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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...
pm2.18OLD 129 Obsolete version of ~ pm2....
pm2.18dOLD 130 Obsolete version of ~ pm2....
pm2.18i 131 Inference associated with ...
notnotr 132 Double negation eliminatio...
notnotri 133 Inference associated with ...
notnotriALT 134 Alternate proof of ~ notno...
notnotrd 135 Deduction associated with ...
con2d 136 A contraposition deduction...
con2 137 Contraposition. Theorem *...
mt2d 138 Modus tollens deduction. ...
mt2i 139 Modus tollens inference. ...
nsyl3 140 A negated syllogism infere...
con2i 141 A contraposition inference...
nsyl 142 A negated syllogism infere...
nsyl2 143 A negated syllogism infere...
notnot 144 Double negation introducti...
notnoti 145 Inference associated with ...
notnotd 146 Deduction associated with ...
con1d 147 A contraposition deduction...
con1 148 Contraposition. Theorem *...
con1i 149 A contraposition inference...
mt3d 150 Modus tollens deduction. ...
mt3i 151 Modus tollens inference. ...
nsyl2OLD 152 Obsolete version of ~ nsyl...
pm2.24i 153 Inference associated with ...
pm2.24d 154 Deduction form of ~ pm2.24...
con3d 155 A contraposition deduction...
con3 156 Contraposition. Theorem *...
con3i 157 A contraposition inference...
con3rr3 158 Rotate through consequent ...
nsyld 159 A negated syllogism deduct...
nsyli 160 A negated syllogism infere...
nsyl4 161 A negated syllogism infere...
pm3.2im 162 Theorem *3.2 of [Whitehead...
mth8 163 Theorem 8 of [Margaris] p....
jc 164 Deduction joining the cons...
impi 165 An importation inference. ...
expi 166 An exportation inference. ...
simprim 167 Simplification. Similar t...
simplim 168 Simplification. Similar t...
pm2.5g 169 General instance of Theore...
pm2.5 170 Theorem *2.5 of [Whitehead...
conax1 171 Contrapositive of ~ ax-1 ....
conax1k 172 Weakening of ~ conax1 . G...
pm2.51 173 Theorem *2.51 of [Whitehea...
pm2.52 174 Theorem *2.52 of [Whitehea...
pm2.521g 175 A general instance of Theo...
pm2.521g2 176 A general instance of Theo...
pm2.521 177 Theorem *2.521 of [Whitehe...
expt 178 Exportation theorem ~ pm3....
impt 179 Importation theorem ~ pm3....
pm2.61d 180 Deduction eliminating an a...
pm2.61d1 181 Inference eliminating an a...
pm2.61d2 182 Inference eliminating an a...
pm2.61i 183 Inference eliminating an a...
pm2.61ii 184 Inference eliminating two ...
pm2.61nii 185 Inference eliminating two ...
pm2.61iii 186 Inference eliminating thre...
ja 187 Inference joining the ante...
jad 188 Deduction form of ~ ja . ...
pm2.61iOLD 189 Obsolete version of ~ pm2....
pm2.01 190 Weak Clavius law. If a fo...
pm2.01d 191 Deduction based on reducti...
pm2.6 192 Theorem *2.6 of [Whitehead...
pm2.61 193 Theorem *2.61 of [Whitehea...
pm2.65 194 Theorem *2.65 of [Whitehea...
pm2.65i 195 Inference for proof by con...
pm2.21dd 196 A contradiction implies an...
pm2.65d 197 Deduction for proof by con...
mto 198 The rule of modus tollens....
mtod 199 Modus tollens deduction. ...
mtoi 200 Modus tollens inference. ...
mt2 201 A rule similar to modus to...
mt3 202 A rule similar to modus to...
peirce 203 Peirce's axiom. A non-int...
looinv 204 The Inversion Axiom of the...
bijust0 205 A self-implication (see ~ ...
bijust 206 Theorem used to justify th...
impbi 209 Property of the biconditio...
impbii 210 Infer an equivalence from ...
impbidd 211 Deduce an equivalence from...
impbid21d 212 Deduce an equivalence from...
impbid 213 Deduce an equivalence from...
dfbi1 214 Relate the biconditional c...
dfbi1ALT 215 Alternate proof of ~ dfbi1...
biimp 216 Property of the biconditio...
biimpi 217 Infer an implication from ...
sylbi 218 A mixed syllogism inferenc...
sylib 219 A mixed syllogism inferenc...
sylbb 220 A mixed syllogism inferenc...
biimpr 221 Property of the biconditio...
bicom1 222 Commutative law for the bi...
bicom 223 Commutative law for the bi...
bicomd 224 Commute two sides of a bic...
bicomi 225 Inference from commutative...
impbid1 226 Infer an equivalence from ...
impbid2 227 Infer an equivalence from ...
impcon4bid 228 A variation on ~ impbid wi...
biimpri 229 Infer a converse implicati...
biimpd 230 Deduce an implication from...
mpbi 231 An inference from a bicond...
mpbir 232 An inference from a bicond...
mpbid 233 A deduction from a bicondi...
mpbii 234 An inference from a nested...
sylibr 235 A mixed syllogism inferenc...
sylbir 236 A mixed syllogism inferenc...
sylbbr 237 A mixed syllogism inferenc...
sylbb1 238 A mixed syllogism inferenc...
sylbb2 239 A mixed syllogism inferenc...
sylibd 240 A syllogism deduction. (C...
sylbid 241 A syllogism deduction. (C...
mpbidi 242 A deduction from a bicondi...
syl5bi 243 A mixed syllogism inferenc...
syl5bir 244 A mixed syllogism inferenc...
syl5ib 245 A mixed syllogism inferenc...
syl5ibcom 246 A mixed syllogism inferenc...
syl5ibr 247 A mixed syllogism inferenc...
syl5ibrcom 248 A mixed syllogism inferenc...
biimprd 249 Deduce a converse implicat...
biimpcd 250 Deduce a commuted implicat...
biimprcd 251 Deduce a converse commuted...
syl6ib 252 A mixed syllogism inferenc...
syl6ibr 253 A mixed syllogism inferenc...
syl6bi 254 A mixed syllogism inferenc...
syl6bir 255 A mixed syllogism inferenc...
syl7bi 256 A mixed syllogism inferenc...
syl8ib 257 A syllogism rule of infere...
mpbird 258 A deduction from a bicondi...
mpbiri 259 An inference from a nested...
sylibrd 260 A syllogism deduction. (C...
sylbird 261 A syllogism deduction. (C...
biid 262 Principle of identity for ...
biidd 263 Principle of identity with...
pm5.1im 264 Two propositions are equiv...
2th 265 Two truths are equivalent....
2thd 266 Two truths are equivalent....
monothetic 267 Two self-implications (see...
ibi 268 Inference that converts a ...
ibir 269 Inference that converts a ...
ibd 270 Deduction that converts a ...
pm5.74 271 Distribution of implicatio...
pm5.74i 272 Distribution of implicatio...
pm5.74ri 273 Distribution of implicatio...
pm5.74d 274 Distribution of implicatio...
pm5.74rd 275 Distribution of implicatio...
bitri 276 An inference from transiti...
bitr2i 277 An inference from transiti...
bitr3i 278 An inference from transiti...
bitr4i 279 An inference from transiti...
bitrd 280 Deduction form of ~ bitri ...
bitr2d 281 Deduction form of ~ bitr2i...
bitr3d 282 Deduction form of ~ bitr3i...
bitr4d 283 Deduction form of ~ bitr4i...
syl5bb 284 A syllogism inference from...
syl5rbb 285 A syllogism inference from...
syl5bbr 286 A syllogism inference from...
syl5rbbr 287 A syllogism inference from...
syl6bb 288 A syllogism inference from...
syl6rbb 289 A syllogism inference from...
syl6bbr 290 A syllogism inference from...
syl6rbbr 291 A syllogism inference from...
3imtr3i 292 A mixed syllogism inferenc...
3imtr4i 293 A mixed syllogism inferenc...
3imtr3d 294 More general version of ~ ...
3imtr4d 295 More general version of ~ ...
3imtr3g 296 More general version of ~ ...
3imtr4g 297 More general version of ~ ...
3bitri 298 A chained inference from t...
3bitrri 299 A chained inference from t...
3bitr2i 300 A chained inference from t...
3bitr2ri 301 A chained inference from t...
3bitr3i 302 A chained inference from t...
3bitr3ri 303 A chained inference from t...
3bitr4i 304 A chained inference from t...
3bitr4ri 305 A chained inference from t...
3bitrd 306 Deduction from transitivit...
3bitrrd 307 Deduction from transitivit...
3bitr2d 308 Deduction from transitivit...
3bitr2rd 309 Deduction from transitivit...
3bitr3d 310 Deduction from transitivit...
3bitr3rd 311 Deduction from transitivit...
3bitr4d 312 Deduction from transitivit...
3bitr4rd 313 Deduction from transitivit...
3bitr3g 314 More general version of ~ ...
3bitr4g 315 More general version of ~ ...
notnotb 316 Double negation. Theorem ...
con34b 317 A biconditional form of co...
con4bid 318 A contraposition deduction...
notbid 319 Deduction negating both si...
notbi 320 Contraposition. Theorem *...
notbii 321 Negate both sides of a log...
con4bii 322 A contraposition inference...
mtbi 323 An inference from a bicond...
mtbir 324 An inference from a bicond...
mtbid 325 A deduction from a bicondi...
mtbird 326 A deduction from a bicondi...
mtbii 327 An inference from a bicond...
mtbiri 328 An inference from a bicond...
sylnib 329 A mixed syllogism inferenc...
sylnibr 330 A mixed syllogism inferenc...
sylnbi 331 A mixed syllogism inferenc...
sylnbir 332 A mixed syllogism inferenc...
xchnxbi 333 Replacement of a subexpres...
xchnxbir 334 Replacement of a subexpres...
xchbinx 335 Replacement of a subexpres...
xchbinxr 336 Replacement of a subexpres...
imbi2i 337 Introduce an antecedent to...
jcn 338 Inference joining the cons...
bibi2i 339 Inference adding a bicondi...
bibi1i 340 Inference adding a bicondi...
bibi12i 341 The equivalence of two equ...
imbi2d 342 Deduction adding an antece...
imbi1d 343 Deduction adding a consequ...
bibi2d 344 Deduction adding a bicondi...
bibi1d 345 Deduction adding a bicondi...
imbi12d 346 Deduction joining two equi...
bibi12d 347 Deduction joining two equi...
imbi12 348 Closed form of ~ imbi12i ....
imbi1 349 Theorem *4.84 of [Whitehea...
imbi2 350 Theorem *4.85 of [Whitehea...
imbi1i 351 Introduce a consequent to ...
imbi12i 352 Join two logical equivalen...
bibi1 353 Theorem *4.86 of [Whitehea...
bitr3 354 Closed nested implication ...
con2bi 355 Contraposition. Theorem *...
con2bid 356 A contraposition deduction...
con1bid 357 A contraposition deduction...
con1bii 358 A contraposition inference...
con2bii 359 A contraposition inference...
con1b 360 Contraposition. Bidirecti...
con2b 361 Contraposition. Bidirecti...
biimt 362 A wff is equivalent to its...
pm5.5 363 Theorem *5.5 of [Whitehead...
a1bi 364 Inference introducing a th...
mt2bi 365 A false consequent falsifi...
mtt 366 Modus-tollens-like theorem...
imnot 367 If a proposition is false,...
pm5.501 368 Theorem *5.501 of [Whitehe...
ibib 369 Implication in terms of im...
ibibr 370 Implication in terms of im...
tbt 371 A wff is equivalent to its...
nbn2 372 The negation of a wff is e...
bibif 373 Transfer negation via an e...
nbn 374 The negation of a wff is e...
nbn3 375 Transfer falsehood via equ...
pm5.21im 376 Two propositions are equiv...
2false 377 Two falsehoods are equival...
2falsed 378 Two falsehoods are equival...
pm5.21ni 379 Two propositions implying ...
pm5.21nii 380 Eliminate an antecedent im...
pm5.21ndd 381 Eliminate an antecedent im...
bija 382 Combine antecedents into a...
pm5.18 383 Theorem *5.18 of [Whitehea...
xor3 384 Two ways to express "exclu...
nbbn 385 Move negation outside of b...
biass 386 Associative law for the bi...
biluk 387 Lukasiewicz's shortest axi...
pm5.19 388 Theorem *5.19 of [Whitehea...
bi2.04 389 Logical equivalence of com...
pm5.4 390 Antecedent absorption impl...
imdi 391 Distributive law for impli...
pm5.41 392 Theorem *5.41 of [Whitehea...
pm4.8 393 Theorem *4.8 of [Whitehead...
pm4.81 394 A formula is equivalent to...
imim21b 395 Simplify an implication be...
pm4.63 398 Theorem *4.63 of [Whitehea...
pm4.67 399 Theorem *4.67 of [Whitehea...
imnan 400 Express an implication in ...
imnani 401 Infer an implication from ...
iman 402 Implication in terms of co...
pm3.24 403 Law of noncontradiction. ...
annim 404 Express a conjunction in t...
pm4.61 405 Theorem *4.61 of [Whitehea...
pm4.65 406 Theorem *4.65 of [Whitehea...
imp 407 Importation inference. (C...
impcom 408 Importation inference with...
con3dimp 409 Variant of ~ con3d with im...
mpnanrd 410 Eliminate the right side o...
impd 411 Importation deduction. (C...
impcomd 412 Importation deduction with...
ex 413 Exportation inference. (T...
expcom 414 Exportation inference with...
expdcom 415 Commuted form of ~ expd . ...
expd 416 Exportation deduction. (C...
expcomd 417 Deduction form of ~ expcom...
imp31 418 An importation inference. ...
imp32 419 An importation inference. ...
exp31 420 An exportation inference. ...
exp32 421 An exportation inference. ...
imp4b 422 An importation inference. ...
imp4a 423 An importation inference. ...
imp4c 424 An importation inference. ...
imp4d 425 An importation inference. ...
imp41 426 An importation inference. ...
imp42 427 An importation inference. ...
imp43 428 An importation inference. ...
imp44 429 An importation inference. ...
imp45 430 An importation inference. ...
exp4b 431 An exportation inference. ...
exp4a 432 An exportation inference. ...
exp4c 433 An exportation inference. ...
exp4d 434 An exportation inference. ...
exp41 435 An exportation inference. ...
exp42 436 An exportation inference. ...
exp43 437 An exportation inference. ...
exp44 438 An exportation inference. ...
exp45 439 An exportation inference. ...
imp5d 440 An importation inference. ...
imp5a 441 An importation inference. ...
imp5g 442 An importation inference. ...
imp55 443 An importation inference. ...
imp511 444 An importation inference. ...
exp5c 445 An exportation inference. ...
exp5j 446 An exportation inference. ...
exp5l 447 An exportation inference. ...
exp53 448 An exportation inference. ...
pm3.3 449 Theorem *3.3 (Exp) of [Whi...
pm3.31 450 Theorem *3.31 (Imp) of [Wh...
impexp 451 Import-export theorem. Pa...
impancom 452 Mixed importation/commutat...
expdimp 453 A deduction version of exp...
expimpd 454 Exportation followed by a ...
impr 455 Import a wff into a right ...
impl 456 Export a wff from a left c...
expr 457 Export a wff from a right ...
expl 458 Export a wff from a left c...
ancoms 459 Inference commuting conjun...
pm3.22 460 Theorem *3.22 of [Whitehea...
ancom 461 Commutative law for conjun...
ancomd 462 Commutation of conjuncts i...
biancomi 463 Commuting conjunction in a...
biancomd 464 Commuting conjunction in a...
ancomst 465 Closed form of ~ ancoms . ...
ancomsd 466 Deduction commuting conjun...
anasss 467 Associative law for conjun...
anassrs 468 Associative law for conjun...
anass 469 Associative law for conjun...
pm3.2 470 Join antecedents with conj...
pm3.2i 471 Infer conjunction of premi...
pm3.21 472 Join antecedents with conj...
pm3.43i 473 Nested conjunction of ante...
pm3.43 474 Theorem *3.43 (Comp) of [W...
dfbi2 475 A theorem similar to the s...
dfbi 476 Definition ~ df-bi rewritt...
biimpa 477 Importation inference from...
biimpar 478 Importation inference from...
biimpac 479 Importation inference from...
biimparc 480 Importation inference from...
adantr 481 Inference adding a conjunc...
adantl 482 Inference adding a conjunc...
simpl 483 Elimination of a conjunct....
simpli 484 Inference eliminating a co...
simpr 485 Elimination of a conjunct....
simpri 486 Inference eliminating a co...
intnan 487 Introduction of conjunct i...
intnanr 488 Introduction of conjunct i...
intnand 489 Introduction of conjunct i...
intnanrd 490 Introduction of conjunct i...
adantld 491 Deduction adding a conjunc...
adantrd 492 Deduction adding a conjunc...
pm3.41 493 Theorem *3.41 of [Whitehea...
pm3.42 494 Theorem *3.42 of [Whitehea...
simpld 495 Deduction eliminating a co...
simprd 496 Deduction eliminating a co...
simprbi 497 Deduction eliminating a co...
simplbi 498 Deduction eliminating a co...
simprbda 499 Deduction eliminating a co...
simplbda 500 Deduction eliminating a co...
simplbi2 501 Deduction eliminating a co...
simplbi2comt 502 Closed form of ~ simplbi2c...
simplbi2com 503 A deduction eliminating a ...
simpl2im 504 Implication from an elimin...
simplbiim 505 Implication from an elimin...
impel 506 An inference for implicati...
mpan9 507 Modus ponens conjoining di...
sylan9 508 Nested syllogism inference...
sylan9r 509 Nested syllogism inference...
sylan9bb 510 Nested syllogism inference...
sylan9bbr 511 Nested syllogism inference...
jca 512 Deduce conjunction of the ...
jcad 513 Deduction conjoining the c...
jca2 514 Inference conjoining the c...
jca31 515 Join three consequents. (...
jca32 516 Join three consequents. (...
jcai 517 Deduction replacing implic...
jcab 518 Distributive law for impli...
pm4.76 519 Theorem *4.76 of [Whitehea...
jctil 520 Inference conjoining a the...
jctir 521 Inference conjoining a the...
jccir 522 Inference conjoining a con...
jccil 523 Inference conjoining a con...
jctl 524 Inference conjoining a the...
jctr 525 Inference conjoining a the...
jctild 526 Deduction conjoining a the...
jctird 527 Deduction conjoining a the...
iba 528 Introduction of antecedent...
ibar 529 Introduction of antecedent...
biantru 530 A wff is equivalent to its...
biantrur 531 A wff is equivalent to its...
biantrud 532 A wff is equivalent to its...
biantrurd 533 A wff is equivalent to its...
bianfi 534 A wff conjoined with false...
bianfd 535 A wff conjoined with false...
baib 536 Move conjunction outside o...
baibr 537 Move conjunction outside o...
rbaibr 538 Move conjunction outside o...
rbaib 539 Move conjunction outside o...
baibd 540 Move conjunction outside o...
rbaibd 541 Move conjunction outside o...
bianabs 542 Absorb a hypothesis into t...
pm5.44 543 Theorem *5.44 of [Whitehea...
pm5.42 544 Theorem *5.42 of [Whitehea...
ancl 545 Conjoin antecedent to left...
anclb 546 Conjoin antecedent to left...
ancr 547 Conjoin antecedent to righ...
ancrb 548 Conjoin antecedent to righ...
ancli 549 Deduction conjoining antec...
ancri 550 Deduction conjoining antec...
ancld 551 Deduction conjoining antec...
ancrd 552 Deduction conjoining antec...
impac 553 Importation with conjuncti...
anc2l 554 Conjoin antecedent to left...
anc2r 555 Conjoin antecedent to righ...
anc2li 556 Deduction conjoining antec...
anc2ri 557 Deduction conjoining antec...
pm4.71 558 Implication in terms of bi...
pm4.71r 559 Implication in terms of bi...
pm4.71i 560 Inference converting an im...
pm4.71ri 561 Inference converting an im...
pm4.71d 562 Deduction converting an im...
pm4.71rd 563 Deduction converting an im...
pm4.24 564 Theorem *4.24 of [Whitehea...
anidm 565 Idempotent law for conjunc...
anidmdbi 566 Conjunction idempotence wi...
anidms 567 Inference from idempotent ...
imdistan 568 Distribution of implicatio...
imdistani 569 Distribution of implicatio...
imdistanri 570 Distribution of implicatio...
imdistand 571 Distribution of implicatio...
imdistanda 572 Distribution of implicatio...
pm5.3 573 Theorem *5.3 of [Whitehead...
pm5.32 574 Distribution of implicatio...
pm5.32i 575 Distribution of implicatio...
pm5.32ri 576 Distribution of implicatio...
pm5.32d 577 Distribution of implicatio...
pm5.32rd 578 Distribution of implicatio...
pm5.32da 579 Distribution of implicatio...
sylan 580 A syllogism inference. (C...
sylanb 581 A syllogism inference. (C...
sylanbr 582 A syllogism inference. (C...
sylanbrc 583 Syllogism inference. (Con...
syl2anc 584 Syllogism inference combin...
syl2anc2 585 Double syllogism inference...
sylancl 586 Syllogism inference combin...
sylancr 587 Syllogism inference combin...
sylancom 588 Syllogism inference with c...
sylanblc 589 Syllogism inference combin...
sylanblrc 590 Syllogism inference combin...
syldan 591 A syllogism deduction with...
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...
pm4.38 634 Theorem *4.38 of [Whitehea...
bi2anan9 635 Deduction joining two equi...
bi2anan9r 636 Deduction joining two equi...
bi2bian9 637 Deduction joining two bico...
bianass 638 An inference to merge two ...
bianassc 639 An inference to merge two ...
an21 640 Swap two conjuncts. (Cont...
an12 641 Swap two conjuncts. Note ...
an32 642 A rearrangement of conjunc...
an13 643 A rearrangement of conjunc...
an31 644 A rearrangement of conjunc...
an12s 645 Swap two conjuncts in ante...
ancom2s 646 Inference commuting a nest...
an13s 647 Swap two conjuncts in ante...
an32s 648 Swap two conjuncts in ante...
ancom1s 649 Inference commuting a nest...
an31s 650 Swap two conjuncts in ante...
anass1rs 651 Commutative-associative la...
an4 652 Rearrangement of 4 conjunc...
an42 653 Rearrangement of 4 conjunc...
an43 654 Rearrangement of 4 conjunc...
an3 655 A rearrangement of conjunc...
an4s 656 Inference rearranging 4 co...
an42s 657 Inference rearranging 4 co...
anabs1 658 Absorption into embedded c...
anabs5 659 Absorption into embedded c...
anabs7 660 Absorption into embedded c...
anabsan 661 Absorption of antecedent w...
anabss1 662 Absorption of antecedent i...
anabss4 663 Absorption of antecedent i...
anabss5 664 Absorption of antecedent i...
anabsi5 665 Absorption of antecedent i...
anabsi6 666 Absorption of antecedent i...
anabsi7 667 Absorption of antecedent i...
anabsi8 668 Absorption of antecedent i...
anabss7 669 Absorption of antecedent i...
anabsan2 670 Absorption of antecedent w...
anabss3 671 Absorption of antecedent i...
anandi 672 Distribution of conjunctio...
anandir 673 Distribution of conjunctio...
anandis 674 Inference that undistribut...
anandirs 675 Inference that undistribut...
sylanl1 676 A syllogism inference. (C...
sylanl2 677 A syllogism inference. (C...
sylanr1 678 A syllogism inference. (C...
sylanr2 679 A syllogism inference. (C...
syl6an 680 A syllogism deduction comb...
syl2an2r 681 ~ syl2anr with antecedents...
syl2an2 682 ~ syl2an with antecedents ...
mpdan 683 An inference based on modu...
mpancom 684 An inference based on modu...
mpidan 685 A deduction which "stacks"...
mpan 686 An inference based on modu...
mpan2 687 An inference based on modu...
mp2an 688 An inference based on modu...
mp4an 689 An inference based on modu...
mpan2d 690 A deduction based on modus...
mpand 691 A deduction based on modus...
mpani 692 An inference based on modu...
mpan2i 693 An inference based on modu...
mp2ani 694 An inference based on modu...
mp2and 695 A deduction based on modus...
mpanl1 696 An inference based on modu...
mpanl2 697 An inference based on modu...
mpanl12 698 An inference based on modu...
mpanr1 699 An inference based on modu...
mpanr2 700 An inference based on modu...
mpanr12 701 An inference based on modu...
mpanlr1 702 An inference based on modu...
mpbirand 703 Detach truth from conjunct...
mpbiran2d 704 Detach truth from conjunct...
mpbiran 705 Detach truth from conjunct...
mpbiran2 706 Detach truth from conjunct...
mpbir2an 707 Detach a conjunction of tr...
mpbi2and 708 Detach a conjunction of tr...
mpbir2and 709 Detach a conjunction of tr...
adantll 710 Deduction adding a conjunc...
adantlr 711 Deduction adding a conjunc...
adantrl 712 Deduction adding a conjunc...
adantrr 713 Deduction adding a conjunc...
adantlll 714 Deduction adding a conjunc...
adantllr 715 Deduction adding a conjunc...
adantlrl 716 Deduction adding a conjunc...
adantlrr 717 Deduction adding a conjunc...
adantrll 718 Deduction adding a conjunc...
adantrlr 719 Deduction adding a conjunc...
adantrrl 720 Deduction adding a conjunc...
adantrrr 721 Deduction adding a conjunc...
ad2antrr 722 Deduction adding two conju...
ad2antlr 723 Deduction adding two conju...
ad2antrl 724 Deduction adding two conju...
ad2antll 725 Deduction adding conjuncts...
ad3antrrr 726 Deduction adding three con...
ad3antlr 727 Deduction adding three con...
ad4antr 728 Deduction adding 4 conjunc...
ad4antlr 729 Deduction adding 4 conjunc...
ad5antr 730 Deduction adding 5 conjunc...
ad5antlr 731 Deduction adding 5 conjunc...
ad6antr 732 Deduction adding 6 conjunc...
ad6antlr 733 Deduction adding 6 conjunc...
ad7antr 734 Deduction adding 7 conjunc...
ad7antlr 735 Deduction adding 7 conjunc...
ad8antr 736 Deduction adding 8 conjunc...
ad8antlr 737 Deduction adding 8 conjunc...
ad9antr 738 Deduction adding 9 conjunc...
ad9antlr 739 Deduction adding 9 conjunc...
ad10antr 740 Deduction adding 10 conjun...
ad10antlr 741 Deduction adding 10 conjun...
ad2ant2l 742 Deduction adding two conju...
ad2ant2r 743 Deduction adding two conju...
ad2ant2lr 744 Deduction adding two conju...
ad2ant2rl 745 Deduction adding two conju...
adantl3r 746 Deduction adding 1 conjunc...
ad4ant13 747 Deduction adding conjuncts...
ad4ant14 748 Deduction adding conjuncts...
ad4ant23 749 Deduction adding conjuncts...
ad4ant24 750 Deduction adding conjuncts...
adantl4r 751 Deduction adding 1 conjunc...
ad5ant12 752 Deduction adding conjuncts...
ad5ant13 753 Deduction adding conjuncts...
ad5ant14 754 Deduction adding conjuncts...
ad5ant15 755 Deduction adding conjuncts...
ad5ant23 756 Deduction adding conjuncts...
ad5ant24 757 Deduction adding conjuncts...
ad5ant25 758 Deduction adding conjuncts...
adantl5r 759 Deduction adding 1 conjunc...
adantl6r 760 Deduction adding 1 conjunc...
pm3.33 761 Theorem *3.33 (Syll) of [W...
pm3.34 762 Theorem *3.34 (Syll) of [W...
simpll 763 Simplification of a conjun...
simplld 764 Deduction form of ~ simpll...
simplr 765 Simplification of a conjun...
simplrd 766 Deduction eliminating a do...
simprl 767 Simplification of a conjun...
simprld 768 Deduction eliminating a do...
simprr 769 Simplification of a conjun...
simprrd 770 Deduction form of ~ simprr...
simplll 771 Simplification of a conjun...
simpllr 772 Simplification of a conjun...
simplrl 773 Simplification of a conjun...
simplrr 774 Simplification of a conjun...
simprll 775 Simplification of a conjun...
simprlr 776 Simplification of a conjun...
simprrl 777 Simplification of a conjun...
simprrr 778 Simplification of a conjun...
simp-4l 779 Simplification of a conjun...
simp-4r 780 Simplification of a conjun...
simp-5l 781 Simplification of a conjun...
simp-5r 782 Simplification of a conjun...
simp-6l 783 Simplification of a conjun...
simp-6r 784 Simplification of a conjun...
simp-7l 785 Simplification of a conjun...
simp-7r 786 Simplification of a conjun...
simp-8l 787 Simplification of a conjun...
simp-8r 788 Simplification of a conjun...
simp-9l 789 Simplification of a conjun...
simp-9r 790 Simplification of a conjun...
simp-10l 791 Simplification of a conjun...
simp-10r 792 Simplification of a conjun...
simp-11l 793 Simplification of a conjun...
simp-11r 794 Simplification of a conjun...
pm2.01da 795 Deduction based on reducti...
pm2.18da 796 Deduction based on reducti...
impbida 797 Deduce an equivalence from...
pm5.21nd 798 Eliminate an antecedent im...
pm3.35 799 Conjunctive detachment. T...
pm5.74da 800 Distribution of implicatio...
bitr 801 Theorem *4.22 of [Whitehea...
biantr 802 A transitive law of equiva...
pm4.14 803 Theorem *4.14 of [Whitehea...
pm3.37 804 Theorem *3.37 (Transp) of ...
anim12 805 Conjoin antecedents and co...
pm3.4 806 Conjunction implies implic...
exbiri 807 Inference form of ~ exbir ...
pm2.61ian 808 Elimination of an antecede...
pm2.61dan 809 Elimination of an antecede...
pm2.61ddan 810 Elimination of two anteced...
pm2.61dda 811 Elimination of two anteced...
mtand 812 A modus tollens deduction....
pm2.65da 813 Deduction for proof by con...
condan 814 Proof by contradiction. (...
biadan 815 An implication is equivale...
biadani 816 Inference associated with ...
biadaniALT 817 Alternate proof of ~ biada...
biadanii 818 Inference associated with ...
pm5.1 819 Two propositions are equiv...
pm5.21 820 Two propositions are equiv...
pm5.35 821 Theorem *5.35 of [Whitehea...
abai 822 Introduce one conjunct as ...
pm4.45im 823 Conjunction with implicati...
impimprbi 824 An implication and its rev...
nan 825 Theorem to move a conjunct...
pm5.31 826 Theorem *5.31 of [Whitehea...
pm5.31r 827 Variant of ~ pm5.31 . (Co...
pm4.15 828 Theorem *4.15 of [Whitehea...
pm5.36 829 Theorem *5.36 of [Whitehea...
annotanannot 830 A conjunction with a negat...
pm5.33 831 Theorem *5.33 of [Whitehea...
syl12anc 832 Syllogism combined with co...
syl21anc 833 Syllogism combined with co...
syl22anc 834 Syllogism combined with co...
syl1111anc 835 Four-hypothesis eliminatio...
mpsyl4anc 836 An elimination deduction. ...
pm4.87 837 Theorem *4.87 of [Whitehea...
bimsc1 838 Removal of conjunct from o...
a2and 839 Deduction distributing a c...
animpimp2impd 840 Deduction deriving nested ...
pm4.64 843 Theorem *4.64 of [Whitehea...
pm4.66 844 Theorem *4.66 of [Whitehea...
pm2.53 845 Theorem *2.53 of [Whitehea...
pm2.54 846 Theorem *2.54 of [Whitehea...
imor 847 Implication in terms of di...
imori 848 Infer disjunction from imp...
imorri 849 Infer implication from dis...
pm4.62 850 Theorem *4.62 of [Whitehea...
jaoi 851 Inference disjoining the a...
jao1i 852 Add a disjunct in the ante...
jaod 853 Deduction disjoining the a...
mpjaod 854 Eliminate a disjunction in...
ori 855 Infer implication from dis...
orri 856 Infer disjunction from imp...
orrd 857 Deduce disjunction from im...
ord 858 Deduce implication from di...
orci 859 Deduction introducing a di...
olci 860 Deduction introducing a di...
orc 861 Introduction of a disjunct...
olc 862 Introduction of a disjunct...
pm1.4 863 Axiom *1.4 of [WhiteheadRu...
orcom 864 Commutative law for disjun...
orcomd 865 Commutation of disjuncts i...
unitresl 866 A lemma for Conjunctive No...
unitresr 867 A lemma for Conjunctive No...
orcoms 868 Commutation of disjuncts i...
orcd 869 Deduction introducing a di...
olcd 870 Deduction introducing a di...
orcs 871 Deduction eliminating disj...
olcs 872 Deduction eliminating disj...
mtord 873 A modus tollens deduction ...
pm3.2ni 874 Infer negated disjunction ...
pm2.45 875 Theorem *2.45 of [Whitehea...
pm2.46 876 Theorem *2.46 of [Whitehea...
pm2.47 877 Theorem *2.47 of [Whitehea...
pm2.48 878 Theorem *2.48 of [Whitehea...
pm2.49 879 Theorem *2.49 of [Whitehea...
norbi 880 If neither of two proposit...
nbior 881 If two propositions are no...
orel1 882 Elimination of disjunction...
pm2.25 883 Theorem *2.25 of [Whitehea...
orel2 884 Elimination of disjunction...
pm2.67-2 885 Slight generalization of T...
pm2.67 886 Theorem *2.67 of [Whitehea...
curryax 887 A non-intuitionistic posit...
exmid 888 Law of excluded middle, al...
exmidd 889 Law of excluded middle in ...
pm2.1 890 Theorem *2.1 of [Whitehead...
pm2.13 891 Theorem *2.13 of [Whitehea...
pm2.621 892 Theorem *2.621 of [Whitehe...
pm2.62 893 Theorem *2.62 of [Whitehea...
pm2.68 894 Theorem *2.68 of [Whitehea...
dfor2 895 Logical 'or' expressed in ...
pm2.07 896 Theorem *2.07 of [Whitehea...
pm1.2 897 Axiom *1.2 of [WhiteheadRu...
oridm 898 Idempotent law for disjunc...
pm4.25 899 Theorem *4.25 of [Whitehea...
pm2.4 900 Theorem *2.4 of [Whitehead...
pm2.41 901 Theorem *2.41 of [Whitehea...
orim12i 902 Disjoin antecedents and co...
orim1i 903 Introduce disjunct to both...
orim2i 904 Introduce disjunct to both...
orim12dALT 905 Alternate proof of ~ orim1...
orbi2i 906 Inference adding a left di...
orbi1i 907 Inference adding a right d...
orbi12i 908 Infer the disjunction of t...
orbi2d 909 Deduction adding a left di...
orbi1d 910 Deduction adding a right d...
orbi1 911 Theorem *4.37 of [Whitehea...
orbi12d 912 Deduction joining two equi...
pm1.5 913 Axiom *1.5 (Assoc) of [Whi...
or12 914 Swap two disjuncts. (Cont...
orass 915 Associative law for disjun...
pm2.31 916 Theorem *2.31 of [Whitehea...
pm2.32 917 Theorem *2.32 of [Whitehea...
pm2.3 918 Theorem *2.3 of [Whitehead...
or32 919 A rearrangement of disjunc...
or4 920 Rearrangement of 4 disjunc...
or42 921 Rearrangement of 4 disjunc...
orordi 922 Distribution of disjunctio...
orordir 923 Distribution of disjunctio...
orimdi 924 Disjunction distributes ov...
pm2.76 925 Theorem *2.76 of [Whitehea...
pm2.85 926 Theorem *2.85 of [Whitehea...
pm2.75 927 Theorem *2.75 of [Whitehea...
pm4.78 928 Implication distributes ov...
biort 929 A wff disjoined with truth...
biorf 930 A wff is equivalent to its...
biortn 931 A wff is equivalent to its...
biorfi 932 A wff is equivalent to its...
pm2.26 933 Theorem *2.26 of [Whitehea...
pm2.63 934 Theorem *2.63 of [Whitehea...
pm2.64 935 Theorem *2.64 of [Whitehea...
pm2.42 936 Theorem *2.42 of [Whitehea...
pm5.11g 937 A general instance of Theo...
pm5.11 938 Theorem *5.11 of [Whitehea...
pm5.12 939 Theorem *5.12 of [Whitehea...
pm5.14 940 Theorem *5.14 of [Whitehea...
pm5.13 941 Theorem *5.13 of [Whitehea...
pm5.55 942 Theorem *5.55 of [Whitehea...
pm4.72 943 Implication in terms of bi...
imimorb 944 Simplify an implication be...
oibabs 945 Absorption of disjunction ...
orbidi 946 Disjunction distributes ov...
pm5.7 947 Disjunction distributes ov...
jaao 948 Inference conjoining and d...
jaoa 949 Inference disjoining and c...
jaoian 950 Inference disjoining the a...
jaodan 951 Deduction disjoining the a...
mpjaodan 952 Eliminate a disjunction in...
pm3.44 953 Theorem *3.44 of [Whitehea...
jao 954 Disjunction of antecedents...
jaob 955 Disjunction of antecedents...
pm4.77 956 Theorem *4.77 of [Whitehea...
pm3.48 957 Theorem *3.48 of [Whitehea...
orim12d 958 Disjoin antecedents and co...
orim1d 959 Disjoin antecedents and co...
orim2d 960 Disjoin antecedents and co...
orim2 961 Axiom *1.6 (Sum) of [White...
pm2.38 962 Theorem *2.38 of [Whitehea...
pm2.36 963 Theorem *2.36 of [Whitehea...
pm2.37 964 Theorem *2.37 of [Whitehea...
pm2.81 965 Theorem *2.81 of [Whitehea...
pm2.8 966 Theorem *2.8 of [Whitehead...
pm2.73 967 Theorem *2.73 of [Whitehea...
pm2.74 968 Theorem *2.74 of [Whitehea...
pm2.82 969 Theorem *2.82 of [Whitehea...
pm4.39 970 Theorem *4.39 of [Whitehea...
animorl 971 Conjunction implies disjun...
animorr 972 Conjunction implies disjun...
animorlr 973 Conjunction implies disjun...
animorrl 974 Conjunction implies disjun...
ianor 975 Negated conjunction in ter...
anor 976 Conjunction in terms of di...
ioran 977 Negated disjunction in ter...
pm4.52 978 Theorem *4.52 of [Whitehea...
pm4.53 979 Theorem *4.53 of [Whitehea...
pm4.54 980 Theorem *4.54 of [Whitehea...
pm4.55 981 Theorem *4.55 of [Whitehea...
pm4.56 982 Theorem *4.56 of [Whitehea...
oran 983 Disjunction in terms of co...
pm4.57 984 Theorem *4.57 of [Whitehea...
pm3.1 985 Theorem *3.1 of [Whitehead...
pm3.11 986 Theorem *3.11 of [Whitehea...
pm3.12 987 Theorem *3.12 of [Whitehea...
pm3.13 988 Theorem *3.13 of [Whitehea...
pm3.14 989 Theorem *3.14 of [Whitehea...
pm4.44 990 Theorem *4.44 of [Whitehea...
pm4.45 991 Theorem *4.45 of [Whitehea...
orabs 992 Absorption of redundant in...
oranabs 993 Absorb a disjunct into a c...
pm5.61 994 Theorem *5.61 of [Whitehea...
pm5.6 995 Conjunction in antecedent ...
orcanai 996 Change disjunction in cons...
pm4.79 997 Theorem *4.79 of [Whitehea...
pm5.53 998 Theorem *5.53 of [Whitehea...
ordi 999 Distributive law for disju...
ordir 1000 Distributive law for disju...
andi 1001 Distributive law for conju...
andir 1002 Distributive law for conju...
orddi 1003 Double distributive law fo...
anddi 1004 Double distributive law fo...
pm5.17 1005 Theorem *5.17 of [Whitehea...
pm5.15 1006 Theorem *5.15 of [Whitehea...
pm5.16 1007 Theorem *5.16 of [Whitehea...
xor 1008 Two ways to express exclus...
nbi2 1009 Two ways to express "exclu...
xordi 1010 Conjunction distributes ov...
pm5.54 1011 Theorem *5.54 of [Whitehea...
pm5.62 1012 Theorem *5.62 of [Whitehea...
pm5.63 1013 Theorem *5.63 of [Whitehea...
niabn 1014 Miscellaneous inference re...
ninba 1015 Miscellaneous inference re...
pm4.43 1016 Theorem *4.43 of [Whitehea...
pm4.82 1017 Theorem *4.82 of [Whitehea...
pm4.83 1018 Theorem *4.83 of [Whitehea...
pclem6 1019 Negation inferred from emb...
bigolden 1020 Dijkstra-Scholten's Golden...
pm5.71 1021 Theorem *5.71 of [Whitehea...
pm5.75 1022 Theorem *5.75 of [Whitehea...
ecase2d 1023 Deduction for elimination ...
ecase3 1024 Inference for elimination ...
ecase 1025 Inference for elimination ...
ecase3d 1026 Deduction for elimination ...
ecased 1027 Deduction for elimination ...
ecase3ad 1028 Deduction for elimination ...
ccase 1029 Inference for combining ca...
ccased 1030 Deduction for combining ca...
ccase2 1031 Inference for combining ca...
4cases 1032 Inference eliminating two ...
4casesdan 1033 Deduction eliminating two ...
cases 1034 Case disjunction according...
dedlem0a 1035 Lemma for an alternate ver...
dedlem0b 1036 Lemma for an alternate ver...
dedlema 1037 Lemma for weak deduction t...
dedlemb 1038 Lemma for weak deduction t...
cases2 1039 Case disjunction according...
cases2ALT 1040 Alternate proof of ~ cases...
dfbi3 1041 An alternate definition of...
pm5.24 1042 Theorem *5.24 of [Whitehea...
4exmid 1043 The disjunction of the fou...
consensus 1044 The consensus theorem. Th...
pm4.42 1045 Theorem *4.42 of [Whitehea...
prlem1 1046 A specialized lemma for se...
prlem2 1047 A specialized lemma for se...
oplem1 1048 A specialized lemma for se...
dn1 1049 A single axiom for Boolean...
bianir 1050 A closed form of ~ mpbir ,...
jaoi2 1051 Inference removing a negat...
jaoi3 1052 Inference separating a dis...
ornld 1053 Selecting one statement fr...
dfifp2 1056 Alternate definition of th...
dfifp3 1057 Alternate definition of th...
dfifp4 1058 Alternate definition of th...
dfifp5 1059 Alternate definition of th...
dfifp6 1060 Alternate definition of th...
dfifp7 1061 Alternate definition of th...
anifp 1062 The conditional operator i...
ifpor 1063 The conditional operator i...
ifpn 1064 Conditional operator for t...
ifptru 1065 Value of the conditional o...
ifpfal 1066 Value of the conditional o...
ifpid 1067 Value of the conditional o...
casesifp 1068 Version of ~ cases express...
ifpbi123d 1069 Equality deduction for con...
ifpimpda 1070 Separation of the values o...
1fpid3 1071 The value of the condition...
elimh 1072 Hypothesis builder for the...
elimhOLD 1073 Obsolete version of ~ elim...
dedt 1074 The weak deduction theorem...
dedtOLD 1075 Obsolete version of ~ dedt...
con3ALT 1076 Proof of ~ con3 from its a...
con3ALTOLD 1077 Obsolete version of ~ con3...
3orass 1082 Associative law for triple...
3orel1 1083 Partial elimination of a t...
3orrot 1084 Rotation law for triple di...
3orcoma 1085 Commutation law for triple...
3orcomb 1086 Commutation law for triple...
3anass 1087 Associative law for triple...
3anan12 1088 Convert triple conjunction...
3anan32 1089 Convert triple conjunction...
3ancoma 1090 Commutation law for triple...
3ancomb 1091 Commutation law for triple...
3anrot 1092 Rotation law for triple co...
3anrev 1093 Reversal law for triple co...
anandi3 1094 Distribution of triple con...
anandi3r 1095 Distribution of triple con...
3anidm 1096 Idempotent law for conjunc...
3an4anass 1097 Associative law for four c...
3ioran 1098 Negated triple disjunction...
3ianor 1099 Negated triple conjunction...
3anor 1100 Triple conjunction express...
3oran 1101 Triple disjunction in term...
3impa 1102 Importation from double to...
3imp 1103 Importation inference. (C...
3imp31 1104 The importation inference ...
3imp231 1105 Importation inference. (C...
3imp21 1106 The importation inference ...
3impb 1107 Importation from double to...
3impib 1108 Importation to triple conj...
3impia 1109 Importation to triple conj...
3expa 1110 Exportation from triple to...
3exp 1111 Exportation inference. (C...
3expb 1112 Exportation from triple to...
3expia 1113 Exportation from triple co...
3expib 1114 Exportation from triple co...
3com12 1115 Commutation in antecedent....
3com13 1116 Commutation in antecedent....
3comr 1117 Commutation in antecedent....
3com23 1118 Commutation in antecedent....
3coml 1119 Commutation in antecedent....
3jca 1120 Join consequents with conj...
3jcad 1121 Deduction conjoining the c...
3adant1 1122 Deduction adding a conjunc...
3adant2 1123 Deduction adding a conjunc...
3adant3 1124 Deduction adding a conjunc...
3ad2ant1 1125 Deduction adding conjuncts...
3ad2ant2 1126 Deduction adding conjuncts...
3ad2ant3 1127 Deduction adding conjuncts...
simp1 1128 Simplification of triple c...
simp2 1129 Simplification of triple c...
simp3 1130 Simplification of triple c...
simp1i 1131 Infer a conjunct from a tr...
simp2i 1132 Infer a conjunct from a tr...
simp3i 1133 Infer a conjunct from a tr...
simp1d 1134 Deduce a conjunct from a t...
simp2d 1135 Deduce a conjunct from a t...
simp3d 1136 Deduce a conjunct from a t...
simp1bi 1137 Deduce a conjunct from a t...
simp2bi 1138 Deduce a conjunct from a t...
simp3bi 1139 Deduce a conjunct from a t...
3simpa 1140 Simplification of triple c...
3simpb 1141 Simplification of triple c...
3simpc 1142 Simplification of triple c...
3anim123i 1143 Join antecedents and conse...
3anim1i 1144 Add two conjuncts to antec...
3anim2i 1145 Add two conjuncts to antec...
3anim3i 1146 Add two conjuncts to antec...
3anbi123i 1147 Join 3 biconditionals with...
3orbi123i 1148 Join 3 biconditionals with...
3anbi1i 1149 Inference adding two conju...
3anbi2i 1150 Inference adding two conju...
3anbi3i 1151 Inference adding two conju...
syl3an 1152 A triple syllogism inferen...
syl3anb 1153 A triple syllogism inferen...
syl3anbr 1154 A triple syllogism inferen...
syl3an1 1155 A syllogism inference. (C...
syl3an2 1156 A syllogism inference. (C...
syl3an3 1157 A syllogism inference. (C...
3adantl1 1158 Deduction adding a conjunc...
3adantl2 1159 Deduction adding a conjunc...
3adantl3 1160 Deduction adding a conjunc...
3adantr1 1161 Deduction adding a conjunc...
3adantr2 1162 Deduction adding a conjunc...
3adantr3 1163 Deduction adding a conjunc...
ad4ant123 1164 Deduction adding conjuncts...
ad4ant124 1165 Deduction adding conjuncts...
ad4ant134 1166 Deduction adding conjuncts...
ad4ant234 1167 Deduction adding conjuncts...
3adant1l 1168 Deduction adding a conjunc...
3adant1r 1169 Deduction adding a conjunc...
3adant2l 1170 Deduction adding a conjunc...
3adant2r 1171 Deduction adding a conjunc...
3adant3l 1172 Deduction adding a conjunc...
3adant3r 1173 Deduction adding a conjunc...
3adant3r1 1174 Deduction adding a conjunc...
3adant3r2 1175 Deduction adding a conjunc...
3adant3r3 1176 Deduction adding a conjunc...
3ad2antl1 1177 Deduction adding conjuncts...
3ad2antl2 1178 Deduction adding conjuncts...
3ad2antl3 1179 Deduction adding conjuncts...
3ad2antr1 1180 Deduction adding conjuncts...
3ad2antr2 1181 Deduction adding conjuncts...
3ad2antr3 1182 Deduction adding conjuncts...
simpl1 1183 Simplification of conjunct...
simpl2 1184 Simplification of conjunct...
simpl3 1185 Simplification of conjunct...
simpr1 1186 Simplification of conjunct...
simpr2 1187 Simplification of conjunct...
simpr3 1188 Simplification of conjunct...
simp1l 1189 Simplification of triple c...
simp1r 1190 Simplification of triple c...
simp2l 1191 Simplification of triple c...
simp2r 1192 Simplification of triple c...
simp3l 1193 Simplification of triple c...
simp3r 1194 Simplification of triple c...
simp11 1195 Simplification of doubly t...
simp12 1196 Simplification of doubly t...
simp13 1197 Simplification of doubly t...
simp21 1198 Simplification of doubly t...
simp22 1199 Simplification of doubly t...
simp23 1200 Simplification of doubly t...
simp31 1201 Simplification of doubly t...
simp32 1202 Simplification of doubly t...
simp33 1203 Simplification of doubly t...
simpll1 1204 Simplification of conjunct...
simpll2 1205 Simplification of conjunct...
simpll3 1206 Simplification of conjunct...
simplr1 1207 Simplification of conjunct...
simplr2 1208 Simplification of conjunct...
simplr3 1209 Simplification of conjunct...
simprl1 1210 Simplification of conjunct...
simprl2 1211 Simplification of conjunct...
simprl3 1212 Simplification of conjunct...
simprr1 1213 Simplification of conjunct...
simprr2 1214 Simplification of conjunct...
simprr3 1215 Simplification of conjunct...
simpl1l 1216 Simplification of conjunct...
simpl1r 1217 Simplification of conjunct...
simpl2l 1218 Simplification of conjunct...
simpl2r 1219 Simplification of conjunct...
simpl3l 1220 Simplification of conjunct...
simpl3r 1221 Simplification of conjunct...
simpr1l 1222 Simplification of conjunct...
simpr1r 1223 Simplification of conjunct...
simpr2l 1224 Simplification of conjunct...
simpr2r 1225 Simplification of conjunct...
simpr3l 1226 Simplification of conjunct...
simpr3r 1227 Simplification of conjunct...
simp1ll 1228 Simplification of conjunct...
simp1lr 1229 Simplification of conjunct...
simp1rl 1230 Simplification of conjunct...
simp1rr 1231 Simplification of conjunct...
simp2ll 1232 Simplification of conjunct...
simp2lr 1233 Simplification of conjunct...
simp2rl 1234 Simplification of conjunct...
simp2rr 1235 Simplification of conjunct...
simp3ll 1236 Simplification of conjunct...
simp3lr 1237 Simplification of conjunct...
simp3rl 1238 Simplification of conjunct...
simp3rr 1239 Simplification of conjunct...
simpl11 1240 Simplification of conjunct...
simpl12 1241 Simplification of conjunct...
simpl13 1242 Simplification of conjunct...
simpl21 1243 Simplification of conjunct...
simpl22 1244 Simplification of conjunct...
simpl23 1245 Simplification of conjunct...
simpl31 1246 Simplification of conjunct...
simpl32 1247 Simplification of conjunct...
simpl33 1248 Simplification of conjunct...
simpr11 1249 Simplification of conjunct...
simpr12 1250 Simplification of conjunct...
simpr13 1251 Simplification of conjunct...
simpr21 1252 Simplification of conjunct...
simpr22 1253 Simplification of conjunct...
simpr23 1254 Simplification of conjunct...
simpr31 1255 Simplification of conjunct...
simpr32 1256 Simplification of conjunct...
simpr33 1257 Simplification of conjunct...
simp1l1 1258 Simplification of conjunct...
simp1l2 1259 Simplification of conjunct...
simp1l3 1260 Simplification of conjunct...
simp1r1 1261 Simplification of conjunct...
simp1r2 1262 Simplification of conjunct...
simp1r3 1263 Simplification of conjunct...
simp2l1 1264 Simplification of conjunct...
simp2l2 1265 Simplification of conjunct...
simp2l3 1266 Simplification of conjunct...
simp2r1 1267 Simplification of conjunct...
simp2r2 1268 Simplification of conjunct...
simp2r3 1269 Simplification of conjunct...
simp3l1 1270 Simplification of conjunct...
simp3l2 1271 Simplification of conjunct...
simp3l3 1272 Simplification of conjunct...
simp3r1 1273 Simplification of conjunct...
simp3r2 1274 Simplification of conjunct...
simp3r3 1275 Simplification of conjunct...
simp11l 1276 Simplification of conjunct...
simp11r 1277 Simplification of conjunct...
simp12l 1278 Simplification of conjunct...
simp12r 1279 Simplification of conjunct...
simp13l 1280 Simplification of conjunct...
simp13r 1281 Simplification of conjunct...
simp21l 1282 Simplification of conjunct...
simp21r 1283 Simplification of conjunct...
simp22l 1284 Simplification of conjunct...
simp22r 1285 Simplification of conjunct...
simp23l 1286 Simplification of conjunct...
simp23r 1287 Simplification of conjunct...
simp31l 1288 Simplification of conjunct...
simp31r 1289 Simplification of conjunct...
simp32l 1290 Simplification of conjunct...
simp32r 1291 Simplification of conjunct...
simp33l 1292 Simplification of conjunct...
simp33r 1293 Simplification of conjunct...
simp111 1294 Simplification of conjunct...
simp112 1295 Simplification of conjunct...
simp113 1296 Simplification of conjunct...
simp121 1297 Simplification of conjunct...
simp122 1298 Simplification of conjunct...
simp123 1299 Simplification of conjunct...
simp131 1300 Simplification of conjunct...
simp132 1301 Simplification of conjunct...
simp133 1302 Simplification of conjunct...
simp211 1303 Simplification of conjunct...
simp212 1304 Simplification of conjunct...
simp213 1305 Simplification of conjunct...
simp221 1306 Simplification of conjunct...
simp222 1307 Simplification of conjunct...
simp223 1308 Simplification of conjunct...
simp231 1309 Simplification of conjunct...
simp232 1310 Simplification of conjunct...
simp233 1311 Simplification of conjunct...
simp311 1312 Simplification of conjunct...
simp312 1313 Simplification of conjunct...
simp313 1314 Simplification of conjunct...
simp321 1315 Simplification of conjunct...
simp322 1316 Simplification of conjunct...
simp323 1317 Simplification of conjunct...
simp331 1318 Simplification of conjunct...
simp332 1319 Simplification of conjunct...
simp333 1320 Simplification of conjunct...
3anibar 1321 Remove a hypothesis from t...
3mix1 1322 Introduction in triple dis...
3mix2 1323 Introduction in triple dis...
3mix3 1324 Introduction in triple dis...
3mix1i 1325 Introduction in triple dis...
3mix2i 1326 Introduction in triple dis...
3mix3i 1327 Introduction in triple dis...
3mix1d 1328 Deduction introducing trip...
3mix2d 1329 Deduction introducing trip...
3mix3d 1330 Deduction introducing trip...
3pm3.2i 1331 Infer conjunction of premi...
pm3.2an3 1332 Version of ~ pm3.2 for a t...
mpbir3an 1333 Detach a conjunction of tr...
mpbir3and 1334 Detach a conjunction of tr...
syl3anbrc 1335 Syllogism inference. (Con...
syl21anbrc 1336 Syllogism inference. (Con...
3imp3i2an 1337 An elimination deduction. ...
ex3 1338 Apply ~ ex to a hypothesis...
3imp1 1339 Importation to left triple...
3impd 1340 Importation deduction for ...
3imp2 1341 Importation to right tripl...
3impdi 1342 Importation inference (und...
3impdir 1343 Importation inference (und...
3exp1 1344 Exportation from left trip...
3expd 1345 Exportation deduction for ...
3exp2 1346 Exportation from right tri...
exp5o 1347 A triple exportation infer...
exp516 1348 A triple exportation infer...
exp520 1349 A triple exportation infer...
3impexp 1350 Version of ~ impexp for a ...
3an1rs 1351 Swap conjuncts. (Contribu...
3anassrs 1352 Associative law for conjun...
ad5ant245 1353 Deduction adding conjuncts...
ad5ant234 1354 Deduction adding conjuncts...
ad5ant235 1355 Deduction adding conjuncts...
ad5ant123 1356 Deduction adding conjuncts...
ad5ant124 1357 Deduction adding conjuncts...
ad5ant125 1358 Deduction adding conjuncts...
ad5ant134 1359 Deduction adding conjuncts...
ad5ant135 1360 Deduction adding conjuncts...
ad5ant145 1361 Deduction adding conjuncts...
ad5ant2345 1362 Deduction adding conjuncts...
syl3anc 1363 Syllogism combined with co...
syl13anc 1364 Syllogism combined with co...
syl31anc 1365 Syllogism combined with co...
syl112anc 1366 Syllogism combined with co...
syl121anc 1367 Syllogism combined with co...
syl211anc 1368 Syllogism combined with co...
syl23anc 1369 Syllogism combined with co...
syl32anc 1370 Syllogism combined with co...
syl122anc 1371 Syllogism combined with co...
syl212anc 1372 Syllogism combined with co...
syl221anc 1373 Syllogism combined with co...
syl113anc 1374 Syllogism combined with co...
syl131anc 1375 Syllogism combined with co...
syl311anc 1376 Syllogism combined with co...
syl33anc 1377 Syllogism combined with co...
syl222anc 1378 Syllogism combined with co...
syl123anc 1379 Syllogism combined with co...
syl132anc 1380 Syllogism combined with co...
syl213anc 1381 Syllogism combined with co...
syl231anc 1382 Syllogism combined with co...
syl312anc 1383 Syllogism combined with co...
syl321anc 1384 Syllogism combined with co...
syl133anc 1385 Syllogism combined with co...
syl313anc 1386 Syllogism combined with co...
syl331anc 1387 Syllogism combined with co...
syl223anc 1388 Syllogism combined with co...
syl232anc 1389 Syllogism combined with co...
syl322anc 1390 Syllogism combined with co...
syl233anc 1391 Syllogism combined with co...
syl323anc 1392 Syllogism combined with co...
syl332anc 1393 Syllogism combined with co...
syl333anc 1394 A syllogism inference comb...
syl3an1b 1395 A syllogism inference. (C...
syl3an2b 1396 A syllogism inference. (C...
syl3an3b 1397 A syllogism inference. (C...
syl3an1br 1398 A syllogism inference. (C...
syl3an2br 1399 A syllogism inference. (C...
syl3an3br 1400 A syllogism inference. (C...
syld3an3 1401 A syllogism inference. (C...
syld3an1 1402 A syllogism inference. (C...
syld3an2 1403 A syllogism inference. (C...
syl3anl1 1404 A syllogism inference. (C...
syl3anl2 1405 A syllogism inference. (C...
syl3anl3 1406 A syllogism inference. (C...
syl3anl 1407 A triple syllogism inferen...
syl3anr1 1408 A syllogism inference. (C...
syl3anr2 1409 A syllogism inference. (C...
syl3anr3 1410 A syllogism inference. (C...
3anidm12 1411 Inference from idempotent ...
3anidm13 1412 Inference from idempotent ...
3anidm23 1413 Inference from idempotent ...
syl2an3an 1414 ~ syl3an with antecedents ...
syl2an23an 1415 Deduction related to ~ syl...
3ori 1416 Infer implication from tri...
3jao 1417 Disjunction of three antec...
3jaob 1418 Disjunction of three antec...
3jaoi 1419 Disjunction of three antec...
3jaod 1420 Disjunction of three antec...
3jaoian 1421 Disjunction of three antec...
3jaodan 1422 Disjunction of three antec...
mpjao3dan 1423 Eliminate a three-way disj...
3jaao 1424 Inference conjoining and d...
syl3an9b 1425 Nested syllogism inference...
3orbi123d 1426 Deduction joining 3 equiva...
3anbi123d 1427 Deduction joining 3 equiva...
3anbi12d 1428 Deduction conjoining and a...
3anbi13d 1429 Deduction conjoining and a...
3anbi23d 1430 Deduction conjoining and a...
3anbi1d 1431 Deduction adding conjuncts...
3anbi2d 1432 Deduction adding conjuncts...
3anbi3d 1433 Deduction adding conjuncts...
3anim123d 1434 Deduction joining 3 implic...
3orim123d 1435 Deduction joining 3 implic...
an6 1436 Rearrangement of 6 conjunc...
3an6 1437 Analogue of ~ an4 for trip...
3or6 1438 Analogue of ~ or4 for trip...
mp3an1 1439 An inference based on modu...
mp3an2 1440 An inference based on modu...
mp3an3 1441 An inference based on modu...
mp3an12 1442 An inference based on modu...
mp3an13 1443 An inference based on modu...
mp3an23 1444 An inference based on modu...
mp3an1i 1445 An inference based on modu...
mp3anl1 1446 An inference based on modu...
mp3anl2 1447 An inference based on modu...
mp3anl3 1448 An inference based on modu...
mp3anr1 1449 An inference based on modu...
mp3anr2 1450 An inference based on modu...
mp3anr3 1451 An inference based on modu...
mp3an 1452 An inference based on modu...
mpd3an3 1453 An inference based on modu...
mpd3an23 1454 An inference based on modu...
mp3and 1455 A deduction based on modus...
mp3an12i 1456 ~ mp3an with antecedents i...
mp3an2i 1457 ~ mp3an with antecedents i...
mp3an3an 1458 ~ mp3an with antecedents i...
mp3an2ani 1459 An elimination deduction. ...
biimp3a 1460 Infer implication from a l...
biimp3ar 1461 Infer implication from a l...
3anandis 1462 Inference that undistribut...
3anandirs 1463 Inference that undistribut...
ecase23d 1464 Deduction for elimination ...
3ecase 1465 Inference for elimination ...
3bior1fd 1466 A disjunction is equivalen...
3bior1fand 1467 A disjunction is equivalen...
3bior2fd 1468 A wff is equivalent to its...
3biant1d 1469 A conjunction is equivalen...
intn3an1d 1470 Introduction of a triple c...
intn3an2d 1471 Introduction of a triple c...
intn3an3d 1472 Introduction of a triple c...
an3andi 1473 Distribution of conjunctio...
an33rean 1474 Rearrange a 9-fold conjunc...
nanan 1477 Conjunction in terms of al...
nanimn 1478 Alternative denial in term...
nanor 1479 Alternative denial in term...
nancom 1480 Alternative denial is comm...
nannan 1481 Nested alternative denials...
nanim 1482 Implication in terms of al...
nannot 1483 Negation in terms of alter...
nanbi 1484 Biconditional in terms of ...
nanbi1 1485 Introduce a right anti-con...
nanbi2 1486 Introduce a left anti-conj...
nanbi12 1487 Join two logical equivalen...
nanbi1i 1488 Introduce a right anti-con...
nanbi2i 1489 Introduce a left anti-conj...
nanbi12i 1490 Join two logical equivalen...
nanbi1d 1491 Introduce a right anti-con...
nanbi2d 1492 Introduce a left anti-conj...
nanbi12d 1493 Join two logical equivalen...
nanass 1494 A characterization of when...
xnor 1497 Two ways to write XNOR. (C...
xorcom 1498 The connector ` \/_ ` is c...
xorass 1499 The connector ` \/_ ` is a...
excxor 1500 This tautology shows that ...
xor2 1501 Two ways to express "exclu...
xoror 1502 XOR implies OR. (Contribut...
xornan 1503 XOR implies NAND. (Contrib...
xornan2 1504 XOR implies NAND (written ...
xorneg2 1505 The connector ` \/_ ` is n...
xorneg1 1506 The connector ` \/_ ` is n...
xorneg 1507 The connector ` \/_ ` is u...
xorbi12i 1508 Equality property for XOR....
xorbi12d 1509 Equality property for XOR....
anxordi 1510 Conjunction distributes ov...
xorexmid 1511 Exclusive-or variant of th...
norcom 1514 The connector ` -\/ ` is c...
nornot 1515 ` -. ` is expressible via ...
nornotOLD 1516 Obsolete version of ~ norn...
noran 1517 ` /\ ` is expressible via ...
noranOLD 1518 Obsolete version of ~ nora...
noror 1519 ` \/ ` is expressible via ...
nororOLD 1520 Obsolete version of ~ noro...
norasslem1 1521 This lemma shows the equiv...
norasslem2 1522 This lemma specializes ~ b...
norasslem3 1523 This lemma specializes ~ b...
norass 1524 A characterization of when...
norassOLD 1525 Obsolete version of ~ nora...
trujust 1530 Soundness justification th...
tru 1532 The truth value ` T. ` is ...
dftru2 1533 An alternate definition of...
trut 1534 A proposition is equivalen...
mptru 1535 Eliminate ` T. ` as an ant...
tbtru 1536 A proposition is equivalen...
bitru 1537 A theorem is equivalent to...
trud 1538 Anything implies ` T. ` . ...
truan 1539 True can be removed from a...
fal 1542 The truth value ` F. ` is ...
nbfal 1543 The negation of a proposit...
bifal 1544 A contradiction is equival...
falim 1545 The truth value ` F. ` imp...
falimd 1546 The truth value ` F. ` imp...
dfnot 1547 Given falsum ` F. ` , we c...
inegd 1548 Negation introduction rule...
efald 1549 Deduction based on reducti...
pm2.21fal 1550 If a wff and its negation ...
truimtru 1551 A ` -> ` identity. (Contr...
truimfal 1552 A ` -> ` identity. (Contr...
falimtru 1553 A ` -> ` identity. (Contr...
falimfal 1554 A ` -> ` identity. (Contr...
nottru 1555 A ` -. ` identity. (Contr...
notfal 1556 A ` -. ` identity. (Contr...
trubitru 1557 A ` <-> ` identity. (Cont...
falbitru 1558 A ` <-> ` identity. (Cont...
trubifal 1559 A ` <-> ` identity. (Cont...
falbifal 1560 A ` <-> ` identity. (Cont...
truantru 1561 A ` /\ ` identity. (Contr...
truanfal 1562 A ` /\ ` identity. (Contr...
falantru 1563 A ` /\ ` identity. (Contr...
falanfal 1564 A ` /\ ` identity. (Contr...
truortru 1565 A ` \/ ` identity. (Contr...
truorfal 1566 A ` \/ ` identity. (Contr...
falortru 1567 A ` \/ ` identity. (Contr...
falorfal 1568 A ` \/ ` identity. (Contr...
trunantru 1569 A ` -/\ ` identity. (Cont...
trunanfal 1570 A ` -/\ ` identity. (Cont...
falnantru 1571 A ` -/\ ` identity. (Cont...
falnanfal 1572 A ` -/\ ` identity. (Cont...
truxortru 1573 A ` \/_ ` identity. (Cont...
truxorfal 1574 A ` \/_ ` identity. (Cont...
falxortru 1575 A ` \/_ ` identity. (Cont...
falxorfal 1576 A ` \/_ ` identity. (Cont...
trunortru 1577 A ` -\/ ` identity. (Cont...
trunortruOLD 1578 Obsolete version of ~ trun...
trunorfal 1579 A ` -\/ ` identity. (Cont...
trunorfalOLD 1580 Obsolete version of ~ trun...
falnortru 1581 A ` -\/ ` identity. (Cont...
falnorfal 1582 A ` -\/ ` identity. (Cont...
falnorfalOLD 1583 Obsolete version of ~ faln...
hadbi123d 1586 Equality theorem for the a...
hadbi123i 1587 Equality theorem for the a...
hadass 1588 Associative law for the ad...
hadbi 1589 The adder sum is the same ...
hadcoma 1590 Commutative law for the ad...
hadcomaOLD 1591 Commutative law for the ad...
hadcomb 1592 Commutative law for the ad...
hadrot 1593 Rotation law for the adder...
hadnot 1594 The adder sum distributes ...
had1 1595 If the first input is true...
had0 1596 If the first input is fals...
hadifp 1597 The value of the adder sum...
cador 1600 The adder carry in disjunc...
cadan 1601 The adder carry in conjunc...
cadbi123d 1602 Equality theorem for the a...
cadbi123i 1603 Equality theorem for the a...
cadcoma 1604 Commutative law for the ad...
cadcomb 1605 Commutative law for the ad...
cadrot 1606 Rotation law for the adder...
cadnot 1607 The adder carry distribute...
cad1 1608 If one input is true, then...
cad0 1609 If one input is false, the...
cadifp 1610 The value of the carry is,...
cad11 1611 If (at least) two inputs a...
cadtru 1612 The adder carry is true as...
minimp 1613 A single axiom for minimal...
minimp-syllsimp 1614 Derivation of Syll-Simp ( ...
minimp-ax1 1615 Derivation of ~ ax-1 from ...
minimp-ax2c 1616 Derivation of a commuted f...
minimp-ax2 1617 Derivation of ~ ax-2 from ...
minimp-pm2.43 1618 Derivation of ~ pm2.43 (al...
impsingle 1619 The shortest single axiom ...
impsingle-step4 1620 Derivation of impsingle-st...
impsingle-step8 1621 Derivation of impsingle-st...
impsingle-ax1 1622 Derivation of impsingle-ax...
impsingle-step15 1623 Derivation of impsingle-st...
impsingle-step18 1624 Derivation of impsingle-st...
impsingle-step19 1625 Derivation of impsingle-st...
impsingle-step20 1626 Derivation of impsingle-st...
impsingle-step21 1627 Derivation of impsingle-st...
impsingle-step22 1628 Derivation of impsingle-st...
impsingle-step25 1629 Derivation of impsingle-st...
impsingle-imim1 1630 Derivation of impsingle-im...
impsingle-peirce 1631 Derivation of impsingle-pe...
tarski-bernays-ax2 1632 Derivation of ~ ax-2 from ...
meredith 1633 Carew Meredith's sole axio...
merlem1 1634 Step 3 of Meredith's proof...
merlem2 1635 Step 4 of Meredith's proof...
merlem3 1636 Step 7 of Meredith's proof...
merlem4 1637 Step 8 of Meredith's proof...
merlem5 1638 Step 11 of Meredith's proo...
merlem6 1639 Step 12 of Meredith's proo...
merlem7 1640 Between steps 14 and 15 of...
merlem8 1641 Step 15 of Meredith's proo...
merlem9 1642 Step 18 of Meredith's proo...
merlem10 1643 Step 19 of Meredith's proo...
merlem11 1644 Step 20 of Meredith's proo...
merlem12 1645 Step 28 of Meredith's proo...
merlem13 1646 Step 35 of Meredith's proo...
luk-1 1647 1 of 3 axioms for proposit...
luk-2 1648 2 of 3 axioms for proposit...
luk-3 1649 3 of 3 axioms for proposit...
luklem1 1650 Used to rederive standard ...
luklem2 1651 Used to rederive standard ...
luklem3 1652 Used to rederive standard ...
luklem4 1653 Used to rederive standard ...
luklem5 1654 Used to rederive standard ...
luklem6 1655 Used to rederive standard ...
luklem7 1656 Used to rederive standard ...
luklem8 1657 Used to rederive standard ...
ax1 1658 Standard propositional axi...
ax2 1659 Standard propositional axi...
ax3 1660 Standard propositional axi...
nic-dfim 1661 This theorem "defines" imp...
nic-dfneg 1662 This theorem "defines" neg...
nic-mp 1663 Derive Nicod's rule of mod...
nic-mpALT 1664 A direct proof of ~ nic-mp...
nic-ax 1665 Nicod's axiom derived from...
nic-axALT 1666 A direct proof of ~ nic-ax...
nic-imp 1667 Inference for ~ nic-mp usi...
nic-idlem1 1668 Lemma for ~ nic-id . (Con...
nic-idlem2 1669 Lemma for ~ nic-id . Infe...
nic-id 1670 Theorem ~ id expressed wit...
nic-swap 1671 The connector ` -/\ ` is s...
nic-isw1 1672 Inference version of ~ nic...
nic-isw2 1673 Inference for swapping nes...
nic-iimp1 1674 Inference version of ~ nic...
nic-iimp2 1675 Inference version of ~ nic...
nic-idel 1676 Inference to remove the tr...
nic-ich 1677 Chained inference. (Contr...
nic-idbl 1678 Double the terms. Since d...
nic-bijust 1679 Biconditional justificatio...
nic-bi1 1680 Inference to extract one s...
nic-bi2 1681 Inference to extract the o...
nic-stdmp 1682 Derive the standard modus ...
nic-luk1 1683 Proof of ~ luk-1 from ~ ni...
nic-luk2 1684 Proof of ~ luk-2 from ~ ni...
nic-luk3 1685 Proof of ~ luk-3 from ~ ni...
lukshef-ax1 1686 This alternative axiom for...
lukshefth1 1687 Lemma for ~ renicax . (Co...
lukshefth2 1688 Lemma for ~ renicax . (Co...
renicax 1689 A rederivation of ~ nic-ax...
tbw-bijust 1690 Justification for ~ tbw-ne...
tbw-negdf 1691 The definition of negation...
tbw-ax1 1692 The first of four axioms i...
tbw-ax2 1693 The second of four axioms ...
tbw-ax3 1694 The third of four axioms i...
tbw-ax4 1695 The fourth of four axioms ...
tbwsyl 1696 Used to rederive the Lukas...
tbwlem1 1697 Used to rederive the Lukas...
tbwlem2 1698 Used to rederive the Lukas...
tbwlem3 1699 Used to rederive the Lukas...
tbwlem4 1700 Used to rederive the Lukas...
tbwlem5 1701 Used to rederive the Lukas...
re1luk1 1702 ~ luk-1 derived from the T...
re1luk2 1703 ~ luk-2 derived from the T...
re1luk3 1704 ~ luk-3 derived from the T...
merco1 1705 A single axiom for proposi...
merco1lem1 1706 Used to rederive the Tarsk...
retbwax4 1707 ~ tbw-ax4 rederived from ~...
retbwax2 1708 ~ tbw-ax2 rederived from ~...
merco1lem2 1709 Used to rederive the Tarsk...
merco1lem3 1710 Used to rederive the Tarsk...
merco1lem4 1711 Used to rederive the Tarsk...
merco1lem5 1712 Used to rederive the Tarsk...
merco1lem6 1713 Used to rederive the Tarsk...
merco1lem7 1714 Used to rederive the Tarsk...
retbwax3 1715 ~ tbw-ax3 rederived from ~...
merco1lem8 1716 Used to rederive the Tarsk...
merco1lem9 1717 Used to rederive the Tarsk...
merco1lem10 1718 Used to rederive the Tarsk...
merco1lem11 1719 Used to rederive the Tarsk...
merco1lem12 1720 Used to rederive the Tarsk...
merco1lem13 1721 Used to rederive the Tarsk...
merco1lem14 1722 Used to rederive the Tarsk...
merco1lem15 1723 Used to rederive the Tarsk...
merco1lem16 1724 Used to rederive the Tarsk...
merco1lem17 1725 Used to rederive the Tarsk...
merco1lem18 1726 Used to rederive the Tarsk...
retbwax1 1727 ~ tbw-ax1 rederived from ~...
merco2 1728 A single axiom for proposi...
mercolem1 1729 Used to rederive the Tarsk...
mercolem2 1730 Used to rederive the Tarsk...
mercolem3 1731 Used to rederive the Tarsk...
mercolem4 1732 Used to rederive the Tarsk...
mercolem5 1733 Used to rederive the Tarsk...
mercolem6 1734 Used to rederive the Tarsk...
mercolem7 1735 Used to rederive the Tarsk...
mercolem8 1736 Used to rederive the Tarsk...
re1tbw1 1737 ~ tbw-ax1 rederived from ~...
re1tbw2 1738 ~ tbw-ax2 rederived from ~...
re1tbw3 1739 ~ tbw-ax3 rederived from ~...
re1tbw4 1740 ~ tbw-ax4 rederived from ~...
rb-bijust 1741 Justification for ~ rb-imd...
rb-imdf 1742 The definition of implicat...
anmp 1743 Modus ponens for ` \/ ` ` ...
rb-ax1 1744 The first of four axioms i...
rb-ax2 1745 The second of four axioms ...
rb-ax3 1746 The third of four axioms i...
rb-ax4 1747 The fourth of four axioms ...
rbsyl 1748 Used to rederive the Lukas...
rblem1 1749 Used to rederive the Lukas...
rblem2 1750 Used to rederive the Lukas...
rblem3 1751 Used to rederive the Lukas...
rblem4 1752 Used to rederive the Lukas...
rblem5 1753 Used to rederive the Lukas...
rblem6 1754 Used to rederive the Lukas...
rblem7 1755 Used to rederive the Lukas...
re1axmp 1756 ~ ax-mp derived from Russe...
re2luk1 1757 ~ luk-1 derived from Russe...
re2luk2 1758 ~ luk-2 derived from Russe...
re2luk3 1759 ~ luk-3 derived from Russe...
mptnan 1760 Modus ponendo tollens 1, o...
mptxor 1761 Modus ponendo tollens 2, o...
mtpor 1762 Modus tollendo ponens (inc...
mtpxor 1763 Modus tollendo ponens (ori...
stoic1a 1764 Stoic logic Thema 1 (part ...
stoic1b 1765 Stoic logic Thema 1 (part ...
stoic2a 1766 Stoic logic Thema 2 versio...
stoic2b 1767 Stoic logic Thema 2 versio...
stoic3 1768 Stoic logic Thema 3. Stat...
stoic4a 1769 Stoic logic Thema 4 versio...
stoic4b 1770 Stoic logic Thema 4 versio...
alnex 1773 Universal quantification o...
eximal 1774 An equivalence between an ...
nf2 1777 Alternate definition of no...
nf3 1778 Alternate definition of no...
nf4 1779 Alternate definition of no...
nfi 1780 Deduce that ` x ` is not f...
nfri 1781 Consequence of the definit...
nfd 1782 Deduce that ` x ` is not f...
nfrd 1783 Consequence of the definit...
nftht 1784 Closed form of ~ nfth . (...
nfntht 1785 Closed form of ~ nfnth . ...
nfntht2 1786 Closed form of ~ nfnth . ...
gen2 1788 Generalization applied twi...
mpg 1789 Modus ponens combined with...
mpgbi 1790 Modus ponens on biconditio...
mpgbir 1791 Modus ponens on biconditio...
nex 1792 Generalization rule for ne...
nfth 1793 No variable is (effectivel...
nfnth 1794 No variable is (effectivel...
hbth 1795 No variable is (effectivel...
nftru 1796 The true constant has no f...
nffal 1797 The false constant has no ...
sptruw 1798 Version of ~ sp when ` ph ...
altru 1799 For all sets, ` T. ` is tr...
alfal 1800 For all sets, ` -. F. ` is...
alim 1802 Restatement of Axiom ~ ax-...
alimi 1803 Inference quantifying both...
2alimi 1804 Inference doubly quantifyi...
ala1 1805 Add an antecedent in a uni...
al2im 1806 Closed form of ~ al2imi . ...
al2imi 1807 Inference quantifying ante...
alanimi 1808 Variant of ~ al2imi with c...
alimdh 1809 Deduction form of Theorem ...
albi 1810 Theorem 19.15 of [Margaris...
albii 1811 Inference adding universal...
2albii 1812 Inference adding two unive...
sylgt 1813 Closed form of ~ sylg . (...
sylg 1814 A syllogism combined with ...
alrimih 1815 Inference form of Theorem ...
hbxfrbi 1816 A utility lemma to transfe...
alex 1817 Universal quantifier in te...
exnal 1818 Existential quantification...
2nalexn 1819 Part of theorem *11.5 in [...
2exnaln 1820 Theorem *11.22 in [Whitehe...
2nexaln 1821 Theorem *11.25 in [Whitehe...
alimex 1822 An equivalence between an ...
aleximi 1823 A variant of ~ al2imi : in...
alexbii 1824 Biconditional form of ~ al...
exim 1825 Theorem 19.22 of [Margaris...
eximi 1826 Inference adding existenti...
2eximi 1827 Inference adding two exist...
eximii 1828 Inference associated with ...
exa1 1829 Add an antecedent in an ex...
19.38 1830 Theorem 19.38 of [Margaris...
19.38a 1831 Under a non-freeness hypot...
19.38b 1832 Under a non-freeness hypot...
imnang 1833 Quantified implication in ...
alinexa 1834 A transformation of quanti...
exnalimn 1835 Existential quantification...
alexn 1836 A relationship between two...
2exnexn 1837 Theorem *11.51 in [Whitehe...
exbi 1838 Theorem 19.18 of [Margaris...
exbii 1839 Inference adding existenti...
2exbii 1840 Inference adding two exist...
3exbii 1841 Inference adding three exi...
nfbiit 1842 Equivalence theorem for th...
nfbii 1843 Equality theorem for the n...
nfxfr 1844 A utility lemma to transfe...
nfxfrd 1845 A utility lemma to transfe...
nfnbi 1846 A variable is non-free in ...
nfnt 1847 If a variable is non-free ...
nfn 1848 Inference associated with ...
nfnd 1849 Deduction associated with ...
exanali 1850 A transformation of quanti...
2exanali 1851 Theorem *11.521 in [Whiteh...
exancom 1852 Commutation of conjunction...
exan 1853 Place a conjunct in the sc...
exanOLD 1854 Obsolete proof of ~ exan a...
alrimdh 1855 Deduction form of Theorem ...
eximdh 1856 Deduction from Theorem 19....
nexdh 1857 Deduction for generalizati...
albidh 1858 Formula-building rule for ...
exbidh 1859 Formula-building rule for ...
exsimpl 1860 Simplification of an exist...
exsimpr 1861 Simplification of an exist...
19.26 1862 Theorem 19.26 of [Margaris...
19.26-2 1863 Theorem ~ 19.26 with two q...
19.26-3an 1864 Theorem ~ 19.26 with tripl...
19.29 1865 Theorem 19.29 of [Margaris...
19.29r 1866 Variation of ~ 19.29 . (C...
19.29r2 1867 Variation of ~ 19.29r with...
19.29x 1868 Variation of ~ 19.29 with ...
19.35 1869 Theorem 19.35 of [Margaris...
19.35i 1870 Inference associated with ...
19.35ri 1871 Inference associated with ...
19.25 1872 Theorem 19.25 of [Margaris...
19.30 1873 Theorem 19.30 of [Margaris...
19.43 1874 Theorem 19.43 of [Margaris...
19.43OLD 1875 Obsolete proof of ~ 19.43 ...
19.33 1876 Theorem 19.33 of [Margaris...
19.33b 1877 The antecedent provides a ...
19.40 1878 Theorem 19.40 of [Margaris...
19.40-2 1879 Theorem *11.42 in [Whitehe...
19.40b 1880 The antecedent provides a ...
albiim 1881 Split a biconditional and ...
2albiim 1882 Split a biconditional and ...
exintrbi 1883 Add/remove a conjunct in t...
exintr 1884 Introduce a conjunct in th...
alsyl 1885 Universally quantified and...
nfimd 1886 If in a context ` x ` is n...
nfimt 1887 Closed form of ~ nfim and ...
nfim 1888 If ` x ` is not free in ` ...
nfand 1889 If in a context ` x ` is n...
nf3and 1890 Deduction form of bound-va...
nfan 1891 If ` x ` is not free in ` ...
nfnan 1892 If ` x ` is not free in ` ...
nf3an 1893 If ` x ` is not free in ` ...
nfbid 1894 If in a context ` x ` is n...
nfbi 1895 If ` x ` is not free in ` ...
nfor 1896 If ` x ` is not free in ` ...
nf3or 1897 If ` x ` is not free in ` ...
empty 1898 Two characterizations of t...
emptyex 1899 On the empty domain, any e...
emptyal 1900 On the empty domain, any u...
emptynf 1901 On the empty domain, any v...
ax5d 1903 Version of ~ ax-5 with ant...
ax5e 1904 A rephrasing of ~ ax-5 usi...
ax5ea 1905 If a formula holds for som...
nfv 1906 If ` x ` is not present in...
nfvd 1907 ~ nfv with antecedent. Us...
alimdv 1908 Deduction form of Theorem ...
eximdv 1909 Deduction form of Theorem ...
2alimdv 1910 Deduction form of Theorem ...
2eximdv 1911 Deduction form of Theorem ...
albidv 1912 Formula-building rule for ...
exbidv 1913 Formula-building rule for ...
nfbidv 1914 An equality theorem for no...
2albidv 1915 Formula-building rule for ...
2exbidv 1916 Formula-building rule for ...
3exbidv 1917 Formula-building rule for ...
4exbidv 1918 Formula-building rule for ...
alrimiv 1919 Inference form of Theorem ...
alrimivv 1920 Inference form of Theorem ...
alrimdv 1921 Deduction form of Theorem ...
exlimiv 1922 Inference form of Theorem ...
exlimiiv 1923 Inference (Rule C) associa...
exlimivv 1924 Inference form of Theorem ...
exlimdv 1925 Deduction form of Theorem ...
exlimdvv 1926 Deduction form of Theorem ...
exlimddv 1927 Existential elimination ru...
nexdv 1928 Deduction for generalizati...
2ax5 1929 Quantification of two vari...
stdpc5v 1930 Version of ~ stdpc5 with a...
19.21v 1931 Version of ~ 19.21 with a ...
19.32v 1932 Version of ~ 19.32 with a ...
19.31v 1933 Version of ~ 19.31 with a ...
19.23v 1934 Version of ~ 19.23 with a ...
19.23vv 1935 Theorem ~ 19.23v extended ...
pm11.53v 1936 Version of ~ pm11.53 with ...
19.36imv 1937 One direction of ~ 19.36v ...
19.36iv 1938 Inference associated with ...
19.37imv 1939 One direction of ~ 19.37v ...
19.37iv 1940 Inference associated with ...
19.41v 1941 Version of ~ 19.41 with a ...
19.41vv 1942 Version of ~ 19.41 with tw...
19.41vvv 1943 Version of ~ 19.41 with th...
19.41vvvv 1944 Version of ~ 19.41 with fo...
19.42v 1945 Version of ~ 19.42 with a ...
exdistr 1946 Distribution of existentia...
exdistrv 1947 Distribute a pair of exist...
4exdistrv 1948 Distribute two pairs of ex...
19.42vv 1949 Version of ~ 19.42 with tw...
exdistr2 1950 Distribution of existentia...
19.42vvv 1951 Version of ~ 19.42 with th...
19.42vvvOLD 1952 Obsolete version of ~ 19.4...
3exdistr 1953 Distribution of existentia...
4exdistr 1954 Distribution of existentia...
weq 1955 Extend wff definition to i...
equs3OLD 1956 Obsolete as of 12-Aug-2023...
speimfw 1957 Specialization, with addit...
speimfwALT 1958 Alternate proof of ~ speim...
spimfw 1959 Specialization, with addit...
ax12i 1960 Inference that has ~ ax-12...
ax6v 1962 Axiom B7 of [Tarski] p. 75...
ax6ev 1963 At least one individual ex...
spimw 1964 Specialization. Lemma 8 o...
spimew 1965 Existential introduction, ...
spimehOLD 1966 Obsolete version of ~ spim...
speiv 1967 Inference from existential...
speivw 1968 Version of ~ spei with a d...
exgen 1969 Rule of existential genera...
exgenOLD 1970 Obsolete version of ~ exge...
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...
19.3vOLD 1980 Obsolete version of ~ 19.3...
spvwOLD 1981 Obsolete version of ~ spvw...
19.39 1982 Theorem 19.39 of [Margaris...
19.24 1983 Theorem 19.24 of [Margaris...
19.34 1984 Theorem 19.34 of [Margaris...
19.36v 1985 Version of ~ 19.36 with a ...
19.12vvv 1986 Version of ~ 19.12vv with ...
19.27v 1987 Version of ~ 19.27 with a ...
19.28v 1988 Version of ~ 19.28 with a ...
19.37v 1989 Version of ~ 19.37 with a ...
19.44v 1990 Version of ~ 19.44 with a ...
19.45v 1991 Version of ~ 19.45 with a ...
spimevw 1992 Existential introduction, ...
spimvw 1993 A weak form of specializat...
spvv 1994 Version of ~ spv with a di...
spfalw 1995 Version of ~ sp when ` ph ...
chvarvv 1996 Version of ~ chvarv with a...
equs4v 1997 Version of ~ equs4 with a ...
alequexv 1998 Version of ~ equs4v with i...
exsbim 1999 One direction of the equiv...
equsv 2000 If a formula does not cont...
equsalvw 2001 Version of ~ equsalv with ...
equsexvw 2002 Version of ~ equsexv with ...
equsexvwOLD 2003 Obsolete version of ~ equs...
cbvaliw 2004 Change bound variable. Us...
cbvalivw 2005 Change bound variable. Us...
ax7v 2007 Weakened version of ~ ax-7...
ax7v1 2008 First of two weakened vers...
ax7v2 2009 Second of two weakened ver...
equid 2010 Identity law for equality....
nfequid 2011 Bound-variable hypothesis ...
equcomiv 2012 Weaker form of ~ equcomi w...
ax6evr 2013 A commuted form of ~ ax6ev...
ax7 2014 Proof of ~ ax-7 from ~ ax7...
equcomi 2015 Commutative law for equali...
equcom 2016 Commutative law for equali...
equcomd 2017 Deduction form of ~ equcom...
equcoms 2018 An inference commuting equ...
equtr 2019 A transitive law for equal...
equtrr 2020 A transitive law for equal...
equeuclr 2021 Commuted version of ~ eque...
equeucl 2022 Equality is a left-Euclide...
equequ1 2023 An equivalence law for equ...
equequ2 2024 An equivalence law for equ...
equtr2 2025 Equality is a left-Euclide...
stdpc6 2026 One of the two equality ax...
equvinv 2027 A variable introduction la...
equvinva 2028 A modified version of the ...
equvelv 2029 A biconditional form of ~ ...
ax13b 2030 An equivalence between two...
spfw 2031 Weak version of ~ sp . Us...
spw 2032 Weak version of the specia...
cbvalw 2033 Change bound variable. Us...
cbvalvw 2034 Change bound variable. Us...
cbvexvw 2035 Change bound variable. Us...
cbvaldvaw 2036 Version of ~ cbvaldva with...
cbvexdvaw 2037 Version of ~ cbvexdva with...
cbval2vw 2038 Version of ~ cbval2vv with...
cbvex2vw 2039 Version of ~ cbvex2vv with...
cbvex4vw 2040 Version of ~ cbvex4v with ...
alcomiw 2041 Weak version of ~ alcom . ...
alcomiwOLD 2042 Obsolete version of ~ alco...
hbn1fw 2043 Weak version of ~ ax-10 fr...
hbn1w 2044 Weak version of ~ hbn1 . ...
hba1w 2045 Weak version of ~ hba1 . ...
hbe1w 2046 Weak version of ~ hbe1 . ...
hbalw 2047 Weak version of ~ hbal . ...
spaev 2048 A special instance of ~ sp...
cbvaev 2049 Change bound variable in a...
aevlem0 2050 Lemma for ~ aevlem . Inst...
aevlem 2051 Lemma for ~ aev and ~ axc1...
aeveq 2052 The antecedent ` A. x x = ...
aev 2053 A "distinctor elimination"...
aev2 2054 A version of ~ aev with tw...
hbaev 2055 Version of ~ hbae with a d...
naev 2056 If some set variables can ...
naev2 2057 Generalization of ~ hbnaev...
hbnaev 2058 Any variable is free in ` ...
sbjust 2059 Justification theorem for ...
sbt 2062 A substitution into a theo...
sbtru 2063 The result of substituting...
stdpc4 2064 The specialization axiom o...
sbtALT 2065 Alternate proof of ~ sbt ,...
