmd1lem1 32359 Lemma for ~ mdslmd1i . (C...
mdslmd1lem2 32360 Lemma for ~ mdslmd1i . (C...
mdslmd1lem3 32361 Lemma for ~ mdslmd1i . (C...
mdslmd1lem4 32362 Lemma for ~ mdslmd1i . (C...
mdslmd1i 32363 Preservation of the modula...
mdslmd2i 32364 Preservation of the modula...
mdsldmd1i 32365 Preservation of the dual m...
mdslmd3i 32366 Modular pair conditions th...
mdslmd4i 32367 Modular pair condition tha...
csmdsymi 32368 Cross-symmetry implies M-s...
mdexchi 32369 An exchange lemma for modu...
cvmd 32370 The covering property impl...
cvdmd 32371 The covering property impl...
ela 32373 Atoms in a Hilbert lattice...
elat2 32374 Expanded membership relati...
elatcv0 32375 A Hilbert lattice element ...
atcv0 32376 An atom covers the zero su...
atssch 32377 Atoms are a subset of the ...
atelch 32378 An atom is a Hilbert latti...
atne0 32379 An atom is not the Hilbert...
atss 32380 A lattice element smaller ...
atsseq 32381 Two atoms in a subset rela...
atcveq0 32382 A Hilbert lattice element ...
h1da 32383 A 1-dimensional subspace i...
spansna 32384 The span of the singleton ...
sh1dle 32385 A 1-dimensional subspace i...
ch1dle 32386 A 1-dimensional subspace i...
atom1d 32387 The 1-dimensional subspace...
superpos 32388 Superposition Principle. ...
chcv1 32389 The Hilbert lattice has th...
chcv2 32390 The Hilbert lattice has th...
chjatom 32391 The join of a closed subsp...
shatomici 32392 The lattice of Hilbert sub...
hatomici 32393 The Hilbert lattice is ato...
hatomic 32394 A Hilbert lattice is atomi...
shatomistici 32395 The lattice of Hilbert sub...
hatomistici 32396 ` CH ` is atomistic, i.e. ...
chpssati 32397 Two Hilbert lattice elemen...
chrelati 32398 The Hilbert lattice is rel...
chrelat2i 32399 A consequence of relative ...
cvati 32400 If a Hilbert lattice eleme...
cvbr4i 32401 An alternate way to expres...
cvexchlem 32402 Lemma for ~ cvexchi . (Co...
cvexchi 32403 The Hilbert lattice satisf...
chrelat2 32404 A consequence of relative ...
chrelat3 32405 A consequence of relative ...
chrelat3i 32406 A consequence of the relat...
chrelat4i 32407 A consequence of relative ...
cvexch 32408 The Hilbert lattice satisf...
cvp 32409 The Hilbert lattice satisf...
atnssm0 32410 The meet of a Hilbert latt...
atnemeq0 32411 The meet of distinct atoms...
atssma 32412 The meet with an atom's su...
atcv0eq 32413 Two atoms covering the zer...
atcv1 32414 Two atoms covering the zer...
atexch 32415 The Hilbert lattice satisf...
atomli 32416 An assertion holding in at...
atoml2i 32417 An assertion holding in at...
atordi 32418 An ordering law for a Hilb...
atcvatlem 32419 Lemma for ~ atcvati . (Co...
atcvati 32420 A nonzero Hilbert lattice ...
atcvat2i 32421 A Hilbert lattice element ...
atord 32422 An ordering law for a Hilb...
atcvat2 32423 A Hilbert lattice element ...
chirredlem1 32424 Lemma for ~ chirredi . (C...
chirredlem2 32425 Lemma for ~ chirredi . (C...
chirredlem3 32426 Lemma for ~ chirredi . (C...
chirredlem4 32427 Lemma for ~ chirredi . (C...
chirredi 32428 The Hilbert lattice is irr...
chirred 32429 The Hilbert lattice is irr...
atcvat3i 32430 A condition implying that ...
atcvat4i 32431 A condition implying exist...
atdmd 32432 Two Hilbert lattice elemen...
atmd 32433 Two Hilbert lattice elemen...
atmd2 32434 Two Hilbert lattice elemen...
atabsi 32435 Absorption of an incompara...
atabs2i 32436 Absorption of an incompara...
mdsymlem1 32437 Lemma for ~ mdsymi . (Con...
mdsymlem2 32438 Lemma for ~ mdsymi . (Con...
mdsymlem3 32439 Lemma for ~ mdsymi . (Con...
mdsymlem4 32440 Lemma for ~ mdsymi . This...
mdsymlem5 32441 Lemma for ~ mdsymi . (Con...
mdsymlem6 32442 Lemma for ~ mdsymi . This...
mdsymlem7 32443 Lemma for ~ mdsymi . Lemm...
mdsymlem8 32444 Lemma for ~ mdsymi . Lemm...
mdsymi 32445 M-symmetry of the Hilbert ...
mdsym 32446 M-symmetry of the Hilbert ...
dmdsym 32447 Dual M-symmetry of the Hil...
atdmd2 32448 Two Hilbert lattice elemen...
sumdmdii 32449 If the subspace sum of two...
cmmdi 32450 Commuting subspaces form a...
cmdmdi 32451 Commuting subspaces form a...
sumdmdlem 32452 Lemma for ~ sumdmdi . The...
sumdmdlem2 32453 Lemma for ~ sumdmdi . (Co...
sumdmdi 32454 The subspace sum of two Hi...
dmdbr4ati 32455 Dual modular pair property...
dmdbr5ati 32456 Dual modular pair property...
dmdbr6ati 32457 Dual modular pair property...
dmdbr7ati 32458 Dual modular pair property...
mdoc1i 32459 Orthocomplements form a mo...
mdoc2i 32460 Orthocomplements form a mo...
dmdoc1i 32461 Orthocomplements form a du...
dmdoc2i 32462 Orthocomplements form a du...
mdcompli 32463 A condition equivalent to ...
dmdcompli 32464 A condition equivalent to ...
mddmdin0i 32465 If dual modular implies mo...
cdjreui 32466 A member of the sum of dis...
cdj1i 32467 Two ways to express " ` A ...
cdj3lem1 32468 A property of " ` A ` and ...
cdj3lem2 32469 Lemma for ~ cdj3i . Value...
cdj3lem2a 32470 Lemma for ~ cdj3i . Closu...
cdj3lem2b 32471 Lemma for ~ cdj3i . The f...
cdj3lem3 32472 Lemma for ~ cdj3i . Value...
cdj3lem3a 32473 Lemma for ~ cdj3i . Closu...
cdj3lem3b 32474 Lemma for ~ cdj3i . The s...
cdj3i 32475 Two ways to express " ` A ...
The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here
mathbox 32476 (_This theorem is a dummy ...
sa-abvi 32477 A theorem about the univer...
xfree 32478 A partial converse to ~ 19...
xfree2 32479 A partial converse to ~ 19...
addltmulALT 32480 A proof readability experi...
an42ds 32481 Inference exchanging the l...
an52ds 32482 Inference exchanging the l...
an62ds 32483 Inference exchanging the l...
an72ds 32484 Inference exchanging the l...
an82ds 32485 Inference exchanging the l...
bian1d 32486 Adding a superfluous conju...
bian1dOLD 32487 Obsolete version of ~ bian...
bibiad 32488 Eliminate an hypothesis ` ...
orim12da 32489 Deduce a disjunction from ...
or3di 32490 Distributive law for disju...
or3dir 32491 Distributive law for disju...
3o1cs 32492 Deduction eliminating disj...
3o2cs 32493 Deduction eliminating disj...
3o3cs 32494 Deduction eliminating disj...
13an22anass 32495 Associative law for four c...
sbc2iedf 32496 Conversion of implicit sub...
rspc2daf 32497 Double restricted speciali...
ralcom4f 32498 Commutation of restricted ...
rexcom4f 32499 Commutation of restricted ...
19.9d2rf 32500 A deduction version of one...
19.9d2r 32501 A deduction version of one...
r19.29ffa 32502 A commonly used pattern ba...
n0limd 32503 Deduction rule for nonempt...
eqtrb 32504 A transposition of equalit...
eqelbid 32505 A variable elimination law...
opsbc2ie 32506 Conversion of implicit sub...
opreu2reuALT 32507 Correspondence between uni...
2reucom 32510 Double restricted existent...
2reu2rex1 32511 Double restricted existent...
2reureurex 32512 Double restricted existent...
2reu2reu2 32513 Double restricted existent...
opreu2reu1 32514 Equivalent definition of t...
sq2reunnltb 32515 There exists a unique deco...
addsqnot2reu 32516 For each complex number ` ...
sbceqbidf 32517 Equality theorem for class...
sbcies 32518 A special version of class...
mo5f 32519 Alternate definition of "a...
nmo 32520 Negation of "at most one"....
reuxfrdf 32521 Transfer existential uniqu...
rexunirn 32522 Restricted existential qua...
rmoxfrd 32523 Transfer "at most one" res...
rmoun 32524 "At most one" restricted e...
rmounid 32525 A case where an "at most o...
riotaeqbidva 32526 Equivalent wff's yield equ...
dmrab 32527 Domain of a restricted cla...
difrab2 32528 Difference of two restrict...
rabexgfGS 32529 Separation Scheme in terms...
rabsnel 32530 Truth implied by equality ...
rabsspr 32531 Conditions for a restricte...
rabsstp 32532 Conditions for a restricte...
3unrab 32533 Union of three restricted ...
foresf1o 32534 From a surjective function...
rabfodom 32535 Domination relation for re...
rabrexfi 32536 Conditions for a class abs...
abrexdomjm 32537 An indexed set is dominate...
abrexdom2jm 32538 An indexed set is dominate...
abrexexd 32539 Existence of a class abstr...
elabreximd 32540 Class substitution in an i...
elabreximdv 32541 Class substitution in an i...
abrexss 32542 A necessary condition for ...
elunsn 32543 Elementhood to a union wit...
nelun 32544 Negated membership for a u...
snsssng 32545 If a singleton is a subset...
n0nsnel 32546 If a class with one elemen...
inin 32547 Intersection with an inter...
inindif 32548 See ~ inundif . (Contribu...
difininv 32549 Condition for the intersec...
difeq 32550 Rewriting an equation with...
eqdif 32551 If both set differences of...
indifbi 32552 Two ways to express equali...
diffib 32553 Case where ~ diffi is a bi...
difxp1ss 32554 Difference law for Cartesi...
difxp2ss 32555 Difference law for Cartesi...
indifundif 32556 A remarkable equation with...
elpwincl1 32557 Closure of intersection wi...
elpwdifcl 32558 Closure of class differenc...
elpwiuncl 32559 Closure of indexed union w...
elpreq 32560 Equality wihin a pair. (C...
nelpr 32561 A set ` A ` not in a pair ...
inpr0 32562 Rewrite an empty intersect...
neldifpr1 32563 The first element of a pai...
neldifpr2 32564 The second element of a pa...
unidifsnel 32565 The other element of a pai...
unidifsnne 32566 The other element of a pai...
ifeqeqx 32567 An equality theorem tailor...
elimifd 32568 Elimination of a condition...
elim2if 32569 Elimination of two conditi...
elim2ifim 32570 Elimination of two conditi...
ifeq3da 32571 Given an expression ` C ` ...
ifnetrue 32572 Deduce truth from a condit...
ifnefals 32573 Deduce falsehood from a co...
ifnebib 32574 The converse of ~ ifbi hol...
uniinn0 32575 Sufficient and necessary c...
uniin1 32576 Union of intersection. Ge...
uniin2 32577 Union of intersection. Ge...
difuncomp 32578 Express a class difference...
elpwunicl 32579 Closure of a set union wit...
cbviunf 32580 Rule used to change the bo...
iuneq12daf 32581 Equality deduction for ind...
iunin1f 32582 Indexed union of intersect...
ssiun3 32583 Subset equivalence for an ...
ssiun2sf 32584 Subset relationship for an...
iuninc 32585 The union of an increasing...
iundifdifd 32586 The intersection of a set ...
iundifdif 32587 The intersection of a set ...
iunrdx 32588 Re-index an indexed union....
iunpreima 32589 Preimage of an indexed uni...
iunrnmptss 32590 A subset relation for an i...
iunxunsn 32591 Appending a set to an inde...
iunxunpr 32592 Appending two sets to an i...
iinabrex 32593 Rewriting an indexed inter...
disjnf 32594 In case ` x ` is not free ...
cbvdisjf 32595 Change bound variables in ...
disjss1f 32596 A subset of a disjoint col...
disjeq1f 32597 Equality theorem for disjo...
disjxun0 32598 Simplify a disjoint union....
disjdifprg 32599 A trivial partition into a...
disjdifprg2 32600 A trivial partition of a s...
disji2f 32601 Property of a disjoint col...
disjif 32602 Property of a disjoint col...
disjorf 32603 Two ways to say that a col...
disjorsf 32604 Two ways to say that a col...
disjif2 32605 Property of a disjoint col...
disjabrex 32606 Rewriting a disjoint colle...
disjabrexf 32607 Rewriting a disjoint colle...
disjpreima 32608 A preimage of a disjoint s...
disjrnmpt 32609 Rewriting a disjoint colle...
disjin 32610 If a collection is disjoin...
disjin2 32611 If a collection is disjoin...
disjxpin 32612 Derive a disjunction over ...
iundisjf 32613 Rewrite a countable union ...
iundisj2f 32614 A disjoint union is disjoi...
disjrdx 32615 Re-index a disjunct collec...
disjex 32616 Two ways to say that two c...
disjexc 32617 A variant of ~ disjex , ap...
disjunsn 32618 Append an element to a dis...
disjun0 32619 Adding the empty element p...
disjiunel 32620 A set of elements B of a d...
disjuniel 32621 A set of elements B of a d...
xpdisjres 32622 Restriction of a constant ...
opeldifid 32623 Ordered pair elementhood o...
difres 32624 Case when class difference...
imadifxp 32625 Image of the difference wi...
relfi 32626 A relation (set) is finite...
0res 32627 Restriction of the empty f...
fcoinver 32628 Build an equivalence relat...
fcoinvbr 32629 Binary relation for the eq...
copsex2dv 32630 Implicit substitution dedu...
brab2d 32631 Expressing that two sets a...
brabgaf 32632 The law of concretion for ...
brelg 32633 Two things in a binary rel...
br8d 32634 Substitution for an eight-...
opabdm 32635 Domain of an ordered-pair ...
opabrn 32636 Range of an ordered-pair c...
opabssi 32637 Sufficient condition for a...
opabid2ss 32638 One direction of ~ opabid2...
ssrelf 32639 A subclass relationship de...
eqrelrd2 32640 A version of ~ eqrelrdv2 w...
erbr3b 32641 Biconditional for equivale...
iunsnima 32642 Image of a singleton by an...
iunsnima2 32643 Version of ~ iunsnima with...
feq2dd 32644 Equality deduction for fun...
feq3dd 32645 Equality deduction for fun...
ac6sf2 32646 Alternate version of ~ ac6...
fnresin 32647 Restriction of a function ...
f1o3d 32648 Describe an implicit one-t...
eldmne0 32649 A function of nonempty dom...
f1rnen 32650 Equinumerosity of the rang...
rinvf1o 32651 Sufficient conditions for ...
fresf1o 32652 Conditions for a restricti...
nfpconfp 32653 The set of fixed points of...
fmptco1f1o 32654 The action of composing (t...
cofmpt2 32655 Express composition of a m...
f1mptrn 32656 Express injection for a ma...
dfimafnf 32657 Alternate definition of th...
funimass4f 32658 Membership relation for th...
suppss2f 32659 Show that the support of a...
ofrn 32660 The range of the function ...
ofrn2 32661 The range of the function ...
off2 32662 The function operation pro...
ofresid 32663 Applying an operation rest...
unipreima 32664 Preimage of a class union....
opfv 32665 Value of a function produc...
xppreima 32666 The preimage of a Cartesia...
2ndimaxp 32667 Image of a cartesian produ...
djussxp2 32668 Stronger version of ~ djus...
2ndresdju 32669 The ` 2nd ` function restr...
2ndresdjuf1o 32670 The ` 2nd ` function restr...
xppreima2 32671 The preimage of a Cartesia...
abfmpunirn 32672 Membership in a union of a...
rabfmpunirn 32673 Membership in a union of a...
abfmpeld 32674 Membership in an element o...
abfmpel 32675 Membership in an element o...
fmptdF 32676 Domain and codomain of the...
fmptcof2 32677 Composition of two functio...
fcomptf 32678 Express composition of two...
acunirnmpt 32679 Axiom of choice for the un...
acunirnmpt2 32680 Axiom of choice for the un...
acunirnmpt2f 32681 Axiom of choice for the un...
aciunf1lem 32682 Choice in an index union. ...
aciunf1 32683 Choice in an index union. ...
ofoprabco 32684 Function operation as a co...
ofpreima 32685 Express the preimage of a ...
ofpreima2 32686 Express the preimage of a ...
funcnvmpt 32687 Condition for a function i...
funcnv5mpt 32688 Two ways to say that a fun...
funcnv4mpt 32689 Two ways to say that a fun...
preimane 32690 Different elements have di...
fnpreimac 32691 Choose a set ` x ` contain...
fgreu 32692 Exactly one point of a fun...
fcnvgreu 32693 If the converse of a relat...
rnmposs 32694 The range of an operation ...
mptssALT 32695 Deduce subset relation of ...
dfcnv2 32696 Alternative definition of ...
mpomptxf 32697 Express a two-argument fun...
of0r 32698 Function operation with th...
suppovss 32699 A bound for the support of...
suppiniseg 32700 Relation between the suppo...
fsuppinisegfi 32701 The initial segment ` ( ``...
fressupp 32702 The restriction of a funct...
fdifsuppconst 32703 A function is a zero const...
ressupprn 32704 The range of a function re...
supppreima 32705 Express the support of a f...
fsupprnfi 32706 Finite support implies fin...
mptiffisupp 32707 Conditions for a mapping f...
cosnopne 32708 Composition of two ordered...
cosnop 32709 Composition of two ordered...
cnvprop 32710 Converse of a pair of orde...
brprop 32711 Binary relation for a pair...
mptprop 32712 Rewrite pairs of ordered p...
coprprop 32713 Composition of two pairs o...
gtiso 32714 Two ways to write a strict...
isoun 32715 Infer an isomorphism from ...
disjdsct 32716 A disjoint collection is d...
df1stres 32717 Definition for a restricti...
df2ndres 32718 Definition for a restricti...
1stpreimas 32719 The preimage of a singleto...
1stpreima 32720 The preimage by ` 1st ` is...
2ndpreima 32721 The preimage by ` 2nd ` is...
curry2ima 32722 The image of a curried fun...
preiman0 32723 The preimage of a nonempty...
intimafv 32724 The intersection of an ima...
supssd 32725 Inequality deduction for s...
infssd 32726 Inequality deduction for i...
imafi2 32727 The image by a finite set ...
unifi3 32728 If a union is finite, then...
snct 32729 A singleton is countable. ...
prct 32730 An unordered pair is count...
mpocti 32731 An operation is countable ...
abrexct 32732 An image set of a countabl...
mptctf 32733 A countable mapping set is...
abrexctf 32734 An image set of a countabl...
padct 32735 Index a countable set with...
cnvoprabOLD 32736 The converse of a class ab...
f1od2 32737 Sufficient condition for a...
fcobij 32738 Composing functions with a...
fcobijfs 32739 Composing finitely support...
suppss3 32740 Deduce a function's suppor...
fsuppcurry1 32741 Finite support of a currie...
fsuppcurry2 32742 Finite support of a currie...
offinsupp1 32743 Finite support for a funct...
ffs2 32744 Rewrite a function's suppo...
ffsrn 32745 The range of a finitely su...
resf1o 32746 Restriction of functions t...
maprnin 32747 Restricting the range of t...
fpwrelmapffslem 32748 Lemma for ~ fpwrelmapffs ....
fpwrelmap 32749 Define a canonical mapping...
fpwrelmapffs 32750 Define a canonical mapping...
creq0 32751 The real representation of...
1nei 32752 The imaginary unit ` _i ` ...
1neg1t1neg1 32753 An integer unit times itse...
nnmulge 32754 Multiplying by a positive ...
submuladdd 32755 The product of a differenc...
muldivdid 32756 Distribution of division o...
cjsubd 32757 Complex conjugate distribu...
re0cj 32758 The conjugate of a pure im...
quad3d 32759 Variant of quadratic equat...
lt2addrd 32760 If the right-hand side of ...
xrlelttric 32761 Trichotomy law for extende...
xaddeq0 32762 Two extended reals which a...
xrinfm 32763 The extended real numbers ...
le2halvesd 32764 A sum is less than the who...
xraddge02 32765 A number is less than or e...
xrge0addge 32766 A number is less than or e...
xlt2addrd 32767 If the right-hand side of ...
xrsupssd 32768 Inequality deduction for s...
xrge0infss 32769 Any subset of nonnegative ...
xrge0infssd 32770 Inequality deduction for i...
xrge0addcld 32771 Nonnegative extended reals...
xrge0subcld 32772 Condition for closure of n...
infxrge0lb 32773 A member of a set of nonne...
infxrge0glb 32774 The infimum of a set of no...
infxrge0gelb 32775 The infimum of a set of no...
xrofsup 32776 The supremum is preserved ...
supxrnemnf 32777 The supremum of a nonempty...
xnn0gt0 32778 Nonzero extended nonnegati...
xnn01gt 32779 An extended nonnegative in...
nn0xmulclb 32780 Finite multiplication in t...
joiniooico 32781 Disjoint joining an open i...
ubico 32782 A right-open interval does...
xeqlelt 32783 Equality in terms of 'less...
eliccelico 32784 Relate elementhood to a cl...
elicoelioo 32785 Relate elementhood to a cl...
iocinioc2 32786 Intersection between two o...
xrdifh 32787 Class difference of a half...
iocinif 32788 Relate intersection of two...
difioo 32789 The difference between two...
difico 32790 The difference between two...
uzssico 32791 Upper integer sets are a s...
fz2ssnn0 32792 A finite set of sequential...
nndiffz1 32793 Upper set of the positive ...
ssnnssfz 32794 For any finite subset of `...
fzne1 32795 Elementhood in a finite se...
fzm1ne1 32796 Elementhood of an integer ...
fzspl 32797 Split the last element of ...
fzdif2 32798 Split the last element of ...
fzodif2 32799 Split the last element of ...
fzodif1 32800 Set difference of two half...
fzsplit3 32801 Split a finite interval of...
bcm1n 32802 The proportion of one bino...
iundisjfi 32803 Rewrite a countable union ...
iundisj2fi 32804 A disjoint union is disjoi...
iundisjcnt 32805 Rewrite a countable union ...
iundisj2cnt 32806 A countable disjoint union...
fzone1 32807 Elementhood in a half-open...
fzom1ne1 32808 Elementhood in a half-open...
f1ocnt 32809 Given a countable set ` A ...
fz1nnct 32810 NN and integer ranges star...
fz1nntr 32811 NN and integer ranges star...
fzo0opth 32812 Equality for a half open i...
nn0difffzod 32813 A nonnegative integer that...
suppssnn0 32814 Show that the support of a...
hashunif 32815 The cardinality of a disjo...
hashxpe 32816 The size of the Cartesian ...
hashgt1 32817 Restate "set contains at l...
znumd 32818 Numerator of an integer. ...
zdend 32819 Denominator of an integer....
numdenneg 32820 Numerator and denominator ...
divnumden2 32821 Calculate the reduced form...
expgt0b 32822 A real number ` A ` raised...
nn0split01 32823 Split 0 and 1 from the non...
nn0disj01 32824 The pair ` { 0 , 1 } ` doe...
nnindf 32825 Principle of Mathematical ...
nn0min 32826 Extracting the minimum pos...
subne0nn 32827 A nonnegative difference i...
ltesubnnd 32828 Subtracting an integer num...
fprodeq02 32829 If one of the factors is z...
pr01ssre 32830 The range of the indicator...
fprodex01 32831 A product of factors equal...
prodpr 32832 A product over a pair is t...
prodtp 32833 A product over a triple is...
fsumub 32834 An upper bound for a term ...
fsumiunle 32835 Upper bound for a sum of n...
dfdec100 32836 Split the hundreds from a ...
dp2eq1 32839 Equality theorem for the d...
dp2eq2 32840 Equality theorem for the d...
dp2eq1i 32841 Equality theorem for the d...
dp2eq2i 32842 Equality theorem for the d...
dp2eq12i 32843 Equality theorem for the d...
dp20u 32844 Add a zero in the tenths (...
dp20h 32845 Add a zero in the unit pla...
dp2cl 32846 Closure for the decimal fr...
dp2clq 32847 Closure for a decimal frac...
rpdp2cl 32848 Closure for a decimal frac...
rpdp2cl2 32849 Closure for a decimal frac...
dp2lt10 32850 Decimal fraction builds re...
dp2lt 32851 Comparing two decimal frac...
dp2ltsuc 32852 Comparing a decimal fracti...
dp2ltc 32853 Comparing two decimal expa...
dpval 32856 Define the value of the de...
dpcl 32857 Prove that the closure of ...
dpfrac1 32858 Prove a simple equivalence...
dpval2 32859 Value of the decimal point...
dpval3 32860 Value of the decimal point...
dpmul10 32861 Multiply by 10 a decimal e...
decdiv10 32862 Divide a decimal number by...
dpmul100 32863 Multiply by 100 a decimal ...
dp3mul10 32864 Multiply by 10 a decimal e...
dpmul1000 32865 Multiply by 1000 a decimal...
dpval3rp 32866 Value of the decimal point...
dp0u 32867 Add a zero in the tenths p...
dp0h 32868 Remove a zero in the units...
rpdpcl 32869 Closure of the decimal poi...
dplt 32870 Comparing two decimal expa...
dplti 32871 Comparing a decimal expans...
dpgti 32872 Comparing a decimal expans...
dpltc 32873 Comparing two decimal inte...
dpexpp1 32874 Add one zero to the mantis...
0dp2dp 32875 Multiply by 10 a decimal e...
dpadd2 32876 Addition with one decimal,...
dpadd 32877 Addition with one decimal....
dpadd3 32878 Addition with two decimals...
dpmul 32879 Multiplication with one de...
dpmul4 32880 An upper bound to multipli...
threehalves 32881 Example theorem demonstrat...
1mhdrd 32882 Example theorem demonstrat...
xdivval 32885 Value of division: the (un...
xrecex 32886 Existence of reciprocal of...
xmulcand 32887 Cancellation law for exten...
xreceu 32888 Existential uniqueness of ...
xdivcld 32889 Closure law for the extend...
xdivcl 32890 Closure law for the extend...
xdivmul 32891 Relationship between divis...
rexdiv 32892 The extended real division...
xdivrec 32893 Relationship between divis...
xdivid 32894 A number divided by itself...
xdiv0 32895 Division into zero is zero...
xdiv0rp 32896 Division into zero is zero...
eliccioo 32897 Membership in a closed int...
elxrge02 32898 Elementhood in the set of ...
xdivpnfrp 32899 Plus infinity divided by a...
rpxdivcld 32900 Closure law for extended d...
xrpxdivcld 32901 Closure law for extended d...
wrdfd 32902 A word is a zero-based seq...
wrdres 32903 Condition for the restrict...
wrdsplex 32904 Existence of a split of a ...
wrdfsupp 32905 A word has finite support....
wrdpmcl 32906 Closure of a word with per...
pfx1s2 32907 The prefix of length 1 of ...
pfxrn2 32908 The range of a prefix of a...
pfxrn3 32909 Express the range of a pre...
pfxf1 32910 Condition for a prefix to ...
s1f1 32911 Conditions for a length 1 ...
s2rnOLD 32912 Obsolete version of ~ s2rn...
s2f1 32913 Conditions for a length 2 ...
s3rnOLD 32914 Obsolete version of ~ s2rn...
s3f1 32915 Conditions for a length 3 ...
s3clhash 32916 Closure of the words of le...
ccatf1 32917 Conditions for a concatena...
ccatdmss 32918 The domain of a concatenat...
pfxlsw2ccat 32919 Reconstruct a word from it...
ccatws1f1o 32920 Conditions for the concate...
ccatws1f1olast 32921 Two ways to reorder symbol...
wrdt2ind 32922 Perform an induction over ...
swrdrn2 32923 The range of a subword is ...
swrdrn3 32924 Express the range of a sub...
swrdf1 32925 Condition for a subword to...
swrdrndisj 32926 Condition for the range of...
splfv3 32927 Symbols to the right of a ...
1cshid 32928 Cyclically shifting a sing...
cshw1s2 32929 Cyclically shifting a leng...
cshwrnid 32930 Cyclically shifting a word...
cshf1o 32931 Condition for the cyclic s...
ressplusf 32932 The group operation functi...
ressnm 32933 The norm in a restricted s...
abvpropd2 32934 Weaker version of ~ abvpro...
oppgle 32935 less-than relation of an o...
oppgleOLD 32936 Obsolete version of ~ oppg...
oppglt 32937 less-than relation of an o...
ressprs 32938 The restriction of a prose...
oduprs 32939 Being a proset is a self-d...
posrasymb 32940 A poset ordering is asymet...
resspos 32941 The restriction of a Poset...
resstos 32942 The restriction of a Toset...
odutos 32943 Being a toset is a self-du...
tlt2 32944 In a Toset, two elements m...
tlt3 32945 In a Toset, two elements m...
trleile 32946 In a Toset, two elements m...
toslublem 32947 Lemma for ~ toslub and ~ x...
toslub 32948 In a toset, the lowest upp...
tosglblem 32949 Lemma for ~ tosglb and ~ x...
tosglb 32950 Same theorem as ~ toslub ,...
clatp0cl 32951 The poset zero of a comple...
clatp1cl 32952 The poset one of a complet...
mntoval 32957 Operation value of the mon...
ismnt 32958 Express the statement " ` ...
ismntd 32959 Property of being a monoto...
mntf 32960 A monotone function is a f...
mgcoval 32961 Operation value of the mon...
mgcval 32962 Monotone Galois connection...
mgcf1 32963 The lower adjoint ` F ` of...
mgcf2 32964 The upper adjoint ` G ` of...
mgccole1 32965 An inequality for the kern...
mgccole2 32966 Inequality for the closure...
mgcmnt1 32967 The lower adjoint ` F ` of...
mgcmnt2 32968 The upper adjoint ` G ` of...
mgcmntco 32969 A Galois connection like s...
dfmgc2lem 32970 Lemma for dfmgc2, backward...
dfmgc2 32971 Alternate definition of th...
mgcmnt1d 32972 Galois connection implies ...
mgcmnt2d 32973 Galois connection implies ...
mgccnv 32974 The inverse Galois connect...
pwrssmgc 32975 Given a function ` F ` , e...
mgcf1olem1 32976 Property of a Galois conne...
mgcf1olem2 32977 Property of a Galois conne...
mgcf1o 32978 Given a Galois connection,...
ischn 32981 Property of being a chain....
chnwrd 32982 A chain is an ordered sequ...
chnltm1 32983 Basic property of a chain....
pfxchn 32984 A prefix of a chain is sti...
chnind 32985 Induction over a chain. S...
chnub 32986 In a chain, the last eleme...
chnlt 32987 Compare any two elements i...
chnso 32988 A chain induces a total or...
xrs0 32991 The zero of the extended r...
xrslt 32992 The "strictly less than" r...
xrsinvgval 32993 The inversion operation in...
xrsmulgzz 32994 The "multiple" function in...
xrstos 32995 The extended real numbers ...
xrsclat 32996 The extended real numbers ...
xrsp0 32997 The poset 0 of the extende...
xrsp1 32998 The poset 1 of the extende...
xrge0base 32999 The base of the extended n...
xrge00 33000 The zero of the extended n...
xrge0plusg 33001 The additive law of the ex...
xrge0le 33002 The "less than or equal to...
xrge0mulgnn0 33003 The group multiple functio...
xrge0addass 33004 Associativity of extended ...
xrge0addgt0 33005 The sum of nonnegative and...
xrge0adddir 33006 Right-distributivity of ex...
xrge0adddi 33007 Left-distributivity of ext...
xrge0npcan 33008 Extended nonnegative real ...
fsumrp0cl 33009 Closure of a finite sum of...
mndcld 33010 Closure of the operation o...
mndassd 33011 A monoid operation is asso...
mndlrinv 33012 In a monoid, if an element...
mndlrinvb 33013 In a monoid, if an element...
mndlactf1 33014 If an element ` X ` of a m...
mndlactfo 33015 An element ` X ` of a mono...
mndractf1 33016 If an element ` X ` of a m...
mndractfo 33017 An element ` X ` of a mono...
mndlactf1o 33018 An element ` X ` of a mono...
mndractf1o 33019 An element ` X ` of a mono...
cmn4d 33020 Commutative/associative la...
cmn246135 33021 Rearrange terms in a commu...
cmn145236 33022 Rearrange terms in a commu...
submcld 33023 Submonoids are closed unde...
abliso 33024 The image of an Abelian gr...
lmhmghmd 33025 A module homomorphism is a...
mhmimasplusg 33026 Value of the operation of ...
lmhmimasvsca 33027 Value of the scalar produc...
grpsubcld 33028 Closure of group subtracti...
subgcld 33029 A subgroup is closed under...
subgsubcld 33030 A subgroup is closed under...
gsumsubg 33031 The group sum in a subgrou...
gsumsra 33032 The group sum in a subring...
gsummpt2co 33033 Split a finite sum into a ...
gsummpt2d 33034 Express a finite sum over ...
lmodvslmhm 33035 Scalar multiplication in a...
gsumvsmul1 33036 Pull a scalar multiplicati...
gsummptres 33037 Extend a finite group sum ...
gsummptres2 33038 Extend a finite group sum ...
gsumzresunsn 33039 Append an element to a fin...
gsumpart 33040 Express a group sum as a d...
gsumtp 33041 Group sum of an unordered ...
gsumhashmul 33042 Express a group sum by gro...
xrge0tsmsd 33043 Any finite or infinite sum...
xrge0tsmsbi 33044 Any limit of a finite or i...
xrge0tsmseq 33045 Any limit of a finite or i...
cntzun 33046 The centralizer of a union...
cntzsnid 33047 The centralizer of the ide...
cntrcrng 33048 The center of a ring is a ...
isomnd 33053 A (left) ordered monoid is...
isogrp 33054 A (left-)ordered group is ...
ogrpgrp 33055 A left-ordered group is a ...
omndmnd 33056 A left-ordered monoid is a...
omndtos 33057 A left-ordered monoid is a...
omndadd 33058 In an ordered monoid, the ...
omndaddr 33059 In a right ordered monoid,...
omndadd2d 33060 In a commutative left orde...
omndadd2rd 33061 In a left- and right- orde...
submomnd 33062 A submonoid of an ordered ...
xrge0omnd 33063 The nonnegative extended r...
omndmul2 33064 In an ordered monoid, the ...
omndmul3 33065 In an ordered monoid, the ...
omndmul 33066 In a commutative ordered m...
ogrpinv0le 33067 In an ordered group, the o...
ogrpsub 33068 In an ordered group, the o...
ogrpaddlt 33069 In an ordered group, stric...
ogrpaddltbi 33070 In a right ordered group, ...
ogrpaddltrd 33071 In a right ordered group, ...
ogrpaddltrbid 33072 In a right ordered group, ...
ogrpsublt 33073 In an ordered group, stric...
ogrpinv0lt 33074 In an ordered group, the o...
ogrpinvlt 33075 In an ordered group, the o...
gsumle 33076 A finite sum in an ordered...
symgfcoeu 33077 Uniqueness property of per...
symgcom 33078 Two permutations ` X ` and...
symgcom2 33079 Two permutations ` X ` and...
symgcntz 33080 All elements of a (finite)...
odpmco 33081 The composition of two odd...
symgsubg 33082 The value of the group sub...
pmtrprfv2 33083 In a transposition of two ...
pmtrcnel 33084 Composing a permutation ` ...
pmtrcnel2 33085 Variation on ~ pmtrcnel . ...
pmtrcnelor 33086 Composing a permutation ` ...
fzo0pmtrlast 33087 Reorder a half-open intege...
wrdpmtrlast 33088 Reorder a word, so that th...
pmtridf1o 33089 Transpositions of ` X ` an...
pmtridfv1 33090 Value at X of the transpos...
pmtridfv2 33091 Value at Y of the transpos...
psgnid 33092 Permutation sign of the id...
psgndmfi 33093 For a finite base set, the...
pmtrto1cl 33094 Useful lemma for the follo...
psgnfzto1stlem 33095 Lemma for ~ psgnfzto1st . ...
fzto1stfv1 33096 Value of our permutation `...
fzto1st1 33097 Special case where the per...
fzto1st 33098 The function moving one el...
fzto1stinvn 33099 Value of the inverse of ou...
psgnfzto1st 33100 The permutation sign for m...
tocycval 33103 Value of the cycle builder...
tocycfv 33104 Function value of a permut...
tocycfvres1 33105 A cyclic permutation is a ...
tocycfvres2 33106 A cyclic permutation is th...
cycpmfvlem 33107 Lemma for ~ cycpmfv1 and ~...
cycpmfv1 33108 Value of a cycle function ...
cycpmfv2 33109 Value of a cycle function ...
cycpmfv3 33110 Values outside of the orbi...
cycpmcl 33111 Cyclic permutations are pe...
tocycf 33112 The permutation cycle buil...
tocyc01 33113 Permutation cycles built f...
cycpm2tr 33114 A cyclic permutation of 2 ...
cycpm2cl 33115 Closure for the 2-cycles. ...
cyc2fv1 33116 Function value of a 2-cycl...
cyc2fv2 33117 Function value of a 2-cycl...
trsp2cyc 33118 Exhibit the word a transpo...
cycpmco2f1 33119 The word U used in ~ cycpm...
cycpmco2rn 33120 The orbit of the compositi...
cycpmco2lem1 33121 Lemma for ~ cycpmco2 . (C...
cycpmco2lem2 33122 Lemma for ~ cycpmco2 . (C...
cycpmco2lem3 33123 Lemma for ~ cycpmco2 . (C...
cycpmco2lem4 33124 Lemma for ~ cycpmco2 . (C...
cycpmco2lem5 33125 Lemma for ~ cycpmco2 . (C...
cycpmco2lem6 33126 Lemma for ~ cycpmco2 . (C...
cycpmco2lem7 33127 Lemma for ~ cycpmco2 . (C...
cycpmco2 33128 The composition of a cycli...
cyc2fvx 33129 Function value of a 2-cycl...
cycpm3cl 33130 Closure of the 3-cycles in...
cycpm3cl2 33131 Closure of the 3-cycles in...
cyc3fv1 33132 Function value of a 3-cycl...
cyc3fv2 33133 Function value of a 3-cycl...
cyc3fv3 33134 Function value of a 3-cycl...
cyc3co2 33135 Represent a 3-cycle as a c...
cycpmconjvlem 33136 Lemma for ~ cycpmconjv . ...
cycpmconjv 33137 A formula for computing co...
cycpmrn 33138 The range of the word used...
tocyccntz 33139 All elements of a (finite)...
evpmval 33140 Value of the set of even p...
cnmsgn0g 33141 The neutral element of the...
evpmsubg 33142 The alternating group is a...
evpmid 33143 The identity is an even pe...
altgnsg 33144 The alternating group ` ( ...
cyc3evpm 33145 3-Cycles are even permutat...
cyc3genpmlem 33146 Lemma for ~ cyc3genpm . (...
cyc3genpm 33147 The alternating group ` A ...
cycpmgcl 33148 Cyclic permutations are pe...
cycpmconjslem1 33149 Lemma for ~ cycpmconjs . ...
cycpmconjslem2 33150 Lemma for ~ cycpmconjs . ...
cycpmconjs 33151 All cycles of the same len...
cyc3conja 33152 All 3-cycles are conjugate...
sgnsv 33155 The sign mapping. (Contri...
sgnsval 33156 The sign value. (Contribu...
sgnsf 33157 The sign function. (Contr...
inftmrel 33162 The infinitesimal relation...
isinftm 33163 Express ` x ` is infinites...
isarchi 33164 Express the predicate " ` ...
pnfinf 33165 Plus infinity is an infini...
xrnarchi 33166 The completed real line is...
isarchi2 33167 Alternative way to express...
submarchi 33168 A submonoid is archimedean...
isarchi3 33169 This is the usual definiti...
archirng 33170 Property of Archimedean or...
archirngz 33171 Property of Archimedean le...
archiexdiv 33172 In an Archimedean group, g...
archiabllem1a 33173 Lemma for ~ archiabl : In...
archiabllem1b 33174 Lemma for ~ archiabl . (C...
archiabllem1 33175 Archimedean ordered groups...
archiabllem2a 33176 Lemma for ~ archiabl , whi...
archiabllem2c 33177 Lemma for ~ archiabl . (C...
archiabllem2b 33178 Lemma for ~ archiabl . (C...
archiabllem2 33179 Archimedean ordered groups...
archiabl 33180 Archimedean left- and righ...
isslmd 33183 The predicate "is a semimo...
slmdlema 33184 Lemma for properties of a ...
lmodslmd 33185 Left semimodules generaliz...
slmdcmn 33186 A semimodule is a commutat...
slmdmnd 33187 A semimodule is a monoid. ...
slmdsrg 33188 The scalar component of a ...
slmdbn0 33189 The base set of a semimodu...
slmdacl 33190 Closure of ring addition f...
slmdmcl 33191 Closure of ring multiplica...
slmdsn0 33192 The set of scalars in a se...
slmdvacl 33193 Closure of vector addition...
slmdass 33194 Semiring left module vecto...
slmdvscl 33195 Closure of scalar product ...
slmdvsdi 33196 Distributive law for scala...
slmdvsdir 33197 Distributive law for scala...
slmdvsass 33198 Associative law for scalar...
slmd0cl 33199 The ring zero in a semimod...
slmd1cl 33200 The ring unity in a semiri...
slmdvs1 33201 Scalar product with ring u...
slmd0vcl 33202 The zero vector is a vecto...
slmd0vlid 33203 Left identity law for the ...
slmd0vrid 33204 Right identity law for the...
slmd0vs 33205 Zero times a vector is the...
slmdvs0 33206 Anything times the zero ve...
gsumvsca1 33207 Scalar product of a finite...
gsumvsca2 33208 Scalar product of a finite...
prmsimpcyc 33209 A group of prime order is ...
cringmul32d 33210 Commutative/associative la...
ringdid 33211 Distributive law for the m...
ringdird 33212 Distributive law for the m...
ringdi22 33213 Expand the product of two ...
urpropd 33214 Sufficient condition for r...
subrgmcld 33215 A subring is closed under ...
ress1r 33216 ` 1r ` is unaffected by re...
ringinvval 33217 The ring inverse expressed...
dvrcan5 33218 Cancellation law for commo...
subrgchr 33219 If ` A ` is a subring of `...
rmfsupp2 33220 A mapping of a multiplicat...
unitnz 33221 In a nonzero ring, a unit ...
isunit2 33222 Alternate definition of be...
isunit3 33223 Alternate definition of be...
irrednzr 33224 A ring with an irreducible...
0ringsubrg 33225 A subring of a zero ring i...
0ringcring 33226 The zero ring is commutati...
reldmrloc 33231 Ring localization is a pro...
erlval 33232 Value of the ring localiza...
rlocval 33233 Expand the value of the ri...
erlcl1 33234 Closure for the ring local...
erlcl2 33235 Closure for the ring local...
erldi 33236 Main property of the ring ...
erlbrd 33237 Deduce the ring localizati...
erlbr2d 33238 Deduce the ring localizati...
erler 33239 The relation used to build...
elrlocbasi 33240 Membership in the basis of...
rlocbas 33241 The base set of a ring loc...
rlocaddval 33242 Value of the addition in t...
rlocmulval 33243 Value of the addition in t...
rloccring 33244 The ring localization ` L ...
rloc0g 33245 The zero of a ring localiz...
rloc1r 33246 The multiplicative identit...
rlocf1 33247 The embedding ` F ` of a r...
domnmuln0rd 33248 In a domain, factors of a ...
domnprodn0 33249 In a domain, a finite prod...
idomrcan 33250 Right-cancellation law for...
domnlcanOLD 33251 Obsolete version of ~ domn...
domnlcanbOLD 33252 Obsolete version of ~ domn...
idomrcanOLD 33253 Obsolete version of ~ idom...
1rrg 33254 The multiplicative identit...
rrgsubm 33255 The left regular elements ...
subrdom 33256 A subring of a domain is a...
subridom 33257 A subring of an integral d...
subrfld 33258 A subring of a field is an...
eufndx 33261 Index value of the Euclide...
eufid 33262 Utility theorem: index-ind...
ringinveu 33265 If a ring unit element ` X...
isdrng4 33266 A division ring is a ring ...
rndrhmcl 33267 The image of a division ri...
sdrgdvcl 33268 A sub-division-ring is clo...
sdrginvcl 33269 A sub-division-ring is clo...
primefldchr 33270 The characteristic of a pr...
fracval 33273 Value of the field of frac...
fracbas 33274 The base of the field of f...
fracerl 33275 Rewrite the ring localizat...
fracf1 33276 The embedding of a commuta...
fracfld 33277 The field of fractions of ...
idomsubr 33278 Every integral domain is i...
fldgenval 33281 Value of the field generat...
fldgenssid 33282 The field generated by a s...
fldgensdrg 33283 A generated subfield is a ...
fldgenssv 33284 A generated subfield is a ...
fldgenss 33285 Generated subfields preser...
fldgenidfld 33286 The subfield generated by ...
fldgenssp 33287 The field generated by a s...
fldgenid 33288 The subfield of a field ` ...
fldgenfld 33289 A generated subfield is a ...
primefldgen1 33290 The prime field of a divis...
1fldgenq 33291 The field of rational numb...
isorng 33296 An ordered ring is a ring ...
orngring 33297 An ordered ring is a ring....
orngogrp 33298 An ordered ring is an orde...
isofld 33299 An ordered field is a fiel...
orngmul 33300 In an ordered ring, the or...
orngsqr 33301 In an ordered ring, all sq...
ornglmulle 33302 In an ordered ring, multip...
orngrmulle 33303 In an ordered ring, multip...
ornglmullt 33304 In an ordered ring, multip...
orngrmullt 33305 In an ordered ring, multip...
orngmullt 33306 In an ordered ring, the st...
ofldfld 33307 An ordered field is a fiel...
ofldtos 33308 An ordered field is a tota...
orng0le1 33309 In an ordered ring, the ri...
ofldlt1 33310 In an ordered field, the r...
ofldchr 33311 The characteristic of an o...
suborng 33312 Every subring of an ordere...
subofld 33313 Every subfield of an order...
isarchiofld 33314 Axiom of Archimedes : a ch...
rhmdvd 33315 A ring homomorphism preser...
kerunit 33316 If a unit element lies in ...
reldmresv 33319 The scalar restriction is ...
resvval 33320 Value of structure restric...
resvid2 33321 General behavior of trivia...
resvval2 33322 Value of nontrivial struct...
resvsca 33323 Base set of a structure re...
resvlem 33324 Other elements of a scalar...
resvlemOLD 33325 Obsolete version of ~ resv...
resvbas 33326 ` Base ` is unaffected by ...
resvbasOLD 33327 Obsolete proof of ~ resvba...
resvplusg 33328 ` +g ` is unaffected by sc...
resvplusgOLD 33329 Obsolete proof of ~ resvpl...
resvvsca 33330 ` .s ` is unaffected by sc...
resvvscaOLD 33331 Obsolete proof of ~ resvvs...
resvmulr 33332 ` .r ` is unaffected by sc...
resvmulrOLD 33333 Obsolete proof of ~ resvmu...
resv0g 33334 ` 0g ` is unaffected by sc...
resv1r 33335 ` 1r ` is unaffected by sc...
resvcmn 33336 Scalar restriction preserv...
gzcrng 33337 The gaussian integers form...
cnfldfld 33338 The complex numbers form a...
reofld 33339 The real numbers form an o...
nn0omnd 33340 The nonnegative integers f...
rearchi 33341 The field of the real numb...
nn0archi 33342 The monoid of the nonnegat...
xrge0slmod 33343 The extended nonnegative r...
qusker 33344 The kernel of a quotient m...
eqgvscpbl 33345 The left coset equivalence...
qusvscpbl 33346 The quotient map distribut...
qusvsval 33347 Value of the scalar multip...
imaslmod 33348 The image structure of a l...
imasmhm 33349 Given a function ` F ` wit...
imasghm 33350 Given a function ` F ` wit...
imasrhm 33351 Given a function ` F ` wit...
imaslmhm 33352 Given a function ` F ` wit...
quslmod 33353 If ` G ` is a submodule in...
quslmhm 33354 If ` G ` is a submodule of...
quslvec 33355 If ` S ` is a vector subsp...
ecxpid 33356 The equivalence class of a...
qsxpid 33357 The quotient set of a cart...
qusxpid 33358 The Group quotient equival...
qustriv 33359 The quotient of a group ` ...
qustrivr 33360 Converse of ~ qustriv . (...
znfermltl 33361 Fermat's little theorem in...
islinds5 33362 A set is linearly independ...
ellspds 33363 Variation on ~ ellspd . (...
0ellsp 33364 Zero is in all spans. (Co...
0nellinds 33365 The group identity cannot ...
rspsnid 33366 A principal ideal contains...
elrsp 33367 Write the elements of a ri...
ellpi 33368 Elementhood in a left prin...
lpirlidllpi 33369 In a principal ideal ring,...
rspidlid 33370 The ideal span of an ideal...
pidlnz 33371 A principal ideal generate...
lbslsp 33372 Any element of a left modu...
lindssn 33373 Any singleton of a nonzero...
lindflbs 33374 Conditions for an independ...
islbs5 33375 An equivalent formulation ...
linds2eq 33376 Deduce equality of element...
lindfpropd 33377 Property deduction for lin...
lindspropd 33378 Property deduction for lin...
dvdsruassoi 33379 If two elements ` X ` and ...
dvdsruasso 33380 Two elements ` X ` and ` Y...
dvdsruasso2 33381 A reformulation of ~ dvdsr...
dvdsrspss 33382 In a ring, an element ` X ...
rspsnasso 33383 Two elements ` X ` and ` Y...
unitprodclb 33384 A finite product is a unit...
elgrplsmsn 33385 Membership in a sumset wit...
lsmsnorb 33386 The sumset of a group with...
lsmsnorb2 33387 The sumset of a single ele...
elringlsm 33388 Membership in a product of...
elringlsmd 33389 Membership in a product of...
ringlsmss 33390 Closure of the product of ...
ringlsmss1 33391 The product of an ideal ` ...
ringlsmss2 33392 The product with an ideal ...
lsmsnpridl 33393 The product of the ring wi...
lsmsnidl 33394 The product of the ring wi...
lsmidllsp 33395 The sum of two ideals is t...
lsmidl 33396 The sum of two ideals is a...
lsmssass 33397 Group sum is associative, ...
grplsm0l 33398 Sumset with the identity s...
grplsmid 33399 The direct sum of an eleme...
quslsm 33400 Express the image by the q...
qusbas2 33401 Alternate definition of th...
qus0g 33402 The identity element of a ...
qusima 33403 The image of a subgroup by...
qusrn 33404 The natural map from eleme...
nsgqus0 33405 A normal subgroup ` N ` is...
nsgmgclem 33406 Lemma for ~ nsgmgc . (Con...
nsgmgc 33407 There is a monotone Galois...
nsgqusf1olem1 33408 Lemma for ~ nsgqusf1o . (...
nsgqusf1olem2 33409 Lemma for ~ nsgqusf1o . (...
nsgqusf1olem3 33410 Lemma for ~ nsgqusf1o . (...
nsgqusf1o 33411 The canonical projection h...
lmhmqusker 33412 A surjective module homomo...
lmicqusker 33413 The image ` H ` of a modul...
lidlmcld 33414 An ideal is closed under l...
intlidl 33415 The intersection of a none...
0ringidl 33416 The zero ideal is the only...
pidlnzb 33417 A principal ideal is nonze...
lidlunitel 33418 If an ideal ` I ` contains...
unitpidl1 33419 The ideal ` I ` generated ...
rhmquskerlem 33420 The mapping ` J ` induced ...
rhmqusker 33421 A surjective ring homomorp...
ricqusker 33422 The image ` H ` of a ring ...
elrspunidl 33423 Elementhood in the span of...
elrspunsn 33424 Membership to the span of ...
lidlincl 33425 Ideals are closed under in...
idlinsubrg 33426 The intersection between a...
rhmimaidl 33427 The image of an ideal ` I ...
drngidl 33428 A nonzero ring is a divisi...
drngidlhash 33429 A ring is a division ring ...
prmidlval 33432 The class of prime ideals ...
isprmidl 33433 The predicate "is a prime ...
prmidlnr 33434 A prime ideal is a proper ...
prmidl 33435 The main property of a pri...
prmidl2 33436 A condition that shows an ...
idlmulssprm 33437 Let ` P ` be a prime ideal...
pridln1 33438 A proper ideal cannot cont...
prmidlidl 33439 A prime ideal is an ideal....
prmidlssidl 33440 Prime ideals as a subset o...
cringm4 33441 Commutative/associative la...
isprmidlc 33442 The predicate "is prime id...
prmidlc 33443 Property of a prime ideal ...
0ringprmidl 33444 The trivial ring does not ...
prmidl0 33445 The zero ideal of a commut...
rhmpreimaprmidl 33446 The preimage of a prime id...
qsidomlem1 33447 If the quotient ring of a ...
qsidomlem2 33448 A quotient by a prime idea...
qsidom 33449 An ideal ` I ` in the comm...
qsnzr 33450 A quotient of a non-zero r...
ssdifidllem 33451 Lemma for ~ ssdifidl : Th...
ssdifidl 33452 Let ` R ` be a ring, and l...
ssdifidlprm 33453 If the set ` S ` of ~ ssdi...
mxidlval 33456 The set of maximal ideals ...
ismxidl 33457 The predicate "is a maxima...
mxidlidl 33458 A maximal ideal is an idea...
mxidlnr 33459 A maximal ideal is proper....
mxidlmax 33460 A maximal ideal is a maxim...
mxidln1 33461 One is not contained in an...
mxidlnzr 33462 A ring with a maximal idea...
mxidlmaxv 33463 An ideal ` I ` strictly co...
crngmxidl 33464 In a commutative ring, max...
mxidlprm 33465 Every maximal ideal is pri...
mxidlirredi 33466 In an integral domain, the...
mxidlirred 33467 In a principal ideal domai...
ssmxidllem 33468 The set ` P ` used in the ...
ssmxidl 33469 Let ` R ` be a ring, and l...
drnglidl1ne0 33470 In a nonzero ring, the zer...
drng0mxidl 33471 In a division ring, the ze...
drngmxidl 33472 The zero ideal is the only...
drngmxidlr 33473 If a ring's only maximal i...
krull 33474 Krull's theorem: Any nonz...
mxidlnzrb 33475 A ring is nonzero if and o...
krullndrng 33476 Krull's theorem for non-di...
opprabs 33477 The opposite ring of the o...
oppreqg 33478 Group coset equivalence re...
opprnsg 33479 Normal subgroups of the op...
opprlidlabs 33480 The ideals of the opposite...
oppr2idl 33481 Two sided ideal of the opp...
opprmxidlabs 33482 The maximal ideal of the o...
opprqusbas 33483 The base of the quotient o...
opprqusplusg 33484 The group operation of the...
opprqus0g 33485 The group identity element...
opprqusmulr 33486 The multiplication operati...
opprqus1r 33487 The ring unity of the quot...
opprqusdrng 33488 The quotient of the opposi...
qsdrngilem 33489 Lemma for ~ qsdrngi . (Co...
qsdrngi 33490 A quotient by a maximal le...
qsdrnglem2 33491 Lemma for ~ qsdrng . (Con...
qsdrng 33492 An ideal ` M ` is both lef...
qsfld 33493 An ideal ` M ` in the comm...
mxidlprmALT 33494 Every maximal ideal is pri...
idlsrgstr 33497 A constructed semiring of ...
idlsrgval 33498 Lemma for ~ idlsrgbas thro...
idlsrgbas 33499 Base of the ideals of a ri...
idlsrgplusg 33500 Additive operation of the ...
idlsrg0g 33501 The zero ideal is the addi...
idlsrgmulr 33502 Multiplicative operation o...
idlsrgtset 33503 Topology component of the ...
idlsrgmulrval 33504 Value of the ring multipli...
idlsrgmulrcl 33505 Ideals of a ring ` R ` are...
idlsrgmulrss1 33506 In a commutative ring, the...
idlsrgmulrss2 33507 The product of two ideals ...
idlsrgmulrssin 33508 In a commutative ring, the...
idlsrgmnd 33509 The ideals of a ring form ...
idlsrgcmnd 33510 The ideals of a ring form ...
rprmval 33511 The prime elements of a ri...
isrprm 33512 Property for ` P ` to be a...
rprmcl 33513 A ring prime is an element...
rprmdvds 33514 If a ring prime ` Q ` divi...
rprmnz 33515 A ring prime is nonzero. ...
rprmnunit 33516 A ring prime is not a unit...
rsprprmprmidl 33517 In a commutative ring, ide...
rsprprmprmidlb 33518 In an integral domain, an ...
rprmndvdsr1 33519 A ring prime element does ...
rprmasso 33520 In an integral domain, the...
rprmasso2 33521 In an integral domain, if ...
rprmasso3 33522 In an integral domain, if ...
unitmulrprm 33523 A ring unit multiplied by ...
rprmndvdsru 33524 A ring prime element does ...
rprmirredlem 33525 Lemma for ~ rprmirred . (...
rprmirred 33526 In an integral domain, rin...
rprmirredb 33527 In a principal ideal domai...
rprmdvdspow 33528 If a prime element divides...
rprmdvdsprod 33529 If a prime element ` Q ` d...
1arithidomlem1 33530 Lemma for ~ 1arithidom . ...
1arithidomlem2 33531 Lemma for ~ 1arithidom : i...
1arithidom 33532 Uniqueness of prime factor...
isufd 33535 The property of being a Un...
ufdprmidl 33536 In a unique factorization ...
ufdidom 33537 A nonzero unique factoriza...
pidufd 33538 Every principal ideal doma...
1arithufdlem1 33539 Lemma for ~ 1arithufd . T...
1arithufdlem2 33540 Lemma for ~ 1arithufd . T...
1arithufdlem3 33541 Lemma for ~ 1arithufd . I...
1arithufdlem4 33542 Lemma for ~ 1arithufd . N...
1arithufd 33543 Existence of a factorizati...
dfufd2lem 33544 Lemma for ~ dfufd2 . (Con...
dfufd2 33545 Alternative definition of ...
zringidom 33546 The ring of integers is an...
zringpid 33547 The ring of integers is a ...
dfprm3 33548 The (positive) prime eleme...
zringfrac 33549 The field of fractions of ...
0ringmon1p 33550 There are no monic polynom...
fply1 33551 Conditions for a function ...
ply1lvec 33552 In a division ring, the un...
evls1fn 33553 Functionality of the subri...
evls1dm 33554 The domain of the subring ...
evls1fvf 33555 The subring evaluation fun...
evl1fvf 33556 The univariate polynomial ...
evl1fpws 33557 Evaluation of a univariate...
ressdeg1 33558 The degree of a univariate...
ressply10g 33559 A restricted polynomial al...
ressply1mon1p 33560 The monic polynomials of a...
ressply1invg 33561 An element of a restricted...
ressply1sub 33562 A restricted polynomial al...
ressasclcl 33563 Closure of the univariate ...
evls1subd 33564 Univariate polynomial eval...
deg1le0eq0 33565 A polynomial with nonposit...
ply1asclunit 33566 A non-zero scalar polynomi...
ply1unit 33567 In a field ` F ` , a polyn...
evl1deg1 33568 Evaluation of a univariate...
evl1deg2 33569 Evaluation of a univariate...
evl1deg3 33570 Evaluation of a univariate...
ply1dg1rt 33571 Express the root ` - B / A...
ply1dg1rtn0 33572 Polynomials of degree 1 ov...
ply1mulrtss 33573 The roots of a factor ` F ...
ply1dg3rt0irred 33574 If a cubic polynomial over...
m1pmeq 33575 If two monic polynomials `...
ply1fermltl 33576 Fermat's little theorem fo...
coe1mon 33577 Coefficient vector of a mo...
ply1moneq 33578 Two monomials are equal if...
coe1zfv 33579 The coefficients of the ze...
coe1vr1 33580 Polynomial coefficient of ...
deg1vr 33581 The degree of the variable...
ply1degltel 33582 Characterize elementhood i...
ply1degleel 33583 Characterize elementhood i...
ply1degltlss 33584 The space ` S ` of the uni...
gsummoncoe1fzo 33585 A coefficient of the polyn...
ply1gsumz 33586 If a polynomial given as a...
deg1addlt 33587 If both factors have degre...
ig1pnunit 33588 The polynomial ideal gener...
ig1pmindeg 33589 The polynomial ideal gener...
q1pdir 33590 Distribution of univariate...
q1pvsca 33591 Scalar multiplication prop...
r1pvsca 33592 Scalar multiplication prop...
r1p0 33593 Polynomial remainder opera...
r1pcyc 33594 The polynomial remainder o...
r1padd1 33595 Addition property of the p...
r1pid2OLD 33596 Obsolete version of ~ r1pi...
r1plmhm 33597 The univariate polynomial ...
r1pquslmic 33598 The univariate polynomial ...
sra1r 33599 The unity element of a sub...
sradrng 33600 Condition for a subring al...
srasubrg 33601 A subring of the original ...
sralvec 33602 Given a sub division ring ...
srafldlvec 33603 Given a subfield ` F ` of ...
resssra 33604 The subring algebra of a r...
lsssra 33605 A subring is a subspace of...
drgext0g 33606 The additive neutral eleme...
drgextvsca 33607 The scalar multiplication ...
drgext0gsca 33608 The additive neutral eleme...
drgextsubrg 33609 The scalar field is a subr...
drgextlsp 33610 The scalar field is a subs...
drgextgsum 33611 Group sum in a division ri...
lvecdimfi 33612 Finite version of ~ lvecdi...
dimval 33615 The dimension of a vector ...
dimvalfi 33616 The dimension of a vector ...
dimcl 33617 Closure of the vector spac...
lmimdim 33618 Module isomorphisms preser...
lmicdim 33619 Module isomorphisms preser...
lvecdim0i 33620 A vector space of dimensio...
lvecdim0 33621 A vector space of dimensio...
lssdimle 33622 The dimension of a linear ...
dimpropd 33623 If two structures have the...
rlmdim 33624 The left vector space indu...
rgmoddimOLD 33625 Obsolete version of ~ rlmd...
frlmdim 33626 Dimension of a free left m...
tnglvec 33627 Augmenting a structure wit...
tngdim 33628 Dimension of a left vector...
rrxdim 33629 Dimension of the generaliz...
matdim 33630 Dimension of the space of ...
lbslsat 33631 A nonzero vector ` X ` is ...
lsatdim 33632 A line, spanned by a nonze...
drngdimgt0 33633 The dimension of a vector ...
lmhmlvec2 33634 A homomorphism of left vec...
kerlmhm 33635 The kernel of a vector spa...
imlmhm 33636 The image of a vector spac...
ply1degltdimlem 33637 Lemma for ~ ply1degltdim ....
ply1degltdim 33638 The space ` S ` of the uni...
lindsunlem 33639 Lemma for ~ lindsun . (Co...
lindsun 33640 Condition for the union of...
lbsdiflsp0 33641 The linear spans of two di...
dimkerim 33642 Given a linear map ` F ` b...
qusdimsum 33643 Let ` W ` be a vector spac...
fedgmullem1 33644 Lemma for ~ fedgmul . (Co...
fedgmullem2 33645 Lemma for ~ fedgmul . (Co...
fedgmul 33646 The multiplicativity formu...
dimlssid 33647 If the dimension of a line...
lvecendof1f1o 33648 If an endomorphism ` U ` o...
lactlmhm 33649 In an associative algebra ...
assalactf1o 33650 In an associative algebra ...
assarrginv 33651 If an element ` X ` of an ...
assafld 33652 If an algebra ` A ` of fin...
relfldext 33661 The field extension is a r...
brfldext 33662 The field extension relati...
ccfldextrr 33663 The field of the complex n...
fldextfld1 33664 A field extension is only ...
fldextfld2 33665 A field extension is only ...
fldextsubrg 33666 Field extension implies a ...
fldextress 33667 Field extension implies a ...
brfinext 33668 The finite field extension...
extdgval 33669 Value of the field extensi...
fldextsralvec 33670 The subring algebra associ...
extdgcl 33671 Closure of the field exten...
extdggt0 33672 Degrees of field extension...
fldexttr 33673 Field extension is a trans...
fldextid 33674 The field extension relati...
extdgid 33675 A trivial field extension ...
extdgmul 33676 The multiplicativity formu...
finexttrb 33677 The extension ` E ` of ` K...
extdg1id 33678 If the degree of the exten...
extdg1b 33679 The degree of the extensio...
fldgenfldext 33680 A subfield ` F ` extended ...
fldextchr 33681 The characteristic of a su...
evls1fldgencl 33682 Closure of the subring pol...
ccfldsrarelvec 33683 The subring algebra of the...
ccfldextdgrr 33684 The degree of the field ex...
irngval 33687 The elements of a field ` ...
elirng 33688 Property for an element ` ...
irngss 33689 All elements of a subring ...
irngssv 33690 An integral element is an ...
0ringirng 33691 A zero ring ` R ` has no i...
irngnzply1lem 33692 In the case of a field ` E...
irngnzply1 33693 In the case of a field ` E...
ply1annidllem 33696 Write the set ` Q ` of pol...
ply1annidl 33697 The set ` Q ` of polynomia...
ply1annnr 33698 The set ` Q ` of polynomia...
ply1annig1p 33699 The ideal ` Q ` of polynom...
minplyval 33700 Expand the value of the mi...
minplycl 33701 The minimal polynomial is ...
ply1annprmidl 33702 The set ` Q ` of polynomia...
minplymindeg 33703 The minimal polynomial of ...
minplyann 33704 The minimal polynomial for...
minplyirredlem 33705 Lemma for ~ minplyirred . ...
minplyirred 33706 A nonzero minimal polynomi...
irngnminplynz 33707 Integral elements have non...
minplym1p 33708 A minimal polynomial is mo...
irredminply 33709 An irreducible, monic, ann...
algextdeglem1 33710 Lemma for ~ algextdeg . (...
algextdeglem2 33711 Lemma for ~ algextdeg . B...
algextdeglem3 33712 Lemma for ~ algextdeg . T...
algextdeglem4 33713 Lemma for ~ algextdeg . B...
algextdeglem5 33714 Lemma for ~ algextdeg . T...
algextdeglem6 33715 Lemma for ~ algextdeg . B...
algextdeglem7 33716 Lemma for ~ algextdeg . T...
algextdeglem8 33717 Lemma for ~ algextdeg . T...
algextdeg 33718 The degree of an algebraic...
rtelextdg2lem 33719 Lemma for ~ rtelextdg2 : ...
rtelextdg2 33720 If an element ` X ` is a s...
fldext2chn 33721 In a non-empty tower ` T `...
constrrtll 33724 In the construction of con...
constrrtlc1 33725 In the construction of con...
constrrtlc2 33726 In the construction of con...
constrrtcclem 33727 In the construction of con...
constrrtcc 33728 In the construction of con...
constr0 33729 The first step of the cons...
constrsuc 33730 Membership in the successo...
constrlim 33731 Limit step of the construc...
constrsscn 33732 Closure of the constructib...
constrsslem 33733 Lemma for ~ constrss . Th...
constr01 33734 ` 0 ` and ` 1 ` are in all...
constrss 33735 Constructed points are in ...
constrmon 33736 The construction of constr...
constrconj 33737 If a point ` X ` of the co...
constrfin 33738 Each step of the construct...
constrelextdg2 33739 If the ` N ` -th step ` ( ...
2sqr3minply 33740 The polynomial ` ( ( X ^ 3...
smatfval 33743 Value of the submatrix. (...
smatrcl 33744 Closure of the rectangular...
smatlem 33745 Lemma for the next theorem...
smattl 33746 Entries of a submatrix, to...
smattr 33747 Entries of a submatrix, to...
smatbl 33748 Entries of a submatrix, bo...
smatbr 33749 Entries of a submatrix, bo...
smatcl 33750 Closure of the square subm...
matmpo 33751 Write a square matrix as a...
1smat1 33752 The submatrix of the ident...
submat1n 33753 One case where the submatr...
submatres 33754 Special case where the sub...
submateqlem1 33755 Lemma for ~ submateq . (C...
submateqlem2 33756 Lemma for ~ submateq . (C...
submateq 33757 Sufficient condition for t...
submatminr1 33758 If we take a submatrix by ...
lmatval 33761 Value of the literal matri...
lmatfval 33762 Entries of a literal matri...
lmatfvlem 33763 Useful lemma to extract li...
lmatcl 33764 Closure of the literal mat...
lmat22lem 33765 Lemma for ~ lmat22e11 and ...
lmat22e11 33766 Entry of a 2x2 literal mat...
lmat22e12 33767 Entry of a 2x2 literal mat...
lmat22e21 33768 Entry of a 2x2 literal mat...
lmat22e22 33769 Entry of a 2x2 literal mat...
lmat22det 33770 The determinant of a liter...
mdetpmtr1 33771 The determinant of a matri...
mdetpmtr2 33772 The determinant of a matri...
mdetpmtr12 33773 The determinant of a matri...
mdetlap1 33774 A Laplace expansion of the...
madjusmdetlem1 33775 Lemma for ~ madjusmdet . ...
madjusmdetlem2 33776 Lemma for ~ madjusmdet . ...
madjusmdetlem3 33777 Lemma for ~ madjusmdet . ...
madjusmdetlem4 33778 Lemma for ~ madjusmdet . ...
madjusmdet 33779 Express the cofactor of th...
mdetlap 33780 Laplace expansion of the d...
ist0cld 33781 The predicate "is a T_0 sp...
txomap 33782 Given two open maps ` F ` ...
qtopt1 33783 If every equivalence class...
qtophaus 33784 If an open map's graph in ...
circtopn 33785 The topology of the unit c...
circcn 33786 The function gluing the re...
reff 33787 For any cover refinement, ...
locfinreflem 33788 A locally finite refinemen...
locfinref 33789 A locally finite refinemen...
iscref 33792 The property that every op...
crefeq 33793 Equality theorem for the "...
creftop 33794 A space where every open c...
crefi 33795 The property that every op...
crefdf 33796 A formulation of ~ crefi e...
crefss 33797 The "every open cover has ...
cmpcref 33798 Equivalent definition of c...
cmpfiref 33799 Every open cover of a Comp...
ldlfcntref 33802 Every open cover of a Lind...
ispcmp 33805 The predicate "is a paraco...
cmppcmp 33806 Every compact space is par...
dispcmp 33807 Every discrete space is pa...
pcmplfin 33808 Given a paracompact topolo...
pcmplfinf 33809 Given a paracompact topolo...
rspecval 33812 Value of the spectrum of t...
rspecbas 33813 The prime ideals form the ...
rspectset 33814 Topology component of the ...
rspectopn 33815 The topology component of ...
zarcls0 33816 The closure of the identit...
zarcls1 33817 The unit ideal ` B ` is th...
zarclsun 33818 The union of two closed se...
zarclsiin 33819 In a Zariski topology, the...
zarclsint 33820 The intersection of a fami...
zarclssn 33821 The closed points of Zaris...
zarcls 33822 The open sets of the Zaris...
zartopn 33823 The Zariski topology is a ...
zartop 33824 The Zariski topology is a ...
zartopon 33825 The points of the Zariski ...
zar0ring 33826 The Zariski Topology of th...
zart0 33827 The Zariski topology is T_...
zarmxt1 33828 The Zariski topology restr...
zarcmplem 33829 Lemma for ~ zarcmp . (Con...
zarcmp 33830 The Zariski topology is co...
rspectps 33831 The spectrum of a ring ` R...
rhmpreimacnlem 33832 Lemma for ~ rhmpreimacn . ...
rhmpreimacn 33833 The function mapping a pri...
metidval 33838 Value of the metric identi...
metidss 33839 As a relation, the metric ...
metidv 33840 ` A ` and ` B ` identify b...
metideq 33841 Basic property of the metr...
metider 33842 The metric identification ...
pstmval 33843 Value of the metric induce...
pstmfval 33844 Function value of the metr...
pstmxmet 33845 The metric induced by a ps...
hauseqcn 33846 In a Hausdorff topology, t...
elunitge0 33847 An element of the closed u...
unitssxrge0 33848 The closed unit interval i...
unitdivcld 33849 Necessary conditions for a...
iistmd 33850 The closed unit interval f...
unicls 33851 The union of the closed se...
tpr2tp 33852 The usual topology on ` ( ...
tpr2uni 33853 The usual topology on ` ( ...
xpinpreima 33854 Rewrite the cartesian prod...
xpinpreima2 33855 Rewrite the cartesian prod...
sqsscirc1 33856 The complex square of side...
sqsscirc2 33857 The complex square of side...
cnre2csqlem 33858 Lemma for ~ cnre2csqima . ...
cnre2csqima 33859 Image of a centered square...
tpr2rico 33860 For any point of an open s...
cnvordtrestixx 33861 The restriction of the 'gr...
prsdm 33862 Domain of the relation of ...
prsrn 33863 Range of the relation of a...
prsss 33864 Relation of a subproset. ...
prsssdm 33865 Domain of a subproset rela...
ordtprsval 33866 Value of the order topolog...
ordtprsuni 33867 Value of the order topolog...
ordtcnvNEW 33868 The order dual generates t...
ordtrestNEW 33869 The subspace topology of a...
ordtrest2NEWlem 33870 Lemma for ~ ordtrest2NEW ....
ordtrest2NEW 33871 An interval-closed set ` A...
ordtconnlem1 33872 Connectedness in the order...
ordtconn 33873 Connectedness in the order...
mndpluscn 33874 A mapping that is both a h...
mhmhmeotmd 33875 Deduce a Topological Monoi...
rmulccn 33876 Multiplication by a real c...
raddcn 33877 Addition in the real numbe...
xrmulc1cn 33878 The operation multiplying ...
fmcncfil 33879 The image of a Cauchy filt...
xrge0hmph 33880 The extended nonnegative r...
xrge0iifcnv 33881 Define a bijection from ` ...
xrge0iifcv 33882 The defined function's val...
xrge0iifiso 33883 The defined bijection from...
xrge0iifhmeo 33884 Expose a homeomorphism fro...
xrge0iifhom 33885 The defined function from ...
xrge0iif1 33886 Condition for the defined ...
xrge0iifmhm 33887 The defined function from ...
xrge0pluscn 33888 The addition operation of ...
xrge0mulc1cn 33889 The operation multiplying ...
xrge0tps 33890 The extended nonnegative r...
xrge0topn 33891 The topology of the extend...
xrge0haus 33892 The topology of the extend...
xrge0tmd 33893 The extended nonnegative r...
xrge0tmdALT 33894 Alternate proof of ~ xrge0...
lmlim 33895 Relate a limit in a given ...
lmlimxrge0 33896 Relate a limit in the nonn...
rge0scvg 33897 Implication of convergence...
fsumcvg4 33898 A serie with finite suppor...
pnfneige0 33899 A neighborhood of ` +oo ` ...
lmxrge0 33900 Express "sequence ` F ` co...
lmdvg 33901 If a monotonic sequence of...
lmdvglim 33902 If a monotonic real number...
pl1cn 33903 A univariate polynomial is...
zringnm 33906 The norm (function) for a ...
zzsnm 33907 The norm of the ring of th...
zlm0 33908 Zero of a ` ZZ ` -module. ...
zlm1 33909 Unity element of a ` ZZ ` ...
zlmds 33910 Distance in a ` ZZ ` -modu...
zlmdsOLD 33911 Obsolete proof of ~ zlmds ...
zlmtset 33912 Topology in a ` ZZ ` -modu...
zlmtsetOLD 33913 Obsolete proof of ~ zlmtse...
zlmnm 33914 Norm of a ` ZZ ` -module (...
zhmnrg 33915 The ` ZZ ` -module built f...
nmmulg 33916 The norm of a group produc...
zrhnm 33917 The norm of the image by `...
cnzh 33918 The ` ZZ ` -module of ` CC...
rezh 33919 The ` ZZ ` -module of ` RR...
qqhval 33922 Value of the canonical hom...
zrhf1ker 33923 The kernel of the homomorp...
zrhchr 33924 The kernel of the homomorp...
zrhker 33925 The kernel of the homomorp...
zrhunitpreima 33926 The preimage by ` ZRHom ` ...
elzrhunit 33927 Condition for the image by...
elzdif0 33928 Lemma for ~ qqhval2 . (Co...
qqhval2lem 33929 Lemma for ~ qqhval2 . (Co...
qqhval2 33930 Value of the canonical hom...
qqhvval 33931 Value of the canonical hom...
qqh0 33932 The image of ` 0 ` by the ...
qqh1 33933 The image of ` 1 ` by the ...
qqhf 33934 ` QQHom ` as a function. ...
qqhvq 33935 The image of a quotient by...
qqhghm 33936 The ` QQHom ` homomorphism...
qqhrhm 33937 The ` QQHom ` homomorphism...
qqhnm 33938 The norm of the image by `...
qqhcn 33939 The ` QQHom ` homomorphism...
qqhucn 33940 The ` QQHom ` homomorphism...
rrhval 33944 Value of the canonical hom...
rrhcn 33945 If the topology of ` R ` i...
rrhf 33946 If the topology of ` R ` i...
isrrext 33948 Express the property " ` R...
rrextnrg 33949 An extension of ` RR ` is ...
rrextdrg 33950 An extension of ` RR ` is ...
rrextnlm 33951 The norm of an extension o...
rrextchr 33952 The ring characteristic of...
rrextcusp 33953 An extension of ` RR ` is ...
rrexttps 33954 An extension of ` RR ` is ...
rrexthaus 33955 The topology of an extensi...
rrextust 33956 The uniformity of an exten...
rerrext 33957 The field of the real numb...
cnrrext 33958 The field of the complex n...
qqtopn 33959 The topology of the field ...
rrhfe 33960 If ` R ` is an extension o...
rrhcne 33961 If ` R ` is an extension o...
rrhqima 33962 The ` RRHom ` homomorphism...
rrh0 33963 The image of ` 0 ` by the ...
xrhval 33966 The value of the embedding...
zrhre 33967 The ` ZRHom ` homomorphism...
qqhre 33968 The ` QQHom ` homomorphism...
rrhre 33969 The ` RRHom ` homomorphism...
relmntop 33972 Manifold is a relation. (...
ismntoplly 33973 Property of being a manifo...
ismntop 33974 Property of being a manifo...
nexple 33975 A lower bound for an expon...
indv 33978 Value of the indicator fun...
indval 33979 Value of the indicator fun...
indval2 33980 Alternate value of the ind...
indf 33981 An indicator function as a...
indfval 33982 Value of the indicator fun...
ind1 33983 Value of the indicator fun...
ind0 33984 Value of the indicator fun...
ind1a 33985 Value of the indicator fun...
indpi1 33986 Preimage of the singleton ...
indsum 33987 Finite sum of a product wi...
indsumin 33988 Finite sum of a product wi...
prodindf 33989 The product of indicators ...
indf1o 33990 The bijection between a po...
indpreima 33991 A function with range ` { ...
indf1ofs 33992 The bijection between fini...
esumex 33995 An extended sum is a set b...
esumcl 33996 Closure for extended sum i...
esumeq12dvaf 33997 Equality deduction for ext...
esumeq12dva 33998 Equality deduction for ext...
esumeq12d 33999 Equality deduction for ext...
esumeq1 34000 Equality theorem for an ex...
esumeq1d 34001 Equality theorem for an ex...
esumeq2 34002 Equality theorem for exten...
esumeq2d 34003 Equality deduction for ext...
esumeq2dv 34004 Equality deduction for ext...
esumeq2sdv 34005 Equality deduction for ext...
nfesum1 34006 Bound-variable hypothesis ...
nfesum2 34007 Bound-variable hypothesis ...
cbvesum 34008 Change bound variable in a...
cbvesumv 34009 Change bound variable in a...
esumid 34010 Identify the extended sum ...
esumgsum 34011 A finite extended sum is t...
esumval 34012 Develop the value of the e...
esumel 34013 The extended sum is a limi...
esumnul 34014 Extended sum over the empt...
esum0 34015 Extended sum of zero. (Co...
esumf1o 34016 Re-index an extended sum u...
esumc 34017 Convert from the collectio...
esumrnmpt 34018 Rewrite an extended sum in...
esumsplit 34019 Split an extended sum into...
esummono 34020 Extended sum is monotonic....
esumpad 34021 Extend an extended sum by ...
esumpad2 34022 Remove zeroes from an exte...
esumadd 34023 Addition of infinite sums....
esumle 34024 If all of the terms of an ...
gsumesum 34025 Relate a group sum on ` ( ...
esumlub 34026 The extended sum is the lo...
esumaddf 34027 Addition of infinite sums....
esumlef 34028 If all of the terms of an ...
esumcst 34029 The extended sum of a cons...
esumsnf 34030 The extended sum of a sing...
esumsn 34031 The extended sum of a sing...
esumpr 34032 Extended sum over a pair. ...
esumpr2 34033 Extended sum over a pair, ...
esumrnmpt2 34034 Rewrite an extended sum in...
esumfzf 34035 Formulating a partial exte...
esumfsup 34036 Formulating an extended su...
esumfsupre 34037 Formulating an extended su...
esumss 34038 Change the index set to a ...
esumpinfval 34039 The value of the extended ...
esumpfinvallem 34040 Lemma for ~ esumpfinval . ...
esumpfinval 34041 The value of the extended ...
esumpfinvalf 34042 Same as ~ esumpfinval , mi...
esumpinfsum 34043 The value of the extended ...
esumpcvgval 34044 The value of the extended ...
esumpmono 34045 The partial sums in an ext...
esumcocn 34046 Lemma for ~ esummulc2 and ...
esummulc1 34047 An extended sum multiplied...
esummulc2 34048 An extended sum multiplied...
esumdivc 34049 An extended sum divided by...
hashf2 34050 Lemma for ~ hasheuni . (C...
hasheuni 34051 The cardinality of a disjo...
esumcvg 34052 The sequence of partial su...
esumcvg2 34053 Simpler version of ~ esumc...
esumcvgsum 34054 The value of the extended ...
esumsup 34055 Express an extended sum as...
esumgect 34056 "Send ` n ` to ` +oo ` " i...
esumcvgre 34057 All terms of a converging ...
esum2dlem 34058 Lemma for ~ esum2d (finite...
esum2d 34059 Write a double extended su...
esumiun 34060 Sum over a nonnecessarily ...
ofceq 34063 Equality theorem for funct...
ofcfval 34064 Value of an operation appl...
ofcval 34065 Evaluate a function/consta...
ofcfn 34066 The function operation pro...
ofcfeqd2 34067 Equality theorem for funct...
ofcfval3 34068 General value of ` ( F oFC...
ofcf 34069 The function/constant oper...
ofcfval2 34070 The function operation exp...
ofcfval4 34071 The function/constant oper...
ofcc 34072 Left operation by a consta...
ofcof 34073 Relate function operation ...
sigaex 34076 Lemma for ~ issiga and ~ i...
sigaval 34077 The set of sigma-algebra w...
issiga 34078 An alternative definition ...
isrnsiga 34079 The property of being a si...
0elsiga 34080 A sigma-algebra contains t...
baselsiga 34081 A sigma-algebra contains i...
sigasspw 34082 A sigma-algebra is a set o...
sigaclcu 34083 A sigma-algebra is closed ...
sigaclcuni 34084 A sigma-algebra is closed ...
sigaclfu 34085 A sigma-algebra is closed ...
sigaclcu2 34086 A sigma-algebra is closed ...
sigaclfu2 34087 A sigma-algebra is closed ...
sigaclcu3 34088 A sigma-algebra is closed ...
issgon 34089 Property of being a sigma-...
sgon 34090 A sigma-algebra is a sigma...
elsigass 34091 An element of a sigma-alge...
elrnsiga 34092 Dropping the base informat...
isrnsigau 34093 The property of being a si...
unielsiga 34094 A sigma-algebra contains i...
dmvlsiga 34095 Lebesgue-measurable subset...
pwsiga 34096 Any power set forms a sigm...
prsiga 34097 The smallest possible sigm...
sigaclci 34098 A sigma-algebra is closed ...
difelsiga 34099 A sigma-algebra is closed ...
unelsiga 34100 A sigma-algebra is closed ...
inelsiga 34101 A sigma-algebra is closed ...
sigainb 34102 Building a sigma-algebra f...
insiga 34103 The intersection of a coll...
sigagenval 34106 Value of the generated sig...
sigagensiga 34107 A generated sigma-algebra ...
sgsiga 34108 A generated sigma-algebra ...
unisg 34109 The sigma-algebra generate...
dmsigagen 34110 A sigma-algebra can be gen...
sssigagen 34111 A set is a subset of the s...
sssigagen2 34112 A subset of the generating...
elsigagen 34113 Any element of a set is al...
elsigagen2 34114 Any countable union of ele...
sigagenss 34115 The generated sigma-algebr...
sigagenss2 34116 Sufficient condition for i...
sigagenid 34117 The sigma-algebra generate...
ispisys 34118 The property of being a pi...
ispisys2 34119 The property of being a pi...
inelpisys 34120 Pi-systems are closed unde...
sigapisys 34121 All sigma-algebras are pi-...
isldsys 34122 The property of being a la...
pwldsys 34123 The power set of the unive...
unelldsys 34124 Lambda-systems are closed ...
sigaldsys 34125 All sigma-algebras are lam...
ldsysgenld 34126 The intersection of all la...
sigapildsyslem 34127 Lemma for ~ sigapildsys . ...
sigapildsys 34128 Sigma-algebra are exactly ...
ldgenpisyslem1 34129 Lemma for ~ ldgenpisys . ...
ldgenpisyslem2 34130 Lemma for ~ ldgenpisys . ...
ldgenpisyslem3 34131 Lemma for ~ ldgenpisys . ...
ldgenpisys 34132 The lambda system ` E ` ge...
dynkin 34133 Dynkin's lambda-pi theorem...
isros 34134 The property of being a ri...
rossspw 34135 A ring of sets is a collec...
0elros 34136 A ring of sets contains th...
unelros 34137 A ring of sets is closed u...
difelros 34138 A ring of sets is closed u...
inelros 34139 A ring of sets is closed u...
fiunelros 34140 A ring of sets is closed u...
issros 34141 The property of being a se...
srossspw 34142 A semiring of sets is a co...
0elsros 34143 A semiring of sets contain...
inelsros 34144 A semiring of sets is clos...
diffiunisros 34145 In semiring of sets, compl...
rossros 34146 Rings of sets are semiring...
brsiga 34149 The Borel Algebra on real ...
brsigarn 34150 The Borel Algebra is a sig...
brsigasspwrn 34151 The Borel Algebra is a set...
unibrsiga 34152 The union of the Borel Alg...
cldssbrsiga 34153 A Borel Algebra contains a...
sxval 34156 Value of the product sigma...
sxsiga 34157 A product sigma-algebra is...
sxsigon 34158 A product sigma-algebra is...
sxuni 34159 The base set of a product ...
elsx 34160 The cartesian product of t...
measbase 34163 The base set of a measure ...
measval 34164 The value of the ` measure...
ismeas 34165 The property of being a me...
isrnmeas 34166 The property of being a me...
dmmeas 34167 The domain of a measure is...
measbasedom 34168 The base set of a measure ...
measfrge0 34169 A measure is a function ov...
measfn 34170 A measure is a function on...
measvxrge0 34171 The values of a measure ar...
measvnul 34172 The measure of the empty s...
measge0 34173 A measure is nonnegative. ...
measle0 34174 If the measure of a given ...
measvun 34175 The measure of a countable...
measxun2 34176 The measure the union of t...
measun 34177 The measure the union of t...
measvunilem 34178 Lemma for ~ measvuni . (C...
measvunilem0 34179 Lemma for ~ measvuni . (C...
measvuni 34180 The measure of a countable...
measssd 34181 A measure is monotone with...
measunl 34182 A measure is sub-additive ...
measiuns 34183 The measure of the union o...
measiun 34184 A measure is sub-additive....
meascnbl 34185 A measure is continuous fr...
measinblem 34186 Lemma for ~ measinb . (Co...
measinb 34187 Building a measure restric...
measres 34188 Building a measure restric...
measinb2 34189 Building a measure restric...
measdivcst 34190 Division of a measure by a...
measdivcstALTV 34191 Alternate version of ~ mea...
cntmeas 34192 The Counting measure is a ...
pwcntmeas 34193 The counting measure is a ...
cntnevol 34194 Counting and Lebesgue meas...
voliune 34195 The Lebesgue measure funct...
volfiniune 34196 The Lebesgue measure funct...
volmeas 34197 The Lebesgue measure is a ...
ddeval1 34200 Value of the delta measure...
ddeval0 34201 Value of the delta measure...
ddemeas 34202 The Dirac delta measure is...
relae 34206 'almost everywhere' is a r...
brae 34207 'almost everywhere' relati...
braew 34208 'almost everywhere' relati...
truae 34209 A truth holds almost every...
aean 34210 A conjunction holds almost...
faeval 34212 Value of the 'almost every...
relfae 34213 The 'almost everywhere' bu...
brfae 34214 'almost everywhere' relati...
ismbfm 34217 The predicate " ` F ` is a...
elunirnmbfm 34218 The property of being a me...
mbfmfun 34219 A measurable function is a...
mbfmf 34220 A measurable function as a...
isanmbfmOLD 34221 Obsolete version of ~ isan...
mbfmcnvima 34222 The preimage by a measurab...
isanmbfm 34223 The predicate to be a meas...
mbfmbfmOLD 34224 A measurable function to a...
mbfmbfm 34225 A measurable function to a...
mbfmcst 34226 A constant function is mea...
1stmbfm 34227 The first projection map i...
2ndmbfm 34228 The second projection map ...
imambfm 34229 If the sigma-algebra in th...
cnmbfm 34230 A continuous function is m...
mbfmco 34231 The composition of two mea...
mbfmco2 34232 The pair building of two m...
mbfmvolf 34233 Measurable functions with ...
elmbfmvol2 34234 Measurable functions with ...
mbfmcnt 34235 All functions are measurab...
br2base 34236 The base set for the gener...
dya2ub 34237 An upper bound for a dyadi...
sxbrsigalem0 34238 The closed half-spaces of ...
sxbrsigalem3 34239 The sigma-algebra generate...
dya2iocival 34240 The function ` I ` returns...
dya2iocress 34241 Dyadic intervals are subse...
dya2iocbrsiga 34242 Dyadic intervals are Borel...
dya2icobrsiga 34243 Dyadic intervals are Borel...
dya2icoseg 34244 For any point and any clos...
dya2icoseg2 34245 For any point and any open...
dya2iocrfn 34246 The function returning dya...
dya2iocct 34247 The dyadic rectangle set i...
dya2iocnrect 34248 For any point of an open r...
dya2iocnei 34249 For any point of an open s...
dya2iocuni 34250 Every open set of ` ( RR X...
dya2iocucvr 34251 The dyadic rectangular set...
sxbrsigalem1 34252 The Borel algebra on ` ( R...
sxbrsigalem2 34253 The sigma-algebra generate...
sxbrsigalem4 34254 The Borel algebra on ` ( R...
sxbrsigalem5 34255 First direction for ~ sxbr...
sxbrsigalem6 34256 First direction for ~ sxbr...
sxbrsiga 34257 The product sigma-algebra ...
omsval 34260 Value of the function mapp...
omsfval 34261 Value of the outer measure...
omscl 34262 A closure lemma for the co...
omsf 34263 A constructed outer measur...
oms0 34264 A constructed outer measur...
omsmon 34265 A constructed outer measur...
omssubaddlem 34266 For any small margin ` E `...
omssubadd 34267 A constructed outer measur...
carsgval 34270 Value of the Caratheodory ...
carsgcl 34271 Closure of the Caratheodor...
elcarsg 34272 Property of being a Carath...
baselcarsg 34273 The universe set, ` O ` , ...
0elcarsg 34274 The empty set is Caratheod...
carsguni 34275 The union of all Caratheod...
elcarsgss 34276 Caratheodory measurable se...
difelcarsg 34277 The Caratheodory measurabl...
inelcarsg 34278 The Caratheodory measurabl...
unelcarsg 34279 The Caratheodory-measurabl...
difelcarsg2 34280 The Caratheodory-measurabl...
carsgmon 34281 Utility lemma: Apply mono...
carsgsigalem 34282 Lemma for the following th...
fiunelcarsg 34283 The Caratheodory measurabl...
carsgclctunlem1 34284 Lemma for ~ carsgclctun . ...
carsggect 34285 The outer measure is count...
carsgclctunlem2 34286 Lemma for ~ carsgclctun . ...
carsgclctunlem3 34287 Lemma for ~ carsgclctun . ...
carsgclctun 34288 The Caratheodory measurabl...
carsgsiga 34289 The Caratheodory measurabl...
omsmeas 34290 The restriction of a const...
pmeasmono 34291 This theorem's hypotheses ...
pmeasadd 34292 A premeasure on a ring of ...
itgeq12dv 34293 Equality theorem for an in...
sitgval 34299 Value of the simple functi...
issibf 34300 The predicate " ` F ` is a...
sibf0 34301 The constant zero function...
sibfmbl 34302 A simple function is measu...
sibff 34303 A simple function is a fun...
sibfrn 34304 A simple function has fini...
sibfima 34305 Any preimage of a singleto...
sibfinima 34306 The measure of the interse...
sibfof 34307 Applying function operatio...
sitgfval 34308 Value of the Bochner integ...
sitgclg 34309 Closure of the Bochner int...
sitgclbn 34310 Closure of the Bochner int...
sitgclcn 34311 Closure of the Bochner int...
sitgclre 34312 Closure of the Bochner int...
sitg0 34313 The integral of the consta...
sitgf 34314 The integral for simple fu...
sitgaddlemb 34315 Lemma for * sitgadd . (Co...
sitmval 34316 Value of the simple functi...
sitmfval 34317 Value of the integral dist...
sitmcl 34318 Closure of the integral di...
sitmf 34319 The integral metric as a f...
oddpwdc 34321 Lemma for ~ eulerpart . T...
oddpwdcv 34322 Lemma for ~ eulerpart : va...
eulerpartlemsv1 34323 Lemma for ~ eulerpart . V...
eulerpartlemelr 34324 Lemma for ~ eulerpart . (...
eulerpartlemsv2 34325 Lemma for ~ eulerpart . V...
eulerpartlemsf 34326 Lemma for ~ eulerpart . (...
eulerpartlems 34327 Lemma for ~ eulerpart . (...
eulerpartlemsv3 34328 Lemma for ~ eulerpart . V...
eulerpartlemgc 34329 Lemma for ~ eulerpart . (...
eulerpartleme 34330 Lemma for ~ eulerpart . (...
eulerpartlemv 34331 Lemma for ~ eulerpart . (...
eulerpartlemo 34332 Lemma for ~ eulerpart : ` ...
eulerpartlemd 34333 Lemma for ~ eulerpart : ` ...
eulerpartlem1 34334 Lemma for ~ eulerpart . (...
eulerpartlemb 34335 Lemma for ~ eulerpart . T...
eulerpartlemt0 34336 Lemma for ~ eulerpart . (...
eulerpartlemf 34337 Lemma for ~ eulerpart : O...
eulerpartlemt 34338 Lemma for ~ eulerpart . (...
eulerpartgbij 34339 Lemma for ~ eulerpart : T...
eulerpartlemgv 34340 Lemma for ~ eulerpart : va...
eulerpartlemr 34341 Lemma for ~ eulerpart . (...
eulerpartlemmf 34342 Lemma for ~ eulerpart . (...
eulerpartlemgvv 34343 Lemma for ~ eulerpart : va...
eulerpartlemgu 34344 Lemma for ~ eulerpart : R...
eulerpartlemgh 34345 Lemma for ~ eulerpart : T...
eulerpartlemgf 34346 Lemma for ~ eulerpart : I...
eulerpartlemgs2 34347 Lemma for ~ eulerpart : T...
eulerpartlemn 34348 Lemma for ~ eulerpart . (...
eulerpart 34349 Euler's theorem on partiti...
subiwrd 34352 Lemma for ~ sseqp1 . (Con...
subiwrdlen 34353 Length of a subword of an ...
iwrdsplit 34354 Lemma for ~ sseqp1 . (Con...
sseqval 34355 Value of the strong sequen...
sseqfv1 34356 Value of the strong sequen...
sseqfn 34357 A strong recursive sequenc...
sseqmw 34358 Lemma for ~ sseqf amd ~ ss...
sseqf 34359 A strong recursive sequenc...
sseqfres 34360 The first elements in the ...
sseqfv2 34361 Value of the strong sequen...
sseqp1 34362 Value of the strong sequen...
fiblem 34365 Lemma for ~ fib0 , ~ fib1 ...
fib0 34366 Value of the Fibonacci seq...
fib1 34367 Value of the Fibonacci seq...
fibp1 34368 Value of the Fibonacci seq...
fib2 34369 Value of the Fibonacci seq...
fib3 34370 Value of the Fibonacci seq...
fib4 34371 Value of the Fibonacci seq...
fib5 34372 Value of the Fibonacci seq...
fib6 34373 Value of the Fibonacci seq...
elprob 34376 The property of being a pr...
domprobmeas 34377 A probability measure is a...
domprobsiga 34378 The domain of a probabilit...
probtot 34379 The probability of the uni...
prob01 34380 A probability is an elemen...
probnul 34381 The probability of the emp...
unveldomd 34382 The universe is an element...
unveldom 34383 The universe is an element...
nuleldmp 34384 The empty set is an elemen...
probcun 34385 The probability of the uni...
probun 34386 The probability of the uni...
probdif 34387 The probability of the dif...
probinc 34388 A probability law is incre...
probdsb 34389 The probability of the com...
probmeasd 34390 A probability measure is a...
probvalrnd 34391 The value of a probability...
probtotrnd 34392 The probability of the uni...
totprobd 34393 Law of total probability, ...
totprob 34394 Law of total probability. ...
probfinmeasb 34395 Build a probability measur...
probfinmeasbALTV 34396 Alternate version of ~ pro...
probmeasb 34397 Build a probability from a...
cndprobval 34400 The value of the condition...
cndprobin 34401 An identity linking condit...
cndprob01 34402 The conditional probabilit...
cndprobtot 34403 The conditional probabilit...
cndprobnul 34404 The conditional probabilit...
cndprobprob 34405 The conditional probabilit...
bayesth 34406 Bayes Theorem. (Contribut...
rrvmbfm 34409 A real-valued random varia...
isrrvv 34410 Elementhood to the set of ...
rrvvf 34411 A real-valued random varia...
rrvfn 34412 A real-valued random varia...
rrvdm 34413 The domain of a random var...
rrvrnss 34414 The range of a random vari...
rrvf2 34415 A real-valued random varia...
rrvdmss 34416 The domain of a random var...
rrvfinvima 34417 For a real-value random va...
