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List of Theorems
RefDescription
idi 1 (_Note_: This inference r...
a1ii 2 (_Note_: This inference r...
mp2 9 A double modus ponens infe...
mp2b 10 A double modus ponens infe...
a1i 11 Inference introducing an a...
2a1i 12 Inference introducing two ...
mp1i 13 Inference detaching an ant...
a2i 14 Inference distributing an ...
mpd 15 A modus ponens deduction. ...
imim2i 16 Inference adding common an...
syl 17 An inference version of th...
3syl 18 Inference chaining two syl...
4syl 19 Inference chaining three s...
mpi 20 A nested modus ponens infe...
mpisyl 21 A syllogism combined with ...
id 22 Principle of identity. Th...
idALT 23 Alternate proof of ~ id . ...
idd 24 Principle of identity ~ id...
a1d 25 Deduction introducing an e...
2a1d 26 Deduction introducing two ...
a1i13 27 Add two antecedents to a w...
2a1 28 A double form of ~ ax-1 . ...
a2d 29 Deduction distributing an ...
sylcom 30 Syllogism inference with c...
syl5com 31 Syllogism inference with c...
com12 32 Inference that swaps (comm...
syl11 33 A syllogism inference. Co...
syl5 34 A syllogism rule of infere...
syl6 35 A syllogism rule of infere...
syl56 36 Combine ~ syl5 and ~ syl6 ...
syl6com 37 Syllogism inference with c...
mpcom 38 Modus ponens inference wit...
syli 39 Syllogism inference with c...
syl2im 40 Replace two antecedents. ...
syl2imc 41 A commuted version of ~ sy...
pm2.27 42 This theorem, sometimes ca...
mpdd 43 A nested modus ponens dedu...
mpid 44 A nested modus ponens dedu...
mpdi 45 A nested modus ponens dedu...
mpii 46 A doubly nested modus pone...
syld 47 Syllogism deduction. Dedu...
syldc 48 Syllogism deduction. Comm...
mp2d 49 A double modus ponens dedu...
a1dd 50 Double deduction introduci...
2a1dd 51 Double deduction introduci...
pm2.43i 52 Inference absorbing redund...
pm2.43d 53 Deduction absorbing redund...
pm2.43a 54 Inference absorbing redund...
pm2.43b 55 Inference absorbing redund...
pm2.43 56 Absorption of redundant an...
imim2d 57 Deduction adding nested an...
imim2 58 A closed form of syllogism...
embantd 59 Deduction embedding an ant...
3syld 60 Triple syllogism deduction...
sylsyld 61 A double syllogism inferen...
imim12i 62 Inference joining two impl...
imim1i 63 Inference adding common co...
imim3i 64 Inference adding three nes...
sylc 65 A syllogism inference comb...
syl3c 66 A syllogism inference comb...
syl6mpi 67 A syllogism inference. (C...
mpsyl 68 Modus ponens combined with...
mpsylsyld 69 Modus ponens combined with...
syl6c 70 Inference combining ~ syl6...
syl6ci 71 A syllogism inference comb...
syldd 72 Nested syllogism deduction...
syl5d 73 A nested syllogism deducti...
syl7 74 A syllogism rule of infere...
syl6d 75 A nested syllogism deducti...
syl8 76 A syllogism rule of infere...
syl9 77 A nested syllogism inferen...
syl9r 78 A nested syllogism inferen...
syl10 79 A nested syllogism inferen...
a1ddd 80 Triple deduction introduci...
imim12d 81 Deduction combining antece...
imim1d 82 Deduction adding nested co...
imim1 83 A closed form of syllogism...
pm2.83 84 Theorem *2.83 of [Whitehea...
peirceroll 85 Over minimal implicational...
com23 86 Commutation of antecedents...
com3r 87 Commutation of antecedents...
com13 88 Commutation of antecedents...
com3l 89 Commutation of antecedents...
pm2.04 90 Swap antecedents. Theorem...
com34 91 Commutation of antecedents...
com4l 92 Commutation of antecedents...
com4t 93 Commutation of antecedents...
com4r 94 Commutation of antecedents...
com24 95 Commutation of antecedents...
com14 96 Commutation of antecedents...
com45 97 Commutation of antecedents...
com35 98 Commutation of antecedents...
com25 99 Commutation of antecedents...
com5l 100 Commutation of antecedents...
com15 101 Commutation of antecedents...
com52l 102 Commutation of antecedents...
com52r 103 Commutation of antecedents...
com5r 104 Commutation of antecedents...
imim12 105 Closed form of ~ imim12i a...
jarr 106 Elimination of a nested an...
jarri 107 Inference associated with ...
pm2.86d 108 Deduction associated with ...
pm2.86 109 Converse of Axiom ~ ax-2 ....
pm2.86i 110 Inference associated with ...
loolin 111 The Linearity Axiom of the...
loowoz 112 An alternate for the Linea...
con4 113 Alias for ~ ax-3 to be use...
con4i 114 Inference associated with ...
con4d 115 Deduction associated with ...
mt4 116 The rule of modus tollens....
mt4d 117 Modus tollens deduction. ...
mt4i 118 Modus tollens inference. ...
pm2.21i 119 A contradiction implies an...
pm2.24ii 120 A contradiction implies an...
pm2.21d 121 A contradiction implies an...
pm2.21ddALT 122 Alternate proof of ~ pm2.2...
pm2.21 123 From a wff and its negatio...
pm2.24 124 Theorem *2.24 of [Whitehea...
jarl 125 Elimination of a nested an...
jarli 126 Inference associated with ...
pm2.18d 127 Deduction form of the Clav...
pm2.18 128 Clavius law, or "consequen...
pm2.18i 129 Inference associated with ...
notnotr 130 Double negation eliminatio...
notnotri 131 Inference associated with ...
notnotriALT 132 Alternate proof of ~ notno...
notnotrd 133 Deduction associated with ...
con2d 134 A contraposition deduction...
con2 135 Contraposition. Theorem *...
mt2d 136 Modus tollens deduction. ...
mt2i 137 Modus tollens inference. ...
nsyl3 138 A negated syllogism infere...
con2i 139 A contraposition inference...
nsyl 140 A negated syllogism infere...
nsyl2 141 A negated syllogism infere...
notnot 142 Double negation introducti...
notnoti 143 Inference associated with ...
notnotd 144 Deduction associated with ...
con1d 145 A contraposition deduction...
con1 146 Contraposition. Theorem *...
con1i 147 A contraposition inference...
mt3d 148 Modus tollens deduction. ...
mt3i 149 Modus tollens inference. ...
pm2.24i 150 Inference associated with ...
pm2.24d 151 Deduction form of ~ pm2.24...
con3d 152 A contraposition deduction...
con3 153 Contraposition. Theorem *...
con3i 154 A contraposition inference...
con3rr3 155 Rotate through consequent ...
nsyld 156 A negated syllogism deduct...
nsyli 157 A negated syllogism infere...
nsyl4 158 A negated syllogism infere...
nsyl5 159 A negated syllogism infere...
pm3.2im 160 Theorem *3.2 of [Whitehead...
jc 161 Deduction joining the cons...
jcn 162 Theorem joining the conseq...
jcnd 163 Deduction joining the cons...
impi 164 An importation inference. ...
expi 165 An exportation inference. ...
simprim 166 Simplification. Similar t...
simplim 167 Simplification. Similar t...
pm2.5g 168 General instance of Theore...
pm2.5 169 Theorem *2.5 of [Whitehead...
conax1 170 Contrapositive of ~ ax-1 ....
conax1k 171 Weakening of ~ conax1 . G...
pm2.51 172 Theorem *2.51 of [Whitehea...
pm2.52 173 Theorem *2.52 of [Whitehea...
pm2.521g 174 A general instance of Theo...
pm2.521g2 175 A general instance of Theo...
pm2.521 176 Theorem *2.521 of [Whitehe...
expt 177 Exportation theorem ~ pm3....
impt 178 Importation theorem ~ pm3....
pm2.61d 179 Deduction eliminating an a...
pm2.61d1 180 Inference eliminating an a...
pm2.61d2 181 Inference eliminating an a...
pm2.61i 182 Inference eliminating an a...
pm2.61ii 183 Inference eliminating two ...
pm2.61nii 184 Inference eliminating two ...
pm2.61iii 185 Inference eliminating thre...
ja 186 Inference joining the ante...
jad 187 Deduction form of ~ ja . ...
pm2.01 188 Weak Clavius law. If a fo...
pm2.01d 189 Deduction based on reducti...
pm2.6 190 Theorem *2.6 of [Whitehead...
pm2.61 191 Theorem *2.61 of [Whitehea...
pm2.65 192 Theorem *2.65 of [Whitehea...
pm2.65i 193 Inference for proof by con...
pm2.21dd 194 A contradiction implies an...
pm2.65d 195 Deduction for proof by con...
mto 196 The rule of modus tollens....
mtod 197 Modus tollens deduction. ...
mtoi 198 Modus tollens inference. ...
mt2 199 A rule similar to modus to...
mt3 200 A rule similar to modus to...
peirce 201 Peirce's axiom. A non-int...
looinv 202 The Inversion Axiom of the...
bijust0 203 A self-implication (see ~ ...
bijust 204 Theorem used to justify th...
impbi 207 Property of the biconditio...
impbii 208 Infer an equivalence from ...
impbidd 209 Deduce an equivalence from...
impbid21d 210 Deduce an equivalence from...
impbid 211 Deduce an equivalence from...
dfbi1 212 Relate the biconditional c...
dfbi1ALT 213 Alternate proof of ~ dfbi1...
biimp 214 Property of the biconditio...
biimpi 215 Infer an implication from ...
sylbi 216 A mixed syllogism inferenc...
sylib 217 A mixed syllogism inferenc...
sylbb 218 A mixed syllogism inferenc...
biimpr 219 Property of the biconditio...
bicom1 220 Commutative law for the bi...
bicom 221 Commutative law for the bi...
bicomd 222 Commute two sides of a bic...
bicomi 223 Inference from commutative...
impbid1 224 Infer an equivalence from ...
impbid2 225 Infer an equivalence from ...
impcon4bid 226 A variation on ~ impbid wi...
biimpri 227 Infer a converse implicati...
biimpd 228 Deduce an implication from...
mpbi 229 An inference from a bicond...
mpbir 230 An inference from a bicond...
mpbid 231 A deduction from a bicondi...
mpbii 232 An inference from a nested...
sylibr 233 A mixed syllogism inferenc...
sylbir 234 A mixed syllogism inferenc...
sylbbr 235 A mixed syllogism inferenc...
sylbb1 236 A mixed syllogism inferenc...
sylbb2 237 A mixed syllogism inferenc...
sylibd 238 A syllogism deduction. (C...
sylbid 239 A syllogism deduction. (C...
mpbidi 240 A deduction from a bicondi...
biimtrid 241 A mixed syllogism inferenc...
biimtrrid 242 A mixed syllogism inferenc...
imbitrid 243 A mixed syllogism inferenc...
syl5ibcom 244 A mixed syllogism inferenc...
imbitrrid 245 A mixed syllogism inferenc...
syl5ibrcom 246 A mixed syllogism inferenc...
biimprd 247 Deduce a converse implicat...
biimpcd 248 Deduce a commuted implicat...
biimprcd 249 Deduce a converse commuted...
imbitrdi 250 A mixed syllogism inferenc...
imbitrrdi 251 A mixed syllogism inferenc...
biimtrdi 252 A mixed syllogism inferenc...
syl6bi 253 A mixed syllogism inferenc...
syl6bir 254 A mixed syllogism inferenc...
syl7bi 255 A mixed syllogism inferenc...
syl8ib 256 A syllogism rule of infere...
mpbird 257 A deduction from a bicondi...
mpbiri 258 An inference from a nested...
sylibrd 259 A syllogism deduction. (C...
sylbird 260 A syllogism deduction. (C...
biid 261 Principle of identity for ...
biidd 262 Principle of identity with...
pm5.1im 263 Two propositions are equiv...
2th 264 Two truths are equivalent....
2thd 265 Two truths are equivalent....
monothetic 266 Two self-implications (see...
ibi 267 Inference that converts a ...
ibir 268 Inference that converts a ...
ibd 269 Deduction that converts a ...
pm5.74 270 Distribution of implicatio...
pm5.74i 271 Distribution of implicatio...
pm5.74ri 272 Distribution of implicatio...
pm5.74d 273 Distribution of implicatio...
pm5.74rd 274 Distribution of implicatio...
bitri 275 An inference from transiti...
bitr2i 276 An inference from transiti...
bitr3i 277 An inference from transiti...
bitr4i 278 An inference from transiti...
bitrd 279 Deduction form of ~ bitri ...
bitr2d 280 Deduction form of ~ bitr2i...
bitr3d 281 Deduction form of ~ bitr3i...
bitr4d 282 Deduction form of ~ bitr4i...
bitrid 283 A syllogism inference from...
bitr2id 284 A syllogism inference from...
bitr3id 285 A syllogism inference from...
bitr3di 286 A syllogism inference from...
bitrdi 287 A syllogism inference from...
bitr2di 288 A syllogism inference from...
bitr4di 289 A syllogism inference from...
bitr4id 290 A syllogism inference from...
3imtr3i 291 A mixed syllogism inferenc...
3imtr4i 292 A mixed syllogism inferenc...
3imtr3d 293 More general version of ~ ...
3imtr4d 294 More general version of ~ ...
3imtr3g 295 More general version of ~ ...
3imtr4g 296 More general version of ~ ...
3bitri 297 A chained inference from t...
3bitrri 298 A chained inference from t...
3bitr2i 299 A chained inference from t...
3bitr2ri 300 A chained inference from t...
3bitr3i 301 A chained inference from t...
3bitr3ri 302 A chained inference from t...
3bitr4i 303 A chained inference from t...
3bitr4ri 304 A chained inference from t...
3bitrd 305 Deduction from transitivit...
3bitrrd 306 Deduction from transitivit...
3bitr2d 307 Deduction from transitivit...
3bitr2rd 308 Deduction from transitivit...
3bitr3d 309 Deduction from transitivit...
3bitr3rd 310 Deduction from transitivit...
3bitr4d 311 Deduction from transitivit...
3bitr4rd 312 Deduction from transitivit...
3bitr3g 313 More general version of ~ ...
3bitr4g 314 More general version of ~ ...
notnotb 315 Double negation. Theorem ...
con34b 316 A biconditional form of co...
con4bid 317 A contraposition deduction...
notbid 318 Deduction negating both si...
notbi 319 Contraposition. Theorem *...
notbii 320 Negate both sides of a log...
con4bii 321 A contraposition inference...
mtbi 322 An inference from a bicond...
mtbir 323 An inference from a bicond...
mtbid 324 A deduction from a bicondi...
mtbird 325 A deduction from a bicondi...
mtbii 326 An inference from a bicond...
mtbiri 327 An inference from a bicond...
sylnib 328 A mixed syllogism inferenc...
sylnibr 329 A mixed syllogism inferenc...
sylnbi 330 A mixed syllogism inferenc...
sylnbir 331 A mixed syllogism inferenc...
xchnxbi 332 Replacement of a subexpres...
xchnxbir 333 Replacement of a subexpres...
xchbinx 334 Replacement of a subexpres...
xchbinxr 335 Replacement of a subexpres...
imbi2i 336 Introduce an antecedent to...
bibi2i 337 Inference adding a bicondi...
bibi1i 338 Inference adding a bicondi...
bibi12i 339 The equivalence of two equ...
imbi2d 340 Deduction adding an antece...
imbi1d 341 Deduction adding a consequ...
bibi2d 342 Deduction adding a bicondi...
bibi1d 343 Deduction adding a bicondi...
imbi12d 344 Deduction joining two equi...
bibi12d 345 Deduction joining two equi...
imbi12 346 Closed form of ~ imbi12i ....
imbi1 347 Theorem *4.84 of [Whitehea...
imbi2 348 Theorem *4.85 of [Whitehea...
imbi1i 349 Introduce a consequent to ...
imbi12i 350 Join two logical equivalen...
bibi1 351 Theorem *4.86 of [Whitehea...
bitr3 352 Closed nested implication ...
con2bi 353 Contraposition. Theorem *...
con2bid 354 A contraposition deduction...
con1bid 355 A contraposition deduction...
con1bii 356 A contraposition inference...
con2bii 357 A contraposition inference...
con1b 358 Contraposition. Bidirecti...
con2b 359 Contraposition. Bidirecti...
biimt 360 A wff is equivalent to its...
pm5.5 361 Theorem *5.5 of [Whitehead...
a1bi 362 Inference introducing a th...
mt2bi 363 A false consequent falsifi...
mtt 364 Modus-tollens-like theorem...
imnot 365 If a proposition is false,...
pm5.501 366 Theorem *5.501 of [Whitehe...
ibib 367 Implication in terms of im...
ibibr 368 Implication in terms of im...
tbt 369 A wff is equivalent to its...
nbn2 370 The negation of a wff is e...
bibif 371 Transfer negation via an e...
nbn 372 The negation of a wff is e...
nbn3 373 Transfer falsehood via equ...
pm5.21im 374 Two propositions are equiv...
2false 375 Two falsehoods are equival...
2falsed 376 Two falsehoods are equival...
pm5.21ni 377 Two propositions implying ...
pm5.21nii 378 Eliminate an antecedent im...
pm5.21ndd 379 Eliminate an antecedent im...
bija 380 Combine antecedents into a...
pm5.18 381 Theorem *5.18 of [Whitehea...
xor3 382 Two ways to express "exclu...
nbbn 383 Move negation outside of b...
biass 384 Associative law for the bi...
biluk 385 Lukasiewicz's shortest axi...
pm5.19 386 Theorem *5.19 of [Whitehea...
bi2.04 387 Logical equivalence of com...
pm5.4 388 Antecedent absorption impl...
imdi 389 Distributive law for impli...
pm5.41 390 Theorem *5.41 of [Whitehea...
imbibi 391 The antecedent of one side...
pm4.8 392 Theorem *4.8 of [Whitehead...
pm4.81 393 A formula is equivalent to...
imim21b 394 Simplify an implication be...
pm4.63 397 Theorem *4.63 of [Whitehea...
pm4.67 398 Theorem *4.67 of [Whitehea...
imnan 399 Express an implication in ...
imnani 400 Infer an implication from ...
iman 401 Implication in terms of co...
pm3.24 402 Law of noncontradiction. ...
annim 403 Express a conjunction in t...
pm4.61 404 Theorem *4.61 of [Whitehea...
pm4.65 405 Theorem *4.65 of [Whitehea...
imp 406 Importation inference. (C...
impcom 407 Importation inference with...
con3dimp 408 Variant of ~ con3d with im...
mpnanrd 409 Eliminate the right side o...
impd 410 Importation deduction. (C...
impcomd 411 Importation deduction with...
ex 412 Exportation inference. (T...
expcom 413 Exportation inference with...
expdcom 414 Commuted form of ~ expd . ...
expd 415 Exportation deduction. (C...
expcomd 416 Deduction form of ~ expcom...
imp31 417 An importation inference. ...
imp32 418 An importation inference. ...
exp31 419 An exportation inference. ...
exp32 420 An exportation inference. ...
imp4b 421 An importation inference. ...
imp4a 422 An importation inference. ...
imp4c 423 An importation inference. ...
imp4d 424 An importation inference. ...
imp41 425 An importation inference. ...
imp42 426 An importation inference. ...
imp43 427 An importation inference. ...
imp44 428 An importation inference. ...
imp45 429 An importation inference. ...
exp4b 430 An exportation inference. ...
exp4a 431 An exportation inference. ...
exp4c 432 An exportation inference. ...
exp4d 433 An exportation inference. ...
exp41 434 An exportation inference. ...
exp42 435 An exportation inference. ...
exp43 436 An exportation inference. ...
exp44 437 An exportation inference. ...
exp45 438 An exportation inference. ...
imp5d 439 An importation inference. ...
imp5a 440 An importation inference. ...
imp5g 441 An importation inference. ...
imp55 442 An importation inference. ...
imp511 443 An importation inference. ...
exp5c 444 An exportation inference. ...
exp5j 445 An exportation inference. ...
exp5l 446 An exportation inference. ...
exp53 447 An exportation inference. ...
pm3.3 448 Theorem *3.3 (Exp) of [Whi...
pm3.31 449 Theorem *3.31 (Imp) of [Wh...
impexp 450 Import-export theorem. Pa...
impancom 451 Mixed importation/commutat...
expdimp 452 A deduction version of exp...
expimpd 453 Exportation followed by a ...
impr 454 Import a wff into a right ...
impl 455 Export a wff from a left c...
expr 456 Export a wff from a right ...
expl 457 Export a wff from a left c...
ancoms 458 Inference commuting conjun...
pm3.22 459 Theorem *3.22 of [Whitehea...
ancom 460 Commutative law for conjun...
ancomd 461 Commutation of conjuncts i...
biancomi 462 Commuting conjunction in a...
biancomd 463 Commuting conjunction in a...
ancomst 464 Closed form of ~ ancoms . ...
ancomsd 465 Deduction commuting conjun...
anasss 466 Associative law for conjun...
anassrs 467 Associative law for conjun...
anass 468 Associative law for conjun...
pm3.2 469 Join antecedents with conj...
pm3.2i 470 Infer conjunction of premi...
pm3.21 471 Join antecedents with conj...
pm3.43i 472 Nested conjunction of ante...
pm3.43 473 Theorem *3.43 (Comp) of [W...
dfbi2 474 A theorem similar to the s...
dfbi 475 Definition ~ df-bi rewritt...
biimpa 476 Importation inference from...
biimpar 477 Importation inference from...
biimpac 478 Importation inference from...
biimparc 479 Importation inference from...
adantr 480 Inference adding a conjunc...
adantl 481 Inference adding a conjunc...
simpl 482 Elimination of a conjunct....
simpli 483 Inference eliminating a co...
simpr 484 Elimination of a conjunct....
simpri 485 Inference eliminating a co...
intnan 486 Introduction of conjunct i...
intnanr 487 Introduction of conjunct i...
intnand 488 Introduction of conjunct i...
intnanrd 489 Introduction of conjunct i...
adantld 490 Deduction adding a conjunc...
adantrd 491 Deduction adding a conjunc...
pm3.41 492 Theorem *3.41 of [Whitehea...
pm3.42 493 Theorem *3.42 of [Whitehea...
simpld 494 Deduction eliminating a co...
simprd 495 Deduction eliminating a co...
simprbi 496 Deduction eliminating a co...
simplbi 497 Deduction eliminating a co...
simprbda 498 Deduction eliminating a co...
simplbda 499 Deduction eliminating a co...
simplbi2 500 Deduction eliminating a co...
simplbi2comt 501 Closed form of ~ simplbi2c...
simplbi2com 502 A deduction eliminating a ...
simpl2im 503 Implication from an elimin...
simplbiim 504 Implication from an elimin...
impel 505 An inference for implicati...
mpan9 506 Modus ponens conjoining di...
sylan9 507 Nested syllogism inference...
sylan9r 508 Nested syllogism inference...
sylan9bb 509 Nested syllogism inference...
sylan9bbr 510 Nested syllogism inference...
jca 511 Deduce conjunction of the ...
jcad 512 Deduction conjoining the c...
jca2 513 Inference conjoining the c...
jca31 514 Join three consequents. (...
jca32 515 Join three consequents. (...
jcai 516 Deduction replacing implic...
jcab 517 Distributive law for impli...
pm4.76 518 Theorem *4.76 of [Whitehea...
jctil 519 Inference conjoining a the...
jctir 520 Inference conjoining a the...
jccir 521 Inference conjoining a con...
jccil 522 Inference conjoining a con...
jctl 523 Inference conjoining a the...
jctr 524 Inference conjoining a the...
jctild 525 Deduction conjoining a the...
jctird 526 Deduction conjoining a the...
iba 527 Introduction of antecedent...
ibar 528 Introduction of antecedent...
biantru 529 A wff is equivalent to its...
biantrur 530 A wff is equivalent to its...
biantrud 531 A wff is equivalent to its...
biantrurd 532 A wff is equivalent to its...
bianfi 533 A wff conjoined with false...
bianfd 534 A wff conjoined with false...
baib 535 Move conjunction outside o...
baibr 536 Move conjunction outside o...
rbaibr 537 Move conjunction outside o...
rbaib 538 Move conjunction outside o...
baibd 539 Move conjunction outside o...
rbaibd 540 Move conjunction outside o...
bianabs 541 Absorb a hypothesis into t...
pm5.44 542 Theorem *5.44 of [Whitehea...
pm5.42 543 Theorem *5.42 of [Whitehea...
ancl 544 Conjoin antecedent to left...
anclb 545 Conjoin antecedent to left...
ancr 546 Conjoin antecedent to righ...
ancrb 547 Conjoin antecedent to righ...
ancli 548 Deduction conjoining antec...
ancri 549 Deduction conjoining antec...
ancld 550 Deduction conjoining antec...
ancrd 551 Deduction conjoining antec...
impac 552 Importation with conjuncti...
anc2l 553 Conjoin antecedent to left...
anc2r 554 Conjoin antecedent to righ...
anc2li 555 Deduction conjoining antec...
anc2ri 556 Deduction conjoining antec...
pm4.71 557 Implication in terms of bi...
pm4.71r 558 Implication in terms of bi...
pm4.71i 559 Inference converting an im...
pm4.71ri 560 Inference converting an im...
pm4.71d 561 Deduction converting an im...
pm4.71rd 562 Deduction converting an im...
pm4.24 563 Theorem *4.24 of [Whitehea...
anidm 564 Idempotent law for conjunc...
anidmdbi 565 Conjunction idempotence wi...
anidms 566 Inference from idempotent ...
imdistan 567 Distribution of implicatio...
imdistani 568 Distribution of implicatio...
imdistanri 569 Distribution of implicatio...
imdistand 570 Distribution of implicatio...
imdistanda 571 Distribution of implicatio...
pm5.3 572 Theorem *5.3 of [Whitehead...
pm5.32 573 Distribution of implicatio...
pm5.32i 574 Distribution of implicatio...
pm5.32ri 575 Distribution of implicatio...
pm5.32d 576 Distribution of implicatio...
pm5.32rd 577 Distribution of implicatio...
pm5.32da 578 Distribution of implicatio...
sylan 579 A syllogism inference. (C...
sylanb 580 A syllogism inference. (C...
sylanbr 581 A syllogism inference. (C...
sylanbrc 582 Syllogism inference. (Con...
syl2anc 583 Syllogism inference combin...
syl2anc2 584 Double syllogism inference...
sylancl 585 Syllogism inference combin...
sylancr 586 Syllogism inference combin...
sylancom 587 Syllogism inference with c...
sylanblc 588 Syllogism inference combin...
sylanblrc 589 Syllogism inference combin...
syldan 590 A syllogism deduction with...
sylbida 591 A syllogism deduction. (C...
sylan2 592 A syllogism inference. (C...
sylan2b 593 A syllogism inference. (C...
sylan2br 594 A syllogism inference. (C...
syl2an 595 A double syllogism inferen...
syl2anr 596 A double syllogism inferen...
syl2anb 597 A double syllogism inferen...
syl2anbr 598 A double syllogism inferen...
sylancb 599 A syllogism inference comb...
sylancbr 600 A syllogism inference comb...
syldanl 601 A syllogism deduction with...
syland 602 A syllogism deduction. (C...
sylani 603 A syllogism inference. (C...
sylan2d 604 A syllogism deduction. (C...
sylan2i 605 A syllogism inference. (C...
syl2ani 606 A syllogism inference. (C...
syl2and 607 A syllogism deduction. (C...
anim12d 608 Conjoin antecedents and co...
anim12d1 609 Variant of ~ anim12d where...
anim1d 610 Add a conjunct to right of...
anim2d 611 Add a conjunct to left of ...
anim12i 612 Conjoin antecedents and co...
anim12ci 613 Variant of ~ anim12i with ...
anim1i 614 Introduce conjunct to both...
anim1ci 615 Introduce conjunct to both...
anim2i 616 Introduce conjunct to both...
anim12ii 617 Conjoin antecedents and co...
anim12dan 618 Conjoin antecedents and co...
im2anan9 619 Deduction joining nested i...
im2anan9r 620 Deduction joining nested i...
pm3.45 621 Theorem *3.45 (Fact) of [W...
anbi2i 622 Introduce a left conjunct ...
anbi1i 623 Introduce a right conjunct...
anbi2ci 624 Variant of ~ anbi2i with c...
anbi1ci 625 Variant of ~ anbi1i with c...
anbi12i 626 Conjoin both sides of two ...
anbi12ci 627 Variant of ~ anbi12i with ...
anbi2d 628 Deduction adding a left co...
anbi1d 629 Deduction adding a right c...
anbi12d 630 Deduction joining two equi...
anbi1 631 Introduce a right conjunct...
anbi2 632 Introduce a left conjunct ...
anbi1cd 633 Introduce a proposition as...
an2anr 634 Double commutation in conj...
pm4.38 635 Theorem *4.38 of [Whitehea...
bi2anan9 636 Deduction joining two equi...
bi2anan9r 637 Deduction joining two equi...
bi2bian9 638 Deduction joining two bico...
bianass 639 An inference to merge two ...
bianassc 640 An inference to merge two ...
an21 641 Swap two conjuncts. (Cont...
an12 642 Swap two conjuncts. Note ...
an32 643 A rearrangement of conjunc...
an13 644 A rearrangement of conjunc...
an31 645 A rearrangement of conjunc...
an12s 646 Swap two conjuncts in ante...
ancom2s 647 Inference commuting a nest...
an13s 648 Swap two conjuncts in ante...
an32s 649 Swap two conjuncts in ante...
ancom1s 650 Inference commuting a nest...
an31s 651 Swap two conjuncts in ante...
anass1rs 652 Commutative-associative la...
an4 653 Rearrangement of 4 conjunc...
an42 654 Rearrangement of 4 conjunc...
an43 655 Rearrangement of 4 conjunc...
an3 656 A rearrangement of conjunc...
an4s 657 Inference rearranging 4 co...
an42s 658 Inference rearranging 4 co...
anabs1 659 Absorption into embedded c...
anabs5 660 Absorption into embedded c...
anabs7 661 Absorption into embedded c...
anabsan 662 Absorption of antecedent w...
anabss1 663 Absorption of antecedent i...
anabss4 664 Absorption of antecedent i...
anabss5 665 Absorption of antecedent i...
anabsi5 666 Absorption of antecedent i...
anabsi6 667 Absorption of antecedent i...
anabsi7 668 Absorption of antecedent i...
anabsi8 669 Absorption of antecedent i...
anabss7 670 Absorption of antecedent i...
anabsan2 671 Absorption of antecedent w...
anabss3 672 Absorption of antecedent i...
anandi 673 Distribution of conjunctio...
anandir 674 Distribution of conjunctio...
anandis 675 Inference that undistribut...
anandirs 676 Inference that undistribut...
sylanl1 677 A syllogism inference. (C...
sylanl2 678 A syllogism inference. (C...
sylanr1 679 A syllogism inference. (C...
sylanr2 680 A syllogism inference. (C...
syl6an 681 A syllogism deduction comb...
syl2an2r 682 ~ syl2anr with antecedents...
syl2an2 683 ~ syl2an with antecedents ...
mpdan 684 An inference based on modu...
mpancom 685 An inference based on modu...
mpidan 686 A deduction which "stacks"...
mpan 687 An inference based on modu...
mpan2 688 An inference based on modu...
mp2an 689 An inference based on modu...
mp4an 690 An inference based on modu...
mpan2d 691 A deduction based on modus...
mpand 692 A deduction based on modus...
mpani 693 An inference based on modu...
mpan2i 694 An inference based on modu...
mp2ani 695 An inference based on modu...
mp2and 696 A deduction based on modus...
mpanl1 697 An inference based on modu...
mpanl2 698 An inference based on modu...
mpanl12 699 An inference based on modu...
mpanr1 700 An inference based on modu...
mpanr2 701 An inference based on modu...
mpanr12 702 An inference based on modu...
mpanlr1 703 An inference based on modu...
mpbirand 704 Detach truth from conjunct...
mpbiran2d 705 Detach truth from conjunct...
mpbiran 706 Detach truth from conjunct...
mpbiran2 707 Detach truth from conjunct...
mpbir2an 708 Detach a conjunction of tr...
mpbi2and 709 Detach a conjunction of tr...
mpbir2and 710 Detach a conjunction of tr...
adantll 711 Deduction adding a conjunc...
adantlr 712 Deduction adding a conjunc...
adantrl 713 Deduction adding a conjunc...
adantrr 714 Deduction adding a conjunc...
adantlll 715 Deduction adding a conjunc...
adantllr 716 Deduction adding a conjunc...
adantlrl 717 Deduction adding a conjunc...
adantlrr 718 Deduction adding a conjunc...
adantrll 719 Deduction adding a conjunc...
adantrlr 720 Deduction adding a conjunc...
adantrrl 721 Deduction adding a conjunc...
adantrrr 722 Deduction adding a conjunc...
ad2antrr 723 Deduction adding two conju...
ad2antlr 724 Deduction adding two conju...
ad2antrl 725 Deduction adding two conju...
ad2antll 726 Deduction adding conjuncts...
ad3antrrr 727 Deduction adding three con...
ad3antlr 728 Deduction adding three con...
ad4antr 729 Deduction adding 4 conjunc...
ad4antlr 730 Deduction adding 4 conjunc...
ad5antr 731 Deduction adding 5 conjunc...
ad5antlr 732 Deduction adding 5 conjunc...
ad6antr 733 Deduction adding 6 conjunc...
ad6antlr 734 Deduction adding 6 conjunc...
ad7antr 735 Deduction adding 7 conjunc...
ad7antlr 736 Deduction adding 7 conjunc...
ad8antr 737 Deduction adding 8 conjunc...
ad8antlr 738 Deduction adding 8 conjunc...
ad9antr 739 Deduction adding 9 conjunc...
ad9antlr 740 Deduction adding 9 conjunc...
ad10antr 741 Deduction adding 10 conjun...
ad10antlr 742 Deduction adding 10 conjun...
ad2ant2l 743 Deduction adding two conju...
ad2ant2r 744 Deduction adding two conju...
ad2ant2lr 745 Deduction adding two conju...
ad2ant2rl 746 Deduction adding two conju...
adantl3r 747 Deduction adding 1 conjunc...
ad4ant13 748 Deduction adding conjuncts...
ad4ant14 749 Deduction adding conjuncts...
ad4ant23 750 Deduction adding conjuncts...
ad4ant24 751 Deduction adding conjuncts...
adantl4r 752 Deduction adding 1 conjunc...
ad5ant12 753 Deduction adding conjuncts...
ad5ant13 754 Deduction adding conjuncts...
ad5ant14 755 Deduction adding conjuncts...
ad5ant15 756 Deduction adding conjuncts...
ad5ant23 757 Deduction adding conjuncts...
ad5ant24 758 Deduction adding conjuncts...
ad5ant25 759 Deduction adding conjuncts...
adantl5r 760 Deduction adding 1 conjunc...
adantl6r 761 Deduction adding 1 conjunc...
pm3.33 762 Theorem *3.33 (Syll) of [W...
pm3.34 763 Theorem *3.34 (Syll) of [W...
simpll 764 Simplification of a conjun...
simplld 765 Deduction form of ~ simpll...
simplr 766 Simplification of a conjun...
simplrd 767 Deduction eliminating a do...
simprl 768 Simplification of a conjun...
simprld 769 Deduction eliminating a do...
simprr 770 Simplification of a conjun...
simprrd 771 Deduction form of ~ simprr...
simplll 772 Simplification of a conjun...
simpllr 773 Simplification of a conjun...
simplrl 774 Simplification of a conjun...
simplrr 775 Simplification of a conjun...
simprll 776 Simplification of a conjun...
simprlr 777 Simplification of a conjun...
simprrl 778 Simplification of a conjun...
simprrr 779 Simplification of a conjun...
simp-4l 780 Simplification of a conjun...
simp-4r 781 Simplification of a conjun...
simp-5l 782 Simplification of a conjun...
simp-5r 783 Simplification of a conjun...
simp-6l 784 Simplification of a conjun...
simp-6r 785 Simplification of a conjun...
simp-7l 786 Simplification of a conjun...
simp-7r 787 Simplification of a conjun...
simp-8l 788 Simplification of a conjun...
simp-8r 789 Simplification of a conjun...
simp-9l 790 Simplification of a conjun...
simp-9r 791 Simplification of a conjun...
simp-10l 792 Simplification of a conjun...
simp-10r 793 Simplification of a conjun...
simp-11l 794 Simplification of a conjun...
simp-11r 795 Simplification of a conjun...
pm2.01da 796 Deduction based on reducti...
pm2.18da 797 Deduction based on reducti...
impbida 798 Deduce an equivalence from...
pm5.21nd 799 Eliminate an antecedent im...
pm3.35 800 Conjunctive detachment. T...
pm5.74da 801 Distribution of implicatio...
bitr 802 Theorem *4.22 of [Whitehea...
biantr 803 A transitive law of equiva...
pm4.14 804 Theorem *4.14 of [Whitehea...
pm3.37 805 Theorem *3.37 (Transp) of ...
anim12 806 Conjoin antecedents and co...
pm3.4 807 Conjunction implies implic...
exbiri 808 Inference form of ~ exbir ...
pm2.61ian 809 Elimination of an antecede...
pm2.61dan 810 Elimination of an antecede...
pm2.61ddan 811 Elimination of two anteced...
pm2.61dda 812 Elimination of two anteced...
mtand 813 A modus tollens deduction....
pm2.65da 814 Deduction for proof by con...
condan 815 Proof by contradiction. (...
biadan 816 An implication is equivale...
biadani 817 Inference associated with ...
biadaniALT 818 Alternate proof of ~ biada...
biadanii 819 Inference associated with ...
biadanid 820 Deduction associated with ...
pm5.1 821 Two propositions are equiv...
pm5.21 822 Two propositions are equiv...
pm5.35 823 Theorem *5.35 of [Whitehea...
abai 824 Introduce one conjunct as ...
pm4.45im 825 Conjunction with implicati...
impimprbi 826 An implication and its rev...
nan 827 Theorem to move a conjunct...
pm5.31 828 Theorem *5.31 of [Whitehea...
pm5.31r 829 Variant of ~ pm5.31 . (Co...
pm4.15 830 Theorem *4.15 of [Whitehea...
pm5.36 831 Theorem *5.36 of [Whitehea...
annotanannot 832 A conjunction with a negat...
pm5.33 833 Theorem *5.33 of [Whitehea...
syl12anc 834 Syllogism combined with co...
syl21anc 835 Syllogism combined with co...
syl22anc 836 Syllogism combined with co...
syl1111anc 837 Four-hypothesis eliminatio...
syldbl2 838 Stacked hypotheseis implie...
mpsyl4anc 839 An elimination deduction. ...
pm4.87 840 Theorem *4.87 of [Whitehea...
bimsc1 841 Removal of conjunct from o...
a2and 842 Deduction distributing a c...
animpimp2impd 843 Deduction deriving nested ...
pm4.64 846 Theorem *4.64 of [Whitehea...
pm4.66 847 Theorem *4.66 of [Whitehea...
pm2.53 848 Theorem *2.53 of [Whitehea...
pm2.54 849 Theorem *2.54 of [Whitehea...
imor 850 Implication in terms of di...
imori 851 Infer disjunction from imp...
imorri 852 Infer implication from dis...
pm4.62 853 Theorem *4.62 of [Whitehea...
jaoi 854 Inference disjoining the a...
jao1i 855 Add a disjunct in the ante...
jaod 856 Deduction disjoining the a...
mpjaod 857 Eliminate a disjunction in...
ori 858 Infer implication from dis...
orri 859 Infer disjunction from imp...
orrd 860 Deduce disjunction from im...
ord 861 Deduce implication from di...
orci 862 Deduction introducing a di...
olci 863 Deduction introducing a di...
orc 864 Introduction of a disjunct...
olc 865 Introduction of a disjunct...
pm1.4 866 Axiom *1.4 of [WhiteheadRu...
orcom 867 Commutative law for disjun...
orcomd 868 Commutation of disjuncts i...
orcoms 869 Commutation of disjuncts i...
orcd 870 Deduction introducing a di...
olcd 871 Deduction introducing a di...
orcs 872 Deduction eliminating disj...
olcs 873 Deduction eliminating disj...
olcnd 874 A lemma for Conjunctive No...
orcnd 875 A lemma for Conjunctive No...
mtord 876 A modus tollens deduction ...
pm3.2ni 877 Infer negated disjunction ...
pm2.45 878 Theorem *2.45 of [Whitehea...
pm2.46 879 Theorem *2.46 of [Whitehea...
pm2.47 880 Theorem *2.47 of [Whitehea...
pm2.48 881 Theorem *2.48 of [Whitehea...
pm2.49 882 Theorem *2.49 of [Whitehea...
norbi 883 If neither of two proposit...
nbior 884 If two propositions are no...
orel1 885 Elimination of disjunction...
pm2.25 886 Theorem *2.25 of [Whitehea...
orel2 887 Elimination of disjunction...
pm2.67-2 888 Slight generalization of T...
pm2.67 889 Theorem *2.67 of [Whitehea...
curryax 890 A non-intuitionistic posit...
exmid 891 Law of excluded middle, al...
exmidd 892 Law of excluded middle in ...
pm2.1 893 Theorem *2.1 of [Whitehead...
pm2.13 894 Theorem *2.13 of [Whitehea...
pm2.621 895 Theorem *2.621 of [Whitehe...
pm2.62 896 Theorem *2.62 of [Whitehea...
pm2.68 897 Theorem *2.68 of [Whitehea...
dfor2 898 Logical 'or' expressed in ...
pm2.07 899 Theorem *2.07 of [Whitehea...
pm1.2 900 Axiom *1.2 of [WhiteheadRu...
oridm 901 Idempotent law for disjunc...
pm4.25 902 Theorem *4.25 of [Whitehea...
pm2.4 903 Theorem *2.4 of [Whitehead...
pm2.41 904 Theorem *2.41 of [Whitehea...
orim12i 905 Disjoin antecedents and co...
orim1i 906 Introduce disjunct to both...
orim2i 907 Introduce disjunct to both...
orim12dALT 908 Alternate proof of ~ orim1...
orbi2i 909 Inference adding a left di...
orbi1i 910 Inference adding a right d...
orbi12i 911 Infer the disjunction of t...
orbi2d 912 Deduction adding a left di...
orbi1d 913 Deduction adding a right d...
orbi1 914 Theorem *4.37 of [Whitehea...
orbi12d 915 Deduction joining two equi...
pm1.5 916 Axiom *1.5 (Assoc) of [Whi...
or12 917 Swap two disjuncts. (Cont...
orass 918 Associative law for disjun...
pm2.31 919 Theorem *2.31 of [Whitehea...
pm2.32 920 Theorem *2.32 of [Whitehea...
pm2.3 921 Theorem *2.3 of [Whitehead...
or32 922 A rearrangement of disjunc...
or4 923 Rearrangement of 4 disjunc...
or42 924 Rearrangement of 4 disjunc...
orordi 925 Distribution of disjunctio...
orordir 926 Distribution of disjunctio...
orimdi 927 Disjunction distributes ov...
pm2.76 928 Theorem *2.76 of [Whitehea...
pm2.85 929 Theorem *2.85 of [Whitehea...
pm2.75 930 Theorem *2.75 of [Whitehea...
pm4.78 931 Implication distributes ov...
biort 932 A disjunction with a true ...
biorf 933 A wff is equivalent to its...
biortn 934 A wff is equivalent to its...
biorfi 935 A wff is equivalent to its...
pm2.26 936 Theorem *2.26 of [Whitehea...
pm2.63 937 Theorem *2.63 of [Whitehea...
pm2.64 938 Theorem *2.64 of [Whitehea...
pm2.42 939 Theorem *2.42 of [Whitehea...
pm5.11g 940 A general instance of Theo...
pm5.11 941 Theorem *5.11 of [Whitehea...
pm5.12 942 Theorem *5.12 of [Whitehea...
pm5.14 943 Theorem *5.14 of [Whitehea...
pm5.13 944 Theorem *5.13 of [Whitehea...
pm5.55 945 Theorem *5.55 of [Whitehea...
pm4.72 946 Implication in terms of bi...
imimorb 947 Simplify an implication be...
oibabs 948 Absorption of disjunction ...
orbidi 949 Disjunction distributes ov...
pm5.7 950 Disjunction distributes ov...
jaao 951 Inference conjoining and d...
jaoa 952 Inference disjoining and c...
jaoian 953 Inference disjoining the a...
jaodan 954 Deduction disjoining the a...
mpjaodan 955 Eliminate a disjunction in...
pm3.44 956 Theorem *3.44 of [Whitehea...
jao 957 Disjunction of antecedents...
jaob 958 Disjunction of antecedents...
pm4.77 959 Theorem *4.77 of [Whitehea...
pm3.48 960 Theorem *3.48 of [Whitehea...
orim12d 961 Disjoin antecedents and co...
orim1d 962 Disjoin antecedents and co...
orim2d 963 Disjoin antecedents and co...
orim2 964 Axiom *1.6 (Sum) of [White...
pm2.38 965 Theorem *2.38 of [Whitehea...
pm2.36 966 Theorem *2.36 of [Whitehea...
pm2.37 967 Theorem *2.37 of [Whitehea...
pm2.81 968 Theorem *2.81 of [Whitehea...
pm2.8 969 Theorem *2.8 of [Whitehead...
pm2.73 970 Theorem *2.73 of [Whitehea...
pm2.74 971 Theorem *2.74 of [Whitehea...
pm2.82 972 Theorem *2.82 of [Whitehea...
pm4.39 973 Theorem *4.39 of [Whitehea...
animorl 974 Conjunction implies disjun...
animorr 975 Conjunction implies disjun...
animorlr 976 Conjunction implies disjun...
animorrl 977 Conjunction implies disjun...
ianor 978 Negated conjunction in ter...
anor 979 Conjunction in terms of di...
ioran 980 Negated disjunction in ter...
pm4.52 981 Theorem *4.52 of [Whitehea...
pm4.53 982 Theorem *4.53 of [Whitehea...
pm4.54 983 Theorem *4.54 of [Whitehea...
pm4.55 984 Theorem *4.55 of [Whitehea...
pm4.56 985 Theorem *4.56 of [Whitehea...
oran 986 Disjunction in terms of co...
pm4.57 987 Theorem *4.57 of [Whitehea...
pm3.1 988 Theorem *3.1 of [Whitehead...
pm3.11 989 Theorem *3.11 of [Whitehea...
pm3.12 990 Theorem *3.12 of [Whitehea...
pm3.13 991 Theorem *3.13 of [Whitehea...
pm3.14 992 Theorem *3.14 of [Whitehea...
pm4.44 993 Theorem *4.44 of [Whitehea...
pm4.45 994 Theorem *4.45 of [Whitehea...
orabs 995 Absorption of redundant in...
oranabs 996 Absorb a disjunct into a c...
pm5.61 997 Theorem *5.61 of [Whitehea...
pm5.6 998 Conjunction in antecedent ...
orcanai 999 Change disjunction in cons...
pm4.79 1000 Theorem *4.79 of [Whitehea...
pm5.53 1001 Theorem *5.53 of [Whitehea...
ordi 1002 Distributive law for disju...
ordir 1003 Distributive law for disju...
andi 1004 Distributive law for conju...
andir 1005 Distributive law for conju...
orddi 1006 Double distributive law fo...
anddi 1007 Double distributive law fo...
pm5.17 1008 Theorem *5.17 of [Whitehea...
pm5.15 1009 Theorem *5.15 of [Whitehea...
pm5.16 1010 Theorem *5.16 of [Whitehea...
xor 1011 Two ways to express exclus...
nbi2 1012 Two ways to express "exclu...
xordi 1013 Conjunction distributes ov...
pm5.54 1014 Theorem *5.54 of [Whitehea...
pm5.62 1015 Theorem *5.62 of [Whitehea...
pm5.63 1016 Theorem *5.63 of [Whitehea...
niabn 1017 Miscellaneous inference re...
ninba 1018 Miscellaneous inference re...
pm4.43 1019 Theorem *4.43 of [Whitehea...
pm4.82 1020 Theorem *4.82 of [Whitehea...
pm4.83 1021 Theorem *4.83 of [Whitehea...
pclem6 1022 Negation inferred from emb...
bigolden 1023 Dijkstra-Scholten's Golden...
pm5.71 1024 Theorem *5.71 of [Whitehea...
pm5.75 1025 Theorem *5.75 of [Whitehea...
ecase2d 1026 Deduction for elimination ...
ecase2dOLD 1027 Obsolete version of ~ ecas...
ecase3 1028 Inference for elimination ...
ecase 1029 Inference for elimination ...
ecase3d 1030 Deduction for elimination ...
ecased 1031 Deduction for elimination ...
ecase3ad 1032 Deduction for elimination ...
ecase3adOLD 1033 Obsolete version of ~ ecas...
ccase 1034 Inference for combining ca...
ccased 1035 Deduction for combining ca...
ccase2 1036 Inference for combining ca...
4cases 1037 Inference eliminating two ...
4casesdan 1038 Deduction eliminating two ...
cases 1039 Case disjunction according...
dedlem0a 1040 Lemma for an alternate ver...
dedlem0b 1041 Lemma for an alternate ver...
dedlema 1042 Lemma for weak deduction t...
dedlemb 1043 Lemma for weak deduction t...
cases2 1044 Case disjunction according...
cases2ALT 1045 Alternate proof of ~ cases...
dfbi3 1046 An alternate definition of...
pm5.24 1047 Theorem *5.24 of [Whitehea...
4exmid 1048 The disjunction of the fou...
consensus 1049 The consensus theorem. Th...
pm4.42 1050 Theorem *4.42 of [Whitehea...
prlem1 1051 A specialized lemma for se...
prlem2 1052 A specialized lemma for se...
oplem1 1053 A specialized lemma for se...
dn1 1054 A single axiom for Boolean...
bianir 1055 A closed form of ~ mpbir ,...
jaoi2 1056 Inference removing a negat...
jaoi3 1057 Inference separating a dis...
ornld 1058 Selecting one statement fr...
dfifp2 1061 Alternate definition of th...
dfifp3 1062 Alternate definition of th...
dfifp4 1063 Alternate definition of th...
dfifp5 1064 Alternate definition of th...
dfifp6 1065 Alternate definition of th...
dfifp7 1066 Alternate definition of th...
ifpdfbi 1067 Define the biconditional a...
anifp 1068 The conditional operator i...
ifpor 1069 The conditional operator i...
ifpn 1070 Conditional operator for t...
ifpnOLD 1071 Obsolete version of ~ ifpn...
ifptru 1072 Value of the conditional o...
ifpfal 1073 Value of the conditional o...
ifpid 1074 Value of the conditional o...
casesifp 1075 Version of ~ cases express...
ifpbi123d 1076 Equivalence deduction for ...
ifpbi23d 1077 Equivalence deduction for ...
ifpimpda 1078 Separation of the values o...
1fpid3 1079 The value of the condition...
elimh 1080 Hypothesis builder for the...
dedt 1081 The weak deduction theorem...
con3ALT 1082 Proof of ~ con3 from its a...
3orass 1087 Associative law for triple...
3orel1 1088 Partial elimination of a t...
3orrot 1089 Rotation law for triple di...
3orcoma 1090 Commutation law for triple...
3orcomb 1091 Commutation law for triple...
3anass 1092 Associative law for triple...
3anan12 1093 Convert triple conjunction...
3anan32 1094 Convert triple conjunction...
3ancoma 1095 Commutation law for triple...
3ancomb 1096 Commutation law for triple...
3anrot 1097 Rotation law for triple co...
3anrev 1098 Reversal law for triple co...
anandi3 1099 Distribution of triple con...
anandi3r 1100 Distribution of triple con...
3anidm 1101 Idempotent law for conjunc...
3an4anass 1102 Associative law for four c...
3ioran 1103 Negated triple disjunction...
3ianor 1104 Negated triple conjunction...
3anor 1105 Triple conjunction express...
3oran 1106 Triple disjunction in term...
3impa 1107 Importation from double to...
3imp 1108 Importation inference. (C...
3imp31 1109 The importation inference ...
3imp231 1110 Importation inference. (C...
3imp21 1111 The importation inference ...
3impb 1112 Importation from double to...
3impib 1113 Importation to triple conj...
3impia 1114 Importation to triple conj...
3expa 1115 Exportation from triple to...
3exp 1116 Exportation inference. (C...
3expb 1117 Exportation from triple to...
3expia 1118 Exportation from triple co...
3expib 1119 Exportation from triple co...
3com12 1120 Commutation in antecedent....
3com13 1121 Commutation in antecedent....
3comr 1122 Commutation in antecedent....
3com23 1123 Commutation in antecedent....
3coml 1124 Commutation in antecedent....
3jca 1125 Join consequents with conj...
3jcad 1126 Deduction conjoining the c...
3adant1 1127 Deduction adding a conjunc...
3adant2 1128 Deduction adding a conjunc...
3adant3 1129 Deduction adding a conjunc...
3ad2ant1 1130 Deduction adding conjuncts...
3ad2ant2 1131 Deduction adding conjuncts...
3ad2ant3 1132 Deduction adding conjuncts...
simp1 1133 Simplification of triple c...
simp2 1134 Simplification of triple c...
simp3 1135 Simplification of triple c...
simp1i 1136 Infer a conjunct from a tr...
simp2i 1137 Infer a conjunct from a tr...
simp3i 1138 Infer a conjunct from a tr...
simp1d 1139 Deduce a conjunct from a t...
simp2d 1140 Deduce a conjunct from a t...
simp3d 1141 Deduce a conjunct from a t...
simp1bi 1142 Deduce a conjunct from a t...
simp2bi 1143 Deduce a conjunct from a t...
simp3bi 1144 Deduce a conjunct from a t...
3simpa 1145 Simplification of triple c...
3simpb 1146 Simplification of triple c...
3simpc 1147 Simplification of triple c...
3anim123i 1148 Join antecedents and conse...
3anim1i 1149 Add two conjuncts to antec...
3anim2i 1150 Add two conjuncts to antec...
3anim3i 1151 Add two conjuncts to antec...
3anbi123i 1152 Join 3 biconditionals with...
3orbi123i 1153 Join 3 biconditionals with...
3anbi1i 1154 Inference adding two conju...
3anbi2i 1155 Inference adding two conju...
3anbi3i 1156 Inference adding two conju...
syl3an 1157 A triple syllogism inferen...
syl3anb 1158 A triple syllogism inferen...
syl3anbr 1159 A triple syllogism inferen...
syl3an1 1160 A syllogism inference. (C...
syl3an2 1161 A syllogism inference. (C...
syl3an3 1162 A syllogism inference. (C...
3adantl1 1163 Deduction adding a conjunc...
3adantl2 1164 Deduction adding a conjunc...
3adantl3 1165 Deduction adding a conjunc...
3adantr1 1166 Deduction adding a conjunc...
3adantr2 1167 Deduction adding a conjunc...
3adantr3 1168 Deduction adding a conjunc...
ad4ant123 1169 Deduction adding conjuncts...
ad4ant124 1170 Deduction adding conjuncts...
ad4ant134 1171 Deduction adding conjuncts...
ad4ant234 1172 Deduction adding conjuncts...
3adant1l 1173 Deduction adding a conjunc...
3adant1r 1174 Deduction adding a conjunc...
3adant2l 1175 Deduction adding a conjunc...
3adant2r 1176 Deduction adding a conjunc...
3adant3l 1177 Deduction adding a conjunc...
3adant3r 1178 Deduction adding a conjunc...
3adant3r1 1179 Deduction adding a conjunc...
3adant3r2 1180 Deduction adding a conjunc...
3adant3r3 1181 Deduction adding a conjunc...
3ad2antl1 1182 Deduction adding conjuncts...
3ad2antl2 1183 Deduction adding conjuncts...
3ad2antl3 1184 Deduction adding conjuncts...
3ad2antr1 1185 Deduction adding conjuncts...
3ad2antr2 1186 Deduction adding conjuncts...
3ad2antr3 1187 Deduction adding conjuncts...
simpl1 1188 Simplification of conjunct...
simpl2 1189 Simplification of conjunct...
simpl3 1190 Simplification of conjunct...
simpr1 1191 Simplification of conjunct...
simpr2 1192 Simplification of conjunct...
simpr3 1193 Simplification of conjunct...
simp1l 1194 Simplification of triple c...
simp1r 1195 Simplification of triple c...
simp2l 1196 Simplification of triple c...
simp2r 1197 Simplification of triple c...
simp3l 1198 Simplification of triple c...
simp3r 1199 Simplification of triple c...
simp11 1200 Simplification of doubly t...
simp12 1201 Simplification of doubly t...
simp13 1202 Simplification of doubly t...
simp21 1203 Simplification of doubly t...
simp22 1204 Simplification of doubly t...
simp23 1205 Simplification of doubly t...
simp31 1206 Simplification of doubly t...
simp32 1207 Simplification of doubly t...
simp33 1208 Simplification of doubly t...
simpll1 1209 Simplification of conjunct...
simpll2 1210 Simplification of conjunct...
simpll3 1211 Simplification of conjunct...
simplr1 1212 Simplification of conjunct...
simplr2 1213 Simplification of conjunct...
simplr3 1214 Simplification of conjunct...
simprl1 1215 Simplification of conjunct...
simprl2 1216 Simplification of conjunct...
simprl3 1217 Simplification of conjunct...
simprr1 1218 Simplification of conjunct...
simprr2 1219 Simplification of conjunct...
simprr3 1220 Simplification of conjunct...
simpl1l 1221 Simplification of conjunct...
simpl1r 1222 Simplification of conjunct...
simpl2l 1223 Simplification of conjunct...
simpl2r 1224 Simplification of conjunct...
simpl3l 1225 Simplification of conjunct...
simpl3r 1226 Simplification of conjunct...
simpr1l 1227 Simplification of conjunct...
simpr1r 1228 Simplification of conjunct...
simpr2l 1229 Simplification of conjunct...
simpr2r 1230 Simplification of conjunct...
simpr3l 1231 Simplification of conjunct...
simpr3r 1232 Simplification of conjunct...
simp1ll 1233 Simplification of conjunct...
simp1lr 1234 Simplification of conjunct...
simp1rl 1235 Simplification of conjunct...
simp1rr 1236 Simplification of conjunct...
simp2ll 1237 Simplification of conjunct...
simp2lr 1238 Simplification of conjunct...
simp2rl 1239 Simplification of conjunct...
simp2rr 1240 Simplification of conjunct...
simp3ll 1241 Simplification of conjunct...
simp3lr 1242 Simplification of conjunct...
simp3rl 1243 Simplification of conjunct...
simp3rr 1244 Simplification of conjunct...
simpl11 1245 Simplification of conjunct...
simpl12 1246 Simplification of conjunct...
simpl13 1247 Simplification of conjunct...
simpl21 1248 Simplification of conjunct...
simpl22 1249 Simplification of conjunct...
simpl23 1250 Simplification of conjunct...
simpl31 1251 Simplification of conjunct...
simpl32 1252 Simplification of conjunct...
simpl33 1253 Simplification of conjunct...
simpr11 1254 Simplification of conjunct...
simpr12 1255 Simplification of conjunct...
simpr13 1256 Simplification of conjunct...
simpr21 1257 Simplification of conjunct...
simpr22 1258 Simplification of conjunct...
simpr23 1259 Simplification of conjunct...
simpr31 1260 Simplification of conjunct...
simpr32 1261 Simplification of conjunct...
simpr33 1262 Simplification of conjunct...
simp1l1 1263 Simplification of conjunct...
simp1l2 1264 Simplification of conjunct...
simp1l3 1265 Simplification of conjunct...
simp1r1 1266 Simplification of conjunct...
simp1r2 1267 Simplification of conjunct...
simp1r3 1268 Simplification of conjunct...
simp2l1 1269 Simplification of conjunct...
simp2l2 1270 Simplification of conjunct...
simp2l3 1271 Simplification of conjunct...
simp2r1 1272 Simplification of conjunct...
simp2r2 1273 Simplification of conjunct...
simp2r3 1274 Simplification of conjunct...
simp3l1 1275 Simplification of conjunct...
simp3l2 1276 Simplification of conjunct...
simp3l3 1277 Simplification of conjunct...
simp3r1 1278 Simplification of conjunct...
simp3r2 1279 Simplification of conjunct...
simp3r3 1280 Simplification of conjunct...
simp11l 1281 Simplification of conjunct...
simp11r 1282 Simplification of conjunct...
simp12l 1283 Simplification of conjunct...
simp12r 1284 Simplification of conjunct...
simp13l 1285 Simplification of conjunct...
simp13r 1286 Simplification of conjunct...
simp21l 1287 Simplification of conjunct...
simp21r 1288 Simplification of conjunct...
simp22l 1289 Simplification of conjunct...
simp22r 1290 Simplification of conjunct...
simp23l 1291 Simplification of conjunct...
simp23r 1292 Simplification of conjunct...
simp31l 1293 Simplification of conjunct...
simp31r 1294 Simplification of conjunct...
simp32l 1295 Simplification of conjunct...
simp32r 1296 Simplification of conjunct...
simp33l 1297 Simplification of conjunct...
simp33r 1298 Simplification of conjunct...
simp111 1299 Simplification of conjunct...
simp112 1300 Simplification of conjunct...
simp113 1301 Simplification of conjunct...
simp121 1302 Simplification of conjunct...
simp122 1303 Simplification of conjunct...
simp123 1304 Simplification of conjunct...
simp131 1305 Simplification of conjunct...
simp132 1306 Simplification of conjunct...
simp133 1307 Simplification of conjunct...
simp211 1308 Simplification of conjunct...
simp212 1309 Simplification of conjunct...
simp213 1310 Simplification of conjunct...
simp221 1311 Simplification of conjunct...
simp222 1312 Simplification of conjunct...
simp223 1313 Simplification of conjunct...
simp231 1314 Simplification of conjunct...
simp232 1315 Simplification of conjunct...
simp233 1316 Simplification of conjunct...
simp311 1317 Simplification of conjunct...
simp312 1318 Simplification of conjunct...
simp313 1319 Simplification of conjunct...
simp321 1320 Simplification of conjunct...
simp322 1321 Simplification of conjunct...
simp323 1322 Simplification of conjunct...
simp331 1323 Simplification of conjunct...
simp332 1324 Simplification of conjunct...
simp333 1325 Simplification of conjunct...
3anibar 1326 Remove a hypothesis from t...
3mix1 1327 Introduction in triple dis...
3mix2 1328 Introduction in triple dis...
3mix3 1329 Introduction in triple dis...
3mix1i 1330 Introduction in triple dis...
3mix2i 1331 Introduction in triple dis...
3mix3i 1332 Introduction in triple dis...
3mix1d 1333 Deduction introducing trip...
3mix2d 1334 Deduction introducing trip...
3mix3d 1335 Deduction introducing trip...
3pm3.2i 1336 Infer conjunction of premi...
pm3.2an3 1337 Version of ~ pm3.2 for a t...
mpbir3an 1338 Detach a conjunction of tr...
mpbir3and 1339 Detach a conjunction of tr...
syl3anbrc 1340 Syllogism inference. (Con...
syl21anbrc 1341 Syllogism inference. (Con...
3imp3i2an 1342 An elimination deduction. ...
ex3 1343 Apply ~ ex to a hypothesis...
3imp1 1344 Importation to left triple...
3impd 1345 Importation deduction for ...
3imp2 1346 Importation to right tripl...
3impdi 1347 Importation inference (und...
3impdir 1348 Importation inference (und...
3exp1 1349 Exportation from left trip...
3expd 1350 Exportation deduction for ...
3exp2 1351 Exportation from right tri...
exp5o 1352 A triple exportation infer...
exp516 1353 A triple exportation infer...
exp520 1354 A triple exportation infer...
3impexp 1355 Version of ~ impexp for a ...
3an1rs 1356 Swap conjuncts. (Contribu...
3anassrs 1357 Associative law for conjun...
ad5ant245 1358 Deduction adding conjuncts...
ad5ant234 1359 Deduction adding conjuncts...
ad5ant235 1360 Deduction adding conjuncts...
ad5ant123 1361 Deduction adding conjuncts...
ad5ant124 1362 Deduction adding conjuncts...
ad5ant125 1363 Deduction adding conjuncts...
ad5ant134 1364 Deduction adding conjuncts...
ad5ant135 1365 Deduction adding conjuncts...
ad5ant145 1366 Deduction adding conjuncts...
ad5ant2345 1367 Deduction adding conjuncts...
syl3anc 1368 Syllogism combined with co...
syl13anc 1369 Syllogism combined with co...
syl31anc 1370 Syllogism combined with co...
syl112anc 1371 Syllogism combined with co...
syl121anc 1372 Syllogism combined with co...
syl211anc 1373 Syllogism combined with co...
syl23anc 1374 Syllogism combined with co...
syl32anc 1375 Syllogism combined with co...
syl122anc 1376 Syllogism combined with co...
syl212anc 1377 Syllogism combined with co...
syl221anc 1378 Syllogism combined with co...
syl113anc 1379 Syllogism combined with co...
syl131anc 1380 Syllogism combined with co...
syl311anc 1381 Syllogism combined with co...
syl33anc 1382 Syllogism combined with co...
syl222anc 1383 Syllogism combined with co...
syl123anc 1384 Syllogism combined with co...
syl132anc 1385 Syllogism combined with co...
syl213anc 1386 Syllogism combined with co...
syl231anc 1387 Syllogism combined with co...
syl312anc 1388 Syllogism combined with co...
syl321anc 1389 Syllogism combined with co...
syl133anc 1390 Syllogism combined with co...
syl313anc 1391 Syllogism combined with co...
syl331anc 1392 Syllogism combined with co...
syl223anc 1393 Syllogism combined with co...
syl232anc 1394 Syllogism combined with co...
syl322anc 1395 Syllogism combined with co...
syl233anc 1396 Syllogism combined with co...
syl323anc 1397 Syllogism combined with co...
syl332anc 1398 Syllogism combined with co...
syl333anc 1399 A syllogism inference comb...
syl3an1b 1400 A syllogism inference. (C...
syl3an2b 1401 A syllogism inference. (C...
syl3an3b 1402 A syllogism inference. (C...
syl3an1br 1403 A syllogism inference. (C...
syl3an2br 1404 A syllogism inference. (C...
syl3an3br 1405 A syllogism inference. (C...
syld3an3 1406 A syllogism inference. (C...
syld3an1 1407 A syllogism inference. (C...
syld3an2 1408 A syllogism inference. (C...
syl3anl1 1409 A syllogism inference. (C...
syl3anl2 1410 A syllogism inference. (C...
syl3anl3 1411 A syllogism inference. (C...
syl3anl 1412 A triple syllogism inferen...
syl3anr1 1413 A syllogism inference. (C...
syl3anr2 1414 A syllogism inference. (C...
syl3anr3 1415 A syllogism inference. (C...
3anidm12 1416 Inference from idempotent ...
3anidm13 1417 Inference from idempotent ...
3anidm23 1418 Inference from idempotent ...
syl2an3an 1419 ~ syl3an with antecedents ...
syl2an23an 1420 Deduction related to ~ syl...
3ori 1421 Infer implication from tri...
3jao 1422 Disjunction of three antec...
3jaob 1423 Disjunction of three antec...
3jaoi 1424 Disjunction of three antec...
3jaod 1425 Disjunction of three antec...
3jaoian 1426 Disjunction of three antec...
3jaodan 1427 Disjunction of three antec...
mpjao3dan 1428 Eliminate a three-way disj...
3jaao 1429 Inference conjoining and d...
syl3an9b 1430 Nested syllogism inference...
3orbi123d 1431 Deduction joining 3 equiva...
3anbi123d 1432 Deduction joining 3 equiva...
3anbi12d 1433 Deduction conjoining and a...
3anbi13d 1434 Deduction conjoining and a...
3anbi23d 1435 Deduction conjoining and a...
3anbi1d 1436 Deduction adding conjuncts...
3anbi2d 1437 Deduction adding conjuncts...
3anbi3d 1438 Deduction adding conjuncts...
3anim123d 1439 Deduction joining 3 implic...
3orim123d 1440 Deduction joining 3 implic...
an6 1441 Rearrangement of 6 conjunc...
3an6 1442 Analogue of ~ an4 for trip...
3or6 1443 Analogue of ~ or4 for trip...
mp3an1 1444 An inference based on modu...
mp3an2 1445 An inference based on modu...
mp3an3 1446 An inference based on modu...
mp3an12 1447 An inference based on modu...
mp3an13 1448 An inference based on modu...
mp3an23 1449 An inference based on modu...
mp3an1i 1450 An inference based on modu...
mp3anl1 1451 An inference based on modu...
mp3anl2 1452 An inference based on modu...
mp3anl3 1453 An inference based on modu...
mp3anr1 1454 An inference based on modu...
mp3anr2 1455 An inference based on modu...
mp3anr3 1456 An inference based on modu...
mp3an 1457 An inference based on modu...
mpd3an3 1458 An inference based on modu...
mpd3an23 1459 An inference based on modu...
mp3and 1460 A deduction based on modus...
mp3an12i 1461 ~ mp3an with antecedents i...
mp3an2i 1462 ~ mp3an with antecedents i...
mp3an3an 1463 ~ mp3an with antecedents i...
mp3an2ani 1464 An elimination deduction. ...
biimp3a 1465 Infer implication from a l...
biimp3ar 1466 Infer implication from a l...
3anandis 1467 Inference that undistribut...
3anandirs 1468 Inference that undistribut...
ecase23d 1469 Deduction for elimination ...
3ecase 1470 Inference for elimination ...
3bior1fd 1471 A disjunction is equivalen...
3bior1fand 1472 A disjunction is equivalen...
3bior2fd 1473 A wff is equivalent to its...
3biant1d 1474 A conjunction is equivalen...
intn3an1d 1475 Introduction of a triple c...
intn3an2d 1476 Introduction of a triple c...
intn3an3d 1477 Introduction of a triple c...
an3andi 1478 Distribution of conjunctio...
an33rean 1479 Rearrange a 9-fold conjunc...
3orel2 1480 Partial elimination of a t...
3orel3 1481 Partial elimination of a t...
3orel13 1482 Elimination of two disjunc...
3pm3.2ni 1483 Triple negated disjunction...
nanan 1486 Conjunction in terms of al...
dfnan2 1487 Alternative denial in term...
nanor 1488 Alternative denial in term...
nancom 1489 Alternative denial is comm...
nannan 1490 Nested alternative denials...
nanim 1491 Implication in terms of al...
nannot 1492 Negation in terms of alter...
nanbi 1493 Biconditional in terms of ...
nanbi1 1494 Introduce a right anti-con...
nanbi2 1495 Introduce a left anti-conj...
nanbi12 1496 Join two logical equivalen...
nanbi1i 1497 Introduce a right anti-con...
nanbi2i 1498 Introduce a left anti-conj...
nanbi12i 1499 Join two logical equivalen...
nanbi1d 1500 Introduce a right anti-con...
nanbi2d 1501 Introduce a left anti-conj...
nanbi12d 1502 Join two logical equivalen...
nanass 1503 A characterization of when...
xnor 1506 Two ways to write XNOR (ex...
xorcom 1507 The connector ` \/_ ` is c...
xorass 1508 The connector ` \/_ ` is a...
excxor 1509 This tautology shows that ...
xor2 1510 Two ways to express "exclu...
xoror 1511 Exclusive disjunction impl...
xornan 1512 Exclusive disjunction impl...
xornan2 1513 XOR implies NAND (written ...
xorneg2 1514 The connector ` \/_ ` is n...
xorneg1 1515 The connector ` \/_ ` is n...
xorneg 1516 The connector ` \/_ ` is u...
xorbi12i 1517 Equality property for excl...
xorbi12d 1518 Equality property for excl...
anxordi 1519 Conjunction distributes ov...
xorexmid 1520 Exclusive-or variant of th...
norcom 1523 The connector ` -\/ ` is c...
nornot 1524 ` -. ` is expressible via ...
noran 1525 ` /\ ` is expressible via ...
noror 1526 ` \/ ` is expressible via ...
norasslem1 1527 This lemma shows the equiv...
norasslem2 1528 This lemma specializes ~ b...
norasslem3 1529 This lemma specializes ~ b...
norass 1530 A characterization of when...
trujust 1535 Soundness justification th...
tru 1537 The truth value ` T. ` is ...
dftru2 1538 An alternate definition of...
trut 1539 A proposition is equivalen...
mptru 1540 Eliminate ` T. ` as an ant...
tbtru 1541 A proposition is equivalen...
bitru 1542 A theorem is equivalent to...
trud 1543 Anything implies ` T. ` . ...
truan 1544 True can be removed from a...
fal 1547 The truth value ` F. ` is ...
nbfal 1548 The negation of a proposit...
bifal 1549 A contradiction is equival...
falim 1550 The truth value ` F. ` imp...
falimd 1551 The truth value ` F. ` imp...
dfnot 1552 Given falsum ` F. ` , we c...
inegd 1553 Negation introduction rule...
efald 1554 Deduction based on reducti...
pm2.21fal 1555 If a wff and its negation ...
truimtru 1556 A ` -> ` identity. (Contr...
truimfal 1557 A ` -> ` identity. (Contr...
falimtru 1558 A ` -> ` identity. (Contr...
falimfal 1559 A ` -> ` identity. (Contr...
nottru 1560 A ` -. ` identity. (Contr...
notfal 1561 A ` -. ` identity. (Contr...
trubitru 1562 A ` <-> ` identity. (Cont...
falbitru 1563 A ` <-> ` identity. (Cont...
trubifal 1564 A ` <-> ` identity. (Cont...
falbifal 1565 A ` <-> ` identity. (Cont...
truantru 1566 A ` /\ ` identity. (Contr...
truanfal 1567 A ` /\ ` identity. (Contr...
falantru 1568 A ` /\ ` identity. (Contr...
falanfal 1569 A ` /\ ` identity. (Contr...
truortru 1570 A ` \/ ` identity. (Contr...
truorfal 1571 A ` \/ ` identity. (Contr...
falortru 1572 A ` \/ ` identity. (Contr...
falorfal 1573 A ` \/ ` identity. (Contr...
trunantru 1574 A ` -/\ ` identity. (Cont...
trunanfal 1575 A ` -/\ ` identity. (Cont...
falnantru 1576 A ` -/\ ` identity. (Cont...
falnanfal 1577 A ` -/\ ` identity. (Cont...
truxortru 1578 A ` \/_ ` identity. (Cont...
truxorfal 1579 A ` \/_ ` identity. (Cont...
falxortru 1580 A ` \/_ ` identity. (Cont...
falxorfal 1581 A ` \/_ ` identity. (Cont...
trunortru 1582 A ` -\/ ` identity. (Cont...
trunorfal 1583 A ` -\/ ` identity. (Cont...
falnortru 1584 A ` -\/ ` identity. (Cont...
falnorfal 1585 A ` -\/ ` identity. (Cont...
hadbi123d 1588 Equality theorem for the a...
hadbi123i 1589 Equality theorem for the a...
hadass 1590 Associative law for the ad...
hadbi 1591 The adder sum is the same ...
hadcoma 1592 Commutative law for the ad...
hadcomb 1593 Commutative law for the ad...
hadrot 1594 Rotation law for the adder...
hadnot 1595 The adder sum distributes ...
had1 1596 If the first input is true...
had0 1597 If the first input is fals...
hadifp 1598 The value of the adder sum...
cador 1601 The adder carry in disjunc...
cadan 1602 The adder carry in conjunc...
cadbi123d 1603 Equality theorem for the a...
cadbi123i 1604 Equality theorem for the a...
cadcoma 1605 Commutative law for the ad...
cadcomb 1606 Commutative law for the ad...
cadrot 1607 Rotation law for the adder...
cadnot 1608 The adder carry distribute...
cad11 1609 If (at least) two inputs a...
cad1 1610 If one input is true, then...
cad0 1611 If one input is false, the...
cad0OLD 1612 Obsolete version of ~ cad0...
cadifp 1613 The value of the carry is,...
cadtru 1614 The adder carry is true as...
minimp 1615 A single axiom for minimal...
minimp-syllsimp 1616 Derivation of Syll-Simp ( ...
minimp-ax1 1617 Derivation of ~ ax-1 from ...
minimp-ax2c 1618 Derivation of a commuted f...
minimp-ax2 1619 Derivation of ~ ax-2 from ...
minimp-pm2.43 1620 Derivation of ~ pm2.43 (al...
impsingle 1621 The shortest single axiom ...
impsingle-step4 1622 Derivation of impsingle-st...
impsingle-step8 1623 Derivation of impsingle-st...
impsingle-ax1 1624 Derivation of impsingle-ax...
impsingle-step15 1625 Derivation of impsingle-st...
impsingle-step18 1626 Derivation of impsingle-st...
impsingle-step19 1627 Derivation of impsingle-st...
impsingle-step20 1628 Derivation of impsingle-st...
impsingle-step21 1629 Derivation of impsingle-st...
impsingle-step22 1630 Derivation of impsingle-st...
impsingle-step25 1631 Derivation of impsingle-st...
impsingle-imim1 1632 Derivation of impsingle-im...
impsingle-peirce 1633 Derivation of impsingle-pe...
tarski-bernays-ax2 1634 Derivation of ~ ax-2 from ...
meredith 1635 Carew Meredith's sole axio...
merlem1 1636 Step 3 of Meredith's proof...
merlem2 1637 Step 4 of Meredith's proof...
merlem3 1638 Step 7 of Meredith's proof...
merlem4 1639 Step 8 of Meredith's proof...
merlem5 1640 Step 11 of Meredith's proo...
merlem6 1641 Step 12 of Meredith's proo...
merlem7 1642 Between steps 14 and 15 of...
merlem8 1643 Step 15 of Meredith's proo...
merlem9 1644 Step 18 of Meredith's proo...
merlem10 1645 Step 19 of Meredith's proo...
merlem11 1646 Step 20 of Meredith's proo...
merlem12 1647 Step 28 of Meredith's proo...
merlem13 1648 Step 35 of Meredith's proo...
luk-1 1649 1 of 3 axioms for proposit...
luk-2 1650 2 of 3 axioms for proposit...
luk-3 1651 3 of 3 axioms for proposit...
luklem1 1652 Used to rederive standard ...
luklem2 1653 Used to rederive standard ...
luklem3 1654 Used to rederive standard ...
luklem4 1655 Used to rederive standard ...
luklem5 1656 Used to rederive standard ...
luklem6 1657 Used to rederive standard ...
luklem7 1658 Used to rederive standard ...
luklem8 1659 Used to rederive standard ...
ax1 1660 Standard propositional axi...
ax2 1661 Standard propositional axi...
ax3 1662 Standard propositional axi...
nic-dfim 1663 This theorem "defines" imp...
nic-dfneg 1664 This theorem "defines" neg...
nic-mp 1665 Derive Nicod's rule of mod...
nic-mpALT 1666 A direct proof of ~ nic-mp...
nic-ax 1667 Nicod's axiom derived from...
nic-axALT 1668 A direct proof of ~ nic-ax...
nic-imp 1669 Inference for ~ nic-mp usi...
nic-idlem1 1670 Lemma for ~ nic-id . (Con...
nic-idlem2 1671 Lemma for ~ nic-id . Infe...
nic-id 1672 Theorem ~ id expressed wit...
nic-swap 1673 The connector ` -/\ ` is s...
nic-isw1 1674 Inference version of ~ nic...
nic-isw2 1675 Inference for swapping nes...
nic-iimp1 1676 Inference version of ~ nic...
nic-iimp2 1677 Inference version of ~ nic...
nic-idel 1678 Inference to remove the tr...
nic-ich 1679 Chained inference. (Contr...
nic-idbl 1680 Double the terms. Since d...
nic-bijust 1681 Biconditional justificatio...
nic-bi1 1682 Inference to extract one s...
nic-bi2 1683 Inference to extract the o...
nic-stdmp 1684 Derive the standard modus ...
nic-luk1 1685 Proof of ~ luk-1 from ~ ni...
nic-luk2 1686 Proof of ~ luk-2 from ~ ni...
nic-luk3 1687 Proof of ~ luk-3 from ~ ni...
lukshef-ax1 1688 This alternative axiom for...
lukshefth1 1689 Lemma for ~ renicax . (Co...
lukshefth2 1690 Lemma for ~ renicax . (Co...
renicax 1691 A rederivation of ~ nic-ax...
tbw-bijust 1692 Justification for ~ tbw-ne...
tbw-negdf 1693 The definition of negation...
tbw-ax1 1694 The first of four axioms i...
tbw-ax2 1695 The second of four axioms ...
tbw-ax3 1696 The third of four axioms i...
tbw-ax4 1697 The fourth of four axioms ...
tbwsyl 1698 Used to rederive the Lukas...
tbwlem1 1699 Used to rederive the Lukas...
tbwlem2 1700 Used to rederive the Lukas...
tbwlem3 1701 Used to rederive the Lukas...
tbwlem4 1702 Used to rederive the Lukas...
tbwlem5 1703 Used to rederive the Lukas...
re1luk1 1704 ~ luk-1 derived from the T...
re1luk2 1705 ~ luk-2 derived from the T...
re1luk3 1706 ~ luk-3 derived from the T...
merco1 1707 A single axiom for proposi...
merco1lem1 1708 Used to rederive the Tarsk...
retbwax4 1709 ~ tbw-ax4 rederived from ~...
retbwax2 1710 ~ tbw-ax2 rederived from ~...
merco1lem2 1711 Used to rederive the Tarsk...
merco1lem3 1712 Used to rederive the Tarsk...
merco1lem4 1713 Used to rederive the Tarsk...
merco1lem5 1714 Used to rederive the Tarsk...
merco1lem6 1715 Used to rederive the Tarsk...
merco1lem7 1716 Used to rederive the Tarsk...
retbwax3 1717 ~ tbw-ax3 rederived from ~...
merco1lem8 1718 Used to rederive the Tarsk...
merco1lem9 1719 Used to rederive the Tarsk...
merco1lem10 1720 Used to rederive the Tarsk...
merco1lem11 1721 Used to rederive the Tarsk...
merco1lem12 1722 Used to rederive the Tarsk...
merco1lem13 1723 Used to rederive the Tarsk...
merco1lem14 1724 Used to rederive the Tarsk...
merco1lem15 1725 Used to rederive the Tarsk...
merco1lem16 1726 Used to rederive the Tarsk...
merco1lem17 1727 Used to rederive the Tarsk...
merco1lem18 1728 Used to rederive the Tarsk...
retbwax1 1729 ~ tbw-ax1 rederived from ~...
merco2 1730 A single axiom for proposi...
mercolem1 1731 Used to rederive the Tarsk...
mercolem2 1732 Used to rederive the Tarsk...
mercolem3 1733 Used to rederive the Tarsk...
mercolem4 1734 Used to rederive the Tarsk...
mercolem5 1735 Used to rederive the Tarsk...
mercolem6 1736 Used to rederive the Tarsk...
mercolem7 1737 Used to rederive the Tarsk...
mercolem8 1738 Used to rederive the Tarsk...
re1tbw1 1739 ~ tbw-ax1 rederived from ~...
re1tbw2 1740 ~ tbw-ax2 rederived from ~...
re1tbw3 1741 ~ tbw-ax3 rederived from ~...
re1tbw4 1742 ~ tbw-ax4 rederived from ~...
rb-bijust 1743 Justification for ~ rb-imd...
rb-imdf 1744 The definition of implicat...
anmp 1745 Modus ponens for ` { \/ , ...
rb-ax1 1746 The first of four axioms i...
rb-ax2 1747 The second of four axioms ...
rb-ax3 1748 The third of four axioms i...
rb-ax4 1749 The fourth of four axioms ...
rbsyl 1750 Used to rederive the Lukas...
rblem1 1751 Used to rederive the Lukas...
rblem2 1752 Used to rederive the Lukas...
rblem3 1753 Used to rederive the Lukas...
rblem4 1754 Used to rederive the Lukas...
rblem5 1755 Used to rederive the Lukas...
rblem6 1756 Used to rederive the Lukas...
rblem7 1757 Used to rederive the Lukas...
re1axmp 1758 ~ ax-mp derived from Russe...
re2luk1 1759 ~ luk-1 derived from Russe...
re2luk2 1760 ~ luk-2 derived from Russe...
re2luk3 1761 ~ luk-3 derived from Russe...
mptnan 1762 Modus ponendo tollens 1, o...
mptxor 1763 Modus ponendo tollens 2, o...
mtpor 1764 Modus tollendo ponens (inc...
mtpxor 1765 Modus tollendo ponens (ori...
stoic1a 1766 Stoic logic Thema 1 (part ...
stoic1b 1767 Stoic logic Thema 1 (part ...
stoic2a 1768 Stoic logic Thema 2 versio...
stoic2b 1769 Stoic logic Thema 2 versio...
stoic3 1770 Stoic logic Thema 3. Stat...
stoic4a 1771 Stoic logic Thema 4 versio...
stoic4b 1772 Stoic logic Thema 4 versio...
alnex 1775 Universal quantification o...
eximal 1776 An equivalence between an ...
nf2 1779 Alternate definition of no...
nf3 1780 Alternate definition of no...
nf4 1781 Alternate definition of no...
nfi 1782 Deduce that ` x ` is not f...
nfri 1783 Consequence of the definit...
nfd 1784 Deduce that ` x ` is not f...
nfrd 1785 Consequence of the definit...
nftht 1786 Closed form of ~ nfth . (...
nfntht 1787 Closed form of ~ nfnth . ...
nfntht2 1788 Closed form of ~ nfnth . ...
gen2 1790 Generalization applied twi...
mpg 1791 Modus ponens combined with...
mpgbi 1792 Modus ponens on biconditio...
mpgbir 1793 Modus ponens on biconditio...
nex 1794 Generalization rule for ne...
nfth 1795 No variable is (effectivel...
nfnth 1796 No variable is (effectivel...
hbth 1797 No variable is (effectivel...
nftru 1798 The true constant has no f...
nffal 1799 The false constant has no ...
sptruw 1800 Version of ~ sp when ` ph ...
altru 1801 For all sets, ` T. ` is tr...
alfal 1802 For all sets, ` -. F. ` is...
alim 1804 Restatement of Axiom ~ ax-...
alimi 1805 Inference quantifying both...
2alimi 1806 Inference doubly quantifyi...
ala1 1807 Add an antecedent in a uni...
al2im 1808 Closed form of ~ al2imi . ...
al2imi 1809 Inference quantifying ante...
alanimi 1810 Variant of ~ al2imi with c...
alimdh 1811 Deduction form of Theorem ...
albi 1812 Theorem 19.15 of [Margaris...
albii 1813 Inference adding universal...
2albii 1814 Inference adding two unive...
3albii 1815 Inference adding three uni...
sylgt 1816 Closed form of ~ sylg . (...
sylg 1817 A syllogism combined with ...
alrimih 1818 Inference form of Theorem ...
hbxfrbi 1819 A utility lemma to transfe...
alex 1820 Universal quantifier in te...
exnal 1821 Existential quantification...
2nalexn 1822 Part of theorem *11.5 in [...
2exnaln 1823 Theorem *11.22 in [Whitehe...
2nexaln 1824 Theorem *11.25 in [Whitehe...
alimex 1825 An equivalence between an ...
aleximi 1826 A variant of ~ al2imi : in...
alexbii 1827 Biconditional form of ~ al...
exim 1828 Theorem 19.22 of [Margaris...
eximi 1829 Inference adding existenti...
2eximi 1830 Inference adding two exist...
eximii 1831 Inference associated with ...
exa1 1832 Add an antecedent in an ex...
19.38 1833 Theorem 19.38 of [Margaris...
19.38a 1834 Under a nonfreeness hypoth...
19.38b 1835 Under a nonfreeness hypoth...
imnang 1836 Quantified implication in ...
alinexa 1837 A transformation of quanti...
exnalimn 1838 Existential quantification...
alexn 1839 A relationship between two...
2exnexn 1840 Theorem *11.51 in [Whitehe...
exbi 1841 Theorem 19.18 of [Margaris...
exbii 1842 Inference adding existenti...
2exbii 1843 Inference adding two exist...
3exbii 1844 Inference adding three exi...
nfbiit 1845 Equivalence theorem for th...
nfbii 1846 Equality theorem for the n...
nfxfr 1847 A utility lemma to transfe...
nfxfrd 1848 A utility lemma to transfe...
nfnbi 1849 A variable is nonfree in a...
nfnbiOLD 1850 Obsolete version of ~ nfnb...
nfnt 1851 If a variable is nonfree i...
nfn 1852 Inference associated with ...
nfnd 1853 Deduction associated with ...
exanali 1854 A transformation of quanti...
2exanali 1855 Theorem *11.521 in [Whiteh...
exancom 1856 Commutation of conjunction...
exan 1857 Place a conjunct in the sc...
alrimdh 1858 Deduction form of Theorem ...
eximdh 1859 Deduction from Theorem 19....
nexdh 1860 Deduction for generalizati...
albidh 1861 Formula-building rule for ...
exbidh 1862 Formula-building rule for ...
exsimpl 1863 Simplification of an exist...
exsimpr 1864 Simplification of an exist...
19.26 1865 Theorem 19.26 of [Margaris...
19.26-2 1866 Theorem ~ 19.26 with two q...
19.26-3an 1867 Theorem ~ 19.26 with tripl...
19.29 1868 Theorem 19.29 of [Margaris...
19.29r 1869 Variation of ~ 19.29 . (C...
19.29r2 1870 Variation of ~ 19.29r with...
19.29x 1871 Variation of ~ 19.29 with ...
19.35 1872 Theorem 19.35 of [Margaris...
19.35i 1873 Inference associated with ...
19.35ri 1874 Inference associated with ...
19.25 1875 Theorem 19.25 of [Margaris...
19.30 1876 Theorem 19.30 of [Margaris...
19.43 1877 Theorem 19.43 of [Margaris...
19.43OLD 1878 Obsolete proof of ~ 19.43 ...
19.33 1879 Theorem 19.33 of [Margaris...
19.33b 1880 The antecedent provides a ...
19.40 1881 Theorem 19.40 of [Margaris...
19.40-2 1882 Theorem *11.42 in [Whitehe...
19.40b 1883 The antecedent provides a ...
albiim 1884 Split a biconditional and ...
2albiim 1885 Split a biconditional and ...
exintrbi 1886 Add/remove a conjunct in t...
exintr 1887 Introduce a conjunct in th...
alsyl 1888 Universally quantified and...
nfimd 1889 If in a context ` x ` is n...
nfimt 1890 Closed form of ~ nfim and ...
nfim 1891 If ` x ` is not free in ` ...
nfand 1892 If in a context ` x ` is n...
nf3and 1893 Deduction form of bound-va...
nfan 1894 If ` x ` is not free in ` ...
nfnan 1895 If ` x ` is not free in ` ...
nf3an 1896 If ` x ` is not free in ` ...
nfbid 1897 If in a context ` x ` is n...
nfbi 1898 If ` x ` is not free in ` ...
nfor 1899 If ` x ` is not free in ` ...
nf3or 1900 If ` x ` is not free in ` ...
empty 1901 Two characterizations of t...
emptyex 1902 On the empty domain, any e...
emptyal 1903 On the empty domain, any u...
emptynf 1904 On the empty domain, any v...
ax5d 1906 Version of ~ ax-5 with ant...
ax5e 1907 A rephrasing of ~ ax-5 usi...
ax5ea 1908 If a formula holds for som...
nfv 1909 If ` x ` is not present in...
nfvd 1910 ~ nfv with antecedent. Us...
alimdv 1911 Deduction form of Theorem ...
eximdv 1912 Deduction form of Theorem ...
2alimdv 1913 Deduction form of Theorem ...
2eximdv 1914 Deduction form of Theorem ...
albidv 1915 Formula-building rule for ...
exbidv 1916 Formula-building rule for ...
nfbidv 1917 An equality theorem for no...
2albidv 1918 Formula-building rule for ...
2exbidv 1919 Formula-building rule for ...
3exbidv 1920 Formula-building rule for ...
4exbidv 1921 Formula-building rule for ...
alrimiv 1922 Inference form of Theorem ...
alrimivv 1923 Inference form of Theorem ...
alrimdv 1924 Deduction form of Theorem ...
exlimiv 1925 Inference form of Theorem ...
exlimiiv 1926 Inference (Rule C) associa...
exlimivv 1927 Inference form of Theorem ...
exlimdv 1928 Deduction form of Theorem ...
exlimdvv 1929 Deduction form of Theorem ...
exlimddv 1930 Existential elimination ru...
nexdv 1931 Deduction for generalizati...
2ax5 1932 Quantification of two vari...
stdpc5v 1933 Version of ~ stdpc5 with a...
19.21v 1934 Version of ~ 19.21 with a ...
19.32v 1935 Version of ~ 19.32 with a ...
19.31v 1936 Version of ~ 19.31 with a ...
19.23v 1937 Version of ~ 19.23 with a ...
19.23vv 1938 Theorem ~ 19.23v extended ...
pm11.53v 1939 Version of ~ pm11.53 with ...
19.36imv 1940 One direction of ~ 19.36v ...
19.36imvOLD 1941 Obsolete version of ~ 19.3...
19.36iv 1942 Inference associated with ...
19.37imv 1943 One direction of ~ 19.37v ...
19.37iv 1944 Inference associated with ...
19.41v 1945 Version of ~ 19.41 with a ...
19.41vv 1946 Version of ~ 19.41 with tw...
19.41vvv 1947 Version of ~ 19.41 with th...
19.41vvvv 1948 Version of ~ 19.41 with fo...
19.42v 1949 Version of ~ 19.42 with a ...
exdistr 1950 Distribution of existentia...
exdistrv 1951 Distribute a pair of exist...
4exdistrv 1952 Distribute two pairs of ex...
19.42vv 1953 Version of ~ 19.42 with tw...
exdistr2 1954 Distribution of existentia...
19.42vvv 1955 Version of ~ 19.42 with th...
3exdistr 1956 Distribution of existentia...
4exdistr 1957 Distribution of existentia...
weq 1958 Extend wff definition to i...
speimfw 1959 Specialization, with addit...
speimfwALT 1960 Alternate proof of ~ speim...
spimfw 1961 Specialization, with addit...
ax12i 1962 Inference that has ~ ax-12...
ax6v 1964 Axiom B7 of [Tarski] p. 75...
ax6ev 1965 At least one individual ex...
spimw 1966 Specialization. Lemma 8 o...
spimew 1967 Existential introduction, ...
speiv 1968 Inference from existential...
speivw 1969 Version of ~ spei with a d...
exgen 1970 Rule of existential genera...
extru 1971 There exists a variable su...
19.2 1972 Theorem 19.2 of [Margaris]...
19.2d 1973 Deduction associated with ...
19.8w 1974 Weak version of ~ 19.8a an...
spnfw 1975 Weak version of ~ sp . Us...
spvw 1976 Version of ~ sp when ` x `...
19.3v 1977 Version of ~ 19.3 with a d...
19.8v 1978 Version of ~ 19.8a with a ...
19.9v 1979 Version of ~ 19.9 with a d...
19.39 1980 Theorem 19.39 of [Margaris...
19.24 1981 Theorem 19.24 of [Margaris...
19.34 1982 Theorem 19.34 of [Margaris...
19.36v 1983 Version of ~ 19.36 with a ...
19.12vvv 1984 Version of ~ 19.12vv with ...
19.27v 1985 Version of ~ 19.27 with a ...
19.28v 1986 Version of ~ 19.28 with a ...
19.37v 1987 Version of ~ 19.37 with a ...
19.44v 1988 Version of ~ 19.44 with a ...
19.45v 1989 Version of ~ 19.45 with a ...
spimevw 1990 Existential introduction, ...
spimvw 1991 A weak form of specializat...
spvv 1992 Specialization, using impl...
spfalw 1993 Version of ~ sp when ` ph ...
chvarvv 1994 Implicit substitution of `...
equs4v 1995 Version of ~ equs4 with a ...
alequexv 1996 Version of ~ equs4v with i...
exsbim 1997 One direction of the equiv...
equsv 1998 If a formula does not cont...
equsalvw 1999 Version of ~ equsalv with ...
equsexvw 2000 Version of ~ equsexv with ...
cbvaliw 2001 Change bound variable. Us...
cbvalivw 2002 Change bound variable. Us...
ax7v 2004 Weakened version of ~ ax-7...
ax7v1 2005 First of two weakened vers...
ax7v2 2006 Second of two weakened ver...
equid 2007 Identity law for equality....
nfequid 2008 Bound-variable hypothesis ...
equcomiv 2009 Weaker form of ~ equcomi w...
ax6evr 2010 A commuted form of ~ ax6ev...
ax7 2011 Proof of ~ ax-7 from ~ ax7...
equcomi 2012 Commutative law for equali...
equcom 2013 Commutative law for equali...
equcomd 2014 Deduction form of ~ equcom...
equcoms 2015 An inference commuting equ...
equtr 2016 A transitive law for equal...
equtrr 2017 A transitive law for equal...
equeuclr 2018 Commuted version of ~ eque...
equeucl 2019 Equality is a left-Euclide...
equequ1 2020 An equivalence law for equ...
equequ2 2021 An equivalence law for equ...
equtr2 2022 Equality is a left-Euclide...
stdpc6 2023 One of the two equality ax...
equvinv 2024 A variable introduction la...
equvinva 2025 A modified version of the ...
equvelv 2026 A biconditional form of ~ ...
ax13b 2027 An equivalence between two...
spfw 2028 Weak version of ~ sp . Us...
spw 2029 Weak version of the specia...
cbvalw 2030 Change bound variable. Us...
cbvalvw 2031 Change bound variable. Us...
cbvexvw 2032 Change bound variable. Us...
cbvaldvaw 2033 Rule used to change the bo...
cbvexdvaw 2034 Rule used to change the bo...
cbval2vw 2035 Rule used to change bound ...
cbvex2vw 2036 Rule used to change bound ...
cbvex4vw 2037 Rule used to change bound ...
alcomiw 2038 Weak version of ~ ax-11 . ...
alcomw 2039 Weak version of ~ alcom an...
hbn1fw 2040 Weak version of ~ ax-10 fr...
hbn1w 2041 Weak version of ~ hbn1 . ...
hba1w 2042 Weak version of ~ hba1 . ...
hbe1w 2043 Weak version of ~ hbe1 . ...
hbalw 2044 Weak version of ~ hbal . ...
19.8aw 2045 If a formula is true, then...
exexw 2046 Existential quantification...
spaev 2047 A special instance of ~ sp...
cbvaev 2048 Change bound variable in a...
aevlem0 2049 Lemma for ~ aevlem . Inst...
aevlem 2050 Lemma for ~ aev and ~ axc1...
aeveq 2051 The antecedent ` A. x x = ...
aev 2052 A "distinctor elimination"...
aev2 2053 A version of ~ aev with tw...
hbaev 2054 All variables are effectiv...
naev 2055 If some set variables can ...
naev2 2056 Generalization of ~ hbnaev...
hbnaev 2057 Any variable is free in ` ...
sbjust 2058 Justification theorem for ...
sbt 2061 A substitution into a theo...
sbtru 2062 The result of substituting...
stdpc4 2063 The specialization axiom o...
sbtALT 2064 Alternate proof of ~ sbt ,...
2stdpc4 2065 A double specialization us...
sbi1 2066 Distribute substitution ov...
spsbim 2067 Distribute substitution ov...
spsbbi 2068 Biconditional property for...
sbimi 2069 Distribute substitution ov...
sb2imi 2070 Distribute substitution ov...
sbbii 2071 Infer substitution into bo...
2sbbii 2072 Infer double substitution ...
sbimdv 2073 Deduction substituting bot...
sbbidv 2074 Deduction substituting bot...
sban 2075 Conjunction inside and out...
sb3an 2076 Threefold conjunction insi...
spsbe 2077 Existential generalization...
sbequ 2078 Equality property for subs...
sbequi 2079 An equality theorem for su...
sb6 2080 Alternate definition of su...
2sb6 2081 Equivalence for double sub...
sb1v 2082 One direction of ~ sb5 , p...
sbv 2083 Substitution for a variabl...
sbcom4 2084 Commutativity law for subs...
pm11.07 2085 Axiom *11.07 in [Whitehead...
sbrimvw 2086 Substitution in an implica...
sbievw 2087 Conversion of implicit sub...
sbiedvw 2088 Conversion of implicit sub...
2sbievw 2089 Conversion of double impli...
sbcom3vv 2090 Substituting ` y ` for ` x...
sbievw2 2091 ~ sbievw applied twice, av...
sbco2vv 2092 A composition law for subs...
equsb3 2093 Substitution in an equalit...
equsb3r 2094 Substitution applied to th...
equsb1v 2095 Substitution applied to an...
nsb 2096 Any substitution in an alw...
sbn1 2097 One direction of ~ sbn , u...
wel 2099 Extend wff definition to i...
ax8v 2101 Weakened version of ~ ax-8...
ax8v1 2102 First of two weakened vers...
ax8v2 2103 Second of two weakened ver...
ax8 2104 Proof of ~ ax-8 from ~ ax8...
elequ1 2105 An identity law for the no...
elsb1 2106 Substitution for the first...
cleljust 2107 When the class variables i...
ax9v 2109 Weakened version of ~ ax-9...
ax9v1 2110 First of two weakened vers...
ax9v2 2111 Second of two weakened ver...
ax9 2112 Proof of ~ ax-9 from ~ ax9...
elequ2 2113 An identity law for the no...
elequ2g 2114 A form of ~ elequ2 with a ...
elsb2 2115 Substitution for the secon...
ax6dgen 2116 Tarski's system uses the w...
ax10w 2117 Weak version of ~ ax-10 fr...
ax11w 2118 Weak version of ~ ax-11 fr...
ax11dgen 2119 Degenerate instance of ~ a...
ax12wlem 2120 Lemma for weak version of ...
ax12w 2121 Weak version of ~ ax-12 fr...
ax12dgen 2122 Degenerate instance of ~ a...
ax12wdemo 2123 Example of an application ...
ax13w 2124 Weak version (principal in...
ax13dgen1 2125 Degenerate instance of ~ a...
ax13dgen2 2126 Degenerate instance of ~ a...
ax13dgen3 2127 Degenerate instance of ~ a...
ax13dgen4 2128 Degenerate instance of ~ a...
hbn1 2130 Alias for ~ ax-10 to be us...
hbe1 2131 The setvar ` x ` is not fr...
hbe1a 2132 Dual statement of ~ hbe1 ....
nf5-1 2133 One direction of ~ nf5 can...
nf5i 2134 Deduce that ` x ` is not f...
nf5dh 2135 Deduce that ` x ` is not f...
nf5dv 2136 Apply the definition of no...
nfnaew 2137 All variables are effectiv...
nfnaewOLD 2138 Obsolete version of ~ nfna...
nfe1 2139 The setvar ` x ` is not fr...
nfa1 2140 The setvar ` x ` is not fr...
nfna1 2141 A convenience theorem part...
nfia1 2142 Lemma 23 of [Monk2] p. 114...
nfnf1 2143 The setvar ` x ` is not fr...
modal5 2144 The analogue in our predic...
nfs1v 2145 The setvar ` x ` is not fr...
alcoms 2147 Swap quantifiers in an ant...
alcom 2148 Theorem 19.5 of [Margaris]...
alrot3 2149 Theorem *11.21 in [Whitehe...
alrot4 2150 Rotate four universal quan...
sbal 2151 Move universal quantifier ...
sbalv 2152 Quantify with new variable...
sbcom2 2153 Commutativity law for subs...
excom 2154 Theorem 19.11 of [Margaris...
excomim 2155 One direction of Theorem 1...
excom13 2156 Swap 1st and 3rd existenti...
exrot3 2157 Rotate existential quantif...
exrot4 2158 Rotate existential quantif...
hbal 2159 If ` x ` is not free in ` ...
hbald 2160 Deduction form of bound-va...
hbsbw 2161 If ` z ` is not free in ` ...
nfa2 2162 Lemma 24 of [Monk2] p. 114...
ax12v 2164 This is essentially Axiom ...
ax12v2 2165 It is possible to remove a...
19.8a 2166 If a wff is true, it is tr...
19.8ad 2167 If a wff is true, it is tr...
sp 2168 Specialization. A univers...
spi 2169 Inference rule of universa...
sps 2170 Generalization of antecede...
2sp 2171 A double specialization (s...
spsd 2172 Deduction generalizing ant...
19.2g 2173 Theorem 19.2 of [Margaris]...
19.21bi 2174 Inference form of ~ 19.21 ...
19.21bbi 2175 Inference removing two uni...
19.23bi 2176 Inference form of Theorem ...
nexr 2177 Inference associated with ...
qexmid 2178 Quantified excluded middle...
nf5r 2179 Consequence of the definit...
nf5ri 2180 Consequence of the definit...
nf5rd 2181 Consequence of the definit...
spimedv 2182 Deduction version of ~ spi...
spimefv 2183 Version of ~ spime with a ...
nfim1 2184 A closed form of ~ nfim . ...
nfan1 2185 A closed form of ~ nfan . ...
19.3t 2186 Closed form of ~ 19.3 and ...
19.3 2187 A wff may be quantified wi...
19.9d 2188 A deduction version of one...
19.9t 2189 Closed form of ~ 19.9 and ...
19.9 2190 A wff may be existentially...
19.21t 2191 Closed form of Theorem 19....
19.21 2192 Theorem 19.21 of [Margaris...
stdpc5 2193 An axiom scheme of standar...
19.21-2 2194 Version of ~ 19.21 with tw...
19.23t 2195 Closed form of Theorem 19....
19.23 2196 Theorem 19.23 of [Margaris...
alimd 2197 Deduction form of Theorem ...
alrimi 2198 Inference form of Theorem ...
alrimdd 2199 Deduction form of Theorem ...
alrimd 2200 Deduction form of Theorem ...
eximd 2201 Deduction form of Theorem ...
exlimi 2202 Inference associated with ...
exlimd 2203 Deduction form of Theorem ...
exlimimdd 2204 Existential elimination ru...
exlimdd 2205 Existential elimination ru...
nexd 2206 Deduction for generalizati...
albid 2207 Formula-building rule for ...
exbid 2208 Formula-building rule for ...
nfbidf 2209 An equality theorem for ef...
19.16 2210 Theorem 19.16 of [Margaris...
19.17 2211 Theorem 19.17 of [Margaris...
19.27 2212 Theorem 19.27 of [Margaris...
19.28 2213 Theorem 19.28 of [Margaris...
19.19 2214 Theorem 19.19 of [Margaris...
19.36 2215 Theorem 19.36 of [Margaris...
19.36i 2216 Inference associated with ...
19.37 2217 Theorem 19.37 of [Margaris...
19.32 2218 Theorem 19.32 of [Margaris...
19.31 2219 Theorem 19.31 of [Margaris...
19.41 2220 Theorem 19.41 of [Margaris...
19.42 2221 Theorem 19.42 of [Margaris...
19.44 2222 Theorem 19.44 of [Margaris...
19.45 2223 Theorem 19.45 of [Margaris...
spimfv 2224 Specialization, using impl...
chvarfv 2225 Implicit substitution of `...
cbv3v2 2226 Version of ~ cbv3 with two...
sbalex 2227 Equivalence of two ways to...
sb4av 2228 Version of ~ sb4a with a d...
sbimd 2229 Deduction substituting bot...
sbbid 2230 Deduction substituting bot...
2sbbid 2231 Deduction doubly substitut...
sbequ1 2232 An equality theorem for su...
sbequ2 2233 An equality theorem for su...
stdpc7 2234 One of the two equality ax...
sbequ12 2235 An equality theorem for su...
sbequ12r 2236 An equality theorem for su...
sbelx 2237 Elimination of substitutio...
sbequ12a 2238 An equality theorem for su...
sbid 2239 An identity theorem for su...
sbcov 2240 A composition law for subs...
sb6a 2241 Equivalence for substituti...
sbid2vw 2242 Reverting substitution yie...
axc16g 2243 Generalization of ~ axc16 ...
axc16 2244 Proof of older axiom ~ ax-...
axc16gb 2245 Biconditional strengthenin...
axc16nf 2246 If ~ dtru is false, then t...
axc11v 2247 Version of ~ axc11 with a ...
axc11rv 2248 Version of ~ axc11r with a...
drsb2 2249 Formula-building lemma for...
equsalv 2250 An equivalence related to ...
equsexv 2251 An equivalence related to ...
equsexvOLD 2252 Obsolete version of ~ equs...
sbft 2253 Substitution has no effect...
sbf 2254 Substitution for a variabl...
sbf2 2255 Substitution has no effect...
sbh 2256 Substitution for a variabl...
hbs1 2257 The setvar ` x ` is not fr...
nfs1f 2258 If ` x ` is not free in ` ...
sb5 2259 Alternate definition of su...
sb5OLD 2260 Obsolete version of ~ sb5 ...
sb56OLD 2261 Obsolete version of ~ sbal...
equs5av 2262 A property related to subs...
2sb5 2263 Equivalence for double sub...
sbco4lem 2264 Lemma for ~ sbco4 . It re...
sbco4lemOLD 2265 Obsolete version of ~ sbco...
sbco4 2266 Two ways of exchanging two...
dfsb7 2267 An alternate definition of...
sbn 2268 Negation inside and outsid...
sbex 2269 Move existential quantifie...
nf5 2270 Alternate definition of ~ ...
nf6 2271 An alternate definition of...
nf5d 2272 Deduce that ` x ` is not f...
nf5di 2273 Since the converse holds b...
19.9h 2274 A wff may be existentially...
19.21h 2275 Theorem 19.21 of [Margaris...
19.23h 2276 Theorem 19.23 of [Margaris...
exlimih 2277 Inference associated with ...
exlimdh 2278 Deduction form of Theorem ...
equsalhw 2279 Version of ~ equsalh with ...
equsexhv 2280 An equivalence related to ...
hba1 2281 The setvar ` x ` is not fr...
hbnt 2282 Closed theorem version of ...
hbn 2283 If ` x ` is not free in ` ...
hbnd 2284 Deduction form of bound-va...
hbim1 2285 A closed form of ~ hbim . ...
hbimd 2286 Deduction form of bound-va...
hbim 2287 If ` x ` is not free in ` ...
hban 2288 If ` x ` is not free in ` ...
hb3an 2289 If ` x ` is not free in ` ...
sbi2 2290 Introduction of implicatio...
sbim 2291 Implication inside and out...
sbrim 2292 Substitution in an implica...
sbrimOLD 2293 Obsolete version of ~ sbri...
sblim 2294 Substitution in an implica...
sbor 2295 Disjunction inside and out...
sbbi 2296 Equivalence inside and out...
sblbis 2297 Introduce left bicondition...
sbrbis 2298 Introduce right biconditio...
sbrbif 2299 Introduce right biconditio...
sbiev 2300 Conversion of implicit sub...
sbiedw 2301 Conversion of implicit sub...
axc7 2302 Show that the original axi...
axc7e 2303 Abbreviated version of ~ a...
modal-b 2304 The analogue in our predic...
19.9ht 2305 A closed version of ~ 19.9...
axc4 2306 Show that the original axi...
axc4i 2307 Inference version of ~ axc...
nfal 2308 If ` x ` is not free in ` ...
nfex 2309 If ` x ` is not free in ` ...
hbex 2310 If ` x ` is not free in ` ...
nfnf 2311 If ` x ` is not free in ` ...
19.12 2312 Theorem 19.12 of [Margaris...
nfald 2313 Deduction form of ~ nfal ....
nfexd 2314 If ` x ` is not free in ` ...
nfsbv 2315 If ` z ` is not free in ` ...
nfsbvOLD 2316 Obsolete version of ~ nfsb...
hbsbwOLD 2317 Obsolete version of ~ hbsb...
sbco2v 2318 A composition law for subs...
aaan 2319 Distribute universal quant...
aaanOLD 2320 Obsolete version of ~ aaan...
eeor 2321 Distribute existential qua...
eeorOLD 2322 Obsolete version of ~ eeor...
cbv3v 2323 Rule used to change bound ...
cbv1v 2324 Rule used to change bound ...
cbv2w 2325 Rule used to change bound ...
cbvaldw 2326 Deduction used to change b...
cbvexdw 2327 Deduction used to change b...
cbv3hv 2328 Rule used to change bound ...
cbvalv1 2329 Rule used to change bound ...
cbvexv1 2330 Rule used to change bound ...
cbval2v 2331 Rule used to change bound ...
cbvex2v 2332 Rule used to change bound ...
dvelimhw 2333 Proof of ~ dvelimh without...
pm11.53 2334 Theorem *11.53 in [Whitehe...
19.12vv 2335 Special case of ~ 19.12 wh...
eean 2336 Distribute existential qua...
eeanv 2337 Distribute a pair of exist...
eeeanv 2338 Distribute three existenti...
ee4anv 2339 Distribute two pairs of ex...
sb8v 2340 Substitution of variable i...
sb8f 2341 Substitution of variable i...
sb8fOLD 2342 Obsolete version of ~ sb8f...
sb8ef 2343 Substitution of variable i...
2sb8ef 2344 An equivalent expression f...
sb6rfv 2345 Reversed substitution. Ve...
sbnf2 2346 Two ways of expressing " `...
exsb 2347 An equivalent expression f...
2exsb 2348 An equivalent expression f...
sbbib 2349 Reversal of substitution. ...
sbbibvv 2350 Reversal of substitution. ...
cbvsbv 2351 Change the bound variable ...
cbvsbvf 2352 Change the bound variable ...
cleljustALT 2353 Alternate proof of ~ clelj...
cleljustALT2 2354 Alternate proof of ~ clelj...
equs5aALT 2355 Alternate proof of ~ equs5...
equs5eALT 2356 Alternate proof of ~ equs5...
axc11r 2357 Same as ~ axc11 but with r...
dral1v 2358 Formula-building lemma for...
dral1vOLD 2359 Obsolete version of ~ dral...
drex1v 2360 Formula-building lemma for...
drnf1v 2361 Formula-building lemma for...
drnf1vOLD 2362 Obsolete version of ~ drnf...
ax13v 2364 A weaker version of ~ ax-1...
ax13lem1 2365 A version of ~ ax13v with ...
ax13 2366 Derive ~ ax-13 from ~ ax13...
ax13lem2 2367 Lemma for ~ nfeqf2 . This...
nfeqf2 2368 An equation between setvar...
dveeq2 2369 Quantifier introduction wh...
nfeqf1 2370 An equation between setvar...
dveeq1 2371 Quantifier introduction wh...
nfeqf 2372 A variable is effectively ...
axc9 2373 Derive set.mm's original ~...
ax6e 2374 At least one individual ex...
ax6 2375 Theorem showing that ~ ax-...
axc10 2376 Show that the original axi...
spimt 2377 Closed theorem form of ~ s...
spim 2378 Specialization, using impl...
spimed 2379 Deduction version of ~ spi...
spime 2380 Existential introduction, ...
spimv 2381 A version of ~ spim with a...
spimvALT 2382 Alternate proof of ~ spimv...
spimev 2383 Distinct-variable version ...
spv 2384 Specialization, using impl...
spei 2385 Inference from existential...
chvar 2386 Implicit substitution of `...
chvarv 2387 Implicit substitution of `...
cbv3 2388 Rule used to change bound ...
cbval 2389 Rule used to change bound ...
cbvex 2390 Rule used to change bound ...
cbvalv 2391 Rule used to change bound ...
cbvexv 2392 Rule used to change bound ...
cbv1 2393 Rule used to change bound ...
cbv2 2394 Rule used to change bound ...
cbv3h 2395 Rule used to change bound ...
cbv1h 2396 Rule used to change bound ...
cbv2h 2397 Rule used to change bound ...
cbvald 2398 Deduction used to change b...
cbvexd 2399 Deduction used to change b...
cbvaldva 2400 Rule used to change the bo...
cbvexdva 2401 Rule used to change the bo...
cbval2 2402 Rule used to change bound ...
cbvex2 2403 Rule used to change bound ...
cbval2vv 2404 Rule used to change bound ...
cbvex2vv 2405 Rule used to change bound ...
cbvex4v 2406 Rule used to change bound ...
equs4 2407 Lemma used in proofs of im...
equsal 2408 An equivalence related to ...
equsex 2409 An equivalence related to ...
equsexALT 2410 Alternate proof of ~ equse...
equsalh 2411 An equivalence related to ...
equsexh 2412 An equivalence related to ...
axc15 2413 Derivation of set.mm's ori...
ax12 2414 Rederivation of Axiom ~ ax...
ax12b 2415 A bidirectional version of...
ax13ALT 2416 Alternate proof of ~ ax13 ...
axc11n 2417 Derive set.mm's original ~...
aecom 2418 Commutation law for identi...
aecoms 2419 A commutation rule for ide...
naecoms 2420 A commutation rule for dis...
axc11 2421 Show that ~ ax-c11 can be ...
hbae 2422 All variables are effectiv...
hbnae 2423 All variables are effectiv...
nfae 2424 All variables are effectiv...
nfnae 2425 All variables are effectiv...
hbnaes 2426 Rule that applies ~ hbnae ...
axc16i 2427 Inference with ~ axc16 as ...
axc16nfALT 2428 Alternate proof of ~ axc16...
dral2 2429 Formula-building lemma for...
dral1 2430 Formula-building lemma for...
dral1ALT 2431 Alternate proof of ~ dral1...
drex1 2432 Formula-building lemma for...
drex2 2433 Formula-building lemma for...
drnf1 2434 Formula-building lemma for...
drnf2 2435 Formula-building lemma for...
nfald2 2436 Variation on ~ nfald which...
nfexd2 2437 Variation on ~ nfexd which...
exdistrf 2438 Distribution of existentia...
dvelimf 2439 Version of ~ dvelimv witho...
dvelimdf 2440 Deduction form of ~ dvelim...
dvelimh 2441 Version of ~ dvelim withou...
dvelim 2442 This theorem can be used t...
dvelimv 2443 Similar to ~ dvelim with f...
dvelimnf 2444 Version of ~ dvelim using ...
dveeq2ALT 2445 Alternate proof of ~ dveeq...
equvini 2446 A variable introduction la...
equvel 2447 A variable elimination law...
equs5a 2448 A property related to subs...
equs5e 2449 A property related to subs...
equs45f 2450 Two ways of expressing sub...
equs5 2451 Lemma used in proofs of su...
dveel1 2452 Quantifier introduction wh...
dveel2 2453 Quantifier introduction wh...
axc14 2454 Axiom ~ ax-c14 is redundan...
sb6x 2455 Equivalence involving subs...
sbequ5 2456 Substitution does not chan...
sbequ6 2457 Substitution does not chan...
sb5rf 2458 Reversed substitution. Us...
sb6rf 2459 Reversed substitution. Fo...
ax12vALT 2460 Alternate proof of ~ ax12v...
2ax6elem 2461 We can always find values ...
2ax6e 2462 We can always find values ...
2sb5rf 2463 Reversed double substituti...
2sb6rf 2464 Reversed double substituti...
sbel2x 2465 Elimination of double subs...
sb4b 2466 Simplified definition of s...
sb3b 2467 Simplified definition of s...
sb3 2468 One direction of a simplif...
sb1 2469 One direction of a simplif...
sb2 2470 One direction of a simplif...
sb4a 2471 A version of one implicati...
dfsb1 2472 Alternate definition of su...
hbsb2 2473 Bound-variable hypothesis ...
nfsb2 2474 Bound-variable hypothesis ...
hbsb2a 2475 Special case of a bound-va...
sb4e 2476 One direction of a simplif...
hbsb2e 2477 Special case of a bound-va...
hbsb3 2478 If ` y ` is not free in ` ...
nfs1 2479 If ` y ` is not free in ` ...
axc16ALT 2480 Alternate proof of ~ axc16...
axc16gALT 2481 Alternate proof of ~ axc16...
equsb1 2482 Substitution applied to an...
equsb2 2483 Substitution applied to an...
dfsb2 2484 An alternate definition of...
dfsb3 2485 An alternate definition of...
drsb1 2486 Formula-building lemma for...
sb2ae 2487 In the case of two success...
sb6f 2488 Equivalence for substituti...
sb5f 2489 Equivalence for substituti...
nfsb4t 2490 A variable not free in a p...
nfsb4 2491 A variable not free in a p...
sbequ8 2492 Elimination of equality fr...
sbie 2493 Conversion of implicit sub...
sbied 2494 Conversion of implicit sub...
sbiedv 2495 Conversion of implicit sub...
2sbiev 2496 Conversion of double impli...
sbcom3 2497 Substituting ` y ` for ` x...
sbco 2498 A composition law for subs...
sbid2 2499 An identity law for substi...
sbid2v 2500 An identity law for substi...
sbidm 2501 An idempotent law for subs...
sbco2 2502 A composition law for subs...
sbco2d 2503 A composition law for subs...
sbco3 2504 A composition law for subs...
sbcom 2505 A commutativity law for su...
sbtrt 2506 Partially closed form of ~...
sbtr 2507 A partial converse to ~ sb...
sb8 2508 Substitution of variable i...
sb8e 2509 Substitution of variable i...
sb9 2510 Commutation of quantificat...
sb9i 2511 Commutation of quantificat...
sbhb 2512 Two ways of expressing " `...
nfsbd 2513 Deduction version of ~ nfs...
nfsb 2514 If ` z ` is not free in ` ...
hbsb 2515 If ` z ` is not free in ` ...
sb7f 2516 This version of ~ dfsb7 do...
sb7h 2517 This version of ~ dfsb7 do...
sb10f 2518 Hao Wang's identity axiom ...
sbal1 2519 Check out ~ sbal for a ver...
sbal2 2520 Move quantifier in and out...
2sb8e 2521 An equivalent expression f...
dfmoeu 2522 An elementary proof of ~ m...
dfeumo 2523 An elementary proof showin...
mojust 2525 Soundness justification th...
nexmo 2527 Nonexistence implies uniqu...
exmo 2528 Any proposition holds for ...
moabs 2529 Absorption of existence co...
moim 2530 The at-most-one quantifier...
moimi 2531 The at-most-one quantifier...
moimdv 2532 The at-most-one quantifier...
mobi 2533 Equivalence theorem for th...
mobii 2534 Formula-building rule for ...
mobidv 2535 Formula-building rule for ...
mobid 2536 Formula-building rule for ...
moa1 2537 If an implication holds fo...
moan 2538 "At most one" is still the...
moani 2539 "At most one" is still tru...
moor 2540 "At most one" is still the...
mooran1 2541 "At most one" imports disj...
mooran2 2542 "At most one" exports disj...
nfmo1 2543 Bound-variable hypothesis ...
nfmod2 2544 Bound-variable hypothesis ...
nfmodv 2545 Bound-variable hypothesis ...
nfmov 2546 Bound-variable hypothesis ...
nfmod 2547 Bound-variable hypothesis ...
nfmo 2548 Bound-variable hypothesis ...
mof 2549 Version of ~ df-mo with di...
mo3 2550 Alternate definition of th...
mo 2551 Equivalent definitions of ...
mo4 2552 At-most-one quantifier exp...
mo4f 2553 At-most-one quantifier exp...
eu3v 2556 An alternate way to expres...
eujust 2557 Soundness justification th...
eujustALT 2558 Alternate proof of ~ eujus...
eu6lem 2559 Lemma of ~ eu6im . A diss...
eu6 2560 Alternate definition of th...
eu6im 2561 One direction of ~ eu6 nee...
euf 2562 Version of ~ eu6 with disj...
euex 2563 Existential uniqueness imp...
eumo 2564 Existential uniqueness imp...
eumoi 2565 Uniqueness inferred from e...
exmoeub 2566 Existence implies that uni...
exmoeu 2567 Existence is equivalent to...
moeuex 2568 Uniqueness implies that ex...
moeu 2569 Uniqueness is equivalent t...
eubi 2570 Equivalence theorem for th...
eubii 2571 Introduce unique existenti...
eubidv 2572 Formula-building rule for ...
eubid 2573 Formula-building rule for ...
nfeu1 2574 Bound-variable hypothesis ...
nfeu1ALT 2575 Alternate proof of ~ nfeu1...
nfeud2 2576 Bound-variable hypothesis ...
nfeudw 2577 Bound-variable hypothesis ...
nfeud 2578 Bound-variable hypothesis ...
nfeuw 2579 Bound-variable hypothesis ...
nfeu 2580 Bound-variable hypothesis ...
dfeu 2581 Rederive ~ df-eu from the ...
dfmo 2582 Rederive ~ df-mo from the ...
euequ 2583 There exists a unique set ...
sb8eulem 2584 Lemma. Factor out the com...
sb8euv 2585 Variable substitution in u...
sb8eu 2586 Variable substitution in u...
sb8mo 2587 Variable substitution for ...
cbvmovw 2588 Change bound variable. Us...
cbvmow 2589 Rule used to change bound ...
cbvmowOLD 2590 Obsolete version of ~ cbvm...
cbvmo 2591 Rule used to change bound ...
cbveuvw 2592 Change bound variable. Us...
cbveuw 2593 Version of ~ cbveu with a ...
cbveuwOLD 2594 Obsolete version of ~ cbve...
cbveu 2595 Rule used to change bound ...
cbveuALT 2596 Alternative proof of ~ cbv...
eu2 2597 An alternate way of defini...
eu1 2598 An alternate way to expres...
euor 2599 Introduce a disjunct into ...
euorv 2600 Introduce a disjunct into ...
euor2 2601 Introduce or eliminate a d...
sbmo 2602 Substitution into an at-mo...
eu4 2603 Uniqueness using implicit ...
euimmo 2604 Existential uniqueness imp...
euim 2605 Add unique existential qua...
moanimlem 2606 Factor out the common proo...
moanimv 2607 Introduction of a conjunct...
moanim 2608 Introduction of a conjunct...
euan 2609 Introduction of a conjunct...
moanmo 2610 Nested at-most-one quantif...
moaneu 2611 Nested at-most-one and uni...
euanv 2612 Introduction of a conjunct...
mopick 2613 "At most one" picks a vari...
moexexlem 2614 Factor out the proof skele...
2moexv 2615 Double quantification with...
moexexvw 2616 "At most one" double quant...
2moswapv 2617 A condition allowing to sw...
2euswapv 2618 A condition allowing to sw...
2euexv 2619 Double quantification with...
2exeuv 2620 Double existential uniquen...
eupick 2621 Existential uniqueness "pi...
eupicka 2622 Version of ~ eupick with c...
eupickb 2623 Existential uniqueness "pi...
eupickbi 2624 Theorem *14.26 in [Whitehe...
mopick2 2625 "At most one" can show the...
moexex 2626 "At most one" double quant...
moexexv 2627 "At most one" double quant...
2moex 2628 Double quantification with...
2euex 2629 Double quantification with...
2eumo 2630 Nested unique existential ...
2eu2ex 2631 Double existential uniquen...
2moswap 2632 A condition allowing to sw...
2euswap 2633 A condition allowing to sw...
2exeu 2634 Double existential uniquen...
2mo2 2635 Two ways of expressing "th...
2mo 2636 Two ways of expressing "th...
2mos 2637 Double "there exists at mo...
2eu1 2638 Double existential uniquen...
2eu1v 2639 Double existential uniquen...
2eu2 2640 Double existential uniquen...
2eu3 2641 Double existential uniquen...
2eu4 2642 This theorem provides us w...
2eu5 2643 An alternate definition of...
2eu6 2644 Two equivalent expressions...
2eu7 2645 Two equivalent expressions...
2eu8 2646 Two equivalent expressions...
euae 2647 Two ways to express "exact...
exists1 2648 Two ways to express "exact...
exists2 2649 A condition implying that ...
barbara 2650 "Barbara", one of the fund...
celarent 2651 "Celarent", one of the syl...
darii 2652 "Darii", one of the syllog...
dariiALT 2653 Alternate proof of ~ darii...
ferio 2654 "Ferio" ("Ferioque"), one ...
barbarilem 2655 Lemma for ~ barbari and th...
barbari 2656 "Barbari", one of the syll...
barbariALT 2657 Alternate proof of ~ barba...
celaront 2658 "Celaront", one of the syl...
cesare 2659 "Cesare", one of the syllo...
camestres 2660 "Camestres", one of the sy...
festino 2661 "Festino", one of the syll...
festinoALT 2662 Alternate proof of ~ festi...
baroco 2663 "Baroco", one of the syllo...
barocoALT 2664 Alternate proof of ~ festi...
cesaro 2665 "Cesaro", one of the syllo...
camestros 2666 "Camestros", one of the sy...
datisi 2667 "Datisi", one of the syllo...
disamis 2668 "Disamis", one of the syll...
ferison 2669 "Ferison", one of the syll...
bocardo 2670 "Bocardo", one of the syll...
darapti 2671 "Darapti", one of the syll...
daraptiALT 2672 Alternate proof of ~ darap...
felapton 2673 "Felapton", one of the syl...
calemes 2674 "Calemes", one of the syll...
dimatis 2675 "Dimatis", one of the syll...
fresison 2676 "Fresison", one of the syl...
calemos 2677 "Calemos", one of the syll...
fesapo 2678 "Fesapo", one of the syllo...
bamalip 2679 "Bamalip", one of the syll...
axia1 2680 Left 'and' elimination (in...
axia2 2681 Right 'and' elimination (i...
axia3 2682 'And' introduction (intuit...
axin1 2683 'Not' introduction (intuit...
axin2 2684 'Not' elimination (intuiti...
axio 2685 Definition of 'or' (intuit...
axi4 2686 Specialization (intuitioni...
axi5r 2687 Converse of ~ axc4 (intuit...
axial 2688 The setvar ` x ` is not fr...
axie1 2689 The setvar ` x ` is not fr...
axie2 2690 A key property of existent...
axi9 2691 Axiom of existence (intuit...
axi10 2692 Axiom of Quantifier Substi...
axi12 2693 Axiom of Quantifier Introd...
axbnd 2694 Axiom of Bundling (intuiti...
axexte 2696 The axiom of extensionalit...
axextg 2697 A generalization of the ax...
axextb 2698 A bidirectional version of...
axextmo 2699 There exists at most one s...
nulmo 2700 There exists at most one e...
eleq1ab 2703 Extension (in the sense of...
cleljustab 2704 Extension of ~ cleljust fr...
abid 2705 Simplification of class ab...
vexwt 2706 A standard theorem of pred...
vexw 2707 If ` ph ` is a theorem, th...
vextru 2708 Every setvar is a member o...
nfsab1 2709 Bound-variable hypothesis ...
hbab1 2710 Bound-variable hypothesis ...
hbab1OLD 2711 Obsolete version of ~ hbab...
hbab 2712 Bound-variable hypothesis ...
hbabg 2713 Bound-variable hypothesis ...
nfsab 2714 Bound-variable hypothesis ...
nfsabg 2715 Bound-variable hypothesis ...
dfcleq 2717 The defining characterizat...
cvjust 2718 Every set is a class. Pro...
ax9ALT 2719 Proof of ~ ax-9 from Tarsk...
eleq2w2 2720 A weaker version of ~ eleq...
eqriv 2721 Infer equality of classes ...
eqrdv 2722 Deduce equality of classes...
eqrdav 2723 Deduce equality of classes...
eqid 2724 Law of identity (reflexivi...
eqidd 2725 Class identity law with an...
eqeq1d 2726 Deduction from equality to...
eqeq1dALT 2727 Alternate proof of ~ eqeq1...
eqeq1 2728 Equality implies equivalen...
eqeq1i 2729 Inference from equality to...
eqcomd 2730 Deduction from commutative...
eqcom 2731 Commutative law for class ...
eqcoms 2732 Inference applying commuta...
eqcomi 2733 Inference from commutative...
neqcomd 2734 Commute an inequality. (C...
eqeq2d 2735 Deduction from equality to...
eqeq2 2736 Equality implies equivalen...
eqeq2i 2737 Inference from equality to...
eqeqan12d 2738 A useful inference for sub...
eqeqan12rd 2739 A useful inference for sub...
eqeq12d 2740 A useful inference for sub...
eqeq12 2741 Equality relationship amon...
eqeq12i 2742 A useful inference for sub...
eqeq12OLD 2743 Obsolete version of ~ eqeq...
eqeq12dOLD 2744 Obsolete version of ~ eqeq...
eqeqan12dOLD 2745 Obsolete version of ~ eqeq...
eqeqan12dALT 2746 Alternate proof of ~ eqeqa...
eqtr 2747 Transitive law for class e...
eqtr2 2748 A transitive law for class...
eqtr2OLD 2749 Obsolete version of eqtr2 ...
eqtr3 2750 A transitive law for class...
eqtr3OLD 2751 Obsolete version of ~ eqtr...
eqtri 2752 An equality transitivity i...
eqtr2i 2753 An equality transitivity i...
eqtr3i 2754 An equality transitivity i...
eqtr4i 2755 An equality transitivity i...
3eqtri 2756 An inference from three ch...
3eqtrri 2757 An inference from three ch...
3eqtr2i 2758 An inference from three ch...
3eqtr2ri 2759 An inference from three ch...
3eqtr3i 2760 An inference from three ch...
3eqtr3ri 2761 An inference from three ch...
3eqtr4i 2762 An inference from three ch...
3eqtr4ri 2763 An inference from three ch...
eqtrd 2764 An equality transitivity d...
eqtr2d 2765 An equality transitivity d...
eqtr3d 2766 An equality transitivity e...
eqtr4d 2767 An equality transitivity e...
3eqtrd 2768 A deduction from three cha...
3eqtrrd 2769 A deduction from three cha...
3eqtr2d 2770 A deduction from three cha...
3eqtr2rd 2771 A deduction from three cha...
3eqtr3d 2772 A deduction from three cha...
3eqtr3rd 2773 A deduction from three cha...
3eqtr4d 2774 A deduction from three cha...
3eqtr4rd 2775 A deduction from three cha...
eqtrid 2776 An equality transitivity d...
eqtr2id 2777 An equality transitivity d...
eqtr3id 2778 An equality transitivity d...
eqtr3di 2779 An equality transitivity d...
eqtrdi 2780 An equality transitivity d...
eqtr2di 2781 An equality transitivity d...
eqtr4di 2782 An equality transitivity d...
eqtr4id 2783 An equality transitivity d...
sylan9eq 2784 An equality transitivity d...
sylan9req 2785 An equality transitivity d...
sylan9eqr 2786 An equality transitivity d...
3eqtr3g 2787 A chained equality inferen...
3eqtr3a 2788 A chained equality inferen...
3eqtr4g 2789 A chained equality inferen...
3eqtr4a 2790 A chained equality inferen...
eq2tri 2791 A compound transitive infe...
abbi 2792 Equivalent formulas yield ...
abbidv 2793 Equivalent wff's yield equ...
abbii 2794 Equivalent wff's yield equ...
abbid 2795 Equivalent wff's yield equ...
abbib 2796 Equal class abstractions r...
cbvabv 2797 Rule used to change bound ...
cbvabw 2798 Rule used to change bound ...
cbvabwOLD 2799 Obsolete version of ~ cbva...
cbvab 2800 Rule used to change bound ...
eqabbw 2801 Version of ~ eqabb using i...
dfclel 2803 Characterization of the el...
elex2 2804 If a class contains anothe...
issetlem 2805 Lemma for ~ elisset and ~ ...
elissetv 2806 An element of a class exis...
elisset 2807 An element of a class exis...
eleq1w 2808 Weaker version of ~ eleq1 ...
eleq2w 2809 Weaker version of ~ eleq2 ...
eleq1d 2810 Deduction from equality to...
eleq2d 2811 Deduction from equality to...
eleq2dALT 2812 Alternate proof of ~ eleq2...
eleq1 2813 Equality implies equivalen...
eleq2 2814 Equality implies equivalen...
eleq12 2815 Equality implies equivalen...
eleq1i 2816 Inference from equality to...
eleq2i 2817 Inference from equality to...
eleq12i 2818 Inference from equality to...
eleq12d 2819 Deduction from equality to...
eleq1a 2820 A transitive-type law rela...
eqeltri 2821 Substitution of equal clas...
eqeltrri 2822 Substitution of equal clas...
eleqtri 2823 Substitution of equal clas...
eleqtrri 2824 Substitution of equal clas...
eqeltrd 2825 Substitution of equal clas...
eqeltrrd 2826 Deduction that substitutes...
eleqtrd 2827 Deduction that substitutes...
eleqtrrd 2828 Deduction that substitutes...
eqeltrid 2829 A membership and equality ...
eqeltrrid 2830 A membership and equality ...
eleqtrid 2831 A membership and equality ...
eleqtrrid 2832 A membership and equality ...
eqeltrdi 2833 A membership and equality ...
eqeltrrdi 2834 A membership and equality ...
eleqtrdi 2835 A membership and equality ...
eleqtrrdi 2836 A membership and equality ...
3eltr3i 2837 Substitution of equal clas...
3eltr4i 2838 Substitution of equal clas...
3eltr3d 2839 Substitution of equal clas...
3eltr4d 2840 Substitution of equal clas...
3eltr3g 2841 Substitution of equal clas...
3eltr4g 2842 Substitution of equal clas...
eleq2s 2843 Substitution of equal clas...
eqneltri 2844 If a class is not an eleme...
eqneltrd 2845 If a class is not an eleme...
eqneltrrd 2846 If a class is not an eleme...
neleqtrd 2847 If a class is not an eleme...
neleqtrrd 2848 If a class is not an eleme...
nelneq 2849 A way of showing two class...
nelneq2 2850 A way of showing two class...
eqsb1 2851 Substitution for the left-...
clelsb1 2852 Substitution for the first...
clelsb2 2853 Substitution for the secon...
clelsb2OLD 2854 Obsolete version of ~ clel...
cleqh 2855 Establish equality between...
hbxfreq 2856 A utility lemma to transfe...
hblem 2857 Change the free variable o...
hblemg 2858 Change the free variable o...
eqabdv 2859 Deduction from a wff to a ...
eqabcdv 2860 Deduction from a wff to a ...
eqabi 2861 Equality of a class variab...
abid1 2862 Every class is equal to a ...
abid2 2863 A simplification of class ...
eqab 2864 One direction of ~ eqabb i...
eqabb 2865 Equality of a class variab...
eqabbOLD 2866 Obsolete version of ~ eqab...
eqabcb 2867 Equality of a class variab...
eqabrd 2868 Equality of a class variab...
eqabri 2869 Equality of a class variab...
eqabcri 2870 Equality of a class variab...
clelab 2871 Membership of a class vari...
clelabOLD 2872 Obsolete version of ~ clel...
clabel 2873 Membership of a class abst...
sbab 2874 The right-hand side of the...
nfcjust 2876 Justification theorem for ...
nfci 2878 Deduce that a class ` A ` ...
nfcii 2879 Deduce that a class ` A ` ...
nfcr 2880 Consequence of the not-fre...
nfcrALT 2881 Alternate version of ~ nfc...
nfcri 2882 Consequence of the not-fre...
nfcd 2883 Deduce that a class ` A ` ...
nfcrd 2884 Consequence of the not-fre...
nfcriOLD 2885 Obsolete version of ~ nfcr...
nfcriOLDOLD 2886 Obsolete version of ~ nfcr...
nfcrii 2887 Consequence of the not-fre...
nfcriiOLD 2888 Obsolete version of ~ nfcr...
nfcriOLDOLDOLD 2889 Obsolete version of ~ nfcr...
nfceqdf 2890 An equality theorem for ef...
nfceqdfOLD 2891 Obsolete version of ~ nfce...
nfceqi 2892 Equality theorem for class...
nfcxfr 2893 A utility lemma to transfe...
nfcxfrd 2894 A utility lemma to transfe...
nfcv 2895 If ` x ` is disjoint from ...
nfcvd 2896 If ` x ` is disjoint from ...
nfab1 2897 Bound-variable hypothesis ...
nfnfc1 2898 The setvar ` x ` is bound ...
clelsb1fw 2899 Substitution for the first...
clelsb1f 2900 Substitution for the first...
nfab 2901 Bound-variable hypothesis ...
nfabg 2902 Bound-variable hypothesis ...
nfaba1 2903 Bound-variable hypothesis ...
nfaba1g 2904 Bound-variable hypothesis ...
nfeqd 2905 Hypothesis builder for equ...
nfeld 2906 Hypothesis builder for ele...
nfnfc 2907 Hypothesis builder for ` F...
nfeq 2908 Hypothesis builder for equ...
nfel 2909 Hypothesis builder for ele...
nfeq1 2910 Hypothesis builder for equ...
nfel1 2911 Hypothesis builder for ele...
nfeq2 2912 Hypothesis builder for equ...
nfel2 2913 Hypothesis builder for ele...
drnfc1 2914 Formula-building lemma for...
drnfc1OLD 2915 Obsolete version of ~ drnf...
drnfc2 2916 Formula-building lemma for...
drnfc2OLD 2917 Obsolete version of ~ drnf...
nfabdw 2918 Bound-variable hypothesis ...
nfabdwOLD 2919 Obsolete version of ~ nfab...
nfabd 2920 Bound-variable hypothesis ...
nfabd2 2921 Bound-variable hypothesis ...
dvelimdc 2922 Deduction form of ~ dvelim...
dvelimc 2923 Version of ~ dvelim for cl...
nfcvf 2924 If ` x ` and ` y ` are dis...
nfcvf2 2925 If ` x ` and ` y ` are dis...
cleqf 2926 Establish equality between...
eqabf 2927 Equality of a class variab...
abid2f 2928 A simplification of class ...
abid2fOLD 2929 Obsolete version of ~ abid...
sbabel 2930 Theorem to move a substitu...
sbabelOLD 2931 Obsolete version of ~ sbab...
neii 2934 Inference associated with ...
neir 2935 Inference associated with ...
nne 2936 Negation of inequality. (...
neneqd 2937 Deduction eliminating ineq...
neneq 2938 From inequality to non-equ...
neqned 2939 If it is not the case that...
neqne 2940 From non-equality to inequ...
neirr 2941 No class is unequal to its...
exmidne 2942 Excluded middle with equal...
eqneqall 2943 A contradiction concerning...
nonconne 2944 Law of noncontradiction wi...
necon3ad 2945 Contrapositive law deducti...
necon3bd 2946 Contrapositive law deducti...
necon2ad 2947 Contrapositive inference f...
necon2bd 2948 Contrapositive inference f...
necon1ad 2949 Contrapositive deduction f...
necon1bd 2950 Contrapositive deduction f...
necon4ad 2951 Contrapositive inference f...
necon4bd 2952 Contrapositive inference f...
necon3d 2953 Contrapositive law deducti...
necon1d 2954 Contrapositive law deducti...
necon2d 2955 Contrapositive inference f...
necon4d 2956 Contrapositive inference f...
necon3ai 2957 Contrapositive inference f...
necon3aiOLD 2958 Obsolete version of ~ neco...
necon3bi 2959 Contrapositive inference f...
necon1ai 2960 Contrapositive inference f...
necon1bi 2961 Contrapositive inference f...
necon2ai 2962 Contrapositive inference f...
necon2bi 2963 Contrapositive inference f...
necon4ai 2964 Contrapositive inference f...
necon3i 2965 Contrapositive inference f...
necon1i 2966 Contrapositive inference f...
necon2i 2967 Contrapositive inference f...
necon4i 2968 Contrapositive inference f...
necon3abid 2969 Deduction from equality to...
necon3bbid 2970 Deduction from equality to...
necon1abid 2971 Contrapositive deduction f...
necon1bbid 2972 Contrapositive inference f...
necon4abid 2973 Contrapositive law deducti...
necon4bbid 2974 Contrapositive law deducti...
necon2abid 2975 Contrapositive deduction f...
necon2bbid 2976 Contrapositive deduction f...
necon3bid 2977 Deduction from equality to...
necon4bid 2978 Contrapositive law deducti...
necon3abii 2979 Deduction from equality to...
necon3bbii 2980 Deduction from equality to...
necon1abii 2981 Contrapositive inference f...
necon1bbii 2982 Contrapositive inference f...
necon2abii 2983 Contrapositive inference f...
necon2bbii 2984 Contrapositive inference f...
necon3bii 2985 Inference from equality to...
necom 2986 Commutation of inequality....
necomi 2987 Inference from commutative...
necomd 2988 Deduction from commutative...
nesym 2989 Characterization of inequa...
nesymi 2990 Inference associated with ...
nesymir 2991 Inference associated with ...
neeq1d 2992 Deduction for inequality. ...
neeq2d 2993 Deduction for inequality. ...
neeq12d 2994 Deduction for inequality. ...
neeq1 2995 Equality theorem for inequ...
neeq2 2996 Equality theorem for inequ...
neeq1i 2997 Inference for inequality. ...
neeq2i 2998 Inference for inequality. ...
neeq12i 2999 Inference for inequality. ...
eqnetrd 3000 Substitution of equal clas...
eqnetrrd 3001 Substitution of equal clas...
neeqtrd 3002 Substitution of equal clas...
eqnetri 3003 Substitution of equal clas...
eqnetrri 3004 Substitution of equal clas...
neeqtri 3005 Substitution of equal clas...
neeqtrri 3006 Substitution of equal clas...
neeqtrrd 3007 Substitution of equal clas...
eqnetrrid 3008 A chained equality inferen...
3netr3d 3009 Substitution of equality i...
3netr4d 3010 Substitution of equality i...
3netr3g 3011 Substitution of equality i...
3netr4g 3012 Substitution of equality i...
nebi 3013 Contraposition law for ine...
pm13.18 3014 Theorem *13.18 in [Whitehe...
pm13.181 3015 Theorem *13.181 in [Whiteh...
pm13.181OLD 3016 Obsolete version of ~ pm13...
pm2.61ine 3017 Inference eliminating an i...
pm2.21ddne 3018 A contradiction implies an...
pm2.61ne 3019 Deduction eliminating an i...
pm2.61dne 3020 Deduction eliminating an i...
pm2.61dane 3021 Deduction eliminating an i...
pm2.61da2ne 3022 Deduction eliminating two ...
pm2.61da3ne 3023 Deduction eliminating thre...
pm2.61iine 3024 Equality version of ~ pm2....
mteqand 3025 A modus tollens deduction ...
neor 3026 Logical OR with an equalit...
neanior 3027 A De Morgan's law for ineq...
ne3anior 3028 A De Morgan's law for ineq...
neorian 3029 A De Morgan's law for ineq...
nemtbir 3030 An inference from an inequ...
nelne1 3031 Two classes are different ...
nelne2 3032 Two classes are different ...
nelelne 3033 Two classes are different ...
neneor 3034 If two classes are differe...
nfne 3035 Bound-variable hypothesis ...
nfned 3036 Bound-variable hypothesis ...
nabbib 3037 Not equivalent wff's corre...
neli 3040 Inference associated with ...
nelir 3041 Inference associated with ...
nelcon3d 3042 Contrapositive law deducti...
neleq12d 3043 Equality theorem for negat...
neleq1 3044 Equality theorem for negat...
neleq2 3045 Equality theorem for negat...
nfnel 3046 Bound-variable hypothesis ...
nfneld 3047 Bound-variable hypothesis ...
nnel 3048 Negation of negated member...
elnelne1 3049 Two classes are different ...
elnelne2 3050 Two classes are different ...
pm2.24nel 3051 A contradiction concerning...
pm2.61danel 3052 Deduction eliminating an e...
rgen 3055 Generalization rule for re...
ralel 3056 All elements of a class ar...
rgenw 3057 Generalization rule for re...
rgen2w 3058 Generalization rule for re...
mprg 3059 Modus ponens combined with...
mprgbir 3060 Modus ponens on biconditio...
raln 3061 Restricted universally qua...
ralnex 3064 Relationship between restr...
dfrex2 3065 Relationship between restr...
nrex 3066 Inference adding restricte...
alral 3067 Universal quantification i...
rexex 3068 Restricted existence impli...
rextru 3069 Two ways of expressing tha...
ralimi2 3070 Inference quantifying both...
reximi2 3071 Inference quantifying both...
ralimia 3072 Inference quantifying both...
reximia 3073 Inference quantifying both...
ralimiaa 3074 Inference quantifying both...
ralimi 3075 Inference quantifying both...
reximi 3076 Inference quantifying both...
ral2imi 3077 Inference quantifying ante...
ralim 3078 Distribution of restricted...
rexim 3079 Theorem 19.22 of [Margaris...
reximiaOLD 3080 Obsolete version of ~ rexi...
ralbii2 3081 Inference adding different...
rexbii2 3082 Inference adding different...
ralbiia 3083 Inference adding restricte...
rexbiia 3084 Inference adding restricte...
ralbii 3085 Inference adding restricte...
rexbii 3086 Inference adding restricte...
ralanid 3087 Cancellation law for restr...
rexanid 3088 Cancellation law for restr...
ralcom3 3089 A commutation law for rest...
ralcom3OLD 3090 Obsolete version of ~ ralc...
dfral2 3091 Relationship between restr...
rexnal 3092 Relationship between restr...
ralinexa 3093 A transformation of restri...
rexanali 3094 A transformation of restri...
ralbi 3095 Distribute a restricted un...
rexbi 3096 Distribute restricted quan...
rexbiOLD 3097 Obsolete version of ~ rexb...
ralrexbid 3098 Formula-building rule for ...
ralrexbidOLD 3099 Obsolete version of ~ ralr...
r19.35 3100 Restricted quantifier vers...
r19.35OLD 3101 Obsolete version of ~ 19.3...
r19.26m 3102 Version of ~ 19.26 and ~ r...
r19.26 3103 Restricted quantifier vers...
r19.26-3 3104 Version of ~ r19.26 with t...
ralbiim 3105 Split a biconditional and ...
r19.29 3106 Restricted quantifier vers...
r19.29OLD 3107 Obsolete version of ~ r19....
r19.29r 3108 Restricted quantifier vers...
r19.29rOLD 3109 Obsolete version of ~ r19....
r19.29imd 3110 Theorem 19.29 of [Margaris...
r19.40 3111 Restricted quantifier vers...
r19.30 3112 Restricted quantifier vers...
r19.30OLD 3113 Obsolete version of ~ 19.3...
r19.43 3114 Restricted quantifier vers...
2ralimi 3115 Inference quantifying both...
3ralimi 3116 Inference quantifying both...
4ralimi 3117 Inference quantifying both...
5ralimi 3118 Inference quantifying both...
6ralimi 3119 Inference quantifying both...
2ralbii 3120 Inference adding two restr...
2rexbii 3121 Inference adding two restr...
3ralbii 3122 Inference adding three res...
4ralbii 3123 Inference adding four rest...
2ralbiim 3124 Split a biconditional and ...
ralnex2 3125 Relationship between two r...
ralnex3 3126 Relationship between three...
rexnal2 3127 Relationship between two r...
rexnal3 3128 Relationship between three...
nrexralim 3129 Negation of a complex pred...
r19.26-2 3130 Restricted quantifier vers...
2r19.29 3131 Theorem ~ r19.29 with two ...
r19.29d2r 3132 Theorem 19.29 of [Margaris...
r19.29d2rOLD 3133 Obsolete version of ~ r19....
r2allem 3134 Lemma factoring out common...
r2exlem 3135 Lemma factoring out common...
hbralrimi 3136 Inference from Theorem 19....
ralrimiv 3137 Inference from Theorem 19....
ralrimiva 3138 Inference from Theorem 19....
rexlimiva 3139 Inference from Theorem 19....
rexlimiv 3140 Inference from Theorem 19....
nrexdv 3141 Deduction adding restricte...
ralrimivw 3142 Inference from Theorem 19....
rexlimivw 3143 Weaker version of ~ rexlim...
ralrimdv 3144 Inference from Theorem 19....
rexlimdv 3145 Inference from Theorem 19....
ralrimdva 3146 Inference from Theorem 19....
rexlimdva 3147 Inference from Theorem 19....
rexlimdvaa 3148 Inference from Theorem 19....
rexlimdva2 3149 Inference from Theorem 19....
r19.29an 3150 A commonly used pattern in...
rexlimdv3a 3151 Inference from Theorem 19....
rexlimdvw 3152 Inference from Theorem 19....
rexlimddv 3153 Restricted existential eli...
r19.29a 3154 A commonly used pattern in...
ralimdv2 3155 Inference quantifying both...
reximdv2 3156 Deduction quantifying both...
reximdvai 3157 Deduction quantifying both...
reximdvaiOLD 3158 Obsolete version of ~ rexi...
ralimdva 3159 Deduction quantifying both...
reximdva 3160 Deduction quantifying both...
ralimdv 3161 Deduction quantifying both...
reximdv 3162 Deduction from Theorem 19....
reximddv 3163 Deduction from Theorem 19....
reximssdv 3164 Derivation of a restricted...
ralbidv2 3165 Formula-building rule for ...
rexbidv2 3166 Formula-building rule for ...
ralbidva 3167 Formula-building rule for ...
rexbidva 3168 Formula-building rule for ...
ralbidv 3169 Formula-building rule for ...
rexbidv 3170 Formula-building rule for ...
r19.21v 3171 Restricted quantifier vers...
r19.21vOLD 3172 Obsolete version of ~ r19....
r19.37v 3173 Restricted quantifier vers...
r19.23v 3174 Restricted quantifier vers...
r19.36v 3175 Restricted quantifier vers...
rexlimivOLD 3176 Obsolete version of ~ rexl...
rexlimivaOLD 3177 Obsolete version of ~ rexl...
rexlimivwOLD 3178 Obsolete version of ~ rexl...
r19.27v 3179 Restricted quantitifer ver...
r19.41v 3180 Restricted quantifier vers...
r19.28v 3181 Restricted quantifier vers...
r19.42v 3182 Restricted quantifier vers...
r19.32v 3183 Restricted quantifier vers...
r19.45v 3184 Restricted quantifier vers...
r19.44v 3185 One direction of a restric...
r2al 3186 Double restricted universa...
r2ex 3187 Double restricted existent...
r3al 3188 Triple restricted universa...
rgen2 3189 Generalization rule for re...
ralrimivv 3190 Inference from Theorem 19....
rexlimivv 3191 Inference from Theorem 19....
ralrimivva 3192 Inference from Theorem 19....
ralrimdvv 3193 Inference from Theorem 19....
rgen3 3194 Generalization rule for re...
ralrimivvva 3195 Inference from Theorem 19....
ralimdvva 3196 Deduction doubly quantifyi...
reximdvva 3197 Deduction doubly quantifyi...
ralimdvv 3198 Deduction doubly quantifyi...
ralimd4v 3199 Deduction quadrupally quan...
ralimd6v 3200 Deduction sextupally quant...
ralrimdvva 3201 Inference from Theorem 19....
rexlimdvv 3202 Inference from Theorem 19....
rexlimdvva 3203 Inference from Theorem 19....
reximddv2 3204 Double deduction from Theo...
r19.29vva 3205 A commonly used pattern ba...
r19.29vvaOLD 3206 Obsolete version of ~ r19....
2rexbiia 3207 Inference adding two restr...
2ralbidva 3208 Formula-building rule for ...
2rexbidva 3209 Formula-building rule for ...
2ralbidv 3210 Formula-building rule for ...
2rexbidv 3211 Formula-building rule for ...
rexralbidv 3212 Formula-building rule for ...
3ralbidv 3213 Formula-building rule for ...
4ralbidv 3214 Formula-building rule for ...
6ralbidv 3215 Formula-building rule for ...
r19.41vv 3216 Version of ~ r19.41v with ...
reeanlem 3217 Lemma factoring out common...
reeanv 3218 Rearrange restricted exist...
3reeanv 3219 Rearrange three restricted...
2ralor 3220 Distribute restricted univ...
2ralorOLD 3221 Obsolete version of ~ 2ral...
risset 3222 Two ways to say " ` A ` be...
nelb 3223 A definition of ` -. A e. ...
nelbOLD 3224 Obsolete version of ~ nelb...
rspw 3225 Restricted specialization....
cbvralvw 3226 Change the bound variable ...
cbvrexvw 3227 Change the bound variable ...
cbvraldva 3228 Rule used to change the bo...
cbvrexdva 3229 Rule used to change the bo...
cbvral2vw 3230 Change bound variables of ...
cbvrex2vw 3231 Change bound variables of ...
cbvral3vw 3232 Change bound variables of ...
cbvral4vw 3233 Change bound variables of ...
cbvral6vw 3234 Change bound variables of ...
cbvral8vw 3235 Change bound variables of ...
rsp 3236 Restricted specialization....
rspa 3237 Restricted specialization....
rspe 3238 Restricted specialization....
rspec 3239 Specialization rule for re...
r19.21bi 3240 Inference from Theorem 19....
r19.21be 3241 Inference from Theorem 19....
r19.21t 3242 Restricted quantifier vers...
r19.21 3243 Restricted quantifier vers...
r19.23t 3244 Closed theorem form of ~ r...
r19.23 3245 Restricted quantifier vers...
ralrimi 3246 Inference from Theorem 19....
ralrimia 3247 Inference from Theorem 19....
rexlimi 3248 Restricted quantifier vers...
ralimdaa 3249 Deduction quantifying both...
reximdai 3250 Deduction from Theorem 19....
r19.37 3251 Restricted quantifier vers...
r19.41 3252 Restricted quantifier vers...
ralrimd 3253 Inference from Theorem 19....
rexlimd2 3254 Version of ~ rexlimd with ...
rexlimd 3255 Deduction form of ~ rexlim...
r19.29af2 3256 A commonly used pattern ba...
r19.29af 3257 A commonly used pattern ba...
reximd2a 3258 Deduction quantifying both...
ralbida 3259 Formula-building rule for ...
ralbidaOLD 3260 Obsolete version of ~ ralb...
rexbida 3261 Formula-building rule for ...
ralbid 3262 Formula-building rule for ...
rexbid 3263 Formula-building rule for ...
rexbidvALT 3264 Alternate proof of ~ rexbi...
rexbidvaALT 3265 Alternate proof of ~ rexbi...
rsp2 3266 Restricted specialization,...
rsp2e 3267 Restricted specialization....
rspec2 3268 Specialization rule for re...
rspec3 3269 Specialization rule for re...
r2alf 3270 Double restricted universa...
r2exf 3271 Double restricted existent...
2ralbida 3272 Formula-building rule for ...
nfra1 3273 The setvar ` x ` is not fr...
nfre1 3274 The setvar ` x ` is not fr...
ralcom4 3275 Commutation of restricted ...
ralcom4OLD 3276 Obsolete version of ~ ralc...
rexcom4 3277 Commutation of restricted ...
ralcom 3278 Commutation of restricted ...
rexcom 3279 Commutation of restricted ...
rexcomOLD 3280 Obsolete version of ~ rexc...
rexcom4a 3281 Specialized existential co...
ralrot3 3282 Rotate three restricted un...
ralcom13 3283 Swap first and third restr...
ralcom13OLD 3284 Obsolete version of ~ ralc...
rexcom13 3285 Swap first and third restr...
rexrot4 3286 Rotate four restricted exi...
2ex2rexrot 3287 Rotate two existential qua...
nfra2w 3288 Similar to Lemma 24 of [Mo...
nfra2wOLD 3289 Obsolete version of ~ nfra...
hbra1 3290 The setvar ` x ` is not fr...
ralcomf 3291 Commutation of restricted ...
rexcomf 3292 Commutation of restricted ...
cbvralfw 3293 Rule used to change bound ...
cbvrexfw 3294 Rule used to change bound ...
cbvralw 3295 Rule used to change bound ...
cbvrexw 3296 Rule used to change bound ...
hbral 3297 Bound-variable hypothesis ...
nfraldw 3298 Deduction version of ~ nfr...
nfrexdw 3299 Deduction version of ~ nfr...
nfralw 3300 Bound-variable hypothesis ...
nfralwOLD 3301 Obsolete version of ~ nfra...
nfrexw 3302 Bound-variable hypothesis ...
r19.12 3303 Restricted quantifier vers...
r19.12OLD 3304 Obsolete version of ~ 19.1...
reean 3305 Rearrange restricted exist...
cbvralsvw 3306 Change bound variable by u...
cbvrexsvw 3307 Change bound variable by u...
cbvralsvwOLD 3308 Obsolete version of ~ cbvr...
cbvrexsvwOLD 3309 Obsolete version of ~ cbvr...
nfraldwOLD 3310 Obsolete version of ~ nfra...
nfra2wOLDOLD 3311 Obsolete version of ~ nfra...
cbvralfwOLD 3312 Obsolete version of ~ cbvr...
rexeq 3313 Equality theorem for restr...
raleq 3314 Equality theorem for restr...
raleqi 3315 Equality inference for res...
rexeqi 3316 Equality inference for res...
raleqdv 3317 Equality deduction for res...
rexeqdv 3318 Equality deduction for res...
raleqbidva 3319 Equality deduction for res...
rexeqbidva 3320 Equality deduction for res...
raleqbidvv 3321 Version of ~ raleqbidv wit...
raleqbidvvOLD 3322 Obsolete version of ~ rale...
rexeqbidvv 3323 Version of ~ rexeqbidv wit...
rexeqbidvvOLD 3324 Obsolete version of ~ rexe...
raleqbi1dv 3325 Equality deduction for res...
rexeqbi1dv 3326 Equality deduction for res...
raleqOLD 3327 Obsolete version of ~ rale...
rexeqOLD 3328 Obsolete version of ~ rale...
raleleq 3329 All elements of a class ar...
raleqbii 3330 Equality deduction for res...
rexeqbii 3331 Equality deduction for res...
raleleqOLD 3332 Obsolete version of ~ rale...
raleleqALT 3333 Alternate proof of ~ ralel...
raleqbidv 3334 Equality deduction for res...
rexeqbidv 3335 Equality deduction for res...
cbvraldva2 3336 Rule used to change the bo...
cbvrexdva2 3337 Rule used to change the bo...
cbvrexdva2OLD 3338 Obsolete version of ~ cbvr...
cbvraldvaOLD 3339 Obsolete version of ~ cbvr...
cbvrexdvaOLD 3340 Obsolete version of ~ cbvr...
raleqf 3341 Equality theorem for restr...
rexeqf 3342 Equality theorem for restr...
rexeqfOLD 3343 Obsolete version of ~ rexe...
raleqbid 3344 Equality deduction for res...
rexeqbid 3345 Equality deduction for res...
sbralie 3346 Implicit to explicit subst...
sbralieALT 3347 Alternative shorter proof ...
cbvralf 3348 Rule used to change bound ...
cbvrexf 3349 Rule used to change bound ...
cbvral 3350 Rule used to change bound ...
cbvrex 3351 Rule used to change bound ...
cbvralv 3352 Change the bound variable ...
cbvrexv 3353 Change the bound variable ...
cbvralsv 3354 Change bound variable by u...
cbvrexsv 3355 Change bound variable by u...
cbvral2v 3356 Change bound variables of ...
cbvrex2v 3357 Change bound variables of ...
cbvral3v 3358 Change bound variables of ...
rgen2a 3359 Generalization rule for re...
nfrald 3360 Deduction version of ~ nfr...
nfrexd 3361 Deduction version of ~ nfr...
nfral 3362 Bound-variable hypothesis ...
nfrex 3363 Bound-variable hypothesis ...
nfra2 3364 Similar to Lemma 24 of [Mo...
ralcom2 3365 Commutation of restricted ...
reu5 3370 Restricted uniqueness in t...
reurmo 3371 Restricted existential uni...
reurex 3372 Restricted unique existenc...
mormo 3373 Unrestricted "at most one"...
rmobiia 3374 Formula-building rule for ...
reubiia 3375 Formula-building rule for ...
rmobii 3376 Formula-building rule for ...
reubii 3377 Formula-building rule for ...
rmoanid 3378 Cancellation law for restr...
reuanid 3379 Cancellation law for restr...
rmoanidOLD 3380 Obsolete version of ~ rmoa...
reuanidOLD 3381 Obsolete version of ~ reua...
2reu2rex 3382 Double restricted existent...
rmobidva 3383 Formula-building rule for ...
reubidva 3384 Formula-building rule for ...
rmobidv 3385 Formula-building rule for ...
reubidv 3386 Formula-building rule for ...
reueubd 3387 Restricted existential uni...
rmo5 3388 Restricted "at most one" i...
nrexrmo 3389 Nonexistence implies restr...
moel 3390 "At most one" element in a...
cbvrmovw 3391 Change the bound variable ...
cbvreuvw 3392 Change the bound variable ...
moelOLD 3393 Obsolete version of ~ moel...
rmobida 3394 Formula-building rule for ...
reubida 3395 Formula-building rule for ...
rmobidvaOLD 3396 Obsolete version of ~ rmob...
cbvrmow 3397 Change the bound variable ...
cbvreuw 3398 Change the bound variable ...
nfrmo1 3399 The setvar ` x ` is not fr...
nfreu1 3400 The setvar ` x ` is not fr...
nfrmow 3401 Bound-variable hypothesis ...
nfreuw 3402 Bound-variable hypothesis ...
cbvrmowOLD 3403 Obsolete version of ~ cbvr...
cbvreuwOLD 3404 Obsolete version of ~ cbvr...
cbvreuvwOLD 3405 Obsolete version of ~ cbvr...
rmoeq1 3406 Equality theorem for restr...
reueq1 3407 Equality theorem for restr...
rmoeq1OLD 3408 Obsolete version of ~ rmoe...
reueq1OLD 3409 Obsolete version of ~ reue...
rmoeqd 3410 Equality deduction for res...
reueqd 3411 Equality deduction for res...
rmoeq1f 3412 Equality theorem for restr...
reueq1f 3413 Equality theorem for restr...
nfreuwOLD 3414 Obsolete version of ~ nfre...
nfrmowOLD 3415 Obsolete version of ~ nfrm...
cbvreu 3416 Change the bound variable ...
cbvrmo 3417 Change the bound variable ...
cbvrmov 3418 Change the bound variable ...
cbvreuv 3419 Change the bound variable ...
nfrmod 3420 Deduction version of ~ nfr...
nfreud 3421 Deduction version of ~ nfr...
nfrmo 3422 Bound-variable hypothesis ...
nfreu 3423 Bound-variable hypothesis ...
rabbidva2 3426 Equivalent wff's yield equ...
rabbia2 3427 Equivalent wff's yield equ...
rabbiia 3428 Equivalent formulas yield ...
rabbiiaOLD 3429 Obsolete version of ~ rabb...
rabbii 3430 Equivalent wff's correspon...
rabbidva 3431 Equivalent wff's yield equ...
rabbidv 3432 Equivalent wff's yield equ...
rabswap 3433 Swap with a membership rel...
cbvrabv 3434 Rule to change the bound v...
rabeqcda 3435 When ` ps ` is always true...
rabeqc 3436 A restricted class abstrac...
rabeqi 3437 Equality theorem for restr...
rabeq 3438 Equality theorem for restr...
rabeqdv 3439 Equality of restricted cla...
rabeqbidva 3440 Equality of restricted cla...
rabeqbidv 3441 Equality of restricted cla...
rabrabi 3442 Abstract builder restricte...
nfrab1 3443 The abstraction variable i...
rabid 3444 An "identity" law of concr...
rabidim1 3445 Membership in a restricted...
reqabi 3446 Inference from equality of...
rabrab 3447 Abstract builder restricte...
rabrabiOLD 3448 Obsolete version of ~ rabr...
rabbida4 3449 Version of ~ rabbidva2 wit...
rabbida 3450 Equivalent wff's yield equ...
rabbid 3451 Version of ~ rabbidv with ...
rabeqd 3452 Deduction form of ~ rabeq ...
rabeqbida 3453 Version of ~ rabeqbidva wi...
rabbi 3454 Equivalent wff's correspon...
rabid2f 3455 An "identity" law for rest...
rabid2 3456 An "identity" law for rest...
rabid2OLD 3457 Obsolete version of ~ rabi...
rabeqf 3458 Equality theorem for restr...
cbvrabw 3459 Rule to change the bound v...
nfrabw 3460 A variable not free in a w...
nfrabwOLD 3461 Obsolete version of ~ nfra...
rabbidaOLD 3462 Obsolete version of ~ rabb...
rabeqiOLD 3463 Obsolete version of ~ rabe...
nfrab 3464 A variable not free in a w...
cbvrab 3465 Rule to change the bound v...
vjust 3467 Justification theorem for ...
dfv2 3469 Alternate definition of th...
vex 3470 All setvar variables are s...
vexOLD 3471 Obsolete version of ~ vex ...
elv 3472 If a proposition is implie...
elvd 3473 If a proposition is implie...
el2v 3474 If a proposition is implie...
eqv 3475 The universe contains ever...
eqvf 3476 The universe contains ever...
abv 3477 The class of sets verifyin...
abvALT 3478 Alternate proof of ~ abv ,...
isset 3479 Two ways to express that "...
issetft 3480 Closed theorem form of ~ i...
issetf 3481 A version of ~ isset that ...
isseti 3482 A way to say " ` A ` is a ...
issetri 3483 A way to say " ` A ` is a ...
eqvisset 3484 A class equal to a variabl...
elex 3485 If a class is a member of ...
elexi 3486 If a class is a member of ...
elexd 3487 If a class is a member of ...
elex2OLD 3488 Obsolete version of ~ elex...
elex22 3489 If two classes each contai...
prcnel 3490 A proper class doesn't bel...
ralv 3491 A universal quantifier res...
rexv 3492 An existential quantifier ...
reuv 3493 A unique existential quant...
rmov 3494 An at-most-one quantifier ...
rabab 3495 A class abstraction restri...
rexcom4b 3496 Specialized existential co...
ceqsal1t 3497 One direction of ~ ceqsalt...
ceqsalt 3498 Closed theorem version of ...
ceqsralt 3499 Restricted quantifier vers...
ceqsalg 3500 A representation of explic...
ceqsalgALT 3501 Alternate proof of ~ ceqsa...
ceqsal 3502 A representation of explic...
ceqsalALT 3503 A representation of explic...
ceqsalv 3504 A representation of explic...
ceqsalvOLD 3505 Obsolete version of ~ ceqs...
ceqsralv 3506 Restricted quantifier vers...
ceqsralvOLD 3507 Obsolete version of ~ ceqs...
gencl 3508 Implicit substitution for ...
2gencl 3509 Implicit substitution for ...
3gencl 3510 Implicit substitution for ...
cgsexg 3511 Implicit substitution infe...
cgsex2g 3512 Implicit substitution infe...
cgsex4g 3513 An implicit substitution i...
cgsex4gOLD 3514 Obsolete version of ~ cgse...
cgsex4gOLDOLD 3515 Obsolete version of ~ cgse...
ceqsex 3516 Elimination of an existent...
ceqsexOLD 3517 Obsolete version of ~ ceqs...
ceqsexv 3518 Elimination of an existent...
ceqsexvOLD 3519 Obsolete version of ~ ceqs...
ceqsexvOLDOLD 3520 Obsolete version of ~ ceqs...
ceqsexv2d 3521 Elimination of an existent...
ceqsex2 3522 Elimination of two existen...
ceqsex2v 3523 Elimination of two existen...
ceqsex3v 3524 Elimination of three exist...
ceqsex4v 3525 Elimination of four existe...
ceqsex6v 3526 Elimination of six existen...
ceqsex8v 3527 Elimination of eight exist...
gencbvex 3528 Change of bound variable u...
gencbvex2 3529 Restatement of ~ gencbvex ...
gencbval 3530 Change of bound variable u...
sbhypf 3531 Introduce an explicit subs...
sbhypfOLD 3532 Obsolete version of ~ sbhy...
vtoclgft 3533 Closed theorem form of ~ v...
vtocleg 3534 Implicit substitution of a...
vtoclg 3535 Implicit substitution of a...
vtocle 3536 Implicit substitution of a...
vtoclbg 3537 Implicit substitution of a...
vtocl 3538 Implicit substitution of a...
vtocldf 3539 Implicit substitution of a...
vtocld 3540 Implicit substitution of a...
vtocldOLD 3541 Obsolete version of ~ vtoc...
vtocl2d 3542 Implicit substitution of t...
vtoclef 3543 Implicit substitution of a...
vtoclf 3544 Implicit substitution of a...
vtoclfOLD 3545 Obsolete version of ~ vtoc...
vtoclALT 3546 Alternate proof of ~ vtocl...
vtocl2 3547 Implicit substitution of c...
vtocl3 3548 Implicit substitution of c...
vtoclb 3549 Implicit substitution of a...
vtoclgf 3550 Implicit substitution of a...
vtoclg1f 3551 Version of ~ vtoclgf with ...
vtoclgOLD 3552 Obsolete version of ~ vtoc...
vtocl2gf 3553 Implicit substitution of a...
vtocl3gf 3554 Implicit substitution of a...
vtocl2g 3555 Implicit substitution of 2...
vtocl3g 3556 Implicit substitution of a...
vtoclgaf 3557 Implicit substitution of a...
vtoclga 3558 Implicit substitution of a...
vtocl2ga 3559 Implicit substitution of 2...
vtocl2gaf 3560 Implicit substitution of 2...
vtocl3gaf 3561 Implicit substitution of 3...
vtocl3ga 3562 Implicit substitution of 3...
vtocl3gaOLD 3563 Obsolete version of ~ vtoc...
vtocl4g 3564 Implicit substitution of 4...
vtocl4ga 3565 Implicit substitution of 4...
vtoclegft 3566 Implicit substitution of a...
vtoclegftOLD 3567 Obsolete version of ~ vtoc...
vtoclri 3568 Implicit substitution of a...
spcimgft 3569 A closed version of ~ spci...
spcgft 3570 A closed version of ~ spcg...
spcimgf 3571 Rule of specialization, us...
spcimegf 3572 Existential specialization...
spcgf 3573 Rule of specialization, us...
spcegf 3574 Existential specialization...
spcimdv 3575 Restricted specialization,...
spcdv 3576 Rule of specialization, us...
spcimedv 3577 Restricted existential spe...
spcgv 3578 Rule of specialization, us...
spcegv 3579 Existential specialization...
spcedv 3580 Existential specialization...
spc2egv 3581 Existential specialization...
spc2gv 3582 Specialization with two qu...
spc2ed 3583 Existential specialization...
spc2d 3584 Specialization with 2 quan...
spc3egv 3585 Existential specialization...
spc3gv 3586 Specialization with three ...
spcv 3587 Rule of specialization, us...
spcev 3588 Existential specialization...
spc2ev 3589 Existential specialization...
rspct 3590 A closed version of ~ rspc...
rspcdf 3591 Restricted specialization,...
rspc 3592 Restricted specialization,...
rspce 3593 Restricted existential spe...
rspcimdv 3594 Restricted specialization,...
rspcimedv 3595 Restricted existential spe...
rspcdv 3596 Restricted specialization,...
rspcedv 3597 Restricted existential spe...
rspcebdv 3598 Restricted existential spe...
rspcdv2 3599 Restricted specialization,...
rspcv 3600 Restricted specialization,...
rspccv 3601 Restricted specialization,...
rspcva 3602 Restricted specialization,...
rspccva 3603 Restricted specialization,...
rspcev 3604 Restricted existential spe...
rspcdva 3605 Restricted specialization,...
rspcedvd 3606 Restricted existential spe...
rspcedvdw 3607 Version of ~ rspcedvd wher...
rspcime 3608 Prove a restricted existen...
rspceaimv 3609 Restricted existential spe...
rspcedeq1vd 3610 Restricted existential spe...
rspcedeq2vd 3611 Restricted existential spe...
rspc2 3612 Restricted specialization ...
rspc2gv 3613 Restricted specialization ...
rspc2v 3614 2-variable restricted spec...
rspc2va 3615 2-variable restricted spec...
rspc2ev 3616 2-variable restricted exis...
2rspcedvdw 3617 Double application of ~ rs...
rspc2dv 3618 2-variable restricted spec...
rspc3v 3619 3-variable restricted spec...
rspc3ev 3620 3-variable restricted exis...
rspc3dv 3621 3-variable restricted spec...
rspc4v 3622 4-variable restricted spec...
rspc6v 3623 6-variable restricted spec...
rspc8v 3624 8-variable restricted spec...
rspceeqv 3625 Restricted existential spe...
ralxpxfr2d 3626 Transfer a universal quant...
rexraleqim 3627 Statement following from e...
eqvincg 3628 A variable introduction la...
eqvinc 3629 A variable introduction la...
eqvincf 3630 A variable introduction la...
alexeqg 3631 Two ways to express substi...
ceqex 3632 Equality implies equivalen...
ceqsexg 3633 A representation of explic...
ceqsexgv 3634 Elimination of an existent...
ceqsrexv 3635 Elimination of a restricte...
ceqsrexbv 3636 Elimination of a restricte...
ceqsralbv 3637 Elimination of a restricte...
ceqsrex2v 3638 Elimination of a restricte...
clel2g 3639 Alternate definition of me...
clel2gOLD 3640 Obsolete version of ~ clel...
clel2 3641 Alternate definition of me...
clel3g 3642 Alternate definition of me...
clel3 3643 Alternate definition of me...
clel4g 3644 Alternate definition of me...
clel4 3645 Alternate definition of me...
clel4OLD 3646 Obsolete version of ~ clel...
clel5 3647 Alternate definition of cl...
pm13.183 3648 Compare theorem *13.183 in...
rr19.3v 3649 Restricted quantifier vers...
rr19.28v 3650 Restricted quantifier vers...
elab6g 3651 Membership in a class abst...
elabd2 3652 Membership in a class abst...
elabd3 3653 Membership in a class abst...
elabgt 3654 Membership in a class abst...
elabgtOLD 3655 Obsolete version of ~ elab...
elabgf 3656 Membership in a class abst...
elabf 3657 Membership in a class abst...
elabg 3658 Membership in a class abst...
elabgOLD 3659 Obsolete version of ~ elab...
elab 3660 Membership in a class abst...
elabOLD 3661 Obsolete version of ~ elab...
elab2g 3662 Membership in a class abst...
elabd 3663 Explicit demonstration the...
elab2 3664 Membership in a class abst...
elab4g 3665 Membership in a class abst...
elab3gf 3666 Membership in a class abst...
elab3g 3667 Membership in a class abst...
elab3 3668 Membership in a class abst...
elrabi 3669 Implication for the member...
elrabiOLD 3670 Obsolete version of ~ elra...
elrabf 3671 Membership in a restricted...
rabtru 3672 Abstract builder using the...
rabeqcOLD 3673 Obsolete version of ~ rabe...
elrab3t 3674 Membership in a restricted...
elrab 3675 Membership in a restricted...
elrab3 3676 Membership in a restricted...
elrabd 3677 Membership in a restricted...
elrab2 3678 Membership in a restricted...
ralab 3679 Universal quantification o...
ralabOLD 3680 Obsolete version of ~ rala...
ralrab 3681 Universal quantification o...
rexab 3682 Existential quantification...
rexabOLD 3683 Obsolete version of ~ rexa...
rexrab 3684 Existential quantification...
ralab2 3685 Universal quantification o...
ralrab2 3686 Universal quantification o...
rexab2 3687 Existential quantification...
rexrab2 3688 Existential quantification...
reurab 3689 Restricted existential uni...
abidnf 3690 Identity used to create cl...
dedhb 3691 A deduction theorem for co...
class2seteq 3692 Writing a set as a class a...
nelrdva 3693 Deduce negative membership...
eqeu 3694 A condition which implies ...
moeq 3695 There exists at most one s...
eueq 3696 A class is a set if and on...
eueqi 3697 There exists a unique set ...
eueq2 3698 Equality has existential u...
eueq3 3699 Equality has existential u...
moeq3 3700 "At most one" property of ...
mosub 3701 "At most one" remains true...
mo2icl 3702 Theorem for inferring "at ...
mob2 3703 Consequence of "at most on...
moi2 3704 Consequence of "at most on...
mob 3705 Equality implied by "at mo...
moi 3706 Equality implied by "at mo...
morex 3707 Derive membership from uni...
euxfr2w 3708 Transfer existential uniqu...
euxfrw 3709 Transfer existential uniqu...
euxfr2 3710 Transfer existential uniqu...
euxfr 3711 Transfer existential uniqu...
euind 3712 Existential uniqueness via...
reu2 3713 A way to express restricte...
reu6 3714 A way to express restricte...
reu3 3715 A way to express restricte...
reu6i 3716 A condition which implies ...
eqreu 3717 A condition which implies ...
rmo4 3718 Restricted "at most one" u...
reu4 3719 Restricted uniqueness usin...
reu7 3720 Restricted uniqueness usin...
reu8 3721 Restricted uniqueness usin...
rmo3f 3722 Restricted "at most one" u...
rmo4f 3723 Restricted "at most one" u...
reu2eqd 3724 Deduce equality from restr...
reueq 3725 Equality has existential u...
rmoeq 3726 Equality's restricted exis...
rmoan 3727 Restricted "at most one" s...
rmoim 3728 Restricted "at most one" i...
rmoimia 3729 Restricted "at most one" i...
rmoimi 3730 Restricted "at most one" i...
rmoimi2 3731 Restricted "at most one" i...
2reu5a 3732 Double restricted existent...
reuimrmo 3733 Restricted uniqueness impl...
2reuswap 3734 A condition allowing swap ...
2reuswap2 3735 A condition allowing swap ...
reuxfrd 3736 Transfer existential uniqu...
reuxfr 3737 Transfer existential uniqu...
reuxfr1d 3738 Transfer existential uniqu...
reuxfr1ds 3739 Transfer existential uniqu...
reuxfr1 3740 Transfer existential uniqu...
reuind 3741 Existential uniqueness via...
2rmorex 3742 Double restricted quantifi...
2reu5lem1 3743 Lemma for ~ 2reu5 . Note ...
2reu5lem2 3744 Lemma for ~ 2reu5 . (Cont...
2reu5lem3 3745 Lemma for ~ 2reu5 . This ...
2reu5 3746 Double restricted existent...
2reurmo 3747 Double restricted quantifi...
2reurex 3748 Double restricted quantifi...
2rmoswap 3749 A condition allowing to sw...
2rexreu 3750 Double restricted existent...
cdeqi 3753 Deduce conditional equalit...
cdeqri 3754 Property of conditional eq...
cdeqth 3755 Deduce conditional equalit...
cdeqnot 3756 Distribute conditional equ...
cdeqal 3757 Distribute conditional equ...
cdeqab 3758 Distribute conditional equ...
cdeqal1 3759 Distribute conditional equ...
cdeqab1 3760 Distribute conditional equ...
cdeqim 3761 Distribute conditional equ...
cdeqcv 3762 Conditional equality for s...
cdeqeq 3763 Distribute conditional equ...
cdeqel 3764 Distribute conditional equ...
nfcdeq 3765 If we have a conditional e...
nfccdeq 3766 Variation of ~ nfcdeq for ...
rru 3767 Relative version of Russel...
ru 3768 Russell's Paradox. Propos...
dfsbcq 3771 Proper substitution of a c...
dfsbcq2 3772 This theorem, which is sim...
sbsbc 3773 Show that ~ df-sb and ~ df...
sbceq1d 3774 Equality theorem for class...
sbceq1dd 3775 Equality theorem for class...
sbceqbid 3776 Equality theorem for class...
sbc8g 3777 This is the closest we can...
sbc2or 3778 The disjunction of two equ...
sbcex 3779 By our definition of prope...
sbceq1a 3780 Equality theorem for class...
sbceq2a 3781 Equality theorem for class...
spsbc 3782 Specialization: if a formu...
spsbcd 3783 Specialization: if a formu...
sbcth 3784 A substitution into a theo...
sbcthdv 3785 Deduction version of ~ sbc...
sbcid 3786 An identity theorem for su...
nfsbc1d 3787 Deduction version of ~ nfs...
nfsbc1 3788 Bound-variable hypothesis ...
nfsbc1v 3789 Bound-variable hypothesis ...
nfsbcdw 3790 Deduction version of ~ nfs...
nfsbcw 3791 Bound-variable hypothesis ...
sbccow 3792 A composition law for clas...
nfsbcd 3793 Deduction version of ~ nfs...
nfsbc 3794 Bound-variable hypothesis ...
sbcco 3795 A composition law for clas...
sbcco2 3796 A composition law for clas...
sbc5 3797 An equivalence for class s...
sbc5ALT 3798 Alternate proof of ~ sbc5 ...
sbc6g 3799 An equivalence for class s...
sbc6gOLD 3800 Obsolete version of ~ sbc6...
sbc6 3801 An equivalence for class s...
sbc7 3802 An equivalence for class s...
cbvsbcw 3803 Change bound variables in ...
cbvsbcvw 3804 Change the bound variable ...
cbvsbc 3805 Change bound variables in ...
cbvsbcv 3806 Change the bound variable ...
sbciegft 3807 Conversion of implicit sub...
sbciegf 3808 Conversion of implicit sub...
sbcieg 3809 Conversion of implicit sub...
sbciegOLD 3810 Obsolete version of ~ sbci...
sbcie2g 3811 Conversion of implicit sub...
sbcie 3812 Conversion of implicit sub...
sbciedf 3813 Conversion of implicit sub...
sbcied 3814 Conversion of implicit sub...
sbciedOLD 3815 Obsolete version of ~ sbci...
sbcied2 3816 Conversion of implicit sub...
elrabsf 3817 Membership in a restricted...
eqsbc1 3818 Substitution for the left-...
sbcng 3819 Move negation in and out o...
sbcimg 3820 Distribution of class subs...
sbcan 3821 Distribution of class subs...
sbcor 3822 Distribution of class subs...
sbcbig 3823 Distribution of class subs...
sbcn1 3824 Move negation in and out o...
sbcim1 3825 Distribution of class subs...
sbcim1OLD 3826 Obsolete version of ~ sbci...
sbcbid 3827 Formula-building deduction...
sbcbidv 3828 Formula-building deduction...
sbcbii 3829 Formula-building inference...
sbcbi1 3830 Distribution of class subs...
sbcbi2 3831 Substituting into equivale...
sbcbi2OLD 3832 Obsolete proof of ~ sbcbi2...
sbcal 3833 Move universal quantifier ...
sbcex2 3834 Move existential quantifie...
sbceqal 3835 Class version of one impli...
sbceqalOLD 3836 Obsolete version of ~ sbce...
sbeqalb 3837 Theorem *14.121 in [Whiteh...
eqsbc2 3838 Substitution for the right...
sbc3an 3839 Distribution of class subs...
sbcel1v 3840 Class substitution into a ...
sbcel2gv 3841 Class substitution into a ...
sbcel21v 3842 Class substitution into a ...
sbcimdv 3843 Substitution analogue of T...
sbcimdvOLD 3844 Obsolete version of ~ sbci...
sbctt 3845 Substitution for a variabl...
sbcgf 3846 Substitution for a variabl...
sbc19.21g 3847 Substitution for a variabl...
sbcg 3848 Substitution for a variabl...
sbcgOLD 3849 Obsolete version of ~ sbcg...
sbcgfi 3850 Substitution for a variabl...
sbc2iegf 3851 Conversion of implicit sub...
sbc2ie 3852 Conversion of implicit sub...
sbc2ieOLD 3853 Obsolete version of ~ sbc2...
sbc2iedv 3854 Conversion of implicit sub...
sbc3ie 3855 Conversion of implicit sub...
sbccomlem 3856 Lemma for ~ sbccom . (Con...
sbccom 3857 Commutative law for double...
sbcralt 3858 Interchange class substitu...
sbcrext 3859 Interchange class substitu...
sbcralg 3860 Interchange class substitu...
sbcrex 3861 Interchange class substitu...
sbcreu 3862 Interchange class substitu...
reu8nf 3863 Restricted uniqueness usin...
sbcabel 3864 Interchange class substitu...
rspsbc 3865 Restricted quantifier vers...
rspsbca 3866 Restricted quantifier vers...
rspesbca 3867 Existence form of ~ rspsbc...
spesbc 3868 Existence form of ~ spsbc ...
spesbcd 3869 form of ~ spsbc . (Contri...
sbcth2 3870 A substitution into a theo...
ra4v 3871 Version of ~ ra4 with a di...
ra4 3872 Restricted quantifier vers...
rmo2 3873 Alternate definition of re...
rmo2i 3874 Condition implying restric...
rmo3 3875 Restricted "at most one" u...
rmob 3876 Consequence of "at most on...
rmoi 3877 Consequence of "at most on...
rmob2 3878 Consequence of "restricted...
rmoi2 3879 Consequence of "restricted...
rmoanim 3880 Introduction of a conjunct...
rmoanimALT 3881 Alternate proof of ~ rmoan...
reuan 3882 Introduction of a conjunct...
2reu1 3883 Double restricted existent...
2reu2 3884 Double restricted existent...
csb2 3887 Alternate expression for t...
csbeq1 3888 Analogue of ~ dfsbcq for p...
csbeq1d 3889 Equality deduction for pro...
csbeq2 3890 Substituting into equivale...
csbeq2d 3891 Formula-building deduction...
csbeq2dv 3892 Formula-building deduction...
csbeq2i 3893 Formula-building inference...
csbeq12dv 3894 Formula-building inference...
cbvcsbw 3895 Change bound variables in ...
cbvcsb 3896 Change bound variables in ...
cbvcsbv 3897 Change the bound variable ...
csbid 3898 Analogue of ~ sbid for pro...
csbeq1a 3899 Equality theorem for prope...
csbcow 3900 Composition law for chaine...
csbco 3901 Composition law for chaine...
csbtt 3902 Substitution doesn't affec...
csbconstgf 3903 Substitution doesn't affec...
csbconstg 3904 Substitution doesn't affec...
csbconstgOLD 3905 Obsolete version of ~ csbc...
csbgfi 3906 Substitution for a variabl...
csbconstgi 3907 The proper substitution of...
nfcsb1d 3908 Bound-variable hypothesis ...
nfcsb1 3909 Bound-variable hypothesis ...
nfcsb1v 3910 Bound-variable hypothesis ...
nfcsbd 3911 Deduction version of ~ nfc...
nfcsbw 3912 Bound-variable hypothesis ...
nfcsb 3913 Bound-variable hypothesis ...
csbhypf 3914 Introduce an explicit subs...
csbiebt 3915 Conversion of implicit sub...
csbiedf 3916 Conversion of implicit sub...
csbieb 3917 Bidirectional conversion b...
csbiebg 3918 Bidirectional conversion b...
csbiegf 3919 Conversion of implicit sub...
csbief 3920 Conversion of implicit sub...
csbie 3921 Conversion of implicit sub...
csbieOLD 3922 Obsolete version of ~ csbi...
csbied 3923 Conversion of implicit sub...
csbiedOLD 3924 Obsolete version of ~ csbi...
csbied2 3925 Conversion of implicit sub...
csbie2t 3926 Conversion of implicit sub...
csbie2 3927 Conversion of implicit sub...
csbie2g 3928 Conversion of implicit sub...
cbvrabcsfw 3929 Version of ~ cbvrabcsf wit...
cbvralcsf 3930 A more general version of ...
cbvrexcsf 3931 A more general version of ...
cbvreucsf 3932 A more general version of ...
cbvrabcsf 3933 A more general version of ...
cbvralv2 3934 Rule used to change the bo...
cbvrexv2 3935 Rule used to change the bo...
rspc2vd 3936 Deduction version of 2-var...
difjust 3942 Soundness justification th...
unjust 3944 Soundness justification th...
injust 3946 Soundness justification th...
dfin5 3948 Alternate definition for t...
dfdif2 3949 Alternate definition of cl...
eldif 3950 Expansion of membership in...
eldifd 3951 If a class is in one class...
eldifad 3952 If a class is in the diffe...
eldifbd 3953 If a class is in the diffe...
elneeldif 3954 The elements of a set diff...
velcomp 3955 Characterization of setvar...
elin 3956 Expansion of membership in...
dfss 3958 Variant of subclass defini...
dfss2 3960 Alternate definition of th...
dfss2OLD 3961 Obsolete version of ~ dfss...
dfss3 3962 Alternate definition of su...
dfss6 3963 Alternate definition of su...
dfss2f 3964 Equivalence for subclass r...
dfss3f 3965 Equivalence for subclass r...
nfss 3966 If ` x ` is not free in ` ...
ssel 3967 Membership relationships f...
sselOLD 3968 Obsolete version of ~ ssel...
ssel2 3969 Membership relationships f...
sseli 3970 Membership implication fro...
sselii 3971 Membership inference from ...
sselid 3972 Membership inference from ...
sseld 3973 Membership deduction from ...
sselda 3974 Membership deduction from ...
sseldd 3975 Membership inference from ...
ssneld 3976 If a class is not in anoth...
ssneldd 3977 If an element is not in a ...
ssriv 3978 Inference based on subclas...
ssrd 3979 Deduction based on subclas...
ssrdv 3980 Deduction based on subclas...
sstr2 3981 Transitivity of subclass r...
sstr 3982 Transitivity of subclass r...
sstri 3983 Subclass transitivity infe...
sstrd 3984 Subclass transitivity dedu...
sstrid 3985 Subclass transitivity dedu...
sstrdi 3986 Subclass transitivity dedu...
sylan9ss 3987 A subclass transitivity de...
sylan9ssr 3988 A subclass transitivity de...
eqss 3989 The subclass relationship ...
eqssi 3990 Infer equality from two su...
eqssd 3991 Equality deduction from tw...
sssseq 3992 If a class is a subclass o...
eqrd 3993 Deduce equality of classes...
eqri 3994 Infer equality of classes ...
eqelssd 3995 Equality deduction from su...
ssid 3996 Any class is a subclass of...
ssidd 3997 Weakening of ~ ssid . (Co...
ssv 3998 Any class is a subclass of...
sseq1 3999 Equality theorem for subcl...
sseq2 4000 Equality theorem for the s...
sseq12 4001 Equality theorem for the s...
sseq1i 4002 An equality inference for ...
sseq2i 4003 An equality inference for ...
sseq12i 4004 An equality inference for ...
sseq1d 4005 An equality deduction for ...
sseq2d 4006 An equality deduction for ...
sseq12d 4007 An equality deduction for ...
eqsstri 4008 Substitution of equality i...
eqsstrri 4009 Substitution of equality i...
sseqtri 4010 Substitution of equality i...
sseqtrri 4011 Substitution of equality i...
eqsstrd 4012 Substitution of equality i...
eqsstrrd 4013 Substitution of equality i...
sseqtrd 4014 Substitution of equality i...
sseqtrrd 4015 Substitution of equality i...
3sstr3i 4016 Substitution of equality i...
3sstr4i 4017 Substitution of equality i...
3sstr3g 4018 Substitution of equality i...
3sstr4g 4019 Substitution of equality i...
3sstr3d 4020 Substitution of equality i...
3sstr4d 4021 Substitution of equality i...
eqsstrid 4022 A chained subclass and equ...
eqsstrrid 4023 A chained subclass and equ...
sseqtrdi 4024 A chained subclass and equ...
sseqtrrdi 4025 A chained subclass and equ...
sseqtrid 4026 Subclass transitivity dedu...
sseqtrrid 4027 Subclass transitivity dedu...
eqsstrdi 4028 A chained subclass and equ...
eqsstrrdi 4029 A chained subclass and equ...
eqimssd 4030 Equality implies inclusion...
eqimsscd 4031 Equality implies inclusion...
eqimss 4032 Equality implies inclusion...
eqimss2 4033 Equality implies inclusion...
eqimssi 4034 Infer subclass relationshi...
eqimss2i 4035 Infer subclass relationshi...
nssne1 4036 Two classes are different ...
nssne2 4037 Two classes are different ...
nss 4038 Negation of subclass relat...
nelss 4039 Demonstrate by witnesses t...
ssrexf 4040 Restricted existential qua...
ssrmof 4041 "At most one" existential ...
ssralv 4042 Quantification restricted ...
ssrexv 4043 Existential quantification...
ss2ralv 4044 Two quantifications restri...
ss2rexv 4045 Two existential quantifica...
ralss 4046 Restricted universal quant...
rexss 4047 Restricted existential qua...
ss2ab 4048 Class abstractions in a su...
abss 4049 Class abstraction in a sub...
ssab 4050 Subclass of a class abstra...
ssabral 4051 The relation for a subclas...
ss2abdv 4052 Deduction of abstraction s...
ss2abdvALT 4053 Alternate proof of ~ ss2ab...
ss2abdvOLD 4054 Obsolete version of ~ ss2a...
ss2abi 4055 Inference of abstraction s...
ss2abiOLD 4056 Obsolete version of ~ ss2a...
abssdv 4057 Deduction of abstraction s...
abssdvOLD 4058 Obsolete version of ~ abss...
abssi 4059 Inference of abstraction s...
ss2rab 4060 Restricted abstraction cla...
rabss 4061 Restricted class abstracti...
ssrab 4062 Subclass of a restricted c...
ssrabdv 4063 Subclass of a restricted c...
rabssdv 4064 Subclass of a restricted c...
ss2rabdv 4065 Deduction of restricted ab...
ss2rabi 4066 Inference of restricted ab...
rabss2 4067 Subclass law for restricte...
ssab2 4068 Subclass relation for the ...
ssrab2 4069 Subclass relation for a re...
ssrab2OLD 4070 Obsolete version of ~ ssra...
rabss3d 4071 Subclass law for restricte...
ssrab3 4072 Subclass relation for a re...
rabssrabd 4073 Subclass of a restricted c...
ssrabeq 4074 If the restricting class o...
rabssab 4075 A restricted class is a su...
uniiunlem 4076 A subset relationship usef...
dfpss2 4077 Alternate definition of pr...
dfpss3 4078 Alternate definition of pr...
psseq1 4079 Equality theorem for prope...
psseq2 4080 Equality theorem for prope...
psseq1i 4081 An equality inference for ...
psseq2i 4082 An equality inference for ...
psseq12i 4083 An equality inference for ...
psseq1d 4084 An equality deduction for ...
psseq2d 4085 An equality deduction for ...
psseq12d 4086 An equality deduction for ...
pssss 4087 A proper subclass is a sub...
pssne 4088 Two classes in a proper su...
pssssd 4089 Deduce subclass from prope...
pssned 4090 Proper subclasses are uneq...
sspss 4091 Subclass in terms of prope...
pssirr 4092 Proper subclass is irrefle...
pssn2lp 4093 Proper subclass has no 2-c...
sspsstri 4094 Two ways of stating tricho...
ssnpss 4095 Partial trichotomy law for...
psstr 4096 Transitive law for proper ...
sspsstr 4097 Transitive law for subclas...
psssstr 4098 Transitive law for subclas...
psstrd 4099 Proper subclass inclusion ...
sspsstrd 4100 Transitivity involving sub...
psssstrd 4101 Transitivity involving sub...
npss 4102 A class is not a proper su...
ssnelpss 4103 A subclass missing a membe...
ssnelpssd 4104 Subclass inclusion with on...
ssexnelpss 4105 If there is an element of ...
dfdif3 4106 Alternate definition of cl...
difeq1 4107 Equality theorem for class...
difeq2 4108 Equality theorem for class...
difeq12 4109 Equality theorem for class...
difeq1i 4110 Inference adding differenc...
difeq2i 4111 Inference adding differenc...
difeq12i 4112 Equality inference for cla...
difeq1d 4113 Deduction adding differenc...
difeq2d 4114 Deduction adding differenc...
difeq12d 4115 Equality deduction for cla...
difeqri 4116 Inference from membership ...
nfdif 4117 Bound-variable hypothesis ...
eldifi 4118 Implication of membership ...
eldifn 4119 Implication of membership ...
elndif 4120 A set does not belong to a...
neldif 4121 Implication of membership ...
difdif 4122 Double class difference. ...
difss 4123 Subclass relationship for ...
difssd 4124 A difference of two classe...
difss2 4125 If a class is contained in...
difss2d 4126 If a class is contained in...
ssdifss 4127 Preservation of a subclass...
ddif 4128 Double complement under un...
ssconb 4129 Contraposition law for sub...
sscon 4130 Contraposition law for sub...
ssdif 4131 Difference law for subsets...
ssdifd 4132 If ` A ` is contained in `...
sscond 4133 If ` A ` is contained in `...
ssdifssd 4134 If ` A ` is contained in `...
ssdif2d 4135 If ` A ` is contained in `...
raldifb 4136 Restricted universal quant...
rexdifi 4137 Restricted existential qua...
complss 4138 Complementation reverses i...
compleq 4139 Two classes are equal if a...
elun 4140 Expansion of membership in...
elunnel1 4141 A member of a union that i...
elunnel2 4142 A member of a union that i...
uneqri 4143 Inference from membership ...
unidm 4144 Idempotent law for union o...
uncom 4145 Commutative law for union ...
equncom 4146 If a class equals the unio...
equncomi 4147 Inference form of ~ equnco...
uneq1 4148 Equality theorem for the u...
uneq2 4149 Equality theorem for the u...
uneq12 4150 Equality theorem for the u...
uneq1i 4151 Inference adding union to ...
uneq2i 4152 Inference adding union to ...
uneq12i 4153 Equality inference for the...
uneq1d 4154 Deduction adding union to ...
uneq2d 4155 Deduction adding union to ...
uneq12d 4156 Equality deduction for the...
nfun 4157 Bound-variable hypothesis ...
unass 4158 Associative law for union ...
un12 4159 A rearrangement of union. ...
un23 4160 A rearrangement of union. ...
un4 4161 A rearrangement of the uni...
unundi 4162 Union distributes over its...
unundir 4163 Union distributes over its...
ssun1 4164 Subclass relationship for ...
ssun2 4165 Subclass relationship for ...
ssun3 4166 Subclass law for union of ...
ssun4 4167 Subclass law for union of ...
elun1 4168 Membership law for union o...
elun2 4169 Membership law for union o...
elunant 4170 A statement is true for ev...
unss1 4171 Subclass law for union of ...
ssequn1 4172 A relationship between sub...
unss2 4173 Subclass law for union of ...
unss12 4174 Subclass law for union of ...
ssequn2 4175 A relationship between sub...
unss 4176 The union of two subclasse...
unssi 4177 An inference showing the u...
unssd 4178 A deduction showing the un...
unssad 4179 If ` ( A u. B ) ` is conta...
unssbd 4180 If ` ( A u. B ) ` is conta...
ssun 4181 A condition that implies i...
rexun 4182 Restricted existential qua...
ralunb 4183 Restricted quantification ...
ralun 4184 Restricted quantification ...
elini 4185 Membership in an intersect...
elind 4186 Deduce membership in an in...
elinel1 4187 Membership in an intersect...
elinel2 4188 Membership in an intersect...
elin2 4189 Membership in a class defi...
elin1d 4190 Elementhood in the first s...
elin2d 4191 Elementhood in the first s...
elin3 4192 Membership in a class defi...
incom 4193 Commutative law for inters...
ineqcom 4194 Two ways of expressing tha...
ineqcomi 4195 Two ways of expressing tha...
ineqri 4196 Inference from membership ...
ineq1 4197 Equality theorem for inter...
ineq2 4198 Equality theorem for inter...
ineq12 4199 Equality theorem for inter...
ineq1i 4200 Equality inference for int...
ineq2i 4201 Equality inference for int...
ineq12i 4202 Equality inference for int...
ineq1d 4203 Equality deduction for int...
ineq2d 4204 Equality deduction for int...
ineq12d 4205 Equality deduction for int...
ineqan12d 4206 Equality deduction for int...
sseqin2 4207 A relationship between sub...
nfin 4208 Bound-variable hypothesis ...
rabbi2dva 4209 Deduction from a wff to a ...
inidm 4210 Idempotent law for interse...
inass 4211 Associative law for inters...
in12 4212 A rearrangement of interse...
in32 4213 A rearrangement of interse...
in13 4214 A rearrangement of interse...
in31 4215 A rearrangement of interse...
inrot 4216 Rotate the intersection of...
in4 4217 Rearrangement of intersect...
inindi 4218 Intersection distributes o...
inindir 4219 Intersection distributes o...
inss1 4220 The intersection of two cl...
inss2 4221 The intersection of two cl...
ssin 4222 Subclass of intersection. ...
ssini 4223 An inference showing that ...
ssind 4224 A deduction showing that a...
ssrin 4225 Add right intersection to ...
sslin 4226 Add left intersection to s...
ssrind 4227 Add right intersection to ...
ss2in 4228 Intersection of subclasses...
ssinss1 4229 Intersection preserves sub...
inss 4230 Inclusion of an intersecti...
rexin 4231 Restricted existential qua...
dfss7 4232 Alternate definition of su...
symdifcom 4235 Symmetric difference commu...
symdifeq1 4236 Equality theorem for symme...
symdifeq2 4237 Equality theorem for symme...
nfsymdif 4238 Hypothesis builder for sym...
elsymdif 4239 Membership in a symmetric ...
dfsymdif4 4240 Alternate definition of th...
elsymdifxor 4241 Membership in a symmetric ...
dfsymdif2 4242 Alternate definition of th...
symdifass 4243 Symmetric difference is as...
difsssymdif 4244 The symmetric difference c...
difsymssdifssd 4245 If the symmetric differenc...
unabs 4246 Absorption law for union. ...
inabs 4247 Absorption law for interse...
nssinpss 4248 Negation of subclass expre...
nsspssun 4249 Negation of subclass expre...
dfss4 4250 Subclass defined in terms ...
dfun2 4251 An alternate definition of...
dfin2 4252 An alternate definition of...
difin 4253 Difference with intersecti...
ssdifim 4254 Implication of a class dif...
ssdifsym 4255 Symmetric class difference...
dfss5 4256 Alternate definition of su...
dfun3 4257 Union defined in terms of ...
dfin3 4258 Intersection defined in te...
dfin4 4259 Alternate definition of th...
invdif 4260 Intersection with universa...
indif 4261 Intersection with class di...
indif2 4262 Bring an intersection in a...
indif1 4263 Bring an intersection in a...
indifcom 4264 Commutation law for inters...
indi 4265 Distributive law for inter...
undi 4266 Distributive law for union...
indir 4267 Distributive law for inter...
undir 4268 Distributive law for union...
unineq 4269 Infer equality from equali...
uneqin 4270 Equality of union and inte...
difundi 4271 Distributive law for class...
difundir 4272 Distributive law for class...
difindi 4273 Distributive law for class...
difindir 4274 Distributive law for class...
indifdi 4275 Distribute intersection ov...
indifdir 4276 Distribute intersection ov...
indifdirOLD 4277 Obsolete version of ~ indi...
difdif2 4278 Class difference by a clas...
undm 4279 De Morgan's law for union....
indm 4280 De Morgan's law for inters...
difun1 4281 A relationship involving d...
undif3 4282 An equality involving clas...
difin2 4283 Represent a class differen...
dif32 4284 Swap second and third argu...
difabs 4285 Absorption-like law for cl...
sscon34b 4286 Relative complementation r...
rcompleq 4287 Two subclasses are equal i...
dfsymdif3 4288 Alternate definition of th...
unabw 4289 Union of two class abstrac...
unab 4290 Union of two class abstrac...
inab 4291 Intersection of two class ...
difab 4292 Difference of two class ab...
abanssl 4293 A class abstraction with a...
abanssr 4294 A class abstraction with a...
notabw 4295 A class abstraction define...
notab 4296 A class abstraction define...
unrab 4297 Union of two restricted cl...
inrab 4298 Intersection of two restri...
inrab2 4299 Intersection with a restri...
difrab 4300 Difference of two restrict...
dfrab3 4301 Alternate definition of re...
dfrab2 4302 Alternate definition of re...
notrab 4303 Complementation of restric...
dfrab3ss 4304 Restricted class abstracti...
rabun2 4305 Abstraction restricted to ...
reuun2 4306 Transfer uniqueness to a s...
reuss2 4307 Transfer uniqueness to a s...
reuss 4308 Transfer uniqueness to a s...
reuun1 4309 Transfer uniqueness to a s...
reupick 4310 Restricted uniqueness "pic...
reupick3 4311 Restricted uniqueness "pic...
reupick2 4312 Restricted uniqueness "pic...
euelss 4313 Transfer uniqueness of an ...
dfnul4 4316 Alternate definition of th...
dfnul2 4317 Alternate definition of th...
dfnul3 4318 Alternate definition of th...
dfnul2OLD 4319 Obsolete version of ~ dfnu...
dfnul3OLD 4320 Obsolete version of ~ dfnu...
dfnul4OLD 4321 Obsolete version of ~ dfnu...
noel 4322 The empty set has no eleme...
noelOLD 4323 Obsolete version of ~ noel...
nel02 4324 The empty set has no eleme...
n0i 4325 If a class has elements, t...
ne0i 4326 If a class has elements, t...
ne0d 4327 Deduction form of ~ ne0i ....
n0ii 4328 If a class has elements, t...
ne0ii 4329 If a class has elements, t...
vn0 4330 The universal class is not...
vn0ALT 4331 Alternate proof of ~ vn0 ....
eq0f 4332 A class is equal to the em...
neq0f 4333 A class is not empty if an...
n0f 4334 A class is nonempty if and...
eq0 4335 A class is equal to the em...
eq0ALT 4336 Alternate proof of ~ eq0 ....
neq0 4337 A class is not empty if an...
n0 4338 A class is nonempty if and...
eq0OLDOLD 4339 Obsolete version of ~ eq0 ...
neq0OLD 4340 Obsolete version of ~ neq0...
n0OLD 4341 Obsolete version of ~ n0 a...
nel0 4342 From the general negation ...
reximdva0 4343 Restricted existence deduc...
rspn0 4344 Specialization for restric...
rspn0OLD 4345 Obsolete version of ~ rspn...
n0rex 4346 There is an element in a n...
ssn0rex 4347 There is an element in a c...
n0moeu 4348 A case of equivalence of "...
rex0 4349 Vacuous restricted existen...
reu0 4350 Vacuous restricted uniquen...
rmo0 4351 Vacuous restricted at-most...
0el 4352 Membership of the empty se...
n0el 4353 Negated membership of the ...
eqeuel 4354 A condition which implies ...
ssdif0 4355 Subclass expressed in term...
difn0 4356 If the difference of two s...
pssdifn0 4357 A proper subclass has a no...
pssdif 4358 A proper subclass has a no...
ndisj 4359 Express that an intersecti...
difin0ss 4360 Difference, intersection, ...
inssdif0 4361 Intersection, subclass, an...
difid 4362 The difference between a c...
difidALT 4363 Alternate proof of ~ difid...
dif0 4364 The difference between a c...
ab0w 4365 The class of sets verifyin...
ab0 4366 The class of sets verifyin...
ab0OLD 4367 Obsolete version of ~ ab0 ...
ab0ALT 4368 Alternate proof of ~ ab0 ,...
dfnf5 4369 Characterization of nonfre...
ab0orv 4370 The class abstraction defi...
ab0orvALT 4371 Alternate proof of ~ ab0or...
abn0 4372 Nonempty class abstraction...
abn0OLD 4373 Obsolete version of ~ abn0...
rab0 4374 Any restricted class abstr...
rabeq0w 4375 Condition for a restricted...
rabeq0 4376 Condition for a restricted...
rabn0 4377 Nonempty restricted class ...
rabxm 4378 Law of excluded middle, in...
rabnc 4379 Law of noncontradiction, i...
elneldisj 4380 The set of elements ` s ` ...
elnelun 4381 The union of the set of el...
un0 4382 The union of a class with ...
in0 4383 The intersection of a clas...
0un 4384 The union of the empty set...
0in 4385 The intersection of the em...
inv1 4386 The intersection of a clas...
unv 4387 The union of a class with ...
0ss 4388 The null set is a subset o...
ss0b 4389 Any subset of the empty se...
ss0 4390 Any subset of the empty se...
sseq0 4391 A subclass of an empty cla...
ssn0 4392 A class with a nonempty su...
0dif 4393 The difference between the...
abf 4394 A class abstraction determ...
abfOLD 4395 Obsolete version of ~ abf ...
eq0rdv 4396 Deduction for equality to ...
eq0rdvALT 4397 Alternate proof of ~ eq0rd...
csbprc 4398 The proper substitution of...
csb0 4399 The proper substitution of...
sbcel12 4400 Distribute proper substitu...
sbceqg 4401 Distribute proper substitu...
sbceqi 4402 Distribution of class subs...
sbcnel12g 4403 Distribute proper substitu...
sbcne12 4404 Distribute proper substitu...
sbcel1g 4405 Move proper substitution i...
sbceq1g 4406 Move proper substitution t...
sbcel2 4407 Move proper substitution i...
sbceq2g 4408 Move proper substitution t...
csbcom 4409 Commutative law for double...
sbcnestgfw 4410 Nest the composition of tw...
csbnestgfw 4411 Nest the composition of tw...
sbcnestgw 4412 Nest the composition of tw...
csbnestgw 4413 Nest the composition of tw...
sbcco3gw 4414 Composition of two substit...
sbcnestgf 4415 Nest the composition of tw...
csbnestgf 4416 Nest the composition of tw...
sbcnestg 4417 Nest the composition of tw...
csbnestg 4418 Nest the composition of tw...
sbcco3g 4419 Composition of two substit...
csbco3g 4420 Composition of two class s...
csbnest1g 4421 Nest the composition of tw...
csbidm 4422 Idempotent law for class s...
csbvarg 4423 The proper substitution of...
csbvargi 4424 The proper substitution of...
sbccsb 4425 Substitution into a wff ex...
sbccsb2 4426 Substitution into a wff ex...
rspcsbela 4427 Special case related to ~ ...
sbnfc2 4428 Two ways of expressing " `...
csbab 4429 Move substitution into a c...
csbun 4430 Distribution of class subs...
csbin 4431 Distribute proper substitu...
csbie2df 4432 Conversion of implicit sub...
2nreu 4433 If there are two different...
un00 4434 Two classes are empty iff ...
vss 4435 Only the universal class h...
0pss 4436 The null set is a proper s...
npss0 4437 No set is a proper subset ...
pssv 4438 Any non-universal class is...
disj 4439 Two ways of saying that tw...
disjOLD 4440 Obsolete version of ~ disj...
disjr 4441 Two ways of saying that tw...
disj1 4442 Two ways of saying that tw...
reldisj 4443 Two ways of saying that tw...
reldisjOLD 4444 Obsolete version of ~ reld...
disj3 4445 Two ways of saying that tw...
disjne 4446 Members of disjoint sets a...
disjeq0 4447 Two disjoint sets are equa...
disjel 4448 A set can't belong to both...
disj2 4449 Two ways of saying that tw...
disj4 4450 Two ways of saying that tw...
ssdisj 4451 Intersection with a subcla...
disjpss 4452 A class is a proper subset...
undisj1 4453 The union of disjoint clas...
undisj2 4454 The union of disjoint clas...
ssindif0 4455 Subclass expressed in term...
inelcm 4456 The intersection of classe...
minel 4457 A minimum element of a cla...
undif4 4458 Distribute union over diff...
disjssun 4459 Subset relation for disjoi...
vdif0 4460 Universal class equality i...
difrab0eq 4461 If the difference between ...
pssnel 4462 A proper subclass has a me...
disjdif 4463 A class and its relative c...
disjdifr 4464 A class and its relative c...
difin0 4465 The difference of a class ...
unvdif 4466 The union of a class and i...
undif1 4467 Absorption of difference b...
undif2 4468 Absorption of difference b...
undifabs 4469 Absorption of difference b...
inundif 4470 The intersection and class...
disjdif2 4471 The difference of a class ...
difun2 4472 Absorption of union by dif...
undif 4473 Union of complementary par...
undifr 4474 Union of complementary par...
undifrOLD 4475 Obsolete version of ~ undi...
undif5 4476 An equality involving clas...
ssdifin0 4477 A subset of a difference d...
ssdifeq0 4478 A class is a subclass of i...
ssundif 4479 A condition equivalent to ...
difcom 4480 Swap the arguments of a cl...
pssdifcom1 4481 Two ways to express overla...
pssdifcom2 4482 Two ways to express non-co...
difdifdir 4483 Distributive law for class...
uneqdifeq 4484 Two ways to say that ` A `...
raldifeq 4485 Equality theorem for restr...
r19.2z 4486 Theorem 19.2 of [Margaris]...
r19.2zb 4487 A response to the notion t...
r19.3rz 4488 Restricted quantification ...
r19.28z 4489 Restricted quantifier vers...
r19.3rzv 4490 Restricted quantification ...
r19.9rzv 4491 Restricted quantification ...
r19.28zv 4492 Restricted quantifier vers...
r19.37zv 4493 Restricted quantifier vers...
r19.45zv 4494 Restricted version of Theo...
r19.44zv 4495 Restricted version of Theo...
r19.27z 4496 Restricted quantifier vers...
r19.27zv 4497 Restricted quantifier vers...
r19.36zv 4498 Restricted quantifier vers...
ralidmw 4499 Idempotent law for restric...
rzal 4500 Vacuous quantification is ...
rzalALT 4501 Alternate proof of ~ rzal ...
rexn0 4502 Restricted existential qua...
ralidm 4503 Idempotent law for restric...
ral0 4504 Vacuous universal quantifi...
ralf0 4505 The quantification of a fa...
rexn0OLD 4506 Obsolete version of ~ rexn...
ralidmOLD 4507 Obsolete version of ~ rali...
ral0OLD 4508 Obsolete version of ~ ral0...
ralf0OLD 4509 Obsolete version of ~ ralf...
ralnralall 4510 A contradiction concerning...
falseral0 4511 A false statement can only...
raaan 4512 Rearrange restricted quant...
raaanv 4513 Rearrange restricted quant...
sbss 4514 Set substitution into the ...
sbcssg 4515 Distribute proper substitu...
raaan2 4516 Rearrange restricted quant...
2reu4lem 4517 Lemma for ~ 2reu4 . (Cont...
2reu4 4518 Definition of double restr...
csbdif 4519 Distribution of class subs...
dfif2 4522 An alternate definition of...
dfif6 4523 An alternate definition of...
ifeq1 4524 Equality theorem for condi...
ifeq2 4525 Equality theorem for condi...
iftrue 4526 Value of the conditional o...
iftruei 4527 Inference associated with ...
iftrued 4528 Value of the conditional o...
iffalse 4529 Value of the conditional o...
iffalsei 4530 Inference associated with ...
iffalsed 4531 Value of the conditional o...
ifnefalse 4532 When values are unequal, b...
ifsb 4533 Distribute a function over...
dfif3 4534 Alternate definition of th...
dfif4 4535 Alternate definition of th...
dfif5 4536 Alternate definition of th...
ifssun 4537 A conditional class is inc...
ifeq12 4538 Equality theorem for condi...
ifeq1d 4539 Equality deduction for con...
ifeq2d 4540 Equality deduction for con...
ifeq12d 4541 Equality deduction for con...
ifbi 4542 Equivalence theorem for co...
ifbid 4543 Equivalence deduction for ...
ifbieq1d 4544 Equivalence/equality deduc...
ifbieq2i 4545 Equivalence/equality infer...
ifbieq2d 4546 Equivalence/equality deduc...
ifbieq12i 4547 Equivalence deduction for ...
ifbieq12d 4548 Equivalence deduction for ...
nfifd 4549 Deduction form of ~ nfif ....
nfif 4550 Bound-variable hypothesis ...
ifeq1da 4551 Conditional equality. (Co...
ifeq2da 4552 Conditional equality. (Co...
ifeq12da 4553 Equivalence deduction for ...
ifbieq12d2 4554 Equivalence deduction for ...
ifclda 4555 Conditional closure. (Con...
ifeqda 4556 Separation of the values o...
elimif 4557 Elimination of a condition...
ifbothda 4558 A wff ` th ` containing a ...
ifboth 4559 A wff ` th ` containing a ...
ifid 4560 Identical true and false a...
eqif 4561 Expansion of an equality w...
ifval 4562 Another expression of the ...
elif 4563 Membership in a conditiona...
ifel 4564 Membership of a conditiona...
ifcl 4565 Membership (closure) of a ...
ifcld 4566 Membership (closure) of a ...
ifcli 4567 Inference associated with ...
ifexd 4568 Existence of the condition...
ifexg 4569 Existence of the condition...
ifex 4570 Existence of the condition...
ifeqor 4571 The possible values of a c...
ifnot 4572 Negating the first argumen...
ifan 4573 Rewrite a conjunction in a...
ifor 4574 Rewrite a disjunction in a...
2if2 4575 Resolve two nested conditi...
ifcomnan 4576 Commute the conditions in ...
csbif 4577 Distribute proper substitu...
dedth 4578 Weak deduction theorem tha...
dedth2h 4579 Weak deduction theorem eli...
dedth3h 4580 Weak deduction theorem eli...
dedth4h 4581 Weak deduction theorem eli...
dedth2v 4582 Weak deduction theorem for...
dedth3v 4583 Weak deduction theorem for...
dedth4v 4584 Weak deduction theorem for...
elimhyp 4585 Eliminate a hypothesis con...
elimhyp2v 4586 Eliminate a hypothesis con...
elimhyp3v 4587 Eliminate a hypothesis con...
elimhyp4v 4588 Eliminate a hypothesis con...
elimel 4589 Eliminate a membership hyp...
elimdhyp 4590 Version of ~ elimhyp where...
keephyp 4591 Transform a hypothesis ` p...
keephyp2v 4592 Keep a hypothesis containi...
keephyp3v 4593 Keep a hypothesis containi...
pwjust 4595 Soundness justification th...
elpwg 4597 Membership in a power clas...
elpw 4598 Membership in a power clas...
velpw 4599 Setvar variable membership...
elpwd 4600 Membership in a power clas...
elpwi 4601 Subset relation implied by...
elpwb 4602 Characterization of the el...
elpwid 4603 An element of a power clas...
elelpwi 4604 If ` A ` belongs to a part...
sspw 4605 The powerclass preserves i...
sspwi 4606 The powerclass preserves i...
sspwd 4607 The powerclass preserves i...
pweq 4608 Equality theorem for power...
pweqALT 4609 Alternate proof of ~ pweq ...
pweqi 4610 Equality inference for pow...
pweqd 4611 Equality deduction for pow...
pwunss 4612 The power class of the uni...
nfpw 4613 Bound-variable hypothesis ...
pwidg 4614 A set is an element of its...
pwidb 4615 A class is an element of i...
pwid 4616 A set is a member of its p...
pwss 4617 Subclass relationship for ...
pwundif 4618 Break up the power class o...
snjust 4619 Soundness justification th...
sneq 4630 Equality theorem for singl...
sneqi 4631 Equality inference for sin...
sneqd 4632 Equality deduction for sin...
dfsn2 4633 Alternate definition of si...
elsng 4634 There is exactly one eleme...
elsn 4635 There is exactly one eleme...
velsn 4636 There is only one element ...
elsni 4637 There is at most one eleme...
absn 4638 Condition for a class abst...
dfpr2 4639 Alternate definition of a ...
dfsn2ALT 4640 Alternate definition of si...
elprg 4641 A member of a pair of clas...
elpri 4642 If a class is an element o...
elpr 4643 A member of a pair of clas...
elpr2g 4644 A member of a pair of sets...
elpr2 4645 A member of a pair of sets...
elpr2OLD 4646 Obsolete version of ~ elpr...
nelpr2 4647 If a class is not an eleme...
nelpr1 4648 If a class is not an eleme...
nelpri 4649 If an element doesn't matc...
prneli 4650 If an element doesn't matc...
nelprd 4651 If an element doesn't matc...
eldifpr 4652 Membership in a set with t...
rexdifpr 4653 Restricted existential qua...
snidg 4654 A set is a member of its s...
snidb 4655 A class is a set iff it is...
snid 4656 A set is a member of its s...
vsnid 4657 A setvar variable is a mem...
elsn2g 4658 There is exactly one eleme...
elsn2 4659 There is exactly one eleme...
nelsn 4660 If a class is not equal to...
rabeqsn 4661 Conditions for a restricte...
rabsssn 4662 Conditions for a restricte...
rabeqsnd 4663 Conditions for a restricte...
ralsnsg 4664 Substitution expressed in ...
rexsns 4665 Restricted existential qua...
rexsngf 4666 Restricted existential qua...
ralsngf 4667 Restricted universal quant...
reusngf 4668 Restricted existential uni...
ralsng 4669 Substitution expressed in ...
rexsng 4670 Restricted existential qua...
reusng 4671 Restricted existential uni...
2ralsng 4672 Substitution expressed in ...
ralsngOLD 4673 Obsolete version of ~ rals...
rexsngOLD 4674 Obsolete version of ~ rexs...
rexreusng 4675 Restricted existential uni...
exsnrex 4676 There is a set being the e...
ralsn 4677 Convert a universal quanti...
rexsn 4678 Convert an existential qua...
elpwunsn 4679 Membership in an extension...
eqoreldif 4680 An element of a set is eit...
eltpg 4681 Members of an unordered tr...
eldiftp 4682 Membership in a set with t...
eltpi 4683 A member of an unordered t...
eltp 4684 A member of an unordered t...
dftp2 4685 Alternate definition of un...
nfpr 4686 Bound-variable hypothesis ...
ifpr 4687 Membership of a conditiona...
ralprgf 4688 Convert a restricted unive...
rexprgf 4689 Convert a restricted exist...
ralprg 4690 Convert a restricted unive...
ralprgOLD 4691 Obsolete version of ~ ralp...
rexprg 4692 Convert a restricted exist...
rexprgOLD 4693 Obsolete version of ~ rexp...
raltpg 4694 Convert a restricted unive...
rextpg 4695 Convert a restricted exist...
ralpr 4696 Convert a restricted unive...
rexpr 4697 Convert a restricted exist...
reuprg0 4698 Convert a restricted exist...
reuprg 4699 Convert a restricted exist...
reurexprg 4700 Convert a restricted exist...
raltp 4701 Convert a universal quanti...
rextp 4702 Convert an existential qua...
nfsn 4703 Bound-variable hypothesis ...
csbsng 4704 Distribute proper substitu...
csbprg 4705 Distribute proper substitu...
elinsn 4706 If the intersection of two...
disjsn 4707 Intersection with the sing...
disjsn2 4708 Two distinct singletons ar...
disjpr2 4709 Two completely distinct un...
disjprsn 4710 The disjoint intersection ...
disjtpsn 4711 The disjoint intersection ...
disjtp2 4712 Two completely distinct un...
snprc 4713 The singleton of a proper ...
snnzb 4714 A singleton is nonempty if...
rmosn 4715 A restricted at-most-one q...
r19.12sn 4716 Special case of ~ r19.12 w...
rabsn 4717 Condition where a restrict...
rabsnifsb 4718 A restricted class abstrac...
rabsnif 4719 A restricted class abstrac...
rabrsn 4720 A restricted class abstrac...
euabsn2 4721 Another way to express exi...
euabsn 4722 Another way to express exi...
reusn 4723 A way to express restricte...
absneu 4724 Restricted existential uni...
rabsneu 4725 Restricted existential uni...
eusn 4726 Two ways to express " ` A ...
rabsnt 4727 Truth implied by equality ...
prcom 4728 Commutative law for unorde...
preq1 4729 Equality theorem for unord...
preq2 4730 Equality theorem for unord...
preq12 4731 Equality theorem for unord...
preq1i 4732 Equality inference for uno...
preq2i 4733 Equality inference for uno...
preq12i 4734 Equality inference for uno...
preq1d 4735 Equality deduction for uno...
preq2d 4736 Equality deduction for uno...
preq12d 4737 Equality deduction for uno...
tpeq1 4738 Equality theorem for unord...
tpeq2 4739 Equality theorem for unord...
tpeq3 4740 Equality theorem for unord...
tpeq1d 4741 Equality theorem for unord...
tpeq2d 4742 Equality theorem for unord...
tpeq3d 4743 Equality theorem for unord...
tpeq123d 4744 Equality theorem for unord...
tprot 4745 Rotation of the elements o...
tpcoma 4746 Swap 1st and 2nd members o...
tpcomb 4747 Swap 2nd and 3rd members o...
tpass 4748 Split off the first elemen...
qdass 4749 Two ways to write an unord...
qdassr 4750 Two ways to write an unord...
tpidm12 4751 Unordered triple ` { A , A...
tpidm13 4752 Unordered triple ` { A , B...
tpidm23 4753 Unordered triple ` { A , B...
tpidm 4754 Unordered triple ` { A , A...
tppreq3 4755 An unordered triple is an ...
prid1g 4756 An unordered pair contains...
prid2g 4757 An unordered pair contains...
prid1 4758 An unordered pair contains...
prid2 4759 An unordered pair contains...
ifpprsnss 4760 An unordered pair is a sin...
prprc1 4761 A proper class vanishes in...
prprc2 4762 A proper class vanishes in...
prprc 4763 An unordered pair containi...
tpid1 4764 One of the three elements ...
tpid1g 4765 Closed theorem form of ~ t...
tpid2 4766 One of the three elements ...
tpid2g 4767 Closed theorem form of ~ t...
tpid3g 4768 Closed theorem form of ~ t...
tpid3 4769 One of the three elements ...
snnzg 4770 The singleton of a set is ...
snn0d 4771 The singleton of a set is ...
snnz 4772 The singleton of a set is ...
prnz 4773 A pair containing a set is...
prnzg 4774 A pair containing a set is...
tpnz 4775 An unordered triple contai...
tpnzd 4776 An unordered triple contai...
raltpd 4777 Convert a universal quanti...
snssb 4778 Characterization of the in...
snssg 4779 The singleton formed on a ...
snssgOLD 4780 Obsolete version of ~ snss...
snss 4781 The singleton of an elemen...
eldifsn 4782 Membership in a set with a...
ssdifsn 4783 Subset of a set with an el...
elpwdifsn 4784 A subset of a set is an el...
eldifsni 4785 Membership in a set with a...
eldifsnneq 4786 An element of a difference...
neldifsn 4787 The class ` A ` is not in ...
neldifsnd 4788 The class ` A ` is not in ...
rexdifsn 4789 Restricted existential qua...
raldifsni 4790 Rearrangement of a propert...
raldifsnb 4791 Restricted universal quant...
eldifvsn 4792 A set is an element of the...
difsn 4793 An element not in a set ca...
difprsnss 4794 Removal of a singleton fro...
difprsn1 4795 Removal of a singleton fro...
difprsn2 4796 Removal of a singleton fro...
diftpsn3 4797 Removal of a singleton fro...
difpr 4798 Removing two elements as p...
tpprceq3 4799 An unordered triple is an ...
tppreqb 4800 An unordered triple is an ...
difsnb 4801 ` ( B \ { A } ) ` equals `...
difsnpss 4802 ` ( B \ { A } ) ` is a pro...
snssi 4803 The singleton of an elemen...
snssd 4804 The singleton of an elemen...
difsnid 4805 If we remove a single elem...
eldifeldifsn 4806 An element of a difference...
pw0 4807 Compute the power set of t...
pwpw0 4808 Compute the power set of t...
snsspr1 4809 A singleton is a subset of...
snsspr2 4810 A singleton is a subset of...
snsstp1 4811 A singleton is a subset of...
snsstp2 4812 A singleton is a subset of...
snsstp3 4813 A singleton is a subset of...
prssg 4814 A pair of elements of a cl...
prss 4815 A pair of elements of a cl...
prssi 4816 A pair of elements of a cl...
prssd 4817 Deduction version of ~ prs...
prsspwg 4818 An unordered pair belongs ...
ssprss 4819 A pair as subset of a pair...
ssprsseq 4820 A proper pair is a subset ...
sssn 4821 The subsets of a singleton...
ssunsn2 4822 The property of being sand...
ssunsn 4823 Possible values for a set ...
eqsn 4824 Two ways to express that a...
issn 4825 A sufficient condition for...
n0snor2el 4826 A nonempty set is either a...
ssunpr 4827 Possible values for a set ...
sspr 4828 The subsets of a pair. (C...
sstp 4829 The subsets of an unordere...
tpss 4830 An unordered triple of ele...
tpssi 4831 An unordered triple of ele...
sneqrg 4832 Closed form of ~ sneqr . ...
sneqr 4833 If the singletons of two s...
snsssn 4834 If a singleton is a subset...
mosneq 4835 There exists at most one s...
sneqbg 4836 Two singletons of sets are...
snsspw 4837 The singleton of a class i...
prsspw 4838 An unordered pair belongs ...
preq1b 4839 Biconditional equality lem...
preq2b 4840 Biconditional equality lem...
preqr1 4841 Reverse equality lemma for...
preqr2 4842 Reverse equality lemma for...
preq12b 4843 Equality relationship for ...
opthpr 4844 An unordered pair has the ...
preqr1g 4845 Reverse equality lemma for...
preq12bg 4846 Closed form of ~ preq12b ....
prneimg 4847 Two pairs are not equal if...
prnebg 4848 A (proper) pair is not equ...
pr1eqbg 4849 A (proper) pair is equal t...
pr1nebg 4850 A (proper) pair is not equ...
preqsnd 4851 Equivalence for a pair equ...
prnesn 4852 A proper unordered pair is...
prneprprc 4853 A proper unordered pair is...
preqsn 4854 Equivalence for a pair equ...
preq12nebg 4855 Equality relationship for ...
prel12g 4856 Equality of two unordered ...
opthprneg 4857 An unordered pair has the ...
elpreqprlem 4858 Lemma for ~ elpreqpr . (C...
elpreqpr 4859 Equality and membership ru...
elpreqprb 4860 A set is an element of an ...
elpr2elpr 4861 For an element ` A ` of an...
dfopif 4862 Rewrite ~ df-op using ` if...
dfopg 4863 Value of the ordered pair ...
dfop 4864 Value of an ordered pair w...
opeq1 4865 Equality theorem for order...
opeq2 4866 Equality theorem for order...
opeq12 4867 Equality theorem for order...
opeq1i 4868 Equality inference for ord...
opeq2i 4869 Equality inference for ord...
opeq12i 4870 Equality inference for ord...
opeq1d 4871 Equality deduction for ord...
opeq2d 4872 Equality deduction for ord...
opeq12d 4873 Equality deduction for ord...
oteq1 4874 Equality theorem for order...
oteq2 4875 Equality theorem for order...
oteq3 4876 Equality theorem for order...
oteq1d 4877 Equality deduction for ord...
oteq2d 4878 Equality deduction for ord...
oteq3d 4879 Equality deduction for ord...
oteq123d 4880 Equality deduction for ord...
nfop 4881 Bound-variable hypothesis ...
nfopd 4882 Deduction version of bound...
csbopg 4883 Distribution of class subs...
opidg 4884 The ordered pair ` <. A , ...
opid 4885 The ordered pair ` <. A , ...
ralunsn 4886 Restricted quantification ...
2ralunsn 4887 Double restricted quantifi...
opprc 4888 Expansion of an ordered pa...
opprc1 4889 Expansion of an ordered pa...
opprc2 4890 Expansion of an ordered pa...
oprcl 4891 If an ordered pair has an ...
pwsn 4892 The power set of a singlet...
pwpr 4893 The power set of an unorde...
pwtp 4894 The power set of an unorde...
pwpwpw0 4895 Compute the power set of t...
pwv 4896 The power class of the uni...
prproe 4897 For an element of a proper...
3elpr2eq 4898 If there are three element...
dfuni2 4901 Alternate definition of cl...
eluni 4902 Membership in class union....
eluni2 4903 Membership in class union....
elunii 4904 Membership in class union....
nfunid 4905 Deduction version of ~ nfu...
nfuni 4906 Bound-variable hypothesis ...
uniss 4907 Subclass relationship for ...
unissi 4908 Subclass relationship for ...
unissd 4909 Subclass relationship for ...
unieq 4910 Equality theorem for class...
unieqi 4911 Inference of equality of t...
unieqd 4912 Deduction of equality of t...
eluniab 4913 Membership in union of a c...
elunirab 4914 Membership in union of a c...
uniprg 4915 The union of a pair is the...
unipr 4916 The union of a pair is the...
uniprOLD 4917 Obsolete version of ~ unip...
uniprgOLD 4918 Obsolete version of ~ unip...
unisng 4919 A set equals the union of ...
unisn 4920 A set equals the union of ...
unisnv 4921 A set equals the union of ...
unisn3 4922 Union of a singleton in th...
dfnfc2 4923 An alternative statement o...
uniun 4924 The class union of the uni...
uniin 4925 The class union of the int...
ssuni 4926 Subclass relationship for ...
uni0b 4927 The union of a set is empt...
uni0c 4928 The union of a set is empt...
uni0 4929 The union of the empty set...
csbuni 4930 Distribute proper substitu...
elssuni 4931 An element of a class is a...
unissel 4932 Condition turning a subcla...
unissb 4933 Relationship involving mem...
unissbOLD 4934 Obsolete version of ~ unis...
uniss2 4935 A subclass condition on th...
unidif 4936 If the difference ` A \ B ...
ssunieq 4937 Relationship implying unio...
unimax 4938 Any member of a class is t...
pwuni 4939 A class is a subclass of t...
dfint2 4942 Alternate definition of cl...
inteq 4943 Equality law for intersect...
inteqi 4944 Equality inference for cla...
inteqd 4945 Equality deduction for cla...
elint 4946 Membership in class inters...
elint2 4947 Membership in class inters...
elintg 4948 Membership in class inters...
elinti 4949 Membership in class inters...
nfint 4950 Bound-variable hypothesis ...
elintabg 4951 Two ways of saying a set i...
elintab 4952 Membership in the intersec...
elintabOLD 4953 Obsolete version of ~ elin...
elintrab 4954 Membership in the intersec...
elintrabg 4955 Membership in the intersec...
int0 4956 The intersection of the em...
intss1 4957 An element of a class incl...
ssint 4958 Subclass of a class inters...
ssintab 4959 Subclass of the intersecti...
ssintub 4960 Subclass of the least uppe...
ssmin 4961 Subclass of the minimum va...
intmin 4962 Any member of a class is t...
intss 4963 Intersection of subclasses...
intssuni 4964 The intersection of a none...
ssintrab 4965 Subclass of the intersecti...
unissint 4966 If the union of a class is...
intssuni2 4967 Subclass relationship for ...
intminss 4968 Under subset ordering, the...
intmin2 4969 Any set is the smallest of...
intmin3 4970 Under subset ordering, the...
intmin4 4971 Elimination of a conjunct ...
intab 4972 The intersection of a spec...
int0el 4973 The intersection of a clas...
intun 4974 The class intersection of ...
intprg 4975 The intersection of a pair...
intpr 4976 The intersection of a pair...
intprOLD 4977 Obsolete version of ~ intp...
intprgOLD 4978 Obsolete version of ~ intp...
intsng 4979 Intersection of a singleto...
intsn 4980 The intersection of a sing...
uniintsn 4981 Two ways to express " ` A ...
uniintab 4982 The union and the intersec...
intunsn 4983 Theorem joining a singleto...
rint0 4984 Relative intersection of a...
elrint 4985 Membership in a restricted...
elrint2 4986 Membership in a restricted...
eliun 4991 Membership in indexed unio...
eliin 4992 Membership in indexed inte...
eliuni 4993 Membership in an indexed u...
iuncom 4994 Commutation of indexed uni...
iuncom4 4995 Commutation of union with ...
iunconst 4996 Indexed union of a constan...
iinconst 4997 Indexed intersection of a ...
iuneqconst 4998 Indexed union of identical...
iuniin 4999 Law combining indexed unio...
iinssiun 5000 An indexed intersection is...
iunss1 5001 Subclass theorem for index...
iinss1 5002 Subclass theorem for index...
iuneq1 5003 Equality theorem for index...
iineq1 5004 Equality theorem for index...
ss2iun 5005 Subclass theorem for index...
iuneq2 5006 Equality theorem for index...
iineq2 5007 Equality theorem for index...
iuneq2i 5008 Equality inference for ind...
iineq2i 5009 Equality inference for ind...
iineq2d 5010 Equality deduction for ind...
iuneq2dv 5011 Equality deduction for ind...
iineq2dv 5012 Equality deduction for ind...
iuneq12df 5013 Equality deduction for ind...
iuneq1d 5014 Equality theorem for index...
iuneq12d 5015 Equality deduction for ind...
iuneq2d 5016 Equality deduction for ind...
nfiun 5017 Bound-variable hypothesis ...
nfiin 5018 Bound-variable hypothesis ...
nfiung 5019 Bound-variable hypothesis ...
nfiing 5020 Bound-variable hypothesis ...
nfiu1 5021 Bound-variable hypothesis ...
nfii1 5022 Bound-variable hypothesis ...
dfiun2g 5023 Alternate definition of in...
dfiun2gOLD 5024 Obsolete version of ~ dfiu...
dfiin2g 5025 Alternate definition of in...
dfiun2 5026 Alternate definition of in...
dfiin2 5027 Alternate definition of in...
dfiunv2 5028 Define double indexed unio...
cbviun 5029 Rule used to change the bo...
cbviin 5030 Change bound variables in ...
cbviung 5031 Rule used to change the bo...
cbviing 5032 Change bound variables in ...
cbviunv 5033 Rule used to change the bo...
cbviinv 5034 Change bound variables in ...
cbviunvg 5035 Rule used to change the bo...
cbviinvg 5036 Change bound variables in ...
iunssf 5037 Subset theorem for an inde...
iunss 5038 Subset theorem for an inde...
ssiun 5039 Subset implication for an ...
ssiun2 5040 Identity law for subset of...
ssiun2s 5041 Subset relationship for an...
iunss2 5042 A subclass condition on th...
iunssd 5043 Subset theorem for an inde...
iunab 5044 The indexed union of a cla...
iunrab 5045 The indexed union of a res...
iunxdif2 5046 Indexed union with a class...
ssiinf 5047 Subset theorem for an inde...
ssiin 5048 Subset theorem for an inde...
iinss 5049 Subset implication for an ...
iinss2 5050 An indexed intersection is...
uniiun 5051 Class union in terms of in...
intiin 5052 Class intersection in term...
iunid 5053 An indexed union of single...
iunidOLD 5054 Obsolete version of ~ iuni...
iun0 5055 An indexed union of the em...
0iun 5056 An empty indexed union is ...
0iin 5057 An empty indexed intersect...
viin 5058 Indexed intersection with ...
iunsn 5059 Indexed union of a singlet...
iunn0 5060 There is a nonempty class ...
iinab 5061 Indexed intersection of a ...
iinrab 5062 Indexed intersection of a ...
iinrab2 5063 Indexed intersection of a ...
iunin2 5064 Indexed union of intersect...
iunin1 5065 Indexed union of intersect...
iinun2 5066 Indexed intersection of un...
iundif2 5067 Indexed union of class dif...
iindif1 5068 Indexed intersection of cl...
2iunin 5069 Rearrange indexed unions o...
iindif2 5070 Indexed intersection of cl...
iinin2 5071 Indexed intersection of in...
iinin1 5072 Indexed intersection of in...
iinvdif 5073 The indexed intersection o...
elriin 5074 Elementhood in a relative ...
riin0 5075 Relative intersection of a...
riinn0 5076 Relative intersection of a...
riinrab 5077 Relative intersection of a...
symdif0 5078 Symmetric difference with ...
symdifv 5079 The symmetric difference w...
symdifid 5080 The symmetric difference o...
iinxsng 5081 A singleton index picks ou...
iinxprg 5082 Indexed intersection with ...
iunxsng 5083 A singleton index picks ou...
iunxsn 5084 A singleton index picks ou...
iunxsngf 5085 A singleton index picks ou...
iunun 5086 Separate a union in an ind...
iunxun 5087 Separate a union in the in...
iunxdif3 5088 An indexed union where som...
iunxprg 5089 A pair index picks out two...
iunxiun 5090 Separate an indexed union ...
iinuni 5091 A relationship involving u...
iununi 5092 A relationship involving u...
sspwuni 5093 Subclass relationship for ...
pwssb 5094 Two ways to express a coll...
elpwpw 5095 Characterization of the el...
pwpwab 5096 The double power class wri...
pwpwssunieq 5097 The class of sets whose un...
elpwuni 5098 Relationship for power cla...
iinpw 5099 The power class of an inte...
iunpwss 5100 Inclusion of an indexed un...
intss2 5101 A nonempty intersection of...
rintn0 5102 Relative intersection of a...
dfdisj2 5105 Alternate definition for d...
disjss2 5106 If each element of a colle...
disjeq2 5107 Equality theorem for disjo...
disjeq2dv 5108 Equality deduction for dis...
disjss1 5109 A subset of a disjoint col...
disjeq1 5110 Equality theorem for disjo...
disjeq1d 5111 Equality theorem for disjo...
disjeq12d 5112 Equality theorem for disjo...
cbvdisj 5113 Change bound variables in ...
cbvdisjv 5114 Change bound variables in ...
nfdisjw 5115 Bound-variable hypothesis ...
nfdisj 5116 Bound-variable hypothesis ...
nfdisj1 5117 Bound-variable hypothesis ...
disjor 5118 Two ways to say that a col...
disjors 5119 Two ways to say that a col...
disji2 5120 Property of a disjoint col...
disji 5121 Property of a disjoint col...
invdisj 5122 If there is a function ` C...
invdisjrabw 5123 Version of ~ invdisjrab wi...
invdisjrab 5124 The restricted class abstr...
disjiun 5125 A disjoint collection yiel...
disjord 5126 Conditions for a collectio...
disjiunb 5127 Two ways to say that a col...
disjiund 5128 Conditions for a collectio...
sndisj 5129 Any collection of singleto...
0disj 5130 Any collection of empty se...
disjxsn 5131 A singleton collection is ...
disjx0 5132 An empty collection is dis...
disjprgw 5133 Version of ~ disjprg with ...
disjprg 5134 A pair collection is disjo...
disjxiun 5135 An indexed union of a disj...
disjxun 5136 The union of two disjoint ...
disjss3 5137 Expand a disjoint collecti...
breq 5140 Equality theorem for binar...
breq1 5141 Equality theorem for a bin...
breq2 5142 Equality theorem for a bin...
breq12 5143 Equality theorem for a bin...
breqi 5144 Equality inference for bin...
breq1i 5145 Equality inference for a b...
breq2i 5146 Equality inference for a b...
breq12i 5147 Equality inference for a b...
breq1d 5148 Equality deduction for a b...
breqd 5149 Equality deduction for a b...
breq2d 5150 Equality deduction for a b...
breq12d 5151 Equality deduction for a b...
breq123d 5152 Equality deduction for a b...
breqdi 5153 Equality deduction for a b...
breqan12d 5154 Equality deduction for a b...
breqan12rd 5155 Equality deduction for a b...
eqnbrtrd 5156 Substitution of equal clas...
nbrne1 5157 Two classes are different ...
nbrne2 5158 Two classes are different ...
eqbrtri 5159 Substitution of equal clas...
eqbrtrd 5160 Substitution of equal clas...
eqbrtrri 5161 Substitution of equal clas...
eqbrtrrd 5162 Substitution of equal clas...
breqtri 5163 Substitution of equal clas...
breqtrd 5164 Substitution of equal clas...
breqtrri 5165 Substitution of equal clas...
breqtrrd 5166 Substitution of equal clas...
3brtr3i 5167 Substitution of equality i...
3brtr4i 5168 Substitution of equality i...
3brtr3d 5169 Substitution of equality i...
3brtr4d 5170 Substitution of equality i...
3brtr3g 5171 Substitution of equality i...
3brtr4g 5172 Substitution of equality i...
eqbrtrid 5173 A chained equality inferen...
eqbrtrrid 5174 A chained equality inferen...
breqtrid 5175 A chained equality inferen...
breqtrrid 5176 A chained equality inferen...
eqbrtrdi 5177 A chained equality inferen...
eqbrtrrdi 5178 A chained equality inferen...
breqtrdi 5179 A chained equality inferen...
breqtrrdi 5180 A chained equality inferen...
ssbrd 5181 Deduction from a subclass ...
ssbr 5182 Implication from a subclas...
ssbri 5183 Inference from a subclass ...
nfbrd 5184 Deduction version of bound...
nfbr 5185 Bound-variable hypothesis ...
brab1 5186 Relationship between a bin...
br0 5187 The empty binary relation ...
brne0 5188 If two sets are in a binar...
brun 5189 The union of two binary re...
brin 5190 The intersection of two re...
brdif 5191 The difference of two bina...
sbcbr123 5192 Move substitution in and o...
sbcbr 5193 Move substitution in and o...
sbcbr12g 5194 Move substitution in and o...
sbcbr1g 5195 Move substitution in and o...
sbcbr2g 5196 Move substitution in and o...
brsymdif 5197 Characterization of the sy...
brralrspcev 5198 Restricted existential spe...
brimralrspcev 5199 Restricted existential spe...
opabss 5202 The collection of ordered ...
opabbid 5203 Equivalent wff's yield equ...
opabbidv 5204 Equivalent wff's yield equ...
opabbii 5205 Equivalent wff's yield equ...
nfopabd 5206 Bound-variable hypothesis ...
nfopab 5207 Bound-variable hypothesis ...
nfopab1 5208 The first abstraction vari...
nfopab2 5209 The second abstraction var...
cbvopab 5210 Rule used to change bound ...
cbvopabv 5211 Rule used to change bound ...
cbvopabvOLD 5212 Obsolete version of ~ cbvo...
cbvopab1 5213 Change first bound variabl...
cbvopab1g 5214 Change first bound variabl...
cbvopab2 5215 Change second bound variab...
cbvopab1s 5216 Change first bound variabl...
cbvopab1v 5217 Rule used to change the fi...
cbvopab1vOLD 5218 Obsolete version of ~ cbvo...
cbvopab2v 5219 Rule used to change the se...
unopab 5220 Union of two ordered pair ...
mpteq12da 5223 An equality inference for ...
mpteq12df 5224 An equality inference for ...
mpteq12dfOLD 5225 Obsolete version of ~ mpte...
mpteq12f 5226 An equality theorem for th...
mpteq12dva 5227 An equality inference for ...
mpteq12dvaOLD 5228 Obsolete version of ~ mpte...
mpteq12dv 5229 An equality inference for ...
mpteq12 5230 An equality theorem for th...
mpteq1 5231 An equality theorem for th...
mpteq1OLD 5232 Obsolete version of ~ mpte...
mpteq1d 5233 An equality theorem for th...
mpteq1i 5234 An equality theorem for th...
mpteq1iOLD 5235 Obsolete version of ~ mpte...
mpteq2da 5236 Slightly more general equa...
mpteq2daOLD 5237 Obsolete version of ~ mpte...
mpteq2dva 5238 Slightly more general equa...
mpteq2dvaOLD 5239 Obsolete version of ~ mpte...
mpteq2dv 5240 An equality inference for ...
mpteq2ia 5241 An equality inference for ...
mpteq2iaOLD 5242 Obsolete version of ~ mpte...
mpteq2i 5243 An equality inference for ...
mpteq12i 5244 An equality inference for ...
nfmpt 5245 Bound-variable hypothesis ...
nfmpt1 5246 Bound-variable hypothesis ...
cbvmptf 5247 Rule to change the bound v...
cbvmptfg 5248 Rule to change the bound v...
cbvmpt 5249 Rule to change the bound v...
cbvmptg 5250 Rule to change the bound v...
cbvmptv 5251 Rule to change the bound v...
cbvmptvOLD 5252 Obsolete version of ~ cbvm...
cbvmptvg 5253 Rule to change the bound v...
mptv 5254 Function with universal do...
dftr2 5257 An alternate way of defini...
dftr2c 5258 Variant of ~ dftr2 with co...
dftr5 5259 An alternate way of defini...
dftr5OLD 5260 Obsolete version of ~ dftr...
dftr3 5261 An alternate way of defini...
dftr4 5262 An alternate way of defini...
treq 5263 Equality theorem for the t...
trel 5264 In a transitive class, the...
trel3 5265 In a transitive class, the...
trss 5266 An element of a transitive...
trin 5267 The intersection of transi...
tr0 5268 The empty set is transitiv...
trv 5269 The universe is transitive...
triun 5270 An indexed union of a clas...
truni 5271 The union of a class of tr...
triin 5272 An indexed intersection of...
trint 5273 The intersection of a clas...
trintss 5274 Any nonempty transitive cl...
axrep1 5276 The version of the Axiom o...
axreplem 5277 Lemma for ~ axrep2 and ~ a...
axrep2 5278 Axiom of Replacement expre...
axrep3 5279 Axiom of Replacement sligh...
axrep4 5280 A more traditional version...
axrep5 5281 Axiom of Replacement (simi...
axrep6 5282 A condensed form of ~ ax-r...
axrep6g 5283 ~ axrep6 in class notation...
zfrepclf 5284 An inference based on the ...
zfrep3cl 5285 An inference based on the ...
zfrep4 5286 A version of Replacement u...
axsepgfromrep 5287 A more general version ~ a...
axsep 5288 Axiom scheme of separation...
axsepg 5290 A more general version of ...
zfauscl 5291 Separation Scheme (Aussond...
bm1.3ii 5292 Convert implication to equ...
ax6vsep 5293 Derive ~ ax6v (a weakened ...
axnulALT 5294 Alternate proof of ~ axnul...
axnul 5295 The Null Set Axiom of ZF s...
0ex 5297 The Null Set Axiom of ZF s...
al0ssb 5298 The empty set is the uniqu...
sseliALT 5299 Alternate proof of ~ sseli...
csbexg 5300 The existence of proper su...
csbex 5301 The existence of proper su...
unisn2 5302 A version of ~ unisn witho...
nalset 5303 No set contains all sets. ...
vnex 5304 The universal class does n...
vprc 5305 The universal class is not...
nvel 5306 The universal class does n...
inex1 5307 Separation Scheme (Aussond...
inex2 5308 Separation Scheme (Aussond...
inex1g 5309 Closed-form, generalized S...
inex2g 5310 Sufficient condition for a...
ssex 5311 The subset of a set is als...
ssexi 5312 The subset of a set is als...
ssexg 5313 The subset of a set is als...
ssexd 5314 A subclass of a set is a s...
prcssprc 5315 The superclass of a proper...
sselpwd 5316 Elementhood to a power set...
difexg 5317 Existence of a difference....
difexi 5318 Existence of a difference,...
difexd 5319 Existence of a difference....
zfausab 5320 Separation Scheme (Aussond...
rabexg 5321 Separation Scheme in terms...
rabex 5322 Separation Scheme in terms...
rabexd 5323 Separation Scheme in terms...
rabex2 5324 Separation Scheme in terms...
rab2ex 5325 A class abstraction based ...
elssabg 5326 Membership in a class abst...
intex 5327 The intersection of a none...
intnex 5328 If a class intersection is...
intexab 5329 The intersection of a none...
intexrab 5330 The intersection of a none...
iinexg 5331 The existence of a class i...
intabs 5332 Absorption of a redundant ...
inuni 5333 The intersection of a unio...
elpw2g 5334 Membership in a power clas...
elpw2 5335 Membership in a power clas...
elpwi2 5336 Membership in a power clas...
elpwi2OLD 5337 Obsolete version of ~ elpw...
axpweq 5338 Two equivalent ways to exp...
pwnss 5339 The power set of a set is ...
pwne 5340 No set equals its power se...
difelpw 5341 A difference is an element...
rabelpw 5342 A restricted class abstrac...
class2set 5343 The class of elements of `...
0elpw 5344 Every power class contains...
pwne0 5345 A power class is never emp...
0nep0 5346 The empty set and its powe...
0inp0 5347 Something cannot be equal ...
unidif0 5348 The removal of the empty s...
eqsnuniex 5349 If a class is equal to the...
iin0 5350 An indexed intersection of...
notzfaus 5351 In the Separation Scheme ~...
intv 5352 The intersection of the un...
zfpow 5354 Axiom of Power Sets expres...
axpow2 5355 A variant of the Axiom of ...
axpow3 5356 A variant of the Axiom of ...
elALT2 5357 Alternate proof of ~ el us...
dtruALT2 5358 Alternate proof of ~ dtru ...
dtrucor 5359 Corollary of ~ dtru . Thi...
dtrucor2 5360 The theorem form of the de...
dvdemo1 5361 Demonstration of a theorem...
dvdemo2 5362 Demonstration of a theorem...
nfnid 5363 A setvar variable is not f...
nfcvb 5364 The "distinctor" expressio...
vpwex 5365 Power set axiom: the power...
pwexg 5366 Power set axiom expressed ...
pwexd 5367 Deduction version of the p...
pwex 5368 Power set axiom expressed ...
pwel 5369 Quantitative version of ~ ...
abssexg 5370 Existence of a class of su...
snexALT 5371 Alternate proof of ~ snex ...
p0ex 5372 The power set of the empty...
p0exALT 5373 Alternate proof of ~ p0ex ...
pp0ex 5374 The power set of the power...
ord3ex 5375 The ordinal number 3 is a ...
dtruALT 5376 Alternate proof of ~ dtru ...
axc16b 5377 This theorem shows that Ax...
eunex 5378 Existential uniqueness imp...
eusv1 5379 Two ways to express single...
eusvnf 5380 Even if ` x ` is free in `...
eusvnfb 5381 Two ways to say that ` A (...
eusv2i 5382 Two ways to express single...
eusv2nf 5383 Two ways to express single...
eusv2 5384 Two ways to express single...
reusv1 5385 Two ways to express single...
reusv2lem1 5386 Lemma for ~ reusv2 . (Con...
reusv2lem2 5387 Lemma for ~ reusv2 . (Con...
reusv2lem3 5388 Lemma for ~ reusv2 . (Con...
reusv2lem4 5389 Lemma for ~ reusv2 . (Con...
reusv2lem5 5390 Lemma for ~ reusv2 . (Con...
reusv2 5391 Two ways to express single...
reusv3i 5392 Two ways of expressing exi...
reusv3 5393 Two ways to express single...
eusv4 5394 Two ways to express single...
alxfr 5395 Transfer universal quantif...
ralxfrd 5396 Transfer universal quantif...
rexxfrd 5397 Transfer universal quantif...
ralxfr2d 5398 Transfer universal quantif...
rexxfr2d 5399 Transfer universal quantif...
ralxfrd2 5400 Transfer universal quantif...
rexxfrd2 5401 Transfer existence from a ...
ralxfr 5402 Transfer universal quantif...
ralxfrALT 5403 Alternate proof of ~ ralxf...
rexxfr 5404 Transfer existence from a ...
rabxfrd 5405 Membership in a restricted...
rabxfr 5406 Membership in a restricted...
reuhypd 5407 A theorem useful for elimi...
reuhyp 5408 A theorem useful for elimi...
zfpair 5409 The Axiom of Pairing of Ze...
axprALT 5410 Alternate proof of ~ axpr ...
axprlem1 5411 Lemma for ~ axpr . There ...
axprlem2 5412 Lemma for ~ axpr . There ...
axprlem3 5413 Lemma for ~ axpr . Elimin...
axprlem4 5414 Lemma for ~ axpr . The fi...
axprlem5 5415 Lemma for ~ axpr . The se...
axpr 5416 Unabbreviated version of t...
zfpair2 5418 Derive the abbreviated ver...
vsnex 5419 A singleton built on a set...
snexg 5420 A singleton built on a set...
snex 5421 A singleton is a set. The...
prex 5422 The Axiom of Pairing using...
exel 5423 There exist two sets, one ...
exexneq 5424 There exist two different ...
exneq 5425 Given any set (the " ` y `...
dtru 5426 Given any set (the " ` y `...
el 5427 Any set is an element of s...
sels 5428 If a class is a set, then ...
selsALT 5429 Alternate proof of ~ sels ...
elALT 5430 Alternate proof of ~ el , ...
dtruOLD 5431 Obsolete proof of ~ dtru a...
snelpwg 5432 A singleton of a set is a ...
snelpwi 5433 If a set is a member of a ...
snelpwiOLD 5434 Obsolete version of ~ snel...
snelpw 5435 A singleton of a set is a ...
prelpw 5436 An unordered pair of two s...
prelpwi 5437 If two sets are members of...
rext 5438 A theorem similar to exten...
sspwb 5439 The powerclass constructio...
unipw 5440 A class equals the union o...
univ 5441 The union of the universe ...
pwtr 5442 A class is transitive iff ...
ssextss 5443 An extensionality-like pri...
ssext 5444 An extensionality-like pri...
nssss 5445 Negation of subclass relat...
pweqb 5446 Classes are equal if and o...
intidg 5447 The intersection of all se...
intidOLD 5448 Obsolete version of ~ inti...
moabex 5449 "At most one" existence im...
rmorabex 5450 Restricted "at most one" e...
euabex 5451 The abstraction of a wff w...
nnullss 5452 A nonempty class (even if ...
exss 5453 Restricted existence in a ...
opex 5454 An ordered pair of classes...
otex 5455 An ordered triple of class...
elopg 5456 Characterization of the el...
elop 5457 Characterization of the el...
opi1 5458 One of the two elements in...
opi2 5459 One of the two elements of...
opeluu 5460 Each member of an ordered ...
op1stb 5461 Extract the first member o...
brv 5462 Two classes are always in ...
opnz 5463 An ordered pair is nonempt...
opnzi 5464 An ordered pair is nonempt...
opth1 5465 Equality of the first memb...
opth 5466 The ordered pair theorem. ...
opthg 5467 Ordered pair theorem. ` C ...
opth1g 5468 Equality of the first memb...
opthg2 5469 Ordered pair theorem. (Co...
opth2 5470 Ordered pair theorem. (Co...
opthneg 5471 Two ordered pairs are not ...
opthne 5472 Two ordered pairs are not ...
otth2 5473 Ordered triple theorem, wi...
otth 5474 Ordered triple theorem. (...
otthg 5475 Ordered triple theorem, cl...
otthne 5476 Contrapositive of the orde...
eqvinop 5477 A variable introduction la...
sbcop1 5478 The proper substitution of...
sbcop 5479 The proper substitution of...
copsexgw 5480 Version of ~ copsexg with ...
copsexg 5481 Substitution of class ` A ...
copsex2t 5482 Closed theorem form of ~ c...
copsex2g 5483 Implicit substitution infe...
copsex2gOLD 5484 Obsolete version of ~ cops...
copsex4g 5485 An implicit substitution i...
0nelop 5486 A property of ordered pair...
opwo0id 5487 An ordered pair is equal t...
opeqex 5488 Equivalence of existence i...
oteqex2 5489 Equivalence of existence i...
oteqex 5490 Equivalence of existence i...
opcom 5491 An ordered pair commutes i...
moop2 5492 "At most one" property of ...
opeqsng 5493 Equivalence for an ordered...
opeqsn 5494 Equivalence for an ordered...
opeqpr 5495 Equivalence for an ordered...
snopeqop 5496 Equivalence for an ordered...
propeqop 5497 Equivalence for an ordered...
propssopi 5498 If a pair of ordered pairs...
snopeqopsnid 5499 Equivalence for an ordered...
mosubopt 5500 "At most one" remains true...
mosubop 5501 "At most one" remains true...
euop2 5502 Transfer existential uniqu...
euotd 5503 Prove existential uniquene...
opthwiener 5504 Justification theorem for ...
uniop 5505 The union of an ordered pa...
uniopel 5506 Ordered pair membership is...
opthhausdorff 5507 Justification theorem for ...
opthhausdorff0 5508 Justification theorem for ...
otsndisj 5509 The singletons consisting ...
otiunsndisj 5510 The union of singletons co...
iunopeqop 5511 Implication of an ordered ...
brsnop 5512 Binary relation for an ord...
brtp 5513 A necessary and sufficient...
opabidw 5514 The law of concretion. Sp...
opabid 5515 The law of concretion. Sp...
elopabw 5516 Membership in a class abst...
elopab 5517 Membership in a class abst...
rexopabb 5518 Restricted existential qua...
vopelopabsb 5519 The law of concretion in t...
opelopabsb 5520 The law of concretion in t...
brabsb 5521 The law of concretion in t...
opelopabt 5522 Closed theorem form of ~ o...
opelopabga 5523 The law of concretion. Th...
brabga 5524 The law of concretion for ...
opelopab2a 5525 Ordered pair membership in...
opelopaba 5526 The law of concretion. Th...
braba 5527 The law of concretion for ...
opelopabg 5528 The law of concretion. Th...
brabg 5529 The law of concretion for ...
opelopabgf 5530 The law of concretion. Th...
opelopab2 5531 Ordered pair membership in...
opelopab 5532 The law of concretion. Th...
brab 5533 The law of concretion for ...
opelopabaf 5534 The law of concretion. Th...
opelopabf 5535 The law of concretion. Th...
ssopab2 5536 Equivalence of ordered pai...
ssopab2bw 5537 Equivalence of ordered pai...
eqopab2bw 5538 Equivalence of ordered pai...
ssopab2b 5539 Equivalence of ordered pai...
ssopab2i 5540 Inference of ordered pair ...
ssopab2dv 5541 Inference of ordered pair ...
eqopab2b 5542 Equivalence of ordered pai...
opabn0 5543 Nonempty ordered pair clas...
opab0 5544 Empty ordered pair class a...
csbopab 5545 Move substitution into a c...
csbopabgALT 5546 Move substitution into a c...
csbmpt12 5547 Move substitution into a m...
csbmpt2 5548 Move substitution into the...
iunopab 5549 Move indexed union inside ...
iunopabOLD 5550 Obsolete version of ~ iuno...
elopabr 5551 Membership in an ordered-p...
elopabran 5552 Membership in an ordered-p...
elopabrOLD 5553 Obsolete version of ~ elop...
rbropapd 5554 Properties of a pair in an...
rbropap 5555 Properties of a pair in a ...
2rbropap 5556 Properties of a pair in a ...
0nelopab 5557 The empty set is never an ...
0nelopabOLD 5558 Obsolete version of ~ 0nel...
brabv 5559 If two classes are in a re...
pwin 5560 The power class of the int...
pwssun 5561 The power class of the uni...
pwun 5562 The power class of the uni...
dfid4 5565 The identity function expr...
dfid2 5566 Alternate definition of th...
dfid3 5567 A stronger version of ~ df...
dfid2OLD 5568 Obsolete version of ~ dfid...
epelg 5571 The membership relation an...
epeli 5572 The membership relation an...
epel 5573 The membership relation an...
0sn0ep 5574 An example for the members...
epn0 5575 The membership relation is...
poss 5580 Subset theorem for the par...
poeq1 5581 Equality theorem for parti...
poeq2 5582 Equality theorem for parti...
nfpo 5583 Bound-variable hypothesis ...
nfso 5584 Bound-variable hypothesis ...
pocl 5585 Characteristic properties ...
poclOLD 5586 Obsolete version of ~ pocl...
ispod 5587 Sufficient conditions for ...
swopolem 5588 Perform the substitutions ...
swopo 5589 A strict weak order is a p...
poirr 5590 A partial order is irrefle...
potr 5591 A partial order is a trans...
po2nr 5592 A partial order has no 2-c...
po3nr 5593 A partial order has no 3-c...
po2ne 5594 Two sets related by a part...
po0 5595 Any relation is a partial ...
pofun 5596 The inverse image of a par...
sopo 5597 A strict linear order is a...
soss 5598 Subset theorem for the str...
soeq1 5599 Equality theorem for the s...
soeq2 5600 Equality theorem for the s...
sonr 5601 A strict order relation is...
sotr 5602 A strict order relation is...
solin 5603 A strict order relation is...
so2nr 5604 A strict order relation ha...
so3nr 5605 A strict order relation ha...
sotric 5606 A strict order relation sa...
sotrieq 5607 Trichotomy law for strict ...
sotrieq2 5608 Trichotomy law for strict ...
soasym 5609 Asymmetry law for strict o...
sotr2 5610 A transitivity relation. ...
issod 5611 An irreflexive, transitive...
issoi 5612 An irreflexive, transitive...
isso2i 5613 Deduce strict ordering fro...
so0 5614 Any relation is a strict o...
somo 5615 A totally ordered set has ...
sotrine 5616 Trichotomy law for strict ...
sotr3 5617 Transitivity law for stric...
dffr6 5624 Alternate definition of ~ ...
frd 5625 A nonempty subset of an ` ...
fri 5626 A nonempty subset of an ` ...
friOLD 5627 Obsolete version of ~ fri ...
seex 5628 The ` R ` -preimage of an ...
exse 5629 Any relation on a set is s...
dffr2 5630 Alternate definition of we...
dffr2ALT 5631 Alternate proof of ~ dffr2...
frc 5632 Property of well-founded r...
frss 5633 Subset theorem for the wel...
sess1 5634 Subset theorem for the set...
sess2 5635 Subset theorem for the set...
freq1 5636 Equality theorem for the w...
freq2 5637 Equality theorem for the w...
seeq1 5638 Equality theorem for the s...
seeq2 5639 Equality theorem for the s...
nffr 5640 Bound-variable hypothesis ...
nfse 5641 Bound-variable hypothesis ...
nfwe 5642 Bound-variable hypothesis ...
frirr 5643 A well-founded relation is...
fr2nr 5644 A well-founded relation ha...
fr0 5645 Any relation is well-found...
frminex 5646 If an element of a well-fo...
efrirr 5647 A well-founded class does ...
efrn2lp 5648 A well-founded class conta...
epse 5649 The membership relation is...
tz7.2 5650 Similar to Theorem 7.2 of ...
dfepfr 5651 An alternate way of saying...
epfrc 5652 A subset of a well-founded...
wess 5653 Subset theorem for the wel...
weeq1 5654 Equality theorem for the w...
weeq2 5655 Equality theorem for the w...
wefr 5656 A well-ordering is well-fo...
weso 5657 A well-ordering is a stric...
wecmpep 5658 The elements of a class we...
wetrep 5659 On a class well-ordered by...
wefrc 5660 A nonempty subclass of a c...
we0 5661 Any relation is a well-ord...
wereu 5662 A nonempty subset of an ` ...
wereu2 5663 A nonempty subclass of an ...
xpeq1 5680 Equality theorem for Carte...
xpss12 5681 Subset theorem for Cartesi...
xpss 5682 A Cartesian product is inc...
inxpssres 5683 Intersection with a Cartes...
relxp 5684 A Cartesian product is a r...
xpss1 5685 Subset relation for Cartes...
xpss2 5686 Subset relation for Cartes...
xpeq2 5687 Equality theorem for Carte...
elxpi 5688 Membership in a Cartesian ...
elxp 5689 Membership in a Cartesian ...
elxp2 5690 Membership in a Cartesian ...
xpeq12 5691 Equality theorem for Carte...
xpeq1i 5692 Equality inference for Car...
xpeq2i 5693 Equality inference for Car...
xpeq12i 5694 Equality inference for Car...
xpeq1d 5695 Equality deduction for Car...
xpeq2d 5696 Equality deduction for Car...
xpeq12d 5697 Equality deduction for Car...
sqxpeqd 5698 Equality deduction for a C...
nfxp 5699 Bound-variable hypothesis ...
0nelxp 5700 The empty set is not a mem...
0nelelxp 5701 A member of a Cartesian pr...
opelxp 5702 Ordered pair membership in...
opelxpi 5703 Ordered pair membership in...
opelxpii 5704 Ordered pair membership in...
opelxpd 5705 Ordered pair membership in...
opelvv 5706 Ordered pair membership in...
opelvvg 5707 Ordered pair membership in...
opelxp1 5708 The first member of an ord...
opelxp2 5709 The second member of an or...
otelxp 5710 Ordered triple membership ...
otelxp1 5711 The first member of an ord...
otel3xp 5712 An ordered triple is an el...
opabssxpd 5713 An ordered-pair class abst...
rabxp 5714 Class abstraction restrict...
brxp 5715 Binary relation on a Carte...
pwvrel 5716 A set is a binary relation...
pwvabrel 5717 The powerclass of the cart...
brrelex12 5718 Two classes related by a b...
brrelex1 5719 If two classes are related...
brrelex2 5720 If two classes are related...
brrelex12i 5721 Two classes that are relat...
brrelex1i 5722 The first argument of a bi...
brrelex2i 5723 The second argument of a b...
nprrel12 5724 Proper classes are not rel...
nprrel 5725 No proper class is related...
0nelrel0 5726 A binary relation does not...
0nelrel 5727 A binary relation does not...
fconstmpt 5728 Representation of a consta...
vtoclr 5729 Variable to class conversi...
opthprc 5730 Justification theorem for ...
brel 5731 Two things in a binary rel...
elxp3 5732 Membership in a Cartesian ...
opeliunxp 5733 Membership in a union of C...
xpundi 5734 Distributive law for Carte...
xpundir 5735 Distributive law for Carte...
xpiundi 5736 Distributive law for Carte...
xpiundir 5737 Distributive law for Carte...
iunxpconst 5738 Membership in a union of C...
xpun 5739 The Cartesian product of t...
elvv 5740 Membership in universal cl...
elvvv 5741 Membership in universal cl...
elvvuni 5742 An ordered pair contains i...
brinxp2 5743 Intersection of binary rel...
brinxp 5744 Intersection of binary rel...
opelinxp 5745 Ordered pair element in an...
poinxp 5746 Intersection of partial or...
soinxp 5747 Intersection of total orde...
frinxp 5748 Intersection of well-found...
seinxp 5749 Intersection of set-like r...
weinxp 5750 Intersection of well-order...
posn 5751 Partial ordering of a sing...
sosn 5752 Strict ordering on a singl...
frsn 5753 Founded relation on a sing...
wesn 5754 Well-ordering of a singlet...
elopaelxp 5755 Membership in an ordered-p...
elopaelxpOLD 5756 Obsolete version of ~ elop...
bropaex12 5757 Two classes related by an ...
opabssxp 5758 An abstraction relation is...
brab2a 5759 The law of concretion for ...
optocl 5760 Implicit substitution of c...
2optocl 5761 Implicit substitution of c...
3optocl 5762 Implicit substitution of c...
opbrop 5763 Ordered pair membership in...
0xp 5764 The Cartesian product with...
csbxp 5765 Distribute proper substitu...
releq 5766 Equality theorem for the r...
releqi 5767 Equality inference for the...
releqd 5768 Equality deduction for the...
nfrel 5769 Bound-variable hypothesis ...
sbcrel 5770 Distribute proper substitu...
relss 5771 Subclass theorem for relat...
ssrel 5772 A subclass relationship de...
ssrelOLD 5773 Obsolete version of ~ ssre...
eqrel 5774 Extensionality principle f...
ssrel2 5775 A subclass relationship de...
ssrel3 5776 Subclass relation in anoth...
relssi 5777 Inference from subclass pr...
relssdv 5778 Deduction from subclass pr...
eqrelriv 5779 Inference from extensional...
eqrelriiv 5780 Inference from extensional...
eqbrriv 5781 Inference from extensional...
eqrelrdv 5782 Deduce equality of relatio...
eqbrrdv 5783 Deduction from extensional...
eqbrrdiv 5784 Deduction from extensional...
eqrelrdv2 5785 A version of ~ eqrelrdv . ...
ssrelrel 5786 A subclass relationship de...
eqrelrel 5787 Extensionality principle f...
elrel 5788 A member of a relation is ...
rel0 5789 The empty set is a relatio...
nrelv 5790 The universal class is not...
relsng 5791 A singleton is a relation ...
relsnb 5792 An at-most-singleton is a ...
relsnopg 5793 A singleton of an ordered ...
relsn 5794 A singleton is a relation ...
relsnop 5795 A singleton of an ordered ...
copsex2gb 5796 Implicit substitution infe...
copsex2ga 5797 Implicit substitution infe...
elopaba 5798 Membership in an ordered-p...
xpsspw 5799 A Cartesian product is inc...
unixpss 5800 The double class union of ...
relun 5801 The union of two relations...
relin1 5802 The intersection with a re...
relin2 5803 The intersection with a re...
relinxp 5804 Intersection with a Cartes...
reldif 5805 A difference cutting down ...
reliun 5806 An indexed union is a rela...
reliin 5807 An indexed intersection is...
reluni 5808 The union of a class is a ...
relint 5809 The intersection of a clas...
relopabiv 5810 A class of ordered pairs i...
relopabv 5811 A class of ordered pairs i...
relopabi 5812 A class of ordered pairs i...
relopabiALT 5813 Alternate proof of ~ relop...
relopab 5814 A class of ordered pairs i...
mptrel 5815 The maps-to notation alway...
reli 5816 The identity relation is a...
rele 5817 The membership relation is...
opabid2 5818 A relation expressed as an...
inopab 5819 Intersection of two ordere...
difopab 5820 Difference of two ordered-...
difopabOLD 5821 Obsolete version of ~ difo...
inxp 5822 Intersection of two Cartes...
xpindi 5823 Distributive law for Carte...
xpindir 5824 Distributive law for Carte...
xpiindi 5825 Distributive law for Carte...
xpriindi 5826 Distributive law for Carte...
eliunxp 5827 Membership in a union of C...
opeliunxp2 5828 Membership in a union of C...
raliunxp 5829 Write a double restricted ...
rexiunxp 5830 Write a double restricted ...
ralxp 5831 Universal quantification r...
rexxp 5832 Existential quantification...
exopxfr 5833 Transfer ordered-pair exis...
exopxfr2 5834 Transfer ordered-pair exis...
djussxp 5835 Disjoint union is a subset...
ralxpf 5836 Version of ~ ralxp with bo...
rexxpf 5837 Version of ~ rexxp with bo...
iunxpf 5838 Indexed union on a Cartesi...
opabbi2dv 5839 Deduce equality of a relat...
relop 5840 A necessary and sufficient...
ideqg 5841 For sets, the identity rel...
ideq 5842 For sets, the identity rel...
ididg 5843 A set is identical to itse...
issetid 5844 Two ways of expressing set...
coss1 5845 Subclass theorem for compo...
coss2 5846 Subclass theorem for compo...
coeq1 5847 Equality theorem for compo...
coeq2 5848 Equality theorem for compo...
coeq1i 5849 Equality inference for com...
coeq2i 5850 Equality inference for com...
coeq1d 5851 Equality deduction for com...
coeq2d 5852 Equality deduction for com...
coeq12i 5853 Equality inference for com...
coeq12d 5854 Equality deduction for com...
nfco 5855 Bound-variable hypothesis ...
brcog 5856 Ordered pair membership in...
opelco2g 5857 Ordered pair membership in...
brcogw 5858 Ordered pair membership in...
eqbrrdva 5859 Deduction from extensional...
brco 5860 Binary relation on a compo...
opelco 5861 Ordered pair membership in...
cnvss 5862 Subset theorem for convers...
cnveq 5863 Equality theorem for conve...
cnveqi 5864 Equality inference for con...
cnveqd 5865 Equality deduction for con...
elcnv 5866 Membership in a converse r...
elcnv2 5867 Membership in a converse r...
nfcnv 5868 Bound-variable hypothesis ...
brcnvg 5869 The converse of a binary r...
opelcnvg 5870 Ordered-pair membership in...
opelcnv 5871 Ordered-pair membership in...
brcnv 5872 The converse of a binary r...
csbcnv 5873 Move class substitution in...
csbcnvgALT 5874 Move class substitution in...
cnvco 5875 Distributive law of conver...
cnvuni 5876 The converse of a class un...
dfdm3 5877 Alternate definition of do...
dfrn2 5878 Alternate definition of ra...
dfrn3 5879 Alternate definition of ra...
elrn2g 5880 Membership in a range. (C...
elrng 5881 Membership in a range. (C...
elrn2 5882 Membership in a range. (C...
elrn 5883 Membership in a range. (C...
ssrelrn 5884 If a relation is a subset ...
dfdm4 5885 Alternate definition of do...
dfdmf 5886 Definition of domain, usin...
csbdm 5887 Distribute proper substitu...
eldmg 5888 Domain membership. Theore...
eldm2g 5889 Domain membership. Theore...
eldm 5890 Membership in a domain. T...
eldm2 5891 Membership in a domain. T...
dmss 5892 Subset theorem for domain....
dmeq 5893 Equality theorem for domai...
dmeqi 5894 Equality inference for dom...
dmeqd 5895 Equality deduction for dom...
opeldmd 5896 Membership of first of an ...
opeldm 5897 Membership of first of an ...
breldm 5898 Membership of first of a b...
breldmg 5899 Membership of first of a b...
dmun 5900 The domain of a union is t...
dmin 5901 The domain of an intersect...
breldmd 5902 Membership of first of a b...
dmiun 5903 The domain of an indexed u...
dmuni 5904 The domain of a union. Pa...
dmopab 5905 The domain of a class of o...
dmopabelb 5906 A set is an element of the...
dmopab2rex 5907 The domain of an ordered p...
dmopabss 5908 Upper bound for the domain...
dmopab3 5909 The domain of a restricted...
dm0 5910 The domain of the empty se...
dmi 5911 The domain of the identity...
dmv 5912 The domain of the universe...
dmep 5913 The domain of the membersh...
dm0rn0 5914 An empty domain is equival...
rn0 5915 The range of the empty set...
rnep 5916 The range of the membershi...
reldm0 5917 A relation is empty iff it...
dmxp 5918 The domain of a Cartesian ...
dmxpid 5919 The domain of a Cartesian ...
dmxpin 5920 The domain of the intersec...
xpid11 5921 The Cartesian square is a ...
dmcnvcnv 5922 The domain of the double c...
rncnvcnv 5923 The range of the double co...
elreldm 5924 The first member of an ord...
rneq 5925 Equality theorem for range...
rneqi 5926 Equality inference for ran...
rneqd 5927 Equality deduction for ran...
rnss 5928 Subset theorem for range. ...
rnssi 5929 Subclass inference for ran...
brelrng 5930 The second argument of a b...
brelrn 5931 The second argument of a b...
opelrn 5932 Membership of second membe...
releldm 5933 The first argument of a bi...
relelrn 5934 The second argument of a b...
releldmb 5935 Membership in a domain. (...
relelrnb 5936 Membership in a range. (C...
releldmi 5937 The first argument of a bi...
relelrni 5938 The second argument of a b...
dfrnf 5939 Definition of range, using...
nfdm 5940 Bound-variable hypothesis ...
nfrn 5941 Bound-variable hypothesis ...
dmiin 5942 Domain of an intersection....
rnopab 5943 The range of a class of or...
rnmpt 5944 The range of a function in...
elrnmpt 5945 The range of a function in...
elrnmpt1s 5946 Elementhood in an image se...
elrnmpt1 5947 Elementhood in an image se...
elrnmptg 5948 Membership in the range of...
elrnmpti 5949 Membership in the range of...
elrnmptd 5950 The range of a function in...
elrnmptdv 5951 Elementhood in the range o...
elrnmpt2d 5952 Elementhood in the range o...
dfiun3g 5953 Alternate definition of in...
dfiin3g 5954 Alternate definition of in...
dfiun3 5955 Alternate definition of in...
dfiin3 5956 Alternate definition of in...
riinint 5957 Express a relative indexed...
relrn0 5958 A relation is empty iff it...
dmrnssfld 5959 The domain and range of a ...
dmcoss 5960 Domain of a composition. ...
rncoss 5961 Range of a composition. (...
dmcosseq 5962 Domain of a composition. ...
dmcoeq 5963 Domain of a composition. ...
rncoeq 5964 Range of a composition. (...
reseq1 5965 Equality theorem for restr...
reseq2 5966 Equality theorem for restr...
reseq1i 5967 Equality inference for res...
reseq2i 5968 Equality inference for res...
reseq12i 5969 Equality inference for res...
reseq1d 5970 Equality deduction for res...
reseq2d 5971 Equality deduction for res...
reseq12d 5972 Equality deduction for res...
nfres 5973 Bound-variable hypothesis ...
csbres 5974 Distribute proper substitu...
res0 5975 A restriction to the empty...
dfres3 5976 Alternate definition of re...
opelres 5977 Ordered pair elementhood i...
brres 5978 Binary relation on a restr...
opelresi 5979 Ordered pair membership in...
brresi 5980 Binary relation on a restr...
opres 5981 Ordered pair membership in...
resieq 5982 A restricted identity rela...
opelidres 5983 ` <. A , A >. ` belongs to...
resres 5984 The restriction of a restr...
resundi 5985 Distributive law for restr...
resundir 5986 Distributive law for restr...
resindi 5987 Class restriction distribu...
resindir 5988 Class restriction distribu...
inres 5989 Move intersection into cla...
resdifcom 5990 Commutative law for restri...
resiun1 5991 Distribution of restrictio...
resiun2 5992 Distribution of restrictio...
dmres 5993 The domain of a restrictio...
ssdmres 5994 A domain restricted to a s...
dmresexg 5995 The domain of a restrictio...
resss 5996 A class includes its restr...
rescom 5997 Commutative law for restri...
ssres 5998 Subclass theorem for restr...
ssres2 5999 Subclass theorem for restr...
relres 6000 A restriction is a relatio...
resabs1 6001 Absorption law for restric...
resabs1d 6002 Absorption law for restric...
resabs2 6003 Absorption law for restric...
residm 6004 Idempotent law for restric...
resima 6005 A restriction to an image....
resima2 6006 Image under a restricted c...
rnresss 6007 The range of a restriction...
xpssres 6008 Restriction of a constant ...
elinxp 6009 Membership in an intersect...
elres 6010 Membership in a restrictio...
elsnres 6011 Membership in restriction ...
relssres 6012 Simplification law for res...
dmressnsn 6013 The domain of a restrictio...
eldmressnsn 6014 The element of the domain ...
eldmeldmressn 6015 An element of the domain (...
resdm 6016 A relation restricted to i...
resexg 6017 The restriction of a set i...
resexd 6018 The restriction of a set i...
resex 6019 The restriction of a set i...
resindm 6020 When restricting a relatio...
resdmdfsn 6021 Restricting a relation to ...
reldisjun 6022 Split a relation into two ...
relresdm1 6023 Restriction of a disjoint ...
resopab 6024 Restriction of a class abs...
iss 6025 A subclass of the identity...
resopab2 6026 Restriction of a class abs...
resmpt 6027 Restriction of the mapping...
resmpt3 6028 Unconditional restriction ...
resmptf 6029 Restriction of the mapping...
resmptd 6030 Restriction of the mapping...
dfres2 6031 Alternate definition of th...
mptss 6032 Sufficient condition for i...
elidinxp 6033 Characterization of the el...
elidinxpid 6034 Characterization of the el...
elrid 6035 Characterization of the el...
idinxpres 6036 The intersection of the id...
idinxpresid 6037 The intersection of the id...
idssxp 6038 A diagonal set as a subset...
opabresid 6039 The restricted identity re...
mptresid 6040 The restricted identity re...
dmresi 6041 The domain of a restricted...
restidsing 6042 Restriction of the identit...
iresn0n0 6043 The identity function rest...
imaeq1 6044 Equality theorem for image...
imaeq2 6045 Equality theorem for image...
imaeq1i 6046 Equality theorem for image...
imaeq2i 6047 Equality theorem for image...
imaeq1d 6048 Equality theorem for image...
imaeq2d 6049 Equality theorem for image...
imaeq12d 6050 Equality theorem for image...
dfima2 6051 Alternate definition of im...
dfima3 6052 Alternate definition of im...
elimag 6053 Membership in an image. T...
elima 6054 Membership in an image. T...
elima2 6055 Membership in an image. T...
elima3 6056 Membership in an image. T...
nfima 6057 Bound-variable hypothesis ...
nfimad 6058 Deduction version of bound...
imadmrn 6059 The image of the domain of...
imassrn 6060 The image of a class is a ...
mptima 6061 Image of a function in map...
mptimass 6062 Image of a function in map...
imai 6063 Image under the identity r...
rnresi 6064 The range of the restricte...
resiima 6065 The image of a restriction...
ima0 6066 Image of the empty set. T...