2stdpc4 2066 A double specialization us...
sbi1 2067 Distribute substitution ov...
spsbim 2068 Distribute substitution ov...
spsbbi 2069 Biconditional property for...
sbimi 2070 Distribute substitution ov...
sb2imi 2071 Distribute substitution ov...
sbbii 2072 Infer substitution into bo...
2sbbii 2073 Infer double substitution ...
sbimdv 2074 Deduction substituting bot...
sbbidv 2075 Deduction substituting bot...
sbbidvOLD 2076 Obsolete version of ~ sbbi...
sban 2077 Conjunction inside and out...
sb3an 2078 Threefold conjunction insi...
spsbe 2079 Existential generalization...
spsbeOLD 2080 Obsolete version of ~ spsb...
sbequ 2081 Equality property for subs...
sbequi 2082 An equality theorem for su...
sbequOLD 2083 Obsolete proof of ~ sbequ ...
sb6 2084 Alternate definition of su...
2sb6 2085 Equivalence for double sub...
sb1v 2086 One direction of ~ sb5 , p...
sb4vOLD 2087 Obsolete as of 30-Jul-2023...
sb2vOLD 2088 Obsolete as of 30-Jul-2023...
sbv 2089 Substitution for a variabl...
sbcom4 2090 Commutativity law for subs...
pm11.07 2091 Axiom *11.07 in [Whitehead...
sbrimvlem 2092 Common proof template for ...
sbrimvw 2093 Substitution in an implica...
sbievw 2094 Version of ~ sbie and ~ sb...
sbiedvw 2095 Version of ~ sbied and ~ s...
2sbievw 2096 Version of ~ 2sbiev with m...
sbcom3vv 2097 Version of ~ sbcom3 with a...
sbievw2 2098 ~ sbievw applied twice, av...
sbco2vv 2099 Version of ~ sbco2 with di...
equsb3 2100 Substitution in an equalit...
equsb3r 2101 Substitution applied to th...
equsb3rOLD 2102 Obsolete version of ~ equs...
equsb1v 2103 Version of ~ equsb1 with a...
equsb1vOLD 2104 Obsolete version of ~ equs...
wel 2106 Extend wff definition to i...
ax8v 2108 Weakened version of ~ ax-8...
ax8v1 2109 First of two weakened vers...
ax8v2 2110 Second of two weakened ver...
ax8 2111 Proof of ~ ax-8 from ~ ax8...
elequ1 2112 An identity law for the no...
elsb3 2113 Substitution applied to an...
cleljust 2114 When the class variables i...
ax9v 2116 Weakened version of ~ ax-9...
ax9v1 2117 First of two weakened vers...
ax9v2 2118 Second of two weakened ver...
ax9 2119 Proof of ~ ax-9 from ~ ax9...
elequ2 2120 An identity law for the no...
elsb4 2121 Substitution applied to an...
elequ2g 2122 A form of ~ elequ2 with a ...
ax6dgen 2123 Tarski's system uses the w...
ax10w 2124 Weak version of ~ ax-10 fr...
ax11w 2125 Weak version of ~ ax-11 fr...
ax11dgen 2126 Degenerate instance of ~ a...
ax12wlem 2127 Lemma for weak version of ...
ax12w 2128 Weak version of ~ ax-12 fr...
ax12dgen 2129 Degenerate instance of ~ a...
ax12wdemo 2130 Example of an application ...
ax13w 2131 Weak version (principal in...
ax13dgen1 2132 Degenerate instance of ~ a...
ax13dgen2 2133 Degenerate instance of ~ a...
ax13dgen3 2134 Degenerate instance of ~ a...
ax13dgen4 2135 Degenerate instance of ~ a...
hbn1 2137 Alias for ~ ax-10 to be us...
hbe1 2138 The setvar ` x ` is not fr...
hbe1a 2139 Dual statement of ~ hbe1 ....
nf5-1 2140 One direction of ~ nf5 can...
nf5i 2141 Deduce that ` x ` is not f...
nf5dh 2142 Deduce that ` x ` is not f...
nf5dv 2143 Apply the definition of no...
nfnaew 2144 Version of ~ nfnae with a ...
nfe1 2145 The setvar ` x ` is not fr...
nfa1 2146 The setvar ` x ` is not fr...
nfna1 2147 A convenience theorem part...
nfia1 2148 Lemma 23 of [Monk2] p. 114...
nfnf1 2149 The setvar ` x ` is not fr...
modal5 2150 The analogue in our predic...
alcoms 2152 Swap quantifiers in an ant...
alcom 2153 Theorem 19.5 of [Margaris]...
alrot3 2154 Theorem *11.21 in [Whitehe...
alrot4 2155 Rotate four universal quan...
sbal 2156 Move universal quantifier ...
sbalv 2157 Quantify with new variable...
sbcom2 2158 Commutativity law for subs...
excom 2159 Theorem 19.11 of [Margaris...
excomim 2160 One direction of Theorem 1...
excom13 2161 Swap 1st and 3rd existenti...
exrot3 2162 Rotate existential quantif...
exrot4 2163 Rotate existential quantif...
hbal 2164 If ` x ` is not free in ` ...
hbald 2165 Deduction form of bound-va...
nfa2 2166 Lemma 24 of [Monk2] p. 114...
ax12v 2168 This is essentially axiom ...
ax12v2 2169 It is possible to remove a...
19.8a 2170 If a wff is true, it is tr...
19.8ad 2171 If a wff is true, it is tr...
sp 2172 Specialization. A univers...
spi 2173 Inference rule of universa...
sps 2174 Generalization of antecede...
2sp 2175 A double specialization (s...
spsd 2176 Deduction generalizing ant...
19.2g 2177 Theorem 19.2 of [Margaris]...
19.21bi 2178 Inference form of ~ 19.21 ...
19.21bbi 2179 Inference removing two uni...
19.23bi 2180 Inference form of Theorem ...
nexr 2181 Inference associated with ...
qexmid 2182 Quantified excluded middle...
nf5r 2183 Consequence of the definit...
nf5rOLD 2184 Obsolete version of ~ nfrd...
nf5ri 2185 Consequence of the definit...
nf5rd 2186 Consequence of the definit...
spimedv 2187 Version of ~ spimed with a...
spimefv 2188 Version of ~ spime with a ...
nfim1 2189 A closed form of ~ nfim . ...
nfan1 2190 A closed form of ~ nfan . ...
19.3t 2191 Closed form of ~ 19.3 and ...
19.3 2192 A wff may be quantified wi...
19.9d 2193 A deduction version of one...
19.9t 2194 Closed form of ~ 19.9 and ...
19.9 2195 A wff may be existentially...
19.21t 2196 Closed form of Theorem 19....
19.21 2197 Theorem 19.21 of [Margaris...
stdpc5 2198 An axiom scheme of standar...
19.21-2 2199 Version of ~ 19.21 with tw...
19.23t 2200 Closed form of Theorem 19....
19.23 2201 Theorem 19.23 of [Margaris...
alimd 2202 Deduction form of Theorem ...
alrimi 2203 Inference form of Theorem ...
alrimdd 2204 Deduction form of Theorem ...
alrimd 2205 Deduction form of Theorem ...
eximd 2206 Deduction form of Theorem ...
exlimi 2207 Inference associated with ...
exlimd 2208 Deduction form of Theorem ...
exlimimdd 2209 Existential elimination ru...
exlimdd 2210 Existential elimination ru...
exlimddOLD 2211 Obsolete version of ~ exli...
exlimimddOLD 2212 Obsolete version of ~ exli...
nexd 2213 Deduction for generalizati...
albid 2214 Formula-building rule for ...
exbid 2215 Formula-building rule for ...
nfbidf 2216 An equality theorem for ef...
19.16 2217 Theorem 19.16 of [Margaris...
19.17 2218 Theorem 19.17 of [Margaris...
19.27 2219 Theorem 19.27 of [Margaris...
19.28 2220 Theorem 19.28 of [Margaris...
19.19 2221 Theorem 19.19 of [Margaris...
19.36 2222 Theorem 19.36 of [Margaris...
19.36i 2223 Inference associated with ...
19.37 2224 Theorem 19.37 of [Margaris...
19.32 2225 Theorem 19.32 of [Margaris...
19.31 2226 Theorem 19.31 of [Margaris...
19.41 2227 Theorem 19.41 of [Margaris...
19.42 2228 Theorem 19.42 of [Margaris...
19.44 2229 Theorem 19.44 of [Margaris...
19.45 2230 Theorem 19.45 of [Margaris...
spimfv 2231 Version of ~ spim with a d...
chvarfv 2232 Version of ~ chvar with a ...
cbv3v2 2233 Version of ~ cbv3 with two...
sb4av 2234 Version of ~ sb4a with a d...
sbimd 2235 Deduction substituting bot...
sbbid 2236 Deduction substituting bot...
2sbbid 2237 Deduction doubly substitut...
sbbidOLD 2238 Obsolete version of ~ sbbi...
sbequ1 2239 An equality theorem for su...
sbequ2 2240 An equality theorem for su...
sbequ2OLD 2241 Obsolete version of ~ sbeq...
stdpc7 2242 One of the two equality ax...
sbequ12 2243 An equality theorem for su...
sbequ12r 2244 An equality theorem for su...
sbelx 2245 Elimination of substitutio...
sbequ12a 2246 An equality theorem for su...
sbid 2247 An identity theorem for su...
sbcov 2248 Version of ~ sbco with a d...
sb6a 2249 Equivalence for substituti...
sbid2vw 2250 Reverting substitution yie...
axc16g 2251 Generalization of ~ axc16 ...
axc16 2252 Proof of older axiom ~ ax-...
axc16gb 2253 Biconditional strengthenin...
axc16nf 2254 If ~ dtru is false, then t...
axc11v 2255 Version of ~ axc11 with a ...
axc11rv 2256 Version of ~ axc11r with a...
drsb2 2257 Formula-building lemma for...
equsalv 2258 Version of ~ equsal with a...
equsexv 2259 Version of ~ equsex with a...
sbft 2260 Substitution has no effect...
sbf 2261 Substitution for a variabl...
sbf2 2262 Substitution has no effect...
sbh 2263 Substitution for a variabl...
nfs1v 2264 The setvar ` x ` is not fr...
hbs1 2265 The setvar ` x ` is not fr...
nfs1f 2266 If ` x ` is not free in ` ...
sb5 2267 Alternate definition of su...
sb56 2268 Two equivalent ways of exp...
sb56OLD 2269 Obsolete version of ~ sb56...
equs5av 2270 Version of ~ equs5a with a...
sb6OLD 2271 Obsolete version of ~ sb6 ...
sb5OLD 2272 Obsolete version of ~ sb5 ...
2sb5 2273 Equivalence for double sub...
sbco4lem 2274 Lemma for ~ sbco4 . It re...
sbco4 2275 Two ways of exchanging two...
dfsb7 2276 An alternate definition of...
dfsb7OLD 2277 Obsolete version of ~ dfsb...
sbn 2278 Negation inside and outsid...
sbex 2279 Move existential quantifie...
sbbibOLD 2280 Obsolete version of ~ sbbi...
nf5 2281 Alternate definition of ~ ...
nf6 2282 An alternate definition of...
nf5d 2283 Deduce that ` x ` is not f...
nf5di 2284 Since the converse holds b...
19.9h 2285 A wff may be existentially...
19.21h 2286 Theorem 19.21 of [Margaris...
19.23h 2287 Theorem 19.23 of [Margaris...
exlimih 2288 Inference associated with ...
exlimdh 2289 Deduction form of Theorem ...
equsalhw 2290 Version of ~ equsalh with ...
equsexhv 2291 Version of ~ equsexh with ...
hba1 2292 The setvar ` x ` is not fr...
hbnt 2293 Closed theorem version of ...
hbn 2294 If ` x ` is not free in ` ...
hbnd 2295 Deduction form of bound-va...
hbim1 2296 A closed form of ~ hbim . ...
hbimd 2297 Deduction form of bound-va...
hbim 2298 If ` x ` is not free in ` ...
hban 2299 If ` x ` is not free in ` ...
hb3an 2300 If ` x ` is not free in ` ...
sbi2 2301 Introduction of implicatio...
sbim 2302 Implication inside and out...
sbanOLD 2303 Obsolete version of ~ sban...
sbrim 2304 Substitution in an implica...
sbrimv 2305 Substitution in an implica...
sblim 2306 Substitution in an implica...
sbor 2307 Disjunction inside and out...
sbbi 2308 Equivalence inside and out...
spsbbiOLD 2309 Obsolete version of ~ spsb...
sblbis 2310 Introduce left bicondition...
sbrbis 2311 Introduce right biconditio...
sbrbif 2312 Introduce right biconditio...
sbnvOLD 2313 Obsolete version of ~ sbn ...
sbi1vOLD 2314 Obsolete version of ~ sbi1...
sbi2vOLD 2315 Obsolete version of ~ sbi2...
sbimvOLD 2316 Obsolete version of ~ sbim...
sbanvOLD 2317 Obsolete version of ~ sban...
sbbivOLD 2318 Obsolete version of ~ sbbi...
spsbimvOLD 2319 Obsolete version of ~ spsb...
sblbisvOLD 2320 Obsolete version of ~ sblb...
sbiev 2321 Conversion of implicit sub...
sbievOLD 2322 Obsolete proof of ~ sbiev ...
sbiedw 2323 Version of ~ sbied with a ...
sbiedwOLD 2324 Obsolete version of ~ sbie...
sbequivvOLD 2325 Obsolete version of ~ sbeq...
sbequvvOLD 2326 Obsolete version of ~ sbeq...
axc7 2327 Show that the original axi...
axc7e 2328 Abbreviated version of ~ a...
modal-b 2329 The analogue in our predic...
19.9ht 2330 A closed version of ~ 19.9...
axc4 2331 Show that the original axi...
axc4i 2332 Inference version of ~ axc...
nfal 2333 If ` x ` is not free in ` ...
nfex 2334 If ` x ` is not free in ` ...
hbex 2335 If ` x ` is not free in ` ...
nfnf 2336 If ` x ` is not free in ` ...
19.12 2337 Theorem 19.12 of [Margaris...
nfald 2338 Deduction form of ~ nfal ....
nfexd 2339 If ` x ` is not free in ` ...
nfsbv 2340 If ` z ` is not free in ` ...
nfsbvOLD 2341 Obsolete version of ~ nfsb...
hbsbw 2342 Version of ~ hbsb with a d...
sbco2v 2343 Version of ~ sbco2 with di...
aaan 2344 Rearrange universal quanti...
eeor 2345 Rearrange existential quan...
cbv3v 2346 Version of ~ cbv3 with a d...
cbv1v 2347 Version of ~ cbv1 with a d...
cbv2w 2348 Version of ~ cbv2 with a d...
cbvaldw 2349 Version of ~ cbvald with a...
cbvexdw 2350 Version of ~ cbvexd with a...
cbv3hv 2351 Version of ~ cbv3h with a ...
cbvalv1 2352 Version of ~ cbval with a ...
cbvexv1 2353 Version of ~ cbvex with a ...
cbval2v 2354 Version of ~ cbval2 with a...
cbval2vOLD 2355 Obsolete version of ~ cbva...
cbvex2v 2356 Version of ~ cbvex2 with a...
dvelimhw 2357 Proof of ~ dvelimh without...
pm11.53 2358 Theorem *11.53 in [Whitehe...
19.12vv 2359 Special case of ~ 19.12 wh...
eean 2360 Rearrange existential quan...
eeanv 2361 Distribute a pair of exist...
eeeanv 2362 Distribute three existenti...
ee4anv 2363 Distribute two pairs of ex...
sb8v 2364 Substitution of variable i...
sb8ev 2365 Substitution of variable i...
2sb8ev 2366 Version of ~ 2sb8e with mo...
sb6rfv 2367 Reversed substitution. Ve...
sbnf2 2368 Two ways of expressing " `...
exsb 2369 An equivalent expression f...
2exsb 2370 An equivalent expression f...
sbbib 2371 Reversal of substitution. ...
sbbibvv 2372 Reversal of substitution. ...
cleljustALT 2373 Alternate proof of ~ clelj...
cleljustALT2 2374 Alternate proof of ~ clelj...
equs5aALT 2375 Alternate proof of ~ equs5...
equs5eALT 2376 Alternate proof of ~ equs5...
axc11r 2377 Same as ~ axc11 but with r...
dral1v 2378 Version of ~ dral1 with a ...
drex1v 2379 Version of ~ drex1 with a ...
drnf1v 2380 Version of ~ drnf1 with a ...
ax13v 2382 A weaker version of ~ ax-1...
ax13lem1 2383 A version of ~ ax13v with ...
ax13 2384 Derive ~ ax-13 from ~ ax13...
ax13lem2 2385 Lemma for ~ nfeqf2 . This...
nfeqf2 2386 An equation between setvar...
dveeq2 2387 Quantifier introduction wh...
nfeqf1 2388 An equation between setvar...
dveeq1 2389 Quantifier introduction wh...
nfeqf 2390 A variable is effectively ...
axc9 2391 Derive set.mm's original ~...
ax6e 2392 At least one individual ex...
ax6 2393 Theorem showing that ~ ax-...
axc10 2394 Show that the original axi...
spimt 2395 Closed theorem form of ~ s...
spim 2396 Specialization, using impl...
spimed 2397 Deduction version of ~ spi...
spime 2398 Existential introduction, ...
spimv 2399 A version of ~ spim with a...
spimvALT 2400 Alternate proof of ~ spimv...
spimev 2401 Distinct-variable version ...
spv 2402 Specialization, using impl...
spei 2403 Inference from existential...
chvar 2404 Implicit substitution of `...
chvarv 2405 Implicit substitution of `...
cbv3 2406 Rule used to change bound ...
cbval 2407 Rule used to change bound ...
cbvex 2408 Rule used to change bound ...
cbvalv 2409 Rule used to change bound ...
cbvexv 2410 Rule used to change bound ...
cbvalvOLD 2411 Obsolete version of ~ cbva...
cbvexvOLD 2412 Obsolete version of ~ cbve...
cbv1 2413 Rule used to change bound ...
cbv2 2414 Rule used to change bound ...
cbv3h 2415 Rule used to change bound ...
cbv1h 2416 Rule used to change bound ...
cbv2h 2417 Rule used to change bound ...
cbv2OLD 2418 Obsolete version of ~ cbv2...
cbvald 2419 Deduction used to change b...
cbvexd 2420 Deduction used to change b...
cbvaldva 2421 Rule used to change the bo...
cbvexdva 2422 Rule used to change the bo...
cbval2 2423 Rule used to change bound ...
cbval2OLD 2424 Obsolete version of ~ cbva...
cbvex2 2425 Rule used to change bound ...
cbval2vv 2426 Rule used to change bound ...
cbvex2vv 2427 Rule used to change bound ...
cbvex4v 2428 Rule used to change bound ...
equs4 2429 Lemma used in proofs of im...
equsal 2430 An equivalence related to ...
equsex 2431 An equivalence related to ...
equsexALT 2432 Alternate proof of ~ equse...
equsalh 2433 An equivalence related to ...
equsexh 2434 An equivalence related to ...
axc15 2435 Derivation of set.mm's ori...
axc15OLD 2436 Obsolete proof of ~ axc15 ...
ax12 2437 Rederivation of axiom ~ ax...
ax12b 2438 A bidirectional version of...
ax13ALT 2439 Alternate proof of ~ ax13 ...
axc11n 2440 Derive set.mm's original ~...
aecom 2441 Commutation law for identi...
aecoms 2442 A commutation rule for ide...
naecoms 2443 A commutation rule for dis...
axc11 2444 Show that ~ ax-c11 can be ...
hbae 2445 All variables are effectiv...
hbnae 2446 All variables are effectiv...
nfae 2447 All variables are effectiv...
nfnae 2448 All variables are effectiv...
hbnaes 2449 Rule that applies ~ hbnae ...
axc16i 2450 Inference with ~ axc16 as ...
axc16nfALT 2451 Alternate proof of ~ axc16...
dral2 2452 Formula-building lemma for...
dral1 2453 Formula-building lemma for...
dral1ALT 2454 Alternate proof of ~ dral1...
drex1 2455 Formula-building lemma for...
drex2 2456 Formula-building lemma for...
drnf1 2457 Formula-building lemma for...
drnf2 2458 Formula-building lemma for...
nfald2 2459 Variation on ~ nfald which...
nfexd2 2460 Variation on ~ nfexd which...
exdistrf 2461 Distribution of existentia...
dvelimf 2462 Version of ~ dvelimv witho...
dvelimdf 2463 Deduction form of ~ dvelim...
dvelimh 2464 Version of ~ dvelim withou...
dvelim 2465 This theorem can be used t...
dvelimv 2466 Similar to ~ dvelim with f...
dvelimnf 2467 Version of ~ dvelim using ...
dveeq2ALT 2468 Alternate proof of ~ dveeq...
equvini 2469 A variable introduction la...
equviniOLD 2470 Obsolete version of ~ equv...
equvel 2471 A variable elimination law...
equs5a 2472 A property related to subs...
equs5e 2473 A property related to subs...
equs45f 2474 Two ways of expressing sub...
equs5 2475 Lemma used in proofs of su...
dveel1 2476 Quantifier introduction wh...
dveel2 2477 Quantifier introduction wh...
axc14 2478 Axiom ~ ax-c14 is redundan...
sb6x 2479 Equivalence involving subs...
sbequ5 2480 Substitution does not chan...
sbequ6 2481 Substitution does not chan...
sb5rf 2482 Reversed substitution. (C...
sb6rf 2483 Reversed substitution. Fo...
ax12vALT 2484 Alternate proof of ~ ax12v...
2ax6elem 2485 We can always find values ...
2ax6e 2486 We can always find values ...
2ax6eOLD 2487 Obsolete version of ~ 2ax6...
2sb5rf 2488 Reversed double substituti...
2sb6rf 2489 Reversed double substituti...
2sb6rfOLD 2490 Obsolete version of ~ 2sb6...
sbel2x 2491 Elimination of double subs...
sb4b 2492 Simplified definition of s...
sb4bOLD 2493 Obsolete version of ~ sb4b...
sb3b 2494 Simplified definition of s...
sb3 2495 One direction of a simplif...
sb1 2496 One direction of a simplif...
sb2 2497 One direction of a simplif...
sb3OLD 2498 Obsolete version of ~ sb3 ...
sb4OLD 2499 Obsolete as of 30-Jul-2023...
sb1OLD 2500 Obsolete version of ~ sb1 ...
sb3bOLD 2501 Obsolete version of ~ sb3b...
sb4a 2502 A version of one implicati...
dfsb1 2503 Alternate definition of su...
spsbeOLDOLD 2504 Obsolete version of ~ spsb...
sb2vOLDOLD 2505 Obsolete version of ~ sb2 ...
sb4vOLDOLD 2506 Obsolete version of ~ sb4v...
sbequ2OLDOLD 2507 Obsolete version of ~ sbeq...
sbimiOLD 2508 Obsolete version of ~ sbim...
sbimdvOLD 2509 Obsolete version of ~ sbim...
equsb1vOLDOLD 2510 Obsolete version of ~ equs...
sbimdOLD 2511 Obsolete version of sbimd ...
sbtvOLD 2512 Obsolete version of ~ sbt ...
sbequ1OLD 2513 Obsolete version of ~ sbeq...
hbsb2 2514 Bound-variable hypothesis ...
nfsb2 2515 Bound-variable hypothesis ...
hbsb2a 2516 Special case of a bound-va...
sb4e 2517 One direction of a simplif...
hbsb2e 2518 Special case of a bound-va...
hbsb3 2519 If ` y ` is not free in ` ...
nfs1 2520 If ` y ` is not free in ` ...
axc16ALT 2521 Alternate proof of ~ axc16...
axc16gALT 2522 Alternate proof of ~ axc16...
equsb1 2523 Substitution applied to an...
equsb2 2524 Substitution applied to an...
dfsb2 2525 An alternate definition of...
dfsb3 2526 An alternate definition of...
sbequiOLD 2527 Obsolete proof of ~ sbequi...
drsb1 2528 Formula-building lemma for...
sb2ae 2529 In the case of two success...
sb6f 2530 Equivalence for substituti...
sb5f 2531 Equivalence for substituti...
nfsb4t 2532 A variable not free in a p...
nfsb4 2533 A variable not free in a p...
sbnOLD 2534 Obsolete version of ~ sbn ...
sbi1OLD 2535 Obsolete version of ~ sbi1...
sbequ8 2536 Elimination of equality fr...
sbie 2537 Conversion of implicit sub...
sbied 2538 Conversion of implicit sub...
sbiedv 2539 Conversion of implicit sub...
2sbiev 2540 Conversion of double impli...
sbcom3 2541 Substituting ` y ` for ` x...
sbco 2542 A composition law for subs...
sbid2 2543 An identity law for substi...
sbid2v 2544 An identity law for substi...
sbidm 2545 An idempotent law for subs...
sbco2 2546 A composition law for subs...
sbco2d 2547 A composition law for subs...
sbco3 2548 A composition law for subs...
sbcom 2549 A commutativity law for su...
sbtrt 2550 Partially closed form of ~...
sbtr 2551 A partial converse to ~ sb...
sb8 2552 Substitution of variable i...
sb8e 2553 Substitution of variable i...
sb9 2554 Commutation of quantificat...
sb9i 2555 Commutation of quantificat...
sbhb 2556 Two ways of expressing " `...
nfsbd 2557 Deduction version of ~ nfs...
nfsb 2558 If ` z ` is not free in ` ...
nfsbOLD 2559 Obsolete version of ~ nfsb...
hbsb 2560 If ` z ` is not free in ` ...
sb7f 2561 This version of ~ dfsb7 do...
sb7h 2562 This version of ~ dfsb7 do...
dfsb7OLDOLD 2563 Obsolete version of ~ dfsb...
sb10f 2564 Hao Wang's identity axiom ...
sbal1 2565 Obsolete version of ~ sbal...
sbal2 2566 Move quantifier in and out...
sbal2OLD 2567 Obsolete version of ~ sbal...
sbalOLD 2568 Obsolete version of ~ sbal...
2sb8e 2569 An equivalent expression f...
sbimiALT 2570 Alternate version of ~ sbi...
sbbiiALT 2571 Alternate version of ~ sbb...
sbequ1ALT 2572 Alternate version of ~ sbe...
sbequ2ALT 2573 Alternate version of ~ sbe...
sbequ12ALT 2574 Alternate version of ~ sbe...
sb1ALT 2575 Alternate version of ~ sb1...
sb2vOLDALT 2576 Alternate version of ~ sb2...
sb4vOLDALT 2577 Alternate version of ~ sb4...
sb6ALT 2578 Alternate version of ~ sb6...
sb5ALT2 2579 Alternate version of ~ sb5...
sb2ALT 2580 Alternate version of ~ sb2...
sb4ALT 2581 Alternate version of one i...
spsbeALT 2582 Alternate version of ~ sps...
stdpc4ALT 2583 Alternate version of ~ std...
dfsb2ALT 2584 Alternate version of ~ dfs...
dfsb3ALT 2585 Alternate version of ~ dfs...
sbftALT 2586 Alternate version of ~ sbf...
sbfALT 2587 Alternate version of ~ sbf...
sbnALT 2588 Alternate version of ~ sbn...
sbequiALT 2589 Alternate version of ~ sbe...
sbequALT 2590 Alternate version of ~ sbe...
sb4aALT 2591 Alternate version of ~ sb4...
hbsb2ALT 2592 Alternate version of ~ hbs...
nfsb2ALT 2593 Alternate version of ~ nfs...
equsb1ALT 2594 Alternate version of ~ equ...
sb6fALT 2595 Alternate version of ~ sb6...
sb5fALT 2596 Alternate version of ~ sb5...
nfsb4tALT 2597 Alternate version of ~ nfs...
nfsb4ALT 2598 Alternate version of ~ nfs...
sbi1ALT 2599 Alternate version of ~ sbi...
sbi2ALT 2600 Alternate version of ~ sbi...
sbimALT 2601 Alternate version of ~ sbi...
sbrimALT 2602 Alternate version of ~ sbr...
sbanALT 2603 Alternate version of ~ sba...
sbbiALT 2604 Alternate version of ~ sbb...
sblbisALT 2605 Alternate version of ~ sbl...
sbieALT 2606 Alternate version of ~ sbi...
sbiedALT 2607 Alternate version of ~ sbi...
sbco2ALT 2608 Alternate version of ~ sbc...
sb7fALT 2609 Alternate version of ~ sb7...
dfsb7ALT 2610 Alternate version of ~ dfs...
dfmoeu 2611 An elementary proof of ~ m...
dfeumo 2612 An elementary proof showin...
mojust 2614 Soundness justification th...
nexmo 2616 Nonexistence implies uniqu...
exmo 2617 Any proposition holds for ...
moabs 2618 Absorption of existence co...
moim 2619 The at-most-one quantifier...
moimi 2620 The at-most-one quantifier...
moimiOLD 2621 Obsolete version of ~ moim...
moimdv 2622 The at-most-one quantifier...
mobi 2623 Equivalence theorem for th...
mobii 2624 Formula-building rule for ...
mobiiOLD 2625 Obsolete version of ~ mobi...
mobidv 2626 Formula-building rule for ...
mobid 2627 Formula-building rule for ...
moa1 2628 If an implication holds fo...
moan 2629 "At most one" is still the...
moani 2630 "At most one" is still tru...
moor 2631 "At most one" is still the...
mooran1 2632 "At most one" imports disj...
mooran2 2633 "At most one" exports disj...
nfmo1 2634 Bound-variable hypothesis ...
nfmod2 2635 Bound-variable hypothesis ...
nfmodv 2636 Bound-variable hypothesis ...
nfmov 2637 Bound-variable hypothesis ...
nfmod 2638 Bound-variable hypothesis ...
nfmo 2639 Bound-variable hypothesis ...
mof 2640 Version of ~ df-mo with di...
mo3 2641 Alternate definition of th...
mo 2642 Equivalent definitions of ...
mo4 2643 At-most-one quantifier exp...
mo4f 2644 At-most-one quantifier exp...
mo4OLD 2645 Obsolete version of ~ mo4 ...
eu3v 2648 An alternate way to expres...
eujust 2649 Soundness justification th...
eujustALT 2650 Alternate proof of ~ eujus...
eu6lem 2651 Lemma of ~ eu6im . A diss...
eu6 2652 Alternate definition of th...
eu6im 2653 One direction of ~ eu6 nee...
euf 2654 Version of ~ eu6 with disj...
euex 2655 Existential uniqueness imp...
eumo 2656 Existential uniqueness imp...
eumoi 2657 Uniqueness inferred from e...
exmoeub 2658 Existence implies that uni...
exmoeu 2659 Existence is equivalent to...
moeuex 2660 Uniqueness implies that ex...
moeu 2661 Uniqueness is equivalent t...
eubi 2662 Equivalence theorem for th...
eubii 2663 Introduce unique existenti...
eubiiOLD 2664 Obsolete version of ~ eubi...
eubidv 2665 Formula-building rule for ...
eubid 2666 Formula-building rule for ...
nfeu1 2667 Bound-variable hypothesis ...
nfeu1ALT 2668 Alternate proof of ~ nfeu1...
nfeud2 2669 Bound-variable hypothesis ...
nfeudw 2670 Version of ~ nfeud with a ...
nfeud 2671 Bound-variable hypothesis ...
nfeuw 2672 Version of ~ nfeu with a d...
nfeu 2673 Bound-variable hypothesis ...
dfeu 2674 Rederive ~ df-eu from the ...
dfmo 2675 Rederive ~ df-mo from the ...
euequ 2676 There exists a unique set ...
sb8eulem 2677 Lemma. Factor out the com...
sb8euv 2678 Variable substitution in u...
sb8eu 2679 Variable substitution in u...
sb8mo 2680 Variable substitution for ...
cbvmow 2681 Version of ~ cbvmo with a ...
cbvmo 2682 Rule used to change bound ...
cbveuw 2683 Version of ~ cbveu with a ...
cbveu 2684 Rule used to change bound ...
cbveuALT 2685 Alternative proof of ~ cbv...
eu2 2686 An alternate way of defini...
eu1 2687 An alternate way to expres...
euor 2688 Introduce a disjunct into ...
euorv 2689 Introduce a disjunct into ...
euor2 2690 Introduce or eliminate a d...
sbmo 2691 Substitution into an at-mo...
eu4 2692 Uniqueness using implicit ...
euimmo 2693 Existential uniqueness imp...
euim 2694 Add unique existential qua...
euimOLD 2695 Obsolete version of ~ euim...
moanimlem 2696 Factor out the common proo...
moanimv 2697 Introduction of a conjunct...
moanim 2698 Introduction of a conjunct...
euan 2699 Introduction of a conjunct...
moanmo 2700 Nested at-most-one quantif...
moaneu 2701 Nested at-most-one and uni...
euanv 2702 Introduction of a conjunct...
mopick 2703 "At most one" picks a vari...
moexexlem 2704 Factor out the proof skele...
2moexv 2705 Double quantification with...
moexexvw 2706 Version of ~ moexexv with ...
2moswapv 2707 Version of ~ 2moswap with ...
2euswapv 2708 Version of ~ 2euswap with ...
2euexv 2709 Version of ~ 2euex with ` ...
2exeuv 2710 Version of ~ 2exeu with ` ...
eupick 2711 Existential uniqueness "pi...
eupicka 2712 Version of ~ eupick with c...
eupickb 2713 Existential uniqueness "pi...
eupickbi 2714 Theorem *14.26 in [Whitehe...
mopick2 2715 "At most one" can show the...
moexex 2716 "At most one" double quant...
moexexv 2717 "At most one" double quant...
2moex 2718 Double quantification with...
2euex 2719 Double quantification with...
2eumo 2720 Nested unique existential ...
2eu2ex 2721 Double existential uniquen...
2moswap 2722 A condition allowing to sw...
2euswap 2723 A condition allowing to sw...
2exeu 2724 Double existential uniquen...
2mo2 2725 Two ways of expressing "th...
2mo 2726 Two ways of expressing "th...
2mos 2727 Double "exists at most one...
2eu1 2728 Double existential uniquen...
2eu1OLD 2729 Obsolete version of ~ 2eu1...
2eu1v 2730 Version of ~ 2eu1 with ` x...
2eu2 2731 Double existential uniquen...
2eu3 2732 Double existential uniquen...
2eu3OLD 2733 Obsolete version of ~ 2eu3...
2eu4 2734 This theorem provides us w...
2eu5 2735 An alternate definition of...
2eu5OLD 2736 Obsolete version of ~ 2eu5...
2eu6 2737 Two equivalent expressions...
2eu7 2738 Two equivalent expressions...
2eu8 2739 Two equivalent expressions...
euae 2740 Two ways to express "exact...
exists1 2741 Two ways to express "exact...
exists2 2742 A condition implying that ...
barbara 2743 "Barbara", one of the fund...
celarent 2744 "Celarent", one of the syl...
darii 2745 "Darii", one of the syllog...
dariiALT 2746 Alternate proof of ~ darii...
ferio 2747 "Ferio" ("Ferioque"), one ...
barbarilem 2748 Lemma for ~ barbari and th...
barbari 2749 "Barbari", one of the syll...
barbariALT 2750 Alternate proof of ~ barba...
celaront 2751 "Celaront", one of the syl...
cesare 2752 "Cesare", one of the syllo...
camestres 2753 "Camestres", one of the sy...
festino 2754 "Festino", one of the syll...
festinoALT 2755 Alternate proof of ~ festi...
baroco 2756 "Baroco", one of the syllo...
barocoALT 2757 Alternate proof of ~ festi...
cesaro 2758 "Cesaro", one of the syllo...
camestros 2759 "Camestros", one of the sy...
datisi 2760 "Datisi", one of the syllo...
disamis 2761 "Disamis", one of the syll...
ferison 2762 "Ferison", one of the syll...
bocardo 2763 "Bocardo", one of the syll...
darapti 2764 "Darapti", one of the syll...
daraptiALT 2765 Alternate proof of ~ darap...
felapton 2766 "Felapton", one of the syl...
calemes 2767 "Calemes", one of the syll...
dimatis 2768 "Dimatis", one of the syll...
fresison 2769 "Fresison", one of the syl...
calemos 2770 "Calemos", one of the syll...
fesapo 2771 "Fesapo", one of the syllo...
bamalip 2772 "Bamalip", one of the syll...
axia1 2773 Left 'and' elimination (in...
axia2 2774 Right 'and' elimination (i...
axia3 2775 'And' introduction (intuit...
axin1 2776 'Not' introduction (intuit...
axin2 2777 'Not' elimination (intuiti...
axio 2778 Definition of 'or' (intuit...
axi4 2779 Specialization (intuitioni...
axi5r 2780 Converse of ~ axc4 (intuit...
axial 2781 The setvar ` x ` is not fr...
axie1 2782 The setvar ` x ` is not fr...
axie2 2783 A key property of existent...
axi9 2784 Axiom of existence (intuit...
axi10 2785 Axiom of Quantifier Substi...
axi12 2786 Axiom of Quantifier Introd...
axi12OLD 2787 Obsolete version of ~ axi1...
axbnd 2788 Axiom of Bundling (intuiti...
axbndOLD 2789 Obsolete version of ~ axbn...
axexte 2791 The axiom of extensionalit...
axextg 2792 A generalization of the ax...
axextb 2793 A bidirectional version of...
axextmo 2794 There exists at most one s...
nulmo 2795 There exists at most one e...
eleq1ab 2798 Extension (in the sense of...
cleljustab 2799 Extension of ~ cleljust fr...
abid 2800 Simplification of class ab...
vexwt 2801 A standard theorem of pred...
vexw 2802 If ` ph ` is a theorem, th...
vextru 2803 Every setvar is a member o...
hbab1 2804 Bound-variable hypothesis ...
nfsab1 2805 Bound-variable hypothesis ...
nfsab1OLD 2806 Obsolete version of ~ nfsa...
hbab 2807 Bound-variable hypothesis ...
hbabg 2808 Bound-variable hypothesis ...
nfsab 2809 Bound-variable hypothesis ...
nfsabg 2810 Bound-variable hypothesis ...
dfcleq 2812 The defining characterizat...
cvjust 2813 Every set is a class. Pro...
ax9ALT 2814 Proof of ~ ax-9 from Tarsk...
eqriv 2815 Infer equality of classes ...
eqrdv 2816 Deduce equality of classes...
eqrdav 2817 Deduce equality of classes...
eqid 2818 Law of identity (reflexivi...
eqidd 2819 Class identity law with an...
eqeq1d 2820 Deduction from equality to...
eqeq1dALT 2821 Shorter proof of ~ eqeq1d ...
eqeq1 2822 Equality implies equivalen...
eqeq1i 2823 Inference from equality to...
eqcomd 2824 Deduction from commutative...
eqcom 2825 Commutative law for class ...
eqcoms 2826 Inference applying commuta...
eqcomi 2827 Inference from commutative...
neqcomd 2828 Commute an inequality. (C...
eqeq2d 2829 Deduction from equality to...
eqeq2 2830 Equality implies equivalen...
eqeq2i 2831 Inference from equality to...
eqeq12 2832 Equality relationship amon...
eqeq12i 2833 A useful inference for sub...
eqeq12d 2834 A useful inference for sub...
eqeqan12d 2835 A useful inference for sub...
eqeqan12dALT 2836 Alternate proof of ~ eqeqa...
eqeqan12rd 2837 A useful inference for sub...
eqtr 2838 Transitive law for class e...
eqtr2 2839 A transitive law for class...
eqtr3 2840 A transitive law for class...
eqtri 2841 An equality transitivity i...
eqtr2i 2842 An equality transitivity i...
eqtr3i 2843 An equality transitivity i...
eqtr4i 2844 An equality transitivity i...
3eqtri 2845 An inference from three ch...
3eqtrri 2846 An inference from three ch...
3eqtr2i 2847 An inference from three ch...
3eqtr2ri 2848 An inference from three ch...
3eqtr3i 2849 An inference from three ch...
3eqtr3ri 2850 An inference from three ch...
3eqtr4i 2851 An inference from three ch...
3eqtr4ri 2852 An inference from three ch...
eqtrd 2853 An equality transitivity d...
eqtr2d 2854 An equality transitivity d...
eqtr3d 2855 An equality transitivity e...
eqtr4d 2856 An equality transitivity e...
3eqtrd 2857 A deduction from three cha...
3eqtrrd 2858 A deduction from three cha...
3eqtr2d 2859 A deduction from three cha...
3eqtr2rd 2860 A deduction from three cha...
3eqtr3d 2861 A deduction from three cha...
3eqtr3rd 2862 A deduction from three cha...
3eqtr4d 2863 A deduction from three cha...
3eqtr4rd 2864 A deduction from three cha...
syl5eq 2865 An equality transitivity d...
syl5req 2866 An equality transitivity d...
syl5eqr 2867 An equality transitivity d...
syl5reqr 2868 An equality transitivity d...
syl6eq 2869 An equality transitivity d...
syl6req 2870 An equality transitivity d...
syl6eqr 2871 An equality transitivity d...
syl6reqr 2872 An equality transitivity d...
sylan9eq 2873 An equality transitivity d...
sylan9req 2874 An equality transitivity d...
sylan9eqr 2875 An equality transitivity d...
3eqtr3g 2876 A chained equality inferen...
3eqtr3a 2877 A chained equality inferen...
3eqtr4g 2878 A chained equality inferen...
3eqtr4a 2879 A chained equality inferen...
eq2tri 2880 A compound transitive infe...
abbi1 2881 Equivalent formulas yield ...
abbidv 2882 Equivalent wff's yield equ...
abbii 2883 Equivalent wff's yield equ...
abbid 2884 Equivalent wff's yield equ...
abbi 2885 Equivalent formulas define...
cbvabv 2886 Rule used to change bound ...
cbvabw 2887 Version of ~ cbvab with a ...
cbvab 2888 Rule used to change bound ...
cbvabvOLD 2889 Obsolete version of ~ cbva...
dfclel 2891 Characterization of the el...
eleq1w 2892 Weaker version of ~ eleq1 ...
eleq2w 2893 Weaker version of ~ eleq2 ...
eleq1d 2894 Deduction from equality to...
eleq2d 2895 Deduction from equality to...
eleq2dALT 2896 Alternate proof of ~ eleq2...
eleq1 2897 Equality implies equivalen...
eleq2 2898 Equality implies equivalen...
eleq12 2899 Equality implies equivalen...
eleq1i 2900 Inference from equality to...
eleq2i 2901 Inference from equality to...
eleq12i 2902 Inference from equality to...
eqneltri 2903 If a class is not an eleme...
eleq12d 2904 Deduction from equality to...
eleq1a 2905 A transitive-type law rela...
eqeltri 2906 Substitution of equal clas...
eqeltrri 2907 Substitution of equal clas...
eleqtri 2908 Substitution of equal clas...
eleqtrri 2909 Substitution of equal clas...
eqeltrd 2910 Substitution of equal clas...
eqeltrrd 2911 Deduction that substitutes...
eleqtrd 2912 Deduction that substitutes...
eleqtrrd 2913 Deduction that substitutes...
eqeltrid 2914 A membership and equality ...
eqeltrrid 2915 A membership and equality ...
eleqtrid 2916 A membership and equality ...
eleqtrrid 2917 A membership and equality ...
syl6eqel 2918 A membership and equality ...
syl6eqelr 2919 A membership and equality ...
eleqtrdi 2920 A membership and equality ...
eleqtrrdi 2921 A membership and equality ...
3eltr3i 2922 Substitution of equal clas...
3eltr4i 2923 Substitution of equal clas...
3eltr3d 2924 Substitution of equal clas...
3eltr4d 2925 Substitution of equal clas...
3eltr3g 2926 Substitution of equal clas...
3eltr4g 2927 Substitution of equal clas...
eleq2s 2928 Substitution of equal clas...
eqneltrd 2929 If a class is not an eleme...
eqneltrrd 2930 If a class is not an eleme...
neleqtrd 2931 If a class is not an eleme...
neleqtrrd 2932 If a class is not an eleme...
cleqh 2933 Establish equality between...
nelneq 2934 A way of showing two class...
nelneq2 2935 A way of showing two class...
eqsb3 2936 Substitution applied to an...
clelsb3 2937 Substitution applied to an...
clelsb3vOLD 2938 Obsolete version of ~ clel...
hbxfreq 2939 A utility lemma to transfe...
hblem 2940 Change the free variable o...
hblemg 2941 Change the free variable o...
abeq2 2942 Equality of a class variab...
abeq1 2943 Equality of a class variab...
abeq2d 2944 Equality of a class variab...
abeq2i 2945 Equality of a class variab...
abeq1i 2946 Equality of a class variab...
abbi2dv 2947 Deduction from a wff to a ...
abbi2dvOLD 2948 Obsolete version of ~ abbi...
abbi1dv 2949 Deduction from a wff to a ...
abbi2i 2950 Equality of a class variab...
abbi2iOLD 2951 Obsolete version of ~ abbi...
abbiOLD 2952 Obsolete proof of ~ abbi a...
abid1 2953 Every class is equal to a ...
abid2 2954 A simplification of class ...
clelab 2955 Membership of a class vari...
clabel 2956 Membership of a class abst...
sbab 2957 The right-hand side of the...
nfcjust 2959 Justification theorem for ...
nfci 2961 Deduce that a class ` A ` ...
nfcii 2962 Deduce that a class ` A ` ...
nfcr 2963 Consequence of the not-fre...
nfcriv 2964 Consequence of the not-fre...
nfcd 2965 Deduce that a class ` A ` ...
nfcrd 2966 Consequence of the not-fre...
nfcrii 2967 Consequence of the not-fre...
nfcri 2968 Consequence of the not-fre...
nfceqdf 2969 An equality theorem for ef...
nfceqi 2970 Equality theorem for class...
nfceqiOLD 2971 Obsolete proof of ~ nfceqi...
nfcxfr 2972 A utility lemma to transfe...
nfcxfrd 2973 A utility lemma to transfe...
nfcv 2974 If ` x ` is disjoint from ...
nfcvd 2975 If ` x ` is disjoint from ...
nfab1 2976 Bound-variable hypothesis ...
nfnfc1 2977 The setvar ` x ` is bound ...
clelsb3fw 2978 Version of ~ clelsb3f with...
clelsb3f 2979 Substitution applied to an...
clelsb3fOLD 2980 Obsolete version of ~ clel...
nfab 2981 Bound-variable hypothesis ...
nfabg 2982 Bound-variable hypothesis ...
nfaba1 2983 Bound-variable hypothesis ...
nfaba1g 2984 Bound-variable hypothesis ...
nfeqd 2985 Hypothesis builder for equ...
nfeld 2986 Hypothesis builder for ele...
nfnfc 2987 Hypothesis builder for ` F...
nfeq 2988 Hypothesis builder for equ...
nfel 2989 Hypothesis builder for ele...
nfeq1 2990 Hypothesis builder for equ...
nfel1 2991 Hypothesis builder for ele...
nfeq2 2992 Hypothesis builder for equ...
nfel2 2993 Hypothesis builder for ele...
drnfc1 2994 Formula-building lemma for...
drnfc1OLD 2995 Obsolete version of ~ drnf...
drnfc2 2996 Formula-building lemma for...
nfabdw 2997 Version of ~ nfabd with a ...
nfabd 2998 Bound-variable hypothesis ...
nfabd2 2999 Bound-variable hypothesis ...
nfabd2OLD 3000 Obsolete version of ~ nfab...
nfabdOLD 3001 Obsolete version of ~ nfab...
dvelimdc 3002 Deduction form of ~ dvelim...
dvelimc 3003 Version of ~ dvelim for cl...
nfcvf 3004 If ` x ` and ` y ` are dis...
nfcvf2 3005 If ` x ` and ` y ` are dis...
nfcvfOLD 3006 Obsolete version of ~ nfcv...
cleqf 3007 Establish equality between...
cleqfOLD 3008 Obsolete version of ~ cleq...
abid2f 3009 A simplification of class ...
abeq2f 3010 Equality of a class variab...
abeq2fOLD 3011 Obsolete version of ~ abeq...
sbabel 3012 Theorem to move a substitu...
neii 3015 Inference associated with ...
neir 3016 Inference associated with ...
nne 3017 Negation of inequality. (...
neneqd 3018 Deduction eliminating ineq...
neneq 3019 From inequality to non-equ...
neqned 3020 If it is not the case that...
neqne 3021 From non-equality to inequ...
neirr 3022 No class is unequal to its...
exmidne 3023 Excluded middle with equal...
eqneqall 3024 A contradiction concerning...
nonconne 3025 Law of noncontradiction wi...
necon3ad 3026 Contrapositive law deducti...
necon3bd 3027 Contrapositive law deducti...
necon2ad 3028 Contrapositive inference f...
necon2bd 3029 Contrapositive inference f...
necon1ad 3030 Contrapositive deduction f...
necon1bd 3031 Contrapositive deduction f...
necon4ad 3032 Contrapositive inference f...
necon4bd 3033 Contrapositive inference f...
necon3d 3034 Contrapositive law deducti...
necon1d 3035 Contrapositive law deducti...
necon2d 3036 Contrapositive inference f...
necon4d 3037 Contrapositive inference f...
necon3ai 3038 Contrapositive inference f...
necon3bi 3039 Contrapositive inference f...
necon1ai 3040 Contrapositive inference f...
necon1bi 3041 Contrapositive inference f...
necon2ai 3042 Contrapositive inference f...
necon2bi 3043 Contrapositive inference f...
necon4ai 3044 Contrapositive inference f...
necon3i 3045 Contrapositive inference f...
necon1i 3046 Contrapositive inference f...
necon2i 3047 Contrapositive inference f...
necon4i 3048 Contrapositive inference f...
necon3abid 3049 Deduction from equality to...
necon3bbid 3050 Deduction from equality to...
necon1abid 3051 Contrapositive deduction f...
necon1bbid 3052 Contrapositive inference f...
necon4abid 3053 Contrapositive law deducti...
necon4bbid 3054 Contrapositive law deducti...
necon2abid 3055 Contrapositive deduction f...
necon2bbid 3056 Contrapositive deduction f...
necon3bid 3057 Deduction from equality to...
necon4bid 3058 Contrapositive law deducti...
necon3abii 3059 Deduction from equality to...
necon3bbii 3060 Deduction from equality to...
necon1abii 3061 Contrapositive inference f...
necon1bbii 3062 Contrapositive inference f...
necon2abii 3063 Contrapositive inference f...
necon2bbii 3064 Contrapositive inference f...
necon3bii 3065 Inference from equality to...
necom 3066 Commutation of inequality....
necomi 3067 Inference from commutative...
necomd 3068 Deduction from commutative...
nesym 3069 Characterization of inequa...
nesymi 3070 Inference associated with ...
nesymir 3071 Inference associated with ...
neeq1d 3072 Deduction for inequality. ...
neeq2d 3073 Deduction for inequality. ...
neeq12d 3074 Deduction for inequality. ...
neeq1 3075 Equality theorem for inequ...
neeq2 3076 Equality theorem for inequ...
neeq1i 3077 Inference for inequality. ...
neeq2i 3078 Inference for inequality. ...
neeq12i 3079 Inference for inequality. ...
eqnetrd 3080 Substitution of equal clas...
eqnetrrd 3081 Substitution of equal clas...
neeqtrd 3082 Substitution of equal clas...
eqnetri 3083 Substitution of equal clas...
eqnetrri 3084 Substitution of equal clas...
neeqtri 3085 Substitution of equal clas...
neeqtrri 3086 Substitution of equal clas...
neeqtrrd 3087 Substitution of equal clas...
eqnetrrid 3088 A chained equality inferen...
3netr3d 3089 Substitution of equality i...
3netr4d 3090 Substitution of equality i...
3netr3g 3091 Substitution of equality i...
3netr4g 3092 Substitution of equality i...
nebi 3093 Contraposition law for ine...
pm13.18 3094 Theorem *13.18 in [Whitehe...
pm13.18OLD 3095 Obsolete version of ~ pm13...
pm13.181 3096 Theorem *13.181 in [Whiteh...
pm2.61ine 3097 Inference eliminating an i...
pm2.21ddne 3098 A contradiction implies an...
pm2.61ne 3099 Deduction eliminating an i...
pm2.61dne 3100 Deduction eliminating an i...
pm2.61dane 3101 Deduction eliminating an i...
pm2.61da2ne 3102 Deduction eliminating two ...
pm2.61da3ne 3103 Deduction eliminating thre...
pm2.61iine 3104 Equality version of ~ pm2....
neor 3105 Logical OR with an equalit...
neanior 3106 A De Morgan's law for ineq...
ne3anior 3107 A De Morgan's law for ineq...
neorian 3108 A De Morgan's law for ineq...
nemtbir 3109 An inference from an inequ...
nelne1 3110 Two classes are different ...
nelne1OLD 3111 Obsolete version of ~ neln...
nelne2 3112 Two classes are different ...
nelne2OLD 3113 Obsolete version of ~ neln...
nelelne 3114 Two classes are different ...
neneor 3115 If two classes are differe...
nfne 3116 Bound-variable hypothesis ...
nfned 3117 Bound-variable hypothesis ...
nabbi 3118 Not equivalent wff's corre...
mteqand 3119 A modus tollens deduction ...
neli 3122 Inference associated with ...
nelir 3123 Inference associated with ...
neleq12d 3124 Equality theorem for negat...
neleq1 3125 Equality theorem for negat...
neleq2 3126 Equality theorem for negat...
nfnel 3127 Bound-variable hypothesis ...
nfneld 3128 Bound-variable hypothesis ...
nnel 3129 Negation of negated member...
elnelne1 3130 Two classes are different ...
elnelne2 3131 Two classes are different ...
nelcon3d 3132 Contrapositive law deducti...
elnelall 3133 A contradiction concerning...
pm2.61danel 3134 Deduction eliminating an e...
rgen 3145 Generalization rule for re...
ralel 3146 All elements of a class ar...
rgenw 3147 Generalization rule for re...
rgen2w 3148 Generalization rule for re...
mprg 3149 Modus ponens combined with...
mprgbir 3150 Modus ponens on biconditio...
alral 3151 Universal quantification i...
raln 3152 Restricted universally qua...
ral2imi 3153 Inference quantifying ante...
ralimi2 3154 Inference quantifying both...
ralimia 3155 Inference quantifying both...
ralimiaa 3156 Inference quantifying both...
ralimi 3157 Inference quantifying both...
2ralimi 3158 Inference quantifying both...
ralim 3159 Distribution of restricted...
ralbii2 3160 Inference adding different...
ralbiia 3161 Inference adding restricte...
ralbii 3162 Inference adding restricte...
2ralbii 3163 Inference adding two restr...
ralbi 3164 Distribute a restricted un...
ralanid 3165 Cancellation law for restr...
ralanidOLD 3166 Obsolete version of ~ rala...
r19.26 3167 Restricted quantifier vers...
r19.26-2 3168 Restricted quantifier vers...
r19.26-3 3169 Version of ~ r19.26 with t...
r19.26m 3170 Version of ~ 19.26 and ~ r...
ralbiim 3171 Split a biconditional and ...
r19.21v 3172 Restricted quantifier vers...
ralimdv2 3173 Inference quantifying both...
ralimdva 3174 Deduction quantifying both...
ralimdv 3175 Deduction quantifying both...
ralimdvva 3176 Deduction doubly quantifyi...
hbralrimi 3177 Inference from Theorem 19....
ralrimiv 3178 Inference from Theorem 19....
ralrimiva 3179 Inference from Theorem 19....
ralrimivw 3180 Inference from Theorem 19....
r19.27v 3181 Restricted quantitifer ver...
r19.27vOLD 3182 Obsolete version of ~ r19....
r19.28v 3183 Restricted quantifier vers...
r19.28vOLD 3184 Obsolete version of ~ r19....
ralrimdv 3185 Inference from Theorem 19....
ralrimdva 3186 Inference from Theorem 19....
ralrimivv 3187 Inference from Theorem 19....
ralrimivva 3188 Inference from Theorem 19....
ralrimivvva 3189 Inference from Theorem 19....
ralrimdvv 3190 Inference from Theorem 19....
ralrimdvva 3191 Inference from Theorem 19....
ralbidv2 3192 Formula-building rule for ...
ralbidva 3193 Formula-building rule for ...
ralbidv 3194 Formula-building rule for ...
2ralbidva 3195 Formula-building rule for ...
2ralbidv 3196 Formula-building rule for ...
r2allem 3197 Lemma factoring out common...
r2al 3198 Double restricted universa...
r3al 3199 Triple restricted universa...
rgen2 3200 Generalization rule for re...
rgen3 3201 Generalization rule for re...
rsp 3202 Restricted specialization....
rspa 3203 Restricted specialization....
rspec 3204 Specialization rule for re...
r19.21bi 3205 Inference from Theorem 19....
r19.21biOLD 3206 Obsolete version of ~ r19....
r19.21be 3207 Inference from Theorem 19....
rspec2 3208 Specialization rule for re...
rspec3 3209 Specialization rule for re...
rsp2 3210 Restricted specialization,...
r19.21t 3211 Restricted quantifier vers...
r19.21 3212 Restricted quantifier vers...
ralrimi 3213 Inference from Theorem 19....
ralimdaa 3214 Deduction quantifying both...
ralrimd 3215 Inference from Theorem 19....
nfra1 3216 The setvar ` x ` is not fr...
hbra1 3217 The setvar ` x ` is not fr...
hbral 3218 Bound-variable hypothesis ...
r2alf 3219 Double restricted universa...
nfraldw 3220 Version of ~ nfrald with a...
nfrald 3221 Deduction version of ~ nfr...
nfralw 3222 Version of ~ nfral with a ...
nfral 3223 Bound-variable hypothesis ...
nfra2w 3224 Version of ~ nfra2 with a ...
nfra2 3225 Similar to Lemma 24 of [Mo...
rgen2a 3226 Generalization rule for re...
ralbida 3227 Formula-building rule for ...
ralbid 3228 Formula-building rule for ...
2ralbida 3229 Formula-building rule for ...
ralbiOLD 3230 Obsolete version of ~ ralb...
raleqbii 3231 Equality deduction for res...
ralcom4 3232 Commutation of restricted ...
ralnex 3233 Relationship between restr...
dfral2 3234 Relationship between restr...
rexnal 3235 Relationship between restr...
dfrex2 3236 Relationship between restr...
rexex 3237 Restricted existence impli...
rexim 3238 Theorem 19.22 of [Margaris...
reximia 3239 Inference quantifying both...
reximi 3240 Inference quantifying both...
reximi2 3241 Inference quantifying both...
rexbii2 3242 Inference adding different...
rexbiia 3243 Inference adding restricte...
rexbii 3244 Inference adding restricte...
2rexbii 3245 Inference adding two restr...
rexcom4 3246 Commutation of restricted ...
2ex2rexrot 3247 Rotate two existential qua...
rexcom4a 3248 Specialized existential co...
rexanid 3249 Cancellation law for restr...
rexanidOLD 3250 Obsolete version of ~ rexa...
r19.29 3251 Restricted quantifier vers...
r19.29r 3252 Restricted quantifier vers...
r19.29rOLD 3253 Obsolete version of ~ r19....
r19.29imd 3254 Theorem 19.29 of [Margaris...
rexnal2 3255 Relationship between two r...
rexnal3 3256 Relationship between three...
ralnex2 3257 Relationship between two r...
ralnex2OLD 3258 Obsolete version of ~ raln...
ralnex3 3259 Relationship between three...
ralnex3OLD 3260 Obsolete version of ~ raln...
ralinexa 3261 A transformation of restri...
rexanali 3262 A transformation of restri...
nrexralim 3263 Negation of a complex pred...
risset 3264 Two ways to say " ` A ` be...
nelb 3265 A definition of ` -. A e. ...
nrex 3266 Inference adding restricte...
nrexdv 3267 Deduction adding restricte...
reximdv2 3268 Deduction quantifying both...
reximdvai 3269 Deduction quantifying both...
reximdv 3270 Deduction from Theorem 19....
reximdva 3271 Deduction quantifying both...
reximddv 3272 Deduction from Theorem 19....
reximssdv 3273 Derivation of a restricted...
reximdvva 3274 Deduction doubly quantifyi...
reximddv2 3275 Double deduction from Theo...
r19.23v 3276 Restricted quantifier vers...
rexlimiv 3277 Inference from Theorem 19....
rexlimiva 3278 Inference from Theorem 19....
rexlimivw 3279 Weaker version of ~ rexlim...
rexlimdv 3280 Inference from Theorem 19....
rexlimdva 3281 Inference from Theorem 19....
rexlimdvaa 3282 Inference from Theorem 19....
rexlimdv3a 3283 Inference from Theorem 19....
rexlimdva2 3284 Inference from Theorem 19....
r19.29an 3285 A commonly used pattern in...
r19.29a 3286 A commonly used pattern in...
rexlimdvw 3287 Inference from Theorem 19....
rexlimddv 3288 Restricted existential eli...
rexlimivv 3289 Inference from Theorem 19....
rexlimdvv 3290 Inference from Theorem 19....
rexlimdvva 3291 Inference from Theorem 19....
rexbidv2 3292 Formula-building rule for ...
rexbidva 3293 Formula-building rule for ...
rexbidv 3294 Formula-building rule for ...
2rexbiia 3295 Inference adding two restr...
2rexbidva 3296 Formula-building rule for ...
2rexbidv 3297 Formula-building rule for ...
rexralbidv 3298 Formula-building rule for ...
r2exlem 3299 Lemma factoring out common...
r2ex 3300 Double restricted existent...
rspe 3301 Restricted specialization....
rsp2e 3302 Restricted specialization....
nfre1 3303 The setvar ` x ` is not fr...
nfrexd 3304 Deduction version of ~ nfr...
nfrexdg 3305 Deduction version of ~ nfr...
nfrex 3306 Bound-variable hypothesis ...
nfrexg 3307 Bound-variable hypothesis ...
reximdai 3308 Deduction from Theorem 19....
reximd2a 3309 Deduction quantifying both...
r19.23t 3310 Closed theorem form of ~ r...
r19.23 3311 Restricted quantifier vers...
rexlimi 3312 Restricted quantifier vers...
rexlimd2 3313 Version of ~ rexlimd with ...
rexlimd 3314 Deduction form of ~ rexlim...
rexbida 3315 Formula-building rule for ...
rexbidvaALT 3316 Alternate proof of ~ rexbi...
rexbid 3317 Formula-building rule for ...
rexbidvALT 3318 Alternate proof of ~ rexbi...
ralrexbid 3319 Formula-building rule for ...
ralrexbidOLD 3320 Obsolete version of ~ ralr...
r19.12 3321 Restricted quantifier vers...
r2exf 3322 Double restricted existent...
rexeqbii 3323 Equality deduction for res...
r19.12OLD 3324 Obsolete version of ~ r19....
reuanid 3325 Cancellation law for restr...
rmoanid 3326 Cancellation law for restr...
r19.29af2 3327 A commonly used pattern ba...
r19.29af 3328 A commonly used pattern ba...
r19.29anOLD 3329 Obsolete version of ~ r19....
r19.29aOLD 3330 Obsolete proof of ~ r19.29...
2r19.29 3331 Theorem ~ r19.29 with two ...
r19.29d2r 3332 Theorem 19.29 of [Margaris...
r19.29vva 3333 A commonly used pattern ba...
r19.29vvaOLD 3334 Obsolete version of ~ r19....
r19.30 3335 Restricted quantifier vers...
r19.30OLD 3336 Obsolete version of ~ r19....
r19.32v 3337 Restricted quantifier vers...
r19.35 3338 Restricted quantifier vers...
r19.36v 3339 Restricted quantifier vers...
r19.37 3340 Restricted quantifier vers...
r19.37v 3341 Restricted quantifier vers...
r19.37vOLD 3342 Obsolete version of ~ r19....
r19.40 3343 Restricted quantifier vers...
r19.41v 3344 Restricted quantifier vers...
r19.41 3345 Restricted quantifier vers...
r19.41vv 3346 Version of ~ r19.41v with ...
r19.42v 3347 Restricted quantifier vers...
r19.43 3348 Restricted quantifier vers...
r19.44v 3349 One direction of a restric...
r19.45v 3350 Restricted quantifier vers...
ralcom 3351 Commutation of restricted ...
rexcom 3352 Commutation of restricted ...
rexcomOLD 3353 Obsolete version of ~ rexc...
ralcomf 3354 Commutation of restricted ...
rexcomf 3355 Commutation of restricted ...
ralcom13 3356 Swap first and third restr...
rexcom13 3357 Swap first and third restr...
ralrot3 3358 Rotate three restricted un...
rexrot4 3359 Rotate four restricted exi...
ralcom2w 3360 Version of ~ ralcom2 with ...
ralcom2 3361 Commutation of restricted ...
ralcom3 3362 A commutation law for rest...
reeanlem 3363 Lemma factoring out common...
reean 3364 Rearrange restricted exist...
reeanv 3365 Rearrange restricted exist...
3reeanv 3366 Rearrange three restricted...
2ralor 3367 Distribute restricted univ...
nfreu1 3368 The setvar ` x ` is not fr...
nfrmo1 3369 The setvar ` x ` is not fr...
nfreud 3370 Deduction version of ~ nfr...
nfrmod 3371 Deduction version of ~ nfr...
nfreuw 3372 Version of ~ nfreu with a ...
nfrmow 3373 Version of ~ nfrmo with a ...
nfreu 3374 Bound-variable hypothesis ...
nfrmo 3375 Bound-variable hypothesis ...
rabid 3376 An "identity" law of concr...
rabrab 3377 Abstract builder restricte...
rabidim1 3378 Membership in a restricted...
rabid2 3379 An "identity" law for rest...
rabid2f 3380 An "identity" law for rest...
rabbi 3381 Equivalent wff's correspon...
nfrab1 3382 The abstraction variable i...
nfrabw 3383 Version of ~ nfrab with a ...
nfrab 3384 A variable not free in a w...
reubida 3385 Formula-building rule for ...
reubidva 3386 Formula-building rule for ...
reubidv 3387 Formula-building rule for ...
reubiia 3388 Formula-building rule for ...
reubii 3389 Formula-building rule for ...
rmobida 3390 Formula-building rule for ...
rmobidva 3391 Formula-building rule for ...
rmobidv 3392 Formula-building rule for ...
rmobiia 3393 Formula-building rule for ...
rmobii 3394 Formula-building rule for ...
raleqf 3395 Equality theorem for restr...
rexeqf 3396 Equality theorem for restr...
reueq1f 3397 Equality theorem for restr...
rmoeq1f 3398 Equality theorem for restr...
raleqbidv 3399 Equality deduction for res...
rexeqbidv 3400 Equality deduction for res...
raleqbi1dv 3401 Equality deduction for res...
rexeqbi1dv 3402 Equality deduction for res...
raleq 3403 Equality theorem for restr...
rexeq 3404 Equality theorem for restr...
reueq1 3405 Equality theorem for restr...
rmoeq1 3406 Equality theorem for restr...
raleqOLD 3407 Obsolete version of ~ rale...
rexeqOLD 3408 Obsolete version of ~ rexe...
reueq1OLD 3409 Obsolete version of ~ reue...
rmoeq1OLD 3410 Obsolete version of ~ rmoe...
raleqi 3411 Equality inference for res...
rexeqi 3412 Equality inference for res...
raleqdv 3413 Equality deduction for res...
rexeqdv 3414 Equality deduction for res...
raleqbi1dvOLD 3415 Obsolete version of ~ rale...
rexeqbi1dvOLD 3416 Obsolete version of ~ rexe...
reueqd 3417 Equality deduction for res...
rmoeqd 3418 Equality deduction for res...
raleqbid 3419 Equality deduction for res...
rexeqbid 3420 Equality deduction for res...
raleqbidvOLD 3421 Obsolete version of ~ rale...
rexeqbidvOLD 3422 Obsolete version of ~ rexe...
raleqbidva 3423 Equality deduction for res...
rexeqbidva 3424 Equality deduction for res...
raleleq 3425 All elements of a class ar...
raleleqALT 3426 Alternate proof of ~ ralel...
mormo 3427 Unrestricted "at most one"...
reu5 3428 Restricted uniqueness in t...
reurex 3429 Restricted unique existenc...
2reu2rex 3430 Double restricted existent...
reurmo 3431 Restricted existential uni...
rmo5 3432 Restricted "at most one" i...
nrexrmo 3433 Nonexistence implies restr...
reueubd 3434 Restricted existential uni...
cbvralfw 3435 Version of ~ cbvralf with ...
cbvrexfw 3436 Version of ~ cbvrexf with ...
cbvralf 3437 Rule used to change bound ...
cbvrexf 3438 Rule used to change bound ...
cbvralw 3439 Version of ~ cbvral with a...
cbvrexw 3440 Version of ~ cbvrex with a...
cbvreuw 3441 Version of ~ cbvreu with a...
cbvrmow 3442 Version of ~ cbvrmo with a...
cbvral 3443 Rule used to change bound ...
cbvrex 3444 Rule used to change bound ...
cbvreu 3445 Change the bound variable ...
cbvrmo 3446 Change the bound variable ...
cbvralvw 3447 Version of ~ cbvralv with ...
cbvrexvw 3448 Version of ~ cbvrexv with ...
cbvreuvw 3449 Version of ~ cbvreuv with ...
cbvralv 3450 Change the bound variable ...
cbvrexv 3451 Change the bound variable ...
cbvreuv 3452 Change the bound variable ...
cbvrmov 3453 Change the bound variable ...
cbvraldva2 3454 Rule used to change the bo...
cbvrexdva2 3455 Rule used to change the bo...
cbvrexdva2OLD 3456 Obsolete version of ~ cbvr...
cbvraldva 3457 Rule used to change the bo...
cbvrexdva 3458 Rule used to change the bo...
cbvral2vw 3459 Version of ~ cbvral2v with...
cbvrex2vw 3460 Version of ~ cbvrex2v with...
cbvral3vw 3461 Version of ~ cbvral3v with...
cbvral2v 3462 Change bound variables of ...
cbvrex2v 3463 Change bound variables of ...
cbvral3v 3464 Change bound variables of ...
cbvralsvw 3465 Version of ~ cbvralsv with...
cbvrexsvw 3466 Version of ~ cbvrexsv with...
cbvralsv 3467 Change bound variable by u...
cbvrexsv 3468 Change bound variable by u...
sbralie 3469 Implicit to explicit subst...
rabbiia 3470 Equivalent wff's yield equ...
rabbii 3471 Equivalent wff's correspon...
rabbida 3472 Equivalent wff's yield equ...
rabbid 3473 Version of ~ rabbidv with ...
rabbidva2 3474 Equivalent wff's yield equ...
rabbia2 3475 Equivalent wff's yield equ...
rabbidva 3476 Equivalent wff's yield equ...
rabbidvaOLD 3477 Obsolete proof of ~ rabbid...
rabbidv 3478 Equivalent wff's yield equ...
rabeqf 3479 Equality theorem for restr...
rabeqi 3480 Equality theorem for restr...
rabeq 3481 Equality theorem for restr...
rabeqdv 3482 Equality of restricted cla...
rabeqbidv 3483 Equality of restricted cla...
rabeqbidva 3484 Equality of restricted cla...
rabeq2i 3485 Inference from equality of...
rabswap 3486 Swap with a membership rel...
cbvrabw 3487 Version of ~ cbvrab with a...
cbvrab 3488 Rule to change the bound v...
cbvrabv 3489 Rule to change the bound v...
cbvrabvOLD 3490 Obsolete version of ~ cbvr...
rabrabi 3491 Abstract builder restricte...
vjust 3493 Soundness justification th...
vex 3495 All setvar variables are s...
vexOLD 3496 Obsolete version of ~ vex ...
elv 3497 If a proposition is implie...
elvd 3498 If a proposition is implie...
el2v 3499 If a proposition is implie...
eqv 3500 The universe contains ever...
eqvf 3501 The universe contains ever...
abv 3502 The class of sets verifyin...
elisset 3503 An element of a class exis...
isset 3504 Two ways to say " ` A ` is...
issetf 3505 A version of ~ isset that ...
isseti 3506 A way to say " ` A ` is a ...
issetiOLD 3507 Obsolete version of ~ isse...
issetri 3508 A way to say " ` A ` is a ...
eqvisset 3509 A class equal to a variabl...
elex 3510 If a class is a member of ...
elexi 3511 If a class is a member of ...
elexd 3512 If a class is a member of ...
elissetOLD 3513 Obsolete version of ~ elis...
elex2 3514 If a class contains anothe...
elex22 3515 If two classes each contai...
prcnel 3516 A proper class doesn't bel...
ralv 3517 A universal quantifier res...
rexv 3518 An existential quantifier ...
reuv 3519 A unique existential quant...
rmov 3520 An at-most-one quantifier ...
rabab 3521 A class abstraction restri...
rexcom4b 3522 Specialized existential co...
ralcom4OLD 3523 Obsolete version of ~ ralc...
rexcom4OLD 3524 Obsolete version of ~ rexc...
ceqsalt 3525 Closed theorem version of ...
ceqsralt 3526 Restricted quantifier vers...
ceqsalg 3527 A representation of explic...
ceqsalgALT 3528 Alternate proof of ~ ceqsa...
ceqsal 3529 A representation of explic...
ceqsalv 3530 A representation of explic...
ceqsralv 3531 Restricted quantifier vers...
gencl 3532 Implicit substitution for ...
2gencl 3533 Implicit substitution for ...
3gencl 3534 Implicit substitution for ...
cgsexg 3535 Implicit substitution infe...
cgsex2g 3536 Implicit substitution infe...
cgsex4g 3537 An implicit substitution i...
ceqsex 3538 Elimination of an existent...
ceqsexv 3539 Elimination of an existent...
ceqsexv2d 3540 Elimination of an existent...
ceqsex2 3541 Elimination of two existen...
ceqsex2v 3542 Elimination of two existen...
ceqsex3v 3543 Elimination of three exist...
ceqsex4v 3544 Elimination of four existe...
ceqsex6v 3545 Elimination of six existen...
ceqsex8v 3546 Elimination of eight exist...
gencbvex 3547 Change of bound variable u...
gencbvex2 3548 Restatement of ~ gencbvex ...
gencbval 3549 Change of bound variable u...
sbhypf 3550 Introduce an explicit subs...
vtoclgft 3551 Closed theorem form of ~ v...
vtoclgftOLD 3552 Obsolete version of ~ vtoc...
vtocldf 3553 Implicit substitution of a...
vtocld 3554 Implicit substitution of a...
vtocl2d 3555 Implicit substitution of t...
vtoclf 3556 Implicit substitution of a...
vtocl 3557 Implicit substitution of a...
vtoclALT 3558 Alternate proof of ~ vtocl...
vtocl2 3559 Implicit substitution of c...
vtocl2OLD 3560 Obsolete proof of ~ vtocl2...
vtocl3 3561 Implicit substitution of c...
vtoclb 3562 Implicit substitution of a...
vtoclgf 3563 Implicit substitution of a...
vtoclg1f 3564 Version of ~ vtoclgf with ...
vtoclg 3565 Implicit substitution of a...
vtoclbg 3566 Implicit substitution of a...
vtocl2gf 3567 Implicit substitution of a...
vtocl3gf 3568 Implicit substitution of a...
vtocl2g 3569 Implicit substitution of 2...
vtoclgaf 3570 Implicit substitution of a...
vtoclga 3571 Implicit substitution of a...
vtocl2ga 3572 Implicit substitution of 2...
vtocl2gaf 3573 Implicit substitution of 2...
vtocl3gaf 3574 Implicit substitution of 3...
vtocl3ga 3575 Implicit substitution of 3...
vtocl4g 3576 Implicit substitution of 4...
vtocl4ga 3577 Implicit substitution of 4...
vtocleg 3578 Implicit substitution of a...
vtoclegft 3579 Implicit substitution of a...
vtoclef 3580 Implicit substitution of a...
vtocle 3581 Implicit substitution of a...
vtoclri 3582 Implicit substitution of a...
spcimgft 3583 A closed version of ~ spci...
spcgft 3584 A closed version of ~ spcg...
spcimgf 3585 Rule of specialization, us...
spcimegf 3586 Existential specialization...
spcgf 3587 Rule of specialization, us...
spcegf 3588 Existential specialization...
spcimdv 3589 Restricted specialization,...
spcdv 3590 Rule of specialization, us...
spcimedv 3591 Restricted existential spe...
spcgv 3592 Rule of specialization, us...
spcgvOLD 3593 Obsolete version of ~ spcg...
spcegv 3594 Existential specialization...
spcegvOLD 3595 Obsolete version of ~ spce...
spcedv 3596 Existential specialization...
spc2egv 3597 Existential specialization...
spc2gv 3598 Specialization with two qu...
spc2ed 3599 Existential specialization...
spc2d 3600 Specialization with 2 quan...
spc3egv 3601 Existential specialization...
spc3gv 3602 Specialization with three ...
spcv 3603 Rule of specialization, us...
spcev 3604 Existential specialization...
spc2ev 3605 Existential specialization...
rspct 3606 A closed version of ~ rspc...
rspcdf 3607 Restricted specialization,...
rspc 3608 Restricted specialization,...
rspce 3609 Restricted existential spe...
rspcimdv 3610 Restricted specialization,...
rspcimedv 3611 Restricted existential spe...
rspcdv 3612 Restricted specialization,...
rspcedv 3613 Restricted existential spe...
rspcebdv 3614 Restricted existential spe...
rspcv 3615 Restricted specialization,...
rspcvOLD 3616 Obsolete version of ~ rspc...
rspccv 3617 Restricted specialization,...
rspcva 3618 Restricted specialization,...
rspccva 3619 Restricted specialization,...
rspcev 3620 Restricted existential spe...
rspcevOLD 3621 Obsolete version of ~ rspc...
rspcdva 3622 Restricted specialization,...
rspcedvd 3623 Restricted existential spe...
rspcime 3624 Prove a restricted existen...
rspceaimv 3625 Restricted existential spe...
rspcedeq1vd 3626 Restricted existential spe...
rspcedeq2vd 3627 Restricted existential spe...
rspc2 3628 Restricted specialization ...
rspc2gv 3629 Restricted specialization ...
rspc2v 3630 2-variable restricted spec...
rspc2va 3631 2-variable restricted spec...
rspc2ev 3632 2-variable restricted exis...
rspc3v 3633 3-variable restricted spec...
rspc3ev 3634 3-variable restricted exis...
rspceeqv 3635 Restricted existential spe...
ralxpxfr2d 3636 Transfer a universal quant...
rexraleqim 3637 Statement following from e...
eqvincg 3638 A variable introduction la...
eqvinc 3639 A variable introduction la...
eqvincf 3640 A variable introduction la...
alexeqg 3641 Two ways to express substi...
ceqex 3642 Equality implies equivalen...
ceqsexg 3643 A representation of explic...
ceqsexgv 3644 Elimination of an existent...
ceqsexgvOLD 3645 Obsolete version of ~ ceqs...
ceqsrexv 3646 Elimination of a restricte...
ceqsrexbv 3647 Elimination of a restricte...
ceqsrex2v 3648 Elimination of a restricte...
clel2g 3649 An alternate definition of...
clel2 3650 An alternate definition of...
clel3g 3651 An alternate definition of...
clel3 3652 An alternate definition of...
clel4 3653 An alternate definition of...
clel5 3654 Alternate definition of cl...
clel5OLD 3655 Obsolete version of ~ clel...
pm13.183 3656 Compare theorem *13.183 in...
pm13.183OLD 3657 Obsolete version of ~ pm13...
rr19.3v 3658 Restricted quantifier vers...
rr19.28v 3659 Restricted quantifier vers...
elabgt 3660 Membership in a class abst...
elabgf 3661 Membership in a class abst...
elabf 3662 Membership in a class abst...
elabg 3663 Membership in a class abst...
elab 3664 Membership in a class abst...
elab2g 3665 Membership in a class abst...
elabd 3666 Explicit demonstration the...
elab2 3667 Membership in a class abst...
elab4g 3668 Membership in a class abst...
elab3gf 3669 Membership in a class abst...
elab3g 3670 Membership in a class abst...
elab3 3671 Membership in a class abst...
elrabi 3672 Implication for the member...
elrabf 3673 Membership in a restricted...
rabtru 3674 Abstract builder using the...
rabeqc 3675 A restricted class abstrac...
elrab3t 3676 Membership in a restricted...
elrab 3677 Membership in a restricted...
elrab3 3678 Membership in a restricted...
elrabd 3679 Membership in a restricted...
elrab2 3680 Membership in a class abst...
ralab 3681 Universal quantification o...
ralrab 3682 Universal quantification o...
rexab 3683 Existential quantification...
rexrab 3684 Existential quantification...
ralab2 3685 Universal quantification o...
ralab2OLD 3686 Obsolete version of ~ rala...
ralrab2 3687 Universal quantification o...
rexab2 3688 Existential quantification...
rexab2OLD 3689 Obsolete version of ~ rexa...
rexrab2 3690 Existential quantification...
abidnf 3691 Identity used to create cl...
dedhb 3692 A deduction theorem for co...
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 Version of ~ euxfr2 with a...
euxfrw 3709 Version of ~ euxfr with a ...
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...
2reurex 3747 Double restricted quantifi...
2reurmo 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 Version of ~ nfsbcd with a...
nfsbcw 3791 Version of ~ nfsbc with a ...
sbccow 3792 Version of ~ sbcco with a ...
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...
sbc6g 3798 An equivalence for class s...
sbc6 3799 An equivalence for class s...
sbc7 3800 An equivalence for class s...
cbvsbcw 3801 Version of ~ cbvsbc with a...
cbvsbcvw 3802 Version of ~ cbvsbcv with ...
cbvsbc 3803 Change bound variables in ...
cbvsbcv 3804 Change the bound variable ...
sbciegft 3805 Conversion of implicit sub...
sbciegf 3806 Conversion of implicit sub...
sbcieg 3807 Conversion of implicit sub...
sbcie2g 3808 Conversion of implicit sub...
sbcie 3809 Conversion of implicit sub...
sbciedf 3810 Conversion of implicit sub...
sbcied 3811 Conversion of implicit sub...
sbcied2 3812 Conversion of implicit sub...
elrabsf 3813 Membership in a restricted...
eqsbc3 3814 Substitution applied to an...
eqsbc3OLD 3815 Obsolete version of ~ eqsb...
sbcng 3816 Move negation in and out o...
sbcimg 3817 Distribution of class subs...
sbcan 3818 Distribution of class subs...
sbcor 3819 Distribution of class subs...
sbcbig 3820 Distribution of class subs...
sbcn1 3821 Move negation in and out o...
sbcim1 3822 Distribution of class subs...
sbcbid 3823 Formula-building deduction...
sbcbidv 3824 Formula-building deduction...
sbcbidvOLD 3825 Obsolete version of ~ sbcb...
sbcbii 3826 Formula-building inference...
sbcbi1 3827 Distribution of class subs...
sbcbi2 3828 Substituting into equivale...
sbcbi2OLD 3829 Obsolete proof of ~ sbcbi2...
sbcal 3830 Move universal quantifier ...
sbcex2 3831 Move existential quantifie...
sbceqal 3832 Class version of one impli...
sbeqalb 3833 Theorem *14.121 in [Whiteh...
eqsbc3r 3834 ~ eqsbc3 with setvar varia...
sbc3an 3835 Distribution of class subs...
sbcel1v 3836 Class substitution into a ...
sbcel1vOLD 3837 Obsolete version of ~ sbce...
sbcel2gv 3838 Class substitution into a ...
sbcel21v 3839 Class substitution into a ...
sbcimdv 3840 Substitution analogue of T...
sbctt 3841 Substitution for a variabl...
sbcgf 3842 Substitution for a variabl...
sbc19.21g 3843 Substitution for a variabl...
sbcg 3844 Substitution for a variabl...
sbcgfi 3845 Substitution for a variabl...
sbc2iegf 3846 Conversion of implicit sub...
sbc2ie 3847 Conversion of implicit sub...
sbc2iedv 3848 Conversion of implicit sub...
sbc3ie 3849 Conversion of implicit sub...
sbccomlem 3850 Lemma for ~ sbccom . (Con...
sbccom 3851 Commutative law for double...
sbcralt 3852 Interchange class substitu...
sbcrext 3853 Interchange class substitu...
sbcralg 3854 Interchange class substitu...
sbcrex 3855 Interchange class substitu...
sbcreu 3856 Interchange class substitu...
reu8nf 3857 Restricted uniqueness usin...
sbcabel 3858 Interchange class substitu...
rspsbc 3859 Restricted quantifier vers...
rspsbca 3860 Restricted quantifier vers...
rspesbca 3861 Existence form of ~ rspsbc...
spesbc 3862 Existence form of ~ spsbc ...
spesbcd 3863 form of ~ spsbc . (Contri...
sbcth2 3864 A substitution into a theo...
ra4v 3865 Version of ~ ra4 with a di...
ra4 3866 Restricted quantifier vers...
rmo2 3867 Alternate definition of re...
rmo2i 3868 Condition implying restric...
rmo3 3869 Restricted "at most one" u...
rmo3OLD 3870 Obsolete version of ~ rmo3...
rmob 3871 Consequence of "at most on...
rmoi 3872 Consequence of "at most on...
rmob2 3873 Consequence of "restricted...
rmoi2 3874 Consequence of "restricted...
rmoanim 3875 Introduction of a conjunct...
rmoanimALT 3876 Alternate proof of ~ rmoan...
reuan 3877 Introduction of a conjunct...
2reu1 3878 Double restricted existent...
2reu2 3879 Double restricted existent...
csb2 3882 Alternate expression for t...
csbeq1 3883 Analogue of ~ dfsbcq for p...
csbeq1d 3884 Equality deduction for pro...
csbeq2 3885 Substituting into equivale...
csbeq2d 3886 Formula-building deduction...
csbeq2dv 3887 Formula-building deduction...
csbeq2i 3888 Formula-building inference...
csbeq12dv 3889 Formula-building inference...
cbvcsbw 3890 Version of ~ cbvcsb with a...
cbvcsb 3891 Change bound variables in ...
cbvcsbv 3892 Change the bound variable ...
csbid 3893 Analogue of ~ sbid for pro...
csbeq1a 3894 Equality theorem for prope...
csbcow 3895 Version of ~ csbco with a ...
csbco 3896 Composition law for chaine...
csbtt 3897 Substitution doesn't affec...
csbconstgf 3898 Substitution doesn't affec...
csbconstg 3899 Substitution doesn't affec...
csbgfi 3900 Substitution for a variabl...
csbconstgi 3901 The proper substitution of...
nfcsb1d 3902 Bound-variable hypothesis ...
nfcsb1 3903 Bound-variable hypothesis ...
nfcsb1v 3904 Bound-variable hypothesis ...
nfcsbd 3905 Deduction version of ~ nfc...
nfcsbw 3906 Version of ~ nfcsb with a ...
nfcsb 3907 Bound-variable hypothesis ...
csbhypf 3908 Introduce an explicit subs...
csbiebt 3909 Conversion of implicit sub...
csbiedf 3910 Conversion of implicit sub...
csbieb 3911 Bidirectional conversion b...
csbiebg 3912 Bidirectional conversion b...
csbiegf 3913 Conversion of implicit sub...
csbief 3914 Conversion of implicit sub...
csbie 3915 Conversion of implicit sub...
csbied 3916 Conversion of implicit sub...
csbied2 3917 Conversion of implicit sub...
csbie2t 3918 Conversion of implicit sub...
csbie2 3919 Conversion of implicit sub...
csbie2g 3920 Conversion of implicit sub...
cbvrabcsfw 3921 Version of ~ cbvrabcsf wit...
cbvralcsf 3922 A more general version of ...
cbvrexcsf 3923 A more general version of ...
cbvreucsf 3924 A more general version of ...
cbvrabcsf 3925 A more general version of ...
cbvralv2 3926 Rule used to change the bo...
cbvrexv2 3927 Rule used to change the bo...
vtocl2dOLD 3928 Obsolete version of ~ vtoc...
rspc2vd 3929 Deduction version of 2-var...
difjust 3935 Soundness justification th...
unjust 3937 Soundness justification th...
injust 3939 Soundness justification th...
dfin5 3941 Alternate definition for t...
dfdif2 3942 Alternate definition of cl...
eldif 3943 Expansion of membership in...
eldifd 3944 If a class is in one class...
eldifad 3945 If a class is in the diffe...
eldifbd 3946 If a class is in the diffe...
elneeldif 3947 The elements of a set diff...
velcomp 3948 Characterization of setvar...
dfss 3950 Variant of subclass defini...
dfss2 3952 Alternate definition of th...
dfss3 3953 Alternate definition of su...
dfss6 3954 Alternate definition of su...
dfss2f 3955 Equivalence for subclass r...
dfss3f 3956 Equivalence for subclass r...
nfss 3957 If ` x ` is not free in ` ...
ssel 3958 Membership relationships f...
ssel2 3959 Membership relationships f...
sseli 3960 Membership implication fro...
sselii 3961 Membership inference from ...
sseldi 3962 Membership inference from ...
sseld 3963 Membership deduction from ...
sselda 3964 Membership deduction from ...
sseldd 3965 Membership inference from ...
ssneld 3966 If a class is not in anoth...
ssneldd 3967 If an element is not in a ...
ssriv 3968 Inference based on subclas...
ssrd 3969 Deduction based on subclas...
ssrdv 3970 Deduction based on subclas...
sstr2 3971 Transitivity of subclass r...
sstr 3972 Transitivity of subclass r...
sstri 3973 Subclass transitivity infe...
sstrd 3974 Subclass transitivity dedu...
sstrid 3975 Subclass transitivity dedu...
sstrdi 3976 Subclass transitivity dedu...
sylan9ss 3977 A subclass transitivity de...
sylan9ssr 3978 A subclass transitivity de...
eqss 3979 The subclass relationship ...
eqssi 3980 Infer equality from two su...
eqssd 3981 Equality deduction from tw...
sssseq 3982 If a class is a subclass o...
eqrd 3983 Deduce equality of classes...
eqri 3984 Infer equality of classes ...
eqelssd 3985 Equality deduction from su...
ssid 3986 Any class is a subclass of...
ssidd 3987 Weakening of ~ ssid . (Co...
ssv 3988 Any class is a subclass of...
sseq1 3989 Equality theorem for subcl...
sseq2 3990 Equality theorem for the s...
sseq12 3991 Equality theorem for the s...
sseq1i 3992 An equality inference for ...
sseq2i 3993 An equality inference for ...
sseq12i 3994 An equality inference for ...
sseq1d 3995 An equality deduction for ...
sseq2d 3996 An equality deduction for ...
sseq12d 3997 An equality deduction for ...
eqsstri 3998 Substitution of equality i...
eqsstrri 3999 Substitution of equality i...
sseqtri 4000 Substitution of equality i...
sseqtrri 4001 Substitution of equality i...
eqsstrd 4002 Substitution of equality i...
eqsstrrd 4003 Substitution of equality i...
sseqtrd 4004 Substitution of equality i...
sseqtrrd 4005 Substitution of equality i...