0rrv 34418 The constant function equa...
rrvadd 34419 The sum of two random vari...
rrvmulc 34420 A random variable multipli...
rrvsum 34421 An indexed sum of random v...
orvcval 34424 Value of the preimage mapp...
orvcval2 34425 Another way to express the...
elorvc 34426 Elementhood of a preimage....
orvcval4 34427 The value of the preimage ...
orvcoel 34428 If the relation produces o...
orvccel 34429 If the relation produces c...
elorrvc 34430 Elementhood of a preimage ...
orrvcval4 34431 The value of the preimage ...
orrvcoel 34432 If the relation produces o...
orrvccel 34433 If the relation produces c...
orvcgteel 34434 Preimage maps produced by ...
orvcelval 34435 Preimage maps produced by ...
orvcelel 34436 Preimage maps produced by ...
dstrvval 34437 The value of the distribut...
dstrvprob 34438 The distribution of a rand...
orvclteel 34439 Preimage maps produced by ...
dstfrvel 34440 Elementhood of preimage ma...
dstfrvunirn 34441 The limit of all preimage ...
orvclteinc 34442 Preimage maps produced by ...
dstfrvinc 34443 A cumulative distribution ...
dstfrvclim1 34444 The limit of the cumulativ...
coinfliplem 34445 Division in the extended r...
coinflipprob 34446 The ` P ` we defined for c...
coinflipspace 34447 The space of our coin-flip...
coinflipuniv 34448 The universe of our coin-f...
coinfliprv 34449 The ` X ` we defined for c...
coinflippv 34450 The probability of heads i...
coinflippvt 34451 The probability of tails i...
ballotlemoex 34452 ` O ` is a set. (Contribu...
ballotlem1 34453 The size of the universe i...
ballotlemelo 34454 Elementhood in ` O ` . (C...
ballotlem2 34455 The probability that the f...
ballotlemfval 34456 The value of ` F ` . (Con...
ballotlemfelz 34457 ` ( F `` C ) ` has values ...
ballotlemfp1 34458 If the ` J ` th ballot is ...
ballotlemfc0 34459 ` F ` takes value 0 betwee...
ballotlemfcc 34460 ` F ` takes value 0 betwee...
ballotlemfmpn 34461 ` ( F `` C ) ` finishes co...
ballotlemfval0 34462 ` ( F `` C ) ` always star...
ballotleme 34463 Elements of ` E ` . (Cont...
ballotlemodife 34464 Elements of ` ( O \ E ) ` ...
ballotlem4 34465 If the first pick is a vot...
ballotlem5 34466 If A is not ahead througho...
ballotlemi 34467 Value of ` I ` for a given...
ballotlemiex 34468 Properties of ` ( I `` C )...
ballotlemi1 34469 The first tie cannot be re...
ballotlemii 34470 The first tie cannot be re...
ballotlemsup 34471 The set of zeroes of ` F `...
ballotlemimin 34472 ` ( I `` C ) ` is the firs...
ballotlemic 34473 If the first vote is for B...
ballotlem1c 34474 If the first vote is for A...
ballotlemsval 34475 Value of ` S ` . (Contrib...
ballotlemsv 34476 Value of ` S ` evaluated a...
ballotlemsgt1 34477 ` S ` maps values less tha...
ballotlemsdom 34478 Domain of ` S ` for a give...
ballotlemsel1i 34479 The range ` ( 1 ... ( I ``...
ballotlemsf1o 34480 The defined ` S ` is a bij...
ballotlemsi 34481 The image by ` S ` of the ...
ballotlemsima 34482 The image by ` S ` of an i...
ballotlemieq 34483 If two countings share the...
ballotlemrval 34484 Value of ` R ` . (Contrib...
ballotlemscr 34485 The image of ` ( R `` C ) ...
ballotlemrv 34486 Value of ` R ` evaluated a...
ballotlemrv1 34487 Value of ` R ` before the ...
ballotlemrv2 34488 Value of ` R ` after the t...
ballotlemro 34489 Range of ` R ` is included...
ballotlemgval 34490 Expand the value of ` .^ `...
ballotlemgun 34491 A property of the defined ...
ballotlemfg 34492 Express the value of ` ( F...
ballotlemfrc 34493 Express the value of ` ( F...
ballotlemfrci 34494 Reverse counting preserves...
ballotlemfrceq 34495 Value of ` F ` for a rever...
ballotlemfrcn0 34496 Value of ` F ` for a rever...
ballotlemrc 34497 Range of ` R ` . (Contrib...
ballotlemirc 34498 Applying ` R ` does not ch...
ballotlemrinv0 34499 Lemma for ~ ballotlemrinv ...
ballotlemrinv 34500 ` R ` is its own inverse :...
ballotlem1ri 34501 When the vote on the first...
ballotlem7 34502 ` R ` is a bijection betwe...
ballotlem8 34503 There are as many counting...
ballotth 34504 Bertrand's ballot problem ...
sgncl 34505 Closure of the signum. (C...
sgnclre 34506 Closure of the signum. (C...
sgnneg 34507 Negation of the signum. (...
sgn3da 34508 A conditional containing a...
sgnmul 34509 Signum of a product. (Con...
sgnmulrp2 34510 Multiplication by a positi...
sgnsub 34511 Subtraction of a number of...
sgnnbi 34512 Negative signum. (Contrib...
sgnpbi 34513 Positive signum. (Contrib...
sgn0bi 34514 Zero signum. (Contributed...
sgnsgn 34515 Signum is idempotent. (Co...
sgnmulsgn 34516 If two real numbers are of...
sgnmulsgp 34517 If two real numbers are of...
fzssfzo 34518 Condition for an integer i...
gsumncl 34519 Closure of a group sum in ...
gsumnunsn 34520 Closure of a group sum in ...
ccatmulgnn0dir 34521 Concatenation of words fol...
ofcccat 34522 Letterwise operations on w...
ofcs1 34523 Letterwise operations on a...
ofcs2 34524 Letterwise operations on a...
plymul02 34525 Product of a polynomial wi...
plymulx0 34526 Coefficients of a polynomi...
plymulx 34527 Coefficients of a polynomi...
plyrecld 34528 Closure of a polynomial wi...
signsplypnf 34529 The quotient of a polynomi...
signsply0 34530 Lemma for the rule of sign...
signspval 34531 The value of the skipping ...
signsw0glem 34532 Neutral element property o...
signswbase 34533 The base of ` W ` is the u...
signswplusg 34534 The operation of ` W ` . ...
signsw0g 34535 The neutral element of ` W...
signswmnd 34536 ` W ` is a monoid structur...
signswrid 34537 The zero-skipping operatio...
signswlid 34538 The zero-skipping operatio...
signswn0 34539 The zero-skipping operatio...
signswch 34540 The zero-skipping operatio...
signslema 34541 Computational part of ~~? ...
signstfv 34542 Value of the zero-skipping...
signstfval 34543 Value of the zero-skipping...
signstcl 34544 Closure of the zero skippi...
signstf 34545 The zero skipping sign wor...
signstlen 34546 Length of the zero skippin...
signstf0 34547 Sign of a single letter wo...
signstfvn 34548 Zero-skipping sign in a wo...
signsvtn0 34549 If the last letter is nonz...
signstfvp 34550 Zero-skipping sign in a wo...
signstfvneq0 34551 In case the first letter i...
signstfvcl 34552 Closure of the zero skippi...
signstfvc 34553 Zero-skipping sign in a wo...
signstres 34554 Restriction of a zero skip...
signstfveq0a 34555 Lemma for ~ signstfveq0 . ...
signstfveq0 34556 In case the last letter is...
signsvvfval 34557 The value of ` V ` , which...
signsvvf 34558 ` V ` is a function. (Con...
signsvf0 34559 There is no change of sign...
signsvf1 34560 In a single-letter word, w...
signsvfn 34561 Number of changes in a wor...
signsvtp 34562 Adding a letter of the sam...
signsvtn 34563 Adding a letter of a diffe...
signsvfpn 34564 Adding a letter of the sam...
signsvfnn 34565 Adding a letter of a diffe...
signlem0 34566 Adding a zero as the highe...
signshf 34567 ` H ` , corresponding to t...
signshwrd 34568 ` H ` , corresponding to t...
signshlen 34569 Length of ` H ` , correspo...
signshnz 34570 ` H ` is not the empty wor...
iblidicc 34571 The identity function is i...
rpsqrtcn 34572 Continuity of the real pos...
divsqrtid 34573 A real number divided by i...
cxpcncf1 34574 The power function on comp...
efmul2picn 34575 Multiplying by ` ( _i x. (...
fct2relem 34576 Lemma for ~ ftc2re . (Con...
ftc2re 34577 The Fundamental Theorem of...
fdvposlt 34578 Functions with a positive ...
fdvneggt 34579 Functions with a negative ...
fdvposle 34580 Functions with a nonnegati...
fdvnegge 34581 Functions with a nonpositi...
prodfzo03 34582 A product of three factors...
actfunsnf1o 34583 The action ` F ` of extend...
actfunsnrndisj 34584 The action ` F ` of extend...
itgexpif 34585 The basis for the circle m...
fsum2dsub 34586 Lemma for ~ breprexp - Re-...
reprval 34589 Value of the representatio...
repr0 34590 There is exactly one repre...
reprf 34591 Members of the representat...
reprsum 34592 Sums of values of the memb...
reprle 34593 Upper bound to the terms i...
reprsuc 34594 Express the representation...
reprfi 34595 Bounded representations ar...
reprss 34596 Representations with terms...
reprinrn 34597 Representations with term ...
reprlt 34598 There are no representatio...
hashreprin 34599 Express a sum of represent...
reprgt 34600 There are no representatio...
reprinfz1 34601 For the representation of ...
reprfi2 34602 Corollary of ~ reprinfz1 ....
reprfz1 34603 Corollary of ~ reprinfz1 ....
hashrepr 34604 Develop the number of repr...
reprpmtf1o 34605 Transposing ` 0 ` and ` X ...
reprdifc 34606 Express the representation...
chpvalz 34607 Value of the second Chebys...
chtvalz 34608 Value of the Chebyshev fun...
breprexplema 34609 Lemma for ~ breprexp (indu...
breprexplemb 34610 Lemma for ~ breprexp (clos...
breprexplemc 34611 Lemma for ~ breprexp (indu...
breprexp 34612 Express the ` S ` th power...
breprexpnat 34613 Express the ` S ` th power...
vtsval 34616 Value of the Vinogradov tr...
vtscl 34617 Closure of the Vinogradov ...
vtsprod 34618 Express the Vinogradov tri...
circlemeth 34619 The Hardy, Littlewood and ...
circlemethnat 34620 The Hardy, Littlewood and ...
circlevma 34621 The Circle Method, where t...
circlemethhgt 34622 The circle method, where t...
hgt750lemc 34626 An upper bound to the summ...
hgt750lemd 34627 An upper bound to the summ...
hgt749d 34628 A deduction version of ~ a...
logdivsqrle 34629 Conditions for ` ( ( log `...
hgt750lem 34630 Lemma for ~ tgoldbachgtd ....
hgt750lem2 34631 Decimal multiplication gal...
hgt750lemf 34632 Lemma for the statement 7....
hgt750lemg 34633 Lemma for the statement 7....
oddprm2 34634 Two ways to write the set ...
hgt750lemb 34635 An upper bound on the cont...
hgt750lema 34636 An upper bound on the cont...
hgt750leme 34637 An upper bound on the cont...
tgoldbachgnn 34638 Lemma for ~ tgoldbachgtd ....
tgoldbachgtde 34639 Lemma for ~ tgoldbachgtd ....
tgoldbachgtda 34640 Lemma for ~ tgoldbachgtd ....
tgoldbachgtd 34641 Odd integers greater than ...
tgoldbachgt 34642 Odd integers greater than ...
istrkg2d 34645 Property of fulfilling dim...
axtglowdim2ALTV 34646 Alternate version of ~ axt...
axtgupdim2ALTV 34647 Alternate version of ~ axt...
afsval 34650 Value of the AFS relation ...
brafs 34651 Binary relation form of th...
tg5segofs 34652 Rephrase ~ axtg5seg using ...
lpadval 34655 Value of the ` leftpad ` f...
lpadlem1 34656 Lemma for the ` leftpad ` ...
lpadlem3 34657 Lemma for ~ lpadlen1 . (C...
lpadlen1 34658 Length of a left-padded wo...
lpadlem2 34659 Lemma for the ` leftpad ` ...
lpadlen2 34660 Length of a left-padded wo...
lpadmax 34661 Length of a left-padded wo...
lpadleft 34662 The contents of prefix of ...
lpadright 34663 The suffix of a left-padde...
bnj170 34676 ` /\ ` -manipulation. (Co...
bnj240 34677 ` /\ ` -manipulation. (Co...
bnj248 34678 ` /\ ` -manipulation. (Co...
bnj250 34679 ` /\ ` -manipulation. (Co...
bnj251 34680 ` /\ ` -manipulation. (Co...
bnj252 34681 ` /\ ` -manipulation. (Co...
bnj253 34682 ` /\ ` -manipulation. (Co...
bnj255 34683 ` /\ ` -manipulation. (Co...
bnj256 34684 ` /\ ` -manipulation. (Co...
bnj257 34685 ` /\ ` -manipulation. (Co...
bnj258 34686 ` /\ ` -manipulation. (Co...
bnj268 34687 ` /\ ` -manipulation. (Co...
bnj290 34688 ` /\ ` -manipulation. (Co...
bnj291 34689 ` /\ ` -manipulation. (Co...
bnj312 34690 ` /\ ` -manipulation. (Co...
bnj334 34691 ` /\ ` -manipulation. (Co...
bnj345 34692 ` /\ ` -manipulation. (Co...
bnj422 34693 ` /\ ` -manipulation. (Co...
bnj432 34694 ` /\ ` -manipulation. (Co...
bnj446 34695 ` /\ ` -manipulation. (Co...
bnj23 34696 First-order logic and set ...
bnj31 34697 First-order logic and set ...
bnj62 34698 First-order logic and set ...
bnj89 34699 First-order logic and set ...
bnj90 34700 First-order logic and set ...
bnj101 34701 First-order logic and set ...
bnj105 34702 First-order logic and set ...
bnj115 34703 First-order logic and set ...
bnj132 34704 First-order logic and set ...
bnj133 34705 First-order logic and set ...
bnj156 34706 First-order logic and set ...
bnj158 34707 First-order logic and set ...
bnj168 34708 First-order logic and set ...
bnj206 34709 First-order logic and set ...
bnj216 34710 First-order logic and set ...
bnj219 34711 First-order logic and set ...
bnj226 34712 First-order logic and set ...
bnj228 34713 First-order logic and set ...
bnj519 34714 First-order logic and set ...
bnj524 34715 First-order logic and set ...
bnj525 34716 First-order logic and set ...
bnj534 34717 First-order logic and set ...
bnj538 34718 First-order logic and set ...
bnj529 34719 First-order logic and set ...
bnj551 34720 First-order logic and set ...
bnj563 34721 First-order logic and set ...
bnj564 34722 First-order logic and set ...
bnj593 34723 First-order logic and set ...
bnj596 34724 First-order logic and set ...
bnj610 34725 Pass from equality ( ` x =...
bnj642 34726 ` /\ ` -manipulation. (Co...
bnj643 34727 ` /\ ` -manipulation. (Co...
bnj645 34728 ` /\ ` -manipulation. (Co...
bnj658 34729 ` /\ ` -manipulation. (Co...
bnj667 34730 ` /\ ` -manipulation. (Co...
bnj705 34731 ` /\ ` -manipulation. (Co...
bnj706 34732 ` /\ ` -manipulation. (Co...
bnj707 34733 ` /\ ` -manipulation. (Co...
bnj708 34734 ` /\ ` -manipulation. (Co...
bnj721 34735 ` /\ ` -manipulation. (Co...
bnj832 34736 ` /\ ` -manipulation. (Co...
bnj835 34737 ` /\ ` -manipulation. (Co...
bnj836 34738 ` /\ ` -manipulation. (Co...
bnj837 34739 ` /\ ` -manipulation. (Co...
bnj769 34740 ` /\ ` -manipulation. (Co...
bnj770 34741 ` /\ ` -manipulation. (Co...
bnj771 34742 ` /\ ` -manipulation. (Co...
bnj887 34743 ` /\ ` -manipulation. (Co...
bnj918 34744 First-order logic and set ...
bnj919 34745 First-order logic and set ...
bnj923 34746 First-order logic and set ...
bnj927 34747 First-order logic and set ...
bnj931 34748 First-order logic and set ...
bnj937 34749 First-order logic and set ...
bnj941 34750 First-order logic and set ...
bnj945 34751 Technical lemma for ~ bnj6...
bnj946 34752 First-order logic and set ...
bnj951 34753 ` /\ ` -manipulation. (Co...
bnj956 34754 First-order logic and set ...
bnj976 34755 First-order logic and set ...
bnj982 34756 First-order logic and set ...
bnj1019 34757 First-order logic and set ...
bnj1023 34758 First-order logic and set ...
bnj1095 34759 First-order logic and set ...
bnj1096 34760 First-order logic and set ...
bnj1098 34761 First-order logic and set ...
bnj1101 34762 First-order logic and set ...
bnj1113 34763 First-order logic and set ...
bnj1109 34764 First-order logic and set ...
bnj1131 34765 First-order logic and set ...
bnj1138 34766 First-order logic and set ...
bnj1142 34767 First-order logic and set ...
bnj1143 34768 First-order logic and set ...
bnj1146 34769 First-order logic and set ...
bnj1149 34770 First-order logic and set ...
bnj1185 34771 First-order logic and set ...
bnj1196 34772 First-order logic and set ...
bnj1198 34773 First-order logic and set ...
bnj1209 34774 First-order logic and set ...
bnj1211 34775 First-order logic and set ...
bnj1213 34776 First-order logic and set ...
bnj1212 34777 First-order logic and set ...
bnj1219 34778 First-order logic and set ...
bnj1224 34779 First-order logic and set ...
bnj1230 34780 First-order logic and set ...
bnj1232 34781 First-order logic and set ...
bnj1235 34782 First-order logic and set ...
bnj1239 34783 First-order logic and set ...
bnj1238 34784 First-order logic and set ...
bnj1241 34785 First-order logic and set ...
bnj1247 34786 First-order logic and set ...
bnj1254 34787 First-order logic and set ...
bnj1262 34788 First-order logic and set ...
bnj1266 34789 First-order logic and set ...
bnj1265 34790 First-order logic and set ...
bnj1275 34791 First-order logic and set ...
bnj1276 34792 First-order logic and set ...
bnj1292 34793 First-order logic and set ...
bnj1293 34794 First-order logic and set ...
bnj1294 34795 First-order logic and set ...
bnj1299 34796 First-order logic and set ...
bnj1304 34797 First-order logic and set ...
bnj1316 34798 First-order logic and set ...
bnj1317 34799 First-order logic and set ...
bnj1322 34800 First-order logic and set ...
bnj1340 34801 First-order logic and set ...
bnj1345 34802 First-order logic and set ...
bnj1350 34803 First-order logic and set ...
bnj1351 34804 First-order logic and set ...
bnj1352 34805 First-order logic and set ...
bnj1361 34806 First-order logic and set ...
bnj1366 34807 First-order logic and set ...
bnj1379 34808 First-order logic and set ...
bnj1383 34809 First-order logic and set ...
bnj1385 34810 First-order logic and set ...
bnj1386 34811 First-order logic and set ...
bnj1397 34812 First-order logic and set ...
bnj1400 34813 First-order logic and set ...
bnj1405 34814 First-order logic and set ...
bnj1422 34815 First-order logic and set ...
bnj1424 34816 First-order logic and set ...
bnj1436 34817 First-order logic and set ...
bnj1441 34818 First-order logic and set ...
bnj1441g 34819 First-order logic and set ...
bnj1454 34820 First-order logic and set ...
bnj1459 34821 First-order logic and set ...
bnj1464 34822 Conversion of implicit sub...
bnj1465 34823 First-order logic and set ...
bnj1468 34824 Conversion of implicit sub...
bnj1476 34825 First-order logic and set ...
bnj1502 34826 First-order logic and set ...
bnj1503 34827 First-order logic and set ...
bnj1517 34828 First-order logic and set ...
bnj1521 34829 First-order logic and set ...
bnj1533 34830 First-order logic and set ...
bnj1534 34831 First-order logic and set ...
bnj1536 34832 First-order logic and set ...
bnj1538 34833 First-order logic and set ...
bnj1541 34834 First-order logic and set ...
bnj1542 34835 First-order logic and set ...
bnj110 34836 Well-founded induction res...
bnj157 34837 Well-founded induction res...
bnj66 34838 Technical lemma for ~ bnj6...
bnj91 34839 First-order logic and set ...
bnj92 34840 First-order logic and set ...
bnj93 34841 Technical lemma for ~ bnj9...
bnj95 34842 Technical lemma for ~ bnj1...
bnj96 34843 Technical lemma for ~ bnj1...
bnj97 34844 Technical lemma for ~ bnj1...
bnj98 34845 Technical lemma for ~ bnj1...
bnj106 34846 First-order logic and set ...
bnj118 34847 First-order logic and set ...
bnj121 34848 First-order logic and set ...
bnj124 34849 Technical lemma for ~ bnj1...
bnj125 34850 Technical lemma for ~ bnj1...
bnj126 34851 Technical lemma for ~ bnj1...
bnj130 34852 Technical lemma for ~ bnj1...
bnj149 34853 Technical lemma for ~ bnj1...
bnj150 34854 Technical lemma for ~ bnj1...
bnj151 34855 Technical lemma for ~ bnj1...
bnj154 34856 Technical lemma for ~ bnj1...
bnj155 34857 Technical lemma for ~ bnj1...
bnj153 34858 Technical lemma for ~ bnj8...
bnj207 34859 Technical lemma for ~ bnj8...
bnj213 34860 First-order logic and set ...
bnj222 34861 Technical lemma for ~ bnj2...
bnj229 34862 Technical lemma for ~ bnj5...
bnj517 34863 Technical lemma for ~ bnj5...
bnj518 34864 Technical lemma for ~ bnj8...
bnj523 34865 Technical lemma for ~ bnj8...
bnj526 34866 Technical lemma for ~ bnj8...
bnj528 34867 Technical lemma for ~ bnj8...
bnj535 34868 Technical lemma for ~ bnj8...
bnj539 34869 Technical lemma for ~ bnj8...
bnj540 34870 Technical lemma for ~ bnj8...
bnj543 34871 Technical lemma for ~ bnj8...
bnj544 34872 Technical lemma for ~ bnj8...
bnj545 34873 Technical lemma for ~ bnj8...
bnj546 34874 Technical lemma for ~ bnj8...
bnj548 34875 Technical lemma for ~ bnj8...
bnj553 34876 Technical lemma for ~ bnj8...
bnj554 34877 Technical lemma for ~ bnj8...
bnj556 34878 Technical lemma for ~ bnj8...
bnj557 34879 Technical lemma for ~ bnj8...
bnj558 34880 Technical lemma for ~ bnj8...
bnj561 34881 Technical lemma for ~ bnj8...
bnj562 34882 Technical lemma for ~ bnj8...
bnj570 34883 Technical lemma for ~ bnj8...
bnj571 34884 Technical lemma for ~ bnj8...
bnj605 34885 Technical lemma. This lem...
bnj581 34886 Technical lemma for ~ bnj5...
bnj589 34887 Technical lemma for ~ bnj8...
bnj590 34888 Technical lemma for ~ bnj8...
bnj591 34889 Technical lemma for ~ bnj8...
bnj594 34890 Technical lemma for ~ bnj8...
bnj580 34891 Technical lemma for ~ bnj5...
bnj579 34892 Technical lemma for ~ bnj8...
bnj602 34893 Equality theorem for the `...
bnj607 34894 Technical lemma for ~ bnj8...
bnj609 34895 Technical lemma for ~ bnj8...
bnj611 34896 Technical lemma for ~ bnj8...
bnj600 34897 Technical lemma for ~ bnj8...
bnj601 34898 Technical lemma for ~ bnj8...
bnj852 34899 Technical lemma for ~ bnj6...
bnj864 34900 Technical lemma for ~ bnj6...
bnj865 34901 Technical lemma for ~ bnj6...
bnj873 34902 Technical lemma for ~ bnj6...
bnj849 34903 Technical lemma for ~ bnj6...
bnj882 34904 Definition (using hypothes...
bnj18eq1 34905 Equality theorem for trans...
bnj893 34906 Property of ` _trCl ` . U...
bnj900 34907 Technical lemma for ~ bnj6...
bnj906 34908 Property of ` _trCl ` . (...
bnj908 34909 Technical lemma for ~ bnj6...
bnj911 34910 Technical lemma for ~ bnj6...
bnj916 34911 Technical lemma for ~ bnj6...
bnj917 34912 Technical lemma for ~ bnj6...
bnj934 34913 Technical lemma for ~ bnj6...
bnj929 34914 Technical lemma for ~ bnj6...
bnj938 34915 Technical lemma for ~ bnj6...
bnj944 34916 Technical lemma for ~ bnj6...
bnj953 34917 Technical lemma for ~ bnj6...
bnj958 34918 Technical lemma for ~ bnj6...
bnj1000 34919 Technical lemma for ~ bnj8...
bnj965 34920 Technical lemma for ~ bnj8...
bnj964 34921 Technical lemma for ~ bnj6...
bnj966 34922 Technical lemma for ~ bnj6...
bnj967 34923 Technical lemma for ~ bnj6...
bnj969 34924 Technical lemma for ~ bnj6...
bnj970 34925 Technical lemma for ~ bnj6...
bnj910 34926 Technical lemma for ~ bnj6...
bnj978 34927 Technical lemma for ~ bnj6...
bnj981 34928 Technical lemma for ~ bnj6...
bnj983 34929 Technical lemma for ~ bnj6...
bnj984 34930 Technical lemma for ~ bnj6...
bnj985v 34931 Version of ~ bnj985 with a...
bnj985 34932 Technical lemma for ~ bnj6...
bnj986 34933 Technical lemma for ~ bnj6...
bnj996 34934 Technical lemma for ~ bnj6...
bnj998 34935 Technical lemma for ~ bnj6...
bnj999 34936 Technical lemma for ~ bnj6...
bnj1001 34937 Technical lemma for ~ bnj6...
bnj1006 34938 Technical lemma for ~ bnj6...
bnj1014 34939 Technical lemma for ~ bnj6...
bnj1015 34940 Technical lemma for ~ bnj6...
bnj1018g 34941 Version of ~ bnj1018 with ...
bnj1018 34942 Technical lemma for ~ bnj6...
bnj1020 34943 Technical lemma for ~ bnj6...
bnj1021 34944 Technical lemma for ~ bnj6...
bnj907 34945 Technical lemma for ~ bnj6...
bnj1029 34946 Property of ` _trCl ` . (...
bnj1033 34947 Technical lemma for ~ bnj6...
bnj1034 34948 Technical lemma for ~ bnj6...
bnj1039 34949 Technical lemma for ~ bnj6...
bnj1040 34950 Technical lemma for ~ bnj6...
bnj1047 34951 Technical lemma for ~ bnj6...
bnj1049 34952 Technical lemma for ~ bnj6...
bnj1052 34953 Technical lemma for ~ bnj6...
bnj1053 34954 Technical lemma for ~ bnj6...
bnj1071 34955 Technical lemma for ~ bnj6...
bnj1083 34956 Technical lemma for ~ bnj6...
bnj1090 34957 Technical lemma for ~ bnj6...
bnj1093 34958 Technical lemma for ~ bnj6...
bnj1097 34959 Technical lemma for ~ bnj6...
bnj1110 34960 Technical lemma for ~ bnj6...
bnj1112 34961 Technical lemma for ~ bnj6...
bnj1118 34962 Technical lemma for ~ bnj6...
bnj1121 34963 Technical lemma for ~ bnj6...
bnj1123 34964 Technical lemma for ~ bnj6...
bnj1030 34965 Technical lemma for ~ bnj6...
bnj1124 34966 Property of ` _trCl ` . (...
bnj1133 34967 Technical lemma for ~ bnj6...
bnj1128 34968 Technical lemma for ~ bnj6...
bnj1127 34969 Property of ` _trCl ` . (...
bnj1125 34970 Property of ` _trCl ` . (...
bnj1145 34971 Technical lemma for ~ bnj6...
bnj1147 34972 Property of ` _trCl ` . (...
bnj1137 34973 Property of ` _trCl ` . (...
bnj1148 34974 Property of ` _pred ` . (...
bnj1136 34975 Technical lemma for ~ bnj6...
bnj1152 34976 Technical lemma for ~ bnj6...
bnj1154 34977 Property of ` Fr ` . (Con...
bnj1171 34978 Technical lemma for ~ bnj6...
bnj1172 34979 Technical lemma for ~ bnj6...
bnj1173 34980 Technical lemma for ~ bnj6...
bnj1174 34981 Technical lemma for ~ bnj6...
bnj1175 34982 Technical lemma for ~ bnj6...
bnj1176 34983 Technical lemma for ~ bnj6...
bnj1177 34984 Technical lemma for ~ bnj6...
bnj1186 34985 Technical lemma for ~ bnj6...
bnj1190 34986 Technical lemma for ~ bnj6...
bnj1189 34987 Technical lemma for ~ bnj6...
bnj69 34988 Existence of a minimal ele...
bnj1228 34989 Existence of a minimal ele...
bnj1204 34990 Well-founded induction. T...
bnj1234 34991 Technical lemma for ~ bnj6...
bnj1245 34992 Technical lemma for ~ bnj6...
bnj1256 34993 Technical lemma for ~ bnj6...
bnj1259 34994 Technical lemma for ~ bnj6...
bnj1253 34995 Technical lemma for ~ bnj6...
bnj1279 34996 Technical lemma for ~ bnj6...
bnj1286 34997 Technical lemma for ~ bnj6...
bnj1280 34998 Technical lemma for ~ bnj6...
bnj1296 34999 Technical lemma for ~ bnj6...
bnj1309 35000 Technical lemma for ~ bnj6...
bnj1307 35001 Technical lemma for ~ bnj6...
bnj1311 35002 Technical lemma for ~ bnj6...
bnj1318 35003 Technical lemma for ~ bnj6...
bnj1326 35004 Technical lemma for ~ bnj6...
bnj1321 35005 Technical lemma for ~ bnj6...
bnj1364 35006 Property of ` _FrSe ` . (...
bnj1371 35007 Technical lemma for ~ bnj6...
bnj1373 35008 Technical lemma for ~ bnj6...
bnj1374 35009 Technical lemma for ~ bnj6...
bnj1384 35010 Technical lemma for ~ bnj6...
bnj1388 35011 Technical lemma for ~ bnj6...
bnj1398 35012 Technical lemma for ~ bnj6...
bnj1413 35013 Property of ` _trCl ` . (...
bnj1408 35014 Technical lemma for ~ bnj1...
bnj1414 35015 Property of ` _trCl ` . (...
bnj1415 35016 Technical lemma for ~ bnj6...
bnj1416 35017 Technical lemma for ~ bnj6...
bnj1418 35018 Property of ` _pred ` . (...
bnj1417 35019 Technical lemma for ~ bnj6...
bnj1421 35020 Technical lemma for ~ bnj6...
bnj1444 35021 Technical lemma for ~ bnj6...
bnj1445 35022 Technical lemma for ~ bnj6...
bnj1446 35023 Technical lemma for ~ bnj6...
bnj1447 35024 Technical lemma for ~ bnj6...
bnj1448 35025 Technical lemma for ~ bnj6...
bnj1449 35026 Technical lemma for ~ bnj6...
bnj1442 35027 Technical lemma for ~ bnj6...
bnj1450 35028 Technical lemma for ~ bnj6...
bnj1423 35029 Technical lemma for ~ bnj6...
bnj1452 35030 Technical lemma for ~ bnj6...
bnj1466 35031 Technical lemma for ~ bnj6...
bnj1467 35032 Technical lemma for ~ bnj6...
bnj1463 35033 Technical lemma for ~ bnj6...
bnj1489 35034 Technical lemma for ~ bnj6...
bnj1491 35035 Technical lemma for ~ bnj6...
bnj1312 35036 Technical lemma for ~ bnj6...
bnj1493 35037 Technical lemma for ~ bnj6...
bnj1497 35038 Technical lemma for ~ bnj6...
bnj1498 35039 Technical lemma for ~ bnj6...
bnj60 35040 Well-founded recursion, pa...
bnj1514 35041 Technical lemma for ~ bnj1...
bnj1518 35042 Technical lemma for ~ bnj1...
bnj1519 35043 Technical lemma for ~ bnj1...
bnj1520 35044 Technical lemma for ~ bnj1...
bnj1501 35045 Technical lemma for ~ bnj1...
bnj1500 35046 Well-founded recursion, pa...
bnj1525 35047 Technical lemma for ~ bnj1...
bnj1529 35048 Technical lemma for ~ bnj1...
bnj1523 35049 Technical lemma for ~ bnj1...
bnj1522 35050 Well-founded recursion, pa...
nfan1c 35051 Variant of ~ nfan and comm...
cbvex1v 35052 Rule used to change bound ...
dvelimalcased 35053 Eliminate a disjoint varia...
dvelimalcasei 35054 Eliminate a disjoint varia...
dvelimexcased 35055 Eliminate a disjoint varia...
dvelimexcasei 35056 Eliminate a disjoint varia...
exdifsn 35057 There exists an element in...
srcmpltd 35058 If a statement is true for...
prsrcmpltd 35059 If a statement is true for...
axsepg2 35060 A generalization of ~ ax-s...
axsepg2ALT 35061 Alternate proof of ~ axsep...
dff15 35062 A one-to-one function in t...
f1resveqaeq 35063 If a function restricted t...
f1resrcmplf1dlem 35064 Lemma for ~ f1resrcmplf1d ...
f1resrcmplf1d 35065 If a function's restrictio...
funen1cnv 35066 If a function is equinumer...
fnrelpredd 35067 A function that preserves ...
cardpred 35068 The cardinality function p...
nummin 35069 Every nonempty class of nu...
axnulg 35070 A generalization of ~ ax-n...
axnulALT2 35071 Alternate proof of ~ axnul...
prcinf 35072 Any proper class is litera...
fineqvrep 35073 If the Axiom of Infinity i...
fineqvpow 35074 If the Axiom of Infinity i...
fineqvac 35075 If the Axiom of Infinity i...
fineqvacALT 35076 Shorter proof of ~ fineqva...
gblacfnacd 35077 If ` F ` is a global choic...
wevgblacfn 35078 If ` R ` is a well-orderin...
zltp1ne 35079 Integer ordering relation....
nnltp1ne 35080 Positive integer ordering ...
nn0ltp1ne 35081 Nonnegative integer orderi...
0nn0m1nnn0 35082 A number is zero if and on...
f1resfz0f1d 35083 If a function with a seque...
fisshasheq 35084 A finite set is equal to i...
revpfxsfxrev 35085 The reverse of a prefix of...
swrdrevpfx 35086 A subword expressed in ter...
lfuhgr 35087 A hypergraph is loop-free ...
lfuhgr2 35088 A hypergraph is loop-free ...
lfuhgr3 35089 A hypergraph is loop-free ...
cplgredgex 35090 Any two (distinct) vertice...
cusgredgex 35091 Any two (distinct) vertice...
cusgredgex2 35092 Any two distinct vertices ...
pfxwlk 35093 A prefix of a walk is a wa...
revwlk 35094 The reverse of a walk is a...
revwlkb 35095 Two words represent a walk...
swrdwlk 35096 Two matching subwords of a...
pthhashvtx 35097 A graph containing a path ...
pthisspthorcycl 35098 A path is either a simple ...
spthcycl 35099 A walk is a trivial path i...
usgrgt2cycl 35100 A non-trivial cycle in a s...
usgrcyclgt2v 35101 A simple graph with a non-...
subgrwlk 35102 If a walk exists in a subg...
subgrtrl 35103 If a trail exists in a sub...
subgrpth 35104 If a path exists in a subg...
subgrcycl 35105 If a cycle exists in a sub...
cusgr3cyclex 35106 Every complete simple grap...
loop1cycl 35107 A hypergraph has a cycle o...
2cycld 35108 Construction of a 2-cycle ...
2cycl2d 35109 Construction of a 2-cycle ...
umgr2cycllem 35110 Lemma for ~ umgr2cycl . (...
umgr2cycl 35111 A multigraph with two dist...
dfacycgr1 35114 An alternate definition of...
isacycgr 35115 The property of being an a...
isacycgr1 35116 The property of being an a...
acycgrcycl 35117 Any cycle in an acyclic gr...
acycgr0v 35118 A null graph (with no vert...
acycgr1v 35119 A multigraph with one vert...
acycgr2v 35120 A simple graph with two ve...
prclisacycgr 35121 A proper class (representi...
acycgrislfgr 35122 An acyclic hypergraph is a...
upgracycumgr 35123 An acyclic pseudograph is ...
umgracycusgr 35124 An acyclic multigraph is a...
upgracycusgr 35125 An acyclic pseudograph is ...
cusgracyclt3v 35126 A complete simple graph is...
pthacycspth 35127 A path in an acyclic graph...
acycgrsubgr 35128 The subgraph of an acyclic...
quartfull 35135 The quartic equation, writ...
deranglem 35136 Lemma for derangements. (...
derangval 35137 Define the derangement fun...
derangf 35138 The derangement number is ...
derang0 35139 The derangement number of ...
derangsn 35140 The derangement number of ...
derangenlem 35141 One half of ~ derangen . ...
derangen 35142 The derangement number is ...
subfacval 35143 The subfactorial is define...
derangen2 35144 Write the derangement numb...
subfacf 35145 The subfactorial is a func...
subfaclefac 35146 The subfactorial is less t...
subfac0 35147 The subfactorial at zero. ...
subfac1 35148 The subfactorial at one. ...
subfacp1lem1 35149 Lemma for ~ subfacp1 . Th...
subfacp1lem2a 35150 Lemma for ~ subfacp1 . Pr...
subfacp1lem2b 35151 Lemma for ~ subfacp1 . Pr...
subfacp1lem3 35152 Lemma for ~ subfacp1 . In...
subfacp1lem4 35153 Lemma for ~ subfacp1 . Th...
subfacp1lem5 35154 Lemma for ~ subfacp1 . In...
subfacp1lem6 35155 Lemma for ~ subfacp1 . By...
subfacp1 35156 A two-term recurrence for ...
subfacval2 35157 A closed-form expression f...
subfaclim 35158 The subfactorial converges...
subfacval3 35159 Another closed form expres...
derangfmla 35160 The derangements formula, ...
erdszelem1 35161 Lemma for ~ erdsze . (Con...
erdszelem2 35162 Lemma for ~ erdsze . (Con...
erdszelem3 35163 Lemma for ~ erdsze . (Con...
erdszelem4 35164 Lemma for ~ erdsze . (Con...
erdszelem5 35165 Lemma for ~ erdsze . (Con...
erdszelem6 35166 Lemma for ~ erdsze . (Con...
erdszelem7 35167 Lemma for ~ erdsze . (Con...
erdszelem8 35168 Lemma for ~ erdsze . (Con...
erdszelem9 35169 Lemma for ~ erdsze . (Con...
erdszelem10 35170 Lemma for ~ erdsze . (Con...
erdszelem11 35171 Lemma for ~ erdsze . (Con...
erdsze 35172 The Erdős-Szekeres th...
erdsze2lem1 35173 Lemma for ~ erdsze2 . (Co...
erdsze2lem2 35174 Lemma for ~ erdsze2 . (Co...
erdsze2 35175 Generalize the statement o...
kur14lem1 35176 Lemma for ~ kur14 . (Cont...
kur14lem2 35177 Lemma for ~ kur14 . Write...
kur14lem3 35178 Lemma for ~ kur14 . A clo...
kur14lem4 35179 Lemma for ~ kur14 . Compl...
kur14lem5 35180 Lemma for ~ kur14 . Closu...
kur14lem6 35181 Lemma for ~ kur14 . If ` ...
kur14lem7 35182 Lemma for ~ kur14 : main p...
kur14lem8 35183 Lemma for ~ kur14 . Show ...
kur14lem9 35184 Lemma for ~ kur14 . Since...
kur14lem10 35185 Lemma for ~ kur14 . Disch...
kur14 35186 Kuratowski's closure-compl...
ispconn 35193 The property of being a pa...
pconncn 35194 The property of being a pa...
pconntop 35195 A simply connected space i...
issconn 35196 The property of being a si...
sconnpconn 35197 A simply connected space i...
sconntop 35198 A simply connected space i...
sconnpht 35199 A closed path in a simply ...
cnpconn 35200 An image of a path-connect...
pconnconn 35201 A path-connected space is ...
txpconn 35202 The topological product of...
ptpconn 35203 The topological product of...
indispconn 35204 The indiscrete topology (o...
connpconn 35205 A connected and locally pa...
qtoppconn 35206 A quotient of a path-conne...
pconnpi1 35207 All fundamental groups in ...
sconnpht2 35208 Any two paths in a simply ...
sconnpi1 35209 A path-connected topologic...
txsconnlem 35210 Lemma for ~ txsconn . (Co...
txsconn 35211 The topological product of...
cvxpconn 35212 A convex subset of the com...
cvxsconn 35213 A convex subset of the com...
blsconn 35214 An open ball in the comple...
cnllysconn 35215 The topology of the comple...
resconn 35216 A subset of ` RR ` is simp...
ioosconn 35217 An open interval is simply...
iccsconn 35218 A closed interval is simpl...
retopsconn 35219 The real numbers are simpl...
iccllysconn 35220 A closed interval is local...
rellysconn 35221 The real numbers are local...
iisconn 35222 The unit interval is simpl...
iillysconn 35223 The unit interval is local...
iinllyconn 35224 The unit interval is local...
fncvm 35227 Lemma for covering maps. ...
cvmscbv 35228 Change bound variables in ...
iscvm 35229 The property of being a co...
cvmtop1 35230 Reverse closure for a cove...
cvmtop2 35231 Reverse closure for a cove...
cvmcn 35232 A covering map is a contin...
cvmcov 35233 Property of a covering map...
cvmsrcl 35234 Reverse closure for an eve...
cvmsi 35235 One direction of ~ cvmsval...
cvmsval 35236 Elementhood in the set ` S...
cvmsss 35237 An even covering is a subs...
cvmsn0 35238 An even covering is nonemp...
cvmsuni 35239 An even covering of ` U ` ...
cvmsdisj 35240 An even covering of ` U ` ...
cvmshmeo 35241 Every element of an even c...
cvmsf1o 35242 ` F ` , localized to an el...
cvmscld 35243 The sets of an even coveri...
cvmsss2 35244 An open subset of an evenl...
cvmcov2 35245 The covering map property ...
cvmseu 35246 Every element in ` U. T ` ...
cvmsiota 35247 Identify the unique elemen...
cvmopnlem 35248 Lemma for ~ cvmopn . (Con...
cvmfolem 35249 Lemma for ~ cvmfo . (Cont...
cvmopn 35250 A covering map is an open ...
cvmliftmolem1 35251 Lemma for ~ cvmliftmo . (...
cvmliftmolem2 35252 Lemma for ~ cvmliftmo . (...
cvmliftmoi 35253 A lift of a continuous fun...
cvmliftmo 35254 A lift of a continuous fun...
cvmliftlem1 35255 Lemma for ~ cvmlift . In ...
cvmliftlem2 35256 Lemma for ~ cvmlift . ` W ...
cvmliftlem3 35257 Lemma for ~ cvmlift . Sin...
cvmliftlem4 35258 Lemma for ~ cvmlift . The...
cvmliftlem5 35259 Lemma for ~ cvmlift . Def...
cvmliftlem6 35260 Lemma for ~ cvmlift . Ind...
cvmliftlem7 35261 Lemma for ~ cvmlift . Pro...
cvmliftlem8 35262 Lemma for ~ cvmlift . The...
cvmliftlem9 35263 Lemma for ~ cvmlift . The...
cvmliftlem10 35264 Lemma for ~ cvmlift . The...
cvmliftlem11 35265 Lemma for ~ cvmlift . (Co...
cvmliftlem13 35266 Lemma for ~ cvmlift . The...
cvmliftlem14 35267 Lemma for ~ cvmlift . Put...
cvmliftlem15 35268 Lemma for ~ cvmlift . Dis...
cvmlift 35269 One of the important prope...
cvmfo 35270 A covering map is an onto ...
cvmliftiota 35271 Write out a function ` H `...
cvmlift2lem1 35272 Lemma for ~ cvmlift2 . (C...
cvmlift2lem9a 35273 Lemma for ~ cvmlift2 and ~...
cvmlift2lem2 35274 Lemma for ~ cvmlift2 . (C...
cvmlift2lem3 35275 Lemma for ~ cvmlift2 . (C...
cvmlift2lem4 35276 Lemma for ~ cvmlift2 . (C...
cvmlift2lem5 35277 Lemma for ~ cvmlift2 . (C...
cvmlift2lem6 35278 Lemma for ~ cvmlift2 . (C...
cvmlift2lem7 35279 Lemma for ~ cvmlift2 . (C...
cvmlift2lem8 35280 Lemma for ~ cvmlift2 . (C...
cvmlift2lem9 35281 Lemma for ~ cvmlift2 . (C...
cvmlift2lem10 35282 Lemma for ~ cvmlift2 . (C...
cvmlift2lem11 35283 Lemma for ~ cvmlift2 . (C...
cvmlift2lem12 35284 Lemma for ~ cvmlift2 . (C...
cvmlift2lem13 35285 Lemma for ~ cvmlift2 . (C...
cvmlift2 35286 A two-dimensional version ...
cvmliftphtlem 35287 Lemma for ~ cvmliftpht . ...
cvmliftpht 35288 If ` G ` and ` H ` are pat...
cvmlift3lem1 35289 Lemma for ~ cvmlift3 . (C...
cvmlift3lem2 35290 Lemma for ~ cvmlift2 . (C...
cvmlift3lem3 35291 Lemma for ~ cvmlift2 . (C...
cvmlift3lem4 35292 Lemma for ~ cvmlift2 . (C...
cvmlift3lem5 35293 Lemma for ~ cvmlift2 . (C...
cvmlift3lem6 35294 Lemma for ~ cvmlift3 . (C...
cvmlift3lem7 35295 Lemma for ~ cvmlift3 . (C...
cvmlift3lem8 35296 Lemma for ~ cvmlift2 . (C...
cvmlift3lem9 35297 Lemma for ~ cvmlift2 . (C...
cvmlift3 35298 A general version of ~ cvm...
snmlff 35299 The function ` F ` from ~ ...
snmlfval 35300 The function ` F ` from ~ ...
snmlval 35301 The property " ` A ` is si...
snmlflim 35302 If ` A ` is simply normal,...
goel 35317 A "Godel-set of membership...
goelel3xp 35318 A "Godel-set of membership...
goeleq12bg 35319 Two "Godel-set of membersh...
gonafv 35320 The "Godel-set for the She...
goaleq12d 35321 Equality of the "Godel-set...
gonanegoal 35322 The Godel-set for the Shef...
satf 35323 The satisfaction predicate...
satfsucom 35324 The satisfaction predicate...
satfn 35325 The satisfaction predicate...
satom 35326 The satisfaction predicate...
satfvsucom 35327 The satisfaction predicate...
satfv0 35328 The value of the satisfact...
satfvsuclem1 35329 Lemma 1 for ~ satfvsuc . ...
satfvsuclem2 35330 Lemma 2 for ~ satfvsuc . ...
satfvsuc 35331 The value of the satisfact...
satfv1lem 35332 Lemma for ~ satfv1 . (Con...
satfv1 35333 The value of the satisfact...
satfsschain 35334 The binary relation of a s...
satfvsucsuc 35335 The satisfaction predicate...
satfbrsuc 35336 The binary relation of a s...
satfrel 35337 The value of the satisfact...
satfdmlem 35338 Lemma for ~ satfdm . (Con...
satfdm 35339 The domain of the satisfac...
satfrnmapom 35340 The range of the satisfact...
satfv0fun 35341 The value of the satisfact...
satf0 35342 The satisfaction predicate...
satf0sucom 35343 The satisfaction predicate...
satf00 35344 The value of the satisfact...
satf0suclem 35345 Lemma for ~ satf0suc , ~ s...
satf0suc 35346 The value of the satisfact...
satf0op 35347 An element of a value of t...
satf0n0 35348 The value of the satisfact...
sat1el2xp 35349 The first component of an ...
fmlafv 35350 The valid Godel formulas o...
fmla 35351 The set of all valid Godel...
fmla0 35352 The valid Godel formulas o...
fmla0xp 35353 The valid Godel formulas o...
fmlasuc0 35354 The valid Godel formulas o...
fmlafvel 35355 A class is a valid Godel f...
fmlasuc 35356 The valid Godel formulas o...
fmla1 35357 The valid Godel formulas o...
isfmlasuc 35358 The characterization of a ...
fmlasssuc 35359 The Godel formulas of heig...
fmlaomn0 35360 The empty set is not a God...
fmlan0 35361 The empty set is not a God...
gonan0 35362 The "Godel-set of NAND" is...
goaln0 35363 The "Godel-set of universa...
gonarlem 35364 Lemma for ~ gonar (inducti...
gonar 35365 If the "Godel-set of NAND"...
goalrlem 35366 Lemma for ~ goalr (inducti...
goalr 35367 If the "Godel-set of unive...
fmla0disjsuc 35368 The set of valid Godel for...
fmlasucdisj 35369 The valid Godel formulas o...
satfdmfmla 35370 The domain of the satisfac...
satffunlem 35371 Lemma for ~ satffunlem1lem...
satffunlem1lem1 35372 Lemma for ~ satffunlem1 . ...
satffunlem1lem2 35373 Lemma 2 for ~ satffunlem1 ...
satffunlem2lem1 35374 Lemma 1 for ~ satffunlem2 ...
dmopab3rexdif 35375 The domain of an ordered p...
satffunlem2lem2 35376 Lemma 2 for ~ satffunlem2 ...
satffunlem1 35377 Lemma 1 for ~ satffun : in...
satffunlem2 35378 Lemma 2 for ~ satffun : in...
satffun 35379 The value of the satisfact...
satff 35380 The satisfaction predicate...
satfun 35381 The satisfaction predicate...
satfvel 35382 An element of the value of...
satfv0fvfmla0 35383 The value of the satisfact...
satefv 35384 The simplified satisfactio...
sate0 35385 The simplified satisfactio...
satef 35386 The simplified satisfactio...
sate0fv0 35387 A simplified satisfaction ...
satefvfmla0 35388 The simplified satisfactio...
sategoelfvb 35389 Characterization of a valu...
sategoelfv 35390 Condition of a valuation `...
ex-sategoelel 35391 Example of a valuation of ...
ex-sategoel 35392 Instance of ~ sategoelfv f...
satfv1fvfmla1 35393 The value of the satisfact...
2goelgoanfmla1 35394 Two Godel-sets of membersh...
satefvfmla1 35395 The simplified satisfactio...
ex-sategoelelomsuc 35396 Example of a valuation of ...
ex-sategoelel12 35397 Example of a valuation of ...
prv 35398 The "proves" relation on a...
elnanelprv 35399 The wff ` ( A e. B -/\ B e...
prv0 35400 Every wff encoded as ` U `...
prv1n 35401 No wff encoded as a Godel-...
mvtval 35470 The set of variable typeco...
mrexval 35471 The set of "raw expression...
mexval 35472 The set of expressions, wh...
mexval2 35473 The set of expressions, wh...
mdvval 35474 The set of disjoint variab...
mvrsval 35475 The set of variables in an...
mvrsfpw 35476 The set of variables in an...
mrsubffval 35477 The substitution of some v...
mrsubfval 35478 The substitution of some v...
mrsubval 35479 The substitution of some v...
mrsubcv 35480 The value of a substituted...
mrsubvr 35481 The value of a substituted...
mrsubff 35482 A substitution is a functi...
mrsubrn 35483 Although it is defined for...
mrsubff1 35484 When restricted to complet...
mrsubff1o 35485 When restricted to complet...
mrsub0 35486 The value of the substitut...
mrsubf 35487 A substitution is a functi...
mrsubccat 35488 Substitution distributes o...
mrsubcn 35489 A substitution does not ch...
elmrsubrn 35490 Characterization of the su...
mrsubco 35491 The composition of two sub...
mrsubvrs 35492 The set of variables in a ...
msubffval 35493 A substitution applied to ...
msubfval 35494 A substitution applied to ...
msubval 35495 A substitution applied to ...
msubrsub 35496 A substitution applied to ...
msubty 35497 The type of a substituted ...
elmsubrn 35498 Characterization of substi...
msubrn 35499 Although it is defined for...
msubff 35500 A substitution is a functi...
msubco 35501 The composition of two sub...
msubf 35502 A substitution is a functi...
mvhfval 35503 Value of the function mapp...
mvhval 35504 Value of the function mapp...
mpstval 35505 A pre-statement is an orde...
elmpst 35506 Property of being a pre-st...
msrfval 35507 Value of the reduct of a p...
msrval 35508 Value of the reduct of a p...
mpstssv 35509 A pre-statement is an orde...
mpst123 35510 Decompose a pre-statement ...
mpstrcl 35511 The elements of a pre-stat...
msrf 35512 The reduct of a pre-statem...
msrrcl 35513 If ` X ` and ` Y ` have th...
mstaval 35514 Value of the set of statem...
msrid 35515 The reduct of a statement ...
msrfo 35516 The reduct of a pre-statem...
mstapst 35517 A statement is a pre-state...
elmsta 35518 Property of being a statem...
ismfs 35519 A formal system is a tuple...
mfsdisj 35520 The constants and variable...
mtyf2 35521 The type function maps var...
mtyf 35522 The type function maps var...
mvtss 35523 The set of variable typeco...
maxsta 35524 An axiom is a statement. ...
mvtinf 35525 Each variable typecode has...
msubff1 35526 When restricted to complet...
msubff1o 35527 When restricted to complet...
mvhf 35528 The function mapping varia...
mvhf1 35529 The function mapping varia...
msubvrs 35530 The set of variables in a ...
mclsrcl 35531 Reverse closure for the cl...
mclsssvlem 35532 Lemma for ~ mclsssv . (Co...
mclsval 35533 The function mapping varia...
mclsssv 35534 The closure of a set of ex...
ssmclslem 35535 Lemma for ~ ssmcls . (Con...
vhmcls 35536 All variable hypotheses ar...
ssmcls 35537 The original expressions a...
ss2mcls 35538 The closure is monotonic u...
mclsax 35539 The closure is closed unde...
mclsind 35540 Induction theorem for clos...
mppspstlem 35541 Lemma for ~ mppspst . (Co...
mppsval 35542 Definition of a provable p...
elmpps 35543 Definition of a provable p...
mppspst 35544 A provable pre-statement i...
mthmval 35545 A theorem is a pre-stateme...
elmthm 35546 A theorem is a pre-stateme...
mthmi 35547 A statement whose reduct i...
mthmsta 35548 A theorem is a pre-stateme...
mppsthm 35549 A provable pre-statement i...
mthmblem 35550 Lemma for ~ mthmb . (Cont...
mthmb 35551 If two statements have the...
mthmpps 35552 Given a theorem, there is ...
mclsppslem 35553 The closure is closed unde...
mclspps 35554 The closure is closed unde...
rexxfr3d 35608 Transfer existential quant...
rexxfr3dALT 35609 Longer proof of ~ rexxfr3d...
rspssbasd 35610 The span of a set of ring ...
ellcsrspsn 35611 Membership in a left coset...
ply1divalg3 35612 Uniqueness of polynomial r...
r1peuqusdeg1 35613 Uniqueness of polynomial r...
problem1 35635 Practice problem 1. Clues...
problem2 35636 Practice problem 2. Clues...
problem3 35637 Practice problem 3. Clues...
problem4 35638 Practice problem 4. Clues...
problem5 35639 Practice problem 5. Clues...
quad3 35640 Variant of quadratic equat...
climuzcnv 35641 Utility lemma to convert b...
sinccvglem 35642 ` ( ( sin `` x ) / x ) ~~>...
sinccvg 35643 ` ( ( sin `` x ) / x ) ~~>...
circum 35644 The circumference of a cir...
elfzm12 35645 Membership in a curtailed ...
nn0seqcvg 35646 A strictly-decreasing nonn...
lediv2aALT 35647 Division of both sides of ...
abs2sqlei 35648 The absolute values of two...
abs2sqlti 35649 The absolute values of two...
abs2sqle 35650 The absolute values of two...
abs2sqlt 35651 The absolute values of two...
abs2difi 35652 Difference of absolute val...
abs2difabsi 35653 Absolute value of differen...
2thALT 35654 Alternate proof of ~ 2th ....
orbi2iALT 35655 Alternate proof of ~ orbi2...
pm3.48ALT 35656 Alternate proof of ~ pm3.4...
3jcadALT 35657 Alternate proof of ~ 3jcad...
currybi 35658 Biconditional version of C...
axextprim 35665 ~ ax-ext without distinct ...
axrepprim 35666 ~ ax-rep without distinct ...
axunprim 35667 ~ ax-un without distinct v...
axpowprim 35668 ~ ax-pow without distinct ...
axregprim 35669 ~ ax-reg without distinct ...
axinfprim 35670 ~ ax-inf without distinct ...
axacprim 35671 ~ ax-ac without distinct v...
untelirr 35672 We call a class "untanged"...
untuni 35673 The union of a class is un...
untsucf 35674 If a class is untangled, t...
unt0 35675 The null set is untangled....
untint 35676 If there is an untangled e...
efrunt 35677 If ` A ` is well-founded b...
untangtr 35678 A transitive class is unta...
3jaodd 35679 Double deduction form of ~...
3orit 35680 Closed form of ~ 3ori . (...
biimpexp 35681 A biconditional in the ant...
nepss 35682 Two classes are unequal if...
3ccased 35683 Triple disjunction form of...
dfso3 35684 Expansion of the definitio...
brtpid1 35685 A binary relation involvin...
brtpid2 35686 A binary relation involvin...
brtpid3 35687 A binary relation involvin...
iota5f 35688 A method for computing iot...
jath 35689 Closed form of ~ ja . Pro...
xpab 35690 Cartesian product of two c...
nnuni 35691 The union of a finite ordi...
sqdivzi 35692 Distribution of square ove...
supfz 35693 The supremum of a finite s...
inffz 35694 The infimum of a finite se...
fz0n 35695 The sequence ` ( 0 ... ( N...
shftvalg 35696 Value of a sequence shifte...
divcnvlin 35697 Limit of the ratio of two ...
climlec3 35698 Comparison of a constant t...
iexpire 35699 ` _i ` raised to itself is...
bcneg1 35700 The binomial coefficient o...
bcm1nt 35701 The proportion of one bino...
bcprod 35702 A product identity for bin...
bccolsum 35703 A column-sum rule for bino...
iprodefisumlem 35704 Lemma for ~ iprodefisum . ...
iprodefisum 35705 Applying the exponential f...
iprodgam 35706 An infinite product versio...
faclimlem1 35707 Lemma for ~ faclim . Clos...
faclimlem2 35708 Lemma for ~ faclim . Show...
faclimlem3 35709 Lemma for ~ faclim . Alge...
faclim 35710 An infinite product expres...
iprodfac 35711 An infinite product expres...
faclim2 35712 Another factorial limit du...
gcd32 35713 Swap the second and third ...
gcdabsorb 35714 Absorption law for gcd. (...
dftr6 35715 A potential definition of ...
coep 35716 Composition with the membe...
coepr 35717 Composition with the conve...
dffr5 35718 A quantifier-free definiti...
dfso2 35719 Quantifier-free definition...
br8 35720 Substitution for an eight-...
br6 35721 Substitution for a six-pla...
br4 35722 Substitution for a four-pl...
cnvco1 35723 Another distributive law o...
cnvco2 35724 Another distributive law o...
eldm3 35725 Quantifier-free definition...
elrn3 35726 Quantifier-free definition...
pocnv 35727 The converse of a partial ...
socnv 35728 The converse of a strict o...
sotrd 35729 Transitivity law for stric...
elintfv 35730 Membership in an intersect...
funpsstri 35731 A condition for subset tri...
fundmpss 35732 If a class ` F ` is a prop...
funsseq 35733 Given two functions with e...
fununiq 35734 The uniqueness condition o...
funbreq 35735 An equality condition for ...
br1steq 35736 Uniqueness condition for t...
br2ndeq 35737 Uniqueness condition for t...
dfdm5 35738 Definition of domain in te...
dfrn5 35739 Definition of range in ter...
opelco3 35740 Alternate way of saying th...
elima4 35741 Quantifier-free expression...
fv1stcnv 35742 The value of the converse ...
fv2ndcnv 35743 The value of the converse ...
setinds 35744 Principle of set induction...
setinds2f 35745 ` _E ` induction schema, u...
setinds2 35746 ` _E ` induction schema, u...
elpotr 35747 A class of transitive sets...
dford5reg 35748 Given ~ ax-reg , an ordina...
dfon2lem1 35749 Lemma for ~ dfon2 . (Cont...
dfon2lem2 35750 Lemma for ~ dfon2 . (Cont...
dfon2lem3 35751 Lemma for ~ dfon2 . All s...
dfon2lem4 35752 Lemma for ~ dfon2 . If tw...
dfon2lem5 35753 Lemma for ~ dfon2 . Two s...
dfon2lem6 35754 Lemma for ~ dfon2 . A tra...
dfon2lem7 35755 Lemma for ~ dfon2 . All e...
dfon2lem8 35756 Lemma for ~ dfon2 . The i...
dfon2lem9 35757 Lemma for ~ dfon2 . A cla...
dfon2 35758 ` On ` consists of all set...
rdgprc0 35759 The value of the recursive...
rdgprc 35760 The value of the recursive...
dfrdg2 35761 Alternate definition of th...
dfrdg3 35762 Generalization of ~ dfrdg2...
axextdfeq 35763 A version of ~ ax-ext for ...
ax8dfeq 35764 A version of ~ ax-8 for us...
axextdist 35765 ~ ax-ext with distinctors ...
axextbdist 35766 ~ axextb with distinctors ...
19.12b 35767 Version of ~ 19.12vv with ...
exnel 35768 There is always a set not ...
distel 35769 Distinctors in terms of me...
axextndbi 35770 ~ axextnd as a bicondition...
hbntg 35771 A more general form of ~ h...
hbimtg 35772 A more general and closed ...
hbaltg 35773 A more general and closed ...
hbng 35774 A more general form of ~ h...
hbimg 35775 A more general form of ~ h...
wsuceq123 35780 Equality theorem for well-...
wsuceq1 35781 Equality theorem for well-...
wsuceq2 35782 Equality theorem for well-...
wsuceq3 35783 Equality theorem for well-...
nfwsuc 35784 Bound-variable hypothesis ...
wlimeq12 35785 Equality theorem for the l...
wlimeq1 35786 Equality theorem for the l...
wlimeq2 35787 Equality theorem for the l...
nfwlim 35788 Bound-variable hypothesis ...
elwlim 35789 Membership in the limit cl...
wzel 35790 The zero of a well-founded...
wsuclem 35791 Lemma for the supremum pro...
wsucex 35792 Existence theorem for well...
wsuccl 35793 If ` X ` is a set with an ...
wsuclb 35794 A well-founded successor i...
wlimss 35795 The class of limit points ...
txpss3v 35844 A tail Cartesian product i...
txprel 35845 A tail Cartesian product i...
brtxp 35846 Characterize a ternary rel...
brtxp2 35847 The binary relation over a...
dfpprod2 35848 Expanded definition of par...
pprodcnveq 35849 A converse law for paralle...
pprodss4v 35850 The parallel product is a ...
brpprod 35851 Characterize a quaternary ...
brpprod3a 35852 Condition for parallel pro...
brpprod3b 35853 Condition for parallel pro...
relsset 35854 The subset class is a bina...
brsset 35855 For sets, the ` SSet ` bin...
idsset 35856 ` _I ` is equal to the int...
eltrans 35857 Membership in the class of...
dfon3 35858 A quantifier-free definiti...
dfon4 35859 Another quantifier-free de...
brtxpsd 35860 Expansion of a common form...
brtxpsd2 35861 Another common abbreviatio...
brtxpsd3 35862 A third common abbreviatio...
relbigcup 35863 The ` Bigcup ` relationshi...
brbigcup 35864 Binary relation over ` Big...
dfbigcup2 35865 ` Bigcup ` using maps-to n...
fobigcup 35866 ` Bigcup ` maps the univer...
fnbigcup 35867 ` Bigcup ` is a function o...
fvbigcup 35868 For sets, ` Bigcup ` yield...
elfix 35869 Membership in the fixpoint...
elfix2 35870 Alternative membership in ...
dffix2 35871 The fixpoints of a class i...
fixssdm 35872 The fixpoints of a class a...
fixssrn 35873 The fixpoints of a class a...
fixcnv 35874 The fixpoints of a class a...
fixun 35875 The fixpoint operator dist...
ellimits 35876 Membership in the class of...
limitssson 35877 The class of all limit ord...
dfom5b 35878 A quantifier-free definiti...
sscoid 35879 A condition for subset and...
dffun10 35880 Another potential definiti...
elfuns 35881 Membership in the class of...
elfunsg 35882 Closed form of ~ elfuns . ...
brsingle 35883 The binary relation form o...
elsingles 35884 Membership in the class of...
fnsingle 35885 The singleton relationship...
fvsingle 35886 The value of the singleton...
dfsingles2 35887 Alternate definition of th...
snelsingles 35888 A singleton is a member of...
dfiota3 35889 A definition of iota using...
dffv5 35890 Another quantifier-free de...
unisnif 35891 Express union of singleton...
brimage 35892 Binary relation form of th...
brimageg 35893 Closed form of ~ brimage ....
funimage 35894 ` Image A ` is a function....
fnimage 35895 ` Image R ` is a function ...
imageval 35896 The image functor in maps-...
fvimage 35897 Value of the image functor...
brcart 35898 Binary relation form of th...
brdomain 35899 Binary relation form of th...
brrange 35900 Binary relation form of th...
brdomaing 35901 Closed form of ~ brdomain ...
brrangeg 35902 Closed form of ~ brrange ....
brimg 35903 Binary relation form of th...
brapply 35904 Binary relation form of th...
brcup 35905 Binary relation form of th...
brcap 35906 Binary relation form of th...
brsuccf 35907 Binary relation form of th...
funpartlem 35908 Lemma for ~ funpartfun . ...
funpartfun 35909 The functional part of ` F...
funpartss 35910 The functional part of ` F...
funpartfv 35911 The function value of the ...
fullfunfnv 35912 The full functional part o...
fullfunfv 35913 The function value of the ...
brfullfun 35914 A binary relation form con...
brrestrict 35915 Binary relation form of th...
dfrecs2 35916 A quantifier-free definiti...
dfrdg4 35917 A quantifier-free definiti...
dfint3 35918 Quantifier-free definition...
imagesset 35919 The Image functor applied ...
brub 35920 Binary relation form of th...
brlb 35921 Binary relation form of th...
altopex 35926 Alternative ordered pairs ...
altopthsn 35927 Two alternate ordered pair...
altopeq12 35928 Equality for alternate ord...
altopeq1 35929 Equality for alternate ord...
altopeq2 35930 Equality for alternate ord...
altopth1 35931 Equality of the first memb...
altopth2 35932 Equality of the second mem...
altopthg 35933 Alternate ordered pair the...
altopthbg 35934 Alternate ordered pair the...
altopth 35935 The alternate ordered pair...
altopthb 35936 Alternate ordered pair the...
altopthc 35937 Alternate ordered pair the...
altopthd 35938 Alternate ordered pair the...
altxpeq1 35939 Equality for alternate Car...
altxpeq2 35940 Equality for alternate Car...
elaltxp 35941 Membership in alternate Ca...
altopelaltxp 35942 Alternate ordered pair mem...
altxpsspw 35943 An inclusion rule for alte...
altxpexg 35944 The alternate Cartesian pr...
rankaltopb 35945 Compute the rank of an alt...
nfaltop 35946 Bound-variable hypothesis ...
sbcaltop 35947 Distribution of class subs...
cgrrflx2d 35950 Deduction form of ~ axcgrr...
cgrtr4d 35951 Deduction form of ~ axcgrt...
cgrtr4and 35952 Deduction form of ~ axcgrt...
cgrrflx 35953 Reflexivity law for congru...
cgrrflxd 35954 Deduction form of ~ cgrrfl...
cgrcomim 35955 Congruence commutes on the...
cgrcom 35956 Congruence commutes betwee...
cgrcomand 35957 Deduction form of ~ cgrcom...
cgrtr 35958 Transitivity law for congr...
cgrtrand 35959 Deduction form of ~ cgrtr ...
cgrtr3 35960 Transitivity law for congr...
cgrtr3and 35961 Deduction form of ~ cgrtr3...
cgrcoml 35962 Congruence commutes on the...
cgrcomr 35963 Congruence commutes on the...
cgrcomlr 35964 Congruence commutes on bot...
cgrcomland 35965 Deduction form of ~ cgrcom...
cgrcomrand 35966 Deduction form of ~ cgrcom...
cgrcomlrand 35967 Deduction form of ~ cgrcom...
cgrtriv 35968 Degenerate segments are co...
cgrid2 35969 Identity law for congruenc...
cgrdegen 35970 Two congruent segments are...
brofs 35971 Binary relation form of th...
5segofs 35972 Rephrase ~ ax5seg using th...
ofscom 35973 The outer five segment pre...
cgrextend 35974 Link congruence over a pai...
cgrextendand 35975 Deduction form of ~ cgrext...
segconeq 35976 Two points that satisfy th...
segconeu 35977 Existential uniqueness ver...
btwntriv2 35978 Betweenness always holds f...
btwncomim 35979 Betweenness commutes. Imp...
btwncom 35980 Betweenness commutes. (Co...
btwncomand 35981 Deduction form of ~ btwnco...
btwntriv1 35982 Betweenness always holds f...
btwnswapid 35983 If you can swap the first ...
btwnswapid2 35984 If you can swap arguments ...
btwnintr 35985 Inner transitivity law for...
btwnexch3 35986 Exchange the first endpoin...
btwnexch3and 35987 Deduction form of ~ btwnex...
btwnouttr2 35988 Outer transitivity law for...
btwnexch2 35989 Exchange the outer point o...
btwnouttr 35990 Outer transitivity law for...
btwnexch 35991 Outer transitivity law for...
btwnexchand 35992 Deduction form of ~ btwnex...
btwndiff 35993 There is always a ` c ` di...
trisegint 35994 A line segment between two...
funtransport 35997 The ` TransportTo ` relati...
fvtransport 35998 Calculate the value of the...
transportcl 35999 Closure law for segment tr...
transportprops 36000 Calculate the defining pro...
brifs 36009 Binary relation form of th...
ifscgr 36010 Inner five segment congrue...
cgrsub 36011 Removing identical parts f...
brcgr3 36012 Binary relation form of th...
cgr3permute3 36013 Permutation law for three-...
cgr3permute1 36014 Permutation law for three-...
cgr3permute2 36015 Permutation law for three-...
cgr3permute4 36016 Permutation law for three-...
cgr3permute5 36017 Permutation law for three-...
cgr3tr4 36018 Transitivity law for three...
cgr3com 36019 Commutativity law for thre...
cgr3rflx 36020 Identity law for three-pla...
cgrxfr 36021 A line segment can be divi...
btwnxfr 36022 A condition for extending ...
colinrel 36023 Colinearity is a relations...
brcolinear2 36024 Alternate colinearity bina...
brcolinear 36025 The binary relation form o...
colinearex 36026 The colinear predicate exi...
colineardim1 36027 If ` A ` is colinear with ...
colinearperm1 36028 Permutation law for coline...
colinearperm3 36029 Permutation law for coline...
colinearperm2 36030 Permutation law for coline...
colinearperm4 36031 Permutation law for coline...
colinearperm5 36032 Permutation law for coline...
colineartriv1 36033 Trivial case of colinearit...
colineartriv2 36034 Trivial case of colinearit...
btwncolinear1 36035 Betweenness implies coline...
btwncolinear2 36036 Betweenness implies coline...
btwncolinear3 36037 Betweenness implies coline...
btwncolinear4 36038 Betweenness implies coline...
btwncolinear5 36039 Betweenness implies coline...
btwncolinear6 36040 Betweenness implies coline...
colinearxfr 36041 Transfer law for colineari...
lineext 36042 Extend a line with a missi...
brofs2 36043 Change some conditions for...
brifs2 36044 Change some conditions for...
brfs 36045 Binary relation form of th...
fscgr 36046 Congruence law for the gen...
linecgr 36047 Congruence rule for lines....
linecgrand 36048 Deduction form of ~ linecg...
lineid 36049 Identity law for points on...
idinside 36050 Law for finding a point in...
endofsegid 36051 If ` A ` , ` B ` , and ` C...
endofsegidand 36052 Deduction form of ~ endofs...
btwnconn1lem1 36053 Lemma for ~ btwnconn1 . T...
btwnconn1lem2 36054 Lemma for ~ btwnconn1 . N...
btwnconn1lem3 36055 Lemma for ~ btwnconn1 . E...
btwnconn1lem4 36056 Lemma for ~ btwnconn1 . A...
btwnconn1lem5 36057 Lemma for ~ btwnconn1 . N...
btwnconn1lem6 36058 Lemma for ~ btwnconn1 . N...
btwnconn1lem7 36059 Lemma for ~ btwnconn1 . U...
btwnconn1lem8 36060 Lemma for ~ btwnconn1 . N...
btwnconn1lem9 36061 Lemma for ~ btwnconn1 . N...
btwnconn1lem10 36062 Lemma for ~ btwnconn1 . N...
btwnconn1lem11 36063 Lemma for ~ btwnconn1 . N...
btwnconn1lem12 36064 Lemma for ~ btwnconn1 . U...
btwnconn1lem13 36065 Lemma for ~ btwnconn1 . B...
btwnconn1lem14 36066 Lemma for ~ btwnconn1 . F...
btwnconn1 36067 Connectitivy law for betwe...
btwnconn2 36068 Another connectivity law f...
btwnconn3 36069 Inner connectivity law for...
midofsegid 36070 If two points fall in the ...
segcon2 36071 Generalization of ~ axsegc...
brsegle 36074 Binary relation form of th...
brsegle2 36075 Alternate characterization...
seglecgr12im 36076 Substitution law for segme...
seglecgr12 36077 Substitution law for segme...
seglerflx 36078 Segment comparison is refl...
seglemin 36079 Any segment is at least as...
segletr 36080 Segment less than is trans...
segleantisym 36081 Antisymmetry law for segme...
seglelin 36082 Linearity law for segment ...
btwnsegle 36083 If ` B ` falls between ` A...
colinbtwnle 36084 Given three colinear point...
broutsideof 36087 Binary relation form of ` ...
broutsideof2 36088 Alternate form of ` Outsid...
outsidene1 36089 Outsideness implies inequa...
outsidene2 36090 Outsideness implies inequa...
btwnoutside 36091 A principle linking outsid...
broutsideof3 36092 Characterization of outsid...
outsideofrflx 36093 Reflexivity of outsideness...
outsideofcom 36094 Commutativity law for outs...
outsideoftr 36095 Transitivity law for outsi...
outsideofeq 36096 Uniqueness law for ` Outsi...
outsideofeu 36097 Given a nondegenerate ray,...
outsidele 36098 Relate ` OutsideOf ` to ` ...
outsideofcol 36099 Outside of implies colinea...
funray 36106 Show that the ` Ray ` rela...
fvray 36107 Calculate the value of the...
funline 36108 Show that the ` Line ` rel...
linedegen 36109 When ` Line ` is applied w...
fvline 36110 Calculate the value of the...
liness 36111 A line is a subset of the ...
fvline2 36112 Alternate definition of a ...
lineunray 36113 A line is composed of a po...
lineelsb2 36114 If ` S ` lies on ` P Q ` ,...
linerflx1 36115 Reflexivity law for line m...
linecom 36116 Commutativity law for line...
linerflx2 36117 Reflexivity law for line m...
ellines 36118 Membership in the set of a...
linethru 36119 If ` A ` is a line contain...
hilbert1.1 36120 There is a line through an...
hilbert1.2 36121 There is at most one line ...
linethrueu 36122 There is a unique line goi...
lineintmo 36123 Two distinct lines interse...
fwddifval 36128 Calculate the value of the...
fwddifnval 36129 The value of the forward d...
fwddifn0 36130 The value of the n-iterate...
fwddifnp1 36131 The value of the n-iterate...
rankung 36132 The rank of the union of t...
ranksng 36133 The rank of a singleton. ...
rankelg 36134 The membership relation is...
rankpwg 36135 The rank of a power set. ...
rank0 36136 The rank of the empty set ...
rankeq1o 36137 The only set with rank ` 1...
elhf 36140 Membership in the heredita...
elhf2 36141 Alternate form of membersh...
elhf2g 36142 Hereditarily finiteness vi...
0hf 36143 The empty set is a heredit...
hfun 36144 The union of two HF sets i...
hfsn 36145 The singleton of an HF set...
hfadj 36146 Adjoining one HF element t...
hfelhf 36147 Any member of an HF set is...
hftr 36148 The class of all hereditar...
hfext 36149 Extensionality for HF sets...
hfuni 36150 The union of an HF set is ...
hfpw 36151 The power class of an HF s...
hfninf 36152 ` _om ` is not hereditaril...
rmoeqi 36153 Equality inference for res...
rmoeqbii 36154 Equality inference for res...
reueqi 36155 Equality inference for res...
reueqbii 36156 Equality inference for res...
sbceqbii 36157 Formula-building inference...
disjeq1i 36158 Equality theorem for disjo...
disjeq12i 36159 Equality theorem for disjo...
rabeqbii 36160 Equality theorem for restr...
iuneq12i 36161 Equality theorem for index...
iineq1i 36162 Equality theorem for index...
iineq12i 36163 Equality theorem for index...
riotaeqbii 36164 Equivalent wff's and equal...
riotaeqi 36165 Equal domains yield equal ...
ixpeq1i 36166 Equality inference for inf...
ixpeq12i 36167 Equality inference for inf...
sumeq2si 36168 Equality inference for sum...
sumeq12si 36169 Equality inference for sum...
prodeq2si 36170 Equality inference for pro...
prodeq12si 36171 Equality inference for pro...
itgeq12i 36172 Equality inference for an ...
itgeq1i 36173 Equality inference for an ...
itgeq2i 36174 Equality inference for an ...
ditgeq123i 36175 Equality inference for the...
ditgeq12i 36176 Equality inference for the...
ditgeq3i 36177 Equality inference for the...
rmoeqdv 36178 Formula-building rule for ...
rmoeqbidv 36179 Formula-building rule for ...
reueqdv 36180 Formula-building rule for ...
reueqbidv 36181 Formula-building rule for ...
sbequbidv 36182 Deduction substituting bot...
disjeq12dv 36183 Equality theorem for disjo...
ixpeq12dv 36184 Equality theorem for infin...
sumeq12sdv 36185 Equality deduction for sum...
prodeq12sdv 36186 Equality deduction for pro...
itgeq12sdv 36187 Equality theorem for an in...
itgeq2sdv 36188 Equality theorem for an in...
ditgeq123dv 36189 Equality theorem for the d...
ditgeq12d 36190 Equality theorem for the d...
ditgeq3sdv 36191 Equality theorem for the d...
in-ax8 36192 A proof of ~ ax-8 that doe...
ss-ax8 36193 A proof of ~ ax-8 that doe...
cbvralvw2 36194 Change bound variable and ...
cbvrexvw2 36195 Change bound variable and ...
cbvrmovw2 36196 Change bound variable and ...
cbvreuvw2 36197 Change bound variable and ...
cbvsbcvw2 36198 Change bound variable of a...
cbvcsbvw2 36199 Change bound variable of a...
cbviunvw2 36200 Change bound variable and ...
cbviinvw2 36201 Change bound variable and ...
cbvmptvw2 36202 Change bound variable and ...
cbvdisjvw2 36203 Change bound variable and ...
cbvriotavw2 36204 Change bound variable and ...
cbvoprab1vw 36205 Change the first bound var...
cbvoprab2vw 36206 Change the second bound va...
cbvoprab123vw 36207 Change all bound variables...
cbvoprab23vw 36208 Change the second and thir...
cbvoprab13vw 36209 Change the first and third...
cbvmpovw2 36210 Change bound variables and...
cbvmpo1vw2 36211 Change domains and the fir...
cbvmpo2vw2 36212 Change domains and the sec...
cbvixpvw2 36213 Change bound variable and ...
cbvsumvw2 36214 Change bound variable and ...
cbvprodvw2 36215 Change bound variable and ...
cbvitgvw2 36216 Change bound variable and ...
cbvditgvw2 36217 Change bound variable and ...
cbvmodavw 36218 Change bound variable in t...
cbveudavw 36219 Change bound variable in t...
cbvrmodavw 36220 Change bound variable in t...
cbvreudavw 36221 Change bound variable in t...
cbvsbdavw 36222 Change bound variable in p...
cbvsbdavw2 36223 Change bound variable in p...
cbvabdavw 36224 Change bound variable in c...
cbvsbcdavw 36225 Change bound variable of a...
cbvsbcdavw2 36226 Change bound variable of a...
cbvcsbdavw 36227 Change bound variable of a...
cbvcsbdavw2 36228 Change bound variable of a...
cbvrabdavw 36229 Change bound variable in r...
cbviundavw 36230 Change bound variable in i...
cbviindavw 36231 Change bound variable in i...
cbvopab1davw 36232 Change the first bound var...
cbvopab2davw 36233 Change the second bound va...
cbvopabdavw 36234 Change bound variables in ...
cbvmptdavw 36235 Change bound variable in a...
cbvdisjdavw 36236 Change bound variable in a...
cbviotadavw 36237 Change bound variable in a...
cbvriotadavw 36238 Change bound variable in a...
cbvoprab1davw 36239 Change the first bound var...
cbvoprab2davw 36240 Change the second bound va...
cbvoprab3davw 36241 Change the third bound var...
cbvoprab123davw 36242 Change all bound variables...
cbvoprab12davw 36243 Change the first and secon...
cbvoprab23davw 36244 Change the second and thir...
cbvoprab13davw 36245 Change the first and third...
cbvixpdavw 36246 Change bound variable in a...
cbvsumdavw 36247 Change bound variable in a...
cbvproddavw 36248 Change bound variable in a...
cbvitgdavw 36249 Change bound variable in a...
cbvditgdavw 36250 Change bound variable in a...
cbvrmodavw2 36251 Change bound variable and ...
cbvreudavw2 36252 Change bound variable and ...
cbvrabdavw2 36253 Change bound variable and ...
cbviundavw2 36254 Change bound variable and ...
cbviindavw2 36255 Change bound variable and ...
cbvmptdavw2 36256 Change bound variable and ...
cbvdisjdavw2 36257 Change bound variable and ...
cbvriotadavw2 36258 Change bound variable and ...
cbvmpodavw2 36259 Change bound variable and ...
cbvmpo1davw2 36260 Change first bound variabl...
cbvmpo2davw2 36261 Change second bound variab...
cbvixpdavw2 36262 Change bound variable and ...
cbvsumdavw2 36263 Change bound variable and ...
cbvproddavw2 36264 Change bound variable and ...
cbvitgdavw2 36265 Change bound variable and ...
cbvditgdavw2 36266 Change bound variable and ...
mpomulnzcnf 36267 Multiplication maps nonzer...
a1i14 36268 Add two antecedents to a w...
a1i24 36269 Add two antecedents to a w...
exp5d 36270 An exportation inference. ...
exp5g 36271 An exportation inference. ...
exp5k 36272 An exportation inference. ...
exp56 36273 An exportation inference. ...
exp58 36274 An exportation inference. ...
exp510 36275 An exportation inference. ...
exp511 36276 An exportation inference. ...
exp512 36277 An exportation inference. ...
3com12d 36278 Commutation in consequent....
imp5p 36279 A triple importation infer...
imp5q 36280 A triple importation infer...
ecase13d 36281 Deduction for elimination ...
subtr 36282 Transitivity of implicit s...
subtr2 36283 Transitivity of implicit s...
trer 36284 A relation intersected wit...
elicc3 36285 An equivalent membership c...
finminlem 36286 A useful lemma about finit...
gtinf 36287 Any number greater than an...
opnrebl 36288 A set is open in the stand...
opnrebl2 36289 A set is open in the stand...
nn0prpwlem 36290 Lemma for ~ nn0prpw . Use...
nn0prpw 36291 Two nonnegative integers a...
topbnd 36292 Two equivalent expressions...
opnbnd 36293 A set is open iff it is di...
cldbnd 36294 A set is closed iff it con...
ntruni 36295 A union of interiors is a ...
clsun 36296 A pairwise union of closur...
clsint2 36297 The closure of an intersec...
opnregcld 36298 A set is regularly closed ...
cldregopn 36299 A set if regularly open if...
neiin 36300 Two neighborhoods intersec...
hmeoclda 36301 Homeomorphisms preserve cl...
hmeocldb 36302 Homeomorphisms preserve cl...
ivthALT 36303 An alternate proof of the ...
fnerel 36306 Fineness is a relation. (...
isfne 36307 The predicate " ` B ` is f...
isfne4 36308 The predicate " ` B ` is f...
isfne4b 36309 A condition for a topology...
isfne2 36310 The predicate " ` B ` is f...
isfne3 36311 The predicate " ` B ` is f...
fnebas 36312 A finer cover covers the s...
fnetg 36313 A finer cover generates a ...
fnessex 36314 If ` B ` is finer than ` A...
fneuni 36315 If ` B ` is finer than ` A...
fneint 36316 If a cover is finer than a...
fness 36317 A cover is finer than its ...
fneref 36318 Reflexivity of the finenes...
fnetr 36319 Transitivity of the finene...
fneval 36320 Two covers are finer than ...
fneer 36321 Fineness intersected with ...
topfne 36322 Fineness for covers corres...
topfneec 36323 A cover is equivalent to a...
topfneec2 36324 A topology is precisely id...
fnessref 36325 A cover is finer iff it ha...
refssfne 36326 A cover is a refinement if...
neibastop1 36327 A collection of neighborho...
neibastop2lem 36328 Lemma for ~ neibastop2 . ...
neibastop2 36329 In the topology generated ...
neibastop3 36330 The topology generated by ...
topmtcl 36331 The meet of a collection o...
topmeet 36332 Two equivalent formulation...
topjoin 36333 Two equivalent formulation...
fnemeet1 36334 The meet of a collection o...
fnemeet2 36335 The meet of equivalence cl...
fnejoin1 36336 Join of equivalence classe...
fnejoin2 36337 Join of equivalence classe...
fgmin 36338 Minimality property of a g...
neifg 36339 The neighborhood filter of...
tailfval 36340 The tail function for a di...
tailval 36341 The tail of an element in ...
eltail 36342 An element of a tail. (Co...
tailf 36343 The tail function of a dir...
tailini 36344 A tail contains its initia...
tailfb 36345 The collection of tails of...
filnetlem1 36346 Lemma for ~ filnet . Chan...
filnetlem2 36347 Lemma for ~ filnet . The ...
filnetlem3 36348 Lemma for ~ filnet . (Con...
filnetlem4 36349 Lemma for ~ filnet . (Con...
filnet 36350 A filter has the same conv...
tb-ax1 36351 The first of three axioms ...
tb-ax2 36352 The second of three axioms...
tb-ax3 36353 The third of three axioms ...
tbsyl 36354 The weak syllogism from Ta...
re1ax2lem 36355 Lemma for ~ re1ax2 . (Con...
re1ax2 36356 ~ ax-2 rederived from the ...
naim1 36357 Constructor theorem for ` ...
naim2 36358 Constructor theorem for ` ...
naim1i 36359 Constructor rule for ` -/\...
naim2i 36360 Constructor rule for ` -/\...
naim12i 36361 Constructor rule for ` -/\...
nabi1i 36362 Constructor rule for ` -/\...
nabi2i 36363 Constructor rule for ` -/\...
nabi12i 36364 Constructor rule for ` -/\...
df3nandALT1 36367 The double nand expressed ...
df3nandALT2 36368 The double nand expressed ...
andnand1 36369 Double and in terms of dou...
imnand2 36370 An ` -> ` nand relation. ...
nalfal 36371 Not all sets hold ` F. ` a...
nexntru 36372 There does not exist a set...
nexfal 36373 There does not exist a set...
neufal 36374 There does not exist exact...
neutru 36375 There does not exist exact...
nmotru 36376 There does not exist at mo...
mofal 36377 There exist at most one se...
nrmo 36378 "At most one" restricted e...
meran1 36379 A single axiom for proposi...
meran2 36380 A single axiom for proposi...
meran3 36381 A single axiom for proposi...
waj-ax 36382 A single axiom for proposi...
lukshef-ax2 36383 A single axiom for proposi...
arg-ax 36384 A single axiom for proposi...
negsym1 36385 In the paper "On Variable ...
imsym1 36386 A symmetry with ` -> ` . ...
bisym1 36387 A symmetry with ` <-> ` . ...
consym1 36388 A symmetry with ` /\ ` . ...
dissym1 36389 A symmetry with ` \/ ` . ...
nandsym1 36390 A symmetry with ` -/\ ` . ...
unisym1 36391 A symmetry with ` A. ` . ...
exisym1 36392 A symmetry with ` E. ` . ...
unqsym1 36393 A symmetry with ` E! ` . ...
amosym1 36394 A symmetry with ` E* ` . ...
subsym1 36395 A symmetry with ` [ x / y ...
ontopbas 36396 An ordinal number is a top...
onsstopbas 36397 The class of ordinal numbe...
onpsstopbas 36398 The class of ordinal numbe...
ontgval 36399 The topology generated fro...
ontgsucval 36400 The topology generated fro...
onsuctop 36401 A successor ordinal number...
onsuctopon 36402 One of the topologies on a...
ordtoplem 36403 Membership of the class of...
ordtop 36404 An ordinal is a topology i...
onsucconni 36405 A successor ordinal number...
onsucconn 36406 A successor ordinal number...
ordtopconn 36407 An ordinal topology is con...
onintopssconn 36408 An ordinal topology is con...
onsuct0 36409 A successor ordinal number...
ordtopt0 36410 An ordinal topology is T_0...
onsucsuccmpi 36411 The successor of a success...
onsucsuccmp 36412 The successor of a success...
limsucncmpi 36413 The successor of a limit o...
limsucncmp 36414 The successor of a limit o...
ordcmp 36415 An ordinal topology is com...
ssoninhaus 36416 The ordinal topologies ` 1...
onint1 36417 The ordinal T_1 spaces are...
oninhaus 36418 The ordinal Hausdorff spac...
fveleq 36419 Please add description her...
findfvcl 36420 Please add description her...
findreccl 36421 Please add description her...
findabrcl 36422 Please add description her...
nnssi2 36423 Convert a theorem for real...
nnssi3 36424 Convert a theorem for real...
nndivsub 36425 Please add description her...
nndivlub 36426 A factor of a positive int...
ee7.2aOLD 36429 Lemma for Euclid's Element...
weiunlem1 36430 Lemma for ~ weiunpo , ~ we...
weiunlem2 36431 Lemma for ~ weiunpo , ~ we...
weiunfrlem 36432 Lemma for ~ weiunfr . (Co...
weiunpo 36433 A partial ordering on an i...
weiunso 36434 A strict ordering on an in...
weiunfr 36435 A well-founded relation on...
weiunse 36436 The relation constructed i...
weiunwe 36437 A well-ordering on an inde...
numiunnum 36438 An indexed union of sets i...
dnival 36439 Value of the "distance to ...
dnicld1 36440 Closure theorem for the "d...
dnicld2 36441 Closure theorem for the "d...
dnif 36442 The "distance to nearest i...
dnizeq0 36443 The distance to nearest in...
dnizphlfeqhlf 36444 The distance to nearest in...
rddif2 36445 Variant of ~ rddif . (Con...
dnibndlem1 36446 Lemma for ~ dnibnd . (Con...
dnibndlem2 36447 Lemma for ~ dnibnd . (Con...
dnibndlem3 36448 Lemma for ~ dnibnd . (Con...
dnibndlem4 36449 Lemma for ~ dnibnd . (Con...
dnibndlem5 36450 Lemma for ~ dnibnd . (Con...
dnibndlem6 36451 Lemma for ~ dnibnd . (Con...
dnibndlem7 36452 Lemma for ~ dnibnd . (Con...
dnibndlem8 36453 Lemma for ~ dnibnd . (Con...
dnibndlem9 36454 Lemma for ~ dnibnd . (Con...
dnibndlem10 36455 Lemma for ~ dnibnd . (Con...
dnibndlem11 36456 Lemma for ~ dnibnd . (Con...
dnibndlem12 36457 Lemma for ~ dnibnd . (Con...
dnibndlem13 36458 Lemma for ~ dnibnd . (Con...
dnibnd 36459 The "distance to nearest i...
dnicn 36460 The "distance to nearest i...
knoppcnlem1 36461 Lemma for ~ knoppcn . (Co...
knoppcnlem2 36462 Lemma for ~ knoppcn . (Co...
knoppcnlem3 36463 Lemma for ~ knoppcn . (Co...
knoppcnlem4 36464 Lemma for ~ knoppcn . (Co...
knoppcnlem5 36465 Lemma for ~ knoppcn . (Co...
knoppcnlem6 36466 Lemma for ~ knoppcn . (Co...
knoppcnlem7 36467 Lemma for ~ knoppcn . (Co...
knoppcnlem8 36468 Lemma for ~ knoppcn . (Co...
knoppcnlem9 36469 Lemma for ~ knoppcn . (Co...
knoppcnlem10 36470 Lemma for ~ knoppcn . (Co...
knoppcnlem11 36471 Lemma for ~ knoppcn . (Co...
knoppcn 36472 The continuous nowhere dif...
knoppcld 36473 Closure theorem for Knopp'...
unblimceq0lem 36474 Lemma for ~ unblimceq0 . ...
unblimceq0 36475 If ` F ` is unbounded near...
unbdqndv1 36476 If the difference quotient...
unbdqndv2lem1 36477 Lemma for ~ unbdqndv2 . (...
unbdqndv2lem2 36478 Lemma for ~ unbdqndv2 . (...
unbdqndv2 36479 Variant of ~ unbdqndv1 wit...
knoppndvlem1 36480 Lemma for ~ knoppndv . (C...
knoppndvlem2 36481 Lemma for ~ knoppndv . (C...
knoppndvlem3 36482 Lemma for ~ knoppndv . (C...
knoppndvlem4 36483 Lemma for ~ knoppndv . (C...
knoppndvlem5 36484 Lemma for ~ knoppndv . (C...
knoppndvlem6 36485 Lemma for ~ knoppndv . (C...
knoppndvlem7 36486 Lemma for ~ knoppndv . (C...
knoppndvlem8 36487 Lemma for ~ knoppndv . (C...
knoppndvlem9 36488 Lemma for ~ knoppndv . (C...
knoppndvlem10 36489 Lemma for ~ knoppndv . (C...
knoppndvlem11 36490 Lemma for ~ knoppndv . (C...
knoppndvlem12 36491 Lemma for ~ knoppndv . (C...
knoppndvlem13 36492 Lemma for ~ knoppndv . (C...
knoppndvlem14 36493 Lemma for ~ knoppndv . (C...
knoppndvlem15 36494 Lemma for ~ knoppndv . (C...
knoppndvlem16 36495 Lemma for ~ knoppndv . (C...
knoppndvlem17 36496 Lemma for ~ knoppndv . (C...
knoppndvlem18 36497 Lemma for ~ knoppndv . (C...
knoppndvlem19 36498 Lemma for ~ knoppndv . (C...
knoppndvlem20 36499 Lemma for ~ knoppndv . (C...
knoppndvlem21 36500 Lemma for ~ knoppndv . (C...
knoppndvlem22 36501 Lemma for ~ knoppndv . (C...
knoppndv 36502 The continuous nowhere dif...
knoppf 36503 Knopp's function is a func...
knoppcn2 36504 Variant of ~ knoppcn with ...
cnndvlem1 36505 Lemma for ~ cnndv . (Cont...
cnndvlem2 36506 Lemma for ~ cnndv . (Cont...
cnndv 36507 There exists a continuous ...
bj-mp2c 36508 A double _modus ponens_ in...
bj-mp2d 36509 A double _modus ponens_ in...
bj-0 36510 A syntactic theorem. See ...
bj-1 36511 In this proof, the use of ...
bj-a1k 36512 Weakening of ~ ax-1 . As ...
bj-poni 36513 Inference associated with ...
bj-nnclav 36514 When ` F. ` is substituted...
bj-nnclavi 36515 Inference associated with ...
bj-nnclavc 36516 Commuted form of ~ bj-nncl...
bj-nnclavci 36517 Inference associated with ...
bj-jarrii 36518 Inference associated with ...
bj-imim21 36519 The propositional function...
bj-imim21i 36520 Inference associated with ...
bj-peircestab 36521 Over minimal implicational...
bj-stabpeirce 36522 This minimal implicational...
bj-syl66ib 36523 A mixed syllogism inferenc...
bj-orim2 36524 Proof of ~ orim2 from the ...
bj-currypeirce 36525 Curry's axiom ~ curryax (a...
bj-peircecurry 36526 Peirce's axiom ~ peirce im...
bj-animbi 36527 Conjunction in terms of im...
bj-currypara 36528 Curry's paradox. Note tha...
bj-con2com 36529 A commuted form of the con...
bj-con2comi 36530 Inference associated with ...
bj-nimn 36531 If a formula is true, then...
bj-nimni 36532 Inference associated with ...
bj-peircei 36533 Inference associated with ...
bj-looinvi 36534 Inference associated with ...
bj-looinvii 36535 Inference associated with ...
bj-mt2bi 36536 Version of ~ mt2 where the...
bj-ntrufal 36537 The negation of a theorem ...
bj-fal 36538 Shortening of ~ fal using ...
bj-jaoi1 36539 Shortens ~ orfa2 (58>53), ...
bj-jaoi2 36540 Shortens ~ consensus (110>...
bj-dfbi4 36541 Alternate definition of th...
bj-dfbi5 36542 Alternate definition of th...
bj-dfbi6 36543 Alternate definition of th...
bj-bijust0ALT 36544 Alternate proof of ~ bijus...
bj-bijust00 36545 A self-implication does no...
bj-consensus 36546 Version of ~ consensus exp...
bj-consensusALT 36547 Alternate proof of ~ bj-co...
bj-df-ifc 36548 Candidate definition for t...
bj-dfif 36549 Alternate definition of th...
bj-ififc 36550 A biconditional connecting...
bj-imbi12 36551 Uncurried (imported) form ...
bj-falor 36552 Dual of ~ truan (which has...
bj-falor2 36553 Dual of ~ truan . (Contri...
bj-bibibi 36554 A property of the bicondit...
bj-imn3ani 36555 Duplication of ~ bnj1224 ....
bj-andnotim 36556 Two ways of expressing a c...
bj-bi3ant 36557 This used to be in the mai...
bj-bisym 36558 This used to be in the mai...
bj-bixor 36559 Equivalence of two ternary...
bj-axdd2 36560 This implication, proved u...
bj-axd2d 36561 This implication, proved u...
bj-axtd 36562 This implication, proved f...
bj-gl4 36563 In a normal modal logic, t...
bj-axc4 36564 Over minimal calculus, the...
prvlem1 36569 An elementary property of ...
prvlem2 36570 An elementary property of ...
bj-babygodel 36571 See the section header com...
bj-babylob 36572 See the section header com...
bj-godellob 36573 Proof of Gödel's theo...
bj-genr 36574 Generalization rule on the...
bj-genl 36575 Generalization rule on the...
bj-genan 36576 Generalization rule on a c...
bj-mpgs 36577 From a closed form theorem...
bj-2alim 36578 Closed form of ~ 2alimi . ...
bj-2exim 36579 Closed form of ~ 2eximi . ...
bj-alanim 36580 Closed form of ~ alanimi ....
bj-2albi 36581 Closed form of ~ 2albii . ...
bj-notalbii 36582 Equivalence of universal q...
bj-2exbi 36583 Closed form of ~ 2exbii . ...
bj-3exbi 36584 Closed form of ~ 3exbii . ...
bj-sylggt 36585 Stronger form of ~ sylgt ,...
bj-sylgt2 36586 Uncurried (imported) form ...
bj-alrimg 36587 The general form of the *a...
bj-alrimd 36588 A slightly more general ~ ...
bj-sylget 36589 Dual statement of ~ sylgt ...
bj-sylget2 36590 Uncurried (imported) form ...
bj-exlimg 36591 The general form of the *e...
bj-sylge 36592 Dual statement of ~ sylg (...
bj-exlimd 36593 A slightly more general ~ ...
bj-nfimexal 36594 A weak from of nonfreeness...
bj-alexim 36595 Closed form of ~ aleximi ....
bj-nexdh 36596 Closed form of ~ nexdh (ac...
bj-nexdh2 36597 Uncurried (imported) form ...
bj-hbxfrbi 36598 Closed form of ~ hbxfrbi ....
bj-hbyfrbi 36599 Version of ~ bj-hbxfrbi wi...
bj-exalim 36600 Distribute quantifiers ove...
bj-exalimi 36601 An inference for distribut...
bj-exalims 36602 Distributing quantifiers o...
bj-exalimsi 36603 An inference for distribut...
bj-ax12ig 36604 A lemma used to prove a we...
bj-ax12i 36605 A weakening of ~ bj-ax12ig...
bj-nfimt 36606 Closed form of ~ nfim and ...
bj-cbvalimt 36607 A lemma in closed form use...
bj-cbveximt 36608 A lemma in closed form use...
bj-eximALT 36609 Alternate proof of ~ exim ...
bj-aleximiALT 36610 Alternate proof of ~ alexi...
bj-eximcom 36611 A commuted form of ~ exim ...
bj-ax12wlem 36612 A lemma used to prove a we...
bj-cbvalim 36613 A lemma used to prove ~ bj...
bj-cbvexim 36614 A lemma used to prove ~ bj...
bj-cbvalimi 36615 An equality-free general i...
bj-cbveximi 36616 An equality-free general i...
bj-cbval 36617 Changing a bound variable ...
bj-cbvex 36618 Changing a bound variable ...
bj-ssbeq 36621 Substitution in an equalit...
bj-ssblem1 36622 A lemma for the definiens ...
bj-ssblem2 36623 An instance of ~ ax-11 pro...
bj-ax12v 36624 A weaker form of ~ ax-12 a...
bj-ax12 36625 Remove a DV condition from...
bj-ax12ssb 36626 Axiom ~ bj-ax12 expressed ...
bj-19.41al 36627 Special case of ~ 19.41 pr...
bj-equsexval 36628 Special case of ~ equsexv ...
bj-subst 36629 Proof of ~ sbalex from cor...
bj-ssbid2 36630 A special case of ~ sbequ2...
bj-ssbid2ALT 36631 Alternate proof of ~ bj-ss...
bj-ssbid1 36632 A special case of ~ sbequ1...
bj-ssbid1ALT 36633 Alternate proof of ~ bj-ss...
bj-ax6elem1 36634 Lemma for ~ bj-ax6e . (Co...
bj-ax6elem2 36635 Lemma for ~ bj-ax6e . (Co...
bj-ax6e 36636 Proof of ~ ax6e (hence ~ a...
bj-spimvwt 36637 Closed form of ~ spimvw . ...
bj-spnfw 36638 Theorem close to a closed ...
bj-cbvexiw 36639 Change bound variable. Th...
bj-cbvexivw 36640 Change bound variable. Th...
bj-modald 36641 A short form of the axiom ...
bj-denot 36642 A weakening of ~ ax-6 and ...
bj-eqs 36643 A lemma for substitutions,...
bj-cbvexw 36644 Change bound variable. Th...
bj-ax12w 36645 The general statement that...
bj-ax89 36646 A theorem which could be u...
bj-cleljusti 36647 One direction of ~ cleljus...
bj-alcomexcom 36648 Commutation of two existen...
bj-hbalt 36649 Closed form of ~ hbal . W...
axc11n11 36650 Proof of ~ axc11n from { ~...
axc11n11r 36651 Proof of ~ axc11n from { ~...
bj-axc16g16 36652 Proof of ~ axc16g from { ~...
bj-ax12v3 36653 A weak version of ~ ax-12 ...
bj-ax12v3ALT 36654 Alternate proof of ~ bj-ax...
bj-sb 36655 A weak variant of ~ sbid2 ...
bj-modalbe 36656 The predicate-calculus ver...
bj-spst 36657 Closed form of ~ sps . On...
bj-19.21bit 36658 Closed form of ~ 19.21bi ....
bj-19.23bit 36659 Closed form of ~ 19.23bi ....
bj-nexrt 36660 Closed form of ~ nexr . C...
bj-alrim 36661 Closed form of ~ alrimi . ...
bj-alrim2 36662 Uncurried (imported) form ...
bj-nfdt0 36663 A theorem close to a close...
bj-nfdt 36664 Closed form of ~ nf5d and ...
bj-nexdt 36665 Closed form of ~ nexd . (...
bj-nexdvt 36666 Closed form of ~ nexdv . ...
bj-alexbiex 36667 Adding a second quantifier...
bj-exexbiex 36668 Adding a second quantifier...
bj-alalbial 36669 Adding a second quantifier...
bj-exalbial 36670 Adding a second quantifier...
bj-19.9htbi 36671 Strengthening ~ 19.9ht by ...
bj-hbntbi 36672 Strengthening ~ hbnt by re...
bj-biexal1 36673 A general FOL biconditiona...
bj-biexal2 36674 When ` ph ` is substituted...
bj-biexal3 36675 When ` ph ` is substituted...
bj-bialal 36676 When ` ph ` is substituted...
bj-biexex 36677 When ` ph ` is substituted...
bj-hbext 36678 Closed form of ~ hbex . (...
bj-nfalt 36679 Closed form of ~ nfal . (...
bj-nfext 36680 Closed form of ~ nfex . (...
bj-eeanvw 36681 Version of ~ exdistrv with...
bj-modal4 36682 First-order logic form of ...
bj-modal4e 36683 First-order logic form of ...
bj-modalb 36684 A short form of the axiom ...
bj-wnf1 36685 When ` ph ` is substituted...
bj-wnf2 36686 When ` ph ` is substituted...
bj-wnfanf 36687 When ` ph ` is substituted...
bj-wnfenf 36688 When ` ph ` is substituted...
bj-substax12 36689 Equivalent form of the axi...
bj-substw 36690 Weak form of the LHS of ~ ...
bj-nnfbi 36693 If two formulas are equiva...
bj-nnfbd 36694 If two formulas are equiva...
bj-nnfbii 36695 If two formulas are equiva...
bj-nnfa 36696 Nonfreeness implies the eq...
bj-nnfad 36697 Nonfreeness implies the eq...
bj-nnfai 36698 Nonfreeness implies the eq...
bj-nnfe 36699 Nonfreeness implies the eq...
bj-nnfed 36700 Nonfreeness implies the eq...
bj-nnfei 36701 Nonfreeness implies the eq...
bj-nnfea 36702 Nonfreeness implies the eq...
bj-nnfead 36703 Nonfreeness implies the eq...
bj-nnfeai 36704 Nonfreeness implies the eq...
bj-dfnnf2 36705 Alternate definition of ~ ...
bj-nnfnfTEMP 36706 New nonfreeness implies ol...
bj-wnfnf 36707 When ` ph ` is substituted...
bj-nnfnt 36708 A variable is nonfree in a...
bj-nnftht 36709 A variable is nonfree in a...
bj-nnfth 36710 A variable is nonfree in a...
bj-nnfnth 36711 A variable is nonfree in t...
bj-nnfim1 36712 A consequence of nonfreene...
bj-nnfim2 36713 A consequence of nonfreene...
bj-nnfim 36714 Nonfreeness in the anteced...
bj-nnfimd 36715 Nonfreeness in the anteced...
bj-nnfan 36716 Nonfreeness in both conjun...
bj-nnfand 36717 Nonfreeness in both conjun...
bj-nnfor 36718 Nonfreeness in both disjun...
bj-nnford 36719 Nonfreeness in both disjun...
bj-nnfbit 36720 Nonfreeness in both sides ...
bj-nnfbid 36721 Nonfreeness in both sides ...
bj-nnfv 36722 A non-occurring variable i...
bj-nnf-alrim 36723 Proof of the closed form o...
bj-nnf-exlim 36724 Proof of the closed form o...
bj-dfnnf3 36725 Alternate definition of no...
bj-nfnnfTEMP 36726 New nonfreeness is equival...
bj-nnfa1 36727 See ~ nfa1 . (Contributed...
bj-nnfe1 36728 See ~ nfe1 . (Contributed...
bj-19.12 36729 See ~ 19.12 . Could be la...
bj-nnflemaa 36730 One of four lemmas for non...
bj-nnflemee 36731 One of four lemmas for non...
bj-nnflemae 36732 One of four lemmas for non...
bj-nnflemea 36733 One of four lemmas for non...
bj-nnfalt 36734 See ~ nfal and ~ bj-nfalt ...
bj-nnfext 36735 See ~ nfex and ~ bj-nfext ...
bj-stdpc5t 36736 Alias of ~ bj-nnf-alrim fo...
bj-19.21t 36737 Statement ~ 19.21t proved ...
bj-19.23t 36738 Statement ~ 19.23t proved ...
bj-19.36im 36739 One direction of ~ 19.36 f...
bj-19.37im 36740 One direction of ~ 19.37 f...
bj-19.42t 36741 Closed form of ~ 19.42 fro...
bj-19.41t 36742 Closed form of ~ 19.41 fro...
bj-sbft 36743 Version of ~ sbft using ` ...
bj-pm11.53vw 36744 Version of ~ pm11.53v with...
bj-pm11.53v 36745 Version of ~ pm11.53v with...
bj-pm11.53a 36746 A variant of ~ pm11.53v . ...
bj-equsvt 36747 A variant of ~ equsv . (C...
bj-equsalvwd 36748 Variant of ~ equsalvw . (...
bj-equsexvwd 36749 Variant of ~ equsexvw . (...
bj-sbievwd 36750 Variant of ~ sbievw . (Co...
bj-axc10 36751 Alternate proof of ~ axc10...
bj-alequex 36752 A fol lemma. See ~ aleque...
bj-spimt2 36753 A step in the proof of ~ s...
bj-cbv3ta 36754 Closed form of ~ cbv3 . (...
bj-cbv3tb 36755 Closed form of ~ cbv3 . (...
bj-hbsb3t 36756 A theorem close to a close...
bj-hbsb3 36757 Shorter proof of ~ hbsb3 ....
bj-nfs1t 36758 A theorem close to a close...
bj-nfs1t2 36759 A theorem close to a close...
bj-nfs1 36760 Shorter proof of ~ nfs1 (t...
bj-axc10v 36761 Version of ~ axc10 with a ...
bj-spimtv 36762 Version of ~ spimt with a ...
bj-cbv3hv2 36763 Version of ~ cbv3h with tw...
bj-cbv1hv 36764 Version of ~ cbv1h with a ...
bj-cbv2hv 36765 Version of ~ cbv2h with a ...
bj-cbv2v 36766 Version of ~ cbv2 with a d...
bj-cbvaldv 36767 Version of ~ cbvald with a...
bj-cbvexdv 36768 Version of ~ cbvexd with a...
bj-cbval2vv 36769 Version of ~ cbval2vv with...
bj-cbvex2vv 36770 Version of ~ cbvex2vv with...
bj-cbvaldvav 36771 Version of ~ cbvaldva with...
bj-cbvexdvav 36772 Version of ~ cbvexdva with...
bj-cbvex4vv 36773 Version of ~ cbvex4v with ...
bj-equsalhv 36774 Version of ~ equsalh with ...
bj-axc11nv 36775 Version of ~ axc11n with a...
bj-aecomsv 36776 Version of ~ aecoms with a...
bj-axc11v 36777 Version of ~ axc11 with a ...
bj-drnf2v 36778 Version of ~ drnf2 with a ...
bj-equs45fv 36779 Version of ~ equs45f with ...
bj-hbs1 36780 Version of ~ hbsb2 with a ...
bj-nfs1v 36781 Version of ~ nfsb2 with a ...
bj-hbsb2av 36782 Version of ~ hbsb2a with a...
bj-hbsb3v 36783 Version of ~ hbsb3 with a ...
bj-nfsab1 36784 Remove dependency on ~ ax-...
bj-dtrucor2v 36785 Version of ~ dtrucor2 with...
bj-hbaeb2 36786 Biconditional version of a...
bj-hbaeb 36787 Biconditional version of ~...
bj-hbnaeb 36788 Biconditional version of ~...
bj-dvv 36789 A special instance of ~ bj...
bj-equsal1t 36790 Duplication of ~ wl-equsal...
bj-equsal1ti 36791 Inference associated with ...
bj-equsal1 36792 One direction of ~ equsal ...
bj-equsal2 36793 One direction of ~ equsal ...
bj-equsal 36794 Shorter proof of ~ equsal ...
stdpc5t 36795 Closed form of ~ stdpc5 . ...
bj-stdpc5 36796 More direct proof of ~ std...