0ima 6067 Image under the empty rela...
csbima12 6068 Move class substitution in...
imadisj 6069 A class whose image under ...
cnvimass 6070 A preimage under any class...
cnvimarndm 6071 The preimage of the range ...
imasng 6072 The image of a singleton. ...
relimasn 6073 The image of a singleton. ...
elrelimasn 6074 Elementhood in the image o...
elimasng1 6075 Membership in an image of ...
elimasn1 6076 Membership in an image of ...
elimasng 6077 Membership in an image of ...
elimasn 6078 Membership in an image of ...
elimasngOLD 6079 Obsolete version of ~ elim...
elimasni 6080 Membership in an image of ...
args 6081 Two ways to express the cl...
elinisegg 6082 Membership in the inverse ...
eliniseg 6083 Membership in the inverse ...
epin 6084 Any set is equal to its pr...
epini 6085 Any set is equal to its pr...
iniseg 6086 An idiom that signifies an...
inisegn0 6087 Nonemptiness of an initial...
dffr3 6088 Alternate definition of we...
dfse2 6089 Alternate definition of se...
imass1 6090 Subset theorem for image. ...
imass2 6091 Subset theorem for image. ...
ndmima 6092 The image of a singleton o...
relcnv 6093 A converse is a relation. ...
relbrcnvg 6094 When ` R ` is a relation, ...
eliniseg2 6095 Eliminate the class existe...
relbrcnv 6096 When ` R ` is a relation, ...
relco 6097 A composition is a relatio...
cotrg 6098 Two ways of saying that th...
cotrgOLD 6099 Obsolete version of ~ cotr...
cotrgOLDOLD 6100 Obsolete version of ~ cotr...
cotr 6101 Two ways of saying a relat...
idrefALT 6102 Alternate proof of ~ idref...
cnvsym 6103 Two ways of saying a relat...
cnvsymOLD 6104 Obsolete proof of ~ cnvsym...
cnvsymOLDOLD 6105 Obsolete proof of ~ cnvsym...
intasym 6106 Two ways of saying a relat...
asymref 6107 Two ways of saying a relat...
asymref2 6108 Two ways of saying a relat...
intirr 6109 Two ways of saying a relat...
brcodir 6110 Two ways of saying that tw...
codir 6111 Two ways of saying a relat...
qfto 6112 A quantifier-free way of e...
xpidtr 6113 A Cartesian square is a tr...
trin2 6114 The intersection of two tr...
poirr2 6115 A partial order is irrefle...
trinxp 6116 The relation induced by a ...
soirri 6117 A strict order relation is...
sotri 6118 A strict order relation is...
son2lpi 6119 A strict order relation ha...
sotri2 6120 A transitivity relation. ...
sotri3 6121 A transitivity relation. ...
poleloe 6122 Express "less than or equa...
poltletr 6123 Transitive law for general...
somin1 6124 Property of a minimum in a...
somincom 6125 Commutativity of minimum i...
somin2 6126 Property of a minimum in a...
soltmin 6127 Being less than a minimum,...
cnvopab 6128 The converse of a class ab...
mptcnv 6129 The converse of a mapping ...
cnv0 6130 The converse of the empty ...
cnvi 6131 The converse of the identi...
cnvun 6132 The converse of a union is...
cnvdif 6133 Distributive law for conve...
cnvin 6134 Distributive law for conve...
rnun 6135 Distributive law for range...
rnin 6136 The range of an intersecti...
rniun 6137 The range of an indexed un...
rnuni 6138 The range of a union. Par...
imaundi 6139 Distributive law for image...
imaundir 6140 The image of a union. (Co...
cnvimassrndm 6141 The preimage of a superset...
dminss 6142 An upper bound for interse...
imainss 6143 An upper bound for interse...
inimass 6144 The image of an intersecti...
inimasn 6145 The intersection of the im...
cnvxp 6146 The converse of a Cartesia...
xp0 6147 The Cartesian product with...
xpnz 6148 The Cartesian product of n...
xpeq0 6149 At least one member of an ...
xpdisj1 6150 Cartesian products with di...
xpdisj2 6151 Cartesian products with di...
xpsndisj 6152 Cartesian products with tw...
difxp 6153 Difference of Cartesian pr...
difxp1 6154 Difference law for Cartesi...
difxp2 6155 Difference law for Cartesi...
djudisj 6156 Disjoint unions with disjo...
xpdifid 6157 The set of distinct couple...
resdisj 6158 A double restriction to di...
rnxp 6159 The range of a Cartesian p...
dmxpss 6160 The domain of a Cartesian ...
rnxpss 6161 The range of a Cartesian p...
rnxpid 6162 The range of a Cartesian s...
ssxpb 6163 A Cartesian product subcla...
xp11 6164 The Cartesian product of n...
xpcan 6165 Cancellation law for Carte...
xpcan2 6166 Cancellation law for Carte...
ssrnres 6167 Two ways to express surjec...
rninxp 6168 Two ways to express surjec...
dminxp 6169 Two ways to express totali...
imainrect 6170 Image by a restricted and ...
xpima 6171 Direct image by a Cartesia...
xpima1 6172 Direct image by a Cartesia...
xpima2 6173 Direct image by a Cartesia...
xpimasn 6174 Direct image of a singleto...
sossfld 6175 The base set of a strict o...
sofld 6176 The base set of a nonempty...
cnvcnv3 6177 The set of all ordered pai...
dfrel2 6178 Alternate definition of re...
dfrel4v 6179 A relation can be expresse...
dfrel4 6180 A relation can be expresse...
cnvcnv 6181 The double converse of a c...
cnvcnv2 6182 The double converse of a c...
cnvcnvss 6183 The double converse of a c...
cnvrescnv 6184 Two ways to express the co...
cnveqb 6185 Equality theorem for conve...
cnveq0 6186 A relation empty iff its c...
dfrel3 6187 Alternate definition of re...
elid 6188 Characterization of the el...
dmresv 6189 The domain of a universal ...
rnresv 6190 The range of a universal r...
dfrn4 6191 Range defined in terms of ...
csbrn 6192 Distribute proper substitu...
rescnvcnv 6193 The restriction of the dou...
cnvcnvres 6194 The double converse of the...
imacnvcnv 6195 The image of the double co...
dmsnn0 6196 The domain of a singleton ...
rnsnn0 6197 The range of a singleton i...
dmsn0 6198 The domain of the singleto...
cnvsn0 6199 The converse of the single...
dmsn0el 6200 The domain of a singleton ...
relsn2 6201 A singleton is a relation ...
dmsnopg 6202 The domain of a singleton ...
dmsnopss 6203 The domain of a singleton ...
dmpropg 6204 The domain of an unordered...
dmsnop 6205 The domain of a singleton ...
dmprop 6206 The domain of an unordered...
dmtpop 6207 The domain of an unordered...
cnvcnvsn 6208 Double converse of a singl...
dmsnsnsn 6209 The domain of the singleto...
rnsnopg 6210 The range of a singleton o...
rnpropg 6211 The range of a pair of ord...
cnvsng 6212 Converse of a singleton of...
rnsnop 6213 The range of a singleton o...
op1sta 6214 Extract the first member o...
cnvsn 6215 Converse of a singleton of...
op2ndb 6216 Extract the second member ...
op2nda 6217 Extract the second member ...
opswap 6218 Swap the members of an ord...
cnvresima 6219 An image under the convers...
resdm2 6220 A class restricted to its ...
resdmres 6221 Restriction to the domain ...
resresdm 6222 A restriction by an arbitr...
imadmres 6223 The image of the domain of...
resdmss 6224 Subset relationship for th...
resdifdi 6225 Distributive law for restr...
resdifdir 6226 Distributive law for restr...
mptpreima 6227 The preimage of a function...
mptiniseg 6228 Converse singleton image o...
dmmpt 6229 The domain of the mapping ...
dmmptss 6230 The domain of a mapping is...
dmmptg 6231 The domain of the mapping ...
rnmpt0f 6232 The range of a function in...
rnmptn0 6233 The range of a function in...
dfco2 6234 Alternate definition of a ...
dfco2a 6235 Generalization of ~ dfco2 ...
coundi 6236 Class composition distribu...
coundir 6237 Class composition distribu...
cores 6238 Restricted first member of...
resco 6239 Associative law for the re...
imaco 6240 Image of the composition o...
rnco 6241 The range of the compositi...
rnco2 6242 The range of the compositi...
dmco 6243 The domain of a compositio...
coeq0 6244 A composition of two relat...
coiun 6245 Composition with an indexe...
cocnvcnv1 6246 A composition is not affec...
cocnvcnv2 6247 A composition is not affec...
cores2 6248 Absorption of a reverse (p...
co02 6249 Composition with the empty...
co01 6250 Composition with the empty...
coi1 6251 Composition with the ident...
coi2 6252 Composition with the ident...
coires1 6253 Composition with a restric...
coass 6254 Associative law for class ...
relcnvtrg 6255 General form of ~ relcnvtr...
relcnvtr 6256 A relation is transitive i...
relssdmrn 6257 A relation is included in ...
relssdmrnOLD 6258 Obsolete version of ~ rels...
resssxp 6259 If the ` R ` -image of a c...
cnvssrndm 6260 The converse is a subset o...
cossxp 6261 Composition as a subset of...
relrelss 6262 Two ways to describe the s...
unielrel 6263 The membership relation fo...
relfld 6264 The double union of a rela...
relresfld 6265 Restriction of a relation ...
relcoi2 6266 Composition with the ident...
relcoi1 6267 Composition with the ident...
unidmrn 6268 The double union of the co...
relcnvfld 6269 if ` R ` is a relation, it...
dfdm2 6270 Alternate definition of do...
unixp 6271 The double class union of ...
unixp0 6272 A Cartesian product is emp...
unixpid 6273 Field of a Cartesian squar...
ressn 6274 Restriction of a class to ...
cnviin 6275 The converse of an interse...
cnvpo 6276 The converse of a partial ...
cnvso 6277 The converse of a strict o...
xpco 6278 Composition of two Cartesi...
xpcoid 6279 Composition of two Cartesi...
elsnxp 6280 Membership in a Cartesian ...
reu3op 6281 There is a unique ordered ...
reuop 6282 There is a unique ordered ...
opreu2reurex 6283 There is a unique ordered ...
opreu2reu 6284 If there is a unique order...
dfpo2 6285 Quantifier-free definition...
csbcog 6286 Distribute proper substitu...
snres0 6287 Condition for restriction ...
imaindm 6288 The image is unaffected by...
predeq123 6291 Equality theorem for the p...
predeq1 6292 Equality theorem for the p...
predeq2 6293 Equality theorem for the p...
predeq3 6294 Equality theorem for the p...
nfpred 6295 Bound-variable hypothesis ...
csbpredg 6296 Move class substitution in...
predpredss 6297 If ` A ` is a subset of ` ...
predss 6298 The predecessor class of `...
sspred 6299 Another subset/predecessor...
dfpred2 6300 An alternate definition of...
dfpred3 6301 An alternate definition of...
dfpred3g 6302 An alternate definition of...
elpredgg 6303 Membership in a predecesso...
elpredg 6304 Membership in a predecesso...
elpredimg 6305 Membership in a predecesso...
elpredim 6306 Membership in a predecesso...
elpred 6307 Membership in a predecesso...
predexg 6308 The predecessor class exis...
predasetexOLD 6309 Obsolete form of ~ predexg...
dffr4 6310 Alternate definition of we...
predel 6311 Membership in the predeces...
predbrg 6312 Closed form of ~ elpredim ...
predtrss 6313 If ` R ` is transitive ove...
predpo 6314 Property of the predecesso...
predso 6315 Property of the predecesso...
setlikespec 6316 If ` R ` is set-like in ` ...
predidm 6317 Idempotent law for the pre...
predin 6318 Intersection law for prede...
predun 6319 Union law for predecessor ...
preddif 6320 Difference law for predece...
predep 6321 The predecessor under the ...
trpred 6322 The class of predecessors ...
preddowncl 6323 A property of classes that...
predpoirr 6324 Given a partial ordering, ...
predfrirr 6325 Given a well-founded relat...
pred0 6326 The predecessor class over...
dfse3 6327 Alternate definition of se...
predrelss 6328 Subset carries from relati...
predprc 6329 The predecessor of a prope...
predres 6330 Predecessor class is unaff...
frpomin 6331 Every nonempty (possibly p...
frpomin2 6332 Every nonempty (possibly p...
frpoind 6333 The principle of well-foun...
frpoinsg 6334 Well-Founded Induction Sch...
frpoins2fg 6335 Well-Founded Induction sch...
frpoins2g 6336 Well-Founded Induction sch...
frpoins3g 6337 Well-Founded Induction sch...
tz6.26 6338 All nonempty subclasses of...
tz6.26OLD 6339 Obsolete proof of ~ tz6.26...
tz6.26i 6340 All nonempty subclasses of...
wfi 6341 The Principle of Well-Orde...
wfiOLD 6342 Obsolete proof of ~ wfi as...
wfii 6343 The Principle of Well-Orde...
wfisg 6344 Well-Ordered Induction Sch...
wfisgOLD 6345 Obsolete version of ~ wfis...
wfis 6346 Well-Ordered Induction Sch...
wfis2fg 6347 Well-Ordered Induction Sch...
wfis2fgOLD 6348 Obsolete version of ~ wfis...
wfis2f 6349 Well-Ordered Induction sch...
wfis2g 6350 Well-Ordered Induction Sch...
wfis2 6351 Well-Ordered Induction sch...
wfis3 6352 Well-Ordered Induction sch...
ordeq 6361 Equality theorem for the o...
elong 6362 An ordinal number is an or...
elon 6363 An ordinal number is an or...
eloni 6364 An ordinal number has the ...
elon2 6365 An ordinal number is an or...
limeq 6366 Equality theorem for the l...
ordwe 6367 Membership well-orders eve...
ordtr 6368 An ordinal class is transi...
ordfr 6369 Membership is well-founded...
ordelss 6370 An element of an ordinal c...
trssord 6371 A transitive subclass of a...
ordirr 6372 No ordinal class is a memb...
nordeq 6373 A member of an ordinal cla...
ordn2lp 6374 An ordinal class cannot be...
tz7.5 6375 A nonempty subclass of an ...
ordelord 6376 An element of an ordinal c...
tron 6377 The class of all ordinal n...
ordelon 6378 An element of an ordinal c...
onelon 6379 An element of an ordinal n...
tz7.7 6380 A transitive class belongs...
ordelssne 6381 For ordinal classes, membe...
ordelpss 6382 For ordinal classes, membe...
ordsseleq 6383 For ordinal classes, inclu...
ordin 6384 The intersection of two or...
onin 6385 The intersection of two or...
ordtri3or 6386 A trichotomy law for ordin...
ordtri1 6387 A trichotomy law for ordin...
ontri1 6388 A trichotomy law for ordin...
ordtri2 6389 A trichotomy law for ordin...
ordtri3 6390 A trichotomy law for ordin...
ordtri4 6391 A trichotomy law for ordin...
orddisj 6392 An ordinal class and its s...
onfr 6393 The ordinal class is well-...
onelpss 6394 Relationship between membe...
onsseleq 6395 Relationship between subse...
onelss 6396 An element of an ordinal n...
ordtr1 6397 Transitive law for ordinal...
ordtr2 6398 Transitive law for ordinal...
ordtr3 6399 Transitive law for ordinal...
ontr1 6400 Transitive law for ordinal...
ontr2 6401 Transitive law for ordinal...
onelssex 6402 Ordinal less than is equiv...
ordunidif 6403 The union of an ordinal st...
ordintdif 6404 If ` B ` is smaller than `...
onintss 6405 If a property is true for ...
oneqmini 6406 A way to show that an ordi...
ord0 6407 The empty set is an ordina...
0elon 6408 The empty set is an ordina...
ord0eln0 6409 A nonempty ordinal contain...
on0eln0 6410 An ordinal number contains...
dflim2 6411 An alternate definition of...
inton 6412 The intersection of the cl...
nlim0 6413 The empty set is not a lim...
limord 6414 A limit ordinal is ordinal...
limuni 6415 A limit ordinal is its own...
limuni2 6416 The union of a limit ordin...
0ellim 6417 A limit ordinal contains t...
limelon 6418 A limit ordinal class that...
onn0 6419 The class of all ordinal n...
suceq 6420 Equality of successors. (...
elsuci 6421 Membership in a successor....
elsucg 6422 Membership in a successor....
elsuc2g 6423 Variant of membership in a...
elsuc 6424 Membership in a successor....
elsuc2 6425 Membership in a successor....
nfsuc 6426 Bound-variable hypothesis ...
elelsuc 6427 Membership in a successor....
sucel 6428 Membership of a successor ...
suc0 6429 The successor of the empty...
sucprc 6430 A proper class is its own ...
unisucs 6431 The union of the successor...
unisucg 6432 A transitive class is equa...
unisuc 6433 A transitive class is equa...
sssucid 6434 A class is included in its...
sucidg 6435 Part of Proposition 7.23 o...
sucid 6436 A set belongs to its succe...
nsuceq0 6437 No successor is empty. (C...
eqelsuc 6438 A set belongs to the succe...
iunsuc 6439 Inductive definition for t...
suctr 6440 The successor of a transit...
trsuc 6441 A set whose successor belo...
trsucss 6442 A member of the successor ...
ordsssuc 6443 An ordinal is a subset of ...
onsssuc 6444 A subset of an ordinal num...
ordsssuc2 6445 An ordinal subset of an or...
onmindif 6446 When its successor is subt...
ordnbtwn 6447 There is no set between an...
onnbtwn 6448 There is no set between an...
sucssel 6449 A set whose successor is a...
orddif 6450 Ordinal derived from its s...
orduniss 6451 An ordinal class includes ...
ordtri2or 6452 A trichotomy law for ordin...
ordtri2or2 6453 A trichotomy law for ordin...
ordtri2or3 6454 A consequence of total ord...
ordelinel 6455 The intersection of two or...
ordssun 6456 Property of a subclass of ...
ordequn 6457 The maximum (i.e. union) o...
ordun 6458 The maximum (i.e., union) ...
onunel 6459 The union of two ordinals ...
ordunisssuc 6460 A subclass relationship fo...
suc11 6461 The successor operation be...
onun2 6462 The union of two ordinals ...
ontr 6463 An ordinal number is a tra...
onunisuc 6464 An ordinal number is equal...
onordi 6465 An ordinal number is an or...
ontrciOLD 6466 Obsolete version of ~ ontr...
onirri 6467 An ordinal number is not a...
oneli 6468 A member of an ordinal num...
onelssi 6469 A member of an ordinal num...
onssneli 6470 An ordering law for ordina...
onssnel2i 6471 An ordering law for ordina...
onelini 6472 An element of an ordinal n...
oneluni 6473 An ordinal number equals i...
onunisuci 6474 An ordinal number is equal...
onsseli 6475 Subset is equivalent to me...
onun2i 6476 The union of two ordinal n...
unizlim 6477 An ordinal equal to its ow...
on0eqel 6478 An ordinal number either e...
snsn0non 6479 The singleton of the singl...
onxpdisj 6480 Ordinal numbers and ordere...
onnev 6481 The class of ordinal numbe...
onnevOLD 6482 Obsolete version of ~ onne...
iotajust 6484 Soundness justification th...
dfiota2 6486 Alternate definition for d...
nfiota1 6487 Bound-variable hypothesis ...
nfiotadw 6488 Deduction version of ~ nfi...
nfiotaw 6489 Bound-variable hypothesis ...
nfiotad 6490 Deduction version of ~ nfi...
nfiota 6491 Bound-variable hypothesis ...
cbviotaw 6492 Change bound variables in ...
cbviotavw 6493 Change bound variables in ...
cbviotavwOLD 6494 Obsolete version of ~ cbvi...
cbviota 6495 Change bound variables in ...
cbviotav 6496 Change bound variables in ...
sb8iota 6497 Variable substitution in d...
iotaeq 6498 Equality theorem for descr...
iotabi 6499 Equivalence theorem for de...
uniabio 6500 Part of Theorem 8.17 in [Q...
iotaval2 6501 Version of ~ iotaval using...
iotauni2 6502 Version of ~ iotauni using...
iotanul2 6503 Version of ~ iotanul using...
iotaval 6504 Theorem 8.19 in [Quine] p....
iotassuni 6505 The ` iota ` class is a su...
iotaex 6506 Theorem 8.23 in [Quine] p....
iotavalOLD 6507 Obsolete version of ~ iota...
iotauni 6508 Equivalence between two di...
iotaint 6509 Equivalence between two di...
iota1 6510 Property of iota. (Contri...
iotanul 6511 Theorem 8.22 in [Quine] p....
iotassuniOLD 6512 Obsolete version of ~ iota...
iotaexOLD 6513 Obsolete version of ~ iota...
iota4 6514 Theorem *14.22 in [Whitehe...
iota4an 6515 Theorem *14.23 in [Whitehe...
iota5 6516 A method for computing iot...
iotabidv 6517 Formula-building deduction...
iotabii 6518 Formula-building deduction...
iotacl 6519 Membership law for descrip...
iota2df 6520 A condition that allows to...
iota2d 6521 A condition that allows to...
iota2 6522 The unique element such th...
iotan0 6523 Representation of "the uni...
sniota 6524 A class abstraction with a...
dfiota4 6525 The ` iota ` operation usi...
csbiota 6526 Class substitution within ...
dffun2 6543 Alternate definition of a ...
dffun2OLD 6544 Obsolete version of ~ dffu...
dffun2OLDOLD 6545 Obsolete version of ~ dffu...
dffun6 6546 Alternate definition of a ...
dffun3 6547 Alternate definition of fu...
dffun3OLD 6548 Obsolete version of ~ dffu...
dffun4 6549 Alternate definition of a ...
dffun5 6550 Alternate definition of fu...
dffun6f 6551 Definition of function, us...
dffun6OLD 6552 Obsolete version of ~ dffu...
funmo 6553 A function has at most one...
funmoOLD 6554 Obsolete version of ~ funm...
funrel 6555 A function is a relation. ...
0nelfun 6556 A function does not contai...
funss 6557 Subclass theorem for funct...
funeq 6558 Equality theorem for funct...
funeqi 6559 Equality inference for the...
funeqd 6560 Equality deduction for the...
nffun 6561 Bound-variable hypothesis ...
sbcfung 6562 Distribute proper substitu...
funeu 6563 There is exactly one value...
funeu2 6564 There is exactly one value...
dffun7 6565 Alternate definition of a ...
dffun8 6566 Alternate definition of a ...
dffun9 6567 Alternate definition of a ...
funfn 6568 A class is a function if a...
funfnd 6569 A function is a function o...
funi 6570 The identity relation is a...
nfunv 6571 The universal class is not...
funopg 6572 A Kuratowski ordered pair ...
funopab 6573 A class of ordered pairs i...
funopabeq 6574 A class of ordered pairs o...
funopab4 6575 A class of ordered pairs o...
funmpt 6576 A function in maps-to nota...
funmpt2 6577 Functionality of a class g...
funco 6578 The composition of two fun...
funresfunco 6579 Composition of two functio...
funres 6580 A restriction of a functio...
funresd 6581 A restriction of a functio...
funssres 6582 The restriction of a funct...
fun2ssres 6583 Equality of restrictions o...
funun 6584 The union of functions wit...
fununmo 6585 If the union of classes is...
fununfun 6586 If the union of classes is...
fundif 6587 A function with removed el...
funcnvsn 6588 The converse singleton of ...
funsng 6589 A singleton of an ordered ...
fnsng 6590 Functionality and domain o...
funsn 6591 A singleton of an ordered ...
funprg 6592 A set of two pairs is a fu...
funtpg 6593 A set of three pairs is a ...
funpr 6594 A function with a domain o...
funtp 6595 A function with a domain o...
fnsn 6596 Functionality and domain o...
fnprg 6597 Function with a domain of ...
fntpg 6598 Function with a domain of ...
fntp 6599 A function with a domain o...
funcnvpr 6600 The converse pair of order...
funcnvtp 6601 The converse triple of ord...
funcnvqp 6602 The converse quadruple of ...
fun0 6603 The empty set is a functio...
funcnv0 6604 The converse of the empty ...
funcnvcnv 6605 The double converse of a f...
funcnv2 6606 A simpler equivalence for ...
funcnv 6607 The converse of a class is...
funcnv3 6608 A condition showing a clas...
fun2cnv 6609 The double converse of a c...
svrelfun 6610 A single-valued relation i...
fncnv 6611 Single-rootedness (see ~ f...
fun11 6612 Two ways of stating that `...
fununi 6613 The union of a chain (with...
funin 6614 The intersection with a fu...
funres11 6615 The restriction of a one-t...
funcnvres 6616 The converse of a restrict...
cnvresid 6617 Converse of a restricted i...
funcnvres2 6618 The converse of a restrict...
funimacnv 6619 The image of the preimage ...
funimass1 6620 A kind of contraposition l...
funimass2 6621 A kind of contraposition l...
imadif 6622 The image of a difference ...
imain 6623 The image of an intersecti...
funimaexg 6624 Axiom of Replacement using...
funimaexgOLD 6625 Obsolete version of ~ funi...
funimaex 6626 The image of a set under a...
isarep1 6627 Part of a study of the Axi...
isarep1OLD 6628 Obsolete version of ~ isar...
isarep2 6629 Part of a study of the Axi...
fneq1 6630 Equality theorem for funct...
fneq2 6631 Equality theorem for funct...
fneq1d 6632 Equality deduction for fun...
fneq2d 6633 Equality deduction for fun...
fneq12d 6634 Equality deduction for fun...
fneq12 6635 Equality theorem for funct...
fneq1i 6636 Equality inference for fun...
fneq2i 6637 Equality inference for fun...
nffn 6638 Bound-variable hypothesis ...
fnfun 6639 A function with domain is ...
fnfund 6640 A function with domain is ...
fnrel 6641 A function with domain is ...
fndm 6642 The domain of a function. ...
fndmi 6643 The domain of a function. ...
fndmd 6644 The domain of a function. ...
funfni 6645 Inference to convert a fun...
fndmu 6646 A function has a unique do...
fnbr 6647 The first argument of bina...
fnop 6648 The first argument of an o...
fneu 6649 There is exactly one value...
fneu2 6650 There is exactly one value...
fnunres1 6651 Restriction of a disjoint ...
fnunres2 6652 Restriction of a disjoint ...
fnun 6653 The union of two functions...
fnund 6654 The union of two functions...
fnunop 6655 Extension of a function wi...
fncofn 6656 Composition of a function ...
fnco 6657 Composition of two functio...
fncoOLD 6658 Obsolete version of ~ fnco...
fnresdm 6659 A function does not change...
fnresdisj 6660 A function restricted to a...
2elresin 6661 Membership in two function...
fnssresb 6662 Restriction of a function ...
fnssres 6663 Restriction of a function ...
fnssresd 6664 Restriction of a function ...
fnresin1 6665 Restriction of a function'...
fnresin2 6666 Restriction of a function'...
fnres 6667 An equivalence for functio...
idfn 6668 The identity relation is a...
fnresi 6669 The restricted identity re...
fnima 6670 The image of a function's ...
fn0 6671 A function with empty doma...
fnimadisj 6672 A class that is disjoint w...
fnimaeq0 6673 Images under a function ne...
dfmpt3 6674 Alternate definition for t...
mptfnf 6675 The maps-to notation defin...
fnmptf 6676 The maps-to notation defin...
fnopabg 6677 Functionality and domain o...
fnopab 6678 Functionality and domain o...
mptfng 6679 The maps-to notation defin...
fnmpt 6680 The maps-to notation defin...
fnmptd 6681 The maps-to notation defin...
mpt0 6682 A mapping operation with e...
fnmpti 6683 Functionality and domain o...
dmmpti 6684 Domain of the mapping oper...
dmmptd 6685 The domain of the mapping ...
mptun 6686 Union of mappings which ar...
partfun 6687 Rewrite a function defined...
feq1 6688 Equality theorem for funct...
feq2 6689 Equality theorem for funct...
feq3 6690 Equality theorem for funct...
feq23 6691 Equality theorem for funct...
feq1d 6692 Equality deduction for fun...
feq2d 6693 Equality deduction for fun...
feq3d 6694 Equality deduction for fun...
feq12d 6695 Equality deduction for fun...
feq123d 6696 Equality deduction for fun...
feq123 6697 Equality theorem for funct...
feq1i 6698 Equality inference for fun...
feq2i 6699 Equality inference for fun...
feq12i 6700 Equality inference for fun...
feq23i 6701 Equality inference for fun...
feq23d 6702 Equality deduction for fun...
nff 6703 Bound-variable hypothesis ...
sbcfng 6704 Distribute proper substitu...
sbcfg 6705 Distribute proper substitu...
elimf 6706 Eliminate a mapping hypoth...
ffn 6707 A mapping is a function wi...
ffnd 6708 A mapping is a function wi...
dffn2 6709 Any function is a mapping ...
ffun 6710 A mapping is a function. ...
ffund 6711 A mapping is a function, d...
frel 6712 A mapping is a relation. ...
freld 6713 A mapping is a relation. ...
frn 6714 The range of a mapping. (...
frnd 6715 Deduction form of ~ frn . ...
fdm 6716 The domain of a mapping. ...
fdmOLD 6717 Obsolete version of ~ fdm ...
fdmd 6718 Deduction form of ~ fdm . ...
fdmi 6719 Inference associated with ...
dffn3 6720 A function maps to its ran...
ffrn 6721 A function maps to its ran...
ffrnb 6722 Characterization of a func...
ffrnbd 6723 A function maps to its ran...
fss 6724 Expanding the codomain of ...
fssd 6725 Expanding the codomain of ...
fssdmd 6726 Expressing that a class is...
fssdm 6727 Expressing that a class is...
fimass 6728 The image of a class under...
fimacnv 6729 The preimage of the codoma...
fcof 6730 Composition of a function ...
fco 6731 Composition of two functio...
fcoOLD 6732 Obsolete version of ~ fco ...
fcod 6733 Composition of two mapping...
fco2 6734 Functionality of a composi...
fssxp 6735 A mapping is a class of or...
funssxp 6736 Two ways of specifying a p...
ffdm 6737 A mapping is a partial fun...
ffdmd 6738 The domain of a function. ...
fdmrn 6739 A different way to write `...
funcofd 6740 Composition of two functio...
fco3OLD 6741 Obsolete version of ~ func...
opelf 6742 The members of an ordered ...
fun 6743 The union of two functions...
fun2 6744 The union of two functions...
fun2d 6745 The union of functions wit...
fnfco 6746 Composition of two functio...
fssres 6747 Restriction of a function ...
fssresd 6748 Restriction of a function ...
fssres2 6749 Restriction of a restricte...
fresin 6750 An identity for the mappin...
resasplit 6751 If two functions agree on ...
fresaun 6752 The union of two functions...
fresaunres2 6753 From the union of two func...
fresaunres1 6754 From the union of two func...
fcoi1 6755 Composition of a mapping a...
fcoi2 6756 Composition of restricted ...
feu 6757 There is exactly one value...
fcnvres 6758 The converse of a restrict...
fimacnvdisj 6759 The preimage of a class di...
fint 6760 Function into an intersect...
fin 6761 Mapping into an intersecti...
f0 6762 The empty function. (Cont...
f00 6763 A class is a function with...
f0bi 6764 A function with empty doma...
f0dom0 6765 A function is empty iff it...
f0rn0 6766 If there is no element in ...
fconst 6767 A Cartesian product with a...
fconstg 6768 A Cartesian product with a...
fnconstg 6769 A Cartesian product with a...
fconst6g 6770 Constant function with loo...
fconst6 6771 A constant function as a m...
f1eq1 6772 Equality theorem for one-t...
f1eq2 6773 Equality theorem for one-t...
f1eq3 6774 Equality theorem for one-t...
nff1 6775 Bound-variable hypothesis ...
dff12 6776 Alternate definition of a ...
f1f 6777 A one-to-one mapping is a ...
f1fn 6778 A one-to-one mapping is a ...
f1fun 6779 A one-to-one mapping is a ...
f1rel 6780 A one-to-one onto mapping ...
f1dm 6781 The domain of a one-to-one...
f1dmOLD 6782 Obsolete version of ~ f1dm...
f1ss 6783 A function that is one-to-...
f1ssr 6784 A function that is one-to-...
f1ssres 6785 A function that is one-to-...
f1resf1 6786 The restriction of an inje...
f1cnvcnv 6787 Two ways to express that a...
f1cof1 6788 Composition of two one-to-...
f1co 6789 Composition of one-to-one ...
f1coOLD 6790 Obsolete version of ~ f1co...
foeq1 6791 Equality theorem for onto ...
foeq2 6792 Equality theorem for onto ...
foeq3 6793 Equality theorem for onto ...
nffo 6794 Bound-variable hypothesis ...
fof 6795 An onto mapping is a mappi...
fofun 6796 An onto mapping is a funct...
fofn 6797 An onto mapping is a funct...
forn 6798 The codomain of an onto fu...
dffo2 6799 Alternate definition of an...
foima 6800 The image of the domain of...
dffn4 6801 A function maps onto its r...
funforn 6802 A function maps its domain...
fodmrnu 6803 An onto function has uniqu...
fimadmfo 6804 A function is a function o...
fores 6805 Restriction of an onto fun...
fimadmfoALT 6806 Alternate proof of ~ fimad...
focnvimacdmdm 6807 The preimage of the codoma...
focofo 6808 Composition of onto functi...
foco 6809 Composition of onto functi...
foconst 6810 A nonzero constant functio...
f1oeq1 6811 Equality theorem for one-t...
f1oeq2 6812 Equality theorem for one-t...
f1oeq3 6813 Equality theorem for one-t...
f1oeq23 6814 Equality theorem for one-t...
f1eq123d 6815 Equality deduction for one...
foeq123d 6816 Equality deduction for ont...
f1oeq123d 6817 Equality deduction for one...
f1oeq1d 6818 Equality deduction for one...
f1oeq2d 6819 Equality deduction for one...
f1oeq3d 6820 Equality deduction for one...
nff1o 6821 Bound-variable hypothesis ...
f1of1 6822 A one-to-one onto mapping ...
f1of 6823 A one-to-one onto mapping ...
f1ofn 6824 A one-to-one onto mapping ...
f1ofun 6825 A one-to-one onto mapping ...
f1orel 6826 A one-to-one onto mapping ...
f1odm 6827 The domain of a one-to-one...
dff1o2 6828 Alternate definition of on...
dff1o3 6829 Alternate definition of on...
f1ofo 6830 A one-to-one onto function...
dff1o4 6831 Alternate definition of on...
dff1o5 6832 Alternate definition of on...
f1orn 6833 A one-to-one function maps...
f1f1orn 6834 A one-to-one function maps...
f1ocnv 6835 The converse of a one-to-o...
f1ocnvb 6836 A relation is a one-to-one...
f1ores 6837 The restriction of a one-t...
f1orescnv 6838 The converse of a one-to-o...
f1imacnv 6839 Preimage of an image. (Co...
foimacnv 6840 A reverse version of ~ f1i...
foun 6841 The union of two onto func...
f1oun 6842 The union of two one-to-on...
f1un 6843 The union of two one-to-on...
resdif 6844 The restriction of a one-t...
resin 6845 The restriction of a one-t...
f1oco 6846 Composition of one-to-one ...
f1cnv 6847 The converse of an injecti...
funcocnv2 6848 Composition with the conve...
fococnv2 6849 The composition of an onto...
f1ococnv2 6850 The composition of a one-t...
f1cocnv2 6851 Composition of an injectiv...
f1ococnv1 6852 The composition of a one-t...
f1cocnv1 6853 Composition of an injectiv...
funcoeqres 6854 Express a constraint on a ...
f1ssf1 6855 A subset of an injective f...
f10 6856 The empty set maps one-to-...
f10d 6857 The empty set maps one-to-...
f1o00 6858 One-to-one onto mapping of...
fo00 6859 Onto mapping of the empty ...
f1o0 6860 One-to-one onto mapping of...
f1oi 6861 A restriction of the ident...
f1ovi 6862 The identity relation is a...
f1osn 6863 A singleton of an ordered ...
f1osng 6864 A singleton of an ordered ...
f1sng 6865 A singleton of an ordered ...
fsnd 6866 A singleton of an ordered ...
f1oprswap 6867 A two-element swap is a bi...
f1oprg 6868 An unordered pair of order...
tz6.12-2 6869 Function value when ` F ` ...
fveu 6870 The value of a function at...
brprcneu 6871 If ` A ` is a proper class...
brprcneuALT 6872 Alternate proof of ~ brprc...
fvprc 6873 A function's value at a pr...
fvprcALT 6874 Alternate proof of ~ fvprc...
rnfvprc 6875 The range of a function va...
fv2 6876 Alternate definition of fu...
dffv3 6877 A definition of function v...
dffv4 6878 The previous definition of...
elfv 6879 Membership in a function v...
fveq1 6880 Equality theorem for funct...
fveq2 6881 Equality theorem for funct...
fveq1i 6882 Equality inference for fun...
fveq1d 6883 Equality deduction for fun...
fveq2i 6884 Equality inference for fun...
fveq2d 6885 Equality deduction for fun...
2fveq3 6886 Equality theorem for neste...
fveq12i 6887 Equality deduction for fun...
fveq12d 6888 Equality deduction for fun...
fveqeq2d 6889 Equality deduction for fun...
fveqeq2 6890 Equality deduction for fun...
nffv 6891 Bound-variable hypothesis ...
nffvmpt1 6892 Bound-variable hypothesis ...
nffvd 6893 Deduction version of bound...
fvex 6894 The value of a class exist...
fvexi 6895 The value of a class exist...
fvexd 6896 The value of a class exist...
fvif 6897 Move a conditional outside...
iffv 6898 Move a conditional outside...
fv3 6899 Alternate definition of th...
fvres 6900 The value of a restricted ...
fvresd 6901 The value of a restricted ...
funssfv 6902 The value of a member of t...
tz6.12c 6903 Corollary of Theorem 6.12(...
tz6.12-1 6904 Function value. Theorem 6...
tz6.12-1OLD 6905 Obsolete version of ~ tz6....
tz6.12 6906 Function value. Theorem 6...
tz6.12f 6907 Function value, using boun...
tz6.12cOLD 6908 Obsolete version of ~ tz6....
tz6.12i 6909 Corollary of Theorem 6.12(...
fvbr0 6910 Two possibilities for the ...
fvrn0 6911 A function value is a memb...
fvn0fvelrn 6912 If the value of a function...
elfvunirn 6913 A function value is a subs...
fvssunirn 6914 The result of a function v...
fvssunirnOLD 6915 Obsolete version of ~ fvss...
ndmfv 6916 The value of a class outsi...
ndmfvrcl 6917 Reverse closure law for fu...
elfvdm 6918 If a function value has a ...
elfvex 6919 If a function value has a ...
elfvexd 6920 If a function value has a ...
eliman0 6921 A nonempty function value ...
nfvres 6922 The value of a non-member ...
nfunsn 6923 If the restriction of a cl...
fvfundmfvn0 6924 If the "value of a class" ...
0fv 6925 Function value of the empt...
fv2prc 6926 A function value of a func...
elfv2ex 6927 If a function value of a f...
fveqres 6928 Equal values imply equal v...
csbfv12 6929 Move class substitution in...
csbfv2g 6930 Move class substitution in...
csbfv 6931 Substitution for a functio...
funbrfv 6932 The second argument of a b...
funopfv 6933 The second element in an o...
fnbrfvb 6934 Equivalence of function va...
fnopfvb 6935 Equivalence of function va...
funbrfvb 6936 Equivalence of function va...
funopfvb 6937 Equivalence of function va...
fnbrfvb2 6938 Version of ~ fnbrfvb for f...
funbrfv2b 6939 Function value in terms of...
dffn5 6940 Representation of a functi...
fnrnfv 6941 The range of a function ex...
fvelrnb 6942 A member of a function's r...
foelcdmi 6943 A member of a surjective f...
dfimafn 6944 Alternate definition of th...
dfimafn2 6945 Alternate definition of th...
funimass4 6946 Membership relation for th...
fvelima 6947 Function value in an image...
funimassd 6948 Sufficient condition for t...
fvelimad 6949 Function value in an image...
feqmptd 6950 Deduction form of ~ dffn5 ...
feqresmpt 6951 Express a restricted funct...
feqmptdf 6952 Deduction form of ~ dffn5f...
dffn5f 6953 Representation of a functi...
fvelimab 6954 Function value in an image...
fvelimabd 6955 Deduction form of ~ fvelim...
unima 6956 Image of a union. (Contri...
fvi 6957 The value of the identity ...
fviss 6958 The value of the identity ...
fniinfv 6959 The indexed intersection o...
fnsnfv 6960 Singleton of function valu...
fnsnfvOLD 6961 Obsolete version of ~ fnsn...
opabiotafun 6962 Define a function whose va...
opabiotadm 6963 Define a function whose va...
opabiota 6964 Define a function whose va...
fnimapr 6965 The image of a pair under ...
ssimaex 6966 The existence of a subimag...
ssimaexg 6967 The existence of a subimag...
funfv 6968 A simplified expression fo...
funfv2 6969 The value of a function. ...
funfv2f 6970 The value of a function. ...
fvun 6971 Value of the union of two ...
fvun1 6972 The value of a union when ...
fvun2 6973 The value of a union when ...
fvun1d 6974 The value of a union when ...
fvun2d 6975 The value of a union when ...
dffv2 6976 Alternate definition of fu...
dmfco 6977 Domains of a function comp...
fvco2 6978 Value of a function compos...
fvco 6979 Value of a function compos...
fvco3 6980 Value of a function compos...
fvco3d 6981 Value of a function compos...
fvco4i 6982 Conditions for a compositi...
fvopab3g 6983 Value of a function given ...
fvopab3ig 6984 Value of a function given ...
brfvopabrbr 6985 The binary relation of a f...
fvmptg 6986 Value of a function given ...
fvmpti 6987 Value of a function given ...
fvmpt 6988 Value of a function given ...
fvmpt2f 6989 Value of a function given ...
fvtresfn 6990 Functionality of a tuple-r...
fvmpts 6991 Value of a function given ...
fvmpt3 6992 Value of a function given ...
fvmpt3i 6993 Value of a function given ...
fvmptdf 6994 Deduction version of ~ fvm...
fvmptd 6995 Deduction version of ~ fvm...
fvmptd2 6996 Deduction version of ~ fvm...
mptrcl 6997 Reverse closure for a mapp...
fvmpt2i 6998 Value of a function given ...
fvmpt2 6999 Value of a function given ...
fvmptss 7000 If all the values of the m...
fvmpt2d 7001 Deduction version of ~ fvm...
fvmptex 7002 Express a function ` F ` w...
fvmptd3f 7003 Alternate deduction versio...
fvmptd2f 7004 Alternate deduction versio...
fvmptdv 7005 Alternate deduction versio...
fvmptdv2 7006 Alternate deduction versio...
mpteqb 7007 Bidirectional equality the...
fvmptt 7008 Closed theorem form of ~ f...
fvmptf 7009 Value of a function given ...
fvmptnf 7010 The value of a function gi...
fvmptd3 7011 Deduction version of ~ fvm...
fvmptn 7012 This somewhat non-intuitiv...
fvmptss2 7013 A mapping always evaluates...
elfvmptrab1w 7014 Implications for the value...
elfvmptrab1 7015 Implications for the value...
elfvmptrab 7016 Implications for the value...
fvopab4ndm 7017 Value of a function given ...
fvmptndm 7018 Value of a function given ...
fvmptrabfv 7019 Value of a function mappin...
fvopab5 7020 The value of a function th...
fvopab6 7021 Value of a function given ...
eqfnfv 7022 Equality of functions is d...
eqfnfv2 7023 Equality of functions is d...
eqfnfv3 7024 Derive equality of functio...
eqfnfvd 7025 Deduction for equality of ...
eqfnfv2f 7026 Equality of functions is d...
eqfunfv 7027 Equality of functions is d...
eqfnun 7028 Two functions on ` A u. B ...
fvreseq0 7029 Equality of restricted fun...
fvreseq1 7030 Equality of a function res...
fvreseq 7031 Equality of restricted fun...
fnmptfvd 7032 A function with a given do...
fndmdif 7033 Two ways to express the lo...
fndmdifcom 7034 The difference set between...
fndmdifeq0 7035 The difference set of two ...
fndmin 7036 Two ways to express the lo...
fneqeql 7037 Two functions are equal if...
fneqeql2 7038 Two functions are equal if...
fnreseql 7039 Two functions are equal on...
chfnrn 7040 The range of a choice func...
funfvop 7041 Ordered pair with function...
funfvbrb 7042 Two ways to say that ` A `...
fvimacnvi 7043 A member of a preimage is ...
fvimacnv 7044 The argument of a function...
funimass3 7045 A kind of contraposition l...
funimass5 7046 A subclass of a preimage i...
funconstss 7047 Two ways of specifying tha...
fvimacnvALT 7048 Alternate proof of ~ fvima...
elpreima 7049 Membership in the preimage...
elpreimad 7050 Membership in the preimage...
fniniseg 7051 Membership in the preimage...
fncnvima2 7052 Inverse images under funct...
fniniseg2 7053 Inverse point images under...
unpreima 7054 Preimage of a union. (Con...
inpreima 7055 Preimage of an intersectio...
difpreima 7056 Preimage of a difference. ...
respreima 7057 The preimage of a restrict...
cnvimainrn 7058 The preimage of the inters...
sspreima 7059 The preimage of a subset i...
iinpreima 7060 Preimage of an intersectio...
intpreima 7061 Preimage of an intersectio...
fimacnvOLD 7062 Obsolete version of ~ fima...
fimacnvinrn 7063 Taking the converse image ...
fimacnvinrn2 7064 Taking the converse image ...
rescnvimafod 7065 The restriction of a funct...
fvn0ssdmfun 7066 If a class' function value...
fnopfv 7067 Ordered pair with function...
fvelrn 7068 A function's value belongs...
nelrnfvne 7069 A function value cannot be...
fveqdmss 7070 If the empty set is not co...
fveqressseq 7071 If the empty set is not co...
fnfvelrn 7072 A function's value belongs...
ffvelcdm 7073 A function's value belongs...
fnfvelrnd 7074 A function's value belongs...
ffvelcdmi 7075 A function's value belongs...
ffvelcdmda 7076 A function's value belongs...
ffvelcdmd 7077 A function's value belongs...
rexrn 7078 Restricted existential qua...
ralrn 7079 Restricted universal quant...
elrnrexdm 7080 For any element in the ran...
elrnrexdmb 7081 For any element in the ran...
eldmrexrn 7082 For any element in the dom...
eldmrexrnb 7083 For any element in the dom...
fvcofneq 7084 The values of two function...
ralrnmptw 7085 A restricted quantifier ov...
rexrnmptw 7086 A restricted quantifier ov...
ralrnmpt 7087 A restricted quantifier ov...
rexrnmpt 7088 A restricted quantifier ov...
f0cli 7089 Unconditional closure of a...
dff2 7090 Alternate definition of a ...
dff3 7091 Alternate definition of a ...
dff4 7092 Alternate definition of a ...
dffo3 7093 An onto mapping expressed ...
dffo4 7094 Alternate definition of an...
dffo5 7095 Alternate definition of an...
exfo 7096 A relation equivalent to t...
dffo3f 7097 An onto mapping expressed ...
foelrn 7098 Property of a surjective f...
foelrnf 7099 Property of a surjective f...
foco2 7100 If a composition of two fu...
fmpt 7101 Functionality of the mappi...
f1ompt 7102 Express bijection for a ma...
fmpti 7103 Functionality of the mappi...
fvmptelcdm 7104 The value of a function at...
fmptd 7105 Domain and codomain of the...
fmpttd 7106 Version of ~ fmptd with in...
fmpt3d 7107 Domain and codomain of the...
fmptdf 7108 A version of ~ fmptd using...
fompt 7109 Express being onto for a m...
ffnfv 7110 A function maps to a class...
ffnfvf 7111 A function maps to a class...
fnfvrnss 7112 An upper bound for range d...
fcdmssb 7113 A function is a function i...
rnmptss 7114 The range of an operation ...
fmpt2d 7115 Domain and codomain of the...
ffvresb 7116 A necessary and sufficient...
f1oresrab 7117 Build a bijection between ...
f1ossf1o 7118 Restricting a bijection, w...
fmptco 7119 Composition of two functio...
fmptcof 7120 Version of ~ fmptco where ...
fmptcos 7121 Composition of two functio...
cofmpt 7122 Express composition of a m...
fcompt 7123 Express composition of two...
fcoconst 7124 Composition with a constan...
fsn 7125 A function maps a singleto...
fsn2 7126 A function that maps a sin...
fsng 7127 A function maps a singleto...
fsn2g 7128 A function that maps a sin...
xpsng 7129 The Cartesian product of t...
xpprsng 7130 The Cartesian product of a...
xpsn 7131 The Cartesian product of t...
f1o2sn 7132 A singleton consisting in ...
residpr 7133 Restriction of the identit...
dfmpt 7134 Alternate definition for t...
fnasrn 7135 A function expressed as th...
idref 7136 Two ways to state that a r...
funiun 7137 A function is a union of s...
funopsn 7138 If a function is an ordere...
funop 7139 An ordered pair is a funct...
funopdmsn 7140 The domain of a function w...
funsndifnop 7141 A singleton of an ordered ...
funsneqopb 7142 A singleton of an ordered ...
ressnop0 7143 If ` A ` is not in ` C ` ,...
fpr 7144 A function with a domain o...
fprg 7145 A function with a domain o...
ftpg 7146 A function with a domain o...
ftp 7147 A function with a domain o...
fnressn 7148 A function restricted to a...
funressn 7149 A function restricted to a...
fressnfv 7150 The value of a function re...
fvrnressn 7151 If the value of a function...
fvressn 7152 The value of a function re...
fvn0fvelrnOLD 7153 Obsolete version of ~ fvn0...
fvconst 7154 The value of a constant fu...
fnsnr 7155 If a class belongs to a fu...
fnsnb 7156 A function whose domain is...
fmptsn 7157 Express a singleton functi...
fmptsng 7158 Express a singleton functi...
fmptsnd 7159 Express a singleton functi...
fmptap 7160 Append an additional value...
fmptapd 7161 Append an additional value...
fmptpr 7162 Express a pair function in...
fvresi 7163 The value of a restricted ...
fninfp 7164 Express the class of fixed...
fnelfp 7165 Property of a fixed point ...
fndifnfp 7166 Express the class of non-f...
fnelnfp 7167 Property of a non-fixed po...
fnnfpeq0 7168 A function is the identity...
fvunsn 7169 Remove an ordered pair not...
fvsng 7170 The value of a singleton o...
fvsn 7171 The value of a singleton o...
fvsnun1 7172 The value of a function wi...
fvsnun2 7173 The value of a function wi...
fnsnsplit 7174 Split a function into a si...
fsnunf 7175 Adjoining a point to a fun...
fsnunf2 7176 Adjoining a point to a pun...
fsnunfv 7177 Recover the added point fr...
fsnunres 7178 Recover the original funct...
funresdfunsn 7179 Restricting a function to ...
fvpr1g 7180 The value of a function wi...
fvpr2g 7181 The value of a function wi...
fvpr2gOLD 7182 Obsolete version of ~ fvpr...
fvpr1 7183 The value of a function wi...
fvpr1OLD 7184 Obsolete version of ~ fvpr...
fvpr2 7185 The value of a function wi...
fvpr2OLD 7186 Obsolete version of ~ fvpr...
fprb 7187 A condition for functionho...
fvtp1 7188 The first value of a funct...
fvtp2 7189 The second value of a func...
fvtp3 7190 The third value of a funct...
fvtp1g 7191 The value of a function wi...
fvtp2g 7192 The value of a function wi...
fvtp3g 7193 The value of a function wi...
tpres 7194 An unordered triple of ord...
fvconst2g 7195 The value of a constant fu...
fconst2g 7196 A constant function expres...
fvconst2 7197 The value of a constant fu...
fconst2 7198 A constant function expres...
fconst5 7199 Two ways to express that a...
rnmptc 7200 Range of a constant functi...
fnprb 7201 A function whose domain ha...
fntpb 7202 A function whose domain ha...
fnpr2g 7203 A function whose domain ha...
fpr2g 7204 A function that maps a pai...
fconstfv 7205 A constant function expres...
fconst3 7206 Two ways to express a cons...
fconst4 7207 Two ways to express a cons...
resfunexg 7208 The restriction of a funct...
resiexd 7209 The restriction of the ide...
fnex 7210 If the domain of a functio...
fnexd 7211 If the domain of a functio...
funex 7212 If the domain of a functio...
opabex 7213 Existence of a function ex...
mptexg 7214 If the domain of a functio...
mptexgf 7215 If the domain of a functio...
mptex 7216 If the domain of a functio...
mptexd 7217 If the domain of a functio...
mptrabex 7218 If the domain of a functio...
fex 7219 If the domain of a mapping...
fexd 7220 If the domain of a mapping...
mptfvmpt 7221 A function in maps-to nota...
eufnfv 7222 A function is uniquely det...
funfvima 7223 A function's value in a pr...
funfvima2 7224 A function's value in an i...
funfvima2d 7225 A function's value in a pr...
fnfvima 7226 The function value of an o...
fnfvimad 7227 A function's value belongs...
resfvresima 7228 The value of the function ...
funfvima3 7229 A class including a functi...
rexima 7230 Existential quantification...
ralima 7231 Universal quantification u...
fvclss 7232 Upper bound for the class ...
elabrex 7233 Elementhood in an image se...
elabrexg 7234 Elementhood in an image se...
abrexco 7235 Composition of two image m...
imaiun 7236 The image of an indexed un...
imauni 7237 The image of a union is th...
fniunfv 7238 The indexed union of a fun...
funiunfv 7239 The indexed union of a fun...
funiunfvf 7240 The indexed union of a fun...
eluniima 7241 Membership in the union of...
elunirn 7242 Membership in the union of...
elunirnALT 7243 Alternate proof of ~ eluni...
elunirn2OLD 7244 Obsolete version of ~ elfv...
fnunirn 7245 Membership in a union of s...
dff13 7246 A one-to-one function in t...
dff13f 7247 A one-to-one function in t...
f1veqaeq 7248 If the values of a one-to-...
f1cofveqaeq 7249 If the values of a composi...
f1cofveqaeqALT 7250 Alternate proof of ~ f1cof...
2f1fvneq 7251 If two one-to-one function...
f1mpt 7252 Express injection for a ma...
f1fveq 7253 Equality of function value...
f1elima 7254 Membership in the image of...
f1imass 7255 Taking images under a one-...
f1imaeq 7256 Taking images under a one-...
f1imapss 7257 Taking images under a one-...
fpropnf1 7258 A function, given by an un...
f1dom3fv3dif 7259 The function values for a ...
f1dom3el3dif 7260 The codomain of a 1-1 func...
dff14a 7261 A one-to-one function in t...
dff14b 7262 A one-to-one function in t...
f12dfv 7263 A one-to-one function with...
f13dfv 7264 A one-to-one function with...
dff1o6 7265 A one-to-one onto function...
f1ocnvfv1 7266 The converse value of the ...
f1ocnvfv2 7267 The value of the converse ...
f1ocnvfv 7268 Relationship between the v...
f1ocnvfvb 7269 Relationship between the v...
nvof1o 7270 An involution is a bijecti...
nvocnv 7271 The converse of an involut...
f1cdmsn 7272 If a one-to-one function w...
fsnex 7273 Relate a function with a s...
f1prex 7274 Relate a one-to-one functi...
f1ocnvdm 7275 The value of the converse ...
f1ocnvfvrneq 7276 If the values of a one-to-...
fcof1 7277 An application is injectiv...
fcofo 7278 An application is surjecti...
cbvfo 7279 Change bound variable betw...
cbvexfo 7280 Change bound variable betw...
cocan1 7281 An injection is left-cance...
cocan2 7282 A surjection is right-canc...
fcof1oinvd 7283 Show that a function is th...
fcof1od 7284 A function is bijective if...
2fcoidinvd 7285 Show that a function is th...
fcof1o 7286 Show that two functions ar...
2fvcoidd 7287 Show that the composition ...
2fvidf1od 7288 A function is bijective if...
2fvidinvd 7289 Show that two functions ar...
foeqcnvco 7290 Condition for function equ...
f1eqcocnv 7291 Condition for function equ...
f1eqcocnvOLD 7292 Obsolete version of ~ f1eq...
fveqf1o 7293 Given a bijection ` F ` , ...
nf1const 7294 A constant function from a...
nf1oconst 7295 A constant function from a...
f1ofvswap 7296 Swapping two values in a b...
fliftrel 7297 ` F ` , a function lift, i...
fliftel 7298 Elementhood in the relatio...
fliftel1 7299 Elementhood in the relatio...
fliftcnv 7300 Converse of the relation `...
fliftfun 7301 The function ` F ` is the ...
fliftfund 7302 The function ` F ` is the ...
fliftfuns 7303 The function ` F ` is the ...
fliftf 7304 The domain and range of th...
fliftval 7305 The value of the function ...
isoeq1 7306 Equality theorem for isomo...
isoeq2 7307 Equality theorem for isomo...
isoeq3 7308 Equality theorem for isomo...
isoeq4 7309 Equality theorem for isomo...
isoeq5 7310 Equality theorem for isomo...
nfiso 7311 Bound-variable hypothesis ...
isof1o 7312 An isomorphism is a one-to...
isof1oidb 7313 A function is a bijection ...
isof1oopb 7314 A function is a bijection ...
isorel 7315 An isomorphism connects bi...
soisores 7316 Express the condition of i...
soisoi 7317 Infer isomorphism from one...
isoid 7318 Identity law for isomorphi...
isocnv 7319 Converse law for isomorphi...
isocnv2 7320 Converse law for isomorphi...
isocnv3 7321 Complementation law for is...
isores2 7322 An isomorphism from one we...
isores1 7323 An isomorphism from one we...
isores3 7324 Induced isomorphism on a s...
isotr 7325 Composition (transitive) l...
isomin 7326 Isomorphisms preserve mini...
isoini 7327 Isomorphisms preserve init...
isoini2 7328 Isomorphisms are isomorphi...
isofrlem 7329 Lemma for ~ isofr . (Cont...
isoselem 7330 Lemma for ~ isose . (Cont...
isofr 7331 An isomorphism preserves w...
isose 7332 An isomorphism preserves s...
isofr2 7333 A weak form of ~ isofr tha...
isopolem 7334 Lemma for ~ isopo . (Cont...
isopo 7335 An isomorphism preserves t...
isosolem 7336 Lemma for ~ isoso . (Cont...
isoso 7337 An isomorphism preserves t...
isowe 7338 An isomorphism preserves t...
isowe2 7339 A weak form of ~ isowe tha...
f1oiso 7340 Any one-to-one onto functi...
f1oiso2 7341 Any one-to-one onto functi...
f1owe 7342 Well-ordering of isomorphi...
weniso 7343 A set-like well-ordering h...
weisoeq 7344 Thus, there is at most one...
weisoeq2 7345 Thus, there is at most one...
knatar 7346 The Knaster-Tarski theorem...
fvresval 7347 The value of a restricted ...
funeldmb 7348 If ` (/) ` is not part of ...
eqfunresadj 7349 Law for adjoining an eleme...
eqfunressuc 7350 Law for equality of restri...
fnssintima 7351 Condition for subset of an...
imaeqsexv 7352 Substitute a function valu...
imaeqsalv 7353 Substitute a function valu...
canth 7354 No set ` A ` is equinumero...
ncanth 7355 Cantor's theorem fails for...
riotaeqdv 7358 Formula-building deduction...
riotabidv 7359 Formula-building deduction...
riotaeqbidv 7360 Equality deduction for res...
riotaex 7361 Restricted iota is a set. ...
riotav 7362 An iota restricted to the ...
riotauni 7363 Restricted iota in terms o...
nfriota1 7364 The abstraction variable i...
nfriotadw 7365 Deduction version of ~ nfr...
cbvriotaw 7366 Change bound variable in a...
cbvriotavw 7367 Change bound variable in a...
cbvriotavwOLD 7368 Obsolete version of ~ cbvr...
nfriotad 7369 Deduction version of ~ nfr...
nfriota 7370 A variable not free in a w...
cbvriota 7371 Change bound variable in a...
cbvriotav 7372 Change bound variable in a...
csbriota 7373 Interchange class substitu...
riotacl2 7374 Membership law for "the un...
riotacl 7375 Closure of restricted iota...
riotasbc 7376 Substitution law for descr...
riotabidva 7377 Equivalent wff's yield equ...
riotabiia 7378 Equivalent wff's yield equ...
riota1 7379 Property of restricted iot...
riota1a 7380 Property of iota. (Contri...
riota2df 7381 A deduction version of ~ r...
riota2f 7382 This theorem shows a condi...
riota2 7383 This theorem shows a condi...
riotaeqimp 7384 If two restricted iota des...
riotaprop 7385 Properties of a restricted...
riota5f 7386 A method for computing res...
riota5 7387 A method for computing res...
riotass2 7388 Restriction of a unique el...
riotass 7389 Restriction of a unique el...
moriotass 7390 Restriction of a unique el...
snriota 7391 A restricted class abstrac...
riotaxfrd 7392 Change the variable ` x ` ...
eusvobj2 7393 Specify the same property ...
eusvobj1 7394 Specify the same object in...
f1ofveu 7395 There is one domain elemen...
f1ocnvfv3 7396 Value of the converse of a...
riotaund 7397 Restricted iota equals the...
riotassuni 7398 The restricted iota class ...
riotaclb 7399 Bidirectional closure of r...
riotarab 7400 Restricted iota of a restr...
oveq 7407 Equality theorem for opera...
oveq1 7408 Equality theorem for opera...
oveq2 7409 Equality theorem for opera...
oveq12 7410 Equality theorem for opera...
oveq1i 7411 Equality inference for ope...
oveq2i 7412 Equality inference for ope...
oveq12i 7413 Equality inference for ope...
oveqi 7414 Equality inference for ope...
oveq123i 7415 Equality inference for ope...
oveq1d 7416 Equality deduction for ope...
oveq2d 7417 Equality deduction for ope...
oveqd 7418 Equality deduction for ope...
oveq12d 7419 Equality deduction for ope...
oveqan12d 7420 Equality deduction for ope...
oveqan12rd 7421 Equality deduction for ope...
oveq123d 7422 Equality deduction for ope...
fvoveq1d 7423 Equality deduction for nes...
fvoveq1 7424 Equality theorem for neste...
ovanraleqv 7425 Equality theorem for a con...
imbrov2fvoveq 7426 Equality theorem for neste...
ovrspc2v 7427 If an operation value is e...
oveqrspc2v 7428 Restricted specialization ...
oveqdr 7429 Equality of two operations...
nfovd 7430 Deduction version of bound...
nfov 7431 Bound-variable hypothesis ...
oprabidw 7432 The law of concretion. Sp...
oprabid 7433 The law of concretion. Sp...
ovex 7434 The result of an operation...
ovexi 7435 The result of an operation...
ovexd 7436 The result of an operation...
ovssunirn 7437 The result of an operation...
0ov 7438 Operation value of the emp...
ovprc 7439 The value of an operation ...
ovprc1 7440 The value of an operation ...
ovprc2 7441 The value of an operation ...
ovrcl 7442 Reverse closure for an ope...
csbov123 7443 Move class substitution in...
csbov 7444 Move class substitution in...
csbov12g 7445 Move class substitution in...
csbov1g 7446 Move class substitution in...
csbov2g 7447 Move class substitution in...
rspceov 7448 A frequently used special ...
elovimad 7449 Elementhood of the image s...
fnbrovb 7450 Value of a binary operatio...
fnotovb 7451 Equivalence of operation v...
opabbrex 7452 A collection of ordered pa...
opabresex2 7453 Restrictions of a collecti...
opabresex2d 7454 Obsolete version of ~ opab...
fvmptopab 7455 The function value of a ma...
fvmptopabOLD 7456 Obsolete version of ~ fvmp...
f1opr 7457 Condition for an operation...
brfvopab 7458 The classes involved in a ...
dfoprab2 7459 Class abstraction for oper...
reloprab 7460 An operation class abstrac...
oprabv 7461 If a pair and a class are ...
nfoprab1 7462 The abstraction variables ...
nfoprab2 7463 The abstraction variables ...
nfoprab3 7464 The abstraction variables ...
nfoprab 7465 Bound-variable hypothesis ...
oprabbid 7466 Equivalent wff's yield equ...
oprabbidv 7467 Equivalent wff's yield equ...
oprabbii 7468 Equivalent wff's yield equ...
ssoprab2 7469 Equivalence of ordered pai...
ssoprab2b 7470 Equivalence of ordered pai...
eqoprab2bw 7471 Equivalence of ordered pai...
eqoprab2b 7472 Equivalence of ordered pai...
mpoeq123 7473 An equality theorem for th...
mpoeq12 7474 An equality theorem for th...
mpoeq123dva 7475 An equality deduction for ...
mpoeq123dv 7476 An equality deduction for ...
mpoeq123i 7477 An equality inference for ...
mpoeq3dva 7478 Slightly more general equa...
mpoeq3ia 7479 An equality inference for ...
mpoeq3dv 7480 An equality deduction for ...
nfmpo1 7481 Bound-variable hypothesis ...
nfmpo2 7482 Bound-variable hypothesis ...
nfmpo 7483 Bound-variable hypothesis ...
0mpo0 7484 A mapping operation with e...
mpo0v 7485 A mapping operation with e...
mpo0 7486 A mapping operation with e...
oprab4 7487 Two ways to state the doma...
cbvoprab1 7488 Rule used to change first ...
cbvoprab2 7489 Change the second bound va...
cbvoprab12 7490 Rule used to change first ...
cbvoprab12v 7491 Rule used to change first ...
cbvoprab3 7492 Rule used to change the th...
cbvoprab3v 7493 Rule used to change the th...
cbvmpox 7494 Rule to change the bound v...
cbvmpo 7495 Rule to change the bound v...
cbvmpov 7496 Rule to change the bound v...
elimdelov 7497 Eliminate a hypothesis whi...
ovif 7498 Move a conditional outside...
ovif2 7499 Move a conditional outside...
ovif12 7500 Move a conditional outside...
ifov 7501 Move a conditional outside...
dmoprab 7502 The domain of an operation...
dmoprabss 7503 The domain of an operation...
rnoprab 7504 The range of an operation ...
rnoprab2 7505 The range of a restricted ...
reldmoprab 7506 The domain of an operation...
oprabss 7507 Structure of an operation ...
eloprabga 7508 The law of concretion for ...
eloprabgaOLD 7509 Obsolete version of ~ elop...
eloprabg 7510 The law of concretion for ...
ssoprab2i 7511 Inference of operation cla...
mpov 7512 Operation with universal d...
mpomptx 7513 Express a two-argument fun...
mpompt 7514 Express a two-argument fun...
mpodifsnif 7515 A mapping with two argumen...
mposnif 7516 A mapping with two argumen...
fconstmpo 7517 Representation of a consta...
resoprab 7518 Restriction of an operatio...
resoprab2 7519 Restriction of an operator...
resmpo 7520 Restriction of the mapping...
funoprabg 7521 "At most one" is a suffici...
funoprab 7522 "At most one" is a suffici...
fnoprabg 7523 Functionality and domain o...
mpofun 7524 The maps-to notation for a...
mpofunOLD 7525 Obsolete version of ~ mpof...
fnoprab 7526 Functionality and domain o...
ffnov 7527 An operation maps to a cla...
fovcld 7528 Closure law for an operati...
fovcl 7529 Closure law for an operati...
eqfnov 7530 Equality of two operations...
eqfnov2 7531 Two operators with the sam...
fnov 7532 Representation of a functi...
mpo2eqb 7533 Bidirectional equality the...
rnmpo 7534 The range of an operation ...
reldmmpo 7535 The domain of an operation...
elrnmpog 7536 Membership in the range of...
elrnmpo 7537 Membership in the range of...
elrnmpores 7538 Membership in the range of...
ralrnmpo 7539 A restricted quantifier ov...
rexrnmpo 7540 A restricted quantifier ov...
ovid 7541 The value of an operation ...
ovidig 7542 The value of an operation ...
ovidi 7543 The value of an operation ...
ov 7544 The value of an operation ...
ovigg 7545 The value of an operation ...
ovig 7546 The value of an operation ...
ovmpt4g 7547 Value of a function given ...
ovmpos 7548 Value of a function given ...
ov2gf 7549 The value of an operation ...
ovmpodxf 7550 Value of an operation give...
ovmpodx 7551 Value of an operation give...
ovmpod 7552 Value of an operation give...
ovmpox 7553 The value of an operation ...
ovmpoga 7554 Value of an operation give...
ovmpoa 7555 Value of an operation give...
ovmpodf 7556 Alternate deduction versio...
ovmpodv 7557 Alternate deduction versio...
ovmpodv2 7558 Alternate deduction versio...
ovmpog 7559 Value of an operation give...
ovmpo 7560 Value of an operation give...
ovmpot 7561 The value of an operation ...
fvmpopr2d 7562 Value of an operation give...
ov3 7563 The value of an operation ...
ov6g 7564 The value of an operation ...
ovg 7565 The value of an operation ...
ovres 7566 The value of a restricted ...
ovresd 7567 Lemma for converting metri...
oprres 7568 The restriction of an oper...
oprssov 7569 The value of a member of t...
fovcdm 7570 An operation's value belon...
fovcdmda 7571 An operation's value belon...
fovcdmd 7572 An operation's value belon...
fnrnov 7573 The range of an operation ...
foov 7574 An onto mapping of an oper...
fnovrn 7575 An operation's value belon...
ovelrn 7576 A member of an operation's...
funimassov 7577 Membership relation for th...
ovelimab 7578 Operation value in an imag...
ovima0 7579 An operation value is a me...
ovconst2 7580 The value of a constant op...
oprssdm 7581 Domain of closure of an op...
nssdmovg 7582 The value of an operation ...
ndmovg 7583 The value of an operation ...
ndmov 7584 The value of an operation ...
ndmovcl 7585 The closure of an operatio...
ndmovrcl 7586 Reverse closure law, when ...
ndmovcom 7587 Any operation is commutati...
ndmovass 7588 Any operation is associati...
ndmovdistr 7589 Any operation is distribut...
ndmovord 7590 Elimination of redundant a...
ndmovordi 7591 Elimination of redundant a...
caovclg 7592 Convert an operation closu...
caovcld 7593 Convert an operation closu...
caovcl 7594 Convert an operation closu...
caovcomg 7595 Convert an operation commu...
caovcomd 7596 Convert an operation commu...
caovcom 7597 Convert an operation commu...
caovassg 7598 Convert an operation assoc...
caovassd 7599 Convert an operation assoc...
caovass 7600 Convert an operation assoc...
caovcang 7601 Convert an operation cance...
caovcand 7602 Convert an operation cance...
caovcanrd 7603 Commute the arguments of a...
caovcan 7604 Convert an operation cance...
caovordig 7605 Convert an operation order...
caovordid 7606 Convert an operation order...
caovordg 7607 Convert an operation order...
caovordd 7608 Convert an operation order...
caovord2d 7609 Operation ordering law wit...
caovord3d 7610 Ordering law. (Contribute...
caovord 7611 Convert an operation order...
caovord2 7612 Operation ordering law wit...
caovord3 7613 Ordering law. (Contribute...
caovdig 7614 Convert an operation distr...
caovdid 7615 Convert an operation distr...
caovdir2d 7616 Convert an operation distr...
caovdirg 7617 Convert an operation rever...
caovdird 7618 Convert an operation distr...
caovdi 7619 Convert an operation distr...
caov32d 7620 Rearrange arguments in a c...
caov12d 7621 Rearrange arguments in a c...
caov31d 7622 Rearrange arguments in a c...
caov13d 7623 Rearrange arguments in a c...
caov4d 7624 Rearrange arguments in a c...
caov411d 7625 Rearrange arguments in a c...
caov42d 7626 Rearrange arguments in a c...
caov32 7627 Rearrange arguments in a c...
caov12 7628 Rearrange arguments in a c...
caov31 7629 Rearrange arguments in a c...
caov13 7630 Rearrange arguments in a c...
caov4 7631 Rearrange arguments in a c...
caov411 7632 Rearrange arguments in a c...
caov42 7633 Rearrange arguments in a c...
caovdir 7634 Reverse distributive law. ...
caovdilem 7635 Lemma used by real number ...
caovlem2 7636 Lemma used in real number ...
caovmo 7637 Uniqueness of inverse elem...
imaeqexov 7638 Substitute an operation va...
imaeqalov 7639 Substitute an operation va...
mpondm0 7640 The value of an operation ...
elmpocl 7641 If a two-parameter class i...
elmpocl1 7642 If a two-parameter class i...
elmpocl2 7643 If a two-parameter class i...
elovmpo 7644 Utility lemma for two-para...
elovmporab 7645 Implications for the value...
elovmporab1w 7646 Implications for the value...
elovmporab1 7647 Implications for the value...
2mpo0 7648 If the operation value of ...
relmptopab 7649 Any function to sets of or...
f1ocnvd 7650 Describe an implicit one-t...
f1od 7651 Describe an implicit one-t...
f1ocnv2d 7652 Describe an implicit one-t...
f1o2d 7653 Describe an implicit one-t...
f1opw2 7654 A one-to-one mapping induc...
f1opw 7655 A one-to-one mapping induc...
elovmpt3imp 7656 If the value of a function...
ovmpt3rab1 7657 The value of an operation ...
ovmpt3rabdm 7658 If the value of a function...
elovmpt3rab1 7659 Implications for the value...
elovmpt3rab 7660 Implications for the value...
ofeqd 7665 Equality theorem for funct...
ofeq 7666 Equality theorem for funct...
ofreq 7667 Equality theorem for funct...
ofexg 7668 A function operation restr...
nfof 7669 Hypothesis builder for fun...
nfofr 7670 Hypothesis builder for fun...
ofrfvalg 7671 Value of a relation applie...
offval 7672 Value of an operation appl...
ofrfval 7673 Value of a relation applie...
ofval 7674 Evaluate a function operat...
ofrval 7675 Exhibit a function relatio...
offn 7676 The function operation pro...
offun 7677 The function operation pro...
offval2f 7678 The function operation exp...
ofmresval 7679 Value of a restriction of ...
fnfvof 7680 Function value of a pointw...
off 7681 The function operation pro...
ofres 7682 Restrict the operands of a...
offval2 7683 The function operation exp...
ofrfval2 7684 The function relation acti...
ofmpteq 7685 Value of a pointwise opera...
ofco 7686 The composition of a funct...
offveq 7687 Convert an identity of the...
offveqb 7688 Equivalent expressions for...
ofc1 7689 Left operation by a consta...
ofc2 7690 Right operation by a const...
ofc12 7691 Function operation on two ...
caofref 7692 Transfer a reflexive law t...
caofinvl 7693 Transfer a left inverse la...
caofid0l 7694 Transfer a left identity l...
caofid0r 7695 Transfer a right identity ...
caofid1 7696 Transfer a right absorptio...
caofid2 7697 Transfer a right absorptio...
caofcom 7698 Transfer a commutative law...
caofrss 7699 Transfer a relation subset...
caofass 7700 Transfer an associative la...
caoftrn 7701 Transfer a transitivity la...
caofdi 7702 Transfer a distributive la...
caofdir 7703 Transfer a reverse distrib...
caonncan 7704 Transfer ~ nncan -shaped l...
relrpss 7707 The proper subset relation...
brrpssg 7708 The proper subset relation...
brrpss 7709 The proper subset relation...
porpss 7710 Every class is partially o...
sorpss 7711 Express strict ordering un...
sorpssi 7712 Property of a chain of set...
sorpssun 7713 A chain of sets is closed ...
sorpssin 7714 A chain of sets is closed ...
sorpssuni 7715 In a chain of sets, a maxi...
sorpssint 7716 In a chain of sets, a mini...
sorpsscmpl 7717 The componentwise compleme...
zfun 7719 Axiom of Union expressed w...
axun2 7720 A variant of the Axiom of ...
uniex2 7721 The Axiom of Union using t...
vuniex 7722 The union of a setvar is a...
uniexg 7723 The ZF Axiom of Union in c...
uniex 7724 The Axiom of Union in clas...
uniexd 7725 Deduction version of the Z...
unex 7726 The union of two sets is a...
tpex 7727 An unordered triple of cla...
unexb 7728 Existence of union is equi...
unexg 7729 A union of two sets is a s...
xpexg 7730 The Cartesian product of t...
xpexd 7731 The Cartesian product of t...
3xpexg 7732 The Cartesian product of t...
xpex 7733 The Cartesian product of t...
unexd 7734 The union of two sets is a...
sqxpexg 7735 The Cartesian square of a ...
abnexg 7736 Sufficient condition for a...
abnex 7737 Sufficient condition for a...
snnex 7738 The class of all singleton...
pwnex 7739 The class of all power set...
difex2 7740 If the subtrahend of a cla...
difsnexi 7741 If the difference of a cla...
uniuni 7742 Expression for double unio...
uniexr 7743 Converse of the Axiom of U...
uniexb 7744 The Axiom of Union and its...
pwexr 7745 Converse of the Axiom of P...
pwexb 7746 The Axiom of Power Sets an...
elpwpwel 7747 A class belongs to a doubl...
eldifpw 7748 Membership in a power clas...
elpwun 7749 Membership in the power cl...
pwuncl 7750 Power classes are closed u...
iunpw 7751 An indexed union of a powe...
fr3nr 7752 A well-founded relation ha...
epne3 7753 A well-founded class conta...
dfwe2 7754 Alternate definition of we...
epweon 7755 The membership relation we...
epweonALT 7756 Alternate proof of ~ epweo...
ordon 7757 The class of all ordinal n...
onprc 7758 No set contains all ordina...
ssorduni 7759 The union of a class of or...
ssonuni 7760 The union of a set of ordi...
ssonunii 7761 The union of a set of ordi...
ordeleqon 7762 A way to express the ordin...
ordsson 7763 Any ordinal class is a sub...
dford5 7764 A class is ordinal iff it ...
onss 7765 An ordinal number is a sub...
predon 7766 The predecessor of an ordi...
predonOLD 7767 Obsolete version of ~ pred...
ssonprc 7768 Two ways of saying a class...
onuni 7769 The union of an ordinal nu...
orduni 7770 The union of an ordinal cl...
onint 7771 The intersection (infimum)...
onint0 7772 The intersection of a clas...
onssmin 7773 A nonempty class of ordina...
onminesb 7774 If a property is true for ...
onminsb 7775 If a property is true for ...
oninton 7776 The intersection of a none...
onintrab 7777 The intersection of a clas...
onintrab2 7778 An existence condition equ...
onnmin 7779 No member of a set of ordi...
onnminsb 7780 An ordinal number smaller ...
oneqmin 7781 A way to show that an ordi...
uniordint 7782 The union of a set of ordi...
onminex 7783 If a wff is true for an or...
sucon 7784 The class of all ordinal n...
sucexb 7785 A successor exists iff its...
sucexg 7786 The successor of a set is ...
sucex 7787 The successor of a set is ...
onmindif2 7788 The minimum of a class of ...
ordsuci 7789 The successor of an ordina...
sucexeloni 7790 If the successor of an ord...
sucexeloniOLD 7791 Obsolete version of ~ suce...
onsuc 7792 The successor of an ordina...
suceloniOLD 7793 Obsolete version of ~ onsu...
ordsuc 7794 A class is ordinal if and ...
ordsucOLD 7795 Obsolete version of ~ ords...
ordpwsuc 7796 The collection of ordinals...
onpwsuc 7797 The collection of ordinal ...
onsucb 7798 A class is an ordinal numb...
ordsucss 7799 The successor of an elemen...
onpsssuc 7800 An ordinal number is a pro...
ordelsuc 7801 A set belongs to an ordina...
onsucmin 7802 The successor of an ordina...
ordsucelsuc 7803 Membership is inherited by...
ordsucsssuc 7804 The subclass relationship ...
ordsucuniel 7805 Given an element ` A ` of ...
ordsucun 7806 The successor of the maxim...
ordunpr 7807 The maximum of two ordinal...
ordunel 7808 The maximum of two ordinal...
onsucuni 7809 A class of ordinal numbers...
ordsucuni 7810 An ordinal class is a subc...
orduniorsuc 7811 An ordinal class is either...
unon 7812 The class of all ordinal n...
ordunisuc 7813 An ordinal class is equal ...
orduniss2 7814 The union of the ordinal s...
onsucuni2 7815 A successor ordinal is the...
0elsuc 7816 The successor of an ordina...
limon 7817 The class of ordinal numbe...
onuniorsuc 7818 An ordinal number is eithe...
onssi 7819 An ordinal number is a sub...
onsuci 7820 The successor of an ordina...
onuniorsuciOLD 7821 Obsolete version of ~ onun...
onuninsuci 7822 An ordinal is equal to its...
onsucssi 7823 A set belongs to an ordina...
nlimsucg 7824 A successor is not a limit...
orduninsuc 7825 An ordinal class is equal ...
ordunisuc2 7826 An ordinal equal to its un...
ordzsl 7827 An ordinal is zero, a succ...
onzsl 7828 An ordinal number is zero,...
dflim3 7829 An alternate definition of...
dflim4 7830 An alternate definition of...
limsuc 7831 The successor of a member ...
limsssuc 7832 A class includes a limit o...
nlimon 7833 Two ways to express the cl...
limuni3 7834 The union of a nonempty cl...
tfi 7835 The Principle of Transfini...
tfisg 7836 A closed form of ~ tfis . ...
tfis 7837 Transfinite Induction Sche...
tfis2f 7838 Transfinite Induction Sche...
tfis2 7839 Transfinite Induction Sche...
tfis3 7840 Transfinite Induction Sche...
tfisi 7841 A transfinite induction sc...
tfinds 7842 Principle of Transfinite I...
tfindsg 7843 Transfinite Induction (inf...
tfindsg2 7844 Transfinite Induction (inf...
tfindes 7845 Transfinite Induction with...
tfinds2 7846 Transfinite Induction (inf...
tfinds3 7847 Principle of Transfinite I...
dfom2 7850 An alternate definition of...
elom 7851 Membership in omega. The ...
omsson 7852 Omega is a subset of ` On ...
limomss 7853 The class of natural numbe...
nnon 7854 A natural number is an ord...
nnoni 7855 A natural number is an ord...
nnord 7856 A natural number is ordina...
trom 7857 The class of finite ordina...
ordom 7858 The class of finite ordina...
elnn 7859 A member of a natural numb...
omon 7860 The class of natural numbe...
omelon2 7861 Omega is an ordinal number...
nnlim 7862 A natural number is not a ...
omssnlim 7863 The class of natural numbe...
limom 7864 Omega is a limit ordinal. ...
peano2b 7865 A class belongs to omega i...
nnsuc 7866 A nonzero natural number i...
omsucne 7867 A natural number is not th...
ssnlim 7868 An ordinal subclass of non...
omsinds 7869 Strong (or "total") induct...
omsindsOLD 7870 Obsolete version of ~ omsi...
omun 7871 The union of two finite or...
peano1 7872 Zero is a natural number. ...
peano1OLD 7873 Obsolete version of ~ pean...
peano2 7874 The successor of any natur...
peano3 7875 The successor of any natur...
peano4 7876 Two natural numbers are eq...
peano5 7877 The induction postulate: a...
peano5OLD 7878 Obsolete version of ~ pean...
nn0suc 7879 A natural number is either...
find 7880 The Principle of Finite In...
findOLD 7881 Obsolete version of ~ find...
finds 7882 Principle of Finite Induct...
findsg 7883 Principle of Finite Induct...
finds2 7884 Principle of Finite Induct...
finds1 7885 Principle of Finite Induct...
findes 7886 Finite induction with expl...
dmexg 7887 The domain of a set is a s...
rnexg 7888 The range of a set is a se...
dmexd 7889 The domain of a set is a s...
fndmexd 7890 If a function is a set, it...
dmfex 7891 If a mapping is a set, its...
fndmexb 7892 The domain of a function i...
fdmexb 7893 The domain of a function i...
dmfexALT 7894 Alternate proof of ~ dmfex...
dmex 7895 The domain of a set is a s...
rnex 7896 The range of a set is a se...
iprc 7897 The identity function is a...
resiexg 7898 The existence of a restric...
imaexg 7899 The image of a set is a se...
imaex 7900 The image of a set is a se...
exse2 7901 Any set relation is set-li...
xpexr 7902 If a Cartesian product is ...
xpexr2 7903 If a nonempty Cartesian pr...
xpexcnv 7904 A condition where the conv...
soex 7905 If the relation in a stric...
elxp4 7906 Membership in a Cartesian ...
elxp5 7907 Membership in a Cartesian ...
cnvexg 7908 The converse of a set is a...
cnvex 7909 The converse of a set is a...
relcnvexb 7910 A relation is a set iff it...
f1oexrnex 7911 If the range of a 1-1 onto...
f1oexbi 7912 There is a one-to-one onto...
coexg 7913 The composition of two set...
coex 7914 The composition of two set...
funcnvuni 7915 The union of a chain (with...
fun11uni 7916 The union of a chain (with...
fex2 7917 A function with bounded do...
fabexg 7918 Existence of a set of func...
fabex 7919 Existence of a set of func...
f1oabexg 7920 The class of all 1-1-onto ...
fiunlem 7921 Lemma for ~ fiun and ~ f1i...
fiun 7922 The union of a chain (with...
f1iun 7923 The union of a chain (with...
fviunfun 7924 The function value of an i...
ffoss 7925 Relationship between a map...
f11o 7926 Relationship between one-t...
resfunexgALT 7927 Alternate proof of ~ resfu...
cofunexg 7928 Existence of a composition...
cofunex2g 7929 Existence of a composition...
fnexALT 7930 Alternate proof of ~ fnex ...
funexw 7931 Weak version of ~ funex th...
mptexw 7932 Weak version of ~ mptex th...
funrnex 7933 If the domain of a functio...
zfrep6 7934 A version of the Axiom of ...
focdmex 7935 If the domain of an onto f...
f1dmex 7936 If the codomain of a one-t...
f1ovv 7937 The codomain/range of a 1-...
fvclex 7938 Existence of the class of ...
fvresex 7939 Existence of the class of ...
abrexexg 7940 Existence of a class abstr...
abrexexgOLD 7941 Obsolete version of ~ abre...
abrexex 7942 Existence of a class abstr...
iunexg 7943 The existence of an indexe...
abrexex2g 7944 Existence of an existentia...
opabex3d 7945 Existence of an ordered pa...
opabex3rd 7946 Existence of an ordered pa...
opabex3 7947 Existence of an ordered pa...
iunex 7948 The existence of an indexe...
abrexex2 7949 Existence of an existentia...
abexssex 7950 Existence of a class abstr...
abexex 7951 A condition where a class ...
f1oweALT 7952 Alternate proof of ~ f1owe...
wemoiso 7953 Thus, there is at most one...
wemoiso2 7954 Thus, there is at most one...
oprabexd 7955 Existence of an operator a...
oprabex 7956 Existence of an operation ...
oprabex3 7957 Existence of an operation ...
oprabrexex2 7958 Existence of an existentia...
ab2rexex 7959 Existence of a class abstr...
ab2rexex2 7960 Existence of an existentia...
xpexgALT 7961 Alternate proof of ~ xpexg...
offval3 7962 General value of ` ( F oF ...
offres 7963 Pointwise combination comm...
ofmres 7964 Equivalent expressions for...
ofmresex 7965 Existence of a restriction...
1stval 7970 The value of the function ...
2ndval 7971 The value of the function ...
1stnpr 7972 Value of the first-member ...
2ndnpr 7973 Value of the second-member...
1st0 7974 The value of the first-mem...
2nd0 7975 The value of the second-me...
op1st 7976 Extract the first member o...
op2nd 7977 Extract the second member ...
op1std 7978 Extract the first member o...
op2ndd 7979 Extract the second member ...
op1stg 7980 Extract the first member o...
op2ndg 7981 Extract the second member ...
ot1stg 7982 Extract the first member o...
ot2ndg 7983 Extract the second member ...
ot3rdg 7984 Extract the third member o...
1stval2 7985 Alternate value of the fun...
2ndval2 7986 Alternate value of the fun...
oteqimp 7987 The components of an order...
fo1st 7988 The ` 1st ` function maps ...
fo2nd 7989 The ` 2nd ` function maps ...
br1steqg 7990 Uniqueness condition for t...
br2ndeqg 7991 Uniqueness condition for t...
f1stres 7992 Mapping of a restriction o...
f2ndres 7993 Mapping of a restriction o...
fo1stres 7994 Onto mapping of a restrict...
fo2ndres 7995 Onto mapping of a restrict...
1st2val 7996 Value of an alternate defi...
2nd2val 7997 Value of an alternate defi...
1stcof 7998 Composition of the first m...
2ndcof 7999 Composition of the second ...
xp1st 8000 Location of the first elem...
xp2nd 8001 Location of the second ele...
elxp6 8002 Membership in a Cartesian ...
elxp7 8003 Membership in a Cartesian ...
eqopi 8004 Equality with an ordered p...
xp2 8005 Representation of Cartesia...
unielxp 8006 The membership relation fo...
1st2nd2 8007 Reconstruction of a member...
1st2ndb 8008 Reconstruction of an order...
xpopth 8009 An ordered pair theorem fo...
eqop 8010 Two ways to express equali...
eqop2 8011 Two ways to express equali...
op1steq 8012 Two ways of expressing tha...
opreuopreu 8013 There is a unique ordered ...
el2xptp 8014 A member of a nested Carte...
el2xptp0 8015 A member of a nested Carte...
el2xpss 8016 Version of ~ elrel for tri...
2nd1st 8017 Swap the members of an ord...
1st2nd 8018 Reconstruction of a member...
1stdm 8019 The first ordered pair com...
2ndrn 8020 The second ordered pair co...
1st2ndbr 8021 Express an element of a re...
releldm2 8022 Two ways of expressing mem...
reldm 8023 An expression for the doma...
releldmdifi 8024 One way of expressing memb...
funfv1st2nd 8025 The function value for the...
funelss 8026 If the first component of ...
funeldmdif 8027 Two ways of expressing mem...
sbcopeq1a 8028 Equality theorem for subst...
csbopeq1a 8029 Equality theorem for subst...
sbcoteq1a 8030 Equality theorem for subst...
dfopab2 8031 A way to define an ordered...
dfoprab3s 8032 A way to define an operati...
dfoprab3 8033 Operation class abstractio...
dfoprab4 8034 Operation class abstractio...
dfoprab4f 8035 Operation class abstractio...
opabex2 8036 Condition for an operation...
opabn1stprc 8037 An ordered-pair class abst...
opiota 8038 The property of a uniquely...
cnvoprab 8039 The converse of a class ab...
dfxp3 8040 Define the Cartesian produ...
elopabi 8041 A consequence of membershi...
eloprabi 8042 A consequence of membershi...
mpomptsx 8043 Express a two-argument fun...
mpompts 8044 Express a two-argument fun...
dmmpossx 8045 The domain of a mapping is...
fmpox 8046 Functionality, domain and ...
fmpo 8047 Functionality, domain and ...
fnmpo 8048 Functionality and domain o...
fnmpoi 8049 Functionality and domain o...
dmmpo 8050 Domain of a class given by...
ovmpoelrn 8051 An operation's value belon...
dmmpoga 8052 Domain of an operation giv...
dmmpogaOLD 8053 Obsolete version of ~ dmmp...
dmmpog 8054 Domain of an operation giv...
mpoexxg 8055 Existence of an operation ...
mpoexg 8056 Existence of an operation ...
mpoexga 8057 If the domain of an operat...
mpoexw 8058 Weak version of ~ mpoex th...
mpoex 8059 If the domain of an operat...
mptmpoopabbrd 8060 The operation value of a f...
mptmpoopabbrdOLD 8061 Obsolete version of ~ mptm...
mptmpoopabovd 8062 The operation value of a f...
mptmpoopabbrdOLDOLD 8063 Obsolete version of ~ mptm...
mptmpoopabovdOLD 8064 Obsolete version of ~ mptm...
el2mpocsbcl 8065 If the operation value of ...
el2mpocl 8066 If the operation value of ...
fnmpoovd 8067 A function with a Cartesia...
offval22 8068 The function operation exp...
brovpreldm 8069 If a binary relation holds...
bropopvvv 8070 If a binary relation holds...
bropfvvvvlem 8071 Lemma for ~ bropfvvvv . (...
bropfvvvv 8072 If a binary relation holds...
ovmptss 8073 If all the values of the m...
relmpoopab 8074 Any function to sets of or...
fmpoco 8075 Composition of two functio...
oprabco 8076 Composition of a function ...
oprab2co 8077 Composition of operator ab...
df1st2 8078 An alternate possible defi...
df2nd2 8079 An alternate possible defi...
1stconst 8080 The mapping of a restricti...
2ndconst 8081 The mapping of a restricti...
dfmpo 8082 Alternate definition for t...
mposn 8083 An operation (in maps-to n...
curry1 8084 Composition with ` ``' ( 2...
curry1val 8085 The value of a curried fun...
curry1f 8086 Functionality of a curried...
curry2 8087 Composition with ` ``' ( 1...
curry2f 8088 Functionality of a curried...
curry2val 8089 The value of a curried fun...
cnvf1olem 8090 Lemma for ~ cnvf1o . (Con...
cnvf1o 8091 Describe a function that m...
fparlem1 8092 Lemma for ~ fpar . (Contr...
fparlem2 8093 Lemma for ~ fpar . (Contr...
fparlem3 8094 Lemma for ~ fpar . (Contr...
fparlem4 8095 Lemma for ~ fpar . (Contr...
fpar 8096 Merge two functions in par...
fsplit 8097 A function that can be use...
fsplitfpar 8098 Merge two functions with a...
offsplitfpar 8099 Express the function opera...
f2ndf 8100 The ` 2nd ` (second compon...
fo2ndf 8101 The ` 2nd ` (second compon...
f1o2ndf1 8102 The ` 2nd ` (second compon...
opco1 8103 Value of an operation prec...
opco2 8104 Value of an operation prec...
opco1i 8105 Inference form of ~ opco1 ...
frxp 8106 A lexicographical ordering...
xporderlem 8107 Lemma for lexicographical ...
poxp 8108 A lexicographical ordering...
soxp 8109 A lexicographical ordering...
wexp 8110 A lexicographical ordering...
fnwelem 8111 Lemma for ~ fnwe . (Contr...
fnwe 8112 A variant on lexicographic...
fnse 8113 Condition for the well-ord...
fvproj 8114 Value of a function on ord...
fimaproj 8115 Image of a cartesian produ...
ralxpes 8116 A version of ~ ralxp with ...
ralxp3f 8117 Restricted for all over a ...
ralxp3 8118 Restricted for all over a ...
ralxp3es 8119 Restricted for-all over a ...
frpoins3xpg 8120 Special case of founded pa...
frpoins3xp3g 8121 Special case of founded pa...
xpord2lem 8122 Lemma for Cartesian produc...
poxp2 8123 Another way of partially o...
frxp2 8124 Another way of giving a we...
xpord2pred 8125 Calculate the predecessor ...
sexp2 8126 Condition for the relation...
xpord2indlem 8127 Induction over the Cartesi...
xpord2ind 8128 Induction over the Cartesi...
xpord3lem 8129 Lemma for triple ordering....
poxp3 8130 Triple Cartesian product p...
frxp3 8131 Give well-foundedness over...
xpord3pred 8132 Calculate the predecsessor...
sexp3 8133 Show that the triple order...
xpord3inddlem 8134 Induction over the triple ...
xpord3indd 8135 Induction over the triple ...
xpord3ind 8136 Induction over the triple ...
orderseqlem 8137 Lemma for ~ poseq and ~ so...
poseq 8138 A partial ordering of ordi...
soseq 8139 A linear ordering of ordin...
suppval 8142 The value of the operation...
supp0prc 8143 The support of a class is ...
suppvalbr 8144 The value of the operation...
supp0 8145 The support of the empty s...
suppval1 8146 The value of the operation...
suppvalfng 8147 The value of the operation...
suppvalfn 8148 The value of the operation...
elsuppfng 8149 An element of the support ...
elsuppfn 8150 An element of the support ...
cnvimadfsn 8151 The support of functions "...
suppimacnvss 8152 The support of functions "...
suppimacnv 8153 Support sets of functions ...
fsuppeq 8154 Two ways of writing the su...
fsuppeqg 8155 Version of ~ fsuppeq avoid...
suppssdm 8156 The support of a function ...
suppsnop 8157 The support of a singleton...
snopsuppss 8158 The support of a singleton...
fvn0elsupp 8159 If the function value for ...
fvn0elsuppb 8160 The function value for a g...
rexsupp 8161 Existential quantification...
ressuppss 8162 The support of the restric...
suppun 8163 The support of a class/fun...
ressuppssdif 8164 The support of the restric...
mptsuppdifd 8165 The support of a function ...
mptsuppd 8166 The support of a function ...
extmptsuppeq 8167 The support of an extended...
suppfnss 8168 The support of a function ...
funsssuppss 8169 The support of a function ...
fnsuppres 8170 Two ways to express restri...
fnsuppeq0 8171 The support of a function ...
fczsupp0 8172 The support of a constant ...
suppss 8173 Show that the support of a...
suppssOLD 8174 Obsolete version of ~ supp...
suppssr 8175 A function is zero outside...
suppssrg 8176 A function is zero outside...
suppssov1 8177 Formula building theorem f...
suppssov2 8178 Formula building theorem f...
suppssof1 8179 Formula building theorem f...
suppss2 8180 Show that the support of a...
suppsssn 8181 Show that the support of a...
suppssfv 8182 Formula building theorem f...
suppofssd 8183 Condition for the support ...
suppofss1d 8184 Condition for the support ...
suppofss2d 8185 Condition for the support ...
suppco 8186 The support of the composi...
suppcoss 8187 The support of the composi...
supp0cosupp0 8188 The support of the composi...
imacosupp 8189 The image of the support o...
opeliunxp2f 8190 Membership in a union of C...
mpoxeldm 8191 If there is an element of ...
mpoxneldm 8192 If the first argument of a...
mpoxopn0yelv 8193 If there is an element of ...
mpoxopynvov0g 8194 If the second argument of ...
mpoxopxnop0 8195 If the first argument of a...
mpoxopx0ov0 8196 If the first argument of a...
mpoxopxprcov0 8197 If the components of the f...
mpoxopynvov0 8198 If the second argument of ...
mpoxopoveq 8199 Value of an operation give...
mpoxopovel 8200 Element of the value of an...
mpoxopoveqd 8201 Value of an operation give...
brovex 8202 A binary relation of the v...
brovmpoex 8203 A binary relation of the v...
sprmpod 8204 The extension of a binary ...
tposss 8207 Subset theorem for transpo...
tposeq 8208 Equality theorem for trans...
tposeqd 8209 Equality theorem for trans...
tposssxp 8210 The transposition is a sub...
reltpos 8211 The transposition is a rel...
brtpos2 8212 Value of the transposition...
brtpos0 8213 The behavior of ` tpos ` w...
reldmtpos 8214 Necessary and sufficient c...
brtpos 8215 The transposition swaps ar...
ottpos 8216 The transposition swaps th...
relbrtpos 8217 The transposition swaps ar...
dmtpos 8218 The domain of ` tpos F ` w...
rntpos 8219 The range of ` tpos F ` wh...
tposexg 8220 The transposition of a set...
ovtpos 8221 The transposition swaps th...
tposfun 8222 The transposition of a fun...
dftpos2 8223 Alternate definition of ` ...
dftpos3 8224 Alternate definition of ` ...
dftpos4 8225 Alternate definition of ` ...
tpostpos 8226 Value of the double transp...
tpostpos2 8227 Value of the double transp...
tposfn2 8228 The domain of a transposit...
tposfo2 8229 Condition for a surjective...
tposf2 8230 The domain and codomain of...
tposf12 8231 Condition for an injective...
tposf1o2 8232 Condition of a bijective t...
tposfo 8233 The domain and codomain/ra...
tposf 8234 The domain and codomain of...
tposfn 8235 Functionality of a transpo...
tpos0 8236 Transposition of the empty...
tposco 8237 Transposition of a composi...
tpossym 8238 Two ways to say a function...
tposeqi 8239 Equality theorem for trans...
tposex 8240 A transposition is a set. ...
nftpos 8241 Hypothesis builder for tra...
tposoprab 8242 Transposition of a class o...
tposmpo 8243 Transposition of a two-arg...
tposconst 8244 The transposition of a con...
mpocurryd 8249 The currying of an operati...
mpocurryvald 8250 The value of a curried ope...
fvmpocurryd 8251 The value of the value of ...
pwuninel2 8254 Direct proof of ~ pwuninel...
pwuninel 8255 The power set of the union...
undefval 8256 Value of the undefined val...
undefnel2 8257 The undefined value genera...
undefnel 8258 The undefined value genera...
undefne0 8259 The undefined value genera...
frecseq123 8262 Equality theorem for the w...
nffrecs 8263 Bound-variable hypothesis ...
csbfrecsg 8264 Move class substitution in...
fpr3g 8265 Functions defined by well-...
frrlem1 8266 Lemma for well-founded rec...
frrlem2 8267 Lemma for well-founded rec...
frrlem3 8268 Lemma for well-founded rec...
frrlem4 8269 Lemma for well-founded rec...
frrlem5 8270 Lemma for well-founded rec...
frrlem6 8271 Lemma for well-founded rec...
frrlem7 8272 Lemma for well-founded rec...
frrlem8 8273 Lemma for well-founded rec...
frrlem9 8274 Lemma for well-founded rec...
frrlem10 8275 Lemma for well-founded rec...
frrlem11 8276 Lemma for well-founded rec...
frrlem12 8277 Lemma for well-founded rec...
frrlem13 8278 Lemma for well-founded rec...
frrlem14 8279 Lemma for well-founded rec...
fprlem1 8280 Lemma for well-founded rec...
fprlem2 8281 Lemma for well-founded rec...
fpr2a 8282 Weak version of ~ fpr2 whi...
fpr1 8283 Law of well-founded recurs...
fpr2 8284 Law of well-founded recurs...
fpr3 8285 Law of well-founded recurs...
frrrel 8286 Show without using the axi...
frrdmss 8287 Show without using the axi...
frrdmcl 8288 Show without using the axi...
fprfung 8289 A "function" defined by we...
fprresex 8290 The restriction of a funct...
dfwrecsOLD 8293 Obsolete definition of the...
wrecseq123 8294 General equality theorem f...
wrecseq123OLD 8295 Obsolete version of ~ wrec...
nfwrecs 8296 Bound-variable hypothesis ...
nfwrecsOLD 8297 Obsolete proof of ~ nfwrec...
wrecseq1 8298 Equality theorem for the w...
wrecseq2 8299 Equality theorem for the w...
wrecseq3 8300 Equality theorem for the w...
csbwrecsg 8301 Move class substitution in...
wfr3g 8302 Functions defined by well-...
wfrlem1OLD 8303 Lemma for well-ordered rec...
wfrlem2OLD 8304 Lemma for well-ordered rec...
wfrlem3OLD 8305 Lemma for well-ordered rec...
wfrlem3OLDa 8306 Lemma for well-ordered rec...
wfrlem4OLD 8307 Lemma for well-ordered rec...
wfrlem5OLD 8308 Lemma for well-ordered rec...
wfrrelOLD 8309 Obsolete proof of ~ wfrrel...
wfrdmssOLD 8310 Obsolete proof of ~ wfrdms...
wfrlem8OLD 8311 Lemma for well-ordered rec...
wfrdmclOLD 8312 Obsolete version of ~ wfrd...
wfrlem10OLD 8313 Lemma for well-ordered rec...
wfrfunOLD 8314 Obsolete proof of ~ wfrfun...
wfrlem12OLD 8315 Lemma for well-ordered rec...
wfrlem13OLD 8316 Lemma for well-ordered rec...
wfrlem14OLD 8317 Lemma for well-ordered rec...
wfrlem15OLD 8318 Lemma for well-ordered rec...
wfrlem16OLD 8319 Lemma for well-ordered rec...
wfrlem17OLD 8320 Without using ~ ax-rep , s...
wfr2aOLD 8321 Obsolete version of ~ wfr2...
wfr1OLD 8322 Obsolete version of ~ wfr1...
wfr2OLD 8323 Obsolete version of ~ wfr2...
wfrrel 8324 The well-ordered recursion...
wfrdmss 8325 The domain of the well-ord...
wfrdmcl 8326 The predecessor class of a...
wfrfun 8327 The "function" generated b...
wfrresex 8328 Show without using the axi...
wfr2a 8329 A weak version of ~ wfr2 w...
wfr1 8330 The Principle of Well-Orde...
wfr2 8331 The Principle of Well-Orde...
wfr3 8332 The principle of Well-Orde...
wfr3OLD 8333 Obsolete form of ~ wfr3 as...
iunon 8334 The indexed union of a set...
iinon 8335 The nonempty indexed inter...
onfununi 8336 A property of functions on...
onovuni 8337 A variant of ~ onfununi fo...
onoviun 8338 A variant of ~ onovuni wit...
onnseq 8339 There are no length ` _om ...
dfsmo2 8342 Alternate definition of a ...
issmo 8343 Conditions for which ` A `...
issmo2 8344 Alternate definition of a ...
smoeq 8345 Equality theorem for stric...
smodm 8346 The domain of a strictly m...
smores 8347 A strictly monotone functi...
smores3 8348 A strictly monotone functi...
smores2 8349 A strictly monotone ordina...
smodm2 8350 The domain of a strictly m...
smofvon2 8351 The function values of a s...
iordsmo 8352 The identity relation rest...
smo0 8353 The null set is a strictly...
smofvon 8354 If ` B ` is a strictly mon...
smoel 8355 If ` x ` is less than ` y ...
smoiun 8356 The value of a strictly mo...
smoiso 8357 If ` F ` is an isomorphism...
smoel2 8358 A strictly monotone ordina...
smo11 8359 A strictly monotone ordina...
smoord 8360 A strictly monotone ordina...
smoword 8361 A strictly monotone ordina...
smogt 8362 A strictly monotone ordina...
smocdmdom 8363 The codomain of a strictly...
smoiso2 8364 The strictly monotone ordi...
dfrecs3 8367 The old definition of tran...
dfrecs3OLD 8368 Obsolete version of ~ dfre...
recseq 8369 Equality theorem for ` rec...
nfrecs 8370 Bound-variable hypothesis ...
tfrlem1 8371 A technical lemma for tran...
tfrlem3a 8372 Lemma for transfinite recu...
tfrlem3 8373 Lemma for transfinite recu...
tfrlem4 8374 Lemma for transfinite recu...
tfrlem5 8375 Lemma for transfinite recu...
recsfval 8376 Lemma for transfinite recu...
tfrlem6 8377 Lemma for transfinite recu...
tfrlem7 8378 Lemma for transfinite recu...
tfrlem8 8379 Lemma for transfinite recu...
tfrlem9 8380 Lemma for transfinite recu...
tfrlem9a 8381 Lemma for transfinite recu...
tfrlem10 8382 Lemma for transfinite recu...
tfrlem11 8383 Lemma for transfinite recu...
tfrlem12 8384 Lemma for transfinite recu...
tfrlem13 8385 Lemma for transfinite recu...
tfrlem14 8386 Lemma for transfinite recu...
tfrlem15 8387 Lemma for transfinite recu...
tfrlem16 8388 Lemma for finite recursion...
tfr1a 8389 A weak version of ~ tfr1 w...
tfr2a 8390 A weak version of ~ tfr2 w...
tfr2b 8391 Without assuming ~ ax-rep ...
tfr1 8392 Principle of Transfinite R...
tfr2 8393 Principle of Transfinite R...
tfr3 8394 Principle of Transfinite R...
tfr1ALT 8395 Alternate proof of ~ tfr1 ...
tfr2ALT 8396 Alternate proof of ~ tfr2 ...
tfr3ALT 8397 Alternate proof of ~ tfr3 ...
recsfnon 8398 Strong transfinite recursi...
recsval 8399 Strong transfinite recursi...
tz7.44lem1 8400 The ordered pair abstracti...
tz7.44-1 8401 The value of ` F ` at ` (/...
tz7.44-2 8402 The value of ` F ` at a su...
tz7.44-3 8403 The value of ` F ` at a li...
rdgeq1 8406 Equality theorem for the r...
rdgeq2 8407 Equality theorem for the r...
rdgeq12 8408 Equality theorem for the r...
nfrdg 8409 Bound-variable hypothesis ...
rdglem1 8410 Lemma used with the recurs...
rdgfun 8411 The recursive definition g...
rdgdmlim 8412 The domain of the recursiv...
rdgfnon 8413 The recursive definition g...
rdgvalg 8414 Value of the recursive def...
rdgval 8415 Value of the recursive def...
rdg0 8416 The initial value of the r...
rdgseg 8417 The initial segments of th...
rdgsucg 8418 The value of the recursive...
rdgsuc 8419 The value of the recursive...
rdglimg 8420 The value of the recursive...
rdglim 8421 The value of the recursive...
rdg0g 8422 The initial value of the r...
rdgsucmptf 8423 The value of the recursive...
rdgsucmptnf 8424 The value of the recursive...
rdgsucmpt2 8425 This version of ~ rdgsucmp...
rdgsucmpt 8426 The value of the recursive...
rdglim2 8427 The value of the recursive...
rdglim2a 8428 The value of the recursive...
rdg0n 8429 If ` A ` is a proper class...
frfnom 8430 The function generated by ...
fr0g 8431 The initial value resultin...
frsuc 8432 The successor value result...
frsucmpt 8433 The successor value result...
frsucmptn 8434 The value of the finite re...
frsucmpt2 8435 The successor value result...
tz7.48lem 8436 A way of showing an ordina...
tz7.48-2 8437 Proposition 7.48(2) of [Ta...
tz7.48-1 8438 Proposition 7.48(1) of [Ta...
tz7.48-3 8439 Proposition 7.48(3) of [Ta...
tz7.49 8440 Proposition 7.49 of [Takeu...
tz7.49c 8441 Corollary of Proposition 7...
seqomlem0 8444 Lemma for ` seqom ` . Cha...
seqomlem1 8445 Lemma for ` seqom ` . The...
seqomlem2 8446 Lemma for ` seqom ` . (Co...
seqomlem3 8447 Lemma for ` seqom ` . (Co...
seqomlem4 8448 Lemma for ` seqom ` . (Co...
seqomeq12 8449 Equality theorem for ` seq...
fnseqom 8450 An index-aware recursive d...
seqom0g 8451 Value of an index-aware re...
seqomsuc 8452 Value of an index-aware re...
omsucelsucb 8453 Membership is inherited by...
df1o2 8468 Expanded value of the ordi...
df2o3 8469 Expanded value of the ordi...
df2o2 8470 Expanded value of the ordi...
1oex 8471 Ordinal 1 is a set. (Cont...
2oex 8472 ` 2o ` is a set. (Contrib...
1on 8473 Ordinal 1 is an ordinal nu...
1onOLD 8474 Obsolete version of ~ 1on ...
2on 8475 Ordinal 2 is an ordinal nu...
2onOLD 8476 Obsolete version of ~ 2on ...
2on0 8477 Ordinal two is not zero. ...
ord3 8478 Ordinal 3 is an ordinal cl...
3on 8479 Ordinal 3 is an ordinal nu...
4on 8480 Ordinal 4 is an ordinal nu...
1oexOLD 8481 Obsolete version of ~ 1oex...
2oexOLD 8482 Obsolete version of ~ 2oex...
1n0 8483 Ordinal one is not equal t...
nlim1 8484 1 is not a limit ordinal. ...
nlim2 8485 2 is not a limit ordinal. ...
xp01disj 8486 Cartesian products with th...
xp01disjl 8487 Cartesian products with th...
ordgt0ge1 8488 Two ways to express that a...
ordge1n0 8489 An ordinal greater than or...
el1o 8490 Membership in ordinal one....
ord1eln01 8491 An ordinal that is not 0 o...
ord2eln012 8492 An ordinal that is not 0, ...
1ellim 8493 A limit ordinal contains 1...
2ellim 8494 A limit ordinal contains 2...
dif1o 8495 Two ways to say that ` A `...
ondif1 8496 Two ways to say that ` A `...
ondif2 8497 Two ways to say that ` A `...
2oconcl 8498 Closure of the pair swappi...
0lt1o 8499 Ordinal zero is less than ...
dif20el 8500 An ordinal greater than on...
0we1 8501 The empty set is a well-or...
brwitnlem 8502 Lemma for relations which ...
fnoa 8503 Functionality and domain o...
fnom 8504 Functionality and domain o...
fnoe 8505 Functionality and domain o...
oav 8506 Value of ordinal addition....
omv 8507 Value of ordinal multiplic...
oe0lem 8508 A helper lemma for ~ oe0 a...
oev 8509 Value of ordinal exponenti...
oevn0 8510 Value of ordinal exponenti...
oa0 8511 Addition with zero. Propo...
om0 8512 Ordinal multiplication wit...
oe0m 8513 Value of zero raised to an...
om0x 8514 Ordinal multiplication wit...
oe0m0 8515 Ordinal exponentiation wit...
oe0m1 8516 Ordinal exponentiation wit...
oe0 8517 Ordinal exponentiation wit...
oev2 8518 Alternate value of ordinal...
oasuc 8519 Addition with successor. ...
oesuclem 8520 Lemma for ~ oesuc . (Cont...
omsuc 8521 Multiplication with succes...
oesuc 8522 Ordinal exponentiation wit...
onasuc 8523 Addition with successor. ...
onmsuc 8524 Multiplication with succes...
onesuc 8525 Exponentiation with a succ...
oa1suc 8526 Addition with 1 is same as...
oalim 8527 Ordinal addition with a li...
omlim 8528 Ordinal multiplication wit...
oelim 8529 Ordinal exponentiation wit...
oacl 8530 Closure law for ordinal ad...
omcl 8531 Closure law for ordinal mu...
oecl 8532 Closure law for ordinal ex...
oa0r 8533 Ordinal addition with zero...
om0r 8534 Ordinal multiplication wit...
o1p1e2 8535 1 + 1 = 2 for ordinal numb...
o2p2e4 8536 2 + 2 = 4 for ordinal numb...
om1 8537 Ordinal multiplication wit...
om1r 8538 Ordinal multiplication wit...
oe1 8539 Ordinal exponentiation wit...
oe1m 8540 Ordinal exponentiation wit...
oaordi 8541 Ordering property of ordin...
oaord 8542 Ordering property of ordin...
oacan 8543 Left cancellation law for ...
oaword 8544 Weak ordering property of ...
oawordri 8545 Weak ordering property of ...
oaord1 8546 An ordinal is less than it...
oaword1 8547 An ordinal is less than or...
oaword2 8548 An ordinal is less than or...
oawordeulem 8549 Lemma for ~ oawordex . (C...
oawordeu 8550 Existence theorem for weak...
oawordexr 8551 Existence theorem for weak...
oawordex 8552 Existence theorem for weak...
oaordex 8553 Existence theorem for orde...
oa00 8554 An ordinal sum is zero iff...
oalimcl 8555 The ordinal sum with a lim...
oaass 8556 Ordinal addition is associ...
oarec 8557 Recursive definition of or...
oaf1o 8558 Left addition by a constan...
oacomf1olem 8559 Lemma for ~ oacomf1o . (C...
oacomf1o 8560 Define a bijection from ` ...
omordi 8561 Ordering property of ordin...
omord2 8562 Ordering property of ordin...
omord 8563 Ordering property of ordin...
omcan 8564 Left cancellation law for ...
omword 8565 Weak ordering property of ...
omwordi 8566 Weak ordering property of ...
omwordri 8567 Weak ordering property of ...
omword1 8568 An ordinal is less than or...
omword2 8569 An ordinal is less than or...
om00 8570 The product of two ordinal...
om00el 8571 The product of two nonzero...
omordlim 8572 Ordering involving the pro...
omlimcl 8573 The product of any nonzero...
odi 8574 Distributive law for ordin...
omass 8575 Multiplication of ordinal ...
oneo 8576 If an ordinal number is ev...
omeulem1 8577 Lemma for ~ omeu : existen...
omeulem2 8578 Lemma for ~ omeu : uniquen...
omopth2 8579 An ordered pair-like theor...
omeu 8580 The division algorithm for...
oen0 8581 Ordinal exponentiation wit...
oeordi 8582 Ordering law for ordinal e...
oeord 8583 Ordering property of ordin...
oecan 8584 Left cancellation law for ...
oeword 8585 Weak ordering property of ...
oewordi 8586 Weak ordering property of ...
oewordri 8587 Weak ordering property of ...
oeworde 8588 Ordinal exponentiation com...
oeordsuc 8589 Ordering property of ordin...
oelim2 8590 Ordinal exponentiation wit...
oeoalem 8591 Lemma for ~ oeoa . (Contr...
oeoa 8592 Sum of exponents law for o...
oeoelem 8593 Lemma for ~ oeoe . (Contr...
oeoe 8594 Product of exponents law f...
oelimcl 8595 The ordinal exponential wi...
oeeulem 8596 Lemma for ~ oeeu . (Contr...
oeeui 8597 The division algorithm for...
oeeu 8598 The division algorithm for...
nna0 8599 Addition with zero. Theor...
nnm0 8600 Multiplication with zero. ...
nnasuc 8601 Addition with successor. ...
nnmsuc 8602 Multiplication with succes...
nnesuc 8603 Exponentiation with a succ...
nna0r 8604 Addition to zero. Remark ...
nnm0r 8605 Multiplication with zero. ...
nnacl 8606 Closure of addition of nat...
nnmcl 8607 Closure of multiplication ...
nnecl 8608 Closure of exponentiation ...
nnacli 8609 ` _om ` is closed under ad...
nnmcli 8610 ` _om ` is closed under mu...
nnarcl 8611 Reverse closure law for ad...
nnacom 8612 Addition of natural number...
nnaordi 8613 Ordering property of addit...
nnaord 8614 Ordering property of addit...
nnaordr 8615 Ordering property of addit...
nnawordi 8616 Adding to both sides of an...
nnaass 8617 Addition of natural number...
nndi 8618 Distributive law for natur...
nnmass 8619 Multiplication of natural ...
nnmsucr 8620 Multiplication with succes...
nnmcom 8621 Multiplication of natural ...
nnaword 8622 Weak ordering property of ...
nnacan 8623 Cancellation law for addit...
nnaword1 8624 Weak ordering property of ...
nnaword2 8625 Weak ordering property of ...
nnmordi 8626 Ordering property of multi...
nnmord 8627 Ordering property of multi...
nnmword 8628 Weak ordering property of ...
nnmcan 8629 Cancellation law for multi...
nnmwordi 8630 Weak ordering property of ...
nnmwordri 8631 Weak ordering property of ...
nnawordex 8632 Equivalence for weak order...
nnaordex 8633 Equivalence for ordering. ...
nnaordex2 8634 Equivalence for ordering. ...
1onn 8635 The ordinal 1 is a natural...
1onnALT 8636 Shorter proof of ~ 1onn us...
2onn 8637 The ordinal 2 is a natural...
2onnALT 8638 Shorter proof of ~ 2onn us...
3onn 8639 The ordinal 3 is a natural...
4onn 8640 The ordinal 4 is a natural...
1one2o 8641 Ordinal one is not ordinal...
oaabslem 8642 Lemma for ~ oaabs . (Cont...
oaabs 8643 Ordinal addition absorbs a...
oaabs2 8644 The absorption law ~ oaabs...
omabslem 8645 Lemma for ~ omabs . (Cont...
omabs 8646 Ordinal multiplication is ...
nnm1 8647 Multiply an element of ` _...
nnm2 8648 Multiply an element of ` _...
nn2m 8649 Multiply an element of ` _...
nnneo 8650 If a natural number is eve...
nneob 8651 A natural number is even i...
omsmolem 8652 Lemma for ~ omsmo . (Cont...
omsmo 8653 A strictly monotonic ordin...
omopthlem1 8654 Lemma for ~ omopthi . (Co...
omopthlem2 8655 Lemma for ~ omopthi . (Co...
omopthi 8656 An ordered pair theorem fo...
omopth 8657 An ordered pair theorem fo...
nnasmo 8658 There is at most one left ...
eldifsucnn 8659 Condition for membership i...
on2recsfn 8662 Show that double recursion...
on2recsov 8663 Calculate the value of the...
on2ind 8664 Double induction over ordi...
on3ind 8665 Triple induction over ordi...
coflton 8666 Cofinality theorem for ord...
cofon1 8667 Cofinality theorem for ord...
cofon2 8668 Cofinality theorem for ord...
cofonr 8669 Inverse cofinality law for...
naddfn 8670 Natural addition is a func...
naddcllem 8671 Lemma for ordinal addition...
naddcl 8672 Closure law for natural ad...
naddov 8673 The value of natural addit...
naddov2 8674 Alternate expression for n...
naddov3 8675 Alternate expression for n...
naddf 8676 Function statement for nat...
naddcom 8677 Natural addition commutes....
naddrid 8678 Ordinal zero is the additi...
naddlid 8679 Ordinal zero is the additi...
naddssim 8680 Ordinal less-than-or-equal...
naddelim 8681 Ordinal less-than is prese...
naddel1 8682 Ordinal less-than is not a...
naddel2 8683 Ordinal less-than is not a...
naddss1 8684 Ordinal less-than-or-equal...
naddss2 8685 Ordinal less-than-or-equal...
naddword1 8686 Weak-ordering principle fo...
naddword2 8687 Weak-ordering principle fo...
naddunif 8688 Uniformity theorem for nat...
naddasslem1 8689 Lemma for ~ naddass . Exp...
naddasslem2 8690 Lemma for ~ naddass . Exp...
naddass 8691 Natural ordinal addition i...
nadd32 8692 Commutative/associative la...
nadd4 8693 Rearragement of terms in a...
nadd42 8694 Rearragement of terms in a...
naddel12 8695 Natural addition to both s...
dfer2 8700 Alternate definition of eq...
dfec2 8702 Alternate definition of ` ...
ecexg 8703 An equivalence class modul...
ecexr 8704 A nonempty equivalence cla...
ereq1 8706 Equality theorem for equiv...
ereq2 8707 Equality theorem for equiv...
errel 8708 An equivalence relation is...
erdm 8709 The domain of an equivalen...
ercl 8710 Elementhood in the field o...
ersym 8711 An equivalence relation is...
ercl2 8712 Elementhood in the field o...
ersymb 8713 An equivalence relation is...
ertr 8714 An equivalence relation is...
ertrd 8715 A transitivity relation fo...
ertr2d 8716 A transitivity relation fo...
ertr3d 8717 A transitivity relation fo...
ertr4d 8718 A transitivity relation fo...
erref 8719 An equivalence relation is...
ercnv 8720 The converse of an equival...
errn 8721 The range and domain of an...
erssxp 8722 An equivalence relation is...
erex 8723 An equivalence relation is...
erexb 8724 An equivalence relation is...
iserd 8725 A reflexive, symmetric, tr...
iseri 8726 A reflexive, symmetric, tr...
iseriALT 8727 Alternate proof of ~ iseri...
brdifun 8728 Evaluate the incomparabili...
swoer 8729 Incomparability under a st...
swoord1 8730 The incomparability equiva...
swoord2 8731 The incomparability equiva...
swoso 8732 If the incomparability rel...
eqerlem 8733 Lemma for ~ eqer . (Contr...
eqer 8734 Equivalence relation invol...
ider 8735 The identity relation is a...
0er 8736 The empty set is an equiva...
eceq1 8737 Equality theorem for equiv...
eceq1d 8738 Equality theorem for equiv...
eceq2 8739 Equality theorem for equiv...
eceq2i 8740 Equality theorem for the `...
eceq2d 8741 Equality theorem for the `...
elecg 8742 Membership in an equivalen...
elec 8743 Membership in an equivalen...
relelec 8744 Membership in an equivalen...
ecss 8745 An equivalence class is a ...
ecdmn0 8746 A representative of a none...
ereldm 8747 Equality of equivalence cl...
erth 8748 Basic property of equivale...
erth2 8749 Basic property of equivale...
erthi 8750 Basic property of equivale...
erdisj 8751 Equivalence classes do not...
ecidsn 8752 An equivalence class modul...
qseq1 8753 Equality theorem for quoti...
qseq2 8754 Equality theorem for quoti...
qseq2i 8755 Equality theorem for quoti...
qseq2d 8756 Equality theorem for quoti...
qseq12 8757 Equality theorem for quoti...
elqsg 8758 Closed form of ~ elqs . (...
elqs 8759 Membership in a quotient s...
elqsi 8760 Membership in a quotient s...
elqsecl 8761 Membership in a quotient s...
ecelqsg 8762 Membership of an equivalen...
ecelqsi 8763 Membership of an equivalen...
ecopqsi 8764 "Closure" law for equivale...
qsexg 8765 A quotient set exists. (C...
qsex 8766 A quotient set exists. (C...
uniqs 8767 The union of a quotient se...
qsss 8768 A quotient set is a set of...
uniqs2 8769 The union of a quotient se...
snec 8770 The singleton of an equiva...
ecqs 8771 Equivalence class in terms...
ecid 8772 A set is equal to its cose...
qsid 8773 A set is equal to its quot...
ectocld 8774 Implicit substitution of c...
ectocl 8775 Implicit substitution of c...
elqsn0 8776 A quotient set does not co...
ecelqsdm 8777 Membership of an equivalen...
xpider 8778 A Cartesian square is an e...
iiner 8779 The intersection of a none...
riiner 8780 The relative intersection ...
erinxp 8781 A restricted equivalence r...
ecinxp 8782 Restrict the relation in a...
qsinxp 8783 Restrict the equivalence r...
qsdisj 8784 Members of a quotient set ...
qsdisj2 8785 A quotient set is a disjoi...
qsel 8786 If an element of a quotien...
uniinqs 8787 Class union distributes ov...
qliftlem 8788 Lemma for theorems about a...
qliftrel 8789 ` F ` , a function lift, i...
qliftel 8790 Elementhood in the relatio...
qliftel1 8791 Elementhood in the relatio...
qliftfun 8792 The function ` F ` is the ...
qliftfund 8793 The function ` F ` is the ...
qliftfuns 8794 The function ` F ` is the ...
qliftf 8795 The domain and codomain of...
qliftval 8796 The value of the function ...
ecoptocl 8797 Implicit substitution of c...
2ecoptocl 8798 Implicit substitution of c...
3ecoptocl 8799 Implicit substitution of c...
brecop 8800 Binary relation on a quoti...
brecop2 8801 Binary relation on a quoti...
eroveu 8802 Lemma for ~ erov and ~ ero...
erovlem 8803 Lemma for ~ erov and ~ ero...
erov 8804 The value of an operation ...
eroprf 8805 Functionality of an operat...
erov2 8806 The value of an operation ...
eroprf2 8807 Functionality of an operat...
ecopoveq 8808 This is the first of sever...
ecopovsym 8809 Assuming the operation ` F...
ecopovtrn 8810 Assuming that operation ` ...
ecopover 8811 Assuming that operation ` ...
eceqoveq 8812 Equality of equivalence re...
ecovcom 8813 Lemma used to transfer a c...
ecovass 8814 Lemma used to transfer an ...
ecovdi 8815 Lemma used to transfer a d...
mapprc 8820 When ` A ` is a proper cla...
pmex 8821 The class of all partial f...
mapex 8822 The class of all functions...
fnmap 8823 Set exponentiation has a u...
fnpm 8824 Partial function exponenti...
reldmmap 8825 Set exponentiation is a we...
mapvalg 8826 The value of set exponenti...
pmvalg 8827 The value of the partial m...
mapval 8828 The value of set exponenti...
elmapg 8829 Membership relation for se...
elmapd 8830 Deduction form of ~ elmapg...
elmapdd 8831 Deduction associated with ...
mapdm0 8832 The empty set is the only ...
elpmg 8833 The predicate "is a partia...
elpm2g 8834 The predicate "is a partia...
elpm2r 8835 Sufficient condition for b...
elpmi 8836 A partial function is a fu...
pmfun 8837 A partial function is a fu...
elmapex 8838 Eliminate antecedent for m...
elmapi 8839 A mapping is a function, f...
mapfset 8840 If ` B ` is a set, the val...
mapssfset 8841 The value of the set expon...
mapfoss 8842 The value of the set expon...
fsetsspwxp 8843 The class of all functions...
fset0 8844 The set of functions from ...
fsetdmprc0 8845 The set of functions with ...
fsetex 8846 The set of functions betwe...
f1setex 8847 The set of injections betw...
fosetex 8848 The set of surjections bet...
f1osetex 8849 The set of bijections betw...
fsetfcdm 8850 The class of functions wit...
fsetfocdm 8851 The class of functions wit...
fsetprcnex 8852 The class of all functions...
fsetcdmex 8853 The class of all functions...
fsetexb 8854 The class of all functions...
elmapfn 8855 A mapping is a function wi...
elmapfun 8856 A mapping is always a func...
elmapssres 8857 A restricted mapping is a ...
fpmg 8858 A total function is a part...
pmss12g 8859 Subset relation for the se...
pmresg 8860 Elementhood of a restricte...
elmap 8861 Membership relation for se...
mapval2 8862 Alternate expression for t...
elpm 8863 The predicate "is a partia...
elpm2 8864 The predicate "is a partia...
fpm 8865 A total function is a part...
mapsspm 8866 Set exponentiation is a su...
pmsspw 8867 Partial maps are a subset ...
mapsspw 8868 Set exponentiation is a su...
mapfvd 8869 The value of a function th...
elmapresaun 8870 ~ fresaun transposed to ma...
fvmptmap 8871 Special case of ~ fvmpt fo...
map0e 8872 Set exponentiation with an...
map0b 8873 Set exponentiation with an...
map0g 8874 Set exponentiation is empt...
0map0sn0 8875 The set of mappings of the...
mapsnd 8876 The value of set exponenti...
map0 8877 Set exponentiation is empt...
mapsn 8878 The value of set exponenti...
mapss 8879 Subset inheritance for set...
fdiagfn 8880 Functionality of the diago...
fvdiagfn 8881 Functionality of the diago...
mapsnconst 8882 Every singleton map is a c...
mapsncnv 8883 Expression for the inverse...
mapsnf1o2 8884 Explicit bijection between...
mapsnf1o3 8885 Explicit bijection in the ...
ralxpmap 8886 Quantification over functi...
dfixp 8889 Eliminate the expression `...
ixpsnval 8890 The value of an infinite C...
elixp2 8891 Membership in an infinite ...
fvixp 8892 Projection of a factor of ...
ixpfn 8893 A nuple is a function. (C...
elixp 8894 Membership in an infinite ...
elixpconst 8895 Membership in an infinite ...
ixpconstg 8896 Infinite Cartesian product...
ixpconst 8897 Infinite Cartesian product...
ixpeq1 8898 Equality theorem for infin...
ixpeq1d 8899 Equality theorem for infin...
ss2ixp 8900 Subclass theorem for infin...
ixpeq2 8901 Equality theorem for infin...
ixpeq2dva 8902 Equality theorem for infin...
ixpeq2dv 8903 Equality theorem for infin...
cbvixp 8904 Change bound variable in a...
cbvixpv 8905 Change bound variable in a...
nfixpw 8906 Bound-variable hypothesis ...
nfixp 8907 Bound-variable hypothesis ...
nfixp1 8908 The index variable in an i...
ixpprc 8909 A cartesian product of pro...
ixpf 8910 A member of an infinite Ca...
uniixp 8911 The union of an infinite C...
ixpexg 8912 The existence of an infini...
ixpin 8913 The intersection of two in...
ixpiin 8914 The indexed intersection o...
ixpint 8915 The intersection of a coll...
ixp0x 8916 An infinite Cartesian prod...
ixpssmap2g 8917 An infinite Cartesian prod...
ixpssmapg 8918 An infinite Cartesian prod...
0elixp 8919 Membership of the empty se...
ixpn0 8920 The infinite Cartesian pro...
ixp0 8921 The infinite Cartesian pro...
ixpssmap 8922 An infinite Cartesian prod...
resixp 8923 Restriction of an element ...
undifixp 8924 Union of two projections o...
mptelixpg 8925 Condition for an explicit ...
resixpfo 8926 Restriction of elements of...
elixpsn 8927 Membership in a class of s...
ixpsnf1o 8928 A bijection between a clas...
mapsnf1o 8929 A bijection between a set ...
boxriin 8930 A rectangular subset of a ...
boxcutc 8931 The relative complement of...
relen 8940 Equinumerosity is a relati...
reldom 8941 Dominance is a relation. ...
relsdom 8942 Strict dominance is a rela...
encv 8943 If two classes are equinum...
breng 8944 Equinumerosity relation. ...
bren 8945 Equinumerosity relation. ...
brenOLD 8946 Obsolete version of ~ bren...
brdom2g 8947 Dominance relation. This ...
brdomg 8948 Dominance relation. (Cont...
brdomgOLD 8949 Obsolete version of ~ brdo...
brdomi 8950 Dominance relation. (Cont...
brdomiOLD 8951 Obsolete version of ~ brdo...
brdom 8952 Dominance relation. (Cont...
domen 8953 Dominance in terms of equi...
domeng 8954 Dominance in terms of equi...
ctex 8955 A countable set is a set. ...
f1oen4g 8956 The domain and range of a ...
f1dom4g 8957 The domain of a one-to-one...
f1oen3g 8958 The domain and range of a ...
f1dom3g 8959 The domain of a one-to-one...
f1oen2g 8960 The domain and range of a ...
f1dom2g 8961 The domain of a one-to-one...
f1dom2gOLD 8962 Obsolete version of ~ f1do...
f1oeng 8963 The domain and range of a ...
f1domg 8964 The domain of a one-to-one...
f1oen 8965 The domain and range of a ...
f1dom 8966 The domain of a one-to-one...
brsdom 8967 Strict dominance relation,...
isfi 8968 Express " ` A ` is finite"...
enssdom 8969 Equinumerosity implies dom...
dfdom2 8970 Alternate definition of do...
endom 8971 Equinumerosity implies dom...
sdomdom 8972 Strict dominance implies d...
sdomnen 8973 Strict dominance implies n...
brdom2 8974 Dominance in terms of stri...
bren2 8975 Equinumerosity expressed i...
enrefg 8976 Equinumerosity is reflexiv...
enref 8977 Equinumerosity is reflexiv...
eqeng 8978 Equality implies equinumer...
domrefg 8979 Dominance is reflexive. (...
en2d 8980 Equinumerosity inference f...
en3d 8981 Equinumerosity inference f...
en2i 8982 Equinumerosity inference f...
en3i 8983 Equinumerosity inference f...
dom2lem 8984 A mapping (first hypothesi...
dom2d 8985 A mapping (first hypothesi...
dom3d 8986 A mapping (first hypothesi...
dom2 8987 A mapping (first hypothesi...
dom3 8988 A mapping (first hypothesi...
idssen 8989 Equality implies equinumer...
domssl 8990 If ` A ` is a subset of ` ...
domssr 8991 If ` C ` is a superset of ...
ssdomg 8992 A set dominates its subset...
ener 8993 Equinumerosity is an equiv...
ensymb 8994 Symmetry of equinumerosity...
ensym 8995 Symmetry of equinumerosity...
ensymi 8996 Symmetry of equinumerosity...
ensymd 8997 Symmetry of equinumerosity...
entr 8998 Transitivity of equinumero...
domtr 8999 Transitivity of dominance ...
entri 9000 A chained equinumerosity i...
entr2i 9001 A chained equinumerosity i...
entr3i 9002 A chained equinumerosity i...
entr4i 9003 A chained equinumerosity i...
endomtr 9004 Transitivity of equinumero...
domentr 9005 Transitivity of dominance ...
f1imaeng 9006 If a function is one-to-on...
f1imaen2g 9007 If a function is one-to-on...
f1imaen 9008 If a function is one-to-on...
en0 9009 The empty set is equinumer...
en0OLD 9010 Obsolete version of ~ en0 ...
en0ALT 9011 Shorter proof of ~ en0 , d...
en0r 9012 The empty set is equinumer...
ensn1 9013 A singleton is equinumerou...
ensn1OLD 9014 Obsolete version of ~ ensn...
ensn1g 9015 A singleton is equinumerou...
enpr1g 9016 ` { A , A } ` has only one...
en1 9017 A set is equinumerous to o...
en1OLD 9018 Obsolete version of ~ en1 ...
en1b 9019 A set is equinumerous to o...
en1bOLD 9020 Obsolete version of ~ en1b...
reuen1 9021 Two ways to express "exact...
euen1 9022 Two ways to express "exact...
euen1b 9023 Two ways to express " ` A ...
en1uniel 9024 A singleton contains its s...
en1unielOLD 9025 Obsolete version of ~ en1u...
2dom 9026 A set that dominates ordin...
fundmen 9027 A function is equinumerous...
fundmeng 9028 A function is equinumerous...
cnven 9029 A relational set is equinu...
cnvct 9030 If a set is countable, so ...
fndmeng 9031 A function is equinumerate...
mapsnend 9032 Set exponentiation to a si...
mapsnen 9033 Set exponentiation to a si...
snmapen 9034 Set exponentiation: a sing...
snmapen1 9035 Set exponentiation: a sing...
map1 9036 Set exponentiation: ordina...
en2sn 9037 Two singletons are equinum...
en2snOLD 9038 Obsolete version of ~ en2s...
en2snOLDOLD 9039 Obsolete version of ~ en2s...
snfi 9040 A singleton is finite. (C...
fiprc 9041 The class of finite sets i...
unen 9042 Equinumerosity of union of...
enrefnn 9043 Equinumerosity is reflexiv...
en2prd 9044 Two unordered pairs are eq...
enpr2d 9045 A pair with distinct eleme...
enpr2dOLD 9046 Obsolete version of ~ enpr...
ssct 9047 Any subset of a countable ...
ssctOLD 9048 Obsolete version of ~ ssct...
difsnen 9049 All decrements of a set ar...
domdifsn 9050 Dominance over a set with ...
xpsnen 9051 A set is equinumerous to i...
xpsneng 9052 A set is equinumerous to i...
xp1en 9053 One times a cardinal numbe...
endisj 9054 Any two sets are equinumer...
undom 9055 Dominance law for union. ...
undomOLD 9056 Obsolete version of ~ undo...
xpcomf1o 9057 The canonical bijection fr...
xpcomco 9058 Composition with the bijec...
xpcomen 9059 Commutative law for equinu...
xpcomeng 9060 Commutative law for equinu...
xpsnen2g 9061 A set is equinumerous to i...
xpassen 9062 Associative law for equinu...
xpdom2 9063 Dominance law for Cartesia...
xpdom2g 9064 Dominance law for Cartesia...
xpdom1g 9065 Dominance law for Cartesia...
xpdom3 9066 A set is dominated by its ...
xpdom1 9067 Dominance law for Cartesia...
domunsncan 9068 A singleton cancellation l...
omxpenlem 9069 Lemma for ~ omxpen . (Con...
omxpen 9070 The cardinal and ordinal p...
omf1o 9071 Construct an explicit bije...
pw2f1olem 9072 Lemma for ~ pw2f1o . (Con...
pw2f1o 9073 The power set of a set is ...
pw2eng 9074 The power set of a set is ...
pw2en 9075 The power set of a set is ...
fopwdom 9076 Covering implies injection...
enfixsn 9077 Given two equipollent sets...
sucdom2OLD 9078 Obsolete version of ~ sucd...
sbthlem1 9079 Lemma for ~ sbth . (Contr...
sbthlem2 9080 Lemma for ~ sbth . (Contr...
sbthlem3 9081 Lemma for ~ sbth . (Contr...
sbthlem4 9082 Lemma for ~ sbth . (Contr...
sbthlem5 9083 Lemma for ~ sbth . (Contr...
sbthlem6 9084 Lemma for ~ sbth . (Contr...
sbthlem7 9085 Lemma for ~ sbth . (Contr...
sbthlem8 9086 Lemma for ~ sbth . (Contr...
sbthlem9 9087 Lemma for ~ sbth . (Contr...
sbthlem10 9088 Lemma for ~ sbth . (Contr...
sbth 9089 Schroeder-Bernstein Theore...
sbthb 9090 Schroeder-Bernstein Theore...
sbthcl 9091 Schroeder-Bernstein Theore...
dfsdom2 9092 Alternate definition of st...
brsdom2 9093 Alternate definition of st...
sdomnsym 9094 Strict dominance is asymme...
domnsym 9095 Theorem 22(i) of [Suppes] ...
0domg 9096 Any set dominates the empt...
0domgOLD 9097 Obsolete version of ~ 0dom...
dom0 9098 A set dominated by the emp...
dom0OLD 9099 Obsolete version of ~ dom0...
0sdomg 9100 A set strictly dominates t...
0sdomgOLD 9101 Obsolete version of ~ 0sdo...
0dom 9102 Any set dominates the empt...
0sdom 9103 A set strictly dominates t...
sdom0 9104 The empty set does not str...
sdom0OLD 9105 Obsolete version of ~ sdom...
sdomdomtr 9106 Transitivity of strict dom...
sdomentr 9107 Transitivity of strict dom...
domsdomtr 9108 Transitivity of dominance ...
ensdomtr 9109 Transitivity of equinumero...
sdomirr 9110 Strict dominance is irrefl...
sdomtr 9111 Strict dominance is transi...
sdomn2lp 9112 Strict dominance has no 2-...
enen1 9113 Equality-like theorem for ...
enen2 9114 Equality-like theorem for ...
domen1 9115 Equality-like theorem for ...
domen2 9116 Equality-like theorem for ...
sdomen1 9117 Equality-like theorem for ...
sdomen2 9118 Equality-like theorem for ...
domtriord 9119 Dominance is trichotomous ...
sdomel 9120 For ordinals, strict domin...
sdomdif 9121 The difference of a set fr...
onsdominel 9122 An ordinal with more eleme...
domunsn 9123 Dominance over a set with ...
fodomr 9124 There exists a mapping fro...
pwdom 9125 Injection of sets implies ...
canth2 9126 Cantor's Theorem. No set ...
canth2g 9127 Cantor's theorem with the ...
2pwuninel 9128 The power set of the power...
2pwne 9129 No set equals the power se...
disjen 9130 A stronger form of ~ pwuni...
disjenex 9131 Existence version of ~ dis...
domss2 9132 A corollary of ~ disjenex ...
domssex2 9133 A corollary of ~ disjenex ...
domssex 9134 Weakening of ~ domssex2 to...
xpf1o 9135 Construct a bijection on a...
xpen 9136 Equinumerosity law for Car...
mapen 9137 Two set exponentiations ar...
mapdom1 9138 Order-preserving property ...
mapxpen 9139 Equinumerosity law for dou...
xpmapenlem 9140 Lemma for ~ xpmapen . (Co...
xpmapen 9141 Equinumerosity law for set...
mapunen 9142 Equinumerosity law for set...
map2xp 9143 A cardinal power with expo...
mapdom2 9144 Order-preserving property ...
mapdom3 9145 Set exponentiation dominat...
pwen 9146 If two sets are equinumero...
ssenen 9147 Equinumerosity of equinume...
limenpsi 9148 A limit ordinal is equinum...
limensuci 9149 A limit ordinal is equinum...
limensuc 9150 A limit ordinal is equinum...
infensuc 9151 Any infinite ordinal is eq...
dif1enlem 9152 Lemma for ~ rexdif1en and ...
dif1enlemOLD 9153 Obsolete version of ~ dif1...
rexdif1en 9154 If a set is equinumerous t...
rexdif1enOLD 9155 Obsolete version of ~ rexd...
dif1en 9156 If a set ` A ` is equinume...
dif1ennn 9157 If a set ` A ` is equinume...
dif1enOLD 9158 Obsolete version of ~ dif1...
findcard 9159 Schema for induction on th...
findcard2 9160 Schema for induction on th...
findcard2s 9161 Variation of ~ findcard2 r...
findcard2d 9162 Deduction version of ~ fin...
nnfi 9163 Natural numbers are finite...
pssnn 9164 A proper subset of a natur...
ssnnfi 9165 A subset of a natural numb...
ssnnfiOLD 9166 Obsolete version of ~ ssnn...
0fin 9167 The empty set is finite. ...
unfi 9168 The union of two finite se...
ssfi 9169 A subset of a finite set i...
ssfiALT 9170 Shorter proof of ~ ssfi us...
imafi 9171 Images of finite sets are ...
pwfir 9172 If the power set of a set ...
pwfilem 9173 Lemma for ~ pwfi . (Contr...
pwfi 9174 The power set of a finite ...
diffi 9175 If ` A ` is finite, ` ( A ...
cnvfi 9176 If a set is finite, its co...
fnfi 9177 A version of ~ fnex for fi...
f1oenfi 9178 If the domain of a one-to-...
f1oenfirn 9179 If the range of a one-to-o...
f1domfi 9180 If the codomain of a one-t...
f1domfi2 9181 If the domain of a one-to-...
enreffi 9182 Equinumerosity is reflexiv...
ensymfib 9183 Symmetry of equinumerosity...
entrfil 9184 Transitivity of equinumero...
enfii 9185 A set equinumerous to a fi...
enfi 9186 Equinumerous sets have the...
enfiALT 9187 Shorter proof of ~ enfi us...
domfi 9188 A set dominated by a finit...
entrfi 9189 Transitivity of equinumero...
entrfir 9190 Transitivity of equinumero...
domtrfil 9191 Transitivity of dominance ...
domtrfi 9192 Transitivity of dominance ...
domtrfir 9193 Transitivity of dominance ...
f1imaenfi 9194 If a function is one-to-on...
ssdomfi 9195 A finite set dominates its...
ssdomfi2 9196 A set dominates its finite...
sbthfilem 9197 Lemma for ~ sbthfi . (Con...
sbthfi 9198 Schroeder-Bernstein Theore...
domnsymfi 9199 If a set dominates a finit...
sdomdomtrfi 9200 Transitivity of strict dom...
domsdomtrfi 9201 Transitivity of dominance ...
sucdom2 9202 Strict dominance of a set ...
phplem1 9203 Lemma for Pigeonhole Princ...
phplem2 9204 Lemma for Pigeonhole Princ...
nneneq 9205 Two equinumerous natural n...
php 9206 Pigeonhole Principle. A n...
php2 9207 Corollary of Pigeonhole Pr...
php3 9208 Corollary of Pigeonhole Pr...
php4 9209 Corollary of the Pigeonhol...
php5 9210 Corollary of the Pigeonhol...
phpeqd 9211 Corollary of the Pigeonhol...
nndomog 9212 Cardinal ordering agrees w...
phplem1OLD 9213 Obsolete lemma for ~ php a...
phplem2OLD 9214 Obsolete lemma for ~ php a...
phplem3OLD 9215 Obsolete version of ~ phpl...
phplem4OLD 9216 Obsolete version of ~ phpl...
nneneqOLD 9217 Obsolete version of ~ nnen...
phpOLD 9218 Obsolete version of ~ php ...
php2OLD 9219 Obsolete version of ~ php2...
php3OLD 9220 Obsolete version of ~ php3...
phpeqdOLD 9221 Obsolete version of ~ phpe...
nndomogOLD 9222 Obsolete version of ~ nndo...
snnen2oOLD 9223 Obsolete version of ~ snne...
onomeneq 9224 An ordinal number equinume...
onomeneqOLD 9225 Obsolete version of ~ onom...
onfin 9226 An ordinal number is finit...
onfin2 9227 A set is a natural number ...
nnfiOLD 9228 Obsolete version of ~ nnfi...
nndomo 9229 Cardinal ordering agrees w...
nnsdomo 9230 Cardinal ordering agrees w...
sucdom 9231 Strict dominance of a set ...
sucdomOLD 9232 Obsolete version of ~ sucd...
snnen2o 9233 A singleton ` { A } ` is n...
0sdom1dom 9234 Strict dominance over 0 is...
0sdom1domALT 9235 Alternate proof of ~ 0sdom...
1sdom2 9236 Ordinal 1 is strictly domi...
1sdom2ALT 9237 Alternate proof of ~ 1sdom...
sdom1 9238 A set has less than one me...
sdom1OLD 9239 Obsolete version of ~ sdom...
modom 9240 Two ways to express "at mo...
modom2 9241 Two ways to express "at mo...
rex2dom 9242 A set that has at least 2 ...
1sdom2dom 9243 Strict dominance over 1 is...
1sdom 9244 A set that strictly domina...
1sdomOLD 9245 Obsolete version of ~ 1sdo...
unxpdomlem1 9246 Lemma for ~ unxpdom . (Tr...
unxpdomlem2 9247 Lemma for ~ unxpdom . (Co...
unxpdomlem3 9248 Lemma for ~ unxpdom . (Co...
unxpdom 9249 Cartesian product dominate...
unxpdom2 9250 Corollary of ~ unxpdom . ...
sucxpdom 9251 Cartesian product dominate...
pssinf 9252 A set equinumerous to a pr...
fisseneq 9253 A finite set is equal to i...
ominf 9254 The set of natural numbers...
ominfOLD 9255 Obsolete version of ~ omin...
isinf 9256 Any set that is not finite...
isinfOLD 9257 Obsolete version of ~ isin...
fineqvlem 9258 Lemma for ~ fineqv . (Con...
fineqv 9259 If the Axiom of Infinity i...
enfiiOLD 9260 Obsolete version of ~ enfi...
pssnnOLD 9261 Obsolete version of ~ pssn...
xpfir 9262 The components of a nonemp...
ssfid 9263 A subset of a finite set i...
infi 9264 The intersection of two se...
rabfi 9265 A restricted class built f...
finresfin 9266 The restriction of a finit...
f1finf1o 9267 Any injection from one fin...
f1finf1oOLD 9268 Obsolete version of ~ f1fi...
nfielex 9269 If a class is not finite, ...
en1eqsn 9270 A set with one element is ...
en1eqsnOLD 9271 Obsolete version of ~ en1e...
en1eqsnbi 9272 A set containing an elemen...
dif1ennnALT 9273 Alternate proof of ~ dif1e...
enp1ilem 9274 Lemma for uses of ~ enp1i ...
enp1i 9275 Proof induction for ~ en2 ...
enp1iOLD 9276 Obsolete version of ~ enp1...
en2 9277 A set equinumerous to ordi...
en3 9278 A set equinumerous to ordi...
en4 9279 A set equinumerous to ordi...
findcard2OLD 9280 Obsolete version of ~ find...
findcard3 9281 Schema for strong inductio...
findcard3OLD 9282 Obsolete version of ~ find...
ac6sfi 9283 A version of ~ ac6s for fi...
frfi 9284 A partial order is well-fo...
fimax2g 9285 A finite set has a maximum...
fimaxg 9286 A finite set has a maximum...
fisupg 9287 Lemma showing existence an...
wofi 9288 A total order on a finite ...
ordunifi 9289 The maximum of a finite co...
nnunifi 9290 The union (supremum) of a ...
unblem1 9291 Lemma for ~ unbnn . After...
unblem2 9292 Lemma for ~ unbnn . The v...
unblem3 9293 Lemma for ~ unbnn . The v...
unblem4 9294 Lemma for ~ unbnn . The f...
unbnn 9295 Any unbounded subset of na...
unbnn2 9296 Version of ~ unbnn that do...
isfinite2 9297 Any set strictly dominated...
nnsdomg 9298 Omega strictly dominates a...
nnsdomgOLD 9299 Obsolete version of ~ nnsd...
isfiniteg 9300 A set is finite iff it is ...
infsdomnn 9301 An infinite set strictly d...
infsdomnnOLD 9302 Obsolete version of ~ infs...
infn0 9303 An infinite set is not emp...
infn0ALT 9304 Shorter proof of ~ infn0 u...
fin2inf 9305 This (useless) theorem, wh...
unfilem1 9306 Lemma for proving that the...
unfilem2 9307 Lemma for proving that the...
unfilem3 9308 Lemma for proving that the...
unfiOLD 9309 Obsolete version of ~ unfi...
unfir 9310 If a union is finite, the ...
unfi2 9311 The union of two finite se...
difinf 9312 An infinite set ` A ` minu...
xpfi 9313 The Cartesian product of t...
xpfiOLD 9314 Obsolete version of ~ xpfi...
3xpfi 9315 The Cartesian product of t...
domunfican 9316 A finite set union cancell...
infcntss 9317 Every infinite set has a d...
prfi 9318 An unordered pair is finit...
tpfi 9319 An unordered triple is fin...
fiint 9320 Equivalent ways of stating...
fodomfi 9321 An onto function implies d...
fodomfib 9322 Equivalence of an onto map...
fofinf1o 9323 Any surjection from one fi...
rneqdmfinf1o 9324 Any function from a finite...
fidomdm 9325 Any finite set dominates i...
dmfi 9326 The domain of a finite set...
fundmfibi 9327 A function is finite if an...
resfnfinfin 9328 The restriction of a funct...
residfi 9329 A restricted identity func...
cnvfiALT 9330 Shorter proof of ~ cnvfi u...
rnfi 9331 The range of a finite set ...
f1dmvrnfibi 9332 A one-to-one function whos...
f1vrnfibi 9333 A one-to-one function whic...
fofi 9334 If an onto function has a ...
f1fi 9335 If a 1-to-1 function has a...
iunfi 9336 The finite union of finite...
unifi 9337 The finite union of finite...
unifi2 9338 The finite union of finite...
infssuni 9339 If an infinite set ` A ` i...
unirnffid 9340 The union of the range of ...
imafiALT 9341 Shorter proof of ~ imafi u...
pwfilemOLD 9342 Obsolete version of ~ pwfi...
pwfiOLD 9343 Obsolete version of ~ pwfi...
mapfi 9344 Set exponentiation of fini...
ixpfi 9345 A Cartesian product of fin...
ixpfi2 9346 A Cartesian product of fin...
mptfi 9347 A finite mapping set is fi...
abrexfi 9348 An image set from a finite...
cnvimamptfin 9349 A preimage of a mapping wi...
elfpw 9350 Membership in a class of f...
unifpw 9351 A set is the union of its ...
f1opwfi 9352 A one-to-one mapping induc...
fissuni 9353 A finite subset of a union...
fipreima 9354 Given a finite subset ` A ...
finsschain 9355 A finite subset of the uni...
indexfi 9356 If for every element of a ...
relfsupp 9359 The property of a function...
relprcnfsupp 9360 A proper class is never fi...
isfsupp 9361 The property of a class to...
isfsuppd 9362 Deduction form of ~ isfsup...
funisfsupp 9363 The property of a function...
fsuppimp 9364 Implications of a class be...
fsuppimpd 9365 A finitely supported funct...
fisuppfi 9366 A function on a finite set...
fidmfisupp 9367 A function with a finite d...
fdmfisuppfi 9368 The support of a function ...
fdmfifsupp 9369 A function with a finite d...
fsuppmptdm 9370 A mapping with a finite do...
fndmfisuppfi 9371 The support of a function ...
fndmfifsupp 9372 A function with a finite d...
suppeqfsuppbi 9373 If two functions have the ...
suppssfifsupp 9374 If the support of a functi...
fsuppsssupp 9375 If the support of a functi...
fsuppxpfi 9376 The cartesian product of t...
fczfsuppd 9377 A constant function with v...
fsuppun 9378 The union of two finitely ...
fsuppunfi 9379 The union of the support o...
fsuppunbi 9380 If the union of two classe...
0fsupp 9381 The empty set is a finitel...
snopfsupp 9382 A singleton containing an ...
funsnfsupp 9383 Finite support for a funct...
fsuppres 9384 The restriction of a finit...
fmptssfisupp 9385 The restriction of a mappi...
ressuppfi 9386 If the support of the rest...
resfsupp 9387 If the restriction of a fu...
resfifsupp 9388 The restriction of a funct...
ffsuppbi 9389 Two ways of saying that a ...
fsuppmptif 9390 A function mapping an argu...
sniffsupp 9391 A function mapping all but...
fsuppcolem 9392 Lemma for ~ fsuppco . For...
fsuppco 9393 The composition of a 1-1 f...
fsuppco2 9394 The composition of a funct...
fsuppcor 9395 The composition of a funct...
mapfienlem1 9396 Lemma 1 for ~ mapfien . (...
mapfienlem2 9397 Lemma 2 for ~ mapfien . (...
mapfienlem3 9398 Lemma 3 for ~ mapfien . (...
mapfien 9399 A bijection of the base se...
mapfien2 9400 Equinumerousity relation f...
fival 9403 The set of all the finite ...
elfi 9404 Specific properties of an ...
elfi2 9405 The empty intersection nee...
elfir 9406 Sufficient condition for a...
intrnfi 9407 Sufficient condition for t...
iinfi 9408 An indexed intersection of...
inelfi 9409 The intersection of two se...
ssfii 9410 Any element of a set ` A `...
fi0 9411 The set of finite intersec...
fieq0 9412 A set is empty iff the cla...
fiin 9413 The elements of ` ( fi `` ...
dffi2 9414 The set of finite intersec...
fiss 9415 Subset relationship for fu...
inficl 9416 A set which is closed unde...
fipwuni 9417 The set of finite intersec...
fisn 9418 A singleton is closed unde...
fiuni 9419 The union of the finite in...
fipwss 9420 If a set is a family of su...
elfiun 9421 A finite intersection of e...
dffi3 9422 The set of finite intersec...
fifo 9423 Describe a surjection from...
marypha1lem 9424 Core induction for Philip ...
marypha1 9425 (Philip) Hall's marriage t...
marypha2lem1 9426 Lemma for ~ marypha2 . Pr...
marypha2lem2 9427 Lemma for ~ marypha2 . Pr...
marypha2lem3 9428 Lemma for ~ marypha2 . Pr...
marypha2lem4 9429 Lemma for ~ marypha2 . Pr...
marypha2 9430 Version of ~ marypha1 usin...
dfsup2 9435 Quantifier-free definition...
supeq1 9436 Equality theorem for supre...
supeq1d 9437 Equality deduction for sup...
supeq1i 9438 Equality inference for sup...
supeq2 9439 Equality theorem for supre...
supeq3 9440 Equality theorem for supre...
supeq123d 9441 Equality deduction for sup...
nfsup 9442 Hypothesis builder for sup...
supmo 9443 Any class ` B ` has at mos...
supexd 9444 A supremum is a set. (Con...
supeu 9445 A supremum is unique. Sim...
supval2 9446 Alternate expression for t...
eqsup 9447 Sufficient condition for a...
eqsupd 9448 Sufficient condition for a...
supcl 9449 A supremum belongs to its ...
supub 9450 A supremum is an upper bou...
suplub 9451 A supremum is the least up...
suplub2 9452 Bidirectional form of ~ su...
supnub 9453 An upper bound is not less...
supex 9454 A supremum is a set. (Con...
sup00 9455 The supremum under an empt...
sup0riota 9456 The supremum of an empty s...
sup0 9457 The supremum of an empty s...
supmax 9458 The greatest element of a ...
fisup2g 9459 A finite set satisfies the...
fisupcl 9460 A nonempty finite set cont...
supgtoreq 9461 The supremum of a finite s...
suppr 9462 The supremum of a pair. (...
supsn 9463 The supremum of a singleto...
supisolem 9464 Lemma for ~ supiso . (Con...
supisoex 9465 Lemma for ~ supiso . (Con...
supiso 9466 Image of a supremum under ...
infeq1 9467 Equality theorem for infim...
infeq1d 9468 Equality deduction for inf...
infeq1i 9469 Equality inference for inf...
infeq2 9470 Equality theorem for infim...
infeq3 9471 Equality theorem for infim...
infeq123d 9472 Equality deduction for inf...
nfinf 9473 Hypothesis builder for inf...
infexd 9474 An infimum is a set. (Con...
eqinf 9475 Sufficient condition for a...
eqinfd 9476 Sufficient condition for a...
infval 9477 Alternate expression for t...
infcllem 9478 Lemma for ~ infcl , ~ infl...
infcl 9479 An infimum belongs to its ...
inflb 9480 An infimum is a lower boun...
infglb 9481 An infimum is the greatest...
infglbb 9482 Bidirectional form of ~ in...
infnlb 9483 A lower bound is not great...
infex 9484 An infimum is a set. (Con...
infmin 9485 The smallest element of a ...
infmo 9486 Any class ` B ` has at mos...
infeu 9487 An infimum is unique. (Co...
fimin2g 9488 A finite set has a minimum...
fiming 9489 A finite set has a minimum...
fiinfg 9490 Lemma showing existence an...
fiinf2g 9491 A finite set satisfies the...
fiinfcl 9492 A nonempty finite set cont...
infltoreq 9493 The infimum of a finite se...
infpr 9494 The infimum of a pair. (C...
infsupprpr 9495 The infimum of a proper pa...
infsn 9496 The infimum of a singleton...
inf00 9497 The infimum regarding an e...
infempty 9498 The infimum of an empty se...
infiso 9499 Image of an infimum under ...
dfoi 9502 Rewrite ~ df-oi with abbre...
oieq1 9503 Equality theorem for ordin...
oieq2 9504 Equality theorem for ordin...
nfoi 9505 Hypothesis builder for ord...
ordiso2 9506 Generalize ~ ordiso to pro...
ordiso 9507 Order-isomorphic ordinal n...
ordtypecbv 9508 Lemma for ~ ordtype . (Co...
ordtypelem1 9509 Lemma for ~ ordtype . (Co...
ordtypelem2 9510 Lemma for ~ ordtype . (Co...
ordtypelem3 9511 Lemma for ~ ordtype . (Co...
ordtypelem4 9512 Lemma for ~ ordtype . (Co...
ordtypelem5 9513 Lemma for ~ ordtype . (Co...
ordtypelem6 9514 Lemma for ~ ordtype . (Co...
ordtypelem7 9515 Lemma for ~ ordtype . ` ra...
ordtypelem8 9516 Lemma for ~ ordtype . (Co...
ordtypelem9 9517 Lemma for ~ ordtype . Eit...
ordtypelem10 9518 Lemma for ~ ordtype . Usi...
oi0 9519 Definition of the ordinal ...
oicl 9520 The order type of the well...
oif 9521 The order isomorphism of t...
oiiso2 9522 The order isomorphism of t...
ordtype 9523 For any set-like well-orde...
oiiniseg 9524 ` ran F ` is an initial se...
ordtype2 9525 For any set-like well-orde...
oiexg 9526 The order isomorphism on a...
oion 9527 The order type of the well...
oiiso 9528 The order isomorphism of t...
oien 9529 The order type of a well-o...
oieu 9530 Uniqueness of the unique o...
oismo 9531 When ` A ` is a subclass o...
oiid 9532 The order type of an ordin...
hartogslem1 9533 Lemma for ~ hartogs . (Co...
hartogslem2 9534 Lemma for ~ hartogs . (Co...
hartogs 9535 The class of ordinals domi...
wofib 9536 The only sets which are we...
wemaplem1 9537 Value of the lexicographic...
wemaplem2 9538 Lemma for ~ wemapso . Tra...
wemaplem3 9539 Lemma for ~ wemapso . Tra...
wemappo 9540 Construct lexicographic or...
wemapsolem 9541 Lemma for ~ wemapso . (Co...
wemapso 9542 Construct lexicographic or...
wemapso2lem 9543 Lemma for ~ wemapso2 . (C...
wemapso2 9544 An alternative to having a...
card2on 9545 The alternate definition o...
card2inf 9546 The alternate definition o...
harf 9549 Functionality of the Harto...
harcl 9550 Values of the Hartogs func...
harval 9551 Function value of the Hart...
elharval 9552 The Hartogs number of a se...
harndom 9553 The Hartogs number of a se...
harword 9554 Weak ordering property of ...
relwdom 9557 Weak dominance is a relati...
brwdom 9558 Property of weak dominance...
brwdomi 9559 Property of weak dominance...
brwdomn0 9560 Weak dominance over nonemp...
0wdom 9561 Any set weakly dominates t...
fowdom 9562 An onto function implies w...
wdomref 9563 Reflexivity of weak domina...
brwdom2 9564 Alternate characterization...
domwdom 9565 Weak dominance is implied ...
wdomtr 9566 Transitivity of weak domin...
wdomen1 9567 Equality-like theorem for ...
wdomen2 9568 Equality-like theorem for ...
wdompwdom 9569 Weak dominance strengthens...
canthwdom 9570 Cantor's Theorem, stated u...
wdom2d 9571 Deduce weak dominance from...
wdomd 9572 Deduce weak dominance from...
brwdom3 9573 Condition for weak dominan...
brwdom3i 9574 Weak dominance implies exi...
unwdomg 9575 Weak dominance of a (disjo...
xpwdomg 9576 Weak dominance of a Cartes...
wdomima2g 9577 A set is weakly dominant o...
wdomimag 9578 A set is weakly dominant o...
unxpwdom2 9579 Lemma for ~ unxpwdom . (C...
unxpwdom 9580 If a Cartesian product is ...
ixpiunwdom 9581 Describe an onto function ...
harwdom 9582 The value of the Hartogs f...
axreg2 9584 Axiom of Regularity expres...
zfregcl 9585 The Axiom of Regularity wi...
zfreg 9586 The Axiom of Regularity us...
elirrv 9587 The membership relation is...
elirr 9588 No class is a member of it...
elneq 9589 A class is not equal to an...
nelaneq 9590 A class is not an element ...
epinid0 9591 The membership relation an...
sucprcreg 9592 A class is equal to its su...
ruv 9593 The Russell class is equal...
ruALT 9594 Alternate proof of ~ ru , ...
disjcsn 9595 A class is disjoint from i...
zfregfr 9596 The membership relation is...
en2lp 9597 No class has 2-cycle membe...
elnanel 9598 Two classes are not elemen...
cnvepnep 9599 The membership (epsilon) r...
epnsym 9600 The membership (epsilon) r...
elnotel 9601 A class cannot be an eleme...
elnel 9602 A class cannot be an eleme...
en3lplem1 9603 Lemma for ~ en3lp . (Cont...
en3lplem2 9604 Lemma for ~ en3lp . (Cont...
en3lp 9605 No class has 3-cycle membe...
preleqg 9606 Equality of two unordered ...
preleq 9607 Equality of two unordered ...
preleqALT 9608 Alternate proof of ~ prele...
opthreg 9609 Theorem for alternate repr...
suc11reg 9610 The successor operation be...
dford2 9611 Assuming ~ ax-reg , an ord...
inf0 9612 Existence of ` _om ` impli...
inf1 9613 Variation of Axiom of Infi...
inf2 9614 Variation of Axiom of Infi...
inf3lema 9615 Lemma for our Axiom of Inf...
inf3lemb 9616 Lemma for our Axiom of Inf...
inf3lemc 9617 Lemma for our Axiom of Inf...
inf3lemd 9618 Lemma for our Axiom of Inf...
inf3lem1 9619 Lemma for our Axiom of Inf...
inf3lem2 9620 Lemma for our Axiom of Inf...
inf3lem3 9621 Lemma for our Axiom of Inf...
inf3lem4 9622 Lemma for our Axiom of Inf...
inf3lem5 9623 Lemma for our Axiom of Inf...
inf3lem6 9624 Lemma for our Axiom of Inf...
inf3lem7 9625 Lemma for our Axiom of Inf...
inf3 9626 Our Axiom of Infinity ~ ax...
infeq5i 9627 Half of ~ infeq5 . (Contr...
infeq5 9628 The statement "there exist...
zfinf 9630 Axiom of Infinity expresse...
axinf2 9631 A standard version of Axio...
zfinf2 9633 A standard version of the ...
omex 9634 The existence of omega (th...
axinf 9635 The first version of the A...
inf5 9636 The statement "there exist...
omelon 9637 Omega is an ordinal number...
dfom3 9638 The class of natural numbe...
elom3 9639 A simplification of ~ elom...
dfom4 9640 A simplification of ~ df-o...
dfom5 9641 ` _om ` is the smallest li...
oancom 9642 Ordinal addition is not co...
isfinite 9643 A set is finite iff it is ...
fict 9644 A finite set is countable ...
nnsdom 9645 A natural number is strict...
omenps 9646 Omega is equinumerous to a...
omensuc 9647 The set of natural numbers...
infdifsn 9648 Removing a singleton from ...
infdiffi 9649 Removing a finite set from...
unbnn3 9650 Any unbounded subset of na...
noinfep 9651 Using the Axiom of Regular...
cantnffval 9654 The value of the Cantor no...
cantnfdm 9655 The domain of the Cantor n...
cantnfvalf 9656 Lemma for ~ cantnf . The ...
cantnfs 9657 Elementhood in the set of ...
cantnfcl 9658 Basic properties of the or...
cantnfval 9659 The value of the Cantor no...
cantnfval2 9660 Alternate expression for t...
cantnfsuc 9661 The value of the recursive...
cantnfle 9662 A lower bound on the ` CNF...
cantnflt 9663 An upper bound on the part...
cantnflt2 9664 An upper bound on the ` CN...
cantnff 9665 The ` CNF ` function is a ...
cantnf0 9666 The value of the zero func...
cantnfrescl 9667 A function is finitely sup...
cantnfres 9668 The ` CNF ` function respe...
cantnfp1lem1 9669 Lemma for ~ cantnfp1 . (C...
cantnfp1lem2 9670 Lemma for ~ cantnfp1 . (C...
cantnfp1lem3 9671 Lemma for ~ cantnfp1 . (C...
cantnfp1 9672 If ` F ` is created by add...
oemapso 9673 The relation ` T ` is a st...
oemapval 9674 Value of the relation ` T ...
oemapvali 9675 If ` F < G ` , then there ...
cantnflem1a 9676 Lemma for ~ cantnf . (Con...
cantnflem1b 9677 Lemma for ~ cantnf . (Con...
cantnflem1c 9678 Lemma for ~ cantnf . (Con...
cantnflem1d 9679 Lemma for ~ cantnf . (Con...
cantnflem1 9680 Lemma for ~ cantnf . This...
cantnflem2 9681 Lemma for ~ cantnf . (Con...
cantnflem3 9682 Lemma for ~ cantnf . Here...
cantnflem4 9683 Lemma for ~ cantnf . Comp...
cantnf 9684 The Cantor Normal Form the...
oemapwe 9685 The lexicographic order on...
cantnffval2 9686 An alternate definition of...
cantnff1o 9687 Simplify the isomorphism o...
wemapwe 9688 Construct lexicographic or...
oef1o 9689 A bijection of the base se...
cnfcomlem 9690 Lemma for ~ cnfcom . (Con...
cnfcom 9691 Any ordinal ` B ` is equin...
cnfcom2lem 9692 Lemma for ~ cnfcom2 . (Co...
cnfcom2 9693 Any nonzero ordinal ` B ` ...
cnfcom3lem 9694 Lemma for ~ cnfcom3 . (Co...
cnfcom3 9695 Any infinite ordinal ` B `...
cnfcom3clem 9696 Lemma for ~ cnfcom3c . (C...
cnfcom3c 9697 Wrap the construction of ~...
ttrcleq 9700 Equality theorem for trans...
nfttrcld 9701 Bound variable hypothesis ...
nfttrcl 9702 Bound variable hypothesis ...
relttrcl 9703 The transitive closure of ...
brttrcl 9704 Characterization of elemen...
brttrcl2 9705 Characterization of elemen...
ssttrcl 9706 If ` R ` is a relation, th...
ttrcltr 9707 The transitive closure of ...
ttrclresv 9708 The transitive closure of ...
ttrclco 9709 Composition law for the tr...
cottrcl 9710 Composition law for the tr...
ttrclss 9711 If ` R ` is a subclass of ...
dmttrcl 9712 The domain of a transitive...
rnttrcl 9713 The range of a transitive ...
ttrclexg 9714 If ` R ` is a set, then so...
dfttrcl2 9715 When ` R ` is a set and a ...
ttrclselem1 9716 Lemma for ~ ttrclse . Sho...
ttrclselem2 9717 Lemma for ~ ttrclse . Sho...
ttrclse 9718 If ` R ` is set-like over ...
trcl 9719 For any set ` A ` , show t...
tz9.1 9720 Every set has a transitive...
tz9.1c 9721 Alternate expression for t...
epfrs 9722 The strong form of the Axi...
zfregs 9723 The strong form of the Axi...
zfregs2 9724 Alternate strong form of t...
setind 9725 Set (epsilon) induction. ...
setind2 9726 Set (epsilon) induction, s...
tcvalg 9729 Value of the transitive cl...
tcid 9730 Defining property of the t...
tctr 9731 Defining property of the t...
tcmin 9732 Defining property of the t...
tc2 9733 A variant of the definitio...
tcsni 9734 The transitive closure of ...
tcss 9735 The transitive closure fun...
tcel 9736 The transitive closure fun...
tcidm 9737 The transitive closure fun...
tc0 9738 The transitive closure of ...
tc00 9739 The transitive closure is ...
frmin 9740 Every (possibly proper) su...
frind 9741 A subclass of a well-found...
frinsg 9742 Well-Founded Induction Sch...
frins 9743 Well-Founded Induction Sch...
frins2f 9744 Well-Founded Induction sch...
frins2 9745 Well-Founded Induction sch...
frins3 9746 Well-Founded Induction sch...
frr3g 9747 Functions defined by well-...
frrlem15 9748 Lemma for general well-fou...
frrlem16 9749 Lemma for general well-fou...
frr1 9750 Law of general well-founde...
frr2 9751 Law of general well-founde...
frr3 9752 Law of general well-founde...
r1funlim 9757 The cumulative hierarchy o...
r1fnon 9758 The cumulative hierarchy o...
r10 9759 Value of the cumulative hi...
r1sucg 9760 Value of the cumulative hi...
r1suc 9761 Value of the cumulative hi...
r1limg 9762 Value of the cumulative hi...
r1lim 9763 Value of the cumulative hi...
r1fin 9764 The first ` _om ` levels o...
r1sdom 9765 Each stage in the cumulati...
r111 9766 The cumulative hierarchy i...
r1tr 9767 The cumulative hierarchy o...
r1tr2 9768 The union of a cumulative ...
r1ordg 9769 Ordering relation for the ...
r1ord3g 9770 Ordering relation for the ...
r1ord 9771 Ordering relation for the ...
r1ord2 9772 Ordering relation for the ...
r1ord3 9773 Ordering relation for the ...
r1sssuc 9774 The value of the cumulativ...
r1pwss 9775 Each set of the cumulative...
r1sscl 9776 Each set of the cumulative...
r1val1 9777 The value of the cumulativ...
tz9.12lem1 9778 Lemma for ~ tz9.12 . (Con...
tz9.12lem2 9779 Lemma for ~ tz9.12 . (Con...
tz9.12lem3 9780 Lemma for ~ tz9.12 . (Con...
tz9.12 9781 A set is well-founded if a...
tz9.13 9782 Every set is well-founded,...
tz9.13g 9783 Every set is well-founded,...
rankwflemb 9784 Two ways of saying a set i...
rankf 9785 The domain and codomain of...
rankon 9786 The rank of a set is an or...
r1elwf 9787 Any member of the cumulati...
rankvalb 9788 Value of the rank function...
rankr1ai 9789 One direction of ~ rankr1a...
rankvaln 9790 Value of the rank function...
rankidb 9791 Identity law for the rank ...
rankdmr1 9792 A rank is a member of the ...
rankr1ag 9793 A version of ~ rankr1a tha...
rankr1bg 9794 A relationship between ran...
r1rankidb 9795 Any set is a subset of the...
r1elssi 9796 The range of the ` R1 ` fu...
r1elss 9797 The range of the ` R1 ` fu...
pwwf 9798 A power set is well-founde...
sswf 9799 A subset of a well-founded...
snwf 9800 A singleton is well-founde...
unwf 9801 A binary union is well-fou...
prwf 9802 An unordered pair is well-...
opwf 9803 An ordered pair is well-fo...
unir1 9804 The cumulative hierarchy o...
jech9.3 9805 Every set belongs to some ...
rankwflem 9806 Every set is well-founded,...
rankval 9807 Value of the rank function...
rankvalg 9808 Value of the rank function...
rankval2 9809 Value of an alternate defi...
uniwf 9810 A union is well-founded if...
rankr1clem 9811 Lemma for ~ rankr1c . (Co...
rankr1c 9812 A relationship between the...
rankidn 9813 A relationship between the...
rankpwi 9814 The rank of a power set. ...
rankelb 9815 The membership relation is...
wfelirr 9816 A well-founded set is not ...
rankval3b 9817 The value of the rank func...
ranksnb 9818 The rank of a singleton. ...
rankonidlem 9819 Lemma for ~ rankonid . (C...
rankonid 9820 The rank of an ordinal num...
onwf 9821 The ordinals are all well-...
onssr1 9822 Initial segments of the or...
rankr1g 9823 A relationship between the...
rankid 9824 Identity law for the rank ...
rankr1 9825 A relationship between the...
ssrankr1 9826 A relationship between an ...
rankr1a 9827 A relationship between ran...
r1val2 9828 The value of the cumulativ...
r1val3 9829 The value of the cumulativ...
rankel 9830 The membership relation is...
rankval3 9831 The value of the rank func...
bndrank 9832 Any class whose elements h...
unbndrank 9833 The elements of a proper c...
rankpw 9834 The rank of a power set. ...
ranklim 9835 The rank of a set belongs ...
r1pw 9836 A stronger property of ` R...
r1pwALT 9837 Alternate shorter proof of...
r1pwcl 9838 The cumulative hierarchy o...
rankssb 9839 The subset relation is inh...
rankss 9840 The subset relation is inh...
rankunb 9841 The rank of the union of t...
rankprb 9842 The rank of an unordered p...
rankopb 9843 The rank of an ordered pai...
rankuni2b 9844 The value of the rank func...
ranksn 9845 The rank of a singleton. ...
rankuni2 9846 The rank of a union. Part...
rankun 9847 The rank of the union of t...
rankpr 9848 The rank of an unordered p...
rankop 9849 The rank of an ordered pai...
r1rankid 9850 Any set is a subset of the...
rankeq0b 9851 A set is empty iff its ran...
rankeq0 9852 A set is empty iff its ran...
rankr1id 9853 The rank of the hierarchy ...
rankuni 9854 The rank of a union. Part...
rankr1b 9855 A relationship between ran...
ranksuc 9856 The rank of a successor. ...
rankuniss 9857 Upper bound of the rank of...
rankval4 9858 The rank of a set is the s...
rankbnd 9859 The rank of a set is bound...
rankbnd2 9860 The rank of a set is bound...
rankc1 9861 A relationship that can be...
rankc2 9862 A relationship that can be...
rankelun 9863 Rank membership is inherit...
rankelpr 9864 Rank membership is inherit...
rankelop 9865 Rank membership is inherit...
rankxpl 9866 A lower bound on the rank ...
rankxpu 9867 An upper bound on the rank...
rankfu 9868 An upper bound on the rank...
rankmapu 9869 An upper bound on the rank...
rankxplim 9870 The rank of a Cartesian pr...
rankxplim2 9871 If the rank of a Cartesian...
rankxplim3 9872 The rank of a Cartesian pr...
rankxpsuc 9873 The rank of a Cartesian pr...
tcwf 9874 The transitive closure fun...
tcrank 9875 This theorem expresses two...
scottex 9876 Scott's trick collects all...
scott0 9877 Scott's trick collects all...
scottexs 9878 Theorem scheme version of ...
scott0s 9879 Theorem scheme version of ...
cplem1 9880 Lemma for the Collection P...
cplem2 9881 Lemma for the Collection P...
cp 9882 Collection Principle. Thi...
bnd 9883 A very strong generalizati...
bnd2 9884 A variant of the Boundedne...
kardex 9885 The collection of all sets...
karden 9886 If we allow the Axiom of R...
htalem 9887 Lemma for defining an emul...
hta 9888 A ZFC emulation of Hilbert...
djueq12 9895 Equality theorem for disjo...
djueq1 9896 Equality theorem for disjo...
djueq2 9897 Equality theorem for disjo...
nfdju 9898 Bound-variable hypothesis ...
djuex 9899 The disjoint union of sets...
djuexb 9900 The disjoint union of two ...
djulcl 9901 Left closure of disjoint u...
djurcl 9902 Right closure of disjoint ...
djulf1o 9903 The left injection functio...
djurf1o 9904 The right injection functi...
inlresf 9905 The left injection restric...
inlresf1 9906 The left injection restric...
inrresf 9907 The right injection restri...
inrresf1 9908 The right injection restri...
djuin 9909 The images of any classes ...
djur 9910 A member of a disjoint uni...
djuss 9911 A disjoint union is a subc...
djuunxp 9912 The union of a disjoint un...
djuexALT 9913 Alternate proof of ~ djuex...
eldju1st 9914 The first component of an ...
eldju2ndl 9915 The second component of an...
eldju2ndr 9916 The second component of an...
djuun 9917 The disjoint union of two ...
1stinl 9918 The first component of the...
2ndinl 9919 The second component of th...
1stinr 9920 The first component of the...
2ndinr 9921 The second component of th...
updjudhf 9922 The mapping of an element ...
updjudhcoinlf 9923 The composition of the map...
updjudhcoinrg 9924 The composition of the map...
updjud 9925 Universal property of the ...
cardf2 9934 The cardinality function i...
cardon 9935 The cardinal number of a s...
isnum2 9936 A way to express well-orde...
isnumi 9937 A set equinumerous to an o...
ennum 9938 Equinumerous sets are equi...
finnum 9939 Every finite set is numera...
onenon 9940 Every ordinal number is nu...
tskwe 9941 A Tarski set is well-order...
xpnum 9942 The cartesian product of n...
cardval3 9943 An alternate definition of...
cardid2 9944 Any numerable set is equin...
isnum3 9945 A set is numerable iff it ...
oncardval 9946 The value of the cardinal ...
oncardid 9947 Any ordinal number is equi...
cardonle 9948 The cardinal of an ordinal...
card0 9949 The cardinality of the emp...
cardidm 9950 The cardinality function i...
oncard 9951 A set is a cardinal number...
ficardom 9952 The cardinal number of a f...
ficardid 9953 A finite set is equinumero...
cardnn 9954 The cardinality of a natur...
cardnueq0 9955 The empty set is the only ...
cardne 9956 No member of a cardinal nu...
carden2a 9957 If two sets have equal non...
carden2b 9958 If two sets are equinumero...
card1 9959 A set has cardinality one ...
cardsn 9960 A singleton has cardinalit...
carddomi2 9961 Two sets have the dominanc...
sdomsdomcardi 9962 A set strictly dominates i...
cardlim 9963 An infinite cardinal is a ...
cardsdomelir 9964 A cardinal strictly domina...
cardsdomel 9965 A cardinal strictly domina...
iscard 9966 Two ways to express the pr...
iscard2 9967 Two ways to express the pr...
carddom2 9968 Two numerable sets have th...
harcard 9969 The class of ordinal numbe...
cardprclem 9970 Lemma for ~ cardprc . (Co...
cardprc 9971 The class of all cardinal ...
carduni 9972 The union of a set of card...
cardiun 9973 The indexed union of a set...
cardennn 9974 If ` A ` is equinumerous t...
cardsucinf 9975 The cardinality of the suc...
cardsucnn 9976 The cardinality of the suc...
cardom 9977 The set of natural numbers...
carden2 9978 Two numerable sets are equ...
cardsdom2 9979 A numerable set is strictl...
domtri2 9980 Trichotomy of dominance fo...
nnsdomel 9981 Strict dominance and eleme...
cardval2 9982 An alternate version of th...
isinffi 9983 An infinite set contains s...
fidomtri 9984 Trichotomy of dominance wi...
fidomtri2 9985 Trichotomy of dominance wi...
harsdom 9986 The Hartogs number of a we...
onsdom 9987 Any well-orderable set is ...
harval2 9988 An alternate expression fo...
harsucnn 9989 The next cardinal after a ...
cardmin2 9990 The smallest ordinal that ...
pm54.43lem 9991 In Theorem *54.43 of [Whit...
pm54.43 9992 Theorem *54.43 of [Whitehe...
enpr2 9993 An unordered pair with dis...
pr2nelemOLD 9994 Obsolete version of ~ enpr...
pr2ne 9995 If an unordered pair has t...
pr2neOLD 9996 Obsolete version of ~ pr2n...
prdom2 9997 An unordered pair has at m...
en2eqpr 9998 Building a set with two el...
en2eleq 9999 Express a set of pair card...
en2other2 10000 Taking the other element t...
dif1card 10001 The cardinality of a nonem...
leweon 10002 Lexicographical order is a...
r0weon 10003 A set-like well-ordering o...
infxpenlem 10004 Lemma for ~ infxpen . (Co...
infxpen 10005 Every infinite ordinal is ...
xpomen 10006 The Cartesian product of o...
xpct 10007 The cartesian product of t...
infxpidm2 10008 Every infinite well-ordera...
infxpenc 10009 A canonical version of ~ i...
infxpenc2lem1 10010 Lemma for ~ infxpenc2 . (...
infxpenc2lem2 10011 Lemma for ~ infxpenc2 . (...
infxpenc2lem3 10012 Lemma for ~ infxpenc2 . (...
infxpenc2 10013 Existence form of ~ infxpe...
iunmapdisj 10014 The union ` U_ n e. C ( A ...
fseqenlem1 10015 Lemma for ~ fseqen . (Con...
fseqenlem2 10016 Lemma for ~ fseqen . (Con...
fseqdom 10017 One half of ~ fseqen . (C...
fseqen 10018 A set that is equinumerous...
infpwfidom 10019 The collection of finite s...
dfac8alem 10020 Lemma for ~ dfac8a . If t...
dfac8a 10021 Numeration theorem: every ...
dfac8b 10022 The well-ordering theorem:...
dfac8clem 10023 Lemma for ~ dfac8c . (Con...
dfac8c 10024 If the union of a set is w...
ac10ct 10025 A proof of the well-orderi...
ween 10026 A set is numerable iff it ...
ac5num 10027 A version of ~ ac5b with t...
ondomen 10028 If a set is dominated by a...
numdom 10029 A set dominated by a numer...
ssnum 10030 A subset of a numerable se...
onssnum 10031 All subsets of the ordinal...
indcardi 10032 Indirect strong induction ...
acnrcl 10033 Reverse closure for the ch...
acneq 10034 Equality theorem for the c...
isacn 10035 The property of being a ch...
acni 10036 The property of being a ch...
acni2 10037 The property of being a ch...
acni3 10038 The property of being a ch...
acnlem 10039 Construct a mapping satisf...
numacn 10040 A well-orderable set has c...
finacn 10041 Every set has finite choic...
acndom 10042 A set with long choice seq...
acnnum 10043 A set ` X ` which has choi...
acnen 10044 The class of choice sets o...
acndom2 10045 A set smaller than one wit...
acnen2 10046 The class of sets with cho...
fodomacn 10047 A version of ~ fodom that ...
fodomnum 10048 A version of ~ fodom that ...
fonum 10049 A surjection maps numerabl...
numwdom 10050 A surjection maps numerabl...
fodomfi2 10051 Onto functions define domi...
wdomfil 10052 Weak dominance agrees with...
infpwfien 10053 Any infinite well-orderabl...
inffien 10054 The set of finite intersec...
wdomnumr 10055 Weak dominance agrees with...
alephfnon 10056 The aleph function is a fu...
aleph0 10057 The first infinite cardina...
alephlim 10058 Value of the aleph functio...
alephsuc 10059 Value of the aleph functio...
alephon 10060 An aleph is an ordinal num...
alephcard 10061 Every aleph is a cardinal ...
alephnbtwn 10062 No cardinal can be sandwic...
alephnbtwn2 10063 No set has equinumerosity ...
alephordilem1 10064 Lemma for ~ alephordi . (...
alephordi 10065 Strict ordering property o...
alephord 10066 Ordering property of the a...
alephord2 10067 Ordering property of the a...
alephord2i 10068 Ordering property of the a...
alephord3 10069 Ordering property of the a...
alephsucdom 10070 A set dominated by an alep...
alephsuc2 10071 An alternate representatio...
alephdom 10072 Relationship between inclu...
alephgeom 10073 Every aleph is greater tha...
alephislim 10074 Every aleph is a limit ord...
aleph11 10075 The aleph function is one-...
alephf1 10076 The aleph function is a on...
alephsdom 10077 If an ordinal is smaller t...
alephdom2 10078 A dominated initial ordina...
alephle 10079 The argument of the aleph ...
cardaleph 10080 Given any transfinite card...
cardalephex 10081 Every transfinite cardinal...
infenaleph 10082 An infinite numerable set ...
isinfcard 10083 Two ways to express the pr...
iscard3 10084 Two ways to express the pr...
cardnum 10085 Two ways to express the cl...
alephinit 10086 An infinite initial ordina...
carduniima 10087 The union of the image of ...
cardinfima 10088 If a mapping to cardinals ...
alephiso 10089 Aleph is an order isomorph...
alephprc 10090 The class of all transfini...
alephsson 10091 The class of transfinite c...
unialeph 10092 The union of the class of ...
alephsmo 10093 The aleph function is stri...
alephf1ALT 10094 Alternate proof of ~ aleph...
alephfplem1 10095 Lemma for ~ alephfp . (Co...
alephfplem2 10096 Lemma for ~ alephfp . (Co...
alephfplem3 10097 Lemma for ~ alephfp . (Co...
alephfplem4 10098 Lemma for ~ alephfp . (Co...
alephfp 10099 The aleph function has a f...
alephfp2 10100 The aleph function has at ...
alephval3 10101 An alternate way to expres...
alephsucpw2 10102 The power set of an aleph ...
mappwen 10103 Power rule for cardinal ar...
finnisoeu 10104 A finite totally ordered s...
iunfictbso 10105 Countability of a countabl...
aceq1 10108 Equivalence of two version...
aceq0 10109 Equivalence of two version...
aceq2 10110 Equivalence of two version...
aceq3lem 10111 Lemma for ~ dfac3 . (Cont...
dfac3 10112 Equivalence of two version...
dfac4 10113 Equivalence of two version...
dfac5lem1 10114 Lemma for ~ dfac5 . (Cont...
dfac5lem2 10115 Lemma for ~ dfac5 . (Cont...
dfac5lem3 10116 Lemma for ~ dfac5 . (Cont...
dfac5lem4 10117 Lemma for ~ dfac5 . (Cont...
dfac5lem5 10118 Lemma for ~ dfac5 . (Cont...
dfac5 10119 Equivalence of two version...
dfac2a 10120 Our Axiom of Choice (in th...
dfac2b 10121 Axiom of Choice (first for...
dfac2 10122 Axiom of Choice (first for...
dfac7 10123 Equivalence of the Axiom o...
dfac0 10124 Equivalence of two version...
dfac1 10125 Equivalence of two version...
dfac8 10126 A proof of the equivalency...
dfac9 10127 Equivalence of the axiom o...
dfac10 10128 Axiom of Choice equivalent...
dfac10c 10129 Axiom of Choice equivalent...
dfac10b 10130 Axiom of Choice equivalent...
acacni 10131 A choice equivalent: every...
dfacacn 10132 A choice equivalent: every...
dfac13 10133 The axiom of choice holds ...
dfac12lem1 10134 Lemma for ~ dfac12 . (Con...
dfac12lem2 10135 Lemma for ~ dfac12 . (Con...
dfac12lem3 10136 Lemma for ~ dfac12 . (Con...
dfac12r 10137 The axiom of choice holds ...
dfac12k 10138 Equivalence of ~ dfac12 an...
dfac12a 10139 The axiom of choice holds ...
dfac12 10140 The axiom of choice holds ...
kmlem1 10141 Lemma for 5-quantifier AC ...
kmlem2 10142 Lemma for 5-quantifier AC ...
kmlem3 10143 Lemma for 5-quantifier AC ...
kmlem4 10144 Lemma for 5-quantifier AC ...
kmlem5 10145 Lemma for 5-quantifier AC ...
kmlem6 10146 Lemma for 5-quantifier AC ...
kmlem7 10147 Lemma for 5-quantifier AC ...
kmlem8 10148 Lemma for 5-quantifier AC ...
kmlem9 10149 Lemma for 5-quantifier AC ...
kmlem10 10150 Lemma for 5-quantifier AC ...
kmlem11 10151 Lemma for 5-quantifier AC ...
kmlem12 10152 Lemma for 5-quantifier AC ...
kmlem13 10153 Lemma for 5-quantifier AC ...
kmlem14 10154 Lemma for 5-quantifier AC ...
kmlem15 10155 Lemma for 5-quantifier AC ...
kmlem16 10156 Lemma for 5-quantifier AC ...
dfackm 10157 Equivalence of the Axiom o...
undjudom 10158 Cardinal addition dominate...
endjudisj 10159 Equinumerosity of a disjoi...
djuen 10160 Disjoint unions of equinum...
djuenun 10161 Disjoint union is equinume...
dju1en 10162 Cardinal addition with car...
dju1dif 10163 Adding and subtracting one...
dju1p1e2 10164 1+1=2 for cardinal number ...
dju1p1e2ALT 10165 Alternate proof of ~ dju1p...
dju0en 10166 Cardinal addition with car...
xp2dju 10167 Two times a cardinal numbe...
djucomen 10168 Commutative law for cardin...
djuassen 10169 Associative law for cardin...
xpdjuen 10170 Cardinal multiplication di...
mapdjuen 10171 Sum of exponents law for c...
pwdjuen 10172 Sum of exponents law for c...
djudom1 10173 Ordering law for cardinal ...
djudom2 10174 Ordering law for cardinal ...
djudoml 10175 A set is dominated by its ...
djuxpdom 10176 Cartesian product dominate...
djufi 10177 The disjoint union of two ...
cdainflem 10178 Any partition of omega int...
djuinf 10179 A set is infinite iff the ...
infdju1 10180 An infinite set is equinum...
pwdju1 10181 The sum of a powerset with...
pwdjuidm 10182 If the natural numbers inj...
djulepw 10183 If ` A ` is idempotent und...
onadju 10184 The cardinal and ordinal s...
cardadju 10185 The cardinal sum is equinu...
djunum 10186 The disjoint union of two ...
unnum 10187 The union of two numerable...
nnadju 10188 The cardinal and ordinal s...
nnadjuALT 10189 Shorter proof of ~ nnadju ...
ficardadju 10190 The disjoint union of fini...
ficardun 10191 The cardinality of the uni...
ficardunOLD 10192 Obsolete version of ~ fica...
ficardun2 10193 The cardinality of the uni...
ficardun2OLD 10194 Obsolete version of ~ fica...
pwsdompw 10195 Lemma for ~ domtriom . Th...
unctb 10196 The union of two countable...
infdjuabs 10197 Absorption law for additio...
infunabs 10198 An infinite set is equinum...
infdju 10199 The sum of two cardinal nu...
infdif 10200 The cardinality of an infi...
infdif2 10201 Cardinality ordering for a...
infxpdom 10202 Dominance law for multipli...
infxpabs 10203 Absorption law for multipl...
infunsdom1 10204 The union of two sets that...
infunsdom 10205 The union of two sets that...
infxp 10206 Absorption law for multipl...
pwdjudom 10207 A property of dominance ov...
infpss 10208 Every infinite set has an ...
infmap2 10209 An exponentiation law for ...
ackbij2lem1 10210 Lemma for ~ ackbij2 . (Co...
ackbij1lem1 10211 Lemma for ~ ackbij2 . (Co...
ackbij1lem2 10212 Lemma for ~ ackbij2 . (Co...
ackbij1lem3 10213 Lemma for ~ ackbij2 . (Co...
ackbij1lem4 10214 Lemma for ~ ackbij2 . (Co...
ackbij1lem5 10215 Lemma for ~ ackbij2 . (Co...
ackbij1lem6 10216 Lemma for ~ ackbij2 . (Co...
ackbij1lem7 10217 Lemma for ~ ackbij1 . (Co...
ackbij1lem8 10218 Lemma for ~ ackbij1 . (Co...
ackbij1lem9 10219 Lemma for ~ ackbij1 . (Co...
ackbij1lem10 10220 Lemma for ~ ackbij1 . (Co...
ackbij1lem11 10221 Lemma for ~ ackbij1 . (Co...
ackbij1lem12 10222 Lemma for ~ ackbij1 . (Co...
ackbij1lem13 10223 Lemma for ~ ackbij1 . (Co...
ackbij1lem14 10224 Lemma for ~ ackbij1 . (Co...
ackbij1lem15 10225 Lemma for ~ ackbij1 . (Co...
ackbij1lem16 10226 Lemma for ~ ackbij1 . (Co...
ackbij1lem17 10227 Lemma for ~ ackbij1 . (Co...
ackbij1lem18 10228 Lemma for ~ ackbij1 . (Co...
ackbij1 10229 The Ackermann bijection, p...
ackbij1b 10230 The Ackermann bijection, p...
ackbij2lem2 10231 Lemma for ~ ackbij2 . (Co...
ackbij2lem3 10232 Lemma for ~ ackbij2 . (Co...
ackbij2lem4 10233 Lemma for ~ ackbij2 . (Co...
ackbij2 10234 The Ackermann bijection, p...
r1om 10235 The set of hereditarily fi...
fictb 10236 A set is countable iff its...
cflem 10237 A lemma used to simplify c...
cfval 10238 Value of the cofinality fu...
cff 10239 Cofinality is a function o...
cfub 10240 An upper bound on cofinali...
cflm 10241 Value of the cofinality fu...
cf0 10242 Value of the cofinality fu...
cardcf 10243 Cofinality is a cardinal n...
cflecard 10244 Cofinality is bounded by t...
cfle 10245 Cofinality is bounded by i...
cfon 10246 The cofinality of any set ...
cfeq0 10247 Only the ordinal zero has ...
cfsuc 10248 Value of the cofinality fu...
cff1 10249 There is always a map from...
cfflb 10250 If there is a cofinal map ...
cfval2 10251 Another expression for the...
coflim 10252 A simpler expression for t...
cflim3 10253 Another expression for the...
cflim2 10254 The cofinality function is...
cfom 10255 Value of the cofinality fu...
cfss 10256 There is a cofinal subset ...
cfslb 10257 Any cofinal subset of ` A ...
cfslbn 10258 Any subset of ` A ` smalle...
cfslb2n 10259 Any small collection of sm...
cofsmo 10260 Any cofinal map implies th...
cfsmolem 10261 Lemma for ~ cfsmo . (Cont...
cfsmo 10262 The map in ~ cff1 can be a...
cfcoflem 10263 Lemma for ~ cfcof , showin...
coftr 10264 If there is a cofinal map ...
cfcof 10265 If there is a cofinal map ...
cfidm 10266 The cofinality function is...
alephsing 10267 The cofinality of a limit ...
sornom 10268 The range of a single-step...
isfin1a 10283 Definition of a Ia-finite ...
fin1ai 10284 Property of a Ia-finite se...
isfin2 10285 Definition of a II-finite ...
fin2i 10286 Property of a II-finite se...
isfin3 10287 Definition of a III-finite...
isfin4 10288 Definition of a IV-finite ...
fin4i 10289 Infer that a set is IV-inf...
isfin5 10290 Definition of a V-finite s...
isfin6 10291 Definition of a VI-finite ...
isfin7 10292 Definition of a VII-finite...
sdom2en01 10293 A set with less than two e...
infpssrlem1 10294 Lemma for ~ infpssr . (Co...
infpssrlem2 10295 Lemma for ~ infpssr . (Co...
infpssrlem3 10296 Lemma for ~ infpssr . (Co...
infpssrlem4 10297 Lemma for ~ infpssr . (Co...
infpssrlem5 10298 Lemma for ~ infpssr . (Co...
infpssr 10299 Dedekind infinity implies ...
fin4en1 10300 Dedekind finite is a cardi...
ssfin4 10301 Dedekind finite sets have ...
domfin4 10302 A set dominated by a Dedek...
ominf4 10303 ` _om ` is Dedekind infini...
infpssALT 10304 Alternate proof of ~ infps...
isfin4-2 10305 Alternate definition of IV...
isfin4p1 10306 Alternate definition of IV...
fin23lem7 10307 Lemma for ~ isfin2-2 . Th...
fin23lem11 10308 Lemma for ~ isfin2-2 . (C...
fin2i2 10309 A II-finite set contains m...
isfin2-2 10310 ` Fin2 ` expressed in term...
ssfin2 10311 A subset of a II-finite se...
enfin2i 10312 II-finiteness is a cardina...
fin23lem24 10313 Lemma for ~ fin23 . In a ...
fincssdom 10314 In a chain of finite sets,...
fin23lem25 10315 Lemma for ~ fin23 . In a ...
fin23lem26 10316 Lemma for ~ fin23lem22 . ...
fin23lem23 10317 Lemma for ~ fin23lem22 . ...
fin23lem22 10318 Lemma for ~ fin23 but coul...
fin23lem27 10319 The mapping constructed in...
isfin3ds 10320 Property of a III-finite s...
ssfin3ds 10321 A subset of a III-finite s...
fin23lem12 10322 The beginning of the proof...
fin23lem13 10323 Lemma for ~ fin23 . Each ...
fin23lem14 10324 Lemma for ~ fin23 . ` U ` ...
fin23lem15 10325 Lemma for ~ fin23 . ` U ` ...
fin23lem16 10326 Lemma for ~ fin23 . ` U ` ...
fin23lem19 10327 Lemma for ~ fin23 . The f...
fin23lem20 10328 Lemma for ~ fin23 . ` X ` ...
fin23lem17 10329 Lemma for ~ fin23 . By ? ...
fin23lem21 10330 Lemma for ~ fin23 . ` X ` ...
fin23lem28 10331 Lemma for ~ fin23 . The r...
fin23lem29 10332 Lemma for ~ fin23 . The r...
fin23lem30 10333 Lemma for ~ fin23 . The r...
fin23lem31 10334 Lemma for ~ fin23 . The r...
fin23lem32 10335 Lemma for ~ fin23 . Wrap ...
fin23lem33 10336 Lemma for ~ fin23 . Disch...
fin23lem34 10337 Lemma for ~ fin23 . Estab...
fin23lem35 10338 Lemma for ~ fin23 . Stric...
fin23lem36 10339 Lemma for ~ fin23 . Weak ...
fin23lem38 10340 Lemma for ~ fin23 . The c...
fin23lem39 10341 Lemma for ~ fin23 . Thus,...
fin23lem40 10342 Lemma for ~ fin23 . ` Fin2...
fin23lem41 10343 Lemma for ~ fin23 . A set...
isf32lem1 10344 Lemma for ~ isfin3-2 . De...
isf32lem2 10345 Lemma for ~ isfin3-2 . No...
isf32lem3 10346 Lemma for ~ isfin3-2 . Be...
isf32lem4 10347 Lemma for ~ isfin3-2 . Be...
isf32lem5 10348 Lemma for ~ isfin3-2 . Th...
isf32lem6 10349 Lemma for ~ isfin3-2 . Ea...
isf32lem7 10350 Lemma for ~ isfin3-2 . Di...
isf32lem8 10351 Lemma for ~ isfin3-2 . K ...
isf32lem9 10352 Lemma for ~ isfin3-2 . Co...
isf32lem10 10353 Lemma for isfin3-2 . Writ...
isf32lem11 10354 Lemma for ~ isfin3-2 . Re...
isf32lem12 10355 Lemma for ~ isfin3-2 . (C...
isfin32i 10356 One half of ~ isfin3-2 . ...
isf33lem 10357 Lemma for ~ isfin3-3 . (C...
isfin3-2 10358 Weakly Dedekind-infinite s...
isfin3-3 10359 Weakly Dedekind-infinite s...
fin33i 10360 Inference from ~ isfin3-3 ...
compsscnvlem 10361 Lemma for ~ compsscnv . (...
compsscnv 10362 Complementation on a power...
isf34lem1 10363 Lemma for ~ isfin3-4 . (C...
isf34lem2 10364 Lemma for ~ isfin3-4 . (C...
compssiso 10365 Complementation is an anti...
isf34lem3 10366 Lemma for ~ isfin3-4 . (C...
compss 10367 Express image under of the...
isf34lem4 10368 Lemma for ~ isfin3-4 . (C...
isf34lem5 10369 Lemma for ~ isfin3-4 . (C...
isf34lem7 10370 Lemma for ~ isfin3-4 . (C...
isf34lem6 10371 Lemma for ~ isfin3-4 . (C...
fin34i 10372 Inference from ~ isfin3-4 ...
isfin3-4 10373 Weakly Dedekind-infinite s...
fin11a 10374 Every I-finite set is Ia-f...
enfin1ai 10375 Ia-finiteness is a cardina...
isfin1-2 10376 A set is finite in the usu...
isfin1-3 10377 A set is I-finite iff ever...
isfin1-4 10378 A set is I-finite iff ever...
dffin1-5 10379 Compact quantifier-free ve...
fin23 10380 Every II-finite set (every...
fin34 10381 Every III-finite set is IV...
isfin5-2 10382 Alternate definition of V-...
fin45 10383 Every IV-finite set is V-f...
fin56 10384 Every V-finite set is VI-f...
fin17 10385 Every I-finite set is VII-...
fin67 10386 Every VI-finite set is VII...
isfin7-2 10387 A set is VII-finite iff it...
fin71num 10388 A well-orderable set is VI...
dffin7-2 10389 Class form of ~ isfin7-2 ....
dfacfin7 10390 Axiom of Choice equivalent...
fin1a2lem1 10391 Lemma for ~ fin1a2 . (Con...
fin1a2lem2 10392 Lemma for ~ fin1a2 . The ...
fin1a2lem3 10393 Lemma for ~ fin1a2 . (Con...
fin1a2lem4 10394 Lemma for ~ fin1a2 . (Con...
fin1a2lem5 10395 Lemma for ~ fin1a2 . (Con...
fin1a2lem6 10396 Lemma for ~ fin1a2 . Esta...
fin1a2lem7 10397 Lemma for ~ fin1a2 . Spli...
fin1a2lem8 10398 Lemma for ~ fin1a2 . Spli...
fin1a2lem9 10399 Lemma for ~ fin1a2 . In a...
fin1a2lem10 10400 Lemma for ~ fin1a2 . A no...
fin1a2lem11 10401 Lemma for ~ fin1a2 . (Con...
fin1a2lem12 10402 Lemma for ~ fin1a2 . (Con...
fin1a2lem13 10403 Lemma for ~ fin1a2 . (Con...
fin12 10404 Weak theorem which skips I...
fin1a2s 10405 An II-infinite set can hav...
fin1a2 10406 Every Ia-finite set is II-...
itunifval 10407 Function value of iterated...
itunifn 10408 Functionality of the itera...
ituni0 10409 A zero-fold iterated union...
itunisuc 10410 Successor iterated union. ...
itunitc1 10411 Each union iterate is a me...
itunitc 10412 The union of all union ite...
ituniiun 10413 Unwrap an iterated union f...
hsmexlem7 10414 Lemma for ~ hsmex . Prope...
hsmexlem8 10415 Lemma for ~ hsmex . Prope...
hsmexlem9 10416 Lemma for ~ hsmex . Prope...
hsmexlem1 10417 Lemma for ~ hsmex . Bound...
hsmexlem2 10418 Lemma for ~ hsmex . Bound...
hsmexlem3 10419 Lemma for ~ hsmex . Clear...
hsmexlem4 10420 Lemma for ~ hsmex . The c...
hsmexlem5 10421 Lemma for ~ hsmex . Combi...
hsmexlem6 10422 Lemma for ~ hsmex . (Cont...
hsmex 10423 The collection of heredita...
hsmex2 10424 The set of hereditary size...
hsmex3 10425 The set of hereditary size...
axcc2lem 10427 Lemma for ~ axcc2 . (Cont...
axcc2 10428 A possibly more useful ver...
axcc3 10429 A possibly more useful ver...
axcc4 10430 A version of ~ axcc3 that ...
acncc 10431 An ~ ax-cc equivalent: eve...
axcc4dom 10432 Relax the constraint on ~ ...
domtriomlem 10433 Lemma for ~ domtriom . (C...
domtriom 10434 Trichotomy of equinumerosi...
fin41 10435 Under countable choice, th...
dominf 10436 A nonempty set that is a s...
dcomex 10438 The Axiom of Dependent Cho...
axdc2lem 10439 Lemma for ~ axdc2 . We co...
axdc2 10440 An apparent strengthening ...
axdc3lem 10441 The class ` S ` of finite ...
axdc3lem2 10442 Lemma for ~ axdc3 . We ha...
axdc3lem3 10443 Simple substitution lemma ...
axdc3lem4 10444 Lemma for ~ axdc3 . We ha...
axdc3 10445 Dependent Choice. Axiom D...
axdc4lem 10446 Lemma for ~ axdc4 . (Cont...
axdc4 10447 A more general version of ...
axcclem 10448 Lemma for ~ axcc . (Contr...
axcc 10449 Although CC can be proven ...
zfac 10451 Axiom of Choice expressed ...
ac2 10452 Axiom of Choice equivalent...
ac3 10453 Axiom of Choice using abbr...
axac3 10455 This theorem asserts that ...
ackm 10456 A remarkable equivalent to...
axac2 10457 Derive ~ ax-ac2 from ~ ax-...
axac 10458 Derive ~ ax-ac from ~ ax-a...
axaci 10459 Apply a choice equivalent....
cardeqv 10460 All sets are well-orderabl...
numth3 10461 All sets are well-orderabl...
numth2 10462 Numeration theorem: any se...
numth 10463 Numeration theorem: every ...
ac7 10464 An Axiom of Choice equival...
ac7g 10465 An Axiom of Choice equival...
ac4 10466 Equivalent of Axiom of Cho...
ac4c 10467 Equivalent of Axiom of Cho...
ac5 10468 An Axiom of Choice equival...
ac5b 10469 Equivalent of Axiom of Cho...
ac6num 10470 A version of ~ ac6 which t...
ac6 10471 Equivalent of Axiom of Cho...
ac6c4 10472 Equivalent of Axiom of Cho...
ac6c5 10473 Equivalent of Axiom of Cho...
ac9 10474 An Axiom of Choice equival...
ac6s 10475 Equivalent of Axiom of Cho...
ac6n 10476 Equivalent of Axiom of Cho...
ac6s2 10477 Generalization of the Axio...
ac6s3 10478 Generalization of the Axio...
ac6sg 10479 ~ ac6s with sethood as ant...
ac6sf 10480 Version of ~ ac6 with boun...
ac6s4 10481 Generalization of the Axio...
ac6s5 10482 Generalization of the Axio...
ac8 10483 An Axiom of Choice equival...
ac9s 10484 An Axiom of Choice equival...
numthcor 10485 Any set is strictly domina...
weth 10486 Well-ordering theorem: any...
zorn2lem1 10487 Lemma for ~ zorn2 . (Cont...
zorn2lem2 10488 Lemma for ~ zorn2 . (Cont...
zorn2lem3 10489 Lemma for ~ zorn2 . (Cont...
zorn2lem4 10490 Lemma for ~ zorn2 . (Cont...
zorn2lem5 10491 Lemma for ~ zorn2 . (Cont...
zorn2lem6 10492 Lemma for ~ zorn2 . (Cont...
zorn2lem7 10493 Lemma for ~ zorn2 . (Cont...
zorn2g 10494 Zorn's Lemma of [Monk1] p....
zorng 10495 Zorn's Lemma. If the unio...
zornn0g 10496 Variant of Zorn's lemma ~ ...
zorn2 10497 Zorn's Lemma of [Monk1] p....
zorn 10498 Zorn's Lemma. If the unio...
zornn0 10499 Variant of Zorn's lemma ~ ...
ttukeylem1 10500 Lemma for ~ ttukey . Expa...
ttukeylem2 10501 Lemma for ~ ttukey . A pr...
ttukeylem3 10502 Lemma for ~ ttukey . (Con...
ttukeylem4 10503 Lemma for ~ ttukey . (Con...
ttukeylem5 10504 Lemma for ~ ttukey . The ...
ttukeylem6 10505 Lemma for ~ ttukey . (Con...
ttukeylem7 10506 Lemma for ~ ttukey . (Con...
ttukey2g 10507 The Teichmüller-Tukey...
ttukeyg 10508 The Teichmüller-Tukey...
ttukey 10509 The Teichmüller-Tukey...
axdclem 10510 Lemma for ~ axdc . (Contr...
axdclem2 10511 Lemma for ~ axdc . Using ...
axdc 10512 This theorem derives ~ ax-...
fodomg 10513 An onto function implies d...
fodom 10514 An onto function implies d...
dmct 10515 The domain of a countable ...
rnct 10516 The range of a countable s...
fodomb 10517 Equivalence of an onto map...
wdomac 10518 When assuming AC, weak and...
brdom3 10519 Equivalence to a dominance...
brdom5 10520 An equivalence to a domina...
brdom4 10521 An equivalence to a domina...
brdom7disj 10522 An equivalence to a domina...
brdom6disj 10523 An equivalence to a domina...
fin71ac 10524 Once we allow AC, the "str...
imadomg 10525 An image of a function und...
fimact 10526 The image by a function of...
fnrndomg 10527 The range of a function is...
fnct 10528 If the domain of a functio...
mptct 10529 A countable mapping set is...
iunfo 10530 Existence of an onto funct...
iundom2g 10531 An upper bound for the car...
iundomg 10532 An upper bound for the car...
iundom 10533 An upper bound for the car...
unidom 10534 An upper bound for the car...
uniimadom 10535 An upper bound for the car...
uniimadomf 10536 An upper bound for the car...
cardval 10537 The value of the cardinal ...
cardid 10538 Any set is equinumerous to...
cardidg 10539 Any set is equinumerous to...
cardidd 10540 Any set is equinumerous to...
cardf 10541 The cardinality function i...
carden 10542 Two sets are equinumerous ...
cardeq0 10543 Only the empty set has car...
unsnen 10544 Equinumerosity of a set wi...
carddom 10545 Two sets have the dominanc...
cardsdom 10546 Two sets have the strict d...
domtri 10547 Trichotomy law for dominan...
entric 10548 Trichotomy of equinumerosi...
entri2 10549 Trichotomy of dominance an...
entri3 10550 Trichotomy of dominance. ...
sdomsdomcard 10551 A set strictly dominates i...
canth3 10552 Cantor's theorem in terms ...
infxpidm 10553 Every infinite class is eq...
ondomon 10554 The class of ordinals domi...
cardmin 10555 The smallest ordinal that ...
ficard 10556 A set is finite iff its ca...
infinf 10557 Equivalence between two in...
unirnfdomd 10558 The union of the range of ...
konigthlem 10559 Lemma for ~ konigth . (Co...
konigth 10560 Konig's Theorem. If ` m (...
alephsucpw 10561 The power set of an aleph ...
aleph1 10562 The set exponentiation of ...
alephval2 10563 An alternate way to expres...
dominfac 10564 A nonempty set that is a s...
iunctb 10565 The countable union of cou...
unictb 10566 The countable union of cou...
infmap 10567 An exponentiation law for ...
alephadd 10568 The sum of two alephs is t...
alephmul 10569 The product of two alephs ...
alephexp1 10570 An exponentiation law for ...
alephsuc3 10571 An alternate representatio...
alephexp2 10572 An expression equinumerous...
alephreg 10573 A successor aleph is regul...
pwcfsdom 10574 A corollary of Konig's The...
cfpwsdom 10575 A corollary of Konig's The...
alephom 10576 From ~ canth2 , we know th...
smobeth 10577 The beth function is stric...
nd1 10578 A lemma for proving condit...
nd2 10579 A lemma for proving condit...
nd3 10580 A lemma for proving condit...
nd4 10581 A lemma for proving condit...
axextnd 10582 A version of the Axiom of ...
axrepndlem1 10583 Lemma for the Axiom of Rep...
axrepndlem2 10584 Lemma for the Axiom of Rep...
axrepnd 10585 A version of the Axiom of ...
axunndlem1 10586 Lemma for the Axiom of Uni...
axunnd 10587 A version of the Axiom of ...
axpowndlem1 10588 Lemma for the Axiom of Pow...
axpowndlem2 10589 Lemma for the Axiom of Pow...
axpowndlem3 10590 Lemma for the Axiom of Pow...
axpowndlem4 10591 Lemma for the Axiom of Pow...
axpownd 10592 A version of the Axiom of ...
axregndlem1 10593 Lemma for the Axiom of Reg...
axregndlem2 10594 Lemma for the Axiom of Reg...
axregnd 10595 A version of the Axiom of ...
axinfndlem1 10596 Lemma for the Axiom of Inf...
axinfnd 10597 A version of the Axiom of ...
axacndlem1 10598 Lemma for the Axiom of Cho...
axacndlem2 10599 Lemma for the Axiom of Cho...
axacndlem3 10600 Lemma for the Axiom of Cho...
axacndlem4 10601 Lemma for the Axiom of Cho...
axacndlem5 10602 Lemma for the Axiom of Cho...
axacnd 10603 A version of the Axiom of ...
zfcndext 10604 Axiom of Extensionality ~ ...
zfcndrep 10605 Axiom of Replacement ~ ax-...
zfcndun 10606 Axiom of Union ~ ax-un , r...
zfcndpow 10607 Axiom of Power Sets ~ ax-p...
zfcndreg 10608 Axiom of Regularity ~ ax-r...
zfcndinf 10609 Axiom of Infinity ~ ax-inf...
zfcndac 10610 Axiom of Choice ~ ax-ac , ...
elgch 10613 Elementhood in the collect...
fingch 10614 A finite set is a GCH-set....
gchi 10615 The only GCH-sets which ha...
gchen1 10616 If ` A <_ B < ~P A ` , and...
gchen2 10617 If ` A < B <_ ~P A ` , and...
gchor 10618 If ` A <_ B <_ ~P A ` , an...
engch 10619 The property of being a GC...
gchdomtri 10620 Under certain conditions, ...
fpwwe2cbv 10621 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem1 10622 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem2 10623 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem3 10624 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem4 10625 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem5 10626 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem6 10627 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem7 10628 Lemma for ~ fpwwe2 . Show...
fpwwe2lem8 10629 Lemma for ~ fpwwe2 . Give...
fpwwe2lem9 10630 Lemma for ~ fpwwe2 . Give...
fpwwe2lem10 10631 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem11 10632 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem12 10633 Lemma for ~ fpwwe2 . (Con...
fpwwe2 10634 Given any function ` F ` f...
fpwwecbv 10635 Lemma for ~ fpwwe . (Cont...
fpwwelem 10636 Lemma for ~ fpwwe . (Cont...
fpwwe 10637 Given any function ` F ` f...
canth4 10638 An "effective" form of Can...
canthnumlem 10639 Lemma for ~ canthnum . (C...
canthnum 10640 The set of well-orderable ...
canthwelem 10641 Lemma for ~ canthwe . (Co...
canthwe 10642 The set of well-orders of ...
canthp1lem1 10643 Lemma for ~ canthp1 . (Co...
canthp1lem2 10644 Lemma for ~ canthp1 . (Co...
canthp1 10645 A slightly stronger form o...
finngch 10646 The exclusion of finite se...
gchdju1 10647 An infinite GCH-set is ide...
gchinf 10648 An infinite GCH-set is Ded...
pwfseqlem1 10649 Lemma for ~ pwfseq . Deri...
pwfseqlem2 10650 Lemma for ~ pwfseq . (Con...
pwfseqlem3 10651 Lemma for ~ pwfseq . Usin...
pwfseqlem4a 10652 Lemma for ~ pwfseqlem4 . ...
pwfseqlem4 10653 Lemma for ~ pwfseq . Deri...
pwfseqlem5 10654 Lemma for ~ pwfseq . Alth...
pwfseq 10655 The powerset of a Dedekind...
pwxpndom2 10656 The powerset of a Dedekind...
pwxpndom 10657 The powerset of a Dedekind...
pwdjundom 10658 The powerset of a Dedekind...
gchdjuidm 10659 An infinite GCH-set is ide...
gchxpidm 10660 An infinite GCH-set is ide...
gchpwdom 10661 A relationship between dom...
gchaleph 10662 If ` ( aleph `` A ) ` is a...
gchaleph2 10663 If ` ( aleph `` A ) ` and ...
hargch 10664 If ` A + ~~ ~P A ` , then ...
alephgch 10665 If ` ( aleph `` suc A ) ` ...
gch2 10666 It is sufficient to requir...
gch3 10667 An equivalent formulation ...
gch-kn 10668 The equivalence of two ver...
gchaclem 10669 Lemma for ~ gchac (obsolet...
gchhar 10670 A "local" form of ~ gchac ...
gchacg 10671 A "local" form of ~ gchac ...
gchac 10672 The Generalized Continuum ...
elwina 10677 Conditions of weak inacces...
elina 10678 Conditions of strong inacc...
winaon 10679 A weakly inaccessible card...
inawinalem 10680 Lemma for ~ inawina . (Co...
inawina 10681 Every strongly inaccessibl...
omina 10682 ` _om ` is a strongly inac...
winacard 10683 A weakly inaccessible card...
winainflem 10684 A weakly inaccessible card...
winainf 10685 A weakly inaccessible card...
winalim 10686 A weakly inaccessible card...
winalim2 10687 A nontrivial weakly inacce...
winafp 10688 A nontrivial weakly inacce...
winafpi 10689 This theorem, which states...
gchina 10690 Assuming the GCH, weakly a...
iswun 10695 Properties of a weak unive...
wuntr 10696 A weak universe is transit...
wununi 10697 A weak universe is closed ...
wunpw 10698 A weak universe is closed ...
wunelss 10699 The elements of a weak uni...
wunpr 10700 A weak universe is closed ...
wunun 10701 A weak universe is closed ...
wuntp 10702 A weak universe is closed ...
wunss 10703 A weak universe is closed ...
wunin 10704 A weak universe is closed ...
wundif 10705 A weak universe is closed ...
wunint 10706 A weak universe is closed ...
wunsn 10707 A weak universe is closed ...
wunsuc 10708 A weak universe is closed ...
wun0 10709 A weak universe contains t...
wunr1om 10710 A weak universe is infinit...
wunom 10711 A weak universe contains a...
wunfi 10712 A weak universe contains a...
wunop 10713 A weak universe is closed ...
wunot 10714 A weak universe is closed ...
wunxp 10715 A weak universe is closed ...
wunpm 10716 A weak universe is closed ...
wunmap 10717 A weak universe is closed ...
wunf 10718 A weak universe is closed ...
wundm 10719 A weak universe is closed ...
wunrn 10720 A weak universe is closed ...
wuncnv 10721 A weak universe is closed ...
wunres 10722 A weak universe is closed ...
wunfv 10723 A weak universe is closed ...
wunco 10724 A weak universe is closed ...
wuntpos 10725 A weak universe is closed ...
intwun 10726 The intersection of a coll...
r1limwun 10727 Each limit stage in the cu...
r1wunlim 10728 The weak universes in the ...
wunex2 10729 Construct a weak universe ...
wunex 10730 Construct a weak universe ...
uniwun 10731 Every set is contained in ...
wunex3 10732 Construct a weak universe ...
wuncval 10733 Value of the weak universe...
wuncid 10734 The weak universe closure ...
wunccl 10735 The weak universe closure ...
wuncss 10736 The weak universe closure ...
wuncidm 10737 The weak universe closure ...
wuncval2 10738 Our earlier expression for...
eltskg 10741 Properties of a Tarski cla...
eltsk2g 10742 Properties of a Tarski cla...
tskpwss 10743 First axiom of a Tarski cl...
tskpw 10744 Second axiom of a Tarski c...
tsken 10745 Third axiom of a Tarski cl...
0tsk 10746 The empty set is a (transi...
tsksdom 10747 An element of a Tarski cla...
tskssel 10748 A part of a Tarski class s...
tskss 10749 The subsets of an element ...
tskin 10750 The intersection of two el...
tsksn 10751 A singleton of an element ...
tsktrss 10752 A transitive element of a ...
tsksuc 10753 If an element of a Tarski ...
tsk0 10754 A nonempty Tarski class co...
tsk1 10755 One is an element of a non...
tsk2 10756 Two is an element of a non...
2domtsk 10757 If a Tarski class is not e...
tskr1om 10758 A nonempty Tarski class is...
tskr1om2 10759 A nonempty Tarski class co...
tskinf 10760 A nonempty Tarski class is...
tskpr 10761 If ` A ` and ` B ` are mem...
tskop 10762 If ` A ` and ` B ` are mem...
tskxpss 10763 A Cartesian product of two...
tskwe2 10764 A Tarski class is well-ord...
inttsk 10765 The intersection of a coll...
inar1 10766 ` ( R1 `` A ) ` for ` A ` ...
r1omALT 10767 Alternate proof of ~ r1om ...
rankcf 10768 Any set must be at least a...
inatsk 10769 ` ( R1 `` A ) ` for ` A ` ...
r1omtsk 10770 The set of hereditarily fi...
tskord 10771 A Tarski class contains al...
tskcard 10772 An even more direct relati...
r1tskina 10773 There is a direct relation...
tskuni 10774 The union of an element of...
tskwun 10775 A nonempty transitive Tars...
tskint 10776 The intersection of an ele...
tskun 10777 The union of two elements ...
tskxp 10778 The Cartesian product of t...
tskmap 10779 Set exponentiation is an e...
tskurn 10780 A transitive Tarski class ...
elgrug 10783 Properties of a Grothendie...
grutr 10784 A Grothendieck universe is...
gruelss 10785 A Grothendieck universe is...
grupw 10786 A Grothendieck universe co...
gruss 10787 Any subset of an element o...
grupr 10788 A Grothendieck universe co...
gruurn 10789 A Grothendieck universe co...
gruiun 10790 If ` B ( x ) ` is a family...
gruuni 10791 A Grothendieck universe co...
grurn 10792 A Grothendieck universe co...
gruima 10793 A Grothendieck universe co...
gruel 10794 Any element of an element ...
grusn 10795 A Grothendieck universe co...
gruop 10796 A Grothendieck universe co...
gruun 10797 A Grothendieck universe co...
gruxp 10798 A Grothendieck universe co...
grumap 10799 A Grothendieck universe co...
gruixp 10800 A Grothendieck universe co...
gruiin 10801 A Grothendieck universe co...
gruf 10802 A Grothendieck universe co...
gruen 10803 A Grothendieck universe co...
gruwun 10804 A nonempty Grothendieck un...
intgru 10805 The intersection of a fami...
ingru 10806 The intersection of a univ...
wfgru 10807 The wellfounded part of a ...
grudomon 10808 Each ordinal that is compa...
gruina 10809 If a Grothendieck universe...
grur1a 10810 A characterization of Grot...
grur1 10811 A characterization of Grot...
grutsk1 10812 Grothendieck universes are...
grutsk 10813 Grothendieck universes are...
axgroth5 10815 The Tarski-Grothendieck ax...
axgroth2 10816 Alternate version of the T...
grothpw 10817 Derive the Axiom of Power ...
grothpwex 10818 Derive the Axiom of Power ...
axgroth6 10819 The Tarski-Grothendieck ax...
grothomex 10820 The Tarski-Grothendieck Ax...
grothac 10821 The Tarski-Grothendieck Ax...
axgroth3 10822 Alternate version of the T...
axgroth4 10823 Alternate version of the T...
grothprimlem 10824 Lemma for ~ grothprim . E...
grothprim 10825 The Tarski-Grothendieck Ax...
grothtsk 10826 The Tarski-Grothendieck Ax...
inaprc 10827 An equivalent to the Tarsk...
tskmval 10830 Value of our tarski map. ...
tskmid 10831 The set ` A ` is an elemen...
tskmcl 10832 A Tarski class that contai...
sstskm 10833 Being a part of ` ( tarski...
eltskm 10834 Belonging to ` ( tarskiMap...
elni 10867 Membership in the class of...
elni2 10868 Membership in the class of...
pinn 10869 A positive integer is a na...
pion 10870 A positive integer is an o...
piord 10871 A positive integer is ordi...
niex 10872 The class of positive inte...
0npi 10873 The empty set is not a pos...
1pi 10874 Ordinal 'one' is a positiv...
addpiord 10875 Positive integer addition ...
mulpiord 10876 Positive integer multiplic...
mulidpi 10877 1 is an identity element f...
ltpiord 10878 Positive integer 'less tha...
ltsopi 10879 Positive integer 'less tha...
ltrelpi 10880 Positive integer 'less tha...
dmaddpi 10881 Domain of addition on posi...
dmmulpi 10882 Domain of multiplication o...
addclpi 10883 Closure of addition of pos...
mulclpi 10884 Closure of multiplication ...
addcompi 10885 Addition of positive integ...
addasspi 10886 Addition of positive integ...
mulcompi 10887 Multiplication of positive...
mulasspi 10888 Multiplication of positive...
distrpi 10889 Multiplication of positive...
addcanpi 10890 Addition cancellation law ...
mulcanpi 10891 Multiplication cancellatio...
addnidpi 10892 There is no identity eleme...
ltexpi 10893 Ordering on positive integ...
ltapi 10894 Ordering property of addit...
ltmpi 10895 Ordering property of multi...
1lt2pi 10896 One is less than two (one ...
nlt1pi 10897 No positive integer is les...
indpi 10898 Principle of Finite Induct...
enqbreq 10910 Equivalence relation for p...
enqbreq2 10911 Equivalence relation for p...
enqer 10912 The equivalence relation f...
enqex 10913 The equivalence relation f...
nqex 10914 The class of positive frac...
0nnq 10915 The empty set is not a pos...
elpqn 10916 Each positive fraction is ...
ltrelnq 10917 Positive fraction 'less th...
pinq 10918 The representatives of pos...
1nq 10919 The positive fraction 'one...
nqereu 10920 There is a unique element ...
nqerf 10921 Corollary of ~ nqereu : th...
nqercl 10922 Corollary of ~ nqereu : cl...
nqerrel 10923 Any member of ` ( N. X. N....
nqerid 10924 Corollary of ~ nqereu : th...
enqeq 10925 Corollary of ~ nqereu : if...
nqereq 10926 The function ` /Q ` acts a...
addpipq2 10927 Addition of positive fract...
addpipq 10928 Addition of positive fract...
addpqnq 10929 Addition of positive fract...
mulpipq2 10930 Multiplication of positive...
mulpipq 10931 Multiplication of positive...
mulpqnq 10932 Multiplication of positive...
ordpipq 10933 Ordering of positive fract...
ordpinq 10934 Ordering of positive fract...
addpqf 10935 Closure of addition on pos...
addclnq 10936 Closure of addition on pos...
mulpqf 10937 Closure of multiplication ...
mulclnq 10938 Closure of multiplication ...
addnqf 10939 Domain of addition on posi...
mulnqf 10940 Domain of multiplication o...
addcompq 10941 Addition of positive fract...
addcomnq 10942 Addition of positive fract...
mulcompq 10943 Multiplication of positive...
mulcomnq 10944 Multiplication of positive...
adderpqlem 10945 Lemma for ~ adderpq . (Co...
mulerpqlem 10946 Lemma for ~ mulerpq . (Co...
adderpq 10947 Addition is compatible wit...
mulerpq 10948 Multiplication is compatib...
addassnq 10949 Addition of positive fract...
mulassnq 10950 Multiplication of positive...
mulcanenq 10951 Lemma for distributive law...
distrnq 10952 Multiplication of positive...
1nqenq 10953 The equivalence class of r...
mulidnq 10954 Multiplication identity el...
recmulnq 10955 Relationship between recip...
recidnq 10956 A positive fraction times ...
recclnq 10957 Closure law for positive f...
recrecnq 10958 Reciprocal of reciprocal o...
dmrecnq 10959 Domain of reciprocal on po...
ltsonq 10960 'Less than' is a strict or...
lterpq 10961 Compatibility of ordering ...
ltanq 10962 Ordering property of addit...
ltmnq 10963 Ordering property of multi...
1lt2nq 10964 One is less than two (one ...
ltaddnq 10965 The sum of two fractions i...
ltexnq 10966 Ordering on positive fract...
halfnq 10967 One-half of any positive f...
nsmallnq 10968 The is no smallest positiv...
ltbtwnnq 10969 There exists a number betw...
ltrnq 10970 Ordering property of recip...
archnq 10971 For any fraction, there is...
npex 10977 The class of positive real...
elnp 10978 Membership in positive rea...
elnpi 10979 Membership in positive rea...
prn0 10980 A positive real is not emp...
prpssnq 10981 A positive real is a subse...
elprnq 10982 A positive real is a set o...
0npr 10983 The empty set is not a pos...
prcdnq 10984 A positive real is closed ...
prub 10985 A positive fraction not in...
prnmax 10986 A positive real has no lar...
npomex 10987 A simplifying observation,...
prnmadd 10988 A positive real has no lar...
ltrelpr 10989 Positive real 'less than' ...
genpv 10990 Value of general operation...
genpelv 10991 Membership in value of gen...
genpprecl 10992 Pre-closure law for genera...
genpdm 10993 Domain of general operatio...
genpn0 10994 The result of an operation...
genpss 10995 The result of an operation...
genpnnp 10996 The result of an operation...
genpcd 10997 Downward closure of an ope...
genpnmax 10998 An operation on positive r...
genpcl 10999 Closure of an operation on...
genpass 11000 Associativity of an operat...
plpv 11001 Value of addition on posit...
mpv 11002 Value of multiplication on...
dmplp 11003 Domain of addition on posi...
dmmp 11004 Domain of multiplication o...
nqpr 11005 The canonical embedding of...
1pr 11006 The positive real number '...
addclprlem1 11007 Lemma to prove downward cl...
addclprlem2 11008 Lemma to prove downward cl...
addclpr 11009 Closure of addition on pos...
mulclprlem 11010 Lemma to prove downward cl...
mulclpr 11011 Closure of multiplication ...
addcompr 11012 Addition of positive reals...
addasspr 11013 Addition of positive reals...
mulcompr 11014 Multiplication of positive...
mulasspr 11015 Multiplication of positive...
distrlem1pr 11016 Lemma for distributive law...
distrlem4pr 11017 Lemma for distributive law...
distrlem5pr 11018 Lemma for distributive law...
distrpr 11019 Multiplication of positive...
1idpr 11020 1 is an identity element f...
ltprord 11021 Positive real 'less than' ...
psslinpr 11022 Proper subset is a linear ...
ltsopr 11023 Positive real 'less than' ...
prlem934 11024 Lemma 9-3.4 of [Gleason] p...
ltaddpr 11025 The sum of two positive re...
ltaddpr2 11026 The sum of two positive re...
ltexprlem1 11027 Lemma for Proposition 9-3....
ltexprlem2 11028 Lemma for Proposition 9-3....
ltexprlem3 11029 Lemma for Proposition 9-3....
ltexprlem4 11030 Lemma for Proposition 9-3....
ltexprlem5 11031 Lemma for Proposition 9-3....
ltexprlem6 11032 Lemma for Proposition 9-3....
ltexprlem7 11033 Lemma for Proposition 9-3....
ltexpri 11034 Proposition 9-3.5(iv) of [...
ltaprlem 11035 Lemma for Proposition 9-3....
ltapr 11036 Ordering property of addit...
addcanpr 11037 Addition cancellation law ...
prlem936 11038 Lemma 9-3.6 of [Gleason] p...
reclem2pr 11039 Lemma for Proposition 9-3....
reclem3pr 11040 Lemma for Proposition 9-3....
reclem4pr 11041 Lemma for Proposition 9-3....
recexpr 11042 The reciprocal of a positi...
suplem1pr 11043 The union of a nonempty, b...
suplem2pr 11044 The union of a set of posi...
supexpr 11045 The union of a nonempty, b...
enrer 11054 The equivalence relation f...
nrex1 11055 The class of signed reals ...
enrbreq 11056 Equivalence relation for s...
enreceq 11057 Equivalence class equality...
enrex 11058 The equivalence relation f...
ltrelsr 11059 Signed real 'less than' is...
addcmpblnr 11060 Lemma showing compatibilit...
mulcmpblnrlem 11061 Lemma used in lemma showin...
mulcmpblnr 11062 Lemma showing compatibilit...
prsrlem1 11063 Decomposing signed reals i...
addsrmo 11064 There is at most one resul...
mulsrmo 11065 There is at most one resul...
addsrpr 11066 Addition of signed reals i...
mulsrpr 11067 Multiplication of signed r...
ltsrpr 11068 Ordering of signed reals i...
gt0srpr 11069 Greater than zero in terms...
0nsr 11070 The empty set is not a sig...
0r 11071 The constant ` 0R ` is a s...
1sr 11072 The constant ` 1R ` is a s...
m1r 11073 The constant ` -1R ` is a ...
addclsr 11074 Closure of addition on sig...
mulclsr 11075 Closure of multiplication ...
dmaddsr 11076 Domain of addition on sign...
dmmulsr 11077 Domain of multiplication o...
addcomsr 11078 Addition of signed reals i...
addasssr 11079 Addition of signed reals i...
mulcomsr 11080 Multiplication of signed r...
mulasssr 11081 Multiplication of signed r...
distrsr 11082 Multiplication of signed r...
m1p1sr 11083 Minus one plus one is zero...
m1m1sr 11084 Minus one times minus one ...
ltsosr 11085 Signed real 'less than' is...
0lt1sr 11086 0 is less than 1 for signe...
1ne0sr 11087 1 and 0 are distinct for s...
0idsr 11088 The signed real number 0 i...
1idsr 11089 1 is an identity element f...
00sr 11090 A signed real times 0 is 0...
ltasr 11091 Ordering property of addit...
pn0sr 11092 A signed real plus its neg...
negexsr 11093 Existence of negative sign...
recexsrlem 11094 The reciprocal of a positi...
addgt0sr 11095 The sum of two positive si...
mulgt0sr 11096 The product of two positiv...
sqgt0sr 11097 The square of a nonzero si...
recexsr 11098 The reciprocal of a nonzer...
mappsrpr 11099 Mapping from positive sign...
ltpsrpr 11100 Mapping of order from posi...
map2psrpr 11101 Equivalence for positive s...
supsrlem 11102 Lemma for supremum theorem...
supsr 11103 A nonempty, bounded set of...
opelcn 11120 Ordered pair membership in...
opelreal 11121 Ordered pair membership in...
elreal 11122 Membership in class of rea...
elreal2 11123 Ordered pair membership in...
0ncn 11124 The empty set is not a com...
ltrelre 11125 'Less than' is a relation ...
addcnsr 11126 Addition of complex number...
mulcnsr 11127 Multiplication of complex ...
eqresr 11128 Equality of real numbers i...
addresr 11129 Addition of real numbers i...
mulresr 11130 Multiplication of real num...
ltresr 11131 Ordering of real subset of...
ltresr2 11132 Ordering of real subset of...
dfcnqs 11133 Technical trick to permit ...
addcnsrec 11134 Technical trick to permit ...
mulcnsrec 11135 Technical trick to permit ...
axaddf 11136 Addition is an operation o...
axmulf 11137 Multiplication is an opera...
axcnex 11138 The complex numbers form a...
axresscn 11139 The real numbers are a sub...
ax1cn 11140 1 is a complex number. Ax...
axicn 11141 ` _i ` is a complex number...
axaddcl 11142 Closure law for addition o...
axaddrcl 11143 Closure law for addition i...
axmulcl 11144 Closure law for multiplica...
axmulrcl 11145 Closure law for multiplica...
axmulcom 11146 Multiplication of complex ...
axaddass 11147 Addition of complex number...
axmulass 11148 Multiplication of complex ...
axdistr 11149 Distributive law for compl...
axi2m1 11150 i-squared equals -1 (expre...
ax1ne0 11151 1 and 0 are distinct. Axi...
ax1rid 11152 ` 1 ` is an identity eleme...
axrnegex 11153 Existence of negative of r...
axrrecex 11154 Existence of reciprocal of...
axcnre 11155 A complex number can be ex...
axpre-lttri 11156 Ordering on reals satisfie...
axpre-lttrn 11157 Ordering on reals is trans...
axpre-ltadd 11158 Ordering property of addit...
axpre-mulgt0 11159 The product of two positiv...
axpre-sup 11160 A nonempty, bounded-above ...
wuncn 11161 A weak universe containing...
cnex 11187 Alias for ~ ax-cnex . See...
addcl 11188 Alias for ~ ax-addcl , for...
readdcl 11189 Alias for ~ ax-addrcl , fo...
mulcl 11190 Alias for ~ ax-mulcl , for...
remulcl 11191 Alias for ~ ax-mulrcl , fo...
mulcom 11192 Alias for ~ ax-mulcom , fo...
addass 11193 Alias for ~ ax-addass , fo...
mulass 11194 Alias for ~ ax-mulass , fo...
adddi 11195 Alias for ~ ax-distr , for...
recn 11196 A real number is a complex...
reex 11197 The real numbers form a se...
reelprrecn 11198 Reals are a subset of the ...
cnelprrecn 11199 Complex numbers are a subs...
mpomulf 11200 Multiplication is an opera...
elimne0 11201 Hypothesis for weak deduct...
adddir 11202 Distributive law for compl...
0cn 11203 Zero is a complex number. ...
0cnd 11204 Zero is a complex number, ...
c0ex 11205 Zero is a set. (Contribut...
1cnd 11206 One is a complex number, d...
1ex 11207 One is a set. (Contribute...
cnre 11208 Alias for ~ ax-cnre , for ...
mulrid 11209 The number 1 is an identit...
mullid 11210 Identity law for multiplic...
1re 11211 The number 1 is real. Thi...
1red 11212 The number 1 is real, dedu...
0re 11213 The number 0 is real. Rem...
0red 11214 The number 0 is real, dedu...
mulridi 11215 Identity law for multiplic...
mullidi 11216 Identity law for multiplic...
addcli 11217 Closure law for addition. ...
mulcli 11218 Closure law for multiplica...
mulcomi 11219 Commutative law for multip...
mulcomli 11220 Commutative law for multip...
addassi 11221 Associative law for additi...
mulassi 11222 Associative law for multip...
adddii 11223 Distributive law (left-dis...
adddiri 11224 Distributive law (right-di...
recni 11225 A real number is a complex...
readdcli 11226 Closure law for addition o...
remulcli 11227 Closure law for multiplica...
mulridd 11228 Identity law for multiplic...
mullidd 11229 Identity law for multiplic...
addcld 11230 Closure law for addition. ...
mulcld 11231 Closure law for multiplica...
mulcomd 11232 Commutative law for multip...
addassd 11233 Associative law for additi...
mulassd 11234 Associative law for multip...
adddid 11235 Distributive law (left-dis...
adddird 11236 Distributive law (right-di...
adddirp1d 11237 Distributive law, plus 1 v...
joinlmuladdmuld 11238 Join AB+CB into (A+C) on L...
recnd 11239 Deduction from real number...
readdcld 11240 Closure law for addition o...
remulcld 11241 Closure law for multiplica...
pnfnre 11252 Plus infinity is not a rea...
pnfnre2 11253 Plus infinity is not a rea...
mnfnre 11254 Minus infinity is not a re...
ressxr 11255 The standard reals are a s...
rexpssxrxp 11256 The Cartesian product of s...
rexr 11257 A standard real is an exte...
0xr 11258 Zero is an extended real. ...
renepnf 11259 No (finite) real equals pl...
renemnf 11260 No real equals minus infin...
rexrd 11261 A standard real is an exte...
renepnfd 11262 No (finite) real equals pl...
renemnfd 11263 No real equals minus infin...
pnfex 11264 Plus infinity exists. (Co...
pnfxr 11265 Plus infinity belongs to t...
pnfnemnf 11266 Plus and minus infinity ar...
mnfnepnf 11267 Minus and plus infinity ar...
mnfxr 11268 Minus infinity belongs to ...
rexri 11269 A standard real is an exte...
1xr 11270 ` 1 ` is an extended real ...
renfdisj 11271 The reals and the infiniti...
ltrelxr 11272 "Less than" is a relation ...
ltrel 11273 "Less than" is a relation....
lerelxr 11274 "Less than or equal to" is...
lerel 11275 "Less than or equal to" is...
xrlenlt 11276 "Less than or equal to" ex...
xrlenltd 11277 "Less than or equal to" ex...
xrltnle 11278 "Less than" expressed in t...
xrnltled 11279 "Not less than" implies "l...
ssxr 11280 The three (non-exclusive) ...
ltxrlt 11281 The standard less-than ` <...
axlttri 11282 Ordering on reals satisfie...
axlttrn 11283 Ordering on reals is trans...
axltadd 11284 Ordering property of addit...
axmulgt0 11285 The product of two positiv...
axsup 11286 A nonempty, bounded-above ...
lttr 11287 Alias for ~ axlttrn , for ...
mulgt0 11288 The product of two positiv...
lenlt 11289 'Less than or equal to' ex...
ltnle 11290 'Less than' expressed in t...
ltso 11291 'Less than' is a strict or...
gtso 11292 'Greater than' is a strict...
lttri2 11293 Consequence of trichotomy....
lttri3 11294 Trichotomy law for 'less t...
lttri4 11295 Trichotomy law for 'less t...
letri3 11296 Trichotomy law. (Contribu...
leloe 11297 'Less than or equal to' ex...
eqlelt 11298 Equality in terms of 'less...
ltle 11299 'Less than' implies 'less ...
leltne 11300 'Less than or equal to' im...
lelttr 11301 Transitive law. (Contribu...
leltletr 11302 Transitive law, weaker for...
ltletr 11303 Transitive law. (Contribu...
ltleletr 11304 Transitive law, weaker for...
letr 11305 Transitive law. (Contribu...
ltnr 11306 'Less than' is irreflexive...
leid 11307 'Less than or equal to' is...
ltne 11308 'Less than' implies not eq...
ltnsym 11309 'Less than' is not symmetr...
ltnsym2 11310 'Less than' is antisymmetr...
letric 11311 Trichotomy law. (Contribu...
ltlen 11312 'Less than' expressed in t...
eqle 11313 Equality implies 'less tha...
eqled 11314 Equality implies 'less tha...
ltadd2 11315 Addition to both sides of ...
ne0gt0 11316 A nonzero nonnegative numb...
lecasei 11317 Ordering elimination by ca...
lelttric 11318 Trichotomy law. (Contribu...
ltlecasei 11319 Ordering elimination by ca...
ltnri 11320 'Less than' is irreflexive...
eqlei 11321 Equality implies 'less tha...
eqlei2 11322 Equality implies 'less tha...
gtneii 11323 'Less than' implies not eq...
ltneii 11324 'Greater than' implies not...
lttri2i 11325 Consequence of trichotomy....
lttri3i 11326 Consequence of trichotomy....
letri3i 11327 Consequence of trichotomy....
leloei 11328 'Less than or equal to' in...
ltleni 11329 'Less than' expressed in t...
ltnsymi 11330 'Less than' is not symmetr...
lenlti 11331 'Less than or equal to' in...
ltnlei 11332 'Less than' in terms of 'l...
ltlei 11333 'Less than' implies 'less ...
ltleii 11334 'Less than' implies 'less ...
ltnei 11335 'Less than' implies not eq...
letrii 11336 Trichotomy law for 'less t...
lttri 11337 'Less than' is transitive....
lelttri 11338 'Less than or equal to', '...
ltletri 11339 'Less than', 'less than or...
letri 11340 'Less than or equal to' is...
le2tri3i 11341 Extended trichotomy law fo...
ltadd2i 11342 Addition to both sides of ...
mulgt0i 11343 The product of two positiv...
mulgt0ii 11344 The product of two positiv...
ltnrd 11345 'Less than' is irreflexive...
gtned 11346 'Less than' implies not eq...
ltned 11347 'Greater than' implies not...
ne0gt0d 11348 A nonzero nonnegative numb...
lttrid 11349 Ordering on reals satisfie...
lttri2d 11350 Consequence of trichotomy....
lttri3d 11351 Consequence of trichotomy....
lttri4d 11352 Trichotomy law for 'less t...
letri3d 11353 Consequence of trichotomy....
leloed 11354 'Less than or equal to' in...
eqleltd 11355 Equality in terms of 'less...
ltlend 11356 'Less than' expressed in t...
lenltd 11357 'Less than or equal to' in...
ltnled 11358 'Less than' in terms of 'l...
ltled 11359 'Less than' implies 'less ...
ltnsymd 11360 'Less than' implies 'less ...
nltled 11361 'Not less than ' implies '...
lensymd 11362 'Less than or equal to' im...
letrid 11363 Trichotomy law for 'less t...
leltned 11364 'Less than or equal to' im...
leneltd 11365 'Less than or equal to' an...
mulgt0d 11366 The product of two positiv...
ltadd2d 11367 Addition to both sides of ...
letrd 11368 Transitive law deduction f...
lelttrd 11369 Transitive law deduction f...
ltadd2dd 11370 Addition to both sides of ...
ltletrd 11371 Transitive law deduction f...
lttrd 11372 Transitive law deduction f...
lelttrdi 11373 If a number is less than a...
dedekind 11374 The Dedekind cut theorem. ...
dedekindle 11375 The Dedekind cut theorem, ...
mul12 11376 Commutative/associative la...
mul32 11377 Commutative/associative la...
mul31 11378 Commutative/associative la...
mul4 11379 Rearrangement of 4 factors...
mul4r 11380 Rearrangement of 4 factors...
muladd11 11381 A simple product of sums e...
1p1times 11382 Two times a number. (Cont...
peano2cn 11383 A theorem for complex numb...
peano2re 11384 A theorem for reals analog...
readdcan 11385 Cancellation law for addit...
00id 11386 ` 0 ` is its own additive ...
mul02lem1 11387 Lemma for ~ mul02 . If an...
mul02lem2 11388 Lemma for ~ mul02 . Zero ...
mul02 11389 Multiplication by ` 0 ` . ...
mul01 11390 Multiplication by ` 0 ` . ...
addrid 11391 ` 0 ` is an additive ident...
cnegex 11392 Existence of the negative ...
cnegex2 11393 Existence of a left invers...
addlid 11394 ` 0 ` is a left identity f...
addcan 11395 Cancellation law for addit...
addcan2 11396 Cancellation law for addit...
addcom 11397 Addition commutes. This u...
addridi 11398 ` 0 ` is an additive ident...
addlidi 11399 ` 0 ` is a left identity f...
mul02i 11400 Multiplication by 0. Theo...
mul01i 11401 Multiplication by ` 0 ` . ...
addcomi 11402 Addition commutes. Based ...
addcomli 11403 Addition commutes. (Contr...
addcani 11404 Cancellation law for addit...
addcan2i 11405 Cancellation law for addit...
mul12i 11406 Commutative/associative la...
mul32i 11407 Commutative/associative la...
mul4i 11408 Rearrangement of 4 factors...
mul02d 11409 Multiplication by 0. Theo...
mul01d 11410 Multiplication by ` 0 ` . ...
addridd 11411 ` 0 ` is an additive ident...
addlidd 11412 ` 0 ` is a left identity f...
addcomd 11413 Addition commutes. Based ...
addcand 11414 Cancellation law for addit...
addcan2d 11415 Cancellation law for addit...
addcanad 11416 Cancelling a term on the l...
addcan2ad 11417 Cancelling a term on the r...
addneintrd 11418 Introducing a term on the ...
addneintr2d 11419 Introducing a term on the ...
mul12d 11420 Commutative/associative la...
mul32d 11421 Commutative/associative la...
mul31d 11422 Commutative/associative la...
mul4d 11423 Rearrangement of 4 factors...
muladd11r 11424 A simple product of sums e...
comraddd 11425 Commute RHS addition, in d...
ltaddneg 11426 Adding a negative number t...
ltaddnegr 11427 Adding a negative number t...
add12 11428 Commutative/associative la...
add32 11429 Commutative/associative la...
add32r 11430 Commutative/associative la...
add4 11431 Rearrangement of 4 terms i...
add42 11432 Rearrangement of 4 terms i...
add12i 11433 Commutative/associative la...
add32i 11434 Commutative/associative la...
add4i 11435 Rearrangement of 4 terms i...
add42i 11436 Rearrangement of 4 terms i...
add12d 11437 Commutative/associative la...
add32d 11438 Commutative/associative la...
add4d 11439 Rearrangement of 4 terms i...
add42d 11440 Rearrangement of 4 terms i...
0cnALT 11445 Alternate proof of ~ 0cn w...
0cnALT2 11446 Alternate proof of ~ 0cnAL...
negeu 11447 Existential uniqueness of ...
subval 11448 Value of subtraction, whic...
negeq 11449 Equality theorem for negat...
negeqi 11450 Equality inference for neg...
negeqd 11451 Equality deduction for neg...
nfnegd 11452 Deduction version of ~ nfn...
nfneg 11453 Bound-variable hypothesis ...
csbnegg 11454 Move class substitution in...
negex 11455 A negative is a set. (Con...
subcl 11456 Closure law for subtractio...
negcl 11457 Closure law for negative. ...
negicn 11458 ` -u _i ` is a complex num...
subf 11459 Subtraction is an operatio...
subadd 11460 Relationship between subtr...
subadd2 11461 Relationship between subtr...
subsub23 11462 Swap subtrahend and result...
pncan 11463 Cancellation law for subtr...
pncan2 11464 Cancellation law for subtr...
pncan3 11465 Subtraction and addition o...
npcan 11466 Cancellation law for subtr...
addsubass 11467 Associative-type law for a...
addsub 11468 Law for addition and subtr...
subadd23 11469 Commutative/associative la...
addsub12 11470 Commutative/associative la...
2addsub 11471 Law for subtraction and ad...
addsubeq4 11472 Relation between sums and ...
pncan3oi 11473 Subtraction and addition o...
mvrraddi 11474 Move the right term in a s...
mvlladdi 11475 Move the left term in a su...
subid 11476 Subtraction of a number fr...
subid1 11477 Identity law for subtracti...
npncan 11478 Cancellation law for subtr...
nppcan 11479 Cancellation law for subtr...
nnpcan 11480 Cancellation law for subtr...
nppcan3 11481 Cancellation law for subtr...
subcan2 11482 Cancellation law for subtr...
subeq0 11483 If the difference between ...
npncan2 11484 Cancellation law for subtr...
subsub2 11485 Law for double subtraction...
nncan 11486 Cancellation law for subtr...
subsub 11487 Law for double subtraction...
nppcan2 11488 Cancellation law for subtr...
subsub3 11489 Law for double subtraction...
subsub4 11490 Law for double subtraction...
sub32 11491 Swap the second and third ...
nnncan 11492 Cancellation law for subtr...
nnncan1 11493 Cancellation law for subtr...
nnncan2 11494 Cancellation law for subtr...
npncan3 11495 Cancellation law for subtr...
pnpcan 11496 Cancellation law for mixed...
pnpcan2 11497 Cancellation law for mixed...
pnncan 11498 Cancellation law for mixed...
ppncan 11499 Cancellation law for mixed...
addsub4 11500 Rearrangement of 4 terms i...
subadd4 11501 Rearrangement of 4 terms i...
sub4 11502 Rearrangement of 4 terms i...
neg0 11503 Minus 0 equals 0. (Contri...
negid 11504 Addition of a number and i...
negsub 11505 Relationship between subtr...
subneg 11506 Relationship between subtr...
negneg 11507 A number is equal to the n...
neg11 11508 Negative is one-to-one. (...
negcon1 11509 Negative contraposition la...
negcon2 11510 Negative contraposition la...
negeq0 11511 A number is zero iff its n...
subcan 11512 Cancellation law for subtr...
negsubdi 11513 Distribution of negative o...
negdi 11514 Distribution of negative o...
negdi2 11515 Distribution of negative o...
negsubdi2 11516 Distribution of negative o...
neg2sub 11517 Relationship between subtr...
renegcli 11518 Closure law for negative o...
resubcli 11519 Closure law for subtractio...
renegcl 11520 Closure law for negative o...
resubcl 11521 Closure law for subtractio...
negreb 11522 The negative of a real is ...
peano2cnm 11523 "Reverse" second Peano pos...
peano2rem 11524 "Reverse" second Peano pos...
negcli 11525 Closure law for negative. ...
negidi 11526 Addition of a number and i...
negnegi 11527 A number is equal to the n...
subidi 11528 Subtraction of a number fr...
subid1i 11529 Identity law for subtracti...
negne0bi 11530 A number is nonzero iff it...
negrebi 11531 The negative of a real is ...
negne0i 11532 The negative of a nonzero ...
subcli 11533 Closure law for subtractio...
pncan3i 11534 Subtraction and addition o...
negsubi 11535 Relationship between subtr...
subnegi 11536 Relationship between subtr...
subeq0i 11537 If the difference between ...
neg11i 11538 Negative is one-to-one. (...
negcon1i 11539 Negative contraposition la...
negcon2i 11540 Negative contraposition la...
negdii 11541 Distribution of negative o...
negsubdii 11542 Distribution of negative o...
negsubdi2i 11543 Distribution of negative o...
subaddi 11544 Relationship between subtr...
subadd2i 11545 Relationship between subtr...
subaddrii 11546 Relationship between subtr...
subsub23i 11547 Swap subtrahend and result...
addsubassi 11548 Associative-type law for s...
addsubi 11549 Law for subtraction and ad...
subcani 11550 Cancellation law for subtr...
subcan2i 11551 Cancellation law for subtr...
pnncani 11552 Cancellation law for mixed...
addsub4i 11553 Rearrangement of 4 terms i...
0reALT 11554 Alternate proof of ~ 0re ....
negcld 11555 Closure law for negative. ...
subidd 11556 Subtraction of a number fr...
subid1d 11557 Identity law for subtracti...
negidd 11558 Addition of a number and i...
negnegd 11559 A number is equal to the n...
negeq0d 11560 A number is zero iff its n...
negne0bd 11561 A number is nonzero iff it...
negcon1d 11562 Contraposition law for una...
negcon1ad 11563 Contraposition law for una...
neg11ad 11564 The negatives of two compl...
negned 11565 If two complex numbers are...
negne0d 11566 The negative of a nonzero ...
negrebd 11567 The negative of a real is ...
subcld 11568 Closure law for subtractio...
pncand 11569 Cancellation law for subtr...
pncan2d 11570 Cancellation law for subtr...
pncan3d 11571 Subtraction and addition o...
npcand 11572 Cancellation law for subtr...
nncand 11573 Cancellation law for subtr...
negsubd 11574 Relationship between subtr...
subnegd 11575 Relationship between subtr...
subeq0d 11576 If the difference between ...
subne0d 11577 Two unequal numbers have n...
subeq0ad 11578 The difference of two comp...
subne0ad 11579 If the difference of two c...
neg11d 11580 If the difference between ...
negdid 11581 Distribution of negative o...
negdi2d 11582 Distribution of negative o...
negsubdid 11583 Distribution of negative o...
negsubdi2d 11584 Distribution of negative o...
neg2subd 11585 Relationship between subtr...
subaddd 11586 Relationship between subtr...
subadd2d 11587 Relationship between subtr...
addsubassd 11588 Associative-type law for s...
addsubd 11589 Law for subtraction and ad...
subadd23d 11590 Commutative/associative la...
addsub12d 11591 Commutative/associative la...
npncand 11592 Cancellation law for subtr...
nppcand 11593 Cancellation law for subtr...
nppcan2d 11594 Cancellation law for subtr...
nppcan3d 11595 Cancellation law for subtr...
subsubd 11596 Law for double subtraction...
subsub2d 11597 Law for double subtraction...
subsub3d 11598 Law for double subtraction...
subsub4d 11599 Law for double subtraction...
sub32d 11600 Swap the second and third ...
nnncand 11601 Cancellation law for subtr...
nnncan1d 11602 Cancellation law for subtr...
nnncan2d 11603 Cancellation law for subtr...
npncan3d 11604 Cancellation law for subtr...
pnpcand 11605 Cancellation law for mixed...
pnpcan2d 11606 Cancellation law for mixed...
pnncand 11607 Cancellation law for mixed...
ppncand 11608 Cancellation law for mixed...
subcand 11609 Cancellation law for subtr...
subcan2d 11610 Cancellation law for subtr...
subcanad 11611 Cancellation law for subtr...
subneintrd 11612 Introducing subtraction on...
subcan2ad 11613 Cancellation law for subtr...
subneintr2d 11614 Introducing subtraction on...
addsub4d 11615 Rearrangement of 4 terms i...
subadd4d 11616 Rearrangement of 4 terms i...
sub4d 11617 Rearrangement of 4 terms i...
2addsubd 11618 Law for subtraction and ad...
addsubeq4d 11619 Relation between sums and ...
subeqxfrd 11620 Transfer two terms of a su...
mvlraddd 11621 Move the right term in a s...
mvlladdd 11622 Move the left term in a su...
mvrraddd 11623 Move the right term in a s...
mvrladdd 11624 Move the left term in a su...
assraddsubd 11625 Associate RHS addition-sub...
subaddeqd 11626 Transfer two terms of a su...
addlsub 11627 Left-subtraction: Subtrac...
addrsub 11628 Right-subtraction: Subtra...
subexsub 11629 A subtraction law: Exchan...
addid0 11630 If adding a number to a an...
addn0nid 11631 Adding a nonzero number to...
pnpncand 11632 Addition/subtraction cance...
subeqrev 11633 Reverse the order of subtr...
addeq0 11634 Two complex numbers add up...
pncan1 11635 Cancellation law for addit...
npcan1 11636 Cancellation law for subtr...
subeq0bd 11637 If two complex numbers are...
renegcld 11638 Closure law for negative o...
resubcld 11639 Closure law for subtractio...
negn0 11640 The image under negation o...
negf1o 11641 Negation is an isomorphism...
kcnktkm1cn 11642 k times k minus 1 is a com...
muladd 11643 Product of two sums. (Con...
subdi 11644 Distribution of multiplica...
subdir 11645 Distribution of multiplica...
ine0 11646 The imaginary unit ` _i ` ...
mulneg1 11647 Product with negative is n...
mulneg2 11648 The product with a negativ...
mulneg12 11649 Swap the negative sign in ...
mul2neg 11650 Product of two negatives. ...
submul2 11651 Convert a subtraction to a...
mulm1 11652 Product with minus one is ...
addneg1mul 11653 Addition with product with...
mulsub 11654 Product of two differences...
mulsub2 11655 Swap the order of subtract...
mulm1i 11656 Product with minus one is ...
mulneg1i 11657 Product with negative is n...
mulneg2i 11658 Product with negative is n...
mul2negi 11659 Product of two negatives. ...
subdii 11660 Distribution of multiplica...
subdiri 11661 Distribution of multiplica...
muladdi 11662 Product of two sums. (Con...
mulm1d 11663 Product with minus one is ...
mulneg1d 11664 Product with negative is n...
mulneg2d 11665 Product with negative is n...
mul2negd 11666 Product of two negatives. ...
subdid 11667 Distribution of multiplica...
subdird 11668 Distribution of multiplica...
muladdd 11669 Product of two sums. (Con...
mulsubd 11670 Product of two differences...
muls1d 11671 Multiplication by one minu...
mulsubfacd 11672 Multiplication followed by...
addmulsub 11673 The product of a sum and a...
subaddmulsub 11674 The difference with a prod...
mulsubaddmulsub 11675 A special difference of a ...
gt0ne0 11676 Positive implies nonzero. ...
lt0ne0 11677 A number which is less tha...
ltadd1 11678 Addition to both sides of ...
leadd1 11679 Addition to both sides of ...
leadd2 11680 Addition to both sides of ...
ltsubadd 11681 'Less than' relationship b...
ltsubadd2 11682 'Less than' relationship b...
lesubadd 11683 'Less than or equal to' re...
lesubadd2 11684 'Less than or equal to' re...
ltaddsub 11685 'Less than' relationship b...
ltaddsub2 11686 'Less than' relationship b...
leaddsub 11687 'Less than or equal to' re...
leaddsub2 11688 'Less than or equal to' re...
suble 11689 Swap subtrahends in an ine...
lesub 11690 Swap subtrahends in an ine...
ltsub23 11691 'Less than' relationship b...
ltsub13 11692 'Less than' relationship b...
le2add 11693 Adding both sides of two '...
ltleadd 11694 Adding both sides of two o...
leltadd 11695 Adding both sides of two o...
lt2add 11696 Adding both sides of two '...
addgt0 11697 The sum of 2 positive numb...
addgegt0 11698 The sum of nonnegative and...
addgtge0 11699 The sum of nonnegative and...
addge0 11700 The sum of 2 nonnegative n...
ltaddpos 11701 Adding a positive number t...
ltaddpos2 11702 Adding a positive number t...
ltsubpos 11703 Subtracting a positive num...
posdif 11704 Comparison of two numbers ...
lesub1 11705 Subtraction from both side...
lesub2 11706 Subtraction of both sides ...
ltsub1 11707 Subtraction from both side...
ltsub2 11708 Subtraction of both sides ...
lt2sub 11709 Subtracting both sides of ...
le2sub 11710 Subtracting both sides of ...
ltneg 11711 Negative of both sides of ...
ltnegcon1 11712 Contraposition of negative...
ltnegcon2 11713 Contraposition of negative...
leneg 11714 Negative of both sides of ...
lenegcon1 11715 Contraposition of negative...
lenegcon2 11716 Contraposition of negative...
lt0neg1 11717 Comparison of a number and...
lt0neg2 11718 Comparison of a number and...
le0neg1 11719 Comparison of a number and...
le0neg2 11720 Comparison of a number and...
addge01 11721 A number is less than or e...
addge02 11722 A number is less than or e...
add20 11723 Two nonnegative numbers ar...
subge0 11724 Nonnegative subtraction. ...
suble0 11725 Nonpositive subtraction. ...
leaddle0 11726 The sum of a real number a...
subge02 11727 Nonnegative subtraction. ...
lesub0 11728 Lemma to show a nonnegativ...
mulge0 11729 The product of two nonnega...
mullt0 11730 The product of two negativ...
msqgt0 11731 A nonzero square is positi...
msqge0 11732 A square is nonnegative. ...
0lt1 11733 0 is less than 1. Theorem...
0le1 11734 0 is less than or equal to...
relin01 11735 An interval law for less t...
ltordlem 11736 Lemma for ~ ltord1 . (Con...
ltord1 11737 Infer an ordering relation...
leord1 11738 Infer an ordering relation...
eqord1 11739 A strictly increasing real...
ltord2 11740 Infer an ordering relation...
leord2 11741 Infer an ordering relation...
eqord2 11742 A strictly decreasing real...
wloglei 11743 Form of ~ wlogle where bot...
wlogle 11744 If the predicate ` ch ( x ...
leidi 11745 'Less than or equal to' is...
gt0ne0i 11746 Positive means nonzero (us...
gt0ne0ii 11747 Positive implies nonzero. ...
msqgt0i 11748 A nonzero square is positi...
msqge0i 11749 A square is nonnegative. ...
addgt0i 11750 Addition of 2 positive num...
addge0i 11751 Addition of 2 nonnegative ...
addgegt0i 11752 Addition of nonnegative an...
addgt0ii 11753 Addition of 2 positive num...
add20i 11754 Two nonnegative numbers ar...
ltnegi 11755 Negative of both sides of ...
lenegi 11756 Negative of both sides of ...
ltnegcon2i 11757 Contraposition of negative...
mulge0i 11758 The product of two nonnega...
lesub0i 11759 Lemma to show a nonnegativ...
ltaddposi 11760 Adding a positive number t...
posdifi 11761 Comparison of two numbers ...
ltnegcon1i 11762 Contraposition of negative...
lenegcon1i 11763 Contraposition of negative...
subge0i 11764 Nonnegative subtraction. ...
ltadd1i 11765 Addition to both sides of ...
leadd1i 11766 Addition to both sides of ...
leadd2i 11767 Addition to both sides of ...
ltsubaddi 11768 'Less than' relationship b...
lesubaddi 11769 'Less than or equal to' re...
ltsubadd2i 11770 'Less than' relationship b...
lesubadd2i 11771 'Less than or equal to' re...
ltaddsubi 11772 'Less than' relationship b...
lt2addi 11773 Adding both side of two in...
le2addi 11774 Adding both side of two in...
gt0ne0d 11775 Positive implies nonzero. ...
lt0ne0d 11776 Something less than zero i...
leidd 11777 'Less than or equal to' is...
msqgt0d 11778 A nonzero square is positi...
msqge0d 11779 A square is nonnegative. ...
lt0neg1d 11780 Comparison of a number and...
lt0neg2d 11781 Comparison of a number and...
le0neg1d 11782 Comparison of a number and...
le0neg2d 11783 Comparison of a number and...
addgegt0d 11784 Addition of nonnegative an...
addgtge0d 11785 Addition of positive and n...
addgt0d 11786 Addition of 2 positive num...
addge0d 11787 Addition of 2 nonnegative ...
mulge0d 11788 The product of two nonnega...
ltnegd 11789 Negative of both sides of ...
lenegd 11790 Negative of both sides of ...
ltnegcon1d 11791 Contraposition of negative...
ltnegcon2d 11792 Contraposition of negative...
lenegcon1d 11793 Contraposition of negative...
lenegcon2d 11794 Contraposition of negative...
ltaddposd 11795 Adding a positive number t...
ltaddpos2d 11796 Adding a positive number t...
ltsubposd 11797 Subtracting a positive num...
posdifd 11798 Comparison of two numbers ...
addge01d 11799 A number is less than or e...
addge02d 11800 A number is less than or e...
subge0d 11801 Nonnegative subtraction. ...
suble0d 11802 Nonpositive subtraction. ...
subge02d 11803 Nonnegative subtraction. ...
ltadd1d 11804 Addition to both sides of ...
leadd1d 11805 Addition to both sides of ...
leadd2d 11806 Addition to both sides of ...
ltsubaddd 11807 'Less than' relationship b...
lesubaddd 11808 'Less than or equal to' re...
ltsubadd2d 11809 'Less than' relationship b...
lesubadd2d 11810 'Less than or equal to' re...
ltaddsubd 11811 'Less than' relationship b...
ltaddsub2d 11812 'Less than' relationship b...
leaddsub2d 11813 'Less than or equal to' re...
subled 11814 Swap subtrahends in an ine...
lesubd 11815 Swap subtrahends in an ine...
ltsub23d 11816 'Less than' relationship b...
ltsub13d 11817 'Less than' relationship b...
lesub1d 11818 Subtraction from both side...
lesub2d 11819 Subtraction of both sides ...
ltsub1d 11820 Subtraction from both side...
ltsub2d 11821 Subtraction of both sides ...
ltadd1dd 11822 Addition to both sides of ...
ltsub1dd 11823 Subtraction from both side...
ltsub2dd 11824 Subtraction of both sides ...
leadd1dd 11825 Addition to both sides of ...
leadd2dd 11826 Addition to both sides of ...
lesub1dd 11827 Subtraction from both side...
lesub2dd 11828 Subtraction of both sides ...
lesub3d 11829 The result of subtracting ...
le2addd 11830 Adding both side of two in...
le2subd 11831 Subtracting both sides of ...
ltleaddd 11832 Adding both sides of two o...
leltaddd 11833 Adding both sides of two o...
lt2addd 11834 Adding both side of two in...
lt2subd 11835 Subtracting both sides of ...
possumd 11836 Condition for a positive s...
sublt0d 11837 When a subtraction gives a...
ltaddsublt 11838 Addition and subtraction o...
1le1 11839 One is less than or equal ...
ixi 11840 ` _i ` times itself is min...
recextlem1 11841 Lemma for ~ recex . (Cont...
recextlem2 11842 Lemma for ~ recex . (Cont...
recex 11843 Existence of reciprocal of...
mulcand 11844 Cancellation law for multi...
mulcan2d 11845 Cancellation law for multi...
mulcanad 11846 Cancellation of a nonzero ...
mulcan2ad 11847 Cancellation of a nonzero ...
mulcan 11848 Cancellation law for multi...
mulcan2 11849 Cancellation law for multi...
mulcani 11850 Cancellation law for multi...
mul0or 11851 If a product is zero, one ...
mulne0b 11852 The product of two nonzero...
mulne0 11853 The product of two nonzero...
mulne0i 11854 The product of two nonzero...
muleqadd 11855 Property of numbers whose ...
receu 11856 Existential uniqueness of ...
mulnzcnf 11857 Multiplication maps nonzer...
msq0i 11858 A number is zero iff its s...
mul0ori 11859 If a product is zero, one ...
msq0d 11860 A number is zero iff its s...
mul0ord 11861 If a product is zero, one ...
mulne0bd 11862 The product of two nonzero...
mulne0d 11863 The product of two nonzero...
mulcan1g 11864 A generalized form of the ...
mulcan2g 11865 A generalized form of the ...
mulne0bad 11866 A factor of a nonzero comp...
mulne0bbd 11867 A factor of a nonzero comp...
1div0 11870 You can't divide by zero, ...
divval 11871 Value of division: if ` A ...
divmul 11872 Relationship between divis...
divmul2 11873 Relationship between divis...
divmul3 11874 Relationship between divis...
divcl 11875 Closure law for division. ...
reccl 11876 Closure law for reciprocal...
divcan2 11877 A cancellation law for div...
divcan1 11878 A cancellation law for div...
diveq0 11879 A ratio is zero iff the nu...
divne0b 11880 The ratio of nonzero numbe...
divne0 11881 The ratio of nonzero numbe...
recne0 11882 The reciprocal of a nonzer...
recid 11883 Multiplication of a number...
recid2 11884 Multiplication of a number...
divrec 11885 Relationship between divis...
divrec2 11886 Relationship between divis...
divass 11887 An associative law for div...
div23 11888 A commutative/associative ...
div32 11889 A commutative/associative ...
div13 11890 A commutative/associative ...
div12 11891 A commutative/associative ...
divmulass 11892 An associative law for div...
divmulasscom 11893 An associative/commutative...
divdir 11894 Distribution of division o...
divcan3 11895 A cancellation law for div...
divcan4 11896 A cancellation law for div...
div11 11897 One-to-one relationship fo...
divid 11898 A number divided by itself...
div0 11899 Division into zero is zero...
div1 11900 A number divided by 1 is i...
1div1e1 11901 1 divided by 1 is 1. (Con...
diveq1 11902 Equality in terms of unit ...
divneg 11903 Move negative sign inside ...
muldivdir 11904 Distribution of division o...
divsubdir 11905 Distribution of division o...
subdivcomb1 11906 Bring a term in a subtract...
subdivcomb2 11907 Bring a term in a subtract...
recrec 11908 A number is equal to the r...
rec11 11909 Reciprocal is one-to-one. ...
rec11r 11910 Mutual reciprocals. (Cont...
divmuldiv 11911 Multiplication of two rati...
divdivdiv 11912 Division of two ratios. T...
divcan5 11913 Cancellation of common fac...
divmul13 11914 Swap the denominators in t...
divmul24 11915 Swap the numerators in the...
divmuleq 11916 Cross-multiply in an equal...
recdiv 11917 The reciprocal of a ratio....
divcan6 11918 Cancellation of inverted f...
divdiv32 11919 Swap denominators in a div...
divcan7 11920 Cancel equal divisors in a...
dmdcan 11921 Cancellation law for divis...
divdiv1 11922 Division into a fraction. ...
divdiv2 11923 Division by a fraction. (...
recdiv2 11924 Division into a reciprocal...
ddcan 11925 Cancellation in a double d...
divadddiv 11926 Addition of two ratios. T...
divsubdiv 11927 Subtraction of two ratios....
conjmul 11928 Two numbers whose reciproc...
rereccl 11929 Closure law for reciprocal...
redivcl 11930 Closure law for division o...
eqneg 11931 A number equal to its nega...
eqnegd 11932 A complex number equals it...
eqnegad 11933 If a complex number equals...
div2neg 11934 Quotient of two negatives....
divneg2 11935 Move negative sign inside ...
recclzi 11936 Closure law for reciprocal...
recne0zi 11937 The reciprocal of a nonzer...
recidzi 11938 Multiplication of a number...
div1i 11939 A number divided by 1 is i...
eqnegi 11940 A number equal to its nega...
reccli 11941 Closure law for reciprocal...
recidi 11942 Multiplication of a number...
recreci 11943 A number is equal to the r...
dividi 11944 A number divided by itself...
div0i 11945 Division into zero is zero...
divclzi 11946 Closure law for division. ...
divcan1zi 11947 A cancellation law for div...
divcan2zi 11948 A cancellation law for div...
divreczi 11949 Relationship between divis...
divcan3zi 11950 A cancellation law for div...
divcan4zi 11951 A cancellation law for div...
rec11i 11952 Reciprocal is one-to-one. ...
divcli 11953 Closure law for division. ...
divcan2i 11954 A cancellation law for div...
divcan1i 11955 A cancellation law for div...
divreci 11956 Relationship between divis...
divcan3i 11957 A cancellation law for div...
divcan4i 11958 A cancellation law for div...
divne0i 11959 The ratio of nonzero numbe...
rec11ii 11960 Reciprocal is one-to-one. ...
divasszi 11961 An associative law for div...
divmulzi 11962 Relationship between divis...
divdirzi 11963 Distribution of division o...
divdiv23zi 11964 Swap denominators in a div...
divmuli 11965 Relationship between divis...
divdiv32i 11966 Swap denominators in a div...
divassi 11967 An associative law for div...
divdiri 11968 Distribution of division o...
div23i 11969 A commutative/associative ...
div11i 11970 One-to-one relationship fo...
divmuldivi 11971 Multiplication of two rati...
divmul13i 11972 Swap denominators of two r...
divadddivi 11973 Addition of two ratios. T...
divdivdivi 11974 Division of two ratios. T...
rerecclzi 11975 Closure law for reciprocal...
rereccli 11976 Closure law for reciprocal...
redivclzi 11977 Closure law for division o...
redivcli 11978 Closure law for division o...
div1d 11979 A number divided by 1 is i...
reccld 11980 Closure law for reciprocal...
recne0d 11981 The reciprocal of a nonzer...
recidd 11982 Multiplication of a number...
recid2d 11983 Multiplication of a number...
recrecd 11984 A number is equal to the r...
dividd 11985 A number divided by itself...
div0d 11986 Division into zero is zero...
divcld 11987 Closure law for division. ...
divcan1d 11988 A cancellation law for div...
divcan2d 11989 A cancellation law for div...
divrecd 11990 Relationship between divis...
divrec2d 11991 Relationship between divis...
divcan3d 11992 A cancellation law for div...
divcan4d 11993 A cancellation law for div...
diveq0d 11994 A ratio is zero iff the nu...
diveq1d 11995 Equality in terms of unit ...
diveq1ad 11996 The quotient of two comple...
diveq0ad 11997 A fraction of complex numb...
divne1d 11998 If two complex numbers are...
divne0bd 11999 A ratio is zero iff the nu...
divnegd 12000 Move negative sign inside ...
divneg2d 12001 Move negative sign inside ...
div2negd 12002 Quotient of two negatives....
divne0d 12003 The ratio of nonzero numbe...
recdivd 12004 The reciprocal of a ratio....
recdiv2d 12005 Division into a reciprocal...
divcan6d 12006 Cancellation of inverted f...
ddcand 12007 Cancellation in a double d...
rec11d 12008 Reciprocal is one-to-one. ...
divmuld 12009 Relationship between divis...
div32d 12010 A commutative/associative ...
div13d 12011 A commutative/associative ...
divdiv32d 12012 Swap denominators in a div...
divcan5d 12013 Cancellation of common fac...
divcan5rd 12014 Cancellation of common fac...
divcan7d 12015 Cancel equal divisors in a...
dmdcand 12016 Cancellation law for divis...
dmdcan2d 12017 Cancellation law for divis...
divdiv1d 12018 Division into a fraction. ...
divdiv2d 12019 Division by a fraction. (...
divmul2d 12020 Relationship between divis...
divmul3d 12021 Relationship between divis...
divassd 12022 An associative law for div...
div12d 12023 A commutative/associative ...
div23d 12024 A commutative/associative ...
divdird 12025 Distribution of division o...
divsubdird 12026 Distribution of division o...
div11d 12027 One-to-one relationship fo...
divmuldivd 12028 Multiplication of two rati...
divmul13d 12029 Swap denominators of two r...
divmul24d 12030 Swap the numerators in the...
divadddivd 12031 Addition of two ratios. T...
divsubdivd 12032 Subtraction of two ratios....
divmuleqd 12033 Cross-multiply in an equal...
divdivdivd 12034 Division of two ratios. T...
diveq1bd 12035 If two complex numbers are...
div2sub 12036 Swap the order of subtract...
div2subd 12037 Swap subtrahend and minuen...
rereccld 12038 Closure law for reciprocal...
redivcld 12039 Closure law for division o...
subrec 12040 Subtraction of reciprocals...
subreci 12041 Subtraction of reciprocals...
subrecd 12042 Subtraction of reciprocals...
mvllmuld 12043 Move the left term in a pr...
mvllmuli 12044 Move the left term in a pr...
ldiv 12045 Left-division. (Contribut...
rdiv 12046 Right-division. (Contribu...
mdiv 12047 A division law. (Contribu...
lineq 12048 Solution of a (scalar) lin...
elimgt0 12049 Hypothesis for weak deduct...
elimge0 12050 Hypothesis for weak deduct...
ltp1 12051 A number is less than itse...
lep1 12052 A number is less than or e...
ltm1 12053 A number minus 1 is less t...
lem1 12054 A number minus 1 is less t...
letrp1 12055 A transitive property of '...
p1le 12056 A transitive property of p...
recgt0 12057 The reciprocal of a positi...
prodgt0 12058 Infer that a multiplicand ...
prodgt02 12059 Infer that a multiplier is...
ltmul1a 12060 Lemma for ~ ltmul1 . Mult...
ltmul1 12061 Multiplication of both sid...
ltmul2 12062 Multiplication of both sid...
lemul1 12063 Multiplication of both sid...
lemul2 12064 Multiplication of both sid...
lemul1a 12065 Multiplication of both sid...
lemul2a 12066 Multiplication of both sid...
ltmul12a 12067 Comparison of product of t...
lemul12b 12068 Comparison of product of t...
lemul12a 12069 Comparison of product of t...
mulgt1 12070 The product of two numbers...
ltmulgt11 12071 Multiplication by a number...
ltmulgt12 12072 Multiplication by a number...
lemulge11 12073 Multiplication by a number...
lemulge12 12074 Multiplication by a number...
ltdiv1 12075 Division of both sides of ...
lediv1 12076 Division of both sides of ...
gt0div 12077 Division of a positive num...
ge0div 12078 Division of a nonnegative ...
divgt0 12079 The ratio of two positive ...
divge0 12080 The ratio of nonnegative a...
mulge0b 12081 A condition for multiplica...
mulle0b 12082 A condition for multiplica...
mulsuble0b 12083 A condition for multiplica...
ltmuldiv 12084 'Less than' relationship b...
ltmuldiv2 12085 'Less than' relationship b...
ltdivmul 12086 'Less than' relationship b...
ledivmul 12087 'Less than or equal to' re...
ltdivmul2 12088 'Less than' relationship b...
lt2mul2div 12089 'Less than' relationship b...
ledivmul2 12090 'Less than or equal to' re...
lemuldiv 12091 'Less than or equal' relat...
lemuldiv2 12092 'Less than or equal' relat...
ltrec 12093 The reciprocal of both sid...
lerec 12094 The reciprocal of both sid...
lt2msq1 12095 Lemma for ~ lt2msq . (Con...
lt2msq 12096 Two nonnegative numbers co...
ltdiv2 12097 Division of a positive num...
ltrec1 12098 Reciprocal swap in a 'less...
lerec2 12099 Reciprocal swap in a 'less...
ledivdiv 12100 Invert ratios of positive ...
lediv2 12101 Division of a positive num...
ltdiv23 12102 Swap denominator with othe...
lediv23 12103 Swap denominator with othe...
lediv12a 12104 Comparison of ratio of two...
lediv2a 12105 Division of both sides of ...
reclt1 12106 The reciprocal of a positi...
recgt1 12107 The reciprocal of a positi...
recgt1i 12108 The reciprocal of a number...
recp1lt1 12109 Construct a number less th...
recreclt 12110 Given a positive number ` ...
le2msq 12111 The square function on non...
msq11 12112 The square of a nonnegativ...
ledivp1 12113 "Less than or equal to" an...
squeeze0 12114 If a nonnegative number is...
ltp1i 12115 A number is less than itse...
recgt0i 12116 The reciprocal of a positi...
recgt0ii 12117 The reciprocal of a positi...
prodgt0i 12118 Infer that a multiplicand ...
divgt0i 12119 The ratio of two positive ...
divge0i 12120 The ratio of nonnegative a...
ltreci 12121 The reciprocal of both sid...
lereci 12122 The reciprocal of both sid...
lt2msqi 12123 The square function on non...
le2msqi 12124 The square function on non...
msq11i 12125 The square of a nonnegativ...
divgt0i2i 12126 The ratio of two positive ...
ltrecii 12127 The reciprocal of both sid...
divgt0ii 12128 The ratio of two positive ...
ltmul1i 12129 Multiplication of both sid...
ltdiv1i 12130 Division of both sides of ...
ltmuldivi 12131 'Less than' relationship b...
ltmul2i 12132 Multiplication of both sid...
lemul1i 12133 Multiplication of both sid...
lemul2i 12134 Multiplication of both sid...
ltdiv23i 12135 Swap denominator with othe...
ledivp1i 12136 "Less than or equal to" an...
ltdivp1i 12137 Less-than and division rel...
ltdiv23ii 12138 Swap denominator with othe...
ltmul1ii 12139 Multiplication of both sid...
ltdiv1ii 12140 Division of both sides of ...
ltp1d 12141 A number is less than itse...
lep1d 12142 A number is less than or e...
ltm1d 12143 A number minus 1 is less t...
lem1d 12144 A number minus 1 is less t...
recgt0d 12145 The reciprocal of a positi...
divgt0d 12146 The ratio of two positive ...
mulgt1d 12147 The product of two numbers...
lemulge11d 12148 Multiplication by a number...
lemulge12d 12149 Multiplication by a number...
lemul1ad 12150 Multiplication of both sid...
lemul2ad 12151 Multiplication of both sid...
ltmul12ad 12152 Comparison of product of t...
lemul12ad 12153 Comparison of product of t...
lemul12bd 12154 Comparison of product of t...
fimaxre 12155 A finite set of real numbe...
fimaxre2 12156 A nonempty finite set of r...
fimaxre3 12157 A nonempty finite set of r...
fiminre 12158 A nonempty finite set of r...
fiminre2 12159 A nonempty finite set of r...
negfi 12160 The negation of a finite s...
lbreu 12161 If a set of reals contains...
lbcl 12162 If a set of reals contains...
lble 12163 If a set of reals contains...
lbinf 12164 If a set of reals contains...
lbinfcl 12165 If a set of reals contains...
lbinfle 12166 If a set of reals contains...
sup2 12167 A nonempty, bounded-above ...
sup3 12168 A version of the completen...
infm3lem 12169 Lemma for ~ infm3 . (Cont...
infm3 12170 The completeness axiom for...
suprcl 12171 Closure of supremum of a n...
suprub 12172 A member of a nonempty bou...
suprubd 12173 Natural deduction form of ...
suprcld 12174 Natural deduction form of ...
suprlub 12175 The supremum of a nonempty...
suprnub 12176 An upper bound is not less...
suprleub 12177 The supremum of a nonempty...
supaddc 12178 The supremum function dist...
supadd 12179 The supremum function dist...
supmul1 12180 The supremum function dist...
supmullem1 12181 Lemma for ~ supmul . (Con...
supmullem2 12182 Lemma for ~ supmul . (Con...
supmul 12183 The supremum function dist...
sup3ii 12184 A version of the completen...
suprclii 12185 Closure of supremum of a n...
suprubii 12186 A member of a nonempty bou...
suprlubii 12187 The supremum of a nonempty...
suprnubii 12188 An upper bound is not less...
suprleubii 12189 The supremum of a nonempty...
riotaneg 12190 The negative of the unique...
negiso 12191 Negation is an order anti-...
dfinfre 12192 The infimum of a set of re...
infrecl 12193 Closure of infimum of a no...
infrenegsup 12194 The infimum of a set of re...
infregelb 12195 Any lower bound of a nonem...
infrelb 12196 If a nonempty set of real ...
infrefilb 12197 The infimum of a finite se...
supfirege 12198 The supremum of a finite s...
inelr 12199 The imaginary unit ` _i ` ...
rimul 12200 A real number times the im...
cru 12201 The representation of comp...
crne0 12202 The real representation of...
creur 12203 The real part of a complex...
creui 12204 The imaginary part of a co...
cju 12205 The complex conjugate of a...
ofsubeq0 12206 Function analogue of ~ sub...
ofnegsub 12207 Function analogue of ~ neg...
ofsubge0 12208 Function analogue of ~ sub...
nnexALT 12211 Alternate proof of ~ nnex ...
peano5nni 12212 Peano's inductive postulat...
nnssre 12213 The positive integers are ...
nnsscn 12214 The positive integers are ...
nnex 12215 The set of positive intege...
nnre 12216 A positive integer is a re...
nncn 12217 A positive integer is a co...
nnrei 12218 A positive integer is a re...
nncni 12219 A positive integer is a co...
1nn 12220 Peano postulate: 1 is a po...
peano2nn 12221 Peano postulate: a success...
dfnn2 12222 Alternate definition of th...
dfnn3 12223 Alternate definition of th...
nnred 12224 A positive integer is a re...
nncnd 12225 A positive integer is a co...
peano2nnd 12226 Peano postulate: a success...
nnind 12227 Principle of Mathematical ...
nnindALT 12228 Principle of Mathematical ...
nnindd 12229 Principle of Mathematical ...
nn1m1nn 12230 Every positive integer is ...
nn1suc 12231 If a statement holds for 1...
nnaddcl 12232 Closure of addition of pos...
nnmulcl 12233 Closure of multiplication ...
nnmulcli 12234 Closure of multiplication ...
nnmtmip 12235 "Minus times minus is plus...
nn2ge 12236 There exists a positive in...
nnge1 12237 A positive integer is one ...
nngt1ne1 12238 A positive integer is grea...
nnle1eq1 12239 A positive integer is less...
nngt0 12240 A positive integer is posi...
nnnlt1 12241 A positive integer is not ...
nnnle0 12242 A positive integer is not ...
nnne0 12243 A positive integer is nonz...
nnneneg 12244 No positive integer is equ...
0nnn 12245 Zero is not a positive int...
0nnnALT 12246 Alternate proof of ~ 0nnn ...
nnne0ALT 12247 Alternate version of ~ nnn...
nngt0i 12248 A positive integer is posi...
nnne0i 12249 A positive integer is nonz...
nndivre 12250 The quotient of a real and...
nnrecre 12251 The reciprocal of a positi...
nnrecgt0 12252 The reciprocal of a positi...
nnsub 12253 Subtraction of positive in...
nnsubi 12254 Subtraction of positive in...
nndiv 12255 Two ways to express " ` A ...
nndivtr 12256 Transitive property of div...
nnge1d 12257 A positive integer is one ...
nngt0d 12258 A positive integer is posi...
nnne0d 12259 A positive integer is nonz...
nnrecred 12260 The reciprocal of a positi...
nnaddcld 12261 Closure of addition of pos...
nnmulcld 12262 Closure of multiplication ...
nndivred 12263 A positive integer is one ...
0ne1 12280 Zero is different from one...
1m1e0 12281 One minus one equals zero....
2nn 12282 2 is a positive integer. ...
2re 12283 The number 2 is real. (Co...
2cn 12284 The number 2 is a complex ...
2cnALT 12285 Alternate proof of ~ 2cn ....
2ex 12286 The number 2 is a set. (C...
2cnd 12287 The number 2 is a complex ...
3nn 12288 3 is a positive integer. ...
3re 12289 The number 3 is real. (Co...
3cn 12290 The number 3 is a complex ...
3ex 12291 The number 3 is a set. (C...
4nn 12292 4 is a positive integer. ...
4re 12293 The number 4 is real. (Co...
4cn 12294 The number 4 is a complex ...
5nn 12295 5 is a positive integer. ...
5re 12296 The number 5 is real. (Co...
5cn 12297 The number 5 is a complex ...
6nn 12298 6 is a positive integer. ...
6re 12299 The number 6 is real. (Co...
6cn 12300 The number 6 is a complex ...
7nn 12301 7 is a positive integer. ...
7re 12302 The number 7 is real. (Co...
7cn 12303 The number 7 is a complex ...
8nn 12304 8 is a positive integer. ...
8re 12305 The number 8 is real. (Co...
8cn 12306 The number 8 is a complex ...
9nn 12307 9 is a positive integer. ...
9re 12308 The number 9 is real. (Co...
9cn 12309 The number 9 is a complex ...
0le0 12310 Zero is nonnegative. (Con...
0le2 12311 The number 0 is less than ...
2pos 12312 The number 2 is positive. ...
2ne0 12313 The number 2 is nonzero. ...
3pos 12314 The number 3 is positive. ...
3ne0 12315 The number 3 is nonzero. ...
4pos 12316 The number 4 is positive. ...
4ne0 12317 The number 4 is nonzero. ...
5pos 12318 The number 5 is positive. ...
6pos 12319 The number 6 is positive. ...
7pos 12320 The number 7 is positive. ...
8pos 12321 The number 8 is positive. ...
9pos 12322 The number 9 is positive. ...
neg1cn 12323 -1 is a complex number. (...
neg1rr 12324 -1 is a real number. (Con...
neg1ne0 12325 -1 is nonzero. (Contribut...
neg1lt0 12326 -1 is less than 0. (Contr...
negneg1e1 12327 ` -u -u 1 ` is 1. (Contri...
1pneg1e0 12328 ` 1 + -u 1 ` is 0. (Contr...
0m0e0 12329 0 minus 0 equals 0. (Cont...
1m0e1 12330 1 - 0 = 1. (Contributed b...
0p1e1 12331 0 + 1 = 1. (Contributed b...
fv0p1e1 12332 Function value at ` N + 1 ...
1p0e1 12333 1 + 0 = 1. (Contributed b...
1p1e2 12334 1 + 1 = 2. (Contributed b...
2m1e1 12335 2 - 1 = 1. The result is ...
1e2m1 12336 1 = 2 - 1. (Contributed b...
3m1e2 12337 3 - 1 = 2. (Contributed b...
4m1e3 12338 4 - 1 = 3. (Contributed b...
5m1e4 12339 5 - 1 = 4. (Contributed b...
6m1e5 12340 6 - 1 = 5. (Contributed b...
7m1e6 12341 7 - 1 = 6. (Contributed b...
8m1e7 12342 8 - 1 = 7. (Contributed b...
9m1e8 12343 9 - 1 = 8. (Contributed b...
2p2e4 12344 Two plus two equals four. ...
2times 12345 Two times a number. (Cont...
times2 12346 A number times 2. (Contri...
2timesi 12347 Two times a number. (Cont...
times2i 12348 A number times 2. (Contri...
2txmxeqx 12349 Two times a complex number...
2div2e1 12350 2 divided by 2 is 1. (Con...
2p1e3 12351 2 + 1 = 3. (Contributed b...
1p2e3 12352 1 + 2 = 3. For a shorter ...
1p2e3ALT 12353 Alternate proof of ~ 1p2e3...
3p1e4 12354 3 + 1 = 4. (Contributed b...
4p1e5 12355 4 + 1 = 5. (Contributed b...
5p1e6 12356 5 + 1 = 6. (Contributed b...
6p1e7 12357 6 + 1 = 7. (Contributed b...
7p1e8 12358 7 + 1 = 8. (Contributed b...
8p1e9 12359 8 + 1 = 9. (Contributed b...
3p2e5 12360 3 + 2 = 5. (Contributed b...
3p3e6 12361 3 + 3 = 6. (Contributed b...
4p2e6 12362 4 + 2 = 6. (Contributed b...
4p3e7 12363 4 + 3 = 7. (Contributed b...
4p4e8 12364 4 + 4 = 8. (Contributed b...
5p2e7 12365 5 + 2 = 7. (Contributed b...
5p3e8 12366 5 + 3 = 8. (Contributed b...
5p4e9 12367 5 + 4 = 9. (Contributed b...
6p2e8 12368 6 + 2 = 8. (Contributed b...
6p3e9 12369 6 + 3 = 9. (Contributed b...
7p2e9 12370 7 + 2 = 9. (Contributed b...
1t1e1 12371 1 times 1 equals 1. (Cont...
2t1e2 12372 2 times 1 equals 2. (Cont...
2t2e4 12373 2 times 2 equals 4. (Cont...