3sstr3i 4006 Substitution of equality i...
3sstr4i 4007 Substitution of equality i...
3sstr3g 4008 Substitution of equality i...
3sstr4g 4009 Substitution of equality i...
3sstr3d 4010 Substitution of equality i...
3sstr4d 4011 Substitution of equality i...
eqsstrid 4012 A chained subclass and equ...
eqsstrrid 4013 A chained subclass and equ...
sseqtrdi 4014 A chained subclass and equ...
sseqtrrdi 4015 A chained subclass and equ...
sseqtrid 4016 Subclass transitivity dedu...
sseqtrrid 4017 Subclass transitivity dedu...
eqsstrdi 4018 A chained subclass and equ...
eqsstrrdi 4019 A chained subclass and equ...
eqimss 4020 Equality implies the subcl...
eqimss2 4021 Equality implies the subcl...
eqimssi 4022 Infer subclass relationshi...
eqimss2i 4023 Infer subclass relationshi...
nssne1 4024 Two classes are different ...
nssne2 4025 Two classes are different ...
nss 4026 Negation of subclass relat...
nelss 4027 Demonstrate by witnesses t...
ssrexf 4028 Restricted existential qua...
ssrmof 4029 "At most one" existential ...
ssralv 4030 Quantification restricted ...
ssrexv 4031 Existential quantification...
ss2ralv 4032 Two quantifications restri...
ss2rexv 4033 Two existential quantifica...
ralss 4034 Restricted universal quant...
rexss 4035 Restricted existential qua...
ss2ab 4036 Class abstractions in a su...
abss 4037 Class abstraction in a sub...
ssab 4038 Subclass of a class abstra...
ssabral 4039 The relation for a subclas...
ss2abi 4040 Inference of abstraction s...
ss2abdv 4041 Deduction of abstraction s...
abssdv 4042 Deduction of abstraction s...
abssi 4043 Inference of abstraction s...
ss2rab 4044 Restricted abstraction cla...
rabss 4045 Restricted class abstracti...
ssrab 4046 Subclass of a restricted c...
ssrabdv 4047 Subclass of a restricted c...
rabssdv 4048 Subclass of a restricted c...
ss2rabdv 4049 Deduction of restricted ab...
ss2rabi 4050 Inference of restricted ab...
rabss2 4051 Subclass law for restricte...
ssab2 4052 Subclass relation for the ...
ssrab2 4053 Subclass relation for a re...
ssrab3 4054 Subclass relation for a re...
rabssrabd 4055 Subclass of a restricted c...
ssrabeq 4056 If the restricting class o...
rabssab 4057 A restricted class is a su...
uniiunlem 4058 A subset relationship usef...
dfpss2 4059 Alternate definition of pr...
dfpss3 4060 Alternate definition of pr...
psseq1 4061 Equality theorem for prope...
psseq2 4062 Equality theorem for prope...
psseq1i 4063 An equality inference for ...
psseq2i 4064 An equality inference for ...
psseq12i 4065 An equality inference for ...
psseq1d 4066 An equality deduction for ...
psseq2d 4067 An equality deduction for ...
psseq12d 4068 An equality deduction for ...
pssss 4069 A proper subclass is a sub...
pssne 4070 Two classes in a proper su...
pssssd 4071 Deduce subclass from prope...
pssned 4072 Proper subclasses are uneq...
sspss 4073 Subclass in terms of prope...
pssirr 4074 Proper subclass is irrefle...
pssn2lp 4075 Proper subclass has no 2-c...
sspsstri 4076 Two ways of stating tricho...
ssnpss 4077 Partial trichotomy law for...
psstr 4078 Transitive law for proper ...
sspsstr 4079 Transitive law for subclas...
psssstr 4080 Transitive law for subclas...
psstrd 4081 Proper subclass inclusion ...
sspsstrd 4082 Transitivity involving sub...
psssstrd 4083 Transitivity involving sub...
npss 4084 A class is not a proper su...
ssnelpss 4085 A subclass missing a membe...
ssnelpssd 4086 Subclass inclusion with on...
ssexnelpss 4087 If there is an element of ...
dfdif3 4088 Alternate definition of cl...
difeq1 4089 Equality theorem for class...
difeq2 4090 Equality theorem for class...
difeq12 4091 Equality theorem for class...
difeq1i 4092 Inference adding differenc...
difeq2i 4093 Inference adding differenc...
difeq12i 4094 Equality inference for cla...
difeq1d 4095 Deduction adding differenc...
difeq2d 4096 Deduction adding differenc...
difeq12d 4097 Equality deduction for cla...
difeqri 4098 Inference from membership ...
nfdif 4099 Bound-variable hypothesis ...
eldifi 4100 Implication of membership ...
eldifn 4101 Implication of membership ...
elndif 4102 A set does not belong to a...
neldif 4103 Implication of membership ...
difdif 4104 Double class difference. ...
difss 4105 Subclass relationship for ...
difssd 4106 A difference of two classe...
difss2 4107 If a class is contained in...
difss2d 4108 If a class is contained in...
ssdifss 4109 Preservation of a subclass...
ddif 4110 Double complement under un...
ssconb 4111 Contraposition law for sub...
sscon 4112 Contraposition law for sub...
ssdif 4113 Difference law for subsets...
ssdifd 4114 If ` A ` is contained in `...
sscond 4115 If ` A ` is contained in `...
ssdifssd 4116 If ` A ` is contained in `...
ssdif2d 4117 If ` A ` is contained in `...
raldifb 4118 Restricted universal quant...
rexdifi 4119 Restricted existential qua...
complss 4120 Complementation reverses i...
compleq 4121 Two classes are equal if a...
elun 4122 Expansion of membership in...
elunnel1 4123 A member of a union that i...
uneqri 4124 Inference from membership ...
unidm 4125 Idempotent law for union o...
uncom 4126 Commutative law for union ...
equncom 4127 If a class equals the unio...
equncomi 4128 Inference form of ~ equnco...
uneq1 4129 Equality theorem for the u...
uneq2 4130 Equality theorem for the u...
uneq12 4131 Equality theorem for the u...
uneq1i 4132 Inference adding union to ...
uneq2i 4133 Inference adding union to ...
uneq12i 4134 Equality inference for the...
uneq1d 4135 Deduction adding union to ...
uneq2d 4136 Deduction adding union to ...
uneq12d 4137 Equality deduction for the...
nfun 4138 Bound-variable hypothesis ...
unass 4139 Associative law for union ...
un12 4140 A rearrangement of union. ...
un23 4141 A rearrangement of union. ...
un4 4142 A rearrangement of the uni...
unundi 4143 Union distributes over its...
unundir 4144 Union distributes over its...
ssun1 4145 Subclass relationship for ...
ssun2 4146 Subclass relationship for ...
ssun3 4147 Subclass law for union of ...
ssun4 4148 Subclass law for union of ...
elun1 4149 Membership law for union o...
elun2 4150 Membership law for union o...
elunant 4151 A statement is true for ev...
unss1 4152 Subclass law for union of ...
ssequn1 4153 A relationship between sub...
unss2 4154 Subclass law for union of ...
unss12 4155 Subclass law for union of ...
ssequn2 4156 A relationship between sub...
unss 4157 The union of two subclasse...
unssi 4158 An inference showing the u...
unssd 4159 A deduction showing the un...
unssad 4160 If ` ( A u. B ) ` is conta...
unssbd 4161 If ` ( A u. B ) ` is conta...
ssun 4162 A condition that implies i...
rexun 4163 Restricted existential qua...
ralunb 4164 Restricted quantification ...
ralun 4165 Restricted quantification ...
elin 4166 Expansion of membership in...
elini 4167 Membership in an intersect...
elind 4168 Deduce membership in an in...
elinel1 4169 Membership in an intersect...
elinel2 4170 Membership in an intersect...
elin2 4171 Membership in a class defi...
elin1d 4172 Elementhood in the first s...
elin2d 4173 Elementhood in the first s...
elin3 4174 Membership in a class defi...
incom 4175 Commutative law for inters...
incomOLD 4176 Obsolete version of ~ inco...
ineqri 4177 Inference from membership ...
ineq1 4178 Equality theorem for inter...
ineq1OLD 4179 Obsolete version of ~ ineq...
ineq2 4180 Equality theorem for inter...
ineq12 4181 Equality theorem for inter...
ineq1i 4182 Equality inference for int...
ineq2i 4183 Equality inference for int...
ineq12i 4184 Equality inference for int...
ineq1d 4185 Equality deduction for int...
ineq2d 4186 Equality deduction for int...
ineq12d 4187 Equality deduction for int...
ineqan12d 4188 Equality deduction for int...
sseqin2 4189 A relationship between sub...
nfin 4190 Bound-variable hypothesis ...
rabbi2dva 4191 Deduction from a wff to a ...
inidm 4192 Idempotent law for interse...
inass 4193 Associative law for inters...
in12 4194 A rearrangement of interse...
in32 4195 A rearrangement of interse...
in13 4196 A rearrangement of interse...
in31 4197 A rearrangement of interse...
inrot 4198 Rotate the intersection of...
in4 4199 Rearrangement of intersect...
inindi 4200 Intersection distributes o...
inindir 4201 Intersection distributes o...
inss1 4202 The intersection of two cl...
inss2 4203 The intersection of two cl...
ssin 4204 Subclass of intersection. ...
ssini 4205 An inference showing that ...
ssind 4206 A deduction showing that a...
ssrin 4207 Add right intersection to ...
sslin 4208 Add left intersection to s...
ssrind 4209 Add right intersection to ...
ss2in 4210 Intersection of subclasses...
ssinss1 4211 Intersection preserves sub...
inss 4212 Inclusion of an intersecti...
rexin 4213 Restricted existential qua...
dfss7 4214 Alternate definition of su...
symdifcom 4217 Symmetric difference commu...
symdifeq1 4218 Equality theorem for symme...
symdifeq2 4219 Equality theorem for symme...
nfsymdif 4220 Hypothesis builder for sym...
elsymdif 4221 Membership in a symmetric ...
dfsymdif4 4222 Alternate definition of th...
elsymdifxor 4223 Membership in a symmetric ...
dfsymdif2 4224 Alternate definition of th...
symdifass 4225 Symmetric difference is as...
difsssymdif 4226 The symmetric difference c...
difsymssdifssd 4227 If the symmetric differenc...
unabs 4228 Absorption law for union. ...
inabs 4229 Absorption law for interse...
nssinpss 4230 Negation of subclass expre...
nsspssun 4231 Negation of subclass expre...
dfss4 4232 Subclass defined in terms ...
dfun2 4233 An alternate definition of...
dfin2 4234 An alternate definition of...
difin 4235 Difference with intersecti...
ssdifim 4236 Implication of a class dif...
ssdifsym 4237 Symmetric class difference...
dfss5 4238 Alternate definition of su...
dfun3 4239 Union defined in terms of ...
dfin3 4240 Intersection defined in te...
dfin4 4241 Alternate definition of th...
invdif 4242 Intersection with universa...
indif 4243 Intersection with class di...
indif2 4244 Bring an intersection in a...
indif1 4245 Bring an intersection in a...
indifcom 4246 Commutation law for inters...
indi 4247 Distributive law for inter...
undi 4248 Distributive law for union...
indir 4249 Distributive law for inter...
undir 4250 Distributive law for union...
unineq 4251 Infer equality from equali...
uneqin 4252 Equality of union and inte...
difundi 4253 Distributive law for class...
difundir 4254 Distributive law for class...
difindi 4255 Distributive law for class...
difindir 4256 Distributive law for class...
indifdir 4257 Distribute intersection ov...
difdif2 4258 Class difference by a clas...
undm 4259 De Morgan's law for union....
indm 4260 De Morgan's law for inters...
difun1 4261 A relationship involving d...
undif3 4262 An equality involving clas...
difin2 4263 Represent a class differen...
dif32 4264 Swap second and third argu...
difabs 4265 Absorption-like law for cl...
dfsymdif3 4266 Alternate definition of th...
unab 4267 Union of two class abstrac...
inab 4268 Intersection of two class ...
difab 4269 Difference of two class ab...
notab 4270 A class builder defined by...
unrab 4271 Union of two restricted cl...
inrab 4272 Intersection of two restri...
inrab2 4273 Intersection with a restri...
difrab 4274 Difference of two restrict...
dfrab3 4275 Alternate definition of re...
dfrab2 4276 Alternate definition of re...
notrab 4277 Complementation of restric...
dfrab3ss 4278 Restricted class abstracti...
rabun2 4279 Abstraction restricted to ...
reuss2 4280 Transfer uniqueness to a s...
reuss 4281 Transfer uniqueness to a s...
reuun1 4282 Transfer uniqueness to a s...
reuun2 4283 Transfer uniqueness to a s...
reupick 4284 Restricted uniqueness "pic...
reupick3 4285 Restricted uniqueness "pic...
reupick2 4286 Restricted uniqueness "pic...
euelss 4287 Transfer uniqueness of an ...
dfnul2 4290 Alternate definition of th...
dfnul2OLD 4291 Obsolete version of ~ dfnu...
dfnul3 4292 Alternate definition of th...
noel 4293 The empty set has no eleme...
noelOLD 4294 Obsolete version of ~ noel...
nel02 4295 The empty set has no eleme...
n0i 4296 If a class has elements, t...
ne0i 4297 If a class has elements, t...
ne0d 4298 Deduction form of ~ ne0i ....
n0ii 4299 If a class has elements, t...
ne0ii 4300 If a class has elements, t...
vn0 4301 The universal class is not...
eq0f 4302 A class is equal to the em...
neq0f 4303 A class is not empty if an...
n0f 4304 A class is nonempty if and...
eq0 4305 A class is equal to the em...
neq0 4306 A class is not empty if an...
n0 4307 A class is nonempty if and...
nel0 4308 From the general negation ...
reximdva0 4309 Restricted existence deduc...
rspn0 4310 Specialization for restric...
n0rex 4311 There is an element in a n...
ssn0rex 4312 There is an element in a c...
n0moeu 4313 A case of equivalence of "...
rex0 4314 Vacuous restricted existen...
reu0 4315 Vacuous restricted uniquen...
rmo0 4316 Vacuous restricted at-most...
0el 4317 Membership of the empty se...
n0el 4318 Negated membership of the ...
eqeuel 4319 A condition which implies ...
ssdif0 4320 Subclass expressed in term...
difn0 4321 If the difference of two s...
pssdifn0 4322 A proper subclass has a no...
pssdif 4323 A proper subclass has a no...
ndisj 4324 Express that an intersecti...
difin0ss 4325 Difference, intersection, ...
inssdif0 4326 Intersection, subclass, an...
difid 4327 The difference between a c...
difidALT 4328 Alternate proof of ~ difid...
dif0 4329 The difference between a c...
ab0 4330 The class of sets verifyin...
dfnf5 4331 Characterization of non-fr...
ab0orv 4332 The class builder of a wff...
abn0 4333 Nonempty class abstraction...
rab0 4334 Any restricted class abstr...
rabeq0 4335 Condition for a restricted...
rabn0 4336 Nonempty restricted class ...
rabxm 4337 Law of excluded middle, in...
rabnc 4338 Law of noncontradiction, i...
elneldisj 4339 The set of elements ` s ` ...
elnelun 4340 The union of the set of el...
un0 4341 The union of a class with ...
in0 4342 The intersection of a clas...
0un 4343 The union of the empty set...
0in 4344 The intersection of the em...
inv1 4345 The intersection of a clas...
unv 4346 The union of a class with ...
0ss 4347 The null set is a subset o...
ss0b 4348 Any subset of the empty se...
ss0 4349 Any subset of the empty se...
sseq0 4350 A subclass of an empty cla...
ssn0 4351 A class with a nonempty su...
0dif 4352 The difference between the...
abf 4353 A class builder with a fal...
eq0rdv 4354 Deduction for equality to ...
csbprc 4355 The proper substitution of...
csb0 4356 The proper substitution of...
sbcel12 4357 Distribute proper substitu...
sbceqg 4358 Distribute proper substitu...
sbceqi 4359 Distribution of class subs...
sbcnel12g 4360 Distribute proper substitu...
sbcne12 4361 Distribute proper substitu...
sbcel1g 4362 Move proper substitution i...
sbceq1g 4363 Move proper substitution t...
sbcel2 4364 Move proper substitution i...
sbceq2g 4365 Move proper substitution t...
csbcom 4366 Commutative law for double...
sbcnestgfw 4367 Version of ~ sbcnestgf wit...
csbnestgfw 4368 Version of ~ csbnestgf wit...
sbcnestgw 4369 Version of ~ sbcnestg with...
csbnestgw 4370 Version of ~ csbnestg with...
sbcco3gw 4371 Version of ~ sbcco3g with ...
sbcnestgf 4372 Nest the composition of tw...
csbnestgf 4373 Nest the composition of tw...
sbcnestg 4374 Nest the composition of tw...
csbnestg 4375 Nest the composition of tw...
sbcco3g 4376 Composition of two substit...
csbco3g 4377 Composition of two class s...
csbnest1g 4378 Nest the composition of tw...
csbidm 4379 Idempotent law for class s...
csbvarg 4380 The proper substitution of...
csbvargi 4381 The proper substitution of...
sbccsb 4382 Substitution into a wff ex...
sbccsb2 4383 Substitution into a wff ex...
rspcsbela 4384 Special case related to ~ ...
sbnfc2 4385 Two ways of expressing " `...
csbab 4386 Move substitution into a c...
csbun 4387 Distribution of class subs...
csbin 4388 Distribute proper substitu...
2nreu 4389 If there are two different...
un00 4390 Two classes are empty iff ...
vss 4391 Only the universal class h...
0pss 4392 The null set is a proper s...
npss0 4393 No set is a proper subset ...
pssv 4394 Any non-universal class is...
disj 4395 Two ways of saying that tw...
disjr 4396 Two ways of saying that tw...
disj1 4397 Two ways of saying that tw...
reldisj 4398 Two ways of saying that tw...
disj3 4399 Two ways of saying that tw...
disjne 4400 Members of disjoint sets a...
disjeq0 4401 Two disjoint sets are equa...
disjel 4402 A set can't belong to both...
disj2 4403 Two ways of saying that tw...
disj4 4404 Two ways of saying that tw...
ssdisj 4405 Intersection with a subcla...
disjpss 4406 A class is a proper subset...
undisj1 4407 The union of disjoint clas...
undisj2 4408 The union of disjoint clas...
ssindif0 4409 Subclass expressed in term...
inelcm 4410 The intersection of classe...
minel 4411 A minimum element of a cla...
undif4 4412 Distribute union over diff...
disjssun 4413 Subset relation for disjoi...
vdif0 4414 Universal class equality i...
difrab0eq 4415 If the difference between ...
pssnel 4416 A proper subclass has a me...
disjdif 4417 A class and its relative c...
difin0 4418 The difference of a class ...
unvdif 4419 The union of a class and i...
undif1 4420 Absorption of difference b...
undif2 4421 Absorption of difference b...
undifabs 4422 Absorption of difference b...
inundif 4423 The intersection and class...
disjdif2 4424 The difference of a class ...
difun2 4425 Absorption of union by dif...
undif 4426 Union of complementary par...
ssdifin0 4427 A subset of a difference d...
ssdifeq0 4428 A class is a subclass of i...
ssundif 4429 A condition equivalent to ...
difcom 4430 Swap the arguments of a cl...
pssdifcom1 4431 Two ways to express overla...
pssdifcom2 4432 Two ways to express non-co...
difdifdir 4433 Distributive law for class...
uneqdifeq 4434 Two ways to say that ` A `...
raldifeq 4435 Equality theorem for restr...
r19.2z 4436 Theorem 19.2 of [Margaris]...
r19.2zb 4437 A response to the notion t...
r19.3rz 4438 Restricted quantification ...
r19.28z 4439 Restricted quantifier vers...
r19.3rzv 4440 Restricted quantification ...
r19.9rzv 4441 Restricted quantification ...
r19.28zv 4442 Restricted quantifier vers...
r19.37zv 4443 Restricted quantifier vers...
r19.45zv 4444 Restricted version of Theo...
r19.44zv 4445 Restricted version of Theo...
r19.27z 4446 Restricted quantifier vers...
r19.27zv 4447 Restricted quantifier vers...
r19.36zv 4448 Restricted quantifier vers...
rzal 4449 Vacuous quantification is ...
rexn0 4450 Restricted existential qua...
ralidm 4451 Idempotent law for restric...
ral0 4452 Vacuous universal quantifi...
ralf0 4453 The quantification of a fa...
ralnralall 4454 A contradiction concerning...
falseral0 4455 A false statement can only...
raaan 4456 Rearrange restricted quant...
raaanv 4457 Rearrange restricted quant...
sbss 4458 Set substitution into the ...
sbcssg 4459 Distribute proper substitu...
raaan2 4460 Rearrange restricted quant...
2reu4lem 4461 Lemma for ~ 2reu4 . (Cont...
2reu4 4462 Definition of double restr...
dfif2 4465 An alternate definition of...
dfif6 4466 An alternate definition of...
ifeq1 4467 Equality theorem for condi...
ifeq2 4468 Equality theorem for condi...
iftrue 4469 Value of the conditional o...
iftruei 4470 Inference associated with ...
iftrued 4471 Value of the conditional o...
iffalse 4472 Value of the conditional o...
iffalsei 4473 Inference associated with ...
iffalsed 4474 Value of the conditional o...
ifnefalse 4475 When values are unequal, b...
ifsb 4476 Distribute a function over...
dfif3 4477 Alternate definition of th...
dfif4 4478 Alternate definition of th...
dfif5 4479 Alternate definition of th...
ifeq12 4480 Equality theorem for condi...
ifeq1d 4481 Equality deduction for con...
ifeq2d 4482 Equality deduction for con...
ifeq12d 4483 Equality deduction for con...
ifbi 4484 Equivalence theorem for co...
ifbid 4485 Equivalence deduction for ...
ifbieq1d 4486 Equivalence/equality deduc...
ifbieq2i 4487 Equivalence/equality infer...
ifbieq2d 4488 Equivalence/equality deduc...
ifbieq12i 4489 Equivalence deduction for ...
ifbieq12d 4490 Equivalence deduction for ...
nfifd 4491 Deduction form of ~ nfif ....
nfif 4492 Bound-variable hypothesis ...
ifeq1da 4493 Conditional equality. (Co...
ifeq2da 4494 Conditional equality. (Co...
ifeq12da 4495 Equivalence deduction for ...
ifbieq12d2 4496 Equivalence deduction for ...
ifclda 4497 Conditional closure. (Con...
ifeqda 4498 Separation of the values o...
elimif 4499 Elimination of a condition...
ifbothda 4500 A wff ` th ` containing a ...
ifboth 4501 A wff ` th ` containing a ...
ifid 4502 Identical true and false a...
eqif 4503 Expansion of an equality w...
ifval 4504 Another expression of the ...
elif 4505 Membership in a conditiona...
ifel 4506 Membership of a conditiona...
ifcl 4507 Membership (closure) of a ...
ifcld 4508 Membership (closure) of a ...
ifcli 4509 Inference associated with ...
ifexg 4510 Conditional operator exist...
ifex 4511 Conditional operator exist...
ifeqor 4512 The possible values of a c...
ifnot 4513 Negating the first argumen...
ifan 4514 Rewrite a conjunction in a...
ifor 4515 Rewrite a disjunction in a...
2if2 4516 Resolve two nested conditi...
ifcomnan 4517 Commute the conditions in ...
csbif 4518 Distribute proper substitu...
dedth 4519 Weak deduction theorem tha...
dedth2h 4520 Weak deduction theorem eli...
dedth3h 4521 Weak deduction theorem eli...
dedth4h 4522 Weak deduction theorem eli...
dedth2v 4523 Weak deduction theorem for...
dedth3v 4524 Weak deduction theorem for...
dedth4v 4525 Weak deduction theorem for...
elimhyp 4526 Eliminate a hypothesis con...
elimhyp2v 4527 Eliminate a hypothesis con...
elimhyp3v 4528 Eliminate a hypothesis con...
elimhyp4v 4529 Eliminate a hypothesis con...
elimel 4530 Eliminate a membership hyp...
elimdhyp 4531 Version of ~ elimhyp where...
keephyp 4532 Transform a hypothesis ` p...
keephyp2v 4533 Keep a hypothesis containi...
keephyp3v 4534 Keep a hypothesis containi...
pwjust 4536 Soundness justification th...
pweq 4538 Equality theorem for power...
pweqi 4539 Equality inference for pow...
pweqd 4540 Equality deduction for pow...
elpwg 4541 Membership in a power clas...
elpw 4542 Membership in a power clas...
velpw 4543 Setvar variable membership...
elpwOLD 4544 Obsolete proof of ~ elpw a...
elpwgOLD 4545 Obsolete proof of ~ elpwg ...
elpwd 4546 Membership in a power clas...
elpwi 4547 Subset relation implied by...
elpwb 4548 Characterization of the el...
elpwid 4549 An element of a power clas...
elelpwi 4550 If ` A ` belongs to a part...
nfpw 4551 Bound-variable hypothesis ...
pwidg 4552 A set is an element of its...
pwidb 4553 A class is an element of i...
pwid 4554 A set is a member of its p...
pwss 4555 Subclass relationship for ...
snjust 4556 Soundness justification th...
sneq 4567 Equality theorem for singl...
sneqi 4568 Equality inference for sin...
sneqd 4569 Equality deduction for sin...
dfsn2 4570 Alternate definition of si...
elsng 4571 There is exactly one eleme...
elsn 4572 There is exactly one eleme...
velsn 4573 There is only one element ...
elsni 4574 There is only one element ...
absn 4575 Condition for a class abst...
dfpr2 4576 Alternate definition of un...
dfsn2ALT 4577 Alternate definition of si...
elprg 4578 A member of an unordered p...
elpri 4579 If a class is an element o...
elpr 4580 A member of an unordered p...
elpr2 4581 A member of an unordered p...
nelpr2 4582 If a class is not an eleme...
nelpr1 4583 If a class is not an eleme...
nelpri 4584 If an element doesn't matc...
prneli 4585 If an element doesn't matc...
nelprd 4586 If an element doesn't matc...
eldifpr 4587 Membership in a set with t...
rexdifpr 4588 Restricted existential qua...
snidg 4589 A set is a member of its s...
snidb 4590 A class is a set iff it is...
snid 4591 A set is a member of its s...
vsnid 4592 A setvar variable is a mem...
elsn2g 4593 There is exactly one eleme...
elsn2 4594 There is exactly one eleme...
nelsn 4595 If a class is not equal to...
rabeqsn 4596 Conditions for a restricte...
rabsssn 4597 Conditions for a restricte...
ralsnsg 4598 Substitution expressed in ...
rexsns 4599 Restricted existential qua...
ralsngOLD 4600 Obsolete proof of ~ ralsng...
rexsngOLD 4601 Obsolete proof of ~ rexsng...
rexsngf 4602 Restricted existential qua...
ralsngf 4603 Restricted universal quant...
reusngf 4604 Restricted existential uni...
ralsng 4605 Substitution expressed in ...
rexsng 4606 Restricted existential qua...
reusng 4607 Restricted existential uni...
2ralsng 4608 Substitution expressed in ...
rexreusng 4609 Restricted existential uni...
exsnrex 4610 There is a set being the e...
ralsn 4611 Convert a quantification o...
rexsn 4612 Restricted existential qua...
elpwunsn 4613 Membership in an extension...
eqoreldif 4614 An element of a set is eit...
eltpg 4615 Members of an unordered tr...
eldiftp 4616 Membership in a set with t...
eltpi 4617 A member of an unordered t...
eltp 4618 A member of an unordered t...
dftp2 4619 Alternate definition of un...
nfpr 4620 Bound-variable hypothesis ...
ifpr 4621 Membership of a conditiona...
ralprgf 4622 Convert a restricted unive...
rexprgf 4623 Convert a restricted exist...
ralprg 4624 Convert a restricted unive...
rexprg 4625 Convert a restricted exist...
raltpg 4626 Convert a restricted unive...
rextpg 4627 Convert a restricted exist...
ralpr 4628 Convert a restricted unive...
rexpr 4629 Convert a restricted exist...
reuprg0 4630 Convert a restricted exist...
reuprg 4631 Convert a restricted exist...
reurexprg 4632 Convert a restricted exist...
raltp 4633 Convert a quantification o...
rextp 4634 Convert a quantification o...
nfsn 4635 Bound-variable hypothesis ...
csbsng 4636 Distribute proper substitu...
csbprg 4637 Distribute proper substitu...
elinsn 4638 If the intersection of two...
disjsn 4639 Intersection with the sing...
disjsn2 4640 Two distinct singletons ar...
disjpr2 4641 Two completely distinct un...
disjprsn 4642 The disjoint intersection ...
disjtpsn 4643 The disjoint intersection ...
disjtp2 4644 Two completely distinct un...
snprc 4645 The singleton of a proper ...
snnzb 4646 A singleton is nonempty if...
rmosn 4647 A restricted at-most-one q...
r19.12sn 4648 Special case of ~ r19.12 w...
rabsn 4649 Condition where a restrict...
rabsnifsb 4650 A restricted class abstrac...
rabsnif 4651 A restricted class abstrac...
rabrsn 4652 A restricted class abstrac...
euabsn2 4653 Another way to express exi...
euabsn 4654 Another way to express exi...
reusn 4655 A way to express restricte...
absneu 4656 Restricted existential uni...
rabsneu 4657 Restricted existential uni...
eusn 4658 Two ways to express " ` A ...
rabsnt 4659 Truth implied by equality ...
prcom 4660 Commutative law for unorde...
preq1 4661 Equality theorem for unord...
preq2 4662 Equality theorem for unord...
preq12 4663 Equality theorem for unord...
preq1i 4664 Equality inference for uno...
preq2i 4665 Equality inference for uno...
preq12i 4666 Equality inference for uno...
preq1d 4667 Equality deduction for uno...
preq2d 4668 Equality deduction for uno...
preq12d 4669 Equality deduction for uno...
tpeq1 4670 Equality theorem for unord...
tpeq2 4671 Equality theorem for unord...
tpeq3 4672 Equality theorem for unord...
tpeq1d 4673 Equality theorem for unord...
tpeq2d 4674 Equality theorem for unord...
tpeq3d 4675 Equality theorem for unord...
tpeq123d 4676 Equality theorem for unord...
tprot 4677 Rotation of the elements o...
tpcoma 4678 Swap 1st and 2nd members o...
tpcomb 4679 Swap 2nd and 3rd members o...
tpass 4680 Split off the first elemen...
qdass 4681 Two ways to write an unord...
qdassr 4682 Two ways to write an unord...
tpidm12 4683 Unordered triple ` { A , A...
tpidm13 4684 Unordered triple ` { A , B...
tpidm23 4685 Unordered triple ` { A , B...
tpidm 4686 Unordered triple ` { A , A...
tppreq3 4687 An unordered triple is an ...
prid1g 4688 An unordered pair contains...
prid2g 4689 An unordered pair contains...
prid1 4690 An unordered pair contains...
prid2 4691 An unordered pair contains...
ifpprsnss 4692 An unordered pair is a sin...
prprc1 4693 A proper class vanishes in...
prprc2 4694 A proper class vanishes in...
prprc 4695 An unordered pair containi...
tpid1 4696 One of the three elements ...
tpid1g 4697 Closed theorem form of ~ t...
tpid2 4698 One of the three elements ...
tpid2g 4699 Closed theorem form of ~ t...
tpid3g 4700 Closed theorem form of ~ t...
tpid3 4701 One of the three elements ...
snnzg 4702 The singleton of a set is ...
snnz 4703 The singleton of a set is ...
prnz 4704 A pair containing a set is...
prnzg 4705 A pair containing a set is...
tpnz 4706 A triplet containing a set...
tpnzd 4707 A triplet containing a set...
raltpd 4708 Convert a quantification o...
snssg 4709 The singleton of an elemen...
snss 4710 The singleton of an elemen...
eldifsn 4711 Membership in a set with a...
ssdifsn 4712 Subset of a set with an el...
elpwdifsn 4713 A subset of a set is an el...
eldifsni 4714 Membership in a set with a...
eldifsnneq 4715 An element of a difference...
eldifsnneqOLD 4716 Obsolete version of ~ eldi...
neldifsn 4717 The class ` A ` is not in ...
neldifsnd 4718 The class ` A ` is not in ...
rexdifsn 4719 Restricted existential qua...
raldifsni 4720 Rearrangement of a propert...
raldifsnb 4721 Restricted universal quant...
eldifvsn 4722 A set is an element of the...
difsn 4723 An element not in a set ca...
difprsnss 4724 Removal of a singleton fro...
difprsn1 4725 Removal of a singleton fro...
difprsn2 4726 Removal of a singleton fro...
diftpsn3 4727 Removal of a singleton fro...
difpr 4728 Removing two elements as p...
tpprceq3 4729 An unordered triple is an ...
tppreqb 4730 An unordered triple is an ...
difsnb 4731 ` ( B \ { A } ) ` equals `...
difsnpss 4732 ` ( B \ { A } ) ` is a pro...
snssi 4733 The singleton of an elemen...
snssd 4734 The singleton of an elemen...
difsnid 4735 If we remove a single elem...
eldifeldifsn 4736 An element of a difference...
pw0 4737 Compute the power set of t...
pwpw0 4738 Compute the power set of t...
snsspr1 4739 A singleton is a subset of...
snsspr2 4740 A singleton is a subset of...
snsstp1 4741 A singleton is a subset of...
snsstp2 4742 A singleton is a subset of...
snsstp3 4743 A singleton is a subset of...
prssg 4744 A pair of elements of a cl...
prss 4745 A pair of elements of a cl...
prssi 4746 A pair of elements of a cl...
prssd 4747 Deduction version of ~ prs...
prsspwg 4748 An unordered pair belongs ...
ssprss 4749 A pair as subset of a pair...
ssprsseq 4750 A proper pair is a subset ...
sssn 4751 The subsets of a singleton...
ssunsn2 4752 The property of being sand...
ssunsn 4753 Possible values for a set ...
eqsn 4754 Two ways to express that a...
issn 4755 A sufficient condition for...
n0snor2el 4756 A nonempty set is either a...
ssunpr 4757 Possible values for a set ...
sspr 4758 The subsets of a pair. (C...
sstp 4759 The subsets of a triple. ...
tpss 4760 A triplet of elements of a...
tpssi 4761 A triple of elements of a ...
sneqrg 4762 Closed form of ~ sneqr . ...
sneqr 4763 If the singletons of two s...
snsssn 4764 If a singleton is a subset...
mosneq 4765 There exists at most one s...
sneqbg 4766 Two singletons of sets are...
snsspw 4767 The singleton of a class i...
prsspw 4768 An unordered pair belongs ...
preq1b 4769 Biconditional equality lem...
preq2b 4770 Biconditional equality lem...
preqr1 4771 Reverse equality lemma for...
preqr2 4772 Reverse equality lemma for...
preq12b 4773 Equality relationship for ...
opthpr 4774 An unordered pair has the ...
preqr1g 4775 Reverse equality lemma for...
preq12bg 4776 Closed form of ~ preq12b ....
prneimg 4777 Two pairs are not equal if...
prnebg 4778 A (proper) pair is not equ...
pr1eqbg 4779 A (proper) pair is equal t...
pr1nebg 4780 A (proper) pair is not equ...
preqsnd 4781 Equivalence for a pair equ...
prnesn 4782 A proper unordered pair is...
prneprprc 4783 A proper unordered pair is...
preqsn 4784 Equivalence for a pair equ...
preq12nebg 4785 Equality relationship for ...
prel12g 4786 Equality of two unordered ...
opthprneg 4787 An unordered pair has the ...
elpreqprlem 4788 Lemma for ~ elpreqpr . (C...
elpreqpr 4789 Equality and membership ru...
elpreqprb 4790 A set is an element of an ...
elpr2elpr 4791 For an element ` A ` of an...
dfopif 4792 Rewrite ~ df-op using ` if...
dfopg 4793 Value of the ordered pair ...
dfop 4794 Value of an ordered pair w...
opeq1 4795 Equality theorem for order...
opeq2 4796 Equality theorem for order...
opeq12 4797 Equality theorem for order...
opeq1i 4798 Equality inference for ord...
opeq2i 4799 Equality inference for ord...
opeq12i 4800 Equality inference for ord...
opeq1d 4801 Equality deduction for ord...
opeq2d 4802 Equality deduction for ord...
opeq12d 4803 Equality deduction for ord...
oteq1 4804 Equality theorem for order...
oteq2 4805 Equality theorem for order...
oteq3 4806 Equality theorem for order...
oteq1d 4807 Equality deduction for ord...
oteq2d 4808 Equality deduction for ord...
oteq3d 4809 Equality deduction for ord...
oteq123d 4810 Equality deduction for ord...
nfop 4811 Bound-variable hypothesis ...
nfopd 4812 Deduction version of bound...
csbopg 4813 Distribution of class subs...
opidg 4814 The ordered pair ` <. A , ...
opid 4815 The ordered pair ` <. A , ...
ralunsn 4816 Restricted quantification ...
2ralunsn 4817 Double restricted quantifi...
opprc 4818 Expansion of an ordered pa...
opprc1 4819 Expansion of an ordered pa...
opprc2 4820 Expansion of an ordered pa...
oprcl 4821 If an ordered pair has an ...
pwsn 4822 The power set of a singlet...
pwsnALT 4823 Alternate proof of ~ pwsn ...
pwpr 4824 The power set of an unorde...
pwtp 4825 The power set of an unorde...
pwpwpw0 4826 Compute the power set of t...
pwv 4827 The power class of the uni...
prproe 4828 For an element of a proper...
3elpr2eq 4829 If there are three element...
dfuni2 4832 Alternate definition of cl...
eluni 4833 Membership in class union....
eluni2 4834 Membership in class union....
elunii 4835 Membership in class union....
nfunid 4836 Deduction version of ~ nfu...
nfuni 4837 Bound-variable hypothesis ...
unieq 4838 Equality theorem for class...
unieqi 4839 Inference of equality of t...
unieqd 4840 Deduction of equality of t...
eluniab 4841 Membership in union of a c...
elunirab 4842 Membership in union of a c...
unipr 4843 The union of a pair is the...
uniprg 4844 The union of a pair is the...
unisng 4845 A set equals the union of ...
unisn 4846 A set equals the union of ...
unisn3 4847 Union of a singleton in th...
dfnfc2 4848 An alternative statement o...
uniun 4849 The class union of the uni...
uniin 4850 The class union of the int...
uniss 4851 Subclass relationship for ...
ssuni 4852 Subclass relationship for ...
unissi 4853 Subclass relationship for ...
unissd 4854 Subclass relationship for ...
uni0b 4855 The union of a set is empt...
uni0c 4856 The union of a set is empt...
uni0 4857 The union of the empty set...
csbuni 4858 Distribute proper substitu...
elssuni 4859 An element of a class is a...
unissel 4860 Condition turning a subcla...
unissb 4861 Relationship involving mem...
uniss2 4862 A subclass condition on th...
unidif 4863 If the difference ` A \ B ...
ssunieq 4864 Relationship implying unio...
unimax 4865 Any member of a class is t...
pwuni 4866 A class is a subclass of t...
dfint2 4869 Alternate definition of cl...
inteq 4870 Equality law for intersect...
inteqi 4871 Equality inference for cla...
inteqd 4872 Equality deduction for cla...
elint 4873 Membership in class inters...
elint2 4874 Membership in class inters...
elintg 4875 Membership in class inters...
elinti 4876 Membership in class inters...
nfint 4877 Bound-variable hypothesis ...
elintab 4878 Membership in the intersec...
elintrab 4879 Membership in the intersec...
elintrabg 4880 Membership in the intersec...
int0 4881 The intersection of the em...
intss1 4882 An element of a class incl...
ssint 4883 Subclass of a class inters...
ssintab 4884 Subclass of the intersecti...
ssintub 4885 Subclass of the least uppe...
ssmin 4886 Subclass of the minimum va...
intmin 4887 Any member of a class is t...
intss 4888 Intersection of subclasses...
intssuni 4889 The intersection of a none...
ssintrab 4890 Subclass of the intersecti...
unissint 4891 If the union of a class is...
intssuni2 4892 Subclass relationship for ...
intminss 4893 Under subset ordering, the...
intmin2 4894 Any set is the smallest of...
intmin3 4895 Under subset ordering, the...
intmin4 4896 Elimination of a conjunct ...
intab 4897 The intersection of a spec...
int0el 4898 The intersection of a clas...
intun 4899 The class intersection of ...
intpr 4900 The intersection of a pair...
intprg 4901 The intersection of a pair...
intsng 4902 Intersection of a singleto...
intsn 4903 The intersection of a sing...
uniintsn 4904 Two ways to express " ` A ...
uniintab 4905 The union and the intersec...
intunsn 4906 Theorem joining a singleto...
rint0 4907 Relative intersection of a...
elrint 4908 Membership in a restricted...
elrint2 4909 Membership in a restricted...
eliun 4914 Membership in indexed unio...
eliin 4915 Membership in indexed inte...
eliuni 4916 Membership in an indexed u...
iuncom 4917 Commutation of indexed uni...
iuncom4 4918 Commutation of union with ...
iunconst 4919 Indexed union of a constan...
iinconst 4920 Indexed intersection of a ...
iuneqconst 4921 Indexed union of identical...
iuniin 4922 Law combining indexed unio...
iinssiun 4923 An indexed intersection is...
iunss1 4924 Subclass theorem for index...
iinss1 4925 Subclass theorem for index...
iuneq1 4926 Equality theorem for index...
iineq1 4927 Equality theorem for index...
ss2iun 4928 Subclass theorem for index...
iuneq2 4929 Equality theorem for index...
iineq2 4930 Equality theorem for index...
iuneq2i 4931 Equality inference for ind...
iineq2i 4932 Equality inference for ind...
iineq2d 4933 Equality deduction for ind...
iuneq2dv 4934 Equality deduction for ind...
iineq2dv 4935 Equality deduction for ind...
iuneq12df 4936 Equality deduction for ind...
iuneq1d 4937 Equality theorem for index...
iuneq12d 4938 Equality deduction for ind...
iuneq2d 4939 Equality deduction for ind...
nfiun 4940 Bound-variable hypothesis ...
nfiin 4941 Bound-variable hypothesis ...
nfiung 4942 Bound-variable hypothesis ...
nfiing 4943 Bound-variable hypothesis ...
nfiu1 4944 Bound-variable hypothesis ...
nfii1 4945 Bound-variable hypothesis ...
dfiun2g 4946 Alternate definition of in...
dfiun2gOLD 4947 Obsolete proof of ~ dfiun2...
dfiin2g 4948 Alternate definition of in...
dfiun2 4949 Alternate definition of in...
dfiin2 4950 Alternate definition of in...
dfiunv2 4951 Define double indexed unio...
cbviun 4952 Rule used to change the bo...
cbviin 4953 Change bound variables in ...
cbviung 4954 Rule used to change the bo...
cbviing 4955 Change bound variables in ...
cbviunv 4956 Rule used to change the bo...
cbviinv 4957 Change bound variables in ...
cbviunvg 4958 Rule used to change the bo...
cbviinvg 4959 Change bound variables in ...
iunss 4960 Subset theorem for an inde...
ssiun 4961 Subset implication for an ...
ssiun2 4962 Identity law for subset of...
ssiun2s 4963 Subset relationship for an...
iunss2 4964 A subclass condition on th...
iunssd 4965 Subset theorem for an inde...
iunab 4966 The indexed union of a cla...
iunrab 4967 The indexed union of a res...
iunxdif2 4968 Indexed union with a class...
ssiinf 4969 Subset theorem for an inde...
ssiin 4970 Subset theorem for an inde...
iinss 4971 Subset implication for an ...
iinss2 4972 An indexed intersection is...
uniiun 4973 Class union in terms of in...
intiin 4974 Class intersection in term...
iunid 4975 An indexed union of single...
iun0 4976 An indexed union of the em...
0iun 4977 An empty indexed union is ...
0iin 4978 An empty indexed intersect...
viin 4979 Indexed intersection with ...
iunn0 4980 There is a nonempty class ...
iinab 4981 Indexed intersection of a ...
iinrab 4982 Indexed intersection of a ...
iinrab2 4983 Indexed intersection of a ...
iunin2 4984 Indexed union of intersect...
iunin1 4985 Indexed union of intersect...
iinun2 4986 Indexed intersection of un...
iundif2 4987 Indexed union of class dif...
iindif1 4988 Indexed intersection of cl...
2iunin 4989 Rearrange indexed unions o...
iindif2 4990 Indexed intersection of cl...
iinin2 4991 Indexed intersection of in...
iinin1 4992 Indexed intersection of in...
iinvdif 4993 The indexed intersection o...
elriin 4994 Elementhood in a relative ...
riin0 4995 Relative intersection of a...
riinn0 4996 Relative intersection of a...
riinrab 4997 Relative intersection of a...
symdif0 4998 Symmetric difference with ...
symdifv 4999 The symmetric difference w...
symdifid 5000 The symmetric difference o...
iinxsng 5001 A singleton index picks ou...
iinxprg 5002 Indexed intersection with ...
iunxsng 5003 A singleton index picks ou...
iunxsn 5004 A singleton index picks ou...
iunxsngf 5005 A singleton index picks ou...
iunun 5006 Separate a union in an ind...
iunxun 5007 Separate a union in the in...
iunxdif3 5008 An indexed union where som...
iunxprg 5009 A pair index picks out two...
iunxiun 5010 Separate an indexed union ...
iinuni 5011 A relationship involving u...
iununi 5012 A relationship involving u...
sspwuni 5013 Subclass relationship for ...
pwssb 5014 Two ways to express a coll...
elpwpw 5015 Characterization of the el...
pwpwab 5016 The double power class wri...
pwpwssunieq 5017 The class of sets whose un...
elpwuni 5018 Relationship for power cla...
iinpw 5019 The power class of an inte...
iunpwss 5020 Inclusion of an indexed un...
rintn0 5021 Relative intersection of a...
dfdisj2 5024 Alternate definition for d...
disjss2 5025 If each element of a colle...
disjeq2 5026 Equality theorem for disjo...
disjeq2dv 5027 Equality deduction for dis...
disjss1 5028 A subset of a disjoint col...
disjeq1 5029 Equality theorem for disjo...
disjeq1d 5030 Equality theorem for disjo...
disjeq12d 5031 Equality theorem for disjo...
cbvdisj 5032 Change bound variables in ...
cbvdisjv 5033 Change bound variables in ...
nfdisjw 5034 Version of ~ nfdisj with a...
nfdisj 5035 Bound-variable hypothesis ...
nfdisj1 5036 Bound-variable hypothesis ...
disjor 5037 Two ways to say that a col...
disjors 5038 Two ways to say that a col...
disji2 5039 Property of a disjoint col...
disji 5040 Property of a disjoint col...
invdisj 5041 If there is a function ` C...
invdisjrabw 5042 Version of ~ invdisjrab wi...
invdisjrab 5043 The restricted class abstr...
disjiun 5044 A disjoint collection yiel...
disjord 5045 Conditions for a collectio...
disjiunb 5046 Two ways to say that a col...
disjiund 5047 Conditions for a collectio...
sndisj 5048 Any collection of singleto...
0disj 5049 Any collection of empty se...
disjxsn 5050 A singleton collection is ...
disjx0 5051 An empty collection is dis...
disjprgw 5052 Version of ~ disjprg with ...
disjprg 5053 A pair collection is disjo...
disjxiun 5054 An indexed union of a disj...
disjxun 5055 The union of two disjoint ...
disjss3 5056 Expand a disjoint collecti...
breq 5059 Equality theorem for binar...
breq1 5060 Equality theorem for a bin...
breq2 5061 Equality theorem for a bin...
breq12 5062 Equality theorem for a bin...
breqi 5063 Equality inference for bin...
breq1i 5064 Equality inference for a b...
breq2i 5065 Equality inference for a b...
breq12i 5066 Equality inference for a b...
breq1d 5067 Equality deduction for a b...
breqd 5068 Equality deduction for a b...
breq2d 5069 Equality deduction for a b...
breq12d 5070 Equality deduction for a b...
breq123d 5071 Equality deduction for a b...
breqdi 5072 Equality deduction for a b...
breqan12d 5073 Equality deduction for a b...
breqan12rd 5074 Equality deduction for a b...
eqnbrtrd 5075 Substitution of equal clas...
nbrne1 5076 Two classes are different ...
nbrne2 5077 Two classes are different ...
eqbrtri 5078 Substitution of equal clas...
eqbrtrd 5079 Substitution of equal clas...
eqbrtrri 5080 Substitution of equal clas...
eqbrtrrd 5081 Substitution of equal clas...
breqtri 5082 Substitution of equal clas...
breqtrd 5083 Substitution of equal clas...
breqtrri 5084 Substitution of equal clas...
breqtrrd 5085 Substitution of equal clas...
3brtr3i 5086 Substitution of equality i...
3brtr4i 5087 Substitution of equality i...
3brtr3d 5088 Substitution of equality i...
3brtr4d 5089 Substitution of equality i...
3brtr3g 5090 Substitution of equality i...
3brtr4g 5091 Substitution of equality i...
eqbrtrid 5092 A chained equality inferen...
eqbrtrrid 5093 A chained equality inferen...
breqtrid 5094 A chained equality inferen...
breqtrrid 5095 A chained equality inferen...
eqbrtrdi 5096 A chained equality inferen...
eqbrtrrdi 5097 A chained equality inferen...
breqtrdi 5098 A chained equality inferen...
breqtrrdi 5099 A chained equality inferen...
ssbrd 5100 Deduction from a subclass ...
ssbr 5101 Implication from a subclas...
ssbri 5102 Inference from a subclass ...
nfbrd 5103 Deduction version of bound...
nfbr 5104 Bound-variable hypothesis ...
brab1 5105 Relationship between a bin...
br0 5106 The empty binary relation ...
brne0 5107 If two sets are in a binar...
brun 5108 The union of two binary re...
brin 5109 The intersection of two re...
brdif 5110 The difference of two bina...
sbcbr123 5111 Move substitution in and o...
sbcbr 5112 Move substitution in and o...
sbcbr12g 5113 Move substitution in and o...
sbcbr1g 5114 Move substitution in and o...
sbcbr2g 5115 Move substitution in and o...
brsymdif 5116 Characterization of the sy...
brralrspcev 5117 Restricted existential spe...
brimralrspcev 5118 Restricted existential spe...
opabss 5121 The collection of ordered ...
opabbid 5122 Equivalent wff's yield equ...
opabbidv 5123 Equivalent wff's yield equ...
opabbii 5124 Equivalent wff's yield equ...
nfopab 5125 Bound-variable hypothesis ...
nfopab1 5126 The first abstraction vari...
nfopab2 5127 The second abstraction var...
cbvopab 5128 Rule used to change bound ...
cbvopabv 5129 Rule used to change bound ...
cbvopab1 5130 Change first bound variabl...
cbvopab1g 5131 Change first bound variabl...
cbvopab2 5132 Change second bound variab...
cbvopab1s 5133 Change first bound variabl...
cbvopab1v 5134 Rule used to change the fi...
cbvopab2v 5135 Rule used to change the se...
unopab 5136 Union of two ordered pair ...
mpteq12df 5139 An equality inference for ...
mpteq12f 5140 An equality theorem for th...
mpteq12dva 5141 An equality inference for ...
mpteq12dv 5142 An equality inference for ...
mpteq12dvOLD 5143 Obsolete version of ~ mpte...
mpteq12 5144 An equality theorem for th...
mpteq1 5145 An equality theorem for th...
mpteq1d 5146 An equality theorem for th...
mpteq1i 5147 An equality theorem for th...
mpteq2ia 5148 An equality inference for ...
mpteq2i 5149 An equality inference for ...
mpteq12i 5150 An equality inference for ...
mpteq2da 5151 Slightly more general equa...
mpteq2dva 5152 Slightly more general equa...
mpteq2dv 5153 An equality inference for ...
nfmpt 5154 Bound-variable hypothesis ...
nfmpt1 5155 Bound-variable hypothesis ...
cbvmptf 5156 Rule to change the bound v...
cbvmptfg 5157 Rule to change the bound v...
cbvmpt 5158 Rule to change the bound v...
cbvmptg 5159 Rule to change the bound v...
cbvmptv 5160 Rule to change the bound v...
cbvmptvg 5161 Rule to change the bound v...
mptv 5162 Function with universal do...
dftr2 5165 An alternate way of defini...
dftr5 5166 An alternate way of defini...
dftr3 5167 An alternate way of defini...
dftr4 5168 An alternate way of defini...
treq 5169 Equality theorem for the t...
trel 5170 In a transitive class, the...
trel3 5171 In a transitive class, the...
trss 5172 An element of a transitive...
trin 5173 The intersection of transi...
tr0 5174 The empty set is transitiv...
trv 5175 The universe is transitive...
triun 5176 An indexed union of a clas...
truni 5177 The union of a class of tr...
triin 5178 An indexed intersection of...
trint 5179 The intersection of a clas...
trintss 5180 Any nonempty transitive cl...
axrep1 5182 The version of the Axiom o...
axreplem 5183 Lemma for ~ axrep2 and ~ a...
axrep2 5184 Axiom of Replacement expre...
axrep3 5185 Axiom of Replacement sligh...
axrep4 5186 A more traditional version...
axrep5 5187 Axiom of Replacement (simi...
axrep6 5188 A condensed form of ~ ax-r...
zfrepclf 5189 An inference based on the ...
zfrep3cl 5190 An inference based on the ...
zfrep4 5191 A version of Replacement u...
axsepgfromrep 5192 A more general version ~ a...
axsep 5193 Axiom scheme of separation...
axsepg 5195 A more general version of ...
zfauscl 5196 Separation Scheme (Aussond...
bm1.3ii 5197 Convert implication to equ...
ax6vsep 5198 Derive ~ ax6v (a weakened ...
axnulALT 5199 Alternate proof of ~ axnul...
axnul 5200 The Null Set Axiom of ZF s...
0ex 5202 The Null Set Axiom of ZF s...
al0ssb 5203 The empty set is the uniqu...
sseliALT 5204 Alternate proof of ~ sseli...
csbexg 5205 The existence of proper su...
csbex 5206 The existence of proper su...
unisn2 5207 A version of ~ unisn witho...
nalset 5208 No set contains all sets. ...
vnex 5209 The universal class does n...
vprc 5210 The universal class is not...
nvel 5211 The universal class does n...
inex1 5212 Separation Scheme (Aussond...
inex2 5213 Separation Scheme (Aussond...
inex1g 5214 Closed-form, generalized S...
inex2g 5215 Sufficient condition for a...
ssex 5216 The subset of a set is als...
ssexi 5217 The subset of a set is als...
ssexg 5218 The subset of a set is als...
ssexd 5219 A subclass of a set is a s...
prcssprc 5220 The superclass of a proper...
sselpwd 5221 Elementhood to a power set...
difexg 5222 Existence of a difference....
difexi 5223 Existence of a difference,...
zfausab 5224 Separation Scheme (Aussond...
rabexg 5225 Separation Scheme in terms...
rabex 5226 Separation Scheme in terms...
rabexd 5227 Separation Scheme in terms...
rabex2 5228 Separation Scheme in terms...
rab2ex 5229 A class abstraction based ...
elssabg 5230 Membership in a class abst...
intex 5231 The intersection of a none...
intnex 5232 If a class intersection is...
intexab 5233 The intersection of a none...
intexrab 5234 The intersection of a none...
iinexg 5235 The existence of a class i...
intabs 5236 Absorption of a redundant ...
inuni 5237 The intersection of a unio...
elpw2g 5238 Membership in a power clas...
elpw2 5239 Membership in a power clas...
elpwi2 5240 Membership in a power clas...
pwnss 5241 The power set of a set is ...
pwne 5242 No set equals its power se...
difelpw 5243 A difference is an element...
rabelpw 5244 A restricted class abstrac...
class2set 5245 Construct, from any class ...
class2seteq 5246 Equality theorem based on ...
0elpw 5247 Every power class contains...
pwne0 5248 A power class is never emp...
0nep0 5249 The empty set and its powe...
0inp0 5250 Something cannot be equal ...
unidif0 5251 The removal of the empty s...
iin0 5252 An indexed intersection of...
notzfaus 5253 In the Separation Scheme ~...
notzfausOLD 5254 Obsolete proof of ~ notzfa...
intv 5255 The intersection of the un...
axpweq 5256 Two equivalent ways to exp...
zfpow 5258 Axiom of Power Sets expres...
axpow2 5259 A variant of the Axiom of ...
axpow3 5260 A variant of the Axiom of ...
el 5261 Every set is an element of...
dtru 5262 At least two sets exist (o...
dtrucor 5263 Corollary of ~ dtru . Thi...
dtrucor2 5264 The theorem form of the de...
dvdemo1 5265 Demonstration of a theorem...
dvdemo2 5266 Demonstration of a theorem...
nfnid 5267 A setvar variable is not f...
nfcvb 5268 The "distinctor" expressio...
vpwex 5269 Power set axiom: the power...
pwexg 5270 Power set axiom expressed ...
pwexd 5271 Deduction version of the p...
pwex 5272 Power set axiom expressed ...
abssexg 5273 Existence of a class of su...
snexALT 5274 Alternate proof of ~ snex ...
p0ex 5275 The power set of the empty...
p0exALT 5276 Alternate proof of ~ p0ex ...
pp0ex 5277 The power set of the power...
ord3ex 5278 The ordinal number 3 is a ...
dtruALT 5279 Alternate proof of ~ dtru ...
axc16b 5280 This theorem shows that ax...
eunex 5281 Existential uniqueness imp...
eusv1 5282 Two ways to express single...
eusvnf 5283 Even if ` x ` is free in `...
eusvnfb 5284 Two ways to say that ` A (...
eusv2i 5285 Two ways to express single...
eusv2nf 5286 Two ways to express single...
eusv2 5287 Two ways to express single...
reusv1 5288 Two ways to express single...
reusv2lem1 5289 Lemma for ~ reusv2 . (Con...
reusv2lem2 5290 Lemma for ~ reusv2 . (Con...
reusv2lem3 5291 Lemma for ~ reusv2 . (Con...
reusv2lem4 5292 Lemma for ~ reusv2 . (Con...
reusv2lem5 5293 Lemma for ~ reusv2 . (Con...
reusv2 5294 Two ways to express single...
reusv3i 5295 Two ways of expressing exi...
reusv3 5296 Two ways to express single...
eusv4 5297 Two ways to express single...
alxfr 5298 Transfer universal quantif...
ralxfrd 5299 Transfer universal quantif...
rexxfrd 5300 Transfer universal quantif...
ralxfr2d 5301 Transfer universal quantif...
rexxfr2d 5302 Transfer universal quantif...
ralxfrd2 5303 Transfer universal quantif...
rexxfrd2 5304 Transfer existence from a ...
ralxfr 5305 Transfer universal quantif...
ralxfrALT 5306 Alternate proof of ~ ralxf...
rexxfr 5307 Transfer existence from a ...
rabxfrd 5308 Class builder membership a...
rabxfr 5309 Class builder membership a...
reuhypd 5310 A theorem useful for elimi...
reuhyp 5311 A theorem useful for elimi...
zfpair 5312 The Axiom of Pairing of Ze...
axprALT 5313 Alternate proof of ~ axpr ...
axprlem1 5314 Lemma for ~ axpr . There ...
axprlem2 5315 Lemma for ~ axpr . There ...
axprlem3 5316 Lemma for ~ axpr . Elimin...
axprlem4 5317 Lemma for ~ axpr . The fi...
axprlem5 5318 Lemma for ~ axpr . The se...
axpr 5319 Unabbreviated version of t...
zfpair2 5321 Derive the abbreviated ver...
snex 5322 A singleton is a set. The...
prex 5323 The Axiom of Pairing using...
sels 5324 If a class is a set, then ...
elALT 5325 Alternate proof of ~ el , ...
dtruALT2 5326 Alternate proof of ~ dtru ...
snelpwi 5327 A singleton of a set belon...
snelpw 5328 A singleton of a set belon...
prelpw 5329 A pair of two sets belongs...
prelpwi 5330 A pair of two sets belongs...
rext 5331 A theorem similar to exten...
sspwb 5332 The powerclass constructio...
unipw 5333 A class equals the union o...
univ 5334 The union of the universe ...
pwel 5335 Membership of a power clas...
pwtr 5336 A class is transitive iff ...
ssextss 5337 An extensionality-like pri...
ssext 5338 An extensionality-like pri...
nssss 5339 Negation of subclass relat...
pweqb 5340 Classes are equal if and o...
intid 5341 The intersection of all se...
moabex 5342 "At most one" existence im...
rmorabex 5343 Restricted "at most one" e...
euabex 5344 The abstraction of a wff w...
nnullss 5345 A nonempty class (even if ...
exss 5346 Restricted existence in a ...
opex 5347 An ordered pair of classes...
otex 5348 An ordered triple of class...
elopg 5349 Characterization of the el...
elop 5350 Characterization of the el...
opi1 5351 One of the two elements in...
opi2 5352 One of the two elements of...
opeluu 5353 Each member of an ordered ...
op1stb 5354 Extract the first member o...
brv 5355 Two classes are always in ...
opnz 5356 An ordered pair is nonempt...
opnzi 5357 An ordered pair is nonempt...
opth1 5358 Equality of the first memb...
opth 5359 The ordered pair theorem. ...
opthg 5360 Ordered pair theorem. ` C ...
opth1g 5361 Equality of the first memb...
opthg2 5362 Ordered pair theorem. (Co...
opth2 5363 Ordered pair theorem. (Co...
opthneg 5364 Two ordered pairs are not ...
opthne 5365 Two ordered pairs are not ...
otth2 5366 Ordered triple theorem, wi...
otth 5367 Ordered triple theorem. (...
otthg 5368 Ordered triple theorem, cl...
eqvinop 5369 A variable introduction la...
sbcop1 5370 The proper substitution of...
sbcop 5371 The proper substitution of...
copsexgw 5372 Version of ~ copsexg with ...
copsexg 5373 Substitution of class ` A ...
copsex2t 5374 Closed theorem form of ~ c...
copsex2g 5375 Implicit substitution infe...
copsex4g 5376 An implicit substitution i...
0nelop 5377 A property of ordered pair...
opwo0id 5378 An ordered pair is equal t...
opeqex 5379 Equivalence of existence i...
oteqex2 5380 Equivalence of existence i...
oteqex 5381 Equivalence of existence i...
opcom 5382 An ordered pair commutes i...
moop2 5383 "At most one" property of ...
opeqsng 5384 Equivalence for an ordered...
opeqsn 5385 Equivalence for an ordered...
opeqpr 5386 Equivalence for an ordered...
snopeqop 5387 Equivalence for an ordered...
propeqop 5388 Equivalence for an ordered...
propssopi 5389 If a pair of ordered pairs...
snopeqopsnid 5390 Equivalence for an ordered...
mosubopt 5391 "At most one" remains true...
mosubop 5392 "At most one" remains true...
euop2 5393 Transfer existential uniqu...
euotd 5394 Prove existential uniquene...
opthwiener 5395 Justification theorem for ...
uniop 5396 The union of an ordered pa...
uniopel 5397 Ordered pair membership is...
opthhausdorff 5398 Justification theorem for ...
opthhausdorff0 5399 Justification theorem for ...
otsndisj 5400 The singletons consisting ...
otiunsndisj 5401 The union of singletons co...
iunopeqop 5402 Implication of an ordered ...
opabidw 5403 Version of ~ opabid with a...
opabid 5404 The law of concretion. Sp...
elopab 5405 Membership in a class abst...
rexopabb 5406 Restricted existential qua...
opelopabsbALT 5407 The law of concretion in t...
opelopabsb 5408 The law of concretion in t...
brabsb 5409 The law of concretion in t...
opelopabt 5410 Closed theorem form of ~ o...
opelopabga 5411 The law of concretion. Th...
brabga 5412 The law of concretion for ...
opelopab2a 5413 Ordered pair membership in...
opelopaba 5414 The law of concretion. Th...
braba 5415 The law of concretion for ...
opelopabg 5416 The law of concretion. Th...
brabg 5417 The law of concretion for ...
opelopabgf 5418 The law of concretion. Th...
opelopab2 5419 Ordered pair membership in...
opelopab 5420 The law of concretion. Th...
brab 5421 The law of concretion for ...
opelopabaf 5422 The law of concretion. Th...
opelopabf 5423 The law of concretion. Th...
ssopab2 5424 Equivalence of ordered pai...
ssopab2bw 5425 Version of ~ ssopab2b with...
eqopab2bw 5426 Version of ~ eqopab2b with...
ssopab2b 5427 Equivalence of ordered pai...
ssopab2i 5428 Inference of ordered pair ...
ssopab2dv 5429 Inference of ordered pair ...
eqopab2b 5430 Equivalence of ordered pai...
opabn0 5431 Nonempty ordered pair clas...
opab0 5432 Empty ordered pair class a...
csbopab 5433 Move substitution into a c...
csbopabgALT 5434 Move substitution into a c...
csbmpt12 5435 Move substitution into a m...
csbmpt2 5436 Move substitution into the...
iunopab 5437 Move indexed union inside ...
elopabr 5438 Membership in a class abst...
elopabran 5439 Membership in a class abst...
rbropapd 5440 Properties of a pair in an...
rbropap 5441 Properties of a pair in a ...
2rbropap 5442 Properties of a pair in a ...
0nelopab 5443 The empty set is never an ...
brabv 5444 If two classes are in a re...
pwin 5445 The power class of the int...
pwunss 5446 The power class of the uni...
pwunssOLD 5447 The power class of the uni...
pwssun 5448 The power class of the uni...
pwundif 5449 Break up the power class o...
pwundifOLD 5450 Obsolete proof of ~ pwundi...
pwun 5451 The power class of the uni...
dfid4 5454 The identity function expr...
dfid3 5455 A stronger version of ~ df...
dfid2 5456 Alternate definition of th...
epelg 5459 The membership relation an...
epelgOLD 5460 Obsolete version of ~ epel...
epeli 5461 The membership relation an...
epel 5462 The membership relation an...
0sn0ep 5463 An example for the members...
epn0 5464 The membership relation is...
poss 5469 Subset theorem for the par...
poeq1 5470 Equality theorem for parti...
poeq2 5471 Equality theorem for parti...
nfpo 5472 Bound-variable hypothesis ...
nfso 5473 Bound-variable hypothesis ...
pocl 5474 Properties of partial orde...
ispod 5475 Sufficient conditions for ...
swopolem 5476 Perform the substitutions ...
swopo 5477 A strict weak order is a p...
poirr 5478 A partial order relation i...
potr 5479 A partial order relation i...
po2nr 5480 A partial order relation h...
po3nr 5481 A partial order relation h...
po2ne 5482 Two classes which are in a...
po0 5483 Any relation is a partial ...
pofun 5484 A function preserves a par...
sopo 5485 A strict linear order is a...
soss 5486 Subset theorem for the str...
soeq1 5487 Equality theorem for the s...
soeq2 5488 Equality theorem for the s...
sonr 5489 A strict order relation is...
sotr 5490 A strict order relation is...
solin 5491 A strict order relation is...
so2nr 5492 A strict order relation ha...
so3nr 5493 A strict order relation ha...
sotric 5494 A strict order relation sa...
sotrieq 5495 Trichotomy law for strict ...
sotrieq2 5496 Trichotomy law for strict ...
soasym 5497 Asymmetry law for strict o...
sotr2 5498 A transitivity relation. ...
issod 5499 An irreflexive, transitive...
issoi 5500 An irreflexive, transitive...
isso2i 5501 Deduce strict ordering fro...
so0 5502 Any relation is a strict o...
somo 5503 A totally ordered set has ...
fri 5510 Property of well-founded r...
seex 5511 The ` R ` -preimage of an ...
exse 5512 Any relation on a set is s...
dffr2 5513 Alternate definition of we...
frc 5514 Property of well-founded r...
frss 5515 Subset theorem for the wel...
sess1 5516 Subset theorem for the set...
sess2 5517 Subset theorem for the set...
freq1 5518 Equality theorem for the w...
freq2 5519 Equality theorem for the w...
seeq1 5520 Equality theorem for the s...
seeq2 5521 Equality theorem for the s...
nffr 5522 Bound-variable hypothesis ...
nfse 5523 Bound-variable hypothesis ...
nfwe 5524 Bound-variable hypothesis ...
frirr 5525 A well-founded relation is...
fr2nr 5526 A well-founded relation ha...
fr0 5527 Any relation is well-found...
frminex 5528 If an element of a well-fo...
efrirr 5529 A well-founded class does ...
efrn2lp 5530 A well-founded class conta...
epse 5531 The membership relation is...
tz7.2 5532 Similar to Theorem 7.2 of ...
dfepfr 5533 An alternate way of saying...
epfrc 5534 A subset of a well-founded...
wess 5535 Subset theorem for the wel...
weeq1 5536 Equality theorem for the w...
weeq2 5537 Equality theorem for the w...
wefr 5538 A well-ordering is well-fo...
weso 5539 A well-ordering is a stric...
wecmpep 5540 The elements of a class we...
wetrep 5541 On a class well-ordered by...
wefrc 5542 A nonempty subclass of a c...
we0 5543 Any relation is a well-ord...
wereu 5544 A subset of a well-ordered...
wereu2 5545 All nonempty subclasses of...
xpeq1 5562 Equality theorem for Carte...
xpss12 5563 Subset theorem for Cartesi...
xpss 5564 A Cartesian product is inc...
inxpssres 5565 Intersection with a Cartes...
relxp 5566 A Cartesian product is a r...
xpss1 5567 Subset relation for Cartes...
xpss2 5568 Subset relation for Cartes...
xpeq2 5569 Equality theorem for Carte...
elxpi 5570 Membership in a Cartesian ...
elxp 5571 Membership in a Cartesian ...
elxp2 5572 Membership in a Cartesian ...
xpeq12 5573 Equality theorem for Carte...
xpeq1i 5574 Equality inference for Car...
xpeq2i 5575 Equality inference for Car...
xpeq12i 5576 Equality inference for Car...
xpeq1d 5577 Equality deduction for Car...
xpeq2d 5578 Equality deduction for Car...
xpeq12d 5579 Equality deduction for Car...
sqxpeqd 5580 Equality deduction for a C...
nfxp 5581 Bound-variable hypothesis ...
0nelxp 5582 The empty set is not a mem...
0nelelxp 5583 A member of a Cartesian pr...
opelxp 5584 Ordered pair membership in...
opelxpi 5585 Ordered pair membership in...
opelxpd 5586 Ordered pair membership in...
opelvv 5587 Ordered pair membership in...
opelvvg 5588 Ordered pair membership in...
opelxp1 5589 The first member of an ord...
opelxp2 5590 The second member of an or...
otelxp1 5591 The first member of an ord...
otel3xp 5592 An ordered triple is an el...
rabxp 5593 Membership in a class buil...
brxp 5594 Binary relation on a Carte...
pwvrel 5595 A set is a binary relation...
pwvabrel 5596 The powerclass of the cart...
brrelex12 5597 Two classes related by a b...
brrelex1 5598 If two classes are related...
brrelex2 5599 If two classes are related...
brrelex12i 5600 Two classes that are relat...
brrelex1i 5601 The first argument of a bi...
brrelex2i 5602 The second argument of a b...
nprrel12 5603 Proper classes are not rel...
nprrel 5604 No proper class is related...
0nelrel0 5605 A binary relation does not...
0nelrel 5606 A binary relation does not...
fconstmpt 5607 Representation of a consta...
vtoclr 5608 Variable to class conversi...
opthprc 5609 Justification theorem for ...
brel 5610 Two things in a binary rel...
elxp3 5611 Membership in a Cartesian ...
opeliunxp 5612 Membership in a union of C...
xpundi 5613 Distributive law for Carte...
xpundir 5614 Distributive law for Carte...
xpiundi 5615 Distributive law for Carte...
xpiundir 5616 Distributive law for Carte...
iunxpconst 5617 Membership in a union of C...
xpun 5618 The Cartesian product of t...
elvv 5619 Membership in universal cl...
elvvv 5620 Membership in universal cl...
elvvuni 5621 An ordered pair contains i...
brinxp2 5622 Intersection with cross pr...
brinxp 5623 Intersection of binary rel...
opelinxp 5624 Ordered pair element in an...
poinxp 5625 Intersection of partial or...
soinxp 5626 Intersection of total orde...
frinxp 5627 Intersection of well-found...
seinxp 5628 Intersection of set-like r...
weinxp 5629 Intersection of well-order...
posn 5630 Partial ordering of a sing...
sosn 5631 Strict ordering on a singl...
frsn 5632 Founded relation on a sing...
wesn 5633 Well-ordering of a singlet...
elopaelxp 5634 Membership in an ordered p...
bropaex12 5635 Two classes related by an ...
opabssxp 5636 An abstraction relation is...
brab2a 5637 The law of concretion for ...
optocl 5638 Implicit substitution of c...
2optocl 5639 Implicit substitution of c...
3optocl 5640 Implicit substitution of c...
opbrop 5641 Ordered pair membership in...
0xp 5642 The Cartesian product with...
csbxp 5643 Distribute proper substitu...
releq 5644 Equality theorem for the r...
releqi 5645 Equality inference for the...
releqd 5646 Equality deduction for the...
nfrel 5647 Bound-variable hypothesis ...
sbcrel 5648 Distribute proper substitu...
relss 5649 Subclass theorem for relat...
ssrel 5650 A subclass relationship de...
eqrel 5651 Extensionality principle f...
ssrel2 5652 A subclass relationship de...
relssi 5653 Inference from subclass pr...
relssdv 5654 Deduction from subclass pr...
eqrelriv 5655 Inference from extensional...
eqrelriiv 5656 Inference from extensional...
eqbrriv 5657 Inference from extensional...
eqrelrdv 5658 Deduce equality of relatio...
eqbrrdv 5659 Deduction from extensional...
eqbrrdiv 5660 Deduction from extensional...
eqrelrdv2 5661 A version of ~ eqrelrdv . ...
ssrelrel 5662 A subclass relationship de...
eqrelrel 5663 Extensionality principle f...
elrel 5664 A member of a relation is ...
rel0 5665 The empty set is a relatio...
nrelv 5666 The universal class is not...
relsng 5667 A singleton is a relation ...
relsnb 5668 An at-most-singleton is a ...
relsnopg 5669 A singleton of an ordered ...
relsn 5670 A singleton is a relation ...
relsnop 5671 A singleton of an ordered ...
copsex2gb 5672 Implicit substitution infe...
copsex2ga 5673 Implicit substitution infe...
elopaba 5674 Membership in an ordered p...
xpsspw 5675 A Cartesian product is inc...
unixpss 5676 The double class union of ...
relun 5677 The union of two relations...
relin1 5678 The intersection with a re...
relin2 5679 The intersection with a re...
relinxp 5680 Intersection with a Cartes...
reldif 5681 A difference cutting down ...
reliun 5682 An indexed union is a rela...
reliin 5683 An indexed intersection is...
reluni 5684 The union of a class is a ...
relint 5685 The intersection of a clas...
relopabiv 5686 A class of ordered pairs i...
relopabi 5687 A class of ordered pairs i...
relopabiALT 5688 Alternate proof of ~ relop...
relopab 5689 A class of ordered pairs i...
mptrel 5690 The maps-to notation alway...
reli 5691 The identity relation is a...
rele 5692 The membership relation is...
opabid2 5693 A relation expressed as an...
inopab 5694 Intersection of two ordere...
difopab 5695 The difference of two orde...
inxp 5696 The intersection of two Ca...
xpindi 5697 Distributive law for Carte...
xpindir 5698 Distributive law for Carte...
xpiindi 5699 Distributive law for Carte...
xpriindi 5700 Distributive law for Carte...
eliunxp 5701 Membership in a union of C...
opeliunxp2 5702 Membership in a union of C...
raliunxp 5703 Write a double restricted ...
rexiunxp 5704 Write a double restricted ...
ralxp 5705 Universal quantification r...
rexxp 5706 Existential quantification...
exopxfr 5707 Transfer ordered-pair exis...
exopxfr2 5708 Transfer ordered-pair exis...
djussxp 5709 Disjoint union is a subset...
ralxpf 5710 Version of ~ ralxp with bo...
rexxpf 5711 Version of ~ rexxp with bo...
iunxpf 5712 Indexed union on a Cartesi...
opabbi2dv 5713 Deduce equality of a relat...
relop 5714 A necessary and sufficient...
ideqg 5715 For sets, the identity rel...
ideq 5716 For sets, the identity rel...
ididg 5717 A set is identical to itse...
issetid 5718 Two ways of expressing set...
coss1 5719 Subclass theorem for compo...
coss2 5720 Subclass theorem for compo...
coeq1 5721 Equality theorem for compo...
coeq2 5722 Equality theorem for compo...
coeq1i 5723 Equality inference for com...
coeq2i 5724 Equality inference for com...
coeq1d 5725 Equality deduction for com...
coeq2d 5726 Equality deduction for com...
coeq12i 5727 Equality inference for com...
coeq12d 5728 Equality deduction for com...
nfco 5729 Bound-variable hypothesis ...
brcog 5730 Ordered pair membership in...
opelco2g 5731 Ordered pair membership in...
brcogw 5732 Ordered pair membership in...
eqbrrdva 5733 Deduction from extensional...
brco 5734 Binary relation on a compo...
opelco 5735 Ordered pair membership in...
cnvss 5736 Subset theorem for convers...
cnveq 5737 Equality theorem for conve...
cnveqi 5738 Equality inference for con...
cnveqd 5739 Equality deduction for con...
elcnv 5740 Membership in a converse r...
elcnv2 5741 Membership in a converse r...
nfcnv 5742 Bound-variable hypothesis ...
brcnvg 5743 The converse of a binary r...
opelcnvg 5744 Ordered-pair membership in...
opelcnv 5745 Ordered-pair membership in...
brcnv 5746 The converse of a binary r...
csbcnv 5747 Move class substitution in...
csbcnvgALT 5748 Move class substitution in...
cnvco 5749 Distributive law of conver...
cnvuni 5750 The converse of a class un...
dfdm3 5751 Alternate definition of do...
dfrn2 5752 Alternate definition of ra...
dfrn3 5753 Alternate definition of ra...
elrn2g 5754 Membership in a range. (C...
elrng 5755 Membership in a range. (C...
ssrelrn 5756 If a relation is a subset ...
dfdm4 5757 Alternate definition of do...
dfdmf 5758 Definition of domain, usin...
csbdm 5759 Distribute proper substitu...
eldmg 5760 Domain membership. Theore...
eldm2g 5761 Domain membership. Theore...
eldm 5762 Membership in a domain. T...
eldm2 5763 Membership in a domain. T...
dmss 5764 Subset theorem for domain....
dmeq 5765 Equality theorem for domai...
dmeqi 5766 Equality inference for dom...
dmeqd 5767 Equality deduction for dom...
opeldmd 5768 Membership of first of an ...
opeldm 5769 Membership of first of an ...
breldm 5770 Membership of first of a b...
breldmg 5771 Membership of first of a b...
dmun 5772 The domain of a union is t...
dmin 5773 The domain of an intersect...
breldmd 5774 Membership of first of a b...
dmiun 5775 The domain of an indexed u...
dmuni 5776 The domain of a union. Pa...
dmopab 5777 The domain of a class of o...
dmopabelb 5778 A set is an element of the...
dmopab2rex 5779 The domain of an ordered p...
dmopabss 5780 Upper bound for the domain...
dmopab3 5781 The domain of a restricted...
opabssxpd 5782 An ordered-pair class abst...
dm0 5783 The domain of the empty se...
dmi 5784 The domain of the identity...
dmv 5785 The domain of the universe...
dmep 5786 The domain of the membersh...
domepOLD 5787 Obsolete proof of ~ dmep a...
dm0rn0 5788 An empty domain is equival...
rn0 5789 The range of the empty set...
rnep 5790 The range of the membershi...
reldm0 5791 A relation is empty iff it...
dmxp 5792 The domain of a Cartesian ...
dmxpid 5793 The domain of a Cartesian ...
dmxpin 5794 The domain of the intersec...
xpid11 5795 The Cartesian square is a ...
dmcnvcnv 5796 The domain of the double c...
rncnvcnv 5797 The range of the double co...
elreldm 5798 The first member of an ord...
rneq 5799 Equality theorem for range...
rneqi 5800 Equality inference for ran...
rneqd 5801 Equality deduction for ran...
rnss 5802 Subset theorem for range. ...
rnssi 5803 Subclass inference for ran...
brelrng 5804 The second argument of a b...
brelrn 5805 The second argument of a b...
opelrn 5806 Membership of second membe...
releldm 5807 The first argument of a bi...
relelrn 5808 The second argument of a b...
releldmb 5809 Membership in a domain. (...
relelrnb 5810 Membership in a range. (C...
releldmi 5811 The first argument of a bi...
relelrni 5812 The second argument of a b...
dfrnf 5813 Definition of range, using...
elrn2 5814 Membership in a range. (C...
elrn 5815 Membership in a range. (C...
nfdm 5816 Bound-variable hypothesis ...
nfrn 5817 Bound-variable hypothesis ...
dmiin 5818 Domain of an intersection....
rnopab 5819 The range of a class of or...
rnmpt 5820 The range of a function in...
elrnmpt 5821 The range of a function in...
elrnmpt1s 5822 Elementhood in an image se...
elrnmpt1 5823 Elementhood in an image se...
elrnmptg 5824 Membership in the range of...
elrnmpti 5825 Membership in the range of...
elrnmptdv 5826 Elementhood in the range o...
elrnmpt2d 5827 Elementhood in the range o...
dfiun3g 5828 Alternate definition of in...
dfiin3g 5829 Alternate definition of in...
dfiun3 5830 Alternate definition of in...
dfiin3 5831 Alternate definition of in...
riinint 5832 Express a relative indexed...
relrn0 5833 A relation is empty iff it...
dmrnssfld 5834 The domain and range of a ...
dmcoss 5835 Domain of a composition. ...
rncoss 5836 Range of a composition. (...
dmcosseq 5837 Domain of a composition. ...
dmcoeq 5838 Domain of a composition. ...
rncoeq 5839 Range of a composition. (...
reseq1 5840 Equality theorem for restr...
reseq2 5841 Equality theorem for restr...
reseq1i 5842 Equality inference for res...
reseq2i 5843 Equality inference for res...
reseq12i 5844 Equality inference for res...
reseq1d 5845 Equality deduction for res...
reseq2d 5846 Equality deduction for res...
reseq12d 5847 Equality deduction for res...
nfres 5848 Bound-variable hypothesis ...
csbres 5849 Distribute proper substitu...
res0 5850 A restriction to the empty...
dfres3 5851 Alternate definition of re...
opelres 5852 Ordered pair elementhood i...
brres 5853 Binary relation on a restr...
opelresi 5854 Ordered pair membership in...
brresi 5855 Binary relation on a restr...
opres 5856 Ordered pair membership in...
resieq 5857 A restricted identity rela...
opelidres 5858 ` <. A , A >. ` belongs to...
resres 5859 The restriction of a restr...
resundi 5860 Distributive law for restr...
resundir 5861 Distributive law for restr...
resindi 5862 Class restriction distribu...
resindir 5863 Class restriction distribu...
inres 5864 Move intersection into cla...
resdifcom 5865 Commutative law for restri...
resiun1 5866 Distribution of restrictio...
resiun2 5867 Distribution of restrictio...
dmres 5868 The domain of a restrictio...
ssdmres 5869 A domain restricted to a s...
dmresexg 5870 The domain of a restrictio...
resss 5871 A class includes its restr...
rescom 5872 Commutative law for restri...
ssres 5873 Subclass theorem for restr...
ssres2 5874 Subclass theorem for restr...
relres 5875 A restriction is a relatio...
resabs1 5876 Absorption law for restric...
resabs1d 5877 Absorption law for restric...
resabs2 5878 Absorption law for restric...
residm 5879 Idempotent law for restric...
resima 5880 A restriction to an image....
resima2 5881 Image under a restricted c...
xpssres 5882 Restriction of a constant ...
elinxp 5883 Membership in an intersect...
elres 5884 Membership in a restrictio...
elsnres 5885 Membership in restriction ...
relssres 5886 Simplification law for res...
dmressnsn 5887 The domain of a restrictio...
eldmressnsn 5888 The element of the domain ...
eldmeldmressn 5889 An element of the domain (...
resdm 5890 A relation restricted to i...
resexg 5891 The restriction of a set i...
resex 5892 The restriction of a set i...
resindm 5893 When restricting a relatio...
resdmdfsn 5894 Restricting a relation to ...
resopab 5895 Restriction of a class abs...
iss 5896 A subclass of the identity...
resopab2 5897 Restriction of a class abs...
resmpt 5898 Restriction of the mapping...
resmpt3 5899 Unconditional restriction ...
resmptf 5900 Restriction of the mapping...
resmptd 5901 Restriction of the mapping...
dfres2 5902 Alternate definition of th...
mptss 5903 Sufficient condition for i...
elidinxp 5904 Characterization of the el...
elidinxpid 5905 Characterization of the el...
elrid 5906 Characterization of the el...
idinxpres 5907 The intersection of the id...
idinxpresid 5908 The intersection of the id...
idssxp 5909 A diagonal set as a subset...
opabresid 5910 The restricted identity re...
mptresid 5911 The restricted identity re...
opabresidOLD 5912 Obsolete version of ~ opab...
mptresidOLD 5913 Obsolete version of ~ mptr...
dmresi 5914 The domain of a restricted...
restidsing 5915 Restriction of the identit...
iresn0n0 5916 The identity function rest...
imaeq1 5917 Equality theorem for image...
imaeq2 5918 Equality theorem for image...
imaeq1i 5919 Equality theorem for image...
imaeq2i 5920 Equality theorem for image...
imaeq1d 5921 Equality theorem for image...
imaeq2d 5922 Equality theorem for image...
imaeq12d 5923 Equality theorem for image...
dfima2 5924 Alternate definition of im...
dfima3 5925 Alternate definition of im...
elimag 5926 Membership in an image. T...
elima 5927 Membership in an image. T...
elima2 5928 Membership in an image. T...
elima3 5929 Membership in an image. T...
nfima 5930 Bound-variable hypothesis ...
nfimad 5931 Deduction version of bound...
imadmrn 5932 The image of the domain of...
imassrn 5933 The image of a class is a ...
mptima 5934 Image of a function in map...
imai 5935 Image under the identity r...
rnresi 5936 The range of the restricte...
resiima 5937 The image of a restriction...
ima0 5938 Image of the empty set. T...