2stdpc5 36797 A double ~ stdpc5 (one dir...
bj-19.21t0 36798 Proof of ~ 19.21t from ~ s...
exlimii 36799 Inference associated with ...
ax11-pm 36800 Proof of ~ ax-11 similar t...
ax6er 36801 Commuted form of ~ ax6e . ...
exlimiieq1 36802 Inferring a theorem when i...
exlimiieq2 36803 Inferring a theorem when i...
ax11-pm2 36804 Proof of ~ ax-11 from the ...
bj-sbsb 36805 Biconditional showing two ...
bj-dfsb2 36806 Alternate (dual) definitio...
bj-sbf3 36807 Substitution has no effect...
bj-sbf4 36808 Substitution has no effect...
bj-eu3f 36809 Version of ~ eu3v where th...
bj-sblem1 36810 Lemma for substitution. (...
bj-sblem2 36811 Lemma for substitution. (...
bj-sblem 36812 Lemma for substitution. (...
bj-sbievw1 36813 Lemma for substitution. (...
bj-sbievw2 36814 Lemma for substitution. (...
bj-sbievw 36815 Lemma for substitution. C...
bj-sbievv 36816 Version of ~ sbie with a s...
bj-moeub 36817 Uniqueness is equivalent t...
bj-sbidmOLD 36818 Obsolete proof of ~ sbidm ...
bj-dvelimdv 36819 Deduction form of ~ dvelim...
bj-dvelimdv1 36820 Curried (exported) form of...
bj-dvelimv 36821 A version of ~ dvelim usin...
bj-nfeel2 36822 Nonfreeness in a membershi...
bj-axc14nf 36823 Proof of a version of ~ ax...
bj-axc14 36824 Alternate proof of ~ axc14...
mobidvALT 36825 Alternate proof of ~ mobid...
sbn1ALT 36826 Alternate proof of ~ sbn1 ...
eliminable1 36827 A theorem used to prove th...
eliminable2a 36828 A theorem used to prove th...
eliminable2b 36829 A theorem used to prove th...
eliminable2c 36830 A theorem used to prove th...
eliminable3a 36831 A theorem used to prove th...
eliminable3b 36832 A theorem used to prove th...
eliminable-velab 36833 A theorem used to prove th...
eliminable-veqab 36834 A theorem used to prove th...
eliminable-abeqv 36835 A theorem used to prove th...
eliminable-abeqab 36836 A theorem used to prove th...
eliminable-abelv 36837 A theorem used to prove th...
eliminable-abelab 36838 A theorem used to prove th...
bj-denoteslem 36839 Duplicate of ~ issettru an...
bj-denotesALTV 36840 Moved to main as ~ iseqset...
bj-issettruALTV 36841 Moved to main as ~ issettr...
bj-elabtru 36842 This is as close as we can...
bj-issetwt 36843 Closed form of ~ bj-issetw...
bj-issetw 36844 The closest one can get to...
bj-issetiv 36845 Version of ~ bj-isseti wit...
bj-isseti 36846 Version of ~ isseti with a...
bj-ralvw 36847 A weak version of ~ ralv n...
bj-rexvw 36848 A weak version of ~ rexv n...
bj-rababw 36849 A weak version of ~ rabab ...
bj-rexcom4bv 36850 Version of ~ rexcom4b and ...
bj-rexcom4b 36851 Remove from ~ rexcom4b dep...
bj-ceqsalt0 36852 The FOL content of ~ ceqsa...
bj-ceqsalt1 36853 The FOL content of ~ ceqsa...
bj-ceqsalt 36854 Remove from ~ ceqsalt depe...
bj-ceqsaltv 36855 Version of ~ bj-ceqsalt wi...
bj-ceqsalg0 36856 The FOL content of ~ ceqsa...
bj-ceqsalg 36857 Remove from ~ ceqsalg depe...
bj-ceqsalgALT 36858 Alternate proof of ~ bj-ce...
bj-ceqsalgv 36859 Version of ~ bj-ceqsalg wi...
bj-ceqsalgvALT 36860 Alternate proof of ~ bj-ce...
bj-ceqsal 36861 Remove from ~ ceqsal depen...
bj-ceqsalv 36862 Remove from ~ ceqsalv depe...
bj-spcimdv 36863 Remove from ~ spcimdv depe...
bj-spcimdvv 36864 Remove from ~ spcimdv depe...
elelb 36865 Equivalence between two co...
bj-pwvrelb 36866 Characterization of the el...
bj-nfcsym 36867 The nonfreeness quantifier...
bj-sbeqALT 36868 Substitution in an equalit...
bj-sbeq 36869 Distribute proper substitu...
bj-sbceqgALT 36870 Distribute proper substitu...
bj-csbsnlem 36871 Lemma for ~ bj-csbsn (in t...
bj-csbsn 36872 Substitution in a singleto...
bj-sbel1 36873 Version of ~ sbcel1g when ...
bj-abv 36874 The class of sets verifyin...
bj-abvALT 36875 Alternate version of ~ bj-...
bj-ab0 36876 The class of sets verifyin...
bj-abf 36877 Shorter proof of ~ abf (wh...
bj-csbprc 36878 More direct proof of ~ csb...
bj-exlimvmpi 36879 A Fol lemma ( ~ exlimiv fo...
bj-exlimmpi 36880 Lemma for ~ bj-vtoclg1f1 (...
bj-exlimmpbi 36881 Lemma for theorems of the ...
bj-exlimmpbir 36882 Lemma for theorems of the ...
bj-vtoclf 36883 Remove dependency on ~ ax-...
bj-vtocl 36884 Remove dependency on ~ ax-...
bj-vtoclg1f1 36885 The FOL content of ~ vtocl...
bj-vtoclg1f 36886 Reprove ~ vtoclg1f from ~ ...
bj-vtoclg1fv 36887 Version of ~ bj-vtoclg1f w...
bj-vtoclg 36888 A version of ~ vtoclg with...
bj-rabeqbid 36889 Version of ~ rabeqbidv wit...
bj-seex 36890 Version of ~ seex with a d...
bj-nfcf 36891 Version of ~ df-nfc with a...
bj-zfauscl 36892 General version of ~ zfaus...
bj-elabd2ALT 36893 Alternate proof of ~ elabd...
bj-unrab 36894 Generalization of ~ unrab ...
bj-inrab 36895 Generalization of ~ inrab ...
bj-inrab2 36896 Shorter proof of ~ inrab ....
bj-inrab3 36897 Generalization of ~ dfrab3...
bj-rabtr 36898 Restricted class abstracti...
bj-rabtrALT 36899 Alternate proof of ~ bj-ra...
bj-rabtrAUTO 36900 Proof of ~ bj-rabtr found ...
bj-gabss 36903 Inclusion of generalized c...
bj-gabssd 36904 Inclusion of generalized c...
bj-gabeqd 36905 Equality of generalized cl...
bj-gabeqis 36906 Equality of generalized cl...
bj-elgab 36907 Elements of a generalized ...
bj-gabima 36908 Generalized class abstract...
bj-ru1 36911 A version of Russell's par...
bj-ru 36912 Remove dependency on ~ ax-...
currysetlem 36913 Lemma for ~ currysetlem , ...
curryset 36914 Curry's paradox in set the...
currysetlem1 36915 Lemma for ~ currysetALT . ...
currysetlem2 36916 Lemma for ~ currysetALT . ...
currysetlem3 36917 Lemma for ~ currysetALT . ...
currysetALT 36918 Alternate proof of ~ curry...
bj-n0i 36919 Inference associated with ...
bj-disjsn01 36920 Disjointness of the single...
bj-0nel1 36921 The empty set does not bel...
bj-1nel0 36922 ` 1o ` does not belong to ...
bj-xpimasn 36923 The image of a singleton, ...
bj-xpima1sn 36924 The image of a singleton b...
bj-xpima1snALT 36925 Alternate proof of ~ bj-xp...
bj-xpima2sn 36926 The image of a singleton b...
bj-xpnzex 36927 If the first factor of a p...
bj-xpexg2 36928 Curried (exported) form of...
bj-xpnzexb 36929 If the first factor of a p...
bj-cleq 36930 Substitution property for ...
bj-snsetex 36931 The class of sets "whose s...
bj-clexab 36932 Sethood of certain classes...
bj-sngleq 36935 Substitution property for ...
bj-elsngl 36936 Characterization of the el...
bj-snglc 36937 Characterization of the el...
bj-snglss 36938 The singletonization of a ...
bj-0nelsngl 36939 The empty set is not a mem...
bj-snglinv 36940 Inverse of singletonizatio...
bj-snglex 36941 A class is a set if and on...
bj-tageq 36944 Substitution property for ...
bj-eltag 36945 Characterization of the el...
bj-0eltag 36946 The empty set belongs to t...
bj-tagn0 36947 The tagging of a class is ...
bj-tagss 36948 The tagging of a class is ...
bj-snglsstag 36949 The singletonization is in...
bj-sngltagi 36950 The singletonization is in...
bj-sngltag 36951 The singletonization and t...
bj-tagci 36952 Characterization of the el...
bj-tagcg 36953 Characterization of the el...
bj-taginv 36954 Inverse of tagging. (Cont...
bj-tagex 36955 A class is a set if and on...
bj-xtageq 36956 The products of a given cl...
bj-xtagex 36957 The product of a set and t...
bj-projeq 36960 Substitution property for ...
bj-projeq2 36961 Substitution property for ...
bj-projun 36962 The class projection on a ...
bj-projex 36963 Sethood of the class proje...
bj-projval 36964 Value of the class project...
bj-1upleq 36967 Substitution property for ...
bj-pr1eq 36970 Substitution property for ...
bj-pr1un 36971 The first projection prese...
bj-pr1val 36972 Value of the first project...
bj-pr11val 36973 Value of the first project...
bj-pr1ex 36974 Sethood of the first proje...
bj-1uplth 36975 The characteristic propert...
bj-1uplex 36976 A monuple is a set if and ...
bj-1upln0 36977 A monuple is nonempty. (C...
bj-2upleq 36980 Substitution property for ...
bj-pr21val 36981 Value of the first project...
bj-pr2eq 36984 Substitution property for ...
bj-pr2un 36985 The second projection pres...
bj-pr2val 36986 Value of the second projec...
bj-pr22val 36987 Value of the second projec...
bj-pr2ex 36988 Sethood of the second proj...
bj-2uplth 36989 The characteristic propert...
bj-2uplex 36990 A couple is a set if and o...
bj-2upln0 36991 A couple is nonempty. (Co...
bj-2upln1upl 36992 A couple is never equal to...
bj-rcleqf 36993 Relative version of ~ cleq...
bj-rcleq 36994 Relative version of ~ dfcl...
bj-reabeq 36995 Relative form of ~ eqabb ....
bj-disj2r 36996 Relative version of ~ ssdi...
bj-sscon 36997 Contraposition law for rel...
bj-abex 36998 Two ways of stating that t...
bj-clex 36999 Two ways of stating that a...
bj-axsn 37000 Two ways of stating the ax...
bj-snexg 37002 A singleton built on a set...
bj-snex 37003 A singleton is a set. See...
bj-axbun 37004 Two ways of stating the ax...
bj-unexg 37006 Existence of binary unions...
bj-prexg 37007 Existence of unordered pai...
bj-prex 37008 Existence of unordered pai...
bj-axadj 37009 Two ways of stating the ax...
bj-adjg1 37011 Existence of the result of...
bj-snfromadj 37012 Singleton from adjunction ...
bj-prfromadj 37013 Unordered pair from adjunc...
bj-adjfrombun 37014 Adjunction from singleton ...
eleq2w2ALT 37015 Alternate proof of ~ eleq2...
bj-clel3gALT 37016 Alternate proof of ~ clel3...
bj-pw0ALT 37017 Alternate proof of ~ pw0 ....
bj-sselpwuni 37018 Quantitative version of ~ ...
bj-unirel 37019 Quantitative version of ~ ...
bj-elpwg 37020 If the intersection of two...
bj-velpwALT 37021 This theorem ~ bj-velpwALT...
bj-elpwgALT 37022 Alternate proof of ~ elpwg...
bj-vjust 37023 Justification theorem for ...
bj-nul 37024 Two formulations of the ax...
bj-nuliota 37025 Definition of the empty se...
bj-nuliotaALT 37026 Alternate proof of ~ bj-nu...
bj-vtoclgfALT 37027 Alternate proof of ~ vtocl...
bj-elsn12g 37028 Join of ~ elsng and ~ elsn...
bj-elsnb 37029 Biconditional version of ~...
bj-pwcfsdom 37030 Remove hypothesis from ~ p...
bj-grur1 37031 Remove hypothesis from ~ g...
bj-bm1.3ii 37032 The extension of a predica...
bj-dfid2ALT 37033 Alternate version of ~ dfi...
bj-0nelopab 37034 The empty set is never an ...
bj-brrelex12ALT 37035 Two classes related by a b...
bj-epelg 37036 The membership relation an...
bj-epelb 37037 Two classes are related by...
bj-nsnid 37038 A set does not contain the...
bj-rdg0gALT 37039 Alternate proof of ~ rdg0g...
bj-evaleq 37040 Equality theorem for the `...
bj-evalfun 37041 The evaluation at a class ...
bj-evalfn 37042 The evaluation at a class ...
bj-evalval 37043 Value of the evaluation at...
bj-evalid 37044 The evaluation at a set of...
bj-ndxarg 37045 Proof of ~ ndxarg from ~ b...
bj-evalidval 37046 Closed general form of ~ s...
bj-rest00 37049 An elementwise intersectio...
bj-restsn 37050 An elementwise intersectio...
bj-restsnss 37051 Special case of ~ bj-rests...
bj-restsnss2 37052 Special case of ~ bj-rests...
bj-restsn0 37053 An elementwise intersectio...
bj-restsn10 37054 Special case of ~ bj-rests...
bj-restsnid 37055 The elementwise intersecti...
bj-rest10 37056 An elementwise intersectio...
bj-rest10b 37057 Alternate version of ~ bj-...
bj-restn0 37058 An elementwise intersectio...
bj-restn0b 37059 Alternate version of ~ bj-...
bj-restpw 37060 The elementwise intersecti...
bj-rest0 37061 An elementwise intersectio...
bj-restb 37062 An elementwise intersectio...
bj-restv 37063 An elementwise intersectio...
bj-resta 37064 An elementwise intersectio...
bj-restuni 37065 The union of an elementwis...
bj-restuni2 37066 The union of an elementwis...
bj-restreg 37067 A reformulation of the axi...
bj-raldifsn 37068 All elements in a set sati...
bj-0int 37069 If ` A ` is a collection o...
bj-mooreset 37070 A Moore collection is a se...
bj-ismoore 37073 Characterization of Moore ...
bj-ismoored0 37074 Necessary condition to be ...
bj-ismoored 37075 Necessary condition to be ...
bj-ismoored2 37076 Necessary condition to be ...
bj-ismooredr 37077 Sufficient condition to be...
bj-ismooredr2 37078 Sufficient condition to be...
bj-discrmoore 37079 The powerclass ` ~P A ` is...
bj-0nmoore 37080 The empty set is not a Moo...
bj-snmoore 37081 A singleton is a Moore col...
bj-snmooreb 37082 A singleton is a Moore col...
bj-prmoore 37083 A pair formed of two neste...
bj-0nelmpt 37084 The empty set is not an el...
bj-mptval 37085 Value of a function given ...
bj-dfmpoa 37086 An equivalent definition o...
bj-mpomptALT 37087 Alternate proof of ~ mpomp...
setsstrset 37104 Relation between ~ df-sets...
bj-nfald 37105 Variant of ~ nfald . (Con...
bj-nfexd 37106 Variant of ~ nfexd . (Con...
copsex2d 37107 Implicit substitution dedu...
copsex2b 37108 Biconditional form of ~ co...
opelopabd 37109 Membership of an ordere pa...
opelopabb 37110 Membership of an ordered p...
opelopabbv 37111 Membership of an ordered p...
bj-opelrelex 37112 The coordinates of an orde...
bj-opelresdm 37113 If an ordered pair is in a...
bj-brresdm 37114 If two classes are related...
brabd0 37115 Expressing that two sets a...
brabd 37116 Expressing that two sets a...
bj-brab2a1 37117 "Unbounded" version of ~ b...
bj-opabssvv 37118 A variant of ~ relopabiv (...
bj-funidres 37119 The restricted identity re...
bj-opelidb 37120 Characterization of the or...
bj-opelidb1 37121 Characterization of the or...
bj-inexeqex 37122 Lemma for ~ bj-opelid (but...
bj-elsn0 37123 If the intersection of two...
bj-opelid 37124 Characterization of the or...
bj-ideqg 37125 Characterization of the cl...
bj-ideqgALT 37126 Alternate proof of ~ bj-id...
bj-ideqb 37127 Characterization of classe...
bj-idres 37128 Alternate expression for t...
bj-opelidres 37129 Characterization of the or...
bj-idreseq 37130 Sufficient condition for t...
bj-idreseqb 37131 Characterization for two c...
bj-ideqg1 37132 For sets, the identity rel...
bj-ideqg1ALT 37133 Alternate proof of bj-ideq...
bj-opelidb1ALT 37134 Characterization of the co...
bj-elid3 37135 Characterization of the co...
bj-elid4 37136 Characterization of the el...
bj-elid5 37137 Characterization of the el...
bj-elid6 37138 Characterization of the el...
bj-elid7 37139 Characterization of the el...
bj-diagval 37142 Value of the functionalize...
bj-diagval2 37143 Value of the functionalize...
bj-eldiag 37144 Characterization of the el...
bj-eldiag2 37145 Characterization of the el...
bj-imdirvallem 37148 Lemma for ~ bj-imdirval an...
bj-imdirval 37149 Value of the functionalize...
bj-imdirval2lem 37150 Lemma for ~ bj-imdirval2 a...
bj-imdirval2 37151 Value of the functionalize...
bj-imdirval3 37152 Value of the functionalize...
bj-imdiridlem 37153 Lemma for ~ bj-imdirid and...
bj-imdirid 37154 Functorial property of the...
bj-opelopabid 37155 Membership in an ordered-p...
bj-opabco 37156 Composition of ordered-pai...
bj-xpcossxp 37157 The composition of two Car...
bj-imdirco 37158 Functorial property of the...
bj-iminvval 37161 Value of the functionalize...
bj-iminvval2 37162 Value of the functionalize...
bj-iminvid 37163 Functorial property of the...
bj-inftyexpitaufo 37170 The function ` inftyexpita...
bj-inftyexpitaudisj 37173 An element of the circle a...
bj-inftyexpiinv 37176 Utility theorem for the in...
bj-inftyexpiinj 37177 Injectivity of the paramet...
bj-inftyexpidisj 37178 An element of the circle a...
bj-ccinftydisj 37181 The circle at infinity is ...
bj-elccinfty 37182 A lemma for infinite exten...
bj-ccssccbar 37185 Complex numbers are extend...
bj-ccinftyssccbar 37186 Infinite extended complex ...
bj-pinftyccb 37189 The class ` pinfty ` is an...
bj-pinftynrr 37190 The extended complex numbe...
bj-minftyccb 37193 The class ` minfty ` is an...
bj-minftynrr 37194 The extended complex numbe...
bj-pinftynminfty 37195 The extended complex numbe...
bj-rrhatsscchat 37204 The real projective line i...
bj-imafv 37219 If the direct image of a s...
bj-funun 37220 Value of a function expres...
bj-fununsn1 37221 Value of a function expres...
bj-fununsn2 37222 Value of a function expres...
bj-fvsnun1 37223 The value of a function wi...
bj-fvsnun2 37224 The value of a function wi...
bj-fvmptunsn1 37225 Value of a function expres...
bj-fvmptunsn2 37226 Value of a function expres...
bj-iomnnom 37227 The canonical bijection fr...
bj-smgrpssmgm 37236 Semigroups are magmas. (C...
bj-smgrpssmgmel 37237 Semigroups are magmas (ele...
bj-mndsssmgrp 37238 Monoids are semigroups. (...
bj-mndsssmgrpel 37239 Monoids are semigroups (el...
bj-cmnssmnd 37240 Commutative monoids are mo...
bj-cmnssmndel 37241 Commutative monoids are mo...
bj-grpssmnd 37242 Groups are monoids. (Cont...
bj-grpssmndel 37243 Groups are monoids (elemen...
bj-ablssgrp 37244 Abelian groups are groups....
bj-ablssgrpel 37245 Abelian groups are groups ...
bj-ablsscmn 37246 Abelian groups are commuta...
bj-ablsscmnel 37247 Abelian groups are commuta...
bj-modssabl 37248 (The additive groups of) m...
bj-vecssmod 37249 Vector spaces are modules....
bj-vecssmodel 37250 Vector spaces are modules ...
bj-finsumval0 37253 Value of a finite sum. (C...
bj-fvimacnv0 37254 Variant of ~ fvimacnv wher...
bj-isvec 37255 The predicate "is a vector...
bj-fldssdrng 37256 Fields are division rings....
bj-flddrng 37257 Fields are division rings ...
bj-rrdrg 37258 The field of real numbers ...
bj-isclm 37259 The predicate "is a subcom...
bj-isrvec 37262 The predicate "is a real v...
bj-rvecmod 37263 Real vector spaces are mod...
bj-rvecssmod 37264 Real vector spaces are mod...
bj-rvecrr 37265 The field of scalars of a ...
bj-isrvecd 37266 The predicate "is a real v...
bj-rvecvec 37267 Real vector spaces are vec...
bj-isrvec2 37268 The predicate "is a real v...
bj-rvecssvec 37269 Real vector spaces are vec...
bj-rveccmod 37270 Real vector spaces are sub...
bj-rvecsscmod 37271 Real vector spaces are sub...
bj-rvecsscvec 37272 Real vector spaces are sub...
bj-rveccvec 37273 Real vector spaces are sub...
bj-rvecssabl 37274 (The additive groups of) r...
bj-rvecabl 37275 (The additive groups of) r...
bj-subcom 37276 A consequence of commutati...
bj-lineqi 37277 Solution of a (scalar) lin...
bj-bary1lem 37278 Lemma for ~ bj-bary1 : exp...
bj-bary1lem1 37279 Lemma for ~ bj-bary1 : com...
bj-bary1 37280 Barycentric coordinates in...
bj-endval 37283 Value of the monoid of end...
bj-endbase 37284 Base set of the monoid of ...
bj-endcomp 37285 Composition law of the mon...
bj-endmnd 37286 The monoid of endomorphism...
taupilem3 37287 Lemma for tau-related theo...
taupilemrplb 37288 A set of positive reals ha...
taupilem1 37289 Lemma for ~ taupi . A pos...
taupilem2 37290 Lemma for ~ taupi . The s...
taupi 37291 Relationship between ` _ta...
dfgcd3 37292 Alternate definition of th...
irrdifflemf 37293 Lemma for ~ irrdiff . The...
irrdiff 37294 The irrationals are exactl...
iccioo01 37295 The closed unit interval i...
csbrecsg 37296 Move class substitution in...
csbrdgg 37297 Move class substitution in...
csboprabg 37298 Move class substitution in...
csbmpo123 37299 Move class substitution in...
con1bii2 37300 A contraposition inference...
con2bii2 37301 A contraposition inference...
vtoclefex 37302 Implicit substitution of a...
rnmptsn 37303 The range of a function ma...
f1omptsnlem 37304 This is the core of the pr...
f1omptsn 37305 A function mapping to sing...
mptsnunlem 37306 This is the core of the pr...
mptsnun 37307 A class ` B ` is equal to ...
dissneqlem 37308 This is the core of the pr...
dissneq 37309 Any topology that contains...
exlimim 37310 Closed form of ~ exlimimd ...
exlimimd 37311 Existential elimination ru...
exellim 37312 Closed form of ~ exellimdd...
exellimddv 37313 Eliminate an antecedent wh...
topdifinfindis 37314 Part of Exercise 3 of [Mun...
topdifinffinlem 37315 This is the core of the pr...
topdifinffin 37316 Part of Exercise 3 of [Mun...
topdifinf 37317 Part of Exercise 3 of [Mun...
topdifinfeq 37318 Two different ways of defi...
icorempo 37319 Closed-below, open-above i...
icoreresf 37320 Closed-below, open-above i...
icoreval 37321 Value of the closed-below,...
icoreelrnab 37322 Elementhood in the set of ...
isbasisrelowllem1 37323 Lemma for ~ isbasisrelowl ...
isbasisrelowllem2 37324 Lemma for ~ isbasisrelowl ...
icoreclin 37325 The set of closed-below, o...
isbasisrelowl 37326 The set of all closed-belo...
icoreunrn 37327 The union of all closed-be...
istoprelowl 37328 The set of all closed-belo...
icoreelrn 37329 A class abstraction which ...
iooelexlt 37330 An element of an open inte...
relowlssretop 37331 The lower limit topology o...
relowlpssretop 37332 The lower limit topology o...
sucneqond 37333 Inequality of an ordinal s...
sucneqoni 37334 Inequality of an ordinal s...
onsucuni3 37335 If an ordinal number has a...
1oequni2o 37336 The ordinal number ` 1o ` ...
rdgsucuni 37337 If an ordinal number has a...
rdgeqoa 37338 If a recursive function wi...
elxp8 37339 Membership in a Cartesian ...
cbveud 37340 Deduction used to change b...
cbvreud 37341 Deduction used to change b...
difunieq 37342 The difference of unions i...
inunissunidif 37343 Theorem about subsets of t...
rdgellim 37344 Elementhood in a recursive...
rdglimss 37345 A recursive definition at ...
rdgssun 37346 In a recursive definition ...
exrecfnlem 37347 Lemma for ~ exrecfn . (Co...
exrecfn 37348 Theorem about the existenc...
exrecfnpw 37349 For any base set, a set wh...
finorwe 37350 If the Axiom of Infinity i...
dffinxpf 37353 This theorem is the same a...
finxpeq1 37354 Equality theorem for Carte...
finxpeq2 37355 Equality theorem for Carte...
csbfinxpg 37356 Distribute proper substitu...
finxpreclem1 37357 Lemma for ` ^^ ` recursion...
finxpreclem2 37358 Lemma for ` ^^ ` recursion...
finxp0 37359 The value of Cartesian exp...
finxp1o 37360 The value of Cartesian exp...
finxpreclem3 37361 Lemma for ` ^^ ` recursion...
finxpreclem4 37362 Lemma for ` ^^ ` recursion...
finxpreclem5 37363 Lemma for ` ^^ ` recursion...
finxpreclem6 37364 Lemma for ` ^^ ` recursion...
finxpsuclem 37365 Lemma for ~ finxpsuc . (C...
finxpsuc 37366 The value of Cartesian exp...
finxp2o 37367 The value of Cartesian exp...
finxp3o 37368 The value of Cartesian exp...
finxpnom 37369 Cartesian exponentiation w...
finxp00 37370 Cartesian exponentiation o...
iunctb2 37371 Using the axiom of countab...
domalom 37372 A class which dominates ev...
isinf2 37373 The converse of ~ isinf . ...
ctbssinf 37374 Using the axiom of choice,...
ralssiun 37375 The index set of an indexe...
nlpineqsn 37376 For every point ` p ` of a...
nlpfvineqsn 37377 Given a subset ` A ` of ` ...
fvineqsnf1 37378 A theorem about functions ...
fvineqsneu 37379 A theorem about functions ...
fvineqsneq 37380 A theorem about functions ...
pibp16 37381 Property P000016 of pi-bas...
pibp19 37382 Property P000019 of pi-bas...
pibp21 37383 Property P000021 of pi-bas...
pibt1 37384 Theorem T000001 of pi-base...
pibt2 37385 Theorem T000002 of pi-base...
wl-section-prop 37386 Intuitionistic logic is no...
wl-section-boot 37390 In this section, I provide...
wl-luk-imim1i 37391 Inference adding common co...
wl-luk-syl 37392 An inference version of th...
wl-luk-imtrid 37393 A syllogism rule of infere...
wl-luk-pm2.18d 37394 Deduction based on reducti...
wl-luk-con4i 37395 Inference rule. Copy of ~...
wl-luk-pm2.24i 37396 Inference rule. Copy of ~...
wl-luk-a1i 37397 Inference rule. Copy of ~...
wl-luk-mpi 37398 A nested _modus ponens_ in...
wl-luk-imim2i 37399 Inference adding common an...
wl-luk-imtrdi 37400 A syllogism rule of infere...
wl-luk-ax3 37401 ~ ax-3 proved from Lukasie...
wl-luk-ax1 37402 ~ ax-1 proved from Lukasie...
wl-luk-pm2.27 37403 This theorem, called "Asse...
wl-luk-com12 37404 Inference that swaps (comm...
wl-luk-pm2.21 37405 From a wff and its negatio...
wl-luk-con1i 37406 A contraposition inference...
wl-luk-ja 37407 Inference joining the ante...
wl-luk-imim2 37408 A closed form of syllogism...
wl-luk-a1d 37409 Deduction introducing an e...
wl-luk-ax2 37410 ~ ax-2 proved from Lukasie...
wl-luk-id 37411 Principle of identity. Th...
wl-luk-notnotr 37412 Converse of double negatio...
wl-luk-pm2.04 37413 Swap antecedents. Theorem...
wl-section-impchain 37414 An implication like ` ( ps...
wl-impchain-mp-x 37415 This series of theorems pr...
wl-impchain-mp-0 37416 This theorem is the start ...
wl-impchain-mp-1 37417 This theorem is in fact a ...
wl-impchain-mp-2 37418 This theorem is in fact a ...
wl-impchain-com-1.x 37419 It is often convenient to ...
wl-impchain-com-1.1 37420 A degenerate form of antec...
wl-impchain-com-1.2 37421 This theorem is in fact a ...
wl-impchain-com-1.3 37422 This theorem is in fact a ...
wl-impchain-com-1.4 37423 This theorem is in fact a ...
wl-impchain-com-n.m 37424 This series of theorems al...
wl-impchain-com-2.3 37425 This theorem is in fact a ...
wl-impchain-com-2.4 37426 This theorem is in fact a ...
wl-impchain-com-3.2.1 37427 This theorem is in fact a ...
wl-impchain-a1-x 37428 If an implication chain is...
wl-impchain-a1-1 37429 Inference rule, a copy of ...
wl-impchain-a1-2 37430 Inference rule, a copy of ...
wl-impchain-a1-3 37431 Inference rule, a copy of ...
wl-ifp-ncond1 37432 If one case of an ` if- ` ...
wl-ifp-ncond2 37433 If one case of an ` if- ` ...
wl-ifpimpr 37434 If one case of an ` if- ` ...
wl-ifp4impr 37435 If one case of an ` if- ` ...
wl-df-3xor 37436 Alternative definition of ...
wl-df3xor2 37437 Alternative definition of ...
wl-df3xor3 37438 Alternative form of ~ wl-d...
wl-3xortru 37439 If the first input is true...
wl-3xorfal 37440 If the first input is fals...
wl-3xorbi 37441 Triple xor can be replaced...
wl-3xorbi2 37442 Alternative form of ~ wl-3...
wl-3xorbi123d 37443 Equivalence theorem for tr...
wl-3xorbi123i 37444 Equivalence theorem for tr...
wl-3xorrot 37445 Rotation law for triple xo...
wl-3xorcoma 37446 Commutative law for triple...
wl-3xorcomb 37447 Commutative law for triple...
wl-3xornot1 37448 Flipping the first input f...
wl-3xornot 37449 Triple xor distributes ove...
wl-1xor 37450 In the recursive scheme ...
wl-2xor 37451 In the recursive scheme ...
wl-df-3mintru2 37452 Alternative definition of ...
wl-df2-3mintru2 37453 The adder carry in disjunc...
wl-df3-3mintru2 37454 The adder carry in conjunc...
wl-df4-3mintru2 37455 An alternative definition ...
wl-1mintru1 37456 Using the recursion formul...
wl-1mintru2 37457 Using the recursion formul...
wl-2mintru1 37458 Using the recursion formul...
wl-2mintru2 37459 Using the recursion formul...
wl-df3maxtru1 37460 Assuming "(n+1)-maxtru1" `...
wl-ax13lem1 37462 A version of ~ ax-wl-13v w...
wl-mps 37463 Replacing a nested consequ...
wl-syls1 37464 Replacing a nested consequ...
wl-syls2 37465 Replacing a nested anteced...
wl-embant 37466 A true wff can always be a...
wl-orel12 37467 In a conjunctive normal fo...
wl-cases2-dnf 37468 A particular instance of ~...
wl-cbvmotv 37469 Change bound variable. Us...
wl-moteq 37470 Change bound variable. Us...
wl-motae 37471 Change bound variable. Us...
wl-moae 37472 Two ways to express "at mo...
wl-euae 37473 Two ways to express "exact...
wl-nax6im 37474 The following series of th...
wl-hbae1 37475 This specialization of ~ h...
wl-naevhba1v 37476 An instance of ~ hbn1w app...
wl-spae 37477 Prove an instance of ~ sp ...
wl-speqv 37478 Under the assumption ` -. ...
wl-19.8eqv 37479 Under the assumption ` -. ...
wl-19.2reqv 37480 Under the assumption ` -. ...
wl-nfalv 37481 If ` x ` is not present in...
wl-nfimf1 37482 An antecedent is irrelevan...
wl-nfae1 37483 Unlike ~ nfae , this speci...
wl-nfnae1 37484 Unlike ~ nfnae , this spec...
wl-aetr 37485 A transitive law for varia...
wl-axc11r 37486 Same as ~ axc11r , but usi...
wl-dral1d 37487 A version of ~ dral1 with ...
wl-cbvalnaed 37488 ~ wl-cbvalnae with a conte...
wl-cbvalnae 37489 A more general version of ...
wl-exeq 37490 The semantics of ` E. x y ...
wl-aleq 37491 The semantics of ` A. x y ...
wl-nfeqfb 37492 Extend ~ nfeqf to an equiv...
wl-nfs1t 37493 If ` y ` is not free in ` ...
wl-equsalvw 37494 Version of ~ equsalv with ...
wl-equsald 37495 Deduction version of ~ equ...
wl-equsaldv 37496 Deduction version of ~ equ...
wl-equsal 37497 A useful equivalence relat...
wl-equsal1t 37498 The expression ` x = y ` i...
wl-equsalcom 37499 This simple equivalence ea...
wl-equsal1i 37500 The antecedent ` x = y ` i...
wl-sbid2ft 37501 A more general version of ...
wl-cbvalsbi 37502 Change bounded variables i...
wl-sbrimt 37503 Substitution with a variab...
wl-sblimt 37504 Substitution with a variab...
wl-sb9v 37505 Commutation of quantificat...
wl-sb8ft 37506 Substitution of variable i...
wl-sb8eft 37507 Substitution of variable i...
wl-sb8t 37508 Substitution of variable i...
wl-sb8et 37509 Substitution of variable i...
wl-sbhbt 37510 Closed form of ~ sbhb . C...
wl-sbnf1 37511 Two ways expressing that `...
wl-equsb3 37512 ~ equsb3 with a distinctor...
wl-equsb4 37513 Substitution applied to an...
wl-2sb6d 37514 Version of ~ 2sb6 with a c...
wl-sbcom2d-lem1 37515 Lemma used to prove ~ wl-s...
wl-sbcom2d-lem2 37516 Lemma used to prove ~ wl-s...
wl-sbcom2d 37517 Version of ~ sbcom2 with a...
wl-sbalnae 37518 A theorem used in eliminat...
wl-sbal1 37519 A theorem used in eliminat...
wl-sbal2 37520 Move quantifier in and out...
wl-2spsbbi 37521 ~ spsbbi applied twice. (...
wl-lem-exsb 37522 This theorem provides a ba...
wl-lem-nexmo 37523 This theorem provides a ba...
wl-lem-moexsb 37524 The antecedent ` A. x ( ph...
wl-alanbii 37525 This theorem extends ~ ala...
wl-mo2df 37526 Version of ~ mof with a co...
wl-mo2tf 37527 Closed form of ~ mof with ...
wl-eudf 37528 Version of ~ eu6 with a co...
wl-eutf 37529 Closed form of ~ eu6 with ...
wl-euequf 37530 ~ euequ proved with a dist...
wl-mo2t 37531 Closed form of ~ mof . (C...
wl-mo3t 37532 Closed form of ~ mo3 . (C...
wl-nfsbtv 37533 Closed form of ~ nfsbv . ...
wl-sb8eut 37534 Substitution of variable i...
wl-sb8eutv 37535 Substitution of variable i...
wl-sb8mot 37536 Substitution of variable i...
wl-sb8motv 37537 Substitution of variable i...
wl-issetft 37538 A closed form of ~ issetf ...
wl-axc11rc11 37539 Proving ~ axc11r from ~ ax...
wl-ax11-lem1 37541 A transitive law for varia...
wl-ax11-lem2 37542 Lemma. (Contributed by Wo...
wl-ax11-lem3 37543 Lemma. (Contributed by Wo...
wl-ax11-lem4 37544 Lemma. (Contributed by Wo...
wl-ax11-lem5 37545 Lemma. (Contributed by Wo...
wl-ax11-lem6 37546 Lemma. (Contributed by Wo...
wl-ax11-lem7 37547 Lemma. (Contributed by Wo...
wl-ax11-lem8 37548 Lemma. (Contributed by Wo...
wl-ax11-lem9 37549 The easy part when ` x ` c...
wl-ax11-lem10 37550 We now have prepared every...
wl-clabv 37551 Variant of ~ df-clab , whe...
wl-dfclab 37552 Rederive ~ df-clab from ~ ...
wl-clabtv 37553 Using class abstraction in...
wl-clabt 37554 Using class abstraction in...
rabiun 37555 Abstraction restricted to ...
iundif1 37556 Indexed union of class dif...
imadifss 37557 The difference of images i...
cureq 37558 Equality theorem for curry...
unceq 37559 Equality theorem for uncur...
curf 37560 Functional property of cur...
uncf 37561 Functional property of unc...
curfv 37562 Value of currying. (Contr...
uncov 37563 Value of uncurrying. (Con...
curunc 37564 Currying of uncurrying. (...
unccur 37565 Uncurrying of currying. (...
phpreu 37566 Theorem related to pigeonh...
finixpnum 37567 A finite Cartesian product...
fin2solem 37568 Lemma for ~ fin2so . (Con...
fin2so 37569 Any totally ordered Tarski...
ltflcei 37570 Theorem to move the floor ...
leceifl 37571 Theorem to move the floor ...
sin2h 37572 Half-angle rule for sine. ...
cos2h 37573 Half-angle rule for cosine...
tan2h 37574 Half-angle rule for tangen...
lindsadd 37575 In a vector space, the uni...
lindsdom 37576 A linearly independent set...
lindsenlbs 37577 A maximal linearly indepen...
matunitlindflem1 37578 One direction of ~ matunit...
matunitlindflem2 37579 One direction of ~ matunit...
matunitlindf 37580 A matrix over a field is i...
ptrest 37581 Expressing a restriction o...
ptrecube 37582 Any point in an open set o...
poimirlem1 37583 Lemma for ~ poimir - the v...
poimirlem2 37584 Lemma for ~ poimir - conse...
poimirlem3 37585 Lemma for ~ poimir to add ...
poimirlem4 37586 Lemma for ~ poimir connect...
poimirlem5 37587 Lemma for ~ poimir to esta...
poimirlem6 37588 Lemma for ~ poimir establi...
poimirlem7 37589 Lemma for ~ poimir , simil...
poimirlem8 37590 Lemma for ~ poimir , estab...
poimirlem9 37591 Lemma for ~ poimir , estab...
poimirlem10 37592 Lemma for ~ poimir establi...
poimirlem11 37593 Lemma for ~ poimir connect...
poimirlem12 37594 Lemma for ~ poimir connect...
poimirlem13 37595 Lemma for ~ poimir - for a...
poimirlem14 37596 Lemma for ~ poimir - for a...
poimirlem15 37597 Lemma for ~ poimir , that ...
poimirlem16 37598 Lemma for ~ poimir establi...
poimirlem17 37599 Lemma for ~ poimir establi...
poimirlem18 37600 Lemma for ~ poimir stating...
poimirlem19 37601 Lemma for ~ poimir establi...
poimirlem20 37602 Lemma for ~ poimir establi...
poimirlem21 37603 Lemma for ~ poimir stating...
poimirlem22 37604 Lemma for ~ poimir , that ...
poimirlem23 37605 Lemma for ~ poimir , two w...
poimirlem24 37606 Lemma for ~ poimir , two w...
poimirlem25 37607 Lemma for ~ poimir stating...
poimirlem26 37608 Lemma for ~ poimir showing...
poimirlem27 37609 Lemma for ~ poimir showing...
poimirlem28 37610 Lemma for ~ poimir , a var...
poimirlem29 37611 Lemma for ~ poimir connect...
poimirlem30 37612 Lemma for ~ poimir combini...
poimirlem31 37613 Lemma for ~ poimir , assig...
poimirlem32 37614 Lemma for ~ poimir , combi...
poimir 37615 Poincare-Miranda theorem. ...
broucube 37616 Brouwer - or as Kulpa call...
heicant 37617 Heine-Cantor theorem: a co...
opnmbllem0 37618 Lemma for ~ ismblfin ; cou...
mblfinlem1 37619 Lemma for ~ ismblfin , ord...
mblfinlem2 37620 Lemma for ~ ismblfin , eff...
mblfinlem3 37621 The difference between two...
mblfinlem4 37622 Backward direction of ~ is...
ismblfin 37623 Measurability in terms of ...
ovoliunnfl 37624 ~ ovoliun is incompatible ...
ex-ovoliunnfl 37625 Demonstration of ~ ovoliun...
voliunnfl 37626 ~ voliun is incompatible w...
volsupnfl 37627 ~ volsup is incompatible w...
mbfresfi 37628 Measurability of a piecewi...
mbfposadd 37629 If the sum of two measurab...
cnambfre 37630 A real-valued, a.e. contin...
dvtanlem 37631 Lemma for ~ dvtan - the do...
dvtan 37632 Derivative of tangent. (C...
itg2addnclem 37633 An alternate expression fo...
itg2addnclem2 37634 Lemma for ~ itg2addnc . T...
itg2addnclem3 37635 Lemma incomprehensible in ...
itg2addnc 37636 Alternate proof of ~ itg2a...
itg2gt0cn 37637 ~ itg2gt0 holds on functio...
ibladdnclem 37638 Lemma for ~ ibladdnc ; cf ...
ibladdnc 37639 Choice-free analogue of ~ ...
itgaddnclem1 37640 Lemma for ~ itgaddnc ; cf....
itgaddnclem2 37641 Lemma for ~ itgaddnc ; cf....
itgaddnc 37642 Choice-free analogue of ~ ...
iblsubnc 37643 Choice-free analogue of ~ ...
itgsubnc 37644 Choice-free analogue of ~ ...
iblabsnclem 37645 Lemma for ~ iblabsnc ; cf....
iblabsnc 37646 Choice-free analogue of ~ ...
iblmulc2nc 37647 Choice-free analogue of ~ ...
itgmulc2nclem1 37648 Lemma for ~ itgmulc2nc ; c...
itgmulc2nclem2 37649 Lemma for ~ itgmulc2nc ; c...
itgmulc2nc 37650 Choice-free analogue of ~ ...
itgabsnc 37651 Choice-free analogue of ~ ...
itggt0cn 37652 ~ itggt0 holds for continu...
ftc1cnnclem 37653 Lemma for ~ ftc1cnnc ; cf....
ftc1cnnc 37654 Choice-free proof of ~ ftc...
ftc1anclem1 37655 Lemma for ~ ftc1anc - the ...
ftc1anclem2 37656 Lemma for ~ ftc1anc - rest...
ftc1anclem3 37657 Lemma for ~ ftc1anc - the ...
ftc1anclem4 37658 Lemma for ~ ftc1anc . (Co...
ftc1anclem5 37659 Lemma for ~ ftc1anc , the ...
ftc1anclem6 37660 Lemma for ~ ftc1anc - cons...
ftc1anclem7 37661 Lemma for ~ ftc1anc . (Co...
ftc1anclem8 37662 Lemma for ~ ftc1anc . (Co...
ftc1anc 37663 ~ ftc1a holds for function...
ftc2nc 37664 Choice-free proof of ~ ftc...
asindmre 37665 Real part of domain of dif...
dvasin 37666 Derivative of arcsine. (C...
dvacos 37667 Derivative of arccosine. ...
dvreasin 37668 Real derivative of arcsine...
dvreacos 37669 Real derivative of arccosi...
areacirclem1 37670 Antiderivative of cross-se...
areacirclem2 37671 Endpoint-inclusive continu...
areacirclem3 37672 Integrability of cross-sec...
areacirclem4 37673 Endpoint-inclusive continu...
areacirclem5 37674 Finding the cross-section ...
areacirc 37675 The area of a circle of ra...
unirep 37676 Define a quantity whose de...
cover2 37677 Two ways of expressing the...
cover2g 37678 Two ways of expressing the...
brabg2 37679 Relation by a binary relat...
opelopab3 37680 Ordered pair membership in...
cocanfo 37681 Cancellation of a surjecti...
brresi2 37682 Restriction of a binary re...
fnopabeqd 37683 Equality deduction for fun...
fvopabf4g 37684 Function value of an opera...
fnopabco 37685 Composition of a function ...
opropabco 37686 Composition of an operator...
cocnv 37687 Composition with a functio...
f1ocan1fv 37688 Cancel a composition by a ...
f1ocan2fv 37689 Cancel a composition by th...
inixp 37690 Intersection of Cartesian ...
upixp 37691 Universal property of the ...
abrexdom 37692 An indexed set is dominate...
abrexdom2 37693 An indexed set is dominate...
ac6gf 37694 Axiom of Choice. (Contrib...
indexa 37695 If for every element of an...
indexdom 37696 If for every element of an...
frinfm 37697 A subset of a well-founded...
welb 37698 A nonempty subset of a wel...
supex2g 37699 Existence of supremum. (C...
supclt 37700 Closure of supremum. (Con...
supubt 37701 Upper bound property of su...
filbcmb 37702 Combine a finite set of lo...
fzmul 37703 Membership of a product in...
sdclem2 37704 Lemma for ~ sdc . (Contri...
sdclem1 37705 Lemma for ~ sdc . (Contri...
sdc 37706 Strong dependent choice. ...
fdc 37707 Finite version of dependen...
fdc1 37708 Variant of ~ fdc with no s...
seqpo 37709 Two ways to say that a seq...
incsequz 37710 An increasing sequence of ...
incsequz2 37711 An increasing sequence of ...
nnubfi 37712 A bounded above set of pos...
nninfnub 37713 An infinite set of positiv...
subspopn 37714 An open set is open in the...
neificl 37715 Neighborhoods are closed u...
lpss2 37716 Limit points of a subset a...
metf1o 37717 Use a bijection with a met...
blssp 37718 A ball in the subspace met...
mettrifi 37719 Generalized triangle inequ...
lmclim2 37720 A sequence in a metric spa...
geomcau 37721 If the distance between co...
caures 37722 The restriction of a Cauch...
caushft 37723 A shifted Cauchy sequence ...
constcncf 37724 A constant function is a c...
cnres2 37725 The restriction of a conti...
cnresima 37726 A continuous function is c...
cncfres 37727 A continuous function on c...
istotbnd 37731 The predicate "is a totall...
istotbnd2 37732 The predicate "is a totall...
istotbnd3 37733 A metric space is totally ...
totbndmet 37734 The predicate "totally bou...
0totbnd 37735 The metric (there is only ...
sstotbnd2 37736 Condition for a subset of ...
sstotbnd 37737 Condition for a subset of ...
sstotbnd3 37738 Use a net that is not nece...
totbndss 37739 A subset of a totally boun...
equivtotbnd 37740 If the metric ` M ` is "st...
isbnd 37742 The predicate "is a bounde...
bndmet 37743 A bounded metric space is ...
isbndx 37744 A "bounded extended metric...
isbnd2 37745 The predicate "is a bounde...
isbnd3 37746 A metric space is bounded ...
isbnd3b 37747 A metric space is bounded ...
bndss 37748 A subset of a bounded metr...
blbnd 37749 A ball is bounded. (Contr...
ssbnd 37750 A subset of a metric space...
totbndbnd 37751 A totally bounded metric s...
equivbnd 37752 If the metric ` M ` is "st...
bnd2lem 37753 Lemma for ~ equivbnd2 and ...
equivbnd2 37754 If balls are totally bound...
prdsbnd 37755 The product metric over fi...
prdstotbnd 37756 The product metric over fi...
prdsbnd2 37757 If balls are totally bound...
cntotbnd 37758 A subset of the complex nu...
cnpwstotbnd 37759 A subset of ` A ^ I ` , wh...
ismtyval 37762 The set of isometries betw...
isismty 37763 The condition "is an isome...
ismtycnv 37764 The inverse of an isometry...
ismtyima 37765 The image of a ball under ...
ismtyhmeolem 37766 Lemma for ~ ismtyhmeo . (...
ismtyhmeo 37767 An isometry is a homeomorp...
ismtybndlem 37768 Lemma for ~ ismtybnd . (C...
ismtybnd 37769 Isometries preserve bounde...
ismtyres 37770 A restriction of an isomet...
heibor1lem 37771 Lemma for ~ heibor1 . A c...
heibor1 37772 One half of ~ heibor , tha...
heiborlem1 37773 Lemma for ~ heibor . We w...
heiborlem2 37774 Lemma for ~ heibor . Subs...
heiborlem3 37775 Lemma for ~ heibor . Usin...
heiborlem4 37776 Lemma for ~ heibor . Usin...
heiborlem5 37777 Lemma for ~ heibor . The ...
heiborlem6 37778 Lemma for ~ heibor . Sinc...
heiborlem7 37779 Lemma for ~ heibor . Sinc...
heiborlem8 37780 Lemma for ~ heibor . The ...
heiborlem9 37781 Lemma for ~ heibor . Disc...
heiborlem10 37782 Lemma for ~ heibor . The ...
heibor 37783 Generalized Heine-Borel Th...
bfplem1 37784 Lemma for ~ bfp . The seq...
bfplem2 37785 Lemma for ~ bfp . Using t...
bfp 37786 Banach fixed point theorem...
rrnval 37789 The n-dimensional Euclidea...
rrnmval 37790 The value of the Euclidean...
rrnmet 37791 Euclidean space is a metri...
rrndstprj1 37792 The distance between two p...
rrndstprj2 37793 Bound on the distance betw...
rrncmslem 37794 Lemma for ~ rrncms . (Con...
rrncms 37795 Euclidean space is complet...
repwsmet 37796 The supremum metric on ` R...
rrnequiv 37797 The supremum metric on ` R...
rrntotbnd 37798 A set in Euclidean space i...
rrnheibor 37799 Heine-Borel theorem for Eu...
ismrer1 37800 An isometry between ` RR `...
reheibor 37801 Heine-Borel theorem for re...
iccbnd 37802 A closed interval in ` RR ...
icccmpALT 37803 A closed interval in ` RR ...
isass 37808 The predicate "is an assoc...
isexid 37809 The predicate ` G ` has a ...
ismgmOLD 37812 Obsolete version of ~ ismg...
clmgmOLD 37813 Obsolete version of ~ mgmc...
opidonOLD 37814 Obsolete version of ~ mndp...
rngopidOLD 37815 Obsolete version of ~ mndp...
opidon2OLD 37816 Obsolete version of ~ mndp...
isexid2 37817 If ` G e. ( Magma i^i ExId...
exidu1 37818 Uniqueness of the left and...
idrval 37819 The value of the identity ...
iorlid 37820 A magma right and left ide...
cmpidelt 37821 A magma right and left ide...
smgrpismgmOLD 37824 Obsolete version of ~ sgrp...
issmgrpOLD 37825 Obsolete version of ~ issg...
smgrpmgm 37826 A semigroup is a magma. (...
smgrpassOLD 37827 Obsolete version of ~ sgrp...
mndoissmgrpOLD 37830 Obsolete version of ~ mnds...
mndoisexid 37831 A monoid has an identity e...
mndoismgmOLD 37832 Obsolete version of ~ mndm...
mndomgmid 37833 A monoid is a magma with a...
ismndo 37834 The predicate "is a monoid...
ismndo1 37835 The predicate "is a monoid...
ismndo2 37836 The predicate "is a monoid...
grpomndo 37837 A group is a monoid. (Con...
exidcl 37838 Closure of the binary oper...
exidreslem 37839 Lemma for ~ exidres and ~ ...
exidres 37840 The restriction of a binar...
exidresid 37841 The restriction of a binar...
ablo4pnp 37842 A commutative/associative ...
grpoeqdivid 37843 Two group elements are equ...
grposnOLD 37844 The group operation for th...
elghomlem1OLD 37847 Obsolete as of 15-Mar-2020...
elghomlem2OLD 37848 Obsolete as of 15-Mar-2020...
elghomOLD 37849 Obsolete version of ~ isgh...
ghomlinOLD 37850 Obsolete version of ~ ghml...
ghomidOLD 37851 Obsolete version of ~ ghmi...
ghomf 37852 Mapping property of a grou...
ghomco 37853 The composition of two gro...
ghomdiv 37854 Group homomorphisms preser...
grpokerinj 37855 A group homomorphism is in...
relrngo 37858 The class of all unital ri...
isrngo 37859 The predicate "is a (unita...
isrngod 37860 Conditions that determine ...
rngoi 37861 The properties of a unital...
rngosm 37862 Functionality of the multi...
rngocl 37863 Closure of the multiplicat...
rngoid 37864 The multiplication operati...
rngoideu 37865 The unity element of a rin...
rngodi 37866 Distributive law for the m...
rngodir 37867 Distributive law for the m...
rngoass 37868 Associative law for the mu...
rngo2 37869 A ring element plus itself...
rngoablo 37870 A ring's addition operatio...
rngoablo2 37871 In a unital ring the addit...
rngogrpo 37872 A ring's addition operatio...
rngone0 37873 The base set of a ring is ...
rngogcl 37874 Closure law for the additi...
rngocom 37875 The addition operation of ...
rngoaass 37876 The addition operation of ...
rngoa32 37877 The addition operation of ...
rngoa4 37878 Rearrangement of 4 terms i...
rngorcan 37879 Right cancellation law for...
rngolcan 37880 Left cancellation law for ...
rngo0cl 37881 A ring has an additive ide...
rngo0rid 37882 The additive identity of a...
rngo0lid 37883 The additive identity of a...
rngolz 37884 The zero of a unital ring ...
rngorz 37885 The zero of a unital ring ...
rngosn3 37886 Obsolete as of 25-Jan-2020...
rngosn4 37887 Obsolete as of 25-Jan-2020...
rngosn6 37888 Obsolete as of 25-Jan-2020...
rngonegcl 37889 A ring is closed under neg...
rngoaddneg1 37890 Adding the negative in a r...
rngoaddneg2 37891 Adding the negative in a r...
rngosub 37892 Subtraction in a ring, in ...
rngmgmbs4 37893 The range of an internal o...
rngodm1dm2 37894 In a unital ring the domai...
rngorn1 37895 In a unital ring the range...
rngorn1eq 37896 In a unital ring the range...
rngomndo 37897 In a unital ring the multi...
rngoidmlem 37898 The unity element of a rin...
rngolidm 37899 The unity element of a rin...
rngoridm 37900 The unity element of a rin...
rngo1cl 37901 The unity element of a rin...
rngoueqz 37902 Obsolete as of 23-Jan-2020...
rngonegmn1l 37903 Negation in a ring is the ...
rngonegmn1r 37904 Negation in a ring is the ...
rngoneglmul 37905 Negation of a product in a...
rngonegrmul 37906 Negation of a product in a...
rngosubdi 37907 Ring multiplication distri...
rngosubdir 37908 Ring multiplication distri...
zerdivemp1x 37909 In a unital ring a left in...
isdivrngo 37912 The predicate "is a divisi...
drngoi 37913 The properties of a divisi...
gidsn 37914 Obsolete as of 23-Jan-2020...
zrdivrng 37915 The zero ring is not a div...
dvrunz 37916 In a division ring the rin...
isgrpda 37917 Properties that determine ...
isdrngo1 37918 The predicate "is a divisi...
divrngcl 37919 The product of two nonzero...
isdrngo2 37920 A division ring is a ring ...
isdrngo3 37921 A division ring is a ring ...
rngohomval 37926 The set of ring homomorphi...
isrngohom 37927 The predicate "is a ring h...
rngohomf 37928 A ring homomorphism is a f...
rngohomcl 37929 Closure law for a ring hom...
rngohom1 37930 A ring homomorphism preser...
rngohomadd 37931 Ring homomorphisms preserv...
rngohommul 37932 Ring homomorphisms preserv...
rngogrphom 37933 A ring homomorphism is a g...
rngohom0 37934 A ring homomorphism preser...
rngohomsub 37935 Ring homomorphisms preserv...
rngohomco 37936 The composition of two rin...
rngokerinj 37937 A ring homomorphism is inj...
rngoisoval 37939 The set of ring isomorphis...
isrngoiso 37940 The predicate "is a ring i...
rngoiso1o 37941 A ring isomorphism is a bi...
rngoisohom 37942 A ring isomorphism is a ri...
rngoisocnv 37943 The inverse of a ring isom...
rngoisoco 37944 The composition of two rin...
isriscg 37946 The ring isomorphism relat...
isrisc 37947 The ring isomorphism relat...
risc 37948 The ring isomorphism relat...
risci 37949 Determine that two rings a...
riscer 37950 Ring isomorphism is an equ...
iscom2 37957 A device to add commutativ...
iscrngo 37958 The predicate "is a commut...
iscrngo2 37959 The predicate "is a commut...
iscringd 37960 Conditions that determine ...
flddivrng 37961 A field is a division ring...
crngorngo 37962 A commutative ring is a ri...
crngocom 37963 The multiplication operati...
crngm23 37964 Commutative/associative la...
crngm4 37965 Commutative/associative la...
fldcrngo 37966 A field is a commutative r...
isfld2 37967 The predicate "is a field"...
crngohomfo 37968 The image of a homomorphis...
idlval 37975 The class of ideals of a r...
isidl 37976 The predicate "is an ideal...
isidlc 37977 The predicate "is an ideal...
idlss 37978 An ideal of ` R ` is a sub...
idlcl 37979 An element of an ideal is ...
idl0cl 37980 An ideal contains ` 0 ` . ...
idladdcl 37981 An ideal is closed under a...
idllmulcl 37982 An ideal is closed under m...
idlrmulcl 37983 An ideal is closed under m...
idlnegcl 37984 An ideal is closed under n...
idlsubcl 37985 An ideal is closed under s...
rngoidl 37986 A ring ` R ` is an ` R ` i...
0idl 37987 The set containing only ` ...
1idl 37988 Two ways of expressing the...
0rngo 37989 In a ring, ` 0 = 1 ` iff t...
divrngidl 37990 The only ideals in a divis...
intidl 37991 The intersection of a none...
inidl 37992 The intersection of two id...
unichnidl 37993 The union of a nonempty ch...
keridl 37994 The kernel of a ring homom...
pridlval 37995 The class of prime ideals ...
ispridl 37996 The predicate "is a prime ...
pridlidl 37997 A prime ideal is an ideal....
pridlnr 37998 A prime ideal is a proper ...
pridl 37999 The main property of a pri...
ispridl2 38000 A condition that shows an ...
maxidlval 38001 The set of maximal ideals ...
ismaxidl 38002 The predicate "is a maxima...
maxidlidl 38003 A maximal ideal is an idea...
maxidlnr 38004 A maximal ideal is proper....
maxidlmax 38005 A maximal ideal is a maxim...
maxidln1 38006 One is not contained in an...
maxidln0 38007 A ring with a maximal idea...
isprrngo 38012 The predicate "is a prime ...
prrngorngo 38013 A prime ring is a ring. (...
smprngopr 38014 A simple ring (one whose o...
divrngpr 38015 A division ring is a prime...
isdmn 38016 The predicate "is a domain...
isdmn2 38017 The predicate "is a domain...
dmncrng 38018 A domain is a commutative ...
dmnrngo 38019 A domain is a ring. (Cont...
flddmn 38020 A field is a domain. (Con...
igenval 38023 The ideal generated by a s...
igenss 38024 A set is a subset of the i...
igenidl 38025 The ideal generated by a s...
igenmin 38026 The ideal generated by a s...
igenidl2 38027 The ideal generated by an ...
igenval2 38028 The ideal generated by a s...
prnc 38029 A principal ideal (an idea...
isfldidl 38030 Determine if a ring is a f...
isfldidl2 38031 Determine if a ring is a f...
ispridlc 38032 The predicate "is a prime ...
pridlc 38033 Property of a prime ideal ...
pridlc2 38034 Property of a prime ideal ...
pridlc3 38035 Property of a prime ideal ...
isdmn3 38036 The predicate "is a domain...
dmnnzd 38037 A domain has no zero-divis...
dmncan1 38038 Cancellation law for domai...
dmncan2 38039 Cancellation law for domai...
efald2 38040 A proof by contradiction. ...
notbinot1 38041 Simplification rule of neg...
bicontr 38042 Biconditional of its own n...
impor 38043 An equivalent formula for ...
orfa 38044 The falsum ` F. ` can be r...
notbinot2 38045 Commutation rule between n...
biimpor 38046 A rewriting rule for bicon...
orfa1 38047 Add a contradicting disjun...
orfa2 38048 Remove a contradicting dis...
bifald 38049 Infer the equivalence to a...
orsild 38050 A lemma for not-or-not eli...
orsird 38051 A lemma for not-or-not eli...
cnf1dd 38052 A lemma for Conjunctive No...
cnf2dd 38053 A lemma for Conjunctive No...
cnfn1dd 38054 A lemma for Conjunctive No...
cnfn2dd 38055 A lemma for Conjunctive No...
or32dd 38056 A rearrangement of disjunc...
notornotel1 38057 A lemma for not-or-not eli...
notornotel2 38058 A lemma for not-or-not eli...
contrd 38059 A proof by contradiction, ...
an12i 38060 An inference from commutin...
exmid2 38061 An excluded middle law. (...
selconj 38062 An inference for selecting...
truconj 38063 Add true as a conjunct. (...
orel 38064 An inference for disjuncti...
negel 38065 An inference for negation ...
botel 38066 An inference for bottom el...
tradd 38067 Add top ad a conjunct. (C...
gm-sbtru 38068 Substitution does not chan...
sbfal 38069 Substitution does not chan...
sbcani 38070 Distribution of class subs...
sbcori 38071 Distribution of class subs...
sbcimi 38072 Distribution of class subs...
sbcni 38073 Move class substitution in...
sbali 38074 Discard class substitution...
sbexi 38075 Discard class substitution...
sbcalf 38076 Move universal quantifier ...
sbcexf 38077 Move existential quantifie...
sbcalfi 38078 Move universal quantifier ...
sbcexfi 38079 Move existential quantifie...
spsbcdi 38080 A lemma for eliminating a ...
alrimii 38081 A lemma for introducing a ...
spesbcdi 38082 A lemma for introducing an...
exlimddvf 38083 A lemma for eliminating an...
exlimddvfi 38084 A lemma for eliminating an...
sbceq1ddi 38085 A lemma for eliminating in...
sbccom2lem 38086 Lemma for ~ sbccom2 . (Co...
sbccom2 38087 Commutative law for double...
sbccom2f 38088 Commutative law for double...
sbccom2fi 38089 Commutative law for double...
csbcom2fi 38090 Commutative law for double...
fald 38091 Refutation of falsity, in ...
tsim1 38092 A Tseitin axiom for logica...
tsim2 38093 A Tseitin axiom for logica...
tsim3 38094 A Tseitin axiom for logica...
tsbi1 38095 A Tseitin axiom for logica...
tsbi2 38096 A Tseitin axiom for logica...
tsbi3 38097 A Tseitin axiom for logica...
tsbi4 38098 A Tseitin axiom for logica...
tsxo1 38099 A Tseitin axiom for logica...
tsxo2 38100 A Tseitin axiom for logica...
tsxo3 38101 A Tseitin axiom for logica...
tsxo4 38102 A Tseitin axiom for logica...
tsan1 38103 A Tseitin axiom for logica...
tsan2 38104 A Tseitin axiom for logica...
tsan3 38105 A Tseitin axiom for logica...
tsna1 38106 A Tseitin axiom for logica...
tsna2 38107 A Tseitin axiom for logica...
tsna3 38108 A Tseitin axiom for logica...
tsor1 38109 A Tseitin axiom for logica...
tsor2 38110 A Tseitin axiom for logica...
tsor3 38111 A Tseitin axiom for logica...
ts3an1 38112 A Tseitin axiom for triple...
ts3an2 38113 A Tseitin axiom for triple...
ts3an3 38114 A Tseitin axiom for triple...
ts3or1 38115 A Tseitin axiom for triple...
ts3or2 38116 A Tseitin axiom for triple...
ts3or3 38117 A Tseitin axiom for triple...
iuneq2f 38118 Equality deduction for ind...
rabeq12f 38119 Equality deduction for res...
csbeq12 38120 Equality deduction for sub...
sbeqi 38121 Equality deduction for sub...
ralbi12f 38122 Equality deduction for res...
oprabbi 38123 Equality deduction for cla...
mpobi123f 38124 Equality deduction for map...
iuneq12f 38125 Equality deduction for ind...
iineq12f 38126 Equality deduction for ind...
opabbi 38127 Equality deduction for cla...
mptbi12f 38128 Equality deduction for map...
orcomdd 38129 Commutativity of logic dis...
scottexf 38130 A version of ~ scottex wit...
scott0f 38131 A version of ~ scott0 with...
scottn0f 38132 A version of ~ scott0f wit...
ac6s3f 38133 Generalization of the Axio...
ac6s6 38134 Generalization of the Axio...
ac6s6f 38135 Generalization of the Axio...
el2v1 38179 New way ( ~ elv , and the ...
el3v1 38180 New way ( ~ elv , and the ...
el3v2 38181 New way ( ~ elv , and the ...
el3v12 38182 New way ( ~ elv , and the ...
el3v13 38183 New way ( ~ elv , and the ...
el3v23 38184 New way ( ~ elv , and the ...
anan 38185 Multiple commutations in c...
triantru3 38186 A wff is equivalent to its...
bianim 38187 Exchanging conjunction in ...
biorfd 38188 A wff is equivalent to its...
eqbrtr 38189 Substitution of equal clas...
eqbrb 38190 Substitution of equal clas...
eqeltr 38191 Substitution of equal clas...
eqelb 38192 Substitution of equal clas...
eqeqan2d 38193 Implication of introducing...
suceqsneq 38194 One-to-one relationship be...
sucdifsn2 38195 Absorption of union with a...
sucdifsn 38196 The difference between the...
disjresin 38197 The restriction to a disjo...
disjresdisj 38198 The intersection of restri...
disjresdif 38199 The difference between res...
disjresundif 38200 Lemma for ~ ressucdifsn2 ....
ressucdifsn2 38201 The difference between res...
ressucdifsn 38202 The difference between res...
inres2 38203 Two ways of expressing the...
coideq 38204 Equality theorem for compo...
nexmo1 38205 If there is no case where ...
ralin 38206 Restricted universal quant...
r2alan 38207 Double restricted universa...
ssrabi 38208 Inference of restricted ab...
rabimbieq 38209 Restricted equivalent wff'...
abeqin 38210 Intersection with class ab...
abeqinbi 38211 Intersection with class ab...
rabeqel 38212 Class element of a restric...
eqrelf 38213 The equality connective be...
br1cnvinxp 38214 Binary relation on the con...
releleccnv 38215 Elementhood in a converse ...
releccnveq 38216 Equality of converse ` R `...
opelvvdif 38217 Negated elementhood of ord...
vvdifopab 38218 Ordered-pair class abstrac...
brvdif 38219 Binary relation with unive...
brvdif2 38220 Binary relation with unive...
brvvdif 38221 Binary relation with the c...
brvbrvvdif 38222 Binary relation with the c...
brcnvep 38223 The converse of the binary...
elecALTV 38224 Elementhood in the ` R ` -...
brcnvepres 38225 Restricted converse epsilo...
brres2 38226 Binary relation on a restr...
br1cnvres 38227 Binary relation on the con...
eldmres 38228 Elementhood in the domain ...
elrnres 38229 Element of the range of a ...
eldmressnALTV 38230 Element of the domain of a...
elrnressn 38231 Element of the range of a ...
eldm4 38232 Elementhood in a domain. ...
eldmres2 38233 Elementhood in the domain ...
eceq1i 38234 Equality theorem for ` C `...
elecres 38235 Elementhood in the restric...
ecres 38236 Restricted coset of ` B ` ...
ecres2 38237 The restricted coset of ` ...
eccnvepres 38238 Restricted converse epsilo...
eleccnvep 38239 Elementhood in the convers...
eccnvep 38240 The converse epsilon coset...
extep 38241 Property of epsilon relati...
disjeccnvep 38242 Property of the epsilon re...
eccnvepres2 38243 The restricted converse ep...
eccnvepres3 38244 Condition for a restricted...
eldmqsres 38245 Elementhood in a restricte...
eldmqsres2 38246 Elementhood in a restricte...
qsss1 38247 Subclass theorem for quoti...
qseq1i 38248 Equality theorem for quoti...
brinxprnres 38249 Binary relation on a restr...
inxprnres 38250 Restriction of a class as ...
dfres4 38251 Alternate definition of th...
exan3 38252 Equivalent expressions wit...
exanres 38253 Equivalent expressions wit...
exanres3 38254 Equivalent expressions wit...
exanres2 38255 Equivalent expressions wit...
cnvepres 38256 Restricted converse epsilo...
eqrel2 38257 Equality of relations. (C...
rncnv 38258 Range of converse is the d...
dfdm6 38259 Alternate definition of do...
dfrn6 38260 Alternate definition of ra...
rncnvepres 38261 The range of the restricte...
dmecd 38262 Equality of the coset of `...
dmec2d 38263 Equality of the coset of `...
brid 38264 Property of the identity b...
ideq2 38265 For sets, the identity bin...
idresssidinxp 38266 Condition for the identity...
idreseqidinxp 38267 Condition for the identity...
extid 38268 Property of identity relat...
inxpss 38269 Two ways to say that an in...
idinxpss 38270 Two ways to say that an in...
ref5 38271 Two ways to say that an in...
inxpss3 38272 Two ways to say that an in...
inxpss2 38273 Two ways to say that inter...
inxpssidinxp 38274 Two ways to say that inter...
idinxpssinxp 38275 Two ways to say that inter...
idinxpssinxp2 38276 Identity intersection with...
idinxpssinxp3 38277 Identity intersection with...
idinxpssinxp4 38278 Identity intersection with...
relcnveq3 38279 Two ways of saying a relat...
relcnveq 38280 Two ways of saying a relat...
relcnveq2 38281 Two ways of saying a relat...
relcnveq4 38282 Two ways of saying a relat...
qsresid 38283 Simplification of a specia...
n0elqs 38284 Two ways of expressing tha...
n0elqs2 38285 Two ways of expressing tha...
ecex2 38286 Condition for a coset to b...
uniqsALTV 38287 The union of a quotient se...
imaexALTV 38288 Existence of an image of a...
ecexALTV 38289 Existence of a coset, like...
rnresequniqs 38290 The range of a restriction...
n0el2 38291 Two ways of expressing tha...
cnvepresex 38292 Sethood condition for the ...
eccnvepex 38293 The converse epsilon coset...
cnvepimaex 38294 The image of converse epsi...
cnvepima 38295 The image of converse epsi...
inex3 38296 Sufficient condition for t...
inxpex 38297 Sufficient condition for a...
eqres 38298 Converting a class constan...
brrabga 38299 The law of concretion for ...
brcnvrabga 38300 The law of concretion for ...
opideq 38301 Equality conditions for or...
iss2 38302 A subclass of the identity...
eldmcnv 38303 Elementhood in a domain of...
dfrel5 38304 Alternate definition of th...
dfrel6 38305 Alternate definition of th...
cnvresrn 38306 Converse restricted to ran...
relssinxpdmrn 38307 Subset of restriction, spe...
cnvref4 38308 Two ways to say that a rel...
cnvref5 38309 Two ways to say that a rel...
ecin0 38310 Two ways of saying that th...
ecinn0 38311 Two ways of saying that th...
ineleq 38312 Equivalence of restricted ...
inecmo 38313 Equivalence of a double re...
inecmo2 38314 Equivalence of a double re...
ineccnvmo 38315 Equivalence of a double re...
alrmomorn 38316 Equivalence of an "at most...
alrmomodm 38317 Equivalence of an "at most...
ineccnvmo2 38318 Equivalence of a double un...
inecmo3 38319 Equivalence of a double un...
moeu2 38320 Uniqueness is equivalent t...
mopickr 38321 "At most one" picks a vari...
moantr 38322 Sufficient condition for t...
brabidgaw 38323 The law of concretion for ...
brabidga 38324 The law of concretion for ...
inxp2 38325 Intersection with a Cartes...
opabf 38326 A class abstraction of a c...
ec0 38327 The empty-coset of a class...
brcnvin 38328 Intersection with a conver...
xrnss3v 38330 A range Cartesian product ...
xrnrel 38331 A range Cartesian product ...
brxrn 38332 Characterize a ternary rel...
brxrn2 38333 A characterization of the ...
dfxrn2 38334 Alternate definition of th...
xrneq1 38335 Equality theorem for the r...
xrneq1i 38336 Equality theorem for the r...
xrneq1d 38337 Equality theorem for the r...
xrneq2 38338 Equality theorem for the r...
xrneq2i 38339 Equality theorem for the r...
xrneq2d 38340 Equality theorem for the r...
xrneq12 38341 Equality theorem for the r...
xrneq12i 38342 Equality theorem for the r...
xrneq12d 38343 Equality theorem for the r...
elecxrn 38344 Elementhood in the ` ( R |...
ecxrn 38345 The ` ( R |X. S ) ` -coset...
disjressuc2 38346 Double restricted quantifi...
disjecxrn 38347 Two ways of saying that ` ...
disjecxrncnvep 38348 Two ways of saying that co...
disjsuc2 38349 Double restricted quantifi...
xrninxp 38350 Intersection of a range Ca...
xrninxp2 38351 Intersection of a range Ca...
xrninxpex 38352 Sufficient condition for t...
inxpxrn 38353 Two ways to express the in...
br1cnvxrn2 38354 The converse of a binary r...
elec1cnvxrn2 38355 Elementhood in the convers...
rnxrn 38356 Range of the range Cartesi...
rnxrnres 38357 Range of a range Cartesian...
rnxrncnvepres 38358 Range of a range Cartesian...
rnxrnidres 38359 Range of a range Cartesian...
xrnres 38360 Two ways to express restri...
xrnres2 38361 Two ways to express restri...
xrnres3 38362 Two ways to express restri...
xrnres4 38363 Two ways to express restri...
xrnresex 38364 Sufficient condition for a...
xrnidresex 38365 Sufficient condition for a...
xrncnvepresex 38366 Sufficient condition for a...
brin2 38367 Binary relation on an inte...
brin3 38368 Binary relation on an inte...
dfcoss2 38371 Alternate definition of th...
dfcoss3 38372 Alternate definition of th...
dfcoss4 38373 Alternate definition of th...
cosscnv 38374 Class of cosets by the con...
coss1cnvres 38375 Class of cosets by the con...
coss2cnvepres 38376 Special case of ~ coss1cnv...
cossex 38377 If ` A ` is a set then the...
cosscnvex 38378 If ` A ` is a set then the...
1cosscnvepresex 38379 Sufficient condition for a...
1cossxrncnvepresex 38380 Sufficient condition for a...
relcoss 38381 Cosets by ` R ` is a relat...
relcoels 38382 Coelements on ` A ` is a r...
cossss 38383 Subclass theorem for the c...
cosseq 38384 Equality theorem for the c...
cosseqi 38385 Equality theorem for the c...
cosseqd 38386 Equality theorem for the c...
1cossres 38387 The class of cosets by a r...
dfcoels 38388 Alternate definition of th...
brcoss 38389 ` A ` and ` B ` are cosets...
brcoss2 38390 Alternate form of the ` A ...
brcoss3 38391 Alternate form of the ` A ...
brcosscnvcoss 38392 For sets, the ` A ` and ` ...
brcoels 38393 ` B ` and ` C ` are coelem...
cocossss 38394 Two ways of saying that co...
cnvcosseq 38395 The converse of cosets by ...
br2coss 38396 Cosets by ` ,~ R ` binary ...
br1cossres 38397 ` B ` and ` C ` are cosets...
br1cossres2 38398 ` B ` and ` C ` are cosets...
brressn 38399 Binary relation on a restr...
ressn2 38400 A class ' R ' restricted t...
refressn 38401 Any class ' R ' restricted...
antisymressn 38402 Every class ' R ' restrict...
trressn 38403 Any class ' R ' restricted...
relbrcoss 38404 ` A ` and ` B ` are cosets...
br1cossinres 38405 ` B ` and ` C ` are cosets...
br1cossxrnres 38406 ` <. B , C >. ` and ` <. D...
br1cossinidres 38407 ` B ` and ` C ` are cosets...
br1cossincnvepres 38408 ` B ` and ` C ` are cosets...
br1cossxrnidres 38409 ` <. B , C >. ` and ` <. D...
br1cossxrncnvepres 38410 ` <. B , C >. ` and ` <. D...
dmcoss3 38411 The domain of cosets is th...
dmcoss2 38412 The domain of cosets is th...
rncossdmcoss 38413 The range of cosets is the...
dm1cosscnvepres 38414 The domain of cosets of th...
dmcoels 38415 The domain of coelements i...
eldmcoss 38416 Elementhood in the domain ...
eldmcoss2 38417 Elementhood in the domain ...
eldm1cossres 38418 Elementhood in the domain ...
eldm1cossres2 38419 Elementhood in the domain ...
refrelcosslem 38420 Lemma for the left side of...
refrelcoss3 38421 The class of cosets by ` R...
refrelcoss2 38422 The class of cosets by ` R...
symrelcoss3 38423 The class of cosets by ` R...
symrelcoss2 38424 The class of cosets by ` R...
cossssid 38425 Equivalent expressions for...
cossssid2 38426 Equivalent expressions for...
cossssid3 38427 Equivalent expressions for...
cossssid4 38428 Equivalent expressions for...
cossssid5 38429 Equivalent expressions for...
brcosscnv 38430 ` A ` and ` B ` are cosets...
brcosscnv2 38431 ` A ` and ` B ` are cosets...
br1cosscnvxrn 38432 ` A ` and ` B ` are cosets...
1cosscnvxrn 38433 Cosets by the converse ran...
cosscnvssid3 38434 Equivalent expressions for...
cosscnvssid4 38435 Equivalent expressions for...
cosscnvssid5 38436 Equivalent expressions for...
coss0 38437 Cosets by the empty set ar...
cossid 38438 Cosets by the identity rel...
cosscnvid 38439 Cosets by the converse ide...
trcoss 38440 Sufficient condition for t...
eleccossin 38441 Two ways of saying that th...
trcoss2 38442 Equivalent expressions for...
elrels2 38444 The element of the relatio...
elrelsrel 38445 The element of the relatio...
elrelsrelim 38446 The element of the relatio...
elrels5 38447 Equivalent expressions for...
elrels6 38448 Equivalent expressions for...
elrelscnveq3 38449 Two ways of saying a relat...
elrelscnveq 38450 Two ways of saying a relat...
elrelscnveq2 38451 Two ways of saying a relat...
elrelscnveq4 38452 Two ways of saying a relat...
cnvelrels 38453 The converse of a set is a...
cosselrels 38454 Cosets of sets are element...
cosscnvelrels 38455 Cosets of converse sets ar...
dfssr2 38457 Alternate definition of th...
relssr 38458 The subset relation is a r...
brssr 38459 The subset relation and su...
brssrid 38460 Any set is a subset of its...
issetssr 38461 Two ways of expressing set...
brssrres 38462 Restricted subset binary r...
br1cnvssrres 38463 Restricted converse subset...
brcnvssr 38464 The converse of a subset r...
brcnvssrid 38465 Any set is a converse subs...
br1cossxrncnvssrres 38466 ` <. B , C >. ` and ` <. D...
extssr 38467 Property of subset relatio...
dfrefrels2 38471 Alternate definition of th...
dfrefrels3 38472 Alternate definition of th...
dfrefrel2 38473 Alternate definition of th...
dfrefrel3 38474 Alternate definition of th...
dfrefrel5 38475 Alternate definition of th...
elrefrels2 38476 Element of the class of re...
elrefrels3 38477 Element of the class of re...
elrefrelsrel 38478 For sets, being an element...
refreleq 38479 Equality theorem for refle...
refrelid 38480 Identity relation is refle...
refrelcoss 38481 The class of cosets by ` R...
refrelressn 38482 Any class ' R ' restricted...
dfcnvrefrels2 38486 Alternate definition of th...
dfcnvrefrels3 38487 Alternate definition of th...
dfcnvrefrel2 38488 Alternate definition of th...
dfcnvrefrel3 38489 Alternate definition of th...
dfcnvrefrel4 38490 Alternate definition of th...
dfcnvrefrel5 38491 Alternate definition of th...
elcnvrefrels2 38492 Element of the class of co...
elcnvrefrels3 38493 Element of the class of co...
elcnvrefrelsrel 38494 For sets, being an element...
cnvrefrelcoss2 38495 Necessary and sufficient c...
cosselcnvrefrels2 38496 Necessary and sufficient c...
cosselcnvrefrels3 38497 Necessary and sufficient c...
cosselcnvrefrels4 38498 Necessary and sufficient c...
cosselcnvrefrels5 38499 Necessary and sufficient c...
dfsymrels2 38503 Alternate definition of th...
dfsymrels3 38504 Alternate definition of th...
dfsymrels4 38505 Alternate definition of th...
dfsymrels5 38506 Alternate definition of th...
dfsymrel2 38507 Alternate definition of th...
dfsymrel3 38508 Alternate definition of th...
dfsymrel4 38509 Alternate definition of th...
dfsymrel5 38510 Alternate definition of th...
elsymrels2 38511 Element of the class of sy...
elsymrels3 38512 Element of the class of sy...
elsymrels4 38513 Element of the class of sy...
elsymrels5 38514 Element of the class of sy...
elsymrelsrel 38515 For sets, being an element...
symreleq 38516 Equality theorem for symme...
symrelim 38517 Symmetric relation implies...
symrelcoss 38518 The class of cosets by ` R...
idsymrel 38519 The identity relation is s...
epnsymrel 38520 The membership (epsilon) r...
symrefref2 38521 Symmetry is a sufficient c...
symrefref3 38522 Symmetry is a sufficient c...
refsymrels2 38523 Elements of the class of r...
refsymrels3 38524 Elements of the class of r...
refsymrel2 38525 A relation which is reflex...
refsymrel3 38526 A relation which is reflex...
elrefsymrels2 38527 Elements of the class of r...
elrefsymrels3 38528 Elements of the class of r...
elrefsymrelsrel 38529 For sets, being an element...
dftrrels2 38533 Alternate definition of th...
dftrrels3 38534 Alternate definition of th...
dftrrel2 38535 Alternate definition of th...
dftrrel3 38536 Alternate definition of th...
eltrrels2 38537 Element of the class of tr...
eltrrels3 38538 Element of the class of tr...
eltrrelsrel 38539 For sets, being an element...
trreleq 38540 Equality theorem for the t...
trrelressn 38541 Any class ' R ' restricted...
dfeqvrels2 38546 Alternate definition of th...
dfeqvrels3 38547 Alternate definition of th...
dfeqvrel2 38548 Alternate definition of th...
dfeqvrel3 38549 Alternate definition of th...
eleqvrels2 38550 Element of the class of eq...
eleqvrels3 38551 Element of the class of eq...
eleqvrelsrel 38552 For sets, being an element...
elcoeleqvrels 38553 Elementhood in the coeleme...
elcoeleqvrelsrel 38554 For sets, being an element...
eqvrelrel 38555 An equivalence relation is...
eqvrelrefrel 38556 An equivalence relation is...
eqvrelsymrel 38557 An equivalence relation is...
eqvreltrrel 38558 An equivalence relation is...
eqvrelim 38559 Equivalence relation impli...
eqvreleq 38560 Equality theorem for equiv...
eqvreleqi 38561 Equality theorem for equiv...
eqvreleqd 38562 Equality theorem for equiv...
eqvrelsym 38563 An equivalence relation is...
eqvrelsymb 38564 An equivalence relation is...
eqvreltr 38565 An equivalence relation is...
eqvreltrd 38566 A transitivity relation fo...
eqvreltr4d 38567 A transitivity relation fo...
eqvrelref 38568 An equivalence relation is...
eqvrelth 38569 Basic property of equivale...
eqvrelcl 38570 Elementhood in the field o...
eqvrelthi 38571 Basic property of equivale...
eqvreldisj 38572 Equivalence classes do not...
qsdisjALTV 38573 Elements of a quotient set...
eqvrelqsel 38574 If an element of a quotien...
eqvrelcoss 38575 Two ways to express equiva...
eqvrelcoss3 38576 Two ways to express equiva...
eqvrelcoss2 38577 Two ways to express equiva...
eqvrelcoss4 38578 Two ways to express equiva...
dfcoeleqvrels 38579 Alternate definition of th...
dfcoeleqvrel 38580 Alternate definition of th...
brredunds 38584 Binary relation on the cla...
brredundsredund 38585 For sets, binary relation ...
redundss3 38586 Implication of redundancy ...
redundeq1 38587 Equivalence of redundancy ...
redundpim3 38588 Implication of redundancy ...
redundpbi1 38589 Equivalence of redundancy ...
refrelsredund4 38590 The naive version of the c...
refrelsredund2 38591 The naive version of the c...
refrelsredund3 38592 The naive version of the c...
refrelredund4 38593 The naive version of the d...
refrelredund2 38594 The naive version of the d...
refrelredund3 38595 The naive version of the d...
dmqseq 38598 Equality theorem for domai...
dmqseqi 38599 Equality theorem for domai...
dmqseqd 38600 Equality theorem for domai...
dmqseqeq1 38601 Equality theorem for domai...
dmqseqeq1i 38602 Equality theorem for domai...
dmqseqeq1d 38603 Equality theorem for domai...
brdmqss 38604 The domain quotient binary...
brdmqssqs 38605 If ` A ` and ` R ` are set...
n0eldmqs 38606 The empty set is not an el...
n0eldmqseq 38607 The empty set is not an el...
n0elim 38608 Implication of that the em...
n0el3 38609 Two ways of expressing tha...
cnvepresdmqss 38610 The domain quotient binary...
cnvepresdmqs 38611 The domain quotient predic...
unidmqs 38612 The range of a relation is...
unidmqseq 38613 The union of the domain qu...
dmqseqim 38614 If the domain quotient of ...
dmqseqim2 38615 Lemma for ~ erimeq2 . (Co...
releldmqs 38616 Elementhood in the domain ...
eldmqs1cossres 38617 Elementhood in the domain ...
releldmqscoss 38618 Elementhood in the domain ...
dmqscoelseq 38619 Two ways to express the eq...
dmqs1cosscnvepreseq 38620 Two ways to express the eq...
brers 38625 Binary equivalence relatio...
dferALTV2 38626 Equivalence relation with ...
erALTVeq1 38627 Equality theorem for equiv...
erALTVeq1i 38628 Equality theorem for equiv...
erALTVeq1d 38629 Equality theorem for equiv...
dfcomember 38630 Alternate definition of th...
dfcomember2 38631 Alternate definition of th...
dfcomember3 38632 Alternate definition of th...
eqvreldmqs 38633 Two ways to express comemb...
eqvreldmqs2 38634 Two ways to express comemb...
brerser 38635 Binary equivalence relatio...
erimeq2 38636 Equivalence relation on it...
erimeq 38637 Equivalence relation on it...
dffunsALTV 38641 Alternate definition of th...
dffunsALTV2 38642 Alternate definition of th...
dffunsALTV3 38643 Alternate definition of th...
dffunsALTV4 38644 Alternate definition of th...
dffunsALTV5 38645 Alternate definition of th...
dffunALTV2 38646 Alternate definition of th...
dffunALTV3 38647 Alternate definition of th...
dffunALTV4 38648 Alternate definition of th...
dffunALTV5 38649 Alternate definition of th...
elfunsALTV 38650 Elementhood in the class o...
elfunsALTV2 38651 Elementhood in the class o...
elfunsALTV3 38652 Elementhood in the class o...
elfunsALTV4 38653 Elementhood in the class o...
elfunsALTV5 38654 Elementhood in the class o...
elfunsALTVfunALTV 38655 The element of the class o...
funALTVfun 38656 Our definition of the func...
funALTVss 38657 Subclass theorem for funct...
funALTVeq 38658 Equality theorem for funct...
funALTVeqi 38659 Equality inference for the...
funALTVeqd 38660 Equality deduction for the...
dfdisjs 38666 Alternate definition of th...
dfdisjs2 38667 Alternate definition of th...
dfdisjs3 38668 Alternate definition of th...
dfdisjs4 38669 Alternate definition of th...
dfdisjs5 38670 Alternate definition of th...
dfdisjALTV 38671 Alternate definition of th...
dfdisjALTV2 38672 Alternate definition of th...
dfdisjALTV3 38673 Alternate definition of th...
dfdisjALTV4 38674 Alternate definition of th...
dfdisjALTV5 38675 Alternate definition of th...
dfeldisj2 38676 Alternate definition of th...
dfeldisj3 38677 Alternate definition of th...
dfeldisj4 38678 Alternate definition of th...
dfeldisj5 38679 Alternate definition of th...
eldisjs 38680 Elementhood in the class o...
eldisjs2 38681 Elementhood in the class o...
eldisjs3 38682 Elementhood in the class o...
eldisjs4 38683 Elementhood in the class o...
eldisjs5 38684 Elementhood in the class o...
eldisjsdisj 38685 The element of the class o...
eleldisjs 38686 Elementhood in the disjoin...
eleldisjseldisj 38687 The element of the disjoin...
disjrel 38688 Disjoint relation is a rel...
disjss 38689 Subclass theorem for disjo...
disjssi 38690 Subclass theorem for disjo...
disjssd 38691 Subclass theorem for disjo...
disjeq 38692 Equality theorem for disjo...
disjeqi 38693 Equality theorem for disjo...
disjeqd 38694 Equality theorem for disjo...
disjdmqseqeq1 38695 Lemma for the equality the...
eldisjss 38696 Subclass theorem for disjo...
eldisjssi 38697 Subclass theorem for disjo...
eldisjssd 38698 Subclass theorem for disjo...
eldisjeq 38699 Equality theorem for disjo...
eldisjeqi 38700 Equality theorem for disjo...
eldisjeqd 38701 Equality theorem for disjo...
disjres 38702 Disjoint restriction. (Co...
eldisjn0elb 38703 Two forms of disjoint elem...
disjxrn 38704 Two ways of saying that a ...
disjxrnres5 38705 Disjoint range Cartesian p...
disjorimxrn 38706 Disjointness condition for...
disjimxrn 38707 Disjointness condition for...
disjimres 38708 Disjointness condition for...
disjimin 38709 Disjointness condition for...
disjiminres 38710 Disjointness condition for...
disjimxrnres 38711 Disjointness condition for...
disjALTV0 38712 The null class is disjoint...
disjALTVid 38713 The class of identity rela...
disjALTVidres 38714 The class of identity rela...
disjALTVinidres 38715 The intersection with rest...
disjALTVxrnidres 38716 The class of range Cartesi...
disjsuc 38717 Disjoint range Cartesian p...
dfantisymrel4 38719 Alternate definition of th...
dfantisymrel5 38720 Alternate definition of th...
antisymrelres 38721 (Contributed by Peter Mazs...
antisymrelressn 38722 (Contributed by Peter Mazs...
dfpart2 38727 Alternate definition of th...
dfmembpart2 38728 Alternate definition of th...
brparts 38729 Binary partitions relation...
brparts2 38730 Binary partitions relation...
brpartspart 38731 Binary partition and the p...
parteq1 38732 Equality theorem for parti...
parteq2 38733 Equality theorem for parti...
parteq12 38734 Equality theorem for parti...
parteq1i 38735 Equality theorem for parti...
parteq1d 38736 Equality theorem for parti...
partsuc2 38737 Property of the partition....
partsuc 38738 Property of the partition....
disjim 38739 The "Divide et Aequivalere...
disjimi 38740 Every disjoint relation ge...
detlem 38741 If a relation is disjoint,...
eldisjim 38742 If the elements of ` A ` a...
eldisjim2 38743 Alternate form of ~ eldisj...
eqvrel0 38744 The null class is an equiv...
det0 38745 The cosets by the null cla...
eqvrelcoss0 38746 The cosets by the null cla...
eqvrelid 38747 The identity relation is a...
eqvrel1cossidres 38748 The cosets by a restricted...
eqvrel1cossinidres 38749 The cosets by an intersect...
eqvrel1cossxrnidres 38750 The cosets by a range Cart...
detid 38751 The cosets by the identity...
eqvrelcossid 38752 The cosets by the identity...
detidres 38753 The cosets by the restrict...
detinidres 38754 The cosets by the intersec...
detxrnidres 38755 The cosets by the range Ca...
disjlem14 38756 Lemma for ~ disjdmqseq , ~...
disjlem17 38757 Lemma for ~ disjdmqseq , ~...
disjlem18 38758 Lemma for ~ disjdmqseq , ~...
disjlem19 38759 Lemma for ~ disjdmqseq , ~...
disjdmqsss 38760 Lemma for ~ disjdmqseq via...
disjdmqscossss 38761 Lemma for ~ disjdmqseq via...
disjdmqs 38762 If a relation is disjoint,...
disjdmqseq 38763 If a relation is disjoint,...
eldisjn0el 38764 Special case of ~ disjdmqs...
partim2 38765 Disjoint relation on its n...
partim 38766 Partition implies equivale...
partimeq 38767 Partition implies that the...
eldisjlem19 38768 Special case of ~ disjlem1...
membpartlem19 38769 Together with ~ disjlem19 ...
petlem 38770 If you can prove that the ...
petlemi 38771 If you can prove disjointn...
pet02 38772 Class ` A ` is a partition...
pet0 38773 Class ` A ` is a partition...
petid2 38774 Class ` A ` is a partition...
petid 38775 A class is a partition by ...
petidres2 38776 Class ` A ` is a partition...
petidres 38777 A class is a partition by ...
petinidres2 38778 Class ` A ` is a partition...
petinidres 38779 A class is a partition by ...
petxrnidres2 38780 Class ` A ` is a partition...
petxrnidres 38781 A class is a partition by ...
eqvreldisj1 38782 The elements of the quotie...
eqvreldisj2 38783 The elements of the quotie...
eqvreldisj3 38784 The elements of the quotie...
eqvreldisj4 38785 Intersection with the conv...
eqvreldisj5 38786 Range Cartesian product wi...
eqvrelqseqdisj2 38787 Implication of ~ eqvreldis...
fences3 38788 Implication of ~ eqvrelqse...
eqvrelqseqdisj3 38789 Implication of ~ eqvreldis...
eqvrelqseqdisj4 38790 Lemma for ~ petincnvepres2...
eqvrelqseqdisj5 38791 Lemma for the Partition-Eq...
mainer 38792 The Main Theorem of Equiva...
partimcomember 38793 Partition with general ` R...
mpet3 38794 Member Partition-Equivalen...
cpet2 38795 The conventional form of t...
cpet 38796 The conventional form of M...
mpet 38797 Member Partition-Equivalen...
mpet2 38798 Member Partition-Equivalen...
mpets2 38799 Member Partition-Equivalen...
mpets 38800 Member Partition-Equivalen...
mainpart 38801 Partition with general ` R...
fences 38802 The Theorem of Fences by E...
fences2 38803 The Theorem of Fences by E...
mainer2 38804 The Main Theorem of Equiva...
mainerim 38805 Every equivalence relation...
petincnvepres2 38806 A partition-equivalence th...
petincnvepres 38807 The shortest form of a par...
pet2 38808 Partition-Equivalence Theo...
pet 38809 Partition-Equivalence Theo...
pets 38810 Partition-Equivalence Theo...
prtlem60 38811 Lemma for ~ prter3 . (Con...
bicomdd 38812 Commute two sides of a bic...
jca2r 38813 Inference conjoining the c...
jca3 38814 Inference conjoining the c...
prtlem70 38815 Lemma for ~ prter3 : a rea...
ibdr 38816 Reverse of ~ ibd . (Contr...
prtlem100 38817 Lemma for ~ prter3 . (Con...
prtlem5 38818 Lemma for ~ prter1 , ~ prt...
prtlem80 38819 Lemma for ~ prter2 . (Con...
brabsb2 38820 A closed form of ~ brabsb ...
eqbrrdv2 38821 Other version of ~ eqbrrdi...
prtlem9 38822 Lemma for ~ prter3 . (Con...
prtlem10 38823 Lemma for ~ prter3 . (Con...
prtlem11 38824 Lemma for ~ prter2 . (Con...
prtlem12 38825 Lemma for ~ prtex and ~ pr...
prtlem13 38826 Lemma for ~ prter1 , ~ prt...
prtlem16 38827 Lemma for ~ prtex , ~ prte...
prtlem400 38828 Lemma for ~ prter2 and als...
erprt 38831 The quotient set of an equ...
prtlem14 38832 Lemma for ~ prter1 , ~ prt...
prtlem15 38833 Lemma for ~ prter1 and ~ p...
prtlem17 38834 Lemma for ~ prter2 . (Con...
prtlem18 38835 Lemma for ~ prter2 . (Con...
prtlem19 38836 Lemma for ~ prter2 . (Con...
prter1 38837 Every partition generates ...
prtex 38838 The equivalence relation g...
prter2 38839 The quotient set of the eq...
prter3 38840 For every partition there ...
axc5 38851 This theorem repeats ~ sp ...
ax4fromc4 38852 Rederivation of Axiom ~ ax...
ax10fromc7 38853 Rederivation of Axiom ~ ax...
ax6fromc10 38854 Rederivation of Axiom ~ ax...
hba1-o 38855 The setvar ` x ` is not fr...
axc4i-o 38856 Inference version of ~ ax-...
equid1 38857 Proof of ~ equid from our ...
equcomi1 38858 Proof of ~ equcomi from ~ ...
aecom-o 38859 Commutation law for identi...
aecoms-o 38860 A commutation rule for ide...
hbae-o 38861 All variables are effectiv...
dral1-o 38862 Formula-building lemma for...
ax12fromc15 38863 Rederivation of Axiom ~ ax...
ax13fromc9 38864 Derive ~ ax-13 from ~ ax-c...
ax5ALT 38865 Axiom to quantify a variab...
sps-o 38866 Generalization of antecede...
hbequid 38867 Bound-variable hypothesis ...
nfequid-o 38868 Bound-variable hypothesis ...
axc5c7 38869 Proof of a single axiom th...
axc5c7toc5 38870 Rederivation of ~ ax-c5 fr...
axc5c7toc7 38871 Rederivation of ~ ax-c7 fr...
axc711 38872 Proof of a single axiom th...
nfa1-o 38873 ` x ` is not free in ` A. ...
axc711toc7 38874 Rederivation of ~ ax-c7 fr...
axc711to11 38875 Rederivation of ~ ax-11 fr...
axc5c711 38876 Proof of a single axiom th...
axc5c711toc5 38877 Rederivation of ~ ax-c5 fr...
axc5c711toc7 38878 Rederivation of ~ ax-c7 fr...
axc5c711to11 38879 Rederivation of ~ ax-11 fr...
equidqe 38880 ~ equid with existential q...
axc5sp1 38881 A special case of ~ ax-c5 ...
equidq 38882 ~ equid with universal qua...
equid1ALT 38883 Alternate proof of ~ equid...
axc11nfromc11 38884 Rederivation of ~ ax-c11n ...
naecoms-o 38885 A commutation rule for dis...
hbnae-o 38886 All variables are effectiv...
dvelimf-o 38887 Proof of ~ dvelimh that us...
dral2-o 38888 Formula-building lemma for...
aev-o 38889 A "distinctor elimination"...
ax5eq 38890 Theorem to add distinct qu...
dveeq2-o 38891 Quantifier introduction wh...
axc16g-o 38892 A generalization of Axiom ...
dveeq1-o 38893 Quantifier introduction wh...
dveeq1-o16 38894 Version of ~ dveeq1 using ...
ax5el 38895 Theorem to add distinct qu...
axc11n-16 38896 This theorem shows that, g...
dveel2ALT 38897 Alternate proof of ~ dveel...
ax12f 38898 Basis step for constructin...
ax12eq 38899 Basis step for constructin...
ax12el 38900 Basis step for constructin...
ax12indn 38901 Induction step for constru...
ax12indi 38902 Induction step for constru...
ax12indalem 38903 Lemma for ~ ax12inda2 and ...
ax12inda2ALT 38904 Alternate proof of ~ ax12i...
ax12inda2 38905 Induction step for constru...
ax12inda 38906 Induction step for constru...
ax12v2-o 38907 Rederivation of ~ ax-c15 f...
ax12a2-o 38908 Derive ~ ax-c15 from a hyp...
axc11-o 38909 Show that ~ ax-c11 can be ...
fsumshftd 38910 Index shift of a finite su...
riotaclbgBAD 38912 Closure of restricted iota...
riotaclbBAD 38913 Closure of restricted iota...
riotasvd 38914 Deduction version of ~ rio...
riotasv2d 38915 Value of description binde...
riotasv2s 38916 The value of description b...
riotasv 38917 Value of description binde...
riotasv3d 38918 A property ` ch ` holding ...
elimhyps 38919 A version of ~ elimhyp usi...
dedths 38920 A version of weak deductio...
renegclALT 38921 Closure law for negative o...
elimhyps2 38922 Generalization of ~ elimhy...
dedths2 38923 Generalization of ~ dedths...
nfcxfrdf 38924 A utility lemma to transfe...
nfded 38925 A deduction theorem that c...
nfded2 38926 A deduction theorem that c...
nfunidALT2 38927 Deduction version of ~ nfu...
nfunidALT 38928 Deduction version of ~ nfu...
nfopdALT 38929 Deduction version of bound...
cnaddcom 38930 Recover the commutative la...
toycom 38931 Show the commutative law f...
lshpset 38936 The set of all hyperplanes...
islshp 38937 The predicate "is a hyperp...
islshpsm 38938 Hyperplane properties expr...
lshplss 38939 A hyperplane is a subspace...
lshpne 38940 A hyperplane is not equal ...
lshpnel 38941 A hyperplane's generating ...
lshpnelb 38942 The subspace sum of a hype...
lshpnel2N 38943 Condition that determines ...
lshpne0 38944 The member of the span in ...
lshpdisj 38945 A hyperplane and the span ...
lshpcmp 38946 If two hyperplanes are com...
lshpinN 38947 The intersection of two di...
lsatset 38948 The set of all 1-dim subsp...
islsat 38949 The predicate "is a 1-dim ...
lsatlspsn2 38950 The span of a nonzero sing...
lsatlspsn 38951 The span of a nonzero sing...
islsati 38952 A 1-dim subspace (atom) (o...
lsateln0 38953 A 1-dim subspace (atom) (o...
lsatlss 38954 The set of 1-dim subspaces...
lsatlssel 38955 An atom is a subspace. (C...
lsatssv 38956 An atom is a set of vector...
lsatn0 38957 A 1-dim subspace (atom) of...
lsatspn0 38958 The span of a vector is an...
lsator0sp 38959 The span of a vector is ei...
lsatssn0 38960 A subspace (or any class) ...
lsatcmp 38961 If two atoms are comparabl...
lsatcmp2 38962 If an atom is included in ...
lsatel 38963 A nonzero vector in an ato...
lsatelbN 38964 A nonzero vector in an ato...
lsat2el 38965 Two atoms sharing a nonzer...
lsmsat 38966 Convert comparison of atom...
lsatfixedN 38967 Show equality with the spa...
lsmsatcv 38968 Subspace sum has the cover...
lssatomic 38969 The lattice of subspaces i...
lssats 38970 The lattice of subspaces i...
lpssat 38971 Two subspaces in a proper ...
lrelat 38972 Subspaces are relatively a...
lssatle 38973 The ordering of two subspa...
lssat 38974 Two subspaces in a proper ...
islshpat 38975 Hyperplane properties expr...
lcvfbr 38978 The covers relation for a ...
lcvbr 38979 The covers relation for a ...
lcvbr2 38980 The covers relation for a ...
lcvbr3 38981 The covers relation for a ...
lcvpss 38982 The covers relation implie...
lcvnbtwn 38983 The covers relation implie...
lcvntr 38984 The covers relation is not...
lcvnbtwn2 38985 The covers relation implie...
lcvnbtwn3 38986 The covers relation implie...
lsmcv2 38987 Subspace sum has the cover...
lcvat 38988 If a subspace covers anoth...
lsatcv0 38989 An atom covers the zero su...
lsatcveq0 38990 A subspace covered by an a...
lsat0cv 38991 A subspace is an atom iff ...
lcvexchlem1 38992 Lemma for ~ lcvexch . (Co...
lcvexchlem2 38993 Lemma for ~ lcvexch . (Co...
lcvexchlem3 38994 Lemma for ~ lcvexch . (Co...
lcvexchlem4 38995 Lemma for ~ lcvexch . (Co...
lcvexchlem5 38996 Lemma for ~ lcvexch . (Co...
lcvexch 38997 Subspaces satisfy the exch...
lcvp 38998 Covering property of Defin...
lcv1 38999 Covering property of a sub...
lcv2 39000 Covering property of a sub...
lsatexch 39001 The atom exchange property...
lsatnle 39002 The meet of a subspace and...
lsatnem0 39003 The meet of distinct atoms...
lsatexch1 39004 The atom exch1ange propert...
lsatcv0eq 39005 If the sum of two atoms co...
lsatcv1 39006 Two atoms covering the zer...
lsatcvatlem 39007 Lemma for ~ lsatcvat . (C...
lsatcvat 39008 A nonzero subspace less th...
lsatcvat2 39009 A subspace covered by the ...
lsatcvat3 39010 A condition implying that ...
islshpcv 39011 Hyperplane properties expr...
l1cvpat 39012 A subspace covered by the ...
l1cvat 39013 Create an atom under an el...
lshpat 39014 Create an atom under a hyp...
lflset 39017 The set of linear function...
islfl 39018 The predicate "is a linear...
lfli 39019 Property of a linear funct...
islfld 39020 Properties that determine ...
lflf 39021 A linear functional is a f...
lflcl 39022 A linear functional value ...
lfl0 39023 A linear functional is zer...
lfladd 39024 Property of a linear funct...
lflsub 39025 Property of a linear funct...
lflmul 39026 Property of a linear funct...
lfl0f 39027 The zero function is a fun...
lfl1 39028 A nonzero functional has a...
lfladdcl 39029 Closure of addition of two...
lfladdcom 39030 Commutativity of functiona...
lfladdass 39031 Associativity of functiona...
lfladd0l 39032 Functional addition with t...
lflnegcl 39033 Closure of the negative of...
lflnegl 39034 A functional plus its nega...
lflvscl 39035 Closure of a scalar produc...
lflvsdi1 39036 Distributive law for (righ...
lflvsdi2 39037 Reverse distributive law f...
lflvsdi2a 39038 Reverse distributive law f...
lflvsass 39039 Associative law for (right...
lfl0sc 39040 The (right vector space) s...
lflsc0N 39041 The scalar product with th...
lfl1sc 39042 The (right vector space) s...
lkrfval 39045 The kernel of a functional...
lkrval 39046 Value of the kernel of a f...
ellkr 39047 Membership in the kernel o...
lkrval2 39048 Value of the kernel of a f...
ellkr2 39049 Membership in the kernel o...
lkrcl 39050 A member of the kernel of ...
lkrf0 39051 The value of a functional ...
lkr0f 39052 The kernel of the zero fun...
lkrlss 39053 The kernel of a linear fun...
lkrssv 39054 The kernel of a linear fun...
lkrsc 39055 The kernel of a nonzero sc...
lkrscss 39056 The kernel of a scalar pro...
eqlkr 39057 Two functionals with the s...
eqlkr2 39058 Two functionals with the s...
eqlkr3 39059 Two functionals with the s...
lkrlsp 39060 The subspace sum of a kern...
lkrlsp2 39061 The subspace sum of a kern...
lkrlsp3 39062 The subspace sum of a kern...
lkrshp 39063 The kernel of a nonzero fu...
lkrshp3 39064 The kernels of nonzero fun...
lkrshpor 39065 The kernel of a functional...
lkrshp4 39066 A kernel is a hyperplane i...
lshpsmreu 39067 Lemma for ~ lshpkrex . Sh...
lshpkrlem1 39068 Lemma for ~ lshpkrex . Th...
lshpkrlem2 39069 Lemma for ~ lshpkrex . Th...
lshpkrlem3 39070 Lemma for ~ lshpkrex . De...
lshpkrlem4 39071 Lemma for ~ lshpkrex . Pa...
lshpkrlem5 39072 Lemma for ~ lshpkrex . Pa...
lshpkrlem6 39073 Lemma for ~ lshpkrex . Sh...
lshpkrcl 39074 The set ` G ` defined by h...
lshpkr 39075 The kernel of functional `...
lshpkrex 39076 There exists a functional ...
lshpset2N 39077 The set of all hyperplanes...
islshpkrN 39078 The predicate "is a hyperp...
lfl1dim 39079 Equivalent expressions for...
lfl1dim2N 39080 Equivalent expressions for...
ldualset 39083 Define the (left) dual of ...
ldualvbase 39084 The vectors of a dual spac...
ldualelvbase 39085 Utility theorem for conver...
ldualfvadd 39086 Vector addition in the dua...
ldualvadd 39087 Vector addition in the dua...
ldualvaddcl 39088 The value of vector additi...
ldualvaddval 39089 The value of the value of ...
ldualsca 39090 The ring of scalars of the...
ldualsbase 39091 Base set of scalar ring fo...
ldualsaddN 39092 Scalar addition for the du...
ldualsmul 39093 Scalar multiplication for ...
ldualfvs 39094 Scalar product operation f...
ldualvs 39095 Scalar product operation v...
ldualvsval 39096 Value of scalar product op...
ldualvscl 39097 The scalar product operati...
ldualvaddcom 39098 Commutative law for vector...
ldualvsass 39099 Associative law for scalar...
ldualvsass2 39100 Associative law for scalar...
ldualvsdi1 39101 Distributive law for scala...
ldualvsdi2 39102 Reverse distributive law f...
ldualgrplem 39103 Lemma for ~ ldualgrp . (C...
ldualgrp 39104 The dual of a vector space...
ldual0 39105 The zero scalar of the dua...
ldual1 39106 The unit scalar of the dua...
ldualneg 39107 The negative of a scalar o...
ldual0v 39108 The zero vector of the dua...
ldual0vcl 39109 The dual zero vector is a ...
lduallmodlem 39110 Lemma for ~ lduallmod . (...
lduallmod 39111 The dual of a left module ...
lduallvec 39112 The dual of a left vector ...
ldualvsub 39113 The value of vector subtra...
ldualvsubcl 39114 Closure of vector subtract...
ldualvsubval 39115 The value of the value of ...
ldualssvscl 39116 Closure of scalar product ...
ldualssvsubcl 39117 Closure of vector subtract...
ldual0vs 39118 Scalar zero times a functi...
lkr0f2 39119 The kernel of the zero fun...
lduallkr3 39120 The kernels of nonzero fun...
lkrpssN 39121 Proper subset relation bet...
lkrin 39122 Intersection of the kernel...
eqlkr4 39123 Two functionals with the s...
ldual1dim 39124 Equivalent expressions for...
ldualkrsc 39125 The kernel of a nonzero sc...
lkrss 39126 The kernel of a scalar pro...
lkrss2N 39127 Two functionals with kerne...
lkreqN 39128 Proportional functionals h...
lkrlspeqN 39129 Condition for colinear fun...
isopos 39138 The predicate "is an ortho...
opposet 39139 Every orthoposet is a pose...
oposlem 39140 Lemma for orthoposet prope...
op01dm 39141 Conditions necessary for z...
op0cl 39142 An orthoposet has a zero e...
op1cl 39143 An orthoposet has a unity ...
op0le 39144 Orthoposet zero is less th...
ople0 39145 An element less than or eq...
opnlen0 39146 An element not less than a...
lub0N 39147 The least upper bound of t...
opltn0 39148 A lattice element greater ...
ople1 39149 Any element is less than t...
op1le 39150 If the orthoposet unity is...
glb0N 39151 The greatest lower bound o...
opoccl 39152 Closure of orthocomplement...
opococ 39153 Double negative law for or...
opcon3b 39154 Contraposition law for ort...
opcon2b 39155 Orthocomplement contraposi...
opcon1b 39156 Orthocomplement contraposi...
oplecon3 39157 Contraposition law for ort...
oplecon3b 39158 Contraposition law for ort...
oplecon1b 39159 Contraposition law for str...
opoc1 39160 Orthocomplement of orthopo...
opoc0 39161 Orthocomplement of orthopo...
opltcon3b 39162 Contraposition law for str...
opltcon1b 39163 Contraposition law for str...
opltcon2b 39164 Contraposition law for str...
opexmid 39165 Law of excluded middle for...
opnoncon 39166 Law of contradiction for o...
riotaocN 39167 The orthocomplement of the...
cmtfvalN 39168 Value of commutes relation...
cmtvalN 39169 Equivalence for commutes r...
isolat 39170 The predicate "is an ortho...
ollat 39171 An ortholattice is a latti...
olop 39172 An ortholattice is an orth...
olposN 39173 An ortholattice is a poset...
isolatiN 39174 Properties that determine ...
oldmm1 39175 De Morgan's law for meet i...
oldmm2 39176 De Morgan's law for meet i...
oldmm3N 39177 De Morgan's law for meet i...
oldmm4 39178 De Morgan's law for meet i...
oldmj1 39179 De Morgan's law for join i...
oldmj2 39180 De Morgan's law for join i...
oldmj3 39181 De Morgan's law for join i...
oldmj4 39182 De Morgan's law for join i...
olj01 39183 An ortholattice element jo...
olj02 39184 An ortholattice element jo...
olm11 39185 The meet of an ortholattic...
olm12 39186 The meet of an ortholattic...
latmassOLD 39187 Ortholattice meet is assoc...
latm12 39188 A rearrangement of lattice...
latm32 39189 A rearrangement of lattice...
latmrot 39190 Rotate lattice meet of 3 c...
latm4 39191 Rearrangement of lattice m...
latmmdiN 39192 Lattice meet distributes o...
latmmdir 39193 Lattice meet distributes o...
olm01 39194 Meet with lattice zero is ...
olm02 39195 Meet with lattice zero is ...
isoml 39196 The predicate "is an ortho...
isomliN 39197 Properties that determine ...
omlol 39198 An orthomodular lattice is...
omlop 39199 An orthomodular lattice is...
omllat 39200 An orthomodular lattice is...
omllaw 39201 The orthomodular law. (Co...
omllaw2N 39202 Variation of orthomodular ...
omllaw3 39203 Orthomodular law equivalen...
omllaw4 39204 Orthomodular law equivalen...
omllaw5N 39205 The orthomodular law. Rem...
cmtcomlemN 39206 Lemma for ~ cmtcomN . ( ~...
cmtcomN 39207 Commutation is symmetric. ...
cmt2N 39208 Commutation with orthocomp...
cmt3N 39209 Commutation with orthocomp...
cmt4N 39210 Commutation with orthocomp...
cmtbr2N 39211 Alternate definition of th...
cmtbr3N 39212 Alternate definition for t...
cmtbr4N 39213 Alternate definition for t...
lecmtN 39214 Ordered elements commute. ...
cmtidN 39215 Any element commutes with ...
omlfh1N 39216 Foulis-Holland Theorem, pa...
omlfh3N 39217 Foulis-Holland Theorem, pa...
omlmod1i2N 39218 Analogue of modular law ~ ...
omlspjN 39219 Contraction of a Sasaki pr...
cvrfval 39226 Value of covers relation "...
cvrval 39227 Binary relation expressing...
cvrlt 39228 The covers relation implie...
cvrnbtwn 39229 There is no element betwee...
ncvr1 39230 No element covers the latt...
cvrletrN 39231 Property of an element abo...
cvrval2 39232 Binary relation expressing...
cvrnbtwn2 39233 The covers relation implie...
cvrnbtwn3 39234 The covers relation implie...
cvrcon3b 39235 Contraposition law for the...
cvrle 39236 The covers relation implie...
cvrnbtwn4 39237 The covers relation implie...
cvrnle 39238 The covers relation implie...
cvrne 39239 The covers relation implie...
cvrnrefN 39240 The covers relation is not...
cvrcmp 39241 If two lattice elements th...
cvrcmp2 39242 If two lattice elements co...
pats 39243 The set of atoms in a pose...
isat 39244 The predicate "is an atom"...
isat2 39245 The predicate "is an atom"...
atcvr0 39246 An atom covers zero. ( ~ ...
atbase 39247 An atom is a member of the...
atssbase 39248 The set of atoms is a subs...