3t1e3 12374 3 times 1 equals 3. (Cont...
3t2e6 12375 3 times 2 equals 6. (Cont...
3t3e9 12376 3 times 3 equals 9. (Cont...
4t2e8 12377 4 times 2 equals 8. (Cont...
2t0e0 12378 2 times 0 equals 0. (Cont...
4d2e2 12379 One half of four is two. ...
1lt2 12380 1 is less than 2. (Contri...
2lt3 12381 2 is less than 3. (Contri...
1lt3 12382 1 is less than 3. (Contri...
3lt4 12383 3 is less than 4. (Contri...
2lt4 12384 2 is less than 4. (Contri...
1lt4 12385 1 is less than 4. (Contri...
4lt5 12386 4 is less than 5. (Contri...
3lt5 12387 3 is less than 5. (Contri...
2lt5 12388 2 is less than 5. (Contri...
1lt5 12389 1 is less than 5. (Contri...
5lt6 12390 5 is less than 6. (Contri...
4lt6 12391 4 is less than 6. (Contri...
3lt6 12392 3 is less than 6. (Contri...
2lt6 12393 2 is less than 6. (Contri...
1lt6 12394 1 is less than 6. (Contri...
6lt7 12395 6 is less than 7. (Contri...
5lt7 12396 5 is less than 7. (Contri...
4lt7 12397 4 is less than 7. (Contri...
3lt7 12398 3 is less than 7. (Contri...
2lt7 12399 2 is less than 7. (Contri...
1lt7 12400 1 is less than 7. (Contri...
7lt8 12401 7 is less than 8. (Contri...
6lt8 12402 6 is less than 8. (Contri...
5lt8 12403 5 is less than 8. (Contri...
4lt8 12404 4 is less than 8. (Contri...
3lt8 12405 3 is less than 8. (Contri...
2lt8 12406 2 is less than 8. (Contri...
1lt8 12407 1 is less than 8. (Contri...
8lt9 12408 8 is less than 9. (Contri...
7lt9 12409 7 is less than 9. (Contri...
6lt9 12410 6 is less than 9. (Contri...
5lt9 12411 5 is less than 9. (Contri...
4lt9 12412 4 is less than 9. (Contri...
3lt9 12413 3 is less than 9. (Contri...
2lt9 12414 2 is less than 9. (Contri...
1lt9 12415 1 is less than 9. (Contri...
0ne2 12416 0 is not equal to 2. (Con...
1ne2 12417 1 is not equal to 2. (Con...
1le2 12418 1 is less than or equal to...
2cnne0 12419 2 is a nonzero complex num...
2rene0 12420 2 is a nonzero real number...
1le3 12421 1 is less than or equal to...
neg1mulneg1e1 12422 ` -u 1 x. -u 1 ` is 1. (C...
halfre 12423 One-half is real. (Contri...
halfcn 12424 One-half is a complex numb...
halfgt0 12425 One-half is greater than z...
halfge0 12426 One-half is not negative. ...
halflt1 12427 One-half is less than one....
1mhlfehlf 12428 Prove that 1 - 1/2 = 1/2. ...
8th4div3 12429 An eighth of four thirds i...
halfpm6th 12430 One half plus or minus one...
it0e0 12431 i times 0 equals 0. (Cont...
2mulicn 12432 ` ( 2 x. _i ) e. CC ` . (...
2muline0 12433 ` ( 2 x. _i ) =/= 0 ` . (...
halfcl 12434 Closure of half of a numbe...
rehalfcl 12435 Real closure of half. (Co...
half0 12436 Half of a number is zero i...
2halves 12437 Two halves make a whole. ...
halfpos2 12438 A number is positive iff i...
halfpos 12439 A positive number is great...
halfnneg2 12440 A number is nonnegative if...
halfaddsubcl 12441 Closure of half-sum and ha...
halfaddsub 12442 Sum and difference of half...
subhalfhalf 12443 Subtracting the half of a ...
lt2halves 12444 A sum is less than the who...
addltmul 12445 Sum is less than product f...
nominpos 12446 There is no smallest posit...
avglt1 12447 Ordering property for aver...
avglt2 12448 Ordering property for aver...
avgle1 12449 Ordering property for aver...
avgle2 12450 Ordering property for aver...
avgle 12451 The average of two numbers...
2timesd 12452 Two times a number. (Cont...
times2d 12453 A number times 2. (Contri...
halfcld 12454 Closure of half of a numbe...
2halvesd 12455 Two halves make a whole. ...
rehalfcld 12456 Real closure of half. (Co...
lt2halvesd 12457 A sum is less than the who...
rehalfcli 12458 Half a real number is real...
lt2addmuld 12459 If two real numbers are le...
add1p1 12460 Adding two times 1 to a nu...
sub1m1 12461 Subtracting two times 1 fr...
cnm2m1cnm3 12462 Subtracting 2 and afterwar...
xp1d2m1eqxm1d2 12463 A complex number increased...
div4p1lem1div2 12464 An integer greater than 5,...
nnunb 12465 The set of positive intege...
arch 12466 Archimedean property of re...
nnrecl 12467 There exists a positive in...
bndndx 12468 A bounded real sequence ` ...
elnn0 12471 Nonnegative integers expre...
nnssnn0 12472 Positive naturals are a su...
nn0ssre 12473 Nonnegative integers are a...
nn0sscn 12474 Nonnegative integers are a...
nn0ex 12475 The set of nonnegative int...
nnnn0 12476 A positive integer is a no...
nnnn0i 12477 A positive integer is a no...
nn0re 12478 A nonnegative integer is a...
nn0cn 12479 A nonnegative integer is a...
nn0rei 12480 A nonnegative integer is a...
nn0cni 12481 A nonnegative integer is a...
dfn2 12482 The set of positive intege...
elnnne0 12483 The positive integer prope...
0nn0 12484 0 is a nonnegative integer...
1nn0 12485 1 is a nonnegative integer...
2nn0 12486 2 is a nonnegative integer...
3nn0 12487 3 is a nonnegative integer...
4nn0 12488 4 is a nonnegative integer...
5nn0 12489 5 is a nonnegative integer...
6nn0 12490 6 is a nonnegative integer...
7nn0 12491 7 is a nonnegative integer...
8nn0 12492 8 is a nonnegative integer...
9nn0 12493 9 is a nonnegative integer...
nn0ge0 12494 A nonnegative integer is g...
nn0nlt0 12495 A nonnegative integer is n...
nn0ge0i 12496 Nonnegative integers are n...
nn0le0eq0 12497 A nonnegative integer is l...
nn0p1gt0 12498 A nonnegative integer incr...
nnnn0addcl 12499 A positive integer plus a ...
nn0nnaddcl 12500 A nonnegative integer plus...
0mnnnnn0 12501 The result of subtracting ...
un0addcl 12502 If ` S ` is closed under a...
un0mulcl 12503 If ` S ` is closed under m...
nn0addcl 12504 Closure of addition of non...
nn0mulcl 12505 Closure of multiplication ...
nn0addcli 12506 Closure of addition of non...
nn0mulcli 12507 Closure of multiplication ...
nn0p1nn 12508 A nonnegative integer plus...
peano2nn0 12509 Second Peano postulate for...
nnm1nn0 12510 A positive integer minus 1...
elnn0nn 12511 The nonnegative integer pr...
elnnnn0 12512 The positive integer prope...
elnnnn0b 12513 The positive integer prope...
elnnnn0c 12514 The positive integer prope...
nn0addge1 12515 A number is less than or e...
nn0addge2 12516 A number is less than or e...
nn0addge1i 12517 A number is less than or e...
nn0addge2i 12518 A number is less than or e...
nn0sub 12519 Subtraction of nonnegative...
ltsubnn0 12520 Subtracting a nonnegative ...
nn0negleid 12521 A nonnegative integer is g...
difgtsumgt 12522 If the difference of a rea...
nn0le2xi 12523 A nonnegative integer is l...
nn0lele2xi 12524 'Less than or equal to' im...
fcdmnn0supp 12525 Two ways to write the supp...
fcdmnn0fsupp 12526 A function into ` NN0 ` is...
fcdmnn0suppg 12527 Version of ~ fcdmnn0supp a...
fcdmnn0fsuppg 12528 Version of ~ fcdmnn0fsupp ...
nnnn0d 12529 A positive integer is a no...
nn0red 12530 A nonnegative integer is a...
nn0cnd 12531 A nonnegative integer is a...
nn0ge0d 12532 A nonnegative integer is g...
nn0addcld 12533 Closure of addition of non...
nn0mulcld 12534 Closure of multiplication ...
nn0readdcl 12535 Closure law for addition o...
nn0n0n1ge2 12536 A nonnegative integer whic...
nn0n0n1ge2b 12537 A nonnegative integer is n...
nn0ge2m1nn 12538 If a nonnegative integer i...
nn0ge2m1nn0 12539 If a nonnegative integer i...
nn0nndivcl 12540 Closure law for dividing o...
elxnn0 12543 An extended nonnegative in...
nn0ssxnn0 12544 The standard nonnegative i...
nn0xnn0 12545 A standard nonnegative int...
xnn0xr 12546 An extended nonnegative in...
0xnn0 12547 Zero is an extended nonneg...
pnf0xnn0 12548 Positive infinity is an ex...
nn0nepnf 12549 No standard nonnegative in...
nn0xnn0d 12550 A standard nonnegative int...
nn0nepnfd 12551 No standard nonnegative in...
xnn0nemnf 12552 No extended nonnegative in...
xnn0xrnemnf 12553 The extended nonnegative i...
xnn0nnn0pnf 12554 An extended nonnegative in...
elz 12557 Membership in the set of i...
nnnegz 12558 The negative of a positive...
zre 12559 An integer is a real. (Co...
zcn 12560 An integer is a complex nu...
zrei 12561 An integer is a real numbe...
zssre 12562 The integers are a subset ...
zsscn 12563 The integers are a subset ...
zex 12564 The set of integers exists...
elnnz 12565 Positive integer property ...
0z 12566 Zero is an integer. (Cont...
0zd 12567 Zero is an integer, deduct...
elnn0z 12568 Nonnegative integer proper...
elznn0nn 12569 Integer property expressed...
elznn0 12570 Integer property expressed...
elznn 12571 Integer property expressed...
zle0orge1 12572 There is no integer in the...
elz2 12573 Membership in the set of i...
dfz2 12574 Alternative definition of ...
zexALT 12575 Alternate proof of ~ zex ....
nnz 12576 A positive integer is an i...
nnssz 12577 Positive integers are a su...
nn0ssz 12578 Nonnegative integers are a...
nnzOLD 12579 Obsolete version of ~ nnz ...
nn0z 12580 A nonnegative integer is a...
nn0zd 12581 A nonnegative integer is a...
nnzd 12582 A positive integer is an i...
nnzi 12583 A positive integer is an i...
nn0zi 12584 A nonnegative integer is a...
elnnz1 12585 Positive integer property ...
znnnlt1 12586 An integer is not a positi...
nnzrab 12587 Positive integers expresse...
nn0zrab 12588 Nonnegative integers expre...
1z 12589 One is an integer. (Contr...
1zzd 12590 One is an integer, deducti...
2z 12591 2 is an integer. (Contrib...
3z 12592 3 is an integer. (Contrib...
4z 12593 4 is an integer. (Contrib...
znegcl 12594 Closure law for negative i...
neg1z 12595 -1 is an integer. (Contri...
znegclb 12596 A complex number is an int...
nn0negz 12597 The negative of a nonnegat...
nn0negzi 12598 The negative of a nonnegat...
zaddcl 12599 Closure of addition of int...
peano2z 12600 Second Peano postulate gen...
zsubcl 12601 Closure of subtraction of ...
peano2zm 12602 "Reverse" second Peano pos...
zletr 12603 Transitive law of ordering...
zrevaddcl 12604 Reverse closure law for ad...
znnsub 12605 The positive difference of...
znn0sub 12606 The nonnegative difference...
nzadd 12607 The sum of a real number n...
zmulcl 12608 Closure of multiplication ...
zltp1le 12609 Integer ordering relation....
zleltp1 12610 Integer ordering relation....
zlem1lt 12611 Integer ordering relation....
zltlem1 12612 Integer ordering relation....
zgt0ge1 12613 An integer greater than ` ...
nnleltp1 12614 Positive integer ordering ...
nnltp1le 12615 Positive integer ordering ...
nnaddm1cl 12616 Closure of addition of pos...
nn0ltp1le 12617 Nonnegative integer orderi...
nn0leltp1 12618 Nonnegative integer orderi...
nn0ltlem1 12619 Nonnegative integer orderi...
nn0sub2 12620 Subtraction of nonnegative...
nn0lt10b 12621 A nonnegative integer less...
nn0lt2 12622 A nonnegative integer less...
nn0le2is012 12623 A nonnegative integer whic...
nn0lem1lt 12624 Nonnegative integer orderi...
nnlem1lt 12625 Positive integer ordering ...
nnltlem1 12626 Positive integer ordering ...
nnm1ge0 12627 A positive integer decreas...
nn0ge0div 12628 Division of a nonnegative ...
zdiv 12629 Two ways to express " ` M ...
zdivadd 12630 Property of divisibility: ...
zdivmul 12631 Property of divisibility: ...
zextle 12632 An extensionality-like pro...
zextlt 12633 An extensionality-like pro...
recnz 12634 The reciprocal of a number...
btwnnz 12635 A number between an intege...
gtndiv 12636 A larger number does not d...
halfnz 12637 One-half is not an integer...
3halfnz 12638 Three halves is not an int...
suprzcl 12639 The supremum of a bounded-...
prime 12640 Two ways to express " ` A ...
msqznn 12641 The square of a nonzero in...
zneo 12642 No even integer equals an ...
nneo 12643 A positive integer is even...
nneoi 12644 A positive integer is even...
zeo 12645 An integer is even or odd....
zeo2 12646 An integer is even or odd ...
peano2uz2 12647 Second Peano postulate for...
peano5uzi 12648 Peano's inductive postulat...
peano5uzti 12649 Peano's inductive postulat...
dfuzi 12650 An expression for the uppe...
uzind 12651 Induction on the upper int...
uzind2 12652 Induction on the upper int...
uzind3 12653 Induction on the upper int...
nn0ind 12654 Principle of Mathematical ...
nn0indALT 12655 Principle of Mathematical ...
nn0indd 12656 Principle of Mathematical ...
fzind 12657 Induction on the integers ...
fnn0ind 12658 Induction on the integers ...
nn0ind-raph 12659 Principle of Mathematical ...
zindd 12660 Principle of Mathematical ...
fzindd 12661 Induction on the integers ...
btwnz 12662 Any real number can be san...
zred 12663 An integer is a real numbe...
zcnd 12664 An integer is a complex nu...
znegcld 12665 Closure law for negative i...
peano2zd 12666 Deduction from second Pean...
zaddcld 12667 Closure of addition of int...
zsubcld 12668 Closure of subtraction of ...
zmulcld 12669 Closure of multiplication ...
znnn0nn 12670 The negative of a negative...
zadd2cl 12671 Increasing an integer by 2...
zriotaneg 12672 The negative of the unique...
suprfinzcl 12673 The supremum of a nonempty...
9p1e10 12676 9 + 1 = 10. (Contributed ...
dfdec10 12677 Version of the definition ...
decex 12678 A decimal number is a set....
deceq1 12679 Equality theorem for the d...
deceq2 12680 Equality theorem for the d...
deceq1i 12681 Equality theorem for the d...
deceq2i 12682 Equality theorem for the d...
deceq12i 12683 Equality theorem for the d...
numnncl 12684 Closure for a numeral (wit...
num0u 12685 Add a zero in the units pl...
num0h 12686 Add a zero in the higher p...
numcl 12687 Closure for a decimal inte...
numsuc 12688 The successor of a decimal...
deccl 12689 Closure for a numeral. (C...
10nn 12690 10 is a positive integer. ...
10pos 12691 The number 10 is positive....
10nn0 12692 10 is a nonnegative intege...
10re 12693 The number 10 is real. (C...
decnncl 12694 Closure for a numeral. (C...
dec0u 12695 Add a zero in the units pl...
dec0h 12696 Add a zero in the higher p...
numnncl2 12697 Closure for a decimal inte...
decnncl2 12698 Closure for a decimal inte...
numlt 12699 Comparing two decimal inte...
numltc 12700 Comparing two decimal inte...
le9lt10 12701 A "decimal digit" (i.e. a ...
declt 12702 Comparing two decimal inte...
decltc 12703 Comparing two decimal inte...
declth 12704 Comparing two decimal inte...
decsuc 12705 The successor of a decimal...
3declth 12706 Comparing two decimal inte...
3decltc 12707 Comparing two decimal inte...
decle 12708 Comparing two decimal inte...
decleh 12709 Comparing two decimal inte...
declei 12710 Comparing a digit to a dec...
numlti 12711 Comparing a digit to a dec...
declti 12712 Comparing a digit to a dec...
decltdi 12713 Comparing a digit to a dec...
numsucc 12714 The successor of a decimal...
decsucc 12715 The successor of a decimal...
1e0p1 12716 The successor of zero. (C...
dec10p 12717 Ten plus an integer. (Con...
numma 12718 Perform a multiply-add of ...
nummac 12719 Perform a multiply-add of ...
numma2c 12720 Perform a multiply-add of ...
numadd 12721 Add two decimal integers `...
numaddc 12722 Add two decimal integers `...
nummul1c 12723 The product of a decimal i...
nummul2c 12724 The product of a decimal i...
decma 12725 Perform a multiply-add of ...
decmac 12726 Perform a multiply-add of ...
decma2c 12727 Perform a multiply-add of ...
decadd 12728 Add two numerals ` M ` and...
decaddc 12729 Add two numerals ` M ` and...
decaddc2 12730 Add two numerals ` M ` and...
decrmanc 12731 Perform a multiply-add of ...
decrmac 12732 Perform a multiply-add of ...
decaddm10 12733 The sum of two multiples o...
decaddi 12734 Add two numerals ` M ` and...
decaddci 12735 Add two numerals ` M ` and...
decaddci2 12736 Add two numerals ` M ` and...
decsubi 12737 Difference between a numer...
decmul1 12738 The product of a numeral w...
decmul1c 12739 The product of a numeral w...
decmul2c 12740 The product of a numeral w...
decmulnc 12741 The product of a numeral w...
11multnc 12742 The product of 11 (as nume...
decmul10add 12743 A multiplication of a numb...
6p5lem 12744 Lemma for ~ 6p5e11 and rel...
5p5e10 12745 5 + 5 = 10. (Contributed ...
6p4e10 12746 6 + 4 = 10. (Contributed ...
6p5e11 12747 6 + 5 = 11. (Contributed ...
6p6e12 12748 6 + 6 = 12. (Contributed ...
7p3e10 12749 7 + 3 = 10. (Contributed ...
7p4e11 12750 7 + 4 = 11. (Contributed ...
7p5e12 12751 7 + 5 = 12. (Contributed ...
7p6e13 12752 7 + 6 = 13. (Contributed ...
7p7e14 12753 7 + 7 = 14. (Contributed ...
8p2e10 12754 8 + 2 = 10. (Contributed ...
8p3e11 12755 8 + 3 = 11. (Contributed ...
8p4e12 12756 8 + 4 = 12. (Contributed ...
8p5e13 12757 8 + 5 = 13. (Contributed ...
8p6e14 12758 8 + 6 = 14. (Contributed ...
8p7e15 12759 8 + 7 = 15. (Contributed ...
8p8e16 12760 8 + 8 = 16. (Contributed ...
9p2e11 12761 9 + 2 = 11. (Contributed ...
9p3e12 12762 9 + 3 = 12. (Contributed ...
9p4e13 12763 9 + 4 = 13. (Contributed ...
9p5e14 12764 9 + 5 = 14. (Contributed ...
9p6e15 12765 9 + 6 = 15. (Contributed ...
9p7e16 12766 9 + 7 = 16. (Contributed ...
9p8e17 12767 9 + 8 = 17. (Contributed ...
9p9e18 12768 9 + 9 = 18. (Contributed ...
10p10e20 12769 10 + 10 = 20. (Contribute...
10m1e9 12770 10 - 1 = 9. (Contributed ...
4t3lem 12771 Lemma for ~ 4t3e12 and rel...
4t3e12 12772 4 times 3 equals 12. (Con...
4t4e16 12773 4 times 4 equals 16. (Con...
5t2e10 12774 5 times 2 equals 10. (Con...
5t3e15 12775 5 times 3 equals 15. (Con...
5t4e20 12776 5 times 4 equals 20. (Con...
5t5e25 12777 5 times 5 equals 25. (Con...
6t2e12 12778 6 times 2 equals 12. (Con...
6t3e18 12779 6 times 3 equals 18. (Con...
6t4e24 12780 6 times 4 equals 24. (Con...
6t5e30 12781 6 times 5 equals 30. (Con...
6t6e36 12782 6 times 6 equals 36. (Con...
7t2e14 12783 7 times 2 equals 14. (Con...
7t3e21 12784 7 times 3 equals 21. (Con...
7t4e28 12785 7 times 4 equals 28. (Con...
7t5e35 12786 7 times 5 equals 35. (Con...
7t6e42 12787 7 times 6 equals 42. (Con...
7t7e49 12788 7 times 7 equals 49. (Con...
8t2e16 12789 8 times 2 equals 16. (Con...
8t3e24 12790 8 times 3 equals 24. (Con...
8t4e32 12791 8 times 4 equals 32. (Con...
8t5e40 12792 8 times 5 equals 40. (Con...
8t6e48 12793 8 times 6 equals 48. (Con...
8t7e56 12794 8 times 7 equals 56. (Con...
8t8e64 12795 8 times 8 equals 64. (Con...
9t2e18 12796 9 times 2 equals 18. (Con...
9t3e27 12797 9 times 3 equals 27. (Con...
9t4e36 12798 9 times 4 equals 36. (Con...
9t5e45 12799 9 times 5 equals 45. (Con...
9t6e54 12800 9 times 6 equals 54. (Con...
9t7e63 12801 9 times 7 equals 63. (Con...
9t8e72 12802 9 times 8 equals 72. (Con...
9t9e81 12803 9 times 9 equals 81. (Con...
9t11e99 12804 9 times 11 equals 99. (Co...
9lt10 12805 9 is less than 10. (Contr...
8lt10 12806 8 is less than 10. (Contr...
7lt10 12807 7 is less than 10. (Contr...
6lt10 12808 6 is less than 10. (Contr...
5lt10 12809 5 is less than 10. (Contr...
4lt10 12810 4 is less than 10. (Contr...
3lt10 12811 3 is less than 10. (Contr...
2lt10 12812 2 is less than 10. (Contr...
1lt10 12813 1 is less than 10. (Contr...
decbin0 12814 Decompose base 4 into base...
decbin2 12815 Decompose base 4 into base...
decbin3 12816 Decompose base 4 into base...
halfthird 12817 Half minus a third. (Cont...
5recm6rec 12818 One fifth minus one sixth....
uzval 12821 The value of the upper int...
uzf 12822 The domain and codomain of...
eluz1 12823 Membership in the upper se...
eluzel2 12824 Implication of membership ...
eluz2 12825 Membership in an upper set...
eluzmn 12826 Membership in an earlier u...
eluz1i 12827 Membership in an upper set...
eluzuzle 12828 An integer in an upper set...
eluzelz 12829 A member of an upper set o...
eluzelre 12830 A member of an upper set o...
eluzelcn 12831 A member of an upper set o...
eluzle 12832 Implication of membership ...
eluz 12833 Membership in an upper set...
uzid 12834 Membership of the least me...
uzidd 12835 Membership of the least me...
uzn0 12836 The upper integers are all...
uztrn 12837 Transitive law for sets of...
uztrn2 12838 Transitive law for sets of...
uzneg 12839 Contraposition law for upp...
uzssz 12840 An upper set of integers i...
uzssre 12841 An upper set of integers i...
uzss 12842 Subset relationship for tw...
uztric 12843 Totality of the ordering r...
uz11 12844 The upper integers functio...
eluzp1m1 12845 Membership in the next upp...
eluzp1l 12846 Strict ordering implied by...
eluzp1p1 12847 Membership in the next upp...
eluzadd 12848 Membership in a later uppe...
eluzsub 12849 Membership in an earlier u...
eluzaddi 12850 Membership in a later uppe...
eluzaddiOLD 12851 Obsolete version of ~ eluz...
eluzsubi 12852 Membership in an earlier u...
eluzsubiOLD 12853 Obsolete version of ~ eluz...
eluzaddOLD 12854 Obsolete version of ~ eluz...
eluzsubOLD 12855 Obsolete version of ~ eluz...
subeluzsub 12856 Membership of a difference...
uzm1 12857 Choices for an element of ...
uznn0sub 12858 The nonnegative difference...
uzin 12859 Intersection of two upper ...
uzp1 12860 Choices for an element of ...
nn0uz 12861 Nonnegative integers expre...
nnuz 12862 Positive integers expresse...
elnnuz 12863 A positive integer express...
elnn0uz 12864 A nonnegative integer expr...
eluz2nn 12865 An integer greater than or...
eluz4eluz2 12866 An integer greater than or...
eluz4nn 12867 An integer greater than or...
eluzge2nn0 12868 If an integer is greater t...
eluz2n0 12869 An integer greater than or...
uzuzle23 12870 An integer in the upper se...
eluzge3nn 12871 If an integer is greater t...
uz3m2nn 12872 An integer greater than or...
1eluzge0 12873 1 is an integer greater th...
2eluzge0 12874 2 is an integer greater th...
2eluzge1 12875 2 is an integer greater th...
uznnssnn 12876 The upper integers startin...
raluz 12877 Restricted universal quant...
raluz2 12878 Restricted universal quant...
rexuz 12879 Restricted existential qua...
rexuz2 12880 Restricted existential qua...
2rexuz 12881 Double existential quantif...
peano2uz 12882 Second Peano postulate for...
peano2uzs 12883 Second Peano postulate for...
peano2uzr 12884 Reversed second Peano axio...
uzaddcl 12885 Addition closure law for a...
nn0pzuz 12886 The sum of a nonnegative i...
uzind4 12887 Induction on the upper set...
uzind4ALT 12888 Induction on the upper set...
uzind4s 12889 Induction on the upper set...
uzind4s2 12890 Induction on the upper set...
uzind4i 12891 Induction on the upper int...
uzwo 12892 Well-ordering principle: a...
uzwo2 12893 Well-ordering principle: a...
nnwo 12894 Well-ordering principle: a...
nnwof 12895 Well-ordering principle: a...
nnwos 12896 Well-ordering principle: a...
indstr 12897 Strong Mathematical Induct...
eluznn0 12898 Membership in a nonnegativ...
eluznn 12899 Membership in a positive u...
eluz2b1 12900 Two ways to say "an intege...
eluz2gt1 12901 An integer greater than or...
eluz2b2 12902 Two ways to say "an intege...
eluz2b3 12903 Two ways to say "an intege...
uz2m1nn 12904 One less than an integer g...
1nuz2 12905 1 is not in ` ( ZZ>= `` 2 ...
elnn1uz2 12906 A positive integer is eith...
uz2mulcl 12907 Closure of multiplication ...
indstr2 12908 Strong Mathematical Induct...
uzinfi 12909 Extract the lower bound of...
nninf 12910 The infimum of the set of ...
nn0inf 12911 The infimum of the set of ...
infssuzle 12912 The infimum of a subset of...
infssuzcl 12913 The infimum of a subset of...
ublbneg 12914 The image under negation o...
eqreznegel 12915 Two ways to express the im...
supminf 12916 The supremum of a bounded-...
lbzbi 12917 If a set of reals is bound...
zsupss 12918 Any nonempty bounded subse...
suprzcl2 12919 The supremum of a bounded-...
suprzub 12920 The supremum of a bounded-...
uzsupss 12921 Any bounded subset of an u...
nn01to3 12922 A (nonnegative) integer be...
nn0ge2m1nnALT 12923 Alternate proof of ~ nn0ge...
uzwo3 12924 Well-ordering principle: a...
zmin 12925 There is a unique smallest...
zmax 12926 There is a unique largest ...
zbtwnre 12927 There is a unique integer ...
rebtwnz 12928 There is a unique greatest...
elq 12931 Membership in the set of r...
qmulz 12932 If ` A ` is rational, then...
znq 12933 The ratio of an integer an...
qre 12934 A rational number is a rea...
zq 12935 An integer is a rational n...
qred 12936 A rational number is a rea...
zssq 12937 The integers are a subset ...
nn0ssq 12938 The nonnegative integers a...
nnssq 12939 The positive integers are ...
qssre 12940 The rationals are a subset...
qsscn 12941 The rationals are a subset...
qex 12942 The set of rational number...
nnq 12943 A positive integer is rati...
qcn 12944 A rational number is a com...
qexALT 12945 Alternate proof of ~ qex ....
qaddcl 12946 Closure of addition of rat...
qnegcl 12947 Closure law for the negati...
qmulcl 12948 Closure of multiplication ...
qsubcl 12949 Closure of subtraction of ...
qreccl 12950 Closure of reciprocal of r...
qdivcl 12951 Closure of division of rat...
qrevaddcl 12952 Reverse closure law for ad...
nnrecq 12953 The reciprocal of a positi...
irradd 12954 The sum of an irrational n...
irrmul 12955 The product of an irration...
elpq 12956 A positive rational is the...
elpqb 12957 A class is a positive rati...
rpnnen1lem2 12958 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem1 12959 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem3 12960 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem4 12961 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem5 12962 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem6 12963 Lemma for ~ rpnnen1 . (Co...
rpnnen1 12964 One half of ~ rpnnen , whe...
reexALT 12965 Alternate proof of ~ reex ...
cnref1o 12966 There is a natural one-to-...
cnexALT 12967 The set of complex numbers...
xrex 12968 The set of extended reals ...
addex 12969 The addition operation is ...
mulex 12970 The multiplication operati...
elrp 12973 Membership in the set of p...
elrpii 12974 Membership in the set of p...
1rp 12975 1 is a positive real. (Co...
2rp 12976 2 is a positive real. (Co...
3rp 12977 3 is a positive real. (Co...
rpssre 12978 The positive reals are a s...
rpre 12979 A positive real is a real....
rpxr 12980 A positive real is an exte...
rpcn 12981 A positive real is a compl...
nnrp 12982 A positive integer is a po...
rpgt0 12983 A positive real is greater...
rpge0 12984 A positive real is greater...
rpregt0 12985 A positive real is a posit...
rprege0 12986 A positive real is a nonne...
rpne0 12987 A positive real is nonzero...
rprene0 12988 A positive real is a nonze...
rpcnne0 12989 A positive real is a nonze...
rpcndif0 12990 A positive real number is ...
ralrp 12991 Quantification over positi...
rexrp 12992 Quantification over positi...
rpaddcl 12993 Closure law for addition o...
rpmulcl 12994 Closure law for multiplica...
rpmtmip 12995 "Minus times minus is plus...
rpdivcl 12996 Closure law for division o...
rpreccl 12997 Closure law for reciprocat...
rphalfcl 12998 Closure law for half of a ...
rpgecl 12999 A number greater than or e...
rphalflt 13000 Half of a positive real is...
rerpdivcl 13001 Closure law for division o...
ge0p1rp 13002 A nonnegative number plus ...
rpneg 13003 Either a nonzero real or i...
negelrp 13004 Elementhood of a negation ...
negelrpd 13005 The negation of a negative...
0nrp 13006 Zero is not a positive rea...
ltsubrp 13007 Subtracting a positive rea...
ltaddrp 13008 Adding a positive number t...
difrp 13009 Two ways to say one number...
elrpd 13010 Membership in the set of p...
nnrpd 13011 A positive integer is a po...
zgt1rpn0n1 13012 An integer greater than 1 ...
rpred 13013 A positive real is a real....
rpxrd 13014 A positive real is an exte...
rpcnd 13015 A positive real is a compl...
rpgt0d 13016 A positive real is greater...
rpge0d 13017 A positive real is greater...
rpne0d 13018 A positive real is nonzero...
rpregt0d 13019 A positive real is real an...
rprege0d 13020 A positive real is real an...
rprene0d 13021 A positive real is a nonze...
rpcnne0d 13022 A positive real is a nonze...
rpreccld 13023 Closure law for reciprocat...
rprecred 13024 Closure law for reciprocat...
rphalfcld 13025 Closure law for half of a ...
reclt1d 13026 The reciprocal of a positi...
recgt1d 13027 The reciprocal of a positi...
rpaddcld 13028 Closure law for addition o...
rpmulcld 13029 Closure law for multiplica...
rpdivcld 13030 Closure law for division o...
ltrecd 13031 The reciprocal of both sid...
lerecd 13032 The reciprocal of both sid...
ltrec1d 13033 Reciprocal swap in a 'less...
lerec2d 13034 Reciprocal swap in a 'less...
lediv2ad 13035 Division of both sides of ...
ltdiv2d 13036 Division of a positive num...
lediv2d 13037 Division of a positive num...
ledivdivd 13038 Invert ratios of positive ...
divge1 13039 The ratio of a number over...
divlt1lt 13040 A real number divided by a...
divle1le 13041 A real number divided by a...
ledivge1le 13042 If a number is less than o...
ge0p1rpd 13043 A nonnegative number plus ...
rerpdivcld 13044 Closure law for division o...
ltsubrpd 13045 Subtracting a positive rea...
ltaddrpd 13046 Adding a positive number t...
ltaddrp2d 13047 Adding a positive number t...
ltmulgt11d 13048 Multiplication by a number...
ltmulgt12d 13049 Multiplication by a number...
gt0divd 13050 Division of a positive num...
ge0divd 13051 Division of a nonnegative ...
rpgecld 13052 A number greater than or e...
divge0d 13053 The ratio of nonnegative a...
ltmul1d 13054 The ratio of nonnegative a...
ltmul2d 13055 Multiplication of both sid...
lemul1d 13056 Multiplication of both sid...
lemul2d 13057 Multiplication of both sid...
ltdiv1d 13058 Division of both sides of ...
lediv1d 13059 Division of both sides of ...
ltmuldivd 13060 'Less than' relationship b...
ltmuldiv2d 13061 'Less than' relationship b...
lemuldivd 13062 'Less than or equal to' re...
lemuldiv2d 13063 'Less than or equal to' re...
ltdivmuld 13064 'Less than' relationship b...
ltdivmul2d 13065 'Less than' relationship b...
ledivmuld 13066 'Less than or equal to' re...
ledivmul2d 13067 'Less than or equal to' re...
ltmul1dd 13068 The ratio of nonnegative a...
ltmul2dd 13069 Multiplication of both sid...
ltdiv1dd 13070 Division of both sides of ...
lediv1dd 13071 Division of both sides of ...
lediv12ad 13072 Comparison of ratio of two...
mul2lt0rlt0 13073 If the result of a multipl...
mul2lt0rgt0 13074 If the result of a multipl...
mul2lt0llt0 13075 If the result of a multipl...
mul2lt0lgt0 13076 If the result of a multipl...
mul2lt0bi 13077 If the result of a multipl...
prodge0rd 13078 Infer that a multiplicand ...
prodge0ld 13079 Infer that a multiplier is...
ltdiv23d 13080 Swap denominator with othe...
lediv23d 13081 Swap denominator with othe...
lt2mul2divd 13082 The ratio of nonnegative a...
nnledivrp 13083 Division of a positive int...
nn0ledivnn 13084 Division of a nonnegative ...
addlelt 13085 If the sum of a real numbe...
ltxr 13092 The 'less than' binary rel...
elxr 13093 Membership in the set of e...
xrnemnf 13094 An extended real other tha...
xrnepnf 13095 An extended real other tha...
xrltnr 13096 The extended real 'less th...
ltpnf 13097 Any (finite) real is less ...
ltpnfd 13098 Any (finite) real is less ...
0ltpnf 13099 Zero is less than plus inf...
mnflt 13100 Minus infinity is less tha...
mnfltd 13101 Minus infinity is less tha...
mnflt0 13102 Minus infinity is less tha...
mnfltpnf 13103 Minus infinity is less tha...
mnfltxr 13104 Minus infinity is less tha...
pnfnlt 13105 No extended real is greate...
nltmnf 13106 No extended real is less t...
pnfge 13107 Plus infinity is an upper ...
xnn0n0n1ge2b 13108 An extended nonnegative in...
0lepnf 13109 0 less than or equal to po...
xnn0ge0 13110 An extended nonnegative in...
mnfle 13111 Minus infinity is less tha...
mnfled 13112 Minus infinity is less tha...
xrltnsym 13113 Ordering on the extended r...
xrltnsym2 13114 'Less than' is antisymmetr...
xrlttri 13115 Ordering on the extended r...
xrlttr 13116 Ordering on the extended r...
xrltso 13117 'Less than' is a strict or...
xrlttri2 13118 Trichotomy law for 'less t...
xrlttri3 13119 Trichotomy law for 'less t...
xrleloe 13120 'Less than or equal' expre...
xrleltne 13121 'Less than or equal to' im...
xrltlen 13122 'Less than' expressed in t...
dfle2 13123 Alternative definition of ...
dflt2 13124 Alternative definition of ...
xrltle 13125 'Less than' implies 'less ...
xrltled 13126 'Less than' implies 'less ...
xrleid 13127 'Less than or equal to' is...
xrleidd 13128 'Less than or equal to' is...
xrletri 13129 Trichotomy law for extende...
xrletri3 13130 Trichotomy law for extende...
xrletrid 13131 Trichotomy law for extende...
xrlelttr 13132 Transitive law for orderin...
xrltletr 13133 Transitive law for orderin...
xrletr 13134 Transitive law for orderin...
xrlttrd 13135 Transitive law for orderin...
xrlelttrd 13136 Transitive law for orderin...
xrltletrd 13137 Transitive law for orderin...
xrletrd 13138 Transitive law for orderin...
xrltne 13139 'Less than' implies not eq...
nltpnft 13140 An extended real is not le...
xgepnf 13141 An extended real which is ...
ngtmnft 13142 An extended real is not gr...
xlemnf 13143 An extended real which is ...
xrrebnd 13144 An extended real is real i...
xrre 13145 A way of proving that an e...
xrre2 13146 An extended real between t...
xrre3 13147 A way of proving that an e...
ge0gtmnf 13148 A nonnegative extended rea...
ge0nemnf 13149 A nonnegative extended rea...
xrrege0 13150 A nonnegative extended rea...
xrmax1 13151 An extended real is less t...
xrmax2 13152 An extended real is less t...
xrmin1 13153 The minimum of two extende...
xrmin2 13154 The minimum of two extende...
xrmaxeq 13155 The maximum of two extende...
xrmineq 13156 The minimum of two extende...
xrmaxlt 13157 Two ways of saying the max...
xrltmin 13158 Two ways of saying an exte...
xrmaxle 13159 Two ways of saying the max...
xrlemin 13160 Two ways of saying a numbe...
max1 13161 A number is less than or e...
max1ALT 13162 A number is less than or e...
max2 13163 A number is less than or e...
2resupmax 13164 The supremum of two real n...
min1 13165 The minimum of two numbers...
min2 13166 The minimum of two numbers...
maxle 13167 Two ways of saying the max...
lemin 13168 Two ways of saying a numbe...
maxlt 13169 Two ways of saying the max...
ltmin 13170 Two ways of saying a numbe...
lemaxle 13171 A real number which is les...
max0sub 13172 Decompose a real number in...
ifle 13173 An if statement transforms...
z2ge 13174 There exists an integer gr...
qbtwnre 13175 The rational numbers are d...
qbtwnxr 13176 The rational numbers are d...
qsqueeze 13177 If a nonnegative real is l...
qextltlem 13178 Lemma for ~ qextlt and qex...
qextlt 13179 An extensionality-like pro...
qextle 13180 An extensionality-like pro...
xralrple 13181 Show that ` A ` is less th...
alrple 13182 Show that ` A ` is less th...
xnegeq 13183 Equality of two extended n...
xnegex 13184 A negative extended real e...
xnegpnf 13185 Minus ` +oo ` . Remark of...
xnegmnf 13186 Minus ` -oo ` . Remark of...
rexneg 13187 Minus a real number. Rema...
xneg0 13188 The negative of zero. (Co...
xnegcl 13189 Closure of extended real n...
xnegneg 13190 Extended real version of ~...
xneg11 13191 Extended real version of ~...
xltnegi 13192 Forward direction of ~ xlt...
xltneg 13193 Extended real version of ~...
xleneg 13194 Extended real version of ~...
xlt0neg1 13195 Extended real version of ~...
xlt0neg2 13196 Extended real version of ~...
xle0neg1 13197 Extended real version of ~...
xle0neg2 13198 Extended real version of ~...
xaddval 13199 Value of the extended real...
xaddf 13200 The extended real addition...
xmulval 13201 Value of the extended real...
xaddpnf1 13202 Addition of positive infin...
xaddpnf2 13203 Addition of positive infin...
xaddmnf1 13204 Addition of negative infin...
xaddmnf2 13205 Addition of negative infin...
pnfaddmnf 13206 Addition of positive and n...
mnfaddpnf 13207 Addition of negative and p...
rexadd 13208 The extended real addition...
rexsub 13209 Extended real subtraction ...
rexaddd 13210 The extended real addition...
xnn0xaddcl 13211 The extended nonnegative i...
xaddnemnf 13212 Closure of extended real a...
xaddnepnf 13213 Closure of extended real a...
xnegid 13214 Extended real version of ~...
xaddcl 13215 The extended real addition...
xaddcom 13216 The extended real addition...
xaddrid 13217 Extended real version of ~...
xaddlid 13218 Extended real version of ~...
xaddridd 13219 ` 0 ` is a right identity ...
xnn0lem1lt 13220 Extended nonnegative integ...
xnn0lenn0nn0 13221 An extended nonnegative in...
xnn0le2is012 13222 An extended nonnegative in...
xnn0xadd0 13223 The sum of two extended no...
xnegdi 13224 Extended real version of ~...
xaddass 13225 Associativity of extended ...
xaddass2 13226 Associativity of extended ...
xpncan 13227 Extended real version of ~...
xnpcan 13228 Extended real version of ~...
xleadd1a 13229 Extended real version of ~...
xleadd2a 13230 Commuted form of ~ xleadd1...
xleadd1 13231 Weakened version of ~ xlea...
xltadd1 13232 Extended real version of ~...
xltadd2 13233 Extended real version of ~...
xaddge0 13234 The sum of nonnegative ext...
xle2add 13235 Extended real version of ~...
xlt2add 13236 Extended real version of ~...
xsubge0 13237 Extended real version of ~...
xposdif 13238 Extended real version of ~...
xlesubadd 13239 Under certain conditions, ...
xmullem 13240 Lemma for ~ rexmul . (Con...
xmullem2 13241 Lemma for ~ xmulneg1 . (C...
xmulcom 13242 Extended real multiplicati...
xmul01 13243 Extended real version of ~...
xmul02 13244 Extended real version of ~...
xmulneg1 13245 Extended real version of ~...
xmulneg2 13246 Extended real version of ~...
rexmul 13247 The extended real multipli...
xmulf 13248 The extended real multipli...
xmulcl 13249 Closure of extended real m...
xmulpnf1 13250 Multiplication by plus inf...
xmulpnf2 13251 Multiplication by plus inf...
xmulmnf1 13252 Multiplication by minus in...
xmulmnf2 13253 Multiplication by minus in...
xmulpnf1n 13254 Multiplication by plus inf...
xmulrid 13255 Extended real version of ~...
xmullid 13256 Extended real version of ~...
xmulm1 13257 Extended real version of ~...
xmulasslem2 13258 Lemma for ~ xmulass . (Co...
xmulgt0 13259 Extended real version of ~...
xmulge0 13260 Extended real version of ~...
xmulasslem 13261 Lemma for ~ xmulass . (Co...
xmulasslem3 13262 Lemma for ~ xmulass . (Co...
xmulass 13263 Associativity of the exten...
xlemul1a 13264 Extended real version of ~...
xlemul2a 13265 Extended real version of ~...
xlemul1 13266 Extended real version of ~...
xlemul2 13267 Extended real version of ~...
xltmul1 13268 Extended real version of ~...
xltmul2 13269 Extended real version of ~...
xadddilem 13270 Lemma for ~ xadddi . (Con...
xadddi 13271 Distributive property for ...
xadddir 13272 Commuted version of ~ xadd...
xadddi2 13273 The assumption that the mu...
xadddi2r 13274 Commuted version of ~ xadd...
x2times 13275 Extended real version of ~...
xnegcld 13276 Closure of extended real n...
xaddcld 13277 The extended real addition...
xmulcld 13278 Closure of extended real m...
xadd4d 13279 Rearrangement of 4 terms i...
xnn0add4d 13280 Rearrangement of 4 terms i...
xrsupexmnf 13281 Adding minus infinity to a...
xrinfmexpnf 13282 Adding plus infinity to a ...
xrsupsslem 13283 Lemma for ~ xrsupss . (Co...
xrinfmsslem 13284 Lemma for ~ xrinfmss . (C...
xrsupss 13285 Any subset of extended rea...
xrinfmss 13286 Any subset of extended rea...
xrinfmss2 13287 Any subset of extended rea...
xrub 13288 By quantifying only over r...
supxr 13289 The supremum of a set of e...
supxr2 13290 The supremum of a set of e...
supxrcl 13291 The supremum of an arbitra...
supxrun 13292 The supremum of the union ...
supxrmnf 13293 Adding minus infinity to a...
supxrpnf 13294 The supremum of a set of e...
supxrunb1 13295 The supremum of an unbound...
supxrunb2 13296 The supremum of an unbound...
supxrbnd1 13297 The supremum of a bounded-...
supxrbnd2 13298 The supremum of a bounded-...
xrsup0 13299 The supremum of an empty s...
supxrub 13300 A member of a set of exten...
supxrlub 13301 The supremum of a set of e...
supxrleub 13302 The supremum of a set of e...
supxrre 13303 The real and extended real...
supxrbnd 13304 The supremum of a bounded-...
supxrgtmnf 13305 The supremum of a nonempty...
supxrre1 13306 The supremum of a nonempty...
supxrre2 13307 The supremum of a nonempty...
supxrss 13308 Smaller sets of extended r...
infxrcl 13309 The infimum of an arbitrar...
infxrlb 13310 A member of a set of exten...
infxrgelb 13311 The infimum of a set of ex...
infxrre 13312 The real and extended real...
infxrmnf 13313 The infinimum of a set of ...
xrinf0 13314 The infimum of the empty s...
infxrss 13315 Larger sets of extended re...
reltre 13316 For all real numbers there...
rpltrp 13317 For all positive real numb...
reltxrnmnf 13318 For all extended real numb...
infmremnf 13319 The infimum of the reals i...
infmrp1 13320 The infimum of the positiv...
ixxval 13329 Value of the interval func...
elixx1 13330 Membership in an interval ...
ixxf 13331 The set of intervals of ex...
ixxex 13332 The set of intervals of ex...
ixxssxr 13333 The set of intervals of ex...
elixx3g 13334 Membership in a set of ope...
ixxssixx 13335 An interval is a subset of...
ixxdisj 13336 Split an interval into dis...
ixxun 13337 Split an interval into two...
ixxin 13338 Intersection of two interv...
ixxss1 13339 Subset relationship for in...
ixxss2 13340 Subset relationship for in...
ixxss12 13341 Subset relationship for in...
ixxub 13342 Extract the upper bound of...
ixxlb 13343 Extract the lower bound of...
iooex 13344 The set of open intervals ...
iooval 13345 Value of the open interval...
ioo0 13346 An empty open interval of ...
ioon0 13347 An open interval of extend...
ndmioo 13348 The open interval function...
iooid 13349 An open interval with iden...
elioo3g 13350 Membership in a set of ope...
elioore 13351 A member of an open interv...
lbioo 13352 An open interval does not ...
ubioo 13353 An open interval does not ...
iooval2 13354 Value of the open interval...
iooin 13355 Intersection of two open i...
iooss1 13356 Subset relationship for op...
iooss2 13357 Subset relationship for op...
iocval 13358 Value of the open-below, c...
icoval 13359 Value of the closed-below,...
iccval 13360 Value of the closed interv...
elioo1 13361 Membership in an open inte...
elioo2 13362 Membership in an open inte...
elioc1 13363 Membership in an open-belo...
elico1 13364 Membership in a closed-bel...
elicc1 13365 Membership in a closed int...
iccid 13366 A closed interval with ide...
ico0 13367 An empty open interval of ...
ioc0 13368 An empty open interval of ...
icc0 13369 An empty closed interval o...
dfrp2 13370 Alternate definition of th...
elicod 13371 Membership in a left-close...
icogelb 13372 An element of a left-close...
elicore 13373 A member of a left-closed ...
ubioc1 13374 The upper bound belongs to...
lbico1 13375 The lower bound belongs to...
iccleub 13376 An element of a closed int...
iccgelb 13377 An element of a closed int...
elioo5 13378 Membership in an open inte...
eliooxr 13379 A nonempty open interval s...
eliooord 13380 Ordering implied by a memb...
elioo4g 13381 Membership in an open inte...
ioossre 13382 An open interval is a set ...
ioosscn 13383 An open interval is a set ...
elioc2 13384 Membership in an open-belo...
elico2 13385 Membership in a closed-bel...
elicc2 13386 Membership in a closed rea...
elicc2i 13387 Inference for membership i...
elicc4 13388 Membership in a closed rea...
iccss 13389 Condition for a closed int...
iccssioo 13390 Condition for a closed int...
icossico 13391 Condition for a closed-bel...
iccss2 13392 Condition for a closed int...
iccssico 13393 Condition for a closed int...
iccssioo2 13394 Condition for a closed int...
iccssico2 13395 Condition for a closed int...
ioomax 13396 The open interval from min...
iccmax 13397 The closed interval from m...
ioopos 13398 The set of positive reals ...
ioorp 13399 The set of positive reals ...
iooshf 13400 Shift the arguments of the...
iocssre 13401 A closed-above interval wi...
icossre 13402 A closed-below interval wi...
iccssre 13403 A closed real interval is ...
iccssxr 13404 A closed interval is a set...
iocssxr 13405 An open-below, closed-abov...
icossxr 13406 A closed-below, open-above...
ioossicc 13407 An open interval is a subs...
iccssred 13408 A closed real interval is ...
eliccxr 13409 A member of a closed inter...
icossicc 13410 A closed-below, open-above...
iocssicc 13411 A closed-above, open-below...
ioossico 13412 An open interval is a subs...
iocssioo 13413 Condition for a closed int...
icossioo 13414 Condition for a closed int...
ioossioo 13415 Condition for an open inte...
iccsupr 13416 A nonempty subset of a clo...
elioopnf 13417 Membership in an unbounded...
elioomnf 13418 Membership in an unbounded...
elicopnf 13419 Membership in a closed unb...
repos 13420 Two ways of saying that a ...
ioof 13421 The set of open intervals ...
iccf 13422 The set of closed interval...
unirnioo 13423 The union of the range of ...
dfioo2 13424 Alternate definition of th...
ioorebas 13425 Open intervals are element...
xrge0neqmnf 13426 A nonnegative extended rea...
xrge0nre 13427 An extended real which is ...
elrege0 13428 The predicate "is a nonneg...
nn0rp0 13429 A nonnegative integer is a...
rge0ssre 13430 Nonnegative real numbers a...
elxrge0 13431 Elementhood in the set of ...
0e0icopnf 13432 0 is a member of ` ( 0 [,)...
0e0iccpnf 13433 0 is a member of ` ( 0 [,]...
ge0addcl 13434 The nonnegative reals are ...
ge0mulcl 13435 The nonnegative reals are ...
ge0xaddcl 13436 The nonnegative reals are ...
ge0xmulcl 13437 The nonnegative extended r...
lbicc2 13438 The lower bound of a close...
ubicc2 13439 The upper bound of a close...
elicc01 13440 Membership in the closed r...
elunitrn 13441 The closed unit interval i...
elunitcn 13442 The closed unit interval i...
0elunit 13443 Zero is an element of the ...
1elunit 13444 One is an element of the c...
iooneg 13445 Membership in a negated op...
iccneg 13446 Membership in a negated cl...
icoshft 13447 A shifted real is a member...
icoshftf1o 13448 Shifting a closed-below, o...
icoun 13449 The union of two adjacent ...
icodisj 13450 Adjacent left-closed right...
ioounsn 13451 The union of an open inter...
snunioo 13452 The closure of one end of ...
snunico 13453 The closure of the open en...
snunioc 13454 The closure of the open en...
prunioo 13455 The closure of an open rea...
ioodisj 13456 If the upper bound of one ...
ioojoin 13457 Join two open intervals to...
difreicc 13458 The class difference of ` ...
iccsplit 13459 Split a closed interval in...
iccshftr 13460 Membership in a shifted in...
iccshftri 13461 Membership in a shifted in...
iccshftl 13462 Membership in a shifted in...
iccshftli 13463 Membership in a shifted in...
iccdil 13464 Membership in a dilated in...
iccdili 13465 Membership in a dilated in...
icccntr 13466 Membership in a contracted...
icccntri 13467 Membership in a contracted...
divelunit 13468 A condition for a ratio to...
lincmb01cmp 13469 A linear combination of tw...
iccf1o 13470 Describe a bijection from ...
iccen 13471 Any nontrivial closed inte...
xov1plusxeqvd 13472 A complex number ` X ` is ...
unitssre 13473 ` ( 0 [,] 1 ) ` is a subse...
unitsscn 13474 The closed unit interval i...
supicc 13475 Supremum of a bounded set ...
supiccub 13476 The supremum of a bounded ...
supicclub 13477 The supremum of a bounded ...
supicclub2 13478 The supremum of a bounded ...
zltaddlt1le 13479 The sum of an integer and ...
xnn0xrge0 13480 An extended nonnegative in...
fzval 13483 The value of a finite set ...
fzval2 13484 An alternative way of expr...
fzf 13485 Establish the domain and c...
elfz1 13486 Membership in a finite set...
elfz 13487 Membership in a finite set...
elfz2 13488 Membership in a finite set...
elfzd 13489 Membership in a finite set...
elfz5 13490 Membership in a finite set...
elfz4 13491 Membership in a finite set...
elfzuzb 13492 Membership in a finite set...
eluzfz 13493 Membership in a finite set...
elfzuz 13494 A member of a finite set o...
elfzuz3 13495 Membership in a finite set...
elfzel2 13496 Membership in a finite set...
elfzel1 13497 Membership in a finite set...
elfzelz 13498 A member of a finite set o...
elfzelzd 13499 A member of a finite set o...
fzssz 13500 A finite sequence of integ...
elfzle1 13501 A member of a finite set o...
elfzle2 13502 A member of a finite set o...
elfzuz2 13503 Implication of membership ...
elfzle3 13504 Membership in a finite set...
eluzfz1 13505 Membership in a finite set...
eluzfz2 13506 Membership in a finite set...
eluzfz2b 13507 Membership in a finite set...
elfz3 13508 Membership in a finite set...
elfz1eq 13509 Membership in a finite set...
elfzubelfz 13510 If there is a member in a ...
peano2fzr 13511 A Peano-postulate-like the...
fzn0 13512 Properties of a finite int...
fz0 13513 A finite set of sequential...
fzn 13514 A finite set of sequential...
fzen 13515 A shifted finite set of se...
fz1n 13516 A 1-based finite set of se...
0nelfz1 13517 0 is not an element of a f...
0fz1 13518 Two ways to say a finite 1...
fz10 13519 There are no integers betw...
uzsubsubfz 13520 Membership of an integer g...
uzsubsubfz1 13521 Membership of an integer g...
ige3m2fz 13522 Membership of an integer g...
fzsplit2 13523 Split a finite interval of...
fzsplit 13524 Split a finite interval of...
fzdisj 13525 Condition for two finite i...
fz01en 13526 0-based and 1-based finite...
elfznn 13527 A member of a finite set o...
elfz1end 13528 A nonempty finite range of...
fz1ssnn 13529 A finite set of positive i...
fznn0sub 13530 Subtraction closure for a ...
fzmmmeqm 13531 Subtracting the difference...
fzaddel 13532 Membership of a sum in a f...
fzadd2 13533 Membership of a sum in a f...
fzsubel 13534 Membership of a difference...
fzopth 13535 A finite set of sequential...
fzass4 13536 Two ways to express a nond...
fzss1 13537 Subset relationship for fi...
fzss2 13538 Subset relationship for fi...
fzssuz 13539 A finite set of sequential...
fzsn 13540 A finite interval of integ...
fzssp1 13541 Subset relationship for fi...
fzssnn 13542 Finite sets of sequential ...
ssfzunsnext 13543 A subset of a finite seque...
ssfzunsn 13544 A subset of a finite seque...
fzsuc 13545 Join a successor to the en...
fzpred 13546 Join a predecessor to the ...
fzpreddisj 13547 A finite set of sequential...
elfzp1 13548 Append an element to a fin...
fzp1ss 13549 Subset relationship for fi...
fzelp1 13550 Membership in a set of seq...
fzp1elp1 13551 Add one to an element of a...
fznatpl1 13552 Shift membership in a fini...
fzpr 13553 A finite interval of integ...
fztp 13554 A finite interval of integ...
fz12pr 13555 An integer range between 1...
fzsuc2 13556 Join a successor to the en...
fzp1disj 13557 ` ( M ... ( N + 1 ) ) ` is...
fzdifsuc 13558 Remove a successor from th...
fzprval 13559 Two ways of defining the f...
fztpval 13560 Two ways of defining the f...
fzrev 13561 Reversal of start and end ...
fzrev2 13562 Reversal of start and end ...
fzrev2i 13563 Reversal of start and end ...
fzrev3 13564 The "complement" of a memb...
fzrev3i 13565 The "complement" of a memb...
fznn 13566 Finite set of sequential i...
elfz1b 13567 Membership in a 1-based fi...
elfz1uz 13568 Membership in a 1-based fi...
elfzm11 13569 Membership in a finite set...
uzsplit 13570 Express an upper integer s...
uzdisj 13571 The first ` N ` elements o...
fseq1p1m1 13572 Add/remove an item to/from...
fseq1m1p1 13573 Add/remove an item to/from...
fz1sbc 13574 Quantification over a one-...
elfzp1b 13575 An integer is a member of ...
elfzm1b 13576 An integer is a member of ...
elfzp12 13577 Options for membership in ...
fzm1 13578 Choices for an element of ...
fzneuz 13579 No finite set of sequentia...
fznuz 13580 Disjointness of the upper ...
uznfz 13581 Disjointness of the upper ...
fzp1nel 13582 One plus the upper bound o...
fzrevral 13583 Reversal of scanning order...
fzrevral2 13584 Reversal of scanning order...
fzrevral3 13585 Reversal of scanning order...
fzshftral 13586 Shift the scanning order i...
ige2m1fz1 13587 Membership of an integer g...
ige2m1fz 13588 Membership in a 0-based fi...
elfz2nn0 13589 Membership in a finite set...
fznn0 13590 Characterization of a fini...
elfznn0 13591 A member of a finite set o...
elfz3nn0 13592 The upper bound of a nonem...
fz0ssnn0 13593 Finite sets of sequential ...
fz1ssfz0 13594 Subset relationship for fi...
0elfz 13595 0 is an element of a finit...
nn0fz0 13596 A nonnegative integer is a...
elfz0add 13597 An element of a finite set...
fz0sn 13598 An integer range from 0 to...
fz0tp 13599 An integer range from 0 to...
fz0to3un2pr 13600 An integer range from 0 to...
fz0to4untppr 13601 An integer range from 0 to...
elfz0ubfz0 13602 An element of a finite set...
elfz0fzfz0 13603 A member of a finite set o...
fz0fzelfz0 13604 If a member of a finite se...
fznn0sub2 13605 Subtraction closure for a ...
uzsubfz0 13606 Membership of an integer g...
fz0fzdiffz0 13607 The difference of an integ...
elfzmlbm 13608 Subtracting the lower boun...
elfzmlbp 13609 Subtracting the lower boun...
fzctr 13610 Lemma for theorems about t...
difelfzle 13611 The difference of two inte...
difelfznle 13612 The difference of two inte...
nn0split 13613 Express the set of nonnega...
nn0disj 13614 The first ` N + 1 ` elemen...
fz0sn0fz1 13615 A finite set of sequential...
fvffz0 13616 The function value of a fu...
1fv 13617 A function on a singleton....
4fvwrd4 13618 The first four function va...
2ffzeq 13619 Two functions over 0-based...
preduz 13620 The value of the predecess...
prednn 13621 The value of the predecess...
prednn0 13622 The value of the predecess...
predfz 13623 Calculate the predecessor ...
fzof 13626 Functionality of the half-...
elfzoel1 13627 Reverse closure for half-o...
elfzoel2 13628 Reverse closure for half-o...
elfzoelz 13629 Reverse closure for half-o...
fzoval 13630 Value of the half-open int...
elfzo 13631 Membership in a half-open ...
elfzo2 13632 Membership in a half-open ...
elfzouz 13633 Membership in a half-open ...
nelfzo 13634 An integer not being a mem...
fzolb 13635 The left endpoint of a hal...
fzolb2 13636 The left endpoint of a hal...
elfzole1 13637 A member in a half-open in...
elfzolt2 13638 A member in a half-open in...
elfzolt3 13639 Membership in a half-open ...
elfzolt2b 13640 A member in a half-open in...
elfzolt3b 13641 Membership in a half-open ...
elfzop1le2 13642 A member in a half-open in...
fzonel 13643 A half-open range does not...
elfzouz2 13644 The upper bound of a half-...
elfzofz 13645 A half-open range is conta...
elfzo3 13646 Express membership in a ha...
fzon0 13647 A half-open integer interv...
fzossfz 13648 A half-open range is conta...
fzossz 13649 A half-open integer interv...
fzon 13650 A half-open set of sequent...
fzo0n 13651 A half-open range of nonne...
fzonlt0 13652 A half-open integer range ...
fzo0 13653 Half-open sets with equal ...
fzonnsub 13654 If ` K < N ` then ` N - K ...
fzonnsub2 13655 If ` M < N ` then ` N - M ...
fzoss1 13656 Subset relationship for ha...
fzoss2 13657 Subset relationship for ha...
fzossrbm1 13658 Subset of a half-open rang...
fzo0ss1 13659 Subset relationship for ha...
fzossnn0 13660 A half-open integer range ...
fzospliti 13661 One direction of splitting...
fzosplit 13662 Split a half-open integer ...
fzodisj 13663 Abutting half-open integer...
fzouzsplit 13664 Split an upper integer set...
fzouzdisj 13665 A half-open integer range ...
fzoun 13666 A half-open integer range ...
fzodisjsn 13667 A half-open integer range ...
prinfzo0 13668 The intersection of a half...
lbfzo0 13669 An integer is strictly gre...
elfzo0 13670 Membership in a half-open ...
elfzo0z 13671 Membership in a half-open ...
nn0p1elfzo 13672 A nonnegative integer incr...
elfzo0le 13673 A member in a half-open ra...
elfzonn0 13674 A member of a half-open ra...
fzonmapblen 13675 The result of subtracting ...
fzofzim 13676 If a nonnegative integer i...
fz1fzo0m1 13677 Translation of one between...
fzossnn 13678 Half-open integer ranges s...
elfzo1 13679 Membership in a half-open ...
fzo1fzo0n0 13680 An integer between 1 and a...
fzo0n0 13681 A half-open integer range ...
fzoaddel 13682 Translate membership in a ...
fzo0addel 13683 Translate membership in a ...
fzo0addelr 13684 Translate membership in a ...
fzoaddel2 13685 Translate membership in a ...
elfzoext 13686 Membership of an integer i...
elincfzoext 13687 Membership of an increased...
fzosubel 13688 Translate membership in a ...
fzosubel2 13689 Membership in a translated...
fzosubel3 13690 Membership in a translated...
eluzgtdifelfzo 13691 Membership of the differen...
ige2m2fzo 13692 Membership of an integer g...
fzocatel 13693 Translate membership in a ...
ubmelfzo 13694 If an integer in a 1-based...
elfzodifsumelfzo 13695 If an integer is in a half...
elfzom1elp1fzo 13696 Membership of an integer i...
elfzom1elfzo 13697 Membership in a half-open ...
fzval3 13698 Expressing a closed intege...
fz0add1fz1 13699 Translate membership in a ...
fzosn 13700 Expressing a singleton as ...
elfzomin 13701 Membership of an integer i...
zpnn0elfzo 13702 Membership of an integer i...
zpnn0elfzo1 13703 Membership of an integer i...
fzosplitsnm1 13704 Removing a singleton from ...
elfzonlteqm1 13705 If an element of a half-op...
fzonn0p1 13706 A nonnegative integer is e...
fzossfzop1 13707 A half-open range of nonne...
fzonn0p1p1 13708 If a nonnegative integer i...
elfzom1p1elfzo 13709 Increasing an element of a...
fzo0ssnn0 13710 Half-open integer ranges s...
fzo01 13711 Expressing the singleton o...
fzo12sn 13712 A 1-based half-open intege...
fzo13pr 13713 A 1-based half-open intege...
fzo0to2pr 13714 A half-open integer range ...
fzo0to3tp 13715 A half-open integer range ...
fzo0to42pr 13716 A half-open integer range ...
fzo1to4tp 13717 A half-open integer range ...
fzo0sn0fzo1 13718 A half-open range of nonne...
elfzo0l 13719 A member of a half-open ra...
fzoend 13720 The endpoint of a half-ope...
fzo0end 13721 The endpoint of a zero-bas...
ssfzo12 13722 Subset relationship for ha...
ssfzoulel 13723 If a half-open integer ran...
ssfzo12bi 13724 Subset relationship for ha...
ubmelm1fzo 13725 The result of subtracting ...
fzofzp1 13726 If a point is in a half-op...
fzofzp1b 13727 If a point is in a half-op...
elfzom1b 13728 An integer is a member of ...
elfzom1elp1fzo1 13729 Membership of a nonnegativ...
elfzo1elm1fzo0 13730 Membership of a positive i...
elfzonelfzo 13731 If an element of a half-op...
fzonfzoufzol 13732 If an element of a half-op...
elfzomelpfzo 13733 An integer increased by an...
elfznelfzo 13734 A value in a finite set of...
elfznelfzob 13735 A value in a finite set of...
peano2fzor 13736 A Peano-postulate-like the...
fzosplitsn 13737 Extending a half-open rang...
fzosplitpr 13738 Extending a half-open inte...
fzosplitprm1 13739 Extending a half-open inte...
fzosplitsni 13740 Membership in a half-open ...
fzisfzounsn 13741 A finite interval of integ...
elfzr 13742 A member of a finite inter...
elfzlmr 13743 A member of a finite inter...
elfz0lmr 13744 A member of a finite inter...
fzostep1 13745 Two possibilities for a nu...
fzoshftral 13746 Shift the scanning order i...
fzind2 13747 Induction on the integers ...
fvinim0ffz 13748 The function values for th...
injresinjlem 13749 Lemma for ~ injresinj . (...
injresinj 13750 A function whose restricti...
subfzo0 13751 The difference between two...
flval 13756 Value of the floor (greate...
flcl 13757 The floor (greatest intege...
reflcl 13758 The floor (greatest intege...
fllelt 13759 A basic property of the fl...
flcld 13760 The floor (greatest intege...
flle 13761 A basic property of the fl...
flltp1 13762 A basic property of the fl...
fllep1 13763 A basic property of the fl...
fraclt1 13764 The fractional part of a r...
fracle1 13765 The fractional part of a r...
fracge0 13766 The fractional part of a r...
flge 13767 The floor function value i...
fllt 13768 The floor function value i...
flflp1 13769 Move floor function betwee...
flid 13770 An integer is its own floo...
flidm 13771 The floor function is idem...
flidz 13772 A real number equals its f...
flltnz 13773 The floor of a non-integer...
flwordi 13774 Ordering relation for the ...
flword2 13775 Ordering relation for the ...
flval2 13776 An alternate way to define...
flval3 13777 An alternate way to define...
flbi 13778 A condition equivalent to ...
flbi2 13779 A condition equivalent to ...
adddivflid 13780 The floor of a sum of an i...
ico01fl0 13781 The floor of a real number...
flge0nn0 13782 The floor of a number grea...
flge1nn 13783 The floor of a number grea...
fldivnn0 13784 The floor function of a di...
refldivcl 13785 The floor function of a di...
divfl0 13786 The floor of a fraction is...
fladdz 13787 An integer can be moved in...
flzadd 13788 An integer can be moved in...
flmulnn0 13789 Move a nonnegative integer...
btwnzge0 13790 A real bounded between an ...
2tnp1ge0ge0 13791 Two times an integer plus ...
flhalf 13792 Ordering relation for the ...
fldivle 13793 The floor function of a di...
fldivnn0le 13794 The floor function of a di...
flltdivnn0lt 13795 The floor function of a di...
ltdifltdiv 13796 If the dividend of a divis...
fldiv4p1lem1div2 13797 The floor of an integer eq...
fldiv4lem1div2uz2 13798 The floor of an integer gr...
fldiv4lem1div2 13799 The floor of a positive in...
ceilval 13800 The value of the ceiling f...
dfceil2 13801 Alternative definition of ...
ceilval2 13802 The value of the ceiling f...
ceicl 13803 The ceiling function retur...
ceilcl 13804 Closure of the ceiling fun...
ceilcld 13805 Closure of the ceiling fun...
ceige 13806 The ceiling of a real numb...
ceilge 13807 The ceiling of a real numb...
ceilged 13808 The ceiling of a real numb...
ceim1l 13809 One less than the ceiling ...
ceilm1lt 13810 One less than the ceiling ...
ceile 13811 The ceiling of a real numb...
ceille 13812 The ceiling of a real numb...
ceilid 13813 An integer is its own ceil...
ceilidz 13814 A real number equals its c...
flleceil 13815 The floor of a real number...
fleqceilz 13816 A real number is an intege...
quoremz 13817 Quotient and remainder of ...
quoremnn0 13818 Quotient and remainder of ...
quoremnn0ALT 13819 Alternate proof of ~ quore...
intfrac2 13820 Decompose a real into inte...
intfracq 13821 Decompose a rational numbe...
fldiv 13822 Cancellation of the embedd...
fldiv2 13823 Cancellation of an embedde...
fznnfl 13824 Finite set of sequential i...
uzsup 13825 An upper set of integers i...
ioopnfsup 13826 An upper set of reals is u...
icopnfsup 13827 An upper set of reals is u...
rpsup 13828 The positive reals are unb...
resup 13829 The real numbers are unbou...
xrsup 13830 The extended real numbers ...
modval 13833 The value of the modulo op...
modvalr 13834 The value of the modulo op...
modcl 13835 Closure law for the modulo...
flpmodeq 13836 Partition of a division in...
modcld 13837 Closure law for the modulo...
mod0 13838 ` A mod B ` is zero iff ` ...
mulmod0 13839 The product of an integer ...
negmod0 13840 ` A ` is divisible by ` B ...
modge0 13841 The modulo operation is no...
modlt 13842 The modulo operation is le...
modelico 13843 Modular reduction produces...
moddiffl 13844 Value of the modulo operat...
moddifz 13845 The modulo operation diffe...
modfrac 13846 The fractional part of a n...
flmod 13847 The floor function express...
intfrac 13848 Break a number into its in...
zmod10 13849 An integer modulo 1 is 0. ...
zmod1congr 13850 Two arbitrary integers are...
modmulnn 13851 Move a positive integer in...
modvalp1 13852 The value of the modulo op...
zmodcl 13853 Closure law for the modulo...
zmodcld 13854 Closure law for the modulo...
zmodfz 13855 An integer mod ` B ` lies ...
zmodfzo 13856 An integer mod ` B ` lies ...
zmodfzp1 13857 An integer mod ` B ` lies ...
modid 13858 Identity law for modulo. ...
modid0 13859 A positive real number mod...
modid2 13860 Identity law for modulo. ...
zmodid2 13861 Identity law for modulo re...
zmodidfzo 13862 Identity law for modulo re...
zmodidfzoimp 13863 Identity law for modulo re...
0mod 13864 Special case: 0 modulo a p...
1mod 13865 Special case: 1 modulo a r...
modabs 13866 Absorption law for modulo....
modabs2 13867 Absorption law for modulo....
modcyc 13868 The modulo operation is pe...
modcyc2 13869 The modulo operation is pe...
modadd1 13870 Addition property of the m...
modaddabs 13871 Absorption law for modulo....
modaddmod 13872 The sum of a real number m...
muladdmodid 13873 The sum of a positive real...
mulp1mod1 13874 The product of an integer ...
modmuladd 13875 Decomposition of an intege...
modmuladdim 13876 Implication of a decomposi...
modmuladdnn0 13877 Implication of a decomposi...
negmod 13878 The negation of a number m...
m1modnnsub1 13879 Minus one modulo a positiv...
m1modge3gt1 13880 Minus one modulo an intege...
addmodid 13881 The sum of a positive inte...
addmodidr 13882 The sum of a positive inte...
modadd2mod 13883 The sum of a real number m...
modm1p1mod0 13884 If a real number modulo a ...
modltm1p1mod 13885 If a real number modulo a ...
modmul1 13886 Multiplication property of...
modmul12d 13887 Multiplication property of...
modnegd 13888 Negation property of the m...
modadd12d 13889 Additive property of the m...
modsub12d 13890 Subtraction property of th...
modsubmod 13891 The difference of a real n...
modsubmodmod 13892 The difference of a real n...
2txmodxeq0 13893 Two times a positive real ...
2submod 13894 If a real number is betwee...
modifeq2int 13895 If a nonnegative integer i...
modaddmodup 13896 The sum of an integer modu...
modaddmodlo 13897 The sum of an integer modu...
modmulmod 13898 The product of a real numb...
modmulmodr 13899 The product of an integer ...
modaddmulmod 13900 The sum of a real number a...
moddi 13901 Distribute multiplication ...
modsubdir 13902 Distribute the modulo oper...
modeqmodmin 13903 A real number equals the d...
modirr 13904 A number modulo an irratio...
modfzo0difsn 13905 For a number within a half...
modsumfzodifsn 13906 The sum of a number within...
modlteq 13907 Two nonnegative integers l...
addmodlteq 13908 Two nonnegative integers l...
om2uz0i 13909 The mapping ` G ` is a one...
om2uzsuci 13910 The value of ` G ` (see ~ ...
om2uzuzi 13911 The value ` G ` (see ~ om2...
om2uzlti 13912 Less-than relation for ` G...
om2uzlt2i 13913 The mapping ` G ` (see ~ o...
om2uzrani 13914 Range of ` G ` (see ~ om2u...
om2uzf1oi 13915 ` G ` (see ~ om2uz0i ) is ...
om2uzisoi 13916 ` G ` (see ~ om2uz0i ) is ...
om2uzoi 13917 An alternative definition ...
om2uzrdg 13918 A helper lemma for the val...
uzrdglem 13919 A helper lemma for the val...
uzrdgfni 13920 The recursive definition g...
uzrdg0i 13921 Initial value of a recursi...
uzrdgsuci 13922 Successor value of a recur...
ltweuz 13923 ` < ` is a well-founded re...
ltwenn 13924 Less than well-orders the ...
ltwefz 13925 Less than well-orders a se...
uzenom 13926 An upper integer set is de...
uzinf 13927 An upper integer set is in...
nnnfi 13928 The set of positive intege...
uzrdgxfr 13929 Transfer the value of the ...
fzennn 13930 The cardinality of a finit...
fzen2 13931 The cardinality of a finit...
cardfz 13932 The cardinality of a finit...
hashgf1o 13933 ` G ` maps ` _om ` one-to-...
fzfi 13934 A finite interval of integ...
fzfid 13935 Commonly used special case...
fzofi 13936 Half-open integer sets are...
fsequb 13937 The values of a finite rea...
fsequb2 13938 The values of a finite rea...
fseqsupcl 13939 The values of a finite rea...
fseqsupubi 13940 The values of a finite rea...
nn0ennn 13941 The nonnegative integers a...
nnenom 13942 The set of positive intege...
nnct 13943 ` NN ` is countable. (Con...
uzindi 13944 Indirect strong induction ...
axdc4uzlem 13945 Lemma for ~ axdc4uz . (Co...
axdc4uz 13946 A version of ~ axdc4 that ...
ssnn0fi 13947 A subset of the nonnegativ...
rabssnn0fi 13948 A subset of the nonnegativ...
uzsinds 13949 Strong (or "total") induct...
nnsinds 13950 Strong (or "total") induct...
nn0sinds 13951 Strong (or "total") induct...
fsuppmapnn0fiublem 13952 Lemma for ~ fsuppmapnn0fiu...
fsuppmapnn0fiub 13953 If all functions of a fini...
fsuppmapnn0fiubex 13954 If all functions of a fini...
fsuppmapnn0fiub0 13955 If all functions of a fini...
suppssfz 13956 Condition for a function o...
fsuppmapnn0ub 13957 If a function over the non...
fsuppmapnn0fz 13958 If a function over the non...
mptnn0fsupp 13959 A mapping from the nonnega...
mptnn0fsuppd 13960 A mapping from the nonnega...
mptnn0fsuppr 13961 A finitely supported mappi...
f13idfv 13962 A one-to-one function with...
seqex 13965 Existence of the sequence ...
seqeq1 13966 Equality theorem for the s...
seqeq2 13967 Equality theorem for the s...
seqeq3 13968 Equality theorem for the s...
seqeq1d 13969 Equality deduction for the...
seqeq2d 13970 Equality deduction for the...
seqeq3d 13971 Equality deduction for the...
seqeq123d 13972 Equality deduction for the...
nfseq 13973 Hypothesis builder for the...
seqval 13974 Value of the sequence buil...
seqfn 13975 The sequence builder funct...
seq1 13976 Value of the sequence buil...
seq1i 13977 Value of the sequence buil...
seqp1 13978 Value of the sequence buil...
seqexw 13979 Weak version of ~ seqex th...
seqp1d 13980 Value of the sequence buil...
seqp1iOLD 13981 Obsolete version of ~ seqp...
seqm1 13982 Value of the sequence buil...
seqcl2 13983 Closure properties of the ...
seqf2 13984 Range of the recursive seq...
seqcl 13985 Closure properties of the ...
seqf 13986 Range of the recursive seq...
seqfveq2 13987 Equality of sequences. (C...
seqfeq2 13988 Equality of sequences. (C...
seqfveq 13989 Equality of sequences. (C...
seqfeq 13990 Equality of sequences. (C...
seqshft2 13991 Shifting the index set of ...
seqres 13992 Restricting its characteri...
serf 13993 An infinite series of comp...
serfre 13994 An infinite series of real...
monoord 13995 Ordering relation for a mo...
monoord2 13996 Ordering relation for a mo...
sermono 13997 The partial sums in an inf...
seqsplit 13998 Split a sequence into two ...
seq1p 13999 Removing the first term fr...
seqcaopr3 14000 Lemma for ~ seqcaopr2 . (...
seqcaopr2 14001 The sum of two infinite se...
seqcaopr 14002 The sum of two infinite se...
seqf1olem2a 14003 Lemma for ~ seqf1o . (Con...
seqf1olem1 14004 Lemma for ~ seqf1o . (Con...
seqf1olem2 14005 Lemma for ~ seqf1o . (Con...
seqf1o 14006 Rearrange a sum via an arb...
seradd 14007 The sum of two infinite se...
sersub 14008 The difference of two infi...
seqid3 14009 A sequence that consists e...
seqid 14010 Discarding the first few t...
seqid2 14011 The last few partial sums ...
seqhomo 14012 Apply a homomorphism to a ...
seqz 14013 If the operation ` .+ ` ha...
seqfeq4 14014 Equality of series under d...
seqfeq3 14015 Equality of series under d...
seqdistr 14016 The distributive property ...
ser0 14017 The value of the partial s...
ser0f 14018 A zero-valued infinite ser...
serge0 14019 A finite sum of nonnegativ...
serle 14020 Comparison of partial sums...
ser1const 14021 Value of the partial serie...
seqof 14022 Distribute function operat...
seqof2 14023 Distribute function operat...
expval 14026 Value of exponentiation to...
expnnval 14027 Value of exponentiation to...
exp0 14028 Value of a complex number ...
0exp0e1 14029 The zeroth power of zero e...
exp1 14030 Value of a complex number ...
expp1 14031 Value of a complex number ...
expneg 14032 Value of a complex number ...
expneg2 14033 Value of a complex number ...
expn1 14034 A complex number raised to...
expcllem 14035 Lemma for proving nonnegat...
expcl2lem 14036 Lemma for proving integer ...
nnexpcl 14037 Closure of exponentiation ...
nn0expcl 14038 Closure of exponentiation ...
zexpcl 14039 Closure of exponentiation ...
qexpcl 14040 Closure of exponentiation ...
reexpcl 14041 Closure of exponentiation ...
expcl 14042 Closure law for nonnegativ...
rpexpcl 14043 Closure law for integer ex...
qexpclz 14044 Closure of integer exponen...
reexpclz 14045 Closure of integer exponen...
expclzlem 14046 Lemma for ~ expclz . (Con...
expclz 14047 Closure law for integer ex...
m1expcl2 14048 Closure of integer exponen...
m1expcl 14049 Closure of exponentiation ...
zexpcld 14050 Closure of exponentiation ...
nn0expcli 14051 Closure of exponentiation ...
nn0sqcl 14052 The square of a nonnegativ...
expm1t 14053 Exponentiation in terms of...
1exp 14054 Value of 1 raised to an in...
expeq0 14055 A positive integer power i...
expne0 14056 A positive integer power i...
expne0i 14057 An integer power is nonzer...
expgt0 14058 A positive real raised to ...
expnegz 14059 Value of a nonzero complex...
0exp 14060 Value of zero raised to a ...
expge0 14061 A nonnegative real raised ...
expge1 14062 A real greater than or equ...
expgt1 14063 A real greater than 1 rais...
mulexp 14064 Nonnegative integer expone...
mulexpz 14065 Integer exponentiation of ...
exprec 14066 Integer exponentiation of ...
expadd 14067 Sum of exponents law for n...
expaddzlem 14068 Lemma for ~ expaddz . (Co...
expaddz 14069 Sum of exponents law for i...
expmul 14070 Product of exponents law f...
expmulz 14071 Product of exponents law f...
m1expeven 14072 Exponentiation of negative...
expsub 14073 Exponent subtraction law f...
expp1z 14074 Value of a nonzero complex...
expm1 14075 Value of a nonzero complex...
expdiv 14076 Nonnegative integer expone...
sqval 14077 Value of the square of a c...
sqneg 14078 The square of the negative...
sqsubswap 14079 Swap the order of subtract...
sqcl 14080 Closure of square. (Contr...
sqmul 14081 Distribution of squaring o...
sqeq0 14082 A complex number is zero i...
sqdiv 14083 Distribution of squaring o...
sqdivid 14084 The square of a nonzero co...
sqne0 14085 A complex number is nonzer...
resqcl 14086 Closure of squaring in rea...
resqcld 14087 Closure of squaring in rea...
sqgt0 14088 The square of a nonzero re...
sqn0rp 14089 The square of a nonzero re...
nnsqcl 14090 The positive naturals are ...
zsqcl 14091 Integers are closed under ...
qsqcl 14092 The square of a rational i...
sq11 14093 The square function is one...
nn0sq11 14094 The square function is one...
lt2sq 14095 The square function is inc...
le2sq 14096 The square function is non...
le2sq2 14097 The square function is non...
sqge0 14098 The square of a real is no...
sqge0d 14099 The square of a real is no...
zsqcl2 14100 The square of an integer i...
0expd 14101 Value of zero raised to a ...
exp0d 14102 Value of a complex number ...
exp1d 14103 Value of a complex number ...
expeq0d 14104 If a positive integer powe...
sqvald 14105 Value of square. Inferenc...
sqcld 14106 Closure of square. (Contr...
sqeq0d 14107 A number is zero iff its s...
expcld 14108 Closure law for nonnegativ...
expp1d 14109 Value of a complex number ...
expaddd 14110 Sum of exponents law for n...
expmuld 14111 Product of exponents law f...
sqrecd 14112 Square of reciprocal is re...
expclzd 14113 Closure law for integer ex...
expne0d 14114 A nonnegative integer powe...
expnegd 14115 Value of a nonzero complex...
exprecd 14116 An integer power of a reci...
expp1zd 14117 Value of a nonzero complex...
expm1d 14118 Value of a nonzero complex...
expsubd 14119 Exponent subtraction law f...
sqmuld 14120 Distribution of squaring o...
sqdivd 14121 Distribution of squaring o...
expdivd 14122 Nonnegative integer expone...
mulexpd 14123 Nonnegative integer expone...
znsqcld 14124 The square of a nonzero in...
reexpcld 14125 Closure of exponentiation ...
expge0d 14126 A nonnegative real raised ...
expge1d 14127 A real greater than or equ...
ltexp2a 14128 Exponent ordering relation...
expmordi 14129 Base ordering relationship...
rpexpmord 14130 Base ordering relationship...
expcan 14131 Cancellation law for integ...
ltexp2 14132 Strict ordering law for ex...
leexp2 14133 Ordering law for exponenti...
leexp2a 14134 Weak ordering relationship...
ltexp2r 14135 The integer powers of a fi...
leexp2r 14136 Weak ordering relationship...
leexp1a 14137 Weak base ordering relatio...
exple1 14138 A real between 0 and 1 inc...
expubnd 14139 An upper bound on ` A ^ N ...
sumsqeq0 14140 The sum of two squres of r...
sqvali 14141 Value of square. Inferenc...
sqcli 14142 Closure of square. (Contr...
sqeq0i 14143 A complex number is zero i...
sqrecii 14144 The square of a reciprocal...
sqmuli 14145 Distribution of squaring o...
sqdivi 14146 Distribution of squaring o...
resqcli 14147 Closure of square in reals...
sqgt0i 14148 The square of a nonzero re...
sqge0i 14149 The square of a real is no...
lt2sqi 14150 The square function on non...
le2sqi 14151 The square function on non...
sq11i 14152 The square function is one...
sq0 14153 The square of 0 is 0. (Co...
sq0i 14154 If a number is zero, then ...
sq0id 14155 If a number is zero, then ...
sq1 14156 The square of 1 is 1. (Co...
neg1sqe1 14157 The square of ` -u 1 ` is ...
sq2 14158 The square of 2 is 4. (Co...
sq3 14159 The square of 3 is 9. (Co...
sq4e2t8 14160 The square of 4 is 2 times...
cu2 14161 The cube of 2 is 8. (Cont...
irec 14162 The reciprocal of ` _i ` ....
i2 14163 ` _i ` squared. (Contribu...
i3 14164 ` _i ` cubed. (Contribute...
i4 14165 ` _i ` to the fourth power...
nnlesq 14166 A positive integer is less...
zzlesq 14167 An integer is less than or...
iexpcyc 14168 Taking ` _i ` to the ` K `...
expnass 14169 A counterexample showing t...
sqlecan 14170 Cancel one factor of a squ...
subsq 14171 Factor the difference of t...
subsq2 14172 Express the difference of ...
binom2i 14173 The square of a binomial. ...
subsqi 14174 Factor the difference of t...
sqeqori 14175 The squares of two complex...
subsq0i 14176 The two solutions to the d...
sqeqor 14177 The squares of two complex...
binom2 14178 The square of a binomial. ...
binom21 14179 Special case of ~ binom2 w...
binom2sub 14180 Expand the square of a sub...
binom2sub1 14181 Special case of ~ binom2su...
binom2subi 14182 Expand the square of a sub...
mulbinom2 14183 The square of a binomial w...
binom3 14184 The cube of a binomial. (...
sq01 14185 If a complex number equals...
zesq 14186 An integer is even iff its...
nnesq 14187 A positive integer is even...
crreczi 14188 Reciprocal of a complex nu...
bernneq 14189 Bernoulli's inequality, du...
bernneq2 14190 Variation of Bernoulli's i...
bernneq3 14191 A corollary of ~ bernneq ....
expnbnd 14192 Exponentiation with a base...
expnlbnd 14193 The reciprocal of exponent...
expnlbnd2 14194 The reciprocal of exponent...
expmulnbnd 14195 Exponentiation with a base...
digit2 14196 Two ways to express the ` ...
digit1 14197 Two ways to express the ` ...
modexp 14198 Exponentiation property of...
discr1 14199 A nonnegative quadratic fo...
discr 14200 If a quadratic polynomial ...
expnngt1 14201 If an integer power with a...
expnngt1b 14202 An integer power with an i...
sqoddm1div8 14203 A squared odd number minus...
nnsqcld 14204 The naturals are closed un...
nnexpcld 14205 Closure of exponentiation ...
nn0expcld 14206 Closure of exponentiation ...
rpexpcld 14207 Closure law for exponentia...
ltexp2rd 14208 The power of a positive nu...
reexpclzd 14209 Closure of exponentiation ...
sqgt0d 14210 The square of a nonzero re...
ltexp2d 14211 Ordering relationship for ...
leexp2d 14212 Ordering law for exponenti...
expcand 14213 Ordering relationship for ...
leexp2ad 14214 Ordering relationship for ...
leexp2rd 14215 Ordering relationship for ...
lt2sqd 14216 The square function on non...
le2sqd 14217 The square function on non...
sq11d 14218 The square function is one...
mulsubdivbinom2 14219 The square of a binomial w...
muldivbinom2 14220 The square of a binomial w...
sq10 14221 The square of 10 is 100. ...
sq10e99m1 14222 The square of 10 is 99 plu...
3dec 14223 A "decimal constructor" wh...
nn0le2msqi 14224 The square function on non...
nn0opthlem1 14225 A rather pretty lemma for ...
nn0opthlem2 14226 Lemma for ~ nn0opthi . (C...
nn0opthi 14227 An ordered pair theorem fo...
nn0opth2i 14228 An ordered pair theorem fo...
nn0opth2 14229 An ordered pair theorem fo...
facnn 14232 Value of the factorial fun...
fac0 14233 The factorial of 0. (Cont...
fac1 14234 The factorial of 1. (Cont...
facp1 14235 The factorial of a success...
fac2 14236 The factorial of 2. (Cont...
fac3 14237 The factorial of 3. (Cont...
fac4 14238 The factorial of 4. (Cont...
facnn2 14239 Value of the factorial fun...
faccl 14240 Closure of the factorial f...
faccld 14241 Closure of the factorial f...
facmapnn 14242 The factorial function res...
facne0 14243 The factorial function is ...
facdiv 14244 A positive integer divides...
facndiv 14245 No positive integer (great...
facwordi 14246 Ordering property of facto...
faclbnd 14247 A lower bound for the fact...
faclbnd2 14248 A lower bound for the fact...
faclbnd3 14249 A lower bound for the fact...
faclbnd4lem1 14250 Lemma for ~ faclbnd4 . Pr...
faclbnd4lem2 14251 Lemma for ~ faclbnd4 . Us...
faclbnd4lem3 14252 Lemma for ~ faclbnd4 . Th...
faclbnd4lem4 14253 Lemma for ~ faclbnd4 . Pr...
faclbnd4 14254 Variant of ~ faclbnd5 prov...
faclbnd5 14255 The factorial function gro...
faclbnd6 14256 Geometric lower bound for ...
facubnd 14257 An upper bound for the fac...
facavg 14258 The product of two factori...
bcval 14261 Value of the binomial coef...
bcval2 14262 Value of the binomial coef...
bcval3 14263 Value of the binomial coef...
bcval4 14264 Value of the binomial coef...
bcrpcl 14265 Closure of the binomial co...
bccmpl 14266 "Complementing" its second...
bcn0 14267 ` N ` choose 0 is 1. Rema...
bc0k 14268 The binomial coefficient "...
bcnn 14269 ` N ` choose ` N ` is 1. ...
bcn1 14270 Binomial coefficient: ` N ...
bcnp1n 14271 Binomial coefficient: ` N ...
bcm1k 14272 The proportion of one bino...
bcp1n 14273 The proportion of one bino...
bcp1nk 14274 The proportion of one bino...
bcval5 14275 Write out the top and bott...
bcn2 14276 Binomial coefficient: ` N ...
bcp1m1 14277 Compute the binomial coeff...
bcpasc 14278 Pascal's rule for the bino...
bccl 14279 A binomial coefficient, in...
bccl2 14280 A binomial coefficient, in...
bcn2m1 14281 Compute the binomial coeff...
bcn2p1 14282 Compute the binomial coeff...
permnn 14283 The number of permutations...
bcnm1 14284 The binomial coefficent of...
4bc3eq4 14285 The value of four choose t...
4bc2eq6 14286 The value of four choose t...
hashkf 14289 The finite part of the siz...
hashgval 14290 The value of the ` # ` fun...
hashginv 14291 The converse of ` G ` maps...
hashinf 14292 The value of the ` # ` fun...
hashbnd 14293 If ` A ` has size bounded ...
hashfxnn0 14294 The size function is a fun...
hashf 14295 The size function maps all...
hashxnn0 14296 The value of the hash func...
hashresfn 14297 Restriction of the domain ...
dmhashres 14298 Restriction of the domain ...
hashnn0pnf 14299 The value of the hash func...
hashnnn0genn0 14300 If the size of a set is no...
hashnemnf 14301 The size of a set is never...
hashv01gt1 14302 The size of a set is eithe...
hashfz1 14303 The set ` ( 1 ... N ) ` ha...
hashen 14304 Two finite sets have the s...
hasheni 14305 Equinumerous sets have the...
hasheqf1o 14306 The size of two finite set...
fiinfnf1o 14307 There is no bijection betw...
hasheqf1oi 14308 The size of two sets is eq...
hashf1rn 14309 The size of a finite set w...
hasheqf1od 14310 The size of two sets is eq...
fz1eqb 14311 Two possibly-empty 1-based...
hashcard 14312 The size function of the c...
hashcl 14313 Closure of the ` # ` funct...
hashxrcl 14314 Extended real closure of t...
hashclb 14315 Reverse closure of the ` #...
nfile 14316 The size of any infinite s...
hashvnfin 14317 A set of finite size is a ...
hashnfinnn0 14318 The size of an infinite se...
isfinite4 14319 A finite set is equinumero...
hasheq0 14320 Two ways of saying a set i...
hashneq0 14321 Two ways of saying a set i...
hashgt0n0 14322 If the size of a set is gr...
hashnncl 14323 Positive natural closure o...
hash0 14324 The empty set has size zer...
hashelne0d 14325 A set with an element has ...
hashsng 14326 The size of a singleton. ...
hashen1 14327 A set has size 1 if and on...
hash1elsn 14328 A set of size 1 with a kno...
hashrabrsn 14329 The size of a restricted c...
hashrabsn01 14330 The size of a restricted c...
hashrabsn1 14331 If the size of a restricte...
hashfn 14332 A function is equinumerous...
fseq1hash 14333 The value of the size func...
hashgadd 14334 ` G ` maps ordinal additio...
hashgval2 14335 A short expression for the...
hashdom 14336 Dominance relation for the...
hashdomi 14337 Non-strict order relation ...
hashsdom 14338 Strict dominance relation ...
hashun 14339 The size of the union of d...
hashun2 14340 The size of the union of f...
hashun3 14341 The size of the union of f...
hashinfxadd 14342 The extended real addition...
hashunx 14343 The size of the union of d...
hashge0 14344 The cardinality of a set i...
hashgt0 14345 The cardinality of a nonem...
hashge1 14346 The cardinality of a nonem...
1elfz0hash 14347 1 is an element of the fin...
hashnn0n0nn 14348 If a nonnegative integer i...
hashunsng 14349 The size of the union of a...
hashunsngx 14350 The size of the union of a...
hashunsnggt 14351 The size of a set is great...
hashprg 14352 The size of an unordered p...
elprchashprn2 14353 If one element of an unord...
hashprb 14354 The size of an unordered p...
hashprdifel 14355 The elements of an unorder...
prhash2ex 14356 There is (at least) one se...
hashle00 14357 If the size of a set is le...
hashgt0elex 14358 If the size of a set is gr...
hashgt0elexb 14359 The size of a set is great...
hashp1i 14360 Size of a finite ordinal. ...
hash1 14361 Size of a finite ordinal. ...
hash2 14362 Size of a finite ordinal. ...
hash3 14363 Size of a finite ordinal. ...
hash4 14364 Size of a finite ordinal. ...
pr0hash2ex 14365 There is (at least) one se...
hashss 14366 The size of a subset is le...
prsshashgt1 14367 The size of a superset of ...
hashin 14368 The size of the intersecti...
hashssdif 14369 The size of the difference...
hashdif 14370 The size of the difference...
hashdifsn 14371 The size of the difference...
hashdifpr 14372 The size of the difference...
hashsn01 14373 The size of a singleton is...
hashsnle1 14374 The size of a singleton is...
hashsnlei 14375 Get an upper bound on a co...
hash1snb 14376 The size of a set is 1 if ...
euhash1 14377 The size of a set is 1 in ...
hash1n0 14378 If the size of a set is 1 ...
hashgt12el 14379 In a set with more than on...
hashgt12el2 14380 In a set with more than on...
hashgt23el 14381 A set with more than two e...
hashunlei 14382 Get an upper bound on a co...
hashsslei 14383 Get an upper bound on a co...
hashfz 14384 Value of the numeric cardi...
fzsdom2 14385 Condition for finite range...
hashfzo 14386 Cardinality of a half-open...
hashfzo0 14387 Cardinality of a half-open...
hashfzp1 14388 Value of the numeric cardi...
hashfz0 14389 Value of the numeric cardi...
hashxplem 14390 Lemma for ~ hashxp . (Con...
hashxp 14391 The size of the Cartesian ...
hashmap 14392 The size of the set expone...
hashpw 14393 The size of the power set ...
hashfun 14394 A finite set is a function...
hashres 14395 The number of elements of ...
hashreshashfun 14396 The number of elements of ...
hashimarn 14397 The size of the image of a...
hashimarni 14398 If the size of the image o...
hashfundm 14399 The size of a set function...
hashf1dmrn 14400 The size of the domain of ...
resunimafz0 14401 TODO-AV: Revise using ` F...
fnfz0hash 14402 The size of a function on ...
ffz0hash 14403 The size of a function on ...
fnfz0hashnn0 14404 The size of a function on ...
ffzo0hash 14405 The size of a function on ...
fnfzo0hash 14406 The size of a function on ...
fnfzo0hashnn0 14407 The value of the size func...
hashbclem 14408 Lemma for ~ hashbc : induc...
hashbc 14409 The binomial coefficient c...
hashfacen 14410 The number of bijections b...
hashfacenOLD 14411 Obsolete version of ~ hash...
hashf1lem1 14412 Lemma for ~ hashf1 . (Con...
hashf1lem1OLD 14413 Obsolete version of ~ hash...
hashf1lem2 14414 Lemma for ~ hashf1 . (Con...
hashf1 14415 The permutation number ` |...
hashfac 14416 A factorial counts the num...
leiso 14417 Two ways to write a strict...
leisorel 14418 Version of ~ isorel for st...
fz1isolem 14419 Lemma for ~ fz1iso . (Con...
fz1iso 14420 Any finite ordered set has...
ishashinf 14421 Any set that is not finite...
seqcoll 14422 The function ` F ` contain...
seqcoll2 14423 The function ` F ` contain...
phphashd 14424 Corollary of the Pigeonhol...
phphashrd 14425 Corollary of the Pigeonhol...
hashprlei 14426 An unordered pair has at m...
hash2pr 14427 A set of size two is an un...
hash2prde 14428 A set of size two is an un...
hash2exprb 14429 A set of size two is an un...
hash2prb 14430 A set of size two is a pro...
prprrab 14431 The set of proper pairs of...
nehash2 14432 The cardinality of a set w...
hash2prd 14433 A set of size two is an un...
hash2pwpr 14434 If the size of a subset of...
hashle2pr 14435 A nonempty set of size les...
hashle2prv 14436 A nonempty subset of a pow...
pr2pwpr 14437 The set of subsets of a pa...
hashge2el2dif 14438 A set with size at least 2...
hashge2el2difr 14439 A set with at least 2 diff...
hashge2el2difb 14440 A set has size at least 2 ...
hashdmpropge2 14441 The size of the domain of ...
hashtplei 14442 An unordered triple has at...
hashtpg 14443 The size of an unordered t...
hashge3el3dif 14444 A set with size at least 3...
elss2prb 14445 An element of the set of s...
hash2sspr 14446 A subset of size two is an...
exprelprel 14447 If there is an element of ...
hash3tr 14448 A set of size three is an ...
hash1to3 14449 If the size of a set is be...
fundmge2nop0 14450 A function with a domain c...
fundmge2nop 14451 A function with a domain c...
fun2dmnop0 14452 A function with a domain c...
fun2dmnop 14453 A function with a domain c...
hashdifsnp1 14454 If the size of a set is a ...
fi1uzind 14455 Properties of an ordered p...
brfi1uzind 14456 Properties of a binary rel...
brfi1ind 14457 Properties of a binary rel...
brfi1indALT 14458 Alternate proof of ~ brfi1...
opfi1uzind 14459 Properties of an ordered p...
opfi1ind 14460 Properties of an ordered p...
iswrd 14463 Property of being a word o...
wrdval 14464 Value of the set of words ...
iswrdi 14465 A zero-based sequence is a...
wrdf 14466 A word is a zero-based seq...
iswrdb 14467 A word over an alphabet is...
wrddm 14468 The indices of a word (i.e...
sswrd 14469 The set of words respects ...
snopiswrd 14470 A singleton of an ordered ...
wrdexg 14471 The set of words over a se...
wrdexb 14472 The set of words over a se...
wrdexi 14473 The set of words over a se...
wrdsymbcl 14474 A symbol within a word ove...
wrdfn 14475 A word is a function with ...
wrdv 14476 A word over an alphabet is...
wrdlndm 14477 The length of a word is no...
iswrdsymb 14478 An arbitrary word is a wor...
wrdfin 14479 A word is a finite set. (...
lencl 14480 The length of a word is a ...
lennncl 14481 The length of a nonempty w...
wrdffz 14482 A word is a function from ...
wrdeq 14483 Equality theorem for the s...
wrdeqi 14484 Equality theorem for the s...
iswrddm0 14485 A function with empty doma...
wrd0 14486 The empty set is a word (t...
0wrd0 14487 The empty word is the only...
ffz0iswrd 14488 A sequence with zero-based...
wrdsymb 14489 A word is a word over the ...
nfwrd 14490 Hypothesis builder for ` W...
csbwrdg 14491 Class substitution for the...
wrdnval 14492 Words of a fixed length ar...
wrdmap 14493 Words as a mapping. (Cont...
hashwrdn 14494 If there is only a finite ...
wrdnfi 14495 If there is only a finite ...
wrdsymb0 14496 A symbol at a position "ou...
wrdlenge1n0 14497 A word with length at leas...
len0nnbi 14498 The length of a word is a ...
wrdlenge2n0 14499 A word with length at leas...
wrdsymb1 14500 The first symbol of a none...
wrdlen1 14501 A word of length 1 starts ...
fstwrdne 14502 The first symbol of a none...
fstwrdne0 14503 The first symbol of a none...
eqwrd 14504 Two words are equal iff th...
elovmpowrd 14505 Implications for the value...
elovmptnn0wrd 14506 Implications for the value...
wrdred1 14507 A word truncated by a symb...
wrdred1hash 14508 The length of a word trunc...
lsw 14511 Extract the last symbol of...
lsw0 14512 The last symbol of an empt...
lsw0g 14513 The last symbol of an empt...
lsw1 14514 The last symbol of a word ...
lswcl 14515 Closure of the last symbol...
lswlgt0cl 14516 The last symbol of a nonem...
ccatfn 14519 The concatenation operator...
ccatfval 14520 Value of the concatenation...
ccatcl 14521 The concatenation of two w...
ccatlen 14522 The length of a concatenat...
ccat0 14523 The concatenation of two w...
ccatval1 14524 Value of a symbol in the l...
ccatval2 14525 Value of a symbol in the r...
ccatval3 14526 Value of a symbol in the r...
elfzelfzccat 14527 An element of a finite set...
ccatvalfn 14528 The concatenation of two w...
ccatsymb 14529 The symbol at a given posi...
ccatfv0 14530 The first symbol of a conc...
ccatval1lsw 14531 The last symbol of the lef...
ccatval21sw 14532 The first symbol of the ri...
ccatlid 14533 Concatenation of a word by...
ccatrid 14534 Concatenation of a word by...
ccatass 14535 Associative law for concat...
ccatrn 14536 The range of a concatenate...
ccatidid 14537 Concatenation of the empty...
lswccatn0lsw 14538 The last symbol of a word ...
lswccat0lsw 14539 The last symbol of a word ...
ccatalpha 14540 A concatenation of two arb...
ccatrcl1 14541 Reverse closure of a conca...
ids1 14544 Identity function protecti...
s1val 14545 Value of a singleton word....
s1rn 14546 The range of a singleton w...
s1eq 14547 Equality theorem for a sin...
s1eqd 14548 Equality theorem for a sin...
s1cl 14549 A singleton word is a word...
s1cld 14550 A singleton word is a word...
s1prc 14551 Value of a singleton word ...
s1cli 14552 A singleton word is a word...
s1len 14553 Length of a singleton word...
s1nz 14554 A singleton word is not th...
s1dm 14555 The domain of a singleton ...
s1dmALT 14556 Alternate version of ~ s1d...
s1fv 14557 Sole symbol of a singleton...
lsws1 14558 The last symbol of a singl...
eqs1 14559 A word of length 1 is a si...
wrdl1exs1 14560 A word of length 1 is a si...
wrdl1s1 14561 A word of length 1 is a si...
s111 14562 The singleton word functio...
ccatws1cl 14563 The concatenation of a wor...
ccatws1clv 14564 The concatenation of a wor...
ccat2s1cl 14565 The concatenation of two s...
ccats1alpha 14566 A concatenation of a word ...
ccatws1len 14567 The length of the concaten...
ccatws1lenp1b 14568 The length of a word is ` ...
wrdlenccats1lenm1 14569 The length of a word is th...
ccat2s1len 14570 The length of the concaten...
ccatw2s1cl 14571 The concatenation of a wor...
ccatw2s1len 14572 The length of the concaten...
ccats1val1 14573 Value of a symbol in the l...
ccats1val2 14574 Value of the symbol concat...
ccat1st1st 14575 The first symbol of a word...
ccat2s1p1 14576 Extract the first of two c...
ccat2s1p2 14577 Extract the second of two ...
ccatw2s1ass 14578 Associative law for a conc...
ccatws1n0 14579 The concatenation of a wor...
ccatws1ls 14580 The last symbol of the con...
lswccats1 14581 The last symbol of a word ...
lswccats1fst 14582 The last symbol of a nonem...
ccatw2s1p1 14583 Extract the symbol of the ...
ccatw2s1p2 14584 Extract the second of two ...
ccat2s1fvw 14585 Extract a symbol of a word...
ccat2s1fst 14586 The first symbol of the co...
swrdnznd 14589 The value of a subword ope...
swrdval 14590 Value of a subword. (Cont...
swrd00 14591 A zero length substring. ...
swrdcl 14592 Closure of the subword ext...
swrdval2 14593 Value of the subword extra...
swrdlen 14594 Length of an extracted sub...
swrdfv 14595 A symbol in an extracted s...
swrdfv0 14596 The first symbol in an ext...
swrdf 14597 A subword of a word is a f...
swrdvalfn 14598 Value of the subword extra...
swrdrn 14599 The range of a subword of ...
swrdlend 14600 The value of the subword e...
swrdnd 14601 The value of the subword e...
swrdnd2 14602 Value of the subword extra...
swrdnnn0nd 14603 The value of a subword ope...
swrdnd0 14604 The value of a subword ope...
swrd0 14605 A subword of an empty set ...
swrdrlen 14606 Length of a right-anchored...
swrdlen2 14607 Length of an extracted sub...
swrdfv2 14608 A symbol in an extracted s...
swrdwrdsymb 14609 A subword is a word over t...
swrdsb0eq 14610 Two subwords with the same...
swrdsbslen 14611 Two subwords with the same...
swrdspsleq 14612 Two words have a common su...
swrds1 14613 Extract a single symbol fr...
swrdlsw 14614 Extract the last single sy...
ccatswrd 14615 Joining two adjacent subwo...
swrdccat2 14616 Recover the right half of ...
pfxnndmnd 14619 The value of a prefix oper...
pfxval 14620 Value of a prefix operatio...
pfx00 14621 The zero length prefix is ...
pfx0 14622 A prefix of an empty set i...
pfxval0 14623 Value of a prefix operatio...
pfxcl 14624 Closure of the prefix extr...
pfxmpt 14625 Value of the prefix extrac...
pfxres 14626 Value of the subword extra...
pfxf 14627 A prefix of a word is a fu...
pfxfn 14628 Value of the prefix extrac...
pfxfv 14629 A symbol in a prefix of a ...
pfxlen 14630 Length of a prefix. (Cont...
pfxid 14631 A word is a prefix of itse...
pfxrn 14632 The range of a prefix of a...
pfxn0 14633 A prefix consisting of at ...
pfxnd 14634 The value of a prefix oper...
pfxnd0 14635 The value of a prefix oper...
pfxwrdsymb 14636 A prefix of a word is a wo...
addlenrevpfx 14637 The sum of the lengths of ...
addlenpfx 14638 The sum of the lengths of ...
pfxfv0 14639 The first symbol of a pref...
pfxtrcfv 14640 A symbol in a word truncat...
pfxtrcfv0 14641 The first symbol in a word...
pfxfvlsw 14642 The last symbol in a nonem...
pfxeq 14643 The prefixes of two words ...
pfxtrcfvl 14644 The last symbol in a word ...
pfxsuffeqwrdeq 14645 Two words are equal if and...
pfxsuff1eqwrdeq 14646 Two (nonempty) words are e...
disjwrdpfx 14647 Sets of words are disjoint...
ccatpfx 14648 Concatenating a prefix wit...
pfxccat1 14649 Recover the left half of a...
pfx1 14650 The prefix of length one o...
swrdswrdlem 14651 Lemma for ~ swrdswrd . (C...
swrdswrd 14652 A subword of a subword is ...
pfxswrd 14653 A prefix of a subword is a...
swrdpfx 14654 A subword of a prefix is a...
pfxpfx 14655 A prefix of a prefix is a ...
pfxpfxid 14656 A prefix of a prefix with ...
pfxcctswrd 14657 The concatenation of the p...
lenpfxcctswrd 14658 The length of the concaten...
lenrevpfxcctswrd 14659 The length of the concaten...
pfxlswccat 14660 Reconstruct a nonempty wor...
ccats1pfxeq 14661 The last symbol of a word ...
ccats1pfxeqrex 14662 There exists a symbol such...
ccatopth 14663 An ~ opth -like theorem fo...
ccatopth2 14664 An ~ opth -like theorem fo...
ccatlcan 14665 Concatenation of words is ...
ccatrcan 14666 Concatenation of words is ...
wrdeqs1cat 14667 Decompose a nonempty word ...
cats1un 14668 Express a word with an ext...
wrdind 14669 Perform induction over the...
wrd2ind 14670 Perform induction over the...
swrdccatfn 14671 The subword of a concatena...
swrdccatin1 14672 The subword of a concatena...
pfxccatin12lem4 14673 Lemma 4 for ~ pfxccatin12 ...
pfxccatin12lem2a 14674 Lemma for ~ pfxccatin12lem...
pfxccatin12lem1 14675 Lemma 1 for ~ pfxccatin12 ...
swrdccatin2 14676 The subword of a concatena...
pfxccatin12lem2c 14677 Lemma for ~ pfxccatin12lem...
pfxccatin12lem2 14678 Lemma 2 for ~ pfxccatin12 ...
pfxccatin12lem3 14679 Lemma 3 for ~ pfxccatin12 ...
pfxccatin12 14680 The subword of a concatena...
pfxccat3 14681 The subword of a concatena...
swrdccat 14682 The subword of a concatena...
pfxccatpfx1 14683 A prefix of a concatenatio...
pfxccatpfx2 14684 A prefix of a concatenatio...
pfxccat3a 14685 A prefix of a concatenatio...
swrdccat3blem 14686 Lemma for ~ swrdccat3b . ...
swrdccat3b 14687 A suffix of a concatenatio...
pfxccatid 14688 A prefix of a concatenatio...
ccats1pfxeqbi 14689 A word is a prefix of a wo...
swrdccatin1d 14690 The subword of a concatena...
swrdccatin2d 14691 The subword of a concatena...
pfxccatin12d 14692 The subword of a concatena...
reuccatpfxs1lem 14693 Lemma for ~ reuccatpfxs1 ....
reuccatpfxs1 14694 There is a unique word hav...
reuccatpfxs1v 14695 There is a unique word hav...
splval 14698 Value of the substring rep...
splcl 14699 Closure of the substring r...
splid 14700 Splicing a subword for the...
spllen 14701 The length of a splice. (...
splfv1 14702 Symbols to the left of a s...
splfv2a 14703 Symbols within the replace...
splval2 14704 Value of a splice, assumin...
revval 14707 Value of the word reversin...
revcl 14708 The reverse of a word is a...
revlen 14709 The reverse of a word has ...
revfv 14710 Reverse of a word at a poi...
rev0 14711 The empty word is its own ...
revs1 14712 Singleton words are their ...
revccat 14713 Antiautomorphic property o...
revrev 14714 Reversal is an involution ...
reps 14717 Construct a function mappi...
repsundef 14718 A function mapping a half-...
repsconst 14719 Construct a function mappi...
repsf 14720 The constructed function m...
repswsymb 14721 The symbols of a "repeated...
repsw 14722 A function mapping a half-...
repswlen 14723 The length of a "repeated ...
repsw0 14724 The "repeated symbol word"...
repsdf2 14725 Alternative definition of ...
repswsymball 14726 All the symbols of a "repe...
repswsymballbi 14727 A word is a "repeated symb...
repswfsts 14728 The first symbol of a none...
repswlsw 14729 The last symbol of a nonem...
repsw1 14730 The "repeated symbol word"...
repswswrd 14731 A subword of a "repeated s...
repswpfx 14732 A prefix of a repeated sym...
repswccat 14733 The concatenation of two "...
repswrevw 14734 The reverse of a "repeated...
cshfn 14737 Perform a cyclical shift f...
cshword 14738 Perform a cyclical shift f...
cshnz 14739 A cyclical shift is the em...
0csh0 14740 Cyclically shifting an emp...
cshw0 14741 A word cyclically shifted ...
cshwmodn 14742 Cyclically shifting a word...
cshwsublen 14743 Cyclically shifting a word...
cshwn 14744 A word cyclically shifted ...
cshwcl 14745 A cyclically shifted word ...
cshwlen 14746 The length of a cyclically...
cshwf 14747 A cyclically shifted word ...
cshwfn 14748 A cyclically shifted word ...
cshwrn 14749 The range of a cyclically ...
cshwidxmod 14750 The symbol at a given inde...
cshwidxmodr 14751 The symbol at a given inde...
cshwidx0mod 14752 The symbol at index 0 of a...
cshwidx0 14753 The symbol at index 0 of a...
cshwidxm1 14754 The symbol at index ((n-N)...
cshwidxm 14755 The symbol at index (n-N) ...
cshwidxn 14756 The symbol at index (n-1) ...
cshf1 14757 Cyclically shifting a word...
cshinj 14758 If a word is injectiv (reg...
repswcshw 14759 A cyclically shifted "repe...
2cshw 14760 Cyclically shifting a word...
2cshwid 14761 Cyclically shifting a word...
lswcshw 14762 The last symbol of a word ...
2cshwcom 14763 Cyclically shifting a word...
cshwleneq 14764 If the results of cyclical...
3cshw 14765 Cyclically shifting a word...
cshweqdif2 14766 If cyclically shifting two...
cshweqdifid 14767 If cyclically shifting a w...
cshweqrep 14768 If cyclically shifting a w...
cshw1 14769 If cyclically shifting a w...
cshw1repsw 14770 If cyclically shifting a w...
cshwsexa 14771 The class of (different!) ...
cshwsexaOLD 14772 Obsolete version of ~ cshw...
2cshwcshw 14773 If a word is a cyclically ...
scshwfzeqfzo 14774 For a nonempty word the se...
cshwcshid 14775 A cyclically shifted word ...
cshwcsh2id 14776 A cyclically shifted word ...
cshimadifsn 14777 The image of a cyclically ...
cshimadifsn0 14778 The image of a cyclically ...
wrdco 14779 Mapping a word by a functi...
lenco 14780 Length of a mapped word is...
s1co 14781 Mapping of a singleton wor...
revco 14782 Mapping of words (i.e., a ...
ccatco 14783 Mapping of words commutes ...
cshco 14784 Mapping of words commutes ...
swrdco 14785 Mapping of words commutes ...
pfxco 14786 Mapping of words commutes ...
lswco 14787 Mapping of (nonempty) word...
repsco 14788 Mapping of words commutes ...
cats1cld 14803 Closure of concatenation w...
cats1co 14804 Closure of concatenation w...
cats1cli 14805 Closure of concatenation w...
cats1fvn 14806 The last symbol of a conca...
cats1fv 14807 A symbol other than the la...
cats1len 14808 The length of concatenatio...
cats1cat 14809 Closure of concatenation w...
cats2cat 14810 Closure of concatenation o...
s2eqd 14811 Equality theorem for a dou...
s3eqd 14812 Equality theorem for a len...
s4eqd 14813 Equality theorem for a len...
s5eqd 14814 Equality theorem for a len...
s6eqd 14815 Equality theorem for a len...
s7eqd 14816 Equality theorem for a len...
s8eqd 14817 Equality theorem for a len...
s3eq2 14818 Equality theorem for a len...
s2cld 14819 A doubleton word is a word...
s3cld 14820 A length 3 string is a wor...
s4cld 14821 A length 4 string is a wor...
s5cld 14822 A length 5 string is a wor...
s6cld 14823 A length 6 string is a wor...
s7cld 14824 A length 7 string is a wor...
s8cld 14825 A length 7 string is a wor...
s2cl 14826 A doubleton word is a word...
s3cl 14827 A length 3 string is a wor...
s2cli 14828 A doubleton word is a word...
s3cli 14829 A length 3 string is a wor...
s4cli 14830 A length 4 string is a wor...
s5cli 14831 A length 5 string is a wor...
s6cli 14832 A length 6 string is a wor...
s7cli 14833 A length 7 string is a wor...
s8cli 14834 A length 8 string is a wor...
s2fv0 14835 Extract the first symbol f...
s2fv1 14836 Extract the second symbol ...
s2len 14837 The length of a doubleton ...
s2dm 14838 The domain of a doubleton ...
s3fv0 14839 Extract the first symbol f...
s3fv1 14840 Extract the second symbol ...
s3fv2 14841 Extract the third symbol f...
s3len 14842 The length of a length 3 s...
s4fv0 14843 Extract the first symbol f...
s4fv1 14844 Extract the second symbol ...
s4fv2 14845 Extract the third symbol f...
s4fv3 14846 Extract the fourth symbol ...
s4len 14847 The length of a length 4 s...
s5len 14848 The length of a length 5 s...
s6len 14849 The length of a length 6 s...
s7len 14850 The length of a length 7 s...
s8len 14851 The length of a length 8 s...
lsws2 14852 The last symbol of a doubl...
lsws3 14853 The last symbol of a 3 let...
lsws4 14854 The last symbol of a 4 let...
s2prop 14855 A length 2 word is an unor...
s2dmALT 14856 Alternate version of ~ s2d...
s3tpop 14857 A length 3 word is an unor...
s4prop 14858 A length 4 word is a union...
s3fn 14859 A length 3 word is a funct...
funcnvs1 14860 The converse of a singleto...
funcnvs2 14861 The converse of a length 2...
funcnvs3 14862 The converse of a length 3...
funcnvs4 14863 The converse of a length 4...
s2f1o 14864 A length 2 word with mutua...
f1oun2prg 14865 A union of unordered pairs...
s4f1o 14866 A length 4 word with mutua...
s4dom 14867 The domain of a length 4 w...
s2co 14868 Mapping a doubleton word b...
s3co 14869 Mapping a length 3 string ...
s0s1 14870 Concatenation of fixed len...
s1s2 14871 Concatenation of fixed len...
s1s3 14872 Concatenation of fixed len...
s1s4 14873 Concatenation of fixed len...
s1s5 14874 Concatenation of fixed len...
s1s6 14875 Concatenation of fixed len...
s1s7 14876 Concatenation of fixed len...
s2s2 14877 Concatenation of fixed len...
s4s2 14878 Concatenation of fixed len...
s4s3 14879 Concatenation of fixed len...
s4s4 14880 Concatenation of fixed len...
s3s4 14881 Concatenation of fixed len...
s2s5 14882 Concatenation of fixed len...
s5s2 14883 Concatenation of fixed len...
s2eq2s1eq 14884 Two length 2 words are equ...
s2eq2seq 14885 Two length 2 words are equ...
s3eqs2s1eq 14886 Two length 3 words are equ...
s3eq3seq 14887 Two length 3 words are equ...
swrds2 14888 Extract two adjacent symbo...
swrds2m 14889 Extract two adjacent symbo...
wrdlen2i 14890 Implications of a word of ...
wrd2pr2op 14891 A word of length two repre...
wrdlen2 14892 A word of length two. (Co...
wrdlen2s2 14893 A word of length two as do...
wrdl2exs2 14894 A word of length two is a ...
pfx2 14895 A prefix of length two. (...
wrd3tpop 14896 A word of length three rep...
wrdlen3s3 14897 A word of length three as ...
repsw2 14898 The "repeated symbol word"...
repsw3 14899 The "repeated symbol word"...
swrd2lsw 14900 Extract the last two symbo...
2swrd2eqwrdeq 14901 Two words of length at lea...
ccatw2s1ccatws2 14902 The concatenation of a wor...
ccat2s1fvwALT 14903 Alternate proof of ~ ccat2...
wwlktovf 14904 Lemma 1 for ~ wrd2f1tovbij...
wwlktovf1 14905 Lemma 2 for ~ wrd2f1tovbij...
wwlktovfo 14906 Lemma 3 for ~ wrd2f1tovbij...
wwlktovf1o 14907 Lemma 4 for ~ wrd2f1tovbij...
wrd2f1tovbij 14908 There is a bijection betwe...
eqwrds3 14909 A word is equal with a len...
wrdl3s3 14910 A word of length 3 is a le...
s3sndisj 14911 The singletons consisting ...
s3iunsndisj 14912 The union of singletons co...
ofccat 14913 Letterwise operations on w...
ofs1 14914 Letterwise operations on a...
ofs2 14915 Letterwise operations on a...
coss12d 14916 Subset deduction for compo...
trrelssd 14917 The composition of subclas...
xpcogend 14918 The most interesting case ...
xpcoidgend 14919 If two classes are not dis...
cotr2g 14920 Two ways of saying that th...
cotr2 14921 Two ways of saying a relat...
cotr3 14922 Two ways of saying a relat...
coemptyd 14923 Deduction about compositio...
xptrrel 14924 The cross product is alway...
0trrel 14925 The empty class is a trans...
cleq1lem 14926 Equality implies bijection...
cleq1 14927 Equality of relations impl...
clsslem 14928 The closure of a subclass ...
trcleq1 14933 Equality of relations impl...
trclsslem 14934 The transitive closure (as...
trcleq2lem 14935 Equality implies bijection...
cvbtrcl 14936 Change of bound variable i...
trcleq12lem 14937 Equality implies bijection...
trclexlem 14938 Existence of relation impl...
trclublem 14939 If a relation exists then ...
trclubi 14940 The Cartesian product of t...
trclubgi 14941 The union with the Cartesi...
trclub 14942 The Cartesian product of t...
trclubg 14943 The union with the Cartesi...
trclfv 14944 The transitive closure of ...
brintclab 14945 Two ways to express a bina...
brtrclfv 14946 Two ways of expressing the...
brcnvtrclfv 14947 Two ways of expressing the...
brtrclfvcnv 14948 Two ways of expressing the...
brcnvtrclfvcnv 14949 Two ways of expressing the...
trclfvss 14950 The transitive closure (as...
trclfvub 14951 The transitive closure of ...
trclfvlb 14952 The transitive closure of ...
trclfvcotr 14953 The transitive closure of ...
trclfvlb2 14954 The transitive closure of ...
trclfvlb3 14955 The transitive closure of ...
cotrtrclfv 14956 The transitive closure of ...
trclidm 14957 The transitive closure of ...
trclun 14958 Transitive closure of a un...
trclfvg 14959 The value of the transitiv...
trclfvcotrg 14960 The value of the transitiv...
reltrclfv 14961 The transitive closure of ...
dmtrclfv 14962 The domain of the transiti...
reldmrelexp 14965 The domain of the repeated...
relexp0g 14966 A relation composed zero t...
relexp0 14967 A relation composed zero t...
relexp0d 14968 A relation composed zero t...
relexpsucnnr 14969 A reduction for relation e...
relexp1g 14970 A relation composed once i...
dfid5 14971 Identity relation is equal...
dfid6 14972 Identity relation expresse...
relexp1d 14973 A relation composed once i...
relexpsucnnl 14974 A reduction for relation e...
relexpsucl 14975 A reduction for relation e...
relexpsucr 14976 A reduction for relation e...
relexpsucrd 14977 A reduction for relation e...
relexpsucld 14978 A reduction for relation e...
relexpcnv 14979 Commutation of converse an...
relexpcnvd 14980 Commutation of converse an...
relexp0rel 14981 The exponentiation of a cl...
relexprelg 14982 The exponentiation of a cl...
relexprel 14983 The exponentiation of a re...
relexpreld 14984 The exponentiation of a re...
relexpnndm 14985 The domain of an exponenti...
relexpdmg 14986 The domain of an exponenti...
relexpdm 14987 The domain of an exponenti...
relexpdmd 14988 The domain of an exponenti...
relexpnnrn 14989 The range of an exponentia...
relexprng 14990 The range of an exponentia...
relexprn 14991 The range of an exponentia...
relexprnd 14992 The range of an exponentia...
relexpfld 14993 The field of an exponentia...
relexpfldd 14994 The field of an exponentia...
relexpaddnn 14995 Relation composition becom...
relexpuzrel 14996 The exponentiation of a cl...
relexpaddg 14997 Relation composition becom...
relexpaddd 14998 Relation composition becom...
rtrclreclem1 15001 The reflexive, transitive ...
dfrtrclrec2 15002 If two elements are connec...
rtrclreclem2 15003 The reflexive, transitive ...
rtrclreclem3 15004 The reflexive, transitive ...
rtrclreclem4 15005 The reflexive, transitive ...
dfrtrcl2 15006 The two definitions ` t* `...
relexpindlem 15007 Principle of transitive in...
relexpind 15008 Principle of transitive in...
rtrclind 15009 Principle of transitive in...
shftlem 15012 Two ways to write a shifte...
shftuz 15013 A shift of the upper integ...
shftfval 15014 The value of the sequence ...
shftdm 15015 Domain of a relation shift...
shftfib 15016 Value of a fiber of the re...
shftfn 15017 Functionality and domain o...
shftval 15018 Value of a sequence shifte...
shftval2 15019 Value of a sequence shifte...
shftval3 15020 Value of a sequence shifte...
shftval4 15021 Value of a sequence shifte...
shftval5 15022 Value of a shifted sequenc...
shftf 15023 Functionality of a shifted...
2shfti 15024 Composite shift operations...
shftidt2 15025 Identity law for the shift...
shftidt 15026 Identity law for the shift...
shftcan1 15027 Cancellation law for the s...
shftcan2 15028 Cancellation law for the s...
seqshft 15029 Shifting the index set of ...
sgnval 15032 Value of the signum functi...
sgn0 15033 The signum of 0 is 0. (Co...
sgnp 15034 The signum of a positive e...
sgnrrp 15035 The signum of a positive r...
sgn1 15036 The signum of 1 is 1. (Co...
sgnpnf 15037 The signum of ` +oo ` is 1...
sgnn 15038 The signum of a negative e...
sgnmnf 15039 The signum of ` -oo ` is -...
cjval 15046 The value of the conjugate...
cjth 15047 The defining property of t...
cjf 15048 Domain and codomain of the...
cjcl 15049 The conjugate of a complex...
reval 15050 The value of the real part...
imval 15051 The value of the imaginary...
imre 15052 The imaginary part of a co...
reim 15053 The real part of a complex...
recl 15054 The real part of a complex...
imcl 15055 The imaginary part of a co...
ref 15056 Domain and codomain of the...
imf 15057 Domain and codomain of the...
crre 15058 The real part of a complex...
crim 15059 The real part of a complex...
replim 15060 Reconstruct a complex numb...
remim 15061 Value of the conjugate of ...
reim0 15062 The imaginary part of a re...
reim0b 15063 A number is real iff its i...
rereb 15064 A number is real iff it eq...
mulre 15065 A product with a nonzero r...
rere 15066 A real number equals its r...
cjreb 15067 A number is real iff it eq...
recj 15068 Real part of a complex con...
reneg 15069 Real part of negative. (C...
readd 15070 Real part distributes over...
resub 15071 Real part distributes over...
remullem 15072 Lemma for ~ remul , ~ immu...
remul 15073 Real part of a product. (...
remul2 15074 Real part of a product. (...
rediv 15075 Real part of a division. ...
imcj 15076 Imaginary part of a comple...
imneg 15077 The imaginary part of a ne...
imadd 15078 Imaginary part distributes...
imsub 15079 Imaginary part distributes...
immul 15080 Imaginary part of a produc...
immul2 15081 Imaginary part of a produc...
imdiv 15082 Imaginary part of a divisi...
cjre 15083 A real number equals its c...
cjcj 15084 The conjugate of the conju...
cjadd 15085 Complex conjugate distribu...
cjmul 15086 Complex conjugate distribu...
ipcnval 15087 Standard inner product on ...
cjmulrcl 15088 A complex number times its...
cjmulval 15089 A complex number times its...
cjmulge0 15090 A complex number times its...
cjneg 15091 Complex conjugate of negat...
addcj 15092 A number plus its conjugat...
cjsub 15093 Complex conjugate distribu...
cjexp 15094 Complex conjugate of posit...
imval2 15095 The imaginary part of a nu...
re0 15096 The real part of zero. (C...
im0 15097 The imaginary part of zero...
re1 15098 The real part of one. (Co...
im1 15099 The imaginary part of one....
rei 15100 The real part of ` _i ` . ...
imi 15101 The imaginary part of ` _i...
cj0 15102 The conjugate of zero. (C...
cji 15103 The complex conjugate of t...
cjreim 15104 The conjugate of a represe...
cjreim2 15105 The conjugate of the repre...
cj11 15106 Complex conjugate is a one...
cjne0 15107 A number is nonzero iff it...
cjdiv 15108 Complex conjugate distribu...
cnrecnv 15109 The inverse to the canonic...
sqeqd 15110 A deduction for showing tw...
recli 15111 The real part of a complex...
imcli 15112 The imaginary part of a co...
cjcli 15113 Closure law for complex co...
replimi 15114 Construct a complex number...
cjcji 15115 The conjugate of the conju...
reim0bi 15116 A number is real iff its i...
rerebi 15117 A real number equals its r...
cjrebi 15118 A number is real iff it eq...
recji 15119 Real part of a complex con...
imcji 15120 Imaginary part of a comple...
cjmulrcli 15121 A complex number times its...
cjmulvali 15122 A complex number times its...
cjmulge0i 15123 A complex number times its...
renegi 15124 Real part of negative. (C...
imnegi 15125 Imaginary part of negative...
cjnegi 15126 Complex conjugate of negat...
addcji 15127 A number plus its conjugat...
readdi 15128 Real part distributes over...
imaddi 15129 Imaginary part distributes...
remuli 15130 Real part of a product. (...
immuli 15131 Imaginary part of a produc...
cjaddi 15132 Complex conjugate distribu...
cjmuli 15133 Complex conjugate distribu...
ipcni 15134 Standard inner product on ...
cjdivi 15135 Complex conjugate distribu...
crrei 15136 The real part of a complex...
crimi 15137 The imaginary part of a co...
recld 15138 The real part of a complex...
imcld 15139 The imaginary part of a co...
cjcld 15140 Closure law for complex co...
replimd 15141 Construct a complex number...
remimd 15142 Value of the conjugate of ...
cjcjd 15143 The conjugate of the conju...
reim0bd 15144 A number is real iff its i...
rerebd 15145 A real number equals its r...
cjrebd 15146 A number is real iff it eq...
cjne0d 15147 A number is nonzero iff it...
recjd 15148 Real part of a complex con...
imcjd 15149 Imaginary part of a comple...
cjmulrcld 15150 A complex number times its...
cjmulvald 15151 A complex number times its...
cjmulge0d 15152 A complex number times its...
renegd 15153 Real part of negative. (C...
imnegd 15154 Imaginary part of negative...
cjnegd 15155 Complex conjugate of negat...
addcjd 15156 A number plus its conjugat...
cjexpd 15157 Complex conjugate of posit...
readdd 15158 Real part distributes over...
imaddd 15159 Imaginary part distributes...
resubd 15160 Real part distributes over...
imsubd 15161 Imaginary part distributes...
remuld 15162 Real part of a product. (...
immuld 15163 Imaginary part of a produc...
cjaddd 15164 Complex conjugate distribu...
cjmuld 15165 Complex conjugate distribu...
ipcnd 15166 Standard inner product on ...
cjdivd 15167 Complex conjugate distribu...
rered 15168 A real number equals its r...
reim0d 15169 The imaginary part of a re...
cjred 15170 A real number equals its c...
remul2d 15171 Real part of a product. (...
immul2d 15172 Imaginary part of a produc...
redivd 15173 Real part of a division. ...
imdivd 15174 Imaginary part of a divisi...
crred 15175 The real part of a complex...
crimd 15176 The imaginary part of a co...
sqrtval 15181 Value of square root funct...
absval 15182 The absolute value (modulu...
rennim 15183 A real number does not lie...
cnpart 15184 The specification of restr...
sqrt0 15185 The square root of zero is...
01sqrexlem1 15186 Lemma for ~ 01sqrex . (Co...
01sqrexlem2 15187 Lemma for ~ 01sqrex . (Co...
01sqrexlem3 15188 Lemma for ~ 01sqrex . (Co...
01sqrexlem4 15189 Lemma for ~ 01sqrex . (Co...
01sqrexlem5 15190 Lemma for ~ 01sqrex . (Co...
01sqrexlem6 15191 Lemma for ~ 01sqrex . (Co...
01sqrexlem7 15192 Lemma for ~ 01sqrex . (Co...
01sqrex 15193 Existence of a square root...
resqrex 15194 Existence of a square root...
sqrmo 15195 Uniqueness for the square ...
resqreu 15196 Existence and uniqueness f...
resqrtcl 15197 Closure of the square root...
resqrtthlem 15198 Lemma for ~ resqrtth . (C...
resqrtth 15199 Square root theorem over t...
remsqsqrt 15200 Square of square root. (C...
sqrtge0 15201 The square root function i...
sqrtgt0 15202 The square root function i...
sqrtmul 15203 Square root distributes ov...
sqrtle 15204 Square root is monotonic. ...
sqrtlt 15205 Square root is strictly mo...
sqrt11 15206 The square root function i...
sqrt00 15207 A square root is zero iff ...
rpsqrtcl 15208 The square root of a posit...
sqrtdiv 15209 Square root distributes ov...
sqrtneglem 15210 The square root of a negat...
sqrtneg 15211 The square root of a negat...
sqrtsq2 15212 Relationship between squar...
sqrtsq 15213 Square root of square. (C...
sqrtmsq 15214 Square root of square. (C...
sqrt1 15215 The square root of 1 is 1....
sqrt4 15216 The square root of 4 is 2....
sqrt9 15217 The square root of 9 is 3....
sqrt2gt1lt2 15218 The square root of 2 is bo...
sqrtm1 15219 The imaginary unit is the ...
nn0sqeq1 15220 A natural number with squa...
absneg 15221 Absolute value of the nega...
abscl 15222 Real closure of absolute v...
abscj 15223 The absolute value of a nu...
absvalsq 15224 Square of value of absolut...
absvalsq2 15225 Square of value of absolut...
sqabsadd 15226 Square of absolute value o...
sqabssub 15227 Square of absolute value o...
absval2 15228 Value of absolute value fu...
abs0 15229 The absolute value of 0. ...
absi 15230 The absolute value of the ...
absge0 15231 Absolute value is nonnegat...
absrpcl 15232 The absolute value of a no...
abs00 15233 The absolute value of a nu...
abs00ad 15234 A complex number is zero i...
abs00bd 15235 If a complex number is zer...
absreimsq 15236 Square of the absolute val...
absreim 15237 Absolute value of a number...
absmul 15238 Absolute value distributes...
absdiv 15239 Absolute value distributes...
absid 15240 A nonnegative number is it...
abs1 15241 The absolute value of one ...
absnid 15242 For a negative number, its...
leabs 15243 A real number is less than...
absor 15244 The absolute value of a re...
absre 15245 Absolute value of a real n...
absresq 15246 Square of the absolute val...
absmod0 15247 ` A ` is divisible by ` B ...
absexp 15248 Absolute value of positive...
absexpz 15249 Absolute value of integer ...
abssq 15250 Square can be moved in and...
sqabs 15251 The squares of two reals a...
absrele 15252 The absolute value of a co...
absimle 15253 The absolute value of a co...
max0add 15254 The sum of the positive an...
absz 15255 A real number is an intege...
nn0abscl 15256 The absolute value of an i...
zabscl 15257 The absolute value of an i...
abslt 15258 Absolute value and 'less t...
absle 15259 Absolute value and 'less t...
abssubne0 15260 If the absolute value of a...
absdiflt 15261 The absolute value of a di...
absdifle 15262 The absolute value of a di...
elicc4abs 15263 Membership in a symmetric ...
lenegsq 15264 Comparison to a nonnegativ...
releabs 15265 The real part of a number ...
recval 15266 Reciprocal expressed with ...
absidm 15267 The absolute value functio...
absgt0 15268 The absolute value of a no...
nnabscl 15269 The absolute value of a no...
abssub 15270 Swapping order of subtract...
abssubge0 15271 Absolute value of a nonneg...
abssuble0 15272 Absolute value of a nonpos...
absmax 15273 The maximum of two numbers...
abstri 15274 Triangle inequality for ab...
abs3dif 15275 Absolute value of differen...
abs2dif 15276 Difference of absolute val...
abs2dif2 15277 Difference of absolute val...
abs2difabs 15278 Absolute value of differen...
abs1m 15279 For any complex number, th...
recan 15280 Cancellation law involving...
absf 15281 Mapping domain and codomai...
abs3lem 15282 Lemma involving absolute v...
abslem2 15283 Lemma involving absolute v...
rddif 15284 The difference between a r...
absrdbnd 15285 Bound on the absolute valu...
fzomaxdiflem 15286 Lemma for ~ fzomaxdif . (...
fzomaxdif 15287 A bound on the separation ...
uzin2 15288 The upper integers are clo...
rexanuz 15289 Combine two different uppe...
rexanre 15290 Combine two different uppe...
rexfiuz 15291 Combine finitely many diff...
rexuz3 15292 Restrict the base of the u...
rexanuz2 15293 Combine two different uppe...
r19.29uz 15294 A version of ~ 19.29 for u...
r19.2uz 15295 A version of ~ r19.2z for ...
rexuzre 15296 Convert an upper real quan...
rexico 15297 Restrict the base of an up...
cau3lem 15298 Lemma for ~ cau3 . (Contr...
cau3 15299 Convert between three-quan...
cau4 15300 Change the base of a Cauch...
caubnd2 15301 A Cauchy sequence of compl...
caubnd 15302 A Cauchy sequence of compl...
sqreulem 15303 Lemma for ~ sqreu : write ...
sqreu 15304 Existence and uniqueness f...
sqrtcl 15305 Closure of the square root...
sqrtthlem 15306 Lemma for ~ sqrtth . (Con...
sqrtf 15307 Mapping domain and codomai...
sqrtth 15308 Square root theorem over t...
sqrtrege0 15309 The square root function m...
eqsqrtor 15310 Solve an equation containi...
eqsqrtd 15311 A deduction for showing th...
eqsqrt2d 15312 A deduction for showing th...
amgm2 15313 Arithmetic-geometric mean ...
sqrtthi 15314 Square root theorem. Theo...
sqrtcli 15315 The square root of a nonne...
sqrtgt0i 15316 The square root of a posit...
sqrtmsqi 15317 Square root of square. (C...
sqrtsqi 15318 Square root of square. (C...
sqsqrti 15319 Square of square root. (C...
sqrtge0i 15320 The square root of a nonne...
absidi 15321 A nonnegative number is it...
absnidi 15322 A negative number is the n...
leabsi 15323 A real number is less than...
absori 15324 The absolute value of a re...
absrei 15325 Absolute value of a real n...
sqrtpclii 15326 The square root of a posit...
sqrtgt0ii 15327 The square root of a posit...
sqrt11i 15328 The square root function i...
sqrtmuli 15329 Square root distributes ov...
sqrtmulii 15330 Square root distributes ov...
sqrtmsq2i 15331 Relationship between squar...
sqrtlei 15332 Square root is monotonic. ...
sqrtlti 15333 Square root is strictly mo...
abslti 15334 Absolute value and 'less t...
abslei 15335 Absolute value and 'less t...
cnsqrt00 15336 A square root of a complex...
absvalsqi 15337 Square of value of absolut...
absvalsq2i 15338 Square of value of absolut...
abscli 15339 Real closure of absolute v...
absge0i 15340 Absolute value is nonnegat...
absval2i 15341 Value of absolute value fu...
abs00i 15342 The absolute value of a nu...
absgt0i 15343 The absolute value of a no...
absnegi 15344 Absolute value of negative...
abscji 15345 The absolute value of a nu...
releabsi 15346 The real part of a number ...
abssubi 15347 Swapping order of subtract...
absmuli 15348 Absolute value distributes...
sqabsaddi 15349 Square of absolute value o...
sqabssubi 15350 Square of absolute value o...
absdivzi 15351 Absolute value distributes...
abstrii 15352 Triangle inequality for ab...
abs3difi 15353 Absolute value of differen...
abs3lemi 15354 Lemma involving absolute v...
rpsqrtcld 15355 The square root of a posit...
sqrtgt0d 15356 The square root of a posit...
absnidd 15357 A negative number is the n...
leabsd 15358 A real number is less than...
absord 15359 The absolute value of a re...
absred 15360 Absolute value of a real n...
resqrtcld 15361 The square root of a nonne...
sqrtmsqd 15362 Square root of square. (C...
sqrtsqd 15363 Square root of square. (C...
sqrtge0d 15364 The square root of a nonne...
sqrtnegd 15365 The square root of a negat...
absidd 15366 A nonnegative number is it...
sqrtdivd 15367 Square root distributes ov...
sqrtmuld 15368 Square root distributes ov...
sqrtsq2d 15369 Relationship between squar...
sqrtled 15370 Square root is monotonic. ...
sqrtltd 15371 Square root is strictly mo...
sqr11d 15372 The square root function i...
absltd 15373 Absolute value and 'less t...
absled 15374 Absolute value and 'less t...
abssubge0d 15375 Absolute value of a nonneg...
abssuble0d 15376 Absolute value of a nonpos...
absdifltd 15377 The absolute value of a di...
absdifled 15378 The absolute value of a di...
icodiamlt 15379 Two elements in a half-ope...
abscld 15380 Real closure of absolute v...
sqrtcld 15381 Closure of the square root...
sqrtrege0d 15382 The real part of the squar...
sqsqrtd 15383 Square root theorem. Theo...
msqsqrtd 15384 Square root theorem. Theo...
sqr00d 15385 A square root is zero iff ...
absvalsqd 15386 Square of value of absolut...
absvalsq2d 15387 Square of value of absolut...
absge0d 15388 Absolute value is nonnegat...
absval2d 15389 Value of absolute value fu...
abs00d 15390 The absolute value of a nu...
absne0d 15391 The absolute value of a nu...
absrpcld 15392 The absolute value of a no...
absnegd 15393 Absolute value of negative...
abscjd 15394 The absolute value of a nu...
releabsd 15395 The real part of a number ...
absexpd 15396 Absolute value of positive...
abssubd 15397 Swapping order of subtract...
absmuld 15398 Absolute value distributes...
absdivd 15399 Absolute value distributes...
abstrid 15400 Triangle inequality for ab...
abs2difd 15401 Difference of absolute val...
abs2dif2d 15402 Difference of absolute val...
abs2difabsd 15403 Absolute value of differen...
abs3difd 15404 Absolute value of differen...
abs3lemd 15405 Lemma involving absolute v...
reusq0 15406 A complex number is the sq...
bhmafibid1cn 15407 The Brahmagupta-Fibonacci ...
bhmafibid2cn 15408 The Brahmagupta-Fibonacci ...
bhmafibid1 15409 The Brahmagupta-Fibonacci ...
bhmafibid2 15410 The Brahmagupta-Fibonacci ...
limsupgord 15413 Ordering property of the s...
limsupcl 15414 Closure of the superior li...
limsupval 15415 The superior limit of an i...
limsupgf 15416 Closure of the superior li...
limsupgval 15417 Value of the superior limi...
limsupgle 15418 The defining property of t...
limsuple 15419 The defining property of t...
limsuplt 15420 The defining property of t...
limsupval2 15421 The superior limit, relati...
limsupgre 15422 If a sequence of real numb...
limsupbnd1 15423 If a sequence is eventuall...
limsupbnd2 15424 If a sequence is eventuall...
climrel 15433 The limit relation is a re...
rlimrel 15434 The limit relation is a re...
clim 15435 Express the predicate: Th...
rlim 15436 Express the predicate: Th...
rlim2 15437 Rewrite ~ rlim for a mappi...
rlim2lt 15438 Use strictly less-than in ...
rlim3 15439 Restrict the range of the ...
climcl 15440 Closure of the limit of a ...
rlimpm 15441 Closure of a function with...
rlimf 15442 Closure of a function with...
rlimss 15443 Domain closure of a functi...
rlimcl 15444 Closure of the limit of a ...
clim2 15445 Express the predicate: Th...
clim2c 15446 Express the predicate ` F ...
clim0 15447 Express the predicate ` F ...
clim0c 15448 Express the predicate ` F ...
rlim0 15449 Express the predicate ` B ...
rlim0lt 15450 Use strictly less-than in ...
climi 15451 Convergence of a sequence ...
climi2 15452 Convergence of a sequence ...
climi0 15453 Convergence of a sequence ...
rlimi 15454 Convergence at infinity of...
rlimi2 15455 Convergence at infinity of...
ello1 15456 Elementhood in the set of ...
ello12 15457 Elementhood in the set of ...
ello12r 15458 Sufficient condition for e...
lo1f 15459 An eventually upper bounde...
lo1dm 15460 An eventually upper bounde...
lo1bdd 15461 The defining property of a...
ello1mpt 15462 Elementhood in the set of ...
ello1mpt2 15463 Elementhood in the set of ...
ello1d 15464 Sufficient condition for e...
lo1bdd2 15465 If an eventually bounded f...
lo1bddrp 15466 Refine ~ o1bdd2 to give a ...
elo1 15467 Elementhood in the set of ...
elo12 15468 Elementhood in the set of ...
elo12r 15469 Sufficient condition for e...
o1f 15470 An eventually bounded func...
o1dm 15471 An eventually bounded func...
o1bdd 15472 The defining property of a...
lo1o1 15473 A function is eventually b...
lo1o12 15474 A function is eventually b...
elo1mpt 15475 Elementhood in the set of ...
elo1mpt2 15476 Elementhood in the set of ...
elo1d 15477 Sufficient condition for e...
o1lo1 15478 A real function is eventua...
o1lo12 15479 A lower bounded real funct...
o1lo1d 15480 A real eventually bounded ...
icco1 15481 Derive eventual boundednes...
o1bdd2 15482 If an eventually bounded f...
o1bddrp 15483 Refine ~ o1bdd2 to give a ...
climconst 15484 An (eventually) constant s...
rlimconst 15485 A constant sequence conver...
rlimclim1 15486 Forward direction of ~ rli...
rlimclim 15487 A sequence on an upper int...
climrlim2 15488 Produce a real limit from ...
climconst2 15489 A constant sequence conver...
climz 15490 The zero sequence converge...
rlimuni 15491 A real function whose doma...
rlimdm 15492 Two ways to express that a...
climuni 15493 An infinite sequence of co...
fclim 15494 The limit relation is func...
climdm 15495 Two ways to express that a...
climeu 15496 An infinite sequence of co...
climreu 15497 An infinite sequence of co...
climmo 15498 An infinite sequence of co...
rlimres 15499 The restriction of a funct...
lo1res 15500 The restriction of an even...
o1res 15501 The restriction of an even...
rlimres2 15502 The restriction of a funct...
lo1res2 15503 The restriction of a funct...
o1res2 15504 The restriction of a funct...
lo1resb 15505 The restriction of a funct...
rlimresb 15506 The restriction of a funct...
o1resb 15507 The restriction of a funct...
climeq 15508 Two functions that are eve...
lo1eq 15509 Two functions that are eve...
rlimeq 15510 Two functions that are eve...
o1eq 15511 Two functions that are eve...
climmpt 15512 Exhibit a function ` G ` w...
2clim 15513 If two sequences converge ...
climmpt2 15514 Relate an integer limit on...
climshftlem 15515 A shifted function converg...
climres 15516 A function restricted to u...
climshft 15517 A shifted function converg...
serclim0 15518 The zero series converges ...
rlimcld2 15519 If ` D ` is a closed set i...
rlimrege0 15520 The limit of a sequence of...
rlimrecl 15521 The limit of a real sequen...
rlimge0 15522 The limit of a sequence of...
climshft2 15523 A shifted function converg...
climrecl 15524 The limit of a convergent ...
climge0 15525 A nonnegative sequence con...
climabs0 15526 Convergence to zero of the...
o1co 15527 Sufficient condition for t...
o1compt 15528 Sufficient condition for t...
rlimcn1 15529 Image of a limit under a c...
rlimcn1b 15530 Image of a limit under a c...
rlimcn3 15531 Image of a limit under a c...
rlimcn2 15532 Image of a limit under a c...
climcn1 15533 Image of a limit under a c...
climcn2 15534 Image of a limit under a c...
addcn2 15535 Complex number addition is...
subcn2 15536 Complex number subtraction...
mulcn2 15537 Complex number multiplicat...
reccn2 15538 The reciprocal function is...
cn1lem 15539 A sufficient condition for...
abscn2 15540 The absolute value functio...
cjcn2 15541 The complex conjugate func...
recn2 15542 The real part function is ...
imcn2 15543 The imaginary part functio...
climcn1lem 15544 The limit of a continuous ...
climabs 15545 Limit of the absolute valu...
climcj 15546 Limit of the complex conju...
climre 15547 Limit of the real part of ...
climim 15548 Limit of the imaginary par...
rlimmptrcl 15549 Reverse closure for a real...
rlimabs 15550 Limit of the absolute valu...
rlimcj 15551 Limit of the complex conju...
rlimre 15552 Limit of the real part of ...
rlimim 15553 Limit of the imaginary par...
o1of2 15554 Show that a binary operati...
o1add 15555 The sum of two eventually ...
o1mul 15556 The product of two eventua...
o1sub 15557 The difference of two even...
rlimo1 15558 Any function with a finite...
rlimdmo1 15559 A convergent function is e...
o1rlimmul 15560 The product of an eventual...
o1const 15561 A constant function is eve...
lo1const 15562 A constant function is eve...
lo1mptrcl 15563 Reverse closure for an eve...
o1mptrcl 15564 Reverse closure for an eve...
o1add2 15565 The sum of two eventually ...
o1mul2 15566 The product of two eventua...
o1sub2 15567 The product of two eventua...
lo1add 15568 The sum of two eventually ...
lo1mul 15569 The product of an eventual...
lo1mul2 15570 The product of an eventual...
o1dif 15571 If the difference of two f...
lo1sub 15572 The difference of an event...
climadd 15573 Limit of the sum of two co...
climmul 15574 Limit of the product of tw...
climsub 15575 Limit of the difference of...
climaddc1 15576 Limit of a constant ` C ` ...
climaddc2 15577 Limit of a constant ` C ` ...
climmulc2 15578 Limit of a sequence multip...
climsubc1 15579 Limit of a constant ` C ` ...
climsubc2 15580 Limit of a constant ` C ` ...
climle 15581 Comparison of the limits o...
climsqz 15582 Convergence of a sequence ...
climsqz2 15583 Convergence of a sequence ...
rlimadd 15584 Limit of the sum of two co...
rlimaddOLD 15585 Obsolete version of ~ rlim...
rlimsub 15586 Limit of the difference of...
rlimmul 15587 Limit of the product of tw...
rlimmulOLD 15588 Obsolete version of ~ rlim...
rlimdiv 15589 Limit of the quotient of t...
rlimneg 15590 Limit of the negative of a...
rlimle 15591 Comparison of the limits o...
rlimsqzlem 15592 Lemma for ~ rlimsqz and ~ ...
rlimsqz 15593 Convergence of a sequence ...
rlimsqz2 15594 Convergence of a sequence ...
lo1le 15595 Transfer eventual upper bo...
o1le 15596 Transfer eventual boundedn...
rlimno1 15597 A function whose inverse c...
clim2ser 15598 The limit of an infinite s...
clim2ser2 15599 The limit of an infinite s...
iserex 15600 An infinite series converg...
isermulc2 15601 Multiplication of an infin...
climlec2 15602 Comparison of a constant t...
iserle 15603 Comparison of the limits o...
iserge0 15604 The limit of an infinite s...
climub 15605 The limit of a monotonic s...
climserle 15606 The partial sums of a conv...
isershft 15607 Index shift of the limit o...
isercolllem1 15608 Lemma for ~ isercoll . (C...
isercolllem2 15609 Lemma for ~ isercoll . (C...
isercolllem3 15610 Lemma for ~ isercoll . (C...
isercoll 15611 Rearrange an infinite seri...
isercoll2 15612 Generalize ~ isercoll so t...
climsup 15613 A bounded monotonic sequen...
climcau 15614 A converging sequence of c...
climbdd 15615 A converging sequence of c...
caucvgrlem 15616 Lemma for ~ caurcvgr . (C...
caurcvgr 15617 A Cauchy sequence of real ...
caucvgrlem2 15618 Lemma for ~ caucvgr . (Co...
caucvgr 15619 A Cauchy sequence of compl...
caurcvg 15620 A Cauchy sequence of real ...
caurcvg2 15621 A Cauchy sequence of real ...
caucvg 15622 A Cauchy sequence of compl...
caucvgb 15623 A function is convergent i...
serf0 15624 If an infinite series conv...
iseraltlem1 15625 Lemma for ~ iseralt . A d...
iseraltlem2 15626 Lemma for ~ iseralt . The...
iseraltlem3 15627 Lemma for ~ iseralt . Fro...
iseralt 15628 The alternating series tes...
sumex 15631 A sum is a set. (Contribu...
sumeq1 15632 Equality theorem for a sum...
nfsum1 15633 Bound-variable hypothesis ...
nfsum 15634 Bound-variable hypothesis ...
sumeq2w 15635 Equality theorem for sum, ...
sumeq2ii 15636 Equality theorem for sum, ...
sumeq2 15637 Equality theorem for sum. ...
cbvsum 15638 Change bound variable in a...
cbvsumv 15639 Change bound variable in a...
cbvsumi 15640 Change bound variable in a...
sumeq1i 15641 Equality inference for sum...
sumeq2i 15642 Equality inference for sum...
sumeq12i 15643 Equality inference for sum...
sumeq1d 15644 Equality deduction for sum...
sumeq2d 15645 Equality deduction for sum...
sumeq2dv 15646 Equality deduction for sum...
sumeq2sdv 15647 Equality deduction for sum...
2sumeq2dv 15648 Equality deduction for dou...
sumeq12dv 15649 Equality deduction for sum...
sumeq12rdv 15650 Equality deduction for sum...
sum2id 15651 The second class argument ...
sumfc 15652 A lemma to facilitate conv...
fz1f1o 15653 A lemma for working with f...
sumrblem 15654 Lemma for ~ sumrb . (Cont...
fsumcvg 15655 The sequence of partial su...
sumrb 15656 Rebase the starting point ...
summolem3 15657 Lemma for ~ summo . (Cont...
summolem2a 15658 Lemma for ~ summo . (Cont...
summolem2 15659 Lemma for ~ summo . (Cont...
summo 15660 A sum has at most one limi...
zsum 15661 Series sum with index set ...
isum 15662 Series sum with an upper i...
fsum 15663 The value of a sum over a ...
sum0 15664 Any sum over the empty set...
sumz 15665 Any sum of zero over a sum...
fsumf1o 15666 Re-index a finite sum usin...
sumss 15667 Change the index set to a ...
fsumss 15668 Change the index set to a ...
sumss2 15669 Change the index set of a ...
fsumcvg2 15670 The sequence of partial su...
fsumsers 15671 Special case of series sum...
fsumcvg3 15672 A finite sum is convergent...
fsumser 15673 A finite sum expressed in ...
fsumcl2lem 15674 - Lemma for finite sum clo...
fsumcllem 15675 - Lemma for finite sum clo...
fsumcl 15676 Closure of a finite sum of...
fsumrecl 15677 Closure of a finite sum of...
fsumzcl 15678 Closure of a finite sum of...
fsumnn0cl 15679 Closure of a finite sum of...
fsumrpcl 15680 Closure of a finite sum of...
fsumclf 15681 Closure of a finite sum of...
fsumzcl2 15682 A finite sum with integer ...
fsumadd 15683 The sum of two finite sums...
fsumsplit 15684 Split a sum into two parts...
fsumsplitf 15685 Split a sum into two parts...
sumsnf 15686 A sum of a singleton is th...
fsumsplitsn 15687 Separate out a term in a f...
fsumsplit1 15688 Separate out a term in a f...
sumsn 15689 A sum of a singleton is th...
fsum1 15690 The finite sum of ` A ( k ...
sumpr 15691 A sum over a pair is the s...
sumtp 15692 A sum over a triple is the...
sumsns 15693 A sum of a singleton is th...
fsumm1 15694 Separate out the last term...
fzosump1 15695 Separate out the last term...
fsum1p 15696 Separate out the first ter...
fsummsnunz 15697 A finite sum all of whose ...
fsumsplitsnun 15698 Separate out a term in a f...
fsump1 15699 The addition of the next t...
isumclim 15700 An infinite sum equals the...
isumclim2 15701 A converging series conver...
isumclim3 15702 The sequence of partial fi...
sumnul 15703 The sum of a non-convergen...
isumcl 15704 The sum of a converging in...
isummulc2 15705 An infinite sum multiplied...
isummulc1 15706 An infinite sum multiplied...
isumdivc 15707 An infinite sum divided by...
isumrecl 15708 The sum of a converging in...
isumge0 15709 An infinite sum of nonnega...
isumadd 15710 Addition of infinite sums....
sumsplit 15711 Split a sum into two parts...
fsump1i 15712 Optimized version of ~ fsu...
fsum2dlem 15713 Lemma for ~ fsum2d - induc...
fsum2d 15714 Write a double sum as a su...
fsumxp 15715 Combine two sums into a si...
fsumcnv 15716 Transform a region of summ...
fsumcom2 15717 Interchange order of summa...
fsumcom 15718 Interchange order of summa...
fsum0diaglem 15719 Lemma for ~ fsum0diag . (...
fsum0diag 15720 Two ways to express "the s...
mptfzshft 15721 1-1 onto function in maps-...
fsumrev 15722 Reversal of a finite sum. ...
fsumshft 15723 Index shift of a finite su...
fsumshftm 15724 Negative index shift of a ...
fsumrev2 15725 Reversal of a finite sum. ...
fsum0diag2 15726 Two ways to express "the s...
fsummulc2 15727 A finite sum multiplied by...
fsummulc1 15728 A finite sum multiplied by...
fsumdivc 15729 A finite sum divided by a ...
fsumneg 15730 Negation of a finite sum. ...
fsumsub 15731 Split a finite sum over a ...
fsum2mul 15732 Separate the nested sum of...
fsumconst 15733 The sum of constant terms ...
fsumdifsnconst 15734 The sum of constant terms ...
modfsummodslem1 15735 Lemma 1 for ~ modfsummods ...
modfsummods 15736 Induction step for ~ modfs...
modfsummod 15737 A finite sum modulo a posi...
fsumge0 15738 If all of the terms of a f...
fsumless 15739 A shorter sum of nonnegati...
fsumge1 15740 A sum of nonnegative numbe...
fsum00 15741 A sum of nonnegative numbe...
fsumle 15742 If all of the terms of fin...
fsumlt 15743 If every term in one finit...
fsumabs 15744 Generalized triangle inequ...
telfsumo 15745 Sum of a telescoping serie...
telfsumo2 15746 Sum of a telescoping serie...
telfsum 15747 Sum of a telescoping serie...
telfsum2 15748 Sum of a telescoping serie...
fsumparts 15749 Summation by parts. (Cont...
fsumrelem 15750 Lemma for ~ fsumre , ~ fsu...
fsumre 15751 The real part of a sum. (...
fsumim 15752 The imaginary part of a su...
fsumcj 15753 The complex conjugate of a...
fsumrlim 15754 Limit of a finite sum of c...
fsumo1 15755 The finite sum of eventual...
o1fsum 15756 If ` A ( k ) ` is O(1), th...
seqabs 15757 Generalized triangle inequ...
iserabs 15758 Generalized triangle inequ...
cvgcmp 15759 A comparison test for conv...
cvgcmpub 15760 An upper bound for the lim...
cvgcmpce 15761 A comparison test for conv...
abscvgcvg 15762 An absolutely convergent s...
climfsum 15763 Limit of a finite sum of c...
fsumiun 15764 Sum over a disjoint indexe...
hashiun 15765 The cardinality of a disjo...
hash2iun 15766 The cardinality of a neste...
hash2iun1dif1 15767 The cardinality of a neste...
hashrabrex 15768 The number of elements in ...
hashuni 15769 The cardinality of a disjo...
qshash 15770 The cardinality of a set w...
ackbijnn 15771 Translate the Ackermann bi...
binomlem 15772 Lemma for ~ binom (binomia...
binom 15773 The binomial theorem: ` ( ...
binom1p 15774 Special case of the binomi...
binom11 15775 Special case of the binomi...
binom1dif 15776 A summation for the differ...
bcxmaslem1 15777 Lemma for ~ bcxmas . (Con...
bcxmas 15778 Parallel summation (Christ...
incexclem 15779 Lemma for ~ incexc . (Con...
incexc 15780 The inclusion/exclusion pr...
incexc2 15781 The inclusion/exclusion pr...
isumshft 15782 Index shift of an infinite...
isumsplit 15783 Split off the first ` N ` ...
isum1p 15784 The infinite sum of a conv...
isumnn0nn 15785 Sum from 0 to infinity in ...
isumrpcl 15786 The infinite sum of positi...
isumle 15787 Comparison of two infinite...
isumless 15788 A finite sum of nonnegativ...
isumsup2 15789 An infinite sum of nonnega...
isumsup 15790 An infinite sum of nonnega...
isumltss 15791 A partial sum of a series ...
climcndslem1 15792 Lemma for ~ climcnds : bou...
climcndslem2 15793 Lemma for ~ climcnds : bou...
climcnds 15794 The Cauchy condensation te...
divrcnv 15795 The sequence of reciprocal...
divcnv 15796 The sequence of reciprocal...
flo1 15797 The floor function satisfi...
divcnvshft 15798 Limit of a ratio function....
supcvg 15799 Extract a sequence ` f ` i...
infcvgaux1i 15800 Auxiliary theorem for appl...
infcvgaux2i 15801 Auxiliary theorem for appl...
harmonic 15802 The harmonic series ` H ` ...
arisum 15803 Arithmetic series sum of t...
arisum2 15804 Arithmetic series sum of t...
trireciplem 15805 Lemma for ~ trirecip . Sh...
trirecip 15806 The sum of the reciprocals...
expcnv 15807 A sequence of powers of a ...
explecnv 15808 A sequence of terms conver...
geoserg 15809 The value of the finite ge...
geoser 15810 The value of the finite ge...
pwdif 15811 The difference of two numb...
pwm1geoser 15812 The n-th power of a number...
geolim 15813 The partial sums in the in...
geolim2 15814 The partial sums in the ge...
georeclim 15815 The limit of a geometric s...
geo2sum 15816 The value of the finite ge...
geo2sum2 15817 The value of the finite ge...
geo2lim 15818 The value of the infinite ...
geomulcvg 15819 The geometric series conve...
geoisum 15820 The infinite sum of ` 1 + ...
geoisumr 15821 The infinite sum of recipr...
geoisum1 15822 The infinite sum of ` A ^ ...
geoisum1c 15823 The infinite sum of ` A x....
0.999... 15824 The recurring decimal 0.99...
geoihalfsum 15825 Prove that the infinite ge...
cvgrat 15826 Ratio test for convergence...
mertenslem1 15827 Lemma for ~ mertens . (Co...
mertenslem2 15828 Lemma for ~ mertens . (Co...
mertens 15829 Mertens' theorem. If ` A ...
prodf 15830 An infinite product of com...
clim2prod 15831 The limit of an infinite p...
clim2div 15832 The limit of an infinite p...
prodfmul 15833 The product of two infinit...
prodf1 15834 The value of the partial p...
prodf1f 15835 A one-valued infinite prod...
prodfclim1 15836 The constant one product c...
prodfn0 15837 No term of a nonzero infin...
prodfrec 15838 The reciprocal of an infin...
prodfdiv 15839 The quotient of two infini...
ntrivcvg 15840 A non-trivially converging...
ntrivcvgn0 15841 A product that converges t...
ntrivcvgfvn0 15842 Any value of a product seq...
ntrivcvgtail 15843 A tail of a non-trivially ...
ntrivcvgmullem 15844 Lemma for ~ ntrivcvgmul . ...
ntrivcvgmul 15845 The product of two non-tri...
prodex 15848 A product is a set. (Cont...
prodeq1f 15849 Equality theorem for a pro...
prodeq1 15850 Equality theorem for a pro...
nfcprod1 15851 Bound-variable hypothesis ...
nfcprod 15852 Bound-variable hypothesis ...
prodeq2w 15853 Equality theorem for produ...
prodeq2ii 15854 Equality theorem for produ...
prodeq2 15855 Equality theorem for produ...
cbvprod 15856 Change bound variable in a...
cbvprodv 15857 Change bound variable in a...
cbvprodi 15858 Change bound variable in a...
prodeq1i 15859 Equality inference for pro...
prodeq2i 15860 Equality inference for pro...
prodeq12i 15861 Equality inference for pro...
prodeq1d 15862 Equality deduction for pro...
prodeq2d 15863 Equality deduction for pro...
prodeq2dv 15864 Equality deduction for pro...
prodeq2sdv 15865 Equality deduction for pro...
2cprodeq2dv 15866 Equality deduction for dou...
prodeq12dv 15867 Equality deduction for pro...
prodeq12rdv 15868 Equality deduction for pro...
prod2id 15869 The second class argument ...
prodrblem 15870 Lemma for ~ prodrb . (Con...
fprodcvg 15871 The sequence of partial pr...
prodrblem2 15872 Lemma for ~ prodrb . (Con...
prodrb 15873 Rebase the starting point ...
prodmolem3 15874 Lemma for ~ prodmo . (Con...
prodmolem2a 15875 Lemma for ~ prodmo . (Con...
prodmolem2 15876 Lemma for ~ prodmo . (Con...
prodmo 15877 A product has at most one ...
zprod 15878 Series product with index ...
iprod 15879 Series product with an upp...
zprodn0 15880 Nonzero series product wit...
iprodn0 15881 Nonzero series product wit...
fprod 15882 The value of a product ove...
fprodntriv 15883 A non-triviality lemma for...
prod0 15884 A product over the empty s...
prod1 15885 Any product of one over a ...
prodfc 15886 A lemma to facilitate conv...
fprodf1o 15887 Re-index a finite product ...
prodss 15888 Change the index set to a ...
fprodss 15889 Change the index set to a ...
fprodser 15890 A finite product expressed...
fprodcl2lem 15891 Finite product closure lem...
fprodcllem 15892 Finite product closure lem...
fprodcl 15893 Closure of a finite produc...
fprodrecl 15894 Closure of a finite produc...
fprodzcl 15895 Closure of a finite produc...
fprodnncl 15896 Closure of a finite produc...
fprodrpcl 15897 Closure of a finite produc...
fprodnn0cl 15898 Closure of a finite produc...
fprodcllemf 15899 Finite product closure lem...
fprodreclf 15900 Closure of a finite produc...
fprodmul 15901 The product of two finite ...
fproddiv 15902 The quotient of two finite...
prodsn 15903 A product of a singleton i...
fprod1 15904 A finite product of only o...
prodsnf 15905 A product of a singleton i...
climprod1 15906 The limit of a product ove...
fprodsplit 15907 Split a finite product int...
fprodm1 15908 Separate out the last term...
fprod1p 15909 Separate out the first ter...
fprodp1 15910 Multiply in the last term ...
fprodm1s 15911 Separate out the last term...
fprodp1s 15912 Multiply in the last term ...
prodsns 15913 A product of the singleton...
fprodfac 15914 Factorial using product no...
fprodabs 15915 The absolute value of a fi...
fprodeq0 15916 Any finite product contain...
fprodshft 15917 Shift the index of a finit...
fprodrev 15918 Reversal of a finite produ...
fprodconst 15919 The product of constant te...
fprodn0 15920 A finite product of nonzer...
fprod2dlem 15921 Lemma for ~ fprod2d - indu...
fprod2d 15922 Write a double product as ...
fprodxp 15923 Combine two products into ...
fprodcnv 15924 Transform a product region...
fprodcom2 15925 Interchange order of multi...
fprodcom 15926 Interchange product order....
fprod0diag 15927 Two ways to express "the p...
fproddivf 15928 The quotient of two finite...
fprodsplitf 15929 Split a finite product int...
fprodsplitsn 15930 Separate out a term in a f...
fprodsplit1f 15931 Separate out a term in a f...
fprodn0f 15932 A finite product of nonzer...
fprodclf 15933 Closure of a finite produc...
fprodge0 15934 If all the terms of a fini...
fprodeq0g 15935 Any finite product contain...
fprodge1 15936 If all of the terms of a f...
fprodle 15937 If all the terms of two fi...
fprodmodd 15938 If all factors of two fini...
iprodclim 15939 An infinite product equals...
iprodclim2 15940 A converging product conve...
iprodclim3 15941 The sequence of partial fi...
iprodcl 15942 The product of a non-trivi...
iprodrecl 15943 The product of a non-trivi...
iprodmul 15944 Multiplication of infinite...
risefacval 15949 The value of the rising fa...
fallfacval 15950 The value of the falling f...
risefacval2 15951 One-based value of rising ...
fallfacval2 15952 One-based value of falling...
fallfacval3 15953 A product representation o...
risefaccllem 15954 Lemma for rising factorial...
fallfaccllem 15955 Lemma for falling factoria...
risefaccl 15956 Closure law for rising fac...
fallfaccl 15957 Closure law for falling fa...
rerisefaccl 15958 Closure law for rising fac...
refallfaccl 15959 Closure law for falling fa...
nnrisefaccl 15960 Closure law for rising fac...
zrisefaccl 15961 Closure law for rising fac...
zfallfaccl 15962 Closure law for falling fa...
nn0risefaccl 15963 Closure law for rising fac...
rprisefaccl 15964 Closure law for rising fac...
risefallfac 15965 A relationship between ris...
fallrisefac 15966 A relationship between fal...
risefall0lem 15967 Lemma for ~ risefac0 and ~...
risefac0 15968 The value of the rising fa...
fallfac0 15969 The value of the falling f...
risefacp1 15970 The value of the rising fa...
fallfacp1 15971 The value of the falling f...
risefacp1d 15972 The value of the rising fa...
fallfacp1d 15973 The value of the falling f...
risefac1 15974 The value of rising factor...
fallfac1 15975 The value of falling facto...
risefacfac 15976 Relate rising factorial to...
fallfacfwd 15977 The forward difference of ...
0fallfac 15978 The value of the zero fall...
0risefac 15979 The value of the zero risi...
binomfallfaclem1 15980 Lemma for ~ binomfallfac ....
binomfallfaclem2 15981 Lemma for ~ binomfallfac ....
binomfallfac 15982 A version of the binomial ...
binomrisefac 15983 A version of the binomial ...
fallfacval4 15984 Represent the falling fact...
bcfallfac 15985 Binomial coefficient in te...
fallfacfac 15986 Relate falling factorial t...
bpolylem 15989 Lemma for ~ bpolyval . (C...
bpolyval 15990 The value of the Bernoulli...
bpoly0 15991 The value of the Bernoulli...
bpoly1 15992 The value of the Bernoulli...
bpolycl 15993 Closure law for Bernoulli ...
bpolysum 15994 A sum for Bernoulli polyno...
bpolydiflem 15995 Lemma for ~ bpolydif . (C...
bpolydif 15996 Calculate the difference b...
fsumkthpow 15997 A closed-form expression f...
bpoly2 15998 The Bernoulli polynomials ...
bpoly3 15999 The Bernoulli polynomials ...
bpoly4 16000 The Bernoulli polynomials ...
fsumcube 16001 Express the sum of cubes i...
eftcl 16014 Closure of a term in the s...
reeftcl 16015 The terms of the series ex...
eftabs 16016 The absolute value of a te...
eftval 16017 The value of a term in the...
efcllem 16018 Lemma for ~ efcl . The se...
ef0lem 16019 The series defining the ex...
efval 16020 Value of the exponential f...
esum 16021 Value of Euler's constant ...
eff 16022 Domain and codomain of the...
efcl 16023 Closure law for the expone...
efval2 16024 Value of the exponential f...
efcvg 16025 The series that defines th...
efcvgfsum 16026 Exponential function conve...
reefcl 16027 The exponential function i...
reefcld 16028 The exponential function i...
ere 16029 Euler's constant ` _e ` = ...
ege2le3 16030 Lemma for ~ egt2lt3 . (Co...
ef0 16031 Value of the exponential f...
efcj 16032 The exponential of a compl...
efaddlem 16033 Lemma for ~ efadd (exponen...
efadd 16034 Sum of exponents law for e...
fprodefsum 16035 Move the exponential funct...
efcan 16036 Cancellation law for expon...
efne0 16037 The exponential of a compl...
efneg 16038 The exponential of the opp...
eff2 16039 The exponential function m...
efsub 16040 Difference of exponents la...
efexp 16041 The exponential of an inte...
efzval 16042 Value of the exponential f...
efgt0 16043 The exponential of a real ...
rpefcl 16044 The exponential of a real ...
rpefcld 16045 The exponential of a real ...
eftlcvg 16046 The tail series of the exp...
eftlcl 16047 Closure of the sum of an i...
reeftlcl 16048 Closure of the sum of an i...
eftlub 16049 An upper bound on the abso...
efsep 16050 Separate out the next term...
effsumlt 16051 The partial sums of the se...
eft0val 16052 The value of the first ter...
ef4p 16053 Separate out the first fou...
efgt1p2 16054 The exponential of a posit...
efgt1p 16055 The exponential of a posit...
efgt1 16056 The exponential of a posit...
eflt 16057 The exponential function o...
efle 16058 The exponential function o...
reef11 16059 The exponential function o...
reeff1 16060 The exponential function m...
eflegeo 16061 The exponential function o...
sinval 16062 Value of the sine function...
cosval 16063 Value of the cosine functi...
sinf 16064 Domain and codomain of the...
cosf 16065 Domain and codomain of the...
sincl 16066 Closure of the sine functi...
coscl 16067 Closure of the cosine func...
tanval 16068 Value of the tangent funct...
tancl 16069 The closure of the tangent...
sincld 16070 Closure of the sine functi...
coscld 16071 Closure of the cosine func...
tancld 16072 Closure of the tangent fun...
tanval2 16073 Express the tangent functi...
tanval3 16074 Express the tangent functi...
resinval 16075 The sine of a real number ...
recosval 16076 The cosine of a real numbe...
efi4p 16077 Separate out the first fou...
resin4p 16078 Separate out the first fou...
recos4p 16079 Separate out the first fou...
resincl 16080 The sine of a real number ...
recoscl 16081 The cosine of a real numbe...
retancl 16082 The closure of the tangent...
resincld 16083 Closure of the sine functi...
recoscld 16084 Closure of the cosine func...
retancld 16085 Closure of the tangent fun...
sinneg 16086 The sine of a negative is ...
cosneg 16087 The cosines of a number an...
tanneg 16088 The tangent of a negative ...
sin0 16089 Value of the sine function...
cos0 16090 Value of the cosine functi...
tan0 16091 The value of the tangent f...
efival 16092 The exponential function i...
efmival 16093 The exponential function i...
sinhval 16094 Value of the hyperbolic si...
coshval 16095 Value of the hyperbolic co...
resinhcl 16096 The hyperbolic sine of a r...
rpcoshcl 16097 The hyperbolic cosine of a...
recoshcl 16098 The hyperbolic cosine of a...
retanhcl 16099 The hyperbolic tangent of ...
tanhlt1 16100 The hyperbolic tangent of ...
tanhbnd 16101 The hyperbolic tangent of ...
efeul 16102 Eulerian representation of...
efieq 16103 The exponentials of two im...
sinadd 16104 Addition formula for sine....
cosadd 16105 Addition formula for cosin...
tanaddlem 16106 A useful intermediate step...
tanadd 16107 Addition formula for tange...
sinsub 16108 Sine of difference. (Cont...
cossub 16109 Cosine of difference. (Co...
addsin 16110 Sum of sines. (Contribute...
subsin 16111 Difference of sines. (Con...
sinmul 16112 Product of sines can be re...
cosmul 16113 Product of cosines can be ...
addcos 16114 Sum of cosines. (Contribu...
subcos 16115 Difference of cosines. (C...
sincossq 16116 Sine squared plus cosine s...
sin2t 16117 Double-angle formula for s...
cos2t 16118 Double-angle formula for c...
cos2tsin 16119 Double-angle formula for c...
sinbnd 16120 The sine of a real number ...
cosbnd 16121 The cosine of a real numbe...
sinbnd2 16122 The sine of a real number ...
cosbnd2 16123 The cosine of a real numbe...
ef01bndlem 16124 Lemma for ~ sin01bnd and ~...
sin01bnd 16125 Bounds on the sine of a po...
cos01bnd 16126 Bounds on the cosine of a ...
cos1bnd 16127 Bounds on the cosine of 1....
cos2bnd 16128 Bounds on the cosine of 2....
sinltx 16129 The sine of a positive rea...
sin01gt0 16130 The sine of a positive rea...
cos01gt0 16131 The cosine of a positive r...
sin02gt0 16132 The sine of a positive rea...
sincos1sgn 16133 The signs of the sine and ...
sincos2sgn 16134 The signs of the sine and ...
sin4lt0 16135 The sine of 4 is negative....
absefi 16136 The absolute value of the ...
absef 16137 The absolute value of the ...
absefib 16138 A complex number is real i...
efieq1re 16139 A number whose imaginary e...
demoivre 16140 De Moivre's Formula. Proo...
demoivreALT 16141 Alternate proof of ~ demoi...
eirrlem 16144 Lemma for ~ eirr . (Contr...
eirr 16145 ` _e ` is irrational. (Co...
egt2lt3 16146 Euler's constant ` _e ` = ...
epos 16147 Euler's constant ` _e ` is...
epr 16148 Euler's constant ` _e ` is...
ene0 16149 ` _e ` is not 0. (Contrib...
ene1 16150 ` _e ` is not 1. (Contrib...
xpnnen 16151 The Cartesian product of t...
znnen 16152 The set of integers and th...
qnnen 16153 The rational numbers are c...
rpnnen2lem1 16154 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem2 16155 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem3 16156 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem4 16157 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem5 16158 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem6 16159 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem7 16160 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem8 16161 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem9 16162 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem10 16163 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem11 16164 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem12 16165 Lemma for ~ rpnnen2 . (Co...
rpnnen2 16166 The other half of ~ rpnnen...
rpnnen 16167 The cardinality of the con...
rexpen 16168 The real numbers are equin...
cpnnen 16169 The complex numbers are eq...
rucALT 16170 Alternate proof of ~ ruc ....
ruclem1 16171 Lemma for ~ ruc (the reals...
ruclem2 16172 Lemma for ~ ruc . Orderin...
ruclem3 16173 Lemma for ~ ruc . The con...
ruclem4 16174 Lemma for ~ ruc . Initial...
ruclem6 16175 Lemma for ~ ruc . Domain ...
ruclem7 16176 Lemma for ~ ruc . Success...
ruclem8 16177 Lemma for ~ ruc . The int...
ruclem9 16178 Lemma for ~ ruc . The fir...
ruclem10 16179 Lemma for ~ ruc . Every f...
ruclem11 16180 Lemma for ~ ruc . Closure...
ruclem12 16181 Lemma for ~ ruc . The sup...
ruclem13 16182 Lemma for ~ ruc . There i...
ruc 16183 The set of positive intege...
resdomq 16184 The set of rationals is st...
aleph1re 16185 There are at least aleph-o...
aleph1irr 16186 There are at least aleph-o...
cnso 16187 The complex numbers can be...
sqrt2irrlem 16188 Lemma for ~ sqrt2irr . Th...
sqrt2irr 16189 The square root of 2 is ir...
sqrt2re 16190 The square root of 2 exist...
sqrt2irr0 16191 The square root of 2 is an...
nthruc 16192 The sequence ` NN ` , ` ZZ...
nthruz 16193 The sequence ` NN ` , ` NN...
divides 16196 Define the divides relatio...
dvdsval2 16197 One nonzero integer divide...
dvdsval3 16198 One nonzero integer divide...
dvdszrcl 16199 Reverse closure for the di...
dvdsmod0 16200 If a positive integer divi...
p1modz1 16201 If a number greater than 1...
dvdsmodexp 16202 If a positive integer divi...
nndivdvds 16203 Strong form of ~ dvdsval2 ...
nndivides 16204 Definition of the divides ...
moddvds 16205 Two ways to say ` A == B `...
modm1div 16206 An integer greater than on...
dvds0lem 16207 A lemma to assist theorems...
dvds1lem 16208 A lemma to assist theorems...
dvds2lem 16209 A lemma to assist theorems...
iddvds 16210 An integer divides itself....
1dvds 16211 1 divides any integer. Th...
dvds0 16212 Any integer divides 0. Th...
negdvdsb 16213 An integer divides another...
dvdsnegb 16214 An integer divides another...
absdvdsb 16215 An integer divides another...
dvdsabsb 16216 An integer divides another...
0dvds 16217 Only 0 is divisible by 0. ...
dvdsmul1 16218 An integer divides a multi...
dvdsmul2 16219 An integer divides a multi...
iddvdsexp 16220 An integer divides a posit...
muldvds1 16221 If a product divides an in...
muldvds2 16222 If a product divides an in...
dvdscmul 16223 Multiplication by a consta...
dvdsmulc 16224 Multiplication by a consta...
dvdscmulr 16225 Cancellation law for the d...
dvdsmulcr 16226 Cancellation law for the d...
summodnegmod 16227 The sum of two integers mo...
modmulconst 16228 Constant multiplication in...
dvds2ln 16229 If an integer divides each...
dvds2add 16230 If an integer divides each...
dvds2sub 16231 If an integer divides each...
dvds2addd 16232 Deduction form of ~ dvds2a...
dvds2subd 16233 Deduction form of ~ dvds2s...
dvdstr 16234 The divides relation is tr...
dvdstrd 16235 The divides relation is tr...
dvdsmultr1 16236 If an integer divides anot...
dvdsmultr1d 16237 Deduction form of ~ dvdsmu...
dvdsmultr2 16238 If an integer divides anot...
dvdsmultr2d 16239 Deduction form of ~ dvdsmu...
ordvdsmul 16240 If an integer divides eith...
dvdssub2 16241 If an integer divides a di...
dvdsadd 16242 An integer divides another...
dvdsaddr 16243 An integer divides another...
dvdssub 16244 An integer divides another...
dvdssubr 16245 An integer divides another...
dvdsadd2b 16246 Adding a multiple of the b...
dvdsaddre2b 16247 Adding a multiple of the b...
fsumdvds 16248 If every term in a sum is ...
dvdslelem 16249 Lemma for ~ dvdsle . (Con...
dvdsle 16250 The divisors of a positive...
dvdsleabs 16251 The divisors of a nonzero ...
dvdsleabs2 16252 Transfer divisibility to a...
dvdsabseq 16253 If two integers divide eac...
dvdseq 16254 If two nonnegative integer...
divconjdvds 16255 If a nonzero integer ` M `...
dvdsdivcl 16256 The complement of a diviso...
dvdsflip 16257 An involution of the divis...
dvdsssfz1 16258 The set of divisors of a n...
dvds1 16259 The only nonnegative integ...
alzdvds 16260 Only 0 is divisible by all...
dvdsext 16261 Poset extensionality for d...
fzm1ndvds 16262 No number between ` 1 ` an...
fzo0dvdseq 16263 Zero is the only one of th...
fzocongeq 16264 Two different elements of ...
addmodlteqALT 16265 Two nonnegative integers l...
dvdsfac 16266 A positive integer divides...
dvdsexp2im 16267 If an integer divides anot...
dvdsexp 16268 A power divides a power wi...
dvdsmod 16269 Any number ` K ` whose mod...
mulmoddvds 16270 If an integer is divisible...
3dvds 16271 A rule for divisibility by...
3dvdsdec 16272 A decimal number is divisi...
3dvds2dec 16273 A decimal number is divisi...
fprodfvdvdsd 16274 A finite product of intege...
fproddvdsd 16275 A finite product of intege...
evenelz 16276 An even number is an integ...
zeo3 16277 An integer is even or odd....
zeo4 16278 An integer is even or odd ...
zeneo 16279 No even integer equals an ...
odd2np1lem 16280 Lemma for ~ odd2np1 . (Co...
odd2np1 16281 An integer is odd iff it i...
even2n 16282 An integer is even iff it ...
oddm1even 16283 An integer is odd iff its ...
oddp1even 16284 An integer is odd iff its ...
oexpneg 16285 The exponential of the neg...
mod2eq0even 16286 An integer is 0 modulo 2 i...
mod2eq1n2dvds 16287 An integer is 1 modulo 2 i...
oddnn02np1 16288 A nonnegative integer is o...
oddge22np1 16289 An integer greater than on...
evennn02n 16290 A nonnegative integer is e...
evennn2n 16291 A positive integer is even...
2tp1odd 16292 A number which is twice an...
mulsucdiv2z 16293 An integer multiplied with...
sqoddm1div8z 16294 A squared odd number minus...
2teven 16295 A number which is twice an...
zeo5 16296 An integer is either even ...
evend2 16297 An integer is even iff its...
oddp1d2 16298 An integer is odd iff its ...
zob 16299 Alternate characterization...
oddm1d2 16300 An integer is odd iff its ...
ltoddhalfle 16301 An integer is less than ha...
halfleoddlt 16302 An integer is greater than...
opoe 16303 The sum of two odds is eve...
omoe 16304 The difference of two odds...
opeo 16305 The sum of an odd and an e...
omeo 16306 The difference of an odd a...
z0even 16307 2 divides 0. That means 0...
n2dvds1 16308 2 does not divide 1. That...
n2dvdsm1 16309 2 does not divide -1. Tha...
z2even 16310 2 divides 2. That means 2...
n2dvds3 16311 2 does not divide 3. That...
z4even 16312 2 divides 4. That means 4...
4dvdseven 16313 An integer which is divisi...
m1expe 16314 Exponentiation of -1 by an...
m1expo 16315 Exponentiation of -1 by an...
m1exp1 16316 Exponentiation of negative...
nn0enne 16317 A positive integer is an e...
nn0ehalf 16318 The half of an even nonneg...
nnehalf 16319 The half of an even positi...
nn0onn 16320 An odd nonnegative integer...
nn0o1gt2 16321 An odd nonnegative integer...
nno 16322 An alternate characterizat...
nn0o 16323 An alternate characterizat...
nn0ob 16324 Alternate characterization...
nn0oddm1d2 16325 A positive integer is odd ...
nnoddm1d2 16326 A positive integer is odd ...
sumeven 16327 If every term in a sum is ...
sumodd 16328 If every term in a sum is ...
evensumodd 16329 If every term in a sum wit...
oddsumodd 16330 If every term in a sum wit...
pwp1fsum 16331 The n-th power of a number...
oddpwp1fsum 16332 An odd power of a number i...
divalglem0 16333 Lemma for ~ divalg . (Con...
divalglem1 16334 Lemma for ~ divalg . (Con...
divalglem2 16335 Lemma for ~ divalg . (Con...
divalglem4 16336 Lemma for ~ divalg . (Con...
divalglem5 16337 Lemma for ~ divalg . (Con...
divalglem6 16338 Lemma for ~ divalg . (Con...
divalglem7 16339 Lemma for ~ divalg . (Con...
divalglem8 16340 Lemma for ~ divalg . (Con...
divalglem9 16341 Lemma for ~ divalg . (Con...
divalglem10 16342 Lemma for ~ divalg . (Con...
divalg 16343 The division algorithm (th...
divalgb 16344 Express the division algor...
divalg2 16345 The division algorithm (th...
divalgmod 16346 The result of the ` mod ` ...
divalgmodcl 16347 The result of the ` mod ` ...
modremain 16348 The result of the modulo o...
ndvdssub 16349 Corollary of the division ...
ndvdsadd 16350 Corollary of the division ...
ndvdsp1 16351 Special case of ~ ndvdsadd...
ndvdsi 16352 A quick test for non-divis...
flodddiv4 16353 The floor of an odd intege...
fldivndvdslt 16354 The floor of an integer di...
flodddiv4lt 16355 The floor of an odd number...
flodddiv4t2lthalf 16356 The floor of an odd number...
bitsfval 16361 Expand the definition of t...
bitsval 16362 Expand the definition of t...
bitsval2 16363 Expand the definition of t...
bitsss 16364 The set of bits of an inte...
bitsf 16365 The ` bits ` function is a...
bits0 16366 Value of the zeroth bit. ...
bits0e 16367 The zeroth bit of an even ...
bits0o 16368 The zeroth bit of an odd n...
bitsp1 16369 The ` M + 1 ` -th bit of `...
bitsp1e 16370 The ` M + 1 ` -th bit of `...
bitsp1o 16371 The ` M + 1 ` -th bit of `...
bitsfzolem 16372 Lemma for ~ bitsfzo . (Co...
bitsfzo 16373 The bits of a number are a...
bitsmod 16374 Truncating the bit sequenc...
bitsfi 16375 Every number is associated...
bitscmp 16376 The bit complement of ` N ...
0bits 16377 The bits of zero. (Contri...
m1bits 16378 The bits of negative one. ...
bitsinv1lem 16379 Lemma for ~ bitsinv1 . (C...
bitsinv1 16380 There is an explicit inver...
bitsinv2 16381 There is an explicit inver...
bitsf1ocnv 16382 The ` bits ` function rest...
bitsf1o 16383 The ` bits ` function rest...
bitsf1 16384 The ` bits ` function is a...
2ebits 16385 The bits of a power of two...
bitsinv 16386 The inverse of the ` bits ...
bitsinvp1 16387 Recursive definition of th...
sadadd2lem2 16388 The core of the proof of ~...
sadfval 16390 Define the addition of two...
sadcf 16391 The carry sequence is a se...
sadc0 16392 The initial element of the...
sadcp1 16393 The carry sequence (which ...
sadval 16394 The full adder sequence is...
sadcaddlem 16395 Lemma for ~ sadcadd . (Co...
sadcadd 16396 Non-recursive definition o...
sadadd2lem 16397 Lemma for ~ sadadd2 . (Co...
sadadd2 16398 Sum of initial segments of...
sadadd3 16399 Sum of initial segments of...
sadcl 16400 The sum of two sequences i...
sadcom 16401 The adder sequence functio...
saddisjlem 16402 Lemma for ~ sadadd . (Con...
saddisj 16403 The sum of disjoint sequen...
sadaddlem 16404 Lemma for ~ sadadd . (Con...
sadadd 16405 For sequences that corresp...
sadid1 16406 The adder sequence functio...
sadid2 16407 The adder sequence functio...
sadasslem 16408 Lemma for ~ sadass . (Con...
sadass 16409 Sequence addition is assoc...
sadeq 16410 Any element of a sequence ...
bitsres 16411 Restrict the bits of a num...
bitsuz 16412 The bits of a number are a...
bitsshft 16413 Shifting a bit sequence to...
smufval 16415 The multiplication of two ...
smupf 16416 The sequence of partial su...
smup0 16417 The initial element of the...
smupp1 16418 The initial element of the...
smuval 16419 Define the addition of two...
smuval2 16420 The partial sum sequence s...
smupvallem 16421 If ` A ` only has elements...
smucl 16422 The product of two sequenc...
smu01lem 16423 Lemma for ~ smu01 and ~ sm...
smu01 16424 Multiplication of a sequen...
smu02 16425 Multiplication of a sequen...
smupval 16426 Rewrite the elements of th...
smup1 16427 Rewrite ~ smupp1 using onl...
smueqlem 16428 Any element of a sequence ...
smueq 16429 Any element of a sequence ...
smumullem 16430 Lemma for ~ smumul . (Con...
smumul 16431 For sequences that corresp...
gcdval 16434 The value of the ` gcd ` o...
gcd0val 16435 The value, by convention, ...
gcdn0val 16436 The value of the ` gcd ` o...
gcdcllem1 16437 Lemma for ~ gcdn0cl , ~ gc...
gcdcllem2 16438 Lemma for ~ gcdn0cl , ~ gc...
gcdcllem3 16439 Lemma for ~ gcdn0cl , ~ gc...
gcdn0cl 16440 Closure of the ` gcd ` ope...
gcddvds 16441 The gcd of two integers di...
dvdslegcd 16442 An integer which divides b...
nndvdslegcd 16443 A positive integer which d...
gcdcl 16444 Closure of the ` gcd ` ope...
gcdnncl 16445 Closure of the ` gcd ` ope...
gcdcld 16446 Closure of the ` gcd ` ope...
gcd2n0cl 16447 Closure of the ` gcd ` ope...
zeqzmulgcd 16448 An integer is the product ...
divgcdz 16449 An integer divided by the ...
gcdf 16450 Domain and codomain of the...
gcdcom 16451 The ` gcd ` operator is co...
gcdcomd 16452 The ` gcd ` operator is co...
divgcdnn 16453 A positive integer divided...
divgcdnnr 16454 A positive integer divided...
gcdeq0 16455 The gcd of two integers is...
gcdn0gt0 16456 The gcd of two integers is...
gcd0id 16457 The gcd of 0 and an intege...
gcdid0 16458 The gcd of an integer and ...
nn0gcdid0 16459 The gcd of a nonnegative i...
gcdneg 16460 Negating one operand of th...
neggcd 16461 Negating one operand of th...
gcdaddmlem 16462 Lemma for ~ gcdaddm . (Co...
gcdaddm 16463 Adding a multiple of one o...
gcdadd 16464 The GCD of two numbers is ...
gcdid 16465 The gcd of a number and it...
gcd1 16466 The gcd of a number with 1...
gcdabs1 16467 ` gcd ` of the absolute va...
gcdabs2 16468 ` gcd ` of the absolute va...
gcdabs 16469 The gcd of two integers is...
gcdabsOLD 16470 Obsolete version of ~ gcda...
modgcd 16471 The gcd remains unchanged ...
1gcd 16472 The GCD of one and an inte...
gcdmultipled 16473 The greatest common diviso...
gcdmultiplez 16474 The GCD of a multiple of a...
gcdmultiple 16475 The GCD of a multiple of a...
dvdsgcdidd 16476 The greatest common diviso...
6gcd4e2 16477 The greatest common diviso...
bezoutlem1 16478 Lemma for ~ bezout . (Con...
bezoutlem2 16479 Lemma for ~ bezout . (Con...
bezoutlem3 16480 Lemma for ~ bezout . (Con...
bezoutlem4 16481 Lemma for ~ bezout . (Con...
bezout 16482 Bézout's identity: ...
dvdsgcd 16483 An integer which divides e...
dvdsgcdb 16484 Biconditional form of ~ dv...
dfgcd2 16485 Alternate definition of th...
gcdass 16486 Associative law for ` gcd ...
mulgcd 16487 Distribute multiplication ...
absmulgcd 16488 Distribute absolute value ...
mulgcdr 16489 Reverse distribution law f...
gcddiv 16490 Division law for GCD. (Con...
gcdzeq 16491 A positive integer ` A ` i...
gcdeq 16492 ` A ` is equal to its gcd ...
dvdssqim 16493 Unidirectional form of ~ d...
dvdsmulgcd 16494 A divisibility equivalent ...
rpmulgcd 16495 If ` K ` and ` M ` are rel...
rplpwr 16496 If ` A ` and ` B ` are rel...
rprpwr 16497 If ` A ` and ` B ` are rel...
rppwr 16498 If ` A ` and ` B ` are rel...
sqgcd 16499 Square distributes over gc...
dvdssqlem 16500 Lemma for ~ dvdssq . (Con...
dvdssq 16501 Two numbers are divisible ...
bezoutr 16502 Partial converse to ~ bezo...
bezoutr1 16503 Converse of ~ bezout for w...
nn0seqcvgd 16504 A strictly-decreasing nonn...
seq1st 16505 A sequence whose iteration...
algr0 16506 The value of the algorithm...
algrf 16507 An algorithm is a step fun...
algrp1 16508 The value of the algorithm...
alginv 16509 If ` I ` is an invariant o...
algcvg 16510 One way to prove that an a...
algcvgblem 16511 Lemma for ~ algcvgb . (Co...
algcvgb 16512 Two ways of expressing tha...
algcvga 16513 The countdown function ` C...
algfx 16514 If ` F ` reaches a fixed p...
eucalgval2 16515 The value of the step func...
eucalgval 16516 Euclid's Algorithm ~ eucal...
eucalgf 16517 Domain and codomain of the...
eucalginv 16518 The invariant of the step ...
eucalglt 16519 The second member of the s...
eucalgcvga 16520 Once Euclid's Algorithm ha...
eucalg 16521 Euclid's Algorithm compute...
lcmval 16526 Value of the ` lcm ` opera...
lcmcom 16527 The ` lcm ` operator is co...
lcm0val 16528 The value, by convention, ...
lcmn0val 16529 The value of the ` lcm ` o...
lcmcllem 16530 Lemma for ~ lcmn0cl and ~ ...
lcmn0cl 16531 Closure of the ` lcm ` ope...
dvdslcm 16532 The lcm of two integers is...
lcmledvds 16533 A positive integer which b...
lcmeq0 16534 The lcm of two integers is...
lcmcl 16535 Closure of the ` lcm ` ope...
gcddvdslcm 16536 The greatest common diviso...
lcmneg 16537 Negating one operand of th...
neglcm 16538 Negating one operand of th...
lcmabs 16539 The lcm of two integers is...
lcmgcdlem 16540 Lemma for ~ lcmgcd and ~ l...
lcmgcd 16541 The product of two numbers...
lcmdvds 16542 The lcm of two integers di...
lcmid 16543 The lcm of an integer and ...
lcm1 16544 The lcm of an integer and ...
lcmgcdnn 16545 The product of two positiv...
lcmgcdeq 16546 Two integers' absolute val...
lcmdvdsb 16547 Biconditional form of ~ lc...
lcmass 16548 Associative law for ` lcm ...
3lcm2e6woprm 16549 The least common multiple ...
6lcm4e12 16550 The least common multiple ...
absproddvds 16551 The absolute value of the ...
absprodnn 16552 The absolute value of the ...
fissn0dvds 16553 For each finite subset of ...
fissn0dvdsn0 16554 For each finite subset of ...
lcmfval 16555 Value of the ` _lcm ` func...
lcmf0val 16556 The value, by convention, ...
lcmfn0val 16557 The value of the ` _lcm ` ...
lcmfnnval 16558 The value of the ` _lcm ` ...
lcmfcllem 16559 Lemma for ~ lcmfn0cl and ~...
lcmfn0cl 16560 Closure of the ` _lcm ` fu...
lcmfpr 16561 The value of the ` _lcm ` ...
lcmfcl 16562 Closure of the ` _lcm ` fu...
lcmfnncl 16563 Closure of the ` _lcm ` fu...
lcmfeq0b 16564 The least common multiple ...
dvdslcmf 16565 The least common multiple ...
lcmfledvds 16566 A positive integer which i...
lcmf 16567 Characterization of the le...
lcmf0 16568 The least common multiple ...
lcmfsn 16569 The least common multiple ...
lcmftp 16570 The least common multiple ...
lcmfunsnlem1 16571 Lemma for ~ lcmfdvds and ~...
lcmfunsnlem2lem1 16572 Lemma 1 for ~ lcmfunsnlem2...
lcmfunsnlem2lem2 16573 Lemma 2 for ~ lcmfunsnlem2...
lcmfunsnlem2 16574 Lemma for ~ lcmfunsn and ~...
lcmfunsnlem 16575 Lemma for ~ lcmfdvds and ~...
lcmfdvds 16576 The least common multiple ...
lcmfdvdsb 16577 Biconditional form of ~ lc...
lcmfunsn 16578 The ` _lcm ` function for ...
lcmfun 16579 The ` _lcm ` function for ...
lcmfass 16580 Associative law for the ` ...
lcmf2a3a4e12 16581 The least common multiple ...
lcmflefac 16582 The least common multiple ...
coprmgcdb 16583 Two positive integers are ...
ncoprmgcdne1b 16584 Two positive integers are ...
ncoprmgcdgt1b 16585 Two positive integers are ...
coprmdvds1 16586 If two positive integers a...
coprmdvds 16587 Euclid's Lemma (see ProofW...
coprmdvds2 16588 If an integer is divisible...
mulgcddvds 16589 One half of ~ rpmulgcd2 , ...
rpmulgcd2 16590 If ` M ` is relatively pri...
qredeq 16591 Two equal reduced fraction...
qredeu 16592 Every rational number has ...
rpmul 16593 If ` K ` is relatively pri...
rpdvds 16594 If ` K ` is relatively pri...
coprmprod 16595 The product of the element...
coprmproddvdslem 16596 Lemma for ~ coprmproddvds ...
coprmproddvds 16597 If a positive integer is d...
congr 16598 Definition of congruence b...
divgcdcoprm0 16599 Integers divided by gcd ar...
divgcdcoprmex 16600 Integers divided by gcd ar...
cncongr1 16601 One direction of the bicon...
cncongr2 16602 The other direction of the...
cncongr 16603 Cancellability of Congruen...
cncongrcoprm 16604 Corollary 1 of Cancellabil...
isprm 16607 The predicate "is a prime ...
prmnn 16608 A prime number is a positi...
prmz 16609 A prime number is an integ...
prmssnn 16610 The prime numbers are a su...
prmex 16611 The set of prime numbers e...
0nprm 16612 0 is not a prime number. ...
1nprm 16613 1 is not a prime number. ...
1idssfct 16614 The positive divisors of a...
isprm2lem 16615 Lemma for ~ isprm2 . (Con...
isprm2 16616 The predicate "is a prime ...
isprm3 16617 The predicate "is a prime ...
isprm4 16618 The predicate "is a prime ...
prmind2 16619 A variation on ~ prmind as...
prmind 16620 Perform induction over the...
dvdsprime 16621 If ` M ` divides a prime, ...
nprm 16622 A product of two integers ...
nprmi 16623 An inference for composite...
dvdsnprmd 16624 If a number is divisible b...
prm2orodd 16625 A prime number is either 2...
2prm 16626 2 is a prime number. (Con...
2mulprm 16627 A multiple of two is prime...
3prm 16628 3 is a prime number. (Con...
4nprm 16629 4 is not a prime number. ...
prmuz2 16630 A prime number is an integ...
prmgt1 16631 A prime number is an integ...
prmm2nn0 16632 Subtracting 2 from a prime...
oddprmgt2 16633 An odd prime is greater th...
oddprmge3 16634 An odd prime is greater th...
ge2nprmge4 16635 A composite integer greate...
sqnprm 16636 A square is never prime. ...
dvdsprm 16637 An integer greater than or...
exprmfct 16638 Every integer greater than...
prmdvdsfz 16639 Each integer greater than ...
nprmdvds1 16640 No prime number divides 1....
isprm5 16641 One need only check prime ...
isprm7 16642 One need only check prime ...
maxprmfct 16643 The set of prime factors o...
divgcdodd 16644 Either ` A / ( A gcd B ) `...
coprm 16645 A prime number either divi...
prmrp 16646 Unequal prime numbers are ...
euclemma 16647 Euclid's lemma. A prime n...
isprm6 16648 A number is prime iff it s...
prmdvdsexp 16649 A prime divides a positive...
prmdvdsexpb 16650 A prime divides a positive...
prmdvdsexpr 16651 If a prime divides a nonne...
prmdvdssq 16652 Condition for a prime divi...
prmdvdssqOLD 16653 Obsolete version of ~ prmd...
prmexpb 16654 Two positive prime powers ...
prmfac1 16655 The factorial of a number ...
dvdszzq 16656 Divisibility for an intege...
rpexp 16657 If two numbers ` A ` and `...
rpexp1i 16658 Relative primality passes ...
rpexp12i 16659 Relative primality passes ...
prmndvdsfaclt 16660 A prime number does not di...
prmdvdsbc 16661 Condition for a prime numb...
prmdvdsncoprmbd 16662 Two positive integers are ...
ncoprmlnprm 16663 If two positive integers a...
cncongrprm 16664 Corollary 2 of Cancellabil...
isevengcd2 16665 The predicate "is an even ...
isoddgcd1 16666 The predicate "is an odd n...
3lcm2e6 16667 The least common multiple ...
qnumval 16672 Value of the canonical num...
qdenval 16673 Value of the canonical den...
qnumdencl 16674 Lemma for ~ qnumcl and ~ q...
qnumcl 16675 The canonical numerator of...
qdencl 16676 The canonical denominator ...
fnum 16677 Canonical numerator define...
fden 16678 Canonical denominator defi...
qnumdenbi 16679 Two numbers are the canoni...
qnumdencoprm 16680 The canonical representati...
qeqnumdivden 16681 Recover a rational number ...
qmuldeneqnum 16682 Multiplying a rational by ...
divnumden 16683 Calculate the reduced form...
divdenle 16684 Reducing a quotient never ...
qnumgt0 16685 A rational is positive iff...
qgt0numnn 16686 A rational is positive iff...
nn0gcdsq 16687 Squaring commutes with GCD...
zgcdsq 16688 ~ nn0gcdsq extended to int...
numdensq 16689 Squaring a rational square...
numsq 16690 Square commutes with canon...
densq 16691 Square commutes with canon...
qden1elz 16692 A rational is an integer i...
zsqrtelqelz 16693 If an integer has a ration...
nonsq 16694 Any integer strictly betwe...
phival 16699 Value of the Euler ` phi `...
phicl2 16700 Bounds and closure for the...
phicl 16701 Closure for the value of t...
phibndlem 16702 Lemma for ~ phibnd . (Con...
phibnd 16703 A slightly tighter bound o...
phicld 16704 Closure for the value of t...
phi1 16705 Value of the Euler ` phi `...
dfphi2 16706 Alternate definition of th...
hashdvds 16707 The number of numbers in a...
phiprmpw 16708 Value of the Euler ` phi `...
phiprm 16709 Value of the Euler ` phi `...
crth 16710 The Chinese Remainder Theo...
phimullem 16711 Lemma for ~ phimul . (Con...
phimul 16712 The Euler ` phi ` function...
eulerthlem1 16713 Lemma for ~ eulerth . (Co...
eulerthlem2 16714 Lemma for ~ eulerth . (Co...
eulerth 16715 Euler's theorem, a general...
fermltl 16716 Fermat's little theorem. ...
prmdiv 16717 Show an explicit expressio...
prmdiveq 16718 The modular inverse of ` A...
prmdivdiv 16719 The (modular) inverse of t...
hashgcdlem 16720 A correspondence between e...
hashgcdeq 16721 Number of initial positive...
phisum 16722 The divisor sum identity o...
odzval 16723 Value of the order functio...
odzcllem 16724 - Lemma for ~ odzcl , show...
odzcl 16725 The order of a group eleme...
odzid 16726 Any element raised to the ...
odzdvds 16727 The only powers of ` A ` t...
odzphi 16728 The order of any group ele...
modprm1div 16729 A prime number divides an ...
m1dvdsndvds 16730 If an integer minus 1 is d...
modprminv 16731 Show an explicit expressio...
modprminveq 16732 The modular inverse of ` A...
vfermltl 16733 Variant of Fermat's little...
vfermltlALT 16734 Alternate proof of ~ vferm...
powm2modprm 16735 If an integer minus 1 is d...
reumodprminv 16736 For any prime number and f...
modprm0 16737 For two positive integers ...
nnnn0modprm0 16738 For a positive integer and...
modprmn0modprm0 16739 For an integer not being 0...
coprimeprodsq 16740 If three numbers are copri...
coprimeprodsq2 16741 If three numbers are copri...
oddprm 16742 A prime not equal to ` 2 `...
nnoddn2prm 16743 A prime not equal to ` 2 `...
oddn2prm 16744 A prime not equal to ` 2 `...
nnoddn2prmb 16745 A number is a prime number...
prm23lt5 16746 A prime less than 5 is eit...
prm23ge5 16747 A prime is either 2 or 3 o...
pythagtriplem1 16748 Lemma for ~ pythagtrip . ...
pythagtriplem2 16749 Lemma for ~ pythagtrip . ...
pythagtriplem3 16750 Lemma for ~ pythagtrip . ...
pythagtriplem4 16751 Lemma for ~ pythagtrip . ...
pythagtriplem10 16752 Lemma for ~ pythagtrip . ...
pythagtriplem6 16753 Lemma for ~ pythagtrip . ...
pythagtriplem7 16754 Lemma for ~ pythagtrip . ...
pythagtriplem8 16755 Lemma for ~ pythagtrip . ...
pythagtriplem9 16756 Lemma for ~ pythagtrip . ...
pythagtriplem11 16757 Lemma for ~ pythagtrip . ...
pythagtriplem12 16758 Lemma for ~ pythagtrip . ...
pythagtriplem13 16759 Lemma for ~ pythagtrip . ...
pythagtriplem14 16760 Lemma for ~ pythagtrip . ...
pythagtriplem15 16761 Lemma for ~ pythagtrip . ...
pythagtriplem16 16762 Lemma for ~ pythagtrip . ...
pythagtriplem17 16763 Lemma for ~ pythagtrip . ...
pythagtriplem18 16764 Lemma for ~ pythagtrip . ...
pythagtriplem19 16765 Lemma for ~ pythagtrip . ...
pythagtrip 16766 Parameterize the Pythagore...
iserodd 16767 Collect the odd terms in a...
pclem 16770 - Lemma for the prime powe...
pcprecl 16771 Closure of the prime power...
pcprendvds 16772 Non-divisibility property ...
pcprendvds2 16773 Non-divisibility property ...
pcpre1 16774 Value of the prime power p...
pcpremul 16775 Multiplicative property of...
pcval 16776 The value of the prime pow...
pceulem 16777 Lemma for ~ pceu . (Contr...
pceu 16778 Uniqueness for the prime p...
pczpre 16779 Connect the prime count pr...
pczcl 16780 Closure of the prime power...
pccl 16781 Closure of the prime power...
pccld 16782 Closure of the prime power...
pcmul 16783 Multiplication property of...
pcdiv 16784 Division property of the p...
pcqmul 16785 Multiplication property of...
pc0 16786 The value of the prime pow...
pc1 16787 Value of the prime count f...
pcqcl 16788 Closure of the general pri...
pcqdiv 16789 Division property of the p...
pcrec 16790 Prime power of a reciproca...
pcexp 16791 Prime power of an exponent...
pcxnn0cl 16792 Extended nonnegative integ...
pcxcl 16793 Extended real closure of t...
pcge0 16794 The prime count of an inte...
pczdvds 16795 Defining property of the p...
pcdvds 16796 Defining property of the p...
pczndvds 16797 Defining property of the p...
pcndvds 16798 Defining property of the p...
pczndvds2 16799 The remainder after dividi...
pcndvds2 16800 The remainder after dividi...
pcdvdsb 16801 ` P ^ A ` divides ` N ` if...
pcelnn 16802 There are a positive numbe...
pceq0 16803 There are zero powers of a...
pcidlem 16804 The prime count of a prime...
pcid 16805 The prime count of a prime...
pcneg 16806 The prime count of a negat...
pcabs 16807 The prime count of an abso...
pcdvdstr 16808 The prime count increases ...
pcgcd1 16809 The prime count of a GCD i...
pcgcd 16810 The prime count of a GCD i...
pc2dvds 16811 A characterization of divi...
pc11 16812 The prime count function, ...
pcz 16813 The prime count function c...
pcprmpw2 16814 Self-referential expressio...
pcprmpw 16815 Self-referential expressio...
dvdsprmpweq 16816 If a positive integer divi...
dvdsprmpweqnn 16817 If an integer greater than...
dvdsprmpweqle 16818 If a positive integer divi...
difsqpwdvds 16819 If the difference of two s...
pcaddlem 16820 Lemma for ~ pcadd . The o...
pcadd 16821 An inequality for the prim...
pcadd2 16822 The inequality of ~ pcadd ...
pcmptcl 16823 Closure for the prime powe...
pcmpt 16824 Construct a function with ...
pcmpt2 16825 Dividing two prime count m...
pcmptdvds 16826 The partial products of th...
pcprod 16827 The product of the primes ...
sumhash 16828 The sum of 1 over a set is...
fldivp1 16829 The difference between the...
pcfaclem 16830 Lemma for ~ pcfac . (Cont...
pcfac 16831 Calculate the prime count ...
pcbc 16832 Calculate the prime count ...
qexpz 16833 If a power of a rational n...
expnprm 16834 A second or higher power o...
oddprmdvds 16835 Every positive integer whi...
prmpwdvds 16836 A relation involving divis...
pockthlem 16837 Lemma for ~ pockthg . (Co...
pockthg 16838 The generalized Pocklingto...
pockthi 16839 Pocklington's theorem, whi...
unbenlem 16840 Lemma for ~ unben . (Cont...
unben 16841 An unbounded set of positi...
infpnlem1 16842 Lemma for ~ infpn . The s...
infpnlem2 16843 Lemma for ~ infpn . For a...
infpn 16844 There exist infinitely man...
infpn2 16845 There exist infinitely man...
prmunb 16846 The primes are unbounded. ...
prminf 16847 There are an infinite numb...
prmreclem1 16848 Lemma for ~ prmrec . Prop...
prmreclem2 16849 Lemma for ~ prmrec . Ther...
prmreclem3 16850 Lemma for ~ prmrec . The ...
prmreclem4 16851 Lemma for ~ prmrec . Show...
prmreclem5 16852 Lemma for ~ prmrec . Here...
prmreclem6 16853 Lemma for ~ prmrec . If t...
prmrec 16854 The sum of the reciprocals...
1arithlem1 16855 Lemma for ~ 1arith . (Con...
1arithlem2 16856 Lemma for ~ 1arith . (Con...
1arithlem3 16857 Lemma for ~ 1arith . (Con...
1arithlem4 16858 Lemma for ~ 1arith . (Con...
1arith 16859 Fundamental theorem of ari...
1arith2 16860 Fundamental theorem of ari...
elgz 16863 Elementhood in the gaussia...
gzcn 16864 A gaussian integer is a co...
zgz 16865 An integer is a gaussian i...
igz 16866 ` _i ` is a gaussian integ...
gznegcl 16867 The gaussian integers are ...
gzcjcl 16868 The gaussian integers are ...
gzaddcl 16869 The gaussian integers are ...
gzmulcl 16870 The gaussian integers are ...
gzreim 16871 Construct a gaussian integ...
gzsubcl 16872 The gaussian integers are ...
gzabssqcl 16873 The squared norm of a gaus...
4sqlem5 16874 Lemma for ~ 4sq . (Contri...
4sqlem6 16875 Lemma for ~ 4sq . (Contri...
4sqlem7 16876 Lemma for ~ 4sq . (Contri...
4sqlem8 16877 Lemma for ~ 4sq . (Contri...
4sqlem9 16878 Lemma for ~ 4sq . (Contri...
4sqlem10 16879 Lemma for ~ 4sq . (Contri...
4sqlem1 16880 Lemma for ~ 4sq . The set...
4sqlem2 16881 Lemma for ~ 4sq . Change ...
4sqlem3 16882 Lemma for ~ 4sq . Suffici...
4sqlem4a 16883 Lemma for ~ 4sqlem4 . (Co...
4sqlem4 16884 Lemma for ~ 4sq . We can ...
mul4sqlem 16885 Lemma for ~ mul4sq : algeb...
mul4sq 16886 Euler's four-square identi...
4sqlem11 16887 Lemma for ~ 4sq . Use the...
4sqlem12 16888 Lemma for ~ 4sq . For any...
4sqlem13 16889 Lemma for ~ 4sq . (Contri...
4sqlem14 16890 Lemma for ~ 4sq . (Contri...
4sqlem15 16891 Lemma for ~ 4sq . (Contri...
4sqlem16 16892 Lemma for ~ 4sq . (Contri...
4sqlem17 16893 Lemma for ~ 4sq . (Contri...
4sqlem18 16894 Lemma for ~ 4sq . Inducti...
4sqlem19 16895 Lemma for ~ 4sq . The pro...
4sq 16896 Lagrange's four-square the...
vdwapfval 16903 Define the arithmetic prog...
vdwapf 16904 The arithmetic progression...
vdwapval 16905 Value of the arithmetic pr...
vdwapun 16906 Remove the first element o...
vdwapid1 16907 The first element of an ar...
vdwap0 16908 Value of a length-1 arithm...
vdwap1 16909 Value of a length-1 arithm...
vdwmc 16910 The predicate " The ` <. R...
vdwmc2 16911 Expand out the definition ...
vdwpc 16912 The predicate " The colori...
vdwlem1 16913 Lemma for ~ vdw . (Contri...
vdwlem2 16914 Lemma for ~ vdw . (Contri...
vdwlem3 16915 Lemma for ~ vdw . (Contri...
vdwlem4 16916 Lemma for ~ vdw . (Contri...
vdwlem5 16917 Lemma for ~ vdw . (Contri...
vdwlem6 16918 Lemma for ~ vdw . (Contri...
vdwlem7 16919 Lemma for ~ vdw . (Contri...
vdwlem8 16920 Lemma for ~ vdw . (Contri...
vdwlem9 16921 Lemma for ~ vdw . (Contri...
vdwlem10 16922 Lemma for ~ vdw . Set up ...
vdwlem11 16923 Lemma for ~ vdw . (Contri...
vdwlem12 16924 Lemma for ~ vdw . ` K = 2 ...
vdwlem13 16925 Lemma for ~ vdw . Main in...
vdw 16926 Van der Waerden's theorem....
vdwnnlem1 16927 Corollary of ~ vdw , and l...
vdwnnlem2 16928 Lemma for ~ vdwnn . The s...
vdwnnlem3 16929 Lemma for ~ vdwnn . (Cont...
vdwnn 16930 Van der Waerden's theorem,...
ramtlecl 16932 The set ` T ` of numbers w...
hashbcval 16934 Value of the "binomial set...
hashbccl 16935 The binomial set is a fini...
hashbcss 16936 Subset relation for the bi...
hashbc0 16937 The set of subsets of size...
hashbc2 16938 The size of the binomial s...
0hashbc 16939 There are no subsets of th...
ramval 16940 The value of the Ramsey nu...
ramcl2lem 16941 Lemma for extended real cl...
ramtcl 16942 The Ramsey number has the ...
ramtcl2 16943 The Ramsey number is an in...
ramtub 16944 The Ramsey number is a low...
ramub 16945 The Ramsey number is a low...
ramub2 16946 It is sufficient to check ...
rami 16947 The defining property of a...
ramcl2 16948 The Ramsey number is eithe...
ramxrcl 16949 The Ramsey number is an ex...
ramubcl 16950 If the Ramsey number is up...
ramlb 16951 Establish a lower bound on...
0ram 16952 The Ramsey number when ` M...
0ram2 16953 The Ramsey number when ` M...
ram0 16954 The Ramsey number when ` R...
0ramcl 16955 Lemma for ~ ramcl : Exist...
ramz2 16956 The Ramsey number when ` F...
ramz 16957 The Ramsey number when ` F...
ramub1lem1 16958 Lemma for ~ ramub1 . (Con...
ramub1lem2 16959 Lemma for ~ ramub1 . (Con...
ramub1 16960 Inductive step for Ramsey'...
ramcl 16961 Ramsey's theorem: the Rams...
ramsey 16962 Ramsey's theorem with the ...
prmoval 16965 Value of the primorial fun...
prmocl 16966 Closure of the primorial f...
prmone0 16967 The primorial function is ...
prmo0 16968 The primorial of 0. (Cont...
prmo1 16969 The primorial of 1. (Cont...
prmop1 16970 The primorial of a success...
prmonn2 16971 Value of the primorial fun...
prmo2 16972 The primorial of 2. (Cont...
prmo3 16973 The primorial of 3. (Cont...
prmdvdsprmo 16974 The primorial of a number ...
prmdvdsprmop 16975 The primorial of a number ...
fvprmselelfz 16976 The value of the prime sel...
fvprmselgcd1 16977 The greatest common diviso...
prmolefac 16978 The primorial of a positiv...
prmodvdslcmf 16979 The primorial of a nonnega...
prmolelcmf 16980 The primorial of a positiv...
prmgaplem1 16981 Lemma for ~ prmgap : The ...
prmgaplem2 16982 Lemma for ~ prmgap : The ...
prmgaplcmlem1 16983 Lemma for ~ prmgaplcm : T...
prmgaplcmlem2 16984 Lemma for ~ prmgaplcm : T...
prmgaplem3 16985 Lemma for ~ prmgap . (Con...
prmgaplem4 16986 Lemma for ~ prmgap . (Con...
prmgaplem5 16987 Lemma for ~ prmgap : for e...
prmgaplem6 16988 Lemma for ~ prmgap : for e...
prmgaplem7 16989 Lemma for ~ prmgap . (Con...
prmgaplem8 16990 Lemma for ~ prmgap . (Con...
prmgap 16991 The prime gap theorem: for...
prmgaplcm 16992 Alternate proof of ~ prmga...
prmgapprmolem 16993 Lemma for ~ prmgapprmo : ...
prmgapprmo 16994 Alternate proof of ~ prmga...
dec2dvds 16995 Divisibility by two is obv...
dec5dvds 16996 Divisibility by five is ob...
dec5dvds2 16997 Divisibility by five is ob...
dec5nprm 16998 Divisibility by five is ob...
dec2nprm 16999 Divisibility by two is obv...
modxai 17000 Add exponents in a power m...
mod2xi 17001 Double exponents in a powe...
modxp1i 17002 Add one to an exponent in ...
mod2xnegi 17003 Version of ~ mod2xi with a...
modsubi 17004 Subtract from within a mod...
gcdi 17005 Calculate a GCD via Euclid...
gcdmodi 17006 Calculate a GCD via Euclid...
decexp2 17007 Calculate a power of two. ...
numexp0 17008 Calculate an integer power...
numexp1 17009 Calculate an integer power...
numexpp1 17010 Calculate an integer power...
numexp2x 17011 Double an integer power. ...
decsplit0b 17012 Split a decimal number int...
decsplit0 17013 Split a decimal number int...
decsplit1 17014 Split a decimal number int...
decsplit 17015 Split a decimal number int...
karatsuba 17016 The Karatsuba multiplicati...
2exp4 17017 Two to the fourth power is...
2exp5 17018 Two to the fifth power is ...
2exp6 17019 Two to the sixth power is ...
2exp7 17020 Two to the seventh power i...
2exp8 17021 Two to the eighth power is...
2exp11 17022 Two to the eleventh power ...
2exp16 17023 Two to the sixteenth power...
3exp3 17024 Three to the third power i...
2expltfac 17025 The factorial grows faster...
cshwsidrepsw 17026 If cyclically shifting a w...
cshwsidrepswmod0 17027 If cyclically shifting a w...
cshwshashlem1 17028 If cyclically shifting a w...
cshwshashlem2 17029 If cyclically shifting a w...
cshwshashlem3 17030 If cyclically shifting a w...
cshwsdisj 17031 The singletons resulting b...
cshwsiun 17032 The set of (different!) wo...
cshwsex 17033 The class of (different!) ...
cshws0 17034 The size of the set of (di...
cshwrepswhash1 17035 The size of the set of (di...
cshwshashnsame 17036 If a word (not consisting ...
cshwshash 17037 If a word has a length bei...
prmlem0 17038 Lemma for ~ prmlem1 and ~ ...
prmlem1a 17039 A quick proof skeleton to ...
prmlem1 17040 A quick proof skeleton to ...
5prm 17041 5 is a prime number. (Con...
6nprm 17042 6 is not a prime number. ...
7prm 17043 7 is a prime number. (Con...
8nprm 17044 8 is not a prime number. ...
9nprm 17045 9 is not a prime number. ...
10nprm 17046 10 is not a prime number. ...
11prm 17047 11 is a prime number. (Co...
13prm 17048 13 is a prime number. (Co...
17prm 17049 17 is a prime number. (Co...
19prm 17050 19 is a prime number. (Co...
23prm 17051 23 is a prime number. (Co...
prmlem2 17052 Our last proving session g...
37prm 17053 37 is a prime number. (Co...
43prm 17054 43 is a prime number. (Co...
83prm 17055 83 is a prime number. (Co...
139prm 17056 139 is a prime number. (C...
163prm 17057 163 is a prime number. (C...
317prm 17058 317 is a prime number. (C...
631prm 17059 631 is a prime number. (C...
prmo4 17060 The primorial of 4. (Cont...
prmo5 17061 The primorial of 5. (Cont...
prmo6 17062 The primorial of 6. (Cont...
1259lem1 17063 Lemma for ~ 1259prm . Cal...
1259lem2 17064 Lemma for ~ 1259prm . Cal...
1259lem3 17065 Lemma for ~ 1259prm . Cal...
1259lem4 17066 Lemma for ~ 1259prm . Cal...
1259lem5 17067 Lemma for ~ 1259prm . Cal...
1259prm 17068 1259 is a prime number. (...
2503lem1 17069 Lemma for ~ 2503prm . Cal...
2503lem2 17070 Lemma for ~ 2503prm . Cal...
2503lem3 17071 Lemma for ~ 2503prm . Cal...
2503prm 17072 2503 is a prime number. (...
4001lem1 17073 Lemma for ~ 4001prm . Cal...
4001lem2 17074 Lemma for ~ 4001prm . Cal...
4001lem3 17075 Lemma for ~ 4001prm . Cal...
4001lem4 17076 Lemma for ~ 4001prm . Cal...
4001prm 17077 4001 is a prime number. (...
brstruct 17080 The structure relation is ...
isstruct2 17081 The property of being a st...
structex 17082 A structure is a set. (Co...
structn0fun 17083 A structure without the em...
isstruct 17084 The property of being a st...
structcnvcnv 17085 Two ways to express the re...
structfung 17086 The converse of the conver...
structfun 17087 Convert between two kinds ...
structfn 17088 Convert between two kinds ...
strleun 17089 Combine two structures int...
strle1 17090 Make a structure from a si...
strle2 17091 Make a structure from a pa...
strle3 17092 Make a structure from a tr...
sbcie2s 17093 A special version of class...
sbcie3s 17094 A special version of class...
reldmsets 17097 The structure override ope...
setsvalg 17098 Value of the structure rep...
setsval 17099 Value of the structure rep...
fvsetsid 17100 The value of the structure...
fsets 17101 The structure replacement ...
setsdm 17102 The domain of a structure ...
setsfun 17103 A structure with replaceme...
setsfun0 17104 A structure with replaceme...
setsn0fun 17105 The value of the structure...
setsstruct2 17106 An extensible structure wi...
setsexstruct2 17107 An extensible structure wi...
setsstruct 17108 An extensible structure wi...
wunsets 17109 Closure of structure repla...
setsres 17110 The structure replacement ...
setsabs 17111 Replacing the same compone...
setscom 17112 Different components can b...
sloteq 17115 Equality theorem for the `...
slotfn 17116 A slot is a function on se...
strfvnd 17117 Deduction version of ~ str...
strfvn 17118 Value of a structure compo...
strfvss 17119 A structure component extr...
wunstr 17120 Closure of a structure ind...
str0 17121 All components of the empt...
strfvi 17122 Structure slot extractors ...
fveqprc 17123 Lemma for showing the equa...
oveqprc 17124 Lemma for showing the equa...
wunndx 17127 Closure of the index extra...
ndxarg 17128 Get the numeric argument f...
ndxid 17129 A structure component extr...
strndxid 17130 The value of a structure c...
setsidvald 17131 Value of the structure rep...
setsidvaldOLD 17132 Obsolete version of ~ sets...
strfvd 17133 Deduction version of ~ str...
strfv2d 17134 Deduction version of ~ str...
strfv2 17135 A variation on ~ strfv to ...
strfv 17136 Extract a structure compon...
strfv3 17137 Variant on ~ strfv for lar...
strssd 17138 Deduction version of ~ str...
strss 17139 Propagate component extrac...
setsid 17140 Value of the structure rep...
setsnid 17141 Value of the structure rep...
setsnidOLD 17142 Obsolete proof of ~ setsni...
baseval 17145 Value of the base set extr...
baseid 17146 Utility theorem: index-ind...
basfn 17147 The base set extractor is ...
base0 17148 The base set of the empty ...
elbasfv 17149 Utility theorem: reverse c...
elbasov 17150 Utility theorem: reverse c...
strov2rcl 17151 Partial reverse closure fo...
basendx 17152 Index value of the base se...
basendxnn 17153 The index value of the bas...
basendxnnOLD 17154 Obsolete proof of ~ basend...
basndxelwund 17155 The index of the base set ...
basprssdmsets 17156 The pair of the base index...
opelstrbas 17157 The base set of a structur...
1strstr 17158 A constructed one-slot str...
1strstr1 17159 A constructed one-slot str...
1strbas 17160 The base set of a construc...
1strbasOLD 17161 Obsolete proof of ~ 1strba...
1strwunbndx 17162 A constructed one-slot str...
1strwun 17163 A constructed one-slot str...
1strwunOLD 17164 Obsolete version of ~ 1str...
2strstr 17165 A constructed two-slot str...
2strbas 17166 The base set of a construc...
2strop 17167 The other slot of a constr...
2strstr1 17168 A constructed two-slot str...
2strstr1OLD 17169 Obsolete version of ~ 2str...
2strbas1 17170 The base set of a construc...
2strop1 17171 The other slot of a constr...
reldmress 17174 The structure restriction ...
ressval 17175 Value of structure restric...
ressid2 17176 General behavior of trivia...
ressval2 17177 Value of nontrivial struct...
ressbas 17178 Base set of a structure re...
ressbasOLD 17179 Obsolete proof of ~ ressba...
ressbasssg 17180 The base set of a restrict...
ressbas2 17181 Base set of a structure re...
ressbasss 17182 The base set of a restrict...
ressbasssOLD 17183 Obsolete proof of ~ ressba...
ressbasss2 17184 The base set of a restrict...
resseqnbas 17185 The components of an exten...
resslemOLD 17186 Obsolete version of ~ ress...
ress0 17187 All restrictions of the nu...
ressid 17188 Behavior of trivial restri...
ressinbas 17189 Restriction only cares abo...
ressval3d 17190 Value of structure restric...
ressval3dOLD 17191 Obsolete version of ~ ress...
ressress 17192 Restriction composition la...
ressabs 17193 Restriction absorption law...
wunress 17194 Closure of structure restr...
wunressOLD 17195 Obsolete proof of ~ wunres...
plusgndx 17222 Index value of the ~ df-pl...
plusgid 17223 Utility theorem: index-ind...
plusgndxnn 17224 The index of the slot for ...
basendxltplusgndx 17225 The index of the slot for ...
basendxnplusgndx 17226 The slot for the base set ...
basendxnplusgndxOLD 17227 Obsolete version of ~ base...
grpstr 17228 A constructed group is a s...
grpstrndx 17229 A constructed group is a s...
grpbase 17230 The base set of a construc...
grpbaseOLD 17231 Obsolete version of ~ grpb...
grpplusg 17232 The operation of a constru...
grpplusgOLD 17233 Obsolete version of ~ grpp...
ressplusg 17234 ` +g ` is unaffected by re...
grpbasex 17235 The base of an explicitly ...
grpplusgx 17236 The operation of an explic...
mulrndx 17237 Index value of the ~ df-mu...
mulridx 17238 Utility theorem: index-ind...
basendxnmulrndx 17239 The slot for the base set ...
basendxnmulrndxOLD 17240 Obsolete proof of ~ basend...
plusgndxnmulrndx 17241 The slot for the group (ad...
rngstr 17242 A constructed ring is a st...
rngbase 17243 The base set of a construc...
rngplusg 17244 The additive operation of ...
rngmulr 17245 The multiplicative operati...
starvndx 17246 Index value of the ~ df-st...
starvid 17247 Utility theorem: index-ind...
starvndxnbasendx 17248 The slot for the involutio...
starvndxnplusgndx 17249 The slot for the involutio...
starvndxnmulrndx 17250 The slot for the involutio...
ressmulr 17251 ` .r ` is unaffected by re...
ressstarv 17252 ` *r ` is unaffected by re...
srngstr 17253 A constructed star ring is...
srngbase 17254 The base set of a construc...
srngplusg 17255 The addition operation of ...
srngmulr 17256 The multiplication operati...
srnginvl 17257 The involution function of...
scandx 17258 Index value of the ~ df-sc...
scaid 17259 Utility theorem: index-ind...
scandxnbasendx 17260 The slot for the scalar is...
scandxnplusgndx 17261 The slot for the scalar fi...
scandxnmulrndx 17262 The slot for the scalar fi...
vscandx 17263 Index value of the ~ df-vs...
vscaid 17264 Utility theorem: index-ind...
vscandxnbasendx 17265 The slot for the scalar pr...
vscandxnplusgndx 17266 The slot for the scalar pr...
vscandxnmulrndx 17267 The slot for the scalar pr...
vscandxnscandx 17268 The slot for the scalar pr...
lmodstr 17269 A constructed left module ...
lmodbase 17270 The base set of a construc...
lmodplusg 17271 The additive operation of ...
lmodsca 17272 The set of scalars of a co...
lmodvsca 17273 The scalar product operati...
ipndx 17274 Index value of the ~ df-ip...
ipid 17275 Utility theorem: index-ind...
ipndxnbasendx 17276 The slot for the inner pro...
ipndxnplusgndx 17277 The slot for the inner pro...
ipndxnmulrndx 17278 The slot for the inner pro...
slotsdifipndx 17279 The slot for the scalar is...
ipsstr 17280 Lemma to shorten proofs of...
ipsbase 17281 The base set of a construc...
ipsaddg 17282 The additive operation of ...
ipsmulr 17283 The multiplicative operati...
ipssca 17284 The set of scalars of a co...
ipsvsca 17285 The scalar product operati...
ipsip 17286 The multiplicative operati...
resssca 17287 ` Scalar ` is unaffected b...
ressvsca 17288 ` .s ` is unaffected by re...
ressip 17289 The inner product is unaff...
phlstr 17290 A constructed pre-Hilbert ...
phlbase 17291 The base set of a construc...
phlplusg 17292 The additive operation of ...
phlsca 17293 The ring of scalars of a c...
phlvsca 17294 The scalar product operati...
phlip 17295 The inner product (Hermiti...
tsetndx 17296 Index value of the ~ df-ts...
tsetid 17297 Utility theorem: index-ind...
tsetndxnn 17298 The index of the slot for ...
basendxlttsetndx 17299 The index of the slot for ...
tsetndxnbasendx 17300 The slot for the topology ...
tsetndxnplusgndx 17301 The slot for the topology ...
tsetndxnmulrndx 17302 The slot for the topology ...
tsetndxnstarvndx 17303 The slot for the topology ...
slotstnscsi 17304 The slots ` Scalar ` , ` ....
topgrpstr 17305 A constructed topological ...
topgrpbas 17306 The base set of a construc...
topgrpplusg 17307 The additive operation of ...
topgrptset 17308 The topology of a construc...
resstset 17309 ` TopSet ` is unaffected b...
plendx 17310 Index value of the ~ df-pl...
pleid 17311 Utility theorem: self-refe...
plendxnn 17312 The index value of the ord...
basendxltplendx 17313 The index value of the ` B...
plendxnbasendx 17314 The slot for the order is ...
plendxnplusgndx 17315 The slot for the "less tha...
plendxnmulrndx 17316 The slot for the "less tha...
plendxnscandx 17317 The slot for the "less tha...
plendxnvscandx 17318 The slot for the "less tha...
slotsdifplendx 17319 The index of the slot for ...
otpsstr 17320 Functionality of a topolog...
otpsbas 17321 The base set of a topologi...
otpstset 17322 The open sets of a topolog...
otpsle 17323 The order of a topological...
ressle 17324 ` le ` is unaffected by re...
ocndx 17325 Index value of the ~ df-oc...
ocid 17326 Utility theorem: index-ind...
basendxnocndx 17327 The slot for the orthocomp...
plendxnocndx 17328 The slot for the orthocomp...
dsndx 17329 Index value of the ~ df-ds...
dsid 17330 Utility theorem: index-ind...
dsndxnn 17331 The index of the slot for ...
basendxltdsndx 17332 The index of the slot for ...
dsndxnbasendx 17333 The slot for the distance ...
dsndxnplusgndx 17334 The slot for the distance ...
dsndxnmulrndx 17335 The slot for the distance ...
slotsdnscsi 17336 The slots ` Scalar ` , ` ....
dsndxntsetndx 17337 The slot for the distance ...
slotsdifdsndx 17338 The index of the slot for ...
unifndx 17339 Index value of the ~ df-un...
unifid 17340 Utility theorem: index-ind...
unifndxnn 17341 The index of the slot for ...
basendxltunifndx 17342 The index of the slot for ...
unifndxnbasendx 17343 The slot for the uniform s...
unifndxntsetndx 17344 The slot for the uniform s...
slotsdifunifndx 17345 The index of the slot for ...
ressunif 17346 ` UnifSet ` is unaffected ...
odrngstr 17347 Functionality of an ordere...
odrngbas 17348 The base set of an ordered...
odrngplusg 17349 The addition operation of ...
odrngmulr 17350 The multiplication operati...
odrngtset 17351 The open sets of an ordere...
odrngle 17352 The order of an ordered me...
odrngds 17353 The metric of an ordered m...
ressds 17354 ` dist ` is unaffected by ...
homndx 17355 Index value of the ~ df-ho...
homid 17356 Utility theorem: index-ind...
ccondx 17357 Index value of the ~ df-cc...
ccoid 17358 Utility theorem: index-ind...
slotsbhcdif 17359 The slots ` Base ` , ` Hom...
slotsbhcdifOLD 17360 Obsolete proof of ~ slotsb...
slotsdifplendx2 17361 The index of the slot for ...
slotsdifocndx 17362 The index of the slot for ...
resshom 17363 ` Hom ` is unaffected by r...
ressco 17364 ` comp ` is unaffected by ...
restfn 17369 The subspace topology oper...
topnfn 17370 The topology extractor fun...
restval 17371 The subspace topology indu...
elrest 17372 The predicate "is an open ...
elrestr 17373 Sufficient condition for b...
0rest 17374 Value of the structure res...
restid2 17375 The subspace topology over...
restsspw 17376 The subspace topology is a...
firest 17377 The finite intersections o...
restid 17378 The subspace topology of t...
topnval 17379 Value of the topology extr...
topnid 17380 Value of the topology extr...
topnpropd 17381 The topology extractor fun...
reldmprds 17393 The structure product is a...
prdsbasex 17395 Lemma for structure produc...
imasvalstr 17396 An image structure value i...
prdsvalstr 17397 Structure product value is...
prdsbaslem 17398 Lemma for ~ prdsbas and si...
prdsvallem 17399 Lemma for ~ prdsval . (Co...
prdsval 17400 Value of the structure pro...
prdssca 17401 Scalar ring of a structure...
prdsbas 17402 Base set of a structure pr...
prdsplusg 17403 Addition in a structure pr...
prdsmulr 17404 Multiplication in a struct...
prdsvsca 17405 Scalar multiplication in a...
prdsip 17406 Inner product in a structu...
prdsle 17407 Structure product weak ord...
prdsless 17408 Closure of the order relat...
prdsds 17409 Structure product distance...
prdsdsfn 17410 Structure product distance...
prdstset 17411 Structure product topology...
prdshom 17412 Structure product hom-sets...
prdsco 17413 Structure product composit...
prdsbas2 17414 The base set of a structur...
prdsbasmpt 17415 A constructed tuple is a p...
prdsbasfn 17416 Points in the structure pr...
prdsbasprj 17417 Each point in a structure ...
prdsplusgval 17418 Value of a componentwise s...
prdsplusgfval 17419 Value of a structure produ...
prdsmulrval 17420 Value of a componentwise r...
prdsmulrfval 17421 Value of a structure produ...
prdsleval 17422 Value of the product order...
prdsdsval 17423 Value of the metric in a s...
prdsvscaval 17424 Scalar multiplication in a...
prdsvscafval 17425 Scalar multiplication of a...
prdsbas3 17426 The base set of an indexed...
prdsbasmpt2 17427 A constructed tuple is a p...
prdsbascl 17428 An element of the base has...
prdsdsval2 17429 Value of the metric in a s...
prdsdsval3 17430 Value of the metric in a s...
pwsval 17431 Value of a structure power...
pwsbas 17432 Base set of a structure po...
pwselbasb 17433 Membership in the base set...
pwselbas 17434 An element of a structure ...
pwsplusgval 17435 Value of addition in a str...
pwsmulrval 17436 Value of multiplication in...
pwsle 17437 Ordering in a structure po...
pwsleval 17438 Ordering in a structure po...
pwsvscafval 17439 Scalar multiplication in a...
pwsvscaval 17440 Scalar multiplication of a...
pwssca 17441 The ring of scalars of a s...
pwsdiagel 17442 Membership of diagonal ele...
pwssnf1o 17443 Triviality of singleton po...
imasval 17456 Value of an image structur...
imasbas 17457 The base set of an image s...
imasds 17458 The distance function of a...
imasdsfn 17459 The distance function is a...
imasdsval 17460 The distance function of a...
imasdsval2 17461 The distance function of a...
imasplusg 17462 The group operation in an ...
imasmulr 17463 The ring multiplication in...
imassca 17464 The scalar field of an ima...
imasvsca 17465 The scalar multiplication ...
imasip 17466 The inner product of an im...
imastset 17467 The topology of an image s...
imasle 17468 The ordering of an image s...
f1ocpbllem 17469 Lemma for ~ f1ocpbl . (Co...
f1ocpbl 17470 An injection is compatible...
f1ovscpbl 17471 An injection is compatible...
f1olecpbl 17472 An injection is compatible...
imasaddfnlem 17473 The image structure operat...
imasaddvallem 17474 The operation of an image ...
imasaddflem 17475 The image set operations a...
imasaddfn 17476 The image structure's grou...
imasaddval 17477 The value of an image stru...
imasaddf 17478 The image structure's grou...
imasmulfn 17479 The image structure's ring...
imasmulval 17480 The value of an image stru...
imasmulf 17481 The image structure's ring...
imasvscafn 17482 The image structure's scal...
imasvscaval 17483 The value of an image stru...
imasvscaf 17484 The image structure's scal...
imasless 17485 The order relation defined...
imasleval 17486 The value of the image str...
qusval 17487 Value of a quotient struct...
quslem 17488 The function in ~ qusval i...
qusin 17489 Restrict the equivalence r...
qusbas 17490 Base set of a quotient str...
quss 17491 The scalar field of a quot...
divsfval 17492 Value of the function in ~...
ercpbllem 17493 Lemma for ~ ercpbl . (Con...
ercpbl 17494 Translate the function com...
erlecpbl 17495 Translate the relation com...
qusaddvallem 17496 Value of an operation defi...
qusaddflem 17497 The operation of a quotien...
qusaddval 17498 The addition in a quotient...
qusaddf 17499 The addition in a quotient...
qusmulval 17500 The multiplication in a qu...
qusmulf 17501 The multiplication in a qu...
fnpr2o 17502 Function with a domain of ...
fnpr2ob 17503 Biconditional version of ~...
fvpr0o 17504 The value of a function wi...
fvpr1o 17505 The value of a function wi...
fvprif 17506 The value of the pair func...
xpsfrnel 17507 Elementhood in the target ...
xpsfeq 17508 A function on ` 2o ` is de...
xpsfrnel2 17509 Elementhood in the target ...
xpscf 17510 Equivalent condition for t...
xpsfval 17511 The value of the function ...
xpsff1o 17512 The function appearing in ...
xpsfrn 17513 A short expression for the...
xpsff1o2 17514 The function appearing in ...
xpsval 17515 Value of the binary struct...
xpsrnbas 17516 The indexed structure prod...
xpsbas 17517 The base set of the binary...
xpsaddlem 17518 Lemma for ~ xpsadd and ~ x...
xpsadd 17519 Value of the addition oper...
xpsmul 17520 Value of the multiplicatio...
xpssca 17521 Value of the scalar field ...
xpsvsca 17522 Value of the scalar multip...
xpsless 17523 Closure of the ordering in...
xpsle 17524 Value of the ordering in a...
ismre 17533 Property of being a Moore ...
fnmre 17534 The Moore collection gener...
mresspw 17535 A Moore collection is a su...
mress 17536 A Moore-closed subset is a...
mre1cl 17537 In any Moore collection th...
mreintcl 17538 A nonempty collection of c...
mreiincl 17539 A nonempty indexed interse...
mrerintcl 17540 The relative intersection ...
mreriincl 17541 The relative intersection ...
mreincl 17542 Two closed sets have a clo...
mreuni 17543 Since the entire base set ...
mreunirn 17544 Two ways to express the no...
ismred 17545 Properties that determine ...
ismred2 17546 Properties that determine ...
mremre 17547 The Moore collections of s...
submre 17548 The subcollection of a clo...
mrcflem 17549 The domain and codomain of...
fnmrc 17550 Moore-closure is a well-be...
mrcfval 17551 Value of the function expr...
mrcf 17552 The Moore closure is a fun...
mrcval 17553 Evaluation of the Moore cl...
mrccl 17554 The Moore closure of a set...
mrcsncl 17555 The Moore closure of a sin...
mrcid 17556 The closure of a closed se...
mrcssv 17557 The closure of a set is a ...
mrcidb 17558 A set is closed iff it is ...
mrcss 17559 Closure preserves subset o...
mrcssid 17560 The closure of a set is a ...
mrcidb2 17561 A set is closed iff it con...
mrcidm 17562 The closure operation is i...
mrcsscl 17563 The closure is the minimal...
mrcuni 17564 Idempotence of closure und...
mrcun 17565 Idempotence of closure und...
mrcssvd 17566 The Moore closure of a set...
mrcssd 17567 Moore closure preserves su...
mrcssidd 17568 A set is contained in its ...
mrcidmd 17569 Moore closure is idempoten...
mressmrcd 17570 In a Moore system, if a se...
submrc 17571 In a closure system which ...
mrieqvlemd 17572 In a Moore system, if ` Y ...
mrisval 17573 Value of the set of indepe...
ismri 17574 Criterion for a set to be ...
ismri2 17575 Criterion for a subset of ...
ismri2d 17576 Criterion for a subset of ...
ismri2dd 17577 Definition of independence...
mriss 17578 An independent set of a Mo...
mrissd 17579 An independent set of a Mo...
ismri2dad 17580 Consequence of a set in a ...
mrieqvd 17581 In a Moore system, a set i...
mrieqv2d 17582 In a Moore system, a set i...
mrissmrcd 17583 In a Moore system, if an i...
mrissmrid 17584 In a Moore system, subsets...
mreexd 17585 In a Moore system, the clo...
mreexmrid 17586 In a Moore system whose cl...
mreexexlemd 17587 This lemma is used to gene...
mreexexlem2d 17588 Used in ~ mreexexlem4d to ...
mreexexlem3d 17589 Base case of the induction...
mreexexlem4d 17590 Induction step of the indu...
mreexexd 17591 Exchange-type theorem. In...
mreexdomd 17592 In a Moore system whose cl...
mreexfidimd 17593 In a Moore system whose cl...
isacs 17594 A set is an algebraic clos...
acsmre 17595 Algebraic closure systems ...
isacs2 17596 In the definition of an al...
acsfiel 17597 A set is closed in an alge...
acsfiel2 17598 A set is closed in an alge...
acsmred 17599 An algebraic closure syste...
isacs1i 17600 A closure system determine...
mreacs 17601 Algebraicity is a composab...
acsfn 17602 Algebraicity of a conditio...
acsfn0 17603 Algebraicity of a point cl...
acsfn1 17604 Algebraicity of a one-argu...
acsfn1c 17605 Algebraicity of a one-argu...
acsfn2 17606 Algebraicity of a two-argu...
iscat 17615 The predicate "is a catego...
iscatd 17616 Properties that determine ...
catidex 17617 Each object in a category ...
catideu 17618 Each object in a category ...
cidfval 17619 Each object in a category ...
cidval 17620 Each object in a category ...
cidffn 17621 The identity arrow constru...
cidfn 17622 The identity arrow operato...
catidd 17623 Deduce the identity arrow ...
iscatd2 17624 Version of ~ iscatd with a...
catidcl 17625 Each object in a category ...
catlid 17626 Left identity property of ...
catrid 17627 Right identity property of...
catcocl 17628 Closure of a composition a...
catass 17629 Associativity of compositi...
catcone0 17630 Composition of non-empty h...
0catg 17631 Any structure with an empt...
0cat 17632 The empty set is a categor...
homffval 17633 Value of the functionalize...
fnhomeqhomf 17634 If the Hom-set operation i...
homfval 17635 Value of the functionalize...
homffn 17636 The functionalized Hom-set...
homfeq 17637 Condition for two categori...
homfeqd 17638 If two structures have the...
homfeqbas 17639 Deduce equality of base se...
homfeqval 17640 Value of the functionalize...
comfffval 17641 Value of the functionalize...
comffval 17642 Value of the functionalize...
comfval 17643 Value of the functionalize...
comfffval2 17644 Value of the functionalize...
comffval2 17645 Value of the functionalize...
comfval2 17646 Value of the functionalize...
comfffn 17647 The functionalized composi...
comffn 17648 The functionalized composi...
comfeq 17649 Condition for two categori...
comfeqd 17650 Condition for two categori...
comfeqval 17651 Equality of two compositio...
catpropd 17652 Two structures with the sa...
cidpropd 17653 Two structures with the sa...
oppcval 17656 Value of the opposite cate...
oppchomfval 17657 Hom-sets of the opposite c...
oppchomfvalOLD 17658 Obsolete proof of ~ oppcho...
oppchom 17659 Hom-sets of the opposite c...
oppccofval 17660 Composition in the opposit...
oppcco 17661 Composition in the opposit...
oppcbas 17662 Base set of an opposite ca...
oppcbasOLD 17663 Obsolete version of ~ oppc...
oppccatid 17664 Lemma for ~ oppccat . (Co...
oppchomf 17665 Hom-sets of the opposite c...
oppcid 17666 Identity function of an op...
oppccat 17667 An opposite category is a ...
2oppcbas 17668 The double opposite catego...
2oppchomf 17669 The double opposite catego...
2oppccomf 17670 The double opposite catego...
oppchomfpropd 17671 If two categories have the...
oppccomfpropd 17672 If two categories have the...
oppccatf 17673 ` oppCat ` restricted to `...
monfval 17678 Definition of a monomorphi...
ismon 17679 Definition of a monomorphi...
ismon2 17680 Write out the monomorphism...
monhom 17681 A monomorphism is a morphi...
moni 17682 Property of a monomorphism...
monpropd 17683 If two categories have the...
oppcmon 17684 A monomorphism in the oppo...
oppcepi 17685 An epimorphism in the oppo...
isepi 17686 Definition of an epimorphi...
isepi2 17687 Write out the epimorphism ...
epihom 17688 An epimorphism is a morphi...
epii 17689 Property of an epimorphism...
sectffval 17696 Value of the section opera...
sectfval 17697 Value of the section relat...
sectss 17698 The section relation is a ...
issect 17699 The property " ` F ` is a ...
issect2 17700 Property of being a sectio...
sectcan 17701 If ` G ` is a section of `...
sectco 17702 Composition of two section...
isofval 17703 Function value of the func...
invffval 17704 Value of the inverse relat...
invfval 17705 Value of the inverse relat...
isinv 17706 Value of the inverse relat...
invss 17707 The inverse relation is a ...
invsym 17708 The inverse relation is sy...
invsym2 17709 The inverse relation is sy...
invfun 17710 The inverse relation is a ...
isoval 17711 The isomorphisms are the d...
inviso1 17712 If ` G ` is an inverse to ...
inviso2 17713 If ` G ` is an inverse to ...
invf 17714 The inverse relation is a ...
invf1o 17715 The inverse relation is a ...
invinv 17716 The inverse of the inverse...
invco 17717 The composition of two iso...
dfiso2 17718 Alternate definition of an...
dfiso3 17719 Alternate definition of an...
inveq 17720 If there are two inverses ...
isofn 17721 The function value of the ...
isohom 17722 An isomorphism is a homomo...
isoco 17723 The composition of two iso...
oppcsect 17724 A section in the opposite ...
oppcsect2 17725 A section in the opposite ...
oppcinv 17726 An inverse in the opposite...
oppciso 17727 An isomorphism in the oppo...
sectmon 17728 If ` F ` is a section of `...
monsect 17729 If ` F ` is a monomorphism...
sectepi 17730 If ` F ` is a section of `...
episect 17731 If ` F ` is an epimorphism...
sectid 17732 The identity is a section ...
invid 17733 The inverse of the identit...
idiso 17734 The identity is an isomorp...
idinv 17735 The inverse of the identit...
invisoinvl 17736 The inverse of an isomorph...
invisoinvr 17737 The inverse of an isomorph...
invcoisoid 17738 The inverse of an isomorph...
isocoinvid 17739 The inverse of an isomorph...
rcaninv 17740 Right cancellation of an i...
cicfval 17743 The set of isomorphic obje...
brcic 17744 The relation "is isomorphi...
cic 17745 Objects ` X ` and ` Y ` in...
brcici 17746 Prove that two objects are...
cicref 17747 Isomorphism is reflexive. ...
ciclcl 17748 Isomorphism implies the le...
cicrcl 17749 Isomorphism implies the ri...
cicsym 17750 Isomorphism is symmetric. ...
cictr 17751 Isomorphism is transitive....
cicer 17752 Isomorphism is an equivale...
sscrel 17759 The subcategory subset rel...
brssc 17760 The subcategory subset rel...
sscpwex 17761 An analogue of ~ pwex for ...
subcrcl 17762 Reverse closure for the su...
sscfn1 17763 The subcategory subset rel...
sscfn2 17764 The subcategory subset rel...
ssclem 17765 Lemma for ~ ssc1 and simil...
isssc 17766 Value of the subcategory s...
ssc1 17767 Infer subset relation on o...
ssc2 17768 Infer subset relation on m...
sscres 17769 Any function restricted to...
sscid 17770 The subcategory subset rel...
ssctr 17771 The subcategory subset rel...
ssceq 17772 The subcategory subset rel...
rescval 17773 Value of the category rest...
rescval2 17774 Value of the category rest...
rescbas 17775 Base set of the category r...
rescbasOLD 17776 Obsolete version of ~ resc...
reschom 17777 Hom-sets of the category r...
reschomf 17778 Hom-sets of the category r...
rescco 17779 Composition in the categor...
resccoOLD 17780 Obsolete proof of ~ rescco...
rescabs 17781 Restriction absorption law...
rescabsOLD 17782 Obsolete proof of ~ seqp1d...
rescabs2 17783 Restriction absorption law...
issubc 17784 Elementhood in the set of ...
issubc2 17785 Elementhood in the set of ...
0ssc 17786 For any category ` C ` , t...
0subcat 17787 For any category ` C ` , t...
catsubcat 17788 For any category ` C ` , `...
subcssc 17789 An element in the set of s...
subcfn 17790 An element in the set of s...
subcss1 17791 The objects of a subcatego...
subcss2 17792 The morphisms of a subcate...
subcidcl 17793 The identity of the origin...
subccocl 17794 A subcategory is closed un...
subccatid 17795 A subcategory is a categor...
subcid 17796 The identity in a subcateg...
subccat 17797 A subcategory is a categor...
issubc3 17798 Alternate definition of a ...
fullsubc 17799 The full subcategory gener...
fullresc 17800 The category formed by str...
resscat 17801 A category restricted to a...
subsubc 17802 A subcategory of a subcate...
relfunc 17811 The set of functors is a r...
funcrcl 17812 Reverse closure for a func...
isfunc 17813 Value of the set of functo...
isfuncd 17814 Deduce that an operation i...
funcf1 17815 The object part of a funct...
funcixp 17816 The morphism part of a fun...
funcf2 17817 The morphism part of a fun...
funcfn2 17818 The morphism part of a fun...
funcid 17819 A functor maps each identi...
funcco 17820 A functor maps composition...
funcsect 17821 The image of a section und...
funcinv 17822 The image of an inverse un...
funciso 17823 The image of an isomorphis...
funcoppc 17824 A functor on categories yi...
idfuval 17825 Value of the identity func...
idfu2nd 17826 Value of the morphism part...
idfu2 17827 Value of the morphism part...
idfu1st 17828 Value of the object part o...
idfu1 17829 Value of the object part o...
idfucl 17830 The identity functor is a ...
cofuval 17831 Value of the composition o...
cofu1st 17832 Value of the object part o...
cofu1 17833 Value of the object part o...
cofu2nd 17834 Value of the morphism part...
cofu2 17835 Value of the morphism part...
cofuval2 17836 Value of the composition o...
cofucl 17837 The composition of two fun...
cofuass 17838 Functor composition is ass...
cofulid 17839 The identity functor is a ...
cofurid 17840 The identity functor is a ...
resfval 17841 Value of the functor restr...
resfval2 17842 Value of the functor restr...
resf1st 17843 Value of the functor restr...
resf2nd 17844 Value of the functor restr...
funcres 17845 A functor restricted to a ...
funcres2b 17846 Condition for a functor to...
funcres2 17847 A functor into a restricte...
idfusubc0 17848 The identity functor for a...
idfusubc 17849 The identity functor for a...
wunfunc 17850 A weak universe is closed ...
wunfuncOLD 17851 Obsolete proof of ~ wunfun...
funcpropd 17852 If two categories have the...
funcres2c 17853 Condition for a functor to...
fullfunc 17858 A full functor is a functo...
fthfunc 17859 A faithful functor is a fu...
relfull 17860 The set of full functors i...
relfth 17861 The set of faithful functo...
isfull 17862 Value of the set of full f...
isfull2 17863 Equivalent condition for a...
fullfo 17864 The morphism map of a full...
fulli 17865 The morphism map of a full...
isfth 17866 Value of the set of faithf...
isfth2 17867 Equivalent condition for a...
isffth2 17868 A fully faithful functor i...
fthf1 17869 The morphism map of a fait...
fthi 17870 The morphism map of a fait...
ffthf1o 17871 The morphism map of a full...
fullpropd 17872 If two categories have the...
fthpropd 17873 If two categories have the...
fulloppc 17874 The opposite functor of a ...
fthoppc 17875 The opposite functor of a ...
ffthoppc 17876 The opposite functor of a ...
fthsect 17877 A faithful functor reflect...
fthinv 17878 A faithful functor reflect...
fthmon 17879 A faithful functor reflect...
fthepi 17880 A faithful functor reflect...
ffthiso 17881 A fully faithful functor r...
fthres2b 17882 Condition for a faithful f...
fthres2c 17883 Condition for a faithful f...
fthres2 17884 A faithful functor into a ...
idffth 17885 The identity functor is a ...
cofull 17886 The composition of two ful...
cofth 17887 The composition of two fai...
coffth 17888 The composition of two ful...
rescfth 17889 The inclusion functor from...
ressffth 17890 The inclusion functor from...
fullres2c 17891 Condition for a full funct...
ffthres2c 17892 Condition for a fully fait...
inclfusubc 17893 The "inclusion functor" fr...
fnfuc 17898 The ` FuncCat ` operation ...
natfval 17899 Value of the function givi...
isnat 17900 Property of being a natura...
isnat2 17901 Property of being a natura...
natffn 17902 The natural transformation...
natrcl 17903 Reverse closure for a natu...
nat1st2nd 17904 Rewrite the natural transf...
natixp 17905 A natural transformation i...
natcl 17906 A component of a natural t...
natfn 17907 A natural transformation i...
nati 17908 Naturality property of a n...
wunnat 17909 A weak universe is closed ...
wunnatOLD 17910 Obsolete proof of ~ wunnat...
catstr 17911 A category structure is a ...
fucval 17912 Value of the functor categ...
fuccofval 17913 Value of the functor categ...
fucbas 17914 The objects of the functor...
fuchom 17915 The morphisms in the funct...
fuchomOLD 17916 Obsolete proof of ~ fuchom...
fucco 17917 Value of the composition o...
fuccoval 17918 Value of the functor categ...
fuccocl 17919 The composition of two nat...
fucidcl 17920 The identity natural trans...
fuclid 17921 Left identity of natural t...
fucrid 17922 Right identity of natural ...
fucass 17923 Associativity of natural t...
fuccatid 17924 The functor category is a ...
fuccat 17925 The functor category is a ...
fucid 17926 The identity morphism in t...
fucsect 17927 Two natural transformation...
fucinv 17928 Two natural transformation...
invfuc 17929 If ` V ( x ) ` is an inver...
fuciso 17930 A natural transformation i...
natpropd 17931 If two categories have the...
fucpropd 17932 If two categories have the...
initofn 17939 ` InitO ` is a function on...
termofn 17940 ` TermO ` is a function on...
zeroofn 17941 ` ZeroO ` is a function on...
initorcl 17942 Reverse closure for an ini...
termorcl 17943 Reverse closure for a term...
zeroorcl 17944 Reverse closure for a zero...
initoval 17945 The value of the initial o...
termoval 17946 The value of the terminal ...
zerooval 17947 The value of the zero obje...
isinito 17948 The predicate "is an initi...
istermo 17949 The predicate "is a termin...
iszeroo 17950 The predicate "is a zero o...
isinitoi 17951 Implication of a class bei...
istermoi 17952 Implication of a class bei...
initoid 17953 For an initial object, the...
termoid 17954 For a terminal object, the...
dfinito2 17955 An initial object is a ter...
dftermo2 17956 A terminal object is an in...
dfinito3 17957 An alternate definition of...
dftermo3 17958 An alternate definition of...
initoo 17959 An initial object is an ob...
termoo 17960 A terminal object is an ob...
iszeroi 17961 Implication of a class bei...
2initoinv 17962 Morphisms between two init...
initoeu1 17963 Initial objects are essent...
initoeu1w 17964 Initial objects are essent...
initoeu2lem0 17965 Lemma 0 for ~ initoeu2 . ...
initoeu2lem1 17966 Lemma 1 for ~ initoeu2 . ...
initoeu2lem2 17967 Lemma 2 for ~ initoeu2 . ...
initoeu2 17968 Initial objects are essent...
2termoinv 17969 Morphisms between two term...
termoeu1 17970 Terminal objects are essen...
termoeu1w 17971 Terminal objects are essen...
homarcl 17980 Reverse closure for an arr...
homafval 17981 Value of the disjointified...
homaf 17982 Functionality of the disjo...
homaval 17983 Value of the disjointified...
elhoma 17984 Value of the disjointified...
elhomai 17985 Produce an arrow from a mo...
elhomai2 17986 Produce an arrow from a mo...
homarcl2 17987 Reverse closure for the do...
homarel 17988 An arrow is an ordered pai...
homa1 17989 The first component of an ...
homahom2 17990 The second component of an...
homahom 17991 The second component of an...
homadm 17992 The domain of an arrow wit...
homacd 17993 The codomain of an arrow w...
homadmcd 17994 Decompose an arrow into do...
arwval 17995 The set of arrows is the u...
arwrcl 17996 The first component of an ...
arwhoma 17997 An arrow is contained in t...
homarw 17998 A hom-set is a subset of t...
arwdm 17999 The domain of an arrow is ...
arwcd 18000 The codomain of an arrow i...
dmaf 18001 The domain function is a f...
cdaf 18002 The codomain function is a...
arwhom 18003 The second component of an...
arwdmcd 18004 Decompose an arrow into do...
idafval 18009 Value of the identity arro...
idaval 18010 Value of the identity arro...
ida2 18011 Morphism part of the ident...
idahom 18012 Domain and codomain of the...
idadm 18013 Domain of the identity arr...
idacd 18014 Codomain of the identity a...
idaf 18015 The identity arrow functio...
coafval 18016 The value of the compositi...
eldmcoa 18017 A pair ` <. G , F >. ` is ...
dmcoass 18018 The domain of composition ...
homdmcoa 18019 If ` F : X --> Y ` and ` G...
coaval 18020 Value of composition for c...
coa2 18021 The morphism part of arrow...
coahom 18022 The composition of two com...
coapm 18023 Composition of arrows is a...
arwlid 18024 Left identity of a categor...
arwrid 18025 Right identity of a catego...
arwass 18026 Associativity of compositi...
setcval 18029 Value of the category of s...
setcbas 18030 Set of objects of the cate...
setchomfval 18031 Set of arrows of the categ...
setchom 18032 Set of arrows of the categ...
elsetchom 18033 A morphism of sets is a fu...
setccofval 18034 Composition in the categor...
setcco 18035 Composition in the categor...
setccatid 18036 Lemma for ~ setccat . (Co...
setccat 18037 The category of sets is a ...
setcid 18038 The identity arrow in the ...
setcmon 18039 A monomorphism of sets is ...
setcepi 18040 An epimorphism of sets is ...
setcsect 18041 A section in the category ...
setcinv 18042 An inverse in the category...
setciso 18043 An isomorphism in the cate...
resssetc 18044 The restriction of the cat...
funcsetcres2 18045 A functor into a smaller c...
setc2obas 18046 ` (/) ` and ` 1o ` are dis...
setc2ohom 18047 ` ( SetCat `` 2o ) ` is a ...
cat1lem 18048 The category of sets in a ...
cat1 18049 The definition of category...
catcval 18052 Value of the category of c...
catcbas 18053 Set of objects of the cate...
catchomfval 18054 Set of arrows of the categ...
catchom 18055 Set of arrows of the categ...
catccofval 18056 Composition in the categor...
catcco 18057 Composition in the categor...
catccatid 18058 Lemma for ~ catccat . (Co...
catcid 18059 The identity arrow in the ...
catccat 18060 The category of categories...
resscatc 18061 The restriction of the cat...
catcisolem 18062 Lemma for ~ catciso . (Co...
catciso 18063 A functor is an isomorphis...
catcbascl 18064 An element of the base set...
catcslotelcl 18065 A slot entry of an element...
catcbaselcl 18066 The base set of an element...
catchomcl 18067 The Hom-set of an element ...
catcccocl 18068 The composition operation ...
catcoppccl 18069 The category of categories...
catcoppcclOLD 18070 Obsolete proof of ~ catcop...
catcfuccl 18071 The category of categories...
catcfucclOLD 18072 Obsolete proof of ~ catcfu...
fncnvimaeqv 18073 The inverse images of the ...
bascnvimaeqv 18074 The inverse image of the u...
estrcval 18077 Value of the category of e...
estrcbas 18078 Set of objects of the cate...
estrchomfval 18079 Set of morphisms ("arrows"...
estrchom 18080 The morphisms between exte...
elestrchom 18081 A morphism between extensi...
estrccofval 18082 Composition in the categor...
estrcco 18083 Composition in the categor...
estrcbasbas 18084 An element of the base set...
estrccatid 18085 Lemma for ~ estrccat . (C...
estrccat 18086 The category of extensible...
estrcid 18087 The identity arrow in the ...
estrchomfn 18088 The Hom-set operation in t...
estrchomfeqhom 18089 The functionalized Hom-set...
estrreslem1 18090 Lemma 1 for ~ estrres . (...
estrreslem1OLD 18091 Obsolete version of ~ estr...
estrreslem2 18092 Lemma 2 for ~ estrres . (...
estrres 18093 Any restriction of a categ...
funcestrcsetclem1 18094 Lemma 1 for ~ funcestrcset...
funcestrcsetclem2 18095 Lemma 2 for ~ funcestrcset...
funcestrcsetclem3 18096 Lemma 3 for ~ funcestrcset...
funcestrcsetclem4 18097 Lemma 4 for ~ funcestrcset...
funcestrcsetclem5 18098 Lemma 5 for ~ funcestrcset...
funcestrcsetclem6 18099 Lemma 6 for ~ funcestrcset...
funcestrcsetclem7 18100 Lemma 7 for ~ funcestrcset...
funcestrcsetclem8 18101 Lemma 8 for ~ funcestrcset...
funcestrcsetclem9 18102 Lemma 9 for ~ funcestrcset...
funcestrcsetc 18103 The "natural forgetful fun...
fthestrcsetc 18104 The "natural forgetful fun...
fullestrcsetc 18105 The "natural forgetful fun...
equivestrcsetc 18106 The "natural forgetful fun...
setc1strwun 18107 A constructed one-slot str...
funcsetcestrclem1 18108 Lemma 1 for ~ funcsetcestr...
funcsetcestrclem2 18109 Lemma 2 for ~ funcsetcestr...
funcsetcestrclem3 18110 Lemma 3 for ~ funcsetcestr...
embedsetcestrclem 18111 Lemma for ~ embedsetcestrc...
funcsetcestrclem4 18112 Lemma 4 for ~ funcsetcestr...
funcsetcestrclem5 18113 Lemma 5 for ~ funcsetcestr...
funcsetcestrclem6 18114 Lemma 6 for ~ funcsetcestr...
funcsetcestrclem7 18115 Lemma 7 for ~ funcsetcestr...
funcsetcestrclem8 18116 Lemma 8 for ~ funcsetcestr...
funcsetcestrclem9 18117 Lemma 9 for ~ funcsetcestr...
funcsetcestrc 18118 The "embedding functor" fr...
fthsetcestrc 18119 The "embedding functor" fr...
fullsetcestrc 18120 The "embedding functor" fr...
embedsetcestrc 18121 The "embedding functor" fr...
fnxpc 18130 The binary product of cate...
xpcval 18131 Value of the binary produc...
xpcbas 18132 Set of objects of the bina...
xpchomfval 18133 Set of morphisms of the bi...
xpchom 18134 Set of morphisms of the bi...
relxpchom 18135 A hom-set in the binary pr...
xpccofval 18136 Value of composition in th...
xpcco 18137 Value of composition in th...
xpcco1st 18138 Value of composition in th...
xpcco2nd 18139 Value of composition in th...
xpchom2 18140 Value of the set of morphi...
xpcco2 18141 Value of composition in th...
xpccatid 18142 The product of two categor...
xpcid 18143 The identity morphism in t...
xpccat 18144 The product of two categor...
1stfval 18145 Value of the first project...
1stf1 18146 Value of the first project...
1stf2 18147 Value of the first project...
2ndfval 18148 Value of the first project...
2ndf1 18149 Value of the first project...
2ndf2 18150 Value of the first project...
1stfcl 18151 The first projection funct...
2ndfcl 18152 The second projection func...
prfval 18153 Value of the pairing funct...
prf1 18154 Value of the pairing funct...
prf2fval 18155 Value of the pairing funct...
prf2 18156 Value of the pairing funct...
prfcl 18157 The pairing of functors ` ...
prf1st 18158 Cancellation of pairing wi...
prf2nd 18159 Cancellation of pairing wi...
1st2ndprf 18160 Break a functor into a pro...
catcxpccl 18161 The category of categories...
catcxpcclOLD 18162 Obsolete proof of ~ catcxp...
xpcpropd 18163 If two categories have the...
evlfval 18172 Value of the evaluation fu...
evlf2 18173 Value of the evaluation fu...
evlf2val 18174 Value of the evaluation na...
evlf1 18175 Value of the evaluation fu...
evlfcllem 18176 Lemma for ~ evlfcl . (Con...
evlfcl 18177 The evaluation functor is ...
curfval 18178 Value of the curry functor...
curf1fval 18179 Value of the object part o...
curf1 18180 Value of the object part o...
curf11 18181 Value of the double evalua...
curf12 18182 The partially evaluated cu...
curf1cl 18183 The partially evaluated cu...
curf2 18184 Value of the curry functor...
curf2val 18185 Value of a component of th...
curf2cl 18186 The curry functor at a mor...
curfcl 18187 The curry functor of a fun...
curfpropd 18188 If two categories have the...
uncfval 18189 Value of the uncurry funct...
uncfcl 18190 The uncurry operation take...
uncf1 18191 Value of the uncurry funct...
uncf2 18192 Value of the uncurry funct...
curfuncf 18193 Cancellation of curry with...
uncfcurf 18194 Cancellation of uncurry wi...
diagval 18195 Define the diagonal functo...
diagcl 18196 The diagonal functor is a ...
diag1cl 18197 The constant functor of ` ...
diag11 18198 Value of the constant func...
diag12 18199 Value of the constant func...
diag2 18200 Value of the diagonal func...
diag2cl 18201 The diagonal functor at a ...
curf2ndf 18202 As shown in ~ diagval , th...
hofval 18207 Value of the Hom functor, ...
hof1fval 18208 The object part of the Hom...
hof1 18209 The object part of the Hom...
hof2fval 18210 The morphism part of the H...
hof2val 18211 The morphism part of the H...
hof2 18212 The morphism part of the H...
hofcllem 18213 Lemma for ~ hofcl . (Cont...
hofcl 18214 Closure of the Hom functor...
oppchofcl 18215 Closure of the opposite Ho...
yonval 18216 Value of the Yoneda embedd...
yoncl 18217 The Yoneda embedding is a ...
yon1cl 18218 The Yoneda embedding at an...
yon11 18219 Value of the Yoneda embedd...
yon12 18220 Value of the Yoneda embedd...
yon2 18221 Value of the Yoneda embedd...
hofpropd 18222 If two categories have the...
yonpropd 18223 If two categories have the...
oppcyon 18224 Value of the opposite Yone...
oyoncl 18225 The opposite Yoneda embedd...
oyon1cl 18226 The opposite Yoneda embedd...
yonedalem1 18227 Lemma for ~ yoneda . (Con...
yonedalem21 18228 Lemma for ~ yoneda . (Con...
yonedalem3a 18229 Lemma for ~ yoneda . (Con...
yonedalem4a 18230 Lemma for ~ yoneda . (Con...
yonedalem4b 18231 Lemma for ~ yoneda . (Con...
yonedalem4c 18232 Lemma for ~ yoneda . (Con...
yonedalem22 18233 Lemma for ~ yoneda . (Con...
yonedalem3b 18234 Lemma for ~ yoneda . (Con...
yonedalem3 18235 Lemma for ~ yoneda . (Con...
yonedainv 18236 The Yoneda Lemma with expl...
yonffthlem 18237 Lemma for ~ yonffth . (Co...
yoneda 18238 The Yoneda Lemma. There i...
yonffth 18239 The Yoneda Lemma. The Yon...
yoniso 18240 If the codomain is recover...
oduval 18243 Value of an order dual str...
oduleval 18244 Value of the less-equal re...
oduleg 18245 Truth of the less-equal re...
odubas 18246 Base set of an order dual ...
odubasOLD 18247 Obsolete proof of ~ odubas...
isprs 18252 Property of being a preord...
prslem 18253 Lemma for ~ prsref and ~ p...
prsref 18254 "Less than or equal to" is...
prstr 18255 "Less than or equal to" is...
isdrs 18256 Property of being a direct...
drsdir 18257 Direction of a directed se...
drsprs 18258 A directed set is a proset...
drsbn0 18259 The base of a directed set...
drsdirfi 18260 Any _finite_ number of ele...
isdrs2 18261 Directed sets may be defin...
ispos 18269 The predicate "is a poset"...
ispos2 18270 A poset is an antisymmetri...
posprs 18271 A poset is a proset. (Con...
posi 18272 Lemma for poset properties...
posref 18273 A poset ordering is reflex...
posasymb 18274 A poset ordering is asymme...
postr 18275 A poset ordering is transi...
0pos 18276 Technical lemma to simplif...
0posOLD 18277 Obsolete proof of ~ 0pos a...
isposd 18278 Properties that determine ...
isposi 18279 Properties that determine ...
isposix 18280 Properties that determine ...
isposixOLD 18281 Obsolete proof of ~ isposi...
pospropd 18282 Posethood is determined on...
odupos 18283 Being a poset is a self-du...
oduposb 18284 Being a poset is a self-du...
pltfval 18286 Value of the less-than rel...
pltval 18287 Less-than relation. ( ~ d...
pltle 18288 "Less than" implies "less ...
pltne 18289 The "less than" relation i...
pltirr 18290 The "less than" relation i...
pleval2i 18291 One direction of ~ pleval2...
pleval2 18292 "Less than or equal to" in...
pltnle 18293 "Less than" implies not co...
pltval3 18294 Alternate expression for t...
pltnlt 18295 The less-than relation imp...
pltn2lp 18296 The less-than relation has...
plttr 18297 The less-than relation is ...
pltletr 18298 Transitive law for chained...
plelttr 18299 Transitive law for chained...
pospo 18300 Write a poset structure in...
lubfval 18305 Value of the least upper b...
lubdm 18306 Domain of the least upper ...
lubfun 18307 The LUB is a function. (C...
lubeldm 18308 Member of the domain of th...
lubelss 18309 A member of the domain of ...
lubeu 18310 Unique existence proper of...
lubval 18311 Value of the least upper b...
lubcl 18312 The least upper bound func...
lubprop 18313 Properties of greatest low...
luble 18314 The greatest lower bound i...
lublecllem 18315 Lemma for ~ lublecl and ~ ...
lublecl 18316 The set of all elements le...
lubid 18317 The LUB of elements less t...
glbfval 18318 Value of the greatest lowe...
glbdm 18319 Domain of the greatest low...
glbfun 18320 The GLB is a function. (C...
glbeldm 18321 Member of the domain of th...
glbelss 18322 A member of the domain of ...
glbeu 18323 Unique existence proper of...
glbval 18324 Value of the greatest lowe...
glbcl 18325 The least upper bound func...
glbprop 18326 Properties of greatest low...
glble 18327 The greatest lower bound i...
joinfval 18328 Value of join function for...
joinfval2 18329 Value of join function for...
joindm 18330 Domain of join function fo...
joindef 18331 Two ways to say that a joi...
joinval 18332 Join value. Since both si...
joincl 18333 Closure of join of element...
joindmss 18334 Subset property of domain ...
joinval2lem 18335 Lemma for ~ joinval2 and ~...
joinval2 18336 Value of join for a poset ...
joineu 18337 Uniqueness of join of elem...
joinlem 18338 Lemma for join properties....
lejoin1 18339 A join's first argument is...
lejoin2 18340 A join's second argument i...
joinle 18341 A join is less than or equ...
meetfval 18342 Value of meet function for...
meetfval2 18343 Value of meet function for...
meetdm 18344 Domain of meet function fo...
meetdef 18345 Two ways to say that a mee...
meetval 18346 Meet value. Since both si...
meetcl 18347 Closure of meet of element...
meetdmss 18348 Subset property of domain ...
meetval2lem 18349 Lemma for ~ meetval2 and ~...
meetval2 18350 Value of meet for a poset ...
meeteu 18351 Uniqueness of meet of elem...
meetlem 18352 Lemma for meet properties....
lemeet1 18353 A meet's first argument is...
lemeet2 18354 A meet's second argument i...
meetle 18355 A meet is less than or equ...
joincomALT 18356 The join of a poset is com...
joincom 18357 The join of a poset is com...
meetcomALT 18358 The meet of a poset is com...
meetcom 18359 The meet of a poset is com...
join0 18360 Lemma for ~ odumeet . (Co...
meet0 18361 Lemma for ~ odujoin . (Co...
odulub 18362 Least upper bounds in a du...
odujoin 18363 Joins in a dual order are ...
oduglb 18364 Greatest lower bounds in a...
odumeet 18365 Meets in a dual order are ...
poslubmo 18366 Least upper bounds in a po...
posglbmo 18367 Greatest lower bounds in a...
poslubd 18368 Properties which determine...
poslubdg 18369 Properties which determine...
posglbdg 18370 Properties which determine...
istos 18373 The predicate "is a toset"...
tosso 18374 Write the totally ordered ...
tospos 18375 A Toset is a Poset. (Cont...
tleile 18376 In a Toset, any two elemen...
tltnle 18377 In a Toset, "less than" is...
p0val 18382 Value of poset zero. (Con...
p1val 18383 Value of poset zero. (Con...
p0le 18384 Any element is less than o...
ple1 18385 Any element is less than o...
islat 18388 The predicate "is a lattic...
odulatb 18389 Being a lattice is self-du...
odulat 18390 Being a lattice is self-du...
latcl2 18391 The join and meet of any t...
latlem 18392 Lemma for lattice properti...
latpos 18393 A lattice is a poset. (Co...
latjcl 18394 Closure of join operation ...
latmcl 18395 Closure of meet operation ...
latref 18396 A lattice ordering is refl...
latasymb 18397 A lattice ordering is asym...
latasym 18398 A lattice ordering is asym...
lattr 18399 A lattice ordering is tran...
latasymd 18400 Deduce equality from latti...
lattrd 18401 A lattice ordering is tran...
latjcom 18402 The join of a lattice comm...
latlej1 18403 A join's first argument is...
latlej2 18404 A join's second argument i...
latjle12 18405 A join is less than or equ...
latleeqj1 18406 "Less than or equal to" in...
latleeqj2 18407 "Less than or equal to" in...
latjlej1 18408 Add join to both sides of ...
latjlej2 18409 Add join to both sides of ...
latjlej12 18410 Add join to both sides of ...
latnlej 18411 An idiom to express that a...
latnlej1l 18412 An idiom to express that a...
latnlej1r 18413 An idiom to express that a...
latnlej2 18414 An idiom to express that a...
latnlej2l 18415 An idiom to express that a...
latnlej2r 18416 An idiom to express that a...
latjidm 18417 Lattice join is idempotent...
latmcom 18418 The join of a lattice comm...
latmle1 18419 A meet is less than or equ...
latmle2 18420 A meet is less than or equ...
latlem12 18421 An element is less than or...
latleeqm1 18422 "Less than or equal to" in...
latleeqm2 18423 "Less than or equal to" in...
latmlem1 18424 Add meet to both sides of ...
latmlem2 18425 Add meet to both sides of ...
latmlem12 18426 Add join to both sides of ...
latnlemlt 18427 Negation of "less than or ...
latnle 18428 Equivalent expressions for...
latmidm 18429 Lattice meet is idempotent...
latabs1 18430 Lattice absorption law. F...
latabs2 18431 Lattice absorption law. F...
latledi 18432 An ortholattice is distrib...
latmlej11 18433 Ordering of a meet and joi...
latmlej12 18434 Ordering of a meet and joi...
latmlej21 18435 Ordering of a meet and joi...
latmlej22 18436 Ordering of a meet and joi...
lubsn 18437 The least upper bound of a...
latjass 18438 Lattice join is associativ...
latj12 18439 Swap 1st and 2nd members o...
latj32 18440 Swap 2nd and 3rd members o...
latj13 18441 Swap 1st and 3rd members o...
latj31 18442 Swap 2nd and 3rd members o...
latjrot 18443 Rotate lattice join of 3 c...
latj4 18444 Rearrangement of lattice j...
latj4rot 18445 Rotate lattice join of 4 c...
latjjdi 18446 Lattice join distributes o...
latjjdir 18447 Lattice join distributes o...
mod1ile 18448 The weak direction of the ...
mod2ile 18449 The weak direction of the ...
latmass 18450 Lattice meet is associativ...
latdisdlem 18451 Lemma for ~ latdisd . (Co...
latdisd 18452 In a lattice, joins distri...
isclat 18455 The predicate "is a comple...
clatpos 18456 A complete lattice is a po...
clatlem 18457 Lemma for properties of a ...
clatlubcl 18458 Any subset of the base set...
clatlubcl2 18459 Any subset of the base set...
clatglbcl 18460 Any subset of the base set...
clatglbcl2 18461 Any subset of the base set...
oduclatb 18462 Being a complete lattice i...
clatl 18463 A complete lattice is a la...
isglbd 18464 Properties that determine ...
lublem 18465 Lemma for the least upper ...
lubub 18466 The LUB of a complete latt...
lubl 18467 The LUB of a complete latt...
lubss 18468 Subset law for least upper...
lubel 18469 An element of a set is les...
lubun 18470 The LUB of a union. (Cont...
clatglb 18471 Properties of greatest low...
clatglble 18472 The greatest lower bound i...
clatleglb 18473 Two ways of expressing "le...
clatglbss 18474 Subset law for greatest lo...
isdlat 18477 Property of being a distri...
dlatmjdi 18478 In a distributive lattice,...
dlatl 18479 A distributive lattice is ...
odudlatb 18480 The dual of a distributive...
dlatjmdi 18481 In a distributive lattice,...
ipostr 18484 The structure of ~ df-ipo ...
ipoval 18485 Value of the inclusion pos...
ipobas 18486 Base set of the inclusion ...
ipolerval 18487 Relation of the inclusion ...
ipotset 18488 Topology of the inclusion ...
ipole 18489 Weak order condition of th...
ipolt 18490 Strict order condition of ...
ipopos 18491 The inclusion poset on a f...
isipodrs 18492 Condition for a family of ...
ipodrscl 18493 Direction by inclusion as ...
ipodrsfi 18494 Finite upper bound propert...
fpwipodrs 18495 The finite subsets of any ...
ipodrsima 18496 The monotone image of a di...
isacs3lem 18497 An algebraic closure syste...
acsdrsel 18498 An algebraic closure syste...
isacs4lem 18499 In a closure system in whi...
isacs5lem 18500 If closure commutes with d...
acsdrscl 18501 In an algebraic closure sy...
acsficl 18502 A closure in an algebraic ...
isacs5 18503 A closure system is algebr...
isacs4 18504 A closure system is algebr...
isacs3 18505 A closure system is algebr...
acsficld 18506 In an algebraic closure sy...
acsficl2d 18507 In an algebraic closure sy...
acsfiindd 18508 In an algebraic closure sy...
acsmapd 18509 In an algebraic closure sy...
acsmap2d 18510 In an algebraic closure sy...
acsinfd 18511 In an algebraic closure sy...
acsdomd 18512 In an algebraic closure sy...
acsinfdimd 18513 In an algebraic closure sy...
acsexdimd 18514 In an algebraic closure sy...
mrelatglb 18515 Greatest lower bounds in a...
mrelatglb0 18516 The empty intersection in ...
mrelatlub 18517 Least upper bounds in a Mo...
mreclatBAD 18518 A Moore space is a complet...
isps 18523 The predicate "is a poset"...
psrel 18524 A poset is a relation. (C...
psref2 18525 A poset is antisymmetric a...
pstr2 18526 A poset is transitive. (C...
pslem 18527 Lemma for ~ psref and othe...
psdmrn 18528 The domain and range of a ...
psref 18529 A poset is reflexive. (Co...
psrn 18530 The range of a poset equal...
psasym 18531 A poset is antisymmetric. ...
pstr 18532 A poset is transitive. (C...
cnvps 18533 The converse of a poset is...
cnvpsb 18534 The converse of a poset is...
psss 18535 Any subset of a partially ...
psssdm2 18536 Field of a subposet. (Con...
psssdm 18537 Field of a subposet. (Con...
istsr 18538 The predicate is a toset. ...
istsr2 18539 The predicate is a toset. ...
tsrlin 18540 A toset is a linear order....
tsrlemax 18541 Two ways of saying a numbe...
tsrps 18542 A toset is a poset. (Cont...
cnvtsr 18543 The converse of a toset is...
tsrss 18544 Any subset of a totally or...
ledm 18545 The domain of ` <_ ` is ` ...
lern 18546 The range of ` <_ ` is ` R...
lefld 18547 The field of the 'less or ...
letsr 18548 The "less than or equal to...
isdir 18553 A condition for a relation...
reldir 18554 A direction is a relation....
dirdm 18555 A direction's domain is eq...
dirref 18556 A direction is reflexive. ...
dirtr 18557 A direction is transitive....
dirge 18558 For any two elements of a ...
tsrdir 18559 A totally ordered set is a...
ismgm 18564 The predicate "is a magma"...
ismgmn0 18565 The predicate "is a magma"...
mgmcl 18566 Closure of the operation o...
isnmgm 18567 A condition for a structur...
mgmsscl 18568 If the base set of a magma...
plusffval 18569 The group addition operati...
plusfval 18570 The group addition operati...
plusfeq 18571 If the addition operation ...
plusffn 18572 The group addition operati...
mgmplusf 18573 The group addition functio...
mgmpropd 18574 If two structures have the...
ismgmd 18575 Deduce a magma from its pr...
issstrmgm 18576 Characterize a substructur...
intopsn 18577 The internal operation for...
mgmb1mgm1 18578 The only magma with a base...
mgm0 18579 Any set with an empty base...
mgm0b 18580 The structure with an empt...
mgm1 18581 The structure with one ele...
opifismgm 18582 A structure with a group a...
mgmidmo 18583 A two-sided identity eleme...
grpidval 18584 The value of the identity ...
grpidpropd 18585 If two structures have the...
fn0g 18586 The group zero extractor i...