0ima 5939 Image under the empty rela...
csbima12 5940 Move class substitution in...
imadisj 5941 A class whose image under ...
cnvimass 5942 A preimage under any class...
cnvimarndm 5943 The preimage of the range ...
imasng 5944 The image of a singleton. ...
relimasn 5945 The image of a singleton. ...
elrelimasn 5946 Elementhood in the image o...
elimasn 5947 Membership in an image of ...
elimasng 5948 Membership in an image of ...
elimasni 5949 Membership in an image of ...
args 5950 Two ways to express the cl...
eliniseg 5951 Membership in an initial s...
epini 5952 Any set is equal to its pr...
iniseg 5953 An idiom that signifies an...
inisegn0 5954 Nonemptiness of an initial...
dffr3 5955 Alternate definition of we...
dfse2 5956 Alternate definition of se...
imass1 5957 Subset theorem for image. ...
imass2 5958 Subset theorem for image. ...
ndmima 5959 The image of a singleton o...
relcnv 5960 A converse is a relation. ...
relbrcnvg 5961 When ` R ` is a relation, ...
eliniseg2 5962 Eliminate the class existe...
relbrcnv 5963 When ` R ` is a relation, ...
cotrg 5964 Two ways of saying that th...
cotr 5965 Two ways of saying a relat...
idrefALT 5966 Alternate proof of ~ idref...
cnvsym 5967 Two ways of saying a relat...
intasym 5968 Two ways of saying a relat...
asymref 5969 Two ways of saying a relat...
asymref2 5970 Two ways of saying a relat...
intirr 5971 Two ways of saying a relat...
brcodir 5972 Two ways of saying that tw...
codir 5973 Two ways of saying a relat...
qfto 5974 A quantifier-free way of e...
xpidtr 5975 A Cartesian square is a tr...
trin2 5976 The intersection of two tr...
poirr2 5977 A partial order relation i...
trinxp 5978 The relation induced by a ...
soirri 5979 A strict order relation is...
sotri 5980 A strict order relation is...
son2lpi 5981 A strict order relation ha...
sotri2 5982 A transitivity relation. ...
sotri3 5983 A transitivity relation. ...
poleloe 5984 Express "less than or equa...
poltletr 5985 Transitive law for general...
somin1 5986 Property of a minimum in a...
somincom 5987 Commutativity of minimum i...
somin2 5988 Property of a minimum in a...
soltmin 5989 Being less than a minimum,...
cnvopab 5990 The converse of a class ab...
mptcnv 5991 The converse of a mapping ...
cnv0 5992 The converse of the empty ...
cnvi 5993 The converse of the identi...
cnvun 5994 The converse of a union is...
cnvdif 5995 Distributive law for conve...
cnvin 5996 Distributive law for conve...
rnun 5997 Distributive law for range...
rnin 5998 The range of an intersecti...
rniun 5999 The range of an indexed un...
rnuni 6000 The range of a union. Par...
imaundi 6001 Distributive law for image...
imaundir 6002 The image of a union. (Co...
dminss 6003 An upper bound for interse...
imainss 6004 An upper bound for interse...
inimass 6005 The image of an intersecti...
inimasn 6006 The intersection of the im...
cnvxp 6007 The converse of a Cartesia...
xp0 6008 The Cartesian product with...
xpnz 6009 The Cartesian product of n...
xpeq0 6010 At least one member of an ...
xpdisj1 6011 Cartesian products with di...
xpdisj2 6012 Cartesian products with di...
xpsndisj 6013 Cartesian products with tw...
difxp 6014 Difference of Cartesian pr...
difxp1 6015 Difference law for Cartesi...
difxp2 6016 Difference law for Cartesi...
djudisj 6017 Disjoint unions with disjo...
xpdifid 6018 The set of distinct couple...
resdisj 6019 A double restriction to di...
rnxp 6020 The range of a Cartesian p...
dmxpss 6021 The domain of a Cartesian ...
rnxpss 6022 The range of a Cartesian p...
rnxpid 6023 The range of a Cartesian s...
ssxpb 6024 A Cartesian product subcla...
xp11 6025 The Cartesian product of n...
xpcan 6026 Cancellation law for Carte...
xpcan2 6027 Cancellation law for Carte...
ssrnres 6028 Two ways to express surjec...
rninxp 6029 Two ways to express surjec...
dminxp 6030 Two ways to express totali...
imainrect 6031 Image by a restricted and ...
xpima 6032 Direct image by a Cartesia...
xpima1 6033 Direct image by a Cartesia...
xpima2 6034 Direct image by a Cartesia...
xpimasn 6035 Direct image of a singleto...
sossfld 6036 The base set of a strict o...
sofld 6037 The base set of a nonempty...
cnvcnv3 6038 The set of all ordered pai...
dfrel2 6039 Alternate definition of re...
dfrel4v 6040 A relation can be expresse...
dfrel4 6041 A relation can be expresse...
cnvcnv 6042 The double converse of a c...
cnvcnv2 6043 The double converse of a c...
cnvcnvss 6044 The double converse of a c...
cnvrescnv 6045 Two ways to express the co...
cnveqb 6046 Equality theorem for conve...
cnveq0 6047 A relation empty iff its c...
dfrel3 6048 Alternate definition of re...
elid 6049 Characterization of the el...
dmresv 6050 The domain of a universal ...
rnresv 6051 The range of a universal r...
dfrn4 6052 Range defined in terms of ...
csbrn 6053 Distribute proper substitu...
rescnvcnv 6054 The restriction of the dou...
cnvcnvres 6055 The double converse of the...
imacnvcnv 6056 The image of the double co...
dmsnn0 6057 The domain of a singleton ...
rnsnn0 6058 The range of a singleton i...
dmsn0 6059 The domain of the singleto...
cnvsn0 6060 The converse of the single...
dmsn0el 6061 The domain of a singleton ...
relsn2 6062 A singleton is a relation ...
dmsnopg 6063 The domain of a singleton ...
dmsnopss 6064 The domain of a singleton ...
dmpropg 6065 The domain of an unordered...
dmsnop 6066 The domain of a singleton ...
dmprop 6067 The domain of an unordered...
dmtpop 6068 The domain of an unordered...
cnvcnvsn 6069 Double converse of a singl...
dmsnsnsn 6070 The domain of the singleto...
rnsnopg 6071 The range of a singleton o...
rnpropg 6072 The range of a pair of ord...
cnvsng 6073 Converse of a singleton of...
rnsnop 6074 The range of a singleton o...
op1sta 6075 Extract the first member o...
cnvsn 6076 Converse of a singleton of...
op2ndb 6077 Extract the second member ...
op2nda 6078 Extract the second member ...
opswap 6079 Swap the members of an ord...
cnvresima 6080 An image under the convers...
resdm2 6081 A class restricted to its ...
resdmres 6082 Restriction to the domain ...
resresdm 6083 A restriction by an arbitr...
imadmres 6084 The image of the domain of...
mptpreima 6085 The preimage of a function...
mptiniseg 6086 Converse singleton image o...
dmmpt 6087 The domain of the mapping ...
dmmptss 6088 The domain of a mapping is...
dmmptg 6089 The domain of the mapping ...
relco 6090 A composition is a relatio...
dfco2 6091 Alternate definition of a ...
dfco2a 6092 Generalization of ~ dfco2 ...
coundi 6093 Class composition distribu...
coundir 6094 Class composition distribu...
cores 6095 Restricted first member of...
resco 6096 Associative law for the re...
imaco 6097 Image of the composition o...
rnco 6098 The range of the compositi...
rnco2 6099 The range of the compositi...
dmco 6100 The domain of a compositio...
coeq0 6101 A composition of two relat...
coiun 6102 Composition with an indexe...
cocnvcnv1 6103 A composition is not affec...
cocnvcnv2 6104 A composition is not affec...
cores2 6105 Absorption of a reverse (p...
co02 6106 Composition with the empty...
co01 6107 Composition with the empty...
coi1 6108 Composition with the ident...
coi2 6109 Composition with the ident...
coires1 6110 Composition with a restric...
coass 6111 Associative law for class ...
relcnvtrg 6112 General form of ~ relcnvtr...
relcnvtr 6113 A relation is transitive i...
relssdmrn 6114 A relation is included in ...
cnvssrndm 6115 The converse is a subset o...
cossxp 6116 Composition as a subset of...
relrelss 6117 Two ways to describe the s...
unielrel 6118 The membership relation fo...
relfld 6119 The double union of a rela...
relresfld 6120 Restriction of a relation ...
relcoi2 6121 Composition with the ident...
relcoi1 6122 Composition with the ident...
unidmrn 6123 The double union of the co...
relcnvfld 6124 if ` R ` is a relation, it...
dfdm2 6125 Alternate definition of do...
unixp 6126 The double class union of ...
unixp0 6127 A Cartesian product is emp...
unixpid 6128 Field of a Cartesian squar...
ressn 6129 Restriction of a class to ...
cnviin 6130 The converse of an interse...
cnvpo 6131 The converse of a partial ...
cnvso 6132 The converse of a strict o...
xpco 6133 Composition of two Cartesi...
xpcoid 6134 Composition of two Cartesi...
elsnxp 6135 Membership in a Cartesian ...
reu3op 6136 There is a unique ordered ...
reuop 6137 There is a unique ordered ...
opreu2reurex 6138 There is a unique ordered ...
opreu2reu 6139 If there is a unique order...
predeq123 6142 Equality theorem for the p...
predeq1 6143 Equality theorem for the p...
predeq2 6144 Equality theorem for the p...
predeq3 6145 Equality theorem for the p...
nfpred 6146 Bound-variable hypothesis ...
predpredss 6147 If ` A ` is a subset of ` ...
predss 6148 The predecessor class of `...
sspred 6149 Another subset/predecessor...
dfpred2 6150 An alternate definition of...
dfpred3 6151 An alternate definition of...
dfpred3g 6152 An alternate definition of...
elpredim 6153 Membership in a predecesso...
elpred 6154 Membership in a predecesso...
elpredg 6155 Membership in a predecesso...
predasetex 6156 The predecessor class exis...
dffr4 6157 Alternate definition of we...
predel 6158 Membership in the predeces...
predpo 6159 Property of the precessor ...
predso 6160 Property of the predecesso...
predbrg 6161 Closed form of ~ elpredim ...
setlikespec 6162 If ` R ` is set-like in ` ...
predidm 6163 Idempotent law for the pre...
predin 6164 Intersection law for prede...
predun 6165 Union law for predecessor ...
preddif 6166 Difference law for predece...
predep 6167 The predecessor under the ...
preddowncl 6168 A property of classes that...
predpoirr 6169 Given a partial ordering, ...
predfrirr 6170 Given a well-founded relat...
pred0 6171 The predecessor class over...
tz6.26 6172 All nonempty subclasses of...
tz6.26i 6173 All nonempty subclasses of...
wfi 6174 The Principle of Well-Foun...
wfii 6175 The Principle of Well-Foun...
wfisg 6176 Well-Founded Induction Sch...
wfis 6177 Well-Founded Induction Sch...
wfis2fg 6178 Well-Founded Induction Sch...
wfis2f 6179 Well Founded Induction sch...
wfis2g 6180 Well-Founded Induction Sch...
wfis2 6181 Well Founded Induction sch...
wfis3 6182 Well Founded Induction sch...
ordeq 6191 Equality theorem for the o...
elong 6192 An ordinal number is an or...
elon 6193 An ordinal number is an or...
eloni 6194 An ordinal number has the ...
elon2 6195 An ordinal number is an or...
limeq 6196 Equality theorem for the l...
ordwe 6197 Membership well-orders eve...
ordtr 6198 An ordinal class is transi...
ordfr 6199 Membership is well-founded...
ordelss 6200 An element of an ordinal c...
trssord 6201 A transitive subclass of a...
ordirr 6202 No ordinal class is a memb...
nordeq 6203 A member of an ordinal cla...
ordn2lp 6204 An ordinal class cannot be...
tz7.5 6205 A nonempty subclass of an ...
ordelord 6206 An element of an ordinal c...
tron 6207 The class of all ordinal n...
ordelon 6208 An element of an ordinal c...
onelon 6209 An element of an ordinal n...
tz7.7 6210 A transitive class belongs...
ordelssne 6211 For ordinal classes, membe...
ordelpss 6212 For ordinal classes, membe...
ordsseleq 6213 For ordinal classes, inclu...
ordin 6214 The intersection of two or...
onin 6215 The intersection of two or...
ordtri3or 6216 A trichotomy law for ordin...
ordtri1 6217 A trichotomy law for ordin...
ontri1 6218 A trichotomy law for ordin...
ordtri2 6219 A trichotomy law for ordin...
ordtri3 6220 A trichotomy law for ordin...
ordtri4 6221 A trichotomy law for ordin...
orddisj 6222 An ordinal class and its s...
onfr 6223 The ordinal class is well-...
onelpss 6224 Relationship between membe...
onsseleq 6225 Relationship between subse...
onelss 6226 An element of an ordinal n...
ordtr1 6227 Transitive law for ordinal...
ordtr2 6228 Transitive law for ordinal...
ordtr3 6229 Transitive law for ordinal...
ontr1 6230 Transitive law for ordinal...
ontr2 6231 Transitive law for ordinal...
ordunidif 6232 The union of an ordinal st...
ordintdif 6233 If ` B ` is smaller than `...
onintss 6234 If a property is true for ...
oneqmini 6235 A way to show that an ordi...
ord0 6236 The empty set is an ordina...
0elon 6237 The empty set is an ordina...
ord0eln0 6238 A nonempty ordinal contain...
on0eln0 6239 An ordinal number contains...
dflim2 6240 An alternate definition of...
inton 6241 The intersection of the cl...
nlim0 6242 The empty set is not a lim...
limord 6243 A limit ordinal is ordinal...
limuni 6244 A limit ordinal is its own...
limuni2 6245 The union of a limit ordin...
0ellim 6246 A limit ordinal contains t...
limelon 6247 A limit ordinal class that...
onn0 6248 The class of all ordinal n...
suceq 6249 Equality of successors. (...
elsuci 6250 Membership in a successor....
elsucg 6251 Membership in a successor....
elsuc2g 6252 Variant of membership in a...
elsuc 6253 Membership in a successor....
elsuc2 6254 Membership in a successor....
nfsuc 6255 Bound-variable hypothesis ...
elelsuc 6256 Membership in a successor....
sucel 6257 Membership of a successor ...
suc0 6258 The successor of the empty...
sucprc 6259 A proper class is its own ...
unisuc 6260 A transitive class is equa...
sssucid 6261 A class is included in its...
sucidg 6262 Part of Proposition 7.23 o...
sucid 6263 A set belongs to its succe...
nsuceq0 6264 No successor is empty. (C...
eqelsuc 6265 A set belongs to the succe...
iunsuc 6266 Inductive definition for t...
suctr 6267 The successor of a transit...
trsuc 6268 A set whose successor belo...
trsucss 6269 A member of the successor ...
ordsssuc 6270 An ordinal is a subset of ...
onsssuc 6271 A subset of an ordinal num...
ordsssuc2 6272 An ordinal subset of an or...
onmindif 6273 When its successor is subt...
ordnbtwn 6274 There is no set between an...
onnbtwn 6275 There is no set between an...
sucssel 6276 A set whose successor is a...
orddif 6277 Ordinal derived from its s...
orduniss 6278 An ordinal class includes ...
ordtri2or 6279 A trichotomy law for ordin...
ordtri2or2 6280 A trichotomy law for ordin...
ordtri2or3 6281 A consequence of total ord...
ordelinel 6282 The intersection of two or...
ordssun 6283 Property of a subclass of ...
ordequn 6284 The maximum (i.e. union) o...
ordun 6285 The maximum (i.e. union) o...
ordunisssuc 6286 A subclass relationship fo...
suc11 6287 The successor operation be...
onordi 6288 An ordinal number is an or...
ontrci 6289 An ordinal number is a tra...
onirri 6290 An ordinal number is not a...
oneli 6291 A member of an ordinal num...
onelssi 6292 A member of an ordinal num...
onssneli 6293 An ordering law for ordina...
onssnel2i 6294 An ordering law for ordina...
onelini 6295 An element of an ordinal n...
oneluni 6296 An ordinal number equals i...
onunisuci 6297 An ordinal number is equal...
onsseli 6298 Subset is equivalent to me...
onun2i 6299 The union of two ordinal n...
unizlim 6300 An ordinal equal to its ow...
on0eqel 6301 An ordinal number either e...
snsn0non 6302 The singleton of the singl...
onxpdisj 6303 Ordinal numbers and ordere...
onnev 6304 The class of ordinal numbe...
iotajust 6306 Soundness justification th...
dfiota2 6308 Alternate definition for d...
nfiota1 6309 Bound-variable hypothesis ...
nfiotadw 6310 Version of ~ nfiotad with ...
nfiotaw 6311 Version of ~ nfiota with a...
nfiotad 6312 Deduction version of ~ nfi...
nfiota 6313 Bound-variable hypothesis ...
cbviotaw 6314 Version of ~ cbviota with ...
cbviotavw 6315 Version of ~ cbviotav with...
cbviota 6316 Change bound variables in ...
cbviotav 6317 Change bound variables in ...
sb8iota 6318 Variable substitution in d...
iotaeq 6319 Equality theorem for descr...
iotabi 6320 Equivalence theorem for de...
uniabio 6321 Part of Theorem 8.17 in [Q...
iotaval 6322 Theorem 8.19 in [Quine] p....
iotauni 6323 Equivalence between two di...
iotaint 6324 Equivalence between two di...
iota1 6325 Property of iota. (Contri...
iotanul 6326 Theorem 8.22 in [Quine] p....
iotassuni 6327 The ` iota ` class is a su...
iotaex 6328 Theorem 8.23 in [Quine] p....
iota4 6329 Theorem *14.22 in [Whitehe...
iota4an 6330 Theorem *14.23 in [Whitehe...
iota5 6331 A method for computing iot...
iotabidv 6332 Formula-building deduction...
iotabii 6333 Formula-building deduction...
iotacl 6334 Membership law for descrip...
iota2df 6335 A condition that allows us...
iota2d 6336 A condition that allows us...
iota2 6337 The unique element such th...
iotan0 6338 Representation of "the uni...
sniota 6339 A class abstraction with a...
dfiota4 6340 The ` iota ` operation usi...
csbiota 6341 Class substitution within ...
dffun2 6358 Alternate definition of a ...
dffun3 6359 Alternate definition of fu...
dffun4 6360 Alternate definition of a ...
dffun5 6361 Alternate definition of fu...
dffun6f 6362 Definition of function, us...
dffun6 6363 Alternate definition of a ...
funmo 6364 A function has at most one...
funrel 6365 A function is a relation. ...
0nelfun 6366 A function does not contai...
funss 6367 Subclass theorem for funct...
funeq 6368 Equality theorem for funct...
funeqi 6369 Equality inference for the...
funeqd 6370 Equality deduction for the...
nffun 6371 Bound-variable hypothesis ...
sbcfung 6372 Distribute proper substitu...
funeu 6373 There is exactly one value...
funeu2 6374 There is exactly one value...
dffun7 6375 Alternate definition of a ...
dffun8 6376 Alternate definition of a ...
dffun9 6377 Alternate definition of a ...
funfn 6378 A class is a function if a...
funfnd 6379 A function is a function o...
funi 6380 The identity relation is a...
nfunv 6381 The universal class is not...
funopg 6382 A Kuratowski ordered pair ...
funopab 6383 A class of ordered pairs i...
funopabeq 6384 A class of ordered pairs o...
funopab4 6385 A class of ordered pairs o...
funmpt 6386 A function in maps-to nota...
funmpt2 6387 Functionality of a class g...
funco 6388 The composition of two fun...
funresfunco 6389 Composition of two functio...
funres 6390 A restriction of a functio...
funssres 6391 The restriction of a funct...
fun2ssres 6392 Equality of restrictions o...
funun 6393 The union of functions wit...
fununmo 6394 If the union of classes is...
fununfun 6395 If the union of classes is...
fundif 6396 A function with removed el...
funcnvsn 6397 The converse singleton of ...
funsng 6398 A singleton of an ordered ...
fnsng 6399 Functionality and domain o...
funsn 6400 A singleton of an ordered ...
funprg 6401 A set of two pairs is a fu...
funtpg 6402 A set of three pairs is a ...
funpr 6403 A function with a domain o...
funtp 6404 A function with a domain o...
fnsn 6405 Functionality and domain o...
fnprg 6406 Function with a domain of ...
fntpg 6407 Function with a domain of ...
fntp 6408 A function with a domain o...
funcnvpr 6409 The converse pair of order...
funcnvtp 6410 The converse triple of ord...
funcnvqp 6411 The converse quadruple of ...
fun0 6412 The empty set is a functio...
funcnv0 6413 The converse of the empty ...
funcnvcnv 6414 The double converse of a f...
funcnv2 6415 A simpler equivalence for ...
funcnv 6416 The converse of a class is...
funcnv3 6417 A condition showing a clas...
fun2cnv 6418 The double converse of a c...
svrelfun 6419 A single-valued relation i...
fncnv 6420 Single-rootedness (see ~ f...
fun11 6421 Two ways of stating that `...
fununi 6422 The union of a chain (with...
funin 6423 The intersection with a fu...
funres11 6424 The restriction of a one-t...
funcnvres 6425 The converse of a restrict...
cnvresid 6426 Converse of a restricted i...
funcnvres2 6427 The converse of a restrict...
funimacnv 6428 The image of the preimage ...
funimass1 6429 A kind of contraposition l...
funimass2 6430 A kind of contraposition l...
imadif 6431 The image of a difference ...
imain 6432 The image of an intersecti...
funimaexg 6433 Axiom of Replacement using...
funimaex 6434 The image of a set under a...
isarep1 6435 Part of a study of the Axi...
isarep2 6436 Part of a study of the Axi...
fneq1 6437 Equality theorem for funct...
fneq2 6438 Equality theorem for funct...
fneq1d 6439 Equality deduction for fun...
fneq2d 6440 Equality deduction for fun...
fneq12d 6441 Equality deduction for fun...
fneq12 6442 Equality theorem for funct...
fneq1i 6443 Equality inference for fun...
fneq2i 6444 Equality inference for fun...
nffn 6445 Bound-variable hypothesis ...
fnfun 6446 A function with domain is ...
fnrel 6447 A function with domain is ...
fndm 6448 The domain of a function. ...
fndmd 6449 The domain of a function. ...
funfni 6450 Inference to convert a fun...
fndmu 6451 A function has a unique do...
fnbr 6452 The first argument of bina...
fnop 6453 The first argument of an o...
fneu 6454 There is exactly one value...
fneu2 6455 There is exactly one value...
fnun 6456 The union of two functions...
fnunsn 6457 Extension of a function wi...
fnco 6458 Composition of two functio...
fnresdm 6459 A function does not change...
fnresdisj 6460 A function restricted to a...
2elresin 6461 Membership in two function...
fnssresb 6462 Restriction of a function ...
fnssres 6463 Restriction of a function ...
fnssresd 6464 Restriction of a function ...
fnresin1 6465 Restriction of a function'...
fnresin2 6466 Restriction of a function'...
fnres 6467 An equivalence for functio...
idfn 6468 The identity relation is a...
fnresi 6469 The restricted identity re...
fnresiOLD 6470 Obsolete proof of ~ fnresi...
fnima 6471 The image of a function's ...
fn0 6472 A function with empty doma...
fnimadisj 6473 A class that is disjoint w...
fnimaeq0 6474 Images under a function ne...
dfmpt3 6475 Alternate definition for t...
mptfnf 6476 The maps-to notation defin...
fnmptf 6477 The maps-to notation defin...
fnopabg 6478 Functionality and domain o...
fnopab 6479 Functionality and domain o...
mptfng 6480 The maps-to notation defin...
fnmpt 6481 The maps-to notation defin...
fnmptd 6482 The maps-to notation defin...
mpt0 6483 A mapping operation with e...
fnmpti 6484 Functionality and domain o...
dmmpti 6485 Domain of the mapping oper...
dmmptd 6486 The domain of the mapping ...
mptun 6487 Union of mappings which ar...
feq1 6488 Equality theorem for funct...
feq2 6489 Equality theorem for funct...
feq3 6490 Equality theorem for funct...
feq23 6491 Equality theorem for funct...
feq1d 6492 Equality deduction for fun...
feq2d 6493 Equality deduction for fun...
feq3d 6494 Equality deduction for fun...
feq12d 6495 Equality deduction for fun...
feq123d 6496 Equality deduction for fun...
feq123 6497 Equality theorem for funct...
feq1i 6498 Equality inference for fun...
feq2i 6499 Equality inference for fun...
feq12i 6500 Equality inference for fun...
feq23i 6501 Equality inference for fun...
feq23d 6502 Equality deduction for fun...
nff 6503 Bound-variable hypothesis ...
sbcfng 6504 Distribute proper substitu...
sbcfg 6505 Distribute proper substitu...
elimf 6506 Eliminate a mapping hypoth...
ffn 6507 A mapping is a function wi...
ffnd 6508 A mapping is a function wi...
dffn2 6509 Any function is a mapping ...
ffun 6510 A mapping is a function. ...
ffund 6511 A mapping is a function, d...
frel 6512 A mapping is a relation. ...
frn 6513 The range of a mapping. (...
frnd 6514 Deduction form of ~ frn . ...
fdm 6515 The domain of a mapping. ...
fdmd 6516 Deduction form of ~ fdm . ...
fdmi 6517 Inference associated with ...
dffn3 6518 A function maps to its ran...
ffrn 6519 A function maps to its ran...
fss 6520 Expanding the codomain of ...
fssd 6521 Expanding the codomain of ...
fssdmd 6522 Expressing that a class is...
fssdm 6523 Expressing that a class is...
fco 6524 Composition of two mapping...
fcod 6525 Composition of two mapping...
fco2 6526 Functionality of a composi...
fssxp 6527 A mapping is a class of or...
funssxp 6528 Two ways of specifying a p...
ffdm 6529 A mapping is a partial fun...
ffdmd 6530 The domain of a function. ...
fdmrn 6531 A different way to write `...
opelf 6532 The members of an ordered ...
fun 6533 The union of two functions...
fun2 6534 The union of two functions...
fun2d 6535 The union of functions wit...
fnfco 6536 Composition of two functio...
fssres 6537 Restriction of a function ...
fssresd 6538 Restriction of a function ...
fssres2 6539 Restriction of a restricte...
fresin 6540 An identity for the mappin...
resasplit 6541 If two functions agree on ...
fresaun 6542 The union of two functions...
fresaunres2 6543 From the union of two func...
fresaunres1 6544 From the union of two func...
fcoi1 6545 Composition of a mapping a...
fcoi2 6546 Composition of restricted ...
feu 6547 There is exactly one value...
fimass 6548 The image of a class is a ...
fcnvres 6549 The converse of a restrict...
fimacnvdisj 6550 The preimage of a class di...
fint 6551 Function into an intersect...
fin 6552 Mapping into an intersecti...
f0 6553 The empty function. (Cont...
f00 6554 A class is a function with...
f0bi 6555 A function with empty doma...
f0dom0 6556 A function is empty iff it...
f0rn0 6557 If there is no element in ...
fconst 6558 A Cartesian product with a...
fconstg 6559 A Cartesian product with a...
fnconstg 6560 A Cartesian product with a...
fconst6g 6561 Constant function with loo...
fconst6 6562 A constant function as a m...
f1eq1 6563 Equality theorem for one-t...
f1eq2 6564 Equality theorem for one-t...
f1eq3 6565 Equality theorem for one-t...
nff1 6566 Bound-variable hypothesis ...
dff12 6567 Alternate definition of a ...
f1f 6568 A one-to-one mapping is a ...
f1fn 6569 A one-to-one mapping is a ...
f1fun 6570 A one-to-one mapping is a ...
f1rel 6571 A one-to-one onto mapping ...
f1dm 6572 The domain of a one-to-one...
f1ss 6573 A function that is one-to-...
f1ssr 6574 A function that is one-to-...
f1ssres 6575 A function that is one-to-...
f1resf1 6576 The restriction of an inje...
f1cnvcnv 6577 Two ways to express that a...
f1co 6578 Composition of one-to-one ...
foeq1 6579 Equality theorem for onto ...
foeq2 6580 Equality theorem for onto ...
foeq3 6581 Equality theorem for onto ...
nffo 6582 Bound-variable hypothesis ...
fof 6583 An onto mapping is a mappi...
fofun 6584 An onto mapping is a funct...
fofn 6585 An onto mapping is a funct...
forn 6586 The codomain of an onto fu...
dffo2 6587 Alternate definition of an...
foima 6588 The image of the domain of...
dffn4 6589 A function maps onto its r...
funforn 6590 A function maps its domain...
fodmrnu 6591 An onto function has uniqu...
fimadmfo 6592 A function is a function o...
fores 6593 Restriction of an onto fun...
fimadmfoALT 6594 Alternate proof of ~ fimad...
foco 6595 Composition of onto functi...
foconst 6596 A nonzero constant functio...
f1oeq1 6597 Equality theorem for one-t...
f1oeq2 6598 Equality theorem for one-t...
f1oeq3 6599 Equality theorem for one-t...
f1oeq23 6600 Equality theorem for one-t...
f1eq123d 6601 Equality deduction for one...
foeq123d 6602 Equality deduction for ont...
f1oeq123d 6603 Equality deduction for one...
f1oeq2d 6604 Equality deduction for one...
f1oeq3d 6605 Equality deduction for one...
nff1o 6606 Bound-variable hypothesis ...
f1of1 6607 A one-to-one onto mapping ...
f1of 6608 A one-to-one onto mapping ...
f1ofn 6609 A one-to-one onto mapping ...
f1ofun 6610 A one-to-one onto mapping ...
f1orel 6611 A one-to-one onto mapping ...
f1odm 6612 The domain of a one-to-one...
dff1o2 6613 Alternate definition of on...
dff1o3 6614 Alternate definition of on...
f1ofo 6615 A one-to-one onto function...
dff1o4 6616 Alternate definition of on...
dff1o5 6617 Alternate definition of on...
f1orn 6618 A one-to-one function maps...
f1f1orn 6619 A one-to-one function maps...
f1ocnv 6620 The converse of a one-to-o...
f1ocnvb 6621 A relation is a one-to-one...
f1ores 6622 The restriction of a one-t...
f1orescnv 6623 The converse of a one-to-o...
f1imacnv 6624 Preimage of an image. (Co...
foimacnv 6625 A reverse version of ~ f1i...
foun 6626 The union of two onto func...
f1oun 6627 The union of two one-to-on...
resdif 6628 The restriction of a one-t...
resin 6629 The restriction of a one-t...
f1oco 6630 Composition of one-to-one ...
f1cnv 6631 The converse of an injecti...
funcocnv2 6632 Composition with the conve...
fococnv2 6633 The composition of an onto...
f1ococnv2 6634 The composition of a one-t...
f1cocnv2 6635 Composition of an injectiv...
f1ococnv1 6636 The composition of a one-t...
f1cocnv1 6637 Composition of an injectiv...
funcoeqres 6638 Express a constraint on a ...
f1ssf1 6639 A subset of an injective f...
f10 6640 The empty set maps one-to-...
f10d 6641 The empty set maps one-to-...
f1o00 6642 One-to-one onto mapping of...
fo00 6643 Onto mapping of the empty ...
f1o0 6644 One-to-one onto mapping of...
f1oi 6645 A restriction of the ident...
f1ovi 6646 The identity relation is a...
f1osn 6647 A singleton of an ordered ...
f1osng 6648 A singleton of an ordered ...
f1sng 6649 A singleton of an ordered ...
fsnd 6650 A singleton of an ordered ...
f1oprswap 6651 A two-element swap is a bi...
f1oprg 6652 An unordered pair of order...
tz6.12-2 6653 Function value when ` F ` ...
fveu 6654 The value of a function at...
brprcneu 6655 If ` A ` is a proper class...
fvprc 6656 A function's value at a pr...
rnfvprc 6657 The range of a function va...
fv2 6658 Alternate definition of fu...
dffv3 6659 A definition of function v...
dffv4 6660 The previous definition of...
elfv 6661 Membership in a function v...
fveq1 6662 Equality theorem for funct...
fveq2 6663 Equality theorem for funct...
fveq1i 6664 Equality inference for fun...
fveq1d 6665 Equality deduction for fun...
fveq2i 6666 Equality inference for fun...
fveq2d 6667 Equality deduction for fun...
2fveq3 6668 Equality theorem for neste...
fveq12i 6669 Equality deduction for fun...
fveq12d 6670 Equality deduction for fun...
fveqeq2d 6671 Equality deduction for fun...
fveqeq2 6672 Equality deduction for fun...
nffv 6673 Bound-variable hypothesis ...
nffvmpt1 6674 Bound-variable hypothesis ...
nffvd 6675 Deduction version of bound...
fvex 6676 The value of a class exist...
fvexi 6677 The value of a class exist...
fvexd 6678 The value of a class exist...
fvif 6679 Move a conditional outside...
iffv 6680 Move a conditional outside...
fv3 6681 Alternate definition of th...
fvres 6682 The value of a restricted ...
fvresd 6683 The value of a restricted ...
funssfv 6684 The value of a member of t...
tz6.12-1 6685 Function value. Theorem 6...
tz6.12 6686 Function value. Theorem 6...
tz6.12f 6687 Function value, using boun...
tz6.12c 6688 Corollary of Theorem 6.12(...
tz6.12i 6689 Corollary of Theorem 6.12(...
fvbr0 6690 Two possibilities for the ...
fvrn0 6691 A function value is a memb...
fvssunirn 6692 The result of a function v...
ndmfv 6693 The value of a class outsi...
ndmfvrcl 6694 Reverse closure law for fu...
elfvdm 6695 If a function value has a ...
elfvex 6696 If a function value has a ...
elfvexd 6697 If a function value has a ...
eliman0 6698 A nonempty function value ...
nfvres 6699 The value of a non-member ...
nfunsn 6700 If the restriction of a cl...
fvfundmfvn0 6701 If the "value of a class" ...
0fv 6702 Function value of the empt...
fv2prc 6703 A function value of a func...
elfv2ex 6704 If a function value of a f...
fveqres 6705 Equal values imply equal v...
csbfv12 6706 Move class substitution in...
csbfv2g 6707 Move class substitution in...
csbfv 6708 Substitution for a functio...
funbrfv 6709 The second argument of a b...
funopfv 6710 The second element in an o...
fnbrfvb 6711 Equivalence of function va...
fnopfvb 6712 Equivalence of function va...
funbrfvb 6713 Equivalence of function va...
funopfvb 6714 Equivalence of function va...
fnbrfvb2 6715 Version of ~ fnbrfvb for f...
funbrfv2b 6716 Function value in terms of...
dffn5 6717 Representation of a functi...
fnrnfv 6718 The range of a function ex...
fvelrnb 6719 A member of a function's r...
foelrni 6720 A member of a surjective f...
dfimafn 6721 Alternate definition of th...
dfimafn2 6722 Alternate definition of th...
funimass4 6723 Membership relation for th...
fvelima 6724 Function value in an image...
fvelimad 6725 Function value in an image...
feqmptd 6726 Deduction form of ~ dffn5 ...
feqresmpt 6727 Express a restricted funct...
feqmptdf 6728 Deduction form of ~ dffn5f...
dffn5f 6729 Representation of a functi...
fvelimab 6730 Function value in an image...
fvelimabd 6731 Deduction form of ~ fvelim...
unima 6732 Image of a union. (Contri...
fvi 6733 The value of the identity ...
fviss 6734 The value of the identity ...
fniinfv 6735 The indexed intersection o...
fnsnfv 6736 Singleton of function valu...
opabiotafun 6737 Define a function whose va...
opabiotadm 6738 Define a function whose va...
opabiota 6739 Define a function whose va...
fnimapr 6740 The image of a pair under ...
ssimaex 6741 The existence of a subimag...
ssimaexg 6742 The existence of a subimag...
funfv 6743 A simplified expression fo...
funfv2 6744 The value of a function. ...
funfv2f 6745 The value of a function. ...
fvun 6746 Value of the union of two ...
fvun1 6747 The value of a union when ...
fvun2 6748 The value of a union when ...
dffv2 6749 Alternate definition of fu...
dmfco 6750 Domains of a function comp...
fvco2 6751 Value of a function compos...
fvco 6752 Value of a function compos...
fvco3 6753 Value of a function compos...
fvco3d 6754 Value of a function compos...
fvco4i 6755 Conditions for a compositi...
fvopab3g 6756 Value of a function given ...
fvopab3ig 6757 Value of a function given ...
brfvopabrbr 6758 The binary relation of a f...
fvmptg 6759 Value of a function given ...
fvmpti 6760 Value of a function given ...
fvmpt 6761 Value of a function given ...
fvmpt2f 6762 Value of a function given ...
fvtresfn 6763 Functionality of a tuple-r...
fvmpts 6764 Value of a function given ...
fvmpt3 6765 Value of a function given ...
fvmpt3i 6766 Value of a function given ...
fvmptd 6767 Deduction version of ~ fvm...
fvmptd2 6768 Deduction version of ~ fvm...
mptrcl 6769 Reverse closure for a mapp...
fvmpt2i 6770 Value of a function given ...
fvmpt2 6771 Value of a function given ...
fvmptss 6772 If all the values of the m...
fvmpt2d 6773 Deduction version of ~ fvm...
fvmptex 6774 Express a function ` F ` w...
fvmptd3f 6775 Alternate deduction versio...
fvmptdf 6776 Alternate deduction versio...
fvmptdv 6777 Alternate deduction versio...
fvmptdv2 6778 Alternate deduction versio...
mpteqb 6779 Bidirectional equality the...
fvmptt 6780 Closed theorem form of ~ f...
fvmptf 6781 Value of a function given ...
fvmptnf 6782 The value of a function gi...
fvmptd3 6783 Deduction version of ~ fvm...
fvmptn 6784 This somewhat non-intuitiv...
fvmptss2 6785 A mapping always evaluates...
elfvmptrab1w 6786 Version of ~ elfvmptrab1 w...
elfvmptrab1 6787 Implications for the value...
elfvmptrab 6788 Implications for the value...
fvopab4ndm 6789 Value of a function given ...
fvmptndm 6790 Value of a function given ...
fvmptrabfv 6791 Value of a function mappin...
fvopab5 6792 The value of a function th...
fvopab6 6793 Value of a function given ...
eqfnfv 6794 Equality of functions is d...
eqfnfv2 6795 Equality of functions is d...
eqfnfv3 6796 Derive equality of functio...
eqfnfvd 6797 Deduction for equality of ...
eqfnfv2f 6798 Equality of functions is d...
eqfunfv 6799 Equality of functions is d...
fvreseq0 6800 Equality of restricted fun...
fvreseq1 6801 Equality of a function res...
fvreseq 6802 Equality of restricted fun...
fnmptfvd 6803 A function with a given do...
fndmdif 6804 Two ways to express the lo...
fndmdifcom 6805 The difference set between...
fndmdifeq0 6806 The difference set of two ...
fndmin 6807 Two ways to express the lo...
fneqeql 6808 Two functions are equal if...
fneqeql2 6809 Two functions are equal if...
fnreseql 6810 Two functions are equal on...
chfnrn 6811 The range of a choice func...
funfvop 6812 Ordered pair with function...
funfvbrb 6813 Two ways to say that ` A `...
fvimacnvi 6814 A member of a preimage is ...
fvimacnv 6815 The argument of a function...
funimass3 6816 A kind of contraposition l...
funimass5 6817 A subclass of a preimage i...
funconstss 6818 Two ways of specifying tha...
fvimacnvALT 6819 Alternate proof of ~ fvima...
elpreima 6820 Membership in the preimage...
elpreimad 6821 Membership in the preimage...
fniniseg 6822 Membership in the preimage...
fncnvima2 6823 Inverse images under funct...
fniniseg2 6824 Inverse point images under...
unpreima 6825 Preimage of a union. (Con...
inpreima 6826 Preimage of an intersectio...
difpreima 6827 Preimage of a difference. ...
respreima 6828 The preimage of a restrict...
iinpreima 6829 Preimage of an intersectio...
intpreima 6830 Preimage of an intersectio...
fimacnv 6831 The preimage of the codoma...
fimacnvinrn 6832 Taking the converse image ...
fimacnvinrn2 6833 Taking the converse image ...
fvn0ssdmfun 6834 If a class' function value...
fnopfv 6835 Ordered pair with function...
fvelrn 6836 A function's value belongs...
nelrnfvne 6837 A function value cannot be...
fveqdmss 6838 If the empty set is not co...
fveqressseq 6839 If the empty set is not co...
fnfvelrn 6840 A function's value belongs...
ffvelrn 6841 A function's value belongs...
ffvelrni 6842 A function's value belongs...
ffvelrnda 6843 A function's value belongs...
ffvelrnd 6844 A function's value belongs...
rexrn 6845 Restricted existential qua...
ralrn 6846 Restricted universal quant...
elrnrexdm 6847 For any element in the ran...
elrnrexdmb 6848 For any element in the ran...
eldmrexrn 6849 For any element in the dom...
eldmrexrnb 6850 For any element in the dom...
fvcofneq 6851 The values of two function...
ralrnmptw 6852 Version of ~ ralrnmpt with...
rexrnmptw 6853 Version of ~ rexrnmpt with...
ralrnmpt 6854 A restricted quantifier ov...
rexrnmpt 6855 A restricted quantifier ov...
f0cli 6856 Unconditional closure of a...
dff2 6857 Alternate definition of a ...
dff3 6858 Alternate definition of a ...
dff4 6859 Alternate definition of a ...
dffo3 6860 An onto mapping expressed ...
dffo4 6861 Alternate definition of an...
dffo5 6862 Alternate definition of an...
exfo 6863 A relation equivalent to t...
foelrn 6864 Property of a surjective f...
foco2 6865 If a composition of two fu...
fmpt 6866 Functionality of the mappi...
f1ompt 6867 Express bijection for a ma...
fmpti 6868 Functionality of the mappi...
fvmptelrn 6869 The value of a function at...
fmptd 6870 Domain and codomain of the...
fmpttd 6871 Version of ~ fmptd with in...
fmpt3d 6872 Domain and codomain of the...
fmptdf 6873 A version of ~ fmptd using...
ffnfv 6874 A function maps to a class...
ffnfvf 6875 A function maps to a class...
fnfvrnss 6876 An upper bound for range d...
frnssb 6877 A function is a function i...
rnmptss 6878 The range of an operation ...
fmpt2d 6879 Domain and codomain of the...
ffvresb 6880 A necessary and sufficient...
f1oresrab 6881 Build a bijection between ...
f1ossf1o 6882 Restricting a bijection, w...
fmptco 6883 Composition of two functio...
fmptcof 6884 Version of ~ fmptco where ...
fmptcos 6885 Composition of two functio...
cofmpt 6886 Express composition of a m...
fcompt 6887 Express composition of two...
fcoconst 6888 Composition with a constan...
fsn 6889 A function maps a singleto...
fsn2 6890 A function that maps a sin...
fsng 6891 A function maps a singleto...
fsn2g 6892 A function that maps a sin...
xpsng 6893 The Cartesian product of t...
xpprsng 6894 The Cartesian product of a...
xpsn 6895 The Cartesian product of t...
f1o2sn 6896 A singleton consisting in ...
residpr 6897 Restriction of the identit...
dfmpt 6898 Alternate definition for t...
fnasrn 6899 A function expressed as th...
idref 6900 Two ways to state that a r...
funiun 6901 A function is a union of s...
funopsn 6902 If a function is an ordere...
funop 6903 An ordered pair is a funct...
funopdmsn 6904 The domain of a function w...
funsndifnop 6905 A singleton of an ordered ...
funsneqopb 6906 A singleton of an ordered ...
ressnop0 6907 If ` A ` is not in ` C ` ,...
fpr 6908 A function with a domain o...
fprg 6909 A function with a domain o...
ftpg 6910 A function with a domain o...
ftp 6911 A function with a domain o...
fnressn 6912 A function restricted to a...
funressn 6913 A function restricted to a...
fressnfv 6914 The value of a function re...
fvrnressn 6915 If the value of a function...
fvressn 6916 The value of a function re...
fvn0fvelrn 6917 If the value of a function...
fvconst 6918 The value of a constant fu...
fnsnr 6919 If a class belongs to a fu...
fnsnb 6920 A function whose domain is...
fmptsn 6921 Express a singleton functi...
fmptsng 6922 Express a singleton functi...
fmptsnd 6923 Express a singleton functi...
fmptap 6924 Append an additional value...
fmptapd 6925 Append an additional value...
fmptpr 6926 Express a pair function in...
fvresi 6927 The value of a restricted ...
fninfp 6928 Express the class of fixed...
fnelfp 6929 Property of a fixed point ...
fndifnfp 6930 Express the class of non-f...
fnelnfp 6931 Property of a non-fixed po...
fnnfpeq0 6932 A function is the identity...
fvunsn 6933 Remove an ordered pair not...
fvsng 6934 The value of a singleton o...
fvsn 6935 The value of a singleton o...
fvsnun1 6936 The value of a function wi...
fvsnun2 6937 The value of a function wi...
fnsnsplit 6938 Split a function into a si...
fsnunf 6939 Adjoining a point to a fun...
fsnunf2 6940 Adjoining a point to a pun...
fsnunfv 6941 Recover the added point fr...
fsnunres 6942 Recover the original funct...
funresdfunsn 6943 Restricting a function to ...
fvpr1 6944 The value of a function wi...
fvpr2 6945 The value of a function wi...
fvpr1g 6946 The value of a function wi...
fvpr2g 6947 The value of a function wi...
fprb 6948 A condition for functionho...
fvtp1 6949 The first value of a funct...
fvtp2 6950 The second value of a func...
fvtp3 6951 The third value of a funct...
fvtp1g 6952 The value of a function wi...
fvtp2g 6953 The value of a function wi...
fvtp3g 6954 The value of a function wi...
tpres 6955 An unordered triple of ord...
fvconst2g 6956 The value of a constant fu...
fconst2g 6957 A constant function expres...
fvconst2 6958 The value of a constant fu...
fconst2 6959 A constant function expres...
fconst5 6960 Two ways to express that a...
rnmptc 6961 Range of a constant functi...
fnprb 6962 A function whose domain ha...
fntpb 6963 A function whose domain ha...
fnpr2g 6964 A function whose domain ha...
fpr2g 6965 A function that maps a pai...
fconstfv 6966 A constant function expres...
fconst3 6967 Two ways to express a cons...
fconst4 6968 Two ways to express a cons...
resfunexg 6969 The restriction of a funct...
resiexd 6970 The restriction of the ide...
fnex 6971 If the domain of a functio...
fnexd 6972 If the domain of a functio...
funex 6973 If the domain of a functio...
opabex 6974 Existence of a function ex...
mptexg 6975 If the domain of a functio...
mptexgf 6976 If the domain of a functio...
mptex 6977 If the domain of a functio...
mptexd 6978 If the domain of a functio...
mptrabex 6979 If the domain of a functio...
fex 6980 If the domain of a mapping...
mptfvmpt 6981 A function in maps-to nota...
eufnfv 6982 A function is uniquely det...
funfvima 6983 A function's value in a pr...
funfvima2 6984 A function's value in an i...
funfvima2d 6985 A function's value in a pr...
fnfvima 6986 The function value of an o...
fnfvimad 6987 A function's value belongs...
resfvresima 6988 The value of the function ...
funfvima3 6989 A class including a functi...
rexima 6990 Existential quantification...
ralima 6991 Universal quantification u...
fvclss 6992 Upper bound for the class ...
elabrex 6993 Elementhood in an image se...
abrexco 6994 Composition of two image m...
imaiun 6995 The image of an indexed un...
imauni 6996 The image of a union is th...
fniunfv 6997 The indexed union of a fun...
funiunfv 6998 The indexed union of a fun...
funiunfvf 6999 The indexed union of a fun...
eluniima 7000 Membership in the union of...
elunirn 7001 Membership in the union of...
elunirnALT 7002 Alternate proof of ~ eluni...
fnunirn 7003 Membership in a union of s...
dff13 7004 A one-to-one function in t...
dff13f 7005 A one-to-one function in t...
f1veqaeq 7006 If the values of a one-to-...
f1cofveqaeq 7007 If the values of a composi...
f1cofveqaeqALT 7008 Alternate proof of ~ f1cof...
2f1fvneq 7009 If two one-to-one function...
f1mpt 7010 Express injection for a ma...
f1fveq 7011 Equality of function value...
f1elima 7012 Membership in the image of...
f1imass 7013 Taking images under a one-...
f1imaeq 7014 Taking images under a one-...
f1imapss 7015 Taking images under a one-...
fpropnf1 7016 A function, given by an un...
f1dom3fv3dif 7017 The function values for a ...
f1dom3el3dif 7018 The range of a 1-1 functio...
dff14a 7019 A one-to-one function in t...
dff14b 7020 A one-to-one function in t...
f12dfv 7021 A one-to-one function with...
f13dfv 7022 A one-to-one function with...
dff1o6 7023 A one-to-one onto function...
f1ocnvfv1 7024 The converse value of the ...
f1ocnvfv2 7025 The value of the converse ...
f1ocnvfv 7026 Relationship between the v...
f1ocnvfvb 7027 Relationship between the v...
nvof1o 7028 An involution is a bijecti...
nvocnv 7029 The converse of an involut...
fsnex 7030 Relate a function with a s...
f1prex 7031 Relate a one-to-one functi...
f1ocnvdm 7032 The value of the converse ...
f1ocnvfvrneq 7033 If the values of a one-to-...
fcof1 7034 An application is injectiv...
fcofo 7035 An application is surjecti...
cbvfo 7036 Change bound variable betw...
cbvexfo 7037 Change bound variable betw...
cocan1 7038 An injection is left-cance...
cocan2 7039 A surjection is right-canc...
fcof1oinvd 7040 Show that a function is th...
fcof1od 7041 A function is bijective if...
2fcoidinvd 7042 Show that a function is th...
fcof1o 7043 Show that two functions ar...
2fvcoidd 7044 Show that the composition ...
2fvidf1od 7045 A function is bijective if...
2fvidinvd 7046 Show that two functions ar...
foeqcnvco 7047 Condition for function equ...
f1eqcocnv 7048 Condition for function equ...
fveqf1o 7049 Given a bijection ` F ` , ...
fliftrel 7050 ` F ` , a function lift, i...
fliftel 7051 Elementhood in the relatio...
fliftel1 7052 Elementhood in the relatio...
fliftcnv 7053 Converse of the relation `...
fliftfun 7054 The function ` F ` is the ...
fliftfund 7055 The function ` F ` is the ...
fliftfuns 7056 The function ` F ` is the ...
fliftf 7057 The domain and range of th...
fliftval 7058 The value of the function ...
isoeq1 7059 Equality theorem for isomo...
isoeq2 7060 Equality theorem for isomo...
isoeq3 7061 Equality theorem for isomo...
isoeq4 7062 Equality theorem for isomo...
isoeq5 7063 Equality theorem for isomo...
nfiso 7064 Bound-variable hypothesis ...
isof1o 7065 An isomorphism is a one-to...
isof1oidb 7066 A function is a bijection ...
isof1oopb 7067 A function is a bijection ...
isorel 7068 An isomorphism connects bi...
soisores 7069 Express the condition of i...
soisoi 7070 Infer isomorphism from one...
isoid 7071 Identity law for isomorphi...
isocnv 7072 Converse law for isomorphi...
isocnv2 7073 Converse law for isomorphi...
isocnv3 7074 Complementation law for is...
isores2 7075 An isomorphism from one we...
isores1 7076 An isomorphism from one we...
isores3 7077 Induced isomorphism on a s...
isotr 7078 Composition (transitive) l...
isomin 7079 Isomorphisms preserve mini...
isoini 7080 Isomorphisms preserve init...
isoini2 7081 Isomorphisms are isomorphi...
isofrlem 7082 Lemma for ~ isofr . (Cont...
isoselem 7083 Lemma for ~ isose . (Cont...
isofr 7084 An isomorphism preserves w...
isose 7085 An isomorphism preserves s...
isofr2 7086 A weak form of ~ isofr tha...
isopolem 7087 Lemma for ~ isopo . (Cont...
isopo 7088 An isomorphism preserves p...
isosolem 7089 Lemma for ~ isoso . (Cont...
isoso 7090 An isomorphism preserves s...
isowe 7091 An isomorphism preserves w...
isowe2 7092 A weak form of ~ isowe tha...
f1oiso 7093 Any one-to-one onto functi...
f1oiso2 7094 Any one-to-one onto functi...
f1owe 7095 Well-ordering of isomorphi...
weniso 7096 A set-like well-ordering h...
weisoeq 7097 Thus, there is at most one...
weisoeq2 7098 Thus, there is at most one...
knatar 7099 The Knaster-Tarski theorem...
canth 7100 No set ` A ` is equinumero...
ncanth 7101 Cantor's theorem fails for...
riotaeqdv 7104 Formula-building deduction...
riotabidv 7105 Formula-building deduction...
riotaeqbidv 7106 Equality deduction for res...
riotaex 7107 Restricted iota is a set. ...
riotav 7108 An iota restricted to the ...
riotauni 7109 Restricted iota in terms o...
nfriota1 7110 The abstraction variable i...
nfriotadw 7111 Version of ~ nfriotad with...
cbvriotaw 7112 Version of ~ cbvriota with...
cbvriotavw 7113 Version of ~ cbvriotav wit...
nfriotad 7114 Deduction version of ~ nfr...
nfriota 7115 A variable not free in a w...
cbvriota 7116 Change bound variable in a...
cbvriotav 7117 Change bound variable in a...
csbriota 7118 Interchange class substitu...
riotacl2 7119 Membership law for "the un...
riotacl 7120 Closure of restricted iota...
riotasbc 7121 Substitution law for descr...
riotabidva 7122 Equivalent wff's yield equ...
riotabiia 7123 Equivalent wff's yield equ...
riota1 7124 Property of restricted iot...
riota1a 7125 Property of iota. (Contri...
riota2df 7126 A deduction version of ~ r...
riota2f 7127 This theorem shows a condi...
riota2 7128 This theorem shows a condi...
riotaeqimp 7129 If two restricted iota des...
riotaprop 7130 Properties of a restricted...
riota5f 7131 A method for computing res...
riota5 7132 A method for computing res...
riotass2 7133 Restriction of a unique el...
riotass 7134 Restriction of a unique el...
moriotass 7135 Restriction of a unique el...
snriota 7136 A restricted class abstrac...
riotaxfrd 7137 Change the variable ` x ` ...
eusvobj2 7138 Specify the same property ...
eusvobj1 7139 Specify the same object in...
f1ofveu 7140 There is one domain elemen...
f1ocnvfv3 7141 Value of the converse of a...
riotaund 7142 Restricted iota equals the...
riotassuni 7143 The restricted iota class ...
riotaclb 7144 Bidirectional closure of r...
oveq 7151 Equality theorem for opera...
oveq1 7152 Equality theorem for opera...
oveq2 7153 Equality theorem for opera...
oveq12 7154 Equality theorem for opera...
oveq1i 7155 Equality inference for ope...
oveq2i 7156 Equality inference for ope...
oveq12i 7157 Equality inference for ope...
oveqi 7158 Equality inference for ope...
oveq123i 7159 Equality inference for ope...
oveq1d 7160 Equality deduction for ope...
oveq2d 7161 Equality deduction for ope...
oveqd 7162 Equality deduction for ope...
oveq12d 7163 Equality deduction for ope...
oveqan12d 7164 Equality deduction for ope...
oveqan12rd 7165 Equality deduction for ope...
oveq123d 7166 Equality deduction for ope...
fvoveq1d 7167 Equality deduction for nes...
fvoveq1 7168 Equality theorem for neste...
ovanraleqv 7169 Equality theorem for a con...
imbrov2fvoveq 7170 Equality theorem for neste...
ovrspc2v 7171 If an operation value is e...
oveqrspc2v 7172 Restricted specialization ...
oveqdr 7173 Equality of two operations...
nfovd 7174 Deduction version of bound...
nfov 7175 Bound-variable hypothesis ...
oprabidw 7176 Version of ~ oprabid with ...
oprabid 7177 The law of concretion. Sp...
ovex 7178 The result of an operation...
ovexi 7179 The result of an operation...
ovexd 7180 The result of an operation...
ovssunirn 7181 The result of an operation...
0ov 7182 Operation value of the emp...
ovprc 7183 The value of an operation ...
ovprc1 7184 The value of an operation ...
ovprc2 7185 The value of an operation ...
ovrcl 7186 Reverse closure for an ope...
csbov123 7187 Move class substitution in...
csbov 7188 Move class substitution in...
csbov12g 7189 Move class substitution in...
csbov1g 7190 Move class substitution in...
csbov2g 7191 Move class substitution in...
rspceov 7192 A frequently used special ...
elovimad 7193 Elementhood of the image s...
fnbrovb 7194 Value of a binary operatio...
fnotovb 7195 Equivalence of operation v...
opabbrex 7196 A collection of ordered pa...
opabresex2d 7197 Restrictions of a collecti...
fvmptopab 7198 The function value of a ma...
f1opr 7199 Condition for an operation...
brfvopab 7200 The classes involved in a ...
dfoprab2 7201 Class abstraction for oper...
reloprab 7202 An operation class abstrac...
oprabv 7203 If a pair and a class are ...
nfoprab1 7204 The abstraction variables ...
nfoprab2 7205 The abstraction variables ...
nfoprab3 7206 The abstraction variables ...
nfoprab 7207 Bound-variable hypothesis ...
oprabbid 7208 Equivalent wff's yield equ...
oprabbidv 7209 Equivalent wff's yield equ...
oprabbii 7210 Equivalent wff's yield equ...
ssoprab2 7211 Equivalence of ordered pai...
ssoprab2b 7212 Equivalence of ordered pai...
eqoprab2bw 7213 Version of ~ eqoprab2b wit...
eqoprab2b 7214 Equivalence of ordered pai...
mpoeq123 7215 An equality theorem for th...
mpoeq12 7216 An equality theorem for th...
mpoeq123dva 7217 An equality deduction for ...
mpoeq123dv 7218 An equality deduction for ...
mpoeq123i 7219 An equality inference for ...
mpoeq3dva 7220 Slightly more general equa...
mpoeq3ia 7221 An equality inference for ...
mpoeq3dv 7222 An equality deduction for ...
nfmpo1 7223 Bound-variable hypothesis ...
nfmpo2 7224 Bound-variable hypothesis ...
nfmpo 7225 Bound-variable hypothesis ...
0mpo0 7226 A mapping operation with e...
mpo0v 7227 A mapping operation with e...
mpo0 7228 A mapping operation with e...
oprab4 7229 Two ways to state the doma...
cbvoprab1 7230 Rule used to change first ...
cbvoprab2 7231 Change the second bound va...
cbvoprab12 7232 Rule used to change first ...
cbvoprab12v 7233 Rule used to change first ...
cbvoprab3 7234 Rule used to change the th...
cbvoprab3v 7235 Rule used to change the th...
cbvmpox 7236 Rule to change the bound v...
cbvmpo 7237 Rule to change the bound v...
cbvmpov 7238 Rule to change the bound v...
elimdelov 7239 Eliminate a hypothesis whi...
ovif 7240 Move a conditional outside...
ovif2 7241 Move a conditional outside...
ovif12 7242 Move a conditional outside...
ifov 7243 Move a conditional outside...
dmoprab 7244 The domain of an operation...
dmoprabss 7245 The domain of an operation...
rnoprab 7246 The range of an operation ...
rnoprab2 7247 The range of a restricted ...
reldmoprab 7248 The domain of an operation...
oprabss 7249 Structure of an operation ...
eloprabga 7250 The law of concretion for ...
eloprabg 7251 The law of concretion for ...
ssoprab2i 7252 Inference of operation cla...
mpov 7253 Operation with universal d...
mpomptx 7254 Express a two-argument fun...
mpompt 7255 Express a two-argument fun...
mpodifsnif 7256 A mapping with two argumen...
mposnif 7257 A mapping with two argumen...
fconstmpo 7258 Representation of a consta...
resoprab 7259 Restriction of an operatio...
resoprab2 7260 Restriction of an operator...
resmpo 7261 Restriction of the mapping...
funoprabg 7262 "At most one" is a suffici...
funoprab 7263 "At most one" is a suffici...
fnoprabg 7264 Functionality and domain o...
mpofun 7265 The maps-to notation for a...
fnoprab 7266 Functionality and domain o...
ffnov 7267 An operation maps to a cla...
fovcl 7268 Closure law for an operati...
eqfnov 7269 Equality of two operations...
eqfnov2 7270 Two operators with the sam...
fnov 7271 Representation of a functi...
mpo2eqb 7272 Bidirectional equality the...
rnmpo 7273 The range of an operation ...
reldmmpo 7274 The domain of an operation...
elrnmpog 7275 Membership in the range of...
elrnmpo 7276 Membership in the range of...
elrnmpores 7277 Membership in the range of...
ralrnmpo 7278 A restricted quantifier ov...
rexrnmpo 7279 A restricted quantifier ov...
ovid 7280 The value of an operation ...
ovidig 7281 The value of an operation ...
ovidi 7282 The value of an operation ...
ov 7283 The value of an operation ...
ovigg 7284 The value of an operation ...
ovig 7285 The value of an operation ...
ovmpt4g 7286 Value of a function given ...
ovmpos 7287 Value of a function given ...
ov2gf 7288 The value of an operation ...
ovmpodxf 7289 Value of an operation give...
ovmpodx 7290 Value of an operation give...
ovmpod 7291 Value of an operation give...
ovmpox 7292 The value of an operation ...
ovmpoga 7293 Value of an operation give...
ovmpoa 7294 Value of an operation give...
ovmpodf 7295 Alternate deduction versio...
ovmpodv 7296 Alternate deduction versio...
ovmpodv2 7297 Alternate deduction versio...
ovmpog 7298 Value of an operation give...
ovmpo 7299 Value of an operation give...
ov3 7300 The value of an operation ...
ov6g 7301 The value of an operation ...
ovg 7302 The value of an operation ...
ovres 7303 The value of a restricted ...
ovresd 7304 Lemma for converting metri...
oprres 7305 The restriction of an oper...
oprssov 7306 The value of a member of t...
fovrn 7307 An operation's value belon...
fovrnda 7308 An operation's value belon...
fovrnd 7309 An operation's value belon...
fnrnov 7310 The range of an operation ...
foov 7311 An onto mapping of an oper...
fnovrn 7312 An operation's value belon...
ovelrn 7313 A member of an operation's...
funimassov 7314 Membership relation for th...
ovelimab 7315 Operation value in an imag...
ovima0 7316 An operation value is a me...
ovconst2 7317 The value of a constant op...
oprssdm 7318 Domain of closure of an op...
nssdmovg 7319 The value of an operation ...
ndmovg 7320 The value of an operation ...
ndmov 7321 The value of an operation ...
ndmovcl 7322 The closure of an operatio...
ndmovrcl 7323 Reverse closure law, when ...
ndmovcom 7324 Any operation is commutati...
ndmovass 7325 Any operation is associati...
ndmovdistr 7326 Any operation is distribut...
ndmovord 7327 Elimination of redundant a...
ndmovordi 7328 Elimination of redundant a...
caovclg 7329 Convert an operation closu...
caovcld 7330 Convert an operation closu...
caovcl 7331 Convert an operation closu...
caovcomg 7332 Convert an operation commu...
caovcomd 7333 Convert an operation commu...
caovcom 7334 Convert an operation commu...
caovassg 7335 Convert an operation assoc...
caovassd 7336 Convert an operation assoc...
caovass 7337 Convert an operation assoc...
caovcang 7338 Convert an operation cance...
caovcand 7339 Convert an operation cance...
caovcanrd 7340 Commute the arguments of a...
caovcan 7341 Convert an operation cance...
caovordig 7342 Convert an operation order...
caovordid 7343 Convert an operation order...
caovordg 7344 Convert an operation order...
caovordd 7345 Convert an operation order...
caovord2d 7346 Operation ordering law wit...
caovord3d 7347 Ordering law. (Contribute...
caovord 7348 Convert an operation order...
caovord2 7349 Operation ordering law wit...
caovord3 7350 Ordering law. (Contribute...
caovdig 7351 Convert an operation distr...
caovdid 7352 Convert an operation distr...
caovdir2d 7353 Convert an operation distr...
caovdirg 7354 Convert an operation rever...
caovdird 7355 Convert an operation distr...
caovdi 7356 Convert an operation distr...
caov32d 7357 Rearrange arguments in a c...
caov12d 7358 Rearrange arguments in a c...
caov31d 7359 Rearrange arguments in a c...
caov13d 7360 Rearrange arguments in a c...
caov4d 7361 Rearrange arguments in a c...
caov411d 7362 Rearrange arguments in a c...
caov42d 7363 Rearrange arguments in a c...
caov32 7364 Rearrange arguments in a c...
caov12 7365 Rearrange arguments in a c...
caov31 7366 Rearrange arguments in a c...
caov13 7367 Rearrange arguments in a c...
caov4 7368 Rearrange arguments in a c...
caov411 7369 Rearrange arguments in a c...
caov42 7370 Rearrange arguments in a c...
caovdir 7371 Reverse distributive law. ...
caovdilem 7372 Lemma used by real number ...
caovlem2 7373 Lemma used in real number ...
caovmo 7374 Uniqueness of inverse elem...
mpondm0 7375 The value of an operation ...
elmpocl 7376 If a two-parameter class i...
elmpocl1 7377 If a two-parameter class i...
elmpocl2 7378 If a two-parameter class i...
elovmpo 7379 Utility lemma for two-para...
elovmporab 7380 Implications for the value...
elovmporab1w 7381 Version of ~ elovmporab1 w...
elovmporab1 7382 Implications for the value...
2mpo0 7383 If the operation value of ...
relmptopab 7384 Any function to sets of or...
f1ocnvd 7385 Describe an implicit one-t...
f1od 7386 Describe an implicit one-t...
f1ocnv2d 7387 Describe an implicit one-t...
f1o2d 7388 Describe an implicit one-t...
f1opw2 7389 A one-to-one mapping induc...
f1opw 7390 A one-to-one mapping induc...
elovmpt3imp 7391 If the value of a function...
ovmpt3rab1 7392 The value of an operation ...
ovmpt3rabdm 7393 If the value of a function...
elovmpt3rab1 7394 Implications for the value...
elovmpt3rab 7395 Implications for the value...
ofeq 7400 Equality theorem for funct...
ofreq 7401 Equality theorem for funct...
ofexg 7402 A function operation restr...
nfof 7403 Hypothesis builder for fun...
nfofr 7404 Hypothesis builder for fun...
offval 7405 Value of an operation appl...
ofrfval 7406 Value of a relation applie...
ofval 7407 Evaluate a function operat...
ofrval 7408 Exhibit a function relatio...
offn 7409 The function operation pro...
offval2f 7410 The function operation exp...
ofmresval 7411 Value of a restriction of ...
fnfvof 7412 Function value of a pointw...
off 7413 The function operation pro...
ofres 7414 Restrict the operands of a...
offval2 7415 The function operation exp...
ofrfval2 7416 The function relation acti...
ofmpteq 7417 Value of a pointwise opera...
ofco 7418 The composition of a funct...
offveq 7419 Convert an identity of the...
offveqb 7420 Equivalent expressions for...
ofc1 7421 Left operation by a consta...
ofc2 7422 Right operation by a const...
ofc12 7423 Function operation on two ...
caofref 7424 Transfer a reflexive law t...
caofinvl 7425 Transfer a left inverse la...
caofid0l 7426 Transfer a left identity l...
caofid0r 7427 Transfer a right identity ...
caofid1 7428 Transfer a right absorptio...
caofid2 7429 Transfer a right absorptio...
caofcom 7430 Transfer a commutative law...
caofrss 7431 Transfer a relation subset...
caofass 7432 Transfer an associative la...
caoftrn 7433 Transfer a transitivity la...
caofdi 7434 Transfer a distributive la...
caofdir 7435 Transfer a reverse distrib...
caonncan 7436 Transfer ~ nncan -shaped l...
relrpss 7439 The proper subset relation...
brrpssg 7440 The proper subset relation...
brrpss 7441 The proper subset relation...
porpss 7442 Every class is partially o...
sorpss 7443 Express strict ordering un...
sorpssi 7444 Property of a chain of set...
sorpssun 7445 A chain of sets is closed ...
sorpssin 7446 A chain of sets is closed ...
sorpssuni 7447 In a chain of sets, a maxi...
sorpssint 7448 In a chain of sets, a mini...
sorpsscmpl 7449 The componentwise compleme...
zfun 7451 Axiom of Union expressed w...
axun2 7452 A variant of the Axiom of ...
uniex2 7453 The Axiom of Union using t...
uniex 7454 The Axiom of Union in clas...
vuniex 7455 The union of a setvar is a...
uniexg 7456 The ZF Axiom of Union in c...
uniexd 7457 Deduction version of the Z...
unex 7458 The union of two sets is a...
tpex 7459 An unordered triple of cla...
unexb 7460 Existence of union is equi...
unexg 7461 A union of two sets is a s...
xpexg 7462 The Cartesian product of t...
xpexd 7463 The Cartesian product of t...
3xpexg 7464 The Cartesian product of t...
xpex 7465 The Cartesian product of t...
sqxpexg 7466 The Cartesian square of a ...
abnexg 7467 Sufficient condition for a...
abnex 7468 Sufficient condition for a...
snnex 7469 The class of all singleton...
pwnex 7470 The class of all power set...
difex2 7471 If the subtrahend of a cla...
difsnexi 7472 If the difference of a cla...
uniuni 7473 Expression for double unio...
uniexr 7474 Converse of the Axiom of U...
uniexb 7475 The Axiom of Union and its...
pwexr 7476 Converse of the Axiom of P...
pwexb 7477 The Axiom of Power Sets an...
elpwpwel 7478 A class belongs to a doubl...
eldifpw 7479 Membership in a power clas...
elpwun 7480 Membership in the power cl...
pwuncl 7481 Power classes are closed u...
iunpw 7482 An indexed union of a powe...
fr3nr 7483 A well-founded relation ha...
epne3 7484 A well-founded class conta...
dfwe2 7485 Alternate definition of we...
epweon 7486 The membership relation we...
ordon 7487 The class of all ordinal n...
onprc 7488 No set contains all ordina...
ssorduni 7489 The union of a class of or...
ssonuni 7490 The union of a set of ordi...
ssonunii 7491 The union of a set of ordi...
ordeleqon 7492 A way to express the ordin...
ordsson 7493 Any ordinal class is a sub...
onss 7494 An ordinal number is a sub...
predon 7495 The predecessor of an ordi...
ssonprc 7496 Two ways of saying a class...
onuni 7497 The union of an ordinal nu...
orduni 7498 The union of an ordinal cl...
onint 7499 The intersection (infimum)...
onint0 7500 The intersection of a clas...
onssmin 7501 A nonempty class of ordina...
onminesb 7502 If a property is true for ...
onminsb 7503 If a property is true for ...
oninton 7504 The intersection of a none...
onintrab 7505 The intersection of a clas...
onintrab2 7506 An existence condition equ...
onnmin 7507 No member of a set of ordi...
onnminsb 7508 An ordinal number smaller ...
oneqmin 7509 A way to show that an ordi...
uniordint 7510 The union of a set of ordi...
onminex 7511 If a wff is true for an or...
sucon 7512 The class of all ordinal n...
sucexb 7513 A successor exists iff its...
sucexg 7514 The successor of a set is ...
sucex 7515 The successor of a set is ...
onmindif2 7516 The minimum of a class of ...
suceloni 7517 The successor of an ordina...
ordsuc 7518 The successor of an ordina...
ordpwsuc 7519 The collection of ordinals...
onpwsuc 7520 The collection of ordinal ...
sucelon 7521 The successor of an ordina...
ordsucss 7522 The successor of an elemen...
onpsssuc 7523 An ordinal number is a pro...
ordelsuc 7524 A set belongs to an ordina...
onsucmin 7525 The successor of an ordina...
ordsucelsuc 7526 Membership is inherited by...
ordsucsssuc 7527 The subclass relationship ...
ordsucuniel 7528 Given an element ` A ` of ...
ordsucun 7529 The successor of the maxim...
ordunpr 7530 The maximum of two ordinal...
ordunel 7531 The maximum of two ordinal...
onsucuni 7532 A class of ordinal numbers...
ordsucuni 7533 An ordinal class is a subc...
orduniorsuc 7534 An ordinal class is either...
unon 7535 The class of all ordinal n...
ordunisuc 7536 An ordinal class is equal ...
orduniss2 7537 The union of the ordinal s...
onsucuni2 7538 A successor ordinal is the...
0elsuc 7539 The successor of an ordina...
limon 7540 The class of ordinal numbe...
onssi 7541 An ordinal number is a sub...
onsuci 7542 The successor of an ordina...
onuniorsuci 7543 An ordinal number is eithe...
onuninsuci 7544 A limit ordinal is not a s...
onsucssi 7545 A set belongs to an ordina...
nlimsucg 7546 A successor is not a limit...
orduninsuc 7547 An ordinal equal to its un...
ordunisuc2 7548 An ordinal equal to its un...
ordzsl 7549 An ordinal is zero, a succ...
onzsl 7550 An ordinal number is zero,...
dflim3 7551 An alternate definition of...
dflim4 7552 An alternate definition of...
limsuc 7553 The successor of a member ...
limsssuc 7554 A class includes a limit o...
nlimon 7555 Two ways to express the cl...
limuni3 7556 The union of a nonempty cl...
tfi 7557 The Principle of Transfini...
tfis 7558 Transfinite Induction Sche...
tfis2f 7559 Transfinite Induction Sche...
tfis2 7560 Transfinite Induction Sche...
tfis3 7561 Transfinite Induction Sche...
tfisi 7562 A transfinite induction sc...
tfinds 7563 Principle of Transfinite I...
tfindsg 7564 Transfinite Induction (inf...
tfindsg2 7565 Transfinite Induction (inf...
tfindes 7566 Transfinite Induction with...
tfinds2 7567 Transfinite Induction (inf...
tfinds3 7568 Principle of Transfinite I...
dfom2 7571 An alternate definition of...
elom 7572 Membership in omega. The ...
omsson 7573 Omega is a subset of ` On ...
limomss 7574 The class of natural numbe...
nnon 7575 A natural number is an ord...
nnoni 7576 A natural number is an ord...
nnord 7577 A natural number is ordina...
ordom 7578 Omega is ordinal. Theorem...
elnn 7579 A member of a natural numb...
omon 7580 The class of natural numbe...
omelon2 7581 Omega is an ordinal number...
nnlim 7582 A natural number is not a ...
omssnlim 7583 The class of natural numbe...
limom 7584 Omega is a limit ordinal. ...
peano2b 7585 A class belongs to omega i...
nnsuc 7586 A nonzero natural number i...
omsucne 7587 A natural number is not th...
ssnlim 7588 An ordinal subclass of non...
omsinds 7589 Strong (or "total") induct...
peano1 7590 Zero is a natural number. ...
peano2 7591 The successor of any natur...
peano3 7592 The successor of any natur...
peano4 7593 Two natural numbers are eq...
peano5 7594 The induction postulate: a...
nn0suc 7595 A natural number is either...
find 7596 The Principle of Finite In...
finds 7597 Principle of Finite Induct...
findsg 7598 Principle of Finite Induct...
finds2 7599 Principle of Finite Induct...
finds1 7600 Principle of Finite Induct...
findes 7601 Finite induction with expl...
dmexg 7602 The domain of a set is a s...
rnexg 7603 The range of a set is a se...
dmexd 7604 The domain of a set is a s...
dmex 7605 The domain of a set is a s...
rnex 7606 The range of a set is a se...
iprc 7607 The identity function is a...
resiexg 7608 The existence of a restric...
imaexg 7609 The image of a set is a se...
imaex 7610 The image of a set is a se...
exse2 7611 Any set relation is set-li...
xpexr 7612 If a Cartesian product is ...
xpexr2 7613 If a nonempty Cartesian pr...
xpexcnv 7614 A condition where the conv...
soex 7615 If the relation in a stric...
elxp4 7616 Membership in a Cartesian ...
elxp5 7617 Membership in a Cartesian ...
cnvexg 7618 The converse of a set is a...
cnvex 7619 The converse of a set is a...
relcnvexb 7620 A relation is a set iff it...
f1oexrnex 7621 If the range of a 1-1 onto...
f1oexbi 7622 There is a one-to-one onto...
coexg 7623 The composition of two set...
coex 7624 The composition of two set...
funcnvuni 7625 The union of a chain (with...
fun11uni 7626 The union of a chain (with...
fex2 7627 A function with bounded do...
fabexg 7628 Existence of a set of func...
fabex 7629 Existence of a set of func...
dmfex 7630 If a mapping is a set, its...
f1oabexg 7631 The class of all 1-1-onto ...
fiunlem 7632 Lemma for ~ fiun and ~ f1i...
fiun 7633 The union of a chain (with...
f1iun 7634 The union of a chain (with...
fviunfun 7635 The function value of an i...
ffoss 7636 Relationship between a map...
f11o 7637 Relationship between one-t...
resfunexgALT 7638 Alternate proof of ~ resfu...
cofunexg 7639 Existence of a composition...
cofunex2g 7640 Existence of a composition...
fnexALT 7641 Alternate proof of ~ fnex ...
funexw 7642 Weak version of ~ funex th...
mptexw 7643 Weak version of ~ mptex th...
funrnex 7644 If the domain of a functio...
zfrep6 7645 A version of the Axiom of ...
fornex 7646 If the domain of an onto f...
f1dmex 7647 If the codomain of a one-t...
f1ovv 7648 The range of a 1-1 onto fu...
fvclex 7649 Existence of the class of ...
fvresex 7650 Existence of the class of ...
abrexexg 7651 Existence of a class abstr...
abrexex 7652 Existence of a class abstr...
iunexg 7653 The existence of an indexe...
abrexex2g 7654 Existence of an existentia...
opabex3d 7655 Existence of an ordered pa...
opabex3rd 7656 Existence of an ordered pa...
opabex3 7657 Existence of an ordered pa...
iunex 7658 The existence of an indexe...
abrexex2 7659 Existence of an existentia...
abexssex 7660 Existence of a class abstr...
abexex 7661 A condition where a class ...
f1oweALT 7662 Alternate proof of ~ f1owe...
wemoiso 7663 Thus, there is at most one...
wemoiso2 7664 Thus, there is at most one...
oprabexd 7665 Existence of an operator a...
oprabex 7666 Existence of an operation ...
oprabex3 7667 Existence of an operation ...
oprabrexex2 7668 Existence of an existentia...
ab2rexex 7669 Existence of a class abstr...
ab2rexex2 7670 Existence of an existentia...
xpexgALT 7671 Alternate proof of ~ xpexg...
offval3 7672 General value of ` ( F oF ...
offres 7673 Pointwise combination comm...
ofmres 7674 Equivalent expressions for...
ofmresex 7675 Existence of a restriction...
1stval 7680 The value of the function ...
2ndval 7681 The value of the function ...
1stnpr 7682 Value of the first-member ...
2ndnpr 7683 Value of the second-member...
1st0 7684 The value of the first-mem...
2nd0 7685 The value of the second-me...
op1st 7686 Extract the first member o...
op2nd 7687 Extract the second member ...
op1std 7688 Extract the first member o...
op2ndd 7689 Extract the second member ...
op1stg 7690 Extract the first member o...
op2ndg 7691 Extract the second member ...
ot1stg 7692 Extract the first member o...
ot2ndg 7693 Extract the second member ...
ot3rdg 7694 Extract the third member o...
1stval2 7695 Alternate value of the fun...
2ndval2 7696 Alternate value of the fun...
oteqimp 7697 The components of an order...
fo1st 7698 The ` 1st ` function maps ...
fo2nd 7699 The ` 2nd ` function maps ...
br1steqg 7700 Uniqueness condition for t...
br2ndeqg 7701 Uniqueness condition for t...
f1stres 7702 Mapping of a restriction o...
f2ndres 7703 Mapping of a restriction o...
fo1stres 7704 Onto mapping of a restrict...
fo2ndres 7705 Onto mapping of a restrict...
1st2val 7706 Value of an alternate defi...
2nd2val 7707 Value of an alternate defi...
1stcof 7708 Composition of the first m...
2ndcof 7709 Composition of the second ...
xp1st 7710 Location of the first elem...
xp2nd 7711 Location of the second ele...
elxp6 7712 Membership in a Cartesian ...
elxp7 7713 Membership in a Cartesian ...
eqopi 7714 Equality with an ordered p...
xp2 7715 Representation of Cartesia...
unielxp 7716 The membership relation fo...
1st2nd2 7717 Reconstruction of a member...
1st2ndb 7718 Reconstruction of an order...
xpopth 7719 An ordered pair theorem fo...
eqop 7720 Two ways to express equali...
eqop2 7721 Two ways to express equali...
op1steq 7722 Two ways of expressing tha...
opreuopreu 7723 There is a unique ordered ...
el2xptp 7724 A member of a nested Carte...
el2xptp0 7725 A member of a nested Carte...
2nd1st 7726 Swap the members of an ord...
1st2nd 7727 Reconstruction of a member...
1stdm 7728 The first ordered pair com...
2ndrn 7729 The second ordered pair co...
1st2ndbr 7730 Express an element of a re...
releldm2 7731 Two ways of expressing mem...
reldm 7732 An expression for the doma...
releldmdifi 7733 One way of expressing memb...
funfv1st2nd 7734 The function value for the...
funelss 7735 If the first component of ...
funeldmdif 7736 Two ways of expressing mem...
sbcopeq1a 7737 Equality theorem for subst...
csbopeq1a 7738 Equality theorem for subst...
dfopab2 7739 A way to define an ordered...
dfoprab3s 7740 A way to define an operati...
dfoprab3 7741 Operation class abstractio...
dfoprab4 7742 Operation class abstractio...
dfoprab4f 7743 Operation class abstractio...
opabex2 7744 Condition for an operation...
opabn1stprc 7745 An ordered-pair class abst...
opiota 7746 The property of a uniquely...
cnvoprab 7747 The converse of a class ab...
dfxp3 7748 Define the Cartesian produ...
elopabi 7749 A consequence of membershi...
eloprabi 7750 A consequence of membershi...
mpomptsx 7751 Express a two-argument fun...
mpompts 7752 Express a two-argument fun...
dmmpossx 7753 The domain of a mapping is...
fmpox 7754 Functionality, domain and ...
fmpo 7755 Functionality, domain and ...
fnmpo 7756 Functionality and domain o...
fnmpoi 7757 Functionality and domain o...
dmmpo 7758 Domain of a class given by...
ovmpoelrn 7759 An operation's value belon...
dmmpoga 7760 Domain of an operation giv...
dmmpog 7761 Domain of an operation giv...
mpoexxg 7762 Existence of an operation ...
mpoexg 7763 Existence of an operation ...
mpoexga 7764 If the domain of an operat...
mpoexw 7765 Weak version of ~ mpoex th...
mpoex 7766 If the domain of an operat...
mptmpoopabbrd 7767 The operation value of a f...
mptmpoopabovd 7768 The operation value of a f...
el2mpocsbcl 7769 If the operation value of ...
el2mpocl 7770 If the operation value of ...
fnmpoovd 7771 A function with a Cartesia...
offval22 7772 The function operation exp...
brovpreldm 7773 If a binary relation holds...
bropopvvv 7774 If a binary relation holds...
bropfvvvvlem 7775 Lemma for ~ bropfvvvv . (...
bropfvvvv 7776 If a binary relation holds...
ovmptss 7777 If all the values of the m...
relmpoopab 7778 Any function to sets of or...
fmpoco 7779 Composition of two functio...
oprabco 7780 Composition of a function ...
oprab2co 7781 Composition of operator ab...
df1st2 7782 An alternate possible defi...
df2nd2 7783 An alternate possible defi...
1stconst 7784 The mapping of a restricti...