0ltat 39249 An atom is greater than ze...
leatb 39250 A poset element less than ...
leat 39251 A poset element less than ...
leat2 39252 A nonzero poset element le...
leat3 39253 A poset element less than ...
meetat 39254 The meet of any element wi...
meetat2 39255 The meet of any element wi...
isatl 39257 The predicate "is an atomi...
atllat 39258 An atomic lattice is a lat...
atlpos 39259 An atomic lattice is a pos...
atl0dm 39260 Condition necessary for ze...
atl0cl 39261 An atomic lattice has a ze...
atl0le 39262 Orthoposet zero is less th...
atlle0 39263 An element less than or eq...
atlltn0 39264 A lattice element greater ...
isat3 39265 The predicate "is an atom"...
atn0 39266 An atom is not zero. ( ~ ...
atnle0 39267 An atom is not less than o...
atlen0 39268 A lattice element is nonze...
atcmp 39269 If two atoms are comparabl...
atncmp 39270 Frequently-used variation ...
atnlt 39271 Two atoms cannot satisfy t...
atcvreq0 39272 An element covered by an a...
atncvrN 39273 Two atoms cannot satisfy t...
atlex 39274 Every nonzero element of a...
atnle 39275 Two ways of expressing "an...
atnem0 39276 The meet of distinct atoms...
atlatmstc 39277 An atomic, complete, ortho...
atlatle 39278 The ordering of two Hilber...
atlrelat1 39279 An atomistic lattice with ...
iscvlat 39281 The predicate "is an atomi...
iscvlat2N 39282 The predicate "is an atomi...
cvlatl 39283 An atomic lattice with the...
cvllat 39284 An atomic lattice with the...
cvlposN 39285 An atomic lattice with the...
cvlexch1 39286 An atomic covering lattice...
cvlexch2 39287 An atomic covering lattice...
cvlexchb1 39288 An atomic covering lattice...
cvlexchb2 39289 An atomic covering lattice...
cvlexch3 39290 An atomic covering lattice...
cvlexch4N 39291 An atomic covering lattice...
cvlatexchb1 39292 A version of ~ cvlexchb1 f...
cvlatexchb2 39293 A version of ~ cvlexchb2 f...
cvlatexch1 39294 Atom exchange property. (...
cvlatexch2 39295 Atom exchange property. (...
cvlatexch3 39296 Atom exchange property. (...
cvlcvr1 39297 The covering property. Pr...
cvlcvrp 39298 A Hilbert lattice satisfie...
cvlatcvr1 39299 An atom is covered by its ...
cvlatcvr2 39300 An atom is covered by its ...
cvlsupr2 39301 Two equivalent ways of exp...
cvlsupr3 39302 Two equivalent ways of exp...
cvlsupr4 39303 Consequence of superpositi...
cvlsupr5 39304 Consequence of superpositi...
cvlsupr6 39305 Consequence of superpositi...
cvlsupr7 39306 Consequence of superpositi...
cvlsupr8 39307 Consequence of superpositi...
ishlat1 39310 The predicate "is a Hilber...
ishlat2 39311 The predicate "is a Hilber...
ishlat3N 39312 The predicate "is a Hilber...
ishlatiN 39313 Properties that determine ...
hlomcmcv 39314 A Hilbert lattice is ortho...
hloml 39315 A Hilbert lattice is ortho...
hlclat 39316 A Hilbert lattice is compl...
hlcvl 39317 A Hilbert lattice is an at...
hlatl 39318 A Hilbert lattice is atomi...
hlol 39319 A Hilbert lattice is an or...
hlop 39320 A Hilbert lattice is an or...
hllat 39321 A Hilbert lattice is a lat...
hllatd 39322 Deduction form of ~ hllat ...
hlomcmat 39323 A Hilbert lattice is ortho...
hlpos 39324 A Hilbert lattice is a pos...
hlatjcl 39325 Closure of join operation....
hlatjcom 39326 Commutatitivity of join op...
hlatjidm 39327 Idempotence of join operat...
hlatjass 39328 Lattice join is associativ...
hlatj12 39329 Swap 1st and 2nd members o...
hlatj32 39330 Swap 2nd and 3rd members o...
hlatjrot 39331 Rotate lattice join of 3 c...
hlatj4 39332 Rearrangement of lattice j...
hlatlej1 39333 A join's first argument is...
hlatlej2 39334 A join's second argument i...
glbconN 39335 De Morgan's law for GLB an...
glbconNOLD 39336 Obsolete version of ~ glbc...
glbconxN 39337 De Morgan's law for GLB an...
atnlej1 39338 If an atom is not less tha...
atnlej2 39339 If an atom is not less tha...
hlsuprexch 39340 A Hilbert lattice has the ...
hlexch1 39341 A Hilbert lattice has the ...
hlexch2 39342 A Hilbert lattice has the ...
hlexchb1 39343 A Hilbert lattice has the ...
hlexchb2 39344 A Hilbert lattice has the ...
hlsupr 39345 A Hilbert lattice has the ...
hlsupr2 39346 A Hilbert lattice has the ...
hlhgt4 39347 A Hilbert lattice has a he...
hlhgt2 39348 A Hilbert lattice has a he...
hl0lt1N 39349 Lattice 0 is less than lat...
hlexch3 39350 A Hilbert lattice has the ...
hlexch4N 39351 A Hilbert lattice has the ...
hlatexchb1 39352 A version of ~ hlexchb1 fo...
hlatexchb2 39353 A version of ~ hlexchb2 fo...
hlatexch1 39354 Atom exchange property. (...
hlatexch2 39355 Atom exchange property. (...
hlatmstcOLDN 39356 An atomic, complete, ortho...
hlatle 39357 The ordering of two Hilber...
hlateq 39358 The equality of two Hilber...
hlrelat1 39359 An atomistic lattice with ...
hlrelat5N 39360 An atomistic lattice with ...
hlrelat 39361 A Hilbert lattice is relat...
hlrelat2 39362 A consequence of relative ...
exatleN 39363 A condition for an atom to...
hl2at 39364 A Hilbert lattice has at l...
atex 39365 At least one atom exists. ...
intnatN 39366 If the intersection with a...
2llnne2N 39367 Condition implying that tw...
2llnneN 39368 Condition implying that tw...
cvr1 39369 A Hilbert lattice has the ...
cvr2N 39370 Less-than and covers equiv...
hlrelat3 39371 The Hilbert lattice is rel...
cvrval3 39372 Binary relation expressing...
cvrval4N 39373 Binary relation expressing...
cvrval5 39374 Binary relation expressing...
cvrp 39375 A Hilbert lattice satisfie...
atcvr1 39376 An atom is covered by its ...
atcvr2 39377 An atom is covered by its ...
cvrexchlem 39378 Lemma for ~ cvrexch . ( ~...
cvrexch 39379 A Hilbert lattice satisfie...
cvratlem 39380 Lemma for ~ cvrat . ( ~ a...
cvrat 39381 A nonzero Hilbert lattice ...
ltltncvr 39382 A chained strong ordering ...
ltcvrntr 39383 Non-transitive condition f...
cvrntr 39384 The covers relation is not...
atcvr0eq 39385 The covers relation is not...
lnnat 39386 A line (the join of two di...
atcvrj0 39387 Two atoms covering the zer...
cvrat2 39388 A Hilbert lattice element ...
atcvrneN 39389 Inequality derived from at...
atcvrj1 39390 Condition for an atom to b...
atcvrj2b 39391 Condition for an atom to b...
atcvrj2 39392 Condition for an atom to b...
atleneN 39393 Inequality derived from at...
atltcvr 39394 An equivalence of less-tha...
atle 39395 Any nonzero element has an...
atlt 39396 Two atoms are unequal iff ...
atlelt 39397 Transfer less-than relatio...
2atlt 39398 Given an atom less than an...
atexchcvrN 39399 Atom exchange property. V...
atexchltN 39400 Atom exchange property. V...
cvrat3 39401 A condition implying that ...
cvrat4 39402 A condition implying exist...
cvrat42 39403 Commuted version of ~ cvra...
2atjm 39404 The meet of a line (expres...
atbtwn 39405 Property of a 3rd atom ` R...
atbtwnexOLDN 39406 There exists a 3rd atom ` ...
atbtwnex 39407 Given atoms ` P ` in ` X `...
3noncolr2 39408 Two ways to express 3 non-...
3noncolr1N 39409 Two ways to express 3 non-...
hlatcon3 39410 Atom exchange combined wit...
hlatcon2 39411 Atom exchange combined wit...
4noncolr3 39412 A way to express 4 non-col...
4noncolr2 39413 A way to express 4 non-col...
4noncolr1 39414 A way to express 4 non-col...
athgt 39415 A Hilbert lattice, whose h...
3dim0 39416 There exists a 3-dimension...
3dimlem1 39417 Lemma for ~ 3dim1 . (Cont...
3dimlem2 39418 Lemma for ~ 3dim1 . (Cont...
3dimlem3a 39419 Lemma for ~ 3dim3 . (Cont...
3dimlem3 39420 Lemma for ~ 3dim1 . (Cont...
3dimlem3OLDN 39421 Lemma for ~ 3dim1 . (Cont...
3dimlem4a 39422 Lemma for ~ 3dim3 . (Cont...
3dimlem4 39423 Lemma for ~ 3dim1 . (Cont...
3dimlem4OLDN 39424 Lemma for ~ 3dim1 . (Cont...
3dim1lem5 39425 Lemma for ~ 3dim1 . (Cont...
3dim1 39426 Construct a 3-dimensional ...
3dim2 39427 Construct 2 new layers on ...
3dim3 39428 Construct a new layer on t...
2dim 39429 Generate a height-3 elemen...
1dimN 39430 An atom is covered by a he...
1cvrco 39431 The orthocomplement of an ...
1cvratex 39432 There exists an atom less ...
1cvratlt 39433 An atom less than or equal...
1cvrjat 39434 An element covered by the ...
1cvrat 39435 Create an atom under an el...
ps-1 39436 The join of two atoms ` R ...
ps-2 39437 Lattice analogue for the p...
2atjlej 39438 Two atoms are different if...
hlatexch3N 39439 Rearrange join of atoms in...
hlatexch4 39440 Exchange 2 atoms. (Contri...
ps-2b 39441 Variation of projective ge...
3atlem1 39442 Lemma for ~ 3at . (Contri...
3atlem2 39443 Lemma for ~ 3at . (Contri...
3atlem3 39444 Lemma for ~ 3at . (Contri...
3atlem4 39445 Lemma for ~ 3at . (Contri...
3atlem5 39446 Lemma for ~ 3at . (Contri...
3atlem6 39447 Lemma for ~ 3at . (Contri...
3atlem7 39448 Lemma for ~ 3at . (Contri...
3at 39449 Any three non-colinear ato...
llnset 39464 The set of lattice lines i...
islln 39465 The predicate "is a lattic...
islln4 39466 The predicate "is a lattic...
llni 39467 Condition implying a latti...
llnbase 39468 A lattice line is a lattic...
islln3 39469 The predicate "is a lattic...
islln2 39470 The predicate "is a lattic...
llni2 39471 The join of two different ...
llnnleat 39472 An atom cannot majorize a ...
llnneat 39473 A lattice line is not an a...
2atneat 39474 The join of two distinct a...
llnn0 39475 A lattice line is nonzero....
islln2a 39476 The predicate "is a lattic...
llnle 39477 Any element greater than 0...
atcvrlln2 39478 An atom under a line is co...
atcvrlln 39479 An element covering an ato...
llnexatN 39480 Given an atom on a line, t...
llncmp 39481 If two lattice lines are c...
llnnlt 39482 Two lattice lines cannot s...
2llnmat 39483 Two intersecting lines int...
2at0mat0 39484 Special case of ~ 2atmat0 ...
2atmat0 39485 The meet of two unequal li...
2atm 39486 An atom majorized by two d...
ps-2c 39487 Variation of projective ge...
lplnset 39488 The set of lattice planes ...
islpln 39489 The predicate "is a lattic...
islpln4 39490 The predicate "is a lattic...
lplni 39491 Condition implying a latti...
islpln3 39492 The predicate "is a lattic...
lplnbase 39493 A lattice plane is a latti...
islpln5 39494 The predicate "is a lattic...
islpln2 39495 The predicate "is a lattic...
lplni2 39496 The join of 3 different at...
lvolex3N 39497 There is an atom outside o...
llnmlplnN 39498 The intersection of a line...
lplnle 39499 Any element greater than 0...
lplnnle2at 39500 A lattice line (or atom) c...
lplnnleat 39501 A lattice plane cannot maj...
lplnnlelln 39502 A lattice plane is not les...
2atnelpln 39503 The join of two atoms is n...
lplnneat 39504 No lattice plane is an ato...
lplnnelln 39505 No lattice plane is a latt...
lplnn0N 39506 A lattice plane is nonzero...
islpln2a 39507 The predicate "is a lattic...
islpln2ah 39508 The predicate "is a lattic...
lplnriaN 39509 Property of a lattice plan...
lplnribN 39510 Property of a lattice plan...
lplnric 39511 Property of a lattice plan...
lplnri1 39512 Property of a lattice plan...
lplnri2N 39513 Property of a lattice plan...
lplnri3N 39514 Property of a lattice plan...
lplnllnneN 39515 Two lattice lines defined ...
llncvrlpln2 39516 A lattice line under a lat...
llncvrlpln 39517 An element covering a latt...
2lplnmN 39518 If the join of two lattice...
2llnmj 39519 The meet of two lattice li...
2atmat 39520 The meet of two intersecti...
lplncmp 39521 If two lattice planes are ...
lplnexatN 39522 Given a lattice line on a ...
lplnexllnN 39523 Given an atom on a lattice...
lplnnlt 39524 Two lattice planes cannot ...
2llnjaN 39525 The join of two different ...
2llnjN 39526 The join of two different ...
2llnm2N 39527 The meet of two different ...
2llnm3N 39528 Two lattice lines in a lat...
2llnm4 39529 Two lattice lines that maj...
2llnmeqat 39530 An atom equals the interse...
lvolset 39531 The set of 3-dim lattice v...
islvol 39532 The predicate "is a 3-dim ...
islvol4 39533 The predicate "is a 3-dim ...
lvoli 39534 Condition implying a 3-dim...
islvol3 39535 The predicate "is a 3-dim ...
lvoli3 39536 Condition implying a 3-dim...
lvolbase 39537 A 3-dim lattice volume is ...
islvol5 39538 The predicate "is a 3-dim ...
islvol2 39539 The predicate "is a 3-dim ...
lvoli2 39540 The join of 4 different at...
lvolnle3at 39541 A lattice plane (or lattic...
lvolnleat 39542 An atom cannot majorize a ...
lvolnlelln 39543 A lattice line cannot majo...
lvolnlelpln 39544 A lattice plane cannot maj...
3atnelvolN 39545 The join of 3 atoms is not...
2atnelvolN 39546 The join of two atoms is n...
lvolneatN 39547 No lattice volume is an at...
lvolnelln 39548 No lattice volume is a lat...
lvolnelpln 39549 No lattice volume is a lat...
lvoln0N 39550 A lattice volume is nonzer...
islvol2aN 39551 The predicate "is a lattic...
4atlem0a 39552 Lemma for ~ 4at . (Contri...
4atlem0ae 39553 Lemma for ~ 4at . (Contri...
4atlem0be 39554 Lemma for ~ 4at . (Contri...
4atlem3 39555 Lemma for ~ 4at . Break i...
4atlem3a 39556 Lemma for ~ 4at . Break i...
4atlem3b 39557 Lemma for ~ 4at . Break i...
4atlem4a 39558 Lemma for ~ 4at . Frequen...
4atlem4b 39559 Lemma for ~ 4at . Frequen...
4atlem4c 39560 Lemma for ~ 4at . Frequen...
4atlem4d 39561 Lemma for ~ 4at . Frequen...
4atlem9 39562 Lemma for ~ 4at . Substit...
4atlem10a 39563 Lemma for ~ 4at . Substit...
4atlem10b 39564 Lemma for ~ 4at . Substit...
4atlem10 39565 Lemma for ~ 4at . Combine...
4atlem11a 39566 Lemma for ~ 4at . Substit...
4atlem11b 39567 Lemma for ~ 4at . Substit...
4atlem11 39568 Lemma for ~ 4at . Combine...
4atlem12a 39569 Lemma for ~ 4at . Substit...
4atlem12b 39570 Lemma for ~ 4at . Substit...
4atlem12 39571 Lemma for ~ 4at . Combine...
4at 39572 Four atoms determine a lat...
4at2 39573 Four atoms determine a lat...
lplncvrlvol2 39574 A lattice line under a lat...
lplncvrlvol 39575 An element covering a latt...
lvolcmp 39576 If two lattice planes are ...
lvolnltN 39577 Two lattice volumes cannot...
2lplnja 39578 The join of two different ...
2lplnj 39579 The join of two different ...
2lplnm2N 39580 The meet of two different ...
2lplnmj 39581 The meet of two lattice pl...
dalemkehl 39582 Lemma for ~ dath . Freque...
dalemkelat 39583 Lemma for ~ dath . Freque...
dalemkeop 39584 Lemma for ~ dath . Freque...
dalempea 39585 Lemma for ~ dath . Freque...
dalemqea 39586 Lemma for ~ dath . Freque...
dalemrea 39587 Lemma for ~ dath . Freque...
dalemsea 39588 Lemma for ~ dath . Freque...
dalemtea 39589 Lemma for ~ dath . Freque...
dalemuea 39590 Lemma for ~ dath . Freque...
dalemyeo 39591 Lemma for ~ dath . Freque...
dalemzeo 39592 Lemma for ~ dath . Freque...
dalemclpjs 39593 Lemma for ~ dath . Freque...
dalemclqjt 39594 Lemma for ~ dath . Freque...
dalemclrju 39595 Lemma for ~ dath . Freque...
dalem-clpjq 39596 Lemma for ~ dath . Freque...
dalemceb 39597 Lemma for ~ dath . Freque...
dalempeb 39598 Lemma for ~ dath . Freque...
dalemqeb 39599 Lemma for ~ dath . Freque...
dalemreb 39600 Lemma for ~ dath . Freque...
dalemseb 39601 Lemma for ~ dath . Freque...
dalemteb 39602 Lemma for ~ dath . Freque...
dalemueb 39603 Lemma for ~ dath . Freque...
dalempjqeb 39604 Lemma for ~ dath . Freque...
dalemsjteb 39605 Lemma for ~ dath . Freque...
dalemtjueb 39606 Lemma for ~ dath . Freque...
dalemqrprot 39607 Lemma for ~ dath . Freque...
dalemyeb 39608 Lemma for ~ dath . Freque...
dalemcnes 39609 Lemma for ~ dath . Freque...
dalempnes 39610 Lemma for ~ dath . Freque...
dalemqnet 39611 Lemma for ~ dath . Freque...
dalempjsen 39612 Lemma for ~ dath . Freque...
dalemply 39613 Lemma for ~ dath . Freque...
dalemsly 39614 Lemma for ~ dath . Freque...
dalemswapyz 39615 Lemma for ~ dath . Swap t...
dalemrot 39616 Lemma for ~ dath . Rotate...
dalemrotyz 39617 Lemma for ~ dath . Rotate...
dalem1 39618 Lemma for ~ dath . Show t...
dalemcea 39619 Lemma for ~ dath . Freque...
dalem2 39620 Lemma for ~ dath . Show t...
dalemdea 39621 Lemma for ~ dath . Freque...
dalemeea 39622 Lemma for ~ dath . Freque...
dalem3 39623 Lemma for ~ dalemdnee . (...
dalem4 39624 Lemma for ~ dalemdnee . (...
dalemdnee 39625 Lemma for ~ dath . Axis o...
dalem5 39626 Lemma for ~ dath . Atom `...
dalem6 39627 Lemma for ~ dath . Analog...
dalem7 39628 Lemma for ~ dath . Analog...
dalem8 39629 Lemma for ~ dath . Plane ...
dalem-cly 39630 Lemma for ~ dalem9 . Cent...
dalem9 39631 Lemma for ~ dath . Since ...
dalem10 39632 Lemma for ~ dath . Atom `...
dalem11 39633 Lemma for ~ dath . Analog...
dalem12 39634 Lemma for ~ dath . Analog...
dalem13 39635 Lemma for ~ dalem14 . (Co...
dalem14 39636 Lemma for ~ dath . Planes...
dalem15 39637 Lemma for ~ dath . The ax...
dalem16 39638 Lemma for ~ dath . The at...
dalem17 39639 Lemma for ~ dath . When p...
dalem18 39640 Lemma for ~ dath . Show t...
dalem19 39641 Lemma for ~ dath . Show t...
dalemccea 39642 Lemma for ~ dath . Freque...
dalemddea 39643 Lemma for ~ dath . Freque...
dalem-ccly 39644 Lemma for ~ dath . Freque...
dalem-ddly 39645 Lemma for ~ dath . Freque...
dalemccnedd 39646 Lemma for ~ dath . Freque...
dalemclccjdd 39647 Lemma for ~ dath . Freque...
dalemcceb 39648 Lemma for ~ dath . Freque...
dalemswapyzps 39649 Lemma for ~ dath . Swap t...
dalemrotps 39650 Lemma for ~ dath . Rotate...
dalemcjden 39651 Lemma for ~ dath . Show t...
dalem20 39652 Lemma for ~ dath . Show t...
dalem21 39653 Lemma for ~ dath . Show t...
dalem22 39654 Lemma for ~ dath . Show t...
dalem23 39655 Lemma for ~ dath . Show t...
dalem24 39656 Lemma for ~ dath . Show t...
dalem25 39657 Lemma for ~ dath . Show t...
dalem27 39658 Lemma for ~ dath . Show t...
dalem28 39659 Lemma for ~ dath . Lemma ...
dalem29 39660 Lemma for ~ dath . Analog...
dalem30 39661 Lemma for ~ dath . Analog...
dalem31N 39662 Lemma for ~ dath . Analog...
dalem32 39663 Lemma for ~ dath . Analog...
dalem33 39664 Lemma for ~ dath . Analog...
dalem34 39665 Lemma for ~ dath . Analog...
dalem35 39666 Lemma for ~ dath . Analog...
dalem36 39667 Lemma for ~ dath . Analog...
dalem37 39668 Lemma for ~ dath . Analog...
dalem38 39669 Lemma for ~ dath . Plane ...
dalem39 39670 Lemma for ~ dath . Auxili...
dalem40 39671 Lemma for ~ dath . Analog...
dalem41 39672 Lemma for ~ dath . (Contr...
dalem42 39673 Lemma for ~ dath . Auxili...
dalem43 39674 Lemma for ~ dath . Planes...
dalem44 39675 Lemma for ~ dath . Dummy ...
dalem45 39676 Lemma for ~ dath . Dummy ...
dalem46 39677 Lemma for ~ dath . Analog...
dalem47 39678 Lemma for ~ dath . Analog...
dalem48 39679 Lemma for ~ dath . Analog...
dalem49 39680 Lemma for ~ dath . Analog...
dalem50 39681 Lemma for ~ dath . Analog...
dalem51 39682 Lemma for ~ dath . Constr...
dalem52 39683 Lemma for ~ dath . Lines ...
dalem53 39684 Lemma for ~ dath . The au...
dalem54 39685 Lemma for ~ dath . Line `...
dalem55 39686 Lemma for ~ dath . Lines ...
dalem56 39687 Lemma for ~ dath . Analog...
dalem57 39688 Lemma for ~ dath . Axis o...
dalem58 39689 Lemma for ~ dath . Analog...
dalem59 39690 Lemma for ~ dath . Analog...
dalem60 39691 Lemma for ~ dath . ` B ` i...
dalem61 39692 Lemma for ~ dath . Show t...
dalem62 39693 Lemma for ~ dath . Elimin...
dalem63 39694 Lemma for ~ dath . Combin...
dath 39695 Desargues's theorem of pro...
dath2 39696 Version of Desargues's the...
lineset 39697 The set of lines in a Hilb...
isline 39698 The predicate "is a line"....
islinei 39699 Condition implying "is a l...
pointsetN 39700 The set of points in a Hil...
ispointN 39701 The predicate "is a point"...
atpointN 39702 The singleton of an atom i...
psubspset 39703 The set of projective subs...
ispsubsp 39704 The predicate "is a projec...
ispsubsp2 39705 The predicate "is a projec...
psubspi 39706 Property of a projective s...
psubspi2N 39707 Property of a projective s...
0psubN 39708 The empty set is a project...
snatpsubN 39709 The singleton of an atom i...
pointpsubN 39710 A point (singleton of an a...
linepsubN 39711 A line is a projective sub...
atpsubN 39712 The set of all atoms is a ...
psubssat 39713 A projective subspace cons...
psubatN 39714 A member of a projective s...
pmapfval 39715 The projective map of a Hi...
pmapval 39716 Value of the projective ma...
elpmap 39717 Member of a projective map...
pmapssat 39718 The projective map of a Hi...
pmapssbaN 39719 A weakening of ~ pmapssat ...
pmaple 39720 The projective map of a Hi...
pmap11 39721 The projective map of a Hi...
pmapat 39722 The projective map of an a...
elpmapat 39723 Member of the projective m...
pmap0 39724 Value of the projective ma...
pmapeq0 39725 A projective map value is ...
pmap1N 39726 Value of the projective ma...
pmapsub 39727 The projective map of a Hi...
pmapglbx 39728 The projective map of the ...
pmapglb 39729 The projective map of the ...
pmapglb2N 39730 The projective map of the ...
pmapglb2xN 39731 The projective map of the ...
pmapmeet 39732 The projective map of a me...
isline2 39733 Definition of line in term...
linepmap 39734 A line described with a pr...
isline3 39735 Definition of line in term...
isline4N 39736 Definition of line in term...
lneq2at 39737 A line equals the join of ...
lnatexN 39738 There is an atom in a line...
lnjatN 39739 Given an atom in a line, t...
lncvrelatN 39740 A lattice element covered ...
lncvrat 39741 A line covers the atoms it...
lncmp 39742 If two lines are comparabl...
2lnat 39743 Two intersecting lines int...
2atm2atN 39744 Two joins with a common at...
2llnma1b 39745 Generalization of ~ 2llnma...
2llnma1 39746 Two different intersecting...
2llnma3r 39747 Two different intersecting...
2llnma2 39748 Two different intersecting...
2llnma2rN 39749 Two different intersecting...
cdlema1N 39750 A condition for required f...
cdlema2N 39751 A condition for required f...
cdlemblem 39752 Lemma for ~ cdlemb . (Con...
cdlemb 39753 Given two atoms not less t...
paddfval 39756 Projective subspace sum op...
paddval 39757 Projective subspace sum op...
elpadd 39758 Member of a projective sub...
elpaddn0 39759 Member of projective subsp...
paddvaln0N 39760 Projective subspace sum op...
elpaddri 39761 Condition implying members...
elpaddatriN 39762 Condition implying members...
elpaddat 39763 Membership in a projective...
elpaddatiN 39764 Consequence of membership ...
elpadd2at 39765 Membership in a projective...
elpadd2at2 39766 Membership in a projective...
paddunssN 39767 Projective subspace sum in...
elpadd0 39768 Member of projective subsp...
paddval0 39769 Projective subspace sum wi...
padd01 39770 Projective subspace sum wi...
padd02 39771 Projective subspace sum wi...
paddcom 39772 Projective subspace sum co...
paddssat 39773 A projective subspace sum ...
sspadd1 39774 A projective subspace sum ...
sspadd2 39775 A projective subspace sum ...
paddss1 39776 Subset law for projective ...
paddss2 39777 Subset law for projective ...
paddss12 39778 Subset law for projective ...
paddasslem1 39779 Lemma for ~ paddass . (Co...
paddasslem2 39780 Lemma for ~ paddass . (Co...
paddasslem3 39781 Lemma for ~ paddass . Res...
paddasslem4 39782 Lemma for ~ paddass . Com...
paddasslem5 39783 Lemma for ~ paddass . Sho...
paddasslem6 39784 Lemma for ~ paddass . (Co...
paddasslem7 39785 Lemma for ~ paddass . Com...
paddasslem8 39786 Lemma for ~ paddass . (Co...
paddasslem9 39787 Lemma for ~ paddass . Com...
paddasslem10 39788 Lemma for ~ paddass . Use...
paddasslem11 39789 Lemma for ~ paddass . The...
paddasslem12 39790 Lemma for ~ paddass . The...
paddasslem13 39791 Lemma for ~ paddass . The...
paddasslem14 39792 Lemma for ~ paddass . Rem...
paddasslem15 39793 Lemma for ~ paddass . Use...
paddasslem16 39794 Lemma for ~ paddass . Use...
paddasslem17 39795 Lemma for ~ paddass . The...
paddasslem18 39796 Lemma for ~ paddass . Com...
paddass 39797 Projective subspace sum is...
padd12N 39798 Commutative/associative la...
padd4N 39799 Rearrangement of 4 terms i...
paddidm 39800 Projective subspace sum is...
paddclN 39801 The projective sum of two ...
paddssw1 39802 Subset law for projective ...
paddssw2 39803 Subset law for projective ...
paddss 39804 Subset law for projective ...
pmodlem1 39805 Lemma for ~ pmod1i . (Con...
pmodlem2 39806 Lemma for ~ pmod1i . (Con...
pmod1i 39807 The modular law holds in a...
pmod2iN 39808 Dual of the modular law. ...
pmodN 39809 The modular law for projec...
pmodl42N 39810 Lemma derived from modular...
pmapjoin 39811 The projective map of the ...
pmapjat1 39812 The projective map of the ...
pmapjat2 39813 The projective map of the ...
pmapjlln1 39814 The projective map of the ...
hlmod1i 39815 A version of the modular l...
atmod1i1 39816 Version of modular law ~ p...
atmod1i1m 39817 Version of modular law ~ p...
atmod1i2 39818 Version of modular law ~ p...
llnmod1i2 39819 Version of modular law ~ p...
atmod2i1 39820 Version of modular law ~ p...
atmod2i2 39821 Version of modular law ~ p...
llnmod2i2 39822 Version of modular law ~ p...
atmod3i1 39823 Version of modular law tha...
atmod3i2 39824 Version of modular law tha...
atmod4i1 39825 Version of modular law tha...
atmod4i2 39826 Version of modular law tha...
llnexchb2lem 39827 Lemma for ~ llnexchb2 . (...
llnexchb2 39828 Line exchange property (co...
llnexch2N 39829 Line exchange property (co...
dalawlem1 39830 Lemma for ~ dalaw . Speci...
dalawlem2 39831 Lemma for ~ dalaw . Utili...
dalawlem3 39832 Lemma for ~ dalaw . First...
dalawlem4 39833 Lemma for ~ dalaw . Secon...
dalawlem5 39834 Lemma for ~ dalaw . Speci...
dalawlem6 39835 Lemma for ~ dalaw . First...
dalawlem7 39836 Lemma for ~ dalaw . Secon...
dalawlem8 39837 Lemma for ~ dalaw . Speci...
dalawlem9 39838 Lemma for ~ dalaw . Speci...
dalawlem10 39839 Lemma for ~ dalaw . Combi...
dalawlem11 39840 Lemma for ~ dalaw . First...
dalawlem12 39841 Lemma for ~ dalaw . Secon...
dalawlem13 39842 Lemma for ~ dalaw . Speci...
dalawlem14 39843 Lemma for ~ dalaw . Combi...
dalawlem15 39844 Lemma for ~ dalaw . Swap ...
dalaw 39845 Desargues's law, derived f...
pclfvalN 39848 The projective subspace cl...
pclvalN 39849 Value of the projective su...
pclclN 39850 Closure of the projective ...
elpclN 39851 Membership in the projecti...
elpcliN 39852 Implication of membership ...
pclssN 39853 Ordering is preserved by s...
pclssidN 39854 A set of atoms is included...
pclidN 39855 The projective subspace cl...
pclbtwnN 39856 A projective subspace sand...
pclunN 39857 The projective subspace cl...
pclun2N 39858 The projective subspace cl...
pclfinN 39859 The projective subspace cl...
pclcmpatN 39860 The set of projective subs...
polfvalN 39863 The projective subspace po...
polvalN 39864 Value of the projective su...
polval2N 39865 Alternate expression for v...
polsubN 39866 The polarity of a set of a...
polssatN 39867 The polarity of a set of a...
pol0N 39868 The polarity of the empty ...
pol1N 39869 The polarity of the whole ...
2pol0N 39870 The closed subspace closur...
polpmapN 39871 The polarity of a projecti...
2polpmapN 39872 Double polarity of a proje...
2polvalN 39873 Value of double polarity. ...
2polssN 39874 A set of atoms is a subset...
3polN 39875 Triple polarity cancels to...
polcon3N 39876 Contraposition law for pol...
2polcon4bN 39877 Contraposition law for pol...
polcon2N 39878 Contraposition law for pol...
polcon2bN 39879 Contraposition law for pol...
pclss2polN 39880 The projective subspace cl...
pcl0N 39881 The projective subspace cl...
pcl0bN 39882 The projective subspace cl...
pmaplubN 39883 The LUB of a projective ma...
sspmaplubN 39884 A set of atoms is a subset...
2pmaplubN 39885 Double projective map of a...
paddunN 39886 The closure of the project...
poldmj1N 39887 De Morgan's law for polari...
pmapj2N 39888 The projective map of the ...
pmapocjN 39889 The projective map of the ...
polatN 39890 The polarity of the single...
2polatN 39891 Double polarity of the sin...
pnonsingN 39892 The intersection of a set ...
psubclsetN 39895 The set of closed projecti...
ispsubclN 39896 The predicate "is a closed...
psubcliN 39897 Property of a closed proje...
psubcli2N 39898 Property of a closed proje...
psubclsubN 39899 A closed projective subspa...
psubclssatN 39900 A closed projective subspa...
pmapidclN 39901 Projective map of the LUB ...
0psubclN 39902 The empty set is a closed ...
1psubclN 39903 The set of all atoms is a ...
atpsubclN 39904 A point (singleton of an a...
pmapsubclN 39905 A projective map value is ...
ispsubcl2N 39906 Alternate predicate for "i...
psubclinN 39907 The intersection of two cl...
paddatclN 39908 The projective sum of a cl...
pclfinclN 39909 The projective subspace cl...
linepsubclN 39910 A line is a closed project...
polsubclN 39911 A polarity is a closed pro...
poml4N 39912 Orthomodular law for proje...
poml5N 39913 Orthomodular law for proje...
poml6N 39914 Orthomodular law for proje...
osumcllem1N 39915 Lemma for ~ osumclN . (Co...
osumcllem2N 39916 Lemma for ~ osumclN . (Co...
osumcllem3N 39917 Lemma for ~ osumclN . (Co...
osumcllem4N 39918 Lemma for ~ osumclN . (Co...
osumcllem5N 39919 Lemma for ~ osumclN . (Co...
osumcllem6N 39920 Lemma for ~ osumclN . Use...
osumcllem7N 39921 Lemma for ~ osumclN . (Co...
osumcllem8N 39922 Lemma for ~ osumclN . (Co...
osumcllem9N 39923 Lemma for ~ osumclN . (Co...
osumcllem10N 39924 Lemma for ~ osumclN . Con...
osumcllem11N 39925 Lemma for ~ osumclN . (Co...
osumclN 39926 Closure of orthogonal sum....
pmapojoinN 39927 For orthogonal elements, p...
pexmidN 39928 Excluded middle law for cl...
pexmidlem1N 39929 Lemma for ~ pexmidN . Hol...
pexmidlem2N 39930 Lemma for ~ pexmidN . (Co...
pexmidlem3N 39931 Lemma for ~ pexmidN . Use...
pexmidlem4N 39932 Lemma for ~ pexmidN . (Co...
pexmidlem5N 39933 Lemma for ~ pexmidN . (Co...
pexmidlem6N 39934 Lemma for ~ pexmidN . (Co...
pexmidlem7N 39935 Lemma for ~ pexmidN . Con...
pexmidlem8N 39936 Lemma for ~ pexmidN . The...
pexmidALTN 39937 Excluded middle law for cl...
pl42lem1N 39938 Lemma for ~ pl42N . (Cont...
pl42lem2N 39939 Lemma for ~ pl42N . (Cont...
pl42lem3N 39940 Lemma for ~ pl42N . (Cont...
pl42lem4N 39941 Lemma for ~ pl42N . (Cont...
pl42N 39942 Law holding in a Hilbert l...
watfvalN 39951 The W atoms function. (Co...
watvalN 39952 Value of the W atoms funct...
iswatN 39953 The predicate "is a W atom...
lhpset 39954 The set of co-atoms (latti...
islhp 39955 The predicate "is a co-ato...
islhp2 39956 The predicate "is a co-ato...
lhpbase 39957 A co-atom is a member of t...
lhp1cvr 39958 The lattice unity covers a...
lhplt 39959 An atom under a co-atom is...
lhp2lt 39960 The join of two atoms unde...
lhpexlt 39961 There exists an atom less ...
lhp0lt 39962 A co-atom is greater than ...
lhpn0 39963 A co-atom is nonzero. TOD...
lhpexle 39964 There exists an atom under...
lhpexnle 39965 There exists an atom not u...
lhpexle1lem 39966 Lemma for ~ lhpexle1 and o...
lhpexle1 39967 There exists an atom under...
lhpexle2lem 39968 Lemma for ~ lhpexle2 . (C...
lhpexle2 39969 There exists atom under a ...
lhpexle3lem 39970 There exists atom under a ...
lhpexle3 39971 There exists atom under a ...
lhpex2leN 39972 There exist at least two d...
lhpoc 39973 The orthocomplement of a c...
lhpoc2N 39974 The orthocomplement of an ...
lhpocnle 39975 The orthocomplement of a c...
lhpocat 39976 The orthocomplement of a c...
lhpocnel 39977 The orthocomplement of a c...
lhpocnel2 39978 The orthocomplement of a c...
lhpjat1 39979 The join of a co-atom (hyp...
lhpjat2 39980 The join of a co-atom (hyp...
lhpj1 39981 The join of a co-atom (hyp...
lhpmcvr 39982 The meet of a lattice hype...
lhpmcvr2 39983 Alternate way to express t...
lhpmcvr3 39984 Specialization of ~ lhpmcv...
lhpmcvr4N 39985 Specialization of ~ lhpmcv...
lhpmcvr5N 39986 Specialization of ~ lhpmcv...
lhpmcvr6N 39987 Specialization of ~ lhpmcv...
lhpm0atN 39988 If the meet of a lattice h...
lhpmat 39989 An element covered by the ...
lhpmatb 39990 An element covered by the ...
lhp2at0 39991 Join and meet with differe...
lhp2atnle 39992 Inequality for 2 different...
lhp2atne 39993 Inequality for joins with ...
lhp2at0nle 39994 Inequality for 2 different...
lhp2at0ne 39995 Inequality for joins with ...
lhpelim 39996 Eliminate an atom not unde...
lhpmod2i2 39997 Modular law for hyperplane...
lhpmod6i1 39998 Modular law for hyperplane...
lhprelat3N 39999 The Hilbert lattice is rel...
cdlemb2 40000 Given two atoms not under ...
lhple 40001 Property of a lattice elem...
lhpat 40002 Create an atom under a co-...
lhpat4N 40003 Property of an atom under ...
lhpat2 40004 Create an atom under a co-...
lhpat3 40005 There is only one atom und...
4atexlemk 40006 Lemma for ~ 4atexlem7 . (...
4atexlemw 40007 Lemma for ~ 4atexlem7 . (...
4atexlempw 40008 Lemma for ~ 4atexlem7 . (...
4atexlemp 40009 Lemma for ~ 4atexlem7 . (...
4atexlemq 40010 Lemma for ~ 4atexlem7 . (...
4atexlems 40011 Lemma for ~ 4atexlem7 . (...
4atexlemt 40012 Lemma for ~ 4atexlem7 . (...
4atexlemutvt 40013 Lemma for ~ 4atexlem7 . (...
4atexlempnq 40014 Lemma for ~ 4atexlem7 . (...
4atexlemnslpq 40015 Lemma for ~ 4atexlem7 . (...
4atexlemkl 40016 Lemma for ~ 4atexlem7 . (...
4atexlemkc 40017 Lemma for ~ 4atexlem7 . (...
4atexlemwb 40018 Lemma for ~ 4atexlem7 . (...
4atexlempsb 40019 Lemma for ~ 4atexlem7 . (...
4atexlemqtb 40020 Lemma for ~ 4atexlem7 . (...
4atexlempns 40021 Lemma for ~ 4atexlem7 . (...
4atexlemswapqr 40022 Lemma for ~ 4atexlem7 . S...
4atexlemu 40023 Lemma for ~ 4atexlem7 . (...
4atexlemv 40024 Lemma for ~ 4atexlem7 . (...
4atexlemunv 40025 Lemma for ~ 4atexlem7 . (...
4atexlemtlw 40026 Lemma for ~ 4atexlem7 . (...
4atexlemntlpq 40027 Lemma for ~ 4atexlem7 . (...
4atexlemc 40028 Lemma for ~ 4atexlem7 . (...
4atexlemnclw 40029 Lemma for ~ 4atexlem7 . (...
4atexlemex2 40030 Lemma for ~ 4atexlem7 . S...
4atexlemcnd 40031 Lemma for ~ 4atexlem7 . (...
4atexlemex4 40032 Lemma for ~ 4atexlem7 . S...
4atexlemex6 40033 Lemma for ~ 4atexlem7 . (...
4atexlem7 40034 Whenever there are at leas...
4atex 40035 Whenever there are at leas...
4atex2 40036 More general version of ~ ...
4atex2-0aOLDN 40037 Same as ~ 4atex2 except th...
4atex2-0bOLDN 40038 Same as ~ 4atex2 except th...
4atex2-0cOLDN 40039 Same as ~ 4atex2 except th...
4atex3 40040 More general version of ~ ...
lautset 40041 The set of lattice automor...
islaut 40042 The predicate "is a lattic...
lautle 40043 Less-than or equal propert...
laut1o 40044 A lattice automorphism is ...
laut11 40045 One-to-one property of a l...
lautcl 40046 A lattice automorphism val...
lautcnvclN 40047 Reverse closure of a latti...
lautcnvle 40048 Less-than or equal propert...
lautcnv 40049 The converse of a lattice ...
lautlt 40050 Less-than property of a la...
lautcvr 40051 Covering property of a lat...
lautj 40052 Meet property of a lattice...
lautm 40053 Meet property of a lattice...
lauteq 40054 A lattice automorphism arg...
idlaut 40055 The identity function is a...
lautco 40056 The composition of two lat...
pautsetN 40057 The set of projective auto...
ispautN 40058 The predicate "is a projec...
ldilfset 40067 The mapping from fiducial ...
ldilset 40068 The set of lattice dilatio...
isldil 40069 The predicate "is a lattic...
ldillaut 40070 A lattice dilation is an a...
ldil1o 40071 A lattice dilation is a on...
ldilval 40072 Value of a lattice dilatio...
idldil 40073 The identity function is a...
ldilcnv 40074 The converse of a lattice ...
ldilco 40075 The composition of two lat...
ltrnfset 40076 The set of all lattice tra...
ltrnset 40077 The set of lattice transla...
isltrn 40078 The predicate "is a lattic...
isltrn2N 40079 The predicate "is a lattic...
ltrnu 40080 Uniqueness property of a l...
ltrnldil 40081 A lattice translation is a...
ltrnlaut 40082 A lattice translation is a...
ltrn1o 40083 A lattice translation is a...
ltrncl 40084 Closure of a lattice trans...
ltrn11 40085 One-to-one property of a l...
ltrncnvnid 40086 If a translation is differ...
ltrncoidN 40087 Two translations are equal...
ltrnle 40088 Less-than or equal propert...
ltrncnvleN 40089 Less-than or equal propert...
ltrnm 40090 Lattice translation of a m...
ltrnj 40091 Lattice translation of a m...
ltrncvr 40092 Covering property of a lat...
ltrnval1 40093 Value of a lattice transla...
ltrnid 40094 A lattice translation is t...
ltrnnid 40095 If a lattice translation i...
ltrnatb 40096 The lattice translation of...
ltrncnvatb 40097 The converse of the lattic...
ltrnel 40098 The lattice translation of...
ltrnat 40099 The lattice translation of...
ltrncnvat 40100 The converse of the lattic...
ltrncnvel 40101 The converse of the lattic...
ltrncoelN 40102 Composition of lattice tra...
ltrncoat 40103 Composition of lattice tra...
ltrncoval 40104 Two ways to express value ...
ltrncnv 40105 The converse of a lattice ...
ltrn11at 40106 Frequently used one-to-one...
ltrneq2 40107 The equality of two transl...
ltrneq 40108 The equality of two transl...
idltrn 40109 The identity function is a...
ltrnmw 40110 Property of lattice transl...
dilfsetN 40111 The mapping from fiducial ...
dilsetN 40112 The set of dilations for a...
isdilN 40113 The predicate "is a dilati...
trnfsetN 40114 The mapping from fiducial ...
trnsetN 40115 The set of translations fo...
istrnN 40116 The predicate "is a transl...
trlfset 40119 The set of all traces of l...
trlset 40120 The set of traces of latti...
trlval 40121 The value of the trace of ...
trlval2 40122 The value of the trace of ...
trlcl 40123 Closure of the trace of a ...
trlcnv 40124 The trace of the converse ...
trljat1 40125 The value of a translation...
trljat2 40126 The value of a translation...
trljat3 40127 The value of a translation...
trlat 40128 If an atom differs from it...
trl0 40129 If an atom not under the f...
trlator0 40130 The trace of a lattice tra...
trlatn0 40131 The trace of a lattice tra...
trlnidat 40132 The trace of a lattice tra...
ltrnnidn 40133 If a lattice translation i...
ltrnideq 40134 Property of the identity l...
trlid0 40135 The trace of the identity ...
trlnidatb 40136 A lattice translation is n...
trlid0b 40137 A lattice translation is t...
trlnid 40138 Different translations wit...
ltrn2ateq 40139 Property of the equality o...
ltrnateq 40140 If any atom (under ` W ` )...
ltrnatneq 40141 If any atom (under ` W ` )...
ltrnatlw 40142 If the value of an atom eq...
trlle 40143 The trace of a lattice tra...
trlne 40144 The trace of a lattice tra...
trlnle 40145 The atom not under the fid...
trlval3 40146 The value of the trace of ...
trlval4 40147 The value of the trace of ...
trlval5 40148 The value of the trace of ...
arglem1N 40149 Lemma for Desargues's law....
cdlemc1 40150 Part of proof of Lemma C i...
cdlemc2 40151 Part of proof of Lemma C i...
cdlemc3 40152 Part of proof of Lemma C i...
cdlemc4 40153 Part of proof of Lemma C i...
cdlemc5 40154 Lemma for ~ cdlemc . (Con...
cdlemc6 40155 Lemma for ~ cdlemc . (Con...
cdlemc 40156 Lemma C in [Crawley] p. 11...
cdlemd1 40157 Part of proof of Lemma D i...
cdlemd2 40158 Part of proof of Lemma D i...
cdlemd3 40159 Part of proof of Lemma D i...
cdlemd4 40160 Part of proof of Lemma D i...
cdlemd5 40161 Part of proof of Lemma D i...
cdlemd6 40162 Part of proof of Lemma D i...
cdlemd7 40163 Part of proof of Lemma D i...
cdlemd8 40164 Part of proof of Lemma D i...
cdlemd9 40165 Part of proof of Lemma D i...
cdlemd 40166 If two translations agree ...
ltrneq3 40167 Two translations agree at ...
cdleme00a 40168 Part of proof of Lemma E i...
cdleme0aa 40169 Part of proof of Lemma E i...
cdleme0a 40170 Part of proof of Lemma E i...
cdleme0b 40171 Part of proof of Lemma E i...
cdleme0c 40172 Part of proof of Lemma E i...
cdleme0cp 40173 Part of proof of Lemma E i...
cdleme0cq 40174 Part of proof of Lemma E i...
cdleme0dN 40175 Part of proof of Lemma E i...
cdleme0e 40176 Part of proof of Lemma E i...
cdleme0fN 40177 Part of proof of Lemma E i...
cdleme0gN 40178 Part of proof of Lemma E i...
cdlemeulpq 40179 Part of proof of Lemma E i...
cdleme01N 40180 Part of proof of Lemma E i...
cdleme02N 40181 Part of proof of Lemma E i...
cdleme0ex1N 40182 Part of proof of Lemma E i...
cdleme0ex2N 40183 Part of proof of Lemma E i...
cdleme0moN 40184 Part of proof of Lemma E i...
cdleme1b 40185 Part of proof of Lemma E i...
cdleme1 40186 Part of proof of Lemma E i...
cdleme2 40187 Part of proof of Lemma E i...
cdleme3b 40188 Part of proof of Lemma E i...
cdleme3c 40189 Part of proof of Lemma E i...
cdleme3d 40190 Part of proof of Lemma E i...
cdleme3e 40191 Part of proof of Lemma E i...
cdleme3fN 40192 Part of proof of Lemma E i...
cdleme3g 40193 Part of proof of Lemma E i...
cdleme3h 40194 Part of proof of Lemma E i...
cdleme3fa 40195 Part of proof of Lemma E i...
cdleme3 40196 Part of proof of Lemma E i...
cdleme4 40197 Part of proof of Lemma E i...
cdleme4a 40198 Part of proof of Lemma E i...
cdleme5 40199 Part of proof of Lemma E i...
cdleme6 40200 Part of proof of Lemma E i...
cdleme7aa 40201 Part of proof of Lemma E i...
cdleme7a 40202 Part of proof of Lemma E i...
cdleme7b 40203 Part of proof of Lemma E i...
cdleme7c 40204 Part of proof of Lemma E i...
cdleme7d 40205 Part of proof of Lemma E i...
cdleme7e 40206 Part of proof of Lemma E i...
cdleme7ga 40207 Part of proof of Lemma E i...
cdleme7 40208 Part of proof of Lemma E i...
cdleme8 40209 Part of proof of Lemma E i...
cdleme9a 40210 Part of proof of Lemma E i...
cdleme9b 40211 Utility lemma for Lemma E ...
cdleme9 40212 Part of proof of Lemma E i...
cdleme10 40213 Part of proof of Lemma E i...
cdleme8tN 40214 Part of proof of Lemma E i...
cdleme9taN 40215 Part of proof of Lemma E i...
cdleme9tN 40216 Part of proof of Lemma E i...
cdleme10tN 40217 Part of proof of Lemma E i...
cdleme16aN 40218 Part of proof of Lemma E i...
cdleme11a 40219 Part of proof of Lemma E i...
cdleme11c 40220 Part of proof of Lemma E i...
cdleme11dN 40221 Part of proof of Lemma E i...
cdleme11e 40222 Part of proof of Lemma E i...
cdleme11fN 40223 Part of proof of Lemma E i...
cdleme11g 40224 Part of proof of Lemma E i...
cdleme11h 40225 Part of proof of Lemma E i...
cdleme11j 40226 Part of proof of Lemma E i...
cdleme11k 40227 Part of proof of Lemma E i...
cdleme11l 40228 Part of proof of Lemma E i...
cdleme11 40229 Part of proof of Lemma E i...
cdleme12 40230 Part of proof of Lemma E i...
cdleme13 40231 Part of proof of Lemma E i...
cdleme14 40232 Part of proof of Lemma E i...
cdleme15a 40233 Part of proof of Lemma E i...
cdleme15b 40234 Part of proof of Lemma E i...
cdleme15c 40235 Part of proof of Lemma E i...
cdleme15d 40236 Part of proof of Lemma E i...
cdleme15 40237 Part of proof of Lemma E i...
cdleme16b 40238 Part of proof of Lemma E i...
cdleme16c 40239 Part of proof of Lemma E i...
cdleme16d 40240 Part of proof of Lemma E i...
cdleme16e 40241 Part of proof of Lemma E i...
cdleme16f 40242 Part of proof of Lemma E i...
cdleme16g 40243 Part of proof of Lemma E i...
cdleme16 40244 Part of proof of Lemma E i...
cdleme17a 40245 Part of proof of Lemma E i...
cdleme17b 40246 Lemma leading to ~ cdleme1...
cdleme17c 40247 Part of proof of Lemma E i...
cdleme17d1 40248 Part of proof of Lemma E i...
cdleme0nex 40249 Part of proof of Lemma E i...
cdleme18a 40250 Part of proof of Lemma E i...
cdleme18b 40251 Part of proof of Lemma E i...
cdleme18c 40252 Part of proof of Lemma E i...
cdleme22gb 40253 Utility lemma for Lemma E ...
cdleme18d 40254 Part of proof of Lemma E i...
cdlemesner 40255 Part of proof of Lemma E i...
cdlemedb 40256 Part of proof of Lemma E i...
cdlemeda 40257 Part of proof of Lemma E i...
cdlemednpq 40258 Part of proof of Lemma E i...
cdlemednuN 40259 Part of proof of Lemma E i...
cdleme20zN 40260 Part of proof of Lemma E i...
cdleme20y 40261 Part of proof of Lemma E i...
cdleme19a 40262 Part of proof of Lemma E i...
cdleme19b 40263 Part of proof of Lemma E i...
cdleme19c 40264 Part of proof of Lemma E i...
cdleme19d 40265 Part of proof of Lemma E i...
cdleme19e 40266 Part of proof of Lemma E i...
cdleme19f 40267 Part of proof of Lemma E i...
cdleme20aN 40268 Part of proof of Lemma E i...
cdleme20bN 40269 Part of proof of Lemma E i...
cdleme20c 40270 Part of proof of Lemma E i...
cdleme20d 40271 Part of proof of Lemma E i...
cdleme20e 40272 Part of proof of Lemma E i...
cdleme20f 40273 Part of proof of Lemma E i...
cdleme20g 40274 Part of proof of Lemma E i...
cdleme20h 40275 Part of proof of Lemma E i...
cdleme20i 40276 Part of proof of Lemma E i...
cdleme20j 40277 Part of proof of Lemma E i...
cdleme20k 40278 Part of proof of Lemma E i...
cdleme20l1 40279 Part of proof of Lemma E i...
cdleme20l2 40280 Part of proof of Lemma E i...
cdleme20l 40281 Part of proof of Lemma E i...
cdleme20m 40282 Part of proof of Lemma E i...
cdleme20 40283 Combine ~ cdleme19f and ~ ...
cdleme21a 40284 Part of proof of Lemma E i...
cdleme21b 40285 Part of proof of Lemma E i...
cdleme21c 40286 Part of proof of Lemma E i...
cdleme21at 40287 Part of proof of Lemma E i...
cdleme21ct 40288 Part of proof of Lemma E i...
cdleme21d 40289 Part of proof of Lemma E i...
cdleme21e 40290 Part of proof of Lemma E i...
cdleme21f 40291 Part of proof of Lemma E i...
cdleme21g 40292 Part of proof of Lemma E i...
cdleme21h 40293 Part of proof of Lemma E i...
cdleme21i 40294 Part of proof of Lemma E i...
cdleme21j 40295 Combine ~ cdleme20 and ~ c...
cdleme21 40296 Part of proof of Lemma E i...
cdleme21k 40297 Eliminate ` S =/= T ` cond...
cdleme22aa 40298 Part of proof of Lemma E i...
cdleme22a 40299 Part of proof of Lemma E i...
cdleme22b 40300 Part of proof of Lemma E i...
cdleme22cN 40301 Part of proof of Lemma E i...
cdleme22d 40302 Part of proof of Lemma E i...
cdleme22e 40303 Part of proof of Lemma E i...
cdleme22eALTN 40304 Part of proof of Lemma E i...
cdleme22f 40305 Part of proof of Lemma E i...
cdleme22f2 40306 Part of proof of Lemma E i...
cdleme22g 40307 Part of proof of Lemma E i...
cdleme23a 40308 Part of proof of Lemma E i...
cdleme23b 40309 Part of proof of Lemma E i...
cdleme23c 40310 Part of proof of Lemma E i...
cdleme24 40311 Quantified version of ~ cd...
cdleme25a 40312 Lemma for ~ cdleme25b . (...
cdleme25b 40313 Transform ~ cdleme24 . TO...
cdleme25c 40314 Transform ~ cdleme25b . (...
cdleme25dN 40315 Transform ~ cdleme25c . (...
cdleme25cl 40316 Show closure of the unique...
cdleme25cv 40317 Change bound variables in ...
cdleme26e 40318 Part of proof of Lemma E i...
cdleme26ee 40319 Part of proof of Lemma E i...
cdleme26eALTN 40320 Part of proof of Lemma E i...
cdleme26fALTN 40321 Part of proof of Lemma E i...
cdleme26f 40322 Part of proof of Lemma E i...
cdleme26f2ALTN 40323 Part of proof of Lemma E i...
cdleme26f2 40324 Part of proof of Lemma E i...
cdleme27cl 40325 Part of proof of Lemma E i...
cdleme27a 40326 Part of proof of Lemma E i...
cdleme27b 40327 Lemma for ~ cdleme27N . (...
cdleme27N 40328 Part of proof of Lemma E i...
cdleme28a 40329 Lemma for ~ cdleme25b . T...
cdleme28b 40330 Lemma for ~ cdleme25b . T...
cdleme28c 40331 Part of proof of Lemma E i...
cdleme28 40332 Quantified version of ~ cd...
cdleme29ex 40333 Lemma for ~ cdleme29b . (...
cdleme29b 40334 Transform ~ cdleme28 . (C...
cdleme29c 40335 Transform ~ cdleme28b . (...
cdleme29cl 40336 Show closure of the unique...
cdleme30a 40337 Part of proof of Lemma E i...
cdleme31so 40338 Part of proof of Lemma E i...
cdleme31sn 40339 Part of proof of Lemma E i...
cdleme31sn1 40340 Part of proof of Lemma E i...
cdleme31se 40341 Part of proof of Lemma D i...
cdleme31se2 40342 Part of proof of Lemma D i...
cdleme31sc 40343 Part of proof of Lemma E i...
cdleme31sde 40344 Part of proof of Lemma D i...
cdleme31snd 40345 Part of proof of Lemma D i...
cdleme31sdnN 40346 Part of proof of Lemma E i...
cdleme31sn1c 40347 Part of proof of Lemma E i...
cdleme31sn2 40348 Part of proof of Lemma E i...
cdleme31fv 40349 Part of proof of Lemma E i...
cdleme31fv1 40350 Part of proof of Lemma E i...
cdleme31fv1s 40351 Part of proof of Lemma E i...
cdleme31fv2 40352 Part of proof of Lemma E i...
cdleme31id 40353 Part of proof of Lemma E i...
cdlemefrs29pre00 40354 ***START OF VALUE AT ATOM ...
cdlemefrs29bpre0 40355 TODO fix comment. (Contri...
cdlemefrs29bpre1 40356 TODO: FIX COMMENT. (Contr...
cdlemefrs29cpre1 40357 TODO: FIX COMMENT. (Contr...
cdlemefrs29clN 40358 TODO: NOT USED? Show clo...
cdlemefrs32fva 40359 Part of proof of Lemma E i...
cdlemefrs32fva1 40360 Part of proof of Lemma E i...
cdlemefr29exN 40361 Lemma for ~ cdlemefs29bpre...
cdlemefr27cl 40362 Part of proof of Lemma E i...
cdlemefr32sn2aw 40363 Show that ` [_ R / s ]_ N ...
cdlemefr32snb 40364 Show closure of ` [_ R / s...
cdlemefr29bpre0N 40365 TODO fix comment. (Contri...
cdlemefr29clN 40366 Show closure of the unique...
cdleme43frv1snN 40367 Value of ` [_ R / s ]_ N `...
cdlemefr32fvaN 40368 Part of proof of Lemma E i...
cdlemefr32fva1 40369 Part of proof of Lemma E i...
cdlemefr31fv1 40370 Value of ` ( F `` R ) ` wh...
cdlemefs29pre00N 40371 FIX COMMENT. TODO: see if ...
cdlemefs27cl 40372 Part of proof of Lemma E i...
cdlemefs32sn1aw 40373 Show that ` [_ R / s ]_ N ...
cdlemefs32snb 40374 Show closure of ` [_ R / s...
cdlemefs29bpre0N 40375 TODO: FIX COMMENT. (Contr...
cdlemefs29bpre1N 40376 TODO: FIX COMMENT. (Contr...
cdlemefs29cpre1N 40377 TODO: FIX COMMENT. (Contr...
cdlemefs29clN 40378 Show closure of the unique...
cdleme43fsv1snlem 40379 Value of ` [_ R / s ]_ N `...
cdleme43fsv1sn 40380 Value of ` [_ R / s ]_ N `...
cdlemefs32fvaN 40381 Part of proof of Lemma E i...
cdlemefs32fva1 40382 Part of proof of Lemma E i...
cdlemefs31fv1 40383 Value of ` ( F `` R ) ` wh...
cdlemefr44 40384 Value of f(r) when r is an...
cdlemefs44 40385 Value of f_s(r) when r is ...
cdlemefr45 40386 Value of f(r) when r is an...
cdlemefr45e 40387 Explicit expansion of ~ cd...
cdlemefs45 40388 Value of f_s(r) when r is ...
cdlemefs45ee 40389 Explicit expansion of ~ cd...
cdlemefs45eN 40390 Explicit expansion of ~ cd...
cdleme32sn1awN 40391 Show that ` [_ R / s ]_ N ...
cdleme41sn3a 40392 Show that ` [_ R / s ]_ N ...
cdleme32sn2awN 40393 Show that ` [_ R / s ]_ N ...
cdleme32snaw 40394 Show that ` [_ R / s ]_ N ...
cdleme32snb 40395 Show closure of ` [_ R / s...
cdleme32fva 40396 Part of proof of Lemma D i...
cdleme32fva1 40397 Part of proof of Lemma D i...
cdleme32fvaw 40398 Show that ` ( F `` R ) ` i...
cdleme32fvcl 40399 Part of proof of Lemma D i...
cdleme32a 40400 Part of proof of Lemma D i...
cdleme32b 40401 Part of proof of Lemma D i...
cdleme32c 40402 Part of proof of Lemma D i...
cdleme32d 40403 Part of proof of Lemma D i...
cdleme32e 40404 Part of proof of Lemma D i...
cdleme32f 40405 Part of proof of Lemma D i...
cdleme32le 40406 Part of proof of Lemma D i...
cdleme35a 40407 Part of proof of Lemma E i...
cdleme35fnpq 40408 Part of proof of Lemma E i...
cdleme35b 40409 Part of proof of Lemma E i...
cdleme35c 40410 Part of proof of Lemma E i...
cdleme35d 40411 Part of proof of Lemma E i...
cdleme35e 40412 Part of proof of Lemma E i...
cdleme35f 40413 Part of proof of Lemma E i...
cdleme35g 40414 Part of proof of Lemma E i...
cdleme35h 40415 Part of proof of Lemma E i...
cdleme35h2 40416 Part of proof of Lemma E i...
cdleme35sn2aw 40417 Part of proof of Lemma E i...
cdleme35sn3a 40418 Part of proof of Lemma E i...
cdleme36a 40419 Part of proof of Lemma E i...
cdleme36m 40420 Part of proof of Lemma E i...
cdleme37m 40421 Part of proof of Lemma E i...
cdleme38m 40422 Part of proof of Lemma E i...
cdleme38n 40423 Part of proof of Lemma E i...
cdleme39a 40424 Part of proof of Lemma E i...
cdleme39n 40425 Part of proof of Lemma E i...
cdleme40m 40426 Part of proof of Lemma E i...
cdleme40n 40427 Part of proof of Lemma E i...
cdleme40v 40428 Part of proof of Lemma E i...
cdleme40w 40429 Part of proof of Lemma E i...
cdleme42a 40430 Part of proof of Lemma E i...
cdleme42c 40431 Part of proof of Lemma E i...
cdleme42d 40432 Part of proof of Lemma E i...
cdleme41sn3aw 40433 Part of proof of Lemma E i...
cdleme41sn4aw 40434 Part of proof of Lemma E i...
cdleme41snaw 40435 Part of proof of Lemma E i...
cdleme41fva11 40436 Part of proof of Lemma E i...
cdleme42b 40437 Part of proof of Lemma E i...
cdleme42e 40438 Part of proof of Lemma E i...
cdleme42f 40439 Part of proof of Lemma E i...
cdleme42g 40440 Part of proof of Lemma E i...
cdleme42h 40441 Part of proof of Lemma E i...
cdleme42i 40442 Part of proof of Lemma E i...
cdleme42k 40443 Part of proof of Lemma E i...
cdleme42ke 40444 Part of proof of Lemma E i...
cdleme42keg 40445 Part of proof of Lemma E i...
cdleme42mN 40446 Part of proof of Lemma E i...
cdleme42mgN 40447 Part of proof of Lemma E i...
cdleme43aN 40448 Part of proof of Lemma E i...
cdleme43bN 40449 Lemma for Lemma E in [Craw...
cdleme43cN 40450 Part of proof of Lemma E i...
cdleme43dN 40451 Part of proof of Lemma E i...
cdleme46f2g2 40452 Conversion for ` G ` to re...
cdleme46f2g1 40453 Conversion for ` G ` to re...
cdleme17d2 40454 Part of proof of Lemma E i...
cdleme17d3 40455 TODO: FIX COMMENT. (Contr...
cdleme17d4 40456 TODO: FIX COMMENT. (Contr...
cdleme17d 40457 Part of proof of Lemma E i...
cdleme48fv 40458 Part of proof of Lemma D i...
cdleme48fvg 40459 Remove ` P =/= Q ` conditi...
cdleme46fvaw 40460 Show that ` ( F `` R ) ` i...
cdleme48bw 40461 TODO: fix comment. TODO: ...
cdleme48b 40462 TODO: fix comment. (Contr...
cdleme46frvlpq 40463 Show that ` ( F `` S ) ` i...
cdleme46fsvlpq 40464 Show that ` ( F `` R ) ` i...
cdlemeg46fvcl 40465 TODO: fix comment. (Contr...
cdleme4gfv 40466 Part of proof of Lemma D i...
cdlemeg47b 40467 TODO: FIX COMMENT. (Contr...
cdlemeg47rv 40468 Value of g_s(r) when r is ...
cdlemeg47rv2 40469 Value of g_s(r) when r is ...
cdlemeg49le 40470 Part of proof of Lemma D i...
cdlemeg46bOLDN 40471 TODO FIX COMMENT. (Contrib...
cdlemeg46c 40472 TODO FIX COMMENT. (Contrib...
cdlemeg46rvOLDN 40473 Value of g_s(r) when r is ...
cdlemeg46rv2OLDN 40474 Value of g_s(r) when r is ...
cdlemeg46fvaw 40475 Show that ` ( F `` R ) ` i...
cdlemeg46nlpq 40476 Show that ` ( G `` S ) ` i...
cdlemeg46ngfr 40477 TODO FIX COMMENT g(f(s))=s...
cdlemeg46nfgr 40478 TODO FIX COMMENT f(g(s))=s...
cdlemeg46sfg 40479 TODO FIX COMMENT f(r) ` \/...
cdlemeg46fjgN 40480 NOT NEEDED? TODO FIX COMM...
cdlemeg46rjgN 40481 NOT NEEDED? TODO FIX COMM...
cdlemeg46fjv 40482 TODO FIX COMMENT f(r) ` \/...
cdlemeg46fsfv 40483 TODO FIX COMMENT f(r) ` \/...
cdlemeg46frv 40484 TODO FIX COMMENT. (f(r) ` ...
cdlemeg46v1v2 40485 TODO FIX COMMENT v_1 = v_2...
cdlemeg46vrg 40486 TODO FIX COMMENT v_1 ` <_ ...
cdlemeg46rgv 40487 TODO FIX COMMENT r ` <_ ` ...
cdlemeg46req 40488 TODO FIX COMMENT r = (v_1 ...
cdlemeg46gfv 40489 TODO FIX COMMENT p. 115 pe...
cdlemeg46gfr 40490 TODO FIX COMMENT p. 116 pe...
cdlemeg46gfre 40491 TODO FIX COMMENT p. 116 pe...
cdlemeg46gf 40492 TODO FIX COMMENT Eliminate...
cdlemeg46fgN 40493 TODO FIX COMMENT p. 116 pe...
cdleme48d 40494 TODO: fix comment. (Contr...
cdleme48gfv1 40495 TODO: fix comment. (Contr...
cdleme48gfv 40496 TODO: fix comment. (Contr...
cdleme48fgv 40497 TODO: fix comment. (Contr...
cdlemeg49lebilem 40498 Part of proof of Lemma D i...
cdleme50lebi 40499 Part of proof of Lemma D i...
cdleme50eq 40500 Part of proof of Lemma D i...
cdleme50f 40501 Part of proof of Lemma D i...
cdleme50f1 40502 Part of proof of Lemma D i...
cdleme50rnlem 40503 Part of proof of Lemma D i...
cdleme50rn 40504 Part of proof of Lemma D i...
cdleme50f1o 40505 Part of proof of Lemma D i...
cdleme50laut 40506 Part of proof of Lemma D i...
cdleme50ldil 40507 Part of proof of Lemma D i...
cdleme50trn1 40508 Part of proof that ` F ` i...
cdleme50trn2a 40509 Part of proof that ` F ` i...
cdleme50trn2 40510 Part of proof that ` F ` i...
cdleme50trn12 40511 Part of proof that ` F ` i...
cdleme50trn3 40512 Part of proof that ` F ` i...
cdleme50trn123 40513 Part of proof that ` F ` i...
cdleme51finvfvN 40514 Part of proof of Lemma E i...
cdleme51finvN 40515 Part of proof of Lemma E i...
cdleme50ltrn 40516 Part of proof of Lemma E i...
cdleme51finvtrN 40517 Part of proof of Lemma E i...
cdleme50ex 40518 Part of Lemma E in [Crawle...
cdleme 40519 Lemma E in [Crawley] p. 11...
cdlemf1 40520 Part of Lemma F in [Crawle...
cdlemf2 40521 Part of Lemma F in [Crawle...
cdlemf 40522 Lemma F in [Crawley] p. 11...
cdlemfnid 40523 ~ cdlemf with additional c...
cdlemftr3 40524 Special case of ~ cdlemf s...
cdlemftr2 40525 Special case of ~ cdlemf s...
cdlemftr1 40526 Part of proof of Lemma G o...
cdlemftr0 40527 Special case of ~ cdlemf s...
trlord 40528 The ordering of two Hilber...
cdlemg1a 40529 Shorter expression for ` G...
cdlemg1b2 40530 This theorem can be used t...
cdlemg1idlemN 40531 Lemma for ~ cdlemg1idN . ...
cdlemg1fvawlemN 40532 Lemma for ~ ltrniotafvawN ...
cdlemg1ltrnlem 40533 Lemma for ~ ltrniotacl . ...
cdlemg1finvtrlemN 40534 Lemma for ~ ltrniotacnvN ....
cdlemg1bOLDN 40535 This theorem can be used t...
cdlemg1idN 40536 Version of ~ cdleme31id wi...
ltrniotafvawN 40537 Version of ~ cdleme46fvaw ...
ltrniotacl 40538 Version of ~ cdleme50ltrn ...
ltrniotacnvN 40539 Version of ~ cdleme51finvt...
ltrniotaval 40540 Value of the unique transl...
ltrniotacnvval 40541 Converse value of the uniq...
ltrniotaidvalN 40542 Value of the unique transl...
ltrniotavalbN 40543 Value of the unique transl...
cdlemeiota 40544 A translation is uniquely ...
cdlemg1ci2 40545 Any function of the form o...
cdlemg1cN 40546 Any translation belongs to...
cdlemg1cex 40547 Any translation is one of ...
cdlemg2cN 40548 Any translation belongs to...
cdlemg2dN 40549 This theorem can be used t...
cdlemg2cex 40550 Any translation is one of ...
cdlemg2ce 40551 Utility theorem to elimina...
cdlemg2jlemOLDN 40552 Part of proof of Lemma E i...
cdlemg2fvlem 40553 Lemma for ~ cdlemg2fv . (...
cdlemg2klem 40554 ~ cdleme42keg with simpler...
cdlemg2idN 40555 Version of ~ cdleme31id wi...
cdlemg3a 40556 Part of proof of Lemma G i...
cdlemg2jOLDN 40557 TODO: Replace this with ~...
cdlemg2fv 40558 Value of a translation in ...
cdlemg2fv2 40559 Value of a translation in ...
cdlemg2k 40560 ~ cdleme42keg with simpler...
cdlemg2kq 40561 ~ cdlemg2k with ` P ` and ...
cdlemg2l 40562 TODO: FIX COMMENT. (Contr...
cdlemg2m 40563 TODO: FIX COMMENT. (Contr...
cdlemg5 40564 TODO: Is there a simpler ...
cdlemb3 40565 Given two atoms not under ...
cdlemg7fvbwN 40566 Properties of a translatio...
cdlemg4a 40567 TODO: FIX COMMENT If fg(p...
cdlemg4b1 40568 TODO: FIX COMMENT. (Contr...
cdlemg4b2 40569 TODO: FIX COMMENT. (Contr...
cdlemg4b12 40570 TODO: FIX COMMENT. (Contr...
cdlemg4c 40571 TODO: FIX COMMENT. (Contr...
cdlemg4d 40572 TODO: FIX COMMENT. (Contr...
cdlemg4e 40573 TODO: FIX COMMENT. (Contr...
cdlemg4f 40574 TODO: FIX COMMENT. (Contr...
cdlemg4g 40575 TODO: FIX COMMENT. (Contr...
cdlemg4 40576 TODO: FIX COMMENT. (Contr...
cdlemg6a 40577 TODO: FIX COMMENT. TODO: ...
cdlemg6b 40578 TODO: FIX COMMENT. TODO: ...
cdlemg6c 40579 TODO: FIX COMMENT. (Contr...
cdlemg6d 40580 TODO: FIX COMMENT. (Contr...
cdlemg6e 40581 TODO: FIX COMMENT. (Contr...
cdlemg6 40582 TODO: FIX COMMENT. (Contr...
cdlemg7fvN 40583 Value of a translation com...
cdlemg7aN 40584 TODO: FIX COMMENT. (Contr...
cdlemg7N 40585 TODO: FIX COMMENT. (Contr...
cdlemg8a 40586 TODO: FIX COMMENT. (Contr...
cdlemg8b 40587 TODO: FIX COMMENT. (Contr...
cdlemg8c 40588 TODO: FIX COMMENT. (Contr...
cdlemg8d 40589 TODO: FIX COMMENT. (Contr...
cdlemg8 40590 TODO: FIX COMMENT. (Contr...
cdlemg9a 40591 TODO: FIX COMMENT. (Contr...
cdlemg9b 40592 The triples ` <. P , ( F `...
cdlemg9 40593 The triples ` <. P , ( F `...
cdlemg10b 40594 TODO: FIX COMMENT. TODO: ...
cdlemg10bALTN 40595 TODO: FIX COMMENT. TODO: ...
cdlemg11a 40596 TODO: FIX COMMENT. (Contr...
cdlemg11aq 40597 TODO: FIX COMMENT. TODO: ...
cdlemg10c 40598 TODO: FIX COMMENT. TODO: ...
cdlemg10a 40599 TODO: FIX COMMENT. (Contr...
cdlemg10 40600 TODO: FIX COMMENT. (Contr...
cdlemg11b 40601 TODO: FIX COMMENT. (Contr...
cdlemg12a 40602 TODO: FIX COMMENT. (Contr...
cdlemg12b 40603 The triples ` <. P , ( F `...
cdlemg12c 40604 The triples ` <. P , ( F `...
cdlemg12d 40605 TODO: FIX COMMENT. (Contr...
cdlemg12e 40606 TODO: FIX COMMENT. (Contr...
cdlemg12f 40607 TODO: FIX COMMENT. (Contr...
cdlemg12g 40608 TODO: FIX COMMENT. TODO: ...
cdlemg12 40609 TODO: FIX COMMENT. (Contr...
cdlemg13a 40610 TODO: FIX COMMENT. (Contr...
cdlemg13 40611 TODO: FIX COMMENT. (Contr...
cdlemg14f 40612 TODO: FIX COMMENT. (Contr...
cdlemg14g 40613 TODO: FIX COMMENT. (Contr...
cdlemg15a 40614 Eliminate the ` ( F `` P )...
cdlemg15 40615 Eliminate the ` ( (...
cdlemg16 40616 Part of proof of Lemma G o...
cdlemg16ALTN 40617 This version of ~ cdlemg16...
cdlemg16z 40618 Eliminate ` ( ( F `...
cdlemg16zz 40619 Eliminate ` P =/= Q ` from...
cdlemg17a 40620 TODO: FIX COMMENT. (Contr...
cdlemg17b 40621 Part of proof of Lemma G i...
cdlemg17dN 40622 TODO: fix comment. (Contr...
cdlemg17dALTN 40623 Same as ~ cdlemg17dN with ...
cdlemg17e 40624 TODO: fix comment. (Contr...
cdlemg17f 40625 TODO: fix comment. (Contr...
cdlemg17g 40626 TODO: fix comment. (Contr...
cdlemg17h 40627 TODO: fix comment. (Contr...
cdlemg17i 40628 TODO: fix comment. (Contr...
cdlemg17ir 40629 TODO: fix comment. (Contr...
cdlemg17j 40630 TODO: fix comment. (Contr...
cdlemg17pq 40631 Utility theorem for swappi...
cdlemg17bq 40632 ~ cdlemg17b with ` P ` and...
cdlemg17iqN 40633 ~ cdlemg17i with ` P ` and...
cdlemg17irq 40634 ~ cdlemg17ir with ` P ` an...
cdlemg17jq 40635 ~ cdlemg17j with ` P ` and...
cdlemg17 40636 Part of Lemma G of [Crawle...
cdlemg18a 40637 Show two lines are differe...
cdlemg18b 40638 Lemma for ~ cdlemg18c . T...
cdlemg18c 40639 Show two lines intersect a...
cdlemg18d 40640 Show two lines intersect a...
cdlemg18 40641 Show two lines intersect a...
cdlemg19a 40642 Show two lines intersect a...
cdlemg19 40643 Show two lines intersect a...
cdlemg20 40644 Show two lines intersect a...
cdlemg21 40645 Version of cdlemg19 with `...
cdlemg22 40646 ~ cdlemg21 with ` ( F `` P...
cdlemg24 40647 Combine ~ cdlemg16z and ~ ...
cdlemg37 40648 Use ~ cdlemg8 to eliminate...
cdlemg25zz 40649 ~ cdlemg16zz restated for ...
cdlemg26zz 40650 ~ cdlemg16zz restated for ...
cdlemg27a 40651 For use with case when ` (...
cdlemg28a 40652 Part of proof of Lemma G o...
cdlemg31b0N 40653 TODO: Fix comment. (Cont...
cdlemg31b0a 40654 TODO: Fix comment. (Cont...
cdlemg27b 40655 TODO: Fix comment. (Cont...
cdlemg31a 40656 TODO: fix comment. (Contr...
cdlemg31b 40657 TODO: fix comment. (Contr...
cdlemg31c 40658 Show that when ` N ` is an...
cdlemg31d 40659 Eliminate ` ( F `` P ) =/=...
cdlemg33b0 40660 TODO: Fix comment. (Cont...
cdlemg33c0 40661 TODO: Fix comment. (Cont...
cdlemg28b 40662 Part of proof of Lemma G o...
cdlemg28 40663 Part of proof of Lemma G o...
cdlemg29 40664 Eliminate ` ( F `` P ) =/=...
cdlemg33a 40665 TODO: Fix comment. (Cont...
cdlemg33b 40666 TODO: Fix comment. (Cont...
cdlemg33c 40667 TODO: Fix comment. (Cont...
cdlemg33d 40668 TODO: Fix comment. (Cont...
cdlemg33e 40669 TODO: Fix comment. (Cont...
cdlemg33 40670 Combine ~ cdlemg33b , ~ cd...
cdlemg34 40671 Use cdlemg33 to eliminate ...
cdlemg35 40672 TODO: Fix comment. TODO:...
cdlemg36 40673 Use cdlemg35 to eliminate ...
cdlemg38 40674 Use ~ cdlemg37 to eliminat...
cdlemg39 40675 Eliminate ` =/= ` conditio...
cdlemg40 40676 Eliminate ` P =/= Q ` cond...
cdlemg41 40677 Convert ~ cdlemg40 to func...
ltrnco 40678 The composition of two tra...
trlcocnv 40679 Swap the arguments of the ...
trlcoabs 40680 Absorption into a composit...
trlcoabs2N 40681 Absorption of the trace of...
trlcoat 40682 The trace of a composition...
trlcocnvat 40683 Commonly used special case...
trlconid 40684 The composition of two dif...
trlcolem 40685 Lemma for ~ trlco . (Cont...
trlco 40686 The trace of a composition...
trlcone 40687 If two translations have d...
cdlemg42 40688 Part of proof of Lemma G o...
cdlemg43 40689 Part of proof of Lemma G o...
cdlemg44a 40690 Part of proof of Lemma G o...
cdlemg44b 40691 Eliminate ` ( F `` P ) =/=...
cdlemg44 40692 Part of proof of Lemma G o...
cdlemg47a 40693 TODO: fix comment. TODO: ...
cdlemg46 40694 Part of proof of Lemma G o...
cdlemg47 40695 Part of proof of Lemma G o...
cdlemg48 40696 Eliminate ` h ` from ~ cdl...
ltrncom 40697 Composition is commutative...
ltrnco4 40698 Rearrange a composition of...
trljco 40699 Trace joined with trace of...
trljco2 40700 Trace joined with trace of...
tgrpfset 40703 The translation group maps...
tgrpset 40704 The translation group for ...
tgrpbase 40705 The base set of the transl...
tgrpopr 40706 The group operation of the...
tgrpov 40707 The group operation value ...
tgrpgrplem 40708 Lemma for ~ tgrpgrp . (Co...
tgrpgrp 40709 The translation group is a...
tgrpabl 40710 The translation group is a...
tendofset 40717 The set of all trace-prese...
tendoset 40718 The set of trace-preservin...
istendo 40719 The predicate "is a trace-...
tendotp 40720 Trace-preserving property ...
istendod 40721 Deduce the predicate "is a...
tendof 40722 Functionality of a trace-p...
tendoeq1 40723 Condition determining equa...
tendovalco 40724 Value of composition of tr...
tendocoval 40725 Value of composition of en...
tendocl 40726 Closure of a trace-preserv...
tendoco2 40727 Distribution of compositio...
tendoidcl 40728 The identity is a trace-pr...
tendo1mul 40729 Multiplicative identity mu...
tendo1mulr 40730 Multiplicative identity mu...
tendococl 40731 The composition of two tra...
tendoid 40732 The identity value of a tr...
tendoeq2 40733 Condition determining equa...
tendoplcbv 40734 Define sum operation for t...
tendopl 40735 Value of endomorphism sum ...
tendopl2 40736 Value of result of endomor...
tendoplcl2 40737 Value of result of endomor...
tendoplco2 40738 Value of result of endomor...
tendopltp 40739 Trace-preserving property ...
tendoplcl 40740 Endomorphism sum is a trac...
tendoplcom 40741 The endomorphism sum opera...
tendoplass 40742 The endomorphism sum opera...
tendodi1 40743 Endomorphism composition d...
tendodi2 40744 Endomorphism composition d...
tendo0cbv 40745 Define additive identity f...
tendo02 40746 Value of additive identity...
tendo0co2 40747 The additive identity trac...
tendo0tp 40748 Trace-preserving property ...
tendo0cl 40749 The additive identity is a...
tendo0pl 40750 Property of the additive i...
tendo0plr 40751 Property of the additive i...
tendoicbv 40752 Define inverse function fo...
tendoi 40753 Value of inverse endomorph...
tendoi2 40754 Value of additive inverse ...
tendoicl 40755 Closure of the additive in...
tendoipl 40756 Property of the additive i...
tendoipl2 40757 Property of the additive i...
erngfset 40758 The division rings on trac...
erngset 40759 The division ring on trace...
erngbase 40760 The base set of the divisi...
erngfplus 40761 Ring addition operation. ...
erngplus 40762 Ring addition operation. ...
erngplus2 40763 Ring addition operation. ...
erngfmul 40764 Ring multiplication operat...
erngmul 40765 Ring addition operation. ...
erngfset-rN 40766 The division rings on trac...
erngset-rN 40767 The division ring on trace...
erngbase-rN 40768 The base set of the divisi...
erngfplus-rN 40769 Ring addition operation. ...
erngplus-rN 40770 Ring addition operation. ...
erngplus2-rN 40771 Ring addition operation. ...
erngfmul-rN 40772 Ring multiplication operat...
erngmul-rN 40773 Ring addition operation. ...
cdlemh1 40774 Part of proof of Lemma H o...
cdlemh2 40775 Part of proof of Lemma H o...
cdlemh 40776 Lemma H of [Crawley] p. 11...
cdlemi1 40777 Part of proof of Lemma I o...
cdlemi2 40778 Part of proof of Lemma I o...
cdlemi 40779 Lemma I of [Crawley] p. 11...
cdlemj1 40780 Part of proof of Lemma J o...
cdlemj2 40781 Part of proof of Lemma J o...
cdlemj3 40782 Part of proof of Lemma J o...
tendocan 40783 Cancellation law: if the v...
tendoid0 40784 A trace-preserving endomor...
tendo0mul 40785 Additive identity multipli...
tendo0mulr 40786 Additive identity multipli...
tendo1ne0 40787 The identity (unity) is no...
tendoconid 40788 The composition (product) ...
tendotr 40789 The trace of the value of ...
cdlemk1 40790 Part of proof of Lemma K o...
cdlemk2 40791 Part of proof of Lemma K o...
cdlemk3 40792 Part of proof of Lemma K o...
cdlemk4 40793 Part of proof of Lemma K o...
cdlemk5a 40794 Part of proof of Lemma K o...
cdlemk5 40795 Part of proof of Lemma K o...
cdlemk6 40796 Part of proof of Lemma K o...
cdlemk8 40797 Part of proof of Lemma K o...
cdlemk9 40798 Part of proof of Lemma K o...
cdlemk9bN 40799 Part of proof of Lemma K o...
cdlemki 40800 Part of proof of Lemma K o...
cdlemkvcl 40801 Part of proof of Lemma K o...
cdlemk10 40802 Part of proof of Lemma K o...
cdlemksv 40803 Part of proof of Lemma K o...
cdlemksel 40804 Part of proof of Lemma K o...
cdlemksat 40805 Part of proof of Lemma K o...
cdlemksv2 40806 Part of proof of Lemma K o...
cdlemk7 40807 Part of proof of Lemma K o...
cdlemk11 40808 Part of proof of Lemma K o...
cdlemk12 40809 Part of proof of Lemma K o...
cdlemkoatnle 40810 Utility lemma. (Contribut...
cdlemk13 40811 Part of proof of Lemma K o...
cdlemkole 40812 Utility lemma. (Contribut...
cdlemk14 40813 Part of proof of Lemma K o...
cdlemk15 40814 Part of proof of Lemma K o...
cdlemk16a 40815 Part of proof of Lemma K o...
cdlemk16 40816 Part of proof of Lemma K o...
cdlemk17 40817 Part of proof of Lemma K o...
cdlemk1u 40818 Part of proof of Lemma K o...
cdlemk5auN 40819 Part of proof of Lemma K o...
cdlemk5u 40820 Part of proof of Lemma K o...
cdlemk6u 40821 Part of proof of Lemma K o...
cdlemkj 40822 Part of proof of Lemma K o...
cdlemkuvN 40823 Part of proof of Lemma K o...
cdlemkuel 40824 Part of proof of Lemma K o...
cdlemkuat 40825 Part of proof of Lemma K o...
cdlemkuv2 40826 Part of proof of Lemma K o...
cdlemk18 40827 Part of proof of Lemma K o...
cdlemk19 40828 Part of proof of Lemma K o...
cdlemk7u 40829 Part of proof of Lemma K o...
cdlemk11u 40830 Part of proof of Lemma K o...
cdlemk12u 40831 Part of proof of Lemma K o...
cdlemk21N 40832 Part of proof of Lemma K o...
cdlemk20 40833 Part of proof of Lemma K o...
cdlemkoatnle-2N 40834 Utility lemma. (Contribut...
cdlemk13-2N 40835 Part of proof of Lemma K o...
cdlemkole-2N 40836 Utility lemma. (Contribut...
cdlemk14-2N 40837 Part of proof of Lemma K o...
cdlemk15-2N 40838 Part of proof of Lemma K o...
cdlemk16-2N 40839 Part of proof of Lemma K o...
cdlemk17-2N 40840 Part of proof of Lemma K o...
cdlemkj-2N 40841 Part of proof of Lemma K o...
cdlemkuv-2N 40842 Part of proof of Lemma K o...
cdlemkuel-2N 40843 Part of proof of Lemma K o...
cdlemkuv2-2 40844 Part of proof of Lemma K o...
cdlemk18-2N 40845 Part of proof of Lemma K o...
cdlemk19-2N 40846 Part of proof of Lemma K o...
cdlemk7u-2N 40847 Part of proof of Lemma K o...
cdlemk11u-2N 40848 Part of proof of Lemma K o...
cdlemk12u-2N 40849 Part of proof of Lemma K o...
cdlemk21-2N 40850 Part of proof of Lemma K o...
cdlemk20-2N 40851 Part of proof of Lemma K o...
cdlemk22 40852 Part of proof of Lemma K o...
cdlemk30 40853 Part of proof of Lemma K o...
cdlemkuu 40854 Convert between function a...
cdlemk31 40855 Part of proof of Lemma K o...