0g0 18587 The identity element funct...
ismgmid 18588 The identity element of a ...
mgmidcl 18589 The identity element of a ...
mgmlrid 18590 The identity element of a ...
ismgmid2 18591 Show that a given element ...
lidrideqd 18592 If there is a left and rig...
lidrididd 18593 If there is a left and rig...
grpidd 18594 Deduce the identity elemen...
mgmidsssn0 18595 Property of the set of ide...
grprinvlem 18596 Lemma for ~ grpinva . (Co...
grpinva 18597 Deduce right inverse from ...
grprida 18598 Deduce right identity from...
gsumvalx 18599 Expand out the substitutio...
gsumval 18600 Expand out the substitutio...
gsumpropd 18601 The group sum depends only...
gsumpropd2lem 18602 Lemma for ~ gsumpropd2 . ...
gsumpropd2 18603 A stronger version of ~ gs...
gsummgmpropd 18604 A stronger version of ~ gs...
gsumress 18605 The group sum in a substru...
gsumval1 18606 Value of the group sum ope...
gsum0 18607 Value of the empty group s...
gsumval2a 18608 Value of the group sum ope...
gsumval2 18609 Value of the group sum ope...
gsumsplit1r 18610 Splitting off the rightmos...
gsumprval 18611 Value of the group sum ope...
gsumpr12val 18612 Value of the group sum ope...
mgmhmrcl 18617 Reverse closure of a magma...
submgmrcl 18618 Reverse closure for submag...
ismgmhm 18619 Property of a magma homomo...
mgmhmf 18620 A magma homomorphism is a ...
mgmhmpropd 18621 Magma homomorphism depends...
mgmhmlin 18622 A magma homomorphism prese...
mgmhmf1o 18623 A magma homomorphism is bi...
idmgmhm 18624 The identity homomorphism ...
issubmgm 18625 Expand definition of a sub...
issubmgm2 18626 Submagmas are subsets that...
rabsubmgmd 18627 Deduction for proving that...
submgmss 18628 Submagmas are subsets of t...
submgmid 18629 Every magma is trivially a...
submgmcl 18630 Submagmas are closed under...
submgmmgm 18631 Submagmas are themselves m...
submgmbas 18632 The base set of a submagma...
subsubmgm 18633 A submagma of a submagma i...
resmgmhm 18634 Restriction of a magma hom...
resmgmhm2 18635 One direction of ~ resmgmh...
resmgmhm2b 18636 Restriction of the codomai...
mgmhmco 18637 The composition of magma h...
mgmhmima 18638 The homomorphic image of a...
mgmhmeql 18639 The equalizer of two magma...
submgmacs 18640 Submagmas are an algebraic...
issgrp 18643 The predicate "is a semigr...
issgrpv 18644 The predicate "is a semigr...
issgrpn0 18645 The predicate "is a semigr...
isnsgrp 18646 A condition for a structur...
sgrpmgm 18647 A semigroup is a magma. (...
sgrpass 18648 A semigroup operation is a...
sgrpcl 18649 Closure of the operation o...
sgrp0 18650 Any set with an empty base...
sgrp0b 18651 The structure with an empt...
sgrp1 18652 The structure with one ele...
issgrpd 18653 Deduce a semigroup from it...
sgrppropd 18654 If two structures are sets...
prdsplusgsgrpcl 18655 Structure product pointwis...
prdssgrpd 18656 The product of a family of...
ismnddef 18659 The predicate "is a monoid...
ismnd 18660 The predicate "is a monoid...
isnmnd 18661 A condition for a structur...
sgrpidmnd 18662 A semigroup with an identi...
mndsgrp 18663 A monoid is a semigroup. ...
mndmgm 18664 A monoid is a magma. (Con...
mndcl 18665 Closure of the operation o...
mndass 18666 A monoid operation is asso...
mndid 18667 A monoid has a two-sided i...
mndideu 18668 The two-sided identity ele...
mnd32g 18669 Commutative/associative la...
mnd12g 18670 Commutative/associative la...
mnd4g 18671 Commutative/associative la...
mndidcl 18672 The identity element of a ...
mndbn0 18673 The base set of a monoid i...
hashfinmndnn 18674 A finite monoid has positi...
mndplusf 18675 The group addition operati...
mndlrid 18676 A monoid's identity elemen...
mndlid 18677 The identity element of a ...
mndrid 18678 The identity element of a ...
ismndd 18679 Deduce a monoid from its p...
mndpfo 18680 The addition operation of ...
mndfo 18681 The addition operation of ...
mndpropd 18682 If two structures have the...
mndprop 18683 If two structures have the...
issubmnd 18684 Characterize a submonoid b...
ress0g 18685 ` 0g ` is unaffected by re...
submnd0 18686 The zero of a submonoid is...
mndinvmod 18687 Uniqueness of an inverse e...
prdsplusgcl 18688 Structure product pointwis...
prdsidlem 18689 Characterization of identi...
prdsmndd 18690 The product of a family of...
prds0g 18691 Zero in a product of monoi...
pwsmnd 18692 The structure power of a m...
pws0g 18693 Zero in a structure power ...
imasmnd2 18694 The image structure of a m...
imasmnd 18695 The image structure of a m...
imasmndf1 18696 The image of a monoid unde...
xpsmnd 18697 The binary product of mono...
xpsmnd0 18698 The identity element of a ...
mnd1 18699 The (smallest) structure r...
mnd1id 18700 The singleton element of a...
ismhm 18705 Property of a monoid homom...
ismhmd 18706 Deduction version of ~ ism...
mhmrcl1 18707 Reverse closure of a monoi...
mhmrcl2 18708 Reverse closure of a monoi...
mhmf 18709 A monoid homomorphism is a...
ismhm0 18710 Property of a monoid homom...
mhmismgmhm 18711 Each monoid homomorphism i...
mhmpropd 18712 Monoid homomorphism depend...
mhmlin 18713 A monoid homomorphism comm...
mhm0 18714 A monoid homomorphism pres...
idmhm 18715 The identity homomorphism ...
mhmf1o 18716 A monoid homomorphism is b...
submrcl 18717 Reverse closure for submon...
issubm 18718 Expand definition of a sub...
issubm2 18719 Submonoids are subsets tha...
issubmndb 18720 The submonoid predicate. ...
issubmd 18721 Deduction for proving a su...
mndissubm 18722 If the base set of a monoi...
resmndismnd 18723 If the base set of a monoi...
submss 18724 Submonoids are subsets of ...
submid 18725 Every monoid is trivially ...
subm0cl 18726 Submonoids contain zero. ...
submcl 18727 Submonoids are closed unde...
submmnd 18728 Submonoids are themselves ...
submbas 18729 The base set of a submonoi...
subm0 18730 Submonoids have the same i...
subsubm 18731 A submonoid of a submonoid...
0subm 18732 The zero submonoid of an a...
insubm 18733 The intersection of two su...
0mhm 18734 The constant zero linear f...
resmhm 18735 Restriction of a monoid ho...
resmhm2 18736 One direction of ~ resmhm2...
resmhm2b 18737 Restriction of the codomai...
mhmco 18738 The composition of monoid ...
mhmimalem 18739 Lemma for ~ mhmima and sim...
mhmima 18740 The homomorphic image of a...
mhmeql 18741 The equalizer of two monoi...
submacs 18742 Submonoids are an algebrai...
mndind 18743 Induction in a monoid. In...
prdspjmhm 18744 A projection from a produc...
pwspjmhm 18745 A projection from a struct...
pwsdiagmhm 18746 Diagonal monoid homomorphi...
pwsco1mhm 18747 Right composition with a f...
pwsco2mhm 18748 Left composition with a mo...
gsumvallem2 18749 Lemma for properties of th...
gsumsubm 18750 Evaluate a group sum in a ...
gsumz 18751 Value of a group sum over ...
gsumwsubmcl 18752 Closure of the composite i...
gsumws1 18753 A singleton composite reco...
gsumwcl 18754 Closure of the composite o...
gsumsgrpccat 18755 Homomorphic property of no...
gsumccat 18756 Homomorphic property of co...
gsumws2 18757 Valuation of a pair in a m...
gsumccatsn 18758 Homomorphic property of co...
gsumspl 18759 The primary purpose of the...
gsumwmhm 18760 Behavior of homomorphisms ...
gsumwspan 18761 The submonoid generated by...
frmdval 18766 Value of the free monoid c...
frmdbas 18767 The base set of a free mon...
frmdelbas 18768 An element of the base set...
frmdplusg 18769 The monoid operation of a ...
frmdadd 18770 Value of the monoid operat...
vrmdfval 18771 The canonical injection fr...
vrmdval 18772 The value of the generatin...
vrmdf 18773 The mapping from the index...
frmdmnd 18774 A free monoid is a monoid....
frmd0 18775 The identity of the free m...
frmdsssubm 18776 The set of words taking va...
frmdgsum 18777 Any word in a free monoid ...
frmdss2 18778 A subset of generators is ...
frmdup1 18779 Any assignment of the gene...
frmdup2 18780 The evaluation map has the...
frmdup3lem 18781 Lemma for ~ frmdup3 . (Co...
frmdup3 18782 Universal property of the ...
efmnd 18785 The monoid of endofunction...
efmndbas 18786 The base set of the monoid...
efmndbasabf 18787 The base set of the monoid...
elefmndbas 18788 Two ways of saying a funct...
elefmndbas2 18789 Two ways of saying a funct...
efmndbasf 18790 Elements in the monoid of ...
efmndhash 18791 The monoid of endofunction...
efmndbasfi 18792 The monoid of endofunction...
efmndfv 18793 The function value of an e...
efmndtset 18794 The topology of the monoid...
efmndplusg 18795 The group operation of a m...
efmndov 18796 The value of the group ope...
efmndcl 18797 The group operation of the...
efmndtopn 18798 The topology of the monoid...
symggrplem 18799 Lemma for ~ symggrp and ~ ...
efmndmgm 18800 The monoid of endofunction...
efmndsgrp 18801 The monoid of endofunction...
ielefmnd 18802 The identity function rest...
efmndid 18803 The identity function rest...
efmndmnd 18804 The monoid of endofunction...
efmnd0nmnd 18805 Even the monoid of endofun...
efmndbas0 18806 The base set of the monoid...
efmnd1hash 18807 The monoid of endofunction...
efmnd1bas 18808 The monoid of endofunction...
efmnd2hash 18809 The monoid of endofunction...
submefmnd 18810 If the base set of a monoi...
sursubmefmnd 18811 The set of surjective endo...
injsubmefmnd 18812 The set of injective endof...
idressubmefmnd 18813 The singleton containing o...
idresefmnd 18814 The structure with the sin...
smndex1ibas 18815 The modulo function ` I ` ...
smndex1iidm 18816 The modulo function ` I ` ...
smndex1gbas 18817 The constant functions ` (...
smndex1gid 18818 The composition of a const...
smndex1igid 18819 The composition of the mod...
smndex1basss 18820 The modulo function ` I ` ...
smndex1bas 18821 The base set of the monoid...
smndex1mgm 18822 The monoid of endofunction...
smndex1sgrp 18823 The monoid of endofunction...
smndex1mndlem 18824 Lemma for ~ smndex1mnd and...
smndex1mnd 18825 The monoid of endofunction...
smndex1id 18826 The modulo function ` I ` ...
smndex1n0mnd 18827 The identity of the monoid...
nsmndex1 18828 The base set ` B ` of the ...
smndex2dbas 18829 The doubling function ` D ...
smndex2dnrinv 18830 The doubling function ` D ...
smndex2hbas 18831 The halving functions ` H ...
smndex2dlinvh 18832 The halving functions ` H ...
mgm2nsgrplem1 18833 Lemma 1 for ~ mgm2nsgrp : ...
mgm2nsgrplem2 18834 Lemma 2 for ~ mgm2nsgrp . ...
mgm2nsgrplem3 18835 Lemma 3 for ~ mgm2nsgrp . ...
mgm2nsgrplem4 18836 Lemma 4 for ~ mgm2nsgrp : ...
mgm2nsgrp 18837 A small magma (with two el...
sgrp2nmndlem1 18838 Lemma 1 for ~ sgrp2nmnd : ...
sgrp2nmndlem2 18839 Lemma 2 for ~ sgrp2nmnd . ...
sgrp2nmndlem3 18840 Lemma 3 for ~ sgrp2nmnd . ...
sgrp2rid2 18841 A small semigroup (with tw...
sgrp2rid2ex 18842 A small semigroup (with tw...
sgrp2nmndlem4 18843 Lemma 4 for ~ sgrp2nmnd : ...
sgrp2nmndlem5 18844 Lemma 5 for ~ sgrp2nmnd : ...
sgrp2nmnd 18845 A small semigroup (with tw...
mgmnsgrpex 18846 There is a magma which is ...
sgrpnmndex 18847 There is a semigroup which...
sgrpssmgm 18848 The class of all semigroup...
mndsssgrp 18849 The class of all monoids i...
pwmndgplus 18850 The operation of the monoi...
pwmndid 18851 The identity of the monoid...
pwmnd 18852 The power set of a class `...
isgrp 18859 The predicate "is a group"...
grpmnd 18860 A group is a monoid. (Con...
grpcl 18861 Closure of the operation o...
grpass 18862 A group operation is assoc...
grpinvex 18863 Every member of a group ha...
grpideu 18864 The two-sided identity ele...
grpassd 18865 A group operation is assoc...
grpmndd 18866 A group is a monoid. (Con...
grpcld 18867 Closure of the operation o...
grpplusf 18868 The group addition operati...
grpplusfo 18869 The group addition operati...
resgrpplusfrn 18870 The underlying set of a gr...
grppropd 18871 If two structures have the...
grpprop 18872 If two structures have the...
grppropstr 18873 Generalize a specific 2-el...
grpss 18874 Show that a structure exte...
isgrpd2e 18875 Deduce a group from its pr...
isgrpd2 18876 Deduce a group from its pr...
isgrpde 18877 Deduce a group from its pr...
isgrpd 18878 Deduce a group from its pr...
isgrpi 18879 Properties that determine ...
grpsgrp 18880 A group is a semigroup. (...
grpmgmd 18881 A group is a magma, deduct...
dfgrp2 18882 Alternate definition of a ...
dfgrp2e 18883 Alternate definition of a ...
isgrpix 18884 Properties that determine ...
grpidcl 18885 The identity element of a ...
grpbn0 18886 The base set of a group is...
grplid 18887 The identity element of a ...
grprid 18888 The identity element of a ...
grplidd 18889 The identity element of a ...
grpridd 18890 The identity element of a ...
grpn0 18891 A group is not empty. (Co...
hashfingrpnn 18892 A finite group has positiv...
grprcan 18893 Right cancellation law for...
grpinveu 18894 The left inverse element o...
grpid 18895 Two ways of saying that an...
isgrpid2 18896 Properties showing that an...
grpidd2 18897 Deduce the identity elemen...
grpinvfval 18898 The inverse function of a ...
grpinvfvalALT 18899 Shorter proof of ~ grpinvf...
grpinvval 18900 The inverse of a group ele...
grpinvfn 18901 Functionality of the group...
grpinvfvi 18902 The group inverse function...
grpsubfval 18903 Group subtraction (divisio...
grpsubfvalALT 18904 Shorter proof of ~ grpsubf...
grpsubval 18905 Group subtraction (divisio...
grpinvf 18906 The group inversion operat...
grpinvcl 18907 A group element's inverse ...
grpinvcld 18908 A group element's inverse ...
grplinv 18909 The left inverse of a grou...
grprinv 18910 The right inverse of a gro...
grpinvid1 18911 The inverse of a group ele...
grpinvid2 18912 The inverse of a group ele...
isgrpinv 18913 Properties showing that a ...
grplinvd 18914 The left inverse of a grou...
grprinvd 18915 The right inverse of a gro...
grplrinv 18916 In a group, every member h...
grpidinv2 18917 A group's properties using...
grpidinv 18918 A group has a left and rig...
grpinvid 18919 The inverse of the identit...
grplcan 18920 Left cancellation law for ...
grpasscan1 18921 An associative cancellatio...
grpasscan2 18922 An associative cancellatio...
grpidrcan 18923 If right adding an element...
grpidlcan 18924 If left adding an element ...
grpinvinv 18925 Double inverse law for gro...
grpinvcnv 18926 The group inverse is its o...
grpinv11 18927 The group inverse is one-t...
grpinvf1o 18928 The group inverse is a one...
grpinvnz 18929 The inverse of a nonzero g...
grpinvnzcl 18930 The inverse of a nonzero g...
grpsubinv 18931 Subtraction of an inverse....
grplmulf1o 18932 Left multiplication by a g...
grpinvpropd 18933 If two structures have the...
grpidssd 18934 If the base set of a group...
grpinvssd 18935 If the base set of a group...
grpinvadd 18936 The inverse of the group o...
grpsubf 18937 Functionality of group sub...
grpsubcl 18938 Closure of group subtracti...
grpsubrcan 18939 Right cancellation law for...
grpinvsub 18940 Inverse of a group subtrac...
grpinvval2 18941 A ~ df-neg -like equation ...
grpsubid 18942 Subtraction of a group ele...
grpsubid1 18943 Subtraction of the identit...
grpsubeq0 18944 If the difference between ...
grpsubadd0sub 18945 Subtraction expressed as a...
grpsubadd 18946 Relationship between group...
grpsubsub 18947 Double group subtraction. ...
grpaddsubass 18948 Associative-type law for g...
grppncan 18949 Cancellation law for subtr...
grpnpcan 18950 Cancellation law for subtr...
grpsubsub4 18951 Double group subtraction (...
grppnpcan2 18952 Cancellation law for mixed...
grpnpncan 18953 Cancellation law for group...
grpnpncan0 18954 Cancellation law for group...
grpnnncan2 18955 Cancellation law for group...
dfgrp3lem 18956 Lemma for ~ dfgrp3 . (Con...
dfgrp3 18957 Alternate definition of a ...
dfgrp3e 18958 Alternate definition of a ...
grplactfval 18959 The left group action of e...
grplactval 18960 The value of the left grou...
grplactcnv 18961 The left group action of e...
grplactf1o 18962 The left group action of e...
grpsubpropd 18963 Weak property deduction fo...
grpsubpropd2 18964 Strong property deduction ...
grp1 18965 The (smallest) structure r...
grp1inv 18966 The inverse function of th...
prdsinvlem 18967 Characterization of invers...
prdsgrpd 18968 The product of a family of...
prdsinvgd 18969 Negation in a product of g...
pwsgrp 18970 A structure power of a gro...
pwsinvg 18971 Negation in a group power....
pwssub 18972 Subtraction in a group pow...
imasgrp2 18973 The image structure of a g...
imasgrp 18974 The image structure of a g...
imasgrpf1 18975 The image of a group under...
qusgrp2 18976 Prove that a quotient stru...
xpsgrp 18977 The binary product of grou...
xpsinv 18978 Value of the negation oper...
xpsgrpsub 18979 Value of the subtraction o...
mhmlem 18980 Lemma for ~ mhmmnd and ~ g...
mhmid 18981 A surjective monoid morphi...
mhmmnd 18982 The image of a monoid ` G ...
mhmfmhm 18983 The function fulfilling th...
ghmgrp 18984 The image of a group ` G `...
mulgfval 18987 Group multiple (exponentia...
mulgfvalALT 18988 Shorter proof of ~ mulgfva...
mulgval 18989 Value of the group multipl...
mulgfn 18990 Functionality of the group...
mulgfvi 18991 The group multiple operati...
mulg0 18992 Group multiple (exponentia...
mulgnn 18993 Group multiple (exponentia...
ressmulgnn 18994 Values for the group multi...
ressmulgnn0 18995 Values for the group multi...
mulgnngsum 18996 Group multiple (exponentia...
mulgnn0gsum 18997 Group multiple (exponentia...
mulg1 18998 Group multiple (exponentia...
mulgnnp1 18999 Group multiple (exponentia...
mulg2 19000 Group multiple (exponentia...
mulgnegnn 19001 Group multiple (exponentia...
mulgnn0p1 19002 Group multiple (exponentia...
mulgnnsubcl 19003 Closure of the group multi...
mulgnn0subcl 19004 Closure of the group multi...
mulgsubcl 19005 Closure of the group multi...
mulgnncl 19006 Closure of the group multi...
mulgnn0cl 19007 Closure of the group multi...
mulgcl 19008 Closure of the group multi...
mulgneg 19009 Group multiple (exponentia...
mulgnegneg 19010 The inverse of a negative ...
mulgm1 19011 Group multiple (exponentia...
mulgnn0cld 19012 Closure of the group multi...
mulgcld 19013 Deduction associated with ...
mulgaddcomlem 19014 Lemma for ~ mulgaddcom . ...
mulgaddcom 19015 The group multiple operato...
mulginvcom 19016 The group multiple operato...
mulginvinv 19017 The group multiple operato...
mulgnn0z 19018 A group multiple of the id...
mulgz 19019 A group multiple of the id...
mulgnndir 19020 Sum of group multiples, fo...
mulgnn0dir 19021 Sum of group multiples, ge...
mulgdirlem 19022 Lemma for ~ mulgdir . (Co...
mulgdir 19023 Sum of group multiples, ge...
mulgp1 19024 Group multiple (exponentia...
mulgneg2 19025 Group multiple (exponentia...
mulgnnass 19026 Product of group multiples...
mulgnn0ass 19027 Product of group multiples...
mulgass 19028 Product of group multiples...
mulgassr 19029 Reversed product of group ...
mulgmodid 19030 Casting out multiples of t...
mulgsubdir 19031 Distribution of group mult...
mhmmulg 19032 A homomorphism of monoids ...
mulgpropd 19033 Two structures with the sa...
submmulgcl 19034 Closure of the group multi...
submmulg 19035 A group multiple is the sa...
pwsmulg 19036 Value of a group multiple ...
issubg 19043 The subgroup predicate. (...
subgss 19044 A subgroup is a subset. (...
subgid 19045 A group is a subgroup of i...
subggrp 19046 A subgroup is a group. (C...
subgbas 19047 The base of the restricted...
subgrcl 19048 Reverse closure for the su...
subg0 19049 A subgroup of a group must...
subginv 19050 The inverse of an element ...
subg0cl 19051 The group identity is an e...
subginvcl 19052 The inverse of an element ...
subgcl 19053 A subgroup is closed under...
subgsubcl 19054 A subgroup is closed under...
subgsub 19055 The subtraction of element...
subgmulgcl 19056 Closure of the group multi...
subgmulg 19057 A group multiple is the sa...
issubg2 19058 Characterize the subgroups...
issubgrpd2 19059 Prove a subgroup by closur...
issubgrpd 19060 Prove a subgroup by closur...
issubg3 19061 A subgroup is a symmetric ...
issubg4 19062 A subgroup is a nonempty s...
grpissubg 19063 If the base set of a group...
resgrpisgrp 19064 If the base set of a group...
subgsubm 19065 A subgroup is a submonoid....
subsubg 19066 A subgroup of a subgroup i...
subgint 19067 The intersection of a none...
0subg 19068 The zero subgroup of an ar...
0subgOLD 19069 Obsolete version of ~ 0sub...
trivsubgd 19070 The only subgroup of a tri...
trivsubgsnd 19071 The only subgroup of a tri...
isnsg 19072 Property of being a normal...
isnsg2 19073 Weaken the condition of ~ ...
nsgbi 19074 Defining property of a nor...
nsgsubg 19075 A normal subgroup is a sub...
nsgconj 19076 The conjugation of an elem...
isnsg3 19077 A subgroup is normal iff t...
subgacs 19078 Subgroups are an algebraic...
nsgacs 19079 Normal subgroups form an a...
elnmz 19080 Elementhood in the normali...
nmzbi 19081 Defining property of the n...
nmzsubg 19082 The normalizer N_G(S) of a...
ssnmz 19083 A subgroup is a subset of ...
isnsg4 19084 A subgroup is normal iff i...
nmznsg 19085 Any subgroup is a normal s...
0nsg 19086 The zero subgroup is norma...
nsgid 19087 The whole group is a norma...
0idnsgd 19088 The whole group and the ze...
trivnsgd 19089 The only normal subgroup o...
triv1nsgd 19090 A trivial group has exactl...
1nsgtrivd 19091 A group with exactly one n...
releqg 19092 The left coset equivalence...
eqgfval 19093 Value of the subgroup left...
eqgval 19094 Value of the subgroup left...
eqger 19095 The subgroup coset equival...
eqglact 19096 A left coset can be expres...
eqgid 19097 The left coset containing ...
eqgen 19098 Each coset is equipotent t...
eqgcpbl 19099 The subgroup coset equival...
quselbas 19100 Membership in the base set...
quseccl0 19101 Closure of the quotient ma...
qusgrp 19102 If ` Y ` is a normal subgr...
quseccl 19103 Closure of the quotient ma...
qusadd 19104 Value of the group operati...
qus0 19105 Value of the group identit...
qusinv 19106 Value of the group inverse...
qussub 19107 Value of the group subtrac...
ecqusaddd 19108 Addition of equivalence cl...
ecqusaddcl 19109 Closure of the addition in...
lagsubg2 19110 Lagrange's theorem for fin...
lagsubg 19111 Lagrange's theorem for Gro...
eqg0subg 19112 The coset equivalence rela...
eqg0subgecsn 19113 The equivalence classes mo...
qus0subgbas 19114 The base set of a quotient...
qus0subgadd 19115 The addition in a quotient...
cycsubmel 19116 Characterization of an ele...
cycsubmcl 19117 The set of nonnegative int...
cycsubm 19118 The set of nonnegative int...
cyccom 19119 Condition for an operation...
cycsubmcom 19120 The operation of a monoid ...
cycsubggend 19121 The cyclic subgroup genera...
cycsubgcl 19122 The set of integer powers ...
cycsubgss 19123 The cyclic subgroup genera...
cycsubg 19124 The cyclic group generated...
cycsubgcld 19125 The cyclic subgroup genera...
cycsubg2 19126 The subgroup generated by ...
cycsubg2cl 19127 Any multiple of an element...
reldmghm 19130 Lemma for group homomorphi...
isghm 19131 Property of being a homomo...
isghm3 19132 Property of a group homomo...
ghmgrp1 19133 A group homomorphism is on...
ghmgrp2 19134 A group homomorphism is on...
ghmf 19135 A group homomorphism is a ...
ghmlin 19136 A homomorphism of groups i...
ghmid 19137 A homomorphism of groups p...
ghminv 19138 A homomorphism of groups p...
ghmsub 19139 Linearity of subtraction t...
isghmd 19140 Deduction for a group homo...
ghmmhm 19141 A group homomorphism is a ...
ghmmhmb 19142 Group homomorphisms and mo...
ghmmulg 19143 A homomorphism of monoids ...
ghmrn 19144 The range of a homomorphis...
0ghm 19145 The constant zero linear f...
idghm 19146 The identity homomorphism ...
resghm 19147 Restriction of a homomorph...
resghm2 19148 One direction of ~ resghm2...
resghm2b 19149 Restriction of the codomai...
ghmghmrn 19150 A group homomorphism from ...
ghmco 19151 The composition of group h...
ghmima 19152 The image of a subgroup un...
ghmpreima 19153 The inverse image of a sub...
ghmeql 19154 The equalizer of two group...
ghmnsgima 19155 The image of a normal subg...
ghmnsgpreima 19156 The inverse image of a nor...
ghmker 19157 The kernel of a homomorphi...
ghmeqker 19158 Two source points map to t...
pwsdiagghm 19159 Diagonal homomorphism into...
f1ghm0to0 19160 If a group homomorphism ` ...
ghmf1 19161 Two ways of saying a group...
kerf1ghm 19162 A group homomorphism ` F `...
ghmf1o 19163 A bijective group homomorp...
conjghm 19164 Conjugation is an automorp...
conjsubg 19165 A conjugated subgroup is a...
conjsubgen 19166 A conjugated subgroup is e...
conjnmz 19167 A subgroup is unchanged un...
conjnmzb 19168 Alternative condition for ...
conjnsg 19169 A normal subgroup is uncha...
qusghm 19170 If ` Y ` is a normal subgr...
ghmpropd 19171 Group homomorphism depends...
gimfn 19176 The group isomorphism func...
isgim 19177 An isomorphism of groups i...
gimf1o 19178 An isomorphism of groups i...
gimghm 19179 An isomorphism of groups i...
isgim2 19180 A group isomorphism is a h...
subggim 19181 Behavior of subgroups unde...
gimcnv 19182 The converse of a bijectiv...
gimco 19183 The composition of group i...
gim0to0 19184 A group isomorphism maps t...
brgic 19185 The relation "is isomorphi...
brgici 19186 Prove isomorphic by an exp...
gicref 19187 Isomorphism is reflexive. ...
giclcl 19188 Isomorphism implies the le...
gicrcl 19189 Isomorphism implies the ri...
gicsym 19190 Isomorphism is symmetric. ...
gictr 19191 Isomorphism is transitive....
gicer 19192 Isomorphism is an equivale...
gicen 19193 Isomorphic groups have equ...
gicsubgen 19194 A less trivial example of ...
isga 19197 The predicate "is a (left)...
gagrp 19198 The left argument of a gro...
gaset 19199 The right argument of a gr...
gagrpid 19200 The identity of the group ...
gaf 19201 The mapping of the group a...
gafo 19202 A group action is onto its...
gaass 19203 An "associative" property ...
ga0 19204 The action of a group on t...
gaid 19205 The trivial action of a gr...
subgga 19206 A subgroup acts on its par...
gass 19207 A subset of a group action...
gasubg 19208 The restriction of a group...
gaid2 19209 A group operation is a lef...
galcan 19210 The action of a particular...
gacan 19211 Group inverses cancel in a...
gapm 19212 The action of a particular...
gaorb 19213 The orbit equivalence rela...
gaorber 19214 The orbit equivalence rela...
gastacl 19215 The stabilizer subgroup in...
gastacos 19216 Write the coset relation f...
orbstafun 19217 Existence and uniqueness f...
orbstaval 19218 Value of the function at a...
orbsta 19219 The Orbit-Stabilizer theor...
orbsta2 19220 Relation between the size ...
cntrval 19225 Substitute definition of t...
cntzfval 19226 First level substitution f...
cntzval 19227 Definition substitution fo...
elcntz 19228 Elementhood in the central...
cntzel 19229 Membership in a centralize...
cntzsnval 19230 Special substitution for t...
elcntzsn 19231 Value of the centralizer o...
sscntz 19232 A centralizer expression f...
cntzrcl 19233 Reverse closure for elemen...
cntzssv 19234 The centralizer is uncondi...
cntzi 19235 Membership in a centralize...
elcntr 19236 Elementhood in the center ...
cntrss 19237 The center is a subset of ...
cntri 19238 Defining property of the c...
resscntz 19239 Centralizer in a substruct...
cntzsgrpcl 19240 Centralizers are closed un...
cntz2ss 19241 Centralizers reverse the s...
cntzrec 19242 Reciprocity relationship f...
cntziinsn 19243 Express any centralizer as...
cntzsubm 19244 Centralizers in a monoid a...
cntzsubg 19245 Centralizers in a group ar...
cntzidss 19246 If the elements of ` S ` c...
cntzmhm 19247 Centralizers in a monoid a...
cntzmhm2 19248 Centralizers in a monoid a...
cntrsubgnsg 19249 A central subgroup is norm...
cntrnsg 19250 The center of a group is a...
oppgval 19253 Value of the opposite grou...
oppgplusfval 19254 Value of the addition oper...
oppgplus 19255 Value of the addition oper...
setsplusg 19256 The other components of an...
oppglemOLD 19257 Obsolete version of ~ sets...
oppgbas 19258 Base set of an opposite gr...
oppgbasOLD 19259 Obsolete version of ~ oppg...
oppgtset 19260 Topology of an opposite gr...
oppgtsetOLD 19261 Obsolete version of ~ oppg...
oppgtopn 19262 Topology of an opposite gr...
oppgmnd 19263 The opposite of a monoid i...
oppgmndb 19264 Bidirectional form of ~ op...
oppgid 19265 Zero in a monoid is a symm...
oppggrp 19266 The opposite of a group is...
oppggrpb 19267 Bidirectional form of ~ op...
oppginv 19268 Inverses in a group are a ...
invoppggim 19269 The inverse is an antiauto...
oppggic 19270 Every group is (naturally)...
oppgsubm 19271 Being a submonoid is a sym...
oppgsubg 19272 Being a subgroup is a symm...
oppgcntz 19273 A centralizer in a group i...
oppgcntr 19274 The center of a group is t...
gsumwrev 19275 A sum in an opposite monoi...
symgval 19278 The value of the symmetric...
permsetexOLD 19279 Obsolete version of ~ f1os...
symgbas 19280 The base set of the symmet...
symgbasexOLD 19281 Obsolete as of 8-Aug-2024....
elsymgbas2 19282 Two ways of saying a funct...
elsymgbas 19283 Two ways of saying a funct...
symgbasf1o 19284 Elements in the symmetric ...
symgbasf 19285 A permutation (element of ...
symgbasmap 19286 A permutation (element of ...
symghash 19287 The symmetric group on ` n...
symgbasfi 19288 The symmetric group on a f...
symgfv 19289 The function value of a pe...
symgfvne 19290 The function values of a p...
symgressbas 19291 The symmetric group on ` A...
symgplusg 19292 The group operation of a s...
symgov 19293 The value of the group ope...
symgcl 19294 The group operation of the...
idresperm 19295 The identity function rest...
symgmov1 19296 For a permutation of a set...
symgmov2 19297 For a permutation of a set...
symgbas0 19298 The base set of the symmet...
symg1hash 19299 The symmetric group on a s...
symg1bas 19300 The symmetric group on a s...
symg2hash 19301 The symmetric group on a (...
symg2bas 19302 The symmetric group on a p...
0symgefmndeq 19303 The symmetric group on the...
snsymgefmndeq 19304 The symmetric group on a s...
symgpssefmnd 19305 For a set ` A ` with more ...
symgvalstruct 19306 The value of the symmetric...
symgvalstructOLD 19307 Obsolete proof of ~ symgva...
symgsubmefmnd 19308 The symmetric group on a s...
symgtset 19309 The topology of the symmet...
symggrp 19310 The symmetric group on a s...
symgid 19311 The group identity element...
symginv 19312 The group inverse in the s...
symgsubmefmndALT 19313 The symmetric group on a s...
galactghm 19314 The currying of a group ac...
lactghmga 19315 The converse of ~ galactgh...
symgtopn 19316 The topology of the symmet...
symgga 19317 The symmetric group induce...
pgrpsubgsymgbi 19318 Every permutation group is...
pgrpsubgsymg 19319 Every permutation group is...
idressubgsymg 19320 The singleton containing o...
idrespermg 19321 The structure with the sin...
cayleylem1 19322 Lemma for ~ cayley . (Con...
cayleylem2 19323 Lemma for ~ cayley . (Con...
cayley 19324 Cayley's Theorem (construc...
cayleyth 19325 Cayley's Theorem (existenc...
symgfix2 19326 If a permutation does not ...
symgextf 19327 The extension of a permuta...
symgextfv 19328 The function value of the ...
symgextfve 19329 The function value of the ...
symgextf1lem 19330 Lemma for ~ symgextf1 . (...
symgextf1 19331 The extension of a permuta...
symgextfo 19332 The extension of a permuta...
symgextf1o 19333 The extension of a permuta...
symgextsymg 19334 The extension of a permuta...
symgextres 19335 The restriction of the ext...
gsumccatsymgsn 19336 Homomorphic property of co...
gsmsymgrfixlem1 19337 Lemma 1 for ~ gsmsymgrfix ...
gsmsymgrfix 19338 The composition of permuta...
fvcosymgeq 19339 The values of two composit...
gsmsymgreqlem1 19340 Lemma 1 for ~ gsmsymgreq ....
gsmsymgreqlem2 19341 Lemma 2 for ~ gsmsymgreq ....
gsmsymgreq 19342 Two combination of permuta...
symgfixelq 19343 A permutation of a set fix...
symgfixels 19344 The restriction of a permu...
symgfixelsi 19345 The restriction of a permu...
symgfixf 19346 The mapping of a permutati...
symgfixf1 19347 The mapping of a permutati...
symgfixfolem1 19348 Lemma 1 for ~ symgfixfo . ...
symgfixfo 19349 The mapping of a permutati...
symgfixf1o 19350 The mapping of a permutati...
f1omvdmvd 19353 A permutation of any class...
f1omvdcnv 19354 A permutation and its inve...
mvdco 19355 Composing two permutations...
f1omvdconj 19356 Conjugation of a permutati...
f1otrspeq 19357 A transposition is charact...
f1omvdco2 19358 If exactly one of two perm...
f1omvdco3 19359 If a point is moved by exa...
pmtrfval 19360 The function generating tr...
pmtrval 19361 A generated transposition,...
pmtrfv 19362 General value of mapping a...
pmtrprfv 19363 In a transposition of two ...
pmtrprfv3 19364 In a transposition of two ...
pmtrf 19365 Functionality of a transpo...
pmtrmvd 19366 A transposition moves prec...
pmtrrn 19367 Transposing two points giv...
pmtrfrn 19368 A transposition (as a kind...
pmtrffv 19369 Mapping of a point under a...
pmtrrn2 19370 For any transposition ther...
pmtrfinv 19371 A transposition function i...
pmtrfmvdn0 19372 A transposition moves at l...
pmtrff1o 19373 A transposition function i...
pmtrfcnv 19374 A transposition function i...
pmtrfb 19375 An intrinsic characterizat...
pmtrfconj 19376 Any conjugate of a transpo...
symgsssg 19377 The symmetric group has su...
symgfisg 19378 The symmetric group has a ...
symgtrf 19379 Transpositions are element...
symggen 19380 The span of the transposit...
symggen2 19381 A finite permutation group...
symgtrinv 19382 To invert a permutation re...
pmtr3ncomlem1 19383 Lemma 1 for ~ pmtr3ncom . ...
pmtr3ncomlem2 19384 Lemma 2 for ~ pmtr3ncom . ...
pmtr3ncom 19385 Transpositions over sets w...
pmtrdifellem1 19386 Lemma 1 for ~ pmtrdifel . ...
pmtrdifellem2 19387 Lemma 2 for ~ pmtrdifel . ...
pmtrdifellem3 19388 Lemma 3 for ~ pmtrdifel . ...
pmtrdifellem4 19389 Lemma 4 for ~ pmtrdifel . ...
pmtrdifel 19390 A transposition of element...
pmtrdifwrdellem1 19391 Lemma 1 for ~ pmtrdifwrdel...
pmtrdifwrdellem2 19392 Lemma 2 for ~ pmtrdifwrdel...
pmtrdifwrdellem3 19393 Lemma 3 for ~ pmtrdifwrdel...
pmtrdifwrdel2lem1 19394 Lemma 1 for ~ pmtrdifwrdel...
pmtrdifwrdel 19395 A sequence of transpositio...
pmtrdifwrdel2 19396 A sequence of transpositio...
pmtrprfval 19397 The transpositions on a pa...
pmtrprfvalrn 19398 The range of the transposi...
psgnunilem1 19403 Lemma for ~ psgnuni . Giv...
psgnunilem5 19404 Lemma for ~ psgnuni . It ...
psgnunilem2 19405 Lemma for ~ psgnuni . Ind...
psgnunilem3 19406 Lemma for ~ psgnuni . Any...
psgnunilem4 19407 Lemma for ~ psgnuni . An ...
m1expaddsub 19408 Addition and subtraction o...
psgnuni 19409 If the same permutation ca...
psgnfval 19410 Function definition of the...
psgnfn 19411 Functionality and domain o...
psgndmsubg 19412 The finitary permutations ...
psgneldm 19413 Property of being a finita...
psgneldm2 19414 The finitary permutations ...
psgneldm2i 19415 A sequence of transpositio...
psgneu 19416 A finitary permutation has...
psgnval 19417 Value of the permutation s...
psgnvali 19418 A finitary permutation has...
psgnvalii 19419 Any representation of a pe...
psgnpmtr 19420 All transpositions are odd...
psgn0fv0 19421 The permutation sign funct...
sygbasnfpfi 19422 The class of non-fixed poi...
psgnfvalfi 19423 Function definition of the...
psgnvalfi 19424 Value of the permutation s...
psgnran 19425 The range of the permutati...
gsmtrcl 19426 The group sum of transposi...
psgnfitr 19427 A permutation of a finite ...
psgnfieu 19428 A permutation of a finite ...
pmtrsn 19429 The value of the transposi...
psgnsn 19430 The permutation sign funct...
psgnprfval 19431 The permutation sign funct...
psgnprfval1 19432 The permutation sign of th...
psgnprfval2 19433 The permutation sign of th...
odfval 19442 Value of the order functio...
odfvalALT 19443 Shorter proof of ~ odfval ...
odval 19444 Second substitution for th...
odlem1 19445 The group element order is...
odcl 19446 The order of a group eleme...
odf 19447 Functionality of the group...
odid 19448 Any element to the power o...
odlem2 19449 Any positive annihilator o...
odmodnn0 19450 Reduce the argument of a g...
mndodconglem 19451 Lemma for ~ mndodcong . (...
mndodcong 19452 If two multipliers are con...
mndodcongi 19453 If two multipliers are con...
oddvdsnn0 19454 The only multiples of ` A ...
odnncl 19455 If a nonzero multiple of a...
odmod 19456 Reduce the argument of a g...
oddvds 19457 The only multiples of ` A ...
oddvdsi 19458 Any group element is annih...
odcong 19459 If two multipliers are con...
odeq 19460 The ~ oddvds property uniq...
odval2 19461 A non-conditional definiti...
odcld 19462 The order of a group eleme...
odm1inv 19463 The (order-1)th multiple o...
odmulgid 19464 A relationship between the...
odmulg2 19465 The order of a multiple di...
odmulg 19466 Relationship between the o...
odmulgeq 19467 A multiple of a point of f...
odbezout 19468 If ` N ` is coprime to the...
od1 19469 The order of the group ide...
odeq1 19470 The group identity is the ...
odinv 19471 The order of the inverse o...
odf1 19472 The multiples of an elemen...
odinf 19473 The multiples of an elemen...
dfod2 19474 An alternative definition ...
odcl2 19475 The order of an element of...
oddvds2 19476 The order of an element of...
finodsubmsubg 19477 A submonoid whose elements...
0subgALT 19478 A shorter proof of ~ 0subg...
submod 19479 The order of an element is...
subgod 19480 The order of an element is...
odsubdvds 19481 The order of an element of...
odf1o1 19482 An element with zero order...
odf1o2 19483 An element with nonzero or...
odhash 19484 An element of zero order g...
odhash2 19485 If an element has nonzero ...
odhash3 19486 An element which generates...
odngen 19487 A cyclic subgroup of size ...
gexval 19488 Value of the exponent of a...
gexlem1 19489 The group element order is...
gexcl 19490 The exponent of a group is...
gexid 19491 Any element to the power o...
gexlem2 19492 Any positive annihilator o...
gexdvdsi 19493 Any group element is annih...
gexdvds 19494 The only ` N ` that annihi...
gexdvds2 19495 An integer divides the gro...
gexod 19496 Any group element is annih...
gexcl3 19497 If the order of every grou...
gexnnod 19498 Every group element has fi...
gexcl2 19499 The exponent of a finite g...
gexdvds3 19500 The exponent of a finite g...
gex1 19501 A group or monoid has expo...
ispgp 19502 A group is a ` P ` -group ...
pgpprm 19503 Reverse closure for the fi...
pgpgrp 19504 Reverse closure for the se...
pgpfi1 19505 A finite group with order ...
pgp0 19506 The identity subgroup is a...
subgpgp 19507 A subgroup of a p-group is...
sylow1lem1 19508 Lemma for ~ sylow1 . The ...
sylow1lem2 19509 Lemma for ~ sylow1 . The ...
sylow1lem3 19510 Lemma for ~ sylow1 . One ...
sylow1lem4 19511 Lemma for ~ sylow1 . The ...
sylow1lem5 19512 Lemma for ~ sylow1 . Usin...
sylow1 19513 Sylow's first theorem. If...
odcau 19514 Cauchy's theorem for the o...
pgpfi 19515 The converse to ~ pgpfi1 ....
pgpfi2 19516 Alternate version of ~ pgp...
pgphash 19517 The order of a p-group. (...
isslw 19518 The property of being a Sy...
slwprm 19519 Reverse closure for the fi...
slwsubg 19520 A Sylow ` P ` -subgroup is...
slwispgp 19521 Defining property of a Syl...
slwpss 19522 A proper superset of a Syl...
slwpgp 19523 A Sylow ` P ` -subgroup is...
pgpssslw 19524 Every ` P ` -subgroup is c...
slwn0 19525 Every finite group contain...
subgslw 19526 A Sylow subgroup that is c...
sylow2alem1 19527 Lemma for ~ sylow2a . An ...
sylow2alem2 19528 Lemma for ~ sylow2a . All...
sylow2a 19529 A named lemma of Sylow's s...
sylow2blem1 19530 Lemma for ~ sylow2b . Eva...
sylow2blem2 19531 Lemma for ~ sylow2b . Lef...
sylow2blem3 19532 Sylow's second theorem. P...
sylow2b 19533 Sylow's second theorem. A...
slwhash 19534 A sylow subgroup has cardi...
fislw 19535 The sylow subgroups of a f...
sylow2 19536 Sylow's second theorem. S...
sylow3lem1 19537 Lemma for ~ sylow3 , first...
sylow3lem2 19538 Lemma for ~ sylow3 , first...
sylow3lem3 19539 Lemma for ~ sylow3 , first...
sylow3lem4 19540 Lemma for ~ sylow3 , first...
sylow3lem5 19541 Lemma for ~ sylow3 , secon...
sylow3lem6 19542 Lemma for ~ sylow3 , secon...
sylow3 19543 Sylow's third theorem. Th...
lsmfval 19548 The subgroup sum function ...
lsmvalx 19549 Subspace sum value (for a ...
lsmelvalx 19550 Subspace sum membership (f...
lsmelvalix 19551 Subspace sum membership (f...
oppglsm 19552 The subspace sum operation...
lsmssv 19553 Subgroup sum is a subset o...
lsmless1x 19554 Subset implies subgroup su...
lsmless2x 19555 Subset implies subgroup su...
lsmub1x 19556 Subgroup sum is an upper b...
lsmub2x 19557 Subgroup sum is an upper b...
lsmval 19558 Subgroup sum value (for a ...
lsmelval 19559 Subgroup sum membership (f...
lsmelvali 19560 Subgroup sum membership (f...
lsmelvalm 19561 Subgroup sum membership an...
lsmelvalmi 19562 Membership of vector subtr...
lsmsubm 19563 The sum of two commuting s...
lsmsubg 19564 The sum of two commuting s...
lsmcom2 19565 Subgroup sum commutes. (C...
smndlsmidm 19566 The direct product is idem...
lsmub1 19567 Subgroup sum is an upper b...
lsmub2 19568 Subgroup sum is an upper b...
lsmunss 19569 Union of subgroups is a su...
lsmless1 19570 Subset implies subgroup su...
lsmless2 19571 Subset implies subgroup su...
lsmless12 19572 Subset implies subgroup su...
lsmidm 19573 Subgroup sum is idempotent...
lsmlub 19574 The least upper bound prop...
lsmss1 19575 Subgroup sum with a subset...
lsmss1b 19576 Subgroup sum with a subset...
lsmss2 19577 Subgroup sum with a subset...
lsmss2b 19578 Subgroup sum with a subset...
lsmass 19579 Subgroup sum is associativ...
mndlsmidm 19580 Subgroup sum is idempotent...
lsm01 19581 Subgroup sum with the zero...
lsm02 19582 Subgroup sum with the zero...
subglsm 19583 The subgroup sum evaluated...
lssnle 19584 Equivalent expressions for...
lsmmod 19585 The modular law holds for ...
lsmmod2 19586 Modular law dual for subgr...
lsmpropd 19587 If two structures have the...
cntzrecd 19588 Commute the "subgroups com...
lsmcntz 19589 The "subgroups commute" pr...
lsmcntzr 19590 The "subgroups commute" pr...
lsmdisj 19591 Disjointness from a subgro...
lsmdisj2 19592 Association of the disjoin...
lsmdisj3 19593 Association of the disjoin...
lsmdisjr 19594 Disjointness from a subgro...
lsmdisj2r 19595 Association of the disjoin...
lsmdisj3r 19596 Association of the disjoin...
lsmdisj2a 19597 Association of the disjoin...
lsmdisj2b 19598 Association of the disjoin...
lsmdisj3a 19599 Association of the disjoin...
lsmdisj3b 19600 Association of the disjoin...
subgdisj1 19601 Vectors belonging to disjo...
subgdisj2 19602 Vectors belonging to disjo...
subgdisjb 19603 Vectors belonging to disjo...
pj1fval 19604 The left projection functi...
pj1val 19605 The left projection functi...
pj1eu 19606 Uniqueness of a left proje...
pj1f 19607 The left projection functi...
pj2f 19608 The right projection funct...
pj1id 19609 Any element of a direct su...
pj1eq 19610 Any element of a direct su...
pj1lid 19611 The left projection functi...
pj1rid 19612 The left projection functi...
pj1ghm 19613 The left projection functi...
pj1ghm2 19614 The left projection functi...
lsmhash 19615 The order of the direct pr...
efgmval 19622 Value of the formal invers...
efgmf 19623 The formal inverse operati...
efgmnvl 19624 The inversion function on ...
efgrcl 19625 Lemma for ~ efgval . (Con...
efglem 19626 Lemma for ~ efgval . (Con...
efgval 19627 Value of the free group co...
efger 19628 Value of the free group co...
efgi 19629 Value of the free group co...
efgi0 19630 Value of the free group co...
efgi1 19631 Value of the free group co...
efgtf 19632 Value of the free group co...
efgtval 19633 Value of the extension fun...
efgval2 19634 Value of the free group co...
efgi2 19635 Value of the free group co...
efgtlen 19636 Value of the free group co...
efginvrel2 19637 The inverse of the reverse...
efginvrel1 19638 The inverse of the reverse...
efgsf 19639 Value of the auxiliary fun...
efgsdm 19640 Elementhood in the domain ...
efgsval 19641 Value of the auxiliary fun...
efgsdmi 19642 Property of the last link ...
efgsval2 19643 Value of the auxiliary fun...
efgsrel 19644 The start and end of any e...
efgs1 19645 A singleton of an irreduci...
efgs1b 19646 Every extension sequence e...
efgsp1 19647 If ` F ` is an extension s...
efgsres 19648 An initial segment of an e...
efgsfo 19649 For any word, there is a s...
efgredlema 19650 The reduced word that form...
efgredlemf 19651 Lemma for ~ efgredleme . ...
efgredlemg 19652 Lemma for ~ efgred . (Con...
efgredleme 19653 Lemma for ~ efgred . (Con...
efgredlemd 19654 The reduced word that form...
efgredlemc 19655 The reduced word that form...
efgredlemb 19656 The reduced word that form...
efgredlem 19657 The reduced word that form...
efgred 19658 The reduced word that form...
efgrelexlema 19659 If two words ` A , B ` are...
efgrelexlemb 19660 If two words ` A , B ` are...
efgrelex 19661 If two words ` A , B ` are...
efgredeu 19662 There is a unique reduced ...
efgred2 19663 Two extension sequences ha...
efgcpbllema 19664 Lemma for ~ efgrelex . De...
efgcpbllemb 19665 Lemma for ~ efgrelex . Sh...
efgcpbl 19666 Two extension sequences ha...
efgcpbl2 19667 Two extension sequences ha...
frgpval 19668 Value of the free group co...
frgpcpbl 19669 Compatibility of the group...
frgp0 19670 The free group is a group....
frgpeccl 19671 Closure of the quotient ma...
frgpgrp 19672 The free group is a group....
frgpadd 19673 Addition in the free group...
frgpinv 19674 The inverse of an element ...
frgpmhm 19675 The "natural map" from wor...
vrgpfval 19676 The canonical injection fr...
vrgpval 19677 The value of the generatin...
vrgpf 19678 The mapping from the index...
vrgpinv 19679 The inverse of a generatin...
frgpuptf 19680 Any assignment of the gene...
frgpuptinv 19681 Any assignment of the gene...
frgpuplem 19682 Any assignment of the gene...
frgpupf 19683 Any assignment of the gene...
frgpupval 19684 Any assignment of the gene...
frgpup1 19685 Any assignment of the gene...
frgpup2 19686 The evaluation map has the...
frgpup3lem 19687 The evaluation map has the...
frgpup3 19688 Universal property of the ...
0frgp 19689 The free group on zero gen...
isabl 19694 The predicate "is an Abeli...
ablgrp 19695 An Abelian group is a grou...
ablgrpd 19696 An Abelian group is a grou...
ablcmn 19697 An Abelian group is a comm...
ablcmnd 19698 An Abelian group is a comm...
iscmn 19699 The predicate "is a commut...
isabl2 19700 The predicate "is an Abeli...
cmnpropd 19701 If two structures have the...
ablpropd 19702 If two structures have the...
ablprop 19703 If two structures have the...
iscmnd 19704 Properties that determine ...
isabld 19705 Properties that determine ...
isabli 19706 Properties that determine ...
cmnmnd 19707 A commutative monoid is a ...
cmncom 19708 A commutative monoid is co...
ablcom 19709 An Abelian group operation...
cmn32 19710 Commutative/associative la...
cmn4 19711 Commutative/associative la...
cmn12 19712 Commutative/associative la...
abl32 19713 Commutative/associative la...
cmnmndd 19714 A commutative monoid is a ...
cmnbascntr 19715 The base set of a commutat...
rinvmod 19716 Uniqueness of a right inve...
ablinvadd 19717 The inverse of an Abelian ...
ablsub2inv 19718 Abelian group subtraction ...
ablsubadd 19719 Relationship between Abeli...
ablsub4 19720 Commutative/associative su...
abladdsub4 19721 Abelian group addition/sub...
abladdsub 19722 Associative-type law for g...
ablsubadd23 19723 Commutative/associative la...
ablsubaddsub 19724 Double subtraction and add...
ablpncan2 19725 Cancellation law for subtr...
ablpncan3 19726 A cancellation law for Abe...
ablsubsub 19727 Law for double subtraction...
ablsubsub4 19728 Law for double subtraction...
ablpnpcan 19729 Cancellation law for mixed...
ablnncan 19730 Cancellation law for group...
ablsub32 19731 Swap the second and third ...
ablnnncan 19732 Cancellation law for group...
ablnnncan1 19733 Cancellation law for group...
ablsubsub23 19734 Swap subtrahend and result...
mulgnn0di 19735 Group multiple of a sum, f...
mulgdi 19736 Group multiple of a sum. ...
mulgmhm 19737 The map from ` x ` to ` n ...
mulgghm 19738 The map from ` x ` to ` n ...
mulgsubdi 19739 Group multiple of a differ...
ghmfghm 19740 The function fulfilling th...
ghmcmn 19741 The image of a commutative...
ghmabl 19742 The image of an abelian gr...
invghm 19743 The inversion map is a gro...
eqgabl 19744 Value of the subgroup cose...
qusecsub 19745 Two subgroup cosets are eq...
subgabl 19746 A subgroup of an abelian g...
subcmn 19747 A submonoid of a commutati...
submcmn 19748 A submonoid of a commutati...
submcmn2 19749 A submonoid is commutative...
cntzcmn 19750 The centralizer of any sub...
cntzcmnss 19751 Any subset in a commutativ...
cntrcmnd 19752 The center of a monoid is ...
cntrabl 19753 The center of a group is a...
cntzspan 19754 If the generators commute,...
cntzcmnf 19755 Discharge the centralizer ...
ghmplusg 19756 The pointwise sum of two l...
ablnsg 19757 Every subgroup of an abeli...
odadd1 19758 The order of a product in ...
odadd2 19759 The order of a product in ...
odadd 19760 The order of a product is ...
gex2abl 19761 A group with exponent 2 (o...
gexexlem 19762 Lemma for ~ gexex . (Cont...
gexex 19763 In an abelian group with f...
torsubg 19764 The set of all elements of...
oddvdssubg 19765 The set of all elements wh...
lsmcomx 19766 Subgroup sum commutes (ext...
ablcntzd 19767 All subgroups in an abelia...
lsmcom 19768 Subgroup sum commutes. (C...
lsmsubg2 19769 The sum of two subgroups i...
lsm4 19770 Commutative/associative la...
prdscmnd 19771 The product of a family of...
prdsabld 19772 The product of a family of...
pwscmn 19773 The structure power on a c...
pwsabl 19774 The structure power on an ...
qusabl 19775 If ` Y ` is a subgroup of ...
abl1 19776 The (smallest) structure r...
abln0 19777 Abelian groups (and theref...
cnaddablx 19778 The complex numbers are an...
cnaddabl 19779 The complex numbers are an...
cnaddid 19780 The group identity element...
cnaddinv 19781 Value of the group inverse...
zaddablx 19782 The integers are an Abelia...
frgpnabllem1 19783 Lemma for ~ frgpnabl . (C...
frgpnabllem2 19784 Lemma for ~ frgpnabl . (C...
frgpnabl 19785 The free group on two or m...
imasabl 19786 The image structure of an ...
iscyg 19789 Definition of a cyclic gro...
iscyggen 19790 The property of being a cy...
iscyggen2 19791 The property of being a cy...
iscyg2 19792 A cyclic group is a group ...
cyggeninv 19793 The inverse of a cyclic ge...
cyggenod 19794 An element is the generato...
cyggenod2 19795 In an infinite cyclic grou...
iscyg3 19796 Definition of a cyclic gro...
iscygd 19797 Definition of a cyclic gro...
iscygodd 19798 Show that a group with an ...
cycsubmcmn 19799 The set of nonnegative int...
cyggrp 19800 A cyclic group is a group....
cygabl 19801 A cyclic group is abelian....
cygctb 19802 A cyclic group is countabl...
0cyg 19803 The trivial group is cycli...
prmcyg 19804 A group with prime order i...
lt6abl 19805 A group with fewer than ` ...
ghmcyg 19806 The image of a cyclic grou...
cyggex2 19807 The exponent of a cyclic g...
cyggex 19808 The exponent of a finite c...
cyggexb 19809 A finite abelian group is ...
giccyg 19810 Cyclicity is a group prope...
cycsubgcyg 19811 The cyclic subgroup genera...
cycsubgcyg2 19812 The cyclic subgroup genera...
gsumval3a 19813 Value of the group sum ope...
gsumval3eu 19814 The group sum as defined i...
gsumval3lem1 19815 Lemma 1 for ~ gsumval3 . ...
gsumval3lem2 19816 Lemma 2 for ~ gsumval3 . ...
gsumval3 19817 Value of the group sum ope...
gsumcllem 19818 Lemma for ~ gsumcl and rel...
gsumzres 19819 Extend a finite group sum ...
gsumzcl2 19820 Closure of a finite group ...
gsumzcl 19821 Closure of a finite group ...
gsumzf1o 19822 Re-index a finite group su...
gsumres 19823 Extend a finite group sum ...
gsumcl2 19824 Closure of a finite group ...
gsumcl 19825 Closure of a finite group ...
gsumf1o 19826 Re-index a finite group su...
gsumreidx 19827 Re-index a finite group su...
gsumzsubmcl 19828 Closure of a group sum in ...
gsumsubmcl 19829 Closure of a group sum in ...
gsumsubgcl 19830 Closure of a group sum in ...
gsumzaddlem 19831 The sum of two group sums....
gsumzadd 19832 The sum of two group sums....
gsumadd 19833 The sum of two group sums....
gsummptfsadd 19834 The sum of two group sums ...
gsummptfidmadd 19835 The sum of two group sums ...
gsummptfidmadd2 19836 The sum of two group sums ...
gsumzsplit 19837 Split a group sum into two...
gsumsplit 19838 Split a group sum into two...
gsumsplit2 19839 Split a group sum into two...
gsummptfidmsplit 19840 Split a group sum expresse...
gsummptfidmsplitres 19841 Split a group sum expresse...
gsummptfzsplit 19842 Split a group sum expresse...
gsummptfzsplitl 19843 Split a group sum expresse...
gsumconst 19844 Sum of a constant series. ...
gsumconstf 19845 Sum of a constant series. ...
gsummptshft 19846 Index shift of a finite gr...
gsumzmhm 19847 Apply a group homomorphism...
gsummhm 19848 Apply a group homomorphism...
gsummhm2 19849 Apply a group homomorphism...
gsummptmhm 19850 Apply a group homomorphism...
gsummulglem 19851 Lemma for ~ gsummulg and ~...
gsummulg 19852 Nonnegative multiple of a ...
gsummulgz 19853 Integer multiple of a grou...
gsumzoppg 19854 The opposite of a group su...
gsumzinv 19855 Inverse of a group sum. (...
gsuminv 19856 Inverse of a group sum. (...
gsummptfidminv 19857 Inverse of a group sum exp...
gsumsub 19858 The difference of two grou...
gsummptfssub 19859 The difference of two grou...
gsummptfidmsub 19860 The difference of two grou...
gsumsnfd 19861 Group sum of a singleton, ...
gsumsnd 19862 Group sum of a singleton, ...
gsumsnf 19863 Group sum of a singleton, ...
gsumsn 19864 Group sum of a singleton. ...
gsumpr 19865 Group sum of a pair. (Con...
gsumzunsnd 19866 Append an element to a fin...
gsumunsnfd 19867 Append an element to a fin...
gsumunsnd 19868 Append an element to a fin...
gsumunsnf 19869 Append an element to a fin...
gsumunsn 19870 Append an element to a fin...
gsumdifsnd 19871 Extract a summand from a f...
gsumpt 19872 Sum of a family that is no...
gsummptf1o 19873 Re-index a finite group su...
gsummptun 19874 Group sum of a disjoint un...
gsummpt1n0 19875 If only one summand in a f...
gsummptif1n0 19876 If only one summand in a f...
gsummptcl 19877 Closure of a finite group ...
gsummptfif1o 19878 Re-index a finite group su...
gsummptfzcl 19879 Closure of a finite group ...
gsum2dlem1 19880 Lemma 1 for ~ gsum2d . (C...
gsum2dlem2 19881 Lemma for ~ gsum2d . (Con...
gsum2d 19882 Write a sum over a two-dim...
gsum2d2lem 19883 Lemma for ~ gsum2d2 : show...
gsum2d2 19884 Write a group sum over a t...
gsumcom2 19885 Two-dimensional commutatio...
gsumxp 19886 Write a group sum over a c...
gsumcom 19887 Commute the arguments of a...
gsumcom3 19888 A commutative law for fini...
gsumcom3fi 19889 A commutative law for fini...
gsumxp2 19890 Write a group sum over a c...
prdsgsum 19891 Finite commutative sums in...
pwsgsum 19892 Finite commutative sums in...
fsfnn0gsumfsffz 19893 Replacing a finitely suppo...
nn0gsumfz 19894 Replacing a finitely suppo...
nn0gsumfz0 19895 Replacing a finitely suppo...
gsummptnn0fz 19896 A final group sum over a f...
gsummptnn0fzfv 19897 A final group sum over a f...
telgsumfzslem 19898 Lemma for ~ telgsumfzs (in...
telgsumfzs 19899 Telescoping group sum rang...
telgsumfz 19900 Telescoping group sum rang...
telgsumfz0s 19901 Telescoping finite group s...
telgsumfz0 19902 Telescoping finite group s...
telgsums 19903 Telescoping finitely suppo...
telgsum 19904 Telescoping finitely suppo...
reldmdprd 19909 The domain of the internal...
dmdprd 19910 The domain of definition o...
dmdprdd 19911 Show that a given family i...
dprddomprc 19912 A family of subgroups inde...
dprddomcld 19913 If a family of subgroups i...
dprdval0prc 19914 The internal direct produc...
dprdval 19915 The value of the internal ...
eldprd 19916 A class ` A ` is an intern...
dprdgrp 19917 Reverse closure for the in...
dprdf 19918 The function ` S ` is a fa...
dprdf2 19919 The function ` S ` is a fa...
dprdcntz 19920 The function ` S ` is a fa...
dprddisj 19921 The function ` S ` is a fa...
dprdw 19922 The property of being a fi...
dprdwd 19923 A mapping being a finitely...
dprdff 19924 A finitely supported funct...
dprdfcl 19925 A finitely supported funct...
dprdffsupp 19926 A finitely supported funct...
dprdfcntz 19927 A function on the elements...
dprdssv 19928 The internal direct produc...
dprdfid 19929 A function mapping all but...
eldprdi 19930 The domain of definition o...
dprdfinv 19931 Take the inverse of a grou...
dprdfadd 19932 Take the sum of group sums...
dprdfsub 19933 Take the difference of gro...
dprdfeq0 19934 The zero function is the o...
dprdf11 19935 Two group sums over a dire...
dprdsubg 19936 The internal direct produc...
dprdub 19937 Each factor is a subset of...
dprdlub 19938 The direct product is smal...
dprdspan 19939 The direct product is the ...
dprdres 19940 Restriction of a direct pr...
dprdss 19941 Create a direct product by...
dprdz 19942 A family consisting entire...
dprd0 19943 The empty family is an int...
dprdf1o 19944 Rearrange the index set of...
dprdf1 19945 Rearrange the index set of...
subgdmdprd 19946 A direct product in a subg...
subgdprd 19947 A direct product in a subg...
dprdsn 19948 A singleton family is an i...
dmdprdsplitlem 19949 Lemma for ~ dmdprdsplit . ...
dprdcntz2 19950 The function ` S ` is a fa...
dprddisj2 19951 The function ` S ` is a fa...
dprd2dlem2 19952 The direct product of a co...
dprd2dlem1 19953 The direct product of a co...
dprd2da 19954 The direct product of a co...
dprd2db 19955 The direct product of a co...
dprd2d2 19956 The direct product of a co...
dmdprdsplit2lem 19957 Lemma for ~ dmdprdsplit . ...
dmdprdsplit2 19958 The direct product splits ...
dmdprdsplit 19959 The direct product splits ...
dprdsplit 19960 The direct product is the ...
dmdprdpr 19961 A singleton family is an i...
dprdpr 19962 A singleton family is an i...
dpjlem 19963 Lemma for theorems about d...
dpjcntz 19964 The two subgroups that app...
dpjdisj 19965 The two subgroups that app...
dpjlsm 19966 The two subgroups that app...
dpjfval 19967 Value of the direct produc...
dpjval 19968 Value of the direct produc...
dpjf 19969 The ` X ` -th index projec...
dpjidcl 19970 The key property of projec...
dpjeq 19971 Decompose a group sum into...
dpjid 19972 The key property of projec...
dpjlid 19973 The ` X ` -th index projec...
dpjrid 19974 The ` Y ` -th index projec...
dpjghm 19975 The direct product is the ...
dpjghm2 19976 The direct product is the ...
ablfacrplem 19977 Lemma for ~ ablfacrp2 . (...
ablfacrp 19978 A finite abelian group who...
ablfacrp2 19979 The factors ` K , L ` of ~...
ablfac1lem 19980 Lemma for ~ ablfac1b . Sa...
ablfac1a 19981 The factors of ~ ablfac1b ...
ablfac1b 19982 Any abelian group is the d...
ablfac1c 19983 The factors of ~ ablfac1b ...
ablfac1eulem 19984 Lemma for ~ ablfac1eu . (...
ablfac1eu 19985 The factorization of ~ abl...
pgpfac1lem1 19986 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem2 19987 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem3a 19988 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem3 19989 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem4 19990 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem5 19991 Lemma for ~ pgpfac1 . (Co...
pgpfac1 19992 Factorization of a finite ...
pgpfaclem1 19993 Lemma for ~ pgpfac . (Con...
pgpfaclem2 19994 Lemma for ~ pgpfac . (Con...
pgpfaclem3 19995 Lemma for ~ pgpfac . (Con...
pgpfac 19996 Full factorization of a fi...
ablfaclem1 19997 Lemma for ~ ablfac . (Con...
ablfaclem2 19998 Lemma for ~ ablfac . (Con...
ablfaclem3 19999 Lemma for ~ ablfac . (Con...
ablfac 20000 The Fundamental Theorem of...
ablfac2 20001 Choose generators for each...
issimpg 20004 The predicate "is a simple...
issimpgd 20005 Deduce a simple group from...
simpggrp 20006 A simple group is a group....
simpggrpd 20007 A simple group is a group....
simpg2nsg 20008 A simple group has two nor...
trivnsimpgd 20009 Trivial groups are not sim...
simpgntrivd 20010 Simple groups are nontrivi...
simpgnideld 20011 A simple group contains a ...
simpgnsgd 20012 The only normal subgroups ...
simpgnsgeqd 20013 A normal subgroup of a sim...
2nsgsimpgd 20014 If any normal subgroup of ...
simpgnsgbid 20015 A nontrivial group is simp...
ablsimpnosubgd 20016 A subgroup of an abelian s...
ablsimpg1gend 20017 An abelian simple group is...
ablsimpgcygd 20018 An abelian simple group is...
ablsimpgfindlem1 20019 Lemma for ~ ablsimpgfind ....
ablsimpgfindlem2 20020 Lemma for ~ ablsimpgfind ....
cycsubggenodd 20021 Relationship between the o...
ablsimpgfind 20022 An abelian simple group is...
fincygsubgd 20023 The subgroup referenced in...
fincygsubgodd 20024 Calculate the order of a s...
fincygsubgodexd 20025 A finite cyclic group has ...
prmgrpsimpgd 20026 A group of prime order is ...
ablsimpgprmd 20027 An abelian simple group ha...
ablsimpgd 20028 An abelian group is simple...
fnmgp 20031 The multiplicative group o...
mgpval 20032 Value of the multiplicatio...
mgpplusg 20033 Value of the group operati...
mgplemOLD 20034 Obsolete version of ~ sets...
mgpbas 20035 Base set of the multiplica...
mgpbasOLD 20036 Obsolete version of ~ mgpb...
mgpsca 20037 The multiplication monoid ...
mgpscaOLD 20038 Obsolete version of ~ mgps...
mgptset 20039 Topology component of the ...
mgptsetOLD 20040 Obsolete version of ~ mgpt...
mgptopn 20041 Topology of the multiplica...
mgpds 20042 Distance function of the m...
mgpdsOLD 20043 Obsolete version of ~ mgpd...
mgpress 20044 Subgroup commutes with the...
mgpressOLD 20045 Obsolete version of ~ mgpr...
prdsmgp 20046 The multiplicative monoid ...
isrng 20049 The predicate "is a non-un...
rngabl 20050 A non-unital ring is an (a...
rngmgp 20051 A non-unital ring is a sem...
rngmgpf 20052 Restricted functionality o...
rnggrp 20053 A non-unital ring is a (ad...
rngass 20054 Associative law for the mu...
rngdi 20055 Distributive law for the m...
rngdir 20056 Distributive law for the m...
rngacl 20057 Closure of the addition op...
rng0cl 20058 The zero element of a non-...
rngcl 20059 Closure of the multiplicat...
rnglz 20060 The zero of a non-unital r...
rngrz 20061 The zero of a non-unital r...
rngmneg1 20062 Negation of a product in a...
rngmneg2 20063 Negation of a product in a...
rngm2neg 20064 Double negation of a produ...
rngansg 20065 Every additive subgroup of...
rngsubdi 20066 Ring multiplication distri...
rngsubdir 20067 Ring multiplication distri...
isrngd 20068 Properties that determine ...
rngpropd 20069 If two structures have the...
prdsmulrngcl 20070 Closure of the multiplicat...
prdsrngd 20071 A product of non-unital ri...
imasrng 20072 The image structure of a n...
imasrngf1 20073 The image of a non-unital ...
xpsrngd 20074 A product of two non-unita...
qusrng 20075 The quotient structure of ...
ringidval 20078 The value of the unity ele...
dfur2 20079 The multiplicative identit...
ringurd 20080 Deduce the unity element o...
issrg 20083 The predicate "is a semiri...
srgcmn 20084 A semiring is a commutativ...
srgmnd 20085 A semiring is a monoid. (...
srgmgp 20086 A semiring is a monoid und...
srgdilem 20087 Lemma for ~ srgdi and ~ sr...
srgcl 20088 Closure of the multiplicat...
srgass 20089 Associative law for the mu...
srgideu 20090 The unity element of a sem...
srgfcl 20091 Functionality of the multi...
srgdi 20092 Distributive law for the m...
srgdir 20093 Distributive law for the m...
srgidcl 20094 The unity element of a sem...
srg0cl 20095 The zero element of a semi...
srgidmlem 20096 Lemma for ~ srglidm and ~ ...
srglidm 20097 The unity element of a sem...
srgridm 20098 The unity element of a sem...
issrgid 20099 Properties showing that an...
srgacl 20100 Closure of the addition op...
srgcom 20101 Commutativity of the addit...
srgrz 20102 The zero of a semiring is ...
srglz 20103 The zero of a semiring is ...
srgisid 20104 In a semiring, the only le...
o2timesd 20105 An element of a ring-like ...
rglcom4d 20106 Restricted commutativity o...
srgo2times 20107 A semiring element plus it...
srgcom4lem 20108 Lemma for ~ srgcom4 . Thi...
srgcom4 20109 Restricted commutativity o...
srg1zr 20110 The only semiring with a b...
srgen1zr 20111 The only semiring with one...
srgmulgass 20112 An associative property be...
srgpcomp 20113 If two elements of a semir...
srgpcompp 20114 If two elements of a semir...
srgpcomppsc 20115 If two elements of a semir...
srglmhm 20116 Left-multiplication in a s...
srgrmhm 20117 Right-multiplication in a ...
srgsummulcr 20118 A finite semiring sum mult...
sgsummulcl 20119 A finite semiring sum mult...
srg1expzeq1 20120 The exponentiation (by a n...
srgbinomlem1 20121 Lemma 1 for ~ srgbinomlem ...
srgbinomlem2 20122 Lemma 2 for ~ srgbinomlem ...
srgbinomlem3 20123 Lemma 3 for ~ srgbinomlem ...
srgbinomlem4 20124 Lemma 4 for ~ srgbinomlem ...
srgbinomlem 20125 Lemma for ~ srgbinom . In...
srgbinom 20126 The binomial theorem for c...
csrgbinom 20127 The binomial theorem for c...
isring 20132 The predicate "is a (unita...
ringgrp 20133 A ring is a group. (Contr...
ringmgp 20134 A ring is a monoid under m...
iscrng 20135 A commutative ring is a ri...
crngmgp 20136 A commutative ring's multi...
ringgrpd 20137 A ring is a group. (Contr...
ringmnd 20138 A ring is a monoid under a...
ringmgm 20139 A ring is a magma. (Contr...
crngring 20140 A commutative ring is a ri...
crngringd 20141 A commutative ring is a ri...
crnggrpd 20142 A commutative ring is a gr...
mgpf 20143 Restricted functionality o...
ringdilem 20144 Properties of a unital rin...
ringcl 20145 Closure of the multiplicat...
crngcom 20146 A commutative ring's multi...
iscrng2 20147 A commutative ring is a ri...
ringass 20148 Associative law for multip...
ringideu 20149 The unity element of a rin...
crngbascntr 20150 The base set of a commutat...
ringassd 20151 Associative law for multip...
ringcld 20152 Closure of the multiplicat...
ringdi 20153 Distributive law for the m...
ringdir 20154 Distributive law for the m...
ringidcl 20155 The unity element of a rin...
ring0cl 20156 The zero element of a ring...
ringidmlem 20157 Lemma for ~ ringlidm and ~...
ringlidm 20158 The unity element of a rin...
ringridm 20159 The unity element of a rin...
isringid 20160 Properties showing that an...
ringlidmd 20161 The unity element of a rin...
ringridmd 20162 The unity element of a rin...
ringid 20163 The multiplication operati...
ringo2times 20164 A ring element plus itself...
ringadd2 20165 A ring element plus itself...
ringidss 20166 A subset of the multiplica...
ringacl 20167 Closure of the addition op...
ringcomlem 20168 Lemma for ~ ringcom . Thi...
ringcom 20169 Commutativity of the addit...
ringabl 20170 A ring is an Abelian group...
ringcmn 20171 A ring is a commutative mo...
ringabld 20172 A ring is an Abelian group...
ringcmnd 20173 A ring is a commutative mo...
ringrng 20174 A unital ring is a non-uni...
ringssrng 20175 The unital rings are non-u...
isringrng 20176 The predicate "is a unital...
ringpropd 20177 If two structures have the...
crngpropd 20178 If two structures have the...
ringprop 20179 If two structures have the...
isringd 20180 Properties that determine ...
iscrngd 20181 Properties that determine ...
ringlz 20182 The zero of a unital ring ...
ringrz 20183 The zero of a unital ring ...
ringlzd 20184 The zero of a unital ring ...
ringrzd 20185 The zero of a unital ring ...
ringsrg 20186 Any ring is also a semirin...
ring1eq0 20187 If one and zero are equal,...
ring1ne0 20188 If a ring has at least two...
ringinvnz1ne0 20189 In a unital ring, a left i...
ringinvnzdiv 20190 In a unital ring, a left i...
ringnegl 20191 Negation in a ring is the ...
ringnegr 20192 Negation in a ring is the ...
ringmneg1 20193 Negation of a product in a...
ringmneg2 20194 Negation of a product in a...
ringm2neg 20195 Double negation of a produ...
ringsubdi 20196 Ring multiplication distri...
ringsubdir 20197 Ring multiplication distri...
mulgass2 20198 An associative property be...
ring1 20199 The (smallest) structure r...
ringn0 20200 Rings exist. (Contributed...
ringlghm 20201 Left-multiplication in a r...
ringrghm 20202 Right-multiplication in a ...
gsummulc1OLD 20203 Obsolete version of ~ gsum...
gsummulc2OLD 20204 Obsolete version of ~ gsum...
gsummulc1 20205 A finite ring sum multipli...
gsummulc2 20206 A finite ring sum multipli...
gsummgp0 20207 If one factor in a finite ...
gsumdixp 20208 Distribute a binary produc...
prdsmulrcl 20209 A structure product of rin...
prdsringd 20210 A product of rings is a ri...
prdscrngd 20211 A product of commutative r...
prds1 20212 Value of the ring unity in...
pwsring 20213 A structure power of a rin...
pws1 20214 Value of the ring unity in...
pwscrng 20215 A structure power of a com...
pwsmgp 20216 The multiplicative group o...
pwspjmhmmgpd 20217 The projection given by ~ ...
pwsexpg 20218 Value of a group exponenti...
imasring 20219 The image structure of a r...
imasringf1 20220 The image of a ring under ...
xpsringd 20221 A product of two rings is ...
xpsring1d 20222 The multiplicative identit...
qusring2 20223 The quotient structure of ...
crngbinom 20224 The binomial theorem for c...
opprval 20227 Value of the opposite ring...
opprmulfval 20228 Value of the multiplicatio...
opprmul 20229 Value of the multiplicatio...
crngoppr 20230 In a commutative ring, the...
opprlem 20231 Lemma for ~ opprbas and ~ ...
opprlemOLD 20232 Obsolete version of ~ oppr...
opprbas 20233 Base set of an opposite ri...
opprbasOLD 20234 Obsolete proof of ~ opprba...
oppradd 20235 Addition operation of an o...
oppraddOLD 20236 Obsolete proof of ~ opprba...
opprrng 20237 An opposite non-unital rin...
opprrngb 20238 A class is a non-unital ri...
opprring 20239 An opposite ring is a ring...
opprringb 20240 Bidirectional form of ~ op...
oppr0 20241 Additive identity of an op...
oppr1 20242 Multiplicative identity of...
opprneg 20243 The negative function in a...
opprsubg 20244 Being a subgroup is a symm...
mulgass3 20245 An associative property be...
reldvdsr 20252 The divides relation is a ...
dvdsrval 20253 Value of the divides relat...
dvdsr 20254 Value of the divides relat...
dvdsr2 20255 Value of the divides relat...
dvdsrmul 20256 A left-multiple of ` X ` i...
dvdsrcl 20257 Closure of a dividing elem...
dvdsrcl2 20258 Closure of a dividing elem...
dvdsrid 20259 An element in a (unital) r...
dvdsrtr 20260 Divisibility is transitive...
dvdsrmul1 20261 The divisibility relation ...
dvdsrneg 20262 An element divides its neg...
dvdsr01 20263 In a ring, zero is divisib...
dvdsr02 20264 Only zero is divisible by ...
isunit 20265 Property of being a unit o...
1unit 20266 The multiplicative identit...
unitcl 20267 A unit is an element of th...
unitss 20268 The set of units is contai...
opprunit 20269 Being a unit is a symmetri...
crngunit 20270 Property of being a unit i...
dvdsunit 20271 A divisor of a unit is a u...
unitmulcl 20272 The product of units is a ...
unitmulclb 20273 Reversal of ~ unitmulcl in...
unitgrpbas 20274 The base set of the group ...
unitgrp 20275 The group of units is a gr...
unitabl 20276 The group of units of a co...
unitgrpid 20277 The identity of the group ...
unitsubm 20278 The group of units is a su...
invrfval 20281 Multiplicative inverse fun...
unitinvcl 20282 The inverse of a unit exis...
unitinvinv 20283 The inverse of the inverse...
ringinvcl 20284 The inverse of a unit is a...
unitlinv 20285 A unit times its inverse i...
unitrinv 20286 A unit times its inverse i...
1rinv 20287 The inverse of the ring un...
0unit 20288 The additive identity is a...
unitnegcl 20289 The negative of a unit is ...
ringunitnzdiv 20290 In a unitary ring, a unit ...
ring1nzdiv 20291 In a unitary ring, the rin...
dvrfval 20294 Division operation in a ri...
dvrval 20295 Division operation in a ri...
dvrcl 20296 Closure of division operat...
unitdvcl 20297 The units are closed under...
dvrid 20298 A ring element divided by ...
dvr1 20299 A ring element divided by ...
dvrass 20300 An associative law for div...
dvrcan1 20301 A cancellation law for div...
dvrcan3 20302 A cancellation law for div...
dvreq1 20303 Equality in terms of ratio...
dvrdir 20304 Distributive law for the d...
rdivmuldivd 20305 Multiplication of two rati...
ringinvdv 20306 Write the inverse function...
rngidpropd 20307 The ring unity depends onl...
dvdsrpropd 20308 The divisibility relation ...
unitpropd 20309 The set of units depends o...
invrpropd 20310 The ring inverse function ...
isirred 20311 An irreducible element of ...
isnirred 20312 The property of being a no...
isirred2 20313 Expand out the class diffe...
opprirred 20314 Irreducibility is symmetri...
irredn0 20315 The additive identity is n...
irredcl 20316 An irreducible element is ...
irrednu 20317 An irreducible element is ...
irredn1 20318 The multiplicative identit...
irredrmul 20319 The product of an irreduci...
irredlmul 20320 The product of a unit and ...
irredmul 20321 If product of two elements...
irredneg 20322 The negative of an irreduc...
irrednegb 20323 An element is irreducible ...
rnghmrcl 20330 Reverse closure of a non-u...
rnghmfn 20331 The mapping of two non-uni...
rnghmval 20332 The set of the non-unital ...
isrnghm 20333 A function is a non-unital...
isrnghmmul 20334 A function is a non-unital...
rnghmmgmhm 20335 A non-unital ring homomorp...
rnghmval2 20336 The non-unital ring homomo...
isrngim 20337 An isomorphism of non-unit...
rngimrcl 20338 Reverse closure for an iso...
rnghmghm 20339 A non-unital ring homomorp...
rnghmf 20340 A ring homomorphism is a f...
rnghmmul 20341 A homomorphism of non-unit...
isrnghm2d 20342 Demonstration of non-unita...
isrnghmd 20343 Demonstration of non-unita...
rnghmf1o 20344 A non-unital ring homomorp...
isrngim2 20345 An isomorphism of non-unit...
rngimf1o 20346 An isomorphism of non-unit...
rngimrnghm 20347 An isomorphism of non-unit...
rngimcnv 20348 The converse of an isomorp...
rnghmco 20349 The composition of non-uni...
idrnghm 20350 The identity homomorphism ...
c0mgm 20351 The constant mapping to ze...
c0mhm 20352 The constant mapping to ze...
c0ghm 20353 The constant mapping to ze...
c0snmgmhm 20354 The constant mapping to ze...
c0snmhm 20355 The constant mapping to ze...
c0snghm 20356 The constant mapping to ze...
rngisomfv1 20357 If there is a non-unital r...
rngisom1 20358 If there is a non-unital r...
rngisomring 20359 If there is a non-unital r...
rngisomring1 20360 If there is a non-unital r...
dfrhm2 20366 The property of a ring hom...
rhmrcl1 20368 Reverse closure of a ring ...
rhmrcl2 20369 Reverse closure of a ring ...
isrhm 20370 A function is a ring homom...
rhmmhm 20371 A ring homomorphism is a h...
rhmisrnghm 20372 Each unital ring homomorph...
isrim0OLD 20373 Obsolete version of ~ isri...
rimrcl 20374 Reverse closure for an iso...
isrim0 20375 A ring isomorphism is a ho...
rhmghm 20376 A ring homomorphism is an ...
rhmf 20377 A ring homomorphism is a f...
rhmmul 20378 A homomorphism of rings pr...
isrhm2d 20379 Demonstration of ring homo...
isrhmd 20380 Demonstration of ring homo...
rhm1 20381 Ring homomorphisms are req...
idrhm 20382 The identity homomorphism ...
rhmf1o 20383 A ring homomorphism is bij...
isrim 20384 An isomorphism of rings is...
isrimOLD 20385 Obsolete version of ~ isri...
rimf1o 20386 An isomorphism of rings is...
rimrhmOLD 20387 Obsolete version of ~ rimr...
rimrhm 20388 A ring isomorphism is a ho...
rimgim 20389 An isomorphism of rings is...
rimisrngim 20390 Each unital ring isomorphi...
rhmfn 20391 The mapping of two rings t...
rhmval 20392 The ring homomorphisms bet...
rhmco 20393 The composition of ring ho...
pwsco1rhm 20394 Right composition with a f...
pwsco2rhm 20395 Left composition with a ri...
brric 20396 The relation "is isomorphi...
brrici 20397 Prove isomorphic by an exp...
brric2 20398 The relation "is isomorphi...
ricgic 20399 If two rings are (ring) is...
rhmdvdsr 20400 A ring homomorphism preser...
rhmopp 20401 A ring homomorphism is als...
elrhmunit 20402 Ring homomorphisms preserv...
rhmunitinv 20403 Ring homomorphisms preserv...
isnzr 20406 Property of a nonzero ring...
nzrnz 20407 One and zero are different...
nzrring 20408 A nonzero ring is a ring. ...
nzrringOLD 20409 Obsolete version of ~ nzrr...
isnzr2 20410 Equivalent characterizatio...
isnzr2hash 20411 Equivalent characterizatio...
opprnzr 20412 The opposite of a nonzero ...
ringelnzr 20413 A ring is nonzero if it ha...
nzrunit 20414 A unit is nonzero in any n...
0ringnnzr 20415 A ring is a zero ring iff ...
0ring 20416 If a ring has only one ele...
0ringdif 20417 A zero ring is a ring whic...
0ringbas 20418 The base set of a zero rin...
0ring01eq 20419 In a ring with only one el...
01eq0ring 20420 If the zero and the identi...
01eq0ringOLD 20421 Obsolete version of ~ 01eq...
0ring01eqbi 20422 In a unital ring the zero ...
0ring1eq0 20423 In a zero ring, a ring whi...
c0rhm 20424 The constant mapping to ze...
c0rnghm 20425 The constant mapping to ze...
zrrnghm 20426 The constant mapping to ze...
nrhmzr 20427 There is no ring homomorph...
islring 20430 The predicate "is a local ...
lringnzr 20431 A local ring is a nonzero ...
lringring 20432 A local ring is a ring. (...
lringnz 20433 A local ring is a nonzero ...
lringuplu 20434 If the sum of two elements...
issubrng 20437 The subring of non-unital ...
subrngss 20438 A subring is a subset. (C...
subrngid 20439 Every non-unital ring is a...
subrngrng 20440 A subring is a non-unital ...
subrngrcl 20441 Reverse closure for a subr...
subrngsubg 20442 A subring is a subgroup. ...
subrngringnsg 20443 A subring is a normal subg...
subrngbas 20444 Base set of a subring stru...
subrng0 20445 A subring always has the s...
subrngacl 20446 A subring is closed under ...
subrngmcl 20447 A subgroup is closed under...
issubrng2 20448 Characterize the subrings ...
opprsubrng 20449 Being a subring is a symme...
subrngint 20450 The intersection of a none...
subrngin 20451 The intersection of two su...
subrngmre 20452 The subrings of a non-unit...
subsubrng 20453 A subring of a subring is ...
subsubrng2 20454 The set of subrings of a s...
rhmimasubrnglem 20455 Lemma for ~ rhmimasubrng :...
rhmimasubrng 20456 The homomorphic image of a...
cntzsubrng 20457 Centralizers in a non-unit...
subrngpropd 20458 If two structures have the...
issubrg 20463 The subring predicate. (C...
subrgss 20464 A subring is a subset. (C...
subrgid 20465 Every ring is a subring of...
subrgring 20466 A subring is a ring. (Con...
subrgcrng 20467 A subring of a commutative...
subrgrcl 20468 Reverse closure for a subr...
subrgsubg 20469 A subring is a subgroup. ...
subrgsubrng 20470 A subring of a unital ring...
subrg0 20471 A subring always has the s...
subrg1cl 20472 A subring contains the mul...
subrgbas 20473 Base set of a subring stru...
subrg1 20474 A subring always has the s...
subrgacl 20475 A subring is closed under ...
subrgmcl 20476 A subgroup is closed under...
subrgsubm 20477 A subring is a submonoid o...
subrgdvds 20478 If an element divides anot...
subrguss 20479 A unit of a subring is a u...
subrginv 20480 A subring always has the s...
subrgdv 20481 A subring always has the s...
subrgunit 20482 An element of a ring is a ...
subrgugrp 20483 The units of a subring for...
issubrg2 20484 Characterize the subrings ...
opprsubrg 20485 Being a subring is a symme...
subrgnzr 20486 A subring of a nonzero rin...
subrgint 20487 The intersection of a none...
subrgin 20488 The intersection of two su...
subrgmre 20489 The subrings of a ring are...
subsubrg 20490 A subring of a subring is ...
subsubrg2 20491 The set of subrings of a s...
issubrg3 20492 A subring is an additive s...
resrhm 20493 Restriction of a ring homo...
resrhm2b 20494 Restriction of the codomai...
rhmeql 20495 The equalizer of two ring ...
rhmima 20496 The homomorphic image of a...
rnrhmsubrg 20497 The range of a ring homomo...
cntzsubr 20498 Centralizers in a ring are...
pwsdiagrhm 20499 Diagonal homomorphism into...
subrgpropd 20500 If two structures have the...
rhmpropd 20501 Ring homomorphism depends ...
rngcval 20504 Value of the category of n...
rnghmresfn 20505 The class of non-unital ri...
rnghmresel 20506 An element of the non-unit...
rngcbas 20507 Set of objects of the cate...
rngchomfval 20508 Set of arrows of the categ...
rngchom 20509 Set of arrows of the categ...
elrngchom 20510 A morphism of non-unital r...
rngchomfeqhom 20511 The functionalized Hom-set...
rngccofval 20512 Composition in the categor...
rngcco 20513 Composition in the categor...
dfrngc2 20514 Alternate definition of th...
rnghmsscmap2 20515 The non-unital ring homomo...
rnghmsscmap 20516 The non-unital ring homomo...
rnghmsubcsetclem1 20517 Lemma 1 for ~ rnghmsubcset...
rnghmsubcsetclem2 20518 Lemma 2 for ~ rnghmsubcset...
rnghmsubcsetc 20519 The non-unital ring homomo...
rngccat 20520 The category of non-unital...
rngcid 20521 The identity arrow in the ...
rngcsect 20522 A section in the category ...
rngcinv 20523 An inverse in the category...
rngciso 20524 An isomorphism in the cate...
rngcifuestrc 20525 The "inclusion functor" fr...
funcrngcsetc 20526 The "natural forgetful fun...
funcrngcsetcALT 20527 Alternate proof of ~ funcr...
zrinitorngc 20528 The zero ring is an initia...
zrtermorngc 20529 The zero ring is a termina...
zrzeroorngc 20530 The zero ring is a zero ob...
ringcval 20533 Value of the category of u...
rhmresfn 20534 The class of unital ring h...
rhmresel 20535 An element of the unital r...
ringcbas 20536 Set of objects of the cate...
ringchomfval 20537 Set of arrows of the categ...
ringchom 20538 Set of arrows of the categ...
elringchom 20539 A morphism of unital rings...
ringchomfeqhom 20540 The functionalized Hom-set...
ringccofval 20541 Composition in the categor...
ringcco 20542 Composition in the categor...
dfringc2 20543 Alternate definition of th...
rhmsscmap2 20544 The unital ring homomorphi...
rhmsscmap 20545 The unital ring homomorphi...
rhmsubcsetclem1 20546 Lemma 1 for ~ rhmsubcsetc ...
rhmsubcsetclem2 20547 Lemma 2 for ~ rhmsubcsetc ...
rhmsubcsetc 20548 The unital ring homomorphi...
ringccat 20549 The category of unital rin...
ringcid 20550 The identity arrow in the ...
rhmsscrnghm 20551 The unital ring homomorphi...
rhmsubcrngclem1 20552 Lemma 1 for ~ rhmsubcrngc ...
rhmsubcrngclem2 20553 Lemma 2 for ~ rhmsubcrngc ...
rhmsubcrngc 20554 The unital ring homomorphi...
rngcresringcat 20555 The restriction of the cat...
ringcsect 20556 A section in the category ...
ringcinv 20557 An inverse in the category...
ringciso 20558 An isomorphism in the cate...
ringcbasbas 20559 An element of the base set...
funcringcsetc 20560 The "natural forgetful fun...
zrtermoringc 20561 The zero ring is a termina...
zrninitoringc 20562 The zero ring is not an in...
srhmsubclem1 20563 Lemma 1 for ~ srhmsubc . ...
srhmsubclem2 20564 Lemma 2 for ~ srhmsubc . ...
srhmsubclem3 20565 Lemma 3 for ~ srhmsubc . ...
srhmsubc 20566 According to ~ df-subc , t...
sringcat 20567 The restriction of the cat...
crhmsubc 20568 According to ~ df-subc , t...
cringcat 20569 The restriction of the cat...
rngcrescrhm 20570 The category of non-unital...
rhmsubclem1 20571 Lemma 1 for ~ rhmsubc . (...
rhmsubclem2 20572 Lemma 2 for ~ rhmsubc . (...
rhmsubclem3 20573 Lemma 3 for ~ rhmsubc . (...
rhmsubclem4 20574 Lemma 4 for ~ rhmsubc . (...
rhmsubc 20575 According to ~ df-subc , t...
rhmsubccat 20576 The restriction of the cat...
isdrng 20581 The predicate "is a divisi...
drngunit 20582 Elementhood in the set of ...
drngui 20583 The set of units of a divi...
drngring 20584 A division ring is a ring....
drngringd 20585 A division ring is a ring....
drnggrpd 20586 A division ring is a group...
drnggrp 20587 A division ring is a group...
isfld 20588 A field is a commutative d...
flddrngd 20589 A field is a division ring...
fldcrngd 20590 A field is a commutative r...
isdrng2 20591 A division ring can equiva...
drngprop 20592 If two structures have the...
drngmgp 20593 A division ring contains a...
drngmcl 20594 The product of two nonzero...
drngid 20595 A division ring's unity is...
drngunz 20596 A division ring's unity is...
drngnzr 20597 All division rings are non...
drngid2 20598 Properties showing that an...
drnginvrcl 20599 Closure of the multiplicat...
drnginvrn0 20600 The multiplicative inverse...
drnginvrcld 20601 Closure of the multiplicat...
drnginvrl 20602 Property of the multiplica...
drnginvrr 20603 Property of the multiplica...
drnginvrld 20604 Property of the multiplica...
drnginvrrd 20605 Property of the multiplica...
drngmul0or 20606 A product is zero iff one ...
drngmulne0 20607 A product is nonzero iff b...
drngmuleq0 20608 An element is zero iff its...
opprdrng 20609 The opposite of a division...
isdrngd 20610 Properties that characteri...
isdrngrd 20611 Properties that characteri...
isdrngdOLD 20612 Obsolete version of ~ isdr...
isdrngrdOLD 20613 Obsolete version of ~ isdr...
drngpropd 20614 If two structures have the...
fldpropd 20615 If two structures have the...
rng1nnzr 20616 The (smallest) structure r...
ring1zr 20617 The only (unital) ring wit...
rngen1zr 20618 The only (unital) ring wit...
ringen1zr 20619 The only unital ring with ...
rng1nfld 20620 The zero ring is not a fie...
issubdrg 20621 Characterize the subfields...
drhmsubc 20622 According to ~ df-subc , t...
drngcat 20623 The restriction of the cat...
fldcat 20624 The restriction of the cat...
fldc 20625 The restriction of the cat...
fldhmsubc 20626 According to ~ df-subc , t...
issdrg 20629 Property of a division sub...
sdrgrcl 20630 Reverse closure for a sub-...
sdrgdrng 20631 A sub-division-ring is a d...
sdrgsubrg 20632 A sub-division-ring is a s...
sdrgid 20633 Every division ring is a d...
sdrgss 20634 A division subring is a su...
sdrgbas 20635 Base set of a sub-division...
issdrg2 20636 Property of a division sub...
sdrgunit 20637 A unit of a sub-division-r...
imadrhmcl 20638 The image of a (nontrivial...
fldsdrgfld 20639 A sub-division-ring of a f...
acsfn1p 20640 Construction of a closure ...
subrgacs 20641 Closure property of subrin...
sdrgacs 20642 Closure property of divisi...
cntzsdrg 20643 Centralizers in division r...
subdrgint 20644 The intersection of a none...
sdrgint 20645 The intersection of a none...
primefld 20646 The smallest sub division ...
primefld0cl 20647 The prime field contains t...
primefld1cl 20648 The prime field contains t...
abvfval 20651 Value of the set of absolu...
isabv 20652 Elementhood in the set of ...
isabvd 20653 Properties that determine ...
abvrcl 20654 Reverse closure for the ab...
abvfge0 20655 An absolute value is a fun...
abvf 20656 An absolute value is a fun...
abvcl 20657 An absolute value is a fun...
abvge0 20658 The absolute value of a nu...
abveq0 20659 The value of an absolute v...
abvne0 20660 The absolute value of a no...
abvgt0 20661 The absolute value of a no...
abvmul 20662 An absolute value distribu...
abvtri 20663 An absolute value satisfie...
abv0 20664 The absolute value of zero...
abv1z 20665 The absolute value of one ...
abv1 20666 The absolute value of one ...
abvneg 20667 The absolute value of a ne...
abvsubtri 20668 An absolute value satisfie...
abvrec 20669 The absolute value distrib...
abvdiv 20670 The absolute value distrib...
abvdom 20671 Any ring with an absolute ...
abvres 20672 The restriction of an abso...
abvtrivd 20673 The trivial absolute value...
abvtriv 20674 The trivial absolute value...
abvpropd 20675 If two structures have the...
staffval 20680 The functionalization of t...
stafval 20681 The functionalization of t...
staffn 20682 The functionalization is e...
issrng 20683 The predicate "is a star r...
srngrhm 20684 The involution function in...
srngring 20685 A star ring is a ring. (C...
srngcnv 20686 The involution function in...
srngf1o 20687 The involution function in...
srngcl 20688 The involution function in...
srngnvl 20689 The involution function in...
srngadd 20690 The involution function in...
srngmul 20691 The involution function in...
srng1 20692 The conjugate of the ring ...
srng0 20693 The conjugate of the ring ...
issrngd 20694 Properties that determine ...
idsrngd 20695 A commutative ring is a st...
islmod 20700 The predicate "is a left m...
lmodlema 20701 Lemma for properties of a ...
islmodd 20702 Properties that determine ...
lmodgrp 20703 A left module is a group. ...
lmodring 20704 The scalar component of a ...
lmodfgrp 20705 The scalar component of a ...
lmodgrpd 20706 A left module is a group. ...
lmodbn0 20707 The base set of a left mod...
lmodacl 20708 Closure of ring addition f...
lmodmcl 20709 Closure of ring multiplica...
lmodsn0 20710 The set of scalars in a le...
lmodvacl 20711 Closure of vector addition...
lmodass 20712 Left module vector sum is ...
lmodlcan 20713 Left cancellation law for ...
lmodvscl 20714 Closure of scalar product ...
lmodvscld 20715 Closure of scalar product ...
scaffval 20716 The scalar multiplication ...
scafval 20717 The scalar multiplication ...
scafeq 20718 If the scalar multiplicati...
scaffn 20719 The scalar multiplication ...
lmodscaf 20720 The scalar multiplication ...
lmodvsdi 20721 Distributive law for scala...
lmodvsdir 20722 Distributive law for scala...
lmodvsass 20723 Associative law for scalar...
lmod0cl 20724 The ring zero in a left mo...
lmod1cl 20725 The ring unity in a left m...
lmodvs1 20726 Scalar product with the ri...
lmod0vcl 20727 The zero vector is a vecto...
lmod0vlid 20728 Left identity law for the ...
lmod0vrid 20729 Right identity law for the...
lmod0vid 20730 Identity equivalent to the...
lmod0vs 20731 Zero times a vector is the...
lmodvs0 20732 Anything times the zero ve...
lmodvsmmulgdi 20733 Distributive law for a gro...
lmodfopnelem1 20734 Lemma 1 for ~ lmodfopne . ...
lmodfopnelem2 20735 Lemma 2 for ~ lmodfopne . ...
lmodfopne 20736 The (functionalized) opera...
lcomf 20737 A linear-combination sum i...
lcomfsupp 20738 A linear-combination sum i...
lmodvnegcl 20739 Closure of vector negative...
lmodvnegid 20740 Addition of a vector with ...
lmodvneg1 20741 Minus 1 times a vector is ...
lmodvsneg 20742 Multiplication of a vector...
lmodvsubcl 20743 Closure of vector subtract...
lmodcom 20744 Left module vector sum is ...
lmodabl 20745 A left module is an abelia...
lmodcmn 20746 A left module is a commuta...
lmodnegadd 20747 Distribute negation throug...
lmod4 20748 Commutative/associative la...
lmodvsubadd 20749 Relationship between vecto...
lmodvaddsub4 20750 Vector addition/subtractio...
lmodvpncan 20751 Addition/subtraction cance...
lmodvnpcan 20752 Cancellation law for vecto...
lmodvsubval2 20753 Value of vector subtractio...
lmodsubvs 20754 Subtraction of a scalar pr...
lmodsubdi 20755 Scalar multiplication dist...
lmodsubdir 20756 Scalar multiplication dist...
lmodsubeq0 20757 If the difference between ...
lmodsubid 20758 Subtraction of a vector fr...
lmodvsghm 20759 Scalar multiplication of t...
lmodprop2d 20760 If two structures have the...
lmodpropd 20761 If two structures have the...
gsumvsmul 20762 Pull a scalar multiplicati...
mptscmfsupp0 20763 A mapping to a scalar prod...
mptscmfsuppd 20764 A function mapping to a sc...
rmodislmodlem 20765 Lemma for ~ rmodislmod . ...
rmodislmod 20766 The right module ` R ` ind...
rmodislmodOLD 20767 Obsolete version of ~ rmod...
lssset 20770 The set of all (not necess...
islss 20771 The predicate "is a subspa...
islssd 20772 Properties that determine ...
lssss 20773 A subspace is a set of vec...
lssel 20774 A subspace member is a vec...
lss1 20775 The set of vectors in a le...
lssuni 20776 The union of all subspaces...
lssn0 20777 A subspace is not empty. ...
00lss 20778 The empty structure has no...
lsscl 20779 Closure property of a subs...
lssvacl 20780 Closure of vector addition...
lssvsubcl 20781 Closure of vector subtract...
lssvancl1 20782 Non-closure: if one vector...
lssvancl2 20783 Non-closure: if one vector...
lss0cl 20784 The zero vector belongs to...
lsssn0 20785 The singleton of the zero ...
lss0ss 20786 The zero subspace is inclu...
lssle0 20787 No subspace is smaller tha...
lssne0 20788 A nonzero subspace has a n...
lssvneln0 20789 A vector ` X ` which doesn...
lssneln0 20790 A vector ` X ` which doesn...
lssssr 20791 Conclude subspace ordering...
lssvscl 20792 Closure of scalar product ...
lssvnegcl 20793 Closure of negative vector...
lsssubg 20794 All subspaces are subgroup...
lsssssubg 20795 All subspaces are subgroup...
islss3 20796 A linear subspace of a mod...
lsslmod 20797 A submodule is a module. ...
lsslss 20798 The subspaces of a subspac...
islss4 20799 A linear subspace is a sub...
lss1d 20800 One-dimensional subspace (...
lssintcl 20801 The intersection of a none...
lssincl 20802 The intersection of two su...
lssmre 20803 The subspaces of a module ...
lssacs 20804 Submodules are an algebrai...
prdsvscacl 20805 Pointwise scalar multiplic...
prdslmodd 20806 The product of a family of...
pwslmod 20807 A structure power of a lef...
lspfval 20810 The span function for a le...
lspf 20811 The span function on a lef...
lspval 20812 The span of a set of vecto...
lspcl 20813 The span of a set of vecto...
lspsncl 20814 The span of a singleton is...
lspprcl 20815 The span of a pair is a su...
lsptpcl 20816 The span of an unordered t...
lspsnsubg 20817 The span of a singleton is...
00lsp 20818 ~ fvco4i lemma for linear ...
lspid 20819 The span of a subspace is ...
lspssv 20820 A span is a set of vectors...
lspss 20821 Span preserves subset orde...
lspssid 20822 A set of vectors is a subs...
lspidm 20823 The span of a set of vecto...
lspun 20824 The span of union is the s...
lspssp 20825 If a set of vectors is a s...
mrclsp 20826 Moore closure generalizes ...
lspsnss 20827 The span of the singleton ...
lspsnel3 20828 A member of the span of th...
lspprss 20829 The span of a pair of vect...
lspsnid 20830 A vector belongs to the sp...
lspsnel6 20831 Relationship between a vec...
lspsnel5 20832 Relationship between a vec...
lspsnel5a 20833 Relationship between a vec...
lspprid1 20834 A member of a pair of vect...
lspprid2 20835 A member of a pair of vect...
lspprvacl 20836 The sum of two vectors bel...
lssats2 20837 A way to express atomistic...
lspsneli 20838 A scalar product with a ve...
lspsn 20839 Span of the singleton of a...
lspsnel 20840 Member of span of the sing...
lspsnvsi 20841 Span of a scalar product o...
lspsnss2 20842 Comparable spans of single...
lspsnneg 20843 Negation does not change t...
lspsnsub 20844 Swapping subtraction order...
lspsn0 20845 Span of the singleton of t...
lsp0 20846 Span of the empty set. (C...
lspuni0 20847 Union of the span of the e...
lspun0 20848 The span of a union with t...
lspsneq0 20849 Span of the singleton is t...
lspsneq0b 20850 Equal singleton spans impl...
lmodindp1 20851 Two independent (non-colin...
lsslsp 20852 Spans in submodules corres...
lsslspOLD 20853 Obsolete version of ~ lssl...
lss0v 20854 The zero vector in a submo...
lsspropd 20855 If two structures have the...
lsppropd 20856 If two structures have the...
reldmlmhm 20863 Lemma for module homomorph...
lmimfn 20864 Lemma for module isomorphi...
islmhm 20865 Property of being a homomo...
islmhm3 20866 Property of a module homom...
lmhmlem 20867 Non-quantified consequence...
lmhmsca 20868 A homomorphism of left mod...
lmghm 20869 A homomorphism of left mod...
lmhmlmod2 20870 A homomorphism of left mod...
lmhmlmod1 20871 A homomorphism of left mod...
lmhmf 20872 A homomorphism of left mod...
lmhmlin 20873 A homomorphism of left mod...
lmodvsinv 20874 Multiplication of a vector...
lmodvsinv2 20875 Multiplying a negated vect...
islmhm2 20876 A one-equation proof of li...
islmhmd 20877 Deduction for a module hom...
0lmhm 20878 The constant zero linear f...
idlmhm 20879 The identity function on a...
invlmhm 20880 The negative function on a...
lmhmco 20881 The composition of two mod...
lmhmplusg 20882 The pointwise sum of two l...
lmhmvsca 20883 The pointwise scalar produ...
lmhmf1o 20884 A bijective module homomor...
lmhmima 20885 The image of a subspace un...
lmhmpreima 20886 The inverse image of a sub...
lmhmlsp 20887 Homomorphisms preserve spa...
lmhmrnlss 20888 The range of a homomorphis...
lmhmkerlss 20889 The kernel of a homomorphi...
reslmhm 20890 Restriction of a homomorph...
reslmhm2 20891 Expansion of the codomain ...
reslmhm2b 20892 Expansion of the codomain ...
lmhmeql 20893 The equalizer of two modul...
lspextmo 20894 A linear function is compl...
pwsdiaglmhm 20895 Diagonal homomorphism into...
pwssplit0 20896 Splitting for structure po...
pwssplit1 20897 Splitting for structure po...
pwssplit2 20898 Splitting for structure po...
pwssplit3 20899 Splitting for structure po...
islmim 20900 An isomorphism of left mod...
lmimf1o 20901 An isomorphism of left mod...
lmimlmhm 20902 An isomorphism of modules ...
lmimgim 20903 An isomorphism of modules ...
islmim2 20904 An isomorphism of left mod...
lmimcnv 20905 The converse of a bijectiv...
brlmic 20906 The relation "is isomorphi...
brlmici 20907 Prove isomorphic by an exp...
lmiclcl 20908 Isomorphism implies the le...
lmicrcl 20909 Isomorphism implies the ri...
lmicsym 20910 Module isomorphism is symm...
lmhmpropd 20911 Module homomorphism depend...
islbs 20914 The predicate " ` B ` is a...
lbsss 20915 A basis is a set of vector...
lbsel 20916 An element of a basis is a...
lbssp 20917 The span of a basis is the...
lbsind 20918 A basis is linearly indepe...
lbsind2 20919 A basis is linearly indepe...
lbspss 20920 No proper subset of a basi...
lsmcl 20921 The sum of two subspaces i...
lsmspsn 20922 Member of subspace sum of ...
lsmelval2 20923 Subspace sum membership in...
lsmsp 20924 Subspace sum in terms of s...
lsmsp2 20925 Subspace sum of spans of s...
lsmssspx 20926 Subspace sum (in its exten...
lsmpr 20927 The span of a pair of vect...
lsppreli 20928 A vector expressed as a su...
lsmelpr 20929 Two ways to say that a vec...
lsppr0 20930 The span of a vector paire...
lsppr 20931 Span of a pair of vectors....
lspprel 20932 Member of the span of a pa...
lspprabs 20933 Absorption of vector sum i...
lspvadd 20934 The span of a vector sum i...
lspsntri 20935 Triangle-type inequality f...
lspsntrim 20936 Triangle-type inequality f...
lbspropd 20937 If two structures have the...
pj1lmhm 20938 The left projection functi...
pj1lmhm2 20939 The left projection functi...
islvec 20942 The predicate "is a left v...
lvecdrng 20943 The set of scalars of a le...
lveclmod 20944 A left vector space is a l...
lveclmodd 20945 A vector space is a left m...
lvecgrpd 20946 A vector space is a group....
lsslvec 20947 A vector subspace is a vec...
lmhmlvec 20948 The property for modules t...
lvecvs0or 20949 If a scalar product is zer...
lvecvsn0 20950 A scalar product is nonzer...
lssvs0or 20951 If a scalar product belong...
lvecvscan 20952 Cancellation law for scala...
lvecvscan2 20953 Cancellation law for scala...
lvecinv 20954 Invert coefficient of scal...
lspsnvs 20955 A nonzero scalar product d...
lspsneleq 20956 Membership relation that i...
lspsncmp 20957 Comparable spans of nonzer...
lspsnne1 20958 Two ways to express that v...
lspsnne2 20959 Two ways to express that v...
lspsnnecom 20960 Swap two vectors with diff...
lspabs2 20961 Absorption law for span of...
lspabs3 20962 Absorption law for span of...
lspsneq 20963 Equal spans of singletons ...
lspsneu 20964 Nonzero vectors with equal...
lspsnel4 20965 A member of the span of th...
lspdisj 20966 The span of a vector not i...
lspdisjb 20967 A nonzero vector is not in...
lspdisj2 20968 Unequal spans are disjoint...
lspfixed 20969 Show membership in the spa...
lspexch 20970 Exchange property for span...
lspexchn1 20971 Exchange property for span...
lspexchn2 20972 Exchange property for span...
lspindpi 20973 Partial independence prope...
lspindp1 20974 Alternate way to say 3 vec...
lspindp2l 20975 Alternate way to say 3 vec...
lspindp2 20976 Alternate way to say 3 vec...
lspindp3 20977 Independence of 2 vectors ...
lspindp4 20978 (Partial) independence of ...
lvecindp 20979 Compute the ` X ` coeffici...
lvecindp2 20980 Sums of independent vector...
lspsnsubn0 20981 Unequal singleton spans im...
lsmcv 20982 Subspace sum has the cover...
lspsolvlem 20983 Lemma for ~ lspsolv . (Co...
lspsolv 20984 If ` X ` is in the span of...
lssacsex 20985 In a vector space, subspac...
lspsnat 20986 There is no subspace stric...
lspsncv0 20987 The span of a singleton co...
lsppratlem1 20988 Lemma for ~ lspprat . Let...
lsppratlem2 20989 Lemma for ~ lspprat . Sho...
lsppratlem3 20990 Lemma for ~ lspprat . In ...
lsppratlem4 20991 Lemma for ~ lspprat . In ...
lsppratlem5 20992 Lemma for ~ lspprat . Com...
lsppratlem6 20993 Lemma for ~ lspprat . Neg...
lspprat 20994 A proper subspace of the s...
islbs2 20995 An equivalent formulation ...
islbs3 20996 An equivalent formulation ...
lbsacsbs 20997 Being a basis in a vector ...
lvecdim 20998 The dimension theorem for ...
lbsextlem1 20999 Lemma for ~ lbsext . The ...
lbsextlem2 21000 Lemma for ~ lbsext . Sinc...
lbsextlem3 21001 Lemma for ~ lbsext . A ch...
lbsextlem4 21002 Lemma for ~ lbsext . ~ lbs...
lbsextg 21003 For any linearly independe...
lbsext 21004 For any linearly independe...
lbsexg 21005 Every vector space has a b...
lbsex 21006 Every vector space has a b...
lvecprop2d 21007 If two structures have the...
lvecpropd 21008 If two structures have the...
sraval 21013 Lemma for ~ srabase throug...
sralem 21014 Lemma for ~ srabase and si...
sralemOLD 21015 Obsolete version of ~ sral...
srabase 21016 Base set of a subring alge...
srabaseOLD 21017 Obsolete proof of ~ srabas...
sraaddg 21018 Additive operation of a su...
sraaddgOLD 21019 Obsolete proof of ~ sraadd...
sramulr 21020 Multiplicative operation o...
sramulrOLD 21021 Obsolete proof of ~ sramul...
srasca 21022 The set of scalars of a su...
srascaOLD 21023 Obsolete proof of ~ srasca...
sravsca 21024 The scalar product operati...
sravscaOLD 21025 Obsolete proof of ~ sravsc...
sraip 21026 The inner product operatio...
sratset 21027 Topology component of a su...
sratsetOLD 21028 Obsolete proof of ~ sratse...
sratopn 21029 Topology component of a su...
srads 21030 Distance function of a sub...
sradsOLD 21031 Obsolete proof of ~ srads ...
sraring 21032 Condition for a subring al...
sralmod 21033 The subring algebra is a l...
sralmod0 21034 The subring module inherit...
issubrgd 21035 Prove a subring by closure...
rlmfn 21036 ` ringLMod ` is a function...
rlmval 21037 Value of the ring module. ...
rlmval2 21038 Value of the ring module e...
rlmbas 21039 Base set of the ring modul...
rlmplusg 21040 Vector addition in the rin...
rlm0 21041 Zero vector in the ring mo...
rlmsub 21042 Subtraction in the ring mo...
rlmmulr 21043 Ring multiplication in the...
rlmsca 21044 Scalars in the ring module...
rlmsca2 21045 Scalars in the ring module...
rlmvsca 21046 Scalar multiplication in t...
rlmtopn 21047 Topology component of the ...
rlmds 21048 Metric component of the ri...
rlmlmod 21049 The ring module is a modul...
rlmlvec 21050 The ring module over a div...
rlmlsm 21051 Subgroup sum of the ring m...
rlmvneg 21052 Vector negation in the rin...
rlmscaf 21053 Functionalized scalar mult...
ixpsnbasval 21054 The value of an infinite C...
lidlval 21059 Value of the set of ring i...
rspval 21060 Value of the ring span fun...
lidlss 21061 An ideal is a subset of th...
lidlssbas 21062 The base set of the restri...
lidlbas 21063 A (left) ideal of a ring i...
islidl 21064 Predicate of being a (left...
rnglidlmcl 21065 A (left) ideal containing ...
rngridlmcl 21066 A right ideal (which is a ...
dflidl2rng 21067 Alternate (the usual textb...
isridlrng 21068 A right ideal is a left id...
lidl0cl 21069 An ideal contains 0. (Con...
lidlacl 21070 An ideal is closed under a...
lidlnegcl 21071 An ideal contains negative...
lidlsubg 21072 An ideal is a subgroup of ...
lidlsubcl 21073 An ideal is closed under s...
lidlmcl 21074 An ideal is closed under l...
lidl1el 21075 An ideal contains 1 iff it...
dflidl2 21076 Alternate (the usual textb...
lidl0ALT 21077 Alternate proof for ~ lidl...
rnglidl0 21078 Every non-unital ring cont...
lidl0 21079 Every ring contains a zero...
lidl1ALT 21080 Alternate proof for ~ lidl...
rnglidl1 21081 The base set of every non-...
lidl1 21082 Every ring contains a unit...
lidlacs 21083 The ideal system is an alg...
rspcl 21084 The span of a set of ring ...
rspssid 21085 The span of a set of ring ...
rsp1 21086 The span of the identity e...
rsp0 21087 The span of the zero eleme...
rspssp 21088 The ideal span of a set of...
mrcrsp 21089 Moore closure generalizes ...
lidlnz 21090 A nonzero ideal contains a...
drngnidl 21091 A division ring has only t...
lidlrsppropd 21092 The left ideals and ring s...
rnglidlmmgm 21093 The multiplicative group o...
rnglidlmsgrp 21094 The multiplicative group o...
rnglidlrng 21095 A (left) ideal of a non-un...
2idlval 21098 Definition of a two-sided ...
isridl 21099 A right ideal is a left id...
2idlelb 21100 Membership in a two-sided ...
2idllidld 21101 A two-sided ideal is a lef...
2idlridld 21102 A two-sided ideal is a rig...
df2idl2rng 21103 Alternate (the usual textb...
df2idl2 21104 Alternate (the usual textb...
ridl0 21105 Every ring contains a zero...
ridl1 21106 Every ring contains a unit...
2idl0 21107 Every ring contains a zero...
2idl1 21108 Every ring contains a unit...
2idlss 21109 A two-sided ideal is a sub...
2idlbas 21110 The base set of a two-side...
2idlelbas 21111 The base set of a two-side...
rng2idlsubrng 21112 A two-sided ideal of a non...
rng2idlnsg 21113 A two-sided ideal of a non...
rng2idl0 21114 The zero (additive identit...
rng2idlsubgsubrng 21115 A two-sided ideal of a non...
rng2idlsubgnsg 21116 A two-sided ideal of a non...
rng2idlsubg0 21117 The zero (additive identit...
2idlcpblrng 21118 The coset equivalence rela...
2idlcpbl 21119 The coset equivalence rela...
qus2idrng 21120 The quotient of a non-unit...
qus1 21121 The multiplicative identit...
qusring 21122 If ` S ` is a two-sided id...
qusrhm 21123 If ` S ` is a two-sided id...
qusmul2 21124 Value of the ring operatio...
crngridl 21125 In a commutative ring, the...
crng2idl 21126 In a commutative ring, a t...
qusmulrng 21127 Value of the multiplicatio...
quscrng 21128 The quotient of a commutat...
rngqiprng1elbas 21129 The ring unity of a two-si...
rngqiprngghmlem1 21130 Lemma 1 for ~ rngqiprngghm...
rngqiprngghmlem2 21131 Lemma 2 for ~ rngqiprngghm...
rngqiprngghmlem3 21132 Lemma 3 for ~ rngqiprngghm...
rngqiprngimfolem 21133 Lemma for ~ rngqiprngimfo ...
rngqiprnglinlem1 21134 Lemma 1 for ~ rngqiprnglin...
rngqiprnglinlem2 21135 Lemma 2 for ~ rngqiprnglin...
rngqiprnglinlem3 21136 Lemma 3 for ~ rngqiprnglin...
rngqiprngimf1lem 21137 Lemma for ~ rngqiprngimf1 ...
rngqipbas 21138 The base set of the produc...
rngqiprng 21139 The product of the quotien...
rngqiprngimf 21140 ` F ` is a function from (...
rngqiprngimfv 21141 The value of the function ...
rngqiprngghm 21142 ` F ` is a homomorphism of...
rngqiprngimf1 21143 ` F ` is a one-to-one func...
rngqiprngimfo 21144 ` F ` is a function from (...
rngqiprnglin 21145 ` F ` is linear with respe...
rngqiprngho 21146 ` F ` is a homomorphism of...
rngqiprngim 21147 ` F ` is an isomorphism of...
rng2idl1cntr 21148 The unity of a two-sided i...
rngringbdlem1 21149 In a unital ring, the quot...
rngringbdlem2 21150 A non-unital ring is unita...
rngringbd 21151 A non-unital ring is unita...
ring2idlqus 21152 For every unital ring ther...
ring2idlqusb 21153 A non-unital ring is unita...
rngqiprngfulem1 21154 Lemma 1 for ~ rngqiprngfu ...
rngqiprngfulem2 21155 Lemma 2 for ~ rngqiprngfu ...
rngqiprngfulem3 21156 Lemma 3 for ~ rngqiprngfu ...
rngqiprngfulem4 21157 Lemma 4 for ~ rngqiprngfu ...
rngqiprngfulem5 21158 Lemma 5 for ~ rngqiprngfu ...
rngqipring1 21159 The ring unity of the prod...
rngqiprngfu 21160 The function value of ` F ...
rngqiprngu 21161 If a non-unital ring has a...
ring2idlqus1 21162 If a non-unital ring has a...
lpival 21167 Value of the set of princi...
islpidl 21168 Property of being a princi...
lpi0 21169 The zero ideal is always p...
lpi1 21170 The unit ideal is always p...
islpir 21171 Principal ideal rings are ...
lpiss 21172 Principal ideals are a sub...
islpir2 21173 Principal ideal rings are ...
lpirring 21174 Principal ideal rings are ...
drnglpir 21175 Division rings are princip...
rspsn 21176 Membership in principal id...
lidldvgen 21177 An element generates an id...
lpigen 21178 An ideal is principal iff ...
rrgval 21187 Value of the set or left-r...
isrrg 21188 Membership in the set of l...
rrgeq0i 21189 Property of a left-regular...
rrgeq0 21190 Left-multiplication by a l...
rrgsupp 21191 Left multiplication by a l...
rrgss 21192 Left-regular elements are ...
unitrrg 21193 Units are regular elements...
isdomn 21194 Expand definition of a dom...
domnnzr 21195 A domain is a nonzero ring...
domnring 21196 A domain is a ring. (Cont...
domneq0 21197 In a domain, a product is ...
domnmuln0 21198 In a domain, a product of ...
isdomn2 21199 A ring is a domain iff all...
domnrrg 21200 In a domain, any nonzero e...
isdomn5 21201 The right conjunct in the ...
isdomn4 21202 A ring is a domain iff it ...
opprdomn 21203 The opposite of a domain i...
abvn0b 21204 Another characterization o...
drngdomn 21205 A division ring is a domai...
isidom 21206 An integral domain is a co...
fldidom 21207 A field is an integral dom...
fldidomOLD 21208 Obsolete version of ~ fldi...
fidomndrnglem 21209 Lemma for ~ fidomndrng . ...
fidomndrng 21210 A finite domain is a divis...
fiidomfld 21211 A finite integral domain i...
cnfldstr 21230 The field of complex numbe...
cnfldex 21231 The field of complex numbe...
cnfldbas 21232 The base set of the field ...
cnfldadd 21233 The addition operation of ...
cnfldmul 21234 The multiplication operati...
cnfldcj 21235 The conjugation operation ...
cnfldtset 21236 The topology component of ...
cnfldle 21237 The ordering of the field ...
cnfldds 21238 The metric of the field of...
cnfldunif 21239 The uniform structure comp...
cnfldfun 21240 The field of complex numbe...
cnfldfunALT 21241 The field of complex numbe...
cnfldfunALTOLD 21242 Obsolete proof of ~ cnfldf...
xrsstr 21243 The extended real structur...
xrsex 21244 The extended real structur...
xrsbas 21245 The base set of the extend...
xrsadd 21246 The addition operation of ...
xrsmul 21247 The multiplication operati...
xrstset 21248 The topology component of ...
xrsle 21249 The ordering of the extend...
cncrng 21250 The complex numbers form a...
cnring 21251 The complex numbers form a...
xrsmcmn 21252 The "multiplicative group"...
cnfld0 21253 Zero is the zero element o...
cnfld1 21254 One is the unity element o...
cnfldneg 21255 The additive inverse in th...
cnfldplusf 21256 The functionalized additio...
cnfldsub 21257 The subtraction operator i...
cndrng 21258 The complex numbers form a...
cnflddiv 21259 The division operation in ...
cnfldinv 21260 The multiplicative inverse...
cnfldmulg 21261 The group multiple functio...
cnfldexp 21262 The exponentiation operato...
cnsrng 21263 The complex numbers form a...
xrsmgm 21264 The "additive group" of th...
xrsnsgrp 21265 The "additive group" of th...
xrsmgmdifsgrp 21266 The "additive group" of th...
xrs1mnd 21267 The extended real numbers,...
xrs10 21268 The zero of the extended r...
xrs1cmn 21269 The extended real numbers ...
xrge0subm 21270 The nonnegative extended r...
xrge0cmn 21271 The nonnegative extended r...
xrsds 21272 The metric of the extended...
xrsdsval 21273 The metric of the extended...
xrsdsreval 21274 The metric of the extended...
xrsdsreclblem 21275 Lemma for ~ xrsdsreclb . ...
xrsdsreclb 21276 The metric of the extended...
cnsubmlem 21277 Lemma for ~ nn0subm and fr...
cnsubglem 21278 Lemma for ~ resubdrg and f...
cnsubrglem 21279 Lemma for ~ resubdrg and f...
cnsubdrglem 21280 Lemma for ~ resubdrg and f...
qsubdrg 21281 The rational numbers form ...
zsubrg 21282 The integers form a subrin...
gzsubrg 21283 The gaussian integers form...
nn0subm 21284 The nonnegative integers f...
rege0subm 21285 The nonnegative reals form...
absabv 21286 The regular absolute value...
zsssubrg 21287 The integers are a subset ...
qsssubdrg 21288 The rational numbers are a...
cnsubrg 21289 There are no subrings of t...
cnmgpabl 21290 The unit group of the comp...
cnmgpid 21291 The group identity element...
cnmsubglem 21292 Lemma for ~ rpmsubg and fr...
rpmsubg 21293 The positive reals form a ...
gzrngunitlem 21294 Lemma for ~ gzrngunit . (...
gzrngunit 21295 The units on ` ZZ [ _i ] `...
gsumfsum 21296 Relate a group sum on ` CC...
regsumfsum 21297 Relate a group sum on ` ( ...
expmhm 21298 Exponentiation is a monoid...
nn0srg 21299 The nonnegative integers f...
rge0srg 21300 The nonnegative real numbe...
zringcrng 21303 The ring of integers is a ...
zringring 21304 The ring of integers is a ...
zringrng 21305 The ring of integers is a ...
zringabl 21306 The ring of integers is an...
zringgrp 21307 The ring of integers is an...
zringbas 21308 The integers are the base ...
zringplusg 21309 The addition operation of ...
zringsub 21310 The subtraction of element...
zringmulg 21311 The multiplication (group ...
zringmulr 21312 The multiplication operati...
zring0 21313 The zero element of the ri...
zring1 21314 The unity element of the r...
zringnzr 21315 The ring of integers is a ...
dvdsrzring 21316 Ring divisibility in the r...
zringlpirlem1 21317 Lemma for ~ zringlpir . A...
zringlpirlem2 21318 Lemma for ~ zringlpir . A...
zringlpirlem3 21319 Lemma for ~ zringlpir . A...
zringinvg 21320 The additive inverse of an...
zringunit 21321 The units of ` ZZ ` are th...
zringlpir 21322 The integers are a princip...
zringndrg 21323 The integers are not a div...
zringcyg 21324 The integers are a cyclic ...
zringsubgval 21325 Subtraction in the ring of...
zringmpg 21326 The multiplicative group o...
prmirredlem 21327 A positive integer is irre...
dfprm2 21328 The positive irreducible e...
prmirred 21329 The irreducible elements o...
expghm 21330 Exponentiation is a group ...
mulgghm2 21331 The powers of a group elem...
mulgrhm 21332 The powers of the element ...
mulgrhm2 21333 The powers of the element ...
irinitoringc 21334 The ring of integers is an...
nzerooringczr 21335 There is no zero object in...
pzriprnglem1 21336 Lemma 1 for ~ pzriprng : `...
pzriprnglem2 21337 Lemma 2 for ~ pzriprng : ...
pzriprnglem3 21338 Lemma 3 for ~ pzriprng : ...
pzriprnglem4 21339 Lemma 4 for ~ pzriprng : `...
pzriprnglem5 21340 Lemma 5 for ~ pzriprng : `...
pzriprnglem6 21341 Lemma 6 for ~ pzriprng : `...
pzriprnglem7 21342 Lemma 7 for ~ pzriprng : `...
pzriprnglem8 21343 Lemma 8 for ~ pzriprng : `...
pzriprnglem9 21344 Lemma 9 for ~ pzriprng : ...
pzriprnglem10 21345 Lemma 10 for ~ pzriprng : ...
pzriprnglem11 21346 Lemma 11 for ~ pzriprng : ...
pzriprnglem12 21347 Lemma 12 for ~ pzriprng : ...
pzriprnglem13 21348 Lemma 13 for ~ pzriprng : ...
pzriprnglem14 21349 Lemma 14 for ~ pzriprng : ...
pzriprngALT 21350 The non-unital ring ` ( ZZ...
pzriprng1ALT 21351 The ring unity of the ring...
pzriprng 21352 The non-unital ring ` ( ZZ...
pzriprng1 21353 The ring unity of the ring...
zrhval 21362 Define the unique homomorp...
zrhval2 21363 Alternate value of the ` Z...
zrhmulg 21364 Value of the ` ZRHom ` hom...
zrhrhmb 21365 The ` ZRHom ` homomorphism...
zrhrhm 21366 The ` ZRHom ` homomorphism...
zrh1 21367 Interpretation of 1 in a r...
zrh0 21368 Interpretation of 0 in a r...
zrhpropd 21369 The ` ZZ ` ring homomorphi...
zlmval 21370 Augment an abelian group w...
zlmlem 21371 Lemma for ~ zlmbas and ~ z...
zlmlemOLD 21372 Obsolete version of ~ zlml...
zlmbas 21373 Base set of a ` ZZ ` -modu...
zlmbasOLD 21374 Obsolete version of ~ zlmb...
zlmplusg 21375 Group operation of a ` ZZ ...
zlmplusgOLD 21376 Obsolete version of ~ zlmb...
zlmmulr 21377 Ring operation of a ` ZZ `...
zlmmulrOLD 21378 Obsolete version of ~ zlmb...
zlmsca 21379 Scalar ring of a ` ZZ ` -m...
zlmvsca 21380 Scalar multiplication oper...
zlmlmod 21381 The ` ZZ ` -module operati...
chrval 21382 Definition substitution of...
chrcl 21383 Closure of the characteris...
chrid 21384 The canonical ` ZZ ` ring ...
chrdvds 21385 The ` ZZ ` ring homomorphi...
chrcong 21386 If two integers are congru...
dvdschrmulg 21387 In a ring, any multiple of...
fermltlchr 21388 A generalization of Fermat...
chrnzr 21389 Nonzero rings are precisel...
chrrhm 21390 The characteristic restric...
domnchr 21391 The characteristic of a do...
znlidl 21392 The set ` n ZZ ` is an ide...
zncrng2 21393 The value of the ` Z/nZ ` ...
znval 21394 The value of the ` Z/nZ ` ...
znle 21395 The value of the ` Z/nZ ` ...
znval2 21396 Self-referential expressio...
znbaslem 21397 Lemma for ~ znbas . (Cont...
znbaslemOLD 21398 Obsolete version of ~ znba...
znbas2 21399 The base set of ` Z/nZ ` i...
znbas2OLD 21400 Obsolete version of ~ znba...
znadd 21401 The additive structure of ...
znaddOLD 21402 Obsolete version of ~ znad...
znmul 21403 The multiplicative structu...
znmulOLD 21404 Obsolete version of ~ znad...
znzrh 21405 The ` ZZ ` ring homomorphi...
znbas 21406 The base set of ` Z/nZ ` s...
zncrng 21407 ` Z/nZ ` is a commutative ...
znzrh2 21408 The ` ZZ ` ring homomorphi...
znzrhval 21409 The ` ZZ ` ring homomorphi...
znzrhfo 21410 The ` ZZ ` ring homomorphi...
zncyg 21411 The group ` ZZ / n ZZ ` is...
zndvds 21412 Express equality of equiva...
zndvds0 21413 Special case of ~ zndvds w...
znf1o 21414 The function ` F ` enumera...
zzngim 21415 The ` ZZ ` ring homomorphi...
znle2 21416 The ordering of the ` Z/nZ...
znleval 21417 The ordering of the ` Z/nZ...
znleval2 21418 The ordering of the ` Z/nZ...
zntoslem 21419 Lemma for ~ zntos . (Cont...
zntos 21420 The ` Z/nZ ` structure is ...
znhash 21421 The ` Z/nZ ` structure has...
znfi 21422 The ` Z/nZ ` structure is ...
znfld 21423 The ` Z/nZ ` structure is ...
znidomb 21424 The ` Z/nZ ` structure is ...
znchr 21425 Cyclic rings are defined b...
znunit 21426 The units of ` Z/nZ ` are ...
znunithash 21427 The size of the unit group...
znrrg 21428 The regular elements of ` ...
cygznlem1 21429 Lemma for ~ cygzn . (Cont...
cygznlem2a 21430 Lemma for ~ cygzn . (Cont...
cygznlem2 21431 Lemma for ~ cygzn . (Cont...
cygznlem3 21432 A cyclic group with ` n ` ...
cygzn 21433 A cyclic group with ` n ` ...
cygth 21434 The "fundamental theorem o...
cyggic 21435 Cyclic groups are isomorph...
frgpcyg 21436 A free group is cyclic iff...
freshmansdream 21437 For a prime number ` P ` ,...
cnmsgnsubg 21438 The signs form a multiplic...
cnmsgnbas 21439 The base set of the sign s...
cnmsgngrp 21440 The group of signs under m...
psgnghm 21441 The sign is a homomorphism...
psgnghm2 21442 The sign is a homomorphism...
psgninv 21443 The sign of a permutation ...
psgnco 21444 Multiplicativity of the pe...
zrhpsgnmhm 21445 Embedding of permutation s...
zrhpsgninv 21446 The embedded sign of a per...
evpmss 21447 Even permutations are perm...
psgnevpmb 21448 A class is an even permuta...
psgnodpm 21449 A permutation which is odd...
psgnevpm 21450 A permutation which is eve...
psgnodpmr 21451 If a permutation has sign ...
zrhpsgnevpm 21452 The sign of an even permut...
zrhpsgnodpm 21453 The sign of an odd permuta...
cofipsgn 21454 Composition of any class `...
zrhpsgnelbas 21455 Embedding of permutation s...
zrhcopsgnelbas 21456 Embedding of permutation s...
evpmodpmf1o 21457 The function for performin...
pmtrodpm 21458 A transposition is an odd ...
psgnfix1 21459 A permutation of a finite ...
psgnfix2 21460 A permutation of a finite ...
psgndiflemB 21461 Lemma 1 for ~ psgndif . (...
psgndiflemA 21462 Lemma 2 for ~ psgndif . (...
psgndif 21463 Embedding of permutation s...
copsgndif 21464 Embedding of permutation s...
rebase 21467 The base of the field of r...
remulg 21468 The multiplication (group ...
resubdrg 21469 The real numbers form a di...
resubgval 21470 Subtraction in the field o...
replusg 21471 The addition operation of ...
remulr 21472 The multiplication operati...
re0g 21473 The zero element of the fi...
re1r 21474 The unity element of the f...
rele2 21475 The ordering relation of t...
relt 21476 The ordering relation of t...
reds 21477 The distance of the field ...
redvr 21478 The division operation of ...
retos 21479 The real numbers are a tot...
refld 21480 The real numbers form a fi...
refldcj 21481 The conjugation operation ...
resrng 21482 The real numbers form a st...
regsumsupp 21483 The group sum over the rea...
rzgrp 21484 The quotient group ` RR / ...
isphl 21489 The predicate "is a genera...
phllvec 21490 A pre-Hilbert space is a l...
phllmod 21491 A pre-Hilbert space is a l...
phlsrng 21492 The scalar ring of a pre-H...
phllmhm 21493 The inner product of a pre...
ipcl 21494 Closure of the inner produ...
ipcj 21495 Conjugate of an inner prod...
iporthcom 21496 Orthogonality (meaning inn...
ip0l 21497 Inner product with a zero ...
ip0r 21498 Inner product with a zero ...
ipeq0 21499 The inner product of a vec...
ipdir 21500 Distributive law for inner...
ipdi 21501 Distributive law for inner...
ip2di 21502 Distributive law for inner...
ipsubdir 21503 Distributive law for inner...
ipsubdi 21504 Distributive law for inner...
ip2subdi 21505 Distributive law for inner...
ipass 21506 Associative law for inner ...
ipassr 21507 "Associative" law for seco...
ipassr2 21508 "Associative" law for inne...
ipffval 21509 The inner product operatio...
ipfval 21510 The inner product operatio...
ipfeq 21511 If the inner product opera...
ipffn 21512 The inner product operatio...
phlipf 21513 The inner product operatio...
ip2eq 21514 Two vectors are equal iff ...
isphld 21515 Properties that determine ...
phlpropd 21516 If two structures have the...
ssipeq 21517 The inner product on a sub...
phssipval 21518 The inner product on a sub...
phssip 21519 The inner product (as a fu...
phlssphl 21520 A subspace of an inner pro...
ocvfval 21527 The orthocomplement operat...
ocvval 21528 Value of the orthocompleme...
elocv 21529 Elementhood in the orthoco...
ocvi 21530 Property of a member of th...
ocvss 21531 The orthocomplement of a s...
ocvocv 21532 A set is contained in its ...
ocvlss 21533 The orthocomplement of a s...
ocv2ss 21534 Orthocomplements reverse s...
ocvin 21535 An orthocomplement has tri...
ocvsscon 21536 Two ways to say that ` S `...
ocvlsp 21537 The orthocomplement of a l...
ocv0 21538 The orthocomplement of the...
ocvz 21539 The orthocomplement of the...
ocv1 21540 The orthocomplement of the...
unocv 21541 The orthocomplement of a u...
iunocv 21542 The orthocomplement of an ...
cssval 21543 The set of closed subspace...
iscss 21544 The predicate "is a closed...
cssi 21545 Property of a closed subsp...
cssss 21546 A closed subspace is a sub...
iscss2 21547 It is sufficient to prove ...
ocvcss 21548 The orthocomplement of any...
cssincl 21549 The zero subspace is a clo...
css0 21550 The zero subspace is a clo...
css1 21551 The whole space is a close...
csslss 21552 A closed subspace of a pre...
lsmcss 21553 A subset of a pre-Hilbert ...
cssmre 21554 The closed subspaces of a ...
mrccss 21555 The Moore closure correspo...
thlval 21556 Value of the Hilbert latti...
thlbas 21557 Base set of the Hilbert la...
thlbasOLD 21558 Obsolete proof of ~ thlbas...
thlle 21559 Ordering on the Hilbert la...
thlleOLD 21560 Obsolete proof of ~ thlle ...
thlleval 21561 Ordering on the Hilbert la...
thloc 21562 Orthocomplement on the Hil...
pjfval 21569 The value of the projectio...
pjdm 21570 A subspace is in the domai...
pjpm 21571 The projection map is a pa...
pjfval2 21572 Value of the projection ma...
pjval 21573 Value of the projection ma...
pjdm2 21574 A subspace is in the domai...
pjff 21575 A projection is a linear o...
pjf 21576 A projection is a function...
pjf2 21577 A projection is a function...
pjfo 21578 A projection is a surjecti...
pjcss 21579 A projection subspace is a...
ocvpj 21580 The orthocomplement of a p...
ishil 21581 The predicate "is a Hilber...
ishil2 21582 The predicate "is a Hilber...
isobs 21583 The predicate "is an ortho...
obsip 21584 The inner product of two e...
obsipid 21585 A basis element has length...
obsrcl 21586 Reverse closure for an ort...
obsss 21587 An orthonormal basis is a ...
obsne0 21588 A basis element is nonzero...
obsocv 21589 An orthonormal basis has t...
obs2ocv 21590 The double orthocomplement...
obselocv 21591 A basis element is in the ...
obs2ss 21592 A basis has no proper subs...
obslbs 21593 An orthogonal basis is a l...
reldmdsmm 21596 The direct sum is a well-b...
dsmmval 21597 Value of the module direct...
dsmmbase 21598 Base set of the module dir...
dsmmval2 21599 Self-referential definitio...
dsmmbas2 21600 Base set of the direct sum...
dsmmfi 21601 For finite products, the d...
dsmmelbas 21602 Membership in the finitely...
dsmm0cl 21603 The all-zero vector is con...
dsmmacl 21604 The finite hull is closed ...
prdsinvgd2 21605 Negation of a single coord...
dsmmsubg 21606 The finite hull of a produ...
dsmmlss 21607 The finite hull of a produ...
dsmmlmod 21608 The direct sum of a family...
frlmval 21611 Value of the "free module"...
frlmlmod 21612 The free module is a modul...
frlmpws 21613 The free module as a restr...
frlmlss 21614 The base set of the free m...
frlmpwsfi 21615 The finite free module is ...
frlmsca 21616 The ring of scalars of a f...
frlm0 21617 Zero in a free module (rin...
frlmbas 21618 Base set of the free modul...
frlmelbas 21619 Membership in the base set...
frlmrcl 21620 If a free module is inhabi...
frlmbasfsupp 21621 Elements of the free modul...
frlmbasmap 21622 Elements of the free modul...
frlmbasf 21623 Elements of the free modul...
frlmlvec 21624 The free module over a div...
frlmfibas 21625 The base set of the finite...
elfrlmbasn0 21626 If the dimension of a free...
frlmplusgval 21627 Addition in a free module....
frlmsubgval 21628 Subtraction in a free modu...
frlmvscafval 21629 Scalar multiplication in a...
frlmvplusgvalc 21630 Coordinates of a sum with ...
frlmvscaval 21631 Coordinates of a scalar mu...
frlmplusgvalb 21632 Addition in a free module ...
frlmvscavalb 21633 Scalar multiplication in a...
frlmvplusgscavalb 21634 Addition combined with sca...
frlmgsum 21635 Finite commutative sums in...
frlmsplit2 21636 Restriction is homomorphic...
frlmsslss 21637 A subset of a free module ...
frlmsslss2 21638 A subset of a free module ...
frlmbas3 21639 An element of the base set...
mpofrlmd 21640 Elements of the free modul...
frlmip 21641 The inner product of a fre...
frlmipval 21642 The inner product of a fre...
frlmphllem 21643 Lemma for ~ frlmphl . (Co...
frlmphl 21644 Conditions for a free modu...
uvcfval 21647 Value of the unit-vector g...
uvcval 21648 Value of a single unit vec...
uvcvval 21649 Value of a unit vector coo...
uvcvvcl 21650 A coordinate of a unit vec...
uvcvvcl2 21651 A unit vector coordinate i...
uvcvv1 21652 The unit vector is one at ...
uvcvv0 21653 The unit vector is zero at...
uvcff 21654 Domain and codomain of the...
uvcf1 21655 In a nonzero ring, each un...
uvcresum 21656 Any element of a free modu...
frlmssuvc1 21657 A scalar multiple of a uni...
frlmssuvc2 21658 A nonzero scalar multiple ...
frlmsslsp 21659 A subset of a free module ...
frlmlbs 21660 The unit vectors comprise ...
frlmup1 21661 Any assignment of unit vec...
frlmup2 21662 The evaluation map has the...
frlmup3 21663 The range of such an evalu...
frlmup4 21664 Universal property of the ...
ellspd 21665 The elements of the span o...
elfilspd 21666 Simplified version of ~ el...
rellindf 21671 The independent-family pre...
islinds 21672 Property of an independent...
linds1 21673 An independent set of vect...
linds2 21674 An independent set of vect...
islindf 21675 Property of an independent...
islinds2 21676 Expanded property of an in...
islindf2 21677 Property of an independent...
lindff 21678 Functional property of a l...
lindfind 21679 A linearly independent fam...
lindsind 21680 A linearly independent set...
lindfind2 21681 In a linearly independent ...
lindsind2 21682 In a linearly independent ...
lindff1 21683 A linearly independent fam...
lindfrn 21684 The range of an independen...
f1lindf 21685 Rearranging and deleting e...
lindfres 21686 Any restriction of an inde...
lindsss 21687 Any subset of an independe...
f1linds 21688 A family constructed from ...
islindf3 21689 In a nonzero ring, indepen...
lindfmm 21690 Linear independence of a f...
lindsmm 21691 Linear independence of a s...
lindsmm2 21692 The monomorphic image of a...
lsslindf 21693 Linear independence is unc...
lsslinds 21694 Linear independence is unc...
islbs4 21695 A basis is an independent ...
lbslinds 21696 A basis is independent. (...
islinds3 21697 A subset is linearly indep...
islinds4 21698 A set is independent in a ...
lmimlbs 21699 The isomorphic image of a ...
lmiclbs 21700 Having a basis is an isomo...
islindf4 21701 A family is independent if...
islindf5 21702 A family is independent if...
indlcim 21703 An independent, spanning f...
lbslcic 21704 A module with a basis is i...
lmisfree 21705 A module has a basis iff i...
lvecisfrlm 21706 Every vector space is isom...
lmimco 21707 The composition of two iso...
lmictra 21708 Module isomorphism is tran...
uvcf1o 21709 In a nonzero ring, the map...
uvcendim 21710 In a nonzero ring, the num...
frlmisfrlm 21711 A free module is isomorphi...
frlmiscvec 21712 Every free module is isomo...
isassa 21719 The properties of an assoc...
assalem 21720 The properties of an assoc...
assaass 21721 Left-associative property ...
assaassr 21722 Right-associative property...
assalmod 21723 An associative algebra is ...
assaring 21724 An associative algebra is ...
assasca 21725 The scalars of an associat...
assa2ass 21726 Left- and right-associativ...
isassad 21727 Sufficient condition for b...
issubassa3 21728 A subring that is also a s...
issubassa 21729 The subalgebras of an asso...
sraassab 21730 A subring algebra is an as...
sraassa 21731 The subring algebra over a...
sraassaOLD 21732 Obsolete version of ~ sraa...
rlmassa 21733 The ring module over a com...
assapropd 21734 If two structures have the...
aspval 21735 Value of the algebraic clo...
asplss 21736 The algebraic span of a se...
aspid 21737 The algebraic span of a su...
aspsubrg 21738 The algebraic span of a se...
aspss 21739 Span preserves subset orde...
aspssid 21740 A set of vectors is a subs...
asclfval 21741 Function value of the alge...
asclval 21742 Value of a mapped algebra ...
asclfn 21743 Unconditional functionalit...
asclf 21744 The algebra scalars functi...
asclghm 21745 The algebra scalars functi...
ascl0 21746 The scalar 0 embedded into...
ascl1 21747 The scalar 1 embedded into...
asclmul1 21748 Left multiplication by a l...
asclmul2 21749 Right multiplication by a ...
ascldimul 21750 The algebra scalars functi...
asclinvg 21751 The group inverse (negatio...
asclrhm 21752 The scalar injection is a ...
rnascl 21753 The set of injected scalar...
issubassa2 21754 A subring of a unital alge...
rnasclsubrg 21755 The scalar multiples of th...
rnasclmulcl 21756 (Vector) multiplication is...
rnasclassa 21757 The scalar multiples of th...
ressascl 21758 The injection of scalars i...
asclpropd 21759 If two structures have the...
aspval2 21760 The algebraic closure is t...
assamulgscmlem1 21761 Lemma 1 for ~ assamulgscm ...
assamulgscmlem2 21762 Lemma for ~ assamulgscm (i...
assamulgscm 21763 Exponentiation of a scalar...
asclmulg 21764 Apply group multiplication...
zlmassa 21765 The ` ZZ ` -module operati...
reldmpsr 21776 The multivariate power ser...
psrval 21777 Value of the multivariate ...
psrvalstr 21778 The multivariate power ser...
psrbag 21779 Elementhood in the set of ...
psrbagf 21780 A finite bag is a function...
psrbagfOLD 21781 Obsolete version of ~ psrb...
psrbagfsupp 21782 Finite bags have finite su...
psrbagfsuppOLD 21783 Obsolete version of ~ psrb...
snifpsrbag 21784 A bag containing one eleme...
fczpsrbag 21785 The constant function equa...
psrbaglesupp 21786 The support of a dominated...
psrbaglesuppOLD 21787 Obsolete version of ~ psrb...
psrbaglecl 21788 The set of finite bags is ...
psrbagleclOLD 21789 Obsolete version of ~ psrb...
psrbagaddcl 21790 The sum of two finite bags...
psrbagaddclOLD 21791 Obsolete version of ~ psrb...
psrbagcon 21792 The analogue of the statem...
psrbagconOLD 21793 Obsolete version of ~ psrb...
psrbaglefi 21794 There are finitely many ba...
psrbaglefiOLD 21795 Obsolete version of ~ psrb...
psrbagconcl 21796 The complement of a bag is...
psrbagconclOLD 21797 Obsolete version of ~ psrb...
psrbagconf1o 21798 Bag complementation is a b...
psrbagconf1oOLD 21799 Obsolete version of ~ psrb...
gsumbagdiaglemOLD 21800 Obsolete version of ~ gsum...
gsumbagdiagOLD 21801 Obsolete version of ~ gsum...
psrass1lemOLD 21802 Obsolete version of ~ psra...
gsumbagdiaglem 21803 Lemma for ~ gsumbagdiag . ...
gsumbagdiag 21804 Two-dimensional commutatio...
psrass1lem 21805 A group sum commutation us...
psrbas 21806 The base set of the multiv...
psrelbas 21807 An element of the set of p...
psrelbasfun 21808 An element of the set of p...
psrplusg 21809 The addition operation of ...
psradd 21810 The addition operation of ...
psraddcl 21811 Closure of the power serie...
psraddclOLD 21812 Obsolete version of ~ psra...
psrmulr 21813 The multiplication operati...
psrmulfval 21814 The multiplication operati...
psrmulval 21815 The multiplication operati...
psrmulcllem 21816 Closure of the power serie...
psrmulcl 21817 Closure of the power serie...
psrsca 21818 The scalar field of the mu...
psrvscafval 21819 The scalar multiplication ...
psrvsca 21820 The scalar multiplication ...
psrvscaval 21821 The scalar multiplication ...
psrvscacl 21822 Closure of the power serie...
psr0cl 21823 The zero element of the ri...
psr0lid 21824 The zero element of the ri...
psrnegcl 21825 The negative function in t...
psrlinv 21826 The negative function in t...
psrgrp 21827 The ring of power series i...
psrgrpOLD 21828 Obsolete proof of ~ psrgrp...
psr0 21829 The zero element of the ri...
psrneg 21830 The negative function of t...
psrlmod 21831 The ring of power series i...
psr1cl 21832 The identity element of th...
psrlidm 21833 The identity element of th...
psrridm 21834 The identity element of th...
psrass1 21835 Associative identity for t...
psrdi 21836 Distributive law for the r...
psrdir 21837 Distributive law for the r...
psrass23l 21838 Associative identity for t...
psrcom 21839 Commutative law for the ri...
psrass23 21840 Associative identities for...
psrring 21841 The ring of power series i...
psr1 21842 The identity element of th...
psrcrng 21843 The ring of power series i...
psrassa 21844 The ring of power series i...
resspsrbas 21845 A restricted power series ...
resspsradd 21846 A restricted power series ...
resspsrmul 21847 A restricted power series ...
resspsrvsca 21848 A restricted power series ...
subrgpsr 21849 A subring of the base ring...
mvrfval 21850 Value of the generating el...
mvrval 21851 Value of the generating el...
mvrval2 21852 Value of the generating el...
mvrid 21853 The ` X i ` -th coefficien...
mvrf 21854 The power series variable ...
mvrf1 21855 The power series variable ...
mvrcl2 21856 A power series variable is...
reldmmpl 21857 The multivariate polynomia...
mplval 21858 Value of the set of multiv...
mplbas 21859 Base set of the set of mul...
mplelbas 21860 Property of being a polyno...
mvrcl 21861 A power series variable is...
mvrf2 21862 The power series/polynomia...
mplrcl 21863 Reverse closure for the po...
mplelsfi 21864 A polynomial treated as a ...
mplval2 21865 Self-referential expressio...
mplbasss 21866 The set of polynomials is ...
mplelf 21867 A polynomial is defined as...
mplsubglem 21868 If ` A ` is an ideal of se...
mpllsslem 21869 If ` A ` is an ideal of su...
mplsubglem2 21870 Lemma for ~ mplsubg and ~ ...
mplsubg 21871 The set of polynomials is ...
mpllss 21872 The set of polynomials is ...
mplsubrglem 21873 Lemma for ~ mplsubrg . (C...
mplsubrg 21874 The set of polynomials is ...
mpl0 21875 The zero polynomial. (Con...
mplplusg 21876 Value of addition in a pol...
mplmulr 21877 Value of multiplication in...
mpladd 21878 The addition operation on ...
mplneg 21879 The negative function on m...
mplmul 21880 The multiplication operati...
mpl1 21881 The identity element of th...
mplsca 21882 The scalar field of a mult...
mplvsca2 21883 The scalar multiplication ...
mplvsca 21884 The scalar multiplication ...
mplvscaval 21885 The scalar multiplication ...
mplgrp 21886 The polynomial ring is a g...
mpllmod 21887 The polynomial ring is a l...
mplring 21888 The polynomial ring is a r...
mpllvec 21889 The polynomial ring is a v...
mplcrng 21890 The polynomial ring is a c...
mplassa 21891 The polynomial ring is an ...
ressmplbas2 21892 The base set of a restrict...
ressmplbas 21893 A restricted polynomial al...
ressmpladd 21894 A restricted polynomial al...
ressmplmul 21895 A restricted polynomial al...
ressmplvsca 21896 A restricted power series ...
subrgmpl 21897 A subring of the base ring...
subrgmvr 21898 The variables in a subring...
subrgmvrf 21899 The variables in a polynom...
mplmon 21900 A monomial is a polynomial...
mplmonmul 21901 The product of two monomia...
mplcoe1 21902 Decompose a polynomial int...
mplcoe3 21903 Decompose a monomial in on...
mplcoe5lem 21904 Lemma for ~ mplcoe4 . (Co...
mplcoe5 21905 Decompose a monomial into ...
mplcoe2 21906 Decompose a monomial into ...
mplbas2 21907 An alternative expression ...
ltbval 21908 Value of the well-order on...
ltbwe 21909 The finite bag order is a ...
reldmopsr 21910 Lemma for ordered power se...
opsrval 21911 The value of the "ordered ...
opsrle 21912 An alternative expression ...
opsrval2 21913 Self-referential expressio...
opsrbaslem 21914 Get a component of the ord...
opsrbaslemOLD 21915 Obsolete version of ~ opsr...
opsrbas 21916 The base set of the ordere...
opsrbasOLD 21917 Obsolete version of ~ opsr...
opsrplusg 21918 The addition operation of ...
opsrplusgOLD 21919 Obsolete version of ~ opsr...
opsrmulr 21920 The multiplication operati...
opsrmulrOLD 21921 Obsolete version of ~ opsr...
opsrvsca 21922 The scalar product operati...
opsrvscaOLD 21923 Obsolete version of ~ opsr...
opsrsca 21924 The scalar ring of the ord...
opsrscaOLD 21925 Obsolete version of ~ opsr...
opsrtoslem1 21926 Lemma for ~ opsrtos . (Co...
opsrtoslem2 21927 Lemma for ~ opsrtos . (Co...
opsrtos 21928 The ordered power series s...
opsrso 21929 The ordered power series s...
opsrcrng 21930 The ring of ordered power ...
opsrassa 21931 The ring of ordered power ...
mplmon2 21932 Express a scaled monomial....
psrbag0 21933 The empty bag is a bag. (...
psrbagsn 21934 A singleton bag is a bag. ...
mplascl 21935 Value of the scalar inject...
mplasclf 21936 The scalar injection is a ...
subrgascl 21937 The scalar injection funct...
subrgasclcl 21938 The scalars in a polynomia...
mplmon2cl 21939 A scaled monomial is a pol...
mplmon2mul 21940 Product of scaled monomial...
mplind 21941 Prove a property of polyno...
mplcoe4 21942 Decompose a polynomial int...
evlslem4 21947 The support of a tensor pr...
psrbagev1 21948 A bag of multipliers provi...
psrbagev1OLD 21949 Obsolete version of ~ psrb...
psrbagev2 21950 Closure of a sum using a b...
psrbagev2OLD 21951 Obsolete version of ~ psrb...
evlslem2 21952 A linear function on the p...
evlslem3 21953 Lemma for ~ evlseu . Poly...
evlslem6 21954 Lemma for ~ evlseu . Fini...
evlslem1 21955 Lemma for ~ evlseu , give ...
evlseu 21956 For a given interpretation...
reldmevls 21957 Well-behaved binary operat...
mpfrcl 21958 Reverse closure for the se...
evlsval 21959 Value of the polynomial ev...
evlsval2 21960 Characterizing properties ...
evlsrhm 21961 Polynomial evaluation is a...
evlssca 21962 Polynomial evaluation maps...
evlsvar 21963 Polynomial evaluation maps...
evlsgsumadd 21964 Polynomial evaluation maps...
evlsgsummul 21965 Polynomial evaluation maps...
evlspw 21966 Polynomial evaluation for ...
evlsvarpw 21967 Polynomial evaluation for ...
evlval 21968 Value of the simple/same r...
evlrhm 21969 The simple evaluation map ...
evlsscasrng 21970 The evaluation of a scalar...
evlsca 21971 Simple polynomial evaluati...
evlsvarsrng 21972 The evaluation of the vari...
evlvar 21973 Simple polynomial evaluati...
mpfconst 21974 Constants are multivariate...
mpfproj 21975 Projections are multivaria...
mpfsubrg 21976 Polynomial functions are a...
mpff 21977 Polynomial functions are f...
mpfaddcl 21978 The sum of multivariate po...
mpfmulcl 21979 The product of multivariat...
mpfind 21980 Prove a property of polyno...
selvffval 21986 Value of the "variable sel...
selvfval 21987 Value of the "variable sel...
selvval 21988 Value of the "variable sel...
mhpfval 21990 Value of the "homogeneous ...
mhpval 21991 Value of the "homogeneous ...
ismhp 21992 Property of being a homoge...
ismhp2 21993 Deduce a homogeneous polyn...
ismhp3 21994 A polynomial is homogeneou...
mhpmpl 21995 A homogeneous polynomial i...
mhpdeg 21996 All nonzero terms of a hom...
mhp0cl 21997 The zero polynomial is hom...
mhpsclcl 21998 A scalar (or constant) pol...
mhpvarcl 21999 A power series variable is...
mhpmulcl 22000 A product of homogeneous p...
mhppwdeg 22001 Degree of a homogeneous po...
mhpaddcl 22002 Homogeneous polynomials ar...
mhpinvcl 22003 Homogeneous polynomials ar...
mhpsubg 22004 Homogeneous polynomials fo...
mhpvscacl 22005 Homogeneous polynomials ar...
mhplss 22006 Homogeneous polynomials fo...
psdffval 22008 Value of the power series ...
psdfval 22009 Give a map between power s...
psdval 22010 Evaluate the partial deriv...
psdcoef 22011 Coefficient of a term of t...
psdcl 22012 The derivative of a power ...
psdmplcl 22013 The derivative of a polyno...
psdadd 22014 The derivative of a sum is...
psdvsca 22015 The derivative of a scaled...
psr1baslem 22027 The set of finite bags on ...
psr1val 22028 Value of the ring of univa...
psr1crng 22029 The ring of univariate pow...
psr1assa 22030 The ring of univariate pow...
psr1tos 22031 The ordered power series s...
psr1bas2 22032 The base set of the ring o...
psr1bas 22033 The base set of the ring o...
vr1val 22034 The value of the generator...
vr1cl2 22035 The variable ` X ` is a me...
ply1val 22036 The value of the set of un...
ply1bas 22037 The value of the base set ...
ply1lss 22038 Univariate polynomials for...
ply1subrg 22039 Univariate polynomials for...
ply1crng 22040 The ring of univariate pol...
ply1assa 22041 The ring of univariate pol...
psr1bascl 22042 A univariate power series ...
psr1basf 22043 Univariate power series ba...
ply1basf 22044 Univariate polynomial base...
ply1bascl 22045 A univariate polynomial is...
ply1bascl2 22046 A univariate polynomial is...
coe1fval 22047 Value of the univariate po...
coe1fv 22048 Value of an evaluated coef...
fvcoe1 22049 Value of a multivariate co...
coe1fval3 22050 Univariate power series co...
coe1f2 22051 Functionality of univariat...
coe1fval2 22052 Univariate polynomial coef...
coe1f 22053 Functionality of univariat...
coe1fvalcl 22054 A coefficient of a univari...
coe1sfi 22055 Finite support of univaria...
coe1fsupp 22056 The coefficient vector of ...
mptcoe1fsupp 22057 A mapping involving coeffi...
coe1ae0 22058 The coefficient vector of ...
vr1cl 22059 The generator of a univari...
opsr0 22060 Zero in the ordered power ...
opsr1 22061 One in the ordered power s...
psr1plusg 22062 Value of addition in a uni...
psr1vsca 22063 Value of scalar multiplica...
psr1mulr 22064 Value of multiplication in...
ply1plusg 22065 Value of addition in a uni...
ply1vsca 22066 Value of scalar multiplica...
ply1mulr 22067 Value of multiplication in...
ply1ass23l 22068 Associative identity with ...
ressply1bas2 22069 The base set of a restrict...
ressply1bas 22070 A restricted polynomial al...
ressply1add 22071 A restricted polynomial al...
ressply1mul 22072 A restricted polynomial al...
ressply1vsca 22073 A restricted power series ...
subrgply1 22074 A subring of the base ring...
gsumply1subr 22075 Evaluate a group sum in a ...
psrbaspropd 22076 Property deduction for pow...
psrplusgpropd 22077 Property deduction for pow...
mplbaspropd 22078 Property deduction for pol...
psropprmul 22079 Reversing multiplication i...
ply1opprmul 22080 Reversing multiplication i...
00ply1bas 22081 Lemma for ~ ply1basfvi and...
ply1basfvi 22082 Protection compatibility o...
ply1plusgfvi 22083 Protection compatibility o...
ply1baspropd 22084 Property deduction for uni...
ply1plusgpropd 22085 Property deduction for uni...
opsrring 22086 Ordered power series form ...
opsrlmod 22087 Ordered power series form ...
psr1ring 22088 Univariate power series fo...
ply1ring 22089 Univariate polynomials for...
psr1lmod 22090 Univariate power series fo...
psr1sca 22091 Scalars of a univariate po...
psr1sca2 22092 Scalars of a univariate po...
ply1lmod 22093 Univariate polynomials for...
ply1sca 22094 Scalars of a univariate po...
ply1sca2 22095 Scalars of a univariate po...
ply1mpl0 22096 The univariate polynomial ...
ply10s0 22097 Zero times a univariate po...
ply1mpl1 22098 The univariate polynomial ...
ply1ascl 22099 The univariate polynomial ...
subrg1ascl 22100 The scalar injection funct...
subrg1asclcl 22101 The scalars in a polynomia...
subrgvr1 22102 The variables in a subring...
subrgvr1cl 22103 The variables in a polynom...
coe1z 22104 The coefficient vector of ...
coe1add 22105 The coefficient vector of ...
coe1addfv 22106 A particular coefficient o...
coe1subfv 22107 A particular coefficient o...
coe1mul2lem1 22108 An equivalence for ~ coe1m...
coe1mul2lem2 22109 An equivalence for ~ coe1m...
coe1mul2 22110 The coefficient vector of ...
coe1mul 22111 The coefficient vector of ...
ply1moncl 22112 Closure of the expression ...
ply1tmcl 22113 Closure of the expression ...
coe1tm 22114 Coefficient vector of a po...
coe1tmfv1 22115 Nonzero coefficient of a p...
coe1tmfv2 22116 Zero coefficient of a poly...
coe1tmmul2 22117 Coefficient vector of a po...
coe1tmmul 22118 Coefficient vector of a po...
coe1tmmul2fv 22119 Function value of a right-...
coe1pwmul 22120 Coefficient vector of a po...
coe1pwmulfv 22121 Function value of a right-...
ply1scltm 22122 A scalar is a term with ze...
coe1sclmul 22123 Coefficient vector of a po...
coe1sclmulfv 22124 A single coefficient of a ...
coe1sclmul2 22125 Coefficient vector of a po...
ply1sclf 22126 A scalar polynomial is a p...
ply1sclcl 22127 The value of the algebra s...
coe1scl 22128 Coefficient vector of a sc...
ply1sclid 22129 Recover the base scalar fr...
ply1sclf1 22130 The polynomial scalar func...
ply1scl0 22131 The zero scalar is zero. ...
ply1scl0OLD 22132 Obsolete version of ~ ply1...
ply1scln0 22133 Nonzero scalars create non...
ply1scl1 22134 The one scalar is the unit...
ply1scl1OLD 22135 Obsolete version of ~ ply1...
ply1idvr1 22136 The identity of a polynomi...
cply1mul 22137 The product of two constan...
ply1coefsupp 22138 The decomposition of a uni...
ply1coe 22139 Decompose a univariate pol...
eqcoe1ply1eq 22140 Two polynomials over the s...
ply1coe1eq 22141 Two polynomials over the s...
cply1coe0 22142 All but the first coeffici...
cply1coe0bi 22143 A polynomial is constant (...
coe1fzgsumdlem 22144 Lemma for ~ coe1fzgsumd (i...
coe1fzgsumd 22145 Value of an evaluated coef...
ply1scleq 22146 Equality of a constant pol...
ply1chr 22147 The characteristic of a po...
gsumsmonply1 22148 A finite group sum of scal...
gsummoncoe1 22149 A coefficient of the polyn...
gsumply1eq 22150 Two univariate polynomials...
lply1binom 22151 The binomial theorem for l...
lply1binomsc 22152 The binomial theorem for l...
ply1fermltlchr 22153 Fermat's little theorem fo...
reldmevls1 22158 Well-behaved binary operat...
ply1frcl 22159 Reverse closure for the se...
evls1fval 22160 Value of the univariate po...
evls1val 22161 Value of the univariate po...
evls1rhmlem 22162 Lemma for ~ evl1rhm and ~ ...
evls1rhm 22163 Polynomial evaluation is a...
evls1sca 22164 Univariate polynomial eval...
evls1gsumadd 22165 Univariate polynomial eval...
evls1gsummul 22166 Univariate polynomial eval...
evls1pw 22167 Univariate polynomial eval...
evls1varpw 22168 Univariate polynomial eval...
evl1fval 22169 Value of the simple/same r...
evl1val 22170 Value of the simple/same r...
evl1fval1lem 22171 Lemma for ~ evl1fval1 . (...
evl1fval1 22172 Value of the simple/same r...
evl1rhm 22173 Polynomial evaluation is a...
fveval1fvcl 22174 The function value of the ...
evl1sca 22175 Polynomial evaluation maps...
evl1scad 22176 Polynomial evaluation buil...
evl1var 22177 Polynomial evaluation maps...
evl1vard 22178 Polynomial evaluation buil...
evls1var 22179 Univariate polynomial eval...
evls1scasrng 22180 The evaluation of a scalar...
evls1varsrng 22181 The evaluation of the vari...
evl1addd 22182 Polynomial evaluation buil...
evl1subd 22183 Polynomial evaluation buil...
evl1muld 22184 Polynomial evaluation buil...
evl1vsd 22185 Polynomial evaluation buil...
evl1expd 22186 Polynomial evaluation buil...
pf1const 22187 Constants are polynomial f...
pf1id 22188 The identity is a polynomi...
pf1subrg 22189 Polynomial functions are a...
pf1rcl 22190 Reverse closure for the se...
pf1f 22191 Polynomial functions are f...
mpfpf1 22192 Convert a multivariate pol...
pf1mpf 22193 Convert a univariate polyn...
pf1addcl 22194 The sum of multivariate po...
pf1mulcl 22195 The product of multivariat...
pf1ind 22196 Prove a property of polyno...
evl1gsumdlem 22197 Lemma for ~ evl1gsumd (ind...
evl1gsumd 22198 Polynomial evaluation buil...
evl1gsumadd 22199 Univariate polynomial eval...
evl1gsumaddval 22200 Value of a univariate poly...
evl1gsummul 22201 Univariate polynomial eval...
evl1varpw 22202 Univariate polynomial eval...
evl1varpwval 22203 Value of a univariate poly...
evl1scvarpw 22204 Univariate polynomial eval...
evl1scvarpwval 22205 Value of a univariate poly...
evl1gsummon 22206 Value of a univariate poly...
mamufval 22209 Functional value of the ma...
mamuval 22210 Multiplication of two matr...
mamufv 22211 A cell in the multiplicati...
mamudm 22212 The domain of the matrix m...
mamufacex 22213 Every solution of the equa...
mamures 22214 Rows in a matrix product a...
mndvcl 22215 Tuple-wise additive closur...
mndvass 22216 Tuple-wise associativity i...
mndvlid 22217 Tuple-wise left identity i...
mndvrid 22218 Tuple-wise right identity ...
grpvlinv 22219 Tuple-wise left inverse in...
grpvrinv 22220 Tuple-wise right inverse i...
mhmvlin 22221 Tuple extension of monoid ...
ringvcl 22222 Tuple-wise multiplication ...
mamucl 22223 Operation closure of matri...
mamuass 22224 Matrix multiplication is a...
mamudi 22225 Matrix multiplication dist...
mamudir 22226 Matrix multiplication dist...
mamuvs1 22227 Matrix multiplication dist...
mamuvs2 22228 Matrix multiplication dist...
matbas0pc 22231 There is no matrix with a ...
matbas0 22232 There is no matrix for a n...
matval 22233 Value of the matrix algebr...
matrcl 22234 Reverse closure for the ma...
matbas 22235 The matrix ring has the sa...
matplusg 22236 The matrix ring has the sa...
matsca 22237 The matrix ring has the sa...
matscaOLD 22238 Obsolete proof of ~ matsca...
matvsca 22239 The matrix ring has the sa...
matvscaOLD 22240 Obsolete proof of ~ matvsc...
mat0 22241 The matrix ring has the sa...
matinvg 22242 The matrix ring has the sa...
mat0op 22243 Value of a zero matrix as ...
matsca2 22244 The scalars of the matrix ...
matbas2 22245 The base set of the matrix...
matbas2i 22246 A matrix is a function. (...
matbas2d 22247 The base set of the matrix...
eqmat 22248 Two square matrices of the...
matecl 22249 Each entry (according to W...
matecld 22250 Each entry (according to W...
matplusg2 22251 Addition in the matrix rin...
matvsca2 22252 Scalar multiplication in t...
matlmod 22253 The matrix ring is a linea...
matgrp 22254 The matrix ring is a group...
matvscl 22255 Closure of the scalar mult...
matsubg 22256 The matrix ring has the sa...
matplusgcell 22257 Addition in the matrix rin...
matsubgcell 22258 Subtraction in the matrix ...
matinvgcell 22259 Additive inversion in the ...
matvscacell 22260 Scalar multiplication in t...
matgsum 22261 Finite commutative sums in...
matmulr 22262 Multiplication in the matr...
mamumat1cl 22263 The identity matrix (as op...
mat1comp 22264 The components of the iden...
mamulid 22265 The identity matrix (as op...
mamurid 22266 The identity matrix (as op...
matring 22267 Existence of the matrix ri...
matassa 22268 Existence of the matrix al...
matmulcell 22269 Multiplication in the matr...
mpomatmul 22270 Multiplication of two N x ...
mat1 22271 Value of an identity matri...
mat1ov 22272 Entries of an identity mat...
mat1bas 22273 The identity matrix is a m...
matsc 22274 The identity matrix multip...
ofco2 22275 Distribution law for the f...
oftpos 22276 The transposition of the v...
mattposcl 22277 The transpose of a square ...
mattpostpos 22278 The transpose of the trans...
mattposvs 22279 The transposition of a mat...
mattpos1 22280 The transposition of the i...
tposmap 22281 The transposition of an I ...
mamutpos 22282 Behavior of transposes in ...
mattposm 22283 Multiplying two transposed...
matgsumcl 22284 Closure of a group sum ove...
madetsumid 22285 The identity summand in th...
matepmcl 22286 Each entry of a matrix wit...
matepm2cl 22287 Each entry of a matrix wit...
madetsmelbas 22288 A summand of the determina...
madetsmelbas2 22289 A summand of the determina...
mat0dimbas0 22290 The empty set is the one a...
mat0dim0 22291 The zero of the algebra of...
mat0dimid 22292 The identity of the algebr...
mat0dimscm 22293 The scalar multiplication ...
mat0dimcrng 22294 The algebra of matrices wi...
mat1dimelbas 22295 A matrix with dimension 1 ...
mat1dimbas 22296 A matrix with dimension 1 ...
mat1dim0 22297 The zero of the algebra of...
mat1dimid 22298 The identity of the algebr...
mat1dimscm 22299 The scalar multiplication ...
mat1dimmul 22300 The ring multiplication in...
mat1dimcrng 22301 The algebra of matrices wi...
mat1f1o 22302 There is a 1-1 function fr...
mat1rhmval 22303 The value of the ring homo...
mat1rhmelval 22304 The value of the ring homo...
mat1rhmcl 22305 The value of the ring homo...
mat1f 22306 There is a function from a...
mat1ghm 22307 There is a group homomorph...
mat1mhm 22308 There is a monoid homomorp...
mat1rhm 22309 There is a ring homomorphi...
mat1rngiso 22310 There is a ring isomorphis...
mat1ric 22311 A ring is isomorphic to th...
dmatval 22316 The set of ` N ` x ` N ` d...
dmatel 22317 A ` N ` x ` N ` diagonal m...
dmatmat 22318 An ` N ` x ` N ` diagonal ...
dmatid 22319 The identity matrix is a d...
dmatelnd 22320 An extradiagonal entry of ...
dmatmul 22321 The product of two diagona...
dmatsubcl 22322 The difference of two diag...
dmatsgrp 22323 The set of diagonal matric...
dmatmulcl 22324 The product of two diagona...
dmatsrng 22325 The set of diagonal matric...
dmatcrng 22326 The subring of diagonal ma...
dmatscmcl 22327 The multiplication of a di...
scmatval 22328 The set of ` N ` x ` N ` s...
scmatel 22329 An ` N ` x ` N ` scalar ma...
scmatscmid 22330 A scalar matrix can be exp...
scmatscmide 22331 An entry of a scalar matri...
scmatscmiddistr 22332 Distributive law for scala...
scmatmat 22333 An ` N ` x ` N ` scalar ma...
scmate 22334 An entry of an ` N ` x ` N...
scmatmats 22335 The set of an ` N ` x ` N ...
scmateALT 22336 Alternate proof of ~ scmat...
scmatscm 22337 The multiplication of a ma...
scmatid 22338 The identity matrix is a s...
scmatdmat 22339 A scalar matrix is a diago...
scmataddcl 22340 The sum of two scalar matr...
scmatsubcl 22341 The difference of two scal...
scmatmulcl 22342 The product of two scalar ...
scmatsgrp 22343 The set of scalar matrices...
scmatsrng 22344 The set of scalar matrices...
scmatcrng 22345 The subring of scalar matr...
scmatsgrp1 22346 The set of scalar matrices...
scmatsrng1 22347 The set of scalar matrices...
smatvscl 22348 Closure of the scalar mult...
scmatlss 22349 The set of scalar matrices...
scmatstrbas 22350 The set of scalar matrices...
scmatrhmval 22351 The value of the ring homo...
scmatrhmcl 22352 The value of the ring homo...
scmatf 22353 There is a function from a...
scmatfo 22354 There is a function from a...
scmatf1 22355 There is a 1-1 function fr...
scmatf1o 22356 There is a bijection betwe...
scmatghm 22357 There is a group homomorph...
scmatmhm 22358 There is a monoid homomorp...
scmatrhm 22359 There is a ring homomorphi...
scmatrngiso 22360 There is a ring isomorphis...
scmatric 22361 A ring is isomorphic to ev...
mat0scmat 22362 The empty matrix over a ri...
mat1scmat 22363 A 1-dimensional matrix ove...
mvmulfval 22366 Functional value of the ma...
mvmulval 22367 Multiplication of a vector...
mvmulfv 22368 A cell/element in the vect...
mavmulval 22369 Multiplication of a vector...
mavmulfv 22370 A cell/element in the vect...
mavmulcl 22371 Multiplication of an NxN m...
1mavmul 22372 Multiplication of the iden...
mavmulass 22373 Associativity of the multi...
mavmuldm 22374 The domain of the matrix v...
mavmulsolcl 22375 Every solution of the equa...
mavmul0 22376 Multiplication of a 0-dime...
mavmul0g 22377 The result of the 0-dimens...
mvmumamul1 22378 The multiplication of an M...
mavmumamul1 22379 The multiplication of an N...
marrepfval 22384 First substitution for the...
marrepval0 22385 Second substitution for th...
marrepval 22386 Third substitution for the...
marrepeval 22387 An entry of a matrix with ...
marrepcl 22388 Closure of the row replace...
marepvfval 22389 First substitution for the...
marepvval0 22390 Second substitution for th...
marepvval 22391 Third substitution for the...
marepveval 22392 An entry of a matrix with ...
marepvcl 22393 Closure of the column repl...
ma1repvcl 22394 Closure of the column repl...
ma1repveval 22395 An entry of an identity ma...
mulmarep1el 22396 Element by element multipl...
mulmarep1gsum1 22397 The sum of element by elem...
mulmarep1gsum2 22398 The sum of element by elem...
1marepvmarrepid 22399 Replacing the ith row by 0...
submabas 22402 Any subset of the index se...
submafval 22403 First substitution for a s...
submaval0 22404 Second substitution for a ...
submaval 22405 Third substitution for a s...
submaeval 22406 An entry of a submatrix of...
1marepvsma1 22407 The submatrix of the ident...
mdetfval 22410 First substitution for the...
mdetleib 22411 Full substitution of our d...
mdetleib2 22412 Leibniz' formula can also ...
nfimdetndef 22413 The determinant is not def...
mdetfval1 22414 First substitution of an a...
mdetleib1 22415 Full substitution of an al...
mdet0pr 22416 The determinant function f...
mdet0f1o 22417 The determinant function f...
mdet0fv0 22418 The determinant of the emp...
mdetf 22419 Functionality of the deter...
mdetcl 22420 The determinant evaluates ...
m1detdiag 22421 The determinant of a 1-dim...
mdetdiaglem 22422 Lemma for ~ mdetdiag . Pr...
mdetdiag 22423 The determinant of a diago...
mdetdiagid 22424 The determinant of a diago...
mdet1 22425 The determinant of the ide...
mdetrlin 22426 The determinant function i...
mdetrsca 22427 The determinant function i...
mdetrsca2 22428 The determinant function i...
mdetr0 22429 The determinant of a matri...
mdet0 22430 The determinant of the zer...
mdetrlin2 22431 The determinant function i...
mdetralt 22432 The determinant function i...
mdetralt2 22433 The determinant function i...
mdetero 22434 The determinant function i...
mdettpos 22435 Determinant is invariant u...
mdetunilem1 22436 Lemma for ~ mdetuni . (Co...
mdetunilem2 22437 Lemma for ~ mdetuni . (Co...
mdetunilem3 22438 Lemma for ~ mdetuni . (Co...
mdetunilem4 22439 Lemma for ~ mdetuni . (Co...
mdetunilem5 22440 Lemma for ~ mdetuni . (Co...
mdetunilem6 22441 Lemma for ~ mdetuni . (Co...
mdetunilem7 22442 Lemma for ~ mdetuni . (Co...
mdetunilem8 22443 Lemma for ~ mdetuni . (Co...
mdetunilem9 22444 Lemma for ~ mdetuni . (Co...
mdetuni0 22445 Lemma for ~ mdetuni . (Co...
mdetuni 22446 According to the definitio...
mdetmul 22447 Multiplicativity of the de...
m2detleiblem1 22448 Lemma 1 for ~ m2detleib . ...
m2detleiblem5 22449 Lemma 5 for ~ m2detleib . ...
m2detleiblem6 22450 Lemma 6 for ~ m2detleib . ...
m2detleiblem7 22451 Lemma 7 for ~ m2detleib . ...
m2detleiblem2 22452 Lemma 2 for ~ m2detleib . ...
m2detleiblem3 22453 Lemma 3 for ~ m2detleib . ...
m2detleiblem4 22454 Lemma 4 for ~ m2detleib . ...
m2detleib 22455 Leibniz' Formula for 2x2-m...
mndifsplit 22460 Lemma for ~ maducoeval2 . ...
madufval 22461 First substitution for the...
maduval 22462 Second substitution for th...
maducoeval 22463 An entry of the adjunct (c...
maducoeval2 22464 An entry of the adjunct (c...
maduf 22465 Creating the adjunct of ma...
madutpos 22466 The adjuct of a transposed...
madugsum 22467 The determinant of a matri...
madurid 22468 Multiplying a matrix with ...
madulid 22469 Multiplying the adjunct of...
minmar1fval 22470 First substitution for the...
minmar1val0 22471 Second substitution for th...
minmar1val 22472 Third substitution for the...
minmar1eval 22473 An entry of a matrix for a...
minmar1marrep 22474 The minor matrix is a spec...
minmar1cl 22475 Closure of the row replace...
maducoevalmin1 22476 The coefficients of an adj...
symgmatr01lem 22477 Lemma for ~ symgmatr01 . ...
symgmatr01 22478 Applying a permutation tha...
gsummatr01lem1 22479 Lemma A for ~ gsummatr01 ....
gsummatr01lem2 22480 Lemma B for ~ gsummatr01 ....
gsummatr01lem3 22481 Lemma 1 for ~ gsummatr01 ....
gsummatr01lem4 22482 Lemma 2 for ~ gsummatr01 ....
gsummatr01 22483 Lemma 1 for ~ smadiadetlem...
marep01ma 22484 Replacing a row of a squar...
smadiadetlem0 22485 Lemma 0 for ~ smadiadet : ...
smadiadetlem1 22486 Lemma 1 for ~ smadiadet : ...
smadiadetlem1a 22487 Lemma 1a for ~ smadiadet :...
smadiadetlem2 22488 Lemma 2 for ~ smadiadet : ...
smadiadetlem3lem0 22489 Lemma 0 for ~ smadiadetlem...
smadiadetlem3lem1 22490 Lemma 1 for ~ smadiadetlem...
smadiadetlem3lem2 22491 Lemma 2 for ~ smadiadetlem...
smadiadetlem3 22492 Lemma 3 for ~ smadiadet . ...
smadiadetlem4 22493 Lemma 4 for ~ smadiadet . ...
smadiadet 22494 The determinant of a subma...
smadiadetglem1 22495 Lemma 1 for ~ smadiadetg ....
smadiadetglem2 22496 Lemma 2 for ~ smadiadetg ....
smadiadetg 22497 The determinant of a squar...
smadiadetg0 22498 Lemma for ~ smadiadetr : v...
smadiadetr 22499 The determinant of a squar...
invrvald 22500 If a matrix multiplied wit...
matinv 22501 The inverse of a matrix is...
matunit 22502 A matrix is a unit in the ...
slesolvec 22503 Every solution of a system...
slesolinv 22504 The solution of a system o...
slesolinvbi 22505 The solution of a system o...
slesolex 22506 Every system of linear equ...
cramerimplem1 22507 Lemma 1 for ~ cramerimp : ...
cramerimplem2 22508 Lemma 2 for ~ cramerimp : ...
cramerimplem3 22509 Lemma 3 for ~ cramerimp : ...
cramerimp 22510 One direction of Cramer's ...
cramerlem1 22511 Lemma 1 for ~ cramer . (C...
cramerlem2 22512 Lemma 2 for ~ cramer . (C...
cramerlem3 22513 Lemma 3 for ~ cramer . (C...
cramer0 22514 Special case of Cramer's r...
cramer 22515 Cramer's rule. According ...
pmatring 22516 The set of polynomial matr...
pmatlmod 22517 The set of polynomial matr...
pmatassa 22518 The set of polynomial matr...
pmat0op 22519 The zero polynomial matrix...
pmat1op 22520 The identity polynomial ma...
pmat1ovd 22521 Entries of the identity po...
pmat0opsc 22522 The zero polynomial matrix...
pmat1opsc 22523 The identity polynomial ma...
pmat1ovscd 22524 Entries of the identity po...
pmatcoe1fsupp 22525 For a polynomial matrix th...
1pmatscmul 22526 The scalar product of the ...
cpmat 22533 Value of the constructor o...
cpmatpmat 22534 A constant polynomial matr...
cpmatel 22535 Property of a constant pol...
cpmatelimp 22536 Implication of a set being...
cpmatel2 22537 Another property of a cons...
cpmatelimp2 22538 Another implication of a s...
1elcpmat 22539 The identity of the ring o...
cpmatacl 22540 The set of all constant po...
cpmatinvcl 22541 The set of all constant po...
cpmatmcllem 22542 Lemma for ~ cpmatmcl . (C...
cpmatmcl 22543 The set of all constant po...
cpmatsubgpmat 22544 The set of all constant po...
cpmatsrgpmat 22545 The set of all constant po...
0elcpmat 22546 The zero of the ring of al...
mat2pmatfval 22547 Value of the matrix transf...
mat2pmatval 22548 The result of a matrix tra...
mat2pmatvalel 22549 A (matrix) element of the ...
mat2pmatbas 22550 The result of a matrix tra...
mat2pmatbas0 22551 The result of a matrix tra...
mat2pmatf 22552 The matrix transformation ...
mat2pmatf1 22553 The matrix transformation ...
mat2pmatghm 22554 The transformation of matr...
mat2pmatmul 22555 The transformation of matr...
mat2pmat1 22556 The transformation of the ...
mat2pmatmhm 22557 The transformation of matr...
mat2pmatrhm 22558 The transformation of matr...
mat2pmatlin 22559 The transformation of matr...
0mat2pmat 22560 The transformed zero matri...
idmatidpmat 22561 The transformed identity m...
d0mat2pmat 22562 The transformed empty set ...
d1mat2pmat 22563 The transformation of a ma...
mat2pmatscmxcl 22564 A transformed matrix multi...
m2cpm 22565 The result of a matrix tra...
m2cpmf 22566 The matrix transformation ...
m2cpmf1 22567 The matrix transformation ...
m2cpmghm 22568 The transformation of matr...
m2cpmmhm 22569 The transformation of matr...
m2cpmrhm 22570 The transformation of matr...
m2pmfzmap 22571 The transformed values of ...
m2pmfzgsumcl 22572 Closure of the sum of scal...
cpm2mfval 22573 Value of the inverse matri...
cpm2mval 22574 The result of an inverse m...
cpm2mvalel 22575 A (matrix) element of the ...
cpm2mf 22576 The inverse matrix transfo...
m2cpminvid 22577 The inverse transformation...
m2cpminvid2lem 22578 Lemma for ~ m2cpminvid2 . ...
m2cpminvid2 22579 The transformation applied...
m2cpmfo 22580 The matrix transformation ...
m2cpmf1o 22581 The matrix transformation ...
m2cpmrngiso 22582 The transformation of matr...
matcpmric 22583 The ring of matrices over ...
m2cpminv 22584 The inverse matrix transfo...
m2cpminv0 22585 The inverse matrix transfo...
decpmatval0 22588 The matrix consisting of t...
decpmatval 22589 The matrix consisting of t...
decpmate 22590 An entry of the matrix con...
decpmatcl 22591 Closure of the decompositi...
decpmataa0 22592 The matrix consisting of t...
decpmatfsupp 22593 The mapping to the matrice...
decpmatid 22594 The matrix consisting of t...
decpmatmullem 22595 Lemma for ~ decpmatmul . ...
decpmatmul 22596 The matrix consisting of t...
decpmatmulsumfsupp 22597 Lemma 0 for ~ pm2mpmhm . ...
pmatcollpw1lem1 22598 Lemma 1 for ~ pmatcollpw1 ...
pmatcollpw1lem2 22599 Lemma 2 for ~ pmatcollpw1 ...
pmatcollpw1 22600 Write a polynomial matrix ...
pmatcollpw2lem 22601 Lemma for ~ pmatcollpw2 . ...
pmatcollpw2 22602 Write a polynomial matrix ...
monmatcollpw 22603 The matrix consisting of t...
pmatcollpwlem 22604 Lemma for ~ pmatcollpw . ...
pmatcollpw 22605 Write a polynomial matrix ...
pmatcollpwfi 22606 Write a polynomial matrix ...
pmatcollpw3lem 22607 Lemma for ~ pmatcollpw3 an...
pmatcollpw3 22608 Write a polynomial matrix ...
pmatcollpw3fi 22609 Write a polynomial matrix ...
pmatcollpw3fi1lem1 22610 Lemma 1 for ~ pmatcollpw3f...
pmatcollpw3fi1lem2 22611 Lemma 2 for ~ pmatcollpw3f...
pmatcollpw3fi1 22612 Write a polynomial matrix ...
pmatcollpwscmatlem1 22613 Lemma 1 for ~ pmatcollpwsc...
pmatcollpwscmatlem2 22614 Lemma 2 for ~ pmatcollpwsc...
pmatcollpwscmat 22615 Write a scalar matrix over...
pm2mpf1lem 22618 Lemma for ~ pm2mpf1 . (Co...
pm2mpval 22619 Value of the transformatio...
pm2mpfval 22620 A polynomial matrix transf...
pm2mpcl 22621 The transformation of poly...
pm2mpf 22622 The transformation of poly...
pm2mpf1 22623 The transformation of poly...
pm2mpcoe1 22624 A coefficient of the polyn...
idpm2idmp 22625 The transformation of the ...
mptcoe1matfsupp 22626 The mapping extracting the...
mply1topmatcllem 22627 Lemma for ~ mply1topmatcl ...
mply1topmatval 22628 A polynomial over matrices...
mply1topmatcl 22629 A polynomial over matrices...
mp2pm2mplem1 22630 Lemma 1 for ~ mp2pm2mp . ...
mp2pm2mplem2 22631 Lemma 2 for ~ mp2pm2mp . ...
mp2pm2mplem3 22632 Lemma 3 for ~ mp2pm2mp . ...
mp2pm2mplem4 22633 Lemma 4 for ~ mp2pm2mp . ...
mp2pm2mplem5 22634 Lemma 5 for ~ mp2pm2mp . ...
mp2pm2mp 22635 A polynomial over matrices...
pm2mpghmlem2 22636 Lemma 2 for ~ pm2mpghm . ...
pm2mpghmlem1 22637 Lemma 1 for pm2mpghm . (C...
pm2mpfo 22638 The transformation of poly...
pm2mpf1o 22639 The transformation of poly...
pm2mpghm 22640 The transformation of poly...
pm2mpgrpiso 22641 The transformation of poly...
pm2mpmhmlem1 22642 Lemma 1 for ~ pm2mpmhm . ...
pm2mpmhmlem2 22643 Lemma 2 for ~ pm2mpmhm . ...
pm2mpmhm 22644 The transformation of poly...
pm2mprhm 22645 The transformation of poly...
pm2mprngiso 22646 The transformation of poly...
pmmpric 22647 The ring of polynomial mat...
monmat2matmon 22648 The transformation of a po...
pm2mp 22649 The transformation of a su...
chmatcl 22652 Closure of the characteris...
chmatval 22653 The entries of the charact...
chpmatfval 22654 Value of the characteristi...
chpmatval 22655 The characteristic polynom...
chpmatply1 22656 The characteristic polynom...
chpmatval2 22657 The characteristic polynom...
chpmat0d 22658 The characteristic polynom...
chpmat1dlem 22659 Lemma for ~ chpmat1d . (C...
chpmat1d 22660 The characteristic polynom...
chpdmatlem0 22661 Lemma 0 for ~ chpdmat . (...
chpdmatlem1 22662 Lemma 1 for ~ chpdmat . (...
chpdmatlem2 22663 Lemma 2 for ~ chpdmat . (...
chpdmatlem3 22664 Lemma 3 for ~ chpdmat . (...
chpdmat 22665 The characteristic polynom...
chpscmat 22666 The characteristic polynom...
chpscmat0 22667 The characteristic polynom...
chpscmatgsumbin 22668 The characteristic polynom...
chpscmatgsummon 22669 The characteristic polynom...
chp0mat 22670 The characteristic polynom...
chpidmat 22671 The characteristic polynom...
chmaidscmat 22672 The characteristic polynom...
fvmptnn04if 22673 The function values of a m...
fvmptnn04ifa 22674 The function value of a ma...
fvmptnn04ifb 22675 The function value of a ma...
fvmptnn04ifc 22676 The function value of a ma...
fvmptnn04ifd 22677 The function value of a ma...
chfacfisf 22678 The "characteristic factor...
chfacfisfcpmat 22679 The "characteristic factor...
chfacffsupp 22680 The "characteristic factor...
chfacfscmulcl 22681 Closure of a scaled value ...
chfacfscmul0 22682 A scaled value of the "cha...
chfacfscmulfsupp 22683 A mapping of scaled values...
chfacfscmulgsum 22684 Breaking up a sum of value...
chfacfpmmulcl 22685 Closure of the value of th...
chfacfpmmul0 22686 The value of the "characte...
chfacfpmmulfsupp 22687 A mapping of values of the...
chfacfpmmulgsum 22688 Breaking up a sum of value...
chfacfpmmulgsum2 22689 Breaking up a sum of value...
cayhamlem1 22690 Lemma 1 for ~ cayleyhamilt...
cpmadurid 22691 The right-hand fundamental...
cpmidgsum 22692 Representation of the iden...
cpmidgsumm2pm 22693 Representation of the iden...
cpmidpmatlem1 22694 Lemma 1 for ~ cpmidpmat . ...
cpmidpmatlem2 22695 Lemma 2 for ~ cpmidpmat . ...
cpmidpmatlem3 22696 Lemma 3 for ~ cpmidpmat . ...
cpmidpmat 22697 Representation of the iden...
cpmadugsumlemB 22698 Lemma B for ~ cpmadugsum ....
cpmadugsumlemC 22699 Lemma C for ~ cpmadugsum ....
cpmadugsumlemF 22700 Lemma F for ~ cpmadugsum ....
cpmadugsumfi 22701 The product of the charact...
cpmadugsum 22702 The product of the charact...
cpmidgsum2 22703 Representation of the iden...
cpmidg2sum 22704 Equality of two sums repre...
cpmadumatpolylem1 22705 Lemma 1 for ~ cpmadumatpol...
cpmadumatpolylem2 22706 Lemma 2 for ~ cpmadumatpol...
cpmadumatpoly 22707 The product of the charact...
cayhamlem2 22708 Lemma for ~ cayhamlem3 . ...
chcoeffeqlem 22709 Lemma for ~ chcoeffeq . (...
chcoeffeq 22710 The coefficients of the ch...
cayhamlem3 22711 Lemma for ~ cayhamlem4 . ...
cayhamlem4 22712 Lemma for ~ cayleyhamilton...
cayleyhamilton0 22713 The Cayley-Hamilton theore...
cayleyhamilton 22714 The Cayley-Hamilton theore...
cayleyhamiltonALT 22715 Alternate proof of ~ cayle...
cayleyhamilton1 22716 The Cayley-Hamilton theore...
istopg 22719 Express the predicate " ` ...
istop2g 22720 Express the predicate " ` ...
uniopn 22721 The union of a subset of a...
iunopn 22722 The indexed union of a sub...
inopn 22723 The intersection of two op...
fitop 22724 A topology is closed under...
fiinopn 22725 The intersection of a none...
iinopn 22726 The intersection of a none...
unopn 22727 The union of two open sets...
0opn 22728 The empty set is an open s...
0ntop 22729 The empty set is not a top...
topopn 22730 The underlying set of a to...
eltopss 22731 A member of a topology is ...
riinopn 22732 A finite indexed relative ...
rintopn 22733 A finite relative intersec...
istopon 22736 Property of being a topolo...
topontop 22737 A topology on a given base...
toponuni 22738 The base set of a topology...
topontopi 22739 A topology on a given base...
toponunii 22740 The base set of a topology...
toptopon 22741 Alternative definition of ...
toptopon2 22742 A topology is the same thi...
topontopon 22743 A topology on a set is a t...
funtopon 22744 The class ` TopOn ` is a f...
toponrestid 22745 Given a topology on a set,...
toponsspwpw 22746 The set of topologies on a...
dmtopon 22747 The domain of ` TopOn ` is...
fntopon 22748 The class ` TopOn ` is a f...
toprntopon 22749 A topology is the same thi...
toponmax 22750 The base set of a topology...
toponss 22751 A member of a topology is ...
toponcom 22752 If ` K ` is a topology on ...
toponcomb 22753 Biconditional form of ~ to...
topgele 22754 The topologies over the sa...
topsn 22755 The only topology on a sin...
istps 22758 Express the predicate "is ...
istps2 22759 Express the predicate "is ...
tpsuni 22760 The base set of a topologi...
tpstop 22761 The topology extractor on ...
tpspropd 22762 A topological space depend...
tpsprop2d 22763 A topological space depend...
topontopn 22764 Express the predicate "is ...
tsettps 22765 If the topology component ...
istpsi 22766 Properties that determine ...
eltpsg 22767 Properties that determine ...
eltpsgOLD 22768 Obsolete version of ~ eltp...
eltpsi 22769 Properties that determine ...
isbasisg 22772 Express the predicate "the...
isbasis2g 22773 Express the predicate "the...
isbasis3g 22774 Express the predicate "the...
basis1 22775 Property of a basis. (Con...
basis2 22776 Property of a basis. (Con...
fiinbas 22777 If a set is closed under f...
basdif0 22778 A basis is not affected by...
baspartn 22779 A disjoint system of sets ...
tgval 22780 The topology generated by ...
tgval2 22781 Definition of a topology g...
eltg 22782 Membership in a topology g...
eltg2 22783 Membership in a topology g...
eltg2b 22784 Membership in a topology g...
eltg4i 22785 An open set in a topology ...
eltg3i 22786 The union of a set of basi...
eltg3 22787 Membership in a topology g...
tgval3 22788 Alternate expression for t...
tg1 22789 Property of a member of a ...
tg2 22790 Property of a member of a ...
bastg 22791 A member of a basis is a s...
unitg 22792 The topology generated by ...
tgss 22793 Subset relation for genera...
tgcl 22794 Show that a basis generate...
tgclb 22795 The property ~ tgcl can be...
tgtopon 22796 A basis generates a topolo...
topbas 22797 A topology is its own basi...
tgtop 22798 A topology is its own basi...
eltop 22799 Membership in a topology, ...
eltop2 22800 Membership in a topology. ...
eltop3 22801 Membership in a topology. ...
fibas 22802 A collection of finite int...
tgdom 22803 A space has no more open s...
tgiun 22804 The indexed union of a set...
tgidm 22805 The topology generator fun...
bastop 22806 Two ways to express that a...
tgtop11 22807 The topology generation fu...
0top 22808 The singleton of the empty...
en1top 22809 ` { (/) } ` is the only to...
en2top 22810 If a topology has two elem...
tgss3 22811 A criterion for determinin...
tgss2 22812 A criterion for determinin...
basgen 22813 Given a topology ` J ` , s...
basgen2 22814 Given a topology ` J ` , s...
2basgen 22815 Conditions that determine ...
tgfiss 22816 If a subbase is included i...
tgdif0 22817 A generated topology is no...
bastop1 22818 A subset of a topology is ...
bastop2 22819 A version of ~ bastop1 tha...
distop 22820 The discrete topology on a...
topnex 22821 The class of all topologie...
distopon 22822 The discrete topology on a...
sn0topon 22823 The singleton of the empty...
sn0top 22824 The singleton of the empty...
indislem 22825 A lemma to eliminate some ...
indistopon 22826 The indiscrete topology on...
indistop 22827 The indiscrete topology on...
indisuni 22828 The base set of the indisc...
fctop 22829 The finite complement topo...
fctop2 22830 The finite complement topo...
cctop 22831 The countable complement t...
ppttop 22832 The particular point topol...
pptbas 22833 The particular point topol...
epttop 22834 The excluded point topolog...
indistpsx 22835 The indiscrete topology on...
indistps 22836 The indiscrete topology on...
indistps2 22837 The indiscrete topology on...
indistpsALT 22838 The indiscrete topology on...
indistpsALTOLD 22839 Obsolete version of ~ indi...
indistps2ALT 22840 The indiscrete topology on...
distps 22841 The discrete topology on a...
fncld 22848 The closed-set generator i...
cldval 22849 The set of closed sets of ...
ntrfval 22850 The interior function on t...
clsfval 22851 The closure function on th...
cldrcl 22852 Reverse closure of the clo...
iscld 22853 The predicate "the class `...
iscld2 22854 A subset of the underlying...
cldss 22855 A closed set is a subset o...
cldss2 22856 The set of closed sets is ...
cldopn 22857 The complement of a closed...
isopn2 22858 A subset of the underlying...
opncld 22859 The complement of an open ...
difopn 22860 The difference of a closed...
topcld 22861 The underlying set of a to...
ntrval 22862 The interior of a subset o...
clsval 22863 The closure of a subset of...
0cld 22864 The empty set is closed. ...
iincld 22865 The indexed intersection o...
intcld 22866 The intersection of a set ...
uncld 22867 The union of two closed se...
cldcls 22868 A closed subset equals its...
incld 22869 The intersection of two cl...
riincld 22870 An indexed relative inters...
iuncld 22871 A finite indexed union of ...
unicld 22872 A finite union of closed s...
clscld 22873 The closure of a subset of...
clsf 22874 The closure function is a ...
ntropn 22875 The interior of a subset o...
clsval2 22876 Express closure in terms o...
ntrval2 22877 Interior expressed in term...
ntrdif 22878 An interior of a complemen...
clsdif 22879 A closure of a complement ...
clsss 22880 Subset relationship for cl...
ntrss 22881 Subset relationship for in...
sscls 22882 A subset of a topology's u...
ntrss2 22883 A subset includes its inte...
ssntr 22884 An open subset of a set is...
clsss3 22885 The closure of a subset of...
ntrss3 22886 The interior of a subset o...
ntrin 22887 A pairwise intersection of...
cmclsopn 22888 The complement of a closur...
cmntrcld 22889 The complement of an inter...
iscld3 22890 A subset is closed iff it ...
iscld4 22891 A subset is closed iff it ...
isopn3 22892 A subset is open iff it eq...
clsidm 22893 The closure operation is i...
ntridm 22894 The interior operation is ...
clstop 22895 The closure of a topology'...
ntrtop 22896 The interior of a topology...
0ntr 22897 A subset with an empty int...
clsss2 22898 If a subset is included in...
elcls 22899 Membership in a closure. ...
elcls2 22900 Membership in a closure. ...
clsndisj 22901 Any open set containing a ...
ntrcls0 22902 A subset whose closure has...
ntreq0 22903 Two ways to say that a sub...
cldmre 22904 The closed sets of a topol...
mrccls 22905 Moore closure generalizes ...
cls0 22906 The closure of the empty s...
ntr0 22907 The interior of the empty ...
isopn3i 22908 An open subset equals its ...
elcls3 22909 Membership in a closure in...
opncldf1 22910 A bijection useful for con...
opncldf2 22911 The values of the open-clo...
opncldf3 22912 The values of the converse...
isclo 22913 A set ` A ` is clopen iff ...
isclo2 22914 A set ` A ` is clopen iff ...
discld 22915 The open sets of a discret...
sn0cld 22916 The closed sets of the top...
indiscld 22917 The closed sets of an indi...
mretopd 22918 A Moore collection which i...
toponmre 22919 The topologies over a give...
cldmreon 22920 The closed sets of a topol...
iscldtop 22921 A family is the closed set...
mreclatdemoBAD 22922 The closed subspaces of a ...
neifval 22925 Value of the neighborhood ...
neif 22926 The neighborhood function ...
neiss2 22927 A set with a neighborhood ...
neival 22928 Value of the set of neighb...
isnei 22929 The predicate "the class `...
neiint 22930 An intuitive definition of...
isneip 22931 The predicate "the class `...
neii1 22932 A neighborhood is included...
neisspw 22933 The neighborhoods of any s...
neii2 22934 Property of a neighborhood...
neiss 22935 Any neighborhood of a set ...
ssnei 22936 A set is included in any o...
elnei 22937 A point belongs to any of ...
0nnei 22938 The empty set is not a nei...
neips 22939 A neighborhood of a set is...
opnneissb 22940 An open set is a neighborh...
opnssneib 22941 Any superset of an open se...
ssnei2 22942 Any subset ` M ` of ` X ` ...
neindisj 22943 Any neighborhood of an ele...
opnneiss 22944 An open set is a neighborh...
opnneip 22945 An open set is a neighborh...
opnnei 22946 A set is open iff it is a ...
tpnei 22947 The underlying set of a to...
neiuni 22948 The union of the neighborh...
neindisj2 22949 A point ` P ` belongs to t...
topssnei 22950 A finer topology has more ...
innei 22951 The intersection of two ne...
opnneiid 22952 Only an open set is a neig...
neissex 22953 For any neighborhood ` N `...
0nei 22954 The empty set is a neighbo...
neipeltop 22955 Lemma for ~ neiptopreu . ...
neiptopuni 22956 Lemma for ~ neiptopreu . ...
neiptoptop 22957 Lemma for ~ neiptopreu . ...
neiptopnei 22958 Lemma for ~ neiptopreu . ...
neiptopreu 22959 If, to each element ` P ` ...
lpfval 22964 The limit point function o...
lpval 22965 The set of limit points of...
islp 22966 The predicate "the class `...
lpsscls 22967 The limit points of a subs...
lpss 22968 The limit points of a subs...
lpdifsn 22969 ` P ` is a limit point of ...
lpss3 22970 Subset relationship for li...
islp2 22971 The predicate " ` P ` is a...
islp3 22972 The predicate " ` P ` is a...
maxlp 22973 A point is a limit point o...
clslp 22974 The closure of a subset of...
islpi 22975 A point belonging to a set...
cldlp 22976 A subset of a topological ...
isperf 22977 Definition of a perfect sp...
isperf2 22978 Definition of a perfect sp...
isperf3 22979 A perfect space is a topol...
perflp 22980 The limit points of a perf...
perfi 22981 Property of a perfect spac...
perftop 22982 A perfect space is a topol...
restrcl 22983 Reverse closure for the su...
restbas 22984 A subspace topology basis ...
tgrest 22985 A subspace can be generate...
resttop 22986 A subspace topology is a t...
resttopon 22987 A subspace topology is a t...
restuni 22988 The underlying set of a su...
stoig 22989 The topological space buil...
restco 22990 Composition of subspaces. ...
restabs 22991 Equivalence of being a sub...
restin 22992 When the subspace region i...
restuni2 22993 The underlying set of a su...
resttopon2 22994 The underlying set of a su...
rest0 22995 The subspace topology indu...
restsn 22996 The only subspace topology...
restsn2 22997 The subspace topology indu...
restcld 22998 A closed set of a subspace...
restcldi 22999 A closed set is closed in ...
restcldr 23000 A set which is closed in t...
restopnb 23001 If ` B ` is an open subset...
ssrest 23002 If ` K ` is a finer topolo...
restopn2 23003 If ` A ` is open, then ` B...
restdis 23004 A subspace of a discrete t...
restfpw 23005 The restriction of the set...
neitr 23006 The neighborhood of a trac...
restcls 23007 A closure in a subspace to...
restntr 23008 An interior in a subspace ...
restlp 23009 The limit points of a subs...
restperf 23010 Perfection of a subspace. ...
perfopn 23011 An open subset of a perfec...
resstopn 23012 The topology of a restrict...
resstps 23013 A restricted topological s...
ordtbaslem 23014 Lemma for ~ ordtbas . In ...
ordtval 23015 Value of the order topolog...
ordtuni 23016 Value of the order topolog...
ordtbas2 23017 Lemma for ~ ordtbas . (Co...
ordtbas 23018 In a total order, the fini...
ordttopon 23019 Value of the order topolog...
ordtopn1 23020 An upward ray ` ( P , +oo ...
ordtopn2 23021 A downward ray ` ( -oo , P...
ordtopn3 23022 An open interval ` ( A , B...
ordtcld1 23023 A downward ray ` ( -oo , P...
ordtcld2 23024 An upward ray ` [ P , +oo ...
ordtcld3 23025 A closed interval ` [ A , ...
ordttop 23026 The order topology is a to...
ordtcnv 23027 The order dual generates t...
ordtrest 23028 The subspace topology of a...
ordtrest2lem 23029 Lemma for ~ ordtrest2 . (...
ordtrest2 23030 An interval-closed set ` A...
letopon 23031 The topology of the extend...
letop 23032 The topology of the extend...
letopuni 23033 The topology of the extend...
xrstopn 23034 The topology component of ...
xrstps 23035 The extended real number s...
leordtvallem1 23036 Lemma for ~ leordtval . (...
leordtvallem2 23037 Lemma for ~ leordtval . (...
leordtval2 23038 The topology of the extend...
leordtval 23039 The topology of the extend...
iccordt 23040 A closed interval is close...
iocpnfordt 23041 An unbounded above open in...
icomnfordt 23042 An unbounded above open in...
iooordt 23043 An open interval is open i...
reordt 23044 The real numbers are an op...
lecldbas 23045 The set of closed interval...
pnfnei 23046 A neighborhood of ` +oo ` ...
mnfnei 23047 A neighborhood of ` -oo ` ...
ordtrestixx 23048 The restriction of the les...
ordtresticc 23049 The restriction of the les...
lmrel 23056 The topological space conv...
lmrcl 23057 Reverse closure for the co...
lmfval 23058 The relation "sequence ` f...
cnfval 23059 The set of all continuous ...
cnpfval 23060 The function mapping the p...
iscn 23061 The predicate "the class `...
cnpval 23062 The set of all functions f...
iscnp 23063 The predicate "the class `...
iscn2 23064 The predicate "the class `...
iscnp2 23065 The predicate "the class `...
cntop1 23066 Reverse closure for a cont...
cntop2 23067 Reverse closure for a cont...
cnptop1 23068 Reverse closure for a func...
cnptop2 23069 Reverse closure for a func...
iscnp3 23070 The predicate "the class `...
cnprcl 23071 Reverse closure for a func...
cnf 23072 A continuous function is a...
cnpf 23073 A continuous function at p...
cnpcl 23074 The value of a continuous ...
cnf2 23075 A continuous function is a...
cnpf2 23076 A continuous function at p...
cnprcl2 23077 Reverse closure for a func...
tgcn 23078 The continuity predicate w...
tgcnp 23079 The "continuous at a point...
subbascn 23080 The continuity predicate w...
ssidcn 23081 The identity function is a...
cnpimaex 23082 Property of a function con...
idcn 23083 A restricted identity func...
lmbr 23084 Express the binary relatio...
lmbr2 23085 Express the binary relatio...
lmbrf 23086 Express the binary relatio...
lmconst 23087 A constant sequence conver...
lmcvg 23088 Convergence property of a ...
iscnp4 23089 The predicate "the class `...
cnpnei 23090 A condition for continuity...
cnima 23091 An open subset of the codo...
cnco 23092 The composition of two con...
cnpco 23093 The composition of a funct...
cnclima 23094 A closed subset of the cod...
iscncl 23095 A characterization of a co...
cncls2i 23096 Property of the preimage o...
cnntri 23097 Property of the preimage o...
cnclsi 23098 Property of the image of a...
cncls2 23099 Continuity in terms of clo...
cncls 23100 Continuity in terms of clo...
cnntr 23101 Continuity in terms of int...
cnss1 23102 If the topology ` K ` is f...
cnss2 23103 If the topology ` K ` is f...
cncnpi 23104 A continuous function is c...
cnsscnp 23105 The set of continuous func...
cncnp 23106 A continuous function is c...
cncnp2 23107 A continuous function is c...
cnnei 23108 Continuity in terms of nei...
cnconst2 23109 A constant function is con...
cnconst 23110 A constant function is con...
cnrest 23111 Continuity of a restrictio...
cnrest2 23112 Equivalence of continuity ...
cnrest2r 23113 Equivalence of continuity ...
cnpresti 23114 One direction of ~ cnprest...
cnprest 23115 Equivalence of continuity ...
cnprest2 23116 Equivalence of point-conti...
cndis 23117 Every function is continuo...
cnindis 23118 Every function is continuo...
cnpdis 23119 If ` A ` is an isolated po...
paste 23120 Pasting lemma. If ` A ` a...
lmfpm 23121 If ` F ` converges, then `...
lmfss 23122 Inclusion of a function ha...
lmcl 23123 Closure of a limit. (Cont...
lmss 23124 Limit on a subspace. (Con...
sslm 23125 A finer topology has fewer...
lmres 23126 A function converges iff i...
lmff 23127 If ` F ` converges, there ...
lmcls 23128 Any convergent sequence of...
lmcld 23129 Any convergent sequence of...
lmcnp 23130 The image of a convergent ...
lmcn 23131 The image of a convergent ...
ist0 23146 The predicate "is a T_0 sp...
ist1 23147 The predicate "is a T_1 sp...
ishaus 23148 The predicate "is a Hausdo...
iscnrm 23149 The property of being comp...
t0sep 23150 Any two topologically indi...
t0dist 23151 Any two distinct points in...
t1sncld 23152 In a T_1 space, singletons...
t1ficld 23153 In a T_1 space, finite set...
hausnei 23154 Neighborhood property of a...
t0top 23155 A T_0 space is a topologic...
t1top 23156 A T_1 space is a topologic...
haustop 23157 A Hausdorff space is a top...
isreg 23158 The predicate "is a regula...
regtop 23159 A regular space is a topol...
regsep 23160 In a regular space, every ...
isnrm 23161 The predicate "is a normal...
nrmtop 23162 A normal space is a topolo...
cnrmtop 23163 A completely normal space ...
iscnrm2 23164 The property of being comp...
ispnrm 23165 The property of being perf...
pnrmnrm 23166 A perfectly normal space i...
pnrmtop 23167 A perfectly normal space i...
pnrmcld 23168 A closed set in a perfectl...
pnrmopn 23169 An open set in a perfectly...
ist0-2 23170 The predicate "is a T_0 sp...
ist0-3 23171 The predicate "is a T_0 sp...
cnt0 23172 The preimage of a T_0 topo...
ist1-2 23173 An alternate characterizat...
t1t0 23174 A T_1 space is a T_0 space...
ist1-3 23175 A space is T_1 iff every p...
cnt1 23176 The preimage of a T_1 topo...
ishaus2 23177 Express the predicate " ` ...
haust1 23178 A Hausdorff space is a T_1...
hausnei2 23179 The Hausdorff condition st...
cnhaus 23180 The preimage of a Hausdorf...
nrmsep3 23181 In a normal space, given a...
nrmsep2 23182 In a normal space, any two...
nrmsep 23183 In a normal space, disjoin...
isnrm2 23184 An alternate characterizat...
isnrm3 23185 A topological space is nor...
cnrmi 23186 A subspace of a completely...
cnrmnrm 23187 A completely normal space ...
restcnrm 23188 A subspace of a completely...
resthauslem 23189 Lemma for ~ resthaus and s...
lpcls 23190 The limit points of the cl...
perfcls 23191 A subset of a perfect spac...
restt0 23192 A subspace of a T_0 topolo...
restt1 23193 A subspace of a T_1 topolo...
resthaus 23194 A subspace of a Hausdorff ...
t1sep2 23195 Any two points in a T_1 sp...
t1sep 23196 Any two distinct points in...
sncld 23197 A singleton is closed in a...
sshauslem 23198 Lemma for ~ sshaus and sim...
sst0 23199 A topology finer than a T_...
sst1 23200 A topology finer than a T_...
sshaus 23201 A topology finer than a Ha...
regsep2 23202 In a regular space, a clos...
isreg2 23203 A topological space is reg...
dnsconst 23204 If a continuous mapping to...
ordtt1 23205 The order topology is T_1 ...
lmmo 23206 A sequence in a Hausdorff ...
lmfun 23207 The convergence relation i...
dishaus 23208 A discrete topology is Hau...
ordthauslem 23209 Lemma for ~ ordthaus . (C...
ordthaus 23210 The order topology of a to...
xrhaus 23211 The topology of the extend...
iscmp 23214 The predicate "is a compac...
cmpcov 23215 An open cover of a compact...
cmpcov2 23216 Rewrite ~ cmpcov for the c...
cmpcovf 23217 Combine ~ cmpcov with ~ ac...
cncmp 23218 Compactness is respected b...
fincmp 23219 A finite topology is compa...
0cmp 23220 The singleton of the empty...
cmptop 23221 A compact topology is a to...
rncmp 23222 The image of a compact set...
imacmp 23223 The image of a compact set...
discmp 23224 A discrete topology is com...
cmpsublem 23225 Lemma for ~ cmpsub . (Con...
cmpsub 23226 Two equivalent ways of des...
tgcmp 23227 A topology generated by a ...
cmpcld 23228 A closed subset of a compa...
uncmp 23229 The union of two compact s...
fiuncmp 23230 A finite union of compact ...
sscmp 23231 A subset of a compact topo...
hauscmplem 23232 Lemma for ~ hauscmp . (Co...
hauscmp 23233 A compact subspace of a T2...
cmpfi 23234 If a topology is compact a...
cmpfii 23235 In a compact topology, a s...
bwth 23236 The glorious Bolzano-Weier...
isconn 23239 The predicate ` J ` is a c...
isconn2 23240 The predicate ` J ` is a c...
connclo 23241 The only nonempty clopen s...
conndisj 23242 If a topology is connected...
conntop 23243 A connected topology is a ...
indisconn 23244 The indiscrete topology (o...
dfconn2 23245 An alternate definition of...
connsuba 23246 Connectedness for a subspa...
connsub 23247 Two equivalent ways of say...
cnconn 23248 Connectedness is respected...
nconnsubb 23249 Disconnectedness for a sub...
connsubclo 23250 If a clopen set meets a co...
connima 23251 The image of a connected s...
conncn 23252 A continuous function from...
iunconnlem 23253 Lemma for ~ iunconn . (Co...
iunconn 23254 The indexed union of conne...
unconn 23255 The union of two connected...
clsconn 23256 The closure of a connected...
conncompid 23257 The connected component co...
conncompconn 23258 The connected component co...
conncompss 23259 The connected component co...
conncompcld 23260 The connected component co...
conncompclo 23261 The connected component co...
t1connperf 23262 A connected T_1 space is p...
is1stc 23267 The predicate "is a first-...
is1stc2 23268 An equivalent way of sayin...
1stctop 23269 A first-countable topology...
1stcclb 23270 A property of points in a ...
1stcfb 23271 For any point ` A ` in a f...
is2ndc 23272 The property of being seco...
2ndctop 23273 A second-countable topolog...
2ndci 23274 A countable basis generate...
2ndcsb 23275 Having a countable subbase...
2ndcredom 23276 A second-countable space h...
2ndc1stc 23277 A second-countable space i...
1stcrestlem 23278 Lemma for ~ 1stcrest . (C...
1stcrest 23279 A subspace of a first-coun...
2ndcrest 23280 A subspace of a second-cou...
2ndcctbss 23281 If a topology is second-co...
2ndcdisj 23282 Any disjoint family of ope...
2ndcdisj2 23283 Any disjoint collection of...
2ndcomap 23284 A surjective continuous op...
2ndcsep 23285 A second-countable topolog...
dis2ndc 23286 A discrete space is second...
1stcelcls 23287 A point belongs to the clo...
1stccnp 23288 A mapping is continuous at...
1stccn 23289 A mapping ` X --> Y ` , wh...
islly 23294 The property of being a lo...
isnlly 23295 The property of being an n...
llyeq 23296 Equality theorem for the `...
nllyeq 23297 Equality theorem for the `...
llytop 23298 A locally ` A ` space is a...
nllytop 23299 A locally ` A ` space is a...
llyi 23300 The property of a locally ...
nllyi 23301 The property of an n-local...
nlly2i 23302 Eliminate the neighborhood...
llynlly 23303 A locally ` A ` space is n...
llyssnlly 23304 A locally ` A ` space is n...
llyss 23305 The "locally" predicate re...
nllyss 23306 The "n-locally" predicate ...
subislly 23307 The property of a subspace...
restnlly 23308 If the property ` A ` pass...
restlly 23309 If the property ` A ` pass...
islly2 23310 An alternative expression ...
llyrest 23311 An open subspace of a loca...
nllyrest 23312 An open subspace of an n-l...
loclly 23313 If ` A ` is a local proper...
llyidm 23314 Idempotence of the "locall...
nllyidm 23315 Idempotence of the "n-loca...
toplly 23316 A topology is locally a to...
topnlly 23317 A topology is n-locally a ...
hauslly 23318 A Hausdorff space is local...
hausnlly 23319 A Hausdorff space is n-loc...
hausllycmp 23320 A compact Hausdorff space ...
cldllycmp 23321 A closed subspace of a loc...
lly1stc 23322 First-countability is a lo...
dislly 23323 The discrete space ` ~P X ...
disllycmp 23324 A discrete space is locall...
dis1stc 23325 A discrete space is first-...
hausmapdom 23326 If ` X ` is a first-counta...
hauspwdom 23327 Simplify the cardinal ` A ...
refrel 23334 Refinement is a relation. ...
isref 23335 The property of being a re...
refbas 23336 A refinement covers the sa...
refssex 23337 Every set in a refinement ...
ssref 23338 A subcover is a refinement...
refref 23339 Reflexivity of refinement....
reftr 23340 Refinement is transitive. ...
refun0 23341 Adding the empty set prese...
isptfin 23342 The statement "is a point-...
islocfin 23343 The statement "is a locall...
finptfin 23344 A finite cover is a point-...
ptfinfin 23345 A point covered by a point...
finlocfin 23346 A finite cover of a topolo...
locfintop 23347 A locally finite cover cov...
locfinbas 23348 A locally finite cover mus...
locfinnei 23349 A point covered by a local...
lfinpfin 23350 A locally finite cover is ...
lfinun 23351 Adding a finite set preser...
locfincmp 23352 For a compact space, the l...
unisngl 23353 Taking the union of the se...
dissnref 23354 The set of singletons is a...
dissnlocfin 23355 The set of singletons is l...
locfindis 23356 The locally finite covers ...
locfincf 23357 A locally finite cover in ...
comppfsc 23358 A space where every open c...
kgenval 23361 Value of the compact gener...
elkgen 23362 Value of the compact gener...
kgeni 23363 Property of the open sets ...
kgentopon 23364 The compact generator gene...
kgenuni 23365 The base set of the compac...
kgenftop 23366 The compact generator gene...
kgenf 23367 The compact generator is a...
kgentop 23368 A compactly generated spac...
kgenss 23369 The compact generator gene...
kgenhaus 23370 The compact generator gene...
kgencmp 23371 The compact generator topo...
kgencmp2 23372 The compact generator topo...
kgenidm 23373 The compact generator is i...
iskgen2 23374 A space is compactly gener...
iskgen3 23375 Derive the usual definitio...
llycmpkgen2 23376 A locally compact space is...
cmpkgen 23377 A compact space is compact...
llycmpkgen 23378 A locally compact space is...
1stckgenlem 23379 The one-point compactifica...
1stckgen 23380 A first-countable space is...
kgen2ss 23381 The compact generator pres...
kgencn 23382 A function from a compactl...
kgencn2 23383 A function ` F : J --> K `...
kgencn3 23384 The set of continuous func...
kgen2cn 23385 A continuous function is a...
txval 23390 Value of the binary topolo...
txuni2 23391 The underlying set of the ...
txbasex 23392 The basis for the product ...
txbas 23393 The set of Cartesian produ...
eltx 23394 A set in a product is open...
txtop 23395 The product of two topolog...
ptval 23396 The value of the product t...
ptpjpre1 23397 The preimage of a projecti...
elpt 23398 Elementhood in the bases o...
elptr 23399 A basic open set in the pr...
elptr2 23400 A basic open set in the pr...
ptbasid 23401 The base set of the produc...
ptuni2 23402 The base set for the produ...
ptbasin 23403 The basis for a product to...
ptbasin2 23404 The basis for a product to...
ptbas 23405 The basis for a product to...
ptpjpre2 23406 The basis for a product to...
ptbasfi 23407 The basis for the product ...
pttop 23408 The product topology is a ...
ptopn 23409 A basic open set in the pr...
ptopn2 23410 A sub-basic open set in th...
xkotf 23411 Functionality of function ...
xkobval 23412 Alternative expression for...
xkoval 23413 Value of the compact-open ...
xkotop 23414 The compact-open topology ...
xkoopn 23415 A basic open set of the co...
txtopi 23416 The product of two topolog...
txtopon 23417 The underlying set of the ...
txuni 23418 The underlying set of the ...
txunii 23419 The underlying set of the ...
ptuni 23420 The base set for the produ...
ptunimpt 23421 Base set of a product topo...
pttopon 23422 The base set for the produ...
pttoponconst 23423 The base set for a product...
ptuniconst 23424 The base set for a product...
xkouni 23425 The base set of the compac...
xkotopon 23426 The base set of the compac...
ptval2 23427 The value of the product t...
txopn 23428 The product of two open se...
txcld 23429 The product of two closed ...
txcls 23430 Closure of a rectangle in ...
txss12 23431 Subset property of the top...
txbasval 23432 It is sufficient to consid...
neitx 23433 The Cartesian product of t...
txcnpi 23434 Continuity of a two-argume...
tx1cn 23435 Continuity of the first pr...
tx2cn 23436 Continuity of the second p...
ptpjcn 23437 Continuity of a projection...
ptpjopn 23438 The projection map is an o...
ptcld 23439 A closed box in the produc...
ptcldmpt 23440 A closed box in the produc...
ptclsg 23441 The closure of a box in th...
ptcls 23442 The closure of a box in th...
dfac14lem 23443 Lemma for ~ dfac14 . By e...
dfac14 23444 Theorem ~ ptcls is an equi...
xkoccn 23445 The "constant function" fu...
txcnp 23446 If two functions are conti...
ptcnplem 23447 Lemma for ~ ptcnp . (Cont...
ptcnp 23448 If every projection of a f...
upxp 23449 Universal property of the ...
txcnmpt 23450 A map into the product of ...
uptx 23451 Universal property of the ...
txcn 23452 A map into the product of ...
ptcn 23453 If every projection of a f...
prdstopn 23454 Topology of a structure pr...
prdstps 23455 A structure product of top...
pwstps 23456 A structure power of a top...
txrest 23457 The subspace of a topologi...
txdis 23458 The topological product of...
txindislem 23459 Lemma for ~ txindis . (Co...
txindis 23460 The topological product of...
txdis1cn 23461 A function is jointly cont...
txlly 23462 If the property ` A ` is p...
txnlly 23463 If the property ` A ` is p...
pthaus 23464 The product of a collectio...
ptrescn 23465 Restriction is a continuou...
txtube 23466 The "tube lemma". If ` X ...
txcmplem1 23467 Lemma for ~ txcmp . (Cont...
txcmplem2 23468 Lemma for ~ txcmp . (Cont...
txcmp 23469 The topological product of...
txcmpb 23470 The topological product of...
hausdiag 23471 A topology is Hausdorff if...
hauseqlcld 23472 In a Hausdorff topology, t...
txhaus 23473 The topological product of...
txlm 23474 Two sequences converge iff...
lmcn2 23475 The image of a convergent ...
tx1stc 23476 The topological product of...
tx2ndc 23477 The topological product of...
txkgen 23478 The topological product of...
xkohaus 23479 If the codomain space is H...
xkoptsub 23480 The compact-open topology ...
xkopt 23481 The compact-open topology ...
xkopjcn 23482 Continuity of a projection...
xkoco1cn 23483 If ` F ` is a continuous f...
xkoco2cn 23484 If ` F ` is a continuous f...
xkococnlem 23485 Continuity of the composit...
xkococn 23486 Continuity of the composit...
cnmptid 23487 The identity function is c...
cnmptc 23488 A constant function is con...
cnmpt11 23489 The composition of continu...
cnmpt11f 23490 The composition of continu...
cnmpt1t 23491 The composition of continu...
cnmpt12f 23492 The composition of continu...
cnmpt12 23493 The composition of continu...
cnmpt1st 23494 The projection onto the fi...
cnmpt2nd 23495 The projection onto the se...
cnmpt2c 23496 A constant function is con...
cnmpt21 23497 The composition of continu...
cnmpt21f 23498 The composition of continu...
cnmpt2t 23499 The composition of continu...
cnmpt22 23500 The composition of continu...
cnmpt22f 23501 The composition of continu...
cnmpt1res 23502 The restriction of a conti...
cnmpt2res 23503 The restriction of a conti...
cnmptcom 23504 The argument converse of a...
cnmptkc 23505 The curried first projecti...
cnmptkp 23506 The evaluation of the inne...
cnmptk1 23507 The composition of a curri...
cnmpt1k 23508 The composition of a one-a...
cnmptkk 23509 The composition of two cur...
xkofvcn 23510 Joint continuity of the fu...
cnmptk1p 23511 The evaluation of a currie...
cnmptk2 23512 The uncurrying of a currie...
xkoinjcn 23513 Continuity of "injection",...
cnmpt2k 23514 The currying of a two-argu...
txconn 23515 The topological product of...
imasnopn 23516 If a relation graph is ope...
imasncld 23517 If a relation graph is clo...
imasncls 23518 If a relation graph is clo...
qtopval 23521 Value of the quotient topo...
qtopval2 23522 Value of the quotient topo...
elqtop 23523 Value of the quotient topo...
qtopres 23524 The quotient topology is u...
qtoptop2 23525 The quotient topology is a...
qtoptop 23526 The quotient topology is a...
elqtop2 23527 Value of the quotient topo...
qtopuni 23528 The base set of the quotie...
elqtop3 23529 Value of the quotient topo...
qtoptopon 23530 The base set of the quotie...
qtopid 23531 A quotient map is a contin...
idqtop 23532 The quotient topology indu...
qtopcmplem 23533 Lemma for ~ qtopcmp and ~ ...
qtopcmp 23534 A quotient of a compact sp...
qtopconn 23535 A quotient of a connected ...
qtopkgen 23536 A quotient of a compactly ...
basqtop 23537 An injection maps bases to...
tgqtop 23538 An injection maps generate...
qtopcld 23539 The property of being a cl...
qtopcn 23540 Universal property of a qu...
qtopss 23541 A surjective continuous fu...
qtopeu 23542 Universal property of the ...
qtoprest 23543 If ` A ` is a saturated op...
qtopomap 23544 If ` F ` is a surjective c...
qtopcmap 23545 If ` F ` is a surjective c...
imastopn 23546 The topology of an image s...
imastps 23547 The image of a topological...
qustps 23548 A quotient structure is a ...
kqfval 23549 Value of the function appe...
kqfeq 23550 Two points in the Kolmogor...
kqffn 23551 The topological indistingu...
kqval 23552 Value of the quotient topo...
kqtopon 23553 The Kolmogorov quotient is...
kqid 23554 The topological indistingu...
ist0-4 23555 The topological indistingu...
kqfvima 23556 When the image set is open...
kqsat 23557 Any open set is saturated ...
kqdisj 23558 A version of ~ imain for t...
kqcldsat 23559 Any closed set is saturate...
kqopn 23560 The topological indistingu...
kqcld 23561 The topological indistingu...
kqt0lem 23562 Lemma for ~ kqt0 . (Contr...
isr0 23563 The property " ` J ` is an...
r0cld 23564 The analogue of the T_1 ax...
regr1lem 23565 Lemma for ~ regr1 . (Cont...
regr1lem2 23566 A Kolmogorov quotient of a...
kqreglem1 23567 A Kolmogorov quotient of a...
kqreglem2 23568 If the Kolmogorov quotient...
kqnrmlem1 23569 A Kolmogorov quotient of a...
kqnrmlem2 23570 If the Kolmogorov quotient...
kqtop 23571 The Kolmogorov quotient is...
kqt0 23572 The Kolmogorov quotient is...
kqf 23573 The Kolmogorov quotient is...
r0sep 23574 The separation property of...
nrmr0reg 23575 A normal R_0 space is also...
regr1 23576 A regular space is R_1, wh...
kqreg 23577 The Kolmogorov quotient of...
kqnrm 23578 The Kolmogorov quotient of...
hmeofn 23583 The set of homeomorphisms ...
hmeofval 23584 The set of all the homeomo...
ishmeo 23585 The predicate F is a homeo...
hmeocn 23586 A homeomorphism is continu...
hmeocnvcn 23587 The converse of a homeomor...
hmeocnv 23588 The converse of a homeomor...
hmeof1o2 23589 A homeomorphism is a 1-1-o...
hmeof1o 23590 A homeomorphism is a 1-1-o...
hmeoima 23591 The image of an open set b...
hmeoopn 23592 Homeomorphisms preserve op...
hmeocld 23593 Homeomorphisms preserve cl...
hmeocls 23594 Homeomorphisms preserve cl...
hmeontr 23595 Homeomorphisms preserve in...
hmeoimaf1o 23596 The function mapping open ...
hmeores 23597 The restriction of a homeo...
hmeoco 23598 The composite of two homeo...
idhmeo 23599 The identity function is a...
hmeocnvb 23600 The converse of a homeomor...
hmeoqtop 23601 A homeomorphism is a quoti...
hmph 23602 Express the predicate ` J ...
hmphi 23603 If there is a homeomorphis...
hmphtop 23604 Reverse closure for the ho...
hmphtop1 23605 The relation "being homeom...
hmphtop2 23606 The relation "being homeom...
hmphref 23607 "Is homeomorphic to" is re...
hmphsym 23608 "Is homeomorphic to" is sy...
hmphtr 23609 "Is homeomorphic to" is tr...
hmpher 23610 "Is homeomorphic to" is an...
hmphen 23611 Homeomorphisms preserve th...
hmphsymb 23612 "Is homeomorphic to" is sy...
haushmphlem 23613 Lemma for ~ haushmph and s...
cmphmph 23614 Compactness is a topologic...
connhmph 23615 Connectedness is a topolog...
t0hmph 23616 T_0 is a topological prope...
t1hmph 23617 T_1 is a topological prope...
haushmph 23618 Hausdorff-ness is a topolo...
reghmph 23619 Regularity is a topologica...
nrmhmph 23620 Normality is a topological...
hmph0 23621 A topology homeomorphic to...
hmphdis 23622 Homeomorphisms preserve to...
hmphindis 23623 Homeomorphisms preserve to...
indishmph 23624 Equinumerous sets equipped...
hmphen2 23625 Homeomorphisms preserve th...
cmphaushmeo 23626 A continuous bijection fro...
ordthmeolem 23627 Lemma for ~ ordthmeo . (C...
ordthmeo 23628 An order isomorphism is a ...
txhmeo 23629 Lift a pair of homeomorphi...
txswaphmeolem 23630 Show inverse for the "swap...
txswaphmeo 23631 There is a homeomorphism f...
pt1hmeo 23632 The canonical homeomorphis...
ptuncnv 23633 Exhibit the converse funct...
ptunhmeo 23634 Define a homeomorphism fro...
xpstopnlem1 23635 The function ` F ` used in...
xpstps 23636 A binary product of topolo...
xpstopnlem2 23637 Lemma for ~ xpstopn . (Co...
xpstopn 23638 The topology on a binary p...
ptcmpfi 23639 A topological product of f...
xkocnv 23640 The inverse of the "curryi...
xkohmeo 23641 The Exponential Law for to...
qtopf1 23642 If a quotient map is injec...
qtophmeo 23643 If two functions on a base...
t0kq 23644 A topological space is T_0...
kqhmph 23645 A topological space is T_0...
ist1-5lem 23646 Lemma for ~ ist1-5 and sim...
t1r0 23647 A T_1 space is R_0. That ...
ist1-5 23648 A topological space is T_1...
ishaus3 23649 A topological space is Hau...
nrmreg 23650 A normal T_1 space is regu...
reghaus 23651 A regular T_0 space is Hau...
nrmhaus 23652 A T_1 normal space is Haus...
elmptrab 23653 Membership in a one-parame...
elmptrab2 23654 Membership in a one-parame...
isfbas 23655 The predicate " ` F ` is a...
fbasne0 23656 There are no empty filter ...
0nelfb 23657 No filter base contains th...
fbsspw 23658 A filter base on a set is ...
fbelss 23659 An element of the filter b...
fbdmn0 23660 The domain of a filter bas...
isfbas2 23661 The predicate " ` F ` is a...
fbasssin 23662 A filter base contains sub...
fbssfi 23663 A filter base contains sub...
fbssint 23664 A filter base contains sub...
fbncp 23665 A filter base does not con...
fbun 23666 A necessary and sufficient...
fbfinnfr 23667 No filter base containing ...
opnfbas 23668 The collection of open sup...
trfbas2 23669 Conditions for the trace o...
trfbas 23670 Conditions for the trace o...
isfil 23673 The predicate "is a filter...
filfbas 23674 A filter is a filter base....
0nelfil 23675 The empty set doesn't belo...
fileln0 23676 An element of a filter is ...
filsspw 23677 A filter is a subset of th...
filelss 23678 An element of a filter is ...
filss 23679 A filter is closed under t...
filin 23680 A filter is closed under t...
filtop 23681 The underlying set belongs...
isfil2 23682 Derive the standard axioms...
isfildlem 23683 Lemma for ~ isfild . (Con...
isfild 23684 Sufficient condition for a...
filfi 23685 A filter is closed under t...
filinn0 23686 The intersection of two el...
filintn0 23687 A filter has the finite in...
filn0 23688 The empty set is not a fil...
infil 23689 The intersection of two fi...
snfil 23690 A singleton is a filter. ...
fbasweak 23691 A filter base on any set i...
snfbas 23692 Condition for a singleton ...
fsubbas 23693 A condition for a set to g...
fbasfip 23694 A filter base has the fini...
fbunfip 23695 A helpful lemma for showin...
fgval 23696 The filter generating clas...
elfg 23697 A condition for elements o...
ssfg 23698 A filter base is a subset ...
fgss 23699 A bigger base generates a ...
fgss2 23700 A condition for a filter t...
fgfil 23701 A filter generates itself....
elfilss 23702 An element belongs to a fi...
filfinnfr 23703 No filter containing a fin...
fgcl 23704 A generated filter is a fi...
fgabs 23705 Absorption law for filter ...
neifil 23706 The neighborhoods of a non...
filunibas 23707 Recover the base set from ...
filunirn 23708 Two ways to express a filt...
filconn 23709 A filter gives rise to a c...
fbasrn 23710 Given a filter on a domain...
filuni 23711 The union of a nonempty se...
trfil1 23712 Conditions for the trace o...
trfil2 23713 Conditions for the trace o...
trfil3 23714 Conditions for the trace o...
trfilss 23715 If ` A ` is a member of th...
fgtr 23716 If ` A ` is a member of th...
trfg 23717 The trace operation and th...
trnei 23718 The trace, over a set ` A ...
cfinfil 23719 Relative complements of th...
csdfil 23720 The set of all elements wh...
supfil 23721 The supersets of a nonempt...
zfbas 23722 The set of upper sets of i...
uzrest 23723 The restriction of the set...
uzfbas 23724 The set of upper sets of i...
isufil 23729 The property of being an u...
ufilfil 23730 An ultrafilter is a filter...
ufilss 23731 For any subset of the base...
ufilb 23732 The complement is in an ul...
ufilmax 23733 Any filter finer than an u...
isufil2 23734 The maximal property of an...
ufprim 23735 An ultrafilter is a prime ...
trufil 23736 Conditions for the trace o...
filssufilg 23737 A filter is contained in s...
filssufil 23738 A filter is contained in s...
isufl 23739 Define the (strong) ultraf...
ufli 23740 Property of a set that sat...
numufl 23741 Consequence of ~ filssufil...
fiufl 23742 A finite set satisfies the...
acufl 23743 The axiom of choice implie...
ssufl 23744 If ` Y ` is a subset of ` ...
ufileu 23745 If the ultrafilter contain...
filufint 23746 A filter is equal to the i...
uffix 23747 Lemma for ~ fixufil and ~ ...
fixufil 23748 The condition describing a...
uffixfr 23749 An ultrafilter is either f...
uffix2 23750 A classification of fixed ...
uffixsn 23751 The singleton of the gener...
ufildom1 23752 An ultrafilter is generate...
uffinfix 23753 An ultrafilter containing ...
cfinufil 23754 An ultrafilter is free iff...
ufinffr 23755 An infinite subset is cont...
ufilen 23756 Any infinite set has an ul...
ufildr 23757 An ultrafilter gives rise ...
fin1aufil 23758 There are no definable fre...
fmval 23769 Introduce a function that ...
fmfil 23770 A mapping filter is a filt...
fmf 23771 Pushing-forward via a func...
fmss 23772 A finer filter produces a ...
elfm 23773 An element of a mapping fi...
elfm2 23774 An element of a mapping fi...
fmfg 23775 The image filter of a filt...
elfm3 23776 An alternate formulation o...
imaelfm 23777 An image of a filter eleme...
rnelfmlem 23778 Lemma for ~ rnelfm . (Con...
rnelfm 23779 A condition for a filter t...
fmfnfmlem1 23780 Lemma for ~ fmfnfm . (Con...
fmfnfmlem2 23781 Lemma for ~ fmfnfm . (Con...
fmfnfmlem3 23782 Lemma for ~ fmfnfm . (Con...
fmfnfmlem4 23783 Lemma for ~ fmfnfm . (Con...
fmfnfm 23784 A filter finer than an ima...
fmufil 23785 An image filter of an ultr...
fmid 23786 The filter map applied to ...
fmco 23787 Composition of image filte...
ufldom 23788 The ultrafilter lemma prop...
flimval 23789 The set of limit points of...
elflim2 23790 The predicate "is a limit ...
flimtop 23791 Reverse closure for the li...
flimneiss 23792 A filter contains the neig...
flimnei 23793 A filter contains all of t...
flimelbas 23794 A limit point of a filter ...
flimfil 23795 Reverse closure for the li...
flimtopon 23796 Reverse closure for the li...
elflim 23797 The predicate "is a limit ...
flimss2 23798 A limit point of a filter ...
flimss1 23799 A limit point of a filter ...
neiflim 23800 A point is a limit point o...
flimopn 23801 The condition for being a ...
fbflim 23802 A condition for a filter t...
fbflim2 23803 A condition for a filter b...
flimclsi 23804 The convergent points of a...
hausflimlem 23805 If ` A ` and ` B ` are bot...
hausflimi 23806 One direction of ~ hausfli...
hausflim 23807 A condition for a topology...
flimcf 23808 Fineness is properly chara...
flimrest 23809 The set of limit points in...
flimclslem 23810 Lemma for ~ flimcls . (Co...
flimcls 23811 Closure in terms of filter...
flimsncls 23812 If ` A ` is a limit point ...
hauspwpwf1 23813 Lemma for ~ hauspwpwdom . ...
hauspwpwdom 23814 If ` X ` is a Hausdorff sp...
flffval 23815 Given a topology and a fil...
flfval 23816 Given a function from a fi...
flfnei 23817 The property of being a li...
flfneii 23818 A neighborhood of a limit ...
isflf 23819 The property of being a li...
flfelbas 23820 A limit point of a functio...
flffbas 23821 Limit points of a function...
flftg 23822 Limit points of a function...
hausflf 23823 If a function has its valu...
hausflf2 23824 If a convergent function h...
cnpflfi 23825 Forward direction of ~ cnp...
cnpflf2 23826 ` F ` is continuous at poi...
cnpflf 23827 Continuity of a function a...
cnflf 23828 A function is continuous i...
cnflf2 23829 A function is continuous i...
flfcnp 23830 A continuous function pres...
lmflf 23831 The topological limit rela...
txflf 23832 Two sequences converge in ...
flfcnp2 23833 The image of a convergent ...
fclsval 23834 The set of all cluster poi...
isfcls 23835 A cluster point of a filte...
fclsfil 23836 Reverse closure for the cl...
fclstop 23837 Reverse closure for the cl...
fclstopon 23838 Reverse closure for the cl...
isfcls2 23839 A cluster point of a filte...
fclsopn 23840 Write the cluster point co...
fclsopni 23841 An open neighborhood of a ...
fclselbas 23842 A cluster point is in the ...
fclsneii 23843 A neighborhood of a cluste...
fclssscls 23844 The set of cluster points ...
fclsnei 23845 Cluster points in terms of...
supnfcls 23846 The filter of supersets of...
fclsbas 23847 Cluster points in terms of...
fclsss1 23848 A finer topology has fewer...
fclsss2 23849 A finer filter has fewer c...
fclsrest 23850 The set of cluster points ...
fclscf 23851 Characterization of finene...
flimfcls 23852 A limit point is a cluster...
fclsfnflim 23853 A filter clusters at a poi...
flimfnfcls 23854 A filter converges to a po...
fclscmpi 23855 Forward direction of ~ fcl...
fclscmp 23856 A space is compact iff eve...
uffclsflim 23857 The cluster points of an u...
ufilcmp 23858 A space is compact iff eve...
fcfval 23859 The set of cluster points ...
isfcf 23860 The property of being a cl...
fcfnei 23861 The property of being a cl...
fcfelbas 23862 A cluster point of a funct...
fcfneii 23863 A neighborhood of a cluste...
flfssfcf 23864 A limit point of a functio...
uffcfflf 23865 If the domain filter is an...
cnpfcfi 23866 Lemma for ~ cnpfcf . If a...
cnpfcf 23867 A function ` F ` is contin...
cnfcf 23868 Continuity of a function i...
flfcntr 23869 A continuous function's va...
alexsublem 23870 Lemma for ~ alexsub . (Co...
alexsub 23871 The Alexander Subbase Theo...
alexsubb 23872 Biconditional form of the ...
alexsubALTlem1 23873 Lemma for ~ alexsubALT . ...
alexsubALTlem2 23874 Lemma for ~ alexsubALT . ...
alexsubALTlem3 23875 Lemma for ~ alexsubALT . ...
alexsubALTlem4 23876 Lemma for ~ alexsubALT . ...
alexsubALT 23877 The Alexander Subbase Theo...
ptcmplem1 23878 Lemma for ~ ptcmp . (Cont...
ptcmplem2 23879 Lemma for ~ ptcmp . (Cont...
ptcmplem3 23880 Lemma for ~ ptcmp . (Cont...
ptcmplem4 23881 Lemma for ~ ptcmp . (Cont...
ptcmplem5 23882 Lemma for ~ ptcmp . (Cont...
ptcmpg 23883 Tychonoff's theorem: The ...
ptcmp 23884 Tychonoff's theorem: The ...
cnextval 23887 The function applying cont...
cnextfval 23888 The continuous extension o...
cnextrel 23889 In the general case, a con...
cnextfun 23890 If the target space is Hau...
cnextfvval 23891 The value of the continuou...
cnextf 23892 Extension by continuity. ...
cnextcn 23893 Extension by continuity. ...
cnextfres1 23894 ` F ` and its extension by...
cnextfres 23895 ` F ` and its extension by...
istmd 23900 The predicate "is a topolo...
tmdmnd 23901 A topological monoid is a ...
tmdtps 23902 A topological monoid is a ...
istgp 23903 The predicate "is a topolo...
tgpgrp 23904 A topological group is a g...
tgptmd 23905 A topological group is a t...
tgptps 23906 A topological group is a t...
tmdtopon 23907 The topology of a topologi...
tgptopon 23908 The topology of a topologi...
tmdcn 23909 In a topological monoid, t...
tgpcn 23910 In a topological group, th...
tgpinv 23911 In a topological group, th...
grpinvhmeo 23912 The inverse function in a ...
cnmpt1plusg 23913 Continuity of the group su...
cnmpt2plusg 23914 Continuity of the group su...
tmdcn2 23915 Write out the definition o...
tgpsubcn 23916 In a topological group, th...
istgp2 23917 A group with a topology is...
tmdmulg 23918 In a topological monoid, t...
tgpmulg 23919 In a topological group, th...
tgpmulg2 23920 In a topological monoid, t...
tmdgsum 23921 In a topological monoid, t...
tmdgsum2 23922 For any neighborhood ` U `...
oppgtmd 23923 The opposite of a topologi...
oppgtgp 23924 The opposite of a topologi...
distgp 23925 Any group equipped with th...
indistgp 23926 Any group equipped with th...
efmndtmd 23927 The monoid of endofunction...
tmdlactcn 23928 The left group action of e...
tgplacthmeo 23929 The left group action of e...
submtmd 23930 A submonoid of a topologic...
subgtgp 23931 A subgroup of a topologica...
symgtgp 23932 The symmetric group is a t...
subgntr 23933 A subgroup of a topologica...
opnsubg 23934 An open subgroup of a topo...
clssubg 23935 The closure of a subgroup ...
clsnsg 23936 The closure of a normal su...
cldsubg 23937 A subgroup of finite index...
tgpconncompeqg 23938 The connected component co...
tgpconncomp 23939 The identity component, th...
tgpconncompss 23940 The identity component is ...
ghmcnp 23941 A group homomorphism on to...
snclseqg 23942 The coset of the closure o...
tgphaus 23943 A topological group is Hau...
tgpt1 23944 Hausdorff and T1 are equiv...
tgpt0 23945 Hausdorff and T0 are equiv...
qustgpopn 23946 A quotient map in a topolo...
qustgplem 23947 Lemma for ~ qustgp . (Con...
qustgp 23948 The quotient of a topologi...
qustgphaus 23949 The quotient of a topologi...
prdstmdd 23950 The product of a family of...
prdstgpd 23951 The product of a family of...
tsmsfbas 23954 The collection of all sets...
tsmslem1 23955 The finite partial sums of...
tsmsval2 23956 Definition of the topologi...
tsmsval 23957 Definition of the topologi...
tsmspropd 23958 The group sum depends only...
eltsms 23959 The property of being a su...
tsmsi 23960 The property of being a su...
tsmscl 23961 A sum in a topological gro...
haustsms 23962 In a Hausdorff topological...
haustsms2 23963 In a Hausdorff topological...
tsmscls 23964 One half of ~ tgptsmscls ,...
tsmsgsum 23965 The convergent points of a...
tsmsid 23966 If a sum is finite, the us...
haustsmsid 23967 In a Hausdorff topological...
tsms0 23968 The sum of zero is zero. ...
tsmssubm 23969 Evaluate an infinite group...
tsmsres 23970 Extend an infinite group s...
tsmsf1o 23971 Re-index an infinite group...
tsmsmhm 23972 Apply a continuous group h...
tsmsadd 23973 The sum of two infinite gr...
tsmsinv 23974 Inverse of an infinite gro...
tsmssub 23975 The difference of two infi...
tgptsmscls 23976 A sum in a topological gro...
tgptsmscld 23977 The set of limit points to...
tsmssplit 23978 Split a topological group ...
tsmsxplem1 23979 Lemma for ~ tsmsxp . (Con...
tsmsxplem2 23980 Lemma for ~ tsmsxp . (Con...
tsmsxp 23981 Write a sum over a two-dim...
istrg 23990 Express the predicate " ` ...
trgtmd 23991 The multiplicative monoid ...
istdrg 23992 Express the predicate " ` ...
tdrgunit 23993 The unit group of a topolo...
trgtgp 23994 A topological ring is a to...
trgtmd2 23995 A topological ring is a to...
trgtps 23996 A topological ring is a to...
trgring 23997 A topological ring is a ri...
trggrp 23998 A topological ring is a gr...
tdrgtrg 23999 A topological division rin...
tdrgdrng 24000 A topological division rin...
tdrgring 24001 A topological division rin...
tdrgtmd 24002 A topological division rin...
tdrgtps 24003 A topological division rin...
istdrg2 24004 A topological-ring divisio...
mulrcn 24005 The functionalization of t...
invrcn2 24006 The multiplicative inverse...
invrcn 24007 The multiplicative inverse...
cnmpt1mulr 24008 Continuity of ring multipl...
cnmpt2mulr 24009 Continuity of ring multipl...
dvrcn 24010 The division function is c...
istlm 24011 The predicate " ` W ` is a...
vscacn 24012 The scalar multiplication ...
tlmtmd 24013 A topological module is a ...
tlmtps 24014 A topological module is a ...
tlmlmod 24015 A topological module is a ...
tlmtrg 24016 The scalar ring of a topol...
tlmscatps 24017 The scalar ring of a topol...
istvc 24018 A topological vector space...
tvctdrg 24019 The scalar field of a topo...
cnmpt1vsca 24020 Continuity of scalar multi...
cnmpt2vsca 24021 Continuity of scalar multi...
tlmtgp 24022 A topological vector space...
tvctlm 24023 A topological vector space...
tvclmod 24024 A topological vector space...
tvclvec 24025 A topological vector space...
ustfn 24028 The defined uniform struct...
ustval 24029 The class of all uniform s...
isust 24030 The predicate " ` U ` is a...
ustssxp 24031 Entourages are subsets of ...
ustssel 24032 A uniform structure is upw...
ustbasel 24033 The full set is always an ...
ustincl 24034 A uniform structure is clo...
ustdiag 24035 The diagonal set is includ...
ustinvel 24036 If ` V ` is an entourage, ...
ustexhalf 24037 For each entourage ` V ` t...
ustrel 24038 The elements of uniform st...
ustfilxp 24039 A uniform structure on a n...
ustne0 24040 A uniform structure cannot...
ustssco 24041 In an uniform structure, a...
ustexsym 24042 In an uniform structure, f...
ustex2sym 24043 In an uniform structure, f...
ustex3sym 24044 In an uniform structure, f...
ustref 24045 Any element of the base se...
ust0 24046 The unique uniform structu...
ustn0 24047 The empty set is not an un...
ustund 24048 If two intersecting sets `...
ustelimasn 24049 Any point ` A ` is near en...
ustneism 24050 For a point ` A ` in ` X `...
elrnustOLD 24051 Obsolete version of ~ elfv...
ustbas2 24052 Second direction for ~ ust...
ustuni 24053 The set union of a uniform...
ustbas 24054 Recover the base of an uni...
ustimasn 24055 Lemma for ~ ustuqtop . (C...
trust 24056 The trace of a uniform str...
utopval 24059 The topology induced by a ...
elutop 24060 Open sets in the topology ...
utoptop 24061 The topology induced by a ...
utopbas 24062 The base of the topology i...
utoptopon 24063 Topology induced by a unif...
restutop 24064 Restriction of a topology ...
restutopopn 24065 The restriction of the top...
ustuqtoplem 24066 Lemma for ~ ustuqtop . (C...
ustuqtop0 24067 Lemma for ~ ustuqtop . (C...
ustuqtop1 24068 Lemma for ~ ustuqtop , sim...
ustuqtop2 24069 Lemma for ~ ustuqtop . (C...
ustuqtop3 24070 Lemma for ~ ustuqtop , sim...
ustuqtop4 24071 Lemma for ~ ustuqtop . (C...
ustuqtop5 24072 Lemma for ~ ustuqtop . (C...
ustuqtop 24073 For a given uniform struct...
utopsnneiplem 24074 The neighborhoods of a poi...
utopsnneip 24075 The neighborhoods of a poi...
utopsnnei 24076 Images of singletons by en...
utop2nei 24077 For any symmetrical entour...
utop3cls 24078 Relation between a topolog...
utopreg 24079 All Hausdorff uniform spac...
ussval 24086 The uniform structure on u...
ussid 24087 In case the base of the ` ...
isusp 24088 The predicate ` W ` is a u...
ressuss 24089 Value of the uniform struc...
ressust 24090 The uniform structure of a...
ressusp 24091 The restriction of a unifo...
tusval 24092 The value of the uniform s...
tuslem 24093 Lemma for ~ tusbas , ~ tus...
tuslemOLD 24094 Obsolete proof of ~ tuslem...
tusbas 24095 The base set of a construc...
tusunif 24096 The uniform structure of a...
tususs 24097 The uniform structure of a...
tustopn 24098 The topology induced by a ...
tususp 24099 A constructed uniform spac...
tustps 24100 A constructed uniform spac...
uspreg 24101 If a uniform space is Haus...
ucnval 24104 The set of all uniformly c...
isucn 24105 The predicate " ` F ` is a...
isucn2 24106 The predicate " ` F ` is a...
ucnimalem 24107 Reformulate the ` G ` func...
ucnima 24108 An equivalent statement of...
ucnprima 24109 The preimage by a uniforml...
iducn 24110 The identity is uniformly ...
cstucnd 24111 A constant function is uni...
ucncn 24112 Uniform continuity implies...
iscfilu 24115 The predicate " ` F ` is a...
cfilufbas 24116 A Cauchy filter base is a ...
cfiluexsm 24117 For a Cauchy filter base a...
fmucndlem 24118 Lemma for ~ fmucnd . (Con...
fmucnd 24119 The image of a Cauchy filt...
cfilufg 24120 The filter generated by a ...
trcfilu 24121 Condition for the trace of...
cfiluweak 24122 A Cauchy filter base is al...
neipcfilu 24123 In an uniform space, a nei...
iscusp 24126 The predicate " ` W ` is a...
cuspusp 24127 A complete uniform space i...
cuspcvg 24128 In a complete uniform spac...
iscusp2 24129 The predicate " ` W ` is a...
cnextucn 24130 Extension by continuity. ...
ucnextcn 24131 Extension by continuity. ...
ispsmet 24132 Express the predicate " ` ...
psmetdmdm 24133 Recover the base set from ...
psmetf 24134 The distance function of a...
psmetcl 24135 Closure of the distance fu...
psmet0 24136 The distance function of a...
psmettri2 24137 Triangle inequality for th...
psmetsym 24138 The distance function of a...
psmettri 24139 Triangle inequality for th...
psmetge0 24140 The distance function of a...
psmetxrge0 24141 The distance function of a...
psmetres2 24142 Restriction of a pseudomet...
psmetlecl 24143 Real closure of an extende...
distspace 24144 A set ` X ` together with ...
ismet 24151 Express the predicate " ` ...
isxmet 24152 Express the predicate " ` ...
ismeti 24153 Properties that determine ...
isxmetd 24154 Properties that determine ...
isxmet2d 24155 It is safe to only require...
metflem 24156 Lemma for ~ metf and other...
xmetf 24157 Mapping of the distance fu...
metf 24158 Mapping of the distance fu...
xmetcl 24159 Closure of the distance fu...
metcl 24160 Closure of the distance fu...
ismet2 24161 An extended metric is a me...
metxmet 24162 A metric is an extended me...
xmetdmdm 24163 Recover the base set from ...
metdmdm 24164 Recover the base set from ...
xmetunirn 24165 Two ways to express an ext...
xmeteq0 24166 The value of an extended m...
meteq0 24167 The value of a metric is z...
xmettri2 24168 Triangle inequality for th...
mettri2 24169 Triangle inequality for th...
xmet0 24170 The distance function of a...
met0 24171 The distance function of a...
xmetge0 24172 The distance function of a...
metge0 24173 The distance function of a...
xmetlecl 24174 Real closure of an extende...
xmetsym 24175 The distance function of a...
xmetpsmet 24176 An extended metric is a ps...
xmettpos 24177 The distance function of a...
metsym 24178 The distance function of a...
xmettri 24179 Triangle inequality for th...
mettri 24180 Triangle inequality for th...
xmettri3 24181 Triangle inequality for th...
mettri3 24182 Triangle inequality for th...
xmetrtri 24183 One half of the reverse tr...
xmetrtri2 24184 The reverse triangle inequ...
metrtri 24185 Reverse triangle inequalit...
xmetgt0 24186 The distance function of a...
metgt0 24187 The distance function of a...
metn0 24188 A metric space is nonempty...
xmetres2 24189 Restriction of an extended...
metreslem 24190 Lemma for ~ metres . (Con...
metres2 24191 Lemma for ~ metres . (Con...
xmetres 24192 A restriction of an extend...
metres 24193 A restriction of a metric ...
0met 24194 The empty metric. (Contri...
prdsdsf 24195 The product metric is a fu...
prdsxmetlem 24196 The product metric is an e...
prdsxmet 24197 The product metric is an e...
prdsmet 24198 The product metric is a me...
ressprdsds 24199 Restriction of a product m...
resspwsds 24200 Restriction of a power met...
imasdsf1olem 24201 Lemma for ~ imasdsf1o . (...
imasdsf1o 24202 The distance function is t...
imasf1oxmet 24203 The image of an extended m...
imasf1omet 24204 The image of a metric is a...
xpsdsfn 24205 Closure of the metric in a...
xpsdsfn2 24206 Closure of the metric in a...
xpsxmetlem 24207 Lemma for ~ xpsxmet . (Co...
xpsxmet 24208 A product metric of extend...
xpsdsval 24209 Value of the metric in a b...
xpsmet 24210 The direct product of two ...
blfvalps 24211 The value of the ball func...
blfval 24212 The value of the ball func...
blvalps 24213 The ball around a point ` ...
blval 24214 The ball around a point ` ...
elblps 24215 Membership in a ball. (Co...
elbl 24216 Membership in a ball. (Co...
elbl2ps 24217 Membership in a ball. (Co...
elbl2 24218 Membership in a ball. (Co...
elbl3ps 24219 Membership in a ball, with...
elbl3 24220 Membership in a ball, with...
blcomps 24221 Commute the arguments to t...
blcom 24222 Commute the arguments to t...
xblpnfps 24223 The infinity ball in an ex...
xblpnf 24224 The infinity ball in an ex...
blpnf 24225 The infinity ball in a sta...
bldisj 24226 Two balls are disjoint if ...
blgt0 24227 A nonempty ball implies th...
bl2in 24228 Two balls are disjoint if ...
xblss2ps 24229 One ball is contained in a...
xblss2 24230 One ball is contained in a...
blss2ps 24231 One ball is contained in a...
blss2 24232 One ball is contained in a...
blhalf 24233 A ball of radius ` R / 2 `...
blfps 24234 Mapping of a ball. (Contr...
blf 24235 Mapping of a ball. (Contr...
blrnps 24236 Membership in the range of...
blrn 24237 Membership in the range of...
xblcntrps 24238 A ball contains its center...
xblcntr 24239 A ball contains its center...
blcntrps 24240 A ball contains its center...
blcntr 24241 A ball contains its center...
xbln0 24242 A ball is nonempty iff the...
bln0 24243 A ball is not empty. (Con...
blelrnps 24244 A ball belongs to the set ...
blelrn 24245 A ball belongs to the set ...
blssm 24246 A ball is a subset of the ...
unirnblps 24247 The union of the set of ba...
unirnbl 24248 The union of the set of ba...
blin 24249 The intersection of two ba...
ssblps 24250 The size of a ball increas...
ssbl 24251 The size of a ball increas...
blssps 24252 Any point ` P ` in a ball ...
blss 24253 Any point ` P ` in a ball ...
blssexps 24254 Two ways to express the ex...
blssex 24255 Two ways to express the ex...
ssblex 24256 A nested ball exists whose...
blin2 24257 Given any two balls and a ...
blbas 24258 The balls of a metric spac...
blres 24259 A ball in a restricted met...
xmeterval 24260 Value of the "finitely sep...
xmeter 24261 The "finitely separated" r...
xmetec 24262 The equivalence classes un...
blssec 24263 A ball centered at ` P ` i...
blpnfctr 24264 The infinity ball in an ex...
xmetresbl 24265 An extended metric restric...
mopnval 24266 An open set is a subset of...
mopntopon 24267 The set of open sets of a ...
mopntop 24268 The set of open sets of a ...
mopnuni 24269 The union of all open sets...
elmopn 24270 The defining property of a...
mopnfss 24271 The family of open sets of...
mopnm 24272 The base set of a metric s...
elmopn2 24273 A defining property of an ...
mopnss 24274 An open set of a metric sp...
isxms 24275 Express the predicate " ` ...
isxms2 24276 Express the predicate " ` ...
isms 24277 Express the predicate " ` ...
isms2 24278 Express the predicate " ` ...
xmstopn 24279 The topology component of ...
mstopn 24280 The topology component of ...
xmstps 24281 An extended metric space i...
msxms 24282 A metric space is an exten...
mstps 24283 A metric space is a topolo...
xmsxmet 24284 The distance function, sui...
msmet 24285 The distance function, sui...
msf 24286 The distance function of a...
xmsxmet2 24287 The distance function, sui...
msmet2 24288 The distance function, sui...
mscl 24289 Closure of the distance fu...
xmscl 24290 Closure of the distance fu...
xmsge0 24291 The distance function in a...
xmseq0 24292 The distance between two p...
xmssym 24293 The distance function in a...
xmstri2 24294 Triangle inequality for th...
mstri2 24295 Triangle inequality for th...
xmstri 24296 Triangle inequality for th...
mstri 24297 Triangle inequality for th...
xmstri3 24298 Triangle inequality for th...
mstri3 24299 Triangle inequality for th...
msrtri 24300 Reverse triangle inequalit...
xmspropd 24301 Property deduction for an ...
mspropd 24302 Property deduction for a m...
setsmsbas 24303 The base set of a construc...
setsmsbasOLD 24304 Obsolete proof of ~ setsms...
setsmsds 24305 The distance function of a...
setsmsdsOLD 24306 Obsolete proof of ~ setsms...
setsmstset 24307 The topology of a construc...
setsmstopn 24308 The topology of a construc...
setsxms 24309 The constructed metric spa...
setsms 24310 The constructed metric spa...
tmsval 24311 For any metric there is an...
tmslem 24312 Lemma for ~ tmsbas , ~ tms...
tmslemOLD 24313 Obsolete version of ~ tmsl...
tmsbas 24314 The base set of a construc...
tmsds 24315 The metric of a constructe...
tmstopn 24316 The topology of a construc...
tmsxms 24317 The constructed metric spa...
tmsms 24318 The constructed metric spa...
imasf1obl 24319 The image of a metric spac...
imasf1oxms 24320 The image of a metric spac...
imasf1oms 24321 The image of a metric spac...
prdsbl 24322 A ball in the product metr...
mopni 24323 An open set of a metric sp...
mopni2 24324 An open set of a metric sp...
mopni3 24325 An open set of a metric sp...
blssopn 24326 The balls of a metric spac...
unimopn 24327 The union of a collection ...
mopnin 24328 The intersection of two op...
mopn0 24329 The empty set is an open s...
rnblopn 24330 A ball of a metric space i...
blopn 24331 A ball of a metric space i...
neibl 24332 The neighborhoods around a...
blnei 24333 A ball around a point is a...
lpbl 24334 Every ball around a limit ...
blsscls2 24335 A smaller closed ball is c...
blcld 24336 A "closed ball" in a metri...
blcls 24337 The closure of an open bal...
blsscls 24338 If two concentric balls ha...
metss 24339 Two ways of saying that me...
metequiv 24340 Two ways of saying that tw...
metequiv2 24341 If there is a sequence of ...
metss2lem 24342 Lemma for ~ metss2 . (Con...
metss2 24343 If the metric ` D ` is "st...
comet 24344 The composition of an exte...
stdbdmetval 24345 Value of the standard boun...
stdbdxmet 24346 The standard bounded metri...
stdbdmet 24347 The standard bounded metri...
stdbdbl 24348 The standard bounded metri...
stdbdmopn 24349 The standard bounded metri...
mopnex 24350 The topology generated by ...
methaus 24351 The topology generated by ...
met1stc 24352 The topology generated by ...
met2ndci 24353 A separable metric space (...
met2ndc 24354 A metric space is second-c...
metrest 24355 Two alternate formulations...
ressxms 24356 The restriction of a metri...
ressms 24357 The restriction of a metri...
prdsmslem1 24358 Lemma for ~ prdsms . The ...
prdsxmslem1 24359 Lemma for ~ prdsms . The ...
prdsxmslem2 24360 Lemma for ~ prdsxms . The...
prdsxms 24361 The indexed product struct...
prdsms 24362 The indexed product struct...
pwsxms 24363 A power of an extended met...
pwsms 24364 A power of a metric space ...
xpsxms 24365 A binary product of metric...
xpsms 24366 A binary product of metric...
tmsxps 24367 Express the product of two...
tmsxpsmopn 24368 Express the product of two...
tmsxpsval 24369 Value of the product of tw...
tmsxpsval2 24370 Value of the product of tw...
metcnp3 24371 Two ways to express that `...
metcnp 24372 Two ways to say a mapping ...
metcnp2 24373 Two ways to say a mapping ...
metcn 24374 Two ways to say a mapping ...
metcnpi 24375 Epsilon-delta property of ...
metcnpi2 24376 Epsilon-delta property of ...
metcnpi3 24377 Epsilon-delta property of ...
txmetcnp 24378 Continuity of a binary ope...
txmetcn 24379 Continuity of a binary ope...
metuval 24380 Value of the uniform struc...
metustel 24381 Define a filter base ` F `...
metustss 24382 Range of the elements of t...
metustrel 24383 Elements of the filter bas...
metustto 24384 Any two elements of the fi...
metustid 24385 The identity diagonal is i...
metustsym 24386 Elements of the filter bas...
metustexhalf 24387 For any element ` A ` of t...
metustfbas 24388 The filter base generated ...
metust 24389 The uniform structure gene...
cfilucfil 24390 Given a metric ` D ` and a...
metuust 24391 The uniform structure gene...
cfilucfil2 24392 Given a metric ` D ` and a...
blval2 24393 The ball around a point ` ...
elbl4 24394 Membership in a ball, alte...
metuel 24395 Elementhood in the uniform...
metuel2 24396 Elementhood in the uniform...
metustbl 24397 The "section" image of an ...
psmetutop 24398 The topology induced by a ...
xmetutop 24399 The topology induced by a ...
xmsusp 24400 If the uniform set of a me...
restmetu 24401 The uniform structure gene...
metucn 24402 Uniform continuity in metr...
dscmet 24403 The discrete metric on any...
dscopn 24404 The discrete metric genera...
nrmmetd 24405 Show that a group norm gen...
abvmet 24406 An absolute value ` F ` ge...
nmfval 24419 The value of the norm func...
nmval 24420 The value of the norm as t...
nmfval0 24421 The value of the norm func...
nmfval2 24422 The value of the norm func...
nmval2 24423 The value of the norm on a...
nmf2 24424 The norm on a metric group...
nmpropd 24425 Weak property deduction fo...
nmpropd2 24426 Strong property deduction ...
isngp 24427 The property of being a no...
isngp2 24428 The property of being a no...
isngp3 24429 The property of being a no...
ngpgrp 24430 A normed group is a group....
ngpms 24431 A normed group is a metric...
ngpxms 24432 A normed group is an exten...
ngptps 24433 A normed group is a topolo...
ngpmet 24434 The (induced) metric of a ...
ngpds 24435 Value of the distance func...
ngpdsr 24436 Value of the distance func...
ngpds2 24437 Write the distance between...
ngpds2r 24438 Write the distance between...
ngpds3 24439 Write the distance between...
ngpds3r 24440 Write the distance between...
ngprcan 24441 Cancel right addition insi...
ngplcan 24442 Cancel left addition insid...
isngp4 24443 Express the property of be...
ngpinvds 24444 Two elements are the same ...
ngpsubcan 24445 Cancel right subtraction i...
nmf 24446 The norm on a normed group...
nmcl 24447 The norm of a normed group...
nmge0 24448 The norm of a normed group...
nmeq0 24449 The identity is the only e...
nmne0 24450 The norm of a nonzero elem...
nmrpcl 24451 The norm of a nonzero elem...
nminv 24452 The norm of a negated elem...
nmmtri 24453 The triangle inequality fo...
nmsub 24454 The norm of the difference...
nmrtri 24455 Reverse triangle inequalit...
nm2dif 24456 Inequality for the differe...
nmtri 24457 The triangle inequality fo...
nmtri2 24458 Triangle inequality for th...
ngpi 24459 The properties of a normed...
nm0 24460 Norm of the identity eleme...
nmgt0 24461 The norm of a nonzero elem...
sgrim 24462 The induced metric on a su...
sgrimval 24463 The induced metric on a su...
subgnm 24464 The norm in a subgroup. (...
subgnm2 24465 A substructure assigns the...
subgngp 24466 A normed group restricted ...
ngptgp 24467 A normed abelian group is ...
ngppropd 24468 Property deduction for a n...
reldmtng 24469 The function ` toNrmGrp ` ...
tngval 24470 Value of the function whic...
tnglem 24471 Lemma for ~ tngbas and sim...
tnglemOLD 24472 Obsolete version of ~ tngl...
tngbas 24473 The base set of a structur...
tngbasOLD 24474 Obsolete proof of ~ tngbas...
tngplusg 24475 The group addition of a st...
tngplusgOLD 24476 Obsolete proof of ~ tngplu...
tng0 24477 The group identity of a st...
tngmulr 24478 The ring multiplication of...
tngmulrOLD 24479 Obsolete proof of ~ tngmul...
tngsca 24480 The scalar ring of a struc...
tngscaOLD 24481 Obsolete proof of ~ tngsca...
tngvsca 24482 The scalar multiplication ...
tngvscaOLD 24483 Obsolete proof of ~ tngvsc...
tngip 24484 The inner product operatio...
tngipOLD 24485 Obsolete proof of ~ tngip ...
tngds 24486 The metric function of a s...
tngdsOLD 24487 Obsolete proof of ~ tngds ...
tngtset 24488 The topology generated by ...
tngtopn 24489 The topology generated by ...
tngnm 24490 The topology generated by ...
tngngp2 24491 A norm turns a group into ...
tngngpd 24492 Derive the axioms for a no...
tngngp 24493 Derive the axioms for a no...
tnggrpr 24494 If a structure equipped wi...
tngngp3 24495 Alternate definition of a ...
nrmtngdist 24496 The augmentation of a norm...
nrmtngnrm 24497 The augmentation of a norm...
tngngpim 24498 The induced metric of a no...
isnrg 24499 A normed ring is a ring wi...
nrgabv 24500 The norm of a normed ring ...
nrgngp 24501 A normed ring is a normed ...
nrgring 24502 A normed ring is a ring. ...
nmmul 24503 The norm of a product in a...
nrgdsdi 24504 Distribute a distance calc...
nrgdsdir 24505 Distribute a distance calc...
nm1 24506 The norm of one in a nonze...
unitnmn0 24507 The norm of a unit is nonz...
nminvr 24508 The norm of an inverse in ...
nmdvr 24509 The norm of a division in ...
nrgdomn 24510 A nonzero normed ring is a...
nrgtgp 24511 A normed ring is a topolog...
subrgnrg 24512 A normed ring restricted t...
tngnrg 24513 Given any absolute value o...
isnlm 24514 A normed (left) module is ...
nmvs 24515 Defining property of a nor...
nlmngp 24516 A normed module is a norme...
nlmlmod 24517 A normed module is a left ...
nlmnrg 24518 The scalar component of a ...
nlmngp2 24519 The scalar component of a ...
nlmdsdi 24520 Distribute a distance calc...
nlmdsdir 24521 Distribute a distance calc...
nlmmul0or 24522 If a scalar product is zer...
sranlm 24523 The subring algebra over a...
nlmvscnlem2 24524 Lemma for ~ nlmvscn . Com...
nlmvscnlem1 24525 Lemma for ~ nlmvscn . (Co...
nlmvscn 24526 The scalar multiplication ...
rlmnlm 24527 The ring module over a nor...
rlmnm 24528 The norm function in the r...
nrgtrg 24529 A normed ring is a topolog...
nrginvrcnlem 24530 Lemma for ~ nrginvrcn . C...
nrginvrcn 24531 The ring inverse function ...
nrgtdrg 24532 A normed division ring is ...
nlmtlm 24533 A normed module is a topol...
isnvc 24534 A normed vector space is j...
nvcnlm 24535 A normed vector space is a...
nvclvec 24536 A normed vector space is a...
nvclmod 24537 A normed vector space is a...
isnvc2 24538 A normed vector space is j...
nvctvc 24539 A normed vector space is a...
lssnlm 24540 A subspace of a normed mod...
lssnvc 24541 A subspace of a normed vec...
rlmnvc 24542 The ring module over a nor...
ngpocelbl 24543 Membership of an off-cente...
nmoffn 24550 The function producing ope...
reldmnghm 24551 Lemma for normed group hom...
reldmnmhm 24552 Lemma for module homomorph...
nmofval 24553 Value of the operator norm...
nmoval 24554 Value of the operator norm...
nmogelb 24555 Property of the operator n...
nmolb 24556 Any upper bound on the val...
nmolb2d 24557 Any upper bound on the val...
nmof 24558 The operator norm is a fun...
nmocl 24559 The operator norm of an op...
nmoge0 24560 The operator norm of an op...
nghmfval 24561 A normed group homomorphis...
isnghm 24562 A normed group homomorphis...
isnghm2 24563 A normed group homomorphis...
isnghm3 24564 A normed group homomorphis...
bddnghm 24565 A bounded group homomorphi...
nghmcl 24566 A normed group homomorphis...
nmoi 24567 The operator norm achieves...
nmoix 24568 The operator norm is a bou...
nmoi2 24569 The operator norm is a bou...
nmoleub 24570 The operator norm, defined...
nghmrcl1 24571 Reverse closure for a norm...
nghmrcl2 24572 Reverse closure for a norm...
nghmghm 24573 A normed group homomorphis...
nmo0 24574 The operator norm of the z...
nmoeq0 24575 The operator norm is zero ...
nmoco 24576 An upper bound on the oper...
nghmco 24577 The composition of normed ...
nmotri 24578 Triangle inequality for th...
nghmplusg 24579 The sum of two bounded lin...
0nghm 24580 The zero operator is a nor...
nmoid 24581 The operator norm of the i...
idnghm 24582 The identity operator is a...
nmods 24583 Upper bound for the distan...
nghmcn 24584 A normed group homomorphis...
isnmhm 24585 A normed module homomorphi...
nmhmrcl1 24586 Reverse closure for a norm...
nmhmrcl2 24587 Reverse closure for a norm...
nmhmlmhm 24588 A normed module homomorphi...
nmhmnghm 24589 A normed module homomorphi...
nmhmghm 24590 A normed module homomorphi...
isnmhm2 24591 A normed module homomorphi...
nmhmcl 24592 A normed module homomorphi...
idnmhm 24593 The identity operator is a...
0nmhm 24594 The zero operator is a bou...
nmhmco 24595 The composition of bounded...
nmhmplusg 24596 The sum of two bounded lin...
qtopbaslem 24597 The set of open intervals ...
qtopbas 24598 The set of open intervals ...
retopbas 24599 A basis for the standard t...
retop 24600 The standard topology on t...
uniretop 24601 The underlying set of the ...
retopon 24602 The standard topology on t...
retps 24603 The standard topological s...
iooretop 24604 Open intervals are open se...
icccld 24605 Closed intervals are close...
icopnfcld 24606 Right-unbounded closed int...
iocmnfcld 24607 Left-unbounded closed inte...
qdensere 24608 ` QQ ` is dense in the sta...
cnmetdval 24609 Value of the distance func...
cnmet 24610 The absolute value metric ...
cnxmet 24611 The absolute value metric ...
cnbl0 24612 Two ways to write the open...
cnblcld 24613 Two ways to write the clos...
cnfldms 24614 The complex number field i...
cnfldxms 24615 The complex number field i...
cnfldtps 24616 The complex number field i...
cnfldnm 24617 The norm of the field of c...
cnngp 24618 The complex numbers form a...
cnnrg 24619 The complex numbers form a...
cnfldtopn 24620 The topology of the comple...
cnfldtopon 24621 The topology of the comple...
cnfldtop 24622 The topology of the comple...
cnfldhaus 24623 The topology of the comple...
unicntop 24624 The underlying set of the ...
cnopn 24625 The set of complex numbers...
zringnrg 24626 The ring of integers is a ...
remetdval 24627 Value of the distance func...
remet 24628 The absolute value metric ...
rexmet 24629 The absolute value metric ...
bl2ioo 24630 A ball in terms of an open...
ioo2bl 24631 An open interval of reals ...
ioo2blex 24632 An open interval of reals ...
blssioo 24633 The balls of the standard ...
tgioo 24634 The topology generated by ...
qdensere2 24635 ` QQ ` is dense in ` RR ` ...
blcvx 24636 An open ball in the comple...
rehaus 24637 The standard topology on t...
tgqioo 24638 The topology generated by ...
re2ndc 24639 The standard topology on t...
resubmet 24640 The subspace topology indu...
tgioo2 24641 The standard topology on t...
rerest 24642 The subspace topology indu...
tgioo3 24643 The standard topology on t...
xrtgioo 24644 The topology on the extend...
xrrest 24645 The subspace topology indu...
xrrest2 24646 The subspace topology indu...
xrsxmet 24647 The metric on the extended...
xrsdsre 24648 The metric on the extended...
xrsblre 24649 Any ball of the metric of ...
xrsmopn 24650 The metric on the extended...
zcld 24651 The integers are a closed ...
recld2 24652 The real numbers are a clo...
zcld2 24653 The integers are a closed ...
zdis 24654 The integers are a discret...
sszcld 24655 Every subset of the intege...
reperflem 24656 A subset of the real numbe...
reperf 24657 The real numbers are a per...
cnperf 24658 The complex numbers are a ...
iccntr 24659 The interior of a closed i...
icccmplem1 24660 Lemma for ~ icccmp . (Con...
icccmplem2 24661 Lemma for ~ icccmp . (Con...
icccmplem3 24662 Lemma for ~ icccmp . (Con...
icccmp 24663 A closed interval in ` RR ...
reconnlem1 24664 Lemma for ~ reconn . Conn...
reconnlem2 24665 Lemma for ~ reconn . (Con...
reconn 24666 A subset of the reals is c...
retopconn 24667 Corollary of ~ reconn . T...
iccconn 24668 A closed interval is conne...
opnreen 24669 Every nonempty open set is...
rectbntr0 24670 A countable subset of the ...
xrge0gsumle 24671 A finite sum in the nonneg...
xrge0tsms 24672 Any finite or infinite sum...
xrge0tsms2 24673 Any finite or infinite sum...
metdcnlem 24674 The metric function of a m...
xmetdcn2 24675 The metric function of an ...
xmetdcn 24676 The metric function of an ...
metdcn2 24677 The metric function of a m...
metdcn 24678 The metric function of a m...
msdcn 24679 The metric function of a m...
cnmpt1ds 24680 Continuity of the metric f...
cnmpt2ds 24681 Continuity of the metric f...
nmcn 24682 The norm of a normed group...
ngnmcncn 24683 The norm of a normed group...
abscn 24684 The absolute value functio...
metdsval 24685 Value of the "distance to ...
metdsf 24686 The distance from a point ...
metdsge 24687 The distance from the poin...
metds0 24688 If a point is in a set, it...
metdstri 24689 A generalization of the tr...
metdsle 24690 The distance from a point ...
metdsre 24691 The distance from a point ...
metdseq0 24692 The distance from a point ...
metdscnlem 24693 Lemma for ~ metdscn . (Co...
metdscn 24694 The function ` F ` which g...
metdscn2 24695 The function ` F ` which g...
metnrmlem1a 24696 Lemma for ~ metnrm . (Con...
metnrmlem1 24697 Lemma for ~ metnrm . (Con...
metnrmlem2 24698 Lemma for ~ metnrm . (Con...
metnrmlem3 24699 Lemma for ~ metnrm . (Con...
metnrm 24700 A metric space is normal. ...
metreg 24701 A metric space is regular....
addcnlem 24702 Lemma for ~ addcn , ~ subc...
addcn 24703 Complex number addition is...
subcn 24704 Complex number subtraction...
mulcn 24705 Complex number multiplicat...
divcnOLD 24706 Obsolete version of ~ divc...
mpomulcn 24707 Complex number multiplicat...
divcn 24708 Complex number division is...
cnfldtgp 24709 The complex numbers form a...
fsumcn 24710 A finite sum of functions ...
fsum2cn 24711 Version of ~ fsumcn for tw...
expcn 24712 The power function on comp...
divccn 24713 Division by a nonzero cons...
expcnOLD 24714 Obsolete version of ~ expc...
divccnOLD 24715 Obsolete version of ~ divc...
sqcn 24716 The square function on com...
iitopon 24721 The unit interval is a top...
iitop 24722 The unit interval is a top...
iiuni 24723 The base set of the unit i...
dfii2 24724 Alternate definition of th...
dfii3 24725 Alternate definition of th...
dfii4 24726 Alternate definition of th...
dfii5 24727 The unit interval expresse...
iicmp 24728 The unit interval is compa...
iiconn 24729 The unit interval is conne...
cncfval 24730 The value of the continuou...
elcncf 24731 Membership in the set of c...
elcncf2 24732 Version of ~ elcncf with a...
cncfrss 24733 Reverse closure of the con...
cncfrss2 24734 Reverse closure of the con...
cncff 24735 A continuous complex funct...
cncfi 24736 Defining property of a con...
elcncf1di 24737 Membership in the set of c...
elcncf1ii 24738 Membership in the set of c...
rescncf 24739 A continuous complex funct...
cncfcdm 24740 Change the codomain of a c...
cncfss 24741 The set of continuous func...
climcncf 24742 Image of a limit under a c...
abscncf 24743 Absolute value is continuo...
recncf 24744 Real part is continuous. ...
imcncf 24745 Imaginary part is continuo...
cjcncf 24746 Complex conjugate is conti...
mulc1cncf 24747 Multiplication by a consta...
divccncf 24748 Division by a constant is ...
cncfco 24749 The composition of two con...
cncfcompt2 24750 Composition of continuous ...
cncfmet 24751 Relate complex function co...
cncfcn 24752 Relate complex function co...
cncfcn1 24753 Relate complex function co...
cncfmptc 24754 A constant function is a c...
cncfmptid 24755 The identity function is a...
cncfmpt1f 24756 Composition of continuous ...
cncfmpt2f 24757 Composition of continuous ...
cncfmpt2ss 24758 Composition of continuous ...
addccncf 24759 Adding a constant is a con...
idcncf 24760 The identity function is a...
sub1cncf 24761 Subtracting a constant is ...
sub2cncf 24762 Subtraction from a constan...
cdivcncf 24763 Division with a constant n...
negcncf 24764 The negative function is c...
negcncfOLD 24765 Obsolete version of ~ negc...
negfcncf 24766 The negative of a continuo...
abscncfALT 24767 Absolute value is continuo...
cncfcnvcn 24768 Rewrite ~ cmphaushmeo for ...
expcncf 24769 The power function on comp...
cnmptre 24770 Lemma for ~ iirevcn and re...
cnmpopc 24771 Piecewise definition of a ...
iirev 24772 Reverse the unit interval....
iirevcn 24773 The reversion function is ...
iihalf1 24774 Map the first half of ` II...
iihalf1cn 24775 The first half function is...
iihalf1cnOLD 24776 Obsolete version of ~ iiha...
iihalf2 24777 Map the second half of ` I...
iihalf2cn 24778 The second half function i...
iihalf2cnOLD 24779 Obsolete version of ~ iiha...
elii1 24780 Divide the unit interval i...
elii2 24781 Divide the unit interval i...
iimulcl 24782 The unit interval is close...
iimulcn 24783 Multiplication is a contin...
iimulcnOLD 24784 Obsolete version of ~ iimu...
icoopnst 24785 A half-open interval start...
iocopnst 24786 A half-open interval endin...
icchmeo 24787 The natural bijection from...
icchmeoOLD 24788 Obsolete version of ~ icch...
icopnfcnv 24789 Define a bijection from ` ...
icopnfhmeo 24790 The defined bijection from...
iccpnfcnv 24791 Define a bijection from ` ...
iccpnfhmeo 24792 The defined bijection from...
xrhmeo 24793 The bijection from ` [ -u ...
xrhmph 24794 The extended reals are hom...
xrcmp 24795 The topology of the extend...
xrconn 24796 The topology of the extend...
icccvx 24797 A linear combination of tw...
oprpiece1res1 24798 Restriction to the first p...
oprpiece1res2 24799 Restriction to the second ...
cnrehmeo 24800 The canonical bijection fr...
cnrehmeoOLD 24801 Obsolete version of ~ cnre...
cnheiborlem 24802 Lemma for ~ cnheibor . (C...
cnheibor 24803 Heine-Borel theorem for co...
cnllycmp 24804 The topology on the comple...
rellycmp 24805 The topology on the reals ...
bndth 24806 The Boundedness Theorem. ...
evth 24807 The Extreme Value Theorem....
evth2 24808 The Extreme Value Theorem,...
lebnumlem1 24809 Lemma for ~ lebnum . The ...
lebnumlem2 24810 Lemma for ~ lebnum . As a...
lebnumlem3 24811 Lemma for ~ lebnum . By t...
lebnum 24812 The Lebesgue number lemma,...
xlebnum 24813 Generalize ~ lebnum to ext...
lebnumii 24814 Specialize the Lebesgue nu...
ishtpy 24820 Membership in the class of...
htpycn 24821 A homotopy is a continuous...
htpyi 24822 A homotopy evaluated at it...
ishtpyd 24823 Deduction for membership i...
htpycom 24824 Given a homotopy from ` F ...
htpyid 24825 A homotopy from a function...
htpyco1 24826 Compose a homotopy with a ...
htpyco2 24827 Compose a homotopy with a ...
htpycc 24828 Concatenate two homotopies...
isphtpy 24829 Membership in the class of...
phtpyhtpy 24830 A path homotopy is a homot...
phtpycn 24831 A path homotopy is a conti...
phtpyi 24832 Membership in the class of...
phtpy01 24833 Two path-homotopic paths h...
isphtpyd 24834 Deduction for membership i...
isphtpy2d 24835 Deduction for membership i...
phtpycom 24836 Given a homotopy from ` F ...
phtpyid 24837 A homotopy from a path to ...
phtpyco2 24838 Compose a path homotopy wi...
phtpycc 24839 Concatenate two path homot...
phtpcrel 24841 The path homotopy relation...
isphtpc 24842 The relation "is path homo...
phtpcer 24843 Path homotopy is an equiva...
phtpc01 24844 Path homotopic paths have ...
reparphti 24845 Lemma for ~ reparpht . (C...
reparphtiOLD 24846 Obsolete version of ~ repa...
reparpht 24847 Reparametrization lemma. ...
phtpcco2 24848 Compose a path homotopy wi...
pcofval 24859 The value of the path conc...
pcoval 24860 The concatenation of two p...
pcovalg 24861 Evaluate the concatenation...
pcoval1 24862 Evaluate the concatenation...
pco0 24863 The starting point of a pa...
pco1 24864 The ending point of a path...
pcoval2 24865 Evaluate the concatenation...
pcocn 24866 The concatenation of two p...
copco 24867 The composition of a conca...
pcohtpylem 24868 Lemma for ~ pcohtpy . (Co...
pcohtpy 24869 Homotopy invariance of pat...
pcoptcl 24870 A constant function is a p...
pcopt 24871 Concatenation with a point...
pcopt2 24872 Concatenation with a point...
pcoass 24873 Order of concatenation doe...
pcorevcl 24874 Closure for a reversed pat...
pcorevlem 24875 Lemma for ~ pcorev . Prov...
pcorev 24876 Concatenation with the rev...
pcorev2 24877 Concatenation with the rev...
pcophtb 24878 The path homotopy equivale...
om1val 24879 The definition of the loop...
om1bas 24880 The base set of the loop s...
om1elbas 24881 Elementhood in the base se...
om1addcl 24882 Closure of the group opera...
om1plusg 24883 The group operation (which...
om1tset 24884 The topology of the loop s...
om1opn 24885 The topology of the loop s...
pi1val 24886 The definition of the fund...
pi1bas 24887 The base set of the fundam...
pi1blem 24888 Lemma for ~ pi1buni . (Co...
pi1buni 24889 Another way to write the l...
pi1bas2 24890 The base set of the fundam...
pi1eluni 24891 Elementhood in the base se...
pi1bas3 24892 The base set of the fundam...
pi1cpbl 24893 The group operation, loop ...
elpi1 24894 The elements of the fundam...
elpi1i 24895 The elements of the fundam...
pi1addf 24896 The group operation of ` p...
pi1addval 24897 The concatenation of two p...
pi1grplem 24898 Lemma for ~ pi1grp . (Con...
pi1grp 24899 The fundamental group is a...
pi1id 24900 The identity element of th...
pi1inv 24901 An inverse in the fundamen...
pi1xfrf 24902 Functionality of the loop ...
pi1xfrval 24903 The value of the loop tran...
pi1xfr 24904 Given a path ` F ` and its...
pi1xfrcnvlem 24905 Given a path ` F ` between...
pi1xfrcnv 24906 Given a path ` F ` between...
pi1xfrgim 24907 The mapping ` G ` between ...
pi1cof 24908 Functionality of the loop ...
pi1coval 24909 The value of the loop tran...
pi1coghm 24910 The mapping ` G ` between ...
isclm 24913 A subcomplex module is a l...
clmsca 24914 The ring of scalars ` F ` ...
clmsubrg 24915 The base set of the ring o...
clmlmod 24916 A subcomplex module is a l...
clmgrp 24917 A subcomplex module is an ...
clmabl 24918 A subcomplex module is an ...
clmring 24919 The scalar ring of a subco...
clmfgrp 24920 The scalar ring of a subco...
clm0 24921 The zero of the scalar rin...
clm1 24922 The identity of the scalar...
clmadd 24923 The addition of the scalar...
clmmul 24924 The multiplication of the ...
clmcj 24925 The conjugation of the sca...
isclmi 24926 Reverse direction of ~ isc...
clmzss 24927 The scalar ring of a subco...
clmsscn 24928 The scalar ring of a subco...
clmsub 24929 Subtraction in the scalar ...
clmneg 24930 Negation in the scalar rin...
clmneg1 24931 Minus one is in the scalar...
clmabs 24932 Norm in the scalar ring of...
clmacl 24933 Closure of ring addition f...
clmmcl 24934 Closure of ring multiplica...
clmsubcl 24935 Closure of ring subtractio...
lmhmclm 24936 The domain of a linear ope...
clmvscl 24937 Closure of scalar product ...
clmvsass 24938 Associative law for scalar...
clmvscom 24939 Commutative law for the sc...
clmvsdir 24940 Distributive law for scala...
clmvsdi 24941 Distributive law for scala...
clmvs1 24942 Scalar product with ring u...
clmvs2 24943 A vector plus itself is tw...
clm0vs 24944 Zero times a vector is the...
clmopfne 24945 The (functionalized) opera...
isclmp 24946 The predicate "is a subcom...
isclmi0 24947 Properties that determine ...
clmvneg1 24948 Minus 1 times a vector is ...
clmvsneg 24949 Multiplication of a vector...
clmmulg 24950 The group multiple functio...
clmsubdir 24951 Scalar multiplication dist...
clmpm1dir 24952 Subtractive distributive l...
clmnegneg 24953 Double negative of a vecto...
clmnegsubdi2 24954 Distribution of negative o...
clmsub4 24955 Rearrangement of 4 terms i...
clmvsrinv 24956 A vector minus itself. (C...
clmvslinv 24957 Minus a vector plus itself...
clmvsubval 24958 Value of vector subtractio...
clmvsubval2 24959 Value of vector subtractio...
clmvz 24960 Two ways to express the ne...
zlmclm 24961 The ` ZZ ` -module operati...
clmzlmvsca 24962 The scalar product of a su...
nmoleub2lem 24963 Lemma for ~ nmoleub2a and ...
nmoleub2lem3 24964 Lemma for ~ nmoleub2a and ...
nmoleub2lem2 24965 Lemma for ~ nmoleub2a and ...
nmoleub2a 24966 The operator norm is the s...
nmoleub2b 24967 The operator norm is the s...
nmoleub3 24968 The operator norm is the s...
nmhmcn 24969 A linear operator over a n...
cmodscexp 24970 The powers of ` _i ` belon...
cmodscmulexp 24971 The scalar product of a ve...
cvslvec 24974 A subcomplex vector space ...
cvsclm 24975 A subcomplex vector space ...
iscvs 24976 A subcomplex vector space ...
iscvsp 24977 The predicate "is a subcom...
iscvsi 24978 Properties that determine ...
cvsi 24979 The properties of a subcom...
cvsunit 24980 Unit group of the scalar r...
cvsdiv 24981 Division of the scalar rin...
cvsdivcl 24982 The scalar field of a subc...
cvsmuleqdivd 24983 An equality involving rati...
cvsdiveqd 24984 An equality involving rati...
cnlmodlem1 24985 Lemma 1 for ~ cnlmod . (C...
cnlmodlem2 24986 Lemma 2 for ~ cnlmod . (C...
cnlmodlem3 24987 Lemma 3 for ~ cnlmod . (C...
cnlmod4 24988 Lemma 4 for ~ cnlmod . (C...
cnlmod 24989 The set of complex numbers...
cnstrcvs 24990 The set of complex numbers...
cnrbas 24991 The set of complex numbers...
cnrlmod 24992 The complex left module of...
cnrlvec 24993 The complex left module of...
cncvs 24994 The complex left module of...
recvs 24995 The field of the real numb...
recvsOLD 24996 Obsolete version of ~ recv...
qcvs 24997 The field of rational numb...
zclmncvs 24998 The ring of integers as le...
isncvsngp 24999 A normed subcomplex vector...
isncvsngpd 25000 Properties that determine ...
ncvsi 25001 The properties of a normed...
ncvsprp 25002 Proportionality property o...
ncvsge0 25003 The norm of a scalar produ...
ncvsm1 25004 The norm of the opposite o...
ncvsdif 25005 The norm of the difference...
ncvspi 25006 The norm of a vector plus ...
ncvs1 25007 From any nonzero vector of...
cnrnvc 25008 The module of complex numb...
cnncvs 25009 The module of complex numb...
cnnm 25010 The norm of the normed sub...
ncvspds 25011 Value of the distance func...
cnindmet 25012 The metric induced on the ...
cnncvsaddassdemo 25013 Derive the associative law...
cnncvsmulassdemo 25014 Derive the associative law...
cnncvsabsnegdemo 25015 Derive the absolute value ...
iscph 25020 A subcomplex pre-Hilbert s...
cphphl 25021 A subcomplex pre-Hilbert s...
cphnlm 25022 A subcomplex pre-Hilbert s...
cphngp 25023 A subcomplex pre-Hilbert s...
cphlmod 25024 A subcomplex pre-Hilbert s...
cphlvec 25025 A subcomplex pre-Hilbert s...
cphnvc 25026 A subcomplex pre-Hilbert s...
cphsubrglem 25027 Lemma for ~ cphsubrg . (C...
cphreccllem 25028 Lemma for ~ cphreccl . (C...
cphsca 25029 A subcomplex pre-Hilbert s...
cphsubrg 25030 The scalar field of a subc...
cphreccl 25031 The scalar field of a subc...
cphdivcl 25032 The scalar field of a subc...
cphcjcl 25033 The scalar field of a subc...
cphsqrtcl 25034 The scalar field of a subc...
cphabscl 25035 The scalar field of a subc...
cphsqrtcl2 25036 The scalar field of a subc...
cphsqrtcl3 25037 If the scalar field of a s...
cphqss 25038 The scalar field of a subc...
cphclm 25039 A subcomplex pre-Hilbert s...
cphnmvs 25040 Norm of a scalar product. ...
cphipcl 25041 An inner product is a memb...
cphnmfval 25042 The value of the norm in a...
cphnm 25043 The square of the norm is ...
nmsq 25044 The square of the norm is ...
cphnmf 25045 The norm of a vector is a ...
cphnmcl 25046 The norm of a vector is a ...
reipcl 25047 An inner product of an ele...
ipge0 25048 The inner product in a sub...
cphipcj 25049 Conjugate of an inner prod...
cphipipcj 25050 An inner product times its...
cphorthcom 25051 Orthogonality (meaning inn...
cphip0l 25052 Inner product with a zero ...
cphip0r 25053 Inner product with a zero ...
cphipeq0 25054 The inner product of a vec...
cphdir 25055 Distributive law for inner...
cphdi 25056 Distributive law for inner...
cph2di 25057 Distributive law for inner...
cphsubdir 25058 Distributive law for inner...
cphsubdi 25059 Distributive law for inner...
cph2subdi 25060 Distributive law for inner...
cphass 25061 Associative law for inner ...
cphassr 25062 "Associative" law for seco...
cph2ass 25063 Move scalar multiplication...
cphassi 25064 Associative law for the fi...
cphassir 25065 "Associative" law for the ...
cphpyth 25066 The pythagorean theorem fo...
tcphex 25067 Lemma for ~ tcphbas and si...
tcphval 25068 Define a function to augme...
tcphbas 25069 The base set of a subcompl...
tchplusg 25070 The addition operation of ...
tcphsub 25071 The subtraction operation ...
tcphmulr 25072 The ring operation of a su...
tcphsca 25073 The scalar field of a subc...
tcphvsca 25074 The scalar multiplication ...
tcphip 25075 The inner product of a sub...
tcphtopn 25076 The topology of a subcompl...
tcphphl 25077 Augmentation of a subcompl...
tchnmfval 25078 The norm of a subcomplex p...
tcphnmval 25079 The norm of a subcomplex p...
cphtcphnm 25080 The norm of a norm-augment...
tcphds 25081 The distance of a pre-Hilb...
phclm 25082 A pre-Hilbert space whose ...
tcphcphlem3 25083 Lemma for ~ tcphcph : real...
ipcau2 25084 The Cauchy-Schwarz inequal...
tcphcphlem1 25085 Lemma for ~ tcphcph : the ...
tcphcphlem2 25086 Lemma for ~ tcphcph : homo...
tcphcph 25087 The standard definition of...
ipcau 25088 The Cauchy-Schwarz inequal...
nmparlem 25089 Lemma for ~ nmpar . (Cont...
nmpar 25090 A subcomplex pre-Hilbert s...
cphipval2 25091 Value of the inner product...