2ndconst 7785 The mapping of a restricti...
dfmpo 7786 Alternate definition for t...
mposn 7787 An operation (in maps-to n...
curry1 7788 Composition with ` ``' ( 2...
curry1val 7789 The value of a curried fun...
curry1f 7790 Functionality of a curried...
curry2 7791 Composition with ` ``' ( 1...
curry2f 7792 Functionality of a curried...
curry2val 7793 The value of a curried fun...
cnvf1olem 7794 Lemma for ~ cnvf1o . (Con...
cnvf1o 7795 Describe a function that m...
fparlem1 7796 Lemma for ~ fpar . (Contr...
fparlem2 7797 Lemma for ~ fpar . (Contr...
fparlem3 7798 Lemma for ~ fpar . (Contr...
fparlem4 7799 Lemma for ~ fpar . (Contr...
fpar 7800 Merge two functions in par...
fsplit 7801 A function that can be use...
fsplitOLD 7802 Obsolete proof of ~ fsplit...
fsplitfpar 7803 Merge two functions with a...
offsplitfpar 7804 Express the function opera...
f2ndf 7805 The ` 2nd ` (second compon...
fo2ndf 7806 The ` 2nd ` (second compon...
f1o2ndf1 7807 The ` 2nd ` (second compon...
algrflem 7808 Lemma for ~ algrf and rela...
frxp 7809 A lexicographical ordering...
xporderlem 7810 Lemma for lexicographical ...
poxp 7811 A lexicographical ordering...
soxp 7812 A lexicographical ordering...
wexp 7813 A lexicographical ordering...
fnwelem 7814 Lemma for ~ fnwe . (Contr...
fnwe 7815 A variant on lexicographic...
fnse 7816 Condition for the well-ord...
fvproj 7817 Value of a function on ord...
fimaproj 7818 Image of a cartesian produ...
suppval 7821 The value of the operation...
supp0prc 7822 The support of a class is ...
suppvalbr 7823 The value of the operation...
supp0 7824 The support of the empty s...
suppval1 7825 The value of the operation...
suppvalfn 7826 The value of the operation...
elsuppfn 7827 An element of the support ...
cnvimadfsn 7828 The support of functions "...
suppimacnvss 7829 The support of functions "...
suppimacnv 7830 Support sets of functions ...
frnsuppeq 7831 Two ways of writing the su...
suppssdm 7832 The support of a function ...
suppsnop 7833 The support of a singleton...
snopsuppss 7834 The support of a singleton...
fvn0elsupp 7835 If the function value for ...
fvn0elsuppb 7836 The function value for a g...
rexsupp 7837 Existential quantification...
ressuppss 7838 The support of the restric...
suppun 7839 The support of a class/fun...
ressuppssdif 7840 The support of the restric...
mptsuppdifd 7841 The support of a function ...
mptsuppd 7842 The support of a function ...
extmptsuppeq 7843 The support of an extended...
suppfnss 7844 The support of a function ...
funsssuppss 7845 The support of a function ...
fnsuppres 7846 Two ways to express restri...
fnsuppeq0 7847 The support of a function ...
fczsupp0 7848 The support of a constant ...
suppss 7849 Show that the support of a...
suppssr 7850 A function is zero outside...
suppssov1 7851 Formula building theorem f...
suppssof1 7852 Formula building theorem f...
suppss2 7853 Show that the support of a...
suppsssn 7854 Show that the support of a...
suppssfv 7855 Formula building theorem f...
suppofssd 7856 Condition for the support ...
suppofss1d 7857 Condition for the support ...
suppofss2d 7858 Condition for the support ...
suppco 7859 The support of the composi...
suppcofnd 7860 The support of the composi...
supp0cosupp0 7861 The support of the composi...
supp0cosupp0OLD 7862 Obsolete version of ~ supp...
imacosupp 7863 The image of the support o...
imacosuppOLD 7864 Obsolete version of ~ imac...
opeliunxp2f 7865 Membership in a union of C...
mpoxeldm 7866 If there is an element of ...
mpoxneldm 7867 If the first argument of a...
mpoxopn0yelv 7868 If there is an element of ...
mpoxopynvov0g 7869 If the second argument of ...
mpoxopxnop0 7870 If the first argument of a...
mpoxopx0ov0 7871 If the first argument of a...
mpoxopxprcov0 7872 If the components of the f...
mpoxopynvov0 7873 If the second argument of ...
mpoxopoveq 7874 Value of an operation give...
mpoxopovel 7875 Element of the value of an...
mpoxopoveqd 7876 Value of an operation give...
brovex 7877 A binary relation of the v...
brovmpoex 7878 A binary relation of the v...
sprmpod 7879 The extension of a binary ...
tposss 7882 Subset theorem for transpo...
tposeq 7883 Equality theorem for trans...
tposeqd 7884 Equality theorem for trans...
tposssxp 7885 The transposition is a sub...
reltpos 7886 The transposition is a rel...
brtpos2 7887 Value of the transposition...
brtpos0 7888 The behavior of ` tpos ` w...
reldmtpos 7889 Necessary and sufficient c...
brtpos 7890 The transposition swaps ar...
ottpos 7891 The transposition swaps th...
relbrtpos 7892 The transposition swaps ar...
dmtpos 7893 The domain of ` tpos F ` w...
rntpos 7894 The range of ` tpos F ` wh...
tposexg 7895 The transposition of a set...
ovtpos 7896 The transposition swaps th...
tposfun 7897 The transposition of a fun...
dftpos2 7898 Alternate definition of ` ...
dftpos3 7899 Alternate definition of ` ...
dftpos4 7900 Alternate definition of ` ...
tpostpos 7901 Value of the double transp...
tpostpos2 7902 Value of the double transp...
tposfn2 7903 The domain of a transposit...
tposfo2 7904 Condition for a surjective...
tposf2 7905 The domain and range of a ...
tposf12 7906 Condition for an injective...
tposf1o2 7907 Condition of a bijective t...
tposfo 7908 The domain and range of a ...
tposf 7909 The domain and range of a ...
tposfn 7910 Functionality of a transpo...
tpos0 7911 Transposition of the empty...
tposco 7912 Transposition of a composi...
tpossym 7913 Two ways to say a function...
tposeqi 7914 Equality theorem for trans...
tposex 7915 A transposition is a set. ...
nftpos 7916 Hypothesis builder for tra...
tposoprab 7917 Transposition of a class o...
tposmpo 7918 Transposition of a two-arg...
tposconst 7919 The transposition of a con...
mpocurryd 7924 The currying of an operati...
mpocurryvald 7925 The value of a curried ope...
fvmpocurryd 7926 The value of the value of ...
pwuninel2 7929 Direct proof of ~ pwuninel...
pwuninel 7930 The power set of the union...
undefval 7931 Value of the undefined val...
undefnel2 7932 The undefined value genera...
undefnel 7933 The undefined value genera...
undefne0 7934 The undefined value genera...
wrecseq123 7937 General equality theorem f...
nfwrecs 7938 Bound-variable hypothesis ...
wrecseq1 7939 Equality theorem for the w...
wrecseq2 7940 Equality theorem for the w...
wrecseq3 7941 Equality theorem for the w...
wfr3g 7942 Functions defined by well-...
wfrlem1 7943 Lemma for well-founded rec...
wfrlem2 7944 Lemma for well-founded rec...
wfrlem3 7945 Lemma for well-founded rec...
wfrlem3a 7946 Lemma for well-founded rec...
wfrlem4 7947 Lemma for well-founded rec...
wfrlem5 7948 Lemma for well-founded rec...
wfrrel 7949 The well-founded recursion...
wfrdmss 7950 The domain of the well-fou...
wfrlem8 7951 Lemma for well-founded rec...
wfrdmcl 7952 Given ` F = wrecs ( R , A ...
wfrlem10 7953 Lemma for well-founded rec...
wfrfun 7954 The well-founded function ...
wfrlem12 7955 Lemma for well-founded rec...
wfrlem13 7956 Lemma for well-founded rec...
wfrlem14 7957 Lemma for well-founded rec...
wfrlem15 7958 Lemma for well-founded rec...
wfrlem16 7959 Lemma for well-founded rec...
wfrlem17 7960 Without using ~ ax-rep , s...
wfr2a 7961 A weak version of ~ wfr2 w...
wfr1 7962 The Principle of Well-Foun...
wfr2 7963 The Principle of Well-Foun...
wfr3 7964 The principle of Well-Foun...
iunon 7965 The indexed union of a set...
iinon 7966 The nonempty indexed inter...
onfununi 7967 A property of functions on...
onovuni 7968 A variant of ~ onfununi fo...
onoviun 7969 A variant of ~ onovuni wit...
onnseq 7970 There are no length ` _om ...
dfsmo2 7973 Alternate definition of a ...
issmo 7974 Conditions for which ` A `...
issmo2 7975 Alternate definition of a ...
smoeq 7976 Equality theorem for stric...
smodm 7977 The domain of a strictly m...
smores 7978 A strictly monotone functi...
smores3 7979 A strictly monotone functi...
smores2 7980 A strictly monotone ordina...
smodm2 7981 The domain of a strictly m...
smofvon2 7982 The function values of a s...
iordsmo 7983 The identity relation rest...
smo0 7984 The null set is a strictly...
smofvon 7985 If ` B ` is a strictly mon...
smoel 7986 If ` x ` is less than ` y ...
smoiun 7987 The value of a strictly mo...
smoiso 7988 If ` F ` is an isomorphism...
smoel2 7989 A strictly monotone ordina...
smo11 7990 A strictly monotone ordina...
smoord 7991 A strictly monotone ordina...
smoword 7992 A strictly monotone ordina...
smogt 7993 A strictly monotone ordina...
smorndom 7994 The range of a strictly mo...
smoiso2 7995 The strictly monotone ordi...
dfrecs3 7998 The old definition of tran...
recseq 7999 Equality theorem for ` rec...
nfrecs 8000 Bound-variable hypothesis ...
tfrlem1 8001 A technical lemma for tran...
tfrlem3a 8002 Lemma for transfinite recu...
tfrlem3 8003 Lemma for transfinite recu...
tfrlem4 8004 Lemma for transfinite recu...
tfrlem5 8005 Lemma for transfinite recu...
recsfval 8006 Lemma for transfinite recu...
tfrlem6 8007 Lemma for transfinite recu...
tfrlem7 8008 Lemma for transfinite recu...
tfrlem8 8009 Lemma for transfinite recu...
tfrlem9 8010 Lemma for transfinite recu...
tfrlem9a 8011 Lemma for transfinite recu...
tfrlem10 8012 Lemma for transfinite recu...
tfrlem11 8013 Lemma for transfinite recu...
tfrlem12 8014 Lemma for transfinite recu...
tfrlem13 8015 Lemma for transfinite recu...
tfrlem14 8016 Lemma for transfinite recu...
tfrlem15 8017 Lemma for transfinite recu...
tfrlem16 8018 Lemma for finite recursion...
tfr1a 8019 A weak version of ~ tfr1 w...
tfr2a 8020 A weak version of ~ tfr2 w...
tfr2b 8021 Without assuming ~ ax-rep ...
tfr1 8022 Principle of Transfinite R...
tfr2 8023 Principle of Transfinite R...
tfr3 8024 Principle of Transfinite R...
tfr1ALT 8025 Alternate proof of ~ tfr1 ...
tfr2ALT 8026 Alternate proof of ~ tfr2 ...
tfr3ALT 8027 Alternate proof of ~ tfr3 ...
recsfnon 8028 Strong transfinite recursi...
recsval 8029 Strong transfinite recursi...
tz7.44lem1 8030 ` G ` is a function. Lemm...
tz7.44-1 8031 The value of ` F ` at ` (/...
tz7.44-2 8032 The value of ` F ` at a su...
tz7.44-3 8033 The value of ` F ` at a li...
rdgeq1 8036 Equality theorem for the r...
rdgeq2 8037 Equality theorem for the r...
rdgeq12 8038 Equality theorem for the r...
nfrdg 8039 Bound-variable hypothesis ...
rdglem1 8040 Lemma used with the recurs...
rdgfun 8041 The recursive definition g...
rdgdmlim 8042 The domain of the recursiv...
rdgfnon 8043 The recursive definition g...
rdgvalg 8044 Value of the recursive def...
rdgval 8045 Value of the recursive def...
rdg0 8046 The initial value of the r...
rdgseg 8047 The initial segments of th...
rdgsucg 8048 The value of the recursive...
rdgsuc 8049 The value of the recursive...
rdglimg 8050 The value of the recursive...
rdglim 8051 The value of the recursive...
rdg0g 8052 The initial value of the r...
rdgsucmptf 8053 The value of the recursive...
rdgsucmptnf 8054 The value of the recursive...
rdgsucmpt2 8055 This version of ~ rdgsucmp...
rdgsucmpt 8056 The value of the recursive...
rdglim2 8057 The value of the recursive...
rdglim2a 8058 The value of the recursive...
frfnom 8059 The function generated by ...
fr0g 8060 The initial value resultin...
frsuc 8061 The successor value result...
frsucmpt 8062 The successor value result...
frsucmptn 8063 The value of the finite re...
frsucmpt2w 8064 Version of ~ frsucmpt2 wit...
frsucmpt2 8065 The successor value result...
tz7.48lem 8066 A way of showing an ordina...
tz7.48-2 8067 Proposition 7.48(2) of [Ta...
tz7.48-1 8068 Proposition 7.48(1) of [Ta...
tz7.48-3 8069 Proposition 7.48(3) of [Ta...
tz7.49 8070 Proposition 7.49 of [Takeu...
tz7.49c 8071 Corollary of Proposition 7...
seqomlem0 8074 Lemma for ` seqom ` . Cha...
seqomlem1 8075 Lemma for ` seqom ` . The...
seqomlem2 8076 Lemma for ` seqom ` . (Co...
seqomlem3 8077 Lemma for ` seqom ` . (Co...
seqomlem4 8078 Lemma for ` seqom ` . (Co...
seqomeq12 8079 Equality theorem for ` seq...
fnseqom 8080 An index-aware recursive d...
seqom0g 8081 Value of an index-aware re...
seqomsuc 8082 Value of an index-aware re...
omsucelsucb 8083 Membership is inherited by...
1on 8098 Ordinal 1 is an ordinal nu...
1oex 8099 Ordinal 1 is a set. (Cont...
2on 8100 Ordinal 2 is an ordinal nu...
2oex 8101 ` 2o ` is a set. (Contrib...
2on0 8102 Ordinal two is not zero. ...
3on 8103 Ordinal 3 is an ordinal nu...
4on 8104 Ordinal 3 is an ordinal nu...
df1o2 8105 Expanded value of the ordi...
df2o3 8106 Expanded value of the ordi...
df2o2 8107 Expanded value of the ordi...
1n0 8108 Ordinal one is not equal t...
xp01disj 8109 Cartesian products with th...
xp01disjl 8110 Cartesian products with th...
ordgt0ge1 8111 Two ways to express that a...
ordge1n0 8112 An ordinal greater than or...
el1o 8113 Membership in ordinal one....
dif1o 8114 Two ways to say that ` A `...
ondif1 8115 Two ways to say that ` A `...
ondif2 8116 Two ways to say that ` A `...
2oconcl 8117 Closure of the pair swappi...
0lt1o 8118 Ordinal zero is less than ...
dif20el 8119 An ordinal greater than on...
0we1 8120 The empty set is a well-or...
brwitnlem 8121 Lemma for relations which ...
fnoa 8122 Functionality and domain o...
fnom 8123 Functionality and domain o...
fnoe 8124 Functionality and domain o...
oav 8125 Value of ordinal addition....
omv 8126 Value of ordinal multiplic...
oe0lem 8127 A helper lemma for ~ oe0 a...
oev 8128 Value of ordinal exponenti...
oevn0 8129 Value of ordinal exponenti...
oa0 8130 Addition with zero. Propo...
om0 8131 Ordinal multiplication wit...
oe0m 8132 Ordinal exponentiation wit...
om0x 8133 Ordinal multiplication wit...
oe0m0 8134 Ordinal exponentiation wit...
oe0m1 8135 Ordinal exponentiation wit...
oe0 8136 Ordinal exponentiation wit...
oev2 8137 Alternate value of ordinal...
oasuc 8138 Addition with successor. ...
oesuclem 8139 Lemma for ~ oesuc . (Cont...
omsuc 8140 Multiplication with succes...
oesuc 8141 Ordinal exponentiation wit...
onasuc 8142 Addition with successor. ...
onmsuc 8143 Multiplication with succes...
onesuc 8144 Exponentiation with a succ...
oa1suc 8145 Addition with 1 is same as...
oalim 8146 Ordinal addition with a li...
omlim 8147 Ordinal multiplication wit...
oelim 8148 Ordinal exponentiation wit...
oacl 8149 Closure law for ordinal ad...
omcl 8150 Closure law for ordinal mu...
oecl 8151 Closure law for ordinal ex...
oa0r 8152 Ordinal addition with zero...
om0r 8153 Ordinal multiplication wit...
o1p1e2 8154 1 + 1 = 2 for ordinal numb...
o2p2e4 8155 2 + 2 = 4 for ordinal numb...
o2p2e4OLD 8156 2 + 2 = 4 for ordinal numb...
om1 8157 Ordinal multiplication wit...
om1r 8158 Ordinal multiplication wit...
oe1 8159 Ordinal exponentiation wit...
oe1m 8160 Ordinal exponentiation wit...
oaordi 8161 Ordering property of ordin...
oaord 8162 Ordering property of ordin...
oacan 8163 Left cancellation law for ...
oaword 8164 Weak ordering property of ...
oawordri 8165 Weak ordering property of ...
oaord1 8166 An ordinal is less than it...
oaword1 8167 An ordinal is less than or...
oaword2 8168 An ordinal is less than or...
oawordeulem 8169 Lemma for ~ oawordex . (C...
oawordeu 8170 Existence theorem for weak...
oawordexr 8171 Existence theorem for weak...
oawordex 8172 Existence theorem for weak...
oaordex 8173 Existence theorem for orde...
oa00 8174 An ordinal sum is zero iff...
oalimcl 8175 The ordinal sum with a lim...
oaass 8176 Ordinal addition is associ...
oarec 8177 Recursive definition of or...
oaf1o 8178 Left addition by a constan...
oacomf1olem 8179 Lemma for ~ oacomf1o . (C...
oacomf1o 8180 Define a bijection from ` ...
omordi 8181 Ordering property of ordin...
omord2 8182 Ordering property of ordin...
omord 8183 Ordering property of ordin...
omcan 8184 Left cancellation law for ...
omword 8185 Weak ordering property of ...
omwordi 8186 Weak ordering property of ...
omwordri 8187 Weak ordering property of ...
omword1 8188 An ordinal is less than or...
omword2 8189 An ordinal is less than or...
om00 8190 The product of two ordinal...
om00el 8191 The product of two nonzero...
omordlim 8192 Ordering involving the pro...
omlimcl 8193 The product of any nonzero...
odi 8194 Distributive law for ordin...
omass 8195 Multiplication of ordinal ...
oneo 8196 If an ordinal number is ev...
omeulem1 8197 Lemma for ~ omeu : existen...
omeulem2 8198 Lemma for ~ omeu : uniquen...
omopth2 8199 An ordered pair-like theor...
omeu 8200 The division algorithm for...
oen0 8201 Ordinal exponentiation wit...
oeordi 8202 Ordering law for ordinal e...
oeord 8203 Ordering property of ordin...
oecan 8204 Left cancellation law for ...
oeword 8205 Weak ordering property of ...
oewordi 8206 Weak ordering property of ...
oewordri 8207 Weak ordering property of ...
oeworde 8208 Ordinal exponentiation com...
oeordsuc 8209 Ordering property of ordin...
oelim2 8210 Ordinal exponentiation wit...
oeoalem 8211 Lemma for ~ oeoa . (Contr...
oeoa 8212 Sum of exponents law for o...
oeoelem 8213 Lemma for ~ oeoe . (Contr...
oeoe 8214 Product of exponents law f...
oelimcl 8215 The ordinal exponential wi...
oeeulem 8216 Lemma for ~ oeeu . (Contr...
oeeui 8217 The division algorithm for...
oeeu 8218 The division algorithm for...
nna0 8219 Addition with zero. Theor...
nnm0 8220 Multiplication with zero. ...
nnasuc 8221 Addition with successor. ...
nnmsuc 8222 Multiplication with succes...
nnesuc 8223 Exponentiation with a succ...
nna0r 8224 Addition to zero. Remark ...
nnm0r 8225 Multiplication with zero. ...
nnacl 8226 Closure of addition of nat...
nnmcl 8227 Closure of multiplication ...
nnecl 8228 Closure of exponentiation ...
nnacli 8229 ` _om ` is closed under ad...
nnmcli 8230 ` _om ` is closed under mu...
nnarcl 8231 Reverse closure law for ad...
nnacom 8232 Addition of natural number...
nnaordi 8233 Ordering property of addit...
nnaord 8234 Ordering property of addit...
nnaordr 8235 Ordering property of addit...
nnawordi 8236 Adding to both sides of an...
nnaass 8237 Addition of natural number...
nndi 8238 Distributive law for natur...
nnmass 8239 Multiplication of natural ...
nnmsucr 8240 Multiplication with succes...
nnmcom 8241 Multiplication of natural ...
nnaword 8242 Weak ordering property of ...
nnacan 8243 Cancellation law for addit...
nnaword1 8244 Weak ordering property of ...
nnaword2 8245 Weak ordering property of ...
nnmordi 8246 Ordering property of multi...
nnmord 8247 Ordering property of multi...
nnmword 8248 Weak ordering property of ...
nnmcan 8249 Cancellation law for multi...
nnmwordi 8250 Weak ordering property of ...
nnmwordri 8251 Weak ordering property of ...
nnawordex 8252 Equivalence for weak order...
nnaordex 8253 Equivalence for ordering. ...
1onn 8254 One is a natural number. ...
2onn 8255 The ordinal 2 is a natural...
3onn 8256 The ordinal 3 is a natural...
4onn 8257 The ordinal 4 is a natural...
1one2o 8258 Ordinal one is not ordinal...
oaabslem 8259 Lemma for ~ oaabs . (Cont...
oaabs 8260 Ordinal addition absorbs a...
oaabs2 8261 The absorption law ~ oaabs...
omabslem 8262 Lemma for ~ omabs . (Cont...
omabs 8263 Ordinal multiplication is ...
nnm1 8264 Multiply an element of ` _...
nnm2 8265 Multiply an element of ` _...
nn2m 8266 Multiply an element of ` _...
nnneo 8267 If a natural number is eve...
nneob 8268 A natural number is even i...
omsmolem 8269 Lemma for ~ omsmo . (Cont...
omsmo 8270 A strictly monotonic ordin...
omopthlem1 8271 Lemma for ~ omopthi . (Co...
omopthlem2 8272 Lemma for ~ omopthi . (Co...
omopthi 8273 An ordered pair theorem fo...
omopth 8274 An ordered pair theorem fo...
dfer2 8279 Alternate definition of eq...
dfec2 8281 Alternate definition of ` ...
ecexg 8282 An equivalence class modul...
ecexr 8283 A nonempty equivalence cla...
ereq1 8285 Equality theorem for equiv...
ereq2 8286 Equality theorem for equiv...
errel 8287 An equivalence relation is...
erdm 8288 The domain of an equivalen...
ercl 8289 Elementhood in the field o...
ersym 8290 An equivalence relation is...
ercl2 8291 Elementhood in the field o...
ersymb 8292 An equivalence relation is...
ertr 8293 An equivalence relation is...
ertrd 8294 A transitivity relation fo...
ertr2d 8295 A transitivity relation fo...
ertr3d 8296 A transitivity relation fo...
ertr4d 8297 A transitivity relation fo...
erref 8298 An equivalence relation is...
ercnv 8299 The converse of an equival...
errn 8300 The range and domain of an...
erssxp 8301 An equivalence relation is...
erex 8302 An equivalence relation is...
erexb 8303 An equivalence relation is...
iserd 8304 A reflexive, symmetric, tr...
iseri 8305 A reflexive, symmetric, tr...
iseriALT 8306 Alternate proof of ~ iseri...
brdifun 8307 Evaluate the incomparabili...
swoer 8308 Incomparability under a st...
swoord1 8309 The incomparability equiva...
swoord2 8310 The incomparability equiva...
swoso 8311 If the incomparability rel...
eqerlem 8312 Lemma for ~ eqer . (Contr...
eqer 8313 Equivalence relation invol...
ider 8314 The identity relation is a...
0er 8315 The empty set is an equiva...
eceq1 8316 Equality theorem for equiv...
eceq1d 8317 Equality theorem for equiv...
eceq2 8318 Equality theorem for equiv...
eceq2i 8319 Equality theorem for the `...
eceq2d 8320 Equality theorem for the `...
elecg 8321 Membership in an equivalen...
elec 8322 Membership in an equivalen...
relelec 8323 Membership in an equivalen...
ecss 8324 An equivalence class is a ...
ecdmn0 8325 A representative of a none...
ereldm 8326 Equality of equivalence cl...
erth 8327 Basic property of equivale...
erth2 8328 Basic property of equivale...
erthi 8329 Basic property of equivale...
erdisj 8330 Equivalence classes do not...
ecidsn 8331 An equivalence class modul...
qseq1 8332 Equality theorem for quoti...
qseq2 8333 Equality theorem for quoti...
qseq2i 8334 Equality theorem for quoti...
qseq2d 8335 Equality theorem for quoti...
qseq12 8336 Equality theorem for quoti...
elqsg 8337 Closed form of ~ elqs . (...
elqs 8338 Membership in a quotient s...
elqsi 8339 Membership in a quotient s...
elqsecl 8340 Membership in a quotient s...
ecelqsg 8341 Membership of an equivalen...
ecelqsi 8342 Membership of an equivalen...
ecopqsi 8343 "Closure" law for equivale...
qsexg 8344 A quotient set exists. (C...
qsex 8345 A quotient set exists. (C...
uniqs 8346 The union of a quotient se...
qsss 8347 A quotient set is a set of...
uniqs2 8348 The union of a quotient se...
snec 8349 The singleton of an equiva...
ecqs 8350 Equivalence class in terms...
ecid 8351 A set is equal to its cose...
qsid 8352 A set is equal to its quot...
ectocld 8353 Implicit substitution of c...
ectocl 8354 Implicit substitution of c...
elqsn0 8355 A quotient set does not co...
ecelqsdm 8356 Membership of an equivalen...
xpider 8357 A Cartesian square is an e...
iiner 8358 The intersection of a none...
riiner 8359 The relative intersection ...
erinxp 8360 A restricted equivalence r...
ecinxp 8361 Restrict the relation in a...
qsinxp 8362 Restrict the equivalence r...
qsdisj 8363 Members of a quotient set ...
qsdisj2 8364 A quotient set is a disjoi...
qsel 8365 If an element of a quotien...
uniinqs 8366 Class union distributes ov...
qliftlem 8367 ` F ` , a function lift, i...
qliftrel 8368 ` F ` , a function lift, i...
qliftel 8369 Elementhood in the relatio...
qliftel1 8370 Elementhood in the relatio...
qliftfun 8371 The function ` F ` is the ...
qliftfund 8372 The function ` F ` is the ...
qliftfuns 8373 The function ` F ` is the ...
qliftf 8374 The domain and range of th...
qliftval 8375 The value of the function ...
ecoptocl 8376 Implicit substitution of c...
2ecoptocl 8377 Implicit substitution of c...
3ecoptocl 8378 Implicit substitution of c...
brecop 8379 Binary relation on a quoti...
brecop2 8380 Binary relation on a quoti...
eroveu 8381 Lemma for ~ erov and ~ ero...
erovlem 8382 Lemma for ~ erov and ~ ero...
erov 8383 The value of an operation ...
eroprf 8384 Functionality of an operat...
erov2 8385 The value of an operation ...
eroprf2 8386 Functionality of an operat...
ecopoveq 8387 This is the first of sever...
ecopovsym 8388 Assuming the operation ` F...
ecopovtrn 8389 Assuming that operation ` ...
ecopover 8390 Assuming that operation ` ...
eceqoveq 8391 Equality of equivalence re...
ecovcom 8392 Lemma used to transfer a c...
ecovass 8393 Lemma used to transfer an ...
ecovdi 8394 Lemma used to transfer a d...
mapprc 8399 When ` A ` is a proper cla...
pmex 8400 The class of all partial f...
mapex 8401 The class of all functions...
fnmap 8402 Set exponentiation has a u...
fnpm 8403 Partial function exponenti...
reldmmap 8404 Set exponentiation is a we...
mapvalg 8405 The value of set exponenti...
pmvalg 8406 The value of the partial m...
mapval 8407 The value of set exponenti...
elmapg 8408 Membership relation for se...
elmapd 8409 Deduction form of ~ elmapg...
mapdm0 8410 The empty set is the only ...
elpmg 8411 The predicate "is a partia...
elpm2g 8412 The predicate "is a partia...
elpm2r 8413 Sufficient condition for b...
elpmi 8414 A partial function is a fu...
pmfun 8415 A partial function is a fu...
elmapex 8416 Eliminate antecedent for m...
elmapi 8417 A mapping is a function, f...
elmapfn 8418 A mapping is a function wi...
elmapfun 8419 A mapping is always a func...
elmapssres 8420 A restricted mapping is a ...
fpmg 8421 A total function is a part...
pmss12g 8422 Subset relation for the se...
pmresg 8423 Elementhood of a restricte...
elmap 8424 Membership relation for se...
mapval2 8425 Alternate expression for t...
elpm 8426 The predicate "is a partia...
elpm2 8427 The predicate "is a partia...
fpm 8428 A total function is a part...
mapsspm 8429 Set exponentiation is a su...
pmsspw 8430 Partial maps are a subset ...
mapsspw 8431 Set exponentiation is a su...
mapfvd 8432 The value of a function th...
elmapresaun 8433 ~ fresaun transposed to ma...
fvmptmap 8434 Special case of ~ fvmpt fo...
map0e 8435 Set exponentiation with an...
map0b 8436 Set exponentiation with an...
map0g 8437 Set exponentiation is empt...
mapsnd 8438 The value of set exponenti...
map0 8439 Set exponentiation is empt...
mapsn 8440 The value of set exponenti...
mapss 8441 Subset inheritance for set...
fdiagfn 8442 Functionality of the diago...
fvdiagfn 8443 Functionality of the diago...
mapsnconst 8444 Every singleton map is a c...
mapsncnv 8445 Expression for the inverse...
mapsnf1o2 8446 Explicit bijection between...
mapsnf1o3 8447 Explicit bijection in the ...
ralxpmap 8448 Quantification over functi...
dfixp 8451 Eliminate the expression `...
ixpsnval 8452 The value of an infinite C...
elixp2 8453 Membership in an infinite ...
fvixp 8454 Projection of a factor of ...
ixpfn 8455 A nuple is a function. (C...
elixp 8456 Membership in an infinite ...
elixpconst 8457 Membership in an infinite ...
ixpconstg 8458 Infinite Cartesian product...
ixpconst 8459 Infinite Cartesian product...
ixpeq1 8460 Equality theorem for infin...
ixpeq1d 8461 Equality theorem for infin...
ss2ixp 8462 Subclass theorem for infin...
ixpeq2 8463 Equality theorem for infin...
ixpeq2dva 8464 Equality theorem for infin...
ixpeq2dv 8465 Equality theorem for infin...
cbvixp 8466 Change bound variable in a...
cbvixpv 8467 Change bound variable in a...
nfixpw 8468 Version of ~ nfixp with a ...
nfixp 8469 Bound-variable hypothesis ...
nfixp1 8470 The index variable in an i...
ixpprc 8471 A cartesian product of pro...
ixpf 8472 A member of an infinite Ca...
uniixp 8473 The union of an infinite C...
ixpexg 8474 The existence of an infini...
ixpin 8475 The intersection of two in...
ixpiin 8476 The indexed intersection o...
ixpint 8477 The intersection of a coll...
ixp0x 8478 An infinite Cartesian prod...
ixpssmap2g 8479 An infinite Cartesian prod...
ixpssmapg 8480 An infinite Cartesian prod...
0elixp 8481 Membership of the empty se...
ixpn0 8482 The infinite Cartesian pro...
ixp0 8483 The infinite Cartesian pro...
ixpssmap 8484 An infinite Cartesian prod...
resixp 8485 Restriction of an element ...
undifixp 8486 Union of two projections o...
mptelixpg 8487 Condition for an explicit ...
resixpfo 8488 Restriction of elements of...
elixpsn 8489 Membership in a class of s...
ixpsnf1o 8490 A bijection between a clas...
mapsnf1o 8491 A bijection between a set ...
boxriin 8492 A rectangular subset of a ...
boxcutc 8493 The relative complement of...
relen 8502 Equinumerosity is a relati...
reldom 8503 Dominance is a relation. ...
relsdom 8504 Strict dominance is a rela...
encv 8505 If two classes are equinum...
bren 8506 Equinumerosity relation. ...
brdomg 8507 Dominance relation. (Cont...
brdomi 8508 Dominance relation. (Cont...
brdom 8509 Dominance relation. (Cont...
domen 8510 Dominance in terms of equi...
domeng 8511 Dominance in terms of equi...
ctex 8512 A countable set is a set. ...
f1oen3g 8513 The domain and range of a ...
f1oen2g 8514 The domain and range of a ...
f1dom2g 8515 The domain of a one-to-one...
f1oeng 8516 The domain and range of a ...
f1domg 8517 The domain of a one-to-one...
f1oen 8518 The domain and range of a ...
f1dom 8519 The domain of a one-to-one...
brsdom 8520 Strict dominance relation,...
isfi 8521 Express " ` A ` is finite....
enssdom 8522 Equinumerosity implies dom...
dfdom2 8523 Alternate definition of do...
endom 8524 Equinumerosity implies dom...
sdomdom 8525 Strict dominance implies d...
sdomnen 8526 Strict dominance implies n...
brdom2 8527 Dominance in terms of stri...
bren2 8528 Equinumerosity expressed i...
enrefg 8529 Equinumerosity is reflexiv...
enref 8530 Equinumerosity is reflexiv...
eqeng 8531 Equality implies equinumer...
domrefg 8532 Dominance is reflexive. (...
en2d 8533 Equinumerosity inference f...
en3d 8534 Equinumerosity inference f...
en2i 8535 Equinumerosity inference f...
en3i 8536 Equinumerosity inference f...
dom2lem 8537 A mapping (first hypothesi...
dom2d 8538 A mapping (first hypothesi...
dom3d 8539 A mapping (first hypothesi...
dom2 8540 A mapping (first hypothesi...
dom3 8541 A mapping (first hypothesi...
idssen 8542 Equality implies equinumer...
ssdomg 8543 A set dominates its subset...
ener 8544 Equinumerosity is an equiv...
ensymb 8545 Symmetry of equinumerosity...
ensym 8546 Symmetry of equinumerosity...
ensymi 8547 Symmetry of equinumerosity...
ensymd 8548 Symmetry of equinumerosity...
entr 8549 Transitivity of equinumero...
domtr 8550 Transitivity of dominance ...
entri 8551 A chained equinumerosity i...
entr2i 8552 A chained equinumerosity i...
entr3i 8553 A chained equinumerosity i...
entr4i 8554 A chained equinumerosity i...
endomtr 8555 Transitivity of equinumero...
domentr 8556 Transitivity of dominance ...
f1imaeng 8557 A one-to-one function's im...
f1imaen2g 8558 A one-to-one function's im...
f1imaen 8559 A one-to-one function's im...
en0 8560 The empty set is equinumer...
ensn1 8561 A singleton is equinumerou...
ensn1g 8562 A singleton is equinumerou...
enpr1g 8563 ` { A , A } ` has only one...
en1 8564 A set is equinumerous to o...
en1b 8565 A set is equinumerous to o...
reuen1 8566 Two ways to express "exact...
euen1 8567 Two ways to express "exact...
euen1b 8568 Two ways to express " ` A ...
en1uniel 8569 A singleton contains its s...
2dom 8570 A set that dominates ordin...
fundmen 8571 A function is equinumerous...
fundmeng 8572 A function is equinumerous...
cnven 8573 A relational set is equinu...
cnvct 8574 If a set is countable, so ...
fndmeng 8575 A function is equinumerate...
mapsnend 8576 Set exponentiation to a si...
mapsnen 8577 Set exponentiation to a si...
snmapen 8578 Set exponentiation: a sing...
snmapen1 8579 Set exponentiation: a sing...
map1 8580 Set exponentiation: ordina...
en2sn 8581 Two singletons are equinum...
snfi 8582 A singleton is finite. (C...
fiprc 8583 The class of finite sets i...
unen 8584 Equinumerosity of union of...
enpr2d 8585 A pair with distinct eleme...
ssct 8586 Any subset of a countable ...
difsnen 8587 All decrements of a set ar...
domdifsn 8588 Dominance over a set with ...
xpsnen 8589 A set is equinumerous to i...
xpsneng 8590 A set is equinumerous to i...
xp1en 8591 One times a cardinal numbe...
endisj 8592 Any two sets are equinumer...
undom 8593 Dominance law for union. ...
xpcomf1o 8594 The canonical bijection fr...
xpcomco 8595 Composition with the bijec...
xpcomen 8596 Commutative law for equinu...
xpcomeng 8597 Commutative law for equinu...
xpsnen2g 8598 A set is equinumerous to i...
xpassen 8599 Associative law for equinu...
xpdom2 8600 Dominance law for Cartesia...
xpdom2g 8601 Dominance law for Cartesia...
xpdom1g 8602 Dominance law for Cartesia...
xpdom3 8603 A set is dominated by its ...
xpdom1 8604 Dominance law for Cartesia...
domunsncan 8605 A singleton cancellation l...
omxpenlem 8606 Lemma for ~ omxpen . (Con...
omxpen 8607 The cardinal and ordinal p...
omf1o 8608 Construct an explicit bije...
pw2f1olem 8609 Lemma for ~ pw2f1o . (Con...
pw2f1o 8610 The power set of a set is ...
pw2eng 8611 The power set of a set is ...
pw2en 8612 The power set of a set is ...
fopwdom 8613 Covering implies injection...
enfixsn 8614 Given two equipollent sets...
sbthlem1 8615 Lemma for ~ sbth . (Contr...
sbthlem2 8616 Lemma for ~ sbth . (Contr...
sbthlem3 8617 Lemma for ~ sbth . (Contr...
sbthlem4 8618 Lemma for ~ sbth . (Contr...
sbthlem5 8619 Lemma for ~ sbth . (Contr...
sbthlem6 8620 Lemma for ~ sbth . (Contr...
sbthlem7 8621 Lemma for ~ sbth . (Contr...
sbthlem8 8622 Lemma for ~ sbth . (Contr...
sbthlem9 8623 Lemma for ~ sbth . (Contr...
sbthlem10 8624 Lemma for ~ sbth . (Contr...
sbth 8625 Schroeder-Bernstein Theore...
sbthb 8626 Schroeder-Bernstein Theore...
sbthcl 8627 Schroeder-Bernstein Theore...
dfsdom2 8628 Alternate definition of st...
brsdom2 8629 Alternate definition of st...
sdomnsym 8630 Strict dominance is asymme...
domnsym 8631 Theorem 22(i) of [Suppes] ...
0domg 8632 Any set dominates the empt...
dom0 8633 A set dominated by the emp...
0sdomg 8634 A set strictly dominates t...
0dom 8635 Any set dominates the empt...
0sdom 8636 A set strictly dominates t...
sdom0 8637 The empty set does not str...
sdomdomtr 8638 Transitivity of strict dom...
sdomentr 8639 Transitivity of strict dom...
domsdomtr 8640 Transitivity of dominance ...
ensdomtr 8641 Transitivity of equinumero...
sdomirr 8642 Strict dominance is irrefl...
sdomtr 8643 Strict dominance is transi...
sdomn2lp 8644 Strict dominance has no 2-...
enen1 8645 Equality-like theorem for ...
enen2 8646 Equality-like theorem for ...
domen1 8647 Equality-like theorem for ...
domen2 8648 Equality-like theorem for ...
sdomen1 8649 Equality-like theorem for ...
sdomen2 8650 Equality-like theorem for ...
domtriord 8651 Dominance is trichotomous ...
sdomel 8652 Strict dominance implies o...
sdomdif 8653 The difference of a set fr...
onsdominel 8654 An ordinal with more eleme...
domunsn 8655 Dominance over a set with ...
fodomr 8656 There exists a mapping fro...
pwdom 8657 Injection of sets implies ...
canth2 8658 Cantor's Theorem. No set ...
canth2g 8659 Cantor's theorem with the ...
2pwuninel 8660 The power set of the power...
2pwne 8661 No set equals the power se...
disjen 8662 A stronger form of ~ pwuni...
disjenex 8663 Existence version of ~ dis...
domss2 8664 A corollary of ~ disjenex ...
domssex2 8665 A corollary of ~ disjenex ...
domssex 8666 Weakening of ~ domssex to ...
xpf1o 8667 Construct a bijection on a...
xpen 8668 Equinumerosity law for Car...
mapen 8669 Two set exponentiations ar...
mapdom1 8670 Order-preserving property ...
mapxpen 8671 Equinumerosity law for dou...
xpmapenlem 8672 Lemma for ~ xpmapen . (Co...
xpmapen 8673 Equinumerosity law for set...
mapunen 8674 Equinumerosity law for set...
map2xp 8675 A cardinal power with expo...
mapdom2 8676 Order-preserving property ...
mapdom3 8677 Set exponentiation dominat...
pwen 8678 If two sets are equinumero...
ssenen 8679 Equinumerosity of equinume...
limenpsi 8680 A limit ordinal is equinum...
limensuci 8681 A limit ordinal is equinum...
limensuc 8682 A limit ordinal is equinum...
infensuc 8683 Any infinite ordinal is eq...
phplem1 8684 Lemma for Pigeonhole Princ...
phplem2 8685 Lemma for Pigeonhole Princ...
phplem3 8686 Lemma for Pigeonhole Princ...
phplem4 8687 Lemma for Pigeonhole Princ...
nneneq 8688 Two equinumerous natural n...
php 8689 Pigeonhole Principle. A n...
php2 8690 Corollary of Pigeonhole Pr...
php3 8691 Corollary of Pigeonhole Pr...
php4 8692 Corollary of the Pigeonhol...
php5 8693 Corollary of the Pigeonhol...
phpeqd 8694 Corollary of the Pigeonhol...
snnen2o 8695 A singleton ` { A } ` is n...
onomeneq 8696 An ordinal number equinume...
onfin 8697 An ordinal number is finit...
onfin2 8698 A set is a natural number ...
nnfi 8699 Natural numbers are finite...
nndomo 8700 Cardinal ordering agrees w...
nnsdomo 8701 Cardinal ordering agrees w...
sucdom2 8702 Strict dominance of a set ...
sucdom 8703 Strict dominance of a set ...
0sdom1dom 8704 Strict dominance over zero...
1sdom2 8705 Ordinal 1 is strictly domi...
sdom1 8706 A set has less than one me...
modom 8707 Two ways to express "at mo...
modom2 8708 Two ways to express "at mo...
1sdom 8709 A set that strictly domina...
unxpdomlem1 8710 Lemma for ~ unxpdom . (Tr...
unxpdomlem2 8711 Lemma for ~ unxpdom . (Co...
unxpdomlem3 8712 Lemma for ~ unxpdom . (Co...
unxpdom 8713 Cartesian product dominate...
unxpdom2 8714 Corollary of ~ unxpdom . ...
sucxpdom 8715 Cartesian product dominate...
pssinf 8716 A set equinumerous to a pr...
fisseneq 8717 A finite set is equal to i...
ominf 8718 The set of natural numbers...
isinf 8719 Any set that is not finite...
fineqvlem 8720 Lemma for ~ fineqv . (Con...
fineqv 8721 If the Axiom of Infinity i...
enfi 8722 Equinumerous sets have the...
enfii 8723 A set equinumerous to a fi...
pssnn 8724 A proper subset of a natur...
ssnnfi 8725 A subset of a natural numb...
ssfi 8726 A subset of a finite set i...
domfi 8727 A set dominated by a finit...
xpfir 8728 The components of a nonemp...
ssfid 8729 A subset of a finite set i...
infi 8730 The intersection of two se...
rabfi 8731 A restricted class built f...
finresfin 8732 The restriction of a finit...
f1finf1o 8733 Any injection from one fin...
0fin 8734 The empty set is finite. ...
nfielex 8735 If a class is not finite, ...
en1eqsn 8736 A set with one element is ...
en1eqsnbi 8737 A set containing an elemen...
diffi 8738 If ` A ` is finite, ` ( A ...
dif1en 8739 If a set ` A ` is equinume...
enp1ilem 8740 Lemma for uses of ~ enp1i ...
enp1i 8741 Proof induction for ~ en2i...
en2 8742 A set equinumerous to ordi...
en3 8743 A set equinumerous to ordi...
en4 8744 A set equinumerous to ordi...
findcard 8745 Schema for induction on th...
findcard2 8746 Schema for induction on th...
findcard2s 8747 Variation of ~ findcard2 r...
findcard2d 8748 Deduction version of ~ fin...
findcard3 8749 Schema for strong inductio...
ac6sfi 8750 A version of ~ ac6s for fi...
frfi 8751 A partial order is well-fo...
fimax2g 8752 A finite set has a maximum...
fimaxg 8753 A finite set has a maximum...
fisupg 8754 Lemma showing existence an...
wofi 8755 A total order on a finite ...
ordunifi 8756 The maximum of a finite co...
nnunifi 8757 The union (supremum) of a ...
unblem1 8758 Lemma for ~ unbnn . After...
unblem2 8759 Lemma for ~ unbnn . The v...
unblem3 8760 Lemma for ~ unbnn . The v...
unblem4 8761 Lemma for ~ unbnn . The f...
unbnn 8762 Any unbounded subset of na...
unbnn2 8763 Version of ~ unbnn that do...
isfinite2 8764 Any set strictly dominated...
nnsdomg 8765 Omega strictly dominates a...
isfiniteg 8766 A set is finite iff it is ...
infsdomnn 8767 An infinite set strictly d...
infn0 8768 An infinite set is not emp...
fin2inf 8769 This (useless) theorem, wh...
unfilem1 8770 Lemma for proving that the...
unfilem2 8771 Lemma for proving that the...
unfilem3 8772 Lemma for proving that the...
unfi 8773 The union of two finite se...
unfir 8774 If a union is finite, the ...
unfi2 8775 The union of two finite se...
difinf 8776 An infinite set ` A ` minu...
xpfi 8777 The Cartesian product of t...
3xpfi 8778 The Cartesian product of t...
domunfican 8779 A finite set union cancell...
infcntss 8780 Every infinite set has a d...
prfi 8781 An unordered pair is finit...
tpfi 8782 An unordered triple is fin...
fiint 8783 Equivalent ways of stating...
fnfi 8784 A version of ~ fnex for fi...
fodomfi 8785 An onto function implies d...
fodomfib 8786 Equivalence of an onto map...
fofinf1o 8787 Any surjection from one fi...
rneqdmfinf1o 8788 Any function from a finite...
fidomdm 8789 Any finite set dominates i...
dmfi 8790 The domain of a finite set...
fundmfibi 8791 A function is finite if an...
resfnfinfin 8792 The restriction of a funct...
residfi 8793 A restricted identity func...
cnvfi 8794 If a set is finite, its co...
rnfi 8795 The range of a finite set ...
f1dmvrnfibi 8796 A one-to-one function whos...
f1vrnfibi 8797 A one-to-one function whic...
fofi 8798 If a function has a finite...
f1fi 8799 If a 1-to-1 function has a...
iunfi 8800 The finite union of finite...
unifi 8801 The finite union of finite...
unifi2 8802 The finite union of finite...
infssuni 8803 If an infinite set ` A ` i...
unirnffid 8804 The union of the range of ...
imafi 8805 Images of finite sets are ...
pwfilem 8806 Lemma for ~ pwfi . (Contr...
pwfi 8807 The power set of a finite ...
mapfi 8808 Set exponentiation of fini...
ixpfi 8809 A Cartesian product of fin...
ixpfi2 8810 A Cartesian product of fin...
mptfi 8811 A finite mapping set is fi...
abrexfi 8812 An image set from a finite...
cnvimamptfin 8813 A preimage of a mapping wi...
elfpw 8814 Membership in a class of f...
unifpw 8815 A set is the union of its ...
f1opwfi 8816 A one-to-one mapping induc...
fissuni 8817 A finite subset of a union...
fipreima 8818 Given a finite subset ` A ...
finsschain 8819 A finite subset of the uni...
indexfi 8820 If for every element of a ...
relfsupp 8823 The property of a function...
relprcnfsupp 8824 A proper class is never fi...
isfsupp 8825 The property of a class to...
funisfsupp 8826 The property of a function...
fsuppimp 8827 Implications of a class be...
fsuppimpd 8828 A finitely supported funct...
fisuppfi 8829 A function on a finite set...
fdmfisuppfi 8830 The support of a function ...
fdmfifsupp 8831 A function with a finite d...
fsuppmptdm 8832 A mapping with a finite do...
fndmfisuppfi 8833 The support of a function ...
fndmfifsupp 8834 A function with a finite d...
suppeqfsuppbi 8835 If two functions have the ...
suppssfifsupp 8836 If the support of a functi...
fsuppsssupp 8837 If the support of a functi...
fsuppxpfi 8838 The cartesian product of t...
fczfsuppd 8839 A constant function with v...
fsuppun 8840 The union of two finitely ...
fsuppunfi 8841 The union of the support o...
fsuppunbi 8842 If the union of two classe...
0fsupp 8843 The empty set is a finitel...
snopfsupp 8844 A singleton containing an ...
funsnfsupp 8845 Finite support for a funct...
fsuppres 8846 The restriction of a finit...
ressuppfi 8847 If the support of the rest...
resfsupp 8848 If the restriction of a fu...
resfifsupp 8849 The restriction of a funct...
frnfsuppbi 8850 Two ways of saying that a ...
fsuppmptif 8851 A function mapping an argu...
fsuppcolem 8852 Lemma for ~ fsuppco . For...
fsuppco 8853 The composition of a 1-1 f...
fsuppco2 8854 The composition of a funct...
fsuppcor 8855 The composition of a funct...
mapfienlem1 8856 Lemma 1 for ~ mapfien . (...
mapfienlem2 8857 Lemma 2 for ~ mapfien . (...
mapfienlem3 8858 Lemma 3 for ~ mapfien . (...
mapfien 8859 A bijection of the base se...
mapfien2 8860 Equinumerousity relation f...
sniffsupp 8861 A function mapping all but...
fival 8864 The set of all the finite ...
elfi 8865 Specific properties of an ...
elfi2 8866 The empty intersection nee...
elfir 8867 Sufficient condition for a...
intrnfi 8868 Sufficient condition for t...
iinfi 8869 An indexed intersection of...
inelfi 8870 The intersection of two se...
ssfii 8871 Any element of a set ` A `...
fi0 8872 The set of finite intersec...
fieq0 8873 A set is empty iff the cla...
fiin 8874 The elements of ` ( fi `` ...
dffi2 8875 The set of finite intersec...
fiss 8876 Subset relationship for fu...
inficl 8877 A set which is closed unde...
fipwuni 8878 The set of finite intersec...
fisn 8879 A singleton is closed unde...
fiuni 8880 The union of the finite in...
fipwss 8881 If a set is a family of su...
elfiun 8882 A finite intersection of e...
dffi3 8883 The set of finite intersec...
fifo 8884 Describe a surjection from...
marypha1lem 8885 Core induction for Philip ...
marypha1 8886 (Philip) Hall's marriage t...
marypha2lem1 8887 Lemma for ~ marypha2 . Pr...
marypha2lem2 8888 Lemma for ~ marypha2 . Pr...
marypha2lem3 8889 Lemma for ~ marypha2 . Pr...
marypha2lem4 8890 Lemma for ~ marypha2 . Pr...
marypha2 8891 Version of ~ marypha1 usin...
dfsup2 8896 Quantifier free definition...
supeq1 8897 Equality theorem for supre...
supeq1d 8898 Equality deduction for sup...
supeq1i 8899 Equality inference for sup...
supeq2 8900 Equality theorem for supre...
supeq3 8901 Equality theorem for supre...
supeq123d 8902 Equality deduction for sup...
nfsup 8903 Hypothesis builder for sup...
supmo 8904 Any class ` B ` has at mos...
supexd 8905 A supremum is a set. (Con...
supeu 8906 A supremum is unique. Sim...
supval2 8907 Alternate expression for t...
eqsup 8908 Sufficient condition for a...
eqsupd 8909 Sufficient condition for a...
supcl 8910 A supremum belongs to its ...
supub 8911 A supremum is an upper bou...
suplub 8912 A supremum is the least up...
suplub2 8913 Bidirectional form of ~ su...
supnub 8914 An upper bound is not less...
supex 8915 A supremum is a set. (Con...
sup00 8916 The supremum under an empt...
sup0riota 8917 The supremum of an empty s...
sup0 8918 The supremum of an empty s...
supmax 8919 The greatest element of a ...
fisup2g 8920 A finite set satisfies the...
fisupcl 8921 A nonempty finite set cont...
supgtoreq 8922 The supremum of a finite s...
suppr 8923 The supremum of a pair. (...
supsn 8924 The supremum of a singleto...
supisolem 8925 Lemma for ~ supiso . (Con...
supisoex 8926 Lemma for ~ supiso . (Con...
supiso 8927 Image of a supremum under ...
infeq1 8928 Equality theorem for infim...
infeq1d 8929 Equality deduction for inf...
infeq1i 8930 Equality inference for inf...
infeq2 8931 Equality theorem for infim...
infeq3 8932 Equality theorem for infim...
infeq123d 8933 Equality deduction for inf...
nfinf 8934 Hypothesis builder for inf...
infexd 8935 An infimum is a set. (Con...
eqinf 8936 Sufficient condition for a...
eqinfd 8937 Sufficient condition for a...
infval 8938 Alternate expression for t...
infcllem 8939 Lemma for ~ infcl , ~ infl...
infcl 8940 An infimum belongs to its ...
inflb 8941 An infimum is a lower boun...
infglb 8942 An infimum is the greatest...
infglbb 8943 Bidirectional form of ~ in...
infnlb 8944 A lower bound is not great...
infex 8945 An infimum is a set. (Con...
infmin 8946 The smallest element of a ...
infmo 8947 Any class ` B ` has at mos...
infeu 8948 An infimum is unique. (Co...
fimin2g 8949 A finite set has a minimum...
fiming 8950 A finite set has a minimum...
fiinfg 8951 Lemma showing existence an...
fiinf2g 8952 A finite set satisfies the...
fiinfcl 8953 A nonempty finite set cont...
infltoreq 8954 The infimum of a finite se...
infpr 8955 The infimum of a pair. (C...
infsupprpr 8956 The infimum of a proper pa...
infsn 8957 The infimum of a singleton...
inf00 8958 The infimum regarding an e...
infempty 8959 The infimum of an empty se...
infiso 8960 Image of an infimum under ...
dfoi 8963 Rewrite ~ df-oi with abbre...
oieq1 8964 Equality theorem for ordin...
oieq2 8965 Equality theorem for ordin...
nfoi 8966 Hypothesis builder for ord...
ordiso2 8967 Generalize ~ ordiso to pro...
ordiso 8968 Order-isomorphic ordinal n...
ordtypecbv 8969 Lemma for ~ ordtype . (Co...
ordtypelem1 8970 Lemma for ~ ordtype . (Co...
ordtypelem2 8971 Lemma for ~ ordtype . (Co...
ordtypelem3 8972 Lemma for ~ ordtype . (Co...
ordtypelem4 8973 Lemma for ~ ordtype . (Co...
ordtypelem5 8974 Lemma for ~ ordtype . (Co...
ordtypelem6 8975 Lemma for ~ ordtype . (Co...
ordtypelem7 8976 Lemma for ~ ordtype . ` ra...
ordtypelem8 8977 Lemma for ~ ordtype . (Co...
ordtypelem9 8978 Lemma for ~ ordtype . Eit...
ordtypelem10 8979 Lemma for ~ ordtype . Usi...
oi0 8980 Definition of the ordinal ...
oicl 8981 The order type of the well...
oif 8982 The order isomorphism of t...
oiiso2 8983 The order isomorphism of t...
ordtype 8984 For any set-like well-orde...
oiiniseg 8985 ` ran F ` is an initial se...
ordtype2 8986 For any set-like well-orde...
oiexg 8987 The order isomorphism on a...
oion 8988 The order type of the well...
oiiso 8989 The order isomorphism of t...
oien 8990 The order type of a well-o...
oieu 8991 Uniqueness of the unique o...
oismo 8992 When ` A ` is a subclass o...
oiid 8993 The order type of an ordin...
hartogslem1 8994 Lemma for ~ hartogs . (Co...
hartogslem2 8995 Lemma for ~ hartogs . (Co...
hartogs 8996 Given any set, the Hartogs...
wofib 8997 The only sets which are we...
wemaplem1 8998 Value of the lexicographic...
wemaplem2 8999 Lemma for ~ wemapso . Tra...
wemaplem3 9000 Lemma for ~ wemapso . Tra...
wemappo 9001 Construct lexicographic or...
wemapsolem 9002 Lemma for ~ wemapso . (Co...
wemapso 9003 Construct lexicographic or...
wemapso2lem 9004 Lemma for ~ wemapso2 . (C...
wemapso2 9005 An alternative to having a...
card2on 9006 The alternate definition o...
card2inf 9007 The alternate definition o...
harf 9012 Functionality of the Harto...
harcl 9013 Closure of the Hartogs fun...
harval 9014 Function value of the Hart...
elharval 9015 The Hartogs number of a se...
harndom 9016 The Hartogs number of a se...
harword 9017 Weak ordering property of ...
relwdom 9018 Weak dominance is a relati...
brwdom 9019 Property of weak dominance...
brwdomi 9020 Property of weak dominance...
brwdomn0 9021 Weak dominance over nonemp...
0wdom 9022 Any set weakly dominates t...
fowdom 9023 An onto function implies w...
wdomref 9024 Reflexivity of weak domina...
brwdom2 9025 Alternate characterization...
domwdom 9026 Weak dominance is implied ...
wdomtr 9027 Transitivity of weak domin...
wdomen1 9028 Equality-like theorem for ...
wdomen2 9029 Equality-like theorem for ...
wdompwdom 9030 Weak dominance strengthens...
canthwdom 9031 Cantor's Theorem, stated u...
wdom2d 9032 Deduce weak dominance from...
wdomd 9033 Deduce weak dominance from...
brwdom3 9034 Condition for weak dominan...
brwdom3i 9035 Weak dominance implies exi...
unwdomg 9036 Weak dominance of a (disjo...
xpwdomg 9037 Weak dominance of a Cartes...
wdomima2g 9038 A set is weakly dominant o...
wdomimag 9039 A set is weakly dominant o...
unxpwdom2 9040 Lemma for ~ unxpwdom . (C...
unxpwdom 9041 If a Cartesian product is ...
harwdom 9042 The Hartogs function is we...
ixpiunwdom 9043 Describe an onto function ...
axreg2 9045 Axiom of Regularity expres...
zfregcl 9046 The Axiom of Regularity wi...
zfreg 9047 The Axiom of Regularity us...
elirrv 9048 The membership relation is...
elirr 9049 No class is a member of it...
elneq 9050 A class is not equal to an...
nelaneq 9051 A class is not an element ...
epinid0 9052 The membership (epsilon) r...
sucprcreg 9053 A class is equal to its su...
ruv 9054 The Russell class is equal...
ruALT 9055 Alternate proof of ~ ru , ...
zfregfr 9056 The membership relation is...
en2lp 9057 No class has 2-cycle membe...
elnanel 9058 Two classes are not elemen...
cnvepnep 9059 The membership (epsilon) r...
epnsym 9060 The membership (epsilon) r...
elnotel 9061 A class cannot be an eleme...
elnel 9062 A class cannot be an eleme...
en3lplem1 9063 Lemma for ~ en3lp . (Cont...
en3lplem2 9064 Lemma for ~ en3lp . (Cont...
en3lp 9065 No class has 3-cycle membe...
preleqg 9066 Equality of two unordered ...
preleq 9067 Equality of two unordered ...
preleqALT 9068 Alternate proof of ~ prele...
opthreg 9069 Theorem for alternate repr...
suc11reg 9070 The successor operation be...
dford2 9071 Assuming ~ ax-reg , an ord...
inf0 9072 Our Axiom of Infinity deri...
inf1 9073 Variation of Axiom of Infi...
inf2 9074 Variation of Axiom of Infi...
inf3lema 9075 Lemma for our Axiom of Inf...
inf3lemb 9076 Lemma for our Axiom of Inf...
inf3lemc 9077 Lemma for our Axiom of Inf...
inf3lemd 9078 Lemma for our Axiom of Inf...
inf3lem1 9079 Lemma for our Axiom of Inf...
inf3lem2 9080 Lemma for our Axiom of Inf...
inf3lem3 9081 Lemma for our Axiom of Inf...
inf3lem4 9082 Lemma for our Axiom of Inf...
inf3lem5 9083 Lemma for our Axiom of Inf...
inf3lem6 9084 Lemma for our Axiom of Inf...
inf3lem7 9085 Lemma for our Axiom of Inf...
inf3 9086 Our Axiom of Infinity ~ ax...
infeq5i 9087 Half of ~ infeq5 . (Contr...
infeq5 9088 The statement "there exist...
zfinf 9090 Axiom of Infinity expresse...
axinf2 9091 A standard version of Axio...
zfinf2 9093 A standard version of the ...
omex 9094 The existence of omega (th...
axinf 9095 The first version of the A...
inf5 9096 The statement "there exist...
omelon 9097 Omega is an ordinal number...
dfom3 9098 The class of natural numbe...
elom3 9099 A simplification of ~ elom...
dfom4 9100 A simplification of ~ df-o...
dfom5 9101 ` _om ` is the smallest li...
oancom 9102 Ordinal addition is not co...
isfinite 9103 A set is finite iff it is ...
fict 9104 A finite set is countable ...
nnsdom 9105 A natural number is strict...
omenps 9106 Omega is equinumerous to a...
omensuc 9107 The set of natural numbers...
infdifsn 9108 Removing a singleton from ...
infdiffi 9109 Removing a finite set from...
unbnn3 9110 Any unbounded subset of na...
noinfep 9111 Using the Axiom of Regular...
cantnffval 9114 The value of the Cantor no...
cantnfdm 9115 The domain of the Cantor n...
cantnfvalf 9116 Lemma for ~ cantnf . The ...
cantnfs 9117 Elementhood in the set of ...
cantnfcl 9118 Basic properties of the or...
cantnfval 9119 The value of the Cantor no...
cantnfval2 9120 Alternate expression for t...
cantnfsuc 9121 The value of the recursive...
cantnfle 9122 A lower bound on the ` CNF...
cantnflt 9123 An upper bound on the part...
cantnflt2 9124 An upper bound on the ` CN...
cantnff 9125 The ` CNF ` function is a ...
cantnf0 9126 The value of the zero func...
cantnfrescl 9127 A function is finitely sup...
cantnfres 9128 The ` CNF ` function respe...
cantnfp1lem1 9129 Lemma for ~ cantnfp1 . (C...
cantnfp1lem2 9130 Lemma for ~ cantnfp1 . (C...
cantnfp1lem3 9131 Lemma for ~ cantnfp1 . (C...
cantnfp1 9132 If ` F ` is created by add...
oemapso 9133 The relation ` T ` is a st...
oemapval 9134 Value of the relation ` T ...
oemapvali 9135 If ` F < G ` , then there ...
cantnflem1a 9136 Lemma for ~ cantnf . (Con...
cantnflem1b 9137 Lemma for ~ cantnf . (Con...
cantnflem1c 9138 Lemma for ~ cantnf . (Con...
cantnflem1d 9139 Lemma for ~ cantnf . (Con...
cantnflem1 9140 Lemma for ~ cantnf . This...
cantnflem2 9141 Lemma for ~ cantnf . (Con...
cantnflem3 9142 Lemma for ~ cantnf . Here...
cantnflem4 9143 Lemma for ~ cantnf . Comp...
cantnf 9144 The Cantor Normal Form the...
oemapwe 9145 The lexicographic order on...
cantnffval2 9146 An alternate definition of...
cantnff1o 9147 Simplify the isomorphism o...
wemapwe 9148 Construct lexicographic or...
oef1o 9149 A bijection of the base se...
cnfcomlem 9150 Lemma for ~ cnfcom . (Con...
cnfcom 9151 Any ordinal ` B ` is equin...
cnfcom2lem 9152 Lemma for ~ cnfcom2 . (Co...
cnfcom2 9153 Any nonzero ordinal ` B ` ...
cnfcom3lem 9154 Lemma for ~ cnfcom3 . (Co...
cnfcom3 9155 Any infinite ordinal ` B `...
cnfcom3clem 9156 Lemma for ~ cnfcom3c . (C...
cnfcom3c 9157 Wrap the construction of ~...
trcl 9158 For any set ` A ` , show t...
tz9.1 9159 Every set has a transitive...
tz9.1c 9160 Alternate expression for t...
epfrs 9161 The strong form of the Axi...
zfregs 9162 The strong form of the Axi...
zfregs2 9163 Alternate strong form of t...
setind 9164 Set (epsilon) induction. ...
setind2 9165 Set (epsilon) induction, s...
tcvalg 9168 Value of the transitive cl...
tcid 9169 Defining property of the t...
tctr 9170 Defining property of the t...
tcmin 9171 Defining property of the t...
tc2 9172 A variant of the definitio...
tcsni 9173 The transitive closure of ...
tcss 9174 The transitive closure fun...
tcel 9175 The transitive closure fun...
tcidm 9176 The transitive closure fun...
tc0 9177 The transitive closure of ...
tc00 9178 The transitive closure is ...
r1funlim 9183 The cumulative hierarchy o...
r1fnon 9184 The cumulative hierarchy o...
r10 9185 Value of the cumulative hi...
r1sucg 9186 Value of the cumulative hi...
r1suc 9187 Value of the cumulative hi...
r1limg 9188 Value of the cumulative hi...
r1lim 9189 Value of the cumulative hi...
r1fin 9190 The first ` _om ` levels o...
r1sdom 9191 Each stage in the cumulati...
r111 9192 The cumulative hierarchy i...
r1tr 9193 The cumulative hierarchy o...
r1tr2 9194 The union of a cumulative ...
r1ordg 9195 Ordering relation for the ...
r1ord3g 9196 Ordering relation for the ...
r1ord 9197 Ordering relation for the ...
r1ord2 9198 Ordering relation for the ...
r1ord3 9199 Ordering relation for the ...
r1sssuc 9200 The value of the cumulativ...
r1pwss 9201 Each set of the cumulative...
r1sscl 9202 Each set of the cumulative...
r1val1 9203 The value of the cumulativ...
tz9.12lem1 9204 Lemma for ~ tz9.12 . (Con...
tz9.12lem2 9205 Lemma for ~ tz9.12 . (Con...
tz9.12lem3 9206 Lemma for ~ tz9.12 . (Con...
tz9.12 9207 A set is well-founded if a...
tz9.13 9208 Every set is well-founded,...
tz9.13g 9209 Every set is well-founded,...
rankwflemb 9210 Two ways of saying a set i...
rankf 9211 The domain and range of th...
rankon 9212 The rank of a set is an or...
r1elwf 9213 Any member of the cumulati...
rankvalb 9214 Value of the rank function...
rankr1ai 9215 One direction of ~ rankr1a...
rankvaln 9216 Value of the rank function...
rankidb 9217 Identity law for the rank ...
rankdmr1 9218 A rank is a member of the ...
rankr1ag 9219 A version of ~ rankr1a tha...
rankr1bg 9220 A relationship between ran...
r1rankidb 9221 Any set is a subset of the...
r1elssi 9222 The range of the ` R1 ` fu...
r1elss 9223 The range of the ` R1 ` fu...
pwwf 9224 A power set is well-founde...
sswf 9225 A subset of a well-founded...
snwf 9226 A singleton is well-founde...
unwf 9227 A binary union is well-fou...
prwf 9228 An unordered pair is well-...
opwf 9229 An ordered pair is well-fo...
unir1 9230 The cumulative hierarchy o...
jech9.3 9231 Every set belongs to some ...
rankwflem 9232 Every set is well-founded,...
rankval 9233 Value of the rank function...
rankvalg 9234 Value of the rank function...
rankval2 9235 Value of an alternate defi...
uniwf 9236 A union is well-founded if...
rankr1clem 9237 Lemma for ~ rankr1c . (Co...
rankr1c 9238 A relationship between the...
rankidn 9239 A relationship between the...
rankpwi 9240 The rank of a power set. ...
rankelb 9241 The membership relation is...
wfelirr 9242 A well-founded set is not ...
rankval3b 9243 The value of the rank func...
ranksnb 9244 The rank of a singleton. ...
rankonidlem 9245 Lemma for ~ rankonid . (C...
rankonid 9246 The rank of an ordinal num...
onwf 9247 The ordinals are all well-...
onssr1 9248 Initial segments of the or...
rankr1g 9249 A relationship between the...
rankid 9250 Identity law for the rank ...
rankr1 9251 A relationship between the...
ssrankr1 9252 A relationship between an ...
rankr1a 9253 A relationship between ran...
r1val2 9254 The value of the cumulativ...
r1val3 9255 The value of the cumulativ...
rankel 9256 The membership relation is...
rankval3 9257 The value of the rank func...
bndrank 9258 Any class whose elements h...
unbndrank 9259 The elements of a proper c...
rankpw 9260 The rank of a power set. ...
ranklim 9261 The rank of a set belongs ...
r1pw 9262 A stronger property of ` R...
r1pwALT 9263 Alternate shorter proof of...
r1pwcl 9264 The cumulative hierarchy o...
rankssb 9265 The subset relation is inh...
rankss 9266 The subset relation is inh...
rankunb 9267 The rank of the union of t...
rankprb 9268 The rank of an unordered p...
rankopb 9269 The rank of an ordered pai...
rankuni2b 9270 The value of the rank func...
ranksn 9271 The rank of a singleton. ...
rankuni2 9272 The rank of a union. Part...
rankun 9273 The rank of the union of t...
rankpr 9274 The rank of an unordered p...
rankop 9275 The rank of an ordered pai...
r1rankid 9276 Any set is a subset of the...
rankeq0b 9277 A set is empty iff its ran...
rankeq0 9278 A set is empty iff its ran...
rankr1id 9279 The rank of the hierarchy ...
rankuni 9280 The rank of a union. Part...
rankr1b 9281 A relationship between ran...
ranksuc 9282 The rank of a successor. ...
rankuniss 9283 Upper bound of the rank of...
rankval4 9284 The rank of a set is the s...
rankbnd 9285 The rank of a set is bound...
rankbnd2 9286 The rank of a set is bound...
rankc1 9287 A relationship that can be...
rankc2 9288 A relationship that can be...
rankelun 9289 Rank membership is inherit...
rankelpr 9290 Rank membership is inherit...
rankelop 9291 Rank membership is inherit...
rankxpl 9292 A lower bound on the rank ...
rankxpu 9293 An upper bound on the rank...
rankfu 9294 An upper bound on the rank...
rankmapu 9295 An upper bound on the rank...
rankxplim 9296 The rank of a Cartesian pr...
rankxplim2 9297 If the rank of a Cartesian...
rankxplim3 9298 The rank of a Cartesian pr...
rankxpsuc 9299 The rank of a Cartesian pr...
tcwf 9300 The transitive closure fun...
tcrank 9301 This theorem expresses two...
scottex 9302 Scott's trick collects all...
scott0 9303 Scott's trick collects all...
scottexs 9304 Theorem scheme version of ...
scott0s 9305 Theorem scheme version of ...
cplem1 9306 Lemma for the Collection P...
cplem2 9307 Lemma for the Collection P...
cp 9308 Collection Principle. Thi...
bnd 9309 A very strong generalizati...
bnd2 9310 A variant of the Boundedne...
kardex 9311 The collection of all sets...
karden 9312 If we allow the Axiom of R...
htalem 9313 Lemma for defining an emul...
hta 9314 A ZFC emulation of Hilbert...
djueq12 9321 Equality theorem for disjo...
djueq1 9322 Equality theorem for disjo...
djueq2 9323 Equality theorem for disjo...
nfdju 9324 Bound-variable hypothesis ...
djuex 9325 The disjoint union of sets...
djuexb 9326 The disjoint union of two ...
djulcl 9327 Left closure of disjoint u...
djurcl 9328 Right closure of disjoint ...
djulf1o 9329 The left injection functio...
djurf1o 9330 The right injection functi...
inlresf 9331 The left injection restric...
inlresf1 9332 The left injection restric...
inrresf 9333 The right injection restri...
inrresf1 9334 The right injection restri...
djuin 9335 The images of any classes ...
djur 9336 A member of a disjoint uni...
djuss 9337 A disjoint union is a subc...
djuunxp 9338 The union of a disjoint un...
djuexALT 9339 Alternate proof of ~ djuex...
eldju1st 9340 The first component of an ...
eldju2ndl 9341 The second component of an...
eldju2ndr 9342 The second component of an...
djuun 9343 The disjoint union of two ...
1stinl 9344 The first component of the...
2ndinl 9345 The second component of th...
1stinr 9346 The first component of the...
2ndinr 9347 The second component of th...
updjudhf 9348 The mapping of an element ...
updjudhcoinlf 9349 The composition of the map...
updjudhcoinrg 9350 The composition of the map...
updjud 9351 Universal property of the ...
cardf2 9360 The cardinality function i...
cardon 9361 The cardinal number of a s...
isnum2 9362 A way to express well-orde...
isnumi 9363 A set equinumerous to an o...
ennum 9364 Equinumerous sets are equi...
finnum 9365 Every finite set is numera...
onenon 9366 Every ordinal number is nu...
tskwe 9367 A Tarski set is well-order...
xpnum 9368 The cartesian product of n...
cardval3 9369 An alternate definition of...
cardid2 9370 Any numerable set is equin...
isnum3 9371 A set is numerable iff it ...
oncardval 9372 The value of the cardinal ...
oncardid 9373 Any ordinal number is equi...
cardonle 9374 The cardinal of an ordinal...
card0 9375 The cardinality of the emp...
cardidm 9376 The cardinality function i...
oncard 9377 A set is a cardinal number...
ficardom 9378 The cardinal number of a f...
ficardid 9379 A finite set is equinumero...
cardnn 9380 The cardinality of a natur...
cardnueq0 9381 The empty set is the only ...
cardne 9382 No member of a cardinal nu...
carden2a 9383 If two sets have equal non...
carden2b 9384 If two sets are equinumero...
card1 9385 A set has cardinality one ...
cardsn 9386 A singleton has cardinalit...
carddomi2 9387 Two sets have the dominanc...
sdomsdomcardi 9388 A set strictly dominates i...
cardlim 9389 An infinite cardinal is a ...
cardsdomelir 9390 A cardinal strictly domina...
cardsdomel 9391 A cardinal strictly domina...
iscard 9392 Two ways to express the pr...
iscard2 9393 Two ways to express the pr...
carddom2 9394 Two numerable sets have th...
harcard 9395 The class of ordinal numbe...
cardprclem 9396 Lemma for ~ cardprc . (Co...
cardprc 9397 The class of all cardinal ...
carduni 9398 The union of a set of card...
cardiun 9399 The indexed union of a set...
cardennn 9400 If ` A ` is equinumerous t...
cardsucinf 9401 The cardinality of the suc...
cardsucnn 9402 The cardinality of the suc...
cardom 9403 The set of natural numbers...
carden2 9404 Two numerable sets are equ...
cardsdom2 9405 A numerable set is strictl...
domtri2 9406 Trichotomy of dominance fo...
nnsdomel 9407 Strict dominance and eleme...
cardval2 9408 An alternate version of th...
isinffi 9409 An infinite set contains s...
fidomtri 9410 Trichotomy of dominance wi...
fidomtri2 9411 Trichotomy of dominance wi...
harsdom 9412 The Hartogs number of a we...
onsdom 9413 Any well-orderable set is ...
harval2 9414 An alternate expression fo...
cardmin2 9415 The smallest ordinal that ...
pm54.43lem 9416 In Theorem *54.43 of [Whit...
pm54.43 9417 Theorem *54.43 of [Whitehe...
pr2nelem 9418 Lemma for ~ pr2ne . (Cont...
pr2ne 9419 If an unordered pair has t...
prdom2 9420 An unordered pair has at m...
en2eqpr 9421 Building a set with two el...
en2eleq 9422 Express a set of pair card...
en2other2 9423 Taking the other element t...
dif1card 9424 The cardinality of a nonem...
leweon 9425 Lexicographical order is a...
r0weon 9426 A set-like well-ordering o...
infxpenlem 9427 Lemma for ~ infxpen . (Co...
infxpen 9428 Every infinite ordinal is ...
xpomen 9429 The Cartesian product of o...
xpct 9430 The cartesian product of t...
infxpidm2 9431 The Cartesian product of a...
infxpenc 9432 A canonical version of ~ i...
infxpenc2lem1 9433 Lemma for ~ infxpenc2 . (...
infxpenc2lem2 9434 Lemma for ~ infxpenc2 . (...
infxpenc2lem3 9435 Lemma for ~ infxpenc2 . (...
infxpenc2 9436 Existence form of ~ infxpe...
iunmapdisj 9437 The union ` U_ n e. C ( A ...
fseqenlem1 9438 Lemma for ~ fseqen . (Con...
fseqenlem2 9439 Lemma for ~ fseqen . (Con...
fseqdom 9440 One half of ~ fseqen . (C...
fseqen 9441 A set that is equinumerous...
infpwfidom 9442 The collection of finite s...
dfac8alem 9443 Lemma for ~ dfac8a . If t...
dfac8a 9444 Numeration theorem: every ...
dfac8b 9445 The well-ordering theorem:...
dfac8clem 9446 Lemma for ~ dfac8c . (Con...
dfac8c 9447 If the union of a set is w...
ac10ct 9448 A proof of the well-orderi...
ween 9449 A set is numerable iff it ...
ac5num 9450 A version of ~ ac5b with t...
ondomen 9451 If a set is dominated by a...
numdom 9452 A set dominated by a numer...
ssnum 9453 A subset of a numerable se...
onssnum 9454 All subsets of the ordinal...
indcardi 9455 Indirect strong induction ...
acnrcl 9456 Reverse closure for the ch...
acneq 9457 Equality theorem for the c...
isacn 9458 The property of being a ch...
acni 9459 The property of being a ch...
acni2 9460 The property of being a ch...
acni3 9461 The property of being a ch...
acnlem 9462 Construct a mapping satisf...
numacn 9463 A well-orderable set has c...
finacn 9464 Every set has finite choic...
acndom 9465 A set with long choice seq...
acnnum 9466 A set ` X ` which has choi...
acnen 9467 The class of choice sets o...
acndom2 9468 A set smaller than one wit...
acnen2 9469 The class of sets with cho...
fodomacn 9470 A version of ~ fodom that ...
fodomnum 9471 A version of ~ fodom that ...
fonum 9472 A surjection maps numerabl...
numwdom 9473 A surjection maps numerabl...
fodomfi2 9474 Onto functions define domi...
wdomfil 9475 Weak dominance agrees with...
infpwfien 9476 Any infinite well-orderabl...
inffien 9477 The set of finite intersec...
wdomnumr 9478 Weak dominance agrees with...
alephfnon 9479 The aleph function is a fu...
aleph0 9480 The first infinite cardina...
alephlim 9481 Value of the aleph functio...
alephsuc 9482 Value of the aleph functio...
alephon 9483 An aleph is an ordinal num...
alephcard 9484 Every aleph is a cardinal ...
alephnbtwn 9485 No cardinal can be sandwic...
alephnbtwn2 9486 No set has equinumerosity ...
alephordilem1 9487 Lemma for ~ alephordi . (...
alephordi 9488 Strict ordering property o...
alephord 9489 Ordering property of the a...
alephord2 9490 Ordering property of the a...
alephord2i 9491 Ordering property of the a...
alephord3 9492 Ordering property of the a...
alephsucdom 9493 A set dominated by an alep...
alephsuc2 9494 An alternate representatio...
alephdom 9495 Relationship between inclu...
alephgeom 9496 Every aleph is greater tha...
alephislim 9497 Every aleph is a limit ord...
aleph11 9498 The aleph function is one-...
alephf1 9499 The aleph function is a on...
alephsdom 9500 If an ordinal is smaller t...
alephdom2 9501 A dominated initial ordina...
alephle 9502 The argument of the aleph ...
cardaleph 9503 Given any transfinite card...
cardalephex 9504 Every transfinite cardinal...
infenaleph 9505 An infinite numerable set ...
isinfcard 9506 Two ways to express the pr...
iscard3 9507 Two ways to express the pr...
cardnum 9508 Two ways to express the cl...
alephinit 9509 An infinite initial ordina...
carduniima 9510 The union of the image of ...
cardinfima 9511 If a mapping to cardinals ...
alephiso 9512 Aleph is an order isomorph...
alephprc 9513 The class of all transfini...
alephsson 9514 The class of transfinite c...
unialeph 9515 The union of the class of ...
alephsmo 9516 The aleph function is stri...
alephf1ALT 9517 Alternate proof of ~ aleph...
alephfplem1 9518 Lemma for ~ alephfp . (Co...
alephfplem2 9519 Lemma for ~ alephfp . (Co...
alephfplem3 9520 Lemma for ~ alephfp . (Co...
alephfplem4 9521 Lemma for ~ alephfp . (Co...
alephfp 9522 The aleph function has a f...
alephfp2 9523 The aleph function has at ...
alephval3 9524 An alternate way to expres...
alephsucpw2 9525 The power set of an aleph ...
mappwen 9526 Power rule for cardinal ar...
finnisoeu 9527 A finite totally ordered s...
iunfictbso 9528 Countability of a countabl...
aceq1 9531 Equivalence of two version...
aceq0 9532 Equivalence of two version...
aceq2 9533 Equivalence of two version...
aceq3lem 9534 Lemma for ~ dfac3 . (Cont...
dfac3 9535 Equivalence of two version...
dfac4 9536 Equivalence of two version...
dfac5lem1 9537 Lemma for ~ dfac5 . (Cont...
dfac5lem2 9538 Lemma for ~ dfac5 . (Cont...
dfac5lem3 9539 Lemma for ~ dfac5 . (Cont...
dfac5lem4 9540 Lemma for ~ dfac5 . (Cont...
dfac5lem5 9541 Lemma for ~ dfac5 . (Cont...
dfac5 9542 Equivalence of two version...
dfac2a 9543 Our Axiom of Choice (in th...
dfac2b 9544 Axiom of Choice (first for...
dfac2 9545 Axiom of Choice (first for...
dfac7 9546 Equivalence of the Axiom o...
dfac0 9547 Equivalence of two version...
dfac1 9548 Equivalence of two version...
dfac8 9549 A proof of the equivalency...
dfac9 9550 Equivalence of the axiom o...
dfac10 9551 Axiom of Choice equivalent...
dfac10c 9552 Axiom of Choice equivalent...
dfac10b 9553 Axiom of Choice equivalent...
acacni 9554 A choice equivalent: every...
dfacacn 9555 A choice equivalent: every...
dfac13 9556 The axiom of choice holds ...
dfac12lem1 9557 Lemma for ~ dfac12 . (Con...
dfac12lem2 9558 Lemma for ~ dfac12 . (Con...
dfac12lem3 9559 Lemma for ~ dfac12 . (Con...
dfac12r 9560 The axiom of choice holds ...
dfac12k 9561 Equivalence of ~ dfac12 an...
dfac12a 9562 The axiom of choice holds ...
dfac12 9563 The axiom of choice holds ...
kmlem1 9564 Lemma for 5-quantifier AC ...
kmlem2 9565 Lemma for 5-quantifier AC ...
kmlem3 9566 Lemma for 5-quantifier AC ...
kmlem4 9567 Lemma for 5-quantifier AC ...
kmlem5 9568 Lemma for 5-quantifier AC ...
kmlem6 9569 Lemma for 5-quantifier AC ...
kmlem7 9570 Lemma for 5-quantifier AC ...
kmlem8 9571 Lemma for 5-quantifier AC ...
kmlem9 9572 Lemma for 5-quantifier AC ...
kmlem10 9573 Lemma for 5-quantifier AC ...
kmlem11 9574 Lemma for 5-quantifier AC ...
kmlem12 9575 Lemma for 5-quantifier AC ...
kmlem13 9576 Lemma for 5-quantifier AC ...
kmlem14 9577 Lemma for 5-quantifier AC ...
kmlem15 9578 Lemma for 5-quantifier AC ...
kmlem16 9579 Lemma for 5-quantifier AC ...
dfackm 9580 Equivalence of the Axiom o...
undjudom 9581 Cardinal addition dominate...
endjudisj 9582 Equinumerosity of a disjoi...
djuen 9583 Disjoint unions of equinum...
djuenun 9584 Disjoint union is equinume...
dju1en 9585 Cardinal addition with car...
dju1dif 9586 Adding and subtracting one...
dju1p1e2 9587 1+1=2 for cardinal number ...
dju1p1e2ALT 9588 Alternate proof of ~ dju1p...
dju0en 9589 Cardinal addition with car...
xp2dju 9590 Two times a cardinal numbe...
djucomen 9591 Commutative law for cardin...
djuassen 9592 Associative law for cardin...
xpdjuen 9593 Cardinal multiplication di...
mapdjuen 9594 Sum of exponents law for c...
pwdjuen 9595 Sum of exponents law for c...
djudom1 9596 Ordering law for cardinal ...
djudom2 9597 Ordering law for cardinal ...
djudoml 9598 A set is dominated by its ...
djuxpdom 9599 Cartesian product dominate...
djufi 9600 The disjoint union of two ...
cdainflem 9601 Any partition of omega int...
djuinf 9602 A set is infinite iff the ...
infdju1 9603 An infinite set is equinum...
pwdju1 9604 The sum of a powerset with...
pwdjuidm 9605 If the natural numbers inj...
djulepw 9606 If ` A ` is idempotent und...
onadju 9607 The cardinal and ordinal s...
cardadju 9608 The cardinal sum is equinu...
djunum 9609 The disjoint union of two ...
unnum 9610 The union of two numerable...
nnadju 9611 The cardinal and ordinal s...
ficardun 9612 The cardinality of the uni...
ficardun2 9613 The cardinality of the uni...
pwsdompw 9614 Lemma for ~ domtriom . Th...
unctb 9615 The union of two countable...
infdjuabs 9616 Absorption law for additio...
infunabs 9617 An infinite set is equinum...
infdju 9618 The sum of two cardinal nu...
infdif 9619 The cardinality of an infi...
infdif2 9620 Cardinality ordering for a...
infxpdom 9621 Dominance law for multipli...
infxpabs 9622 Absorption law for multipl...
infunsdom1 9623 The union of two sets that...
infunsdom 9624 The union of two sets that...
infxp 9625 Absorption law for multipl...
pwdjudom 9626 A property of dominance ov...
infpss 9627 Every infinite set has an ...
infmap2 9628 An exponentiation law for ...
ackbij2lem1 9629 Lemma for ~ ackbij2 . (Co...
ackbij1lem1 9630 Lemma for ~ ackbij2 . (Co...
ackbij1lem2 9631 Lemma for ~ ackbij2 . (Co...
ackbij1lem3 9632 Lemma for ~ ackbij2 . (Co...
ackbij1lem4 9633 Lemma for ~ ackbij2 . (Co...
ackbij1lem5 9634 Lemma for ~ ackbij2 . (Co...
ackbij1lem6 9635 Lemma for ~ ackbij2 . (Co...
ackbij1lem7 9636 Lemma for ~ ackbij1 . (Co...
ackbij1lem8 9637 Lemma for ~ ackbij1 . (Co...
ackbij1lem9 9638 Lemma for ~ ackbij1 . (Co...
ackbij1lem10 9639 Lemma for ~ ackbij1 . (Co...
ackbij1lem11 9640 Lemma for ~ ackbij1 . (Co...
ackbij1lem12 9641 Lemma for ~ ackbij1 . (Co...
ackbij1lem13 9642 Lemma for ~ ackbij1 . (Co...
ackbij1lem14 9643 Lemma for ~ ackbij1 . (Co...
ackbij1lem15 9644 Lemma for ~ ackbij1 . (Co...
ackbij1lem16 9645 Lemma for ~ ackbij1 . (Co...
ackbij1lem17 9646 Lemma for ~ ackbij1 . (Co...
ackbij1lem18 9647 Lemma for ~ ackbij1 . (Co...
ackbij1 9648 The Ackermann bijection, p...
ackbij1b 9649 The Ackermann bijection, p...
ackbij2lem2 9650 Lemma for ~ ackbij2 . (Co...
ackbij2lem3 9651 Lemma for ~ ackbij2 . (Co...
ackbij2lem4 9652 Lemma for ~ ackbij2 . (Co...
ackbij2 9653 The Ackermann bijection, p...
r1om 9654 The set of hereditarily fi...
fictb 9655 A set is countable iff its...
cflem 9656 A lemma used to simplify c...
cfval 9657 Value of the cofinality fu...
cff 9658 Cofinality is a function o...
cfub 9659 An upper bound on cofinali...
cflm 9660 Value of the cofinality fu...
cf0 9661 Value of the cofinality fu...
cardcf 9662 Cofinality is a cardinal n...
cflecard 9663 Cofinality is bounded by t...
cfle 9664 Cofinality is bounded by i...
cfon 9665 The cofinality of any set ...
cfeq0 9666 Only the ordinal zero has ...
cfsuc 9667 Value of the cofinality fu...
cff1 9668 There is always a map from...
cfflb 9669 If there is a cofinal map ...
cfval2 9670 Another expression for the...
coflim 9671 A simpler expression for t...
cflim3 9672 Another expression for the...
cflim2 9673 The cofinality function is...
cfom 9674 Value of the cofinality fu...
cfss 9675 There is a cofinal subset ...
cfslb 9676 Any cofinal subset of ` A ...
cfslbn 9677 Any subset of ` A ` smalle...
cfslb2n 9678 Any small collection of sm...
cofsmo 9679 Any cofinal map implies th...
cfsmolem 9680 Lemma for ~ cfsmo . (Cont...
cfsmo 9681 The map in ~ cff1 can be a...
cfcoflem 9682 Lemma for ~ cfcof , showin...
coftr 9683 If there is a cofinal map ...
cfcof 9684 If there is a cofinal map ...
cfidm 9685 The cofinality function is...
alephsing 9686 The cofinality of a limit ...
sornom 9687 The range of a single-step...
isfin1a 9702 Definition of a Ia-finite ...
fin1ai 9703 Property of a Ia-finite se...
isfin2 9704 Definition of a II-finite ...
fin2i 9705 Property of a II-finite se...
isfin3 9706 Definition of a III-finite...
isfin4 9707 Definition of a IV-finite ...
fin4i 9708 Infer that a set is IV-inf...
isfin5 9709 Definition of a V-finite s...
isfin6 9710 Definition of a VI-finite ...
isfin7 9711 Definition of a VII-finite...
sdom2en01 9712 A set with less than two e...
infpssrlem1 9713 Lemma for ~ infpssr . (Co...
infpssrlem2 9714 Lemma for ~ infpssr . (Co...
infpssrlem3 9715 Lemma for ~ infpssr . (Co...
infpssrlem4 9716 Lemma for ~ infpssr . (Co...
infpssrlem5 9717 Lemma for ~ infpssr . (Co...
infpssr 9718 Dedekind infinity implies ...
fin4en1 9719 Dedekind finite is a cardi...
ssfin4 9720 Dedekind finite sets have ...
domfin4 9721 A set dominated by a Dedek...
ominf4 9722 ` _om ` is Dedekind infini...
infpssALT 9723 Alternate proof of ~ infps...
isfin4-2 9724 Alternate definition of IV...
isfin4p1 9725 Alternate definition of IV...
fin23lem7 9726 Lemma for ~ isfin2-2 . Th...
fin23lem11 9727 Lemma for ~ isfin2-2 . (C...
fin2i2 9728 A II-finite set contains m...
isfin2-2 9729 ` Fin2 ` expressed in term...
ssfin2 9730 A subset of a II-finite se...
enfin2i 9731 II-finiteness is a cardina...
fin23lem24 9732 Lemma for ~ fin23 . In a ...
fincssdom 9733 In a chain of finite sets,...
fin23lem25 9734 Lemma for ~ fin23 . In a ...
fin23lem26 9735 Lemma for ~ fin23lem22 . ...
fin23lem23 9736 Lemma for ~ fin23lem22 . ...
fin23lem22 9737 Lemma for ~ fin23 but coul...
fin23lem27 9738 The mapping constructed in...
isfin3ds 9739 Property of a III-finite s...
ssfin3ds 9740 A subset of a III-finite s...
fin23lem12 9741 The beginning of the proof...
fin23lem13 9742 Lemma for ~ fin23 . Each ...
fin23lem14 9743 Lemma for ~ fin23 . ` U ` ...
fin23lem15 9744 Lemma for ~ fin23 . ` U ` ...
fin23lem16 9745 Lemma for ~ fin23 . ` U ` ...
fin23lem19 9746 Lemma for ~ fin23 . The f...
fin23lem20 9747 Lemma for ~ fin23 . ` X ` ...
fin23lem17 9748 Lemma for ~ fin23 . By ? ...
fin23lem21 9749 Lemma for ~ fin23 . ` X ` ...
fin23lem28 9750 Lemma for ~ fin23 . The r...
fin23lem29 9751 Lemma for ~ fin23 . The r...
fin23lem30 9752 Lemma for ~ fin23 . The r...
fin23lem31 9753 Lemma for ~ fin23 . The r...
fin23lem32 9754 Lemma for ~ fin23 . Wrap ...
fin23lem33 9755 Lemma for ~ fin23 . Disch...
fin23lem34 9756 Lemma for ~ fin23 . Estab...
fin23lem35 9757 Lemma for ~ fin23 . Stric...
fin23lem36 9758 Lemma for ~ fin23 . Weak ...
fin23lem38 9759 Lemma for ~ fin23 . The c...
fin23lem39 9760 Lemma for ~ fin23 . Thus,...
fin23lem40 9761 Lemma for ~ fin23 . ` Fin2...
fin23lem41 9762 Lemma for ~ fin23 . A set...
isf32lem1 9763 Lemma for ~ isfin3-2 . De...
isf32lem2 9764 Lemma for ~ isfin3-2 . No...
isf32lem3 9765 Lemma for ~ isfin3-2 . Be...
isf32lem4 9766 Lemma for ~ isfin3-2 . Be...
isf32lem5 9767 Lemma for ~ isfin3-2 . Th...
isf32lem6 9768 Lemma for ~ isfin3-2 . Ea...
isf32lem7 9769 Lemma for ~ isfin3-2 . Di...
isf32lem8 9770 Lemma for ~ isfin3-2 . K ...
isf32lem9 9771 Lemma for ~ isfin3-2 . Co...
isf32lem10 9772 Lemma for isfin3-2 . Writ...
isf32lem11 9773 Lemma for ~ isfin3-2 . Re...
isf32lem12 9774 Lemma for ~ isfin3-2 . (C...
isfin32i 9775 One half of ~ isfin3-2 . ...
isf33lem 9776 Lemma for ~ isfin3-3 . (C...
isfin3-2 9777 Weakly Dedekind-infinite s...
isfin3-3 9778 Weakly Dedekind-infinite s...
fin33i 9779 Inference from ~ isfin3-3 ...
compsscnvlem 9780 Lemma for ~ compsscnv . (...
compsscnv 9781 Complementation on a power...
isf34lem1 9782 Lemma for ~ isfin3-4 . (C...
isf34lem2 9783 Lemma for ~ isfin3-4 . (C...
compssiso 9784 Complementation is an anti...
isf34lem3 9785 Lemma for ~ isfin3-4 . (C...
compss 9786 Express image under of the...
isf34lem4 9787 Lemma for ~ isfin3-4 . (C...
isf34lem5 9788 Lemma for ~ isfin3-4 . (C...
isf34lem7 9789 Lemma for ~ isfin3-4 . (C...
isf34lem6 9790 Lemma for ~ isfin3-4 . (C...
fin34i 9791 Inference from ~ isfin3-4 ...
isfin3-4 9792 Weakly Dedekind-infinite s...
fin11a 9793 Every I-finite set is Ia-f...
enfin1ai 9794 Ia-finiteness is a cardina...
isfin1-2 9795 A set is finite in the usu...
isfin1-3 9796 A set is I-finite iff ever...
isfin1-4 9797 A set is I-finite iff ever...
dffin1-5 9798 Compact quantifier-free ve...
fin23 9799 Every II-finite set (every...
fin34 9800 Every III-finite set is IV...
isfin5-2 9801 Alternate definition of V-...
fin45 9802 Every IV-finite set is V-f...
fin56 9803 Every V-finite set is VI-f...