cdlemk32 40856 Part of proof of Lemma K o...
cdlemkuel-3 40857 Part of proof of Lemma K o...
cdlemkuv2-3N 40858 Part of proof of Lemma K o...
cdlemk18-3N 40859 Part of proof of Lemma K o...
cdlemk22-3 40860 Part of proof of Lemma K o...
cdlemk23-3 40861 Part of proof of Lemma K o...
cdlemk24-3 40862 Part of proof of Lemma K o...
cdlemk25-3 40863 Part of proof of Lemma K o...
cdlemk26b-3 40864 Part of proof of Lemma K o...
cdlemk26-3 40865 Part of proof of Lemma K o...
cdlemk27-3 40866 Part of proof of Lemma K o...
cdlemk28-3 40867 Part of proof of Lemma K o...
cdlemk33N 40868 Part of proof of Lemma K o...
cdlemk34 40869 Part of proof of Lemma K o...
cdlemk29-3 40870 Part of proof of Lemma K o...
cdlemk35 40871 Part of proof of Lemma K o...
cdlemk36 40872 Part of proof of Lemma K o...
cdlemk37 40873 Part of proof of Lemma K o...
cdlemk38 40874 Part of proof of Lemma K o...
cdlemk39 40875 Part of proof of Lemma K o...
cdlemk40 40876 TODO: fix comment. (Contr...
cdlemk40t 40877 TODO: fix comment. (Contr...
cdlemk40f 40878 TODO: fix comment. (Contr...
cdlemk41 40879 Part of proof of Lemma K o...
cdlemkfid1N 40880 Lemma for ~ cdlemkfid3N . ...
cdlemkid1 40881 Lemma for ~ cdlemkid . (C...
cdlemkfid2N 40882 Lemma for ~ cdlemkfid3N . ...
cdlemkid2 40883 Lemma for ~ cdlemkid . (C...
cdlemkfid3N 40884 TODO: is this useful or sh...
cdlemky 40885 Part of proof of Lemma K o...
cdlemkyu 40886 Convert between function a...
cdlemkyuu 40887 ~ cdlemkyu with some hypot...
cdlemk11ta 40888 Part of proof of Lemma K o...
cdlemk19ylem 40889 Lemma for ~ cdlemk19y . (...
cdlemk11tb 40890 Part of proof of Lemma K o...
cdlemk19y 40891 ~ cdlemk19 with simpler hy...
cdlemkid3N 40892 Lemma for ~ cdlemkid . (C...
cdlemkid4 40893 Lemma for ~ cdlemkid . (C...
cdlemkid5 40894 Lemma for ~ cdlemkid . (C...
cdlemkid 40895 The value of the tau funct...
cdlemk35s 40896 Substitution version of ~ ...
cdlemk35s-id 40897 Substitution version of ~ ...
cdlemk39s 40898 Substitution version of ~ ...
cdlemk39s-id 40899 Substitution version of ~ ...
cdlemk42 40900 Part of proof of Lemma K o...
cdlemk19xlem 40901 Lemma for ~ cdlemk19x . (...
cdlemk19x 40902 ~ cdlemk19 with simpler hy...
cdlemk42yN 40903 Part of proof of Lemma K o...
cdlemk11tc 40904 Part of proof of Lemma K o...
cdlemk11t 40905 Part of proof of Lemma K o...
cdlemk45 40906 Part of proof of Lemma K o...
cdlemk46 40907 Part of proof of Lemma K o...
cdlemk47 40908 Part of proof of Lemma K o...
cdlemk48 40909 Part of proof of Lemma K o...
cdlemk49 40910 Part of proof of Lemma K o...
cdlemk50 40911 Part of proof of Lemma K o...
cdlemk51 40912 Part of proof of Lemma K o...
cdlemk52 40913 Part of proof of Lemma K o...
cdlemk53a 40914 Lemma for ~ cdlemk53 . (C...
cdlemk53b 40915 Lemma for ~ cdlemk53 . (C...
cdlemk53 40916 Part of proof of Lemma K o...
cdlemk54 40917 Part of proof of Lemma K o...
cdlemk55a 40918 Lemma for ~ cdlemk55 . (C...
cdlemk55b 40919 Lemma for ~ cdlemk55 . (C...
cdlemk55 40920 Part of proof of Lemma K o...
cdlemkyyN 40921 Part of proof of Lemma K o...
cdlemk43N 40922 Part of proof of Lemma K o...
cdlemk35u 40923 Substitution version of ~ ...
cdlemk55u1 40924 Lemma for ~ cdlemk55u . (...
cdlemk55u 40925 Part of proof of Lemma K o...
cdlemk39u1 40926 Lemma for ~ cdlemk39u . (...
cdlemk39u 40927 Part of proof of Lemma K o...
cdlemk19u1 40928 ~ cdlemk19 with simpler hy...
cdlemk19u 40929 Part of Lemma K of [Crawle...
cdlemk56 40930 Part of Lemma K of [Crawle...
cdlemk19w 40931 Use a fixed element to eli...
cdlemk56w 40932 Use a fixed element to eli...
cdlemk 40933 Lemma K of [Crawley] p. 11...
tendoex 40934 Generalization of Lemma K ...
cdleml1N 40935 Part of proof of Lemma L o...
cdleml2N 40936 Part of proof of Lemma L o...
cdleml3N 40937 Part of proof of Lemma L o...
cdleml4N 40938 Part of proof of Lemma L o...
cdleml5N 40939 Part of proof of Lemma L o...
cdleml6 40940 Part of proof of Lemma L o...
cdleml7 40941 Part of proof of Lemma L o...
cdleml8 40942 Part of proof of Lemma L o...
cdleml9 40943 Part of proof of Lemma L o...
dva1dim 40944 Two expressions for the 1-...
dvhb1dimN 40945 Two expressions for the 1-...
erng1lem 40946 Value of the endomorphism ...
erngdvlem1 40947 Lemma for ~ eringring . (...
erngdvlem2N 40948 Lemma for ~ eringring . (...
erngdvlem3 40949 Lemma for ~ eringring . (...
erngdvlem4 40950 Lemma for ~ erngdv . (Con...
eringring 40951 An endomorphism ring is a ...
erngdv 40952 An endomorphism ring is a ...
erng0g 40953 The division ring zero of ...
erng1r 40954 The division ring unity of...
erngdvlem1-rN 40955 Lemma for ~ eringring . (...
erngdvlem2-rN 40956 Lemma for ~ eringring . (...
erngdvlem3-rN 40957 Lemma for ~ eringring . (...
erngdvlem4-rN 40958 Lemma for ~ erngdv . (Con...
erngring-rN 40959 An endomorphism ring is a ...
erngdv-rN 40960 An endomorphism ring is a ...
dvafset 40963 The constructed partial ve...
dvaset 40964 The constructed partial ve...
dvasca 40965 The ring base set of the c...
dvabase 40966 The ring base set of the c...
dvafplusg 40967 Ring addition operation fo...
dvaplusg 40968 Ring addition operation fo...
dvaplusgv 40969 Ring addition operation fo...
dvafmulr 40970 Ring multiplication operat...
dvamulr 40971 Ring multiplication operat...
dvavbase 40972 The vectors (vector base s...
dvafvadd 40973 The vector sum operation f...
dvavadd 40974 Ring addition operation fo...
dvafvsca 40975 Ring addition operation fo...
dvavsca 40976 Ring addition operation fo...
tendospcl 40977 Closure of endomorphism sc...
tendospass 40978 Associative law for endomo...
tendospdi1 40979 Forward distributive law f...
tendocnv 40980 Converse of a trace-preser...
tendospdi2 40981 Reverse distributive law f...
tendospcanN 40982 Cancellation law for trace...
dvaabl 40983 The constructed partial ve...
dvalveclem 40984 Lemma for ~ dvalvec . (Co...
dvalvec 40985 The constructed partial ve...
dva0g 40986 The zero vector of partial...
diaffval 40989 The partial isomorphism A ...
diafval 40990 The partial isomorphism A ...
diaval 40991 The partial isomorphism A ...
diaelval 40992 Member of the partial isom...
diafn 40993 Functionality and domain o...
diadm 40994 Domain of the partial isom...
diaeldm 40995 Member of domain of the pa...
diadmclN 40996 A member of domain of the ...
diadmleN 40997 A member of domain of the ...
dian0 40998 The value of the partial i...
dia0eldmN 40999 The lattice zero belongs t...
dia1eldmN 41000 The fiducial hyperplane (t...
diass 41001 The value of the partial i...
diael 41002 A member of the value of t...
diatrl 41003 Trace of a member of the p...
diaelrnN 41004 Any value of the partial i...
dialss 41005 The value of partial isomo...
diaord 41006 The partial isomorphism A ...
dia11N 41007 The partial isomorphism A ...
diaf11N 41008 The partial isomorphism A ...
diaclN 41009 Closure of partial isomorp...
diacnvclN 41010 Closure of partial isomorp...
dia0 41011 The value of the partial i...
dia1N 41012 The value of the partial i...
dia1elN 41013 The largest subspace in th...
diaglbN 41014 Partial isomorphism A of a...
diameetN 41015 Partial isomorphism A of a...
diainN 41016 Inverse partial isomorphis...
diaintclN 41017 The intersection of partia...
diasslssN 41018 The partial isomorphism A ...
diassdvaN 41019 The partial isomorphism A ...
dia1dim 41020 Two expressions for the 1-...
dia1dim2 41021 Two expressions for a 1-di...
dia1dimid 41022 A vector (translation) bel...
dia2dimlem1 41023 Lemma for ~ dia2dim . Sho...
dia2dimlem2 41024 Lemma for ~ dia2dim . Def...
dia2dimlem3 41025 Lemma for ~ dia2dim . Def...
dia2dimlem4 41026 Lemma for ~ dia2dim . Sho...
dia2dimlem5 41027 Lemma for ~ dia2dim . The...
dia2dimlem6 41028 Lemma for ~ dia2dim . Eli...
dia2dimlem7 41029 Lemma for ~ dia2dim . Eli...
dia2dimlem8 41030 Lemma for ~ dia2dim . Eli...
dia2dimlem9 41031 Lemma for ~ dia2dim . Eli...
dia2dimlem10 41032 Lemma for ~ dia2dim . Con...
dia2dimlem11 41033 Lemma for ~ dia2dim . Con...
dia2dimlem12 41034 Lemma for ~ dia2dim . Obt...
dia2dimlem13 41035 Lemma for ~ dia2dim . Eli...
dia2dim 41036 A two-dimensional subspace...
dvhfset 41039 The constructed full vecto...
dvhset 41040 The constructed full vecto...
dvhsca 41041 The ring of scalars of the...
dvhbase 41042 The ring base set of the c...
dvhfplusr 41043 Ring addition operation fo...
dvhfmulr 41044 Ring multiplication operat...
dvhmulr 41045 Ring multiplication operat...
dvhvbase 41046 The vectors (vector base s...
dvhelvbasei 41047 Vector membership in the c...
dvhvaddcbv 41048 Change bound variables to ...
dvhvaddval 41049 The vector sum operation f...
dvhfvadd 41050 The vector sum operation f...
dvhvadd 41051 The vector sum operation f...
dvhopvadd 41052 The vector sum operation f...
dvhopvadd2 41053 The vector sum operation f...
dvhvaddcl 41054 Closure of the vector sum ...
dvhvaddcomN 41055 Commutativity of vector su...
dvhvaddass 41056 Associativity of vector su...
dvhvscacbv 41057 Change bound variables to ...
dvhvscaval 41058 The scalar product operati...
dvhfvsca 41059 Scalar product operation f...
dvhvsca 41060 Scalar product operation f...
dvhopvsca 41061 Scalar product operation f...
dvhvscacl 41062 Closure of the scalar prod...
tendoinvcl 41063 Closure of multiplicative ...
tendolinv 41064 Left multiplicative invers...
tendorinv 41065 Right multiplicative inver...
dvhgrp 41066 The full vector space ` U ...
dvhlveclem 41067 Lemma for ~ dvhlvec . TOD...
dvhlvec 41068 The full vector space ` U ...
dvhlmod 41069 The full vector space ` U ...
dvh0g 41070 The zero vector of vector ...
dvheveccl 41071 Properties of a unit vecto...
dvhopclN 41072 Closure of a ` DVecH ` vec...
dvhopaddN 41073 Sum of ` DVecH ` vectors e...
dvhopspN 41074 Scalar product of ` DVecH ...
dvhopN 41075 Decompose a ` DVecH ` vect...
dvhopellsm 41076 Ordered pair membership in...
cdlemm10N 41077 The image of the map ` G `...
docaffvalN 41080 Subspace orthocomplement f...
docafvalN 41081 Subspace orthocomplement f...
docavalN 41082 Subspace orthocomplement f...
docaclN 41083 Closure of subspace orthoc...
diaocN 41084 Value of partial isomorphi...
doca2N 41085 Double orthocomplement of ...
doca3N 41086 Double orthocomplement of ...
dvadiaN 41087 Any closed subspace is a m...
diarnN 41088 Partial isomorphism A maps...
diaf1oN 41089 The partial isomorphism A ...
djaffvalN 41092 Subspace join for ` DVecA ...
djafvalN 41093 Subspace join for ` DVecA ...
djavalN 41094 Subspace join for ` DVecA ...
djaclN 41095 Closure of subspace join f...
djajN 41096 Transfer lattice join to `...
dibffval 41099 The partial isomorphism B ...
dibfval 41100 The partial isomorphism B ...
dibval 41101 The partial isomorphism B ...
dibopelvalN 41102 Member of the partial isom...
dibval2 41103 Value of the partial isomo...
dibopelval2 41104 Member of the partial isom...
dibval3N 41105 Value of the partial isomo...
dibelval3 41106 Member of the partial isom...
dibopelval3 41107 Member of the partial isom...
dibelval1st 41108 Membership in value of the...
dibelval1st1 41109 Membership in value of the...
dibelval1st2N 41110 Membership in value of the...
dibelval2nd 41111 Membership in value of the...
dibn0 41112 The value of the partial i...
dibfna 41113 Functionality and domain o...
dibdiadm 41114 Domain of the partial isom...
dibfnN 41115 Functionality and domain o...
dibdmN 41116 Domain of the partial isom...
dibeldmN 41117 Member of domain of the pa...
dibord 41118 The isomorphism B for a la...
dib11N 41119 The isomorphism B for a la...
dibf11N 41120 The partial isomorphism A ...
dibclN 41121 Closure of partial isomorp...
dibvalrel 41122 The value of partial isomo...
dib0 41123 The value of partial isomo...
dib1dim 41124 Two expressions for the 1-...
dibglbN 41125 Partial isomorphism B of a...
dibintclN 41126 The intersection of partia...
dib1dim2 41127 Two expressions for a 1-di...
dibss 41128 The partial isomorphism B ...
diblss 41129 The value of partial isomo...
diblsmopel 41130 Membership in subspace sum...
dicffval 41133 The partial isomorphism C ...
dicfval 41134 The partial isomorphism C ...
dicval 41135 The partial isomorphism C ...
dicopelval 41136 Membership in value of the...
dicelvalN 41137 Membership in value of the...
dicval2 41138 The partial isomorphism C ...
dicelval3 41139 Member of the partial isom...
dicopelval2 41140 Membership in value of the...
dicelval2N 41141 Membership in value of the...
dicfnN 41142 Functionality and domain o...
dicdmN 41143 Domain of the partial isom...
dicvalrelN 41144 The value of partial isomo...
dicssdvh 41145 The partial isomorphism C ...
dicelval1sta 41146 Membership in value of the...
dicelval1stN 41147 Membership in value of the...
dicelval2nd 41148 Membership in value of the...
dicvaddcl 41149 Membership in value of the...
dicvscacl 41150 Membership in value of the...
dicn0 41151 The value of the partial i...
diclss 41152 The value of partial isomo...
diclspsn 41153 The value of isomorphism C...
cdlemn2 41154 Part of proof of Lemma N o...
cdlemn2a 41155 Part of proof of Lemma N o...
cdlemn3 41156 Part of proof of Lemma N o...
cdlemn4 41157 Part of proof of Lemma N o...
cdlemn4a 41158 Part of proof of Lemma N o...
cdlemn5pre 41159 Part of proof of Lemma N o...
cdlemn5 41160 Part of proof of Lemma N o...
cdlemn6 41161 Part of proof of Lemma N o...
cdlemn7 41162 Part of proof of Lemma N o...
cdlemn8 41163 Part of proof of Lemma N o...
cdlemn9 41164 Part of proof of Lemma N o...
cdlemn10 41165 Part of proof of Lemma N o...
cdlemn11a 41166 Part of proof of Lemma N o...
cdlemn11b 41167 Part of proof of Lemma N o...
cdlemn11c 41168 Part of proof of Lemma N o...
cdlemn11pre 41169 Part of proof of Lemma N o...
cdlemn11 41170 Part of proof of Lemma N o...
cdlemn 41171 Lemma N of [Crawley] p. 12...
dihordlem6 41172 Part of proof of Lemma N o...
dihordlem7 41173 Part of proof of Lemma N o...
dihordlem7b 41174 Part of proof of Lemma N o...
dihjustlem 41175 Part of proof after Lemma ...
dihjust 41176 Part of proof after Lemma ...
dihord1 41177 Part of proof after Lemma ...
dihord2a 41178 Part of proof after Lemma ...
dihord2b 41179 Part of proof after Lemma ...
dihord2cN 41180 Part of proof after Lemma ...
dihord11b 41181 Part of proof after Lemma ...
dihord10 41182 Part of proof after Lemma ...
dihord11c 41183 Part of proof after Lemma ...
dihord2pre 41184 Part of proof after Lemma ...
dihord2pre2 41185 Part of proof after Lemma ...
dihord2 41186 Part of proof after Lemma ...
dihffval 41189 The isomorphism H for a la...
dihfval 41190 Isomorphism H for a lattic...
dihval 41191 Value of isomorphism H for...
dihvalc 41192 Value of isomorphism H for...
dihlsscpre 41193 Closure of isomorphism H f...
dihvalcqpre 41194 Value of isomorphism H for...
dihvalcq 41195 Value of isomorphism H for...
dihvalb 41196 Value of isomorphism H for...
dihopelvalbN 41197 Ordered pair member of the...
dihvalcqat 41198 Value of isomorphism H for...
dih1dimb 41199 Two expressions for a 1-di...
dih1dimb2 41200 Isomorphism H at an atom u...
dih1dimc 41201 Isomorphism H at an atom n...
dib2dim 41202 Extend ~ dia2dim to partia...
dih2dimb 41203 Extend ~ dib2dim to isomor...
dih2dimbALTN 41204 Extend ~ dia2dim to isomor...
dihopelvalcqat 41205 Ordered pair member of the...
dihvalcq2 41206 Value of isomorphism H for...
dihopelvalcpre 41207 Member of value of isomorp...
dihopelvalc 41208 Member of value of isomorp...
dihlss 41209 The value of isomorphism H...
dihss 41210 The value of isomorphism H...
dihssxp 41211 An isomorphism H value is ...
dihopcl 41212 Closure of an ordered pair...
xihopellsmN 41213 Ordered pair membership in...
dihopellsm 41214 Ordered pair membership in...
dihord6apre 41215 Part of proof that isomorp...
dihord3 41216 The isomorphism H for a la...
dihord4 41217 The isomorphism H for a la...
dihord5b 41218 Part of proof that isomorp...
dihord6b 41219 Part of proof that isomorp...
dihord6a 41220 Part of proof that isomorp...
dihord5apre 41221 Part of proof that isomorp...
dihord5a 41222 Part of proof that isomorp...
dihord 41223 The isomorphism H is order...
dih11 41224 The isomorphism H is one-t...
dihf11lem 41225 Functionality of the isomo...
dihf11 41226 The isomorphism H for a la...
dihfn 41227 Functionality and domain o...
dihdm 41228 Domain of isomorphism H. (...
dihcl 41229 Closure of isomorphism H. ...
dihcnvcl 41230 Closure of isomorphism H c...
dihcnvid1 41231 The converse isomorphism o...
dihcnvid2 41232 The isomorphism of a conve...
dihcnvord 41233 Ordering property for conv...
dihcnv11 41234 The converse of isomorphis...
dihsslss 41235 The isomorphism H maps to ...
dihrnlss 41236 The isomorphism H maps to ...
dihrnss 41237 The isomorphism H maps to ...
dihvalrel 41238 The value of isomorphism H...
dih0 41239 The value of isomorphism H...
dih0bN 41240 A lattice element is zero ...
dih0vbN 41241 A vector is zero iff its s...
dih0cnv 41242 The isomorphism H converse...
dih0rn 41243 The zero subspace belongs ...
dih0sb 41244 A subspace is zero iff the...
dih1 41245 The value of isomorphism H...
dih1rn 41246 The full vector space belo...
dih1cnv 41247 The isomorphism H converse...
dihwN 41248 Value of isomorphism H at ...
dihmeetlem1N 41249 Isomorphism H of a conjunc...
dihglblem5apreN 41250 A conjunction property of ...
dihglblem5aN 41251 A conjunction property of ...
dihglblem2aN 41252 Lemma for isomorphism H of...
dihglblem2N 41253 The GLB of a set of lattic...
dihglblem3N 41254 Isomorphism H of a lattice...
dihglblem3aN 41255 Isomorphism H of a lattice...
dihglblem4 41256 Isomorphism H of a lattice...
dihglblem5 41257 Isomorphism H of a lattice...
dihmeetlem2N 41258 Isomorphism H of a conjunc...
dihglbcpreN 41259 Isomorphism H of a lattice...
dihglbcN 41260 Isomorphism H of a lattice...
dihmeetcN 41261 Isomorphism H of a lattice...
dihmeetbN 41262 Isomorphism H of a lattice...
dihmeetbclemN 41263 Lemma for isomorphism H of...
dihmeetlem3N 41264 Lemma for isomorphism H of...
dihmeetlem4preN 41265 Lemma for isomorphism H of...
dihmeetlem4N 41266 Lemma for isomorphism H of...
dihmeetlem5 41267 Part of proof that isomorp...
dihmeetlem6 41268 Lemma for isomorphism H of...
dihmeetlem7N 41269 Lemma for isomorphism H of...
dihjatc1 41270 Lemma for isomorphism H of...
dihjatc2N 41271 Isomorphism H of join with...
dihjatc3 41272 Isomorphism H of join with...
dihmeetlem8N 41273 Lemma for isomorphism H of...
dihmeetlem9N 41274 Lemma for isomorphism H of...
dihmeetlem10N 41275 Lemma for isomorphism H of...
dihmeetlem11N 41276 Lemma for isomorphism H of...
dihmeetlem12N 41277 Lemma for isomorphism H of...
dihmeetlem13N 41278 Lemma for isomorphism H of...
dihmeetlem14N 41279 Lemma for isomorphism H of...
dihmeetlem15N 41280 Lemma for isomorphism H of...
dihmeetlem16N 41281 Lemma for isomorphism H of...
dihmeetlem17N 41282 Lemma for isomorphism H of...
dihmeetlem18N 41283 Lemma for isomorphism H of...
dihmeetlem19N 41284 Lemma for isomorphism H of...
dihmeetlem20N 41285 Lemma for isomorphism H of...
dihmeetALTN 41286 Isomorphism H of a lattice...
dih1dimatlem0 41287 Lemma for ~ dih1dimat . (...
dih1dimatlem 41288 Lemma for ~ dih1dimat . (...
dih1dimat 41289 Any 1-dimensional subspace...
dihlsprn 41290 The span of a vector belon...
dihlspsnssN 41291 A subspace included in a 1...
dihlspsnat 41292 The inverse isomorphism H ...
dihatlat 41293 The isomorphism H of an at...
dihat 41294 There exists at least one ...
dihpN 41295 The value of isomorphism H...
dihlatat 41296 The reverse isomorphism H ...
dihatexv 41297 There is a nonzero vector ...
dihatexv2 41298 There is a nonzero vector ...
dihglblem6 41299 Isomorphism H of a lattice...
dihglb 41300 Isomorphism H of a lattice...
dihglb2 41301 Isomorphism H of a lattice...
dihmeet 41302 Isomorphism H of a lattice...
dihintcl 41303 The intersection of closed...
dihmeetcl 41304 Closure of closed subspace...
dihmeet2 41305 Reverse isomorphism H of a...
dochffval 41308 Subspace orthocomplement f...
dochfval 41309 Subspace orthocomplement f...
dochval 41310 Subspace orthocomplement f...
dochval2 41311 Subspace orthocomplement f...
dochcl 41312 Closure of subspace orthoc...
dochlss 41313 A subspace orthocomplement...
dochssv 41314 A subspace orthocomplement...
dochfN 41315 Domain and codomain of the...
dochvalr 41316 Orthocomplement of a close...
doch0 41317 Orthocomplement of the zer...
doch1 41318 Orthocomplement of the uni...
dochoc0 41319 The zero subspace is close...
dochoc1 41320 The unit subspace (all vec...
dochvalr2 41321 Orthocomplement of a close...
dochvalr3 41322 Orthocomplement of a close...
doch2val2 41323 Double orthocomplement for...
dochss 41324 Subset law for orthocomple...
dochocss 41325 Double negative law for or...
dochoc 41326 Double negative law for or...
dochsscl 41327 If a set of vectors is inc...
dochoccl 41328 A set of vectors is closed...
dochord 41329 Ordering law for orthocomp...
dochord2N 41330 Ordering law for orthocomp...
dochord3 41331 Ordering law for orthocomp...
doch11 41332 Orthocomplement is one-to-...
dochsordN 41333 Strict ordering law for or...
dochn0nv 41334 An orthocomplement is nonz...
dihoml4c 41335 Version of ~ dihoml4 with ...
dihoml4 41336 Orthomodular law for const...
dochspss 41337 The span of a set of vecto...
dochocsp 41338 The span of an orthocomple...
dochspocN 41339 The span of an orthocomple...
dochocsn 41340 The double orthocomplement...
dochsncom 41341 Swap vectors in an orthoco...
dochsat 41342 The double orthocomplement...
dochshpncl 41343 If a hyperplane is not clo...
dochlkr 41344 Equivalent conditions for ...
dochkrshp 41345 The closure of a kernel is...
dochkrshp2 41346 Properties of the closure ...
dochkrshp3 41347 Properties of the closure ...
dochkrshp4 41348 Properties of the closure ...
dochdmj1 41349 De Morgan-like law for sub...
dochnoncon 41350 Law of noncontradiction. ...
dochnel2 41351 A nonzero member of a subs...
dochnel 41352 A nonzero vector doesn't b...
djhffval 41355 Subspace join for ` DVecH ...
djhfval 41356 Subspace join for ` DVecH ...
djhval 41357 Subspace join for ` DVecH ...
djhval2 41358 Value of subspace join for...
djhcl 41359 Closure of subspace join f...
djhlj 41360 Transfer lattice join to `...
djhljjN 41361 Lattice join in terms of `...
djhjlj 41362 ` DVecH ` vector space clo...
djhj 41363 ` DVecH ` vector space clo...
djhcom 41364 Subspace join commutes. (...
djhspss 41365 Subspace span of union is ...
djhsumss 41366 Subspace sum is a subset o...
dihsumssj 41367 The subspace sum of two is...
djhunssN 41368 Subspace union is a subset...
dochdmm1 41369 De Morgan-like law for clo...
djhexmid 41370 Excluded middle property o...
djh01 41371 Closed subspace join with ...
djh02 41372 Closed subspace join with ...
djhlsmcl 41373 A closed subspace sum equa...
djhcvat42 41374 A covering property. ( ~ ...
dihjatb 41375 Isomorphism H of lattice j...
dihjatc 41376 Isomorphism H of lattice j...
dihjatcclem1 41377 Lemma for isomorphism H of...
dihjatcclem2 41378 Lemma for isomorphism H of...
dihjatcclem3 41379 Lemma for ~ dihjatcc . (C...
dihjatcclem4 41380 Lemma for isomorphism H of...
dihjatcc 41381 Isomorphism H of lattice j...
dihjat 41382 Isomorphism H of lattice j...
dihprrnlem1N 41383 Lemma for ~ dihprrn , show...
dihprrnlem2 41384 Lemma for ~ dihprrn . (Co...
dihprrn 41385 The span of a vector pair ...
djhlsmat 41386 The sum of two subspace at...
dihjat1lem 41387 Subspace sum of a closed s...
dihjat1 41388 Subspace sum of a closed s...
dihsmsprn 41389 Subspace sum of a closed s...
dihjat2 41390 The subspace sum of a clos...
dihjat3 41391 Isomorphism H of lattice j...
dihjat4 41392 Transfer the subspace sum ...
dihjat6 41393 Transfer the subspace sum ...
dihsmsnrn 41394 The subspace sum of two si...
dihsmatrn 41395 The subspace sum of a clos...
dihjat5N 41396 Transfer lattice join with...
dvh4dimat 41397 There is an atom that is o...
dvh3dimatN 41398 There is an atom that is o...
dvh2dimatN 41399 Given an atom, there exist...
dvh1dimat 41400 There exists an atom. (Co...
dvh1dim 41401 There exists a nonzero vec...
dvh4dimlem 41402 Lemma for ~ dvh4dimN . (C...
dvhdimlem 41403 Lemma for ~ dvh2dim and ~ ...
dvh2dim 41404 There is a vector that is ...
dvh3dim 41405 There is a vector that is ...
dvh4dimN 41406 There is a vector that is ...
dvh3dim2 41407 There is a vector that is ...
dvh3dim3N 41408 There is a vector that is ...
dochsnnz 41409 The orthocomplement of a s...
dochsatshp 41410 The orthocomplement of a s...
dochsatshpb 41411 The orthocomplement of a s...
dochsnshp 41412 The orthocomplement of a n...
dochshpsat 41413 A hyperplane is closed iff...
dochkrsat 41414 The orthocomplement of a k...
dochkrsat2 41415 The orthocomplement of a k...
dochsat0 41416 The orthocomplement of a k...
dochkrsm 41417 The subspace sum of a clos...
dochexmidat 41418 Special case of excluded m...
dochexmidlem1 41419 Lemma for ~ dochexmid . H...
dochexmidlem2 41420 Lemma for ~ dochexmid . (...
dochexmidlem3 41421 Lemma for ~ dochexmid . U...
dochexmidlem4 41422 Lemma for ~ dochexmid . (...
dochexmidlem5 41423 Lemma for ~ dochexmid . (...
dochexmidlem6 41424 Lemma for ~ dochexmid . (...
dochexmidlem7 41425 Lemma for ~ dochexmid . C...
dochexmidlem8 41426 Lemma for ~ dochexmid . T...
dochexmid 41427 Excluded middle law for cl...
dochsnkrlem1 41428 Lemma for ~ dochsnkr . (C...
dochsnkrlem2 41429 Lemma for ~ dochsnkr . (C...
dochsnkrlem3 41430 Lemma for ~ dochsnkr . (C...
dochsnkr 41431 A (closed) kernel expresse...
dochsnkr2 41432 Kernel of the explicit fun...
dochsnkr2cl 41433 The ` X ` determining func...
dochflcl 41434 Closure of the explicit fu...
dochfl1 41435 The value of the explicit ...
dochfln0 41436 The value of a functional ...
dochkr1 41437 A nonzero functional has a...
dochkr1OLDN 41438 A nonzero functional has a...
lpolsetN 41441 The set of polarities of a...
islpolN 41442 The predicate "is a polari...
islpoldN 41443 Properties that determine ...
lpolfN 41444 Functionality of a polarit...
lpolvN 41445 The polarity of the whole ...
lpolconN 41446 Contraposition property of...
lpolsatN 41447 The polarity of an atomic ...
lpolpolsatN 41448 Property of a polarity. (...
dochpolN 41449 The subspace orthocompleme...
lcfl1lem 41450 Property of a functional w...
lcfl1 41451 Property of a functional w...
lcfl2 41452 Property of a functional w...
lcfl3 41453 Property of a functional w...
lcfl4N 41454 Property of a functional w...
lcfl5 41455 Property of a functional w...
lcfl5a 41456 Property of a functional w...
lcfl6lem 41457 Lemma for ~ lcfl6 . A fun...
lcfl7lem 41458 Lemma for ~ lcfl7N . If t...
lcfl6 41459 Property of a functional w...
lcfl7N 41460 Property of a functional w...
lcfl8 41461 Property of a functional w...
lcfl8a 41462 Property of a functional w...
lcfl8b 41463 Property of a nonzero func...
lcfl9a 41464 Property implying that a f...
lclkrlem1 41465 The set of functionals hav...
lclkrlem2a 41466 Lemma for ~ lclkr . Use ~...
lclkrlem2b 41467 Lemma for ~ lclkr . (Cont...
lclkrlem2c 41468 Lemma for ~ lclkr . (Cont...
lclkrlem2d 41469 Lemma for ~ lclkr . (Cont...
lclkrlem2e 41470 Lemma for ~ lclkr . The k...
lclkrlem2f 41471 Lemma for ~ lclkr . Const...
lclkrlem2g 41472 Lemma for ~ lclkr . Compa...
lclkrlem2h 41473 Lemma for ~ lclkr . Elimi...
lclkrlem2i 41474 Lemma for ~ lclkr . Elimi...
lclkrlem2j 41475 Lemma for ~ lclkr . Kerne...
lclkrlem2k 41476 Lemma for ~ lclkr . Kerne...
lclkrlem2l 41477 Lemma for ~ lclkr . Elimi...
lclkrlem2m 41478 Lemma for ~ lclkr . Const...
lclkrlem2n 41479 Lemma for ~ lclkr . (Cont...
lclkrlem2o 41480 Lemma for ~ lclkr . When ...
lclkrlem2p 41481 Lemma for ~ lclkr . When ...
lclkrlem2q 41482 Lemma for ~ lclkr . The s...
lclkrlem2r 41483 Lemma for ~ lclkr . When ...
lclkrlem2s 41484 Lemma for ~ lclkr . Thus,...
lclkrlem2t 41485 Lemma for ~ lclkr . We el...
lclkrlem2u 41486 Lemma for ~ lclkr . ~ lclk...
lclkrlem2v 41487 Lemma for ~ lclkr . When ...
lclkrlem2w 41488 Lemma for ~ lclkr . This ...
lclkrlem2x 41489 Lemma for ~ lclkr . Elimi...
lclkrlem2y 41490 Lemma for ~ lclkr . Resta...
lclkrlem2 41491 The set of functionals hav...
lclkr 41492 The set of functionals wit...
lcfls1lem 41493 Property of a functional w...
lcfls1N 41494 Property of a functional w...
lcfls1c 41495 Property of a functional w...
lclkrslem1 41496 The set of functionals hav...
lclkrslem2 41497 The set of functionals hav...
lclkrs 41498 The set of functionals hav...
lclkrs2 41499 The set of functionals wit...
lcfrvalsnN 41500 Reconstruction from the du...
lcfrlem1 41501 Lemma for ~ lcfr . Note t...
lcfrlem2 41502 Lemma for ~ lcfr . (Contr...
lcfrlem3 41503 Lemma for ~ lcfr . (Contr...
lcfrlem4 41504 Lemma for ~ lcfr . (Contr...
lcfrlem5 41505 Lemma for ~ lcfr . The se...
lcfrlem6 41506 Lemma for ~ lcfr . Closur...
lcfrlem7 41507 Lemma for ~ lcfr . Closur...
lcfrlem8 41508 Lemma for ~ lcf1o and ~ lc...
lcfrlem9 41509 Lemma for ~ lcf1o . (This...
lcf1o 41510 Define a function ` J ` th...
lcfrlem10 41511 Lemma for ~ lcfr . (Contr...
lcfrlem11 41512 Lemma for ~ lcfr . (Contr...
lcfrlem12N 41513 Lemma for ~ lcfr . (Contr...
lcfrlem13 41514 Lemma for ~ lcfr . (Contr...
lcfrlem14 41515 Lemma for ~ lcfr . (Contr...
lcfrlem15 41516 Lemma for ~ lcfr . (Contr...
lcfrlem16 41517 Lemma for ~ lcfr . (Contr...
lcfrlem17 41518 Lemma for ~ lcfr . Condit...
lcfrlem18 41519 Lemma for ~ lcfr . (Contr...
lcfrlem19 41520 Lemma for ~ lcfr . (Contr...
lcfrlem20 41521 Lemma for ~ lcfr . (Contr...
lcfrlem21 41522 Lemma for ~ lcfr . (Contr...
lcfrlem22 41523 Lemma for ~ lcfr . (Contr...
lcfrlem23 41524 Lemma for ~ lcfr . TODO: ...
lcfrlem24 41525 Lemma for ~ lcfr . (Contr...
lcfrlem25 41526 Lemma for ~ lcfr . Specia...
lcfrlem26 41527 Lemma for ~ lcfr . Specia...
lcfrlem27 41528 Lemma for ~ lcfr . Specia...
lcfrlem28 41529 Lemma for ~ lcfr . TODO: ...
lcfrlem29 41530 Lemma for ~ lcfr . (Contr...
lcfrlem30 41531 Lemma for ~ lcfr . (Contr...
lcfrlem31 41532 Lemma for ~ lcfr . (Contr...
lcfrlem32 41533 Lemma for ~ lcfr . (Contr...
lcfrlem33 41534 Lemma for ~ lcfr . (Contr...
lcfrlem34 41535 Lemma for ~ lcfr . (Contr...
lcfrlem35 41536 Lemma for ~ lcfr . (Contr...
lcfrlem36 41537 Lemma for ~ lcfr . (Contr...
lcfrlem37 41538 Lemma for ~ lcfr . (Contr...
lcfrlem38 41539 Lemma for ~ lcfr . Combin...
lcfrlem39 41540 Lemma for ~ lcfr . Elimin...
lcfrlem40 41541 Lemma for ~ lcfr . Elimin...
lcfrlem41 41542 Lemma for ~ lcfr . Elimin...
lcfrlem42 41543 Lemma for ~ lcfr . Elimin...
lcfr 41544 Reconstruction of a subspa...
lcdfval 41547 Dual vector space of funct...
lcdval 41548 Dual vector space of funct...
lcdval2 41549 Dual vector space of funct...
lcdlvec 41550 The dual vector space of f...
lcdlmod 41551 The dual vector space of f...
lcdvbase 41552 Vector base set of a dual ...
lcdvbasess 41553 The vector base set of the...
lcdvbaselfl 41554 A vector in the base set o...
lcdvbasecl 41555 Closure of the value of a ...
lcdvadd 41556 Vector addition for the cl...
lcdvaddval 41557 The value of the value of ...
lcdsca 41558 The ring of scalars of the...
lcdsbase 41559 Base set of scalar ring fo...
lcdsadd 41560 Scalar addition for the cl...
lcdsmul 41561 Scalar multiplication for ...
lcdvs 41562 Scalar product for the clo...
lcdvsval 41563 Value of scalar product op...
lcdvscl 41564 The scalar product operati...
lcdlssvscl 41565 Closure of scalar product ...
lcdvsass 41566 Associative law for scalar...
lcd0 41567 The zero scalar of the clo...
lcd1 41568 The unit scalar of the clo...
lcdneg 41569 The unit scalar of the clo...
lcd0v 41570 The zero functional in the...
lcd0v2 41571 The zero functional in the...
lcd0vvalN 41572 Value of the zero function...
lcd0vcl 41573 Closure of the zero functi...
lcd0vs 41574 A scalar zero times a func...
lcdvs0N 41575 A scalar times the zero fu...
lcdvsub 41576 The value of vector subtra...
lcdvsubval 41577 The value of the value of ...
lcdlss 41578 Subspaces of a dual vector...
lcdlss2N 41579 Subspaces of a dual vector...
lcdlsp 41580 Span in the set of functio...
lcdlkreqN 41581 Colinear functionals have ...
lcdlkreq2N 41582 Colinear functionals have ...
mapdffval 41585 Projectivity from vector s...
mapdfval 41586 Projectivity from vector s...
mapdval 41587 Value of projectivity from...
mapdvalc 41588 Value of projectivity from...
mapdval2N 41589 Value of projectivity from...
mapdval3N 41590 Value of projectivity from...
mapdval4N 41591 Value of projectivity from...
mapdval5N 41592 Value of projectivity from...
mapdordlem1a 41593 Lemma for ~ mapdord . (Co...
mapdordlem1bN 41594 Lemma for ~ mapdord . (Co...
mapdordlem1 41595 Lemma for ~ mapdord . (Co...
mapdordlem2 41596 Lemma for ~ mapdord . Ord...
mapdord 41597 Ordering property of the m...
mapd11 41598 The map defined by ~ df-ma...
mapddlssN 41599 The mapping of a subspace ...
mapdsn 41600 Value of the map defined b...
mapdsn2 41601 Value of the map defined b...
mapdsn3 41602 Value of the map defined b...
mapd1dim2lem1N 41603 Value of the map defined b...
mapdrvallem2 41604 Lemma for ~ mapdrval . TO...
mapdrvallem3 41605 Lemma for ~ mapdrval . (C...
mapdrval 41606 Given a dual subspace ` R ...
mapd1o 41607 The map defined by ~ df-ma...
mapdrn 41608 Range of the map defined b...
mapdunirnN 41609 Union of the range of the ...
mapdrn2 41610 Range of the map defined b...
mapdcnvcl 41611 Closure of the converse of...
mapdcl 41612 Closure the value of the m...
mapdcnvid1N 41613 Converse of the value of t...
mapdsord 41614 Strong ordering property o...
mapdcl2 41615 The mapping of a subspace ...
mapdcnvid2 41616 Value of the converse of t...
mapdcnvordN 41617 Ordering property of the c...
mapdcnv11N 41618 The converse of the map de...
mapdcv 41619 Covering property of the c...
mapdincl 41620 Closure of dual subspace i...
mapdin 41621 Subspace intersection is p...
mapdlsmcl 41622 Closure of dual subspace s...
mapdlsm 41623 Subspace sum is preserved ...
mapd0 41624 Projectivity map of the ze...
mapdcnvatN 41625 Atoms are preserved by the...
mapdat 41626 Atoms are preserved by the...
mapdspex 41627 The map of a span equals t...
mapdn0 41628 Transfer nonzero property ...
mapdncol 41629 Transfer non-colinearity f...
mapdindp 41630 Transfer (part of) vector ...
mapdpglem1 41631 Lemma for ~ mapdpg . Baer...
mapdpglem2 41632 Lemma for ~ mapdpg . Baer...
mapdpglem2a 41633 Lemma for ~ mapdpg . (Con...
mapdpglem3 41634 Lemma for ~ mapdpg . Baer...
mapdpglem4N 41635 Lemma for ~ mapdpg . (Con...
mapdpglem5N 41636 Lemma for ~ mapdpg . (Con...
mapdpglem6 41637 Lemma for ~ mapdpg . Baer...
mapdpglem8 41638 Lemma for ~ mapdpg . Baer...
mapdpglem9 41639 Lemma for ~ mapdpg . Baer...
mapdpglem10 41640 Lemma for ~ mapdpg . Baer...
mapdpglem11 41641 Lemma for ~ mapdpg . (Con...
mapdpglem12 41642 Lemma for ~ mapdpg . TODO...
mapdpglem13 41643 Lemma for ~ mapdpg . (Con...
mapdpglem14 41644 Lemma for ~ mapdpg . (Con...
mapdpglem15 41645 Lemma for ~ mapdpg . (Con...
mapdpglem16 41646 Lemma for ~ mapdpg . Baer...
mapdpglem17N 41647 Lemma for ~ mapdpg . Baer...
mapdpglem18 41648 Lemma for ~ mapdpg . Baer...
mapdpglem19 41649 Lemma for ~ mapdpg . Baer...
mapdpglem20 41650 Lemma for ~ mapdpg . Baer...
mapdpglem21 41651 Lemma for ~ mapdpg . (Con...
mapdpglem22 41652 Lemma for ~ mapdpg . Baer...
mapdpglem23 41653 Lemma for ~ mapdpg . Baer...
mapdpglem30a 41654 Lemma for ~ mapdpg . (Con...
mapdpglem30b 41655 Lemma for ~ mapdpg . (Con...
mapdpglem25 41656 Lemma for ~ mapdpg . Baer...
mapdpglem26 41657 Lemma for ~ mapdpg . Baer...
mapdpglem27 41658 Lemma for ~ mapdpg . Baer...
mapdpglem29 41659 Lemma for ~ mapdpg . Baer...
mapdpglem28 41660 Lemma for ~ mapdpg . Baer...
mapdpglem30 41661 Lemma for ~ mapdpg . Baer...
mapdpglem31 41662 Lemma for ~ mapdpg . Baer...
mapdpglem24 41663 Lemma for ~ mapdpg . Exis...
mapdpglem32 41664 Lemma for ~ mapdpg . Uniq...
mapdpg 41665 Part 1 of proof of the fir...
baerlem3lem1 41666 Lemma for ~ baerlem3 . (C...
baerlem5alem1 41667 Lemma for ~ baerlem5a . (...
baerlem5blem1 41668 Lemma for ~ baerlem5b . (...
baerlem3lem2 41669 Lemma for ~ baerlem3 . (C...
baerlem5alem2 41670 Lemma for ~ baerlem5a . (...
baerlem5blem2 41671 Lemma for ~ baerlem5b . (...
baerlem3 41672 An equality that holds whe...
baerlem5a 41673 An equality that holds whe...
baerlem5b 41674 An equality that holds whe...
baerlem5amN 41675 An equality that holds whe...
baerlem5bmN 41676 An equality that holds whe...
baerlem5abmN 41677 An equality that holds whe...
mapdindp0 41678 Vector independence lemma....
mapdindp1 41679 Vector independence lemma....
mapdindp2 41680 Vector independence lemma....
mapdindp3 41681 Vector independence lemma....
mapdindp4 41682 Vector independence lemma....
mapdhval 41683 Lemmma for ~~? mapdh . (C...
mapdhval0 41684 Lemmma for ~~? mapdh . (C...
mapdhval2 41685 Lemmma for ~~? mapdh . (C...
mapdhcl 41686 Lemmma for ~~? mapdh . (C...
mapdheq 41687 Lemmma for ~~? mapdh . Th...
mapdheq2 41688 Lemmma for ~~? mapdh . On...
mapdheq2biN 41689 Lemmma for ~~? mapdh . Pa...
mapdheq4lem 41690 Lemma for ~ mapdheq4 . Pa...
mapdheq4 41691 Lemma for ~~? mapdh . Par...
mapdh6lem1N 41692 Lemma for ~ mapdh6N . Par...
mapdh6lem2N 41693 Lemma for ~ mapdh6N . Par...
mapdh6aN 41694 Lemma for ~ mapdh6N . Par...
mapdh6b0N 41695 Lemmma for ~ mapdh6N . (C...
mapdh6bN 41696 Lemmma for ~ mapdh6N . (C...
mapdh6cN 41697 Lemmma for ~ mapdh6N . (C...
mapdh6dN 41698 Lemmma for ~ mapdh6N . (C...
mapdh6eN 41699 Lemmma for ~ mapdh6N . Pa...
mapdh6fN 41700 Lemmma for ~ mapdh6N . Pa...
mapdh6gN 41701 Lemmma for ~ mapdh6N . Pa...
mapdh6hN 41702 Lemmma for ~ mapdh6N . Pa...
mapdh6iN 41703 Lemmma for ~ mapdh6N . El...
mapdh6jN 41704 Lemmma for ~ mapdh6N . El...
mapdh6kN 41705 Lemmma for ~ mapdh6N . El...
mapdh6N 41706 Part (6) of [Baer] p. 47 l...
mapdh7eN 41707 Part (7) of [Baer] p. 48 l...
mapdh7cN 41708 Part (7) of [Baer] p. 48 l...
mapdh7dN 41709 Part (7) of [Baer] p. 48 l...
mapdh7fN 41710 Part (7) of [Baer] p. 48 l...
mapdh75e 41711 Part (7) of [Baer] p. 48 l...
mapdh75cN 41712 Part (7) of [Baer] p. 48 l...
mapdh75d 41713 Part (7) of [Baer] p. 48 l...
mapdh75fN 41714 Part (7) of [Baer] p. 48 l...
hvmapffval 41717 Map from nonzero vectors t...
hvmapfval 41718 Map from nonzero vectors t...
hvmapval 41719 Value of map from nonzero ...
hvmapvalvalN 41720 Value of value of map (i.e...
hvmapidN 41721 The value of the vector to...
hvmap1o 41722 The vector to functional m...
hvmapclN 41723 Closure of the vector to f...
hvmap1o2 41724 The vector to functional m...
hvmapcl2 41725 Closure of the vector to f...
hvmaplfl 41726 The vector to functional m...
hvmaplkr 41727 Kernel of the vector to fu...
mapdhvmap 41728 Relationship between ` map...
lspindp5 41729 Obtain an independent vect...
hdmaplem1 41730 Lemma to convert a frequen...
hdmaplem2N 41731 Lemma to convert a frequen...
hdmaplem3 41732 Lemma to convert a frequen...
hdmaplem4 41733 Lemma to convert a frequen...
mapdh8a 41734 Part of Part (8) in [Baer]...
mapdh8aa 41735 Part of Part (8) in [Baer]...
mapdh8ab 41736 Part of Part (8) in [Baer]...
mapdh8ac 41737 Part of Part (8) in [Baer]...
mapdh8ad 41738 Part of Part (8) in [Baer]...
mapdh8b 41739 Part of Part (8) in [Baer]...
mapdh8c 41740 Part of Part (8) in [Baer]...
mapdh8d0N 41741 Part of Part (8) in [Baer]...
mapdh8d 41742 Part of Part (8) in [Baer]...
mapdh8e 41743 Part of Part (8) in [Baer]...
mapdh8g 41744 Part of Part (8) in [Baer]...
mapdh8i 41745 Part of Part (8) in [Baer]...
mapdh8j 41746 Part of Part (8) in [Baer]...
mapdh8 41747 Part (8) in [Baer] p. 48. ...
mapdh9a 41748 Lemma for part (9) in [Bae...
mapdh9aOLDN 41749 Lemma for part (9) in [Bae...
hdmap1ffval 41754 Preliminary map from vecto...
hdmap1fval 41755 Preliminary map from vecto...
hdmap1vallem 41756 Value of preliminary map f...
hdmap1val 41757 Value of preliminary map f...
hdmap1val0 41758 Value of preliminary map f...
hdmap1val2 41759 Value of preliminary map f...
hdmap1eq 41760 The defining equation for ...
hdmap1cbv 41761 Frequently used lemma to c...
hdmap1valc 41762 Connect the value of the p...
hdmap1cl 41763 Convert closure theorem ~ ...
hdmap1eq2 41764 Convert ~ mapdheq2 to use ...
hdmap1eq4N 41765 Convert ~ mapdheq4 to use ...
hdmap1l6lem1 41766 Lemma for ~ hdmap1l6 . Pa...
hdmap1l6lem2 41767 Lemma for ~ hdmap1l6 . Pa...
hdmap1l6a 41768 Lemma for ~ hdmap1l6 . Pa...
hdmap1l6b0N 41769 Lemmma for ~ hdmap1l6 . (...
hdmap1l6b 41770 Lemmma for ~ hdmap1l6 . (...
hdmap1l6c 41771 Lemmma for ~ hdmap1l6 . (...
hdmap1l6d 41772 Lemmma for ~ hdmap1l6 . (...
hdmap1l6e 41773 Lemmma for ~ hdmap1l6 . P...
hdmap1l6f 41774 Lemmma for ~ hdmap1l6 . P...
hdmap1l6g 41775 Lemmma for ~ hdmap1l6 . P...
hdmap1l6h 41776 Lemmma for ~ hdmap1l6 . P...
hdmap1l6i 41777 Lemmma for ~ hdmap1l6 . E...
hdmap1l6j 41778 Lemmma for ~ hdmap1l6 . E...
hdmap1l6k 41779 Lemmma for ~ hdmap1l6 . E...
hdmap1l6 41780 Part (6) of [Baer] p. 47 l...
hdmap1eulem 41781 Lemma for ~ hdmap1eu . TO...
hdmap1eulemOLDN 41782 Lemma for ~ hdmap1euOLDN ....
hdmap1eu 41783 Convert ~ mapdh9a to use t...
hdmap1euOLDN 41784 Convert ~ mapdh9aOLDN to u...
hdmapffval 41785 Map from vectors to functi...
hdmapfval 41786 Map from vectors to functi...
hdmapval 41787 Value of map from vectors ...
hdmapfnN 41788 Functionality of map from ...
hdmapcl 41789 Closure of map from vector...
hdmapval2lem 41790 Lemma for ~ hdmapval2 . (...
hdmapval2 41791 Value of map from vectors ...
hdmapval0 41792 Value of map from vectors ...
hdmapeveclem 41793 Lemma for ~ hdmapevec . T...
hdmapevec 41794 Value of map from vectors ...
hdmapevec2 41795 The inner product of the r...
hdmapval3lemN 41796 Value of map from vectors ...
hdmapval3N 41797 Value of map from vectors ...
hdmap10lem 41798 Lemma for ~ hdmap10 . (Co...
hdmap10 41799 Part 10 in [Baer] p. 48 li...
hdmap11lem1 41800 Lemma for ~ hdmapadd . (C...
hdmap11lem2 41801 Lemma for ~ hdmapadd . (C...
hdmapadd 41802 Part 11 in [Baer] p. 48 li...
hdmapeq0 41803 Part of proof of part 12 i...
hdmapnzcl 41804 Nonzero vector closure of ...
hdmapneg 41805 Part of proof of part 12 i...
hdmapsub 41806 Part of proof of part 12 i...
hdmap11 41807 Part of proof of part 12 i...
hdmaprnlem1N 41808 Part of proof of part 12 i...
hdmaprnlem3N 41809 Part of proof of part 12 i...
hdmaprnlem3uN 41810 Part of proof of part 12 i...
hdmaprnlem4tN 41811 Lemma for ~ hdmaprnN . TO...
hdmaprnlem4N 41812 Part of proof of part 12 i...
hdmaprnlem6N 41813 Part of proof of part 12 i...
hdmaprnlem7N 41814 Part of proof of part 12 i...
hdmaprnlem8N 41815 Part of proof of part 12 i...
hdmaprnlem9N 41816 Part of proof of part 12 i...
hdmaprnlem3eN 41817 Lemma for ~ hdmaprnN . (C...
hdmaprnlem10N 41818 Lemma for ~ hdmaprnN . Sh...
hdmaprnlem11N 41819 Lemma for ~ hdmaprnN . Sh...
hdmaprnlem15N 41820 Lemma for ~ hdmaprnN . El...
hdmaprnlem16N 41821 Lemma for ~ hdmaprnN . El...
hdmaprnlem17N 41822 Lemma for ~ hdmaprnN . In...
hdmaprnN 41823 Part of proof of part 12 i...
hdmapf1oN 41824 Part 12 in [Baer] p. 49. ...
hdmap14lem1a 41825 Prior to part 14 in [Baer]...
hdmap14lem2a 41826 Prior to part 14 in [Baer]...
hdmap14lem1 41827 Prior to part 14 in [Baer]...
hdmap14lem2N 41828 Prior to part 14 in [Baer]...
hdmap14lem3 41829 Prior to part 14 in [Baer]...
hdmap14lem4a 41830 Simplify ` ( A \ { Q } ) `...
hdmap14lem4 41831 Simplify ` ( A \ { Q } ) `...
hdmap14lem6 41832 Case where ` F ` is zero. ...
hdmap14lem7 41833 Combine cases of ` F ` . ...
hdmap14lem8 41834 Part of proof of part 14 i...
hdmap14lem9 41835 Part of proof of part 14 i...
hdmap14lem10 41836 Part of proof of part 14 i...
hdmap14lem11 41837 Part of proof of part 14 i...
hdmap14lem12 41838 Lemma for proof of part 14...
hdmap14lem13 41839 Lemma for proof of part 14...
hdmap14lem14 41840 Part of proof of part 14 i...
hdmap14lem15 41841 Part of proof of part 14 i...
hgmapffval 41844 Map from the scalar divisi...
hgmapfval 41845 Map from the scalar divisi...
hgmapval 41846 Value of map from the scal...
hgmapfnN 41847 Functionality of scalar si...
hgmapcl 41848 Closure of scalar sigma ma...
hgmapdcl 41849 Closure of the vector spac...
hgmapvs 41850 Part 15 of [Baer] p. 50 li...
hgmapval0 41851 Value of the scalar sigma ...
hgmapval1 41852 Value of the scalar sigma ...
hgmapadd 41853 Part 15 of [Baer] p. 50 li...
hgmapmul 41854 Part 15 of [Baer] p. 50 li...
hgmaprnlem1N 41855 Lemma for ~ hgmaprnN . (C...
hgmaprnlem2N 41856 Lemma for ~ hgmaprnN . Pa...
hgmaprnlem3N 41857 Lemma for ~ hgmaprnN . El...
hgmaprnlem4N 41858 Lemma for ~ hgmaprnN . El...
hgmaprnlem5N 41859 Lemma for ~ hgmaprnN . El...
hgmaprnN 41860 Part of proof of part 16 i...
hgmap11 41861 The scalar sigma map is on...
hgmapf1oN 41862 The scalar sigma map is a ...
hgmapeq0 41863 The scalar sigma map is ze...
hdmapipcl 41864 The inner product (Hermiti...
hdmapln1 41865 Linearity property that wi...
hdmaplna1 41866 Additive property of first...
hdmaplns1 41867 Subtraction property of fi...
hdmaplnm1 41868 Multiplicative property of...
hdmaplna2 41869 Additive property of secon...
hdmapglnm2 41870 g-linear property of secon...
hdmapgln2 41871 g-linear property that wil...
hdmaplkr 41872 Kernel of the vector to du...
hdmapellkr 41873 Membership in the kernel (...
hdmapip0 41874 Zero property that will be...
hdmapip1 41875 Construct a proportional v...
hdmapip0com 41876 Commutation property of Ba...
hdmapinvlem1 41877 Line 27 in [Baer] p. 110. ...
hdmapinvlem2 41878 Line 28 in [Baer] p. 110, ...
hdmapinvlem3 41879 Line 30 in [Baer] p. 110, ...
hdmapinvlem4 41880 Part 1.1 of Proposition 1 ...
hdmapglem5 41881 Part 1.2 in [Baer] p. 110 ...
hgmapvvlem1 41882 Involution property of sca...
hgmapvvlem2 41883 Lemma for ~ hgmapvv . Eli...
hgmapvvlem3 41884 Lemma for ~ hgmapvv . Eli...
hgmapvv 41885 Value of a double involuti...
hdmapglem7a 41886 Lemma for ~ hdmapg . (Con...
hdmapglem7b 41887 Lemma for ~ hdmapg . (Con...
hdmapglem7 41888 Lemma for ~ hdmapg . Line...
hdmapg 41889 Apply the scalar sigma fun...
hdmapoc 41890 Express our constructed or...
hlhilset 41893 The final Hilbert space co...
hlhilsca 41894 The scalar of the final co...
hlhilbase 41895 The base set of the final ...
hlhilplus 41896 The vector addition for th...
hlhilslem 41897 Lemma for ~ hlhilsbase etc...
hlhilslemOLD 41898 Obsolete version of ~ hlhi...
hlhilsbase 41899 The scalar base set of the...
hlhilsbaseOLD 41900 Obsolete version of ~ hlhi...
hlhilsplus 41901 Scalar addition for the fi...
hlhilsplusOLD 41902 Obsolete version of ~ hlhi...
hlhilsmul 41903 Scalar multiplication for ...
hlhilsmulOLD 41904 Obsolete version of ~ hlhi...
hlhilsbase2 41905 The scalar base set of the...
hlhilsplus2 41906 Scalar addition for the fi...
hlhilsmul2 41907 Scalar multiplication for ...
hlhils0 41908 The scalar ring zero for t...
hlhils1N 41909 The scalar ring unity for ...
hlhilvsca 41910 The scalar product for the...
hlhilip 41911 Inner product operation fo...
hlhilipval 41912 Value of inner product ope...
hlhilnvl 41913 The involution operation o...
hlhillvec 41914 The final constructed Hilb...
hlhildrng 41915 The star division ring for...
hlhilsrnglem 41916 Lemma for ~ hlhilsrng . (...
hlhilsrng 41917 The star division ring for...
hlhil0 41918 The zero vector for the fi...
hlhillsm 41919 The vector sum operation f...
hlhilocv 41920 The orthocomplement for th...
hlhillcs 41921 The closed subspaces of th...
hlhilphllem 41922 Lemma for ~ hlhil . (Cont...
hlhilhillem 41923 Lemma for ~ hlhil . (Cont...
hlathil 41924 Construction of a Hilbert ...
iscsrg 41927 A commutative semiring is ...
rhmzrhval 41928 Evaluation of integers acr...
zndvdchrrhm 41929 Construction of a ring hom...
leexp1ad 41930 Weak base ordering relatio...
relogbcld 41931 Closure of the general log...
relogbexpd 41932 Identity law for general l...
relogbzexpd 41933 Power law for the general ...
logblebd 41934 The general logarithm is m...
uzindd 41935 Induction on the upper int...
fzadd2d 41936 Membership of a sum in a f...
zltlem1d 41937 Integer ordering relation,...
zltp1led 41938 Integer ordering relation,...
fzne2d 41939 Elementhood in a finite se...
eqfnfv2d2 41940 Equality of functions is d...
fzsplitnd 41941 Split a finite interval of...
fzsplitnr 41942 Split a finite interval of...
addassnni 41943 Associative law for additi...
addcomnni 41944 Commutative law for additi...
mulassnni 41945 Associative law for multip...
mulcomnni 41946 Commutative law for multip...
gcdcomnni 41947 Commutative law for gcd. ...
gcdnegnni 41948 Negation invariance for gc...
neggcdnni 41949 Negation invariance for gc...
bccl2d 41950 Closure of the binomial co...
recbothd 41951 Take reciprocal on both si...
gcdmultiplei 41952 The GCD of a multiple of a...
gcdaddmzz2nni 41953 Adding a multiple of one o...
gcdaddmzz2nncomi 41954 Adding a multiple of one o...
gcdnncli 41955 Closure of the gcd operato...
muldvds1d 41956 If a product divides an in...
muldvds2d 41957 If a product divides an in...
nndivdvdsd 41958 A positive integer divides...
nnproddivdvdsd 41959 A product of natural numbe...
coprmdvds2d 41960 If an integer is divisible...
imadomfi 41961 An image of a function und...
12gcd5e1 41962 The gcd of 12 and 5 is 1. ...
60gcd6e6 41963 The gcd of 60 and 6 is 6. ...
60gcd7e1 41964 The gcd of 60 and 7 is 1. ...
420gcd8e4 41965 The gcd of 420 and 8 is 4....
lcmeprodgcdi 41966 Calculate the least common...
12lcm5e60 41967 The lcm of 12 and 5 is 60....
60lcm6e60 41968 The lcm of 60 and 6 is 60....
60lcm7e420 41969 The lcm of 60 and 7 is 420...
420lcm8e840 41970 The lcm of 420 and 8 is 84...
lcmfunnnd 41971 Useful equation to calcula...
lcm1un 41972 Least common multiple of n...
lcm2un 41973 Least common multiple of n...
lcm3un 41974 Least common multiple of n...
lcm4un 41975 Least common multiple of n...
lcm5un 41976 Least common multiple of n...
lcm6un 41977 Least common multiple of n...
lcm7un 41978 Least common multiple of n...
lcm8un 41979 Least common multiple of n...
3factsumint1 41980 Move constants out of inte...
3factsumint2 41981 Move constants out of inte...
3factsumint3 41982 Move constants out of inte...
3factsumint4 41983 Move constants out of inte...
3factsumint 41984 Helpful equation for lcm i...
resopunitintvd 41985 Restrict continuous functi...
resclunitintvd 41986 Restrict continuous functi...
resdvopclptsd 41987 Restrict derivative on uni...
lcmineqlem1 41988 Part of lcm inequality lem...
lcmineqlem2 41989 Part of lcm inequality lem...
lcmineqlem3 41990 Part of lcm inequality lem...
lcmineqlem4 41991 Part of lcm inequality lem...
lcmineqlem5 41992 Technical lemma for recipr...
lcmineqlem6 41993 Part of lcm inequality lem...
lcmineqlem7 41994 Derivative of 1-x for chai...
lcmineqlem8 41995 Derivative of (1-x)^(N-M)....
lcmineqlem9 41996 (1-x)^(N-M) is continuous....
lcmineqlem10 41997 Induction step of ~ lcmine...
lcmineqlem11 41998 Induction step, continuati...
lcmineqlem12 41999 Base case for induction. ...
lcmineqlem13 42000 Induction proof for lcm in...
lcmineqlem14 42001 Technical lemma for inequa...
lcmineqlem15 42002 F times the least common m...
lcmineqlem16 42003 Technical divisibility lem...
lcmineqlem17 42004 Inequality of 2^{2n}. (Co...
lcmineqlem18 42005 Technical lemma to shift f...
lcmineqlem19 42006 Dividing implies inequalit...
lcmineqlem20 42007 Inequality for lcm lemma. ...
lcmineqlem21 42008 The lcm inequality lemma w...
lcmineqlem22 42009 The lcm inequality lemma w...
lcmineqlem23 42010 Penultimate step to the lc...
lcmineqlem 42011 The least common multiple ...
3exp7 42012 3 to the power of 7 equals...
3lexlogpow5ineq1 42013 First inequality in inequa...
3lexlogpow5ineq2 42014 Second inequality in inequ...
3lexlogpow5ineq4 42015 Sharper logarithm inequali...
3lexlogpow5ineq3 42016 Combined inequality chain ...
3lexlogpow2ineq1 42017 Result for bound in AKS in...
3lexlogpow2ineq2 42018 Result for bound in AKS in...
3lexlogpow5ineq5 42019 Result for bound in AKS in...
intlewftc 42020 Inequality inference by in...
aks4d1lem1 42021 Technical lemma to reduce ...
aks4d1p1p1 42022 Exponential law for finite...
dvrelog2 42023 The derivative of the loga...
dvrelog3 42024 The derivative of the loga...
dvrelog2b 42025 Derivative of the binary l...
0nonelalab 42026 Technical lemma for open i...
dvrelogpow2b 42027 Derivative of the power of...
aks4d1p1p3 42028 Bound of a ceiling of the ...
aks4d1p1p2 42029 Rewrite ` A ` in more suit...
aks4d1p1p4 42030 Technical step for inequal...
dvle2 42031 Collapsed ~ dvle . (Contr...
aks4d1p1p6 42032 Inequality lift to differe...
aks4d1p1p7 42033 Bound of intermediary of i...
aks4d1p1p5 42034 Show inequality for existe...
aks4d1p1 42035 Show inequality for existe...
aks4d1p2 42036 Technical lemma for existe...
aks4d1p3 42037 There exists a small enoug...
aks4d1p4 42038 There exists a small enoug...
aks4d1p5 42039 Show that ` N ` and ` R ` ...
aks4d1p6 42040 The maximal prime power ex...
aks4d1p7d1 42041 Technical step in AKS lemm...
aks4d1p7 42042 Technical step in AKS lemm...
aks4d1p8d1 42043 If a prime divides one num...
aks4d1p8d2 42044 Any prime power dividing a...
aks4d1p8d3 42045 The remainder of a divisio...
aks4d1p8 42046 Show that ` N ` and ` R ` ...
aks4d1p9 42047 Show that the order is bou...
aks4d1 42048 Lemma 4.1 from ~ https://w...
fldhmf1 42049 A field homomorphism is in...
isprimroot 42052 The value of a primitive r...
isprimroot2 42053 Alternative way of creatin...
mndmolinv 42054 An element of a monoid tha...
linvh 42055 If an element has a unique...
primrootsunit1 42056 Primitive roots have left ...
primrootsunit 42057 Primitive roots have left ...
primrootscoprmpow 42058 Coprime powers of primitiv...
posbezout 42059 Bezout's identity restrict...
primrootscoprf 42060 Coprime powers of primitiv...
primrootscoprbij 42061 A bijection between coprim...
primrootscoprbij2 42062 A bijection between coprim...
remexz 42063 Division with rest. (Cont...
primrootlekpowne0 42064 There is no smaller power ...
primrootspoweq0 42065 The power of a ` R ` -th p...
aks6d1c1p1 42066 Definition of the introspe...
aks6d1c1p1rcl 42067 Reverse closure of the int...
aks6d1c1p2 42068 ` P ` and linear factors a...
aks6d1c1p3 42069 In a field with a Frobeniu...
aks6d1c1p4 42070 The product of polynomials...
aks6d1c1p5 42071 The product of exponents i...
aks6d1c1p7 42072 ` X ` is introspective to ...
aks6d1c1p6 42073 If a polynomials ` F ` is ...
aks6d1c1p8 42074 If a number ` E ` is intro...
aks6d1c1 42075 Claim 1 of Theorem 6.1 ~ h...
evl1gprodd 42076 Polynomial evaluation buil...
aks6d1c2p1 42077 In the AKS-theorem the sub...
aks6d1c2p2 42078 Injective condition for co...
hashscontpowcl 42079 Closure of E for ~ https:/...
hashscontpow1 42080 Helper lemma for to prove ...
hashscontpow 42081 If a set contains all ` N ...
aks6d1c3 42082 Claim 3 of Theorem 6.1 of ...
aks6d1c4 42083 Claim 4 of Theorem 6.1 of ...
aks6d1c1rh 42084 Claim 1 of AKS primality p...
aks6d1c2lem3 42085 Lemma for ~ aks6d1c2 to si...
aks6d1c2lem4 42086 Claim 2 of Theorem 6.1 AKS...
hashnexinj 42087 If the number of elements ...
hashnexinjle 42088 If the number of elements ...
aks6d1c2 42089 Claim 2 of Theorem 6.1 of ...
rspcsbnea 42090 Special case related to ~ ...
idomnnzpownz 42091 A non-zero power in an int...
idomnnzgmulnz 42092 A finite product of non-ze...
ringexp0nn 42093 Zero to the power of a pos...
aks6d1c5lem0 42094 Lemma for Claim 5 of Theor...
aks6d1c5lem1 42095 Lemma for claim 5, evaluat...
aks6d1c5lem3 42096 Lemma for Claim 5, polynom...
aks6d1c5lem2 42097 Lemma for Claim 5, contrad...
aks6d1c5 42098 Claim 5 of Theorem 6.1 ~ h...
deg1gprod 42099 Degree multiplication is a...
deg1pow 42100 Exact degree of a power of...
5bc2eq10 42101 The value of 5 choose 2. ...
facp2 42102 The factorial of a success...
2np3bcnp1 42103 Part of induction step for...
2ap1caineq 42104 Inequality for Theorem 6.6...
sticksstones1 42105 Different strictly monoton...
sticksstones2 42106 The range function on stri...
sticksstones3 42107 The range function on stri...
sticksstones4 42108 Equinumerosity lemma for s...
sticksstones5 42109 Count the number of strict...
sticksstones6 42110 Function induces an order ...
sticksstones7 42111 Closure property of sticks...
sticksstones8 42112 Establish mapping between ...
sticksstones9 42113 Establish mapping between ...
sticksstones10 42114 Establish mapping between ...
sticksstones11 42115 Establish bijective mappin...
sticksstones12a 42116 Establish bijective mappin...
sticksstones12 42117 Establish bijective mappin...
sticksstones13 42118 Establish bijective mappin...
sticksstones14 42119 Sticks and stones with def...
sticksstones15 42120 Sticks and stones with alm...
sticksstones16 42121 Sticks and stones with col...
sticksstones17 42122 Extend sticks and stones t...
sticksstones18 42123 Extend sticks and stones t...
sticksstones19 42124 Extend sticks and stones t...
sticksstones20 42125 Lift sticks and stones to ...
sticksstones21 42126 Lift sticks and stones to ...
sticksstones22 42127 Non-exhaustive sticks and ...
sticksstones23 42128 Non-exhaustive sticks and ...
aks6d1c6lem1 42129 Lemma for claim 6, deduce ...
aks6d1c6lem2 42130 Every primitive root is ro...
aks6d1c6lem3 42131 Claim 6 of Theorem 6.1 of ...
aks6d1c6lem4 42132 Claim 6 of Theorem 6.1 of ...
aks6d1c6isolem1 42133 Lemma to construct the map...
aks6d1c6isolem2 42134 Lemma to construct the gro...
aks6d1c6isolem3 42135 The preimage of a map send...
aks6d1c6lem5 42136 Eliminate the size hypothe...
bcled 42137 Inequality for binomial co...
bcle2d 42138 Inequality for binomial co...
aks6d1c7lem1 42139 The last set of inequaliti...
aks6d1c7lem2 42140 Contradiction to Claim 2 a...
aks6d1c7lem3 42141 Remove lots of hypotheses ...
aks6d1c7lem4 42142 In the AKS algorithm there...
aks6d1c7 42143 ` N ` is a prime power if ...
rhmqusspan 42144 Ring homomorphism out of a...
aks5lem1 42145 Section 5 of ~ https://www...
aks5lem2 42146 Lemma for section 5 ~ http...
ply1asclzrhval 42147 Transfer results from alge...
aks5lem3a 42148 Lemma for AKS section 5. ...
aks5lem4a 42149 Lemma for AKS section 5, r...
aks5lem5a 42150 Lemma for AKS, section 5, ...
aks5lem6 42151 Connect results of section...
indstrd 42152 Strong induction, deductio...
grpods 42153 Relate sums of elements of...
unitscyglem1 42154 Lemma for unitscyg. (Cont...
unitscyglem2 42155 Lemma for unitscyg. (Cont...
unitscyglem3 42156 Lemma for unitscyg. (Cont...
unitscyglem4 42157 Lemma for unitscyg (Contri...
unitscyglem5 42158 Lemma for unitscyg (Contri...
aks5lem7 42159 Lemma for aks5. We clean ...
aks5lem8 42160 Lemma for aks5. Clean up ...
exfinfldd 42162 For any prime ` P ` and an...
aks5 42163 The AKS Primality test, gi...
metakunt1 42164 A is an endomapping. (Con...
metakunt2 42165 A is an endomapping. (Con...
metakunt3 42166 Value of A. (Contributed b...
metakunt4 42167 Value of A. (Contributed b...
metakunt5 42168 C is the left inverse for ...
metakunt6 42169 C is the left inverse for ...
metakunt7 42170 C is the left inverse for ...
metakunt8 42171 C is the left inverse for ...
metakunt9 42172 C is the left inverse for ...
metakunt10 42173 C is the right inverse for...
metakunt11 42174 C is the right inverse for...
metakunt12 42175 C is the right inverse for...
metakunt13 42176 C is the right inverse for...
metakunt14 42177 A is a primitive permutati...
metakunt15 42178 Construction of another pe...
metakunt16 42179 Construction of another pe...
metakunt17 42180 The union of three disjoin...
metakunt18 42181 Disjoint domains and codom...
metakunt19 42182 Domains on restrictions of...
metakunt20 42183 Show that B coincides on t...
metakunt21 42184 Show that B coincides on t...
metakunt22 42185 Show that B coincides on t...
metakunt23 42186 B coincides on the union o...
metakunt24 42187 Technical condition such t...
metakunt25 42188 B is a permutation. (Cont...
metakunt26 42189 Construction of one soluti...
metakunt27 42190 Construction of one soluti...
metakunt28 42191 Construction of one soluti...
metakunt29 42192 Construction of one soluti...
metakunt30 42193 Construction of one soluti...
metakunt31 42194 Construction of one soluti...
metakunt32 42195 Construction of one soluti...
metakunt33 42196 Construction of one soluti...
metakunt34 42197 ` D ` is a permutation. (...
fac2xp3 42198 Factorial of 2x+3, sublemm...
prodsplit 42199 Product split into two fac...
2xp3dxp2ge1d 42200 2x+3 is greater than or eq...
factwoffsmonot 42201 A factorial with offset is...
intnanrt 42202 Introduction of conjunct i...
ioin9i8 42203 Miscellaneous inference cr...
jaodd 42204 Double deduction form of ~...
syl3an12 42205 A double syllogism inferen...
sbtd 42206 A true statement is true u...
sbor2 42207 One direction of ~ sbor , ...
sbalexi 42208 Inference form of ~ sbalex...
19.9dev 42209 ~ 19.9d in the case of an ...
3rspcedvd 42210 Triple application of ~ rs...
sn-axrep5v 42211 A condensed form of ~ axre...
sn-axprlem3 42212 ~ axprlem3 using only Tars...
sn-exelALT 42213 Alternate proof of ~ exel ...
ss2ab1 42214 Class abstractions in a su...
ssabdv 42215 Deduction of abstraction s...
sn-iotalem 42216 An unused lemma showing th...
sn-iotalemcor 42217 Corollary of ~ sn-iotalem ...
abbi1sn 42218 Originally part of ~ uniab...
brif2 42219 Move a relation inside and...
brif12 42220 Move a relation inside and...
pssexg 42221 The proper subset of a set...
pssn0 42222 A proper superset is nonem...
psspwb 42223 Classes are proper subclas...
xppss12 42224 Proper subset theorem for ...
elpwbi 42225 Membership in a power set,...
imaopab 42226 The image of a class of or...
fnsnbt 42227 A function's domain is a s...
fnimasnd 42228 The image of a function by...
eqresfnbd 42229 Property of being the rest...
f1o2d2 42230 Sufficient condition for a...
fmpocos 42231 Composition of two functio...
ovmpogad 42232 Value of an operation give...
ofun 42233 A function operation of un...
dfqs2 42234 Alternate definition of qu...
dfqs3 42235 Alternate definition of qu...
qseq12d 42236 Equality theorem for quoti...
qsalrel 42237 The quotient set is equal ...
elmapssresd 42238 A restricted mapping is a ...
supinf 42239 The supremum is the infimu...
mapcod 42240 Compose two mappings. (Co...
fisdomnn 42241 A finite set is dominated ...
ltex 42242 The less-than relation is ...
leex 42243 The less-than-or-equal-to ...
subex 42244 The subtraction operation ...
absex 42245 The absolute value functio...
cjex 42246 The conjugate function is ...
fzosumm1 42247 Separate out the last term...
ccatcan2d 42248 Cancellation law for conca...
c0exALT 42249 Alternate proof of ~ c0ex ...
0cnALT3 42250 Alternate proof of ~ 0cn u...
elre0re 42251 Specialized version of ~ 0...
1t1e1ALT 42252 Alternate proof of ~ 1t1e1...
lttrii 42253 'Less than' is transitive....
remulcan2d 42254 ~ mulcan2d for real number...
readdridaddlidd 42255 Given some real number ` B...
sn-1ne2 42256 A proof of ~ 1ne2 without ...
nnn1suc 42257 A positive integer that is...
nnadd1com 42258 Addition with 1 is commuta...
nnaddcom 42259 Addition is commutative fo...
nnaddcomli 42260 Version of ~ addcomli for ...
nnadddir 42261 Right-distributivity for n...
nnmul1com 42262 Multiplication with 1 is c...
nnmulcom 42263 Multiplication is commutat...
readdrcl2d 42264 Reverse closure for additi...
mvrrsubd 42265 Move a subtraction in the ...
laddrotrd 42266 Rotate the variables right...
raddswap12d 42267 Swap the first two variabl...
lsubrotld 42268 Rotate the variables left ...
rsubrotld 42269 Rotate the variables left ...
lsubswap23d 42270 Swap the second and third ...
addsubeq4com 42271 Relation between sums and ...
sqsumi 42272 A sum squared. (Contribut...
negn0nposznnd 42273 Lemma for ~ dffltz . (Con...
sqmid3api 42274 Value of the square of the...
decaddcom 42275 Commute ones place in addi...
sqn5i 42276 The square of a number end...
sqn5ii 42277 The square of a number end...
decpmulnc 42278 Partial products algorithm...
decpmul 42279 Partial products algorithm...
sqdeccom12 42280 The square of a number in ...
sq3deccom12 42281 Variant of ~ sqdeccom12 wi...
4t5e20 42282 4 times 5 equals 20. (Con...
sq4 42283 The square of 4 is 16. (C...
sq5 42284 The square of 5 is 25. (C...
sq6 42285 The square of 6 is 36. (C...
sq7 42286 The square of 7 is 49. (C...
sq8 42287 The square of 8 is 64. (C...
sq9 42288 The square of 9 is 81. (C...
4rp 42289 4 is a positive real. (Co...
5rp 42290 5 is a positive real. (Co...
6rp 42291 6 is a positive real. (Co...
7rp 42292 7 is a positive real. (Co...
8rp 42293 8 is a positive real. (Co...
9rp 42294 9 is a positive real. (Co...
235t711 42295 Calculate a product by lon...
ex-decpmul 42296 Example usage of ~ decpmul...
eluzp1 42297 Membership in a successor ...
sn-eluzp1l 42298 Shorter proof of ~ eluzp1l...
fz1sumconst 42299 The sum of ` N ` constant ...
fz1sump1 42300 Add one more term to a sum...
oddnumth 42301 The Odd Number Theorem. T...
nicomachus 42302 Nicomachus's Theorem. The...
sumcubes 42303 The sum of the first ` N `...
pine0 42304 ` _pi ` is nonzero. (Cont...
ine1 42305 ` _i ` is not 1. (Contrib...
0tie0 42306 0 times ` _i ` equals 0. ...
it1ei 42307 ` _i ` times 1 equals ` _i...
1tiei 42308 1 times ` _i ` equals ` _i...
itrere 42309 ` _i ` times a real is rea...
retire 42310 A real times ` _i ` is rea...
oexpreposd 42311 Lemma for ~ dffltz . TODO...
explt1d 42312 A nonnegative real number ...
expeq1d 42313 A nonnegative real number ...
expeqidd 42314 A nonnegative real number ...
exp11d 42315 ~ exp11nnd for nonzero int...
0dvds0 42316 0 divides 0. (Contributed...
absdvdsabsb 42317 Divisibility is invariant ...
gcdnn0id 42318 The ` gcd ` of a nonnegati...
gcdle1d 42319 The greatest common diviso...
gcdle2d 42320 The greatest common diviso...
dvdsexpad 42321 Deduction associated with ...
dvdsexpnn 42322 ~ dvdssqlem generalized to...
dvdsexpnn0 42323 ~ dvdsexpnn generalized to...
dvdsexpb 42324 ~ dvdssq generalized to po...
posqsqznn 42325 When a positive rational s...
zdivgd 42326 Two ways to express " ` N ...
efne0d 42327 The exponential of a compl...
efsubd 42328 Difference of exponents la...
ef11d 42329 General condition for the ...
logccne0d 42330 The logarithm isn't 0 if i...
cxp112d 42331 General condition for comp...
cxp111d 42332 General condition for comp...
cxpi11d 42333 ` _i ` to the powers of ` ...
logne0d 42334 Deduction form of ~ logne0...
rxp112d 42335 Real exponentiation is one...
log11d 42336 The natural logarithm is o...
rplog11d 42337 The natural logarithm is o...
rxp11d 42338 Real exponentiation is one...
tanhalfpim 42339 The tangent of ` _pi / 2 `...
tan3rdpi 42340 The tangent of ` _pi / 3 `...
asin1half 42341 The arcsine of ` 1 / 2 ` i...
acos1half 42342 The arccosine of ` 1 / 2 `...
resubval 42345 Value of real subtraction,...
renegeulemv 42346 Lemma for ~ renegeu and si...
renegeulem 42347 Lemma for ~ renegeu and si...
renegeu 42348 Existential uniqueness of ...
rernegcl 42349 Closure law for negative r...
renegadd 42350 Relationship between real ...
renegid 42351 Addition of a real number ...
reneg0addlid 42352 Negative zero is a left ad...
resubeulem1 42353 Lemma for ~ resubeu . A v...
resubeulem2 42354 Lemma for ~ resubeu . A v...
resubeu 42355 Existential uniqueness of ...
rersubcl 42356 Closure for real subtracti...
resubadd 42357 Relation between real subt...
resubaddd 42358 Relationship between subtr...
resubf 42359 Real subtraction is an ope...
repncan2 42360 Addition and subtraction o...
repncan3 42361 Addition and subtraction o...
readdsub 42362 Law for addition and subtr...
reladdrsub 42363 Move LHS of a sum into RHS...
reltsub1 42364 Subtraction from both side...
reltsubadd2 42365 'Less than' relationship b...
resubcan2 42366 Cancellation law for real ...
resubsub4 42367 Law for double subtraction...
rennncan2 42368 Cancellation law for real ...
renpncan3 42369 Cancellation law for real ...
repnpcan 42370 Cancellation law for addit...
reppncan 42371 Cancellation law for mixed...
resubidaddlidlem 42372 Lemma for ~ resubidaddlid ...
resubidaddlid 42373 Any real number subtracted...
resubdi 42374 Distribution of multiplica...
re1m1e0m0 42375 Equality of two left-addit...
sn-00idlem1 42376 Lemma for ~ sn-00id . (Co...
sn-00idlem2 42377 Lemma for ~ sn-00id . (Co...
sn-00idlem3 42378 Lemma for ~ sn-00id . (Co...
sn-00id 42379 ~ 00id proven without ~ ax...
re0m0e0 42380 Real number version of ~ 0...
readdlid 42381 Real number version of ~ a...
sn-addlid 42382 ~ addlid without ~ ax-mulc...
remul02 42383 Real number version of ~ m...
sn-0ne2 42384 ~ 0ne2 without ~ ax-mulcom...
remul01 42385 Real number version of ~ m...
resubid 42386 Subtraction of a real numb...
readdrid 42387 Real number version of ~ a...
resubid1 42388 Real number version of ~ s...
renegneg 42389 A real number is equal to ...
readdcan2 42390 Commuted version of ~ read...
renegid2 42391 Commuted version of ~ rene...
remulneg2d 42392 Product with negative is n...
sn-it0e0 42393 Proof of ~ it0e0 without ~...
sn-negex12 42394 A combination of ~ cnegex ...
sn-negex 42395 Proof of ~ cnegex without ...
sn-negex2 42396 Proof of ~ cnegex2 without...
sn-addcand 42397 ~ addcand without ~ ax-mul...
sn-addrid 42398 ~ addrid without ~ ax-mulc...
sn-addcan2d 42399 ~ addcan2d without ~ ax-mu...
reixi 42400 ~ ixi without ~ ax-mulcom ...
rei4 42401 ~ i4 without ~ ax-mulcom ....
sn-addid0 42402 A number that sums to itse...
sn-mul01 42403 ~ mul01 without ~ ax-mulco...
sn-subeu 42404 ~ negeu without ~ ax-mulco...
sn-subcl 42405 ~ subcl without ~ ax-mulco...
sn-subf 42406 ~ subf without ~ ax-mulcom...
resubeqsub 42407 Equivalence between real s...
subresre 42408 Subtraction restricted to ...
addinvcom 42409 A number commutes with its...
remulinvcom 42410 A left multiplicative inve...
remullid 42411 Commuted version of ~ ax-1...
sn-1ticom 42412 Lemma for ~ sn-mullid and ...
sn-mullid 42413 ~ mullid without ~ ax-mulc...
sn-it1ei 42414 ~ it1ei without ~ ax-mulco...
ipiiie0 42415 The multiplicative inverse...
remulcand 42416 Commuted version of ~ remu...
sn-0tie0 42417 Lemma for ~ sn-mul02 . Co...
sn-mul02 42418 ~ mul02 without ~ ax-mulco...
sn-ltaddpos 42419 ~ ltaddpos without ~ ax-mu...
sn-ltaddneg 42420 ~ ltaddneg without ~ ax-mu...
reposdif 42421 Comparison of two numbers ...
relt0neg1 42422 Comparison of a real and i...
relt0neg2 42423 Comparison of a real and i...
sn-addlt0d 42424 The sum of negative number...
sn-addgt0d 42425 The sum of positive number...
sn-nnne0 42426 ~ nnne0 without ~ ax-mulco...
reelznn0nn 42427 ~ elznn0nn restated using ...
nn0addcom 42428 Addition is commutative fo...
zaddcomlem 42429 Lemma for ~ zaddcom . (Co...
zaddcom 42430 Addition is commutative fo...
renegmulnnass 42431 Move multiplication by a n...
nn0mulcom 42432 Multiplication is commutat...
zmulcomlem 42433 Lemma for ~ zmulcom . (Co...
zmulcom 42434 Multiplication is commutat...
mulgt0con1dlem 42435 Lemma for ~ mulgt0con1d . ...
mulgt0con1d 42436 Counterpart to ~ mulgt0con...
mulgt0con2d 42437 Lemma for ~ mulgt0b2d and ...
mulgt0b2d 42438 Biconditional, deductive f...
sn-ltmul2d 42439 ~ ltmul2d without ~ ax-mul...
sn-ltmulgt11d 42440 ~ ltmulgt11d without ~ ax-...
sn-0lt1 42441 ~ 0lt1 without ~ ax-mulcom...
sn-ltp1 42442 ~ ltp1 without ~ ax-mulcom...
sn-mulgt1d 42443 ~ mulgt1d without ~ ax-mul...
reneg1lt0 42444 Lemma for ~ sn-inelr . (C...
sn-inelr 42445 ~ inelr without ~ ax-mulco...
sn-itrere 42446 ` _i ` times a real is rea...
sn-retire 42447 Commuted version of ~ sn-i...
cnreeu 42448 The reals in the expressio...
sn-sup2 42449 ~ sup2 with exactly the sa...
sn-sup3d 42450 ~ sup3 without ~ ax-mulcom...
sn-suprcld 42451 ~ suprcld without ~ ax-mul...
sn-suprubd 42452 ~ suprubd without ~ ax-mul...
nelsubginvcld 42453 The inverse of a non-subgr...
nelsubgcld 42454 A non-subgroup-member plus...
nelsubgsubcld 42455 A non-subgroup-member minu...
rnasclg 42456 The set of injected scalar...
frlmfielbas 42457 The vectors of a finite fr...
frlmfzwrd 42458 A vector of a module with ...
frlmfzowrd 42459 A vector of a module with ...
frlmfzolen 42460 The dimension of a vector ...
frlmfzowrdb 42461 The vectors of a module wi...
frlmfzoccat 42462 The concatenation of two v...
frlmvscadiccat 42463 Scalar multiplication dist...
grpasscan2d 42464 An associative cancellatio...
grpcominv1 42465 If two elements commute, t...
grpcominv2 42466 If two elements commute, t...
finsubmsubg 42467 A submonoid of a finite gr...
opprmndb 42468 A class is a monoid if and...
opprgrpb 42469 A class is a group if and ...
opprablb 42470 A class is an Abelian grou...
imacrhmcl 42471 The image of a commutative...
rimrcl1 42472 Reverse closure of a ring ...
rimrcl2 42473 Reverse closure of a ring ...
rimcnv 42474 The converse of a ring iso...
rimco 42475 The composition of ring is...
ricsym 42476 Ring isomorphism is symmet...
rictr 42477 Ring isomorphism is transi...
riccrng1 42478 Ring isomorphism preserves...
riccrng 42479 A ring is commutative if a...
domnexpgn0cl 42480 In a domain, a (nonnegativ...
drnginvrn0d 42481 A multiplicative inverse i...
drngmullcan 42482 Cancellation of a nonzero ...
drngmulrcan 42483 Cancellation of a nonzero ...
drnginvmuld 42484 Inverse of a nonzero produ...
ricdrng1 42485 A ring isomorphism maps a ...
ricdrng 42486 A ring is a division ring ...
ricfld 42487 A ring is a field if and o...
asclf1 42488 Two ways of saying the sca...
abvexp 42489 Move exponentiation in and...
fimgmcyclem 42490 Lemma for ~ fimgmcyc . (C...
fimgmcyc 42491 Version of ~ odcl2 for fin...
fidomncyc 42492 Version of ~ odcl2 for mul...
fiabv 42493 In a finite domain (a fini...
lvecgrp 42494 A vector space is a group....
lvecring 42495 The scalar component of a ...
frlm0vald 42496 All coordinates of the zer...
frlmsnic 42497 Given a free module with a...
uvccl 42498 A unit vector is a vector....
uvcn0 42499 A unit vector is nonzero. ...
pwselbasr 42500 The reverse direction of ~...
pwsgprod 42501 Finite products in a power...
psrmnd 42502 The ring of power series i...
psrbagres 42503 Restrict a bag of variable...
mplcrngd 42504 The polynomial ring is a c...
mplsubrgcl 42505 An element of a polynomial...
mhmcopsr 42506 The composition of a monoi...
mhmcoaddpsr 42507 Show that the ring homomor...
rhmcomulpsr 42508 Show that the ring homomor...
rhmpsr 42509 Provide a ring homomorphis...
rhmpsr1 42510 Provide a ring homomorphis...
mplascl0 42511 The zero scalar as a polyn...
mplascl1 42512 The one scalar as a polyno...
mplmapghm 42513 The function ` H ` mapping...
evl0 42514 The zero polynomial evalua...
evlscl 42515 A polynomial over the ring...
evlsval3 42516 Give a formula for the pol...
evlsvval 42517 Give a formula for the eva...
evlsvvvallem 42518 Lemma for ~ evlsvvval akin...
evlsvvvallem2 42519 Lemma for theorems using ~...
evlsvvval 42520 Give a formula for the eva...
evlsscaval 42521 Polynomial evaluation buil...
evlsvarval 42522 Polynomial evaluation buil...
evlsbagval 42523 Polynomial evaluation buil...
evlsexpval 42524 Polynomial evaluation buil...
evlsaddval 42525 Polynomial evaluation buil...
evlsmulval 42526 Polynomial evaluation buil...
evlsmaprhm 42527 The function ` F ` mapping...
evlsevl 42528 Evaluation in a subring is...
evlcl 42529 A polynomial over the ring...
evlvvval 42530 Give a formula for the eva...
evlvvvallem 42531 Lemma for theorems using ~...
evladdval 42532 Polynomial evaluation buil...
evlmulval 42533 Polynomial evaluation buil...
selvcllem1 42534 ` T ` is an associative al...
selvcllem2 42535 ` D ` is a ring homomorphi...
selvcllem3 42536 The third argument passed ...
selvcllemh 42537 Apply the third argument (...
selvcllem4 42538 The fourth argument passed...
selvcllem5 42539 The fifth argument passed ...
selvcl 42540 Closure of the "variable s...
selvval2 42541 Value of the "variable sel...
selvvvval 42542 Recover the original polyn...
evlselvlem 42543 Lemma for ~ evlselv . Use...
evlselv 42544 Evaluating a selection of ...
selvadd 42545 The "variable selection" f...
selvmul 42546 The "variable selection" f...
fsuppind 42547 Induction on functions ` F...
fsuppssindlem1 42548 Lemma for ~ fsuppssind . ...
fsuppssindlem2 42549 Lemma for ~ fsuppssind . ...
fsuppssind 42550 Induction on functions ` F...
mhpind 42551 The homogeneous polynomial...
evlsmhpvvval 42552 Give a formula for the eva...
mhphflem 42553 Lemma for ~ mhphf . Add s...
mhphf 42554 A homogeneous polynomial d...
mhphf2 42555 A homogeneous polynomial d...
mhphf3 42556 A homogeneous polynomial d...
mhphf4 42557 A homogeneous polynomial d...
prjspval 42560 Value of the projective sp...
prjsprel 42561 Utility theorem regarding ...
prjspertr 42562 The relation in ` PrjSp ` ...
prjsperref 42563 The relation in ` PrjSp ` ...
prjspersym 42564 The relation in ` PrjSp ` ...
prjsper 42565 The relation used to defin...
prjspreln0 42566 Two nonzero vectors are eq...
prjspvs 42567 A nonzero multiple of a ve...
prjsprellsp 42568 Two vectors are equivalent...
prjspeclsp 42569 The vectors equivalent to ...
prjspval2 42570 Alternate definition of pr...
prjspnval 42573 Value of the n-dimensional...
prjspnerlem 42574 A lemma showing that the e...
prjspnval2 42575 Value of the n-dimensional...
prjspner 42576 The relation used to defin...
prjspnvs 42577 A nonzero multiple of a ve...
prjspnssbas 42578 A projective point spans a...
prjspnn0 42579 A projective point is none...
0prjspnlem 42580 Lemma for ~ 0prjspn . The...
prjspnfv01 42581 Any vector is equivalent t...
prjspner01 42582 Any vector is equivalent t...
prjspner1 42583 Two vectors whose zeroth c...
0prjspnrel 42584 In the zero-dimensional pr...
0prjspn 42585 A zero-dimensional project...
prjcrvfval 42588 Value of the projective cu...
prjcrvval 42589 Value of the projective cu...
prjcrv0 42590 The "curve" (zero set) cor...
dffltz 42591 Fermat's Last Theorem (FLT...
fltmul 42592 A counterexample to FLT st...
fltdiv 42593 A counterexample to FLT st...
flt0 42594 A counterexample for FLT d...
fltdvdsabdvdsc 42595 Any factor of both ` A ` a...
fltabcoprmex 42596 A counterexample to FLT im...
fltaccoprm 42597 A counterexample to FLT wi...
fltbccoprm 42598 A counterexample to FLT wi...
fltabcoprm 42599 A counterexample to FLT wi...
infdesc 42600 Infinite descent. The hyp...
fltne 42601 If a counterexample to FLT...
flt4lem 42602 Raising a number to the fo...
flt4lem1 42603 Satisfy the antecedent use...
flt4lem2 42604 If ` A ` is even, ` B ` is...
flt4lem3 42605 Equivalent to ~ pythagtrip...
flt4lem4 42606 If the product of two copr...
flt4lem5 42607 In the context of the lemm...
flt4lem5elem 42608 Version of ~ fltaccoprm an...
flt4lem5a 42609 Part 1 of Equation 1 of ...
flt4lem5b 42610 Part 2 of Equation 1 of ...
flt4lem5c 42611 Part 2 of Equation 2 of ...
flt4lem5d 42612 Part 3 of Equation 2 of ...
flt4lem5e 42613 Satisfy the hypotheses of ...
flt4lem5f 42614 Final equation of ~...
flt4lem6 42615 Remove shared factors in a...
flt4lem7 42616 Convert ~ flt4lem5f into a...
nna4b4nsq 42617 Strengthening of Fermat's ...
fltltc 42618 ` ( C ^ N ) ` is the large...
fltnltalem 42619 Lemma for ~ fltnlta . A l...
fltnlta 42620 In a Fermat counterexample...
iddii 42621 Version of ~ a1ii with the...
bicomdALT 42622 Alternate proof of ~ bicom...
alan 42623 Alias for ~ 19.26 for easi...
exor 42624 Alias for ~ 19.43 for easi...
rexor 42625 Alias for ~ r19.43 for eas...
ruvALT 42626 Alternate proof of ~ ruv w...
sn-wcdeq 42627 Alternative to ~ wcdeq and...
sq45 42628 45 squared is 2025. (Cont...
sum9cubes 42629 The sum of the first nine ...
sn-isghm 42630 Longer proof of ~ isghm , ...
aprilfools2025 42631 An abuse of notation. (Co...
nfa1w 42632 Replace ~ ax-10 in ~ nfa1 ...
eu6w 42633 Replace ~ ax-10 , ~ ax-12 ...
abbibw 42634 Replace ~ ax-10 , ~ ax-11 ...
absnw 42635 Replace ~ ax-10 , ~ ax-11 ...
euabsn2w 42636 Replace ~ ax-10 , ~ ax-11 ...
sn-tz6.12-2 42637 ~ tz6.12-2 without ~ ax-10...
cu3addd 42638 Cube of sum of three numbe...
sqnegd 42639 The square of the negative...
negexpidd 42640 The sum of a real number t...
rexlimdv3d 42641 An extended version of ~ r...
3cubeslem1 42642 Lemma for ~ 3cubes . (Con...
3cubeslem2 42643 Lemma for ~ 3cubes . Used...
3cubeslem3l 42644 Lemma for ~ 3cubes . (Con...
3cubeslem3r 42645 Lemma for ~ 3cubes . (Con...
3cubeslem3 42646 Lemma for ~ 3cubes . (Con...
3cubeslem4 42647 Lemma for ~ 3cubes . This...
3cubes 42648 Every rational number is a...
rntrclfvOAI 42649 The range of the transitiv...
moxfr 42650 Transfer at-most-one betwe...
imaiinfv 42651 Indexed intersection of an...
elrfi 42652 Elementhood in a set of re...
elrfirn 42653 Elementhood in a set of re...
elrfirn2 42654 Elementhood in a set of re...
cmpfiiin 42655 In a compact topology, a s...
ismrcd1 42656 Any function from the subs...
ismrcd2 42657 Second half of ~ ismrcd1 ....
istopclsd 42658 A closure function which s...
ismrc 42659 A function is a Moore clos...
isnacs 42662 Expand definition of Noeth...
nacsfg 42663 In a Noetherian-type closu...
isnacs2 42664 Express Noetherian-type cl...
mrefg2 42665 Slight variation on finite...
mrefg3 42666 Slight variation on finite...
nacsacs 42667 A closure system of Noethe...
isnacs3 42668 A choice-free order equiva...
incssnn0 42669 Transitivity induction of ...
nacsfix 42670 An increasing sequence of ...
constmap 42671 A constant (represented wi...
mapco2g 42672 Renaming indices in a tupl...
mapco2 42673 Post-composition (renaming...
mapfzcons 42674 Extending a one-based mapp...
mapfzcons1 42675 Recover prefix mapping fro...
mapfzcons1cl 42676 A nonempty mapping has a p...
mapfzcons2 42677 Recover added element from...
mptfcl 42678 Interpret range of a maps-...
mzpclval 42683 Substitution lemma for ` m...
elmzpcl 42684 Double substitution lemma ...
mzpclall 42685 The set of all functions w...
mzpcln0 42686 Corollary of ~ mzpclall : ...
mzpcl1 42687 Defining property 1 of a p...
mzpcl2 42688 Defining property 2 of a p...
mzpcl34 42689 Defining properties 3 and ...
mzpval 42690 Value of the ` mzPoly ` fu...
dmmzp 42691 ` mzPoly ` is defined for ...
mzpincl 42692 Polynomial closedness is a...
mzpconst 42693 Constant functions are pol...
mzpf 42694 A polynomial function is a...
mzpproj 42695 A projection function is p...
mzpadd 42696 The pointwise sum of two p...
mzpmul 42697 The pointwise product of t...
mzpconstmpt 42698 A constant function expres...
mzpaddmpt 42699 Sum of polynomial function...
mzpmulmpt 42700 Product of polynomial func...
mzpsubmpt 42701 The difference of two poly...
mzpnegmpt 42702 Negation of a polynomial f...
mzpexpmpt 42703 Raise a polynomial functio...
mzpindd 42704 "Structural" induction to ...
mzpmfp 42705 Relationship between multi...
mzpsubst 42706 Substituting polynomials f...
mzprename 42707 Simplified version of ~ mz...
mzpresrename 42708 A polynomial is a polynomi...
mzpcompact2lem 42709 Lemma for ~ mzpcompact2 . ...
mzpcompact2 42710 Polynomials are finitary o...
coeq0i 42711 ~ coeq0 but without explic...
fzsplit1nn0 42712 Split a finite 1-based set...
eldiophb 42715 Initial expression of Diop...
eldioph 42716 Condition for a set to be ...
diophrw 42717 Renaming and adding unused...
eldioph2lem1 42718 Lemma for ~ eldioph2 . Co...
eldioph2lem2 42719 Lemma for ~ eldioph2 . Co...
eldioph2 42720 Construct a Diophantine se...
eldioph2b 42721 While Diophantine sets wer...
eldiophelnn0 42722 Remove antecedent on ` B `...
eldioph3b 42723 Define Diophantine sets in...
eldioph3 42724 Inference version of ~ eld...
ellz1 42725 Membership in a lower set ...
lzunuz 42726 The union of a lower set o...
fz1eqin 42727 Express a one-based finite...
lzenom 42728 Lower integers are countab...
elmapresaunres2 42729 ~ fresaunres2 transposed t...
diophin 42730 If two sets are Diophantin...
diophun 42731 If two sets are Diophantin...
eldiophss 42732 Diophantine sets are sets ...
diophrex 42733 Projecting a Diophantine s...
eq0rabdioph 42734 This is the first of a num...
eqrabdioph 42735 Diophantine set builder fo...
0dioph 42736 The null set is Diophantin...
vdioph 42737 The "universal" set (as la...
anrabdioph 42738 Diophantine set builder fo...
orrabdioph 42739 Diophantine set builder fo...
3anrabdioph 42740 Diophantine set builder fo...
3orrabdioph 42741 Diophantine set builder fo...
2sbcrex 42742 Exchange an existential qu...
sbcrexgOLD 42743 Interchange class substitu...