4cphipval2 25092 Four times the inner produ...
cphipval 25093 Value of the inner product...
ipcnlem2 25094 The inner product operatio...
ipcnlem1 25095 The inner product operatio...
ipcn 25096 The inner product operatio...
cnmpt1ip 25097 Continuity of inner produc...
cnmpt2ip 25098 Continuity of inner produc...
csscld 25099 A "closed subspace" in a s...
clsocv 25100 The orthogonal complement ...
cphsscph 25101 A subspace of a subcomplex...
lmmbr 25108 Express the binary relatio...
lmmbr2 25109 Express the binary relatio...
lmmbr3 25110 Express the binary relatio...
lmmcvg 25111 Convergence property of a ...
lmmbrf 25112 Express the binary relatio...
lmnn 25113 A condition that implies c...
cfilfval 25114 The set of Cauchy filters ...
iscfil 25115 The property of being a Ca...
iscfil2 25116 The property of being a Ca...
cfilfil 25117 A Cauchy filter is a filte...
cfili 25118 Property of a Cauchy filte...
cfil3i 25119 A Cauchy filter contains b...
cfilss 25120 A filter finer than a Cauc...
fgcfil 25121 The Cauchy filter conditio...
fmcfil 25122 The Cauchy filter conditio...
iscfil3 25123 A filter is Cauchy iff it ...
cfilfcls 25124 Similar to ultrafilters ( ...
caufval 25125 The set of Cauchy sequence...
iscau 25126 Express the property " ` F...
iscau2 25127 Express the property " ` F...
iscau3 25128 Express the Cauchy sequenc...
iscau4 25129 Express the property " ` F...
iscauf 25130 Express the property " ` F...
caun0 25131 A metric with a Cauchy seq...
caufpm 25132 Inclusion of a Cauchy sequ...
caucfil 25133 A Cauchy sequence predicat...
iscmet 25134 The property " ` D ` is a ...
cmetcvg 25135 The convergence of a Cauch...
cmetmet 25136 A complete metric space is...
cmetmeti 25137 A complete metric space is...
cmetcaulem 25138 Lemma for ~ cmetcau . (Co...
cmetcau 25139 The convergence of a Cauch...
iscmet3lem3 25140 Lemma for ~ iscmet3 . (Co...
iscmet3lem1 25141 Lemma for ~ iscmet3 . (Co...
iscmet3lem2 25142 Lemma for ~ iscmet3 . (Co...
iscmet3 25143 The property " ` D ` is a ...
iscmet2 25144 A metric ` D ` is complete...
cfilresi 25145 A Cauchy filter on a metri...
cfilres 25146 Cauchy filter on a metric ...
caussi 25147 Cauchy sequence on a metri...
causs 25148 Cauchy sequence on a metri...
equivcfil 25149 If the metric ` D ` is "st...
equivcau 25150 If the metric ` D ` is "st...
lmle 25151 If the distance from each ...
nglmle 25152 If the norm of each member...
lmclim 25153 Relate a limit on the metr...
lmclimf 25154 Relate a limit on the metr...
metelcls 25155 A point belongs to the clo...
metcld 25156 A subset of a metric space...
metcld2 25157 A subset of a metric space...
caubl 25158 Sufficient condition to en...
caublcls 25159 The convergent point of a ...
metcnp4 25160 Two ways to say a mapping ...
metcn4 25161 Two ways to say a mapping ...
iscmet3i 25162 Properties that determine ...
lmcau 25163 Every convergent sequence ...
flimcfil 25164 Every convergent filter in...
metsscmetcld 25165 A complete subspace of a m...
cmetss 25166 A subspace of a complete m...
equivcmet 25167 If two metrics are strongl...
relcmpcmet 25168 If ` D ` is a metric space...
cmpcmet 25169 A compact metric space is ...
cfilucfil3 25170 Given a metric ` D ` and a...
cfilucfil4 25171 Given a metric ` D ` and a...
cncmet 25172 The set of complex numbers...
recmet 25173 The real numbers are a com...
bcthlem1 25174 Lemma for ~ bcth . Substi...
bcthlem2 25175 Lemma for ~ bcth . The ba...
bcthlem3 25176 Lemma for ~ bcth . The li...
bcthlem4 25177 Lemma for ~ bcth . Given ...
bcthlem5 25178 Lemma for ~ bcth . The pr...
bcth 25179 Baire's Category Theorem. ...
bcth2 25180 Baire's Category Theorem, ...
bcth3 25181 Baire's Category Theorem, ...
isbn 25188 A Banach space is a normed...
bnsca 25189 The scalar field of a Bana...
bnnvc 25190 A Banach space is a normed...
bnnlm 25191 A Banach space is a normed...
bnngp 25192 A Banach space is a normed...
bnlmod 25193 A Banach space is a left m...
bncms 25194 A Banach space is a comple...
iscms 25195 A complete metric space is...
cmscmet 25196 The induced metric on a co...
bncmet 25197 The induced metric on Bana...
cmsms 25198 A complete metric space is...
cmspropd 25199 Property deduction for a c...
cmssmscld 25200 The restriction of a metri...
cmsss 25201 The restriction of a compl...
lssbn 25202 A subspace of a Banach spa...
cmetcusp1 25203 If the uniform set of a co...
cmetcusp 25204 The uniform space generate...
cncms 25205 The field of complex numbe...
cnflduss 25206 The uniform structure of t...
cnfldcusp 25207 The field of complex numbe...
resscdrg 25208 The real numbers are a sub...
cncdrg 25209 The only complete subfield...
srabn 25210 The subring algebra over a...
rlmbn 25211 The ring module over a com...
ishl 25212 The predicate "is a subcom...
hlbn 25213 Every subcomplex Hilbert s...
hlcph 25214 Every subcomplex Hilbert s...
hlphl 25215 Every subcomplex Hilbert s...
hlcms 25216 Every subcomplex Hilbert s...
hlprlem 25217 Lemma for ~ hlpr . (Contr...
hlress 25218 The scalar field of a subc...
hlpr 25219 The scalar field of a subc...
ishl2 25220 A Hilbert space is a compl...
cphssphl 25221 A Banach subspace of a sub...
cmslssbn 25222 A complete linear subspace...
cmscsscms 25223 A closed subspace of a com...
bncssbn 25224 A closed subspace of a Ban...
cssbn 25225 A complete subspace of a n...
csschl 25226 A complete subspace of a c...
cmslsschl 25227 A complete linear subspace...
chlcsschl 25228 A closed subspace of a sub...
retopn 25229 The topology of the real n...
recms 25230 The real numbers form a co...
reust 25231 The Uniform structure of t...
recusp 25232 The real numbers form a co...
rrxval 25237 Value of the generalized E...
rrxbase 25238 The base of the generalize...
rrxprds 25239 Expand the definition of t...
rrxip 25240 The inner product of the g...
rrxnm 25241 The norm of the generalize...
rrxcph 25242 Generalized Euclidean real...
rrxds 25243 The distance over generali...
rrxvsca 25244 The scalar product over ge...
rrxplusgvscavalb 25245 The result of the addition...
rrxsca 25246 The field of real numbers ...
rrx0 25247 The zero ("origin") in a g...
rrx0el 25248 The zero ("origin") in a g...
csbren 25249 Cauchy-Schwarz-Bunjakovsky...
trirn 25250 Triangle inequality in R^n...
rrxf 25251 Euclidean vectors as funct...
rrxfsupp 25252 Euclidean vectors are of f...
rrxsuppss 25253 Support of Euclidean vecto...
rrxmvallem 25254 Support of the function us...
rrxmval 25255 The value of the Euclidean...
rrxmfval 25256 The value of the Euclidean...
rrxmetlem 25257 Lemma for ~ rrxmet . (Con...
rrxmet 25258 Euclidean space is a metri...
rrxdstprj1 25259 The distance between two p...
rrxbasefi 25260 The base of the generalize...
rrxdsfi 25261 The distance over generali...
rrxmetfi 25262 Euclidean space is a metri...
rrxdsfival 25263 The value of the Euclidean...
ehlval 25264 Value of the Euclidean spa...
ehlbase 25265 The base of the Euclidean ...
ehl0base 25266 The base of the Euclidean ...
ehl0 25267 The Euclidean space of dim...
ehleudis 25268 The Euclidean distance fun...
ehleudisval 25269 The value of the Euclidean...
ehl1eudis 25270 The Euclidean distance fun...
ehl1eudisval 25271 The value of the Euclidean...
ehl2eudis 25272 The Euclidean distance fun...
ehl2eudisval 25273 The value of the Euclidean...
minveclem1 25274 Lemma for ~ minvec . The ...
minveclem4c 25275 Lemma for ~ minvec . The ...
minveclem2 25276 Lemma for ~ minvec . Any ...
minveclem3a 25277 Lemma for ~ minvec . ` D `...
minveclem3b 25278 Lemma for ~ minvec . The ...
minveclem3 25279 Lemma for ~ minvec . The ...
minveclem4a 25280 Lemma for ~ minvec . ` F `...
minveclem4b 25281 Lemma for ~ minvec . The ...
minveclem4 25282 Lemma for ~ minvec . The ...
minveclem5 25283 Lemma for ~ minvec . Disc...
minveclem6 25284 Lemma for ~ minvec . Any ...
minveclem7 25285 Lemma for ~ minvec . Sinc...
minvec 25286 Minimizing vector theorem,...
pjthlem1 25287 Lemma for ~ pjth . (Contr...
pjthlem2 25288 Lemma for ~ pjth . (Contr...
pjth 25289 Projection Theorem: Any H...
pjth2 25290 Projection Theorem with ab...
cldcss 25291 Corollary of the Projectio...
cldcss2 25292 Corollary of the Projectio...
hlhil 25293 Corollary of the Projectio...
addcncf 25294 The addition of two contin...
subcncf 25295 The addition of two contin...
mulcncf 25296 The multiplication of two ...
mulcncfOLD 25297 Obsolete version of ~ mulc...
divcncf 25298 The quotient of two contin...
pmltpclem1 25299 Lemma for ~ pmltpc . (Con...
pmltpclem2 25300 Lemma for ~ pmltpc . (Con...
pmltpc 25301 Any function on the reals ...
ivthlem1 25302 Lemma for ~ ivth . The se...
ivthlem2 25303 Lemma for ~ ivth . Show t...
ivthlem3 25304 Lemma for ~ ivth , the int...
ivth 25305 The intermediate value the...
ivth2 25306 The intermediate value the...
ivthle 25307 The intermediate value the...
ivthle2 25308 The intermediate value the...
ivthicc 25309 The interval between any t...
evthicc 25310 Specialization of the Extr...
evthicc2 25311 Combine ~ ivthicc with ~ e...
cniccbdd 25312 A continuous function on a...
ovolfcl 25317 Closure for the interval e...
ovolfioo 25318 Unpack the interval coveri...
ovolficc 25319 Unpack the interval coveri...
ovolficcss 25320 Any (closed) interval cove...
ovolfsval 25321 The value of the interval ...
ovolfsf 25322 Closure for the interval l...
ovolsf 25323 Closure for the partial su...
ovolval 25324 The value of the outer mea...
elovolmlem 25325 Lemma for ~ elovolm and re...
elovolm 25326 Elementhood in the set ` M...
elovolmr 25327 Sufficient condition for e...
ovolmge0 25328 The set ` M ` is composed ...
ovolcl 25329 The volume of a set is an ...
ovollb 25330 The outer volume is a lowe...
ovolgelb 25331 The outer volume is the gr...
ovolge0 25332 The volume of a set is alw...
ovolf 25333 The domain and codomain of...
ovollecl 25334 If an outer volume is boun...
ovolsslem 25335 Lemma for ~ ovolss . (Con...
ovolss 25336 The volume of a set is mon...
ovolsscl 25337 If a set is contained in a...
ovolssnul 25338 A subset of a nullset is n...
ovollb2lem 25339 Lemma for ~ ovollb2 . (Co...
ovollb2 25340 It is often more convenien...
ovolctb 25341 The volume of a denumerabl...
ovolq 25342 The rational numbers have ...
ovolctb2 25343 The volume of a countable ...
ovol0 25344 The empty set has 0 outer ...
ovolfi 25345 A finite set has 0 outer L...
ovolsn 25346 A singleton has 0 outer Le...
ovolunlem1a 25347 Lemma for ~ ovolun . (Con...
ovolunlem1 25348 Lemma for ~ ovolun . (Con...
ovolunlem2 25349 Lemma for ~ ovolun . (Con...
ovolun 25350 The Lebesgue outer measure...
ovolunnul 25351 Adding a nullset does not ...
ovolfiniun 25352 The Lebesgue outer measure...
ovoliunlem1 25353 Lemma for ~ ovoliun . (Co...
ovoliunlem2 25354 Lemma for ~ ovoliun . (Co...
ovoliunlem3 25355 Lemma for ~ ovoliun . (Co...
ovoliun 25356 The Lebesgue outer measure...
ovoliun2 25357 The Lebesgue outer measure...
ovoliunnul 25358 A countable union of nulls...
shft2rab 25359 If ` B ` is a shift of ` A...
ovolshftlem1 25360 Lemma for ~ ovolshft . (C...
ovolshftlem2 25361 Lemma for ~ ovolshft . (C...
ovolshft 25362 The Lebesgue outer measure...
sca2rab 25363 If ` B ` is a scale of ` A...
ovolscalem1 25364 Lemma for ~ ovolsca . (Co...
ovolscalem2 25365 Lemma for ~ ovolshft . (C...
ovolsca 25366 The Lebesgue outer measure...
ovolicc1 25367 The measure of a closed in...
ovolicc2lem1 25368 Lemma for ~ ovolicc2 . (C...
ovolicc2lem2 25369 Lemma for ~ ovolicc2 . (C...
ovolicc2lem3 25370 Lemma for ~ ovolicc2 . (C...
ovolicc2lem4 25371 Lemma for ~ ovolicc2 . (C...
ovolicc2lem5 25372 Lemma for ~ ovolicc2 . (C...
ovolicc2 25373 The measure of a closed in...
ovolicc 25374 The measure of a closed in...
ovolicopnf 25375 The measure of a right-unb...
ovolre 25376 The measure of the real nu...
ismbl 25377 The predicate " ` A ` is L...
ismbl2 25378 From ~ ovolun , it suffice...
volres 25379 A self-referencing abbrevi...
volf 25380 The domain and codomain of...
mblvol 25381 The volume of a measurable...
mblss 25382 A measurable set is a subs...
mblsplit 25383 The defining property of m...
volss 25384 The Lebesgue measure is mo...
cmmbl 25385 The complement of a measur...
nulmbl 25386 A nullset is measurable. ...
nulmbl2 25387 A set of outer measure zer...
unmbl 25388 A union of measurable sets...
shftmbl 25389 A shift of a measurable se...
0mbl 25390 The empty set is measurabl...
rembl 25391 The set of all real number...
unidmvol 25392 The union of the Lebesgue ...
inmbl 25393 An intersection of measura...
difmbl 25394 A difference of measurable...
finiunmbl 25395 A finite union of measurab...
volun 25396 The Lebesgue measure funct...
volinun 25397 Addition of non-disjoint s...
volfiniun 25398 The volume of a disjoint f...
iundisj 25399 Rewrite a countable union ...
iundisj2 25400 A disjoint union is disjoi...
voliunlem1 25401 Lemma for ~ voliun . (Con...
voliunlem2 25402 Lemma for ~ voliun . (Con...
voliunlem3 25403 Lemma for ~ voliun . (Con...
iunmbl 25404 The measurable sets are cl...
voliun 25405 The Lebesgue measure funct...
volsuplem 25406 Lemma for ~ volsup . (Con...
volsup 25407 The volume of the limit of...
iunmbl2 25408 The measurable sets are cl...
ioombl1lem1 25409 Lemma for ~ ioombl1 . (Co...
ioombl1lem2 25410 Lemma for ~ ioombl1 . (Co...
ioombl1lem3 25411 Lemma for ~ ioombl1 . (Co...
ioombl1lem4 25412 Lemma for ~ ioombl1 . (Co...
ioombl1 25413 An open right-unbounded in...
icombl1 25414 A closed unbounded-above i...
icombl 25415 A closed-below, open-above...
ioombl 25416 An open real interval is m...
iccmbl 25417 A closed real interval is ...
iccvolcl 25418 A closed real interval has...
ovolioo 25419 The measure of an open int...
volioo 25420 The measure of an open int...
ioovolcl 25421 An open real interval has ...
ovolfs2 25422 Alternative expression for...
ioorcl2 25423 An open interval with fini...
ioorf 25424 Define a function from ope...
ioorval 25425 Define a function from ope...
ioorinv2 25426 The function ` F ` is an "...
ioorinv 25427 The function ` F ` is an "...
ioorcl 25428 The function ` F ` does no...
uniiccdif 25429 A union of closed interval...
uniioovol 25430 A disjoint union of open i...
uniiccvol 25431 An almost-disjoint union o...
uniioombllem1 25432 Lemma for ~ uniioombl . (...
uniioombllem2a 25433 Lemma for ~ uniioombl . (...
uniioombllem2 25434 Lemma for ~ uniioombl . (...
uniioombllem3a 25435 Lemma for ~ uniioombl . (...
uniioombllem3 25436 Lemma for ~ uniioombl . (...
uniioombllem4 25437 Lemma for ~ uniioombl . (...
uniioombllem5 25438 Lemma for ~ uniioombl . (...
uniioombllem6 25439 Lemma for ~ uniioombl . (...
uniioombl 25440 A disjoint union of open i...
uniiccmbl 25441 An almost-disjoint union o...
dyadf 25442 The function ` F ` returns...
dyadval 25443 Value of the dyadic ration...
dyadovol 25444 Volume of a dyadic rationa...
dyadss 25445 Two closed dyadic rational...
dyaddisjlem 25446 Lemma for ~ dyaddisj . (C...
dyaddisj 25447 Two closed dyadic rational...
dyadmaxlem 25448 Lemma for ~ dyadmax . (Co...
dyadmax 25449 Any nonempty set of dyadic...
dyadmbllem 25450 Lemma for ~ dyadmbl . (Co...
dyadmbl 25451 Any union of dyadic ration...
opnmbllem 25452 Lemma for ~ opnmbl . (Con...
opnmbl 25453 All open sets are measurab...
opnmblALT 25454 All open sets are measurab...
subopnmbl 25455 Sets which are open in a m...
volsup2 25456 The volume of ` A ` is the...
volcn 25457 The function formed by res...
volivth 25458 The Intermediate Value The...
vitalilem1 25459 Lemma for ~ vitali . (Con...
vitalilem2 25460 Lemma for ~ vitali . (Con...
vitalilem3 25461 Lemma for ~ vitali . (Con...
vitalilem4 25462 Lemma for ~ vitali . (Con...
vitalilem5 25463 Lemma for ~ vitali . (Con...
vitali 25464 If the reals can be well-o...
ismbf1 25475 The predicate " ` F ` is a...
mbff 25476 A measurable function is a...
mbfdm 25477 The domain of a measurable...
mbfconstlem 25478 Lemma for ~ mbfconst and r...
ismbf 25479 The predicate " ` F ` is a...
ismbfcn 25480 A complex function is meas...
mbfima 25481 Definitional property of a...
mbfimaicc 25482 The preimage of any closed...
mbfimasn 25483 The preimage of a point un...
mbfconst 25484 A constant function is mea...
mbf0 25485 The empty function is meas...
mbfid 25486 The identity function is m...
mbfmptcl 25487 Lemma for the ` MblFn ` pr...
mbfdm2 25488 The domain of a measurable...
ismbfcn2 25489 A complex function is meas...
ismbfd 25490 Deduction to prove measura...
ismbf2d 25491 Deduction to prove measura...
mbfeqalem1 25492 Lemma for ~ mbfeqalem2 . ...
mbfeqalem2 25493 Lemma for ~ mbfeqa . (Con...
mbfeqa 25494 If two functions are equal...
mbfres 25495 The restriction of a measu...
mbfres2 25496 Measurability of a piecewi...
mbfss 25497 Change the domain of a mea...
mbfmulc2lem 25498 Multiplication by a consta...
mbfmulc2re 25499 Multiplication by a consta...
mbfmax 25500 The maximum of two functio...
mbfneg 25501 The negative of a measurab...
mbfpos 25502 The positive part of a mea...
mbfposr 25503 Converse to ~ mbfpos . (C...
mbfposb 25504 A function is measurable i...
ismbf3d 25505 Simplified form of ~ ismbf...
mbfimaopnlem 25506 Lemma for ~ mbfimaopn . (...
mbfimaopn 25507 The preimage of any open s...
mbfimaopn2 25508 The preimage of any set op...
cncombf 25509 The composition of a conti...
cnmbf 25510 A continuous function is m...
mbfaddlem 25511 The sum of two measurable ...
mbfadd 25512 The sum of two measurable ...
mbfsub 25513 The difference of two meas...
mbfmulc2 25514 A complex constant times a...
mbfsup 25515 The supremum of a sequence...
mbfinf 25516 The infimum of a sequence ...
mbflimsup 25517 The limit supremum of a se...
mbflimlem 25518 The pointwise limit of a s...
mbflim 25519 The pointwise limit of a s...
0pval 25522 The zero function evaluate...
0plef 25523 Two ways to say that the f...
0pledm 25524 Adjust the domain of the l...
isi1f 25525 The predicate " ` F ` is a...
i1fmbf 25526 Simple functions are measu...
i1ff 25527 A simple function is a fun...
i1frn 25528 A simple function has fini...
i1fima 25529 Any preimage of a simple f...
i1fima2 25530 Any preimage of a simple f...
i1fima2sn 25531 Preimage of a singleton. ...
i1fd 25532 A simplified set of assump...
i1f0rn 25533 Any simple function takes ...
itg1val 25534 The value of the integral ...
itg1val2 25535 The value of the integral ...
itg1cl 25536 Closure of the integral on...
itg1ge0 25537 Closure of the integral on...
i1f0 25538 The zero function is simpl...
itg10 25539 The zero function has zero...
i1f1lem 25540 Lemma for ~ i1f1 and ~ itg...
i1f1 25541 Base case simple functions...
itg11 25542 The integral of an indicat...
itg1addlem1 25543 Decompose a preimage, whic...
i1faddlem 25544 Decompose the preimage of ...
i1fmullem 25545 Decompose the preimage of ...
i1fadd 25546 The sum of two simple func...
i1fmul 25547 The pointwise product of t...
itg1addlem2 25548 Lemma for ~ itg1add . The...
itg1addlem3 25549 Lemma for ~ itg1add . (Co...
itg1addlem4 25550 Lemma for ~ itg1add . (Co...
itg1addlem4OLD 25551 Obsolete version of ~ itg1...
itg1addlem5 25552 Lemma for ~ itg1add . (Co...
itg1add 25553 The integral of a sum of s...
i1fmulclem 25554 Decompose the preimage of ...
i1fmulc 25555 A nonnegative constant tim...
itg1mulc 25556 The integral of a constant...
i1fres 25557 The "restriction" of a sim...
i1fpos 25558 The positive part of a sim...
i1fposd 25559 Deduction form of ~ i1fpos...
i1fsub 25560 The difference of two simp...
itg1sub 25561 The integral of a differen...
itg10a 25562 The integral of a simple f...
itg1ge0a 25563 The integral of an almost ...
itg1lea 25564 Approximate version of ~ i...
itg1le 25565 If one simple function dom...
itg1climres 25566 Restricting the simple fun...
mbfi1fseqlem1 25567 Lemma for ~ mbfi1fseq . (...
mbfi1fseqlem2 25568 Lemma for ~ mbfi1fseq . (...
mbfi1fseqlem3 25569 Lemma for ~ mbfi1fseq . (...
mbfi1fseqlem4 25570 Lemma for ~ mbfi1fseq . T...
mbfi1fseqlem5 25571 Lemma for ~ mbfi1fseq . V...
mbfi1fseqlem6 25572 Lemma for ~ mbfi1fseq . V...
mbfi1fseq 25573 A characterization of meas...
mbfi1flimlem 25574 Lemma for ~ mbfi1flim . (...
mbfi1flim 25575 Any real measurable functi...
mbfmullem2 25576 Lemma for ~ mbfmul . (Con...
mbfmullem 25577 Lemma for ~ mbfmul . (Con...
mbfmul 25578 The product of two measura...
itg2lcl 25579 The set of lower sums is a...
itg2val 25580 Value of the integral on n...
itg2l 25581 Elementhood in the set ` L...
itg2lr 25582 Sufficient condition for e...
xrge0f 25583 A real function is a nonne...
itg2cl 25584 The integral of a nonnegat...
itg2ub 25585 The integral of a nonnegat...
itg2leub 25586 Any upper bound on the int...
itg2ge0 25587 The integral of a nonnegat...
itg2itg1 25588 The integral of a nonnegat...
itg20 25589 The integral of the zero f...
itg2lecl 25590 If an ` S.2 ` integral is ...
itg2le 25591 If one function dominates ...
itg2const 25592 Integral of a constant fun...
itg2const2 25593 When the base set of a con...
itg2seq 25594 Definitional property of t...
itg2uba 25595 Approximate version of ~ i...
itg2lea 25596 Approximate version of ~ i...
itg2eqa 25597 Approximate equality of in...
itg2mulclem 25598 Lemma for ~ itg2mulc . (C...
itg2mulc 25599 The integral of a nonnegat...
itg2splitlem 25600 Lemma for ~ itg2split . (...
itg2split 25601 The ` S.2 ` integral split...
itg2monolem1 25602 Lemma for ~ itg2mono . We...
itg2monolem2 25603 Lemma for ~ itg2mono . (C...
itg2monolem3 25604 Lemma for ~ itg2mono . (C...
itg2mono 25605 The Monotone Convergence T...
itg2i1fseqle 25606 Subject to the conditions ...
itg2i1fseq 25607 Subject to the conditions ...
itg2i1fseq2 25608 In an extension to the res...
itg2i1fseq3 25609 Special case of ~ itg2i1fs...
itg2addlem 25610 Lemma for ~ itg2add . (Co...
itg2add 25611 The ` S.2 ` integral is li...
itg2gt0 25612 If the function ` F ` is s...
itg2cnlem1 25613 Lemma for ~ itgcn . (Cont...
itg2cnlem2 25614 Lemma for ~ itgcn . (Cont...
itg2cn 25615 A sort of absolute continu...
ibllem 25616 Conditioned equality theor...
isibl 25617 The predicate " ` F ` is i...
isibl2 25618 The predicate " ` F ` is i...
iblmbf 25619 An integrable function is ...
iblitg 25620 If a function is integrabl...
dfitg 25621 Evaluate the class substit...
itgex 25622 An integral is a set. (Co...
itgeq1f 25623 Equality theorem for an in...
itgeq1 25624 Equality theorem for an in...
nfitg1 25625 Bound-variable hypothesis ...
nfitg 25626 Bound-variable hypothesis ...
cbvitg 25627 Change bound variable in a...
cbvitgv 25628 Change bound variable in a...
itgeq2 25629 Equality theorem for an in...
itgresr 25630 The domain of an integral ...
itg0 25631 The integral of anything o...
itgz 25632 The integral of zero on an...
itgeq2dv 25633 Equality theorem for an in...
itgmpt 25634 Change bound variable in a...
itgcl 25635 The integral of an integra...
itgvallem 25636 Substitution lemma. (Cont...
itgvallem3 25637 Lemma for ~ itgposval and ...
ibl0 25638 The zero function is integ...
iblcnlem1 25639 Lemma for ~ iblcnlem . (C...
iblcnlem 25640 Expand out the universal q...
itgcnlem 25641 Expand out the sum in ~ df...
iblrelem 25642 Integrability of a real fu...
iblposlem 25643 Lemma for ~ iblpos . (Con...
iblpos 25644 Integrability of a nonnega...
iblre 25645 Integrability of a real fu...
itgrevallem1 25646 Lemma for ~ itgposval and ...
itgposval 25647 The integral of a nonnegat...
itgreval 25648 Decompose the integral of ...
itgrecl 25649 Real closure of an integra...
iblcn 25650 Integrability of a complex...
itgcnval 25651 Decompose the integral of ...
itgre 25652 Real part of an integral. ...
itgim 25653 Imaginary part of an integ...
iblneg 25654 The negative of an integra...
itgneg 25655 Negation of an integral. ...
iblss 25656 A subset of an integrable ...
iblss2 25657 Change the domain of an in...
itgitg2 25658 Transfer an integral using...
i1fibl 25659 A simple function is integ...
itgitg1 25660 Transfer an integral using...
itgle 25661 Monotonicity of an integra...
itgge0 25662 The integral of a positive...
itgss 25663 Expand the set of an integ...
itgss2 25664 Expand the set of an integ...
itgeqa 25665 Approximate equality of in...
itgss3 25666 Expand the set of an integ...
itgioo 25667 Equality of integrals on o...
itgless 25668 Expand the integral of a n...
iblconst 25669 A constant function is int...
itgconst 25670 Integral of a constant fun...
ibladdlem 25671 Lemma for ~ ibladd . (Con...
ibladd 25672 Add two integrals over the...
iblsub 25673 Subtract two integrals ove...
itgaddlem1 25674 Lemma for ~ itgadd . (Con...
itgaddlem2 25675 Lemma for ~ itgadd . (Con...
itgadd 25676 Add two integrals over the...
itgsub 25677 Subtract two integrals ove...
itgfsum 25678 Take a finite sum of integ...
iblabslem 25679 Lemma for ~ iblabs . (Con...
iblabs 25680 The absolute value of an i...
iblabsr 25681 A measurable function is i...
iblmulc2 25682 Multiply an integral by a ...
itgmulc2lem1 25683 Lemma for ~ itgmulc2 : pos...
itgmulc2lem2 25684 Lemma for ~ itgmulc2 : rea...
itgmulc2 25685 Multiply an integral by a ...
itgabs 25686 The triangle inequality fo...
itgsplit 25687 The ` S. ` integral splits...
itgspliticc 25688 The ` S. ` integral splits...
itgsplitioo 25689 The ` S. ` integral splits...
bddmulibl 25690 A bounded function times a...
bddibl 25691 A bounded function is inte...
cniccibl 25692 A continuous function on a...
bddiblnc 25693 Choice-free proof of ~ bdd...
cnicciblnc 25694 Choice-free proof of ~ cni...
itggt0 25695 The integral of a strictly...
itgcn 25696 Transfer ~ itg2cn to the f...
ditgeq1 25699 Equality theorem for the d...
ditgeq2 25700 Equality theorem for the d...
ditgeq3 25701 Equality theorem for the d...
ditgeq3dv 25702 Equality theorem for the d...
ditgex 25703 A directed integral is a s...
ditg0 25704 Value of the directed inte...
cbvditg 25705 Change bound variable in a...
cbvditgv 25706 Change bound variable in a...
ditgpos 25707 Value of the directed inte...
ditgneg 25708 Value of the directed inte...
ditgcl 25709 Closure of a directed inte...
ditgswap 25710 Reverse a directed integra...
ditgsplitlem 25711 Lemma for ~ ditgsplit . (...
ditgsplit 25712 This theorem is the raison...
reldv 25721 The derivative function is...
limcvallem 25722 Lemma for ~ ellimc . (Con...
limcfval 25723 Value and set bounds on th...
ellimc 25724 Value of the limit predica...
limcrcl 25725 Reverse closure for the li...
limccl 25726 Closure of the limit opera...
limcdif 25727 It suffices to consider fu...
ellimc2 25728 Write the definition of a ...
limcnlp 25729 If ` B ` is not a limit po...
ellimc3 25730 Write the epsilon-delta de...
limcflflem 25731 Lemma for ~ limcflf . (Co...
limcflf 25732 The limit operator can be ...
limcmo 25733 If ` B ` is a limit point ...
limcmpt 25734 Express the limit operator...
limcmpt2 25735 Express the limit operator...
limcresi 25736 Any limit of ` F ` is also...
limcres 25737 If ` B ` is an interior po...
cnplimc 25738 A function is continuous a...
cnlimc 25739 ` F ` is a continuous func...
cnlimci 25740 If ` F ` is a continuous f...
cnmptlimc 25741 If ` F ` is a continuous f...
limccnp 25742 If the limit of ` F ` at `...
limccnp2 25743 The image of a convergent ...
limcco 25744 Composition of two limits....
limciun 25745 A point is a limit of ` F ...
limcun 25746 A point is a limit of ` F ...
dvlem 25747 Closure for a difference q...
dvfval 25748 Value and set bounds on th...
eldv 25749 The differentiable predica...
dvcl 25750 The derivative function ta...
dvbssntr 25751 The set of differentiable ...
dvbss 25752 The set of differentiable ...
dvbsss 25753 The set of differentiable ...
perfdvf 25754 The derivative is a functi...
recnprss 25755 Both ` RR ` and ` CC ` are...
recnperf 25756 Both ` RR ` and ` CC ` are...
dvfg 25757 Explicitly write out the f...
dvf 25758 The derivative is a functi...
dvfcn 25759 The derivative is a functi...
dvreslem 25760 Lemma for ~ dvres . (Cont...
dvres2lem 25761 Lemma for ~ dvres2 . (Con...
dvres 25762 Restriction of a derivativ...
dvres2 25763 Restriction of the base se...
dvres3 25764 Restriction of a complex d...
dvres3a 25765 Restriction of a complex d...
dvidlem 25766 Lemma for ~ dvid and ~ dvc...
dvmptresicc 25767 Derivative of a function r...
dvconst 25768 Derivative of a constant f...
dvid 25769 Derivative of the identity...
dvcnp 25770 The difference quotient is...
dvcnp2 25771 A function is continuous a...
dvcnp2OLD 25772 Obsolete version of ~ dvcn...
dvcn 25773 A differentiable function ...
dvnfval 25774 Value of the iterated deri...
dvnff 25775 The iterated derivative is...
dvn0 25776 Zero times iterated deriva...
dvnp1 25777 Successor iterated derivat...
dvn1 25778 One times iterated derivat...
dvnf 25779 The N-times derivative is ...
dvnbss 25780 The set of N-times differe...
dvnadd 25781 The ` N ` -th derivative o...
dvn2bss 25782 An N-times differentiable ...
dvnres 25783 Multiple derivative versio...
cpnfval 25784 Condition for n-times cont...
fncpn 25785 The ` C^n ` object is a fu...
elcpn 25786 Condition for n-times cont...
cpnord 25787 ` C^n ` conditions are ord...
cpncn 25788 A ` C^n ` function is cont...
cpnres 25789 The restriction of a ` C^n...
dvaddbr 25790 The sum rule for derivativ...
dvmulbr 25791 The product rule for deriv...
dvmulbrOLD 25792 Obsolete version of ~ dvmu...
dvadd 25793 The sum rule for derivativ...
dvmul 25794 The product rule for deriv...
dvaddf 25795 The sum rule for everywher...
dvmulf 25796 The product rule for every...
dvcmul 25797 The product rule when one ...
dvcmulf 25798 The product rule when one ...
dvcobr 25799 The chain rule for derivat...
dvcobrOLD 25800 Obsolete version of ~ dvco...
dvco 25801 The chain rule for derivat...
dvcof 25802 The chain rule for everywh...
dvcjbr 25803 The derivative of the conj...
dvcj 25804 The derivative of the conj...
dvfre 25805 The derivative of a real f...
dvnfre 25806 The ` N ` -th derivative o...
dvexp 25807 Derivative of a power func...
dvexp2 25808 Derivative of an exponenti...
dvrec 25809 Derivative of the reciproc...
dvmptres3 25810 Function-builder for deriv...
dvmptid 25811 Function-builder for deriv...
dvmptc 25812 Function-builder for deriv...
dvmptcl 25813 Closure lemma for ~ dvmptc...
dvmptadd 25814 Function-builder for deriv...
dvmptmul 25815 Function-builder for deriv...
dvmptres2 25816 Function-builder for deriv...
dvmptres 25817 Function-builder for deriv...
dvmptcmul 25818 Function-builder for deriv...
dvmptdivc 25819 Function-builder for deriv...
dvmptneg 25820 Function-builder for deriv...
dvmptsub 25821 Function-builder for deriv...
dvmptcj 25822 Function-builder for deriv...
dvmptre 25823 Function-builder for deriv...
dvmptim 25824 Function-builder for deriv...
dvmptntr 25825 Function-builder for deriv...
dvmptco 25826 Function-builder for deriv...
dvrecg 25827 Derivative of the reciproc...
dvmptdiv 25828 Function-builder for deriv...
dvmptfsum 25829 Function-builder for deriv...
dvcnvlem 25830 Lemma for ~ dvcnvre . (Co...
dvcnv 25831 A weak version of ~ dvcnvr...
dvexp3 25832 Derivative of an exponenti...
dveflem 25833 Derivative of the exponent...
dvef 25834 Derivative of the exponent...
dvsincos 25835 Derivative of the sine and...
dvsin 25836 Derivative of the sine fun...
dvcos 25837 Derivative of the cosine f...
dvferm1lem 25838 Lemma for ~ dvferm . (Con...
dvferm1 25839 One-sided version of ~ dvf...
dvferm2lem 25840 Lemma for ~ dvferm . (Con...
dvferm2 25841 One-sided version of ~ dvf...
dvferm 25842 Fermat's theorem on statio...
rollelem 25843 Lemma for ~ rolle . (Cont...
rolle 25844 Rolle's theorem. If ` F `...
cmvth 25845 Cauchy's Mean Value Theore...
cmvthOLD 25846 Obsolete version of ~ cmvt...
mvth 25847 The Mean Value Theorem. I...
dvlip 25848 A function with derivative...
dvlipcn 25849 A complex function with de...
dvlip2 25850 Combine the results of ~ d...
c1liplem1 25851 Lemma for ~ c1lip1 . (Con...
c1lip1 25852 C^1 functions are Lipschit...
c1lip2 25853 C^1 functions are Lipschit...
c1lip3 25854 C^1 functions are Lipschit...
dveq0 25855 If a continuous function h...
dv11cn 25856 Two functions defined on a...
dvgt0lem1 25857 Lemma for ~ dvgt0 and ~ dv...
dvgt0lem2 25858 Lemma for ~ dvgt0 and ~ dv...
dvgt0 25859 A function on a closed int...
dvlt0 25860 A function on a closed int...
dvge0 25861 A function on a closed int...
dvle 25862 If ` A ( x ) , C ( x ) ` a...
dvivthlem1 25863 Lemma for ~ dvivth . (Con...
dvivthlem2 25864 Lemma for ~ dvivth . (Con...
dvivth 25865 Darboux' theorem, or the i...
dvne0 25866 A function on a closed int...
dvne0f1 25867 A function on a closed int...
lhop1lem 25868 Lemma for ~ lhop1 . (Cont...
lhop1 25869 L'Hôpital's Rule for...
lhop2 25870 L'Hôpital's Rule for...
lhop 25871 L'Hôpital's Rule. I...
dvcnvrelem1 25872 Lemma for ~ dvcnvre . (Co...
dvcnvrelem2 25873 Lemma for ~ dvcnvre . (Co...
dvcnvre 25874 The derivative rule for in...
dvcvx 25875 A real function with stric...
dvfsumle 25876 Compare a finite sum to an...
dvfsumleOLD 25877 Obsolete version of ~ dvfs...
dvfsumge 25878 Compare a finite sum to an...
dvfsumabs 25879 Compare a finite sum to an...
dvmptrecl 25880 Real closure of a derivati...
dvfsumrlimf 25881 Lemma for ~ dvfsumrlim . ...
dvfsumlem1 25882 Lemma for ~ dvfsumrlim . ...
dvfsumlem2 25883 Lemma for ~ dvfsumrlim . ...
dvfsumlem2OLD 25884 Obsolete version of ~ dvfs...
dvfsumlem3 25885 Lemma for ~ dvfsumrlim . ...
dvfsumlem4 25886 Lemma for ~ dvfsumrlim . ...
dvfsumrlimge0 25887 Lemma for ~ dvfsumrlim . ...
dvfsumrlim 25888 Compare a finite sum to an...
dvfsumrlim2 25889 Compare a finite sum to an...
dvfsumrlim3 25890 Conjoin the statements of ...
dvfsum2 25891 The reverse of ~ dvfsumrli...
ftc1lem1 25892 Lemma for ~ ftc1a and ~ ft...
ftc1lem2 25893 Lemma for ~ ftc1 . (Contr...
ftc1a 25894 The Fundamental Theorem of...
ftc1lem3 25895 Lemma for ~ ftc1 . (Contr...
ftc1lem4 25896 Lemma for ~ ftc1 . (Contr...
ftc1lem5 25897 Lemma for ~ ftc1 . (Contr...
ftc1lem6 25898 Lemma for ~ ftc1 . (Contr...
ftc1 25899 The Fundamental Theorem of...
ftc1cn 25900 Strengthen the assumptions...
ftc2 25901 The Fundamental Theorem of...
ftc2ditglem 25902 Lemma for ~ ftc2ditg . (C...
ftc2ditg 25903 Directed integral analogue...
itgparts 25904 Integration by parts. If ...
itgsubstlem 25905 Lemma for ~ itgsubst . (C...
itgsubst 25906 Integration by ` u ` -subs...
itgpowd 25907 The integral of a monomial...
reldmmdeg 25912 Multivariate degree is a b...
tdeglem1 25913 Functionality of the total...
tdeglem1OLD 25914 Obsolete version of ~ tdeg...
tdeglem3 25915 Additivity of the total de...
tdeglem3OLD 25916 Obsolete version of ~ tdeg...
tdeglem4 25917 There is only one multi-in...
tdeglem4OLD 25918 Obsolete version of ~ tdeg...
tdeglem2 25919 Simplification of total de...
mdegfval 25920 Value of the multivariate ...
mdegval 25921 Value of the multivariate ...
mdegleb 25922 Property of being of limit...
mdeglt 25923 If there is an upper limit...
mdegldg 25924 A nonzero polynomial has s...
mdegxrcl 25925 Closure of polynomial degr...
mdegxrf 25926 Functionality of polynomia...
mdegcl 25927 Sharp closure for multivar...
mdeg0 25928 Degree of the zero polynom...
mdegnn0cl 25929 Degree of a nonzero polyno...
degltlem1 25930 Theorem on arithmetic of e...
degltp1le 25931 Theorem on arithmetic of e...
mdegaddle 25932 The degree of a sum is at ...
mdegvscale 25933 The degree of a scalar mul...
mdegvsca 25934 The degree of a scalar mul...
mdegle0 25935 A polynomial has nonpositi...
mdegmullem 25936 Lemma for ~ mdegmulle2 . ...
mdegmulle2 25937 The multivariate degree of...
deg1fval 25938 Relate univariate polynomi...
deg1xrf 25939 Functionality of univariat...
deg1xrcl 25940 Closure of univariate poly...
deg1cl 25941 Sharp closure of univariat...
mdegpropd 25942 Property deduction for pol...
deg1fvi 25943 Univariate polynomial degr...
deg1propd 25944 Property deduction for pol...
deg1z 25945 Degree of the zero univari...
deg1nn0cl 25946 Degree of a nonzero univar...
deg1n0ima 25947 Degree image of a set of p...
deg1nn0clb 25948 A polynomial is nonzero if...
deg1lt0 25949 A polynomial is zero iff i...
deg1ldg 25950 A nonzero univariate polyn...
deg1ldgn 25951 An index at which a polyno...
deg1ldgdomn 25952 A nonzero univariate polyn...
deg1leb 25953 Property of being of limit...
deg1val 25954 Value of the univariate de...
deg1lt 25955 If the degree of a univari...
deg1ge 25956 Conversely, a nonzero coef...
coe1mul3 25957 The coefficient vector of ...
coe1mul4 25958 Value of the "leading" coe...
deg1addle 25959 The degree of a sum is at ...
deg1addle2 25960 If both factors have degre...
deg1add 25961 Exact degree of a sum of t...
deg1vscale 25962 The degree of a scalar tim...
deg1vsca 25963 The degree of a scalar tim...
deg1invg 25964 The degree of the negated ...
deg1suble 25965 The degree of a difference...
deg1sub 25966 Exact degree of a differen...
deg1mulle2 25967 Produce a bound on the pro...
deg1sublt 25968 Subtraction of two polynom...
deg1le0 25969 A polynomial has nonpositi...
deg1sclle 25970 A scalar polynomial has no...
deg1scl 25971 A nonzero scalar polynomia...
deg1mul2 25972 Degree of multiplication o...
deg1mul3 25973 Degree of multiplication o...
deg1mul3le 25974 Degree of multiplication o...
deg1tmle 25975 Limiting degree of a polyn...
deg1tm 25976 Exact degree of a polynomi...
deg1pwle 25977 Limiting degree of a varia...
deg1pw 25978 Exact degree of a variable...
ply1nz 25979 Univariate polynomials ove...
ply1nzb 25980 Univariate polynomials are...
ply1domn 25981 Corollary of ~ deg1mul2 : ...
ply1idom 25982 The ring of univariate pol...
ply1divmo 25993 Uniqueness of a quotient i...
ply1divex 25994 Lemma for ~ ply1divalg : e...
ply1divalg 25995 The division algorithm for...
ply1divalg2 25996 Reverse the order of multi...
uc1pval 25997 Value of the set of unitic...
isuc1p 25998 Being a unitic polynomial....
mon1pval 25999 Value of the set of monic ...
ismon1p 26000 Being a monic polynomial. ...
uc1pcl 26001 Unitic polynomials are pol...
mon1pcl 26002 Monic polynomials are poly...
uc1pn0 26003 Unitic polynomials are not...
mon1pn0 26004 Monic polynomials are not ...
uc1pdeg 26005 Unitic polynomials have no...
uc1pldg 26006 Unitic polynomials have un...
mon1pldg 26007 Unitic polynomials have on...
mon1puc1p 26008 Monic polynomials are unit...
uc1pmon1p 26009 Make a unitic polynomial m...
deg1submon1p 26010 The difference of two moni...
q1pval 26011 Value of the univariate po...
q1peqb 26012 Characterizing property of...
q1pcl 26013 Closure of the quotient by...
r1pval 26014 Value of the polynomial re...
r1pcl 26015 Closure of remainder follo...
r1pdeglt 26016 The remainder has a degree...
r1pid 26017 Express the original polyn...
dvdsq1p 26018 Divisibility in a polynomi...
dvdsr1p 26019 Divisibility in a polynomi...
ply1remlem 26020 A term of the form ` x - N...
ply1rem 26021 The polynomial remainder t...
facth1 26022 The factor theorem and its...
fta1glem1 26023 Lemma for ~ fta1g . (Cont...
fta1glem2 26024 Lemma for ~ fta1g . (Cont...
fta1g 26025 The one-sided fundamental ...
fta1blem 26026 Lemma for ~ fta1b . (Cont...
fta1b 26027 The assumption that ` R ` ...
drnguc1p 26028 Over a division ring, all ...
ig1peu 26029 There is a unique monic po...
ig1pval 26030 Substitutions for the poly...
ig1pval2 26031 Generator of the zero idea...
ig1pval3 26032 Characterizing properties ...
ig1pcl 26033 The monic generator of an ...
ig1pdvds 26034 The monic generator of an ...
ig1prsp 26035 Any ideal of polynomials o...
ply1lpir 26036 The ring of polynomials ov...
ply1pid 26037 The polynomials over a fie...
plyco0 26046 Two ways to say that a fun...
plyval 26047 Value of the polynomial se...
plybss 26048 Reverse closure of the par...
elply 26049 Definition of a polynomial...
elply2 26050 The coefficient function c...
plyun0 26051 The set of polynomials is ...
plyf 26052 The polynomial is a functi...
plyss 26053 The polynomial set functio...
plyssc 26054 Every polynomial ring is c...
elplyr 26055 Sufficient condition for e...
elplyd 26056 Sufficient condition for e...
ply1termlem 26057 Lemma for ~ ply1term . (C...
ply1term 26058 A one-term polynomial. (C...
plypow 26059 A power is a polynomial. ...
plyconst 26060 A constant function is a p...
ne0p 26061 A test to show that a poly...
ply0 26062 The zero function is a pol...
plyid 26063 The identity function is a...
plyeq0lem 26064 Lemma for ~ plyeq0 . If `...
plyeq0 26065 If a polynomial is zero at...
plypf1 26066 Write the set of complex p...
plyaddlem1 26067 Derive the coefficient fun...
plymullem1 26068 Derive the coefficient fun...
plyaddlem 26069 Lemma for ~ plyadd . (Con...
plymullem 26070 Lemma for ~ plymul . (Con...
plyadd 26071 The sum of two polynomials...
plymul 26072 The product of two polynom...
plysub 26073 The difference of two poly...
plyaddcl 26074 The sum of two polynomials...
plymulcl 26075 The product of two polynom...
plysubcl 26076 The difference of two poly...
coeval 26077 Value of the coefficient f...
coeeulem 26078 Lemma for ~ coeeu . (Cont...
coeeu 26079 Uniqueness of the coeffici...
coelem 26080 Lemma for properties of th...
coeeq 26081 If ` A ` satisfies the pro...
dgrval 26082 Value of the degree functi...
dgrlem 26083 Lemma for ~ dgrcl and simi...
coef 26084 The domain and codomain of...
coef2 26085 The domain and codomain of...
coef3 26086 The domain and codomain of...
dgrcl 26087 The degree of any polynomi...
dgrub 26088 If the ` M ` -th coefficie...
dgrub2 26089 All the coefficients above...
dgrlb 26090 If all the coefficients ab...
coeidlem 26091 Lemma for ~ coeid . (Cont...
coeid 26092 Reconstruct a polynomial a...
coeid2 26093 Reconstruct a polynomial a...
coeid3 26094 Reconstruct a polynomial a...
plyco 26095 The composition of two pol...
coeeq2 26096 Compute the coefficient fu...
dgrle 26097 Given an explicit expressi...
dgreq 26098 If the highest term in a p...
0dgr 26099 A constant function has de...
0dgrb 26100 A function has degree zero...
dgrnznn 26101 A nonzero polynomial with ...
coefv0 26102 The result of evaluating a...
coeaddlem 26103 Lemma for ~ coeadd and ~ d...
coemullem 26104 Lemma for ~ coemul and ~ d...
coeadd 26105 The coefficient function o...
coemul 26106 A coefficient of a product...
coe11 26107 The coefficient function i...
coemulhi 26108 The leading coefficient of...
coemulc 26109 The coefficient function i...
coe0 26110 The coefficients of the ze...
coesub 26111 The coefficient function o...
coe1termlem 26112 The coefficient function o...
coe1term 26113 The coefficient function o...
dgr1term 26114 The degree of a monomial. ...
plycn 26115 A polynomial is a continuo...
plycnOLD 26116 Obsolete version of ~ plyc...
dgr0 26117 The degree of the zero pol...
coeidp 26118 The coefficients of the id...
dgrid 26119 The degree of the identity...
dgreq0 26120 The leading coefficient of...
dgrlt 26121 Two ways to say that the d...
dgradd 26122 The degree of a sum of pol...
dgradd2 26123 The degree of a sum of pol...
dgrmul2 26124 The degree of a product of...
dgrmul 26125 The degree of a product of...
dgrmulc 26126 Scalar multiplication by a...
dgrsub 26127 The degree of a difference...
dgrcolem1 26128 The degree of a compositio...
dgrcolem2 26129 Lemma for ~ dgrco . (Cont...
dgrco 26130 The degree of a compositio...
plycjlem 26131 Lemma for ~ plycj and ~ co...
plycj 26132 The double conjugation of ...
coecj 26133 Double conjugation of a po...
plyrecj 26134 A polynomial with real coe...
plymul0or 26135 Polynomial multiplication ...
ofmulrt 26136 The set of roots of a prod...
plyreres 26137 Real-coefficient polynomia...
dvply1 26138 Derivative of a polynomial...
dvply2g 26139 The derivative of a polyno...
dvply2 26140 The derivative of a polyno...
dvnply2 26141 Polynomials have polynomia...
dvnply 26142 Polynomials have polynomia...
plycpn 26143 Polynomials are smooth. (...
quotval 26146 Value of the quotient func...
plydivlem1 26147 Lemma for ~ plydivalg . (...
plydivlem2 26148 Lemma for ~ plydivalg . (...
plydivlem3 26149 Lemma for ~ plydivex . Ba...
plydivlem4 26150 Lemma for ~ plydivex . In...
plydivex 26151 Lemma for ~ plydivalg . (...
plydiveu 26152 Lemma for ~ plydivalg . (...
plydivalg 26153 The division algorithm on ...
quotlem 26154 Lemma for properties of th...
quotcl 26155 The quotient of two polyno...
quotcl2 26156 Closure of the quotient fu...
quotdgr 26157 Remainder property of the ...
plyremlem 26158 Closure of a linear factor...
plyrem 26159 The polynomial remainder t...
facth 26160 The factor theorem. If a ...
fta1lem 26161 Lemma for ~ fta1 . (Contr...
fta1 26162 The easy direction of the ...
quotcan 26163 Exact division with a mult...
vieta1lem1 26164 Lemma for ~ vieta1 . (Con...
vieta1lem2 26165 Lemma for ~ vieta1 : induc...
vieta1 26166 The first-order Vieta's fo...
plyexmo 26167 An infinite set of values ...
elaa 26170 Elementhood in the set of ...
aacn 26171 An algebraic number is a c...
aasscn 26172 The algebraic numbers are ...
elqaalem1 26173 Lemma for ~ elqaa . The f...
elqaalem2 26174 Lemma for ~ elqaa . (Cont...
elqaalem3 26175 Lemma for ~ elqaa . (Cont...
elqaa 26176 The set of numbers generat...
qaa 26177 Every rational number is a...
qssaa 26178 The rational numbers are c...
iaa 26179 The imaginary unit is alge...
aareccl 26180 The reciprocal of an algeb...
aacjcl 26181 The conjugate of an algebr...
aannenlem1 26182 Lemma for ~ aannen . (Con...
aannenlem2 26183 Lemma for ~ aannen . (Con...
aannenlem3 26184 The algebraic numbers are ...
aannen 26185 The algebraic numbers are ...
aalioulem1 26186 Lemma for ~ aaliou . An i...
aalioulem2 26187 Lemma for ~ aaliou . (Con...
aalioulem3 26188 Lemma for ~ aaliou . (Con...
aalioulem4 26189 Lemma for ~ aaliou . (Con...
aalioulem5 26190 Lemma for ~ aaliou . (Con...
aalioulem6 26191 Lemma for ~ aaliou . (Con...
aaliou 26192 Liouville's theorem on dio...
geolim3 26193 Geometric series convergen...
aaliou2 26194 Liouville's approximation ...
aaliou2b 26195 Liouville's approximation ...
aaliou3lem1 26196 Lemma for ~ aaliou3 . (Co...
aaliou3lem2 26197 Lemma for ~ aaliou3 . (Co...
aaliou3lem3 26198 Lemma for ~ aaliou3 . (Co...
aaliou3lem8 26199 Lemma for ~ aaliou3 . (Co...
aaliou3lem4 26200 Lemma for ~ aaliou3 . (Co...
aaliou3lem5 26201 Lemma for ~ aaliou3 . (Co...
aaliou3lem6 26202 Lemma for ~ aaliou3 . (Co...
aaliou3lem7 26203 Lemma for ~ aaliou3 . (Co...
aaliou3lem9 26204 Example of a "Liouville nu...
aaliou3 26205 Example of a "Liouville nu...
taylfvallem1 26210 Lemma for ~ taylfval . (C...
taylfvallem 26211 Lemma for ~ taylfval . (C...
taylfval 26212 Define the Taylor polynomi...
eltayl 26213 Value of the Taylor series...
taylf 26214 The Taylor series defines ...
tayl0 26215 The Taylor series is alway...
taylplem1 26216 Lemma for ~ taylpfval and ...
taylplem2 26217 Lemma for ~ taylpfval and ...
taylpfval 26218 Define the Taylor polynomi...
taylpf 26219 The Taylor polynomial is a...
taylpval 26220 Value of the Taylor polyno...
taylply2 26221 The Taylor polynomial is a...
taylply 26222 The Taylor polynomial is a...
dvtaylp 26223 The derivative of the Tayl...
dvntaylp 26224 The ` M ` -th derivative o...
dvntaylp0 26225 The first ` N ` derivative...
taylthlem1 26226 Lemma for ~ taylth . This...
taylthlem2 26227 Lemma for ~ taylth . (Con...
taylth 26228 Taylor's theorem. The Tay...
ulmrel 26231 The uniform limit relation...
ulmscl 26232 Closure of the base set in...
ulmval 26233 Express the predicate: Th...
ulmcl 26234 Closure of a uniform limit...
ulmf 26235 Closure of a uniform limit...
ulmpm 26236 Closure of a uniform limit...
ulmf2 26237 Closure of a uniform limit...
ulm2 26238 Simplify ~ ulmval when ` F...
ulmi 26239 The uniform limit property...
ulmclm 26240 A uniform limit of functio...
ulmres 26241 A sequence of functions co...
ulmshftlem 26242 Lemma for ~ ulmshft . (Co...
ulmshft 26243 A sequence of functions co...
ulm0 26244 Every function converges u...
ulmuni 26245 A sequence of functions un...
ulmdm 26246 Two ways to express that a...
ulmcaulem 26247 Lemma for ~ ulmcau and ~ u...
ulmcau 26248 A sequence of functions co...
ulmcau2 26249 A sequence of functions co...
ulmss 26250 A uniform limit of functio...
ulmbdd 26251 A uniform limit of bounded...
ulmcn 26252 A uniform limit of continu...
ulmdvlem1 26253 Lemma for ~ ulmdv . (Cont...
ulmdvlem2 26254 Lemma for ~ ulmdv . (Cont...
ulmdvlem3 26255 Lemma for ~ ulmdv . (Cont...
ulmdv 26256 If ` F ` is a sequence of ...
mtest 26257 The Weierstrass M-test. I...
mtestbdd 26258 Given the hypotheses of th...
mbfulm 26259 A uniform limit of measura...
iblulm 26260 A uniform limit of integra...
itgulm 26261 A uniform limit of integra...
itgulm2 26262 A uniform limit of integra...
pserval 26263 Value of the function ` G ...
pserval2 26264 Value of the function ` G ...
psergf 26265 The sequence of terms in t...
radcnvlem1 26266 Lemma for ~ radcnvlt1 , ~ ...
radcnvlem2 26267 Lemma for ~ radcnvlt1 , ~ ...
radcnvlem3 26268 Lemma for ~ radcnvlt1 , ~ ...
radcnv0 26269 Zero is always a convergen...
radcnvcl 26270 The radius of convergence ...
radcnvlt1 26271 If ` X ` is within the ope...
radcnvlt2 26272 If ` X ` is within the ope...
radcnvle 26273 If ` X ` is a convergent p...
dvradcnv 26274 The radius of convergence ...
pserulm 26275 If ` S ` is a region conta...
psercn2 26276 Since by ~ pserulm the ser...
psercn2OLD 26277 Obsolete version of ~ pser...
psercnlem2 26278 Lemma for ~ psercn . (Con...
psercnlem1 26279 Lemma for ~ psercn . (Con...
psercn 26280 An infinite series converg...
pserdvlem1 26281 Lemma for ~ pserdv . (Con...
pserdvlem2 26282 Lemma for ~ pserdv . (Con...
pserdv 26283 The derivative of a power ...
pserdv2 26284 The derivative of a power ...
abelthlem1 26285 Lemma for ~ abelth . (Con...
abelthlem2 26286 Lemma for ~ abelth . The ...
abelthlem3 26287 Lemma for ~ abelth . (Con...
abelthlem4 26288 Lemma for ~ abelth . (Con...
abelthlem5 26289 Lemma for ~ abelth . (Con...
abelthlem6 26290 Lemma for ~ abelth . (Con...
abelthlem7a 26291 Lemma for ~ abelth . (Con...
abelthlem7 26292 Lemma for ~ abelth . (Con...
abelthlem8 26293 Lemma for ~ abelth . (Con...
abelthlem9 26294 Lemma for ~ abelth . By a...
abelth 26295 Abel's theorem. If the po...
abelth2 26296 Abel's theorem, restricted...
efcn 26297 The exponential function i...
sincn 26298 Sine is continuous. (Cont...
coscn 26299 Cosine is continuous. (Co...
reeff1olem 26300 Lemma for ~ reeff1o . (Co...
reeff1o 26301 The real exponential funct...
reefiso 26302 The exponential function o...
efcvx 26303 The exponential function o...
reefgim 26304 The exponential function i...
pilem1 26305 Lemma for ~ pire , ~ pigt2...
pilem2 26306 Lemma for ~ pire , ~ pigt2...
pilem3 26307 Lemma for ~ pire , ~ pigt2...
pigt2lt4 26308 ` _pi ` is between 2 and 4...
sinpi 26309 The sine of ` _pi ` is 0. ...
pire 26310 ` _pi ` is a real number. ...
picn 26311 ` _pi ` is a complex numbe...
pipos 26312 ` _pi ` is positive. (Con...
pirp 26313 ` _pi ` is a positive real...
negpicn 26314 ` -u _pi ` is a real numbe...
sinhalfpilem 26315 Lemma for ~ sinhalfpi and ...
halfpire 26316 ` _pi / 2 ` is real. (Con...
neghalfpire 26317 ` -u _pi / 2 ` is real. (...
neghalfpirx 26318 ` -u _pi / 2 ` is an exten...
pidiv2halves 26319 Adding ` _pi / 2 ` to itse...
sinhalfpi 26320 The sine of ` _pi / 2 ` is...
coshalfpi 26321 The cosine of ` _pi / 2 ` ...
cosneghalfpi 26322 The cosine of ` -u _pi / 2...
efhalfpi 26323 The exponential of ` _i _p...
cospi 26324 The cosine of ` _pi ` is `...
efipi 26325 The exponential of ` _i x....
eulerid 26326 Euler's identity. (Contri...
sin2pi 26327 The sine of ` 2 _pi ` is 0...
cos2pi 26328 The cosine of ` 2 _pi ` is...
ef2pi 26329 The exponential of ` 2 _pi...
ef2kpi 26330 If ` K ` is an integer, th...
efper 26331 The exponential function i...
sinperlem 26332 Lemma for ~ sinper and ~ c...
sinper 26333 The sine function is perio...
cosper 26334 The cosine function is per...
sin2kpi 26335 If ` K ` is an integer, th...
cos2kpi 26336 If ` K ` is an integer, th...
sin2pim 26337 Sine of a number subtracte...
cos2pim 26338 Cosine of a number subtrac...
sinmpi 26339 Sine of a number less ` _p...
cosmpi 26340 Cosine of a number less ` ...
sinppi 26341 Sine of a number plus ` _p...
cosppi 26342 Cosine of a number plus ` ...
efimpi 26343 The exponential function a...
sinhalfpip 26344 The sine of ` _pi / 2 ` pl...
sinhalfpim 26345 The sine of ` _pi / 2 ` mi...
coshalfpip 26346 The cosine of ` _pi / 2 ` ...
coshalfpim 26347 The cosine of ` _pi / 2 ` ...
ptolemy 26348 Ptolemy's Theorem. This t...
sincosq1lem 26349 Lemma for ~ sincosq1sgn . ...
sincosq1sgn 26350 The signs of the sine and ...
sincosq2sgn 26351 The signs of the sine and ...
sincosq3sgn 26352 The signs of the sine and ...
sincosq4sgn 26353 The signs of the sine and ...
coseq00topi 26354 Location of the zeroes of ...
coseq0negpitopi 26355 Location of the zeroes of ...
tanrpcl 26356 Positive real closure of t...
tangtx 26357 The tangent function is gr...
tanabsge 26358 The tangent function is gr...
sinq12gt0 26359 The sine of a number stric...
sinq12ge0 26360 The sine of a number betwe...
sinq34lt0t 26361 The sine of a number stric...
cosq14gt0 26362 The cosine of a number str...
cosq14ge0 26363 The cosine of a number bet...
sincosq1eq 26364 Complementarity of the sin...
sincos4thpi 26365 The sine and cosine of ` _...
tan4thpi 26366 The tangent of ` _pi / 4 `...
sincos6thpi 26367 The sine and cosine of ` _...
sincos3rdpi 26368 The sine and cosine of ` _...
pigt3 26369 ` _pi ` is greater than 3....
pige3 26370 ` _pi ` is greater than or...
pige3ALT 26371 Alternate proof of ~ pige3...
abssinper 26372 The absolute value of sine...
sinkpi 26373 The sine of an integer mul...
coskpi 26374 The absolute value of the ...
sineq0 26375 A complex number whose sin...
coseq1 26376 A complex number whose cos...
cos02pilt1 26377 Cosine is less than one be...
cosq34lt1 26378 Cosine is less than one in...
efeq1 26379 A complex number whose exp...
cosne0 26380 The cosine function has no...
cosordlem 26381 Lemma for ~ cosord . (Con...
cosord 26382 Cosine is decreasing over ...
cos0pilt1 26383 Cosine is between minus on...
cos11 26384 Cosine is one-to-one over ...
sinord 26385 Sine is increasing over th...
recosf1o 26386 The cosine function is a b...
resinf1o 26387 The sine function is a bij...
tanord1 26388 The tangent function is st...
tanord 26389 The tangent function is st...
tanregt0 26390 The real part of the tange...
negpitopissre 26391 The interval ` ( -u _pi (,...
efgh 26392 The exponential function o...
efif1olem1 26393 Lemma for ~ efif1o . (Con...
efif1olem2 26394 Lemma for ~ efif1o . (Con...
efif1olem3 26395 Lemma for ~ efif1o . (Con...
efif1olem4 26396 The exponential function o...
efif1o 26397 The exponential function o...
efifo 26398 The exponential function o...
eff1olem 26399 The exponential function m...
eff1o 26400 The exponential function m...
efabl 26401 The image of a subgroup of...
efsubm 26402 The image of a subgroup of...
circgrp 26403 The circle group ` T ` is ...
circsubm 26404 The circle group ` T ` is ...
logrn 26409 The range of the natural l...
ellogrn 26410 Write out the property ` A...
dflog2 26411 The natural logarithm func...
relogrn 26412 The range of the natural l...
logrncn 26413 The range of the natural l...
eff1o2 26414 The exponential function r...
logf1o 26415 The natural logarithm func...
dfrelog 26416 The natural logarithm func...
relogf1o 26417 The natural logarithm func...
logrncl 26418 Closure of the natural log...
logcl 26419 Closure of the natural log...
logimcl 26420 Closure of the imaginary p...
logcld 26421 The logarithm of a nonzero...
logimcld 26422 The imaginary part of the ...
logimclad 26423 The imaginary part of the ...
abslogimle 26424 The imaginary part of the ...
logrnaddcl 26425 The range of the natural l...
relogcl 26426 Closure of the natural log...
eflog 26427 Relationship between the n...
logeq0im1 26428 If the logarithm of a numb...
logccne0 26429 The logarithm isn't 0 if i...
logne0 26430 Logarithm of a non-1 posit...
reeflog 26431 Relationship between the n...
logef 26432 Relationship between the n...
relogef 26433 Relationship between the n...
logeftb 26434 Relationship between the n...
relogeftb 26435 Relationship between the n...
log1 26436 The natural logarithm of `...
loge 26437 The natural logarithm of `...
logneg 26438 The natural logarithm of a...
logm1 26439 The natural logarithm of n...
lognegb 26440 If a number has imaginary ...
relogoprlem 26441 Lemma for ~ relogmul and ~...
relogmul 26442 The natural logarithm of t...
relogdiv 26443 The natural logarithm of t...
explog 26444 Exponentiation of a nonzer...
reexplog 26445 Exponentiation of a positi...
relogexp 26446 The natural logarithm of p...
relog 26447 Real part of a logarithm. ...
relogiso 26448 The natural logarithm func...
reloggim 26449 The natural logarithm is a...
logltb 26450 The natural logarithm func...
logfac 26451 The logarithm of a factori...
eflogeq 26452 Solve an equation involvin...
logleb 26453 Natural logarithm preserve...
rplogcl 26454 Closure of the logarithm f...
logge0 26455 The logarithm of a number ...
logcj 26456 The natural logarithm dist...
efiarg 26457 The exponential of the "ar...
cosargd 26458 The cosine of the argument...
cosarg0d 26459 The cosine of the argument...
argregt0 26460 Closure of the argument of...
argrege0 26461 Closure of the argument of...
argimgt0 26462 Closure of the argument of...
argimlt0 26463 Closure of the argument of...
logimul 26464 Multiplying a number by ` ...
logneg2 26465 The logarithm of the negat...
logmul2 26466 Generalization of ~ relogm...
logdiv2 26467 Generalization of ~ relogd...
abslogle 26468 Bound on the magnitude of ...
tanarg 26469 The basic relation between...
logdivlti 26470 The ` log x / x ` function...
logdivlt 26471 The ` log x / x ` function...
logdivle 26472 The ` log x / x ` function...
relogcld 26473 Closure of the natural log...
reeflogd 26474 Relationship between the n...
relogmuld 26475 The natural logarithm of t...
relogdivd 26476 The natural logarithm of t...
logled 26477 Natural logarithm preserve...
relogefd 26478 Relationship between the n...
rplogcld 26479 Closure of the logarithm f...
logge0d 26480 The logarithm of a number ...
logge0b 26481 The logarithm of a number ...
loggt0b 26482 The logarithm of a number ...
logle1b 26483 The logarithm of a number ...
loglt1b 26484 The logarithm of a number ...
divlogrlim 26485 The inverse logarithm func...
logno1 26486 The logarithm function is ...
dvrelog 26487 The derivative of the real...
relogcn 26488 The real logarithm functio...
ellogdm 26489 Elementhood in the "contin...
logdmn0 26490 A number in the continuous...
logdmnrp 26491 A number in the continuous...
logdmss 26492 The continuity domain of `...
logcnlem2 26493 Lemma for ~ logcn . (Cont...
logcnlem3 26494 Lemma for ~ logcn . (Cont...
logcnlem4 26495 Lemma for ~ logcn . (Cont...
logcnlem5 26496 Lemma for ~ logcn . (Cont...
logcn 26497 The logarithm function is ...
dvloglem 26498 Lemma for ~ dvlog . (Cont...
logdmopn 26499 The "continuous domain" of...
logf1o2 26500 The logarithm maps its con...
dvlog 26501 The derivative of the comp...
dvlog2lem 26502 Lemma for ~ dvlog2 . (Con...
dvlog2 26503 The derivative of the comp...
advlog 26504 The antiderivative of the ...
advlogexp 26505 The antiderivative of a po...
efopnlem1 26506 Lemma for ~ efopn . (Cont...
efopnlem2 26507 Lemma for ~ efopn . (Cont...
efopn 26508 The exponential map is an ...
logtayllem 26509 Lemma for ~ logtayl . (Co...
logtayl 26510 The Taylor series for ` -u...
logtaylsum 26511 The Taylor series for ` -u...
logtayl2 26512 Power series expression fo...
logccv 26513 The natural logarithm func...
cxpval 26514 Value of the complex power...
cxpef 26515 Value of the complex power...
0cxp 26516 Value of the complex power...
cxpexpz 26517 Relate the complex power f...
cxpexp 26518 Relate the complex power f...
logcxp 26519 Logarithm of a complex pow...
cxp0 26520 Value of the complex power...
cxp1 26521 Value of the complex power...
1cxp 26522 Value of the complex power...
ecxp 26523 Write the exponential func...
cxpcl 26524 Closure of the complex pow...
recxpcl 26525 Real closure of the comple...
rpcxpcl 26526 Positive real closure of t...
cxpne0 26527 Complex exponentiation is ...
cxpeq0 26528 Complex exponentiation is ...
cxpadd 26529 Sum of exponents law for c...
cxpp1 26530 Value of a nonzero complex...
cxpneg 26531 Value of a complex number ...
cxpsub 26532 Exponent subtraction law f...
cxpge0 26533 Nonnegative exponentiation...
mulcxplem 26534 Lemma for ~ mulcxp . (Con...
mulcxp 26535 Complex exponentiation of ...
cxprec 26536 Complex exponentiation of ...
divcxp 26537 Complex exponentiation of ...
cxpmul 26538 Product of exponents law f...
cxpmul2 26539 Product of exponents law f...
cxproot 26540 The complex power function...
cxpmul2z 26541 Generalize ~ cxpmul2 to ne...
abscxp 26542 Absolute value of a power,...
abscxp2 26543 Absolute value of a power,...
cxplt 26544 Ordering property for comp...
cxple 26545 Ordering property for comp...
cxplea 26546 Ordering property for comp...
cxple2 26547 Ordering property for comp...
cxplt2 26548 Ordering property for comp...
cxple2a 26549 Ordering property for comp...
cxplt3 26550 Ordering property for comp...
cxple3 26551 Ordering property for comp...
cxpsqrtlem 26552 Lemma for ~ cxpsqrt . (Co...
cxpsqrt 26553 The complex exponential fu...
logsqrt 26554 Logarithm of a square root...
cxp0d 26555 Value of the complex power...
cxp1d 26556 Value of the complex power...
1cxpd 26557 Value of the complex power...
cxpcld 26558 Closure of the complex pow...
cxpmul2d 26559 Product of exponents law f...
0cxpd 26560 Value of the complex power...
cxpexpzd 26561 Relate the complex power f...
cxpefd 26562 Value of the complex power...
cxpne0d 26563 Complex exponentiation is ...
cxpp1d 26564 Value of a nonzero complex...
cxpnegd 26565 Value of a complex number ...
cxpmul2zd 26566 Generalize ~ cxpmul2 to ne...
cxpaddd 26567 Sum of exponents law for c...
cxpsubd 26568 Exponent subtraction law f...
cxpltd 26569 Ordering property for comp...
cxpled 26570 Ordering property for comp...
cxplead 26571 Ordering property for comp...
divcxpd 26572 Complex exponentiation of ...
recxpcld 26573 Positive real closure of t...
cxpge0d 26574 Nonnegative exponentiation...
cxple2ad 26575 Ordering property for comp...
cxplt2d 26576 Ordering property for comp...
cxple2d 26577 Ordering property for comp...
mulcxpd 26578 Complex exponentiation of ...
recxpf1lem 26579 Complex exponentiation on ...
cxpsqrtth 26580 Square root theorem over t...
2irrexpq 26581 There exist irrational num...
cxprecd 26582 Complex exponentiation of ...
rpcxpcld 26583 Positive real closure of t...
logcxpd 26584 Logarithm of a complex pow...
cxplt3d 26585 Ordering property for comp...
cxple3d 26586 Ordering property for comp...
cxpmuld 26587 Product of exponents law f...
cxpgt0d 26588 A positive real raised to ...
cxpcom 26589 Commutative law for real e...
dvcxp1 26590 The derivative of a comple...
dvcxp2 26591 The derivative of a comple...
dvsqrt 26592 The derivative of the real...
dvcncxp1 26593 Derivative of complex powe...
dvcnsqrt 26594 Derivative of square root ...
cxpcn 26595 Domain of continuity of th...
cxpcnOLD 26596 Obsolete version of ~ cxpc...
cxpcn2 26597 Continuity of the complex ...
cxpcn3lem 26598 Lemma for ~ cxpcn3 . (Con...
cxpcn3 26599 Extend continuity of the c...
resqrtcn 26600 Continuity of the real squ...
sqrtcn 26601 Continuity of the square r...
cxpaddlelem 26602 Lemma for ~ cxpaddle . (C...
cxpaddle 26603 Ordering property for comp...
abscxpbnd 26604 Bound on the absolute valu...
root1id 26605 Property of an ` N ` -th r...
root1eq1 26606 The only powers of an ` N ...
root1cj 26607 Within the ` N ` -th roots...
cxpeq 26608 Solve an equation involvin...
loglesqrt 26609 An upper bound on the loga...
logreclem 26610 Symmetry of the natural lo...
logrec 26611 Logarithm of a reciprocal ...
logbval 26614 Define the value of the ` ...
logbcl 26615 General logarithm closure....
logbid1 26616 General logarithm is 1 whe...
logb1 26617 The logarithm of ` 1 ` to ...
elogb 26618 The general logarithm of a...
logbchbase 26619 Change of base for logarit...
relogbval 26620 Value of the general logar...
relogbcl 26621 Closure of the general log...
relogbzcl 26622 Closure of the general log...
relogbreexp 26623 Power law for the general ...
relogbzexp 26624 Power law for the general ...
relogbmul 26625 The logarithm of the produ...
relogbmulexp 26626 The logarithm of the produ...
relogbdiv 26627 The logarithm of the quoti...
relogbexp 26628 Identity law for general l...
nnlogbexp 26629 Identity law for general l...
logbrec 26630 Logarithm of a reciprocal ...
logbleb 26631 The general logarithm func...
logblt 26632 The general logarithm func...
relogbcxp 26633 Identity law for the gener...
cxplogb 26634 Identity law for the gener...
relogbcxpb 26635 The logarithm is the inver...
logbmpt 26636 The general logarithm to a...
logbf 26637 The general logarithm to a...
logbfval 26638 The general logarithm of a...
relogbf 26639 The general logarithm to a...
logblog 26640 The general logarithm to t...
logbgt0b 26641 The logarithm of a positiv...
logbgcd1irr 26642 The logarithm of an intege...
2logb9irr 26643 Example for ~ logbgcd1irr ...
logbprmirr 26644 The logarithm of a prime t...
2logb3irr 26645 Example for ~ logbprmirr ....
2logb9irrALT 26646 Alternate proof of ~ 2logb...
sqrt2cxp2logb9e3 26647 The square root of two to ...
2irrexpqALT 26648 Alternate proof of ~ 2irre...
angval 26649 Define the angle function,...
angcan 26650 Cancel a constant multipli...
angneg 26651 Cancel a negative sign in ...
angvald 26652 The (signed) angle between...
angcld 26653 The (signed) angle between...
angrteqvd 26654 Two vectors are at a right...
cosangneg2d 26655 The cosine of the angle be...
angrtmuld 26656 Perpendicularity of two ve...
ang180lem1 26657 Lemma for ~ ang180 . Show...
ang180lem2 26658 Lemma for ~ ang180 . Show...
ang180lem3 26659 Lemma for ~ ang180 . Sinc...
ang180lem4 26660 Lemma for ~ ang180 . Redu...
ang180lem5 26661 Lemma for ~ ang180 : Redu...
ang180 26662 The sum of angles ` m A B ...
lawcoslem1 26663 Lemma for ~ lawcos . Here...
lawcos 26664 Law of cosines (also known...
pythag 26665 Pythagorean theorem. Give...
isosctrlem1 26666 Lemma for ~ isosctr . (Co...
isosctrlem2 26667 Lemma for ~ isosctr . Cor...
isosctrlem3 26668 Lemma for ~ isosctr . Cor...
isosctr 26669 Isosceles triangle theorem...
ssscongptld 26670 If two triangles have equa...
affineequiv 26671 Equivalence between two wa...
affineequiv2 26672 Equivalence between two wa...
affineequiv3 26673 Equivalence between two wa...
affineequiv4 26674 Equivalence between two wa...
affineequivne 26675 Equivalence between two wa...
angpieqvdlem 26676 Equivalence used in the pr...
angpieqvdlem2 26677 Equivalence used in ~ angp...
angpined 26678 If the angle at ABC is ` _...
angpieqvd 26679 The angle ABC is ` _pi ` i...
chordthmlem 26680 If ` M ` is the midpoint o...
chordthmlem2 26681 If M is the midpoint of AB...
chordthmlem3 26682 If M is the midpoint of AB...
chordthmlem4 26683 If P is on the segment AB ...
chordthmlem5 26684 If P is on the segment AB ...
chordthm 26685 The intersecting chords th...
heron 26686 Heron's formula gives the ...
quad2 26687 The quadratic equation, wi...
quad 26688 The quadratic equation. (...
1cubrlem 26689 The cube roots of unity. ...
1cubr 26690 The cube roots of unity. ...
dcubic1lem 26691 Lemma for ~ dcubic1 and ~ ...
dcubic2 26692 Reverse direction of ~ dcu...
dcubic1 26693 Forward direction of ~ dcu...
dcubic 26694 Solutions to the depressed...
mcubic 26695 Solutions to a monic cubic...
cubic2 26696 The solution to the genera...
cubic 26697 The cubic equation, which ...
binom4 26698 Work out a quartic binomia...
dquartlem1 26699 Lemma for ~ dquart . (Con...
dquartlem2 26700 Lemma for ~ dquart . (Con...
dquart 26701 Solve a depressed quartic ...
quart1cl 26702 Closure lemmas for ~ quart...
quart1lem 26703 Lemma for ~ quart1 . (Con...
quart1 26704 Depress a quartic equation...
quartlem1 26705 Lemma for ~ quart . (Cont...
quartlem2 26706 Closure lemmas for ~ quart...
quartlem3 26707 Closure lemmas for ~ quart...
quartlem4 26708 Closure lemmas for ~ quart...
quart 26709 The quartic equation, writ...
asinlem 26716 The argument to the logari...
asinlem2 26717 The argument to the logari...
asinlem3a 26718 Lemma for ~ asinlem3 . (C...
asinlem3 26719 The argument to the logari...
asinf 26720 Domain and codomain of the...
asincl 26721 Closure for the arcsin fun...
acosf 26722 Domain and codoamin of the...
acoscl 26723 Closure for the arccos fun...
atandm 26724 Since the property is a li...
atandm2 26725 This form of ~ atandm is a...
atandm3 26726 A compact form of ~ atandm...
atandm4 26727 A compact form of ~ atandm...
atanf 26728 Domain and codoamin of the...
atancl 26729 Closure for the arctan fun...
asinval 26730 Value of the arcsin functi...
acosval 26731 Value of the arccos functi...
atanval 26732 Value of the arctan functi...
atanre 26733 A real number is in the do...
asinneg 26734 The arcsine function is od...
acosneg 26735 The negative symmetry rela...
efiasin 26736 The exponential of the arc...
sinasin 26737 The arcsine function is an...
cosacos 26738 The arccosine function is ...
asinsinlem 26739 Lemma for ~ asinsin . (Co...
asinsin 26740 The arcsine function compo...
acoscos 26741 The arccosine function is ...
asin1 26742 The arcsine of ` 1 ` is ` ...
acos1 26743 The arccosine of ` 1 ` is ...
reasinsin 26744 The arcsine function compo...
asinsinb 26745 Relationship between sine ...
acoscosb 26746 Relationship between cosin...
asinbnd 26747 The arcsine function has r...
acosbnd 26748 The arccosine function has...
asinrebnd 26749 Bounds on the arcsine func...
asinrecl 26750 The arcsine function is re...
acosrecl 26751 The arccosine function is ...
cosasin 26752 The cosine of the arcsine ...
sinacos 26753 The sine of the arccosine ...
atandmneg 26754 The domain of the arctange...
atanneg 26755 The arctangent function is...
atan0 26756 The arctangent of zero is ...
atandmcj 26757 The arctangent function di...
atancj 26758 The arctangent function di...
atanrecl 26759 The arctangent function is...
efiatan 26760 Value of the exponential o...
atanlogaddlem 26761 Lemma for ~ atanlogadd . ...
atanlogadd 26762 The rule ` sqrt ( z w ) = ...
atanlogsublem 26763 Lemma for ~ atanlogsub . ...
atanlogsub 26764 A variation on ~ atanlogad...
efiatan2 26765 Value of the exponential o...
2efiatan 26766 Value of the exponential o...
tanatan 26767 The arctangent function is...
atandmtan 26768 The tangent function has r...
cosatan 26769 The cosine of an arctangen...
cosatanne0 26770 The arctangent function ha...
atantan 26771 The arctangent function is...
atantanb 26772 Relationship between tange...
atanbndlem 26773 Lemma for ~ atanbnd . (Co...
atanbnd 26774 The arctangent function is...
atanord 26775 The arctangent function is...
atan1 26776 The arctangent of ` 1 ` is...
bndatandm 26777 A point in the open unit d...
atans 26778 The "domain of continuity"...
atans2 26779 It suffices to show that `...
atansopn 26780 The domain of continuity o...
atansssdm 26781 The domain of continuity o...
ressatans 26782 The real number line is a ...
dvatan 26783 The derivative of the arct...
atancn 26784 The arctangent is a contin...
atantayl 26785 The Taylor series for ` ar...
atantayl2 26786 The Taylor series for ` ar...
atantayl3 26787 The Taylor series for ` ar...
leibpilem1 26788 Lemma for ~ leibpi . (Con...
leibpilem2 26789 The Leibniz formula for ` ...
leibpi 26790 The Leibniz formula for ` ...
leibpisum 26791 The Leibniz formula for ` ...
log2cnv 26792 Using the Taylor series fo...
log2tlbnd 26793 Bound the error term in th...
log2ublem1 26794 Lemma for ~ log2ub . The ...
log2ublem2 26795 Lemma for ~ log2ub . (Con...
log2ublem3 26796 Lemma for ~ log2ub . In d...
log2ub 26797 ` log 2 ` is less than ` 2...
log2le1 26798 ` log 2 ` is less than ` 1...
birthdaylem1 26799 Lemma for ~ birthday . (C...
birthdaylem2 26800 For general ` N ` and ` K ...
birthdaylem3 26801 For general ` N ` and ` K ...
birthday 26802 The Birthday Problem. The...
dmarea 26805 The domain of the area fun...
areambl 26806 The fibers of a measurable...
areass 26807 A measurable region is a s...
dfarea 26808 Rewrite ~ df-area self-ref...
areaf 26809 Area measurement is a func...
areacl 26810 The area of a measurable r...
areage0 26811 The area of a measurable r...
areaval 26812 The area of a measurable r...
rlimcnp 26813 Relate a limit of a real-v...
rlimcnp2 26814 Relate a limit of a real-v...
rlimcnp3 26815 Relate a limit of a real-v...
xrlimcnp 26816 Relate a limit of a real-v...
efrlim 26817 The limit of the sequence ...
efrlimOLD 26818 Obsolete version of ~ efrl...
dfef2 26819 The limit of the sequence ...
cxplim 26820 A power to a negative expo...
sqrtlim 26821 The inverse square root fu...
rlimcxp 26822 Any power to a positive ex...
o1cxp 26823 An eventually bounded func...
cxp2limlem 26824 A linear factor grows slow...
cxp2lim 26825 Any power grows slower tha...
cxploglim 26826 The logarithm grows slower...
cxploglim2 26827 Every power of the logarit...
divsqrtsumlem 26828 Lemma for ~ divsqrsum and ...
divsqrsumf 26829 The function ` F ` used in...
divsqrsum 26830 The sum ` sum_ n <_ x ( 1 ...
divsqrtsum2 26831 A bound on the distance of...
divsqrtsumo1 26832 The sum ` sum_ n <_ x ( 1 ...
cvxcl 26833 Closure of a 0-1 linear co...
scvxcvx 26834 A strictly convex function...
jensenlem1 26835 Lemma for ~ jensen . (Con...
jensenlem2 26836 Lemma for ~ jensen . (Con...
jensen 26837 Jensen's inequality, a fin...
amgmlem 26838 Lemma for ~ amgm . (Contr...
amgm 26839 Inequality of arithmetic a...
logdifbnd 26842 Bound on the difference of...
logdiflbnd 26843 Lower bound on the differe...
emcllem1 26844 Lemma for ~ emcl . The se...
emcllem2 26845 Lemma for ~ emcl . ` F ` i...
emcllem3 26846 Lemma for ~ emcl . The fu...
emcllem4 26847 Lemma for ~ emcl . The di...
emcllem5 26848 Lemma for ~ emcl . The pa...
emcllem6 26849 Lemma for ~ emcl . By the...
emcllem7 26850 Lemma for ~ emcl and ~ har...
emcl 26851 Closure and bounds for the...
harmonicbnd 26852 A bound on the harmonic se...
harmonicbnd2 26853 A bound on the harmonic se...
emre 26854 The Euler-Mascheroni const...
emgt0 26855 The Euler-Mascheroni const...
harmonicbnd3 26856 A bound on the harmonic se...
harmoniclbnd 26857 A bound on the harmonic se...
harmonicubnd 26858 A bound on the harmonic se...
harmonicbnd4 26859 The asymptotic behavior of...
fsumharmonic 26860 Bound a finite sum based o...
zetacvg 26863 The zeta series is converg...
eldmgm 26870 Elementhood in the set of ...
dmgmaddn0 26871 If ` A ` is not a nonposit...
dmlogdmgm 26872 If ` A ` is in the continu...
rpdmgm 26873 A positive real number is ...
dmgmn0 26874 If ` A ` is not a nonposit...
dmgmaddnn0 26875 If ` A ` is not a nonposit...
dmgmdivn0 26876 Lemma for ~ lgamf . (Cont...
lgamgulmlem1 26877 Lemma for ~ lgamgulm . (C...
lgamgulmlem2 26878 Lemma for ~ lgamgulm . (C...
lgamgulmlem3 26879 Lemma for ~ lgamgulm . (C...
lgamgulmlem4 26880 Lemma for ~ lgamgulm . (C...
lgamgulmlem5 26881 Lemma for ~ lgamgulm . (C...
lgamgulmlem6 26882 The series ` G ` is unifor...
lgamgulm 26883 The series ` G ` is unifor...
lgamgulm2 26884 Rewrite the limit of the s...
lgambdd 26885 The log-Gamma function is ...
lgamucov 26886 The ` U ` regions used in ...
lgamucov2 26887 The ` U ` regions used in ...
lgamcvglem 26888 Lemma for ~ lgamf and ~ lg...
lgamcl 26889 The log-Gamma function is ...
lgamf 26890 The log-Gamma function is ...
gamf 26891 The Gamma function is a co...
gamcl 26892 The exponential of the log...
eflgam 26893 The exponential of the log...
gamne0 26894 The Gamma function is neve...
igamval 26895 Value of the inverse Gamma...
igamz 26896 Value of the inverse Gamma...
igamgam 26897 Value of the inverse Gamma...
igamlgam 26898 Value of the inverse Gamma...
igamf 26899 Closure of the inverse Gam...
igamcl 26900 Closure of the inverse Gam...
gamigam 26901 The Gamma function is the ...
lgamcvg 26902 The series ` G ` converges...
lgamcvg2 26903 The series ` G ` converges...
gamcvg 26904 The pointwise exponential ...
lgamp1 26905 The functional equation of...
gamp1 26906 The functional equation of...
gamcvg2lem 26907 Lemma for ~ gamcvg2 . (Co...
gamcvg2 26908 An infinite product expres...
regamcl 26909 The Gamma function is real...
relgamcl 26910 The log-Gamma function is ...
rpgamcl 26911 The log-Gamma function is ...
lgam1 26912 The log-Gamma function at ...
gam1 26913 The log-Gamma function at ...
facgam 26914 The Gamma function general...
gamfac 26915 The Gamma function general...
wilthlem1 26916 The only elements that are...
wilthlem2 26917 Lemma for ~ wilth : induct...
wilthlem3 26918 Lemma for ~ wilth . Here ...
wilth 26919 Wilson's theorem. A numbe...
wilthimp 26920 The forward implication of...
ftalem1 26921 Lemma for ~ fta : "growth...
ftalem2 26922 Lemma for ~ fta . There e...
ftalem3 26923 Lemma for ~ fta . There e...
ftalem4 26924 Lemma for ~ fta : Closure...
ftalem5 26925 Lemma for ~ fta : Main pr...
ftalem6 26926 Lemma for ~ fta : Dischar...
ftalem7 26927 Lemma for ~ fta . Shift t...
fta 26928 The Fundamental Theorem of...
basellem1 26929 Lemma for ~ basel . Closu...
basellem2 26930 Lemma for ~ basel . Show ...
basellem3 26931 Lemma for ~ basel . Using...
basellem4 26932 Lemma for ~ basel . By ~ ...
basellem5 26933 Lemma for ~ basel . Using...
basellem6 26934 Lemma for ~ basel . The f...
basellem7 26935 Lemma for ~ basel . The f...
basellem8 26936 Lemma for ~ basel . The f...
basellem9 26937 Lemma for ~ basel . Since...
basel 26938 The sum of the inverse squ...
efnnfsumcl 26951 Finite sum closure in the ...
ppisval 26952 The set of primes less tha...
ppisval2 26953 The set of primes less tha...
ppifi 26954 The set of primes less tha...
prmdvdsfi 26955 The set of prime divisors ...
chtf 26956 Domain and codoamin of the...
chtcl 26957 Real closure of the Chebys...
chtval 26958 Value of the Chebyshev fun...
efchtcl 26959 The Chebyshev function is ...
chtge0 26960 The Chebyshev function is ...
vmaval 26961 Value of the von Mangoldt ...
isppw 26962 Two ways to say that ` A `...
isppw2 26963 Two ways to say that ` A `...
vmappw 26964 Value of the von Mangoldt ...
vmaprm 26965 Value of the von Mangoldt ...
vmacl 26966 Closure for the von Mangol...
vmaf 26967 Functionality of the von M...
efvmacl 26968 The von Mangoldt is closed...
vmage0 26969 The von Mangoldt function ...
chpval 26970 Value of the second Chebys...
chpf 26971 Functionality of the secon...
chpcl 26972 Closure for the second Che...
efchpcl 26973 The second Chebyshev funct...
chpge0 26974 The second Chebyshev funct...
ppival 26975 Value of the prime-countin...
ppival2 26976 Value of the prime-countin...
ppival2g 26977 Value of the prime-countin...
ppif 26978 Domain and codomain of the...
ppicl 26979 Real closure of the prime-...
muval 26980 The value of the Möbi...
muval1 26981 The value of the Möbi...
muval2 26982 The value of the Möbi...
isnsqf 26983 Two ways to say that a num...
issqf 26984 Two ways to say that a num...
sqfpc 26985 The prime count of a squar...
dvdssqf 26986 A divisor of a squarefree ...
sqf11 26987 A squarefree number is com...
muf 26988 The Möbius function i...
mucl 26989 Closure of the Möbius...
sgmval 26990 The value of the divisor f...
sgmval2 26991 The value of the divisor f...
0sgm 26992 The value of the sum-of-di...
sgmf 26993 The divisor function is a ...
sgmcl 26994 Closure of the divisor fun...
sgmnncl 26995 Closure of the divisor fun...
mule1 26996 The Möbius function t...
chtfl 26997 The Chebyshev function doe...
chpfl 26998 The second Chebyshev funct...
ppiprm 26999 The prime-counting functio...
ppinprm 27000 The prime-counting functio...
chtprm 27001 The Chebyshev function at ...
chtnprm 27002 The Chebyshev function at ...
chpp1 27003 The second Chebyshev funct...
chtwordi 27004 The Chebyshev function is ...
chpwordi 27005 The second Chebyshev funct...
chtdif 27006 The difference of the Cheb...
efchtdvds 27007 The exponentiated Chebyshe...
ppifl 27008 The prime-counting functio...
ppip1le 27009 The prime-counting functio...
ppiwordi 27010 The prime-counting functio...
ppidif 27011 The difference of the prim...
ppi1 27012 The prime-counting functio...
cht1 27013 The Chebyshev function at ...
vma1 27014 The von Mangoldt function ...
chp1 27015 The second Chebyshev funct...
ppi1i 27016 Inference form of ~ ppiprm...
ppi2i 27017 Inference form of ~ ppinpr...
ppi2 27018 The prime-counting functio...
ppi3 27019 The prime-counting functio...
cht2 27020 The Chebyshev function at ...
cht3 27021 The Chebyshev function at ...
ppinncl 27022 Closure of the prime-count...
chtrpcl 27023 Closure of the Chebyshev f...
ppieq0 27024 The prime-counting functio...
ppiltx 27025 The prime-counting functio...
prmorcht 27026 Relate the primorial (prod...
mumullem1 27027 Lemma for ~ mumul . A mul...
mumullem2 27028 Lemma for ~ mumul . The p...
mumul 27029 The Möbius function i...
sqff1o 27030 There is a bijection from ...
fsumdvdsdiaglem 27031 A "diagonal commutation" o...
fsumdvdsdiag 27032 A "diagonal commutation" o...
fsumdvdscom 27033 A double commutation of di...
dvdsppwf1o 27034 A bijection from the divis...
dvdsflf1o 27035 A bijection from the numbe...
dvdsflsumcom 27036 A sum commutation from ` s...
fsumfldivdiaglem 27037 Lemma for ~ fsumfldivdiag ...
fsumfldivdiag 27038 The right-hand side of ~ d...
musum 27039 The sum of the Möbius...
musumsum 27040 Evaluate a collapsing sum ...
muinv 27041 The Möbius inversion ...
mpodvdsmulf1o 27042 If ` M ` and ` N ` are two...
fsumdvdsmul 27043 Product of two divisor sum...
dvdsmulf1o 27044 If ` M ` and ` N ` are two...
fsumdvdsmulOLD 27045 Obsolete version of ~ fsum...
sgmppw 27046 The value of the divisor f...
0sgmppw 27047 A prime power ` P ^ K ` ha...
1sgmprm 27048 The sum of divisors for a ...
1sgm2ppw 27049 The sum of the divisors of...
sgmmul 27050 The divisor function for f...
ppiublem1 27051 Lemma for ~ ppiub . (Cont...
ppiublem2 27052 A prime greater than ` 3 `...
ppiub 27053 An upper bound on the prim...
vmalelog 27054 The von Mangoldt function ...
chtlepsi 27055 The first Chebyshev functi...
chprpcl 27056 Closure of the second Cheb...
chpeq0 27057 The second Chebyshev funct...
chteq0 27058 The first Chebyshev functi...
chtleppi 27059 Upper bound on the ` theta...
chtublem 27060 Lemma for ~ chtub . (Cont...
chtub 27061 An upper bound on the Cheb...
fsumvma 27062 Rewrite a sum over the von...
fsumvma2 27063 Apply ~ fsumvma for the co...
pclogsum 27064 The logarithmic analogue o...
vmasum 27065 The sum of the von Mangold...
logfac2 27066 Another expression for the...
chpval2 27067 Express the second Chebysh...
chpchtsum 27068 The second Chebyshev funct...
chpub 27069 An upper bound on the seco...
logfacubnd 27070 A simple upper bound on th...
logfaclbnd 27071 A lower bound on the logar...
logfacbnd3 27072 Show the stronger statemen...
logfacrlim 27073 Combine the estimates ~ lo...
logexprlim 27074 The sum ` sum_ n <_ x , lo...
logfacrlim2 27075 Write out ~ logfacrlim as ...
mersenne 27076 A Mersenne prime is a prim...
perfect1 27077 Euclid's contribution to t...
perfectlem1 27078 Lemma for ~ perfect . (Co...
perfectlem2 27079 Lemma for ~ perfect . (Co...
perfect 27080 The Euclid-Euler theorem, ...
dchrval 27083 Value of the group of Diri...
dchrbas 27084 Base set of the group of D...
dchrelbas 27085 A Dirichlet character is a...
dchrelbas2 27086 A Dirichlet character is a...
dchrelbas3 27087 A Dirichlet character is a...
dchrelbasd 27088 A Dirichlet character is a...
dchrrcl 27089 Reverse closure for a Diri...
dchrmhm 27090 A Dirichlet character is a...
dchrf 27091 A Dirichlet character is a...
dchrelbas4 27092 A Dirichlet character is a...
dchrzrh1 27093 Value of a Dirichlet chara...
dchrzrhcl 27094 A Dirichlet character take...
dchrzrhmul 27095 A Dirichlet character is c...
dchrplusg 27096 Group operation on the gro...
dchrmul 27097 Group operation on the gro...
dchrmulcl 27098 Closure of the group opera...
dchrn0 27099 A Dirichlet character is n...
dchr1cl 27100 Closure of the principal D...
dchrmullid 27101 Left identity for the prin...
dchrinvcl 27102 Closure of the group inver...
dchrabl 27103 The set of Dirichlet chara...
dchrfi 27104 The group of Dirichlet cha...
dchrghm 27105 A Dirichlet character rest...
dchr1 27106 Value of the principal Dir...
dchreq 27107 A Dirichlet character is d...
dchrresb 27108 A Dirichlet character is d...
dchrabs 27109 A Dirichlet character take...
dchrinv 27110 The inverse of a Dirichlet...
dchrabs2 27111 A Dirichlet character take...
dchr1re 27112 The principal Dirichlet ch...
dchrptlem1 27113 Lemma for ~ dchrpt . (Con...
dchrptlem2 27114 Lemma for ~ dchrpt . (Con...
dchrptlem3 27115 Lemma for ~ dchrpt . (Con...
dchrpt 27116 For any element other than...
dchrsum2 27117 An orthogonality relation ...
dchrsum 27118 An orthogonality relation ...
sumdchr2 27119 Lemma for ~ sumdchr . (Co...
dchrhash 27120 There are exactly ` phi ( ...
sumdchr 27121 An orthogonality relation ...
dchr2sum 27122 An orthogonality relation ...
sum2dchr 27123 An orthogonality relation ...
bcctr 27124 Value of the central binom...
pcbcctr 27125 Prime count of a central b...
bcmono 27126 The binomial coefficient i...
bcmax 27127 The binomial coefficient t...
bcp1ctr 27128 Ratio of two central binom...
bclbnd 27129 A bound on the binomial co...
efexple 27130 Convert a bound on a power...
bpos1lem 27131 Lemma for ~ bpos1 . (Cont...
bpos1 27132 Bertrand's postulate, chec...
bposlem1 27133 An upper bound on the prim...
bposlem2 27134 There are no odd primes in...
bposlem3 27135 Lemma for ~ bpos . Since ...
bposlem4 27136 Lemma for ~ bpos . (Contr...
bposlem5 27137 Lemma for ~ bpos . Bound ...
bposlem6 27138 Lemma for ~ bpos . By usi...
bposlem7 27139 Lemma for ~ bpos . The fu...
bposlem8 27140 Lemma for ~ bpos . Evalua...
bposlem9 27141 Lemma for ~ bpos . Derive...
bpos 27142 Bertrand's postulate: ther...
zabsle1 27145 ` { -u 1 , 0 , 1 } ` is th...
lgslem1 27146 When ` a ` is coprime to t...
lgslem2 27147 The set ` Z ` of all integ...
lgslem3 27148 The set ` Z ` of all integ...
lgslem4 27149 Lemma for ~ lgsfcl2 . (Co...
lgsval 27150 Value of the Legendre symb...
lgsfval 27151 Value of the function ` F ...
lgsfcl2 27152 The function ` F ` is clos...
lgscllem 27153 The Legendre symbol is an ...
lgsfcl 27154 Closure of the function ` ...
lgsfle1 27155 The function ` F ` has mag...
lgsval2lem 27156 Lemma for ~ lgsval2 . (Co...
lgsval4lem 27157 Lemma for ~ lgsval4 . (Co...
lgscl2 27158 The Legendre symbol is an ...
lgs0 27159 The Legendre symbol when t...
lgscl 27160 The Legendre symbol is an ...
lgsle1 27161 The Legendre symbol has ab...
lgsval2 27162 The Legendre symbol at a p...
lgs2 27163 The Legendre symbol at ` 2...
lgsval3 27164 The Legendre symbol at an ...
lgsvalmod 27165 The Legendre symbol is equ...
lgsval4 27166 Restate ~ lgsval for nonze...
lgsfcl3 27167 Closure of the function ` ...
lgsval4a 27168 Same as ~ lgsval4 for posi...
lgscl1 27169 The value of the Legendre ...
lgsneg 27170 The Legendre symbol is eit...
lgsneg1 27171 The Legendre symbol for no...
lgsmod 27172 The Legendre (Jacobi) symb...
lgsdilem 27173 Lemma for ~ lgsdi and ~ lg...
lgsdir2lem1 27174 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem2 27175 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem3 27176 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem4 27177 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem5 27178 Lemma for ~ lgsdir2 . (Co...
lgsdir2 27179 The Legendre symbol is com...
lgsdirprm 27180 The Legendre symbol is com...
lgsdir 27181 The Legendre symbol is com...
lgsdilem2 27182 Lemma for ~ lgsdi . (Cont...
lgsdi 27183 The Legendre symbol is com...
lgsne0 27184 The Legendre symbol is non...
lgsabs1 27185 The Legendre symbol is non...
lgssq 27186 The Legendre symbol at a s...
lgssq2 27187 The Legendre symbol at a s...
lgsprme0 27188 The Legendre symbol at any...
1lgs 27189 The Legendre symbol at ` 1...
lgs1 27190 The Legendre symbol at ` 1...
lgsmodeq 27191 The Legendre (Jacobi) symb...
lgsmulsqcoprm 27192 The Legendre (Jacobi) symb...
lgsdirnn0 27193 Variation on ~ lgsdir vali...
lgsdinn0 27194 Variation on ~ lgsdi valid...
lgsqrlem1 27195 Lemma for ~ lgsqr . (Cont...
lgsqrlem2 27196 Lemma for ~ lgsqr . (Cont...
lgsqrlem3 27197 Lemma for ~ lgsqr . (Cont...
lgsqrlem4 27198 Lemma for ~ lgsqr . (Cont...
lgsqrlem5 27199 Lemma for ~ lgsqr . (Cont...
lgsqr 27200 The Legendre symbol for od...
lgsqrmod 27201 If the Legendre symbol of ...
lgsqrmodndvds 27202 If the Legendre symbol of ...
lgsdchrval 27203 The Legendre symbol functi...
lgsdchr 27204 The Legendre symbol functi...
gausslemma2dlem0a 27205 Auxiliary lemma 1 for ~ ga...
gausslemma2dlem0b 27206 Auxiliary lemma 2 for ~ ga...
gausslemma2dlem0c 27207 Auxiliary lemma 3 for ~ ga...
gausslemma2dlem0d 27208 Auxiliary lemma 4 for ~ ga...
gausslemma2dlem0e 27209 Auxiliary lemma 5 for ~ ga...
gausslemma2dlem0f 27210 Auxiliary lemma 6 for ~ ga...
gausslemma2dlem0g 27211 Auxiliary lemma 7 for ~ ga...
gausslemma2dlem0h 27212 Auxiliary lemma 8 for ~ ga...
gausslemma2dlem0i 27213 Auxiliary lemma 9 for ~ ga...
gausslemma2dlem1a 27214 Lemma for ~ gausslemma2dle...
gausslemma2dlem1 27215 Lemma 1 for ~ gausslemma2d...
gausslemma2dlem2 27216 Lemma 2 for ~ gausslemma2d...
gausslemma2dlem3 27217 Lemma 3 for ~ gausslemma2d...
gausslemma2dlem4 27218 Lemma 4 for ~ gausslemma2d...
gausslemma2dlem5a 27219 Lemma for ~ gausslemma2dle...
gausslemma2dlem5 27220 Lemma 5 for ~ gausslemma2d...
gausslemma2dlem6 27221 Lemma 6 for ~ gausslemma2d...
gausslemma2dlem7 27222 Lemma 7 for ~ gausslemma2d...
gausslemma2d 27223 Gauss' Lemma (see also the...
lgseisenlem1 27224 Lemma for ~ lgseisen . If...
lgseisenlem2 27225 Lemma for ~ lgseisen . Th...
lgseisenlem3 27226 Lemma for ~ lgseisen . (C...
lgseisenlem4 27227 Lemma for ~ lgseisen . Th...
lgseisen 27228 Eisenstein's lemma, an exp...
lgsquadlem1 27229 Lemma for ~ lgsquad . Cou...
lgsquadlem2 27230 Lemma for ~ lgsquad . Cou...
lgsquadlem3 27231 Lemma for ~ lgsquad . (Co...
lgsquad 27232 The Law of Quadratic Recip...
lgsquad2lem1 27233 Lemma for ~ lgsquad2 . (C...
lgsquad2lem2 27234 Lemma for ~ lgsquad2 . (C...
lgsquad2 27235 Extend ~ lgsquad to coprim...
lgsquad3 27236 Extend ~ lgsquad2 to integ...
m1lgs 27237 The first supplement to th...
2lgslem1a1 27238 Lemma 1 for ~ 2lgslem1a . ...
2lgslem1a2 27239 Lemma 2 for ~ 2lgslem1a . ...
2lgslem1a 27240 Lemma 1 for ~ 2lgslem1 . ...
2lgslem1b 27241 Lemma 2 for ~ 2lgslem1 . ...
2lgslem1c 27242 Lemma 3 for ~ 2lgslem1 . ...
2lgslem1 27243 Lemma 1 for ~ 2lgs . (Con...
2lgslem2 27244 Lemma 2 for ~ 2lgs . (Con...
2lgslem3a 27245 Lemma for ~ 2lgslem3a1 . ...
2lgslem3b 27246 Lemma for ~ 2lgslem3b1 . ...
2lgslem3c 27247 Lemma for ~ 2lgslem3c1 . ...
2lgslem3d 27248 Lemma for ~ 2lgslem3d1 . ...
2lgslem3a1 27249 Lemma 1 for ~ 2lgslem3 . ...
2lgslem3b1 27250 Lemma 2 for ~ 2lgslem3 . ...
2lgslem3c1 27251 Lemma 3 for ~ 2lgslem3 . ...
2lgslem3d1 27252 Lemma 4 for ~ 2lgslem3 . ...
2lgslem3 27253 Lemma 3 for ~ 2lgs . (Con...
2lgs2 27254 The Legendre symbol for ` ...
2lgslem4 27255 Lemma 4 for ~ 2lgs : speci...
2lgs 27256 The second supplement to t...
2lgsoddprmlem1 27257 Lemma 1 for ~ 2lgsoddprm ....
2lgsoddprmlem2 27258 Lemma 2 for ~ 2lgsoddprm ....
2lgsoddprmlem3a 27259 Lemma 1 for ~ 2lgsoddprmle...
2lgsoddprmlem3b 27260 Lemma 2 for ~ 2lgsoddprmle...
2lgsoddprmlem3c 27261 Lemma 3 for ~ 2lgsoddprmle...
2lgsoddprmlem3d 27262 Lemma 4 for ~ 2lgsoddprmle...
2lgsoddprmlem3 27263 Lemma 3 for ~ 2lgsoddprm ....
2lgsoddprmlem4 27264 Lemma 4 for ~ 2lgsoddprm ....
2lgsoddprm 27265 The second supplement to t...
2sqlem1 27266 Lemma for ~ 2sq . (Contri...
2sqlem2 27267 Lemma for ~ 2sq . (Contri...
mul2sq 27268 Fibonacci's identity (actu...
2sqlem3 27269 Lemma for ~ 2sqlem5 . (Co...
2sqlem4 27270 Lemma for ~ 2sqlem5 . (Co...
2sqlem5 27271 Lemma for ~ 2sq . If a nu...
2sqlem6 27272 Lemma for ~ 2sq . If a nu...
2sqlem7 27273 Lemma for ~ 2sq . (Contri...
2sqlem8a 27274 Lemma for ~ 2sqlem8 . (Co...
2sqlem8 27275 Lemma for ~ 2sq . (Contri...
2sqlem9 27276 Lemma for ~ 2sq . (Contri...
2sqlem10 27277 Lemma for ~ 2sq . Every f...
2sqlem11 27278 Lemma for ~ 2sq . (Contri...
2sq 27279 All primes of the form ` 4...
2sqblem 27280 Lemma for ~ 2sqb . (Contr...
2sqb 27281 The converse to ~ 2sq . (...
2sq2 27282 ` 2 ` is the sum of square...
2sqn0 27283 If the sum of two squares ...
2sqcoprm 27284 If the sum of two squares ...
2sqmod 27285 Given two decompositions o...
2sqmo 27286 There exists at most one d...
2sqnn0 27287 All primes of the form ` 4...
2sqnn 27288 All primes of the form ` 4...
addsq2reu 27289 For each complex number ` ...
addsqn2reu 27290 For each complex number ` ...
addsqrexnreu 27291 For each complex number, t...
addsqnreup 27292 There is no unique decompo...
addsq2nreurex 27293 For each complex number ` ...
addsqn2reurex2 27294 For each complex number ` ...
2sqreulem1 27295 Lemma 1 for ~ 2sqreu . (C...
2sqreultlem 27296 Lemma for ~ 2sqreult . (C...
2sqreultblem 27297 Lemma for ~ 2sqreultb . (...
2sqreunnlem1 27298 Lemma 1 for ~ 2sqreunn . ...
2sqreunnltlem 27299 Lemma for ~ 2sqreunnlt . ...
2sqreunnltblem 27300 Lemma for ~ 2sqreunnltb . ...
2sqreulem2 27301 Lemma 2 for ~ 2sqreu etc. ...
2sqreulem3 27302 Lemma 3 for ~ 2sqreu etc. ...
2sqreulem4 27303 Lemma 4 for ~ 2sqreu et. ...
2sqreunnlem2 27304 Lemma 2 for ~ 2sqreunn . ...
2sqreu 27305 There exists a unique deco...
2sqreunn 27306 There exists a unique deco...
2sqreult 27307 There exists a unique deco...
2sqreultb 27308 There exists a unique deco...
2sqreunnlt 27309 There exists a unique deco...
2sqreunnltb 27310 There exists a unique deco...
2sqreuop 27311 There exists a unique deco...
2sqreuopnn 27312 There exists a unique deco...
2sqreuoplt 27313 There exists a unique deco...
2sqreuopltb 27314 There exists a unique deco...
2sqreuopnnlt 27315 There exists a unique deco...
2sqreuopnnltb 27316 There exists a unique deco...
2sqreuopb 27317 There exists a unique deco...
chebbnd1lem1 27318 Lemma for ~ chebbnd1 : sho...
chebbnd1lem2 27319 Lemma for ~ chebbnd1 : Sh...
chebbnd1lem3 27320 Lemma for ~ chebbnd1 : get...
chebbnd1 27321 The Chebyshev bound: The ...
chtppilimlem1 27322 Lemma for ~ chtppilim . (...
chtppilimlem2 27323 Lemma for ~ chtppilim . (...
chtppilim 27324 The ` theta ` function is ...
chto1ub 27325 The ` theta ` function is ...
chebbnd2 27326 The Chebyshev bound, part ...
chto1lb 27327 The ` theta ` function is ...
chpchtlim 27328 The ` psi ` and ` theta ` ...
chpo1ub 27329 The ` psi ` function is up...
chpo1ubb 27330 The ` psi ` function is up...
vmadivsum 27331 The sum of the von Mangold...
vmadivsumb 27332 Give a total bound on the ...
rplogsumlem1 27333 Lemma for ~ rplogsum . (C...
rplogsumlem2 27334 Lemma for ~ rplogsum . Eq...
dchrisum0lem1a 27335 Lemma for ~ dchrisum0lem1 ...
rpvmasumlem 27336 Lemma for ~ rpvmasum . Ca...
dchrisumlema 27337 Lemma for ~ dchrisum . Le...
dchrisumlem1 27338 Lemma for ~ dchrisum . Le...
dchrisumlem2 27339 Lemma for ~ dchrisum . Le...
dchrisumlem3 27340 Lemma for ~ dchrisum . Le...
dchrisum 27341 If ` n e. [ M , +oo ) |-> ...
dchrmusumlema 27342 Lemma for ~ dchrmusum and ...
dchrmusum2 27343 The sum of the Möbius...
dchrvmasumlem1 27344 An alternative expression ...
dchrvmasum2lem 27345 Give an expression for ` l...
dchrvmasum2if 27346 Combine the results of ~ d...
dchrvmasumlem2 27347 Lemma for ~ dchrvmasum . ...
dchrvmasumlem3 27348 Lemma for ~ dchrvmasum . ...
dchrvmasumlema 27349 Lemma for ~ dchrvmasum and...
dchrvmasumiflem1 27350 Lemma for ~ dchrvmasumif ....
dchrvmasumiflem2 27351 Lemma for ~ dchrvmasum . ...
dchrvmasumif 27352 An asymptotic approximatio...
dchrvmaeq0 27353 The set ` W ` is the colle...
dchrisum0fval 27354 Value of the function ` F ...
dchrisum0fmul 27355 The function ` F ` , the d...
dchrisum0ff 27356 The function ` F ` is a re...
dchrisum0flblem1 27357 Lemma for ~ dchrisum0flb ....
dchrisum0flblem2 27358 Lemma for ~ dchrisum0flb ....
dchrisum0flb 27359 The divisor sum of a real ...
dchrisum0fno1 27360 The sum ` sum_ k <_ x , F ...
rpvmasum2 27361 A partial result along the...
dchrisum0re 27362 Suppose ` X ` is a non-pri...
dchrisum0lema 27363 Lemma for ~ dchrisum0 . A...
dchrisum0lem1b 27364 Lemma for ~ dchrisum0lem1 ...
dchrisum0lem1 27365 Lemma for ~ dchrisum0 . (...
dchrisum0lem2a 27366 Lemma for ~ dchrisum0 . (...
dchrisum0lem2 27367 Lemma for ~ dchrisum0 . (...
dchrisum0lem3 27368 Lemma for ~ dchrisum0 . (...
dchrisum0 27369 The sum ` sum_ n e. NN , X...
dchrisumn0 27370 The sum ` sum_ n e. NN , X...
dchrmusumlem 27371 The sum of the Möbius...
dchrvmasumlem 27372 The sum of the Möbius...
dchrmusum 27373 The sum of the Möbius...
dchrvmasum 27374 The sum of the von Mangold...
rpvmasum 27375 The sum of the von Mangold...
rplogsum 27376 The sum of ` log p / p ` o...
dirith2 27377 Dirichlet's theorem: there...
dirith 27378 Dirichlet's theorem: there...
mudivsum 27379 Asymptotic formula for ` s...
mulogsumlem 27380 Lemma for ~ mulogsum . (C...
mulogsum 27381 Asymptotic formula for ...
logdivsum 27382 Asymptotic analysis of ...
mulog2sumlem1 27383 Asymptotic formula for ...
mulog2sumlem2 27384 Lemma for ~ mulog2sum . (...
mulog2sumlem3 27385 Lemma for ~ mulog2sum . (...
mulog2sum 27386 Asymptotic formula for ...
vmalogdivsum2 27387 The sum ` sum_ n <_ x , La...
vmalogdivsum 27388 The sum ` sum_ n <_ x , La...
2vmadivsumlem 27389 Lemma for ~ 2vmadivsum . ...
2vmadivsum 27390 The sum ` sum_ m n <_ x , ...
logsqvma 27391 A formula for ` log ^ 2 ( ...
logsqvma2 27392 The Möbius inverse of...
log2sumbnd 27393 Bound on the difference be...
selberglem1 27394 Lemma for ~ selberg . Est...
selberglem2 27395 Lemma for ~ selberg . (Co...
selberglem3 27396 Lemma for ~ selberg . Est...
selberg 27397 Selberg's symmetry formula...
selbergb 27398 Convert eventual boundedne...
selberg2lem 27399 Lemma for ~ selberg2 . Eq...
selberg2 27400 Selberg's symmetry formula...
selberg2b 27401 Convert eventual boundedne...
chpdifbndlem1 27402 Lemma for ~ chpdifbnd . (...
chpdifbndlem2 27403 Lemma for ~ chpdifbnd . (...
chpdifbnd 27404 A bound on the difference ...
logdivbnd 27405 A bound on a sum of logs, ...
selberg3lem1 27406 Introduce a log weighting ...
selberg3lem2 27407 Lemma for ~ selberg3 . Eq...
selberg3 27408 Introduce a log weighting ...
selberg4lem1 27409 Lemma for ~ selberg4 . Eq...
selberg4 27410 The Selberg symmetry formu...
pntrval 27411 Define the residual of the...
pntrf 27412 Functionality of the resid...
pntrmax 27413 There is a bound on the re...
pntrsumo1 27414 A bound on a sum over ` R ...
pntrsumbnd 27415 A bound on a sum over ` R ...
pntrsumbnd2 27416 A bound on a sum over ` R ...
selbergr 27417 Selberg's symmetry formula...
selberg3r 27418 Selberg's symmetry formula...
selberg4r 27419 Selberg's symmetry formula...
selberg34r 27420 The sum of ~ selberg3r and...
pntsval 27421 Define the "Selberg functi...
pntsf 27422 Functionality of the Selbe...
selbergs 27423 Selberg's symmetry formula...
selbergsb 27424 Selberg's symmetry formula...
pntsval2 27425 The Selberg function can b...
pntrlog2bndlem1 27426 The sum of ~ selberg3r and...
pntrlog2bndlem2 27427 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem3 27428 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem4 27429 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem5 27430 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem6a 27431 Lemma for ~ pntrlog2bndlem...
pntrlog2bndlem6 27432 Lemma for ~ pntrlog2bnd . ...
pntrlog2bnd 27433 A bound on ` R ( x ) log ^...
pntpbnd1a 27434 Lemma for ~ pntpbnd . (Co...
pntpbnd1 27435 Lemma for ~ pntpbnd . (Co...
pntpbnd2 27436 Lemma for ~ pntpbnd . (Co...
pntpbnd 27437 Lemma for ~ pnt . Establi...
pntibndlem1 27438 Lemma for ~ pntibnd . (Co...
pntibndlem2a 27439 Lemma for ~ pntibndlem2 . ...
pntibndlem2 27440 Lemma for ~ pntibnd . The...
pntibndlem3 27441 Lemma for ~ pntibnd . Pac...
pntibnd 27442 Lemma for ~ pnt . Establi...
pntlemd 27443 Lemma for ~ pnt . Closure...
pntlemc 27444 Lemma for ~ pnt . Closure...
pntlema 27445 Lemma for ~ pnt . Closure...
pntlemb 27446 Lemma for ~ pnt . Unpack ...
pntlemg 27447 Lemma for ~ pnt . Closure...
pntlemh 27448 Lemma for ~ pnt . Bounds ...
pntlemn 27449 Lemma for ~ pnt . The "na...
pntlemq 27450 Lemma for ~ pntlemj . (Co...
pntlemr 27451 Lemma for ~ pntlemj . (Co...
pntlemj 27452 Lemma for ~ pnt . The ind...
pntlemi 27453 Lemma for ~ pnt . Elimina...
pntlemf 27454 Lemma for ~ pnt . Add up ...
pntlemk 27455 Lemma for ~ pnt . Evaluat...
pntlemo 27456 Lemma for ~ pnt . Combine...
pntleme 27457 Lemma for ~ pnt . Package...
pntlem3 27458 Lemma for ~ pnt . Equatio...
pntlemp 27459 Lemma for ~ pnt . Wrappin...
pntleml 27460 Lemma for ~ pnt . Equatio...
pnt3 27461 The Prime Number Theorem, ...
pnt2 27462 The Prime Number Theorem, ...
pnt 27463 The Prime Number Theorem: ...
abvcxp 27464 Raising an absolute value ...
padicfval 27465 Value of the p-adic absolu...
padicval 27466 Value of the p-adic absolu...
ostth2lem1 27467 Lemma for ~ ostth2 , altho...
qrngbas 27468 The base set of the field ...
qdrng 27469 The rationals form a divis...
qrng0 27470 The zero element of the fi...
qrng1 27471 The unity element of the f...
qrngneg 27472 The additive inverse in th...
qrngdiv 27473 The division operation in ...
qabvle 27474 By using induction on ` N ...
qabvexp 27475 Induct the product rule ~ ...
ostthlem1 27476 Lemma for ~ ostth . If tw...
ostthlem2 27477 Lemma for ~ ostth . Refin...
qabsabv 27478 The regular absolute value...
padicabv 27479 The p-adic absolute value ...
padicabvf 27480 The p-adic absolute value ...
padicabvcxp 27481 All positive powers of the...
ostth1 27482 - Lemma for ~ ostth : triv...
ostth2lem2 27483 Lemma for ~ ostth2 . (Con...
ostth2lem3 27484 Lemma for ~ ostth2 . (Con...
ostth2lem4 27485 Lemma for ~ ostth2 . (Con...
ostth2 27486 - Lemma for ~ ostth : regu...
ostth3 27487 - Lemma for ~ ostth : p-ad...
ostth 27488 Ostrowski's theorem, which...
elno 27495 Membership in the surreals...
sltval 27496 The value of the surreal l...
bdayval 27497 The value of the birthday ...
nofun 27498 A surreal is a function. ...
nodmon 27499 The domain of a surreal is...
norn 27500 The range of a surreal is ...
nofnbday 27501 A surreal is a function ov...
nodmord 27502 The domain of a surreal ha...
elno2 27503 An alternative condition f...
elno3 27504 Another condition for memb...
sltval2 27505 Alternate expression for s...
nofv 27506 The function value of a su...
nosgnn0 27507 ` (/) ` is not a surreal s...
nosgnn0i 27508 If ` X ` is a surreal sign...
noreson 27509 The restriction of a surre...
sltintdifex 27510 If ` A
sltres 27511 If the restrictions of two...
noxp1o 27512 The Cartesian product of a...
noseponlem 27513 Lemma for ~ nosepon . Con...
nosepon 27514 Given two unequal surreals...
noextend 27515 Extending a surreal by one...
noextendseq 27516 Extend a surreal by a sequ...
noextenddif 27517 Calculate the place where ...
noextendlt 27518 Extending a surreal with a...
noextendgt 27519 Extending a surreal with a...
nolesgn2o 27520 Given ` A ` less-than or e...
nolesgn2ores 27521 Given ` A ` less-than or e...
nogesgn1o 27522 Given ` A ` greater than o...
nogesgn1ores 27523 Given ` A ` greater than o...
sltsolem1 27524 Lemma for ~ sltso . The "...
sltso 27525 Less-than totally orders t...
bdayfo 27526 The birthday function maps...
fvnobday 27527 The value of a surreal at ...
nosepnelem 27528 Lemma for ~ nosepne . (Co...
nosepne 27529 The value of two non-equal...
nosep1o 27530 If the value of a surreal ...
nosep2o 27531 If the value of a surreal ...
nosepdmlem 27532 Lemma for ~ nosepdm . (Co...
nosepdm 27533 The first place two surrea...
nosepeq 27534 The values of two surreals...
nosepssdm 27535 Given two non-equal surrea...
nodenselem4 27536 Lemma for ~ nodense . Sho...
nodenselem5 27537 Lemma for ~ nodense . If ...
nodenselem6 27538 The restriction of a surre...
nodenselem7 27539 Lemma for ~ nodense . ` A ...
nodenselem8 27540 Lemma for ~ nodense . Giv...
nodense 27541 Given two distinct surreal...
bdayimaon 27542 Lemma for full-eta propert...
nolt02olem 27543 Lemma for ~ nolt02o . If ...
nolt02o 27544 Given ` A ` less-than ` B ...
nogt01o 27545 Given ` A ` greater than `...
noresle 27546 Restriction law for surrea...
nomaxmo 27547 A class of surreals has at...
nominmo 27548 A class of surreals has at...
nosupprefixmo 27549 In any class of surreals, ...
noinfprefixmo 27550 In any class of surreals, ...
nosupcbv 27551 Lemma to change bound vari...
nosupno 27552 The next several theorems ...
nosupdm 27553 The domain of the surreal ...
nosupbday 27554 Birthday bounding law for ...
nosupfv 27555 The value of surreal supre...
nosupres 27556 A restriction law for surr...
nosupbnd1lem1 27557 Lemma for ~ nosupbnd1 . E...
nosupbnd1lem2 27558 Lemma for ~ nosupbnd1 . W...
nosupbnd1lem3 27559 Lemma for ~ nosupbnd1 . I...
nosupbnd1lem4 27560 Lemma for ~ nosupbnd1 . I...
nosupbnd1lem5 27561 Lemma for ~ nosupbnd1 . I...
nosupbnd1lem6 27562 Lemma for ~ nosupbnd1 . E...
nosupbnd1 27563 Bounding law from below fo...
nosupbnd2lem1 27564 Bounding law from above wh...
nosupbnd2 27565 Bounding law from above fo...
noinfcbv 27566 Change bound variables for...
noinfno 27567 The next several theorems ...
noinfdm 27568 Next, we calculate the dom...
noinfbday 27569 Birthday bounding law for ...
noinffv 27570 The value of surreal infim...
noinfres 27571 The restriction of surreal...
noinfbnd1lem1 27572 Lemma for ~ noinfbnd1 . E...
noinfbnd1lem2 27573 Lemma for ~ noinfbnd1 . W...
noinfbnd1lem3 27574 Lemma for ~ noinfbnd1 . I...
noinfbnd1lem4 27575 Lemma for ~ noinfbnd1 . I...
noinfbnd1lem5 27576 Lemma for ~ noinfbnd1 . I...
noinfbnd1lem6 27577 Lemma for ~ noinfbnd1 . E...
noinfbnd1 27578 Bounding law from above fo...
noinfbnd2lem1 27579 Bounding law from below wh...
noinfbnd2 27580 Bounding law from below fo...
nosupinfsep 27581 Given two sets of surreals...
noetasuplem1 27582 Lemma for ~ noeta . Estab...
noetasuplem2 27583 Lemma for ~ noeta . The r...
noetasuplem3 27584 Lemma for ~ noeta . ` Z ` ...
noetasuplem4 27585 Lemma for ~ noeta . When ...
noetainflem1 27586 Lemma for ~ noeta . Estab...
noetainflem2 27587 Lemma for ~ noeta . The r...
noetainflem3 27588 Lemma for ~ noeta . ` W ` ...
noetainflem4 27589 Lemma for ~ noeta . If ` ...
noetalem1 27590 Lemma for ~ noeta . Eithe...
noetalem2 27591 Lemma for ~ noeta . The f...
noeta 27592 The full-eta axiom for the...
sltirr 27595 Surreal less-than is irref...
slttr 27596 Surreal less-than is trans...
sltasym 27597 Surreal less-than is asymm...
sltlin 27598 Surreal less-than obeys tr...
slttrieq2 27599 Trichotomy law for surreal...
slttrine 27600 Trichotomy law for surreal...
slenlt 27601 Surreal less-than or equal...
sltnle 27602 Surreal less-than in terms...
sleloe 27603 Surreal less-than or equal...
sletri3 27604 Trichotomy law for surreal...
sltletr 27605 Surreal transitive law. (...
slelttr 27606 Surreal transitive law. (...
sletr 27607 Surreal transitive law. (...
slttrd 27608 Surreal less-than is trans...
sltletrd 27609 Surreal less-than is trans...
slelttrd 27610 Surreal less-than is trans...
sletrd 27611 Surreal less-than or equal...
slerflex 27612 Surreal less-than or equal...
sletric 27613 Surreal trichotomy law. (...
maxs1 27614 A surreal is less than or ...
maxs2 27615 A surreal is less than or ...
mins1 27616 The minimum of two surreal...
mins2 27617 The minimum of two surreal...
sltled 27618 Surreal less-than implies ...
sltne 27619 Surreal less-than implies ...
sltlend 27620 Surreal less-than in terms...
bdayfun 27621 The birthday function is a...
bdayfn 27622 The birthday function is a...
bdaydm 27623 The birthday function's do...
bdayrn 27624 The birthday function's ra...
bdayelon 27625 The value of the birthday ...
nocvxminlem 27626 Lemma for ~ nocvxmin . Gi...
nocvxmin 27627 Given a nonempty convex cl...
noprc 27628 The surreal numbers are a ...
noeta2 27633 A version of ~ noeta with ...
brsslt 27634 Binary relation form of th...
ssltex1 27635 The first argument of surr...
ssltex2 27636 The second argument of sur...
ssltss1 27637 The first argument of surr...
ssltss2 27638 The second argument of sur...
ssltsep 27639 The separation property of...
ssltd 27640 Deduce surreal set less-th...
ssltsn 27641 Surreal set less-than of t...
ssltsepc 27642 Two elements of separated ...
ssltsepcd 27643 Two elements of separated ...
sssslt1 27644 Relation between surreal s...
sssslt2 27645 Relation between surreal s...
nulsslt 27646 The empty set is less-than...
nulssgt 27647 The empty set is greater t...
conway 27648 Conway's Simplicity Theore...
scutval 27649 The value of the surreal c...
scutcut 27650 Cut properties of the surr...
scutcl 27651 Closure law for surreal cu...
scutcld 27652 Closure law for surreal cu...
scutbday 27653 The birthday of the surrea...
eqscut 27654 Condition for equality to ...
eqscut2 27655 Condition for equality to ...
sslttr 27656 Transitive law for surreal...
ssltun1 27657 Union law for surreal set ...
ssltun2 27658 Union law for surreal set ...
scutun12 27659 Union law for surreal cuts...
dmscut 27660 The domain of the surreal ...
scutf 27661 Functionality statement fo...
etasslt 27662 A restatement of ~ noeta u...
etasslt2 27663 A version of ~ etasslt wit...
scutbdaybnd 27664 An upper bound on the birt...
scutbdaybnd2 27665 An upper bound on the birt...
scutbdaybnd2lim 27666 An upper bound on the birt...
scutbdaylt 27667 If a surreal lies in a gap...
slerec 27668 A comparison law for surre...
sltrec 27669 A comparison law for surre...
ssltdisj 27670 If ` A ` preceeds ` B ` , ...
0sno 27675 Surreal zero is a surreal....
1sno 27676 Surreal one is a surreal. ...
bday0s 27677 Calculate the birthday of ...
0slt1s 27678 Surreal zero is less than ...
bday0b 27679 The only surreal with birt...
bday1s 27680 The birthday of surreal on...
cuteq0 27681 Condition for a surreal cu...
cuteq1 27682 Condition for a surreal cu...
sgt0ne0 27683 A positive surreal is not ...
sgt0ne0d 27684 A positive surreal is not ...
madeval 27695 The value of the made by f...
madeval2 27696 Alternative characterizati...
oldval 27697 The value of the old optio...
newval 27698 The value of the new optio...
madef 27699 The made function is a fun...
oldf 27700 The older function is a fu...
newf 27701 The new function is a func...
old0 27702 No surreal is older than `...
madessno 27703 Made sets are surreals. (...
oldssno 27704 Old sets are surreals. (C...
newssno 27705 New sets are surreals. (C...
leftval 27706 The value of the left opti...
rightval 27707 The value of the right opt...
leftf 27708 The functionality of the l...
rightf 27709 The functionality of the r...
elmade 27710 Membership in the made fun...
elmade2 27711 Membership in the made fun...
elold 27712 Membership in an old set. ...
ssltleft 27713 A surreal is greater than ...
ssltright 27714 A surreal is less than its...
lltropt 27715 The left options of a surr...
made0 27716 The only surreal made on d...
new0 27717 The only surreal new on da...
old1 27718 The only surreal older tha...
madess 27719 If ` A ` is less than or e...
oldssmade 27720 The older-than set is a su...
leftssold 27721 The left options are a sub...
rightssold 27722 The right options are a su...
leftssno 27723 The left set of a surreal ...
rightssno 27724 The right set of a surreal...
madecut 27725 Given a section that is a ...
madeun 27726 The made set is the union ...
madeoldsuc 27727 The made set is the old se...
oldsuc 27728 The value of the old set a...
oldlim 27729 The value of the old set a...
madebdayim 27730 If a surreal is a member o...
oldbdayim 27731 If ` X ` is in the old set...
oldirr 27732 No surreal is a member of ...
leftirr 27733 No surreal is a member of ...
rightirr 27734 No surreal is a member of ...
left0s 27735 The left set of ` 0s ` is ...
right0s 27736 The right set of ` 0s ` is...
left1s 27737 The left set of ` 1s ` is ...
right1s 27738 The right set of ` 1s ` is...
lrold 27739 The union of the left and ...
madebdaylemold 27740 Lemma for ~ madebday . If...
madebdaylemlrcut 27741 Lemma for ~ madebday . If...
madebday 27742 A surreal is part of the s...
oldbday 27743 A surreal is part of the s...
newbday 27744 A surreal is an element of...
lrcut 27745 A surreal is equal to the ...
scutfo 27746 The surreal cut function i...
sltn0 27747 If ` X ` is less than ` Y ...
lruneq 27748 If two surreals share a bi...
sltlpss 27749 If two surreals share a bi...
slelss 27750 If two surreals ` A ` and ...
0elold 27751 Zero is in the old set of ...
0elleft 27752 Zero is in the left set of...
0elright 27753 Zero is in the right set o...
cofsslt 27754 If every element of ` A ` ...
coinitsslt 27755 If ` B ` is coinitial with...
cofcut1 27756 If ` C ` is cofinal with `...
cofcut1d 27757 If ` C ` is cofinal with `...
cofcut2 27758 If ` A ` and ` C ` are mut...
cofcut2d 27759 If ` A ` and ` C ` are mut...
cofcutr 27760 If ` X ` is the cut of ` A...
cofcutr1d 27761 If ` X ` is the cut of ` A...
cofcutr2d 27762 If ` X ` is the cut of ` A...
cofcutrtime 27763 If ` X ` is the cut of ` A...
cofcutrtime1d 27764 If ` X ` is a timely cut o...
cofcutrtime2d 27765 If ` X ` is a timely cut o...
cofss 27766 Cofinality for a subset. ...
coiniss 27767 Coinitiality for a subset....
cutlt 27768 Eliminating all elements b...
cutpos 27769 Reduce the elements of a c...
lrrecval 27772 The next step in the devel...
lrrecval2 27773 Next, we establish an alte...
lrrecpo 27774 Now, we establish that ` R...
lrrecse 27775 Next, we show that ` R ` i...
lrrecfr 27776 Now we show that ` R ` is ...
lrrecpred 27777 Finally, we calculate the ...
noinds 27778 Induction principle for a ...
norecfn 27779 Surreal recursion over one...
norecov 27780 Calculate the value of the...
noxpordpo 27783 To get through most of the...
noxpordfr 27784 Next we establish the foun...
noxpordse 27785 Next we establish the set-...
noxpordpred 27786 Next we calculate the pred...
no2indslem 27787 Double induction on surrea...
no2inds 27788 Double induction on surrea...
norec2fn 27789 The double-recursion opera...
norec2ov 27790 The value of the double-re...
no3inds 27791 Triple induction over surr...
addsfn 27794 Surreal addition is a func...
addsval 27795 The value of surreal addit...
addsval2 27796 The value of surreal addit...
addsrid 27797 Surreal addition to zero i...
addsridd 27798 Surreal addition to zero i...
addscom 27799 Surreal addition commutes....
addscomd 27800 Surreal addition commutes....
addslid 27801 Surreal addition to zero i...
addsproplem1 27802 Lemma for surreal addition...
addsproplem2 27803 Lemma for surreal addition...
addsproplem3 27804 Lemma for surreal addition...
addsproplem4 27805 Lemma for surreal addition...
addsproplem5 27806 Lemma for surreal addition...
addsproplem6 27807 Lemma for surreal addition...
addsproplem7 27808 Lemma for surreal addition...
addsprop 27809 Inductively show that surr...
addscutlem 27810 Lemma for ~ addscut . Sho...
addscut 27811 Demonstrate the cut proper...
addscut2 27812 Show that the cut involved...
addscld 27813 Surreal numbers are closed...
addscl 27814 Surreal numbers are closed...
addsf 27815 Function statement for sur...
addsfo 27816 Surreal addition is onto. ...
peano2no 27817 A theorem for surreals tha...
sltadd1im 27818 Surreal less-than is prese...
sltadd2im 27819 Surreal less-than is prese...
sleadd1im 27820 Surreal less-than or equal...
sleadd2im 27821 Surreal less-than or equal...
sleadd1 27822 Addition to both sides of ...
sleadd2 27823 Addition to both sides of ...
sltadd2 27824 Addition to both sides of ...
sltadd1 27825 Addition to both sides of ...
addscan2 27826 Cancellation law for surre...
addscan1 27827 Cancellation law for surre...
sleadd1d 27828 Addition to both sides of ...
sleadd2d 27829 Addition to both sides of ...
sltadd2d 27830 Addition to both sides of ...
sltadd1d 27831 Addition to both sides of ...
addscan2d 27832 Cancellation law for surre...
addscan1d 27833 Cancellation law for surre...
addsuniflem 27834 Lemma for ~ addsunif . St...
addsunif 27835 Uniformity theorem for sur...
addsasslem1 27836 Lemma for addition associa...
addsasslem2 27837 Lemma for addition associa...
addsass 27838 Surreal addition is associ...
addsassd 27839 Surreal addition is associ...
adds32d 27840 Commutative/associative la...
adds12d 27841 Commutative/associative la...
adds4d 27842 Rearrangement of four term...
adds42d 27843 Rearrangement of four term...
sltaddpos1d 27844 Addition of a positive num...
sltaddpos2d 27845 Addition of a positive num...
slt2addd 27846 Adding both sides of two s...
addsgt0d 27847 The sum of two positive su...
negsfn 27852 Surreal negation is a func...
subsfn 27853 Surreal subtraction is a f...
negsval 27854 The value of the surreal n...
negs0s 27855 Negative surreal zero is s...
negsproplem1 27856 Lemma for surreal negation...
negsproplem2 27857 Lemma for surreal negation...
negsproplem3 27858 Lemma for surreal negation...
negsproplem4 27859 Lemma for surreal negation...
negsproplem5 27860 Lemma for surreal negation...
negsproplem6 27861 Lemma for surreal negation...
negsproplem7 27862 Lemma for surreal negation...
negsprop 27863 Show closure and ordering ...
negscl 27864 The surreals are closed un...
negscld 27865 The surreals are closed un...
sltnegim 27866 The forward direction of t...
negscut 27867 The cut properties of surr...
negscut2 27868 The cut that defines surre...
negsid 27869 Surreal addition of a numb...
negsidd 27870 Surreal addition of a numb...
negsex 27871 Every surreal has a negati...
negnegs 27872 A surreal is equal to the ...
sltneg 27873 Negative of both sides of ...
sleneg 27874 Negative of both sides of ...
sltnegd 27875 Negative of both sides of ...
slenegd 27876 Negative of both sides of ...
negs11 27877 Surreal negation is one-to...
negsdi 27878 Distribution of surreal ne...
slt0neg2d 27879 Comparison of a surreal an...
negsf 27880 Function statement for sur...
negsfo 27881 Function statement for sur...
negsf1o 27882 Surreal negation is a bije...
negsunif 27883 Uniformity property for su...
negsbdaylem 27884 Lemma for ~ negsbday . Bo...
negsbday 27885 Negation of a surreal numb...
subsval 27886 The value of surreal subtr...
subsvald 27887 The value of surreal subtr...
subscl 27888 Closure law for surreal su...
subscld 27889 Closure law for surreal su...
negsval2 27890 Surreal negation in terms ...
negsval2d 27891 Surreal negation in terms ...
subsid1 27892 Identity law for subtracti...
subsid 27893 Subtraction of a surreal f...
subadds 27894 Relationship between addit...
subaddsd 27895 Relationship between addit...
pncans 27896 Cancellation law for surre...
pncan3s 27897 Subtraction and addition o...
pncan2s 27898 Cancellation law for surre...
npcans 27899 Cancellation law for surre...
sltsub1 27900 Subtraction from both side...
sltsub2 27901 Subtraction from both side...
sltsub1d 27902 Subtraction from both side...
sltsub2d 27903 Subtraction from both side...
negsubsdi2d 27904 Distribution of negative o...
addsubsassd 27905 Associative-type law for s...
addsubsd 27906 Law for surreal addition a...
sltsubsubbd 27907 Equivalence for the surrea...
sltsubsub2bd 27908 Equivalence for the surrea...
sltsubsub3bd 27909 Equivalence for the surrea...
slesubsubbd 27910 Equivalence for the surrea...
slesubsub2bd 27911 Equivalence for the surrea...
slesubsub3bd 27912 Equivalence for the surrea...
sltsubaddd 27913 Surreal less-than relation...
sltsubadd2d 27914 Surreal less-than relation...
sltaddsubd 27915 Surreal less-than relation...
sltaddsub2d 27916 Surreal less-than relation...
subsubs4d 27917 Law for double surreal sub...
subsubs2d 27918 Law for double surreal sub...
nncansd 27919 Cancellation law for surre...
posdifsd 27920 Comparison of two surreals...
sltsubposd 27921 Subtraction of a positive ...
mulsfn 27924 Surreal multiplication is ...
mulsval 27925 The value of surreal multi...
mulsval2lem 27926 Lemma for ~ mulsval2 . Ch...
mulsval2 27927 The value of surreal multi...
muls01 27928 Surreal multiplication by ...
mulsrid 27929 Surreal one is a right ide...
mulsridd 27930 Surreal one is a right ide...
mulsproplemcbv 27931 Lemma for surreal multipli...
mulsproplem1 27932 Lemma for surreal multipli...
mulsproplem2 27933 Lemma for surreal multipli...
mulsproplem3 27934 Lemma for surreal multipli...
mulsproplem4 27935 Lemma for surreal multipli...
mulsproplem5 27936 Lemma for surreal multipli...
mulsproplem6 27937 Lemma for surreal multipli...
mulsproplem7 27938 Lemma for surreal multipli...
mulsproplem8 27939 Lemma for surreal multipli...
mulsproplem9 27940 Lemma for surreal multipli...
mulsproplem10 27941 Lemma for surreal multipli...
mulsproplem11 27942 Lemma for surreal multipli...
mulsproplem12 27943 Lemma for surreal multipli...
mulsproplem13 27944 Lemma for surreal multipli...
mulsproplem14 27945 Lemma for surreal multipli...
mulsprop 27946 Surreals are closed under ...
mulscutlem 27947 Lemma for ~ mulscut . Sta...
mulscut 27948 Show the cut properties of...
mulscut2 27949 Show that the cut involved...
mulscl 27950 The surreals are closed un...
mulscld 27951 The surreals are closed un...
sltmul 27952 An ordering relationship f...
sltmuld 27953 An ordering relationship f...
slemuld 27954 An ordering relationship f...
mulscom 27955 Surreal multiplication com...
mulscomd 27956 Surreal multiplication com...
muls02 27957 Surreal multiplication by ...
mulslid 27958 Surreal one is a left iden...
mulslidd 27959 Surreal one is a left iden...
mulsgt0 27960 The product of two positiv...
mulsgt0d 27961 The product of two positiv...
mulsge0d 27962 The product of two non-neg...
ssltmul1 27963 One surreal set less-than ...
ssltmul2 27964 One surreal set less-than ...
mulsuniflem 27965 Lemma for ~ mulsunif . St...
mulsunif 27966 Surreal multiplication has...
addsdilem1 27967 Lemma for surreal distribu...
addsdilem2 27968 Lemma for surreal distribu...
addsdilem3 27969 Lemma for ~ addsdi . Show...
addsdilem4 27970 Lemma for ~ addsdi . Show...
addsdi 27971 Distributive law for surre...
addsdid 27972 Distributive law for surre...
addsdird 27973 Distributive law for surre...
subsdid 27974 Distribution of surreal mu...
subsdird 27975 Distribution of surreal mu...
mulnegs1d 27976 Product with negative is n...
mulnegs2d 27977 Product with negative is n...
mul2negsd 27978 Surreal product of two neg...
mulsasslem1 27979 Lemma for ~ mulsass . Exp...
mulsasslem2 27980 Lemma for ~ mulsass . Exp...
mulsasslem3 27981 Lemma for ~ mulsass . Dem...
mulsass 27982 Associative law for surrea...
mulsassd 27983 Associative law for surrea...
muls4d 27984 Rearrangement of four surr...
mulsunif2lem 27985 Lemma for ~ mulsunif2 . S...
mulsunif2 27986 Alternate expression for s...
sltmul2 27987 Multiplication of both sid...
sltmul2d 27988 Multiplication of both sid...
sltmul1d 27989 Multiplication of both sid...
slemul2d 27990 Multiplication of both sid...
slemul1d 27991 Multiplication of both sid...
sltmulneg1d 27992 Multiplication of both sid...
sltmulneg2d 27993 Multiplication of both sid...
mulscan2dlem 27994 Lemma for ~ mulscan2d . C...
mulscan2d 27995 Cancellation of surreal mu...
mulscan1d 27996 Cancellation of surreal mu...
muls12d 27997 Commutative/associative la...
slemul1ad 27998 Multiplication of both sid...
sltmul12ad 27999 Comparison of the product ...
divsmo 28000 Uniqueness of surreal inve...
muls0ord 28001 If a surreal product is ze...
mulsne0bd 28002 The product of two non-zer...
divsval 28005 The value of surreal divis...
norecdiv 28006 If a surreal has a recipro...
noreceuw 28007 If a surreal has a recipro...
divsmulw 28008 Relationship between surre...
divsmulwd 28009 Relationship between surre...
divsclw 28010 Weak division closure law....
divsclwd 28011 Weak division closure law....
divscan2wd 28012 A weak cancellation law fo...
divscan1wd 28013 A weak cancellation law fo...
sltdivmulwd 28014 Surreal less-than relation...
sltdivmul2wd 28015 Surreal less-than relation...
sltmuldivwd 28016 Surreal less-than relation...
sltmuldiv2wd 28017 Surreal less-than relation...
divsasswd 28018 An associative law for sur...
divs1 28019 A surreal divided by one i...
precsexlemcbv 28020 Lemma for surreal reciproc...
precsexlem1 28021 Lemma for surreal reciproc...
precsexlem2 28022 Lemma for surreal reciproc...
precsexlem3 28023 Lemma for surreal reciproc...
precsexlem4 28024 Lemma for surreal reciproc...
precsexlem5 28025 Lemma for surreal reciproc...
precsexlem6 28026 Lemma for surreal reciproc...
precsexlem7 28027 Lemma for surreal reciproc...
precsexlem8 28028 Lemma for surreal reciproc...
precsexlem9 28029 Lemma for surreal reciproc...
precsexlem10 28030 Lemma for surreal reciproc...
precsexlem11 28031 Lemma for surreal reciproc...
precsex 28032 Every positive surreal has...
recsex 28033 A non-zero surreal has a r...
recsexd 28034 A non-zero surreal has a r...
divsmul 28035 Relationship between surre...
divsmuld 28036 Relationship between surre...
divscl 28037 Surreal division closure l...
divscld 28038 Surreal division closure l...
divscan2d 28039 A cancellation law for sur...
divscan1d 28040 A cancellation law for sur...
sltdivmuld 28041 Surreal less-than relation...
sltdivmul2d 28042 Surreal less-than relation...
sltmuldivd 28043 Surreal less-than relation...
sltmuldiv2d 28044 Surreal less-than relation...
divsassd 28045 An associative law for sur...
divmuldivsd 28046 Multiplication of two surr...
abssval 28049 The value of surreal absol...
absscl 28050 Closure law for surreal ab...
abssid 28051 The absolute value of a no...
abs0s 28052 The absolute value of surr...
abssnid 28053 For a negative surreal, it...
absmuls 28054 Surreal absolute value dis...
abssge0 28055 The absolute value of a su...
abssor 28056 The absolute value of a su...
abssneg 28057 Surreal absolute value of ...
sleabs 28058 A surreal is less than or ...
absslt 28059 Surreal absolute value and...
elons 28062 Membership in the class of...
onssno 28063 The surreal ordinals are a...
onsno 28064 A surreal ordinal is a sur...
0ons 28065 Surreal zero is a surreal ...
1ons 28066 Surreal one is a surreal o...
elons2 28067 A surreal is ordinal iff i...
elons2d 28068 The cut of any set of surr...
sltonold 28069 The class of ordinals less...
sltonex 28070 The class of ordinals less...
onscutleft 28071 A surreal ordinal is equal...
seqsex 28074 Existence of the surreal s...
seqseq123d 28075 Equality deduction for the...
nfseqs 28076 Hypothesis builder for the...
seqsval 28077 The value of the surreal s...
noseqex 28078 The next several theorems ...
noseq0 28079 The surreal ` A ` is a mem...
noseqp1 28080 One plus an element of ` Z...
noseqind 28081 Peano's inductive postulat...
noseqinds 28082 Induction schema for surre...
noseqssno 28083 A surreal sequence is a su...
noseqno 28084 An element of a surreal se...
om2noseq0 28085 The mapping ` G ` is a one...
om2noseqsuc 28086 The value of ` G ` at a su...
om2noseqfo 28087 Function statement for ` G...
om2noseqlt 28088 Surreal less-than relation...
om2noseqlt2 28089 The mapping ` G ` preserve...
om2noseqf1o 28090 ` G ` is a bijection. (Co...
om2noseqiso 28091 ` G ` is an isomorphism fr...
om2noseqoi 28092 An alternative definition ...
om2noseqrdg 28093 A helper lemma for the val...
noseqrdglem 28094 A helper lemma for the val...
noseqrdgfn 28095 The recursive definition g...
noseqrdg0 28096 Initial value of a recursi...
noseqrdgsuc 28097 Successor value of a recur...
seqsfn 28098 The surreal sequence build...
seqs1 28099 The value of the surreal s...
seqsp1 28100 The value of the surreal s...
n0sex 28105 The set of all non-negativ...
nnsex 28106 The set of all positive su...
peano5n0s 28107 Peano's inductive postulat...
n0ssno 28108 The non-negative surreal i...
nnssn0s 28109 The positive surreal integ...
nnssno 28110 The positive surreal integ...
n0sno 28111 A non-negative surreal int...
nnsno 28112 A positive surreal integer...
n0snod 28113 A non-negative surreal int...
nnsnod 28114 A positive surreal integer...
0n0s 28115 Peano postulate: ` 0s ` is...
peano2n0s 28116 Peano postulate: the succe...
dfn0s2 28117 Alternate definition of th...
n0sind 28118 Principle of Mathematical ...
n0scut 28119 A cut form for surreal nat...
n0ons 28120 A surreal natural is a sur...
nnne0s 28121 A surreal positive integer...
n0sge0 28122 A non-negative integer is ...
nnsgt0 28123 A positive integer is grea...
elnns 28124 Membership in the positive...
elnns2 28125 A positive surreal integer...
n0addscl 28126 The non-negative surreal i...
n0mulscl 28127 The non-negative surreal i...
nnaddscl 28128 The positive surreal integ...
nnmulscl 28129 The positive surreal integ...
1n0s 28130 Surreal one is a non-negat...
1nns 28131 Surreal one is a positive ...
peano2nns 28132 Peano postulate for positi...
n0sbday 28133 A non-negative surreal int...
n0ssold 28134 The non-negative surreal i...
nnsrecgt0d 28135 The reciprocal of a positi...
seqn0sfn 28136 The surreal sequence build...
elreno 28139 Membership in the set of s...
recut 28140 The cut involved in defini...
0reno 28141 Surreal zero is a surreal ...
renegscl 28142 The surreal reals are clos...
readdscl 28143 The surreal reals are clos...
remulscllem1 28144 Lemma for ~ remulscl . Sp...
remulscllem2 28145 Lemma for ~ remulscl . Bo...
remulscl 28146 The surreal reals are clos...
itvndx 28157 Index value of the Interva...
lngndx 28158 Index value of the "line" ...
itvid 28159 Utility theorem: index-ind...
lngid 28160 Utility theorem: index-ind...
slotsinbpsd 28161 The slots ` Base ` , ` +g ...
slotslnbpsd 28162 The slots ` Base ` , ` +g ...
lngndxnitvndx 28163 The slot for the line is n...
trkgstr 28164 Functionality of a Tarski ...
trkgbas 28165 The base set of a Tarski g...
trkgdist 28166 The measure of a distance ...
trkgitv 28167 The congruence relation in...
istrkgc 28174 Property of being a Tarski...
istrkgb 28175 Property of being a Tarski...
istrkgcb 28176 Property of being a Tarski...
istrkge 28177 Property of fulfilling Euc...
istrkgl 28178 Building lines from the se...
istrkgld 28179 Property of fulfilling the...
istrkg2ld 28180 Property of fulfilling the...
istrkg3ld 28181 Property of fulfilling the...
axtgcgrrflx 28182 Axiom of reflexivity of co...
axtgcgrid 28183 Axiom of identity of congr...
axtgsegcon 28184 Axiom of segment construct...
axtg5seg 28185 Five segments axiom, Axiom...
axtgbtwnid 28186 Identity of Betweenness. ...
axtgpasch 28187 Axiom of (Inner) Pasch, Ax...
axtgcont1 28188 Axiom of Continuity. Axio...
axtgcont 28189 Axiom of Continuity. Axio...
axtglowdim2 28190 Lower dimension axiom for ...
axtgupdim2 28191 Upper dimension axiom for ...
axtgeucl 28192 Euclid's Axiom. Axiom A10...
tgjustf 28193 Given any function ` F ` ,...
tgjustr 28194 Given any equivalence rela...
tgjustc1 28195 A justification for using ...
tgjustc2 28196 A justification for using ...
tgcgrcomimp 28197 Congruence commutes on the...
tgcgrcomr 28198 Congruence commutes on the...
tgcgrcoml 28199 Congruence commutes on the...
tgcgrcomlr 28200 Congruence commutes on bot...
tgcgreqb 28201 Congruence and equality. ...
tgcgreq 28202 Congruence and equality. ...
tgcgrneq 28203 Congruence and equality. ...
tgcgrtriv 28204 Degenerate segments are co...
tgcgrextend 28205 Link congruence over a pai...
tgsegconeq 28206 Two points that satisfy th...
tgbtwntriv2 28207 Betweenness always holds f...
tgbtwncom 28208 Betweenness commutes. The...
tgbtwncomb 28209 Betweenness commutes, bico...
tgbtwnne 28210 Betweenness and inequality...
tgbtwntriv1 28211 Betweenness always holds f...
tgbtwnswapid 28212 If you can swap the first ...
tgbtwnintr 28213 Inner transitivity law for...
tgbtwnexch3 28214 Exchange the first endpoin...
tgbtwnouttr2 28215 Outer transitivity law for...
tgbtwnexch2 28216 Exchange the outer point o...
tgbtwnouttr 28217 Outer transitivity law for...
tgbtwnexch 28218 Outer transitivity law for...
tgtrisegint 28219 A line segment between two...
tglowdim1 28220 Lower dimension axiom for ...
tglowdim1i 28221 Lower dimension axiom for ...
tgldimor 28222 Excluded-middle like state...
tgldim0eq 28223 In dimension zero, any two...
tgldim0itv 28224 In dimension zero, any two...
tgldim0cgr 28225 In dimension zero, any two...
tgbtwndiff 28226 There is always a ` c ` di...
tgdim01 28227 In geometries of dimension...
tgifscgr 28228 Inner five segment congrue...
tgcgrsub 28229 Removing identical parts f...
iscgrg 28232 The congruence property fo...
iscgrgd 28233 The property for two seque...
iscgrglt 28234 The property for two seque...
trgcgrg 28235 The property for two trian...
trgcgr 28236 Triangle congruence. (Con...
ercgrg 28237 The shape congruence relat...
tgcgrxfr 28238 A line segment can be divi...
cgr3id 28239 Reflexivity law for three-...
cgr3simp1 28240 Deduce segment congruence ...
cgr3simp2 28241 Deduce segment congruence ...
cgr3simp3 28242 Deduce segment congruence ...
cgr3swap12 28243 Permutation law for three-...
cgr3swap23 28244 Permutation law for three-...
cgr3swap13 28245 Permutation law for three-...
cgr3rotr 28246 Permutation law for three-...
cgr3rotl 28247 Permutation law for three-...
trgcgrcom 28248 Commutative law for three-...
cgr3tr 28249 Transitivity law for three...
tgbtwnxfr 28250 A condition for extending ...
tgcgr4 28251 Two quadrilaterals to be c...
isismt 28254 Property of being an isome...
ismot 28255 Property of being an isome...
motcgr 28256 Property of a motion: dist...
idmot 28257 The identity is a motion. ...
motf1o 28258 Motions are bijections. (...
motcl 28259 Closure of motions. (Cont...
motco 28260 The composition of two mot...
cnvmot 28261 The converse of a motion i...
motplusg 28262 The operation for motions ...
motgrp 28263 The motions of a geometry ...
motcgrg 28264 Property of a motion: dist...
motcgr3 28265 Property of a motion: dist...
tglng 28266 Lines of a Tarski Geometry...
tglnfn 28267 Lines as functions. (Cont...
tglnunirn 28268 Lines are sets of points. ...
tglnpt 28269 Lines are sets of points. ...
tglngne 28270 It takes two different poi...
tglngval 28271 The line going through poi...
tglnssp 28272 Lines are subset of the ge...
tgellng 28273 Property of lying on the l...
tgcolg 28274 We choose the notation ` (...
btwncolg1 28275 Betweenness implies coline...
btwncolg2 28276 Betweenness implies coline...
btwncolg3 28277 Betweenness implies coline...
colcom 28278 Swapping the points defini...
colrot1 28279 Rotating the points defini...
colrot2 28280 Rotating the points defini...
ncolcom 28281 Swapping non-colinear poin...
ncolrot1 28282 Rotating non-colinear poin...
ncolrot2 28283 Rotating non-colinear poin...
tgdim01ln 28284 In geometries of dimension...
ncoltgdim2 28285 If there are three non-col...
lnxfr 28286 Transfer law for colineari...
lnext 28287 Extend a line with a missi...
tgfscgr 28288 Congruence law for the gen...
lncgr 28289 Congruence rule for lines....
lnid 28290 Identity law for points on...
tgidinside 28291 Law for finding a point in...
tgbtwnconn1lem1 28292 Lemma for ~ tgbtwnconn1 . ...
tgbtwnconn1lem2 28293 Lemma for ~ tgbtwnconn1 . ...
tgbtwnconn1lem3 28294 Lemma for ~ tgbtwnconn1 . ...
tgbtwnconn1 28295 Connectivity law for betwe...
tgbtwnconn2 28296 Another connectivity law f...
tgbtwnconn3 28297 Inner connectivity law for...
tgbtwnconnln3 28298 Derive colinearity from be...
tgbtwnconn22 28299 Double connectivity law fo...
tgbtwnconnln1 28300 Derive colinearity from be...
tgbtwnconnln2 28301 Derive colinearity from be...
legval 28304 Value of the less-than rel...
legov 28305 Value of the less-than rel...
legov2 28306 An equivalent definition o...
legid 28307 Reflexivity of the less-th...
btwnleg 28308 Betweenness implies less-t...
legtrd 28309 Transitivity of the less-t...
legtri3 28310 Equality from the less-tha...
legtrid 28311 Trichotomy law for the les...
leg0 28312 Degenerated (zero-length) ...
legeq 28313 Deduce equality from "less...
legbtwn 28314 Deduce betweenness from "l...
tgcgrsub2 28315 Removing identical parts f...
ltgseg 28316 The set ` E ` denotes the ...
ltgov 28317 Strict "shorter than" geom...
legov3 28318 An equivalent definition o...
legso 28319 The "shorter than" relatio...
ishlg 28322 Rays : Definition 6.1 of ...
hlcomb 28323 The half-line relation com...
hlcomd 28324 The half-line relation com...
hlne1 28325 The half-line relation imp...
hlne2 28326 The half-line relation imp...
hlln 28327 The half-line relation imp...
hleqnid 28328 The endpoint does not belo...
hlid 28329 The half-line relation is ...
hltr 28330 The half-line relation is ...
hlbtwn 28331 Betweenness is a sufficien...
btwnhl1 28332 Deduce half-line from betw...
btwnhl2 28333 Deduce half-line from betw...
btwnhl 28334 Swap betweenness for a hal...
lnhl 28335 Either a point ` C ` on th...
hlcgrex 28336 Construct a point on a hal...
hlcgreulem 28337 Lemma for ~ hlcgreu . (Co...
hlcgreu 28338 The point constructed in ~...
btwnlng1 28339 Betweenness implies coline...
btwnlng2 28340 Betweenness implies coline...
btwnlng3 28341 Betweenness implies coline...
lncom 28342 Swapping the points defini...
lnrot1 28343 Rotating the points defini...
lnrot2 28344 Rotating the points defini...
ncolne1 28345 Non-colinear points are di...
ncolne2 28346 Non-colinear points are di...
tgisline 28347 The property of being a pr...
tglnne 28348 It takes two different poi...
tglndim0 28349 There are no lines in dime...
tgelrnln 28350 The property of being a pr...
tglineeltr 28351 Transitivity law for lines...
tglineelsb2 28352 If ` S ` lies on PQ , then...
tglinerflx1 28353 Reflexivity law for line m...
tglinerflx2 28354 Reflexivity law for line m...
tglinecom 28355 Commutativity law for line...
tglinethru 28356 If ` A ` is a line contain...
tghilberti1 28357 There is a line through an...
tghilberti2 28358 There is at most one line ...
tglinethrueu 28359 There is a unique line goi...
tglnne0 28360 A line ` A ` has at least ...
tglnpt2 28361 Find a second point on a l...
tglineintmo 28362 Two distinct lines interse...
tglineineq 28363 Two distinct lines interse...
tglineneq 28364 Given three non-colinear p...
tglineinteq 28365 Two distinct lines interse...
ncolncol 28366 Deduce non-colinearity fro...
coltr 28367 A transitivity law for col...
coltr3 28368 A transitivity law for col...
colline 28369 Three points are colinear ...
tglowdim2l 28370 Reformulation of the lower...
tglowdim2ln 28371 There is always one point ...
mirreu3 28374 Existential uniqueness of ...
mirval 28375 Value of the point inversi...
mirfv 28376 Value of the point inversi...
mircgr 28377 Property of the image by t...
mirbtwn 28378 Property of the image by t...
ismir 28379 Property of the image by t...
mirf 28380 Point inversion as functio...
mircl 28381 Closure of the point inver...
mirmir 28382 The point inversion functi...
mircom 28383 Variation on ~ mirmir . (...
mirreu 28384 Any point has a unique ant...
mireq 28385 Equality deduction for poi...
mirinv 28386 The only invariant point o...
mirne 28387 Mirror of non-center point...
mircinv 28388 The center point is invari...
mirf1o 28389 The point inversion functi...
miriso 28390 The point inversion functi...
mirbtwni 28391 Point inversion preserves ...
mirbtwnb 28392 Point inversion preserves ...
mircgrs 28393 Point inversion preserves ...
mirmir2 28394 Point inversion of a point...
mirmot 28395 Point investion is a motio...
mirln 28396 If two points are on the s...
mirln2 28397 If a point and its mirror ...
mirconn 28398 Point inversion of connect...
mirhl 28399 If two points ` X ` and ` ...
mirbtwnhl 28400 If the center of the point...
mirhl2 28401 Deduce half-line relation ...
mircgrextend 28402 Link congruence over a pai...
mirtrcgr 28403 Point inversion of one poi...
mirauto 28404 Point inversion preserves ...
miduniq 28405 Uniqueness of the middle p...
miduniq1 28406 Uniqueness of the middle p...
miduniq2 28407 If two point inversions co...
colmid 28408 Colinearity and equidistan...
symquadlem 28409 Lemma of the symetrial qua...
krippenlem 28410 Lemma for ~ krippen . We ...
krippen 28411 Krippenlemma (German for c...
midexlem 28412 Lemma for the existence of...
israg 28417 Property for 3 points A, B...
ragcom 28418 Commutative rule for right...
ragcol 28419 The right angle property i...
ragmir 28420 Right angle property is pr...
mirrag 28421 Right angle is conserved b...
ragtrivb 28422 Trivial right angle. Theo...
ragflat2 28423 Deduce equality from two r...
ragflat 28424 Deduce equality from two r...
ragtriva 28425 Trivial right angle. Theo...
ragflat3 28426 Right angle and colinearit...
ragcgr 28427 Right angle and colinearit...
motrag 28428 Right angles are preserved...
ragncol 28429 Right angle implies non-co...
perpln1 28430 Derive a line from perpend...
perpln2 28431 Derive a line from perpend...
isperp 28432 Property for 2 lines A, B ...
perpcom 28433 The "perpendicular" relati...
perpneq 28434 Two perpendicular lines ar...
isperp2 28435 Property for 2 lines A, B,...
isperp2d 28436 One direction of ~ isperp2...
ragperp 28437 Deduce that two lines are ...
footexALT 28438 Alternative version of ~ f...
footexlem1 28439 Lemma for ~ footex . (Con...
footexlem2 28440 Lemma for ~ footex . (Con...
footex 28441 From a point ` C ` outside...
foot 28442 From a point ` C ` outside...
footne 28443 Uniqueness of the foot poi...
footeq 28444 Uniqueness of the foot poi...
hlperpnel 28445 A point on a half-line whi...
perprag 28446 Deduce a right angle from ...
perpdragALT 28447 Deduce a right angle from ...
perpdrag 28448 Deduce a right angle from ...
colperp 28449 Deduce a perpendicularity ...
colperpexlem1 28450 Lemma for ~ colperp . Fir...
colperpexlem2 28451 Lemma for ~ colperpex . S...
colperpexlem3 28452 Lemma for ~ colperpex . C...
colperpex 28453 In dimension 2 and above, ...
mideulem2 28454 Lemma for ~ opphllem , whi...
opphllem 28455 Lemma 8.24 of [Schwabhause...
mideulem 28456 Lemma for ~ mideu . We ca...
midex 28457 Existence of the midpoint,...
mideu 28458 Existence and uniqueness o...
islnopp 28459 The property for two point...
islnoppd 28460 Deduce that ` A ` and ` B ...
oppne1 28461 Points lying on opposite s...
oppne2 28462 Points lying on opposite s...
oppne3 28463 Points lying on opposite s...
oppcom 28464 Commutativity rule for "op...
opptgdim2 28465 If two points opposite to ...
oppnid 28466 The "opposite to a line" r...
opphllem1 28467 Lemma for ~ opphl . (Cont...
opphllem2 28468 Lemma for ~ opphl . Lemma...
opphllem3 28469 Lemma for ~ opphl : We as...
opphllem4 28470 Lemma for ~ opphl . (Cont...
opphllem5 28471 Second part of Lemma 9.4 o...
opphllem6 28472 First part of Lemma 9.4 of...
oppperpex 28473 Restating ~ colperpex usin...
opphl 28474 If two points ` A ` and ` ...
outpasch 28475 Axiom of Pasch, outer form...
hlpasch 28476 An application of the axio...
ishpg 28479 Value of the half-plane re...
hpgbr 28480 Half-planes : property for...
hpgne1 28481 Points on the open half pl...
hpgne2 28482 Points on the open half pl...
lnopp2hpgb 28483 Theorem 9.8 of [Schwabhaus...
lnoppnhpg 28484 If two points lie on the o...
hpgerlem 28485 Lemma for the proof that t...
hpgid 28486 The half-plane relation is...
hpgcom 28487 The half-plane relation co...
hpgtr 28488 The half-plane relation is...
colopp 28489 Opposite sides of a line f...
colhp 28490 Half-plane relation for co...
hphl 28491 If two points are on the s...
midf 28496 Midpoint as a function. (...
midcl 28497 Closure of the midpoint. ...
ismidb 28498 Property of the midpoint. ...
midbtwn 28499 Betweenness of midpoint. ...
midcgr 28500 Congruence of midpoint. (...
midid 28501 Midpoint of a null segment...
midcom 28502 Commutativity rule for the...
mirmid 28503 Point inversion preserves ...
lmieu 28504 Uniqueness of the line mir...
lmif 28505 Line mirror as a function....
lmicl 28506 Closure of the line mirror...
islmib 28507 Property of the line mirro...
lmicom 28508 The line mirroring functio...
lmilmi 28509 Line mirroring is an invol...
lmireu 28510 Any point has a unique ant...
lmieq 28511 Equality deduction for lin...
lmiinv 28512 The invariants of the line...
lmicinv 28513 The mirroring line is an i...
lmimid 28514 If we have a right angle, ...
lmif1o 28515 The line mirroring functio...
lmiisolem 28516 Lemma for ~ lmiiso . (Con...
lmiiso 28517 The line mirroring functio...
lmimot 28518 Line mirroring is a motion...
hypcgrlem1 28519 Lemma for ~ hypcgr , case ...
hypcgrlem2 28520 Lemma for ~ hypcgr , case ...
hypcgr 28521 If the catheti of two righ...
lmiopp 28522 Line mirroring produces po...
lnperpex 28523 Existence of a perpendicul...
trgcopy 28524 Triangle construction: a c...
trgcopyeulem 28525 Lemma for ~ trgcopyeu . (...
trgcopyeu 28526 Triangle construction: a c...
iscgra 28529 Property for two angles AB...
iscgra1 28530 A special version of ~ isc...
iscgrad 28531 Sufficient conditions for ...
cgrane1 28532 Angles imply inequality. ...
cgrane2 28533 Angles imply inequality. ...
cgrane3 28534 Angles imply inequality. ...
cgrane4 28535 Angles imply inequality. ...
cgrahl1 28536 Angle congruence is indepe...
cgrahl2 28537 Angle congruence is indepe...
cgracgr 28538 First direction of proposi...
cgraid 28539 Angle congruence is reflex...
cgraswap 28540 Swap rays in a congruence ...
cgrcgra 28541 Triangle congruence implie...
cgracom 28542 Angle congruence commutes....
cgratr 28543 Angle congruence is transi...
flatcgra 28544 Flat angles are congruent....
cgraswaplr 28545 Swap both side of angle co...
cgrabtwn 28546 Angle congruence preserves...
cgrahl 28547 Angle congruence preserves...
cgracol 28548 Angle congruence preserves...
cgrancol 28549 Angle congruence preserves...
dfcgra2 28550 This is the full statement...
sacgr 28551 Supplementary angles of co...
oacgr 28552 Vertical angle theorem. V...
acopy 28553 Angle construction. Theor...
acopyeu 28554 Angle construction. Theor...
isinag 28558 Property for point ` X ` t...
isinagd 28559 Sufficient conditions for ...
inagflat 28560 Any point lies in a flat a...
inagswap 28561 Swap the order of the half...
inagne1 28562 Deduce inequality from the...
inagne2 28563 Deduce inequality from the...
inagne3 28564 Deduce inequality from the...
inaghl 28565 The "point lie in angle" r...
isleag 28567 Geometrical "less than" pr...
isleagd 28568 Sufficient condition for "...
leagne1 28569 Deduce inequality from the...
leagne2 28570 Deduce inequality from the...
leagne3 28571 Deduce inequality from the...
leagne4 28572 Deduce inequality from the...
cgrg3col4 28573 Lemma 11.28 of [Schwabhaus...
tgsas1 28574 First congruence theorem: ...
tgsas 28575 First congruence theorem: ...
tgsas2 28576 First congruence theorem: ...
tgsas3 28577 First congruence theorem: ...
tgasa1 28578 Second congruence theorem:...
tgasa 28579 Second congruence theorem:...
tgsss1 28580 Third congruence theorem: ...
tgsss2 28581 Third congruence theorem: ...
tgsss3 28582 Third congruence theorem: ...
dfcgrg2 28583 Congruence for two triangl...
isoas 28584 Congruence theorem for iso...
iseqlg 28587 Property of a triangle bei...
iseqlgd 28588 Condition for a triangle t...
f1otrgds 28589 Convenient lemma for ~ f1o...
f1otrgitv 28590 Convenient lemma for ~ f1o...
f1otrg 28591 A bijection between bases ...
f1otrge 28592 A bijection between bases ...
ttgval 28595 Define a function to augme...
ttgvalOLD 28596 Obsolete proof of ~ ttgval...
ttglem 28597 Lemma for ~ ttgbas , ~ ttg...
ttglemOLD 28598 Obsolete version of ~ ttgl...
ttgbas 28599 The base set of a subcompl...
ttgbasOLD 28600 Obsolete proof of ~ ttgbas...
ttgplusg 28601 The addition operation of ...
ttgplusgOLD 28602 Obsolete proof of ~ ttgplu...
ttgsub 28603 The subtraction operation ...
ttgvsca 28604 The scalar product of a su...
ttgvscaOLD 28605 Obsolete proof of ~ ttgvsc...
ttgds 28606 The metric of a subcomplex...
ttgdsOLD 28607 Obsolete proof of ~ ttgds ...
ttgitvval 28608 Betweenness for a subcompl...
ttgelitv 28609 Betweenness for a subcompl...
ttgbtwnid 28610 Any subcomplex module equi...
ttgcontlem1 28611 Lemma for % ttgcont . (Co...
xmstrkgc 28612 Any metric space fulfills ...
cchhllem 28613 Lemma for chlbas and chlvs...
cchhllemOLD 28614 Obsolete version of ~ cchh...
elee 28621 Membership in a Euclidean ...
mptelee 28622 A condition for a mapping ...
eleenn 28623 If ` A ` is in ` ( EE `` N...
eleei 28624 The forward direction of ~...
eedimeq 28625 A point belongs to at most...
brbtwn 28626 The binary relation form o...
brcgr 28627 The binary relation form o...
fveere 28628 The function value of a po...
fveecn 28629 The function value of a po...
eqeefv 28630 Two points are equal iff t...
eqeelen 28631 Two points are equal iff t...
brbtwn2 28632 Alternate characterization...
colinearalglem1 28633 Lemma for ~ colinearalg . ...
colinearalglem2 28634 Lemma for ~ colinearalg . ...
colinearalglem3 28635 Lemma for ~ colinearalg . ...
colinearalglem4 28636 Lemma for ~ colinearalg . ...
colinearalg 28637 An algebraic characterizat...
eleesub 28638 Membership of a subtractio...
eleesubd 28639 Membership of a subtractio...
axdimuniq 28640 The unique dimension axiom...
axcgrrflx 28641 ` A ` is as far from ` B `...
axcgrtr 28642 Congruence is transitive. ...
axcgrid 28643 If there is no distance be...
axsegconlem1 28644 Lemma for ~ axsegcon . Ha...
axsegconlem2 28645 Lemma for ~ axsegcon . Sh...
axsegconlem3 28646 Lemma for ~ axsegcon . Sh...
axsegconlem4 28647 Lemma for ~ axsegcon . Sh...
axsegconlem5 28648 Lemma for ~ axsegcon . Sh...
axsegconlem6 28649 Lemma for ~ axsegcon . Sh...
axsegconlem7 28650 Lemma for ~ axsegcon . Sh...
axsegconlem8 28651 Lemma for ~ axsegcon . Sh...
axsegconlem9 28652 Lemma for ~ axsegcon . Sh...
axsegconlem10 28653 Lemma for ~ axsegcon . Sh...
axsegcon 28654 Any segment ` A B ` can be...
ax5seglem1 28655 Lemma for ~ ax5seg . Rexp...
ax5seglem2 28656 Lemma for ~ ax5seg . Rexp...
ax5seglem3a 28657 Lemma for ~ ax5seg . (Con...
ax5seglem3 28658 Lemma for ~ ax5seg . Comb...
ax5seglem4 28659 Lemma for ~ ax5seg . Give...
ax5seglem5 28660 Lemma for ~ ax5seg . If `...
ax5seglem6 28661 Lemma for ~ ax5seg . Give...
ax5seglem7 28662 Lemma for ~ ax5seg . An a...
ax5seglem8 28663 Lemma for ~ ax5seg . Use ...
ax5seglem9 28664 Lemma for ~ ax5seg . Take...
ax5seg 28665 The five segment axiom. T...
axbtwnid 28666 Points are indivisible. T...
axpaschlem 28667 Lemma for ~ axpasch . Set...
axpasch 28668 The inner Pasch axiom. Ta...
axlowdimlem1 28669 Lemma for ~ axlowdim . Es...
axlowdimlem2 28670 Lemma for ~ axlowdim . Sh...
axlowdimlem3 28671 Lemma for ~ axlowdim . Se...
axlowdimlem4 28672 Lemma for ~ axlowdim . Se...
axlowdimlem5 28673 Lemma for ~ axlowdim . Sh...
axlowdimlem6 28674 Lemma for ~ axlowdim . Sh...
axlowdimlem7 28675 Lemma for ~ axlowdim . Se...
axlowdimlem8 28676 Lemma for ~ axlowdim . Ca...
axlowdimlem9 28677 Lemma for ~ axlowdim . Ca...
axlowdimlem10 28678 Lemma for ~ axlowdim . Se...
axlowdimlem11 28679 Lemma for ~ axlowdim . Ca...
axlowdimlem12 28680 Lemma for ~ axlowdim . Ca...
axlowdimlem13 28681 Lemma for ~ axlowdim . Es...
axlowdimlem14 28682 Lemma for ~ axlowdim . Ta...
axlowdimlem15 28683 Lemma for ~ axlowdim . Se...
axlowdimlem16 28684 Lemma for ~ axlowdim . Se...
axlowdimlem17 28685 Lemma for ~ axlowdim . Es...
axlowdim1 28686 The lower dimension axiom ...
axlowdim2 28687 The lower two-dimensional ...
axlowdim 28688 The general lower dimensio...
axeuclidlem 28689 Lemma for ~ axeuclid . Ha...
axeuclid 28690 Euclid's axiom. Take an a...
axcontlem1 28691 Lemma for ~ axcont . Chan...
axcontlem2 28692 Lemma for ~ axcont . The ...
axcontlem3 28693 Lemma for ~ axcont . Give...
axcontlem4 28694 Lemma for ~ axcont . Give...
axcontlem5 28695 Lemma for ~ axcont . Comp...
axcontlem6 28696 Lemma for ~ axcont . Stat...
axcontlem7 28697 Lemma for ~ axcont . Give...
axcontlem8 28698 Lemma for ~ axcont . A po...
axcontlem9 28699 Lemma for ~ axcont . Give...
axcontlem10 28700 Lemma for ~ axcont . Give...
axcontlem11 28701 Lemma for ~ axcont . Elim...
axcontlem12 28702 Lemma for ~ axcont . Elim...
axcont 28703 The axiom of continuity. ...
eengv 28706 The value of the Euclidean...
eengstr 28707 The Euclidean geometry as ...
eengbas 28708 The Base of the Euclidean ...
ebtwntg 28709 The betweenness relation u...
ecgrtg 28710 The congruence relation us...
elntg 28711 The line definition in the...
elntg2 28712 The line definition in the...
eengtrkg 28713 The geometry structure for...
eengtrkge 28714 The geometry structure for...
edgfid 28717 Utility theorem: index-ind...
edgfndx 28718 Index value of the ~ df-ed...
edgfndxnn 28719 The index value of the edg...
edgfndxid 28720 The value of the edge func...
edgfndxidOLD 28721 Obsolete version of ~ edgf...
basendxltedgfndx 28722 The index value of the ` B...
baseltedgfOLD 28723 Obsolete proof of ~ basend...
basendxnedgfndx 28724 The slots ` Base ` and ` ....
vtxval 28729 The set of vertices of a g...
iedgval 28730 The set of indexed edges o...
1vgrex 28731 A graph with at least one ...
opvtxval 28732 The set of vertices of a g...
opvtxfv 28733 The set of vertices of a g...
opvtxov 28734 The set of vertices of a g...
opiedgval 28735 The set of indexed edges o...
opiedgfv 28736 The set of indexed edges o...
opiedgov 28737 The set of indexed edges o...
opvtxfvi 28738 The set of vertices of a g...
opiedgfvi 28739 The set of indexed edges o...
funvtxdmge2val 28740 The set of vertices of an ...
funiedgdmge2val 28741 The set of indexed edges o...
funvtxdm2val 28742 The set of vertices of an ...
funiedgdm2val 28743 The set of indexed edges o...
funvtxval0 28744 The set of vertices of an ...
basvtxval 28745 The set of vertices of a g...
edgfiedgval 28746 The set of indexed edges o...
funvtxval 28747 The set of vertices of a g...
funiedgval 28748 The set of indexed edges o...
structvtxvallem 28749 Lemma for ~ structvtxval a...
structvtxval 28750 The set of vertices of an ...
structiedg0val 28751 The set of indexed edges o...
structgrssvtxlem 28752 Lemma for ~ structgrssvtx ...
structgrssvtx 28753 The set of vertices of a g...
structgrssiedg 28754 The set of indexed edges o...
struct2grstr 28755 A graph represented as an ...
struct2grvtx 28756 The set of vertices of a g...
struct2griedg 28757 The set of indexed edges o...
graop 28758 Any representation of a gr...
grastruct 28759 Any representation of a gr...
gropd 28760 If any representation of a...
grstructd 28761 If any representation of a...
gropeld 28762 If any representation of a...
grstructeld 28763 If any representation of a...
setsvtx 28764 The vertices of a structur...
setsiedg 28765 The (indexed) edges of a s...
snstrvtxval 28766 The set of vertices of a g...
snstriedgval 28767 The set of indexed edges o...
vtxval0 28768 Degenerated case 1 for ver...
iedgval0 28769 Degenerated case 1 for edg...
vtxvalsnop 28770 Degenerated case 2 for ver...
iedgvalsnop 28771 Degenerated case 2 for edg...
vtxval3sn 28772 Degenerated case 3 for ver...
iedgval3sn 28773 Degenerated case 3 for edg...
vtxvalprc 28774 Degenerated case 4 for ver...
iedgvalprc 28775 Degenerated case 4 for edg...
edgval 28778 The edges of a graph. (Co...
iedgedg 28779 An indexed edge is an edge...
edgopval 28780 The edges of a graph repre...
edgov 28781 The edges of a graph repre...
edgstruct 28782 The edges of a graph repre...
edgiedgb 28783 A set is an edge iff it is...
edg0iedg0 28784 There is no edge in a grap...
isuhgr 28789 The predicate "is an undir...
isushgr 28790 The predicate "is an undir...
uhgrf 28791 The edge function of an un...
ushgrf 28792 The edge function of an un...
uhgrss 28793 An edge is a subset of ver...
uhgreq12g 28794 If two sets have the same ...
uhgrfun 28795 The edge function of an un...
uhgrn0 28796 An edge is a nonempty subs...
lpvtx 28797 The endpoints of a loop (w...
ushgruhgr 28798 An undirected simple hyper...
isuhgrop 28799 The property of being an u...
uhgr0e 28800 The empty graph, with vert...
uhgr0vb 28801 The null graph, with no ve...
uhgr0 28802 The null graph represented...
uhgrun 28803 The union ` U ` of two (un...
uhgrunop 28804 The union of two (undirect...
ushgrun 28805 The union ` U ` of two (un...
ushgrunop 28806 The union of two (undirect...
uhgrstrrepe 28807 Replacing (or adding) the ...
incistruhgr 28808 An _incidence structure_ `...
isupgr 28813 The property of being an u...
wrdupgr 28814 The property of being an u...
upgrf 28815 The edge function of an un...
upgrfn 28816 The edge function of an un...
upgrss 28817 An edge is a subset of ver...
upgrn0 28818 An edge is a nonempty subs...
upgrle 28819 An edge of an undirected p...
upgrfi 28820 An edge is a finite subset...
upgrex 28821 An edge is an unordered pa...
upgrbi 28822 Show that an unordered pai...
upgrop 28823 A pseudograph represented ...
isumgr 28824 The property of being an u...
isumgrs 28825 The simplified property of...
wrdumgr 28826 The property of being an u...
umgrf 28827 The edge function of an un...
umgrfn 28828 The edge function of an un...
umgredg2 28829 An edge of a multigraph ha...
umgrbi 28830 Show that an unordered pai...
upgruhgr 28831 An undirected pseudograph ...
umgrupgr 28832 An undirected multigraph i...
umgruhgr 28833 An undirected multigraph i...
upgrle2 28834 An edge of an undirected p...
umgrnloopv 28835 In a multigraph, there is ...
umgredgprv 28836 In a multigraph, an edge i...
umgrnloop 28837 In a multigraph, there is ...
umgrnloop0 28838 A multigraph has no loops....
umgr0e 28839 The empty graph, with vert...
upgr0e 28840 The empty graph, with vert...
upgr1elem 28841 Lemma for ~ upgr1e and ~ u...
upgr1e 28842 A pseudograph with one edg...
upgr0eop 28843 The empty graph, with vert...
upgr1eop 28844 A pseudograph with one edg...
upgr0eopALT 28845 Alternate proof of ~ upgr0...
upgr1eopALT 28846 Alternate proof of ~ upgr1...
upgrun 28847 The union ` U ` of two pse...
upgrunop 28848 The union of two pseudogra...
umgrun 28849 The union ` U ` of two mul...
umgrunop 28850 The union of two multigrap...
umgrislfupgrlem 28851 Lemma for ~ umgrislfupgr a...
umgrislfupgr 28852 A multigraph is a loop-fre...
lfgredgge2 28853 An edge of a loop-free gra...
lfgrnloop 28854 A loop-free graph has no l...
uhgredgiedgb 28855 In a hypergraph, a set is ...
uhgriedg0edg0 28856 A hypergraph has no edges ...
uhgredgn0 28857 An edge of a hypergraph is...
edguhgr 28858 An edge of a hypergraph is...
uhgredgrnv 28859 An edge of a hypergraph co...
uhgredgss 28860 The set of edges of a hype...
upgredgss 28861 The set of edges of a pseu...
umgredgss 28862 The set of edges of a mult...
edgupgr 28863 Properties of an edge of a...
edgumgr 28864 Properties of an edge of a...
uhgrvtxedgiedgb 28865 In a hypergraph, a vertex ...
upgredg 28866 For each edge in a pseudog...
umgredg 28867 For each edge in a multigr...
upgrpredgv 28868 An edge of a pseudograph a...
umgrpredgv 28869 An edge of a multigraph al...
upgredg2vtx 28870 For a vertex incident to a...
upgredgpr 28871 If a proper pair (of verti...
edglnl 28872 The edges incident with a ...
numedglnl 28873 The number of edges incide...
umgredgne 28874 An edge of a multigraph al...
umgrnloop2 28875 A multigraph has no loops....
umgredgnlp 28876 An edge of a multigraph is...
isuspgr 28881 The property of being a si...
isusgr 28882 The property of being a si...
uspgrf 28883 The edge function of a sim...
usgrf 28884 The edge function of a sim...
isusgrs 28885 The property of being a si...
usgrfs 28886 The edge function of a sim...
usgrfun 28887 The edge function of a sim...
usgredgss 28888 The set of edges of a simp...
edgusgr 28889 An edge of a simple graph ...
isuspgrop 28890 The property of being an u...
isusgrop 28891 The property of being an u...
usgrop 28892 A simple graph represented...
isausgr 28893 The property of an unorder...
ausgrusgrb 28894 The equivalence of the def...
usgrausgri 28895 A simple graph represented...
ausgrumgri 28896 If an alternatively define...
ausgrusgri 28897 The equivalence of the def...
usgrausgrb 28898 The equivalence of the def...
usgredgop 28899 An edge of a simple graph ...
usgrf1o 28900 The edge function of a sim...
usgrf1 28901 The edge function of a sim...
uspgrf1oedg 28902 The edge function of a sim...
usgrss 28903 An edge is a subset of ver...
uspgrushgr 28904 A simple pseudograph is an...
uspgrupgr 28905 A simple pseudograph is an...
uspgrupgrushgr 28906 A graph is a simple pseudo...
usgruspgr 28907 A simple graph is a simple...
usgrumgr 28908 A simple graph is an undir...
usgrumgruspgr 28909 A graph is a simple graph ...
usgruspgrb 28910 A class is a simple graph ...
usgrupgr 28911 A simple graph is an undir...
usgruhgr 28912 A simple graph is an undir...
usgrislfuspgr 28913 A simple graph is a loop-f...
uspgrun 28914 The union ` U ` of two sim...
uspgrunop 28915 The union of two simple ps...
usgrun 28916 The union ` U ` of two sim...
usgrunop 28917 The union of two simple gr...
usgredg2 28918 The value of the "edge fun...
usgredg2ALT 28919 Alternate proof of ~ usgre...
usgredgprv 28920 In a simple graph, an edge...
usgredgprvALT 28921 Alternate proof of ~ usgre...
usgredgppr 28922 An edge of a simple graph ...
usgrpredgv 28923 An edge of a simple graph ...
edgssv2 28924 An edge of a simple graph ...
usgredg 28925 For each edge in a simple ...
usgrnloopv 28926 In a simple graph, there i...
usgrnloopvALT 28927 Alternate proof of ~ usgrn...
usgrnloop 28928 In a simple graph, there i...
usgrnloopALT 28929 Alternate proof of ~ usgrn...
usgrnloop0 28930 A simple graph has no loop...
usgrnloop0ALT 28931 Alternate proof of ~ usgrn...
usgredgne 28932 An edge of a simple graph ...
usgrf1oedg 28933 The edge function of a sim...
uhgr2edg 28934 If a vertex is adjacent to...
umgr2edg 28935 If a vertex is adjacent to...
usgr2edg 28936 If a vertex is adjacent to...
umgr2edg1 28937 If a vertex is adjacent to...
usgr2edg1 28938 If a vertex is adjacent to...
umgrvad2edg 28939 If a vertex is adjacent to...
umgr2edgneu 28940 If a vertex is adjacent to...
usgrsizedg 28941 In a simple graph, the siz...
usgredg3 28942 The value of the "edge fun...
usgredg4 28943 For a vertex incident to a...
usgredgreu 28944 For a vertex incident to a...
usgredg2vtx 28945 For a vertex incident to a...
uspgredg2vtxeu 28946 For a vertex incident to a...
usgredg2vtxeu 28947 For a vertex incident to a...
usgredg2vtxeuALT 28948 Alternate proof of ~ usgre...
uspgredg2vlem 28949 Lemma for ~ uspgredg2v . ...
uspgredg2v 28950 In a simple pseudograph, t...
usgredg2vlem1 28951 Lemma 1 for ~ usgredg2v . ...
usgredg2vlem2 28952 Lemma 2 for ~ usgredg2v . ...
usgredg2v 28953 In a simple graph, the map...
usgriedgleord 28954 Alternate version of ~ usg...
ushgredgedg 28955 In a simple hypergraph the...
usgredgedg 28956 In a simple graph there is...
ushgredgedgloop 28957 In a simple hypergraph the...
uspgredgleord 28958 In a simple pseudograph th...
usgredgleord 28959 In a simple graph the numb...
usgredgleordALT 28960 Alternate proof for ~ usgr...
usgrstrrepe 28961 Replacing (or adding) the ...
usgr0e 28962 The empty graph, with vert...
usgr0vb 28963 The null graph, with no ve...
uhgr0v0e 28964 The null graph, with no ve...
uhgr0vsize0 28965 The size of a hypergraph w...
uhgr0edgfi 28966 A graph of order 0 (i.e. w...
usgr0v 28967 The null graph, with no ve...
uhgr0vusgr 28968 The null graph, with no ve...
usgr0 28969 The null graph represented...
uspgr1e 28970 A simple pseudograph with ...
usgr1e 28971 A simple graph with one ed...
usgr0eop 28972 The empty graph, with vert...
uspgr1eop 28973 A simple pseudograph with ...
uspgr1ewop 28974 A simple pseudograph with ...
uspgr1v1eop 28975 A simple pseudograph with ...
usgr1eop 28976 A simple graph with (at le...
uspgr2v1e2w 28977 A simple pseudograph with ...
usgr2v1e2w 28978 A simple graph with two ve...
edg0usgr 28979 A class without edges is a...
lfuhgr1v0e 28980 A loop-free hypergraph wit...
usgr1vr 28981 A simple graph with one ve...
usgr1v 28982 A class with one (or no) v...
usgr1v0edg 28983 A class with one (or no) v...
usgrexmpldifpr 28984 Lemma for ~ usgrexmpledg :...
usgrexmplef 28985 Lemma for ~ usgrexmpl . (...
usgrexmpllem 28986 Lemma for ~ usgrexmpl . (...
usgrexmplvtx 28987 The vertices ` 0 , 1 , 2 ,...
usgrexmpledg 28988 The edges ` { 0 , 1 } , { ...
usgrexmpl 28989 ` G ` is a simple graph of...
griedg0prc 28990 The class of empty graphs ...
griedg0ssusgr 28991 The class of all simple gr...
usgrprc 28992 The class of simple graphs...
relsubgr 28995 The class of the subgraph ...
subgrv 28996 If a class is a subgraph o...
issubgr 28997 The property of a set to b...
issubgr2 28998 The property of a set to b...
subgrprop 28999 The properties of a subgra...
subgrprop2 29000 The properties of a subgra...
uhgrissubgr 29001 The property of a hypergra...
subgrprop3 29002 The properties of a subgra...
egrsubgr 29003 An empty graph consisting ...
0grsubgr 29004 The null graph (represente...
0uhgrsubgr 29005 The null graph (as hypergr...
uhgrsubgrself 29006 A hypergraph is a subgraph...
subgrfun 29007 The edge function of a sub...
subgruhgrfun 29008 The edge function of a sub...
subgreldmiedg 29009 An element of the domain o...
subgruhgredgd 29010 An edge of a subgraph of a...
subumgredg2 29011 An edge of a subgraph of a...
subuhgr 29012 A subgraph of a hypergraph...
subupgr 29013 A subgraph of a pseudograp...
subumgr 29014 A subgraph of a multigraph...
subusgr 29015 A subgraph of a simple gra...
uhgrspansubgrlem 29016 Lemma for ~ uhgrspansubgr ...
uhgrspansubgr 29017 A spanning subgraph ` S ` ...
uhgrspan 29018 A spanning subgraph ` S ` ...
upgrspan 29019 A spanning subgraph ` S ` ...
umgrspan 29020 A spanning subgraph ` S ` ...
usgrspan 29021 A spanning subgraph ` S ` ...
uhgrspanop 29022 A spanning subgraph of a h...
upgrspanop 29023 A spanning subgraph of a p...
umgrspanop 29024 A spanning subgraph of a m...
usgrspanop 29025 A spanning subgraph of a s...
uhgrspan1lem1 29026 Lemma 1 for ~ uhgrspan1 . ...
uhgrspan1lem2 29027 Lemma 2 for ~ uhgrspan1 . ...
uhgrspan1lem3 29028 Lemma 3 for ~ uhgrspan1 . ...
uhgrspan1 29029 The induced subgraph ` S `...
upgrreslem 29030 Lemma for ~ upgrres . (Co...
umgrreslem 29031 Lemma for ~ umgrres and ~ ...
upgrres 29032 A subgraph obtained by rem...
umgrres 29033 A subgraph obtained by rem...
usgrres 29034 A subgraph obtained by rem...
upgrres1lem1 29035 Lemma 1 for ~ upgrres1 . ...
umgrres1lem 29036 Lemma for ~ umgrres1 . (C...
upgrres1lem2 29037 Lemma 2 for ~ upgrres1 . ...
upgrres1lem3 29038 Lemma 3 for ~ upgrres1 . ...
upgrres1 29039 A pseudograph obtained by ...
umgrres1 29040 A multigraph obtained by r...
usgrres1 29041 Restricting a simple graph...
isfusgr 29044 The property of being a fi...
fusgrvtxfi 29045 A finite simple graph has ...
isfusgrf1 29046 The property of being a fi...
isfusgrcl 29047 The property of being a fi...
fusgrusgr 29048 A finite simple graph is a...
opfusgr 29049 A finite simple graph repr...
usgredgffibi 29050 The number of edges in a s...
fusgredgfi 29051 In a finite simple graph t...
usgr1v0e 29052 The size of a (finite) sim...
usgrfilem 29053 In a finite simple graph, ...
fusgrfisbase 29054 Induction base for ~ fusgr...
fusgrfisstep 29055 Induction step in ~ fusgrf...
fusgrfis 29056 A finite simple graph is o...
fusgrfupgrfs 29057 A finite simple graph is a...
nbgrprc0 29060 The set of neighbors is em...
nbgrcl 29061 If a class ` X ` has at le...
nbgrval 29062 The set of neighbors of a ...
dfnbgr2 29063 Alternate definition of th...
dfnbgr3 29064 Alternate definition of th...
nbgrnvtx0 29065 If a class ` X ` is not a ...
nbgrel 29066 Characterization of a neig...
nbgrisvtx 29067 Every neighbor ` N ` of a ...
nbgrssvtx 29068 The neighbors of a vertex ...
nbuhgr 29069 The set of neighbors of a ...
nbupgr 29070 The set of neighbors of a ...
nbupgrel 29071 A neighbor of a vertex in ...
nbumgrvtx 29072 The set of neighbors of a ...
nbumgr 29073 The set of neighbors of an...
nbusgrvtx 29074 The set of neighbors of a ...
nbusgr 29075 The set of neighbors of an...
nbgr2vtx1edg 29076 If a graph has two vertice...
nbuhgr2vtx1edgblem 29077 Lemma for ~ nbuhgr2vtx1edg...
nbuhgr2vtx1edgb 29078 If a hypergraph has two ve...
nbusgreledg 29079 A class/vertex is a neighb...
uhgrnbgr0nb 29080 A vertex which is not endp...
nbgr0vtxlem 29081 Lemma for ~ nbgr0vtx and ~...
nbgr0vtx 29082 In a null graph (with no v...
nbgr0edg 29083 In an empty graph (with no...
nbgr1vtx 29084 In a graph with one vertex...
nbgrnself 29085 A vertex in a graph is not...
nbgrnself2 29086 A class ` X ` is not a nei...
nbgrssovtx 29087 The neighbors of a vertex ...
nbgrssvwo2 29088 The neighbors of a vertex ...
nbgrsym 29089 In a graph, the neighborho...
nbupgrres 29090 The neighborhood of a vert...
usgrnbcnvfv 29091 Applying the edge function...
nbusgredgeu 29092 For each neighbor of a ver...
edgnbusgreu 29093 For each edge incident to ...
nbusgredgeu0 29094 For each neighbor of a ver...
nbusgrf1o0 29095 The mapping of neighbors o...
nbusgrf1o1 29096 The set of neighbors of a ...
nbusgrf1o 29097 The set of neighbors of a ...
nbedgusgr 29098 The number of neighbors of...
edgusgrnbfin 29099 The number of neighbors of...
nbusgrfi 29100 The class of neighbors of ...
nbfiusgrfi 29101 The class of neighbors of ...
hashnbusgrnn0 29102 The number of neighbors of...
nbfusgrlevtxm1 29103 The number of neighbors of...
nbfusgrlevtxm2 29104 If there is a vertex which...
nbusgrvtxm1 29105 If the number of neighbors...
nb3grprlem1 29106 Lemma 1 for ~ nb3grpr . (...
nb3grprlem2 29107 Lemma 2 for ~ nb3grpr . (...
nb3grpr 29108 The neighbors of a vertex ...
nb3grpr2 29109 The neighbors of a vertex ...
nb3gr2nb 29110 If the neighbors of two ve...
uvtxval 29113 The set of all universal v...
uvtxel 29114 A universal vertex, i.e. a...
uvtxisvtx 29115 A universal vertex is a ve...
uvtxssvtx 29116 The set of the universal v...
vtxnbuvtx 29117 A universal vertex has all...
uvtxnbgrss 29118 A universal vertex has all...
uvtxnbgrvtx 29119 A universal vertex is neig...
uvtx0 29120 There is no universal vert...
isuvtx 29121 The set of all universal v...
uvtxel1 29122 Characterization of a univ...
uvtx01vtx 29123 If a graph/class has no ed...
uvtx2vtx1edg 29124 If a graph has two vertice...
uvtx2vtx1edgb 29125 If a hypergraph has two ve...
uvtxnbgr 29126 A universal vertex has all...
uvtxnbgrb 29127 A vertex is universal iff ...
uvtxusgr 29128 The set of all universal v...
uvtxusgrel 29129 A universal vertex, i.e. a...
uvtxnm1nbgr 29130 A universal vertex has ` n...
nbusgrvtxm1uvtx 29131 If the number of neighbors...
uvtxnbvtxm1 29132 A universal vertex has ` n...
nbupgruvtxres 29133 The neighborhood of a univ...
uvtxupgrres 29134 A universal vertex is univ...
cplgruvtxb 29139 A graph ` G ` is complete ...
prcliscplgr 29140 A proper class (representi...
iscplgr 29141 The property of being a co...
iscplgrnb 29142 A graph is complete iff al...
iscplgredg 29143 A graph ` G ` is complete ...
iscusgr 29144 The property of being a co...
cusgrusgr 29145 A complete simple graph is...
cusgrcplgr 29146 A complete simple graph is...
iscusgrvtx 29147 A simple graph is complete...
cusgruvtxb 29148 A simple graph is complete...
iscusgredg 29149 A simple graph is complete...
cusgredg 29150 In a complete simple graph...
cplgr0 29151 The null graph (with no ve...
cusgr0 29152 The null graph (with no ve...
cplgr0v 29153 A null graph (with no vert...
cusgr0v 29154 A graph with no vertices a...
cplgr1vlem 29155 Lemma for ~ cplgr1v and ~ ...
cplgr1v 29156 A graph with one vertex is...
cusgr1v 29157 A graph with one vertex an...
cplgr2v 29158 An undirected hypergraph w...
cplgr2vpr 29159 An undirected hypergraph w...
nbcplgr 29160 In a complete graph, each ...
cplgr3v 29161 A pseudograph with three (...
cusgr3vnbpr 29162 The neighbors of a vertex ...
cplgrop 29163 A complete graph represent...
cusgrop 29164 A complete simple graph re...
cusgrexilem1 29165 Lemma 1 for ~ cusgrexi . ...
usgrexilem 29166 Lemma for ~ usgrexi . (Co...
usgrexi 29167 An arbitrary set regarded ...
cusgrexilem2 29168 Lemma 2 for ~ cusgrexi . ...
cusgrexi 29169 An arbitrary set ` V ` reg...
cusgrexg 29170 For each set there is a se...
structtousgr 29171 Any (extensible) structure...
structtocusgr 29172 Any (extensible) structure...
cffldtocusgr 29173 The field of complex numbe...
cusgrres 29174 Restricting a complete sim...
cusgrsizeindb0 29175 Base case of the induction...
cusgrsizeindb1 29176 Base case of the induction...
cusgrsizeindslem 29177 Lemma for ~ cusgrsizeinds ...
cusgrsizeinds 29178 Part 1 of induction step i...
cusgrsize2inds 29179 Induction step in ~ cusgrs...
cusgrsize 29180 The size of a finite compl...
cusgrfilem1 29181 Lemma 1 for ~ cusgrfi . (...
cusgrfilem2 29182 Lemma 2 for ~ cusgrfi . (...
cusgrfilem3 29183 Lemma 3 for ~ cusgrfi . (...
cusgrfi 29184 If the size of a complete ...
usgredgsscusgredg 29185 A simple graph is a subgra...
usgrsscusgr 29186 A simple graph is a subgra...
sizusglecusglem1 29187 Lemma 1 for ~ sizusglecusg...
sizusglecusglem2 29188 Lemma 2 for ~ sizusglecusg...
sizusglecusg 29189 The size of a simple graph...
fusgrmaxsize 29190 The maximum size of a fini...
vtxdgfval 29193 The value of the vertex de...
vtxdgval 29194 The degree of a vertex. (...
vtxdgfival 29195 The degree of a vertex for...
vtxdgop 29196 The vertex degree expresse...
vtxdgf 29197 The vertex degree function...
vtxdgelxnn0 29198 The degree of a vertex is ...
vtxdg0v 29199 The degree of a vertex in ...
vtxdg0e 29200 The degree of a vertex in ...
vtxdgfisnn0 29201 The degree of a vertex in ...
vtxdgfisf 29202 The vertex degree function...
vtxdeqd 29203 Equality theorem for the v...
vtxduhgr0e 29204 The degree of a vertex in ...
vtxdlfuhgr1v 29205 The degree of the vertex i...
vdumgr0 29206 A vertex in a multigraph h...
vtxdun 29207 The degree of a vertex in ...
vtxdfiun 29208 The degree of a vertex in ...
vtxduhgrun 29209 The degree of a vertex in ...
vtxduhgrfiun 29210 The degree of a vertex in ...
vtxdlfgrval 29211 The value of the vertex de...
vtxdumgrval 29212 The value of the vertex de...
vtxdusgrval 29213 The value of the vertex de...
vtxd0nedgb 29214 A vertex has degree 0 iff ...
vtxdushgrfvedglem 29215 Lemma for ~ vtxdushgrfvedg...
vtxdushgrfvedg 29216 The value of the vertex de...
vtxdusgrfvedg 29217 The value of the vertex de...
vtxduhgr0nedg 29218 If a vertex in a hypergrap...
vtxdumgr0nedg 29219 If a vertex in a multigrap...
vtxduhgr0edgnel 29220 A vertex in a hypergraph h...
vtxdusgr0edgnel 29221 A vertex in a simple graph...
vtxdusgr0edgnelALT 29222 Alternate proof of ~ vtxdu...
vtxdgfusgrf 29223 The vertex degree function...
vtxdgfusgr 29224 In a finite simple graph, ...
fusgrn0degnn0 29225 In a nonempty, finite grap...
1loopgruspgr 29226 A graph with one edge whic...
1loopgredg 29227 The set of edges in a grap...
1loopgrnb0 29228 In a graph (simple pseudog...
1loopgrvd2 29229 The vertex degree of a one...
1loopgrvd0 29230 The vertex degree of a one...
1hevtxdg0 29231 The vertex degree of verte...
1hevtxdg1 29232 The vertex degree of verte...
1hegrvtxdg1 29233 The vertex degree of a gra...
1hegrvtxdg1r 29234 The vertex degree of a gra...
1egrvtxdg1 29235 The vertex degree of a one...
1egrvtxdg1r 29236 The vertex degree of a one...
1egrvtxdg0 29237 The vertex degree of a one...
p1evtxdeqlem 29238 Lemma for ~ p1evtxdeq and ...
p1evtxdeq 29239 If an edge ` E ` which doe...
p1evtxdp1 29240 If an edge ` E ` (not bein...
uspgrloopvtx 29241 The set of vertices in a g...
uspgrloopvtxel 29242 A vertex in a graph (simpl...
uspgrloopiedg 29243 The set of edges in a grap...
uspgrloopedg 29244 The set of edges in a grap...
uspgrloopnb0 29245 In a graph (simple pseudog...
uspgrloopvd2 29246 The vertex degree of a one...
umgr2v2evtx 29247 The set of vertices in a m...
umgr2v2evtxel 29248 A vertex in a multigraph w...
umgr2v2eiedg 29249 The edge function in a mul...
umgr2v2eedg 29250 The set of edges in a mult...
umgr2v2e 29251 A multigraph with two edge...
umgr2v2enb1 29252 In a multigraph with two e...
umgr2v2evd2 29253 In a multigraph with two e...
hashnbusgrvd 29254 In a simple graph, the num...
usgruvtxvdb 29255 In a finite simple graph w...
vdiscusgrb 29256 A finite simple graph with...
vdiscusgr 29257 In a finite complete simpl...
vtxdusgradjvtx 29258 The degree of a vertex in ...
usgrvd0nedg 29259 If a vertex in a simple gr...
uhgrvd00 29260 If every vertex in a hyper...
usgrvd00 29261 If every vertex in a simpl...
vdegp1ai 29262 The induction step for a v...
vdegp1bi 29263 The induction step for a v...
vdegp1ci 29264 The induction step for a v...
vtxdginducedm1lem1 29265 Lemma 1 for ~ vtxdginduced...
vtxdginducedm1lem2 29266 Lemma 2 for ~ vtxdginduced...
vtxdginducedm1lem3 29267 Lemma 3 for ~ vtxdginduced...
vtxdginducedm1lem4 29268 Lemma 4 for ~ vtxdginduced...
vtxdginducedm1 29269 The degree of a vertex ` v...
vtxdginducedm1fi 29270 The degree of a vertex ` v...
finsumvtxdg2ssteplem1 29271 Lemma for ~ finsumvtxdg2ss...
finsumvtxdg2ssteplem2 29272 Lemma for ~ finsumvtxdg2ss...
finsumvtxdg2ssteplem3 29273 Lemma for ~ finsumvtxdg2ss...
finsumvtxdg2ssteplem4 29274 Lemma for ~ finsumvtxdg2ss...
finsumvtxdg2sstep 29275 Induction step of ~ finsum...
finsumvtxdg2size 29276 The sum of the degrees of ...
fusgr1th 29277 The sum of the degrees of ...
finsumvtxdgeven 29278 The sum of the degrees of ...
vtxdgoddnumeven 29279 The number of vertices of ...
fusgrvtxdgonume 29280 The number of vertices of ...
isrgr 29285 The property of a class be...
rgrprop 29286 The properties of a k-regu...
isrusgr 29287 The property of being a k-...
rusgrprop 29288 The properties of a k-regu...
rusgrrgr 29289 A k-regular simple graph i...
rusgrusgr 29290 A k-regular simple graph i...
finrusgrfusgr 29291 A finite regular simple gr...
isrusgr0 29292 The property of being a k-...
rusgrprop0 29293 The properties of a k-regu...
usgreqdrusgr 29294 If all vertices in a simpl...
fusgrregdegfi 29295 In a nonempty finite simpl...
fusgrn0eqdrusgr 29296 If all vertices in a nonem...
frusgrnn0 29297 In a nonempty finite k-reg...
0edg0rgr 29298 A graph is 0-regular if it...
uhgr0edg0rgr 29299 A hypergraph is 0-regular ...
uhgr0edg0rgrb 29300 A hypergraph is 0-regular ...
usgr0edg0rusgr 29301 A simple graph is 0-regula...
0vtxrgr 29302 A null graph (with no vert...
0vtxrusgr 29303 A graph with no vertices a...
0uhgrrusgr 29304 The null graph as hypergra...
0grrusgr 29305 The null graph represented...
0grrgr 29306 The null graph represented...
cusgrrusgr 29307 A complete simple graph wi...
cusgrm1rusgr 29308 A finite simple graph with...
rusgrpropnb 29309 The properties of a k-regu...
rusgrpropedg 29310 The properties of a k-regu...
rusgrpropadjvtx 29311 The properties of a k-regu...
rusgrnumwrdl2 29312 In a k-regular simple grap...
rusgr1vtxlem 29313 Lemma for ~ rusgr1vtx . (...
rusgr1vtx 29314 If a k-regular simple grap...
rgrusgrprc 29315 The class of 0-regular sim...
rusgrprc 29316 The class of 0-regular sim...
rgrprc 29317 The class of 0-regular gra...
rgrprcx 29318 The class of 0-regular gra...
rgrx0ndm 29319 0 is not in the domain of ...
rgrx0nd 29320 The potentially alternativ...
ewlksfval 29327 The set of s-walks of edge...
isewlk 29328 Conditions for a function ...
ewlkprop 29329 Properties of an s-walk of...
ewlkinedg 29330 The intersection (common v...
ewlkle 29331 An s-walk of edges is also...
upgrewlkle2 29332 In a pseudograph, there is...
wkslem1 29333 Lemma 1 for walks to subst...
wkslem2 29334 Lemma 2 for walks to subst...
wksfval 29335 The set of walks (in an un...
iswlk 29336 Properties of a pair of fu...
wlkprop 29337 Properties of a walk. (Co...
wlkv 29338 The classes involved in a ...
iswlkg 29339 Generalization of ~ iswlk ...
wlkf 29340 The mapping enumerating th...
wlkcl 29341 A walk has length ` # ( F ...
wlkp 29342 The mapping enumerating th...
wlkpwrd 29343 The sequence of vertices o...
wlklenvp1 29344 The number of vertices of ...
wksv 29345 The class of walks is a se...
wksvOLD 29346 Obsolete version of ~ wksv...
wlkn0 29347 The sequence of vertices o...
wlklenvm1 29348 The number of edges of a w...
ifpsnprss 29349 Lemma for ~ wlkvtxeledg : ...
wlkvtxeledg 29350 Each pair of adjacent vert...
wlkvtxiedg 29351 The vertices of a walk are...
relwlk 29352 The set ` ( Walks `` G ) `...
wlkvv 29353 If there is at least one w...
wlkop 29354 A walk is an ordered pair....
wlkcpr 29355 A walk as class with two c...
wlk2f 29356 If there is a walk ` W ` t...
wlkcomp 29357 A walk expressed by proper...
wlkcompim 29358 Implications for the prope...
wlkelwrd 29359 The components of a walk a...
wlkeq 29360 Conditions for two walks (...
edginwlk 29361 The value of the edge func...
upgredginwlk 29362 The value of the edge func...
iedginwlk 29363 The value of the edge func...
wlkl1loop 29364 A walk of length 1 from a ...
wlk1walk 29365 A walk is a 1-walk "on the...
wlk1ewlk 29366 A walk is an s-walk "on th...
upgriswlk 29367 Properties of a pair of fu...
upgrwlkedg 29368 The edges of a walk in a p...
upgrwlkcompim 29369 Implications for the prope...
wlkvtxedg 29370 The vertices of a walk are...
upgrwlkvtxedg 29371 The pairs of connected ver...
uspgr2wlkeq 29372 Conditions for two walks w...
uspgr2wlkeq2 29373 Conditions for two walks w...
uspgr2wlkeqi 29374 Conditions for two walks w...
umgrwlknloop 29375 In a multigraph, each walk...
wlkResOLD 29376 Obsolete version of ~ opab...
wlkv0 29377 If there is a walk in the ...
g0wlk0 29378 There is no walk in a null...
0wlk0 29379 There is no walk for the e...
wlk0prc 29380 There is no walk in a null...
wlklenvclwlk 29381 The number of vertices in ...
wlkson 29382 The set of walks between t...
iswlkon 29383 Properties of a pair of fu...
wlkonprop 29384 Properties of a walk betwe...
wlkpvtx 29385 A walk connects vertices. ...
wlkepvtx 29386 The endpoints of a walk ar...
wlkoniswlk 29387 A walk between two vertice...
wlkonwlk 29388 A walk is a walk between i...
wlkonwlk1l 29389 A walk is a walk from its ...
wlksoneq1eq2 29390 Two walks with identical s...
wlkonl1iedg 29391 If there is a walk between...
wlkon2n0 29392 The length of a walk betwe...
2wlklem 29393 Lemma for theorems for wal...
upgr2wlk 29394 Properties of a pair of fu...
wlkreslem 29395 Lemma for ~ wlkres . (Con...
wlkres 29396 The restriction ` <. H , Q...
redwlklem 29397 Lemma for ~ redwlk . (Con...
redwlk 29398 A walk ending at the last ...
wlkp1lem1 29399 Lemma for ~ wlkp1 . (Cont...
wlkp1lem2 29400 Lemma for ~ wlkp1 . (Cont...
wlkp1lem3 29401 Lemma for ~ wlkp1 . (Cont...
wlkp1lem4 29402 Lemma for ~ wlkp1 . (Cont...
wlkp1lem5 29403 Lemma for ~ wlkp1 . (Cont...
wlkp1lem6 29404 Lemma for ~ wlkp1 . (Cont...
wlkp1lem7 29405 Lemma for ~ wlkp1 . (Cont...
wlkp1lem8 29406 Lemma for ~ wlkp1 . (Cont...
wlkp1 29407 Append one path segment (e...
wlkdlem1 29408 Lemma 1 for ~ wlkd . (Con...
wlkdlem2 29409 Lemma 2 for ~ wlkd . (Con...
wlkdlem3 29410 Lemma 3 for ~ wlkd . (Con...
wlkdlem4 29411 Lemma 4 for ~ wlkd . (Con...
wlkd 29412 Two words representing a w...
lfgrwlkprop 29413 Two adjacent vertices in a...
lfgriswlk 29414 Conditions for a pair of f...
lfgrwlknloop 29415 In a loop-free graph, each...
reltrls 29420 The set ` ( Trails `` G ) ...
trlsfval 29421 The set of trails (in an u...
istrl 29422 Conditions for a pair of c...
trliswlk 29423 A trail is a walk. (Contr...
trlf1 29424 The enumeration ` F ` of a...
trlreslem 29425 Lemma for ~ trlres . Form...
trlres 29426 The restriction ` <. H , Q...
upgrtrls 29427 The set of trails in a pse...
upgristrl 29428 Properties of a pair of fu...
upgrf1istrl 29429 Properties of a pair of a ...
wksonproplem 29430 Lemma for theorems for pro...
wksonproplemOLD 29431 Obsolete version of ~ wkso...
trlsonfval 29432 The set of trails between ...
istrlson 29433 Properties of a pair of fu...
trlsonprop 29434 Properties of a trail betw...
trlsonistrl 29435 A trail between two vertic...
trlsonwlkon 29436 A trail between two vertic...
trlontrl 29437 A trail is a trail between...
relpths 29446 The set ` ( Paths `` G ) `...
pthsfval 29447 The set of paths (in an un...
spthsfval 29448 The set of simple paths (i...
ispth 29449 Conditions for a pair of c...
isspth 29450 Conditions for a pair of c...
pthistrl 29451 A path is a trail (in an u...
spthispth 29452 A simple path is a path (i...
pthiswlk 29453 A path is a walk (in an un...
spthiswlk 29454 A simple path is a walk (i...
pthdivtx 29455 The inner vertices of a pa...
pthdadjvtx 29456 The adjacent vertices of a...
2pthnloop 29457 A path of length at least ...
upgr2pthnlp 29458 A path of length at least ...
spthdifv 29459 The vertices of a simple p...
spthdep 29460 A simple path (at least of...
pthdepisspth 29461 A path with different star...
upgrwlkdvdelem 29462 Lemma for ~ upgrwlkdvde . ...
upgrwlkdvde 29463 In a pseudograph, all edge...
upgrspthswlk 29464 The set of simple paths in...
upgrwlkdvspth 29465 A walk consisting of diffe...
pthsonfval 29466 The set of paths between t...
spthson 29467 The set of simple paths be...
ispthson 29468 Properties of a pair of fu...
isspthson 29469 Properties of a pair of fu...
pthsonprop 29470 Properties of a path betwe...
spthonprop 29471 Properties of a simple pat...
pthonispth 29472 A path between two vertice...
pthontrlon 29473 A path between two vertice...
pthonpth 29474 A path is a path between i...
isspthonpth 29475 A pair of functions is a s...
spthonisspth 29476 A simple path between to v...
spthonpthon 29477 A simple path between two ...
spthonepeq 29478 The endpoints of a simple ...
uhgrwkspthlem1 29479 Lemma 1 for ~ uhgrwkspth ....
uhgrwkspthlem2 29480 Lemma 2 for ~ uhgrwkspth ....
uhgrwkspth 29481 Any walk of length 1 betwe...
usgr2wlkneq 29482 The vertices and edges are...
usgr2wlkspthlem1 29483 Lemma 1 for ~ usgr2wlkspth...
usgr2wlkspthlem2 29484 Lemma 2 for ~ usgr2wlkspth...
usgr2wlkspth 29485 In a simple graph, any wal...
usgr2trlncl 29486 In a simple graph, any tra...
usgr2trlspth 29487 In a simple graph, any tra...
usgr2pthspth 29488 In a simple graph, any pat...
usgr2pthlem 29489 Lemma for ~ usgr2pth . (C...
usgr2pth 29490 In a simple graph, there i...
usgr2pth0 29491 In a simply graph, there i...
pthdlem1 29492 Lemma 1 for ~ pthd . (Con...
pthdlem2lem 29493 Lemma for ~ pthdlem2 . (C...
pthdlem2 29494 Lemma 2 for ~ pthd . (Con...
pthd 29495 Two words representing a t...
clwlks 29498 The set of closed walks (i...
isclwlk 29499 A pair of functions repres...
clwlkiswlk 29500 A closed walk is a walk (i...
clwlkwlk 29501 Closed walks are walks (in...
clwlkswks 29502 Closed walks are walks (in...
isclwlke 29503 Properties of a pair of fu...
isclwlkupgr 29504 Properties of a pair of fu...
clwlkcomp 29505 A closed walk expressed by...
clwlkcompim 29506 Implications for the prope...
upgrclwlkcompim 29507 Implications for the prope...
clwlkcompbp 29508 Basic properties of the co...
clwlkl1loop 29509 A closed walk of length 1 ...
crcts 29514 The set of circuits (in an...
cycls 29515 The set of cycles (in an u...
iscrct 29516 Sufficient and necessary c...
iscycl 29517 Sufficient and necessary c...
crctprop 29518 The properties of a circui...
cyclprop 29519 The properties of a cycle:...
crctisclwlk 29520 A circuit is a closed walk...
crctistrl 29521 A circuit is a trail. (Co...
crctiswlk 29522 A circuit is a walk. (Con...
cyclispth 29523 A cycle is a path. (Contr...
cycliswlk 29524 A cycle is a walk. (Contr...
cycliscrct 29525 A cycle is a circuit. (Co...
cyclnspth 29526 A (non-trivial) cycle is n...
cyclispthon 29527 A cycle is a path starting...
lfgrn1cycl 29528 In a loop-free graph there...
usgr2trlncrct 29529 In a simple graph, any tra...
umgrn1cycl 29530 In a multigraph graph (wit...
uspgrn2crct 29531 In a simple pseudograph th...
usgrn2cycl 29532 In a simple graph there ar...
crctcshwlkn0lem1 29533 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem2 29534 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem3 29535 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem4 29536 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem5 29537 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem6 29538 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem7 29539 Lemma for ~ crctcshwlkn0 ....
crctcshlem1 29540 Lemma for ~ crctcsh . (Co...
crctcshlem2 29541 Lemma for ~ crctcsh . (Co...
crctcshlem3 29542 Lemma for ~ crctcsh . (Co...
crctcshlem4 29543 Lemma for ~ crctcsh . (Co...
crctcshwlkn0 29544 Cyclically shifting the in...
crctcshwlk 29545 Cyclically shifting the in...
crctcshtrl 29546 Cyclically shifting the in...
crctcsh 29547 Cyclically shifting the in...
wwlks 29558 The set of walks (in an un...
iswwlks 29559 A word over the set of ver...
wwlksn 29560 The set of walks (in an un...
iswwlksn 29561 A word over the set of ver...
wwlksnprcl 29562 Derivation of the length o...
iswwlksnx 29563 Properties of a word to re...
wwlkbp 29564 Basic properties of a walk...
wwlknbp 29565 Basic properties of a walk...
wwlknp 29566 Properties of a set being ...
wwlknbp1 29567 Other basic properties of ...
wwlknvtx 29568 The symbols of a word ` W ...
wwlknllvtx 29569 If a word ` W ` represents...
wwlknlsw 29570 If a word represents a wal...
wspthsn 29571 The set of simple paths of...
iswspthn 29572 An element of the set of s...
wspthnp 29573 Properties of a set being ...
wwlksnon 29574 The set of walks of a fixe...
wspthsnon 29575 The set of simple paths of...
iswwlksnon 29576 The set of walks of a fixe...
wwlksnon0 29577 Sufficient conditions for ...
wwlksonvtx 29578 If a word ` W ` represents...
iswspthsnon 29579 The set of simple paths of...
wwlknon 29580 An element of the set of w...
wspthnon 29581 An element of the set of s...
wspthnonp 29582 Properties of a set being ...
wspthneq1eq2 29583 Two simple paths with iden...
wwlksn0s 29584 The set of all walks as wo...
wwlkssswrd 29585 Walks (represented by word...
wwlksn0 29586 A walk of length 0 is repr...
0enwwlksnge1 29587 In graphs without edges, t...
wwlkswwlksn 29588 A walk of a fixed length a...
wwlkssswwlksn 29589 The walks of a fixed lengt...
wlkiswwlks1 29590 The sequence of vertices i...
wlklnwwlkln1 29591 The sequence of vertices i...
wlkiswwlks2lem1 29592 Lemma 1 for ~ wlkiswwlks2 ...
wlkiswwlks2lem2 29593 Lemma 2 for ~ wlkiswwlks2 ...
wlkiswwlks2lem3 29594 Lemma 3 for ~ wlkiswwlks2 ...
wlkiswwlks2lem4 29595 Lemma 4 for ~ wlkiswwlks2 ...
wlkiswwlks2lem5 29596 Lemma 5 for ~ wlkiswwlks2 ...
wlkiswwlks2lem6 29597 Lemma 6 for ~ wlkiswwlks2 ...
wlkiswwlks2 29598 A walk as word corresponds...
wlkiswwlks 29599 A walk as word corresponds...
wlkiswwlksupgr2 29600 A walk as word corresponds...
wlkiswwlkupgr 29601 A walk as word corresponds...
wlkswwlksf1o 29602 The mapping of (ordinary) ...
wlkswwlksen 29603 The set of walks as words ...
wwlksm1edg 29604 Removing the trailing edge...
wlklnwwlkln2lem 29605 Lemma for ~ wlklnwwlkln2 a...
wlklnwwlkln2 29606 A walk of length ` N ` as ...
wlklnwwlkn 29607 A walk of length ` N ` as ...
wlklnwwlklnupgr2 29608 A walk of length ` N ` as ...
wlklnwwlknupgr 29609 A walk of length ` N ` as ...
wlknewwlksn 29610 If a walk in a pseudograph...
wlknwwlksnbij 29611 The mapping ` ( t e. T |->...
wlknwwlksnen 29612 In a simple pseudograph, t...
wlknwwlksneqs 29613 The set of walks of a fixe...
wwlkseq 29614 Equality of two walks (as ...
wwlksnred 29615 Reduction of a walk (as wo...
wwlksnext 29616 Extension of a walk (as wo...
wwlksnextbi 29617 Extension of a walk (as wo...
wwlksnredwwlkn 29618 For each walk (as word) of...
wwlksnredwwlkn0 29619 For each walk (as word) of...
wwlksnextwrd 29620 Lemma for ~ wwlksnextbij ....
wwlksnextfun 29621 Lemma for ~ wwlksnextbij ....
wwlksnextinj 29622 Lemma for ~ wwlksnextbij ....
wwlksnextsurj 29623 Lemma for ~ wwlksnextbij ....
wwlksnextbij0 29624 Lemma for ~ wwlksnextbij ....
wwlksnextbij 29625 There is a bijection betwe...
wwlksnexthasheq 29626 The number of the extensio...
disjxwwlksn 29627 Sets of walks (as words) e...
wwlksnndef 29628 Conditions for ` WWalksN `...
wwlksnfi 29629 The number of walks repres...
wlksnfi 29630 The number of walks of fix...
wlksnwwlknvbij 29631 There is a bijection betwe...
wwlksnextproplem1 29632 Lemma 1 for ~ wwlksnextpro...
wwlksnextproplem2 29633 Lemma 2 for ~ wwlksnextpro...
wwlksnextproplem3 29634 Lemma 3 for ~ wwlksnextpro...
wwlksnextprop 29635 Adding additional properti...
disjxwwlkn 29636 Sets of walks (as words) e...
hashwwlksnext 29637 Number of walks (as words)...
wwlksnwwlksnon 29638 A walk of fixed length is ...
wspthsnwspthsnon 29639 A simple path of fixed len...
wspthsnonn0vne 29640 If the set of simple paths...
wspthsswwlkn 29641 The set of simple paths of...
wspthnfi 29642 In a finite graph, the set...
wwlksnonfi 29643 In a finite graph, the set...
wspthsswwlknon 29644 The set of simple paths of...
wspthnonfi 29645 In a finite graph, the set...
wspniunwspnon 29646 The set of nonempty simple...
wspn0 29647 If there are no vertices, ...
2wlkdlem1 29648 Lemma 1 for ~ 2wlkd . (Co...
2wlkdlem2 29649 Lemma 2 for ~ 2wlkd . (Co...
2wlkdlem3 29650 Lemma 3 for ~ 2wlkd . (Co...
2wlkdlem4 29651 Lemma 4 for ~ 2wlkd . (Co...
2wlkdlem5 29652 Lemma 5 for ~ 2wlkd . (Co...
2pthdlem1 29653 Lemma 1 for ~ 2pthd . (Co...
2wlkdlem6 29654 Lemma 6 for ~ 2wlkd . (Co...
2wlkdlem7 29655 Lemma 7 for ~ 2wlkd . (Co...
2wlkdlem8 29656 Lemma 8 for ~ 2wlkd . (Co...
2wlkdlem9 29657 Lemma 9 for ~ 2wlkd . (Co...
2wlkdlem10 29658 Lemma 10 for ~ 3wlkd . (C...
2wlkd 29659 Construction of a walk fro...
2wlkond 29660 A walk of length 2 from on...
2trld 29661 Construction of a trail fr...
2trlond 29662 A trail of length 2 from o...
2pthd 29663 A path of length 2 from on...
2spthd 29664 A simple path of length 2 ...
2pthond 29665 A simple path of length 2 ...
2pthon3v 29666 For a vertex adjacent to t...
umgr2adedgwlklem 29667 Lemma for ~ umgr2adedgwlk ...
umgr2adedgwlk 29668 In a multigraph, two adjac...
umgr2adedgwlkon 29669 In a multigraph, two adjac...
umgr2adedgwlkonALT 29670 Alternate proof for ~ umgr...
umgr2adedgspth 29671 In a multigraph, two adjac...
umgr2wlk 29672 In a multigraph, there is ...
umgr2wlkon 29673 For each pair of adjacent ...
elwwlks2s3 29674 A walk of length 2 as word...
midwwlks2s3 29675 There is a vertex between ...
wwlks2onv 29676 If a length 3 string repre...
elwwlks2ons3im 29677 A walk as word of length 2...
elwwlks2ons3 29678 For each walk of length 2 ...
s3wwlks2on 29679 A length 3 string which re...
umgrwwlks2on 29680 A walk of length 2 between...
wwlks2onsym 29681 There is a walk of length ...
elwwlks2on 29682 A walk of length 2 between...
elwspths2on 29683 A simple path of length 2 ...
wpthswwlks2on 29684 For two different vertices...
2wspdisj 29685 All simple paths of length...
2wspiundisj 29686 All simple paths of length...
usgr2wspthons3 29687 A simple path of length 2 ...
usgr2wspthon 29688 A simple path of length 2 ...
elwwlks2 29689 A walk of length 2 between...
elwspths2spth 29690 A simple path of length 2 ...
rusgrnumwwlkl1 29691 In a k-regular graph, ther...
rusgrnumwwlkslem 29692 Lemma for ~ rusgrnumwwlks ...
rusgrnumwwlklem 29693 Lemma for ~ rusgrnumwwlk e...
rusgrnumwwlkb0 29694 Induction base 0 for ~ rus...
rusgrnumwwlkb1 29695 Induction base 1 for ~ rus...
rusgr0edg 29696 Special case for graphs wi...
rusgrnumwwlks 29697 Induction step for ~ rusgr...
rusgrnumwwlk 29698 In a ` K `-regular graph, ...
rusgrnumwwlkg 29699 In a ` K `-regular graph, ...
rusgrnumwlkg 29700 In a k-regular graph, the ...
clwwlknclwwlkdif 29701 The set ` A ` of walks of ...
clwwlknclwwlkdifnum 29702 In a ` K `-regular graph, ...
clwwlk 29705 The set of closed walks (i...
isclwwlk 29706 Properties of a word to re...
clwwlkbp 29707 Basic properties of a clos...
clwwlkgt0 29708 There is no empty closed w...
clwwlksswrd 29709 Closed walks (represented ...
clwwlk1loop 29710 A closed walk of length 1 ...
clwwlkccatlem 29711 Lemma for ~ clwwlkccat : i...
clwwlkccat 29712 The concatenation of two w...
umgrclwwlkge2 29713 A closed walk in a multigr...
clwlkclwwlklem2a1 29714 Lemma 1 for ~ clwlkclwwlkl...
clwlkclwwlklem2a2 29715 Lemma 2 for ~ clwlkclwwlkl...
clwlkclwwlklem2a3 29716 Lemma 3 for ~ clwlkclwwlkl...
clwlkclwwlklem2fv1 29717 Lemma 4a for ~ clwlkclwwlk...
clwlkclwwlklem2fv2 29718 Lemma 4b for ~ clwlkclwwlk...
clwlkclwwlklem2a4 29719 Lemma 4 for ~ clwlkclwwlkl...
clwlkclwwlklem2a 29720 Lemma for ~ clwlkclwwlklem...
clwlkclwwlklem1 29721 Lemma 1 for ~ clwlkclwwlk ...
clwlkclwwlklem2 29722 Lemma 2 for ~ clwlkclwwlk ...
clwlkclwwlklem3 29723 Lemma 3 for ~ clwlkclwwlk ...
clwlkclwwlk 29724 A closed walk as word of l...
clwlkclwwlk2 29725 A closed walk corresponds ...
clwlkclwwlkflem 29726 Lemma for ~ clwlkclwwlkf ....
clwlkclwwlkf1lem2 29727 Lemma 2 for ~ clwlkclwwlkf...
clwlkclwwlkf1lem3 29728 Lemma 3 for ~ clwlkclwwlkf...
clwlkclwwlkfolem 29729 Lemma for ~ clwlkclwwlkfo ...
clwlkclwwlkf 29730 ` F ` is a function from t...
clwlkclwwlkfo 29731 ` F ` is a function from t...
clwlkclwwlkf1 29732 ` F ` is a one-to-one func...
clwlkclwwlkf1o 29733 ` F ` is a bijection betwe...
clwlkclwwlken 29734 The set of the nonempty cl...
clwwisshclwwslemlem 29735 Lemma for ~ clwwisshclwwsl...
clwwisshclwwslem 29736 Lemma for ~ clwwisshclwws ...
clwwisshclwws 29737 Cyclically shifting a clos...
clwwisshclwwsn 29738 Cyclically shifting a clos...
erclwwlkrel 29739 ` .~ ` is a relation. (Co...
erclwwlkeq 29740 Two classes are equivalent...
erclwwlkeqlen 29741 If two classes are equival...
erclwwlkref 29742 ` .~ ` is a reflexive rela...
erclwwlksym 29743 ` .~ ` is a symmetric rela...
erclwwlktr 29744 ` .~ ` is a transitive rel...
erclwwlk 29745 ` .~ ` is an equivalence r...
clwwlkn 29748 The set of closed walks of...
isclwwlkn 29749 A word over the set of ver...
clwwlkn0 29750 There is no closed walk of...
clwwlkneq0 29751 Sufficient conditions for ...
clwwlkclwwlkn 29752 A closed walk of a fixed l...
clwwlksclwwlkn 29753 The closed walks of a fixe...
clwwlknlen 29754 The length of a word repre...
clwwlknnn 29755 The length of a closed wal...
clwwlknwrd 29756 A closed walk of a fixed l...
clwwlknbp 29757 Basic properties of a clos...
isclwwlknx 29758 Characterization of a word...
clwwlknp 29759 Properties of a set being ...
clwwlknwwlksn 29760 A word representing a clos...
clwwlknlbonbgr1 29761 The last but one vertex in...
clwwlkinwwlk 29762 If the initial vertex of a...
clwwlkn1 29763 A closed walk of length 1 ...
loopclwwlkn1b 29764 The singleton word consist...
clwwlkn1loopb 29765 A word represents a closed...
clwwlkn2 29766 A closed walk of length 2 ...
clwwlknfi 29767 If there is only a finite ...
clwwlkel 29768 Obtaining a closed walk (a...
clwwlkf 29769 Lemma 1 for ~ clwwlkf1o : ...
clwwlkfv 29770 Lemma 2 for ~ clwwlkf1o : ...
clwwlkf1 29771 Lemma 3 for ~ clwwlkf1o : ...
clwwlkfo 29772 Lemma 4 for ~ clwwlkf1o : ...
clwwlkf1o 29773 F is a 1-1 onto function, ...
clwwlken 29774 The set of closed walks of...
clwwlknwwlkncl 29775 Obtaining a closed walk (a...
clwwlkwwlksb 29776 A nonempty word over verti...
clwwlknwwlksnb 29777 A word over vertices repre...
clwwlkext2edg 29778 If a word concatenated wit...
wwlksext2clwwlk 29779 If a word represents a wal...
wwlksubclwwlk 29780 Any prefix of a word repre...
clwwnisshclwwsn 29781 Cyclically shifting a clos...
eleclclwwlknlem1 29782 Lemma 1 for ~ eleclclwwlkn...
eleclclwwlknlem2 29783 Lemma 2 for ~ eleclclwwlkn...
clwwlknscsh 29784 The set of cyclical shifts...
clwwlknccat 29785 The concatenation of two w...
umgr2cwwk2dif 29786 If a word represents a clo...
umgr2cwwkdifex 29787 If a word represents a clo...
erclwwlknrel 29788 ` .~ ` is a relation. (Co...
erclwwlkneq 29789 Two classes are equivalent...
erclwwlkneqlen 29790 If two classes are equival...
erclwwlknref 29791 ` .~ ` is a reflexive rela...
erclwwlknsym 29792 ` .~ ` is a symmetric rela...
erclwwlkntr 29793 ` .~ ` is a transitive rel...
erclwwlkn 29794 ` .~ ` is an equivalence r...
qerclwwlknfi 29795 The quotient set of the se...
hashclwwlkn0 29796 The number of closed walks...
eclclwwlkn1 29797 An equivalence class accor...
eleclclwwlkn 29798 A member of an equivalence...
hashecclwwlkn1 29799 The size of every equivale...
umgrhashecclwwlk 29800 The size of every equivale...
fusgrhashclwwlkn 29801 The size of the set of clo...
clwwlkndivn 29802 The size of the set of clo...
clwlknf1oclwwlknlem1 29803 Lemma 1 for ~ clwlknf1oclw...
clwlknf1oclwwlknlem2 29804 Lemma 2 for ~ clwlknf1oclw...
clwlknf1oclwwlknlem3 29805 Lemma 3 for ~ clwlknf1oclw...
clwlknf1oclwwlkn 29806 There is a one-to-one onto...
clwlkssizeeq 29807 The size of the set of clo...
clwlksndivn 29808 The size of the set of clo...
clwwlknonmpo 29811 ` ( ClWWalksNOn `` G ) ` i...
clwwlknon 29812 The set of closed walks on...
isclwwlknon 29813 A word over the set of ver...
clwwlk0on0 29814 There is no word over the ...
clwwlknon0 29815 Sufficient conditions for ...
clwwlknonfin 29816 In a finite graph ` G ` , ...
clwwlknonel 29817 Characterization of a word...
clwwlknonccat 29818 The concatenation of two w...
clwwlknon1 29819 The set of closed walks on...
clwwlknon1loop 29820 If there is a loop at vert...
clwwlknon1nloop 29821 If there is no loop at ver...
clwwlknon1sn 29822 The set of (closed) walks ...
clwwlknon1le1 29823 There is at most one (clos...
clwwlknon2 29824 The set of closed walks on...
clwwlknon2x 29825 The set of closed walks on...
s2elclwwlknon2 29826 Sufficient conditions of a...
clwwlknon2num 29827 In a ` K `-regular graph `...
clwwlknonwwlknonb 29828 A word over vertices repre...
clwwlknonex2lem1 29829 Lemma 1 for ~ clwwlknonex2...
clwwlknonex2lem2 29830 Lemma 2 for ~ clwwlknonex2...
clwwlknonex2 29831 Extending a closed walk ` ...
clwwlknonex2e 29832 Extending a closed walk ` ...
clwwlknondisj 29833 The sets of closed walks o...
clwwlknun 29834 The set of closed walks of...
clwwlkvbij 29835 There is a bijection betwe...
0ewlk 29836 The empty set (empty seque...
1ewlk 29837 A sequence of 1 edge is an...
0wlk 29838 A pair of an empty set (of...
is0wlk 29839 A pair of an empty set (of...
0wlkonlem1 29840 Lemma 1 for ~ 0wlkon and ~...
0wlkonlem2 29841 Lemma 2 for ~ 0wlkon and ~...
0wlkon 29842 A walk of length 0 from a ...
0wlkons1 29843 A walk of length 0 from a ...
0trl 29844 A pair of an empty set (of...
is0trl 29845 A pair of an empty set (of...
0trlon 29846 A trail of length 0 from a...
0pth 29847 A pair of an empty set (of...
0spth 29848 A pair of an empty set (of...
0pthon 29849 A path of length 0 from a ...
0pthon1 29850 A path of length 0 from a ...
0pthonv 29851 For each vertex there is a...
0clwlk 29852 A pair of an empty set (of...
0clwlkv 29853 Any vertex (more precisely...
0clwlk0 29854 There is no closed walk in...
0crct 29855 A pair of an empty set (of...
0cycl 29856 A pair of an empty set (of...
1pthdlem1 29857 Lemma 1 for ~ 1pthd . (Co...
1pthdlem2 29858 Lemma 2 for ~ 1pthd . (Co...
1wlkdlem1 29859 Lemma 1 for ~ 1wlkd . (Co...
1wlkdlem2 29860 Lemma 2 for ~ 1wlkd . (Co...
1wlkdlem3 29861 Lemma 3 for ~ 1wlkd . (Co...
1wlkdlem4 29862 Lemma 4 for ~ 1wlkd . (Co...
1wlkd 29863 In a graph with two vertic...
1trld 29864 In a graph with two vertic...
1pthd 29865 In a graph with two vertic...
1pthond 29866 In a graph with two vertic...
upgr1wlkdlem1 29867 Lemma 1 for ~ upgr1wlkd . ...
upgr1wlkdlem2 29868 Lemma 2 for ~ upgr1wlkd . ...
upgr1wlkd 29869 In a pseudograph with two ...
upgr1trld 29870 In a pseudograph with two ...
upgr1pthd 29871 In a pseudograph with two ...
upgr1pthond 29872 In a pseudograph with two ...
lppthon 29873 A loop (which is an edge a...
lp1cycl 29874 A loop (which is an edge a...
1pthon2v 29875 For each pair of adjacent ...
1pthon2ve 29876 For each pair of adjacent ...
wlk2v2elem1 29877 Lemma 1 for ~ wlk2v2e : ` ...
wlk2v2elem2 29878 Lemma 2 for ~ wlk2v2e : T...
wlk2v2e 29879 In a graph with two vertic...
ntrl2v2e 29880 A walk which is not a trai...
3wlkdlem1 29881 Lemma 1 for ~ 3wlkd . (Co...
3wlkdlem2 29882 Lemma 2 for ~ 3wlkd . (Co...
3wlkdlem3 29883 Lemma 3 for ~ 3wlkd . (Co...
3wlkdlem4 29884 Lemma 4 for ~ 3wlkd . (Co...
3wlkdlem5 29885 Lemma 5 for ~ 3wlkd . (Co...
3pthdlem1 29886 Lemma 1 for ~ 3pthd . (Co...
3wlkdlem6 29887 Lemma 6 for ~ 3wlkd . (Co...
3wlkdlem7 29888 Lemma 7 for ~ 3wlkd . (Co...
3wlkdlem8 29889 Lemma 8 for ~ 3wlkd . (Co...
3wlkdlem9 29890 Lemma 9 for ~ 3wlkd . (Co...
3wlkdlem10 29891 Lemma 10 for ~ 3wlkd . (C...
3wlkd 29892 Construction of a walk fro...
3wlkond 29893 A walk of length 3 from on...
3trld 29894 Construction of a trail fr...
3trlond 29895 A trail of length 3 from o...
3pthd 29896 A path of length 3 from on...
3pthond 29897 A path of length 3 from on...
3spthd 29898 A simple path of length 3 ...
3spthond 29899 A simple path of length 3 ...
3cycld 29900 Construction of a 3-cycle ...
3cyclpd 29901 Construction of a 3-cycle ...
upgr3v3e3cycl 29902 If there is a cycle of len...
uhgr3cyclexlem 29903 Lemma for ~ uhgr3cyclex . ...
uhgr3cyclex 29904 If there are three differe...
umgr3cyclex 29905 If there are three (differ...
umgr3v3e3cycl 29906 If and only if there is a ...
upgr4cycl4dv4e 29907 If there is a cycle of len...
dfconngr1 29910 Alternative definition of ...
isconngr 29911 The property of being a co...
isconngr1 29912 The property of being a co...
cusconngr 29913 A complete hypergraph is c...
0conngr 29914 A graph without vertices i...
0vconngr 29915 A graph without vertices i...
1conngr 29916 A graph with (at most) one...
conngrv2edg 29917 A vertex in a connected gr...
vdn0conngrumgrv2 29918 A vertex in a connected mu...
releupth 29921 The set ` ( EulerPaths `` ...
eupths 29922 The Eulerian paths on the ...
iseupth 29923 The property " ` <. F , P ...
iseupthf1o 29924 The property " ` <. F , P ...
eupthi 29925 Properties of an Eulerian ...
eupthf1o 29926 The ` F ` function in an E...
eupthfi 29927 Any graph with an Eulerian...
eupthseg 29928 The ` N ` -th edge in an e...
upgriseupth 29929 The property " ` <. F , P ...
upgreupthi 29930 Properties of an Eulerian ...
upgreupthseg 29931 The ` N ` -th edge in an e...
eupthcl 29932 An Eulerian path has lengt...
eupthistrl 29933 An Eulerian path is a trai...
eupthiswlk 29934 An Eulerian path is a walk...
eupthpf 29935 The ` P ` function in an E...
eupth0 29936 There is an Eulerian path ...
eupthres 29937 The restriction ` <. H , Q...
eupthp1 29938 Append one path segment to...
eupth2eucrct 29939 Append one path segment to...
eupth2lem1 29940 Lemma for ~ eupth2 . (Con...
eupth2lem2 29941 Lemma for ~ eupth2 . (Con...
trlsegvdeglem1 29942 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem2 29943 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem3 29944 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem4 29945 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem5 29946 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem6 29947 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem7 29948 Lemma for ~ trlsegvdeg . ...
trlsegvdeg 29949 Formerly part of proof of ...
eupth2lem3lem1 29950 Lemma for ~ eupth2lem3 . ...
eupth2lem3lem2 29951 Lemma for ~ eupth2lem3 . ...
eupth2lem3lem3 29952 Lemma for ~ eupth2lem3 , f...
eupth2lem3lem4 29953 Lemma for ~ eupth2lem3 , f...
eupth2lem3lem5 29954 Lemma for ~ eupth2 . (Con...
eupth2lem3lem6 29955 Formerly part of proof of ...
eupth2lem3lem7 29956 Lemma for ~ eupth2lem3 : ...
eupthvdres 29957 Formerly part of proof of ...
eupth2lem3 29958 Lemma for ~ eupth2 . (Con...
eupth2lemb 29959 Lemma for ~ eupth2 (induct...
eupth2lems 29960 Lemma for ~ eupth2 (induct...
eupth2 29961 The only vertices of odd d...
eulerpathpr 29962 A graph with an Eulerian p...
eulerpath 29963 A pseudograph with an Eule...
eulercrct 29964 A pseudograph with an Eule...
eucrctshift 29965 Cyclically shifting the in...
eucrct2eupth1 29966 Removing one edge ` ( I ``...
eucrct2eupth 29967 Removing one edge ` ( I ``...
konigsbergvtx 29968 The set of vertices of the...
konigsbergiedg 29969 The indexed edges of the K...
konigsbergiedgw 29970 The indexed edges of the K...
konigsbergssiedgwpr 29971 Each subset of the indexed...
konigsbergssiedgw 29972 Each subset of the indexed...
konigsbergumgr 29973 The Königsberg graph ...
konigsberglem1 29974 Lemma 1 for ~ konigsberg :...
konigsberglem2 29975 Lemma 2 for ~ konigsberg :...
konigsberglem3 29976 Lemma 3 for ~ konigsberg :...
konigsberglem4 29977 Lemma 4 for ~ konigsberg :...
konigsberglem5 29978 Lemma 5 for ~ konigsberg :...
konigsberg 29979 The Königsberg Bridge...
isfrgr 29982 The property of being a fr...
frgrusgr 29983 A friendship graph is a si...
frgr0v 29984 Any null graph (set with n...
frgr0vb 29985 Any null graph (without ve...
frgruhgr0v 29986 Any null graph (without ve...
frgr0 29987 The null graph (graph with...
frcond1 29988 The friendship condition: ...
frcond2 29989 The friendship condition: ...
frgreu 29990 Variant of ~ frcond2 : An...
frcond3 29991 The friendship condition, ...
frcond4 29992 The friendship condition, ...
frgr1v 29993 Any graph with (at most) o...
nfrgr2v 29994 Any graph with two (differ...
frgr3vlem1 29995 Lemma 1 for ~ frgr3v . (C...
frgr3vlem2 29996 Lemma 2 for ~ frgr3v . (C...
frgr3v 29997 Any graph with three verti...
1vwmgr 29998 Every graph with one verte...
3vfriswmgrlem 29999 Lemma for ~ 3vfriswmgr . ...
3vfriswmgr 30000 Every friendship graph wit...
1to2vfriswmgr 30001 Every friendship graph wit...
1to3vfriswmgr 30002 Every friendship graph wit...
1to3vfriendship 30003 The friendship theorem for...
2pthfrgrrn 30004 Between any two (different...
2pthfrgrrn2 30005 Between any two (different...
2pthfrgr 30006 Between any two (different...
3cyclfrgrrn1 30007 Every vertex in a friendsh...
3cyclfrgrrn 30008 Every vertex in a friendsh...
3cyclfrgrrn2 30009 Every vertex in a friendsh...
3cyclfrgr 30010 Every vertex in a friendsh...
4cycl2v2nb 30011 In a (maybe degenerate) 4-...
4cycl2vnunb 30012 In a 4-cycle, two distinct...
n4cyclfrgr 30013 There is no 4-cycle in a f...
4cyclusnfrgr 30014 A graph with a 4-cycle is ...
frgrnbnb 30015 If two neighbors ` U ` and...
frgrconngr 30016 A friendship graph is conn...
vdgn0frgrv2 30017 A vertex in a friendship g...
vdgn1frgrv2 30018 Any vertex in a friendship...
vdgn1frgrv3 30019 Any vertex in a friendship...
vdgfrgrgt2 30020 Any vertex in a friendship...
frgrncvvdeqlem1 30021 Lemma 1 for ~ frgrncvvdeq ...
frgrncvvdeqlem2 30022 Lemma 2 for ~ frgrncvvdeq ...
frgrncvvdeqlem3 30023 Lemma 3 for ~ frgrncvvdeq ...
frgrncvvdeqlem4 30024 Lemma 4 for ~ frgrncvvdeq ...
frgrncvvdeqlem5 30025 Lemma 5 for ~ frgrncvvdeq ...
frgrncvvdeqlem6 30026 Lemma 6 for ~ frgrncvvdeq ...
frgrncvvdeqlem7 30027 Lemma 7 for ~ frgrncvvdeq ...
frgrncvvdeqlem8 30028 Lemma 8 for ~ frgrncvvdeq ...
frgrncvvdeqlem9 30029 Lemma 9 for ~ frgrncvvdeq ...
frgrncvvdeqlem10 30030 Lemma 10 for ~ frgrncvvdeq...
frgrncvvdeq 30031 In a friendship graph, two...
frgrwopreglem4a 30032 In a friendship graph any ...
frgrwopreglem5a 30033 If a friendship graph has ...
frgrwopreglem1 30034 Lemma 1 for ~ frgrwopreg :...
frgrwopreglem2 30035 Lemma 2 for ~ frgrwopreg ....
frgrwopreglem3 30036 Lemma 3 for ~ frgrwopreg ....
frgrwopreglem4 30037 Lemma 4 for ~ frgrwopreg ....
frgrwopregasn 30038 According to statement 5 i...
frgrwopregbsn 30039 According to statement 5 i...
frgrwopreg1 30040 According to statement 5 i...
frgrwopreg2 30041 According to statement 5 i...
frgrwopreglem5lem 30042 Lemma for ~ frgrwopreglem5...
frgrwopreglem5 30043 Lemma 5 for ~ frgrwopreg ....
frgrwopreglem5ALT 30044 Alternate direct proof of ...
frgrwopreg 30045 In a friendship graph ther...
frgrregorufr0 30046 In a friendship graph ther...
frgrregorufr 30047 If there is a vertex havin...
frgrregorufrg 30048 If there is a vertex havin...
frgr2wwlkeu 30049 For two different vertices...
frgr2wwlkn0 30050 In a friendship graph, the...
frgr2wwlk1 30051 In a friendship graph, the...
frgr2wsp1 30052 In a friendship graph, the...
frgr2wwlkeqm 30053 If there is a (simple) pat...
frgrhash2wsp 30054 The number of simple paths...
fusgreg2wsplem 30055 Lemma for ~ fusgreg2wsp an...
fusgr2wsp2nb 30056 The set of paths of length...
fusgreghash2wspv 30057 According to statement 7 i...
fusgreg2wsp 30058 In a finite simple graph, ...
2wspmdisj 30059 The sets of paths of lengt...
fusgreghash2wsp 30060 In a finite k-regular grap...
frrusgrord0lem 30061 Lemma for ~ frrusgrord0 . ...
frrusgrord0 30062 If a nonempty finite frien...
frrusgrord 30063 If a nonempty finite frien...
numclwwlk2lem1lem 30064 Lemma for ~ numclwwlk2lem1...
2clwwlklem 30065 Lemma for ~ clwwnonrepclww...
clwwnrepclwwn 30066 If the initial vertex of a...
clwwnonrepclwwnon 30067 If the initial vertex of a...
2clwwlk2clwwlklem 30068 Lemma for ~ 2clwwlk2clwwlk...
2clwwlk 30069 Value of operation ` C ` ,...
2clwwlk2 30070 The set ` ( X C 2 ) ` of d...
2clwwlkel 30071 Characterization of an ele...
2clwwlk2clwwlk 30072 An element of the value of...
numclwwlk1lem2foalem 30073 Lemma for ~ numclwwlk1lem2...
extwwlkfab 30074 The set ` ( X C N ) ` of d...
extwwlkfabel 30075 Characterization of an ele...
numclwwlk1lem2foa 30076 Going forth and back from ...
numclwwlk1lem2f 30077 ` T ` is a function, mappi...
numclwwlk1lem2fv 30078 Value of the function ` T ...
numclwwlk1lem2f1 30079 ` T ` is a 1-1 function. ...
numclwwlk1lem2fo 30080 ` T ` is an onto function....
numclwwlk1lem2f1o 30081 ` T ` is a 1-1 onto functi...
numclwwlk1lem2 30082 The set of double loops of...
numclwwlk1 30083 Statement 9 in [Huneke] p....
clwwlknonclwlknonf1o 30084 ` F ` is a bijection betwe...
clwwlknonclwlknonen 30085 The sets of the two repres...
dlwwlknondlwlknonf1olem1 30086 Lemma 1 for ~ dlwwlknondlw...
dlwwlknondlwlknonf1o 30087 ` F ` is a bijection betwe...
dlwwlknondlwlknonen 30088 The sets of the two repres...
wlkl0 30089 There is exactly one walk ...
clwlknon2num 30090 There are k walks of lengt...
numclwlk1lem1 30091 Lemma 1 for ~ numclwlk1 (S...
numclwlk1lem2 30092 Lemma 2 for ~ numclwlk1 (S...
numclwlk1 30093 Statement 9 in [Huneke] p....
numclwwlkovh0 30094 Value of operation ` H ` ,...
numclwwlkovh 30095 Value of operation ` H ` ,...
numclwwlkovq 30096 Value of operation ` Q ` ,...
numclwwlkqhash 30097 In a ` K `-regular graph, ...
numclwwlk2lem1 30098 In a friendship graph, for...
numclwlk2lem2f 30099 ` R ` is a function mappin...
numclwlk2lem2fv 30100 Value of the function ` R ...
numclwlk2lem2f1o 30101 ` R ` is a 1-1 onto functi...
numclwwlk2lem3 30102 In a friendship graph, the...
numclwwlk2 30103 Statement 10 in [Huneke] p...
numclwwlk3lem1 30104 Lemma 2 for ~ numclwwlk3 ....
numclwwlk3lem2lem 30105 Lemma for ~ numclwwlk3lem2...
numclwwlk3lem2 30106 Lemma 1 for ~ numclwwlk3 :...
numclwwlk3 30107 Statement 12 in [Huneke] p...
numclwwlk4 30108 The total number of closed...
numclwwlk5lem 30109 Lemma for ~ numclwwlk5 . ...
numclwwlk5 30110 Statement 13 in [Huneke] p...
numclwwlk7lem 30111 Lemma for ~ numclwwlk7 , ~...
numclwwlk6 30112 For a prime divisor ` P ` ...
numclwwlk7 30113 Statement 14 in [Huneke] p...
numclwwlk8 30114 The size of the set of clo...
frgrreggt1 30115 If a finite nonempty frien...
frgrreg 30116 If a finite nonempty frien...
frgrregord013 30117 If a finite friendship gra...
frgrregord13 30118 If a nonempty finite frien...
frgrogt3nreg 30119 If a finite friendship gra...
friendshipgt3 30120 The friendship theorem for...
friendship 30121 The friendship theorem: I...
conventions 30122

H...

conventions-labels 30123

...

conventions-comments 30124

...

natded 30125 Here are typical n...
ex-natded5.2 30126 Theorem 5.2 of [Clemente] ...
ex-natded5.2-2 30127 A more efficient proof of ...
ex-natded5.2i 30128 The same as ~ ex-natded5.2...
ex-natded5.3 30129 Theorem 5.3 of [Clemente] ...
ex-natded5.3-2 30130 A more efficient proof of ...
ex-natded5.3i 30131 The same as ~ ex-natded5.3...
ex-natded5.5 30132 Theorem 5.5 of [Clemente] ...
ex-natded5.7 30133 Theorem 5.7 of [Clemente] ...
ex-natded5.7-2 30134 A more efficient proof of ...
ex-natded5.8 30135 Theorem 5.8 of [Clemente] ...
ex-natded5.8-2 30136 A more efficient proof of ...
ex-natded5.13 30137 Theorem 5.13 of [Clemente]...
ex-natded5.13-2 30138 A more efficient proof of ...
ex-natded9.20 30139 Theorem 9.20 of [Clemente]...
ex-natded9.20-2 30140 A more efficient proof of ...
ex-natded9.26 30141 Theorem 9.26 of [Clemente]...
ex-natded9.26-2 30142 A more efficient proof of ...
ex-or 30143 Example for ~ df-or . Exa...
ex-an 30144 Example for ~ df-an . Exa...
ex-dif 30145 Example for ~ df-dif . Ex...
ex-un 30146 Example for ~ df-un . Exa...
ex-in 30147 Example for ~ df-in . Exa...
ex-uni 30148 Example for ~ df-uni . Ex...
ex-ss 30149 Example for ~ df-ss . Exa...
ex-pss 30150 Example for ~ df-pss . Ex...
ex-pw 30151 Example for ~ df-pw . Exa...
ex-pr 30152 Example for ~ df-pr . (Co...
ex-br 30153 Example for ~ df-br . Exa...
ex-opab 30154 Example for ~ df-opab . E...
ex-eprel 30155 Example for ~ df-eprel . ...
ex-id 30156 Example for ~ df-id . Exa...
ex-po 30157 Example for ~ df-po . Exa...
ex-xp 30158 Example for ~ df-xp . Exa...
ex-cnv 30159 Example for ~ df-cnv . Ex...
ex-co 30160 Example for ~ df-co . Exa...
ex-dm 30161 Example for ~ df-dm . Exa...
ex-rn 30162 Example for ~ df-rn . Exa...
ex-res 30163 Example for ~ df-res . Ex...
ex-ima 30164 Example for ~ df-ima . Ex...
ex-fv 30165 Example for ~ df-fv . Exa...
ex-1st 30166 Example for ~ df-1st . Ex...
ex-2nd 30167 Example for ~ df-2nd . Ex...
1kp2ke3k 30168 Example for ~ df-dec , 100...
ex-fl 30169 Example for ~ df-fl . Exa...
ex-ceil 30170 Example for ~ df-ceil . (...
ex-mod 30171 Example for ~ df-mod . (C...
ex-exp 30172 Example for ~ df-exp . (C...
ex-fac 30173 Example for ~ df-fac . (C...
ex-bc 30174 Example for ~ df-bc . (Co...
ex-hash 30175 Example for ~ df-hash . (...
ex-sqrt 30176 Example for ~ df-sqrt . (...
ex-abs 30177 Example for ~ df-abs . (C...
ex-dvds 30178 Example for ~ df-dvds : 3 ...
ex-gcd 30179 Example for ~ df-gcd . (C...
ex-lcm 30180 Example for ~ df-lcm . (C...
ex-prmo 30181 Example for ~ df-prmo : ` ...
aevdemo 30182 Proof illustrating the com...
ex-ind-dvds 30183 Example of a proof by indu...
ex-fpar 30184 Formalized example provide...
avril1 30185 Poisson d'Avril's Theorem....
2bornot2b 30186 The law of excluded middle...
helloworld 30187 The classic "Hello world" ...
1p1e2apr1 30188 One plus one equals two. ...
eqid1 30189 Law of identity (reflexivi...
1div0apr 30190 Division by zero is forbid...
topnfbey 30191 Nothing seems to be imposs...
9p10ne21 30192 9 + 10 is not equal to 21....
9p10ne21fool 30193 9 + 10 equals 21. This as...
nrt2irr 30195 The ` N ` -th root of 2 is...
isplig 30198 The predicate "is a planar...
ispligb 30199 The predicate "is a planar...
tncp 30200 In any planar incidence ge...
l2p 30201 For any line in a planar i...
lpni 30202 For any line in a planar i...
nsnlplig 30203 There is no "one-point lin...
nsnlpligALT 30204 Alternate version of ~ nsn...
n0lplig 30205 There is no "empty line" i...
n0lpligALT 30206 Alternate version of ~ n0l...
eulplig 30207 Through two distinct point...
pliguhgr 30208 Any planar incidence geome...
dummylink 30209 Alias for ~ a1ii that may ...
id1 30210 Alias for ~ idALT that may...
isgrpo 30219 The predicate "is a group ...
isgrpoi 30220 Properties that determine ...
grpofo 30221 A group operation maps ont...
grpocl 30222 Closure law for a group op...
grpolidinv 30223 A group has a left identit...
grpon0 30224 The base set of a group is...
grpoass 30225 A group operation is assoc...
grpoidinvlem1 30226 Lemma for ~ grpoidinv . (...
grpoidinvlem2 30227 Lemma for ~ grpoidinv . (...
grpoidinvlem3 30228 Lemma for ~ grpoidinv . (...
grpoidinvlem4 30229 Lemma for ~ grpoidinv . (...
grpoidinv 30230 A group has a left and rig...
grpoideu 30231 The left identity element ...
grporndm 30232 A group's range in terms o...
0ngrp 30233 The empty set is not a gro...
gidval 30234 The value of the identity ...
grpoidval 30235 Lemma for ~ grpoidcl and o...
grpoidcl 30236 The identity element of a ...
grpoidinv2 30237 A group's properties using...
grpolid 30238 The identity element of a ...
grporid 30239 The identity element of a ...
grporcan 30240 Right cancellation law for...
grpoinveu 30241 The left inverse element o...
grpoid 30242 Two ways of saying that an...
grporn 30243 The range of a group opera...
grpoinvfval 30244 The inverse function of a ...
grpoinvval 30245 The inverse of a group ele...
grpoinvcl 30246 A group element's inverse ...
grpoinv 30247 The properties of a group ...
grpolinv 30248 The left inverse of a grou...
grporinv 30249 The right inverse of a gro...
grpoinvid1 30250 The inverse of a group ele...
grpoinvid2 30251 The inverse of a group ele...
grpolcan 30252 Left cancellation law for ...
grpo2inv 30253 Double inverse law for gro...
grpoinvf 30254 Mapping of the inverse fun...
grpoinvop 30255 The inverse of the group o...
grpodivfval 30256 Group division (or subtrac...
grpodivval 30257 Group division (or subtrac...
grpodivinv 30258 Group division by an inver...
grpoinvdiv 30259 Inverse of a group divisio...
grpodivf 30260 Mapping for group division...
grpodivcl 30261 Closure of group division ...
grpodivdiv 30262 Double group division. (C...
grpomuldivass 30263 Associative-type law for m...
grpodivid 30264 Division of a group member...
grponpcan 30265 Cancellation law for group...
isablo 30268 The predicate "is an Abeli...
ablogrpo 30269 An Abelian group operation...
ablocom 30270 An Abelian group operation...
ablo32 30271 Commutative/associative la...
ablo4 30272 Commutative/associative la...
isabloi 30273 Properties that determine ...
ablomuldiv 30274 Law for group multiplicati...
ablodivdiv 30275 Law for double group divis...
ablodivdiv4 30276 Law for double group divis...
ablodiv32 30277 Swap the second and third ...
ablonncan 30278 Cancellation law for group...
ablonnncan1 30279 Cancellation law for group...
vcrel 30282 The class of all complex v...
vciOLD 30283 Obsolete version of ~ cvsi...
vcsm 30284 Functionality of th scalar...
vccl 30285 Closure of the scalar prod...
vcidOLD 30286 Identity element for the s...
vcdi 30287 Distributive law for the s...
vcdir 30288 Distributive law for the s...
vcass 30289 Associative law for the sc...
vc2OLD 30290 A vector plus itself is tw...
vcablo 30291 Vector addition is an Abel...
vcgrp 30292 Vector addition is a group...
vclcan 30293 Left cancellation law for ...
vczcl 30294 The zero vector is a vecto...
vc0rid 30295 The zero vector is a right...
vc0 30296 Zero times a vector is the...
vcz 30297 Anything times the zero ve...
vcm 30298 Minus 1 times a vector is ...
isvclem 30299 Lemma for ~ isvcOLD . (Co...
vcex 30300 The components of a comple...
isvcOLD 30301 The predicate "is a comple...
isvciOLD 30302 Properties that determine ...
cnaddabloOLD 30303 Obsolete version of ~ cnad...
cnidOLD 30304 Obsolete version of ~ cnad...
cncvcOLD 30305 Obsolete version of ~ cncv...
nvss 30315 Structure of the class of ...
nvvcop 30316 A normed complex vector sp...
nvrel 30324 The class of all normed co...
vafval 30325 Value of the function for ...
bafval 30326 Value of the function for ...
smfval 30327 Value of the function for ...
0vfval 30328 Value of the function for ...
nmcvfval 30329 Value of the norm function...
nvop2 30330 A normed complex vector sp...
nvvop 30331 The vector space component...
isnvlem 30332 Lemma for ~ isnv . (Contr...
nvex 30333 The components of a normed...
isnv 30334 The predicate "is a normed...
isnvi 30335 Properties that determine ...
nvi 30336 The properties of a normed...
nvvc 30337 The vector space component...
nvablo 30338 The vector addition operat...
nvgrp 30339 The vector addition operat...
nvgf 30340 Mapping for the vector add...
nvsf 30341 Mapping for the scalar mul...
nvgcl 30342 Closure law for the vector...
nvcom 30343 The vector addition (group...
nvass 30344 The vector addition (group...
nvadd32 30345 Commutative/associative la...
nvrcan 30346 Right cancellation law for...
nvadd4 30347 Rearrangement of 4 terms i...
nvscl 30348 Closure law for the scalar...
nvsid 30349 Identity element for the s...
nvsass 30350 Associative law for the sc...
nvscom 30351 Commutative law for the sc...
nvdi 30352 Distributive law for the s...
nvdir 30353 Distributive law for the s...
nv2 30354 A vector plus itself is tw...
vsfval 30355 Value of the function for ...
nvzcl 30356 Closure law for the zero v...
nv0rid 30357 The zero vector is a right...
nv0lid 30358 The zero vector is a left ...
nv0 30359 Zero times a vector is the...
nvsz 30360 Anything times the zero ve...
nvinv 30361 Minus 1 times a vector is ...
nvinvfval 30362 Function for the negative ...
nvm 30363 Vector subtraction in term...
nvmval 30364 Value of vector subtractio...
nvmval2 30365 Value of vector subtractio...
nvmfval 30366 Value of the function for ...
nvmf 30367 Mapping for the vector sub...
nvmcl 30368 Closure law for the vector...
nvnnncan1 30369 Cancellation law for vecto...
nvmdi 30370 Distributive law for scala...
nvnegneg 30371 Double negative of a vecto...
nvmul0or 30372 If a scalar product is zer...
nvrinv 30373 A vector minus itself. (C...
nvlinv 30374 Minus a vector plus itself...
nvpncan2 30375 Cancellation law for vecto...
nvpncan 30376 Cancellation law for vecto...
nvaddsub 30377 Commutative/associative la...
nvnpcan 30378 Cancellation law for a nor...
nvaddsub4 30379 Rearrangement of 4 terms i...
nvmeq0 30380 The difference between two...
nvmid 30381 A vector minus itself is t...
nvf 30382 Mapping for the norm funct...
nvcl 30383 The norm of a normed compl...
nvcli 30384 The norm of a normed compl...
nvs 30385 Proportionality property o...
nvsge0 30386 The norm of a scalar produ...
nvm1 30387 The norm of the negative o...
nvdif 30388 The norm of the difference...
nvpi 30389 The norm of a vector plus ...
nvz0 30390 The norm of a zero vector ...
nvz 30391 The norm of a vector is ze...
nvtri 30392 Triangle inequality for th...
nvmtri 30393 Triangle inequality for th...
nvabs 30394 Norm difference property o...
nvge0 30395 The norm of a normed compl...
nvgt0 30396 A nonzero norm is positive...
nv1 30397 From any nonzero vector, c...
nvop 30398 A complex inner product sp...
cnnv 30399 The set of complex numbers...
cnnvg 30400 The vector addition (group...
cnnvba 30401 The base set of the normed...
cnnvs 30402 The scalar product operati...
cnnvnm 30403 The norm operation of the ...
cnnvm 30404 The vector subtraction ope...
elimnv 30405 Hypothesis elimination lem...
elimnvu 30406 Hypothesis elimination lem...
imsval 30407 Value of the induced metri...
imsdval 30408 Value of the induced metri...
imsdval2 30409 Value of the distance func...
nvnd 30410 The norm of a normed compl...
imsdf 30411 Mapping for the induced me...
imsmetlem 30412 Lemma for ~ imsmet . (Con...
imsmet 30413 The induced metric of a no...
imsxmet 30414 The induced metric of a no...
cnims 30415 The metric induced on the ...
vacn 30416 Vector addition is jointly...
nmcvcn 30417 The norm of a normed compl...
nmcnc 30418 The norm of a normed compl...
smcnlem 30419 Lemma for ~ smcn . (Contr...
smcn 30420 Scalar multiplication is j...
vmcn 30421 Vector subtraction is join...
dipfval 30424 The inner product function...
ipval 30425 Value of the inner product...
ipval2lem2 30426 Lemma for ~ ipval3 . (Con...
ipval2lem3 30427 Lemma for ~ ipval3 . (Con...
ipval2lem4 30428 Lemma for ~ ipval3 . (Con...
ipval2 30429 Expansion of the inner pro...
4ipval2 30430 Four times the inner produ...
ipval3 30431 Expansion of the inner pro...
ipidsq 30432 The inner product of a vec...
ipnm 30433 Norm expressed in terms of...
dipcl 30434 An inner product is a comp...
ipf 30435 Mapping for the inner prod...
dipcj 30436 The complex conjugate of a...
ipipcj 30437 An inner product times its...
diporthcom 30438 Orthogonality (meaning inn...
dip0r 30439 Inner product with a zero ...
dip0l 30440 Inner product with a zero ...
ipz 30441 The inner product of a vec...
dipcn 30442 Inner product is jointly c...
sspval 30445 The set of all subspaces o...
isssp 30446 The predicate "is a subspa...
sspid 30447 A normed complex vector sp...
sspnv 30448 A subspace is a normed com...
sspba 30449 The base set of a subspace...
sspg 30450 Vector addition on a subsp...
sspgval 30451 Vector addition on a subsp...
ssps 30452 Scalar multiplication on a...
sspsval 30453 Scalar multiplication on a...
sspmlem 30454 Lemma for ~ sspm and other...
sspmval 30455 Vector addition on a subsp...
sspm 30456 Vector subtraction on a su...
sspz 30457 The zero vector of a subsp...
sspn 30458 The norm on a subspace is ...
sspnval 30459 The norm on a subspace in ...
sspimsval 30460 The induced metric on a su...
sspims 30461 The induced metric on a su...
lnoval 30474 The set of linear operator...
islno 30475 The predicate "is a linear...
lnolin 30476 Basic linearity property o...
lnof 30477 A linear operator is a map...
lno0 30478 The value of a linear oper...
lnocoi 30479 The composition of two lin...
lnoadd 30480 Addition property of a lin...
lnosub 30481 Subtraction property of a ...
lnomul 30482 Scalar multiplication prop...
nvo00 30483 Two ways to express a zero...
nmoofval 30484 The operator norm function...
nmooval 30485 The operator norm function...
nmosetre 30486 The set in the supremum of...
nmosetn0 30487 The set in the supremum of...
nmoxr 30488 The norm of an operator is...
nmooge0 30489 The norm of an operator is...
nmorepnf 30490 The norm of an operator is...
nmoreltpnf 30491 The norm of any operator i...
nmogtmnf 30492 The norm of an operator is...
nmoolb 30493 A lower bound for an opera...
nmoubi 30494 An upper bound for an oper...
nmoub3i 30495 An upper bound for an oper...
nmoub2i 30496 An upper bound for an oper...
nmobndi 30497 Two ways to express that a...
nmounbi 30498 Two ways two express that ...
nmounbseqi 30499 An unbounded operator dete...
nmounbseqiALT 30500 Alternate shorter proof of...
nmobndseqi 30501 A bounded sequence determi...
nmobndseqiALT 30502 Alternate shorter proof of...
bloval 30503 The class of bounded linea...
isblo 30504 The predicate "is a bounde...
isblo2 30505 The predicate "is a bounde...
bloln 30506 A bounded operator is a li...
blof 30507 A bounded operator is an o...
nmblore 30508 The norm of a bounded oper...
0ofval 30509 The zero operator between ...
0oval 30510 Value of the zero operator...
0oo 30511 The zero operator is an op...
0lno 30512 The zero operator is linea...
nmoo0 30513 The operator norm of the z...
0blo 30514 The zero operator is a bou...
nmlno0lem 30515 Lemma for ~ nmlno0i . (Co...
nmlno0i 30516 The norm of a linear opera...
nmlno0 30517 The norm of a linear opera...
nmlnoubi 30518 An upper bound for the ope...
nmlnogt0 30519 The norm of a nonzero line...
lnon0 30520 The domain of a nonzero li...
nmblolbii 30521 A lower bound for the norm...
nmblolbi 30522 A lower bound for the norm...
isblo3i 30523 The predicate "is a bounde...
blo3i 30524 Properties that determine ...
blometi 30525 Upper bound for the distan...
blocnilem 30526 Lemma for ~ blocni and ~ l...
blocni 30527 A linear operator is conti...
lnocni 30528 If a linear operator is co...
blocn 30529 A linear operator is conti...
blocn2 30530 A bounded linear operator ...
ajfval 30531 The adjoint function. (Co...
hmoval 30532 The set of Hermitian (self...
ishmo 30533 The predicate "is a hermit...
phnv 30536 Every complex inner produc...
phrel 30537 The class of all complex i...
phnvi 30538 Every complex inner produc...
isphg 30539 The predicate "is a comple...
phop 30540 A complex inner product sp...
cncph 30541 The set of complex numbers...
elimph 30542 Hypothesis elimination lem...
elimphu 30543 Hypothesis elimination lem...
isph 30544 The predicate "is an inner...
phpar2 30545 The parallelogram law for ...
phpar 30546 The parallelogram law for ...
ip0i 30547 A slight variant of Equati...
ip1ilem 30548 Lemma for ~ ip1i . (Contr...
ip1i 30549 Equation 6.47 of [Ponnusam...
ip2i 30550 Equation 6.48 of [Ponnusam...
ipdirilem 30551 Lemma for ~ ipdiri . (Con...
ipdiri 30552 Distributive law for inner...
ipasslem1 30553 Lemma for ~ ipassi . Show...
ipasslem2 30554 Lemma for ~ ipassi . Show...
ipasslem3 30555 Lemma for ~ ipassi . Show...
ipasslem4 30556 Lemma for ~ ipassi . Show...
ipasslem5 30557 Lemma for ~ ipassi . Show...
ipasslem7 30558 Lemma for ~ ipassi . Show...
ipasslem8 30559 Lemma for ~ ipassi . By ~...
ipasslem9 30560 Lemma for ~ ipassi . Conc...
ipasslem10 30561 Lemma for ~ ipassi . Show...
ipasslem11 30562 Lemma for ~ ipassi . Show...
ipassi 30563 Associative law for inner ...
dipdir 30564 Distributive law for inner...
dipdi 30565 Distributive law for inner...
ip2dii 30566 Inner product of two sums....
dipass 30567 Associative law for inner ...
dipassr 30568 "Associative" law for seco...
dipassr2 30569 "Associative" law for inne...
dipsubdir 30570 Distributive law for inner...
dipsubdi 30571 Distributive law for inner...
pythi 30572 The Pythagorean theorem fo...
siilem1 30573 Lemma for ~ sii . (Contri...
siilem2 30574 Lemma for ~ sii . (Contri...
siii 30575 Inference from ~ sii . (C...
sii 30576 Obsolete version of ~ ipca...
ipblnfi 30577 A function ` F ` generated...
ip2eqi 30578 Two vectors are equal iff ...
phoeqi 30579 A condition implying that ...
ajmoi 30580 Every operator has at most...
ajfuni 30581 The adjoint function is a ...
ajfun 30582 The adjoint function is a ...
ajval 30583 Value of the adjoint funct...
iscbn 30586 A complex Banach space is ...
cbncms 30587 The induced metric on comp...
bnnv 30588 Every complex Banach space...
bnrel 30589 The class of all complex B...
bnsscmcl 30590 A subspace of a Banach spa...
cnbn 30591 The set of complex numbers...
ubthlem1 30592 Lemma for ~ ubth . The fu...
ubthlem2 30593 Lemma for ~ ubth . Given ...
ubthlem3 30594 Lemma for ~ ubth . Prove ...
ubth 30595 Uniform Boundedness Theore...
minvecolem1 30596 Lemma for ~ minveco . The...
minvecolem2 30597 Lemma for ~ minveco . Any...
minvecolem3 30598 Lemma for ~ minveco . The...
minvecolem4a 30599 Lemma for ~ minveco . ` F ...
minvecolem4b 30600 Lemma for ~ minveco . The...
minvecolem4c 30601 Lemma for ~ minveco . The...
minvecolem4 30602 Lemma for ~ minveco . The...
minvecolem5 30603 Lemma for ~ minveco . Dis...
minvecolem6 30604 Lemma for ~ minveco . Any...
minvecolem7 30605 Lemma for ~ minveco . Sin...
minveco 30606 Minimizing vector theorem,...
ishlo 30609 The predicate "is a comple...
hlobn 30610 Every complex Hilbert spac...
hlph 30611 Every complex Hilbert spac...
hlrel 30612 The class of all complex H...
hlnv 30613 Every complex Hilbert spac...
hlnvi 30614 Every complex Hilbert spac...
hlvc 30615 Every complex Hilbert spac...
hlcmet 30616 The induced metric on a co...
hlmet 30617 The induced metric on a co...
hlpar2 30618 The parallelogram law sati...
hlpar 30619 The parallelogram law sati...
hlex 30620 The base set of a Hilbert ...
hladdf 30621 Mapping for Hilbert space ...
hlcom 30622 Hilbert space vector addit...
hlass 30623 Hilbert space vector addit...
hl0cl 30624 The Hilbert space zero vec...
hladdid 30625 Hilbert space addition wit...
hlmulf 30626 Mapping for Hilbert space ...
hlmulid 30627 Hilbert space scalar multi...
hlmulass 30628 Hilbert space scalar multi...
hldi 30629 Hilbert space scalar multi...
hldir 30630 Hilbert space scalar multi...
hlmul0 30631 Hilbert space scalar multi...
hlipf 30632 Mapping for Hilbert space ...
hlipcj 30633 Conjugate law for Hilbert ...
hlipdir 30634 Distributive law for Hilbe...
hlipass 30635 Associative law for Hilber...
hlipgt0 30636 The inner product of a Hil...
hlcompl 30637 Completeness of a Hilbert ...
cnchl 30638 The set of complex numbers...
htthlem 30639 Lemma for ~ htth . The co...
htth 30640 Hellinger-Toeplitz Theorem...