fin17 9804 Every I-finite set is VII-...
fin67 9805 Every VI-finite set is VII...
isfin7-2 9806 A set is VII-finite iff it...
fin71num 9807 A well-orderable set is VI...
dffin7-2 9808 Class form of ~ isfin7-2 ....
dfacfin7 9809 Axiom of Choice equivalent...
fin1a2lem1 9810 Lemma for ~ fin1a2 . (Con...
fin1a2lem2 9811 Lemma for ~ fin1a2 . (Con...
fin1a2lem3 9812 Lemma for ~ fin1a2 . (Con...
fin1a2lem4 9813 Lemma for ~ fin1a2 . (Con...
fin1a2lem5 9814 Lemma for ~ fin1a2 . (Con...
fin1a2lem6 9815 Lemma for ~ fin1a2 . Esta...
fin1a2lem7 9816 Lemma for ~ fin1a2 . Spli...
fin1a2lem8 9817 Lemma for ~ fin1a2 . Spli...
fin1a2lem9 9818 Lemma for ~ fin1a2 . In a...
fin1a2lem10 9819 Lemma for ~ fin1a2 . A no...
fin1a2lem11 9820 Lemma for ~ fin1a2 . (Con...
fin1a2lem12 9821 Lemma for ~ fin1a2 . (Con...
fin1a2lem13 9822 Lemma for ~ fin1a2 . (Con...
fin12 9823 Weak theorem which skips I...
fin1a2s 9824 An II-infinite set can hav...
fin1a2 9825 Every Ia-finite set is II-...
itunifval 9826 Function value of iterated...
itunifn 9827 Functionality of the itera...
ituni0 9828 A zero-fold iterated union...
itunisuc 9829 Successor iterated union. ...
itunitc1 9830 Each union iterate is a me...
itunitc 9831 The union of all union ite...
ituniiun 9832 Unwrap an iterated union f...
hsmexlem7 9833 Lemma for ~ hsmex . Prope...
hsmexlem8 9834 Lemma for ~ hsmex . Prope...
hsmexlem9 9835 Lemma for ~ hsmex . Prope...
hsmexlem1 9836 Lemma for ~ hsmex . Bound...
hsmexlem2 9837 Lemma for ~ hsmex . Bound...
hsmexlem3 9838 Lemma for ~ hsmex . Clear...
hsmexlem4 9839 Lemma for ~ hsmex . The c...
hsmexlem5 9840 Lemma for ~ hsmex . Combi...
hsmexlem6 9841 Lemma for ~ hsmex . (Cont...
hsmex 9842 The collection of heredita...
hsmex2 9843 The set of hereditary size...
hsmex3 9844 The set of hereditary size...
axcc2lem 9846 Lemma for ~ axcc2 . (Cont...
axcc2 9847 A possibly more useful ver...
axcc3 9848 A possibly more useful ver...
axcc4 9849 A version of ~ axcc3 that ...
acncc 9850 An ~ ax-cc equivalent: eve...
axcc4dom 9851 Relax the constraint on ~ ...
domtriomlem 9852 Lemma for ~ domtriom . (C...
domtriom 9853 Trichotomy of equinumerosi...
fin41 9854 Under countable choice, th...
dominf 9855 A nonempty set that is a s...
dcomex 9857 The Axiom of Dependent Cho...
axdc2lem 9858 Lemma for ~ axdc2 . We co...
axdc2 9859 An apparent strengthening ...
axdc3lem 9860 The class ` S ` of finite ...
axdc3lem2 9861 Lemma for ~ axdc3 . We ha...
axdc3lem3 9862 Simple substitution lemma ...
axdc3lem4 9863 Lemma for ~ axdc3 . We ha...
axdc3 9864 Dependent Choice. Axiom D...
axdc4lem 9865 Lemma for ~ axdc4 . (Cont...
axdc4 9866 A more general version of ...
axcclem 9867 Lemma for ~ axcc . (Contr...
axcc 9868 Although CC can be proven ...
zfac 9870 Axiom of Choice expressed ...
ac2 9871 Axiom of Choice equivalent...
ac3 9872 Axiom of Choice using abbr...
axac3 9874 This theorem asserts that ...
ackm 9875 A remarkable equivalent to...
axac2 9876 Derive ~ ax-ac2 from ~ ax-...
axac 9877 Derive ~ ax-ac from ~ ax-a...
axaci 9878 Apply a choice equivalent....
cardeqv 9879 All sets are well-orderabl...
numth3 9880 All sets are well-orderabl...
numth2 9881 Numeration theorem: any se...
numth 9882 Numeration theorem: every ...
ac7 9883 An Axiom of Choice equival...
ac7g 9884 An Axiom of Choice equival...
ac4 9885 Equivalent of Axiom of Cho...
ac4c 9886 Equivalent of Axiom of Cho...
ac5 9887 An Axiom of Choice equival...
ac5b 9888 Equivalent of Axiom of Cho...
ac6num 9889 A version of ~ ac6 which t...
ac6 9890 Equivalent of Axiom of Cho...
ac6c4 9891 Equivalent of Axiom of Cho...
ac6c5 9892 Equivalent of Axiom of Cho...
ac9 9893 An Axiom of Choice equival...
ac6s 9894 Equivalent of Axiom of Cho...
ac6n 9895 Equivalent of Axiom of Cho...
ac6s2 9896 Generalization of the Axio...
ac6s3 9897 Generalization of the Axio...
ac6sg 9898 ~ ac6s with sethood as ant...
ac6sf 9899 Version of ~ ac6 with boun...
ac6s4 9900 Generalization of the Axio...
ac6s5 9901 Generalization of the Axio...
ac8 9902 An Axiom of Choice equival...
ac9s 9903 An Axiom of Choice equival...
numthcor 9904 Any set is strictly domina...
weth 9905 Well-ordering theorem: any...
zorn2lem1 9906 Lemma for ~ zorn2 . (Cont...
zorn2lem2 9907 Lemma for ~ zorn2 . (Cont...
zorn2lem3 9908 Lemma for ~ zorn2 . (Cont...
zorn2lem4 9909 Lemma for ~ zorn2 . (Cont...
zorn2lem5 9910 Lemma for ~ zorn2 . (Cont...
zorn2lem6 9911 Lemma for ~ zorn2 . (Cont...
zorn2lem7 9912 Lemma for ~ zorn2 . (Cont...
zorn2g 9913 Zorn's Lemma of [Monk1] p....
zorng 9914 Zorn's Lemma. If the unio...
zornn0g 9915 Variant of Zorn's lemma ~ ...
zorn2 9916 Zorn's Lemma of [Monk1] p....
zorn 9917 Zorn's Lemma. If the unio...
zornn0 9918 Variant of Zorn's lemma ~ ...
ttukeylem1 9919 Lemma for ~ ttukey . Expa...
ttukeylem2 9920 Lemma for ~ ttukey . A pr...
ttukeylem3 9921 Lemma for ~ ttukey . (Con...
ttukeylem4 9922 Lemma for ~ ttukey . (Con...
ttukeylem5 9923 Lemma for ~ ttukey . The ...
ttukeylem6 9924 Lemma for ~ ttukey . (Con...
ttukeylem7 9925 Lemma for ~ ttukey . (Con...
ttukey2g 9926 The Teichmüller-Tukey...
ttukeyg 9927 The Teichmüller-Tukey...
ttukey 9928 The Teichmüller-Tukey...
axdclem 9929 Lemma for ~ axdc . (Contr...
axdclem2 9930 Lemma for ~ axdc . Using ...
axdc 9931 This theorem derives ~ ax-...
fodom 9932 An onto function implies d...
fodomg 9933 An onto function implies d...
dmct 9934 The domain of a countable ...
rnct 9935 The range of a countable s...
fodomb 9936 Equivalence of an onto map...
wdomac 9937 When assuming AC, weak and...
brdom3 9938 Equivalence to a dominance...
brdom5 9939 An equivalence to a domina...
brdom4 9940 An equivalence to a domina...
brdom7disj 9941 An equivalence to a domina...
brdom6disj 9942 An equivalence to a domina...
fin71ac 9943 Once we allow AC, the "str...
imadomg 9944 An image of a function und...
fimact 9945 The image by a function of...
fnrndomg 9946 The range of a function is...
fnct 9947 If the domain of a functio...
mptct 9948 A countable mapping set is...
iunfo 9949 Existence of an onto funct...
iundom2g 9950 An upper bound for the car...
iundomg 9951 An upper bound for the car...
iundom 9952 An upper bound for the car...
unidom 9953 An upper bound for the car...
uniimadom 9954 An upper bound for the car...
uniimadomf 9955 An upper bound for the car...
cardval 9956 The value of the cardinal ...
cardid 9957 Any set is equinumerous to...
cardidg 9958 Any set is equinumerous to...
cardidd 9959 Any set is equinumerous to...
cardf 9960 The cardinality function i...
carden 9961 Two sets are equinumerous ...
cardeq0 9962 Only the empty set has car...
unsnen 9963 Equinumerosity of a set wi...
carddom 9964 Two sets have the dominanc...
cardsdom 9965 Two sets have the strict d...
domtri 9966 Trichotomy law for dominan...
entric 9967 Trichotomy of equinumerosi...
entri2 9968 Trichotomy of dominance an...
entri3 9969 Trichotomy of dominance. ...
sdomsdomcard 9970 A set strictly dominates i...
canth3 9971 Cantor's theorem in terms ...
infxpidm 9972 The Cartesian product of a...
ondomon 9973 The collection of ordinal ...
cardmin 9974 The smallest ordinal that ...
ficard 9975 A set is finite iff its ca...
infinf 9976 Equivalence between two in...
unirnfdomd 9977 The union of the range of ...
konigthlem 9978 Lemma for ~ konigth . (Co...
konigth 9979 Konig's Theorem. If ` m (...
alephsucpw 9980 The power set of an aleph ...
aleph1 9981 The set exponentiation of ...
alephval2 9982 An alternate way to expres...
dominfac 9983 A nonempty set that is a s...
iunctb 9984 The countable union of cou...
unictb 9985 The countable union of cou...
infmap 9986 An exponentiation law for ...
alephadd 9987 The sum of two alephs is t...
alephmul 9988 The product of two alephs ...
alephexp1 9989 An exponentiation law for ...
alephsuc3 9990 An alternate representatio...
alephexp2 9991 An expression equinumerous...
alephreg 9992 A successor aleph is regul...
pwcfsdom 9993 A corollary of Konig's The...
cfpwsdom 9994 A corollary of Konig's The...
alephom 9995 From ~ canth2 , we know th...
smobeth 9996 The beth function is stric...
nd1 9997 A lemma for proving condit...
nd2 9998 A lemma for proving condit...
nd3 9999 A lemma for proving condit...
nd4 10000 A lemma for proving condit...
axextnd 10001 A version of the Axiom of ...
axrepndlem1 10002 Lemma for the Axiom of Rep...
axrepndlem2 10003 Lemma for the Axiom of Rep...
axrepnd 10004 A version of the Axiom of ...
axunndlem1 10005 Lemma for the Axiom of Uni...
axunnd 10006 A version of the Axiom of ...
axpowndlem1 10007 Lemma for the Axiom of Pow...
axpowndlem2 10008 Lemma for the Axiom of Pow...
axpowndlem3 10009 Lemma for the Axiom of Pow...
axpowndlem4 10010 Lemma for the Axiom of Pow...
axpownd 10011 A version of the Axiom of ...
axregndlem1 10012 Lemma for the Axiom of Reg...
axregndlem2 10013 Lemma for the Axiom of Reg...
axregnd 10014 A version of the Axiom of ...
axinfndlem1 10015 Lemma for the Axiom of Inf...
axinfnd 10016 A version of the Axiom of ...
axacndlem1 10017 Lemma for the Axiom of Cho...
axacndlem2 10018 Lemma for the Axiom of Cho...
axacndlem3 10019 Lemma for the Axiom of Cho...
axacndlem4 10020 Lemma for the Axiom of Cho...
axacndlem5 10021 Lemma for the Axiom of Cho...
axacnd 10022 A version of the Axiom of ...
zfcndext 10023 Axiom of Extensionality ~ ...
zfcndrep 10024 Axiom of Replacement ~ ax-...
zfcndun 10025 Axiom of Union ~ ax-un , r...
zfcndpow 10026 Axiom of Power Sets ~ ax-p...
zfcndreg 10027 Axiom of Regularity ~ ax-r...
zfcndinf 10028 Axiom of Infinity ~ ax-inf...
zfcndac 10029 Axiom of Choice ~ ax-ac , ...
elgch 10032 Elementhood in the collect...
fingch 10033 A finite set is a GCH-set....
gchi 10034 The only GCH-sets which ha...
gchen1 10035 If ` A <_ B < ~P A ` , and...
gchen2 10036 If ` A < B <_ ~P A ` , and...
gchor 10037 If ` A <_ B <_ ~P A ` , an...
engch 10038 The property of being a GC...
gchdomtri 10039 Under certain conditions, ...
fpwwe2cbv 10040 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem1 10041 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem2 10042 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem3 10043 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem5 10044 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem6 10045 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem7 10046 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem8 10047 Lemma for ~ fpwwe2 . Show...
fpwwe2lem9 10048 Lemma for ~ fpwwe2 . Give...
fpwwe2lem10 10049 Lemma for ~ fpwwe2 . Give...
fpwwe2lem11 10050 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem12 10051 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem13 10052 Lemma for ~ fpwwe2 . (Con...
fpwwe2 10053 Given any function ` F ` f...
fpwwecbv 10054 Lemma for ~ fpwwe . (Cont...
fpwwelem 10055 Lemma for ~ fpwwe . (Cont...
fpwwe 10056 Given any function ` F ` f...
canth4 10057 An "effective" form of Can...
canthnumlem 10058 Lemma for ~ canthnum . (C...
canthnum 10059 The set of well-orderable ...
canthwelem 10060 Lemma for ~ canthwe . (Co...
canthwe 10061 The set of well-orders of ...
canthp1lem1 10062 Lemma for ~ canthp1 . (Co...
canthp1lem2 10063 Lemma for ~ canthp1 . (Co...
canthp1 10064 A slightly stronger form o...
finngch 10065 The exclusion of finite se...
gchdju1 10066 An infinite GCH-set is ide...
gchinf 10067 An infinite GCH-set is Ded...
pwfseqlem1 10068 Lemma for ~ pwfseq . Deri...
pwfseqlem2 10069 Lemma for ~ pwfseq . (Con...
pwfseqlem3 10070 Lemma for ~ pwfseq . Usin...
pwfseqlem4a 10071 Lemma for ~ pwfseqlem4 . ...
pwfseqlem4 10072 Lemma for ~ pwfseq . Deri...
pwfseqlem5 10073 Lemma for ~ pwfseq . Alth...
pwfseq 10074 The powerset of a Dedekind...
pwxpndom2 10075 The powerset of a Dedekind...
pwxpndom 10076 The powerset of a Dedekind...
pwdjundom 10077 The powerset of a Dedekind...
gchdjuidm 10078 An infinite GCH-set is ide...
gchxpidm 10079 An infinite GCH-set is ide...
gchpwdom 10080 A relationship between dom...
gchaleph 10081 If ` ( aleph `` A ) ` is a...
gchaleph2 10082 If ` ( aleph `` A ) ` and ...
hargch 10083 If ` A + ~~ ~P A ` , then ...
alephgch 10084 If ` ( aleph `` suc A ) ` ...
gch2 10085 It is sufficient to requir...
gch3 10086 An equivalent formulation ...
gch-kn 10087 The equivalence of two ver...
gchaclem 10088 Lemma for ~ gchac (obsolet...
gchhar 10089 A "local" form of ~ gchac ...
gchacg 10090 A "local" form of ~ gchac ...
gchac 10091 The Generalized Continuum ...
elwina 10096 Conditions of weak inacces...
elina 10097 Conditions of strong inacc...
winaon 10098 A weakly inaccessible card...
inawinalem 10099 Lemma for ~ inawina . (Co...
inawina 10100 Every strongly inaccessibl...
omina 10101 ` _om ` is a strongly inac...
winacard 10102 A weakly inaccessible card...
winainflem 10103 A weakly inaccessible card...
winainf 10104 A weakly inaccessible card...
winalim 10105 A weakly inaccessible card...
winalim2 10106 A nontrivial weakly inacce...
winafp 10107 A nontrivial weakly inacce...
winafpi 10108 This theorem, which states...
gchina 10109 Assuming the GCH, weakly a...
iswun 10114 Properties of a weak unive...
wuntr 10115 A weak universe is transit...
wununi 10116 A weak universe is closed ...
wunpw 10117 A weak universe is closed ...
wunelss 10118 The elements of a weak uni...
wunpr 10119 A weak universe is closed ...
wunun 10120 A weak universe is closed ...
wuntp 10121 A weak universe is closed ...
wunss 10122 A weak universe is closed ...
wunin 10123 A weak universe is closed ...
wundif 10124 A weak universe is closed ...
wunint 10125 A weak universe is closed ...
wunsn 10126 A weak universe is closed ...
wunsuc 10127 A weak universe is closed ...
wun0 10128 A weak universe contains t...
wunr1om 10129 A weak universe is infinit...
wunom 10130 A weak universe contains a...
wunfi 10131 A weak universe contains a...
wunop 10132 A weak universe is closed ...
wunot 10133 A weak universe is closed ...
wunxp 10134 A weak universe is closed ...
wunpm 10135 A weak universe is closed ...
wunmap 10136 A weak universe is closed ...
wunf 10137 A weak universe is closed ...
wundm 10138 A weak universe is closed ...
wunrn 10139 A weak universe is closed ...
wuncnv 10140 A weak universe is closed ...
wunres 10141 A weak universe is closed ...
wunfv 10142 A weak universe is closed ...
wunco 10143 A weak universe is closed ...
wuntpos 10144 A weak universe is closed ...
intwun 10145 The intersection of a coll...
r1limwun 10146 Each limit stage in the cu...
r1wunlim 10147 The weak universes in the ...
wunex2 10148 Construct a weak universe ...
wunex 10149 Construct a weak universe ...
uniwun 10150 Every set is contained in ...
wunex3 10151 Construct a weak universe ...
wuncval 10152 Value of the weak universe...
wuncid 10153 The weak universe closure ...
wunccl 10154 The weak universe closure ...
wuncss 10155 The weak universe closure ...
wuncidm 10156 The weak universe closure ...
wuncval2 10157 Our earlier expression for...
eltskg 10160 Properties of a Tarski cla...
eltsk2g 10161 Properties of a Tarski cla...
tskpwss 10162 First axiom of a Tarski cl...
tskpw 10163 Second axiom of a Tarski c...
tsken 10164 Third axiom of a Tarski cl...
0tsk 10165 The empty set is a (transi...
tsksdom 10166 An element of a Tarski cla...
tskssel 10167 A part of a Tarski class s...
tskss 10168 The subsets of an element ...
tskin 10169 The intersection of two el...
tsksn 10170 A singleton of an element ...
tsktrss 10171 A transitive element of a ...
tsksuc 10172 If an element of a Tarski ...
tsk0 10173 A nonempty Tarski class co...
tsk1 10174 One is an element of a non...
tsk2 10175 Two is an element of a non...
2domtsk 10176 If a Tarski class is not e...
tskr1om 10177 A nonempty Tarski class is...
tskr1om2 10178 A nonempty Tarski class co...
tskinf 10179 A nonempty Tarski class is...
tskpr 10180 If ` A ` and ` B ` are mem...
tskop 10181 If ` A ` and ` B ` are mem...
tskxpss 10182 A Cartesian product of two...
tskwe2 10183 A Tarski class is well-ord...
inttsk 10184 The intersection of a coll...
inar1 10185 ` ( R1 `` A ) ` for ` A ` ...
r1omALT 10186 Alternate proof of ~ r1om ...
rankcf 10187 Any set must be at least a...
inatsk 10188 ` ( R1 `` A ) ` for ` A ` ...
r1omtsk 10189 The set of hereditarily fi...
tskord 10190 A Tarski class contains al...
tskcard 10191 An even more direct relati...
r1tskina 10192 There is a direct relation...
tskuni 10193 The union of an element of...
tskwun 10194 A nonempty transitive Tars...
tskint 10195 The intersection of an ele...
tskun 10196 The union of two elements ...
tskxp 10197 The Cartesian product of t...
tskmap 10198 Set exponentiation is an e...
tskurn 10199 A transitive Tarski class ...
elgrug 10202 Properties of a Grothendie...
grutr 10203 A Grothendieck universe is...
gruelss 10204 A Grothendieck universe is...
grupw 10205 A Grothendieck universe co...
gruss 10206 Any subset of an element o...
grupr 10207 A Grothendieck universe co...
gruurn 10208 A Grothendieck universe co...
gruiun 10209 If ` B ( x ) ` is a family...
gruuni 10210 A Grothendieck universe co...
grurn 10211 A Grothendieck universe co...
gruima 10212 A Grothendieck universe co...
gruel 10213 Any element of an element ...
grusn 10214 A Grothendieck universe co...
gruop 10215 A Grothendieck universe co...
gruun 10216 A Grothendieck universe co...
gruxp 10217 A Grothendieck universe co...
grumap 10218 A Grothendieck universe co...
gruixp 10219 A Grothendieck universe co...
gruiin 10220 A Grothendieck universe co...
gruf 10221 A Grothendieck universe co...
gruen 10222 A Grothendieck universe co...
gruwun 10223 A nonempty Grothendieck un...
intgru 10224 The intersection of a fami...
ingru 10225 The intersection of a univ...
wfgru 10226 The wellfounded part of a ...
grudomon 10227 Each ordinal that is compa...
gruina 10228 If a Grothendieck universe...
grur1a 10229 A characterization of Grot...
grur1 10230 A characterization of Grot...
grutsk1 10231 Grothendieck universes are...
grutsk 10232 Grothendieck universes are...
axgroth5 10234 The Tarski-Grothendieck ax...
axgroth2 10235 Alternate version of the T...
grothpw 10236 Derive the Axiom of Power ...
grothpwex 10237 Derive the Axiom of Power ...
axgroth6 10238 The Tarski-Grothendieck ax...
grothomex 10239 The Tarski-Grothendieck Ax...
grothac 10240 The Tarski-Grothendieck Ax...
axgroth3 10241 Alternate version of the T...
axgroth4 10242 Alternate version of the T...
grothprimlem 10243 Lemma for ~ grothprim . E...
grothprim 10244 The Tarski-Grothendieck Ax...
grothtsk 10245 The Tarski-Grothendieck Ax...
inaprc 10246 An equivalent to the Tarsk...
tskmval 10249 Value of our tarski map. ...
tskmid 10250 The set ` A ` is an elemen...
tskmcl 10251 A Tarski class that contai...
sstskm 10252 Being a part of ` ( tarski...
eltskm 10253 Belonging to ` ( tarskiMap...
elni 10286 Membership in the class of...
elni2 10287 Membership in the class of...
pinn 10288 A positive integer is a na...
pion 10289 A positive integer is an o...
piord 10290 A positive integer is ordi...
niex 10291 The class of positive inte...
0npi 10292 The empty set is not a pos...
1pi 10293 Ordinal 'one' is a positiv...
addpiord 10294 Positive integer addition ...
mulpiord 10295 Positive integer multiplic...
mulidpi 10296 1 is an identity element f...
ltpiord 10297 Positive integer 'less tha...
ltsopi 10298 Positive integer 'less tha...
ltrelpi 10299 Positive integer 'less tha...
dmaddpi 10300 Domain of addition on posi...
dmmulpi 10301 Domain of multiplication o...
addclpi 10302 Closure of addition of pos...
mulclpi 10303 Closure of multiplication ...
addcompi 10304 Addition of positive integ...
addasspi 10305 Addition of positive integ...
mulcompi 10306 Multiplication of positive...
mulasspi 10307 Multiplication of positive...
distrpi 10308 Multiplication of positive...
addcanpi 10309 Addition cancellation law ...
mulcanpi 10310 Multiplication cancellatio...
addnidpi 10311 There is no identity eleme...
ltexpi 10312 Ordering on positive integ...
ltapi 10313 Ordering property of addit...
ltmpi 10314 Ordering property of multi...
1lt2pi 10315 One is less than two (one ...
nlt1pi 10316 No positive integer is les...
indpi 10317 Principle of Finite Induct...
enqbreq 10329 Equivalence relation for p...
enqbreq2 10330 Equivalence relation for p...
enqer 10331 The equivalence relation f...
enqex 10332 The equivalence relation f...
nqex 10333 The class of positive frac...
0nnq 10334 The empty set is not a pos...
elpqn 10335 Each positive fraction is ...
ltrelnq 10336 Positive fraction 'less th...
pinq 10337 The representatives of pos...
1nq 10338 The positive fraction 'one...
nqereu 10339 There is a unique element ...
nqerf 10340 Corollary of ~ nqereu : th...
nqercl 10341 Corollary of ~ nqereu : cl...
nqerrel 10342 Any member of ` ( N. X. N....
nqerid 10343 Corollary of ~ nqereu : th...
enqeq 10344 Corollary of ~ nqereu : if...
nqereq 10345 The function ` /Q ` acts a...
addpipq2 10346 Addition of positive fract...
addpipq 10347 Addition of positive fract...
addpqnq 10348 Addition of positive fract...
mulpipq2 10349 Multiplication of positive...
mulpipq 10350 Multiplication of positive...
mulpqnq 10351 Multiplication of positive...
ordpipq 10352 Ordering of positive fract...
ordpinq 10353 Ordering of positive fract...
addpqf 10354 Closure of addition on pos...
addclnq 10355 Closure of addition on pos...
mulpqf 10356 Closure of multiplication ...
mulclnq 10357 Closure of multiplication ...
addnqf 10358 Domain of addition on posi...
mulnqf 10359 Domain of multiplication o...
addcompq 10360 Addition of positive fract...
addcomnq 10361 Addition of positive fract...
mulcompq 10362 Multiplication of positive...
mulcomnq 10363 Multiplication of positive...
adderpqlem 10364 Lemma for ~ adderpq . (Co...
mulerpqlem 10365 Lemma for ~ mulerpq . (Co...
adderpq 10366 Addition is compatible wit...
mulerpq 10367 Multiplication is compatib...
addassnq 10368 Addition of positive fract...
mulassnq 10369 Multiplication of positive...
mulcanenq 10370 Lemma for distributive law...
distrnq 10371 Multiplication of positive...
1nqenq 10372 The equivalence class of r...
mulidnq 10373 Multiplication identity el...
recmulnq 10374 Relationship between recip...
recidnq 10375 A positive fraction times ...
recclnq 10376 Closure law for positive f...
recrecnq 10377 Reciprocal of reciprocal o...
dmrecnq 10378 Domain of reciprocal on po...
ltsonq 10379 'Less than' is a strict or...
lterpq 10380 Compatibility of ordering ...
ltanq 10381 Ordering property of addit...
ltmnq 10382 Ordering property of multi...
1lt2nq 10383 One is less than two (one ...
ltaddnq 10384 The sum of two fractions i...
ltexnq 10385 Ordering on positive fract...
halfnq 10386 One-half of any positive f...
nsmallnq 10387 The is no smallest positiv...
ltbtwnnq 10388 There exists a number betw...
ltrnq 10389 Ordering property of recip...
archnq 10390 For any fraction, there is...
npex 10396 The class of positive real...
elnp 10397 Membership in positive rea...
elnpi 10398 Membership in positive rea...
prn0 10399 A positive real is not emp...
prpssnq 10400 A positive real is a subse...
elprnq 10401 A positive real is a set o...
0npr 10402 The empty set is not a pos...
prcdnq 10403 A positive real is closed ...
prub 10404 A positive fraction not in...
prnmax 10405 A positive real has no lar...
npomex 10406 A simplifying observation,...
prnmadd 10407 A positive real has no lar...
ltrelpr 10408 Positive real 'less than' ...
genpv 10409 Value of general operation...
genpelv 10410 Membership in value of gen...
genpprecl 10411 Pre-closure law for genera...
genpdm 10412 Domain of general operatio...
genpn0 10413 The result of an operation...
genpss 10414 The result of an operation...
genpnnp 10415 The result of an operation...
genpcd 10416 Downward closure of an ope...
genpnmax 10417 An operation on positive r...
genpcl 10418 Closure of an operation on...
genpass 10419 Associativity of an operat...
plpv 10420 Value of addition on posit...
mpv 10421 Value of multiplication on...
dmplp 10422 Domain of addition on posi...
dmmp 10423 Domain of multiplication o...
nqpr 10424 The canonical embedding of...
1pr 10425 The positive real number '...
addclprlem1 10426 Lemma to prove downward cl...
addclprlem2 10427 Lemma to prove downward cl...
addclpr 10428 Closure of addition on pos...
mulclprlem 10429 Lemma to prove downward cl...
mulclpr 10430 Closure of multiplication ...
addcompr 10431 Addition of positive reals...
addasspr 10432 Addition of positive reals...
mulcompr 10433 Multiplication of positive...
mulasspr 10434 Multiplication of positive...
distrlem1pr 10435 Lemma for distributive law...
distrlem4pr 10436 Lemma for distributive law...
distrlem5pr 10437 Lemma for distributive law...
distrpr 10438 Multiplication of positive...
1idpr 10439 1 is an identity element f...
ltprord 10440 Positive real 'less than' ...
psslinpr 10441 Proper subset is a linear ...
ltsopr 10442 Positive real 'less than' ...
prlem934 10443 Lemma 9-3.4 of [Gleason] p...
ltaddpr 10444 The sum of two positive re...
ltaddpr2 10445 The sum of two positive re...
ltexprlem1 10446 Lemma for Proposition 9-3....
ltexprlem2 10447 Lemma for Proposition 9-3....
ltexprlem3 10448 Lemma for Proposition 9-3....
ltexprlem4 10449 Lemma for Proposition 9-3....
ltexprlem5 10450 Lemma for Proposition 9-3....
ltexprlem6 10451 Lemma for Proposition 9-3....
ltexprlem7 10452 Lemma for Proposition 9-3....
ltexpri 10453 Proposition 9-3.5(iv) of [...
ltaprlem 10454 Lemma for Proposition 9-3....
ltapr 10455 Ordering property of addit...
addcanpr 10456 Addition cancellation law ...
prlem936 10457 Lemma 9-3.6 of [Gleason] p...
reclem2pr 10458 Lemma for Proposition 9-3....
reclem3pr 10459 Lemma for Proposition 9-3....
reclem4pr 10460 Lemma for Proposition 9-3....
recexpr 10461 The reciprocal of a positi...
suplem1pr 10462 The union of a nonempty, b...
suplem2pr 10463 The union of a set of posi...
supexpr 10464 The union of a nonempty, b...
enrer 10473 The equivalence relation f...
nrex1 10474 The class of signed reals ...
enrbreq 10475 Equivalence relation for s...
enreceq 10476 Equivalence class equality...
enrex 10477 The equivalence relation f...
ltrelsr 10478 Signed real 'less than' is...
addcmpblnr 10479 Lemma showing compatibilit...
mulcmpblnrlem 10480 Lemma used in lemma showin...
mulcmpblnr 10481 Lemma showing compatibilit...
prsrlem1 10482 Decomposing signed reals i...
addsrmo 10483 There is at most one resul...
mulsrmo 10484 There is at most one resul...
addsrpr 10485 Addition of signed reals i...
mulsrpr 10486 Multiplication of signed r...
ltsrpr 10487 Ordering of signed reals i...
gt0srpr 10488 Greater than zero in terms...
0nsr 10489 The empty set is not a sig...
0r 10490 The constant ` 0R ` is a s...
1sr 10491 The constant ` 1R ` is a s...
m1r 10492 The constant ` -1R ` is a ...
addclsr 10493 Closure of addition on sig...
mulclsr 10494 Closure of multiplication ...
dmaddsr 10495 Domain of addition on sign...
dmmulsr 10496 Domain of multiplication o...
addcomsr 10497 Addition of signed reals i...
addasssr 10498 Addition of signed reals i...
mulcomsr 10499 Multiplication of signed r...
mulasssr 10500 Multiplication of signed r...
distrsr 10501 Multiplication of signed r...
m1p1sr 10502 Minus one plus one is zero...
m1m1sr 10503 Minus one times minus one ...
ltsosr 10504 Signed real 'less than' is...
0lt1sr 10505 0 is less than 1 for signe...
1ne0sr 10506 1 and 0 are distinct for s...
0idsr 10507 The signed real number 0 i...
1idsr 10508 1 is an identity element f...
00sr 10509 A signed real times 0 is 0...
ltasr 10510 Ordering property of addit...
pn0sr 10511 A signed real plus its neg...
negexsr 10512 Existence of negative sign...
recexsrlem 10513 The reciprocal of a positi...
addgt0sr 10514 The sum of two positive si...
mulgt0sr 10515 The product of two positiv...
sqgt0sr 10516 The square of a nonzero si...
recexsr 10517 The reciprocal of a nonzer...
mappsrpr 10518 Mapping from positive sign...
ltpsrpr 10519 Mapping of order from posi...
map2psrpr 10520 Equivalence for positive s...
supsrlem 10521 Lemma for supremum theorem...
supsr 10522 A nonempty, bounded set of...
opelcn 10539 Ordered pair membership in...
opelreal 10540 Ordered pair membership in...
elreal 10541 Membership in class of rea...
elreal2 10542 Ordered pair membership in...
0ncn 10543 The empty set is not a com...
ltrelre 10544 'Less than' is a relation ...
addcnsr 10545 Addition of complex number...
mulcnsr 10546 Multiplication of complex ...
eqresr 10547 Equality of real numbers i...
addresr 10548 Addition of real numbers i...
mulresr 10549 Multiplication of real num...
ltresr 10550 Ordering of real subset of...
ltresr2 10551 Ordering of real subset of...
dfcnqs 10552 Technical trick to permit ...
addcnsrec 10553 Technical trick to permit ...
mulcnsrec 10554 Technical trick to permit ...
axaddf 10555 Addition is an operation o...
axmulf 10556 Multiplication is an opera...
axcnex 10557 The complex numbers form a...
axresscn 10558 The real numbers are a sub...
ax1cn 10559 1 is a complex number. Ax...
axicn 10560 ` _i ` is a complex number...
axaddcl 10561 Closure law for addition o...
axaddrcl 10562 Closure law for addition i...
axmulcl 10563 Closure law for multiplica...
axmulrcl 10564 Closure law for multiplica...
axmulcom 10565 Multiplication of complex ...
axaddass 10566 Addition of complex number...
axmulass 10567 Multiplication of complex ...
axdistr 10568 Distributive law for compl...
axi2m1 10569 i-squared equals -1 (expre...
ax1ne0 10570 1 and 0 are distinct. Axi...
ax1rid 10571 ` 1 ` is an identity eleme...
axrnegex 10572 Existence of negative of r...
axrrecex 10573 Existence of reciprocal of...
axcnre 10574 A complex number can be ex...
axpre-lttri 10575 Ordering on reals satisfie...
axpre-lttrn 10576 Ordering on reals is trans...
axpre-ltadd 10577 Ordering property of addit...
axpre-mulgt0 10578 The product of two positiv...
axpre-sup 10579 A nonempty, bounded-above ...
wuncn 10580 A weak universe containing...
cnex 10606 Alias for ~ ax-cnex . See...
addcl 10607 Alias for ~ ax-addcl , for...
readdcl 10608 Alias for ~ ax-addrcl , fo...
mulcl 10609 Alias for ~ ax-mulcl , for...
remulcl 10610 Alias for ~ ax-mulrcl , fo...
mulcom 10611 Alias for ~ ax-mulcom , fo...
addass 10612 Alias for ~ ax-addass , fo...
mulass 10613 Alias for ~ ax-mulass , fo...
adddi 10614 Alias for ~ ax-distr , for...
recn 10615 A real number is a complex...
reex 10616 The real numbers form a se...
reelprrecn 10617 Reals are a subset of the ...
cnelprrecn 10618 Complex numbers are a subs...
elimne0 10619 Hypothesis for weak deduct...
adddir 10620 Distributive law for compl...
0cn 10621 Zero is a complex number. ...
0cnd 10622 Zero is a complex number, ...
c0ex 10623 Zero is a set. (Contribut...
1cnd 10624 One is a complex number, d...
1ex 10625 One is a set. (Contribute...
cnre 10626 Alias for ~ ax-cnre , for ...
mulid1 10627 The number 1 is an identit...
mulid2 10628 Identity law for multiplic...
1re 10629 The number 1 is real. Thi...
1red 10630 The number 1 is real, dedu...
0re 10631 The number 0 is real. Rem...
0red 10632 The number 0 is real, dedu...
mulid1i 10633 Identity law for multiplic...
mulid2i 10634 Identity law for multiplic...
addcli 10635 Closure law for addition. ...
mulcli 10636 Closure law for multiplica...
mulcomi 10637 Commutative law for multip...
mulcomli 10638 Commutative law for multip...
addassi 10639 Associative law for additi...
mulassi 10640 Associative law for multip...
adddii 10641 Distributive law (left-dis...
adddiri 10642 Distributive law (right-di...
recni 10643 A real number is a complex...
readdcli 10644 Closure law for addition o...
remulcli 10645 Closure law for multiplica...
mulid1d 10646 Identity law for multiplic...
mulid2d 10647 Identity law for multiplic...
addcld 10648 Closure law for addition. ...
mulcld 10649 Closure law for multiplica...
mulcomd 10650 Commutative law for multip...
addassd 10651 Associative law for additi...
mulassd 10652 Associative law for multip...
adddid 10653 Distributive law (left-dis...
adddird 10654 Distributive law (right-di...
adddirp1d 10655 Distributive law, plus 1 v...
joinlmuladdmuld 10656 Join AB+CB into (A+C) on L...
recnd 10657 Deduction from real number...
readdcld 10658 Closure law for addition o...
remulcld 10659 Closure law for multiplica...
pnfnre 10670 Plus infinity is not a rea...
pnfnre2 10671 Plus infinity is not a rea...
mnfnre 10672 Minus infinity is not a re...
ressxr 10673 The standard reals are a s...
rexpssxrxp 10674 The Cartesian product of s...
rexr 10675 A standard real is an exte...
0xr 10676 Zero is an extended real. ...
renepnf 10677 No (finite) real equals pl...
renemnf 10678 No real equals minus infin...
rexrd 10679 A standard real is an exte...
renepnfd 10680 No (finite) real equals pl...
renemnfd 10681 No real equals minus infin...
pnfex 10682 Plus infinity exists. (Co...
pnfxr 10683 Plus infinity belongs to t...
pnfnemnf 10684 Plus and minus infinity ar...
mnfnepnf 10685 Minus and plus infinity ar...
mnfxr 10686 Minus infinity belongs to ...
rexri 10687 A standard real is an exte...
1xr 10688 ` 1 ` is an extended real ...
renfdisj 10689 The reals and the infiniti...
ltrelxr 10690 "Less than" is a relation ...
ltrel 10691 "Less than" is a relation....
lerelxr 10692 "Less than or equal to" is...
lerel 10693 "Less than or equal to" is...
xrlenlt 10694 "Less than or equal to" ex...
xrlenltd 10695 "Less than or equal to" ex...
xrltnle 10696 "Less than" expressed in t...
xrnltled 10697 "Not less than" implies "l...
ssxr 10698 The three (non-exclusive) ...
ltxrlt 10699 The standard less-than ` <...
axlttri 10700 Ordering on reals satisfie...
axlttrn 10701 Ordering on reals is trans...
axltadd 10702 Ordering property of addit...
axmulgt0 10703 The product of two positiv...
axsup 10704 A nonempty, bounded-above ...
lttr 10705 Alias for ~ axlttrn , for ...
mulgt0 10706 The product of two positiv...
lenlt 10707 'Less than or equal to' ex...
ltnle 10708 'Less than' expressed in t...
ltso 10709 'Less than' is a strict or...
gtso 10710 'Greater than' is a strict...
lttri2 10711 Consequence of trichotomy....
lttri3 10712 Trichotomy law for 'less t...
lttri4 10713 Trichotomy law for 'less t...
letri3 10714 Trichotomy law. (Contribu...
leloe 10715 'Less than or equal to' ex...
eqlelt 10716 Equality in terms of 'less...
ltle 10717 'Less than' implies 'less ...
leltne 10718 'Less than or equal to' im...
lelttr 10719 Transitive law. (Contribu...
ltletr 10720 Transitive law. (Contribu...
ltleletr 10721 Transitive law, weaker for...
letr 10722 Transitive law. (Contribu...
ltnr 10723 'Less than' is irreflexive...
leid 10724 'Less than or equal to' is...
ltne 10725 'Less than' implies not eq...
ltnsym 10726 'Less than' is not symmetr...
ltnsym2 10727 'Less than' is antisymmetr...
letric 10728 Trichotomy law. (Contribu...
ltlen 10729 'Less than' expressed in t...
eqle 10730 Equality implies 'less tha...
eqled 10731 Equality implies 'less tha...
ltadd2 10732 Addition to both sides of ...
ne0gt0 10733 A nonzero nonnegative numb...
lecasei 10734 Ordering elimination by ca...
lelttric 10735 Trichotomy law. (Contribu...
ltlecasei 10736 Ordering elimination by ca...
ltnri 10737 'Less than' is irreflexive...
eqlei 10738 Equality implies 'less tha...
eqlei2 10739 Equality implies 'less tha...
gtneii 10740 'Less than' implies not eq...
ltneii 10741 'Greater than' implies not...
lttri2i 10742 Consequence of trichotomy....
lttri3i 10743 Consequence of trichotomy....
letri3i 10744 Consequence of trichotomy....
leloei 10745 'Less than or equal to' in...
ltleni 10746 'Less than' expressed in t...
ltnsymi 10747 'Less than' is not symmetr...
lenlti 10748 'Less than or equal to' in...
ltnlei 10749 'Less than' in terms of 'l...
ltlei 10750 'Less than' implies 'less ...
ltleii 10751 'Less than' implies 'less ...
ltnei 10752 'Less than' implies not eq...
letrii 10753 Trichotomy law for 'less t...
lttri 10754 'Less than' is transitive....
lelttri 10755 'Less than or equal to', '...
ltletri 10756 'Less than', 'less than or...
letri 10757 'Less than or equal to' is...
le2tri3i 10758 Extended trichotomy law fo...
ltadd2i 10759 Addition to both sides of ...
mulgt0i 10760 The product of two positiv...
mulgt0ii 10761 The product of two positiv...
ltnrd 10762 'Less than' is irreflexive...
gtned 10763 'Less than' implies not eq...
ltned 10764 'Greater than' implies not...
ne0gt0d 10765 A nonzero nonnegative numb...
lttrid 10766 Ordering on reals satisfie...
lttri2d 10767 Consequence of trichotomy....
lttri3d 10768 Consequence of trichotomy....
lttri4d 10769 Trichotomy law for 'less t...
letri3d 10770 Consequence of trichotomy....
leloed 10771 'Less than or equal to' in...
eqleltd 10772 Equality in terms of 'less...
ltlend 10773 'Less than' expressed in t...
lenltd 10774 'Less than or equal to' in...
ltnled 10775 'Less than' in terms of 'l...
ltled 10776 'Less than' implies 'less ...
ltnsymd 10777 'Less than' implies 'less ...
nltled 10778 'Not less than ' implies '...
lensymd 10779 'Less than or equal to' im...
letrid 10780 Trichotomy law for 'less t...
leltned 10781 'Less than or equal to' im...
leneltd 10782 'Less than or equal to' an...
mulgt0d 10783 The product of two positiv...
ltadd2d 10784 Addition to both sides of ...
letrd 10785 Transitive law deduction f...
lelttrd 10786 Transitive law deduction f...
ltadd2dd 10787 Addition to both sides of ...
ltletrd 10788 Transitive law deduction f...
lttrd 10789 Transitive law deduction f...
lelttrdi 10790 If a number is less than a...
dedekind 10791 The Dedekind cut theorem. ...
dedekindle 10792 The Dedekind cut theorem, ...
mul12 10793 Commutative/associative la...
mul32 10794 Commutative/associative la...
mul31 10795 Commutative/associative la...
mul4 10796 Rearrangement of 4 factors...
mul4r 10797 Rearrangement of 4 factors...
muladd11 10798 A simple product of sums e...
1p1times 10799 Two times a number. (Cont...
peano2cn 10800 A theorem for complex numb...
peano2re 10801 A theorem for reals analog...
readdcan 10802 Cancellation law for addit...
00id 10803 ` 0 ` is its own additive ...
mul02lem1 10804 Lemma for ~ mul02 . If an...
mul02lem2 10805 Lemma for ~ mul02 . Zero ...
mul02 10806 Multiplication by ` 0 ` . ...
mul01 10807 Multiplication by ` 0 ` . ...
addid1 10808 ` 0 ` is an additive ident...
cnegex 10809 Existence of the negative ...
cnegex2 10810 Existence of a left invers...
addid2 10811 ` 0 ` is a left identity f...
addcan 10812 Cancellation law for addit...
addcan2 10813 Cancellation law for addit...
addcom 10814 Addition commutes. This u...
addid1i 10815 ` 0 ` is an additive ident...
addid2i 10816 ` 0 ` is a left identity f...
mul02i 10817 Multiplication by 0. Theo...
mul01i 10818 Multiplication by ` 0 ` . ...
addcomi 10819 Addition commutes. Based ...
addcomli 10820 Addition commutes. (Contr...
addcani 10821 Cancellation law for addit...
addcan2i 10822 Cancellation law for addit...
mul12i 10823 Commutative/associative la...
mul32i 10824 Commutative/associative la...
mul4i 10825 Rearrangement of 4 factors...
mul02d 10826 Multiplication by 0. Theo...
mul01d 10827 Multiplication by ` 0 ` . ...
addid1d 10828 ` 0 ` is an additive ident...
addid2d 10829 ` 0 ` is a left identity f...
addcomd 10830 Addition commutes. Based ...
addcand 10831 Cancellation law for addit...
addcan2d 10832 Cancellation law for addit...
addcanad 10833 Cancelling a term on the l...
addcan2ad 10834 Cancelling a term on the r...
addneintrd 10835 Introducing a term on the ...
addneintr2d 10836 Introducing a term on the ...
mul12d 10837 Commutative/associative la...
mul32d 10838 Commutative/associative la...
mul31d 10839 Commutative/associative la...
mul4d 10840 Rearrangement of 4 factors...
muladd11r 10841 A simple product of sums e...
comraddd 10842 Commute RHS addition, in d...
ltaddneg 10843 Adding a negative number t...
ltaddnegr 10844 Adding a negative number t...
add12 10845 Commutative/associative la...
add32 10846 Commutative/associative la...
add32r 10847 Commutative/associative la...
add4 10848 Rearrangement of 4 terms i...
add42 10849 Rearrangement of 4 terms i...
add12i 10850 Commutative/associative la...
add32i 10851 Commutative/associative la...
add4i 10852 Rearrangement of 4 terms i...
add42i 10853 Rearrangement of 4 terms i...
add12d 10854 Commutative/associative la...
add32d 10855 Commutative/associative la...
add4d 10856 Rearrangement of 4 terms i...
add42d 10857 Rearrangement of 4 terms i...
0cnALT 10862 Alternate proof of ~ 0cn w...
0cnALT2 10863 Alternate proof of ~ 0cnAL...
negeu 10864 Existential uniqueness of ...
subval 10865 Value of subtraction, whic...
negeq 10866 Equality theorem for negat...
negeqi 10867 Equality inference for neg...
negeqd 10868 Equality deduction for neg...
nfnegd 10869 Deduction version of ~ nfn...
nfneg 10870 Bound-variable hypothesis ...
csbnegg 10871 Move class substitution in...
negex 10872 A negative is a set. (Con...
subcl 10873 Closure law for subtractio...
negcl 10874 Closure law for negative. ...
negicn 10875 ` -u _i ` is a complex num...
subf 10876 Subtraction is an operatio...
subadd 10877 Relationship between subtr...
subadd2 10878 Relationship between subtr...
subsub23 10879 Swap subtrahend and result...
pncan 10880 Cancellation law for subtr...
pncan2 10881 Cancellation law for subtr...
pncan3 10882 Subtraction and addition o...
npcan 10883 Cancellation law for subtr...
addsubass 10884 Associative-type law for a...
addsub 10885 Law for addition and subtr...
subadd23 10886 Commutative/associative la...
addsub12 10887 Commutative/associative la...
2addsub 10888 Law for subtraction and ad...
addsubeq4 10889 Relation between sums and ...
pncan3oi 10890 Subtraction and addition o...
mvrraddi 10891 Move RHS right addition to...
mvlladdi 10892 Move LHS left addition to ...
subid 10893 Subtraction of a number fr...
subid1 10894 Identity law for subtracti...
npncan 10895 Cancellation law for subtr...
nppcan 10896 Cancellation law for subtr...
nnpcan 10897 Cancellation law for subtr...
nppcan3 10898 Cancellation law for subtr...
subcan2 10899 Cancellation law for subtr...
subeq0 10900 If the difference between ...
npncan2 10901 Cancellation law for subtr...
subsub2 10902 Law for double subtraction...
nncan 10903 Cancellation law for subtr...
subsub 10904 Law for double subtraction...
nppcan2 10905 Cancellation law for subtr...
subsub3 10906 Law for double subtraction...
subsub4 10907 Law for double subtraction...
sub32 10908 Swap the second and third ...
nnncan 10909 Cancellation law for subtr...
nnncan1 10910 Cancellation law for subtr...
nnncan2 10911 Cancellation law for subtr...
npncan3 10912 Cancellation law for subtr...
pnpcan 10913 Cancellation law for mixed...
pnpcan2 10914 Cancellation law for mixed...
pnncan 10915 Cancellation law for mixed...
ppncan 10916 Cancellation law for mixed...
addsub4 10917 Rearrangement of 4 terms i...
subadd4 10918 Rearrangement of 4 terms i...
sub4 10919 Rearrangement of 4 terms i...
neg0 10920 Minus 0 equals 0. (Contri...
negid 10921 Addition of a number and i...
negsub 10922 Relationship between subtr...
subneg 10923 Relationship between subtr...
negneg 10924 A number is equal to the n...
neg11 10925 Negative is one-to-one. (...
negcon1 10926 Negative contraposition la...
negcon2 10927 Negative contraposition la...
negeq0 10928 A number is zero iff its n...
subcan 10929 Cancellation law for subtr...
negsubdi 10930 Distribution of negative o...
negdi 10931 Distribution of negative o...
negdi2 10932 Distribution of negative o...
negsubdi2 10933 Distribution of negative o...
neg2sub 10934 Relationship between subtr...
renegcli 10935 Closure law for negative o...
resubcli 10936 Closure law for subtractio...
renegcl 10937 Closure law for negative o...
resubcl 10938 Closure law for subtractio...
negreb 10939 The negative of a real is ...
peano2cnm 10940 "Reverse" second Peano pos...
peano2rem 10941 "Reverse" second Peano pos...
negcli 10942 Closure law for negative. ...
negidi 10943 Addition of a number and i...
negnegi 10944 A number is equal to the n...
subidi 10945 Subtraction of a number fr...
subid1i 10946 Identity law for subtracti...
negne0bi 10947 A number is nonzero iff it...
negrebi 10948 The negative of a real is ...
negne0i 10949 The negative of a nonzero ...
subcli 10950 Closure law for subtractio...
pncan3i 10951 Subtraction and addition o...
negsubi 10952 Relationship between subtr...
subnegi 10953 Relationship between subtr...
subeq0i 10954 If the difference between ...
neg11i 10955 Negative is one-to-one. (...
negcon1i 10956 Negative contraposition la...
negcon2i 10957 Negative contraposition la...
negdii 10958 Distribution of negative o...
negsubdii 10959 Distribution of negative o...
negsubdi2i 10960 Distribution of negative o...
subaddi 10961 Relationship between subtr...
subadd2i 10962 Relationship between subtr...
subaddrii 10963 Relationship between subtr...
subsub23i 10964 Swap subtrahend and result...
addsubassi 10965 Associative-type law for s...
addsubi 10966 Law for subtraction and ad...
subcani 10967 Cancellation law for subtr...
subcan2i 10968 Cancellation law for subtr...
pnncani 10969 Cancellation law for mixed...
addsub4i 10970 Rearrangement of 4 terms i...
0reALT 10971 Alternate proof of ~ 0re ....
negcld 10972 Closure law for negative. ...
subidd 10973 Subtraction of a number fr...
subid1d 10974 Identity law for subtracti...
negidd 10975 Addition of a number and i...
negnegd 10976 A number is equal to the n...
negeq0d 10977 A number is zero iff its n...
negne0bd 10978 A number is nonzero iff it...
negcon1d 10979 Contraposition law for una...
negcon1ad 10980 Contraposition law for una...
neg11ad 10981 The negatives of two compl...
negned 10982 If two complex numbers are...
negne0d 10983 The negative of a nonzero ...
negrebd 10984 The negative of a real is ...
subcld 10985 Closure law for subtractio...
pncand 10986 Cancellation law for subtr...
pncan2d 10987 Cancellation law for subtr...
pncan3d 10988 Subtraction and addition o...
npcand 10989 Cancellation law for subtr...
nncand 10990 Cancellation law for subtr...
negsubd 10991 Relationship between subtr...
subnegd 10992 Relationship between subtr...
subeq0d 10993 If the difference between ...
subne0d 10994 Two unequal numbers have n...
subeq0ad 10995 The difference of two comp...
subne0ad 10996 If the difference of two c...
neg11d 10997 If the difference between ...
negdid 10998 Distribution of negative o...
negdi2d 10999 Distribution of negative o...
negsubdid 11000 Distribution of negative o...
negsubdi2d 11001 Distribution of negative o...
neg2subd 11002 Relationship between subtr...
subaddd 11003 Relationship between subtr...
subadd2d 11004 Relationship between subtr...
addsubassd 11005 Associative-type law for s...
addsubd 11006 Law for subtraction and ad...
subadd23d 11007 Commutative/associative la...
addsub12d 11008 Commutative/associative la...
npncand 11009 Cancellation law for subtr...
nppcand 11010 Cancellation law for subtr...
nppcan2d 11011 Cancellation law for subtr...
nppcan3d 11012 Cancellation law for subtr...
subsubd 11013 Law for double subtraction...
subsub2d 11014 Law for double subtraction...
subsub3d 11015 Law for double subtraction...
subsub4d 11016 Law for double subtraction...
sub32d 11017 Swap the second and third ...
nnncand 11018 Cancellation law for subtr...
nnncan1d 11019 Cancellation law for subtr...
nnncan2d 11020 Cancellation law for subtr...
npncan3d 11021 Cancellation law for subtr...
pnpcand 11022 Cancellation law for mixed...
pnpcan2d 11023 Cancellation law for mixed...
pnncand 11024 Cancellation law for mixed...
ppncand 11025 Cancellation law for mixed...
subcand 11026 Cancellation law for subtr...
subcan2d 11027 Cancellation law for subtr...
subcanad 11028 Cancellation law for subtr...
subneintrd 11029 Introducing subtraction on...
subcan2ad 11030 Cancellation law for subtr...
subneintr2d 11031 Introducing subtraction on...
addsub4d 11032 Rearrangement of 4 terms i...
subadd4d 11033 Rearrangement of 4 terms i...
sub4d 11034 Rearrangement of 4 terms i...
2addsubd 11035 Law for subtraction and ad...
addsubeq4d 11036 Relation between sums and ...
subeqxfrd 11037 Transfer two terms of a su...
mvlraddd 11038 Move LHS right addition to...
mvlladdd 11039 Move LHS left addition to ...
mvrraddd 11040 Move RHS right addition to...
mvrladdd 11041 Move RHS left addition to ...
assraddsubd 11042 Associate RHS addition-sub...
subaddeqd 11043 Transfer two terms of a su...
addlsub 11044 Left-subtraction: Subtrac...
addrsub 11045 Right-subtraction: Subtra...
subexsub 11046 A subtraction law: Exchan...
addid0 11047 If adding a number to a an...
addn0nid 11048 Adding a nonzero number to...
pnpncand 11049 Addition/subtraction cance...
subeqrev 11050 Reverse the order of subtr...
addeq0 11051 Two complex numbers add up...
pncan1 11052 Cancellation law for addit...
npcan1 11053 Cancellation law for subtr...
subeq0bd 11054 If two complex numbers are...
renegcld 11055 Closure law for negative o...
resubcld 11056 Closure law for subtractio...
negn0 11057 The image under negation o...
negf1o 11058 Negation is an isomorphism...
kcnktkm1cn 11059 k times k minus 1 is a com...
muladd 11060 Product of two sums. (Con...
subdi 11061 Distribution of multiplica...
subdir 11062 Distribution of multiplica...
ine0 11063 The imaginary unit ` _i ` ...
mulneg1 11064 Product with negative is n...
mulneg2 11065 The product with a negativ...
mulneg12 11066 Swap the negative sign in ...
mul2neg 11067 Product of two negatives. ...
submul2 11068 Convert a subtraction to a...
mulm1 11069 Product with minus one is ...
addneg1mul 11070 Addition with product with...
mulsub 11071 Product of two differences...
mulsub2 11072 Swap the order of subtract...
mulm1i 11073 Product with minus one is ...
mulneg1i 11074 Product with negative is n...
mulneg2i 11075 Product with negative is n...
mul2negi 11076 Product of two negatives. ...
subdii 11077 Distribution of multiplica...
subdiri 11078 Distribution of multiplica...
muladdi 11079 Product of two sums. (Con...
mulm1d 11080 Product with minus one is ...
mulneg1d 11081 Product with negative is n...
mulneg2d 11082 Product with negative is n...
mul2negd 11083 Product of two negatives. ...
subdid 11084 Distribution of multiplica...
subdird 11085 Distribution of multiplica...
muladdd 11086 Product of two sums. (Con...
mulsubd 11087 Product of two differences...
muls1d 11088 Multiplication by one minu...
mulsubfacd 11089 Multiplication followed by...
addmulsub 11090 The product of a sum and a...
subaddmulsub 11091 The difference with a prod...
mulsubaddmulsub 11092 A special difference of a ...
gt0ne0 11093 Positive implies nonzero. ...
lt0ne0 11094 A number which is less tha...
ltadd1 11095 Addition to both sides of ...
leadd1 11096 Addition to both sides of ...
leadd2 11097 Addition to both sides of ...
ltsubadd 11098 'Less than' relationship b...
ltsubadd2 11099 'Less than' relationship b...
lesubadd 11100 'Less than or equal to' re...
lesubadd2 11101 'Less than or equal to' re...
ltaddsub 11102 'Less than' relationship b...
ltaddsub2 11103 'Less than' relationship b...
leaddsub 11104 'Less than or equal to' re...
leaddsub2 11105 'Less than or equal to' re...
suble 11106 Swap subtrahends in an ine...
lesub 11107 Swap subtrahends in an ine...
ltsub23 11108 'Less than' relationship b...
ltsub13 11109 'Less than' relationship b...
le2add 11110 Adding both sides of two '...
ltleadd 11111 Adding both sides of two o...
leltadd 11112 Adding both sides of two o...
lt2add 11113 Adding both sides of two '...
addgt0 11114 The sum of 2 positive numb...
addgegt0 11115 The sum of nonnegative and...
addgtge0 11116 The sum of nonnegative and...
addge0 11117 The sum of 2 nonnegative n...
ltaddpos 11118 Adding a positive number t...
ltaddpos2 11119 Adding a positive number t...
ltsubpos 11120 Subtracting a positive num...
posdif 11121 Comparison of two numbers ...
lesub1 11122 Subtraction from both side...
lesub2 11123 Subtraction of both sides ...
ltsub1 11124 Subtraction from both side...
ltsub2 11125 Subtraction of both sides ...
lt2sub 11126 Subtracting both sides of ...
le2sub 11127 Subtracting both sides of ...
ltneg 11128 Negative of both sides of ...
ltnegcon1 11129 Contraposition of negative...
ltnegcon2 11130 Contraposition of negative...
leneg 11131 Negative of both sides of ...
lenegcon1 11132 Contraposition of negative...
lenegcon2 11133 Contraposition of negative...
lt0neg1 11134 Comparison of a number and...
lt0neg2 11135 Comparison of a number and...
le0neg1 11136 Comparison of a number and...
le0neg2 11137 Comparison of a number and...
addge01 11138 A number is less than or e...
addge02 11139 A number is less than or e...
add20 11140 Two nonnegative numbers ar...
subge0 11141 Nonnegative subtraction. ...
suble0 11142 Nonpositive subtraction. ...
leaddle0 11143 The sum of a real number a...
subge02 11144 Nonnegative subtraction. ...
lesub0 11145 Lemma to show a nonnegativ...
mulge0 11146 The product of two nonnega...
mullt0 11147 The product of two negativ...
msqgt0 11148 A nonzero square is positi...
msqge0 11149 A square is nonnegative. ...
0lt1 11150 0 is less than 1. Theorem...
0le1 11151 0 is less than or equal to...
relin01 11152 An interval law for less t...
ltordlem 11153 Lemma for ~ ltord1 . (Con...
ltord1 11154 Infer an ordering relation...
leord1 11155 Infer an ordering relation...
eqord1 11156 A strictly increasing real...
ltord2 11157 Infer an ordering relation...
leord2 11158 Infer an ordering relation...
eqord2 11159 A strictly decreasing real...
wloglei 11160 Form of ~ wlogle where bot...
wlogle 11161 If the predicate ` ch ( x ...
leidi 11162 'Less than or equal to' is...
gt0ne0i 11163 Positive means nonzero (us...
gt0ne0ii 11164 Positive implies nonzero. ...
msqgt0i 11165 A nonzero square is positi...
msqge0i 11166 A square is nonnegative. ...
addgt0i 11167 Addition of 2 positive num...
addge0i 11168 Addition of 2 nonnegative ...
addgegt0i 11169 Addition of nonnegative an...
addgt0ii 11170 Addition of 2 positive num...
add20i 11171 Two nonnegative numbers ar...
ltnegi 11172 Negative of both sides of ...
lenegi 11173 Negative of both sides of ...
ltnegcon2i 11174 Contraposition of negative...
mulge0i 11175 The product of two nonnega...
lesub0i 11176 Lemma to show a nonnegativ...
ltaddposi 11177 Adding a positive number t...
posdifi 11178 Comparison of two numbers ...
ltnegcon1i 11179 Contraposition of negative...
lenegcon1i 11180 Contraposition of negative...
subge0i 11181 Nonnegative subtraction. ...
ltadd1i 11182 Addition to both sides of ...
leadd1i 11183 Addition to both sides of ...
leadd2i 11184 Addition to both sides of ...
ltsubaddi 11185 'Less than' relationship b...
lesubaddi 11186 'Less than or equal to' re...
ltsubadd2i 11187 'Less than' relationship b...
lesubadd2i 11188 'Less than or equal to' re...
ltaddsubi 11189 'Less than' relationship b...
lt2addi 11190 Adding both side of two in...
le2addi 11191 Adding both side of two in...
gt0ne0d 11192 Positive implies nonzero. ...
lt0ne0d 11193 Something less than zero i...
leidd 11194 'Less than or equal to' is...
msqgt0d 11195 A nonzero square is positi...
msqge0d 11196 A square is nonnegative. ...
lt0neg1d 11197 Comparison of a number and...
lt0neg2d 11198 Comparison of a number and...
le0neg1d 11199 Comparison of a number and...
le0neg2d 11200 Comparison of a number and...
addgegt0d 11201 Addition of nonnegative an...
addgtge0d 11202 Addition of positive and n...
addgt0d 11203 Addition of 2 positive num...
addge0d 11204 Addition of 2 nonnegative ...
mulge0d 11205 The product of two nonnega...
ltnegd 11206 Negative of both sides of ...
lenegd 11207 Negative of both sides of ...
ltnegcon1d 11208 Contraposition of negative...
ltnegcon2d 11209 Contraposition of negative...
lenegcon1d 11210 Contraposition of negative...
lenegcon2d 11211 Contraposition of negative...
ltaddposd 11212 Adding a positive number t...
ltaddpos2d 11213 Adding a positive number t...
ltsubposd 11214 Subtracting a positive num...
posdifd 11215 Comparison of two numbers ...
addge01d 11216 A number is less than or e...
addge02d 11217 A number is less than or e...
subge0d 11218 Nonnegative subtraction. ...
suble0d 11219 Nonpositive subtraction. ...
subge02d 11220 Nonnegative subtraction. ...
ltadd1d 11221 Addition to both sides of ...
leadd1d 11222 Addition to both sides of ...
leadd2d 11223 Addition to both sides of ...
ltsubaddd 11224 'Less than' relationship b...
lesubaddd 11225 'Less than or equal to' re...
ltsubadd2d 11226 'Less than' relationship b...
lesubadd2d 11227 'Less than or equal to' re...
ltaddsubd 11228 'Less than' relationship b...
ltaddsub2d 11229 'Less than' relationship b...
leaddsub2d 11230 'Less than or equal to' re...
subled 11231 Swap subtrahends in an ine...
lesubd 11232 Swap subtrahends in an ine...
ltsub23d 11233 'Less than' relationship b...
ltsub13d 11234 'Less than' relationship b...
lesub1d 11235 Subtraction from both side...
lesub2d 11236 Subtraction of both sides ...
ltsub1d 11237 Subtraction from both side...
ltsub2d 11238 Subtraction of both sides ...
ltadd1dd 11239 Addition to both sides of ...
ltsub1dd 11240 Subtraction from both side...
ltsub2dd 11241 Subtraction of both sides ...
leadd1dd 11242 Addition to both sides of ...
leadd2dd 11243 Addition to both sides of ...
lesub1dd 11244 Subtraction from both side...
lesub2dd 11245 Subtraction of both sides ...
lesub3d 11246 The result of subtracting ...
le2addd 11247 Adding both side of two in...
le2subd 11248 Subtracting both sides of ...
ltleaddd 11249 Adding both sides of two o...
leltaddd 11250 Adding both sides of two o...
lt2addd 11251 Adding both side of two in...
lt2subd 11252 Subtracting both sides of ...
possumd 11253 Condition for a positive s...
sublt0d 11254 When a subtraction gives a...
ltaddsublt 11255 Addition and subtraction o...
1le1 11256 One is less than or equal ...
ixi 11257 ` _i ` times itself is min...
recextlem1 11258 Lemma for ~ recex . (Cont...
recextlem2 11259 Lemma for ~ recex . (Cont...
recex 11260 Existence of reciprocal of...
mulcand 11261 Cancellation law for multi...
mulcan2d 11262 Cancellation law for multi...
mulcanad 11263 Cancellation of a nonzero ...
mulcan2ad 11264 Cancellation of a nonzero ...
mulcan 11265 Cancellation law for multi...
mulcan2 11266 Cancellation law for multi...
mulcani 11267 Cancellation law for multi...
mul0or 11268 If a product is zero, one ...
mulne0b 11269 The product of two nonzero...
mulne0 11270 The product of two nonzero...
mulne0i 11271 The product of two nonzero...
muleqadd 11272 Property of numbers whose ...
receu 11273 Existential uniqueness of ...
mulnzcnopr 11274 Multiplication maps nonzer...
msq0i 11275 A number is zero iff its s...
mul0ori 11276 If a product is zero, one ...
msq0d 11277 A number is zero iff its s...
mul0ord 11278 If a product is zero, one ...
mulne0bd 11279 The product of two nonzero...
mulne0d 11280 The product of two nonzero...
mulcan1g 11281 A generalized form of the ...
mulcan2g 11282 A generalized form of the ...
mulne0bad 11283 A factor of a nonzero comp...
mulne0bbd 11284 A factor of a nonzero comp...
1div0 11287 You can't divide by zero, ...
divval 11288 Value of division: if ` A ...
divmul 11289 Relationship between divis...
divmul2 11290 Relationship between divis...
divmul3 11291 Relationship between divis...
divcl 11292 Closure law for division. ...
reccl 11293 Closure law for reciprocal...
divcan2 11294 A cancellation law for div...
divcan1 11295 A cancellation law for div...
diveq0 11296 A ratio is zero iff the nu...
divne0b 11297 The ratio of nonzero numbe...
divne0 11298 The ratio of nonzero numbe...
recne0 11299 The reciprocal of a nonzer...
recid 11300 Multiplication of a number...
recid2 11301 Multiplication of a number...
divrec 11302 Relationship between divis...
divrec2 11303 Relationship between divis...
divass 11304 An associative law for div...
div23 11305 A commutative/associative ...
div32 11306 A commutative/associative ...
div13 11307 A commutative/associative ...
div12 11308 A commutative/associative ...
divmulass 11309 An associative law for div...
divmulasscom 11310 An associative/commutative...
divdir 11311 Distribution of division o...
divcan3 11312 A cancellation law for div...
divcan4 11313 A cancellation law for div...
div11 11314 One-to-one relationship fo...
divid 11315 A number divided by itself...
div0 11316 Division into zero is zero...
div1 11317 A number divided by 1 is i...
1div1e1 11318 1 divided by 1 is 1. (Con...
diveq1 11319 Equality in terms of unit ...
divneg 11320 Move negative sign inside ...
muldivdir 11321 Distribution of division o...
divsubdir 11322 Distribution of division o...
subdivcomb1 11323 Bring a term in a subtract...
subdivcomb2 11324 Bring a term in a subtract...
recrec 11325 A number is equal to the r...
rec11 11326 Reciprocal is one-to-one. ...
rec11r 11327 Mutual reciprocals. (Cont...
divmuldiv 11328 Multiplication of two rati...
divdivdiv 11329 Division of two ratios. T...
divcan5 11330 Cancellation of common fac...
divmul13 11331 Swap the denominators in t...
divmul24 11332 Swap the numerators in the...
divmuleq 11333 Cross-multiply in an equal...
recdiv 11334 The reciprocal of a ratio....
divcan6 11335 Cancellation of inverted f...
divdiv32 11336 Swap denominators in a div...
divcan7 11337 Cancel equal divisors in a...
dmdcan 11338 Cancellation law for divis...
divdiv1 11339 Division into a fraction. ...
divdiv2 11340 Division by a fraction. (...
recdiv2 11341 Division into a reciprocal...
ddcan 11342 Cancellation in a double d...
divadddiv 11343 Addition of two ratios. T...
divsubdiv 11344 Subtraction of two ratios....
conjmul 11345 Two numbers whose reciproc...
rereccl 11346 Closure law for reciprocal...
redivcl 11347 Closure law for division o...
eqneg 11348 A number equal to its nega...
eqnegd 11349 A complex number equals it...
eqnegad 11350 If a complex number equals...
div2neg 11351 Quotient of two negatives....
divneg2 11352 Move negative sign inside ...
recclzi 11353 Closure law for reciprocal...
recne0zi 11354 The reciprocal of a nonzer...
recidzi 11355 Multiplication of a number...
div1i 11356 A number divided by 1 is i...
eqnegi 11357 A number equal to its nega...
reccli 11358 Closure law for reciprocal...
recidi 11359 Multiplication of a number...
recreci 11360 A number is equal to the r...
dividi 11361 A number divided by itself...
div0i 11362 Division into zero is zero...
divclzi 11363 Closure law for division. ...
divcan1zi 11364 A cancellation law for div...
divcan2zi 11365 A cancellation law for div...
divreczi 11366 Relationship between divis...
divcan3zi 11367 A cancellation law for div...
divcan4zi 11368 A cancellation law for div...
rec11i 11369 Reciprocal is one-to-one. ...
divcli 11370 Closure law for division. ...
divcan2i 11371 A cancellation law for div...
divcan1i 11372 A cancellation law for div...
divreci 11373 Relationship between divis...
divcan3i 11374 A cancellation law for div...
divcan4i 11375 A cancellation law for div...
divne0i 11376 The ratio of nonzero numbe...
rec11ii 11377 Reciprocal is one-to-one. ...
divasszi 11378 An associative law for div...
divmulzi 11379 Relationship between divis...
divdirzi 11380 Distribution of division o...
divdiv23zi 11381 Swap denominators in a div...
divmuli 11382 Relationship between divis...
divdiv32i 11383 Swap denominators in a div...
divassi 11384 An associative law for div...
divdiri 11385 Distribution of division o...
div23i 11386 A commutative/associative ...
div11i 11387 One-to-one relationship fo...
divmuldivi 11388 Multiplication of two rati...
divmul13i 11389 Swap denominators of two r...
divadddivi 11390 Addition of two ratios. T...
divdivdivi 11391 Division of two ratios. T...
rerecclzi 11392 Closure law for reciprocal...
rereccli 11393 Closure law for reciprocal...
redivclzi 11394 Closure law for division o...
redivcli 11395 Closure law for division o...
div1d 11396 A number divided by 1 is i...
reccld 11397 Closure law for reciprocal...
recne0d 11398 The reciprocal of a nonzer...
recidd 11399 Multiplication of a number...
recid2d 11400 Multiplication of a number...
recrecd 11401 A number is equal to the r...
dividd 11402 A number divided by itself...
div0d 11403 Division into zero is zero...
divcld 11404 Closure law for division. ...
divcan1d 11405 A cancellation law for div...
divcan2d 11406 A cancellation law for div...
divrecd 11407 Relationship between divis...
divrec2d 11408 Relationship between divis...
divcan3d 11409 A cancellation law for div...
divcan4d 11410 A cancellation law for div...
diveq0d 11411 A ratio is zero iff the nu...
diveq1d 11412 Equality in terms of unit ...
diveq1ad 11413 The quotient of two comple...
diveq0ad 11414 A fraction of complex numb...
divne1d 11415 If two complex numbers are...
divne0bd 11416 A ratio is zero iff the nu...
divnegd 11417 Move negative sign inside ...
divneg2d 11418 Move negative sign inside ...
div2negd 11419 Quotient of two negatives....
divne0d 11420 The ratio of nonzero numbe...
recdivd 11421 The reciprocal of a ratio....
recdiv2d 11422 Division into a reciprocal...
divcan6d 11423 Cancellation of inverted f...
ddcand 11424 Cancellation in a double d...
rec11d 11425 Reciprocal is one-to-one. ...
divmuld 11426 Relationship between divis...
div32d 11427 A commutative/associative ...
div13d 11428 A commutative/associative ...
divdiv32d 11429 Swap denominators in a div...
divcan5d 11430 Cancellation of common fac...
divcan5rd 11431 Cancellation of common fac...
divcan7d 11432 Cancel equal divisors in a...
dmdcand 11433 Cancellation law for divis...
dmdcan2d 11434 Cancellation law for divis...
divdiv1d 11435 Division into a fraction. ...
divdiv2d 11436 Division by a fraction. (...
divmul2d 11437 Relationship between divis...
divmul3d 11438 Relationship between divis...
divassd 11439 An associative law for div...
div12d 11440 A commutative/associative ...
div23d 11441 A commutative/associative ...
divdird 11442 Distribution of division o...
divsubdird 11443 Distribution of division o...
div11d 11444 One-to-one relationship fo...
divmuldivd 11445 Multiplication of two rati...
divmul13d 11446 Swap denominators of two r...
divmul24d 11447 Swap the numerators in the...
divadddivd 11448 Addition of two ratios. T...
divsubdivd 11449 Subtraction of two ratios....
divmuleqd 11450 Cross-multiply in an equal...
divdivdivd 11451 Division of two ratios. T...
diveq1bd 11452 If two complex numbers are...
div2sub 11453 Swap the order of subtract...
div2subd 11454 Swap subtrahend and minuen...
rereccld 11455 Closure law for reciprocal...
redivcld 11456 Closure law for division o...
subrec 11457 Subtraction of reciprocals...
subreci 11458 Subtraction of reciprocals...
subrecd 11459 Subtraction of reciprocals...
mvllmuld 11460 Move LHS left multiplicati...
mvllmuli 11461 Move LHS left multiplicati...
ldiv 11462 Left-division. (Contribut...
rdiv 11463 Right-division. (Contribu...
mdiv 11464 A division law. (Contribu...
lineq 11465 Solution of a (scalar) lin...
elimgt0 11466 Hypothesis for weak deduct...
elimge0 11467 Hypothesis for weak deduct...
ltp1 11468 A number is less than itse...
lep1 11469 A number is less than or e...
ltm1 11470 A number minus 1 is less t...
lem1 11471 A number minus 1 is less t...
letrp1 11472 A transitive property of '...
p1le 11473 A transitive property of p...
recgt0 11474 The reciprocal of a positi...
prodgt0 11475 Infer that a multiplicand ...
prodgt02 11476 Infer that a multiplier is...
ltmul1a 11477 Lemma for ~ ltmul1 . Mult...
ltmul1 11478 Multiplication of both sid...
ltmul2 11479 Multiplication of both sid...
lemul1 11480 Multiplication of both sid...
lemul2 11481 Multiplication of both sid...
lemul1a 11482 Multiplication of both sid...
lemul2a 11483 Multiplication of both sid...
ltmul12a 11484 Comparison of product of t...
lemul12b 11485 Comparison of product of t...
lemul12a 11486 Comparison of product of t...
mulgt1 11487 The product of two numbers...
ltmulgt11 11488 Multiplication by a number...
ltmulgt12 11489 Multiplication by a number...
lemulge11 11490 Multiplication by a number...
lemulge12 11491 Multiplication by a number...
ltdiv1 11492 Division of both sides of ...
lediv1 11493 Division of both sides of ...
gt0div 11494 Division of a positive num...
ge0div 11495 Division of a nonnegative ...
divgt0 11496 The ratio of two positive ...
divge0 11497 The ratio of nonnegative a...
mulge0b 11498 A condition for multiplica...
mulle0b 11499 A condition for multiplica...
mulsuble0b 11500 A condition for multiplica...
ltmuldiv 11501 'Less than' relationship b...
ltmuldiv2 11502 'Less than' relationship b...
ltdivmul 11503 'Less than' relationship b...
ledivmul 11504 'Less than or equal to' re...
ltdivmul2 11505 'Less than' relationship b...
lt2mul2div 11506 'Less than' relationship b...
ledivmul2 11507 'Less than or equal to' re...
lemuldiv 11508 'Less than or equal' relat...
lemuldiv2 11509 'Less than or equal' relat...
ltrec 11510 The reciprocal of both sid...
lerec 11511 The reciprocal of both sid...
lt2msq1 11512 Lemma for ~ lt2msq . (Con...
lt2msq 11513 Two nonnegative numbers co...
ltdiv2 11514 Division of a positive num...
ltrec1 11515 Reciprocal swap in a 'less...
lerec2 11516 Reciprocal swap in a 'less...
ledivdiv 11517 Invert ratios of positive ...
lediv2 11518 Division of a positive num...
ltdiv23 11519 Swap denominator with othe...
lediv23 11520 Swap denominator with othe...
lediv12a 11521 Comparison of ratio of two...
lediv2a 11522 Division of both sides of ...
reclt1 11523 The reciprocal of a positi...
recgt1 11524 The reciprocal of a positi...
recgt1i 11525 The reciprocal of a number...
recp1lt1 11526 Construct a number less th...
recreclt 11527 Given a positive number ` ...
le2msq 11528 The square function on non...
msq11 11529 The square of a nonnegativ...
ledivp1 11530 "Less than or equal to" an...
squeeze0 11531 If a nonnegative number is...
ltp1i 11532 A number is less than itse...
recgt0i 11533 The reciprocal of a positi...
recgt0ii 11534 The reciprocal of a positi...
prodgt0i 11535 Infer that a multiplicand ...
divgt0i 11536 The ratio of two positive ...
divge0i 11537 The ratio of nonnegative a...
ltreci 11538 The reciprocal of both sid...
lereci 11539 The reciprocal of both sid...
lt2msqi 11540 The square function on non...
le2msqi 11541 The square function on non...
msq11i 11542 The square of a nonnegativ...
divgt0i2i 11543 The ratio of two positive ...
ltrecii 11544 The reciprocal of both sid...
divgt0ii 11545 The ratio of two positive ...
ltmul1i 11546 Multiplication of both sid...
ltdiv1i 11547 Division of both sides of ...
ltmuldivi 11548 'Less than' relationship b...
ltmul2i 11549 Multiplication of both sid...
lemul1i 11550 Multiplication of both sid...
lemul2i 11551 Multiplication of both sid...
ltdiv23i 11552 Swap denominator with othe...
ledivp1i 11553 "Less than or equal to" an...
ltdivp1i 11554 Less-than and division rel...
ltdiv23ii 11555 Swap denominator with othe...
ltmul1ii 11556 Multiplication of both sid...
ltdiv1ii 11557 Division of both sides of ...
ltp1d 11558 A number is less than itse...
lep1d 11559 A number is less than or e...
ltm1d 11560 A number minus 1 is less t...
lem1d 11561 A number minus 1 is less t...
recgt0d 11562 The reciprocal of a positi...
divgt0d 11563 The ratio of two positive ...
mulgt1d 11564 The product of two numbers...
lemulge11d 11565 Multiplication by a number...
lemulge12d 11566 Multiplication by a number...
lemul1ad 11567 Multiplication of both sid...
lemul2ad 11568 Multiplication of both sid...
ltmul12ad 11569 Comparison of product of t...
lemul12ad 11570 Comparison of product of t...
lemul12bd 11571 Comparison of product of t...
fimaxre 11572 A finite set of real numbe...
fimaxreOLD 11573 Obsolete version of ~ fima...
fimaxre2 11574 A nonempty finite set of r...
fimaxre3 11575 A nonempty finite set of r...
fiminre 11576 A nonempty finite set of r...
negfi 11577 The negation of a finite s...
fiminreOLD 11578 Obsolete version of ~ fimi...
lbreu 11579 If a set of reals contains...
lbcl 11580 If a set of reals contains...
lble 11581 If a set of reals contains...
lbinf 11582 If a set of reals contains...
lbinfcl 11583 If a set of reals contains...
lbinfle 11584 If a set of reals contains...
sup2 11585 A nonempty, bounded-above ...
sup3 11586 A version of the completen...
infm3lem 11587 Lemma for ~ infm3 . (Cont...
infm3 11588 The completeness axiom for...
suprcl 11589 Closure of supremum of a n...
suprub 11590 A member of a nonempty bou...
suprubd 11591 Natural deduction form of ...
suprcld 11592 Natural deduction form of ...
suprlub 11593 The supremum of a nonempty...
suprnub 11594 An upper bound is not less...
suprleub 11595 The supremum of a nonempty...
supaddc 11596 The supremum function dist...
supadd 11597 The supremum function dist...
supmul1 11598 The supremum function dist...
supmullem1 11599 Lemma for ~ supmul . (Con...
supmullem2 11600 Lemma for ~ supmul . (Con...
supmul 11601 The supremum function dist...
sup3ii 11602 A version of the completen...
suprclii 11603 Closure of supremum of a n...
suprubii 11604 A member of a nonempty bou...
suprlubii 11605 The supremum of a nonempty...
suprnubii 11606 An upper bound is not less...
suprleubii 11607 The supremum of a nonempty...
riotaneg 11608 The negative of the unique...
negiso 11609 Negation is an order anti-...
dfinfre 11610 The infimum of a set of re...
infrecl 11611 Closure of infimum of a no...
infrenegsup 11612 The infimum of a set of re...
infregelb 11613 Any lower bound of a nonem...
infrelb 11614 If a nonempty set of real ...
supfirege 11615 The supremum of a finite s...
inelr 11616 The imaginary unit ` _i ` ...
rimul 11617 A real number times the im...
cru 11618 The representation of comp...
crne0 11619 The real representation of...
creur 11620 The real part of a complex...
creui 11621 The imaginary part of a co...
cju 11622 The complex conjugate of a...
ofsubeq0 11623 Function analogue of ~ sub...
ofnegsub 11624 Function analogue of ~ neg...
ofsubge0 11625 Function analogue of ~ sub...
nnexALT 11628 Alternate proof of ~ nnex ...
peano5nni 11629 Peano's inductive postulat...
nnssre 11630 The positive integers are ...
nnsscn 11631 The positive integers are ...
nnex 11632 The set of positive intege...
nnre 11633 A positive integer is a re...
nncn 11634 A positive integer is a co...
nnrei 11635 A positive integer is a re...
nncni 11636 A positive integer is a co...
1nn 11637 Peano postulate: 1 is a po...
peano2nn 11638 Peano postulate: a success...
dfnn2 11639 Alternate definition of th...
dfnn3 11640 Alternate definition of th...
nnred 11641 A positive integer is a re...
nncnd 11642 A positive integer is a co...
peano2nnd 11643 Peano postulate: a success...
nnind 11644 Principle of Mathematical ...
nnindALT 11645 Principle of Mathematical ...
nn1m1nn 11646 Every positive integer is ...
nn1suc 11647 If a statement holds for 1...
nnaddcl 11648 Closure of addition of pos...
nnmulcl 11649 Closure of multiplication ...
nnmulcli 11650 Closure of multiplication ...
nnmtmip 11651 "Minus times minus is plus...
nn2ge 11652 There exists a positive in...
nnge1 11653 A positive integer is one ...
nngt1ne1 11654 A positive integer is grea...
nnle1eq1 11655 A positive integer is less...
nngt0 11656 A positive integer is posi...
nnnlt1 11657 A positive integer is not ...
nnnle0 11658 A positive integer is not ...
nnne0 11659 A positive integer is nonz...
nnneneg 11660 No positive integer is equ...
0nnn 11661 Zero is not a positive int...
0nnnALT 11662 Alternate proof of ~ 0nnn ...
nnne0ALT 11663 Alternate version of ~ nnn...
nngt0i 11664 A positive integer is posi...
nnne0i 11665 A positive integer is nonz...
nndivre 11666 The quotient of a real and...
nnrecre 11667 The reciprocal of a positi...
nnrecgt0 11668 The reciprocal of a positi...
nnsub 11669 Subtraction of positive in...
nnsubi 11670 Subtraction of positive in...
nndiv 11671 Two ways to express " ` A ...
nndivtr 11672 Transitive property of div...
nnge1d 11673 A positive integer is one ...
nngt0d 11674 A positive integer is posi...
nnne0d 11675 A positive integer is nonz...
nnrecred 11676 The reciprocal of a positi...
nnaddcld 11677 Closure of addition of pos...
nnmulcld 11678 Closure of multiplication ...
nndivred 11679 A positive integer is one ...
0ne1 11696 Zero is different from one...
1m1e0 11697 One minus one equals zero....
2nn 11698 2 is a positive integer. ...
2re 11699 The number 2 is real. (Co...
2cn 11700 The number 2 is a complex ...
2cnALT 11701 Alternate proof of ~ 2cn ....
2ex 11702 The number 2 is a set. (C...
2cnd 11703 The number 2 is a complex ...
3nn 11704 3 is a positive integer. ...
3re 11705 The number 3 is real. (Co...
3cn 11706 The number 3 is a complex ...
3ex 11707 The number 3 is a set. (C...
4nn 11708 4 is a positive integer. ...
4re 11709 The number 4 is real. (Co...
4cn 11710 The number 4 is a complex ...
5nn 11711 5 is a positive integer. ...
5re 11712 The number 5 is real. (Co...
5cn 11713 The number 5 is a complex ...
6nn 11714 6 is a positive integer. ...
6re 11715 The number 6 is real. (Co...
6cn 11716 The number 6 is a complex ...
7nn 11717 7 is a positive integer. ...
7re 11718 The number 7 is real. (Co...
7cn 11719 The number 7 is a complex ...
8nn 11720 8 is a positive integer. ...
8re 11721 The number 8 is real. (Co...
8cn 11722 The number 8 is a complex ...
9nn 11723 9 is a positive integer. ...
9re 11724 The number 9 is real. (Co...
9cn 11725 The number 9 is a complex ...
0le0 11726 Zero is nonnegative. (Con...
0le2 11727 The number 0 is less than ...
2pos 11728 The number 2 is positive. ...
2ne0 11729 The number 2 is nonzero. ...
3pos 11730 The number 3 is positive. ...
3ne0 11731 The number 3 is nonzero. ...
4pos 11732 The number 4 is positive. ...
4ne0 11733 The number 4 is nonzero. ...
5pos 11734 The number 5 is positive. ...
6pos 11735 The number 6 is positive. ...
7pos 11736 The number 7 is positive. ...
8pos 11737 The number 8 is positive. ...
9pos 11738 The number 9 is positive. ...
neg1cn 11739 -1 is a complex number. (...
neg1rr 11740 -1 is a real number. (Con...
neg1ne0 11741 -1 is nonzero. (Contribut...
neg1lt0 11742 -1 is less than 0. (Contr...
negneg1e1 11743 ` -u -u 1 ` is 1. (Contri...
1pneg1e0 11744 ` 1 + -u 1 ` is 0. (Contr...
0m0e0 11745 0 minus 0 equals 0. (Cont...
1m0e1 11746 1 - 0 = 1. (Contributed b...
0p1e1 11747 0 + 1 = 1. (Contributed b...
fv0p1e1 11748 Function value at ` N + 1 ...
1p0e1 11749 1 + 0 = 1. (Contributed b...
1p1e2 11750 1 + 1 = 2. (Contributed b...
2m1e1 11751 2 - 1 = 1. The result is ...
1e2m1 11752 1 = 2 - 1. (Contributed b...
3m1e2 11753 3 - 1 = 2. (Contributed b...
4m1e3 11754 4 - 1 = 3. (Contributed b...
5m1e4 11755 5 - 1 = 4. (Contributed b...
6m1e5 11756 6 - 1 = 5. (Contributed b...
7m1e6 11757 7 - 1 = 6. (Contributed b...
8m1e7 11758 8 - 1 = 7. (Contributed b...
9m1e8 11759 9 - 1 = 8. (Contributed b...
2p2e4 11760 Two plus two equals four. ...
2times 11761 Two times a number. (Cont...
times2 11762 A number times 2. (Contri...
2timesi 11763 Two times a number. (Cont...
times2i 11764 A number times 2. (Contri...
2txmxeqx 11765 Two times a complex number...
2div2e1 11766 2 divided by 2 is 1. (Con...
2p1e3 11767 2 + 1 = 3. (Contributed b...
1p2e3 11768 1 + 2 = 3. For a shorter ...
1p2e3ALT 11769 Alternate proof of ~ 1p2e3...
3p1e4 11770 3 + 1 = 4. (Contributed b...