2sbcrexOLD 42744 Exchange an existential qu...
sbc2rex 42745 Exchange a substitution wi...
sbc2rexgOLD 42746 Exchange a substitution wi...
sbc4rex 42747 Exchange a substitution wi...
sbc4rexgOLD 42748 Exchange a substitution wi...
sbcrot3 42749 Rotate a sequence of three...
sbcrot5 42750 Rotate a sequence of five ...
sbccomieg 42751 Commute two explicit subst...
rexrabdioph 42752 Diophantine set builder fo...
rexfrabdioph 42753 Diophantine set builder fo...
2rexfrabdioph 42754 Diophantine set builder fo...
3rexfrabdioph 42755 Diophantine set builder fo...
4rexfrabdioph 42756 Diophantine set builder fo...
6rexfrabdioph 42757 Diophantine set builder fo...
7rexfrabdioph 42758 Diophantine set builder fo...
rabdiophlem1 42759 Lemma for arithmetic dioph...
rabdiophlem2 42760 Lemma for arithmetic dioph...
elnn0rabdioph 42761 Diophantine set builder fo...
rexzrexnn0 42762 Rewrite an existential qua...
lerabdioph 42763 Diophantine set builder fo...
eluzrabdioph 42764 Diophantine set builder fo...
elnnrabdioph 42765 Diophantine set builder fo...
ltrabdioph 42766 Diophantine set builder fo...
nerabdioph 42767 Diophantine set builder fo...
dvdsrabdioph 42768 Divisibility is a Diophant...
eldioph4b 42769 Membership in ` Dioph ` ex...
eldioph4i 42770 Forward-only version of ~ ...
diophren 42771 Change variables in a Diop...
rabrenfdioph 42772 Change variable numbers in...
rabren3dioph 42773 Change variable numbers in...
fphpd 42774 Pigeonhole principle expre...
fphpdo 42775 Pigeonhole principle for s...
ctbnfien 42776 An infinite subset of a co...
fiphp3d 42777 Infinite pigeonhole princi...
rencldnfilem 42778 Lemma for ~ rencldnfi . (...
rencldnfi 42779 A set of real numbers whic...
irrapxlem1 42780 Lemma for ~ irrapx1 . Div...
irrapxlem2 42781 Lemma for ~ irrapx1 . Two...
irrapxlem3 42782 Lemma for ~ irrapx1 . By ...
irrapxlem4 42783 Lemma for ~ irrapx1 . Eli...
irrapxlem5 42784 Lemma for ~ irrapx1 . Swi...
irrapxlem6 42785 Lemma for ~ irrapx1 . Exp...
irrapx1 42786 Dirichlet's approximation ...
pellexlem1 42787 Lemma for ~ pellex . Arit...
pellexlem2 42788 Lemma for ~ pellex . Arit...
pellexlem3 42789 Lemma for ~ pellex . To e...
pellexlem4 42790 Lemma for ~ pellex . Invo...
pellexlem5 42791 Lemma for ~ pellex . Invo...
pellexlem6 42792 Lemma for ~ pellex . Doin...
pellex 42793 Every Pell equation has a ...
pell1qrval 42804 Value of the set of first-...
elpell1qr 42805 Membership in a first-quad...
pell14qrval 42806 Value of the set of positi...
elpell14qr 42807 Membership in the set of p...
pell1234qrval 42808 Value of the set of genera...
elpell1234qr 42809 Membership in the set of g...
pell1234qrre 42810 General Pell solutions are...
pell1234qrne0 42811 No solution to a Pell equa...
pell1234qrreccl 42812 General solutions of the P...
pell1234qrmulcl 42813 General solutions of the P...
pell14qrss1234 42814 A positive Pell solution i...
pell14qrre 42815 A positive Pell solution i...
pell14qrne0 42816 A positive Pell solution i...
pell14qrgt0 42817 A positive Pell solution i...
pell14qrrp 42818 A positive Pell solution i...
pell1234qrdich 42819 A general Pell solution is...
elpell14qr2 42820 A number is a positive Pel...
pell14qrmulcl 42821 Positive Pell solutions ar...
pell14qrreccl 42822 Positive Pell solutions ar...
pell14qrdivcl 42823 Positive Pell solutions ar...
pell14qrexpclnn0 42824 Lemma for ~ pell14qrexpcl ...
pell14qrexpcl 42825 Positive Pell solutions ar...
pell1qrss14 42826 First-quadrant Pell soluti...
pell14qrdich 42827 A positive Pell solution i...
pell1qrge1 42828 A Pell solution in the fir...
pell1qr1 42829 1 is a Pell solution and i...
elpell1qr2 42830 The first quadrant solutio...
pell1qrgaplem 42831 Lemma for ~ pell1qrgap . ...
pell1qrgap 42832 First-quadrant Pell soluti...
pell14qrgap 42833 Positive Pell solutions ar...
pell14qrgapw 42834 Positive Pell solutions ar...
pellqrexplicit 42835 Condition for a calculated...
infmrgelbi 42836 Any lower bound of a nonem...
pellqrex 42837 There is a nontrivial solu...
pellfundval 42838 Value of the fundamental s...
pellfundre 42839 The fundamental solution o...
pellfundge 42840 Lower bound on the fundame...
pellfundgt1 42841 Weak lower bound on the Pe...
pellfundlb 42842 A nontrivial first quadran...
pellfundglb 42843 If a real is larger than t...
pellfundex 42844 The fundamental solution a...
pellfund14gap 42845 There are no solutions bet...
pellfundrp 42846 The fundamental Pell solut...
pellfundne1 42847 The fundamental Pell solut...
reglogcl 42848 General logarithm is a rea...
reglogltb 42849 General logarithm preserve...
reglogleb 42850 General logarithm preserve...
reglogmul 42851 Multiplication law for gen...
reglogexp 42852 Power law for general log....
reglogbas 42853 General log of the base is...
reglog1 42854 General log of 1 is 0. (C...
reglogexpbas 42855 General log of a power of ...
pellfund14 42856 Every positive Pell soluti...
pellfund14b 42857 The positive Pell solution...
rmxfval 42862 Value of the X sequence. ...
rmyfval 42863 Value of the Y sequence. ...
rmspecsqrtnq 42864 The discriminant used to d...
rmspecnonsq 42865 The discriminant used to d...
qirropth 42866 This lemma implements the ...
rmspecfund 42867 The base of exponent used ...
rmxyelqirr 42868 The solutions used to cons...
rmxyelqirrOLD 42869 Obsolete version of ~ rmxy...
rmxypairf1o 42870 The function used to extra...
rmxyelxp 42871 Lemma for ~ frmx and ~ frm...
frmx 42872 The X sequence is a nonneg...
frmy 42873 The Y sequence is an integ...
rmxyval 42874 Main definition of the X a...
rmspecpos 42875 The discriminant used to d...
rmxycomplete 42876 The X and Y sequences take...
rmxynorm 42877 The X and Y sequences defi...
rmbaserp 42878 The base of exponentiation...
rmxyneg 42879 Negation law for X and Y s...
rmxyadd 42880 Addition formula for X and...
rmxy1 42881 Value of the X and Y seque...
rmxy0 42882 Value of the X and Y seque...
rmxneg 42883 Negation law (even functio...
rmx0 42884 Value of X sequence at 0. ...
rmx1 42885 Value of X sequence at 1. ...
rmxadd 42886 Addition formula for X seq...
rmyneg 42887 Negation formula for Y seq...
rmy0 42888 Value of Y sequence at 0. ...
rmy1 42889 Value of Y sequence at 1. ...
rmyadd 42890 Addition formula for Y seq...
rmxp1 42891 Special addition-of-1 form...
rmyp1 42892 Special addition of 1 form...
rmxm1 42893 Subtraction of 1 formula f...
rmym1 42894 Subtraction of 1 formula f...
rmxluc 42895 The X sequence is a Lucas ...
rmyluc 42896 The Y sequence is a Lucas ...
rmyluc2 42897 Lucas sequence property of...
rmxdbl 42898 "Double-angle formula" for...
rmydbl 42899 "Double-angle formula" for...
monotuz 42900 A function defined on an u...
monotoddzzfi 42901 A function which is odd an...
monotoddzz 42902 A function (given implicit...
oddcomabszz 42903 An odd function which take...
2nn0ind 42904 Induction on nonnegative i...
zindbi 42905 Inductively transfer a pro...
rmxypos 42906 For all nonnegative indice...
ltrmynn0 42907 The Y-sequence is strictly...
ltrmxnn0 42908 The X-sequence is strictly...
lermxnn0 42909 The X-sequence is monotoni...
rmxnn 42910 The X-sequence is defined ...
ltrmy 42911 The Y-sequence is strictly...
rmyeq0 42912 Y is zero only at zero. (...
rmyeq 42913 Y is one-to-one. (Contrib...
lermy 42914 Y is monotonic (non-strict...
rmynn 42915 ` rmY ` is positive for po...
rmynn0 42916 ` rmY ` is nonnegative for...
rmyabs 42917 ` rmY ` commutes with ` ab...
jm2.24nn 42918 X(n) is strictly greater t...
jm2.17a 42919 First half of lemma 2.17 o...
jm2.17b 42920 Weak form of the second ha...
jm2.17c 42921 Second half of lemma 2.17 ...
jm2.24 42922 Lemma 2.24 of [JonesMatija...
rmygeid 42923 Y(n) increases faster than...
congtr 42924 A wff of the form ` A || (...
congadd 42925 If two pairs of numbers ar...
congmul 42926 If two pairs of numbers ar...
congsym 42927 Congruence mod ` A ` is a ...
congneg 42928 If two integers are congru...
congsub 42929 If two pairs of numbers ar...
congid 42930 Every integer is congruent...
mzpcong 42931 Polynomials commute with c...
congrep 42932 Every integer is congruent...
congabseq 42933 If two integers are congru...
acongid 42934 A wff like that in this th...
acongsym 42935 Symmetry of alternating co...
acongneg2 42936 Negate right side of alter...
acongtr 42937 Transitivity of alternatin...
acongeq12d 42938 Substitution deduction for...
acongrep 42939 Every integer is alternati...
fzmaxdif 42940 Bound on the difference be...
fzneg 42941 Reflection of a finite ran...
acongeq 42942 Two numbers in the fundame...
dvdsacongtr 42943 Alternating congruence pas...
coprmdvdsb 42944 Multiplication by a coprim...
modabsdifz 42945 Divisibility in terms of m...
dvdsabsmod0 42946 Divisibility in terms of m...
jm2.18 42947 Theorem 2.18 of [JonesMati...
jm2.19lem1 42948 Lemma for ~ jm2.19 . X an...
jm2.19lem2 42949 Lemma for ~ jm2.19 . (Con...
jm2.19lem3 42950 Lemma for ~ jm2.19 . (Con...
jm2.19lem4 42951 Lemma for ~ jm2.19 . Exte...
jm2.19 42952 Lemma 2.19 of [JonesMatija...
jm2.21 42953 Lemma for ~ jm2.20nn . Ex...
jm2.22 42954 Lemma for ~ jm2.20nn . Ap...
jm2.23 42955 Lemma for ~ jm2.20nn . Tr...
jm2.20nn 42956 Lemma 2.20 of [JonesMatija...
jm2.25lem1 42957 Lemma for ~ jm2.26 . (Con...
jm2.25 42958 Lemma for ~ jm2.26 . Rema...
jm2.26a 42959 Lemma for ~ jm2.26 . Reve...
jm2.26lem3 42960 Lemma for ~ jm2.26 . Use ...
jm2.26 42961 Lemma 2.26 of [JonesMatija...
jm2.15nn0 42962 Lemma 2.15 of [JonesMatija...
jm2.16nn0 42963 Lemma 2.16 of [JonesMatija...
jm2.27a 42964 Lemma for ~ jm2.27 . Reve...
jm2.27b 42965 Lemma for ~ jm2.27 . Expa...
jm2.27c 42966 Lemma for ~ jm2.27 . Forw...
jm2.27 42967 Lemma 2.27 of [JonesMatija...
jm2.27dlem1 42968 Lemma for ~ rmydioph . Su...
jm2.27dlem2 42969 Lemma for ~ rmydioph . Th...
jm2.27dlem3 42970 Lemma for ~ rmydioph . In...
jm2.27dlem4 42971 Lemma for ~ rmydioph . In...
jm2.27dlem5 42972 Lemma for ~ rmydioph . Us...
rmydioph 42973 ~ jm2.27 restated in terms...
rmxdiophlem 42974 X can be expressed in term...
rmxdioph 42975 X is a Diophantine functio...
jm3.1lem1 42976 Lemma for ~ jm3.1 . (Cont...
jm3.1lem2 42977 Lemma for ~ jm3.1 . (Cont...
jm3.1lem3 42978 Lemma for ~ jm3.1 . (Cont...
jm3.1 42979 Diophantine expression for...
expdiophlem1 42980 Lemma for ~ expdioph . Fu...
expdiophlem2 42981 Lemma for ~ expdioph . Ex...
expdioph 42982 The exponential function i...
setindtr 42983 Set induction for sets con...
setindtrs 42984 Set induction scheme witho...
dford3lem1 42985 Lemma for ~ dford3 . (Con...
dford3lem2 42986 Lemma for ~ dford3 . (Con...
dford3 42987 Ordinals are precisely the...
dford4 42988 ~ dford3 expressed in prim...
wopprc 42989 Unrelated: Wiener pairs t...
rpnnen3lem 42990 Lemma for ~ rpnnen3 . (Co...
rpnnen3 42991 Dedekind cut injection of ...
axac10 42992 Characterization of choice...
harinf 42993 The Hartogs number of an i...
wdom2d2 42994 Deduction for weak dominan...
ttac 42995 Tarski's theorem about cho...
pw2f1ocnv 42996 Define a bijection between...
pw2f1o2 42997 Define a bijection between...
pw2f1o2val 42998 Function value of the ~ pw...
pw2f1o2val2 42999 Membership in a mapped set...
limsuc2 43000 Limit ordinals in the sens...
wepwsolem 43001 Transfer an ordering on ch...
wepwso 43002 A well-ordering induces a ...
dnnumch1 43003 Define an enumeration of a...
dnnumch2 43004 Define an enumeration (wea...
dnnumch3lem 43005 Value of the ordinal injec...
dnnumch3 43006 Define an injection from a...
dnwech 43007 Define a well-ordering fro...
fnwe2val 43008 Lemma for ~ fnwe2 . Subst...
fnwe2lem1 43009 Lemma for ~ fnwe2 . Subst...
fnwe2lem2 43010 Lemma for ~ fnwe2 . An el...
fnwe2lem3 43011 Lemma for ~ fnwe2 . Trich...
fnwe2 43012 A well-ordering can be con...
aomclem1 43013 Lemma for ~ dfac11 . This...
aomclem2 43014 Lemma for ~ dfac11 . Succ...
aomclem3 43015 Lemma for ~ dfac11 . Succ...
aomclem4 43016 Lemma for ~ dfac11 . Limi...
aomclem5 43017 Lemma for ~ dfac11 . Comb...
aomclem6 43018 Lemma for ~ dfac11 . Tran...
aomclem7 43019 Lemma for ~ dfac11 . ` ( R...
aomclem8 43020 Lemma for ~ dfac11 . Perf...
dfac11 43021 The right-hand side of thi...
kelac1 43022 Kelley's choice, basic for...
kelac2lem 43023 Lemma for ~ kelac2 and ~ d...
kelac2 43024 Kelley's choice, most comm...
dfac21 43025 Tychonoff's theorem is a c...
islmodfg 43028 Property of a finitely gen...
islssfg 43029 Property of a finitely gen...
islssfg2 43030 Property of a finitely gen...
islssfgi 43031 Finitely spanned subspaces...
fglmod 43032 Finitely generated left mo...
lsmfgcl 43033 The sum of two finitely ge...
islnm 43036 Property of being a Noethe...
islnm2 43037 Property of being a Noethe...
lnmlmod 43038 A Noetherian left module i...
lnmlssfg 43039 A submodule of Noetherian ...
lnmlsslnm 43040 All submodules of a Noethe...
lnmfg 43041 A Noetherian left module i...
kercvrlsm 43042 The domain of a linear fun...
lmhmfgima 43043 A homomorphism maps finite...
lnmepi 43044 Epimorphic images of Noeth...
lmhmfgsplit 43045 If the kernel and range of...
lmhmlnmsplit 43046 If the kernel and range of...
lnmlmic 43047 Noetherian is an invariant...
pwssplit4 43048 Splitting for structure po...
filnm 43049 Finite left modules are No...
pwslnmlem0 43050 Zeroeth powers are Noether...
pwslnmlem1 43051 First powers are Noetheria...
pwslnmlem2 43052 A sum of powers is Noether...
pwslnm 43053 Finite powers of Noetheria...
unxpwdom3 43054 Weaker version of ~ unxpwd...
pwfi2f1o 43055 The ~ pw2f1o bijection rel...
pwfi2en 43056 Finitely supported indicat...
frlmpwfi 43057 Formal linear combinations...
gicabl 43058 Being Abelian is a group i...
imasgim 43059 A relabeling of the elemen...
isnumbasgrplem1 43060 A set which is equipollent...
harn0 43061 The Hartogs number of a se...
numinfctb 43062 A numerable infinite set c...
isnumbasgrplem2 43063 If the (to be thought of a...
isnumbasgrplem3 43064 Every nonempty numerable s...
isnumbasabl 43065 A set is numerable iff it ...
isnumbasgrp 43066 A set is numerable iff it ...
dfacbasgrp 43067 A choice equivalent in abs...
islnr 43070 Property of a left-Noether...
lnrring 43071 Left-Noetherian rings are ...
lnrlnm 43072 Left-Noetherian rings have...
islnr2 43073 Property of being a left-N...
islnr3 43074 Relate left-Noetherian rin...
lnr2i 43075 Given an ideal in a left-N...
lpirlnr 43076 Left principal ideal rings...
lnrfrlm 43077 Finite-dimensional free mo...
lnrfg 43078 Finitely-generated modules...
lnrfgtr 43079 A submodule of a finitely ...
hbtlem1 43082 Value of the leading coeff...
hbtlem2 43083 Leading coefficient ideals...
hbtlem7 43084 Functionality of leading c...
hbtlem4 43085 The leading ideal function...
hbtlem3 43086 The leading ideal function...
hbtlem5 43087 The leading ideal function...
hbtlem6 43088 There is a finite set of p...
hbt 43089 The Hilbert Basis Theorem ...
dgrsub2 43094 Subtracting two polynomial...
elmnc 43095 Property of a monic polyno...
mncply 43096 A monic polynomial is a po...
mnccoe 43097 A monic polynomial has lea...
mncn0 43098 A monic polynomial is not ...
dgraaval 43103 Value of the degree functi...
dgraalem 43104 Properties of the degree o...
dgraacl 43105 Closure of the degree func...
dgraaf 43106 Degree function on algebra...
dgraaub 43107 Upper bound on degree of a...
dgraa0p 43108 A rational polynomial of d...
mpaaeu 43109 An algebraic number has ex...
mpaaval 43110 Value of the minimal polyn...
mpaalem 43111 Properties of the minimal ...
mpaacl 43112 Minimal polynomial is a po...
mpaadgr 43113 Minimal polynomial has deg...
mpaaroot 43114 The minimal polynomial of ...
mpaamn 43115 Minimal polynomial is moni...
itgoval 43120 Value of the integral-over...
aaitgo 43121 The standard algebraic num...
itgoss 43122 An integral element is int...
itgocn 43123 All integral elements are ...
cnsrexpcl 43124 Exponentiation is closed i...
fsumcnsrcl 43125 Finite sums are closed in ...
cnsrplycl 43126 Polynomials are closed in ...
rgspnval 43127 Value of the ring-span of ...
rgspncl 43128 The ring-span of a set is ...
rgspnssid 43129 The ring-span of a set con...
rgspnmin 43130 The ring-span is contained...
rgspnid 43131 The span of a subring is i...
rngunsnply 43132 Adjoining one element to a...
flcidc 43133 Finite linear combinations...
algstr 43136 Lemma to shorten proofs of...
algbase 43137 The base set of a construc...
algaddg 43138 The additive operation of ...
algmulr 43139 The multiplicative operati...
algsca 43140 The set of scalars of a co...
algvsca 43141 The scalar product operati...
mendval 43142 Value of the module endomo...
mendbas 43143 Base set of the module end...
mendplusgfval 43144 Addition in the module end...
mendplusg 43145 A specific addition in the...
mendmulrfval 43146 Multiplication in the modu...
mendmulr 43147 A specific multiplication ...
mendsca 43148 The module endomorphism al...
mendvscafval 43149 Scalar multiplication in t...
mendvsca 43150 A specific scalar multipli...
mendring 43151 The module endomorphism al...
mendlmod 43152 The module endomorphism al...
mendassa 43153 The module endomorphism al...
idomodle 43154 Limit on the number of ` N...
fiuneneq 43155 Two finite sets of equal s...
idomsubgmo 43156 The units of an integral d...
proot1mul 43157 Any primitive ` N ` -th ro...
proot1hash 43158 If an integral domain has ...
proot1ex 43159 The complex field has prim...
mon1psubm 43162 Monic polynomials are a mu...
deg1mhm 43163 Homomorphic property of th...
cytpfn 43164 Functionality of the cyclo...
cytpval 43165 Substitutions for the Nth ...
fgraphopab 43166 Express a function as a su...
fgraphxp 43167 Express a function as a su...
hausgraph 43168 The graph of a continuous ...
r1sssucd 43173 Deductive form of ~ r1sssu...
iocunico 43174 Split an open interval int...
iocinico 43175 The intersection of two se...
iocmbl 43176 An open-below, closed-abov...
cnioobibld 43177 A bounded, continuous func...
arearect 43178 The area of a rectangle wh...
areaquad 43179 The area of a quadrilatera...
uniel 43180 Two ways to say a union is...
unielss 43181 Two ways to say the union ...
unielid 43182 Two ways to say the union ...
ssunib 43183 Two ways to say a class is...
rp-intrabeq 43184 Equality theorem for supre...
rp-unirabeq 43185 Equality theorem for infim...
onmaxnelsup 43186 Two ways to say the maximu...
onsupneqmaxlim0 43187 If the supremum of a class...
onsupcl2 43188 The supremum of a set of o...
onuniintrab 43189 The union of a set of ordi...
onintunirab 43190 The intersection of a non-...
onsupnmax 43191 If the union of a class of...
onsupuni 43192 The supremum of a set of o...
onsupuni2 43193 The supremum of a set of o...
onsupintrab 43194 The supremum of a set of o...
onsupintrab2 43195 The supremum of a set of o...
onsupcl3 43196 The supremum of a set of o...
onsupex3 43197 The supremum of a set of o...
onuniintrab2 43198 The union of a set of ordi...
oninfint 43199 The infimum of a non-empty...
oninfunirab 43200 The infimum of a non-empty...
oninfcl2 43201 The infimum of a non-empty...
onsupmaxb 43202 The union of a class of or...
onexgt 43203 For any ordinal, there is ...
onexomgt 43204 For any ordinal, there is ...
omlimcl2 43205 The product of a limit ord...
onexlimgt 43206 For any ordinal, there is ...
onexoegt 43207 For any ordinal, there is ...
oninfex2 43208 The infimum of a non-empty...
onsupeqmax 43209 Condition when the supremu...
onsupeqnmax 43210 Condition when the supremu...
onsuplub 43211 The supremum of a set of o...
onsupnub 43212 An upper bound of a set of...
onfisupcl 43213 Sufficient condition when ...
onelord 43214 Every element of a ordinal...
onepsuc 43215 Every ordinal is less than...
epsoon 43216 The ordinals are strictly ...
epirron 43217 The strict order on the or...
oneptr 43218 The strict order on the or...
oneltr 43219 The elementhood relation o...
oneptri 43220 The strict, complete (line...
oneltri 43221 The elementhood relation o...
ordeldif 43222 Membership in the differen...
ordeldifsucon 43223 Membership in the differen...
ordeldif1o 43224 Membership in the differen...
ordne0gt0 43225 Ordinal zero is less than ...
ondif1i 43226 Ordinal zero is less than ...
onsucelab 43227 The successor of every ord...
dflim6 43228 A limit ordinal is a non-z...
limnsuc 43229 A limit ordinal is not an ...
onsucss 43230 If one ordinal is less tha...
ordnexbtwnsuc 43231 For any distinct pair of o...
orddif0suc 43232 For any distinct pair of o...
onsucf1lem 43233 For ordinals, the successo...
onsucf1olem 43234 The successor operation is...
onsucrn 43235 The successor operation is...
onsucf1o 43236 The successor operation is...
dflim7 43237 A limit ordinal is a non-z...
onov0suclim 43238 Compactly express rules fo...
oa0suclim 43239 Closed form expression of ...
om0suclim 43240 Closed form expression of ...
oe0suclim 43241 Closed form expression of ...
oaomoecl 43242 The operations of addition...
onsupsucismax 43243 If the union of a set of o...
onsssupeqcond 43244 If for every element of a ...
limexissup 43245 An ordinal which is a limi...
limiun 43246 A limit ordinal is the uni...
limexissupab 43247 An ordinal which is a limi...
om1om1r 43248 Ordinal one is both a left...
oe0rif 43249 Ordinal zero raised to any...
oasubex 43250 While subtraction can't be...
nnamecl 43251 Natural numbers are closed...
onsucwordi 43252 The successor operation pr...
oalim2cl 43253 The ordinal sum of any ord...
oaltublim 43254 Given ` C ` is a limit ord...
oaordi3 43255 Ordinal addition of the sa...
oaord3 43256 When the same ordinal is a...
1oaomeqom 43257 Ordinal one plus omega is ...
oaabsb 43258 The right addend absorbs t...
oaordnrex 43259 When omega is added on the...
oaordnr 43260 When the same ordinal is a...
omge1 43261 Any non-zero ordinal produ...
omge2 43262 Any non-zero ordinal produ...
omlim2 43263 The non-zero product with ...
omord2lim 43264 Given a limit ordinal, the...
omord2i 43265 Ordinal multiplication of ...
omord2com 43266 When the same non-zero ord...
2omomeqom 43267 Ordinal two times omega is...
omnord1ex 43268 When omega is multiplied o...
omnord1 43269 When the same non-zero ord...
oege1 43270 Any non-zero ordinal power...
oege2 43271 Any power of an ordinal at...
rp-oelim2 43272 The power of an ordinal at...
oeord2lim 43273 Given a limit ordinal, the...
oeord2i 43274 Ordinal exponentiation of ...
oeord2com 43275 When the same base at leas...
nnoeomeqom 43276 Any natural number at leas...
df3o2 43277 Ordinal 3 is the unordered...
df3o3 43278 Ordinal 3, fully expanded....
oenord1ex 43279 When ordinals two and thre...
oenord1 43280 When two ordinals (both at...
oaomoencom 43281 Ordinal addition, multipli...
oenassex 43282 Ordinal two raised to two ...
oenass 43283 Ordinal exponentiation is ...
cantnftermord 43284 For terms of the form of a...
cantnfub 43285 Given a finite number of t...
cantnfub2 43286 Given a finite number of t...
bropabg 43287 Equivalence for two classe...
cantnfresb 43288 A Cantor normal form which...
cantnf2 43289 For every ordinal, ` A ` ,...
oawordex2 43290 If ` C ` is between ` A ` ...
nnawordexg 43291 If an ordinal, ` B ` , is ...
succlg 43292 Closure law for ordinal su...
dflim5 43293 A limit ordinal is either ...
oacl2g 43294 Closure law for ordinal ad...
onmcl 43295 If an ordinal is less than...
omabs2 43296 Ordinal multiplication by ...
omcl2 43297 Closure law for ordinal mu...
omcl3g 43298 Closure law for ordinal mu...
ordsssucb 43299 An ordinal number is less ...
tfsconcatlem 43300 Lemma for ~ tfsconcatun . ...
tfsconcatun 43301 The concatenation of two t...
tfsconcatfn 43302 The concatenation of two t...
tfsconcatfv1 43303 An early value of the conc...
tfsconcatfv2 43304 A latter value of the conc...
tfsconcatfv 43305 The value of the concatena...
tfsconcatrn 43306 The range of the concatena...
tfsconcatfo 43307 The concatenation of two t...
tfsconcatb0 43308 The concatentation with th...
tfsconcat0i 43309 The concatentation with th...
tfsconcat0b 43310 The concatentation with th...
tfsconcat00 43311 The concatentation of two ...
tfsconcatrev 43312 If the domain of a transfi...
tfsconcatrnss12 43313 The range of the concatena...
tfsconcatrnss 43314 The concatenation of trans...
tfsconcatrnsson 43315 The concatenation of trans...
tfsnfin 43316 A transfinite sequence is ...
rp-tfslim 43317 The limit of a sequence of...
ofoafg 43318 Addition operator for func...
ofoaf 43319 Addition operator for func...
ofoafo 43320 Addition operator for func...
ofoacl 43321 Closure law for component ...
ofoaid1 43322 Identity law for component...
ofoaid2 43323 Identity law for component...
ofoaass 43324 Component-wise addition of...
ofoacom 43325 Component-wise addition of...
naddcnff 43326 Addition operator for Cant...
naddcnffn 43327 Addition operator for Cant...
naddcnffo 43328 Addition of Cantor normal ...
naddcnfcl 43329 Closure law for component-...
naddcnfcom 43330 Component-wise ordinal add...
naddcnfid1 43331 Identity law for component...
naddcnfid2 43332 Identity law for component...
naddcnfass 43333 Component-wise addition of...
onsucunifi 43334 The successor to the union...
sucunisn 43335 The successor to the union...
onsucunipr 43336 The successor to the union...
onsucunitp 43337 The successor to the union...
oaun3lem1 43338 The class of all ordinal s...
oaun3lem2 43339 The class of all ordinal s...
oaun3lem3 43340 The class of all ordinal s...
oaun3lem4 43341 The class of all ordinal s...
rp-abid 43342 Two ways to express a clas...
oadif1lem 43343 Express the set difference...
oadif1 43344 Express the set difference...
oaun2 43345 Ordinal addition as a unio...
oaun3 43346 Ordinal addition as a unio...
naddov4 43347 Alternate expression for n...
nadd2rabtr 43348 The set of ordinals which ...
nadd2rabord 43349 The set of ordinals which ...
nadd2rabex 43350 The class of ordinals whic...
nadd2rabon 43351 The set of ordinals which ...
nadd1rabtr 43352 The set of ordinals which ...
nadd1rabord 43353 The set of ordinals which ...
nadd1rabex 43354 The class of ordinals whic...
nadd1rabon 43355 The set of ordinals which ...
nadd1suc 43356 Natural addition with 1 is...
naddass1 43357 Natural addition of ordina...
naddgeoa 43358 Natural addition results i...
naddonnn 43359 Natural addition with a na...
naddwordnexlem0 43360 When ` A ` is the sum of a...
naddwordnexlem1 43361 When ` A ` is the sum of a...
naddwordnexlem2 43362 When ` A ` is the sum of a...
naddwordnexlem3 43363 When ` A ` is the sum of a...
oawordex3 43364 When ` A ` is the sum of a...
naddwordnexlem4 43365 When ` A ` is the sum of a...
ordsssucim 43366 If an ordinal is less than...
insucid 43367 The intersection of a clas...
om2 43368 Two ways to double an ordi...
oaltom 43369 Multiplication eventually ...
oe2 43370 Two ways to square an ordi...
omltoe 43371 Exponentiation eventually ...
abeqabi 43372 Generalized condition for ...
abpr 43373 Condition for a class abst...
abtp 43374 Condition for a class abst...
ralopabb 43375 Restricted universal quant...
fpwfvss 43376 Functions into a powerset ...
sdomne0 43377 A class that strictly domi...
sdomne0d 43378 A class that strictly domi...
safesnsupfiss 43379 If ` B ` is a finite subse...
safesnsupfiub 43380 If ` B ` is a finite subse...
safesnsupfidom1o 43381 If ` B ` is a finite subse...
safesnsupfilb 43382 If ` B ` is a finite subse...
isoeq145d 43383 Equality deduction for iso...
resisoeq45d 43384 Equality deduction for equ...
negslem1 43385 An equivalence between ide...
nvocnvb 43386 Equivalence to saying the ...
rp-brsslt 43387 Binary relation form of a ...
nla0002 43388 Extending a linear order t...
nla0003 43389 Extending a linear order t...
nla0001 43390 Extending a linear order t...
faosnf0.11b 43391 ` B ` is called a non-limi...
dfno2 43392 A surreal number, in the f...
onnog 43393 Every ordinal maps to a su...
onnobdayg 43394 Every ordinal maps to a su...
bdaybndex 43395 Bounds formed from the bir...
bdaybndbday 43396 Bounds formed from the bir...
onno 43397 Every ordinal maps to a su...
onnoi 43398 Every ordinal maps to a su...
0no 43399 Ordinal zero maps to a sur...
1no 43400 Ordinal one maps to a surr...
2no 43401 Ordinal two maps to a surr...
3no 43402 Ordinal three maps to a su...
4no 43403 Ordinal four maps to a sur...
fnimafnex 43404 The functional image of a ...
nlimsuc 43405 A successor is not a limit...
nlim1NEW 43406 1 is not a limit ordinal. ...
nlim2NEW 43407 2 is not a limit ordinal. ...
nlim3 43408 3 is not a limit ordinal. ...
nlim4 43409 4 is not a limit ordinal. ...
oa1un 43410 Given ` A e. On ` , let ` ...
oa1cl 43411 ` A +o 1o ` is in ` On ` ....
0finon 43412 0 is a finite ordinal. Se...
1finon 43413 1 is a finite ordinal. Se...
2finon 43414 2 is a finite ordinal. Se...
3finon 43415 3 is a finite ordinal. Se...
4finon 43416 4 is a finite ordinal. Se...
finona1cl 43417 The finite ordinals are cl...
finonex 43418 The finite ordinals are a ...
fzunt 43419 Union of two adjacent fini...
fzuntd 43420 Union of two adjacent fini...
fzunt1d 43421 Union of two overlapping f...
fzuntgd 43422 Union of two adjacent or o...
ifpan123g 43423 Conjunction of conditional...
ifpan23 43424 Conjunction of conditional...
ifpdfor2 43425 Define or in terms of cond...
ifporcor 43426 Corollary of commutation o...
ifpdfan2 43427 Define and with conditiona...
ifpancor 43428 Corollary of commutation o...
ifpdfor 43429 Define or in terms of cond...
ifpdfan 43430 Define and with conditiona...
ifpbi2 43431 Equivalence theorem for co...
ifpbi3 43432 Equivalence theorem for co...
ifpim1 43433 Restate implication as con...
ifpnot 43434 Restate negated wff as con...
ifpid2 43435 Restate wff as conditional...
ifpim2 43436 Restate implication as con...
ifpbi23 43437 Equivalence theorem for co...
ifpbiidcor 43438 Restatement of ~ biid . (...
ifpbicor 43439 Corollary of commutation o...
ifpxorcor 43440 Corollary of commutation o...
ifpbi1 43441 Equivalence theorem for co...
ifpnot23 43442 Negation of conditional lo...
ifpnotnotb 43443 Factor conditional logic o...
ifpnorcor 43444 Corollary of commutation o...
ifpnancor 43445 Corollary of commutation o...
ifpnot23b 43446 Negation of conditional lo...
ifpbiidcor2 43447 Restatement of ~ biid . (...
ifpnot23c 43448 Negation of conditional lo...
ifpnot23d 43449 Negation of conditional lo...
ifpdfnan 43450 Define nand as conditional...
ifpdfxor 43451 Define xor as conditional ...
ifpbi12 43452 Equivalence theorem for co...
ifpbi13 43453 Equivalence theorem for co...
ifpbi123 43454 Equivalence theorem for co...
ifpidg 43455 Restate wff as conditional...
ifpid3g 43456 Restate wff as conditional...
ifpid2g 43457 Restate wff as conditional...
ifpid1g 43458 Restate wff as conditional...
ifpim23g 43459 Restate implication as con...
ifpim3 43460 Restate implication as con...
ifpnim1 43461 Restate negated implicatio...
ifpim4 43462 Restate implication as con...
ifpnim2 43463 Restate negated implicatio...
ifpim123g 43464 Implication of conditional...
ifpim1g 43465 Implication of conditional...
ifp1bi 43466 Substitute the first eleme...
ifpbi1b 43467 When the first variable is...
ifpimimb 43468 Factor conditional logic o...
ifpororb 43469 Factor conditional logic o...
ifpananb 43470 Factor conditional logic o...
ifpnannanb 43471 Factor conditional logic o...
ifpor123g 43472 Disjunction of conditional...
ifpimim 43473 Consequnce of implication....
ifpbibib 43474 Factor conditional logic o...
ifpxorxorb 43475 Factor conditional logic o...
rp-fakeimass 43476 A special case where impli...
rp-fakeanorass 43477 A special case where a mix...
rp-fakeoranass 43478 A special case where a mix...
rp-fakeinunass 43479 A special case where a mix...
rp-fakeuninass 43480 A special case where a mix...
rp-isfinite5 43481 A set is said to be finite...
rp-isfinite6 43482 A set is said to be finite...
intabssd 43483 When for each element ` y ...
eu0 43484 There is only one empty se...
epelon2 43485 Over the ordinal numbers, ...
ontric3g 43486 For all ` x , y e. On ` , ...
dfsucon 43487 ` A ` is called a successo...
snen1g 43488 A singleton is equinumerou...
snen1el 43489 A singleton is equinumerou...
sn1dom 43490 A singleton is dominated b...
pr2dom 43491 An unordered pair is domin...
tr3dom 43492 An unordered triple is dom...
ensucne0 43493 A class equinumerous to a ...
ensucne0OLD 43494 A class equinumerous to a ...
dfom6 43495 Let ` _om ` be defined to ...
infordmin 43496 ` _om ` is the smallest in...
iscard4 43497 Two ways to express the pr...
minregex 43498 Given any cardinal number ...
minregex2 43499 Given any cardinal number ...
iscard5 43500 Two ways to express the pr...
elrncard 43501 Let us define a cardinal n...
harval3 43502 ` ( har `` A ) ` is the le...
harval3on 43503 For any ordinal number ` A...
omssrncard 43504 All natural numbers are ca...
0iscard 43505 0 is a cardinal number. (...
1iscard 43506 1 is a cardinal number. (...
omiscard 43507 ` _om ` is a cardinal numb...
sucomisnotcard 43508 ` _om +o 1o ` is not a car...
nna1iscard 43509 For any natural number, th...
har2o 43510 The least cardinal greater...
en2pr 43511 A class is equinumerous to...
pr2cv 43512 If an unordered pair is eq...
pr2el1 43513 If an unordered pair is eq...
pr2cv1 43514 If an unordered pair is eq...
pr2el2 43515 If an unordered pair is eq...
pr2cv2 43516 If an unordered pair is eq...
pren2 43517 An unordered pair is equin...
pr2eldif1 43518 If an unordered pair is eq...
pr2eldif2 43519 If an unordered pair is eq...
pren2d 43520 A pair of two distinct set...
aleph1min 43521 ` ( aleph `` 1o ) ` is the...
alephiso2 43522 ` aleph ` is a strictly or...
alephiso3 43523 ` aleph ` is a strictly or...
pwelg 43524 The powerclass is an eleme...
pwinfig 43525 The powerclass of an infin...
pwinfi2 43526 The powerclass of an infin...
pwinfi3 43527 The powerclass of an infin...
pwinfi 43528 The powerclass of an infin...
fipjust 43529 A definition of the finite...
cllem0 43530 The class of all sets with...
superficl 43531 The class of all supersets...
superuncl 43532 The class of all supersets...
ssficl 43533 The class of all subsets o...
ssuncl 43534 The class of all subsets o...
ssdifcl 43535 The class of all subsets o...
sssymdifcl 43536 The class of all subsets o...
fiinfi 43537 If two classes have the fi...
rababg 43538 Condition when restricted ...
elinintab 43539 Two ways of saying a set i...
elmapintrab 43540 Two ways to say a set is a...
elinintrab 43541 Two ways of saying a set i...
inintabss 43542 Upper bound on intersectio...
inintabd 43543 Value of the intersection ...
xpinintabd 43544 Value of the intersection ...
relintabex 43545 If the intersection of a c...
elcnvcnvintab 43546 Two ways of saying a set i...
relintab 43547 Value of the intersection ...
nonrel 43548 A non-relation is equal to...
elnonrel 43549 Only an ordered pair where...
cnvssb 43550 Subclass theorem for conve...
relnonrel 43551 The non-relation part of a...
cnvnonrel 43552 The converse of the non-re...
brnonrel 43553 A non-relation cannot rela...
dmnonrel 43554 The domain of the non-rela...
rnnonrel 43555 The range of the non-relat...
resnonrel 43556 A restriction of the non-r...
imanonrel 43557 An image under the non-rel...
cononrel1 43558 Composition with the non-r...
cononrel2 43559 Composition with the non-r...
elmapintab 43560 Two ways to say a set is a...
fvnonrel 43561 The function value of any ...
elinlem 43562 Two ways to say a set is a...
elcnvcnvlem 43563 Two ways to say a set is a...
cnvcnvintabd 43564 Value of the relationship ...
elcnvlem 43565 Two ways to say a set is a...
elcnvintab 43566 Two ways of saying a set i...
cnvintabd 43567 Value of the converse of t...
undmrnresiss 43568 Two ways of saying the ide...
reflexg 43569 Two ways of saying a relat...
cnvssco 43570 A condition weaker than re...
refimssco 43571 Reflexive relations are su...
cleq2lem 43572 Equality implies bijection...
cbvcllem 43573 Change of bound variable i...
clublem 43574 If a superset ` Y ` of ` X...
clss2lem 43575 The closure of a property ...
dfid7 43576 Definition of identity rel...
mptrcllem 43577 Show two versions of a clo...
cotrintab 43578 The intersection of a clas...
rclexi 43579 The reflexive closure of a...
rtrclexlem 43580 Existence of relation impl...
rtrclex 43581 The reflexive-transitive c...
trclubgNEW 43582 If a relation exists then ...
trclubNEW 43583 If a relation exists then ...
trclexi 43584 The transitive closure of ...
rtrclexi 43585 The reflexive-transitive c...
clrellem 43586 When the property ` ps ` h...
clcnvlem 43587 When ` A ` , an upper boun...
cnvtrucl0 43588 The converse of the trivia...
cnvrcl0 43589 The converse of the reflex...
cnvtrcl0 43590 The converse of the transi...
dmtrcl 43591 The domain of the transiti...
rntrcl 43592 The range of the transitiv...
dfrtrcl5 43593 Definition of reflexive-tr...
trcleq2lemRP 43594 Equality implies bijection...
sqrtcvallem1 43595 Two ways of saying a compl...
reabsifneg 43596 Alternate expression for t...
reabsifnpos 43597 Alternate expression for t...
reabsifpos 43598 Alternate expression for t...
reabsifnneg 43599 Alternate expression for t...
reabssgn 43600 Alternate expression for t...
sqrtcvallem2 43601 Equivalent to saying that ...
sqrtcvallem3 43602 Equivalent to saying that ...
sqrtcvallem4 43603 Equivalent to saying that ...
sqrtcvallem5 43604 Equivalent to saying that ...
sqrtcval 43605 Explicit formula for the c...
sqrtcval2 43606 Explicit formula for the c...
resqrtval 43607 Real part of the complex s...
imsqrtval 43608 Imaginary part of the comp...
resqrtvalex 43609 Example for ~ resqrtval . ...
imsqrtvalex 43610 Example for ~ imsqrtval . ...
al3im 43611 Version of ~ ax-4 for a ne...
intima0 43612 Two ways of expressing the...
elimaint 43613 Element of image of inters...
cnviun 43614 Converse of indexed union....
imaiun1 43615 The image of an indexed un...
coiun1 43616 Composition with an indexe...
elintima 43617 Element of intersection of...
intimass 43618 The image under the inters...
intimass2 43619 The image under the inters...
intimag 43620 Requirement for the image ...
intimasn 43621 Two ways to express the im...
intimasn2 43622 Two ways to express the im...
ss2iundf 43623 Subclass theorem for index...
ss2iundv 43624 Subclass theorem for index...
cbviuneq12df 43625 Rule used to change the bo...
cbviuneq12dv 43626 Rule used to change the bo...
conrel1d 43627 Deduction about compositio...
conrel2d 43628 Deduction about compositio...
trrelind 43629 The intersection of transi...
xpintrreld 43630 The intersection of a tran...
restrreld 43631 The restriction of a trans...
trrelsuperreldg 43632 Concrete construction of a...
trficl 43633 The class of all transitiv...
cnvtrrel 43634 The converse of a transiti...
trrelsuperrel2dg 43635 Concrete construction of a...
dfrcl2 43638 Reflexive closure of a rel...
dfrcl3 43639 Reflexive closure of a rel...
dfrcl4 43640 Reflexive closure of a rel...
relexp2 43641 A set operated on by the r...
relexpnul 43642 If the domain and range of...
eliunov2 43643 Membership in the indexed ...
eltrclrec 43644 Membership in the indexed ...
elrtrclrec 43645 Membership in the indexed ...
briunov2 43646 Two classes related by the...
brmptiunrelexpd 43647 If two elements are connec...
fvmptiunrelexplb0d 43648 If the indexed union range...
fvmptiunrelexplb0da 43649 If the indexed union range...
fvmptiunrelexplb1d 43650 If the indexed union range...
brfvid 43651 If two elements are connec...
brfvidRP 43652 If two elements are connec...
fvilbd 43653 A set is a subset of its i...
fvilbdRP 43654 A set is a subset of its i...
brfvrcld 43655 If two elements are connec...
brfvrcld2 43656 If two elements are connec...
fvrcllb0d 43657 A restriction of the ident...
fvrcllb0da 43658 A restriction of the ident...
fvrcllb1d 43659 A set is a subset of its i...
brtrclrec 43660 Two classes related by the...
brrtrclrec 43661 Two classes related by the...
briunov2uz 43662 Two classes related by the...
eliunov2uz 43663 Membership in the indexed ...
ov2ssiunov2 43664 Any particular operator va...
relexp0eq 43665 The zeroth power of relati...
iunrelexp0 43666 Simplification of zeroth p...
relexpxpnnidm 43667 Any positive power of a Ca...
relexpiidm 43668 Any power of any restricti...
relexpss1d 43669 The relational power of a ...
comptiunov2i 43670 The composition two indexe...
corclrcl 43671 The reflexive closure is i...
iunrelexpmin1 43672 The indexed union of relat...
relexpmulnn 43673 With exponents limited to ...
relexpmulg 43674 With ordered exponents, th...
trclrelexplem 43675 The union of relational po...
iunrelexpmin2 43676 The indexed union of relat...
relexp01min 43677 With exponents limited to ...
relexp1idm 43678 Repeated raising a relatio...
relexp0idm 43679 Repeated raising a relatio...
relexp0a 43680 Absorption law for zeroth ...
relexpxpmin 43681 The composition of powers ...
relexpaddss 43682 The composition of two pow...
iunrelexpuztr 43683 The indexed union of relat...
dftrcl3 43684 Transitive closure of a re...
brfvtrcld 43685 If two elements are connec...
fvtrcllb1d 43686 A set is a subset of its i...
trclfvcom 43687 The transitive closure of ...
cnvtrclfv 43688 The converse of the transi...
cotrcltrcl 43689 The transitive closure is ...
trclimalb2 43690 Lower bound for image unde...
brtrclfv2 43691 Two ways to indicate two e...
trclfvdecomr 43692 The transitive closure of ...
trclfvdecoml 43693 The transitive closure of ...
dmtrclfvRP 43694 The domain of the transiti...
rntrclfvRP 43695 The range of the transitiv...
rntrclfv 43696 The range of the transitiv...
dfrtrcl3 43697 Reflexive-transitive closu...
brfvrtrcld 43698 If two elements are connec...
fvrtrcllb0d 43699 A restriction of the ident...
fvrtrcllb0da 43700 A restriction of the ident...
fvrtrcllb1d 43701 A set is a subset of its i...
dfrtrcl4 43702 Reflexive-transitive closu...
corcltrcl 43703 The composition of the ref...
cortrcltrcl 43704 Composition with the refle...
corclrtrcl 43705 Composition with the refle...
cotrclrcl 43706 The composition of the ref...
cortrclrcl 43707 Composition with the refle...
cotrclrtrcl 43708 Composition with the refle...
cortrclrtrcl 43709 The reflexive-transitive c...
frege77d 43710 If the images of both ` { ...
frege81d 43711 If the image of ` U ` is a...
frege83d 43712 If the image of the union ...
frege96d 43713 If ` C ` follows ` A ` in ...
frege87d 43714 If the images of both ` { ...
frege91d 43715 If ` B ` follows ` A ` in ...
frege97d 43716 If ` A ` contains all elem...
frege98d 43717 If ` C ` follows ` A ` and...
frege102d 43718 If either ` A ` and ` C ` ...
frege106d 43719 If ` B ` follows ` A ` in ...
frege108d 43720 If either ` A ` and ` C ` ...
frege109d 43721 If ` A ` contains all elem...
frege114d 43722 If either ` R ` relates ` ...
frege111d 43723 If either ` A ` and ` C ` ...
frege122d 43724 If ` F ` is a function, ` ...
frege124d 43725 If ` F ` is a function, ` ...
frege126d 43726 If ` F ` is a function, ` ...
frege129d 43727 If ` F ` is a function and...
frege131d 43728 If ` F ` is a function and...
frege133d 43729 If ` F ` is a function and...
dfxor4 43730 Express exclusive-or in te...
dfxor5 43731 Express exclusive-or in te...
df3or2 43732 Express triple-or in terms...
df3an2 43733 Express triple-and in term...
nev 43734 Express that not every set...
0pssin 43735 Express that an intersecti...
dfhe2 43738 The property of relation `...
dfhe3 43739 The property of relation `...
heeq12 43740 Equality law for relations...
heeq1 43741 Equality law for relations...
heeq2 43742 Equality law for relations...
sbcheg 43743 Distribute proper substitu...
hess 43744 Subclass law for relations...
xphe 43745 Any Cartesian product is h...
0he 43746 The empty relation is here...
0heALT 43747 The empty relation is here...
he0 43748 Any relation is hereditary...
unhe1 43749 The union of two relations...
snhesn 43750 Any singleton is hereditar...
idhe 43751 The identity relation is h...
psshepw 43752 The relation between sets ...
sshepw 43753 The relation between sets ...
rp-simp2-frege 43756 Simplification of triple c...
rp-simp2 43757 Simplification of triple c...
rp-frege3g 43758 Add antecedent to ~ ax-fre...
frege3 43759 Add antecedent to ~ ax-fre...
rp-misc1-frege 43760 Double-use of ~ ax-frege2 ...
rp-frege24 43761 Introducing an embedded an...
rp-frege4g 43762 Deduction related to distr...
frege4 43763 Special case of closed for...
frege5 43764 A closed form of ~ syl . ...
rp-7frege 43765 Distribute antecedent and ...
rp-4frege 43766 Elimination of a nested an...
rp-6frege 43767 Elimination of a nested an...
rp-8frege 43768 Eliminate antecedent when ...
rp-frege25 43769 Closed form for ~ a1dd . ...
frege6 43770 A closed form of ~ imim2d ...
axfrege8 43771 Swap antecedents. Identic...
frege7 43772 A closed form of ~ syl6 . ...
frege26 43774 Identical to ~ idd . Prop...
frege27 43775 We cannot (at the same tim...
frege9 43776 Closed form of ~ syl with ...
frege12 43777 A closed form of ~ com23 ....
frege11 43778 Elimination of a nested an...
frege24 43779 Closed form for ~ a1d . D...
frege16 43780 A closed form of ~ com34 ....
frege25 43781 Closed form for ~ a1dd . ...
frege18 43782 Closed form of a syllogism...
frege22 43783 A closed form of ~ com45 ....
frege10 43784 Result commuting anteceden...
frege17 43785 A closed form of ~ com3l ....
frege13 43786 A closed form of ~ com3r ....
frege14 43787 Closed form of a deduction...
frege19 43788 A closed form of ~ syl6 . ...
frege23 43789 Syllogism followed by rota...
frege15 43790 A closed form of ~ com4r ....
frege21 43791 Replace antecedent in ante...
frege20 43792 A closed form of ~ syl8 . ...
axfrege28 43793 Contraposition. Identical...
frege29 43795 Closed form of ~ con3d . ...
frege30 43796 Commuted, closed form of ~...
axfrege31 43797 Identical to ~ notnotr . ...
frege32 43799 Deduce ~ con1 from ~ con3 ...
frege33 43800 If ` ph ` or ` ps ` takes ...
frege34 43801 If as a consequence of the...
frege35 43802 Commuted, closed form of ~...
frege36 43803 The case in which ` ps ` i...
frege37 43804 If ` ch ` is a necessary c...
frege38 43805 Identical to ~ pm2.21 . P...
frege39 43806 Syllogism between ~ pm2.18...
frege40 43807 Anything implies ~ pm2.18 ...
axfrege41 43808 Identical to ~ notnot . A...
frege42 43810 Not not ~ id . Propositio...
frege43 43811 If there is a choice only ...
frege44 43812 Similar to a commuted ~ pm...
frege45 43813 Deduce ~ pm2.6 from ~ con1...
frege46 43814 If ` ps ` holds when ` ph ...
frege47 43815 Deduce consequence follows...
frege48 43816 Closed form of syllogism w...
frege49 43817 Closed form of deduction w...
frege50 43818 Closed form of ~ jaoi . P...
frege51 43819 Compare with ~ jaod . Pro...
axfrege52a 43820 Justification for ~ ax-fre...
frege52aid 43822 The case when the content ...
frege53aid 43823 Specialization of ~ frege5...
frege53a 43824 Lemma for ~ frege55a . Pr...
axfrege54a 43825 Justification for ~ ax-fre...
frege54cor0a 43827 Synonym for logical equiva...
frege54cor1a 43828 Reflexive equality. (Cont...
frege55aid 43829 Lemma for ~ frege57aid . ...
frege55lem1a 43830 Necessary deduction regard...
frege55lem2a 43831 Core proof of Proposition ...
frege55a 43832 Proposition 55 of [Frege18...
frege55cor1a 43833 Proposition 55 of [Frege18...
frege56aid 43834 Lemma for ~ frege57aid . ...
frege56a 43835 Proposition 56 of [Frege18...
frege57aid 43836 This is the all important ...
frege57a 43837 Analogue of ~ frege57aid ....
axfrege58a 43838 Identical to ~ anifp . Ju...
frege58acor 43840 Lemma for ~ frege59a . (C...
frege59a 43841 A kind of Aristotelian inf...
frege60a 43842 Swap antecedents of ~ ax-f...
frege61a 43843 Lemma for ~ frege65a . Pr...
frege62a 43844 A kind of Aristotelian inf...
frege63a 43845 Proposition 63 of [Frege18...
frege64a 43846 Lemma for ~ frege65a . Pr...
frege65a 43847 A kind of Aristotelian inf...
frege66a 43848 Swap antecedents of ~ freg...
frege67a 43849 Lemma for ~ frege68a . Pr...
frege68a 43850 Combination of applying a ...
axfrege52c 43851 Justification for ~ ax-fre...
frege52b 43853 The case when the content ...
frege53b 43854 Lemma for frege102 (via ~ ...
axfrege54c 43855 Reflexive equality of clas...
frege54b 43857 Reflexive equality of sets...
frege54cor1b 43858 Reflexive equality. (Cont...
frege55lem1b 43859 Necessary deduction regard...
frege55lem2b 43860 Lemma for ~ frege55b . Co...
frege55b 43861 Lemma for ~ frege57b . Pr...
frege56b 43862 Lemma for ~ frege57b . Pr...
frege57b 43863 Analogue of ~ frege57aid ....
axfrege58b 43864 If ` A. x ph ` is affirmed...
frege58bid 43866 If ` A. x ph ` is affirmed...
frege58bcor 43867 Lemma for ~ frege59b . (C...
frege59b 43868 A kind of Aristotelian inf...
frege60b 43869 Swap antecedents of ~ ax-f...
frege61b 43870 Lemma for ~ frege65b . Pr...
frege62b 43871 A kind of Aristotelian inf...
frege63b 43872 Lemma for ~ frege91 . Pro...
frege64b 43873 Lemma for ~ frege65b . Pr...
frege65b 43874 A kind of Aristotelian inf...
frege66b 43875 Swap antecedents of ~ freg...
frege67b 43876 Lemma for ~ frege68b . Pr...
frege68b 43877 Combination of applying a ...
frege53c 43878 Proposition 53 of [Frege18...
frege54cor1c 43879 Reflexive equality. (Cont...
frege55lem1c 43880 Necessary deduction regard...
frege55lem2c 43881 Core proof of Proposition ...
frege55c 43882 Proposition 55 of [Frege18...
frege56c 43883 Lemma for ~ frege57c . Pr...
frege57c 43884 Swap order of implication ...
frege58c 43885 Principle related to ~ sp ...
frege59c 43886 A kind of Aristotelian inf...
frege60c 43887 Swap antecedents of ~ freg...
frege61c 43888 Lemma for ~ frege65c . Pr...
frege62c 43889 A kind of Aristotelian inf...
frege63c 43890 Analogue of ~ frege63b . ...
frege64c 43891 Lemma for ~ frege65c . Pr...
frege65c 43892 A kind of Aristotelian inf...
frege66c 43893 Swap antecedents of ~ freg...
frege67c 43894 Lemma for ~ frege68c . Pr...
frege68c 43895 Combination of applying a ...
dffrege69 43896 If from the proposition th...
frege70 43897 Lemma for ~ frege72 . Pro...
frege71 43898 Lemma for ~ frege72 . Pro...
frege72 43899 If property ` A ` is hered...
frege73 43900 Lemma for ~ frege87 . Pro...
frege74 43901 If ` X ` has a property ` ...
frege75 43902 If from the proposition th...
dffrege76 43903 If from the two propositio...
frege77 43904 If ` Y ` follows ` X ` in ...
frege78 43905 Commuted form of ~ frege77...
frege79 43906 Distributed form of ~ freg...
frege80 43907 Add additional condition t...
frege81 43908 If ` X ` has a property ` ...
frege82 43909 Closed-form deduction base...
frege83 43910 Apply commuted form of ~ f...
frege84 43911 Commuted form of ~ frege81...
frege85 43912 Commuted form of ~ frege77...
frege86 43913 Conclusion about element o...
frege87 43914 If ` Z ` is a result of an...
frege88 43915 Commuted form of ~ frege87...
frege89 43916 One direction of ~ dffrege...
frege90 43917 Add antecedent to ~ frege8...
frege91 43918 Every result of an applica...
frege92 43919 Inference from ~ frege91 ....
frege93 43920 Necessary condition for tw...
frege94 43921 Looking one past a pair re...
frege95 43922 Looking one past a pair re...
frege96 43923 Every result of an applica...
frege97 43924 The property of following ...
frege98 43925 If ` Y ` follows ` X ` and...
dffrege99 43926 If ` Z ` is identical with...
frege100 43927 One direction of ~ dffrege...
frege101 43928 Lemma for ~ frege102 . Pr...
frege102 43929 If ` Z ` belongs to the ` ...
frege103 43930 Proposition 103 of [Frege1...
frege104 43931 Proposition 104 of [Frege1...
frege105 43932 Proposition 105 of [Frege1...
frege106 43933 Whatever follows ` X ` in ...
frege107 43934 Proposition 107 of [Frege1...
frege108 43935 If ` Y ` belongs to the ` ...
frege109 43936 The property of belonging ...
frege110 43937 Proposition 110 of [Frege1...
frege111 43938 If ` Y ` belongs to the ` ...
frege112 43939 Identity implies belonging...
frege113 43940 Proposition 113 of [Frege1...
frege114 43941 If ` X ` belongs to the ` ...
dffrege115 43942 If from the circumstance t...
frege116 43943 One direction of ~ dffrege...
frege117 43944 Lemma for ~ frege118 . Pr...
frege118 43945 Simplified application of ...
frege119 43946 Lemma for ~ frege120 . Pr...
frege120 43947 Simplified application of ...
frege121 43948 Lemma for ~ frege122 . Pr...
frege122 43949 If ` X ` is a result of an...
frege123 43950 Lemma for ~ frege124 . Pr...
frege124 43951 If ` X ` is a result of an...
frege125 43952 Lemma for ~ frege126 . Pr...
frege126 43953 If ` M ` follows ` Y ` in ...
frege127 43954 Communte antecedents of ~ ...
frege128 43955 Lemma for ~ frege129 . Pr...
frege129 43956 If the procedure ` R ` is ...
frege130 43957 Lemma for ~ frege131 . Pr...
frege131 43958 If the procedure ` R ` is ...
frege132 43959 Lemma for ~ frege133 . Pr...
frege133 43960 If the procedure ` R ` is ...
enrelmap 43961 The set of all possible re...
enrelmapr 43962 The set of all possible re...
enmappw 43963 The set of all mappings fr...
enmappwid 43964 The set of all mappings fr...
rfovd 43965 Value of the operator, ` (...
rfovfvd 43966 Value of the operator, ` (...
rfovfvfvd 43967 Value of the operator, ` (...
rfovcnvf1od 43968 Properties of the operator...
rfovcnvd 43969 Value of the converse of t...
rfovf1od 43970 The value of the operator,...
rfovcnvfvd 43971 Value of the converse of t...
fsovd 43972 Value of the operator, ` (...
fsovrfovd 43973 The operator which gives a...
fsovfvd 43974 Value of the operator, ` (...
fsovfvfvd 43975 Value of the operator, ` (...
fsovfd 43976 The operator, ` ( A O B ) ...
fsovcnvlem 43977 The ` O ` operator, which ...
fsovcnvd 43978 The value of the converse ...
fsovcnvfvd 43979 The value of the converse ...
fsovf1od 43980 The value of ` ( A O B ) `...
dssmapfvd 43981 Value of the duality opera...
dssmapfv2d 43982 Value of the duality opera...
dssmapfv3d 43983 Value of the duality opera...
dssmapnvod 43984 For any base set ` B ` the...
dssmapf1od 43985 For any base set ` B ` the...
dssmap2d 43986 For any base set ` B ` the...
or3or 43987 Decompose disjunction into...
andi3or 43988 Distribute over triple dis...
uneqsn 43989 If a union of classes is e...
brfvimex 43990 If a binary relation holds...
brovmptimex 43991 If a binary relation holds...
brovmptimex1 43992 If a binary relation holds...
brovmptimex2 43993 If a binary relation holds...
brcoffn 43994 Conditions allowing the de...
brcofffn 43995 Conditions allowing the de...
brco2f1o 43996 Conditions allowing the de...
brco3f1o 43997 Conditions allowing the de...
ntrclsbex 43998 If (pseudo-)interior and (...
ntrclsrcomplex 43999 The relative complement of...
neik0imk0p 44000 Kuratowski's K0 axiom impl...
ntrk2imkb 44001 If an interior function is...
ntrkbimka 44002 If the interiors of disjoi...
ntrk0kbimka 44003 If the interiors of disjoi...
clsk3nimkb 44004 If the base set is not emp...
clsk1indlem0 44005 The ansatz closure functio...
clsk1indlem2 44006 The ansatz closure functio...
clsk1indlem3 44007 The ansatz closure functio...
clsk1indlem4 44008 The ansatz closure functio...
clsk1indlem1 44009 The ansatz closure functio...
clsk1independent 44010 For generalized closure fu...
neik0pk1imk0 44011 Kuratowski's K0' and K1 ax...
isotone1 44012 Two different ways to say ...
isotone2 44013 Two different ways to say ...
ntrk1k3eqk13 44014 An interior function is bo...
ntrclsf1o 44015 If (pseudo-)interior and (...
ntrclsnvobr 44016 If (pseudo-)interior and (...
ntrclsiex 44017 If (pseudo-)interior and (...
ntrclskex 44018 If (pseudo-)interior and (...
ntrclsfv1 44019 If (pseudo-)interior and (...
ntrclsfv2 44020 If (pseudo-)interior and (...
ntrclselnel1 44021 If (pseudo-)interior and (...
ntrclselnel2 44022 If (pseudo-)interior and (...
ntrclsfv 44023 The value of the interior ...
ntrclsfveq1 44024 If interior and closure fu...
ntrclsfveq2 44025 If interior and closure fu...
ntrclsfveq 44026 If interior and closure fu...
ntrclsss 44027 If interior and closure fu...
ntrclsneine0lem 44028 If (pseudo-)interior and (...
ntrclsneine0 44029 If (pseudo-)interior and (...
ntrclscls00 44030 If (pseudo-)interior and (...
ntrclsiso 44031 If (pseudo-)interior and (...
ntrclsk2 44032 An interior function is co...
ntrclskb 44033 The interiors of disjoint ...
ntrclsk3 44034 The intersection of interi...
ntrclsk13 44035 The interior of the inters...
ntrclsk4 44036 Idempotence of the interio...
ntrneibex 44037 If (pseudo-)interior and (...
ntrneircomplex 44038 The relative complement of...
ntrneif1o 44039 If (pseudo-)interior and (...
ntrneiiex 44040 If (pseudo-)interior and (...
ntrneinex 44041 If (pseudo-)interior and (...
ntrneicnv 44042 If (pseudo-)interior and (...
ntrneifv1 44043 If (pseudo-)interior and (...
ntrneifv2 44044 If (pseudo-)interior and (...
ntrneiel 44045 If (pseudo-)interior and (...
ntrneifv3 44046 The value of the neighbors...
ntrneineine0lem 44047 If (pseudo-)interior and (...
ntrneineine1lem 44048 If (pseudo-)interior and (...
ntrneifv4 44049 The value of the interior ...
ntrneiel2 44050 Membership in iterated int...
ntrneineine0 44051 If (pseudo-)interior and (...
ntrneineine1 44052 If (pseudo-)interior and (...
ntrneicls00 44053 If (pseudo-)interior and (...
ntrneicls11 44054 If (pseudo-)interior and (...
ntrneiiso 44055 If (pseudo-)interior and (...
ntrneik2 44056 An interior function is co...
ntrneix2 44057 An interior (closure) func...
ntrneikb 44058 The interiors of disjoint ...
ntrneixb 44059 The interiors (closures) o...
ntrneik3 44060 The intersection of interi...
ntrneix3 44061 The closure of the union o...
ntrneik13 44062 The interior of the inters...
ntrneix13 44063 The closure of the union o...
ntrneik4w 44064 Idempotence of the interio...
ntrneik4 44065 Idempotence of the interio...
clsneibex 44066 If (pseudo-)closure and (p...
clsneircomplex 44067 The relative complement of...
clsneif1o 44068 If a (pseudo-)closure func...
clsneicnv 44069 If a (pseudo-)closure func...
clsneikex 44070 If closure and neighborhoo...
clsneinex 44071 If closure and neighborhoo...
clsneiel1 44072 If a (pseudo-)closure func...
clsneiel2 44073 If a (pseudo-)closure func...
clsneifv3 44074 Value of the neighborhoods...
clsneifv4 44075 Value of the closure (inte...
neicvgbex 44076 If (pseudo-)neighborhood a...
neicvgrcomplex 44077 The relative complement of...
neicvgf1o 44078 If neighborhood and conver...
neicvgnvo 44079 If neighborhood and conver...
neicvgnvor 44080 If neighborhood and conver...
neicvgmex 44081 If the neighborhoods and c...
neicvgnex 44082 If the neighborhoods and c...
neicvgel1 44083 A subset being an element ...
neicvgel2 44084 The complement of a subset...
neicvgfv 44085 The value of the neighborh...
ntrrn 44086 The range of the interior ...
ntrf 44087 The interior function of a...
ntrf2 44088 The interior function is a...
ntrelmap 44089 The interior function is a...
clsf2 44090 The closure function is a ...
clselmap 44091 The closure function is a ...
dssmapntrcls 44092 The interior and closure o...
dssmapclsntr 44093 The closure and interior o...
gneispa 44094 Each point ` p ` of the ne...
gneispb 44095 Given a neighborhood ` N `...
gneispace2 44096 The predicate that ` F ` i...
gneispace3 44097 The predicate that ` F ` i...
gneispace 44098 The predicate that ` F ` i...
gneispacef 44099 A generic neighborhood spa...
gneispacef2 44100 A generic neighborhood spa...
gneispacefun 44101 A generic neighborhood spa...
gneispacern 44102 A generic neighborhood spa...
gneispacern2 44103 A generic neighborhood spa...
gneispace0nelrn 44104 A generic neighborhood spa...
gneispace0nelrn2 44105 A generic neighborhood spa...
gneispace0nelrn3 44106 A generic neighborhood spa...
gneispaceel 44107 Every neighborhood of a po...
gneispaceel2 44108 Every neighborhood of a po...
gneispacess 44109 All supersets of a neighbo...
gneispacess2 44110 All supersets of a neighbo...
k0004lem1 44111 Application of ~ ssin to r...
k0004lem2 44112 A mapping with a particula...
k0004lem3 44113 When the value of a mappin...
k0004val 44114 The topological simplex of...
k0004ss1 44115 The topological simplex of...
k0004ss2 44116 The topological simplex of...
k0004ss3 44117 The topological simplex of...
k0004val0 44118 The topological simplex of...
inductionexd 44119 Simple induction example. ...
wwlemuld 44120 Natural deduction form of ...
leeq1d 44121 Specialization of ~ breq1d...
leeq2d 44122 Specialization of ~ breq2d...
absmulrposd 44123 Specialization of absmuld ...
imadisjld 44124 Natural dduction form of o...
wnefimgd 44125 The image of a mapping fro...
fco2d 44126 Natural deduction form of ...
wfximgfd 44127 The value of a function on...
extoimad 44128 If |f(x)| <= C for all x t...
imo72b2lem0 44129 Lemma for ~ imo72b2 . (Co...
suprleubrd 44130 Natural deduction form of ...
imo72b2lem2 44131 Lemma for ~ imo72b2 . (Co...
suprlubrd 44132 Natural deduction form of ...
imo72b2lem1 44133 Lemma for ~ imo72b2 . (Co...
lemuldiv3d 44134 'Less than or equal to' re...
lemuldiv4d 44135 'Less than or equal to' re...
imo72b2 44136 IMO 1972 B2. (14th Intern...
int-addcomd 44137 AdditionCommutativity gene...
int-addassocd 44138 AdditionAssociativity gene...
int-addsimpd 44139 AdditionSimplification gen...
int-mulcomd 44140 MultiplicationCommutativit...
int-mulassocd 44141 MultiplicationAssociativit...
int-mulsimpd 44142 MultiplicationSimplificati...
int-leftdistd 44143 AdditionMultiplicationLeft...
int-rightdistd 44144 AdditionMultiplicationRigh...
int-sqdefd 44145 SquareDefinition generator...
int-mul11d 44146 First MultiplicationOne ge...
int-mul12d 44147 Second MultiplicationOne g...
int-add01d 44148 First AdditionZero generat...
int-add02d 44149 Second AdditionZero genera...
int-sqgeq0d 44150 SquareGEQZero generator ru...
int-eqprincd 44151 PrincipleOfEquality genera...
int-eqtransd 44152 EqualityTransitivity gener...
int-eqmvtd 44153 EquMoveTerm generator rule...
int-eqineqd 44154 EquivalenceImpliesDoubleIn...
int-ineqmvtd 44155 IneqMoveTerm generator rul...
int-ineq1stprincd 44156 FirstPrincipleOfInequality...
int-ineq2ndprincd 44157 SecondPrincipleOfInequalit...
int-ineqtransd 44158 InequalityTransitivity gen...
unitadd 44159 Theorem used in conjunctio...
gsumws3 44160 Valuation of a length 3 wo...
gsumws4 44161 Valuation of a length 4 wo...
amgm2d 44162 Arithmetic-geometric mean ...
amgm3d 44163 Arithmetic-geometric mean ...
amgm4d 44164 Arithmetic-geometric mean ...
spALT 44165 ~ sp can be proven from th...
elnelneqd 44166 Two classes are not equal ...
elnelneq2d 44167 Two classes are not equal ...
rr-spce 44168 Prove an existential. (Co...
rexlimdvaacbv 44169 Unpack a restricted existe...
rexlimddvcbvw 44170 Unpack a restricted existe...
rexlimddvcbv 44171 Unpack a restricted existe...
rr-elrnmpt3d 44172 Elementhood in an image se...
finnzfsuppd 44173 If a function is zero outs...
rr-phpd 44174 Equivalent of ~ php withou...
suceqd 44175 Deduction associated with ...
tfindsd 44176 Deduction associated with ...
mnringvald 44179 Value of the monoid ring f...
mnringnmulrd 44180 Components of a monoid rin...
mnringnmulrdOLD 44181 Obsolete version of ~ mnri...
mnringbased 44182 The base set of a monoid r...
mnringbasedOLD 44183 Obsolete version of ~ mnri...
mnringbaserd 44184 The base set of a monoid r...
mnringelbased 44185 Membership in the base set...
mnringbasefd 44186 Elements of a monoid ring ...
mnringbasefsuppd 44187 Elements of a monoid ring ...
mnringaddgd 44188 The additive operation of ...
mnringaddgdOLD 44189 Obsolete version of ~ mnri...
mnring0gd 44190 The additive identity of a...
mnring0g2d 44191 The additive identity of a...
mnringmulrd 44192 The ring product of a mono...
mnringscad 44193 The scalar ring of a monoi...
mnringscadOLD 44194 Obsolete version of ~ mnri...
mnringvscad 44195 The scalar product of a mo...
mnringvscadOLD 44196 Obsolete version of ~ mnri...
mnringlmodd 44197 Monoid rings are left modu...
mnringmulrvald 44198 Value of multiplication in...
mnringmulrcld 44199 Monoid rings are closed un...
gru0eld 44200 A nonempty Grothendieck un...
grusucd 44201 Grothendieck universes are...
r1rankcld 44202 Any rank of the cumulative...
grur1cld 44203 Grothendieck universes are...
grurankcld 44204 Grothendieck universes are...
grurankrcld 44205 If a Grothendieck universe...
scotteqd 44208 Equality theorem for the S...
scotteq 44209 Closed form of ~ scotteqd ...
nfscott 44210 Bound-variable hypothesis ...
scottabf 44211 Value of the Scott operati...
scottab 44212 Value of the Scott operati...
scottabes 44213 Value of the Scott operati...
scottss 44214 Scott's trick produces a s...
elscottab 44215 An element of the output o...
scottex2 44216 ~ scottex expressed using ...
scotteld 44217 The Scott operation sends ...
scottelrankd 44218 Property of a Scott's tric...
scottrankd 44219 Rank of a nonempty Scott's...
gruscottcld 44220 If a Grothendieck universe...
dfcoll2 44223 Alternate definition of th...
colleq12d 44224 Equality theorem for the c...
colleq1 44225 Equality theorem for the c...
colleq2 44226 Equality theorem for the c...
nfcoll 44227 Bound-variable hypothesis ...
collexd 44228 The output of the collecti...
cpcolld 44229 Property of the collection...
cpcoll2d 44230 ~ cpcolld with an extra ex...
grucollcld 44231 A Grothendieck universe co...
ismnu 44232 The hypothesis of this the...
mnuop123d 44233 Operations of a minimal un...
mnussd 44234 Minimal universes are clos...
mnuss2d 44235 ~ mnussd with arguments pr...
mnu0eld 44236 A nonempty minimal univers...
mnuop23d 44237 Second and third operation...
mnupwd 44238 Minimal universes are clos...
mnusnd 44239 Minimal universes are clos...
mnuprssd 44240 A minimal universe contain...
mnuprss2d 44241 Special case of ~ mnuprssd...
mnuop3d 44242 Third operation of a minim...
mnuprdlem1 44243 Lemma for ~ mnuprd . (Con...
mnuprdlem2 44244 Lemma for ~ mnuprd . (Con...
mnuprdlem3 44245 Lemma for ~ mnuprd . (Con...
mnuprdlem4 44246 Lemma for ~ mnuprd . Gene...
mnuprd 44247 Minimal universes are clos...
mnuunid 44248 Minimal universes are clos...
mnuund 44249 Minimal universes are clos...
mnutrcld 44250 Minimal universes contain ...
mnutrd 44251 Minimal universes are tran...
mnurndlem1 44252 Lemma for ~ mnurnd . (Con...
mnurndlem2 44253 Lemma for ~ mnurnd . Dedu...
mnurnd 44254 Minimal universes contain ...
mnugrud 44255 Minimal universes are Grot...
grumnudlem 44256 Lemma for ~ grumnud . (Co...
grumnud 44257 Grothendieck universes are...
grumnueq 44258 The class of Grothendieck ...
expandan 44259 Expand conjunction to prim...
expandexn 44260 Expand an existential quan...
expandral 44261 Expand a restricted univer...
expandrexn 44262 Expand a restricted existe...
expandrex 44263 Expand a restricted existe...
expanduniss 44264 Expand ` U. A C_ B ` to pr...
ismnuprim 44265 Express the predicate on `...
rr-grothprimbi 44266 Express "every set is cont...
inagrud 44267 Inaccessible levels of the...
inaex 44268 Assuming the Tarski-Grothe...
gruex 44269 Assuming the Tarski-Grothe...
rr-groth 44270 An equivalent of ~ ax-grot...
rr-grothprim 44271 An equivalent of ~ ax-grot...
ismnushort 44272 Express the predicate on `...
dfuniv2 44273 Alternative definition of ...
rr-grothshortbi 44274 Express "every set is cont...
rr-grothshort 44275 A shorter equivalent of ~ ...
nanorxor 44276 'nand' is equivalent to th...
undisjrab 44277 Union of two disjoint rest...
iso0 44278 The empty set is an ` R , ...
ssrecnpr 44279 ` RR ` is a subset of both...
seff 44280 Let set ` S ` be the real ...
sblpnf 44281 The infinity ball in the a...
prmunb2 44282 The primes are unbounded. ...
dvgrat 44283 Ratio test for divergence ...
cvgdvgrat 44284 Ratio test for convergence...
radcnvrat 44285 Let ` L ` be the limit, if...
reldvds 44286 The divides relation is in...
nznngen 44287 All positive integers in t...
nzss 44288 The set of multiples of _m...
nzin 44289 The intersection of the se...
nzprmdif 44290 Subtract one prime's multi...
hashnzfz 44291 Special case of ~ hashdvds...
hashnzfz2 44292 Special case of ~ hashnzfz...
hashnzfzclim 44293 As the upper bound ` K ` o...
caofcan 44294 Transfer a cancellation la...
ofsubid 44295 Function analogue of ~ sub...
ofmul12 44296 Function analogue of ~ mul...
ofdivrec 44297 Function analogue of ~ div...
ofdivcan4 44298 Function analogue of ~ div...
ofdivdiv2 44299 Function analogue of ~ div...
lhe4.4ex1a 44300 Example of the Fundamental...
dvsconst 44301 Derivative of a constant f...
dvsid 44302 Derivative of the identity...
dvsef 44303 Derivative of the exponent...
expgrowthi 44304 Exponential growth and dec...
dvconstbi 44305 The derivative of a functi...
expgrowth 44306 Exponential growth and dec...
bccval 44309 Value of the generalized b...
bcccl 44310 Closure of the generalized...
bcc0 44311 The generalized binomial c...
bccp1k 44312 Generalized binomial coeff...
bccm1k 44313 Generalized binomial coeff...
bccn0 44314 Generalized binomial coeff...
bccn1 44315 Generalized binomial coeff...
bccbc 44316 The binomial coefficient a...
uzmptshftfval 44317 When ` F ` is a maps-to fu...
dvradcnv2 44318 The radius of convergence ...
binomcxplemwb 44319 Lemma for ~ binomcxp . Th...
binomcxplemnn0 44320 Lemma for ~ binomcxp . Wh...
binomcxplemrat 44321 Lemma for ~ binomcxp . As...
binomcxplemfrat 44322 Lemma for ~ binomcxp . ~ b...
binomcxplemradcnv 44323 Lemma for ~ binomcxp . By...
binomcxplemdvbinom 44324 Lemma for ~ binomcxp . By...
binomcxplemcvg 44325 Lemma for ~ binomcxp . Th...
binomcxplemdvsum 44326 Lemma for ~ binomcxp . Th...
binomcxplemnotnn0 44327 Lemma for ~ binomcxp . Wh...
binomcxp 44328 Generalize the binomial th...
pm10.12 44329 Theorem *10.12 in [Whitehe...
pm10.14 44330 Theorem *10.14 in [Whitehe...
pm10.251 44331 Theorem *10.251 in [Whiteh...
pm10.252 44332 Theorem *10.252 in [Whiteh...
pm10.253 44333 Theorem *10.253 in [Whiteh...
albitr 44334 Theorem *10.301 in [Whiteh...
pm10.42 44335 Theorem *10.42 in [Whitehe...
pm10.52 44336 Theorem *10.52 in [Whitehe...
pm10.53 44337 Theorem *10.53 in [Whitehe...
pm10.541 44338 Theorem *10.541 in [Whiteh...
pm10.542 44339 Theorem *10.542 in [Whiteh...
pm10.55 44340 Theorem *10.55 in [Whitehe...
pm10.56 44341 Theorem *10.56 in [Whitehe...
pm10.57 44342 Theorem *10.57 in [Whitehe...
2alanimi 44343 Removes two universal quan...
2al2imi 44344 Removes two universal quan...
pm11.11 44345 Theorem *11.11 in [Whitehe...
pm11.12 44346 Theorem *11.12 in [Whitehe...
19.21vv 44347 Compare Theorem *11.3 in [...
2alim 44348 Theorem *11.32 in [Whitehe...
2albi 44349 Theorem *11.33 in [Whitehe...
2exim 44350 Theorem *11.34 in [Whitehe...
2exbi 44351 Theorem *11.341 in [Whiteh...
spsbce-2 44352 Theorem *11.36 in [Whitehe...
19.33-2 44353 Theorem *11.421 in [Whiteh...
19.36vv 44354 Theorem *11.43 in [Whitehe...
19.31vv 44355 Theorem *11.44 in [Whitehe...
19.37vv 44356 Theorem *11.46 in [Whitehe...
19.28vv 44357 Theorem *11.47 in [Whitehe...
pm11.52 44358 Theorem *11.52 in [Whitehe...
aaanv 44359 Theorem *11.56 in [Whitehe...
pm11.57 44360 Theorem *11.57 in [Whitehe...
pm11.58 44361 Theorem *11.58 in [Whitehe...
pm11.59 44362 Theorem *11.59 in [Whitehe...
pm11.6 44363 Theorem *11.6 in [Whitehea...
pm11.61 44364 Theorem *11.61 in [Whitehe...
pm11.62 44365 Theorem *11.62 in [Whitehe...
pm11.63 44366 Theorem *11.63 in [Whitehe...
pm11.7 44367 Theorem *11.7 in [Whitehea...
pm11.71 44368 Theorem *11.71 in [Whitehe...
sbeqal1 44369 If ` x = y ` always implie...
sbeqal1i 44370 Suppose you know ` x = y `...
sbeqal2i 44371 If ` x = y ` implies ` x =...
axc5c4c711 44372 Proof of a theorem that ca...
axc5c4c711toc5 44373 Rederivation of ~ sp from ...
axc5c4c711toc4 44374 Rederivation of ~ axc4 fro...
axc5c4c711toc7 44375 Rederivation of ~ axc7 fro...
axc5c4c711to11 44376 Rederivation of ~ ax-11 fr...
axc11next 44377 This theorem shows that, g...
pm13.13a 44378 One result of theorem *13....
pm13.13b 44379 Theorem *13.13 in [Whitehe...
pm13.14 44380 Theorem *13.14 in [Whitehe...
pm13.192 44381 Theorem *13.192 in [Whiteh...
pm13.193 44382 Theorem *13.193 in [Whiteh...
pm13.194 44383 Theorem *13.194 in [Whiteh...
pm13.195 44384 Theorem *13.195 in [Whiteh...
pm13.196a 44385 Theorem *13.196 in [Whiteh...
2sbc6g 44386 Theorem *13.21 in [Whitehe...
2sbc5g 44387 Theorem *13.22 in [Whitehe...
iotain 44388 Equivalence between two di...
iotaexeu 44389 The iota class exists. Th...
iotasbc 44390 Definition *14.01 in [Whit...
iotasbc2 44391 Theorem *14.111 in [Whiteh...
pm14.12 44392 Theorem *14.12 in [Whitehe...
pm14.122a 44393 Theorem *14.122 in [Whiteh...
pm14.122b 44394 Theorem *14.122 in [Whiteh...
pm14.122c 44395 Theorem *14.122 in [Whiteh...
pm14.123a 44396 Theorem *14.123 in [Whiteh...
pm14.123b 44397 Theorem *14.123 in [Whiteh...
pm14.123c 44398 Theorem *14.123 in [Whiteh...
pm14.18 44399 Theorem *14.18 in [Whitehe...
iotaequ 44400 Theorem *14.2 in [Whitehea...
iotavalb 44401 Theorem *14.202 in [Whiteh...
iotasbc5 44402 Theorem *14.205 in [Whiteh...
pm14.24 44403 Theorem *14.24 in [Whitehe...
iotavalsb 44404 Theorem *14.242 in [Whiteh...
sbiota1 44405 Theorem *14.25 in [Whitehe...
sbaniota 44406 Theorem *14.26 in [Whitehe...
eubiOLD 44407 Obsolete proof of ~ eubi a...
iotasbcq 44408 Theorem *14.272 in [Whiteh...
elnev 44409 Any set that contains one ...
rusbcALT 44410 A version of Russell's par...
compeq 44411 Equality between two ways ...
compne 44412 The complement of ` A ` is...
compab 44413 Two ways of saying "the co...
conss2 44414 Contrapositive law for sub...
conss1 44415 Contrapositive law for sub...
ralbidar 44416 More general form of ~ ral...
rexbidar 44417 More general form of ~ rex...
dropab1 44418 Theorem to aid use of the ...
dropab2 44419 Theorem to aid use of the ...
ipo0 44420 If the identity relation p...
ifr0 44421 A class that is founded by...
ordpss 44422 ~ ordelpss with an anteced...
fvsb 44423 Explicit substitution of a...
fveqsb 44424 Implicit substitution of a...
xpexb 44425 A Cartesian product exists...
trelpss 44426 An element of a transitive...
addcomgi 44427 Generalization of commutat...
addrval 44437 Value of the operation of ...
subrval 44438 Value of the operation of ...
mulvval 44439 Value of the operation of ...
addrfv 44440 Vector addition at a value...
subrfv 44441 Vector subtraction at a va...
mulvfv 44442 Scalar multiplication at a...
addrfn 44443 Vector addition produces a...
subrfn 44444 Vector subtraction produce...
mulvfn 44445 Scalar multiplication prod...
addrcom 44446 Vector addition is commuta...
idiALT 44450 Placeholder for ~ idi . T...
exbir 44451 Exportation implication al...
3impexpbicom 44452 Version of ~ 3impexp where...
3impexpbicomi 44453 Inference associated with ...
bi1imp 44454 Importation inference simi...
bi2imp 44455 Importation inference simi...
bi3impb 44456 Similar to ~ 3impb with im...
bi3impa 44457 Similar to ~ 3impa with im...
bi23impib 44458 ~ 3impib with the inner im...
bi13impib 44459 ~ 3impib with the outer im...
bi123impib 44460 ~ 3impib with the implicat...
bi13impia 44461 ~ 3impia with the outer im...
bi123impia 44462 ~ 3impia with the implicat...
bi33imp12 44463 ~ 3imp with innermost impl...
bi23imp13 44464 ~ 3imp with middle implica...
bi13imp23 44465 ~ 3imp with outermost impl...
bi13imp2 44466 Similar to ~ 3imp except t...
bi12imp3 44467 Similar to ~ 3imp except a...
bi23imp1 44468 Similar to ~ 3imp except a...
bi123imp0 44469 Similar to ~ 3imp except a...
4animp1 44470 A single hypothesis unific...
4an31 44471 A rearrangement of conjunc...
4an4132 44472 A rearrangement of conjunc...
expcomdg 44473 Biconditional form of ~ ex...
iidn3 44474 ~ idn3 without virtual ded...
ee222 44475 ~ e222 without virtual ded...
ee3bir 44476 Right-biconditional form o...
ee13 44477 ~ e13 without virtual dedu...
ee121 44478 ~ e121 without virtual ded...
ee122 44479 ~ e122 without virtual ded...
ee333 44480 ~ e333 without virtual ded...
ee323 44481 ~ e323 without virtual ded...
3ornot23 44482 If the second and third di...
orbi1r 44483 ~ orbi1 with order of disj...
3orbi123 44484 ~ pm4.39 with a 3-conjunct...
syl5imp 44485 Closed form of ~ syl5 . D...
impexpd 44486 The following User's Proof...
com3rgbi 44487 The following User's Proof...
impexpdcom 44488 The following User's Proof...
ee1111 44489 Non-virtual deduction form...
pm2.43bgbi 44490 Logical equivalence of a 2...
pm2.43cbi 44491 Logical equivalence of a 3...
ee233 44492 Non-virtual deduction form...
imbi13 44493 Join three logical equival...
ee33 44494 Non-virtual deduction form...
con5 44495 Biconditional contrapositi...
con5i 44496 Inference form of ~ con5 ....
exlimexi 44497 Inference similar to Theor...
sb5ALT 44498 Equivalence for substituti...
eexinst01 44499 ~ exinst01 without virtual...
eexinst11 44500 ~ exinst11 without virtual...
vk15.4j 44501 Excercise 4j of Unit 15 of...
notnotrALT 44502 Converse of double negatio...
con3ALT2 44503 Contraposition. Alternate...
ssralv2 44504 Quantification restricted ...
sbc3or 44505 ~ sbcor with a 3-disjuncts...
alrim3con13v 44506 Closed form of ~ alrimi wi...
rspsbc2 44507 ~ rspsbc with two quantify...
sbcoreleleq 44508 Substitution of a setvar v...
tratrb 44509 If a class is transitive a...
ordelordALT 44510 An element of an ordinal c...
sbcim2g 44511 Distribution of class subs...
sbcbi 44512 Implication form of ~ sbcb...
trsbc 44513 Formula-building inference...
truniALT 44514 The union of a class of tr...
onfrALTlem5 44515 Lemma for ~ onfrALT . (Co...
onfrALTlem4 44516 Lemma for ~ onfrALT . (Co...
onfrALTlem3 44517 Lemma for ~ onfrALT . (Co...
ggen31 44518 ~ gen31 without virtual de...
onfrALTlem2 44519 Lemma for ~ onfrALT . (Co...
cbvexsv 44520 A theorem pertaining to th...
onfrALTlem1 44521 Lemma for ~ onfrALT . (Co...
onfrALT 44522 The membership relation is...
19.41rg 44523 Closed form of right-to-le...
opelopab4 44524 Ordered pair membership in...
2pm13.193 44525 ~ pm13.193 for two variabl...
hbntal 44526 A closed form of ~ hbn . ~...
hbimpg 44527 A closed form of ~ hbim . ...
hbalg 44528 Closed form of ~ hbal . D...
hbexg 44529 Closed form of ~ nfex . D...
ax6e2eq 44530 Alternate form of ~ ax6e f...
ax6e2nd 44531 If at least two sets exist...
ax6e2ndeq 44532 "At least two sets exist" ...
2sb5nd 44533 Equivalence for double sub...
2uasbanh 44534 Distribute the unabbreviat...
2uasban 44535 Distribute the unabbreviat...
e2ebind 44536 Absorption of an existenti...
elpwgded 44537 ~ elpwgdedVD in convention...
trelded 44538 Deduction form of ~ trel ....
jaoded 44539 Deduction form of ~ jao . ...
sbtT 44540 A substitution into a theo...
not12an2impnot1 44541 If a double conjunction is...
in1 44544 Inference form of ~ df-vd1...
iin1 44545 ~ in1 without virtual dedu...
dfvd1ir 44546 Inference form of ~ df-vd1...
idn1 44547 Virtual deduction identity...
dfvd1imp 44548 Left-to-right part of defi...
dfvd1impr 44549 Right-to-left part of defi...
dfvd2 44552 Definition of a 2-hypothes...
dfvd2an 44555 Definition of a 2-hypothes...
dfvd2ani 44556 Inference form of ~ dfvd2a...
dfvd2anir 44557 Right-to-left inference fo...
dfvd2i 44558 Inference form of ~ dfvd2 ...
dfvd2ir 44559 Right-to-left inference fo...
dfvd3 44564 Definition of a 3-hypothes...
dfvd3i 44565 Inference form of ~ dfvd3 ...
dfvd3ir 44566 Right-to-left inference fo...
dfvd3an 44567 Definition of a 3-hypothes...
dfvd3ani 44568 Inference form of ~ dfvd3a...
dfvd3anir 44569 Right-to-left inference fo...
vd01 44570 A virtual hypothesis virtu...
vd02 44571 Two virtual hypotheses vir...
vd03 44572 A theorem is virtually inf...
vd12 44573 A virtual deduction with 1...
vd13 44574 A virtual deduction with 1...
vd23 44575 A virtual deduction with 2...
dfvd2imp 44576 The virtual deduction form...
dfvd2impr 44577 A 2-antecedent nested impl...
in2 44578 The virtual deduction intr...
int2 44579 The virtual deduction intr...
iin2 44580 ~ in2 without virtual dedu...
in2an 44581 The virtual deduction intr...
in3 44582 The virtual deduction intr...
iin3 44583 ~ in3 without virtual dedu...
in3an 44584 The virtual deduction intr...
int3 44585 The virtual deduction intr...
idn2 44586 Virtual deduction identity...
iden2 44587 Virtual deduction identity...
idn3 44588 Virtual deduction identity...
gen11 44589 Virtual deduction generali...
gen11nv 44590 Virtual deduction generali...
gen12 44591 Virtual deduction generali...
gen21 44592 Virtual deduction generali...
gen21nv 44593 Virtual deduction form of ...
gen31 44594 Virtual deduction generali...
gen22 44595 Virtual deduction generali...
ggen22 44596 ~ gen22 without virtual de...
exinst 44597 Existential Instantiation....
exinst01 44598 Existential Instantiation....
exinst11 44599 Existential Instantiation....
e1a 44600 A Virtual deduction elimin...
el1 44601 A Virtual deduction elimin...
e1bi 44602 Biconditional form of ~ e1...
e1bir 44603 Right biconditional form o...
e2 44604 A virtual deduction elimin...
e2bi 44605 Biconditional form of ~ e2...
e2bir 44606 Right biconditional form o...
ee223 44607 ~ e223 without virtual ded...
e223 44608 A virtual deduction elimin...
e222 44609 A virtual deduction elimin...
e220 44610 A virtual deduction elimin...
ee220 44611 ~ e220 without virtual ded...
e202 44612 A virtual deduction elimin...
ee202 44613 ~ e202 without virtual ded...
e022 44614 A virtual deduction elimin...
ee022 44615 ~ e022 without virtual ded...
e002 44616 A virtual deduction elimin...
ee002 44617 ~ e002 without virtual ded...
e020 44618 A virtual deduction elimin...
ee020 44619 ~ e020 without virtual ded...
e200 44620 A virtual deduction elimin...
ee200 44621 ~ e200 without virtual ded...
e221 44622 A virtual deduction elimin...
ee221 44623 ~ e221 without virtual ded...
e212 44624 A virtual deduction elimin...
ee212 44625 ~ e212 without virtual ded...
e122 44626 A virtual deduction elimin...
e112 44627 A virtual deduction elimin...
ee112 44628 ~ e112 without virtual ded...
e121 44629 A virtual deduction elimin...
e211 44630 A virtual deduction elimin...
ee211 44631 ~ e211 without virtual ded...
e210 44632 A virtual deduction elimin...
ee210 44633 ~ e210 without virtual ded...
e201 44634 A virtual deduction elimin...
ee201 44635 ~ e201 without virtual ded...
e120 44636 A virtual deduction elimin...
ee120 44637 Virtual deduction rule ~ e...
e021 44638 A virtual deduction elimin...
ee021 44639 ~ e021 without virtual ded...
e012 44640 A virtual deduction elimin...
ee012 44641 ~ e012 without virtual ded...
e102 44642 A virtual deduction elimin...
ee102 44643 ~ e102 without virtual ded...
e22 44644 A virtual deduction elimin...
e22an 44645 Conjunction form of ~ e22 ...
ee22an 44646 ~ e22an without virtual de...
e111 44647 A virtual deduction elimin...
e1111 44648 A virtual deduction elimin...
e110 44649 A virtual deduction elimin...
ee110 44650 ~ e110 without virtual ded...
e101 44651 A virtual deduction elimin...
ee101 44652 ~ e101 without virtual ded...
e011 44653 A virtual deduction elimin...
ee011 44654 ~ e011 without virtual ded...
e100 44655 A virtual deduction elimin...
ee100 44656 ~ e100 without virtual ded...
e010 44657 A virtual deduction elimin...
ee010 44658 ~ e010 without virtual ded...
e001 44659 A virtual deduction elimin...
ee001 44660 ~ e001 without virtual ded...
e11 44661 A virtual deduction elimin...
e11an 44662 Conjunction form of ~ e11 ...
ee11an 44663 ~ e11an without virtual de...
e01 44664 A virtual deduction elimin...
e01an 44665 Conjunction form of ~ e01 ...
ee01an 44666 ~ e01an without virtual de...
e10 44667 A virtual deduction elimin...
e10an 44668 Conjunction form of ~ e10 ...
ee10an 44669 ~ e10an without virtual de...
e02 44670 A virtual deduction elimin...
e02an 44671 Conjunction form of ~ e02 ...
ee02an 44672 ~ e02an without virtual de...
eel021old 44673 ~ el021old without virtual...
el021old 44674 A virtual deduction elimin...
eel132 44675 ~ syl2an with antecedents ...
eel000cT 44676 An elimination deduction. ...
eel0TT 44677 An elimination deduction. ...
eelT00 44678 An elimination deduction. ...
eelTTT 44679 An elimination deduction. ...
eelT11 44680 An elimination deduction. ...
eelT1 44681 Syllogism inference combin...
eelT12 44682 An elimination deduction. ...
eelTT1 44683 An elimination deduction. ...
eelT01 44684 An elimination deduction. ...
eel0T1 44685 An elimination deduction. ...
eel12131 44686 An elimination deduction. ...
eel2131 44687 ~ syl2an with antecedents ...
eel3132 44688 ~ syl2an with antecedents ...
eel0321old 44689 ~ el0321old without virtua...
el0321old 44690 A virtual deduction elimin...
eel2122old 44691 ~ el2122old without virtua...
el2122old 44692 A virtual deduction elimin...
eel0000 44693 Elimination rule similar t...
eel00001 44694 An elimination deduction. ...
eel00000 44695 Elimination rule similar ~...
eel11111 44696 Five-hypothesis eliminatio...
e12 44697 A virtual deduction elimin...
e12an 44698 Conjunction form of ~ e12 ...
el12 44699 Virtual deduction form of ...
e20 44700 A virtual deduction elimin...
e20an 44701 Conjunction form of ~ e20 ...
ee20an 44702 ~ e20an without virtual de...
e21 44703 A virtual deduction elimin...
e21an 44704 Conjunction form of ~ e21 ...
ee21an 44705 ~ e21an without virtual de...
e333 44706 A virtual deduction elimin...
e33 44707 A virtual deduction elimin...
e33an 44708 Conjunction form of ~ e33 ...
ee33an 44709 ~ e33an without virtual de...
e3 44710 Meta-connective form of ~ ...
e3bi 44711 Biconditional form of ~ e3...
e3bir 44712 Right biconditional form o...
e03 44713 A virtual deduction elimin...
ee03 44714 ~ e03 without virtual dedu...
e03an 44715 Conjunction form of ~ e03 ...
ee03an 44716 Conjunction form of ~ ee03...
e30 44717 A virtual deduction elimin...
ee30 44718 ~ e30 without virtual dedu...
e30an 44719 A virtual deduction elimin...
ee30an 44720 Conjunction form of ~ ee30...
e13 44721 A virtual deduction elimin...
e13an 44722 A virtual deduction elimin...
ee13an 44723 ~ e13an without virtual de...
e31 44724 A virtual deduction elimin...
ee31 44725 ~ e31 without virtual dedu...
e31an 44726 A virtual deduction elimin...
ee31an 44727 ~ e31an without virtual de...
e23 44728 A virtual deduction elimin...
e23an 44729 A virtual deduction elimin...
ee23an 44730 ~ e23an without virtual de...
e32 44731 A virtual deduction elimin...
ee32 44732 ~ e32 without virtual dedu...
e32an 44733 A virtual deduction elimin...
ee32an 44734 ~ e33an without virtual de...
e123 44735 A virtual deduction elimin...
ee123 44736 ~ e123 without virtual ded...
el123 44737 A virtual deduction elimin...
e233 44738 A virtual deduction elimin...
e323 44739 A virtual deduction elimin...
e000 44740 A virtual deduction elimin...
e00 44741 Elimination rule identical...
e00an 44742 Elimination rule identical...
eel00cT 44743 An elimination deduction. ...
eelTT 44744 An elimination deduction. ...
e0a 44745 Elimination rule identical...
eelT 44746 An elimination deduction. ...
eel0cT 44747 An elimination deduction. ...
eelT0 44748 An elimination deduction. ...
e0bi 44749 Elimination rule identical...
e0bir 44750 Elimination rule identical...
uun0.1 44751 Convention notation form o...
un0.1 44752 ` T. ` is the constant tru...
uunT1 44753 A deduction unionizing a n...
uunT1p1 44754 A deduction unionizing a n...
uunT21 44755 A deduction unionizing a n...
uun121 44756 A deduction unionizing a n...
uun121p1 44757 A deduction unionizing a n...
uun132 44758 A deduction unionizing a n...
uun132p1 44759 A deduction unionizing a n...
anabss7p1 44760 A deduction unionizing a n...
un10 44761 A unionizing deduction. (...
un01 44762 A unionizing deduction. (...
un2122 44763 A deduction unionizing a n...
uun2131 44764 A deduction unionizing a n...
uun2131p1 44765 A deduction unionizing a n...
uunTT1 44766 A deduction unionizing a n...
uunTT1p1 44767 A deduction unionizing a n...
uunTT1p2 44768 A deduction unionizing a n...
uunT11 44769 A deduction unionizing a n...
uunT11p1 44770 A deduction unionizing a n...
uunT11p2 44771 A deduction unionizing a n...
uunT12 44772 A deduction unionizing a n...
uunT12p1 44773 A deduction unionizing a n...
uunT12p2 44774 A deduction unionizing a n...
uunT12p3 44775 A deduction unionizing a n...
uunT12p4 44776 A deduction unionizing a n...
uunT12p5 44777 A deduction unionizing a n...
uun111 44778 A deduction unionizing a n...
3anidm12p1 44779 A deduction unionizing a n...
3anidm12p2 44780 A deduction unionizing a n...
uun123 44781 A deduction unionizing a n...
uun123p1 44782 A deduction unionizing a n...
uun123p2 44783 A deduction unionizing a n...
uun123p3 44784 A deduction unionizing a n...
uun123p4 44785 A deduction unionizing a n...
uun2221 44786 A deduction unionizing a n...
uun2221p1 44787 A deduction unionizing a n...
uun2221p2 44788 A deduction unionizing a n...
3impdirp1 44789 A deduction unionizing a n...
3impcombi 44790 A 1-hypothesis proposition...
trsspwALT 44791 Virtual deduction proof of...
trsspwALT2 44792 Virtual deduction proof of...
trsspwALT3 44793 Short predicate calculus p...
sspwtr 44794 Virtual deduction proof of...
sspwtrALT 44795 Virtual deduction proof of...
sspwtrALT2 44796 Short predicate calculus p...
pwtrVD 44797 Virtual deduction proof of...
pwtrrVD 44798 Virtual deduction proof of...
suctrALT 44799 The successor of a transit...
snssiALTVD 44800 Virtual deduction proof of...
snssiALT 44801 If a class is an element o...
snsslVD 44802 Virtual deduction proof of...
snssl 44803 If a singleton is a subcla...
snelpwrVD 44804 Virtual deduction proof of...
unipwrVD 44805 Virtual deduction proof of...
unipwr 44806 A class is a subclass of t...
sstrALT2VD 44807 Virtual deduction proof of...
sstrALT2 44808 Virtual deduction proof of...
suctrALT2VD 44809 Virtual deduction proof of...
suctrALT2 44810 Virtual deduction proof of...
elex2VD 44811 Virtual deduction proof of...
elex22VD 44812 Virtual deduction proof of...
eqsbc2VD 44813 Virtual deduction proof of...
zfregs2VD 44814 Virtual deduction proof of...
tpid3gVD 44815 Virtual deduction proof of...
en3lplem1VD 44816 Virtual deduction proof of...
en3lplem2VD 44817 Virtual deduction proof of...
en3lpVD 44818 Virtual deduction proof of...
simplbi2VD 44819 Virtual deduction proof of...
3ornot23VD 44820 Virtual deduction proof of...
orbi1rVD 44821 Virtual deduction proof of...
bitr3VD 44822 Virtual deduction proof of...