The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here
h2hva 30696 The group (addition) opera...
h2hsm 30697 The scalar product operati...
h2hnm 30698 The norm function of Hilbe...
h2hvs 30699 The vector subtraction ope...
h2hmetdval 30700 Value of the distance func...
h2hcau 30701 The Cauchy sequences of Hi...
h2hlm 30702 The limit sequences of Hil...
axhilex-zf 30703 Derive Axiom ~ ax-hilex fr...
axhfvadd-zf 30704 Derive Axiom ~ ax-hfvadd f...
axhvcom-zf 30705 Derive Axiom ~ ax-hvcom fr...
axhvass-zf 30706 Derive Axiom ~ ax-hvass fr...
axhv0cl-zf 30707 Derive Axiom ~ ax-hv0cl fr...
axhvaddid-zf 30708 Derive Axiom ~ ax-hvaddid ...
axhfvmul-zf 30709 Derive Axiom ~ ax-hfvmul f...
axhvmulid-zf 30710 Derive Axiom ~ ax-hvmulid ...
axhvmulass-zf 30711 Derive Axiom ~ ax-hvmulass...
axhvdistr1-zf 30712 Derive Axiom ~ ax-hvdistr1...
axhvdistr2-zf 30713 Derive Axiom ~ ax-hvdistr2...
axhvmul0-zf 30714 Derive Axiom ~ ax-hvmul0 f...
axhfi-zf 30715 Derive Axiom ~ ax-hfi from...
axhis1-zf 30716 Derive Axiom ~ ax-his1 fro...
axhis2-zf 30717 Derive Axiom ~ ax-his2 fro...
axhis3-zf 30718 Derive Axiom ~ ax-his3 fro...
axhis4-zf 30719 Derive Axiom ~ ax-his4 fro...
axhcompl-zf 30720 Derive Axiom ~ ax-hcompl f...
hvmulex 30733 The Hilbert space scalar p...
hvaddcl 30734 Closure of vector addition...
hvmulcl 30735 Closure of scalar multipli...
hvmulcli 30736 Closure inference for scal...
hvsubf 30737 Mapping domain and codomai...
hvsubval 30738 Value of vector subtractio...
hvsubcl 30739 Closure of vector subtract...
hvaddcli 30740 Closure of vector addition...
hvcomi 30741 Commutation of vector addi...
hvsubvali 30742 Value of vector subtractio...
hvsubcli 30743 Closure of vector subtract...
ifhvhv0 30744 Prove ` if ( A e. ~H , A ,...
hvaddlid 30745 Addition with the zero vec...
hvmul0 30746 Scalar multiplication with...
hvmul0or 30747 If a scalar product is zer...
hvsubid 30748 Subtraction of a vector fr...
hvnegid 30749 Addition of negative of a ...
hv2neg 30750 Two ways to express the ne...
hvaddlidi 30751 Addition with the zero vec...
hvnegidi 30752 Addition of negative of a ...
hv2negi 30753 Two ways to express the ne...
hvm1neg 30754 Convert minus one times a ...
hvaddsubval 30755 Value of vector addition i...
hvadd32 30756 Commutative/associative la...
hvadd12 30757 Commutative/associative la...
hvadd4 30758 Hilbert vector space addit...
hvsub4 30759 Hilbert vector space addit...
hvaddsub12 30760 Commutative/associative la...
hvpncan 30761 Addition/subtraction cance...
hvpncan2 30762 Addition/subtraction cance...
hvaddsubass 30763 Associativity of sum and d...
hvpncan3 30764 Subtraction and addition o...
hvmulcom 30765 Scalar multiplication comm...
hvsubass 30766 Hilbert vector space assoc...
hvsub32 30767 Hilbert vector space commu...
hvmulassi 30768 Scalar multiplication asso...
hvmulcomi 30769 Scalar multiplication comm...
hvmul2negi 30770 Double negative in scalar ...
hvsubdistr1 30771 Scalar multiplication dist...
hvsubdistr2 30772 Scalar multiplication dist...
hvdistr1i 30773 Scalar multiplication dist...
hvsubdistr1i 30774 Scalar multiplication dist...
hvassi 30775 Hilbert vector space assoc...
hvadd32i 30776 Hilbert vector space commu...
hvsubassi 30777 Hilbert vector space assoc...
hvsub32i 30778 Hilbert vector space commu...
hvadd12i 30779 Hilbert vector space commu...
hvadd4i 30780 Hilbert vector space addit...
hvsubsub4i 30781 Hilbert vector space addit...
hvsubsub4 30782 Hilbert vector space addit...
hv2times 30783 Two times a vector. (Cont...
hvnegdii 30784 Distribution of negative o...
hvsubeq0i 30785 If the difference between ...
hvsubcan2i 30786 Vector cancellation law. ...
hvaddcani 30787 Cancellation law for vecto...
hvsubaddi 30788 Relationship between vecto...
hvnegdi 30789 Distribution of negative o...
hvsubeq0 30790 If the difference between ...
hvaddeq0 30791 If the sum of two vectors ...
hvaddcan 30792 Cancellation law for vecto...
hvaddcan2 30793 Cancellation law for vecto...
hvmulcan 30794 Cancellation law for scala...
hvmulcan2 30795 Cancellation law for scala...
hvsubcan 30796 Cancellation law for vecto...
hvsubcan2 30797 Cancellation law for vecto...
hvsub0 30798 Subtraction of a zero vect...
hvsubadd 30799 Relationship between vecto...
hvaddsub4 30800 Hilbert vector space addit...
hicl 30802 Closure of inner product. ...
hicli 30803 Closure inference for inne...
his5 30808 Associative law for inner ...
his52 30809 Associative law for inner ...
his35 30810 Move scalar multiplication...
his35i 30811 Move scalar multiplication...
his7 30812 Distributive law for inner...
hiassdi 30813 Distributive/associative l...
his2sub 30814 Distributive law for inner...
his2sub2 30815 Distributive law for inner...
hire 30816 A necessary and sufficient...
hiidrcl 30817 Real closure of inner prod...
hi01 30818 Inner product with the 0 v...
hi02 30819 Inner product with the 0 v...
hiidge0 30820 Inner product with self is...
his6 30821 Zero inner product with se...
his1i 30822 Conjugate law for inner pr...
abshicom 30823 Commuted inner products ha...
hial0 30824 A vector whose inner produ...
hial02 30825 A vector whose inner produ...
hisubcomi 30826 Two vector subtractions si...
hi2eq 30827 Lemma used to prove equali...
hial2eq 30828 Two vectors whose inner pr...
hial2eq2 30829 Two vectors whose inner pr...
orthcom 30830 Orthogonality commutes. (...
normlem0 30831 Lemma used to derive prope...
normlem1 30832 Lemma used to derive prope...
normlem2 30833 Lemma used to derive prope...
normlem3 30834 Lemma used to derive prope...
normlem4 30835 Lemma used to derive prope...
normlem5 30836 Lemma used to derive prope...
normlem6 30837 Lemma used to derive prope...
normlem7 30838 Lemma used to derive prope...
normlem8 30839 Lemma used to derive prope...
normlem9 30840 Lemma used to derive prope...
normlem7tALT 30841 Lemma used to derive prope...
bcseqi 30842 Equality case of Bunjakova...
normlem9at 30843 Lemma used to derive prope...
dfhnorm2 30844 Alternate definition of th...
normf 30845 The norm function maps fro...
normval 30846 The value of the norm of a...
normcl 30847 Real closure of the norm o...
normge0 30848 The norm of a vector is no...
normgt0 30849 The norm of nonzero vector...
norm0 30850 The norm of a zero vector....
norm-i 30851 Theorem 3.3(i) of [Beran] ...
normne0 30852 A norm is nonzero iff its ...
normcli 30853 Real closure of the norm o...
normsqi 30854 The square of a norm. (Co...
norm-i-i 30855 Theorem 3.3(i) of [Beran] ...
normsq 30856 The square of a norm. (Co...
normsub0i 30857 Two vectors are equal iff ...
normsub0 30858 Two vectors are equal iff ...
norm-ii-i 30859 Triangle inequality for no...
norm-ii 30860 Triangle inequality for no...
norm-iii-i 30861 Theorem 3.3(iii) of [Beran...
norm-iii 30862 Theorem 3.3(iii) of [Beran...
normsubi 30863 Negative doesn't change th...
normpythi 30864 Analogy to Pythagorean the...
normsub 30865 Swapping order of subtract...
normneg 30866 The norm of a vector equal...
normpyth 30867 Analogy to Pythagorean the...
normpyc 30868 Corollary to Pythagorean t...
norm3difi 30869 Norm of differences around...
norm3adifii 30870 Norm of differences around...
norm3lem 30871 Lemma involving norm of di...
norm3dif 30872 Norm of differences around...
norm3dif2 30873 Norm of differences around...
norm3lemt 30874 Lemma involving norm of di...
norm3adifi 30875 Norm of differences around...
normpari 30876 Parallelogram law for norm...
normpar 30877 Parallelogram law for norm...
normpar2i 30878 Corollary of parallelogram...
polid2i 30879 Generalized polarization i...
polidi 30880 Polarization identity. Re...
polid 30881 Polarization identity. Re...
hilablo 30882 Hilbert space vector addit...
hilid 30883 The group identity element...
hilvc 30884 Hilbert space is a complex...
hilnormi 30885 Hilbert space norm in term...
hilhhi 30886 Deduce the structure of Hi...
hhnv 30887 Hilbert space is a normed ...
hhva 30888 The group (addition) opera...
hhba 30889 The base set of Hilbert sp...
hh0v 30890 The zero vector of Hilbert...
hhsm 30891 The scalar product operati...
hhvs 30892 The vector subtraction ope...
hhnm 30893 The norm function of Hilbe...
hhims 30894 The induced metric of Hilb...
hhims2 30895 Hilbert space distance met...
hhmet 30896 The induced metric of Hilb...
hhxmet 30897 The induced metric of Hilb...
hhmetdval 30898 Value of the distance func...
hhip 30899 The inner product operatio...
hhph 30900 The Hilbert space of the H...
bcsiALT 30901 Bunjakovaskij-Cauchy-Schwa...
bcsiHIL 30902 Bunjakovaskij-Cauchy-Schwa...
bcs 30903 Bunjakovaskij-Cauchy-Schwa...
bcs2 30904 Corollary of the Bunjakova...
bcs3 30905 Corollary of the Bunjakova...
hcau 30906 Member of the set of Cauch...
hcauseq 30907 A Cauchy sequences on a Hi...
hcaucvg 30908 A Cauchy sequence on a Hil...
seq1hcau 30909 A sequence on a Hilbert sp...
hlimi 30910 Express the predicate: Th...
hlimseqi 30911 A sequence with a limit on...
hlimveci 30912 Closure of the limit of a ...
hlimconvi 30913 Convergence of a sequence ...
hlim2 30914 The limit of a sequence on...
hlimadd 30915 Limit of the sum of two se...
hilmet 30916 The Hilbert space norm det...
hilxmet 30917 The Hilbert space norm det...
hilmetdval 30918 Value of the distance func...
hilims 30919 Hilbert space distance met...
hhcau 30920 The Cauchy sequences of Hi...
hhlm 30921 The limit sequences of Hil...
hhcmpl 30922 Lemma used for derivation ...
hilcompl 30923 Lemma used for derivation ...
hhcms 30925 The Hilbert space induced ...
hhhl 30926 The Hilbert space structur...
hilcms 30927 The Hilbert space norm det...
hilhl 30928 The Hilbert space of the H...
issh 30930 Subspace ` H ` of a Hilber...
issh2 30931 Subspace ` H ` of a Hilber...
shss 30932 A subspace is a subset of ...
shel 30933 A member of a subspace of ...
shex 30934 The set of subspaces of a ...
shssii 30935 A closed subspace of a Hil...
sheli 30936 A member of a subspace of ...
shelii 30937 A member of a subspace of ...
sh0 30938 The zero vector belongs to...
shaddcl 30939 Closure of vector addition...
shmulcl 30940 Closure of vector scalar m...
issh3 30941 Subspace ` H ` of a Hilber...
shsubcl 30942 Closure of vector subtract...
isch 30944 Closed subspace ` H ` of a...
isch2 30945 Closed subspace ` H ` of a...
chsh 30946 A closed subspace is a sub...
chsssh 30947 Closed subspaces are subsp...
chex 30948 The set of closed subspace...
chshii 30949 A closed subspace is a sub...
ch0 30950 The zero vector belongs to...
chss 30951 A closed subspace of a Hil...
chel 30952 A member of a closed subsp...
chssii 30953 A closed subspace of a Hil...
cheli 30954 A member of a closed subsp...
chelii 30955 A member of a closed subsp...
chlimi 30956 The limit property of a cl...
hlim0 30957 The zero sequence in Hilbe...
hlimcaui 30958 If a sequence in Hilbert s...
hlimf 30959 Function-like behavior of ...
hlimuni 30960 A Hilbert space sequence c...
hlimreui 30961 The limit of a Hilbert spa...
hlimeui 30962 The limit of a Hilbert spa...
isch3 30963 A Hilbert subspace is clos...
chcompl 30964 Completeness of a closed s...
helch 30965 The Hilbert lattice one (w...
ifchhv 30966 Prove ` if ( A e. CH , A ,...
helsh 30967 Hilbert space is a subspac...
shsspwh 30968 Subspaces are subsets of H...
chsspwh 30969 Closed subspaces are subse...
hsn0elch 30970 The zero subspace belongs ...
norm1 30971 From any nonzero Hilbert s...
norm1exi 30972 A normalized vector exists...
norm1hex 30973 A normalized vector can ex...
elch0 30976 Membership in zero for clo...
h0elch 30977 The zero subspace is a clo...
h0elsh 30978 The zero subspace is a sub...
hhssva 30979 The vector addition operat...
hhsssm 30980 The scalar multiplication ...
hhssnm 30981 The norm operation on a su...
issubgoilem 30982 Lemma for ~ hhssabloilem ....
hhssabloilem 30983 Lemma for ~ hhssabloi . F...
hhssabloi 30984 Abelian group property of ...
hhssablo 30985 Abelian group property of ...
hhssnv 30986 Normed complex vector spac...
hhssnvt 30987 Normed complex vector spac...
hhsst 30988 A member of ` SH ` is a su...
hhshsslem1 30989 Lemma for ~ hhsssh . (Con...
hhshsslem2 30990 Lemma for ~ hhsssh . (Con...
hhsssh 30991 The predicate " ` H ` is a...
hhsssh2 30992 The predicate " ` H ` is a...
hhssba 30993 The base set of a subspace...
hhssvs 30994 The vector subtraction ope...
hhssvsf 30995 Mapping of the vector subt...
hhssims 30996 Induced metric of a subspa...
hhssims2 30997 Induced metric of a subspa...
hhssmet 30998 Induced metric of a subspa...
hhssmetdval 30999 Value of the distance func...
hhsscms 31000 The induced metric of a cl...
hhssbnOLD 31001 Obsolete version of ~ cssb...
ocval 31002 Value of orthogonal comple...
ocel 31003 Membership in orthogonal c...
shocel 31004 Membership in orthogonal c...
ocsh 31005 The orthogonal complement ...
shocsh 31006 The orthogonal complement ...
ocss 31007 An orthogonal complement i...
shocss 31008 An orthogonal complement i...
occon 31009 Contraposition law for ort...
occon2 31010 Double contraposition for ...
occon2i 31011 Double contraposition for ...
oc0 31012 The zero vector belongs to...
ocorth 31013 Members of a subset and it...
shocorth 31014 Members of a subspace and ...
ococss 31015 Inclusion in complement of...
shococss 31016 Inclusion in complement of...
shorth 31017 Members of orthogonal subs...
ocin 31018 Intersection of a Hilbert ...
occon3 31019 Hilbert lattice contraposi...
ocnel 31020 A nonzero vector in the co...
chocvali 31021 Value of the orthogonal co...
shuni 31022 Two subspaces with trivial...
chocunii 31023 Lemma for uniqueness part ...
pjhthmo 31024 Projection Theorem, unique...
occllem 31025 Lemma for ~ occl . (Contr...
occl 31026 Closure of complement of H...
shoccl 31027 Closure of complement of H...
choccl 31028 Closure of complement of H...
choccli 31029 Closure of ` CH ` orthocom...
shsval 31034 Value of subspace sum of t...
shsss 31035 The subspace sum is a subs...
shsel 31036 Membership in the subspace...
shsel3 31037 Membership in the subspace...
shseli 31038 Membership in subspace sum...
shscli 31039 Closure of subspace sum. ...
shscl 31040 Closure of subspace sum. ...
shscom 31041 Commutative law for subspa...
shsva 31042 Vector sum belongs to subs...
shsel1 31043 A subspace sum contains a ...
shsel2 31044 A subspace sum contains a ...
shsvs 31045 Vector subtraction belongs...
shsub1 31046 Subspace sum is an upper b...
shsub2 31047 Subspace sum is an upper b...
choc0 31048 The orthocomplement of the...
choc1 31049 The orthocomplement of the...
chocnul 31050 Orthogonal complement of t...
shintcli 31051 Closure of intersection of...
shintcl 31052 The intersection of a none...
chintcli 31053 The intersection of a none...
chintcl 31054 The intersection (infimum)...
spanval 31055 Value of the linear span o...
hsupval 31056 Value of supremum of set o...
chsupval 31057 The value of the supremum ...
spancl 31058 The span of a subset of Hi...
elspancl 31059 A member of a span is a ve...
shsupcl 31060 Closure of the subspace su...
hsupcl 31061 Closure of supremum of set...
chsupcl 31062 Closure of supremum of sub...
hsupss 31063 Subset relation for suprem...
chsupss 31064 Subset relation for suprem...
hsupunss 31065 The union of a set of Hilb...
chsupunss 31066 The union of a set of clos...
spanss2 31067 A subset of Hilbert space ...
shsupunss 31068 The union of a set of subs...
spanid 31069 A subspace of Hilbert spac...
spanss 31070 Ordering relationship for ...
spanssoc 31071 The span of a subset of Hi...
sshjval 31072 Value of join for subsets ...
shjval 31073 Value of join in ` SH ` . ...
chjval 31074 Value of join in ` CH ` . ...
chjvali 31075 Value of join in ` CH ` . ...
sshjval3 31076 Value of join for subsets ...
sshjcl 31077 Closure of join for subset...
shjcl 31078 Closure of join in ` SH ` ...
chjcl 31079 Closure of join in ` CH ` ...
shjcom 31080 Commutative law for Hilber...
shless 31081 Subset implies subset of s...
shlej1 31082 Add disjunct to both sides...
shlej2 31083 Add disjunct to both sides...
shincli 31084 Closure of intersection of...
shscomi 31085 Commutative law for subspa...
shsvai 31086 Vector sum belongs to subs...
shsel1i 31087 A subspace sum contains a ...
shsel2i 31088 A subspace sum contains a ...
shsvsi 31089 Vector subtraction belongs...
shunssi 31090 Union is smaller than subs...
shunssji 31091 Union is smaller than Hilb...
shsleji 31092 Subspace sum is smaller th...
shjcomi 31093 Commutative law for join i...
shsub1i 31094 Subspace sum is an upper b...
shsub2i 31095 Subspace sum is an upper b...
shub1i 31096 Hilbert lattice join is an...
shjcli 31097 Closure of ` CH ` join. (...
shjshcli 31098 ` SH ` closure of join. (...
shlessi 31099 Subset implies subset of s...
shlej1i 31100 Add disjunct to both sides...
shlej2i 31101 Add disjunct to both sides...
shslej 31102 Subspace sum is smaller th...
shincl 31103 Closure of intersection of...
shub1 31104 Hilbert lattice join is an...
shub2 31105 A subspace is a subset of ...
shsidmi 31106 Idempotent law for Hilbert...
shslubi 31107 The least upper bound law ...
shlesb1i 31108 Hilbert lattice ordering i...
shsval2i 31109 An alternate way to expres...
shsval3i 31110 An alternate way to expres...
shmodsi 31111 The modular law holds for ...
shmodi 31112 The modular law is implied...
pjhthlem1 31113 Lemma for ~ pjhth . (Cont...
pjhthlem2 31114 Lemma for ~ pjhth . (Cont...
pjhth 31115 Projection Theorem: Any H...
pjhtheu 31116 Projection Theorem: Any H...
pjhfval 31118 The value of the projectio...
pjhval 31119 Value of a projection. (C...
pjpreeq 31120 Equality with a projection...
pjeq 31121 Equality with a projection...
axpjcl 31122 Closure of a projection in...
pjhcl 31123 Closure of a projection in...
omlsilem 31124 Lemma for orthomodular law...
omlsii 31125 Subspace inference form of...
omlsi 31126 Subspace form of orthomodu...
ococi 31127 Complement of complement o...
ococ 31128 Complement of complement o...
dfch2 31129 Alternate definition of th...
ococin 31130 The double complement is t...
hsupval2 31131 Alternate definition of su...
chsupval2 31132 The value of the supremum ...
sshjval2 31133 Value of join in the set o...
chsupid 31134 A subspace is the supremum...
chsupsn 31135 Value of supremum of subse...
shlub 31136 Hilbert lattice join is th...
shlubi 31137 Hilbert lattice join is th...
pjhtheu2 31138 Uniqueness of ` y ` for th...
pjcli 31139 Closure of a projection in...
pjhcli 31140 Closure of a projection in...
pjpjpre 31141 Decomposition of a vector ...
axpjpj 31142 Decomposition of a vector ...
pjclii 31143 Closure of a projection in...
pjhclii 31144 Closure of a projection in...
pjpj0i 31145 Decomposition of a vector ...
pjpji 31146 Decomposition of a vector ...
pjpjhth 31147 Projection Theorem: Any H...
pjpjhthi 31148 Projection Theorem: Any H...
pjop 31149 Orthocomplement projection...
pjpo 31150 Projection in terms of ort...
pjopi 31151 Orthocomplement projection...
pjpoi 31152 Projection in terms of ort...
pjoc1i 31153 Projection of a vector in ...
pjchi 31154 Projection of a vector in ...
pjoccl 31155 The part of a vector that ...
pjoc1 31156 Projection of a vector in ...
pjomli 31157 Subspace form of orthomodu...
pjoml 31158 Subspace form of orthomodu...
pjococi 31159 Proof of orthocomplement t...
pjoc2i 31160 Projection of a vector in ...
pjoc2 31161 Projection of a vector in ...
sh0le 31162 The zero subspace is the s...
ch0le 31163 The zero subspace is the s...
shle0 31164 No subspace is smaller tha...
chle0 31165 No Hilbert lattice element...
chnlen0 31166 A Hilbert lattice element ...
ch0pss 31167 The zero subspace is a pro...
orthin 31168 The intersection of orthog...
ssjo 31169 The lattice join of a subs...
shne0i 31170 A nonzero subspace has a n...
shs0i 31171 Hilbert subspace sum with ...
shs00i 31172 Two subspaces are zero iff...
ch0lei 31173 The closed subspace zero i...
chle0i 31174 No Hilbert closed subspace...
chne0i 31175 A nonzero closed subspace ...
chocini 31176 Intersection of a closed s...
chj0i 31177 Join with lattice zero in ...
chm1i 31178 Meet with lattice one in `...
chjcli 31179 Closure of ` CH ` join. (...
chsleji 31180 Subspace sum is smaller th...
chseli 31181 Membership in subspace sum...
chincli 31182 Closure of Hilbert lattice...
chsscon3i 31183 Hilbert lattice contraposi...
chsscon1i 31184 Hilbert lattice contraposi...
chsscon2i 31185 Hilbert lattice contraposi...
chcon2i 31186 Hilbert lattice contraposi...
chcon1i 31187 Hilbert lattice contraposi...
chcon3i 31188 Hilbert lattice contraposi...
chunssji 31189 Union is smaller than ` CH...
chjcomi 31190 Commutative law for join i...
chub1i 31191 ` CH ` join is an upper bo...
chub2i 31192 ` CH ` join is an upper bo...
chlubi 31193 Hilbert lattice join is th...
chlubii 31194 Hilbert lattice join is th...
chlej1i 31195 Add join to both sides of ...
chlej2i 31196 Add join to both sides of ...
chlej12i 31197 Add join to both sides of ...
chlejb1i 31198 Hilbert lattice ordering i...
chdmm1i 31199 De Morgan's law for meet i...
chdmm2i 31200 De Morgan's law for meet i...
chdmm3i 31201 De Morgan's law for meet i...
chdmm4i 31202 De Morgan's law for meet i...
chdmj1i 31203 De Morgan's law for join i...
chdmj2i 31204 De Morgan's law for join i...
chdmj3i 31205 De Morgan's law for join i...
chdmj4i 31206 De Morgan's law for join i...
chnlei 31207 Equivalent expressions for...
chjassi 31208 Associative law for Hilber...
chj00i 31209 Two Hilbert lattice elemen...
chjoi 31210 The join of a closed subsp...
chj1i 31211 Join with Hilbert lattice ...
chm0i 31212 Meet with Hilbert lattice ...
chm0 31213 Meet with Hilbert lattice ...
shjshsi 31214 Hilbert lattice join equal...
shjshseli 31215 A closed subspace sum equa...
chne0 31216 A nonzero closed subspace ...
chocin 31217 Intersection of a closed s...
chssoc 31218 A closed subspace less tha...
chj0 31219 Join with Hilbert lattice ...
chslej 31220 Subspace sum is smaller th...
chincl 31221 Closure of Hilbert lattice...
chsscon3 31222 Hilbert lattice contraposi...
chsscon1 31223 Hilbert lattice contraposi...
chsscon2 31224 Hilbert lattice contraposi...
chpsscon3 31225 Hilbert lattice contraposi...
chpsscon1 31226 Hilbert lattice contraposi...
chpsscon2 31227 Hilbert lattice contraposi...
chjcom 31228 Commutative law for Hilber...
chub1 31229 Hilbert lattice join is gr...
chub2 31230 Hilbert lattice join is gr...
chlub 31231 Hilbert lattice join is th...
chlej1 31232 Add join to both sides of ...
chlej2 31233 Add join to both sides of ...
chlejb1 31234 Hilbert lattice ordering i...
chlejb2 31235 Hilbert lattice ordering i...
chnle 31236 Equivalent expressions for...
chjo 31237 The join of a closed subsp...
chabs1 31238 Hilbert lattice absorption...
chabs2 31239 Hilbert lattice absorption...
chabs1i 31240 Hilbert lattice absorption...
chabs2i 31241 Hilbert lattice absorption...
chjidm 31242 Idempotent law for Hilbert...
chjidmi 31243 Idempotent law for Hilbert...
chj12i 31244 A rearrangement of Hilbert...
chj4i 31245 Rearrangement of the join ...
chjjdiri 31246 Hilbert lattice join distr...
chdmm1 31247 De Morgan's law for meet i...
chdmm2 31248 De Morgan's law for meet i...
chdmm3 31249 De Morgan's law for meet i...
chdmm4 31250 De Morgan's law for meet i...
chdmj1 31251 De Morgan's law for join i...
chdmj2 31252 De Morgan's law for join i...
chdmj3 31253 De Morgan's law for join i...
chdmj4 31254 De Morgan's law for join i...
chjass 31255 Associative law for Hilber...
chj12 31256 A rearrangement of Hilbert...
chj4 31257 Rearrangement of the join ...
ledii 31258 An ortholattice is distrib...
lediri 31259 An ortholattice is distrib...
lejdii 31260 An ortholattice is distrib...
lejdiri 31261 An ortholattice is distrib...
ledi 31262 An ortholattice is distrib...
spansn0 31263 The span of the singleton ...
span0 31264 The span of the empty set ...
elspani 31265 Membership in the span of ...
spanuni 31266 The span of a union is the...
spanun 31267 The span of a union is the...
sshhococi 31268 The join of two Hilbert sp...
hne0 31269 Hilbert space has a nonzer...
chsup0 31270 The supremum of the empty ...
h1deoi 31271 Membership in orthocomplem...
h1dei 31272 Membership in 1-dimensiona...
h1did 31273 A generating vector belong...
h1dn0 31274 A nonzero vector generates...
h1de2i 31275 Membership in 1-dimensiona...
h1de2bi 31276 Membership in 1-dimensiona...
h1de2ctlem 31277 Lemma for ~ h1de2ci . (Co...
h1de2ci 31278 Membership in 1-dimensiona...
spansni 31279 The span of a singleton in...
elspansni 31280 Membership in the span of ...
spansn 31281 The span of a singleton in...
spansnch 31282 The span of a Hilbert spac...
spansnsh 31283 The span of a Hilbert spac...
spansnchi 31284 The span of a singleton in...
spansnid 31285 A vector belongs to the sp...
spansnmul 31286 A scalar product with a ve...
elspansncl 31287 A member of a span of a si...
elspansn 31288 Membership in the span of ...
elspansn2 31289 Membership in the span of ...
spansncol 31290 The singletons of collinea...
spansneleqi 31291 Membership relation implie...
spansneleq 31292 Membership relation that i...
spansnss 31293 The span of the singleton ...
elspansn3 31294 A member of the span of th...
elspansn4 31295 A span membership conditio...
elspansn5 31296 A vector belonging to both...
spansnss2 31297 The span of the singleton ...
normcan 31298 Cancellation-type law that...
pjspansn 31299 A projection on the span o...
spansnpji 31300 A subset of Hilbert space ...
spanunsni 31301 The span of the union of a...
spanpr 31302 The span of a pair of vect...
h1datomi 31303 A 1-dimensional subspace i...
h1datom 31304 A 1-dimensional subspace i...
cmbr 31306 Binary relation expressing...
pjoml2i 31307 Variation of orthomodular ...
pjoml3i 31308 Variation of orthomodular ...
pjoml4i 31309 Variation of orthomodular ...
pjoml5i 31310 The orthomodular law. Rem...
pjoml6i 31311 An equivalent of the ortho...
cmbri 31312 Binary relation expressing...
cmcmlem 31313 Commutation is symmetric. ...
cmcmi 31314 Commutation is symmetric. ...
cmcm2i 31315 Commutation with orthocomp...
cmcm3i 31316 Commutation with orthocomp...
cmcm4i 31317 Commutation with orthocomp...
cmbr2i 31318 Alternate definition of th...
cmcmii 31319 Commutation is symmetric. ...
cmcm2ii 31320 Commutation with orthocomp...
cmcm3ii 31321 Commutation with orthocomp...
cmbr3i 31322 Alternate definition for t...
cmbr4i 31323 Alternate definition for t...
lecmi 31324 Comparable Hilbert lattice...
lecmii 31325 Comparable Hilbert lattice...
cmj1i 31326 A Hilbert lattice element ...
cmj2i 31327 A Hilbert lattice element ...
cmm1i 31328 A Hilbert lattice element ...
cmm2i 31329 A Hilbert lattice element ...
cmbr3 31330 Alternate definition for t...
cm0 31331 The zero Hilbert lattice e...
cmidi 31332 The commutes relation is r...
pjoml2 31333 Variation of orthomodular ...
pjoml3 31334 Variation of orthomodular ...
pjoml5 31335 The orthomodular law. Rem...
cmcm 31336 Commutation is symmetric. ...
cmcm3 31337 Commutation with orthocomp...
cmcm2 31338 Commutation with orthocomp...
lecm 31339 Comparable Hilbert lattice...
fh1 31340 Foulis-Holland Theorem. I...
fh2 31341 Foulis-Holland Theorem. I...
cm2j 31342 A lattice element that com...
fh1i 31343 Foulis-Holland Theorem. I...
fh2i 31344 Foulis-Holland Theorem. I...
fh3i 31345 Variation of the Foulis-Ho...
fh4i 31346 Variation of the Foulis-Ho...
cm2ji 31347 A lattice element that com...
cm2mi 31348 A lattice element that com...
qlax1i 31349 One of the equations showi...
qlax2i 31350 One of the equations showi...
qlax3i 31351 One of the equations showi...
qlax4i 31352 One of the equations showi...
qlax5i 31353 One of the equations showi...
qlaxr1i 31354 One of the conditions show...
qlaxr2i 31355 One of the conditions show...
qlaxr4i 31356 One of the conditions show...
qlaxr5i 31357 One of the conditions show...
qlaxr3i 31358 A variation of the orthomo...
chscllem1 31359 Lemma for ~ chscl . (Cont...
chscllem2 31360 Lemma for ~ chscl . (Cont...
chscllem3 31361 Lemma for ~ chscl . (Cont...
chscllem4 31362 Lemma for ~ chscl . (Cont...
chscl 31363 The subspace sum of two cl...
osumi 31364 If two closed subspaces of...
osumcori 31365 Corollary of ~ osumi . (C...
osumcor2i 31366 Corollary of ~ osumi , sho...
osum 31367 If two closed subspaces of...
spansnji 31368 The subspace sum of a clos...
spansnj 31369 The subspace sum of a clos...
spansnscl 31370 The subspace sum of a clos...
sumspansn 31371 The sum of two vectors bel...
spansnm0i 31372 The meet of different one-...
nonbooli 31373 A Hilbert lattice with two...
spansncvi 31374 Hilbert space has the cove...
spansncv 31375 Hilbert space has the cove...
5oalem1 31376 Lemma for orthoarguesian l...
5oalem2 31377 Lemma for orthoarguesian l...
5oalem3 31378 Lemma for orthoarguesian l...
5oalem4 31379 Lemma for orthoarguesian l...
5oalem5 31380 Lemma for orthoarguesian l...
5oalem6 31381 Lemma for orthoarguesian l...
5oalem7 31382 Lemma for orthoarguesian l...
5oai 31383 Orthoarguesian law 5OA. Th...
3oalem1 31384 Lemma for 3OA (weak) ortho...
3oalem2 31385 Lemma for 3OA (weak) ortho...
3oalem3 31386 Lemma for 3OA (weak) ortho...
3oalem4 31387 Lemma for 3OA (weak) ortho...
3oalem5 31388 Lemma for 3OA (weak) ortho...
3oalem6 31389 Lemma for 3OA (weak) ortho...
3oai 31390 3OA (weak) orthoarguesian ...
pjorthi 31391 Projection components on o...
pjch1 31392 Property of identity proje...
pjo 31393 The orthogonal projection....
pjcompi 31394 Component of a projection....
pjidmi 31395 A projection is idempotent...
pjadjii 31396 A projection is self-adjoi...
pjaddii 31397 Projection of vector sum i...
pjinormii 31398 The inner product of a pro...
pjmulii 31399 Projection of (scalar) pro...
pjsubii 31400 Projection of vector diffe...
pjsslem 31401 Lemma for subset relations...
pjss2i 31402 Subset relationship for pr...
pjssmii 31403 Projection meet property. ...
pjssge0ii 31404 Theorem 4.5(iv)->(v) of [B...
pjdifnormii 31405 Theorem 4.5(v)<->(vi) of [...
pjcji 31406 The projection on a subspa...
pjadji 31407 A projection is self-adjoi...
pjaddi 31408 Projection of vector sum i...
pjinormi 31409 The inner product of a pro...
pjsubi 31410 Projection of vector diffe...
pjmuli 31411 Projection of scalar produ...
pjige0i 31412 The inner product of a pro...
pjige0 31413 The inner product of a pro...
pjcjt2 31414 The projection on a subspa...
pj0i 31415 The projection of the zero...
pjch 31416 Projection of a vector in ...
pjid 31417 The projection of a vector...
pjvec 31418 The set of vectors belongi...
pjocvec 31419 The set of vectors belongi...
pjocini 31420 Membership of projection i...
pjini 31421 Membership of projection i...
pjjsi 31422 A sufficient condition for...
pjfni 31423 Functionality of a project...
pjrni 31424 The range of a projection....
pjfoi 31425 A projection maps onto its...
pjfi 31426 The mapping of a projectio...
pjvi 31427 The value of a projection ...
pjhfo 31428 A projection maps onto its...
pjrn 31429 The range of a projection....
pjhf 31430 The mapping of a projectio...
pjfn 31431 Functionality of a project...
pjsumi 31432 The projection on a subspa...
pj11i 31433 One-to-one correspondence ...
pjdsi 31434 Vector decomposition into ...
pjds3i 31435 Vector decomposition into ...
pj11 31436 One-to-one correspondence ...
pjmfn 31437 Functionality of the proje...
pjmf1 31438 The projector function map...
pjoi0 31439 The inner product of proje...
pjoi0i 31440 The inner product of proje...
pjopythi 31441 Pythagorean theorem for pr...
pjopyth 31442 Pythagorean theorem for pr...
pjnormi 31443 The norm of the projection...
pjpythi 31444 Pythagorean theorem for pr...
pjneli 31445 If a vector does not belon...
pjnorm 31446 The norm of the projection...
pjpyth 31447 Pythagorean theorem for pr...
pjnel 31448 If a vector does not belon...
pjnorm2 31449 A vector belongs to the su...
mayete3i 31450 Mayet's equation E_3. Par...
mayetes3i 31451 Mayet's equation E^*_3, de...
hosmval 31457 Value of the sum of two Hi...
hommval 31458 Value of the scalar produc...
hodmval 31459 Value of the difference of...
hfsmval 31460 Value of the sum of two Hi...
hfmmval 31461 Value of the scalar produc...
hosval 31462 Value of the sum of two Hi...
homval 31463 Value of the scalar produc...
hodval 31464 Value of the difference of...
hfsval 31465 Value of the sum of two Hi...
hfmval 31466 Value of the scalar produc...
hoscl 31467 Closure of the sum of two ...
homcl 31468 Closure of the scalar prod...
hodcl 31469 Closure of the difference ...
ho0val 31472 Value of the zero Hilbert ...
ho0f 31473 Functionality of the zero ...
df0op2 31474 Alternate definition of Hi...
dfiop2 31475 Alternate definition of Hi...
hoif 31476 Functionality of the Hilbe...
hoival 31477 The value of the Hilbert s...
hoico1 31478 Composition with the Hilbe...
hoico2 31479 Composition with the Hilbe...
hoaddcl 31480 The sum of Hilbert space o...
homulcl 31481 The scalar product of a Hi...
hoeq 31482 Equality of Hilbert space ...
hoeqi 31483 Equality of Hilbert space ...
hoscli 31484 Closure of Hilbert space o...
hodcli 31485 Closure of Hilbert space o...
hocoi 31486 Composition of Hilbert spa...
hococli 31487 Closure of composition of ...
hocofi 31488 Mapping of composition of ...
hocofni 31489 Functionality of compositi...
hoaddcli 31490 Mapping of sum of Hilbert ...
hosubcli 31491 Mapping of difference of H...
hoaddfni 31492 Functionality of sum of Hi...
hosubfni 31493 Functionality of differenc...
hoaddcomi 31494 Commutativity of sum of Hi...
hosubcl 31495 Mapping of difference of H...
hoaddcom 31496 Commutativity of sum of Hi...
hodsi 31497 Relationship between Hilbe...
hoaddassi 31498 Associativity of sum of Hi...
hoadd12i 31499 Commutative/associative la...
hoadd32i 31500 Commutative/associative la...
hocadddiri 31501 Distributive law for Hilbe...
hocsubdiri 31502 Distributive law for Hilbe...
ho2coi 31503 Double composition of Hilb...
hoaddass 31504 Associativity of sum of Hi...
hoadd32 31505 Commutative/associative la...
hoadd4 31506 Rearrangement of 4 terms i...
hocsubdir 31507 Distributive law for Hilbe...
hoaddridi 31508 Sum of a Hilbert space ope...
hodidi 31509 Difference of a Hilbert sp...
ho0coi 31510 Composition of the zero op...
hoid1i 31511 Composition of Hilbert spa...
hoid1ri 31512 Composition of Hilbert spa...
hoaddrid 31513 Sum of a Hilbert space ope...
hodid 31514 Difference of a Hilbert sp...
hon0 31515 A Hilbert space operator i...
hodseqi 31516 Subtraction and addition o...
ho0subi 31517 Subtraction of Hilbert spa...
honegsubi 31518 Relationship between Hilbe...
ho0sub 31519 Subtraction of Hilbert spa...
hosubid1 31520 The zero operator subtract...
honegsub 31521 Relationship between Hilbe...
homullid 31522 An operator equals its sca...
homco1 31523 Associative law for scalar...
homulass 31524 Scalar product associative...
hoadddi 31525 Scalar product distributiv...
hoadddir 31526 Scalar product reverse dis...
homul12 31527 Swap first and second fact...
honegneg 31528 Double negative of a Hilbe...
hosubneg 31529 Relationship between opera...
hosubdi 31530 Scalar product distributiv...
honegdi 31531 Distribution of negative o...
honegsubdi 31532 Distribution of negative o...
honegsubdi2 31533 Distribution of negative o...
hosubsub2 31534 Law for double subtraction...
hosub4 31535 Rearrangement of 4 terms i...
hosubadd4 31536 Rearrangement of 4 terms i...
hoaddsubass 31537 Associative-type law for a...
hoaddsub 31538 Law for operator addition ...
hosubsub 31539 Law for double subtraction...
hosubsub4 31540 Law for double subtraction...
ho2times 31541 Two times a Hilbert space ...
hoaddsubassi 31542 Associativity of sum and d...
hoaddsubi 31543 Law for sum and difference...
hosd1i 31544 Hilbert space operator sum...
hosd2i 31545 Hilbert space operator sum...
hopncani 31546 Hilbert space operator can...
honpcani 31547 Hilbert space operator can...
hosubeq0i 31548 If the difference between ...
honpncani 31549 Hilbert space operator can...
ho01i 31550 A condition implying that ...
ho02i 31551 A condition implying that ...
hoeq1 31552 A condition implying that ...
hoeq2 31553 A condition implying that ...
adjmo 31554 Every Hilbert space operat...
adjsym 31555 Symmetry property of an ad...
eigrei 31556 A necessary and sufficient...
eigre 31557 A necessary and sufficient...
eigposi 31558 A sufficient condition (fi...
eigorthi 31559 A necessary and sufficient...
eigorth 31560 A necessary and sufficient...
nmopval 31578 Value of the norm of a Hil...
elcnop 31579 Property defining a contin...
ellnop 31580 Property defining a linear...
lnopf 31581 A linear Hilbert space ope...
elbdop 31582 Property defining a bounde...
bdopln 31583 A bounded linear Hilbert s...
bdopf 31584 A bounded linear Hilbert s...
nmopsetretALT 31585 The set in the supremum of...
nmopsetretHIL 31586 The set in the supremum of...
nmopsetn0 31587 The set in the supremum of...
nmopxr 31588 The norm of a Hilbert spac...
nmoprepnf 31589 The norm of a Hilbert spac...
nmopgtmnf 31590 The norm of a Hilbert spac...
nmopreltpnf 31591 The norm of a Hilbert spac...
nmopre 31592 The norm of a bounded oper...
elbdop2 31593 Property defining a bounde...
elunop 31594 Property defining a unitar...
elhmop 31595 Property defining a Hermit...
hmopf 31596 A Hermitian operator is a ...
hmopex 31597 The class of Hermitian ope...
nmfnval 31598 Value of the norm of a Hil...
nmfnsetre 31599 The set in the supremum of...
nmfnsetn0 31600 The set in the supremum of...
nmfnxr 31601 The norm of any Hilbert sp...
nmfnrepnf 31602 The norm of a Hilbert spac...
nlfnval 31603 Value of the null space of...
elcnfn 31604 Property defining a contin...
ellnfn 31605 Property defining a linear...
lnfnf 31606 A linear Hilbert space fun...
dfadj2 31607 Alternate definition of th...
funadj 31608 Functionality of the adjoi...
dmadjss 31609 The domain of the adjoint ...
dmadjop 31610 A member of the domain of ...
adjeu 31611 Elementhood in the domain ...
adjval 31612 Value of the adjoint funct...
adjval2 31613 Value of the adjoint funct...
cnvadj 31614 The adjoint function equal...
funcnvadj 31615 The converse of the adjoin...
adj1o 31616 The adjoint function maps ...
dmadjrn 31617 The adjoint of an operator...
eigvecval 31618 The set of eigenvectors of...
eigvalfval 31619 The eigenvalues of eigenve...
specval 31620 The value of the spectrum ...
speccl 31621 The spectrum of an operato...
hhlnoi 31622 The linear operators of Hi...
hhnmoi 31623 The norm of an operator in...
hhbloi 31624 A bounded linear operator ...
hh0oi 31625 The zero operator in Hilbe...
hhcno 31626 The continuous operators o...
hhcnf 31627 The continuous functionals...
dmadjrnb 31628 The adjoint of an operator...
nmoplb 31629 A lower bound for an opera...
nmopub 31630 An upper bound for an oper...
nmopub2tALT 31631 An upper bound for an oper...
nmopub2tHIL 31632 An upper bound for an oper...
nmopge0 31633 The norm of any Hilbert sp...
nmopgt0 31634 A linear Hilbert space ope...
cnopc 31635 Basic continuity property ...
lnopl 31636 Basic linearity property o...
unop 31637 Basic inner product proper...
unopf1o 31638 A unitary operator in Hilb...
unopnorm 31639 A unitary operator is idem...
cnvunop 31640 The inverse (converse) of ...
unopadj 31641 The inverse (converse) of ...
unoplin 31642 A unitary operator is line...
counop 31643 The composition of two uni...
hmop 31644 Basic inner product proper...
hmopre 31645 The inner product of the v...
nmfnlb 31646 A lower bound for a functi...
nmfnleub 31647 An upper bound for the nor...
nmfnleub2 31648 An upper bound for the nor...
nmfnge0 31649 The norm of any Hilbert sp...
elnlfn 31650 Membership in the null spa...
elnlfn2 31651 Membership in the null spa...
cnfnc 31652 Basic continuity property ...
lnfnl 31653 Basic linearity property o...
adjcl 31654 Closure of the adjoint of ...
adj1 31655 Property of an adjoint Hil...
adj2 31656 Property of an adjoint Hil...
adjeq 31657 A property that determines...
adjadj 31658 Double adjoint. Theorem 3...
adjvalval 31659 Value of the value of the ...
unopadj2 31660 The adjoint of a unitary o...
hmopadj 31661 A Hermitian operator is se...
hmdmadj 31662 Every Hermitian operator h...
hmopadj2 31663 An operator is Hermitian i...
hmoplin 31664 A Hermitian operator is li...
brafval 31665 The bra of a vector, expre...
braval 31666 A bra-ket juxtaposition, e...
braadd 31667 Linearity property of bra ...
bramul 31668 Linearity property of bra ...
brafn 31669 The bra function is a func...
bralnfn 31670 The Dirac bra function is ...
bracl 31671 Closure of the bra functio...
bra0 31672 The Dirac bra of the zero ...
brafnmul 31673 Anti-linearity property of...
kbfval 31674 The outer product of two v...
kbop 31675 The outer product of two v...
kbval 31676 The value of the operator ...
kbmul 31677 Multiplication property of...
kbpj 31678 If a vector ` A ` has norm...
eleigvec 31679 Membership in the set of e...
eleigvec2 31680 Membership in the set of e...
eleigveccl 31681 Closure of an eigenvector ...
eigvalval 31682 The eigenvalue of an eigen...
eigvalcl 31683 An eigenvalue is a complex...
eigvec1 31684 Property of an eigenvector...
eighmre 31685 The eigenvalues of a Hermi...
eighmorth 31686 Eigenvectors of a Hermitia...
nmopnegi 31687 Value of the norm of the n...
lnop0 31688 The value of a linear Hilb...
lnopmul 31689 Multiplicative property of...
lnopli 31690 Basic scalar product prope...
lnopfi 31691 A linear Hilbert space ope...
lnop0i 31692 The value of a linear Hilb...
lnopaddi 31693 Additive property of a lin...
lnopmuli 31694 Multiplicative property of...
lnopaddmuli 31695 Sum/product property of a ...
lnopsubi 31696 Subtraction property for a...
lnopsubmuli 31697 Subtraction/product proper...
lnopmulsubi 31698 Product/subtraction proper...
homco2 31699 Move a scalar product out ...
idunop 31700 The identity function (res...
0cnop 31701 The identically zero funct...
0cnfn 31702 The identically zero funct...
idcnop 31703 The identity function (res...
idhmop 31704 The Hilbert space identity...
0hmop 31705 The identically zero funct...
0lnop 31706 The identically zero funct...
0lnfn 31707 The identically zero funct...
nmop0 31708 The norm of the zero opera...
nmfn0 31709 The norm of the identicall...
hmopbdoptHIL 31710 A Hermitian operator is a ...
hoddii 31711 Distributive law for Hilbe...
hoddi 31712 Distributive law for Hilbe...
nmop0h 31713 The norm of any operator o...
idlnop 31714 The identity function (res...
0bdop 31715 The identically zero opera...
adj0 31716 Adjoint of the zero operat...
nmlnop0iALT 31717 A linear operator with a z...
nmlnop0iHIL 31718 A linear operator with a z...
nmlnopgt0i 31719 A linear Hilbert space ope...
nmlnop0 31720 A linear operator with a z...
nmlnopne0 31721 A linear operator with a n...
lnopmi 31722 The scalar product of a li...
lnophsi 31723 The sum of two linear oper...
lnophdi 31724 The difference of two line...
lnopcoi 31725 The composition of two lin...
lnopco0i 31726 The composition of a linea...
lnopeq0lem1 31727 Lemma for ~ lnopeq0i . Ap...
lnopeq0lem2 31728 Lemma for ~ lnopeq0i . (C...
lnopeq0i 31729 A condition implying that ...
lnopeqi 31730 Two linear Hilbert space o...
lnopeq 31731 Two linear Hilbert space o...
lnopunilem1 31732 Lemma for ~ lnopunii . (C...
lnopunilem2 31733 Lemma for ~ lnopunii . (C...
lnopunii 31734 If a linear operator (whos...
elunop2 31735 An operator is unitary iff...
nmopun 31736 Norm of a unitary Hilbert ...
unopbd 31737 A unitary operator is a bo...
lnophmlem1 31738 Lemma for ~ lnophmi . (Co...
lnophmlem2 31739 Lemma for ~ lnophmi . (Co...
lnophmi 31740 A linear operator is Hermi...
lnophm 31741 A linear operator is Hermi...
hmops 31742 The sum of two Hermitian o...
hmopm 31743 The scalar product of a He...
hmopd 31744 The difference of two Herm...
hmopco 31745 The composition of two com...
nmbdoplbi 31746 A lower bound for the norm...
nmbdoplb 31747 A lower bound for the norm...
nmcexi 31748 Lemma for ~ nmcopexi and ~...
nmcopexi 31749 The norm of a continuous l...
nmcoplbi 31750 A lower bound for the norm...
nmcopex 31751 The norm of a continuous l...
nmcoplb 31752 A lower bound for the norm...
nmophmi 31753 The norm of the scalar pro...
bdophmi 31754 The scalar product of a bo...
lnconi 31755 Lemma for ~ lnopconi and ~...
lnopconi 31756 A condition equivalent to ...
lnopcon 31757 A condition equivalent to ...
lnopcnbd 31758 A linear operator is conti...
lncnopbd 31759 A continuous linear operat...
lncnbd 31760 A continuous linear operat...
lnopcnre 31761 A linear operator is conti...
lnfnli 31762 Basic property of a linear...
lnfnfi 31763 A linear Hilbert space fun...
lnfn0i 31764 The value of a linear Hilb...
lnfnaddi 31765 Additive property of a lin...
lnfnmuli 31766 Multiplicative property of...
lnfnaddmuli 31767 Sum/product property of a ...
lnfnsubi 31768 Subtraction property for a...
lnfn0 31769 The value of a linear Hilb...
lnfnmul 31770 Multiplicative property of...
nmbdfnlbi 31771 A lower bound for the norm...
nmbdfnlb 31772 A lower bound for the norm...
nmcfnexi 31773 The norm of a continuous l...
nmcfnlbi 31774 A lower bound for the norm...
nmcfnex 31775 The norm of a continuous l...
nmcfnlb 31776 A lower bound of the norm ...
lnfnconi 31777 A condition equivalent to ...
lnfncon 31778 A condition equivalent to ...
lnfncnbd 31779 A linear functional is con...
imaelshi 31780 The image of a subspace un...
rnelshi 31781 The range of a linear oper...
nlelshi 31782 The null space of a linear...
nlelchi 31783 The null space of a contin...
riesz3i 31784 A continuous linear functi...
riesz4i 31785 A continuous linear functi...
riesz4 31786 A continuous linear functi...
riesz1 31787 Part 1 of the Riesz repres...
riesz2 31788 Part 2 of the Riesz repres...
cnlnadjlem1 31789 Lemma for ~ cnlnadji (Theo...
cnlnadjlem2 31790 Lemma for ~ cnlnadji . ` G...
cnlnadjlem3 31791 Lemma for ~ cnlnadji . By...
cnlnadjlem4 31792 Lemma for ~ cnlnadji . Th...
cnlnadjlem5 31793 Lemma for ~ cnlnadji . ` F...
cnlnadjlem6 31794 Lemma for ~ cnlnadji . ` F...
cnlnadjlem7 31795 Lemma for ~ cnlnadji . He...
cnlnadjlem8 31796 Lemma for ~ cnlnadji . ` F...
cnlnadjlem9 31797 Lemma for ~ cnlnadji . ` F...
cnlnadji 31798 Every continuous linear op...
cnlnadjeui 31799 Every continuous linear op...
cnlnadjeu 31800 Every continuous linear op...
cnlnadj 31801 Every continuous linear op...
cnlnssadj 31802 Every continuous linear Hi...
bdopssadj 31803 Every bounded linear Hilbe...
bdopadj 31804 Every bounded linear Hilbe...
adjbdln 31805 The adjoint of a bounded l...
adjbdlnb 31806 An operator is bounded and...
adjbd1o 31807 The mapping of adjoints of...
adjlnop 31808 The adjoint of an operator...
adjsslnop 31809 Every operator with an adj...
nmopadjlei 31810 Property of the norm of an...
nmopadjlem 31811 Lemma for ~ nmopadji . (C...
nmopadji 31812 Property of the norm of an...
adjeq0 31813 An operator is zero iff it...
adjmul 31814 The adjoint of the scalar ...
adjadd 31815 The adjoint of the sum of ...
nmoptrii 31816 Triangle inequality for th...
nmopcoi 31817 Upper bound for the norm o...
bdophsi 31818 The sum of two bounded lin...
bdophdi 31819 The difference between two...
bdopcoi 31820 The composition of two bou...
nmoptri2i 31821 Triangle-type inequality f...
adjcoi 31822 The adjoint of a compositi...
nmopcoadji 31823 The norm of an operator co...
nmopcoadj2i 31824 The norm of an operator co...
nmopcoadj0i 31825 An operator composed with ...
unierri 31826 If we approximate a chain ...
branmfn 31827 The norm of the bra functi...
brabn 31828 The bra of a vector is a b...
rnbra 31829 The set of bras equals the...
bra11 31830 The bra function maps vect...
bracnln 31831 A bra is a continuous line...
cnvbraval 31832 Value of the converse of t...
cnvbracl 31833 Closure of the converse of...
cnvbrabra 31834 The converse bra of the br...
bracnvbra 31835 The bra of the converse br...
bracnlnval 31836 The vector that a continuo...
cnvbramul 31837 Multiplication property of...
kbass1 31838 Dirac bra-ket associative ...
kbass2 31839 Dirac bra-ket associative ...
kbass3 31840 Dirac bra-ket associative ...
kbass4 31841 Dirac bra-ket associative ...
kbass5 31842 Dirac bra-ket associative ...
kbass6 31843 Dirac bra-ket associative ...
leopg 31844 Ordering relation for posi...
leop 31845 Ordering relation for oper...
leop2 31846 Ordering relation for oper...
leop3 31847 Operator ordering in terms...
leoppos 31848 Binary relation defining a...
leoprf2 31849 The ordering relation for ...
leoprf 31850 The ordering relation for ...
leopsq 31851 The square of a Hermitian ...
0leop 31852 The zero operator is a pos...
idleop 31853 The identity operator is a...
leopadd 31854 The sum of two positive op...
leopmuli 31855 The scalar product of a no...
leopmul 31856 The scalar product of a po...
leopmul2i 31857 Scalar product applied to ...
leoptri 31858 The positive operator orde...
leoptr 31859 The positive operator orde...
leopnmid 31860 A bounded Hermitian operat...
nmopleid 31861 A nonzero, bounded Hermiti...
opsqrlem1 31862 Lemma for opsqri . (Contr...
opsqrlem2 31863 Lemma for opsqri . ` F `` ...
opsqrlem3 31864 Lemma for opsqri . (Contr...
opsqrlem4 31865 Lemma for opsqri . (Contr...
opsqrlem5 31866 Lemma for opsqri . (Contr...
opsqrlem6 31867 Lemma for opsqri . (Contr...
pjhmopi 31868 A projector is a Hermitian...
pjlnopi 31869 A projector is a linear op...
pjnmopi 31870 The operator norm of a pro...
pjbdlni 31871 A projector is a bounded l...
pjhmop 31872 A projection is a Hermitia...
hmopidmchi 31873 An idempotent Hermitian op...
hmopidmpji 31874 An idempotent Hermitian op...
hmopidmch 31875 An idempotent Hermitian op...
hmopidmpj 31876 An idempotent Hermitian op...
pjsdii 31877 Distributive law for Hilbe...
pjddii 31878 Distributive law for Hilbe...
pjsdi2i 31879 Chained distributive law f...
pjcoi 31880 Composition of projections...
pjcocli 31881 Closure of composition of ...
pjcohcli 31882 Closure of composition of ...
pjadjcoi 31883 Adjoint of composition of ...
pjcofni 31884 Functionality of compositi...
pjss1coi 31885 Subset relationship for pr...
pjss2coi 31886 Subset relationship for pr...
pjssmi 31887 Projection meet property. ...
pjssge0i 31888 Theorem 4.5(iv)->(v) of [B...
pjdifnormi 31889 Theorem 4.5(v)<->(vi) of [...
pjnormssi 31890 Theorem 4.5(i)<->(vi) of [...
pjorthcoi 31891 Composition of projections...
pjscji 31892 The projection of orthogon...
pjssumi 31893 The projection on a subspa...
pjssposi 31894 Projector ordering can be ...
pjordi 31895 The definition of projecto...
pjssdif2i 31896 The projection subspace of...
pjssdif1i 31897 A necessary and sufficient...
pjimai 31898 The image of a projection....
pjidmcoi 31899 A projection is idempotent...
pjoccoi 31900 Composition of projections...
pjtoi 31901 Subspace sum of projection...
pjoci 31902 Projection of orthocomplem...
pjidmco 31903 A projection operator is i...
dfpjop 31904 Definition of projection o...
pjhmopidm 31905 Two ways to express the se...
elpjidm 31906 A projection operator is i...
elpjhmop 31907 A projection operator is H...
0leopj 31908 A projector is a positive ...
pjadj2 31909 A projector is self-adjoin...
pjadj3 31910 A projector is self-adjoin...
elpjch 31911 Reconstruction of the subs...
elpjrn 31912 Reconstruction of the subs...
pjinvari 31913 A closed subspace ` H ` wi...
pjin1i 31914 Lemma for Theorem 1.22 of ...
pjin2i 31915 Lemma for Theorem 1.22 of ...
pjin3i 31916 Lemma for Theorem 1.22 of ...
pjclem1 31917 Lemma for projection commu...
pjclem2 31918 Lemma for projection commu...
pjclem3 31919 Lemma for projection commu...
pjclem4a 31920 Lemma for projection commu...
pjclem4 31921 Lemma for projection commu...
pjci 31922 Two subspaces commute iff ...
pjcmul1i 31923 A necessary and sufficient...
pjcmul2i 31924 The projection subspace of...
pjcohocli 31925 Closure of composition of ...
pjadj2coi 31926 Adjoint of double composit...
pj2cocli 31927 Closure of double composit...
pj3lem1 31928 Lemma for projection tripl...
pj3si 31929 Stronger projection triple...
pj3i 31930 Projection triplet theorem...
pj3cor1i 31931 Projection triplet corolla...
pjs14i 31932 Theorem S-14 of Watanabe, ...
isst 31935 Property of a state. (Con...
ishst 31936 Property of a complex Hilb...
sticl 31937 ` [ 0 , 1 ] ` closure of t...
stcl 31938 Real closure of the value ...
hstcl 31939 Closure of the value of a ...
hst1a 31940 Unit value of a Hilbert-sp...
hstel2 31941 Properties of a Hilbert-sp...
hstorth 31942 Orthogonality property of ...
hstosum 31943 Orthogonal sum property of...
hstoc 31944 Sum of a Hilbert-space-val...
hstnmoc 31945 Sum of norms of a Hilbert-...
stge0 31946 The value of a state is no...
stle1 31947 The value of a state is le...
hstle1 31948 The norm of the value of a...
hst1h 31949 The norm of a Hilbert-spac...
hst0h 31950 The norm of a Hilbert-spac...
hstpyth 31951 Pythagorean property of a ...
hstle 31952 Ordering property of a Hil...
hstles 31953 Ordering property of a Hil...
hstoh 31954 A Hilbert-space-valued sta...
hst0 31955 A Hilbert-space-valued sta...
sthil 31956 The value of a state at th...
stj 31957 The value of a state on a ...
sto1i 31958 The state of a subspace pl...
sto2i 31959 The state of the orthocomp...
stge1i 31960 If a state is greater than...
stle0i 31961 If a state is less than or...
stlei 31962 Ordering law for states. ...
stlesi 31963 Ordering law for states. ...
stji1i 31964 Join of components of Sasa...
stm1i 31965 State of component of unit...
stm1ri 31966 State of component of unit...
stm1addi 31967 Sum of states whose meet i...
staddi 31968 If the sum of 2 states is ...
stm1add3i 31969 Sum of states whose meet i...
stadd3i 31970 If the sum of 3 states is ...
st0 31971 The state of the zero subs...
strlem1 31972 Lemma for strong state the...
strlem2 31973 Lemma for strong state the...
strlem3a 31974 Lemma for strong state the...
strlem3 31975 Lemma for strong state the...
strlem4 31976 Lemma for strong state the...
strlem5 31977 Lemma for strong state the...
strlem6 31978 Lemma for strong state the...
stri 31979 Strong state theorem. The...
strb 31980 Strong state theorem (bidi...
hstrlem2 31981 Lemma for strong set of CH...
hstrlem3a 31982 Lemma for strong set of CH...
hstrlem3 31983 Lemma for strong set of CH...
hstrlem4 31984 Lemma for strong set of CH...
hstrlem5 31985 Lemma for strong set of CH...
hstrlem6 31986 Lemma for strong set of CH...
hstri 31987 Hilbert space admits a str...
hstrbi 31988 Strong CH-state theorem (b...
largei 31989 A Hilbert lattice admits a...
jplem1 31990 Lemma for Jauch-Piron theo...
jplem2 31991 Lemma for Jauch-Piron theo...
jpi 31992 The function ` S ` , that ...
golem1 31993 Lemma for Godowski's equat...
golem2 31994 Lemma for Godowski's equat...
goeqi 31995 Godowski's equation, shown...
stcltr1i 31996 Property of a strong class...
stcltr2i 31997 Property of a strong class...
stcltrlem1 31998 Lemma for strong classical...
stcltrlem2 31999 Lemma for strong classical...
stcltrthi 32000 Theorem for classically st...
cvbr 32004 Binary relation expressing...
cvbr2 32005 Binary relation expressing...
cvcon3 32006 Contraposition law for the...
cvpss 32007 The covers relation implie...
cvnbtwn 32008 The covers relation implie...
cvnbtwn2 32009 The covers relation implie...
cvnbtwn3 32010 The covers relation implie...
cvnbtwn4 32011 The covers relation implie...
cvnsym 32012 The covers relation is not...
cvnref 32013 The covers relation is not...
cvntr 32014 The covers relation is not...
spansncv2 32015 Hilbert space has the cove...
mdbr 32016 Binary relation expressing...
mdi 32017 Consequence of the modular...
mdbr2 32018 Binary relation expressing...
mdbr3 32019 Binary relation expressing...
mdbr4 32020 Binary relation expressing...
dmdbr 32021 Binary relation expressing...
dmdmd 32022 The dual modular pair prop...
mddmd 32023 The modular pair property ...
dmdi 32024 Consequence of the dual mo...
dmdbr2 32025 Binary relation expressing...
dmdi2 32026 Consequence of the dual mo...
dmdbr3 32027 Binary relation expressing...
dmdbr4 32028 Binary relation expressing...
dmdi4 32029 Consequence of the dual mo...
dmdbr5 32030 Binary relation expressing...
mddmd2 32031 Relationship between modul...
mdsl0 32032 A sublattice condition tha...
ssmd1 32033 Ordering implies the modul...
ssmd2 32034 Ordering implies the modul...
ssdmd1 32035 Ordering implies the dual ...
ssdmd2 32036 Ordering implies the dual ...
dmdsl3 32037 Sublattice mapping for a d...
mdsl3 32038 Sublattice mapping for a m...
mdslle1i 32039 Order preservation of the ...
mdslle2i 32040 Order preservation of the ...
mdslj1i 32041 Join preservation of the o...
mdslj2i 32042 Meet preservation of the r...
mdsl1i 32043 If the modular pair proper...
mdsl2i 32044 If the modular pair proper...
mdsl2bi 32045 If the modular pair proper...
cvmdi 32046 The covering property impl...
mdslmd1lem1 32047 Lemma for ~ mdslmd1i . (C...
mdslmd1lem2 32048 Lemma for ~ mdslmd1i . (C...
mdslmd1lem3 32049 Lemma for ~ mdslmd1i . (C...
mdslmd1lem4 32050 Lemma for ~ mdslmd1i . (C...
mdslmd1i 32051 Preservation of the modula...
mdslmd2i 32052 Preservation of the modula...
mdsldmd1i 32053 Preservation of the dual m...
mdslmd3i 32054 Modular pair conditions th...
mdslmd4i 32055 Modular pair condition tha...
csmdsymi 32056 Cross-symmetry implies M-s...
mdexchi 32057 An exchange lemma for modu...
cvmd 32058 The covering property impl...
cvdmd 32059 The covering property impl...
ela 32061 Atoms in a Hilbert lattice...
elat2 32062 Expanded membership relati...
elatcv0 32063 A Hilbert lattice element ...
atcv0 32064 An atom covers the zero su...
atssch 32065 Atoms are a subset of the ...
atelch 32066 An atom is a Hilbert latti...
atne0 32067 An atom is not the Hilbert...
atss 32068 A lattice element smaller ...
atsseq 32069 Two atoms in a subset rela...
atcveq0 32070 A Hilbert lattice element ...
h1da 32071 A 1-dimensional subspace i...
spansna 32072 The span of the singleton ...
sh1dle 32073 A 1-dimensional subspace i...
ch1dle 32074 A 1-dimensional subspace i...
atom1d 32075 The 1-dimensional subspace...
superpos 32076 Superposition Principle. ...
chcv1 32077 The Hilbert lattice has th...
chcv2 32078 The Hilbert lattice has th...
chjatom 32079 The join of a closed subsp...
shatomici 32080 The lattice of Hilbert sub...
hatomici 32081 The Hilbert lattice is ato...
hatomic 32082 A Hilbert lattice is atomi...
shatomistici 32083 The lattice of Hilbert sub...
hatomistici 32084 ` CH ` is atomistic, i.e. ...
chpssati 32085 Two Hilbert lattice elemen...
chrelati 32086 The Hilbert lattice is rel...
chrelat2i 32087 A consequence of relative ...
cvati 32088 If a Hilbert lattice eleme...
cvbr4i 32089 An alternate way to expres...
cvexchlem 32090 Lemma for ~ cvexchi . (Co...
cvexchi 32091 The Hilbert lattice satisf...
chrelat2 32092 A consequence of relative ...
chrelat3 32093 A consequence of relative ...
chrelat3i 32094 A consequence of the relat...
chrelat4i 32095 A consequence of relative ...
cvexch 32096 The Hilbert lattice satisf...
cvp 32097 The Hilbert lattice satisf...
atnssm0 32098 The meet of a Hilbert latt...
atnemeq0 32099 The meet of distinct atoms...
atssma 32100 The meet with an atom's su...
atcv0eq 32101 Two atoms covering the zer...
atcv1 32102 Two atoms covering the zer...
atexch 32103 The Hilbert lattice satisf...
atomli 32104 An assertion holding in at...
atoml2i 32105 An assertion holding in at...
atordi 32106 An ordering law for a Hilb...
atcvatlem 32107 Lemma for ~ atcvati . (Co...
atcvati 32108 A nonzero Hilbert lattice ...
atcvat2i 32109 A Hilbert lattice element ...
atord 32110 An ordering law for a Hilb...
atcvat2 32111 A Hilbert lattice element ...
chirredlem1 32112 Lemma for ~ chirredi . (C...
chirredlem2 32113 Lemma for ~ chirredi . (C...
chirredlem3 32114 Lemma for ~ chirredi . (C...
chirredlem4 32115 Lemma for ~ chirredi . (C...
chirredi 32116 The Hilbert lattice is irr...
chirred 32117 The Hilbert lattice is irr...
atcvat3i 32118 A condition implying that ...
atcvat4i 32119 A condition implying exist...
atdmd 32120 Two Hilbert lattice elemen...
atmd 32121 Two Hilbert lattice elemen...
atmd2 32122 Two Hilbert lattice elemen...
atabsi 32123 Absorption of an incompara...
atabs2i 32124 Absorption of an incompara...
mdsymlem1 32125 Lemma for ~ mdsymi . (Con...
mdsymlem2 32126 Lemma for ~ mdsymi . (Con...
mdsymlem3 32127 Lemma for ~ mdsymi . (Con...
mdsymlem4 32128 Lemma for ~ mdsymi . This...
mdsymlem5 32129 Lemma for ~ mdsymi . (Con...
mdsymlem6 32130 Lemma for ~ mdsymi . This...
mdsymlem7 32131 Lemma for ~ mdsymi . Lemm...
mdsymlem8 32132 Lemma for ~ mdsymi . Lemm...
mdsymi 32133 M-symmetry of the Hilbert ...
mdsym 32134 M-symmetry of the Hilbert ...
dmdsym 32135 Dual M-symmetry of the Hil...
atdmd2 32136 Two Hilbert lattice elemen...
sumdmdii 32137 If the subspace sum of two...
cmmdi 32138 Commuting subspaces form a...
cmdmdi 32139 Commuting subspaces form a...
sumdmdlem 32140 Lemma for ~ sumdmdi . The...
sumdmdlem2 32141 Lemma for ~ sumdmdi . (Co...
sumdmdi 32142 The subspace sum of two Hi...
dmdbr4ati 32143 Dual modular pair property...
dmdbr5ati 32144 Dual modular pair property...
dmdbr6ati 32145 Dual modular pair property...
dmdbr7ati 32146 Dual modular pair property...
mdoc1i 32147 Orthocomplements form a mo...
mdoc2i 32148 Orthocomplements form a mo...
dmdoc1i 32149 Orthocomplements form a du...
dmdoc2i 32150 Orthocomplements form a du...
mdcompli 32151 A condition equivalent to ...
dmdcompli 32152 A condition equivalent to ...
mddmdin0i 32153 If dual modular implies mo...
cdjreui 32154 A member of the sum of dis...
cdj1i 32155 Two ways to express " ` A ...
cdj3lem1 32156 A property of " ` A ` and ...
cdj3lem2 32157 Lemma for ~ cdj3i . Value...
cdj3lem2a 32158 Lemma for ~ cdj3i . Closu...
cdj3lem2b 32159 Lemma for ~ cdj3i . The f...
cdj3lem3 32160 Lemma for ~ cdj3i . Value...
cdj3lem3a 32161 Lemma for ~ cdj3i . Closu...
cdj3lem3b 32162 Lemma for ~ cdj3i . The s...
cdj3i 32163 Two ways to express " ` A ...
The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here
mathbox 32164 (_This theorem is a dummy ...
sa-abvi 32165 A theorem about the univer...
xfree 32166 A partial converse to ~ 19...
xfree2 32167 A partial converse to ~ 19...
addltmulALT 32168 A proof readability experi...
bian1d 32169 Adding a superfluous conju...
or3di 32170 Distributive law for disju...
or3dir 32171 Distributive law for disju...
3o1cs 32172 Deduction eliminating disj...
3o2cs 32173 Deduction eliminating disj...
3o3cs 32174 Deduction eliminating disj...
13an22anass 32175 Associative law for four c...
sbc2iedf 32176 Conversion of implicit sub...
rspc2daf 32177 Double restricted speciali...
ralcom4f 32178 Commutation of restricted ...
rexcom4f 32179 Commutation of restricted ...
19.9d2rf 32180 A deduction version of one...
19.9d2r 32181 A deduction version of one...
r19.29ffa 32182 A commonly used pattern ba...
eqtrb 32183 A transposition of equalit...
eqelbid 32184 A variable elimination law...
opsbc2ie 32185 Conversion of implicit sub...
opreu2reuALT 32186 Correspondence between uni...
2reucom 32189 Double restricted existent...
2reu2rex1 32190 Double restricted existent...
2reureurex 32191 Double restricted existent...
2reu2reu2 32192 Double restricted existent...
opreu2reu1 32193 Equivalent definition of t...
sq2reunnltb 32194 There exists a unique deco...
addsqnot2reu 32195 For each complex number ` ...
sbceqbidf 32196 Equality theorem for class...
sbcies 32197 A special version of class...
mo5f 32198 Alternate definition of "a...
nmo 32199 Negation of "at most one"....
reuxfrdf 32200 Transfer existential uniqu...
rexunirn 32201 Restricted existential qua...
rmoxfrd 32202 Transfer "at most one" res...
rmoun 32203 "At most one" restricted e...
rmounid 32204 A case where an "at most o...
riotaeqbidva 32205 Equivalent wff's yield equ...
dmrab 32206 Domain of a restricted cla...
difrab2 32207 Difference of two restrict...
rabexgfGS 32208 Separation Scheme in terms...
rabsnel 32209 Truth implied by equality ...
eqrrabd 32210 Deduce equality with a res...
foresf1o 32211 From a surjective function...
rabfodom 32212 Domination relation for re...
abrexdomjm 32213 An indexed set is dominate...
abrexdom2jm 32214 An indexed set is dominate...
abrexexd 32215 Existence of a class abstr...
elabreximd 32216 Class substitution in an i...
elabreximdv 32217 Class substitution in an i...
abrexss 32218 A necessary condition for ...
elunsn 32219 Elementhood to a union wit...
nelun 32220 Negated membership for a u...
snsssng 32221 If a singleton is a subset...
inin 32222 Intersection with an inter...
inindif 32223 See ~ inundif . (Contribu...
difininv 32224 Condition for the intersec...
difeq 32225 Rewriting an equation with...
eqdif 32226 If both set differences of...
indifbi 32227 Two ways to express equali...
diffib 32228 Case where ~ diffi is a bi...
difxp1ss 32229 Difference law for Cartesi...
difxp2ss 32230 Difference law for Cartesi...
indifundif 32231 A remarkable equation with...
elpwincl1 32232 Closure of intersection wi...
elpwdifcl 32233 Closure of class differenc...
elpwiuncl 32234 Closure of indexed union w...
eqsnd 32235 Deduce that a set is a sin...
elpreq 32236 Equality wihin a pair. (C...
nelpr 32237 A set ` A ` not in a pair ...
inpr0 32238 Rewrite an empty intersect...
neldifpr1 32239 The first element of a pai...
neldifpr2 32240 The second element of a pa...
unidifsnel 32241 The other element of a pai...
unidifsnne 32242 The other element of a pai...
ifeqeqx 32243 An equality theorem tailor...
elimifd 32244 Elimination of a condition...
elim2if 32245 Elimination of two conditi...
elim2ifim 32246 Elimination of two conditi...
ifeq3da 32247 Given an expression ` C ` ...
ifnetrue 32248 Deduce truth from a condit...
ifnefals 32249 Deduce falsehood from a co...
ifnebib 32250 The converse of ~ ifbi hol...
uniinn0 32251 Sufficient and necessary c...
uniin1 32252 Union of intersection. Ge...
uniin2 32253 Union of intersection. Ge...
difuncomp 32254 Express a class difference...
elpwunicl 32255 Closure of a set union wit...
cbviunf 32256 Rule used to change the bo...
iuneq12daf 32257 Equality deduction for ind...
iunin1f 32258 Indexed union of intersect...
ssiun3 32259 Subset equivalence for an ...
ssiun2sf 32260 Subset relationship for an...
iuninc 32261 The union of an increasing...
iundifdifd 32262 The intersection of a set ...
iundifdif 32263 The intersection of a set ...
iunrdx 32264 Re-index an indexed union....
iunpreima 32265 Preimage of an indexed uni...
iunrnmptss 32266 A subset relation for an i...
iunxunsn 32267 Appending a set to an inde...
iunxunpr 32268 Appending two sets to an i...
iinabrex 32269 Rewriting an indexed inter...
disjnf 32270 In case ` x ` is not free ...
cbvdisjf 32271 Change bound variables in ...
disjss1f 32272 A subset of a disjoint col...
disjeq1f 32273 Equality theorem for disjo...
disjxun0 32274 Simplify a disjoint union....
disjdifprg 32275 A trivial partition into a...
disjdifprg2 32276 A trivial partition of a s...
disji2f 32277 Property of a disjoint col...
disjif 32278 Property of a disjoint col...
disjorf 32279 Two ways to say that a col...
disjorsf 32280 Two ways to say that a col...
disjif2 32281 Property of a disjoint col...
disjabrex 32282 Rewriting a disjoint colle...
disjabrexf 32283 Rewriting a disjoint colle...
disjpreima 32284 A preimage of a disjoint s...
disjrnmpt 32285 Rewriting a disjoint colle...
disjin 32286 If a collection is disjoin...
disjin2 32287 If a collection is disjoin...
disjxpin 32288 Derive a disjunction over ...
iundisjf 32289 Rewrite a countable union ...
iundisj2f 32290 A disjoint union is disjoi...
disjrdx 32291 Re-index a disjunct collec...
disjex 32292 Two ways to say that two c...
disjexc 32293 A variant of ~ disjex , ap...
disjunsn 32294 Append an element to a dis...
disjun0 32295 Adding the empty element p...
disjiunel 32296 A set of elements B of a d...
disjuniel 32297 A set of elements B of a d...
xpdisjres 32298 Restriction of a constant ...
opeldifid 32299 Ordered pair elementhood o...
difres 32300 Case when class difference...
imadifxp 32301 Image of the difference wi...
relfi 32302 A relation (set) is finite...
0res 32303 Restriction of the empty f...
fcoinver 32304 Build an equivalence relat...
fcoinvbr 32305 Binary relation for the eq...
brabgaf 32306 The law of concretion for ...
brelg 32307 Two things in a binary rel...
br8d 32308 Substitution for an eight-...
opabdm 32309 Domain of an ordered-pair ...
opabrn 32310 Range of an ordered-pair c...
opabssi 32311 Sufficient condition for a...
opabid2ss 32312 One direction of ~ opabid2...
ssrelf 32313 A subclass relationship de...
eqrelrd2 32314 A version of ~ eqrelrdv2 w...
erbr3b 32315 Biconditional for equivale...
iunsnima 32316 Image of a singleton by an...
iunsnima2 32317 Version of ~ iunsnima with...
ac6sf2 32318 Alternate version of ~ ac6...
fnresin 32319 Restriction of a function ...
f1o3d 32320 Describe an implicit one-t...
eldmne0 32321 A function of nonempty dom...
f1rnen 32322 Equinumerosity of the rang...
rinvf1o 32323 Sufficient conditions for ...
fresf1o 32324 Conditions for a restricti...
nfpconfp 32325 The set of fixed points of...
fmptco1f1o 32326 The action of composing (t...
cofmpt2 32327 Express composition of a m...
f1mptrn 32328 Express injection for a ma...
dfimafnf 32329 Alternate definition of th...
funimass4f 32330 Membership relation for th...
elimampt 32331 Membership in the image of...
suppss2f 32332 Show that the support of a...
ofrn 32333 The range of the function ...
ofrn2 32334 The range of the function ...
off2 32335 The function operation pro...
ofresid 32336 Applying an operation rest...
fimarab 32337 Expressing the image of a ...
unipreima 32338 Preimage of a class union....
opfv 32339 Value of a function produc...
xppreima 32340 The preimage of a Cartesia...
2ndimaxp 32341 Image of a cartesian produ...
djussxp2 32342 Stronger version of ~ djus...
2ndresdju 32343 The ` 2nd ` function restr...
2ndresdjuf1o 32344 The ` 2nd ` function restr...
xppreima2 32345 The preimage of a Cartesia...
abfmpunirn 32346 Membership in a union of a...
rabfmpunirn 32347 Membership in a union of a...
abfmpeld 32348 Membership in an element o...
abfmpel 32349 Membership in an element o...
fmptdF 32350 Domain and codomain of the...
fmptcof2 32351 Composition of two functio...
fcomptf 32352 Express composition of two...
acunirnmpt 32353 Axiom of choice for the un...
acunirnmpt2 32354 Axiom of choice for the un...
acunirnmpt2f 32355 Axiom of choice for the un...
aciunf1lem 32356 Choice in an index union. ...
aciunf1 32357 Choice in an index union. ...
ofoprabco 32358 Function operation as a co...
ofpreima 32359 Express the preimage of a ...
ofpreima2 32360 Express the preimage of a ...
funcnvmpt 32361 Condition for a function i...
funcnv5mpt 32362 Two ways to say that a fun...
funcnv4mpt 32363 Two ways to say that a fun...
preimane 32364 Different elements have di...
fnpreimac 32365 Choose a set ` x ` contain...
fgreu 32366 Exactly one point of a fun...
fcnvgreu 32367 If the converse of a relat...
rnmposs 32368 The range of an operation ...
mptssALT 32369 Deduce subset relation of ...
dfcnv2 32370 Alternative definition of ...
fnimatp 32371 The image of an unordered ...
rnexd 32372 The range of a set is a se...
imaexd 32373 The image of a set is a se...
mpomptxf 32374 Express a two-argument fun...
suppovss 32375 A bound for the support of...
fvdifsupp 32376 Function value is zero out...
suppiniseg 32377 Relation between the suppo...
fsuppinisegfi 32378 The initial segment ` ( ``...
fressupp 32379 The restriction of a funct...
fdifsuppconst 32380 A function is a zero const...
ressupprn 32381 The range of a function re...
supppreima 32382 Express the support of a f...
fsupprnfi 32383 Finite support implies fin...
mptiffisupp 32384 Conditions for a mapping f...
cosnopne 32385 Composition of two ordered...
cosnop 32386 Composition of two ordered...
cnvprop 32387 Converse of a pair of orde...
brprop 32388 Binary relation for a pair...
mptprop 32389 Rewrite pairs of ordered p...
coprprop 32390 Composition of two pairs o...
gtiso 32391 Two ways to write a strict...
isoun 32392 Infer an isomorphism from ...
disjdsct 32393 A disjoint collection is d...
df1stres 32394 Definition for a restricti...
df2ndres 32395 Definition for a restricti...
1stpreimas 32396 The preimage of a singleto...
1stpreima 32397 The preimage by ` 1st ` is...
2ndpreima 32398 The preimage by ` 2nd ` is...
curry2ima 32399 The image of a curried fun...
preiman0 32400 The preimage of a nonempty...
intimafv 32401 The intersection of an ima...
ecref 32402 All elements are in their ...
supssd 32403 Inequality deduction for s...
infssd 32404 Inequality deduction for i...
imafi2 32405 The image by a finite set ...
unifi3 32406 If a union is finite, then...
snct 32407 A singleton is countable. ...
prct 32408 An unordered pair is count...
mpocti 32409 An operation is countable ...
abrexct 32410 An image set of a countabl...
mptctf 32411 A countable mapping set is...
abrexctf 32412 An image set of a countabl...
padct 32413 Index a countable set with...
cnvoprabOLD 32414 The converse of a class ab...
f1od2 32415 Sufficient condition for a...
fcobij 32416 Composing functions with a...
fcobijfs 32417 Composing finitely support...
suppss3 32418 Deduce a function's suppor...
fsuppcurry1 32419 Finite support of a currie...
fsuppcurry2 32420 Finite support of a currie...
offinsupp1 32421 Finite support for a funct...
ffs2 32422 Rewrite a function's suppo...
ffsrn 32423 The range of a finitely su...
resf1o 32424 Restriction of functions t...
maprnin 32425 Restricting the range of t...
fpwrelmapffslem 32426 Lemma for ~ fpwrelmapffs ....
fpwrelmap 32427 Define a canonical mapping...
fpwrelmapffs 32428 Define a canonical mapping...
creq0 32429 The real representation of...
1nei 32430 The imaginary unit ` _i ` ...
1neg1t1neg1 32431 An integer unit times itse...
nnmulge 32432 Multiplying by a positive ...
lt2addrd 32433 If the right-hand side of ...
xrlelttric 32434 Trichotomy law for extende...
xaddeq0 32435 Two extended reals which a...
xrinfm 32436 The extended real numbers ...
le2halvesd 32437 A sum is less than the who...
xraddge02 32438 A number is less than or e...
xrge0addge 32439 A number is less than or e...
xlt2addrd 32440 If the right-hand side of ...
xrsupssd 32441 Inequality deduction for s...
xrge0infss 32442 Any subset of nonnegative ...
xrge0infssd 32443 Inequality deduction for i...
xrge0addcld 32444 Nonnegative extended reals...
xrge0subcld 32445 Condition for closure of n...
infxrge0lb 32446 A member of a set of nonne...
infxrge0glb 32447 The infimum of a set of no...
infxrge0gelb 32448 The infimum of a set of no...
xrofsup 32449 The supremum is preserved ...
supxrnemnf 32450 The supremum of a nonempty...
xnn0gt0 32451 Nonzero extended nonnegati...
xnn01gt 32452 An extended nonnegative in...
nn0xmulclb 32453 Finite multiplication in t...
joiniooico 32454 Disjoint joining an open i...
ubico 32455 A right-open interval does...
xeqlelt 32456 Equality in terms of 'less...
eliccelico 32457 Relate elementhood to a cl...
elicoelioo 32458 Relate elementhood to a cl...
iocinioc2 32459 Intersection between two o...
xrdifh 32460 Class difference of a half...
iocinif 32461 Relate intersection of two...
difioo 32462 The difference between two...
difico 32463 The difference between two...
uzssico 32464 Upper integer sets are a s...
fz2ssnn0 32465 A finite set of sequential...
nndiffz1 32466 Upper set of the positive ...
ssnnssfz 32467 For any finite subset of `...
fzne1 32468 Elementhood in a finite se...
fzm1ne1 32469 Elementhood of an integer ...
fzspl 32470 Split the last element of ...
fzdif2 32471 Split the last element of ...
fzodif2 32472 Split the last element of ...
fzodif1 32473 Set difference of two half...
fzsplit3 32474 Split a finite interval of...
bcm1n 32475 The proportion of one bino...
iundisjfi 32476 Rewrite a countable union ...
iundisj2fi 32477 A disjoint union is disjoi...
iundisjcnt 32478 Rewrite a countable union ...
iundisj2cnt 32479 A countable disjoint union...
fzone1 32480 Elementhood in a half-open...
fzom1ne1 32481 Elementhood in a half-open...
f1ocnt 32482 Given a countable set ` A ...
fz1nnct 32483 NN and integer ranges star...
fz1nntr 32484 NN and integer ranges star...
nn0difffzod 32485 A nonnegative integer that...
suppssnn0 32486 Show that the support of a...
hashunif 32487 The cardinality of a disjo...
hashxpe 32488 The size of the Cartesian ...
hashgt1 32489 Restate "set contains at l...
numdenneg 32490 Numerator and denominator ...
divnumden2 32491 Calculate the reduced form...
nnindf 32492 Principle of Mathematical ...
nn0min 32493 Extracting the minimum pos...
subne0nn 32494 A nonnegative difference i...
ltesubnnd 32495 Subtracting an integer num...
fprodeq02 32496 If one of the factors is z...
pr01ssre 32497 The range of the indicator...
fprodex01 32498 A product of factors equal...
prodpr 32499 A product over a pair is t...
prodtp 32500 A product over a triple is...
fsumub 32501 An upper bound for a term ...
fsumiunle 32502 Upper bound for a sum of n...
dfdec100 32503 Split the hundreds from a ...
dp2eq1 32506 Equality theorem for the d...
dp2eq2 32507 Equality theorem for the d...
dp2eq1i 32508 Equality theorem for the d...
dp2eq2i 32509 Equality theorem for the d...
dp2eq12i 32510 Equality theorem for the d...
dp20u 32511 Add a zero in the tenths (...
dp20h 32512 Add a zero in the unit pla...
dp2cl 32513 Closure for the decimal fr...
dp2clq 32514 Closure for a decimal frac...
rpdp2cl 32515 Closure for a decimal frac...
rpdp2cl2 32516 Closure for a decimal frac...
dp2lt10 32517 Decimal fraction builds re...
dp2lt 32518 Comparing two decimal frac...
dp2ltsuc 32519 Comparing a decimal fracti...
dp2ltc 32520 Comparing two decimal expa...
dpval 32523 Define the value of the de...
dpcl 32524 Prove that the closure of ...
dpfrac1 32525 Prove a simple equivalence...
dpval2 32526 Value of the decimal point...
dpval3 32527 Value of the decimal point...
dpmul10 32528 Multiply by 10 a decimal e...
decdiv10 32529 Divide a decimal number by...
dpmul100 32530 Multiply by 100 a decimal ...
dp3mul10 32531 Multiply by 10 a decimal e...
dpmul1000 32532 Multiply by 1000 a decimal...
dpval3rp 32533 Value of the decimal point...
dp0u 32534 Add a zero in the tenths p...
dp0h 32535 Remove a zero in the units...
rpdpcl 32536 Closure of the decimal poi...
dplt 32537 Comparing two decimal expa...
dplti 32538 Comparing a decimal expans...
dpgti 32539 Comparing a decimal expans...
dpltc 32540 Comparing two decimal inte...
dpexpp1 32541 Add one zero to the mantis...
0dp2dp 32542 Multiply by 10 a decimal e...
dpadd2 32543 Addition with one decimal,...
dpadd 32544 Addition with one decimal....
dpadd3 32545 Addition with two decimals...
dpmul 32546 Multiplication with one de...
dpmul4 32547 An upper bound to multipli...
threehalves 32548 Example theorem demonstrat...
1mhdrd 32549 Example theorem demonstrat...
xdivval 32552 Value of division: the (un...
xrecex 32553 Existence of reciprocal of...
xmulcand 32554 Cancellation law for exten...
xreceu 32555 Existential uniqueness of ...
xdivcld 32556 Closure law for the extend...
xdivcl 32557 Closure law for the extend...
xdivmul 32558 Relationship between divis...
rexdiv 32559 The extended real division...
xdivrec 32560 Relationship between divis...
xdivid 32561 A number divided by itself...
xdiv0 32562 Division into zero is zero...
xdiv0rp 32563 Division into zero is zero...
eliccioo 32564 Membership in a closed int...
elxrge02 32565 Elementhood in the set of ...
xdivpnfrp 32566 Plus infinity divided by a...
rpxdivcld 32567 Closure law for extended d...
xrpxdivcld 32568 Closure law for extended d...
wrdfd 32569 A word is a zero-based seq...
wrdres 32570 Condition for the restrict...
wrdsplex 32571 Existence of a split of a ...
pfx1s2 32572 The prefix of length 1 of ...
pfxrn2 32573 The range of a prefix of a...
pfxrn3 32574 Express the range of a pre...
pfxf1 32575 Condition for a prefix to ...
s1f1 32576 Conditions for a length 1 ...
s2rn 32577 Range of a length 2 string...
s2f1 32578 Conditions for a length 2 ...
s3rn 32579 Range of a length 3 string...
s3f1 32580 Conditions for a length 3 ...
s3clhash 32581 Closure of the words of le...
ccatf1 32582 Conditions for a concatena...
pfxlsw2ccat 32583 Reconstruct a word from it...
wrdt2ind 32584 Perform an induction over ...
swrdrn2 32585 The range of a subword is ...
swrdrn3 32586 Express the range of a sub...
swrdf1 32587 Condition for a subword to...
swrdrndisj 32588 Condition for the range of...
splfv3 32589 Symbols to the right of a ...
1cshid 32590 Cyclically shifting a sing...
cshw1s2 32591 Cyclically shifting a leng...
cshwrnid 32592 Cyclically shifting a word...
cshf1o 32593 Condition for the cyclic s...
ressplusf 32594 The group operation functi...
ressnm 32595 The norm in a restricted s...
abvpropd2 32596 Weaker version of ~ abvpro...
oppgle 32597 less-than relation of an o...
oppgleOLD 32598 Obsolete version of ~ oppg...
oppglt 32599 less-than relation of an o...
ressprs 32600 The restriction of a prose...
oduprs 32601 Being a proset is a self-d...
posrasymb 32602 A poset ordering is asymet...
resspos 32603 The restriction of a Poset...
resstos 32604 The restriction of a Toset...
odutos 32605 Being a toset is a self-du...
tlt2 32606 In a Toset, two elements m...
tlt3 32607 In a Toset, two elements m...
trleile 32608 In a Toset, two elements m...
toslublem 32609 Lemma for ~ toslub and ~ x...
toslub 32610 In a toset, the lowest upp...
tosglblem 32611 Lemma for ~ tosglb and ~ x...
tosglb 32612 Same theorem as ~ toslub ,...
clatp0cl 32613 The poset zero of a comple...
clatp1cl 32614 The poset one of a complet...
mntoval 32619 Operation value of the mon...
ismnt 32620 Express the statement " ` ...
ismntd 32621 Property of being a monoto...
mntf 32622 A monotone function is a f...
mgcoval 32623 Operation value of the mon...
mgcval 32624 Monotone Galois connection...
mgcf1 32625 The lower adjoint ` F ` of...
mgcf2 32626 The upper adjoint ` G ` of...
mgccole1 32627 An inequality for the kern...
mgccole2 32628 Inequality for the closure...
mgcmnt1 32629 The lower adjoint ` F ` of...
mgcmnt2 32630 The upper adjoint ` G ` of...
mgcmntco 32631 A Galois connection like s...
dfmgc2lem 32632 Lemma for dfmgc2, backward...
dfmgc2 32633 Alternate definition of th...
mgcmnt1d 32634 Galois connection implies ...
mgcmnt2d 32635 Galois connection implies ...
mgccnv 32636 The inverse Galois connect...
pwrssmgc 32637 Given a function ` F ` , e...
mgcf1olem1 32638 Property of a Galois conne...
mgcf1olem2 32639 Property of a Galois conne...
mgcf1o 32640 Given a Galois connection,...
xrs0 32643 The zero of the extended r...
xrslt 32644 The "strictly less than" r...
xrsinvgval 32645 The inversion operation in...
xrsmulgzz 32646 The "multiple" function in...
xrstos 32647 The extended real numbers ...
xrsclat 32648 The extended real numbers ...
xrsp0 32649 The poset 0 of the extende...
xrsp1 32650 The poset 1 of the extende...
xrge0base 32651 The base of the extended n...
xrge00 32652 The zero of the extended n...
xrge0plusg 32653 The additive law of the ex...
xrge0le 32654 The "less than or equal to...
xrge0mulgnn0 32655 The group multiple functio...
xrge0addass 32656 Associativity of extended ...
xrge0addgt0 32657 The sum of nonnegative and...
xrge0adddir 32658 Right-distributivity of ex...
xrge0adddi 32659 Left-distributivity of ext...
xrge0npcan 32660 Extended nonnegative real ...
fsumrp0cl 32661 Closure of a finite sum of...
abliso 32662 The image of an Abelian gr...
lmhmghmd 32663 A module homomorphism is a...
mhmimasplusg 32664 Value of the operation of ...
lmhmimasvsca 32665 Value of the scalar produc...
gsumsubg 32666 The group sum in a subgrou...
gsumsra 32667 The group sum in a subring...
gsummpt2co 32668 Split a finite sum into a ...
gsummpt2d 32669 Express a finite sum over ...
lmodvslmhm 32670 Scalar multiplication in a...
gsumvsmul1 32671 Pull a scalar multiplicati...
gsummptres 32672 Extend a finite group sum ...
gsummptres2 32673 Extend a finite group sum ...
gsumzresunsn 32674 Append an element to a fin...
gsumpart 32675 Express a group sum as a d...
gsumhashmul 32676 Express a group sum by gro...
xrge0tsmsd 32677 Any finite or infinite sum...
xrge0tsmsbi 32678 Any limit of a finite or i...
xrge0tsmseq 32679 Any limit of a finite or i...
cntzun 32680 The centralizer of a union...
cntzsnid 32681 The centralizer of the ide...
cntrcrng 32682 The center of a ring is a ...
isomnd 32687 A (left) ordered monoid is...
isogrp 32688 A (left-)ordered group is ...
ogrpgrp 32689 A left-ordered group is a ...
omndmnd 32690 A left-ordered monoid is a...
omndtos 32691 A left-ordered monoid is a...
omndadd 32692 In an ordered monoid, the ...
omndaddr 32693 In a right ordered monoid,...
omndadd2d 32694 In a commutative left orde...
omndadd2rd 32695 In a left- and right- orde...
submomnd 32696 A submonoid of an ordered ...
xrge0omnd 32697 The nonnegative extended r...
omndmul2 32698 In an ordered monoid, the ...
omndmul3 32699 In an ordered monoid, the ...
omndmul 32700 In a commutative ordered m...
ogrpinv0le 32701 In an ordered group, the o...
ogrpsub 32702 In an ordered group, the o...
ogrpaddlt 32703 In an ordered group, stric...
ogrpaddltbi 32704 In a right ordered group, ...
ogrpaddltrd 32705 In a right ordered group, ...
ogrpaddltrbid 32706 In a right ordered group, ...
ogrpsublt 32707 In an ordered group, stric...
ogrpinv0lt 32708 In an ordered group, the o...
ogrpinvlt 32709 In an ordered group, the o...
gsumle 32710 A finite sum in an ordered...
symgfcoeu 32711 Uniqueness property of per...
symgcom 32712 Two permutations ` X ` and...
symgcom2 32713 Two permutations ` X ` and...
symgcntz 32714 All elements of a (finite)...
odpmco 32715 The composition of two odd...
symgsubg 32716 The value of the group sub...
pmtrprfv2 32717 In a transposition of two ...
pmtrcnel 32718 Composing a permutation ` ...
pmtrcnel2 32719 Variation on ~ pmtrcnel . ...
pmtrcnelor 32720 Composing a permutation ` ...
pmtridf1o 32721 Transpositions of ` X ` an...
pmtridfv1 32722 Value at X of the transpos...
pmtridfv2 32723 Value at Y of the transpos...
psgnid 32724 Permutation sign of the id...
psgndmfi 32725 For a finite base set, the...
pmtrto1cl 32726 Useful lemma for the follo...
psgnfzto1stlem 32727 Lemma for ~ psgnfzto1st . ...
fzto1stfv1 32728 Value of our permutation `...
fzto1st1 32729 Special case where the per...
fzto1st 32730 The function moving one el...
fzto1stinvn 32731 Value of the inverse of ou...
psgnfzto1st 32732 The permutation sign for m...
tocycval 32735 Value of the cycle builder...
tocycfv 32736 Function value of a permut...
tocycfvres1 32737 A cyclic permutation is a ...
tocycfvres2 32738 A cyclic permutation is th...
cycpmfvlem 32739 Lemma for ~ cycpmfv1 and ~...
cycpmfv1 32740 Value of a cycle function ...
cycpmfv2 32741 Value of a cycle function ...
cycpmfv3 32742 Values outside of the orbi...
cycpmcl 32743 Cyclic permutations are pe...
tocycf 32744 The permutation cycle buil...
tocyc01 32745 Permutation cycles built f...
cycpm2tr 32746 A cyclic permutation of 2 ...
cycpm2cl 32747 Closure for the 2-cycles. ...
cyc2fv1 32748 Function value of a 2-cycl...
cyc2fv2 32749 Function value of a 2-cycl...
trsp2cyc 32750 Exhibit the word a transpo...
cycpmco2f1 32751 The word U used in ~ cycpm...
cycpmco2rn 32752 The orbit of the compositi...
cycpmco2lem1 32753 Lemma for ~ cycpmco2 . (C...
cycpmco2lem2 32754 Lemma for ~ cycpmco2 . (C...
cycpmco2lem3 32755 Lemma for ~ cycpmco2 . (C...
cycpmco2lem4 32756 Lemma for ~ cycpmco2 . (C...
cycpmco2lem5 32757 Lemma for ~ cycpmco2 . (C...
cycpmco2lem6 32758 Lemma for ~ cycpmco2 . (C...
cycpmco2lem7 32759 Lemma for ~ cycpmco2 . (C...
cycpmco2 32760 The composition of a cycli...
cyc2fvx 32761 Function value of a 2-cycl...
cycpm3cl 32762 Closure of the 3-cycles in...
cycpm3cl2 32763 Closure of the 3-cycles in...
cyc3fv1 32764 Function value of a 3-cycl...
cyc3fv2 32765 Function value of a 3-cycl...
cyc3fv3 32766 Function value of a 3-cycl...
cyc3co2 32767 Represent a 3-cycle as a c...
cycpmconjvlem 32768 Lemma for ~ cycpmconjv . ...
cycpmconjv 32769 A formula for computing co...
cycpmrn 32770 The range of the word used...
tocyccntz 32771 All elements of a (finite)...
evpmval 32772 Value of the set of even p...
cnmsgn0g 32773 The neutral element of the...
evpmsubg 32774 The alternating group is a...
evpmid 32775 The identity is an even pe...
altgnsg 32776 The alternating group ` ( ...
cyc3evpm 32777 3-Cycles are even permutat...
cyc3genpmlem 32778 Lemma for ~ cyc3genpm . (...
cyc3genpm 32779 The alternating group ` A ...
cycpmgcl 32780 Cyclic permutations are pe...
cycpmconjslem1 32781 Lemma for ~ cycpmconjs . ...
cycpmconjslem2 32782 Lemma for ~ cycpmconjs . ...
cycpmconjs 32783 All cycles of the same len...
cyc3conja 32784 All 3-cycles are conjugate...
sgnsv 32787 The sign mapping. (Contri...
sgnsval 32788 The sign value. (Contribu...
sgnsf 32789 The sign function. (Contr...
inftmrel 32794 The infinitesimal relation...
isinftm 32795 Express ` x ` is infinites...
isarchi 32796 Express the predicate " ` ...
pnfinf 32797 Plus infinity is an infini...
xrnarchi 32798 The completed real line is...
isarchi2 32799 Alternative way to express...
submarchi 32800 A submonoid is archimedean...
isarchi3 32801 This is the usual definiti...
archirng 32802 Property of Archimedean or...
archirngz 32803 Property of Archimedean le...
archiexdiv 32804 In an Archimedean group, g...
archiabllem1a 32805 Lemma for ~ archiabl : In...
archiabllem1b 32806 Lemma for ~ archiabl . (C...
archiabllem1 32807 Archimedean ordered groups...
archiabllem2a 32808 Lemma for ~ archiabl , whi...
archiabllem2c 32809 Lemma for ~ archiabl . (C...
archiabllem2b 32810 Lemma for ~ archiabl . (C...
archiabllem2 32811 Archimedean ordered groups...
archiabl 32812 Archimedean left- and righ...
isslmd 32815 The predicate "is a semimo...
slmdlema 32816 Lemma for properties of a ...
lmodslmd 32817 Left semimodules generaliz...
slmdcmn 32818 A semimodule is a commutat...
slmdmnd 32819 A semimodule is a monoid. ...
slmdsrg 32820 The scalar component of a ...
slmdbn0 32821 The base set of a semimodu...
slmdacl 32822 Closure of ring addition f...
slmdmcl 32823 Closure of ring multiplica...
slmdsn0 32824 The set of scalars in a se...
slmdvacl 32825 Closure of vector addition...
slmdass 32826 Semiring left module vecto...
slmdvscl 32827 Closure of scalar product ...
slmdvsdi 32828 Distributive law for scala...
slmdvsdir 32829 Distributive law for scala...
slmdvsass 32830 Associative law for scalar...
slmd0cl 32831 The ring zero in a semimod...
slmd1cl 32832 The ring unity in a semiri...
slmdvs1 32833 Scalar product with ring u...
slmd0vcl 32834 The zero vector is a vecto...
slmd0vlid 32835 Left identity law for the ...
slmd0vrid 32836 Right identity law for the...
slmd0vs 32837 Zero times a vector is the...
slmdvs0 32838 Anything times the zero ve...
gsumvsca1 32839 Scalar product of a finite...
gsumvsca2 32840 Scalar product of a finite...
prmsimpcyc 32841 A group of prime order is ...
idomdomd 32842 An integral domain is a do...
idomringd 32843 An integral domain is a ri...
domnlcan 32844 Left-cancellation law for ...
idomrcan 32845 Right-cancellation law for...
urpropd 32846 Sufficient condition for r...
0ringsubrg 32847 A subring of a zero ring i...
frobrhm 32848 In a commutative ring with...
ress1r 32849 ` 1r ` is unaffected by re...
ringinvval 32850 The ring inverse expressed...
dvrcan5 32851 Cancellation law for commo...
subrgchr 32852 If ` A ` is a subring of `...
rmfsupp2 32853 A mapping of a multiplicat...
eufndx 32856 Index value of the Euclide...
eufid 32857 Utility theorem: index-ind...
ringinveu 32860 If a ring unit element ` X...
isdrng4 32861 A division ring is a ring ...
rndrhmcl 32862 The image of a division ri...
sdrgdvcl 32863 A sub-division-ring is clo...
sdrginvcl 32864 A sub-division-ring is clo...
primefldchr 32865 The characteristic of a pr...
fldgenval 32868 Value of the field generat...
fldgenssid 32869 The field generated by a s...
fldgensdrg 32870 A generated subfield is a ...
fldgenssv 32871 A generated subfield is a ...
fldgenss 32872 Generated subfields preser...
fldgenidfld 32873 The subfield generated by ...
fldgenssp 32874 The field generated by a s...
fldgenid 32875 The subfield of a field ` ...
fldgenfld 32876 A generated subfield is a ...
primefldgen1 32877 The prime field of a divis...
1fldgenq 32878 The field of rational numb...
isorng 32883 An ordered ring is a ring ...
orngring 32884 An ordered ring is a ring....
orngogrp 32885 An ordered ring is an orde...
isofld 32886 An ordered field is a fiel...
orngmul 32887 In an ordered ring, the or...
orngsqr 32888 In an ordered ring, all sq...
ornglmulle 32889 In an ordered ring, multip...
orngrmulle 32890 In an ordered ring, multip...
ornglmullt 32891 In an ordered ring, multip...
orngrmullt 32892 In an ordered ring, multip...
orngmullt 32893 In an ordered ring, the st...
ofldfld 32894 An ordered field is a fiel...
ofldtos 32895 An ordered field is a tota...
orng0le1 32896 In an ordered ring, the ri...
ofldlt1 32897 In an ordered field, the r...
ofldchr 32898 The characteristic of an o...
suborng 32899 Every subring of an ordere...
subofld 32900 Every subfield of an order...
isarchiofld 32901 Axiom of Archimedes : a ch...
rhmdvd 32902 A ring homomorphism preser...
kerunit 32903 If a unit element lies in ...
reldmresv 32906 The scalar restriction is ...
resvval 32907 Value of structure restric...
resvid2 32908 General behavior of trivia...
resvval2 32909 Value of nontrivial struct...
resvsca 32910 Base set of a structure re...
resvlem 32911 Other elements of a scalar...
resvlemOLD 32912 Obsolete version of ~ resv...
resvbas 32913 ` Base ` is unaffected by ...
resvbasOLD 32914 Obsolete proof of ~ resvba...
resvplusg 32915 ` +g ` is unaffected by sc...
resvplusgOLD 32916 Obsolete proof of ~ resvpl...
resvvsca 32917 ` .s ` is unaffected by sc...
resvvscaOLD 32918 Obsolete proof of ~ resvvs...
resvmulr 32919 ` .r ` is unaffected by sc...
resvmulrOLD 32920 Obsolete proof of ~ resvmu...
resv0g 32921 ` 0g ` is unaffected by sc...
resv1r 32922 ` 1r ` is unaffected by sc...
resvcmn 32923 Scalar restriction preserv...
gzcrng 32924 The gaussian integers form...
reofld 32925 The real numbers form an o...
nn0omnd 32926 The nonnegative integers f...
rearchi 32927 The field of the real numb...
nn0archi 32928 The monoid of the nonnegat...
xrge0slmod 32929 The extended nonnegative r...
qusker 32930 The kernel of a quotient m...
eqgvscpbl 32931 The left coset equivalence...
qusvscpbl 32932 The quotient map distribut...
qusvsval 32933 Value of the scalar multip...
imaslmod 32934 The image structure of a l...
imasmhm 32935 Given a function ` F ` wit...
imasghm 32936 Given a function ` F ` wit...
imasrhm 32937 Given a function ` F ` wit...
imaslmhm 32938 Given a function ` F ` wit...
quslmod 32939 If ` G ` is a submodule in...
quslmhm 32940 If ` G ` is a submodule of...
quslvec 32941 If ` S ` is a vector subsp...
ecxpid 32942 The equivalence class of a...
eqg0el 32943 Equivalence class of a quo...
qsxpid 32944 The quotient set of a cart...
qusxpid 32945 The Group quotient equival...
qustriv 32946 The quotient of a group ` ...
qustrivr 32947 Converse of ~ qustriv . (...
znfermltl 32948 Fermat's little theorem in...
islinds5 32949 A set is linearly independ...
ellspds 32950 Variation on ~ ellspd . (...
0ellsp 32951 Zero is in all spans. (Co...
0nellinds 32952 The group identity cannot ...
rspsnel 32953 Membership in a principal ...
rspsnid 32954 A principal ideal contains...
elrsp 32955 Write the elements of a ri...
rspidlid 32956 The ideal span of an ideal...
pidlnz 32957 A principal ideal generate...
dvdsruassoi 32958 If two elements ` X ` and ...
dvdsruasso 32959 Two elements ` X ` and ` Y...
dvdsrspss 32960 In a ring, an element ` X ...
rspsnasso 32961 Two elements ` X ` and ` Y...
lbslsp 32962 Any element of a left modu...
lindssn 32963 Any singleton of a nonzero...
lindflbs 32964 Conditions for an independ...
islbs5 32965 An equivalent formulation ...
linds2eq 32966 Deduce equality of element...
lindfpropd 32967 Property deduction for lin...
lindspropd 32968 Property deduction for lin...
elgrplsmsn 32969 Membership in a sumset wit...
lsmsnorb 32970 The sumset of a group with...
lsmsnorb2 32971 The sumset of a single ele...
elringlsm 32972 Membership in a product of...
elringlsmd 32973 Membership in a product of...
ringlsmss 32974 Closure of the product of ...
ringlsmss1 32975 The product of an ideal ` ...
ringlsmss2 32976 The product with an ideal ...
lsmsnpridl 32977 The product of the ring wi...
lsmsnidl 32978 The product of the ring wi...
lsmidllsp 32979 The sum of two ideals is t...
lsmidl 32980 The sum of two ideals is a...
lsmssass 32981 Group sum is associative, ...
grplsm0l 32982 Sumset with the identity s...
grplsmid 32983 The direct sum of an eleme...
qusmul 32984 Value of the ring operatio...
quslsm 32985 Express the image by the q...
qusbas2 32986 Alternate definition of th...
qus0g 32987 The identity element of a ...
qusima 32988 The image of a subgroup by...
qusrn 32989 The natural map from eleme...
nsgqus0 32990 A normal subgroup ` N ` is...
nsgmgclem 32991 Lemma for ~ nsgmgc . (Con...
nsgmgc 32992 There is a monotone Galois...
nsgqusf1olem1 32993 Lemma for ~ nsgqusf1o . (...
nsgqusf1olem2 32994 Lemma for ~ nsgqusf1o . (...
nsgqusf1olem3 32995 Lemma for ~ nsgqusf1o . (...
nsgqusf1o 32996 The canonical projection h...
ghmquskerlem1 32997 Lemma for ~ ghmqusker (Con...
ghmquskerco 32998 In the case of theorem ~ g...
ghmquskerlem2 32999 Lemma for ~ ghmqusker . (...
ghmquskerlem3 33000 The mapping ` H ` induced ...
ghmqusker 33001 A surjective group homomor...
gicqusker 33002 The image ` H ` of a group...
lmhmqusker 33003 A surjective module homomo...
lmicqusker 33004 The image ` H ` of a modul...
intlidl 33005 The intersection of a none...
rhmpreimaidl 33006 The preimage of an ideal b...
kerlidl 33007 The kernel of a ring homom...
0ringidl 33008 The zero ideal is the only...
pidlnzb 33009 A principal ideal is nonze...
lidlunitel 33010 If an ideal ` I ` contains...
unitpidl1 33011 The ideal ` I ` generated ...
rhmquskerlem 33012 The mapping ` J ` induced ...
rhmqusker 33013 A surjective ring homomorp...
ricqusker 33014 The image ` H ` of a ring ...
elrspunidl 33015 Elementhood in the span of...
elrspunsn 33016 Membership to the span of ...
lidlincl 33017 Ideals are closed under in...
idlinsubrg 33018 The intersection between a...
rhmimaidl 33019 The image of an ideal ` I ...
drngidl 33020 A nonzero ring is a divisi...
drngidlhash 33021 A ring is a division ring ...
prmidlval 33024 The class of prime ideals ...
isprmidl 33025 The predicate "is a prime ...
prmidlnr 33026 A prime ideal is a proper ...
prmidl 33027 The main property of a pri...
prmidl2 33028 A condition that shows an ...
idlmulssprm 33029 Let ` P ` be a prime ideal...
pridln1 33030 A proper ideal cannot cont...
prmidlidl 33031 A prime ideal is an ideal....
prmidlssidl 33032 Prime ideals as a subset o...
lidlnsg 33033 An ideal is a normal subgr...
cringm4 33034 Commutative/associative la...
isprmidlc 33035 The predicate "is prime id...
prmidlc 33036 Property of a prime ideal ...
0ringprmidl 33037 The trivial ring does not ...
prmidl0 33038 The zero ideal of a commut...
rhmpreimaprmidl 33039 The preimage of a prime id...
qsidomlem1 33040 If the quotient ring of a ...
qsidomlem2 33041 A quotient by a prime idea...
qsidom 33042 An ideal ` I ` in the comm...
qsnzr 33043 A quotient of a non-zero r...
mxidlval 33046 The set of maximal ideals ...
ismxidl 33047 The predicate "is a maxima...
mxidlidl 33048 A maximal ideal is an idea...
mxidlnr 33049 A maximal ideal is proper....
mxidlmax 33050 A maximal ideal is a maxim...
mxidln1 33051 One is not contained in an...
mxidlnzr 33052 A ring with a maximal idea...
mxidlmaxv 33053 An ideal ` I ` strictly co...
crngmxidl 33054 In a commutative ring, max...
mxidlprm 33055 Every maximal ideal is pri...
mxidlirredi 33056 In an integral domain, the...
mxidlirred 33057 In a principal ideal domai...
ssmxidllem 33058 The set ` P ` used in the ...
ssmxidl 33059 Let ` R ` be a ring, and l...
drnglidl1ne0 33060 In a nonzero ring, the zer...
drng0mxidl 33061 In a division ring, the ze...
drngmxidl 33062 The zero ideal is the only...
krull 33063 Krull's theorem: Any nonz...
mxidlnzrb 33064 A ring is nonzero if and o...
opprabs 33065 The opposite ring of the o...
oppreqg 33066 Group coset equivalence re...
opprnsg 33067 Normal subgroups of the op...
opprlidlabs 33068 The ideals of the opposite...
oppr2idl 33069 Two sided ideal of the opp...
opprmxidlabs 33070 The maximal ideal of the o...
opprqusbas 33071 The base of the quotient o...
opprqusplusg 33072 The group operation of the...
opprqus0g 33073 The group identity element...
opprqusmulr 33074 The multiplication operati...
opprqus1r 33075 The ring unity of the quot...
opprqusdrng 33076 The quotient of the opposi...
qsdrngilem 33077 Lemma for ~ qsdrngi . (Co...
qsdrngi 33078 A quotient by a maximal le...
qsdrnglem2 33079 Lemma for ~ qsdrng . (Con...
qsdrng 33080 An ideal ` M ` is both lef...
qsfld 33081 An ideal ` M ` in the comm...
mxidlprmALT 33082 Every maximal ideal is pri...
idlsrgstr 33085 A constructed semiring of ...
idlsrgval 33086 Lemma for ~ idlsrgbas thro...
idlsrgbas 33087 Base of the ideals of a ri...
idlsrgplusg 33088 Additive operation of the ...
idlsrg0g 33089 The zero ideal is the addi...
idlsrgmulr 33090 Multiplicative operation o...
idlsrgtset 33091 Topology component of the ...
idlsrgmulrval 33092 Value of the ring multipli...
idlsrgmulrcl 33093 Ideals of a ring ` R ` are...
idlsrgmulrss1 33094 In a commutative ring, the...
idlsrgmulrss2 33095 The product of two ideals ...
idlsrgmulrssin 33096 In a commutative ring, the...
idlsrgmnd 33097 The ideals of a ring form ...
idlsrgcmnd 33098 The ideals of a ring form ...
isufd 33101 The property of being a Un...
rprmval 33102 The prime elements of a ri...
isrprm 33103 Property for ` P ` to be a...
0ringmon1p 33104 There are no monic polynom...
fply1 33105 Conditions for a function ...
ply1lvec 33106 In a division ring, the un...
evls1fn 33107 Functionality of the subri...
evls1dm 33108 The domain of the subring ...
evls1fvf 33109 The subring evaluation fun...
evls1scafv 33110 Value of the univariate po...
evls1expd 33111 Univariate polynomial eval...
evls1varpwval 33112 Univariate polynomial eval...
evls1fpws 33113 Evaluation of a univariate...
ressply1evl 33114 Evaluation of a univariate...
evls1addd 33115 Univariate polynomial eval...
evls1muld 33116 Univariate polynomial eval...
evls1vsca 33117 Univariate polynomial eval...
ressdeg1 33118 The degree of a univariate...
ply1ascl0 33119 The zero scalar as a polyn...
ply1ascl1 33120 The multiplicative unit sc...
ply1asclunit 33121 A non-zero scalar polynomi...
deg1le0eq0 33122 A polynomial with nonposit...
ressply10g 33123 A restricted polynomial al...
ressply1mon1p 33124 The monic polynomials of a...
ressply1invg 33125 An element of a restricted...
ressply1sub 33126 A restricted polynomial al...
asclply1subcl 33127 Closure of the algebra sca...
ply1fermltl 33128 Fermat's little theorem fo...
coe1mon 33129 Coefficient vector of a mo...
ply1moneq 33130 Two monomials are equal if...
ply1degltel 33131 Characterize elementhood i...
ply1degleel 33132 Characterize elementhood i...
ply1degltlss 33133 The space ` S ` of the uni...
gsummoncoe1fzo 33134 A coefficient of the polyn...
ply1gsumz 33135 If a polynomial given as a...
deg1addlt 33136 If both factors have degre...
ig1pnunit 33137 The polynomial ideal gener...
ig1pmindeg 33138 The polynomial ideal gener...
q1pdir 33139 Distribution of univariate...
q1pvsca 33140 Scalar multiplication prop...
r1pvsca 33141 Scalar multiplication prop...
r1p0 33142 Polynomial remainder opera...
r1pcyc 33143 The polynomial remainder o...
r1padd1 33144 Addition property of the p...
r1pid2 33145 Identity law for polynomia...
r1plmhm 33146 The univariate polynomial ...
r1pquslmic 33147 The univariate polynomial ...
sra1r 33148 The unity element of a sub...
sradrng 33149 Condition for a subring al...
srasubrg 33150 A subring of the original ...
sralvec 33151 Given a sub division ring ...
srafldlvec 33152 Given a subfield ` F ` of ...
resssra 33153 The subring algebra of a r...
lsssra 33154 A subring is a subspace of...
drgext0g 33155 The additive neutral eleme...
drgextvsca 33156 The scalar multiplication ...
drgext0gsca 33157 The additive neutral eleme...
drgextsubrg 33158 The scalar field is a subr...
drgextlsp 33159 The scalar field is a subs...
drgextgsum 33160 Group sum in a division ri...
lvecdimfi 33161 Finite version of ~ lvecdi...
dimval 33164 The dimension of a vector ...
dimvalfi 33165 The dimension of a vector ...
dimcl 33166 Closure of the vector spac...
lmimdim 33167 Module isomorphisms preser...
lmicdim 33168 Module isomorphisms preser...
lvecdim0i 33169 A vector space of dimensio...
lvecdim0 33170 A vector space of dimensio...
lssdimle 33171 The dimension of a linear ...
dimpropd 33172 If two structures have the...
rlmdim 33173 The left vector space indu...
rgmoddimOLD 33174 Obsolete version of ~ rlmd...
frlmdim 33175 Dimension of a free left m...
tnglvec 33176 Augmenting a structure wit...
tngdim 33177 Dimension of a left vector...
rrxdim 33178 Dimension of the generaliz...
matdim 33179 Dimension of the space of ...
lbslsat 33180 A nonzero vector ` X ` is ...
lsatdim 33181 A line, spanned by a nonze...
drngdimgt0 33182 The dimension of a vector ...
lmhmlvec2 33183 A homomorphism of left vec...
kerlmhm 33184 The kernel of a vector spa...
imlmhm 33185 The image of a vector spac...
ply1degltdimlem 33186 Lemma for ~ ply1degltdim ....
ply1degltdim 33187 The space ` S ` of the uni...
lindsunlem 33188 Lemma for ~ lindsun . (Co...
lindsun 33189 Condition for the union of...
lbsdiflsp0 33190 The linear spans of two di...
dimkerim 33191 Given a linear map ` F ` b...
qusdimsum 33192 Let ` W ` be a vector spac...
fedgmullem1 33193 Lemma for ~ fedgmul . (Co...
fedgmullem2 33194 Lemma for ~ fedgmul . (Co...
fedgmul 33195 The multiplicativity formu...
relfldext 33204 The field extension is a r...
brfldext 33205 The field extension relati...
ccfldextrr 33206 The field of the complex n...
fldextfld1 33207 A field extension is only ...
fldextfld2 33208 A field extension is only ...
fldextsubrg 33209 Field extension implies a ...
fldextress 33210 Field extension implies a ...
brfinext 33211 The finite field extension...
extdgval 33212 Value of the field extensi...
fldextsralvec 33213 The subring algebra associ...
extdgcl 33214 Closure of the field exten...
extdggt0 33215 Degrees of field extension...
fldexttr 33216 Field extension is a trans...
fldextid 33217 The field extension relati...
extdgid 33218 A trivial field extension ...
extdgmul 33219 The multiplicativity formu...
finexttrb 33220 The extension ` E ` of ` K...
extdg1id 33221 If the degree of the exten...
extdg1b 33222 The degree of the extensio...
fldextchr 33223 The characteristic of a su...
evls1fldgencl 33224 Closure of the subring pol...
ccfldsrarelvec 33225 The subring algebra of the...
ccfldextdgrr 33226 The degree of the field ex...
irngval 33229 The elements of a field ` ...
elirng 33230 Property for an element ` ...
irngss 33231 All elements of a subring ...
irngssv 33232 An integral element is an ...
0ringirng 33233 A zero ring ` R ` has no i...
irngnzply1lem 33234 In the case of a field ` E...
irngnzply1 33235 In the case of a field ` E...
evls1fvcl 33238 Variant of ~ fveval1fvcl f...
evls1maprhm 33239 The function ` F ` mapping...
evls1maplmhm 33240 The function ` F ` mapping...
evls1maprnss 33241 The function ` F ` mapping...
ply1annidllem 33242 Write the set ` Q ` of pol...
ply1annidl 33243 The set ` Q ` of polynomia...
ply1annnr 33244 The set ` Q ` of polynomia...
ply1annig1p 33245 The ideal ` Q ` of polynom...
minplyval 33246 Expand the value of the mi...
minplycl 33247 The minimal polynomial is ...
ply1annprmidl 33248 The set ` Q ` of polynomia...
minplyirredlem 33249 Lemma for ~ minplyirred . ...
minplyirred 33250 A nonzero minimal polynomi...
irngnminplynz 33251 Integral elements have non...
minplym1p 33252 A minimal polynomial is mo...
algextdeglem1 33253 Lemma for ~ algextdeg . (...
algextdeglem2 33254 Lemma for ~ algextdeg . (...
algextdeglem3 33255 Lemma for ~ algextdeg . (...
algextdeglem4 33256 Lemma for ~ algextdeg . (...
algextdeglem5 33257 Lemma for ~ algextdeg . (...
algextdeglem6 33258 Lemma for ~ algextdeg . (...
algextdeglem7 33259 Lemma for ~ algextdeg . (...
algextdeglem8 33260 Lemma for ~ algextdeg . (...
algextdeg 33261 The degree of an algebraic...
smatfval 33264 Value of the submatrix. (...
smatrcl 33265 Closure of the rectangular...
smatlem 33266 Lemma for the next theorem...
smattl 33267 Entries of a submatrix, to...
smattr 33268 Entries of a submatrix, to...
smatbl 33269 Entries of a submatrix, bo...
smatbr 33270 Entries of a submatrix, bo...
smatcl 33271 Closure of the square subm...
matmpo 33272 Write a square matrix as a...
1smat1 33273 The submatrix of the ident...
submat1n 33274 One case where the submatr...
submatres 33275 Special case where the sub...
submateqlem1 33276 Lemma for ~ submateq . (C...
submateqlem2 33277 Lemma for ~ submateq . (C...
submateq 33278 Sufficient condition for t...
submatminr1 33279 If we take a submatrix by ...
lmatval 33282 Value of the literal matri...
lmatfval 33283 Entries of a literal matri...
lmatfvlem 33284 Useful lemma to extract li...
lmatcl 33285 Closure of the literal mat...
lmat22lem 33286 Lemma for ~ lmat22e11 and ...
lmat22e11 33287 Entry of a 2x2 literal mat...
lmat22e12 33288 Entry of a 2x2 literal mat...
lmat22e21 33289 Entry of a 2x2 literal mat...
lmat22e22 33290 Entry of a 2x2 literal mat...
lmat22det 33291 The determinant of a liter...
mdetpmtr1 33292 The determinant of a matri...
mdetpmtr2 33293 The determinant of a matri...
mdetpmtr12 33294 The determinant of a matri...
mdetlap1 33295 A Laplace expansion of the...
madjusmdetlem1 33296 Lemma for ~ madjusmdet . ...
madjusmdetlem2 33297 Lemma for ~ madjusmdet . ...
madjusmdetlem3 33298 Lemma for ~ madjusmdet . ...
madjusmdetlem4 33299 Lemma for ~ madjusmdet . ...
madjusmdet 33300 Express the cofactor of th...
mdetlap 33301 Laplace expansion of the d...
ist0cld 33302 The predicate "is a T_0 sp...
txomap 33303 Given two open maps ` F ` ...
qtopt1 33304 If every equivalence class...
qtophaus 33305 If an open map's graph in ...
circtopn 33306 The topology of the unit c...
circcn 33307 The function gluing the re...
reff 33308 For any cover refinement, ...
locfinreflem 33309 A locally finite refinemen...
locfinref 33310 A locally finite refinemen...
iscref 33313 The property that every op...
crefeq 33314 Equality theorem for the "...
creftop 33315 A space where every open c...
crefi 33316 The property that every op...
crefdf 33317 A formulation of ~ crefi e...
crefss 33318 The "every open cover has ...
cmpcref 33319 Equivalent definition of c...
cmpfiref 33320 Every open cover of a Comp...
ldlfcntref 33323 Every open cover of a Lind...
ispcmp 33326 The predicate "is a paraco...
cmppcmp 33327 Every compact space is par...
dispcmp 33328 Every discrete space is pa...
pcmplfin 33329 Given a paracompact topolo...
pcmplfinf 33330 Given a paracompact topolo...
rspecval 33333 Value of the spectrum of t...
rspecbas 33334 The prime ideals form the ...
rspectset 33335 Topology component of the ...
rspectopn 33336 The topology component of ...
zarcls0 33337 The closure of the identit...
zarcls1 33338 The unit ideal ` B ` is th...
zarclsun 33339 The union of two closed se...
zarclsiin 33340 In a Zariski topology, the...
zarclsint 33341 The intersection of a fami...
zarclssn 33342 The closed points of Zaris...
zarcls 33343 The open sets of the Zaris...
zartopn 33344 The Zariski topology is a ...
zartop 33345 The Zariski topology is a ...
zartopon 33346 The points of the Zariski ...
zar0ring 33347 The Zariski Topology of th...
zart0 33348 The Zariski topology is T_...
zarmxt1 33349 The Zariski topology restr...
zarcmplem 33350 Lemma for ~ zarcmp . (Con...
zarcmp 33351 The Zariski topology is co...
rspectps 33352 The spectrum of a ring ` R...
rhmpreimacnlem 33353 Lemma for ~ rhmpreimacn . ...
rhmpreimacn 33354 The function mapping a pri...
metidval 33359 Value of the metric identi...
metidss 33360 As a relation, the metric ...
metidv 33361 ` A ` and ` B ` identify b...
metideq 33362 Basic property of the metr...
metider 33363 The metric identification ...
pstmval 33364 Value of the metric induce...
pstmfval 33365 Function value of the metr...
pstmxmet 33366 The metric induced by a ps...
hauseqcn 33367 In a Hausdorff topology, t...
elunitge0 33368 An element of the closed u...
unitssxrge0 33369 The closed unit interval i...
unitdivcld 33370 Necessary conditions for a...
iistmd 33371 The closed unit interval f...
unicls 33372 The union of the closed se...
tpr2tp 33373 The usual topology on ` ( ...
tpr2uni 33374 The usual topology on ` ( ...
xpinpreima 33375 Rewrite the cartesian prod...
xpinpreima2 33376 Rewrite the cartesian prod...
sqsscirc1 33377 The complex square of side...
sqsscirc2 33378 The complex square of side...
cnre2csqlem 33379 Lemma for ~ cnre2csqima . ...
cnre2csqima 33380 Image of a centered square...
tpr2rico 33381 For any point of an open s...
cnvordtrestixx 33382 The restriction of the 'gr...
prsdm 33383 Domain of the relation of ...
prsrn 33384 Range of the relation of a...
prsss 33385 Relation of a subproset. ...
prsssdm 33386 Domain of a subproset rela...
ordtprsval 33387 Value of the order topolog...
ordtprsuni 33388 Value of the order topolog...
ordtcnvNEW 33389 The order dual generates t...
ordtrestNEW 33390 The subspace topology of a...
ordtrest2NEWlem 33391 Lemma for ~ ordtrest2NEW ....
ordtrest2NEW 33392 An interval-closed set ` A...
ordtconnlem1 33393 Connectedness in the order...
ordtconn 33394 Connectedness in the order...
mndpluscn 33395 A mapping that is both a h...
mhmhmeotmd 33396 Deduce a Topological Monoi...
rmulccn 33397 Multiplication by a real c...
raddcn 33398 Addition in the real numbe...
xrmulc1cn 33399 The operation multiplying ...
fmcncfil 33400 The image of a Cauchy filt...
xrge0hmph 33401 The extended nonnegative r...
xrge0iifcnv 33402 Define a bijection from ` ...
xrge0iifcv 33403 The defined function's val...
xrge0iifiso 33404 The defined bijection from...
xrge0iifhmeo 33405 Expose a homeomorphism fro...
xrge0iifhom 33406 The defined function from ...
xrge0iif1 33407 Condition for the defined ...
xrge0iifmhm 33408 The defined function from ...
xrge0pluscn 33409 The addition operation of ...
xrge0mulc1cn 33410 The operation multiplying ...
xrge0tps 33411 The extended nonnegative r...
xrge0topn 33412 The topology of the extend...
xrge0haus 33413 The topology of the extend...
xrge0tmd 33414 The extended nonnegative r...
xrge0tmdALT 33415 Alternate proof of ~ xrge0...
lmlim 33416 Relate a limit in a given ...
lmlimxrge0 33417 Relate a limit in the nonn...
rge0scvg 33418 Implication of convergence...
fsumcvg4 33419 A serie with finite suppor...
pnfneige0 33420 A neighborhood of ` +oo ` ...
lmxrge0 33421 Express "sequence ` F ` co...
lmdvg 33422 If a monotonic sequence of...
lmdvglim 33423 If a monotonic real number...
pl1cn 33424 A univariate polynomial is...
zringnm 33427 The norm (function) for a ...
zzsnm 33428 The norm of the ring of th...
zlm0 33429 Zero of a ` ZZ ` -module. ...
zlm1 33430 Unity element of a ` ZZ ` ...
zlmds 33431 Distance in a ` ZZ ` -modu...
zlmdsOLD 33432 Obsolete proof of ~ zlmds ...
zlmtset 33433 Topology in a ` ZZ ` -modu...
zlmtsetOLD 33434 Obsolete proof of ~ zlmtse...
zlmnm 33435 Norm of a ` ZZ ` -module (...
zhmnrg 33436 The ` ZZ ` -module built f...
nmmulg 33437 The norm of a group produc...
zrhnm 33438 The norm of the image by `...
cnzh 33439 The ` ZZ ` -module of ` CC...
rezh 33440 The ` ZZ ` -module of ` RR...
qqhval 33443 Value of the canonical hom...
zrhf1ker 33444 The kernel of the homomorp...
zrhchr 33445 The kernel of the homomorp...
zrhker 33446 The kernel of the homomorp...
zrhunitpreima 33447 The preimage by ` ZRHom ` ...
elzrhunit 33448 Condition for the image by...
elzdif0 33449 Lemma for ~ qqhval2 . (Co...
qqhval2lem 33450 Lemma for ~ qqhval2 . (Co...
qqhval2 33451 Value of the canonical hom...
qqhvval 33452 Value of the canonical hom...
qqh0 33453 The image of ` 0 ` by the ...
qqh1 33454 The image of ` 1 ` by the ...
qqhf 33455 ` QQHom ` as a function. ...
qqhvq 33456 The image of a quotient by...
qqhghm 33457 The ` QQHom ` homomorphism...
qqhrhm 33458 The ` QQHom ` homomorphism...
qqhnm 33459 The norm of the image by `...
qqhcn 33460 The ` QQHom ` homomorphism...
qqhucn 33461 The ` QQHom ` homomorphism...
rrhval 33465 Value of the canonical hom...
rrhcn 33466 If the topology of ` R ` i...
rrhf 33467 If the topology of ` R ` i...
isrrext 33469 Express the property " ` R...
rrextnrg 33470 An extension of ` RR ` is ...
rrextdrg 33471 An extension of ` RR ` is ...
rrextnlm 33472 The norm of an extension o...
rrextchr 33473 The ring characteristic of...
rrextcusp 33474 An extension of ` RR ` is ...
rrexttps 33475 An extension of ` RR ` is ...
rrexthaus 33476 The topology of an extensi...
rrextust 33477 The uniformity of an exten...
rerrext 33478 The field of the real numb...
cnrrext 33479 The field of the complex n...
qqtopn 33480 The topology of the field ...
rrhfe 33481 If ` R ` is an extension o...
rrhcne 33482 If ` R ` is an extension o...
rrhqima 33483 The ` RRHom ` homomorphism...
rrh0 33484 The image of ` 0 ` by the ...
xrhval 33487 The value of the embedding...
zrhre 33488 The ` ZRHom ` homomorphism...
qqhre 33489 The ` QQHom ` homomorphism...
rrhre 33490 The ` RRHom ` homomorphism...
relmntop 33493 Manifold is a relation. (...
ismntoplly 33494 Property of being a manifo...
ismntop 33495 Property of being a manifo...
nexple 33496 A lower bound for an expon...
indv 33499 Value of the indicator fun...
indval 33500 Value of the indicator fun...
indval2 33501 Alternate value of the ind...
indf 33502 An indicator function as a...
indfval 33503 Value of the indicator fun...
ind1 33504 Value of the indicator fun...
ind0 33505 Value of the indicator fun...
ind1a 33506 Value of the indicator fun...
indpi1 33507 Preimage of the singleton ...
indsum 33508 Finite sum of a product wi...
indsumin 33509 Finite sum of a product wi...
prodindf 33510 The product of indicators ...
indf1o 33511 The bijection between a po...
indpreima 33512 A function with range ` { ...
indf1ofs 33513 The bijection between fini...
esumex 33516 An extended sum is a set b...
esumcl 33517 Closure for extended sum i...
esumeq12dvaf 33518 Equality deduction for ext...
esumeq12dva 33519 Equality deduction for ext...
esumeq12d 33520 Equality deduction for ext...
esumeq1 33521 Equality theorem for an ex...
esumeq1d 33522 Equality theorem for an ex...
esumeq2 33523 Equality theorem for exten...
esumeq2d 33524 Equality deduction for ext...
esumeq2dv 33525 Equality deduction for ext...
esumeq2sdv 33526 Equality deduction for ext...
nfesum1 33527 Bound-variable hypothesis ...
nfesum2 33528 Bound-variable hypothesis ...
cbvesum 33529 Change bound variable in a...
cbvesumv 33530 Change bound variable in a...
esumid 33531 Identify the extended sum ...
esumgsum 33532 A finite extended sum is t...
esumval 33533 Develop the value of the e...
esumel 33534 The extended sum is a limi...
esumnul 33535 Extended sum over the empt...
esum0 33536 Extended sum of zero. (Co...
esumf1o 33537 Re-index an extended sum u...
esumc 33538 Convert from the collectio...
esumrnmpt 33539 Rewrite an extended sum in...
esumsplit 33540 Split an extended sum into...
esummono 33541 Extended sum is monotonic....
esumpad 33542 Extend an extended sum by ...
esumpad2 33543 Remove zeroes from an exte...
esumadd 33544 Addition of infinite sums....
esumle 33545 If all of the terms of an ...
gsumesum 33546 Relate a group sum on ` ( ...
esumlub 33547 The extended sum is the lo...
esumaddf 33548 Addition of infinite sums....
esumlef 33549 If all of the terms of an ...
esumcst 33550 The extended sum of a cons...
esumsnf 33551 The extended sum of a sing...
esumsn 33552 The extended sum of a sing...
esumpr 33553 Extended sum over a pair. ...
esumpr2 33554 Extended sum over a pair, ...
esumrnmpt2 33555 Rewrite an extended sum in...
esumfzf 33556 Formulating a partial exte...
esumfsup 33557 Formulating an extended su...
esumfsupre 33558 Formulating an extended su...
esumss 33559 Change the index set to a ...
esumpinfval 33560 The value of the extended ...
esumpfinvallem 33561 Lemma for ~ esumpfinval . ...
esumpfinval 33562 The value of the extended ...
esumpfinvalf 33563 Same as ~ esumpfinval , mi...
esumpinfsum 33564 The value of the extended ...
esumpcvgval 33565 The value of the extended ...
esumpmono 33566 The partial sums in an ext...
esumcocn 33567 Lemma for ~ esummulc2 and ...
esummulc1 33568 An extended sum multiplied...
esummulc2 33569 An extended sum multiplied...
esumdivc 33570 An extended sum divided by...
hashf2 33571 Lemma for ~ hasheuni . (C...
hasheuni 33572 The cardinality of a disjo...
esumcvg 33573 The sequence of partial su...
esumcvg2 33574 Simpler version of ~ esumc...
esumcvgsum 33575 The value of the extended ...
esumsup 33576 Express an extended sum as...
esumgect 33577 "Send ` n ` to ` +oo ` " i...
esumcvgre 33578 All terms of a converging ...
esum2dlem 33579 Lemma for ~ esum2d (finite...
esum2d 33580 Write a double extended su...
esumiun 33581 Sum over a nonnecessarily ...
ofceq 33584 Equality theorem for funct...
ofcfval 33585 Value of an operation appl...
ofcval 33586 Evaluate a function/consta...
ofcfn 33587 The function operation pro...
ofcfeqd2 33588 Equality theorem for funct...
ofcfval3 33589 General value of ` ( F oFC...
ofcf 33590 The function/constant oper...
ofcfval2 33591 The function operation exp...
ofcfval4 33592 The function/constant oper...
ofcc 33593 Left operation by a consta...
ofcof 33594 Relate function operation ...
sigaex 33597 Lemma for ~ issiga and ~ i...
sigaval 33598 The set of sigma-algebra w...
issiga 33599 An alternative definition ...
isrnsiga 33600 The property of being a si...
0elsiga 33601 A sigma-algebra contains t...
baselsiga 33602 A sigma-algebra contains i...
sigasspw 33603 A sigma-algebra is a set o...
sigaclcu 33604 A sigma-algebra is closed ...
sigaclcuni 33605 A sigma-algebra is closed ...
sigaclfu 33606 A sigma-algebra is closed ...
sigaclcu2 33607 A sigma-algebra is closed ...
sigaclfu2 33608 A sigma-algebra is closed ...
sigaclcu3 33609 A sigma-algebra is closed ...
issgon 33610 Property of being a sigma-...
sgon 33611 A sigma-algebra is a sigma...
elsigass 33612 An element of a sigma-alge...
elrnsiga 33613 Dropping the base informat...
isrnsigau 33614 The property of being a si...
unielsiga 33615 A sigma-algebra contains i...
dmvlsiga 33616 Lebesgue-measurable subset...
pwsiga 33617 Any power set forms a sigm...
prsiga 33618 The smallest possible sigm...
sigaclci 33619 A sigma-algebra is closed ...
difelsiga 33620 A sigma-algebra is closed ...
unelsiga 33621 A sigma-algebra is closed ...
inelsiga 33622 A sigma-algebra is closed ...
sigainb 33623 Building a sigma-algebra f...
insiga 33624 The intersection of a coll...
sigagenval 33627 Value of the generated sig...
sigagensiga 33628 A generated sigma-algebra ...
sgsiga 33629 A generated sigma-algebra ...
unisg 33630 The sigma-algebra generate...
dmsigagen 33631 A sigma-algebra can be gen...
sssigagen 33632 A set is a subset of the s...
sssigagen2 33633 A subset of the generating...
elsigagen 33634 Any element of a set is al...
elsigagen2 33635 Any countable union of ele...
sigagenss 33636 The generated sigma-algebr...
sigagenss2 33637 Sufficient condition for i...
sigagenid 33638 The sigma-algebra generate...
ispisys 33639 The property of being a pi...
ispisys2 33640 The property of being a pi...
inelpisys 33641 Pi-systems are closed unde...
sigapisys 33642 All sigma-algebras are pi-...
isldsys 33643 The property of being a la...
pwldsys 33644 The power set of the unive...
unelldsys 33645 Lambda-systems are closed ...
sigaldsys 33646 All sigma-algebras are lam...
ldsysgenld 33647 The intersection of all la...
sigapildsyslem 33648 Lemma for ~ sigapildsys . ...
sigapildsys 33649 Sigma-algebra are exactly ...
ldgenpisyslem1 33650 Lemma for ~ ldgenpisys . ...
ldgenpisyslem2 33651 Lemma for ~ ldgenpisys . ...
ldgenpisyslem3 33652 Lemma for ~ ldgenpisys . ...
ldgenpisys 33653 The lambda system ` E ` ge...
dynkin 33654 Dynkin's lambda-pi theorem...
isros 33655 The property of being a ri...
rossspw 33656 A ring of sets is a collec...
0elros 33657 A ring of sets contains th...
unelros 33658 A ring of sets is closed u...
difelros 33659 A ring of sets is closed u...
inelros 33660 A ring of sets is closed u...
fiunelros 33661 A ring of sets is closed u...
issros 33662 The property of being a se...
srossspw 33663 A semiring of sets is a co...
0elsros 33664 A semiring of sets contain...
inelsros 33665 A semiring of sets is clos...
diffiunisros 33666 In semiring of sets, compl...
rossros 33667 Rings of sets are semiring...
brsiga 33670 The Borel Algebra on real ...
brsigarn 33671 The Borel Algebra is a sig...
brsigasspwrn 33672 The Borel Algebra is a set...
unibrsiga 33673 The union of the Borel Alg...
cldssbrsiga 33674 A Borel Algebra contains a...
sxval 33677 Value of the product sigma...
sxsiga 33678 A product sigma-algebra is...
sxsigon 33679 A product sigma-algebra is...
sxuni 33680 The base set of a product ...
elsx 33681 The cartesian product of t...
measbase 33684 The base set of a measure ...
measval 33685 The value of the ` measure...
ismeas 33686 The property of being a me...
isrnmeas 33687 The property of being a me...
dmmeas 33688 The domain of a measure is...
measbasedom 33689 The base set of a measure ...
measfrge0 33690 A measure is a function ov...
measfn 33691 A measure is a function on...
measvxrge0 33692 The values of a measure ar...
measvnul 33693 The measure of the empty s...
measge0 33694 A measure is nonnegative. ...
measle0 33695 If the measure of a given ...
measvun 33696 The measure of a countable...
measxun2 33697 The measure the union of t...
measun 33698 The measure the union of t...
measvunilem 33699 Lemma for ~ measvuni . (C...
measvunilem0 33700 Lemma for ~ measvuni . (C...
measvuni 33701 The measure of a countable...
measssd 33702 A measure is monotone with...
measunl 33703 A measure is sub-additive ...
measiuns 33704 The measure of the union o...
measiun 33705 A measure is sub-additive....
meascnbl 33706 A measure is continuous fr...
measinblem 33707 Lemma for ~ measinb . (Co...
measinb 33708 Building a measure restric...
measres 33709 Building a measure restric...
measinb2 33710 Building a measure restric...
measdivcst 33711 Division of a measure by a...
measdivcstALTV 33712 Alternate version of ~ mea...
cntmeas 33713 The Counting measure is a ...
pwcntmeas 33714 The counting measure is a ...
cntnevol 33715 Counting and Lebesgue meas...
voliune 33716 The Lebesgue measure funct...
volfiniune 33717 The Lebesgue measure funct...
volmeas 33718 The Lebesgue measure is a ...
ddeval1 33721 Value of the delta measure...
ddeval0 33722 Value of the delta measure...
ddemeas 33723 The Dirac delta measure is...
relae 33727 'almost everywhere' is a r...
brae 33728 'almost everywhere' relati...
braew 33729 'almost everywhere' relati...
truae 33730 A truth holds almost every...
aean 33731 A conjunction holds almost...
faeval 33733 Value of the 'almost every...
relfae 33734 The 'almost everywhere' bu...
brfae 33735 'almost everywhere' relati...
ismbfm 33738 The predicate " ` F ` is a...
elunirnmbfm 33739 The property of being a me...
mbfmfun 33740 A measurable function is a...
mbfmf 33741 A measurable function as a...
isanmbfmOLD 33742 Obsolete version of ~ isan...
mbfmcnvima 33743 The preimage by a measurab...
isanmbfm 33744 The predicate to be a meas...
mbfmbfmOLD 33745 A measurable function to a...
mbfmbfm 33746 A measurable function to a...
mbfmcst 33747 A constant function is mea...
1stmbfm 33748 The first projection map i...
2ndmbfm 33749 The second projection map ...
imambfm 33750 If the sigma-algebra in th...
cnmbfm 33751 A continuous function is m...
mbfmco 33752 The composition of two mea...
mbfmco2 33753 The pair building of two m...
mbfmvolf 33754 Measurable functions with ...
elmbfmvol2 33755 Measurable functions with ...
mbfmcnt 33756 All functions are measurab...
br2base 33757 The base set for the gener...
dya2ub 33758 An upper bound for a dyadi...
sxbrsigalem0 33759 The closed half-spaces of ...
sxbrsigalem3 33760 The sigma-algebra generate...
dya2iocival 33761 The function ` I ` returns...
dya2iocress 33762 Dyadic intervals are subse...
dya2iocbrsiga 33763 Dyadic intervals are Borel...
dya2icobrsiga 33764 Dyadic intervals are Borel...
dya2icoseg 33765 For any point and any clos...
dya2icoseg2 33766 For any point and any open...
dya2iocrfn 33767 The function returning dya...
dya2iocct 33768 The dyadic rectangle set i...
dya2iocnrect 33769 For any point of an open r...
dya2iocnei 33770 For any point of an open s...
dya2iocuni 33771 Every open set of ` ( RR X...
dya2iocucvr 33772 The dyadic rectangular set...
sxbrsigalem1 33773 The Borel algebra on ` ( R...
sxbrsigalem2 33774 The sigma-algebra generate...
sxbrsigalem4 33775 The Borel algebra on ` ( R...
sxbrsigalem5 33776 First direction for ~ sxbr...
sxbrsigalem6 33777 First direction for ~ sxbr...
sxbrsiga 33778 The product sigma-algebra ...
omsval 33781 Value of the function mapp...
omsfval 33782 Value of the outer measure...
omscl 33783 A closure lemma for the co...
omsf 33784 A constructed outer measur...
oms0 33785 A constructed outer measur...
omsmon 33786 A constructed outer measur...
omssubaddlem 33787 For any small margin ` E `...
omssubadd 33788 A constructed outer measur...
carsgval 33791 Value of the Caratheodory ...
carsgcl 33792 Closure of the Caratheodor...
elcarsg 33793 Property of being a Carath...
baselcarsg 33794 The universe set, ` O ` , ...
0elcarsg 33795 The empty set is Caratheod...
carsguni 33796 The union of all Caratheod...
elcarsgss 33797 Caratheodory measurable se...
difelcarsg 33798 The Caratheodory measurabl...
inelcarsg 33799 The Caratheodory measurabl...
unelcarsg 33800 The Caratheodory-measurabl...
difelcarsg2 33801 The Caratheodory-measurabl...
carsgmon 33802 Utility lemma: Apply mono...
carsgsigalem 33803 Lemma for the following th...
fiunelcarsg 33804 The Caratheodory measurabl...
carsgclctunlem1 33805 Lemma for ~ carsgclctun . ...
carsggect 33806 The outer measure is count...
carsgclctunlem2 33807 Lemma for ~ carsgclctun . ...
carsgclctunlem3 33808 Lemma for ~ carsgclctun . ...
carsgclctun 33809 The Caratheodory measurabl...
carsgsiga 33810 The Caratheodory measurabl...
omsmeas 33811 The restriction of a const...
pmeasmono 33812 This theorem's hypotheses ...
pmeasadd 33813 A premeasure on a ring of ...
itgeq12dv 33814 Equality theorem for an in...
sitgval 33820 Value of the simple functi...
issibf 33821 The predicate " ` F ` is a...
sibf0 33822 The constant zero function...
sibfmbl 33823 A simple function is measu...
sibff 33824 A simple function is a fun...
sibfrn 33825 A simple function has fini...
sibfima 33826 Any preimage of a singleto...
sibfinima 33827 The measure of the interse...
sibfof 33828 Applying function operatio...
sitgfval 33829 Value of the Bochner integ...
sitgclg 33830 Closure of the Bochner int...
sitgclbn 33831 Closure of the Bochner int...
sitgclcn 33832 Closure of the Bochner int...
sitgclre 33833 Closure of the Bochner int...
sitg0 33834 The integral of the consta...
sitgf 33835 The integral for simple fu...
sitgaddlemb 33836 Lemma for * sitgadd . (Co...
sitmval 33837 Value of the simple functi...
sitmfval 33838 Value of the integral dist...
sitmcl 33839 Closure of the integral di...
sitmf 33840 The integral metric as a f...
oddpwdc 33842 Lemma for ~ eulerpart . T...
oddpwdcv 33843 Lemma for ~ eulerpart : va...
eulerpartlemsv1 33844 Lemma for ~ eulerpart . V...
eulerpartlemelr 33845 Lemma for ~ eulerpart . (...
eulerpartlemsv2 33846 Lemma for ~ eulerpart . V...
eulerpartlemsf 33847 Lemma for ~ eulerpart . (...
eulerpartlems 33848 Lemma for ~ eulerpart . (...
eulerpartlemsv3 33849 Lemma for ~ eulerpart . V...
eulerpartlemgc 33850 Lemma for ~ eulerpart . (...
eulerpartleme 33851 Lemma for ~ eulerpart . (...
eulerpartlemv 33852 Lemma for ~ eulerpart . (...
eulerpartlemo 33853 Lemma for ~ eulerpart : ` ...
eulerpartlemd 33854 Lemma for ~ eulerpart : ` ...
eulerpartlem1 33855 Lemma for ~ eulerpart . (...
eulerpartlemb 33856 Lemma for ~ eulerpart . T...
eulerpartlemt0 33857 Lemma for ~ eulerpart . (...
eulerpartlemf 33858 Lemma for ~ eulerpart : O...
eulerpartlemt 33859 Lemma for ~ eulerpart . (...
eulerpartgbij 33860 Lemma for ~ eulerpart : T...
eulerpartlemgv 33861 Lemma for ~ eulerpart : va...
eulerpartlemr 33862 Lemma for ~ eulerpart . (...
eulerpartlemmf 33863 Lemma for ~ eulerpart . (...
eulerpartlemgvv 33864 Lemma for ~ eulerpart : va...
eulerpartlemgu 33865 Lemma for ~ eulerpart : R...
eulerpartlemgh 33866 Lemma for ~ eulerpart : T...
eulerpartlemgf 33867 Lemma for ~ eulerpart : I...
eulerpartlemgs2 33868 Lemma for ~ eulerpart : T...
eulerpartlemn 33869 Lemma for ~ eulerpart . (...
eulerpart 33870 Euler's theorem on partiti...
subiwrd 33873 Lemma for ~ sseqp1 . (Con...
subiwrdlen 33874 Length of a subword of an ...
iwrdsplit 33875 Lemma for ~ sseqp1 . (Con...
sseqval 33876 Value of the strong sequen...
sseqfv1 33877 Value of the strong sequen...
sseqfn 33878 A strong recursive sequenc...
sseqmw 33879 Lemma for ~ sseqf amd ~ ss...
sseqf 33880 A strong recursive sequenc...
sseqfres 33881 The first elements in the ...
sseqfv2 33882 Value of the strong sequen...
sseqp1 33883 Value of the strong sequen...
fiblem 33886 Lemma for ~ fib0 , ~ fib1 ...
fib0 33887 Value of the Fibonacci seq...
fib1 33888 Value of the Fibonacci seq...
fibp1 33889 Value of the Fibonacci seq...
fib2 33890 Value of the Fibonacci seq...
fib3 33891 Value of the Fibonacci seq...
fib4 33892 Value of the Fibonacci seq...
fib5 33893 Value of the Fibonacci seq...
fib6 33894 Value of the Fibonacci seq...
elprob 33897 The property of being a pr...
domprobmeas 33898 A probability measure is a...
domprobsiga 33899 The domain of a probabilit...
probtot 33900 The probability of the uni...
prob01 33901 A probability is an elemen...
probnul 33902 The probability of the emp...
unveldomd 33903 The universe is an element...
unveldom 33904 The universe is an element...
nuleldmp 33905 The empty set is an elemen...
probcun 33906 The probability of the uni...
probun 33907 The probability of the uni...
probdif 33908 The probability of the dif...
probinc 33909 A probability law is incre...
probdsb 33910 The probability of the com...
probmeasd 33911 A probability measure is a...
probvalrnd 33912 The value of a probability...
probtotrnd 33913 The probability of the uni...
totprobd 33914 Law of total probability, ...
totprob 33915 Law of total probability. ...
probfinmeasb 33916 Build a probability measur...
probfinmeasbALTV 33917 Alternate version of ~ pro...
probmeasb 33918 Build a probability from a...
cndprobval 33921 The value of the condition...
cndprobin 33922 An identity linking condit...
cndprob01 33923 The conditional probabilit...
cndprobtot 33924 The conditional probabilit...
cndprobnul 33925 The conditional probabilit...
cndprobprob 33926 The conditional probabilit...
bayesth 33927 Bayes Theorem. (Contribut...
rrvmbfm 33930 A real-valued random varia...
isrrvv 33931 Elementhood to the set of ...
rrvvf 33932 A real-valued random varia...
rrvfn 33933 A real-valued random varia...
rrvdm 33934 The domain of a random var...
rrvrnss 33935 The range of a random vari...
rrvf2 33936 A real-valued random varia...
rrvdmss 33937 The domain of a random var...
rrvfinvima 33938 For a real-value random va...
0rrv 33939 The constant function equa...
rrvadd 33940 The sum of two random vari...
rrvmulc 33941 A random variable multipli...
rrvsum 33942 An indexed sum of random v...
orvcval 33945 Value of the preimage mapp...
orvcval2 33946 Another way to express the...
elorvc 33947 Elementhood of a preimage....
orvcval4 33948 The value of the preimage ...
orvcoel 33949 If the relation produces o...
orvccel 33950 If the relation produces c...
elorrvc 33951 Elementhood of a preimage ...
orrvcval4 33952 The value of the preimage ...
orrvcoel 33953 If the relation produces o...
orrvccel 33954 If the relation produces c...
orvcgteel 33955 Preimage maps produced by ...
orvcelval 33956 Preimage maps produced by ...
orvcelel 33957 Preimage maps produced by ...
dstrvval 33958 The value of the distribut...
dstrvprob 33959 The distribution of a rand...
orvclteel 33960 Preimage maps produced by ...
dstfrvel 33961 Elementhood of preimage ma...
dstfrvunirn 33962 The limit of all preimage ...
orvclteinc 33963 Preimage maps produced by ...
dstfrvinc 33964 A cumulative distribution ...
dstfrvclim1 33965 The limit of the cumulativ...
coinfliplem 33966 Division in the extended r...
coinflipprob 33967 The ` P ` we defined for c...
coinflipspace 33968 The space of our coin-flip...
coinflipuniv 33969 The universe of our coin-f...
coinfliprv 33970 The ` X ` we defined for c...
coinflippv 33971 The probability of heads i...
coinflippvt 33972 The probability of tails i...
ballotlemoex 33973 ` O ` is a set. (Contribu...
ballotlem1 33974 The size of the universe i...
ballotlemelo 33975 Elementhood in ` O ` . (C...
ballotlem2 33976 The probability that the f...
ballotlemfval 33977 The value of ` F ` . (Con...
ballotlemfelz 33978 ` ( F `` C ) ` has values ...
ballotlemfp1 33979 If the ` J ` th ballot is ...
ballotlemfc0 33980 ` F ` takes value 0 betwee...
ballotlemfcc 33981 ` F ` takes value 0 betwee...
ballotlemfmpn 33982 ` ( F `` C ) ` finishes co...
ballotlemfval0 33983 ` ( F `` C ) ` always star...
ballotleme 33984 Elements of ` E ` . (Cont...
ballotlemodife 33985 Elements of ` ( O \ E ) ` ...
ballotlem4 33986 If the first pick is a vot...
ballotlem5 33987 If A is not ahead througho...
ballotlemi 33988 Value of ` I ` for a given...
ballotlemiex 33989 Properties of ` ( I `` C )...
ballotlemi1 33990 The first tie cannot be re...
ballotlemii 33991 The first tie cannot be re...
ballotlemsup 33992 The set of zeroes of ` F `...
ballotlemimin 33993 ` ( I `` C ) ` is the firs...
ballotlemic 33994 If the first vote is for B...
ballotlem1c 33995 If the first vote is for A...
ballotlemsval 33996 Value of ` S ` . (Contrib...
ballotlemsv 33997 Value of ` S ` evaluated a...
ballotlemsgt1 33998 ` S ` maps values less tha...
ballotlemsdom 33999 Domain of ` S ` for a give...
ballotlemsel1i 34000 The range ` ( 1 ... ( I ``...
ballotlemsf1o 34001 The defined ` S ` is a bij...
ballotlemsi 34002 The image by ` S ` of the ...
ballotlemsima 34003 The image by ` S ` of an i...
ballotlemieq 34004 If two countings share the...
ballotlemrval 34005 Value of ` R ` . (Contrib...
ballotlemscr 34006 The image of ` ( R `` C ) ...
ballotlemrv 34007 Value of ` R ` evaluated a...
ballotlemrv1 34008 Value of ` R ` before the ...
ballotlemrv2 34009 Value of ` R ` after the t...
ballotlemro 34010 Range of ` R ` is included...
ballotlemgval 34011 Expand the value of ` .^ `...
ballotlemgun 34012 A property of the defined ...
ballotlemfg 34013 Express the value of ` ( F...
ballotlemfrc 34014 Express the value of ` ( F...
ballotlemfrci 34015 Reverse counting preserves...
ballotlemfrceq 34016 Value of ` F ` for a rever...
ballotlemfrcn0 34017 Value of ` F ` for a rever...
ballotlemrc 34018 Range of ` R ` . (Contrib...
ballotlemirc 34019 Applying ` R ` does not ch...
ballotlemrinv0 34020 Lemma for ~ ballotlemrinv ...
ballotlemrinv 34021 ` R ` is its own inverse :...
ballotlem1ri 34022 When the vote on the first...
ballotlem7 34023 ` R ` is a bijection betwe...
ballotlem8 34024 There are as many counting...
ballotth 34025 Bertrand's ballot problem ...
sgncl 34026 Closure of the signum. (C...
sgnclre 34027 Closure of the signum. (C...
sgnneg 34028 Negation of the signum. (...
sgn3da 34029 A conditional containing a...
sgnmul 34030 Signum of a product. (Con...
sgnmulrp2 34031 Multiplication by a positi...
sgnsub 34032 Subtraction of a number of...
sgnnbi 34033 Negative signum. (Contrib...
sgnpbi 34034 Positive signum. (Contrib...
sgn0bi 34035 Zero signum. (Contributed...
sgnsgn 34036 Signum is idempotent. (Co...
sgnmulsgn 34037 If two real numbers are of...
sgnmulsgp 34038 If two real numbers are of...
fzssfzo 34039 Condition for an integer i...
gsumncl 34040 Closure of a group sum in ...
gsumnunsn 34041 Closure of a group sum in ...
ccatmulgnn0dir 34042 Concatenation of words fol...
ofcccat 34043 Letterwise operations on w...
ofcs1 34044 Letterwise operations on a...
ofcs2 34045 Letterwise operations on a...
plymul02 34046 Product of a polynomial wi...
plymulx0 34047 Coefficients of a polynomi...
plymulx 34048 Coefficients of a polynomi...
plyrecld 34049 Closure of a polynomial wi...
signsplypnf 34050 The quotient of a polynomi...
signsply0 34051 Lemma for the rule of sign...
signspval 34052 The value of the skipping ...
signsw0glem 34053 Neutral element property o...
signswbase 34054 The base of ` W ` is the u...
signswplusg 34055 The operation of ` W ` . ...
signsw0g 34056 The neutral element of ` W...
signswmnd 34057 ` W ` is a monoid structur...
signswrid 34058 The zero-skipping operatio...
signswlid 34059 The zero-skipping operatio...
signswn0 34060 The zero-skipping operatio...
signswch 34061 The zero-skipping operatio...
signslema 34062 Computational part of ~~? ...
signstfv 34063 Value of the zero-skipping...
signstfval 34064 Value of the zero-skipping...
signstcl 34065 Closure of the zero skippi...
signstf 34066 The zero skipping sign wor...
signstlen 34067 Length of the zero skippin...
signstf0 34068 Sign of a single letter wo...
signstfvn 34069 Zero-skipping sign in a wo...
signsvtn0 34070 If the last letter is nonz...
signstfvp 34071 Zero-skipping sign in a wo...
signstfvneq0 34072 In case the first letter i...
signstfvcl 34073 Closure of the zero skippi...
signstfvc 34074 Zero-skipping sign in a wo...
signstres 34075 Restriction of a zero skip...
signstfveq0a 34076 Lemma for ~ signstfveq0 . ...
signstfveq0 34077 In case the last letter is...
signsvvfval 34078 The value of ` V ` , which...
signsvvf 34079 ` V ` is a function. (Con...
signsvf0 34080 There is no change of sign...
signsvf1 34081 In a single-letter word, w...
signsvfn 34082 Number of changes in a wor...
signsvtp 34083 Adding a letter of the sam...
signsvtn 34084 Adding a letter of a diffe...
signsvfpn 34085 Adding a letter of the sam...
signsvfnn 34086 Adding a letter of a diffe...
signlem0 34087 Adding a zero as the highe...
signshf 34088 ` H ` , corresponding to t...
signshwrd 34089 ` H ` , corresponding to t...
signshlen 34090 Length of ` H ` , correspo...
signshnz 34091 ` H ` is not the empty wor...
efcld 34092 Closure law for the expone...
iblidicc 34093 The identity function is i...
rpsqrtcn 34094 Continuity of the real pos...
divsqrtid 34095 A real number divided by i...
cxpcncf1 34096 The power function on comp...
efmul2picn 34097 Multiplying by ` ( _i x. (...
fct2relem 34098 Lemma for ~ ftc2re . (Con...
ftc2re 34099 The Fundamental Theorem of...
fdvposlt 34100 Functions with a positive ...
fdvneggt 34101 Functions with a negative ...
fdvposle 34102 Functions with a nonnegati...
fdvnegge 34103 Functions with a nonpositi...
prodfzo03 34104 A product of three factors...
actfunsnf1o 34105 The action ` F ` of extend...
actfunsnrndisj 34106 The action ` F ` of extend...
itgexpif 34107 The basis for the circle m...
fsum2dsub 34108 Lemma for ~ breprexp - Re-...
reprval 34111 Value of the representatio...
repr0 34112 There is exactly one repre...
reprf 34113 Members of the representat...
reprsum 34114 Sums of values of the memb...
reprle 34115 Upper bound to the terms i...
reprsuc 34116 Express the representation...
reprfi 34117 Bounded representations ar...
reprss 34118 Representations with terms...
reprinrn 34119 Representations with term ...
reprlt 34120 There are no representatio...
hashreprin 34121 Express a sum of represent...
reprgt 34122 There are no representatio...
reprinfz1 34123 For the representation of ...
reprfi2 34124 Corollary of ~ reprinfz1 ....
reprfz1 34125 Corollary of ~ reprinfz1 ....
hashrepr 34126 Develop the number of repr...
reprpmtf1o 34127 Transposing ` 0 ` and ` X ...
reprdifc 34128 Express the representation...
chpvalz 34129 Value of the second Chebys...
chtvalz 34130 Value of the Chebyshev fun...
breprexplema 34131 Lemma for ~ breprexp (indu...
breprexplemb 34132 Lemma for ~ breprexp (clos...
breprexplemc 34133 Lemma for ~ breprexp (indu...
breprexp 34134 Express the ` S ` th power...
breprexpnat 34135 Express the ` S ` th power...
vtsval 34138 Value of the Vinogradov tr...
vtscl 34139 Closure of the Vinogradov ...
vtsprod 34140 Express the Vinogradov tri...
circlemeth 34141 The Hardy, Littlewood and ...
circlemethnat 34142 The Hardy, Littlewood and ...
circlevma 34143 The Circle Method, where t...
circlemethhgt 34144 The circle method, where t...
hgt750lemc 34148 An upper bound to the summ...
hgt750lemd 34149 An upper bound to the summ...
hgt749d 34150 A deduction version of ~ a...
logdivsqrle 34151 Conditions for ` ( ( log `...
hgt750lem 34152 Lemma for ~ tgoldbachgtd ....
hgt750lem2 34153 Decimal multiplication gal...
hgt750lemf 34154 Lemma for the statement 7....
hgt750lemg 34155 Lemma for the statement 7....
oddprm2 34156 Two ways to write the set ...
hgt750lemb 34157 An upper bound on the cont...
hgt750lema 34158 An upper bound on the cont...
hgt750leme 34159 An upper bound on the cont...
tgoldbachgnn 34160 Lemma for ~ tgoldbachgtd ....
tgoldbachgtde 34161 Lemma for ~ tgoldbachgtd ....
tgoldbachgtda 34162 Lemma for ~ tgoldbachgtd ....
tgoldbachgtd 34163 Odd integers greater than ...
tgoldbachgt 34164 Odd integers greater than ...
istrkg2d 34167 Property of fulfilling dim...
axtglowdim2ALTV 34168 Alternate version of ~ axt...
axtgupdim2ALTV 34169 Alternate version of ~ axt...
afsval 34172 Value of the AFS relation ...
brafs 34173 Binary relation form of th...
tg5segofs 34174 Rephrase ~ axtg5seg using ...
lpadval 34177 Value of the ` leftpad ` f...
lpadlem1 34178 Lemma for the ` leftpad ` ...
lpadlem3 34179 Lemma for ~ lpadlen1 . (C...
lpadlen1 34180 Length of a left-padded wo...
lpadlem2 34181 Lemma for the ` leftpad ` ...
lpadlen2 34182 Length of a left-padded wo...
lpadmax 34183 Length of a left-padded wo...
lpadleft 34184 The contents of prefix of ...
lpadright 34185 The suffix of a left-padde...
bnj170 34198 ` /\ ` -manipulation. (Co...
bnj240 34199 ` /\ ` -manipulation. (Co...
bnj248 34200 ` /\ ` -manipulation. (Co...
bnj250 34201 ` /\ ` -manipulation. (Co...
bnj251 34202 ` /\ ` -manipulation. (Co...
bnj252 34203 ` /\ ` -manipulation. (Co...
bnj253 34204 ` /\ ` -manipulation. (Co...
bnj255 34205 ` /\ ` -manipulation. (Co...
bnj256 34206 ` /\ ` -manipulation. (Co...
bnj257 34207 ` /\ ` -manipulation. (Co...
bnj258 34208 ` /\ ` -manipulation. (Co...
bnj268 34209 ` /\ ` -manipulation. (Co...
bnj290 34210 ` /\ ` -manipulation. (Co...
bnj291 34211 ` /\ ` -manipulation. (Co...
bnj312 34212 ` /\ ` -manipulation. (Co...
bnj334 34213 ` /\ ` -manipulation. (Co...
bnj345 34214 ` /\ ` -manipulation. (Co...
bnj422 34215 ` /\ ` -manipulation. (Co...
bnj432 34216 ` /\ ` -manipulation. (Co...
bnj446 34217 ` /\ ` -manipulation. (Co...
bnj23 34218 First-order logic and set ...
bnj31 34219 First-order logic and set ...
bnj62 34220 First-order logic and set ...
bnj89 34221 First-order logic and set ...
bnj90 34222 First-order logic and set ...
bnj101 34223 First-order logic and set ...
bnj105 34224 First-order logic and set ...
bnj115 34225 First-order logic and set ...
bnj132 34226 First-order logic and set ...
bnj133 34227 First-order logic and set ...
bnj156 34228 First-order logic and set ...
bnj158 34229 First-order logic and set ...
bnj168 34230 First-order logic and set ...
bnj206 34231 First-order logic and set ...
bnj216 34232 First-order logic and set ...
bnj219 34233 First-order logic and set ...
bnj226 34234 First-order logic and set ...
bnj228 34235 First-order logic and set ...
bnj519 34236 First-order logic and set ...
bnj524 34237 First-order logic and set ...
bnj525 34238 First-order logic and set ...
bnj534 34239 First-order logic and set ...
bnj538 34240 First-order logic and set ...
bnj529 34241 First-order logic and set ...
bnj551 34242 First-order logic and set ...
bnj563 34243 First-order logic and set ...
bnj564 34244 First-order logic and set ...
bnj593 34245 First-order logic and set ...
bnj596 34246 First-order logic and set ...
bnj610 34247 Pass from equality ( ` x =...
bnj642 34248 ` /\ ` -manipulation. (Co...
bnj643 34249 ` /\ ` -manipulation. (Co...
bnj645 34250 ` /\ ` -manipulation. (Co...
bnj658 34251 ` /\ ` -manipulation. (Co...
bnj667 34252 ` /\ ` -manipulation. (Co...
bnj705 34253 ` /\ ` -manipulation. (Co...
bnj706 34254 ` /\ ` -manipulation. (Co...
bnj707 34255 ` /\ ` -manipulation. (Co...
bnj708 34256 ` /\ ` -manipulation. (Co...
bnj721 34257 ` /\ ` -manipulation. (Co...
bnj832 34258 ` /\ ` -manipulation. (Co...
bnj835 34259 ` /\ ` -manipulation. (Co...
bnj836 34260 ` /\ ` -manipulation. (Co...
bnj837 34261 ` /\ ` -manipulation. (Co...
bnj769 34262 ` /\ ` -manipulation. (Co...
bnj770 34263 ` /\ ` -manipulation. (Co...
bnj771 34264 ` /\ ` -manipulation. (Co...
bnj887 34265 ` /\ ` -manipulation. (Co...
bnj918 34266 First-order logic and set ...
bnj919 34267 First-order logic and set ...
bnj923 34268 First-order logic and set ...
bnj927 34269 First-order logic and set ...
bnj931 34270 First-order logic and set ...
bnj937 34271 First-order logic and set ...
bnj941 34272 First-order logic and set ...
bnj945 34273 Technical lemma for ~ bnj6...
bnj946 34274 First-order logic and set ...
bnj951 34275 ` /\ ` -manipulation. (Co...
bnj956 34276 First-order logic and set ...
bnj976 34277 First-order logic and set ...
bnj982 34278 First-order logic and set ...
bnj1019 34279 First-order logic and set ...
bnj1023 34280 First-order logic and set ...
bnj1095 34281 First-order logic and set ...
bnj1096 34282 First-order logic and set ...
bnj1098 34283 First-order logic and set ...
bnj1101 34284 First-order logic and set ...
bnj1113 34285 First-order logic and set ...
bnj1109 34286 First-order logic and set ...
bnj1131 34287 First-order logic and set ...
bnj1138 34288 First-order logic and set ...
bnj1142 34289 First-order logic and set ...
bnj1143 34290 First-order logic and set ...
bnj1146 34291 First-order logic and set ...
bnj1149 34292 First-order logic and set ...
bnj1185 34293 First-order logic and set ...
bnj1196 34294 First-order logic and set ...
bnj1198 34295 First-order logic and set ...
bnj1209 34296 First-order logic and set ...
bnj1211 34297 First-order logic and set ...
bnj1213 34298 First-order logic and set ...
bnj1212 34299 First-order logic and set ...
bnj1219 34300 First-order logic and set ...
bnj1224 34301 First-order logic and set ...
bnj1230 34302 First-order logic and set ...
bnj1232 34303 First-order logic and set ...
bnj1235 34304 First-order logic and set ...
bnj1239 34305 First-order logic and set ...
bnj1238 34306 First-order logic and set ...
bnj1241 34307 First-order logic and set ...
bnj1247 34308 First-order logic and set ...
bnj1254 34309 First-order logic and set ...
bnj1262 34310 First-order logic and set ...
bnj1266 34311 First-order logic and set ...
bnj1265 34312 First-order logic and set ...
bnj1275 34313 First-order logic and set ...
bnj1276 34314 First-order logic and set ...
bnj1292 34315 First-order logic and set ...
bnj1293 34316 First-order logic and set ...
bnj1294 34317 First-order logic and set ...
bnj1299 34318 First-order logic and set ...
bnj1304 34319 First-order logic and set ...
bnj1316 34320 First-order logic and set ...
bnj1317 34321 First-order logic and set ...
bnj1322 34322 First-order logic and set ...
bnj1340 34323 First-order logic and set ...
bnj1345 34324 First-order logic and set ...
bnj1350 34325 First-order logic and set ...
bnj1351 34326 First-order logic and set ...
bnj1352 34327 First-order logic and set ...
bnj1361 34328 First-order logic and set ...
bnj1366 34329 First-order logic and set ...
bnj1379 34330 First-order logic and set ...
bnj1383 34331 First-order logic and set ...
bnj1385 34332 First-order logic and set ...
bnj1386 34333 First-order logic and set ...
bnj1397 34334 First-order logic and set ...
bnj1400 34335 First-order logic and set ...
bnj1405 34336 First-order logic and set ...
bnj1422 34337 First-order logic and set ...
bnj1424 34338 First-order logic and set ...
bnj1436 34339 First-order logic and set ...
bnj1441 34340 First-order logic and set ...
bnj1441g 34341 First-order logic and set ...
bnj1454 34342 First-order logic and set ...
bnj1459 34343 First-order logic and set ...
bnj1464 34344 Conversion of implicit sub...
bnj1465 34345 First-order logic and set ...
bnj1468 34346 Conversion of implicit sub...
bnj1476 34347 First-order logic and set ...
bnj1502 34348 First-order logic and set ...
bnj1503 34349 First-order logic and set ...
bnj1517 34350 First-order logic and set ...
bnj1521 34351 First-order logic and set ...
bnj1533 34352 First-order logic and set ...
bnj1534 34353 First-order logic and set ...
bnj1536 34354 First-order logic and set ...
bnj1538 34355 First-order logic and set ...
bnj1541 34356 First-order logic and set ...
bnj1542 34357 First-order logic and set ...
bnj110 34358 Well-founded induction res...
bnj157 34359 Well-founded induction res...
bnj66 34360 Technical lemma for ~ bnj6...
bnj91 34361 First-order logic and set ...
bnj92 34362 First-order logic and set ...
bnj93 34363 Technical lemma for ~ bnj9...
bnj95 34364 Technical lemma for ~ bnj1...
bnj96 34365 Technical lemma for ~ bnj1...
bnj97 34366 Technical lemma for ~ bnj1...
bnj98 34367 Technical lemma for ~ bnj1...
bnj106 34368 First-order logic and set ...
bnj118 34369 First-order logic and set ...
bnj121 34370 First-order logic and set ...
bnj124 34371 Technical lemma for ~ bnj1...
bnj125 34372 Technical lemma for ~ bnj1...
bnj126 34373 Technical lemma for ~ bnj1...
bnj130 34374 Technical lemma for ~ bnj1...
bnj149 34375 Technical lemma for ~ bnj1...
bnj150 34376 Technical lemma for ~ bnj1...
bnj151 34377 Technical lemma for ~ bnj1...
bnj154 34378 Technical lemma for ~ bnj1...
bnj155 34379 Technical lemma for ~ bnj1...
bnj153 34380 Technical lemma for ~ bnj8...
bnj207 34381 Technical lemma for ~ bnj8...
bnj213 34382 First-order logic and set ...
bnj222 34383 Technical lemma for ~ bnj2...
bnj229 34384 Technical lemma for ~ bnj5...
bnj517 34385 Technical lemma for ~ bnj5...
bnj518 34386 Technical lemma for ~ bnj8...
bnj523 34387 Technical lemma for ~ bnj8...
bnj526 34388 Technical lemma for ~ bnj8...
bnj528 34389 Technical lemma for ~ bnj8...
bnj535 34390 Technical lemma for ~ bnj8...
bnj539 34391 Technical lemma for ~ bnj8...
bnj540 34392 Technical lemma for ~ bnj8...
bnj543 34393 Technical lemma for ~ bnj8...
bnj544 34394 Technical lemma for ~ bnj8...
bnj545 34395 Technical lemma for ~ bnj8...
bnj546 34396 Technical lemma for ~ bnj8...
bnj548 34397 Technical lemma for ~ bnj8...
bnj553 34398 Technical lemma for ~ bnj8...
bnj554 34399 Technical lemma for ~ bnj8...
bnj556 34400 Technical lemma for ~ bnj8...
bnj557 34401 Technical lemma for ~ bnj8...
bnj558 34402 Technical lemma for ~ bnj8...
bnj561 34403 Technical lemma for ~ bnj8...
bnj562 34404 Technical lemma for ~ bnj8...
bnj570 34405 Technical lemma for ~ bnj8...
bnj571 34406 Technical lemma for ~ bnj8...
bnj605 34407 Technical lemma. This lem...
bnj581 34408 Technical lemma for ~ bnj5...
bnj589 34409 Technical lemma for ~ bnj8...
bnj590 34410 Technical lemma for ~ bnj8...
bnj591 34411 Technical lemma for ~ bnj8...
bnj594 34412 Technical lemma for ~ bnj8...
bnj580 34413 Technical lemma for ~ bnj5...
bnj579 34414 Technical lemma for ~ bnj8...
bnj602 34415 Equality theorem for the `...
bnj607 34416 Technical lemma for ~ bnj8...
bnj609 34417 Technical lemma for ~ bnj8...
bnj611 34418 Technical lemma for ~ bnj8...
bnj600 34419 Technical lemma for ~ bnj8...
bnj601 34420 Technical lemma for ~ bnj8...
bnj852 34421 Technical lemma for ~ bnj6...
bnj864 34422 Technical lemma for ~ bnj6...
bnj865 34423 Technical lemma for ~ bnj6...
bnj873 34424 Technical lemma for ~ bnj6...
bnj849 34425 Technical lemma for ~ bnj6...
bnj882 34426 Definition (using hypothes...
bnj18eq1 34427 Equality theorem for trans...
bnj893 34428 Property of ` _trCl ` . U...
bnj900 34429 Technical lemma for ~ bnj6...
bnj906 34430 Property of ` _trCl ` . (...
bnj908 34431 Technical lemma for ~ bnj6...
bnj911 34432 Technical lemma for ~ bnj6...
bnj916 34433 Technical lemma for ~ bnj6...
bnj917 34434 Technical lemma for ~ bnj6...
bnj934 34435 Technical lemma for ~ bnj6...
bnj929 34436 Technical lemma for ~ bnj6...
bnj938 34437 Technical lemma for ~ bnj6...
bnj944 34438 Technical lemma for ~ bnj6...
bnj953 34439 Technical lemma for ~ bnj6...
bnj958 34440 Technical lemma for ~ bnj6...
bnj1000 34441 Technical lemma for ~ bnj8...
bnj965 34442 Technical lemma for ~ bnj8...
bnj964 34443 Technical lemma for ~ bnj6...
bnj966 34444 Technical lemma for ~ bnj6...
bnj967 34445 Technical lemma for ~ bnj6...
bnj969 34446 Technical lemma for ~ bnj6...
bnj970 34447 Technical lemma for ~ bnj6...
bnj910 34448 Technical lemma for ~ bnj6...
bnj978 34449 Technical lemma for ~ bnj6...
bnj981 34450 Technical lemma for ~ bnj6...
bnj983 34451 Technical lemma for ~ bnj6...
bnj984 34452 Technical lemma for ~ bnj6...
bnj985v 34453 Version of ~ bnj985 with a...
bnj985 34454 Technical lemma for ~ bnj6...
bnj986 34455 Technical lemma for ~ bnj6...
bnj996 34456 Technical lemma for ~ bnj6...
bnj998 34457 Technical lemma for ~ bnj6...
bnj999 34458 Technical lemma for ~ bnj6...
bnj1001 34459 Technical lemma for ~ bnj6...
bnj1006 34460 Technical lemma for ~ bnj6...
bnj1014 34461 Technical lemma for ~ bnj6...
bnj1015 34462 Technical lemma for ~ bnj6...
bnj1018g 34463 Version of ~ bnj1018 with ...
bnj1018 34464 Technical lemma for ~ bnj6...
bnj1020 34465 Technical lemma for ~ bnj6...
bnj1021 34466 Technical lemma for ~ bnj6...
bnj907 34467 Technical lemma for ~ bnj6...
bnj1029 34468 Property of ` _trCl ` . (...
bnj1033 34469 Technical lemma for ~ bnj6...
bnj1034 34470 Technical lemma for ~ bnj6...
bnj1039 34471 Technical lemma for ~ bnj6...
bnj1040 34472 Technical lemma for ~ bnj6...
bnj1047 34473 Technical lemma for ~ bnj6...
bnj1049 34474 Technical lemma for ~ bnj6...
bnj1052 34475 Technical lemma for ~ bnj6...
bnj1053 34476 Technical lemma for ~ bnj6...
bnj1071 34477 Technical lemma for ~ bnj6...
bnj1083 34478 Technical lemma for ~ bnj6...
bnj1090 34479 Technical lemma for ~ bnj6...
bnj1093 34480 Technical lemma for ~ bnj6...
bnj1097 34481 Technical lemma for ~ bnj6...
bnj1110 34482 Technical lemma for ~ bnj6...
bnj1112 34483 Technical lemma for ~ bnj6...
bnj1118 34484 Technical lemma for ~ bnj6...
bnj1121 34485 Technical lemma for ~ bnj6...
bnj1123 34486 Technical lemma for ~ bnj6...
bnj1030 34487 Technical lemma for ~ bnj6...
bnj1124 34488 Property of ` _trCl ` . (...
bnj1133 34489 Technical lemma for ~ bnj6...
bnj1128 34490 Technical lemma for ~ bnj6...
bnj1127 34491 Property of ` _trCl ` . (...
bnj1125 34492 Property of ` _trCl ` . (...
bnj1145 34493 Technical lemma for ~ bnj6...
bnj1147 34494 Property of ` _trCl ` . (...
bnj1137 34495 Property of ` _trCl ` . (...
bnj1148 34496 Property of ` _pred ` . (...
bnj1136 34497 Technical lemma for ~ bnj6...
bnj1152 34498 Technical lemma for ~ bnj6...
bnj1154 34499 Property of ` Fr ` . (Con...
bnj1171 34500 Technical lemma for ~ bnj6...
bnj1172 34501 Technical lemma for ~ bnj6...
bnj1173 34502 Technical lemma for ~ bnj6...
bnj1174 34503 Technical lemma for ~ bnj6...
bnj1175 34504 Technical lemma for ~ bnj6...
bnj1176 34505 Technical lemma for ~ bnj6...
bnj1177 34506 Technical lemma for ~ bnj6...
bnj1186 34507 Technical lemma for ~ bnj6...
bnj1190 34508 Technical lemma for ~ bnj6...
bnj1189 34509 Technical lemma for ~ bnj6...
bnj69 34510 Existence of a minimal ele...
bnj1228 34511 Existence of a minimal ele...
bnj1204 34512 Well-founded induction. T...
bnj1234 34513 Technical lemma for ~ bnj6...
bnj1245 34514 Technical lemma for ~ bnj6...
bnj1256 34515 Technical lemma for ~ bnj6...
bnj1259 34516 Technical lemma for ~ bnj6...
bnj1253 34517 Technical lemma for ~ bnj6...
bnj1279 34518 Technical lemma for ~ bnj6...
bnj1286 34519 Technical lemma for ~ bnj6...
bnj1280 34520 Technical lemma for ~ bnj6...
bnj1296 34521 Technical lemma for ~ bnj6...
bnj1309 34522 Technical lemma for ~ bnj6...
bnj1307 34523 Technical lemma for ~ bnj6...
bnj1311 34524 Technical lemma for ~ bnj6...
bnj1318 34525 Technical lemma for ~ bnj6...
bnj1326 34526 Technical lemma for ~ bnj6...
bnj1321 34527 Technical lemma for ~ bnj6...
bnj1364 34528 Property of ` _FrSe ` . (...
bnj1371 34529 Technical lemma for ~ bnj6...
bnj1373 34530 Technical lemma for ~ bnj6...
bnj1374 34531 Technical lemma for ~ bnj6...
bnj1384 34532 Technical lemma for ~ bnj6...
bnj1388 34533 Technical lemma for ~ bnj6...
bnj1398 34534 Technical lemma for ~ bnj6...
bnj1413 34535 Property of ` _trCl ` . (...
bnj1408 34536 Technical lemma for ~ bnj1...
bnj1414 34537 Property of ` _trCl ` . (...
bnj1415 34538 Technical lemma for ~ bnj6...
bnj1416 34539 Technical lemma for ~ bnj6...
bnj1418 34540 Property of ` _pred ` . (...
bnj1417 34541 Technical lemma for ~ bnj6...
bnj1421 34542 Technical lemma for ~ bnj6...
bnj1444 34543 Technical lemma for ~ bnj6...
bnj1445 34544 Technical lemma for ~ bnj6...
bnj1446 34545 Technical lemma for ~ bnj6...
bnj1447 34546 Technical lemma for ~ bnj6...
bnj1448 34547 Technical lemma for ~ bnj6...
bnj1449 34548 Technical lemma for ~ bnj6...
bnj1442 34549 Technical lemma for ~ bnj6...
bnj1450 34550 Technical lemma for ~ bnj6...
bnj1423 34551 Technical lemma for ~ bnj6...
bnj1452 34552 Technical lemma for ~ bnj6...
bnj1466 34553 Technical lemma for ~ bnj6...
bnj1467 34554 Technical lemma for ~ bnj6...
bnj1463 34555 Technical lemma for ~ bnj6...
bnj1489 34556 Technical lemma for ~ bnj6...
bnj1491 34557 Technical lemma for ~ bnj6...
bnj1312 34558 Technical lemma for ~ bnj6...
bnj1493 34559 Technical lemma for ~ bnj6...
bnj1497 34560 Technical lemma for ~ bnj6...
bnj1498 34561 Technical lemma for ~ bnj6...
bnj60 34562 Well-founded recursion, pa...
bnj1514 34563 Technical lemma for ~ bnj1...
bnj1518 34564 Technical lemma for ~ bnj1...
bnj1519 34565 Technical lemma for ~ bnj1...
bnj1520 34566 Technical lemma for ~ bnj1...
bnj1501 34567 Technical lemma for ~ bnj1...
bnj1500 34568 Well-founded recursion, pa...
bnj1525 34569 Technical lemma for ~ bnj1...
bnj1529 34570 Technical lemma for ~ bnj1...
bnj1523 34571 Technical lemma for ~ bnj1...
bnj1522 34572 Well-founded recursion, pa...
exdifsn 34573 There exists an element in...
srcmpltd 34574 If a statement is true for...
prsrcmpltd 34575 If a statement is true for...
dff15 34576 A one-to-one function in t...
f1resveqaeq 34577 If a function restricted t...
f1resrcmplf1dlem 34578 Lemma for ~ f1resrcmplf1d ...
f1resrcmplf1d 34579 If a function's restrictio...
funen1cnv 34580 If a function is equinumer...
fnrelpredd 34581 A function that preserves ...
cardpred 34582 The cardinality function p...
nummin 34583 Every nonempty class of nu...
fineqvrep 34584 If the Axiom of Infinity i...
fineqvpow 34585 If the Axiom of Infinity i...
fineqvac 34586 If the Axiom of Infinity i...
fineqvacALT 34587 Shorter proof of ~ fineqva...
zltp1ne 34588 Integer ordering relation....
nnltp1ne 34589 Positive integer ordering ...
nn0ltp1ne 34590 Nonnegative integer orderi...
0nn0m1nnn0 34591 A number is zero if and on...
f1resfz0f1d 34592 If a function with a seque...
fisshasheq 34593 A finite set is equal to i...
hashf1dmcdm 34594 The size of the domain of ...
revpfxsfxrev 34595 The reverse of a prefix of...
swrdrevpfx 34596 A subword expressed in ter...
lfuhgr 34597 A hypergraph is loop-free ...
lfuhgr2 34598 A hypergraph is loop-free ...
lfuhgr3 34599 A hypergraph is loop-free ...
cplgredgex 34600 Any two (distinct) vertice...
cusgredgex 34601 Any two (distinct) vertice...
cusgredgex2 34602 Any two distinct vertices ...
pfxwlk 34603 A prefix of a walk is a wa...
revwlk 34604 The reverse of a walk is a...
revwlkb 34605 Two words represent a walk...
swrdwlk 34606 Two matching subwords of a...
pthhashvtx 34607 A graph containing a path ...
pthisspthorcycl 34608 A path is either a simple ...
spthcycl 34609 A walk is a trivial path i...
usgrgt2cycl 34610 A non-trivial cycle in a s...
usgrcyclgt2v 34611 A simple graph with a non-...
subgrwlk 34612 If a walk exists in a subg...
subgrtrl 34613 If a trail exists in a sub...
subgrpth 34614 If a path exists in a subg...
subgrcycl 34615 If a cycle exists in a sub...
cusgr3cyclex 34616 Every complete simple grap...
loop1cycl 34617 A hypergraph has a cycle o...
2cycld 34618 Construction of a 2-cycle ...
2cycl2d 34619 Construction of a 2-cycle ...
umgr2cycllem 34620 Lemma for ~ umgr2cycl . (...
umgr2cycl 34621 A multigraph with two dist...
dfacycgr1 34624 An alternate definition of...
isacycgr 34625 The property of being an a...
isacycgr1 34626 The property of being an a...
acycgrcycl 34627 Any cycle in an acyclic gr...
acycgr0v 34628 A null graph (with no vert...
acycgr1v 34629 A multigraph with one vert...
acycgr2v 34630 A simple graph with two ve...
prclisacycgr 34631 A proper class (representi...
acycgrislfgr 34632 An acyclic hypergraph is a...
upgracycumgr 34633 An acyclic pseudograph is ...
umgracycusgr 34634 An acyclic multigraph is a...
upgracycusgr 34635 An acyclic pseudograph is ...
cusgracyclt3v 34636 A complete simple graph is...
pthacycspth 34637 A path in an acyclic graph...
acycgrsubgr 34638 The subgraph of an acyclic...
quartfull 34645 The quartic equation, writ...
deranglem 34646 Lemma for derangements. (...
derangval 34647 Define the derangement fun...
derangf 34648 The derangement number is ...
derang0 34649 The derangement number of ...
derangsn 34650 The derangement number of ...
derangenlem 34651 One half of ~ derangen . ...
derangen 34652 The derangement number is ...
subfacval 34653 The subfactorial is define...
derangen2 34654 Write the derangement numb...
subfacf 34655 The subfactorial is a func...
subfaclefac 34656 The subfactorial is less t...
subfac0 34657 The subfactorial at zero. ...
subfac1 34658 The subfactorial at one. ...
subfacp1lem1 34659 Lemma for ~ subfacp1 . Th...
subfacp1lem2a 34660 Lemma for ~ subfacp1 . Pr...
subfacp1lem2b 34661 Lemma for ~ subfacp1 . Pr...
subfacp1lem3 34662 Lemma for ~ subfacp1 . In...
subfacp1lem4 34663 Lemma for ~ subfacp1 . Th...
subfacp1lem5 34664 Lemma for ~ subfacp1 . In...
subfacp1lem6 34665 Lemma for ~ subfacp1 . By...
subfacp1 34666 A two-term recurrence for ...
subfacval2 34667 A closed-form expression f...
subfaclim 34668 The subfactorial converges...
subfacval3 34669 Another closed form expres...
derangfmla 34670 The derangements formula, ...
erdszelem1 34671 Lemma for ~ erdsze . (Con...
erdszelem2 34672 Lemma for ~ erdsze . (Con...
erdszelem3 34673 Lemma for ~ erdsze . (Con...
erdszelem4 34674 Lemma for ~ erdsze . (Con...
erdszelem5 34675 Lemma for ~ erdsze . (Con...
erdszelem6 34676 Lemma for ~ erdsze . (Con...
erdszelem7 34677 Lemma for ~ erdsze . (Con...
erdszelem8 34678 Lemma for ~ erdsze . (Con...
erdszelem9 34679 Lemma for ~ erdsze . (Con...
erdszelem10 34680 Lemma for ~ erdsze . (Con...
erdszelem11 34681 Lemma for ~ erdsze . (Con...
erdsze 34682 The ErdÅ‘s-Szekeres th...
erdsze2lem1 34683 Lemma for ~ erdsze2 . (Co...
erdsze2lem2 34684 Lemma for ~ erdsze2 . (Co...
erdsze2 34685 Generalize the statement o...
kur14lem1 34686 Lemma for ~ kur14 . (Cont...
kur14lem2 34687 Lemma for ~ kur14 . Write...
kur14lem3 34688 Lemma for ~ kur14 . A clo...
kur14lem4 34689 Lemma for ~ kur14 . Compl...
kur14lem5 34690 Lemma for ~ kur14 . Closu...
kur14lem6 34691 Lemma for ~ kur14 . If ` ...
kur14lem7 34692 Lemma for ~ kur14 : main p...
kur14lem8 34693 Lemma for ~ kur14 . Show ...
kur14lem9 34694 Lemma for ~ kur14 . Since...
kur14lem10 34695 Lemma for ~ kur14 . Disch...
kur14 34696 Kuratowski's closure-compl...
ispconn 34703 The property of being a pa...
pconncn 34704 The property of being a pa...
pconntop 34705 A simply connected space i...
issconn 34706 The property of being a si...
sconnpconn 34707 A simply connected space i...
sconntop 34708 A simply connected space i...
sconnpht 34709 A closed path in a simply ...
cnpconn 34710 An image of a path-connect...
pconnconn 34711 A path-connected space is ...
txpconn 34712 The topological product of...
ptpconn 34713 The topological product of...
indispconn 34714 The indiscrete topology (o...
connpconn 34715 A connected and locally pa...
qtoppconn 34716 A quotient of a path-conne...
pconnpi1 34717 All fundamental groups in ...
sconnpht2 34718 Any two paths in a simply ...
sconnpi1 34719 A path-connected topologic...
txsconnlem 34720 Lemma for ~ txsconn . (Co...
txsconn 34721 The topological product of...
cvxpconn 34722 A convex subset of the com...
cvxsconn 34723 A convex subset of the com...
blsconn 34724 An open ball in the comple...
cnllysconn 34725 The topology of the comple...
resconn 34726 A subset of ` RR ` is simp...
ioosconn 34727 An open interval is simply...
iccsconn 34728 A closed interval is simpl...
retopsconn 34729 The real numbers are simpl...
iccllysconn 34730 A closed interval is local...
rellysconn 34731 The real numbers are local...
iisconn 34732 The unit interval is simpl...
iillysconn 34733 The unit interval is local...
iinllyconn 34734 The unit interval is local...
fncvm 34737 Lemma for covering maps. ...
cvmscbv 34738 Change bound variables in ...
iscvm 34739 The property of being a co...
cvmtop1 34740 Reverse closure for a cove...
cvmtop2 34741 Reverse closure for a cove...
cvmcn 34742 A covering map is a contin...
cvmcov 34743 Property of a covering map...
cvmsrcl 34744 Reverse closure for an eve...
cvmsi 34745 One direction of ~ cvmsval...
cvmsval 34746 Elementhood in the set ` S...
cvmsss 34747 An even covering is a subs...
cvmsn0 34748 An even covering is nonemp...
cvmsuni 34749 An even covering of ` U ` ...
cvmsdisj 34750 An even covering of ` U ` ...
cvmshmeo 34751 Every element of an even c...
cvmsf1o 34752 ` F ` , localized to an el...
cvmscld 34753 The sets of an even coveri...
cvmsss2 34754 An open subset of an evenl...
cvmcov2 34755 The covering map property ...
cvmseu 34756 Every element in ` U. T ` ...
cvmsiota 34757 Identify the unique elemen...
cvmopnlem 34758 Lemma for ~ cvmopn . (Con...
cvmfolem 34759 Lemma for ~ cvmfo . (Cont...
cvmopn 34760 A covering map is an open ...
cvmliftmolem1 34761 Lemma for ~ cvmliftmo . (...
cvmliftmolem2 34762 Lemma for ~ cvmliftmo . (...
cvmliftmoi 34763 A lift of a continuous fun...
cvmliftmo 34764 A lift of a continuous fun...
cvmliftlem1 34765 Lemma for ~ cvmlift . In ...
cvmliftlem2 34766 Lemma for ~ cvmlift . ` W ...
cvmliftlem3 34767 Lemma for ~ cvmlift . Sin...
cvmliftlem4 34768 Lemma for ~ cvmlift . The...
cvmliftlem5 34769 Lemma for ~ cvmlift . Def...
cvmliftlem6 34770 Lemma for ~ cvmlift . Ind...
cvmliftlem7 34771 Lemma for ~ cvmlift . Pro...
cvmliftlem8 34772 Lemma for ~ cvmlift . The...
cvmliftlem9 34773 Lemma for ~ cvmlift . The...
cvmliftlem10 34774 Lemma for ~ cvmlift . The...
cvmliftlem11 34775 Lemma for ~ cvmlift . (Co...
cvmliftlem13 34776 Lemma for ~ cvmlift . The...
cvmliftlem14 34777 Lemma for ~ cvmlift . Put...
cvmliftlem15 34778 Lemma for ~ cvmlift . Dis...
cvmlift 34779 One of the important prope...
cvmfo 34780 A covering map is an onto ...
cvmliftiota 34781 Write out a function ` H `...
cvmlift2lem1 34782 Lemma for ~ cvmlift2 . (C...
cvmlift2lem9a 34783 Lemma for ~ cvmlift2 and ~...
cvmlift2lem2 34784 Lemma for ~ cvmlift2 . (C...
cvmlift2lem3 34785 Lemma for ~ cvmlift2 . (C...
cvmlift2lem4 34786 Lemma for ~ cvmlift2 . (C...
cvmlift2lem5 34787 Lemma for ~ cvmlift2 . (C...
cvmlift2lem6 34788 Lemma for ~ cvmlift2 . (C...
cvmlift2lem7 34789 Lemma for ~ cvmlift2 . (C...
cvmlift2lem8 34790 Lemma for ~ cvmlift2 . (C...
cvmlift2lem9 34791 Lemma for ~ cvmlift2 . (C...
cvmlift2lem10 34792 Lemma for ~ cvmlift2 . (C...
cvmlift2lem11 34793 Lemma for ~ cvmlift2 . (C...
cvmlift2lem12 34794 Lemma for ~ cvmlift2 . (C...
cvmlift2lem13 34795 Lemma for ~ cvmlift2 . (C...
cvmlift2 34796 A two-dimensional version ...
cvmliftphtlem 34797 Lemma for ~ cvmliftpht . ...
cvmliftpht 34798 If ` G ` and ` H ` are pat...
cvmlift3lem1 34799 Lemma for ~ cvmlift3 . (C...
cvmlift3lem2 34800 Lemma for ~ cvmlift2 . (C...
cvmlift3lem3 34801 Lemma for ~ cvmlift2 . (C...
cvmlift3lem4 34802 Lemma for ~ cvmlift2 . (C...
cvmlift3lem5 34803 Lemma for ~ cvmlift2 . (C...
cvmlift3lem6 34804 Lemma for ~ cvmlift3 . (C...
cvmlift3lem7 34805 Lemma for ~ cvmlift3 . (C...
cvmlift3lem8 34806 Lemma for ~ cvmlift2 . (C...
cvmlift3lem9 34807 Lemma for ~ cvmlift2 . (C...
cvmlift3 34808 A general version of ~ cvm...
snmlff 34809 The function ` F ` from ~ ...
snmlfval 34810 The function ` F ` from ~ ...
snmlval 34811 The property " ` A ` is si...
snmlflim 34812 If ` A ` is simply normal,...
goel 34827 A "Godel-set of membership...
goelel3xp 34828 A "Godel-set of membership...
goeleq12bg 34829 Two "Godel-set of membersh...
gonafv 34830 The "Godel-set for the She...
goaleq12d 34831 Equality of the "Godel-set...
gonanegoal 34832 The Godel-set for the Shef...
satf 34833 The satisfaction predicate...
satfsucom 34834 The satisfaction predicate...
satfn 34835 The satisfaction predicate...
satom 34836 The satisfaction predicate...
satfvsucom 34837 The satisfaction predicate...
satfv0 34838 The value of the satisfact...
satfvsuclem1 34839 Lemma 1 for ~ satfvsuc . ...
satfvsuclem2 34840 Lemma 2 for ~ satfvsuc . ...
satfvsuc 34841 The value of the satisfact...
satfv1lem 34842 Lemma for ~ satfv1 . (Con...
satfv1 34843 The value of the satisfact...
satfsschain 34844 The binary relation of a s...
satfvsucsuc 34845 The satisfaction predicate...
satfbrsuc 34846 The binary relation of a s...
satfrel 34847 The value of the satisfact...
satfdmlem 34848 Lemma for ~ satfdm . (Con...
satfdm 34849 The domain of the satisfac...
satfrnmapom 34850 The range of the satisfact...
satfv0fun 34851 The value of the satisfact...
satf0 34852 The satisfaction predicate...
satf0sucom 34853 The satisfaction predicate...
satf00 34854 The value of the satisfact...
satf0suclem 34855 Lemma for ~ satf0suc , ~ s...
satf0suc 34856 The value of the satisfact...
satf0op 34857 An element of a value of t...
satf0n0 34858 The value of the satisfact...
sat1el2xp 34859 The first component of an ...
fmlafv 34860 The valid Godel formulas o...
fmla 34861 The set of all valid Godel...
fmla0 34862 The valid Godel formulas o...
fmla0xp 34863 The valid Godel formulas o...
fmlasuc0 34864 The valid Godel formulas o...
fmlafvel 34865 A class is a valid Godel f...
fmlasuc 34866 The valid Godel formulas o...
fmla1 34867 The valid Godel formulas o...
isfmlasuc 34868 The characterization of a ...
fmlasssuc 34869 The Godel formulas of heig...
fmlaomn0 34870 The empty set is not a God...
fmlan0 34871 The empty set is not a God...
gonan0 34872 The "Godel-set of NAND" is...
goaln0 34873 The "Godel-set of universa...
gonarlem 34874 Lemma for ~ gonar (inducti...
gonar 34875 If the "Godel-set of NAND"...
goalrlem 34876 Lemma for ~ goalr (inducti...
goalr 34877 If the "Godel-set of unive...
fmla0disjsuc 34878 The set of valid Godel for...
fmlasucdisj 34879 The valid Godel formulas o...
satfdmfmla 34880 The domain of the satisfac...
satffunlem 34881 Lemma for ~ satffunlem1lem...
satffunlem1lem1 34882 Lemma for ~ satffunlem1 . ...
satffunlem1lem2 34883 Lemma 2 for ~ satffunlem1 ...
satffunlem2lem1 34884 Lemma 1 for ~ satffunlem2 ...
dmopab3rexdif 34885 The domain of an ordered p...
satffunlem2lem2 34886 Lemma 2 for ~ satffunlem2 ...
satffunlem1 34887 Lemma 1 for ~ satffun : in...
satffunlem2 34888 Lemma 2 for ~ satffun : in...
satffun 34889 The value of the satisfact...
satff 34890 The satisfaction predicate...
satfun 34891 The satisfaction predicate...
satfvel 34892 An element of the value of...
satfv0fvfmla0 34893 The value of the satisfact...
satefv 34894 The simplified satisfactio...
sate0 34895 The simplified satisfactio...
satef 34896 The simplified satisfactio...
sate0fv0 34897 A simplified satisfaction ...
satefvfmla0 34898 The simplified satisfactio...
sategoelfvb 34899 Characterization of a valu...
sategoelfv 34900 Condition of a valuation `...
ex-sategoelel 34901 Example of a valuation of ...
ex-sategoel 34902 Instance of ~ sategoelfv f...
satfv1fvfmla1 34903 The value of the satisfact...
2goelgoanfmla1 34904 Two Godel-sets of membersh...
satefvfmla1 34905 The simplified satisfactio...
ex-sategoelelomsuc 34906 Example of a valuation of ...
ex-sategoelel12 34907 Example of a valuation of ...
prv 34908 The "proves" relation on a...
elnanelprv 34909 The wff ` ( A e. B -/\ B e...
prv0 34910 Every wff encoded as ` U `...
prv1n 34911 No wff encoded as a Godel-...
mvtval 34980 The set of variable typeco...
mrexval 34981 The set of "raw expression...
mexval 34982 The set of expressions, wh...
mexval2 34983 The set of expressions, wh...
mdvval 34984 The set of disjoint variab...
mvrsval 34985 The set of variables in an...
mvrsfpw 34986 The set of variables in an...
mrsubffval 34987 The substitution of some v...
mrsubfval 34988 The substitution of some v...
mrsubval 34989 The substitution of some v...
mrsubcv 34990 The value of a substituted...
mrsubvr 34991 The value of a substituted...
mrsubff 34992 A substitution is a functi...
mrsubrn 34993 Although it is defined for...
mrsubff1 34994 When restricted to complet...
mrsubff1o 34995 When restricted to complet...
mrsub0 34996 The value of the substitut...
mrsubf 34997 A substitution is a functi...
mrsubccat 34998 Substitution distributes o...
mrsubcn 34999 A substitution does not ch...
elmrsubrn 35000 Characterization of the su...
mrsubco 35001 The composition of two sub...
mrsubvrs 35002 The set of variables in a ...
msubffval 35003 A substitution applied to ...
msubfval 35004 A substitution applied to ...
msubval 35005 A substitution applied to ...
msubrsub 35006 A substitution applied to ...
msubty 35007 The type of a substituted ...
elmsubrn 35008 Characterization of substi...
msubrn 35009 Although it is defined for...
msubff 35010 A substitution is a functi...
msubco 35011 The composition of two sub...
msubf 35012 A substitution is a functi...
mvhfval 35013 Value of the function mapp...
mvhval 35014 Value of the function mapp...
mpstval 35015 A pre-statement is an orde...
elmpst 35016 Property of being a pre-st...
msrfval 35017 Value of the reduct of a p...
msrval 35018 Value of the reduct of a p...
mpstssv 35019 A pre-statement is an orde...
mpst123 35020 Decompose a pre-statement ...
mpstrcl 35021 The elements of a pre-stat...
msrf 35022 The reduct of a pre-statem...
msrrcl 35023 If ` X ` and ` Y ` have th...
mstaval 35024 Value of the set of statem...
msrid 35025 The reduct of a statement ...
msrfo 35026 The reduct of a pre-statem...
mstapst 35027 A statement is a pre-state...
elmsta 35028 Property of being a statem...
ismfs 35029 A formal system is a tuple...
mfsdisj 35030 The constants and variable...
mtyf2 35031 The type function maps var...
mtyf 35032 The type function maps var...
mvtss 35033 The set of variable typeco...
maxsta 35034 An axiom is a statement. ...
mvtinf 35035 Each variable typecode has...
msubff1 35036 When restricted to complet...
msubff1o 35037 When restricted to complet...
mvhf 35038 The function mapping varia...
mvhf1 35039 The function mapping varia...
msubvrs 35040 The set of variables in a ...
mclsrcl 35041 Reverse closure for the cl...
mclsssvlem 35042 Lemma for ~ mclsssv . (Co...
mclsval 35043 The function mapping varia...
mclsssv 35044 The closure of a set of ex...
ssmclslem 35045 Lemma for ~ ssmcls . (Con...
vhmcls 35046 All variable hypotheses ar...
ssmcls 35047 The original expressions a...
ss2mcls 35048 The closure is monotonic u...
mclsax 35049 The closure is closed unde...
mclsind 35050 Induction theorem for clos...
mppspstlem 35051 Lemma for ~ mppspst . (Co...
mppsval 35052 Definition of a provable p...
elmpps 35053 Definition of a provable p...
mppspst 35054 A provable pre-statement i...
mthmval 35055 A theorem is a pre-stateme...
elmthm 35056 A theorem is a pre-stateme...
mthmi 35057 A statement whose reduct i...
mthmsta 35058 A theorem is a pre-stateme...
mppsthm 35059 A provable pre-statement i...
mthmblem 35060 Lemma for ~ mthmb . (Cont...
mthmb 35061 If two statements have the...
mthmpps 35062 Given a theorem, there is ...
mclsppslem 35063 The closure is closed unde...
mclspps 35064 The closure is closed unde...
problem1 35139 Practice problem 1. Clues...
problem2 35140 Practice problem 2. Clues...
problem3 35141 Practice problem 3. Clues...
problem4 35142 Practice problem 4. Clues...
problem5 35143 Practice problem 5. Clues...
quad3 35144 Variant of quadratic equat...
climuzcnv 35145 Utility lemma to convert b...
sinccvglem 35146 ` ( ( sin `` x ) / x ) ~~>...
sinccvg 35147 ` ( ( sin `` x ) / x ) ~~>...
circum 35148 The circumference of a cir...
elfzm12 35149 Membership in a curtailed ...
nn0seqcvg 35150 A strictly-decreasing nonn...
lediv2aALT 35151 Division of both sides of ...
abs2sqlei 35152 The absolute values of two...
abs2sqlti 35153 The absolute values of two...
abs2sqle 35154 The absolute values of two...
abs2sqlt 35155 The absolute values of two...
abs2difi 35156 Difference of absolute val...
abs2difabsi 35157 Absolute value of differen...
currybi 35158 Biconditional version of C...
axextprim 35165 ~ ax-ext without distinct ...
axrepprim 35166 ~ ax-rep without distinct ...
axunprim 35167 ~ ax-un without distinct v...
axpowprim 35168 ~ ax-pow without distinct ...
axregprim 35169 ~ ax-reg without distinct ...
axinfprim 35170 ~ ax-inf without distinct ...
axacprim 35171 ~ ax-ac without distinct v...
untelirr 35172 We call a class "untanged"...
untuni 35173 The union of a class is un...
untsucf 35174 If a class is untangled, t...
unt0 35175 The null set is untangled....
untint 35176 If there is an untangled e...
efrunt 35177 If ` A ` is well-founded b...
untangtr 35178 A transitive class is unta...
3jaodd 35179 Double deduction form of ~...
3orit 35180 Closed form of ~ 3ori . (...
biimpexp 35181 A biconditional in the ant...
nepss 35182 Two classes are unequal if...
3ccased 35183 Triple disjunction form of...
dfso3 35184 Expansion of the definitio...
brtpid1 35185 A binary relation involvin...
brtpid2 35186 A binary relation involvin...
brtpid3 35187 A binary relation involvin...
iota5f 35188 A method for computing iot...
jath 35189 Closed form of ~ ja . Pro...
xpab 35190 Cartesian product of two c...
nnuni 35191 The union of a finite ordi...
sqdivzi 35192 Distribution of square ove...
supfz 35193 The supremum of a finite s...
inffz 35194 The infimum of a finite se...
fz0n 35195 The sequence ` ( 0 ... ( N...
shftvalg 35196 Value of a sequence shifte...
divcnvlin 35197 Limit of the ratio of two ...
climlec3 35198 Comparison of a constant t...
logi 35199 Calculate the logarithm of...
iexpire 35200 ` _i ` raised to itself is...
bcneg1 35201 The binomial coefficent ov...
bcm1nt 35202 The proportion of one bion...
bcprod 35203 A product identity for bin...
bccolsum 35204 A column-sum rule for bino...
iprodefisumlem 35205 Lemma for ~ iprodefisum . ...
iprodefisum 35206 Applying the exponential f...
iprodgam 35207 An infinite product versio...
faclimlem1 35208 Lemma for ~ faclim . Clos...
faclimlem2 35209 Lemma for ~ faclim . Show...
faclimlem3 35210 Lemma for ~ faclim . Alge...
faclim 35211 An infinite product expres...
iprodfac 35212 An infinite product expres...
faclim2 35213 Another factorial limit du...
gcd32 35214 Swap the second and third ...
gcdabsorb 35215 Absorption law for gcd. (...
dftr6 35216 A potential definition of ...
coep 35217 Composition with the membe...
coepr 35218 Composition with the conve...
dffr5 35219 A quantifier-free definiti...
dfso2 35220 Quantifier-free definition...
br8 35221 Substitution for an eight-...
br6 35222 Substitution for a six-pla...
br4 35223 Substitution for a four-pl...
cnvco1 35224 Another distributive law o...
cnvco2 35225 Another distributive law o...
eldm3 35226 Quantifier-free definition...
elrn3 35227 Quantifier-free definition...
pocnv 35228 The converse of a partial ...
socnv 35229 The converse of a strict o...
sotrd 35230 Transitivity law for stric...
elintfv 35231 Membership in an intersect...
funpsstri 35232 A condition for subset tri...
fundmpss 35233 If a class ` F ` is a prop...
funsseq 35234 Given two functions with e...
fununiq 35235 The uniqueness condition o...
funbreq 35236 An equality condition for ...
br1steq 35237 Uniqueness condition for t...
br2ndeq 35238 Uniqueness condition for t...
dfdm5 35239 Definition of domain in te...
dfrn5 35240 Definition of range in ter...
opelco3 35241 Alternate way of saying th...
elima4 35242 Quantifier-free expression...
fv1stcnv 35243 The value of the converse ...
fv2ndcnv 35244 The value of the converse ...
setinds 35245 Principle of set induction...
setinds2f 35246 ` _E ` induction schema, u...
setinds2 35247 ` _E ` induction schema, u...
elpotr 35248 A class of transitive sets...
dford5reg 35249 Given ~ ax-reg , an ordina...
dfon2lem1 35250 Lemma for ~ dfon2 . (Cont...
dfon2lem2 35251 Lemma for ~ dfon2 . (Cont...
dfon2lem3 35252 Lemma for ~ dfon2 . All s...
dfon2lem4 35253 Lemma for ~ dfon2 . If tw...
dfon2lem5 35254 Lemma for ~ dfon2 . Two s...
dfon2lem6 35255 Lemma for ~ dfon2 . A tra...
dfon2lem7 35256 Lemma for ~ dfon2 . All e...
dfon2lem8 35257 Lemma for ~ dfon2 . The i...
dfon2lem9 35258 Lemma for ~ dfon2 . A cla...
dfon2 35259 ` On ` consists of all set...
rdgprc0 35260 The value of the recursive...
rdgprc 35261 The value of the recursive...
dfrdg2 35262 Alternate definition of th...
dfrdg3 35263 Generalization of ~ dfrdg2...
axextdfeq 35264 A version of ~ ax-ext for ...
ax8dfeq 35265 A version of ~ ax-8 for us...
axextdist 35266 ~ ax-ext with distinctors ...
axextbdist 35267 ~ axextb with distinctors ...
19.12b 35268 Version of ~ 19.12vv with ...
exnel 35269 There is always a set not ...
distel 35270 Distinctors in terms of me...
axextndbi 35271 ~ axextnd as a bicondition...
hbntg 35272 A more general form of ~ h...
hbimtg 35273 A more general and closed ...
hbaltg 35274 A more general and closed ...
hbng 35275 A more general form of ~ h...
hbimg 35276 A more general form of ~ h...
wsuceq123 35281 Equality theorem for well-...
wsuceq1 35282 Equality theorem for well-...
wsuceq2 35283 Equality theorem for well-...
wsuceq3 35284 Equality theorem for well-...
nfwsuc 35285 Bound-variable hypothesis ...
wlimeq12 35286 Equality theorem for the l...
wlimeq1 35287 Equality theorem for the l...
wlimeq2 35288 Equality theorem for the l...
nfwlim 35289 Bound-variable hypothesis ...
elwlim 35290 Membership in the limit cl...
wzel 35291 The zero of a well-founded...
wsuclem 35292 Lemma for the supremum pro...
wsucex 35293 Existence theorem for well...
wsuccl 35294 If ` X ` is a set with an ...
wsuclb 35295 A well-founded successor i...
wlimss 35296 The class of limit points ...
txpss3v 35345 A tail Cartesian product i...
txprel 35346 A tail Cartesian product i...
brtxp 35347 Characterize a ternary rel...
brtxp2 35348 The binary relation over a...
dfpprod2 35349 Expanded definition of par...
pprodcnveq 35350 A converse law for paralle...
pprodss4v 35351 The parallel product is a ...
brpprod 35352 Characterize a quaternary ...
brpprod3a 35353 Condition for parallel pro...
brpprod3b 35354 Condition for parallel pro...
relsset 35355 The subset class is a bina...
brsset 35356 For sets, the ` SSet ` bin...
idsset 35357 ` _I ` is equal to the int...
eltrans 35358 Membership in the class of...
dfon3 35359 A quantifier-free definiti...
dfon4 35360 Another quantifier-free de...
brtxpsd 35361 Expansion of a common form...
brtxpsd2 35362 Another common abbreviatio...
brtxpsd3 35363 A third common abbreviatio...
relbigcup 35364 The ` Bigcup ` relationshi...
brbigcup 35365 Binary relation over ` Big...
dfbigcup2 35366 ` Bigcup ` using maps-to n...
fobigcup 35367 ` Bigcup ` maps the univer...
fnbigcup 35368 ` Bigcup ` is a function o...
fvbigcup 35369 For sets, ` Bigcup ` yield...
elfix 35370 Membership in the fixpoint...
elfix2 35371 Alternative membership in ...
dffix2 35372 The fixpoints of a class i...
fixssdm 35373 The fixpoints of a class a...
fixssrn 35374 The fixpoints of a class a...
fixcnv 35375 The fixpoints of a class a...
fixun 35376 The fixpoint operator dist...
ellimits 35377 Membership in the class of...
limitssson 35378 The class of all limit ord...
dfom5b 35379 A quantifier-free definiti...
sscoid 35380 A condition for subset and...
dffun10 35381 Another potential definiti...
elfuns 35382 Membership in the class of...
elfunsg 35383 Closed form of ~ elfuns . ...
brsingle 35384 The binary relation form o...
elsingles 35385 Membership in the class of...
fnsingle 35386 The singleton relationship...
fvsingle 35387 The value of the singleton...
dfsingles2 35388 Alternate definition of th...
snelsingles 35389 A singleton is a member of...
dfiota3 35390 A definition of iota using...
dffv5 35391 Another quantifier-free de...
unisnif 35392 Express union of singleton...
brimage 35393 Binary relation form of th...
brimageg 35394 Closed form of ~ brimage ....
funimage 35395 ` Image A ` is a function....
fnimage 35396 ` Image R ` is a function ...
imageval 35397 The image functor in maps-...
fvimage 35398 Value of the image functor...
brcart 35399 Binary relation form of th...
brdomain 35400 Binary relation form of th...
brrange 35401 Binary relation form of th...
brdomaing 35402 Closed form of ~ brdomain ...
brrangeg 35403 Closed form of ~ brrange ....
brimg 35404 Binary relation form of th...
brapply 35405 Binary relation form of th...
brcup 35406 Binary relation form of th...
brcap 35407 Binary relation form of th...
brsuccf 35408 Binary relation form of th...
funpartlem 35409 Lemma for ~ funpartfun . ...
funpartfun 35410 The functional part of ` F...
funpartss 35411 The functional part of ` F...
funpartfv 35412 The function value of the ...
fullfunfnv 35413 The full functional part o...
fullfunfv 35414 The function value of the ...
brfullfun 35415 A binary relation form con...
brrestrict 35416 Binary relation form of th...
dfrecs2 35417 A quantifier-free definiti...
dfrdg4 35418 A quantifier-free definiti...
dfint3 35419 Quantifier-free definition...
imagesset 35420 The Image functor applied ...
brub 35421 Binary relation form of th...
brlb 35422 Binary relation form of th...
altopex 35427 Alternative ordered pairs ...
altopthsn 35428 Two alternate ordered pair...
altopeq12 35429 Equality for alternate ord...
altopeq1 35430 Equality for alternate ord...
altopeq2 35431 Equality for alternate ord...
altopth1 35432 Equality of the first memb...
altopth2 35433 Equality of the second mem...
altopthg 35434 Alternate ordered pair the...
altopthbg 35435 Alternate ordered pair the...
altopth 35436 The alternate ordered pair...
altopthb 35437 Alternate ordered pair the...
altopthc 35438 Alternate ordered pair the...
altopthd 35439 Alternate ordered pair the...
altxpeq1 35440 Equality for alternate Car...
altxpeq2 35441 Equality for alternate Car...
elaltxp 35442 Membership in alternate Ca...
altopelaltxp 35443 Alternate ordered pair mem...
altxpsspw 35444 An inclusion rule for alte...
altxpexg 35445 The alternate Cartesian pr...
rankaltopb 35446 Compute the rank of an alt...
nfaltop 35447 Bound-variable hypothesis ...
sbcaltop 35448 Distribution of class subs...
cgrrflx2d 35451 Deduction form of ~ axcgrr...
cgrtr4d 35452 Deduction form of ~ axcgrt...
cgrtr4and 35453 Deduction form of ~ axcgrt...
cgrrflx 35454 Reflexivity law for congru...
cgrrflxd 35455 Deduction form of ~ cgrrfl...
cgrcomim 35456 Congruence commutes on the...
cgrcom 35457 Congruence commutes betwee...
cgrcomand 35458 Deduction form of ~ cgrcom...
cgrtr 35459 Transitivity law for congr...
cgrtrand 35460 Deduction form of ~ cgrtr ...
cgrtr3 35461 Transitivity law for congr...
cgrtr3and 35462 Deduction form of ~ cgrtr3...
cgrcoml 35463 Congruence commutes on the...
cgrcomr 35464 Congruence commutes on the...
cgrcomlr 35465 Congruence commutes on bot...
cgrcomland 35466 Deduction form of ~ cgrcom...
cgrcomrand 35467 Deduction form of ~ cgrcom...
cgrcomlrand 35468 Deduction form of ~ cgrcom...
cgrtriv 35469 Degenerate segments are co...
cgrid2 35470 Identity law for congruenc...
cgrdegen 35471 Two congruent segments are...
brofs 35472 Binary relation form of th...
5segofs 35473 Rephrase ~ ax5seg using th...
ofscom 35474 The outer five segment pre...
cgrextend 35475 Link congruence over a pai...
cgrextendand 35476 Deduction form of ~ cgrext...
segconeq 35477 Two points that satisfy th...
segconeu 35478 Existential uniqueness ver...
btwntriv2 35479 Betweenness always holds f...
btwncomim 35480 Betweenness commutes. Imp...
btwncom 35481 Betweenness commutes. (Co...
btwncomand 35482 Deduction form of ~ btwnco...
btwntriv1 35483 Betweenness always holds f...
btwnswapid 35484 If you can swap the first ...
btwnswapid2 35485 If you can swap arguments ...
btwnintr 35486 Inner transitivity law for...
btwnexch3 35487 Exchange the first endpoin...
btwnexch3and 35488 Deduction form of ~ btwnex...
btwnouttr2 35489 Outer transitivity law for...
btwnexch2 35490 Exchange the outer point o...
btwnouttr 35491 Outer transitivity law for...
btwnexch 35492 Outer transitivity law for...
btwnexchand 35493 Deduction form of ~ btwnex...
btwndiff 35494 There is always a ` c ` di...
trisegint 35495 A line segment between two...
funtransport 35498 The ` TransportTo ` relati...
fvtransport 35499 Calculate the value of the...
transportcl 35500 Closure law for segment tr...
transportprops 35501 Calculate the defining pro...
brifs 35510 Binary relation form of th...
ifscgr 35511 Inner five segment congrue...
cgrsub 35512 Removing identical parts f...
brcgr3 35513 Binary relation form of th...
cgr3permute3 35514 Permutation law for three-...
cgr3permute1 35515 Permutation law for three-...
cgr3permute2 35516 Permutation law for three-...
cgr3permute4 35517 Permutation law for three-...
cgr3permute5 35518 Permutation law for three-...
cgr3tr4 35519 Transitivity law for three...
cgr3com 35520 Commutativity law for thre...
cgr3rflx 35521 Identity law for three-pla...
cgrxfr 35522 A line segment can be divi...
btwnxfr 35523 A condition for extending ...
colinrel 35524 Colinearity is a relations...
brcolinear2 35525 Alternate colinearity bina...
brcolinear 35526 The binary relation form o...
colinearex 35527 The colinear predicate exi...
colineardim1 35528 If ` A ` is colinear with ...
colinearperm1 35529 Permutation law for coline...
colinearperm3 35530 Permutation law for coline...
colinearperm2 35531 Permutation law for coline...
colinearperm4 35532 Permutation law for coline...
colinearperm5 35533 Permutation law for coline...
colineartriv1 35534 Trivial case of colinearit...
colineartriv2 35535 Trivial case of colinearit...
btwncolinear1 35536 Betweenness implies coline...
btwncolinear2 35537 Betweenness implies coline...
btwncolinear3 35538 Betweenness implies coline...
btwncolinear4 35539 Betweenness implies coline...
btwncolinear5 35540 Betweenness implies coline...
btwncolinear6 35541 Betweenness implies coline...
colinearxfr 35542 Transfer law for colineari...
lineext 35543 Extend a line with a missi...
brofs2 35544 Change some conditions for...
brifs2 35545 Change some conditions for...
brfs 35546 Binary relation form of th...
fscgr 35547 Congruence law for the gen...
linecgr 35548 Congruence rule for lines....
linecgrand 35549 Deduction form of ~ linecg...
lineid 35550 Identity law for points on...
idinside 35551 Law for finding a point in...
endofsegid 35552 If ` A ` , ` B ` , and ` C...
endofsegidand 35553 Deduction form of ~ endofs...
btwnconn1lem1 35554 Lemma for ~ btwnconn1 . T...
btwnconn1lem2 35555 Lemma for ~ btwnconn1 . N...
btwnconn1lem3 35556 Lemma for ~ btwnconn1 . E...
btwnconn1lem4 35557 Lemma for ~ btwnconn1 . A...
btwnconn1lem5 35558 Lemma for ~ btwnconn1 . N...
btwnconn1lem6 35559 Lemma for ~ btwnconn1 . N...
btwnconn1lem7 35560 Lemma for ~ btwnconn1 . U...
btwnconn1lem8 35561 Lemma for ~ btwnconn1 . N...
btwnconn1lem9 35562 Lemma for ~ btwnconn1 . N...
btwnconn1lem10 35563 Lemma for ~ btwnconn1 . N...
btwnconn1lem11 35564 Lemma for ~ btwnconn1 . N...
btwnconn1lem12 35565 Lemma for ~ btwnconn1 . U...
btwnconn1lem13 35566 Lemma for ~ btwnconn1 . B...
btwnconn1lem14 35567 Lemma for ~ btwnconn1 . F...
btwnconn1 35568 Connectitivy law for betwe...
btwnconn2 35569 Another connectivity law f...
btwnconn3 35570 Inner connectivity law for...
midofsegid 35571 If two points fall in the ...
segcon2 35572 Generalization of ~ axsegc...
brsegle 35575 Binary relation form of th...
brsegle2 35576 Alternate characterization...
seglecgr12im 35577 Substitution law for segme...
seglecgr12 35578 Substitution law for segme...
seglerflx 35579 Segment comparison is refl...
seglemin 35580 Any segment is at least as...
segletr 35581 Segment less than is trans...
segleantisym 35582 Antisymmetry law for segme...
seglelin 35583 Linearity law for segment ...
btwnsegle 35584 If ` B ` falls between ` A...
colinbtwnle 35585 Given three colinear point...
broutsideof 35588 Binary relation form of ` ...
broutsideof2 35589 Alternate form of ` Outsid...
outsidene1 35590 Outsideness implies inequa...
outsidene2 35591 Outsideness implies inequa...
btwnoutside 35592 A principle linking outsid...
broutsideof3 35593 Characterization of outsid...
outsideofrflx 35594 Reflexivity of outsideness...
outsideofcom 35595 Commutativity law for outs...
outsideoftr 35596 Transitivity law for outsi...
outsideofeq 35597 Uniqueness law for ` Outsi...
outsideofeu 35598 Given a nondegenerate ray,...
outsidele 35599 Relate ` OutsideOf ` to ` ...
outsideofcol 35600 Outside of implies colinea...
funray 35607 Show that the ` Ray ` rela...
fvray 35608 Calculate the value of the...
funline 35609 Show that the ` Line ` rel...
linedegen 35610 When ` Line ` is applied w...
fvline 35611 Calculate the value of the...
liness 35612 A line is a subset of the ...
fvline2 35613 Alternate definition of a ...
lineunray 35614 A line is composed of a po...
lineelsb2 35615 If ` S ` lies on ` P Q ` ,...
linerflx1 35616 Reflexivity law for line m...
linecom 35617 Commutativity law for line...
linerflx2 35618 Reflexivity law for line m...
ellines 35619 Membership in the set of a...
linethru 35620 If ` A ` is a line contain...
hilbert1.1 35621 There is a line through an...
hilbert1.2 35622 There is at most one line ...
linethrueu 35623 There is a unique line goi...
lineintmo 35624 Two distinct lines interse...
fwddifval 35629 Calculate the value of the...
fwddifnval 35630 The value of the forward d...
fwddifn0 35631 The value of the n-iterate...
fwddifnp1 35632 The value of the n-iterate...
rankung 35633 The rank of the union of t...
ranksng 35634 The rank of a singleton. ...
rankelg 35635 The membership relation is...
rankpwg 35636 The rank of a power set. ...
rank0 35637 The rank of the empty set ...
rankeq1o 35638 The only set with rank ` 1...
elhf 35641 Membership in the heredita...
elhf2 35642 Alternate form of membersh...
elhf2g 35643 Hereditarily finiteness vi...
0hf 35644 The empty set is a heredit...
hfun 35645 The union of two HF sets i...
hfsn 35646 The singleton of an HF set...
hfadj 35647 Adjoining one HF element t...
hfelhf 35648 Any member of an HF set is...
hftr 35649 The class of all hereditar...
hfext 35650 Extensionality for HF sets...
hfuni 35651 The union of an HF set is ...
hfpw 35652 The power class of an HF s...
hfninf 35653 ` _om ` is not hereditaril...
mpomulnzcnf 35654 Multiplication maps nonzer...
mpomulex 35655 The multiplication operati...
gg-cnfldex 35656 The field of complex numbe...
gg-taylthlem2 35657 Lemma for ~ taylth . (Con...
mpoaddf 35658 Addition is an operation o...
mpoaddex 35659 The addition operation is ...
gg-dfcnfld 35660 Alternative definition of ...
gg-cnfldstr 35661 The field of complex numbe...
gg-cnfldbas 35662 The base set of the field ...
mpocnfldadd 35663 The addition operation of ...
mpocnfldmul 35664 The multiplication operati...
gg-cnfldcj 35665 The conjugation operation ...
gg-cnfldtset 35666 The topology component of ...
gg-cnfldle 35667 The ordering of the field ...
gg-cnfldds 35668 The metric of the field of...
gg-cnfldunif 35669 The uniform structure comp...
gg-cnfldfun 35670 The field of complex numbe...
gg-cnfldfunALT 35671 The field of complex numbe...
gg-cffldtocusgr 35672 The field of complex numbe...
gg-cncrng 35673 The complex numbers form a...
gg-cnfld1 35674 One is the unity element o...
a1i14 35675 Add two antecedents to a w...
a1i24 35676 Add two antecedents to a w...
exp5d 35677 An exportation inference. ...
exp5g 35678 An exportation inference. ...
exp5k 35679 An exportation inference. ...
exp56 35680 An exportation inference. ...
exp58 35681 An exportation inference. ...
exp510 35682 An exportation inference. ...
exp511 35683 An exportation inference. ...
exp512 35684 An exportation inference. ...
3com12d 35685 Commutation in consequent....
imp5p 35686 A triple importation infer...
imp5q 35687 A triple importation infer...
ecase13d 35688 Deduction for elimination ...
subtr 35689 Transitivity of implicit s...
subtr2 35690 Transitivity of implicit s...
trer 35691 A relation intersected wit...
elicc3 35692 An equivalent membership c...
finminlem 35693 A useful lemma about finit...
gtinf 35694 Any number greater than an...
opnrebl 35695 A set is open in the stand...
opnrebl2 35696 A set is open in the stand...
nn0prpwlem 35697 Lemma for ~ nn0prpw . Use...
nn0prpw 35698 Two nonnegative integers a...
topbnd 35699 Two equivalent expressions...
opnbnd 35700 A set is open iff it is di...
cldbnd 35701 A set is closed iff it con...
ntruni 35702 A union of interiors is a ...
clsun 35703 A pairwise union of closur...
clsint2 35704 The closure of an intersec...
opnregcld 35705 A set is regularly closed ...
cldregopn 35706 A set if regularly open if...
neiin 35707 Two neighborhoods intersec...
hmeoclda 35708 Homeomorphisms preserve cl...
hmeocldb 35709 Homeomorphisms preserve cl...
ivthALT 35710 An alternate proof of the ...
fnerel 35713 Fineness is a relation. (...
isfne 35714 The predicate " ` B ` is f...
isfne4 35715 The predicate " ` B ` is f...
isfne4b 35716 A condition for a topology...
isfne2 35717 The predicate " ` B ` is f...
isfne3 35718 The predicate " ` B ` is f...
fnebas 35719 A finer cover covers the s...
fnetg 35720 A finer cover generates a ...
fnessex 35721 If ` B ` is finer than ` A...
fneuni 35722 If ` B ` is finer than ` A...
fneint 35723 If a cover is finer than a...
fness 35724 A cover is finer than its ...
fneref 35725 Reflexivity of the finenes...
fnetr 35726 Transitivity of the finene...
fneval 35727 Two covers are finer than ...
fneer 35728 Fineness intersected with ...
topfne 35729 Fineness for covers corres...
topfneec 35730 A cover is equivalent to a...
topfneec2 35731 A topology is precisely id...
fnessref 35732 A cover is finer iff it ha...
refssfne 35733 A cover is a refinement if...
neibastop1 35734 A collection of neighborho...
neibastop2lem 35735 Lemma for ~ neibastop2 . ...
neibastop2 35736 In the topology generated ...
neibastop3 35737 The topology generated by ...
topmtcl 35738 The meet of a collection o...
topmeet 35739 Two equivalent formulation...
topjoin 35740 Two equivalent formulation...
fnemeet1 35741 The meet of a collection o...
fnemeet2 35742 The meet of equivalence cl...
fnejoin1 35743 Join of equivalence classe...
fnejoin2 35744 Join of equivalence classe...
fgmin 35745 Minimality property of a g...
neifg 35746 The neighborhood filter of...
tailfval 35747 The tail function for a di...
tailval 35748 The tail of an element in ...
eltail 35749 An element of a tail. (Co...
tailf 35750 The tail function of a dir...
tailini 35751 A tail contains its initia...
tailfb 35752 The collection of tails of...
filnetlem1 35753 Lemma for ~ filnet . Chan...
filnetlem2 35754 Lemma for ~ filnet . The ...
filnetlem3 35755 Lemma for ~ filnet . (Con...
filnetlem4 35756 Lemma for ~ filnet . (Con...
filnet 35757 A filter has the same conv...
tb-ax1 35758 The first of three axioms ...
tb-ax2 35759 The second of three axioms...
tb-ax3 35760 The third of three axioms ...
tbsyl 35761 The weak syllogism from Ta...
re1ax2lem 35762 Lemma for ~ re1ax2 . (Con...
re1ax2 35763 ~ ax-2 rederived from the ...
naim1 35764 Constructor theorem for ` ...
naim2 35765 Constructor theorem for ` ...
naim1i 35766 Constructor rule for ` -/\...
naim2i 35767 Constructor rule for ` -/\...
naim12i 35768 Constructor rule for ` -/\...
nabi1i 35769 Constructor rule for ` -/\...
nabi2i 35770 Constructor rule for ` -/\...
nabi12i 35771 Constructor rule for ` -/\...
df3nandALT1 35774 The double nand expressed ...
df3nandALT2 35775 The double nand expressed ...
andnand1 35776 Double and in terms of dou...
imnand2 35777 An ` -> ` nand relation. ...
nalfal 35778 Not all sets hold ` F. ` a...
nexntru 35779 There does not exist a set...
nexfal 35780 There does not exist a set...
neufal 35781 There does not exist exact...
neutru 35782 There does not exist exact...
nmotru 35783 There does not exist at mo...
mofal 35784 There exist at most one se...
nrmo 35785 "At most one" restricted e...
meran1 35786 A single axiom for proposi...
meran2 35787 A single axiom for proposi...
meran3 35788 A single axiom for proposi...
waj-ax 35789 A single axiom for proposi...
lukshef-ax2 35790 A single axiom for proposi...
arg-ax 35791 A single axiom for proposi...
negsym1 35792 In the paper "On Variable ...
imsym1 35793 A symmetry with ` -> ` . ...
bisym1 35794 A symmetry with ` <-> ` . ...
consym1 35795 A symmetry with ` /\ ` . ...
dissym1 35796 A symmetry with ` \/ ` . ...
nandsym1 35797 A symmetry with ` -/\ ` . ...
unisym1 35798 A symmetry with ` A. ` . ...
exisym1 35799 A symmetry with ` E. ` . ...
unqsym1 35800 A symmetry with ` E! ` . ...
amosym1 35801 A symmetry with ` E* ` . ...
subsym1 35802 A symmetry with ` [ x / y ...
ontopbas 35803 An ordinal number is a top...
onsstopbas 35804 The class of ordinal numbe...
onpsstopbas 35805 The class of ordinal numbe...
ontgval 35806 The topology generated fro...
ontgsucval 35807 The topology generated fro...
onsuctop 35808 A successor ordinal number...
onsuctopon 35809 One of the topologies on a...
ordtoplem 35810 Membership of the class of...
ordtop 35811 An ordinal is a topology i...
onsucconni 35812 A successor ordinal number...
onsucconn 35813 A successor ordinal number...
ordtopconn 35814 An ordinal topology is con...
onintopssconn 35815 An ordinal topology is con...
onsuct0 35816 A successor ordinal number...
ordtopt0 35817 An ordinal topology is T_0...
onsucsuccmpi 35818 The successor of a success...
onsucsuccmp 35819 The successor of a success...
limsucncmpi 35820 The successor of a limit o...
limsucncmp 35821 The successor of a limit o...
ordcmp 35822 An ordinal topology is com...
ssoninhaus 35823 The ordinal topologies ` 1...
onint1 35824 The ordinal T_1 spaces are...
oninhaus 35825 The ordinal Hausdorff spac...
fveleq 35826 Please add description her...
findfvcl 35827 Please add description her...
findreccl 35828 Please add description her...
findabrcl 35829 Please add description her...
nnssi2 35830 Convert a theorem for real...
nnssi3 35831 Convert a theorem for real...
nndivsub 35832 Please add description her...
nndivlub 35833 A factor of a positive int...
ee7.2aOLD 35836 Lemma for Euclid's Element...
dnival 35837 Value of the "distance to ...
dnicld1 35838 Closure theorem for the "d...
dnicld2 35839 Closure theorem for the "d...
dnif 35840 The "distance to nearest i...
dnizeq0 35841 The distance to nearest in...
dnizphlfeqhlf 35842 The distance to nearest in...
rddif2 35843 Variant of ~ rddif . (Con...
dnibndlem1 35844 Lemma for ~ dnibnd . (Con...
dnibndlem2 35845 Lemma for ~ dnibnd . (Con...
dnibndlem3 35846 Lemma for ~ dnibnd . (Con...
dnibndlem4 35847 Lemma for ~ dnibnd . (Con...
dnibndlem5 35848 Lemma for ~ dnibnd . (Con...
dnibndlem6 35849 Lemma for ~ dnibnd . (Con...
dnibndlem7 35850 Lemma for ~ dnibnd . (Con...
dnibndlem8 35851 Lemma for ~ dnibnd . (Con...
dnibndlem9 35852 Lemma for ~ dnibnd . (Con...
dnibndlem10 35853 Lemma for ~ dnibnd . (Con...
dnibndlem11 35854 Lemma for ~ dnibnd . (Con...
dnibndlem12 35855 Lemma for ~ dnibnd . (Con...
dnibndlem13 35856 Lemma for ~ dnibnd . (Con...
dnibnd 35857 The "distance to nearest i...
dnicn 35858 The "distance to nearest i...
knoppcnlem1 35859 Lemma for ~ knoppcn . (Co...
knoppcnlem2 35860 Lemma for ~ knoppcn . (Co...
knoppcnlem3 35861 Lemma for ~ knoppcn . (Co...
knoppcnlem4 35862 Lemma for ~ knoppcn . (Co...
knoppcnlem5 35863 Lemma for ~ knoppcn . (Co...
knoppcnlem6 35864 Lemma for ~ knoppcn . (Co...
knoppcnlem7 35865 Lemma for ~ knoppcn . (Co...
knoppcnlem8 35866 Lemma for ~ knoppcn . (Co...
knoppcnlem9 35867 Lemma for ~ knoppcn . (Co...
knoppcnlem10 35868 Lemma for ~ knoppcn . (Co...
knoppcnlem11 35869 Lemma for ~ knoppcn . (Co...
knoppcn 35870 The continuous nowhere dif...
knoppcld 35871 Closure theorem for Knopp'...
unblimceq0lem 35872 Lemma for ~ unblimceq0 . ...
unblimceq0 35873 If ` F ` is unbounded near...
unbdqndv1 35874 If the difference quotient...
unbdqndv2lem1 35875 Lemma for ~ unbdqndv2 . (...
unbdqndv2lem2 35876 Lemma for ~ unbdqndv2 . (...
unbdqndv2 35877 Variant of ~ unbdqndv1 wit...
knoppndvlem1 35878 Lemma for ~ knoppndv . (C...
knoppndvlem2 35879 Lemma for ~ knoppndv . (C...
knoppndvlem3 35880 Lemma for ~ knoppndv . (C...
knoppndvlem4 35881 Lemma for ~ knoppndv . (C...
knoppndvlem5 35882 Lemma for ~ knoppndv . (C...
knoppndvlem6 35883 Lemma for ~ knoppndv . (C...
knoppndvlem7 35884 Lemma for ~ knoppndv . (C...
knoppndvlem8 35885 Lemma for ~ knoppndv . (C...
knoppndvlem9 35886 Lemma for ~ knoppndv . (C...
knoppndvlem10 35887 Lemma for ~ knoppndv . (C...
knoppndvlem11 35888 Lemma for ~ knoppndv . (C...
knoppndvlem12 35889 Lemma for ~ knoppndv . (C...
knoppndvlem13 35890 Lemma for ~ knoppndv . (C...
knoppndvlem14 35891 Lemma for ~ knoppndv . (C...
knoppndvlem15 35892 Lemma for ~ knoppndv . (C...
knoppndvlem16 35893 Lemma for ~ knoppndv . (C...
knoppndvlem17 35894 Lemma for ~ knoppndv . (C...
knoppndvlem18 35895 Lemma for ~ knoppndv . (C...
knoppndvlem19 35896 Lemma for ~ knoppndv . (C...
knoppndvlem20 35897 Lemma for ~ knoppndv . (C...
knoppndvlem21 35898 Lemma for ~ knoppndv . (C...
knoppndvlem22 35899 Lemma for ~ knoppndv . (C...
knoppndv 35900 The continuous nowhere dif...
knoppf 35901 Knopp's function is a func...
knoppcn2 35902 Variant of ~ knoppcn with ...
cnndvlem1 35903 Lemma for ~ cnndv . (Cont...
cnndvlem2 35904 Lemma for ~ cnndv . (Cont...
cnndv 35905 There exists a continuous ...
bj-mp2c 35906 A double modus ponens infe...
bj-mp2d 35907 A double modus ponens infe...
bj-0 35908 A syntactic theorem. See ...
bj-1 35909 In this proof, the use of ...
bj-a1k 35910 Weakening of ~ ax-1 . As ...
bj-poni 35911 Inference associated with ...
bj-nnclav 35912 When ` F. ` is substituted...
bj-nnclavi 35913 Inference associated with ...
bj-nnclavc 35914 Commuted form of ~ bj-nncl...
bj-nnclavci 35915 Inference associated with ...
bj-jarrii 35916 Inference associated with ...
bj-imim21 35917 The propositional function...
bj-imim21i 35918 Inference associated with ...
bj-peircestab 35919 Over minimal implicational...
bj-stabpeirce 35920 This minimal implicational...
bj-syl66ib 35921 A mixed syllogism inferenc...
bj-orim2 35922 Proof of ~ orim2 from the ...
bj-currypeirce 35923 Curry's axiom ~ curryax (a...
bj-peircecurry 35924 Peirce's axiom ~ peirce im...
bj-animbi 35925 Conjunction in terms of im...
bj-currypara 35926 Curry's paradox. Note tha...
bj-con2com 35927 A commuted form of the con...
bj-con2comi 35928 Inference associated with ...
bj-pm2.01i 35929 Inference associated with ...
bj-nimn 35930 If a formula is true, then...
bj-nimni 35931 Inference associated with ...
bj-peircei 35932 Inference associated with ...
bj-looinvi 35933 Inference associated with ...
bj-looinvii 35934 Inference associated with ...
bj-mt2bi 35935 Version of ~ mt2 where the...
bj-ntrufal 35936 The negation of a theorem ...
bj-fal 35937 Shortening of ~ fal using ...
bj-jaoi1 35938 Shortens ~ orfa2 (58>53), ...
bj-jaoi2 35939 Shortens ~ consensus (110>...
bj-dfbi4 35940 Alternate definition of th...
bj-dfbi5 35941 Alternate definition of th...
bj-dfbi6 35942 Alternate definition of th...
bj-bijust0ALT 35943 Alternate proof of ~ bijus...
bj-bijust00 35944 A self-implication does no...
bj-consensus 35945 Version of ~ consensus exp...
bj-consensusALT 35946 Alternate proof of ~ bj-co...
bj-df-ifc 35947 Candidate definition for t...
bj-dfif 35948 Alternate definition of th...
bj-ififc 35949 A biconditional connecting...
bj-imbi12 35950 Uncurried (imported) form ...
bj-biorfi 35951 This should be labeled "bi...
bj-falor 35952 Dual of ~ truan (which has...
bj-falor2 35953 Dual of ~ truan . (Contri...
bj-bibibi 35954 A property of the bicondit...
bj-imn3ani 35955 Duplication of ~ bnj1224 ....
bj-andnotim 35956 Two ways of expressing a c...
bj-bi3ant 35957 This used to be in the mai...
bj-bisym 35958 This used to be in the mai...
bj-bixor 35959 Equivalence of two ternary...
bj-axdd2 35960 This implication, proved u...
bj-axd2d 35961 This implication, proved u...
bj-axtd 35962 This implication, proved f...
bj-gl4 35963 In a normal modal logic, t...
bj-axc4 35964 Over minimal calculus, the...
prvlem1 35969 An elementary property of ...
prvlem2 35970 An elementary property of ...
bj-babygodel 35971 See the section header com...
bj-babylob 35972 See the section header com...
bj-godellob 35973 Proof of Gödel's theo...
bj-genr 35974 Generalization rule on the...
bj-genl 35975 Generalization rule on the...
bj-genan 35976 Generalization rule on a c...
bj-mpgs 35977 From a closed form theorem...
bj-2alim 35978 Closed form of ~ 2alimi . ...
bj-2exim 35979 Closed form of ~ 2eximi . ...
bj-alanim 35980 Closed form of ~ alanimi ....
bj-2albi 35981 Closed form of ~ 2albii . ...
bj-notalbii 35982 Equivalence of universal q...
bj-2exbi 35983 Closed form of ~ 2exbii . ...
bj-3exbi 35984 Closed form of ~ 3exbii . ...
bj-sylgt2 35985 Uncurried (imported) form ...
bj-alrimg 35986 The general form of the *a...
bj-alrimd 35987 A slightly more general ~ ...
bj-sylget 35988 Dual statement of ~ sylgt ...
bj-sylget2 35989 Uncurried (imported) form ...
bj-exlimg 35990 The general form of the *e...
bj-sylge 35991 Dual statement of ~ sylg (...
bj-exlimd 35992 A slightly more general ~ ...
bj-nfimexal 35993 A weak from of nonfreeness...
bj-alexim 35994 Closed form of ~ aleximi ....
bj-nexdh 35995 Closed form of ~ nexdh (ac...
bj-nexdh2 35996 Uncurried (imported) form ...
bj-hbxfrbi 35997 Closed form of ~ hbxfrbi ....
bj-hbyfrbi 35998 Version of ~ bj-hbxfrbi wi...
bj-exalim 35999 Distribute quantifiers ove...
bj-exalimi 36000 An inference for distribut...
bj-exalims 36001 Distributing quantifiers o...
bj-exalimsi 36002 An inference for distribut...
bj-ax12ig 36003 A lemma used to prove a we...
bj-ax12i 36004 A weakening of ~ bj-ax12ig...
bj-nfimt 36005 Closed form of ~ nfim and ...
bj-cbvalimt 36006 A lemma in closed form use...
bj-cbveximt 36007 A lemma in closed form use...
bj-eximALT 36008 Alternate proof of ~ exim ...
bj-aleximiALT 36009 Alternate proof of ~ alexi...
bj-eximcom 36010 A commuted form of ~ exim ...
bj-ax12wlem 36011 A lemma used to prove a we...
bj-cbvalim 36012 A lemma used to prove ~ bj...
bj-cbvexim 36013 A lemma used to prove ~ bj...
bj-cbvalimi 36014 An equality-free general i...
bj-cbveximi 36015 An equality-free general i...
bj-cbval 36016 Changing a bound variable ...
bj-cbvex 36017 Changing a bound variable ...
bj-ssbeq 36020 Substitution in an equalit...
bj-ssblem1 36021 A lemma for the definiens ...
bj-ssblem2 36022 An instance of ~ ax-11 pro...
bj-ax12v 36023 A weaker form of ~ ax-12 a...
bj-ax12 36024 Remove a DV condition from...
bj-ax12ssb 36025 Axiom ~ bj-ax12 expressed ...
bj-19.41al 36026 Special case of ~ 19.41 pr...
bj-equsexval 36027 Special case of ~ equsexv ...
bj-subst 36028 Proof of ~ sbalex from cor...
bj-ssbid2 36029 A special case of ~ sbequ2...
bj-ssbid2ALT 36030 Alternate proof of ~ bj-ss...
bj-ssbid1 36031 A special case of ~ sbequ1...
bj-ssbid1ALT 36032 Alternate proof of ~ bj-ss...
bj-ax6elem1 36033 Lemma for ~ bj-ax6e . (Co...
bj-ax6elem2 36034 Lemma for ~ bj-ax6e . (Co...
bj-ax6e 36035 Proof of ~ ax6e (hence ~ a...
bj-spimvwt 36036 Closed form of ~ spimvw . ...
bj-spnfw 36037 Theorem close to a closed ...
bj-cbvexiw 36038 Change bound variable. Th...
bj-cbvexivw 36039 Change bound variable. Th...
bj-modald 36040 A short form of the axiom ...
bj-denot 36041 A weakening of ~ ax-6 and ...
bj-eqs 36042 A lemma for substitutions,...
bj-cbvexw 36043 Change bound variable. Th...
bj-ax12w 36044 The general statement that...
bj-ax89 36045 A theorem which could be u...
bj-elequ12 36046 An identity law for the no...
bj-cleljusti 36047 One direction of ~ cleljus...
bj-alcomexcom 36048 Commutation of two existen...
bj-hbalt 36049 Closed form of ~ hbal . W...
axc11n11 36050 Proof of ~ axc11n from { ~...
axc11n11r 36051 Proof of ~ axc11n from { ~...
bj-axc16g16 36052 Proof of ~ axc16g from { ~...
bj-ax12v3 36053 A weak version of ~ ax-12 ...
bj-ax12v3ALT 36054 Alternate proof of ~ bj-ax...
bj-sb 36055 A weak variant of ~ sbid2 ...
bj-modalbe 36056 The predicate-calculus ver...
bj-spst 36057 Closed form of ~ sps . On...
bj-19.21bit 36058 Closed form of ~ 19.21bi ....
bj-19.23bit 36059 Closed form of ~ 19.23bi ....
bj-nexrt 36060 Closed form of ~ nexr . C...
bj-alrim 36061 Closed form of ~ alrimi . ...
bj-alrim2 36062 Uncurried (imported) form ...
bj-nfdt0 36063 A theorem close to a close...
bj-nfdt 36064 Closed form of ~ nf5d and ...
bj-nexdt 36065 Closed form of ~ nexd . (...
bj-nexdvt 36066 Closed form of ~ nexdv . ...
bj-alexbiex 36067 Adding a second quantifier...
bj-exexbiex 36068 Adding a second quantifier...
bj-alalbial 36069 Adding a second quantifier...
bj-exalbial 36070 Adding a second quantifier...
bj-19.9htbi 36071 Strengthening ~ 19.9ht by ...
bj-hbntbi 36072 Strengthening ~ hbnt by re...
bj-biexal1 36073 A general FOL biconditiona...
bj-biexal2 36074 When ` ph ` is substituted...
bj-biexal3 36075 When ` ph ` is substituted...
bj-bialal 36076 When ` ph ` is substituted...
bj-biexex 36077 When ` ph ` is substituted...
bj-hbext 36078 Closed form of ~ hbex . (...
bj-nfalt 36079 Closed form of ~ nfal . (...
bj-nfext 36080 Closed form of ~ nfex . (...
bj-eeanvw 36081 Version of ~ exdistrv with...
bj-modal4 36082 First-order logic form of ...
bj-modal4e 36083 First-order logic form of ...
bj-modalb 36084 A short form of the axiom ...
bj-wnf1 36085 When ` ph ` is substituted...
bj-wnf2 36086 When ` ph ` is substituted...
bj-wnfanf 36087 When ` ph ` is substituted...
bj-wnfenf 36088 When ` ph ` is substituted...
bj-substax12 36089 Equivalent form of the axi...
bj-substw 36090 Weak form of the LHS of ~ ...
bj-nnfbi 36093 If two formulas are equiva...
bj-nnfbd 36094 If two formulas are equiva...
bj-nnfbii 36095 If two formulas are equiva...
bj-nnfa 36096 Nonfreeness implies the eq...
bj-nnfad 36097 Nonfreeness implies the eq...
bj-nnfai 36098 Nonfreeness implies the eq...
bj-nnfe 36099 Nonfreeness implies the eq...
bj-nnfed 36100 Nonfreeness implies the eq...
bj-nnfei 36101 Nonfreeness implies the eq...
bj-nnfea 36102 Nonfreeness implies the eq...
bj-nnfead 36103 Nonfreeness implies the eq...
bj-nnfeai 36104 Nonfreeness implies the eq...
bj-dfnnf2 36105 Alternate definition of ~ ...
bj-nnfnfTEMP 36106 New nonfreeness implies ol...
bj-wnfnf 36107 When ` ph ` is substituted...
bj-nnfnt 36108 A variable is nonfree in a...
bj-nnftht 36109 A variable is nonfree in a...
bj-nnfth 36110 A variable is nonfree in a...
bj-nnfnth 36111 A variable is nonfree in t...
bj-nnfim1 36112 A consequence of nonfreene...
bj-nnfim2 36113 A consequence of nonfreene...
bj-nnfim 36114 Nonfreeness in the anteced...
bj-nnfimd 36115 Nonfreeness in the anteced...
bj-nnfan 36116 Nonfreeness in both conjun...
bj-nnfand 36117 Nonfreeness in both conjun...
bj-nnfor 36118 Nonfreeness in both disjun...
bj-nnford 36119 Nonfreeness in both disjun...
bj-nnfbit 36120 Nonfreeness in both sides ...
bj-nnfbid 36121 Nonfreeness in both sides ...
bj-nnfv 36122 A non-occurring variable i...
bj-nnf-alrim 36123 Proof of the closed form o...
bj-nnf-exlim 36124 Proof of the closed form o...
bj-dfnnf3 36125 Alternate definition of no...
bj-nfnnfTEMP 36126 New nonfreeness is equival...
bj-nnfa1 36127 See ~ nfa1 . (Contributed...
bj-nnfe1 36128 See ~ nfe1 . (Contributed...
bj-19.12 36129 See ~ 19.12 . Could be la...
bj-nnflemaa 36130 One of four lemmas for non...
bj-nnflemee 36131 One of four lemmas for non...
bj-nnflemae 36132 One of four lemmas for non...
bj-nnflemea 36133 One of four lemmas for non...
bj-nnfalt 36134 See ~ nfal and ~ bj-nfalt ...
bj-nnfext 36135 See ~ nfex and ~ bj-nfext ...
bj-stdpc5t 36136 Alias of ~ bj-nnf-alrim fo...
bj-19.21t 36137 Statement ~ 19.21t proved ...
bj-19.23t 36138 Statement ~ 19.23t proved ...
bj-19.36im 36139 One direction of ~ 19.36 f...
bj-19.37im 36140 One direction of ~ 19.37 f...
bj-19.42t 36141 Closed form of ~ 19.42 fro...
bj-19.41t 36142 Closed form of ~ 19.41 fro...
bj-sbft 36143 Version of ~ sbft using ` ...
bj-pm11.53vw 36144 Version of ~ pm11.53v with...
bj-pm11.53v 36145 Version of ~ pm11.53v with...
bj-pm11.53a 36146 A variant of ~ pm11.53v . ...
bj-equsvt 36147 A variant of ~ equsv . (C...
bj-equsalvwd 36148 Variant of ~ equsalvw . (...
bj-equsexvwd 36149 Variant of ~ equsexvw . (...
bj-sbievwd 36150 Variant of ~ sbievw . (Co...
bj-axc10 36151 Alternate proof of ~ axc10...
bj-alequex 36152 A fol lemma. See ~ aleque...
bj-spimt2 36153 A step in the proof of ~ s...
bj-cbv3ta 36154 Closed form of ~ cbv3 . (...
bj-cbv3tb 36155 Closed form of ~ cbv3 . (...
bj-hbsb3t 36156 A theorem close to a close...
bj-hbsb3 36157 Shorter proof of ~ hbsb3 ....
bj-nfs1t 36158 A theorem close to a close...
bj-nfs1t2 36159 A theorem close to a close...
bj-nfs1 36160 Shorter proof of ~ nfs1 (t...
bj-axc10v 36161 Version of ~ axc10 with a ...
bj-spimtv 36162 Version of ~ spimt with a ...
bj-cbv3hv2 36163 Version of ~ cbv3h with tw...
bj-cbv1hv 36164 Version of ~ cbv1h with a ...
bj-cbv2hv 36165 Version of ~ cbv2h with a ...
bj-cbv2v 36166 Version of ~ cbv2 with a d...
bj-cbvaldv 36167 Version of ~ cbvald with a...
bj-cbvexdv 36168 Version of ~ cbvexd with a...
bj-cbval2vv 36169 Version of ~ cbval2vv with...
bj-cbvex2vv 36170 Version of ~ cbvex2vv with...
bj-cbvaldvav 36171 Version of ~ cbvaldva with...
bj-cbvexdvav 36172 Version of ~ cbvexdva with...
bj-cbvex4vv 36173 Version of ~ cbvex4v with ...
bj-equsalhv 36174 Version of ~ equsalh with ...
bj-axc11nv 36175 Version of ~ axc11n with a...
bj-aecomsv 36176 Version of ~ aecoms with a...
bj-axc11v 36177 Version of ~ axc11 with a ...
bj-drnf2v 36178 Version of ~ drnf2 with a ...
bj-equs45fv 36179 Version of ~ equs45f with ...
bj-hbs1 36180 Version of ~ hbsb2 with a ...
bj-nfs1v 36181 Version of ~ nfsb2 with a ...
bj-hbsb2av 36182 Version of ~ hbsb2a with a...
bj-hbsb3v 36183 Version of ~ hbsb3 with a ...
bj-nfsab1 36184 Remove dependency on ~ ax-...
bj-dtrucor2v 36185 Version of ~ dtrucor2 with...
bj-hbaeb2 36186 Biconditional version of a...
bj-hbaeb 36187 Biconditional version of ~...
bj-hbnaeb 36188 Biconditional version of ~...
bj-dvv 36189 A special instance of ~ bj...
bj-equsal1t 36190 Duplication of ~ wl-equsal...
bj-equsal1ti 36191 Inference associated with ...
bj-equsal1 36192 One direction of ~ equsal ...
bj-equsal2 36193 One direction of ~ equsal ...
bj-equsal 36194 Shorter proof of ~ equsal ...
stdpc5t 36195 Closed form of ~ stdpc5 . ...
bj-stdpc5 36196 More direct proof of ~ std...
2stdpc5 36197 A double ~ stdpc5 (one dir...
bj-19.21t0 36198 Proof of ~ 19.21t from ~ s...
exlimii 36199 Inference associated with ...
ax11-pm 36200 Proof of ~ ax-11 similar t...
ax6er 36201 Commuted form of ~ ax6e . ...
exlimiieq1 36202 Inferring a theorem when i...
exlimiieq2 36203 Inferring a theorem when i...
ax11-pm2 36204 Proof of ~ ax-11 from the ...
bj-sbsb 36205 Biconditional showing two ...
bj-dfsb2 36206 Alternate (dual) definitio...
bj-sbf3 36207 Substitution has no effect...
bj-sbf4 36208 Substitution has no effect...
bj-sbnf 36209 Move nonfree predicate in ...
bj-eu3f 36210 Version of ~ eu3v where th...
bj-sblem1 36211 Lemma for substitution. (...
bj-sblem2 36212 Lemma for substitution. (...
bj-sblem 36213 Lemma for substitution. (...
bj-sbievw1 36214 Lemma for substitution. (...
bj-sbievw2 36215 Lemma for substitution. (...
bj-sbievw 36216 Lemma for substitution. C...
bj-sbievv 36217 Version of ~ sbie with a s...
bj-moeub 36218 Uniqueness is equivalent t...
bj-sbidmOLD 36219 Obsolete proof of ~ sbidm ...
bj-dvelimdv 36220 Deduction form of ~ dvelim...
bj-dvelimdv1 36221 Curried (exported) form of...
bj-dvelimv 36222 A version of ~ dvelim usin...
bj-nfeel2 36223 Nonfreeness in a membershi...
bj-axc14nf 36224 Proof of a version of ~ ax...
bj-axc14 36225 Alternate proof of ~ axc14...
mobidvALT 36226 Alternate proof of ~ mobid...
sbn1ALT 36227 Alternate proof of ~ sbn1 ...
eliminable1 36228 A theorem used to prove th...
eliminable2a 36229 A theorem used to prove th...
eliminable2b 36230 A theorem used to prove th...
eliminable2c 36231 A theorem used to prove th...
eliminable3a 36232 A theorem used to prove th...
eliminable3b 36233 A theorem used to prove th...
eliminable-velab 36234 A theorem used to prove th...
eliminable-veqab 36235 A theorem used to prove th...
eliminable-abeqv 36236 A theorem used to prove th...
eliminable-abeqab 36237 A theorem used to prove th...
eliminable-abelv 36238 A theorem used to prove th...
eliminable-abelab 36239 A theorem used to prove th...
bj-denoteslem 36240 Lemma for ~ bj-denotes . ...
bj-denotes 36241 This would be the justific...
bj-issettru 36242 Weak version of ~ isset wi...
bj-elabtru 36243 This is as close as we can...
bj-issetwt 36244 Closed form of ~ bj-issetw...
bj-issetw 36245 The closest one can get to...
bj-elissetALT 36246 Alternate proof of ~ eliss...
bj-issetiv 36247 Version of ~ bj-isseti wit...
bj-isseti 36248 Version of ~ isseti with a...
bj-ralvw 36249 A weak version of ~ ralv n...
bj-rexvw 36250 A weak version of ~ rexv n...
bj-rababw 36251 A weak version of ~ rabab ...
bj-rexcom4bv 36252 Version of ~ rexcom4b and ...
bj-rexcom4b 36253 Remove from ~ rexcom4b dep...
bj-ceqsalt0 36254 The FOL content of ~ ceqsa...
bj-ceqsalt1 36255 The FOL content of ~ ceqsa...
bj-ceqsalt 36256 Remove from ~ ceqsalt depe...
bj-ceqsaltv 36257 Version of ~ bj-ceqsalt wi...
bj-ceqsalg0 36258 The FOL content of ~ ceqsa...
bj-ceqsalg 36259 Remove from ~ ceqsalg depe...
bj-ceqsalgALT 36260 Alternate proof of ~ bj-ce...
bj-ceqsalgv 36261 Version of ~ bj-ceqsalg wi...
bj-ceqsalgvALT 36262 Alternate proof of ~ bj-ce...
bj-ceqsal 36263 Remove from ~ ceqsal depen...
bj-ceqsalv 36264 Remove from ~ ceqsalv depe...
bj-spcimdv 36265 Remove from ~ spcimdv depe...
bj-spcimdvv 36266 Remove from ~ spcimdv depe...
elelb 36267 Equivalence between two co...
bj-pwvrelb 36268 Characterization of the el...
bj-nfcsym 36269 The nonfreeness quantifier...
bj-sbeqALT 36270 Substitution in an equalit...
bj-sbeq 36271 Distribute proper substitu...
bj-sbceqgALT 36272 Distribute proper substitu...
bj-csbsnlem 36273 Lemma for ~ bj-csbsn (in t...
bj-csbsn 36274 Substitution in a singleto...
bj-sbel1 36275 Version of ~ sbcel1g when ...
bj-abv 36276 The class of sets verifyin...
bj-abvALT 36277 Alternate version of ~ bj-...
bj-ab0 36278 The class of sets verifyin...
bj-abf 36279 Shorter proof of ~ abf (wh...
bj-csbprc 36280 More direct proof of ~ csb...
bj-exlimvmpi 36281 A Fol lemma ( ~ exlimiv fo...
bj-exlimmpi 36282 Lemma for ~ bj-vtoclg1f1 (...
bj-exlimmpbi 36283 Lemma for theorems of the ...
bj-exlimmpbir 36284 Lemma for theorems of the ...
bj-vtoclf 36285 Remove dependency on ~ ax-...
bj-vtocl 36286 Remove dependency on ~ ax-...
bj-vtoclg1f1 36287 The FOL content of ~ vtocl...
bj-vtoclg1f 36288 Reprove ~ vtoclg1f from ~ ...
bj-vtoclg1fv 36289 Version of ~ bj-vtoclg1f w...
bj-vtoclg 36290 A version of ~ vtoclg with...
bj-rabeqbid 36291 Version of ~ rabeqbidv wit...
bj-seex 36292 Version of ~ seex with a d...
bj-nfcf 36293 Version of ~ df-nfc with a...
bj-zfauscl 36294 General version of ~ zfaus...
bj-elabd2ALT 36295 Alternate proof of ~ elabd...
bj-unrab 36296 Generalization of ~ unrab ...
bj-inrab 36297 Generalization of ~ inrab ...
bj-inrab2 36298 Shorter proof of ~ inrab ....
bj-inrab3 36299 Generalization of ~ dfrab3...
bj-rabtr 36300 Restricted class abstracti...
bj-rabtrALT 36301 Alternate proof of ~ bj-ra...
bj-rabtrAUTO 36302 Proof of ~ bj-rabtr found ...
bj-gabss 36305 Inclusion of generalized c...
bj-gabssd 36306 Inclusion of generalized c...
bj-gabeqd 36307 Equality of generalized cl...
bj-gabeqis 36308 Equality of generalized cl...
bj-elgab 36309 Elements of a generalized ...
bj-gabima 36310 Generalized class abstract...
bj-ru0 36313 The FOL part of Russell's ...
bj-ru1 36314 A version of Russell's par...
bj-ru 36315 Remove dependency on ~ ax-...
currysetlem 36316 Lemma for ~ currysetlem , ...
curryset 36317 Curry's paradox in set the...
currysetlem1 36318 Lemma for ~ currysetALT . ...
currysetlem2 36319 Lemma for ~ currysetALT . ...
currysetlem3 36320 Lemma for ~ currysetALT . ...
currysetALT 36321 Alternate proof of ~ curry...
bj-n0i 36322 Inference associated with ...
bj-disjsn01 36323 Disjointness of the single...
bj-0nel1 36324 The empty set does not bel...
bj-1nel0 36325 ` 1o ` does not belong to ...
bj-xpimasn 36326 The image of a singleton, ...
bj-xpima1sn 36327 The image of a singleton b...
bj-xpima1snALT 36328 Alternate proof of ~ bj-xp...
bj-xpima2sn 36329 The image of a singleton b...
bj-xpnzex 36330 If the first factor of a p...
bj-xpexg2 36331 Curried (exported) form of...
bj-xpnzexb 36332 If the first factor of a p...
bj-cleq 36333 Substitution property for ...
bj-snsetex 36334 The class of sets "whose s...
bj-clexab 36335 Sethood of certain classes...
bj-sngleq 36338 Substitution property for ...
bj-elsngl 36339 Characterization of the el...
bj-snglc 36340 Characterization of the el...
bj-snglss 36341 The singletonization of a ...
bj-0nelsngl 36342 The empty set is not a mem...
bj-snglinv 36343 Inverse of singletonizatio...
bj-snglex 36344 A class is a set if and on...
bj-tageq 36347 Substitution property for ...
bj-eltag 36348 Characterization of the el...
bj-0eltag 36349 The empty set belongs to t...
bj-tagn0 36350 The tagging of a class is ...
bj-tagss 36351 The tagging of a class is ...
bj-snglsstag 36352 The singletonization is in...
bj-sngltagi 36353 The singletonization is in...
bj-sngltag 36354 The singletonization and t...
bj-tagci 36355 Characterization of the el...
bj-tagcg 36356 Characterization of the el...
bj-taginv 36357 Inverse of tagging. (Cont...
bj-tagex 36358 A class is a set if and on...
bj-xtageq 36359 The products of a given cl...
bj-xtagex 36360 The product of a set and t...
bj-projeq 36363 Substitution property for ...
bj-projeq2 36364 Substitution property for ...
bj-projun 36365 The class projection on a ...
bj-projex 36366 Sethood of the class proje...
bj-projval 36367 Value of the class project...
bj-1upleq 36370 Substitution property for ...
bj-pr1eq 36373 Substitution property for ...
bj-pr1un 36374 The first projection prese...
bj-pr1val 36375 Value of the first project...
bj-pr11val 36376 Value of the first project...
bj-pr1ex 36377 Sethood of the first proje...
bj-1uplth 36378 The characteristic propert...
bj-1uplex 36379 A monuple is a set if and ...
bj-1upln0 36380 A monuple is nonempty. (C...
bj-2upleq 36383 Substitution property for ...
bj-pr21val 36384 Value of the first project...
bj-pr2eq 36387 Substitution property for ...
bj-pr2un 36388 The second projection pres...
bj-pr2val 36389 Value of the second projec...
bj-pr22val 36390 Value of the second projec...
bj-pr2ex 36391 Sethood of the second proj...
bj-2uplth 36392 The characteristic propert...
bj-2uplex 36393 A couple is a set if and o...
bj-2upln0 36394 A couple is nonempty. (Co...
bj-2upln1upl 36395 A couple is never equal to...
bj-rcleqf 36396 Relative version of ~ cleq...
bj-rcleq 36397 Relative version of ~ dfcl...
bj-reabeq 36398 Relative form of ~ eqabb ....
bj-disj2r 36399 Relative version of ~ ssdi...
bj-sscon 36400 Contraposition law for rel...
bj-abex 36401 Two ways of stating that t...
bj-clex 36402 Two ways of stating that a...
bj-axsn 36403 Two ways of stating the ax...
bj-snexg 36405 A singleton built on a set...
bj-snex 36406 A singleton is a set. See...
bj-axbun 36407 Two ways of stating the ax...
bj-unexg 36409 Existence of binary unions...
bj-prexg 36410 Existence of unordered pai...
bj-prex 36411 Existence of unordered pai...
bj-axadj 36412 Two ways of stating the ax...
bj-adjg1 36414 Existence of the result of...
bj-snfromadj 36415 Singleton from adjunction ...
bj-prfromadj 36416 Unordered pair from adjunc...
bj-adjfrombun 36417 Adjunction from singleton ...
eleq2w2ALT 36418 Alternate proof of ~ eleq2...
bj-clel3gALT 36419 Alternate proof of ~ clel3...
bj-pw0ALT 36420 Alternate proof of ~ pw0 ....
bj-sselpwuni 36421 Quantitative version of ~ ...
bj-unirel 36422 Quantitative version of ~ ...
bj-elpwg 36423 If the intersection of two...
bj-velpwALT 36424 This theorem ~ bj-velpwALT...
bj-elpwgALT 36425 Alternate proof of ~ elpwg...
bj-vjust 36426 Justification theorem for ...
bj-nul 36427 Two formulations of the ax...
bj-nuliota 36428 Definition of the empty se...
bj-nuliotaALT 36429 Alternate proof of ~ bj-nu...
bj-vtoclgfALT 36430 Alternate proof of ~ vtocl...
bj-elsn12g 36431 Join of ~ elsng and ~ elsn...
bj-elsnb 36432 Biconditional version of ~...
bj-pwcfsdom 36433 Remove hypothesis from ~ p...
bj-grur1 36434 Remove hypothesis from ~ g...
bj-bm1.3ii 36435 The extension of a predica...
bj-dfid2ALT 36436 Alternate version of ~ dfi...
bj-0nelopab 36437 The empty set is never an ...
bj-brrelex12ALT 36438 Two classes related by a b...
bj-epelg 36439 The membership relation an...
bj-epelb 36440 Two classes are related by...
bj-nsnid 36441 A set does not contain the...
bj-rdg0gALT 36442 Alternate proof of ~ rdg0g...
bj-evaleq 36443 Equality theorem for the `...
bj-evalfun 36444 The evaluation at a class ...
bj-evalfn 36445 The evaluation at a class ...
bj-evalval 36446 Value of the evaluation at...
bj-evalid 36447 The evaluation at a set of...
bj-ndxarg 36448 Proof of ~ ndxarg from ~ b...
bj-evalidval 36449 Closed general form of ~ s...
bj-rest00 36452 An elementwise intersectio...
bj-restsn 36453 An elementwise intersectio...
bj-restsnss 36454 Special case of ~ bj-rests...
bj-restsnss2 36455 Special case of ~ bj-rests...
bj-restsn0 36456 An elementwise intersectio...
bj-restsn10 36457 Special case of ~ bj-rests...
bj-restsnid 36458 The elementwise intersecti...
bj-rest10 36459 An elementwise intersectio...
bj-rest10b 36460 Alternate version of ~ bj-...
bj-restn0 36461 An elementwise intersectio...
bj-restn0b 36462 Alternate version of ~ bj-...
bj-restpw 36463 The elementwise intersecti...
bj-rest0 36464 An elementwise intersectio...
bj-restb 36465 An elementwise intersectio...
bj-restv 36466 An elementwise intersectio...
bj-resta 36467 An elementwise intersectio...
bj-restuni 36468 The union of an elementwis...
bj-restuni2 36469 The union of an elementwis...
bj-restreg 36470 A reformulation of the axi...
bj-raldifsn 36471 All elements in a set sati...
bj-0int 36472 If ` A ` is a collection o...
bj-mooreset 36473 A Moore collection is a se...
bj-ismoore 36476 Characterization of Moore ...
bj-ismoored0 36477 Necessary condition to be ...
bj-ismoored 36478 Necessary condition to be ...
bj-ismoored2 36479 Necessary condition to be ...
bj-ismooredr 36480 Sufficient condition to be...
bj-ismooredr2 36481 Sufficient condition to be...
bj-discrmoore 36482 The powerclass ` ~P A ` is...
bj-0nmoore 36483 The empty set is not a Moo...
bj-snmoore 36484 A singleton is a Moore col...
bj-snmooreb 36485 A singleton is a Moore col...
bj-prmoore 36486 A pair formed of two neste...
bj-0nelmpt 36487 The empty set is not an el...
bj-mptval 36488 Value of a function given ...
bj-dfmpoa 36489 An equivalent definition o...
bj-mpomptALT 36490 Alternate proof of ~ mpomp...
setsstrset 36507 Relation between ~ df-sets...
bj-nfald 36508 Variant of ~ nfald . (Con...
bj-nfexd 36509 Variant of ~ nfexd . (Con...
copsex2d 36510 Implicit substitution dedu...
copsex2b 36511 Biconditional form of ~ co...
opelopabd 36512 Membership of an ordere pa...
opelopabb 36513 Membership of an ordered p...
opelopabbv 36514 Membership of an ordered p...
bj-opelrelex 36515 The coordinates of an orde...
bj-opelresdm 36516 If an ordered pair is in a...
bj-brresdm 36517 If two classes are related...
brabd0 36518 Expressing that two sets a...
brabd 36519 Expressing that two sets a...
bj-brab2a1 36520 "Unbounded" version of ~ b...
bj-opabssvv 36521 A variant of ~ relopabiv (...
bj-funidres 36522 The restricted identity re...
bj-opelidb 36523 Characterization of the or...
bj-opelidb1 36524 Characterization of the or...
bj-inexeqex 36525 Lemma for ~ bj-opelid (but...
bj-elsn0 36526 If the intersection of two...
bj-opelid 36527 Characterization of the or...
bj-ideqg 36528 Characterization of the cl...
bj-ideqgALT 36529 Alternate proof of ~ bj-id...
bj-ideqb 36530 Characterization of classe...
bj-idres 36531 Alternate expression for t...
bj-opelidres 36532 Characterization of the or...
bj-idreseq 36533 Sufficient condition for t...
bj-idreseqb 36534 Characterization for two c...
bj-ideqg1 36535 For sets, the identity rel...
bj-ideqg1ALT 36536 Alternate proof of bj-ideq...
bj-opelidb1ALT 36537 Characterization of the co...
bj-elid3 36538 Characterization of the co...
bj-elid4 36539 Characterization of the el...
bj-elid5 36540 Characterization of the el...
bj-elid6 36541 Characterization of the el...
bj-elid7 36542 Characterization of the el...
bj-diagval 36545 Value of the functionalize...
bj-diagval2 36546 Value of the functionalize...
bj-eldiag 36547 Characterization of the el...
bj-eldiag2 36548 Characterization of the el...
bj-imdirvallem 36551 Lemma for ~ bj-imdirval an...
bj-imdirval 36552 Value of the functionalize...
bj-imdirval2lem 36553 Lemma for ~ bj-imdirval2 a...
bj-imdirval2 36554 Value of the functionalize...
bj-imdirval3 36555 Value of the functionalize...
bj-imdiridlem 36556 Lemma for ~ bj-imdirid and...
bj-imdirid 36557 Functorial property of the...
bj-opelopabid 36558 Membership in an ordered-p...
bj-opabco 36559 Composition of ordered-pai...
bj-xpcossxp 36560 The composition of two Car...
bj-imdirco 36561 Functorial property of the...
bj-iminvval 36564 Value of the functionalize...
bj-iminvval2 36565 Value of the functionalize...
bj-iminvid 36566 Functorial property of the...
bj-inftyexpitaufo 36573 The function ` inftyexpita...
bj-inftyexpitaudisj 36576 An element of the circle a...
bj-inftyexpiinv 36579 Utility theorem for the in...
bj-inftyexpiinj 36580 Injectivity of the paramet...
bj-inftyexpidisj 36581 An element of the circle a...
bj-ccinftydisj 36584 The circle at infinity is ...
bj-elccinfty 36585 A lemma for infinite exten...
bj-ccssccbar 36588 Complex numbers are extend...
bj-ccinftyssccbar 36589 Infinite extended complex ...
bj-pinftyccb 36592 The class ` pinfty ` is an...
bj-pinftynrr 36593 The extended complex numbe...
bj-minftyccb 36596 The class ` minfty ` is an...
bj-minftynrr 36597 The extended complex numbe...
bj-pinftynminfty 36598 The extended complex numbe...
bj-rrhatsscchat 36607 The real projective line i...
bj-imafv 36622 If the direct image of a s...
bj-funun 36623 Value of a function expres...
bj-fununsn1 36624 Value of a function expres...
bj-fununsn2 36625 Value of a function expres...
bj-fvsnun1 36626 The value of a function wi...
bj-fvsnun2 36627 The value of a function wi...
bj-fvmptunsn1 36628 Value of a function expres...
bj-fvmptunsn2 36629 Value of a function expres...
bj-iomnnom 36630 The canonical bijection fr...
bj-smgrpssmgm 36639 Semigroups are magmas. (C...
bj-smgrpssmgmel 36640 Semigroups are magmas (ele...
bj-mndsssmgrp 36641 Monoids are semigroups. (...
bj-mndsssmgrpel 36642 Monoids are semigroups (el...
bj-cmnssmnd 36643 Commutative monoids are mo...
bj-cmnssmndel 36644 Commutative monoids are mo...
bj-grpssmnd 36645 Groups are monoids. (Cont...
bj-grpssmndel 36646 Groups are monoids (elemen...
bj-ablssgrp 36647 Abelian groups are groups....
bj-ablssgrpel 36648 Abelian groups are groups ...
bj-ablsscmn 36649 Abelian groups are commuta...
bj-ablsscmnel 36650 Abelian groups are commuta...
bj-modssabl 36651 (The additive groups of) m...
bj-vecssmod 36652 Vector spaces are modules....
bj-vecssmodel 36653 Vector spaces are modules ...
bj-finsumval0 36656 Value of a finite sum. (C...
bj-fvimacnv0 36657 Variant of ~ fvimacnv wher...
bj-isvec 36658 The predicate "is a vector...
bj-fldssdrng 36659 Fields are division rings....
bj-flddrng 36660 Fields are division rings ...
bj-rrdrg 36661 The field of real numbers ...
bj-isclm 36662 The predicate "is a subcom...
bj-isrvec 36665 The predicate "is a real v...
bj-rvecmod 36666 Real vector spaces are mod...
bj-rvecssmod 36667 Real vector spaces are mod...
bj-rvecrr 36668 The field of scalars of a ...
bj-isrvecd 36669 The predicate "is a real v...
bj-rvecvec 36670 Real vector spaces are vec...
bj-isrvec2 36671 The predicate "is a real v...
bj-rvecssvec 36672 Real vector spaces are vec...
bj-rveccmod 36673 Real vector spaces are sub...
bj-rvecsscmod 36674 Real vector spaces are sub...
bj-rvecsscvec 36675 Real vector spaces are sub...
bj-rveccvec 36676 Real vector spaces are sub...
bj-rvecssabl 36677 (The additive groups of) r...
bj-rvecabl 36678 (The additive groups of) r...
bj-subcom 36679 A consequence of commutati...
bj-lineqi 36680 Solution of a (scalar) lin...
bj-bary1lem 36681 Lemma for ~ bj-bary1 : exp...
bj-bary1lem1 36682 Lemma for bj-bary1: comput...
bj-bary1 36683 Barycentric coordinates in...
bj-endval 36686 Value of the monoid of end...
bj-endbase 36687 Base set of the monoid of ...
bj-endcomp 36688 Composition law of the mon...
bj-endmnd 36689 The monoid of endomorphism...
taupilem3 36690 Lemma for tau-related theo...
taupilemrplb 36691 A set of positive reals ha...
taupilem1 36692 Lemma for ~ taupi . A pos...
taupilem2 36693 Lemma for ~ taupi . The s...
taupi 36694 Relationship between ` _ta...
dfgcd3 36695 Alternate definition of th...
irrdifflemf 36696 Lemma for ~ irrdiff . The...
irrdiff 36697 The irrationals are exactl...
iccioo01 36698 The closed unit interval i...
csbrecsg 36699 Move class substitution in...
csbrdgg 36700 Move class substitution in...
csboprabg 36701 Move class substitution in...
csbmpo123 36702 Move class substitution in...
con1bii2 36703 A contraposition inference...
con2bii2 36704 A contraposition inference...
vtoclefex 36705 Implicit substitution of a...
rnmptsn 36706 The range of a function ma...
f1omptsnlem 36707 This is the core of the pr...
f1omptsn 36708 A function mapping to sing...
mptsnunlem 36709 This is the core of the pr...
mptsnun 36710 A class ` B ` is equal to ...
dissneqlem 36711 This is the core of the pr...
dissneq 36712 Any topology that contains...
exlimim 36713 Closed form of ~ exlimimd ...
exlimimd 36714 Existential elimination ru...
exellim 36715 Closed form of ~ exellimdd...
exellimddv 36716 Eliminate an antecedent wh...
topdifinfindis 36717 Part of Exercise 3 of [Mun...
topdifinffinlem 36718 This is the core of the pr...
topdifinffin 36719 Part of Exercise 3 of [Mun...
topdifinf 36720 Part of Exercise 3 of [Mun...
topdifinfeq 36721 Two different ways of defi...
icorempo 36722 Closed-below, open-above i...
icoreresf 36723 Closed-below, open-above i...
icoreval 36724 Value of the closed-below,...
icoreelrnab 36725 Elementhood in the set of ...
isbasisrelowllem1 36726 Lemma for ~ isbasisrelowl ...
isbasisrelowllem2 36727 Lemma for ~ isbasisrelowl ...
icoreclin 36728 The set of closed-below, o...
isbasisrelowl 36729 The set of all closed-belo...
icoreunrn 36730 The union of all closed-be...
istoprelowl 36731 The set of all closed-belo...
icoreelrn 36732 A class abstraction which ...
iooelexlt 36733 An element of an open inte...
relowlssretop 36734 The lower limit topology o...
relowlpssretop 36735 The lower limit topology o...
sucneqond 36736 Inequality of an ordinal s...
sucneqoni 36737 Inequality of an ordinal s...
onsucuni3 36738 If an ordinal number has a...