4p1e5 11771 4 + 1 = 5. (Contributed b...
5p1e6 11772 5 + 1 = 6. (Contributed b...
6p1e7 11773 6 + 1 = 7. (Contributed b...
7p1e8 11774 7 + 1 = 8. (Contributed b...
8p1e9 11775 8 + 1 = 9. (Contributed b...
3p2e5 11776 3 + 2 = 5. (Contributed b...
3p3e6 11777 3 + 3 = 6. (Contributed b...
4p2e6 11778 4 + 2 = 6. (Contributed b...
4p3e7 11779 4 + 3 = 7. (Contributed b...
4p4e8 11780 4 + 4 = 8. (Contributed b...
5p2e7 11781 5 + 2 = 7. (Contributed b...
5p3e8 11782 5 + 3 = 8. (Contributed b...
5p4e9 11783 5 + 4 = 9. (Contributed b...
6p2e8 11784 6 + 2 = 8. (Contributed b...
6p3e9 11785 6 + 3 = 9. (Contributed b...
7p2e9 11786 7 + 2 = 9. (Contributed b...
1t1e1 11787 1 times 1 equals 1. (Cont...
2t1e2 11788 2 times 1 equals 2. (Cont...
2t2e4 11789 2 times 2 equals 4. (Cont...
3t1e3 11790 3 times 1 equals 3. (Cont...
3t2e6 11791 3 times 2 equals 6. (Cont...
3t3e9 11792 3 times 3 equals 9. (Cont...
4t2e8 11793 4 times 2 equals 8. (Cont...
2t0e0 11794 2 times 0 equals 0. (Cont...
4d2e2 11795 One half of four is two. ...
1lt2 11796 1 is less than 2. (Contri...
2lt3 11797 2 is less than 3. (Contri...
1lt3 11798 1 is less than 3. (Contri...
3lt4 11799 3 is less than 4. (Contri...
2lt4 11800 2 is less than 4. (Contri...
1lt4 11801 1 is less than 4. (Contri...
4lt5 11802 4 is less than 5. (Contri...
3lt5 11803 3 is less than 5. (Contri...
2lt5 11804 2 is less than 5. (Contri...
1lt5 11805 1 is less than 5. (Contri...
5lt6 11806 5 is less than 6. (Contri...
4lt6 11807 4 is less than 6. (Contri...
3lt6 11808 3 is less than 6. (Contri...
2lt6 11809 2 is less than 6. (Contri...
1lt6 11810 1 is less than 6. (Contri...
6lt7 11811 6 is less than 7. (Contri...
5lt7 11812 5 is less than 7. (Contri...
4lt7 11813 4 is less than 7. (Contri...
3lt7 11814 3 is less than 7. (Contri...
2lt7 11815 2 is less than 7. (Contri...
1lt7 11816 1 is less than 7. (Contri...
7lt8 11817 7 is less than 8. (Contri...
6lt8 11818 6 is less than 8. (Contri...
5lt8 11819 5 is less than 8. (Contri...
4lt8 11820 4 is less than 8. (Contri...
3lt8 11821 3 is less than 8. (Contri...
2lt8 11822 2 is less than 8. (Contri...
1lt8 11823 1 is less than 8. (Contri...
8lt9 11824 8 is less than 9. (Contri...
7lt9 11825 7 is less than 9. (Contri...
6lt9 11826 6 is less than 9. (Contri...
5lt9 11827 5 is less than 9. (Contri...
4lt9 11828 4 is less than 9. (Contri...
3lt9 11829 3 is less than 9. (Contri...
2lt9 11830 2 is less than 9. (Contri...
1lt9 11831 1 is less than 9. (Contri...
0ne2 11832 0 is not equal to 2. (Con...
1ne2 11833 1 is not equal to 2. (Con...
1le2 11834 1 is less than or equal to...
2cnne0 11835 2 is a nonzero complex num...
2rene0 11836 2 is a nonzero real number...
1le3 11837 1 is less than or equal to...
neg1mulneg1e1 11838 ` -u 1 x. -u 1 ` is 1. (C...
halfre 11839 One-half is real. (Contri...
halfcn 11840 One-half is a complex numb...
halfgt0 11841 One-half is greater than z...
halfge0 11842 One-half is not negative. ...
halflt1 11843 One-half is less than one....
1mhlfehlf 11844 Prove that 1 - 1/2 = 1/2. ...
8th4div3 11845 An eighth of four thirds i...
halfpm6th 11846 One half plus or minus one...
it0e0 11847 i times 0 equals 0. (Cont...
2mulicn 11848 ` ( 2 x. _i ) e. CC ` . (...
2muline0 11849 ` ( 2 x. _i ) =/= 0 ` . (...
halfcl 11850 Closure of half of a numbe...
rehalfcl 11851 Real closure of half. (Co...
half0 11852 Half of a number is zero i...
2halves 11853 Two halves make a whole. ...
halfpos2 11854 A number is positive iff i...
halfpos 11855 A positive number is great...
halfnneg2 11856 A number is nonnegative if...
halfaddsubcl 11857 Closure of half-sum and ha...
halfaddsub 11858 Sum and difference of half...
subhalfhalf 11859 Subtracting the half of a ...
lt2halves 11860 A sum is less than the who...
addltmul 11861 Sum is less than product f...
nominpos 11862 There is no smallest posit...
avglt1 11863 Ordering property for aver...
avglt2 11864 Ordering property for aver...
avgle1 11865 Ordering property for aver...
avgle2 11866 Ordering property for aver...
avgle 11867 The average of two numbers...
2timesd 11868 Two times a number. (Cont...
times2d 11869 A number times 2. (Contri...
halfcld 11870 Closure of half of a numbe...
2halvesd 11871 Two halves make a whole. ...
rehalfcld 11872 Real closure of half. (Co...
lt2halvesd 11873 A sum is less than the who...
rehalfcli 11874 Half a real number is real...
lt2addmuld 11875 If two real numbers are le...
add1p1 11876 Adding two times 1 to a nu...
sub1m1 11877 Subtracting two times 1 fr...
cnm2m1cnm3 11878 Subtracting 2 and afterwar...
xp1d2m1eqxm1d2 11879 A complex number increased...
div4p1lem1div2 11880 An integer greater than 5,...
nnunb 11881 The set of positive intege...
arch 11882 Archimedean property of re...
nnrecl 11883 There exists a positive in...
bndndx 11884 A bounded real sequence ` ...
elnn0 11887 Nonnegative integers expre...
nnssnn0 11888 Positive naturals are a su...
nn0ssre 11889 Nonnegative integers are a...
nn0sscn 11890 Nonnegative integers are a...
nn0ex 11891 The set of nonnegative int...
nnnn0 11892 A positive integer is a no...
nnnn0i 11893 A positive integer is a no...
nn0re 11894 A nonnegative integer is a...
nn0cn 11895 A nonnegative integer is a...
nn0rei 11896 A nonnegative integer is a...
nn0cni 11897 A nonnegative integer is a...
dfn2 11898 The set of positive intege...
elnnne0 11899 The positive integer prope...
0nn0 11900 0 is a nonnegative integer...
1nn0 11901 1 is a nonnegative integer...
2nn0 11902 2 is a nonnegative integer...
3nn0 11903 3 is a nonnegative integer...
4nn0 11904 4 is a nonnegative integer...
5nn0 11905 5 is a nonnegative integer...
6nn0 11906 6 is a nonnegative integer...
7nn0 11907 7 is a nonnegative integer...
8nn0 11908 8 is a nonnegative integer...
9nn0 11909 9 is a nonnegative integer...
nn0ge0 11910 A nonnegative integer is g...
nn0nlt0 11911 A nonnegative integer is n...
nn0ge0i 11912 Nonnegative integers are n...
nn0le0eq0 11913 A nonnegative integer is l...
nn0p1gt0 11914 A nonnegative integer incr...
nnnn0addcl 11915 A positive integer plus a ...
nn0nnaddcl 11916 A nonnegative integer plus...
0mnnnnn0 11917 The result of subtracting ...
un0addcl 11918 If ` S ` is closed under a...
un0mulcl 11919 If ` S ` is closed under m...
nn0addcl 11920 Closure of addition of non...
nn0mulcl 11921 Closure of multiplication ...
nn0addcli 11922 Closure of addition of non...
nn0mulcli 11923 Closure of multiplication ...
nn0p1nn 11924 A nonnegative integer plus...
peano2nn0 11925 Second Peano postulate for...
nnm1nn0 11926 A positive integer minus 1...
elnn0nn 11927 The nonnegative integer pr...
elnnnn0 11928 The positive integer prope...
elnnnn0b 11929 The positive integer prope...
elnnnn0c 11930 The positive integer prope...
nn0addge1 11931 A number is less than or e...
nn0addge2 11932 A number is less than or e...
nn0addge1i 11933 A number is less than or e...
nn0addge2i 11934 A number is less than or e...
nn0sub 11935 Subtraction of nonnegative...
ltsubnn0 11936 Subtracting a nonnegative ...
nn0negleid 11937 A nonnegative integer is g...
difgtsumgt 11938 If the difference of a rea...
nn0le2xi 11939 A nonnegative integer is l...
nn0lele2xi 11940 'Less than or equal to' im...
frnnn0supp 11941 Two ways to write the supp...
frnnn0fsupp 11942 A function on ` NN0 ` is f...
nnnn0d 11943 A positive integer is a no...
nn0red 11944 A nonnegative integer is a...
nn0cnd 11945 A nonnegative integer is a...
nn0ge0d 11946 A nonnegative integer is g...
nn0addcld 11947 Closure of addition of non...
nn0mulcld 11948 Closure of multiplication ...
nn0readdcl 11949 Closure law for addition o...
nn0n0n1ge2 11950 A nonnegative integer whic...
nn0n0n1ge2b 11951 A nonnegative integer is n...
nn0ge2m1nn 11952 If a nonnegative integer i...
nn0ge2m1nn0 11953 If a nonnegative integer i...
nn0nndivcl 11954 Closure law for dividing o...
elxnn0 11957 An extended nonnegative in...
nn0ssxnn0 11958 The standard nonnegative i...
nn0xnn0 11959 A standard nonnegative int...
xnn0xr 11960 An extended nonnegative in...
0xnn0 11961 Zero is an extended nonneg...
pnf0xnn0 11962 Positive infinity is an ex...
nn0nepnf 11963 No standard nonnegative in...
nn0xnn0d 11964 A standard nonnegative int...
nn0nepnfd 11965 No standard nonnegative in...
xnn0nemnf 11966 No extended nonnegative in...
xnn0xrnemnf 11967 The extended nonnegative i...
xnn0nnn0pnf 11968 An extended nonnegative in...
elz 11971 Membership in the set of i...
nnnegz 11972 The negative of a positive...
zre 11973 An integer is a real. (Co...
zcn 11974 An integer is a complex nu...
zrei 11975 An integer is a real numbe...
zssre 11976 The integers are a subset ...
zsscn 11977 The integers are a subset ...
zex 11978 The set of integers exists...
elnnz 11979 Positive integer property ...
0z 11980 Zero is an integer. (Cont...
0zd 11981 Zero is an integer, deduct...
elnn0z 11982 Nonnegative integer proper...
elznn0nn 11983 Integer property expressed...
elznn0 11984 Integer property expressed...
elznn 11985 Integer property expressed...
zle0orge1 11986 There is no integer in the...
elz2 11987 Membership in the set of i...
dfz2 11988 Alternative definition of ...
zexALT 11989 Alternate proof of ~ zex ....
nnssz 11990 Positive integers are a su...
nn0ssz 11991 Nonnegative integers are a...
nnz 11992 A positive integer is an i...
nn0z 11993 A nonnegative integer is a...
nnzi 11994 A positive integer is an i...
nn0zi 11995 A nonnegative integer is a...
elnnz1 11996 Positive integer property ...
znnnlt1 11997 An integer is not a positi...
nnzrab 11998 Positive integers expresse...
nn0zrab 11999 Nonnegative integers expre...
1z 12000 One is an integer. (Contr...
1zzd 12001 One is an integer, deducti...
2z 12002 2 is an integer. (Contrib...
3z 12003 3 is an integer. (Contrib...
4z 12004 4 is an integer. (Contrib...
znegcl 12005 Closure law for negative i...
neg1z 12006 -1 is an integer. (Contri...
znegclb 12007 A complex number is an int...
nn0negz 12008 The negative of a nonnegat...
nn0negzi 12009 The negative of a nonnegat...
zaddcl 12010 Closure of addition of int...
peano2z 12011 Second Peano postulate gen...
zsubcl 12012 Closure of subtraction of ...
peano2zm 12013 "Reverse" second Peano pos...
zletr 12014 Transitive law of ordering...
zrevaddcl 12015 Reverse closure law for ad...
znnsub 12016 The positive difference of...
znn0sub 12017 The nonnegative difference...
nzadd 12018 The sum of a real number n...
zmulcl 12019 Closure of multiplication ...
zltp1le 12020 Integer ordering relation....
zleltp1 12021 Integer ordering relation....
zlem1lt 12022 Integer ordering relation....
zltlem1 12023 Integer ordering relation....
zgt0ge1 12024 An integer greater than ` ...
nnleltp1 12025 Positive integer ordering ...
nnltp1le 12026 Positive integer ordering ...
nnaddm1cl 12027 Closure of addition of pos...
nn0ltp1le 12028 Nonnegative integer orderi...
nn0leltp1 12029 Nonnegative integer orderi...
nn0ltlem1 12030 Nonnegative integer orderi...
nn0sub2 12031 Subtraction of nonnegative...
nn0lt10b 12032 A nonnegative integer less...
nn0lt2 12033 A nonnegative integer less...
nn0le2is012 12034 A nonnegative integer whic...
nn0lem1lt 12035 Nonnegative integer orderi...
nnlem1lt 12036 Positive integer ordering ...
nnltlem1 12037 Positive integer ordering ...
nnm1ge0 12038 A positive integer decreas...
nn0ge0div 12039 Division of a nonnegative ...
zdiv 12040 Two ways to express " ` M ...
zdivadd 12041 Property of divisibility: ...
zdivmul 12042 Property of divisibility: ...
zextle 12043 An extensionality-like pro...
zextlt 12044 An extensionality-like pro...
recnz 12045 The reciprocal of a number...
btwnnz 12046 A number between an intege...
gtndiv 12047 A larger number does not d...
halfnz 12048 One-half is not an integer...
3halfnz 12049 Three halves is not an int...
suprzcl 12050 The supremum of a bounded-...
prime 12051 Two ways to express " ` A ...
msqznn 12052 The square of a nonzero in...
zneo 12053 No even integer equals an ...
nneo 12054 A positive integer is even...
nneoi 12055 A positive integer is even...
zeo 12056 An integer is even or odd....
zeo2 12057 An integer is even or odd ...
peano2uz2 12058 Second Peano postulate for...
peano5uzi 12059 Peano's inductive postulat...
peano5uzti 12060 Peano's inductive postulat...
dfuzi 12061 An expression for the uppe...
uzind 12062 Induction on the upper int...
uzind2 12063 Induction on the upper int...
uzind3 12064 Induction on the upper int...
nn0ind 12065 Principle of Mathematical ...
nn0indALT 12066 Principle of Mathematical ...
nn0indd 12067 Principle of Mathematical ...
fzind 12068 Induction on the integers ...
fnn0ind 12069 Induction on the integers ...
nn0ind-raph 12070 Principle of Mathematical ...
zindd 12071 Principle of Mathematical ...
btwnz 12072 Any real number can be san...
nn0zd 12073 A positive integer is an i...
nnzd 12074 A nonnegative integer is a...
zred 12075 An integer is a real numbe...
zcnd 12076 An integer is a complex nu...
znegcld 12077 Closure law for negative i...
peano2zd 12078 Deduction from second Pean...
zaddcld 12079 Closure of addition of int...
zsubcld 12080 Closure of subtraction of ...
zmulcld 12081 Closure of multiplication ...
znnn0nn 12082 The negative of a negative...
zadd2cl 12083 Increasing an integer by 2...
zriotaneg 12084 The negative of the unique...
suprfinzcl 12085 The supremum of a nonempty...
9p1e10 12088 9 + 1 = 10. (Contributed ...
dfdec10 12089 Version of the definition ...
decex 12090 A decimal number is a set....
deceq1 12091 Equality theorem for the d...
deceq2 12092 Equality theorem for the d...
deceq1i 12093 Equality theorem for the d...
deceq2i 12094 Equality theorem for the d...
deceq12i 12095 Equality theorem for the d...
numnncl 12096 Closure for a numeral (wit...
num0u 12097 Add a zero in the units pl...
num0h 12098 Add a zero in the higher p...
numcl 12099 Closure for a decimal inte...
numsuc 12100 The successor of a decimal...
deccl 12101 Closure for a numeral. (C...
10nn 12102 10 is a positive integer. ...
10pos 12103 The number 10 is positive....
10nn0 12104 10 is a nonnegative intege...
10re 12105 The number 10 is real. (C...
decnncl 12106 Closure for a numeral. (C...
dec0u 12107 Add a zero in the units pl...
dec0h 12108 Add a zero in the higher p...
numnncl2 12109 Closure for a decimal inte...
decnncl2 12110 Closure for a decimal inte...
numlt 12111 Comparing two decimal inte...
numltc 12112 Comparing two decimal inte...
le9lt10 12113 A "decimal digit" (i.e. a ...
declt 12114 Comparing two decimal inte...
decltc 12115 Comparing two decimal inte...
declth 12116 Comparing two decimal inte...
decsuc 12117 The successor of a decimal...
3declth 12118 Comparing two decimal inte...
3decltc 12119 Comparing two decimal inte...
decle 12120 Comparing two decimal inte...
decleh 12121 Comparing two decimal inte...
declei 12122 Comparing a digit to a dec...
numlti 12123 Comparing a digit to a dec...
declti 12124 Comparing a digit to a dec...
decltdi 12125 Comparing a digit to a dec...
numsucc 12126 The successor of a decimal...
decsucc 12127 The successor of a decimal...
1e0p1 12128 The successor of zero. (C...
dec10p 12129 Ten plus an integer. (Con...
numma 12130 Perform a multiply-add of ...
nummac 12131 Perform a multiply-add of ...
numma2c 12132 Perform a multiply-add of ...
numadd 12133 Add two decimal integers `...
numaddc 12134 Add two decimal integers `...
nummul1c 12135 The product of a decimal i...
nummul2c 12136 The product of a decimal i...
decma 12137 Perform a multiply-add of ...
decmac 12138 Perform a multiply-add of ...
decma2c 12139 Perform a multiply-add of ...
decadd 12140 Add two numerals ` M ` and...
decaddc 12141 Add two numerals ` M ` and...
decaddc2 12142 Add two numerals ` M ` and...
decrmanc 12143 Perform a multiply-add of ...
decrmac 12144 Perform a multiply-add of ...
decaddm10 12145 The sum of two multiples o...
decaddi 12146 Add two numerals ` M ` and...
decaddci 12147 Add two numerals ` M ` and...
decaddci2 12148 Add two numerals ` M ` and...
decsubi 12149 Difference between a numer...
decmul1 12150 The product of a numeral w...
decmul1c 12151 The product of a numeral w...
decmul2c 12152 The product of a numeral w...
decmulnc 12153 The product of a numeral w...
11multnc 12154 The product of 11 (as nume...
decmul10add 12155 A multiplication of a numb...
6p5lem 12156 Lemma for ~ 6p5e11 and rel...
5p5e10 12157 5 + 5 = 10. (Contributed ...
6p4e10 12158 6 + 4 = 10. (Contributed ...
6p5e11 12159 6 + 5 = 11. (Contributed ...
6p6e12 12160 6 + 6 = 12. (Contributed ...
7p3e10 12161 7 + 3 = 10. (Contributed ...
7p4e11 12162 7 + 4 = 11. (Contributed ...
7p5e12 12163 7 + 5 = 12. (Contributed ...
7p6e13 12164 7 + 6 = 13. (Contributed ...
7p7e14 12165 7 + 7 = 14. (Contributed ...
8p2e10 12166 8 + 2 = 10. (Contributed ...
8p3e11 12167 8 + 3 = 11. (Contributed ...
8p4e12 12168 8 + 4 = 12. (Contributed ...
8p5e13 12169 8 + 5 = 13. (Contributed ...
8p6e14 12170 8 + 6 = 14. (Contributed ...
8p7e15 12171 8 + 7 = 15. (Contributed ...
8p8e16 12172 8 + 8 = 16. (Contributed ...
9p2e11 12173 9 + 2 = 11. (Contributed ...
9p3e12 12174 9 + 3 = 12. (Contributed ...
9p4e13 12175 9 + 4 = 13. (Contributed ...
9p5e14 12176 9 + 5 = 14. (Contributed ...
9p6e15 12177 9 + 6 = 15. (Contributed ...
9p7e16 12178 9 + 7 = 16. (Contributed ...
9p8e17 12179 9 + 8 = 17. (Contributed ...
9p9e18 12180 9 + 9 = 18. (Contributed ...
10p10e20 12181 10 + 10 = 20. (Contribute...
10m1e9 12182 10 - 1 = 9. (Contributed ...
4t3lem 12183 Lemma for ~ 4t3e12 and rel...
4t3e12 12184 4 times 3 equals 12. (Con...
4t4e16 12185 4 times 4 equals 16. (Con...
5t2e10 12186 5 times 2 equals 10. (Con...
5t3e15 12187 5 times 3 equals 15. (Con...
5t4e20 12188 5 times 4 equals 20. (Con...
5t5e25 12189 5 times 5 equals 25. (Con...
6t2e12 12190 6 times 2 equals 12. (Con...
6t3e18 12191 6 times 3 equals 18. (Con...
6t4e24 12192 6 times 4 equals 24. (Con...
6t5e30 12193 6 times 5 equals 30. (Con...
6t6e36 12194 6 times 6 equals 36. (Con...
7t2e14 12195 7 times 2 equals 14. (Con...
7t3e21 12196 7 times 3 equals 21. (Con...
7t4e28 12197 7 times 4 equals 28. (Con...
7t5e35 12198 7 times 5 equals 35. (Con...
7t6e42 12199 7 times 6 equals 42. (Con...
7t7e49 12200 7 times 7 equals 49. (Con...
8t2e16 12201 8 times 2 equals 16. (Con...
8t3e24 12202 8 times 3 equals 24. (Con...
8t4e32 12203 8 times 4 equals 32. (Con...
8t5e40 12204 8 times 5 equals 40. (Con...
8t6e48 12205 8 times 6 equals 48. (Con...
8t7e56 12206 8 times 7 equals 56. (Con...
8t8e64 12207 8 times 8 equals 64. (Con...
9t2e18 12208 9 times 2 equals 18. (Con...
9t3e27 12209 9 times 3 equals 27. (Con...
9t4e36 12210 9 times 4 equals 36. (Con...
9t5e45 12211 9 times 5 equals 45. (Con...
9t6e54 12212 9 times 6 equals 54. (Con...
9t7e63 12213 9 times 7 equals 63. (Con...
9t8e72 12214 9 times 8 equals 72. (Con...
9t9e81 12215 9 times 9 equals 81. (Con...
9t11e99 12216 9 times 11 equals 99. (Co...
9lt10 12217 9 is less than 10. (Contr...
8lt10 12218 8 is less than 10. (Contr...
7lt10 12219 7 is less than 10. (Contr...
6lt10 12220 6 is less than 10. (Contr...
5lt10 12221 5 is less than 10. (Contr...
4lt10 12222 4 is less than 10. (Contr...
3lt10 12223 3 is less than 10. (Contr...
2lt10 12224 2 is less than 10. (Contr...
1lt10 12225 1 is less than 10. (Contr...
decbin0 12226 Decompose base 4 into base...
decbin2 12227 Decompose base 4 into base...
decbin3 12228 Decompose base 4 into base...
halfthird 12229 Half minus a third. (Cont...
5recm6rec 12230 One fifth minus one sixth....
uzval 12233 The value of the upper int...
uzf 12234 The domain and range of th...
eluz1 12235 Membership in the upper se...
eluzel2 12236 Implication of membership ...
eluz2 12237 Membership in an upper set...
eluzmn 12238 Membership in an earlier u...
eluz1i 12239 Membership in an upper set...
eluzuzle 12240 An integer in an upper set...
eluzelz 12241 A member of an upper set o...
eluzelre 12242 A member of an upper set o...
eluzelcn 12243 A member of an upper set o...
eluzle 12244 Implication of membership ...
eluz 12245 Membership in an upper set...
uzid 12246 Membership of the least me...
uzidd 12247 Membership of the least me...
uzn0 12248 The upper integers are all...
uztrn 12249 Transitive law for sets of...
uztrn2 12250 Transitive law for sets of...
uzneg 12251 Contraposition law for upp...
uzssz 12252 An upper set of integers i...
uzss 12253 Subset relationship for tw...
uztric 12254 Totality of the ordering r...
uz11 12255 The upper integers functio...
eluzp1m1 12256 Membership in the next upp...
eluzp1l 12257 Strict ordering implied by...
eluzp1p1 12258 Membership in the next upp...
eluzaddi 12259 Membership in a later uppe...
eluzsubi 12260 Membership in an earlier u...
eluzadd 12261 Membership in a later uppe...
eluzsub 12262 Membership in an earlier u...
subeluzsub 12263 Membership of a difference...
uzm1 12264 Choices for an element of ...
uznn0sub 12265 The nonnegative difference...
uzin 12266 Intersection of two upper ...
uzp1 12267 Choices for an element of ...
nn0uz 12268 Nonnegative integers expre...
nnuz 12269 Positive integers expresse...
elnnuz 12270 A positive integer express...
elnn0uz 12271 A nonnegative integer expr...
eluz2nn 12272 An integer greater than or...
eluz4eluz2 12273 An integer greater than or...
eluz4nn 12274 An integer greater than or...
eluzge2nn0 12275 If an integer is greater t...
eluz2n0 12276 An integer greater than or...
uzuzle23 12277 An integer in the upper se...
eluzge3nn 12278 If an integer is greater t...
uz3m2nn 12279 An integer greater than or...
1eluzge0 12280 1 is an integer greater th...
2eluzge0 12281 2 is an integer greater th...
2eluzge1 12282 2 is an integer greater th...
uznnssnn 12283 The upper integers startin...
raluz 12284 Restricted universal quant...
raluz2 12285 Restricted universal quant...
rexuz 12286 Restricted existential qua...
rexuz2 12287 Restricted existential qua...
2rexuz 12288 Double existential quantif...
peano2uz 12289 Second Peano postulate for...
peano2uzs 12290 Second Peano postulate for...
peano2uzr 12291 Reversed second Peano axio...
uzaddcl 12292 Addition closure law for a...
nn0pzuz 12293 The sum of a nonnegative i...
uzind4 12294 Induction on the upper set...
uzind4ALT 12295 Induction on the upper set...
uzind4s 12296 Induction on the upper set...
uzind4s2 12297 Induction on the upper set...
uzind4i 12298 Induction on the upper int...
uzwo 12299 Well-ordering principle: a...
uzwo2 12300 Well-ordering principle: a...
nnwo 12301 Well-ordering principle: a...
nnwof 12302 Well-ordering principle: a...
nnwos 12303 Well-ordering principle: a...
indstr 12304 Strong Mathematical Induct...
eluznn0 12305 Membership in a nonnegativ...
eluznn 12306 Membership in a positive u...
eluz2b1 12307 Two ways to say "an intege...
eluz2gt1 12308 An integer greater than or...
eluz2b2 12309 Two ways to say "an intege...
eluz2b3 12310 Two ways to say "an intege...
uz2m1nn 12311 One less than an integer g...
1nuz2 12312 1 is not in ` ( ZZ>= `` 2 ...
elnn1uz2 12313 A positive integer is eith...
uz2mulcl 12314 Closure of multiplication ...
indstr2 12315 Strong Mathematical Induct...
uzinfi 12316 Extract the lower bound of...
nninf 12317 The infimum of the set of ...
nn0inf 12318 The infimum of the set of ...
infssuzle 12319 The infimum of a subset of...
infssuzcl 12320 The infimum of a subset of...
ublbneg 12321 The image under negation o...
eqreznegel 12322 Two ways to express the im...
supminf 12323 The supremum of a bounded-...
lbzbi 12324 If a set of reals is bound...
zsupss 12325 Any nonempty bounded subse...
suprzcl2 12326 The supremum of a bounded-...
suprzub 12327 The supremum of a bounded-...
uzsupss 12328 Any bounded subset of an u...
nn01to3 12329 A (nonnegative) integer be...
nn0ge2m1nnALT 12330 Alternate proof of ~ nn0ge...
uzwo3 12331 Well-ordering principle: a...
zmin 12332 There is a unique smallest...
zmax 12333 There is a unique largest ...
zbtwnre 12334 There is a unique integer ...
rebtwnz 12335 There is a unique greatest...
elq 12338 Membership in the set of r...
qmulz 12339 If ` A ` is rational, then...
znq 12340 The ratio of an integer an...
qre 12341 A rational number is a rea...
zq 12342 An integer is a rational n...
zssq 12343 The integers are a subset ...
nn0ssq 12344 The nonnegative integers a...
nnssq 12345 The positive integers are ...
qssre 12346 The rationals are a subset...
qsscn 12347 The rationals are a subset...
qex 12348 The set of rational number...
nnq 12349 A positive integer is rati...
qcn 12350 A rational number is a com...
qexALT 12351 Alternate proof of ~ qex ....
qaddcl 12352 Closure of addition of rat...
qnegcl 12353 Closure law for the negati...
qmulcl 12354 Closure of multiplication ...
qsubcl 12355 Closure of subtraction of ...
qreccl 12356 Closure of reciprocal of r...
qdivcl 12357 Closure of division of rat...
qrevaddcl 12358 Reverse closure law for ad...
nnrecq 12359 The reciprocal of a positi...
irradd 12360 The sum of an irrational n...
irrmul 12361 The product of an irration...
elpq 12362 A positive rational is the...
elpqb 12363 A class is a positive rati...
rpnnen1lem2 12364 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem1 12365 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem3 12366 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem4 12367 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem5 12368 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem6 12369 Lemma for ~ rpnnen1 . (Co...
rpnnen1 12370 One half of ~ rpnnen , whe...
reexALT 12371 Alternate proof of ~ reex ...
cnref1o 12372 There is a natural one-to-...
cnexALT 12373 The set of complex numbers...
xrex 12374 The set of extended reals ...
addex 12375 The addition operation is ...
mulex 12376 The multiplication operati...
elrp 12379 Membership in the set of p...
elrpii 12380 Membership in the set of p...
1rp 12381 1 is a positive real. (Co...
2rp 12382 2 is a positive real. (Co...
3rp 12383 3 is a positive real. (Co...
rpssre 12384 The positive reals are a s...
rpre 12385 A positive real is a real....
rpxr 12386 A positive real is an exte...
rpcn 12387 A positive real is a compl...
nnrp 12388 A positive integer is a po...
rpgt0 12389 A positive real is greater...
rpge0 12390 A positive real is greater...
rpregt0 12391 A positive real is a posit...
rprege0 12392 A positive real is a nonne...
rpne0 12393 A positive real is nonzero...
rprene0 12394 A positive real is a nonze...
rpcnne0 12395 A positive real is a nonze...
rpcndif0 12396 A positive real number is ...
ralrp 12397 Quantification over positi...
rexrp 12398 Quantification over positi...
rpaddcl 12399 Closure law for addition o...
rpmulcl 12400 Closure law for multiplica...
rpmtmip 12401 "Minus times minus is plus...
rpdivcl 12402 Closure law for division o...
rpreccl 12403 Closure law for reciprocat...
rphalfcl 12404 Closure law for half of a ...
rpgecl 12405 A number greater than or e...
rphalflt 12406 Half of a positive real is...
rerpdivcl 12407 Closure law for division o...
ge0p1rp 12408 A nonnegative number plus ...
rpneg 12409 Either a nonzero real or i...
negelrp 12410 Elementhood of a negation ...
negelrpd 12411 The negation of a negative...
0nrp 12412 Zero is not a positive rea...
ltsubrp 12413 Subtracting a positive rea...
ltaddrp 12414 Adding a positive number t...
difrp 12415 Two ways to say one number...
elrpd 12416 Membership in the set of p...
nnrpd 12417 A positive integer is a po...
zgt1rpn0n1 12418 An integer greater than 1 ...
rpred 12419 A positive real is a real....
rpxrd 12420 A positive real is an exte...
rpcnd 12421 A positive real is a compl...
rpgt0d 12422 A positive real is greater...
rpge0d 12423 A positive real is greater...
rpne0d 12424 A positive real is nonzero...
rpregt0d 12425 A positive real is real an...
rprege0d 12426 A positive real is real an...
rprene0d 12427 A positive real is a nonze...
rpcnne0d 12428 A positive real is a nonze...
rpreccld 12429 Closure law for reciprocat...
rprecred 12430 Closure law for reciprocat...
rphalfcld 12431 Closure law for half of a ...
reclt1d 12432 The reciprocal of a positi...
recgt1d 12433 The reciprocal of a positi...
rpaddcld 12434 Closure law for addition o...
rpmulcld 12435 Closure law for multiplica...
rpdivcld 12436 Closure law for division o...
ltrecd 12437 The reciprocal of both sid...
lerecd 12438 The reciprocal of both sid...
ltrec1d 12439 Reciprocal swap in a 'less...
lerec2d 12440 Reciprocal swap in a 'less...
lediv2ad 12441 Division of both sides of ...
ltdiv2d 12442 Division of a positive num...
lediv2d 12443 Division of a positive num...
ledivdivd 12444 Invert ratios of positive ...
divge1 12445 The ratio of a number over...
divlt1lt 12446 A real number divided by a...
divle1le 12447 A real number divided by a...
ledivge1le 12448 If a number is less than o...
ge0p1rpd 12449 A nonnegative number plus ...
rerpdivcld 12450 Closure law for division o...
ltsubrpd 12451 Subtracting a positive rea...
ltaddrpd 12452 Adding a positive number t...
ltaddrp2d 12453 Adding a positive number t...
ltmulgt11d 12454 Multiplication by a number...
ltmulgt12d 12455 Multiplication by a number...
gt0divd 12456 Division of a positive num...
ge0divd 12457 Division of a nonnegative ...
rpgecld 12458 A number greater than or e...
divge0d 12459 The ratio of nonnegative a...
ltmul1d 12460 The ratio of nonnegative a...
ltmul2d 12461 Multiplication of both sid...
lemul1d 12462 Multiplication of both sid...
lemul2d 12463 Multiplication of both sid...
ltdiv1d 12464 Division of both sides of ...
lediv1d 12465 Division of both sides of ...
ltmuldivd 12466 'Less than' relationship b...
ltmuldiv2d 12467 'Less than' relationship b...
lemuldivd 12468 'Less than or equal to' re...
lemuldiv2d 12469 'Less than or equal to' re...
ltdivmuld 12470 'Less than' relationship b...
ltdivmul2d 12471 'Less than' relationship b...
ledivmuld 12472 'Less than or equal to' re...
ledivmul2d 12473 'Less than or equal to' re...
ltmul1dd 12474 The ratio of nonnegative a...
ltmul2dd 12475 Multiplication of both sid...
ltdiv1dd 12476 Division of both sides of ...
lediv1dd 12477 Division of both sides of ...
lediv12ad 12478 Comparison of ratio of two...
mul2lt0rlt0 12479 If the result of a multipl...
mul2lt0rgt0 12480 If the result of a multipl...
mul2lt0llt0 12481 If the result of a multipl...
mul2lt0lgt0 12482 If the result of a multipl...
mul2lt0bi 12483 If the result of a multipl...
prodge0rd 12484 Infer that a multiplicand ...
prodge0ld 12485 Infer that a multiplier is...
ltdiv23d 12486 Swap denominator with othe...
lediv23d 12487 Swap denominator with othe...
lt2mul2divd 12488 The ratio of nonnegative a...
nnledivrp 12489 Division of a positive int...
nn0ledivnn 12490 Division of a nonnegative ...
addlelt 12491 If the sum of a real numbe...
ltxr 12498 The 'less than' binary rel...
elxr 12499 Membership in the set of e...
xrnemnf 12500 An extended real other tha...
xrnepnf 12501 An extended real other tha...
xrltnr 12502 The extended real 'less th...
ltpnf 12503 Any (finite) real is less ...
ltpnfd 12504 Any (finite) real is less ...
0ltpnf 12505 Zero is less than plus inf...
mnflt 12506 Minus infinity is less tha...
mnfltd 12507 Minus infinity is less tha...
mnflt0 12508 Minus infinity is less tha...
mnfltpnf 12509 Minus infinity is less tha...
mnfltxr 12510 Minus infinity is less tha...
pnfnlt 12511 No extended real is greate...
nltmnf 12512 No extended real is less t...
pnfge 12513 Plus infinity is an upper ...
xnn0n0n1ge2b 12514 An extended nonnegative in...
0lepnf 12515 0 less than or equal to po...
xnn0ge0 12516 An extended nonnegative in...
mnfle 12517 Minus infinity is less tha...
xrltnsym 12518 Ordering on the extended r...
xrltnsym2 12519 'Less than' is antisymmetr...
xrlttri 12520 Ordering on the extended r...
xrlttr 12521 Ordering on the extended r...
xrltso 12522 'Less than' is a strict or...
xrlttri2 12523 Trichotomy law for 'less t...
xrlttri3 12524 Trichotomy law for 'less t...
xrleloe 12525 'Less than or equal' expre...
xrleltne 12526 'Less than or equal to' im...
xrltlen 12527 'Less than' expressed in t...
dfle2 12528 Alternative definition of ...
dflt2 12529 Alternative definition of ...
xrltle 12530 'Less than' implies 'less ...
xrltled 12531 'Less than' implies 'less ...
xrleid 12532 'Less than or equal to' is...
xrleidd 12533 'Less than or equal to' is...
xrletri 12534 Trichotomy law for extende...
xrletri3 12535 Trichotomy law for extende...
xrletrid 12536 Trichotomy law for extende...
xrlelttr 12537 Transitive law for orderin...
xrltletr 12538 Transitive law for orderin...
xrletr 12539 Transitive law for orderin...
xrlttrd 12540 Transitive law for orderin...
xrlelttrd 12541 Transitive law for orderin...
xrltletrd 12542 Transitive law for orderin...
xrletrd 12543 Transitive law for orderin...
xrltne 12544 'Less than' implies not eq...
nltpnft 12545 An extended real is not le...
xgepnf 12546 An extended real which is ...
ngtmnft 12547 An extended real is not gr...
xlemnf 12548 An extended real which is ...
xrrebnd 12549 An extended real is real i...
xrre 12550 A way of proving that an e...
xrre2 12551 An extended real between t...
xrre3 12552 A way of proving that an e...
ge0gtmnf 12553 A nonnegative extended rea...
ge0nemnf 12554 A nonnegative extended rea...
xrrege0 12555 A nonnegative extended rea...
xrmax1 12556 An extended real is less t...
xrmax2 12557 An extended real is less t...
xrmin1 12558 The minimum of two extende...
xrmin2 12559 The minimum of two extende...
xrmaxeq 12560 The maximum of two extende...
xrmineq 12561 The minimum of two extende...
xrmaxlt 12562 Two ways of saying the max...
xrltmin 12563 Two ways of saying an exte...
xrmaxle 12564 Two ways of saying the max...
xrlemin 12565 Two ways of saying a numbe...
max1 12566 A number is less than or e...
max1ALT 12567 A number is less than or e...
max2 12568 A number is less than or e...
2resupmax 12569 The supremum of two real n...
min1 12570 The minimum of two numbers...
min2 12571 The minimum of two numbers...
maxle 12572 Two ways of saying the max...
lemin 12573 Two ways of saying a numbe...
maxlt 12574 Two ways of saying the max...
ltmin 12575 Two ways of saying a numbe...
lemaxle 12576 A real number which is les...
max0sub 12577 Decompose a real number in...
ifle 12578 An if statement transforms...
z2ge 12579 There exists an integer gr...
qbtwnre 12580 The rational numbers are d...
qbtwnxr 12581 The rational numbers are d...
qsqueeze 12582 If a nonnegative real is l...
qextltlem 12583 Lemma for ~ qextlt and qex...
qextlt 12584 An extensionality-like pro...
qextle 12585 An extensionality-like pro...
xralrple 12586 Show that ` A ` is less th...
alrple 12587 Show that ` A ` is less th...
xnegeq 12588 Equality of two extended n...
xnegex 12589 A negative extended real e...
xnegpnf 12590 Minus ` +oo ` . Remark of...
xnegmnf 12591 Minus ` -oo ` . Remark of...
rexneg 12592 Minus a real number. Rema...
xneg0 12593 The negative of zero. (Co...
xnegcl 12594 Closure of extended real n...
xnegneg 12595 Extended real version of ~...
xneg11 12596 Extended real version of ~...
xltnegi 12597 Forward direction of ~ xlt...
xltneg 12598 Extended real version of ~...
xleneg 12599 Extended real version of ~...
xlt0neg1 12600 Extended real version of ~...
xlt0neg2 12601 Extended real version of ~...
xle0neg1 12602 Extended real version of ~...
xle0neg2 12603 Extended real version of ~...
xaddval 12604 Value of the extended real...
xaddf 12605 The extended real addition...
xmulval 12606 Value of the extended real...
xaddpnf1 12607 Addition of positive infin...
xaddpnf2 12608 Addition of positive infin...
xaddmnf1 12609 Addition of negative infin...
xaddmnf2 12610 Addition of negative infin...
pnfaddmnf 12611 Addition of positive and n...
mnfaddpnf 12612 Addition of negative and p...
rexadd 12613 The extended real addition...
rexsub 12614 Extended real subtraction ...
rexaddd 12615 The extended real addition...
xnn0xaddcl 12616 The extended nonnegative i...
xaddnemnf 12617 Closure of extended real a...
xaddnepnf 12618 Closure of extended real a...
xnegid 12619 Extended real version of ~...
xaddcl 12620 The extended real addition...
xaddcom 12621 The extended real addition...
xaddid1 12622 Extended real version of ~...
xaddid2 12623 Extended real version of ~...
xaddid1d 12624 ` 0 ` is a right identity ...
xnn0lem1lt 12625 Extended nonnegative integ...
xnn0lenn0nn0 12626 An extended nonnegative in...
xnn0le2is012 12627 An extended nonnegative in...
xnn0xadd0 12628 The sum of two extended no...
xnegdi 12629 Extended real version of ~...
xaddass 12630 Associativity of extended ...
xaddass2 12631 Associativity of extended ...
xpncan 12632 Extended real version of ~...
xnpcan 12633 Extended real version of ~...
xleadd1a 12634 Extended real version of ~...
xleadd2a 12635 Commuted form of ~ xleadd1...
xleadd1 12636 Weakened version of ~ xlea...
xltadd1 12637 Extended real version of ~...
xltadd2 12638 Extended real version of ~...
xaddge0 12639 The sum of nonnegative ext...
xle2add 12640 Extended real version of ~...
xlt2add 12641 Extended real version of ~...
xsubge0 12642 Extended real version of ~...
xposdif 12643 Extended real version of ~...
xlesubadd 12644 Under certain conditions, ...
xmullem 12645 Lemma for ~ rexmul . (Con...
xmullem2 12646 Lemma for ~ xmulneg1 . (C...
xmulcom 12647 Extended real multiplicati...
xmul01 12648 Extended real version of ~...
xmul02 12649 Extended real version of ~...
xmulneg1 12650 Extended real version of ~...
xmulneg2 12651 Extended real version of ~...
rexmul 12652 The extended real multipli...
xmulf 12653 The extended real multipli...
xmulcl 12654 Closure of extended real m...
xmulpnf1 12655 Multiplication by plus inf...
xmulpnf2 12656 Multiplication by plus inf...
xmulmnf1 12657 Multiplication by minus in...
xmulmnf2 12658 Multiplication by minus in...
xmulpnf1n 12659 Multiplication by plus inf...
xmulid1 12660 Extended real version of ~...
xmulid2 12661 Extended real version of ~...
xmulm1 12662 Extended real version of ~...
xmulasslem2 12663 Lemma for ~ xmulass . (Co...
xmulgt0 12664 Extended real version of ~...
xmulge0 12665 Extended real version of ~...
xmulasslem 12666 Lemma for ~ xmulass . (Co...
xmulasslem3 12667 Lemma for ~ xmulass . (Co...
xmulass 12668 Associativity of the exten...
xlemul1a 12669 Extended real version of ~...
xlemul2a 12670 Extended real version of ~...
xlemul1 12671 Extended real version of ~...
xlemul2 12672 Extended real version of ~...
xltmul1 12673 Extended real version of ~...
xltmul2 12674 Extended real version of ~...
xadddilem 12675 Lemma for ~ xadddi . (Con...
xadddi 12676 Distributive property for ...
xadddir 12677 Commuted version of ~ xadd...
xadddi2 12678 The assumption that the mu...
xadddi2r 12679 Commuted version of ~ xadd...
x2times 12680 Extended real version of ~...
xnegcld 12681 Closure of extended real n...
xaddcld 12682 The extended real addition...
xmulcld 12683 Closure of extended real m...
xadd4d 12684 Rearrangement of 4 terms i...
xnn0add4d 12685 Rearrangement of 4 terms i...
xrsupexmnf 12686 Adding minus infinity to a...
xrinfmexpnf 12687 Adding plus infinity to a ...
xrsupsslem 12688 Lemma for ~ xrsupss . (Co...
xrinfmsslem 12689 Lemma for ~ xrinfmss . (C...
xrsupss 12690 Any subset of extended rea...
xrinfmss 12691 Any subset of extended rea...
xrinfmss2 12692 Any subset of extended rea...
xrub 12693 By quantifying only over r...
supxr 12694 The supremum of a set of e...
supxr2 12695 The supremum of a set of e...
supxrcl 12696 The supremum of an arbitra...
supxrun 12697 The supremum of the union ...
supxrmnf 12698 Adding minus infinity to a...
supxrpnf 12699 The supremum of a set of e...
supxrunb1 12700 The supremum of an unbound...
supxrunb2 12701 The supremum of an unbound...
supxrbnd1 12702 The supremum of a bounded-...
supxrbnd2 12703 The supremum of a bounded-...
xrsup0 12704 The supremum of an empty s...
supxrub 12705 A member of a set of exten...
supxrlub 12706 The supremum of a set of e...
supxrleub 12707 The supremum of a set of e...
supxrre 12708 The real and extended real...
supxrbnd 12709 The supremum of a bounded-...
supxrgtmnf 12710 The supremum of a nonempty...
supxrre1 12711 The supremum of a nonempty...
supxrre2 12712 The supremum of a nonempty...
supxrss 12713 Smaller sets of extended r...
infxrcl 12714 The infimum of an arbitrar...
infxrlb 12715 A member of a set of exten...
infxrgelb 12716 The infimum of a set of ex...
infxrre 12717 The real and extended real...
infxrmnf 12718 The infinimum of a set of ...
xrinf0 12719 The infimum of the empty s...
infxrss 12720 Larger sets of extended re...
reltre 12721 For all real numbers there...
rpltrp 12722 For all positive real numb...
reltxrnmnf 12723 For all extended real numb...
infmremnf 12724 The infimum of the reals i...
infmrp1 12725 The infimum of the positiv...
ixxval 12734 Value of the interval func...
elixx1 12735 Membership in an interval ...
ixxf 12736 The set of intervals of ex...
ixxex 12737 The set of intervals of ex...
ixxssxr 12738 The set of intervals of ex...
elixx3g 12739 Membership in a set of ope...
ixxssixx 12740 An interval is a subset of...
ixxdisj 12741 Split an interval into dis...
ixxun 12742 Split an interval into two...
ixxin 12743 Intersection of two interv...
ixxss1 12744 Subset relationship for in...
ixxss2 12745 Subset relationship for in...
ixxss12 12746 Subset relationship for in...
ixxub 12747 Extract the upper bound of...
ixxlb 12748 Extract the lower bound of...
iooex 12749 The set of open intervals ...
iooval 12750 Value of the open interval...
ioo0 12751 An empty open interval of ...
ioon0 12752 An open interval of extend...
ndmioo 12753 The open interval function...
iooid 12754 An open interval with iden...
elioo3g 12755 Membership in a set of ope...
elioore 12756 A member of an open interv...
lbioo 12757 An open interval does not ...
ubioo 12758 An open interval does not ...
iooval2 12759 Value of the open interval...
iooin 12760 Intersection of two open i...
iooss1 12761 Subset relationship for op...
iooss2 12762 Subset relationship for op...
iocval 12763 Value of the open-below, c...
icoval 12764 Value of the closed-below,...
iccval 12765 Value of the closed interv...
elioo1 12766 Membership in an open inte...
elioo2 12767 Membership in an open inte...
elioc1 12768 Membership in an open-belo...
elico1 12769 Membership in a closed-bel...
elicc1 12770 Membership in a closed int...
iccid 12771 A closed interval with ide...
ico0 12772 An empty open interval of ...
ioc0 12773 An empty open interval of ...
icc0 12774 An empty closed interval o...
elicod 12775 Membership in a left-close...
icogelb 12776 An element of a left-close...
elicore 12777 A member of a left-closed ...
ubioc1 12778 The upper bound belongs to...
lbico1 12779 The lower bound belongs to...
iccleub 12780 An element of a closed int...
iccgelb 12781 An element of a closed int...
elioo5 12782 Membership in an open inte...
eliooxr 12783 A nonempty open interval s...
eliooord 12784 Ordering implied by a memb...
elioo4g 12785 Membership in an open inte...
ioossre 12786 An open interval is a set ...
elioc2 12787 Membership in an open-belo...
elico2 12788 Membership in a closed-bel...
elicc2 12789 Membership in a closed rea...
elicc2i 12790 Inference for membership i...
elicc4 12791 Membership in a closed rea...
iccss 12792 Condition for a closed int...
iccssioo 12793 Condition for a closed int...
icossico 12794 Condition for a closed-bel...
iccss2 12795 Condition for a closed int...
iccssico 12796 Condition for a closed int...
iccssioo2 12797 Condition for a closed int...
iccssico2 12798 Condition for a closed int...
ioomax 12799 The open interval from min...
iccmax 12800 The closed interval from m...
ioopos 12801 The set of positive reals ...
ioorp 12802 The set of positive reals ...
iooshf 12803 Shift the arguments of the...
iocssre 12804 A closed-above interval wi...
icossre 12805 A closed-below interval wi...
iccssre 12806 A closed real interval is ...
iccssxr 12807 A closed interval is a set...
iocssxr 12808 An open-below, closed-abov...
icossxr 12809 A closed-below, open-above...
ioossicc 12810 An open interval is a subs...
eliccxr 12811 A member of a closed inter...
icossicc 12812 A closed-below, open-above...
iocssicc 12813 A closed-above, open-below...
ioossico 12814 An open interval is a subs...
iocssioo 12815 Condition for a closed int...
icossioo 12816 Condition for a closed int...
ioossioo 12817 Condition for an open inte...
iccsupr 12818 A nonempty subset of a clo...
elioopnf 12819 Membership in an unbounded...
elioomnf 12820 Membership in an unbounded...
elicopnf 12821 Membership in a closed unb...
repos 12822 Two ways of saying that a ...
ioof 12823 The set of open intervals ...
iccf 12824 The set of closed interval...
unirnioo 12825 The union of the range of ...
dfioo2 12826 Alternate definition of th...
ioorebas 12827 Open intervals are element...
xrge0neqmnf 12828 A nonnegative extended rea...
xrge0nre 12829 An extended real which is ...
elrege0 12830 The predicate "is a nonneg...
nn0rp0 12831 A nonnegative integer is a...
rge0ssre 12832 Nonnegative real numbers a...
elxrge0 12833 Elementhood in the set of ...
0e0icopnf 12834 0 is a member of ` ( 0 [,)...
0e0iccpnf 12835 0 is a member of ` ( 0 [,]...
ge0addcl 12836 The nonnegative reals are ...
ge0mulcl 12837 The nonnegative reals are ...
ge0xaddcl 12838 The nonnegative reals are ...
ge0xmulcl 12839 The nonnegative extended r...
lbicc2 12840 The lower bound of a close...
ubicc2 12841 The upper bound of a close...
elicc01 12842 Membership in the closed r...
0elunit 12843 Zero is an element of the ...
1elunit 12844 One is an element of the c...
iooneg 12845 Membership in a negated op...
iccneg 12846 Membership in a negated cl...
icoshft 12847 A shifted real is a member...
icoshftf1o 12848 Shifting a closed-below, o...
icoun 12849 The union of two adjacent ...
icodisj 12850 Adjacent left-closed right...
ioounsn 12851 The union of an open inter...
snunioo 12852 The closure of one end of ...
snunico 12853 The closure of the open en...
snunioc 12854 The closure of the open en...
prunioo 12855 The closure of an open rea...
ioodisj 12856 If the upper bound of one ...
ioojoin 12857 Join two open intervals to...
difreicc 12858 The class difference of ` ...
iccsplit 12859 Split a closed interval in...
iccshftr 12860 Membership in a shifted in...
iccshftri 12861 Membership in a shifted in...
iccshftl 12862 Membership in a shifted in...
iccshftli 12863 Membership in a shifted in...
iccdil 12864 Membership in a dilated in...
iccdili 12865 Membership in a dilated in...
icccntr 12866 Membership in a contracted...
icccntri 12867 Membership in a contracted...
divelunit 12868 A condition for a ratio to...
lincmb01cmp 12869 A linear combination of tw...
iccf1o 12870 Describe a bijection from ...
iccen 12871 Any nontrivial closed inte...
xov1plusxeqvd 12872 A complex number ` X ` is ...
unitssre 12873 ` ( 0 [,] 1 ) ` is a subse...
supicc 12874 Supremum of a bounded set ...
supiccub 12875 The supremum of a bounded ...
supicclub 12876 The supremum of a bounded ...
supicclub2 12877 The supremum of a bounded ...
zltaddlt1le 12878 The sum of an integer and ...
xnn0xrge0 12879 An extended nonnegative in...
fzval 12882 The value of a finite set ...
fzval2 12883 An alternative way of expr...
fzf 12884 Establish the domain and c...
elfz1 12885 Membership in a finite set...
elfz 12886 Membership in a finite set...
elfz2 12887 Membership in a finite set...
elfz5 12888 Membership in a finite set...
elfz4 12889 Membership in a finite set...
elfzuzb 12890 Membership in a finite set...
eluzfz 12891 Membership in a finite set...
elfzuz 12892 A member of a finite set o...
elfzuz3 12893 Membership in a finite set...
elfzel2 12894 Membership in a finite set...
elfzel1 12895 Membership in a finite set...
elfzelz 12896 A member of a finite set o...
fzssz 12897 A finite sequence of integ...
elfzle1 12898 A member of a finite set o...
elfzle2 12899 A member of a finite set o...
elfzuz2 12900 Implication of membership ...
elfzle3 12901 Membership in a finite set...
eluzfz1 12902 Membership in a finite set...
eluzfz2 12903 Membership in a finite set...
eluzfz2b 12904 Membership in a finite set...
elfz3 12905 Membership in a finite set...
elfz1eq 12906 Membership in a finite set...
elfzubelfz 12907 If there is a member in a ...
peano2fzr 12908 A Peano-postulate-like the...
fzn0 12909 Properties of a finite int...
fz0 12910 A finite set of sequential...
fzn 12911 A finite set of sequential...
fzen 12912 A shifted finite set of se...
fz1n 12913 A 1-based finite set of se...
0nelfz1 12914 0 is not an element of a f...
0fz1 12915 Two ways to say a finite 1...
fz10 12916 There are no integers betw...
uzsubsubfz 12917 Membership of an integer g...
uzsubsubfz1 12918 Membership of an integer g...
ige3m2fz 12919 Membership of an integer g...
fzsplit2 12920 Split a finite interval of...
fzsplit 12921 Split a finite interval of...
fzdisj 12922 Condition for two finite i...
fz01en 12923 0-based and 1-based finite...
elfznn 12924 A member of a finite set o...
elfz1end 12925 A nonempty finite range of...
fz1ssnn 12926 A finite set of positive i...
fznn0sub 12927 Subtraction closure for a ...
fzmmmeqm 12928 Subtracting the difference...
fzaddel 12929 Membership of a sum in a f...
fzadd2 12930 Membership of a sum in a f...
fzsubel 12931 Membership of a difference...
fzopth 12932 A finite set of sequential...
fzass4 12933 Two ways to express a nond...
fzss1 12934 Subset relationship for fi...
fzss2 12935 Subset relationship for fi...
fzssuz 12936 A finite set of sequential...
fzsn 12937 A finite interval of integ...
fzssp1 12938 Subset relationship for fi...
fzssnn 12939 Finite sets of sequential ...
ssfzunsnext 12940 A subset of a finite seque...
ssfzunsn 12941 A subset of a finite seque...
fzsuc 12942 Join a successor to the en...
fzpred 12943 Join a predecessor to the ...
fzpreddisj 12944 A finite set of sequential...
elfzp1 12945 Append an element to a fin...
fzp1ss 12946 Subset relationship for fi...
fzelp1 12947 Membership in a set of seq...
fzp1elp1 12948 Add one to an element of a...
fznatpl1 12949 Shift membership in a fini...
fzpr 12950 A finite interval of integ...
fztp 12951 A finite interval of integ...
fz12pr 12952 An integer range between 1...
fzsuc2 12953 Join a successor to the en...
fzp1disj 12954 ` ( M ... ( N + 1 ) ) ` is...
fzdifsuc 12955 Remove a successor from th...
fzprval 12956 Two ways of defining the f...
fztpval 12957 Two ways of defining the f...
fzrev 12958 Reversal of start and end ...
fzrev2 12959 Reversal of start and end ...
fzrev2i 12960 Reversal of start and end ...
fzrev3 12961 The "complement" of a memb...
fzrev3i 12962 The "complement" of a memb...
fznn 12963 Finite set of sequential i...
elfz1b 12964 Membership in a 1-based fi...
elfz1uz 12965 Membership in a 1-based fi...
elfzm11 12966 Membership in a finite set...
uzsplit 12967 Express an upper integer s...
uzdisj 12968 The first ` N ` elements o...
fseq1p1m1 12969 Add/remove an item to/from...
fseq1m1p1 12970 Add/remove an item to/from...
fz1sbc 12971 Quantification over a one-...
elfzp1b 12972 An integer is a member of ...
elfzm1b 12973 An integer is a member of ...
elfzp12 12974 Options for membership in ...
fzm1 12975 Choices for an element of ...
fzneuz 12976 No finite set of sequentia...
fznuz 12977 Disjointness of the upper ...
uznfz 12978 Disjointness of the upper ...
fzp1nel 12979 One plus the upper bound o...
fzrevral 12980 Reversal of scanning order...
fzrevral2 12981 Reversal of scanning order...
fzrevral3 12982 Reversal of scanning order...
fzshftral 12983 Shift the scanning order i...
ige2m1fz1 12984 Membership of an integer g...
ige2m1fz 12985 Membership in a 0-based fi...
elfz2nn0 12986 Membership in a finite set...
fznn0 12987 Characterization of a fini...
elfznn0 12988 A member of a finite set o...
elfz3nn0 12989 The upper bound of a nonem...
fz0ssnn0 12990 Finite sets of sequential ...
fz1ssfz0 12991 Subset relationship for fi...
0elfz 12992 0 is an element of a finit...
nn0fz0 12993 A nonnegative integer is a...
elfz0add 12994 An element of a finite set...
fz0sn 12995 An integer range from 0 to...
fz0tp 12996 An integer range from 0 to...
fz0to3un2pr 12997 An integer range from 0 to...
fz0to4untppr 12998 An integer range from 0 to...
elfz0ubfz0 12999 An element of a finite set...
elfz0fzfz0 13000 A member of a finite set o...
fz0fzelfz0 13001 If a member of a finite se...
fznn0sub2 13002 Subtraction closure for a ...
uzsubfz0 13003 Membership of an integer g...
fz0fzdiffz0 13004 The difference of an integ...
elfzmlbm 13005 Subtracting the lower boun...
elfzmlbp 13006 Subtracting the lower boun...
fzctr 13007 Lemma for theorems about t...
difelfzle 13008 The difference of two inte...
difelfznle 13009 The difference of two inte...
nn0split 13010 Express the set of nonnega...
nn0disj 13011 The first ` N + 1 ` elemen...
fz0sn0fz1 13012 A finite set of sequential...
fvffz0 13013 The function value of a fu...
1fv 13014 A function on a singleton....
4fvwrd4 13015 The first four function va...
2ffzeq 13016 Two functions over 0-based...
preduz 13017 The value of the predecess...
prednn 13018 The value of the predecess...
prednn0 13019 The value of the predecess...
predfz 13020 Calculate the predecessor ...
fzof 13023 Functionality of the half-...
elfzoel1 13024 Reverse closure for half-o...
elfzoel2 13025 Reverse closure for half-o...
elfzoelz 13026 Reverse closure for half-o...
fzoval 13027 Value of the half-open int...
elfzo 13028 Membership in a half-open ...
elfzo2 13029 Membership in a half-open ...
elfzouz 13030 Membership in a half-open ...
nelfzo 13031 An integer not being a mem...
fzolb 13032 The left endpoint of a hal...
fzolb2 13033 The left endpoint of a hal...
elfzole1 13034 A member in a half-open in...
elfzolt2 13035 A member in a half-open in...
elfzolt3 13036 Membership in a half-open ...
elfzolt2b 13037 A member in a half-open in...
elfzolt3b 13038 Membership in a half-open ...
fzonel 13039 A half-open range does not...
elfzouz2 13040 The upper bound of a half-...
elfzofz 13041 A half-open range is conta...
elfzo3 13042 Express membership in a ha...
fzon0 13043 A half-open integer interv...
fzossfz 13044 A half-open range is conta...
fzossz 13045 A half-open integer interv...
fzon 13046 A half-open set of sequent...
fzo0n 13047 A half-open range of nonne...
fzonlt0 13048 A half-open integer range ...
fzo0 13049 Half-open sets with equal ...
fzonnsub 13050 If ` K < N ` then ` N - K ...
fzonnsub2 13051 If ` M < N ` then ` N - M ...
fzoss1 13052 Subset relationship for ha...
fzoss2 13053 Subset relationship for ha...
fzossrbm1 13054 Subset of a half-open rang...
fzo0ss1 13055 Subset relationship for ha...
fzossnn0 13056 A half-open integer range ...
fzospliti 13057 One direction of splitting...
fzosplit 13058 Split a half-open integer ...
fzodisj 13059 Abutting half-open integer...
fzouzsplit 13060 Split an upper integer set...
fzouzdisj 13061 A half-open integer range ...
fzoun 13062 A half-open integer range ...
fzodisjsn 13063 A half-open integer range ...
prinfzo0 13064 The intersection of a half...
lbfzo0 13065 An integer is strictly gre...
elfzo0 13066 Membership in a half-open ...
elfzo0z 13067 Membership in a half-open ...
nn0p1elfzo 13068 A nonnegative integer incr...
elfzo0le 13069 A member in a half-open ra...
elfzonn0 13070 A member of a half-open ra...
fzonmapblen 13071 The result of subtracting ...
fzofzim 13072 If a nonnegative integer i...
fz1fzo0m1 13073 Translation of one between...
fzossnn 13074 Half-open integer ranges s...
elfzo1 13075 Membership in a half-open ...
fzo1fzo0n0 13076 An integer between 1 and a...
fzo0n0 13077 A half-open integer range ...
fzoaddel 13078 Translate membership in a ...
fzo0addel 13079 Translate membership in a ...
fzo0addelr 13080 Translate membership in a ...
fzoaddel2 13081 Translate membership in a ...
elfzoext 13082 Membership of an integer i...
elincfzoext 13083 Membership of an increased...
fzosubel 13084 Translate membership in a ...
fzosubel2 13085 Membership in a translated...
fzosubel3 13086 Membership in a translated...
eluzgtdifelfzo 13087 Membership of the differen...
ige2m2fzo 13088 Membership of an integer g...
fzocatel 13089 Translate membership in a ...
ubmelfzo 13090 If an integer in a 1-based...
elfzodifsumelfzo 13091 If an integer is in a half...
elfzom1elp1fzo 13092 Membership of an integer i...
elfzom1elfzo 13093 Membership in a half-open ...
fzval3 13094 Expressing a closed intege...
fz0add1fz1 13095 Translate membership in a ...
fzosn 13096 Expressing a singleton as ...
elfzomin 13097 Membership of an integer i...
zpnn0elfzo 13098 Membership of an integer i...
zpnn0elfzo1 13099 Membership of an integer i...
fzosplitsnm1 13100 Removing a singleton from ...
elfzonlteqm1 13101 If an element of a half-op...
fzonn0p1 13102 A nonnegative integer is e...
fzossfzop1 13103 A half-open range of nonne...
fzonn0p1p1 13104 If a nonnegative integer i...
elfzom1p1elfzo 13105 Increasing an element of a...
fzo0ssnn0 13106 Half-open integer ranges s...
fzo01 13107 Expressing the singleton o...
fzo12sn 13108 A 1-based half-open intege...
fzo13pr 13109 A 1-based half-open intege...
fzo0to2pr 13110 A half-open integer range ...
fzo0to3tp 13111 A half-open integer range ...
fzo0to42pr 13112 A half-open integer range ...
fzo1to4tp 13113 A half-open integer range ...
fzo0sn0fzo1 13114 A half-open range of nonne...
elfzo0l 13115 A member of a half-open ra...
fzoend 13116 The endpoint of a half-ope...
fzo0end 13117 The endpoint of a zero-bas...
ssfzo12 13118 Subset relationship for ha...
ssfzoulel 13119 If a half-open integer ran...
ssfzo12bi 13120 Subset relationship for ha...
ubmelm1fzo 13121 The result of subtracting ...
fzofzp1 13122 If a point is in a half-op...
fzofzp1b 13123 If a point is in a half-op...
elfzom1b 13124 An integer is a member of ...
elfzom1elp1fzo1 13125 Membership of a nonnegativ...
elfzo1elm1fzo0 13126 Membership of a positive i...
elfzonelfzo 13127 If an element of a half-op...
fzonfzoufzol 13128 If an element of a half-op...
elfzomelpfzo 13129 An integer increased by an...
elfznelfzo 13130 A value in a finite set of...
elfznelfzob 13131 A value in a finite set of...
peano2fzor 13132 A Peano-postulate-like the...
fzosplitsn 13133 Extending a half-open rang...
fzosplitpr 13134 Extending a half-open inte...
fzosplitprm1 13135 Extending a half-open inte...
fzosplitsni 13136 Membership in a half-open ...
fzisfzounsn 13137 A finite interval of integ...
elfzr 13138 A member of a finite inter...
elfzlmr 13139 A member of a finite inter...
elfz0lmr 13140 A member of a finite inter...
fzostep1 13141 Two possibilities for a nu...
fzoshftral 13142 Shift the scanning order i...
fzind2 13143 Induction on the integers ...
fvinim0ffz 13144 The function values for th...
injresinjlem 13145 Lemma for ~ injresinj . (...
injresinj 13146 A function whose restricti...
subfzo0 13147 The difference between two...
flval 13152 Value of the floor (greate...
flcl 13153 The floor (greatest intege...
reflcl 13154 The floor (greatest intege...
fllelt 13155 A basic property of the fl...
flcld 13156 The floor (greatest intege...
flle 13157 A basic property of the fl...
flltp1 13158 A basic property of the fl...
fllep1 13159 A basic property of the fl...
fraclt1 13160 The fractional part of a r...
fracle1 13161 The fractional part of a r...
fracge0 13162 The fractional part of a r...
flge 13163 The floor function value i...
fllt 13164 The floor function value i...
flflp1 13165 Move floor function betwee...
flid 13166 An integer is its own floo...
flidm 13167 The floor function is idem...
flidz 13168 A real number equals its f...
flltnz 13169 If A is not an integer, th...
flwordi 13170 Ordering relationship for ...
flword2 13171 Ordering relationship for ...
flval2 13172 An alternate way to define...
flval3 13173 An alternate way to define...
flbi 13174 A condition equivalent to ...
flbi2 13175 A condition equivalent to ...
adddivflid 13176 The floor of a sum of an i...
ico01fl0 13177 The floor of a real number...
flge0nn0 13178 The floor of a number grea...
flge1nn 13179 The floor of a number grea...
fldivnn0 13180 The floor function of a di...
refldivcl 13181 The floor function of a di...
divfl0 13182 The floor of a fraction is...
fladdz 13183 An integer can be moved in...
flzadd 13184 An integer can be moved in...
flmulnn0 13185 Move a nonnegative integer...
btwnzge0 13186 A real bounded between an ...
2tnp1ge0ge0 13187 Two times an integer plus ...
flhalf 13188 Ordering relation for the ...
fldivle 13189 The floor function of a di...
fldivnn0le 13190 The floor function of a di...
flltdivnn0lt 13191 The floor function of a di...
ltdifltdiv 13192 If the dividend of a divis...
fldiv4p1lem1div2 13193 The floor of an integer eq...
fldiv4lem1div2uz2 13194 The floor of an integer gr...
fldiv4lem1div2 13195 The floor of a positive in...
ceilval 13196 The value of the ceiling f...
dfceil2 13197 Alternative definition of ...
ceilval2 13198 The value of the ceiling f...
ceicl 13199 The ceiling function retur...
ceilcl 13200 Closure of the ceiling fun...
ceige 13201 The ceiling of a real numb...
ceilge 13202 The ceiling of a real numb...
ceim1l 13203 One less than the ceiling ...
ceilm1lt 13204 One less than the ceiling ...
ceile 13205 The ceiling of a real numb...
ceille 13206 The ceiling of a real numb...
ceilid 13207 An integer is its own ceil...
ceilidz 13208 A real number equals its c...
flleceil 13209 The floor of a real number...
fleqceilz 13210 A real number is an intege...
quoremz 13211 Quotient and remainder of ...
quoremnn0 13212 Quotient and remainder of ...
quoremnn0ALT 13213 Alternate proof of ~ quore...
intfrac2 13214 Decompose a real into inte...
intfracq 13215 Decompose a rational numbe...
fldiv 13216 Cancellation of the embedd...
fldiv2 13217 Cancellation of an embedde...
fznnfl 13218 Finite set of sequential i...
uzsup 13219 An upper set of integers i...
ioopnfsup 13220 An upper set of reals is u...
icopnfsup 13221 An upper set of reals is u...
rpsup 13222 The positive reals are unb...
resup 13223 The real numbers are unbou...
xrsup 13224 The extended real numbers ...
modval 13227 The value of the modulo op...
modvalr 13228 The value of the modulo op...
modcl 13229 Closure law for the modulo...
flpmodeq 13230 Partition of a division in...
modcld 13231 Closure law for the modulo...
mod0 13232 ` A mod B ` is zero iff ` ...
mulmod0 13233 The product of an integer ...
negmod0 13234 ` A ` is divisible by ` B ...
modge0 13235 The modulo operation is no...
modlt 13236 The modulo operation is le...
modelico 13237 Modular reduction produces...
moddiffl 13238 Value of the modulo operat...
moddifz 13239 The modulo operation diffe...
modfrac 13240 The fractional part of a n...
flmod 13241 The floor function express...
intfrac 13242 Break a number into its in...
zmod10 13243 An integer modulo 1 is 0. ...
zmod1congr 13244 Two arbitrary integers are...
modmulnn 13245 Move a positive integer in...
modvalp1 13246 The value of the modulo op...
zmodcl 13247 Closure law for the modulo...
zmodcld 13248 Closure law for the modulo...
zmodfz 13249 An integer mod ` B ` lies ...
zmodfzo 13250 An integer mod ` B ` lies ...
zmodfzp1 13251 An integer mod ` B ` lies ...
modid 13252 Identity law for modulo. ...
modid0 13253 A positive real number mod...
modid2 13254 Identity law for modulo. ...
zmodid2 13255 Identity law for modulo re...
zmodidfzo 13256 Identity law for modulo re...
zmodidfzoimp 13257 Identity law for modulo re...
0mod 13258 Special case: 0 modulo a p...
1mod 13259 Special case: 1 modulo a r...
modabs 13260 Absorption law for modulo....
modabs2 13261 Absorption law for modulo....
modcyc 13262 The modulo operation is pe...
modcyc2 13263 The modulo operation is pe...
modadd1 13264 Addition property of the m...
modaddabs 13265 Absorption law for modulo....
modaddmod 13266 The sum of a real number m...
muladdmodid 13267 The sum of a positive real...
mulp1mod1 13268 The product of an integer ...
modmuladd 13269 Decomposition of an intege...
modmuladdim 13270 Implication of a decomposi...
modmuladdnn0 13271 Implication of a decomposi...
negmod 13272 The negation of a number m...
m1modnnsub1 13273 Minus one modulo a positiv...
m1modge3gt1 13274 Minus one modulo an intege...
addmodid 13275 The sum of a positive inte...
addmodidr 13276 The sum of a positive inte...
modadd2mod 13277 The sum of a real number m...
modm1p1mod0 13278 If a real number modulo a ...
modltm1p1mod 13279 If a real number modulo a ...
modmul1 13280 Multiplication property of...
modmul12d 13281 Multiplication property of...
modnegd 13282 Negation property of the m...
modadd12d 13283 Additive property of the m...
modsub12d 13284 Subtraction property of th...
modsubmod 13285 The difference of a real n...
modsubmodmod 13286 The difference of a real n...
2txmodxeq0 13287 Two times a positive real ...
2submod 13288 If a real number is betwee...
modifeq2int 13289 If a nonnegative integer i...
modaddmodup 13290 The sum of an integer modu...
modaddmodlo 13291 The sum of an integer modu...
modmulmod 13292 The product of a real numb...
modmulmodr 13293 The product of an integer ...
modaddmulmod 13294 The sum of a real number a...
moddi 13295 Distribute multiplication ...
modsubdir 13296 Distribute the modulo oper...
modeqmodmin 13297 A real number equals the d...
modirr 13298 A number modulo an irratio...
modfzo0difsn 13299 For a number within a half...
modsumfzodifsn 13300 The sum of a number within...
modlteq 13301 Two nonnegative integers l...
addmodlteq 13302 Two nonnegative integers l...
om2uz0i 13303 The mapping ` G ` is a one...
om2uzsuci 13304 The value of ` G ` (see ~ ...
om2uzuzi 13305 The value ` G ` (see ~ om2...
om2uzlti 13306 Less-than relation for ` G...
om2uzlt2i 13307 The mapping ` G ` (see ~ o...
om2uzrani 13308 Range of ` G ` (see ~ om2u...
om2uzf1oi 13309 ` G ` (see ~ om2uz0i ) is ...
om2uzisoi 13310 ` G ` (see ~ om2uz0i ) is ...
om2uzoi 13311 An alternative definition ...
om2uzrdg 13312 A helper lemma for the val...
uzrdglem 13313 A helper lemma for the val...
uzrdgfni 13314 The recursive definition g...
uzrdg0i 13315 Initial value of a recursi...
uzrdgsuci 13316 Successor value of a recur...
ltweuz 13317 ` < ` is a well-founded re...
ltwenn 13318 Less than well-orders the ...
ltwefz 13319 Less than well-orders a se...
uzenom 13320 An upper integer set is de...
uzinf 13321 An upper integer set is in...
nnnfi 13322 The set of positive intege...
uzrdgxfr 13323 Transfer the value of the ...
fzennn 13324 The cardinality of a finit...
fzen2 13325 The cardinality of a finit...
cardfz 13326 The cardinality of a finit...
hashgf1o 13327 ` G ` maps ` _om ` one-to-...
fzfi 13328 A finite interval of integ...
fzfid 13329 Commonly used special case...
fzofi 13330 Half-open integer sets are...
fsequb 13331 The values of a finite rea...
fsequb2 13332 The values of a finite rea...
fseqsupcl 13333 The values of a finite rea...
fseqsupubi 13334 The values of a finite rea...
nn0ennn 13335 The nonnegative integers a...
nnenom 13336 The set of positive intege...
nnct 13337 ` NN ` is countable. (Con...
uzindi 13338 Indirect strong induction ...
axdc4uzlem 13339 Lemma for ~ axdc4uz . (Co...
axdc4uz 13340 A version of ~ axdc4 that ...
ssnn0fi 13341 A subset of the nonnegativ...
rabssnn0fi 13342 A subset of the nonnegativ...
uzsinds 13343 Strong (or "total") induct...
nnsinds 13344 Strong (or "total") induct...
nn0sinds 13345 Strong (or "total") induct...
fsuppmapnn0fiublem 13346 Lemma for ~ fsuppmapnn0fiu...
fsuppmapnn0fiub 13347 If all functions of a fini...
fsuppmapnn0fiubex 13348 If all functions of a fini...
fsuppmapnn0fiub0 13349 If all functions of a fini...
suppssfz 13350 Condition for a function o...
fsuppmapnn0ub 13351 If a function over the non...
fsuppmapnn0fz 13352 If a function over the non...
mptnn0fsupp 13353 A mapping from the nonnega...
mptnn0fsuppd 13354 A mapping from the nonnega...
mptnn0fsuppr 13355 A finitely supported mappi...
f13idfv 13356 A one-to-one function with...
seqex 13359 Existence of the sequence ...
seqeq1 13360 Equality theorem for the s...
seqeq2 13361 Equality theorem for the s...
seqeq3 13362 Equality theorem for the s...
seqeq1d 13363 Equality deduction for the...
seqeq2d 13364 Equality deduction for the...
seqeq3d 13365 Equality deduction for the...
seqeq123d 13366 Equality deduction for the...
nfseq 13367 Hypothesis builder for the...
seqval 13368 Value of the sequence buil...
seqfn 13369 The sequence builder funct...
seq1 13370 Value of the sequence buil...
seq1i 13371 Value of the sequence buil...
seqp1 13372 Value of the sequence buil...
seqexw 13373 Weak version of ~ seqex th...
seqp1i 13374 Value of the sequence buil...
seqm1 13375 Value of the sequence buil...
seqcl2 13376 Closure properties of the ...
seqf2 13377 Range of the recursive seq...
seqcl 13378 Closure properties of the ...
seqf 13379 Range of the recursive seq...
seqfveq2 13380 Equality of sequences. (C...
seqfeq2 13381 Equality of sequences. (C...
seqfveq 13382 Equality of sequences. (C...
seqfeq 13383 Equality of sequences. (C...
seqshft2 13384 Shifting the index set of ...
seqres 13385 Restricting its characteri...
serf 13386 An infinite series of comp...
serfre 13387 An infinite series of real...
monoord 13388 Ordering relation for a mo...
monoord2 13389 Ordering relation for a mo...
sermono 13390 The partial sums in an inf...
seqsplit 13391 Split a sequence into two ...
seq1p 13392 Removing the first term fr...
seqcaopr3 13393 Lemma for ~ seqcaopr2 . (...
seqcaopr2 13394 The sum of two infinite se...
seqcaopr 13395 The sum of two infinite se...
seqf1olem2a 13396 Lemma for ~ seqf1o . (Con...
seqf1olem1 13397 Lemma for ~ seqf1o . (Con...
seqf1olem2 13398 Lemma for ~ seqf1o . (Con...
seqf1o 13399 Rearrange a sum via an arb...
seradd 13400 The sum of two infinite se...
sersub 13401 The difference of two infi...
seqid3 13402 A sequence that consists e...
seqid 13403 Discarding the first few t...
seqid2 13404 The last few partial sums ...
seqhomo 13405 Apply a homomorphism to a ...
seqz 13406 If the operation ` .+ ` ha...
seqfeq4 13407 Equality of series under d...
seqfeq3 13408 Equality of series under d...
seqdistr 13409 The distributive property ...
ser0 13410 The value of the partial s...
ser0f 13411 A zero-valued infinite ser...
serge0 13412 A finite sum of nonnegativ...
serle 13413 Comparison of partial sums...
ser1const 13414 Value of the partial serie...
seqof 13415 Distribute function operat...
seqof2 13416 Distribute function operat...
expval 13419 Value of exponentiation to...
expnnval 13420 Value of exponentiation to...
exp0 13421 Value of a complex number ...
0exp0e1 13422 ` 0 ^ 0 = 1 ` . This is o...
exp1 13423 Value of a complex number ...
expp1 13424 Value of a complex number ...
expneg 13425 Value of a complex number ...
expneg2 13426 Value of a complex number ...
expn1 13427 A number to the negative o...
expcllem 13428 Lemma for proving nonnegat...
expcl2lem 13429 Lemma for proving integer ...
nnexpcl 13430 Closure of exponentiation ...
nn0expcl 13431 Closure of exponentiation ...
zexpcl 13432 Closure of exponentiation ...
qexpcl 13433 Closure of exponentiation ...
reexpcl 13434 Closure of exponentiation ...
expcl 13435 Closure law for nonnegativ...
rpexpcl 13436 Closure law for exponentia...
reexpclz 13437 Closure of exponentiation ...
qexpclz 13438 Closure of exponentiation ...
m1expcl2 13439 Closure of exponentiation ...
m1expcl 13440 Closure of exponentiation ...
expclzlem 13441 Closure law for integer ex...
expclz 13442 Closure law for integer ex...
nn0expcli 13443 Closure of exponentiation ...
nn0sqcl 13444 The square of a nonnegativ...
expm1t 13445 Exponentiation in terms of...
1exp 13446 Value of one raised to a n...
expeq0 13447 Positive integer exponenti...
expne0 13448 Positive integer exponenti...
expne0i 13449 Nonnegative integer expone...
expgt0 13450 Nonnegative integer expone...
expnegz 13451 Value of a complex number ...
0exp 13452 Value of zero raised to a ...
expge0 13453 Nonnegative integer expone...
expge1 13454 Nonnegative integer expone...
expgt1 13455 Positive integer exponenti...
mulexp 13456 Positive integer exponenti...
mulexpz 13457 Integer exponentiation of ...
exprec 13458 Nonnegative integer expone...
expadd 13459 Sum of exponents law for n...
expaddzlem 13460 Lemma for ~ expaddz . (Co...
expaddz 13461 Sum of exponents law for i...
expmul 13462 Product of exponents law f...
expmulz 13463 Product of exponents law f...
m1expeven 13464 Exponentiation of negative...
expsub 13465 Exponent subtraction law f...
expp1z 13466 Value of a nonzero complex...
expm1 13467 Value of a complex number ...
expdiv 13468 Nonnegative integer expone...
sqval 13469 Value of the square of a c...
sqneg 13470 The square of the negative...
sqsubswap 13471 Swap the order of subtract...
sqcl 13472 Closure of square. (Contr...
sqmul 13473 Distribution of square ove...
sqeq0 13474 A number is zero iff its s...
sqdiv 13475 Distribution of square ove...
sqdivid 13476 The square of a nonzero nu...
sqne0 13477 A number is nonzero iff it...
resqcl 13478 Closure of the square of a...
sqgt0 13479 The square of a nonzero re...
sqn0rp 13480 The square of a nonzero re...
nnsqcl 13481 The naturals are closed un...
zsqcl 13482 Integers are closed under ...
qsqcl 13483 The square of a rational i...
sq11 13484 The square function is one...
nn0sq11 13485 The square function is one...
lt2sq 13486 The square function on non...
le2sq 13487 The square function on non...
le2sq2 13488 The square of a 'less than...
sqge0 13489 A square of a real is nonn...
zsqcl2 13490 The square of an integer i...
0expd 13491 Value of zero raised to a ...
exp0d 13492 Value of a complex number ...
exp1d 13493 Value of a complex number ...
expeq0d 13494 Positive integer exponenti...
sqvald 13495 Value of square. Inferenc...
sqcld 13496 Closure of square. (Contr...
sqeq0d 13497 A number is zero iff its s...
expcld 13498 Closure law for nonnegativ...
expp1d 13499 Value of a complex number ...
expaddd 13500 Sum of exponents law for n...
expmuld 13501 Product of exponents law f...
sqrecd 13502 Square of reciprocal. (Co...
expclzd 13503 Closure law for integer ex...
expne0d 13504 Nonnegative integer expone...
expnegd 13505 Value of a complex number ...
exprecd 13506 Nonnegative integer expone...
expp1zd 13507 Value of a nonzero complex...
expm1d 13508 Value of a complex number ...
expsubd 13509 Exponent subtraction law f...
sqmuld 13510 Distribution of square ove...
sqdivd 13511 Distribution of square ove...
expdivd 13512 Nonnegative integer expone...
mulexpd 13513 Positive integer exponenti...
znsqcld 13514 The square of a nonzero in...
reexpcld 13515 Closure of exponentiation ...
expge0d 13516 Nonnegative integer expone...
expge1d 13517 Nonnegative integer expone...
ltexp2a 13518 Ordering relationship for ...
expmordi 13519 Mantissa ordering relation...
rpexpmord 13520 Mantissa ordering relation...
expcan 13521 Cancellation law for expon...
ltexp2 13522 Ordering law for exponenti...
leexp2 13523 Ordering law for exponenti...
leexp2a 13524 Weak ordering relationship...
ltexp2r 13525 The power of a positive nu...
leexp2r 13526 Weak ordering relationship...
leexp1a 13527 Weak mantissa ordering rel...
exple1 13528 Nonnegative integer expone...
expubnd 13529 An upper bound on ` A ^ N ...
sumsqeq0 13530 Two real numbers are equal...
sqvali 13531 Value of square. Inferenc...
sqcli 13532 Closure of square. (Contr...
sqeq0i 13533 A number is zero iff its s...
sqrecii 13534 Square of reciprocal. (Co...
sqmuli 13535 Distribution of square ove...
sqdivi 13536 Distribution of square ove...
resqcli 13537 Closure of square in reals...
sqgt0i 13538 The square of a nonzero re...
sqge0i 13539 A square of a real is nonn...
lt2sqi 13540 The square function on non...
le2sqi 13541 The square function on non...
sq11i 13542 The square function is one...
sq0 13543 The square of 0 is 0. (Co...
sq0i 13544 If a number is zero, its s...
sq0id 13545 If a number is zero, its s...
sq1 13546 The square of 1 is 1. (Co...
neg1sqe1 13547 ` -u 1 ` squared is 1. (C...
sq2 13548 The square of 2 is 4. (Co...
sq3 13549 The square of 3 is 9. (Co...
sq4e2t8 13550 The square of 4 is 2 times...
cu2 13551 The cube of 2 is 8. (Cont...
irec 13552 The reciprocal of ` _i ` ....
i2 13553 ` _i ` squared. (Contribu...
i3 13554 ` _i ` cubed. (Contribute...
i4 13555 ` _i ` to the fourth power...
nnlesq 13556 A positive integer is less...
iexpcyc 13557 Taking ` _i ` to the ` K `...
expnass 13558 A counterexample showing t...
sqlecan 13559 Cancel one factor of a squ...
subsq 13560 Factor the difference of t...
subsq2 13561 Express the difference of ...
binom2i 13562 The square of a binomial. ...
subsqi 13563 Factor the difference of t...
sqeqori 13564 The squares of two complex...
subsq0i 13565 The two solutions to the d...
sqeqor 13566 The squares of two complex...
binom2 13567 The square of a binomial. ...
binom21 13568 Special case of ~ binom2 w...
binom2sub 13569 Expand the square of a sub...
binom2sub1 13570 Special case of ~ binom2su...
binom2subi 13571 Expand the square of a sub...
mulbinom2 13572 The square of a binomial w...
binom3 13573 The cube of a binomial. (...
sq01 13574 If a complex number equals...
zesq 13575 An integer is even iff its...
nnesq 13576 A positive integer is even...
crreczi 13577 Reciprocal of a complex nu...
bernneq 13578 Bernoulli's inequality, du...
bernneq2 13579 Variation of Bernoulli's i...
bernneq3 13580 A corollary of ~ bernneq ....
expnbnd 13581 Exponentiation with a mant...
expnlbnd 13582 The reciprocal of exponent...
expnlbnd2 13583 The reciprocal of exponent...
expmulnbnd 13584 Exponentiation with a mant...
digit2 13585 Two ways to express the ` ...
digit1 13586 Two ways to express the ` ...
modexp 13587 Exponentiation property of...
discr1 13588 A nonnegative quadratic fo...
discr 13589 If a quadratic polynomial ...
expnngt1 13590 If an integer power with a...
expnngt1b 13591 An integer power with an i...
sqoddm1div8 13592 A squared odd number minus...
nnsqcld 13593 The naturals are closed un...
nnexpcld 13594 Closure of exponentiation ...
nn0expcld 13595 Closure of exponentiation ...
rpexpcld 13596 Closure law for exponentia...
ltexp2rd 13597 The power of a positive nu...
reexpclzd 13598 Closure of exponentiation ...
resqcld 13599 Closure of square in reals...
sqge0d 13600 A square of a real is nonn...
sqgt0d 13601 The square of a nonzero re...
ltexp2d 13602 Ordering relationship for ...
leexp2d 13603 Ordering law for exponenti...
expcand 13604 Ordering relationship for ...
leexp2ad 13605 Ordering relationship for ...
leexp2rd 13606 Ordering relationship for ...
lt2sqd 13607 The square function on non...
le2sqd 13608 The square function on non...
sq11d 13609 The square function is one...
mulsubdivbinom2 13610 The square of a binomial w...
muldivbinom2 13611 The square of a binomial w...
sq10 13612 The square of 10 is 100. ...
sq10e99m1 13613 The square of 10 is 99 plu...