3orbi123VD 44823 Virtual deduction proof of...
sbc3orgVD 44824 Virtual deduction proof of...
19.21a3con13vVD 44825 Virtual deduction proof of...
exbirVD 44826 Virtual deduction proof of...
exbiriVD 44827 Virtual deduction proof of...
rspsbc2VD 44828 Virtual deduction proof of...
3impexpVD 44829 Virtual deduction proof of...
3impexpbicomVD 44830 Virtual deduction proof of...
3impexpbicomiVD 44831 Virtual deduction proof of...
sbcoreleleqVD 44832 Virtual deduction proof of...
hbra2VD 44833 Virtual deduction proof of...
tratrbVD 44834 Virtual deduction proof of...
al2imVD 44835 Virtual deduction proof of...
syl5impVD 44836 Virtual deduction proof of...
idiVD 44837 Virtual deduction proof of...
ancomstVD 44838 Closed form of ~ ancoms . ...
ssralv2VD 44839 Quantification restricted ...
ordelordALTVD 44840 An element of an ordinal c...
equncomVD 44841 If a class equals the unio...
equncomiVD 44842 Inference form of ~ equnco...
sucidALTVD 44843 A set belongs to its succe...
sucidALT 44844 A set belongs to its succe...
sucidVD 44845 A set belongs to its succe...
imbi12VD 44846 Implication form of ~ imbi...
imbi13VD 44847 Join three logical equival...
sbcim2gVD 44848 Distribution of class subs...
sbcbiVD 44849 Implication form of ~ sbcb...
trsbcVD 44850 Formula-building inference...
truniALTVD 44851 The union of a class of tr...
ee33VD 44852 Non-virtual deduction form...
trintALTVD 44853 The intersection of a clas...
trintALT 44854 The intersection of a clas...
undif3VD 44855 The first equality of Exer...
sbcssgVD 44856 Virtual deduction proof of...
csbingVD 44857 Virtual deduction proof of...
onfrALTlem5VD 44858 Virtual deduction proof of...
onfrALTlem4VD 44859 Virtual deduction proof of...
onfrALTlem3VD 44860 Virtual deduction proof of...
simplbi2comtVD 44861 Virtual deduction proof of...
onfrALTlem2VD 44862 Virtual deduction proof of...
onfrALTlem1VD 44863 Virtual deduction proof of...
onfrALTVD 44864 Virtual deduction proof of...
csbeq2gVD 44865 Virtual deduction proof of...
csbsngVD 44866 Virtual deduction proof of...
csbxpgVD 44867 Virtual deduction proof of...
csbresgVD 44868 Virtual deduction proof of...
csbrngVD 44869 Virtual deduction proof of...
csbima12gALTVD 44870 Virtual deduction proof of...
csbunigVD 44871 Virtual deduction proof of...
csbfv12gALTVD 44872 Virtual deduction proof of...
con5VD 44873 Virtual deduction proof of...
relopabVD 44874 Virtual deduction proof of...
19.41rgVD 44875 Virtual deduction proof of...
2pm13.193VD 44876 Virtual deduction proof of...
hbimpgVD 44877 Virtual deduction proof of...
hbalgVD 44878 Virtual deduction proof of...
hbexgVD 44879 Virtual deduction proof of...
ax6e2eqVD 44880 The following User's Proof...
ax6e2ndVD 44881 The following User's Proof...
ax6e2ndeqVD 44882 The following User's Proof...
2sb5ndVD 44883 The following User's Proof...
2uasbanhVD 44884 The following User's Proof...
e2ebindVD 44885 The following User's Proof...
sb5ALTVD 44886 The following User's Proof...
vk15.4jVD 44887 The following User's Proof...
notnotrALTVD 44888 The following User's Proof...
con3ALTVD 44889 The following User's Proof...
elpwgdedVD 44890 Membership in a power clas...
sspwimp 44891 If a class is a subclass o...
sspwimpVD 44892 The following User's Proof...
sspwimpcf 44893 If a class is a subclass o...
sspwimpcfVD 44894 The following User's Proof...
suctrALTcf 44895 The successor of a transit...
suctrALTcfVD 44896 The following User's Proof...
suctrALT3 44897 The successor of a transit...
sspwimpALT 44898 If a class is a subclass o...
unisnALT 44899 A set equals the union of ...
notnotrALT2 44900 Converse of double negatio...
sspwimpALT2 44901 If a class is a subclass o...
e2ebindALT 44902 Absorption of an existenti...
ax6e2ndALT 44903 If at least two sets exist...
ax6e2ndeqALT 44904 "At least two sets exist" ...
2sb5ndALT 44905 Equivalence for double sub...
chordthmALT 44906 The intersecting chords th...
isosctrlem1ALT 44907 Lemma for ~ isosctr . Thi...
iunconnlem2 44908 The indexed union of conne...
iunconnALT 44909 The indexed union of conne...
sineq0ALT 44910 A complex number whose sin...
trwf 44911 The class of well-founded ...
traxext 44912 A transitive class models ...
wfaxext 44913 The class of well-founded ...
xpwf 44914 The Cartesian product of t...
dmwf 44915 The domain of a well-found...
rnwf 44916 The range of a well-founde...
evth2f 44917 A version of ~ evth2 using...
elunif 44918 A version of ~ eluni using...
rzalf 44919 A version of ~ rzal using ...
fvelrnbf 44920 A version of ~ fvelrnb usi...
rfcnpre1 44921 If F is a continuous funct...
ubelsupr 44922 If U belongs to A and U is...
fsumcnf 44923 A finite sum of functions ...
mulltgt0 44924 The product of a negative ...
rspcegf 44925 A version of ~ rspcev usin...
rabexgf 44926 A version of ~ rabexg usin...
fcnre 44927 A function continuous with...
sumsnd 44928 A sum of a singleton is th...
evthf 44929 A version of ~ evth using ...
cnfex 44930 The class of continuous fu...
fnchoice 44931 For a finite set, a choice...
refsumcn 44932 A finite sum of continuous...
rfcnpre2 44933 If ` F ` is a continuous f...
cncmpmax 44934 When the hypothesis for th...
rfcnpre3 44935 If F is a continuous funct...
rfcnpre4 44936 If F is a continuous funct...
sumpair 44937 Sum of two distinct comple...
rfcnnnub 44938 Given a real continuous fu...
refsum2cnlem1 44939 This is the core Lemma for...
refsum2cn 44940 The sum of two continuus r...
adantlllr 44941 Deduction adding a conjunc...
3adantlr3 44942 Deduction adding a conjunc...
3adantll2 44943 Deduction adding a conjunc...
3adantll3 44944 Deduction adding a conjunc...
ssnel 44945 If not element of a set, t...
sncldre 44946 A singleton is closed w.r....
n0p 44947 A polynomial with a nonzer...
pm2.65ni 44948 Inference rule for proof b...
pwssfi 44949 Every element of the power...
iuneq2df 44950 Equality deduction for ind...
nnfoctb 44951 There exists a mapping fro...
ssinss1d 44952 Intersection preserves sub...
elpwinss 44953 An element of the powerset...
unidmex 44954 If ` F ` is a set, then ` ...
ndisj2 44955 A non-disjointness conditi...
zenom 44956 The set of integer numbers...
uzwo4 44957 Well-ordering principle: a...
unisn0 44958 The union of the singleton...
ssin0 44959 If two classes are disjoin...
inabs3 44960 Absorption law for interse...
pwpwuni 44961 Relationship between power...
disjiun2 44962 In a disjoint collection, ...
0pwfi 44963 The empty set is in any po...
ssinss2d 44964 Intersection preserves sub...
zct 44965 The set of integer numbers...
pwfin0 44966 A finite set always belong...
uzct 44967 An upper integer set is co...
iunxsnf 44968 A singleton index picks ou...
fiiuncl 44969 If a set is closed under t...
iunp1 44970 The addition of the next s...
fiunicl 44971 If a set is closed under t...
ixpeq2d 44972 Equality theorem for infin...
disjxp1 44973 The sets of a cartesian pr...
disjsnxp 44974 The sets in the cartesian ...
eliind 44975 Membership in indexed inte...
rspcef 44976 Restricted existential spe...
ixpssmapc 44977 An infinite Cartesian prod...
elintd 44978 Membership in class inters...
ssdf 44979 A sufficient condition for...
brneqtrd 44980 Substitution of equal clas...
ssnct 44981 A set containing an uncoun...
ssuniint 44982 Sufficient condition for b...
elintdv 44983 Membership in class inters...
ssd 44984 A sufficient condition for...
ralimralim 44985 Introducing any antecedent...
snelmap 44986 Membership of the element ...
xrnmnfpnf 44987 An extended real that is n...
nelrnmpt 44988 Non-membership in the rang...
iuneq1i 44989 Equality theorem for index...
nssrex 44990 Negation of subclass relat...
ssinc 44991 Inclusion relation for a m...
ssdec 44992 Inclusion relation for a m...
elixpconstg 44993 Membership in an infinite ...
iineq1d 44994 Equality theorem for index...
metpsmet 44995 A metric is a pseudometric...
ixpssixp 44996 Subclass theorem for infin...
ballss3 44997 A sufficient condition for...
iunincfi 44998 Given a sequence of increa...
nsstr 44999 If it's not a subclass, it...
rexanuz3 45000 Combine two different uppe...
cbvmpo2 45001 Rule to change the second ...
cbvmpo1 45002 Rule to change the first b...
eliuniin 45003 Indexed union of indexed i...
ssabf 45004 Subclass of a class abstra...
pssnssi 45005 A proper subclass does not...
rabidim2 45006 Membership in a restricted...
eluni2f 45007 Membership in class union....
eliin2f 45008 Membership in indexed inte...
nssd 45009 Negation of subclass relat...
iineq12dv 45010 Equality deduction for ind...
supxrcld 45011 The supremum of an arbitra...
elrestd 45012 A sufficient condition for...
eliuniincex 45013 Counterexample to show tha...
eliincex 45014 Counterexample to show tha...
eliinid 45015 Membership in an indexed i...
abssf 45016 Class abstraction in a sub...
supxrubd 45017 A member of a set of exten...
ssrabf 45018 Subclass of a restricted c...
ssrabdf 45019 Subclass of a restricted c...
eliin2 45020 Membership in indexed inte...
ssrab2f 45021 Subclass relation for a re...
restuni3 45022 The underlying set of a su...
rabssf 45023 Restricted class abstracti...
eliuniin2 45024 Indexed union of indexed i...
restuni4 45025 The underlying set of a su...
restuni6 45026 The underlying set of a su...
restuni5 45027 The underlying set of a su...
unirestss 45028 The union of an elementwis...
iniin1 45029 Indexed intersection of in...
iniin2 45030 Indexed intersection of in...
cbvrabv2 45031 A more general version of ...
cbvrabv2w 45032 A more general version of ...
iinssiin 45033 Subset implication for an ...
eliind2 45034 Membership in indexed inte...
iinssd 45035 Subset implication for an ...
rabbida2 45036 Equivalent wff's yield equ...
iinexd 45037 The existence of an indexe...
rabexf 45038 Separation Scheme in terms...
rabbida3 45039 Equivalent wff's yield equ...
r19.36vf 45040 Restricted quantifier vers...
raleqd 45041 Equality deduction for res...
iinssf 45042 Subset implication for an ...
iinssdf 45043 Subset implication for an ...
resabs2i 45044 Absorption law for restric...
ssdf2 45045 A sufficient condition for...
rabssd 45046 Restricted class abstracti...
rexnegd 45047 Minus a real number. (Con...
rexlimd3 45048 * Inference from Theorem 1...
resabs1i 45049 Absorption law for restric...
nel1nelin 45050 Membership in an intersect...
nel2nelin 45051 Membership in an intersect...
nel1nelini 45052 Membership in an intersect...
nel2nelini 45053 Membership in an intersect...
eliunid 45054 Membership in indexed unio...
reximdd 45055 Deduction from Theorem 19....
inopnd 45056 The intersection of two op...
ss2rabdf 45057 Deduction of restricted ab...
restopn3 45058 If ` A ` is open, then ` A...
restopnssd 45059 A topology restricted to a...
restsubel 45060 A subset belongs in the sp...
toprestsubel 45061 A subset is open in the to...
rabidd 45062 An "identity" law of concr...
iunssdf 45063 Subset theorem for an inde...
iinss2d 45064 Subset implication for an ...
r19.3rzf 45065 Restricted quantification ...
r19.28zf 45066 Restricted quantifier vers...
iindif2f 45067 Indexed intersection of cl...
ralfal 45068 Two ways of expressing emp...
archd 45069 Archimedean property of re...
eliund 45070 Membership in indexed unio...
nimnbi 45071 If an implication is false...
nimnbi2 45072 If an implication is false...
notbicom 45073 Commutative law for the ne...
rexeqif 45074 Equality inference for res...
rspced 45075 Restricted existential spe...
feq1dd 45076 Equality deduction for fun...
fnresdmss 45077 A function does not change...
fmptsnxp 45078 Maps-to notation and Carte...
fvmpt2bd 45079 Value of a function given ...
rnmptfi 45080 The range of a function wi...
fresin2 45081 Restriction of a function ...
ffi 45082 A function with finite dom...
suprnmpt 45083 An explicit bound for the ...
rnffi 45084 The range of a function wi...
mptelpm 45085 A function in maps-to nota...
rnmptpr 45086 Range of a function define...
resmpti 45087 Restriction of the mapping...
founiiun 45088 Union expressed as an inde...
rnresun 45089 Distribution law for range...
elrnmptf 45090 The range of a function in...
rnmptssrn 45091 Inclusion relation for two...
disjf1 45092 A 1 to 1 mapping built fro...
rnsnf 45093 The range of a function wh...
wessf1ornlem 45094 Given a function ` F ` on ...
wessf1orn 45095 Given a function ` F ` on ...
nelrnres 45096 If ` A ` is not in the ran...
disjrnmpt2 45097 Disjointness of the range ...
elrnmpt1sf 45098 Elementhood in an image se...
founiiun0 45099 Union expressed as an inde...
disjf1o 45100 A bijection built from dis...
disjinfi 45101 Only a finite number of di...
fvovco 45102 Value of the composition o...
ssnnf1octb 45103 There exists a bijection b...
nnf1oxpnn 45104 There is a bijection betwe...
rnmptssd 45105 The range of a function gi...
projf1o 45106 A biijection from a set to...
fvmap 45107 Function value for a membe...
fvixp2 45108 Projection of a factor of ...
choicefi 45109 For a finite set, a choice...
mpct 45110 The exponentiation of a co...
cnmetcoval 45111 Value of the distance func...
fcomptss 45112 Express composition of two...
elmapsnd 45113 Membership in a set expone...
mapss2 45114 Subset inheritance for set...
fsneq 45115 Equality condition for two...
difmap 45116 Difference of two sets exp...
unirnmap 45117 Given a subset of a set ex...
inmap 45118 Intersection of two sets e...
fcoss 45119 Composition of two mapping...
fsneqrn 45120 Equality condition for two...
difmapsn 45121 Difference of two sets exp...
mapssbi 45122 Subset inheritance for set...
unirnmapsn 45123 Equality theorem for a sub...
iunmapss 45124 The indexed union of set e...
ssmapsn 45125 A subset ` C ` of a set ex...
iunmapsn 45126 The indexed union of set e...
absfico 45127 Mapping domain and codomai...
icof 45128 The set of left-closed rig...
elpmrn 45129 The range of a partial fun...
imaexi 45130 The image of a set is a se...
axccdom 45131 Relax the constraint on ax...
dmmptdff 45132 The domain of the mapping ...
dmmptdf 45133 The domain of the mapping ...
elpmi2 45134 The domain of a partial fu...
dmrelrnrel 45135 A relation preserving func...
fvcod 45136 Value of a function compos...
elrnmpoid 45137 Membership in the range of...
axccd 45138 An alternative version of ...
axccd2 45139 An alternative version of ...
feqresmptf 45140 Express a restricted funct...
dmmptssf 45141 The domain of a mapping is...
dmmptdf2 45142 The domain of the mapping ...
dmuz 45143 Domain of the upper intege...
fmptd2f 45144 Domain and codomain of the...
mpteq1df 45145 An equality theorem for th...
mpteq1dfOLD 45146 Obsolete version of ~ mpte...
mptexf 45147 If the domain of a functio...
fvmpt4 45148 Value of a function given ...
fmptf 45149 Functionality of the mappi...
resimass 45150 The image of a restriction...
mptssid 45151 The mapping operation expr...
mptfnd 45152 The maps-to notation defin...
mpteq12daOLD 45153 Obsolete version of ~ mpte...
rnmptlb 45154 Boundness below of the ran...
rnmptbddlem 45155 Boundness of the range of ...
rnmptbdd 45156 Boundness of the range of ...
funimaeq 45157 Membership relation for th...
rnmptssf 45158 The range of a function gi...
rnmptbd2lem 45159 Boundness below of the ran...
rnmptbd2 45160 Boundness below of the ran...
infnsuprnmpt 45161 The indexed infimum of rea...
suprclrnmpt 45162 Closure of the indexed sup...
suprubrnmpt2 45163 A member of a nonempty ind...
suprubrnmpt 45164 A member of a nonempty ind...
rnmptssdf 45165 The range of a function gi...
rnmptbdlem 45166 Boundness above of the ran...
rnmptbd 45167 Boundness above of the ran...
rnmptss2 45168 The range of a function gi...
elmptima 45169 The image of a function in...
ralrnmpt3 45170 A restricted quantifier ov...
fvelima2 45171 Function value in an image...
rnmptssbi 45172 The range of a function gi...
imass2d 45173 Subset theorem for image. ...
imassmpt 45174 Membership relation for th...
fpmd 45175 A total function is a part...
fconst7 45176 An alternative way to expr...
fnmptif 45177 Functionality and domain o...
dmmptif 45178 Domain of the mapping oper...
mpteq2dfa 45179 Slightly more general equa...
dmmpt1 45180 The domain of the mapping ...
fmptff 45181 Functionality of the mappi...
fvmptelcdmf 45182 The value of a function at...
fmptdff 45183 A version of ~ fmptd using...
fvmpt2df 45184 Deduction version of ~ fvm...
rn1st 45185 The range of a function wi...
rnmptssff 45186 The range of a function gi...
rnmptssdff 45187 The range of a function gi...
fvmpt4d 45188 Value of a function given ...
sub2times 45189 Subtracting from a number,...
nnxrd 45190 A natural number is an ext...
nnxr 45191 A natural number is an ext...
abssubrp 45192 The distance of two distin...
elfzfzo 45193 Relationship between membe...
oddfl 45194 Odd number representation ...
abscosbd 45195 Bound for the absolute val...
mul13d 45196 Commutative/associative la...
negpilt0 45197 Negative ` _pi ` is negati...
dstregt0 45198 A complex number ` A ` tha...
subadd4b 45199 Rearrangement of 4 terms i...
xrlttri5d 45200 Not equal and not larger i...
neglt 45201 The negative of a positive...
zltlesub 45202 If an integer ` N ` is les...
divlt0gt0d 45203 The ratio of a negative nu...
subsub23d 45204 Swap subtrahend and result...
2timesgt 45205 Double of a positive real ...
reopn 45206 The reals are open with re...
sub31 45207 Swap the first and third t...
nnne1ge2 45208 A positive integer which i...
lefldiveq 45209 A closed enough, smaller r...
negsubdi3d 45210 Distribution of negative o...
ltdiv2dd 45211 Division of a positive num...
abssinbd 45212 Bound for the absolute val...
halffl 45213 Floor of ` ( 1 / 2 ) ` . ...
monoords 45214 Ordering relation for a st...
hashssle 45215 The size of a subset of a ...
lttri5d 45216 Not equal and not larger i...
fzisoeu 45217 A finite ordered set has a...
lt3addmuld 45218 If three real numbers are ...
absnpncan2d 45219 Triangular inequality, com...
fperiodmullem 45220 A function with period ` T...
fperiodmul 45221 A function with period T i...
upbdrech 45222 Choice of an upper bound f...
lt4addmuld 45223 If four real numbers are l...
absnpncan3d 45224 Triangular inequality, com...
upbdrech2 45225 Choice of an upper bound f...
ssfiunibd 45226 A finite union of bounded ...
fzdifsuc2 45227 Remove a successor from th...
fzsscn 45228 A finite sequence of integ...
divcan8d 45229 A cancellation law for div...
dmmcand 45230 Cancellation law for divis...
fzssre 45231 A finite sequence of integ...
bccld 45232 A binomial coefficient, in...
leadd12dd 45233 Addition to both sides of ...
fzssnn0 45234 A finite set of sequential...
xreqle 45235 Equality implies 'less tha...
xaddlidd 45236 ` 0 ` is a left identity f...
xadd0ge 45237 A number is less than or e...
elfzolem1 45238 A member in a half-open in...
xrgtned 45239 'Greater than' implies not...
xrleneltd 45240 'Less than or equal to' an...
xaddcomd 45241 The extended real addition...
supxrre3 45242 The supremum of a nonempty...
uzfissfz 45243 For any finite subset of t...
xleadd2d 45244 Addition of extended reals...
suprltrp 45245 The supremum of a nonempty...
xleadd1d 45246 Addition of extended reals...
xreqled 45247 Equality implies 'less tha...
xrgepnfd 45248 An extended real greater t...
xrge0nemnfd 45249 A nonnegative extended rea...
supxrgere 45250 If a real number can be ap...
iuneqfzuzlem 45251 Lemma for ~ iuneqfzuz : he...
iuneqfzuz 45252 If two unions indexed by u...
xle2addd 45253 Adding both side of two in...
supxrgelem 45254 If an extended real number...
supxrge 45255 If an extended real number...
suplesup 45256 If any element of ` A ` ca...
infxrglb 45257 The infimum of a set of ex...
xadd0ge2 45258 A number is less than or e...
nepnfltpnf 45259 An extended real that is n...
ltadd12dd 45260 Addition to both sides of ...
nemnftgtmnft 45261 An extended real that is n...
xrgtso 45262 'Greater than' is a strict...
rpex 45263 The positive reals form a ...
xrge0ge0 45264 A nonnegative extended rea...
xrssre 45265 A subset of extended reals...
ssuzfz 45266 A finite subset of the upp...
absfun 45267 The absolute value is a fu...
infrpge 45268 The infimum of a nonempty,...
xrlexaddrp 45269 If an extended real number...
supsubc 45270 The supremum function dist...
xralrple2 45271 Show that ` A ` is less th...
nnuzdisj 45272 The first ` N ` elements o...
ltdivgt1 45273 Divsion by a number greate...
xrltned 45274 'Less than' implies not eq...
nnsplit 45275 Express the set of positiv...
divdiv3d 45276 Division into a fraction. ...
abslt2sqd 45277 Comparison of the square o...
qenom 45278 The set of rational number...
qct 45279 The set of rational number...
xrltnled 45280 'Less than' in terms of 'l...
lenlteq 45281 'less than or equal to' bu...
xrred 45282 An extended real that is n...
rr2sscn2 45283 The cartesian square of ` ...
infxr 45284 The infimum of a set of ex...
infxrunb2 45285 The infimum of an unbounde...
infxrbnd2 45286 The infimum of a bounded-b...
infleinflem1 45287 Lemma for ~ infleinf , cas...
infleinflem2 45288 Lemma for ~ infleinf , whe...
infleinf 45289 If any element of ` B ` ca...
xralrple4 45290 Show that ` A ` is less th...
xralrple3 45291 Show that ` A ` is less th...
eluzelzd 45292 A member of an upper set o...
suplesup2 45293 If any element of ` A ` is...
recnnltrp 45294 ` N ` is a natural number ...
nnn0 45295 The set of positive intege...
fzct 45296 A finite set of sequential...
rpgtrecnn 45297 Any positive real number i...
fzossuz 45298 A half-open integer interv...
infxrrefi 45299 The real and extended real...
xrralrecnnle 45300 Show that ` A ` is less th...
fzoct 45301 A finite set of sequential...
frexr 45302 A function taking real val...
nnrecrp 45303 The reciprocal of a positi...
reclt0d 45304 The reciprocal of a negati...
lt0neg1dd 45305 If a number is negative, i...
infxrcld 45306 The infimum of an arbitrar...
xrralrecnnge 45307 Show that ` A ` is less th...
reclt0 45308 The reciprocal of a negati...
ltmulneg 45309 Multiplying by a negative ...
allbutfi 45310 For all but finitely many....
ltdiv23neg 45311 Swap denominator with othe...
xreqnltd 45312 A consequence of trichotom...
mnfnre2 45313 Minus infinity is not a re...
zssxr 45314 The integers are a subset ...
fisupclrnmpt 45315 A nonempty finite indexed ...
supxrunb3 45316 The supremum of an unbound...
elfzod 45317 Membership in a half-open ...
fimaxre4 45318 A nonempty finite set of r...
ren0 45319 The set of reals is nonemp...
eluzelz2 45320 A member of an upper set o...
resabs2d 45321 Absorption law for restric...
uzid2 45322 Membership of the least me...
supxrleubrnmpt 45323 The supremum of a nonempty...
uzssre2 45324 An upper set of integers i...
uzssd 45325 Subset relationship for tw...
eluzd 45326 Membership in an upper set...
infxrlbrnmpt2 45327 A member of a nonempty ind...
xrre4 45328 An extended real is real i...
uz0 45329 The upper integers functio...
eluzelz2d 45330 A member of an upper set o...
infleinf2 45331 If any element in ` B ` is...
unb2ltle 45332 "Unbounded below" expresse...
uzidd2 45333 Membership of the least me...
uzssd2 45334 Subset relationship for tw...
rexabslelem 45335 An indexed set of absolute...
rexabsle 45336 An indexed set of absolute...
allbutfiinf 45337 Given a "for all but finit...
supxrrernmpt 45338 The real and extended real...
suprleubrnmpt 45339 The supremum of a nonempty...
infrnmptle 45340 An indexed infimum of exte...
infxrunb3 45341 The infimum of an unbounde...
uzn0d 45342 The upper integers are all...
uzssd3 45343 Subset relationship for tw...
rexabsle2 45344 An indexed set of absolute...
infxrunb3rnmpt 45345 The infimum of an unbounde...
supxrre3rnmpt 45346 The indexed supremum of a ...
uzublem 45347 A set of reals, indexed by...
uzub 45348 A set of reals, indexed by...
ssrexr 45349 A subset of the reals is a...
supxrmnf2 45350 Removing minus infinity fr...
supxrcli 45351 The supremum of an arbitra...
uzid3 45352 Membership of the least me...
infxrlesupxr 45353 The supremum of a nonempty...
xnegeqd 45354 Equality of two extended n...
xnegrecl 45355 The extended real negative...
xnegnegi 45356 Extended real version of ~...
xnegeqi 45357 Equality of two extended n...
nfxnegd 45358 Deduction version of ~ nfx...
xnegnegd 45359 Extended real version of ~...
uzred 45360 An upper integer is a real...
xnegcli 45361 Closure of extended real n...
supminfrnmpt 45362 The indexed supremum of a ...
infxrpnf 45363 Adding plus infinity to a ...
infxrrnmptcl 45364 The infimum of an arbitrar...
leneg2d 45365 Negative of one side of 'l...
supxrltinfxr 45366 The supremum of the empty ...
max1d 45367 A number is less than or e...
supxrleubrnmptf 45368 The supremum of a nonempty...
nleltd 45369 'Not less than or equal to...
zxrd 45370 An integer is an extended ...
infxrgelbrnmpt 45371 The infimum of an indexed ...
rphalfltd 45372 Half of a positive real is...
uzssz2 45373 An upper set of integers i...
leneg3d 45374 Negative of one side of 'l...
max2d 45375 A number is less than or e...
uzn0bi 45376 The upper integers functio...
xnegrecl2 45377 If the extended real negat...
nfxneg 45378 Bound-variable hypothesis ...
uzxrd 45379 An upper integer is an ext...
infxrpnf2 45380 Removing plus infinity fro...
supminfxr 45381 The extended real suprema ...
infrpgernmpt 45382 The infimum of a nonempty,...
xnegre 45383 An extended real is real i...
xnegrecl2d 45384 If the extended real negat...
uzxr 45385 An upper integer is an ext...
supminfxr2 45386 The extended real suprema ...
xnegred 45387 An extended real is real i...
supminfxrrnmpt 45388 The indexed supremum of a ...
min1d 45389 The minimum of two numbers...
min2d 45390 The minimum of two numbers...
pnfged 45391 Plus infinity is an upper ...
xrnpnfmnf 45392 An extended real that is n...
uzsscn 45393 An upper set of integers i...
absimnre 45394 The absolute value of the ...
uzsscn2 45395 An upper set of integers i...
xrtgcntopre 45396 The standard topologies on...
absimlere 45397 The absolute value of the ...
rpssxr 45398 The positive reals are a s...
monoordxrv 45399 Ordering relation for a mo...
monoordxr 45400 Ordering relation for a mo...
monoord2xrv 45401 Ordering relation for a mo...
monoord2xr 45402 Ordering relation for a mo...
xrpnf 45403 An extended real is plus i...
xlenegcon1 45404 Extended real version of ~...
xlenegcon2 45405 Extended real version of ~...
pimxrneun 45406 The preimage of a set of e...
caucvgbf 45407 A function is convergent i...
cvgcau 45408 A convergent function is C...
cvgcaule 45409 A convergent function is C...
rexanuz2nf 45410 A simple counterexample re...
gtnelioc 45411 A real number larger than ...
ioossioc 45412 An open interval is a subs...
ioondisj2 45413 A condition for two open i...
ioondisj1 45414 A condition for two open i...
ioogtlb 45415 An element of a closed int...
evthiccabs 45416 Extreme Value Theorem on y...
ltnelicc 45417 A real number smaller than...
eliood 45418 Membership in an open real...
iooabslt 45419 An upper bound for the dis...
gtnelicc 45420 A real number greater than...
iooinlbub 45421 An open interval has empty...
iocgtlb 45422 An element of a left-open ...
iocleub 45423 An element of a left-open ...
eliccd 45424 Membership in a closed rea...
eliccre 45425 A member of a closed inter...
eliooshift 45426 Element of an open interva...
eliocd 45427 Membership in a left-open ...
icoltub 45428 An element of a left-close...
eliocre 45429 A member of a left-open ri...
iooltub 45430 An element of an open inte...
ioontr 45431 The interior of an interva...
snunioo1 45432 The closure of one end of ...
lbioc 45433 A left-open right-closed i...
ioomidp 45434 The midpoint is an element...
iccdifioo 45435 If the open inverval is re...
iccdifprioo 45436 An open interval is the cl...
ioossioobi 45437 Biconditional form of ~ io...
iccshift 45438 A closed interval shifted ...
iccsuble 45439 An upper bound to the dist...
iocopn 45440 A left-open right-closed i...
eliccelioc 45441 Membership in a closed int...
iooshift 45442 An open interval shifted b...
iccintsng 45443 Intersection of two adiace...
icoiccdif 45444 Left-closed right-open int...
icoopn 45445 A left-closed right-open i...
icoub 45446 A left-closed, right-open ...
eliccxrd 45447 Membership in a closed rea...
pnfel0pnf 45448 ` +oo ` is a nonnegative e...
eliccnelico 45449 An element of a closed int...
eliccelicod 45450 A member of a closed inter...
ge0xrre 45451 A nonnegative extended rea...
ge0lere 45452 A nonnegative extended Rea...
elicores 45453 Membership in a left-close...
inficc 45454 The infimum of a nonempty ...
qinioo 45455 The rational numbers are d...
lenelioc 45456 A real number smaller than...
ioonct 45457 A nonempty open interval i...
xrgtnelicc 45458 A real number greater than...
iccdificc 45459 The difference of two clos...
iocnct 45460 A nonempty left-open, righ...
iccnct 45461 A closed interval, with mo...
iooiinicc 45462 A closed interval expresse...
iccgelbd 45463 An element of a closed int...
iooltubd 45464 An element of an open inte...
icoltubd 45465 An element of a left-close...
qelioo 45466 The rational numbers are d...
tgqioo2 45467 Every open set of reals is...
iccleubd 45468 An element of a closed int...
elioored 45469 A member of an open interv...
ioogtlbd 45470 An element of a closed int...
ioofun 45471 ` (,) ` is a function. (C...
icomnfinre 45472 A left-closed, right-open,...
sqrlearg 45473 The square compared with i...
ressiocsup 45474 If the supremum belongs to...
ressioosup 45475 If the supremum does not b...
iooiinioc 45476 A left-open, right-closed ...
ressiooinf 45477 If the infimum does not be...
icogelbd 45478 An element of a left-close...
iocleubd 45479 An element of a left-open ...
uzinico 45480 An upper interval of integ...
preimaiocmnf 45481 Preimage of a right-closed...
uzinico2 45482 An upper interval of integ...
uzinico3 45483 An upper interval of integ...
icossico2 45484 Condition for a closed-bel...
dmico 45485 The domain of the closed-b...
ndmico 45486 The closed-below, open-abo...
uzubioo 45487 The upper integers are unb...
uzubico 45488 The upper integers are unb...
uzubioo2 45489 The upper integers are unb...
uzubico2 45490 The upper integers are unb...
iocgtlbd 45491 An element of a left-open ...
xrtgioo2 45492 The topology on the extend...
tgioo4 45493 The standard topology on t...
fsummulc1f 45494 Closure of a finite sum of...
fsumnncl 45495 Closure of a nonempty, fin...
fsumge0cl 45496 The finite sum of nonnegat...
fsumf1of 45497 Re-index a finite sum usin...
fsumiunss 45498 Sum over a disjoint indexe...
fsumreclf 45499 Closure of a finite sum of...
fsumlessf 45500 A shorter sum of nonnegati...
fsumsupp0 45501 Finite sum of function val...
fsumsermpt 45502 A finite sum expressed in ...
fmul01 45503 Multiplying a finite numbe...
fmulcl 45504 If ' Y ' is closed under t...
fmuldfeqlem1 45505 induction step for the pro...
fmuldfeq 45506 X and Z are two equivalent...
fmul01lt1lem1 45507 Given a finite multiplicat...
fmul01lt1lem2 45508 Given a finite multiplicat...
fmul01lt1 45509 Given a finite multiplicat...
cncfmptss 45510 A continuous complex funct...
rrpsscn 45511 The positive reals are a s...
mulc1cncfg 45512 A version of ~ mulc1cncf u...
infrglb 45513 The infimum of a nonempty ...
expcnfg 45514 If ` F ` is a complex cont...
prodeq2ad 45515 Equality deduction for pro...
fprodsplit1 45516 Separate out a term in a f...
fprodexp 45517 Positive integer exponenti...
fprodabs2 45518 The absolute value of a fi...
fprod0 45519 A finite product with a ze...
mccllem 45520 * Induction step for ~ mcc...
mccl 45521 A multinomial coefficient,...
fprodcnlem 45522 A finite product of functi...
fprodcn 45523 A finite product of functi...
clim1fr1 45524 A class of sequences of fr...
isumneg 45525 Negation of a converging s...
climrec 45526 Limit of the reciprocal of...
climmulf 45527 A version of ~ climmul usi...
climexp 45528 The limit of natural power...
climinf 45529 A bounded monotonic noninc...
climsuselem1 45530 The subsequence index ` I ...
climsuse 45531 A subsequence ` G ` of a c...
climrecf 45532 A version of ~ climrec usi...
climneg 45533 Complex limit of the negat...
climinff 45534 A version of ~ climinf usi...
climdivf 45535 Limit of the ratio of two ...
climreeq 45536 If ` F ` is a real functio...
ellimciota 45537 An explicit value for the ...
climaddf 45538 A version of ~ climadd usi...
mullimc 45539 Limit of the product of tw...
ellimcabssub0 45540 An equivalent condition fo...
limcdm0 45541 If a function has empty do...
islptre 45542 An equivalence condition f...
limccog 45543 Limit of the composition o...
limciccioolb 45544 The limit of a function at...
climf 45545 Express the predicate: Th...
mullimcf 45546 Limit of the multiplicatio...
constlimc 45547 Limit of constant function...
rexlim2d 45548 Inference removing two res...
idlimc 45549 Limit of the identity func...
divcnvg 45550 The sequence of reciprocal...
limcperiod 45551 If ` F ` is a periodic fun...
limcrecl 45552 If ` F ` is a real-valued ...
sumnnodd 45553 A series indexed by ` NN `...
lptioo2 45554 The upper bound of an open...
lptioo1 45555 The lower bound of an open...
elprn1 45556 A member of an unordered p...
elprn2 45557 A member of an unordered p...
limcmptdm 45558 The domain of a maps-to fu...
clim2f 45559 Express the predicate: Th...
limcicciooub 45560 The limit of a function at...
ltmod 45561 A sufficient condition for...
islpcn 45562 A characterization for a l...
lptre2pt 45563 If a set in the real line ...
limsupre 45564 If a sequence is bounded, ...
limcresiooub 45565 The left limit doesn't cha...
limcresioolb 45566 The right limit doesn't ch...
limcleqr 45567 If the left and the right ...
lptioo2cn 45568 The upper bound of an open...
lptioo1cn 45569 The lower bound of an open...
neglimc 45570 Limit of the negative func...
addlimc 45571 Sum of two limits. (Contr...
0ellimcdiv 45572 If the numerator converges...
clim2cf 45573 Express the predicate ` F ...
limclner 45574 For a limit point, both fr...
sublimc 45575 Subtraction of two limits....
reclimc 45576 Limit of the reciprocal of...
clim0cf 45577 Express the predicate ` F ...
limclr 45578 For a limit point, both fr...
divlimc 45579 Limit of the quotient of t...
expfac 45580 Factorial grows faster tha...
climconstmpt 45581 A constant sequence conver...
climresmpt 45582 A function restricted to u...
climsubmpt 45583 Limit of the difference of...
climsubc2mpt 45584 Limit of the difference of...
climsubc1mpt 45585 Limit of the difference of...
fnlimfv 45586 The value of the limit fun...
climreclf 45587 The limit of a convergent ...
climeldmeq 45588 Two functions that are eve...
climf2 45589 Express the predicate: Th...
fnlimcnv 45590 The sequence of function v...
climeldmeqmpt 45591 Two functions that are eve...
climfveq 45592 Two functions that are eve...
clim2f2 45593 Express the predicate: Th...
climfveqmpt 45594 Two functions that are eve...
climd 45595 Express the predicate: Th...
clim2d 45596 The limit of complex numbe...
fnlimfvre 45597 The limit function of real...
allbutfifvre 45598 Given a sequence of real-v...
climleltrp 45599 The limit of complex numbe...
fnlimfvre2 45600 The limit function of real...
fnlimf 45601 The limit function of real...
fnlimabslt 45602 A sequence of function val...
climfveqf 45603 Two functions that are eve...
climmptf 45604 Exhibit a function ` G ` w...
climfveqmpt3 45605 Two functions that are eve...
climeldmeqf 45606 Two functions that are eve...
climreclmpt 45607 The limit of B convergent ...
limsupref 45608 If a sequence is bounded, ...
limsupbnd1f 45609 If a sequence is eventuall...
climbddf 45610 A converging sequence of c...
climeqf 45611 Two functions that are eve...
climeldmeqmpt3 45612 Two functions that are eve...
limsupcld 45613 Closure of the superior li...
climfv 45614 The limit of a convergent ...
limsupval3 45615 The superior limit of an i...
climfveqmpt2 45616 Two functions that are eve...
limsup0 45617 The superior limit of the ...
climeldmeqmpt2 45618 Two functions that are eve...
limsupresre 45619 The supremum limit of a fu...
climeqmpt 45620 Two functions that are eve...
climfvd 45621 The limit of a convergent ...
limsuplesup 45622 An upper bound for the sup...
limsupresico 45623 The superior limit doesn't...
limsuppnfdlem 45624 If the restriction of a fu...
limsuppnfd 45625 If the restriction of a fu...
limsupresuz 45626 If the real part of the do...
limsupub 45627 If the limsup is not ` +oo...
limsupres 45628 The superior limit of a re...
climinf2lem 45629 A convergent, nonincreasin...
climinf2 45630 A convergent, nonincreasin...
limsupvaluz 45631 The superior limit, when t...
limsupresuz2 45632 If the domain of a functio...
limsuppnflem 45633 If the restriction of a fu...
limsuppnf 45634 If the restriction of a fu...
limsupubuzlem 45635 If the limsup is not ` +oo...
limsupubuz 45636 For a real-valued function...
climinf2mpt 45637 A bounded below, monotonic...
climinfmpt 45638 A bounded below, monotonic...
climinf3 45639 A convergent, nonincreasin...
limsupvaluzmpt 45640 The superior limit, when t...
limsupequzmpt2 45641 Two functions that are eve...
limsupubuzmpt 45642 If the limsup is not ` +oo...
limsupmnflem 45643 The superior limit of a fu...
limsupmnf 45644 The superior limit of a fu...
limsupequzlem 45645 Two functions that are eve...
limsupequz 45646 Two functions that are eve...
limsupre2lem 45647 Given a function on the ex...
limsupre2 45648 Given a function on the ex...
limsupmnfuzlem 45649 The superior limit of a fu...
limsupmnfuz 45650 The superior limit of a fu...
limsupequzmptlem 45651 Two functions that are eve...
limsupequzmpt 45652 Two functions that are eve...
limsupre2mpt 45653 Given a function on the ex...
limsupequzmptf 45654 Two functions that are eve...
limsupre3lem 45655 Given a function on the ex...
limsupre3 45656 Given a function on the ex...
limsupre3mpt 45657 Given a function on the ex...
limsupre3uzlem 45658 Given a function on the ex...
limsupre3uz 45659 Given a function on the ex...
limsupreuz 45660 Given a function on the re...
limsupvaluz2 45661 The superior limit, when t...
limsupreuzmpt 45662 Given a function on the re...
supcnvlimsup 45663 If a function on a set of ...
supcnvlimsupmpt 45664 If a function on a set of ...
0cnv 45665 If ` (/) ` is a complex nu...
climuzlem 45666 Express the predicate: Th...
climuz 45667 Express the predicate: Th...
lmbr3v 45668 Express the binary relatio...
climisp 45669 If a sequence converges to...
lmbr3 45670 Express the binary relatio...
climrescn 45671 A sequence converging w.r....
climxrrelem 45672 If a sequence ranging over...
climxrre 45673 If a sequence ranging over...
limsuplt2 45676 The defining property of t...
liminfgord 45677 Ordering property of the i...
limsupvald 45678 The superior limit of a se...
limsupresicompt 45679 The superior limit doesn't...
limsupcli 45680 Closure of the superior li...
liminfgf 45681 Closure of the inferior li...
liminfval 45682 The inferior limit of a se...
climlimsup 45683 A sequence of real numbers...
limsupge 45684 The defining property of t...
liminfgval 45685 Value of the inferior limi...
liminfcl 45686 Closure of the inferior li...
liminfvald 45687 The inferior limit of a se...
liminfval5 45688 The inferior limit of an i...
limsupresxr 45689 The superior limit of a fu...
liminfresxr 45690 The inferior limit of a fu...
liminfval2 45691 The superior limit, relati...
climlimsupcex 45692 Counterexample for ~ climl...
liminfcld 45693 Closure of the inferior li...
liminfresico 45694 The inferior limit doesn't...
limsup10exlem 45695 The range of the given fun...
limsup10ex 45696 The superior limit of a fu...
liminf10ex 45697 The inferior limit of a fu...
liminflelimsuplem 45698 The superior limit is grea...
liminflelimsup 45699 The superior limit is grea...
limsupgtlem 45700 For any positive real, the...
limsupgt 45701 Given a sequence of real n...
liminfresre 45702 The inferior limit of a fu...
liminfresicompt 45703 The inferior limit doesn't...
liminfltlimsupex 45704 An example where the ` lim...
liminfgelimsup 45705 The inferior limit is grea...
liminfvalxr 45706 Alternate definition of ` ...
liminfresuz 45707 If the real part of the do...
liminflelimsupuz 45708 The superior limit is grea...
liminfvalxrmpt 45709 Alternate definition of ` ...
liminfresuz2 45710 If the domain of a functio...
liminfgelimsupuz 45711 The inferior limit is grea...
liminfval4 45712 Alternate definition of ` ...
liminfval3 45713 Alternate definition of ` ...
liminfequzmpt2 45714 Two functions that are eve...
liminfvaluz 45715 Alternate definition of ` ...
liminf0 45716 The inferior limit of the ...
limsupval4 45717 Alternate definition of ` ...
liminfvaluz2 45718 Alternate definition of ` ...
liminfvaluz3 45719 Alternate definition of ` ...
liminflelimsupcex 45720 A counterexample for ~ lim...
limsupvaluz3 45721 Alternate definition of ` ...
liminfvaluz4 45722 Alternate definition of ` ...
limsupvaluz4 45723 Alternate definition of ` ...
climliminflimsupd 45724 If a sequence of real numb...
liminfreuzlem 45725 Given a function on the re...
liminfreuz 45726 Given a function on the re...
liminfltlem 45727 Given a sequence of real n...
liminflt 45728 Given a sequence of real n...
climliminf 45729 A sequence of real numbers...
liminflimsupclim 45730 A sequence of real numbers...
climliminflimsup 45731 A sequence of real numbers...
climliminflimsup2 45732 A sequence of real numbers...
climliminflimsup3 45733 A sequence of real numbers...
climliminflimsup4 45734 A sequence of real numbers...
limsupub2 45735 A extended real valued fun...
limsupubuz2 45736 A sequence with values in ...
xlimpnfxnegmnf 45737 A sequence converges to ` ...
liminflbuz2 45738 A sequence with values in ...
liminfpnfuz 45739 The inferior limit of a fu...
liminflimsupxrre 45740 A sequence with values in ...
xlimrel 45743 The limit on extended real...
xlimres 45744 A function converges iff i...
xlimcl 45745 The limit of a sequence of...
rexlimddv2 45746 Restricted existential eli...
xlimclim 45747 Given a sequence of reals,...
xlimconst 45748 A constant sequence conver...
climxlim 45749 A converging sequence in t...
xlimbr 45750 Express the binary relatio...
fuzxrpmcn 45751 A function mapping from an...
cnrefiisplem 45752 Lemma for ~ cnrefiisp (som...
cnrefiisp 45753 A non-real, complex number...
xlimxrre 45754 If a sequence ranging over...
xlimmnfvlem1 45755 Lemma for ~ xlimmnfv : the...
xlimmnfvlem2 45756 Lemma for ~ xlimmnf : the ...
xlimmnfv 45757 A function converges to mi...
xlimconst2 45758 A sequence that eventually...
xlimpnfvlem1 45759 Lemma for ~ xlimpnfv : the...
xlimpnfvlem2 45760 Lemma for ~ xlimpnfv : the...
xlimpnfv 45761 A function converges to pl...
xlimclim2lem 45762 Lemma for ~ xlimclim2 . H...
xlimclim2 45763 Given a sequence of extend...
xlimmnf 45764 A function converges to mi...
xlimpnf 45765 A function converges to pl...
xlimmnfmpt 45766 A function converges to pl...
xlimpnfmpt 45767 A function converges to pl...
climxlim2lem 45768 In this lemma for ~ climxl...
climxlim2 45769 A sequence of extended rea...
dfxlim2v 45770 An alternative definition ...
dfxlim2 45771 An alternative definition ...
climresd 45772 A function restricted to u...
climresdm 45773 A real function converges ...
dmclimxlim 45774 A real valued sequence tha...
xlimmnflimsup2 45775 A sequence of extended rea...
xlimuni 45776 An infinite sequence conve...
xlimclimdm 45777 A sequence of extended rea...
xlimfun 45778 The convergence relation o...
xlimmnflimsup 45779 If a sequence of extended ...
xlimdm 45780 Two ways to express that a...
xlimpnfxnegmnf2 45781 A sequence converges to ` ...
xlimresdm 45782 A function converges in th...
xlimpnfliminf 45783 If a sequence of extended ...
xlimpnfliminf2 45784 A sequence of extended rea...
xlimliminflimsup 45785 A sequence of extended rea...
xlimlimsupleliminf 45786 A sequence of extended rea...
coseq0 45787 A complex number whose cos...
sinmulcos 45788 Multiplication formula for...
coskpi2 45789 The cosine of an integer m...
cosnegpi 45790 The cosine of negative ` _...
sinaover2ne0 45791 If ` A ` in ` ( 0 , 2 _pi ...
cosknegpi 45792 The cosine of an integer m...
mulcncff 45793 The multiplication of two ...
cncfmptssg 45794 A continuous complex funct...
constcncfg 45795 A constant function is a c...
idcncfg 45796 The identity function is a...
cncfshift 45797 A periodic continuous func...
resincncf 45798 ` sin ` restricted to real...
addccncf2 45799 Adding a constant is a con...
0cnf 45800 The empty set is a continu...
fsumcncf 45801 The finite sum of continuo...
cncfperiod 45802 A periodic continuous func...
subcncff 45803 The subtraction of two con...
negcncfg 45804 The opposite of a continuo...
cnfdmsn 45805 A function with a singleto...
cncfcompt 45806 Composition of continuous ...
addcncff 45807 The sum of two continuous ...
ioccncflimc 45808 Limit at the upper bound o...
cncfuni 45809 A complex function on a su...
icccncfext 45810 A continuous function on a...
cncficcgt0 45811 A the absolute value of a ...
icocncflimc 45812 Limit at the lower bound, ...
cncfdmsn 45813 A complex function with a ...
divcncff 45814 The quotient of two contin...
cncfshiftioo 45815 A periodic continuous func...
cncfiooicclem1 45816 A continuous function ` F ...
cncfiooicc 45817 A continuous function ` F ...
cncfiooiccre 45818 A continuous function ` F ...
cncfioobdlem 45819 ` G ` actually extends ` F...
cncfioobd 45820 A continuous function ` F ...
jumpncnp 45821 Jump discontinuity or disc...
cxpcncf2 45822 The complex power function...
fprodcncf 45823 The finite product of cont...
add1cncf 45824 Addition to a constant is ...
add2cncf 45825 Addition to a constant is ...
sub1cncfd 45826 Subtracting a constant is ...
sub2cncfd 45827 Subtraction from a constan...
fprodsub2cncf 45828 ` F ` is continuous. (Con...
fprodadd2cncf 45829 ` F ` is continuous. (Con...
fprodsubrecnncnvlem 45830 The sequence ` S ` of fini...
fprodsubrecnncnv 45831 The sequence ` S ` of fini...
fprodaddrecnncnvlem 45832 The sequence ` S ` of fini...
fprodaddrecnncnv 45833 The sequence ` S ` of fini...
dvsinexp 45834 The derivative of sin^N . ...
dvcosre 45835 The real derivative of the...
dvsinax 45836 Derivative exercise: the d...
dvsubf 45837 The subtraction rule for e...
dvmptconst 45838 Function-builder for deriv...
dvcnre 45839 From complex differentiati...
dvmptidg 45840 Function-builder for deriv...
dvresntr 45841 Function-builder for deriv...
fperdvper 45842 The derivative of a period...
dvasinbx 45843 Derivative exercise: the d...
dvresioo 45844 Restriction of a derivativ...
dvdivf 45845 The quotient rule for ever...
dvdivbd 45846 A sufficient condition for...
dvsubcncf 45847 A sufficient condition for...
dvmulcncf 45848 A sufficient condition for...
dvcosax 45849 Derivative exercise: the d...
dvdivcncf 45850 A sufficient condition for...
dvbdfbdioolem1 45851 Given a function with boun...
dvbdfbdioolem2 45852 A function on an open inte...
dvbdfbdioo 45853 A function on an open inte...
ioodvbdlimc1lem1 45854 If ` F ` has bounded deriv...
ioodvbdlimc1lem2 45855 Limit at the lower bound o...
ioodvbdlimc1 45856 A real function with bound...
ioodvbdlimc2lem 45857 Limit at the upper bound o...
ioodvbdlimc2 45858 A real function with bound...
dvdmsscn 45859 ` X ` is a subset of ` CC ...
dvmptmulf 45860 Function-builder for deriv...
dvnmptdivc 45861 Function-builder for itera...
dvdsn1add 45862 If ` K ` divides ` N ` but...
dvxpaek 45863 Derivative of the polynomi...
dvnmptconst 45864 The ` N ` -th derivative o...
dvnxpaek 45865 The ` n ` -th derivative o...
dvnmul 45866 Function-builder for the `...
dvmptfprodlem 45867 Induction step for ~ dvmpt...
dvmptfprod 45868 Function-builder for deriv...
dvnprodlem1 45869 ` D ` is bijective. (Cont...
dvnprodlem2 45870 Induction step for ~ dvnpr...
dvnprodlem3 45871 The multinomial formula fo...
dvnprod 45872 The multinomial formula fo...
itgsin0pilem1 45873 Calculation of the integra...
ibliccsinexp 45874 sin^n on a closed interval...
itgsin0pi 45875 Calculation of the integra...
iblioosinexp 45876 sin^n on an open integral ...
itgsinexplem1 45877 Integration by parts is ap...
itgsinexp 45878 A recursive formula for th...
iblconstmpt 45879 A constant function is int...
itgeq1d 45880 Equality theorem for an in...
mbfres2cn 45881 Measurability of a piecewi...
vol0 45882 The measure of the empty s...
ditgeqiooicc 45883 A function ` F ` on an ope...
volge0 45884 The volume of a set is alw...
cnbdibl 45885 A continuous bounded funct...
snmbl 45886 A singleton is measurable....
ditgeq3d 45887 Equality theorem for the d...
iblempty 45888 The empty function is inte...
iblsplit 45889 The union of two integrabl...
volsn 45890 A singleton has 0 Lebesgue...
itgvol0 45891 If the domani is negligibl...
itgcoscmulx 45892 Exercise: the integral of ...
iblsplitf 45893 A version of ~ iblsplit us...
ibliooicc 45894 If a function is integrabl...
volioc 45895 The measure of a left-open...
iblspltprt 45896 If a function is integrabl...
itgsincmulx 45897 Exercise: the integral of ...
itgsubsticclem 45898 lemma for ~ itgsubsticc . ...
itgsubsticc 45899 Integration by u-substitut...
itgioocnicc 45900 The integral of a piecewis...
iblcncfioo 45901 A continuous function ` F ...
itgspltprt 45902 The ` S. ` integral splits...
itgiccshift 45903 The integral of a function...
itgperiod 45904 The integral of a periodic...
itgsbtaddcnst 45905 Integral substitution, add...
volico 45906 The measure of left-closed...
sublevolico 45907 The Lebesgue measure of a ...
dmvolss 45908 Lebesgue measurable sets a...
ismbl3 45909 The predicate " ` A ` is L...
volioof 45910 The function that assigns ...
ovolsplit 45911 The Lebesgue outer measure...
fvvolioof 45912 The function value of the ...
volioore 45913 The measure of an open int...
fvvolicof 45914 The function value of the ...
voliooico 45915 An open interval and a lef...
ismbl4 45916 The predicate " ` A ` is L...
volioofmpt 45917 ` ( ( vol o. (,) ) o. F ) ...
volicoff 45918 ` ( ( vol o. [,) ) o. F ) ...
voliooicof 45919 The Lebesgue measure of op...
volicofmpt 45920 ` ( ( vol o. [,) ) o. F ) ...
volicc 45921 The Lebesgue measure of a ...
voliccico 45922 A closed interval and a le...
mbfdmssre 45923 The domain of a measurable...
stoweidlem1 45924 Lemma for ~ stoweid . Thi...
stoweidlem2 45925 lemma for ~ stoweid : here...
stoweidlem3 45926 Lemma for ~ stoweid : if `...
stoweidlem4 45927 Lemma for ~ stoweid : a cl...
stoweidlem5 45928 There exists a δ as ...
stoweidlem6 45929 Lemma for ~ stoweid : two ...
stoweidlem7 45930 This lemma is used to prov...
stoweidlem8 45931 Lemma for ~ stoweid : two ...
stoweidlem9 45932 Lemma for ~ stoweid : here...
stoweidlem10 45933 Lemma for ~ stoweid . Thi...
stoweidlem11 45934 This lemma is used to prov...
stoweidlem12 45935 Lemma for ~ stoweid . Thi...
stoweidlem13 45936 Lemma for ~ stoweid . Thi...
stoweidlem14 45937 There exists a ` k ` as in...
stoweidlem15 45938 This lemma is used to prov...
stoweidlem16 45939 Lemma for ~ stoweid . The...
stoweidlem17 45940 This lemma proves that the...
stoweidlem18 45941 This theorem proves Lemma ...
stoweidlem19 45942 If a set of real functions...
stoweidlem20 45943 If a set A of real functio...
stoweidlem21 45944 Once the Stone Weierstrass...
stoweidlem22 45945 If a set of real functions...
stoweidlem23 45946 This lemma is used to prov...
stoweidlem24 45947 This lemma proves that for...
stoweidlem25 45948 This lemma proves that for...
stoweidlem26 45949 This lemma is used to prov...
stoweidlem27 45950 This lemma is used to prov...
stoweidlem28 45951 There exists a δ as ...
stoweidlem29 45952 When the hypothesis for th...
stoweidlem30 45953 This lemma is used to prov...
stoweidlem31 45954 This lemma is used to prov...
stoweidlem32 45955 If a set A of real functio...
stoweidlem33 45956 If a set of real functions...
stoweidlem34 45957 This lemma proves that for...
stoweidlem35 45958 This lemma is used to prov...
stoweidlem36 45959 This lemma is used to prov...
stoweidlem37 45960 This lemma is used to prov...
stoweidlem38 45961 This lemma is used to prov...
stoweidlem39 45962 This lemma is used to prov...
stoweidlem40 45963 This lemma proves that q_n...
stoweidlem41 45964 This lemma is used to prov...
stoweidlem42 45965 This lemma is used to prov...
stoweidlem43 45966 This lemma is used to prov...
stoweidlem44 45967 This lemma is used to prov...
stoweidlem45 45968 This lemma proves that, gi...
stoweidlem46 45969 This lemma proves that set...
stoweidlem47 45970 Subtracting a constant fro...
stoweidlem48 45971 This lemma is used to prov...
stoweidlem49 45972 There exists a function q_...
stoweidlem50 45973 This lemma proves that set...
stoweidlem51 45974 There exists a function x ...
stoweidlem52 45975 There exists a neighborhoo...
stoweidlem53 45976 This lemma is used to prov...
stoweidlem54 45977 There exists a function ` ...
stoweidlem55 45978 This lemma proves the exis...
stoweidlem56 45979 This theorem proves Lemma ...
stoweidlem57 45980 There exists a function x ...
stoweidlem58 45981 This theorem proves Lemma ...
stoweidlem59 45982 This lemma proves that the...
stoweidlem60 45983 This lemma proves that the...
stoweidlem61 45984 This lemma proves that the...
stoweidlem62 45985 This theorem proves the St...
stoweid 45986 This theorem proves the St...
stowei 45987 This theorem proves the St...
wallispilem1 45988 ` I ` is monotone: increas...
wallispilem2 45989 A first set of properties ...
wallispilem3 45990 I maps to real values. (C...
wallispilem4 45991 ` F ` maps to explicit exp...
wallispilem5 45992 The sequence ` H ` converg...
wallispi 45993 Wallis' formula for π :...
wallispi2lem1 45994 An intermediate step betwe...
wallispi2lem2 45995 Two expressions are proven...
wallispi2 45996 An alternative version of ...
stirlinglem1 45997 A simple limit of fraction...
stirlinglem2 45998 ` A ` maps to positive rea...
stirlinglem3 45999 Long but simple algebraic ...
stirlinglem4 46000 Algebraic manipulation of ...
stirlinglem5 46001 If ` T ` is between ` 0 ` ...
stirlinglem6 46002 A series that converges to...
stirlinglem7 46003 Algebraic manipulation of ...
stirlinglem8 46004 If ` A ` converges to ` C ...
stirlinglem9 46005 ` ( ( B `` N ) - ( B `` ( ...
stirlinglem10 46006 A bound for any B(N)-B(N +...
stirlinglem11 46007 ` B ` is decreasing. (Con...
stirlinglem12 46008 The sequence ` B ` is boun...
stirlinglem13 46009 ` B ` is decreasing and ha...
stirlinglem14 46010 The sequence ` A ` converg...
stirlinglem15 46011 The Stirling's formula is ...
stirling 46012 Stirling's approximation f...
stirlingr 46013 Stirling's approximation f...
dirkerval 46014 The N_th Dirichlet Kernel....
dirker2re 46015 The Dirichlet Kernel value...
dirkerdenne0 46016 The Dirichlet Kernel denom...
dirkerval2 46017 The N_th Dirichlet Kernel ...
dirkerre 46018 The Dirichlet Kernel at an...
dirkerper 46019 the Dirichlet Kernel has p...
dirkerf 46020 For any natural number ` N...
dirkertrigeqlem1 46021 Sum of an even number of a...
dirkertrigeqlem2 46022 Trigonomic equality lemma ...
dirkertrigeqlem3 46023 Trigonometric equality lem...
dirkertrigeq 46024 Trigonometric equality for...
dirkeritg 46025 The definite integral of t...
dirkercncflem1 46026 If ` Y ` is a multiple of ...
dirkercncflem2 46027 Lemma used to prove that t...
dirkercncflem3 46028 The Dirichlet Kernel is co...
dirkercncflem4 46029 The Dirichlet Kernel is co...
dirkercncf 46030 For any natural number ` N...
fourierdlem1 46031 A partition interval is a ...
fourierdlem2 46032 Membership in a partition....
fourierdlem3 46033 Membership in a partition....
fourierdlem4 46034 ` E ` is a function that m...
fourierdlem5 46035 ` S ` is a function. (Con...
fourierdlem6 46036 ` X ` is in the periodic p...
fourierdlem7 46037 The difference between the...
fourierdlem8 46038 A partition interval is a ...
fourierdlem9 46039 ` H ` is a complex functio...
fourierdlem10 46040 Condition on the bounds of...
fourierdlem11 46041 If there is a partition, t...
fourierdlem12 46042 A point of a partition is ...
fourierdlem13 46043 Value of ` V ` in terms of...
fourierdlem14 46044 Given the partition ` V ` ...
fourierdlem15 46045 The range of the partition...
fourierdlem16 46046 The coefficients of the fo...
fourierdlem17 46047 The defined ` L ` is actua...
fourierdlem18 46048 The function ` S ` is cont...
fourierdlem19 46049 If two elements of ` D ` h...
fourierdlem20 46050 Every interval in the part...
fourierdlem21 46051 The coefficients of the fo...
fourierdlem22 46052 The coefficients of the fo...
fourierdlem23 46053 If ` F ` is continuous and...
fourierdlem24 46054 A sufficient condition for...
fourierdlem25 46055 If ` C ` is not in the ran...
fourierdlem26 46056 Periodic image of a point ...
fourierdlem27 46057 A partition open interval ...
fourierdlem28 46058 Derivative of ` ( F `` ( X...
fourierdlem29 46059 Explicit function value fo...
fourierdlem30 46060 Sum of three small pieces ...
fourierdlem31 46061 If ` A ` is finite and for...
fourierdlem32 46062 Limit of a continuous func...
fourierdlem33 46063 Limit of a continuous func...
fourierdlem34 46064 A partition is one to one....
fourierdlem35 46065 There is a single point in...
fourierdlem36 46066 ` F ` is an isomorphism. ...
fourierdlem37 46067 ` I ` is a function that m...
fourierdlem38 46068 The function ` F ` is cont...
fourierdlem39 46069 Integration by parts of ...
fourierdlem40 46070 ` H ` is a continuous func...
fourierdlem41 46071 Lemma used to prove that e...
fourierdlem42 46072 The set of points in a mov...
fourierdlem43 46073 ` K ` is a real function. ...
fourierdlem44 46074 A condition for having ` (...
fourierdlem46 46075 The function ` F ` has a l...
fourierdlem47 46076 For ` r ` large enough, th...
fourierdlem48 46077 The given periodic functio...
fourierdlem49 46078 The given periodic functio...
fourierdlem50 46079 Continuity of ` O ` and it...
fourierdlem51 46080 ` X ` is in the periodic p...
fourierdlem52 46081 d16:d17,d18:jca |- ( ph ->...
fourierdlem53 46082 The limit of ` F ( s ) ` a...
fourierdlem54 46083 Given a partition ` Q ` an...
fourierdlem55 46084 ` U ` is a real function. ...
fourierdlem56 46085 Derivative of the ` K ` fu...
fourierdlem57 46086 The derivative of ` O ` . ...
fourierdlem58 46087 The derivative of ` K ` is...
fourierdlem59 46088 The derivative of ` H ` is...
fourierdlem60 46089 Given a differentiable fun...
fourierdlem61 46090 Given a differentiable fun...
fourierdlem62 46091 The function ` K ` is cont...
fourierdlem63 46092 The upper bound of interva...
fourierdlem64 46093 The partition ` V ` is fin...
fourierdlem65 46094 The distance of two adjace...
fourierdlem66 46095 Value of the ` G ` functio...
fourierdlem67 46096 ` G ` is a function. (Con...
fourierdlem68 46097 The derivative of ` O ` is...
fourierdlem69 46098 A piecewise continuous fun...
fourierdlem70 46099 A piecewise continuous fun...
fourierdlem71 46100 A periodic piecewise conti...
fourierdlem72 46101 The derivative of ` O ` is...
fourierdlem73 46102 A version of the Riemann L...
fourierdlem74 46103 Given a piecewise smooth f...
fourierdlem75 46104 Given a piecewise smooth f...
fourierdlem76 46105 Continuity of ` O ` and it...
fourierdlem77 46106 If ` H ` is bounded, then ...
fourierdlem78 46107 ` G ` is continuous when r...
fourierdlem79 46108 ` E ` projects every inter...
fourierdlem80 46109 The derivative of ` O ` is...
fourierdlem81 46110 The integral of a piecewis...
fourierdlem82 46111 Integral by substitution, ...
fourierdlem83 46112 The fourier partial sum fo...
fourierdlem84 46113 If ` F ` is piecewise cont...
fourierdlem85 46114 Limit of the function ` G ...
fourierdlem86 46115 Continuity of ` O ` and it...
fourierdlem87 46116 The integral of ` G ` goes...
fourierdlem88 46117 Given a piecewise continuo...
fourierdlem89 46118 Given a piecewise continuo...
fourierdlem90 46119 Given a piecewise continuo...
fourierdlem91 46120 Given a piecewise continuo...
fourierdlem92 46121 The integral of a piecewis...
fourierdlem93 46122 Integral by substitution (...
fourierdlem94 46123 For a piecewise smooth fun...
fourierdlem95 46124 Algebraic manipulation of ...
fourierdlem96 46125 limit for ` F ` at the low...
fourierdlem97 46126 ` F ` is continuous on the...
fourierdlem98 46127 ` F ` is continuous on the...
fourierdlem99 46128 limit for ` F ` at the upp...
fourierdlem100 46129 A piecewise continuous fun...
fourierdlem101 46130 Integral by substitution f...
fourierdlem102 46131 For a piecewise smooth fun...
fourierdlem103 46132 The half lower part of the...
fourierdlem104 46133 The half upper part of the...
fourierdlem105 46134 A piecewise continuous fun...
fourierdlem106 46135 For a piecewise smooth fun...
fourierdlem107 46136 The integral of a piecewis...
fourierdlem108 46137 The integral of a piecewis...
fourierdlem109 46138 The integral of a piecewis...
fourierdlem110 46139 The integral of a piecewis...
fourierdlem111 46140 The fourier partial sum fo...
fourierdlem112 46141 Here abbreviations (local ...
fourierdlem113 46142 Fourier series convergence...
fourierdlem114 46143 Fourier series convergence...
fourierdlem115 46144 Fourier serier convergence...
fourierd 46145 Fourier series convergence...
fourierclimd 46146 Fourier series convergence...
fourierclim 46147 Fourier series convergence...
fourier 46148 Fourier series convergence...
fouriercnp 46149 If ` F ` is continuous at ...
fourier2 46150 Fourier series convergence...
sqwvfoura 46151 Fourier coefficients for t...
sqwvfourb 46152 Fourier series ` B ` coeff...
fourierswlem 46153 The Fourier series for the...
fouriersw 46154 Fourier series convergence...
fouriercn 46155 If the derivative of ` F `...
elaa2lem 46156 Elementhood in the set of ...
elaa2 46157 Elementhood in the set of ...
etransclem1 46158 ` H ` is a function. (Con...
etransclem2 46159 Derivative of ` G ` . (Co...
etransclem3 46160 The given ` if ` term is a...
etransclem4 46161 ` F ` expressed as a finit...
etransclem5 46162 A change of bound variable...
etransclem6 46163 A change of bound variable...
etransclem7 46164 The given product is an in...
etransclem8 46165 ` F ` is a function. (Con...
etransclem9 46166 If ` K ` divides ` N ` but...
etransclem10 46167 The given ` if ` term is a...
etransclem11 46168 A change of bound variable...
etransclem12 46169 ` C ` applied to ` N ` . ...
etransclem13 46170 ` F ` applied to ` Y ` . ...
etransclem14 46171 Value of the term ` T ` , ...
etransclem15 46172 Value of the term ` T ` , ...
etransclem16 46173 Every element in the range...
etransclem17 46174 The ` N ` -th derivative o...
etransclem18 46175 The given function is inte...
etransclem19 46176 The ` N ` -th derivative o...
etransclem20 46177 ` H ` is smooth. (Contrib...
etransclem21 46178 The ` N ` -th derivative o...
etransclem22 46179 The ` N ` -th derivative o...
etransclem23 46180 This is the claim proof in...
etransclem24 46181 ` P ` divides the I -th de...
etransclem25 46182 ` P ` factorial divides th...
etransclem26 46183 Every term in the sum of t...
etransclem27 46184 The ` N ` -th derivative o...
etransclem28 46185 ` ( P - 1 ) ` factorial di...
etransclem29 46186 The ` N ` -th derivative o...
etransclem30 46187 The ` N ` -th derivative o...
etransclem31 46188 The ` N ` -th derivative o...
etransclem32 46189 This is the proof for the ...
etransclem33 46190 ` F ` is smooth. (Contrib...
etransclem34 46191 The ` N ` -th derivative o...
etransclem35 46192 ` P ` does not divide the ...
etransclem36 46193 The ` N ` -th derivative o...
etransclem37 46194 ` ( P - 1 ) ` factorial di...
etransclem38 46195 ` P ` divides the I -th de...
etransclem39 46196 ` G ` is a function. (Con...
etransclem40 46197 The ` N ` -th derivative o...
etransclem41 46198 ` P ` does not divide the ...
etransclem42 46199 The ` N ` -th derivative o...
etransclem43 46200 ` G ` is a continuous func...
etransclem44 46201 The given finite sum is no...
etransclem45 46202 ` K ` is an integer. (Con...
etransclem46 46203 This is the proof for equa...
etransclem47 46204 ` _e ` is transcendental. ...
etransclem48 46205 ` _e ` is transcendental. ...
etransc 46206 ` _e ` is transcendental. ...
rrxtopn 46207 The topology of the genera...
rrxngp 46208 Generalized Euclidean real...
rrxtps 46209 Generalized Euclidean real...
rrxtopnfi 46210 The topology of the n-dime...
rrxtopon 46211 The topology on generalize...
rrxtop 46212 The topology on generalize...
rrndistlt 46213 Given two points in the sp...
rrxtoponfi 46214 The topology on n-dimensio...
rrxunitopnfi 46215 The base set of the standa...
rrxtopn0 46216 The topology of the zero-d...
qndenserrnbllem 46217 n-dimensional rational num...
qndenserrnbl 46218 n-dimensional rational num...
rrxtopn0b 46219 The topology of the zero-d...
qndenserrnopnlem 46220 n-dimensional rational num...
qndenserrnopn 46221 n-dimensional rational num...
qndenserrn 46222 n-dimensional rational num...
rrxsnicc 46223 A multidimensional singlet...
rrnprjdstle 46224 The distance between two p...
rrndsmet 46225 ` D ` is a metric for the ...
rrndsxmet 46226 ` D ` is an extended metri...
ioorrnopnlem 46227 The a point in an indexed ...
ioorrnopn 46228 The indexed product of ope...
ioorrnopnxrlem 46229 Given a point ` F ` that b...
ioorrnopnxr 46230 The indexed product of ope...
issal 46237 Express the predicate " ` ...
pwsal 46238 The power set of a given s...
salunicl 46239 SAlg sigma-algebra is clos...
saluncl 46240 The union of two sets in a...
prsal 46241 The pair of the empty set ...
saldifcl 46242 The complement of an eleme...
0sal 46243 The empty set belongs to e...
salgenval 46244 The sigma-algebra generate...
saliunclf 46245 SAlg sigma-algebra is clos...
saliuncl 46246 SAlg sigma-algebra is clos...
salincl 46247 The intersection of two se...
saluni 46248 A set is an element of any...
saliinclf 46249 SAlg sigma-algebra is clos...
saliincl 46250 SAlg sigma-algebra is clos...
saldifcl2 46251 The difference of two elem...
intsaluni 46252 The union of an arbitrary ...
intsal 46253 The arbitrary intersection...
salgenn0 46254 The set used in the defini...
salgencl 46255 ` SalGen ` actually genera...
issald 46256 Sufficient condition to pr...
salexct 46257 An example of nontrivial s...
sssalgen 46258 A set is a subset of the s...
salgenss 46259 The sigma-algebra generate...
salgenuni 46260 The base set of the sigma-...
issalgend 46261 One side of ~ dfsalgen2 . ...
salexct2 46262 An example of a subset tha...
unisalgen 46263 The union of a set belongs...
dfsalgen2 46264 Alternate characterization...
salexct3 46265 An example of a sigma-alge...
salgencntex 46266 This counterexample shows ...
salgensscntex 46267 This counterexample shows ...
issalnnd 46268 Sufficient condition to pr...
dmvolsal 46269 Lebesgue measurable sets f...
saldifcld 46270 The complement of an eleme...
saluncld 46271 The union of two sets in a...
salgencld 46272 ` SalGen ` actually genera...
0sald 46273 The empty set belongs to e...
iooborel 46274 An open interval is a Bore...
salincld 46275 The intersection of two se...
salunid 46276 A set is an element of any...
unisalgen2 46277 The union of a set belongs...
bor1sal 46278 The Borel sigma-algebra on...
iocborel 46279 A left-open, right-closed ...
subsaliuncllem 46280 A subspace sigma-algebra i...
subsaliuncl 46281 A subspace sigma-algebra i...
subsalsal 46282 A subspace sigma-algebra i...
subsaluni 46283 A set belongs to the subsp...
salrestss 46284 A sigma-algebra restricted...
sge0rnre 46287 When ` sum^ ` is applied t...
fge0icoicc 46288 If ` F ` maps to nonnegati...
sge0val 46289 The value of the sum of no...
fge0npnf 46290 If ` F ` maps to nonnegati...
sge0rnn0 46291 The range used in the defi...
sge0vald 46292 The value of the sum of no...
fge0iccico 46293 A range of nonnegative ext...
gsumge0cl 46294 Closure of group sum, for ...
sge0reval 46295 Value of the sum of nonneg...
sge0pnfval 46296 If a term in the sum of no...
fge0iccre 46297 A range of nonnegative ext...
sge0z 46298 Any nonnegative extended s...
sge00 46299 The sum of nonnegative ext...
fsumlesge0 46300 Every finite subsum of non...
sge0revalmpt 46301 Value of the sum of nonneg...
sge0sn 46302 A sum of a nonnegative ext...
sge0tsms 46303 ` sum^ ` applied to a nonn...
sge0cl 46304 The arbitrary sum of nonne...
sge0f1o 46305 Re-index a nonnegative ext...
sge0snmpt 46306 A sum of a nonnegative ext...
sge0ge0 46307 The sum of nonnegative ext...
sge0xrcl 46308 The arbitrary sum of nonne...
sge0repnf 46309 The of nonnegative extende...
sge0fsum 46310 The arbitrary sum of a fin...
sge0rern 46311 If the sum of nonnegative ...
sge0supre 46312 If the arbitrary sum of no...
sge0fsummpt 46313 The arbitrary sum of a fin...
sge0sup 46314 The arbitrary sum of nonne...
sge0less 46315 A shorter sum of nonnegati...
sge0rnbnd 46316 The range used in the defi...
sge0pr 46317 Sum of a pair of nonnegati...
sge0gerp 46318 The arbitrary sum of nonne...
sge0pnffigt 46319 If the sum of nonnegative ...
sge0ssre 46320 If a sum of nonnegative ex...
sge0lefi 46321 A sum of nonnegative exten...
sge0lessmpt 46322 A shorter sum of nonnegati...
sge0ltfirp 46323 If the sum of nonnegative ...
sge0prle 46324 The sum of a pair of nonne...
sge0gerpmpt 46325 The arbitrary sum of nonne...
sge0resrnlem 46326 The sum of nonnegative ext...
sge0resrn 46327 The sum of nonnegative ext...
sge0ssrempt 46328 If a sum of nonnegative ex...
sge0resplit 46329 ` sum^ ` splits into two p...
sge0le 46330 If all of the terms of sum...
sge0ltfirpmpt 46331 If the extended sum of non...
sge0split 46332 Split a sum of nonnegative...
sge0lempt 46333 If all of the terms of sum...
sge0splitmpt 46334 Split a sum of nonnegative...
sge0ss 46335 Change the index set to a ...
sge0iunmptlemfi 46336 Sum of nonnegative extende...
sge0p1 46337 The addition of the next t...
sge0iunmptlemre 46338 Sum of nonnegative extende...
sge0fodjrnlem 46339 Re-index a nonnegative ext...
sge0fodjrn 46340 Re-index a nonnegative ext...
sge0iunmpt 46341 Sum of nonnegative extende...
sge0iun 46342 Sum of nonnegative extende...
sge0nemnf 46343 The generalized sum of non...
sge0rpcpnf 46344 The sum of an infinite num...
sge0rernmpt 46345 If the sum of nonnegative ...
sge0lefimpt 46346 A sum of nonnegative exten...
nn0ssge0 46347 Nonnegative integers are n...
sge0clmpt 46348 The generalized sum of non...
sge0ltfirpmpt2 46349 If the extended sum of non...
sge0isum 46350 If a series of nonnegative...
sge0xrclmpt 46351 The generalized sum of non...
sge0xp 46352 Combine two generalized su...
sge0isummpt 46353 If a series of nonnegative...
sge0ad2en 46354 The value of the infinite ...
sge0isummpt2 46355 If a series of nonnegative...
sge0xaddlem1 46356 The extended addition of t...
sge0xaddlem2 46357 The extended addition of t...
sge0xadd 46358 The extended addition of t...
sge0fsummptf 46359 The generalized sum of a f...
sge0snmptf 46360 A sum of a nonnegative ext...
sge0ge0mpt 46361 The sum of nonnegative ext...
sge0repnfmpt 46362 The of nonnegative extende...
sge0pnffigtmpt 46363 If the generalized sum of ...
sge0splitsn 46364 Separate out a term in a g...
sge0pnffsumgt 46365 If the sum of nonnegative ...
sge0gtfsumgt 46366 If the generalized sum of ...
sge0uzfsumgt 46367 If a real number is smalle...
sge0pnfmpt 46368 If a term in the sum of no...
sge0seq 46369 A series of nonnegative re...
sge0reuz 46370 Value of the generalized s...
sge0reuzb 46371 Value of the generalized s...
ismea 46374 Express the predicate " ` ...
dmmeasal 46375 The domain of a measure is...
meaf 46376 A measure is a function th...
mea0 46377 The measure of the empty s...
nnfoctbdjlem 46378 There exists a mapping fro...
nnfoctbdj 46379 There exists a mapping fro...
meadjuni 46380 The measure of the disjoin...
meacl 46381 The measure of a set is a ...
iundjiunlem 46382 The sets in the sequence `...
iundjiun 46383 Given a sequence ` E ` of ...
meaxrcl 46384 The measure of a set is an...
meadjun 46385 The measure of the union o...
meassle 46386 The measure of a set is gr...
meaunle 46387 The measure of the union o...
meadjiunlem 46388 The sum of nonnegative ext...
meadjiun 46389 The measure of the disjoin...
ismeannd 46390 Sufficient condition to pr...
meaiunlelem 46391 The measure of the union o...
meaiunle 46392 The measure of the union o...
psmeasurelem 46393 ` M ` applied to a disjoin...
psmeasure 46394 Point supported measure, R...
voliunsge0lem 46395 The Lebesgue measure funct...
voliunsge0 46396 The Lebesgue measure funct...
volmea 46397 The Lebesgue measure on th...
meage0 46398 If the measure of a measur...
meadjunre 46399 The measure of the union o...
meassre 46400 If the measure of a measur...
meale0eq0 46401 A measure that is less tha...
meadif 46402 The measure of the differe...
meaiuninclem 46403 Measures are continuous fr...
meaiuninc 46404 Measures are continuous fr...
meaiuninc2 46405 Measures are continuous fr...
meaiunincf 46406 Measures are continuous fr...
meaiuninc3v 46407 Measures are continuous fr...
meaiuninc3 46408 Measures are continuous fr...
meaiininclem 46409 Measures are continuous fr...
meaiininc 46410 Measures are continuous fr...
meaiininc2 46411 Measures are continuous fr...
caragenval 46416 The sigma-algebra generate...
isome 46417 Express the predicate " ` ...
caragenel 46418 Membership in the Caratheo...
omef 46419 An outer measure is a func...
ome0 46420 The outer measure of the e...
omessle 46421 The outer measure of a set...
omedm 46422 The domain of an outer mea...
caragensplit 46423 If ` E ` is in the set gen...
caragenelss 46424 An element of the Caratheo...
carageneld 46425 Membership in the Caratheo...
omecl 46426 The outer measure of a set...
caragenss 46427 The sigma-algebra generate...
omeunile 46428 The outer measure of the u...
caragen0 46429 The empty set belongs to a...
omexrcl 46430 The outer measure of a set...
caragenunidm 46431 The base set of an outer m...
caragensspw 46432 The sigma-algebra generate...
omessre 46433 If the outer measure of a ...
caragenuni 46434 The base set of the sigma-...
caragenuncllem 46435 The Caratheodory's constru...
caragenuncl 46436 The Caratheodory's constru...
caragendifcl 46437 The Caratheodory's constru...
caragenfiiuncl 46438 The Caratheodory's constru...
omeunle 46439 The outer measure of the u...
omeiunle 46440 The outer measure of the i...
omelesplit 46441 The outer measure of a set...
omeiunltfirp 46442 If the outer measure of a ...
omeiunlempt 46443 The outer measure of the i...
carageniuncllem1 46444 The outer measure of ` A i...
carageniuncllem2 46445 The Caratheodory's constru...
carageniuncl 46446 The Caratheodory's constru...
caragenunicl 46447 The Caratheodory's constru...
caragensal 46448 Caratheodory's method gene...
caratheodorylem1 46449 Lemma used to prove that C...
caratheodorylem2 46450 Caratheodory's constructio...
caratheodory 46451 Caratheodory's constructio...
0ome 46452 The map that assigns 0 to ...
isomenndlem 46453 ` O ` is sub-additive w.r....
isomennd 46454 Sufficient condition to pr...
caragenel2d 46455 Membership in the Caratheo...
omege0 46456 If the outer measure of a ...
omess0 46457 If the outer measure of a ...
caragencmpl 46458 A measure built with the C...
vonval 46463 Value of the Lebesgue meas...
ovnval 46464 Value of the Lebesgue oute...
elhoi 46465 Membership in a multidimen...
icoresmbl 46466 A closed-below, open-above...
hoissre 46467 The projection of a half-o...
ovnval2 46468 Value of the Lebesgue oute...
volicorecl 46469 The Lebesgue measure of a ...
hoiprodcl 46470 The pre-measure of half-op...
hoicvr 46471 ` I ` is a countable set o...
hoissrrn 46472 A half-open interval is a ...
ovn0val 46473 The Lebesgue outer measure...
ovnn0val 46474 The value of a (multidimen...
ovnval2b 46475 Value of the Lebesgue oute...
volicorescl 46476 The Lebesgue measure of a ...
ovnprodcl 46477 The product used in the de...
hoiprodcl2 46478 The pre-measure of half-op...
hoicvrrex 46479 Any subset of the multidim...
ovnsupge0 46480 The set used in the defini...
ovnlecvr 46481 Given a subset of multidim...
ovnpnfelsup 46482 ` +oo ` is an element of t...
ovnsslelem 46483 The (multidimensional, non...
ovnssle 46484 The (multidimensional) Leb...
ovnlerp 46485 The Lebesgue outer measure...
ovnf 46486 The Lebesgue outer measure...
ovncvrrp 46487 The Lebesgue outer measure...
ovn0lem 46488 For any finite dimension, ...
ovn0 46489 For any finite dimension, ...
ovncl 46490 The Lebesgue outer measure...
ovn02 46491 For the zero-dimensional s...
ovnxrcl 46492 The Lebesgue outer measure...
ovnsubaddlem1 46493 The Lebesgue outer measure...
ovnsubaddlem2 46494 ` ( voln* `` X ) ` is suba...
ovnsubadd 46495 ` ( voln* `` X ) ` is suba...
ovnome 46496 ` ( voln* `` X ) ` is an o...
vonmea 46497 ` ( voln `` X ) ` is a mea...
volicon0 46498 The measure of a nonempty ...
hsphoif 46499 ` H ` is a function (that ...
hoidmvval 46500 The dimensional volume of ...
hoissrrn2 46501 A half-open interval is a ...
hsphoival 46502 ` H ` is a function (that ...
hoiprodcl3 46503 The pre-measure of half-op...
volicore 46504 The Lebesgue measure of a ...
hoidmvcl 46505 The dimensional volume of ...
hoidmv0val 46506 The dimensional volume of ...
hoidmvn0val 46507 The dimensional volume of ...
hsphoidmvle2 46508 The dimensional volume of ...
hsphoidmvle 46509 The dimensional volume of ...
hoidmvval0 46510 The dimensional volume of ...
hoiprodp1 46511 The dimensional volume of ...
sge0hsphoire 46512 If the generalized sum of ...
hoidmvval0b 46513 The dimensional volume of ...
hoidmv1lelem1 46514 The supremum of ` U ` belo...
hoidmv1lelem2 46515 This is the contradiction ...
hoidmv1lelem3 46516 The dimensional volume of ...
hoidmv1le 46517 The dimensional volume of ...
hoidmvlelem1 46518 The supremum of ` U ` belo...
hoidmvlelem2 46519 This is the contradiction ...
hoidmvlelem3 46520 This is the contradiction ...
hoidmvlelem4 46521 The dimensional volume of ...
hoidmvlelem5 46522 The dimensional volume of ...
hoidmvle 46523 The dimensional volume of ...
ovnhoilem1 46524 The Lebesgue outer measure...
ovnhoilem2 46525 The Lebesgue outer measure...
ovnhoi 46526 The Lebesgue outer measure...
dmovn 46527 The domain of the Lebesgue...
hoicoto2 46528 The half-open interval exp...
dmvon 46529 Lebesgue measurable n-dime...
hoi2toco 46530 The half-open interval exp...
hoidifhspval 46531 ` D ` is a function that r...
hspval 46532 The value of the half-spac...
ovnlecvr2 46533 Given a subset of multidim...
ovncvr2 46534 ` B ` and ` T ` are the le...
dmovnsal 46535 The domain of the Lebesgue...
unidmovn 46536 Base set of the n-dimensio...
rrnmbl 46537 The set of n-dimensional R...
hoidifhspval2 46538 ` D ` is a function that r...
hspdifhsp 46539 A n-dimensional half-open ...
unidmvon 46540 Base set of the n-dimensio...
hoidifhspf 46541 ` D ` is a function that r...
hoidifhspval3 46542 ` D ` is a function that r...
hoidifhspdmvle 46543 The dimensional volume of ...
voncmpl 46544 The Lebesgue measure is co...
hoiqssbllem1 46545 The center of the n-dimens...
hoiqssbllem2 46546 The center of the n-dimens...
hoiqssbllem3 46547 A n-dimensional ball conta...
hoiqssbl 46548 A n-dimensional ball conta...
hspmbllem1 46549 Any half-space of the n-di...
hspmbllem2 46550 Any half-space of the n-di...
hspmbllem3 46551 Any half-space of the n-di...
hspmbl 46552 Any half-space of the n-di...
hoimbllem 46553 Any n-dimensional half-ope...
hoimbl 46554 Any n-dimensional half-ope...
opnvonmbllem1 46555 The half-open interval exp...
opnvonmbllem2 46556 An open subset of the n-di...
opnvonmbl 46557 An open subset of the n-di...
opnssborel 46558 Open sets of a generalized...
borelmbl 46559 All Borel subsets of the n...
volicorege0 46560 The Lebesgue measure of a ...
isvonmbl 46561 The predicate " ` A ` is m...
mblvon 46562 The n-dimensional Lebesgue...
vonmblss 46563 n-dimensional Lebesgue mea...
volico2 46564 The measure of left-closed...
vonmblss2 46565 n-dimensional Lebesgue mea...
ovolval2lem 46566 The value of the Lebesgue ...
ovolval2 46567 The value of the Lebesgue ...
ovnsubadd2lem 46568 ` ( voln* `` X ) ` is suba...
ovnsubadd2 46569 ` ( voln* `` X ) ` is suba...
ovolval3 46570 The value of the Lebesgue ...
ovnsplit 46571 The n-dimensional Lebesgue...
ovolval4lem1 46572 |- ( ( ph /\ n e. A ) -> ...
ovolval4lem2 46573 The value of the Lebesgue ...
ovolval4 46574 The value of the Lebesgue ...
ovolval5lem1 46575 ` |- ( ph -> ( sum^ `` ( n...
ovolval5lem2 46576 ` |- ( ( ph /\ n e. NN ) -...
ovolval5lem3 46577 The value of the Lebesgue ...
ovolval5 46578 The value of the Lebesgue ...
ovnovollem1 46579 if ` F ` is a cover of ` B...
ovnovollem2 46580 if ` I ` is a cover of ` (...
ovnovollem3 46581 The 1-dimensional Lebesgue...
ovnovol 46582 The 1-dimensional Lebesgue...
vonvolmbllem 46583 If a subset ` B ` of real ...
vonvolmbl 46584 A subset of Real numbers i...
vonvol 46585 The 1-dimensional Lebesgue...
vonvolmbl2 46586 A subset ` X ` of the spac...
vonvol2 46587 The 1-dimensional Lebesgue...
hoimbl2 46588 Any n-dimensional half-ope...
voncl 46589 The Lebesgue measure of a ...
vonhoi 46590 The Lebesgue outer measure...
vonxrcl 46591 The Lebesgue measure of a ...
ioosshoi 46592 A n-dimensional open inter...
vonn0hoi 46593 The Lebesgue outer measure...
von0val 46594 The Lebesgue measure (for ...
vonhoire 46595 The Lebesgue measure of a ...
iinhoiicclem 46596 A n-dimensional closed int...
iinhoiicc 46597 A n-dimensional closed int...
iunhoiioolem 46598 A n-dimensional open inter...
iunhoiioo 46599 A n-dimensional open inter...
ioovonmbl 46600 Any n-dimensional open int...
iccvonmbllem 46601 Any n-dimensional closed i...
iccvonmbl 46602 Any n-dimensional closed i...
vonioolem1 46603 The sequence of the measur...
vonioolem2 46604 The n-dimensional Lebesgue...
vonioo 46605 The n-dimensional Lebesgue...
vonicclem1 46606 The sequence of the measur...
vonicclem2 46607 The n-dimensional Lebesgue...
vonicc 46608 The n-dimensional Lebesgue...
snvonmbl 46609 A n-dimensional singleton ...
vonn0ioo 46610 The n-dimensional Lebesgue...
vonn0icc 46611 The n-dimensional Lebesgue...
ctvonmbl 46612 Any n-dimensional countabl...
vonn0ioo2 46613 The n-dimensional Lebesgue...
vonsn 46614 The n-dimensional Lebesgue...
vonn0icc2 46615 The n-dimensional Lebesgue...
vonct 46616 The n-dimensional Lebesgue...
vitali2 46617 There are non-measurable s...
pimltmnf2f 46620 Given a real-valued functi...
pimltmnf2 46621 Given a real-valued functi...
preimagelt 46622 The preimage of a right-op...
preimalegt 46623 The preimage of a left-ope...
pimconstlt0 46624 Given a constant function,...
pimconstlt1 46625 Given a constant function,...
pimltpnff 46626 Given a real-valued functi...
pimltpnf 46627 Given a real-valued functi...
pimgtpnf2f 46628 Given a real-valued functi...
pimgtpnf2 46629 Given a real-valued functi...
salpreimagelt 46630 If all the preimages of le...
pimrecltpos 46631 The preimage of an unbound...
salpreimalegt 46632 If all the preimages of ri...
pimiooltgt 46633 The preimage of an open in...
preimaicomnf 46634 Preimage of an open interv...
pimltpnf2f 46635 Given a real-valued functi...
pimltpnf2 46636 Given a real-valued functi...
pimgtmnf2 46637 Given a real-valued functi...
pimdecfgtioc 46638 Given a nonincreasing func...
pimincfltioc 46639 Given a nondecreasing func...
pimdecfgtioo 46640 Given a nondecreasing func...
pimincfltioo 46641 Given a nondecreasing func...
preimaioomnf 46642 Preimage of an open interv...
preimageiingt 46643 A preimage of a left-close...
preimaleiinlt 46644 A preimage of a left-open,...
pimgtmnff 46645 Given a real-valued functi...
pimgtmnf 46646 Given a real-valued functi...
pimrecltneg 46647 The preimage of an unbound...
salpreimagtge 46648 If all the preimages of le...
salpreimaltle 46649 If all the preimages of ri...
issmflem 46650 The predicate " ` F ` is a...
issmf 46651 The predicate " ` F ` is a...
salpreimalelt 46652 If all the preimages of ri...
salpreimagtlt 46653 If all the preimages of le...
smfpreimalt 46654 Given a function measurabl...
smff 46655 A function measurable w.r....
smfdmss 46656 The domain of a function m...
issmff 46657 The predicate " ` F ` is a...
issmfd 46658 A sufficient condition for...
smfpreimaltf 46659 Given a function measurabl...
issmfdf 46660 A sufficient condition for...
sssmf 46661 The restriction of a sigma...
mbfresmf 46662 A real-valued measurable f...
cnfsmf 46663 A continuous function is m...
incsmflem 46664 A nondecreasing function i...
incsmf 46665 A real-valued, nondecreasi...
smfsssmf 46666 If a function is measurabl...
issmflelem 46667 The predicate " ` F ` is a...
issmfle 46668 The predicate " ` F ` is a...
smfpimltmpt 46669 Given a function measurabl...
smfpimltxr 46670 Given a function measurabl...
issmfdmpt 46671 A sufficient condition for...
smfconst 46672 Given a sigma-algebra over...
sssmfmpt 46673 The restriction of a sigma...
cnfrrnsmf 46674 A function, continuous fro...
smfid 46675 The identity function is B...
bormflebmf 46676 A Borel measurable functio...
smfpreimale 46677 Given a function measurabl...
issmfgtlem 46678 The predicate " ` F ` is a...
issmfgt 46679 The predicate " ` F ` is a...
issmfled 46680 A sufficient condition for...
smfpimltxrmptf 46681 Given a function measurabl...
smfpimltxrmpt 46682 Given a function measurabl...
smfmbfcex 46683 A constant function, with ...
issmfgtd 46684 A sufficient condition for...
smfpreimagt 46685 Given a function measurabl...
smfaddlem1 46686 Given the sum of two funct...
smfaddlem2 46687 The sum of two sigma-measu...
smfadd 46688 The sum of two sigma-measu...
decsmflem 46689 A nonincreasing function i...
decsmf 46690 A real-valued, nonincreasi...
smfpreimagtf 46691 Given a function measurabl...
issmfgelem 46692 The predicate " ` F ` is a...
issmfge 46693 The predicate " ` F ` is a...
smflimlem1 46694 Lemma for the proof that t...
smflimlem2 46695 Lemma for the proof that t...
smflimlem3 46696 The limit of sigma-measura...
smflimlem4 46697 Lemma for the proof that t...
smflimlem5 46698 Lemma for the proof that t...
smflimlem6 46699 Lemma for the proof that t...
smflim 46700 The limit of sigma-measura...
nsssmfmbflem 46701 The sigma-measurable funct...
nsssmfmbf 46702 The sigma-measurable funct...
smfpimgtxr 46703 Given a function measurabl...
smfpimgtmpt 46704 Given a function measurabl...
smfpreimage 46705 Given a function measurabl...
mbfpsssmf 46706 Real-valued measurable fun...
smfpimgtxrmptf 46707 Given a function measurabl...
smfpimgtxrmpt 46708 Given a function measurabl...
smfpimioompt 46709 Given a function measurabl...
smfpimioo 46710 Given a function measurabl...
smfresal 46711 Given a sigma-measurable f...
smfrec 46712 The reciprocal of a sigma-...
smfres 46713 The restriction of sigma-m...
smfmullem1 46714 The multiplication of two ...
smfmullem2 46715 The multiplication of two ...
smfmullem3 46716 The multiplication of two ...
smfmullem4 46717 The multiplication of two ...
smfmul 46718 The multiplication of two ...
smfmulc1 46719 A sigma-measurable functio...
smfdiv 46720 The fraction of two sigma-...
smfpimbor1lem1 46721 Every open set belongs to ...
smfpimbor1lem2 46722 Given a sigma-measurable f...
smfpimbor1 46723 Given a sigma-measurable f...
smf2id 46724 Twice the identity functio...
smfco 46725 The composition of a Borel...
smfneg 46726 The negative of a sigma-me...
smffmptf 46727 A function measurable w.r....
smffmpt 46728 A function measurable w.r....
smflim2 46729 The limit of a sequence of...
smfpimcclem 46730 Lemma for ~ smfpimcc given...
smfpimcc 46731 Given a countable set of s...
issmfle2d 46732 A sufficient condition for...
smflimmpt 46733 The limit of a sequence of...
smfsuplem1 46734 The supremum of a countabl...
smfsuplem2 46735 The supremum of a countabl...
smfsuplem3 46736 The supremum of a countabl...
smfsup 46737 The supremum of a countabl...
smfsupmpt 46738 The supremum of a countabl...
smfsupxr 46739 The supremum of a countabl...
smfinflem 46740 The infimum of a countable...
smfinf 46741 The infimum of a countable...
smfinfmpt 46742 The infimum of a countable...
smflimsuplem1 46743 If ` H ` converges, the ` ...
smflimsuplem2 46744 The superior limit of a se...
smflimsuplem3 46745 The limit of the ` ( H `` ...
smflimsuplem4 46746 If ` H ` converges, the ` ...
smflimsuplem5 46747 ` H ` converges to the sup...
smflimsuplem6 46748 The superior limit of a se...
smflimsuplem7 46749 The superior limit of a se...
smflimsuplem8 46750 The superior limit of a se...
smflimsup 46751 The superior limit of a se...
smflimsupmpt 46752 The superior limit of a se...
smfliminflem 46753 The inferior limit of a co...
smfliminf 46754 The inferior limit of a co...
smfliminfmpt 46755 The inferior limit of a co...
adddmmbl 46756 If two functions have doma...
adddmmbl2 46757 If two functions have doma...
muldmmbl 46758 If two functions have doma...
muldmmbl2 46759 If two functions have doma...
smfdmmblpimne 46760 If a measurable function w...
smfdivdmmbl 46761 If a functions and a sigma...
smfpimne 46762 Given a function measurabl...
smfpimne2 46763 Given a function measurabl...
smfdivdmmbl2 46764 If a functions and a sigma...
fsupdm 46765 The domain of the sup func...
fsupdm2 46766 The domain of the sup func...
smfsupdmmbllem 46767 If a countable set of sigm...
smfsupdmmbl 46768 If a countable set of sigm...
finfdm 46769 The domain of the inf func...
finfdm2 46770 The domain of the inf func...
smfinfdmmbllem 46771 If a countable set of sigm...
smfinfdmmbl 46772 If a countable set of sigm...
sigarval 46773 Define the signed area by ...
sigarim 46774 Signed area takes value in...
sigarac 46775 Signed area is anticommuta...
sigaraf 46776 Signed area is additive by...
sigarmf 46777 Signed area is additive (w...
sigaras 46778 Signed area is additive by...
sigarms 46779 Signed area is additive (w...
sigarls 46780 Signed area is linear by t...
sigarid 46781 Signed area of a flat para...
sigarexp 46782 Expand the signed area for...
sigarperm 46783 Signed area ` ( A - C ) G ...
sigardiv 46784 If signed area between vec...
sigarimcd 46785 Signed area takes value in...
sigariz 46786 If signed area is zero, th...
sigarcol 46787 Given three points ` A ` ,...
sharhght 46788 Let ` A B C ` be a triangl...
sigaradd 46789 Subtracting (double) area ...
cevathlem1 46790 Ceva's theorem first lemma...
cevathlem2 46791 Ceva's theorem second lemm...
cevath 46792 Ceva's theorem. Let ` A B...
simpcntrab 46793 The center of a simple gro...
et-ltneverrefl 46794 Less-than class is never r...
et-equeucl 46795 Alternative proof that equ...
et-sqrtnegnre 46796 The square root of a negat...
natlocalincr 46797 Global monotonicity on hal...
natglobalincr 46798 Local monotonicity on half...
upwordnul 46801 Empty set is an increasing...
upwordisword 46802 Any increasing sequence is...
singoutnword 46803 Singleton with character o...
singoutnupword 46804 Singleton with character o...
upwordsing 46805 Singleton is an increasing...
upwordsseti 46806 Strictly increasing sequen...
tworepnotupword 46807 Concatenation of identical...
upwrdfi 46808 There is a finite number o...
hirstL-ax3 46809 The third axiom of a syste...
ax3h 46810 Recover ~ ax-3 from ~ hirs...
aibandbiaiffaiffb 46811 A closed form showing (a i...
aibandbiaiaiffb 46812 A closed form showing (a i...
notatnand 46813 Do not use. Use intnanr i...
aistia 46814 Given a is equivalent to `...
aisfina 46815 Given a is equivalent to `...
bothtbothsame 46816 Given both a, b are equiva...
bothfbothsame 46817 Given both a, b are equiva...
aiffbbtat 46818 Given a is equivalent to b...
aisbbisfaisf 46819 Given a is equivalent to b...
axorbtnotaiffb 46820 Given a is exclusive to b,...
aiffnbandciffatnotciffb 46821 Given a is equivalent to (...
axorbciffatcxorb 46822 Given a is equivalent to (...
aibnbna 46823 Given a implies b, (not b)...
aibnbaif 46824 Given a implies b, not b, ...
aiffbtbat 46825 Given a is equivalent to b...
astbstanbst 46826 Given a is equivalent to T...
aistbistaandb 46827 Given a is equivalent to T...
aisbnaxb 46828 Given a is equivalent to b...
atbiffatnnb 46829 If a implies b, then a imp...
bisaiaisb 46830 Application of bicom1 with...
atbiffatnnbalt 46831 If a implies b, then a imp...
abnotbtaxb 46832 Assuming a, not b, there e...
abnotataxb 46833 Assuming not a, b, there e...
conimpf 46834 Assuming a, not b, and a i...
conimpfalt 46835 Assuming a, not b, and a i...
aistbisfiaxb 46836 Given a is equivalent to T...
aisfbistiaxb 46837 Given a is equivalent to F...
aifftbifffaibif 46838 Given a is equivalent to T...
aifftbifffaibifff 46839 Given a is equivalent to T...
atnaiana 46840 Given a, it is not the cas...
ainaiaandna 46841 Given a, a implies it is n...
abcdta 46842 Given (((a and b) and c) a...
abcdtb 46843 Given (((a and b) and c) a...
abcdtc 46844 Given (((a and b) and c) a...
abcdtd 46845 Given (((a and b) and c) a...
abciffcbatnabciffncba 46846 Operands in a biconditiona...
abciffcbatnabciffncbai 46847 Operands in a biconditiona...
nabctnabc 46848 not ( a -> ( b /\ c ) ) we...
jabtaib 46849 For when pm3.4 lacks a pm3...
onenotinotbothi 46850 From one negated implicati...
twonotinotbothi 46851 From these two negated imp...
clifte 46852 show d is the same as an i...
cliftet 46853 show d is the same as an i...
clifteta 46854 show d is the same as an i...
cliftetb 46855 show d is the same as an i...
confun 46856 Given the hypotheses there...
confun2 46857 Confun simplified to two p...
confun3 46858 Confun's more complex form...
confun4 46859 An attempt at derivative. ...
confun5 46860 An attempt at derivative. ...
plcofph 46861 Given, a,b and a "definiti...
pldofph 46862 Given, a,b c, d, "definiti...
plvcofph 46863 Given, a,b,d, and "definit...
plvcofphax 46864 Given, a,b,d, and "definit...
plvofpos 46865 rh is derivable because ON...
mdandyv0 46866 Given the equivalences set...
mdandyv1 46867 Given the equivalences set...
mdandyv2 46868 Given the equivalences set...
mdandyv3 46869 Given the equivalences set...
mdandyv4 46870 Given the equivalences set...
mdandyv5 46871 Given the equivalences set...
mdandyv6 46872 Given the equivalences set...
mdandyv7 46873 Given the equivalences set...
mdandyv8 46874 Given the equivalences set...
mdandyv9 46875 Given the equivalences set...
mdandyv10 46876 Given the equivalences set...
mdandyv11 46877 Given the equivalences set...
mdandyv12 46878 Given the equivalences set...
mdandyv13 46879 Given the equivalences set...
mdandyv14 46880 Given the equivalences set...
mdandyv15 46881 Given the equivalences set...
mdandyvr0 46882 Given the equivalences set...
mdandyvr1 46883 Given the equivalences set...
mdandyvr2 46884 Given the equivalences set...
mdandyvr3 46885 Given the equivalences set...
mdandyvr4 46886 Given the equivalences set...
mdandyvr5 46887 Given the equivalences set...
mdandyvr6 46888 Given the equivalences set...
mdandyvr7 46889 Given the equivalences set...
mdandyvr8 46890 Given the equivalences set...
mdandyvr9 46891 Given the equivalences set...
mdandyvr10 46892 Given the equivalences set...
mdandyvr11 46893 Given the equivalences set...
mdandyvr12 46894 Given the equivalences set...
mdandyvr13 46895 Given the equivalences set...
mdandyvr14 46896 Given the equivalences set...
mdandyvr15 46897 Given the equivalences set...
mdandyvrx0 46898 Given the exclusivities se...
mdandyvrx1 46899 Given the exclusivities se...
mdandyvrx2 46900 Given the exclusivities se...
mdandyvrx3 46901 Given the exclusivities se...
mdandyvrx4 46902 Given the exclusivities se...
mdandyvrx5 46903 Given the exclusivities se...
mdandyvrx6 46904 Given the exclusivities se...
mdandyvrx7 46905 Given the exclusivities se...
mdandyvrx8 46906 Given the exclusivities se...
mdandyvrx9 46907 Given the exclusivities se...
mdandyvrx10 46908 Given the exclusivities se...
mdandyvrx11 46909 Given the exclusivities se...
mdandyvrx12 46910 Given the exclusivities se...
mdandyvrx13 46911 Given the exclusivities se...
mdandyvrx14 46912 Given the exclusivities se...
mdandyvrx15 46913 Given the exclusivities se...