1oequni2o 36739 The ordinal number ` 1o ` ...
rdgsucuni 36740 If an ordinal number has a...
rdgeqoa 36741 If a recursive function wi...
elxp8 36742 Membership in a Cartesian ...
cbveud 36743 Deduction used to change b...
cbvreud 36744 Deduction used to change b...
difunieq 36745 The difference of unions i...
inunissunidif 36746 Theorem about subsets of t...
rdgellim 36747 Elementhood in a recursive...
rdglimss 36748 A recursive definition at ...
rdgssun 36749 In a recursive definition ...
exrecfnlem 36750 Lemma for ~ exrecfn . (Co...
exrecfn 36751 Theorem about the existenc...
exrecfnpw 36752 For any base set, a set wh...
finorwe 36753 If the Axiom of Infinity i...
dffinxpf 36756 This theorem is the same a...
finxpeq1 36757 Equality theorem for Carte...
finxpeq2 36758 Equality theorem for Carte...
csbfinxpg 36759 Distribute proper substitu...
finxpreclem1 36760 Lemma for ` ^^ ` recursion...
finxpreclem2 36761 Lemma for ` ^^ ` recursion...
finxp0 36762 The value of Cartesian exp...
finxp1o 36763 The value of Cartesian exp...
finxpreclem3 36764 Lemma for ` ^^ ` recursion...
finxpreclem4 36765 Lemma for ` ^^ ` recursion...
finxpreclem5 36766 Lemma for ` ^^ ` recursion...
finxpreclem6 36767 Lemma for ` ^^ ` recursion...
finxpsuclem 36768 Lemma for ~ finxpsuc . (C...
finxpsuc 36769 The value of Cartesian exp...
finxp2o 36770 The value of Cartesian exp...
finxp3o 36771 The value of Cartesian exp...
finxpnom 36772 Cartesian exponentiation w...
finxp00 36773 Cartesian exponentiation o...
iunctb2 36774 Using the axiom of countab...
domalom 36775 A class which dominates ev...
isinf2 36776 The converse of ~ isinf . ...
ctbssinf 36777 Using the axiom of choice,...
ralssiun 36778 The index set of an indexe...
nlpineqsn 36779 For every point ` p ` of a...
nlpfvineqsn 36780 Given a subset ` A ` of ` ...
fvineqsnf1 36781 A theorem about functions ...
fvineqsneu 36782 A theorem about functions ...
fvineqsneq 36783 A theorem about functions ...
pibp16 36784 Property P000016 of pi-bas...
pibp19 36785 Property P000019 of pi-bas...
pibp21 36786 Property P000021 of pi-bas...
pibt1 36787 Theorem T000001 of pi-base...
pibt2 36788 Theorem T000002 of pi-base...
wl-section-prop 36789 Intuitionistic logic is no...
wl-section-boot 36793 In this section, I provide...
wl-luk-imim1i 36794 Inference adding common co...
wl-luk-syl 36795 An inference version of th...
wl-luk-imtrid 36796 A syllogism rule of infere...
wl-luk-pm2.18d 36797 Deduction based on reducti...
wl-luk-con4i 36798 Inference rule. Copy of ~...
wl-luk-pm2.24i 36799 Inference rule. Copy of ~...
wl-luk-a1i 36800 Inference rule. Copy of ~...
wl-luk-mpi 36801 A nested modus ponens infe...
wl-luk-imim2i 36802 Inference adding common an...
wl-luk-imtrdi 36803 A syllogism rule of infere...
wl-luk-ax3 36804 ~ ax-3 proved from Lukasie...
wl-luk-ax1 36805 ~ ax-1 proved from Lukasie...
wl-luk-pm2.27 36806 This theorem, called "Asse...
wl-luk-com12 36807 Inference that swaps (comm...
wl-luk-pm2.21 36808 From a wff and its negatio...
wl-luk-con1i 36809 A contraposition inference...
wl-luk-ja 36810 Inference joining the ante...
wl-luk-imim2 36811 A closed form of syllogism...
wl-luk-a1d 36812 Deduction introducing an e...
wl-luk-ax2 36813 ~ ax-2 proved from Lukasie...
wl-luk-id 36814 Principle of identity. Th...
wl-luk-notnotr 36815 Converse of double negatio...
wl-luk-pm2.04 36816 Swap antecedents. Theorem...
wl-section-impchain 36817 An implication like ` ( ps...
wl-impchain-mp-x 36818 This series of theorems pr...
wl-impchain-mp-0 36819 This theorem is the start ...
wl-impchain-mp-1 36820 This theorem is in fact a ...
wl-impchain-mp-2 36821 This theorem is in fact a ...
wl-impchain-com-1.x 36822 It is often convenient to ...
wl-impchain-com-1.1 36823 A degenerate form of antec...
wl-impchain-com-1.2 36824 This theorem is in fact a ...
wl-impchain-com-1.3 36825 This theorem is in fact a ...
wl-impchain-com-1.4 36826 This theorem is in fact a ...
wl-impchain-com-n.m 36827 This series of theorems al...
wl-impchain-com-2.3 36828 This theorem is in fact a ...
wl-impchain-com-2.4 36829 This theorem is in fact a ...
wl-impchain-com-3.2.1 36830 This theorem is in fact a ...
wl-impchain-a1-x 36831 If an implication chain is...
wl-impchain-a1-1 36832 Inference rule, a copy of ...
wl-impchain-a1-2 36833 Inference rule, a copy of ...
wl-impchain-a1-3 36834 Inference rule, a copy of ...
wl-ifp-ncond1 36835 If one case of an ` if- ` ...
wl-ifp-ncond2 36836 If one case of an ` if- ` ...
wl-ifpimpr 36837 If one case of an ` if- ` ...
wl-ifp4impr 36838 If one case of an ` if- ` ...
wl-df-3xor 36839 Alternative definition of ...
wl-df3xor2 36840 Alternative definition of ...
wl-df3xor3 36841 Alternative form of ~ wl-d...
wl-3xortru 36842 If the first input is true...
wl-3xorfal 36843 If the first input is fals...
wl-3xorbi 36844 Triple xor can be replaced...
wl-3xorbi2 36845 Alternative form of ~ wl-3...
wl-3xorbi123d 36846 Equivalence theorem for tr...
wl-3xorbi123i 36847 Equivalence theorem for tr...
wl-3xorrot 36848 Rotation law for triple xo...
wl-3xorcoma 36849 Commutative law for triple...
wl-3xorcomb 36850 Commutative law for triple...
wl-3xornot1 36851 Flipping the first input f...
wl-3xornot 36852 Triple xor distributes ove...
wl-1xor 36853 In the recursive scheme ...
wl-2xor 36854 In the recursive scheme ...
wl-df-3mintru2 36855 Alternative definition of ...
wl-df2-3mintru2 36856 The adder carry in disjunc...
wl-df3-3mintru2 36857 The adder carry in conjunc...
wl-df4-3mintru2 36858 An alternative definition ...
wl-1mintru1 36859 Using the recursion formul...
wl-1mintru2 36860 Using the recursion formul...
wl-2mintru1 36861 Using the recursion formul...
wl-2mintru2 36862 Using the recursion formul...
wl-df3maxtru1 36863 Assuming "(n+1)-maxtru1" `...
wl-ax13lem1 36865 A version of ~ ax-wl-13v w...
wl-mps 36866 Replacing a nested consequ...
wl-syls1 36867 Replacing a nested consequ...
wl-syls2 36868 Replacing a nested anteced...
wl-embant 36869 A true wff can always be a...
wl-orel12 36870 In a conjunctive normal fo...
wl-cases2-dnf 36871 A particular instance of ~...
wl-cbvmotv 36872 Change bound variable. Us...
wl-moteq 36873 Change bound variable. Us...
wl-motae 36874 Change bound variable. Us...
wl-moae 36875 Two ways to express "at mo...
wl-euae 36876 Two ways to express "exact...
wl-nax6im 36877 The following series of th...
wl-hbae1 36878 This specialization of ~ h...
wl-naevhba1v 36879 An instance of ~ hbn1w app...
wl-spae 36880 Prove an instance of ~ sp ...
wl-speqv 36881 Under the assumption ` -. ...
wl-19.8eqv 36882 Under the assumption ` -. ...
wl-19.2reqv 36883 Under the assumption ` -. ...
wl-nfalv 36884 If ` x ` is not present in...
wl-nfimf1 36885 An antecedent is irrelevan...
wl-nfae1 36886 Unlike ~ nfae , this speci...
wl-nfnae1 36887 Unlike ~ nfnae , this spec...
wl-aetr 36888 A transitive law for varia...
wl-axc11r 36889 Same as ~ axc11r , but usi...
wl-dral1d 36890 A version of ~ dral1 with ...
wl-cbvalnaed 36891 ~ wl-cbvalnae with a conte...
wl-cbvalnae 36892 A more general version of ...
wl-exeq 36893 The semantics of ` E. x y ...
wl-aleq 36894 The semantics of ` A. x y ...
wl-nfeqfb 36895 Extend ~ nfeqf to an equiv...
wl-nfs1t 36896 If ` y ` is not free in ` ...
wl-equsalvw 36897 Version of ~ equsalv with ...
wl-equsald 36898 Deduction version of ~ equ...
wl-equsal 36899 A useful equivalence relat...
wl-equsal1t 36900 The expression ` x = y ` i...
wl-equsalcom 36901 This simple equivalence ea...
wl-equsal1i 36902 The antecedent ` x = y ` i...
wl-sb6rft 36903 A specialization of ~ wl-e...
wl-cbvalsbi 36904 Change bounded variables i...
wl-sbrimt 36905 Substitution with a variab...
wl-sblimt 36906 Substitution with a variab...
wl-sb8t 36907 Substitution of variable i...
wl-sb8et 36908 Substitution of variable i...
wl-sbhbt 36909 Closed form of ~ sbhb . C...
wl-sbnf1 36910 Two ways expressing that `...
wl-equsb3 36911 ~ equsb3 with a distinctor...
wl-equsb4 36912 Substitution applied to an...
wl-2sb6d 36913 Version of ~ 2sb6 with a c...
wl-sbcom2d-lem1 36914 Lemma used to prove ~ wl-s...
wl-sbcom2d-lem2 36915 Lemma used to prove ~ wl-s...
wl-sbcom2d 36916 Version of ~ sbcom2 with a...
wl-sbalnae 36917 A theorem used in eliminat...
wl-sbal1 36918 A theorem used in eliminat...
wl-sbal2 36919 Move quantifier in and out...
wl-2spsbbi 36920 ~ spsbbi applied twice. (...
wl-lem-exsb 36921 This theorem provides a ba...
wl-lem-nexmo 36922 This theorem provides a ba...
wl-lem-moexsb 36923 The antecedent ` A. x ( ph...
wl-alanbii 36924 This theorem extends ~ ala...
wl-mo2df 36925 Version of ~ mof with a co...
wl-mo2tf 36926 Closed form of ~ mof with ...
wl-eudf 36927 Version of ~ eu6 with a co...
wl-eutf 36928 Closed form of ~ eu6 with ...
wl-euequf 36929 ~ euequ proved with a dist...
wl-mo2t 36930 Closed form of ~ mof . (C...
wl-mo3t 36931 Closed form of ~ mo3 . (C...
wl-sb8eut 36932 Substitution of variable i...
wl-sb8mot 36933 Substitution of variable i...
wl-issetft 36934 A closed form of ~ issetf ...
wl-axc11rc11 36935 Proving ~ axc11r from ~ ax...
wl-ax11-lem1 36937 A transitive law for varia...
wl-ax11-lem2 36938 Lemma. (Contributed by Wo...
wl-ax11-lem3 36939 Lemma. (Contributed by Wo...
wl-ax11-lem4 36940 Lemma. (Contributed by Wo...
wl-ax11-lem5 36941 Lemma. (Contributed by Wo...
wl-ax11-lem6 36942 Lemma. (Contributed by Wo...
wl-ax11-lem7 36943 Lemma. (Contributed by Wo...
wl-ax11-lem8 36944 Lemma. (Contributed by Wo...
wl-ax11-lem9 36945 The easy part when ` x ` c...
wl-ax11-lem10 36946 We now have prepared every...
wl-clabv 36947 Variant of ~ df-clab , whe...
wl-dfclab 36948 Rederive ~ df-clab from ~ ...
wl-clabtv 36949 Using class abstraction in...
wl-clabt 36950 Using class abstraction in...
rabiun 36951 Abstraction restricted to ...
iundif1 36952 Indexed union of class dif...
imadifss 36953 The difference of images i...
cureq 36954 Equality theorem for curry...
unceq 36955 Equality theorem for uncur...
curf 36956 Functional property of cur...
uncf 36957 Functional property of unc...
curfv 36958 Value of currying. (Contr...
uncov 36959 Value of uncurrying. (Con...
curunc 36960 Currying of uncurrying. (...
unccur 36961 Uncurrying of currying. (...
phpreu 36962 Theorem related to pigeonh...
finixpnum 36963 A finite Cartesian product...
fin2solem 36964 Lemma for ~ fin2so . (Con...
fin2so 36965 Any totally ordered Tarski...
ltflcei 36966 Theorem to move the floor ...
leceifl 36967 Theorem to move the floor ...
sin2h 36968 Half-angle rule for sine. ...
cos2h 36969 Half-angle rule for cosine...
tan2h 36970 Half-angle rule for tangen...
lindsadd 36971 In a vector space, the uni...
lindsdom 36972 A linearly independent set...
lindsenlbs 36973 A maximal linearly indepen...
matunitlindflem1 36974 One direction of ~ matunit...
matunitlindflem2 36975 One direction of ~ matunit...
matunitlindf 36976 A matrix over a field is i...
ptrest 36977 Expressing a restriction o...
ptrecube 36978 Any point in an open set o...
poimirlem1 36979 Lemma for ~ poimir - the v...
poimirlem2 36980 Lemma for ~ poimir - conse...
poimirlem3 36981 Lemma for ~ poimir to add ...
poimirlem4 36982 Lemma for ~ poimir connect...
poimirlem5 36983 Lemma for ~ poimir to esta...
poimirlem6 36984 Lemma for ~ poimir establi...
poimirlem7 36985 Lemma for ~ poimir , simil...
poimirlem8 36986 Lemma for ~ poimir , estab...
poimirlem9 36987 Lemma for ~ poimir , estab...
poimirlem10 36988 Lemma for ~ poimir establi...
poimirlem11 36989 Lemma for ~ poimir connect...
poimirlem12 36990 Lemma for ~ poimir connect...
poimirlem13 36991 Lemma for ~ poimir - for a...
poimirlem14 36992 Lemma for ~ poimir - for a...
poimirlem15 36993 Lemma for ~ poimir , that ...
poimirlem16 36994 Lemma for ~ poimir establi...
poimirlem17 36995 Lemma for ~ poimir establi...
poimirlem18 36996 Lemma for ~ poimir stating...
poimirlem19 36997 Lemma for ~ poimir establi...
poimirlem20 36998 Lemma for ~ poimir establi...
poimirlem21 36999 Lemma for ~ poimir stating...
poimirlem22 37000 Lemma for ~ poimir , that ...
poimirlem23 37001 Lemma for ~ poimir , two w...
poimirlem24 37002 Lemma for ~ poimir , two w...
poimirlem25 37003 Lemma for ~ poimir stating...
poimirlem26 37004 Lemma for ~ poimir showing...
poimirlem27 37005 Lemma for ~ poimir showing...
poimirlem28 37006 Lemma for ~ poimir , a var...
poimirlem29 37007 Lemma for ~ poimir connect...
poimirlem30 37008 Lemma for ~ poimir combini...
poimirlem31 37009 Lemma for ~ poimir , assig...
poimirlem32 37010 Lemma for ~ poimir , combi...
poimir 37011 Poincare-Miranda theorem. ...
broucube 37012 Brouwer - or as Kulpa call...
heicant 37013 Heine-Cantor theorem: a co...
opnmbllem0 37014 Lemma for ~ ismblfin ; cou...
mblfinlem1 37015 Lemma for ~ ismblfin , ord...
mblfinlem2 37016 Lemma for ~ ismblfin , eff...
mblfinlem3 37017 The difference between two...
mblfinlem4 37018 Backward direction of ~ is...
ismblfin 37019 Measurability in terms of ...
ovoliunnfl 37020 ~ ovoliun is incompatible ...
ex-ovoliunnfl 37021 Demonstration of ~ ovoliun...
voliunnfl 37022 ~ voliun is incompatible w...
volsupnfl 37023 ~ volsup is incompatible w...
mbfresfi 37024 Measurability of a piecewi...
mbfposadd 37025 If the sum of two measurab...
cnambfre 37026 A real-valued, a.e. contin...
dvtanlem 37027 Lemma for ~ dvtan - the do...
dvtan 37028 Derivative of tangent. (C...
itg2addnclem 37029 An alternate expression fo...
itg2addnclem2 37030 Lemma for ~ itg2addnc . T...
itg2addnclem3 37031 Lemma incomprehensible in ...
itg2addnc 37032 Alternate proof of ~ itg2a...
itg2gt0cn 37033 ~ itg2gt0 holds on functio...
ibladdnclem 37034 Lemma for ~ ibladdnc ; cf ...
ibladdnc 37035 Choice-free analogue of ~ ...
itgaddnclem1 37036 Lemma for ~ itgaddnc ; cf....
itgaddnclem2 37037 Lemma for ~ itgaddnc ; cf....
itgaddnc 37038 Choice-free analogue of ~ ...
iblsubnc 37039 Choice-free analogue of ~ ...
itgsubnc 37040 Choice-free analogue of ~ ...
iblabsnclem 37041 Lemma for ~ iblabsnc ; cf....
iblabsnc 37042 Choice-free analogue of ~ ...
iblmulc2nc 37043 Choice-free analogue of ~ ...
itgmulc2nclem1 37044 Lemma for ~ itgmulc2nc ; c...
itgmulc2nclem2 37045 Lemma for ~ itgmulc2nc ; c...
itgmulc2nc 37046 Choice-free analogue of ~ ...
itgabsnc 37047 Choice-free analogue of ~ ...
itggt0cn 37048 ~ itggt0 holds for continu...
ftc1cnnclem 37049 Lemma for ~ ftc1cnnc ; cf....
ftc1cnnc 37050 Choice-free proof of ~ ftc...
ftc1anclem1 37051 Lemma for ~ ftc1anc - the ...
ftc1anclem2 37052 Lemma for ~ ftc1anc - rest...
ftc1anclem3 37053 Lemma for ~ ftc1anc - the ...
ftc1anclem4 37054 Lemma for ~ ftc1anc . (Co...
ftc1anclem5 37055 Lemma for ~ ftc1anc , the ...
ftc1anclem6 37056 Lemma for ~ ftc1anc - cons...
ftc1anclem7 37057 Lemma for ~ ftc1anc . (Co...
ftc1anclem8 37058 Lemma for ~ ftc1anc . (Co...
ftc1anc 37059 ~ ftc1a holds for function...
ftc2nc 37060 Choice-free proof of ~ ftc...
asindmre 37061 Real part of domain of dif...
dvasin 37062 Derivative of arcsine. (C...
dvacos 37063 Derivative of arccosine. ...
dvreasin 37064 Real derivative of arcsine...
dvreacos 37065 Real derivative of arccosi...
areacirclem1 37066 Antiderivative of cross-se...
areacirclem2 37067 Endpoint-inclusive continu...
areacirclem3 37068 Integrability of cross-sec...
areacirclem4 37069 Endpoint-inclusive continu...
areacirclem5 37070 Finding the cross-section ...
areacirc 37071 The area of a circle of ra...
unirep 37072 Define a quantity whose de...
cover2 37073 Two ways of expressing the...
cover2g 37074 Two ways of expressing the...
brabg2 37075 Relation by a binary relat...
opelopab3 37076 Ordered pair membership in...
cocanfo 37077 Cancellation of a surjecti...
brresi2 37078 Restriction of a binary re...
fnopabeqd 37079 Equality deduction for fun...
fvopabf4g 37080 Function value of an opera...
fnopabco 37081 Composition of a function ...
opropabco 37082 Composition of an operator...
cocnv 37083 Composition with a functio...
f1ocan1fv 37084 Cancel a composition by a ...
f1ocan2fv 37085 Cancel a composition by th...
inixp 37086 Intersection of Cartesian ...
upixp 37087 Universal property of the ...
abrexdom 37088 An indexed set is dominate...
abrexdom2 37089 An indexed set is dominate...
ac6gf 37090 Axiom of Choice. (Contrib...
indexa 37091 If for every element of an...
indexdom 37092 If for every element of an...
frinfm 37093 A subset of a well-founded...
welb 37094 A nonempty subset of a wel...
supex2g 37095 Existence of supremum. (C...
supclt 37096 Closure of supremum. (Con...
supubt 37097 Upper bound property of su...
filbcmb 37098 Combine a finite set of lo...
fzmul 37099 Membership of a product in...
sdclem2 37100 Lemma for ~ sdc . (Contri...
sdclem1 37101 Lemma for ~ sdc . (Contri...
sdc 37102 Strong dependent choice. ...
fdc 37103 Finite version of dependen...
fdc1 37104 Variant of ~ fdc with no s...
seqpo 37105 Two ways to say that a seq...
incsequz 37106 An increasing sequence of ...
incsequz2 37107 An increasing sequence of ...
nnubfi 37108 A bounded above set of pos...
nninfnub 37109 An infinite set of positiv...
subspopn 37110 An open set is open in the...
neificl 37111 Neighborhoods are closed u...
lpss2 37112 Limit points of a subset a...
metf1o 37113 Use a bijection with a met...
blssp 37114 A ball in the subspace met...
mettrifi 37115 Generalized triangle inequ...
lmclim2 37116 A sequence in a metric spa...
geomcau 37117 If the distance between co...
caures 37118 The restriction of a Cauch...
caushft 37119 A shifted Cauchy sequence ...
constcncf 37120 A constant function is a c...
cnres2 37121 The restriction of a conti...
cnresima 37122 A continuous function is c...
cncfres 37123 A continuous function on c...
istotbnd 37127 The predicate "is a totall...
istotbnd2 37128 The predicate "is a totall...
istotbnd3 37129 A metric space is totally ...
totbndmet 37130 The predicate "totally bou...
0totbnd 37131 The metric (there is only ...
sstotbnd2 37132 Condition for a subset of ...
sstotbnd 37133 Condition for a subset of ...
sstotbnd3 37134 Use a net that is not nece...
totbndss 37135 A subset of a totally boun...
equivtotbnd 37136 If the metric ` M ` is "st...
isbnd 37138 The predicate "is a bounde...
bndmet 37139 A bounded metric space is ...
isbndx 37140 A "bounded extended metric...
isbnd2 37141 The predicate "is a bounde...
isbnd3 37142 A metric space is bounded ...
isbnd3b 37143 A metric space is bounded ...
bndss 37144 A subset of a bounded metr...
blbnd 37145 A ball is bounded. (Contr...
ssbnd 37146 A subset of a metric space...
totbndbnd 37147 A totally bounded metric s...
equivbnd 37148 If the metric ` M ` is "st...
bnd2lem 37149 Lemma for ~ equivbnd2 and ...
equivbnd2 37150 If balls are totally bound...
prdsbnd 37151 The product metric over fi...
prdstotbnd 37152 The product metric over fi...
prdsbnd2 37153 If balls are totally bound...
cntotbnd 37154 A subset of the complex nu...
cnpwstotbnd 37155 A subset of ` A ^ I ` , wh...
ismtyval 37158 The set of isometries betw...
isismty 37159 The condition "is an isome...
ismtycnv 37160 The inverse of an isometry...
ismtyima 37161 The image of a ball under ...
ismtyhmeolem 37162 Lemma for ~ ismtyhmeo . (...
ismtyhmeo 37163 An isometry is a homeomorp...
ismtybndlem 37164 Lemma for ~ ismtybnd . (C...
ismtybnd 37165 Isometries preserve bounde...
ismtyres 37166 A restriction of an isomet...
heibor1lem 37167 Lemma for ~ heibor1 . A c...
heibor1 37168 One half of ~ heibor , tha...
heiborlem1 37169 Lemma for ~ heibor . We w...
heiborlem2 37170 Lemma for ~ heibor . Subs...
heiborlem3 37171 Lemma for ~ heibor . Usin...
heiborlem4 37172 Lemma for ~ heibor . Usin...
heiborlem5 37173 Lemma for ~ heibor . The ...
heiborlem6 37174 Lemma for ~ heibor . Sinc...
heiborlem7 37175 Lemma for ~ heibor . Sinc...
heiborlem8 37176 Lemma for ~ heibor . The ...
heiborlem9 37177 Lemma for ~ heibor . Disc...
heiborlem10 37178 Lemma for ~ heibor . The ...
heibor 37179 Generalized Heine-Borel Th...
bfplem1 37180 Lemma for ~ bfp . The seq...
bfplem2 37181 Lemma for ~ bfp . Using t...
bfp 37182 Banach fixed point theorem...
rrnval 37185 The n-dimensional Euclidea...
rrnmval 37186 The value of the Euclidean...
rrnmet 37187 Euclidean space is a metri...
rrndstprj1 37188 The distance between two p...
rrndstprj2 37189 Bound on the distance betw...
rrncmslem 37190 Lemma for ~ rrncms . (Con...
rrncms 37191 Euclidean space is complet...
repwsmet 37192 The supremum metric on ` R...
rrnequiv 37193 The supremum metric on ` R...
rrntotbnd 37194 A set in Euclidean space i...
rrnheibor 37195 Heine-Borel theorem for Eu...
ismrer1 37196 An isometry between ` RR `...
reheibor 37197 Heine-Borel theorem for re...
iccbnd 37198 A closed interval in ` RR ...
icccmpALT 37199 A closed interval in ` RR ...
isass 37204 The predicate "is an assoc...
isexid 37205 The predicate ` G ` has a ...
ismgmOLD 37208 Obsolete version of ~ ismg...
clmgmOLD 37209 Obsolete version of ~ mgmc...
opidonOLD 37210 Obsolete version of ~ mndp...
rngopidOLD 37211 Obsolete version of ~ mndp...
opidon2OLD 37212 Obsolete version of ~ mndp...
isexid2 37213 If ` G e. ( Magma i^i ExId...
exidu1 37214 Uniqueness of the left and...
idrval 37215 The value of the identity ...
iorlid 37216 A magma right and left ide...
cmpidelt 37217 A magma right and left ide...
smgrpismgmOLD 37220 Obsolete version of ~ sgrp...
issmgrpOLD 37221 Obsolete version of ~ issg...
smgrpmgm 37222 A semigroup is a magma. (...
smgrpassOLD 37223 Obsolete version of ~ sgrp...
mndoissmgrpOLD 37226 Obsolete version of ~ mnds...
mndoisexid 37227 A monoid has an identity e...
mndoismgmOLD 37228 Obsolete version of ~ mndm...
mndomgmid 37229 A monoid is a magma with a...
ismndo 37230 The predicate "is a monoid...
ismndo1 37231 The predicate "is a monoid...
ismndo2 37232 The predicate "is a monoid...
grpomndo 37233 A group is a monoid. (Con...
exidcl 37234 Closure of the binary oper...
exidreslem 37235 Lemma for ~ exidres and ~ ...
exidres 37236 The restriction of a binar...
exidresid 37237 The restriction of a binar...
ablo4pnp 37238 A commutative/associative ...
grpoeqdivid 37239 Two group elements are equ...
grposnOLD 37240 The group operation for th...
elghomlem1OLD 37243 Obsolete as of 15-Mar-2020...
elghomlem2OLD 37244 Obsolete as of 15-Mar-2020...
elghomOLD 37245 Obsolete version of ~ isgh...
ghomlinOLD 37246 Obsolete version of ~ ghml...
ghomidOLD 37247 Obsolete version of ~ ghmi...
ghomf 37248 Mapping property of a grou...
ghomco 37249 The composition of two gro...
ghomdiv 37250 Group homomorphisms preser...
grpokerinj 37251 A group homomorphism is in...
relrngo 37254 The class of all unital ri...
isrngo 37255 The predicate "is a (unita...
isrngod 37256 Conditions that determine ...
rngoi 37257 The properties of a unital...
rngosm 37258 Functionality of the multi...
rngocl 37259 Closure of the multiplicat...
rngoid 37260 The multiplication operati...
rngoideu 37261 The unity element of a rin...
rngodi 37262 Distributive law for the m...
rngodir 37263 Distributive law for the m...
rngoass 37264 Associative law for the mu...
rngo2 37265 A ring element plus itself...
rngoablo 37266 A ring's addition operatio...
rngoablo2 37267 In a unital ring the addit...
rngogrpo 37268 A ring's addition operatio...
rngone0 37269 The base set of a ring is ...
rngogcl 37270 Closure law for the additi...
rngocom 37271 The addition operation of ...
rngoaass 37272 The addition operation of ...
rngoa32 37273 The addition operation of ...
rngoa4 37274 Rearrangement of 4 terms i...
rngorcan 37275 Right cancellation law for...
rngolcan 37276 Left cancellation law for ...
rngo0cl 37277 A ring has an additive ide...
rngo0rid 37278 The additive identity of a...
rngo0lid 37279 The additive identity of a...
rngolz 37280 The zero of a unital ring ...
rngorz 37281 The zero of a unital ring ...
rngosn3 37282 Obsolete as of 25-Jan-2020...
rngosn4 37283 Obsolete as of 25-Jan-2020...
rngosn6 37284 Obsolete as of 25-Jan-2020...
rngonegcl 37285 A ring is closed under neg...
rngoaddneg1 37286 Adding the negative in a r...
rngoaddneg2 37287 Adding the negative in a r...
rngosub 37288 Subtraction in a ring, in ...
rngmgmbs4 37289 The range of an internal o...
rngodm1dm2 37290 In a unital ring the domai...
rngorn1 37291 In a unital ring the range...
rngorn1eq 37292 In a unital ring the range...
rngomndo 37293 In a unital ring the multi...
rngoidmlem 37294 The unity element of a rin...
rngolidm 37295 The unity element of a rin...
rngoridm 37296 The unity element of a rin...
rngo1cl 37297 The unity element of a rin...
rngoueqz 37298 Obsolete as of 23-Jan-2020...
rngonegmn1l 37299 Negation in a ring is the ...
rngonegmn1r 37300 Negation in a ring is the ...
rngoneglmul 37301 Negation of a product in a...
rngonegrmul 37302 Negation of a product in a...
rngosubdi 37303 Ring multiplication distri...
rngosubdir 37304 Ring multiplication distri...
zerdivemp1x 37305 In a unital ring a left in...
isdivrngo 37308 The predicate "is a divisi...
drngoi 37309 The properties of a divisi...
gidsn 37310 Obsolete as of 23-Jan-2020...
zrdivrng 37311 The zero ring is not a div...
dvrunz 37312 In a division ring the rin...
isgrpda 37313 Properties that determine ...
isdrngo1 37314 The predicate "is a divisi...
divrngcl 37315 The product of two nonzero...
isdrngo2 37316 A division ring is a ring ...
isdrngo3 37317 A division ring is a ring ...
rngohomval 37322 The set of ring homomorphi...
isrngohom 37323 The predicate "is a ring h...
rngohomf 37324 A ring homomorphism is a f...
rngohomcl 37325 Closure law for a ring hom...
rngohom1 37326 A ring homomorphism preser...
rngohomadd 37327 Ring homomorphisms preserv...
rngohommul 37328 Ring homomorphisms preserv...
rngogrphom 37329 A ring homomorphism is a g...
rngohom0 37330 A ring homomorphism preser...
rngohomsub 37331 Ring homomorphisms preserv...
rngohomco 37332 The composition of two rin...
rngokerinj 37333 A ring homomorphism is inj...
rngoisoval 37335 The set of ring isomorphis...
isrngoiso 37336 The predicate "is a ring i...
rngoiso1o 37337 A ring isomorphism is a bi...
rngoisohom 37338 A ring isomorphism is a ri...
rngoisocnv 37339 The inverse of a ring isom...
rngoisoco 37340 The composition of two rin...
isriscg 37342 The ring isomorphism relat...
isrisc 37343 The ring isomorphism relat...
risc 37344 The ring isomorphism relat...
risci 37345 Determine that two rings a...
riscer 37346 Ring isomorphism is an equ...
iscom2 37353 A device to add commutativ...
iscrngo 37354 The predicate "is a commut...
iscrngo2 37355 The predicate "is a commut...
iscringd 37356 Conditions that determine ...
flddivrng 37357 A field is a division ring...
crngorngo 37358 A commutative ring is a ri...
crngocom 37359 The multiplication operati...
crngm23 37360 Commutative/associative la...
crngm4 37361 Commutative/associative la...
fldcrngo 37362 A field is a commutative r...
isfld2 37363 The predicate "is a field"...
crngohomfo 37364 The image of a homomorphis...
idlval 37371 The class of ideals of a r...
isidl 37372 The predicate "is an ideal...
isidlc 37373 The predicate "is an ideal...
idlss 37374 An ideal of ` R ` is a sub...
idlcl 37375 An element of an ideal is ...
idl0cl 37376 An ideal contains ` 0 ` . ...
idladdcl 37377 An ideal is closed under a...
idllmulcl 37378 An ideal is closed under m...
idlrmulcl 37379 An ideal is closed under m...
idlnegcl 37380 An ideal is closed under n...
idlsubcl 37381 An ideal is closed under s...
rngoidl 37382 A ring ` R ` is an ` R ` i...
0idl 37383 The set containing only ` ...
1idl 37384 Two ways of expressing the...
0rngo 37385 In a ring, ` 0 = 1 ` iff t...
divrngidl 37386 The only ideals in a divis...
intidl 37387 The intersection of a none...
inidl 37388 The intersection of two id...
unichnidl 37389 The union of a nonempty ch...
keridl 37390 The kernel of a ring homom...
pridlval 37391 The class of prime ideals ...
ispridl 37392 The predicate "is a prime ...
pridlidl 37393 A prime ideal is an ideal....
pridlnr 37394 A prime ideal is a proper ...
pridl 37395 The main property of a pri...
ispridl2 37396 A condition that shows an ...
maxidlval 37397 The set of maximal ideals ...
ismaxidl 37398 The predicate "is a maxima...
maxidlidl 37399 A maximal ideal is an idea...
maxidlnr 37400 A maximal ideal is proper....
maxidlmax 37401 A maximal ideal is a maxim...
maxidln1 37402 One is not contained in an...
maxidln0 37403 A ring with a maximal idea...
isprrngo 37408 The predicate "is a prime ...
prrngorngo 37409 A prime ring is a ring. (...
smprngopr 37410 A simple ring (one whose o...
divrngpr 37411 A division ring is a prime...
isdmn 37412 The predicate "is a domain...
isdmn2 37413 The predicate "is a domain...
dmncrng 37414 A domain is a commutative ...
dmnrngo 37415 A domain is a ring. (Cont...
flddmn 37416 A field is a domain. (Con...
igenval 37419 The ideal generated by a s...
igenss 37420 A set is a subset of the i...
igenidl 37421 The ideal generated by a s...
igenmin 37422 The ideal generated by a s...
igenidl2 37423 The ideal generated by an ...
igenval2 37424 The ideal generated by a s...
prnc 37425 A principal ideal (an idea...
isfldidl 37426 Determine if a ring is a f...
isfldidl2 37427 Determine if a ring is a f...
ispridlc 37428 The predicate "is a prime ...
pridlc 37429 Property of a prime ideal ...
pridlc2 37430 Property of a prime ideal ...
pridlc3 37431 Property of a prime ideal ...
isdmn3 37432 The predicate "is a domain...
dmnnzd 37433 A domain has no zero-divis...
dmncan1 37434 Cancellation law for domai...
dmncan2 37435 Cancellation law for domai...
efald2 37436 A proof by contradiction. ...
notbinot1 37437 Simplification rule of neg...
bicontr 37438 Biconditional of its own n...
impor 37439 An equivalent formula for ...
orfa 37440 The falsum ` F. ` can be r...
notbinot2 37441 Commutation rule between n...
biimpor 37442 A rewriting rule for bicon...
orfa1 37443 Add a contradicting disjun...
orfa2 37444 Remove a contradicting dis...
bifald 37445 Infer the equivalence to a...
orsild 37446 A lemma for not-or-not eli...
orsird 37447 A lemma for not-or-not eli...
cnf1dd 37448 A lemma for Conjunctive No...
cnf2dd 37449 A lemma for Conjunctive No...
cnfn1dd 37450 A lemma for Conjunctive No...
cnfn2dd 37451 A lemma for Conjunctive No...
or32dd 37452 A rearrangement of disjunc...
notornotel1 37453 A lemma for not-or-not eli...
notornotel2 37454 A lemma for not-or-not eli...
contrd 37455 A proof by contradiction, ...
an12i 37456 An inference from commutin...
exmid2 37457 An excluded middle law. (...
selconj 37458 An inference for selecting...
truconj 37459 Add true as a conjunct. (...
orel 37460 An inference for disjuncti...
negel 37461 An inference for negation ...
botel 37462 An inference for bottom el...
tradd 37463 Add top ad a conjunct. (C...
gm-sbtru 37464 Substitution does not chan...
sbfal 37465 Substitution does not chan...
sbcani 37466 Distribution of class subs...
sbcori 37467 Distribution of class subs...
sbcimi 37468 Distribution of class subs...
sbcni 37469 Move class substitution in...
sbali 37470 Discard class substitution...
sbexi 37471 Discard class substitution...
sbcalf 37472 Move universal quantifier ...
sbcexf 37473 Move existential quantifie...
sbcalfi 37474 Move universal quantifier ...
sbcexfi 37475 Move existential quantifie...
spsbcdi 37476 A lemma for eliminating a ...
alrimii 37477 A lemma for introducing a ...
spesbcdi 37478 A lemma for introducing an...
exlimddvf 37479 A lemma for eliminating an...
exlimddvfi 37480 A lemma for eliminating an...
sbceq1ddi 37481 A lemma for eliminating in...
sbccom2lem 37482 Lemma for ~ sbccom2 . (Co...
sbccom2 37483 Commutative law for double...
sbccom2f 37484 Commutative law for double...
sbccom2fi 37485 Commutative law for double...
csbcom2fi 37486 Commutative law for double...
fald 37487 Refutation of falsity, in ...
tsim1 37488 A Tseitin axiom for logica...
tsim2 37489 A Tseitin axiom for logica...
tsim3 37490 A Tseitin axiom for logica...
tsbi1 37491 A Tseitin axiom for logica...
tsbi2 37492 A Tseitin axiom for logica...
tsbi3 37493 A Tseitin axiom for logica...
tsbi4 37494 A Tseitin axiom for logica...
tsxo1 37495 A Tseitin axiom for logica...
tsxo2 37496 A Tseitin axiom for logica...
tsxo3 37497 A Tseitin axiom for logica...
tsxo4 37498 A Tseitin axiom for logica...
tsan1 37499 A Tseitin axiom for logica...
tsan2 37500 A Tseitin axiom for logica...
tsan3 37501 A Tseitin axiom for logica...
tsna1 37502 A Tseitin axiom for logica...
tsna2 37503 A Tseitin axiom for logica...
tsna3 37504 A Tseitin axiom for logica...
tsor1 37505 A Tseitin axiom for logica...
tsor2 37506 A Tseitin axiom for logica...
tsor3 37507 A Tseitin axiom for logica...
ts3an1 37508 A Tseitin axiom for triple...
ts3an2 37509 A Tseitin axiom for triple...
ts3an3 37510 A Tseitin axiom for triple...
ts3or1 37511 A Tseitin axiom for triple...
ts3or2 37512 A Tseitin axiom for triple...
ts3or3 37513 A Tseitin axiom for triple...
iuneq2f 37514 Equality deduction for ind...
rabeq12f 37515 Equality deduction for res...
csbeq12 37516 Equality deduction for sub...
sbeqi 37517 Equality deduction for sub...
ralbi12f 37518 Equality deduction for res...
oprabbi 37519 Equality deduction for cla...
mpobi123f 37520 Equality deduction for map...
iuneq12f 37521 Equality deduction for ind...
iineq12f 37522 Equality deduction for ind...
opabbi 37523 Equality deduction for cla...
mptbi12f 37524 Equality deduction for map...
orcomdd 37525 Commutativity of logic dis...
scottexf 37526 A version of ~ scottex wit...
scott0f 37527 A version of ~ scott0 with...
scottn0f 37528 A version of ~ scott0f wit...
ac6s3f 37529 Generalization of the Axio...
ac6s6 37530 Generalization of the Axio...
ac6s6f 37531 Generalization of the Axio...
el2v1 37575 New way ( ~ elv , and the ...
el3v 37576 New way ( ~ elv , and the ...
el3v1 37577 New way ( ~ elv , and the ...
el3v2 37578 New way ( ~ elv , and the ...
el3v3 37579 New way ( ~ elv , and the ...
el3v12 37580 New way ( ~ elv , and the ...
el3v13 37581 New way ( ~ elv , and the ...
el3v23 37582 New way ( ~ elv , and the ...
anan 37583 Multiple commutations in c...
triantru3 37584 A wff is equivalent to its...
bianbi 37585 Exchanging conjunction in ...
bianim 37586 Exchanging conjunction in ...
biorfd 37587 A wff is equivalent to its...
eqbrtr 37588 Substitution of equal clas...
eqbrb 37589 Substitution of equal clas...
eqeltr 37590 Substitution of equal clas...
eqelb 37591 Substitution of equal clas...
eqeqan2d 37592 Implication of introducing...
suceqsneq 37593 One-to-one relationship be...
sucdifsn2 37594 Absorption of union with a...
sucdifsn 37595 The difference between the...
disjresin 37596 The restriction to a disjo...
disjresdisj 37597 The intersection of restri...
disjresdif 37598 The difference between res...
disjresundif 37599 Lemma for ~ ressucdifsn2 ....
ressucdifsn2 37600 The difference between res...
ressucdifsn 37601 The difference between res...
inres2 37602 Two ways of expressing the...
coideq 37603 Equality theorem for compo...
nexmo1 37604 If there is no case where ...
ralin 37605 Restricted universal quant...
r2alan 37606 Double restricted universa...
ssrabi 37607 Inference of restricted ab...
rabbieq 37608 Equivalent wff's correspon...
rabimbieq 37609 Restricted equivalent wff'...
abeqin 37610 Intersection with class ab...
abeqinbi 37611 Intersection with class ab...
rabeqel 37612 Class element of a restric...
eqrelf 37613 The equality connective be...
br1cnvinxp 37614 Binary relation on the con...
releleccnv 37615 Elementhood in a converse ...
releccnveq 37616 Equality of converse ` R `...
opelvvdif 37617 Negated elementhood of ord...
vvdifopab 37618 Ordered-pair class abstrac...
brvdif 37619 Binary relation with unive...
brvdif2 37620 Binary relation with unive...
brvvdif 37621 Binary relation with the c...
brvbrvvdif 37622 Binary relation with the c...
brcnvep 37623 The converse of the binary...
elecALTV 37624 Elementhood in the ` R ` -...
brcnvepres 37625 Restricted converse epsilo...
brres2 37626 Binary relation on a restr...
br1cnvres 37627 Binary relation on the con...
eldmres 37628 Elementhood in the domain ...
elrnres 37629 Element of the range of a ...
eldmressnALTV 37630 Element of the domain of a...
elrnressn 37631 Element of the range of a ...
eldm4 37632 Elementhood in a domain. ...
eldmres2 37633 Elementhood in the domain ...
eceq1i 37634 Equality theorem for ` C `...
elecres 37635 Elementhood in the restric...
ecres 37636 Restricted coset of ` B ` ...
ecres2 37637 The restricted coset of ` ...
eccnvepres 37638 Restricted converse epsilo...
eleccnvep 37639 Elementhood in the convers...
eccnvep 37640 The converse epsilon coset...
extep 37641 Property of epsilon relati...
disjeccnvep 37642 Property of the epsilon re...
eccnvepres2 37643 The restricted converse ep...
eccnvepres3 37644 Condition for a restricted...
eldmqsres 37645 Elementhood in a restricte...
eldmqsres2 37646 Elementhood in a restricte...
qsss1 37647 Subclass theorem for quoti...
qseq1i 37648 Equality theorem for quoti...
qseq1d 37649 Equality theorem for quoti...
brinxprnres 37650 Binary relation on a restr...
inxprnres 37651 Restriction of a class as ...
dfres4 37652 Alternate definition of th...
exan3 37653 Equivalent expressions wit...
exanres 37654 Equivalent expressions wit...
exanres3 37655 Equivalent expressions wit...
exanres2 37656 Equivalent expressions wit...
cnvepres 37657 Restricted converse epsilo...
eqrel2 37658 Equality of relations. (C...
rncnv 37659 Range of converse is the d...
dfdm6 37660 Alternate definition of do...
dfrn6 37661 Alternate definition of ra...
rncnvepres 37662 The range of the restricte...
dmecd 37663 Equality of the coset of `...
dmec2d 37664 Equality of the coset of `...
brid 37665 Property of the identity b...
ideq2 37666 For sets, the identity bin...
idresssidinxp 37667 Condition for the identity...
idreseqidinxp 37668 Condition for the identity...
extid 37669 Property of identity relat...
inxpss 37670 Two ways to say that an in...
idinxpss 37671 Two ways to say that an in...
ref5 37672 Two ways to say that an in...
inxpss3 37673 Two ways to say that an in...
inxpss2 37674 Two ways to say that inter...
inxpssidinxp 37675 Two ways to say that inter...
idinxpssinxp 37676 Two ways to say that inter...
idinxpssinxp2 37677 Identity intersection with...
idinxpssinxp3 37678 Identity intersection with...
idinxpssinxp4 37679 Identity intersection with...
relcnveq3 37680 Two ways of saying a relat...
relcnveq 37681 Two ways of saying a relat...
relcnveq2 37682 Two ways of saying a relat...
relcnveq4 37683 Two ways of saying a relat...
qsresid 37684 Simplification of a specia...
n0elqs 37685 Two ways of expressing tha...
n0elqs2 37686 Two ways of expressing tha...
ecex2 37687 Condition for a coset to b...
uniqsALTV 37688 The union of a quotient se...
imaexALTV 37689 Existence of an image of a...
ecexALTV 37690 Existence of a coset, like...
rnresequniqs 37691 The range of a restriction...
n0el2 37692 Two ways of expressing tha...
cnvepresex 37693 Sethood condition for the ...
eccnvepex 37694 The converse epsilon coset...
cnvepimaex 37695 The image of converse epsi...
cnvepima 37696 The image of converse epsi...
inex3 37697 Sufficient condition for t...
inxpex 37698 Sufficient condition for a...
eqres 37699 Converting a class constan...
brrabga 37700 The law of concretion for ...
brcnvrabga 37701 The law of concretion for ...
opideq 37702 Equality conditions for or...
iss2 37703 A subclass of the identity...
eldmcnv 37704 Elementhood in a domain of...
dfrel5 37705 Alternate definition of th...
dfrel6 37706 Alternate definition of th...
cnvresrn 37707 Converse restricted to ran...
relssinxpdmrn 37708 Subset of restriction, spe...
cnvref4 37709 Two ways to say that a rel...
cnvref5 37710 Two ways to say that a rel...
ecin0 37711 Two ways of saying that th...
ecinn0 37712 Two ways of saying that th...
ineleq 37713 Equivalence of restricted ...
inecmo 37714 Equivalence of a double re...
inecmo2 37715 Equivalence of a double re...
ineccnvmo 37716 Equivalence of a double re...
alrmomorn 37717 Equivalence of an "at most...
alrmomodm 37718 Equivalence of an "at most...
ineccnvmo2 37719 Equivalence of a double un...
inecmo3 37720 Equivalence of a double un...
moeu2 37721 Uniqueness is equivalent t...
mopickr 37722 "At most one" picks a vari...
moantr 37723 Sufficient condition for t...
brabidgaw 37724 The law of concretion for ...
brabidga 37725 The law of concretion for ...
inxp2 37726 Intersection with a Cartes...
opabf 37727 A class abstraction of a c...
ec0 37728 The empty-coset of a class...
0qs 37729 Quotient set with the empt...
brcnvin 37730 Intersection with a conver...
xrnss3v 37732 A range Cartesian product ...
xrnrel 37733 A range Cartesian product ...
brxrn 37734 Characterize a ternary rel...
brxrn2 37735 A characterization of the ...
dfxrn2 37736 Alternate definition of th...
xrneq1 37737 Equality theorem for the r...
xrneq1i 37738 Equality theorem for the r...
xrneq1d 37739 Equality theorem for the r...
xrneq2 37740 Equality theorem for the r...
xrneq2i 37741 Equality theorem for the r...
xrneq2d 37742 Equality theorem for the r...
xrneq12 37743 Equality theorem for the r...
xrneq12i 37744 Equality theorem for the r...
xrneq12d 37745 Equality theorem for the r...
elecxrn 37746 Elementhood in the ` ( R |...
ecxrn 37747 The ` ( R |X. S ) ` -coset...
disjressuc2 37748 Double restricted quantifi...
disjecxrn 37749 Two ways of saying that ` ...
disjecxrncnvep 37750 Two ways of saying that co...
disjsuc2 37751 Double restricted quantifi...
xrninxp 37752 Intersection of a range Ca...
xrninxp2 37753 Intersection of a range Ca...
xrninxpex 37754 Sufficient condition for t...
inxpxrn 37755 Two ways to express the in...
br1cnvxrn2 37756 The converse of a binary r...
elec1cnvxrn2 37757 Elementhood in the convers...
rnxrn 37758 Range of the range Cartesi...
rnxrnres 37759 Range of a range Cartesian...
rnxrncnvepres 37760 Range of a range Cartesian...
rnxrnidres 37761 Range of a range Cartesian...
xrnres 37762 Two ways to express restri...
xrnres2 37763 Two ways to express restri...
xrnres3 37764 Two ways to express restri...
xrnres4 37765 Two ways to express restri...
xrnresex 37766 Sufficient condition for a...
xrnidresex 37767 Sufficient condition for a...
xrncnvepresex 37768 Sufficient condition for a...
brin2 37769 Binary relation on an inte...
brin3 37770 Binary relation on an inte...
dfcoss2 37773 Alternate definition of th...
dfcoss3 37774 Alternate definition of th...
dfcoss4 37775 Alternate definition of th...
cosscnv 37776 Class of cosets by the con...
coss1cnvres 37777 Class of cosets by the con...
coss2cnvepres 37778 Special case of ~ coss1cnv...
cossex 37779 If ` A ` is a set then the...
cosscnvex 37780 If ` A ` is a set then the...
1cosscnvepresex 37781 Sufficient condition for a...
1cossxrncnvepresex 37782 Sufficient condition for a...
relcoss 37783 Cosets by ` R ` is a relat...
relcoels 37784 Coelements on ` A ` is a r...
cossss 37785 Subclass theorem for the c...
cosseq 37786 Equality theorem for the c...
cosseqi 37787 Equality theorem for the c...
cosseqd 37788 Equality theorem for the c...
1cossres 37789 The class of cosets by a r...
dfcoels 37790 Alternate definition of th...
brcoss 37791 ` A ` and ` B ` are cosets...
brcoss2 37792 Alternate form of the ` A ...
brcoss3 37793 Alternate form of the ` A ...
brcosscnvcoss 37794 For sets, the ` A ` and ` ...
brcoels 37795 ` B ` and ` C ` are coelem...
cocossss 37796 Two ways of saying that co...
cnvcosseq 37797 The converse of cosets by ...
br2coss 37798 Cosets by ` ,~ R ` binary ...
br1cossres 37799 ` B ` and ` C ` are cosets...
br1cossres2 37800 ` B ` and ` C ` are cosets...
brressn 37801 Binary relation on a restr...
ressn2 37802 A class ' R ' restricted t...
refressn 37803 Any class ' R ' restricted...
antisymressn 37804 Every class ' R ' restrict...
trressn 37805 Any class ' R ' restricted...
relbrcoss 37806 ` A ` and ` B ` are cosets...
br1cossinres 37807 ` B ` and ` C ` are cosets...
br1cossxrnres 37808 ` <. B , C >. ` and ` <. D...
br1cossinidres 37809 ` B ` and ` C ` are cosets...
br1cossincnvepres 37810 ` B ` and ` C ` are cosets...
br1cossxrnidres 37811 ` <. B , C >. ` and ` <. D...
br1cossxrncnvepres 37812 ` <. B , C >. ` and ` <. D...
dmcoss3 37813 The domain of cosets is th...
dmcoss2 37814 The domain of cosets is th...
rncossdmcoss 37815 The range of cosets is the...
dm1cosscnvepres 37816 The domain of cosets of th...
dmcoels 37817 The domain of coelements i...
eldmcoss 37818 Elementhood in the domain ...
eldmcoss2 37819 Elementhood in the domain ...
eldm1cossres 37820 Elementhood in the domain ...
eldm1cossres2 37821 Elementhood in the domain ...
refrelcosslem 37822 Lemma for the left side of...
refrelcoss3 37823 The class of cosets by ` R...
refrelcoss2 37824 The class of cosets by ` R...
symrelcoss3 37825 The class of cosets by ` R...
symrelcoss2 37826 The class of cosets by ` R...
cossssid 37827 Equivalent expressions for...
cossssid2 37828 Equivalent expressions for...
cossssid3 37829 Equivalent expressions for...
cossssid4 37830 Equivalent expressions for...
cossssid5 37831 Equivalent expressions for...
brcosscnv 37832 ` A ` and ` B ` are cosets...
brcosscnv2 37833 ` A ` and ` B ` are cosets...
br1cosscnvxrn 37834 ` A ` and ` B ` are cosets...
1cosscnvxrn 37835 Cosets by the converse ran...
cosscnvssid3 37836 Equivalent expressions for...
cosscnvssid4 37837 Equivalent expressions for...
cosscnvssid5 37838 Equivalent expressions for...
coss0 37839 Cosets by the empty set ar...
cossid 37840 Cosets by the identity rel...
cosscnvid 37841 Cosets by the converse ide...
trcoss 37842 Sufficient condition for t...
eleccossin 37843 Two ways of saying that th...
trcoss2 37844 Equivalent expressions for...
elrels2 37846 The element of the relatio...
elrelsrel 37847 The element of the relatio...
elrelsrelim 37848 The element of the relatio...
elrels5 37849 Equivalent expressions for...
elrels6 37850 Equivalent expressions for...
elrelscnveq3 37851 Two ways of saying a relat...
elrelscnveq 37852 Two ways of saying a relat...
elrelscnveq2 37853 Two ways of saying a relat...
elrelscnveq4 37854 Two ways of saying a relat...
cnvelrels 37855 The converse of a set is a...
cosselrels 37856 Cosets of sets are element...
cosscnvelrels 37857 Cosets of converse sets ar...
dfssr2 37859 Alternate definition of th...
relssr 37860 The subset relation is a r...
brssr 37861 The subset relation and su...
brssrid 37862 Any set is a subset of its...
issetssr 37863 Two ways of expressing set...
brssrres 37864 Restricted subset binary r...
br1cnvssrres 37865 Restricted converse subset...
brcnvssr 37866 The converse of a subset r...
brcnvssrid 37867 Any set is a converse subs...
br1cossxrncnvssrres 37868 ` <. B , C >. ` and ` <. D...
extssr 37869 Property of subset relatio...
dfrefrels2 37873 Alternate definition of th...
dfrefrels3 37874 Alternate definition of th...
dfrefrel2 37875 Alternate definition of th...
dfrefrel3 37876 Alternate definition of th...
dfrefrel5 37877 Alternate definition of th...
elrefrels2 37878 Element of the class of re...
elrefrels3 37879 Element of the class of re...
elrefrelsrel 37880 For sets, being an element...
refreleq 37881 Equality theorem for refle...
refrelid 37882 Identity relation is refle...
refrelcoss 37883 The class of cosets by ` R...
refrelressn 37884 Any class ' R ' restricted...
dfcnvrefrels2 37888 Alternate definition of th...
dfcnvrefrels3 37889 Alternate definition of th...
dfcnvrefrel2 37890 Alternate definition of th...
dfcnvrefrel3 37891 Alternate definition of th...
dfcnvrefrel4 37892 Alternate definition of th...
dfcnvrefrel5 37893 Alternate definition of th...
elcnvrefrels2 37894 Element of the class of co...
elcnvrefrels3 37895 Element of the class of co...
elcnvrefrelsrel 37896 For sets, being an element...
cnvrefrelcoss2 37897 Necessary and sufficient c...
cosselcnvrefrels2 37898 Necessary and sufficient c...
cosselcnvrefrels3 37899 Necessary and sufficient c...
cosselcnvrefrels4 37900 Necessary and sufficient c...
cosselcnvrefrels5 37901 Necessary and sufficient c...
dfsymrels2 37905 Alternate definition of th...
dfsymrels3 37906 Alternate definition of th...
dfsymrels4 37907 Alternate definition of th...
dfsymrels5 37908 Alternate definition of th...
dfsymrel2 37909 Alternate definition of th...
dfsymrel3 37910 Alternate definition of th...
dfsymrel4 37911 Alternate definition of th...
dfsymrel5 37912 Alternate definition of th...
elsymrels2 37913 Element of the class of sy...
elsymrels3 37914 Element of the class of sy...
elsymrels4 37915 Element of the class of sy...
elsymrels5 37916 Element of the class of sy...
elsymrelsrel 37917 For sets, being an element...
symreleq 37918 Equality theorem for symme...
symrelim 37919 Symmetric relation implies...
symrelcoss 37920 The class of cosets by ` R...
idsymrel 37921 The identity relation is s...
epnsymrel 37922 The membership (epsilon) r...
symrefref2 37923 Symmetry is a sufficient c...
symrefref3 37924 Symmetry is a sufficient c...
refsymrels2 37925 Elements of the class of r...
refsymrels3 37926 Elements of the class of r...
refsymrel2 37927 A relation which is reflex...
refsymrel3 37928 A relation which is reflex...
elrefsymrels2 37929 Elements of the class of r...
elrefsymrels3 37930 Elements of the class of r...
elrefsymrelsrel 37931 For sets, being an element...
dftrrels2 37935 Alternate definition of th...
dftrrels3 37936 Alternate definition of th...
dftrrel2 37937 Alternate definition of th...
dftrrel3 37938 Alternate definition of th...
eltrrels2 37939 Element of the class of tr...
eltrrels3 37940 Element of the class of tr...
eltrrelsrel 37941 For sets, being an element...
trreleq 37942 Equality theorem for the t...
trrelressn 37943 Any class ' R ' restricted...
dfeqvrels2 37948 Alternate definition of th...
dfeqvrels3 37949 Alternate definition of th...
dfeqvrel2 37950 Alternate definition of th...
dfeqvrel3 37951 Alternate definition of th...
eleqvrels2 37952 Element of the class of eq...
eleqvrels3 37953 Element of the class of eq...
eleqvrelsrel 37954 For sets, being an element...
elcoeleqvrels 37955 Elementhood in the coeleme...
elcoeleqvrelsrel 37956 For sets, being an element...
eqvrelrel 37957 An equivalence relation is...
eqvrelrefrel 37958 An equivalence relation is...
eqvrelsymrel 37959 An equivalence relation is...
eqvreltrrel 37960 An equivalence relation is...
eqvrelim 37961 Equivalence relation impli...
eqvreleq 37962 Equality theorem for equiv...
eqvreleqi 37963 Equality theorem for equiv...
eqvreleqd 37964 Equality theorem for equiv...
eqvrelsym 37965 An equivalence relation is...
eqvrelsymb 37966 An equivalence relation is...
eqvreltr 37967 An equivalence relation is...
eqvreltrd 37968 A transitivity relation fo...
eqvreltr4d 37969 A transitivity relation fo...
eqvrelref 37970 An equivalence relation is...
eqvrelth 37971 Basic property of equivale...
eqvrelcl 37972 Elementhood in the field o...
eqvrelthi 37973 Basic property of equivale...
eqvreldisj 37974 Equivalence classes do not...
qsdisjALTV 37975 Elements of a quotient set...
eqvrelqsel 37976 If an element of a quotien...
eqvrelcoss 37977 Two ways to express equiva...
eqvrelcoss3 37978 Two ways to express equiva...
eqvrelcoss2 37979 Two ways to express equiva...
eqvrelcoss4 37980 Two ways to express equiva...
dfcoeleqvrels 37981 Alternate definition of th...
dfcoeleqvrel 37982 Alternate definition of th...
brredunds 37986 Binary relation on the cla...
brredundsredund 37987 For sets, binary relation ...
redundss3 37988 Implication of redundancy ...
redundeq1 37989 Equivalence of redundancy ...
redundpim3 37990 Implication of redundancy ...
redundpbi1 37991 Equivalence of redundancy ...
refrelsredund4 37992 The naive version of the c...
refrelsredund2 37993 The naive version of the c...
refrelsredund3 37994 The naive version of the c...
refrelredund4 37995 The naive version of the d...
refrelredund2 37996 The naive version of the d...
refrelredund3 37997 The naive version of the d...
dmqseq 38000 Equality theorem for domai...
dmqseqi 38001 Equality theorem for domai...
dmqseqd 38002 Equality theorem for domai...
dmqseqeq1 38003 Equality theorem for domai...
dmqseqeq1i 38004 Equality theorem for domai...
dmqseqeq1d 38005 Equality theorem for domai...
brdmqss 38006 The domain quotient binary...
brdmqssqs 38007 If ` A ` and ` R ` are set...
n0eldmqs 38008 The empty set is not an el...
n0eldmqseq 38009 The empty set is not an el...
n0elim 38010 Implication of that the em...
n0el3 38011 Two ways of expressing tha...
cnvepresdmqss 38012 The domain quotient binary...
cnvepresdmqs 38013 The domain quotient predic...
unidmqs 38014 The range of a relation is...
unidmqseq 38015 The union of the domain qu...
dmqseqim 38016 If the domain quotient of ...
dmqseqim2 38017 Lemma for ~ erimeq2 . (Co...
releldmqs 38018 Elementhood in the domain ...
eldmqs1cossres 38019 Elementhood in the domain ...
releldmqscoss 38020 Elementhood in the domain ...
dmqscoelseq 38021 Two ways to express the eq...
dmqs1cosscnvepreseq 38022 Two ways to express the eq...
brers 38027 Binary equivalence relatio...
dferALTV2 38028 Equivalence relation with ...
erALTVeq1 38029 Equality theorem for equiv...
erALTVeq1i 38030 Equality theorem for equiv...
erALTVeq1d 38031 Equality theorem for equiv...
dfcomember 38032 Alternate definition of th...
dfcomember2 38033 Alternate definition of th...
dfcomember3 38034 Alternate definition of th...
eqvreldmqs 38035 Two ways to express comemb...
eqvreldmqs2 38036 Two ways to express comemb...
brerser 38037 Binary equivalence relatio...
erimeq2 38038 Equivalence relation on it...
erimeq 38039 Equivalence relation on it...
dffunsALTV 38043 Alternate definition of th...
dffunsALTV2 38044 Alternate definition of th...
dffunsALTV3 38045 Alternate definition of th...
dffunsALTV4 38046 Alternate definition of th...
dffunsALTV5 38047 Alternate definition of th...
dffunALTV2 38048 Alternate definition of th...
dffunALTV3 38049 Alternate definition of th...
dffunALTV4 38050 Alternate definition of th...
dffunALTV5 38051 Alternate definition of th...
elfunsALTV 38052 Elementhood in the class o...
elfunsALTV2 38053 Elementhood in the class o...
elfunsALTV3 38054 Elementhood in the class o...
elfunsALTV4 38055 Elementhood in the class o...
elfunsALTV5 38056 Elementhood in the class o...
elfunsALTVfunALTV 38057 The element of the class o...
funALTVfun 38058 Our definition of the func...
funALTVss 38059 Subclass theorem for funct...
funALTVeq 38060 Equality theorem for funct...
funALTVeqi 38061 Equality inference for the...
funALTVeqd 38062 Equality deduction for the...
dfdisjs 38068 Alternate definition of th...
dfdisjs2 38069 Alternate definition of th...
dfdisjs3 38070 Alternate definition of th...
dfdisjs4 38071 Alternate definition of th...
dfdisjs5 38072 Alternate definition of th...
dfdisjALTV 38073 Alternate definition of th...
dfdisjALTV2 38074 Alternate definition of th...
dfdisjALTV3 38075 Alternate definition of th...
dfdisjALTV4 38076 Alternate definition of th...
dfdisjALTV5 38077 Alternate definition of th...
dfeldisj2 38078 Alternate definition of th...
dfeldisj3 38079 Alternate definition of th...
dfeldisj4 38080 Alternate definition of th...
dfeldisj5 38081 Alternate definition of th...
eldisjs 38082 Elementhood in the class o...
eldisjs2 38083 Elementhood in the class o...
eldisjs3 38084 Elementhood in the class o...
eldisjs4 38085 Elementhood in the class o...
eldisjs5 38086 Elementhood in the class o...
eldisjsdisj 38087 The element of the class o...
eleldisjs 38088 Elementhood in the disjoin...
eleldisjseldisj 38089 The element of the disjoin...
disjrel 38090 Disjoint relation is a rel...
disjss 38091 Subclass theorem for disjo...
disjssi 38092 Subclass theorem for disjo...
disjssd 38093 Subclass theorem for disjo...
disjeq 38094 Equality theorem for disjo...
disjeqi 38095 Equality theorem for disjo...
disjeqd 38096 Equality theorem for disjo...
disjdmqseqeq1 38097 Lemma for the equality the...
eldisjss 38098 Subclass theorem for disjo...
eldisjssi 38099 Subclass theorem for disjo...
eldisjssd 38100 Subclass theorem for disjo...
eldisjeq 38101 Equality theorem for disjo...
eldisjeqi 38102 Equality theorem for disjo...
eldisjeqd 38103 Equality theorem for disjo...
disjres 38104 Disjoint restriction. (Co...
eldisjn0elb 38105 Two forms of disjoint elem...
disjxrn 38106 Two ways of saying that a ...
disjxrnres5 38107 Disjoint range Cartesian p...
disjorimxrn 38108 Disjointness condition for...
disjimxrn 38109 Disjointness condition for...
disjimres 38110 Disjointness condition for...
disjimin 38111 Disjointness condition for...
disjiminres 38112 Disjointness condition for...
disjimxrnres 38113 Disjointness condition for...
disjALTV0 38114 The null class is disjoint...
disjALTVid 38115 The class of identity rela...
disjALTVidres 38116 The class of identity rela...
disjALTVinidres 38117 The intersection with rest...
disjALTVxrnidres 38118 The class of range Cartesi...
disjsuc 38119 Disjoint range Cartesian p...
dfantisymrel4 38121 Alternate definition of th...
dfantisymrel5 38122 Alternate definition of th...
antisymrelres 38123 (Contributed by Peter Mazs...
antisymrelressn 38124 (Contributed by Peter Mazs...
dfpart2 38129 Alternate definition of th...
dfmembpart2 38130 Alternate definition of th...
brparts 38131 Binary partitions relation...
brparts2 38132 Binary partitions relation...
brpartspart 38133 Binary partition and the p...
parteq1 38134 Equality theorem for parti...
parteq2 38135 Equality theorem for parti...
parteq12 38136 Equality theorem for parti...
parteq1i 38137 Equality theorem for parti...
parteq1d 38138 Equality theorem for parti...
partsuc2 38139 Property of the partition....
partsuc 38140 Property of the partition....
disjim 38141 The "Divide et Aequivalere...
disjimi 38142 Every disjoint relation ge...
detlem 38143 If a relation is disjoint,...
eldisjim 38144 If the elements of ` A ` a...
eldisjim2 38145 Alternate form of ~ eldisj...
eqvrel0 38146 The null class is an equiv...
det0 38147 The cosets by the null cla...
eqvrelcoss0 38148 The cosets by the null cla...
eqvrelid 38149 The identity relation is a...
eqvrel1cossidres 38150 The cosets by a restricted...
eqvrel1cossinidres 38151 The cosets by an intersect...
eqvrel1cossxrnidres 38152 The cosets by a range Cart...
detid 38153 The cosets by the identity...
eqvrelcossid 38154 The cosets by the identity...
detidres 38155 The cosets by the restrict...
detinidres 38156 The cosets by the intersec...
detxrnidres 38157 The cosets by the range Ca...
disjlem14 38158 Lemma for ~ disjdmqseq , ~...
disjlem17 38159 Lemma for ~ disjdmqseq , ~...
disjlem18 38160 Lemma for ~ disjdmqseq , ~...
disjlem19 38161 Lemma for ~ disjdmqseq , ~...
disjdmqsss 38162 Lemma for ~ disjdmqseq via...
disjdmqscossss 38163 Lemma for ~ disjdmqseq via...
disjdmqs 38164 If a relation is disjoint,...
disjdmqseq 38165 If a relation is disjoint,...
eldisjn0el 38166 Special case of ~ disjdmqs...
partim2 38167 Disjoint relation on its n...
partim 38168 Partition implies equivale...
partimeq 38169 Partition implies that the...
eldisjlem19 38170 Special case of ~ disjlem1...
membpartlem19 38171 Together with ~ disjlem19 ...
petlem 38172 If you can prove that the ...
petlemi 38173 If you can prove disjointn...
pet02 38174 Class ` A ` is a partition...
pet0 38175 Class ` A ` is a partition...
petid2 38176 Class ` A ` is a partition...
petid 38177 A class is a partition by ...
petidres2 38178 Class ` A ` is a partition...
petidres 38179 A class is a partition by ...
petinidres2 38180 Class ` A ` is a partition...
petinidres 38181 A class is a partition by ...
petxrnidres2 38182 Class ` A ` is a partition...
petxrnidres 38183 A class is a partition by ...
eqvreldisj1 38184 The elements of the quotie...
eqvreldisj2 38185 The elements of the quotie...
eqvreldisj3 38186 The elements of the quotie...
eqvreldisj4 38187 Intersection with the conv...
eqvreldisj5 38188 Range Cartesian product wi...
eqvrelqseqdisj2 38189 Implication of ~ eqvreldis...
fences3 38190 Implication of ~ eqvrelqse...
eqvrelqseqdisj3 38191 Implication of ~ eqvreldis...
eqvrelqseqdisj4 38192 Lemma for ~ petincnvepres2...
eqvrelqseqdisj5 38193 Lemma for the Partition-Eq...
mainer 38194 The Main Theorem of Equiva...
partimcomember 38195 Partition with general ` R...
mpet3 38196 Member Partition-Equivalen...
cpet2 38197 The conventional form of t...
cpet 38198 The conventional form of M...
mpet 38199 Member Partition-Equivalen...
mpet2 38200 Member Partition-Equivalen...
mpets2 38201 Member Partition-Equivalen...
mpets 38202 Member Partition-Equivalen...
mainpart 38203 Partition with general ` R...
fences 38204 The Theorem of Fences by E...
fences2 38205 The Theorem of Fences by E...
mainer2 38206 The Main Theorem of Equiva...
mainerim 38207 Every equivalence relation...
petincnvepres2 38208 A partition-equivalence th...
petincnvepres 38209 The shortest form of a par...
pet2 38210 Partition-Equivalence Theo...
pet 38211 Partition-Equivalence Theo...
pets 38212 Partition-Equivalence Theo...
prtlem60 38213 Lemma for ~ prter3 . (Con...
bicomdd 38214 Commute two sides of a bic...
jca2r 38215 Inference conjoining the c...
jca3 38216 Inference conjoining the c...
prtlem70 38217 Lemma for ~ prter3 : a rea...
ibdr 38218 Reverse of ~ ibd . (Contr...
prtlem100 38219 Lemma for ~ prter3 . (Con...
prtlem5 38220 Lemma for ~ prter1 , ~ prt...
prtlem80 38221 Lemma for ~ prter2 . (Con...
brabsb2 38222 A closed form of ~ brabsb ...
eqbrrdv2 38223 Other version of ~ eqbrrdi...
prtlem9 38224 Lemma for ~ prter3 . (Con...
prtlem10 38225 Lemma for ~ prter3 . (Con...
prtlem11 38226 Lemma for ~ prter2 . (Con...
prtlem12 38227 Lemma for ~ prtex and ~ pr...
prtlem13 38228 Lemma for ~ prter1 , ~ prt...
prtlem16 38229 Lemma for ~ prtex , ~ prte...
prtlem400 38230 Lemma for ~ prter2 and als...
erprt 38233 The quotient set of an equ...
prtlem14 38234 Lemma for ~ prter1 , ~ prt...
prtlem15 38235 Lemma for ~ prter1 and ~ p...
prtlem17 38236 Lemma for ~ prter2 . (Con...
prtlem18 38237 Lemma for ~ prter2 . (Con...
prtlem19 38238 Lemma for ~ prter2 . (Con...
prter1 38239 Every partition generates ...
prtex 38240 The equivalence relation g...
prter2 38241 The quotient set of the eq...
prter3 38242 For every partition there ...
axc5 38253 This theorem repeats ~ sp ...
ax4fromc4 38254 Rederivation of Axiom ~ ax...
ax10fromc7 38255 Rederivation of Axiom ~ ax...
ax6fromc10 38256 Rederivation of Axiom ~ ax...
hba1-o 38257 The setvar ` x ` is not fr...
axc4i-o 38258 Inference version of ~ ax-...
equid1 38259 Proof of ~ equid from our ...
equcomi1 38260 Proof of ~ equcomi from ~ ...
aecom-o 38261 Commutation law for identi...
aecoms-o 38262 A commutation rule for ide...
hbae-o 38263 All variables are effectiv...
dral1-o 38264 Formula-building lemma for...
ax12fromc15 38265 Rederivation of Axiom ~ ax...
ax13fromc9 38266 Derive ~ ax-13 from ~ ax-c...
ax5ALT 38267 Axiom to quantify a variab...
sps-o 38268 Generalization of antecede...
hbequid 38269 Bound-variable hypothesis ...
nfequid-o 38270 Bound-variable hypothesis ...
axc5c7 38271 Proof of a single axiom th...
axc5c7toc5 38272 Rederivation of ~ ax-c5 fr...
axc5c7toc7 38273 Rederivation of ~ ax-c7 fr...
axc711 38274 Proof of a single axiom th...
nfa1-o 38275 ` x ` is not free in ` A. ...
axc711toc7 38276 Rederivation of ~ ax-c7 fr...
axc711to11 38277 Rederivation of ~ ax-11 fr...
axc5c711 38278 Proof of a single axiom th...
axc5c711toc5 38279 Rederivation of ~ ax-c5 fr...
axc5c711toc7 38280 Rederivation of ~ ax-c7 fr...
axc5c711to11 38281 Rederivation of ~ ax-11 fr...
equidqe 38282 ~ equid with existential q...
axc5sp1 38283 A special case of ~ ax-c5 ...
equidq 38284 ~ equid with universal qua...
equid1ALT 38285 Alternate proof of ~ equid...
axc11nfromc11 38286 Rederivation of ~ ax-c11n ...
naecoms-o 38287 A commutation rule for dis...
hbnae-o 38288 All variables are effectiv...
dvelimf-o 38289 Proof of ~ dvelimh that us...
dral2-o 38290 Formula-building lemma for...
aev-o 38291 A "distinctor elimination"...
ax5eq 38292 Theorem to add distinct qu...
dveeq2-o 38293 Quantifier introduction wh...
axc16g-o 38294 A generalization of Axiom ...
dveeq1-o 38295 Quantifier introduction wh...
dveeq1-o16 38296 Version of ~ dveeq1 using ...
ax5el 38297 Theorem to add distinct qu...
axc11n-16 38298 This theorem shows that, g...
dveel2ALT 38299 Alternate proof of ~ dveel...
ax12f 38300 Basis step for constructin...
ax12eq 38301 Basis step for constructin...
ax12el 38302 Basis step for constructin...
ax12indn 38303 Induction step for constru...
ax12indi 38304 Induction step for constru...
ax12indalem 38305 Lemma for ~ ax12inda2 and ...
ax12inda2ALT 38306 Alternate proof of ~ ax12i...
ax12inda2 38307 Induction step for constru...
ax12inda 38308 Induction step for constru...
ax12v2-o 38309 Rederivation of ~ ax-c15 f...
ax12a2-o 38310 Derive ~ ax-c15 from a hyp...
axc11-o 38311 Show that ~ ax-c11 can be ...
fsumshftd 38312 Index shift of a finite su...
riotaclbgBAD 38314 Closure of restricted iota...
riotaclbBAD 38315 Closure of restricted iota...
riotasvd 38316 Deduction version of ~ rio...
riotasv2d 38317 Value of description binde...
riotasv2s 38318 The value of description b...
riotasv 38319 Value of description binde...
riotasv3d 38320 A property ` ch ` holding ...
elimhyps 38321 A version of ~ elimhyp usi...
dedths 38322 A version of weak deductio...
renegclALT 38323 Closure law for negative o...
elimhyps2 38324 Generalization of ~ elimhy...
dedths2 38325 Generalization of ~ dedths...
nfcxfrdf 38326 A utility lemma to transfe...
nfded 38327 A deduction theorem that c...
nfded2 38328 A deduction theorem that c...
nfunidALT2 38329 Deduction version of ~ nfu...
nfunidALT 38330 Deduction version of ~ nfu...
nfopdALT 38331 Deduction version of bound...
cnaddcom 38332 Recover the commutative la...
toycom 38333 Show the commutative law f...
lshpset 38338 The set of all hyperplanes...
islshp 38339 The predicate "is a hyperp...
islshpsm 38340 Hyperplane properties expr...
lshplss 38341 A hyperplane is a subspace...
lshpne 38342 A hyperplane is not equal ...
lshpnel 38343 A hyperplane's generating ...
lshpnelb 38344 The subspace sum of a hype...
lshpnel2N 38345 Condition that determines ...
lshpne0 38346 The member of the span in ...
lshpdisj 38347 A hyperplane and the span ...
lshpcmp 38348 If two hyperplanes are com...
lshpinN 38349 The intersection of two di...
lsatset 38350 The set of all 1-dim subsp...
islsat 38351 The predicate "is a 1-dim ...
lsatlspsn2 38352 The span of a nonzero sing...
lsatlspsn 38353 The span of a nonzero sing...
islsati 38354 A 1-dim subspace (atom) (o...
lsateln0 38355 A 1-dim subspace (atom) (o...
lsatlss 38356 The set of 1-dim subspaces...
lsatlssel 38357 An atom is a subspace. (C...
lsatssv 38358 An atom is a set of vector...
lsatn0 38359 A 1-dim subspace (atom) of...
lsatspn0 38360 The span of a vector is an...
lsator0sp 38361 The span of a vector is ei...
lsatssn0 38362 A subspace (or any class) ...
lsatcmp 38363 If two atoms are comparabl...
lsatcmp2 38364 If an atom is included in ...
lsatel 38365 A nonzero vector in an ato...
lsatelbN 38366 A nonzero vector in an ato...
lsat2el 38367 Two atoms sharing a nonzer...
lsmsat 38368 Convert comparison of atom...
lsatfixedN 38369 Show equality with the spa...
lsmsatcv 38370 Subspace sum has the cover...
lssatomic 38371 The lattice of subspaces i...
lssats 38372 The lattice of subspaces i...
lpssat 38373 Two subspaces in a proper ...
lrelat 38374 Subspaces are relatively a...
lssatle 38375 The ordering of two subspa...
lssat 38376 Two subspaces in a proper ...
islshpat 38377 Hyperplane properties expr...
lcvfbr 38380 The covers relation for a ...
lcvbr 38381 The covers relation for a ...
lcvbr2 38382 The covers relation for a ...
lcvbr3 38383 The covers relation for a ...
lcvpss 38384 The covers relation implie...
lcvnbtwn 38385 The covers relation implie...
lcvntr 38386 The covers relation is not...
lcvnbtwn2 38387 The covers relation implie...
lcvnbtwn3 38388 The covers relation implie...
lsmcv2 38389 Subspace sum has the cover...
lcvat 38390 If a subspace covers anoth...
lsatcv0 38391 An atom covers the zero su...
lsatcveq0 38392 A subspace covered by an a...
lsat0cv 38393 A subspace is an atom iff ...
lcvexchlem1 38394 Lemma for ~ lcvexch . (Co...
lcvexchlem2 38395 Lemma for ~ lcvexch . (Co...
lcvexchlem3 38396 Lemma for ~ lcvexch . (Co...
lcvexchlem4 38397 Lemma for ~ lcvexch . (Co...
lcvexchlem5 38398 Lemma for ~ lcvexch . (Co...
lcvexch 38399 Subspaces satisfy the exch...
lcvp 38400 Covering property of Defin...
lcv1 38401 Covering property of a sub...
lcv2 38402 Covering property of a sub...
lsatexch 38403 The atom exchange property...
lsatnle 38404 The meet of a subspace and...
lsatnem0 38405 The meet of distinct atoms...
lsatexch1 38406 The atom exch1ange propert...
lsatcv0eq 38407 If the sum of two atoms co...
lsatcv1 38408 Two atoms covering the zer...
lsatcvatlem 38409 Lemma for ~ lsatcvat . (C...
lsatcvat 38410 A nonzero subspace less th...
lsatcvat2 38411 A subspace covered by the ...
lsatcvat3 38412 A condition implying that ...
islshpcv 38413 Hyperplane properties expr...
l1cvpat 38414 A subspace covered by the ...
l1cvat 38415 Create an atom under an el...
lshpat 38416 Create an atom under a hyp...
lflset 38419 The set of linear function...
islfl 38420 The predicate "is a linear...
lfli 38421 Property of a linear funct...
islfld 38422 Properties that determine ...
lflf 38423 A linear functional is a f...
lflcl 38424 A linear functional value ...
lfl0 38425 A linear functional is zer...
lfladd 38426 Property of a linear funct...
lflsub 38427 Property of a linear funct...
lflmul 38428 Property of a linear funct...
lfl0f 38429 The zero function is a fun...
lfl1 38430 A nonzero functional has a...
lfladdcl 38431 Closure of addition of two...
lfladdcom 38432 Commutativity of functiona...
lfladdass 38433 Associativity of functiona...
lfladd0l 38434 Functional addition with t...
lflnegcl 38435 Closure of the negative of...
lflnegl 38436 A functional plus its nega...
lflvscl 38437 Closure of a scalar produc...
lflvsdi1 38438 Distributive law for (righ...
lflvsdi2 38439 Reverse distributive law f...
lflvsdi2a 38440 Reverse distributive law f...
lflvsass 38441 Associative law for (right...
lfl0sc 38442 The (right vector space) s...
lflsc0N 38443 The scalar product with th...
lfl1sc 38444 The (right vector space) s...
lkrfval 38447 The kernel of a functional...
lkrval 38448 Value of the kernel of a f...
ellkr 38449 Membership in the kernel o...
lkrval2 38450 Value of the kernel of a f...
ellkr2 38451 Membership in the kernel o...
lkrcl 38452 A member of the kernel of ...
lkrf0 38453 The value of a functional ...
lkr0f 38454 The kernel of the zero fun...
lkrlss 38455 The kernel of a linear fun...
lkrssv 38456 The kernel of a linear fun...
lkrsc 38457 The kernel of a nonzero sc...
lkrscss 38458 The kernel of a scalar pro...
eqlkr 38459 Two functionals with the s...
eqlkr2 38460 Two functionals with the s...
eqlkr3 38461 Two functionals with the s...
lkrlsp 38462 The subspace sum of a kern...
lkrlsp2 38463 The subspace sum of a kern...
lkrlsp3 38464 The subspace sum of a kern...
lkrshp 38465 The kernel of a nonzero fu...
lkrshp3 38466 The kernels of nonzero fun...
lkrshpor 38467 The kernel of a functional...
lkrshp4 38468 A kernel is a hyperplane i...
lshpsmreu 38469 Lemma for ~ lshpkrex . Sh...
lshpkrlem1 38470 Lemma for ~ lshpkrex . Th...
lshpkrlem2 38471 Lemma for ~ lshpkrex . Th...
lshpkrlem3 38472 Lemma for ~ lshpkrex . De...
lshpkrlem4 38473 Lemma for ~ lshpkrex . Pa...
lshpkrlem5 38474 Lemma for ~ lshpkrex . Pa...
lshpkrlem6 38475 Lemma for ~ lshpkrex . Sh...
lshpkrcl 38476 The set ` G ` defined by h...
lshpkr 38477 The kernel of functional `...
lshpkrex 38478 There exists a functional ...
lshpset2N 38479 The set of all hyperplanes...
islshpkrN 38480 The predicate "is a hyperp...
lfl1dim 38481 Equivalent expressions for...
lfl1dim2N 38482 Equivalent expressions for...
ldualset 38485 Define the (left) dual of ...
ldualvbase 38486 The vectors of a dual spac...
ldualelvbase 38487 Utility theorem for conver...
ldualfvadd 38488 Vector addition in the dua...
ldualvadd 38489 Vector addition in the dua...
ldualvaddcl 38490 The value of vector additi...
ldualvaddval 38491 The value of the value of ...
ldualsca 38492 The ring of scalars of the...
ldualsbase 38493 Base set of scalar ring fo...
ldualsaddN 38494 Scalar addition for the du...
ldualsmul 38495 Scalar multiplication for ...
ldualfvs 38496 Scalar product operation f...
ldualvs 38497 Scalar product operation v...
ldualvsval 38498 Value of scalar product op...
ldualvscl 38499 The scalar product operati...
ldualvaddcom 38500 Commutative law for vector...
ldualvsass 38501 Associative law for scalar...
ldualvsass2 38502 Associative law for scalar...
ldualvsdi1 38503 Distributive law for scala...
ldualvsdi2 38504 Reverse distributive law f...
ldualgrplem 38505 Lemma for ~ ldualgrp . (C...
ldualgrp 38506 The dual of a vector space...
ldual0 38507 The zero scalar of the dua...
ldual1 38508 The unit scalar of the dua...
ldualneg 38509 The negative of a scalar o...
ldual0v 38510 The zero vector of the dua...
ldual0vcl 38511 The dual zero vector is a ...
lduallmodlem 38512 Lemma for ~ lduallmod . (...
lduallmod 38513 The dual of a left module ...
lduallvec 38514 The dual of a left vector ...
ldualvsub 38515 The value of vector subtra...
ldualvsubcl 38516 Closure of vector subtract...
ldualvsubval 38517 The value of the value of ...
ldualssvscl 38518 Closure of scalar product ...
ldualssvsubcl 38519 Closure of vector subtract...
ldual0vs 38520 Scalar zero times a functi...
lkr0f2 38521 The kernel of the zero fun...
lduallkr3 38522 The kernels of nonzero fun...
lkrpssN 38523 Proper subset relation bet...
lkrin 38524 Intersection of the kernel...
eqlkr4 38525 Two functionals with the s...
ldual1dim 38526 Equivalent expressions for...
ldualkrsc 38527 The kernel of a nonzero sc...
lkrss 38528 The kernel of a scalar pro...
lkrss2N 38529 Two functionals with kerne...
lkreqN 38530 Proportional functionals h...
lkrlspeqN 38531 Condition for colinear fun...
isopos 38540 The predicate "is an ortho...
opposet 38541 Every orthoposet is a pose...
oposlem 38542 Lemma for orthoposet prope...
op01dm 38543 Conditions necessary for z...
op0cl 38544 An orthoposet has a zero e...
op1cl 38545 An orthoposet has a unity ...
op0le 38546 Orthoposet zero is less th...
ople0 38547 An element less than or eq...
opnlen0 38548 An element not less than a...
lub0N 38549 The least upper bound of t...
opltn0 38550 A lattice element greater ...
ople1 38551 Any element is less than t...
op1le 38552 If the orthoposet unity is...
glb0N 38553 The greatest lower bound o...
opoccl 38554 Closure of orthocomplement...
opococ 38555 Double negative law for or...
opcon3b 38556 Contraposition law for ort...
opcon2b 38557 Orthocomplement contraposi...
opcon1b 38558 Orthocomplement contraposi...
oplecon3 38559 Contraposition law for ort...
oplecon3b 38560 Contraposition law for ort...
oplecon1b 38561 Contraposition law for str...
opoc1 38562 Orthocomplement of orthopo...
opoc0 38563 Orthocomplement of orthopo...
opltcon3b 38564 Contraposition law for str...
opltcon1b 38565 Contraposition law for str...
opltcon2b 38566 Contraposition law for str...
opexmid 38567 Law of excluded middle for...
opnoncon 38568 Law of contradiction for o...
riotaocN 38569 The orthocomplement of the...
cmtfvalN 38570 Value of commutes relation...
cmtvalN 38571 Equivalence for commutes r...
isolat 38572 The predicate "is an ortho...
ollat 38573 An ortholattice is a latti...
olop 38574 An ortholattice is an orth...
olposN 38575 An ortholattice is a poset...
isolatiN 38576 Properties that determine ...
oldmm1 38577 De Morgan's law for meet i...
oldmm2 38578 De Morgan's law for meet i...
oldmm3N 38579 De Morgan's law for meet i...
oldmm4 38580 De Morgan's law for meet i...
oldmj1 38581 De Morgan's law for join i...
oldmj2 38582 De Morgan's law for join i...
oldmj3 38583 De Morgan's law for join i...
oldmj4 38584 De Morgan's law for join i...
olj01 38585 An ortholattice element jo...
olj02 38586 An ortholattice element jo...
olm11 38587 The meet of an ortholattic...
olm12 38588 The meet of an ortholattic...
latmassOLD 38589 Ortholattice meet is assoc...
latm12 38590 A rearrangement of lattice...
latm32 38591 A rearrangement of lattice...
latmrot 38592 Rotate lattice meet of 3 c...
latm4 38593 Rearrangement of lattice m...
latmmdiN 38594 Lattice meet distributes o...
latmmdir 38595 Lattice meet distributes o...
olm01 38596 Meet with lattice zero is ...
olm02 38597 Meet with lattice zero is ...
isoml 38598 The predicate "is an ortho...
isomliN 38599 Properties that determine ...
omlol 38600 An orthomodular lattice is...
omlop 38601 An orthomodular lattice is...
omllat 38602 An orthomodular lattice is...
omllaw 38603 The orthomodular law. (Co...
omllaw2N 38604 Variation of orthomodular ...
omllaw3 38605 Orthomodular law equivalen...
omllaw4 38606 Orthomodular law equivalen...
omllaw5N 38607 The orthomodular law. Rem...
cmtcomlemN 38608 Lemma for ~ cmtcomN . ( ~...
cmtcomN 38609 Commutation is symmetric. ...
cmt2N 38610 Commutation with orthocomp...
cmt3N 38611 Commutation with orthocomp...
cmt4N 38612 Commutation with orthocomp...
cmtbr2N 38613 Alternate definition of th...
cmtbr3N 38614 Alternate definition for t...
cmtbr4N 38615 Alternate definition for t...
lecmtN 38616 Ordered elements commute. ...
cmtidN 38617 Any element commutes with ...
omlfh1N 38618 Foulis-Holland Theorem, pa...
omlfh3N 38619 Foulis-Holland Theorem, pa...
omlmod1i2N 38620 Analogue of modular law ~ ...
omlspjN 38621 Contraction of a Sasaki pr...
cvrfval 38628 Value of covers relation "...
cvrval 38629 Binary relation expressing...
cvrlt 38630 The covers relation implie...
cvrnbtwn 38631 There is no element betwee...
ncvr1 38632 No element covers the latt...
cvrletrN 38633 Property of an element abo...
cvrval2 38634 Binary relation expressing...
cvrnbtwn2 38635 The covers relation implie...
cvrnbtwn3 38636 The covers relation implie...
cvrcon3b 38637 Contraposition law for the...
cvrle 38638 The covers relation implie...
cvrnbtwn4 38639 The covers relation implie...
cvrnle 38640 The covers relation implie...
cvrne 38641 The covers relation implie...
cvrnrefN 38642 The covers relation is not...
cvrcmp 38643 If two lattice elements th...
cvrcmp2 38644 If two lattice elements co...
pats 38645 The set of atoms in a pose...
isat 38646 The predicate "is an atom"...
isat2 38647 The predicate "is an atom"...
atcvr0 38648 An atom covers zero. ( ~ ...
atbase 38649 An atom is a member of the...
atssbase 38650 The set of atoms is a subs...
0ltat 38651 An atom is greater than ze...
leatb 38652 A poset element less than ...
leat 38653 A poset element less than ...
leat2 38654 A nonzero poset element le...
leat3 38655 A poset element less than ...
meetat 38656 The meet of any element wi...
meetat2 38657 The meet of any element wi...
isatl 38659 The predicate "is an atomi...
atllat 38660 An atomic lattice is a lat...
atlpos 38661 An atomic lattice is a pos...
atl0dm 38662 Condition necessary for ze...
atl0cl 38663 An atomic lattice has a ze...
atl0le 38664 Orthoposet zero is less th...
atlle0 38665 An element less than or eq...
atlltn0 38666 A lattice element greater ...
isat3 38667 The predicate "is an atom"...
atn0 38668 An atom is not zero. ( ~ ...
atnle0 38669 An atom is not less than o...
atlen0 38670 A lattice element is nonze...
atcmp 38671 If two atoms are comparabl...
atncmp 38672 Frequently-used variation ...
atnlt 38673 Two atoms cannot satisfy t...
atcvreq0 38674 An element covered by an a...
atncvrN 38675 Two atoms cannot satisfy t...
atlex 38676 Every nonzero element of a...
atnle 38677 Two ways of expressing "an...
atnem0 38678 The meet of distinct atoms...
atlatmstc 38679 An atomic, complete, ortho...
atlatle 38680 The ordering of two Hilber...
atlrelat1 38681 An atomistic lattice with ...
iscvlat 38683 The predicate "is an atomi...
iscvlat2N 38684 The predicate "is an atomi...
cvlatl 38685 An atomic lattice with the...
cvllat 38686 An atomic lattice with the...
cvlposN 38687 An atomic lattice with the...
cvlexch1 38688 An atomic covering lattice...
cvlexch2 38689 An atomic covering lattice...
cvlexchb1 38690 An atomic covering lattice...
cvlexchb2 38691 An atomic covering lattice...
cvlexch3 38692 An atomic covering lattice...
cvlexch4N 38693 An atomic covering lattice...
cvlatexchb1 38694 A version of ~ cvlexchb1 f...
cvlatexchb2 38695 A version of ~ cvlexchb2 f...
cvlatexch1 38696 Atom exchange property. (...
cvlatexch2 38697 Atom exchange property. (...
cvlatexch3 38698 Atom exchange property. (...
cvlcvr1 38699 The covering property. Pr...
cvlcvrp 38700 A Hilbert lattice satisfie...
cvlatcvr1 38701 An atom is covered by its ...
cvlatcvr2 38702 An atom is covered by its ...
cvlsupr2 38703 Two equivalent ways of exp...
cvlsupr3 38704 Two equivalent ways of exp...
cvlsupr4 38705 Consequence of superpositi...
cvlsupr5 38706 Consequence of superpositi...
cvlsupr6 38707 Consequence of superpositi...
cvlsupr7 38708 Consequence of superpositi...
cvlsupr8 38709 Consequence of superpositi...
ishlat1 38712 The predicate "is a Hilber...
ishlat2 38713 The predicate "is a Hilber...
ishlat3N 38714 The predicate "is a Hilber...
ishlatiN 38715 Properties that determine ...
hlomcmcv 38716 A Hilbert lattice is ortho...
hloml 38717 A Hilbert lattice is ortho...
hlclat 38718 A Hilbert lattice is compl...
hlcvl 38719 A Hilbert lattice is an at...
hlatl 38720 A Hilbert lattice is atomi...
hlol 38721 A Hilbert lattice is an or...
hlop 38722 A Hilbert lattice is an or...
hllat 38723 A Hilbert lattice is a lat...
hllatd 38724 Deduction form of ~ hllat ...
hlomcmat 38725 A Hilbert lattice is ortho...
hlpos 38726 A Hilbert lattice is a pos...
hlatjcl 38727 Closure of join operation....
hlatjcom 38728 Commutatitivity of join op...
hlatjidm 38729 Idempotence of join operat...
hlatjass 38730 Lattice join is associativ...
hlatj12 38731 Swap 1st and 2nd members o...
hlatj32 38732 Swap 2nd and 3rd members o...
hlatjrot 38733 Rotate lattice join of 3 c...
hlatj4 38734 Rearrangement of lattice j...
hlatlej1 38735 A join's first argument is...
hlatlej2 38736 A join's second argument i...
glbconN 38737 De Morgan's law for GLB an...
glbconNOLD 38738 Obsolete version of ~ glbc...
glbconxN 38739 De Morgan's law for GLB an...
atnlej1 38740 If an atom is not less tha...
atnlej2 38741 If an atom is not less tha...
hlsuprexch 38742 A Hilbert lattice has the ...
hlexch1 38743 A Hilbert lattice has the ...
hlexch2 38744 A Hilbert lattice has the ...
hlexchb1 38745 A Hilbert lattice has the ...
hlexchb2 38746 A Hilbert lattice has the ...
hlsupr 38747 A Hilbert lattice has the ...
hlsupr2 38748 A Hilbert lattice has the ...
hlhgt4 38749 A Hilbert lattice has a he...
hlhgt2 38750 A Hilbert lattice has a he...
hl0lt1N 38751 Lattice 0 is less than lat...
hlexch3 38752 A Hilbert lattice has the ...
hlexch4N 38753 A Hilbert lattice has the ...
hlatexchb1 38754 A version of ~ hlexchb1 fo...
hlatexchb2 38755 A version of ~ hlexchb2 fo...
hlatexch1 38756 Atom exchange property. (...
hlatexch2 38757 Atom exchange property. (...
hlatmstcOLDN 38758 An atomic, complete, ortho...
hlatle 38759 The ordering of two Hilber...
hlateq 38760 The equality of two Hilber...
hlrelat1 38761 An atomistic lattice with ...
hlrelat5N 38762 An atomistic lattice with ...
hlrelat 38763 A Hilbert lattice is relat...
hlrelat2 38764 A consequence of relative ...
exatleN 38765 A condition for an atom to...
hl2at 38766 A Hilbert lattice has at l...
atex 38767 At least one atom exists. ...
intnatN 38768 If the intersection with a...
2llnne2N 38769 Condition implying that tw...
2llnneN 38770 Condition implying that tw...
cvr1 38771 A Hilbert lattice has the ...
cvr2N 38772 Less-than and covers equiv...
hlrelat3 38773 The Hilbert lattice is rel...
cvrval3 38774 Binary relation expressing...
cvrval4N 38775 Binary relation expressing...
cvrval5 38776 Binary relation expressing...
cvrp 38777 A Hilbert lattice satisfie...
atcvr1 38778 An atom is covered by its ...
atcvr2 38779 An atom is covered by its ...
cvrexchlem 38780 Lemma for ~ cvrexch . ( ~...
cvrexch 38781 A Hilbert lattice satisfie...
cvratlem 38782 Lemma for ~ cvrat . ( ~ a...
cvrat 38783 A nonzero Hilbert lattice ...
ltltncvr 38784 A chained strong ordering ...
ltcvrntr 38785 Non-transitive condition f...
cvrntr 38786 The covers relation is not...
atcvr0eq 38787 The covers relation is not...
lnnat 38788 A line (the join of two di...
atcvrj0 38789 Two atoms covering the zer...
cvrat2 38790 A Hilbert lattice element ...
atcvrneN 38791 Inequality derived from at...
atcvrj1 38792 Condition for an atom to b...
atcvrj2b 38793 Condition for an atom to b...
atcvrj2 38794 Condition for an atom to b...
atleneN 38795 Inequality derived from at...
atltcvr 38796 An equivalence of less-tha...
atle 38797 Any nonzero element has an...
atlt 38798 Two atoms are unequal iff ...
atlelt 38799 Transfer less-than relatio...
2atlt 38800 Given an atom less than an...
atexchcvrN 38801 Atom exchange property. V...
atexchltN 38802 Atom exchange property. V...
cvrat3 38803 A condition implying that ...
cvrat4 38804 A condition implying exist...
cvrat42 38805 Commuted version of ~ cvra...
2atjm 38806 The meet of a line (expres...
atbtwn 38807 Property of a 3rd atom ` R...
atbtwnexOLDN 38808 There exists a 3rd atom ` ...
atbtwnex 38809 Given atoms ` P ` in ` X `...
3noncolr2 38810 Two ways to express 3 non-...
3noncolr1N 38811 Two ways to express 3 non-...
hlatcon3 38812 Atom exchange combined wit...
hlatcon2 38813 Atom exchange combined wit...
4noncolr3 38814 A way to express 4 non-col...
4noncolr2 38815 A way to express 4 non-col...
4noncolr1 38816 A way to express 4 non-col...
athgt 38817 A Hilbert lattice, whose h...
3dim0 38818 There exists a 3-dimension...
3dimlem1 38819 Lemma for ~ 3dim1 . (Cont...
3dimlem2 38820 Lemma for ~ 3dim1 . (Cont...
3dimlem3a 38821 Lemma for ~ 3dim3 . (Cont...
3dimlem3 38822 Lemma for ~ 3dim1 . (Cont...
3dimlem3OLDN 38823 Lemma for ~ 3dim1 . (Cont...
3dimlem4a 38824 Lemma for ~ 3dim3 . (Cont...
3dimlem4 38825 Lemma for ~ 3dim1 . (Cont...
3dimlem4OLDN 38826 Lemma for ~ 3dim1 . (Cont...
3dim1lem5 38827 Lemma for ~ 3dim1 . (Cont...
3dim1 38828 Construct a 3-dimensional ...
3dim2 38829 Construct 2 new layers on ...
3dim3 38830 Construct a new layer on t...
2dim 38831 Generate a height-3 elemen...
1dimN 38832 An atom is covered by a he...
1cvrco 38833 The orthocomplement of an ...
1cvratex 38834 There exists an atom less ...
1cvratlt 38835 An atom less than or equal...
1cvrjat 38836 An element covered by the ...
1cvrat 38837 Create an atom under an el...
ps-1 38838 The join of two atoms ` R ...
ps-2 38839 Lattice analogue for the p...
2atjlej 38840 Two atoms are different if...
hlatexch3N 38841 Rearrange join of atoms in...
hlatexch4 38842 Exchange 2 atoms. (Contri...
ps-2b 38843 Variation of projective ge...
3atlem1 38844 Lemma for ~ 3at . (Contri...
3atlem2 38845 Lemma for ~ 3at . (Contri...
3atlem3 38846 Lemma for ~ 3at . (Contri...
3atlem4 38847 Lemma for ~ 3at . (Contri...
3atlem5 38848 Lemma for ~ 3at . (Contri...
3atlem6 38849 Lemma for ~ 3at . (Contri...
3atlem7 38850 Lemma for ~ 3at . (Contri...
3at 38851 Any three non-colinear ato...
llnset 38866 The set of lattice lines i...
islln 38867 The predicate "is a lattic...
islln4 38868 The predicate "is a lattic...
llni 38869 Condition implying a latti...
llnbase 38870 A lattice line is a lattic...
islln3 38871 The predicate "is a lattic...
islln2 38872 The predicate "is a lattic...
llni2 38873 The join of two different ...
llnnleat 38874 An atom cannot majorize a ...
llnneat 38875 A lattice line is not an a...
2atneat 38876 The join of two distinct a...
llnn0 38877 A lattice line is nonzero....
islln2a 38878 The predicate "is a lattic...
llnle 38879 Any element greater than 0...
atcvrlln2 38880 An atom under a line is co...
atcvrlln 38881 An element covering an ato...
llnexatN 38882 Given an atom on a line, t...
llncmp 38883 If two lattice lines are c...
llnnlt 38884 Two lattice lines cannot s...
2llnmat 38885 Two intersecting lines int...
2at0mat0 38886 Special case of ~ 2atmat0 ...
2atmat0 38887 The meet of two unequal li...
2atm 38888 An atom majorized by two d...
ps-2c 38889 Variation of projective ge...
lplnset 38890 The set of lattice planes ...
islpln 38891 The predicate "is a lattic...
islpln4 38892 The predicate "is a lattic...
lplni 38893 Condition implying a latti...
islpln3 38894 The predicate "is a lattic...
lplnbase 38895 A lattice plane is a latti...
islpln5 38896 The predicate "is a lattic...
islpln2 38897 The predicate "is a lattic...
lplni2 38898 The join of 3 different at...
lvolex3N 38899 There is an atom outside o...
llnmlplnN 38900 The intersection of a line...
lplnle 38901 Any element greater than 0...
lplnnle2at 38902 A lattice line (or atom) c...
lplnnleat 38903 A lattice plane cannot maj...
lplnnlelln 38904 A lattice plane is not les...
2atnelpln 38905 The join of two atoms is n...
lplnneat 38906 No lattice plane is an ato...
lplnnelln 38907 No lattice plane is a latt...
lplnn0N 38908 A lattice plane is nonzero...
islpln2a 38909 The predicate "is a lattic...
islpln2ah 38910 The predicate "is a lattic...
lplnriaN 38911 Property of a lattice plan...
lplnribN 38912 Property of a lattice plan...
lplnric 38913 Property of a lattice plan...
lplnri1 38914 Property of a lattice plan...
lplnri2N 38915 Property of a lattice plan...
lplnri3N 38916 Property of a lattice plan...
lplnllnneN 38917 Two lattice lines defined ...
llncvrlpln2 38918 A lattice line under a lat...
llncvrlpln 38919 An element covering a latt...
2lplnmN 38920 If the join of two lattice...
2llnmj 38921 The meet of two lattice li...
2atmat 38922 The meet of two intersecti...
lplncmp 38923 If two lattice planes are ...
lplnexatN 38924 Given a lattice line on a ...
lplnexllnN 38925 Given an atom on a lattice...
lplnnlt 38926 Two lattice planes cannot ...
2llnjaN 38927 The join of two different ...
2llnjN 38928 The join of two different ...
2llnm2N 38929 The meet of two different ...
2llnm3N 38930 Two lattice lines in a lat...
2llnm4 38931 Two lattice lines that maj...
2llnmeqat 38932 An atom equals the interse...
lvolset 38933 The set of 3-dim lattice v...
islvol 38934 The predicate "is a 3-dim ...
islvol4 38935 The predicate "is a 3-dim ...
lvoli 38936 Condition implying a 3-dim...
islvol3 38937 The predicate "is a 3-dim ...
lvoli3 38938 Condition implying a 3-dim...
lvolbase 38939 A 3-dim lattice volume is ...
islvol5 38940 The predicate "is a 3-dim ...
islvol2 38941 The predicate "is a 3-dim ...
lvoli2 38942 The join of 4 different at...
lvolnle3at 38943 A lattice plane (or lattic...
lvolnleat 38944 An atom cannot majorize a ...
lvolnlelln 38945 A lattice line cannot majo...
lvolnlelpln 38946 A lattice plane cannot maj...
3atnelvolN 38947 The join of 3 atoms is not...
2atnelvolN 38948 The join of two atoms is n...
lvolneatN 38949 No lattice volume is an at...
lvolnelln 38950 No lattice volume is a lat...
lvolnelpln 38951 No lattice volume is a lat...
lvoln0N 38952 A lattice volume is nonzer...
islvol2aN 38953 The predicate "is a lattic...
4atlem0a 38954 Lemma for ~ 4at . (Contri...
4atlem0ae 38955 Lemma for ~ 4at . (Contri...
4atlem0be 38956 Lemma for ~ 4at . (Contri...
4atlem3 38957 Lemma for ~ 4at . Break i...
4atlem3a 38958 Lemma for ~ 4at . Break i...
4atlem3b 38959 Lemma for ~ 4at . Break i...
4atlem4a 38960 Lemma for ~ 4at . Frequen...
4atlem4b 38961 Lemma for ~ 4at . Frequen...
4atlem4c 38962 Lemma for ~ 4at . Frequen...
4atlem4d 38963 Lemma for ~ 4at . Frequen...
4atlem9 38964 Lemma for ~ 4at . Substit...
4atlem10a 38965 Lemma for ~ 4at . Substit...
4atlem10b 38966 Lemma for ~ 4at . Substit...
4atlem10 38967 Lemma for ~ 4at . Combine...
4atlem11a 38968 Lemma for ~ 4at . Substit...
4atlem11b 38969 Lemma for ~ 4at . Substit...
4atlem11 38970 Lemma for ~ 4at . Combine...
4atlem12a 38971 Lemma for ~ 4at . Substit...
4atlem12b 38972 Lemma for ~ 4at . Substit...
4atlem12 38973 Lemma for ~ 4at . Combine...
4at 38974 Four atoms determine a lat...
4at2 38975 Four atoms determine a lat...
lplncvrlvol2 38976 A lattice line under a lat...
lplncvrlvol 38977 An element covering a latt...
lvolcmp 38978 If two lattice planes are ...
lvolnltN 38979 Two lattice volumes cannot...
2lplnja 38980 The join of two different ...
2lplnj 38981 The join of two different ...
2lplnm2N 38982 The meet of two different ...
2lplnmj 38983 The meet of two lattice pl...
dalemkehl 38984 Lemma for ~ dath . Freque...
dalemkelat 38985 Lemma for ~ dath . Freque...
dalemkeop 38986 Lemma for ~ dath . Freque...
dalempea 38987 Lemma for ~ dath . Freque...
dalemqea 38988 Lemma for ~ dath . Freque...
dalemrea 38989 Lemma for ~ dath . Freque...
dalemsea 38990 Lemma for ~ dath . Freque...
dalemtea 38991 Lemma for ~ dath . Freque...
dalemuea 38992 Lemma for ~ dath . Freque...
dalemyeo 38993 Lemma for ~ dath . Freque...
dalemzeo 38994 Lemma for ~ dath . Freque...
dalemclpjs 38995 Lemma for ~ dath . Freque...
dalemclqjt 38996 Lemma for ~ dath . Freque...
dalemclrju 38997 Lemma for ~ dath . Freque...
dalem-clpjq 38998 Lemma for ~ dath . Freque...
dalemceb 38999 Lemma for ~ dath . Freque...
dalempeb 39000 Lemma for ~ dath . Freque...
dalemqeb 39001 Lemma for ~ dath . Freque...
dalemreb 39002 Lemma for ~ dath . Freque...
dalemseb 39003 Lemma for ~ dath . Freque...
dalemteb 39004 Lemma for ~ dath . Freque...
dalemueb 39005 Lemma for ~ dath . Freque...
dalempjqeb 39006 Lemma for ~ dath . Freque...
dalemsjteb 39007 Lemma for ~ dath . Freque...
dalemtjueb 39008 Lemma for ~ dath . Freque...
dalemqrprot 39009 Lemma for ~ dath . Freque...
dalemyeb 39010 Lemma for ~ dath . Freque...
dalemcnes 39011 Lemma for ~ dath . Freque...
dalempnes 39012 Lemma for ~ dath . Freque...
dalemqnet 39013 Lemma for ~ dath . Freque...
dalempjsen 39014 Lemma for ~ dath . Freque...
dalemply 39015 Lemma for ~ dath . Freque...
dalemsly 39016 Lemma for ~ dath . Freque...
dalemswapyz 39017 Lemma for ~ dath . Swap t...
dalemrot 39018 Lemma for ~ dath . Rotate...
dalemrotyz 39019 Lemma for ~ dath . Rotate...
dalem1 39020 Lemma for ~ dath . Show t...
dalemcea 39021 Lemma for ~ dath . Freque...
dalem2 39022 Lemma for ~ dath . Show t...
dalemdea 39023 Lemma for ~ dath . Freque...
dalemeea 39024 Lemma for ~ dath . Freque...
dalem3 39025 Lemma for ~ dalemdnee . (...
dalem4 39026 Lemma for ~ dalemdnee . (...
dalemdnee 39027 Lemma for ~ dath . Axis o...
dalem5 39028 Lemma for ~ dath . Atom `...
dalem6 39029 Lemma for ~ dath . Analog...
dalem7 39030 Lemma for ~ dath . Analog...
dalem8 39031 Lemma for ~ dath . Plane ...
dalem-cly 39032 Lemma for ~ dalem9 . Cent...
dalem9 39033 Lemma for ~ dath . Since ...
dalem10 39034 Lemma for ~ dath . Atom `...
dalem11 39035 Lemma for ~ dath . Analog...
dalem12 39036 Lemma for ~ dath . Analog...
dalem13 39037 Lemma for ~ dalem14 . (Co...
dalem14 39038 Lemma for ~ dath . Planes...
dalem15 39039 Lemma for ~ dath . The ax...
dalem16 39040 Lemma for ~ dath . The at...
dalem17 39041 Lemma for ~ dath . When p...
dalem18 39042 Lemma for ~ dath . Show t...
dalem19 39043 Lemma for ~ dath . Show t...
dalemccea 39044 Lemma for ~ dath . Freque...
dalemddea 39045 Lemma for ~ dath . Freque...
dalem-ccly 39046 Lemma for ~ dath . Freque...
dalem-ddly 39047 Lemma for ~ dath . Freque...
dalemccnedd 39048 Lemma for ~ dath . Freque...
dalemclccjdd 39049 Lemma for ~ dath . Freque...
dalemcceb 39050 Lemma for ~ dath . Freque...
dalemswapyzps 39051 Lemma for ~ dath . Swap t...
dalemrotps 39052 Lemma for ~ dath . Rotate...
dalemcjden 39053 Lemma for ~ dath . Show t...
dalem20 39054 Lemma for ~ dath . Show t...
dalem21 39055 Lemma for ~ dath . Show t...
dalem22 39056 Lemma for ~ dath . Show t...
dalem23 39057 Lemma for ~ dath . Show t...
dalem24 39058 Lemma for ~ dath . Show t...
dalem25 39059 Lemma for ~ dath . Show t...
dalem27 39060 Lemma for ~ dath . Show t...
dalem28 39061 Lemma for ~ dath . Lemma ...
dalem29 39062 Lemma for ~ dath . Analog...
dalem30 39063 Lemma for ~ dath . Analog...
dalem31N 39064 Lemma for ~ dath . Analog...
dalem32 39065 Lemma for ~ dath . Analog...
dalem33 39066 Lemma for ~ dath . Analog...
dalem34 39067 Lemma for ~ dath . Analog...
dalem35 39068 Lemma for ~ dath . Analog...
dalem36 39069 Lemma for ~ dath . Analog...
dalem37 39070 Lemma for ~ dath . Analog...
dalem38 39071 Lemma for ~ dath . Plane ...
dalem39 39072 Lemma for ~ dath . Auxili...
dalem40 39073 Lemma for ~ dath . Analog...
dalem41 39074 Lemma for ~ dath . (Contr...
dalem42 39075 Lemma for ~ dath . Auxili...
dalem43 39076 Lemma for ~ dath . Planes...
dalem44 39077 Lemma for ~ dath . Dummy ...
dalem45 39078 Lemma for ~ dath . Dummy ...
dalem46 39079 Lemma for ~ dath . Analog...
dalem47 39080 Lemma for ~ dath . Analog...
dalem48 39081 Lemma for ~ dath . Analog...
dalem49 39082 Lemma for ~ dath . Analog...
dalem50 39083 Lemma for ~ dath . Analog...
dalem51 39084 Lemma for ~ dath . Constr...
dalem52 39085 Lemma for ~ dath . Lines ...
dalem53 39086 Lemma for ~ dath . The au...
dalem54 39087 Lemma for ~ dath . Line `...
dalem55 39088 Lemma for ~ dath . Lines ...
dalem56 39089 Lemma for ~ dath . Analog...
dalem57 39090 Lemma for ~ dath . Axis o...
dalem58 39091 Lemma for ~ dath . Analog...
dalem59 39092 Lemma for ~ dath . Analog...
dalem60 39093 Lemma for ~ dath . ` B ` i...
dalem61 39094 Lemma for ~ dath . Show t...
dalem62 39095 Lemma for ~ dath . Elimin...
dalem63 39096 Lemma for ~ dath . Combin...
dath 39097 Desargues's theorem of pro...
dath2 39098 Version of Desargues's the...
lineset 39099 The set of lines in a Hilb...
isline 39100 The predicate "is a line"....
islinei 39101 Condition implying "is a l...
pointsetN 39102 The set of points in a Hil...
ispointN 39103 The predicate "is a point"...
atpointN 39104 The singleton of an atom i...
psubspset 39105 The set of projective subs...
ispsubsp 39106 The predicate "is a projec...
ispsubsp2 39107 The predicate "is a projec...
psubspi 39108 Property of a projective s...
psubspi2N 39109 Property of a projective s...
0psubN 39110 The empty set is a project...
snatpsubN 39111 The singleton of an atom i...
pointpsubN 39112 A point (singleton of an a...
linepsubN 39113 A line is a projective sub...
atpsubN 39114 The set of all atoms is a ...
psubssat 39115 A projective subspace cons...
psubatN 39116 A member of a projective s...
pmapfval 39117 The projective map of a Hi...
pmapval 39118 Value of the projective ma...
elpmap 39119 Member of a projective map...
pmapssat 39120 The projective map of a Hi...
pmapssbaN 39121 A weakening of ~ pmapssat ...
pmaple 39122 The projective map of a Hi...
pmap11 39123 The projective map of a Hi...
pmapat 39124 The projective map of an a...
elpmapat 39125 Member of the projective m...
pmap0 39126 Value of the projective ma...
pmapeq0 39127 A projective map value is ...
pmap1N 39128 Value of the projective ma...
pmapsub 39129 The projective map of a Hi...
pmapglbx 39130 The projective map of the ...
pmapglb 39131 The projective map of the ...
pmapglb2N 39132 The projective map of the ...
pmapglb2xN 39133 The projective map of the ...
pmapmeet 39134 The projective map of a me...
isline2 39135 Definition of line in term...
linepmap 39136 A line described with a pr...
isline3 39137 Definition of line in term...
isline4N 39138 Definition of line in term...
lneq2at 39139 A line equals the join of ...
lnatexN 39140 There is an atom in a line...
lnjatN 39141 Given an atom in a line, t...
lncvrelatN 39142 A lattice element covered ...
lncvrat 39143 A line covers the atoms it...
lncmp 39144 If two lines are comparabl...
2lnat 39145 Two intersecting lines int...
2atm2atN 39146 Two joins with a common at...
2llnma1b 39147 Generalization of ~ 2llnma...
2llnma1 39148 Two different intersecting...
2llnma3r 39149 Two different intersecting...
2llnma2 39150 Two different intersecting...
2llnma2rN 39151 Two different intersecting...
cdlema1N 39152 A condition for required f...
cdlema2N 39153 A condition for required f...
cdlemblem 39154 Lemma for ~ cdlemb . (Con...
cdlemb 39155 Given two atoms not less t...
paddfval 39158 Projective subspace sum op...
paddval 39159 Projective subspace sum op...
elpadd 39160 Member of a projective sub...
elpaddn0 39161 Member of projective subsp...
paddvaln0N 39162 Projective subspace sum op...
elpaddri 39163 Condition implying members...
elpaddatriN 39164 Condition implying members...
elpaddat 39165 Membership in a projective...
elpaddatiN 39166 Consequence of membership ...
elpadd2at 39167 Membership in a projective...
elpadd2at2 39168 Membership in a projective...
paddunssN 39169 Projective subspace sum in...
elpadd0 39170 Member of projective subsp...
paddval0 39171 Projective subspace sum wi...
padd01 39172 Projective subspace sum wi...
padd02 39173 Projective subspace sum wi...
paddcom 39174 Projective subspace sum co...
paddssat 39175 A projective subspace sum ...
sspadd1 39176 A projective subspace sum ...
sspadd2 39177 A projective subspace sum ...
paddss1 39178 Subset law for projective ...
paddss2 39179 Subset law for projective ...
paddss12 39180 Subset law for projective ...
paddasslem1 39181 Lemma for ~ paddass . (Co...
paddasslem2 39182 Lemma for ~ paddass . (Co...
paddasslem3 39183 Lemma for ~ paddass . Res...
paddasslem4 39184 Lemma for ~ paddass . Com...
paddasslem5 39185 Lemma for ~ paddass . Sho...
paddasslem6 39186 Lemma for ~ paddass . (Co...
paddasslem7 39187 Lemma for ~ paddass . Com...
paddasslem8 39188 Lemma for ~ paddass . (Co...
paddasslem9 39189 Lemma for ~ paddass . Com...
paddasslem10 39190 Lemma for ~ paddass . Use...
paddasslem11 39191 Lemma for ~ paddass . The...
paddasslem12 39192 Lemma for ~ paddass . The...
paddasslem13 39193 Lemma for ~ paddass . The...
paddasslem14 39194 Lemma for ~ paddass . Rem...
paddasslem15 39195 Lemma for ~ paddass . Use...
paddasslem16 39196 Lemma for ~ paddass . Use...
paddasslem17 39197 Lemma for ~ paddass . The...
paddasslem18 39198 Lemma for ~ paddass . Com...
paddass 39199 Projective subspace sum is...
padd12N 39200 Commutative/associative la...
padd4N 39201 Rearrangement of 4 terms i...
paddidm 39202 Projective subspace sum is...
paddclN 39203 The projective sum of two ...
paddssw1 39204 Subset law for projective ...
paddssw2 39205 Subset law for projective ...
paddss 39206 Subset law for projective ...
pmodlem1 39207 Lemma for ~ pmod1i . (Con...
pmodlem2 39208 Lemma for ~ pmod1i . (Con...
pmod1i 39209 The modular law holds in a...
pmod2iN 39210 Dual of the modular law. ...
pmodN 39211 The modular law for projec...
pmodl42N 39212 Lemma derived from modular...
pmapjoin 39213 The projective map of the ...
pmapjat1 39214 The projective map of the ...
pmapjat2 39215 The projective map of the ...
pmapjlln1 39216 The projective map of the ...
hlmod1i 39217 A version of the modular l...
atmod1i1 39218 Version of modular law ~ p...
atmod1i1m 39219 Version of modular law ~ p...
atmod1i2 39220 Version of modular law ~ p...
llnmod1i2 39221 Version of modular law ~ p...
atmod2i1 39222 Version of modular law ~ p...
atmod2i2 39223 Version of modular law ~ p...
llnmod2i2 39224 Version of modular law ~ p...
atmod3i1 39225 Version of modular law tha...
atmod3i2 39226 Version of modular law tha...
atmod4i1 39227 Version of modular law tha...
atmod4i2 39228 Version of modular law tha...
llnexchb2lem 39229 Lemma for ~ llnexchb2 . (...
llnexchb2 39230 Line exchange property (co...
llnexch2N 39231 Line exchange property (co...
dalawlem1 39232 Lemma for ~ dalaw . Speci...
dalawlem2 39233 Lemma for ~ dalaw . Utili...
dalawlem3 39234 Lemma for ~ dalaw . First...
dalawlem4 39235 Lemma for ~ dalaw . Secon...
dalawlem5 39236 Lemma for ~ dalaw . Speci...
dalawlem6 39237 Lemma for ~ dalaw . First...
dalawlem7 39238 Lemma for ~ dalaw . Secon...
dalawlem8 39239 Lemma for ~ dalaw . Speci...
dalawlem9 39240 Lemma for ~ dalaw . Speci...
dalawlem10 39241 Lemma for ~ dalaw . Combi...
dalawlem11 39242 Lemma for ~ dalaw . First...
dalawlem12 39243 Lemma for ~ dalaw . Secon...
dalawlem13 39244 Lemma for ~ dalaw . Speci...
dalawlem14 39245 Lemma for ~ dalaw . Combi...
dalawlem15 39246 Lemma for ~ dalaw . Swap ...
dalaw 39247 Desargues's law, derived f...
pclfvalN 39250 The projective subspace cl...
pclvalN 39251 Value of the projective su...
pclclN 39252 Closure of the projective ...
elpclN 39253 Membership in the projecti...
elpcliN 39254 Implication of membership ...
pclssN 39255 Ordering is preserved by s...
pclssidN 39256 A set of atoms is included...
pclidN 39257 The projective subspace cl...
pclbtwnN 39258 A projective subspace sand...
pclunN 39259 The projective subspace cl...
pclun2N 39260 The projective subspace cl...
pclfinN 39261 The projective subspace cl...
pclcmpatN 39262 The set of projective subs...
polfvalN 39265 The projective subspace po...
polvalN 39266 Value of the projective su...
polval2N 39267 Alternate expression for v...
polsubN 39268 The polarity of a set of a...
polssatN 39269 The polarity of a set of a...
pol0N 39270 The polarity of the empty ...
pol1N 39271 The polarity of the whole ...
2pol0N 39272 The closed subspace closur...
polpmapN 39273 The polarity of a projecti...
2polpmapN 39274 Double polarity of a proje...
2polvalN 39275 Value of double polarity. ...
2polssN 39276 A set of atoms is a subset...
3polN 39277 Triple polarity cancels to...
polcon3N 39278 Contraposition law for pol...
2polcon4bN 39279 Contraposition law for pol...
polcon2N 39280 Contraposition law for pol...
polcon2bN 39281 Contraposition law for pol...
pclss2polN 39282 The projective subspace cl...
pcl0N 39283 The projective subspace cl...
pcl0bN 39284 The projective subspace cl...
pmaplubN 39285 The LUB of a projective ma...
sspmaplubN 39286 A set of atoms is a subset...
2pmaplubN 39287 Double projective map of a...
paddunN 39288 The closure of the project...
poldmj1N 39289 De Morgan's law for polari...
pmapj2N 39290 The projective map of the ...
pmapocjN 39291 The projective map of the ...
polatN 39292 The polarity of the single...
2polatN 39293 Double polarity of the sin...
pnonsingN 39294 The intersection of a set ...
psubclsetN 39297 The set of closed projecti...
ispsubclN 39298 The predicate "is a closed...
psubcliN 39299 Property of a closed proje...
psubcli2N 39300 Property of a closed proje...
psubclsubN 39301 A closed projective subspa...
psubclssatN 39302 A closed projective subspa...
pmapidclN 39303 Projective map of the LUB ...
0psubclN 39304 The empty set is a closed ...
1psubclN 39305 The set of all atoms is a ...
atpsubclN 39306 A point (singleton of an a...
pmapsubclN 39307 A projective map value is ...
ispsubcl2N 39308 Alternate predicate for "i...
psubclinN 39309 The intersection of two cl...
paddatclN 39310 The projective sum of a cl...
pclfinclN 39311 The projective subspace cl...
linepsubclN 39312 A line is a closed project...
polsubclN 39313 A polarity is a closed pro...
poml4N 39314 Orthomodular law for proje...
poml5N 39315 Orthomodular law for proje...
poml6N 39316 Orthomodular law for proje...
osumcllem1N 39317 Lemma for ~ osumclN . (Co...
osumcllem2N 39318 Lemma for ~ osumclN . (Co...
osumcllem3N 39319 Lemma for ~ osumclN . (Co...
osumcllem4N 39320 Lemma for ~ osumclN . (Co...
osumcllem5N 39321 Lemma for ~ osumclN . (Co...
osumcllem6N 39322 Lemma for ~ osumclN . Use...
osumcllem7N 39323 Lemma for ~ osumclN . (Co...
osumcllem8N 39324 Lemma for ~ osumclN . (Co...
osumcllem9N 39325 Lemma for ~ osumclN . (Co...
osumcllem10N 39326 Lemma for ~ osumclN . Con...
osumcllem11N 39327 Lemma for ~ osumclN . (Co...
osumclN 39328 Closure of orthogonal sum....
pmapojoinN 39329 For orthogonal elements, p...
pexmidN 39330 Excluded middle law for cl...
pexmidlem1N 39331 Lemma for ~ pexmidN . Hol...
pexmidlem2N 39332 Lemma for ~ pexmidN . (Co...
pexmidlem3N 39333 Lemma for ~ pexmidN . Use...
pexmidlem4N 39334 Lemma for ~ pexmidN . (Co...
pexmidlem5N 39335 Lemma for ~ pexmidN . (Co...
pexmidlem6N 39336 Lemma for ~ pexmidN . (Co...
pexmidlem7N 39337 Lemma for ~ pexmidN . Con...
pexmidlem8N 39338 Lemma for ~ pexmidN . The...
pexmidALTN 39339 Excluded middle law for cl...
pl42lem1N 39340 Lemma for ~ pl42N . (Cont...
pl42lem2N 39341 Lemma for ~ pl42N . (Cont...
pl42lem3N 39342 Lemma for ~ pl42N . (Cont...
pl42lem4N 39343 Lemma for ~ pl42N . (Cont...
pl42N 39344 Law holding in a Hilbert l...
watfvalN 39353 The W atoms function. (Co...
watvalN 39354 Value of the W atoms funct...
iswatN 39355 The predicate "is a W atom...
lhpset 39356 The set of co-atoms (latti...
islhp 39357 The predicate "is a co-ato...
islhp2 39358 The predicate "is a co-ato...
lhpbase 39359 A co-atom is a member of t...
lhp1cvr 39360 The lattice unity covers a...
lhplt 39361 An atom under a co-atom is...
lhp2lt 39362 The join of two atoms unde...
lhpexlt 39363 There exists an atom less ...
lhp0lt 39364 A co-atom is greater than ...
lhpn0 39365 A co-atom is nonzero. TOD...
lhpexle 39366 There exists an atom under...
lhpexnle 39367 There exists an atom not u...
lhpexle1lem 39368 Lemma for ~ lhpexle1 and o...
lhpexle1 39369 There exists an atom under...
lhpexle2lem 39370 Lemma for ~ lhpexle2 . (C...
lhpexle2 39371 There exists atom under a ...
lhpexle3lem 39372 There exists atom under a ...
lhpexle3 39373 There exists atom under a ...
lhpex2leN 39374 There exist at least two d...
lhpoc 39375 The orthocomplement of a c...
lhpoc2N 39376 The orthocomplement of an ...
lhpocnle 39377 The orthocomplement of a c...
lhpocat 39378 The orthocomplement of a c...
lhpocnel 39379 The orthocomplement of a c...
lhpocnel2 39380 The orthocomplement of a c...
lhpjat1 39381 The join of a co-atom (hyp...
lhpjat2 39382 The join of a co-atom (hyp...
lhpj1 39383 The join of a co-atom (hyp...
lhpmcvr 39384 The meet of a lattice hype...
lhpmcvr2 39385 Alternate way to express t...
lhpmcvr3 39386 Specialization of ~ lhpmcv...
lhpmcvr4N 39387 Specialization of ~ lhpmcv...
lhpmcvr5N 39388 Specialization of ~ lhpmcv...
lhpmcvr6N 39389 Specialization of ~ lhpmcv...
lhpm0atN 39390 If the meet of a lattice h...
lhpmat 39391 An element covered by the ...
lhpmatb 39392 An element covered by the ...
lhp2at0 39393 Join and meet with differe...
lhp2atnle 39394 Inequality for 2 different...
lhp2atne 39395 Inequality for joins with ...
lhp2at0nle 39396 Inequality for 2 different...
lhp2at0ne 39397 Inequality for joins with ...
lhpelim 39398 Eliminate an atom not unde...
lhpmod2i2 39399 Modular law for hyperplane...
lhpmod6i1 39400 Modular law for hyperplane...
lhprelat3N 39401 The Hilbert lattice is rel...
cdlemb2 39402 Given two atoms not under ...
lhple 39403 Property of a lattice elem...
lhpat 39404 Create an atom under a co-...
lhpat4N 39405 Property of an atom under ...
lhpat2 39406 Create an atom under a co-...
lhpat3 39407 There is only one atom und...
4atexlemk 39408 Lemma for ~ 4atexlem7 . (...
4atexlemw 39409 Lemma for ~ 4atexlem7 . (...
4atexlempw 39410 Lemma for ~ 4atexlem7 . (...
4atexlemp 39411 Lemma for ~ 4atexlem7 . (...
4atexlemq 39412 Lemma for ~ 4atexlem7 . (...
4atexlems 39413 Lemma for ~ 4atexlem7 . (...
4atexlemt 39414 Lemma for ~ 4atexlem7 . (...
4atexlemutvt 39415 Lemma for ~ 4atexlem7 . (...
4atexlempnq 39416 Lemma for ~ 4atexlem7 . (...
4atexlemnslpq 39417 Lemma for ~ 4atexlem7 . (...
4atexlemkl 39418 Lemma for ~ 4atexlem7 . (...
4atexlemkc 39419 Lemma for ~ 4atexlem7 . (...
4atexlemwb 39420 Lemma for ~ 4atexlem7 . (...
4atexlempsb 39421 Lemma for ~ 4atexlem7 . (...
4atexlemqtb 39422 Lemma for ~ 4atexlem7 . (...
4atexlempns 39423 Lemma for ~ 4atexlem7 . (...
4atexlemswapqr 39424 Lemma for ~ 4atexlem7 . S...
4atexlemu 39425 Lemma for ~ 4atexlem7 . (...
4atexlemv 39426 Lemma for ~ 4atexlem7 . (...
4atexlemunv 39427 Lemma for ~ 4atexlem7 . (...
4atexlemtlw 39428 Lemma for ~ 4atexlem7 . (...
4atexlemntlpq 39429 Lemma for ~ 4atexlem7 . (...
4atexlemc 39430 Lemma for ~ 4atexlem7 . (...
4atexlemnclw 39431 Lemma for ~ 4atexlem7 . (...
4atexlemex2 39432 Lemma for ~ 4atexlem7 . S...
4atexlemcnd 39433 Lemma for ~ 4atexlem7 . (...
4atexlemex4 39434 Lemma for ~ 4atexlem7 . S...
4atexlemex6 39435 Lemma for ~ 4atexlem7 . (...
4atexlem7 39436 Whenever there are at leas...
4atex 39437 Whenever there are at leas...
4atex2 39438 More general version of ~ ...
4atex2-0aOLDN 39439 Same as ~ 4atex2 except th...
4atex2-0bOLDN 39440 Same as ~ 4atex2 except th...
4atex2-0cOLDN 39441 Same as ~ 4atex2 except th...
4atex3 39442 More general version of ~ ...
lautset 39443 The set of lattice automor...
islaut 39444 The predicate "is a lattic...
lautle 39445 Less-than or equal propert...
laut1o 39446 A lattice automorphism is ...
laut11 39447 One-to-one property of a l...
lautcl 39448 A lattice automorphism val...
lautcnvclN 39449 Reverse closure of a latti...
lautcnvle 39450 Less-than or equal propert...
lautcnv 39451 The converse of a lattice ...
lautlt 39452 Less-than property of a la...
lautcvr 39453 Covering property of a lat...
lautj 39454 Meet property of a lattice...
lautm 39455 Meet property of a lattice...
lauteq 39456 A lattice automorphism arg...
idlaut 39457 The identity function is a...
lautco 39458 The composition of two lat...
pautsetN 39459 The set of projective auto...
ispautN 39460 The predicate "is a projec...
ldilfset 39469 The mapping from fiducial ...
ldilset 39470 The set of lattice dilatio...
isldil 39471 The predicate "is a lattic...
ldillaut 39472 A lattice dilation is an a...
ldil1o 39473 A lattice dilation is a on...
ldilval 39474 Value of a lattice dilatio...
idldil 39475 The identity function is a...
ldilcnv 39476 The converse of a lattice ...
ldilco 39477 The composition of two lat...
ltrnfset 39478 The set of all lattice tra...
ltrnset 39479 The set of lattice transla...
isltrn 39480 The predicate "is a lattic...
isltrn2N 39481 The predicate "is a lattic...
ltrnu 39482 Uniqueness property of a l...
ltrnldil 39483 A lattice translation is a...
ltrnlaut 39484 A lattice translation is a...
ltrn1o 39485 A lattice translation is a...
ltrncl 39486 Closure of a lattice trans...
ltrn11 39487 One-to-one property of a l...
ltrncnvnid 39488 If a translation is differ...
ltrncoidN 39489 Two translations are equal...
ltrnle 39490 Less-than or equal propert...
ltrncnvleN 39491 Less-than or equal propert...
ltrnm 39492 Lattice translation of a m...
ltrnj 39493 Lattice translation of a m...
ltrncvr 39494 Covering property of a lat...
ltrnval1 39495 Value of a lattice transla...
ltrnid 39496 A lattice translation is t...
ltrnnid 39497 If a lattice translation i...
ltrnatb 39498 The lattice translation of...
ltrncnvatb 39499 The converse of the lattic...
ltrnel 39500 The lattice translation of...
ltrnat 39501 The lattice translation of...
ltrncnvat 39502 The converse of the lattic...
ltrncnvel 39503 The converse of the lattic...
ltrncoelN 39504 Composition of lattice tra...
ltrncoat 39505 Composition of lattice tra...
ltrncoval 39506 Two ways to express value ...
ltrncnv 39507 The converse of a lattice ...
ltrn11at 39508 Frequently used one-to-one...
ltrneq2 39509 The equality of two transl...
ltrneq 39510 The equality of two transl...
idltrn 39511 The identity function is a...
ltrnmw 39512 Property of lattice transl...
dilfsetN 39513 The mapping from fiducial ...
dilsetN 39514 The set of dilations for a...
isdilN 39515 The predicate "is a dilati...
trnfsetN 39516 The mapping from fiducial ...
trnsetN 39517 The set of translations fo...
istrnN 39518 The predicate "is a transl...
trlfset 39521 The set of all traces of l...
trlset 39522 The set of traces of latti...
trlval 39523 The value of the trace of ...
trlval2 39524 The value of the trace of ...
trlcl 39525 Closure of the trace of a ...
trlcnv 39526 The trace of the converse ...
trljat1 39527 The value of a translation...
trljat2 39528 The value of a translation...
trljat3 39529 The value of a translation...
trlat 39530 If an atom differs from it...
trl0 39531 If an atom not under the f...
trlator0 39532 The trace of a lattice tra...
trlatn0 39533 The trace of a lattice tra...
trlnidat 39534 The trace of a lattice tra...
ltrnnidn 39535 If a lattice translation i...
ltrnideq 39536 Property of the identity l...
trlid0 39537 The trace of the identity ...
trlnidatb 39538 A lattice translation is n...
trlid0b 39539 A lattice translation is t...
trlnid 39540 Different translations wit...
ltrn2ateq 39541 Property of the equality o...
ltrnateq 39542 If any atom (under ` W ` )...
ltrnatneq 39543 If any atom (under ` W ` )...
ltrnatlw 39544 If the value of an atom eq...
trlle 39545 The trace of a lattice tra...
trlne 39546 The trace of a lattice tra...
trlnle 39547 The atom not under the fid...
trlval3 39548 The value of the trace of ...
trlval4 39549 The value of the trace of ...
trlval5 39550 The value of the trace of ...
arglem1N 39551 Lemma for Desargues's law....
cdlemc1 39552 Part of proof of Lemma C i...
cdlemc2 39553 Part of proof of Lemma C i...
cdlemc3 39554 Part of proof of Lemma C i...
cdlemc4 39555 Part of proof of Lemma C i...
cdlemc5 39556 Lemma for ~ cdlemc . (Con...
cdlemc6 39557 Lemma for ~ cdlemc . (Con...
cdlemc 39558 Lemma C in [Crawley] p. 11...
cdlemd1 39559 Part of proof of Lemma D i...
cdlemd2 39560 Part of proof of Lemma D i...
cdlemd3 39561 Part of proof of Lemma D i...
cdlemd4 39562 Part of proof of Lemma D i...
cdlemd5 39563 Part of proof of Lemma D i...
cdlemd6 39564 Part of proof of Lemma D i...
cdlemd7 39565 Part of proof of Lemma D i...
cdlemd8 39566 Part of proof of Lemma D i...
cdlemd9 39567 Part of proof of Lemma D i...
cdlemd 39568 If two translations agree ...
ltrneq3 39569 Two translations agree at ...
cdleme00a 39570 Part of proof of Lemma E i...
cdleme0aa 39571 Part of proof of Lemma E i...
cdleme0a 39572 Part of proof of Lemma E i...
cdleme0b 39573 Part of proof of Lemma E i...
cdleme0c 39574 Part of proof of Lemma E i...
cdleme0cp 39575 Part of proof of Lemma E i...
cdleme0cq 39576 Part of proof of Lemma E i...
cdleme0dN 39577 Part of proof of Lemma E i...
cdleme0e 39578 Part of proof of Lemma E i...
cdleme0fN 39579 Part of proof of Lemma E i...
cdleme0gN 39580 Part of proof of Lemma E i...
cdlemeulpq 39581 Part of proof of Lemma E i...
cdleme01N 39582 Part of proof of Lemma E i...
cdleme02N 39583 Part of proof of Lemma E i...
cdleme0ex1N 39584 Part of proof of Lemma E i...
cdleme0ex2N 39585 Part of proof of Lemma E i...
cdleme0moN 39586 Part of proof of Lemma E i...
cdleme1b 39587 Part of proof of Lemma E i...
cdleme1 39588 Part of proof of Lemma E i...
cdleme2 39589 Part of proof of Lemma E i...
cdleme3b 39590 Part of proof of Lemma E i...
cdleme3c 39591 Part of proof of Lemma E i...
cdleme3d 39592 Part of proof of Lemma E i...
cdleme3e 39593 Part of proof of Lemma E i...
cdleme3fN 39594 Part of proof of Lemma E i...
cdleme3g 39595 Part of proof of Lemma E i...
cdleme3h 39596 Part of proof of Lemma E i...
cdleme3fa 39597 Part of proof of Lemma E i...
cdleme3 39598 Part of proof of Lemma E i...
cdleme4 39599 Part of proof of Lemma E i...
cdleme4a 39600 Part of proof of Lemma E i...
cdleme5 39601 Part of proof of Lemma E i...
cdleme6 39602 Part of proof of Lemma E i...
cdleme7aa 39603 Part of proof of Lemma E i...
cdleme7a 39604 Part of proof of Lemma E i...
cdleme7b 39605 Part of proof of Lemma E i...
cdleme7c 39606 Part of proof of Lemma E i...
cdleme7d 39607 Part of proof of Lemma E i...
cdleme7e 39608 Part of proof of Lemma E i...
cdleme7ga 39609 Part of proof of Lemma E i...
cdleme7 39610 Part of proof of Lemma E i...
cdleme8 39611 Part of proof of Lemma E i...
cdleme9a 39612 Part of proof of Lemma E i...
cdleme9b 39613 Utility lemma for Lemma E ...
cdleme9 39614 Part of proof of Lemma E i...
cdleme10 39615 Part of proof of Lemma E i...
cdleme8tN 39616 Part of proof of Lemma E i...
cdleme9taN 39617 Part of proof of Lemma E i...
cdleme9tN 39618 Part of proof of Lemma E i...
cdleme10tN 39619 Part of proof of Lemma E i...
cdleme16aN 39620 Part of proof of Lemma E i...
cdleme11a 39621 Part of proof of Lemma E i...
cdleme11c 39622 Part of proof of Lemma E i...
cdleme11dN 39623 Part of proof of Lemma E i...
cdleme11e 39624 Part of proof of Lemma E i...
cdleme11fN 39625 Part of proof of Lemma E i...
cdleme11g 39626 Part of proof of Lemma E i...
cdleme11h 39627 Part of proof of Lemma E i...
cdleme11j 39628 Part of proof of Lemma E i...
cdleme11k 39629 Part of proof of Lemma E i...
cdleme11l 39630 Part of proof of Lemma E i...
cdleme11 39631 Part of proof of Lemma E i...
cdleme12 39632 Part of proof of Lemma E i...
cdleme13 39633 Part of proof of Lemma E i...
cdleme14 39634 Part of proof of Lemma E i...
cdleme15a 39635 Part of proof of Lemma E i...
cdleme15b 39636 Part of proof of Lemma E i...
cdleme15c 39637 Part of proof of Lemma E i...
cdleme15d 39638 Part of proof of Lemma E i...
cdleme15 39639 Part of proof of Lemma E i...
cdleme16b 39640 Part of proof of Lemma E i...
cdleme16c 39641 Part of proof of Lemma E i...
cdleme16d 39642 Part of proof of Lemma E i...
cdleme16e 39643 Part of proof of Lemma E i...
cdleme16f 39644 Part of proof of Lemma E i...
cdleme16g 39645 Part of proof of Lemma E i...
cdleme16 39646 Part of proof of Lemma E i...
cdleme17a 39647 Part of proof of Lemma E i...
cdleme17b 39648 Lemma leading to ~ cdleme1...
cdleme17c 39649 Part of proof of Lemma E i...
cdleme17d1 39650 Part of proof of Lemma E i...
cdleme0nex 39651 Part of proof of Lemma E i...
cdleme18a 39652 Part of proof of Lemma E i...
cdleme18b 39653 Part of proof of Lemma E i...
cdleme18c 39654 Part of proof of Lemma E i...
cdleme22gb 39655 Utility lemma for Lemma E ...
cdleme18d 39656 Part of proof of Lemma E i...
cdlemesner 39657 Part of proof of Lemma E i...
cdlemedb 39658 Part of proof of Lemma E i...
cdlemeda 39659 Part of proof of Lemma E i...
cdlemednpq 39660 Part of proof of Lemma E i...
cdlemednuN 39661 Part of proof of Lemma E i...
cdleme20zN 39662 Part of proof of Lemma E i...
cdleme20y 39663 Part of proof of Lemma E i...
cdleme19a 39664 Part of proof of Lemma E i...
cdleme19b 39665 Part of proof of Lemma E i...
cdleme19c 39666 Part of proof of Lemma E i...
cdleme19d 39667 Part of proof of Lemma E i...
cdleme19e 39668 Part of proof of Lemma E i...
cdleme19f 39669 Part of proof of Lemma E i...
cdleme20aN 39670 Part of proof of Lemma E i...
cdleme20bN 39671 Part of proof of Lemma E i...
cdleme20c 39672 Part of proof of Lemma E i...
cdleme20d 39673 Part of proof of Lemma E i...
cdleme20e 39674 Part of proof of Lemma E i...
cdleme20f 39675 Part of proof of Lemma E i...
cdleme20g 39676 Part of proof of Lemma E i...
cdleme20h 39677 Part of proof of Lemma E i...
cdleme20i 39678 Part of proof of Lemma E i...
cdleme20j 39679 Part of proof of Lemma E i...
cdleme20k 39680 Part of proof of Lemma E i...
cdleme20l1 39681 Part of proof of Lemma E i...
cdleme20l2 39682 Part of proof of Lemma E i...
cdleme20l 39683 Part of proof of Lemma E i...
cdleme20m 39684 Part of proof of Lemma E i...
cdleme20 39685 Combine ~ cdleme19f and ~ ...
cdleme21a 39686 Part of proof of Lemma E i...
cdleme21b 39687 Part of proof of Lemma E i...
cdleme21c 39688 Part of proof of Lemma E i...
cdleme21at 39689 Part of proof of Lemma E i...
cdleme21ct 39690 Part of proof of Lemma E i...
cdleme21d 39691 Part of proof of Lemma E i...
cdleme21e 39692 Part of proof of Lemma E i...
cdleme21f 39693 Part of proof of Lemma E i...
cdleme21g 39694 Part of proof of Lemma E i...
cdleme21h 39695 Part of proof of Lemma E i...
cdleme21i 39696 Part of proof of Lemma E i...
cdleme21j 39697 Combine ~ cdleme20 and ~ c...
cdleme21 39698 Part of proof of Lemma E i...
cdleme21k 39699 Eliminate ` S =/= T ` cond...
cdleme22aa 39700 Part of proof of Lemma E i...
cdleme22a 39701 Part of proof of Lemma E i...
cdleme22b 39702 Part of proof of Lemma E i...
cdleme22cN 39703 Part of proof of Lemma E i...
cdleme22d 39704 Part of proof of Lemma E i...
cdleme22e 39705 Part of proof of Lemma E i...
cdleme22eALTN 39706 Part of proof of Lemma E i...
cdleme22f 39707 Part of proof of Lemma E i...
cdleme22f2 39708 Part of proof of Lemma E i...
cdleme22g 39709 Part of proof of Lemma E i...
cdleme23a 39710 Part of proof of Lemma E i...
cdleme23b 39711 Part of proof of Lemma E i...
cdleme23c 39712 Part of proof of Lemma E i...
cdleme24 39713 Quantified version of ~ cd...
cdleme25a 39714 Lemma for ~ cdleme25b . (...
cdleme25b 39715 Transform ~ cdleme24 . TO...
cdleme25c 39716 Transform ~ cdleme25b . (...
cdleme25dN 39717 Transform ~ cdleme25c . (...
cdleme25cl 39718 Show closure of the unique...
cdleme25cv 39719 Change bound variables in ...
cdleme26e 39720 Part of proof of Lemma E i...
cdleme26ee 39721 Part of proof of Lemma E i...
cdleme26eALTN 39722 Part of proof of Lemma E i...
cdleme26fALTN 39723 Part of proof of Lemma E i...
cdleme26f 39724 Part of proof of Lemma E i...
cdleme26f2ALTN 39725 Part of proof of Lemma E i...
cdleme26f2 39726 Part of proof of Lemma E i...
cdleme27cl 39727 Part of proof of Lemma E i...
cdleme27a 39728 Part of proof of Lemma E i...
cdleme27b 39729 Lemma for ~ cdleme27N . (...
cdleme27N 39730 Part of proof of Lemma E i...
cdleme28a 39731 Lemma for ~ cdleme25b . T...
cdleme28b 39732 Lemma for ~ cdleme25b . T...
cdleme28c 39733 Part of proof of Lemma E i...
cdleme28 39734 Quantified version of ~ cd...
cdleme29ex 39735 Lemma for ~ cdleme29b . (...
cdleme29b 39736 Transform ~ cdleme28 . (C...
cdleme29c 39737 Transform ~ cdleme28b . (...
cdleme29cl 39738 Show closure of the unique...
cdleme30a 39739 Part of proof of Lemma E i...
cdleme31so 39740 Part of proof of Lemma E i...
cdleme31sn 39741 Part of proof of Lemma E i...
cdleme31sn1 39742 Part of proof of Lemma E i...
cdleme31se 39743 Part of proof of Lemma D i...
cdleme31se2 39744 Part of proof of Lemma D i...
cdleme31sc 39745 Part of proof of Lemma E i...
cdleme31sde 39746 Part of proof of Lemma D i...
cdleme31snd 39747 Part of proof of Lemma D i...
cdleme31sdnN 39748 Part of proof of Lemma E i...
cdleme31sn1c 39749 Part of proof of Lemma E i...
cdleme31sn2 39750 Part of proof of Lemma E i...
cdleme31fv 39751 Part of proof of Lemma E i...
cdleme31fv1 39752 Part of proof of Lemma E i...
cdleme31fv1s 39753 Part of proof of Lemma E i...
cdleme31fv2 39754 Part of proof of Lemma E i...
cdleme31id 39755 Part of proof of Lemma E i...
cdlemefrs29pre00 39756 ***START OF VALUE AT ATOM ...
cdlemefrs29bpre0 39757 TODO fix comment. (Contri...
cdlemefrs29bpre1 39758 TODO: FIX COMMENT. (Contr...
cdlemefrs29cpre1 39759 TODO: FIX COMMENT. (Contr...
cdlemefrs29clN 39760 TODO: NOT USED? Show clo...
cdlemefrs32fva 39761 Part of proof of Lemma E i...
cdlemefrs32fva1 39762 Part of proof of Lemma E i...
cdlemefr29exN 39763 Lemma for ~ cdlemefs29bpre...
cdlemefr27cl 39764 Part of proof of Lemma E i...
cdlemefr32sn2aw 39765 Show that ` [_ R / s ]_ N ...
cdlemefr32snb 39766 Show closure of ` [_ R / s...
cdlemefr29bpre0N 39767 TODO fix comment. (Contri...
cdlemefr29clN 39768 Show closure of the unique...
cdleme43frv1snN 39769 Value of ` [_ R / s ]_ N `...
cdlemefr32fvaN 39770 Part of proof of Lemma E i...
cdlemefr32fva1 39771 Part of proof of Lemma E i...
cdlemefr31fv1 39772 Value of ` ( F `` R ) ` wh...
cdlemefs29pre00N 39773 FIX COMMENT. TODO: see if ...
cdlemefs27cl 39774 Part of proof of Lemma E i...
cdlemefs32sn1aw 39775 Show that ` [_ R / s ]_ N ...
cdlemefs32snb 39776 Show closure of ` [_ R / s...
cdlemefs29bpre0N 39777 TODO: FIX COMMENT. (Contr...
cdlemefs29bpre1N 39778 TODO: FIX COMMENT. (Contr...
cdlemefs29cpre1N 39779 TODO: FIX COMMENT. (Contr...
cdlemefs29clN 39780 Show closure of the unique...
cdleme43fsv1snlem 39781 Value of ` [_ R / s ]_ N `...
cdleme43fsv1sn 39782 Value of ` [_ R / s ]_ N `...
cdlemefs32fvaN 39783 Part of proof of Lemma E i...
cdlemefs32fva1 39784 Part of proof of Lemma E i...
cdlemefs31fv1 39785 Value of ` ( F `` R ) ` wh...
cdlemefr44 39786 Value of f(r) when r is an...
cdlemefs44 39787 Value of f_s(r) when r is ...
cdlemefr45 39788 Value of f(r) when r is an...
cdlemefr45e 39789 Explicit expansion of ~ cd...
cdlemefs45 39790 Value of f_s(r) when r is ...
cdlemefs45ee 39791 Explicit expansion of ~ cd...
cdlemefs45eN 39792 Explicit expansion of ~ cd...
cdleme32sn1awN 39793 Show that ` [_ R / s ]_ N ...
cdleme41sn3a 39794 Show that ` [_ R / s ]_ N ...
cdleme32sn2awN 39795 Show that ` [_ R / s ]_ N ...
cdleme32snaw 39796 Show that ` [_ R / s ]_ N ...
cdleme32snb 39797 Show closure of ` [_ R / s...
cdleme32fva 39798 Part of proof of Lemma D i...
cdleme32fva1 39799 Part of proof of Lemma D i...
cdleme32fvaw 39800 Show that ` ( F `` R ) ` i...
cdleme32fvcl 39801 Part of proof of Lemma D i...
cdleme32a 39802 Part of proof of Lemma D i...
cdleme32b 39803 Part of proof of Lemma D i...
cdleme32c 39804 Part of proof of Lemma D i...
cdleme32d 39805 Part of proof of Lemma D i...
cdleme32e 39806 Part of proof of Lemma D i...
cdleme32f 39807 Part of proof of Lemma D i...
cdleme32le 39808 Part of proof of Lemma D i...
cdleme35a 39809 Part of proof of Lemma E i...
cdleme35fnpq 39810 Part of proof of Lemma E i...
cdleme35b 39811 Part of proof of Lemma E i...
cdleme35c 39812 Part of proof of Lemma E i...
cdleme35d 39813 Part of proof of Lemma E i...
cdleme35e 39814 Part of proof of Lemma E i...
cdleme35f 39815 Part of proof of Lemma E i...
cdleme35g 39816 Part of proof of Lemma E i...
cdleme35h 39817 Part of proof of Lemma E i...
cdleme35h2 39818 Part of proof of Lemma E i...
cdleme35sn2aw 39819 Part of proof of Lemma E i...
cdleme35sn3a 39820 Part of proof of Lemma E i...
cdleme36a 39821 Part of proof of Lemma E i...
cdleme36m 39822 Part of proof of Lemma E i...
cdleme37m 39823 Part of proof of Lemma E i...
cdleme38m 39824 Part of proof of Lemma E i...
cdleme38n 39825 Part of proof of Lemma E i...
cdleme39a 39826 Part of proof of Lemma E i...
cdleme39n 39827 Part of proof of Lemma E i...
cdleme40m 39828 Part of proof of Lemma E i...
cdleme40n 39829 Part of proof of Lemma E i...
cdleme40v 39830 Part of proof of Lemma E i...
cdleme40w 39831 Part of proof of Lemma E i...
cdleme42a 39832 Part of proof of Lemma E i...
cdleme42c 39833 Part of proof of Lemma E i...
cdleme42d 39834 Part of proof of Lemma E i...
cdleme41sn3aw 39835 Part of proof of Lemma E i...
cdleme41sn4aw 39836 Part of proof of Lemma E i...
cdleme41snaw 39837 Part of proof of Lemma E i...
cdleme41fva11 39838 Part of proof of Lemma E i...
cdleme42b 39839 Part of proof of Lemma E i...
cdleme42e 39840 Part of proof of Lemma E i...
cdleme42f 39841 Part of proof of Lemma E i...
cdleme42g 39842 Part of proof of Lemma E i...
cdleme42h 39843 Part of proof of Lemma E i...
cdleme42i 39844 Part of proof of Lemma E i...
cdleme42k 39845 Part of proof of Lemma E i...
cdleme42ke 39846 Part of proof of Lemma E i...
cdleme42keg 39847 Part of proof of Lemma E i...
cdleme42mN 39848 Part of proof of Lemma E i...
cdleme42mgN 39849 Part of proof of Lemma E i...
cdleme43aN 39850 Part of proof of Lemma E i...
cdleme43bN 39851 Lemma for Lemma E in [Craw...
cdleme43cN 39852 Part of proof of Lemma E i...
cdleme43dN 39853 Part of proof of Lemma E i...
cdleme46f2g2 39854 Conversion for ` G ` to re...
cdleme46f2g1 39855 Conversion for ` G ` to re...
cdleme17d2 39856 Part of proof of Lemma E i...
cdleme17d3 39857 TODO: FIX COMMENT. (Contr...
cdleme17d4 39858 TODO: FIX COMMENT. (Contr...
cdleme17d 39859 Part of proof of Lemma E i...
cdleme48fv 39860 Part of proof of Lemma D i...
cdleme48fvg 39861 Remove ` P =/= Q ` conditi...
cdleme46fvaw 39862 Show that ` ( F `` R ) ` i...
cdleme48bw 39863 TODO: fix comment. TODO: ...
cdleme48b 39864 TODO: fix comment. (Contr...
cdleme46frvlpq 39865 Show that ` ( F `` S ) ` i...
cdleme46fsvlpq 39866 Show that ` ( F `` R ) ` i...
cdlemeg46fvcl 39867 TODO: fix comment. (Contr...
cdleme4gfv 39868 Part of proof of Lemma D i...
cdlemeg47b 39869 TODO: FIX COMMENT. (Contr...
cdlemeg47rv 39870 Value of g_s(r) when r is ...
cdlemeg47rv2 39871 Value of g_s(r) when r is ...
cdlemeg49le 39872 Part of proof of Lemma D i...
cdlemeg46bOLDN 39873 TODO FIX COMMENT. (Contrib...
cdlemeg46c 39874 TODO FIX COMMENT. (Contrib...
cdlemeg46rvOLDN 39875 Value of g_s(r) when r is ...
cdlemeg46rv2OLDN 39876 Value of g_s(r) when r is ...
cdlemeg46fvaw 39877 Show that ` ( F `` R ) ` i...
cdlemeg46nlpq 39878 Show that ` ( G `` S ) ` i...
cdlemeg46ngfr 39879 TODO FIX COMMENT g(f(s))=s...
cdlemeg46nfgr 39880 TODO FIX COMMENT f(g(s))=s...
cdlemeg46sfg 39881 TODO FIX COMMENT f(r) ` \/...
cdlemeg46fjgN 39882 NOT NEEDED? TODO FIX COMM...
cdlemeg46rjgN 39883 NOT NEEDED? TODO FIX COMM...
cdlemeg46fjv 39884 TODO FIX COMMENT f(r) ` \/...
cdlemeg46fsfv 39885 TODO FIX COMMENT f(r) ` \/...
cdlemeg46frv 39886 TODO FIX COMMENT. (f(r) ` ...
cdlemeg46v1v2 39887 TODO FIX COMMENT v_1 = v_2...
cdlemeg46vrg 39888 TODO FIX COMMENT v_1 ` <_ ...
cdlemeg46rgv 39889 TODO FIX COMMENT r ` <_ ` ...
cdlemeg46req 39890 TODO FIX COMMENT r = (v_1 ...
cdlemeg46gfv 39891 TODO FIX COMMENT p. 115 pe...
cdlemeg46gfr 39892 TODO FIX COMMENT p. 116 pe...
cdlemeg46gfre 39893 TODO FIX COMMENT p. 116 pe...
cdlemeg46gf 39894 TODO FIX COMMENT Eliminate...
cdlemeg46fgN 39895 TODO FIX COMMENT p. 116 pe...
cdleme48d 39896 TODO: fix comment. (Contr...
cdleme48gfv1 39897 TODO: fix comment. (Contr...
cdleme48gfv 39898 TODO: fix comment. (Contr...
cdleme48fgv 39899 TODO: fix comment. (Contr...
cdlemeg49lebilem 39900 Part of proof of Lemma D i...
cdleme50lebi 39901 Part of proof of Lemma D i...
cdleme50eq 39902 Part of proof of Lemma D i...
cdleme50f 39903 Part of proof of Lemma D i...
cdleme50f1 39904 Part of proof of Lemma D i...
cdleme50rnlem 39905 Part of proof of Lemma D i...
cdleme50rn 39906 Part of proof of Lemma D i...
cdleme50f1o 39907 Part of proof of Lemma D i...
cdleme50laut 39908 Part of proof of Lemma D i...
cdleme50ldil 39909 Part of proof of Lemma D i...
cdleme50trn1 39910 Part of proof that ` F ` i...
cdleme50trn2a 39911 Part of proof that ` F ` i...
cdleme50trn2 39912 Part of proof that ` F ` i...
cdleme50trn12 39913 Part of proof that ` F ` i...
cdleme50trn3 39914 Part of proof that ` F ` i...
cdleme50trn123 39915 Part of proof that ` F ` i...
cdleme51finvfvN 39916 Part of proof of Lemma E i...
cdleme51finvN 39917 Part of proof of Lemma E i...
cdleme50ltrn 39918 Part of proof of Lemma E i...
cdleme51finvtrN 39919 Part of proof of Lemma E i...
cdleme50ex 39920 Part of Lemma E in [Crawle...
cdleme 39921 Lemma E in [Crawley] p. 11...
cdlemf1 39922 Part of Lemma F in [Crawle...
cdlemf2 39923 Part of Lemma F in [Crawle...
cdlemf 39924 Lemma F in [Crawley] p. 11...
cdlemfnid 39925 ~ cdlemf with additional c...
cdlemftr3 39926 Special case of ~ cdlemf s...
cdlemftr2 39927 Special case of ~ cdlemf s...
cdlemftr1 39928 Part of proof of Lemma G o...
cdlemftr0 39929 Special case of ~ cdlemf s...
trlord 39930 The ordering of two Hilber...
cdlemg1a 39931 Shorter expression for ` G...
cdlemg1b2 39932 This theorem can be used t...
cdlemg1idlemN 39933 Lemma for ~ cdlemg1idN . ...
cdlemg1fvawlemN 39934 Lemma for ~ ltrniotafvawN ...
cdlemg1ltrnlem 39935 Lemma for ~ ltrniotacl . ...
cdlemg1finvtrlemN 39936 Lemma for ~ ltrniotacnvN ....
cdlemg1bOLDN 39937 This theorem can be used t...
cdlemg1idN 39938 Version of ~ cdleme31id wi...
ltrniotafvawN 39939 Version of ~ cdleme46fvaw ...
ltrniotacl 39940 Version of ~ cdleme50ltrn ...
ltrniotacnvN 39941 Version of ~ cdleme51finvt...
ltrniotaval 39942 Value of the unique transl...
ltrniotacnvval 39943 Converse value of the uniq...
ltrniotaidvalN 39944 Value of the unique transl...
ltrniotavalbN 39945 Value of the unique transl...
cdlemeiota 39946 A translation is uniquely ...
cdlemg1ci2 39947 Any function of the form o...
cdlemg1cN 39948 Any translation belongs to...
cdlemg1cex 39949 Any translation is one of ...
cdlemg2cN 39950 Any translation belongs to...
cdlemg2dN 39951 This theorem can be used t...
cdlemg2cex 39952 Any translation is one of ...
cdlemg2ce 39953 Utility theorem to elimina...
cdlemg2jlemOLDN 39954 Part of proof of Lemma E i...
cdlemg2fvlem 39955 Lemma for ~ cdlemg2fv . (...
cdlemg2klem 39956 ~ cdleme42keg with simpler...
cdlemg2idN 39957 Version of ~ cdleme31id wi...
cdlemg3a 39958 Part of proof of Lemma G i...
cdlemg2jOLDN 39959 TODO: Replace this with ~...
cdlemg2fv 39960 Value of a translation in ...
cdlemg2fv2 39961 Value of a translation in ...
cdlemg2k 39962 ~ cdleme42keg with simpler...
cdlemg2kq 39963 ~ cdlemg2k with ` P ` and ...
cdlemg2l 39964 TODO: FIX COMMENT. (Contr...
cdlemg2m 39965 TODO: FIX COMMENT. (Contr...
cdlemg5 39966 TODO: Is there a simpler ...
cdlemb3 39967 Given two atoms not under ...
cdlemg7fvbwN 39968 Properties of a translatio...
cdlemg4a 39969 TODO: FIX COMMENT If fg(p...
cdlemg4b1 39970 TODO: FIX COMMENT. (Contr...
cdlemg4b2 39971 TODO: FIX COMMENT. (Contr...
cdlemg4b12 39972 TODO: FIX COMMENT. (Contr...
cdlemg4c 39973 TODO: FIX COMMENT. (Contr...
cdlemg4d 39974 TODO: FIX COMMENT. (Contr...
cdlemg4e 39975 TODO: FIX COMMENT. (Contr...
cdlemg4f 39976 TODO: FIX COMMENT. (Contr...
cdlemg4g 39977 TODO: FIX COMMENT. (Contr...
cdlemg4 39978 TODO: FIX COMMENT. (Contr...
cdlemg6a 39979 TODO: FIX COMMENT. TODO: ...
cdlemg6b 39980 TODO: FIX COMMENT. TODO: ...
cdlemg6c 39981 TODO: FIX COMMENT. (Contr...
cdlemg6d 39982 TODO: FIX COMMENT. (Contr...
cdlemg6e 39983 TODO: FIX COMMENT. (Contr...
cdlemg6 39984 TODO: FIX COMMENT. (Contr...
cdlemg7fvN 39985 Value of a translation com...
cdlemg7aN 39986 TODO: FIX COMMENT. (Contr...
cdlemg7N 39987 TODO: FIX COMMENT. (Contr...
cdlemg8a 39988 TODO: FIX COMMENT. (Contr...
cdlemg8b 39989 TODO: FIX COMMENT. (Contr...
cdlemg8c 39990 TODO: FIX COMMENT. (Contr...
cdlemg8d 39991 TODO: FIX COMMENT. (Contr...
cdlemg8 39992 TODO: FIX COMMENT. (Contr...
cdlemg9a 39993 TODO: FIX COMMENT. (Contr...
cdlemg9b 39994 The triples ` <. P , ( F `...
cdlemg9 39995 The triples ` <. P , ( F `...
cdlemg10b 39996 TODO: FIX COMMENT. TODO: ...
cdlemg10bALTN 39997 TODO: FIX COMMENT. TODO: ...
cdlemg11a 39998 TODO: FIX COMMENT. (Contr...
cdlemg11aq 39999 TODO: FIX COMMENT. TODO: ...
cdlemg10c 40000 TODO: FIX COMMENT. TODO: ...
cdlemg10a 40001 TODO: FIX COMMENT. (Contr...
cdlemg10 40002 TODO: FIX COMMENT. (Contr...
cdlemg11b 40003 TODO: FIX COMMENT. (Contr...
cdlemg12a 40004 TODO: FIX COMMENT. (Contr...
cdlemg12b 40005 The triples ` <. P , ( F `...
cdlemg12c 40006 The triples ` <. P , ( F `...
cdlemg12d 40007 TODO: FIX COMMENT. (Contr...
cdlemg12e 40008 TODO: FIX COMMENT. (Contr...
cdlemg12f 40009 TODO: FIX COMMENT. (Contr...
cdlemg12g 40010 TODO: FIX COMMENT. TODO: ...
cdlemg12 40011 TODO: FIX COMMENT. (Contr...
cdlemg13a 40012 TODO: FIX COMMENT. (Contr...
cdlemg13 40013 TODO: FIX COMMENT. (Contr...
cdlemg14f 40014 TODO: FIX COMMENT. (Contr...
cdlemg14g 40015 TODO: FIX COMMENT. (Contr...
cdlemg15a 40016 Eliminate the ` ( F `` P )...
cdlemg15 40017 Eliminate the ` ( (...
cdlemg16 40018 Part of proof of Lemma G o...
cdlemg16ALTN 40019 This version of ~ cdlemg16...
cdlemg16z 40020 Eliminate ` ( ( F `...
cdlemg16zz 40021 Eliminate ` P =/= Q ` from...
cdlemg17a 40022 TODO: FIX COMMENT. (Contr...
cdlemg17b 40023 Part of proof of Lemma G i...
cdlemg17dN 40024 TODO: fix comment. (Contr...
cdlemg17dALTN 40025 Same as ~ cdlemg17dN with ...
cdlemg17e 40026 TODO: fix comment. (Contr...
cdlemg17f 40027 TODO: fix comment. (Contr...
cdlemg17g 40028 TODO: fix comment. (Contr...
cdlemg17h 40029 TODO: fix comment. (Contr...
cdlemg17i 40030 TODO: fix comment. (Contr...
cdlemg17ir 40031 TODO: fix comment. (Contr...
cdlemg17j 40032 TODO: fix comment. (Contr...
cdlemg17pq 40033 Utility theorem for swappi...
cdlemg17bq 40034 ~ cdlemg17b with ` P ` and...
cdlemg17iqN 40035 ~ cdlemg17i with ` P ` and...
cdlemg17irq 40036 ~ cdlemg17ir with ` P ` an...
cdlemg17jq 40037 ~ cdlemg17j with ` P ` and...
cdlemg17 40038 Part of Lemma G of [Crawle...
cdlemg18a 40039 Show two lines are differe...
cdlemg18b 40040 Lemma for ~ cdlemg18c . T...
cdlemg18c 40041 Show two lines intersect a...
cdlemg18d 40042 Show two lines intersect a...
cdlemg18 40043 Show two lines intersect a...
cdlemg19a 40044 Show two lines intersect a...
cdlemg19 40045 Show two lines intersect a...
cdlemg20 40046 Show two lines intersect a...
cdlemg21 40047 Version of cdlemg19 with `...
cdlemg22 40048 ~ cdlemg21 with ` ( F `` P...
cdlemg24 40049 Combine ~ cdlemg16z and ~ ...
cdlemg37 40050 Use ~ cdlemg8 to eliminate...
cdlemg25zz 40051 ~ cdlemg16zz restated for ...
cdlemg26zz 40052 ~ cdlemg16zz restated for ...
cdlemg27a 40053 For use with case when ` (...
cdlemg28a 40054 Part of proof of Lemma G o...
cdlemg31b0N 40055 TODO: Fix comment. (Cont...
cdlemg31b0a 40056 TODO: Fix comment. (Cont...
cdlemg27b 40057 TODO: Fix comment. (Cont...
cdlemg31a 40058 TODO: fix comment. (Contr...
cdlemg31b 40059 TODO: fix comment. (Contr...
cdlemg31c 40060 Show that when ` N ` is an...
cdlemg31d 40061 Eliminate ` ( F `` P ) =/=...
cdlemg33b0 40062 TODO: Fix comment. (Cont...
cdlemg33c0 40063 TODO: Fix comment. (Cont...
cdlemg28b 40064 Part of proof of Lemma G o...
cdlemg28 40065 Part of proof of Lemma G o...
cdlemg29 40066 Eliminate ` ( F `` P ) =/=...
cdlemg33a 40067 TODO: Fix comment. (Cont...
cdlemg33b 40068 TODO: Fix comment. (Cont...
cdlemg33c 40069 TODO: Fix comment. (Cont...
cdlemg33d 40070 TODO: Fix comment. (Cont...
cdlemg33e 40071 TODO: Fix comment. (Cont...
cdlemg33 40072 Combine ~ cdlemg33b , ~ cd...
cdlemg34 40073 Use cdlemg33 to eliminate ...
cdlemg35 40074 TODO: Fix comment. TODO:...
cdlemg36 40075 Use cdlemg35 to eliminate ...
cdlemg38 40076 Use ~ cdlemg37 to eliminat...
cdlemg39 40077 Eliminate ` =/= ` conditio...
cdlemg40 40078 Eliminate ` P =/= Q ` cond...
cdlemg41 40079 Convert ~ cdlemg40 to func...
ltrnco 40080 The composition of two tra...
trlcocnv 40081 Swap the arguments of the ...
trlcoabs 40082 Absorption into a composit...
trlcoabs2N 40083 Absorption of the trace of...
trlcoat 40084 The trace of a composition...
trlcocnvat 40085 Commonly used special case...
trlconid 40086 The composition of two dif...
trlcolem 40087 Lemma for ~ trlco . (Cont...
trlco 40088 The trace of a composition...
trlcone 40089 If two translations have d...
cdlemg42 40090 Part of proof of Lemma G o...
cdlemg43 40091 Part of proof of Lemma G o...
cdlemg44a 40092 Part of proof of Lemma G o...
cdlemg44b 40093 Eliminate ` ( F `` P ) =/=...
cdlemg44 40094 Part of proof of Lemma G o...
cdlemg47a 40095 TODO: fix comment. TODO: ...
cdlemg46 40096 Part of proof of Lemma G o...
cdlemg47 40097 Part of proof of Lemma G o...
cdlemg48 40098 Eliminate ` h ` from ~ cdl...
ltrncom 40099 Composition is commutative...
ltrnco4 40100 Rearrange a composition of...
trljco 40101 Trace joined with trace of...
trljco2 40102 Trace joined with trace of...
tgrpfset 40105 The translation group maps...
tgrpset 40106 The translation group for ...
tgrpbase 40107 The base set of the transl...
tgrpopr 40108 The group operation of the...
tgrpov 40109 The group operation value ...
tgrpgrplem 40110 Lemma for ~ tgrpgrp . (Co...
tgrpgrp 40111 The translation group is a...
tgrpabl 40112 The translation group is a...
tendofset 40119 The set of all trace-prese...
tendoset 40120 The set of trace-preservin...
istendo 40121 The predicate "is a trace-...
tendotp 40122 Trace-preserving property ...
istendod 40123 Deduce the predicate "is a...
tendof 40124 Functionality of a trace-p...
tendoeq1 40125 Condition determining equa...
tendovalco 40126 Value of composition of tr...
tendocoval 40127 Value of composition of en...
tendocl 40128 Closure of a trace-preserv...
tendoco2 40129 Distribution of compositio...
tendoidcl 40130 The identity is a trace-pr...
tendo1mul 40131 Multiplicative identity mu...
tendo1mulr 40132 Multiplicative identity mu...
tendococl 40133 The composition of two tra...
tendoid 40134 The identity value of a tr...
tendoeq2 40135 Condition determining equa...
tendoplcbv 40136 Define sum operation for t...
tendopl 40137 Value of endomorphism sum ...
tendopl2 40138 Value of result of endomor...
tendoplcl2 40139 Value of result of endomor...
tendoplco2 40140 Value of result of endomor...
tendopltp 40141 Trace-preserving property ...
tendoplcl 40142 Endomorphism sum is a trac...
tendoplcom 40143 The endomorphism sum opera...
tendoplass 40144 The endomorphism sum opera...
tendodi1 40145 Endomorphism composition d...
tendodi2 40146 Endomorphism composition d...
tendo0cbv 40147 Define additive identity f...
tendo02 40148 Value of additive identity...
tendo0co2 40149 The additive identity trac...
tendo0tp 40150 Trace-preserving property ...
tendo0cl 40151 The additive identity is a...
tendo0pl 40152 Property of the additive i...
tendo0plr 40153 Property of the additive i...
tendoicbv 40154 Define inverse function fo...
tendoi 40155 Value of inverse endomorph...
tendoi2 40156 Value of additive inverse ...
tendoicl 40157 Closure of the additive in...
tendoipl 40158 Property of the additive i...
tendoipl2 40159 Property of the additive i...
erngfset 40160 The division rings on trac...
erngset 40161 The division ring on trace...
erngbase 40162 The base set of the divisi...
erngfplus 40163 Ring addition operation. ...
erngplus 40164 Ring addition operation. ...
erngplus2 40165 Ring addition operation. ...
erngfmul 40166 Ring multiplication operat...
erngmul 40167 Ring addition operation. ...
erngfset-rN 40168 The division rings on trac...
erngset-rN 40169 The division ring on trace...
erngbase-rN 40170 The base set of the divisi...
erngfplus-rN 40171 Ring addition operation. ...
erngplus-rN 40172 Ring addition operation. ...
erngplus2-rN 40173 Ring addition operation. ...
erngfmul-rN 40174 Ring multiplication operat...
erngmul-rN 40175 Ring addition operation. ...
cdlemh1 40176 Part of proof of Lemma H o...
cdlemh2 40177 Part of proof of Lemma H o...
cdlemh 40178 Lemma H of [Crawley] p. 11...
cdlemi1 40179 Part of proof of Lemma I o...
cdlemi2 40180 Part of proof of Lemma I o...
cdlemi 40181 Lemma I of [Crawley] p. 11...
cdlemj1 40182 Part of proof of Lemma J o...
cdlemj2 40183 Part of proof of Lemma J o...
cdlemj3 40184 Part of proof of Lemma J o...
tendocan 40185 Cancellation law: if the v...
tendoid0 40186 A trace-preserving endomor...
tendo0mul 40187 Additive identity multipli...
tendo0mulr 40188 Additive identity multipli...
tendo1ne0 40189 The identity (unity) is no...
tendoconid 40190 The composition (product) ...
tendotr 40191 The trace of the value of ...
cdlemk1 40192 Part of proof of Lemma K o...
cdlemk2 40193 Part of proof of Lemma K o...
cdlemk3 40194 Part of proof of Lemma K o...
cdlemk4 40195 Part of proof of Lemma K o...
cdlemk5a 40196 Part of proof of Lemma K o...
cdlemk5 40197 Part of proof of Lemma K o...
cdlemk6 40198 Part of proof of Lemma K o...
cdlemk8 40199 Part of proof of Lemma K o...
cdlemk9 40200 Part of proof of Lemma K o...
cdlemk9bN 40201 Part of proof of Lemma K o...
cdlemki 40202 Part of proof of Lemma K o...
cdlemkvcl 40203 Part of proof of Lemma K o...
cdlemk10 40204 Part of proof of Lemma K o...
cdlemksv 40205 Part of proof of Lemma K o...
cdlemksel 40206 Part of proof of Lemma K o...
cdlemksat 40207 Part of proof of Lemma K o...
cdlemksv2 40208 Part of proof of Lemma K o...
cdlemk7 40209 Part of proof of Lemma K o...
cdlemk11 40210 Part of proof of Lemma K o...
cdlemk12 40211 Part of proof of Lemma K o...
cdlemkoatnle 40212 Utility lemma. (Contribut...
cdlemk13 40213 Part of proof of Lemma K o...
cdlemkole 40214 Utility lemma. (Contribut...
cdlemk14 40215 Part of proof of Lemma K o...
cdlemk15 40216 Part of proof of Lemma K o...
cdlemk16a 40217 Part of proof of Lemma K o...
cdlemk16 40218 Part of proof of Lemma K o...
cdlemk17 40219 Part of proof of Lemma K o...
cdlemk1u 40220 Part of proof of Lemma K o...
cdlemk5auN 40221 Part of proof of Lemma K o...
cdlemk5u 40222 Part of proof of Lemma K o...
cdlemk6u 40223 Part of proof of Lemma K o...
cdlemkj 40224 Part of proof of Lemma K o...
cdlemkuvN 40225 Part of proof of Lemma K o...
cdlemkuel 40226 Part of proof of Lemma K o...
cdlemkuat 40227 Part of proof of Lemma K o...
cdlemkuv2 40228 Part of proof of Lemma K o...
cdlemk18 40229 Part of proof of Lemma K o...
cdlemk19 40230 Part of proof of Lemma K o...
cdlemk7u 40231 Part of proof of Lemma K o...
cdlemk11u 40232 Part of proof of Lemma K o...
cdlemk12u 40233 Part of proof of Lemma K o...
cdlemk21N 40234 Part of proof of Lemma K o...
cdlemk20 40235 Part of proof of Lemma K o...
cdlemkoatnle-2N 40236 Utility lemma. (Contribut...
cdlemk13-2N 40237 Part of proof of Lemma K o...
cdlemkole-2N 40238 Utility lemma. (Contribut...
cdlemk14-2N 40239 Part of proof of Lemma K o...
cdlemk15-2N 40240 Part of proof of Lemma K o...
cdlemk16-2N 40241 Part of proof of Lemma K o...
cdlemk17-2N 40242 Part of proof of Lemma K o...
cdlemkj-2N 40243 Part of proof of Lemma K o...
cdlemkuv-2N 40244 Part of proof of Lemma K o...
cdlemkuel-2N 40245 Part of proof of Lemma K o...
cdlemkuv2-2 40246 Part of proof of Lemma K o...
cdlemk18-2N 40247 Part of proof of Lemma K o...
cdlemk19-2N 40248 Part of proof of Lemma K o...
cdlemk7u-2N 40249 Part of proof of Lemma K o...
cdlemk11u-2N 40250 Part of proof of Lemma K o...
cdlemk12u-2N 40251 Part of proof of Lemma K o...
cdlemk21-2N 40252 Part of proof of Lemma K o...
cdlemk20-2N 40253 Part of proof of Lemma K o...
cdlemk22 40254 Part of proof of Lemma K o...
cdlemk30 40255 Part of proof of Lemma K o...
cdlemkuu 40256 Convert between function a...
cdlemk31 40257 Part of proof of Lemma K o...
cdlemk32 40258 Part of proof of Lemma K o...
cdlemkuel-3 40259 Part of proof of Lemma K o...
cdlemkuv2-3N 40260 Part of proof of Lemma K o...
cdlemk18-3N 40261 Part of proof of Lemma K o...
cdlemk22-3 40262 Part of proof of Lemma K o...
cdlemk23-3 40263 Part of proof of Lemma K o...
cdlemk24-3 40264 Part of proof of Lemma K o...
cdlemk25-3 40265 Part of proof of Lemma K o...
cdlemk26b-3 40266 Part of proof of Lemma K o...
cdlemk26-3 40267 Part of proof of Lemma K o...
cdlemk27-3 40268 Part of proof of Lemma K o...
cdlemk28-3 40269 Part of proof of Lemma K o...
cdlemk33N 40270 Part of proof of Lemma K o...
cdlemk34 40271 Part of proof of Lemma K o...
cdlemk29-3 40272 Part of proof of Lemma K o...
cdlemk35 40273 Part of proof of Lemma K o...
cdlemk36 40274 Part of proof of Lemma K o...
cdlemk37 40275 Part of proof of Lemma K o...
cdlemk38 40276 Part of proof of Lemma K o...
cdlemk39 40277 Part of proof of Lemma K o...
cdlemk40 40278 TODO: fix comment. (Contr...
cdlemk40t 40279 TODO: fix comment. (Contr...
cdlemk40f 40280 TODO: fix comment. (Contr...
cdlemk41 40281 Part of proof of Lemma K o...
cdlemkfid1N 40282 Lemma for ~ cdlemkfid3N . ...
cdlemkid1 40283 Lemma for ~ cdlemkid . (C...
cdlemkfid2N 40284 Lemma for ~ cdlemkfid3N . ...
cdlemkid2 40285 Lemma for ~ cdlemkid . (C...
cdlemkfid3N 40286 TODO: is this useful or sh...
cdlemky 40287 Part of proof of Lemma K o...
cdlemkyu 40288 Convert between function a...
cdlemkyuu 40289 ~ cdlemkyu with some hypot...
cdlemk11ta 40290 Part of proof of Lemma K o...
cdlemk19ylem 40291 Lemma for ~ cdlemk19y . (...
cdlemk11tb 40292 Part of proof of Lemma K o...
cdlemk19y 40293 ~ cdlemk19 with simpler hy...
cdlemkid3N 40294 Lemma for ~ cdlemkid . (C...
cdlemkid4 40295 Lemma for ~ cdlemkid . (C...
cdlemkid5 40296 Lemma for ~ cdlemkid . (C...
cdlemkid 40297 The value of the tau funct...
cdlemk35s 40298 Substitution version of ~ ...
cdlemk35s-id 40299 Substitution version of ~ ...
cdlemk39s 40300 Substitution version of ~ ...
cdlemk39s-id 40301 Substitution version of ~ ...
cdlemk42 40302 Part of proof of Lemma K o...
cdlemk19xlem 40303 Lemma for ~ cdlemk19x . (...
cdlemk19x 40304 ~ cdlemk19 with simpler hy...
cdlemk42yN 40305 Part of proof of Lemma K o...
cdlemk11tc 40306 Part of proof of Lemma K o...
cdlemk11t 40307 Part of proof of Lemma K o...
cdlemk45 40308 Part of proof of Lemma K o...
cdlemk46 40309 Part of proof of Lemma K o...
cdlemk47 40310 Part of proof of Lemma K o...
cdlemk48 40311 Part of proof of Lemma K o...
cdlemk49 40312 Part of proof of Lemma K o...
cdlemk50 40313 Part of proof of Lemma K o...
cdlemk51 40314 Part of proof of Lemma K o...
cdlemk52 40315 Part of proof of Lemma K o...
cdlemk53a 40316 Lemma for ~ cdlemk53 . (C...
cdlemk53b 40317 Lemma for ~ cdlemk53 . (C...
cdlemk53 40318 Part of proof of Lemma K o...
cdlemk54 40319 Part of proof of Lemma K o...
cdlemk55a 40320 Lemma for ~ cdlemk55 . (C...
cdlemk55b 40321 Lemma for ~ cdlemk55 . (C...
cdlemk55 40322 Part of proof of Lemma K o...
cdlemkyyN 40323 Part of proof of Lemma K o...
cdlemk43N 40324 Part of proof of Lemma K o...
cdlemk35u 40325 Substitution version of ~ ...
cdlemk55u1 40326 Lemma for ~ cdlemk55u . (...
cdlemk55u 40327 Part of proof of Lemma K o...
cdlemk39u1 40328 Lemma for ~ cdlemk39u . (...
cdlemk39u 40329 Part of proof of Lemma K o...
cdlemk19u1 40330 ~ cdlemk19 with simpler hy...
cdlemk19u 40331 Part of Lemma K of [Crawle...
cdlemk56 40332 Part of Lemma K of [Crawle...
cdlemk19w 40333 Use a fixed element to eli...
cdlemk56w 40334 Use a fixed element to eli...
cdlemk 40335 Lemma K of [Crawley] p. 11...
tendoex 40336 Generalization of Lemma K ...
cdleml1N 40337 Part of proof of Lemma L o...
cdleml2N 40338 Part of proof of Lemma L o...
cdleml3N 40339 Part of proof of Lemma L o...
cdleml4N 40340 Part of proof of Lemma L o...
cdleml5N 40341 Part of proof of Lemma L o...
cdleml6 40342 Part of proof of Lemma L o...
cdleml7 40343 Part of proof of Lemma L o...
cdleml8 40344 Part of proof of Lemma L o...
cdleml9 40345 Part of proof of Lemma L o...
dva1dim 40346 Two expressions for the 1-...
dvhb1dimN 40347 Two expressions for the 1-...
erng1lem 40348 Value of the endomorphism ...
erngdvlem1 40349 Lemma for ~ eringring . (...
erngdvlem2N 40350 Lemma for ~ eringring . (...
erngdvlem3 40351 Lemma for ~ eringring . (...
erngdvlem4 40352 Lemma for ~ erngdv . (Con...
eringring 40353 An endomorphism ring is a ...
erngdv 40354 An endomorphism ring is a ...
erng0g 40355 The division ring zero of ...
erng1r 40356 The division ring unity of...
erngdvlem1-rN 40357 Lemma for ~ eringring . (...
erngdvlem2-rN 40358 Lemma for ~ eringring . (...
erngdvlem3-rN 40359 Lemma for ~ eringring . (...
erngdvlem4-rN 40360 Lemma for ~ erngdv . (Con...
erngring-rN 40361 An endomorphism ring is a ...
erngdv-rN 40362 An endomorphism ring is a ...
dvafset 40365 The constructed partial ve...
dvaset 40366 The constructed partial ve...
dvasca 40367 The ring base set of the c...
dvabase 40368 The ring base set of the c...
dvafplusg 40369 Ring addition operation fo...
dvaplusg 40370 Ring addition operation fo...
dvaplusgv 40371 Ring addition operation fo...
dvafmulr 40372 Ring multiplication operat...
dvamulr 40373 Ring multiplication operat...
dvavbase 40374 The vectors (vector base s...
dvafvadd 40375 The vector sum operation f...
dvavadd 40376 Ring addition operation fo...
dvafvsca 40377 Ring addition operation fo...
dvavsca 40378 Ring addition operation fo...
tendospcl 40379 Closure of endomorphism sc...
tendospass 40380 Associative law for endomo...
tendospdi1 40381 Forward distributive law f...
tendocnv 40382 Converse of a trace-preser...
tendospdi2 40383 Reverse distributive law f...
tendospcanN 40384 Cancellation law for trace...
dvaabl 40385 The constructed partial ve...
dvalveclem 40386 Lemma for ~ dvalvec . (Co...
dvalvec 40387 The constructed partial ve...
dva0g 40388 The zero vector of partial...
diaffval 40391 The partial isomorphism A ...
diafval 40392 The partial isomorphism A ...
diaval 40393 The partial isomorphism A ...
diaelval 40394 Member of the partial isom...
diafn 40395 Functionality and domain o...
diadm 40396 Domain of the partial isom...
diaeldm 40397 Member of domain of the pa...
diadmclN 40398 A member of domain of the ...
diadmleN 40399 A member of domain of the ...
dian0 40400 The value of the partial i...
dia0eldmN 40401 The lattice zero belongs t...
dia1eldmN 40402 The fiducial hyperplane (t...
diass 40403 The value of the partial i...
diael 40404 A member of the value of t...
diatrl 40405 Trace of a member of the p...
diaelrnN 40406 Any value of the partial i...
dialss 40407 The value of partial isomo...
diaord 40408 The partial isomorphism A ...
dia11N 40409 The partial isomorphism A ...
diaf11N 40410 The partial isomorphism A ...
diaclN 40411 Closure of partial isomorp...
diacnvclN 40412 Closure of partial isomorp...
dia0 40413 The value of the partial i...
dia1N 40414 The value of the partial i...
dia1elN 40415 The largest subspace in th...
diaglbN 40416 Partial isomorphism A of a...
diameetN 40417 Partial isomorphism A of a...
diainN 40418 Inverse partial isomorphis...
diaintclN 40419 The intersection of partia...
diasslssN 40420 The partial isomorphism A ...
diassdvaN 40421 The partial isomorphism A ...
dia1dim 40422 Two expressions for the 1-...
dia1dim2 40423 Two expressions for a 1-di...
dia1dimid 40424 A vector (translation) bel...
dia2dimlem1 40425 Lemma for ~ dia2dim . Sho...
dia2dimlem2 40426 Lemma for ~ dia2dim . Def...
dia2dimlem3 40427 Lemma for ~ dia2dim . Def...
dia2dimlem4 40428 Lemma for ~ dia2dim . Sho...
dia2dimlem5 40429 Lemma for ~ dia2dim . The...
dia2dimlem6 40430 Lemma for ~ dia2dim . Eli...
dia2dimlem7 40431 Lemma for ~ dia2dim . Eli...
dia2dimlem8 40432 Lemma for ~ dia2dim . Eli...
dia2dimlem9 40433 Lemma for ~ dia2dim . Eli...
dia2dimlem10 40434 Lemma for ~ dia2dim . Con...
dia2dimlem11 40435 Lemma for ~ dia2dim . Con...
dia2dimlem12 40436 Lemma for ~ dia2dim . Obt...
dia2dimlem13 40437 Lemma for ~ dia2dim . Eli...
dia2dim 40438 A two-dimensional subspace...
dvhfset 40441 The constructed full vecto...
dvhset 40442 The constructed full vecto...
dvhsca 40443 The ring of scalars of the...
dvhbase 40444 The ring base set of the c...
dvhfplusr 40445 Ring addition operation fo...
dvhfmulr 40446 Ring multiplication operat...
dvhmulr 40447 Ring multiplication operat...
dvhvbase 40448 The vectors (vector base s...
dvhelvbasei 40449 Vector membership in the c...
dvhvaddcbv 40450 Change bound variables to ...
dvhvaddval 40451 The vector sum operation f...
dvhfvadd 40452 The vector sum operation f...
dvhvadd 40453 The vector sum operation f...
dvhopvadd 40454 The vector sum operation f...
dvhopvadd2 40455 The vector sum operation f...
dvhvaddcl 40456 Closure of the vector sum ...
dvhvaddcomN 40457 Commutativity of vector su...
dvhvaddass 40458 Associativity of vector su...
dvhvscacbv 40459 Change bound variables to ...
dvhvscaval 40460 The scalar product operati...
dvhfvsca 40461 Scalar product operation f...
dvhvsca 40462 Scalar product operation f...
dvhopvsca 40463 Scalar product operation f...
dvhvscacl 40464 Closure of the scalar prod...
tendoinvcl 40465 Closure of multiplicative ...
tendolinv 40466 Left multiplicative invers...
tendorinv 40467 Right multiplicative inver...
dvhgrp 40468 The full vector space ` U ...
dvhlveclem 40469 Lemma for ~ dvhlvec . TOD...
dvhlvec 40470 The full vector space ` U ...
dvhlmod 40471 The full vector space ` U ...
dvh0g 40472 The zero vector of vector ...
dvheveccl 40473 Properties of a unit vecto...
dvhopclN 40474 Closure of a ` DVecH ` vec...
dvhopaddN 40475 Sum of ` DVecH ` vectors e...
dvhopspN 40476 Scalar product of ` DVecH ...
dvhopN 40477 Decompose a ` DVecH ` vect...
dvhopellsm 40478 Ordered pair membership in...
cdlemm10N 40479 The image of the map ` G `...
docaffvalN 40482 Subspace orthocomplement f...
docafvalN 40483 Subspace orthocomplement f...
docavalN 40484 Subspace orthocomplement f...
docaclN 40485 Closure of subspace orthoc...
diaocN 40486 Value of partial isomorphi...
doca2N 40487 Double orthocomplement of ...
doca3N 40488 Double orthocomplement of ...
dvadiaN 40489 Any closed subspace is a m...
diarnN 40490 Partial isomorphism A maps...
diaf1oN 40491 The partial isomorphism A ...
djaffvalN 40494 Subspace join for ` DVecA ...
djafvalN 40495 Subspace join for ` DVecA ...
djavalN 40496 Subspace join for ` DVecA ...
djaclN 40497 Closure of subspace join f...
djajN 40498 Transfer lattice join to `...
dibffval 40501 The partial isomorphism B ...
dibfval 40502 The partial isomorphism B ...
dibval 40503 The partial isomorphism B ...
dibopelvalN 40504 Member of the partial isom...
dibval2 40505 Value of the partial isomo...
dibopelval2 40506 Member of the partial isom...
dibval3N 40507 Value of the partial isomo...
dibelval3 40508 Member of the partial isom...
dibopelval3 40509 Member of the partial isom...
dibelval1st 40510 Membership in value of the...
dibelval1st1 40511 Membership in value of the...
dibelval1st2N 40512 Membership in value of the...
dibelval2nd 40513 Membership in value of the...
dibn0 40514 The value of the partial i...
dibfna 40515 Functionality and domain o...
dibdiadm 40516 Domain of the partial isom...
dibfnN 40517 Functionality and domain o...
dibdmN 40518 Domain of the partial isom...
dibeldmN 40519 Member of domain of the pa...
dibord 40520 The isomorphism B for a la...
dib11N 40521 The isomorphism B for a la...
dibf11N 40522 The partial isomorphism A ...
dibclN 40523 Closure of partial isomorp...
dibvalrel 40524 The value of partial isomo...
dib0 40525 The value of partial isomo...
dib1dim 40526 Two expressions for the 1-...
dibglbN 40527 Partial isomorphism B of a...
dibintclN 40528 The intersection of partia...
dib1dim2 40529 Two expressions for a 1-di...
dibss 40530 The partial isomorphism B ...
diblss 40531 The value of partial isomo...
diblsmopel 40532 Membership in subspace sum...
dicffval 40535 The partial isomorphism C ...
dicfval 40536 The partial isomorphism C ...
dicval 40537 The partial isomorphism C ...
dicopelval 40538 Membership in value of the...
dicelvalN 40539 Membership in value of the...
dicval2 40540 The partial isomorphism C ...
dicelval3 40541 Member of the partial isom...
dicopelval2 40542 Membership in value of the...
dicelval2N 40543 Membership in value of the...
dicfnN 40544 Functionality and domain o...
dicdmN 40545 Domain of the partial isom...
dicvalrelN 40546 The value of partial isomo...
dicssdvh 40547 The partial isomorphism C ...
dicelval1sta 40548 Membership in value of the...
dicelval1stN 40549 Membership in value of the...
dicelval2nd 40550 Membership in value of the...
dicvaddcl 40551 Membership in value of the...
dicvscacl 40552 Membership in value of the...
dicn0 40553 The value of the partial i...
diclss 40554 The value of partial isomo...
diclspsn 40555 The value of isomorphism C...
cdlemn2 40556 Part of proof of Lemma N o...
cdlemn2a 40557 Part of proof of Lemma N o...
cdlemn3 40558 Part of proof of Lemma N o...
cdlemn4 40559 Part of proof of Lemma N o...
cdlemn4a 40560 Part of proof of Lemma N o...
cdlemn5pre 40561 Part of proof of Lemma N o...
cdlemn5 40562 Part of proof of Lemma N o...
cdlemn6 40563 Part of proof of Lemma N o...
cdlemn7 40564 Part of proof of Lemma N o...
cdlemn8 40565 Part of proof of Lemma N o...
cdlemn9 40566 Part of proof of Lemma N o...
cdlemn10 40567 Part of proof of Lemma N o...
cdlemn11a 40568 Part of proof of Lemma N o...
cdlemn11b 40569 Part of proof of Lemma N o...
cdlemn11c 40570 Part of proof of Lemma N o...
cdlemn11pre 40571 Part of proof of Lemma N o...
cdlemn11 40572 Part of proof of Lemma N o...
cdlemn 40573 Lemma N of [Crawley] p. 12...
dihordlem6 40574 Part of proof of Lemma N o...
dihordlem7 40575 Part of proof of Lemma N o...
dihordlem7b 40576 Part of proof of Lemma N o...
dihjustlem 40577 Part of proof after Lemma ...
dihjust 40578 Part of proof after Lemma ...
dihord1 40579 Part of proof after Lemma ...
dihord2a 40580 Part of proof after Lemma ...
dihord2b 40581 Part of proof after Lemma ...
dihord2cN 40582 Part of proof after Lemma ...
dihord11b 40583 Part of proof after Lemma ...
dihord10 40584 Part of proof after Lemma ...
dihord11c 40585 Part of proof after Lemma ...
dihord2pre 40586 Part of proof after Lemma ...
dihord2pre2 40587 Part of proof after Lemma ...
dihord2 40588 Part of proof after Lemma ...
dihffval 40591 The isomorphism H for a la...
dihfval 40592 Isomorphism H for a lattic...
dihval 40593 Value of isomorphism H for...
dihvalc 40594 Value of isomorphism H for...
dihlsscpre 40595 Closure of isomorphism H f...
dihvalcqpre 40596 Value of isomorphism H for...
dihvalcq 40597 Value of isomorphism H for...
dihvalb 40598 Value of isomorphism H for...
dihopelvalbN 40599 Ordered pair member of the...
dihvalcqat 40600 Value of isomorphism H for...
dih1dimb 40601 Two expressions for a 1-di...
dih1dimb2 40602 Isomorphism H at an atom u...
dih1dimc 40603 Isomorphism H at an atom n...
dib2dim 40604 Extend ~ dia2dim to partia...
dih2dimb 40605 Extend ~ dib2dim to isomor...
dih2dimbALTN 40606 Extend ~ dia2dim to isomor...
dihopelvalcqat 40607 Ordered pair member of the...
dihvalcq2 40608 Value of isomorphism H for...
dihopelvalcpre 40609 Member of value of isomorp...
dihopelvalc 40610 Member of value of isomorp...
dihlss 40611 The value of isomorphism H...
dihss 40612 The value of isomorphism H...
dihssxp 40613 An isomorphism H value is ...
dihopcl 40614 Closure of an ordered pair...
xihopellsmN 40615 Ordered pair membership in...
dihopellsm 40616 Ordered pair membership in...
dihord6apre 40617 Part of proof that isomorp...
dihord3 40618 The isomorphism H for a la...
dihord4 40619 The isomorphism H for a la...
dihord5b 40620 Part of proof that isomorp...
dihord6b 40621 Part of proof that isomorp...
dihord6a 40622 Part of proof that isomorp...
dihord5apre 40623 Part of proof that isomorp...
dihord5a 40624 Part of proof that isomorp...
dihord 40625 The isomorphism H is order...
dih11 40626 The isomorphism H is one-t...
dihf11lem 40627 Functionality of the isomo...
dihf11 40628 The isomorphism H for a la...
dihfn 40629 Functionality and domain o...
dihdm 40630 Domain of isomorphism H. (...
dihcl 40631 Closure of isomorphism H. ...
dihcnvcl 40632 Closure of isomorphism H c...
dihcnvid1 40633 The converse isomorphism o...
dihcnvid2 40634 The isomorphism of a conve...
dihcnvord 40635 Ordering property for conv...
dihcnv11 40636 The converse of isomorphis...
dihsslss 40637 The isomorphism H maps to ...
dihrnlss 40638 The isomorphism H maps to ...
dihrnss 40639 The isomorphism H maps to ...
dihvalrel 40640 The value of isomorphism H...
dih0 40641 The value of isomorphism H...
dih0bN 40642 A lattice element is zero ...
dih0vbN 40643 A vector is zero iff its s...
dih0cnv 40644 The isomorphism H converse...
dih0rn 40645 The zero subspace belongs ...
dih0sb 40646 A subspace is zero iff the...
dih1 40647 The value of isomorphism H...
dih1rn 40648 The full vector space belo...
dih1cnv 40649 The isomorphism H converse...
dihwN 40650 Value of isomorphism H at ...
dihmeetlem1N 40651 Isomorphism H of a conjunc...
dihglblem5apreN 40652 A conjunction property of ...
dihglblem5aN 40653 A conjunction property of ...
dihglblem2aN 40654 Lemma for isomorphism H of...
dihglblem2N 40655 The GLB of a set of lattic...
dihglblem3N 40656 Isomorphism H of a lattice...
dihglblem3aN 40657 Isomorphism H of a lattice...
dihglblem4 40658 Isomorphism H of a lattice...
dihglblem5 40659 Isomorphism H of a lattice...
dihmeetlem2N 40660 Isomorphism H of a conjunc...
dihglbcpreN 40661 Isomorphism H of a lattice...
dihglbcN 40662 Isomorphism H of a lattice...
dihmeetcN 40663 Isomorphism H of a lattice...
dihmeetbN 40664 Isomorphism H of a lattice...
dihmeetbclemN 40665 Lemma for isomorphism H of...
dihmeetlem3N 40666 Lemma for isomorphism H of...
dihmeetlem4preN 40667 Lemma for isomorphism H of...
dihmeetlem4N 40668 Lemma for isomorphism H of...
dihmeetlem5 40669 Part of proof that isomorp...
dihmeetlem6 40670 Lemma for isomorphism H of...
dihmeetlem7N 40671 Lemma for isomorphism H of...
dihjatc1 40672 Lemma for isomorphism H of...
dihjatc2N 40673 Isomorphism H of join with...
dihjatc3 40674 Isomorphism H of join with...
dihmeetlem8N 40675 Lemma for isomorphism H of...
dihmeetlem9N 40676 Lemma for isomorphism H of...
dihmeetlem10N 40677 Lemma for isomorphism H of...
dihmeetlem11N 40678 Lemma for isomorphism H of...
dihmeetlem12N 40679 Lemma for isomorphism H of...
dihmeetlem13N 40680 Lemma for isomorphism H of...
dihmeetlem14N 40681 Lemma for isomorphism H of...
dihmeetlem15N 40682 Lemma for isomorphism H of...
dihmeetlem16N 40683 Lemma for isomorphism H of...
dihmeetlem17N 40684 Lemma for isomorphism H of...
dihmeetlem18N 40685 Lemma for isomorphism H of...
dihmeetlem19N 40686 Lemma for isomorphism H of...
dihmeetlem20N 40687 Lemma for isomorphism H of...
dihmeetALTN 40688 Isomorphism H of a lattice...
dih1dimatlem0 40689 Lemma for ~ dih1dimat . (...
dih1dimatlem 40690 Lemma for ~ dih1dimat . (...
dih1dimat 40691 Any 1-dimensional subspace...
dihlsprn 40692 The span of a vector belon...
dihlspsnssN 40693 A subspace included in a 1...
dihlspsnat 40694 The inverse isomorphism H ...
dihatlat 40695 The isomorphism H of an at...
dihat 40696 There exists at least one ...
dihpN 40697 The value of isomorphism H...
dihlatat 40698 The reverse isomorphism H ...
dihatexv 40699 There is a nonzero vector ...
dihatexv2 40700 There is a nonzero vector ...
dihglblem6 40701 Isomorphism H of a lattice...
dihglb 40702 Isomorphism H of a lattice...
dihglb2 40703 Isomorphism H of a lattice...
dihmeet 40704 Isomorphism H of a lattice...
dihintcl 40705 The intersection of closed...
dihmeetcl 40706 Closure of closed subspace...
dihmeet2 40707 Reverse isomorphism H of a...
dochffval 40710 Subspace orthocomplement f...
dochfval 40711 Subspace orthocomplement f...
dochval 40712 Subspace orthocomplement f...
dochval2 40713 Subspace orthocomplement f...
dochcl 40714 Closure of subspace orthoc...
dochlss 40715 A subspace orthocomplement...
dochssv 40716 A subspace orthocomplement...
dochfN 40717 Domain and codomain of the...
dochvalr 40718 Orthocomplement of a close...
doch0 40719 Orthocomplement of the zer...
doch1 40720 Orthocomplement of the uni...
dochoc0 40721 The zero subspace is close...
dochoc1 40722 The unit subspace (all vec...
dochvalr2 40723 Orthocomplement of a close...
dochvalr3 40724 Orthocomplement of a close...
doch2val2 40725 Double orthocomplement for...
dochss 40726 Subset law for orthocomple...
dochocss 40727 Double negative law for or...
dochoc 40728 Double negative law for or...
dochsscl 40729 If a set of vectors is inc...
dochoccl 40730 A set of vectors is closed...
dochord 40731 Ordering law for orthocomp...
dochord2N 40732 Ordering law for orthocomp...
dochord3 40733 Ordering law for orthocomp...
doch11 40734 Orthocomplement is one-to-...
dochsordN 40735 Strict ordering law for or...
dochn0nv 40736 An orthocomplement is nonz...
dihoml4c 40737 Version of ~ dihoml4 with ...
dihoml4 40738 Orthomodular law for const...
dochspss 40739 The span of a set of vecto...
dochocsp 40740 The span of an orthocomple...
dochspocN 40741 The span of an orthocomple...
dochocsn 40742 The double orthocomplement...
dochsncom 40743 Swap vectors in an orthoco...
dochsat 40744 The double orthocomplement...
dochshpncl 40745 If a hyperplane is not clo...
dochlkr 40746 Equivalent conditions for ...
dochkrshp 40747 The closure of a kernel is...
dochkrshp2 40748 Properties of the closure ...
dochkrshp3 40749 Properties of the closure ...
dochkrshp4 40750 Properties of the closure ...
dochdmj1 40751 De Morgan-like law for sub...
dochnoncon 40752 Law of noncontradiction. ...
dochnel2 40753 A nonzero member of a subs...
dochnel 40754 A nonzero vector doesn't b...
djhffval 40757 Subspace join for ` DVecH ...
djhfval 40758 Subspace join for ` DVecH ...
djhval 40759 Subspace join for ` DVecH ...
djhval2 40760 Value of subspace join for...
djhcl 40761 Closure of subspace join f...
djhlj 40762 Transfer lattice join to `...
djhljjN 40763 Lattice join in terms of `...
djhjlj 40764 ` DVecH ` vector space clo...
djhj 40765 ` DVecH ` vector space clo...
djhcom 40766 Subspace join commutes. (...
djhspss 40767 Subspace span of union is ...
djhsumss 40768 Subspace sum is a subset o...
dihsumssj 40769 The subspace sum of two is...
djhunssN 40770 Subspace union is a subset...
dochdmm1 40771 De Morgan-like law for clo...
djhexmid 40772 Excluded middle property o...
djh01 40773 Closed subspace join with ...
djh02 40774 Closed subspace join with ...
djhlsmcl 40775 A closed subspace sum equa...
djhcvat42 40776 A covering property. ( ~ ...
dihjatb 40777 Isomorphism H of lattice j...
dihjatc 40778 Isomorphism H of lattice j...
dihjatcclem1 40779 Lemma for isomorphism H of...
dihjatcclem2 40780 Lemma for isomorphism H of...
dihjatcclem3 40781 Lemma for ~ dihjatcc . (C...
dihjatcclem4 40782 Lemma for isomorphism H of...
dihjatcc 40783 Isomorphism H of lattice j...
dihjat 40784 Isomorphism H of lattice j...
dihprrnlem1N 40785 Lemma for ~ dihprrn , show...
dihprrnlem2 40786 Lemma for ~ dihprrn . (Co...
dihprrn 40787 The span of a vector pair ...
djhlsmat 40788 The sum of two subspace at...
dihjat1lem 40789 Subspace sum of a closed s...
dihjat1 40790 Subspace sum of a closed s...
dihsmsprn 40791 Subspace sum of a closed s...
dihjat2 40792 The subspace sum of a clos...
dihjat3 40793 Isomorphism H of lattice j...
dihjat4 40794 Transfer the subspace sum ...
dihjat6 40795 Transfer the subspace sum ...
dihsmsnrn 40796 The subspace sum of two si...
dihsmatrn 40797 The subspace sum of a clos...
dihjat5N 40798 Transfer lattice join with...
dvh4dimat 40799 There is an atom that is o...
dvh3dimatN 40800 There is an atom that is o...
dvh2dimatN 40801 Given an atom, there exist...
dvh1dimat 40802 There exists an atom. (Co...
dvh1dim 40803 There exists a nonzero vec...
dvh4dimlem 40804 Lemma for ~ dvh4dimN . (C...
dvhdimlem 40805 Lemma for ~ dvh2dim and ~ ...
dvh2dim 40806 There is a vector that is ...
dvh3dim 40807 There is a vector that is ...
dvh4dimN 40808 There is a vector that is ...
dvh3dim2 40809 There is a vector that is ...
dvh3dim3N 40810 There is a vector that is ...
dochsnnz 40811 The orthocomplement of a s...
dochsatshp 40812 The orthocomplement of a s...
dochsatshpb 40813 The orthocomplement of a s...
dochsnshp 40814 The orthocomplement of a n...
dochshpsat 40815 A hyperplane is closed iff...
dochkrsat 40816 The orthocomplement of a k...
dochkrsat2 40817 The orthocomplement of a k...
dochsat0 40818 The orthocomplement of a k...
dochkrsm 40819 The subspace sum of a clos...
dochexmidat 40820 Special case of excluded m...
dochexmidlem1 40821 Lemma for ~ dochexmid . H...
dochexmidlem2 40822 Lemma for ~ dochexmid . (...
dochexmidlem3 40823 Lemma for ~ dochexmid . U...
dochexmidlem4 40824 Lemma for ~ dochexmid . (...
dochexmidlem5 40825 Lemma for ~ dochexmid . (...
dochexmidlem6 40826 Lemma for ~ dochexmid . (...
dochexmidlem7 40827 Lemma for ~ dochexmid . C...
dochexmidlem8 40828 Lemma for ~ dochexmid . T...
dochexmid 40829 Excluded middle law for cl...
dochsnkrlem1 40830 Lemma for ~ dochsnkr . (C...
dochsnkrlem2 40831 Lemma for ~ dochsnkr . (C...
dochsnkrlem3 40832 Lemma for ~ dochsnkr . (C...
dochsnkr 40833 A (closed) kernel expresse...
dochsnkr2 40834 Kernel of the explicit fun...
dochsnkr2cl 40835 The ` X ` determining func...
dochflcl 40836 Closure of the explicit fu...
dochfl1 40837 The value of the explicit ...
dochfln0 40838 The value of a functional ...
dochkr1 40839 A nonzero functional has a...
dochkr1OLDN 40840 A nonzero functional has a...
lpolsetN 40843 The set of polarities of a...
islpolN 40844 The predicate "is a polari...
islpoldN 40845 Properties that determine ...
lpolfN 40846 Functionality of a polarit...
lpolvN 40847 The polarity of the whole ...
lpolconN 40848 Contraposition property of...
lpolsatN 40849 The polarity of an atomic ...
lpolpolsatN 40850 Property of a polarity. (...
dochpolN 40851 The subspace orthocompleme...
lcfl1lem 40852 Property of a functional w...
lcfl1 40853 Property of a functional w...
lcfl2 40854 Property of a functional w...
lcfl3 40855 Property of a functional w...
lcfl4N 40856 Property of a functional w...
lcfl5 40857 Property of a functional w...
lcfl5a 40858 Property of a functional w...
lcfl6lem 40859 Lemma for ~ lcfl6 . A fun...
lcfl7lem 40860 Lemma for ~ lcfl7N . If t...
lcfl6 40861 Property of a functional w...
lcfl7N 40862 Property of a functional w...
lcfl8 40863 Property of a functional w...
lcfl8a 40864 Property of a functional w...
lcfl8b 40865 Property of a nonzero func...
lcfl9a 40866 Property implying that a f...
lclkrlem1 40867 The set of functionals hav...
lclkrlem2a 40868 Lemma for ~ lclkr . Use ~...
lclkrlem2b 40869 Lemma for ~ lclkr . (Cont...
lclkrlem2c 40870 Lemma for ~ lclkr . (Cont...
lclkrlem2d 40871 Lemma for ~ lclkr . (Cont...
lclkrlem2e 40872 Lemma for ~ lclkr . The k...
lclkrlem2f 40873 Lemma for ~ lclkr . Const...
lclkrlem2g 40874 Lemma for ~ lclkr . Compa...
lclkrlem2h 40875 Lemma for ~ lclkr . Elimi...
lclkrlem2i 40876 Lemma for ~ lclkr . Elimi...
lclkrlem2j 40877 Lemma for ~ lclkr . Kerne...
lclkrlem2k 40878 Lemma for ~ lclkr . Kerne...
lclkrlem2l 40879 Lemma for ~ lclkr . Elimi...
lclkrlem2m 40880 Lemma for ~ lclkr . Const...
lclkrlem2n 40881 Lemma for ~ lclkr . (Cont...
lclkrlem2o 40882 Lemma for ~ lclkr . When ...
lclkrlem2p 40883 Lemma for ~ lclkr . When ...
lclkrlem2q 40884 Lemma for ~ lclkr . The s...
lclkrlem2r 40885 Lemma for ~ lclkr . When ...
lclkrlem2s 40886 Lemma for ~ lclkr . Thus,...
lclkrlem2t 40887 Lemma for ~ lclkr . We el...
lclkrlem2u 40888 Lemma for ~ lclkr . ~ lclk...
lclkrlem2v 40889 Lemma for ~ lclkr . When ...
lclkrlem2w 40890 Lemma for ~ lclkr . This ...
lclkrlem2x 40891 Lemma for ~ lclkr . Elimi...
lclkrlem2y 40892 Lemma for ~ lclkr . Resta...
lclkrlem2 40893 The set of functionals hav...
lclkr 40894 The set of functionals wit...
lcfls1lem 40895 Property of a functional w...
lcfls1N 40896 Property of a functional w...
lcfls1c 40897 Property of a functional w...
lclkrslem1 40898 The set of functionals hav...
lclkrslem2 40899 The set of functionals hav...
lclkrs 40900 The set of functionals hav...
lclkrs2 40901 The set of functionals wit...
lcfrvalsnN 40902 Reconstruction from the du...
lcfrlem1 40903 Lemma for ~ lcfr . Note t...
lcfrlem2 40904 Lemma for ~ lcfr . (Contr...
lcfrlem3 40905 Lemma for ~ lcfr . (Contr...
lcfrlem4 40906 Lemma for ~ lcfr . (Contr...
lcfrlem5 40907 Lemma for ~ lcfr . The se...
lcfrlem6 40908 Lemma for ~ lcfr . Closur...
lcfrlem7 40909 Lemma for ~ lcfr . Closur...
lcfrlem8 40910 Lemma for ~ lcf1o and ~ lc...
lcfrlem9 40911 Lemma for ~ lcf1o . (This...
lcf1o 40912 Define a function ` J ` th...
lcfrlem10 40913 Lemma for ~ lcfr . (Contr...
lcfrlem11 40914 Lemma for ~ lcfr . (Contr...
lcfrlem12N 40915 Lemma for ~ lcfr . (Contr...
lcfrlem13 40916 Lemma for ~ lcfr . (Contr...
lcfrlem14 40917 Lemma for ~ lcfr . (Contr...
lcfrlem15 40918 Lemma for ~ lcfr . (Contr...
lcfrlem16 40919 Lemma for ~ lcfr . (Contr...
lcfrlem17 40920 Lemma for ~ lcfr . Condit...
lcfrlem18 40921 Lemma for ~ lcfr . (Contr...
lcfrlem19 40922 Lemma for ~ lcfr . (Contr...
lcfrlem20 40923 Lemma for ~ lcfr . (Contr...
lcfrlem21 40924 Lemma for ~ lcfr . (Contr...
lcfrlem22 40925 Lemma for ~ lcfr . (Contr...
lcfrlem23 40926 Lemma for ~ lcfr . TODO: ...
lcfrlem24 40927 Lemma for ~ lcfr . (Contr...
lcfrlem25 40928 Lemma for ~ lcfr . Specia...
lcfrlem26 40929 Lemma for ~ lcfr . Specia...
lcfrlem27 40930 Lemma for ~ lcfr . Specia...
lcfrlem28 40931 Lemma for ~ lcfr . TODO: ...
lcfrlem29 40932 Lemma for ~ lcfr . (Contr...
lcfrlem30 40933 Lemma for ~ lcfr . (Contr...
lcfrlem31 40934 Lemma for ~ lcfr . (Contr...
lcfrlem32 40935 Lemma for ~ lcfr . (Contr...
lcfrlem33 40936 Lemma for ~ lcfr . (Contr...
lcfrlem34 40937 Lemma for ~ lcfr . (Contr...
lcfrlem35 40938 Lemma for ~ lcfr . (Contr...
lcfrlem36 40939 Lemma for ~ lcfr . (Contr...
lcfrlem37 40940 Lemma for ~ lcfr . (Contr...
lcfrlem38 40941 Lemma for ~ lcfr . Combin...
lcfrlem39 40942 Lemma for ~ lcfr . Elimin...
lcfrlem40 40943 Lemma for ~ lcfr . Elimin...
lcfrlem41 40944 Lemma for ~ lcfr . Elimin...
lcfrlem42 40945 Lemma for ~ lcfr . Elimin...
lcfr 40946 Reconstruction of a subspa...
lcdfval 40949 Dual vector space of funct...
lcdval 40950 Dual vector space of funct...
lcdval2 40951 Dual vector space of funct...
lcdlvec 40952 The dual vector space of f...
lcdlmod 40953 The dual vector space of f...
lcdvbase 40954 Vector base set of a dual ...
lcdvbasess 40955 The vector base set of the...
lcdvbaselfl 40956 A vector in the base set o...
lcdvbasecl 40957 Closure of the value of a ...
lcdvadd 40958 Vector addition for the cl...
lcdvaddval 40959 The value of the value of ...
lcdsca 40960 The ring of scalars of the...
lcdsbase 40961 Base set of scalar ring fo...
lcdsadd 40962 Scalar addition for the cl...
lcdsmul 40963 Scalar multiplication for ...
lcdvs 40964 Scalar product for the clo...
lcdvsval 40965 Value of scalar product op...
lcdvscl 40966 The scalar product operati...
lcdlssvscl 40967 Closure of scalar product ...
lcdvsass 40968 Associative law for scalar...
lcd0 40969 The zero scalar of the clo...
lcd1 40970 The unit scalar of the clo...
lcdneg 40971 The unit scalar of the clo...
lcd0v 40972 The zero functional in the...
lcd0v2 40973 The zero functional in the...
lcd0vvalN 40974 Value of the zero function...
lcd0vcl 40975 Closure of the zero functi...
lcd0vs 40976 A scalar zero times a func...
lcdvs0N 40977 A scalar times the zero fu...
lcdvsub 40978 The value of vector subtra...
lcdvsubval 40979 The value of the value of ...
lcdlss 40980 Subspaces of a dual vector...
lcdlss2N 40981 Subspaces of a dual vector...
lcdlsp 40982 Span in the set of functio...
lcdlkreqN 40983 Colinear functionals have ...
lcdlkreq2N 40984 Colinear functionals have ...
mapdffval 40987 Projectivity from vector s...
mapdfval 40988 Projectivity from vector s...
mapdval 40989 Value of projectivity from...
mapdvalc 40990 Value of projectivity from...
mapdval2N 40991 Value of projectivity from...
mapdval3N 40992 Value of projectivity from...
mapdval4N 40993 Value of projectivity from...
mapdval5N 40994 Value of projectivity from...
mapdordlem1a 40995 Lemma for ~ mapdord . (Co...
mapdordlem1bN 40996 Lemma for ~ mapdord . (Co...
mapdordlem1 40997 Lemma for ~ mapdord . (Co...
mapdordlem2 40998 Lemma for ~ mapdord . Ord...
mapdord 40999 Ordering property of the m...
mapd11 41000 The map defined by ~ df-ma...
mapddlssN 41001 The mapping of a subspace ...
mapdsn 41002 Value of the map defined b...
mapdsn2 41003 Value of the map defined b...
mapdsn3 41004 Value of the map defined b...
mapd1dim2lem1N 41005 Value of the map defined b...
mapdrvallem2 41006 Lemma for ~ mapdrval . TO...
mapdrvallem3 41007 Lemma for ~ mapdrval . (C...
mapdrval 41008 Given a dual subspace ` R ...
mapd1o 41009 The map defined by ~ df-ma...
mapdrn 41010 Range of the map defined b...
mapdunirnN 41011 Union of the range of the ...
mapdrn2 41012 Range of the map defined b...
mapdcnvcl 41013 Closure of the converse of...
mapdcl 41014 Closure the value of the m...
mapdcnvid1N 41015 Converse of the value of t...
mapdsord 41016 Strong ordering property o...
mapdcl2 41017 The mapping of a subspace ...
mapdcnvid2 41018 Value of the converse of t...
mapdcnvordN 41019 Ordering property of the c...
mapdcnv11N 41020 The converse of the map de...
mapdcv 41021 Covering property of the c...
mapdincl 41022 Closure of dual subspace i...
mapdin 41023 Subspace intersection is p...
mapdlsmcl 41024 Closure of dual subspace s...
mapdlsm 41025 Subspace sum is preserved ...
mapd0 41026 Projectivity map of the ze...
mapdcnvatN 41027 Atoms are preserved by the...
mapdat 41028 Atoms are preserved by the...
mapdspex 41029 The map of a span equals t...
mapdn0 41030 Transfer nonzero property ...
mapdncol 41031 Transfer non-colinearity f...
mapdindp 41032 Transfer (part of) vector ...
mapdpglem1 41033 Lemma for ~ mapdpg . Baer...
mapdpglem2 41034 Lemma for ~ mapdpg . Baer...
mapdpglem2a 41035 Lemma for ~ mapdpg . (Con...
mapdpglem3 41036 Lemma for ~ mapdpg . Baer...
mapdpglem4N 41037 Lemma for ~ mapdpg . (Con...
mapdpglem5N 41038 Lemma for ~ mapdpg . (Con...
mapdpglem6 41039 Lemma for ~ mapdpg . Baer...
mapdpglem8 41040 Lemma for ~ mapdpg . Baer...
mapdpglem9 41041 Lemma for ~ mapdpg . Baer...
mapdpglem10 41042 Lemma for ~ mapdpg . Baer...
mapdpglem11 41043 Lemma for ~ mapdpg . (Con...
mapdpglem12 41044 Lemma for ~ mapdpg . TODO...
mapdpglem13 41045 Lemma for ~ mapdpg . (Con...
mapdpglem14 41046 Lemma for ~ mapdpg . (Con...
mapdpglem15 41047 Lemma for ~ mapdpg . (Con...
mapdpglem16 41048 Lemma for ~ mapdpg . Baer...
mapdpglem17N 41049 Lemma for ~ mapdpg . Baer...
mapdpglem18 41050 Lemma for ~ mapdpg . Baer...
mapdpglem19 41051 Lemma for ~ mapdpg . Baer...
mapdpglem20 41052 Lemma for ~ mapdpg . Baer...
mapdpglem21 41053 Lemma for ~ mapdpg . (Con...
mapdpglem22 41054 Lemma for ~ mapdpg . Baer...
mapdpglem23 41055 Lemma for ~ mapdpg . Baer...
mapdpglem30a 41056 Lemma for ~ mapdpg . (Con...
mapdpglem30b 41057 Lemma for ~ mapdpg . (Con...
mapdpglem25 41058 Lemma for ~ mapdpg . Baer...
mapdpglem26 41059 Lemma for ~ mapdpg . Baer...
mapdpglem27 41060 Lemma for ~ mapdpg . Baer...
mapdpglem29 41061 Lemma for ~ mapdpg . Baer...
mapdpglem28 41062 Lemma for ~ mapdpg . Baer...
mapdpglem30 41063 Lemma for ~ mapdpg . Baer...
mapdpglem31 41064 Lemma for ~ mapdpg . Baer...
mapdpglem24 41065 Lemma for ~ mapdpg . Exis...
mapdpglem32 41066 Lemma for ~ mapdpg . Uniq...
mapdpg 41067 Part 1 of proof of the fir...
baerlem3lem1 41068 Lemma for ~ baerlem3 . (C...
baerlem5alem1 41069 Lemma for ~ baerlem5a . (...
baerlem5blem1 41070 Lemma for ~ baerlem5b . (...
baerlem3lem2 41071 Lemma for ~ baerlem3 . (C...
baerlem5alem2 41072 Lemma for ~ baerlem5a . (...
baerlem5blem2 41073 Lemma for ~ baerlem5b . (...
baerlem3 41074 An equality that holds whe...
baerlem5a 41075 An equality that holds whe...
baerlem5b 41076 An equality that holds whe...
baerlem5amN 41077 An equality that holds whe...
baerlem5bmN 41078 An equality that holds whe...
baerlem5abmN 41079 An equality that holds whe...
mapdindp0 41080 Vector independence lemma....
mapdindp1 41081 Vector independence lemma....
mapdindp2 41082 Vector independence lemma....
mapdindp3 41083 Vector independence lemma....
mapdindp4 41084 Vector independence lemma....
mapdhval 41085 Lemmma for ~~? mapdh . (C...
mapdhval0 41086 Lemmma for ~~? mapdh . (C...
mapdhval2 41087 Lemmma for ~~? mapdh . (C...
mapdhcl 41088 Lemmma for ~~? mapdh . (C...
mapdheq 41089 Lemmma for ~~? mapdh . Th...
mapdheq2 41090 Lemmma for ~~? mapdh . On...
mapdheq2biN 41091 Lemmma for ~~? mapdh . Pa...
mapdheq4lem 41092 Lemma for ~ mapdheq4 . Pa...
mapdheq4 41093 Lemma for ~~? mapdh . Par...
mapdh6lem1N 41094 Lemma for ~ mapdh6N . Par...
mapdh6lem2N 41095 Lemma for ~ mapdh6N . Par...
mapdh6aN 41096 Lemma for ~ mapdh6N . Par...
mapdh6b0N 41097 Lemmma for ~ mapdh6N . (C...
mapdh6bN 41098 Lemmma for ~ mapdh6N . (C...
mapdh6cN 41099 Lemmma for ~ mapdh6N . (C...
mapdh6dN 41100 Lemmma for ~ mapdh6N . (C...
mapdh6eN 41101 Lemmma for ~ mapdh6N . Pa...
mapdh6fN 41102 Lemmma for ~ mapdh6N . Pa...
mapdh6gN 41103 Lemmma for ~ mapdh6N . Pa...
mapdh6hN 41104 Lemmma for ~ mapdh6N . Pa...
mapdh6iN 41105 Lemmma for ~ mapdh6N . El...
mapdh6jN 41106 Lemmma for ~ mapdh6N . El...
mapdh6kN 41107 Lemmma for ~ mapdh6N . El...
mapdh6N 41108 Part (6) of [Baer] p. 47 l...
mapdh7eN 41109 Part (7) of [Baer] p. 48 l...
mapdh7cN 41110 Part (7) of [Baer] p. 48 l...
mapdh7dN 41111 Part (7) of [Baer] p. 48 l...
mapdh7fN 41112 Part (7) of [Baer] p. 48 l...
mapdh75e 41113 Part (7) of [Baer] p. 48 l...
mapdh75cN 41114 Part (7) of [Baer] p. 48 l...
mapdh75d 41115 Part (7) of [Baer] p. 48 l...
mapdh75fN 41116 Part (7) of [Baer] p. 48 l...
hvmapffval 41119 Map from nonzero vectors t...
hvmapfval 41120 Map from nonzero vectors t...
hvmapval 41121 Value of map from nonzero ...
hvmapvalvalN 41122 Value of value of map (i.e...
hvmapidN 41123 The value of the vector to...
hvmap1o 41124 The vector to functional m...
hvmapclN 41125 Closure of the vector to f...
hvmap1o2 41126 The vector to functional m...
hvmapcl2 41127 Closure of the vector to f...
hvmaplfl 41128 The vector to functional m...
hvmaplkr 41129 Kernel of the vector to fu...
mapdhvmap 41130 Relationship between ` map...
lspindp5 41131 Obtain an independent vect...
hdmaplem1 41132 Lemma to convert a frequen...
hdmaplem2N 41133 Lemma to convert a frequen...
hdmaplem3 41134 Lemma to convert a frequen...
hdmaplem4 41135 Lemma to convert a frequen...
mapdh8a 41136 Part of Part (8) in [Baer]...
mapdh8aa 41137 Part of Part (8) in [Baer]...
mapdh8ab 41138 Part of Part (8) in [Baer]...
mapdh8ac 41139 Part of Part (8) in [Baer]...
mapdh8ad 41140 Part of Part (8) in [Baer]...
mapdh8b 41141 Part of Part (8) in [Baer]...
mapdh8c 41142 Part of Part (8) in [Baer]...
mapdh8d0N 41143 Part of Part (8) in [Baer]...
mapdh8d 41144 Part of Part (8) in [Baer]...
mapdh8e 41145 Part of Part (8) in [Baer]...
mapdh8g 41146 Part of Part (8) in [Baer]...
mapdh8i 41147 Part of Part (8) in [Baer]...
mapdh8j 41148 Part of Part (8) in [Baer]...
mapdh8 41149 Part (8) in [Baer] p. 48. ...
mapdh9a 41150 Lemma for part (9) in [Bae...
mapdh9aOLDN 41151 Lemma for part (9) in [Bae...
hdmap1ffval 41156 Preliminary map from vecto...
hdmap1fval 41157 Preliminary map from vecto...
hdmap1vallem 41158 Value of preliminary map f...
hdmap1val 41159 Value of preliminary map f...
hdmap1val0 41160 Value of preliminary map f...
hdmap1val2 41161 Value of preliminary map f...
hdmap1eq 41162 The defining equation for ...
hdmap1cbv 41163 Frequently used lemma to c...
hdmap1valc 41164 Connect the value of the p...
hdmap1cl 41165 Convert closure theorem ~ ...
hdmap1eq2 41166 Convert ~ mapdheq2 to use ...
hdmap1eq4N 41167 Convert ~ mapdheq4 to use ...
hdmap1l6lem1 41168 Lemma for ~ hdmap1l6 . Pa...
hdmap1l6lem2 41169 Lemma for ~ hdmap1l6 . Pa...
hdmap1l6a 41170 Lemma for ~ hdmap1l6 . Pa...
hdmap1l6b0N 41171 Lemmma for ~ hdmap1l6 . (...
hdmap1l6b 41172 Lemmma for ~ hdmap1l6 . (...
hdmap1l6c 41173 Lemmma for ~ hdmap1l6 . (...
hdmap1l6d 41174 Lemmma for ~ hdmap1l6 . (...
hdmap1l6e 41175 Lemmma for ~ hdmap1l6 . P...
hdmap1l6f 41176 Lemmma for ~ hdmap1l6 . P...
hdmap1l6g 41177 Lemmma for ~ hdmap1l6 . P...
hdmap1l6h 41178 Lemmma for ~ hdmap1l6 . P...
hdmap1l6i 41179 Lemmma for ~ hdmap1l6 . E...
hdmap1l6j 41180 Lemmma for ~ hdmap1l6 . E...
hdmap1l6k 41181 Lemmma for ~ hdmap1l6 . E...
hdmap1l6 41182 Part (6) of [Baer] p. 47 l...
hdmap1eulem 41183 Lemma for ~ hdmap1eu . TO...
hdmap1eulemOLDN 41184 Lemma for ~ hdmap1euOLDN ....
hdmap1eu 41185 Convert ~ mapdh9a to use t...
hdmap1euOLDN 41186 Convert ~ mapdh9aOLDN to u...
hdmapffval 41187 Map from vectors to functi...
hdmapfval 41188 Map from vectors to functi...
hdmapval 41189 Value of map from vectors ...
hdmapfnN 41190 Functionality of map from ...
hdmapcl 41191 Closure of map from vector...
hdmapval2lem 41192 Lemma for ~ hdmapval2 . (...
hdmapval2 41193 Value of map from vectors ...
hdmapval0 41194 Value of map from vectors ...
hdmapeveclem 41195 Lemma for ~ hdmapevec . T...
hdmapevec 41196 Value of map from vectors ...
hdmapevec2 41197 The inner product of the r...
hdmapval3lemN 41198 Value of map from vectors ...
hdmapval3N 41199 Value of map from vectors ...
hdmap10lem 41200 Lemma for ~ hdmap10 . (Co...
hdmap10 41201 Part 10 in [Baer] p. 48 li...
hdmap11lem1 41202 Lemma for ~ hdmapadd . (C...
hdmap11lem2 41203 Lemma for ~ hdmapadd . (C...
hdmapadd 41204 Part 11 in [Baer] p. 48 li...
hdmapeq0 41205 Part of proof of part 12 i...
hdmapnzcl 41206 Nonzero vector closure of ...
hdmapneg 41207 Part of proof of part 12 i...
hdmapsub 41208 Part of proof of part 12 i...
hdmap11 41209 Part of proof of part 12 i...
hdmaprnlem1N 41210 Part of proof of part 12 i...
hdmaprnlem3N 41211 Part of proof of part 12 i...
hdmaprnlem3uN 41212 Part of proof of part 12 i...
hdmaprnlem4tN 41213 Lemma for ~ hdmaprnN . TO...
hdmaprnlem4N 41214 Part of proof of part 12 i...
hdmaprnlem6N 41215 Part of proof of part 12 i...
hdmaprnlem7N 41216 Part of proof of part 12 i...
hdmaprnlem8N 41217 Part of proof of part 12 i...
hdmaprnlem9N 41218 Part of proof of part 12 i...
hdmaprnlem3eN 41219 Lemma for ~ hdmaprnN . (C...
hdmaprnlem10N 41220 Lemma for ~ hdmaprnN . Sh...
hdmaprnlem11N 41221 Lemma for ~ hdmaprnN . Sh...
hdmaprnlem15N 41222 Lemma for ~ hdmaprnN . El...
hdmaprnlem16N 41223 Lemma for ~ hdmaprnN . El...
hdmaprnlem17N 41224 Lemma for ~ hdmaprnN . In...
hdmaprnN 41225 Part of proof of part 12 i...
hdmapf1oN 41226 Part 12 in [Baer] p. 49. ...
hdmap14lem1a 41227 Prior to part 14 in [Baer]...
hdmap14lem2a 41228 Prior to part 14 in [Baer]...
hdmap14lem1 41229 Prior to part 14 in [Baer]...
hdmap14lem2N 41230 Prior to part 14 in [Baer]...
hdmap14lem3 41231 Prior to part 14 in [Baer]...
hdmap14lem4a 41232 Simplify ` ( A \ { Q } ) `...
hdmap14lem4 41233 Simplify ` ( A \ { Q } ) `...
hdmap14lem6 41234 Case where ` F ` is zero. ...
hdmap14lem7 41235 Combine cases of ` F ` . ...
hdmap14lem8 41236 Part of proof of part 14 i...
hdmap14lem9 41237 Part of proof of part 14 i...
hdmap14lem10 41238 Part of proof of part 14 i...
hdmap14lem11 41239 Part of proof of part 14 i...
hdmap14lem12 41240 Lemma for proof of part 14...
hdmap14lem13 41241 Lemma for proof of part 14...
hdmap14lem14 41242 Part of proof of part 14 i...
hdmap14lem15 41243 Part of proof of part 14 i...
hgmapffval 41246 Map from the scalar divisi...
hgmapfval 41247 Map from the scalar divisi...
hgmapval 41248 Value of map from the scal...
hgmapfnN 41249 Functionality of scalar si...
hgmapcl 41250 Closure of scalar sigma ma...
hgmapdcl 41251 Closure of the vector spac...
hgmapvs 41252 Part 15 of [Baer] p. 50 li...
hgmapval0 41253 Value of the scalar sigma ...
hgmapval1 41254 Value of the scalar sigma ...
hgmapadd 41255 Part 15 of [Baer] p. 50 li...
hgmapmul 41256 Part 15 of [Baer] p. 50 li...
hgmaprnlem1N 41257 Lemma for ~ hgmaprnN . (C...
hgmaprnlem2N 41258 Lemma for ~ hgmaprnN . Pa...
hgmaprnlem3N 41259 Lemma for ~ hgmaprnN . El...
hgmaprnlem4N 41260 Lemma for ~ hgmaprnN . El...
hgmaprnlem5N 41261 Lemma for ~ hgmaprnN . El...
hgmaprnN 41262 Part of proof of part 16 i...
hgmap11 41263 The scalar sigma map is on...
hgmapf1oN 41264 The scalar sigma map is a ...
hgmapeq0 41265 The scalar sigma map is ze...
hdmapipcl 41266 The inner product (Hermiti...
hdmapln1 41267 Linearity property that wi...
hdmaplna1 41268 Additive property of first...
hdmaplns1 41269 Subtraction property of fi...
hdmaplnm1 41270 Multiplicative property of...
hdmaplna2 41271 Additive property of secon...
hdmapglnm2 41272 g-linear property of secon...
hdmapgln2 41273 g-linear property that wil...
hdmaplkr 41274 Kernel of the vector to du...
hdmapellkr 41275 Membership in the kernel (...
hdmapip0 41276 Zero property that will be...
hdmapip1 41277 Construct a proportional v...
hdmapip0com 41278 Commutation property of Ba...
hdmapinvlem1 41279 Line 27 in [Baer] p. 110. ...
hdmapinvlem2 41280 Line 28 in [Baer] p. 110, ...
hdmapinvlem3 41281 Line 30 in [Baer] p. 110, ...
hdmapinvlem4 41282 Part 1.1 of Proposition 1 ...
hdmapglem5 41283 Part 1.2 in [Baer] p. 110 ...
hgmapvvlem1 41284 Involution property of sca...
hgmapvvlem2 41285 Lemma for ~ hgmapvv . Eli...
hgmapvvlem3 41286 Lemma for ~ hgmapvv . Eli...
hgmapvv 41287 Value of a double involuti...
hdmapglem7a 41288 Lemma for ~ hdmapg . (Con...
hdmapglem7b 41289 Lemma for ~ hdmapg . (Con...
hdmapglem7 41290 Lemma for ~ hdmapg . Line...
hdmapg 41291 Apply the scalar sigma fun...
hdmapoc 41292 Express our constructed or...
hlhilset 41295 The final Hilbert space co...
hlhilsca 41296 The scalar of the final co...
hlhilbase 41297 The base set of the final ...
hlhilplus 41298 The vector addition for th...
hlhilslem 41299 Lemma for ~ hlhilsbase etc...
hlhilslemOLD 41300 Obsolete version of ~ hlhi...
hlhilsbase 41301 The scalar base set of the...
hlhilsbaseOLD 41302 Obsolete version of ~ hlhi...
hlhilsplus 41303 Scalar addition for the fi...
hlhilsplusOLD 41304 Obsolete version of ~ hlhi...
hlhilsmul 41305 Scalar multiplication for ...
hlhilsmulOLD 41306 Obsolete version of ~ hlhi...
hlhilsbase2 41307 The scalar base set of the...
hlhilsplus2 41308 Scalar addition for the fi...
hlhilsmul2 41309 Scalar multiplication for ...
hlhils0 41310 The scalar ring zero for t...
hlhils1N 41311 The scalar ring unity for ...
hlhilvsca 41312 The scalar product for the...
hlhilip 41313 Inner product operation fo...
hlhilipval 41314 Value of inner product ope...
hlhilnvl 41315 The involution operation o...
hlhillvec 41316 The final constructed Hilb...
hlhildrng 41317 The star division ring for...
hlhilsrnglem 41318 Lemma for ~ hlhilsrng . (...
hlhilsrng 41319 The star division ring for...
hlhil0 41320 The zero vector for the fi...
hlhillsm 41321 The vector sum operation f...
hlhilocv 41322 The orthocomplement for th...
hlhillcs 41323 The closed subspaces of th...
hlhilphllem 41324 Lemma for ~ hlhil . (Cont...
hlhilhillem 41325 Lemma for ~ hlhil . (Cont...
hlathil 41326 Construction of a Hilbert ...
iscsrg 41329 A commutative semiring is ...
leexp1ad 41330 Weak base ordering relatio...
relogbcld 41331 Closure of the general log...
relogbexpd 41332 Identity law for general l...
relogbzexpd 41333 Power law for the general ...
logblebd 41334 The general logarithm is m...
uzindd 41335 Induction on the upper int...
fzadd2d 41336 Membership of a sum in a f...
zltlem1d 41337 Integer ordering relation,...
zltp1led 41338 Integer ordering relation,...
fzne2d 41339 Elementhood in a finite se...
eqfnfv2d2 41340 Equality of functions is d...
fzsplitnd 41341 Split a finite interval of...
fzsplitnr 41342 Split a finite interval of...
addassnni 41343 Associative law for additi...
addcomnni 41344 Commutative law for additi...
mulassnni 41345 Associative law for multip...
mulcomnni 41346 Commutative law for multip...
gcdcomnni 41347 Commutative law for gcd. ...
gcdnegnni 41348 Negation invariance for gc...
neggcdnni 41349 Negation invariance for gc...
bccl2d 41350 Closure of the binomial co...
recbothd 41351 Take reciprocal on both si...
gcdmultiplei 41352 The GCD of a multiple of a...
gcdaddmzz2nni 41353 Adding a multiple of one o...
gcdaddmzz2nncomi 41354 Adding a multiple of one o...
gcdnncli 41355 Closure of the gcd operato...
muldvds1d 41356 If a product divides an in...
muldvds2d 41357 If a product divides an in...
nndivdvdsd 41358 A positive integer divides...
nnproddivdvdsd 41359 A product of natural numbe...
coprmdvds2d 41360 If an integer is divisible...
12gcd5e1 41361 The gcd of 12 and 5 is 1. ...
60gcd6e6 41362 The gcd of 60 and 6 is 6. ...
60gcd7e1 41363 The gcd of 60 and 7 is 1. ...
420gcd8e4 41364 The gcd of 420 and 8 is 4....
lcmeprodgcdi 41365 Calculate the least common...
12lcm5e60 41366 The lcm of 12 and 5 is 60....
60lcm6e60 41367 The lcm of 60 and 6 is 60....
60lcm7e420 41368 The lcm of 60 and 7 is 420...
420lcm8e840 41369 The lcm of 420 and 8 is 84...
lcmfunnnd 41370 Useful equation to calcula...
lcm1un 41371 Least common multiple of n...
lcm2un 41372 Least common multiple of n...
lcm3un 41373 Least common multiple of n...
lcm4un 41374 Least common multiple of n...
lcm5un 41375 Least common multiple of n...
lcm6un 41376 Least common multiple of n...
lcm7un 41377 Least common multiple of n...
lcm8un 41378 Least common multiple of n...
3factsumint1 41379 Move constants out of inte...
3factsumint2 41380 Move constants out of inte...
3factsumint3 41381 Move constants out of inte...
3factsumint4 41382 Move constants out of inte...
3factsumint 41383 Helpful equation for lcm i...
resopunitintvd 41384 Restrict continuous functi...
resclunitintvd 41385 Restrict continuous functi...
resdvopclptsd 41386 Restrict derivative on uni...
lcmineqlem1 41387 Part of lcm inequality lem...
lcmineqlem2 41388 Part of lcm inequality lem...
lcmineqlem3 41389 Part of lcm inequality lem...
lcmineqlem4 41390 Part of lcm inequality lem...
lcmineqlem5 41391 Technical lemma for recipr...
lcmineqlem6 41392 Part of lcm inequality lem...
lcmineqlem7 41393 Derivative of 1-x for chai...
lcmineqlem8 41394 Derivative of (1-x)^(N-M)....
lcmineqlem9 41395 (1-x)^(N-M) is continuous....
lcmineqlem10 41396 Induction step of ~ lcmine...
lcmineqlem11 41397 Induction step, continuati...
lcmineqlem12 41398 Base case for induction. ...
lcmineqlem13 41399 Induction proof for lcm in...
lcmineqlem14 41400 Technical lemma for inequa...
lcmineqlem15 41401 F times the least common m...
lcmineqlem16 41402 Technical divisibility lem...
lcmineqlem17 41403 Inequality of 2^{2n}. (Co...
lcmineqlem18 41404 Technical lemma to shift f...
lcmineqlem19 41405 Dividing implies inequalit...
lcmineqlem20 41406 Inequality for lcm lemma. ...
lcmineqlem21 41407 The lcm inequality lemma w...
lcmineqlem22 41408 The lcm inequality lemma w...
lcmineqlem23 41409 Penultimate step to the lc...
lcmineqlem 41410 The least common multiple ...
3exp7 41411 3 to the power of 7 equals...
3lexlogpow5ineq1 41412 First inequality in inequa...
3lexlogpow5ineq2 41413 Second inequality in inequ...
3lexlogpow5ineq4 41414 Sharper logarithm inequali...
3lexlogpow5ineq3 41415 Combined inequality chain ...
3lexlogpow2ineq1 41416 Result for bound in AKS in...
3lexlogpow2ineq2 41417 Result for bound in AKS in...
3lexlogpow5ineq5 41418 Result for bound in AKS in...
intlewftc 41419 Inequality inference by in...
aks4d1lem1 41420 Technical lemma to reduce ...
aks4d1p1p1 41421 Exponential law for finite...
dvrelog2 41422 The derivative of the loga...
dvrelog3 41423 The derivative of the loga...
dvrelog2b 41424 Derivative of the binary l...
0nonelalab 41425 Technical lemma for open i...
dvrelogpow2b 41426 Derivative of the power of...
aks4d1p1p3 41427 Bound of a ceiling of the ...
aks4d1p1p2 41428 Rewrite ` A ` in more suit...
aks4d1p1p4 41429 Technical step for inequal...
dvle2 41430 Collapsed ~ dvle . (Contr...
aks4d1p1p6 41431 Inequality lift to differe...
aks4d1p1p7 41432 Bound of intermediary of i...
aks4d1p1p5 41433 Show inequality for existe...
aks4d1p1 41434 Show inequality for existe...
aks4d1p2 41435 Technical lemma for existe...
aks4d1p3 41436 There exists a small enoug...
aks4d1p4 41437 There exists a small enoug...
aks4d1p5 41438 Show that ` N ` and ` R ` ...
aks4d1p6 41439 The maximal prime power ex...
aks4d1p7d1 41440 Technical step in AKS lemm...
aks4d1p7 41441 Technical step in AKS lemm...
aks4d1p8d1 41442 If a prime divides one num...
aks4d1p8d2 41443 Any prime power dividing a...
aks4d1p8d3 41444 The remainder of a divisio...
aks4d1p8 41445 Show that ` N ` and ` R ` ...
aks4d1p9 41446 Show that the order is bou...
aks4d1 41447 Lemma 4.1 from ~ https://w...
fldhmf1 41448 A field homomorphism is in...
aks6d1c2p1 41449 In the AKS-theorem the sub...
aks6d1c2p2 41450 Injective condition for co...
5bc2eq10 41451 The value of 5 choose 2. ...
facp2 41452 The factorial of a success...
2np3bcnp1 41453 Part of induction step for...
2ap1caineq 41454 Inequality for Theorem 6.6...
sticksstones1 41455 Different strictly monoton...
sticksstones2 41456 The range function on stri...
sticksstones3 41457 The range function on stri...
sticksstones4 41458 Equinumerosity lemma for s...
sticksstones5 41459 Count the number of strict...
sticksstones6 41460 Function induces an order ...
sticksstones7 41461 Closure property of sticks...
sticksstones8 41462 Establish mapping between ...
sticksstones9 41463 Establish mapping between ...
sticksstones10 41464 Establish mapping between ...
sticksstones11 41465 Establish bijective mappin...
sticksstones12a 41466 Establish bijective mappin...
sticksstones12 41467 Establish bijective mappin...
sticksstones13 41468 Establish bijective mappin...
sticksstones14 41469 Sticks and stones with def...
sticksstones15 41470 Sticks and stones with alm...
sticksstones16 41471 Sticks and stones with col...
sticksstones17 41472 Extend sticks and stones t...
sticksstones18 41473 Extend sticks and stones t...
sticksstones19 41474 Extend sticks and stones t...
sticksstones20 41475 Lift sticks and stones to ...
sticksstones21 41476 Lift sticks and stones to ...
sticksstones22 41477 Non-exhaustive sticks and ...
metakunt1 41478 A is an endomapping. (Con...
metakunt2 41479 A is an endomapping. (Con...
metakunt3 41480 Value of A. (Contributed b...
metakunt4 41481 Value of A. (Contributed b...
metakunt5 41482 C is the left inverse for ...
metakunt6 41483 C is the left inverse for ...
metakunt7 41484 C is the left inverse for ...
metakunt8 41485 C is the left inverse for ...
metakunt9 41486 C is the left inverse for ...
metakunt10 41487 C is the right inverse for...
metakunt11 41488 C is the right inverse for...
metakunt12 41489 C is the right inverse for...
metakunt13 41490 C is the right inverse for...
metakunt14 41491 A is a primitive permutati...
metakunt15 41492 Construction of another pe...
metakunt16 41493 Construction of another pe...
metakunt17 41494 The union of three disjoin...
metakunt18 41495 Disjoint domains and codom...
metakunt19 41496 Domains on restrictions of...
metakunt20 41497 Show that B coincides on t...
metakunt21 41498 Show that B coincides on t...
metakunt22 41499 Show that B coincides on t...
metakunt23 41500 B coincides on the union o...
metakunt24 41501 Technical condition such t...
metakunt25 41502 B is a permutation. (Cont...
metakunt26 41503 Construction of one soluti...
metakunt27 41504 Construction of one soluti...
metakunt28 41505 Construction of one soluti...
metakunt29 41506 Construction of one soluti...
metakunt30 41507 Construction of one soluti...
metakunt31 41508 Construction of one soluti...
metakunt32 41509 Construction of one soluti...
metakunt33 41510 Construction of one soluti...
metakunt34 41511 ` D ` is a permutation. (...
andiff 41512 Adding biconditional when ...
fac2xp3 41513 Factorial of 2x+3, sublemm...
prodsplit 41514 Product split into two fac...
2xp3dxp2ge1d 41515 2x+3 is greater than or eq...
factwoffsmonot 41516 A factorial with offset is...
ioin9i8 41517 Miscellaneous inference cr...
jaodd 41518 Double deduction form of ~...
syl3an12 41519 A double syllogism inferen...
sbtd 41520 A true statement is true u...
sbor2 41521 One direction of ~ sbor , ...
19.9dev 41522 ~ 19.9d in the case of an ...
3rspcedvdw 41523 Triple application of ~ rs...
3rspcedvd 41524 Triple application of ~ rs...
rabdif 41525 Move difference in and out...
sn-axrep5v 41526 A condensed form of ~ axre...
sn-axprlem3 41527 ~ axprlem3 using only Tars...
sn-exelALT 41528 Alternate proof of ~ exel ...
ss2ab1 41529 Class abstractions in a su...
ssabdv 41530 Deduction of abstraction s...
sn-iotalem 41531 An unused lemma showing th...
sn-iotalemcor 41532 Corollary of ~ sn-iotalem ...
abbi1sn 41533 Originally part of ~ uniab...
brif1 41534 Move a relation inside and...
brif2 41535 Move a relation inside and...
brif12 41536 Move a relation inside and...
pssexg 41537 The proper subset of a set...
pssn0 41538 A proper superset is nonem...
psspwb 41539 Classes are proper subclas...
xppss12 41540 Proper subset theorem for ...
coexd 41541 The composition of two set...
elpwbi 41542 Membership in a power set,...
imaopab 41543 The image of a class of or...
fnsnbt 41544 A function's domain is a s...
fnimasnd 41545 The image of a function by...
fvmptd4 41546 Deduction version of ~ fvm...
eqresfnbd 41547 Property of being the rest...
f1o2d2 41548 Sufficient condition for a...
fmpocos 41549 Composition of two functio...
ovmpogad 41550 Value of an operation give...
ofun 41551 A function operation of un...
dfqs2 41552 Alternate definition of qu...
dfqs3 41553 Alternate definition of qu...
qseq12d 41554 Equality theorem for quoti...
qsalrel 41555 The quotient set is equal ...
fsuppfund 41556 A finitely supported funct...
fsuppsssuppgd 41557 If the support of a functi...
fsuppss 41558 A subset of a finitely sup...
elmapssresd 41559 A restricted mapping is a ...
mapcod 41560 Compose two mappings. (Co...
fzosumm1 41561 Separate out the last term...
ccatcan2d 41562 Cancellation law for conca...
nelsubginvcld 41563 The inverse of a non-subgr...
nelsubgcld 41564 A non-subgroup-member plus...
nelsubgsubcld 41565 A non-subgroup-member minu...
rnasclg 41566 The set of injected scalar...
frlmfielbas 41567 The vectors of a finite fr...
frlmfzwrd 41568 A vector of a module with ...
frlmfzowrd 41569 A vector of a module with ...
frlmfzolen 41570 The dimension of a vector ...
frlmfzowrdb 41571 The vectors of a module wi...
frlmfzoccat 41572 The concatenation of two v...
frlmvscadiccat 41573 Scalar multiplication dist...
grpasscan2d 41574 An associative cancellatio...
grpcominv1 41575 If two elements commute, t...
grpcominv2 41576 If two elements commute, t...
finsubmsubg 41577 A submonoid of a finite gr...
crngcomd 41578 Multiplication is commutat...
crng12d 41579 Commutative/associative la...
imacrhmcl 41580 The image of a commutative...
rimrcl1 41581 Reverse closure of a ring ...
rimrcl2 41582 Reverse closure of a ring ...
rimcnv 41583 The converse of a ring iso...
rimco 41584 The composition of ring is...
ricsym 41585 Ring isomorphism is symmet...
rictr 41586 Ring isomorphism is transi...
riccrng1 41587 Ring isomorphism preserves...
riccrng 41588 A ring is commutative if a...
drnginvrn0d 41589 A multiplicative inverse i...
drngmulcanad 41590 Cancellation of a nonzero ...
drngmulcan2ad 41591 Cancellation of a nonzero ...
drnginvmuld 41592 Inverse of a nonzero produ...
ricdrng1 41593 A ring isomorphism maps a ...
ricdrng 41594 A ring is a division ring ...
ricfld 41595 A ring is a field if and o...
lvecgrp 41596 A vector space is a group....
lvecring 41597 The scalar component of a ...
frlm0vald 41598 All coordinates of the zer...
frlmsnic 41599 Given a free module with a...
uvccl 41600 A unit vector is a vector....
uvcn0 41601 A unit vector is nonzero. ...
pwselbasr 41602 The reverse direction of ~...
pwsgprod 41603 Finite products in a power...
psrbagres 41604 Restrict a bag of variable...
mpllmodd 41605 The polynomial ring is a l...
mplringd 41606 The polynomial ring is a r...
mplcrngd 41607 The polynomial ring is a c...
mplsubrgcl 41608 An element of a polynomial...
mhmcompl 41609 The composition of a monoi...
rhmmpllem1 41610 Lemma for ~ rhmmpl . A su...
rhmmpllem2 41611 Lemma for ~ rhmmpl . A su...
mhmcoaddmpl 41612 Show that the ring homomor...
rhmcomulmpl 41613 Show that the ring homomor...
rhmmpl 41614 Provide a ring homomorphis...
mplascl0 41615 The zero scalar as a polyn...
mplascl1 41616 The one scalar as a polyno...
mplmapghm 41617 The function ` H ` mapping...
evl0 41618 The zero polynomial evalua...
evlscl 41619 A polynomial over the ring...
evlsval3 41620 Give a formula for the pol...
evlsvval 41621 Give a formula for the eva...
evlsvvvallem 41622 Lemma for ~ evlsvvval akin...
evlsvvvallem2 41623 Lemma for theorems using ~...
evlsvvval 41624 Give a formula for the eva...
evlsscaval 41625 Polynomial evaluation buil...
evlsvarval 41626 Polynomial evaluation buil...
evlsbagval 41627 Polynomial evaluation buil...
evlsexpval 41628 Polynomial evaluation buil...
evlsaddval 41629 Polynomial evaluation buil...
evlsmulval 41630 Polynomial evaluation buil...
evlsmaprhm 41631 The function ` F ` mapping...
evlsevl 41632 Evaluation in a subring is...
evlcl 41633 A polynomial over the ring...
evlvvval 41634 Give a formula for the eva...
evlvvvallem 41635 Lemma for theorems using ~...
evladdval 41636 Polynomial evaluation buil...
evlmulval 41637 Polynomial evaluation buil...
selvcllem1 41638 ` T ` is an associative al...
selvcllem2 41639 ` D ` is a ring homomorphi...
selvcllem3 41640 The third argument passed ...
selvcllemh 41641 Apply the third argument (...
selvcllem4 41642 The fourth argument passed...
selvcllem5 41643 The fifth argument passed ...
selvcl 41644 Closure of the "variable s...
selvval2 41645 Value of the "variable sel...
selvvvval 41646 Recover the original polyn...
evlselvlem 41647 Lemma for ~ evlselv . Use...
evlselv 41648 Evaluating a selection of ...
selvadd 41649 The "variable selection" f...
selvmul 41650 The "variable selection" f...
fsuppind 41651 Induction on functions ` F...
fsuppssindlem1 41652 Lemma for ~ fsuppssind . ...
fsuppssindlem2 41653 Lemma for ~ fsuppssind . ...
fsuppssind 41654 Induction on functions ` F...
mhpind 41655 The homogeneous polynomial...
evlsmhpvvval 41656 Give a formula for the eva...
mhphflem 41657 Lemma for ~ mhphf . Add s...
mhphf 41658 A homogeneous polynomial d...
mhphf2 41659 A homogeneous polynomial d...
mhphf3 41660 A homogeneous polynomial d...
mhphf4 41661 A homogeneous polynomial d...
c0exALT 41662 Alternate proof of ~ c0ex ...
0cnALT3 41663 Alternate proof of ~ 0cn u...
elre0re 41664 Specialized version of ~ 0...
1t1e1ALT 41665 Alternate proof of ~ 1t1e1...
remulcan2d 41666 ~ mulcan2d for real number...
readdridaddlidd 41667 Given some real number ` B...
sn-1ne2 41668 A proof of ~ 1ne2 without ...
nnn1suc 41669 A positive integer that is...
nnadd1com 41670 Addition with 1 is commuta...
nnaddcom 41671 Addition is commutative fo...
nnaddcomli 41672 Version of ~ addcomli for ...
nnadddir 41673 Right-distributivity for n...
nnmul1com 41674 Multiplication with 1 is c...
nnmulcom 41675 Multiplication is commutat...
mvrrsubd 41676 Move a subtraction in the ...
laddrotrd 41677 Rotate the variables right...
raddcom12d 41678 Swap the first two variabl...
lsubrotld 41679 Rotate the variables left ...
lsubcom23d 41680 Swap the second and third ...
addsubeq4com 41681 Relation between sums and ...
sqsumi 41682 A sum squared. (Contribut...
negn0nposznnd 41683 Lemma for ~ dffltz . (Con...
sqmid3api 41684 Value of the square of the...
decaddcom 41685 Commute ones place in addi...
sqn5i 41686 The square of a number end...
sqn5ii 41687 The square of a number end...
decpmulnc 41688 Partial products algorithm...
decpmul 41689 Partial products algorithm...
sqdeccom12 41690 The square of a number in ...
sq3deccom12 41691 Variant of ~ sqdeccom12 wi...
4t5e20 41692 4 times 5 equals 20. (Con...
sq9 41693 The square of 9 is 81. (C...
235t711 41694 Calculate a product by lon...
ex-decpmul 41695 Example usage of ~ decpmul...
fz1sumconst 41696 The sum of ` N ` constant ...
fz1sump1 41697 Add one more term to a sum...
oddnumth 41698 The Odd Number Theorem. T...
nicomachus 41699 Nicomachus's Theorem. The...
sumcubes 41700 The sum of the first ` N `...
oexpreposd 41701 Lemma for ~ dffltz . TODO...
ltexp1d 41702 ~ ltmul1d for exponentiati...
ltexp1dd 41703 Raising both sides of 'les...
exp11nnd 41704 ~ sq11d for positive real ...
exp11d 41705 ~ exp11nnd for nonzero int...
0dvds0 41706 0 divides 0. (Contributed...
absdvdsabsb 41707 Divisibility is invariant ...
dvdsexpim 41708 ~ dvdssqim generalized to ...
gcdnn0id 41709 The ` gcd ` of a nonnegati...
gcdle1d 41710 The greatest common diviso...
gcdle2d 41711 The greatest common diviso...
dvdsexpad 41712 Deduction associated with ...
nn0rppwr 41713 If ` A ` and ` B ` are rel...
expgcd 41714 Exponentiation distributes...
nn0expgcd 41715 Exponentiation distributes...
zexpgcd 41716 Exponentiation distributes...
numdenexp 41717 ~ numdensq extended to non...
numexp 41718 ~ numsq extended to nonneg...
denexp 41719 ~ densq extended to nonneg...
dvdsexpnn 41720 ~ dvdssqlem generalized to...
dvdsexpnn0 41721 ~ dvdsexpnn generalized to...
dvdsexpb 41722 ~ dvdssq generalized to po...
posqsqznn 41723 When a positive rational s...
zrtelqelz 41724 ~ zsqrtelqelz generalized ...
zrtdvds 41725 A positive integer root di...
rtprmirr 41726 The root of a prime number...
resubval 41729 Value of real subtraction,...
renegeulemv 41730 Lemma for ~ renegeu and si...
renegeulem 41731 Lemma for ~ renegeu and si...
renegeu 41732 Existential uniqueness of ...
rernegcl 41733 Closure law for negative r...
renegadd 41734 Relationship between real ...
renegid 41735 Addition of a real number ...
reneg0addlid 41736 Negative zero is a left ad...
resubeulem1 41737 Lemma for ~ resubeu . A v...
resubeulem2 41738 Lemma for ~ resubeu . A v...
resubeu 41739 Existential uniqueness of ...
rersubcl 41740 Closure for real subtracti...
resubadd 41741 Relation between real subt...
resubaddd 41742 Relationship between subtr...
resubf 41743 Real subtraction is an ope...
repncan2 41744 Addition and subtraction o...
repncan3 41745 Addition and subtraction o...
readdsub 41746 Law for addition and subtr...
reladdrsub 41747 Move LHS of a sum into RHS...
reltsub1 41748 Subtraction from both side...
reltsubadd2 41749 'Less than' relationship b...
resubcan2 41750 Cancellation law for real ...
resubsub4 41751 Law for double subtraction...
rennncan2 41752 Cancellation law for real ...
renpncan3 41753 Cancellation law for real ...
repnpcan 41754 Cancellation law for addit...
reppncan 41755 Cancellation law for mixed...
resubidaddlidlem 41756 Lemma for ~ resubidaddlid ...
resubidaddlid 41757 Any real number subtracted...
resubdi 41758 Distribution of multiplica...
re1m1e0m0 41759 Equality of two left-addit...
sn-00idlem1 41760 Lemma for ~ sn-00id . (Co...
sn-00idlem2 41761 Lemma for ~ sn-00id . (Co...
sn-00idlem3 41762 Lemma for ~ sn-00id . (Co...
sn-00id 41763 ~ 00id proven without ~ ax...
re0m0e0 41764 Real number version of ~ 0...
readdlid 41765 Real number version of ~ a...
sn-addlid 41766 ~ addlid without ~ ax-mulc...
remul02 41767 Real number version of ~ m...
sn-0ne2 41768 ~ 0ne2 without ~ ax-mulcom...
remul01 41769 Real number version of ~ m...
resubid 41770 Subtraction of a real numb...
readdrid 41771 Real number version of ~ a...
resubid1 41772 Real number version of ~ s...
renegneg 41773 A real number is equal to ...
readdcan2 41774 Commuted version of ~ read...
renegid2 41775 Commuted version of ~ rene...
remulneg2d 41776 Product with negative is n...
sn-it0e0 41777 Proof of ~ it0e0 without ~...
sn-negex12 41778 A combination of ~ cnegex ...
sn-negex 41779 Proof of ~ cnegex without ...
sn-negex2 41780 Proof of ~ cnegex2 without...
sn-addcand 41781 ~ addcand without ~ ax-mul...
sn-addrid 41782 ~ addrid without ~ ax-mulc...
sn-addcan2d 41783 ~ addcan2d without ~ ax-mu...
reixi 41784 ~ ixi without ~ ax-mulcom ...
rei4 41785 ~ i4 without ~ ax-mulcom ....
sn-addid0 41786 A number that sums to itse...
sn-mul01 41787 ~ mul01 without ~ ax-mulco...
sn-subeu 41788 ~ negeu without ~ ax-mulco...
sn-subcl 41789 ~ subcl without ~ ax-mulco...
sn-subf 41790 ~ subf without ~ ax-mulcom...
resubeqsub 41791 Equivalence between real s...
subresre 41792 Subtraction restricted to ...
addinvcom 41793 A number commutes with its...
remulinvcom 41794 A left multiplicative inve...
remullid 41795 Commuted version of ~ ax-1...
sn-1ticom 41796 Lemma for ~ sn-mullid and ...
sn-mullid 41797 ~ mullid without ~ ax-mulc...
it1ei 41798 ` 1 ` is a multiplicative ...
ipiiie0 41799 The multiplicative inverse...
remulcand 41800 Commuted version of ~ remu...
sn-0tie0 41801 Lemma for ~ sn-mul02 . Co...
sn-mul02 41802 ~ mul02 without ~ ax-mulco...
sn-ltaddpos 41803 ~ ltaddpos without ~ ax-mu...
sn-ltaddneg 41804 ~ ltaddneg without ~ ax-mu...
reposdif 41805 Comparison of two numbers ...
relt0neg1 41806 Comparison of a real and i...
relt0neg2 41807 Comparison of a real and i...
sn-addlt0d 41808 The sum of negative number...
sn-addgt0d 41809 The sum of positive number...
sn-nnne0 41810 ~ nnne0 without ~ ax-mulco...
reelznn0nn 41811 ~ elznn0nn restated using ...
nn0addcom 41812 Addition is commutative fo...
zaddcomlem 41813 Lemma for ~ zaddcom . (Co...
zaddcom 41814 Addition is commutative fo...
renegmulnnass 41815 Move multiplication by a n...
nn0mulcom 41816 Multiplication is commutat...
zmulcomlem 41817 Lemma for ~ zmulcom . (Co...
zmulcom 41818 Multiplication is commutat...
mulgt0con1dlem 41819 Lemma for ~ mulgt0con1d . ...
mulgt0con1d 41820 Counterpart to ~ mulgt0con...
mulgt0con2d 41821 Lemma for ~ mulgt0b2d and ...
mulgt0b2d 41822 Biconditional, deductive f...
sn-ltmul2d 41823 ~ ltmul2d without ~ ax-mul...
sn-0lt1 41824 ~ 0lt1 without ~ ax-mulcom...
sn-ltp1 41825 ~ ltp1 without ~ ax-mulcom...
reneg1lt0 41826 Lemma for ~ sn-inelr . (C...
sn-inelr 41827 ~ inelr without ~ ax-mulco...
itrere 41828 ` _i ` times a real is rea...
retire 41829 Commuted version of ~ itre...
cnreeu 41830 The reals in the expressio...
sn-sup2 41831 ~ sup2 with exactly the sa...
prjspval 41834 Value of the projective sp...
prjsprel 41835 Utility theorem regarding ...
prjspertr 41836 The relation in ` PrjSp ` ...
prjsperref 41837 The relation in ` PrjSp ` ...
prjspersym 41838 The relation in ` PrjSp ` ...
prjsper 41839 The relation used to defin...
prjspreln0 41840 Two nonzero vectors are eq...
prjspvs 41841 A nonzero multiple of a ve...
prjsprellsp 41842 Two vectors are equivalent...
prjspeclsp 41843 The vectors equivalent to ...
prjspval2 41844 Alternate definition of pr...
prjspnval 41847 Value of the n-dimensional...
prjspnerlem 41848 A lemma showing that the e...
prjspnval2 41849 Value of the n-dimensional...
prjspner 41850 The relation used to defin...
prjspnvs 41851 A nonzero multiple of a ve...
prjspnssbas 41852 A projective point spans a...
prjspnn0 41853 A projective point is none...
0prjspnlem 41854 Lemma for ~ 0prjspn . The...
prjspnfv01 41855 Any vector is equivalent t...
prjspner01 41856 Any vector is equivalent t...
prjspner1 41857 Two vectors whose zeroth c...
0prjspnrel 41858 In the zero-dimensional pr...
0prjspn 41859 A zero-dimensional project...
prjcrvfval 41862 Value of the projective cu...
prjcrvval 41863 Value of the projective cu...
prjcrv0 41864 The "curve" (zero set) cor...
dffltz 41865 Fermat's Last Theorem (FLT...
fltmul 41866 A counterexample to FLT st...
fltdiv 41867 A counterexample to FLT st...
flt0 41868 A counterexample for FLT d...
fltdvdsabdvdsc 41869 Any factor of both ` A ` a...
fltabcoprmex 41870 A counterexample to FLT im...
fltaccoprm 41871 A counterexample to FLT wi...
fltbccoprm 41872 A counterexample to FLT wi...
fltabcoprm 41873 A counterexample to FLT wi...
infdesc 41874 Infinite descent. The hyp...
fltne 41875 If a counterexample to FLT...
flt4lem 41876 Raising a number to the fo...
flt4lem1 41877 Satisfy the antecedent use...
flt4lem2 41878 If ` A ` is even, ` B ` is...
flt4lem3 41879 Equivalent to ~ pythagtrip...
flt4lem4 41880 If the product of two copr...
flt4lem5 41881 In the context of the lemm...
flt4lem5elem 41882 Version of ~ fltaccoprm an...
flt4lem5a 41883 Part 1 of Equation 1 of ...
flt4lem5b 41884 Part 2 of Equation 1 of ...
flt4lem5c 41885 Part 2 of Equation 2 of ...
flt4lem5d 41886 Part 3 of Equation 2 of ...
flt4lem5e 41887 Satisfy the hypotheses of ...
flt4lem5f 41888 Final equation of ~...
flt4lem6 41889 Remove shared factors in a...
flt4lem7 41890 Convert ~ flt4lem5f into a...
nna4b4nsq 41891 Strengthening of Fermat's ...
fltltc 41892 ` ( C ^ N ) ` is the large...
fltnltalem 41893 Lemma for ~ fltnlta . A l...
fltnlta 41894 In a Fermat counterexample...
iddii 41895 Version of ~ a1ii with the...
bicomdALT 41896 Alternate proof of ~ bicom...
elabgw 41897 Membership in a class abst...
elab2gw 41898 Membership in a class abst...
elrab2w 41899 Membership in a restricted...
ruvALT 41900 Alternate proof of ~ ruv w...
sn-wcdeq 41901 Alternative to ~ wcdeq and...
sq45 41902 45 squared is 2025. (Cont...
sum9cubes 41903 The sum of the first nine ...
acos1half 41904 The arccosine of ` 1 / 2 `...
aprilfools2025 41905 An abuse of notation. (Co...
binom2d 41906 Deduction form of binom2. ...
cu3addd 41907 Cube of sum of three numbe...
sqnegd 41908 The square of the negative...
negexpidd 41909 The sum of a real number t...
rexlimdv3d 41910 An extended version of ~ r...
3cubeslem1 41911 Lemma for ~ 3cubes . (Con...
3cubeslem2 41912 Lemma for ~ 3cubes . Used...
3cubeslem3l 41913 Lemma for ~ 3cubes . (Con...
3cubeslem3r 41914 Lemma for ~ 3cubes . (Con...
3cubeslem3 41915 Lemma for ~ 3cubes . (Con...
3cubeslem4 41916 Lemma for ~ 3cubes . This...
3cubes 41917 Every rational number is a...
rntrclfvOAI 41918 The range of the transitiv...
moxfr 41919 Transfer at-most-one betwe...
imaiinfv 41920 Indexed intersection of an...
elrfi 41921 Elementhood in a set of re...
elrfirn 41922 Elementhood in a set of re...
elrfirn2 41923 Elementhood in a set of re...
cmpfiiin 41924 In a compact topology, a s...
ismrcd1 41925 Any function from the subs...
ismrcd2 41926 Second half of ~ ismrcd1 ....
istopclsd 41927 A closure function which s...
ismrc 41928 A function is a Moore clos...
isnacs 41931 Expand definition of Noeth...
nacsfg 41932 In a Noetherian-type closu...
isnacs2 41933 Express Noetherian-type cl...
mrefg2 41934 Slight variation on finite...
mrefg3 41935 Slight variation on finite...
nacsacs 41936 A closure system of Noethe...
isnacs3 41937 A choice-free order equiva...
incssnn0 41938 Transitivity induction of ...
nacsfix 41939 An increasing sequence of ...
constmap 41940 A constant (represented wi...
mapco2g 41941 Renaming indices in a tupl...
mapco2 41942 Post-composition (renaming...
mapfzcons 41943 Extending a one-based mapp...
mapfzcons1 41944 Recover prefix mapping fro...
mapfzcons1cl 41945 A nonempty mapping has a p...
mapfzcons2 41946 Recover added element from...
mptfcl 41947 Interpret range of a maps-...
mzpclval 41952 Substitution lemma for ` m...
elmzpcl 41953 Double substitution lemma ...
mzpclall 41954 The set of all functions w...
mzpcln0 41955 Corollary of ~ mzpclall : ...
mzpcl1 41956 Defining property 1 of a p...
mzpcl2 41957 Defining property 2 of a p...
mzpcl34 41958 Defining properties 3 and ...
mzpval 41959 Value of the ` mzPoly ` fu...
dmmzp 41960 ` mzPoly ` is defined for ...
mzpincl 41961 Polynomial closedness is a...
mzpconst 41962 Constant functions are pol...
mzpf 41963 A polynomial function is a...
mzpproj 41964 A projection function is p...
mzpadd 41965 The pointwise sum of two p...
mzpmul 41966 The pointwise product of t...
mzpconstmpt 41967 A constant function expres...
mzpaddmpt 41968 Sum of polynomial function...
mzpmulmpt 41969 Product of polynomial func...
mzpsubmpt 41970 The difference of two poly...
mzpnegmpt 41971 Negation of a polynomial f...
mzpexpmpt 41972 Raise a polynomial functio...
mzpindd 41973 "Structural" induction to ...
mzpmfp 41974 Relationship between multi...
mzpsubst 41975 Substituting polynomials f...
mzprename 41976 Simplified version of ~ mz...
mzpresrename 41977 A polynomial is a polynomi...
mzpcompact2lem 41978 Lemma for ~ mzpcompact2 . ...
mzpcompact2 41979 Polynomials are finitary o...
coeq0i 41980 ~ coeq0 but without explic...
fzsplit1nn0 41981 Split a finite 1-based set...
eldiophb 41984 Initial expression of Diop...
eldioph 41985 Condition for a set to be ...
diophrw 41986 Renaming and adding unused...
eldioph2lem1 41987 Lemma for ~ eldioph2 . Co...
eldioph2lem2 41988 Lemma for ~ eldioph2 . Co...
eldioph2 41989 Construct a Diophantine se...
eldioph2b 41990 While Diophantine sets wer...
eldiophelnn0 41991 Remove antecedent on ` B `...
eldioph3b 41992 Define Diophantine sets in...
eldioph3 41993 Inference version of ~ eld...
ellz1 41994 Membership in a lower set ...
lzunuz 41995 The union of a lower set o...
fz1eqin 41996 Express a one-based finite...
lzenom 41997 Lower integers are countab...
elmapresaunres2 41998 ~ fresaunres2 transposed t...
diophin 41999 If two sets are Diophantin...
diophun 42000 If two sets are Diophantin...
eldiophss 42001 Diophantine sets are sets ...
diophrex 42002 Projecting a Diophantine s...
eq0rabdioph 42003 This is the first of a num...
eqrabdioph 42004 Diophantine set builder fo...
0dioph 42005 The null set is Diophantin...
vdioph 42006 The "universal" set (as la...
anrabdioph 42007 Diophantine set builder fo...
orrabdioph 42008 Diophantine set builder fo...
3anrabdioph 42009 Diophantine set builder fo...
3orrabdioph 42010 Diophantine set builder fo...
2sbcrex 42011 Exchange an existential qu...
sbcrexgOLD 42012 Interchange class substitu...
2sbcrexOLD 42013 Exchange an existential qu...
sbc2rex 42014 Exchange a substitution wi...
sbc2rexgOLD 42015 Exchange a substitution wi...
sbc4rex 42016 Exchange a substitution wi...
sbc4rexgOLD 42017 Exchange a substitution wi...
sbcrot3 42018 Rotate a sequence of three...
sbcrot5 42019 Rotate a sequence of five ...
sbccomieg 42020 Commute two explicit subst...
rexrabdioph 42021 Diophantine set builder fo...
rexfrabdioph 42022 Diophantine set builder fo...
2rexfrabdioph 42023 Diophantine set builder fo...
3rexfrabdioph 42024 Diophantine set builder fo...
4rexfrabdioph 42025 Diophantine set builder fo...
6rexfrabdioph 42026 Diophantine set builder fo...
7rexfrabdioph 42027 Diophantine set builder fo...
rabdiophlem1 42028 Lemma for arithmetic dioph...
rabdiophlem2 42029 Lemma for arithmetic dioph...
elnn0rabdioph 42030 Diophantine set builder fo...
rexzrexnn0 42031 Rewrite an existential qua...
lerabdioph 42032 Diophantine set builder fo...
eluzrabdioph 42033 Diophantine set builder fo...
elnnrabdioph 42034 Diophantine set builder fo...
ltrabdioph 42035 Diophantine set builder fo...
nerabdioph 42036 Diophantine set builder fo...
dvdsrabdioph 42037 Divisibility is a Diophant...
eldioph4b 42038 Membership in ` Dioph ` ex...
eldioph4i 42039 Forward-only version of ~ ...
diophren 42040 Change variables in a Diop...
rabrenfdioph 42041 Change variable numbers in...
rabren3dioph 42042 Change variable numbers in...
fphpd 42043 Pigeonhole principle expre...
fphpdo 42044 Pigeonhole principle for s...
ctbnfien 42045 An infinite subset of a co...
fiphp3d 42046 Infinite pigeonhole princi...
rencldnfilem 42047 Lemma for ~ rencldnfi . (...
rencldnfi 42048 A set of real numbers whic...
irrapxlem1 42049 Lemma for ~ irrapx1 . Div...
irrapxlem2 42050 Lemma for ~ irrapx1 . Two...
irrapxlem3 42051 Lemma for ~ irrapx1 . By ...
irrapxlem4 42052 Lemma for ~ irrapx1 . Eli...
irrapxlem5 42053 Lemma for ~ irrapx1 . Swi...
irrapxlem6 42054 Lemma for ~ irrapx1 . Exp...
irrapx1 42055 Dirichlet's approximation ...
pellexlem1 42056 Lemma for ~ pellex . Arit...
pellexlem2 42057 Lemma for ~ pellex . Arit...
pellexlem3 42058 Lemma for ~ pellex . To e...
pellexlem4 42059 Lemma for ~ pellex . Invo...
pellexlem5 42060 Lemma for ~ pellex . Invo...
pellexlem6 42061 Lemma for ~ pellex . Doin...
pellex 42062 Every Pell equation has a ...
pell1qrval 42073 Value of the set of first-...
elpell1qr 42074 Membership in a first-quad...
pell14qrval 42075 Value of the set of positi...
elpell14qr 42076 Membership in the set of p...
pell1234qrval 42077 Value of the set of genera...
elpell1234qr 42078 Membership in the set of g...
pell1234qrre 42079 General Pell solutions are...
pell1234qrne0 42080 No solution to a Pell equa...
pell1234qrreccl 42081 General solutions of the P...
pell1234qrmulcl 42082 General solutions of the P...
pell14qrss1234 42083 A positive Pell solution i...
pell14qrre 42084 A positive Pell solution i...
pell14qrne0 42085 A positive Pell solution i...
pell14qrgt0 42086 A positive Pell solution i...
pell14qrrp 42087 A positive Pell solution i...
pell1234qrdich 42088 A general Pell solution is...
elpell14qr2 42089 A number is a positive Pel...
pell14qrmulcl 42090 Positive Pell solutions ar...
pell14qrreccl 42091 Positive Pell solutions ar...
pell14qrdivcl 42092 Positive Pell solutions ar...
pell14qrexpclnn0 42093 Lemma for ~ pell14qrexpcl ...
pell14qrexpcl 42094 Positive Pell solutions ar...
pell1qrss14 42095 First-quadrant Pell soluti...
pell14qrdich 42096 A positive Pell solution i...
pell1qrge1 42097 A Pell solution in the fir...
pell1qr1 42098 1 is a Pell solution and i...
elpell1qr2 42099 The first quadrant solutio...
pell1qrgaplem 42100 Lemma for ~ pell1qrgap . ...
pell1qrgap 42101 First-quadrant Pell soluti...
pell14qrgap 42102 Positive Pell solutions ar...
pell14qrgapw 42103 Positive Pell solutions ar...
pellqrexplicit 42104 Condition for a calculated...
infmrgelbi 42105 Any lower bound of a nonem...
pellqrex 42106 There is a nontrivial solu...
pellfundval 42107 Value of the fundamental s...
pellfundre 42108 The fundamental solution o...
pellfundge 42109 Lower bound on the fundame...
pellfundgt1 42110 Weak lower bound on the Pe...
pellfundlb 42111 A nontrivial first quadran...
pellfundglb 42112 If a real is larger than t...
pellfundex 42113 The fundamental solution a...
pellfund14gap 42114 There are no solutions bet...
pellfundrp 42115 The fundamental Pell solut...
pellfundne1 42116 The fundamental Pell solut...
reglogcl 42117 General logarithm is a rea...
reglogltb 42118 General logarithm preserve...
reglogleb 42119 General logarithm preserve...
reglogmul 42120 Multiplication law for gen...
reglogexp 42121 Power law for general log....
reglogbas 42122 General log of the base is...
reglog1 42123 General log of 1 is 0. (C...
reglogexpbas 42124 General log of a power of ...
pellfund14 42125 Every positive Pell soluti...
pellfund14b 42126 The positive Pell solution...
rmxfval 42131 Value of the X sequence. ...
rmyfval 42132 Value of the Y sequence. ...
rmspecsqrtnq 42133 The discriminant used to d...
rmspecnonsq 42134 The discriminant used to d...
qirropth 42135 This lemma implements the ...
rmspecfund 42136 The base of exponent used ...
rmxyelqirr 42137 The solutions used to cons...
rmxyelqirrOLD 42138 Obsolete version of ~ rmxy...
rmxypairf1o 42139 The function used to extra...
rmxyelxp 42140 Lemma for ~ frmx and ~ frm...
frmx 42141 The X sequence is a nonneg...
frmy 42142 The Y sequence is an integ...
rmxyval 42143 Main definition of the X a...
rmspecpos 42144 The discriminant used to d...
rmxycomplete 42145 The X and Y sequences take...
rmxynorm 42146 The X and Y sequences defi...
rmbaserp 42147 The base of exponentiation...
rmxyneg 42148 Negation law for X and Y s...
rmxyadd 42149 Addition formula for X and...
rmxy1 42150 Value of the X and Y seque...
rmxy0 42151 Value of the X and Y seque...
rmxneg 42152 Negation law (even functio...
rmx0 42153 Value of X sequence at 0. ...
rmx1 42154 Value of X sequence at 1. ...
rmxadd 42155 Addition formula for X seq...
rmyneg 42156 Negation formula for Y seq...
rmy0 42157 Value of Y sequence at 0. ...
rmy1 42158 Value of Y sequence at 1. ...
rmyadd 42159 Addition formula for Y seq...
rmxp1 42160 Special addition-of-1 form...
rmyp1 42161 Special addition of 1 form...
rmxm1 42162 Subtraction of 1 formula f...
rmym1 42163 Subtraction of 1 formula f...
rmxluc 42164 The X sequence is a Lucas ...
rmyluc 42165 The Y sequence is a Lucas ...
rmyluc2 42166 Lucas sequence property of...
rmxdbl 42167 "Double-angle formula" for...
rmydbl 42168 "Double-angle formula" for...
monotuz 42169 A function defined on an u...
monotoddzzfi 42170 A function which is odd an...
monotoddzz 42171 A function (given implicit...
oddcomabszz 42172 An odd function which take...