H15NH16TH15IH16 46914 Given 15 hypotheses and a ...
dandysum2p2e4 46915 CONTRADICTION PROVED AT 1 ...
mdandysum2p2e4 46916 CONTRADICTION PROVED AT 1 ...
adh-jarrsc 46917 Replacement of a nested an...
adh-minim 46918 A single axiom for minimal...
adh-minim-ax1-ax2-lem1 46919 First lemma for the deriva...
adh-minim-ax1-ax2-lem2 46920 Second lemma for the deriv...
adh-minim-ax1-ax2-lem3 46921 Third lemma for the deriva...
adh-minim-ax1-ax2-lem4 46922 Fourth lemma for the deriv...
adh-minim-ax1 46923 Derivation of ~ ax-1 from ...
adh-minim-ax2-lem5 46924 Fifth lemma for the deriva...
adh-minim-ax2-lem6 46925 Sixth lemma for the deriva...
adh-minim-ax2c 46926 Derivation of a commuted f...
adh-minim-ax2 46927 Derivation of ~ ax-2 from ...
adh-minim-idALT 46928 Derivation of ~ id (reflex...
adh-minim-pm2.43 46929 Derivation of ~ pm2.43 Whi...
adh-minimp 46930 Another single axiom for m...
adh-minimp-jarr-imim1-ax2c-lem1 46931 First lemma for the deriva...
adh-minimp-jarr-lem2 46932 Second lemma for the deriv...
adh-minimp-jarr-ax2c-lem3 46933 Third lemma for the deriva...
adh-minimp-sylsimp 46934 Derivation of ~ jarr (also...
adh-minimp-ax1 46935 Derivation of ~ ax-1 from ...
adh-minimp-imim1 46936 Derivation of ~ imim1 ("le...
adh-minimp-ax2c 46937 Derivation of a commuted f...
adh-minimp-ax2-lem4 46938 Fourth lemma for the deriv...
adh-minimp-ax2 46939 Derivation of ~ ax-2 from ...
adh-minimp-idALT 46940 Derivation of ~ id (reflex...
adh-minimp-pm2.43 46941 Derivation of ~ pm2.43 Whi...
n0nsn2el 46942 If a class with one elemen...
eusnsn 46943 There is a unique element ...
absnsb 46944 If the class abstraction `...
euabsneu 46945 Another way to express exi...
elprneb 46946 An element of a proper uno...
oppr 46947 Equality for ordered pairs...
opprb 46948 Equality for unordered pai...
or2expropbilem1 46949 Lemma 1 for ~ or2expropbi ...
or2expropbilem2 46950 Lemma 2 for ~ or2expropbi ...
or2expropbi 46951 If two classes are strictl...
eubrv 46952 If there is a unique set w...
eubrdm 46953 If there is a unique set w...
eldmressn 46954 Element of the domain of a...
iota0def 46955 Example for a defined iota...
iota0ndef 46956 Example for an undefined i...
fveqvfvv 46957 If a function's value at a...
fnresfnco 46958 Composition of two functio...
funcoressn 46959 A composition restricted t...
funressnfv 46960 A restriction to a singlet...
funressndmfvrn 46961 The value of a function ` ...
funressnvmo 46962 A function restricted to a...
funressnmo 46963 A function restricted to a...
funressneu 46964 There is exactly one value...
fresfo 46965 Conditions for a restricti...
fsetsniunop 46966 The class of all functions...
fsetabsnop 46967 The class of all functions...
fsetsnf 46968 The mapping of an element ...
fsetsnf1 46969 The mapping of an element ...
fsetsnfo 46970 The mapping of an element ...
fsetsnf1o 46971 The mapping of an element ...
fsetsnprcnex 46972 The class of all functions...
cfsetssfset 46973 The class of constant func...
cfsetsnfsetfv 46974 The function value of the ...
cfsetsnfsetf 46975 The mapping of the class o...
cfsetsnfsetf1 46976 The mapping of the class o...
cfsetsnfsetfo 46977 The mapping of the class o...
cfsetsnfsetf1o 46978 The mapping of the class o...
fsetprcnexALT 46979 First version of proof for...
fcoreslem1 46980 Lemma 1 for ~ fcores . (C...
fcoreslem2 46981 Lemma 2 for ~ fcores . (C...
fcoreslem3 46982 Lemma 3 for ~ fcores . (C...
fcoreslem4 46983 Lemma 4 for ~ fcores . (C...
fcores 46984 Every composite function `...
fcoresf1lem 46985 Lemma for ~ fcoresf1 . (C...
fcoresf1 46986 If a composition is inject...
fcoresf1b 46987 A composition is injective...
fcoresfo 46988 If a composition is surjec...
fcoresfob 46989 A composition is surjectiv...
fcoresf1ob 46990 A composition is bijective...
f1cof1blem 46991 Lemma for ~ f1cof1b and ~ ...
3f1oss1 46992 The composition of three b...
3f1oss2 46993 The composition of three b...
f1cof1b 46994 If the range of ` F ` equa...
funfocofob 46995 If the domain of a functio...
fnfocofob 46996 If the domain of a functio...
focofob 46997 If the domain of a functio...
f1ocof1ob 46998 If the range of ` F ` equa...
f1ocof1ob2 46999 If the range of ` F ` equa...
aiotajust 47001 Soundness justification th...
dfaiota2 47003 Alternate definition of th...
reuabaiotaiota 47004 The iota and the alternate...
reuaiotaiota 47005 The iota and the alternate...
aiotaexb 47006 The alternate iota over a ...
aiotavb 47007 The alternate iota over a ...
aiotaint 47008 This is to ~ df-aiota what...
dfaiota3 47009 Alternate definition of ` ...
iotan0aiotaex 47010 If the iota over a wff ` p...
aiotaexaiotaiota 47011 The alternate iota over a ...
aiotaval 47012 Theorem 8.19 in [Quine] p....
aiota0def 47013 Example for a defined alte...
aiota0ndef 47014 Example for an undefined a...
r19.32 47015 Theorem 19.32 of [Margaris...
rexsb 47016 An equivalent expression f...
rexrsb 47017 An equivalent expression f...
2rexsb 47018 An equivalent expression f...
2rexrsb 47019 An equivalent expression f...
cbvral2 47020 Change bound variables of ...
cbvrex2 47021 Change bound variables of ...
ralndv1 47022 Example for a theorem abou...
ralndv2 47023 Second example for a theor...
reuf1odnf 47024 There is exactly one eleme...
reuf1od 47025 There is exactly one eleme...
euoreqb 47026 There is a set which is eq...
2reu3 47027 Double restricted existent...
2reu7 47028 Two equivalent expressions...
2reu8 47029 Two equivalent expressions...
2reu8i 47030 Implication of a double re...
2reuimp0 47031 Implication of a double re...
2reuimp 47032 Implication of a double re...
ralbinrald 47039 Elemination of a restricte...
nvelim 47040 If a class is the universa...
alneu 47041 If a statement holds for a...
eu2ndop1stv 47042 If there is a unique secon...
dfateq12d 47043 Equality deduction for "de...
nfdfat 47044 Bound-variable hypothesis ...
dfdfat2 47045 Alternate definition of th...
fundmdfat 47046 A function is defined at a...
dfatprc 47047 A function is not defined ...
dfatelrn 47048 The value of a function ` ...
dfafv2 47049 Alternative definition of ...
afveq12d 47050 Equality deduction for fun...
afveq1 47051 Equality theorem for funct...
afveq2 47052 Equality theorem for funct...
nfafv 47053 Bound-variable hypothesis ...
csbafv12g 47054 Move class substitution in...
afvfundmfveq 47055 If a class is a function r...
afvnfundmuv 47056 If a set is not in the dom...
ndmafv 47057 The value of a class outsi...
afvvdm 47058 If the function value of a...
nfunsnafv 47059 If the restriction of a cl...
afvvfunressn 47060 If the function value of a...
afvprc 47061 A function's value at a pr...
afvvv 47062 If a function's value at a...
afvpcfv0 47063 If the value of the altern...
afvnufveq 47064 The value of the alternati...
afvvfveq 47065 The value of the alternati...
afv0fv0 47066 If the value of the altern...
afvfvn0fveq 47067 If the function's value at...
afv0nbfvbi 47068 The function's value at an...
afvfv0bi 47069 The function's value at an...
afveu 47070 The value of a function at...
fnbrafvb 47071 Equivalence of function va...
fnopafvb 47072 Equivalence of function va...
funbrafvb 47073 Equivalence of function va...
funopafvb 47074 Equivalence of function va...
funbrafv 47075 The second argument of a b...
funbrafv2b 47076 Function value in terms of...
dfafn5a 47077 Representation of a functi...
dfafn5b 47078 Representation of a functi...
fnrnafv 47079 The range of a function ex...
afvelrnb 47080 A member of a function's r...
afvelrnb0 47081 A member of a function's r...
dfaimafn 47082 Alternate definition of th...
dfaimafn2 47083 Alternate definition of th...
afvelima 47084 Function value in an image...
afvelrn 47085 A function's value belongs...
fnafvelrn 47086 A function's value belongs...
fafvelcdm 47087 A function's value belongs...
ffnafv 47088 A function maps to a class...
afvres 47089 The value of a restricted ...
tz6.12-afv 47090 Function value. Theorem 6...
tz6.12-1-afv 47091 Function value (Theorem 6....
dmfcoafv 47092 Domains of a function comp...
afvco2 47093 Value of a function compos...
rlimdmafv 47094 Two ways to express that a...
aoveq123d 47095 Equality deduction for ope...
nfaov 47096 Bound-variable hypothesis ...
csbaovg 47097 Move class substitution in...
aovfundmoveq 47098 If a class is a function r...
aovnfundmuv 47099 If an ordered pair is not ...
ndmaov 47100 The value of an operation ...
ndmaovg 47101 The value of an operation ...
aovvdm 47102 If the operation value of ...
nfunsnaov 47103 If the restriction of a cl...
aovvfunressn 47104 If the operation value of ...
aovprc 47105 The value of an operation ...
aovrcl 47106 Reverse closure for an ope...
aovpcov0 47107 If the alternative value o...
aovnuoveq 47108 The alternative value of t...
aovvoveq 47109 The alternative value of t...
aov0ov0 47110 If the alternative value o...
aovovn0oveq 47111 If the operation's value a...
aov0nbovbi 47112 The operation's value on a...
aovov0bi 47113 The operation's value on a...
rspceaov 47114 A frequently used special ...
fnotaovb 47115 Equivalence of operation v...
ffnaov 47116 An operation maps to a cla...
faovcl 47117 Closure law for an operati...
aovmpt4g 47118 Value of a function given ...
aoprssdm 47119 Domain of closure of an op...
ndmaovcl 47120 The "closure" of an operat...
ndmaovrcl 47121 Reverse closure law, in co...
ndmaovcom 47122 Any operation is commutati...
ndmaovass 47123 Any operation is associati...
ndmaovdistr 47124 Any operation is distribut...
dfatafv2iota 47127 If a function is defined a...
ndfatafv2 47128 The alternate function val...
ndfatafv2undef 47129 The alternate function val...
dfatafv2ex 47130 The alternate function val...
afv2ex 47131 The alternate function val...
afv2eq12d 47132 Equality deduction for fun...
afv2eq1 47133 Equality theorem for funct...
afv2eq2 47134 Equality theorem for funct...
nfafv2 47135 Bound-variable hypothesis ...
csbafv212g 47136 Move class substitution in...
fexafv2ex 47137 The alternate function val...
ndfatafv2nrn 47138 The alternate function val...
ndmafv2nrn 47139 The value of a class outsi...
funressndmafv2rn 47140 The alternate function val...
afv2ndefb 47141 Two ways to say that an al...
nfunsnafv2 47142 If the restriction of a cl...
afv2prc 47143 A function's value at a pr...
dfatafv2rnb 47144 The alternate function val...
afv2orxorb 47145 If a set is in the range o...
dmafv2rnb 47146 The alternate function val...
fundmafv2rnb 47147 The alternate function val...
afv2elrn 47148 An alternate function valu...
afv20defat 47149 If the alternate function ...
fnafv2elrn 47150 An alternate function valu...
fafv2elcdm 47151 An alternate function valu...
fafv2elrnb 47152 An alternate function valu...
fcdmvafv2v 47153 If the codomain of a funct...
tz6.12-2-afv2 47154 Function value when ` F ` ...
afv2eu 47155 The value of a function at...
afv2res 47156 The value of a restricted ...
tz6.12-afv2 47157 Function value (Theorem 6....
tz6.12-1-afv2 47158 Function value (Theorem 6....
tz6.12c-afv2 47159 Corollary of Theorem 6.12(...
tz6.12i-afv2 47160 Corollary of Theorem 6.12(...
funressnbrafv2 47161 The second argument of a b...
dfatbrafv2b 47162 Equivalence of function va...
dfatopafv2b 47163 Equivalence of function va...
funbrafv2 47164 The second argument of a b...
fnbrafv2b 47165 Equivalence of function va...
fnopafv2b 47166 Equivalence of function va...
funbrafv22b 47167 Equivalence of function va...
funopafv2b 47168 Equivalence of function va...
dfatsnafv2 47169 Singleton of function valu...
dfafv23 47170 A definition of function v...
dfatdmfcoafv2 47171 Domain of a function compo...
dfatcolem 47172 Lemma for ~ dfatco . (Con...
dfatco 47173 The predicate "defined at"...
afv2co2 47174 Value of a function compos...
rlimdmafv2 47175 Two ways to express that a...
dfafv22 47176 Alternate definition of ` ...
afv2ndeffv0 47177 If the alternate function ...
dfatafv2eqfv 47178 If a function is defined a...
afv2rnfveq 47179 If the alternate function ...
afv20fv0 47180 If the alternate function ...
afv2fvn0fveq 47181 If the function's value at...
afv2fv0 47182 If the function's value at...
afv2fv0b 47183 The function's value at an...
afv2fv0xorb 47184 If a set is in the range o...
an4com24 47185 Rearrangement of 4 conjunc...
3an4ancom24 47186 Commutative law for a conj...
4an21 47187 Rearrangement of 4 conjunc...
dfnelbr2 47190 Alternate definition of th...
nelbr 47191 The binary relation of a s...
nelbrim 47192 If a set is related to ano...
nelbrnel 47193 A set is related to anothe...
nelbrnelim 47194 If a set is related to ano...
ralralimp 47195 Selecting one of two alter...
otiunsndisjX 47196 The union of singletons co...
fvifeq 47197 Equality of function value...
rnfdmpr 47198 The range of a one-to-one ...
imarnf1pr 47199 The image of the range of ...
funop1 47200 A function is an ordered p...
fun2dmnopgexmpl 47201 A function with a domain c...
opabresex0d 47202 A collection of ordered pa...
opabbrfex0d 47203 A collection of ordered pa...
opabresexd 47204 A collection of ordered pa...
opabbrfexd 47205 A collection of ordered pa...
f1oresf1orab 47206 Build a bijection by restr...
f1oresf1o 47207 Build a bijection by restr...
f1oresf1o2 47208 Build a bijection by restr...
fvmptrab 47209 Value of a function mappin...
fvmptrabdm 47210 Value of a function mappin...
cnambpcma 47211 ((a-b)+c)-a = c-a holds fo...
cnapbmcpd 47212 ((a+b)-c)+d = ((a+d)+b)-c ...
addsubeq0 47213 The sum of two complex num...
leaddsuble 47214 Addition and subtraction o...
2leaddle2 47215 If two real numbers are le...
ltnltne 47216 Variant of trichotomy law ...
p1lep2 47217 A real number increasd by ...
ltsubsubaddltsub 47218 If the result of subtracti...
zm1nn 47219 An integer minus 1 is posi...
readdcnnred 47220 The sum of a real number a...
resubcnnred 47221 The difference of a real n...
recnmulnred 47222 The product of a real numb...
cndivrenred 47223 The quotient of an imagina...
sqrtnegnre 47224 The square root of a negat...
nn0resubcl 47225 Closure law for subtractio...
zgeltp1eq 47226 If an integer is between a...
1t10e1p1e11 47227 11 is 1 times 10 to the po...
deccarry 47228 Add 1 to a 2 digit number ...
eluzge0nn0 47229 If an integer is greater t...
nltle2tri 47230 Negated extended trichotom...
ssfz12 47231 Subset relationship for fi...
elfz2z 47232 Membership of an integer i...
2elfz3nn0 47233 If there are two elements ...
fz0addcom 47234 The addition of two member...
2elfz2melfz 47235 If the sum of two integers...
fz0addge0 47236 The sum of two integers in...
elfzlble 47237 Membership of an integer i...
elfzelfzlble 47238 Membership of an element o...
fzopred 47239 Join a predecessor to the ...
fzopredsuc 47240 Join a predecessor and a s...
1fzopredsuc 47241 Join 0 and a successor to ...
el1fzopredsuc 47242 An element of an open inte...
subsubelfzo0 47243 Subtracting a difference f...
2ffzoeq 47244 Two functions over a half-...
m1mod0mod1 47245 An integer decreased by 1 ...
elmod2 47246 An integer modulo 2 is eit...
smonoord 47247 Ordering relation for a st...
fsummsndifre 47248 A finite sum with one of i...
fsumsplitsndif 47249 Separate out a term in a f...
fsummmodsndifre 47250 A finite sum of summands m...
fsummmodsnunz 47251 A finite sum of summands m...
setsidel 47252 The injected slot is an el...
setsnidel 47253 The injected slot is an el...
setsv 47254 The value of the structure...
preimafvsnel 47255 The preimage of a function...
preimafvn0 47256 The preimage of a function...
uniimafveqt 47257 The union of the image of ...
uniimaprimaeqfv 47258 The union of the image of ...
setpreimafvex 47259 The class ` P ` of all pre...
elsetpreimafvb 47260 The characterization of an...
elsetpreimafv 47261 An element of the class ` ...
elsetpreimafvssdm 47262 An element of the class ` ...
fvelsetpreimafv 47263 There is an element in a p...
preimafvelsetpreimafv 47264 The preimage of a function...
preimafvsspwdm 47265 The class ` P ` of all pre...
0nelsetpreimafv 47266 The empty set is not an el...
elsetpreimafvbi 47267 An element of the preimage...
elsetpreimafveqfv 47268 The elements of the preima...
eqfvelsetpreimafv 47269 If an element of the domai...
elsetpreimafvrab 47270 An element of the preimage...
imaelsetpreimafv 47271 The image of an element of...
uniimaelsetpreimafv 47272 The union of the image of ...
elsetpreimafveq 47273 If two preimages of functi...
fundcmpsurinjlem1 47274 Lemma 1 for ~ fundcmpsurin...
fundcmpsurinjlem2 47275 Lemma 2 for ~ fundcmpsurin...
fundcmpsurinjlem3 47276 Lemma 3 for ~ fundcmpsurin...
imasetpreimafvbijlemf 47277 Lemma for ~ imasetpreimafv...
imasetpreimafvbijlemfv 47278 Lemma for ~ imasetpreimafv...
imasetpreimafvbijlemfv1 47279 Lemma for ~ imasetpreimafv...
imasetpreimafvbijlemf1 47280 Lemma for ~ imasetpreimafv...
imasetpreimafvbijlemfo 47281 Lemma for ~ imasetpreimafv...
imasetpreimafvbij 47282 The mapping ` H ` is a bij...
fundcmpsurbijinjpreimafv 47283 Every function ` F : A -->...
fundcmpsurinjpreimafv 47284 Every function ` F : A -->...
fundcmpsurinj 47285 Every function ` F : A -->...
fundcmpsurbijinj 47286 Every function ` F : A -->...
fundcmpsurinjimaid 47287 Every function ` F : A -->...
fundcmpsurinjALT 47288 Alternate proof of ~ fundc...
iccpval 47291 Partition consisting of a ...
iccpart 47292 A special partition. Corr...
iccpartimp 47293 Implications for a class b...
iccpartres 47294 The restriction of a parti...
iccpartxr 47295 If there is a partition, t...
iccpartgtprec 47296 If there is a partition, t...
iccpartipre 47297 If there is a partition, t...
iccpartiltu 47298 If there is a partition, t...
iccpartigtl 47299 If there is a partition, t...
iccpartlt 47300 If there is a partition, t...
iccpartltu 47301 If there is a partition, t...
iccpartgtl 47302 If there is a partition, t...
iccpartgt 47303 If there is a partition, t...
iccpartleu 47304 If there is a partition, t...
iccpartgel 47305 If there is a partition, t...
iccpartrn 47306 If there is a partition, t...
iccpartf 47307 The range of the partition...
iccpartel 47308 If there is a partition, t...
iccelpart 47309 An element of any partitio...
iccpartiun 47310 A half-open interval of ex...
icceuelpartlem 47311 Lemma for ~ icceuelpart . ...
icceuelpart 47312 An element of a partitione...
iccpartdisj 47313 The segments of a partitio...
iccpartnel 47314 A point of a partition is ...
fargshiftfv 47315 If a class is a function, ...
fargshiftf 47316 If a class is a function, ...
fargshiftf1 47317 If a function is 1-1, then...
fargshiftfo 47318 If a function is onto, the...
fargshiftfva 47319 The values of a shifted fu...
lswn0 47320 The last symbol of a not e...
nfich1 47323 The first interchangeable ...
nfich2 47324 The second interchangeable...
ichv 47325 Setvar variables are inter...
ichf 47326 Setvar variables are inter...
ichid 47327 A setvar variable is alway...
icht 47328 A theorem is interchangeab...
ichbidv 47329 Formula building rule for ...
ichcircshi 47330 The setvar variables are i...
ichan 47331 If two setvar variables ar...
ichn 47332 Negation does not affect i...
ichim 47333 Formula building rule for ...
dfich2 47334 Alternate definition of th...
ichcom 47335 The interchangeability of ...
ichbi12i 47336 Equivalence for interchang...
icheqid 47337 In an equality for the sam...
icheq 47338 In an equality of setvar v...
ichnfimlem 47339 Lemma for ~ ichnfim : A s...
ichnfim 47340 If in an interchangeabilit...
ichnfb 47341 If ` x ` and ` y ` are int...
ichal 47342 Move a universal quantifie...
ich2al 47343 Two setvar variables are a...
ich2ex 47344 Two setvar variables are a...
ichexmpl1 47345 Example for interchangeabl...
ichexmpl2 47346 Example for interchangeabl...
ich2exprop 47347 If the setvar variables ar...
ichnreuop 47348 If the setvar variables ar...
ichreuopeq 47349 If the setvar variables ar...
sprid 47350 Two identical representati...
elsprel 47351 An unordered pair is an el...
spr0nelg 47352 The empty set is not an el...
sprval 47355 The set of all unordered p...
sprvalpw 47356 The set of all unordered p...
sprssspr 47357 The set of all unordered p...
spr0el 47358 The empty set is not an un...
sprvalpwn0 47359 The set of all unordered p...
sprel 47360 An element of the set of a...
prssspr 47361 An element of a subset of ...
prelspr 47362 An unordered pair of eleme...
prsprel 47363 The elements of a pair fro...
prsssprel 47364 The elements of a pair fro...
sprvalpwle2 47365 The set of all unordered p...
sprsymrelfvlem 47366 Lemma for ~ sprsymrelf and...
sprsymrelf1lem 47367 Lemma for ~ sprsymrelf1 . ...
sprsymrelfolem1 47368 Lemma 1 for ~ sprsymrelfo ...
sprsymrelfolem2 47369 Lemma 2 for ~ sprsymrelfo ...
sprsymrelfv 47370 The value of the function ...
sprsymrelf 47371 The mapping ` F ` is a fun...
sprsymrelf1 47372 The mapping ` F ` is a one...
sprsymrelfo 47373 The mapping ` F ` is a fun...
sprsymrelf1o 47374 The mapping ` F ` is a bij...
sprbisymrel 47375 There is a bijection betwe...
sprsymrelen 47376 The class ` P ` of subsets...
prpair 47377 Characterization of a prop...
prproropf1olem0 47378 Lemma 0 for ~ prproropf1o ...
prproropf1olem1 47379 Lemma 1 for ~ prproropf1o ...
prproropf1olem2 47380 Lemma 2 for ~ prproropf1o ...
prproropf1olem3 47381 Lemma 3 for ~ prproropf1o ...
prproropf1olem4 47382 Lemma 4 for ~ prproropf1o ...
prproropf1o 47383 There is a bijection betwe...
prproropen 47384 The set of proper pairs an...
prproropreud 47385 There is exactly one order...
pairreueq 47386 Two equivalent representat...
paireqne 47387 Two sets are not equal iff...
prprval 47390 The set of all proper unor...
prprvalpw 47391 The set of all proper unor...
prprelb 47392 An element of the set of a...
prprelprb 47393 A set is an element of the...
prprspr2 47394 The set of all proper unor...
prprsprreu 47395 There is a unique proper u...
prprreueq 47396 There is a unique proper u...
sbcpr 47397 The proper substitution of...
reupr 47398 There is a unique unordere...
reuprpr 47399 There is a unique proper u...
poprelb 47400 Equality for unordered pai...
2exopprim 47401 The existence of an ordere...
reuopreuprim 47402 There is a unique unordere...
fmtno 47405 The ` N ` th Fermat number...
fmtnoge3 47406 Each Fermat number is grea...
fmtnonn 47407 Each Fermat number is a po...
fmtnom1nn 47408 A Fermat number minus one ...
fmtnoodd 47409 Each Fermat number is odd....
fmtnorn 47410 A Fermat number is a funct...
fmtnof1 47411 The enumeration of the Fer...
fmtnoinf 47412 The set of Fermat numbers ...
fmtnorec1 47413 The first recurrence relat...
sqrtpwpw2p 47414 The floor of the square ro...
fmtnosqrt 47415 The floor of the square ro...
fmtno0 47416 The ` 0 ` th Fermat number...
fmtno1 47417 The ` 1 ` st Fermat number...
fmtnorec2lem 47418 Lemma for ~ fmtnorec2 (ind...
fmtnorec2 47419 The second recurrence rela...
fmtnodvds 47420 Any Fermat number divides ...
goldbachthlem1 47421 Lemma 1 for ~ goldbachth ....
goldbachthlem2 47422 Lemma 2 for ~ goldbachth ....
goldbachth 47423 Goldbach's theorem: Two d...
fmtnorec3 47424 The third recurrence relat...
fmtnorec4 47425 The fourth recurrence rela...
fmtno2 47426 The ` 2 ` nd Fermat number...
fmtno3 47427 The ` 3 ` rd Fermat number...
fmtno4 47428 The ` 4 ` th Fermat number...
fmtno5lem1 47429 Lemma 1 for ~ fmtno5 . (C...
fmtno5lem2 47430 Lemma 2 for ~ fmtno5 . (C...
fmtno5lem3 47431 Lemma 3 for ~ fmtno5 . (C...
fmtno5lem4 47432 Lemma 4 for ~ fmtno5 . (C...
fmtno5 47433 The ` 5 ` th Fermat number...
fmtno0prm 47434 The ` 0 ` th Fermat number...
fmtno1prm 47435 The ` 1 ` st Fermat number...
fmtno2prm 47436 The ` 2 ` nd Fermat number...
257prm 47437 257 is a prime number (the...
fmtno3prm 47438 The ` 3 ` rd Fermat number...
odz2prm2pw 47439 Any power of two is coprim...
fmtnoprmfac1lem 47440 Lemma for ~ fmtnoprmfac1 :...
fmtnoprmfac1 47441 Divisor of Fermat number (...
fmtnoprmfac2lem1 47442 Lemma for ~ fmtnoprmfac2 ....
fmtnoprmfac2 47443 Divisor of Fermat number (...
fmtnofac2lem 47444 Lemma for ~ fmtnofac2 (Ind...
fmtnofac2 47445 Divisor of Fermat number (...
fmtnofac1 47446 Divisor of Fermat number (...
fmtno4sqrt 47447 The floor of the square ro...
fmtno4prmfac 47448 If P was a (prime) factor ...
fmtno4prmfac193 47449 If P was a (prime) factor ...
fmtno4nprmfac193 47450 193 is not a (prime) facto...
fmtno4prm 47451 The ` 4 `-th Fermat number...
65537prm 47452 65537 is a prime number (t...
fmtnofz04prm 47453 The first five Fermat numb...
fmtnole4prm 47454 The first five Fermat numb...
fmtno5faclem1 47455 Lemma 1 for ~ fmtno5fac . ...
fmtno5faclem2 47456 Lemma 2 for ~ fmtno5fac . ...
fmtno5faclem3 47457 Lemma 3 for ~ fmtno5fac . ...
fmtno5fac 47458 The factorization of the `...
fmtno5nprm 47459 The ` 5 ` th Fermat number...
prmdvdsfmtnof1lem1 47460 Lemma 1 for ~ prmdvdsfmtno...
prmdvdsfmtnof1lem2 47461 Lemma 2 for ~ prmdvdsfmtno...
prmdvdsfmtnof 47462 The mapping of a Fermat nu...
prmdvdsfmtnof1 47463 The mapping of a Fermat nu...
prminf2 47464 The set of prime numbers i...
2pwp1prm 47465 For ` ( ( 2 ^ k ) + 1 ) ` ...
2pwp1prmfmtno 47466 Every prime number of the ...
m2prm 47467 The second Mersenne number...
m3prm 47468 The third Mersenne number ...
flsqrt 47469 A condition equivalent to ...
flsqrt5 47470 The floor of the square ro...
3ndvds4 47471 3 does not divide 4. (Con...
139prmALT 47472 139 is a prime number. In...
31prm 47473 31 is a prime number. In ...
m5prm 47474 The fifth Mersenne number ...
127prm 47475 127 is a prime number. (C...
m7prm 47476 The seventh Mersenne numbe...
m11nprm 47477 The eleventh Mersenne numb...
mod42tp1mod8 47478 If a number is ` 3 ` modul...
sfprmdvdsmersenne 47479 If ` Q ` is a safe prime (...
sgprmdvdsmersenne 47480 If ` P ` is a Sophie Germa...
lighneallem1 47481 Lemma 1 for ~ lighneal . ...
lighneallem2 47482 Lemma 2 for ~ lighneal . ...
lighneallem3 47483 Lemma 3 for ~ lighneal . ...
lighneallem4a 47484 Lemma 1 for ~ lighneallem4...
lighneallem4b 47485 Lemma 2 for ~ lighneallem4...
lighneallem4 47486 Lemma 3 for ~ lighneal . ...
lighneal 47487 If a power of a prime ` P ...
modexp2m1d 47488 The square of an integer w...
proththdlem 47489 Lemma for ~ proththd . (C...
proththd 47490 Proth's theorem (1878). I...
5tcu2e40 47491 5 times the cube of 2 is 4...
3exp4mod41 47492 3 to the fourth power is -...
41prothprmlem1 47493 Lemma 1 for ~ 41prothprm ....
41prothprmlem2 47494 Lemma 2 for ~ 41prothprm ....
41prothprm 47495 41 is a _Proth prime_. (C...
quad1 47496 A condition for a quadrati...
requad01 47497 A condition for a quadrati...
requad1 47498 A condition for a quadrati...
requad2 47499 A condition for a quadrati...
iseven 47504 The predicate "is an even ...
isodd 47505 The predicate "is an odd n...
evenz 47506 An even number is an integ...
oddz 47507 An odd number is an intege...
evendiv2z 47508 The result of dividing an ...
oddp1div2z 47509 The result of dividing an ...
oddm1div2z 47510 The result of dividing an ...
isodd2 47511 The predicate "is an odd n...
dfodd2 47512 Alternate definition for o...
dfodd6 47513 Alternate definition for o...
dfeven4 47514 Alternate definition for e...
evenm1odd 47515 The predecessor of an even...
evenp1odd 47516 The successor of an even n...
oddp1eveni 47517 The successor of an odd nu...
oddm1eveni 47518 The predecessor of an odd ...
evennodd 47519 An even number is not an o...
oddneven 47520 An odd number is not an ev...
enege 47521 The negative of an even nu...
onego 47522 The negative of an odd num...
m1expevenALTV 47523 Exponentiation of -1 by an...
m1expoddALTV 47524 Exponentiation of -1 by an...
dfeven2 47525 Alternate definition for e...
dfodd3 47526 Alternate definition for o...
iseven2 47527 The predicate "is an even ...
isodd3 47528 The predicate "is an odd n...
2dvdseven 47529 2 divides an even number. ...
m2even 47530 A multiple of 2 is an even...
2ndvdsodd 47531 2 does not divide an odd n...
2dvdsoddp1 47532 2 divides an odd number in...
2dvdsoddm1 47533 2 divides an odd number de...
dfeven3 47534 Alternate definition for e...
dfodd4 47535 Alternate definition for o...
dfodd5 47536 Alternate definition for o...
zefldiv2ALTV 47537 The floor of an even numbe...
zofldiv2ALTV 47538 The floor of an odd numer ...
oddflALTV 47539 Odd number representation ...
iseven5 47540 The predicate "is an even ...
isodd7 47541 The predicate "is an odd n...
dfeven5 47542 Alternate definition for e...
dfodd7 47543 Alternate definition for o...
gcd2odd1 47544 The greatest common diviso...
zneoALTV 47545 No even integer equals an ...
zeoALTV 47546 An integer is even or odd....
zeo2ALTV 47547 An integer is even or odd ...
nneoALTV 47548 A positive integer is even...
nneoiALTV 47549 A positive integer is even...
odd2np1ALTV 47550 An integer is odd iff it i...
oddm1evenALTV 47551 An integer is odd iff its ...
oddp1evenALTV 47552 An integer is odd iff its ...
oexpnegALTV 47553 The exponential of the neg...
oexpnegnz 47554 The exponential of the neg...
bits0ALTV 47555 Value of the zeroth bit. ...
bits0eALTV 47556 The zeroth bit of an even ...
bits0oALTV 47557 The zeroth bit of an odd n...
divgcdoddALTV 47558 Either ` A / ( A gcd B ) `...
opoeALTV 47559 The sum of two odds is eve...
opeoALTV 47560 The sum of an odd and an e...
omoeALTV 47561 The difference of two odds...
omeoALTV 47562 The difference of an odd a...
oddprmALTV 47563 A prime not equal to ` 2 `...
0evenALTV 47564 0 is an even number. (Con...
0noddALTV 47565 0 is not an odd number. (...
1oddALTV 47566 1 is an odd number. (Cont...
1nevenALTV 47567 1 is not an even number. ...
2evenALTV 47568 2 is an even number. (Con...
2noddALTV 47569 2 is not an odd number. (...
nn0o1gt2ALTV 47570 An odd nonnegative integer...
nnoALTV 47571 An alternate characterizat...
nn0oALTV 47572 An alternate characterizat...
nn0e 47573 An alternate characterizat...
nneven 47574 An alternate characterizat...
nn0onn0exALTV 47575 For each odd nonnegative i...
nn0enn0exALTV 47576 For each even nonnegative ...
nnennexALTV 47577 For each even positive int...
nnpw2evenALTV 47578 2 to the power of a positi...
epoo 47579 The sum of an even and an ...
emoo 47580 The difference of an even ...
epee 47581 The sum of two even number...
emee 47582 The difference of two even...
evensumeven 47583 If a summand is even, the ...
3odd 47584 3 is an odd number. (Cont...
4even 47585 4 is an even number. (Con...
5odd 47586 5 is an odd number. (Cont...
6even 47587 6 is an even number. (Con...
7odd 47588 7 is an odd number. (Cont...
8even 47589 8 is an even number. (Con...
evenprm2 47590 A prime number is even iff...
oddprmne2 47591 Every prime number not bei...
oddprmuzge3 47592 A prime number which is od...
evenltle 47593 If an even number is great...
odd2prm2 47594 If an odd number is the su...
even3prm2 47595 If an even number is the s...
mogoldbblem 47596 Lemma for ~ mogoldbb . (C...
perfectALTVlem1 47597 Lemma for ~ perfectALTV . ...
perfectALTVlem2 47598 Lemma for ~ perfectALTV . ...
perfectALTV 47599 The Euclid-Euler theorem, ...
fppr 47602 The set of Fermat pseudopr...
fpprmod 47603 The set of Fermat pseudopr...
fpprel 47604 A Fermat pseudoprime to th...
fpprbasnn 47605 The base of a Fermat pseud...
fpprnn 47606 A Fermat pseudoprime to th...
fppr2odd 47607 A Fermat pseudoprime to th...
11t31e341 47608 341 is the product of 11 a...
2exp340mod341 47609 Eight to the eighth power ...
341fppr2 47610 341 is the (smallest) _Pou...
4fppr1 47611 4 is the (smallest) Fermat...
8exp8mod9 47612 Eight to the eighth power ...
9fppr8 47613 9 is the (smallest) Fermat...
dfwppr 47614 Alternate definition of a ...
fpprwppr 47615 A Fermat pseudoprime to th...
fpprwpprb 47616 An integer ` X ` which is ...
fpprel2 47617 An alternate definition fo...
nfermltl8rev 47618 Fermat's little theorem wi...
nfermltl2rev 47619 Fermat's little theorem wi...
nfermltlrev 47620 Fermat's little theorem re...
isgbe 47627 The predicate "is an even ...
isgbow 47628 The predicate "is a weak o...
isgbo 47629 The predicate "is an odd G...
gbeeven 47630 An even Goldbach number is...
gbowodd 47631 A weak odd Goldbach number...
gbogbow 47632 A (strong) odd Goldbach nu...
gboodd 47633 An odd Goldbach number is ...
gbepos 47634 Any even Goldbach number i...
gbowpos 47635 Any weak odd Goldbach numb...
gbopos 47636 Any odd Goldbach number is...
gbegt5 47637 Any even Goldbach number i...
gbowgt5 47638 Any weak odd Goldbach numb...
gbowge7 47639 Any weak odd Goldbach numb...
gboge9 47640 Any odd Goldbach number is...
gbege6 47641 Any even Goldbach number i...
gbpart6 47642 The Goldbach partition of ...
gbpart7 47643 The (weak) Goldbach partit...
gbpart8 47644 The Goldbach partition of ...
gbpart9 47645 The (strong) Goldbach part...
gbpart11 47646 The (strong) Goldbach part...
6gbe 47647 6 is an even Goldbach numb...
7gbow 47648 7 is a weak odd Goldbach n...
8gbe 47649 8 is an even Goldbach numb...
9gbo 47650 9 is an odd Goldbach numbe...
11gbo 47651 11 is an odd Goldbach numb...
stgoldbwt 47652 If the strong ternary Gold...
sbgoldbwt 47653 If the strong binary Goldb...
sbgoldbst 47654 If the strong binary Goldb...
sbgoldbaltlem1 47655 Lemma 1 for ~ sbgoldbalt :...
sbgoldbaltlem2 47656 Lemma 2 for ~ sbgoldbalt :...
sbgoldbalt 47657 An alternate (related to t...
sbgoldbb 47658 If the strong binary Goldb...
sgoldbeven3prm 47659 If the binary Goldbach con...
sbgoldbm 47660 If the strong binary Goldb...
mogoldbb 47661 If the modern version of t...
sbgoldbmb 47662 The strong binary Goldbach...
sbgoldbo 47663 If the strong binary Goldb...
nnsum3primes4 47664 4 is the sum of at most 3 ...
nnsum4primes4 47665 4 is the sum of at most 4 ...
nnsum3primesprm 47666 Every prime is "the sum of...
nnsum4primesprm 47667 Every prime is "the sum of...
nnsum3primesgbe 47668 Any even Goldbach number i...
nnsum4primesgbe 47669 Any even Goldbach number i...
nnsum3primesle9 47670 Every integer greater than...
nnsum4primesle9 47671 Every integer greater than...
nnsum4primesodd 47672 If the (weak) ternary Gold...
nnsum4primesoddALTV 47673 If the (strong) ternary Go...
evengpop3 47674 If the (weak) ternary Gold...
evengpoap3 47675 If the (strong) ternary Go...
nnsum4primeseven 47676 If the (weak) ternary Gold...
nnsum4primesevenALTV 47677 If the (strong) ternary Go...
wtgoldbnnsum4prm 47678 If the (weak) ternary Gold...
stgoldbnnsum4prm 47679 If the (strong) ternary Go...
bgoldbnnsum3prm 47680 If the binary Goldbach con...
bgoldbtbndlem1 47681 Lemma 1 for ~ bgoldbtbnd :...
bgoldbtbndlem2 47682 Lemma 2 for ~ bgoldbtbnd ....
bgoldbtbndlem3 47683 Lemma 3 for ~ bgoldbtbnd ....
bgoldbtbndlem4 47684 Lemma 4 for ~ bgoldbtbnd ....
bgoldbtbnd 47685 If the binary Goldbach con...
tgoldbachgtALTV 47688 Variant of Thierry Arnoux'...
bgoldbachlt 47689 The binary Goldbach conjec...
tgblthelfgott 47691 The ternary Goldbach conje...
tgoldbachlt 47692 The ternary Goldbach conje...
tgoldbach 47693 The ternary Goldbach conje...
clnbgrprc0 47696 The closed neighborhood is...
clnbgrcl 47697 If a class ` X ` has at le...
clnbgrval 47698 The closed neighborhood of...
dfclnbgr2 47699 Alternate definition of th...
dfclnbgr4 47700 Alternate definition of th...
dfclnbgr3 47701 Alternate definition of th...
clnbgrnvtx0 47702 If a class ` X ` is not a ...
clnbgrel 47703 Characterization of a memb...
clnbgrvtxel 47704 Every vertex ` K ` is a me...
clnbgrisvtx 47705 Every member ` N ` of the ...
clnbgrssvtx 47706 The closed neighborhood of...
clnbgrn0 47707 The closed neighborhood of...
clnbupgr 47708 The closed neighborhood of...
clnbupgrel 47709 A member of the closed nei...
clnbgr0vtx 47710 In a null graph (with no v...
clnbgr0edg 47711 In an empty graph (with no...
clnbgrsym 47712 In a graph, the closed nei...
predgclnbgrel 47713 If a (not necessarily prop...
clnbgredg 47714 A vertices connected by an...
clnbgrssedg 47715 The vertices connected by ...
edgusgrclnbfin 47716 The size of the closed nei...
clnbusgrfi 47717 The closed neighborhood of...
clnbfiusgrfi 47718 The closed neighborhood of...
clnbgrlevtx 47719 The size of the closed nei...
dfsclnbgr2 47720 Alternate definition of th...
sclnbgrel 47721 Characterization of a memb...
sclnbgrelself 47722 A vertex ` N ` is a member...
sclnbgrisvtx 47723 Every member ` X ` of the ...
dfclnbgr5 47724 Alternate definition of th...
dfnbgr5 47725 Alternate definition of th...
dfnbgrss 47726 Subset chain for different...
dfvopnbgr2 47727 Alternate definition of th...
vopnbgrel 47728 Characterization of a memb...
vopnbgrelself 47729 A vertex ` N ` is a member...
dfclnbgr6 47730 Alternate definition of th...
dfnbgr6 47731 Alternate definition of th...
dfsclnbgr6 47732 Alternate definition of a ...
dfnbgrss2 47733 Subset chain for different...
isisubgr 47736 The subgraph induced by a ...
isubgriedg 47737 The edges of an induced su...
isubgrvtxuhgr 47738 The subgraph induced by th...
isubgrvtx 47739 The vertices of an induced...
isubgruhgr 47740 An induced subgraph of a h...
isubgrsubgr 47741 An induced subgraph of a h...
isubgrupgr 47742 An induced subgraph of a p...
isubgrumgr 47743 An induced subgraph of a m...
isubgrusgr 47744 An induced subgraph of a s...
isubgr0uhgr 47745 The subgraph induced by an...
grimfn 47751 The graph isomorphism func...
grimdmrel 47752 The domain of the graph is...
isgrim 47754 An isomorphism of graphs i...
grimprop 47755 Properties of an isomorphi...
grimf1o 47756 An isomorphism of graphs i...
isuspgrim0lem 47757 An isomorphism of simple p...
isuspgrim0 47758 An isomorphism of simple p...
uspgrimprop 47759 An isomorphism of simple p...
isuspgrimlem 47760 Lemma for ~ isuspgrim . (...
isuspgrim 47761 A class is an isomorphism ...
grimidvtxedg 47762 The identity relation rest...
grimid 47763 The identity relation rest...
grimuhgr 47764 If there is a graph isomor...
grimcnv 47765 The converse of a graph is...
grimco 47766 The composition of graph i...
brgric 47767 The relation "is isomorphi...
brgrici 47768 Prove that two graphs are ...
gricrcl 47769 Reverse closure of the "is...
dfgric2 47770 Alternate, explicit defini...
gricbri 47771 Implications of two graphs...
gricushgr 47772 The "is isomorphic to" rel...
gricuspgr 47773 The "is isomorphic to" rel...
gricrel 47774 The "is isomorphic to" rel...
gricref 47775 Graph isomorphism is refle...
gricsym 47776 Graph isomorphism is symme...
gricsymb 47777 Graph isomorphism is symme...
grictr 47778 Graph isomorphism is trans...
gricer 47779 Isomorphism is an equivale...
gricen 47780 Isomorphic graphs have equ...
opstrgric 47781 A graph represented as an ...
ushggricedg 47782 A simple hypergraph (with ...
isubgrgrim 47783 Isomorphic subgraphs induc...
uhgrimisgrgriclem 47784 Lemma for ~ uhgrimisgrgric...
uhgrimisgrgric 47785 For isomorphic hypergraphs...
clnbgrisubgrgrim 47786 Isomorphic subgraphs induc...
clnbgrgrimlem 47787 Lemma for ~ clnbgrgrim : ...
clnbgrgrim 47788 Graph isomorphisms between...
grimedg 47789 Graph isomorphisms map edg...
grtriproplem 47792 Lemma for ~ grtriprop . (...
grtri 47793 The triangles in a graph. ...
grtriprop 47794 The properties of a triang...
grtrif1o 47795 Any bijection onto a trian...
isgrtri 47796 A triangle in a graph. (C...
grtrissvtx 47797 A triangle is a subset of ...
grtriclwlk3 47798 A triangle induces a close...
grtrimap 47799 Conditions for mapping tri...
grimgrtri 47800 Graph isomorphisms map tri...
usgrgrtrirex 47801 Conditions for a simple gr...
grlimfn 47805 The graph local isomorphis...
grlimdmrel 47806 The domain of the graph lo...
isgrlim 47808 A local isomorphism of gra...
isgrlim2 47809 A local isomorphism of gra...
grlimprop 47810 Properties of a local isom...
grlimf1o 47811 A local isomorphism of gra...
grlimprop2 47812 Properties of a local isom...
uhgrimgrlim 47813 An isomorphism of hypergra...
uspgrlimlem1 47814 Lemma 1 for ~ uspgrlim . ...
uspgrlimlem2 47815 Lemma 2 for ~ uspgrlim . ...
uspgrlimlem3 47816 Lemma 3 for ~ uspgrlim . ...
uspgrlimlem4 47817 Lemma 4 for ~ uspgrlim . ...
uspgrlim 47818 A local isomorphism of sim...
usgrlimprop 47819 Properties of a local isom...
grlimgrtrilem1 47820 Lemma 3 for ~ grlimgrtri ....
grlimgrtrilem2 47821 Lemma 3 for ~ grlimgrtri ....
grlimgrtri 47822 Local isomorphisms between...
brgrlic 47823 The relation "is locally i...
brgrilci 47824 Prove that two graphs are ...
grlicrel 47825 The "is locally isomorphic...
grlicrcl 47826 Reverse closure of the "is...
dfgrlic2 47827 Alternate, explicit defini...
grilcbri 47828 Implications of two graphs...
dfgrlic3 47829 Alternate, explicit defini...
grilcbri2 47830 Implications of two graphs...
grlicref 47831 Graph local isomorphism is...
grlicsym 47832 Graph local isomorphism is...
grlicsymb 47833 Graph local isomorphism is...
grlictr 47834 Graph local isomorphism is...
grlicer 47835 Local isomorphism is an eq...
grlicen 47836 Locally isomorphic graphs ...
gricgrlic 47837 Isomorphic hypergraphs are...
usgrexmpl1lem 47838 Lemma for ~ usgrexmpl1 . ...
usgrexmpl1 47839 ` G ` is a simple graph of...
usgrexmpl1vtx 47840 The vertices ` 0 , 1 , 2 ,...
usgrexmpl1edg 47841 The edges ` { 0 , 1 } , { ...
usgrexmpl1tri 47842 ` G ` contains a triangle ...
usgrexmpl2lem 47843 Lemma for ~ usgrexmpl2 . ...
usgrexmpl2 47844 ` G ` is a simple graph of...
usgrexmpl2vtx 47845 The vertices ` 0 , 1 , 2 ,...
usgrexmpl2edg 47846 The edges ` { 0 , 1 } , { ...
usgrexmpl2nblem 47847 Lemma for ~ usgrexmpl2nb0 ...
usgrexmpl2nb0 47848 The neighborhood of the fi...
usgrexmpl2nb1 47849 The neighborhood of the se...
usgrexmpl2nb2 47850 The neighborhood of the th...
usgrexmpl2nb3 47851 The neighborhood of the fo...
usgrexmpl2nb4 47852 The neighborhood of the fi...
usgrexmpl2nb5 47853 The neighborhood of the si...
usgrexmpl2trifr 47854 ` G ` is triangle-free. (...
usgrexmpl12ngric 47855 The graphs ` H ` and ` G `...
usgrexmpl12ngrlic 47856 The graphs ` H ` and ` G `...
1hegrlfgr 47857 A graph ` G ` with one hyp...
upwlksfval 47860 The set of simple walks (i...
isupwlk 47861 Properties of a pair of fu...
isupwlkg 47862 Generalization of ~ isupwl...
upwlkbprop 47863 Basic properties of a simp...
upwlkwlk 47864 A simple walk is a walk. ...
upgrwlkupwlk 47865 In a pseudograph, a walk i...
upgrwlkupwlkb 47866 In a pseudograph, the defi...
upgrisupwlkALT 47867 Alternate proof of ~ upgri...
upgredgssspr 47868 The set of edges of a pseu...
uspgropssxp 47869 The set ` G ` of "simple p...
uspgrsprfv 47870 The value of the function ...
uspgrsprf 47871 The mapping ` F ` is a fun...
uspgrsprf1 47872 The mapping ` F ` is a one...
uspgrsprfo 47873 The mapping ` F ` is a fun...
uspgrsprf1o 47874 The mapping ` F ` is a bij...
uspgrex 47875 The class ` G ` of all "si...
uspgrbispr 47876 There is a bijection betwe...
uspgrspren 47877 The set ` G ` of the "simp...
uspgrymrelen 47878 The set ` G ` of the "simp...
uspgrbisymrel 47879 There is a bijection betwe...
uspgrbisymrelALT 47880 Alternate proof of ~ uspgr...
ovn0dmfun 47881 If a class operation value...
xpsnopab 47882 A Cartesian product with a...
xpiun 47883 A Cartesian product expres...
ovn0ssdmfun 47884 If a class' operation valu...
fnxpdmdm 47885 The domain of the domain o...
cnfldsrngbas 47886 The base set of a subring ...
cnfldsrngadd 47887 The group addition operati...
cnfldsrngmul 47888 The ring multiplication op...
plusfreseq 47889 If the empty set is not co...
mgmplusfreseq 47890 If the empty set is not co...
0mgm 47891 A set with an empty base s...
opmpoismgm 47892 A structure with a group a...
copissgrp 47893 A structure with a constan...
copisnmnd 47894 A structure with a constan...
0nodd 47895 0 is not an odd integer. ...
1odd 47896 1 is an odd integer. (Con...
2nodd 47897 2 is not an odd integer. ...
oddibas 47898 Lemma 1 for ~ oddinmgm : ...
oddiadd 47899 Lemma 2 for ~ oddinmgm : ...
oddinmgm 47900 The structure of all odd i...
nnsgrpmgm 47901 The structure of positive ...
nnsgrp 47902 The structure of positive ...
nnsgrpnmnd 47903 The structure of positive ...
nn0mnd 47904 The set of nonnegative int...
gsumsplit2f 47905 Split a group sum into two...
gsumdifsndf 47906 Extract a summand from a f...
gsumfsupp 47907 A group sum of a family ca...
iscllaw 47914 The predicate "is a closed...
iscomlaw 47915 The predicate "is a commut...
clcllaw 47916 Closure of a closed operat...
isasslaw 47917 The predicate "is an assoc...
asslawass 47918 Associativity of an associ...
mgmplusgiopALT 47919 Slot 2 (group operation) o...
sgrpplusgaopALT 47920 Slot 2 (group operation) o...
intopval 47927 The internal (binary) oper...
intop 47928 An internal (binary) opera...
clintopval 47929 The closed (internal binar...
assintopval 47930 The associative (closed in...
assintopmap 47931 The associative (closed in...
isclintop 47932 The predicate "is a closed...
clintop 47933 A closed (internal binary)...
assintop 47934 An associative (closed int...
isassintop 47935 The predicate "is an assoc...
clintopcllaw 47936 The closure law holds for ...
assintopcllaw 47937 The closure low holds for ...
assintopasslaw 47938 The associative low holds ...
assintopass 47939 An associative (closed int...
ismgmALT 47948 The predicate "is a magma"...
iscmgmALT 47949 The predicate "is a commut...
issgrpALT 47950 The predicate "is a semigr...
iscsgrpALT 47951 The predicate "is a commut...
mgm2mgm 47952 Equivalence of the two def...
sgrp2sgrp 47953 Equivalence of the two def...
lmod0rng 47954 If the scalar ring of a mo...
nzrneg1ne0 47955 The additive inverse of th...
lidldomn1 47956 If a (left) ideal (which i...
lidlabl 47957 A (left) ideal of a ring i...
lidlrng 47958 A (left) ideal of a ring i...
zlidlring 47959 The zero (left) ideal of a...
uzlidlring 47960 Only the zero (left) ideal...
lidldomnnring 47961 A (left) ideal of a domain...
0even 47962 0 is an even integer. (Co...
1neven 47963 1 is not an even integer. ...
2even 47964 2 is an even integer. (Co...
2zlidl 47965 The even integers are a (l...
2zrng 47966 The ring of integers restr...
2zrngbas 47967 The base set of R is the s...
2zrngadd 47968 The group addition operati...
2zrng0 47969 The additive identity of R...
2zrngamgm 47970 R is an (additive) magma. ...
2zrngasgrp 47971 R is an (additive) semigro...
2zrngamnd 47972 R is an (additive) monoid....
2zrngacmnd 47973 R is a commutative (additi...
2zrngagrp 47974 R is an (additive) group. ...
2zrngaabl 47975 R is an (additive) abelian...
2zrngmul 47976 The ring multiplication op...
2zrngmmgm 47977 R is a (multiplicative) ma...
2zrngmsgrp 47978 R is a (multiplicative) se...
2zrngALT 47979 The ring of integers restr...
2zrngnmlid 47980 R has no multiplicative (l...
2zrngnmrid 47981 R has no multiplicative (r...
2zrngnmlid2 47982 R has no multiplicative (l...
2zrngnring 47983 R is not a unital ring. (...
cznrnglem 47984 Lemma for ~ cznrng : The ...
cznabel 47985 The ring constructed from ...
cznrng 47986 The ring constructed from ...
cznnring 47987 The ring constructed from ...
rngcvalALTV 47990 Value of the category of n...
rngcbasALTV 47991 Set of objects of the cate...
rngchomfvalALTV 47992 Set of arrows of the categ...
rngchomALTV 47993 Set of arrows of the categ...
elrngchomALTV 47994 A morphism of non-unital r...
rngccofvalALTV 47995 Composition in the categor...
rngccoALTV 47996 Composition in the categor...
rngccatidALTV 47997 Lemma for ~ rngccatALTV . ...
rngccatALTV 47998 The category of non-unital...
rngcidALTV 47999 The identity arrow in the ...
rngcsectALTV 48000 A section in the category ...
rngcinvALTV 48001 An inverse in the category...
rngcisoALTV 48002 An isomorphism in the cate...
rngchomffvalALTV 48003 The value of the functiona...
rngchomrnghmresALTV 48004 The value of the functiona...
rngcrescrhmALTV 48005 The category of non-unital...
rhmsubcALTVlem1 48006 Lemma 1 for ~ rhmsubcALTV ...
rhmsubcALTVlem2 48007 Lemma 2 for ~ rhmsubcALTV ...
rhmsubcALTVlem3 48008 Lemma 3 for ~ rhmsubcALTV ...
rhmsubcALTVlem4 48009 Lemma 4 for ~ rhmsubcALTV ...
rhmsubcALTV 48010 According to ~ df-subc , t...
rhmsubcALTVcat 48011 The restriction of the cat...
ringcvalALTV 48014 Value of the category of r...
funcringcsetcALTV2lem1 48015 Lemma 1 for ~ funcringcset...
funcringcsetcALTV2lem2 48016 Lemma 2 for ~ funcringcset...
funcringcsetcALTV2lem3 48017 Lemma 3 for ~ funcringcset...
funcringcsetcALTV2lem4 48018 Lemma 4 for ~ funcringcset...
funcringcsetcALTV2lem5 48019 Lemma 5 for ~ funcringcset...
funcringcsetcALTV2lem6 48020 Lemma 6 for ~ funcringcset...
funcringcsetcALTV2lem7 48021 Lemma 7 for ~ funcringcset...
funcringcsetcALTV2lem8 48022 Lemma 8 for ~ funcringcset...
funcringcsetcALTV2lem9 48023 Lemma 9 for ~ funcringcset...
funcringcsetcALTV2 48024 The "natural forgetful fun...
ringcbasALTV 48025 Set of objects of the cate...
ringchomfvalALTV 48026 Set of arrows of the categ...
ringchomALTV 48027 Set of arrows of the categ...
elringchomALTV 48028 A morphism of rings is a f...
ringccofvalALTV 48029 Composition in the categor...
ringccoALTV 48030 Composition in the categor...
ringccatidALTV 48031 Lemma for ~ ringccatALTV ....
ringccatALTV 48032 The category of rings is a...
ringcidALTV 48033 The identity arrow in the ...
ringcsectALTV 48034 A section in the category ...
ringcinvALTV 48035 An inverse in the category...
ringcisoALTV 48036 An isomorphism in the cate...
ringcbasbasALTV 48037 An element of the base set...
funcringcsetclem1ALTV 48038 Lemma 1 for ~ funcringcset...
funcringcsetclem2ALTV 48039 Lemma 2 for ~ funcringcset...
funcringcsetclem3ALTV 48040 Lemma 3 for ~ funcringcset...
funcringcsetclem4ALTV 48041 Lemma 4 for ~ funcringcset...
funcringcsetclem5ALTV 48042 Lemma 5 for ~ funcringcset...
funcringcsetclem6ALTV 48043 Lemma 6 for ~ funcringcset...
funcringcsetclem7ALTV 48044 Lemma 7 for ~ funcringcset...
funcringcsetclem8ALTV 48045 Lemma 8 for ~ funcringcset...
funcringcsetclem9ALTV 48046 Lemma 9 for ~ funcringcset...
funcringcsetcALTV 48047 The "natural forgetful fun...
srhmsubcALTVlem1 48048 Lemma 1 for ~ srhmsubcALTV...
srhmsubcALTVlem2 48049 Lemma 2 for ~ srhmsubcALTV...
srhmsubcALTV 48050 According to ~ df-subc , t...
sringcatALTV 48051 The restriction of the cat...
crhmsubcALTV 48052 According to ~ df-subc , t...
cringcatALTV 48053 The restriction of the cat...
drhmsubcALTV 48054 According to ~ df-subc , t...
drngcatALTV 48055 The restriction of the cat...
fldcatALTV 48056 The restriction of the cat...
fldcALTV 48057 The restriction of the cat...
fldhmsubcALTV 48058 According to ~ df-subc , t...
opeliun2xp 48059 Membership of an ordered p...
eliunxp2 48060 Membership in a union of C...
mpomptx2 48061 Express a two-argument fun...
cbvmpox2 48062 Rule to change the bound v...
dmmpossx2 48063 The domain of a mapping is...
mpoexxg2 48064 Existence of an operation ...
ovmpordxf 48065 Value of an operation give...
ovmpordx 48066 Value of an operation give...
ovmpox2 48067 The value of an operation ...
fdmdifeqresdif 48068 The restriction of a condi...
offvalfv 48069 The function operation exp...
ofaddmndmap 48070 The function operation app...
mapsnop 48071 A singleton of an ordered ...
fprmappr 48072 A function with a domain o...
mapprop 48073 An unordered pair containi...
ztprmneprm 48074 A prime is not an integer ...
2t6m3t4e0 48075 2 times 6 minus 3 times 4 ...
ssnn0ssfz 48076 For any finite subset of `...
nn0sumltlt 48077 If the sum of two nonnegat...
bcpascm1 48078 Pascal's rule for the bino...
altgsumbc 48079 The sum of binomial coeffi...
altgsumbcALT 48080 Alternate proof of ~ altgs...
zlmodzxzlmod 48081 The ` ZZ `-module ` ZZ X. ...
zlmodzxzel 48082 An element of the (base se...
zlmodzxz0 48083 The ` 0 ` of the ` ZZ `-mo...
zlmodzxzscm 48084 The scalar multiplication ...
zlmodzxzadd 48085 The addition of the ` ZZ `...
zlmodzxzsubm 48086 The subtraction of the ` Z...
zlmodzxzsub 48087 The subtraction of the ` Z...
mgpsumunsn 48088 Extract a summand/factor f...
mgpsumz 48089 If the group sum for the m...
mgpsumn 48090 If the group sum for the m...
exple2lt6 48091 A nonnegative integer to t...
pgrple2abl 48092 Every symmetric group on a...
pgrpgt2nabl 48093 Every symmetric group on a...
invginvrid 48094 Identity for a multiplicat...
rmsupp0 48095 The support of a mapping o...
domnmsuppn0 48096 The support of a mapping o...
rmsuppss 48097 The support of a mapping o...
mndpsuppss 48098 The support of a mapping o...
scmsuppss 48099 The support of a mapping o...
rmsuppfi 48100 The support of a mapping o...
rmfsupp 48101 A mapping of a multiplicat...
mndpsuppfi 48102 The support of a mapping o...
mndpfsupp 48103 A mapping of a scalar mult...
scmsuppfi 48104 The support of a mapping o...
scmfsupp 48105 A mapping of a scalar mult...
suppmptcfin 48106 The support of a mapping w...
mptcfsupp 48107 A mapping with value 0 exc...
fsuppmptdmf 48108 A mapping with a finite do...
lmodvsmdi 48109 Multiple distributive law ...
gsumlsscl 48110 Closure of a group sum in ...
assaascl0 48111 The scalar 0 embedded into...
assaascl1 48112 The scalar 1 embedded into...
ply1vr1smo 48113 The variable in a polynomi...
ply1sclrmsm 48114 The ring multiplication of...
coe1id 48115 Coefficient vector of the ...
coe1sclmulval 48116 The value of the coefficie...
ply1mulgsumlem1 48117 Lemma 1 for ~ ply1mulgsum ...
ply1mulgsumlem2 48118 Lemma 2 for ~ ply1mulgsum ...
ply1mulgsumlem3 48119 Lemma 3 for ~ ply1mulgsum ...
ply1mulgsumlem4 48120 Lemma 4 for ~ ply1mulgsum ...
ply1mulgsum 48121 The product of two polynom...
evl1at0 48122 Polynomial evaluation for ...
evl1at1 48123 Polynomial evaluation for ...
linply1 48124 A term of the form ` x - C...
lineval 48125 A term of the form ` x - C...
linevalexample 48126 The polynomial ` x - 3 ` o...
dmatALTval 48131 The algebra of ` N ` x ` N...
dmatALTbas 48132 The base set of the algebr...
dmatALTbasel 48133 An element of the base set...
dmatbas 48134 The set of all ` N ` x ` N...
lincop 48139 A linear combination as op...
lincval 48140 The value of a linear comb...
dflinc2 48141 Alternative definition of ...
lcoop 48142 A linear combination as op...
lcoval 48143 The value of a linear comb...
lincfsuppcl 48144 A linear combination of ve...
linccl 48145 A linear combination of ve...
lincval0 48146 The value of an empty line...
lincvalsng 48147 The linear combination ove...
lincvalsn 48148 The linear combination ove...
lincvalpr 48149 The linear combination ove...
lincval1 48150 The linear combination ove...
lcosn0 48151 Properties of a linear com...
lincvalsc0 48152 The linear combination whe...
lcoc0 48153 Properties of a linear com...
linc0scn0 48154 If a set contains the zero...
lincdifsn 48155 A vector is a linear combi...
linc1 48156 A vector is a linear combi...
lincellss 48157 A linear combination of a ...
lco0 48158 The set of empty linear co...
lcoel0 48159 The zero vector is always ...
lincsum 48160 The sum of two linear comb...
lincscm 48161 A linear combinations mult...
lincsumcl 48162 The sum of two linear comb...
lincscmcl 48163 The multiplication of a li...
lincsumscmcl 48164 The sum of a linear combin...
lincolss 48165 According to the statement...
ellcoellss 48166 Every linear combination o...
lcoss 48167 A set of vectors of a modu...
lspsslco 48168 Lemma for ~ lspeqlco . (C...
lcosslsp 48169 Lemma for ~ lspeqlco . (C...
lspeqlco 48170 Equivalence of a _span_ of...
rellininds 48174 The class defining the rel...
linindsv 48176 The classes of the module ...
islininds 48177 The property of being a li...
linindsi 48178 The implications of being ...
linindslinci 48179 The implications of being ...
islinindfis 48180 The property of being a li...
islinindfiss 48181 The property of being a li...
linindscl 48182 A linearly independent set...
lindepsnlininds 48183 A linearly dependent subse...
islindeps 48184 The property of being a li...
lincext1 48185 Property 1 of an extension...
lincext2 48186 Property 2 of an extension...
lincext3 48187 Property 3 of an extension...
lindslinindsimp1 48188 Implication 1 for ~ lindsl...
lindslinindimp2lem1 48189 Lemma 1 for ~ lindslininds...
lindslinindimp2lem2 48190 Lemma 2 for ~ lindslininds...
lindslinindimp2lem3 48191 Lemma 3 for ~ lindslininds...
lindslinindimp2lem4 48192 Lemma 4 for ~ lindslininds...
lindslinindsimp2lem5 48193 Lemma 5 for ~ lindslininds...
lindslinindsimp2 48194 Implication 2 for ~ lindsl...
lindslininds 48195 Equivalence of definitions...
linds0 48196 The empty set is always a ...
el0ldep 48197 A set containing the zero ...
el0ldepsnzr 48198 A set containing the zero ...
lindsrng01 48199 Any subset of a module is ...
lindszr 48200 Any subset of a module ove...
snlindsntorlem 48201 Lemma for ~ snlindsntor . ...
snlindsntor 48202 A singleton is linearly in...
ldepsprlem 48203 Lemma for ~ ldepspr . (Co...
ldepspr 48204 If a vector is a scalar mu...
lincresunit3lem3 48205 Lemma 3 for ~ lincresunit3...
lincresunitlem1 48206 Lemma 1 for properties of ...
lincresunitlem2 48207 Lemma for properties of a ...
lincresunit1 48208 Property 1 of a specially ...
lincresunit2 48209 Property 2 of a specially ...
lincresunit3lem1 48210 Lemma 1 for ~ lincresunit3...
lincresunit3lem2 48211 Lemma 2 for ~ lincresunit3...
lincresunit3 48212 Property 3 of a specially ...
lincreslvec3 48213 Property 3 of a specially ...
islindeps2 48214 Conditions for being a lin...
islininds2 48215 Implication of being a lin...
isldepslvec2 48216 Alternative definition of ...
lindssnlvec 48217 A singleton not containing...
lmod1lem1 48218 Lemma 1 for ~ lmod1 . (Co...
lmod1lem2 48219 Lemma 2 for ~ lmod1 . (Co...
lmod1lem3 48220 Lemma 3 for ~ lmod1 . (Co...
lmod1lem4 48221 Lemma 4 for ~ lmod1 . (Co...
lmod1lem5 48222 Lemma 5 for ~ lmod1 . (Co...
lmod1 48223 The (smallest) structure r...
lmod1zr 48224 The (smallest) structure r...
lmod1zrnlvec 48225 There is a (left) module (...
lmodn0 48226 Left modules exist. (Cont...
zlmodzxzequa 48227 Example of an equation wit...
zlmodzxznm 48228 Example of a linearly depe...
zlmodzxzldeplem 48229 A and B are not equal. (C...
zlmodzxzequap 48230 Example of an equation wit...
zlmodzxzldeplem1 48231 Lemma 1 for ~ zlmodzxzldep...
zlmodzxzldeplem2 48232 Lemma 2 for ~ zlmodzxzldep...
zlmodzxzldeplem3 48233 Lemma 3 for ~ zlmodzxzldep...
zlmodzxzldeplem4 48234 Lemma 4 for ~ zlmodzxzldep...
zlmodzxzldep 48235 { A , B } is a linearly de...
ldepsnlinclem1 48236 Lemma 1 for ~ ldepsnlinc ....
ldepsnlinclem2 48237 Lemma 2 for ~ ldepsnlinc ....
lvecpsslmod 48238 The class of all (left) ve...
ldepsnlinc 48239 The reverse implication of...
ldepslinc 48240 For (left) vector spaces, ...
suppdm 48241 If the range of a function...
eluz2cnn0n1 48242 An integer greater than 1 ...
divge1b 48243 The ratio of a real number...
divgt1b 48244 The ratio of a real number...
ltsubaddb 48245 Equivalence for the "less ...
ltsubsubb 48246 Equivalence for the "less ...
ltsubadd2b 48247 Equivalence for the "less ...
divsub1dir 48248 Distribution of division o...
expnegico01 48249 An integer greater than 1 ...
elfzolborelfzop1 48250 An element of a half-open ...
pw2m1lepw2m1 48251 2 to the power of a positi...
zgtp1leeq 48252 If an integer is between a...
flsubz 48253 An integer can be moved in...
fldivmod 48254 Expressing the floor of a ...
mod0mul 48255 If an integer is 0 modulo ...
modn0mul 48256 If an integer is not 0 mod...
m1modmmod 48257 An integer decreased by 1 ...
difmodm1lt 48258 The difference between an ...
nn0onn0ex 48259 For each odd nonnegative i...
nn0enn0ex 48260 For each even nonnegative ...
nnennex 48261 For each even positive int...
nneop 48262 A positive integer is even...
nneom 48263 A positive integer is even...
nn0eo 48264 A nonnegative integer is e...
nnpw2even 48265 2 to the power of a positi...
zefldiv2 48266 The floor of an even integ...
zofldiv2 48267 The floor of an odd intege...
nn0ofldiv2 48268 The floor of an odd nonneg...
flnn0div2ge 48269 The floor of a positive in...
flnn0ohalf 48270 The floor of the half of a...
logcxp0 48271 Logarithm of a complex pow...
regt1loggt0 48272 The natural logarithm for ...
fdivval 48275 The quotient of two functi...
fdivmpt 48276 The quotient of two functi...
fdivmptf 48277 The quotient of two functi...
refdivmptf 48278 The quotient of two functi...
fdivpm 48279 The quotient of two functi...
refdivpm 48280 The quotient of two functi...
fdivmptfv 48281 The function value of a qu...
refdivmptfv 48282 The function value of a qu...
bigoval 48285 Set of functions of order ...
elbigofrcl 48286 Reverse closure of the "bi...
elbigo 48287 Properties of a function o...
elbigo2 48288 Properties of a function o...
elbigo2r 48289 Sufficient condition for a...
elbigof 48290 A function of order G(x) i...
elbigodm 48291 The domain of a function o...
elbigoimp 48292 The defining property of a...
elbigolo1 48293 A function (into the posit...
rege1logbrege0 48294 The general logarithm, wit...
rege1logbzge0 48295 The general logarithm, wit...
fllogbd 48296 A real number is between t...
relogbmulbexp 48297 The logarithm of the produ...
relogbdivb 48298 The logarithm of the quoti...
logbge0b 48299 The logarithm of a number ...
logblt1b 48300 The logarithm of a number ...
fldivexpfllog2 48301 The floor of a positive re...
nnlog2ge0lt1 48302 A positive integer is 1 if...
logbpw2m1 48303 The floor of the binary lo...
fllog2 48304 The floor of the binary lo...
blenval 48307 The binary length of an in...
blen0 48308 The binary length of 0. (...
blenn0 48309 The binary length of a "nu...
blenre 48310 The binary length of a pos...
blennn 48311 The binary length of a pos...
blennnelnn 48312 The binary length of a pos...
blennn0elnn 48313 The binary length of a non...
blenpw2 48314 The binary length of a pow...
blenpw2m1 48315 The binary length of a pow...
nnpw2blen 48316 A positive integer is betw...
nnpw2blenfzo 48317 A positive integer is betw...
nnpw2blenfzo2 48318 A positive integer is eith...
nnpw2pmod 48319 Every positive integer can...
blen1 48320 The binary length of 1. (...
blen2 48321 The binary length of 2. (...
nnpw2p 48322 Every positive integer can...
nnpw2pb 48323 A number is a positive int...
blen1b 48324 The binary length of a non...
blennnt2 48325 The binary length of a pos...
nnolog2flm1 48326 The floor of the binary lo...
blennn0em1 48327 The binary length of the h...
blennngt2o2 48328 The binary length of an od...
blengt1fldiv2p1 48329 The binary length of an in...
blennn0e2 48330 The binary length of an ev...
digfval 48333 Operation to obtain the ` ...
digval 48334 The ` K ` th digit of a no...
digvalnn0 48335 The ` K ` th digit of a no...
nn0digval 48336 The ` K ` th digit of a no...
dignn0fr 48337 The digits of the fraction...
dignn0ldlem 48338 Lemma for ~ dignnld . (Co...
dignnld 48339 The leading digits of a po...
dig2nn0ld 48340 The leading digits of a po...
dig2nn1st 48341 The first (relevant) digit...
dig0 48342 All digits of 0 are 0. (C...
digexp 48343 The ` K ` th digit of a po...
dig1 48344 All but one digits of 1 ar...
0dig1 48345 The ` 0 ` th digit of 1 is...
0dig2pr01 48346 The integers 0 and 1 corre...
dig2nn0 48347 A digit of a nonnegative i...
0dig2nn0e 48348 The last bit of an even in...
0dig2nn0o 48349 The last bit of an odd int...
dig2bits 48350 The ` K ` th digit of a no...
dignn0flhalflem1 48351 Lemma 1 for ~ dignn0flhalf...
dignn0flhalflem2 48352 Lemma 2 for ~ dignn0flhalf...
dignn0ehalf 48353 The digits of the half of ...
dignn0flhalf 48354 The digits of the rounded ...
nn0sumshdiglemA 48355 Lemma for ~ nn0sumshdig (i...
nn0sumshdiglemB 48356 Lemma for ~ nn0sumshdig (i...
nn0sumshdiglem1 48357 Lemma 1 for ~ nn0sumshdig ...
nn0sumshdiglem2 48358 Lemma 2 for ~ nn0sumshdig ...
nn0sumshdig 48359 A nonnegative integer can ...
nn0mulfsum 48360 Trivial algorithm to calcu...
nn0mullong 48361 Standard algorithm (also k...
naryfval 48364 The set of the n-ary (endo...
naryfvalixp 48365 The set of the n-ary (endo...
naryfvalel 48366 An n-ary (endo)function on...
naryrcl 48367 Reverse closure for n-ary ...
naryfvalelfv 48368 The value of an n-ary (end...
naryfvalelwrdf 48369 An n-ary (endo)function on...
0aryfvalel 48370 A nullary (endo)function o...
0aryfvalelfv 48371 The value of a nullary (en...
1aryfvalel 48372 A unary (endo)function on ...
fv1arycl 48373 Closure of a unary (endo)f...
1arympt1 48374 A unary (endo)function in ...
1arympt1fv 48375 The value of a unary (endo...
1arymaptfv 48376 The value of the mapping o...
1arymaptf 48377 The mapping of unary (endo...
1arymaptf1 48378 The mapping of unary (endo...
1arymaptfo 48379 The mapping of unary (endo...
1arymaptf1o 48380 The mapping of unary (endo...
1aryenef 48381 The set of unary (endo)fun...
1aryenefmnd 48382 The set of unary (endo)fun...
2aryfvalel 48383 A binary (endo)function on...
fv2arycl 48384 Closure of a binary (endo)...
2arympt 48385 A binary (endo)function in...
2arymptfv 48386 The value of a binary (end...
2arymaptfv 48387 The value of the mapping o...
2arymaptf 48388 The mapping of binary (end...
2arymaptf1 48389 The mapping of binary (end...
2arymaptfo 48390 The mapping of binary (end...
2arymaptf1o 48391 The mapping of binary (end...
2aryenef 48392 The set of binary (endo)fu...
itcoval 48397 The value of the function ...
itcoval0 48398 A function iterated zero t...
itcoval1 48399 A function iterated once. ...
itcoval2 48400 A function iterated twice....
itcoval3 48401 A function iterated three ...
itcoval0mpt 48402 A mapping iterated zero ti...
itcovalsuc 48403 The value of the function ...
itcovalsucov 48404 The value of the function ...
itcovalendof 48405 The n-th iterate of an end...
itcovalpclem1 48406 Lemma 1 for ~ itcovalpc : ...
itcovalpclem2 48407 Lemma 2 for ~ itcovalpc : ...
itcovalpc 48408 The value of the function ...
itcovalt2lem2lem1 48409 Lemma 1 for ~ itcovalt2lem...
itcovalt2lem2lem2 48410 Lemma 2 for ~ itcovalt2lem...
itcovalt2lem1 48411 Lemma 1 for ~ itcovalt2 : ...
itcovalt2lem2 48412 Lemma 2 for ~ itcovalt2 : ...
itcovalt2 48413 The value of the function ...
ackvalsuc1mpt 48414 The Ackermann function at ...
ackvalsuc1 48415 The Ackermann function at ...
ackval0 48416 The Ackermann function at ...
ackval1 48417 The Ackermann function at ...
ackval2 48418 The Ackermann function at ...
ackval3 48419 The Ackermann function at ...
ackendofnn0 48420 The Ackermann function at ...
ackfnnn0 48421 The Ackermann function at ...
ackval0val 48422 The Ackermann function at ...
ackvalsuc0val 48423 The Ackermann function at ...
ackvalsucsucval 48424 The Ackermann function at ...
ackval0012 48425 The Ackermann function at ...
ackval1012 48426 The Ackermann function at ...
ackval2012 48427 The Ackermann function at ...
ackval3012 48428 The Ackermann function at ...
ackval40 48429 The Ackermann function at ...
ackval41a 48430 The Ackermann function at ...
ackval41 48431 The Ackermann function at ...
ackval42 48432 The Ackermann function at ...
ackval42a 48433 The Ackermann function at ...
ackval50 48434 The Ackermann function at ...
fv1prop 48435 The function value of unor...
fv2prop 48436 The function value of unor...
submuladdmuld 48437 Transformation of a sum of...
affinecomb1 48438 Combination of two real af...
affinecomb2 48439 Combination of two real af...
affineid 48440 Identity of an affine comb...
1subrec1sub 48441 Subtract the reciprocal of...
resum2sqcl 48442 The sum of two squares of ...
resum2sqgt0 48443 The sum of the square of a...
resum2sqrp 48444 The sum of the square of a...
resum2sqorgt0 48445 The sum of the square of t...
reorelicc 48446 Membership in and outside ...
rrx2pxel 48447 The x-coordinate of a poin...
rrx2pyel 48448 The y-coordinate of a poin...
prelrrx2 48449 An unordered pair of order...
prelrrx2b 48450 An unordered pair of order...
rrx2pnecoorneor 48451 If two different points ` ...
rrx2pnedifcoorneor 48452 If two different points ` ...
rrx2pnedifcoorneorr 48453 If two different points ` ...
rrx2xpref1o 48454 There is a bijection betwe...
rrx2xpreen 48455 The set of points in the t...
rrx2plord 48456 The lexicographical orderi...
rrx2plord1 48457 The lexicographical orderi...
rrx2plord2 48458 The lexicographical orderi...
rrx2plordisom 48459 The set of points in the t...
rrx2plordso 48460 The lexicographical orderi...
ehl2eudisval0 48461 The Euclidean distance of ...
ehl2eudis0lt 48462 An upper bound of the Eucl...
lines 48467 The lines passing through ...
line 48468 The line passing through t...
rrxlines 48469 Definition of lines passin...
rrxline 48470 The line passing through t...
rrxlinesc 48471 Definition of lines passin...
rrxlinec 48472 The line passing through t...
eenglngeehlnmlem1 48473 Lemma 1 for ~ eenglngeehln...
eenglngeehlnmlem2 48474 Lemma 2 for ~ eenglngeehln...
eenglngeehlnm 48475 The line definition in the...
rrx2line 48476 The line passing through t...
rrx2vlinest 48477 The vertical line passing ...
rrx2linest 48478 The line passing through t...
rrx2linesl 48479 The line passing through t...
rrx2linest2 48480 The line passing through t...
elrrx2linest2 48481 The line passing through t...
spheres 48482 The spheres for given cent...
sphere 48483 A sphere with center ` X `...
rrxsphere 48484 The sphere with center ` M...
2sphere 48485 The sphere with center ` M...
2sphere0 48486 The sphere around the orig...
line2ylem 48487 Lemma for ~ line2y . This...
line2 48488 Example for a line ` G ` p...
line2xlem 48489 Lemma for ~ line2x . This...
line2x 48490 Example for a horizontal l...
line2y 48491 Example for a vertical lin...
itsclc0lem1 48492 Lemma for theorems about i...
itsclc0lem2 48493 Lemma for theorems about i...
itsclc0lem3 48494 Lemma for theorems about i...
itscnhlc0yqe 48495 Lemma for ~ itsclc0 . Qua...
itschlc0yqe 48496 Lemma for ~ itsclc0 . Qua...
itsclc0yqe 48497 Lemma for ~ itsclc0 . Qua...
itsclc0yqsollem1 48498 Lemma 1 for ~ itsclc0yqsol...
itsclc0yqsollem2 48499 Lemma 2 for ~ itsclc0yqsol...
itsclc0yqsol 48500 Lemma for ~ itsclc0 . Sol...
itscnhlc0xyqsol 48501 Lemma for ~ itsclc0 . Sol...
itschlc0xyqsol1 48502 Lemma for ~ itsclc0 . Sol...
itschlc0xyqsol 48503 Lemma for ~ itsclc0 . Sol...
itsclc0xyqsol 48504 Lemma for ~ itsclc0 . Sol...
itsclc0xyqsolr 48505 Lemma for ~ itsclc0 . Sol...
itsclc0xyqsolb 48506 Lemma for ~ itsclc0 . Sol...
itsclc0 48507 The intersection points of...
itsclc0b 48508 The intersection points of...
itsclinecirc0 48509 The intersection points of...
itsclinecirc0b 48510 The intersection points of...
itsclinecirc0in 48511 The intersection points of...
itsclquadb 48512 Quadratic equation for the...
itsclquadeu 48513 Quadratic equation for the...
2itscplem1 48514 Lemma 1 for ~ 2itscp . (C...
2itscplem2 48515 Lemma 2 for ~ 2itscp . (C...
2itscplem3 48516 Lemma D for ~ 2itscp . (C...
2itscp 48517 A condition for a quadrati...
itscnhlinecirc02plem1 48518 Lemma 1 for ~ itscnhlineci...
itscnhlinecirc02plem2 48519 Lemma 2 for ~ itscnhlineci...
itscnhlinecirc02plem3 48520 Lemma 3 for ~ itscnhlineci...
itscnhlinecirc02p 48521 Intersection of a nonhoriz...
inlinecirc02plem 48522 Lemma for ~ inlinecirc02p ...
inlinecirc02p 48523 Intersection of a line wit...
inlinecirc02preu 48524 Intersection of a line wit...
pm4.71da 48525 Deduction converting a bic...
logic1 48526 Distribution of implicatio...
logic1a 48527 Variant of ~ logic1 . (Co...
logic2 48528 Variant of ~ logic1 . (Co...
pm5.32dav 48529 Distribution of implicatio...
pm5.32dra 48530 Reverse distribution of im...
exp12bd 48531 The import-export theorem ...
mpbiran3d 48532 Equivalence with a conjunc...
mpbiran4d 48533 Equivalence with a conjunc...
dtrucor3 48534 An example of how ~ ax-5 w...
ralbidb 48535 Formula-building rule for ...
ralbidc 48536 Formula-building rule for ...
r19.41dv 48537 A complex deduction form o...
rmotru 48538 Two ways of expressing "at...
reutru 48539 Two ways of expressing "ex...
reutruALT 48540 Alternate proof for ~ reut...
ssdisjd 48541 Subset preserves disjointn...
ssdisjdr 48542 Subset preserves disjointn...
disjdifb 48543 Relative complement is ant...
predisj 48544 Preimages of disjoint sets...
vsn 48545 The singleton of the unive...
mosn 48546 "At most one" element in a...
mo0 48547 "At most one" element in a...
mosssn 48548 "At most one" element in a...
mo0sn 48549 Two ways of expressing "at...
mosssn2 48550 Two ways of expressing "at...
unilbss 48551 Superclass of the greatest...
inpw 48552 Two ways of expressing a c...
mof0 48553 There is at most one funct...
mof02 48554 A variant of ~ mof0 . (Co...
mof0ALT 48555 Alternate proof for ~ mof0...
eufsnlem 48556 There is exactly one funct...
eufsn 48557 There is exactly one funct...
eufsn2 48558 There is exactly one funct...
mofsn 48559 There is at most one funct...
mofsn2 48560 There is at most one funct...
mofsssn 48561 There is at most one funct...
mofmo 48562 There is at most one funct...
mofeu 48563 The uniqueness of a functi...
elfvne0 48564 If a function value has a ...
fdomne0 48565 A function with non-empty ...
f1sn2g 48566 A function that maps a sin...
f102g 48567 A function that maps the e...
f1mo 48568 A function that maps a set...
f002 48569 A function with an empty c...
map0cor 48570 A function exists iff an e...
fvconstr 48571 Two ways of expressing ` A...
fvconstrn0 48572 Two ways of expressing ` A...
fvconstr2 48573 Two ways of expressing ` A...
fvconst0ci 48574 A constant function's valu...
fvconstdomi 48575 A constant function's valu...
f1omo 48576 There is at most one eleme...
f1omoALT 48577 There is at most one eleme...
iccin 48578 Intersection of two closed...
iccdisj2 48579 If the upper bound of one ...
iccdisj 48580 If the upper bound of one ...
mreuniss 48581 The union of a collection ...
clduni 48582 The union of closed sets i...
opncldeqv 48583 Conditions on open sets ar...
opndisj 48584 Two ways of saying that tw...
clddisj 48585 Two ways of saying that tw...
neircl 48586 Reverse closure of the nei...
opnneilem 48587 Lemma factoring out common...
opnneir 48588 If something is true for a...
opnneirv 48589 A variant of ~ opnneir wit...
opnneilv 48590 The converse of ~ opnneir ...
opnneil 48591 A variant of ~ opnneilv . ...
opnneieqv 48592 The equivalence between ne...
opnneieqvv 48593 The equivalence between ne...
restcls2lem 48594 A closed set in a subspace...
restcls2 48595 A closed set in a subspace...
restclsseplem 48596 Lemma for ~ restclssep . ...
restclssep 48597 Two disjoint closed sets i...
cnneiima 48598 Given a continuous functio...
iooii 48599 Open intervals are open se...
icccldii 48600 Closed intervals are close...
i0oii 48601 ` ( 0 [,) A ) ` is open in...
io1ii 48602 ` ( A (,] 1 ) ` is open in...
sepnsepolem1 48603 Lemma for ~ sepnsepo . (C...
sepnsepolem2 48604 Open neighborhood and neig...
sepnsepo 48605 Open neighborhood and neig...
sepdisj 48606 Separated sets are disjoin...
seposep 48607 If two sets are separated ...
sepcsepo 48608 If two sets are separated ...
sepfsepc 48609 If two sets are separated ...
seppsepf 48610 If two sets are precisely ...
seppcld 48611 If two sets are precisely ...
isnrm4 48612 A topological space is nor...
dfnrm2 48613 A topological space is nor...
dfnrm3 48614 A topological space is nor...
iscnrm3lem1 48615 Lemma for ~ iscnrm3 . Sub...
iscnrm3lem2 48616 Lemma for ~ iscnrm3 provin...
iscnrm3lem3 48617 Lemma for ~ iscnrm3lem4 . ...
iscnrm3lem4 48618 Lemma for ~ iscnrm3lem5 an...
iscnrm3lem5 48619 Lemma for ~ iscnrm3l . (C...
iscnrm3lem6 48620 Lemma for ~ iscnrm3lem7 . ...
iscnrm3lem7 48621 Lemma for ~ iscnrm3rlem8 a...
iscnrm3rlem1 48622 Lemma for ~ iscnrm3rlem2 ....
iscnrm3rlem2 48623 Lemma for ~ iscnrm3rlem3 ....
iscnrm3rlem3 48624 Lemma for ~ iscnrm3r . Th...
iscnrm3rlem4 48625 Lemma for ~ iscnrm3rlem8 ....
iscnrm3rlem5 48626 Lemma for ~ iscnrm3rlem6 ....
iscnrm3rlem6 48627 Lemma for ~ iscnrm3rlem7 ....
iscnrm3rlem7 48628 Lemma for ~ iscnrm3rlem8 ....
iscnrm3rlem8 48629 Lemma for ~ iscnrm3r . Di...
iscnrm3r 48630 Lemma for ~ iscnrm3 . If ...
iscnrm3llem1 48631 Lemma for ~ iscnrm3l . Cl...
iscnrm3llem2 48632 Lemma for ~ iscnrm3l . If...
iscnrm3l 48633 Lemma for ~ iscnrm3 . Giv...
iscnrm3 48634 A completely normal topolo...
iscnrm3v 48635 A topology is completely n...
iscnrm4 48636 A completely normal topolo...
isprsd 48637 Property of being a preord...
lubeldm2 48638 Member of the domain of th...
glbeldm2 48639 Member of the domain of th...
lubeldm2d 48640 Member of the domain of th...
glbeldm2d 48641 Member of the domain of th...
lubsscl 48642 If a subset of ` S ` conta...
glbsscl 48643 If a subset of ` S ` conta...
lubprlem 48644 Lemma for ~ lubprdm and ~ ...
lubprdm 48645 The set of two comparable ...
lubpr 48646 The LUB of the set of two ...
glbprlem 48647 Lemma for ~ glbprdm and ~ ...
glbprdm 48648 The set of two comparable ...
glbpr 48649 The GLB of the set of two ...
joindm2 48650 The join of any two elemen...
joindm3 48651 The join of any two elemen...
meetdm2 48652 The meet of any two elemen...
meetdm3 48653 The meet of any two elemen...
posjidm 48654 Poset join is idempotent. ...
posmidm 48655 Poset meet is idempotent. ...
toslat 48656 A toset is a lattice. (Co...
isclatd 48657 The predicate "is a comple...
intubeu 48658 Existential uniqueness of ...
unilbeu 48659 Existential uniqueness of ...
ipolublem 48660 Lemma for ~ ipolubdm and ~...
ipolubdm 48661 The domain of the LUB of t...
ipolub 48662 The LUB of the inclusion p...
ipoglblem 48663 Lemma for ~ ipoglbdm and ~...
ipoglbdm 48664 The domain of the GLB of t...
ipoglb 48665 The GLB of the inclusion p...
ipolub0 48666 The LUB of the empty set i...
ipolub00 48667 The LUB of the empty set i...
ipoglb0 48668 The GLB of the empty set i...
mrelatlubALT 48669 Least upper bounds in a Mo...
mrelatglbALT 48670 Greatest lower bounds in a...
mreclat 48671 A Moore space is a complet...
topclat 48672 A topology is a complete l...
toplatglb0 48673 The empty intersection in ...
toplatlub 48674 Least upper bounds in a to...
toplatglb 48675 Greatest lower bounds in a...
toplatjoin 48676 Joins in a topology are re...
toplatmeet 48677 Meets in a topology are re...
topdlat 48678 A topology is a distributi...
elmgpcntrd 48679 The center of a ring. (Co...
asclelbas 48680 Lifted scalars are in the ...
asclcntr 48681 The algebra scalars functi...
asclcom 48682 Scalars are commutative af...
catprslem 48683 Lemma for ~ catprs . (Con...
catprs 48684 A preorder can be extracte...
catprs2 48685 A category equipped with t...
catprsc 48686 A construction of the preo...
catprsc2 48687 An alternate construction ...
endmndlem 48688 A diagonal hom-set in a ca...
idmon 48689 An identity arrow, or an i...
idepi 48690 An identity arrow, or an i...
funcf2lem 48691 A utility theorem for prov...
isthinc 48694 The predicate "is a thin c...
isthinc2 48695 A thin category is a categ...
isthinc3 48696 A thin category is a categ...
thincc 48697 A thin category is a categ...
thinccd 48698 A thin category is a categ...
thincssc 48699 A thin category is a categ...
isthincd2lem1 48700 Lemma for ~ isthincd2 and ...
thincmo2 48701 Morphisms in the same hom-...
thincmo 48702 There is at most one morph...
thincmoALT 48703 Alternate proof for ~ thin...
thincmod 48704 At most one morphism in ea...
thincn0eu 48705 In a thin category, a hom-...
thincid 48706 In a thin category, a morp...
thincmon 48707 In a thin category, all mo...
thincepi 48708 In a thin category, all mo...
isthincd2lem2 48709 Lemma for ~ isthincd2 . (...
isthincd 48710 The predicate "is a thin c...
isthincd2 48711 The predicate " ` C ` is a...
oppcthin 48712 The opposite category of a...
subthinc 48713 A subcategory of a thin ca...
functhinclem1 48714 Lemma for ~ functhinc . G...
functhinclem2 48715 Lemma for ~ functhinc . (...
functhinclem3 48716 Lemma for ~ functhinc . T...
functhinclem4 48717 Lemma for ~ functhinc . O...
functhinc 48718 A functor to a thin catego...
fullthinc 48719 A functor to a thin catego...
fullthinc2 48720 A full functor to a thin c...
thincfth 48721 A functor from a thin cate...
thincciso 48722 Two thin categories are is...
0thincg 48723 Any structure with an empt...
0thinc 48724 The empty category (see ~ ...
indthinc 48725 An indiscrete category in ...
indthincALT 48726 An alternate proof for ~ i...
prsthinc 48727 Preordered sets as categor...
setcthin 48728 A category of sets all of ...
setc2othin 48729 The category ` ( SetCat ``...
thincsect 48730 In a thin category, one mo...
thincsect2 48731 In a thin category, ` F ` ...
thincinv 48732 In a thin category, ` F ` ...
thinciso 48733 In a thin category, ` F : ...
thinccic 48734 In a thin category, two ob...
prstcval 48737 Lemma for ~ prstcnidlem an...
prstcnidlem 48738 Lemma for ~ prstcnid and ~...
prstcnid 48739 Components other than ` Ho...
prstcbas 48740 The base set is unchanged....
prstcleval 48741 Value of the less-than-or-...
prstclevalOLD 48742 Obsolete proof of ~ prstcl...
prstcle 48743 Value of the less-than-or-...
prstcocval 48744 Orthocomplementation is un...
prstcocvalOLD 48745 Obsolete proof of ~ prstco...
prstcoc 48746 Orthocomplementation is un...
prstchomval 48747 Hom-sets of the constructe...
prstcprs 48748 The category is a preorder...
prstcthin 48749 The preordered set is equi...
prstchom 48750 Hom-sets of the constructe...
prstchom2 48751 Hom-sets of the constructe...
prstchom2ALT 48752 Hom-sets of the constructe...
postcpos 48753 The converted category is ...
postcposALT 48754 Alternate proof for ~ post...
postc 48755 The converted category is ...
mndtcval 48758 Value of the category buil...
mndtcbasval 48759 The base set of the catego...
mndtcbas 48760 The category built from a ...
mndtcob 48761 Lemma for ~ mndtchom and ~...
mndtcbas2 48762 Two objects in a category ...
mndtchom 48763 The only hom-set of the ca...
mndtcco 48764 The composition of the cat...
mndtcco2 48765 The composition of the cat...
mndtccatid 48766 Lemma for ~ mndtccat and ~...
mndtccat 48767 The function value is a ca...
mndtcid 48768 The identity morphism, or ...
grptcmon 48769 All morphisms in a categor...
grptcepi 48770 All morphisms in a categor...
nfintd 48771 Bound-variable hypothesis ...
nfiund 48772 Bound-variable hypothesis ...
nfiundg 48773 Bound-variable hypothesis ...
iunord 48774 The indexed union of a col...
iunordi 48775 The indexed union of a col...
spd 48776 Specialization deduction, ...
spcdvw 48777 A version of ~ spcdv where...
tfis2d 48778 Transfinite Induction Sche...
bnd2d 48779 Deduction form of ~ bnd2 ....
dffun3f 48780 Alternate definition of fu...
setrecseq 48783 Equality theorem for set r...
nfsetrecs 48784 Bound-variable hypothesis ...
setrec1lem1 48785 Lemma for ~ setrec1 . Thi...
setrec1lem2 48786 Lemma for ~ setrec1 . If ...
setrec1lem3 48787 Lemma for ~ setrec1 . If ...
setrec1lem4 48788 Lemma for ~ setrec1 . If ...
setrec1 48789 This is the first of two f...
setrec2fun 48790 This is the second of two ...
setrec2lem1 48791 Lemma for ~ setrec2 . The...
setrec2lem2 48792 Lemma for ~ setrec2 . The...
setrec2 48793 This is the second of two ...
setrec2v 48794 Version of ~ setrec2 with ...
setrec2mpt 48795 Version of ~ setrec2 where...
setis 48796 Version of ~ setrec2 expre...
elsetrecslem 48797 Lemma for ~ elsetrecs . A...
elsetrecs 48798 A set ` A ` is an element ...
setrecsss 48799 The ` setrecs ` operator r...
setrecsres 48800 A recursively generated cl...
vsetrec 48801 Construct ` _V ` using set...
0setrec 48802 If a function sends the em...
onsetreclem1 48803 Lemma for ~ onsetrec . (C...
onsetreclem2 48804 Lemma for ~ onsetrec . (C...
onsetreclem3 48805 Lemma for ~ onsetrec . (C...
onsetrec 48806 Construct ` On ` using set...
elpglem1 48809 Lemma for ~ elpg . (Contr...
elpglem2 48810 Lemma for ~ elpg . (Contr...
elpglem3 48811 Lemma for ~ elpg . (Contr...
elpg 48812 Membership in the class of...
pgindlem 48813 Lemma for ~ pgind . (Cont...
pgindnf 48814 Version of ~ pgind with ex...
pgind 48815 Induction on partizan game...
sbidd 48816 An identity theorem for su...
sbidd-misc 48817 An identity theorem for su...
gte-lte 48822 Simple relationship betwee...
gt-lt 48823 Simple relationship betwee...
gte-lteh 48824 Relationship between ` <_ ...
gt-lth 48825 Relationship between ` < `...
ex-gt 48826 Simple example of ` > ` , ...
ex-gte 48827 Simple example of ` >_ ` ,...
sinhval-named 48834 Value of the named sinh fu...
coshval-named 48835 Value of the named cosh fu...
tanhval-named 48836 Value of the named tanh fu...
sinh-conventional 48837 Conventional definition of...
sinhpcosh 48838 Prove that ` ( sinh `` A )...
secval 48845 Value of the secant functi...
cscval 48846 Value of the cosecant func...
cotval 48847 Value of the cotangent fun...
seccl 48848 The closure of the secant ...
csccl 48849 The closure of the cosecan...
cotcl 48850 The closure of the cotange...
reseccl 48851 The closure of the secant ...
recsccl 48852 The closure of the cosecan...
recotcl 48853 The closure of the cotange...
recsec 48854 The reciprocal of secant i...
reccsc 48855 The reciprocal of cosecant...
reccot 48856 The reciprocal of cotangen...
rectan 48857 The reciprocal of tangent ...
sec0 48858 The value of the secant fu...
onetansqsecsq 48859 Prove the tangent squared ...
cotsqcscsq 48860 Prove the tangent squared ...
ifnmfalse 48861 If A is not a member of B,...
logb2aval 48862 Define the value of the ` ...
comraddi 48869 Commute RHS addition. See...
mvlraddi 48870 Move the right term in a s...
mvrladdi 48871 Move the left term in a su...
assraddsubi 48872 Associate RHS addition-sub...
joinlmuladdmuli 48873 Join AB+CB into (A+C) on L...
joinlmulsubmuld 48874 Join AB-CB into (A-C) on L...
joinlmulsubmuli 48875 Join AB-CB into (A-C) on L...
mvlrmuld 48876 Move the right term in a p...
mvlrmuli 48877 Move the right term in a p...
i2linesi 48878 Solve for the intersection...
i2linesd 48879 Solve for the intersection...
alimp-surprise 48880 Demonstrate that when usin...
alimp-no-surprise 48881 There is no "surprise" in ...
empty-surprise 48882 Demonstrate that when usin...
empty-surprise2 48883 "Prove" that false is true...
eximp-surprise 48884 Show what implication insi...
eximp-surprise2 48885 Show that "there exists" w...
alsconv 48890 There is an equivalence be...
alsi1d 48891 Deduction rule: Given "al...
alsi2d 48892 Deduction rule: Given "al...
alsc1d 48893 Deduction rule: Given "al...
alsc2d 48894 Deduction rule: Given "al...
alscn0d 48895 Deduction rule: Given "al...
alsi-no-surprise 48896 Demonstrate that there is ...
5m4e1 48897 Prove that 5 - 4 = 1. (Co...
2p2ne5 48898 Prove that ` 2 + 2 =/= 5 `...
resolution 48899 Resolution rule. This is ...
testable 48900 In classical logic all wff...
aacllem 48901 Lemma for other theorems a...
amgmwlem 48902 Weighted version of ~ amgm...
amgmlemALT 48903 Alternate proof of ~ amgml...
amgmw2d 48904 Weighted arithmetic-geomet...
young2d 48905 Young's inequality for ` n...
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