2nn0ind 42173 Induction on nonnegative i...
zindbi 42174 Inductively transfer a pro...
rmxypos 42175 For all nonnegative indice...
ltrmynn0 42176 The Y-sequence is strictly...
ltrmxnn0 42177 The X-sequence is strictly...
lermxnn0 42178 The X-sequence is monotoni...
rmxnn 42179 The X-sequence is defined ...
ltrmy 42180 The Y-sequence is strictly...
rmyeq0 42181 Y is zero only at zero. (...
rmyeq 42182 Y is one-to-one. (Contrib...
lermy 42183 Y is monotonic (non-strict...
rmynn 42184 ` rmY ` is positive for po...
rmynn0 42185 ` rmY ` is nonnegative for...
rmyabs 42186 ` rmY ` commutes with ` ab...
jm2.24nn 42187 X(n) is strictly greater t...
jm2.17a 42188 First half of lemma 2.17 o...
jm2.17b 42189 Weak form of the second ha...
jm2.17c 42190 Second half of lemma 2.17 ...
jm2.24 42191 Lemma 2.24 of [JonesMatija...
rmygeid 42192 Y(n) increases faster than...
congtr 42193 A wff of the form ` A || (...
congadd 42194 If two pairs of numbers ar...
congmul 42195 If two pairs of numbers ar...
congsym 42196 Congruence mod ` A ` is a ...
congneg 42197 If two integers are congru...
congsub 42198 If two pairs of numbers ar...
congid 42199 Every integer is congruent...
mzpcong 42200 Polynomials commute with c...
congrep 42201 Every integer is congruent...
congabseq 42202 If two integers are congru...
acongid 42203 A wff like that in this th...
acongsym 42204 Symmetry of alternating co...
acongneg2 42205 Negate right side of alter...
acongtr 42206 Transitivity of alternatin...
acongeq12d 42207 Substitution deduction for...
acongrep 42208 Every integer is alternati...
fzmaxdif 42209 Bound on the difference be...
fzneg 42210 Reflection of a finite ran...
acongeq 42211 Two numbers in the fundame...
dvdsacongtr 42212 Alternating congruence pas...
coprmdvdsb 42213 Multiplication by a coprim...
modabsdifz 42214 Divisibility in terms of m...
dvdsabsmod0 42215 Divisibility in terms of m...
jm2.18 42216 Theorem 2.18 of [JonesMati...
jm2.19lem1 42217 Lemma for ~ jm2.19 . X an...
jm2.19lem2 42218 Lemma for ~ jm2.19 . (Con...
jm2.19lem3 42219 Lemma for ~ jm2.19 . (Con...
jm2.19lem4 42220 Lemma for ~ jm2.19 . Exte...
jm2.19 42221 Lemma 2.19 of [JonesMatija...
jm2.21 42222 Lemma for ~ jm2.20nn . Ex...
jm2.22 42223 Lemma for ~ jm2.20nn . Ap...
jm2.23 42224 Lemma for ~ jm2.20nn . Tr...
jm2.20nn 42225 Lemma 2.20 of [JonesMatija...
jm2.25lem1 42226 Lemma for ~ jm2.26 . (Con...
jm2.25 42227 Lemma for ~ jm2.26 . Rema...
jm2.26a 42228 Lemma for ~ jm2.26 . Reve...
jm2.26lem3 42229 Lemma for ~ jm2.26 . Use ...
jm2.26 42230 Lemma 2.26 of [JonesMatija...
jm2.15nn0 42231 Lemma 2.15 of [JonesMatija...
jm2.16nn0 42232 Lemma 2.16 of [JonesMatija...
jm2.27a 42233 Lemma for ~ jm2.27 . Reve...
jm2.27b 42234 Lemma for ~ jm2.27 . Expa...
jm2.27c 42235 Lemma for ~ jm2.27 . Forw...
jm2.27 42236 Lemma 2.27 of [JonesMatija...
jm2.27dlem1 42237 Lemma for ~ rmydioph . Su...
jm2.27dlem2 42238 Lemma for ~ rmydioph . Th...
jm2.27dlem3 42239 Lemma for ~ rmydioph . In...
jm2.27dlem4 42240 Lemma for ~ rmydioph . In...
jm2.27dlem5 42241 Lemma for ~ rmydioph . Us...
rmydioph 42242 ~ jm2.27 restated in terms...
rmxdiophlem 42243 X can be expressed in term...
rmxdioph 42244 X is a Diophantine functio...
jm3.1lem1 42245 Lemma for ~ jm3.1 . (Cont...
jm3.1lem2 42246 Lemma for ~ jm3.1 . (Cont...
jm3.1lem3 42247 Lemma for ~ jm3.1 . (Cont...
jm3.1 42248 Diophantine expression for...
expdiophlem1 42249 Lemma for ~ expdioph . Fu...
expdiophlem2 42250 Lemma for ~ expdioph . Ex...
expdioph 42251 The exponential function i...
setindtr 42252 Set induction for sets con...
setindtrs 42253 Set induction scheme witho...
dford3lem1 42254 Lemma for ~ dford3 . (Con...
dford3lem2 42255 Lemma for ~ dford3 . (Con...
dford3 42256 Ordinals are precisely the...
dford4 42257 ~ dford3 expressed in prim...
wopprc 42258 Unrelated: Wiener pairs t...
rpnnen3lem 42259 Lemma for ~ rpnnen3 . (Co...
rpnnen3 42260 Dedekind cut injection of ...
axac10 42261 Characterization of choice...
harinf 42262 The Hartogs number of an i...
wdom2d2 42263 Deduction for weak dominan...
ttac 42264 Tarski's theorem about cho...
pw2f1ocnv 42265 Define a bijection between...
pw2f1o2 42266 Define a bijection between...
pw2f1o2val 42267 Function value of the ~ pw...
pw2f1o2val2 42268 Membership in a mapped set...
soeq12d 42269 Equality deduction for tot...
freq12d 42270 Equality deduction for fou...
weeq12d 42271 Equality deduction for wel...
limsuc2 42272 Limit ordinals in the sens...
wepwsolem 42273 Transfer an ordering on ch...
wepwso 42274 A well-ordering induces a ...
dnnumch1 42275 Define an enumeration of a...
dnnumch2 42276 Define an enumeration (wea...
dnnumch3lem 42277 Value of the ordinal injec...
dnnumch3 42278 Define an injection from a...
dnwech 42279 Define a well-ordering fro...
fnwe2val 42280 Lemma for ~ fnwe2 . Subst...
fnwe2lem1 42281 Lemma for ~ fnwe2 . Subst...
fnwe2lem2 42282 Lemma for ~ fnwe2 . An el...
fnwe2lem3 42283 Lemma for ~ fnwe2 . Trich...
fnwe2 42284 A well-ordering can be con...
aomclem1 42285 Lemma for ~ dfac11 . This...
aomclem2 42286 Lemma for ~ dfac11 . Succ...
aomclem3 42287 Lemma for ~ dfac11 . Succ...
aomclem4 42288 Lemma for ~ dfac11 . Limi...
aomclem5 42289 Lemma for ~ dfac11 . Comb...
aomclem6 42290 Lemma for ~ dfac11 . Tran...
aomclem7 42291 Lemma for ~ dfac11 . ` ( R...
aomclem8 42292 Lemma for ~ dfac11 . Perf...
dfac11 42293 The right-hand side of thi...
kelac1 42294 Kelley's choice, basic for...
kelac2lem 42295 Lemma for ~ kelac2 and ~ d...
kelac2 42296 Kelley's choice, most comm...
dfac21 42297 Tychonoff's theorem is a c...
islmodfg 42300 Property of a finitely gen...
islssfg 42301 Property of a finitely gen...
islssfg2 42302 Property of a finitely gen...
islssfgi 42303 Finitely spanned subspaces...
fglmod 42304 Finitely generated left mo...
lsmfgcl 42305 The sum of two finitely ge...
islnm 42308 Property of being a Noethe...
islnm2 42309 Property of being a Noethe...
lnmlmod 42310 A Noetherian left module i...
lnmlssfg 42311 A submodule of Noetherian ...
lnmlsslnm 42312 All submodules of a Noethe...
lnmfg 42313 A Noetherian left module i...
kercvrlsm 42314 The domain of a linear fun...
lmhmfgima 42315 A homomorphism maps finite...
lnmepi 42316 Epimorphic images of Noeth...
lmhmfgsplit 42317 If the kernel and range of...
lmhmlnmsplit 42318 If the kernel and range of...
lnmlmic 42319 Noetherian is an invariant...
pwssplit4 42320 Splitting for structure po...
filnm 42321 Finite left modules are No...
pwslnmlem0 42322 Zeroeth powers are Noether...
pwslnmlem1 42323 First powers are Noetheria...
pwslnmlem2 42324 A sum of powers is Noether...
pwslnm 42325 Finite powers of Noetheria...
unxpwdom3 42326 Weaker version of ~ unxpwd...
pwfi2f1o 42327 The ~ pw2f1o bijection rel...
pwfi2en 42328 Finitely supported indicat...
frlmpwfi 42329 Formal linear combinations...
gicabl 42330 Being Abelian is a group i...
imasgim 42331 A relabeling of the elemen...
isnumbasgrplem1 42332 A set which is equipollent...
harn0 42333 The Hartogs number of a se...
numinfctb 42334 A numerable infinite set c...
isnumbasgrplem2 42335 If the (to be thought of a...
isnumbasgrplem3 42336 Every nonempty numerable s...
isnumbasabl 42337 A set is numerable iff it ...
isnumbasgrp 42338 A set is numerable iff it ...
dfacbasgrp 42339 A choice equivalent in abs...
islnr 42342 Property of a left-Noether...
lnrring 42343 Left-Noetherian rings are ...
lnrlnm 42344 Left-Noetherian rings have...
islnr2 42345 Property of being a left-N...
islnr3 42346 Relate left-Noetherian rin...
lnr2i 42347 Given an ideal in a left-N...
lpirlnr 42348 Left principal ideal rings...
lnrfrlm 42349 Finite-dimensional free mo...
lnrfg 42350 Finitely-generated modules...
lnrfgtr 42351 A submodule of a finitely ...
hbtlem1 42354 Value of the leading coeff...
hbtlem2 42355 Leading coefficient ideals...
hbtlem7 42356 Functionality of leading c...
hbtlem4 42357 The leading ideal function...
hbtlem3 42358 The leading ideal function...
hbtlem5 42359 The leading ideal function...
hbtlem6 42360 There is a finite set of p...
hbt 42361 The Hilbert Basis Theorem ...
dgrsub2 42366 Subtracting two polynomial...
elmnc 42367 Property of a monic polyno...
mncply 42368 A monic polynomial is a po...
mnccoe 42369 A monic polynomial has lea...
mncn0 42370 A monic polynomial is not ...
dgraaval 42375 Value of the degree functi...
dgraalem 42376 Properties of the degree o...
dgraacl 42377 Closure of the degree func...
dgraaf 42378 Degree function on algebra...
dgraaub 42379 Upper bound on degree of a...
dgraa0p 42380 A rational polynomial of d...
mpaaeu 42381 An algebraic number has ex...
mpaaval 42382 Value of the minimal polyn...
mpaalem 42383 Properties of the minimal ...
mpaacl 42384 Minimal polynomial is a po...
mpaadgr 42385 Minimal polynomial has deg...
mpaaroot 42386 The minimal polynomial of ...
mpaamn 42387 Minimal polynomial is moni...
itgoval 42392 Value of the integral-over...
aaitgo 42393 The standard algebraic num...
itgoss 42394 An integral element is int...
itgocn 42395 All integral elements are ...
cnsrexpcl 42396 Exponentiation is closed i...
fsumcnsrcl 42397 Finite sums are closed in ...
cnsrplycl 42398 Polynomials are closed in ...
rgspnval 42399 Value of the ring-span of ...
rgspncl 42400 The ring-span of a set is ...
rgspnssid 42401 The ring-span of a set con...
rgspnmin 42402 The ring-span is contained...
rgspnid 42403 The span of a subring is i...
rngunsnply 42404 Adjoining one element to a...
flcidc 42405 Finite linear combinations...
algstr 42408 Lemma to shorten proofs of...
algbase 42409 The base set of a construc...
algaddg 42410 The additive operation of ...
algmulr 42411 The multiplicative operati...
algsca 42412 The set of scalars of a co...
algvsca 42413 The scalar product operati...
mendval 42414 Value of the module endomo...
mendbas 42415 Base set of the module end...
mendplusgfval 42416 Addition in the module end...
mendplusg 42417 A specific addition in the...
mendmulrfval 42418 Multiplication in the modu...
mendmulr 42419 A specific multiplication ...
mendsca 42420 The module endomorphism al...
mendvscafval 42421 Scalar multiplication in t...
mendvsca 42422 A specific scalar multipli...
mendring 42423 The module endomorphism al...
mendlmod 42424 The module endomorphism al...
mendassa 42425 The module endomorphism al...
idomrootle 42426 No element of an integral ...
idomodle 42427 Limit on the number of ` N...
fiuneneq 42428 Two finite sets of equal s...
idomsubgmo 42429 The units of an integral d...
proot1mul 42430 Any primitive ` N ` -th ro...
proot1hash 42431 If an integral domain has ...
proot1ex 42432 The complex field has prim...
isdomn3 42435 Nonzero elements form a mu...
mon1pid 42436 Monicity and degree of the...
mon1psubm 42437 Monic polynomials are a mu...
deg1mhm 42438 Homomorphic property of th...
cytpfn 42439 Functionality of the cyclo...
cytpval 42440 Substitutions for the Nth ...
fgraphopab 42441 Express a function as a su...
fgraphxp 42442 Express a function as a su...
hausgraph 42443 The graph of a continuous ...
r1sssucd 42448 Deductive form of ~ r1sssu...
iocunico 42449 Split an open interval int...
iocinico 42450 The intersection of two se...
iocmbl 42451 An open-below, closed-abov...
cnioobibld 42452 A bounded, continuous func...
arearect 42453 The area of a rectangle wh...
areaquad 42454 The area of a quadrilatera...
uniel 42455 Two ways to say a union is...
unielss 42456 Two ways to say the union ...
unielid 42457 Two ways to say the union ...
ssunib 42458 Two ways to say a class is...
rp-intrabeq 42459 Equality theorem for supre...
rp-unirabeq 42460 Equality theorem for infim...
onmaxnelsup 42461 Two ways to say the maximu...
onsupneqmaxlim0 42462 If the supremum of a class...
onsupcl2 42463 The supremum of a set of o...
onuniintrab 42464 The union of a set of ordi...
onintunirab 42465 The intersection of a non-...
onsupnmax 42466 If the union of a class of...
onsupuni 42467 The supremum of a set of o...
onsupuni2 42468 The supremum of a set of o...
onsupintrab 42469 The supremum of a set of o...
onsupintrab2 42470 The supremum of a set of o...
onsupcl3 42471 The supremum of a set of o...
onsupex3 42472 The supremum of a set of o...
onuniintrab2 42473 The union of a set of ordi...
oninfint 42474 The infimum of a non-empty...
oninfunirab 42475 The infimum of a non-empty...
oninfcl2 42476 The infimum of a non-empty...
onsupmaxb 42477 The union of a class of or...
onexgt 42478 For any ordinal, there is ...
onexomgt 42479 For any ordinal, there is ...
omlimcl2 42480 The product of a limit ord...
onexlimgt 42481 For any ordinal, there is ...
onexoegt 42482 For any ordinal, there is ...
oninfex2 42483 The infimum of a non-empty...
onsupeqmax 42484 Condition when the supremu...
onsupeqnmax 42485 Condition when the supremu...
onsuplub 42486 The supremum of a set of o...
onsupnub 42487 An upper bound of a set of...
onfisupcl 42488 Sufficient condition when ...
onelord 42489 Every element of a ordinal...
onepsuc 42490 Every ordinal is less than...
epsoon 42491 The ordinals are strictly ...
epirron 42492 The strict order on the or...
oneptr 42493 The strict order on the or...
oneltr 42494 The elementhood relation o...
oneptri 42495 The strict, complete (line...
oneltri 42496 The elementhood relation o...
ordeldif 42497 Membership in the differen...
ordeldifsucon 42498 Membership in the differen...
ordeldif1o 42499 Membership in the differen...
ordne0gt0 42500 Ordinal zero is less than ...
ondif1i 42501 Ordinal zero is less than ...
onsucelab 42502 The successor of every ord...
dflim6 42503 A limit ordinal is a non-z...
limnsuc 42504 A limit ordinal is not an ...
onsucss 42505 If one ordinal is less tha...
ordnexbtwnsuc 42506 For any distinct pair of o...
orddif0suc 42507 For any distinct pair of o...
onsucf1lem 42508 For ordinals, the successo...
onsucf1olem 42509 The successor operation is...
onsucrn 42510 The successor operation is...
onsucf1o 42511 The successor operation is...
dflim7 42512 A limit ordinal is a non-z...
onov0suclim 42513 Compactly express rules fo...
oa0suclim 42514 Closed form expression of ...
om0suclim 42515 Closed form expression of ...
oe0suclim 42516 Closed form expression of ...
oaomoecl 42517 The operations of addition...
onsupsucismax 42518 If the union of a set of o...
onsssupeqcond 42519 If for every element of a ...
limexissup 42520 An ordinal which is a limi...
limiun 42521 A limit ordinal is the uni...
limexissupab 42522 An ordinal which is a limi...
om1om1r 42523 Ordinal one is both a left...
oe0rif 42524 Ordinal zero raised to any...
oasubex 42525 While subtraction can't be...
nnamecl 42526 Natural numbers are closed...
onsucwordi 42527 The successor operation pr...
oalim2cl 42528 The ordinal sum of any ord...
oaltublim 42529 Given ` C ` is a limit ord...
oaordi3 42530 Ordinal addition of the sa...
oaord3 42531 When the same ordinal is a...
1oaomeqom 42532 Ordinal one plus omega is ...
oaabsb 42533 The right addend absorbs t...
oaordnrex 42534 When omega is added on the...
oaordnr 42535 When the same ordinal is a...
omge1 42536 Any non-zero ordinal produ...
omge2 42537 Any non-zero ordinal produ...
omlim2 42538 The non-zero product with ...
omord2lim 42539 Given a limit ordinal, the...
omord2i 42540 Ordinal multiplication of ...
omord2com 42541 When the same non-zero ord...
2omomeqom 42542 Ordinal two times omega is...
omnord1ex 42543 When omega is multiplied o...
omnord1 42544 When the same non-zero ord...
oege1 42545 Any non-zero ordinal power...
oege2 42546 Any power of an ordinal at...
rp-oelim2 42547 The power of an ordinal at...
oeord2lim 42548 Given a limit ordinal, the...
oeord2i 42549 Ordinal exponentiation of ...
oeord2com 42550 When the same base at leas...
nnoeomeqom 42551 Any natural number at leas...
df3o2 42552 Ordinal 3 is the unordered...
df3o3 42553 Ordinal 3, fully expanded....
oenord1ex 42554 When ordinals two and thre...
oenord1 42555 When two ordinals (both at...
oaomoencom 42556 Ordinal addition, multipli...
oenassex 42557 Ordinal two raised to two ...
oenass 42558 Ordinal exponentiation is ...
cantnftermord 42559 For terms of the form of a...
cantnfub 42560 Given a finite number of t...
cantnfub2 42561 Given a finite number of t...
bropabg 42562 Equivalence for two classe...
cantnfresb 42563 A Cantor normal form which...
cantnf2 42564 For every ordinal, ` A ` ,...
oawordex2 42565 If ` C ` is between ` A ` ...
nnawordexg 42566 If an ordinal, ` B ` , is ...
succlg 42567 Closure law for ordinal su...
dflim5 42568 A limit ordinal is either ...
oacl2g 42569 Closure law for ordinal ad...
onmcl 42570 If an ordinal is less than...
omabs2 42571 Ordinal multiplication by ...
omcl2 42572 Closure law for ordinal mu...
omcl3g 42573 Closure law for ordinal mu...
ordsssucb 42574 An ordinal number is less ...
tfsconcatlem 42575 Lemma for ~ tfsconcatun . ...
tfsconcatun 42576 The concatenation of two t...
tfsconcatfn 42577 The concatenation of two t...
tfsconcatfv1 42578 An early value of the conc...
tfsconcatfv2 42579 A latter value of the conc...
tfsconcatfv 42580 The value of the concatena...
tfsconcatrn 42581 The range of the concatena...
tfsconcatfo 42582 The concatenation of two t...
tfsconcatb0 42583 The concatentation with th...
tfsconcat0i 42584 The concatentation with th...
tfsconcat0b 42585 The concatentation with th...
tfsconcat00 42586 The concatentation of two ...
tfsconcatrev 42587 If the domain of a transfi...
tfsconcatrnss12 42588 The range of the concatena...
tfsconcatrnss 42589 The concatenation of trans...
tfsconcatrnsson 42590 The concatenation of trans...
tfsnfin 42591 A transfinite sequence is ...
rp-tfslim 42592 The limit of a sequence of...
ofoafg 42593 Addition operator for func...
ofoaf 42594 Addition operator for func...
ofoafo 42595 Addition operator for func...
ofoacl 42596 Closure law for component ...
ofoaid1 42597 Identity law for component...
ofoaid2 42598 Identity law for component...
ofoaass 42599 Component-wise addition of...
ofoacom 42600 Component-wise addition of...
naddcnff 42601 Addition operator for Cant...
naddcnffn 42602 Addition operator for Cant...
naddcnffo 42603 Addition of Cantor normal ...
naddcnfcl 42604 Closure law for component-...
naddcnfcom 42605 Component-wise ordinal add...
naddcnfid1 42606 Identity law for component...
naddcnfid2 42607 Identity law for component...
naddcnfass 42608 Component-wise addition of...
onsucunifi 42609 The successor to the union...
sucunisn 42610 The successor to the union...
onsucunipr 42611 The successor to the union...
onsucunitp 42612 The successor to the union...
oaun3lem1 42613 The class of all ordinal s...
oaun3lem2 42614 The class of all ordinal s...
oaun3lem3 42615 The class of all ordinal s...
oaun3lem4 42616 The class of all ordinal s...
rp-abid 42617 Two ways to express a clas...
oadif1lem 42618 Express the set difference...
oadif1 42619 Express the set difference...
oaun2 42620 Ordinal addition as a unio...
oaun3 42621 Ordinal addition as a unio...
naddov4 42622 Alternate expression for n...
nadd2rabtr 42623 The set of ordinals which ...
nadd2rabord 42624 The set of ordinals which ...
nadd2rabex 42625 The class of ordinals whic...
nadd2rabon 42626 The set of ordinals which ...
nadd1rabtr 42627 The set of ordinals which ...
nadd1rabord 42628 The set of ordinals which ...
nadd1rabex 42629 The class of ordinals whic...
nadd1rabon 42630 The set of ordinals which ...
nadd1suc 42631 Natural addition with 1 is...
naddsuc2 42632 Natural addition with succ...
naddass1 42633 Natural addition of ordina...
naddgeoa 42634 Natural addition results i...
naddonnn 42635 Natural addition with a na...
naddwordnexlem0 42636 When ` A ` is the sum of a...
naddwordnexlem1 42637 When ` A ` is the sum of a...
naddwordnexlem2 42638 When ` A ` is the sum of a...
naddwordnexlem3 42639 When ` A ` is the sum of a...
oawordex3 42640 When ` A ` is the sum of a...
naddwordnexlem4 42641 When ` A ` is the sum of a...
ordsssucim 42642 If an ordinal is less than...
insucid 42643 The intersection of a clas...
om2 42644 Two ways to double an ordi...
oaltom 42645 Multiplication eventually ...
oe2 42646 Two ways to square an ordi...
omltoe 42647 Exponentiation eventually ...
abeqabi 42648 Generalized condition for ...
abpr 42649 Condition for a class abst...
abtp 42650 Condition for a class abst...
ralopabb 42651 Restricted universal quant...
fpwfvss 42652 Functions into a powerset ...
sdomne0 42653 A class that strictly domi...
sdomne0d 42654 A class that strictly domi...
safesnsupfiss 42655 If ` B ` is a finite subse...
safesnsupfiub 42656 If ` B ` is a finite subse...
safesnsupfidom1o 42657 If ` B ` is a finite subse...
safesnsupfilb 42658 If ` B ` is a finite subse...
isoeq145d 42659 Equality deduction for iso...
resisoeq45d 42660 Equality deduction for equ...
negslem1 42661 An equivalence between ide...
nvocnvb 42662 Equivalence to saying the ...
rp-brsslt 42663 Binary relation form of a ...
nla0002 42664 Extending a linear order t...
nla0003 42665 Extending a linear order t...
nla0001 42666 Extending a linear order t...
faosnf0.11b 42667 ` B ` is called a non-limi...
dfno2 42668 A surreal number, in the f...
onnog 42669 Every ordinal maps to a su...
onnobdayg 42670 Every ordinal maps to a su...
bdaybndex 42671 Bounds formed from the bir...
bdaybndbday 42672 Bounds formed from the bir...
onno 42673 Every ordinal maps to a su...
onnoi 42674 Every ordinal maps to a su...
0no 42675 Ordinal zero maps to a sur...
1no 42676 Ordinal one maps to a surr...
2no 42677 Ordinal two maps to a surr...
3no 42678 Ordinal three maps to a su...
4no 42679 Ordinal four maps to a sur...
fnimafnex 42680 The functional image of a ...
nlimsuc 42681 A successor is not a limit...
nlim1NEW 42682 1 is not a limit ordinal. ...
nlim2NEW 42683 2 is not a limit ordinal. ...
nlim3 42684 3 is not a limit ordinal. ...
nlim4 42685 4 is not a limit ordinal. ...
oa1un 42686 Given ` A e. On ` , let ` ...
oa1cl 42687 ` A +o 1o ` is in ` On ` ....
0finon 42688 0 is a finite ordinal. Se...
1finon 42689 1 is a finite ordinal. Se...
2finon 42690 2 is a finite ordinal. Se...
3finon 42691 3 is a finite ordinal. Se...
4finon 42692 4 is a finite ordinal. Se...
finona1cl 42693 The finite ordinals are cl...
finonex 42694 The finite ordinals are a ...
fzunt 42695 Union of two adjacent fini...
fzuntd 42696 Union of two adjacent fini...
fzunt1d 42697 Union of two overlapping f...
fzuntgd 42698 Union of two adjacent or o...
ifpan123g 42699 Conjunction of conditional...
ifpan23 42700 Conjunction of conditional...
ifpdfor2 42701 Define or in terms of cond...
ifporcor 42702 Corollary of commutation o...
ifpdfan2 42703 Define and with conditiona...
ifpancor 42704 Corollary of commutation o...
ifpdfor 42705 Define or in terms of cond...
ifpdfan 42706 Define and with conditiona...
ifpbi2 42707 Equivalence theorem for co...
ifpbi3 42708 Equivalence theorem for co...
ifpim1 42709 Restate implication as con...
ifpnot 42710 Restate negated wff as con...
ifpid2 42711 Restate wff as conditional...
ifpim2 42712 Restate implication as con...
ifpbi23 42713 Equivalence theorem for co...
ifpbiidcor 42714 Restatement of ~ biid . (...
ifpbicor 42715 Corollary of commutation o...
ifpxorcor 42716 Corollary of commutation o...
ifpbi1 42717 Equivalence theorem for co...
ifpnot23 42718 Negation of conditional lo...
ifpnotnotb 42719 Factor conditional logic o...
ifpnorcor 42720 Corollary of commutation o...
ifpnancor 42721 Corollary of commutation o...
ifpnot23b 42722 Negation of conditional lo...
ifpbiidcor2 42723 Restatement of ~ biid . (...
ifpnot23c 42724 Negation of conditional lo...
ifpnot23d 42725 Negation of conditional lo...
ifpdfnan 42726 Define nand as conditional...
ifpdfxor 42727 Define xor as conditional ...
ifpbi12 42728 Equivalence theorem for co...
ifpbi13 42729 Equivalence theorem for co...
ifpbi123 42730 Equivalence theorem for co...
ifpidg 42731 Restate wff as conditional...
ifpid3g 42732 Restate wff as conditional...
ifpid2g 42733 Restate wff as conditional...
ifpid1g 42734 Restate wff as conditional...
ifpim23g 42735 Restate implication as con...
ifpim3 42736 Restate implication as con...
ifpnim1 42737 Restate negated implicatio...
ifpim4 42738 Restate implication as con...
ifpnim2 42739 Restate negated implicatio...
ifpim123g 42740 Implication of conditional...
ifpim1g 42741 Implication of conditional...
ifp1bi 42742 Substitute the first eleme...
ifpbi1b 42743 When the first variable is...
ifpimimb 42744 Factor conditional logic o...
ifpororb 42745 Factor conditional logic o...
ifpananb 42746 Factor conditional logic o...
ifpnannanb 42747 Factor conditional logic o...
ifpor123g 42748 Disjunction of conditional...
ifpimim 42749 Consequnce of implication....
ifpbibib 42750 Factor conditional logic o...
ifpxorxorb 42751 Factor conditional logic o...
rp-fakeimass 42752 A special case where impli...
rp-fakeanorass 42753 A special case where a mix...
rp-fakeoranass 42754 A special case where a mix...
rp-fakeinunass 42755 A special case where a mix...
rp-fakeuninass 42756 A special case where a mix...
rp-isfinite5 42757 A set is said to be finite...
rp-isfinite6 42758 A set is said to be finite...
intabssd 42759 When for each element ` y ...
eu0 42760 There is only one empty se...
epelon2 42761 Over the ordinal numbers, ...
ontric3g 42762 For all ` x , y e. On ` , ...
dfsucon 42763 ` A ` is called a successo...
snen1g 42764 A singleton is equinumerou...
snen1el 42765 A singleton is equinumerou...
sn1dom 42766 A singleton is dominated b...
pr2dom 42767 An unordered pair is domin...
tr3dom 42768 An unordered triple is dom...
ensucne0 42769 A class equinumerous to a ...
ensucne0OLD 42770 A class equinumerous to a ...
dfom6 42771 Let ` _om ` be defined to ...
infordmin 42772 ` _om ` is the smallest in...
iscard4 42773 Two ways to express the pr...
minregex 42774 Given any cardinal number ...
minregex2 42775 Given any cardinal number ...
iscard5 42776 Two ways to express the pr...
elrncard 42777 Let us define a cardinal n...
harval3 42778 ` ( har `` A ) ` is the le...
harval3on 42779 For any ordinal number ` A...
omssrncard 42780 All natural numbers are ca...
0iscard 42781 0 is a cardinal number. (...
1iscard 42782 1 is a cardinal number. (...
omiscard 42783 ` _om ` is a cardinal numb...
sucomisnotcard 42784 ` _om +o 1o ` is not a car...
nna1iscard 42785 For any natural number, th...
har2o 42786 The least cardinal greater...
en2pr 42787 A class is equinumerous to...
pr2cv 42788 If an unordered pair is eq...
pr2el1 42789 If an unordered pair is eq...
pr2cv1 42790 If an unordered pair is eq...
pr2el2 42791 If an unordered pair is eq...
pr2cv2 42792 If an unordered pair is eq...
pren2 42793 An unordered pair is equin...
pr2eldif1 42794 If an unordered pair is eq...
pr2eldif2 42795 If an unordered pair is eq...
pren2d 42796 A pair of two distinct set...
aleph1min 42797 ` ( aleph `` 1o ) ` is the...
alephiso2 42798 ` aleph ` is a strictly or...
alephiso3 42799 ` aleph ` is a strictly or...
pwelg 42800 The powerclass is an eleme...
pwinfig 42801 The powerclass of an infin...
pwinfi2 42802 The powerclass of an infin...
pwinfi3 42803 The powerclass of an infin...
pwinfi 42804 The powerclass of an infin...
fipjust 42805 A definition of the finite...
cllem0 42806 The class of all sets with...
superficl 42807 The class of all supersets...
superuncl 42808 The class of all supersets...
ssficl 42809 The class of all subsets o...
ssuncl 42810 The class of all subsets o...
ssdifcl 42811 The class of all subsets o...
sssymdifcl 42812 The class of all subsets o...
fiinfi 42813 If two classes have the fi...
rababg 42814 Condition when restricted ...
elinintab 42815 Two ways of saying a set i...
elmapintrab 42816 Two ways to say a set is a...
elinintrab 42817 Two ways of saying a set i...
inintabss 42818 Upper bound on intersectio...
inintabd 42819 Value of the intersection ...
xpinintabd 42820 Value of the intersection ...
relintabex 42821 If the intersection of a c...
elcnvcnvintab 42822 Two ways of saying a set i...
relintab 42823 Value of the intersection ...
nonrel 42824 A non-relation is equal to...
elnonrel 42825 Only an ordered pair where...
cnvssb 42826 Subclass theorem for conve...
relnonrel 42827 The non-relation part of a...
cnvnonrel 42828 The converse of the non-re...
brnonrel 42829 A non-relation cannot rela...
dmnonrel 42830 The domain of the non-rela...
rnnonrel 42831 The range of the non-relat...
resnonrel 42832 A restriction of the non-r...
imanonrel 42833 An image under the non-rel...
cononrel1 42834 Composition with the non-r...
cononrel2 42835 Composition with the non-r...
elmapintab 42836 Two ways to say a set is a...
fvnonrel 42837 The function value of any ...
elinlem 42838 Two ways to say a set is a...
elcnvcnvlem 42839 Two ways to say a set is a...
cnvcnvintabd 42840 Value of the relationship ...
elcnvlem 42841 Two ways to say a set is a...
elcnvintab 42842 Two ways of saying a set i...
cnvintabd 42843 Value of the converse of t...
undmrnresiss 42844 Two ways of saying the ide...
reflexg 42845 Two ways of saying a relat...
cnvssco 42846 A condition weaker than re...
refimssco 42847 Reflexive relations are su...
cleq2lem 42848 Equality implies bijection...
cbvcllem 42849 Change of bound variable i...
clublem 42850 If a superset ` Y ` of ` X...
clss2lem 42851 The closure of a property ...
dfid7 42852 Definition of identity rel...
mptrcllem 42853 Show two versions of a clo...
cotrintab 42854 The intersection of a clas...
rclexi 42855 The reflexive closure of a...
rtrclexlem 42856 Existence of relation impl...
rtrclex 42857 The reflexive-transitive c...
trclubgNEW 42858 If a relation exists then ...
trclubNEW 42859 If a relation exists then ...
trclexi 42860 The transitive closure of ...
rtrclexi 42861 The reflexive-transitive c...
clrellem 42862 When the property ` ps ` h...
clcnvlem 42863 When ` A ` , an upper boun...
cnvtrucl0 42864 The converse of the trivia...
cnvrcl0 42865 The converse of the reflex...
cnvtrcl0 42866 The converse of the transi...
dmtrcl 42867 The domain of the transiti...
rntrcl 42868 The range of the transitiv...
dfrtrcl5 42869 Definition of reflexive-tr...
trcleq2lemRP 42870 Equality implies bijection...
sqrtcvallem1 42871 Two ways of saying a compl...
reabsifneg 42872 Alternate expression for t...
reabsifnpos 42873 Alternate expression for t...
reabsifpos 42874 Alternate expression for t...
reabsifnneg 42875 Alternate expression for t...
reabssgn 42876 Alternate expression for t...
sqrtcvallem2 42877 Equivalent to saying that ...
sqrtcvallem3 42878 Equivalent to saying that ...
sqrtcvallem4 42879 Equivalent to saying that ...
sqrtcvallem5 42880 Equivalent to saying that ...
sqrtcval 42881 Explicit formula for the c...
sqrtcval2 42882 Explicit formula for the c...
resqrtval 42883 Real part of the complex s...
imsqrtval 42884 Imaginary part of the comp...
resqrtvalex 42885 Example for ~ resqrtval . ...
imsqrtvalex 42886 Example for ~ imsqrtval . ...
al3im 42887 Version of ~ ax-4 for a ne...
intima0 42888 Two ways of expressing the...
elimaint 42889 Element of image of inters...
cnviun 42890 Converse of indexed union....
imaiun1 42891 The image of an indexed un...
coiun1 42892 Composition with an indexe...
elintima 42893 Element of intersection of...
intimass 42894 The image under the inters...
intimass2 42895 The image under the inters...
intimag 42896 Requirement for the image ...
intimasn 42897 Two ways to express the im...
intimasn2 42898 Two ways to express the im...
ss2iundf 42899 Subclass theorem for index...
ss2iundv 42900 Subclass theorem for index...
cbviuneq12df 42901 Rule used to change the bo...
cbviuneq12dv 42902 Rule used to change the bo...
conrel1d 42903 Deduction about compositio...
conrel2d 42904 Deduction about compositio...
trrelind 42905 The intersection of transi...
xpintrreld 42906 The intersection of a tran...
restrreld 42907 The restriction of a trans...
trrelsuperreldg 42908 Concrete construction of a...
trficl 42909 The class of all transitiv...
cnvtrrel 42910 The converse of a transiti...
trrelsuperrel2dg 42911 Concrete construction of a...
dfrcl2 42914 Reflexive closure of a rel...
dfrcl3 42915 Reflexive closure of a rel...
dfrcl4 42916 Reflexive closure of a rel...
relexp2 42917 A set operated on by the r...
relexpnul 42918 If the domain and range of...
eliunov2 42919 Membership in the indexed ...
eltrclrec 42920 Membership in the indexed ...
elrtrclrec 42921 Membership in the indexed ...
briunov2 42922 Two classes related by the...
brmptiunrelexpd 42923 If two elements are connec...
fvmptiunrelexplb0d 42924 If the indexed union range...
fvmptiunrelexplb0da 42925 If the indexed union range...
fvmptiunrelexplb1d 42926 If the indexed union range...
brfvid 42927 If two elements are connec...
brfvidRP 42928 If two elements are connec...
fvilbd 42929 A set is a subset of its i...
fvilbdRP 42930 A set is a subset of its i...
brfvrcld 42931 If two elements are connec...
brfvrcld2 42932 If two elements are connec...
fvrcllb0d 42933 A restriction of the ident...
fvrcllb0da 42934 A restriction of the ident...
fvrcllb1d 42935 A set is a subset of its i...
brtrclrec 42936 Two classes related by the...
brrtrclrec 42937 Two classes related by the...
briunov2uz 42938 Two classes related by the...
eliunov2uz 42939 Membership in the indexed ...
ov2ssiunov2 42940 Any particular operator va...
relexp0eq 42941 The zeroth power of relati...
iunrelexp0 42942 Simplification of zeroth p...
relexpxpnnidm 42943 Any positive power of a Ca...
relexpiidm 42944 Any power of any restricti...
relexpss1d 42945 The relational power of a ...
comptiunov2i 42946 The composition two indexe...
corclrcl 42947 The reflexive closure is i...
iunrelexpmin1 42948 The indexed union of relat...
relexpmulnn 42949 With exponents limited to ...
relexpmulg 42950 With ordered exponents, th...
trclrelexplem 42951 The union of relational po...
iunrelexpmin2 42952 The indexed union of relat...
relexp01min 42953 With exponents limited to ...
relexp1idm 42954 Repeated raising a relatio...
relexp0idm 42955 Repeated raising a relatio...
relexp0a 42956 Absorption law for zeroth ...
relexpxpmin 42957 The composition of powers ...
relexpaddss 42958 The composition of two pow...
iunrelexpuztr 42959 The indexed union of relat...
dftrcl3 42960 Transitive closure of a re...
brfvtrcld 42961 If two elements are connec...
fvtrcllb1d 42962 A set is a subset of its i...
trclfvcom 42963 The transitive closure of ...
cnvtrclfv 42964 The converse of the transi...
cotrcltrcl 42965 The transitive closure is ...
trclimalb2 42966 Lower bound for image unde...
brtrclfv2 42967 Two ways to indicate two e...
trclfvdecomr 42968 The transitive closure of ...
trclfvdecoml 42969 The transitive closure of ...
dmtrclfvRP 42970 The domain of the transiti...
rntrclfvRP 42971 The range of the transitiv...
rntrclfv 42972 The range of the transitiv...
dfrtrcl3 42973 Reflexive-transitive closu...
brfvrtrcld 42974 If two elements are connec...
fvrtrcllb0d 42975 A restriction of the ident...
fvrtrcllb0da 42976 A restriction of the ident...
fvrtrcllb1d 42977 A set is a subset of its i...
dfrtrcl4 42978 Reflexive-transitive closu...
corcltrcl 42979 The composition of the ref...
cortrcltrcl 42980 Composition with the refle...
corclrtrcl 42981 Composition with the refle...
cotrclrcl 42982 The composition of the ref...
cortrclrcl 42983 Composition with the refle...
cotrclrtrcl 42984 Composition with the refle...
cortrclrtrcl 42985 The reflexive-transitive c...
frege77d 42986 If the images of both ` { ...
frege81d 42987 If the image of ` U ` is a...
frege83d 42988 If the image of the union ...
frege96d 42989 If ` C ` follows ` A ` in ...
frege87d 42990 If the images of both ` { ...
frege91d 42991 If ` B ` follows ` A ` in ...
frege97d 42992 If ` A ` contains all elem...
frege98d 42993 If ` C ` follows ` A ` and...
frege102d 42994 If either ` A ` and ` C ` ...
frege106d 42995 If ` B ` follows ` A ` in ...
frege108d 42996 If either ` A ` and ` C ` ...
frege109d 42997 If ` A ` contains all elem...
frege114d 42998 If either ` R ` relates ` ...
frege111d 42999 If either ` A ` and ` C ` ...
frege122d 43000 If ` F ` is a function, ` ...
frege124d 43001 If ` F ` is a function, ` ...
frege126d 43002 If ` F ` is a function, ` ...
frege129d 43003 If ` F ` is a function and...
frege131d 43004 If ` F ` is a function and...
frege133d 43005 If ` F ` is a function and...
dfxor4 43006 Express exclusive-or in te...
dfxor5 43007 Express exclusive-or in te...
df3or2 43008 Express triple-or in terms...
df3an2 43009 Express triple-and in term...
nev 43010 Express that not every set...
0pssin 43011 Express that an intersecti...
dfhe2 43014 The property of relation `...
dfhe3 43015 The property of relation `...
heeq12 43016 Equality law for relations...
heeq1 43017 Equality law for relations...
heeq2 43018 Equality law for relations...
sbcheg 43019 Distribute proper substitu...
hess 43020 Subclass law for relations...
xphe 43021 Any Cartesian product is h...
0he 43022 The empty relation is here...
0heALT 43023 The empty relation is here...
he0 43024 Any relation is hereditary...
unhe1 43025 The union of two relations...
snhesn 43026 Any singleton is hereditar...
idhe 43027 The identity relation is h...
psshepw 43028 The relation between sets ...
sshepw 43029 The relation between sets ...
rp-simp2-frege 43032 Simplification of triple c...
rp-simp2 43033 Simplification of triple c...
rp-frege3g 43034 Add antecedent to ~ ax-fre...
frege3 43035 Add antecedent to ~ ax-fre...
rp-misc1-frege 43036 Double-use of ~ ax-frege2 ...
rp-frege24 43037 Introducing an embedded an...
rp-frege4g 43038 Deduction related to distr...
frege4 43039 Special case of closed for...
frege5 43040 A closed form of ~ syl . ...
rp-7frege 43041 Distribute antecedent and ...
rp-4frege 43042 Elimination of a nested an...
rp-6frege 43043 Elimination of a nested an...
rp-8frege 43044 Eliminate antecedent when ...
rp-frege25 43045 Closed form for ~ a1dd . ...
frege6 43046 A closed form of ~ imim2d ...
axfrege8 43047 Swap antecedents. Identic...
frege7 43048 A closed form of ~ syl6 . ...
frege26 43050 Identical to ~ idd . Prop...
frege27 43051 We cannot (at the same tim...
frege9 43052 Closed form of ~ syl with ...
frege12 43053 A closed form of ~ com23 ....
frege11 43054 Elimination of a nested an...
frege24 43055 Closed form for ~ a1d . D...
frege16 43056 A closed form of ~ com34 ....
frege25 43057 Closed form for ~ a1dd . ...
frege18 43058 Closed form of a syllogism...
frege22 43059 A closed form of ~ com45 ....
frege10 43060 Result commuting anteceden...
frege17 43061 A closed form of ~ com3l ....
frege13 43062 A closed form of ~ com3r ....
frege14 43063 Closed form of a deduction...
frege19 43064 A closed form of ~ syl6 . ...
frege23 43065 Syllogism followed by rota...
frege15 43066 A closed form of ~ com4r ....
frege21 43067 Replace antecedent in ante...
frege20 43068 A closed form of ~ syl8 . ...
axfrege28 43069 Contraposition. Identical...
frege29 43071 Closed form of ~ con3d . ...
frege30 43072 Commuted, closed form of ~...
axfrege31 43073 Identical to ~ notnotr . ...
frege32 43075 Deduce ~ con1 from ~ con3 ...
frege33 43076 If ` ph ` or ` ps ` takes ...
frege34 43077 If as a consequence of the...
frege35 43078 Commuted, closed form of ~...
frege36 43079 The case in which ` ps ` i...
frege37 43080 If ` ch ` is a necessary c...
frege38 43081 Identical to ~ pm2.21 . P...
frege39 43082 Syllogism between ~ pm2.18...
frege40 43083 Anything implies ~ pm2.18 ...
axfrege41 43084 Identical to ~ notnot . A...
frege42 43086 Not not ~ id . Propositio...
frege43 43087 If there is a choice only ...
frege44 43088 Similar to a commuted ~ pm...
frege45 43089 Deduce ~ pm2.6 from ~ con1...
frege46 43090 If ` ps ` holds when ` ph ...
frege47 43091 Deduce consequence follows...
frege48 43092 Closed form of syllogism w...
frege49 43093 Closed form of deduction w...
frege50 43094 Closed form of ~ jaoi . P...
frege51 43095 Compare with ~ jaod . Pro...
axfrege52a 43096 Justification for ~ ax-fre...
frege52aid 43098 The case when the content ...
frege53aid 43099 Specialization of ~ frege5...
frege53a 43100 Lemma for ~ frege55a . Pr...
axfrege54a 43101 Justification for ~ ax-fre...
frege54cor0a 43103 Synonym for logical equiva...
frege54cor1a 43104 Reflexive equality. (Cont...
frege55aid 43105 Lemma for ~ frege57aid . ...
frege55lem1a 43106 Necessary deduction regard...
frege55lem2a 43107 Core proof of Proposition ...
frege55a 43108 Proposition 55 of [Frege18...
frege55cor1a 43109 Proposition 55 of [Frege18...
frege56aid 43110 Lemma for ~ frege57aid . ...
frege56a 43111 Proposition 56 of [Frege18...
frege57aid 43112 This is the all imporant f...
frege57a 43113 Analogue of ~ frege57aid ....
axfrege58a 43114 Identical to ~ anifp . Ju...
frege58acor 43116 Lemma for ~ frege59a . (C...
frege59a 43117 A kind of Aristotelian inf...
frege60a 43118 Swap antecedents of ~ ax-f...
frege61a 43119 Lemma for ~ frege65a . Pr...
frege62a 43120 A kind of Aristotelian inf...
frege63a 43121 Proposition 63 of [Frege18...
frege64a 43122 Lemma for ~ frege65a . Pr...
frege65a 43123 A kind of Aristotelian inf...
frege66a 43124 Swap antecedents of ~ freg...
frege67a 43125 Lemma for ~ frege68a . Pr...
frege68a 43126 Combination of applying a ...
axfrege52c 43127 Justification for ~ ax-fre...
frege52b 43129 The case when the content ...
frege53b 43130 Lemma for frege102 (via ~ ...
axfrege54c 43131 Reflexive equality of clas...
frege54b 43133 Reflexive equality of sets...
frege54cor1b 43134 Reflexive equality. (Cont...
frege55lem1b 43135 Necessary deduction regard...
frege55lem2b 43136 Lemma for ~ frege55b . Co...
frege55b 43137 Lemma for ~ frege57b . Pr...
frege56b 43138 Lemma for ~ frege57b . Pr...
frege57b 43139 Analogue of ~ frege57aid ....
axfrege58b 43140 If ` A. x ph ` is affirmed...
frege58bid 43142 If ` A. x ph ` is affirmed...
frege58bcor 43143 Lemma for ~ frege59b . (C...
frege59b 43144 A kind of Aristotelian inf...
frege60b 43145 Swap antecedents of ~ ax-f...
frege61b 43146 Lemma for ~ frege65b . Pr...
frege62b 43147 A kind of Aristotelian inf...
frege63b 43148 Lemma for ~ frege91 . Pro...
frege64b 43149 Lemma for ~ frege65b . Pr...
frege65b 43150 A kind of Aristotelian inf...
frege66b 43151 Swap antecedents of ~ freg...
frege67b 43152 Lemma for ~ frege68b . Pr...
frege68b 43153 Combination of applying a ...
frege53c 43154 Proposition 53 of [Frege18...
frege54cor1c 43155 Reflexive equality. (Cont...
frege55lem1c 43156 Necessary deduction regard...
frege55lem2c 43157 Core proof of Proposition ...
frege55c 43158 Proposition 55 of [Frege18...
frege56c 43159 Lemma for ~ frege57c . Pr...
frege57c 43160 Swap order of implication ...
frege58c 43161 Principle related to ~ sp ...
frege59c 43162 A kind of Aristotelian inf...
frege60c 43163 Swap antecedents of ~ freg...
frege61c 43164 Lemma for ~ frege65c . Pr...
frege62c 43165 A kind of Aristotelian inf...
frege63c 43166 Analogue of ~ frege63b . ...
frege64c 43167 Lemma for ~ frege65c . Pr...
frege65c 43168 A kind of Aristotelian inf...
frege66c 43169 Swap antecedents of ~ freg...
frege67c 43170 Lemma for ~ frege68c . Pr...
frege68c 43171 Combination of applying a ...
dffrege69 43172 If from the proposition th...
frege70 43173 Lemma for ~ frege72 . Pro...
frege71 43174 Lemma for ~ frege72 . Pro...
frege72 43175 If property ` A ` is hered...
frege73 43176 Lemma for ~ frege87 . Pro...
frege74 43177 If ` X ` has a property ` ...
frege75 43178 If from the proposition th...
dffrege76 43179 If from the two propositio...
frege77 43180 If ` Y ` follows ` X ` in ...
frege78 43181 Commuted form of ~ frege77...
frege79 43182 Distributed form of ~ freg...
frege80 43183 Add additional condition t...
frege81 43184 If ` X ` has a property ` ...
frege82 43185 Closed-form deduction base...
frege83 43186 Apply commuted form of ~ f...
frege84 43187 Commuted form of ~ frege81...
frege85 43188 Commuted form of ~ frege77...
frege86 43189 Conclusion about element o...
frege87 43190 If ` Z ` is a result of an...
frege88 43191 Commuted form of ~ frege87...
frege89 43192 One direction of ~ dffrege...
frege90 43193 Add antecedent to ~ frege8...
frege91 43194 Every result of an applica...
frege92 43195 Inference from ~ frege91 ....
frege93 43196 Necessary condition for tw...
frege94 43197 Looking one past a pair re...
frege95 43198 Looking one past a pair re...
frege96 43199 Every result of an applica...
frege97 43200 The property of following ...
frege98 43201 If ` Y ` follows ` X ` and...
dffrege99 43202 If ` Z ` is identical with...
frege100 43203 One direction of ~ dffrege...
frege101 43204 Lemma for ~ frege102 . Pr...
frege102 43205 If ` Z ` belongs to the ` ...
frege103 43206 Proposition 103 of [Frege1...
frege104 43207 Proposition 104 of [Frege1...
frege105 43208 Proposition 105 of [Frege1...
frege106 43209 Whatever follows ` X ` in ...
frege107 43210 Proposition 107 of [Frege1...
frege108 43211 If ` Y ` belongs to the ` ...
frege109 43212 The property of belonging ...
frege110 43213 Proposition 110 of [Frege1...
frege111 43214 If ` Y ` belongs to the ` ...
frege112 43215 Identity implies belonging...
frege113 43216 Proposition 113 of [Frege1...
frege114 43217 If ` X ` belongs to the ` ...
dffrege115 43218 If from the circumstance t...
frege116 43219 One direction of ~ dffrege...
frege117 43220 Lemma for ~ frege118 . Pr...
frege118 43221 Simplified application of ...
frege119 43222 Lemma for ~ frege120 . Pr...
frege120 43223 Simplified application of ...
frege121 43224 Lemma for ~ frege122 . Pr...
frege122 43225 If ` X ` is a result of an...
frege123 43226 Lemma for ~ frege124 . Pr...
frege124 43227 If ` X ` is a result of an...
frege125 43228 Lemma for ~ frege126 . Pr...
frege126 43229 If ` M ` follows ` Y ` in ...
frege127 43230 Communte antecedents of ~ ...
frege128 43231 Lemma for ~ frege129 . Pr...
frege129 43232 If the procedure ` R ` is ...
frege130 43233 Lemma for ~ frege131 . Pr...
frege131 43234 If the procedure ` R ` is ...
frege132 43235 Lemma for ~ frege133 . Pr...
frege133 43236 If the procedure ` R ` is ...
enrelmap 43237 The set of all possible re...
enrelmapr 43238 The set of all possible re...
enmappw 43239 The set of all mappings fr...
enmappwid 43240 The set of all mappings fr...
rfovd 43241 Value of the operator, ` (...
rfovfvd 43242 Value of the operator, ` (...
rfovfvfvd 43243 Value of the operator, ` (...
rfovcnvf1od 43244 Properties of the operator...
rfovcnvd 43245 Value of the converse of t...
rfovf1od 43246 The value of the operator,...
rfovcnvfvd 43247 Value of the converse of t...
fsovd 43248 Value of the operator, ` (...
fsovrfovd 43249 The operator which gives a...
fsovfvd 43250 Value of the operator, ` (...
fsovfvfvd 43251 Value of the operator, ` (...
fsovfd 43252 The operator, ` ( A O B ) ...
fsovcnvlem 43253 The ` O ` operator, which ...
fsovcnvd 43254 The value of the converse ...
fsovcnvfvd 43255 The value of the converse ...
fsovf1od 43256 The value of ` ( A O B ) `...
dssmapfvd 43257 Value of the duality opera...
dssmapfv2d 43258 Value of the duality opera...
dssmapfv3d 43259 Value of the duality opera...
dssmapnvod 43260 For any base set ` B ` the...
dssmapf1od 43261 For any base set ` B ` the...
dssmap2d 43262 For any base set ` B ` the...
or3or 43263 Decompose disjunction into...
andi3or 43264 Distribute over triple dis...
uneqsn 43265 If a union of classes is e...
brfvimex 43266 If a binary relation holds...
brovmptimex 43267 If a binary relation holds...
brovmptimex1 43268 If a binary relation holds...
brovmptimex2 43269 If a binary relation holds...
brcoffn 43270 Conditions allowing the de...
brcofffn 43271 Conditions allowing the de...
brco2f1o 43272 Conditions allowing the de...
brco3f1o 43273 Conditions allowing the de...
ntrclsbex 43274 If (pseudo-)interior and (...
ntrclsrcomplex 43275 The relative complement of...
neik0imk0p 43276 Kuratowski's K0 axiom impl...
ntrk2imkb 43277 If an interior function is...
ntrkbimka 43278 If the interiors of disjoi...
ntrk0kbimka 43279 If the interiors of disjoi...
clsk3nimkb 43280 If the base set is not emp...
clsk1indlem0 43281 The ansatz closure functio...
clsk1indlem2 43282 The ansatz closure functio...
clsk1indlem3 43283 The ansatz closure functio...
clsk1indlem4 43284 The ansatz closure functio...
clsk1indlem1 43285 The ansatz closure functio...
clsk1independent 43286 For generalized closure fu...
neik0pk1imk0 43287 Kuratowski's K0' and K1 ax...
isotone1 43288 Two different ways to say ...
isotone2 43289 Two different ways to say ...
ntrk1k3eqk13 43290 An interior function is bo...
ntrclsf1o 43291 If (pseudo-)interior and (...
ntrclsnvobr 43292 If (pseudo-)interior and (...
ntrclsiex 43293 If (pseudo-)interior and (...
ntrclskex 43294 If (pseudo-)interior and (...
ntrclsfv1 43295 If (pseudo-)interior and (...
ntrclsfv2 43296 If (pseudo-)interior and (...
ntrclselnel1 43297 If (pseudo-)interior and (...
ntrclselnel2 43298 If (pseudo-)interior and (...
ntrclsfv 43299 The value of the interior ...
ntrclsfveq1 43300 If interior and closure fu...
ntrclsfveq2 43301 If interior and closure fu...
ntrclsfveq 43302 If interior and closure fu...
ntrclsss 43303 If interior and closure fu...
ntrclsneine0lem 43304 If (pseudo-)interior and (...
ntrclsneine0 43305 If (pseudo-)interior and (...
ntrclscls00 43306 If (pseudo-)interior and (...
ntrclsiso 43307 If (pseudo-)interior and (...
ntrclsk2 43308 An interior function is co...
ntrclskb 43309 The interiors of disjoint ...
ntrclsk3 43310 The intersection of interi...
ntrclsk13 43311 The interior of the inters...
ntrclsk4 43312 Idempotence of the interio...
ntrneibex 43313 If (pseudo-)interior and (...
ntrneircomplex 43314 The relative complement of...
ntrneif1o 43315 If (pseudo-)interior and (...
ntrneiiex 43316 If (pseudo-)interior and (...
ntrneinex 43317 If (pseudo-)interior and (...
ntrneicnv 43318 If (pseudo-)interior and (...
ntrneifv1 43319 If (pseudo-)interior and (...
ntrneifv2 43320 If (pseudo-)interior and (...
ntrneiel 43321 If (pseudo-)interior and (...
ntrneifv3 43322 The value of the neighbors...
ntrneineine0lem 43323 If (pseudo-)interior and (...
ntrneineine1lem 43324 If (pseudo-)interior and (...
ntrneifv4 43325 The value of the interior ...
ntrneiel2 43326 Membership in iterated int...
ntrneineine0 43327 If (pseudo-)interior and (...
ntrneineine1 43328 If (pseudo-)interior and (...
ntrneicls00 43329 If (pseudo-)interior and (...
ntrneicls11 43330 If (pseudo-)interior and (...
ntrneiiso 43331 If (pseudo-)interior and (...
ntrneik2 43332 An interior function is co...
ntrneix2 43333 An interior (closure) func...
ntrneikb 43334 The interiors of disjoint ...
ntrneixb 43335 The interiors (closures) o...
ntrneik3 43336 The intersection of interi...
ntrneix3 43337 The closure of the union o...
ntrneik13 43338 The interior of the inters...
ntrneix13 43339 The closure of the union o...
ntrneik4w 43340 Idempotence of the interio...
ntrneik4 43341 Idempotence of the interio...
clsneibex 43342 If (pseudo-)closure and (p...
clsneircomplex 43343 The relative complement of...
clsneif1o 43344 If a (pseudo-)closure func...
clsneicnv 43345 If a (pseudo-)closure func...
clsneikex 43346 If closure and neighborhoo...
clsneinex 43347 If closure and neighborhoo...
clsneiel1 43348 If a (pseudo-)closure func...
clsneiel2 43349 If a (pseudo-)closure func...
clsneifv3 43350 Value of the neighborhoods...
clsneifv4 43351 Value of the closure (inte...
neicvgbex 43352 If (pseudo-)neighborhood a...
neicvgrcomplex 43353 The relative complement of...
neicvgf1o 43354 If neighborhood and conver...
neicvgnvo 43355 If neighborhood and conver...
neicvgnvor 43356 If neighborhood and conver...
neicvgmex 43357 If the neighborhoods and c...
neicvgnex 43358 If the neighborhoods and c...
neicvgel1 43359 A subset being an element ...
neicvgel2 43360 The complement of a subset...
neicvgfv 43361 The value of the neighborh...
ntrrn 43362 The range of the interior ...
ntrf 43363 The interior function of a...
ntrf2 43364 The interior function is a...
ntrelmap 43365 The interior function is a...
clsf2 43366 The closure function is a ...
clselmap 43367 The closure function is a ...
dssmapntrcls 43368 The interior and closure o...
dssmapclsntr 43369 The closure and interior o...
gneispa 43370 Each point ` p ` of the ne...
gneispb 43371 Given a neighborhood ` N `...
gneispace2 43372 The predicate that ` F ` i...
gneispace3 43373 The predicate that ` F ` i...
gneispace 43374 The predicate that ` F ` i...
gneispacef 43375 A generic neighborhood spa...
gneispacef2 43376 A generic neighborhood spa...
gneispacefun 43377 A generic neighborhood spa...
gneispacern 43378 A generic neighborhood spa...
gneispacern2 43379 A generic neighborhood spa...
gneispace0nelrn 43380 A generic neighborhood spa...
gneispace0nelrn2 43381 A generic neighborhood spa...
gneispace0nelrn3 43382 A generic neighborhood spa...
gneispaceel 43383 Every neighborhood of a po...
gneispaceel2 43384 Every neighborhood of a po...
gneispacess 43385 All supersets of a neighbo...
gneispacess2 43386 All supersets of a neighbo...
k0004lem1 43387 Application of ~ ssin to r...
k0004lem2 43388 A mapping with a particula...
k0004lem3 43389 When the value of a mappin...
k0004val 43390 The topological simplex of...
k0004ss1 43391 The topological simplex of...
k0004ss2 43392 The topological simplex of...
k0004ss3 43393 The topological simplex of...
k0004val0 43394 The topological simplex of...
inductionexd 43395 Simple induction example. ...
wwlemuld 43396 Natural deduction form of ...
leeq1d 43397 Specialization of ~ breq1d...
leeq2d 43398 Specialization of ~ breq2d...
absmulrposd 43399 Specialization of absmuld ...
imadisjld 43400 Natural dduction form of o...
imadisjlnd 43401 Natural deduction form of ...
wnefimgd 43402 The image of a mapping fro...
fco2d 43403 Natural deduction form of ...
wfximgfd 43404 The value of a function on...
extoimad 43405 If |f(x)| <= C for all x t...
imo72b2lem0 43406 Lemma for ~ imo72b2 . (Co...
suprleubrd 43407 Natural deduction form of ...
imo72b2lem2 43408 Lemma for ~ imo72b2 . (Co...
suprlubrd 43409 Natural deduction form of ...
imo72b2lem1 43410 Lemma for ~ imo72b2 . (Co...
lemuldiv3d 43411 'Less than or equal to' re...
lemuldiv4d 43412 'Less than or equal to' re...
imo72b2 43413 IMO 1972 B2. (14th Intern...
int-addcomd 43414 AdditionCommutativity gene...
int-addassocd 43415 AdditionAssociativity gene...
int-addsimpd 43416 AdditionSimplification gen...
int-mulcomd 43417 MultiplicationCommutativit...
int-mulassocd 43418 MultiplicationAssociativit...
int-mulsimpd 43419 MultiplicationSimplificati...
int-leftdistd 43420 AdditionMultiplicationLeft...
int-rightdistd 43421 AdditionMultiplicationRigh...
int-sqdefd 43422 SquareDefinition generator...
int-mul11d 43423 First MultiplicationOne ge...
int-mul12d 43424 Second MultiplicationOne g...
int-add01d 43425 First AdditionZero generat...
int-add02d 43426 Second AdditionZero genera...
int-sqgeq0d 43427 SquareGEQZero generator ru...
int-eqprincd 43428 PrincipleOfEquality genera...
int-eqtransd 43429 EqualityTransitivity gener...
int-eqmvtd 43430 EquMoveTerm generator rule...
int-eqineqd 43431 EquivalenceImpliesDoubleIn...
int-ineqmvtd 43432 IneqMoveTerm generator rul...
int-ineq1stprincd 43433 FirstPrincipleOfInequality...
int-ineq2ndprincd 43434 SecondPrincipleOfInequalit...
int-ineqtransd 43435 InequalityTransitivity gen...
unitadd 43436 Theorem used in conjunctio...
gsumws3 43437 Valuation of a length 3 wo...
gsumws4 43438 Valuation of a length 4 wo...
amgm2d 43439 Arithmetic-geometric mean ...
amgm3d 43440 Arithmetic-geometric mean ...
amgm4d 43441 Arithmetic-geometric mean ...
spALT 43442 ~ sp can be proven from th...
elnelneqd 43443 Two classes are not equal ...
elnelneq2d 43444 Two classes are not equal ...
rr-spce 43445 Prove an existential. (Co...
rexlimdvaacbv 43446 Unpack a restricted existe...
rexlimddvcbvw 43447 Unpack a restricted existe...
rexlimddvcbv 43448 Unpack a restricted existe...
rr-elrnmpt3d 43449 Elementhood in an image se...
finnzfsuppd 43450 If a function is zero outs...
rr-phpd 43451 Equivalent of ~ php withou...
suceqd 43452 Deduction associated with ...
tfindsd 43453 Deduction associated with ...
mnringvald 43456 Value of the monoid ring f...
mnringnmulrd 43457 Components of a monoid rin...
mnringnmulrdOLD 43458 Obsolete version of ~ mnri...
mnringbased 43459 The base set of a monoid r...
mnringbasedOLD 43460 Obsolete version of ~ mnri...
mnringbaserd 43461 The base set of a monoid r...
mnringelbased 43462 Membership in the base set...
mnringbasefd 43463 Elements of a monoid ring ...
mnringbasefsuppd 43464 Elements of a monoid ring ...
mnringaddgd 43465 The additive operation of ...
mnringaddgdOLD 43466 Obsolete version of ~ mnri...
mnring0gd 43467 The additive identity of a...
mnring0g2d 43468 The additive identity of a...
mnringmulrd 43469 The ring product of a mono...
mnringscad 43470 The scalar ring of a monoi...
mnringscadOLD 43471 Obsolete version of ~ mnri...
mnringvscad 43472 The scalar product of a mo...
mnringvscadOLD 43473 Obsolete version of ~ mnri...
mnringlmodd 43474 Monoid rings are left modu...
mnringmulrvald 43475 Value of multiplication in...
mnringmulrcld 43476 Monoid rings are closed un...
gru0eld 43477 A nonempty Grothendieck un...
grusucd 43478 Grothendieck universes are...
r1rankcld 43479 Any rank of the cumulative...
grur1cld 43480 Grothendieck universes are...
grurankcld 43481 Grothendieck universes are...
grurankrcld 43482 If a Grothendieck universe...
scotteqd 43485 Equality theorem for the S...
scotteq 43486 Closed form of ~ scotteqd ...
nfscott 43487 Bound-variable hypothesis ...
scottabf 43488 Value of the Scott operati...
scottab 43489 Value of the Scott operati...
scottabes 43490 Value of the Scott operati...
scottss 43491 Scott's trick produces a s...
elscottab 43492 An element of the output o...
scottex2 43493 ~ scottex expressed using ...
scotteld 43494 The Scott operation sends ...
scottelrankd 43495 Property of a Scott's tric...
scottrankd 43496 Rank of a nonempty Scott's...
gruscottcld 43497 If a Grothendieck universe...
dfcoll2 43500 Alternate definition of th...
colleq12d 43501 Equality theorem for the c...
colleq1 43502 Equality theorem for the c...
colleq2 43503 Equality theorem for the c...
nfcoll 43504 Bound-variable hypothesis ...
collexd 43505 The output of the collecti...
cpcolld 43506 Property of the collection...
cpcoll2d 43507 ~ cpcolld with an extra ex...
grucollcld 43508 A Grothendieck universe co...
ismnu 43509 The hypothesis of this the...
mnuop123d 43510 Operations of a minimal un...
mnussd 43511 Minimal universes are clos...
mnuss2d 43512 ~ mnussd with arguments pr...
mnu0eld 43513 A nonempty minimal univers...
mnuop23d 43514 Second and third operation...
mnupwd 43515 Minimal universes are clos...
mnusnd 43516 Minimal universes are clos...
mnuprssd 43517 A minimal universe contain...
mnuprss2d 43518 Special case of ~ mnuprssd...
mnuop3d 43519 Third operation of a minim...
mnuprdlem1 43520 Lemma for ~ mnuprd . (Con...
mnuprdlem2 43521 Lemma for ~ mnuprd . (Con...
mnuprdlem3 43522 Lemma for ~ mnuprd . (Con...
mnuprdlem4 43523 Lemma for ~ mnuprd . Gene...
mnuprd 43524 Minimal universes are clos...
mnuunid 43525 Minimal universes are clos...
mnuund 43526 Minimal universes are clos...
mnutrcld 43527 Minimal universes contain ...
mnutrd 43528 Minimal universes are tran...
mnurndlem1 43529 Lemma for ~ mnurnd . (Con...
mnurndlem2 43530 Lemma for ~ mnurnd . Dedu...
mnurnd 43531 Minimal universes contain ...
mnugrud 43532 Minimal universes are Grot...
grumnudlem 43533 Lemma for ~ grumnud . (Co...
grumnud 43534 Grothendieck universes are...
grumnueq 43535 The class of Grothendieck ...
expandan 43536 Expand conjunction to prim...
expandexn 43537 Expand an existential quan...
expandral 43538 Expand a restricted univer...
expandrexn 43539 Expand a restricted existe...
expandrex 43540 Expand a restricted existe...
expanduniss 43541 Expand ` U. A C_ B ` to pr...
ismnuprim 43542 Express the predicate on `...
rr-grothprimbi 43543 Express "every set is cont...
inagrud 43544 Inaccessible levels of the...
inaex 43545 Assuming the Tarski-Grothe...
gruex 43546 Assuming the Tarski-Grothe...
rr-groth 43547 An equivalent of ~ ax-grot...
rr-grothprim 43548 An equivalent of ~ ax-grot...
ismnushort 43549 Express the predicate on `...
dfuniv2 43550 Alternative definition of ...
rr-grothshortbi 43551 Express "every set is cont...
rr-grothshort 43552 A shorter equivalent of ~ ...
nanorxor 43553 'nand' is equivalent to th...
undisjrab 43554 Union of two disjoint rest...
iso0 43555 The empty set is an ` R , ...
ssrecnpr 43556 ` RR ` is a subset of both...
seff 43557 Let set ` S ` be the real ...
sblpnf 43558 The infinity ball in the a...
prmunb2 43559 The primes are unbounded. ...
dvgrat 43560 Ratio test for divergence ...
cvgdvgrat 43561 Ratio test for convergence...
radcnvrat 43562 Let ` L ` be the limit, if...
reldvds 43563 The divides relation is in...
nznngen 43564 All positive integers in t...
nzss 43565 The set of multiples of _m...
nzin 43566 The intersection of the se...
nzprmdif 43567 Subtract one prime's multi...
hashnzfz 43568 Special case of ~ hashdvds...
hashnzfz2 43569 Special case of ~ hashnzfz...
hashnzfzclim 43570 As the upper bound ` K ` o...
caofcan 43571 Transfer a cancellation la...
ofsubid 43572 Function analogue of ~ sub...
ofmul12 43573 Function analogue of ~ mul...
ofdivrec 43574 Function analogue of ~ div...
ofdivcan4 43575 Function analogue of ~ div...
ofdivdiv2 43576 Function analogue of ~ div...
lhe4.4ex1a 43577 Example of the Fundamental...
dvsconst 43578 Derivative of a constant f...
dvsid 43579 Derivative of the identity...
dvsef 43580 Derivative of the exponent...
expgrowthi 43581 Exponential growth and dec...
dvconstbi 43582 The derivative of a functi...
expgrowth 43583 Exponential growth and dec...
bccval 43586 Value of the generalized b...
bcccl 43587 Closure of the generalized...
bcc0 43588 The generalized binomial c...
bccp1k 43589 Generalized binomial coeff...
bccm1k 43590 Generalized binomial coeff...
bccn0 43591 Generalized binomial coeff...
bccn1 43592 Generalized binomial coeff...
bccbc 43593 The binomial coefficient a...
uzmptshftfval 43594 When ` F ` is a maps-to fu...
dvradcnv2 43595 The radius of convergence ...
binomcxplemwb 43596 Lemma for ~ binomcxp . Th...
binomcxplemnn0 43597 Lemma for ~ binomcxp . Wh...
binomcxplemrat 43598 Lemma for ~ binomcxp . As...
binomcxplemfrat 43599 Lemma for ~ binomcxp . ~ b...
binomcxplemradcnv 43600 Lemma for ~ binomcxp . By...
binomcxplemdvbinom 43601 Lemma for ~ binomcxp . By...
binomcxplemcvg 43602 Lemma for ~ binomcxp . Th...
binomcxplemdvsum 43603 Lemma for ~ binomcxp . Th...
binomcxplemnotnn0 43604 Lemma for ~ binomcxp . Wh...
binomcxp 43605 Generalize the binomial th...
pm10.12 43606 Theorem *10.12 in [Whitehe...
pm10.14 43607 Theorem *10.14 in [Whitehe...
pm10.251 43608 Theorem *10.251 in [Whiteh...
pm10.252 43609 Theorem *10.252 in [Whiteh...
pm10.253 43610 Theorem *10.253 in [Whiteh...
albitr 43611 Theorem *10.301 in [Whiteh...
pm10.42 43612 Theorem *10.42 in [Whitehe...
pm10.52 43613 Theorem *10.52 in [Whitehe...
pm10.53 43614 Theorem *10.53 in [Whitehe...
pm10.541 43615 Theorem *10.541 in [Whiteh...
pm10.542 43616 Theorem *10.542 in [Whiteh...
pm10.55 43617 Theorem *10.55 in [Whitehe...
pm10.56 43618 Theorem *10.56 in [Whitehe...
pm10.57 43619 Theorem *10.57 in [Whitehe...
2alanimi 43620 Removes two universal quan...
2al2imi 43621 Removes two universal quan...
pm11.11 43622 Theorem *11.11 in [Whitehe...
pm11.12 43623 Theorem *11.12 in [Whitehe...
19.21vv 43624 Compare Theorem *11.3 in [...
2alim 43625 Theorem *11.32 in [Whitehe...
2albi 43626 Theorem *11.33 in [Whitehe...
2exim 43627 Theorem *11.34 in [Whitehe...
2exbi 43628 Theorem *11.341 in [Whiteh...
spsbce-2 43629 Theorem *11.36 in [Whitehe...
19.33-2 43630 Theorem *11.421 in [Whiteh...
19.36vv 43631 Theorem *11.43 in [Whitehe...
19.31vv 43632 Theorem *11.44 in [Whitehe...
19.37vv 43633 Theorem *11.46 in [Whitehe...
19.28vv 43634 Theorem *11.47 in [Whitehe...
pm11.52 43635 Theorem *11.52 in [Whitehe...
aaanv 43636 Theorem *11.56 in [Whitehe...
pm11.57 43637 Theorem *11.57 in [Whitehe...
pm11.58 43638 Theorem *11.58 in [Whitehe...
pm11.59 43639 Theorem *11.59 in [Whitehe...
pm11.6 43640 Theorem *11.6 in [Whitehea...
pm11.61 43641 Theorem *11.61 in [Whitehe...
pm11.62 43642 Theorem *11.62 in [Whitehe...
pm11.63 43643 Theorem *11.63 in [Whitehe...
pm11.7 43644 Theorem *11.7 in [Whitehea...
pm11.71 43645 Theorem *11.71 in [Whitehe...
sbeqal1 43646 If ` x = y ` always implie...
sbeqal1i 43647 Suppose you know ` x = y `...
sbeqal2i 43648 If ` x = y ` implies ` x =...
axc5c4c711 43649 Proof of a theorem that ca...
axc5c4c711toc5 43650 Rederivation of ~ sp from ...
axc5c4c711toc4 43651 Rederivation of ~ axc4 fro...
axc5c4c711toc7 43652 Rederivation of ~ axc7 fro...
axc5c4c711to11 43653 Rederivation of ~ ax-11 fr...
axc11next 43654 This theorem shows that, g...
pm13.13a 43655 One result of theorem *13....
pm13.13b 43656 Theorem *13.13 in [Whitehe...
pm13.14 43657 Theorem *13.14 in [Whitehe...
pm13.192 43658 Theorem *13.192 in [Whiteh...
pm13.193 43659 Theorem *13.193 in [Whiteh...
pm13.194 43660 Theorem *13.194 in [Whiteh...
pm13.195 43661 Theorem *13.195 in [Whiteh...
pm13.196a 43662 Theorem *13.196 in [Whiteh...
2sbc6g 43663 Theorem *13.21 in [Whitehe...
2sbc5g 43664 Theorem *13.22 in [Whitehe...
iotain 43665 Equivalence between two di...
iotaexeu 43666 The iota class exists. Th...
iotasbc 43667 Definition *14.01 in [Whit...
iotasbc2 43668 Theorem *14.111 in [Whiteh...
pm14.12 43669 Theorem *14.12 in [Whitehe...
pm14.122a 43670 Theorem *14.122 in [Whiteh...
pm14.122b 43671 Theorem *14.122 in [Whiteh...
pm14.122c 43672 Theorem *14.122 in [Whiteh...
pm14.123a 43673 Theorem *14.123 in [Whiteh...
pm14.123b 43674 Theorem *14.123 in [Whiteh...
pm14.123c 43675 Theorem *14.123 in [Whiteh...
pm14.18 43676 Theorem *14.18 in [Whitehe...
iotaequ 43677 Theorem *14.2 in [Whitehea...
iotavalb 43678 Theorem *14.202 in [Whiteh...
iotasbc5 43679 Theorem *14.205 in [Whiteh...
pm14.24 43680 Theorem *14.24 in [Whitehe...
iotavalsb 43681 Theorem *14.242 in [Whiteh...
sbiota1 43682 Theorem *14.25 in [Whitehe...
sbaniota 43683 Theorem *14.26 in [Whitehe...
eubiOLD 43684 Obsolete proof of ~ eubi a...
iotasbcq 43685 Theorem *14.272 in [Whiteh...
elnev 43686 Any set that contains one ...
rusbcALT 43687 A version of Russell's par...
compeq 43688 Equality between two ways ...
compne 43689 The complement of ` A ` is...
compab 43690 Two ways of saying "the co...
conss2 43691 Contrapositive law for sub...
conss1 43692 Contrapositive law for sub...
ralbidar 43693 More general form of ~ ral...
rexbidar 43694 More general form of ~ rex...
dropab1 43695 Theorem to aid use of the ...
dropab2 43696 Theorem to aid use of the ...
ipo0 43697 If the identity relation p...
ifr0 43698 A class that is founded by...
ordpss 43699 ~ ordelpss with an anteced...
fvsb 43700 Explicit substitution of a...
fveqsb 43701 Implicit substitution of a...
xpexb 43702 A Cartesian product exists...
trelpss 43703 An element of a transitive...
addcomgi 43704 Generalization of commutat...
addrval 43714 Value of the operation of ...
subrval 43715 Value of the operation of ...
mulvval 43716 Value of the operation of ...
addrfv 43717 Vector addition at a value...
subrfv 43718 Vector subtraction at a va...
mulvfv 43719 Scalar multiplication at a...
addrfn 43720 Vector addition produces a...
subrfn 43721 Vector subtraction produce...
mulvfn 43722 Scalar multiplication prod...
addrcom 43723 Vector addition is commuta...
idiALT 43727 Placeholder for ~ idi . T...
exbir 43728 Exportation implication al...
3impexpbicom 43729 Version of ~ 3impexp where...
3impexpbicomi 43730 Inference associated with ...
bi1imp 43731 Importation inference simi...
bi2imp 43732 Importation inference simi...
bi3impb 43733 Similar to ~ 3impb with im...
bi3impa 43734 Similar to ~ 3impa with im...
bi23impib 43735 ~ 3impib with the inner im...
bi13impib 43736 ~ 3impib with the outer im...
bi123impib 43737 ~ 3impib with the implicat...
bi13impia 43738 ~ 3impia with the outer im...
bi123impia 43739 ~ 3impia with the implicat...
bi33imp12 43740 ~ 3imp with innermost impl...
bi23imp13 43741 ~ 3imp with middle implica...
bi13imp23 43742 ~ 3imp with outermost impl...
bi13imp2 43743 Similar to ~ 3imp except t...
bi12imp3 43744 Similar to ~ 3imp except a...
bi23imp1 43745 Similar to ~ 3imp except a...
bi123imp0 43746 Similar to ~ 3imp except a...
4animp1 43747 A single hypothesis unific...
4an31 43748 A rearrangement of conjunc...
4an4132 43749 A rearrangement of conjunc...
expcomdg 43750 Biconditional form of ~ ex...
iidn3 43751 ~ idn3 without virtual ded...
ee222 43752 ~ e222 without virtual ded...
ee3bir 43753 Right-biconditional form o...
ee13 43754 ~ e13 without virtual dedu...
ee121 43755 ~ e121 without virtual ded...
ee122 43756 ~ e122 without virtual ded...
ee333 43757 ~ e333 without virtual ded...
ee323 43758 ~ e323 without virtual ded...
3ornot23 43759 If the second and third di...
orbi1r 43760 ~ orbi1 with order of disj...
3orbi123 43761 ~ pm4.39 with a 3-conjunct...
syl5imp 43762 Closed form of ~ syl5 . D...
impexpd 43763 The following User's Proof...
com3rgbi 43764 The following User's Proof...
impexpdcom 43765 The following User's Proof...
ee1111 43766 Non-virtual deduction form...
pm2.43bgbi 43767 Logical equivalence of a 2...
pm2.43cbi 43768 Logical equivalence of a 3...
ee233 43769 Non-virtual deduction form...
imbi13 43770 Join three logical equival...
ee33 43771 Non-virtual deduction form...
con5 43772 Biconditional contrapositi...
con5i 43773 Inference form of ~ con5 ....
exlimexi 43774 Inference similar to Theor...
sb5ALT 43775 Equivalence for substituti...
eexinst01 43776 ~ exinst01 without virtual...
eexinst11 43777 ~ exinst11 without virtual...
vk15.4j 43778 Excercise 4j of Unit 15 of...
notnotrALT 43779 Converse of double negatio...
con3ALT2 43780 Contraposition. Alternate...
ssralv2 43781 Quantification restricted ...
sbc3or 43782 ~ sbcor with a 3-disjuncts...
alrim3con13v 43783 Closed form of ~ alrimi wi...
rspsbc2 43784 ~ rspsbc with two quantify...
sbcoreleleq 43785 Substitution of a setvar v...
tratrb 43786 If a class is transitive a...
ordelordALT 43787 An element of an ordinal c...
sbcim2g 43788 Distribution of class subs...
sbcbi 43789 Implication form of ~ sbcb...
trsbc 43790 Formula-building inference...
truniALT 43791 The union of a class of tr...
onfrALTlem5 43792 Lemma for ~ onfrALT . (Co...
onfrALTlem4 43793 Lemma for ~ onfrALT . (Co...
onfrALTlem3 43794 Lemma for ~ onfrALT . (Co...
ggen31 43795 ~ gen31 without virtual de...
onfrALTlem2 43796 Lemma for ~ onfrALT . (Co...
cbvexsv 43797 A theorem pertaining to th...
onfrALTlem1 43798 Lemma for ~ onfrALT . (Co...
onfrALT 43799 The membership relation is...
19.41rg 43800 Closed form of right-to-le...
opelopab4 43801 Ordered pair membership in...
2pm13.193 43802 ~ pm13.193 for two variabl...
hbntal 43803 A closed form of ~ hbn . ~...
hbimpg 43804 A closed form of ~ hbim . ...
hbalg 43805 Closed form of ~ hbal . D...
hbexg 43806 Closed form of ~ nfex . D...
ax6e2eq 43807 Alternate form of ~ ax6e f...
ax6e2nd 43808 If at least two sets exist...
ax6e2ndeq 43809 "At least two sets exist" ...
2sb5nd 43810 Equivalence for double sub...
2uasbanh 43811 Distribute the unabbreviat...
2uasban 43812 Distribute the unabbreviat...
e2ebind 43813 Absorption of an existenti...
elpwgded 43814 ~ elpwgdedVD in convention...
trelded 43815 Deduction form of ~ trel ....
jaoded 43816 Deduction form of ~ jao . ...
sbtT 43817 A substitution into a theo...
not12an2impnot1 43818 If a double conjunction is...
in1 43821 Inference form of ~ df-vd1...
iin1 43822 ~ in1 without virtual dedu...
dfvd1ir 43823 Inference form of ~ df-vd1...
idn1 43824 Virtual deduction identity...
dfvd1imp 43825 Left-to-right part of defi...
dfvd1impr 43826 Right-to-left part of defi...
dfvd2 43829 Definition of a 2-hypothes...
dfvd2an 43832 Definition of a 2-hypothes...
dfvd2ani 43833 Inference form of ~ dfvd2a...
dfvd2anir 43834 Right-to-left inference fo...
dfvd2i 43835 Inference form of ~ dfvd2 ...
dfvd2ir 43836 Right-to-left inference fo...
dfvd3 43841 Definition of a 3-hypothes...
dfvd3i 43842 Inference form of ~ dfvd3 ...
dfvd3ir 43843 Right-to-left inference fo...
dfvd3an 43844 Definition of a 3-hypothes...
dfvd3ani 43845 Inference form of ~ dfvd3a...
dfvd3anir 43846 Right-to-left inference fo...
vd01 43847 A virtual hypothesis virtu...
vd02 43848 Two virtual hypotheses vir...
vd03 43849 A theorem is virtually inf...
vd12 43850 A virtual deduction with 1...
vd13 43851 A virtual deduction with 1...
vd23 43852 A virtual deduction with 2...
dfvd2imp 43853 The virtual deduction form...
dfvd2impr 43854 A 2-antecedent nested impl...
in2 43855 The virtual deduction intr...
int2 43856 The virtual deduction intr...
iin2 43857 ~ in2 without virtual dedu...
in2an 43858 The virtual deduction intr...
in3 43859 The virtual deduction intr...
iin3 43860 ~ in3 without virtual dedu...
in3an 43861 The virtual deduction intr...
int3 43862 The virtual deduction intr...
idn2 43863 Virtual deduction identity...
iden2 43864 Virtual deduction identity...
idn3 43865 Virtual deduction identity...
gen11 43866 Virtual deduction generali...
gen11nv 43867 Virtual deduction generali...
gen12 43868 Virtual deduction generali...
gen21 43869 Virtual deduction generali...
gen21nv 43870 Virtual deduction form of ...
gen31 43871 Virtual deduction generali...
gen22 43872 Virtual deduction generali...
ggen22 43873 ~ gen22 without virtual de...
exinst 43874 Existential Instantiation....
exinst01 43875 Existential Instantiation....
exinst11 43876 Existential Instantiation....
e1a 43877 A Virtual deduction elimin...
el1 43878 A Virtual deduction elimin...
e1bi 43879 Biconditional form of ~ e1...
e1bir 43880 Right biconditional form o...
e2 43881 A virtual deduction elimin...
e2bi 43882 Biconditional form of ~ e2...
e2bir 43883 Right biconditional form o...
ee223 43884 ~ e223 without virtual ded...
e223 43885 A virtual deduction elimin...
e222 43886 A virtual deduction elimin...
e220 43887 A virtual deduction elimin...
ee220 43888 ~ e220 without virtual ded...
e202 43889 A virtual deduction elimin...
ee202 43890 ~ e202 without virtual ded...
e022 43891 A virtual deduction elimin...
ee022 43892 ~ e022 without virtual ded...
e002 43893 A virtual deduction elimin...
ee002 43894 ~ e002 without virtual ded...
e020 43895 A virtual deduction elimin...
ee020 43896 ~ e020 without virtual ded...
e200 43897 A virtual deduction elimin...
ee200 43898 ~ e200 without virtual ded...
e221 43899 A virtual deduction elimin...
ee221 43900 ~ e221 without virtual ded...
e212 43901 A virtual deduction elimin...
ee212 43902 ~ e212 without virtual ded...
e122 43903 A virtual deduction elimin...
e112 43904 A virtual deduction elimin...
ee112 43905 ~ e112 without virtual ded...
e121 43906 A virtual deduction elimin...
e211 43907 A virtual deduction elimin...
ee211 43908 ~ e211 without virtual ded...
e210 43909 A virtual deduction elimin...
ee210 43910 ~ e210 without virtual ded...
e201 43911 A virtual deduction elimin...
ee201 43912 ~ e201 without virtual ded...
e120 43913 A virtual deduction elimin...
ee120 43914 Virtual deduction rule ~ e...
e021 43915 A virtual deduction elimin...
ee021 43916 ~ e021 without virtual ded...
e012 43917 A virtual deduction elimin...
ee012 43918 ~ e012 without virtual ded...
e102 43919 A virtual deduction elimin...
ee102 43920 ~ e102 without virtual ded...
e22 43921 A virtual deduction elimin...
e22an 43922 Conjunction form of ~ e22 ...
ee22an 43923 ~ e22an without virtual de...
e111 43924 A virtual deduction elimin...
e1111 43925 A virtual deduction elimin...
e110 43926 A virtual deduction elimin...
ee110 43927 ~ e110 without virtual ded...
e101 43928 A virtual deduction elimin...
ee101 43929 ~ e101 without virtual ded...
e011 43930 A virtual deduction elimin...
ee011 43931 ~ e011 without virtual ded...
e100 43932 A virtual deduction elimin...
ee100 43933 ~ e100 without virtual ded...
e010 43934 A virtual deduction elimin...
ee010 43935 ~ e010 without virtual ded...
e001 43936 A virtual deduction elimin...
ee001 43937 ~ e001 without virtual ded...
e11 43938 A virtual deduction elimin...
e11an 43939 Conjunction form of ~ e11 ...
ee11an 43940 ~ e11an without virtual de...
e01 43941 A virtual deduction elimin...
e01an 43942 Conjunction form of ~ e01 ...
ee01an 43943 ~ e01an without virtual de...
e10 43944 A virtual deduction elimin...
e10an 43945 Conjunction form of ~ e10 ...
ee10an 43946 ~ e10an without virtual de...
e02 43947 A virtual deduction elimin...
e02an 43948 Conjunction form of ~ e02 ...
ee02an 43949 ~ e02an without virtual de...
eel021old 43950 ~ el021old without virtual...
el021old 43951 A virtual deduction elimin...
eel132 43952 ~ syl2an with antecedents ...
eel000cT 43953 An elimination deduction. ...
eel0TT 43954 An elimination deduction. ...
eelT00 43955 An elimination deduction. ...
eelTTT 43956 An elimination deduction. ...
eelT11 43957 An elimination deduction. ...
eelT1 43958 Syllogism inference combin...
eelT12 43959 An elimination deduction. ...
eelTT1 43960 An elimination deduction. ...
eelT01 43961 An elimination deduction. ...
eel0T1 43962 An elimination deduction. ...
eel12131 43963 An elimination deduction. ...
eel2131 43964 ~ syl2an with antecedents ...
eel3132 43965 ~ syl2an with antecedents ...
eel0321old 43966 ~ el0321old without virtua...
el0321old 43967 A virtual deduction elimin...
eel2122old 43968 ~ el2122old without virtua...
el2122old 43969 A virtual deduction elimin...
eel0000 43970 Elimination rule similar t...
eel00001 43971 An elimination deduction. ...
eel00000 43972 Elimination rule similar ~...
eel11111 43973 Five-hypothesis eliminatio...
e12 43974 A virtual deduction elimin...
e12an 43975 Conjunction form of ~ e12 ...
el12 43976 Virtual deduction form of ...
e20 43977 A virtual deduction elimin...
e20an 43978 Conjunction form of ~ e20 ...
ee20an 43979 ~ e20an without virtual de...
e21 43980 A virtual deduction elimin...
e21an 43981 Conjunction form of ~ e21 ...
ee21an 43982 ~ e21an without virtual de...
e333 43983 A virtual deduction elimin...
e33 43984 A virtual deduction elimin...
e33an 43985 Conjunction form of ~ e33 ...
ee33an 43986 ~ e33an without virtual de...
e3 43987 Meta-connective form of ~ ...
e3bi 43988 Biconditional form of ~ e3...
e3bir 43989 Right biconditional form o...
e03 43990 A virtual deduction elimin...
ee03 43991 ~ e03 without virtual dedu...
e03an 43992 Conjunction form of ~ e03 ...
ee03an 43993 Conjunction form of ~ ee03...
e30 43994 A virtual deduction elimin...
ee30 43995 ~ e30 without virtual dedu...
e30an 43996 A virtual deduction elimin...
ee30an 43997 Conjunction form of ~ ee30...
e13 43998 A virtual deduction elimin...
e13an 43999 A virtual deduction elimin...
ee13an 44000 ~ e13an without virtual de...
e31 44001 A virtual deduction elimin...
ee31 44002 ~ e31 without virtual dedu...
e31an 44003 A virtual deduction elimin...
ee31an 44004 ~ e31an without virtual de...
e23 44005 A virtual deduction elimin...
e23an 44006 A virtual deduction elimin...
ee23an 44007 ~ e23an without virtual de...
e32 44008 A virtual deduction elimin...
ee32 44009 ~ e32 without virtual dedu...
e32an 44010 A virtual deduction elimin...
ee32an 44011 ~ e33an without virtual de...
e123 44012 A virtual deduction elimin...
ee123 44013 ~ e123 without virtual ded...
el123 44014 A virtual deduction elimin...
e233 44015 A virtual deduction elimin...
e323 44016 A virtual deduction elimin...
e000 44017 A virtual deduction elimin...
e00 44018 Elimination rule identical...
e00an 44019 Elimination rule identical...
eel00cT 44020 An elimination deduction. ...
eelTT 44021 An elimination deduction. ...
e0a 44022 Elimination rule identical...
eelT 44023 An elimination deduction. ...
eel0cT 44024 An elimination deduction. ...
eelT0 44025 An elimination deduction. ...
e0bi 44026 Elimination rule identical...
e0bir 44027 Elimination rule identical...
uun0.1 44028 Convention notation form o...
un0.1 44029 ` T. ` is the constant tru...
uunT1 44030 A deduction unionizing a n...
uunT1p1 44031 A deduction unionizing a n...
uunT21 44032 A deduction unionizing a n...
uun121 44033 A deduction unionizing a n...
uun121p1 44034 A deduction unionizing a n...
uun132 44035 A deduction unionizing a n...
uun132p1 44036 A deduction unionizing a n...
anabss7p1 44037 A deduction unionizing a n...
un10 44038 A unionizing deduction. (...
un01 44039 A unionizing deduction. (...
un2122 44040 A deduction unionizing a n...
uun2131 44041 A deduction unionizing a n...
uun2131p1 44042 A deduction unionizing a n...
uunTT1 44043 A deduction unionizing a n...
uunTT1p1 44044 A deduction unionizing a n...
uunTT1p2 44045 A deduction unionizing a n...
uunT11 44046 A deduction unionizing a n...
uunT11p1 44047 A deduction unionizing a n...
uunT11p2 44048 A deduction unionizing a n...
uunT12 44049 A deduction unionizing a n...
uunT12p1 44050 A deduction unionizing a n...
uunT12p2 44051 A deduction unionizing a n...
uunT12p3 44052 A deduction unionizing a n...
uunT12p4 44053 A deduction unionizing a n...
uunT12p5 44054 A deduction unionizing a n...
uun111 44055 A deduction unionizing a n...
3anidm12p1 44056 A deduction unionizing a n...
3anidm12p2 44057 A deduction unionizing a n...
uun123 44058 A deduction unionizing a n...
uun123p1 44059 A deduction unionizing a n...
uun123p2 44060 A deduction unionizing a n...
uun123p3 44061 A deduction unionizing a n...
uun123p4 44062 A deduction unionizing a n...
uun2221 44063 A deduction unionizing a n...
uun2221p1 44064 A deduction unionizing a n...
uun2221p2 44065 A deduction unionizing a n...
3impdirp1 44066 A deduction unionizing a n...
3impcombi 44067 A 1-hypothesis proposition...
trsspwALT 44068 Virtual deduction proof of...
trsspwALT2 44069 Virtual deduction proof of...
trsspwALT3 44070 Short predicate calculus p...
sspwtr 44071 Virtual deduction proof of...
sspwtrALT 44072 Virtual deduction proof of...
sspwtrALT2 44073 Short predicate calculus p...
pwtrVD 44074 Virtual deduction proof of...
pwtrrVD 44075 Virtual deduction proof of...
suctrALT 44076 The successor of a transit...
snssiALTVD 44077 Virtual deduction proof of...
snssiALT 44078 If a class is an element o...
snsslVD 44079 Virtual deduction proof of...
snssl 44080 If a singleton is a subcla...
snelpwrVD 44081 Virtual deduction proof of...
unipwrVD 44082 Virtual deduction proof of...
unipwr 44083 A class is a subclass of t...
sstrALT2VD 44084 Virtual deduction proof of...
sstrALT2 44085 Virtual deduction proof of...
suctrALT2VD 44086 Virtual deduction proof of...
suctrALT2 44087 Virtual deduction proof of...
elex2VD 44088 Virtual deduction proof of...
elex22VD 44089 Virtual deduction proof of...
eqsbc2VD 44090 Virtual deduction proof of...
zfregs2VD 44091 Virtual deduction proof of...
tpid3gVD 44092 Virtual deduction proof of...
en3lplem1VD 44093 Virtual deduction proof of...
en3lplem2VD 44094 Virtual deduction proof of...
en3lpVD 44095 Virtual deduction proof of...
simplbi2VD 44096 Virtual deduction proof of...
3ornot23VD 44097 Virtual deduction proof of...
orbi1rVD 44098 Virtual deduction proof of...
bitr3VD 44099 Virtual deduction proof of...
3orbi123VD 44100 Virtual deduction proof of...
sbc3orgVD 44101 Virtual deduction proof of...
19.21a3con13vVD 44102 Virtual deduction proof of...
exbirVD 44103 Virtual deduction proof of...
exbiriVD 44104 Virtual deduction proof of...
rspsbc2VD 44105 Virtual deduction proof of...
3impexpVD 44106 Virtual deduction proof of...
3impexpbicomVD 44107 Virtual deduction proof of...
3impexpbicomiVD 44108 Virtual deduction proof of...
sbcoreleleqVD 44109 Virtual deduction proof of...
hbra2VD 44110 Virtual deduction proof of...
tratrbVD 44111 Virtual deduction proof of...
al2imVD 44112 Virtual deduction proof of...
syl5impVD 44113 Virtual deduction proof of...
idiVD 44114 Virtual deduction proof of...
ancomstVD 44115 Closed form of ~ ancoms . ...
ssralv2VD 44116 Quantification restricted ...
ordelordALTVD 44117 An element of an ordinal c...
equncomVD 44118 If a class equals the unio...
equncomiVD 44119 Inference form of ~ equnco...
sucidALTVD 44120 A set belongs to its succe...
sucidALT 44121 A set belongs to its succe...
sucidVD 44122 A set belongs to its succe...
imbi12VD 44123 Implication form of ~ imbi...
imbi13VD 44124 Join three logical equival...
sbcim2gVD 44125 Distribution of class subs...
sbcbiVD 44126 Implication form of ~ sbcb...
trsbcVD 44127 Formula-building inference...
truniALTVD 44128 The union of a class of tr...
ee33VD 44129 Non-virtual deduction form...
trintALTVD 44130 The intersection of a clas...
trintALT 44131 The intersection of a clas...
undif3VD 44132 The first equality of Exer...
sbcssgVD 44133 Virtual deduction proof of...
csbingVD 44134 Virtual deduction proof of...
onfrALTlem5VD 44135 Virtual deduction proof of...
onfrALTlem4VD 44136 Virtual deduction proof of...
onfrALTlem3VD 44137 Virtual deduction proof of...
simplbi2comtVD 44138 Virtual deduction proof of...
onfrALTlem2VD 44139 Virtual deduction proof of...
onfrALTlem1VD 44140 Virtual deduction proof of...
onfrALTVD 44141 Virtual deduction proof of...
csbeq2gVD 44142 Virtual deduction proof of...
csbsngVD 44143 Virtual deduction proof of...
csbxpgVD 44144 Virtual deduction proof of...
csbresgVD 44145 Virtual deduction proof of...
csbrngVD 44146 Virtual deduction proof of...
csbima12gALTVD 44147 Virtual deduction proof of...
csbunigVD 44148 Virtual deduction proof of...
csbfv12gALTVD 44149 Virtual deduction proof of...
con5VD 44150 Virtual deduction proof of...
relopabVD 44151 Virtual deduction proof of...
19.41rgVD 44152 Virtual deduction proof of...
2pm13.193VD 44153 Virtual deduction proof of...
hbimpgVD 44154 Virtual deduction proof of...
hbalgVD 44155 Virtual deduction proof of...
hbexgVD 44156 Virtual deduction proof of...
ax6e2eqVD 44157 The following User's Proof...
ax6e2ndVD 44158 The following User's Proof...
ax6e2ndeqVD 44159 The following User's Proof...
2sb5ndVD 44160 The following User's Proof...
2uasbanhVD 44161 The following User's Proof...
e2ebindVD 44162 The following User's Proof...
sb5ALTVD 44163 The following User's Proof...
vk15.4jVD 44164 The following User's Proof...
notnotrALTVD 44165 The following User's Proof...
con3ALTVD 44166 The following User's Proof...
elpwgdedVD 44167 Membership in a power clas...
sspwimp 44168 If a class is a subclass o...
sspwimpVD 44169 The following User's Proof...
sspwimpcf 44170 If a class is a subclass o...
sspwimpcfVD 44171 The following User's Proof...
suctrALTcf 44172 The sucessor of a transiti...
suctrALTcfVD 44173 The following User's Proof...
suctrALT3 44174 The successor of a transit...
sspwimpALT 44175 If a class is a subclass o...
unisnALT 44176 A set equals the union of ...
notnotrALT2 44177 Converse of double negatio...
sspwimpALT2 44178 If a class is a subclass o...
e2ebindALT 44179 Absorption of an existenti...
ax6e2ndALT 44180 If at least two sets exist...
ax6e2ndeqALT 44181 "At least two sets exist" ...
2sb5ndALT 44182 Equivalence for double sub...
chordthmALT 44183 The intersecting chords th...
isosctrlem1ALT 44184 Lemma for ~ isosctr . Thi...
iunconnlem2 44185 The indexed union of conne...
iunconnALT 44186 The indexed union of conne...
sineq0ALT 44187 A complex number whose sin...
evth2f 44188 A version of ~ evth2 using...
elunif 44189 A version of ~ eluni using...
rzalf 44190 A version of ~ rzal using ...
fvelrnbf 44191 A version of ~ fvelrnb usi...
rfcnpre1 44192 If F is a continuous funct...
ubelsupr 44193 If U belongs to A and U is...
fsumcnf 44194 A finite sum of functions ...
mulltgt0 44195 The product of a negative ...
rspcegf 44196 A version of ~ rspcev usin...
rabexgf 44197 A version of ~ rabexg usin...
fcnre 44198 A function continuous with...
sumsnd 44199 A sum of a singleton is th...
evthf 44200 A version of ~ evth using ...
cnfex 44201 The class of continuous fu...
fnchoice 44202 For a finite set, a choice...
refsumcn 44203 A finite sum of continuous...
rfcnpre2 44204 If ` F ` is a continuous f...
cncmpmax 44205 When the hypothesis for th...
rfcnpre3 44206 If F is a continuous funct...
rfcnpre4 44207 If F is a continuous funct...
sumpair 44208 Sum of two distinct comple...
rfcnnnub 44209 Given a real continuous fu...
refsum2cnlem1 44210 This is the core Lemma for...
refsum2cn 44211 The sum of two continuus r...
adantlllr 44212 Deduction adding a conjunc...
3adantlr3 44213 Deduction adding a conjunc...
3adantll2 44214 Deduction adding a conjunc...
3adantll3 44215 Deduction adding a conjunc...
ssnel 44216 If not element of a set, t...
sncldre 44217 A singleton is closed w.r....
n0p 44218 A polynomial with a nonzer...
pm2.65ni 44219 Inference rule for proof b...
pwssfi 44220 Every element of the power...
iuneq2df 44221 Equality deduction for ind...
nnfoctb 44222 There exists a mapping fro...
ssinss1d 44223 Intersection preserves sub...
elpwinss 44224 An element of the powerset...
unidmex 44225 If ` F ` is a set, then ` ...
ndisj2 44226 A non-disjointness conditi...
zenom 44227 The set of integer numbers...
uzwo4 44228 Well-ordering principle: a...
unisn0 44229 The union of the singleton...
ssin0 44230 If two classes are disjoin...
inabs3 44231 Absorption law for interse...
pwpwuni 44232 Relationship between power...
disjiun2 44233 In a disjoint collection, ...
0pwfi 44234 The empty set is in any po...
ssinss2d 44235 Intersection preserves sub...
zct 44236 The set of integer numbers...
pwfin0 44237 A finite set always belong...
uzct 44238 An upper integer set is co...
iunxsnf 44239 A singleton index picks ou...
fiiuncl 44240 If a set is closed under t...
iunp1 44241 The addition of the next s...
fiunicl 44242 If a set is closed under t...
ixpeq2d 44243 Equality theorem for infin...
disjxp1 44244 The sets of a cartesian pr...
disjsnxp 44245 The sets in the cartesian ...
eliind 44246 Membership in indexed inte...
rspcef 44247 Restricted existential spe...
inn0f 44248 A nonempty intersection. ...
ixpssmapc 44249 An infinite Cartesian prod...
inn0 44250 A nonempty intersection. ...
elintd 44251 Membership in class inters...
ssdf 44252 A sufficient condition for...
brneqtrd 44253 Substitution of equal clas...
ssnct 44254 A set containing an uncoun...
ssuniint 44255 Sufficient condition for b...
elintdv 44256 Membership in class inters...
ssd 44257 A sufficient condition for...
ralimralim 44258 Introducing any antecedent...
snelmap 44259 Membership of the element ...
xrnmnfpnf 44260 An extended real that is n...
nelrnmpt 44261 Non-membership in the rang...
iuneq1i 44262 Equality theorem for index...
nssrex 44263 Negation of subclass relat...
ssinc 44264 Inclusion relation for a m...
ssdec 44265 Inclusion relation for a m...
elixpconstg 44266 Membership in an infinite ...
iineq1d 44267 Equality theorem for index...
metpsmet 44268 A metric is a pseudometric...
ixpssixp 44269 Subclass theorem for infin...
ballss3 44270 A sufficient condition for...
iunincfi 44271 Given a sequence of increa...
nsstr 44272 If it's not a subclass, it...
rexanuz3 44273 Combine two different uppe...
cbvmpo2 44274 Rule to change the second ...
cbvmpo1 44275 Rule to change the first b...
eliuniin 44276 Indexed union of indexed i...
ssabf 44277 Subclass of a class abstra...
pssnssi 44278 A proper subclass does not...
rabidim2 44279 Membership in a restricted...
eluni2f 44280 Membership in class union....
eliin2f 44281 Membership in indexed inte...
nssd 44282 Negation of subclass relat...
iineq12dv 44283 Equality deduction for ind...
supxrcld 44284 The supremum of an arbitra...
elrestd 44285 A sufficient condition for...
eliuniincex 44286 Counterexample to show tha...
eliincex 44287 Counterexample to show tha...
eliinid 44288 Membership in an indexed i...
abssf 44289 Class abstraction in a sub...
supxrubd 44290 A member of a set of exten...
ssrabf 44291 Subclass of a restricted c...
ssrabdf 44292 Subclass of a restricted c...
eliin2 44293 Membership in indexed inte...
ssrab2f 44294 Subclass relation for a re...
restuni3 44295 The underlying set of a su...
rabssf 44296 Restricted class abstracti...
eliuniin2 44297 Indexed union of indexed i...
restuni4 44298 The underlying set of a su...
restuni6 44299 The underlying set of a su...
restuni5 44300 The underlying set of a su...
unirestss 44301 The union of an elementwis...
iniin1 44302 Indexed intersection of in...
iniin2 44303 Indexed intersection of in...
cbvrabv2 44304 A more general version of ...
cbvrabv2w 44305 A more general version of ...
iinssiin 44306 Subset implication for an ...
eliind2 44307 Membership in indexed inte...
iinssd 44308 Subset implication for an ...
rabbida2 44309 Equivalent wff's yield equ...
iinexd 44310 The existence of an indexe...
rabexf 44311 Separation Scheme in terms...
rabbida3 44312 Equivalent wff's yield equ...
r19.36vf 44313 Restricted quantifier vers...
raleqd 44314 Equality deduction for res...
iinssf 44315 Subset implication for an ...
iinssdf 44316 Subset implication for an ...
resabs2i 44317 Absorption law for restric...
ssdf2 44318 A sufficient condition for...
rabssd 44319 Restricted class abstracti...
rexnegd 44320 Minus a real number. (Con...
rexlimd3 44321 * Inference from Theorem 1...
resabs1i 44322 Absorption law for restric...
nel1nelin 44323 Membership in an intersect...
nel2nelin 44324 Membership in an intersect...
nel1nelini 44325 Membership in an intersect...
nel2nelini 44326 Membership in an intersect...
eliunid 44327 Membership in indexed unio...
reximddv3 44328 Deduction from Theorem 19....
reximdd 44329 Deduction from Theorem 19....
unfid 44330 The union of two finite se...
inopnd 44331 The intersection of two op...
ss2rabdf 44332 Deduction of restricted ab...
restopn3 44333 If ` A ` is open, then ` A...
restopnssd 44334 A topology restricted to a...
restsubel 44335 A subset belongs in the sp...
toprestsubel 44336 A subset is open in the to...
rabidd 44337 An "identity" law of concr...
iunssdf 44338 Subset theorem for an inde...
iinss2d 44339 Subset implication for an ...
r19.3rzf 44340 Restricted quantification ...
r19.28zf 44341 Restricted quantifier vers...
iindif2f 44342 Indexed intersection of cl...
ralfal 44343 Two ways of expressing emp...
archd 44344 Archimedean property of re...
eliund 44345 Membership in indexed unio...
nimnbi 44346 If an implication is false...
nimnbi2 44347 If an implication is false...
notbicom 44348 Commutative law for the ne...
rexeqif 44349 Equality inference for res...
rspced 44350 Restricted existential spe...
feq1dd 44351 Equality deduction for fun...
fnresdmss 44352 A function does not change...
fmptsnxp 44353 Maps-to notation and Carte...
fvmpt2bd 44354 Value of a function given ...
rnmptfi 44355 The range of a function wi...
fresin2 44356 Restriction of a function ...
ffi 44357 A function with finite dom...
suprnmpt 44358 An explicit bound for the ...
rnffi 44359 The range of a function wi...
mptelpm 44360 A function in maps-to nota...
rnmptpr 44361 Range of a function define...
resmpti 44362 Restriction of the mapping...
founiiun 44363 Union expressed as an inde...
rnresun 44364 Distribution law for range...
elrnmptf 44365 The range of a function in...
rnmptssrn 44366 Inclusion relation for two...
disjf1 44367 A 1 to 1 mapping built fro...
rnsnf 44368 The range of a function wh...
wessf1ornlem 44369 Given a function ` F ` on ...
wessf1orn 44370 Given a function ` F ` on ...
nelrnres 44371 If ` A ` is not in the ran...
disjrnmpt2 44372 Disjointness of the range ...
elrnmpt1sf 44373 Elementhood in an image se...
founiiun0 44374 Union expressed as an inde...
disjf1o 44375 A bijection built from dis...
disjinfi 44376 Only a finite number of di...
fvovco 44377 Value of the composition o...
ssnnf1octb 44378 There exists a bijection b...
nnf1oxpnn 44379 There is a bijection betwe...
rnmptssd 44380 The range of a function gi...
projf1o 44381 A biijection from a set to...
fvmap 44382 Function value for a membe...
fvixp2 44383 Projection of a factor of ...
choicefi 44384 For a finite set, a choice...
mpct 44385 The exponentiation of a co...
cnmetcoval 44386 Value of the distance func...
fcomptss 44387 Express composition of two...
elmapsnd 44388 Membership in a set expone...
mapss2 44389 Subset inheritance for set...
fsneq 44390 Equality condition for two...
difmap 44391 Difference of two sets exp...
unirnmap 44392 Given a subset of a set ex...
inmap 44393 Intersection of two sets e...
fcoss 44394 Composition of two mapping...
fsneqrn 44395 Equality condition for two...
difmapsn 44396 Difference of two sets exp...
mapssbi 44397 Subset inheritance for set...
unirnmapsn 44398 Equality theorem for a sub...
iunmapss 44399 The indexed union of set e...
ssmapsn 44400 A subset ` C ` of a set ex...
iunmapsn 44401 The indexed union of set e...
absfico 44402 Mapping domain and codomai...
icof 44403 The set of left-closed rig...
elpmrn 44404 The range of a partial fun...
imaexi 44405 The image of a set is a se...
axccdom 44406 Relax the constraint on ax...
dmmptdff 44407 The domain of the mapping ...
dmmptdf 44408 The domain of the mapping ...
elpmi2 44409 The domain of a partial fu...
dmrelrnrel 44410 A relation preserving func...
fvcod 44411 Value of a function compos...
elrnmpoid 44412 Membership in the range of...
axccd 44413 An alternative version of ...
axccd2 44414 An alternative version of ...
fimassd 44415 The image of a class is a ...
feqresmptf 44416 Express a restricted funct...
elrnmpt1d 44417 Elementhood in an image se...
dmresss 44418 The domain of a restrictio...
dmmptssf 44419 The domain of a mapping is...
dmmptdf2 44420 The domain of the mapping ...
dmuz 44421 Domain of the upper intege...
fmptd2f 44422 Domain and codomain of the...
mpteq1df 44423 An equality theorem for th...
mpteq1dfOLD 44424 Obsolete version of ~ mpte...
mptexf 44425 If the domain of a functio...
fvmpt4 44426 Value of a function given ...
fmptf 44427 Functionality of the mappi...
resimass 44428 The image of a restriction...
mptssid 44429 The mapping operation expr...
mptfnd 44430 The maps-to notation defin...
mpteq12daOLD 44431 Obsolete version of ~ mpte...
rnmptlb 44432 Boundness below of the ran...
rnmptbddlem 44433 Boundness of the range of ...
rnmptbdd 44434 Boundness of the range of ...
funimaeq 44435 Membership relation for th...
rnmptssf 44436 The range of a function gi...
rnmptbd2lem 44437 Boundness below of the ran...
rnmptbd2 44438 Boundness below of the ran...
infnsuprnmpt 44439 The indexed infimum of rea...
suprclrnmpt 44440 Closure of the indexed sup...
suprubrnmpt2 44441 A member of a nonempty ind...
suprubrnmpt 44442 A member of a nonempty ind...
rnmptssdf 44443 The range of a function gi...
rnmptbdlem 44444 Boundness above of the ran...
rnmptbd 44445 Boundness above of the ran...
rnmptss2 44446 The range of a function gi...
elmptima 44447 The image of a function in...
ralrnmpt3 44448 A restricted quantifier ov...
fvelima2 44449 Function value in an image...
rnmptssbi 44450 The range of a function gi...
imass2d 44451 Subset theorem for image. ...
imassmpt 44452 Membership relation for th...
fpmd 44453 A total function is a part...
fconst7 44454 An alternative way to expr...
fnmptif 44455 Functionality and domain o...
dmmptif 44456 Domain of the mapping oper...
mpteq2dfa 44457 Slightly more general equa...
dmmpt1 44458 The domain of the mapping ...
fmptff 44459 Functionality of the mappi...
fvmptelcdmf 44460 The value of a function at...
fmptdff 44461 A version of ~ fmptd using...
fvmpt2df 44462 Deduction version of ~ fvm...
rn1st 44463 The range of a function wi...
rnmptssff 44464 The range of a function gi...
rnmptssdff 44465 The range of a function gi...
fvmpt4d 44466 Value of a function given ...
sub2times 44467 Subtracting from a number,...
nnxrd 44468 A natural number is an ext...
nnxr 44469 A natural number is an ext...
abssubrp 44470 The distance of two distin...
elfzfzo 44471 Relationship between membe...
oddfl 44472 Odd number representation ...
abscosbd 44473 Bound for the absolute val...
mul13d 44474 Commutative/associative la...
negpilt0 44475 Negative ` _pi ` is negati...
dstregt0 44476 A complex number ` A ` tha...
subadd4b 44477 Rearrangement of 4 terms i...
xrlttri5d 44478 Not equal and not larger i...
neglt 44479 The negative of a positive...
zltlesub 44480 If an integer ` N ` is les...
divlt0gt0d 44481 The ratio of a negative nu...
subsub23d 44482 Swap subtrahend and result...
2timesgt 44483 Double of a positive real ...
reopn 44484 The reals are open with re...
sub31 44485 Swap the first and third t...
nnne1ge2 44486 A positive integer which i...
lefldiveq 44487 A closed enough, smaller r...
negsubdi3d 44488 Distribution of negative o...
ltdiv2dd 44489 Division of a positive num...
abssinbd 44490 Bound for the absolute val...
halffl 44491 Floor of ` ( 1 / 2 ) ` . ...
monoords 44492 Ordering relation for a st...
hashssle 44493 The size of a subset of a ...
lttri5d 44494 Not equal and not larger i...
fzisoeu 44495 A finite ordered set has a...
lt3addmuld 44496 If three real numbers are ...
absnpncan2d 44497 Triangular inequality, com...
fperiodmullem 44498 A function with period ` T...
fperiodmul 44499 A function with period T i...
upbdrech 44500 Choice of an upper bound f...
lt4addmuld 44501 If four real numbers are l...
absnpncan3d 44502 Triangular inequality, com...
upbdrech2 44503 Choice of an upper bound f...
ssfiunibd 44504 A finite union of bounded ...
fzdifsuc2 44505 Remove a successor from th...
fzsscn 44506 A finite sequence of integ...
divcan8d 44507 A cancellation law for div...
dmmcand 44508 Cancellation law for divis...
fzssre 44509 A finite sequence of integ...
bccld 44510 A binomial coefficient, in...
leadd12dd 44511 Addition to both sides of ...
fzssnn0 44512 A finite set of sequential...
xreqle 44513 Equality implies 'less tha...
xaddlidd 44514 ` 0 ` is a left identity f...
xadd0ge 44515 A number is less than or e...
elfzolem1 44516 A member in a half-open in...
xrgtned 44517 'Greater than' implies not...
xrleneltd 44518 'Less than or equal to' an...
xaddcomd 44519 The extended real addition...
supxrre3 44520 The supremum of a nonempty...
uzfissfz 44521 For any finite subset of t...
xleadd2d 44522 Addition of extended reals...
suprltrp 44523 The supremum of a nonempty...
xleadd1d 44524 Addition of extended reals...
xreqled 44525 Equality implies 'less tha...
xrgepnfd 44526 An extended real greater t...
xrge0nemnfd 44527 A nonnegative extended rea...
supxrgere 44528 If a real number can be ap...
iuneqfzuzlem 44529 Lemma for ~ iuneqfzuz : he...
iuneqfzuz 44530 If two unions indexed by u...
xle2addd 44531 Adding both side of two in...
supxrgelem 44532 If an extended real number...
supxrge 44533 If an extended real number...
suplesup 44534 If any element of ` A ` ca...
infxrglb 44535 The infimum of a set of ex...
xadd0ge2 44536 A number is less than or e...
nepnfltpnf 44537 An extended real that is n...
ltadd12dd 44538 Addition to both sides of ...
nemnftgtmnft 44539 An extended real that is n...
xrgtso 44540 'Greater than' is a strict...
rpex 44541 The positive reals form a ...
xrge0ge0 44542 A nonnegative extended rea...
xrssre 44543 A subset of extended reals...
ssuzfz 44544 A finite subset of the upp...
absfun 44545 The absolute value is a fu...
infrpge 44546 The infimum of a nonempty,...
xrlexaddrp 44547 If an extended real number...
supsubc 44548 The supremum function dist...
xralrple2 44549 Show that ` A ` is less th...
nnuzdisj 44550 The first ` N ` elements o...
ltdivgt1 44551 Divsion by a number greate...
xrltned 44552 'Less than' implies not eq...
nnsplit 44553 Express the set of positiv...
divdiv3d 44554 Division into a fraction. ...
abslt2sqd 44555 Comparison of the square o...
qenom 44556 The set of rational number...
qct 44557 The set of rational number...
xrltnled 44558 'Less than' in terms of 'l...
lenlteq 44559 'less than or equal to' bu...
xrred 44560 An extended real that is n...
rr2sscn2 44561 The cartesian square of ` ...
infxr 44562 The infimum of a set of ex...
infxrunb2 44563 The infimum of an unbounde...
infxrbnd2 44564 The infimum of a bounded-b...
infleinflem1 44565 Lemma for ~ infleinf , cas...
infleinflem2 44566 Lemma for ~ infleinf , whe...
infleinf 44567 If any element of ` B ` ca...
xralrple4 44568 Show that ` A ` is less th...
xralrple3 44569 Show that ` A ` is less th...
eluzelzd 44570 A member of an upper set o...
suplesup2 44571 If any element of ` A ` is...
recnnltrp 44572 ` N ` is a natural number ...
nnn0 44573 The set of positive intege...
fzct 44574 A finite set of sequential...
rpgtrecnn 44575 Any positive real number i...
fzossuz 44576 A half-open integer interv...
infxrrefi 44577 The real and extended real...
xrralrecnnle 44578 Show that ` A ` is less th...
fzoct 44579 A finite set of sequential...
frexr 44580 A function taking real val...
nnrecrp 44581 The reciprocal of a positi...
reclt0d 44582 The reciprocal of a negati...
lt0neg1dd 44583 If a number is negative, i...
infxrcld 44584 The infimum of an arbitrar...
xrralrecnnge 44585 Show that ` A ` is less th...
reclt0 44586 The reciprocal of a negati...
ltmulneg 44587 Multiplying by a negative ...
allbutfi 44588 For all but finitely many....
ltdiv23neg 44589 Swap denominator with othe...
xreqnltd 44590 A consequence of trichotom...
mnfnre2 44591 Minus infinity is not a re...
zssxr 44592 The integers are a subset ...
fisupclrnmpt 44593 A nonempty finite indexed ...
supxrunb3 44594 The supremum of an unbound...
elfzod 44595 Membership in a half-open ...
fimaxre4 44596 A nonempty finite set of r...
ren0 44597 The set of reals is nonemp...
eluzelz2 44598 A member of an upper set o...
resabs2d 44599 Absorption law for restric...
uzid2 44600 Membership of the least me...
supxrleubrnmpt 44601 The supremum of a nonempty...
uzssre2 44602 An upper set of integers i...
uzssd 44603 Subset relationship for tw...
eluzd 44604 Membership in an upper set...
infxrlbrnmpt2 44605 A member of a nonempty ind...
xrre4 44606 An extended real is real i...
uz0 44607 The upper integers functio...
eluzelz2d 44608 A member of an upper set o...
infleinf2 44609 If any element in ` B ` is...
unb2ltle 44610 "Unbounded below" expresse...
uzidd2 44611 Membership of the least me...
uzssd2 44612 Subset relationship for tw...
rexabslelem 44613 An indexed set of absolute...
rexabsle 44614 An indexed set of absolute...
allbutfiinf 44615 Given a "for all but finit...
supxrrernmpt 44616 The real and extended real...
suprleubrnmpt 44617 The supremum of a nonempty...
infrnmptle 44618 An indexed infimum of exte...
infxrunb3 44619 The infimum of an unbounde...
uzn0d 44620 The upper integers are all...
uzssd3 44621 Subset relationship for tw...
rexabsle2 44622 An indexed set of absolute...
infxrunb3rnmpt 44623 The infimum of an unbounde...
supxrre3rnmpt 44624 The indexed supremum of a ...
uzublem 44625 A set of reals, indexed by...
uzub 44626 A set of reals, indexed by...
ssrexr 44627 A subset of the reals is a...
supxrmnf2 44628 Removing minus infinity fr...
supxrcli 44629 The supremum of an arbitra...
uzid3 44630 Membership of the least me...
infxrlesupxr 44631 The supremum of a nonempty...
xnegeqd 44632 Equality of two extended n...
xnegrecl 44633 The extended real negative...
xnegnegi 44634 Extended real version of ~...
xnegeqi 44635 Equality of two extended n...
nfxnegd 44636 Deduction version of ~ nfx...
xnegnegd 44637 Extended real version of ~...
uzred 44638 An upper integer is a real...
xnegcli 44639 Closure of extended real n...
supminfrnmpt 44640 The indexed supremum of a ...
infxrpnf 44641 Adding plus infinity to a ...
infxrrnmptcl 44642 The infimum of an arbitrar...
leneg2d 44643 Negative of one side of 'l...
supxrltinfxr 44644 The supremum of the empty ...
max1d 44645 A number is less than or e...
supxrleubrnmptf 44646 The supremum of a nonempty...
nleltd 44647 'Not less than or equal to...
zxrd 44648 An integer is an extended ...
infxrgelbrnmpt 44649 The infimum of an indexed ...
rphalfltd 44650 Half of a positive real is...
uzssz2 44651 An upper set of integers i...
leneg3d 44652 Negative of one side of 'l...
max2d 44653 A number is less than or e...
uzn0bi 44654 The upper integers functio...
xnegrecl2 44655 If the extended real negat...
nfxneg 44656 Bound-variable hypothesis ...
uzxrd 44657 An upper integer is an ext...
infxrpnf2 44658 Removing plus infinity fro...
supminfxr 44659 The extended real suprema ...
infrpgernmpt 44660 The infimum of a nonempty,...
xnegre 44661 An extended real is real i...
xnegrecl2d 44662 If the extended real negat...
uzxr 44663 An upper integer is an ext...
supminfxr2 44664 The extended real suprema ...
xnegred 44665 An extended real is real i...
supminfxrrnmpt 44666 The indexed supremum of a ...
min1d 44667 The minimum of two numbers...
min2d 44668 The minimum of two numbers...
pnfged 44669 Plus infinity is an upper ...
xrnpnfmnf 44670 An extended real that is n...
uzsscn 44671 An upper set of integers i...
absimnre 44672 The absolute value of the ...
uzsscn2 44673 An upper set of integers i...
xrtgcntopre 44674 The standard topologies on...
absimlere 44675 The absolute value of the ...
rpssxr 44676 The positive reals are a s...
monoordxrv 44677 Ordering relation for a mo...
monoordxr 44678 Ordering relation for a mo...
monoord2xrv 44679 Ordering relation for a mo...
monoord2xr 44680 Ordering relation for a mo...
xrpnf 44681 An extended real is plus i...
xlenegcon1 44682 Extended real version of ~...
xlenegcon2 44683 Extended real version of ~...
pimxrneun 44684 The preimage of a set of e...
caucvgbf 44685 A function is convergent i...
cvgcau 44686 A convergent function is C...
cvgcaule 44687 A convergent function is C...
rexanuz2nf 44688 A simple counterexample re...
gtnelioc 44689 A real number larger than ...
ioossioc 44690 An open interval is a subs...
ioondisj2 44691 A condition for two open i...
ioondisj1 44692 A condition for two open i...
ioogtlb 44693 An element of a closed int...
evthiccabs 44694 Extreme Value Theorem on y...
ltnelicc 44695 A real number smaller than...
eliood 44696 Membership in an open real...
iooabslt 44697 An upper bound for the dis...
gtnelicc 44698 A real number greater than...
iooinlbub 44699 An open interval has empty...
iocgtlb 44700 An element of a left-open ...
iocleub 44701 An element of a left-open ...
eliccd 44702 Membership in a closed rea...
eliccre 44703 A member of a closed inter...
eliooshift 44704 Element of an open interva...
eliocd 44705 Membership in a left-open ...
icoltub 44706 An element of a left-close...
eliocre 44707 A member of a left-open ri...
iooltub 44708 An element of an open inte...
ioontr 44709 The interior of an interva...
snunioo1 44710 The closure of one end of ...
lbioc 44711 A left-open right-closed i...
ioomidp 44712 The midpoint is an element...
iccdifioo 44713 If the open inverval is re...
iccdifprioo 44714 An open interval is the cl...
ioossioobi 44715 Biconditional form of ~ io...
iccshift 44716 A closed interval shifted ...
iccsuble 44717 An upper bound to the dist...
iocopn 44718 A left-open right-closed i...
eliccelioc 44719 Membership in a closed int...
iooshift 44720 An open interval shifted b...
iccintsng 44721 Intersection of two adiace...
icoiccdif 44722 Left-closed right-open int...
icoopn 44723 A left-closed right-open i...
icoub 44724 A left-closed, right-open ...
eliccxrd 44725 Membership in a closed rea...
pnfel0pnf 44726 ` +oo ` is a nonnegative e...
eliccnelico 44727 An element of a closed int...
eliccelicod 44728 A member of a closed inter...
ge0xrre 44729 A nonnegative extended rea...
ge0lere 44730 A nonnegative extended Rea...
elicores 44731 Membership in a left-close...
inficc 44732 The infimum of a nonempty ...
qinioo 44733 The rational numbers are d...
lenelioc 44734 A real number smaller than...
ioonct 44735 A nonempty open interval i...
xrgtnelicc 44736 A real number greater than...
iccdificc 44737 The difference of two clos...
iocnct 44738 A nonempty left-open, righ...
iccnct 44739 A closed interval, with mo...
iooiinicc 44740 A closed interval expresse...
iccgelbd 44741 An element of a closed int...
iooltubd 44742 An element of an open inte...
icoltubd 44743 An element of a left-close...
qelioo 44744 The rational numbers are d...
tgqioo2 44745 Every open set of reals is...
iccleubd 44746 An element of a closed int...
elioored 44747 A member of an open interv...
ioogtlbd 44748 An element of a closed int...
ioofun 44749 ` (,) ` is a function. (C...
icomnfinre 44750 A left-closed, right-open,...
sqrlearg 44751 The square compared with i...
ressiocsup 44752 If the supremum belongs to...
ressioosup 44753 If the supremum does not b...
iooiinioc 44754 A left-open, right-closed ...
ressiooinf 44755 If the infimum does not be...
icogelbd 44756 An element of a left-close...
iocleubd 44757 An element of a left-open ...
uzinico 44758 An upper interval of integ...
preimaiocmnf 44759 Preimage of a right-closed...
uzinico2 44760 An upper interval of integ...
uzinico3 44761 An upper interval of integ...
icossico2 44762 Condition for a closed-bel...
dmico 44763 The domain of the closed-b...
ndmico 44764 The closed-below, open-abo...
uzubioo 44765 The upper integers are unb...
uzubico 44766 The upper integers are unb...
uzubioo2 44767 The upper integers are unb...
uzubico2 44768 The upper integers are unb...
iocgtlbd 44769 An element of a left-open ...
xrtgioo2 44770 The topology on the extend...
tgioo4 44771 The standard topology on t...
fsummulc1f 44772 Closure of a finite sum of...
fsumnncl 44773 Closure of a nonempty, fin...
fsumge0cl 44774 The finite sum of nonnegat...
fsumf1of 44775 Re-index a finite sum usin...
fsumiunss 44776 Sum over a disjoint indexe...
fsumreclf 44777 Closure of a finite sum of...
fsumlessf 44778 A shorter sum of nonnegati...
fsumsupp0 44779 Finite sum of function val...
fsumsermpt 44780 A finite sum expressed in ...
fmul01 44781 Multiplying a finite numbe...
fmulcl 44782 If ' Y ' is closed under t...
fmuldfeqlem1 44783 induction step for the pro...
fmuldfeq 44784 X and Z are two equivalent...
fmul01lt1lem1 44785 Given a finite multiplicat...
fmul01lt1lem2 44786 Given a finite multiplicat...
fmul01lt1 44787 Given a finite multiplicat...
cncfmptss 44788 A continuous complex funct...
rrpsscn 44789 The positive reals are a s...
mulc1cncfg 44790 A version of ~ mulc1cncf u...
infrglb 44791 The infimum of a nonempty ...
expcnfg 44792 If ` F ` is a complex cont...
prodeq2ad 44793 Equality deduction for pro...
fprodsplit1 44794 Separate out a term in a f...
fprodexp 44795 Positive integer exponenti...
fprodabs2 44796 The absolute value of a fi...
fprod0 44797 A finite product with a ze...
mccllem 44798 * Induction step for ~ mcc...
mccl 44799 A multinomial coefficient,...
fprodcnlem 44800 A finite product of functi...
fprodcn 44801 A finite product of functi...
clim1fr1 44802 A class of sequences of fr...
isumneg 44803 Negation of a converging s...
climrec 44804 Limit of the reciprocal of...
climmulf 44805 A version of ~ climmul usi...
climexp 44806 The limit of natural power...
climinf 44807 A bounded monotonic noninc...
climsuselem1 44808 The subsequence index ` I ...
climsuse 44809 A subsequence ` G ` of a c...
climrecf 44810 A version of ~ climrec usi...
climneg 44811 Complex limit of the negat...
climinff 44812 A version of ~ climinf usi...
climdivf 44813 Limit of the ratio of two ...
climreeq 44814 If ` F ` is a real functio...
ellimciota 44815 An explicit value for the ...
climaddf 44816 A version of ~ climadd usi...
mullimc 44817 Limit of the product of tw...
ellimcabssub0 44818 An equivalent condition fo...
limcdm0 44819 If a function has empty do...
islptre 44820 An equivalence condition f...
limccog 44821 Limit of the composition o...
limciccioolb 44822 The limit of a function at...
climf 44823 Express the predicate: Th...
mullimcf 44824 Limit of the multiplicatio...
constlimc 44825 Limit of constant function...
rexlim2d 44826 Inference removing two res...
idlimc 44827 Limit of the identity func...
divcnvg 44828 The sequence of reciprocal...
limcperiod 44829 If ` F ` is a periodic fun...
limcrecl 44830 If ` F ` is a real-valued ...
sumnnodd 44831 A series indexed by ` NN `...
lptioo2 44832 The upper bound of an open...
lptioo1 44833 The lower bound of an open...
elprn1 44834 A member of an unordered p...
elprn2 44835 A member of an unordered p...
limcmptdm 44836 The domain of a maps-to fu...
clim2f 44837 Express the predicate: Th...
limcicciooub 44838 The limit of a function at...
ltmod 44839 A sufficient condition for...
islpcn 44840 A characterization for a l...
lptre2pt 44841 If a set in the real line ...
limsupre 44842 If a sequence is bounded, ...
limcresiooub 44843 The left limit doesn't cha...
limcresioolb 44844 The right limit doesn't ch...
limcleqr 44845 If the left and the right ...
lptioo2cn 44846 The upper bound of an open...
lptioo1cn 44847 The lower bound of an open...
neglimc 44848 Limit of the negative func...
addlimc 44849 Sum of two limits. (Contr...
0ellimcdiv 44850 If the numerator converges...
clim2cf 44851 Express the predicate ` F ...
limclner 44852 For a limit point, both fr...
sublimc 44853 Subtraction of two limits....
reclimc 44854 Limit of the reciprocal of...
clim0cf 44855 Express the predicate ` F ...
limclr 44856 For a limit point, both fr...
divlimc 44857 Limit of the quotient of t...
expfac 44858 Factorial grows faster tha...
climconstmpt 44859 A constant sequence conver...
climresmpt 44860 A function restricted to u...
climsubmpt 44861 Limit of the difference of...
climsubc2mpt 44862 Limit of the difference of...
climsubc1mpt 44863 Limit of the difference of...
fnlimfv 44864 The value of the limit fun...
climreclf 44865 The limit of a convergent ...
climeldmeq 44866 Two functions that are eve...
climf2 44867 Express the predicate: Th...
fnlimcnv 44868 The sequence of function v...
climeldmeqmpt 44869 Two functions that are eve...
climfveq 44870 Two functions that are eve...
clim2f2 44871 Express the predicate: Th...
climfveqmpt 44872 Two functions that are eve...
climd 44873 Express the predicate: Th...
clim2d 44874 The limit of complex numbe...
fnlimfvre 44875 The limit function of real...
allbutfifvre 44876 Given a sequence of real-v...
climleltrp 44877 The limit of complex numbe...
fnlimfvre2 44878 The limit function of real...
fnlimf 44879 The limit function of real...
fnlimabslt 44880 A sequence of function val...
climfveqf 44881 Two functions that are eve...
climmptf 44882 Exhibit a function ` G ` w...
climfveqmpt3 44883 Two functions that are eve...
climeldmeqf 44884 Two functions that are eve...
climreclmpt 44885 The limit of B convergent ...
limsupref 44886 If a sequence is bounded, ...
limsupbnd1f 44887 If a sequence is eventuall...
climbddf 44888 A converging sequence of c...
climeqf 44889 Two functions that are eve...
climeldmeqmpt3 44890 Two functions that are eve...
limsupcld 44891 Closure of the superior li...
climfv 44892 The limit of a convergent ...
limsupval3 44893 The superior limit of an i...
climfveqmpt2 44894 Two functions that are eve...
limsup0 44895 The superior limit of the ...
climeldmeqmpt2 44896 Two functions that are eve...
limsupresre 44897 The supremum limit of a fu...
climeqmpt 44898 Two functions that are eve...
climfvd 44899 The limit of a convergent ...
limsuplesup 44900 An upper bound for the sup...
limsupresico 44901 The superior limit doesn't...
limsuppnfdlem 44902 If the restriction of a fu...
limsuppnfd 44903 If the restriction of a fu...
limsupresuz 44904 If the real part of the do...
limsupub 44905 If the limsup is not ` +oo...
limsupres 44906 The superior limit of a re...
climinf2lem 44907 A convergent, nonincreasin...
climinf2 44908 A convergent, nonincreasin...
limsupvaluz 44909 The superior limit, when t...
limsupresuz2 44910 If the domain of a functio...
limsuppnflem 44911 If the restriction of a fu...
limsuppnf 44912 If the restriction of a fu...
limsupubuzlem 44913 If the limsup is not ` +oo...
limsupubuz 44914 For a real-valued function...
climinf2mpt 44915 A bounded below, monotonic...
climinfmpt 44916 A bounded below, monotonic...
climinf3 44917 A convergent, nonincreasin...
limsupvaluzmpt 44918 The superior limit, when t...
limsupequzmpt2 44919 Two functions that are eve...
limsupubuzmpt 44920 If the limsup is not ` +oo...
limsupmnflem 44921 The superior limit of a fu...
limsupmnf 44922 The superior limit of a fu...
limsupequzlem 44923 Two functions that are eve...
limsupequz 44924 Two functions that are eve...
limsupre2lem 44925 Given a function on the ex...
limsupre2 44926 Given a function on the ex...
limsupmnfuzlem 44927 The superior limit of a fu...
limsupmnfuz 44928 The superior limit of a fu...
limsupequzmptlem 44929 Two functions that are eve...
limsupequzmpt 44930 Two functions that are eve...
limsupre2mpt 44931 Given a function on the ex...
limsupequzmptf 44932 Two functions that are eve...
limsupre3lem 44933 Given a function on the ex...
limsupre3 44934 Given a function on the ex...
limsupre3mpt 44935 Given a function on the ex...
limsupre3uzlem 44936 Given a function on the ex...
limsupre3uz 44937 Given a function on the ex...
limsupreuz 44938 Given a function on the re...
limsupvaluz2 44939 The superior limit, when t...
limsupreuzmpt 44940 Given a function on the re...
supcnvlimsup 44941 If a function on a set of ...
supcnvlimsupmpt 44942 If a function on a set of ...
0cnv 44943 If ` (/) ` is a complex nu...
climuzlem 44944 Express the predicate: Th...
climuz 44945 Express the predicate: Th...
lmbr3v 44946 Express the binary relatio...
climisp 44947 If a sequence converges to...
lmbr3 44948 Express the binary relatio...
climrescn 44949 A sequence converging w.r....
climxrrelem 44950 If a sequence ranging over...
climxrre 44951 If a sequence ranging over...
limsuplt2 44954 The defining property of t...
liminfgord 44955 Ordering property of the i...
limsupvald 44956 The superior limit of a se...
limsupresicompt 44957 The superior limit doesn't...
limsupcli 44958 Closure of the superior li...
liminfgf 44959 Closure of the inferior li...
liminfval 44960 The inferior limit of a se...
climlimsup 44961 A sequence of real numbers...
limsupge 44962 The defining property of t...
liminfgval 44963 Value of the inferior limi...
liminfcl 44964 Closure of the inferior li...
liminfvald 44965 The inferior limit of a se...
liminfval5 44966 The inferior limit of an i...
limsupresxr 44967 The superior limit of a fu...
liminfresxr 44968 The inferior limit of a fu...
liminfval2 44969 The superior limit, relati...
climlimsupcex 44970 Counterexample for ~ climl...
liminfcld 44971 Closure of the inferior li...
liminfresico 44972 The inferior limit doesn't...
limsup10exlem 44973 The range of the given fun...
limsup10ex 44974 The superior limit of a fu...
liminf10ex 44975 The inferior limit of a fu...
liminflelimsuplem 44976 The superior limit is grea...
liminflelimsup 44977 The superior limit is grea...
limsupgtlem 44978 For any positive real, the...
limsupgt 44979 Given a sequence of real n...
liminfresre 44980 The inferior limit of a fu...
liminfresicompt 44981 The inferior limit doesn't...
liminfltlimsupex 44982 An example where the ` lim...
liminfgelimsup 44983 The inferior limit is grea...
liminfvalxr 44984 Alternate definition of ` ...
liminfresuz 44985 If the real part of the do...
liminflelimsupuz 44986 The superior limit is grea...
liminfvalxrmpt 44987 Alternate definition of ` ...
liminfresuz2 44988 If the domain of a functio...
liminfgelimsupuz 44989 The inferior limit is grea...
liminfval4 44990 Alternate definition of ` ...
liminfval3 44991 Alternate definition of ` ...
liminfequzmpt2 44992 Two functions that are eve...
liminfvaluz 44993 Alternate definition of ` ...
liminf0 44994 The inferior limit of the ...
limsupval4 44995 Alternate definition of ` ...
liminfvaluz2 44996 Alternate definition of ` ...
liminfvaluz3 44997 Alternate definition of ` ...
liminflelimsupcex 44998 A counterexample for ~ lim...
limsupvaluz3 44999 Alternate definition of ` ...
liminfvaluz4 45000 Alternate definition of ` ...
limsupvaluz4 45001 Alternate definition of ` ...
climliminflimsupd 45002 If a sequence of real numb...
liminfreuzlem 45003 Given a function on the re...
liminfreuz 45004 Given a function on the re...
liminfltlem 45005 Given a sequence of real n...
liminflt 45006 Given a sequence of real n...
climliminf 45007 A sequence of real numbers...
liminflimsupclim 45008 A sequence of real numbers...
climliminflimsup 45009 A sequence of real numbers...
climliminflimsup2 45010 A sequence of real numbers...
climliminflimsup3 45011 A sequence of real numbers...
climliminflimsup4 45012 A sequence of real numbers...
limsupub2 45013 A extended real valued fun...
limsupubuz2 45014 A sequence with values in ...
xlimpnfxnegmnf 45015 A sequence converges to ` ...
liminflbuz2 45016 A sequence with values in ...
liminfpnfuz 45017 The inferior limit of a fu...
liminflimsupxrre 45018 A sequence with values in ...
xlimrel 45021 The limit on extended real...
xlimres 45022 A function converges iff i...
xlimcl 45023 The limit of a sequence of...
rexlimddv2 45024 Restricted existential eli...
xlimclim 45025 Given a sequence of reals,...
xlimconst 45026 A constant sequence conver...
climxlim 45027 A converging sequence in t...
xlimbr 45028 Express the binary relatio...
fuzxrpmcn 45029 A function mapping from an...
cnrefiisplem 45030 Lemma for ~ cnrefiisp (som...
cnrefiisp 45031 A non-real, complex number...
xlimxrre 45032 If a sequence ranging over...
xlimmnfvlem1 45033 Lemma for ~ xlimmnfv : the...
xlimmnfvlem2 45034 Lemma for ~ xlimmnf : the ...
xlimmnfv 45035 A function converges to mi...
xlimconst2 45036 A sequence that eventually...
xlimpnfvlem1 45037 Lemma for ~ xlimpnfv : the...
xlimpnfvlem2 45038 Lemma for ~ xlimpnfv : the...
xlimpnfv 45039 A function converges to pl...
xlimclim2lem 45040 Lemma for ~ xlimclim2 . H...
xlimclim2 45041 Given a sequence of extend...
xlimmnf 45042 A function converges to mi...
xlimpnf 45043 A function converges to pl...
xlimmnfmpt 45044 A function converges to pl...
xlimpnfmpt 45045 A function converges to pl...
climxlim2lem 45046 In this lemma for ~ climxl...
climxlim2 45047 A sequence of extended rea...
dfxlim2v 45048 An alternative definition ...
dfxlim2 45049 An alternative definition ...
climresd 45050 A function restricted to u...
climresdm 45051 A real function converges ...
dmclimxlim 45052 A real valued sequence tha...
xlimmnflimsup2 45053 A sequence of extended rea...
xlimuni 45054 An infinite sequence conve...
xlimclimdm 45055 A sequence of extended rea...
xlimfun 45056 The convergence relation o...
xlimmnflimsup 45057 If a sequence of extended ...
xlimdm 45058 Two ways to express that a...
xlimpnfxnegmnf2 45059 A sequence converges to ` ...
xlimresdm 45060 A function converges in th...
xlimpnfliminf 45061 If a sequence of extended ...
xlimpnfliminf2 45062 A sequence of extended rea...
xlimliminflimsup 45063 A sequence of extended rea...
xlimlimsupleliminf 45064 A sequence of extended rea...
coseq0 45065 A complex number whose cos...
sinmulcos 45066 Multiplication formula for...
coskpi2 45067 The cosine of an integer m...
cosnegpi 45068 The cosine of negative ` _...
sinaover2ne0 45069 If ` A ` in ` ( 0 , 2 _pi ...
cosknegpi 45070 The cosine of an integer m...
mulcncff 45071 The multiplication of two ...
cncfmptssg 45072 A continuous complex funct...
constcncfg 45073 A constant function is a c...
idcncfg 45074 The identity function is a...
cncfshift 45075 A periodic continuous func...
resincncf 45076 ` sin ` restricted to real...
addccncf2 45077 Adding a constant is a con...
0cnf 45078 The empty set is a continu...
fsumcncf 45079 The finite sum of continuo...
cncfperiod 45080 A periodic continuous func...
subcncff 45081 The subtraction of two con...
negcncfg 45082 The opposite of a continuo...
cnfdmsn 45083 A function with a singleto...
cncfcompt 45084 Composition of continuous ...
addcncff 45085 The sum of two continuous ...
ioccncflimc 45086 Limit at the upper bound o...
cncfuni 45087 A complex function on a su...
icccncfext 45088 A continuous function on a...
cncficcgt0 45089 A the absolute value of a ...
icocncflimc 45090 Limit at the lower bound, ...
cncfdmsn 45091 A complex function with a ...
divcncff 45092 The quotient of two contin...
cncfshiftioo 45093 A periodic continuous func...
cncfiooicclem1 45094 A continuous function ` F ...
cncfiooicc 45095 A continuous function ` F ...
cncfiooiccre 45096 A continuous function ` F ...
cncfioobdlem 45097 ` G ` actually extends ` F...
cncfioobd 45098 A continuous function ` F ...
jumpncnp 45099 Jump discontinuity or disc...
cxpcncf2 45100 The complex power function...
fprodcncf 45101 The finite product of cont...
add1cncf 45102 Addition to a constant is ...
add2cncf 45103 Addition to a constant is ...
sub1cncfd 45104 Subtracting a constant is ...
sub2cncfd 45105 Subtraction from a constan...
fprodsub2cncf 45106 ` F ` is continuous. (Con...
fprodadd2cncf 45107 ` F ` is continuous. (Con...
fprodsubrecnncnvlem 45108 The sequence ` S ` of fini...
fprodsubrecnncnv 45109 The sequence ` S ` of fini...
fprodaddrecnncnvlem 45110 The sequence ` S ` of fini...
fprodaddrecnncnv 45111 The sequence ` S ` of fini...
dvsinexp 45112 The derivative of sin^N . ...
dvcosre 45113 The real derivative of the...
dvsinax 45114 Derivative exercise: the d...
dvsubf 45115 The subtraction rule for e...
dvmptconst 45116 Function-builder for deriv...
dvcnre 45117 From complex differentiati...
dvmptidg 45118 Function-builder for deriv...
dvresntr 45119 Function-builder for deriv...
fperdvper 45120 The derivative of a period...
dvasinbx 45121 Derivative exercise: the d...
dvresioo 45122 Restriction of a derivativ...
dvdivf 45123 The quotient rule for ever...
dvdivbd 45124 A sufficient condition for...
dvsubcncf 45125 A sufficient condition for...
dvmulcncf 45126 A sufficient condition for...
dvcosax 45127 Derivative exercise: the d...
dvdivcncf 45128 A sufficient condition for...
dvbdfbdioolem1 45129 Given a function with boun...
dvbdfbdioolem2 45130 A function on an open inte...
dvbdfbdioo 45131 A function on an open inte...
ioodvbdlimc1lem1 45132 If ` F ` has bounded deriv...
ioodvbdlimc1lem2 45133 Limit at the lower bound o...
ioodvbdlimc1 45134 A real function with bound...
ioodvbdlimc2lem 45135 Limit at the upper bound o...
ioodvbdlimc2 45136 A real function with bound...
dvdmsscn 45137 ` X ` is a subset of ` CC ...
dvmptmulf 45138 Function-builder for deriv...
dvnmptdivc 45139 Function-builder for itera...
dvdsn1add 45140 If ` K ` divides ` N ` but...
dvxpaek 45141 Derivative of the polynomi...
dvnmptconst 45142 The ` N ` -th derivative o...
dvnxpaek 45143 The ` n ` -th derivative o...
dvnmul 45144 Function-builder for the `...
dvmptfprodlem 45145 Induction step for ~ dvmpt...
dvmptfprod 45146 Function-builder for deriv...
dvnprodlem1 45147 ` D ` is bijective. (Cont...
dvnprodlem2 45148 Induction step for ~ dvnpr...
dvnprodlem3 45149 The multinomial formula fo...
dvnprod 45150 The multinomial formula fo...
itgsin0pilem1 45151 Calculation of the integra...
ibliccsinexp 45152 sin^n on a closed interval...
itgsin0pi 45153 Calculation of the integra...
iblioosinexp 45154 sin^n on an open integral ...
itgsinexplem1 45155 Integration by parts is ap...
itgsinexp 45156 A recursive formula for th...
iblconstmpt 45157 A constant function is int...
itgeq1d 45158 Equality theorem for an in...
mbfres2cn 45159 Measurability of a piecewi...
vol0 45160 The measure of the empty s...
ditgeqiooicc 45161 A function ` F ` on an ope...
volge0 45162 The volume of a set is alw...
cnbdibl 45163 A continuous bounded funct...
snmbl 45164 A singleton is measurable....
ditgeq3d 45165 Equality theorem for the d...
iblempty 45166 The empty function is inte...
iblsplit 45167 The union of two integrabl...
volsn 45168 A singleton has 0 Lebesgue...
itgvol0 45169 If the domani is negligibl...
itgcoscmulx 45170 Exercise: the integral of ...
iblsplitf 45171 A version of ~ iblsplit us...
ibliooicc 45172 If a function is integrabl...
volioc 45173 The measure of a left-open...
iblspltprt 45174 If a function is integrabl...
itgsincmulx 45175 Exercise: the integral of ...
itgsubsticclem 45176 lemma for ~ itgsubsticc . ...
itgsubsticc 45177 Integration by u-substitut...
itgioocnicc 45178 The integral of a piecewis...
iblcncfioo 45179 A continuous function ` F ...
itgspltprt 45180 The ` S. ` integral splits...
itgiccshift 45181 The integral of a function...
itgperiod 45182 The integral of a periodic...
itgsbtaddcnst 45183 Integral substitution, add...
volico 45184 The measure of left-closed...
sublevolico 45185 The Lebesgue measure of a ...
dmvolss 45186 Lebesgue measurable sets a...
ismbl3 45187 The predicate " ` A ` is L...
volioof 45188 The function that assigns ...
ovolsplit 45189 The Lebesgue outer measure...
fvvolioof 45190 The function value of the ...
volioore 45191 The measure of an open int...
fvvolicof 45192 The function value of the ...
voliooico 45193 An open interval and a lef...
ismbl4 45194 The predicate " ` A ` is L...
volioofmpt 45195 ` ( ( vol o. (,) ) o. F ) ...
volicoff 45196 ` ( ( vol o. [,) ) o. F ) ...
voliooicof 45197 The Lebesgue measure of op...
volicofmpt 45198 ` ( ( vol o. [,) ) o. F ) ...
volicc 45199 The Lebesgue measure of a ...
voliccico 45200 A closed interval and a le...
mbfdmssre 45201 The domain of a measurable...
stoweidlem1 45202 Lemma for ~ stoweid . Thi...
stoweidlem2 45203 lemma for ~ stoweid : here...
stoweidlem3 45204 Lemma for ~ stoweid : if `...
stoweidlem4 45205 Lemma for ~ stoweid : a cl...
stoweidlem5 45206 There exists a δ as ...
stoweidlem6 45207 Lemma for ~ stoweid : two ...
stoweidlem7 45208 This lemma is used to prov...
stoweidlem8 45209 Lemma for ~ stoweid : two ...
stoweidlem9 45210 Lemma for ~ stoweid : here...
stoweidlem10 45211 Lemma for ~ stoweid . Thi...
stoweidlem11 45212 This lemma is used to prov...
stoweidlem12 45213 Lemma for ~ stoweid . Thi...
stoweidlem13 45214 Lemma for ~ stoweid . Thi...
stoweidlem14 45215 There exists a ` k ` as in...
stoweidlem15 45216 This lemma is used to prov...
stoweidlem16 45217 Lemma for ~ stoweid . The...
stoweidlem17 45218 This lemma proves that the...
stoweidlem18 45219 This theorem proves Lemma ...
stoweidlem19 45220 If a set of real functions...
stoweidlem20 45221 If a set A of real functio...
stoweidlem21 45222 Once the Stone Weierstrass...
stoweidlem22 45223 If a set of real functions...
stoweidlem23 45224 This lemma is used to prov...
stoweidlem24 45225 This lemma proves that for...
stoweidlem25 45226 This lemma proves that for...
stoweidlem26 45227 This lemma is used to prov...
stoweidlem27 45228 This lemma is used to prov...
stoweidlem28 45229 There exists a δ as ...
stoweidlem29 45230 When the hypothesis for th...
stoweidlem30 45231 This lemma is used to prov...
stoweidlem31 45232 This lemma is used to prov...
stoweidlem32 45233 If a set A of real functio...
stoweidlem33 45234 If a set of real functions...
stoweidlem34 45235 This lemma proves that for...
stoweidlem35 45236 This lemma is used to prov...
stoweidlem36 45237 This lemma is used to prov...
stoweidlem37 45238 This lemma is used to prov...
stoweidlem38 45239 This lemma is used to prov...
stoweidlem39 45240 This lemma is used to prov...
stoweidlem40 45241 This lemma proves that q_n...
stoweidlem41 45242 This lemma is used to prov...
stoweidlem42 45243 This lemma is used to prov...
stoweidlem43 45244 This lemma is used to prov...
stoweidlem44 45245 This lemma is used to prov...
stoweidlem45 45246 This lemma proves that, gi...
stoweidlem46 45247 This lemma proves that set...
stoweidlem47 45248 Subtracting a constant fro...
stoweidlem48 45249 This lemma is used to prov...
stoweidlem49 45250 There exists a function q_...
stoweidlem50 45251 This lemma proves that set...
stoweidlem51 45252 There exists a function x ...
stoweidlem52 45253 There exists a neighborhoo...
stoweidlem53 45254 This lemma is used to prov...
stoweidlem54 45255 There exists a function ` ...
stoweidlem55 45256 This lemma proves the exis...
stoweidlem56 45257 This theorem proves Lemma ...
stoweidlem57 45258 There exists a function x ...
stoweidlem58 45259 This theorem proves Lemma ...
stoweidlem59 45260 This lemma proves that the...
stoweidlem60 45261 This lemma proves that the...
stoweidlem61 45262 This lemma proves that the...
stoweidlem62 45263 This theorem proves the St...
stoweid 45264 This theorem proves the St...
stowei 45265 This theorem proves the St...
wallispilem1 45266 ` I ` is monotone: increas...
wallispilem2 45267 A first set of properties ...
wallispilem3 45268 I maps to real values. (C...
wallispilem4 45269 ` F ` maps to explicit exp...
wallispilem5 45270 The sequence ` H ` converg...
wallispi 45271 Wallis' formula for Ï€ :...
wallispi2lem1 45272 An intermediate step betwe...
wallispi2lem2 45273 Two expressions are proven...
wallispi2 45274 An alternative version of ...
stirlinglem1 45275 A simple limit of fraction...
stirlinglem2 45276 ` A ` maps to positive rea...
stirlinglem3 45277 Long but simple algebraic ...
stirlinglem4 45278 Algebraic manipulation of ...
stirlinglem5 45279 If ` T ` is between ` 0 ` ...
stirlinglem6 45280 A series that converges to...
stirlinglem7 45281 Algebraic manipulation of ...
stirlinglem8 45282 If ` A ` converges to ` C ...
stirlinglem9 45283 ` ( ( B `` N ) - ( B `` ( ...
stirlinglem10 45284 A bound for any B(N)-B(N +...
stirlinglem11 45285 ` B ` is decreasing. (Con...
stirlinglem12 45286 The sequence ` B ` is boun...
stirlinglem13 45287 ` B ` is decreasing and ha...
stirlinglem14 45288 The sequence ` A ` converg...
stirlinglem15 45289 The Stirling's formula is ...
stirling 45290 Stirling's approximation f...
stirlingr 45291 Stirling's approximation f...
dirkerval 45292 The N_th Dirichlet Kernel....
dirker2re 45293 The Dirichlet Kernel value...
dirkerdenne0 45294 The Dirichlet Kernel denom...
dirkerval2 45295 The N_th Dirichlet Kernel ...
dirkerre 45296 The Dirichlet Kernel at an...
dirkerper 45297 the Dirichlet Kernel has p...
dirkerf 45298 For any natural number ` N...
dirkertrigeqlem1 45299 Sum of an even number of a...
dirkertrigeqlem2 45300 Trigonomic equality lemma ...
dirkertrigeqlem3 45301 Trigonometric equality lem...
dirkertrigeq 45302 Trigonometric equality for...
dirkeritg 45303 The definite integral of t...
dirkercncflem1 45304 If ` Y ` is a multiple of ...
dirkercncflem2 45305 Lemma used to prove that t...
dirkercncflem3 45306 The Dirichlet Kernel is co...
dirkercncflem4 45307 The Dirichlet Kernel is co...
dirkercncf 45308 For any natural number ` N...
fourierdlem1 45309 A partition interval is a ...
fourierdlem2 45310 Membership in a partition....
fourierdlem3 45311 Membership in a partition....
fourierdlem4 45312 ` E ` is a function that m...
fourierdlem5 45313 ` S ` is a function. (Con...
fourierdlem6 45314 ` X ` is in the periodic p...
fourierdlem7 45315 The difference between the...
fourierdlem8 45316 A partition interval is a ...
fourierdlem9 45317 ` H ` is a complex functio...
fourierdlem10 45318 Condition on the bounds of...
fourierdlem11 45319 If there is a partition, t...
fourierdlem12 45320 A point of a partition is ...
fourierdlem13 45321 Value of ` V ` in terms of...
fourierdlem14 45322 Given the partition ` V ` ...
fourierdlem15 45323 The range of the partition...
fourierdlem16 45324 The coefficients of the fo...
fourierdlem17 45325 The defined ` L ` is actua...
fourierdlem18 45326 The function ` S ` is cont...
fourierdlem19 45327 If two elements of ` D ` h...
fourierdlem20 45328 Every interval in the part...
fourierdlem21 45329 The coefficients of the fo...
fourierdlem22 45330 The coefficients of the fo...
fourierdlem23 45331 If ` F ` is continuous and...
fourierdlem24 45332 A sufficient condition for...
fourierdlem25 45333 If ` C ` is not in the ran...
fourierdlem26 45334 Periodic image of a point ...
fourierdlem27 45335 A partition open interval ...
fourierdlem28 45336 Derivative of ` ( F `` ( X...
fourierdlem29 45337 Explicit function value fo...
fourierdlem30 45338 Sum of three small pieces ...
fourierdlem31 45339 If ` A ` is finite and for...
fourierdlem32 45340 Limit of a continuous func...
fourierdlem33 45341 Limit of a continuous func...
fourierdlem34 45342 A partition is one to one....
fourierdlem35 45343 There is a single point in...
fourierdlem36 45344 ` F ` is an isomorphism. ...
fourierdlem37 45345 ` I ` is a function that m...
fourierdlem38 45346 The function ` F ` is cont...
fourierdlem39 45347 Integration by parts of ...
fourierdlem40 45348 ` H ` is a continuous func...
fourierdlem41 45349 Lemma used to prove that e...
fourierdlem42 45350 The set of points in a mov...
fourierdlem43 45351 ` K ` is a real function. ...
fourierdlem44 45352 A condition for having ` (...
fourierdlem46 45353 The function ` F ` has a l...
fourierdlem47 45354 For ` r ` large enough, th...
fourierdlem48 45355 The given periodic functio...
fourierdlem49 45356 The given periodic functio...
fourierdlem50 45357 Continuity of ` O ` and it...
fourierdlem51 45358 ` X ` is in the periodic p...
fourierdlem52 45359 d16:d17,d18:jca |- ( ph ->...
fourierdlem53 45360 The limit of ` F ( s ) ` a...
fourierdlem54 45361 Given a partition ` Q ` an...
fourierdlem55 45362 ` U ` is a real function. ...
fourierdlem56 45363 Derivative of the ` K ` fu...
fourierdlem57 45364 The derivative of ` O ` . ...
fourierdlem58 45365 The derivative of ` K ` is...
fourierdlem59 45366 The derivative of ` H ` is...
fourierdlem60 45367 Given a differentiable fun...
fourierdlem61 45368 Given a differentiable fun...
fourierdlem62 45369 The function ` K ` is cont...
fourierdlem63 45370 The upper bound of interva...
fourierdlem64 45371 The partition ` V ` is fin...
fourierdlem65 45372 The distance of two adjace...
fourierdlem66 45373 Value of the ` G ` functio...
fourierdlem67 45374 ` G ` is a function. (Con...
fourierdlem68 45375 The derivative of ` O ` is...
fourierdlem69 45376 A piecewise continuous fun...
fourierdlem70 45377 A piecewise continuous fun...
fourierdlem71 45378 A periodic piecewise conti...
fourierdlem72 45379 The derivative of ` O ` is...
fourierdlem73 45380 A version of the Riemann L...
fourierdlem74 45381 Given a piecewise smooth f...
fourierdlem75 45382 Given a piecewise smooth f...
fourierdlem76 45383 Continuity of ` O ` and it...
fourierdlem77 45384 If ` H ` is bounded, then ...
fourierdlem78 45385 ` G ` is continuous when r...
fourierdlem79 45386 ` E ` projects every inter...
fourierdlem80 45387 The derivative of ` O ` is...
fourierdlem81 45388 The integral of a piecewis...
fourierdlem82 45389 Integral by substitution, ...
fourierdlem83 45390 The fourier partial sum fo...
fourierdlem84 45391 If ` F ` is piecewise coni...
fourierdlem85 45392 Limit of the function ` G ...
fourierdlem86 45393 Continuity of ` O ` and it...
fourierdlem87 45394 The integral of ` G ` goes...
fourierdlem88 45395 Given a piecewise continuo...
fourierdlem89 45396 Given a piecewise continuo...
fourierdlem90 45397 Given a piecewise continuo...
fourierdlem91 45398 Given a piecewise continuo...
fourierdlem92 45399 The integral of a piecewis...
fourierdlem93 45400 Integral by substitution (...
fourierdlem94 45401 For a piecewise smooth fun...
fourierdlem95 45402 Algebraic manipulation of ...
fourierdlem96 45403 limit for ` F ` at the low...
fourierdlem97 45404 ` F ` is continuous on the...
fourierdlem98 45405 ` F ` is continuous on the...
fourierdlem99 45406 limit for ` F ` at the upp...
fourierdlem100 45407 A piecewise continuous fun...
fourierdlem101 45408 Integral by substitution f...
fourierdlem102 45409 For a piecewise smooth fun...
fourierdlem103 45410 The half lower part of the...
fourierdlem104 45411 The half upper part of the...
fourierdlem105 45412 A piecewise continuous fun...
fourierdlem106 45413 For a piecewise smooth fun...
fourierdlem107 45414 The integral of a piecewis...
fourierdlem108 45415 The integral of a piecewis...
fourierdlem109 45416 The integral of a piecewis...
fourierdlem110 45417 The integral of a piecewis...
fourierdlem111 45418 The fourier partial sum fo...
fourierdlem112 45419 Here abbreviations (local ...
fourierdlem113 45420 Fourier series convergence...
fourierdlem114 45421 Fourier series convergence...
fourierdlem115 45422 Fourier serier convergence...
fourierd 45423 Fourier series convergence...
fourierclimd 45424 Fourier series convergence...
fourierclim 45425 Fourier series convergence...
fourier 45426 Fourier series convergence...
fouriercnp 45427 If ` F ` is continuous at ...
fourier2 45428 Fourier series convergence...
sqwvfoura 45429 Fourier coefficients for t...
sqwvfourb 45430 Fourier series ` B ` coeff...
fourierswlem 45431 The Fourier series for the...
fouriersw 45432 Fourier series convergence...
fouriercn 45433 If the derivative of ` F `...
elaa2lem 45434 Elementhood in the set of ...
elaa2 45435 Elementhood in the set of ...
etransclem1 45436 ` H ` is a function. (Con...
etransclem2 45437 Derivative of ` G ` . (Co...
etransclem3 45438 The given ` if ` term is a...
etransclem4 45439 ` F ` expressed as a finit...
etransclem5 45440 A change of bound variable...
etransclem6 45441 A change of bound variable...
etransclem7 45442 The given product is an in...
etransclem8 45443 ` F ` is a function. (Con...
etransclem9 45444 If ` K ` divides ` N ` but...
etransclem10 45445 The given ` if ` term is a...
etransclem11 45446 A change of bound variable...
etransclem12 45447 ` C ` applied to ` N ` . ...
etransclem13 45448 ` F ` applied to ` Y ` . ...
etransclem14 45449 Value of the term ` T ` , ...
etransclem15 45450 Value of the term ` T ` , ...
etransclem16 45451 Every element in the range...
etransclem17 45452 The ` N ` -th derivative o...
etransclem18 45453 The given function is inte...
etransclem19 45454 The ` N ` -th derivative o...
etransclem20 45455 ` H ` is smooth. (Contrib...
etransclem21 45456 The ` N ` -th derivative o...
etransclem22 45457 The ` N ` -th derivative o...
etransclem23 45458 This is the claim proof in...
etransclem24 45459 ` P ` divides the I -th de...
etransclem25 45460 ` P ` factorial divides th...
etransclem26 45461 Every term in the sum of t...
etransclem27 45462 The ` N ` -th derivative o...
etransclem28 45463 ` ( P - 1 ) ` factorial di...
etransclem29 45464 The ` N ` -th derivative o...
etransclem30 45465 The ` N ` -th derivative o...
etransclem31 45466 The ` N ` -th derivative o...
etransclem32 45467 This is the proof for the ...
etransclem33 45468 ` F ` is smooth. (Contrib...
etransclem34 45469 The ` N ` -th derivative o...
etransclem35 45470 ` P ` does not divide the ...
etransclem36 45471 The ` N ` -th derivative o...
etransclem37 45472 ` ( P - 1 ) ` factorial di...
etransclem38 45473 ` P ` divides the I -th de...
etransclem39 45474 ` G ` is a function. (Con...
etransclem40 45475 The ` N ` -th derivative o...
etransclem41 45476 ` P ` does not divide the ...
etransclem42 45477 The ` N ` -th derivative o...
etransclem43 45478 ` G ` is a continuous func...
etransclem44 45479 The given finite sum is no...
etransclem45 45480 ` K ` is an integer. (Con...
etransclem46 45481 This is the proof for equa...
etransclem47 45482 ` _e ` is transcendental. ...
etransclem48 45483 ` _e ` is transcendental. ...
etransc 45484 ` _e ` is transcendental. ...
rrxtopn 45485 The topology of the genera...
rrxngp 45486 Generalized Euclidean real...
rrxtps 45487 Generalized Euclidean real...
rrxtopnfi 45488 The topology of the n-dime...
rrxtopon 45489 The topology on generalize...
rrxtop 45490 The topology on generalize...
rrndistlt 45491 Given two points in the sp...
rrxtoponfi 45492 The topology on n-dimensio...
rrxunitopnfi 45493 The base set of the standa...
rrxtopn0 45494 The topology of the zero-d...
qndenserrnbllem 45495 n-dimensional rational num...
qndenserrnbl 45496 n-dimensional rational num...
rrxtopn0b 45497 The topology of the zero-d...
qndenserrnopnlem 45498 n-dimensional rational num...
qndenserrnopn 45499 n-dimensional rational num...
qndenserrn 45500 n-dimensional rational num...
rrxsnicc 45501 A multidimensional singlet...
rrnprjdstle 45502 The distance between two p...
rrndsmet 45503 ` D ` is a metric for the ...
rrndsxmet 45504 ` D ` is an extended metri...
ioorrnopnlem 45505 The a point in an indexed ...
ioorrnopn 45506 The indexed product of ope...
ioorrnopnxrlem 45507 Given a point ` F ` that b...
ioorrnopnxr 45508 The indexed product of ope...
issal 45515 Express the predicate " ` ...
pwsal 45516 The power set of a given s...
salunicl 45517 SAlg sigma-algebra is clos...
saluncl 45518 The union of two sets in a...
prsal 45519 The pair of the empty set ...
saldifcl 45520 The complement of an eleme...
0sal 45521 The empty set belongs to e...
salgenval 45522 The sigma-algebra generate...
saliunclf 45523 SAlg sigma-algebra is clos...
saliuncl 45524 SAlg sigma-algebra is clos...
salincl 45525 The intersection of two se...
saluni 45526 A set is an element of any...
saliinclf 45527 SAlg sigma-algebra is clos...
saliincl 45528 SAlg sigma-algebra is clos...
saldifcl2 45529 The difference of two elem...
intsaluni 45530 The union of an arbitrary ...
intsal 45531 The arbitrary intersection...
salgenn0 45532 The set used in the defini...
salgencl 45533 ` SalGen ` actually genera...
issald 45534 Sufficient condition to pr...
salexct 45535 An example of nontrivial s...
sssalgen 45536 A set is a subset of the s...
salgenss 45537 The sigma-algebra generate...
salgenuni 45538 The base set of the sigma-...
issalgend 45539 One side of ~ dfsalgen2 . ...
salexct2 45540 An example of a subset tha...
unisalgen 45541 The union of a set belongs...
dfsalgen2 45542 Alternate characterization...
salexct3 45543 An example of a sigma-alge...
salgencntex 45544 This counterexample shows ...
salgensscntex 45545 This counterexample shows ...
issalnnd 45546 Sufficient condition to pr...
dmvolsal 45547 Lebesgue measurable sets f...
saldifcld 45548 The complement of an eleme...
saluncld 45549 The union of two sets in a...
salgencld 45550 ` SalGen ` actually genera...
0sald 45551 The empty set belongs to e...
iooborel 45552 An open interval is a Bore...
salincld 45553 The intersection of two se...
salunid 45554 A set is an element of any...
unisalgen2 45555 The union of a set belongs...
bor1sal 45556 The Borel sigma-algebra on...
iocborel 45557 A left-open, right-closed ...
subsaliuncllem 45558 A subspace sigma-algebra i...
subsaliuncl 45559 A subspace sigma-algebra i...
subsalsal 45560 A subspace sigma-algebra i...
subsaluni 45561 A set belongs to the subsp...
salrestss 45562 A sigma-algebra restricted...
sge0rnre 45565 When ` sum^ ` is applied t...
fge0icoicc 45566 If ` F ` maps to nonnegati...
sge0val 45567 The value of the sum of no...
fge0npnf 45568 If ` F ` maps to nonnegati...
sge0rnn0 45569 The range used in the defi...
sge0vald 45570 The value of the sum of no...
fge0iccico 45571 A range of nonnegative ext...
gsumge0cl 45572 Closure of group sum, for ...
sge0reval 45573 Value of the sum of nonneg...
sge0pnfval 45574 If a term in the sum of no...
fge0iccre 45575 A range of nonnegative ext...
sge0z 45576 Any nonnegative extended s...
sge00 45577 The sum of nonnegative ext...
fsumlesge0 45578 Every finite subsum of non...
sge0revalmpt 45579 Value of the sum of nonneg...
sge0sn 45580 A sum of a nonnegative ext...
sge0tsms 45581 ` sum^ ` applied to a nonn...
sge0cl 45582 The arbitrary sum of nonne...
sge0f1o 45583 Re-index a nonnegative ext...
sge0snmpt 45584 A sum of a nonnegative ext...
sge0ge0 45585 The sum of nonnegative ext...
sge0xrcl 45586 The arbitrary sum of nonne...
sge0repnf 45587 The of nonnegative extende...
sge0fsum 45588 The arbitrary sum of a fin...
sge0rern 45589 If the sum of nonnegative ...
sge0supre 45590 If the arbitrary sum of no...
sge0fsummpt 45591 The arbitrary sum of a fin...
sge0sup 45592 The arbitrary sum of nonne...
sge0less 45593 A shorter sum of nonnegati...
sge0rnbnd 45594 The range used in the defi...
sge0pr 45595 Sum of a pair of nonnegati...
sge0gerp 45596 The arbitrary sum of nonne...
sge0pnffigt 45597 If the sum of nonnegative ...
sge0ssre 45598 If a sum of nonnegative ex...
sge0lefi 45599 A sum of nonnegative exten...
sge0lessmpt 45600 A shorter sum of nonnegati...
sge0ltfirp 45601 If the sum of nonnegative ...
sge0prle 45602 The sum of a pair of nonne...
sge0gerpmpt 45603 The arbitrary sum of nonne...
sge0resrnlem 45604 The sum of nonnegative ext...
sge0resrn 45605 The sum of nonnegative ext...
sge0ssrempt 45606 If a sum of nonnegative ex...
sge0resplit 45607 ` sum^ ` splits into two p...
sge0le 45608 If all of the terms of sum...
sge0ltfirpmpt 45609 If the extended sum of non...
sge0split 45610 Split a sum of nonnegative...
sge0lempt 45611 If all of the terms of sum...
sge0splitmpt 45612 Split a sum of nonnegative...
sge0ss 45613 Change the index set to a ...
sge0iunmptlemfi 45614 Sum of nonnegative extende...
sge0p1 45615 The addition of the next t...
sge0iunmptlemre 45616 Sum of nonnegative extende...
sge0fodjrnlem 45617 Re-index a nonnegative ext...
sge0fodjrn 45618 Re-index a nonnegative ext...
sge0iunmpt 45619 Sum of nonnegative extende...
sge0iun 45620 Sum of nonnegative extende...
sge0nemnf 45621 The generalized sum of non...
sge0rpcpnf 45622 The sum of an infinite num...
sge0rernmpt 45623 If the sum of nonnegative ...
sge0lefimpt 45624 A sum of nonnegative exten...
nn0ssge0 45625 Nonnegative integers are n...
sge0clmpt 45626 The generalized sum of non...
sge0ltfirpmpt2 45627 If the extended sum of non...
sge0isum 45628 If a series of nonnegative...
sge0xrclmpt 45629 The generalized sum of non...
sge0xp 45630 Combine two generalized su...
sge0isummpt 45631 If a series of nonnegative...
sge0ad2en 45632 The value of the infinite ...
sge0isummpt2 45633 If a series of nonnegative...
sge0xaddlem1 45634 The extended addition of t...
sge0xaddlem2 45635 The extended addition of t...
sge0xadd 45636 The extended addition of t...
sge0fsummptf 45637 The generalized sum of a f...
sge0snmptf 45638 A sum of a nonnegative ext...
sge0ge0mpt 45639 The sum of nonnegative ext...
sge0repnfmpt 45640 The of nonnegative extende...
sge0pnffigtmpt 45641 If the generalized sum of ...
sge0splitsn 45642 Separate out a term in a g...
sge0pnffsumgt 45643 If the sum of nonnegative ...
sge0gtfsumgt 45644 If the generalized sum of ...
sge0uzfsumgt 45645 If a real number is smalle...
sge0pnfmpt 45646 If a term in the sum of no...
sge0seq 45647 A series of nonnegative re...
sge0reuz 45648 Value of the generalized s...
sge0reuzb 45649 Value of the generalized s...
ismea 45652 Express the predicate " ` ...
dmmeasal 45653 The domain of a measure is...
meaf 45654 A measure is a function th...
mea0 45655 The measure of the empty s...
nnfoctbdjlem 45656 There exists a mapping fro...
nnfoctbdj 45657 There exists a mapping fro...
meadjuni 45658 The measure of the disjoin...
meacl 45659 The measure of a set is a ...
iundjiunlem 45660 The sets in the sequence `...
iundjiun 45661 Given a sequence ` E ` of ...
meaxrcl 45662 The measure of a set is an...
meadjun 45663 The measure of the union o...
meassle 45664 The measure of a set is gr...
meaunle 45665 The measure of the union o...
meadjiunlem 45666 The sum of nonnegative ext...
meadjiun 45667 The measure of the disjoin...
ismeannd 45668 Sufficient condition to pr...
meaiunlelem 45669 The measure of the union o...
meaiunle 45670 The measure of the union o...
psmeasurelem 45671 ` M ` applied to a disjoin...
psmeasure 45672 Point supported measure, R...
voliunsge0lem 45673 The Lebesgue measure funct...
voliunsge0 45674 The Lebesgue measure funct...
volmea 45675 The Lebesgue measure on th...
meage0 45676 If the measure of a measur...
meadjunre 45677 The measure of the union o...
meassre 45678 If the measure of a measur...
meale0eq0 45679 A measure that is less tha...
meadif 45680 The measure of the differe...
meaiuninclem 45681 Measures are continuous fr...
meaiuninc 45682 Measures are continuous fr...
meaiuninc2 45683 Measures are continuous fr...
meaiunincf 45684 Measures are continuous fr...
meaiuninc3v 45685 Measures are continuous fr...
meaiuninc3 45686 Measures are continuous fr...
meaiininclem 45687 Measures are continuous fr...
meaiininc 45688 Measures are continuous fr...
meaiininc2 45689 Measures are continuous fr...
caragenval 45694 The sigma-algebra generate...
isome 45695 Express the predicate " ` ...
caragenel 45696 Membership in the Caratheo...
omef 45697 An outer measure is a func...
ome0 45698 The outer measure of the e...
omessle 45699 The outer measure of a set...
omedm 45700 The domain of an outer mea...
caragensplit 45701 If ` E ` is in the set gen...
caragenelss 45702 An element of the Caratheo...
carageneld 45703 Membership in the Caratheo...
omecl 45704 The outer measure of a set...
caragenss 45705 The sigma-algebra generate...
omeunile 45706 The outer measure of the u...
caragen0 45707 The empty set belongs to a...
omexrcl 45708 The outer measure of a set...
caragenunidm 45709 The base set of an outer m...
caragensspw 45710 The sigma-algebra generate...
omessre 45711 If the outer measure of a ...
caragenuni 45712 The base set of the sigma-...
caragenuncllem 45713 The Caratheodory's constru...
caragenuncl 45714 The Caratheodory's constru...
caragendifcl 45715 The Caratheodory's constru...
caragenfiiuncl 45716 The Caratheodory's constru...
omeunle 45717 The outer measure of the u...
omeiunle 45718 The outer measure of the i...
omelesplit 45719 The outer measure of a set...
omeiunltfirp 45720 If the outer measure of a ...
omeiunlempt 45721 The outer measure of the i...
carageniuncllem1 45722 The outer measure of ` A i...
carageniuncllem2 45723 The Caratheodory's constru...
carageniuncl 45724 The Caratheodory's constru...
caragenunicl 45725 The Caratheodory's constru...
caragensal 45726 Caratheodory's method gene...
caratheodorylem1 45727 Lemma used to prove that C...
caratheodorylem2 45728 Caratheodory's constructio...
caratheodory 45729 Caratheodory's constructio...
0ome 45730 The map that assigns 0 to ...
isomenndlem 45731 ` O ` is sub-additive w.r....
isomennd 45732 Sufficient condition to pr...
caragenel2d 45733 Membership in the Caratheo...
omege0 45734 If the outer measure of a ...
omess0 45735 If the outer measure of a ...
caragencmpl 45736 A measure built with the C...
vonval 45741 Value of the Lebesgue meas...
ovnval 45742 Value of the Lebesgue oute...
elhoi 45743 Membership in a multidimen...
icoresmbl 45744 A closed-below, open-above...
hoissre 45745 The projection of a half-o...
ovnval2 45746 Value of the Lebesgue oute...
volicorecl 45747 The Lebesgue measure of a ...
hoiprodcl 45748 The pre-measure of half-op...
hoicvr 45749 ` I ` is a countable set o...
hoissrrn 45750 A half-open interval is a ...
ovn0val 45751 The Lebesgue outer measure...
ovnn0val 45752 The value of a (multidimen...
ovnval2b 45753 Value of the Lebesgue oute...
volicorescl 45754 The Lebesgue measure of a ...
ovnprodcl 45755 The product used in the de...
hoiprodcl2 45756 The pre-measure of half-op...
hoicvrrex 45757 Any subset of the multidim...
ovnsupge0 45758 The set used in the defini...
ovnlecvr 45759 Given a subset of multidim...
ovnpnfelsup 45760 ` +oo ` is an element of t...
ovnsslelem 45761 The (multidimensional, non...
ovnssle 45762 The (multidimensional) Leb...
ovnlerp 45763 The Lebesgue outer measure...
ovnf 45764 The Lebesgue outer measure...
ovncvrrp 45765 The Lebesgue outer measure...
ovn0lem 45766 For any finite dimension, ...
ovn0 45767 For any finite dimension, ...
ovncl 45768 The Lebesgue outer measure...
ovn02 45769 For the zero-dimensional s...
ovnxrcl 45770 The Lebesgue outer measure...
ovnsubaddlem1 45771 The Lebesgue outer measure...
ovnsubaddlem2 45772 ` ( voln* `` X ) ` is suba...
ovnsubadd 45773 ` ( voln* `` X ) ` is suba...
ovnome 45774 ` ( voln* `` X ) ` is an o...
vonmea 45775 ` ( voln `` X ) ` is a mea...
volicon0 45776 The measure of a nonempty ...
hsphoif 45777 ` H ` is a function (that ...
hoidmvval 45778 The dimensional volume of ...
hoissrrn2 45779 A half-open interval is a ...
hsphoival 45780 ` H ` is a function (that ...
hoiprodcl3 45781 The pre-measure of half-op...
volicore 45782 The Lebesgue measure of a ...
hoidmvcl 45783 The dimensional volume of ...
hoidmv0val 45784 The dimensional volume of ...
hoidmvn0val 45785 The dimensional volume of ...
hsphoidmvle2 45786 The dimensional volume of ...
hsphoidmvle 45787 The dimensional volume of ...
hoidmvval0 45788 The dimensional volume of ...
hoiprodp1 45789 The dimensional volume of ...
sge0hsphoire 45790 If the generalized sum of ...
hoidmvval0b 45791 The dimensional volume of ...
hoidmv1lelem1 45792 The supremum of ` U ` belo...
hoidmv1lelem2 45793 This is the contradiction ...
hoidmv1lelem3 45794 The dimensional volume of ...
hoidmv1le 45795 The dimensional volume of ...
hoidmvlelem1 45796 The supremum of ` U ` belo...
hoidmvlelem2 45797 This is the contradiction ...
hoidmvlelem3 45798 This is the contradiction ...
hoidmvlelem4 45799 The dimensional volume of ...
hoidmvlelem5 45800 The dimensional volume of ...
hoidmvle 45801 The dimensional volume of ...
ovnhoilem1 45802 The Lebesgue outer measure...
ovnhoilem2 45803 The Lebesgue outer measure...
ovnhoi 45804 The Lebesgue outer measure...
dmovn 45805 The domain of the Lebesgue...
hoicoto2 45806 The half-open interval exp...
dmvon 45807 Lebesgue measurable n-dime...
hoi2toco 45808 The half-open interval exp...
hoidifhspval 45809 ` D ` is a function that r...
hspval 45810 The value of the half-spac...
ovnlecvr2 45811 Given a subset of multidim...
ovncvr2 45812 ` B ` and ` T ` are the le...
dmovnsal 45813 The domain of the Lebesgue...
unidmovn 45814 Base set of the n-dimensio...
rrnmbl 45815 The set of n-dimensional R...
hoidifhspval2 45816 ` D ` is a function that r...
hspdifhsp 45817 A n-dimensional half-open ...
unidmvon 45818 Base set of the n-dimensio...
hoidifhspf 45819 ` D ` is a function that r...
hoidifhspval3 45820 ` D ` is a function that r...
hoidifhspdmvle 45821 The dimensional volume of ...
voncmpl 45822 The Lebesgue measure is co...
hoiqssbllem1 45823 The center of the n-dimens...
hoiqssbllem2 45824 The center of the n-dimens...
hoiqssbllem3 45825 A n-dimensional ball conta...
hoiqssbl 45826 A n-dimensional ball conta...
hspmbllem1 45827 Any half-space of the n-di...
hspmbllem2 45828 Any half-space of the n-di...
hspmbllem3 45829 Any half-space of the n-di...
hspmbl 45830 Any half-space of the n-di...
hoimbllem 45831 Any n-dimensional half-ope...
hoimbl 45832 Any n-dimensional half-ope...
opnvonmbllem1 45833 The half-open interval exp...
opnvonmbllem2 45834 An open subset of the n-di...
opnvonmbl 45835 An open subset of the n-di...
opnssborel 45836 Open sets of a generalized...
borelmbl 45837 All Borel subsets of the n...
volicorege0 45838 The Lebesgue measure of a ...
isvonmbl 45839 The predicate " ` A ` is m...
mblvon 45840 The n-dimensional Lebesgue...
vonmblss 45841 n-dimensional Lebesgue mea...
volico2 45842 The measure of left-closed...
vonmblss2 45843 n-dimensional Lebesgue mea...
ovolval2lem 45844 The value of the Lebesgue ...
ovolval2 45845 The value of the Lebesgue ...
ovnsubadd2lem 45846 ` ( voln* `` X ) ` is suba...
ovnsubadd2 45847 ` ( voln* `` X ) ` is suba...
ovolval3 45848 The value of the Lebesgue ...
ovnsplit 45849 The n-dimensional Lebesgue...
ovolval4lem1 45850 |- ( ( ph /\ n e. A ) -> ...
ovolval4lem2 45851 The value of the Lebesgue ...
ovolval4 45852 The value of the Lebesgue ...
ovolval5lem1 45853 ` |- ( ph -> ( sum^ `` ( n...
ovolval5lem2 45854 ` |- ( ( ph /\ n e. NN ) -...
ovolval5lem3 45855 The value of the Lebesgue ...
ovolval5 45856 The value of the Lebesgue ...
ovnovollem1 45857 if ` F ` is a cover of ` B...
ovnovollem2 45858 if ` I ` is a cover of ` (...
ovnovollem3 45859 The 1-dimensional Lebesgue...
ovnovol 45860 The 1-dimensional Lebesgue...
vonvolmbllem 45861 If a subset ` B ` of real ...
vonvolmbl 45862 A subset of Real numbers i...
vonvol 45863 The 1-dimensional Lebesgue...
vonvolmbl2 45864 A subset ` X ` of the spac...
vonvol2 45865 The 1-dimensional Lebesgue...
hoimbl2 45866 Any n-dimensional half-ope...
voncl 45867 The Lebesgue measure of a ...
vonhoi 45868 The Lebesgue outer measure...
vonxrcl 45869 The Lebesgue measure of a ...
ioosshoi 45870 A n-dimensional open inter...
vonn0hoi 45871 The Lebesgue outer measure...
von0val 45872 The Lebesgue measure (for ...
vonhoire 45873 The Lebesgue measure of a ...
iinhoiicclem 45874 A n-dimensional closed int...
iinhoiicc 45875 A n-dimensional closed int...
iunhoiioolem 45876 A n-dimensional open inter...
iunhoiioo 45877 A n-dimensional open inter...
ioovonmbl 45878 Any n-dimensional open int...
iccvonmbllem 45879 Any n-dimensional closed i...
iccvonmbl 45880 Any n-dimensional closed i...
vonioolem1 45881 The sequence of the measur...
vonioolem2 45882 The n-dimensional Lebesgue...
vonioo 45883 The n-dimensional Lebesgue...
vonicclem1 45884 The sequence of the measur...
vonicclem2 45885 The n-dimensional Lebesgue...
vonicc 45886 The n-dimensional Lebesgue...
snvonmbl 45887 A n-dimensional singleton ...
vonn0ioo 45888 The n-dimensional Lebesgue...
vonn0icc 45889 The n-dimensional Lebesgue...
ctvonmbl 45890 Any n-dimensional countabl...
vonn0ioo2 45891 The n-dimensional Lebesgue...
vonsn 45892 The n-dimensional Lebesgue...
vonn0icc2 45893 The n-dimensional Lebesgue...
vonct 45894 The n-dimensional Lebesgue...
vitali2 45895 There are non-measurable s...
pimltmnf2f 45898 Given a real-valued functi...
pimltmnf2 45899 Given a real-valued functi...
preimagelt 45900 The preimage of a right-op...
preimalegt 45901 The preimage of a left-ope...
pimconstlt0 45902 Given a constant function,...
pimconstlt1 45903 Given a constant function,...
pimltpnff 45904 Given a real-valued functi...
pimltpnf 45905 Given a real-valued functi...
pimgtpnf2f 45906 Given a real-valued functi...
pimgtpnf2 45907 Given a real-valued functi...
salpreimagelt 45908 If all the preimages of le...
pimrecltpos 45909 The preimage of an unbound...
salpreimalegt 45910 If all the preimages of ri...
pimiooltgt 45911 The preimage of an open in...
preimaicomnf 45912 Preimage of an open interv...
pimltpnf2f 45913 Given a real-valued functi...
pimltpnf2 45914 Given a real-valued functi...
pimgtmnf2 45915 Given a real-valued functi...
pimdecfgtioc 45916 Given a nonincreasing func...
pimincfltioc 45917 Given a nondecreasing func...
pimdecfgtioo 45918 Given a nondecreasing func...
pimincfltioo 45919 Given a nondecreasing func...
preimaioomnf 45920 Preimage of an open interv...
preimageiingt 45921 A preimage of a left-close...
preimaleiinlt 45922 A preimage of a left-open,...
pimgtmnff 45923 Given a real-valued functi...
pimgtmnf 45924 Given a real-valued functi...
pimrecltneg 45925 The preimage of an unbound...
salpreimagtge 45926 If all the preimages of le...
salpreimaltle 45927 If all the preimages of ri...
issmflem 45928 The predicate " ` F ` is a...
issmf 45929 The predicate " ` F ` is a...
salpreimalelt 45930 If all the preimages of ri...
salpreimagtlt 45931 If all the preimages of le...
smfpreimalt 45932 Given a function measurabl...
smff 45933 A function measurable w.r....
smfdmss 45934 The domain of a function m...
issmff 45935 The predicate " ` F ` is a...
issmfd 45936 A sufficient condition for...
smfpreimaltf 45937 Given a function measurabl...
issmfdf 45938 A sufficient condition for...
sssmf 45939 The restriction of a sigma...
mbfresmf 45940 A real-valued measurable f...
cnfsmf 45941 A continuous function is m...
incsmflem 45942 A nondecreasing function i...
incsmf 45943 A real-valued, nondecreasi...
smfsssmf 45944 If a function is measurabl...
issmflelem 45945 The predicate " ` F ` is a...
issmfle 45946 The predicate " ` F ` is a...
smfpimltmpt 45947 Given a function measurabl...
smfpimltxr 45948 Given a function measurabl...
issmfdmpt 45949 A sufficient condition for...
smfconst 45950 Given a sigma-algebra over...
sssmfmpt 45951 The restriction of a sigma...
cnfrrnsmf 45952 A function, continuous fro...
smfid 45953 The identity function is B...
bormflebmf 45954 A Borel measurable functio...
smfpreimale 45955 Given a function measurabl...
issmfgtlem 45956 The predicate " ` F ` is a...
issmfgt 45957 The predicate " ` F ` is a...
issmfled 45958 A sufficient condition for...
smfpimltxrmptf 45959 Given a function measurabl...
smfpimltxrmpt 45960 Given a function measurabl...
smfmbfcex 45961 A constant function, with ...
issmfgtd 45962 A sufficient condition for...
smfpreimagt 45963 Given a function measurabl...
smfaddlem1 45964 Given the sum of two funct...
smfaddlem2 45965 The sum of two sigma-measu...
smfadd 45966 The sum of two sigma-measu...
decsmflem 45967 A nonincreasing function i...
decsmf 45968 A real-valued, nonincreasi...
smfpreimagtf 45969 Given a function measurabl...
issmfgelem 45970 The predicate " ` F ` is a...
issmfge 45971 The predicate " ` F ` is a...
smflimlem1 45972 Lemma for the proof that t...
smflimlem2 45973 Lemma for the proof that t...
smflimlem3 45974 The limit of sigma-measura...
smflimlem4 45975 Lemma for the proof that t...
smflimlem5 45976 Lemma for the proof that t...
smflimlem6 45977 Lemma for the proof that t...
smflim 45978 The limit of sigma-measura...
nsssmfmbflem 45979 The sigma-measurable funct...
nsssmfmbf 45980 The sigma-measurable funct...
smfpimgtxr 45981 Given a function measurabl...
smfpimgtmpt 45982 Given a function measurabl...
smfpreimage 45983 Given a function measurabl...
mbfpsssmf 45984 Real-valued measurable fun...
smfpimgtxrmptf 45985 Given a function measurabl...
smfpimgtxrmpt 45986 Given a function measurabl...
smfpimioompt 45987 Given a function measurabl...
smfpimioo 45988 Given a function measurabl...
smfresal 45989 Given a sigma-measurable f...
smfrec 45990 The reciprocal of a sigma-...
smfres 45991 The restriction of sigma-m...
smfmullem1 45992 The multiplication of two ...
smfmullem2 45993 The multiplication of two ...
smfmullem3 45994 The multiplication of two ...
smfmullem4 45995 The multiplication of two ...
smfmul 45996 The multiplication of two ...
smfmulc1 45997 A sigma-measurable functio...
smfdiv 45998 The fraction of two sigma-...
smfpimbor1lem1 45999 Every open set belongs to ...
smfpimbor1lem2 46000 Given a sigma-measurable f...
smfpimbor1 46001 Given a sigma-measurable f...
smf2id 46002 Twice the identity functio...
smfco 46003 The composition of a Borel...
smfneg 46004 The negative of a sigma-me...
smffmptf 46005 A function measurable w.r....
smffmpt 46006 A function measurable w.r....
smflim2 46007 The limit of a sequence of...
smfpimcclem 46008 Lemma for ~ smfpimcc given...
smfpimcc 46009 Given a countable set of s...
issmfle2d 46010 A sufficient condition for...
smflimmpt 46011 The limit of a sequence of...
smfsuplem1 46012 The supremum of a countabl...
smfsuplem2 46013 The supremum of a countabl...
smfsuplem3 46014 The supremum of a countabl...
smfsup 46015 The supremum of a countabl...
smfsupmpt 46016 The supremum of a countabl...
smfsupxr 46017 The supremum of a countabl...
smfinflem 46018 The infimum of a countable...
smfinf 46019 The infimum of a countable...
smfinfmpt 46020 The infimum of a countable...
smflimsuplem1 46021 If ` H ` converges, the ` ...
smflimsuplem2 46022 The superior limit of a se...
smflimsuplem3 46023 The limit of the ` ( H `` ...
smflimsuplem4 46024 If ` H ` converges, the ` ...
smflimsuplem5 46025 ` H ` converges to the sup...
smflimsuplem6 46026 The superior limit of a se...
smflimsuplem7 46027 The superior limit of a se...
smflimsuplem8 46028 The superior limit of a se...
smflimsup 46029 The superior limit of a se...
smflimsupmpt 46030 The superior limit of a se...
smfliminflem 46031 The inferior limit of a co...
smfliminf 46032 The inferior limit of a co...
smfliminfmpt 46033 The inferior limit of a co...
adddmmbl 46034 If two functions have doma...
adddmmbl2 46035 If two functions have doma...
muldmmbl 46036 If two functions have doma...
muldmmbl2 46037 If two functions have doma...
smfdmmblpimne 46038 If a measurable function w...
smfdivdmmbl 46039 If a functions and a sigma...
smfpimne 46040 Given a function measurabl...
smfpimne2 46041 Given a function measurabl...
smfdivdmmbl2 46042 If a functions and a sigma...
fsupdm 46043 The domain of the sup func...
fsupdm2 46044 The domain of the sup func...
smfsupdmmbllem 46045 If a countable set of sigm...
smfsupdmmbl 46046 If a countable set of sigm...
finfdm 46047 The domain of the inf func...
finfdm2 46048 The domain of the inf func...
smfinfdmmbllem 46049 If a countable set of sigm...
smfinfdmmbl 46050 If a countable set of sigm...
sigarval 46051 Define the signed area by ...
sigarim 46052 Signed area takes value in...
sigarac 46053 Signed area is anticommuta...
sigaraf 46054 Signed area is additive by...
sigarmf 46055 Signed area is additive (w...
sigaras 46056 Signed area is additive by...
sigarms 46057 Signed area is additive (w...
sigarls 46058 Signed area is linear by t...
sigarid 46059 Signed area of a flat para...
sigarexp 46060 Expand the signed area for...
sigarperm 46061 Signed area ` ( A - C ) G ...
sigardiv 46062 If signed area between vec...
sigarimcd 46063 Signed area takes value in...
sigariz 46064 If signed area is zero, th...
sigarcol 46065 Given three points ` A ` ,...
sharhght 46066 Let ` A B C ` be a triangl...
sigaradd 46067 Subtracting (double) area ...
cevathlem1 46068 Ceva's theorem first lemma...
cevathlem2 46069 Ceva's theorem second lemm...
cevath 46070 Ceva's theorem. Let ` A B...
simpcntrab 46071 The center of a simple gro...
et-ltneverrefl 46072 Less-than class is never r...
et-equeucl 46073 Alternative proof that equ...
et-sqrtnegnre 46074 The square root of a negat...
natlocalincr 46075 Global monotonicity on hal...
natglobalincr 46076 Local monotonicity on half...
upwordnul 46079 Empty set is an increasing...
upwordisword 46080 Any increasing sequence is...
singoutnword 46081 Singleton with character o...
singoutnupword 46082 Singleton with character o...
upwordsing 46083 Singleton is an increasing...
upwordsseti 46084 Strictly increasing sequen...
tworepnotupword 46085 Concatenation of identical...
upwrdfi 46086 There is a finite number o...
hirstL-ax3 46087 The third axiom of a syste...
ax3h 46088 Recover ~ ax-3 from ~ hirs...
aibandbiaiffaiffb 46089 A closed form showing (a i...
aibandbiaiaiffb 46090 A closed form showing (a i...
notatnand 46091 Do not use. Use intnanr i...
aistia 46092 Given a is equivalent to `...
aisfina 46093 Given a is equivalent to `...
bothtbothsame 46094 Given both a, b are equiva...
bothfbothsame 46095 Given both a, b are equiva...
aiffbbtat 46096 Given a is equivalent to b...
aisbbisfaisf 46097 Given a is equivalent to b...
axorbtnotaiffb 46098 Given a is exclusive to b,...
aiffnbandciffatnotciffb 46099 Given a is equivalent to (...
axorbciffatcxorb 46100 Given a is equivalent to (...
aibnbna 46101 Given a implies b, (not b)...
aibnbaif 46102 Given a implies b, not b, ...
aiffbtbat 46103 Given a is equivalent to b...
astbstanbst 46104 Given a is equivalent to T...
aistbistaandb 46105 Given a is equivalent to T...
aisbnaxb 46106 Given a is equivalent to b...
atbiffatnnb 46107 If a implies b, then a imp...
bisaiaisb 46108 Application of bicom1 with...
atbiffatnnbalt 46109 If a implies b, then a imp...
abnotbtaxb 46110 Assuming a, not b, there e...
abnotataxb 46111 Assuming not a, b, there e...
conimpf 46112 Assuming a, not b, and a i...
conimpfalt 46113 Assuming a, not b, and a i...
aistbisfiaxb 46114 Given a is equivalent to T...
aisfbistiaxb 46115 Given a is equivalent to F...
aifftbifffaibif 46116 Given a is equivalent to T...
aifftbifffaibifff 46117 Given a is equivalent to T...
atnaiana 46118 Given a, it is not the cas...
ainaiaandna 46119 Given a, a implies it is n...
abcdta 46120 Given (((a and b) and c) a...
abcdtb 46121 Given (((a and b) and c) a...
abcdtc 46122 Given (((a and b) and c) a...
abcdtd 46123 Given (((a and b) and c) a...
abciffcbatnabciffncba 46124 Operands in a biconditiona...
abciffcbatnabciffncbai 46125 Operands in a biconditiona...
nabctnabc 46126 not ( a -> ( b /\ c ) ) we...
jabtaib 46127 For when pm3.4 lacks a pm3...
onenotinotbothi 46128 From one negated implicati...
twonotinotbothi 46129 From these two negated imp...
clifte 46130 show d is the same as an i...
cliftet 46131 show d is the same as an i...
clifteta 46132 show d is the same as an i...
cliftetb 46133 show d is the same as an i...
confun 46134 Given the hypotheses there...
confun2 46135 Confun simplified to two p...
confun3 46136 Confun's more complex form...
confun4 46137 An attempt at derivative. ...
confun5 46138 An attempt at derivative. ...
plcofph 46139 Given, a,b and a "definiti...
pldofph 46140 Given, a,b c, d, "definiti...
plvcofph 46141 Given, a,b,d, and "definit...
plvcofphax 46142 Given, a,b,d, and "definit...
plvofpos 46143 rh is derivable because ON...
mdandyv0 46144 Given the equivalences set...
mdandyv1 46145 Given the equivalences set...
mdandyv2 46146 Given the equivalences set...
mdandyv3 46147 Given the equivalences set...
mdandyv4 46148 Given the equivalences set...
mdandyv5 46149 Given the equivalences set...
mdandyv6 46150 Given the equivalences set...
mdandyv7 46151 Given the equivalences set...
mdandyv8 46152 Given the equivalences set...
mdandyv9 46153 Given the equivalences set...
mdandyv10 46154 Given the equivalences set...
mdandyv11 46155 Given the equivalences set...
mdandyv12 46156 Given the equivalences set...
mdandyv13 46157 Given the equivalences set...
mdandyv14 46158 Given the equivalences set...
mdandyv15 46159 Given the equivalences set...
mdandyvr0 46160 Given the equivalences set...
mdandyvr1 46161 Given the equivalences set...
mdandyvr2 46162 Given the equivalences set...
mdandyvr3 46163 Given the equivalences set...
mdandyvr4 46164 Given the equivalences set...
mdandyvr5 46165 Given the equivalences set...
mdandyvr6 46166 Given the equivalences set...
mdandyvr7 46167 Given the equivalences set...
mdandyvr8 46168 Given the equivalences set...
mdandyvr9 46169 Given the equivalences set...
mdandyvr10 46170 Given the equivalences set...
mdandyvr11 46171 Given the equivalences set...
mdandyvr12 46172 Given the equivalences set...
mdandyvr13 46173 Given the equivalences set...
mdandyvr14 46174 Given the equivalences set...
mdandyvr15 46175 Given the equivalences set...
mdandyvrx0 46176 Given the exclusivities se...
mdandyvrx1 46177 Given the exclusivities se...
mdandyvrx2 46178 Given the exclusivities se...
mdandyvrx3 46179 Given the exclusivities se...
mdandyvrx4 46180 Given the exclusivities se...
mdandyvrx5 46181 Given the exclusivities se...
mdandyvrx6 46182 Given the exclusivities se...
mdandyvrx7 46183 Given the exclusivities se...
mdandyvrx8 46184 Given the exclusivities se...
mdandyvrx9 46185 Given the exclusivities se...
mdandyvrx10 46186 Given the exclusivities se...
mdandyvrx11 46187 Given the exclusivities se...
mdandyvrx12 46188 Given the exclusivities se...
mdandyvrx13 46189 Given the exclusivities se...
mdandyvrx14 46190 Given the exclusivities se...
mdandyvrx15 46191 Given the exclusivities se...
H15NH16TH15IH16 46192 Given 15 hypotheses and a ...
dandysum2p2e4 46193 CONTRADICTION PROVED AT 1 ...
mdandysum2p2e4 46194 CONTRADICTION PROVED AT 1 ...
adh-jarrsc 46195 Replacement of a nested an...
adh-minim 46196 A single axiom for minimal...
adh-minim-ax1-ax2-lem1 46197 First lemma for the deriva...
adh-minim-ax1-ax2-lem2 46198 Second lemma for the deriv...
adh-minim-ax1-ax2-lem3 46199 Third lemma for the deriva...
adh-minim-ax1-ax2-lem4 46200 Fourth lemma for the deriv...
adh-minim-ax1 46201 Derivation of ~ ax-1 from ...
adh-minim-ax2-lem5 46202 Fifth lemma for the deriva...
adh-minim-ax2-lem6 46203 Sixth lemma for the deriva...
adh-minim-ax2c 46204 Derivation of a commuted f...
adh-minim-ax2 46205 Derivation of ~ ax-2 from ...
adh-minim-idALT 46206 Derivation of ~ id (reflex...
adh-minim-pm2.43 46207 Derivation of ~ pm2.43 Whi...
adh-minimp 46208 Another single axiom for m...
adh-minimp-jarr-imim1-ax2c-lem1 46209 First lemma for the deriva...
adh-minimp-jarr-lem2 46210 Second lemma for the deriv...
adh-minimp-jarr-ax2c-lem3 46211 Third lemma for the deriva...
adh-minimp-sylsimp 46212 Derivation of ~ jarr (also...
adh-minimp-ax1 46213 Derivation of ~ ax-1 from ...
adh-minimp-imim1 46214 Derivation of ~ imim1 ("le...
adh-minimp-ax2c 46215 Derivation of a commuted f...
adh-minimp-ax2-lem4 46216 Fourth lemma for the deriv...
adh-minimp-ax2 46217 Derivation of ~ ax-2 from ...
adh-minimp-idALT 46218 Derivation of ~ id (reflex...
adh-minimp-pm2.43 46219 Derivation of ~ pm2.43 Whi...
n0nsn2el 46220 If a class with one elemen...
eusnsn 46221 There is a unique element ...
absnsb 46222 If the class abstraction `...
euabsneu 46223 Another way to express exi...
elprneb 46224 An element of a proper uno...
oppr 46225 Equality for ordered pairs...
opprb 46226 Equality for unordered pai...
or2expropbilem1 46227 Lemma 1 for ~ or2expropbi ...
or2expropbilem2 46228 Lemma 2 for ~ or2expropbi ...
or2expropbi 46229 If two classes are strictl...
eubrv 46230 If there is a unique set w...
eubrdm 46231 If there is a unique set w...
eldmressn 46232 Element of the domain of a...
iota0def 46233 Example for a defined iota...
iota0ndef 46234 Example for an undefined i...
fveqvfvv 46235 If a function's value at a...
fnresfnco 46236 Composition of two functio...
funcoressn 46237 A composition restricted t...
funressnfv 46238 A restriction to a singlet...
funressndmfvrn 46239 The value of a function ` ...
funressnvmo 46240 A function restricted to a...
funressnmo 46241 A function restricted to a...
funressneu 46242 There is exactly one value...
fresfo 46243 Conditions for a restricti...
fsetsniunop 46244 The class of all functions...
fsetabsnop 46245 The class of all functions...
fsetsnf 46246 The mapping of an element ...
fsetsnf1 46247 The mapping of an element ...
fsetsnfo 46248 The mapping of an element ...
fsetsnf1o 46249 The mapping of an element ...
fsetsnprcnex 46250 The class of all functions...
cfsetssfset 46251 The class of constant func...
cfsetsnfsetfv 46252 The function value of the ...
cfsetsnfsetf 46253 The mapping of the class o...
cfsetsnfsetf1 46254 The mapping of the class o...
cfsetsnfsetfo 46255 The mapping of the class o...
cfsetsnfsetf1o 46256 The mapping of the class o...
fsetprcnexALT 46257 First version of proof for...
fcoreslem1 46258 Lemma 1 for ~ fcores . (C...
fcoreslem2 46259 Lemma 2 for ~ fcores . (C...
fcoreslem3 46260 Lemma 3 for ~ fcores . (C...
fcoreslem4 46261 Lemma 4 for ~ fcores . (C...
fcores 46262 Every composite function `...
fcoresf1lem 46263 Lemma for ~ fcoresf1 . (C...
fcoresf1 46264 If a composition is inject...
fcoresf1b 46265 A composition is injective...
fcoresfo 46266 If a composition is surjec...
fcoresfob 46267 A composition is surjectiv...
fcoresf1ob 46268 A composition is bijective...
f1cof1blem 46269 Lemma for ~ f1cof1b and ~ ...
f1cof1b 46270 If the range of ` F ` equa...
funfocofob 46271 If the domain of a functio...
fnfocofob 46272 If the domain of a functio...
focofob 46273 If the domain of a functio...
f1ocof1ob 46274 If the range of ` F ` equa...
f1ocof1ob2 46275 If the range of ` F ` equa...
aiotajust 46277 Soundness justification th...
dfaiota2 46279 Alternate definition of th...
reuabaiotaiota 46280 The iota and the alternate...
reuaiotaiota 46281 The iota and the alternate...
aiotaexb 46282 The alternate iota over a ...
aiotavb 46283 The alternate iota over a ...
aiotaint 46284 This is to ~ df-aiota what...
dfaiota3 46285 Alternate definition of ` ...
iotan0aiotaex 46286 If the iota over a wff ` p...
aiotaexaiotaiota 46287 The alternate iota over a ...
aiotaval 46288 Theorem 8.19 in [Quine] p....
aiota0def 46289 Example for a defined alte...
aiota0ndef 46290 Example for an undefined a...
r19.32 46291 Theorem 19.32 of [Margaris...
rexsb 46292 An equivalent expression f...
rexrsb 46293 An equivalent expression f...
2rexsb 46294 An equivalent expression f...
2rexrsb 46295 An equivalent expression f...
cbvral2 46296 Change bound variables of ...
cbvrex2 46297 Change bound variables of ...
ralndv1 46298 Example for a theorem abou...
ralndv2 46299 Second example for a theor...
reuf1odnf 46300 There is exactly one eleme...
reuf1od 46301 There is exactly one eleme...
euoreqb 46302 There is a set which is eq...
2reu3 46303 Double restricted existent...
2reu7 46304 Two equivalent expressions...
2reu8 46305 Two equivalent expressions...
2reu8i 46306 Implication of a double re...
2reuimp0 46307 Implication of a double re...
2reuimp 46308 Implication of a double re...
ralbinrald 46315 Elemination of a restricte...
nvelim 46316 If a class is the universa...
alneu 46317 If a statement holds for a...
eu2ndop1stv 46318 If there is a unique secon...
dfateq12d 46319 Equality deduction for "de...
nfdfat 46320 Bound-variable hypothesis ...
dfdfat2 46321 Alternate definition of th...
fundmdfat 46322 A function is defined at a...
dfatprc 46323 A function is not defined ...
dfatelrn 46324 The value of a function ` ...
dfafv2 46325 Alternative definition of ...
afveq12d 46326 Equality deduction for fun...
afveq1 46327 Equality theorem for funct...
afveq2 46328 Equality theorem for funct...
nfafv 46329 Bound-variable hypothesis ...
csbafv12g 46330 Move class substitution in...
afvfundmfveq 46331 If a class is a function r...
afvnfundmuv 46332 If a set is not in the dom...
ndmafv 46333 The value of a class outsi...
afvvdm 46334 If the function value of a...
nfunsnafv 46335 If the restriction of a cl...
afvvfunressn 46336 If the function value of a...
afvprc 46337 A function's value at a pr...
afvvv 46338 If a function's value at a...
afvpcfv0 46339 If the value of the altern...
afvnufveq 46340 The value of the alternati...
afvvfveq 46341 The value of the alternati...
afv0fv0 46342 If the value of the altern...
afvfvn0fveq 46343 If the function's value at...
afv0nbfvbi 46344 The function's value at an...
afvfv0bi 46345 The function's value at an...
afveu 46346 The value of a function at...
fnbrafvb 46347 Equivalence of function va...
fnopafvb 46348 Equivalence of function va...
funbrafvb 46349 Equivalence of function va...
funopafvb 46350 Equivalence of function va...
funbrafv 46351 The second argument of a b...
funbrafv2b 46352 Function value in terms of...
dfafn5a 46353 Representation of a functi...
dfafn5b 46354 Representation of a functi...
fnrnafv 46355 The range of a function ex...
afvelrnb 46356 A member of a function's r...
afvelrnb0 46357 A member of a function's r...
dfaimafn 46358 Alternate definition of th...
dfaimafn2 46359 Alternate definition of th...
afvelima 46360 Function value in an image...
afvelrn 46361 A function's value belongs...
fnafvelrn 46362 A function's value belongs...
fafvelcdm 46363 A function's value belongs...
ffnafv 46364 A function maps to a class...
afvres 46365 The value of a restricted ...
tz6.12-afv 46366 Function value. Theorem 6...
tz6.12-1-afv 46367 Function value (Theorem 6....
dmfcoafv 46368 Domains of a function comp...
afvco2 46369 Value of a function compos...
rlimdmafv 46370 Two ways to express that a...
aoveq123d 46371 Equality deduction for ope...
nfaov 46372 Bound-variable hypothesis ...
csbaovg 46373 Move class substitution in...
aovfundmoveq 46374 If a class is a function r...
aovnfundmuv 46375 If an ordered pair is not ...
ndmaov 46376 The value of an operation ...
ndmaovg 46377 The value of an operation ...
aovvdm 46378 If the operation value of ...
nfunsnaov 46379 If the restriction of a cl...
aovvfunressn 46380 If the operation value of ...
aovprc 46381 The value of an operation ...
aovrcl 46382 Reverse closure for an ope...
aovpcov0 46383 If the alternative value o...
aovnuoveq 46384 The alternative value of t...
aovvoveq 46385 The alternative value of t...
aov0ov0 46386 If the alternative value o...
aovovn0oveq 46387 If the operation's value a...
aov0nbovbi 46388 The operation's value on a...
aovov0bi 46389 The operation's value on a...
rspceaov 46390 A frequently used special ...
fnotaovb 46391 Equivalence of operation v...
ffnaov 46392 An operation maps to a cla...
faovcl 46393 Closure law for an operati...
aovmpt4g 46394 Value of a function given ...
aoprssdm 46395 Domain of closure of an op...
ndmaovcl 46396 The "closure" of an operat...
ndmaovrcl 46397 Reverse closure law, in co...
ndmaovcom 46398 Any operation is commutati...
ndmaovass 46399 Any operation is associati...
ndmaovdistr 46400 Any operation is distribut...
dfatafv2iota 46403 If a function is defined a...
ndfatafv2 46404 The alternate function val...
ndfatafv2undef 46405 The alternate function val...
dfatafv2ex 46406 The alternate function val...
afv2ex 46407 The alternate function val...
afv2eq12d 46408 Equality deduction for fun...
afv2eq1 46409 Equality theorem for funct...
afv2eq2 46410 Equality theorem for funct...
nfafv2 46411 Bound-variable hypothesis ...
csbafv212g 46412 Move class substitution in...
fexafv2ex 46413 The alternate function val...
ndfatafv2nrn 46414 The alternate function val...
ndmafv2nrn 46415 The value of a class outsi...
funressndmafv2rn 46416 The alternate function val...
afv2ndefb 46417 Two ways to say that an al...
nfunsnafv2 46418 If the restriction of a cl...
afv2prc 46419 A function's value at a pr...
dfatafv2rnb 46420 The alternate function val...
afv2orxorb 46421 If a set is in the range o...
dmafv2rnb 46422 The alternate function val...
fundmafv2rnb 46423 The alternate function val...
afv2elrn 46424 An alternate function valu...
afv20defat 46425 If the alternate function ...
fnafv2elrn 46426 An alternate function valu...
fafv2elcdm 46427 An alternate function valu...
fafv2elrnb 46428 An alternate function valu...
fcdmvafv2v 46429 If the codomain of a funct...
tz6.12-2-afv2 46430 Function value when ` F ` ...
afv2eu 46431 The value of a function at...
afv2res 46432 The value of a restricted ...
tz6.12-afv2 46433 Function value (Theorem 6....
tz6.12-1-afv2 46434 Function value (Theorem 6....
tz6.12c-afv2 46435 Corollary of Theorem 6.12(...
tz6.12i-afv2 46436 Corollary of Theorem 6.12(...
funressnbrafv2 46437 The second argument of a b...
dfatbrafv2b 46438 Equivalence of function va...
dfatopafv2b 46439 Equivalence of function va...
funbrafv2 46440 The second argument of a b...
fnbrafv2b 46441 Equivalence of function va...
fnopafv2b 46442 Equivalence of function va...
funbrafv22b 46443 Equivalence of function va...
funopafv2b 46444 Equivalence of function va...
dfatsnafv2 46445 Singleton of function valu...
dfafv23 46446 A definition of function v...
dfatdmfcoafv2 46447 Domain of a function compo...
dfatcolem 46448 Lemma for ~ dfatco . (Con...
dfatco 46449 The predicate "defined at"...
afv2co2 46450 Value of a function compos...
rlimdmafv2 46451 Two ways to express that a...
dfafv22 46452 Alternate definition of ` ...
afv2ndeffv0 46453 If the alternate function ...
dfatafv2eqfv 46454 If a function is defined a...
afv2rnfveq 46455 If the alternate function ...
afv20fv0 46456 If the alternate function ...
afv2fvn0fveq 46457 If the function's value at...
afv2fv0 46458 If the function's value at...
afv2fv0b 46459 The function's value at an...
afv2fv0xorb 46460 If a set is in the range o...
an4com24 46461 Rearrangement of 4 conjunc...
3an4ancom24 46462 Commutative law for a conj...
4an21 46463 Rearrangement of 4 conjunc...
dfnelbr2 46466 Alternate definition of th...
nelbr 46467 The binary relation of a s...
nelbrim 46468 If a set is related to ano...
nelbrnel 46469 A set is related to anothe...
nelbrnelim 46470 If a set is related to ano...
ralralimp 46471 Selecting one of two alter...
otiunsndisjX 46472 The union of singletons co...
fvifeq 46473 Equality of function value...
rnfdmpr 46474 The range of a one-to-one ...
imarnf1pr 46475 The image of the range of ...
funop1 46476 A function is an ordered p...
fun2dmnopgexmpl 46477 A function with a domain c...
opabresex0d 46478 A collection of ordered pa...
opabbrfex0d 46479 A collection of ordered pa...
opabresexd 46480 A collection of ordered pa...
opabbrfexd 46481 A collection of ordered pa...
f1oresf1orab 46482 Build a bijection by restr...
f1oresf1o 46483 Build a bijection by restr...
f1oresf1o2 46484 Build a bijection by restr...
fvmptrab 46485 Value of a function mappin...
fvmptrabdm 46486 Value of a function mappin...
cnambpcma 46487 ((a-b)+c)-a = c-a holds fo...
cnapbmcpd 46488 ((a+b)-c)+d = ((a+d)+b)-c ...
addsubeq0 46489 The sum of two complex num...
leaddsuble 46490 Addition and subtraction o...
2leaddle2 46491 If two real numbers are le...
ltnltne 46492 Variant of trichotomy law ...
p1lep2 46493 A real number increasd by ...
ltsubsubaddltsub 46494 If the result of subtracti...
zm1nn 46495 An integer minus 1 is posi...
readdcnnred 46496 The sum of a real number a...
resubcnnred 46497 The difference of a real n...
recnmulnred 46498 The product of a real numb...
cndivrenred 46499 The quotient of an imagina...
sqrtnegnre 46500 The square root of a negat...
nn0resubcl 46501 Closure law for subtractio...
zgeltp1eq 46502 If an integer is between a...
1t10e1p1e11 46503 11 is 1 times 10 to the po...
deccarry 46504 Add 1 to a 2 digit number ...
eluzge0nn0 46505 If an integer is greater t...
nltle2tri 46506 Negated extended trichotom...
ssfz12 46507 Subset relationship for fi...
elfz2z 46508 Membership of an integer i...
2elfz3nn0 46509 If there are two elements ...
fz0addcom 46510 The addition of two member...
2elfz2melfz 46511 If the sum of two integers...
fz0addge0 46512 The sum of two integers in...
elfzlble 46513 Membership of an integer i...
elfzelfzlble 46514 Membership of an element o...
fzopred 46515 Join a predecessor to the ...
fzopredsuc 46516 Join a predecessor and a s...
1fzopredsuc 46517 Join 0 and a successor to ...
el1fzopredsuc 46518 An element of an open inte...
subsubelfzo0 46519 Subtracting a difference f...
fzoopth 46520 A half-open integer range ...
2ffzoeq 46521 Two functions over a half-...
m1mod0mod1 46522 An integer decreased by 1 ...
elmod2 46523 An integer modulo 2 is eit...
smonoord 46524 Ordering relation for a st...
fsummsndifre 46525 A finite sum with one of i...
fsumsplitsndif 46526 Separate out a term in a f...
fsummmodsndifre 46527 A finite sum of summands m...
fsummmodsnunz 46528 A finite sum of summands m...
setsidel 46529 The injected slot is an el...
setsnidel 46530 The injected slot is an el...
setsv 46531 The value of the structure...
preimafvsnel 46532 The preimage of a function...
preimafvn0 46533 The preimage of a function...
uniimafveqt 46534 The union of the image of ...
uniimaprimaeqfv 46535 The union of the image of ...
setpreimafvex 46536 The class ` P ` of all pre...
elsetpreimafvb 46537 The characterization of an...
elsetpreimafv 46538 An element of the class ` ...
elsetpreimafvssdm 46539 An element of the class ` ...
fvelsetpreimafv 46540 There is an element in a p...
preimafvelsetpreimafv 46541 The preimage of a function...
preimafvsspwdm 46542 The class ` P ` of all pre...
0nelsetpreimafv 46543 The empty set is not an el...
elsetpreimafvbi 46544 An element of the preimage...
elsetpreimafveqfv 46545 The elements of the preima...
eqfvelsetpreimafv 46546 If an element of the domai...
elsetpreimafvrab 46547 An element of the preimage...
imaelsetpreimafv 46548 The image of an element of...
uniimaelsetpreimafv 46549 The union of the image of ...
elsetpreimafveq 46550 If two preimages of functi...
fundcmpsurinjlem1 46551 Lemma 1 for ~ fundcmpsurin...
fundcmpsurinjlem2 46552 Lemma 2 for ~ fundcmpsurin...
fundcmpsurinjlem3 46553 Lemma 3 for ~ fundcmpsurin...
imasetpreimafvbijlemf 46554 Lemma for ~ imasetpreimafv...
imasetpreimafvbijlemfv 46555 Lemma for ~ imasetpreimafv...
imasetpreimafvbijlemfv1 46556 Lemma for ~ imasetpreimafv...
imasetpreimafvbijlemf1 46557 Lemma for ~ imasetpreimafv...
imasetpreimafvbijlemfo 46558 Lemma for ~ imasetpreimafv...
imasetpreimafvbij 46559 The mapping ` H ` is a bij...
fundcmpsurbijinjpreimafv 46560 Every function ` F : A -->...
fundcmpsurinjpreimafv 46561 Every function ` F : A -->...
fundcmpsurinj 46562 Every function ` F : A -->...
fundcmpsurbijinj 46563 Every function ` F : A -->...
fundcmpsurinjimaid 46564 Every function ` F : A -->...
fundcmpsurinjALT 46565 Alternate proof of ~ fundc...
iccpval 46568 Partition consisting of a ...
iccpart 46569 A special partition. Corr...
iccpartimp 46570 Implications for a class b...
iccpartres 46571 The restriction of a parti...
iccpartxr 46572 If there is a partition, t...
iccpartgtprec 46573 If there is a partition, t...
iccpartipre 46574 If there is a partition, t...
iccpartiltu 46575 If there is a partition, t...
iccpartigtl 46576 If there is a partition, t...
iccpartlt 46577 If there is a partition, t...
iccpartltu 46578 If there is a partition, t...
iccpartgtl 46579 If there is a partition, t...
iccpartgt 46580 If there is a partition, t...
iccpartleu 46581 If there is a partition, t...
iccpartgel 46582 If there is a partition, t...
iccpartrn 46583 If there is a partition, t...
iccpartf 46584 The range of the partition...
iccpartel 46585 If there is a partition, t...
iccelpart 46586 An element of any partitio...
iccpartiun 46587 A half-open interval of ex...
icceuelpartlem 46588 Lemma for ~ icceuelpart . ...
icceuelpart 46589 An element of a partitione...
iccpartdisj 46590 The segments of a partitio...
iccpartnel 46591 A point of a partition is ...
fargshiftfv 46592 If a class is a function, ...
fargshiftf 46593 If a class is a function, ...
fargshiftf1 46594 If a function is 1-1, then...
fargshiftfo 46595 If a function is onto, the...
fargshiftfva 46596 The values of a shifted fu...
lswn0 46597 The last symbol of a not e...
nfich1 46600 The first interchangeable ...
nfich2 46601 The second interchangeable...
ichv 46602 Setvar variables are inter...
ichf 46603 Setvar variables are inter...
ichid 46604 A setvar variable is alway...
icht 46605 A theorem is interchangeab...
ichbidv 46606 Formula building rule for ...
ichcircshi 46607 The setvar variables are i...
ichan 46608 If two setvar variables ar...
ichn 46609 Negation does not affect i...
ichim 46610 Formula building rule for ...
dfich2 46611 Alternate definition of th...
ichcom 46612 The interchangeability of ...
ichbi12i 46613 Equivalence for interchang...
icheqid 46614 In an equality for the sam...
icheq 46615 In an equality of setvar v...
ichnfimlem 46616 Lemma for ~ ichnfim : A s...
ichnfim 46617 If in an interchangeabilit...
ichnfb 46618 If ` x ` and ` y ` are int...
ichal 46619 Move a universal quantifie...
ich2al 46620 Two setvar variables are a...
ich2ex 46621 Two setvar variables are a...
ichexmpl1 46622 Example for interchangeabl...
ichexmpl2 46623 Example for interchangeabl...
ich2exprop 46624 If the setvar variables ar...
ichnreuop 46625 If the setvar variables ar...
ichreuopeq 46626 If the setvar variables ar...
sprid 46627 Two identical representati...
elsprel 46628 An unordered pair is an el...
spr0nelg 46629 The empty set is not an el...
sprval 46632 The set of all unordered p...
sprvalpw 46633 The set of all unordered p...
sprssspr 46634 The set of all unordered p...
spr0el 46635 The empty set is not an un...
sprvalpwn0 46636 The set of all unordered p...
sprel 46637 An element of the set of a...
prssspr 46638 An element of a subset of ...
prelspr 46639 An unordered pair of eleme...
prsprel 46640 The elements of a pair fro...
prsssprel 46641 The elements of a pair fro...
sprvalpwle2 46642 The set of all unordered p...
sprsymrelfvlem 46643 Lemma for ~ sprsymrelf and...
sprsymrelf1lem 46644 Lemma for ~ sprsymrelf1 . ...
sprsymrelfolem1 46645 Lemma 1 for ~ sprsymrelfo ...
sprsymrelfolem2 46646 Lemma 2 for ~ sprsymrelfo ...
sprsymrelfv 46647 The value of the function ...
sprsymrelf 46648 The mapping ` F ` is a fun...
sprsymrelf1 46649 The mapping ` F ` is a one...
sprsymrelfo 46650 The mapping ` F ` is a fun...
sprsymrelf1o 46651 The mapping ` F ` is a bij...
sprbisymrel 46652 There is a bijection betwe...
sprsymrelen 46653 The class ` P ` of subsets...
prpair 46654 Characterization of a prop...
prproropf1olem0 46655 Lemma 0 for ~ prproropf1o ...
prproropf1olem1 46656 Lemma 1 for ~ prproropf1o ...
prproropf1olem2 46657 Lemma 2 for ~ prproropf1o ...
prproropf1olem3 46658 Lemma 3 for ~ prproropf1o ...
prproropf1olem4 46659 Lemma 4 for ~ prproropf1o ...
prproropf1o 46660 There is a bijection betwe...
prproropen 46661 The set of proper pairs an...
prproropreud 46662 There is exactly one order...
pairreueq 46663 Two equivalent representat...
paireqne 46664 Two sets are not equal iff...
prprval 46667 The set of all proper unor...
prprvalpw 46668 The set of all proper unor...
prprelb 46669 An element of the set of a...
prprelprb 46670 A set is an element of the...
prprspr2 46671 The set of all proper unor...
prprsprreu 46672 There is a unique proper u...
prprreueq 46673 There is a unique proper u...
sbcpr 46674 The proper substitution of...
reupr 46675 There is a unique unordere...
reuprpr 46676 There is a unique proper u...
poprelb 46677 Equality for unordered pai...
2exopprim 46678 The existence of an ordere...
reuopreuprim 46679 There is a unique unordere...
fmtno 46682 The ` N ` th Fermat number...
fmtnoge3 46683 Each Fermat number is grea...
fmtnonn 46684 Each Fermat number is a po...
fmtnom1nn 46685 A Fermat number minus one ...
fmtnoodd 46686 Each Fermat number is odd....
fmtnorn 46687 A Fermat number is a funct...
fmtnof1 46688 The enumeration of the Fer...
fmtnoinf 46689 The set of Fermat numbers ...
fmtnorec1 46690 The first recurrence relat...
sqrtpwpw2p 46691 The floor of the square ro...
fmtnosqrt 46692 The floor of the square ro...
fmtno0 46693 The ` 0 ` th Fermat number...
fmtno1 46694 The ` 1 ` st Fermat number...
fmtnorec2lem 46695 Lemma for ~ fmtnorec2 (ind...
fmtnorec2 46696 The second recurrence rela...
fmtnodvds 46697 Any Fermat number divides ...
goldbachthlem1 46698 Lemma 1 for ~ goldbachth ....
goldbachthlem2 46699 Lemma 2 for ~ goldbachth ....
goldbachth 46700 Goldbach's theorem: Two d...
fmtnorec3 46701 The third recurrence relat...
fmtnorec4 46702 The fourth recurrence rela...
fmtno2 46703 The ` 2 ` nd Fermat number...
fmtno3 46704 The ` 3 ` rd Fermat number...
fmtno4 46705 The ` 4 ` th Fermat number...
fmtno5lem1 46706 Lemma 1 for ~ fmtno5 . (C...
fmtno5lem2 46707 Lemma 2 for ~ fmtno5 . (C...
fmtno5lem3 46708 Lemma 3 for ~ fmtno5 . (C...
fmtno5lem4 46709 Lemma 4 for ~ fmtno5 . (C...
fmtno5 46710 The ` 5 ` th Fermat number...
fmtno0prm 46711 The ` 0 ` th Fermat number...
fmtno1prm 46712 The ` 1 ` st Fermat number...
fmtno2prm 46713 The ` 2 ` nd Fermat number...
257prm 46714 257 is a prime number (the...
fmtno3prm 46715 The ` 3 ` rd Fermat number...
odz2prm2pw 46716 Any power of two is coprim...
fmtnoprmfac1lem 46717 Lemma for ~ fmtnoprmfac1 :...
fmtnoprmfac1 46718 Divisor of Fermat number (...
fmtnoprmfac2lem1 46719 Lemma for ~ fmtnoprmfac2 ....
fmtnoprmfac2 46720 Divisor of Fermat number (...
fmtnofac2lem 46721 Lemma for ~ fmtnofac2 (Ind...
fmtnofac2 46722 Divisor of Fermat number (...
fmtnofac1 46723 Divisor of Fermat number (...
fmtno4sqrt 46724 The floor of the square ro...
fmtno4prmfac 46725 If P was a (prime) factor ...
fmtno4prmfac193 46726 If P was a (prime) factor ...
fmtno4nprmfac193 46727 193 is not a (prime) facto...
fmtno4prm 46728 The ` 4 `-th Fermat number...
65537prm 46729 65537 is a prime number (t...
fmtnofz04prm 46730 The first five Fermat numb...
fmtnole4prm 46731 The first five Fermat numb...
fmtno5faclem1 46732 Lemma 1 for ~ fmtno5fac . ...
fmtno5faclem2 46733 Lemma 2 for ~ fmtno5fac . ...
fmtno5faclem3 46734 Lemma 3 for ~ fmtno5fac . ...
fmtno5fac 46735 The factorisation of the `...
fmtno5nprm 46736 The ` 5 ` th Fermat number...
prmdvdsfmtnof1lem1 46737 Lemma 1 for ~ prmdvdsfmtno...
prmdvdsfmtnof1lem2 46738 Lemma 2 for ~ prmdvdsfmtno...
prmdvdsfmtnof 46739 The mapping of a Fermat nu...
prmdvdsfmtnof1 46740 The mapping of a Fermat nu...
prminf2 46741 The set of prime numbers i...
2pwp1prm 46742 For ` ( ( 2 ^ k ) + 1 ) ` ...
2pwp1prmfmtno 46743 Every prime number of the ...
m2prm 46744 The second Mersenne number...
m3prm 46745 The third Mersenne number ...
flsqrt 46746 A condition equivalent to ...
flsqrt5 46747 The floor of the square ro...
3ndvds4 46748 3 does not divide 4. (Con...
139prmALT 46749 139 is a prime number. In...
31prm 46750 31 is a prime number. In ...
m5prm 46751 The fifth Mersenne number ...
127prm 46752 127 is a prime number. (C...
m7prm 46753 The seventh Mersenne numbe...
m11nprm 46754 The eleventh Mersenne numb...
mod42tp1mod8 46755 If a number is ` 3 ` modul...
sfprmdvdsmersenne 46756 If ` Q ` is a safe prime (...
sgprmdvdsmersenne 46757 If ` P ` is a Sophie Germa...
lighneallem1 46758 Lemma 1 for ~ lighneal . ...
lighneallem2 46759 Lemma 2 for ~ lighneal . ...
lighneallem3 46760 Lemma 3 for ~ lighneal . ...
lighneallem4a 46761 Lemma 1 for ~ lighneallem4...
lighneallem4b 46762 Lemma 2 for ~ lighneallem4...
lighneallem4 46763 Lemma 3 for ~ lighneal . ...
lighneal 46764 If a power of a prime ` P ...
modexp2m1d 46765 The square of an integer w...
proththdlem 46766 Lemma for ~ proththd . (C...
proththd 46767 Proth's theorem (1878). I...
5tcu2e40 46768 5 times the cube of 2 is 4...
3exp4mod41 46769 3 to the fourth power is -...
41prothprmlem1 46770 Lemma 1 for ~ 41prothprm ....
41prothprmlem2 46771 Lemma 2 for ~ 41prothprm ....
41prothprm 46772 41 is a _Proth prime_. (C...
quad1 46773 A condition for a quadrati...
requad01 46774 A condition for a quadrati...
requad1 46775 A condition for a quadrati...
requad2 46776 A condition for a quadrati...
iseven 46781 The predicate "is an even ...
isodd 46782 The predicate "is an odd n...
evenz 46783 An even number is an integ...
oddz 46784 An odd number is an intege...
evendiv2z 46785 The result of dividing an ...
oddp1div2z 46786 The result of dividing an ...
oddm1div2z 46787 The result of dividing an ...
isodd2 46788 The predicate "is an odd n...
dfodd2 46789 Alternate definition for o...
dfodd6 46790 Alternate definition for o...
dfeven4 46791 Alternate definition for e...
evenm1odd 46792 The predecessor of an even...
evenp1odd 46793 The successor of an even n...
oddp1eveni 46794 The successor of an odd nu...
oddm1eveni 46795 The predecessor of an odd ...
evennodd 46796 An even number is not an o...
oddneven 46797 An odd number is not an ev...
enege 46798 The negative of an even nu...
onego 46799 The negative of an odd num...
m1expevenALTV 46800 Exponentiation of -1 by an...
m1expoddALTV 46801 Exponentiation of -1 by an...
dfeven2 46802 Alternate definition for e...
dfodd3 46803 Alternate definition for o...
iseven2 46804 The predicate "is an even ...
isodd3 46805 The predicate "is an odd n...
2dvdseven 46806 2 divides an even number. ...
m2even 46807 A multiple of 2 is an even...
2ndvdsodd 46808 2 does not divide an odd n...
2dvdsoddp1 46809 2 divides an odd number in...
2dvdsoddm1 46810 2 divides an odd number de...
dfeven3 46811 Alternate definition for e...
dfodd4 46812 Alternate definition for o...
dfodd5 46813 Alternate definition for o...
zefldiv2ALTV 46814 The floor of an even numbe...
zofldiv2ALTV 46815 The floor of an odd numer ...
oddflALTV 46816 Odd number representation ...
iseven5 46817 The predicate "is an even ...
isodd7 46818 The predicate "is an odd n...
dfeven5 46819 Alternate definition for e...
dfodd7 46820 Alternate definition for o...
gcd2odd1 46821 The greatest common diviso...
zneoALTV 46822 No even integer equals an ...
zeoALTV 46823 An integer is even or odd....
zeo2ALTV 46824 An integer is even or odd ...
nneoALTV 46825 A positive integer is even...
nneoiALTV 46826 A positive integer is even...
odd2np1ALTV 46827 An integer is odd iff it i...
oddm1evenALTV 46828 An integer is odd iff its ...
oddp1evenALTV 46829 An integer is odd iff its ...
oexpnegALTV 46830 The exponential of the neg...
oexpnegnz 46831 The exponential of the neg...
bits0ALTV 46832 Value of the zeroth bit. ...
bits0eALTV 46833 The zeroth bit of an even ...
bits0oALTV 46834 The zeroth bit of an odd n...
divgcdoddALTV 46835 Either ` A / ( A gcd B ) `...
opoeALTV 46836 The sum of two odds is eve...
opeoALTV 46837 The sum of an odd and an e...
omoeALTV 46838 The difference of two odds...
omeoALTV 46839 The difference of an odd a...
oddprmALTV 46840 A prime not equal to ` 2 `...
0evenALTV 46841 0 is an even number. (Con...
0noddALTV 46842 0 is not an odd number. (...
1oddALTV 46843 1 is an odd number. (Cont...
1nevenALTV 46844 1 is not an even number. ...
2evenALTV 46845 2 is an even number. (Con...
2noddALTV 46846 2 is not an odd number. (...
nn0o1gt2ALTV 46847 An odd nonnegative integer...
nnoALTV 46848 An alternate characterizat...
nn0oALTV 46849 An alternate characterizat...
nn0e 46850 An alternate characterizat...
nneven 46851 An alternate characterizat...
nn0onn0exALTV 46852 For each odd nonnegative i...
nn0enn0exALTV 46853 For each even nonnegative ...
nnennexALTV 46854 For each even positive int...
nnpw2evenALTV 46855 2 to the power of a positi...
epoo 46856 The sum of an even and an ...
emoo 46857 The difference of an even ...
epee 46858 The sum of two even number...
emee 46859 The difference of two even...
evensumeven 46860 If a summand is even, the ...
3odd 46861 3 is an odd number. (Cont...
4even 46862 4 is an even number. (Con...
5odd 46863 5 is an odd number. (Cont...
6even 46864 6 is an even number. (Con...
7odd 46865 7 is an odd number. (Cont...
8even 46866 8 is an even number. (Con...
evenprm2 46867 A prime number is even iff...
oddprmne2 46868 Every prime number not bei...
oddprmuzge3 46869 A prime number which is od...
evenltle 46870 If an even number is great...
odd2prm2 46871 If an odd number is the su...
even3prm2 46872 If an even number is the s...
mogoldbblem 46873 Lemma for ~ mogoldbb . (C...
perfectALTVlem1 46874 Lemma for ~ perfectALTV . ...
perfectALTVlem2 46875 Lemma for ~ perfectALTV . ...
perfectALTV 46876 The Euclid-Euler theorem, ...
fppr 46879 The set of Fermat pseudopr...
fpprmod 46880 The set of Fermat pseudopr...
fpprel 46881 A Fermat pseudoprime to th...
fpprbasnn 46882 The base of a Fermat pseud...
fpprnn 46883 A Fermat pseudoprime to th...
fppr2odd 46884 A Fermat pseudoprime to th...
11t31e341 46885 341 is the product of 11 a...
2exp340mod341 46886 Eight to the eighth power ...
341fppr2 46887 341 is the (smallest) _Pou...
4fppr1 46888 4 is the (smallest) Fermat...
8exp8mod9 46889 Eight to the eighth power ...
9fppr8 46890 9 is the (smallest) Fermat...
dfwppr 46891 Alternate definition of a ...
fpprwppr 46892 A Fermat pseudoprime to th...
fpprwpprb 46893 An integer ` X ` which is ...
fpprel2 46894 An alternate definition fo...
nfermltl8rev 46895 Fermat's little theorem wi...
nfermltl2rev 46896 Fermat's little theorem wi...
nfermltlrev 46897 Fermat's little theorem re...
isgbe 46904 The predicate "is an even ...
isgbow 46905 The predicate "is a weak o...
isgbo 46906 The predicate "is an odd G...
gbeeven 46907 An even Goldbach number is...
gbowodd 46908 A weak odd Goldbach number...
gbogbow 46909 A (strong) odd Goldbach nu...
gboodd 46910 An odd Goldbach number is ...
gbepos 46911 Any even Goldbach number i...
gbowpos 46912 Any weak odd Goldbach numb...
gbopos 46913 Any odd Goldbach number is...
gbegt5 46914 Any even Goldbach number i...
gbowgt5 46915 Any weak odd Goldbach numb...
gbowge7 46916 Any weak odd Goldbach numb...
gboge9 46917 Any odd Goldbach number is...
gbege6 46918 Any even Goldbach number i...
gbpart6 46919 The Goldbach partition of ...
gbpart7 46920 The (weak) Goldbach partit...
gbpart8 46921 The Goldbach partition of ...
gbpart9 46922 The (strong) Goldbach part...
gbpart11 46923 The (strong) Goldbach part...
6gbe 46924 6 is an even Goldbach numb...
7gbow 46925 7 is a weak odd Goldbach n...
8gbe 46926 8 is an even Goldbach numb...
9gbo 46927 9 is an odd Goldbach numbe...
11gbo 46928 11 is an odd Goldbach numb...
stgoldbwt 46929 If the strong ternary Gold...
sbgoldbwt 46930 If the strong binary Goldb...
sbgoldbst 46931 If the strong binary Goldb...
sbgoldbaltlem1 46932 Lemma 1 for ~ sbgoldbalt :...
sbgoldbaltlem2 46933 Lemma 2 for ~ sbgoldbalt :...
sbgoldbalt 46934 An alternate (related to t...
sbgoldbb 46935 If the strong binary Goldb...
sgoldbeven3prm 46936 If the binary Goldbach con...
sbgoldbm 46937 If the strong binary Goldb...
mogoldbb 46938 If the modern version of t...
sbgoldbmb 46939 The strong binary Goldbach...
sbgoldbo 46940 If the strong binary Goldb...
nnsum3primes4 46941 4 is the sum of at most 3 ...
nnsum4primes4 46942 4 is the sum of at most 4 ...
nnsum3primesprm 46943 Every prime is "the sum of...
nnsum4primesprm 46944 Every prime is "the sum of...
nnsum3primesgbe 46945 Any even Goldbach number i...
nnsum4primesgbe 46946 Any even Goldbach number i...
nnsum3primesle9 46947 Every integer greater than...
nnsum4primesle9 46948 Every integer greater than...
nnsum4primesodd 46949 If the (weak) ternary Gold...
nnsum4primesoddALTV 46950 If the (strong) ternary Go...
evengpop3 46951 If the (weak) ternary Gold...
evengpoap3 46952 If the (strong) ternary Go...
nnsum4primeseven 46953 If the (weak) ternary Gold...
nnsum4primesevenALTV 46954 If the (strong) ternary Go...
wtgoldbnnsum4prm 46955 If the (weak) ternary Gold...
stgoldbnnsum4prm 46956 If the (strong) ternary Go...
bgoldbnnsum3prm 46957 If the binary Goldbach con...
bgoldbtbndlem1 46958 Lemma 1 for ~ bgoldbtbnd :...
bgoldbtbndlem2 46959 Lemma 2 for ~ bgoldbtbnd ....
bgoldbtbndlem3 46960 Lemma 3 for ~ bgoldbtbnd ....
bgoldbtbndlem4 46961 Lemma 4 for ~ bgoldbtbnd ....
bgoldbtbnd 46962 If the binary Goldbach con...
tgoldbachgtALTV 46965 Variant of Thierry Arnoux'...
bgoldbachlt 46966 The binary Goldbach conjec...
tgblthelfgott 46968 The ternary Goldbach conje...
tgoldbachlt 46969 The ternary Goldbach conje...
tgoldbach 46970 The ternary Goldbach conje...
isomgrrel 46975 The isomorphy relation for...
isomgr 46976 The isomorphy relation for...
isisomgr 46977 Implications of two graphs...
isomgreqve 46978 A set is isomorphic to a h...
isomushgr 46979 The isomorphy relation for...
isomuspgrlem1 46980 Lemma 1 for ~ isomuspgr . ...
isomuspgrlem2a 46981 Lemma 1 for ~ isomuspgrlem...
isomuspgrlem2b 46982 Lemma 2 for ~ isomuspgrlem...
isomuspgrlem2c 46983 Lemma 3 for ~ isomuspgrlem...
isomuspgrlem2d 46984 Lemma 4 for ~ isomuspgrlem...
isomuspgrlem2e 46985 Lemma 5 for ~ isomuspgrlem...
isomuspgrlem2 46986 Lemma 2 for ~ isomuspgr . ...
isomuspgr 46987 The isomorphy relation for...
isomgrref 46988 The isomorphy relation is ...
isomgrsym 46989 The isomorphy relation is ...
isomgrsymb 46990 The isomorphy relation is ...
isomgrtrlem 46991 Lemma for ~ isomgrtr . (C...
isomgrtr 46992 The isomorphy relation is ...
strisomgrop 46993 A graph represented as an ...
ushrisomgr 46994 A simple hypergraph (with ...
1hegrlfgr 46995 A graph ` G ` with one hyp...
upwlksfval 46998 The set of simple walks (i...
isupwlk 46999 Properties of a pair of fu...
isupwlkg 47000 Generalization of ~ isupwl...
upwlkbprop 47001 Basic properties of a simp...
upwlkwlk 47002 A simple walk is a walk. ...
upgrwlkupwlk 47003 In a pseudograph, a walk i...
upgrwlkupwlkb 47004 In a pseudograph, the defi...
upgrisupwlkALT 47005 Alternate proof of ~ upgri...
upgredgssspr 47006 The set of edges of a pseu...
uspgropssxp 47007 The set ` G ` of "simple p...
uspgrsprfv 47008 The value of the function ...
uspgrsprf 47009 The mapping ` F ` is a fun...
uspgrsprf1 47010 The mapping ` F ` is a one...
uspgrsprfo 47011 The mapping ` F ` is a fun...
uspgrsprf1o 47012 The mapping ` F ` is a bij...
uspgrex 47013 The class ` G ` of all "si...
uspgrbispr 47014 There is a bijection betwe...
uspgrspren 47015 The set ` G ` of the "simp...
uspgrymrelen 47016 The set ` G ` of the "simp...
uspgrbisymrel 47017 There is a bijection betwe...
uspgrbisymrelALT 47018 Alternate proof of ~ uspgr...
ovn0dmfun 47019 If a class operation value...
xpsnopab 47020 A Cartesian product with a...
xpiun 47021 A Cartesian product expres...
ovn0ssdmfun 47022 If a class' operation valu...
fnxpdmdm 47023 The domain of the domain o...
cnfldsrngbas 47024 The base set of a subring ...
cnfldsrngadd 47025 The group addition operati...
cnfldsrngmul 47026 The ring multiplication op...
plusfreseq 47027 If the empty set is not co...
mgmplusfreseq 47028 If the empty set is not co...
0mgm 47029 A set with an empty base s...
opmpoismgm 47030 A structure with a group a...
copissgrp 47031 A structure with a constan...
copisnmnd 47032 A structure with a constan...
0nodd 47033 0 is not an odd integer. ...
1odd 47034 1 is an odd integer. (Con...
2nodd 47035 2 is not an odd integer. ...
oddibas 47036 Lemma 1 for ~ oddinmgm : ...
oddiadd 47037 Lemma 2 for ~ oddinmgm : ...
oddinmgm 47038 The structure of all odd i...
nnsgrpmgm 47039 The structure of positive ...
nnsgrp 47040 The structure of positive ...
nnsgrpnmnd 47041 The structure of positive ...
nn0mnd 47042 The set of nonnegative int...
gsumsplit2f 47043 Split a group sum into two...
gsumdifsndf 47044 Extract a summand from a f...
gsumfsupp 47045 A group sum of a family ca...
iscllaw 47052 The predicate "is a closed...
iscomlaw 47053 The predicate "is a commut...
clcllaw 47054 Closure of a closed operat...
isasslaw 47055 The predicate "is an assoc...
asslawass 47056 Associativity of an associ...
mgmplusgiopALT 47057 Slot 2 (group operation) o...
sgrpplusgaopALT 47058 Slot 2 (group operation) o...
intopval 47065 The internal (binary) oper...
intop 47066 An internal (binary) opera...
clintopval 47067 The closed (internal binar...
assintopval 47068 The associative (closed in...
assintopmap 47069 The associative (closed in...
isclintop 47070 The predicate "is a closed...
clintop 47071 A closed (internal binary)...
assintop 47072 An associative (closed int...
isassintop 47073 The predicate "is an assoc...
clintopcllaw 47074 The closure law holds for ...
assintopcllaw 47075 The closure low holds for ...
assintopasslaw 47076 The associative low holds ...
assintopass 47077 An associative (closed int...
ismgmALT 47086 The predicate "is a magma"...
iscmgmALT 47087 The predicate "is a commut...
issgrpALT 47088 The predicate "is a semigr...
iscsgrpALT 47089 The predicate "is a commut...
mgm2mgm 47090 Equivalence of the two def...
sgrp2sgrp 47091 Equivalence of the two def...
lmod0rng 47092 If the scalar ring of a mo...
nzrneg1ne0 47093 The additive inverse of th...
lidldomn1 47094 If a (left) ideal (which i...
lidlabl 47095 A (left) ideal of a ring i...
lidlrng 47096 A (left) ideal of a ring i...
zlidlring 47097 The zero (left) ideal of a...
uzlidlring 47098 Only the zero (left) ideal...
lidldomnnring 47099 A (left) ideal of a domain...
0even 47100 0 is an even integer. (Co...
1neven 47101 1 is not an even integer. ...
2even 47102 2 is an even integer. (Co...
2zlidl 47103 The even integers are a (l...
2zrng 47104 The ring of integers restr...
2zrngbas 47105 The base set of R is the s...
2zrngadd 47106 The group addition operati...
2zrng0 47107 The additive identity of R...
2zrngamgm 47108 R is an (additive) magma. ...
2zrngasgrp 47109 R is an (additive) semigro...
2zrngamnd 47110 R is an (additive) monoid....
2zrngacmnd 47111 R is a commutative (additi...
2zrngagrp 47112 R is an (additive) group. ...
2zrngaabl 47113 R is an (additive) abelian...
2zrngmul 47114 The ring multiplication op...
2zrngmmgm 47115 R is a (multiplicative) ma...
2zrngmsgrp 47116 R is a (multiplicative) se...
2zrngALT 47117 The ring of integers restr...
2zrngnmlid 47118 R has no multiplicative (l...
2zrngnmrid 47119 R has no multiplicative (r...
2zrngnmlid2 47120 R has no multiplicative (l...
2zrngnring 47121 R is not a unital ring. (...
cznrnglem 47122 Lemma for ~ cznrng : The ...
cznabel 47123 The ring constructed from ...
cznrng 47124 The ring constructed from ...
cznnring 47125 The ring constructed from ...
rngcvalALTV 47128 Value of the category of n...
rngcbasALTV 47129 Set of objects of the cate...
rngchomfvalALTV 47130 Set of arrows of the categ...
rngchomALTV 47131 Set of arrows of the categ...
elrngchomALTV 47132 A morphism of non-unital r...
rngccofvalALTV 47133 Composition in the categor...
rngccoALTV 47134 Composition in the categor...
rngccatidALTV 47135 Lemma for ~ rngccatALTV . ...
rngccatALTV 47136 The category of non-unital...
rngcidALTV 47137 The identity arrow in the ...
rngcsectALTV 47138 A section in the category ...
rngcinvALTV 47139 An inverse in the category...
rngcisoALTV 47140 An isomorphism in the cate...
rngchomffvalALTV 47141 The value of the functiona...
rngchomrnghmresALTV 47142 The value of the functiona...
rngcrescrhmALTV 47143 The category of non-unital...
rhmsubcALTVlem1 47144 Lemma 1 for ~ rhmsubcALTV ...
rhmsubcALTVlem2 47145 Lemma 2 for ~ rhmsubcALTV ...
rhmsubcALTVlem3 47146 Lemma 3 for ~ rhmsubcALTV ...
rhmsubcALTVlem4 47147 Lemma 4 for ~ rhmsubcALTV ...
rhmsubcALTV 47148 According to ~ df-subc , t...
rhmsubcALTVcat 47149 The restriction of the cat...
ringcvalALTV 47152 Value of the category of r...
funcringcsetcALTV2lem1 47153 Lemma 1 for ~ funcringcset...
funcringcsetcALTV2lem2 47154 Lemma 2 for ~ funcringcset...
funcringcsetcALTV2lem3 47155 Lemma 3 for ~ funcringcset...
funcringcsetcALTV2lem4 47156 Lemma 4 for ~ funcringcset...
funcringcsetcALTV2lem5 47157 Lemma 5 for ~ funcringcset...
funcringcsetcALTV2lem6 47158 Lemma 6 for ~ funcringcset...
funcringcsetcALTV2lem7 47159 Lemma 7 for ~ funcringcset...
funcringcsetcALTV2lem8 47160 Lemma 8 for ~ funcringcset...
funcringcsetcALTV2lem9 47161 Lemma 9 for ~ funcringcset...
funcringcsetcALTV2 47162 The "natural forgetful fun...
ringcbasALTV 47163 Set of objects of the cate...
ringchomfvalALTV 47164 Set of arrows of the categ...
ringchomALTV 47165 Set of arrows of the categ...
elringchomALTV 47166 A morphism of rings is a f...
ringccofvalALTV 47167 Composition in the categor...
ringccoALTV 47168 Composition in the categor...
ringccatidALTV 47169 Lemma for ~ ringccatALTV ....
ringccatALTV 47170 The category of rings is a...
ringcidALTV 47171 The identity arrow in the ...
ringcsectALTV 47172 A section in the category ...
ringcinvALTV 47173 An inverse in the category...
ringcisoALTV 47174 An isomorphism in the cate...
ringcbasbasALTV 47175 An element of the base set...
funcringcsetclem1ALTV 47176 Lemma 1 for ~ funcringcset...
funcringcsetclem2ALTV 47177 Lemma 2 for ~ funcringcset...
funcringcsetclem3ALTV 47178 Lemma 3 for ~ funcringcset...
funcringcsetclem4ALTV 47179 Lemma 4 for ~ funcringcset...
funcringcsetclem5ALTV 47180 Lemma 5 for ~ funcringcset...
funcringcsetclem6ALTV 47181 Lemma 6 for ~ funcringcset...
funcringcsetclem7ALTV 47182 Lemma 7 for ~ funcringcset...
funcringcsetclem8ALTV 47183 Lemma 8 for ~ funcringcset...
funcringcsetclem9ALTV 47184 Lemma 9 for ~ funcringcset...
funcringcsetcALTV 47185 The "natural forgetful fun...
srhmsubcALTVlem1 47186 Lemma 1 for ~ srhmsubcALTV...
srhmsubcALTVlem2 47187 Lemma 2 for ~ srhmsubcALTV...
srhmsubcALTV 47188 According to ~ df-subc , t...
sringcatALTV 47189 The restriction of the cat...
crhmsubcALTV 47190 According to ~ df-subc , t...
cringcatALTV 47191 The restriction of the cat...
drhmsubcALTV 47192 According to ~ df-subc , t...
drngcatALTV 47193 The restriction of the cat...
fldcatALTV 47194 The restriction of the cat...
fldcALTV 47195 The restriction of the cat...
fldhmsubcALTV 47196 According to ~ df-subc , t...
opeliun2xp 47197 Membership of an ordered p...
eliunxp2 47198 Membership in a union of C...
mpomptx2 47199 Express a two-argument fun...
cbvmpox2 47200 Rule to change the bound v...
dmmpossx2 47201 The domain of a mapping is...
mpoexxg2 47202 Existence of an operation ...
ovmpordxf 47203 Value of an operation give...
ovmpordx 47204 Value of an operation give...
ovmpox2 47205 The value of an operation ...
fdmdifeqresdif 47206 The restriction of a condi...
offvalfv 47207 The function operation exp...
ofaddmndmap 47208 The function operation app...
mapsnop 47209 A singleton of an ordered ...
fprmappr 47210 A function with a domain o...
mapprop 47211 An unordered pair containi...
ztprmneprm 47212 A prime is not an integer ...
2t6m3t4e0 47213 2 times 6 minus 3 times 4 ...
ssnn0ssfz 47214 For any finite subset of `...
nn0sumltlt 47215 If the sum of two nonnegat...
bcpascm1 47216 Pascal's rule for the bino...
altgsumbc 47217 The sum of binomial coeffi...
altgsumbcALT 47218 Alternate proof of ~ altgs...
zlmodzxzlmod 47219 The ` ZZ `-module ` ZZ X. ...
zlmodzxzel 47220 An element of the (base se...
zlmodzxz0 47221 The ` 0 ` of the ` ZZ `-mo...
zlmodzxzscm 47222 The scalar multiplication ...
zlmodzxzadd 47223 The addition of the ` ZZ `...
zlmodzxzsubm 47224 The subtraction of the ` Z...
zlmodzxzsub 47225 The subtraction of the ` Z...
mgpsumunsn 47226 Extract a summand/factor f...
mgpsumz 47227 If the group sum for the m...
mgpsumn 47228 If the group sum for the m...
exple2lt6 47229 A nonnegative integer to t...
pgrple2abl 47230 Every symmetric group on a...
pgrpgt2nabl 47231 Every symmetric group on a...
invginvrid 47232 Identity for a multiplicat...
rmsupp0 47233 The support of a mapping o...
domnmsuppn0 47234 The support of a mapping o...
rmsuppss 47235 The support of a mapping o...
mndpsuppss 47236 The support of a mapping o...
scmsuppss 47237 The support of a mapping o...
rmsuppfi 47238 The support of a mapping o...
rmfsupp 47239 A mapping of a multiplicat...
mndpsuppfi 47240 The support of a mapping o...
mndpfsupp 47241 A mapping of a scalar mult...
scmsuppfi 47242 The support of a mapping o...
scmfsupp 47243 A mapping of a scalar mult...
suppmptcfin 47244 The support of a mapping w...
mptcfsupp 47245 A mapping with value 0 exc...
fsuppmptdmf 47246 A mapping with a finite do...
lmodvsmdi 47247 Multiple distributive law ...
gsumlsscl 47248 Closure of a group sum in ...
assaascl0 47249 The scalar 0 embedded into...
assaascl1 47250 The scalar 1 embedded into...
ply1vr1smo 47251 The variable in a polynomi...
ply1sclrmsm 47252 The ring multiplication of...
coe1id 47253 Coefficient vector of the ...
coe1sclmulval 47254 The value of the coefficie...
ply1mulgsumlem1 47255 Lemma 1 for ~ ply1mulgsum ...
ply1mulgsumlem2 47256 Lemma 2 for ~ ply1mulgsum ...
ply1mulgsumlem3 47257 Lemma 3 for ~ ply1mulgsum ...
ply1mulgsumlem4 47258 Lemma 4 for ~ ply1mulgsum ...
ply1mulgsum 47259 The product of two polynom...
evl1at0 47260 Polynomial evaluation for ...
evl1at1 47261 Polynomial evaluation for ...
linply1 47262 A term of the form ` x - C...
lineval 47263 A term of the form ` x - C...
linevalexample 47264 The polynomial ` x - 3 ` o...
dmatALTval 47269 The algebra of ` N ` x ` N...
dmatALTbas 47270 The base set of the algebr...
dmatALTbasel 47271 An element of the base set...
dmatbas 47272 The set of all ` N ` x ` N...
lincop 47277 A linear combination as op...
lincval 47278 The value of a linear comb...
dflinc2 47279 Alternative definition of ...
lcoop 47280 A linear combination as op...
lcoval 47281 The value of a linear comb...
lincfsuppcl 47282 A linear combination of ve...
linccl 47283 A linear combination of ve...
lincval0 47284 The value of an empty line...
lincvalsng 47285 The linear combination ove...
lincvalsn 47286 The linear combination ove...
lincvalpr 47287 The linear combination ove...
lincval1 47288 The linear combination ove...
lcosn0 47289 Properties of a linear com...
lincvalsc0 47290 The linear combination whe...
lcoc0 47291 Properties of a linear com...
linc0scn0 47292 If a set contains the zero...
lincdifsn 47293 A vector is a linear combi...
linc1 47294 A vector is a linear combi...
lincellss 47295 A linear combination of a ...
lco0 47296 The set of empty linear co...
lcoel0 47297 The zero vector is always ...
lincsum 47298 The sum of two linear comb...
lincscm 47299 A linear combinations mult...
lincsumcl 47300 The sum of two linear comb...
lincscmcl 47301 The multiplication of a li...
lincsumscmcl 47302 The sum of a linear combin...
lincolss 47303 According to the statement...
ellcoellss 47304 Every linear combination o...
lcoss 47305 A set of vectors of a modu...
lspsslco 47306 Lemma for ~ lspeqlco . (C...
lcosslsp 47307 Lemma for ~ lspeqlco . (C...
lspeqlco 47308 Equivalence of a _span_ of...
rellininds 47312 The class defining the rel...
linindsv 47314 The classes of the module ...
islininds 47315 The property of being a li...
linindsi 47316 The implications of being ...
linindslinci 47317 The implications of being ...
islinindfis 47318 The property of being a li...
islinindfiss 47319 The property of being a li...
linindscl 47320 A linearly independent set...
lindepsnlininds 47321 A linearly dependent subse...
islindeps 47322 The property of being a li...
lincext1 47323 Property 1 of an extension...
lincext2 47324 Property 2 of an extension...
lincext3 47325 Property 3 of an extension...
lindslinindsimp1 47326 Implication 1 for ~ lindsl...
lindslinindimp2lem1 47327 Lemma 1 for ~ lindslininds...
lindslinindimp2lem2 47328 Lemma 2 for ~ lindslininds...
lindslinindimp2lem3 47329 Lemma 3 for ~ lindslininds...
lindslinindimp2lem4 47330 Lemma 4 for ~ lindslininds...
lindslinindsimp2lem5 47331 Lemma 5 for ~ lindslininds...
lindslinindsimp2 47332 Implication 2 for ~ lindsl...
lindslininds 47333 Equivalence of definitions...
linds0 47334 The empty set is always a ...
el0ldep 47335 A set containing the zero ...
el0ldepsnzr 47336 A set containing the zero ...
lindsrng01 47337 Any subset of a module is ...
lindszr 47338 Any subset of a module ove...
snlindsntorlem 47339 Lemma for ~ snlindsntor . ...
snlindsntor 47340 A singleton is linearly in...
ldepsprlem 47341 Lemma for ~ ldepspr . (Co...
ldepspr 47342 If a vector is a scalar mu...
lincresunit3lem3 47343 Lemma 3 for ~ lincresunit3...
lincresunitlem1 47344 Lemma 1 for properties of ...
lincresunitlem2 47345 Lemma for properties of a ...
lincresunit1 47346 Property 1 of a specially ...
lincresunit2 47347 Property 2 of a specially ...
lincresunit3lem1 47348 Lemma 1 for ~ lincresunit3...
lincresunit3lem2 47349 Lemma 2 for ~ lincresunit3...
lincresunit3 47350 Property 3 of a specially ...
lincreslvec3 47351 Property 3 of a specially ...
islindeps2 47352 Conditions for being a lin...
islininds2 47353 Implication of being a lin...
isldepslvec2 47354 Alternative definition of ...
lindssnlvec 47355 A singleton not containing...
lmod1lem1 47356 Lemma 1 for ~ lmod1 . (Co...
lmod1lem2 47357 Lemma 2 for ~ lmod1 . (Co...
lmod1lem3 47358 Lemma 3 for ~ lmod1 . (Co...
lmod1lem4 47359 Lemma 4 for ~ lmod1 . (Co...
lmod1lem5 47360 Lemma 5 for ~ lmod1 . (Co...
lmod1 47361 The (smallest) structure r...
lmod1zr 47362 The (smallest) structure r...
lmod1zrnlvec 47363 There is a (left) module (...
lmodn0 47364 Left modules exist. (Cont...
zlmodzxzequa 47365 Example of an equation wit...
zlmodzxznm 47366 Example of a linearly depe...
zlmodzxzldeplem 47367 A and B are not equal. (C...
zlmodzxzequap 47368 Example of an equation wit...
zlmodzxzldeplem1 47369 Lemma 1 for ~ zlmodzxzldep...
zlmodzxzldeplem2 47370 Lemma 2 for ~ zlmodzxzldep...
zlmodzxzldeplem3 47371 Lemma 3 for ~ zlmodzxzldep...
zlmodzxzldeplem4 47372 Lemma 4 for ~ zlmodzxzldep...
zlmodzxzldep 47373 { A , B } is a linearly de...
ldepsnlinclem1 47374 Lemma 1 for ~ ldepsnlinc ....
ldepsnlinclem2 47375 Lemma 2 for ~ ldepsnlinc ....
lvecpsslmod 47376 The class of all (left) ve...
ldepsnlinc 47377 The reverse implication of...
ldepslinc 47378 For (left) vector spaces, ...
suppdm 47379 If the range of a function...
eluz2cnn0n1 47380 An integer greater than 1 ...
divge1b 47381 The ratio of a real number...
divgt1b 47382 The ratio of a real number...
ltsubaddb 47383 Equivalence for the "less ...
ltsubsubb 47384 Equivalence for the "less ...
ltsubadd2b 47385 Equivalence for the "less ...
divsub1dir 47386 Distribution of division o...
expnegico01 47387 An integer greater than 1 ...
elfzolborelfzop1 47388 An element of a half-open ...
pw2m1lepw2m1 47389 2 to the power of a positi...
zgtp1leeq 47390 If an integer is between a...
flsubz 47391 An integer can be moved in...
fldivmod 47392 Expressing the floor of a ...
mod0mul 47393 If an integer is 0 modulo ...
modn0mul 47394 If an integer is not 0 mod...
m1modmmod 47395 An integer decreased by 1 ...
difmodm1lt 47396 The difference between an ...
nn0onn0ex 47397 For each odd nonnegative i...
nn0enn0ex 47398 For each even nonnegative ...
nnennex 47399 For each even positive int...
nneop 47400 A positive integer is even...
nneom 47401 A positive integer is even...
nn0eo 47402 A nonnegative integer is e...
nnpw2even 47403 2 to the power of a positi...
zefldiv2 47404 The floor of an even integ...
zofldiv2 47405 The floor of an odd intege...
nn0ofldiv2 47406 The floor of an odd nonneg...
flnn0div2ge 47407 The floor of a positive in...
flnn0ohalf 47408 The floor of the half of a...
logcxp0 47409 Logarithm of a complex pow...
regt1loggt0 47410 The natural logarithm for ...
fdivval 47413 The quotient of two functi...
fdivmpt 47414 The quotient of two functi...
fdivmptf 47415 The quotient of two functi...
refdivmptf 47416 The quotient of two functi...
fdivpm 47417 The quotient of two functi...
refdivpm 47418 The quotient of two functi...
fdivmptfv 47419 The function value of a qu...
refdivmptfv 47420 The function value of a qu...
bigoval 47423 Set of functions of order ...
elbigofrcl 47424 Reverse closure of the "bi...
elbigo 47425 Properties of a function o...
elbigo2 47426 Properties of a function o...
elbigo2r 47427 Sufficient condition for a...
elbigof 47428 A function of order G(x) i...
elbigodm 47429 The domain of a function o...
elbigoimp 47430 The defining property of a...
elbigolo1 47431 A function (into the posit...
rege1logbrege0 47432 The general logarithm, wit...
rege1logbzge0 47433 The general logarithm, wit...
fllogbd 47434 A real number is between t...
relogbmulbexp 47435 The logarithm of the produ...
relogbdivb 47436 The logarithm of the quoti...
logbge0b 47437 The logarithm of a number ...
logblt1b 47438 The logarithm of a number ...
fldivexpfllog2 47439 The floor of a positive re...
nnlog2ge0lt1 47440 A positive integer is 1 if...
logbpw2m1 47441 The floor of the binary lo...
fllog2 47442 The floor of the binary lo...
blenval 47445 The binary length of an in...
blen0 47446 The binary length of 0. (...
blenn0 47447 The binary length of a "nu...
blenre 47448 The binary length of a pos...
blennn 47449 The binary length of a pos...
blennnelnn 47450 The binary length of a pos...
blennn0elnn 47451 The binary length of a non...
blenpw2 47452 The binary length of a pow...
blenpw2m1 47453 The binary length of a pow...
nnpw2blen 47454 A positive integer is betw...
nnpw2blenfzo 47455 A positive integer is betw...
nnpw2blenfzo2 47456 A positive integer is eith...
nnpw2pmod 47457 Every positive integer can...
blen1 47458 The binary length of 1. (...
blen2 47459 The binary length of 2. (...
nnpw2p 47460 Every positive integer can...
nnpw2pb 47461 A number is a positive int...
blen1b 47462 The binary length of a non...
blennnt2 47463 The binary length of a pos...
nnolog2flm1 47464 The floor of the binary lo...
blennn0em1 47465 The binary length of the h...
blennngt2o2 47466 The binary length of an od...
blengt1fldiv2p1 47467 The binary length of an in...
blennn0e2 47468 The binary length of an ev...
digfval 47471 Operation to obtain the ` ...
digval 47472 The ` K ` th digit of a no...
digvalnn0 47473 The ` K ` th digit of a no...
nn0digval 47474 The ` K ` th digit of a no...
dignn0fr 47475 The digits of the fraction...
dignn0ldlem 47476 Lemma for ~ dignnld . (Co...
dignnld 47477 The leading digits of a po...
dig2nn0ld 47478 The leading digits of a po...
dig2nn1st 47479 The first (relevant) digit...
dig0 47480 All digits of 0 are 0. (C...
digexp 47481 The ` K ` th digit of a po...
dig1 47482 All but one digits of 1 ar...
0dig1 47483 The ` 0 ` th digit of 1 is...
0dig2pr01 47484 The integers 0 and 1 corre...
dig2nn0 47485 A digit of a nonnegative i...
0dig2nn0e 47486 The last bit of an even in...
0dig2nn0o 47487 The last bit of an odd int...
dig2bits 47488 The ` K ` th digit of a no...
dignn0flhalflem1 47489 Lemma 1 for ~ dignn0flhalf...
dignn0flhalflem2 47490 Lemma 2 for ~ dignn0flhalf...
dignn0ehalf 47491 The digits of the half of ...
dignn0flhalf 47492 The digits of the rounded ...
nn0sumshdiglemA 47493 Lemma for ~ nn0sumshdig (i...
nn0sumshdiglemB 47494 Lemma for ~ nn0sumshdig (i...
nn0sumshdiglem1 47495 Lemma 1 for ~ nn0sumshdig ...
nn0sumshdiglem2 47496 Lemma 2 for ~ nn0sumshdig ...
nn0sumshdig 47497 A nonnegative integer can ...
nn0mulfsum 47498 Trivial algorithm to calcu...
nn0mullong 47499 Standard algorithm (also k...
naryfval 47502 The set of the n-ary (endo...
naryfvalixp 47503 The set of the n-ary (endo...
naryfvalel 47504 An n-ary (endo)function on...
naryrcl 47505 Reverse closure for n-ary ...
naryfvalelfv 47506 The value of an n-ary (end...
naryfvalelwrdf 47507 An n-ary (endo)function on...
0aryfvalel 47508 A nullary (endo)function o...
0aryfvalelfv 47509 The value of a nullary (en...
1aryfvalel 47510 A unary (endo)function on ...
fv1arycl 47511 Closure of a unary (endo)f...
1arympt1 47512 A unary (endo)function in ...
1arympt1fv 47513 The value of a unary (endo...
1arymaptfv 47514 The value of the mapping o...
1arymaptf 47515 The mapping of unary (endo...
1arymaptf1 47516 The mapping of unary (endo...
1arymaptfo 47517 The mapping of unary (endo...
1arymaptf1o 47518 The mapping of unary (endo...
1aryenef 47519 The set of unary (endo)fun...
1aryenefmnd 47520 The set of unary (endo)fun...
2aryfvalel 47521 A binary (endo)function on...
fv2arycl 47522 Closure of a binary (endo)...
2arympt 47523 A binary (endo)function in...
2arymptfv 47524 The value of a binary (end...
2arymaptfv 47525 The value of the mapping o...
2arymaptf 47526 The mapping of binary (end...
2arymaptf1 47527 The mapping of binary (end...
2arymaptfo 47528 The mapping of binary (end...
2arymaptf1o 47529 The mapping of binary (end...
2aryenef 47530 The set of binary (endo)fu...
itcoval 47535 The value of the function ...
itcoval0 47536 A function iterated zero t...
itcoval1 47537 A function iterated once. ...
itcoval2 47538 A function iterated twice....
itcoval3 47539 A function iterated three ...
itcoval0mpt 47540 A mapping iterated zero ti...
itcovalsuc 47541 The value of the function ...
itcovalsucov 47542 The value of the function ...
itcovalendof 47543 The n-th iterate of an end...
itcovalpclem1 47544 Lemma 1 for ~ itcovalpc : ...
itcovalpclem2 47545 Lemma 2 for ~ itcovalpc : ...
itcovalpc 47546 The value of the function ...
itcovalt2lem2lem1 47547 Lemma 1 for ~ itcovalt2lem...
itcovalt2lem2lem2 47548 Lemma 2 for ~ itcovalt2lem...
itcovalt2lem1 47549 Lemma 1 for ~ itcovalt2 : ...
itcovalt2lem2 47550 Lemma 2 for ~ itcovalt2 : ...
itcovalt2 47551 The value of the function ...
ackvalsuc1mpt 47552 The Ackermann function at ...
ackvalsuc1 47553 The Ackermann function at ...
ackval0 47554 The Ackermann function at ...
ackval1 47555 The Ackermann function at ...
ackval2 47556 The Ackermann function at ...
ackval3 47557 The Ackermann function at ...
ackendofnn0 47558 The Ackermann function at ...
ackfnnn0 47559 The Ackermann function at ...
ackval0val 47560 The Ackermann function at ...
ackvalsuc0val 47561 The Ackermann function at ...
ackvalsucsucval 47562 The Ackermann function at ...
ackval0012 47563 The Ackermann function at ...
ackval1012 47564 The Ackermann function at ...
ackval2012 47565 The Ackermann function at ...
ackval3012 47566 The Ackermann function at ...
ackval40 47567 The Ackermann function at ...
ackval41a 47568 The Ackermann function at ...
ackval41 47569 The Ackermann function at ...
ackval42 47570 The Ackermann function at ...
ackval42a 47571 The Ackermann function at ...
ackval50 47572 The Ackermann function at ...
fv1prop 47573 The function value of unor...
fv2prop 47574 The function value of unor...
submuladdmuld 47575 Transformation of a sum of...
affinecomb1 47576 Combination of two real af...
affinecomb2 47577 Combination of two real af...
affineid 47578 Identity of an affine comb...
1subrec1sub 47579 Subtract the reciprocal of...
resum2sqcl 47580 The sum of two squares of ...
resum2sqgt0 47581 The sum of the square of a...
resum2sqrp 47582 The sum of the square of a...
resum2sqorgt0 47583 The sum of the square of t...
reorelicc 47584 Membership in and outside ...
rrx2pxel 47585 The x-coordinate of a poin...
rrx2pyel 47586 The y-coordinate of a poin...
prelrrx2 47587 An unordered pair of order...
prelrrx2b 47588 An unordered pair of order...
rrx2pnecoorneor 47589 If two different points ` ...
rrx2pnedifcoorneor 47590 If two different points ` ...
rrx2pnedifcoorneorr 47591 If two different points ` ...
rrx2xpref1o 47592 There is a bijection betwe...
rrx2xpreen 47593 The set of points in the t...
rrx2plord 47594 The lexicographical orderi...
rrx2plord1 47595 The lexicographical orderi...
rrx2plord2 47596 The lexicographical orderi...
rrx2plordisom 47597 The set of points in the t...
rrx2plordso 47598 The lexicographical orderi...
ehl2eudisval0 47599 The Euclidean distance of ...
ehl2eudis0lt 47600 An upper bound of the Eucl...
lines 47605 The lines passing through ...
line 47606 The line passing through t...
rrxlines 47607 Definition of lines passin...
rrxline 47608 The line passing through t...
rrxlinesc 47609 Definition of lines passin...
rrxlinec 47610 The line passing through t...
eenglngeehlnmlem1 47611 Lemma 1 for ~ eenglngeehln...
eenglngeehlnmlem2 47612 Lemma 2 for ~ eenglngeehln...
eenglngeehlnm 47613 The line definition in the...
rrx2line 47614 The line passing through t...
rrx2vlinest 47615 The vertical line passing ...
rrx2linest 47616 The line passing through t...
rrx2linesl 47617 The line passing through t...
rrx2linest2 47618 The line passing through t...
elrrx2linest2 47619 The line passing through t...
spheres 47620 The spheres for given cent...
sphere 47621 A sphere with center ` X `...
rrxsphere 47622 The sphere with center ` M...
2sphere 47623 The sphere with center ` M...
2sphere0 47624 The sphere around the orig...
line2ylem 47625 Lemma for ~ line2y . This...
line2 47626 Example for a line ` G ` p...
line2xlem 47627 Lemma for ~ line2x . This...
line2x 47628 Example for a horizontal l...
line2y 47629 Example for a vertical lin...
itsclc0lem1 47630 Lemma for theorems about i...
itsclc0lem2 47631 Lemma for theorems about i...
itsclc0lem3 47632 Lemma for theorems about i...
itscnhlc0yqe 47633 Lemma for ~ itsclc0 . Qua...
itschlc0yqe 47634 Lemma for ~ itsclc0 . Qua...
itsclc0yqe 47635 Lemma for ~ itsclc0 . Qua...
itsclc0yqsollem1 47636 Lemma 1 for ~ itsclc0yqsol...
itsclc0yqsollem2 47637 Lemma 2 for ~ itsclc0yqsol...
itsclc0yqsol 47638 Lemma for ~ itsclc0 . Sol...
itscnhlc0xyqsol 47639 Lemma for ~ itsclc0 . Sol...
itschlc0xyqsol1 47640 Lemma for ~ itsclc0 . Sol...
itschlc0xyqsol 47641 Lemma for ~ itsclc0 . Sol...
itsclc0xyqsol 47642 Lemma for ~ itsclc0 . Sol...
itsclc0xyqsolr 47643 Lemma for ~ itsclc0 . Sol...
itsclc0xyqsolb 47644 Lemma for ~ itsclc0 . Sol...
itsclc0 47645 The intersection points of...
itsclc0b 47646 The intersection points of...
itsclinecirc0 47647 The intersection points of...
itsclinecirc0b 47648 The intersection points of...
itsclinecirc0in 47649 The intersection points of...
itsclquadb 47650 Quadratic equation for the...
itsclquadeu 47651 Quadratic equation for the...
2itscplem1 47652 Lemma 1 for ~ 2itscp . (C...
2itscplem2 47653 Lemma 2 for ~ 2itscp . (C...
2itscplem3 47654 Lemma D for ~ 2itscp . (C...
2itscp 47655 A condition for a quadrati...
itscnhlinecirc02plem1 47656 Lemma 1 for ~ itscnhlineci...
itscnhlinecirc02plem2 47657 Lemma 2 for ~ itscnhlineci...
itscnhlinecirc02plem3 47658 Lemma 3 for ~ itscnhlineci...
itscnhlinecirc02p 47659 Intersection of a nonhoriz...
inlinecirc02plem 47660 Lemma for ~ inlinecirc02p ...
inlinecirc02p 47661 Intersection of a line wit...
inlinecirc02preu 47662 Intersection of a line wit...
pm4.71da 47663 Deduction converting a bic...
logic1 47664 Distribution of implicatio...
logic1a 47665 Variant of ~ logic1 . (Co...
logic2 47666 Variant of ~ logic1 . (Co...
pm5.32dav 47667 Distribution of implicatio...
pm5.32dra 47668 Reverse distribution of im...
exp12bd 47669 The import-export theorem ...
mpbiran3d 47670 Equivalence with a conjunc...
mpbiran4d 47671 Equivalence with a conjunc...
dtrucor3 47672 An example of how ~ ax-5 w...
ralbidb 47673 Formula-building rule for ...
ralbidc 47674 Formula-building rule for ...
r19.41dv 47675 A complex deduction form o...
rspceb2dv 47676 Restricted existential spe...
rmotru 47677 Two ways of expressing "at...
reutru 47678 Two ways of expressing "ex...
reutruALT 47679 Alternate proof for ~ reut...
ssdisjd 47680 Subset preserves disjointn...
ssdisjdr 47681 Subset preserves disjointn...
disjdifb 47682 Relative complement is ant...
predisj 47683 Preimages of disjoint sets...
vsn 47684 The singleton of the unive...
mosn 47685 "At most one" element in a...
mo0 47686 "At most one" element in a...
mosssn 47687 "At most one" element in a...
mo0sn 47688 Two ways of expressing "at...
mosssn2 47689 Two ways of expressing "at...
unilbss 47690 Superclass of the greatest...
inpw 47691 Two ways of expressing a c...
mof0 47692 There is at most one funct...
mof02 47693 A variant of ~ mof0 . (Co...
mof0ALT 47694 Alternate proof for ~ mof0...
eufsnlem 47695 There is exactly one funct...
eufsn 47696 There is exactly one funct...
eufsn2 47697 There is exactly one funct...
mofsn 47698 There is at most one funct...
mofsn2 47699 There is at most one funct...
mofsssn 47700 There is at most one funct...
mofmo 47701 There is at most one funct...
mofeu 47702 The uniqueness of a functi...
elfvne0 47703 If a function value has a ...
fdomne0 47704 A function with non-empty ...
f1sn2g 47705 A function that maps a sin...
f102g 47706 A function that maps the e...
f1mo 47707 A function that maps a set...
f002 47708 A function with an empty c...
map0cor 47709 A function exists iff an e...
fvconstr 47710 Two ways of expressing ` A...
fvconstrn0 47711 Two ways of expressing ` A...
fvconstr2 47712 Two ways of expressing ` A...
fvconst0ci 47713 A constant function's valu...
fvconstdomi 47714 A constant function's valu...
f1omo 47715 There is at most one eleme...
f1omoALT 47716 There is at most one eleme...
iccin 47717 Intersection of two closed...
iccdisj2 47718 If the upper bound of one ...
iccdisj 47719 If the upper bound of one ...
mreuniss 47720 The union of a collection ...
clduni 47721 The union of closed sets i...
opncldeqv 47722 Conditions on open sets ar...
opndisj 47723 Two ways of saying that tw...
clddisj 47724 Two ways of saying that tw...
neircl 47725 Reverse closure of the nei...
opnneilem 47726 Lemma factoring out common...
opnneir 47727 If something is true for a...
opnneirv 47728 A variant of ~ opnneir wit...
opnneilv 47729 The converse of ~ opnneir ...
opnneil 47730 A variant of ~ opnneilv . ...
opnneieqv 47731 The equivalence between ne...
opnneieqvv 47732 The equivalence between ne...
restcls2lem 47733 A closed set in a subspace...
restcls2 47734 A closed set in a subspace...
restclsseplem 47735 Lemma for ~ restclssep . ...
restclssep 47736 Two disjoint closed sets i...
cnneiima 47737 Given a continuous functio...
iooii 47738 Open intervals are open se...
icccldii 47739 Closed intervals are close...
i0oii 47740 ` ( 0 [,) A ) ` is open in...
io1ii 47741 ` ( A (,] 1 ) ` is open in...
sepnsepolem1 47742 Lemma for ~ sepnsepo . (C...
sepnsepolem2 47743 Open neighborhood and neig...
sepnsepo 47744 Open neighborhood and neig...
sepdisj 47745 Separated sets are disjoin...
seposep 47746 If two sets are separated ...
sepcsepo 47747 If two sets are separated ...
sepfsepc 47748 If two sets are separated ...
seppsepf 47749 If two sets are precisely ...
seppcld 47750 If two sets are precisely ...
isnrm4 47751 A topological space is nor...
dfnrm2 47752 A topological space is nor...
dfnrm3 47753 A topological space is nor...
iscnrm3lem1 47754 Lemma for ~ iscnrm3 . Sub...
iscnrm3lem2 47755 Lemma for ~ iscnrm3 provin...
iscnrm3lem3 47756 Lemma for ~ iscnrm3lem4 . ...
iscnrm3lem4 47757 Lemma for ~ iscnrm3lem5 an...
iscnrm3lem5 47758 Lemma for ~ iscnrm3l . (C...
iscnrm3lem6 47759 Lemma for ~ iscnrm3lem7 . ...
iscnrm3lem7 47760 Lemma for ~ iscnrm3rlem8 a...
iscnrm3rlem1 47761 Lemma for ~ iscnrm3rlem2 ....
iscnrm3rlem2 47762 Lemma for ~ iscnrm3rlem3 ....
iscnrm3rlem3 47763 Lemma for ~ iscnrm3r . Th...
iscnrm3rlem4 47764 Lemma for ~ iscnrm3rlem8 ....
iscnrm3rlem5 47765 Lemma for ~ iscnrm3rlem6 ....
iscnrm3rlem6 47766 Lemma for ~ iscnrm3rlem7 ....
iscnrm3rlem7 47767 Lemma for ~ iscnrm3rlem8 ....
iscnrm3rlem8 47768 Lemma for ~ iscnrm3r . Di...
iscnrm3r 47769 Lemma for ~ iscnrm3 . If ...
iscnrm3llem1 47770 Lemma for ~ iscnrm3l . Cl...
iscnrm3llem2 47771 Lemma for ~ iscnrm3l . If...
iscnrm3l 47772 Lemma for ~ iscnrm3 . Giv...
iscnrm3 47773 A completely normal topolo...
iscnrm3v 47774 A topology is completely n...
iscnrm4 47775 A completely normal topolo...
isprsd 47776 Property of being a preord...
lubeldm2 47777 Member of the domain of th...
glbeldm2 47778 Member of the domain of th...
lubeldm2d 47779 Member of the domain of th...
glbeldm2d 47780 Member of the domain of th...
lubsscl 47781 If a subset of ` S ` conta...
glbsscl 47782 If a subset of ` S ` conta...
lubprlem 47783 Lemma for ~ lubprdm and ~ ...
lubprdm 47784 The set of two comparable ...
lubpr 47785 The LUB of the set of two ...
glbprlem 47786 Lemma for ~ glbprdm and ~ ...
glbprdm 47787 The set of two comparable ...
glbpr 47788 The GLB of the set of two ...
joindm2 47789 The join of any two elemen...
joindm3 47790 The join of any two elemen...
meetdm2 47791 The meet of any two elemen...
meetdm3 47792 The meet of any two elemen...
posjidm 47793 Poset join is idempotent. ...
posmidm 47794 Poset meet is idempotent. ...
toslat 47795 A toset is a lattice. (Co...
isclatd 47796 The predicate "is a comple...
intubeu 47797 Existential uniqueness of ...
unilbeu 47798 Existential uniqueness of ...
ipolublem 47799 Lemma for ~ ipolubdm and ~...
ipolubdm 47800 The domain of the LUB of t...
ipolub 47801 The LUB of the inclusion p...
ipoglblem 47802 Lemma for ~ ipoglbdm and ~...
ipoglbdm 47803 The domain of the GLB of t...
ipoglb 47804 The GLB of the inclusion p...
ipolub0 47805 The LUB of the empty set i...
ipolub00 47806 The LUB of the empty set i...
ipoglb0 47807 The GLB of the empty set i...
mrelatlubALT 47808 Least upper bounds in a Mo...
mrelatglbALT 47809 Greatest lower bounds in a...
mreclat 47810 A Moore space is a complet...
topclat 47811 A topology is a complete l...
toplatglb0 47812 The empty intersection in ...
toplatlub 47813 Least upper bounds in a to...
toplatglb 47814 Greatest lower bounds in a...
toplatjoin 47815 Joins in a topology are re...
toplatmeet 47816 Meets in a topology are re...
topdlat 47817 A topology is a distributi...
catprslem 47818 Lemma for ~ catprs . (Con...
catprs 47819 A preorder can be extracte...
catprs2 47820 A category equipped with t...
catprsc 47821 A construction of the preo...
catprsc2 47822 An alternate construction ...
endmndlem 47823 A diagonal hom-set in a ca...
idmon 47824 An identity arrow, or an i...
idepi 47825 An identity arrow, or an i...
funcf2lem 47826 A utility theorem for prov...
isthinc 47829 The predicate "is a thin c...
isthinc2 47830 A thin category is a categ...
isthinc3 47831 A thin category is a categ...
thincc 47832 A thin category is a categ...
thinccd 47833 A thin category is a categ...
thincssc 47834 A thin category is a categ...
isthincd2lem1 47835 Lemma for ~ isthincd2 and ...
thincmo2 47836 Morphisms in the same hom-...
thincmo 47837 There is at most one morph...
thincmoALT 47838 Alternate proof for ~ thin...
thincmod 47839 At most one morphism in ea...
thincn0eu 47840 In a thin category, a hom-...
thincid 47841 In a thin category, a morp...
thincmon 47842 In a thin category, all mo...
thincepi 47843 In a thin category, all mo...
isthincd2lem2 47844 Lemma for ~ isthincd2 . (...
isthincd 47845 The predicate "is a thin c...
isthincd2 47846 The predicate " ` C ` is a...
oppcthin 47847 The opposite category of a...
subthinc 47848 A subcategory of a thin ca...
functhinclem1 47849 Lemma for ~ functhinc . G...
functhinclem2 47850 Lemma for ~ functhinc . (...
functhinclem3 47851 Lemma for ~ functhinc . T...
functhinclem4 47852 Lemma for ~ functhinc . O...
functhinc 47853 A functor to a thin catego...
fullthinc 47854 A functor to a thin catego...
fullthinc2 47855 A full functor to a thin c...
thincfth 47856 A functor from a thin cate...
thincciso 47857 Two thin categories are is...
0thincg 47858 Any structure with an empt...
0thinc 47859 The empty category (see ~ ...
indthinc 47860 An indiscrete category in ...
indthincALT 47861 An alternate proof for ~ i...
prsthinc 47862 Preordered sets as categor...
setcthin 47863 A category of sets all of ...
setc2othin 47864 The category ` ( SetCat ``...
thincsect 47865 In a thin category, one mo...
thincsect2 47866 In a thin category, ` F ` ...
thincinv 47867 In a thin category, ` F ` ...
thinciso 47868 In a thin category, ` F : ...
thinccic 47869 In a thin category, two ob...
prstcval 47872 Lemma for ~ prstcnidlem an...
prstcnidlem 47873 Lemma for ~ prstcnid and ~...
prstcnid 47874 Components other than ` Ho...
prstcbas 47875 The base set is unchanged....
prstcleval 47876 Value of the less-than-or-...
prstclevalOLD 47877 Obsolete proof of ~ prstcl...
prstcle 47878 Value of the less-than-or-...
prstcocval 47879 Orthocomplementation is un...
prstcocvalOLD 47880 Obsolete proof of ~ prstco...
prstcoc 47881 Orthocomplementation is un...
prstchomval 47882 Hom-sets of the constructe...
prstcprs 47883 The category is a preorder...
prstcthin 47884 The preordered set is equi...
prstchom 47885 Hom-sets of the constructe...
prstchom2 47886 Hom-sets of the constructe...
prstchom2ALT 47887 Hom-sets of the constructe...
postcpos 47888 The converted category is ...
postcposALT 47889 Alternate proof for ~ post...
postc 47890 The converted category is ...
mndtcval 47893 Value of the category buil...
mndtcbasval 47894 The base set of the catego...
mndtcbas 47895 The category built from a ...
mndtcob 47896 Lemma for ~ mndtchom and ~...
mndtcbas2 47897 Two objects in a category ...
mndtchom 47898 The only hom-set of the ca...
mndtcco 47899 The composition of the cat...
mndtcco2 47900 The composition of the cat...
mndtccatid 47901 Lemma for ~ mndtccat and ~...
mndtccat 47902 The function value is a ca...
mndtcid 47903 The identity morphism, or ...
grptcmon 47904 All morphisms in a categor...
grptcepi 47905 All morphisms in a categor...
nfintd 47906 Bound-variable hypothesis ...
nfiund 47907 Bound-variable hypothesis ...
nfiundg 47908 Bound-variable hypothesis ...
iunord 47909 The indexed union of a col...
iunordi 47910 The indexed union of a col...
spd 47911 Specialization deduction, ...
spcdvw 47912 A version of ~ spcdv where...
tfis2d 47913 Transfinite Induction Sche...
bnd2d 47914 Deduction form of ~ bnd2 ....
dffun3f 47915 Alternate definition of fu...
setrecseq 47918 Equality theorem for set r...
nfsetrecs 47919 Bound-variable hypothesis ...
setrec1lem1 47920 Lemma for ~ setrec1 . Thi...
setrec1lem2 47921 Lemma for ~ setrec1 . If ...
setrec1lem3 47922 Lemma for ~ setrec1 . If ...
setrec1lem4 47923 Lemma for ~ setrec1 . If ...
setrec1 47924 This is the first of two f...
setrec2fun 47925 This is the second of two ...
setrec2lem1 47926 Lemma for ~ setrec2 . The...
setrec2lem2 47927 Lemma for ~ setrec2 . The...
setrec2 47928 This is the second of two ...
setrec2v 47929 Version of ~ setrec2 with ...
setrec2mpt 47930 Version of ~ setrec2 where...
setis 47931 Version of ~ setrec2 expre...
elsetrecslem 47932 Lemma for ~ elsetrecs . A...
elsetrecs 47933 A set ` A ` is an element ...
setrecsss 47934 The ` setrecs ` operator r...
setrecsres 47935 A recursively generated cl...
vsetrec 47936 Construct ` _V ` using set...
0setrec 47937 If a function sends the em...
onsetreclem1 47938 Lemma for ~ onsetrec . (C...
onsetreclem2 47939 Lemma for ~ onsetrec . (C...
onsetreclem3 47940 Lemma for ~ onsetrec . (C...
onsetrec 47941 Construct ` On ` using set...
elpglem1 47944 Lemma for ~ elpg . (Contr...
elpglem2 47945 Lemma for ~ elpg . (Contr...
elpglem3 47946 Lemma for ~ elpg . (Contr...
elpg 47947 Membership in the class of...
pgindlem 47948 Lemma for ~ pgind . (Cont...
pgindnf 47949 Version of ~ pgind with ex...
pgind 47950 Induction on partizan game...
sbidd 47951 An identity theorem for su...
sbidd-misc 47952 An identity theorem for su...
gte-lte 47957 Simple relationship betwee...
gt-lt 47958 Simple relationship betwee...
gte-lteh 47959 Relationship between ` <_ ...
gt-lth 47960 Relationship between ` < `...
ex-gt 47961 Simple example of ` > ` , ...
ex-gte 47962 Simple example of ` >_ ` ,...
sinhval-named 47969 Value of the named sinh fu...
coshval-named 47970 Value of the named cosh fu...
tanhval-named 47971 Value of the named tanh fu...
sinh-conventional 47972 Conventional definition of...
sinhpcosh 47973 Prove that ` ( sinh `` A )...
secval 47980 Value of the secant functi...
cscval 47981 Value of the cosecant func...
cotval 47982 Value of the cotangent fun...
seccl 47983 The closure of the secant ...
csccl 47984 The closure of the cosecan...
cotcl 47985 The closure of the cotange...
reseccl 47986 The closure of the secant ...
recsccl 47987 The closure of the cosecan...
recotcl 47988 The closure of the cotange...
recsec 47989 The reciprocal of secant i...
reccsc 47990 The reciprocal of cosecant...
reccot 47991 The reciprocal of cotangen...
rectan 47992 The reciprocal of tangent ...
sec0 47993 The value of the secant fu...
onetansqsecsq 47994 Prove the tangent squared ...
cotsqcscsq 47995 Prove the tangent squared ...
ifnmfalse 47996 If A is not a member of B,...
logb2aval 47997 Define the value of the ` ...
comraddi 48004 Commute RHS addition. See...
mvlraddi 48005 Move the right term in a s...
mvrladdi 48006 Move the left term in a su...
assraddsubi 48007 Associate RHS addition-sub...
joinlmuladdmuli 48008 Join AB+CB into (A+C) on L...
joinlmulsubmuld 48009 Join AB-CB into (A-C) on L...
joinlmulsubmuli 48010 Join AB-CB into (A-C) on L...
mvlrmuld 48011 Move the right term in a p...
mvlrmuli 48012 Move the right term in a p...
i2linesi 48013 Solve for the intersection...
i2linesd 48014 Solve for the intersection...
alimp-surprise 48015 Demonstrate that when usin...
alimp-no-surprise 48016 There is no "surprise" in ...
empty-surprise 48017 Demonstrate that when usin...
empty-surprise2 48018 "Prove" that false is true...
eximp-surprise 48019 Show what implication insi...
eximp-surprise2 48020 Show that "there exists" w...
alsconv 48025 There is an equivalence be...
alsi1d 48026 Deduction rule: Given "al...
alsi2d 48027 Deduction rule: Given "al...
alsc1d 48028 Deduction rule: Given "al...
alsc2d 48029 Deduction rule: Given "al...
alscn0d 48030 Deduction rule: Given "al...
alsi-no-surprise 48031 Demonstrate that there is ...
5m4e1 48032 Prove that 5 - 4 = 1. (Co...
2p2ne5 48033 Prove that ` 2 + 2 =/= 5 `...
resolution 48034 Resolution rule. This is ...
testable 48035 In classical logic all wff...
aacllem 48036 Lemma for other theorems a...
amgmwlem 48037 Weighted version of ~ amgm...
amgmlemALT 48038 Alternate proof of ~ amgml...
amgmw2d 48039 Weighted arithmetic-geomet...
young2d 48040 Young's inequality for ` n...
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