3dec 13614 A "decimal constructor" wh...
nn0le2msqi 13615 The square function on non...
nn0opthlem1 13616 A rather pretty lemma for ...
nn0opthlem2 13617 Lemma for ~ nn0opthi . (C...
nn0opthi 13618 An ordered pair theorem fo...
nn0opth2i 13619 An ordered pair theorem fo...
nn0opth2 13620 An ordered pair theorem fo...
facnn 13623 Value of the factorial fun...
fac0 13624 The factorial of 0. (Cont...
fac1 13625 The factorial of 1. (Cont...
facp1 13626 The factorial of a success...
fac2 13627 The factorial of 2. (Cont...
fac3 13628 The factorial of 3. (Cont...
fac4 13629 The factorial of 4. (Cont...
facnn2 13630 Value of the factorial fun...
faccl 13631 Closure of the factorial f...
faccld 13632 Closure of the factorial f...
facmapnn 13633 The factorial function res...
facne0 13634 The factorial function is ...
facdiv 13635 A positive integer divides...
facndiv 13636 No positive integer (great...
facwordi 13637 Ordering property of facto...
faclbnd 13638 A lower bound for the fact...
faclbnd2 13639 A lower bound for the fact...
faclbnd3 13640 A lower bound for the fact...
faclbnd4lem1 13641 Lemma for ~ faclbnd4 . Pr...
faclbnd4lem2 13642 Lemma for ~ faclbnd4 . Us...
faclbnd4lem3 13643 Lemma for ~ faclbnd4 . Th...
faclbnd4lem4 13644 Lemma for ~ faclbnd4 . Pr...
faclbnd4 13645 Variant of ~ faclbnd5 prov...
faclbnd5 13646 The factorial function gro...
faclbnd6 13647 Geometric lower bound for ...
facubnd 13648 An upper bound for the fac...
facavg 13649 The product of two factori...
bcval 13652 Value of the binomial coef...
bcval2 13653 Value of the binomial coef...
bcval3 13654 Value of the binomial coef...
bcval4 13655 Value of the binomial coef...
bcrpcl 13656 Closure of the binomial co...
bccmpl 13657 "Complementing" its second...
bcn0 13658 ` N ` choose 0 is 1. Rema...
bc0k 13659 The binomial coefficient "...
bcnn 13660 ` N ` choose ` N ` is 1. ...
bcn1 13661 Binomial coefficient: ` N ...
bcnp1n 13662 Binomial coefficient: ` N ...
bcm1k 13663 The proportion of one bino...
bcp1n 13664 The proportion of one bino...
bcp1nk 13665 The proportion of one bino...
bcval5 13666 Write out the top and bott...
bcn2 13667 Binomial coefficient: ` N ...
bcp1m1 13668 Compute the binomial coeff...
bcpasc 13669 Pascal's rule for the bino...
bccl 13670 A binomial coefficient, in...
bccl2 13671 A binomial coefficient, in...
bcn2m1 13672 Compute the binomial coeff...
bcn2p1 13673 Compute the binomial coeff...
permnn 13674 The number of permutations...
bcnm1 13675 The binomial coefficent of...
4bc3eq4 13676 The value of four choose t...
4bc2eq6 13677 The value of four choose t...
hashkf 13680 The finite part of the siz...
hashgval 13681 The value of the ` # ` fun...
hashginv 13682 ` ``' G ` maps the size fu...
hashinf 13683 The value of the ` # ` fun...
hashbnd 13684 If ` A ` has size bounded ...
hashfxnn0 13685 The size function is a fun...
hashf 13686 The size function maps all...
hashxnn0 13687 The value of the hash func...
hashresfn 13688 Restriction of the domain ...
dmhashres 13689 Restriction of the domain ...
hashnn0pnf 13690 The value of the hash func...
hashnnn0genn0 13691 If the size of a set is no...
hashnemnf 13692 The size of a set is never...
hashv01gt1 13693 The size of a set is eithe...
hashfz1 13694 The set ` ( 1 ... N ) ` ha...
hashen 13695 Two finite sets have the s...
hasheni 13696 Equinumerous sets have the...
hasheqf1o 13697 The size of two finite set...
fiinfnf1o 13698 There is no bijection betw...
focdmex 13699 The codomain of an onto fu...
hasheqf1oi 13700 The size of two sets is eq...
hashf1rn 13701 The size of a finite set w...
hasheqf1od 13702 The size of two sets is eq...
fz1eqb 13703 Two possibly-empty 1-based...
hashcard 13704 The size function of the c...
hashcl 13705 Closure of the ` # ` funct...
hashxrcl 13706 Extended real closure of t...
hashclb 13707 Reverse closure of the ` #...
nfile 13708 The size of any infinite s...
hashvnfin 13709 A set of finite size is a ...
hashnfinnn0 13710 The size of an infinite se...
isfinite4 13711 A finite set is equinumero...
hasheq0 13712 Two ways of saying a finit...
hashneq0 13713 Two ways of saying a set i...
hashgt0n0 13714 If the size of a set is gr...
hashnncl 13715 Positive natural closure o...
hash0 13716 The empty set has size zer...
hashelne0d 13717 A set with an element has ...
hashsng 13718 The size of a singleton. ...
hashen1 13719 A set has size 1 if and on...
hash1elsn 13720 A set of size 1 with a kno...
hashrabrsn 13721 The size of a restricted c...
hashrabsn01 13722 The size of a restricted c...
hashrabsn1 13723 If the size of a restricte...
hashfn 13724 A function is equinumerous...
fseq1hash 13725 The value of the size func...
hashgadd 13726 ` G ` maps ordinal additio...
hashgval2 13727 A short expression for the...
hashdom 13728 Dominance relation for the...
hashdomi 13729 Non-strict order relation ...
hashsdom 13730 Strict dominance relation ...
hashun 13731 The size of the union of d...
hashun2 13732 The size of the union of f...
hashun3 13733 The size of the union of f...
hashinfxadd 13734 The extended real addition...
hashunx 13735 The size of the union of d...
hashge0 13736 The cardinality of a set i...
hashgt0 13737 The cardinality of a nonem...
hashge1 13738 The cardinality of a nonem...
1elfz0hash 13739 1 is an element of the fin...
hashnn0n0nn 13740 If a nonnegative integer i...
hashunsng 13741 The size of the union of a...
hashunsngx 13742 The size of the union of a...
hashunsnggt 13743 The size of a set is great...
hashprg 13744 The size of an unordered p...
elprchashprn2 13745 If one element of an unord...
hashprb 13746 The size of an unordered p...
hashprdifel 13747 The elements of an unorder...
prhash2ex 13748 There is (at least) one se...
hashle00 13749 If the size of a set is le...
hashgt0elex 13750 If the size of a set is gr...
hashgt0elexb 13751 The size of a set is great...
hashp1i 13752 Size of a finite ordinal. ...
hash1 13753 Size of a finite ordinal. ...
hash2 13754 Size of a finite ordinal. ...
hash3 13755 Size of a finite ordinal. ...
hash4 13756 Size of a finite ordinal. ...
pr0hash2ex 13757 There is (at least) one se...
hashss 13758 The size of a subset is le...
prsshashgt1 13759 The size of a superset of ...
hashin 13760 The size of the intersecti...
hashssdif 13761 The size of the difference...
hashdif 13762 The size of the difference...
hashdifsn 13763 The size of the difference...
hashdifpr 13764 The size of the difference...
hashsn01 13765 The size of a singleton is...
hashsnle1 13766 The size of a singleton is...
hashsnlei 13767 Get an upper bound on a co...
hash1snb 13768 The size of a set is 1 if ...
euhash1 13769 The size of a set is 1 in ...
hash1n0 13770 If the size of a set is 1 ...
hashgt12el 13771 In a set with more than on...
hashgt12el2 13772 In a set with more than on...
hashgt23el 13773 A set with more than two e...
hashunlei 13774 Get an upper bound on a co...
hashsslei 13775 Get an upper bound on a co...
hashfz 13776 Value of the numeric cardi...
fzsdom2 13777 Condition for finite range...
hashfzo 13778 Cardinality of a half-open...
hashfzo0 13779 Cardinality of a half-open...
hashfzp1 13780 Value of the numeric cardi...
hashfz0 13781 Value of the numeric cardi...
hashxplem 13782 Lemma for ~ hashxp . (Con...
hashxp 13783 The size of the Cartesian ...
hashmap 13784 The size of the set expone...
hashpw 13785 The size of the power set ...
hashfun 13786 A finite set is a function...
hashres 13787 The number of elements of ...
hashreshashfun 13788 The number of elements of ...
hashimarn 13789 The size of the image of a...
hashimarni 13790 If the size of the image o...
resunimafz0 13791 TODO-AV: Revise using ` F...
fnfz0hash 13792 The size of a function on ...
ffz0hash 13793 The size of a function on ...
fnfz0hashnn0 13794 The size of a function on ...
ffzo0hash 13795 The size of a function on ...
fnfzo0hash 13796 The size of a function on ...
fnfzo0hashnn0 13797 The value of the size func...
hashbclem 13798 Lemma for ~ hashbc : induc...
hashbc 13799 The binomial coefficient c...
hashfacen 13800 The number of bijections b...
hashf1lem1 13801 Lemma for ~ hashf1 . (Con...
hashf1lem2 13802 Lemma for ~ hashf1 . (Con...
hashf1 13803 The permutation number ` |...
hashfac 13804 A factorial counts the num...
leiso 13805 Two ways to write a strict...
leisorel 13806 Version of ~ isorel for st...
fz1isolem 13807 Lemma for ~ fz1iso . (Con...
fz1iso 13808 Any finite ordered set has...
ishashinf 13809 Any set that is not finite...
seqcoll 13810 The function ` F ` contain...
seqcoll2 13811 The function ` F ` contain...
phphashd 13812 Corollary of the Pigeonhol...
phphashrd 13813 Corollary of the Pigeonhol...
hashprlei 13814 An unordered pair has at m...
hash2pr 13815 A set of size two is an un...
hash2prde 13816 A set of size two is an un...
hash2exprb 13817 A set of size two is an un...
hash2prb 13818 A set of size two is a pro...
prprrab 13819 The set of proper pairs of...
nehash2 13820 The cardinality of a set w...
hash2prd 13821 A set of size two is an un...
hash2pwpr 13822 If the size of a subset of...
hashle2pr 13823 A nonempty set of size les...
hashle2prv 13824 A nonempty subset of a pow...
pr2pwpr 13825 The set of subsets of a pa...
hashge2el2dif 13826 A set with size at least 2...
hashge2el2difr 13827 A set with at least 2 diff...
hashge2el2difb 13828 A set has size at least 2 ...
hashdmpropge2 13829 The size of the domain of ...
hashtplei 13830 An unordered triple has at...
hashtpg 13831 The size of an unordered t...
hashge3el3dif 13832 A set with size at least 3...
elss2prb 13833 An element of the set of s...
hash2sspr 13834 A subset of size two is an...
exprelprel 13835 If there is an element of ...
hash3tr 13836 A set of size three is an ...
hash1to3 13837 If the size of a set is be...
fundmge2nop0 13838 A function with a domain c...
fundmge2nop 13839 A function with a domain c...
fun2dmnop0 13840 A function with a domain c...
fun2dmnop 13841 A function with a domain c...
hashdifsnp1 13842 If the size of a set is a ...
fi1uzind 13843 Properties of an ordered p...
brfi1uzind 13844 Properties of a binary rel...
brfi1ind 13845 Properties of a binary rel...
brfi1indALT 13846 Alternate proof of ~ brfi1...
opfi1uzind 13847 Properties of an ordered p...
opfi1ind 13848 Properties of an ordered p...
iswrd 13851 Property of being a word o...
wrdval 13852 Value of the set of words ...
iswrdi 13853 A zero-based sequence is a...
wrdf 13854 A word is a zero-based seq...
iswrdb 13855 A word over an alphabet is...
wrddm 13856 The indices of a word (i.e...
sswrd 13857 The set of words respects ...
snopiswrd 13858 A singleton of an ordered ...
wrdexg 13859 The set of words over a se...
wrdexgOLD 13860 Obsolete proof of ~ wrdexg...
wrdexb 13861 The set of words over a se...
wrdexi 13862 The set of words over a se...
wrdsymbcl 13863 A symbol within a word ove...
wrdfn 13864 A word is a function with ...
wrdv 13865 A word over an alphabet is...
wrdvOLD 13866 Obsolete proof of ~ wrdv a...
wrdlndm 13867 The length of a word is no...
wrdlndmOLD 13868 Obsolete proof of ~ wrdlnd...
iswrdsymb 13869 An arbitrary word is a wor...
wrdfin 13870 A word is a finite set. (...
lencl 13871 The length of a word is a ...
lennncl 13872 The length of a nonempty w...
wrdffz 13873 A word is a function from ...
wrdeq 13874 Equality theorem for the s...
wrdeqi 13875 Equality theorem for the s...
iswrddm0 13876 A function with empty doma...
wrd0 13877 The empty set is a word (t...
0wrd0 13878 The empty word is the only...
ffz0iswrd 13879 A sequence with zero-based...
ffz0iswrdOLD 13880 Obsolete proof of ~ ffz0is...
wrdsymb 13881 A word is a word over the ...
nfwrd 13882 Hypothesis builder for ` W...
csbwrdg 13883 Class substitution for the...
wrdnval 13884 Words of a fixed length ar...
wrdmap 13885 Words as a mapping. (Cont...
hashwrdn 13886 If there is only a finite ...
wrdnfi 13887 If there is only a finite ...
wrdnfiOLD 13888 Obsolete version of ~ wrdn...
wrdsymb0 13889 A symbol at a position "ou...
wrdlenge1n0 13890 A word with length at leas...
len0nnbi 13891 The length of a word is a ...
wrdlenge2n0 13892 A word with length at leas...
wrdsymb1 13893 The first symbol of a none...
wrdlen1 13894 A word of length 1 starts ...
fstwrdne 13895 The first symbol of a none...
fstwrdne0 13896 The first symbol of a none...
eqwrd 13897 Two words are equal iff th...
elovmpowrd 13898 Implications for the value...
elovmptnn0wrd 13899 Implications for the value...
wrdred1 13900 A word truncated by a symb...
wrdred1hash 13901 The length of a word trunc...
lsw 13904 Extract the last symbol of...
lsw0 13905 The last symbol of an empt...
lsw0g 13906 The last symbol of an empt...
lsw1 13907 The last symbol of a word ...
lswcl 13908 Closure of the last symbol...
lswlgt0cl 13909 The last symbol of a nonem...
ccatfn 13912 The concatenation operator...
ccatfval 13913 Value of the concatenation...
ccatcl 13914 The concatenation of two w...
ccatlen 13915 The length of a concatenat...
ccatlenOLD 13916 Obsolete version of ~ ccat...
ccat0 13917 The concatenation of two w...
ccatval1 13918 Value of a symbol in the l...
ccatval1OLD 13919 Obsolete version of ~ ccat...
ccatval2 13920 Value of a symbol in the r...
ccatval3 13921 Value of a symbol in the r...
elfzelfzccat 13922 An element of a finite set...
ccatvalfn 13923 The concatenation of two w...
ccatsymb 13924 The symbol at a given posi...
ccatfv0 13925 The first symbol of a conc...
ccatval1lsw 13926 The last symbol of the lef...
ccatval21sw 13927 The first symbol of the ri...
ccatlid 13928 Concatenation of a word by...
ccatrid 13929 Concatenation of a word by...
ccatass 13930 Associative law for concat...
ccatrn 13931 The range of a concatenate...
ccatidid 13932 Concatenation of the empty...
lswccatn0lsw 13933 The last symbol of a word ...
lswccat0lsw 13934 The last symbol of a word ...
ccatalpha 13935 A concatenation of two arb...
ccatrcl1 13936 Reverse closure of a conca...
ids1 13939 Identity function protecti...
s1val 13940 Value of a singleton word....
s1rn 13941 The range of a singleton w...
s1eq 13942 Equality theorem for a sin...
s1eqd 13943 Equality theorem for a sin...
s1cl 13944 A singleton word is a word...
s1cld 13945 A singleton word is a word...
s1prc 13946 Value of a singleton word ...
s1cli 13947 A singleton word is a word...
s1len 13948 Length of a singleton word...
s1nz 13949 A singleton word is not th...
s1dm 13950 The domain of a singleton ...
s1dmALT 13951 Alternate version of ~ s1d...
s1fv 13952 Sole symbol of a singleton...
lsws1 13953 The last symbol of a singl...
eqs1 13954 A word of length 1 is a si...
wrdl1exs1 13955 A word of length 1 is a si...
wrdl1s1 13956 A word of length 1 is a si...
s111 13957 The singleton word functio...
ccatws1cl 13958 The concatenation of a wor...
ccatws1clv 13959 The concatenation of a wor...
ccat2s1cl 13960 The concatenation of two s...
ccats1alpha 13961 A concatenation of a word ...
ccatws1len 13962 The length of the concaten...
ccatws1lenp1b 13963 The length of a word is ` ...
wrdlenccats1lenm1 13964 The length of a word is th...
ccat2s1len 13965 The length of the concaten...
ccat2s1lenOLD 13966 Obsolete version of ~ ccat...
ccatw2s1cl 13967 The concatenation of a wor...
ccatw2s1len 13968 The length of the concaten...
ccats1val1 13969 Value of a symbol in the l...
ccats1val1OLD 13970 Obsolete version of ~ ccat...
ccats1val2 13971 Value of the symbol concat...
ccat1st1st 13972 The first symbol of a word...
ccat2s1p1 13973 Extract the first of two c...
ccat2s1p2 13974 Extract the second of two ...
ccat2s1p1OLD 13975 Obsolete version of ~ ccat...
ccat2s1p2OLD 13976 Obsolete version of ~ ccat...
ccatw2s1ass 13977 Associative law for a conc...
ccatw2s1assOLD 13978 Obsolete version of ~ ccat...
ccatws1n0 13979 The concatenation of a wor...
ccatws1ls 13980 The last symbol of the con...
lswccats1 13981 The last symbol of a word ...
lswccats1fst 13982 The last symbol of a nonem...
ccatw2s1p1 13983 Extract the symbol of the ...
ccatw2s1p1OLD 13984 Obsolete version of ~ ccat...
ccatw2s1p2 13985 Extract the second of two ...
ccat2s1fvw 13986 Extract a symbol of a word...
ccat2s1fvwOLD 13987 Obsolete version of ~ ccat...
ccat2s1fst 13988 The first symbol of the co...
ccat2s1fstOLD 13989 Obsolete version of ~ ccat...
swrdnznd 13992 The value of a subword ope...
swrdval 13993 Value of a subword. (Cont...
swrd00 13994 A zero length substring. ...
swrdcl 13995 Closure of the subword ext...
swrdval2 13996 Value of the subword extra...
swrdlen 13997 Length of an extracted sub...
swrdfv 13998 A symbol in an extracted s...
swrdfv0 13999 The first symbol in an ext...
swrdf 14000 A subword of a word is a f...
swrdvalfn 14001 Value of the subword extra...
swrdrn 14002 The range of a subword of ...
swrdlend 14003 The value of the subword e...
swrdnd 14004 The value of the subword e...
swrdnd2 14005 Value of the subword extra...
swrdnnn0nd 14006 The value of a subword ope...
swrdnd0 14007 The value of a subword ope...
swrd0 14008 A subword of an empty set ...
swrdrlen 14009 Length of a right-anchored...
swrdlen2 14010 Length of an extracted sub...
swrdfv2 14011 A symbol in an extracted s...
swrdwrdsymb 14012 A subword is a word over t...
swrdsb0eq 14013 Two subwords with the same...
swrdsbslen 14014 Two subwords with the same...
swrdspsleq 14015 Two words have a common su...
swrds1 14016 Extract a single symbol fr...
swrdlsw 14017 Extract the last single sy...
ccatswrd 14018 Joining two adjacent subwo...
swrdccat2 14019 Recover the right half of ...
pfxnndmnd 14022 The value of a prefix oper...
pfxval 14023 Value of a prefix operatio...
pfx00 14024 The zero length prefix is ...
pfx0 14025 A prefix of an empty set i...
pfxval0 14026 Value of a prefix operatio...
pfxcl 14027 Closure of the prefix extr...
pfxmpt 14028 Value of the prefix extrac...
pfxres 14029 Value of the subword extra...
pfxf 14030 A prefix of a word is a fu...
pfxfn 14031 Value of the prefix extrac...
pfxfv 14032 A symbol in a prefix of a ...
pfxlen 14033 Length of a prefix. (Cont...
pfxid 14034 A word is a prefix of itse...
pfxrn 14035 The range of a prefix of a...
pfxn0 14036 A prefix consisting of at ...
pfxnd 14037 The value of a prefix oper...
pfxnd0 14038 The value of a prefix oper...
pfxwrdsymb 14039 A prefix of a word is a wo...
addlenrevpfx 14040 The sum of the lengths of ...
addlenpfx 14041 The sum of the lengths of ...
pfxfv0 14042 The first symbol of a pref...
pfxtrcfv 14043 A symbol in a word truncat...
pfxtrcfv0 14044 The first symbol in a word...
pfxfvlsw 14045 The last symbol in a nonem...
pfxeq 14046 The prefixes of two words ...
pfxtrcfvl 14047 The last symbol in a word ...
pfxsuffeqwrdeq 14048 Two words are equal if and...
pfxsuff1eqwrdeq 14049 Two (nonempty) words are e...
disjwrdpfx 14050 Sets of words are disjoint...
ccatpfx 14051 Concatenating a prefix wit...
pfxccat1 14052 Recover the left half of a...
pfx1 14053 The prefix of length one o...
swrdswrdlem 14054 Lemma for ~ swrdswrd . (C...
swrdswrd 14055 A subword of a subword is ...
pfxswrd 14056 A prefix of a subword is a...
swrdpfx 14057 A subword of a prefix is a...
pfxpfx 14058 A prefix of a prefix is a ...
pfxpfxid 14059 A prefix of a prefix with ...
pfxcctswrd 14060 The concatenation of the p...
lenpfxcctswrd 14061 The length of the concaten...
lenrevpfxcctswrd 14062 The length of the concaten...
pfxlswccat 14063 Reconstruct a nonempty wor...
ccats1pfxeq 14064 The last symbol of a word ...
ccats1pfxeqrex 14065 There exists a symbol such...
ccatopth 14066 An ~ opth -like theorem fo...
ccatopth2 14067 An ~ opth -like theorem fo...
ccatlcan 14068 Concatenation of words is ...
ccatrcan 14069 Concatenation of words is ...
wrdeqs1cat 14070 Decompose a nonempty word ...
cats1un 14071 Express a word with an ext...
wrdind 14072 Perform induction over the...
wrd2ind 14073 Perform induction over the...
swrdccatfn 14074 The subword of a concatena...
swrdccatin1 14075 The subword of a concatena...
pfxccatin12lem4 14076 Lemma 4 for ~ pfxccatin12 ...
pfxccatin12lem2a 14077 Lemma for ~ pfxccatin12lem...
pfxccatin12lem1 14078 Lemma 1 for ~ pfxccatin12 ...
swrdccatin2 14079 The subword of a concatena...
pfxccatin12lem2c 14080 Lemma for ~ pfxccatin12lem...
pfxccatin12lem2 14081 Lemma 2 for ~ pfxccatin12 ...
pfxccatin12lem3 14082 Lemma 3 for ~ pfxccatin12 ...
pfxccatin12 14083 The subword of a concatena...
pfxccat3 14084 The subword of a concatena...
swrdccat 14085 The subword of a concatena...
pfxccatpfx1 14086 A prefix of a concatenatio...
pfxccatpfx2 14087 A prefix of a concatenatio...
pfxccat3a 14088 A prefix of a concatenatio...
swrdccat3blem 14089 Lemma for ~ swrdccat3b . ...
swrdccat3b 14090 A suffix of a concatenatio...
pfxccatid 14091 A prefix of a concatenatio...
ccats1pfxeqbi 14092 A word is a prefix of a wo...
swrdccatin1d 14093 The subword of a concatena...
swrdccatin2d 14094 The subword of a concatena...
pfxccatin12d 14095 The subword of a concatena...
reuccatpfxs1lem 14096 Lemma for ~ reuccatpfxs1 ....
reuccatpfxs1 14097 There is a unique word hav...
reuccatpfxs1v 14098 There is a unique word hav...
splval 14101 Value of the substring rep...
splcl 14102 Closure of the substring r...
splid 14103 Splicing a subword for the...
spllen 14104 The length of a splice. (...
splfv1 14105 Symbols to the left of a s...
splfv2a 14106 Symbols within the replace...
splval2 14107 Value of a splice, assumin...
revval 14110 Value of the word reversin...
revcl 14111 The reverse of a word is a...
revlen 14112 The reverse of a word has ...
revfv 14113 Reverse of a word at a poi...
rev0 14114 The empty word is its own ...
revs1 14115 Singleton words are their ...
revccat 14116 Antiautomorphic property o...
revrev 14117 Reversal is an involution ...
reps 14120 Construct a function mappi...
repsundef 14121 A function mapping a half-...
repsconst 14122 Construct a function mappi...
repsf 14123 The constructed function m...
repswsymb 14124 The symbols of a "repeated...
repsw 14125 A function mapping a half-...
repswlen 14126 The length of a "repeated ...
repsw0 14127 The "repeated symbol word"...
repsdf2 14128 Alternative definition of ...
repswsymball 14129 All the symbols of a "repe...
repswsymballbi 14130 A word is a "repeated symb...
repswfsts 14131 The first symbol of a none...
repswlsw 14132 The last symbol of a nonem...
repsw1 14133 The "repeated symbol word"...
repswswrd 14134 A subword of a "repeated s...
repswpfx 14135 A prefix of a repeated sym...
repswccat 14136 The concatenation of two "...
repswrevw 14137 The reverse of a "repeated...
cshfn 14140 Perform a cyclical shift f...
cshword 14141 Perform a cyclical shift f...
cshnz 14142 A cyclical shift is the em...
0csh0 14143 Cyclically shifting an emp...
cshw0 14144 A word cyclically shifted ...
cshwmodn 14145 Cyclically shifting a word...
cshwsublen 14146 Cyclically shifting a word...
cshwn 14147 A word cyclically shifted ...
cshwcl 14148 A cyclically shifted word ...
cshwlen 14149 The length of a cyclically...
cshwf 14150 A cyclically shifted word ...
cshwfn 14151 A cyclically shifted word ...
cshwrn 14152 The range of a cyclically ...
cshwidxmod 14153 The symbol at a given inde...
cshwidxmodr 14154 The symbol at a given inde...
cshwidx0mod 14155 The symbol at index 0 of a...
cshwidx0 14156 The symbol at index 0 of a...
cshwidxm1 14157 The symbol at index ((n-N)...
cshwidxm 14158 The symbol at index (n-N) ...
cshwidxn 14159 The symbol at index (n-1) ...
cshf1 14160 Cyclically shifting a word...
cshinj 14161 If a word is injectiv (reg...
repswcshw 14162 A cyclically shifted "repe...
2cshw 14163 Cyclically shifting a word...
2cshwid 14164 Cyclically shifting a word...
lswcshw 14165 The last symbol of a word ...
2cshwcom 14166 Cyclically shifting a word...
cshwleneq 14167 If the results of cyclical...
3cshw 14168 Cyclically shifting a word...
cshweqdif2 14169 If cyclically shifting two...
cshweqdifid 14170 If cyclically shifting a w...
cshweqrep 14171 If cyclically shifting a w...
cshw1 14172 If cyclically shifting a w...
cshw1repsw 14173 If cyclically shifting a w...
cshwsexa 14174 The class of (different!) ...
2cshwcshw 14175 If a word is a cyclically ...
scshwfzeqfzo 14176 For a nonempty word the se...
cshwcshid 14177 A cyclically shifted word ...
cshwcsh2id 14178 A cyclically shifted word ...
cshimadifsn 14179 The image of a cyclically ...
cshimadifsn0 14180 The image of a cyclically ...
wrdco 14181 Mapping a word by a functi...
lenco 14182 Length of a mapped word is...
s1co 14183 Mapping of a singleton wor...
revco 14184 Mapping of words (i.e., a ...
ccatco 14185 Mapping of words commutes ...
cshco 14186 Mapping of words commutes ...
swrdco 14187 Mapping of words commutes ...
pfxco 14188 Mapping of words commutes ...
lswco 14189 Mapping of (nonempty) word...
repsco 14190 Mapping of words commutes ...
cats1cld 14205 Closure of concatenation w...
cats1co 14206 Closure of concatenation w...
cats1cli 14207 Closure of concatenation w...
cats1fvn 14208 The last symbol of a conca...
cats1fv 14209 A symbol other than the la...
cats1len 14210 The length of concatenatio...
cats1cat 14211 Closure of concatenation w...
cats2cat 14212 Closure of concatenation o...
s2eqd 14213 Equality theorem for a dou...
s3eqd 14214 Equality theorem for a len...
s4eqd 14215 Equality theorem for a len...
s5eqd 14216 Equality theorem for a len...
s6eqd 14217 Equality theorem for a len...
s7eqd 14218 Equality theorem for a len...
s8eqd 14219 Equality theorem for a len...
s3eq2 14220 Equality theorem for a len...
s2cld 14221 A doubleton word is a word...
s3cld 14222 A length 3 string is a wor...
s4cld 14223 A length 4 string is a wor...
s5cld 14224 A length 5 string is a wor...
s6cld 14225 A length 6 string is a wor...
s7cld 14226 A length 7 string is a wor...
s8cld 14227 A length 7 string is a wor...
s2cl 14228 A doubleton word is a word...
s3cl 14229 A length 3 string is a wor...
s2cli 14230 A doubleton word is a word...
s3cli 14231 A length 3 string is a wor...
s4cli 14232 A length 4 string is a wor...
s5cli 14233 A length 5 string is a wor...
s6cli 14234 A length 6 string is a wor...
s7cli 14235 A length 7 string is a wor...
s8cli 14236 A length 8 string is a wor...
s2fv0 14237 Extract the first symbol f...
s2fv1 14238 Extract the second symbol ...
s2len 14239 The length of a doubleton ...
s2dm 14240 The domain of a doubleton ...
s3fv0 14241 Extract the first symbol f...
s3fv1 14242 Extract the second symbol ...
s3fv2 14243 Extract the third symbol f...
s3len 14244 The length of a length 3 s...
s4fv0 14245 Extract the first symbol f...
s4fv1 14246 Extract the second symbol ...
s4fv2 14247 Extract the third symbol f...
s4fv3 14248 Extract the fourth symbol ...
s4len 14249 The length of a length 4 s...
s5len 14250 The length of a length 5 s...
s6len 14251 The length of a length 6 s...
s7len 14252 The length of a length 7 s...
s8len 14253 The length of a length 8 s...
lsws2 14254 The last symbol of a doubl...
lsws3 14255 The last symbol of a 3 let...
lsws4 14256 The last symbol of a 4 let...
s2prop 14257 A length 2 word is an unor...
s2dmALT 14258 Alternate version of ~ s2d...
s3tpop 14259 A length 3 word is an unor...
s4prop 14260 A length 4 word is a union...
s3fn 14261 A length 3 word is a funct...
funcnvs1 14262 The converse of a singleto...
funcnvs2 14263 The converse of a length 2...
funcnvs3 14264 The converse of a length 3...
funcnvs4 14265 The converse of a length 4...
s2f1o 14266 A length 2 word with mutua...
f1oun2prg 14267 A union of unordered pairs...
s4f1o 14268 A length 4 word with mutua...
s4dom 14269 The domain of a length 4 w...
s2co 14270 Mapping a doubleton word b...
s3co 14271 Mapping a length 3 string ...
s0s1 14272 Concatenation of fixed len...
s1s2 14273 Concatenation of fixed len...
s1s3 14274 Concatenation of fixed len...
s1s4 14275 Concatenation of fixed len...
s1s5 14276 Concatenation of fixed len...
s1s6 14277 Concatenation of fixed len...
s1s7 14278 Concatenation of fixed len...
s2s2 14279 Concatenation of fixed len...
s4s2 14280 Concatenation of fixed len...
s4s3 14281 Concatenation of fixed len...
s4s4 14282 Concatenation of fixed len...
s3s4 14283 Concatenation of fixed len...
s2s5 14284 Concatenation of fixed len...
s5s2 14285 Concatenation of fixed len...
s2eq2s1eq 14286 Two length 2 words are equ...
s2eq2seq 14287 Two length 2 words are equ...
s3eqs2s1eq 14288 Two length 3 words are equ...
s3eq3seq 14289 Two length 3 words are equ...
swrds2 14290 Extract two adjacent symbo...
swrds2m 14291 Extract two adjacent symbo...
wrdlen2i 14292 Implications of a word of ...
wrd2pr2op 14293 A word of length two repre...
wrdlen2 14294 A word of length two. (Co...
wrdlen2s2 14295 A word of length two as do...
wrdl2exs2 14296 A word of length two is a ...
pfx2 14297 A prefix of length two. (...
wrd3tpop 14298 A word of length three rep...
wrdlen3s3 14299 A word of length three as ...
repsw2 14300 The "repeated symbol word"...
repsw3 14301 The "repeated symbol word"...
swrd2lsw 14302 Extract the last two symbo...
2swrd2eqwrdeq 14303 Two words of length at lea...
ccatw2s1ccatws2 14304 The concatenation of a wor...
ccatw2s1ccatws2OLD 14305 Obsolete version of ~ ccat...
ccat2s1fvwALT 14306 Alternate proof of ~ ccat2...
ccat2s1fvwALTOLD 14307 Obsolete version of ~ ccat...
wwlktovf 14308 Lemma 1 for ~ wrd2f1tovbij...
wwlktovf1 14309 Lemma 2 for ~ wrd2f1tovbij...
wwlktovfo 14310 Lemma 3 for ~ wrd2f1tovbij...
wwlktovf1o 14311 Lemma 4 for ~ wrd2f1tovbij...
wrd2f1tovbij 14312 There is a bijection betwe...
eqwrds3 14313 A word is equal with a len...
wrdl3s3 14314 A word of length 3 is a le...
s3sndisj 14315 The singletons consisting ...
s3iunsndisj 14316 The union of singletons co...
ofccat 14317 Letterwise operations on w...
ofs1 14318 Letterwise operations on a...
ofs2 14319 Letterwise operations on a...
coss12d 14320 Subset deduction for compo...
trrelssd 14321 The composition of subclas...
xpcogend 14322 The most interesting case ...
xpcoidgend 14323 If two classes are not dis...
cotr2g 14324 Two ways of saying that th...
cotr2 14325 Two ways of saying a relat...
cotr3 14326 Two ways of saying a relat...
coemptyd 14327 Deduction about compositio...
xptrrel 14328 The cross product is alway...
0trrel 14329 The empty class is a trans...
cleq1lem 14330 Equality implies bijection...
cleq1 14331 Equality of relations impl...
clsslem 14332 The closure of a subclass ...
trcleq1 14337 Equality of relations impl...
trclsslem 14338 The transitive closure (as...
trcleq2lem 14339 Equality implies bijection...
cvbtrcl 14340 Change of bound variable i...
trcleq12lem 14341 Equality implies bijection...
trclexlem 14342 Existence of relation impl...
trclublem 14343 If a relation exists then ...
trclubi 14344 The Cartesian product of t...
trclubgi 14345 The union with the Cartesi...
trclub 14346 The Cartesian product of t...
trclubg 14347 The union with the Cartesi...
trclfv 14348 The transitive closure of ...
brintclab 14349 Two ways to express a bina...
brtrclfv 14350 Two ways of expressing the...
brcnvtrclfv 14351 Two ways of expressing the...
brtrclfvcnv 14352 Two ways of expressing the...
brcnvtrclfvcnv 14353 Two ways of expressing the...
trclfvss 14354 The transitive closure (as...
trclfvub 14355 The transitive closure of ...
trclfvlb 14356 The transitive closure of ...
trclfvcotr 14357 The transitive closure of ...
trclfvlb2 14358 The transitive closure of ...
trclfvlb3 14359 The transitive closure of ...
cotrtrclfv 14360 The transitive closure of ...
trclidm 14361 The transitive closure of ...
trclun 14362 Transitive closure of a un...
trclfvg 14363 The value of the transitiv...
trclfvcotrg 14364 The value of the transitiv...
reltrclfv 14365 The transitive closure of ...
dmtrclfv 14366 The domain of the transiti...
relexp0g 14369 A relation composed zero t...
relexp0 14370 A relation composed zero t...
relexp0d 14371 A relation composed zero t...
relexpsucnnr 14372 A reduction for relation e...
relexp1g 14373 A relation composed once i...
dfid5 14374 Identity relation is equal...
dfid6 14375 Identity relation expresse...
relexpsucr 14376 A reduction for relation e...
relexpsucrd 14377 A reduction for relation e...
relexp1d 14378 A relation composed once i...
relexpsucnnl 14379 A reduction for relation e...
relexpsucl 14380 A reduction for relation e...
relexpsucld 14381 A reduction for relation e...
relexpcnv 14382 Commutation of converse an...
relexpcnvd 14383 Commutation of converse an...
relexp0rel 14384 The exponentiation of a cl...
relexprelg 14385 The exponentiation of a cl...
relexprel 14386 The exponentiation of a re...
relexpreld 14387 The exponentiation of a re...
relexpnndm 14388 The domain of an exponenti...
relexpdmg 14389 The domain of an exponenti...
relexpdm 14390 The domain of an exponenti...
relexpdmd 14391 The domain of an exponenti...
relexpnnrn 14392 The range of an exponentia...
relexprng 14393 The range of an exponentia...
relexprn 14394 The range of an exponentia...
relexprnd 14395 The range of an exponentia...
relexpfld 14396 The field of an exponentia...
relexpfldd 14397 The field of an exponentia...
relexpaddnn 14398 Relation composition becom...
relexpuzrel 14399 The exponentiation of a cl...
relexpaddg 14400 Relation composition becom...
relexpaddd 14401 Relation composition becom...
dfrtrclrec2 14404 If two elements are connec...
rtrclreclem1 14405 The reflexive, transitive ...
rtrclreclem2 14406 The reflexive, transitive ...
rtrclreclem3 14407 The reflexive, transitive ...
rtrclreclem4 14408 The reflexive, transitive ...
dfrtrcl2 14409 The two definitions ` t* `...
relexpindlem 14410 Principle of transitive in...
relexpind 14411 Principle of transitive in...
rtrclind 14412 Principle of transitive in...
shftlem 14415 Two ways to write a shifte...
shftuz 14416 A shift of the upper integ...
shftfval 14417 The value of the sequence ...
shftdm 14418 Domain of a relation shift...
shftfib 14419 Value of a fiber of the re...
shftfn 14420 Functionality and domain o...
shftval 14421 Value of a sequence shifte...
shftval2 14422 Value of a sequence shifte...
shftval3 14423 Value of a sequence shifte...
shftval4 14424 Value of a sequence shifte...
shftval5 14425 Value of a shifted sequenc...
shftf 14426 Functionality of a shifted...
2shfti 14427 Composite shift operations...
shftidt2 14428 Identity law for the shift...
shftidt 14429 Identity law for the shift...
shftcan1 14430 Cancellation law for the s...
shftcan2 14431 Cancellation law for the s...
seqshft 14432 Shifting the index set of ...
sgnval 14435 Value of the signum functi...
sgn0 14436 The signum of 0 is 0. (Co...
sgnp 14437 The signum of a positive e...
sgnrrp 14438 The signum of a positive r...
sgn1 14439 The signum of 1 is 1. (Co...
sgnpnf 14440 The signum of ` +oo ` is 1...
sgnn 14441 The signum of a negative e...
sgnmnf 14442 The signum of ` -oo ` is -...
cjval 14449 The value of the conjugate...
cjth 14450 The defining property of t...
cjf 14451 Domain and codomain of the...
cjcl 14452 The conjugate of a complex...
reval 14453 The value of the real part...
imval 14454 The value of the imaginary...
imre 14455 The imaginary part of a co...
reim 14456 The real part of a complex...
recl 14457 The real part of a complex...
imcl 14458 The imaginary part of a co...
ref 14459 Domain and codomain of the...
imf 14460 Domain and codomain of the...
crre 14461 The real part of a complex...
crim 14462 The real part of a complex...
replim 14463 Reconstruct a complex numb...
remim 14464 Value of the conjugate of ...
reim0 14465 The imaginary part of a re...
reim0b 14466 A number is real iff its i...
rereb 14467 A number is real iff it eq...
mulre 14468 A product with a nonzero r...
rere 14469 A real number equals its r...
cjreb 14470 A number is real iff it eq...
recj 14471 Real part of a complex con...
reneg 14472 Real part of negative. (C...
readd 14473 Real part distributes over...
resub 14474 Real part distributes over...
remullem 14475 Lemma for ~ remul , ~ immu...
remul 14476 Real part of a product. (...
remul2 14477 Real part of a product. (...
rediv 14478 Real part of a division. ...
imcj 14479 Imaginary part of a comple...
imneg 14480 The imaginary part of a ne...
imadd 14481 Imaginary part distributes...
imsub 14482 Imaginary part distributes...
immul 14483 Imaginary part of a produc...
immul2 14484 Imaginary part of a produc...
imdiv 14485 Imaginary part of a divisi...
cjre 14486 A real number equals its c...
cjcj 14487 The conjugate of the conju...
cjadd 14488 Complex conjugate distribu...
cjmul 14489 Complex conjugate distribu...
ipcnval 14490 Standard inner product on ...
cjmulrcl 14491 A complex number times its...
cjmulval 14492 A complex number times its...
cjmulge0 14493 A complex number times its...
cjneg 14494 Complex conjugate of negat...
addcj 14495 A number plus its conjugat...
cjsub 14496 Complex conjugate distribu...
cjexp 14497 Complex conjugate of posit...
imval2 14498 The imaginary part of a nu...
re0 14499 The real part of zero. (C...
im0 14500 The imaginary part of zero...
re1 14501 The real part of one. (Co...
im1 14502 The imaginary part of one....
rei 14503 The real part of ` _i ` . ...
imi 14504 The imaginary part of ` _i...
cj0 14505 The conjugate of zero. (C...
cji 14506 The complex conjugate of t...
cjreim 14507 The conjugate of a represe...
cjreim2 14508 The conjugate of the repre...
cj11 14509 Complex conjugate is a one...
cjne0 14510 A number is nonzero iff it...
cjdiv 14511 Complex conjugate distribu...
cnrecnv 14512 The inverse to the canonic...
sqeqd 14513 A deduction for showing tw...
recli 14514 The real part of a complex...
imcli 14515 The imaginary part of a co...
cjcli 14516 Closure law for complex co...
replimi 14517 Construct a complex number...
cjcji 14518 The conjugate of the conju...
reim0bi 14519 A number is real iff its i...
rerebi 14520 A real number equals its r...
cjrebi 14521 A number is real iff it eq...
recji 14522 Real part of a complex con...
imcji 14523 Imaginary part of a comple...
cjmulrcli 14524 A complex number times its...
cjmulvali 14525 A complex number times its...
cjmulge0i 14526 A complex number times its...
renegi 14527 Real part of negative. (C...
imnegi 14528 Imaginary part of negative...
cjnegi 14529 Complex conjugate of negat...
addcji 14530 A number plus its conjugat...
readdi 14531 Real part distributes over...
imaddi 14532 Imaginary part distributes...
remuli 14533 Real part of a product. (...
immuli 14534 Imaginary part of a produc...
cjaddi 14535 Complex conjugate distribu...
cjmuli 14536 Complex conjugate distribu...
ipcni 14537 Standard inner product on ...
cjdivi 14538 Complex conjugate distribu...
crrei 14539 The real part of a complex...
crimi 14540 The imaginary part of a co...
recld 14541 The real part of a complex...
imcld 14542 The imaginary part of a co...
cjcld 14543 Closure law for complex co...
replimd 14544 Construct a complex number...
remimd 14545 Value of the conjugate of ...
cjcjd 14546 The conjugate of the conju...
reim0bd 14547 A number is real iff its i...
rerebd 14548 A real number equals its r...
cjrebd 14549 A number is real iff it eq...
cjne0d 14550 A number is nonzero iff it...
recjd 14551 Real part of a complex con...
imcjd 14552 Imaginary part of a comple...
cjmulrcld 14553 A complex number times its...
cjmulvald 14554 A complex number times its...
cjmulge0d 14555 A complex number times its...
renegd 14556 Real part of negative. (C...
imnegd 14557 Imaginary part of negative...
cjnegd 14558 Complex conjugate of negat...
addcjd 14559 A number plus its conjugat...
cjexpd 14560 Complex conjugate of posit...
readdd 14561 Real part distributes over...
imaddd 14562 Imaginary part distributes...
resubd 14563 Real part distributes over...
imsubd 14564 Imaginary part distributes...
remuld 14565 Real part of a product. (...
immuld 14566 Imaginary part of a produc...
cjaddd 14567 Complex conjugate distribu...
cjmuld 14568 Complex conjugate distribu...
ipcnd 14569 Standard inner product on ...
cjdivd 14570 Complex conjugate distribu...
rered 14571 A real number equals its r...
reim0d 14572 The imaginary part of a re...
cjred 14573 A real number equals its c...
remul2d 14574 Real part of a product. (...
immul2d 14575 Imaginary part of a produc...
redivd 14576 Real part of a division. ...
imdivd 14577 Imaginary part of a divisi...
crred 14578 The real part of a complex...
crimd 14579 The imaginary part of a co...
sqrtval 14584 Value of square root funct...
absval 14585 The absolute value (modulu...
rennim 14586 A real number does not lie...
cnpart 14587 The specification of restr...
sqr0lem 14588 Square root of zero. (Con...
sqrt0 14589 Square root of zero. (Con...
sqrlem1 14590 Lemma for ~ 01sqrex . (Co...
sqrlem2 14591 Lemma for ~ 01sqrex . (Co...
sqrlem3 14592 Lemma for ~ 01sqrex . (Co...
sqrlem4 14593 Lemma for ~ 01sqrex . (Co...
sqrlem5 14594 Lemma for ~ 01sqrex . (Co...
sqrlem6 14595 Lemma for ~ 01sqrex . (Co...
sqrlem7 14596 Lemma for ~ 01sqrex . (Co...
01sqrex 14597 Existence of a square root...
resqrex 14598 Existence of a square root...
sqrmo 14599 Uniqueness for the square ...
resqreu 14600 Existence and uniqueness f...
resqrtcl 14601 Closure of the square root...
resqrtthlem 14602 Lemma for ~ resqrtth . (C...
resqrtth 14603 Square root theorem over t...
remsqsqrt 14604 Square of square root. (C...
sqrtge0 14605 The square root function i...
sqrtgt0 14606 The square root function i...
sqrtmul 14607 Square root distributes ov...
sqrtle 14608 Square root is monotonic. ...
sqrtlt 14609 Square root is strictly mo...
sqrt11 14610 The square root function i...
sqrt00 14611 A square root is zero iff ...
rpsqrtcl 14612 The square root of a posit...
sqrtdiv 14613 Square root distributes ov...
sqrtneglem 14614 The square root of a negat...
sqrtneg 14615 The square root of a negat...
sqrtsq2 14616 Relationship between squar...
sqrtsq 14617 Square root of square. (C...
sqrtmsq 14618 Square root of square. (C...
sqrt1 14619 The square root of 1 is 1....
sqrt4 14620 The square root of 4 is 2....
sqrt9 14621 The square root of 9 is 3....
sqrt2gt1lt2 14622 The square root of 2 is bo...
sqrtm1 14623 The imaginary unit is the ...
nn0sqeq1 14624 A natural number with squa...
absneg 14625 Absolute value of the oppo...
abscl 14626 Real closure of absolute v...
abscj 14627 The absolute value of a nu...
absvalsq 14628 Square of value of absolut...
absvalsq2 14629 Square of value of absolut...
sqabsadd 14630 Square of absolute value o...
sqabssub 14631 Square of absolute value o...
absval2 14632 Value of absolute value fu...
abs0 14633 The absolute value of 0. ...
absi 14634 The absolute value of the ...
absge0 14635 Absolute value is nonnegat...
absrpcl 14636 The absolute value of a no...
abs00 14637 The absolute value of a nu...
abs00ad 14638 A complex number is zero i...
abs00bd 14639 If a complex number is zer...
absreimsq 14640 Square of the absolute val...
absreim 14641 Absolute value of a number...
absmul 14642 Absolute value distributes...
absdiv 14643 Absolute value distributes...
absid 14644 A nonnegative number is it...
abs1 14645 The absolute value of one ...
absnid 14646 A negative number is the n...
leabs 14647 A real number is less than...
absor 14648 The absolute value of a re...
absre 14649 Absolute value of a real n...
absresq 14650 Square of the absolute val...
absmod0 14651 ` A ` is divisible by ` B ...
absexp 14652 Absolute value of positive...
absexpz 14653 Absolute value of integer ...
abssq 14654 Square can be moved in and...
sqabs 14655 The squares of two reals a...
absrele 14656 The absolute value of a co...
absimle 14657 The absolute value of a co...
max0add 14658 The sum of the positive an...
absz 14659 A real number is an intege...
nn0abscl 14660 The absolute value of an i...
zabscl 14661 The absolute value of an i...
abslt 14662 Absolute value and 'less t...
absle 14663 Absolute value and 'less t...
abssubne0 14664 If the absolute value of a...
absdiflt 14665 The absolute value of a di...
absdifle 14666 The absolute value of a di...
elicc4abs 14667 Membership in a symmetric ...
lenegsq 14668 Comparison to a nonnegativ...
releabs 14669 The real part of a number ...
recval 14670 Reciprocal expressed with ...
absidm 14671 The absolute value functio...
absgt0 14672 The absolute value of a no...
nnabscl 14673 The absolute value of a no...
abssub 14674 Swapping order of subtract...
abssubge0 14675 Absolute value of a nonneg...
abssuble0 14676 Absolute value of a nonpos...
absmax 14677 The maximum of two numbers...
abstri 14678 Triangle inequality for ab...
abs3dif 14679 Absolute value of differen...
abs2dif 14680 Difference of absolute val...
abs2dif2 14681 Difference of absolute val...
abs2difabs 14682 Absolute value of differen...
abs1m 14683 For any complex number, th...
recan 14684 Cancellation law involving...
absf 14685 Mapping domain and codomai...
abs3lem 14686 Lemma involving absolute v...
abslem2 14687 Lemma involving absolute v...
rddif 14688 The difference between a r...
absrdbnd 14689 Bound on the absolute valu...
fzomaxdiflem 14690 Lemma for ~ fzomaxdif . (...
fzomaxdif 14691 A bound on the separation ...
uzin2 14692 The upper integers are clo...
rexanuz 14693 Combine two different uppe...
rexanre 14694 Combine two different uppe...
rexfiuz 14695 Combine finitely many diff...
rexuz3 14696 Restrict the base of the u...
rexanuz2 14697 Combine two different uppe...
r19.29uz 14698 A version of ~ 19.29 for u...
r19.2uz 14699 A version of ~ r19.2z for ...
rexuzre 14700 Convert an upper real quan...
rexico 14701 Restrict the base of an up...
cau3lem 14702 Lemma for ~ cau3 . (Contr...
cau3 14703 Convert between three-quan...
cau4 14704 Change the base of a Cauch...
caubnd2 14705 A Cauchy sequence of compl...
caubnd 14706 A Cauchy sequence of compl...
sqreulem 14707 Lemma for ~ sqreu : write ...
sqreu 14708 Existence and uniqueness f...
sqrtcl 14709 Closure of the square root...
sqrtthlem 14710 Lemma for ~ sqrtth . (Con...
sqrtf 14711 Mapping domain and codomai...
sqrtth 14712 Square root theorem over t...
sqrtrege0 14713 The square root function m...
eqsqrtor 14714 Solve an equation containi...
eqsqrtd 14715 A deduction for showing th...
eqsqrt2d 14716 A deduction for showing th...
amgm2 14717 Arithmetic-geometric mean ...
sqrtthi 14718 Square root theorem. Theo...
sqrtcli 14719 The square root of a nonne...
sqrtgt0i 14720 The square root of a posit...
sqrtmsqi 14721 Square root of square. (C...
sqrtsqi 14722 Square root of square. (C...
sqsqrti 14723 Square of square root. (C...
sqrtge0i 14724 The square root of a nonne...
absidi 14725 A nonnegative number is it...
absnidi 14726 A negative number is the n...
leabsi 14727 A real number is less than...
absori 14728 The absolute value of a re...
absrei 14729 Absolute value of a real n...
sqrtpclii 14730 The square root of a posit...
sqrtgt0ii 14731 The square root of a posit...
sqrt11i 14732 The square root function i...
sqrtmuli 14733 Square root distributes ov...
sqrtmulii 14734 Square root distributes ov...
sqrtmsq2i 14735 Relationship between squar...
sqrtlei 14736 Square root is monotonic. ...
sqrtlti 14737 Square root is strictly mo...
abslti 14738 Absolute value and 'less t...
abslei 14739 Absolute value and 'less t...
cnsqrt00 14740 A square root of a complex...
absvalsqi 14741 Square of value of absolut...
absvalsq2i 14742 Square of value of absolut...
abscli 14743 Real closure of absolute v...
absge0i 14744 Absolute value is nonnegat...
absval2i 14745 Value of absolute value fu...
abs00i 14746 The absolute value of a nu...
absgt0i 14747 The absolute value of a no...
absnegi 14748 Absolute value of negative...
abscji 14749 The absolute value of a nu...
releabsi 14750 The real part of a number ...
abssubi 14751 Swapping order of subtract...
absmuli 14752 Absolute value distributes...
sqabsaddi 14753 Square of absolute value o...
sqabssubi 14754 Square of absolute value o...
absdivzi 14755 Absolute value distributes...
abstrii 14756 Triangle inequality for ab...
abs3difi 14757 Absolute value of differen...
abs3lemi 14758 Lemma involving absolute v...
rpsqrtcld 14759 The square root of a posit...
sqrtgt0d 14760 The square root of a posit...
absnidd 14761 A negative number is the n...
leabsd 14762 A real number is less than...
absord 14763 The absolute value of a re...
absred 14764 Absolute value of a real n...
resqrtcld 14765 The square root of a nonne...
sqrtmsqd 14766 Square root of square. (C...
sqrtsqd 14767 Square root of square. (C...
sqrtge0d 14768 The square root of a nonne...
sqrtnegd 14769 The square root of a negat...
absidd 14770 A nonnegative number is it...
sqrtdivd 14771 Square root distributes ov...
sqrtmuld 14772 Square root distributes ov...
sqrtsq2d 14773 Relationship between squar...
sqrtled 14774 Square root is monotonic. ...
sqrtltd 14775 Square root is strictly mo...
sqr11d 14776 The square root function i...
absltd 14777 Absolute value and 'less t...
absled 14778 Absolute value and 'less t...
abssubge0d 14779 Absolute value of a nonneg...
abssuble0d 14780 Absolute value of a nonpos...
absdifltd 14781 The absolute value of a di...
absdifled 14782 The absolute value of a di...
icodiamlt 14783 Two elements in a half-ope...
abscld 14784 Real closure of absolute v...
sqrtcld 14785 Closure of the square root...
sqrtrege0d 14786 The real part of the squar...
sqsqrtd 14787 Square root theorem. Theo...
msqsqrtd 14788 Square root theorem. Theo...
sqr00d 14789 A square root is zero iff ...
absvalsqd 14790 Square of value of absolut...
absvalsq2d 14791 Square of value of absolut...
absge0d 14792 Absolute value is nonnegat...
absval2d 14793 Value of absolute value fu...
abs00d 14794 The absolute value of a nu...
absne0d 14795 The absolute value of a nu...
absrpcld 14796 The absolute value of a no...
absnegd 14797 Absolute value of negative...
abscjd 14798 The absolute value of a nu...
releabsd 14799 The real part of a number ...
absexpd 14800 Absolute value of positive...
abssubd 14801 Swapping order of subtract...
absmuld 14802 Absolute value distributes...
absdivd 14803 Absolute value distributes...
abstrid 14804 Triangle inequality for ab...
abs2difd 14805 Difference of absolute val...
abs2dif2d 14806 Difference of absolute val...
abs2difabsd 14807 Absolute value of differen...
abs3difd 14808 Absolute value of differen...
abs3lemd 14809 Lemma involving absolute v...
reusq0 14810 A complex number is the sq...
bhmafibid1cn 14811 The Brahmagupta-Fibonacci ...
bhmafibid2cn 14812 The Brahmagupta-Fibonacci ...
bhmafibid1 14813 The Brahmagupta-Fibonacci ...
bhmafibid2 14814 The Brahmagupta-Fibonacci ...
limsupgord 14817 Ordering property of the s...
limsupcl 14818 Closure of the superior li...
limsupval 14819 The superior limit of an i...
limsupgf 14820 Closure of the superior li...
limsupgval 14821 Value of the superior limi...
limsupgle 14822 The defining property of t...
limsuple 14823 The defining property of t...
limsuplt 14824 The defining property of t...
limsupval2 14825 The superior limit, relati...
limsupgre 14826 If a sequence of real numb...
limsupbnd1 14827 If a sequence is eventuall...
limsupbnd2 14828 If a sequence is eventuall...
climrel 14837 The limit relation is a re...
rlimrel 14838 The limit relation is a re...
clim 14839 Express the predicate: Th...
rlim 14840 Express the predicate: Th...
rlim2 14841 Rewrite ~ rlim for a mappi...
rlim2lt 14842 Use strictly less-than in ...
rlim3 14843 Restrict the range of the ...
climcl 14844 Closure of the limit of a ...
rlimpm 14845 Closure of a function with...
rlimf 14846 Closure of a function with...
rlimss 14847 Domain closure of a functi...
rlimcl 14848 Closure of the limit of a ...
clim2 14849 Express the predicate: Th...
clim2c 14850 Express the predicate ` F ...
clim0 14851 Express the predicate ` F ...
clim0c 14852 Express the predicate ` F ...
rlim0 14853 Express the predicate ` B ...
rlim0lt 14854 Use strictly less-than in ...
climi 14855 Convergence of a sequence ...
climi2 14856 Convergence of a sequence ...
climi0 14857 Convergence of a sequence ...
rlimi 14858 Convergence at infinity of...
rlimi2 14859 Convergence at infinity of...
ello1 14860 Elementhood in the set of ...
ello12 14861 Elementhood in the set of ...
ello12r 14862 Sufficient condition for e...
lo1f 14863 An eventually upper bounde...
lo1dm 14864 An eventually upper bounde...
lo1bdd 14865 The defining property of a...
ello1mpt 14866 Elementhood in the set of ...
ello1mpt2 14867 Elementhood in the set of ...
ello1d 14868 Sufficient condition for e...
lo1bdd2 14869 If an eventually bounded f...
lo1bddrp 14870 Refine ~ o1bdd2 to give a ...
elo1 14871 Elementhood in the set of ...
elo12 14872 Elementhood in the set of ...
elo12r 14873 Sufficient condition for e...
o1f 14874 An eventually bounded func...
o1dm 14875 An eventually bounded func...
o1bdd 14876 The defining property of a...
lo1o1 14877 A function is eventually b...
lo1o12 14878 A function is eventually b...
elo1mpt 14879 Elementhood in the set of ...
elo1mpt2 14880 Elementhood in the set of ...
elo1d 14881 Sufficient condition for e...
o1lo1 14882 A real function is eventua...
o1lo12 14883 A lower bounded real funct...
o1lo1d 14884 A real eventually bounded ...
icco1 14885 Derive eventual boundednes...
o1bdd2 14886 If an eventually bounded f...
o1bddrp 14887 Refine ~ o1bdd2 to give a ...
climconst 14888 An (eventually) constant s...
rlimconst 14889 A constant sequence conver...
rlimclim1 14890 Forward direction of ~ rli...
rlimclim 14891 A sequence on an upper int...
climrlim2 14892 Produce a real limit from ...
climconst2 14893 A constant sequence conver...
climz 14894 The zero sequence converge...
rlimuni 14895 A real function whose doma...
rlimdm 14896 Two ways to express that a...
climuni 14897 An infinite sequence of co...
fclim 14898 The limit relation is func...
climdm 14899 Two ways to express that a...
climeu 14900 An infinite sequence of co...
climreu 14901 An infinite sequence of co...
climmo 14902 An infinite sequence of co...
rlimres 14903 The restriction of a funct...
lo1res 14904 The restriction of an even...
o1res 14905 The restriction of an even...
rlimres2 14906 The restriction of a funct...
lo1res2 14907 The restriction of a funct...
o1res2 14908 The restriction of a funct...
lo1resb 14909 The restriction of a funct...
rlimresb 14910 The restriction of a funct...
o1resb 14911 The restriction of a funct...
climeq 14912 Two functions that are eve...
lo1eq 14913 Two functions that are eve...
rlimeq 14914 Two functions that are eve...
o1eq 14915 Two functions that are eve...
climmpt 14916 Exhibit a function ` G ` w...
2clim 14917 If two sequences converge ...
climmpt2 14918 Relate an integer limit on...
climshftlem 14919 A shifted function converg...
climres 14920 A function restricted to u...
climshft 14921 A shifted function converg...
serclim0 14922 The zero series converges ...
rlimcld2 14923 If ` D ` is a closed set i...
rlimrege0 14924 The limit of a sequence of...
rlimrecl 14925 The limit of a real sequen...
rlimge0 14926 The limit of a sequence of...
climshft2 14927 A shifted function converg...
climrecl 14928 The limit of a convergent ...
climge0 14929 A nonnegative sequence con...
climabs0 14930 Convergence to zero of the...
o1co 14931 Sufficient condition for t...
o1compt 14932 Sufficient condition for t...
rlimcn1 14933 Image of a limit under a c...
rlimcn1b 14934 Image of a limit under a c...
rlimcn2 14935 Image of a limit under a c...
climcn1 14936 Image of a limit under a c...
climcn2 14937 Image of a limit under a c...
addcn2 14938 Complex number addition is...
subcn2 14939 Complex number subtraction...
mulcn2 14940 Complex number multiplicat...
reccn2 14941 The reciprocal function is...
cn1lem 14942 A sufficient condition for...
abscn2 14943 The absolute value functio...
cjcn2 14944 The complex conjugate func...
recn2 14945 The real part function is ...
imcn2 14946 The imaginary part functio...
climcn1lem 14947 The limit of a continuous ...
climabs 14948 Limit of the absolute valu...
climcj 14949 Limit of the complex conju...
climre 14950 Limit of the real part of ...
climim 14951 Limit of the imaginary par...
rlimmptrcl 14952 Reverse closure for a real...
rlimabs 14953 Limit of the absolute valu...
rlimcj 14954 Limit of the complex conju...
rlimre 14955 Limit of the real part of ...
rlimim 14956 Limit of the imaginary par...
o1of2 14957 Show that a binary operati...
o1add 14958 The sum of two eventually ...
o1mul 14959 The product of two eventua...
o1sub 14960 The difference of two even...
rlimo1 14961 Any function with a finite...
rlimdmo1 14962 A convergent function is e...
o1rlimmul 14963 The product of an eventual...
o1const 14964 A constant function is eve...
lo1const 14965 A constant function is eve...
lo1mptrcl 14966 Reverse closure for an eve...
o1mptrcl 14967 Reverse closure for an eve...
o1add2 14968 The sum of two eventually ...
o1mul2 14969 The product of two eventua...
o1sub2 14970 The product of two eventua...
lo1add 14971 The sum of two eventually ...
lo1mul 14972 The product of an eventual...
lo1mul2 14973 The product of an eventual...
o1dif 14974 If the difference of two f...
lo1sub 14975 The difference of an event...
climadd 14976 Limit of the sum of two co...
climmul 14977 Limit of the product of tw...
climsub 14978 Limit of the difference of...
climaddc1 14979 Limit of a constant ` C ` ...
climaddc2 14980 Limit of a constant ` C ` ...
climmulc2 14981 Limit of a sequence multip...
climsubc1 14982 Limit of a constant ` C ` ...
climsubc2 14983 Limit of a constant ` C ` ...
climle 14984 Comparison of the limits o...
climsqz 14985 Convergence of a sequence ...
climsqz2 14986 Convergence of a sequence ...
rlimadd 14987 Limit of the sum of two co...
rlimsub 14988 Limit of the difference of...
rlimmul 14989 Limit of the product of tw...
rlimdiv 14990 Limit of the quotient of t...
rlimneg 14991 Limit of the negative of a...
rlimle 14992 Comparison of the limits o...
rlimsqzlem 14993 Lemma for ~ rlimsqz and ~ ...
rlimsqz 14994 Convergence of a sequence ...
rlimsqz2 14995 Convergence of a sequence ...
lo1le 14996 Transfer eventual upper bo...
o1le 14997 Transfer eventual boundedn...
rlimno1 14998 A function whose inverse c...
clim2ser 14999 The limit of an infinite s...
clim2ser2 15000 The limit of an infinite s...
iserex 15001 An infinite series converg...
isermulc2 15002 Multiplication of an infin...
climlec2 15003 Comparison of a constant t...
iserle 15004 Comparison of the limits o...
iserge0 15005 The limit of an infinite s...
climub 15006 The limit of a monotonic s...
climserle 15007 The partial sums of a conv...
isershft 15008 Index shift of the limit o...
isercolllem1 15009 Lemma for ~ isercoll . (C...
isercolllem2 15010 Lemma for ~ isercoll . (C...
isercolllem3 15011 Lemma for ~ isercoll . (C...
isercoll 15012 Rearrange an infinite seri...
isercoll2 15013 Generalize ~ isercoll so t...
climsup 15014 A bounded monotonic sequen...
climcau 15015 A converging sequence of c...
climbdd 15016 A converging sequence of c...
caucvgrlem 15017 Lemma for ~ caurcvgr . (C...
caurcvgr 15018 A Cauchy sequence of real ...
caucvgrlem2 15019 Lemma for ~ caucvgr . (Co...
caucvgr 15020 A Cauchy sequence of compl...
caurcvg 15021 A Cauchy sequence of real ...
caurcvg2 15022 A Cauchy sequence of real ...
caucvg 15023 A Cauchy sequence of compl...
caucvgb 15024 A function is convergent i...
serf0 15025 If an infinite series conv...
iseraltlem1 15026 Lemma for ~ iseralt . A d...
iseraltlem2 15027 Lemma for ~ iseralt . The...
iseraltlem3 15028 Lemma for ~ iseralt . Fro...
iseralt 15029 The alternating series tes...
sumex 15032 A sum is a set. (Contribu...
sumeq1 15033 Equality theorem for a sum...
nfsum1 15034 Bound-variable hypothesis ...
nfsumw 15035 Version of ~ nfsum with a ...
nfsum 15036 Bound-variable hypothesis ...
sumeq2w 15037 Equality theorem for sum, ...
sumeq2ii 15038 Equality theorem for sum, ...
sumeq2 15039 Equality theorem for sum. ...
cbvsum 15040 Change bound variable in a...
cbvsumv 15041 Change bound variable in a...
cbvsumi 15042 Change bound variable in a...
sumeq1i 15043 Equality inference for sum...
sumeq2i 15044 Equality inference for sum...
sumeq12i 15045 Equality inference for sum...
sumeq1d 15046 Equality deduction for sum...
sumeq2d 15047 Equality deduction for sum...
sumeq2dv 15048 Equality deduction for sum...
sumeq2sdv 15049 Equality deduction for sum...
2sumeq2dv 15050 Equality deduction for dou...
sumeq12dv 15051 Equality deduction for sum...
sumeq12rdv 15052 Equality deduction for sum...
sum2id 15053 The second class argument ...
sumfc 15054 A lemma to facilitate conv...
fz1f1o 15055 A lemma for working with f...
sumrblem 15056 Lemma for ~ sumrb . (Cont...
fsumcvg 15057 The sequence of partial su...
sumrb 15058 Rebase the starting point ...
summolem3 15059 Lemma for ~ summo . (Cont...
summolem2a 15060 Lemma for ~ summo . (Cont...
summolem2 15061 Lemma for ~ summo . (Cont...
summo 15062 A sum has at most one limi...
zsum 15063 Series sum with index set ...
isum 15064 Series sum with an upper i...
fsum 15065 The value of a sum over a ...
sum0 15066 Any sum over the empty set...
sumz 15067 Any sum of zero over a sum...
fsumf1o 15068 Re-index a finite sum usin...
sumss 15069 Change the index set to a ...
fsumss 15070 Change the index set to a ...
sumss2 15071 Change the index set of a ...
fsumcvg2 15072 The sequence of partial su...
fsumsers 15073 Special case of series sum...
fsumcvg3 15074 A finite sum is convergent...
fsumser 15075 A finite sum expressed in ...
fsumcl2lem 15076 - Lemma for finite sum clo...
fsumcllem 15077 - Lemma for finite sum clo...
fsumcl 15078 Closure of a finite sum of...
fsumrecl 15079 Closure of a finite sum of...
fsumzcl 15080 Closure of a finite sum of...
fsumnn0cl 15081 Closure of a finite sum of...
fsumrpcl 15082 Closure of a finite sum of...
fsumzcl2 15083 A finite sum with integer ...
fsumadd 15084 The sum of two finite sums...
fsumsplit 15085 Split a sum into two parts...
fsumsplitf 15086 Split a sum into two parts...
sumsnf 15087 A sum of a singleton is th...
fsumsplitsn 15088 Separate out a term in a f...
sumsn 15089 A sum of a singleton is th...
fsum1 15090 The finite sum of ` A ( k ...
sumpr 15091 A sum over a pair is the s...
sumtp 15092 A sum over a triple is the...
sumsns 15093 A sum of a singleton is th...
fsumm1 15094 Separate out the last term...
fzosump1 15095 Separate out the last term...
fsum1p 15096 Separate out the first ter...
fsummsnunz 15097 A finite sum all of whose ...
fsumsplitsnun 15098 Separate out a term in a f...
fsump1 15099 The addition of the next t...
isumclim 15100 An infinite sum equals the...
isumclim2 15101 A converging series conver...
isumclim3 15102 The sequence of partial fi...
sumnul 15103 The sum of a non-convergen...
isumcl 15104 The sum of a converging in...
isummulc2 15105 An infinite sum multiplied...
isummulc1 15106 An infinite sum multiplied...
isumdivc 15107 An infinite sum divided by...
isumrecl 15108 The sum of a converging in...
isumge0 15109 An infinite sum of nonnega...
isumadd 15110 Addition of infinite sums....
sumsplit 15111 Split a sum into two parts...
fsump1i 15112 Optimized version of ~ fsu...
fsum2dlem 15113 Lemma for ~ fsum2d - induc...
fsum2d 15114 Write a double sum as a su...
fsumxp 15115 Combine two sums into a si...
fsumcnv 15116 Transform a region of summ...
fsumcom2 15117 Interchange order of summa...
fsumcom 15118 Interchange order of summa...
fsum0diaglem 15119 Lemma for ~ fsum0diag . (...
fsum0diag 15120 Two ways to express "the s...
mptfzshft 15121 1-1 onto function in maps-...
fsumrev 15122 Reversal of a finite sum. ...
fsumshft 15123 Index shift of a finite su...
fsumshftm 15124 Negative index shift of a ...
fsumrev2 15125 Reversal of a finite sum. ...
fsum0diag2 15126 Two ways to express "the s...
fsummulc2 15127 A finite sum multiplied by...
fsummulc1 15128 A finite sum multiplied by...
fsumdivc 15129 A finite sum divided by a ...
fsumneg 15130 Negation of a finite sum. ...
fsumsub 15131 Split a finite sum over a ...
fsum2mul 15132 Separate the nested sum of...
fsumconst 15133 The sum of constant terms ...
fsumdifsnconst 15134 The sum of constant terms ...
modfsummodslem1 15135 Lemma 1 for ~ modfsummods ...
modfsummods 15136 Induction step for ~ modfs...
modfsummod 15137 A finite sum modulo a posi...
fsumge0 15138 If all of the terms of a f...
fsumless 15139 A shorter sum of nonnegati...
fsumge1 15140 A sum of nonnegative numbe...
fsum00 15141 A sum of nonnegative numbe...
fsumle 15142 If all of the terms of fin...
fsumlt 15143 If every term in one finit...
fsumabs 15144 Generalized triangle inequ...
telfsumo 15145 Sum of a telescoping serie...
telfsumo2 15146 Sum of a telescoping serie...
telfsum 15147 Sum of a telescoping serie...
telfsum2 15148 Sum of a telescoping serie...
fsumparts 15149 Summation by parts. (Cont...
fsumrelem 15150 Lemma for ~ fsumre , ~ fsu...
fsumre 15151 The real part of a sum. (...
fsumim 15152 The imaginary part of a su...
fsumcj 15153 The complex conjugate of a...
fsumrlim 15154 Limit of a finite sum of c...
fsumo1 15155 The finite sum of eventual...
o1fsum 15156 If ` A ( k ) ` is O(1), th...
seqabs 15157 Generalized triangle inequ...
iserabs 15158 Generalized triangle inequ...
cvgcmp 15159 A comparison test for conv...
cvgcmpub 15160 An upper bound for the lim...
cvgcmpce 15161 A comparison test for conv...
abscvgcvg 15162 An absolutely convergent s...
climfsum 15163 Limit of a finite sum of c...
fsumiun 15164 Sum over a disjoint indexe...
hashiun 15165 The cardinality of a disjo...
hash2iun 15166 The cardinality of a neste...
hash2iun1dif1 15167 The cardinality of a neste...
hashrabrex 15168 The number of elements in ...
hashuni 15169 The cardinality of a disjo...
qshash 15170 The cardinality of a set w...
ackbijnn 15171 Translate the Ackermann bi...
binomlem 15172 Lemma for ~ binom (binomia...
binom 15173 The binomial theorem: ` ( ...
binom1p 15174 Special case of the binomi...
binom11 15175 Special case of the binomi...
binom1dif 15176 A summation for the differ...
bcxmaslem1 15177 Lemma for ~ bcxmas . (Con...
bcxmas 15178 Parallel summation (Christ...
incexclem 15179 Lemma for ~ incexc . (Con...
incexc 15180 The inclusion/exclusion pr...
incexc2 15181 The inclusion/exclusion pr...
isumshft 15182 Index shift of an infinite...
isumsplit 15183 Split off the first ` N ` ...
isum1p 15184 The infinite sum of a conv...
isumnn0nn 15185 Sum from 0 to infinity in ...
isumrpcl 15186 The infinite sum of positi...
isumle 15187 Comparison of two infinite...
isumless 15188 A finite sum of nonnegativ...
isumsup2 15189 An infinite sum of nonnega...
isumsup 15190 An infinite sum of nonnega...
isumltss 15191 A partial sum of a series ...
climcndslem1 15192 Lemma for ~ climcnds : bou...
climcndslem2 15193 Lemma for ~ climcnds : bou...
climcnds 15194 The Cauchy condensation te...
divrcnv 15195 The sequence of reciprocal...
divcnv 15196 The sequence of reciprocal...
flo1 15197 The floor function satisfi...
divcnvshft 15198 Limit of a ratio function....
supcvg 15199 Extract a sequence ` f ` i...
infcvgaux1i 15200 Auxiliary theorem for appl...
infcvgaux2i 15201 Auxiliary theorem for appl...
harmonic 15202 The harmonic series ` H ` ...
arisum 15203 Arithmetic series sum of t...
arisum2 15204 Arithmetic series sum of t...
trireciplem 15205 Lemma for ~ trirecip . Sh...
trirecip 15206 The sum of the reciprocals...
expcnv 15207 A sequence of powers of a ...
explecnv 15208 A sequence of terms conver...
geoserg 15209 The value of the finite ge...
geoser 15210 The value of the finite ge...
pwdif 15211 The difference of two numb...
pwm1geoser 15212 The n-th power of a number...
pwm1geoserOLD 15213 Obsolete version of ~ pwm1...
geolim 15214 The partial sums in the in...
geolim2 15215 The partial sums in the ge...
georeclim 15216 The limit of a geometric s...
geo2sum 15217 The value of the finite ge...
geo2sum2 15218 The value of the finite ge...
geo2lim 15219 The value of the infinite ...
geomulcvg 15220 The geometric series conve...
geoisum 15221 The infinite sum of ` 1 + ...
geoisumr 15222 The infinite sum of recipr...
geoisum1 15223 The infinite sum of ` A ^ ...
geoisum1c 15224 The infinite sum of ` A x....
0.999... 15225 The recurring decimal 0.99...
geoihalfsum 15226 Prove that the infinite ge...
cvgrat 15227 Ratio test for convergence...
mertenslem1 15228 Lemma for ~ mertens . (Co...
mertenslem2 15229 Lemma for ~ mertens . (Co...
mertens 15230 Mertens' theorem. If ` A ...
prodf 15231 An infinite product of com...
clim2prod 15232 The limit of an infinite p...
clim2div 15233 The limit of an infinite p...
prodfmul 15234 The product of two infinit...
prodf1 15235 The value of the partial p...
prodf1f 15236 A one-valued infinite prod...
prodfclim1 15237 The constant one product c...
prodfn0 15238 No term of a nonzero infin...
prodfrec 15239 The reciprocal of an infin...
prodfdiv 15240 The quotient of two infini...
ntrivcvg 15241 A non-trivially converging...
ntrivcvgn0 15242 A product that converges t...
ntrivcvgfvn0 15243 Any value of a product seq...
ntrivcvgtail 15244 A tail of a non-trivially ...
ntrivcvgmullem 15245 Lemma for ~ ntrivcvgmul . ...
ntrivcvgmul 15246 The product of two non-tri...
prodex 15249 A product is a set. (Cont...
prodeq1f 15250 Equality theorem for a pro...
prodeq1 15251 Equality theorem for a pro...
nfcprod1 15252 Bound-variable hypothesis ...
nfcprod 15253 Bound-variable hypothesis ...
prodeq2w 15254 Equality theorem for produ...
prodeq2ii 15255 Equality theorem for produ...
prodeq2 15256 Equality theorem for produ...
cbvprod 15257 Change bound variable in a...
cbvprodv 15258 Change bound variable in a...
cbvprodi 15259 Change bound variable in a...
prodeq1i 15260 Equality inference for pro...
prodeq2i 15261 Equality inference for pro...
prodeq12i 15262 Equality inference for pro...
prodeq1d 15263 Equality deduction for pro...
prodeq2d 15264 Equality deduction for pro...
prodeq2dv 15265 Equality deduction for pro...
prodeq2sdv 15266 Equality deduction for pro...
2cprodeq2dv 15267 Equality deduction for dou...
prodeq12dv 15268 Equality deduction for pro...
prodeq12rdv 15269 Equality deduction for pro...
prod2id 15270 The second class argument ...
prodrblem 15271 Lemma for ~ prodrb . (Con...
fprodcvg 15272 The sequence of partial pr...
prodrblem2 15273 Lemma for ~ prodrb . (Con...
prodrb 15274 Rebase the starting point ...
prodmolem3 15275 Lemma for ~ prodmo . (Con...
prodmolem2a 15276 Lemma for ~ prodmo . (Con...
prodmolem2 15277 Lemma for ~ prodmo . (Con...
prodmo 15278 A product has at most one ...
zprod 15279 Series product with index ...
iprod 15280 Series product with an upp...
zprodn0 15281 Nonzero series product wit...
iprodn0 15282 Nonzero series product wit...
fprod 15283 The value of a product ove...
fprodntriv 15284 A non-triviality lemma for...
prod0 15285 A product over the empty s...
prod1 15286 Any product of one over a ...
prodfc 15287 A lemma to facilitate conv...
fprodf1o 15288 Re-index a finite product ...
prodss 15289 Change the index set to a ...
fprodss 15290 Change the index set to a ...
fprodser 15291 A finite product expressed...
fprodcl2lem 15292 Finite product closure lem...
fprodcllem 15293 Finite product closure lem...
fprodcl 15294 Closure of a finite produc...
fprodrecl 15295 Closure of a finite produc...
fprodzcl 15296 Closure of a finite produc...
fprodnncl 15297 Closure of a finite produc...
fprodrpcl 15298 Closure of a finite produc...
fprodnn0cl 15299 Closure of a finite produc...
fprodcllemf 15300 Finite product closure lem...
fprodreclf 15301 Closure of a finite produc...
fprodmul 15302 The product of two finite ...
fproddiv 15303 The quotient of two finite...
prodsn 15304 A product of a singleton i...
fprod1 15305 A finite product of only o...
prodsnf 15306 A product of a singleton i...
climprod1 15307 The limit of a product ove...
fprodsplit 15308 Split a finite product int...
fprodm1 15309 Separate out the last term...
fprod1p 15310 Separate out the first ter...
fprodp1 15311 Multiply in the last term ...
fprodm1s 15312 Separate out the last term...
fprodp1s 15313 Multiply in the last term ...
prodsns 15314 A product of the singleton...
fprodfac 15315 Factorial using product no...
fprodabs 15316 The absolute value of a fi...
fprodeq0 15317 Anything finite product co...
fprodshft 15318 Shift the index of a finit...
fprodrev 15319 Reversal of a finite produ...
fprodconst 15320 The product of constant te...
fprodn0 15321 A finite product of nonzer...
fprod2dlem 15322 Lemma for ~ fprod2d - indu...
fprod2d 15323 Write a double product as ...
fprodxp 15324 Combine two products into ...
fprodcnv 15325 Transform a product region...
fprodcom2 15326 Interchange order of multi...
fprodcom 15327 Interchange product order....
fprod0diag 15328 Two ways to express "the p...
fproddivf 15329 The quotient of two finite...
fprodsplitf 15330 Split a finite product int...
fprodsplitsn 15331 Separate out a term in a f...
fprodsplit1f 15332 Separate out a term in a f...
fprodn0f 15333 A finite product of nonzer...
fprodclf 15334 Closure of a finite produc...
fprodge0 15335 If all the terms of a fini...
fprodeq0g 15336 Any finite product contain...
fprodge1 15337 If all of the terms of a f...
fprodle 15338 If all the terms of two fi...
fprodmodd 15339 If all factors of two fini...
iprodclim 15340 An infinite product equals...
iprodclim2 15341 A converging product conve...
iprodclim3 15342 The sequence of partial fi...
iprodcl 15343 The product of a non-trivi...
iprodrecl 15344 The product of a non-trivi...
iprodmul 15345 Multiplication of infinite...
risefacval 15350 The value of the rising fa...
fallfacval 15351 The value of the falling f...
risefacval2 15352 One-based value of rising ...
fallfacval2 15353 One-based value of falling...
fallfacval3 15354 A product representation o...
risefaccllem 15355 Lemma for rising factorial...
fallfaccllem 15356 Lemma for falling factoria...
risefaccl 15357 Closure law for rising fac...
fallfaccl 15358 Closure law for falling fa...
rerisefaccl 15359 Closure law for rising fac...
refallfaccl 15360 Closure law for falling fa...
nnrisefaccl 15361 Closure law for rising fac...
zrisefaccl 15362 Closure law for rising fac...
zfallfaccl 15363 Closure law for falling fa...
nn0risefaccl 15364 Closure law for rising fac...
rprisefaccl 15365 Closure law for rising fac...
risefallfac 15366 A relationship between ris...
fallrisefac 15367 A relationship between fal...
risefall0lem 15368 Lemma for ~ risefac0 and ~...
risefac0 15369 The value of the rising fa...
fallfac0 15370 The value of the falling f...
risefacp1 15371 The value of the rising fa...
fallfacp1 15372 The value of the falling f...
risefacp1d 15373 The value of the rising fa...
fallfacp1d 15374 The value of the falling f...
risefac1 15375 The value of rising factor...
fallfac1 15376 The value of falling facto...
risefacfac 15377 Relate rising factorial to...
fallfacfwd 15378 The forward difference of ...
0fallfac 15379 The value of the zero fall...
0risefac 15380 The value of the zero risi...
binomfallfaclem1 15381 Lemma for ~ binomfallfac ....
binomfallfaclem2 15382 Lemma for ~ binomfallfac ....
binomfallfac 15383 A version of the binomial ...
binomrisefac 15384 A version of the binomial ...
fallfacval4 15385 Represent the falling fact...
bcfallfac 15386 Binomial coefficient in te...
fallfacfac 15387 Relate falling factorial t...
bpolylem 15390 Lemma for ~ bpolyval . (C...
bpolyval 15391 The value of the Bernoulli...
bpoly0 15392 The value of the Bernoulli...
bpoly1 15393 The value of the Bernoulli...
bpolycl 15394 Closure law for Bernoulli ...
bpolysum 15395 A sum for Bernoulli polyno...
bpolydiflem 15396 Lemma for ~ bpolydif . (C...
bpolydif 15397 Calculate the difference b...
fsumkthpow 15398 A closed-form expression f...
bpoly2 15399 The Bernoulli polynomials ...
bpoly3 15400 The Bernoulli polynomials ...
bpoly4 15401 The Bernoulli polynomials ...
fsumcube 15402 Express the sum of cubes i...
eftcl 15415 Closure of a term in the s...
reeftcl 15416 The terms of the series ex...
eftabs 15417 The absolute value of a te...
eftval 15418 The value of a term in the...
efcllem 15419 Lemma for ~ efcl . The se...
ef0lem 15420 The series defining the ex...
efval 15421 Value of the exponential f...
esum 15422 Value of Euler's constant ...
eff 15423 Domain and codomain of the...
efcl 15424 Closure law for the expone...
efval2 15425 Value of the exponential f...
efcvg 15426 The series that defines th...
efcvgfsum 15427 Exponential function conve...
reefcl 15428 The exponential function i...
reefcld 15429 The exponential function i...
ere 15430 Euler's constant ` _e ` = ...
ege2le3 15431 Lemma for ~ egt2lt3 . (Co...
ef0 15432 Value of the exponential f...
efcj 15433 The exponential of a compl...
efaddlem 15434 Lemma for ~ efadd (exponen...
efadd 15435 Sum of exponents law for e...
fprodefsum 15436 Move the exponential funct...
efcan 15437 Cancellation law for expon...
efne0 15438 The exponential of a compl...
efneg 15439 The exponential of the opp...
eff2 15440 The exponential function m...
efsub 15441 Difference of exponents la...
efexp 15442 The exponential of an inte...
efzval 15443 Value of the exponential f...
efgt0 15444 The exponential of a real ...
rpefcl 15445 The exponential of a real ...
rpefcld 15446 The exponential of a real ...
eftlcvg 15447 The tail series of the exp...
eftlcl 15448 Closure of the sum of an i...
reeftlcl 15449 Closure of the sum of an i...
eftlub 15450 An upper bound on the abso...
efsep 15451 Separate out the next term...
effsumlt 15452 The partial sums of the se...
eft0val 15453 The value of the first ter...
ef4p 15454 Separate out the first fou...
efgt1p2 15455 The exponential of a posit...
efgt1p 15456 The exponential of a posit...
efgt1 15457 The exponential of a posit...
eflt 15458 The exponential function o...
efle 15459 The exponential function o...
reef11 15460 The exponential function o...
reeff1 15461 The exponential function m...
eflegeo 15462 The exponential function o...
sinval 15463 Value of the sine function...
cosval 15464 Value of the cosine functi...
sinf 15465 Domain and codomain of the...
cosf 15466 Domain and codomain of the...
sincl 15467 Closure of the sine functi...
coscl 15468 Closure of the cosine func...
tanval 15469 Value of the tangent funct...
tancl 15470 The closure of the tangent...
sincld 15471 Closure of the sine functi...
coscld 15472 Closure of the cosine func...
tancld 15473 Closure of the tangent fun...
tanval2 15474 Express the tangent functi...
tanval3 15475 Express the tangent functi...
resinval 15476 The sine of a real number ...
recosval 15477 The cosine of a real numbe...
efi4p 15478 Separate out the first fou...
resin4p 15479 Separate out the first fou...
recos4p 15480 Separate out the first fou...
resincl 15481 The sine of a real number ...
recoscl 15482 The cosine of a real numbe...
retancl 15483 The closure of the tangent...
resincld 15484 Closure of the sine functi...
recoscld 15485 Closure of the cosine func...
retancld 15486 Closure of the tangent fun...
sinneg 15487 The sine of a negative is ...
cosneg 15488 The cosines of a number an...
tanneg 15489 The tangent of a negative ...
sin0 15490 Value of the sine function...
cos0 15491 Value of the cosine functi...
tan0 15492 The value of the tangent f...
efival 15493 The exponential function i...
efmival 15494 The exponential function i...
sinhval 15495 Value of the hyperbolic si...
coshval 15496 Value of the hyperbolic co...
resinhcl 15497 The hyperbolic sine of a r...
rpcoshcl 15498 The hyperbolic cosine of a...
recoshcl 15499 The hyperbolic cosine of a...
retanhcl 15500 The hyperbolic tangent of ...
tanhlt1 15501 The hyperbolic tangent of ...
tanhbnd 15502 The hyperbolic tangent of ...
efeul 15503 Eulerian representation of...
efieq 15504 The exponentials of two im...
sinadd 15505 Addition formula for sine....
cosadd 15506 Addition formula for cosin...
tanaddlem 15507 A useful intermediate step...
tanadd 15508 Addition formula for tange...
sinsub 15509 Sine of difference. (Cont...
cossub 15510 Cosine of difference. (Co...
addsin 15511 Sum of sines. (Contribute...
subsin 15512 Difference of sines. (Con...
sinmul 15513 Product of sines can be re...
cosmul 15514 Product of cosines can be ...
addcos 15515 Sum of cosines. (Contribu...
subcos 15516 Difference of cosines. (C...
sincossq 15517 Sine squared plus cosine s...
sin2t 15518 Double-angle formula for s...
cos2t 15519 Double-angle formula for c...
cos2tsin 15520 Double-angle formula for c...
sinbnd 15521 The sine of a real number ...
cosbnd 15522 The cosine of a real numbe...
sinbnd2 15523 The sine of a real number ...
cosbnd2 15524 The cosine of a real numbe...
ef01bndlem 15525 Lemma for ~ sin01bnd and ~...
sin01bnd 15526 Bounds on the sine of a po...
cos01bnd 15527 Bounds on the cosine of a ...
cos1bnd 15528 Bounds on the cosine of 1....
cos2bnd 15529 Bounds on the cosine of 2....
sinltx 15530 The sine of a positive rea...
sin01gt0 15531 The sine of a positive rea...
cos01gt0 15532 The cosine of a positive r...
sin02gt0 15533 The sine of a positive rea...
sincos1sgn 15534 The signs of the sine and ...
sincos2sgn 15535 The signs of the sine and ...
sin4lt0 15536 The sine of 4 is negative....
absefi 15537 The absolute value of the ...
absef 15538 The absolute value of the ...
absefib 15539 A complex number is real i...
efieq1re 15540 A number whose imaginary e...
demoivre 15541 De Moivre's Formula. Proo...
demoivreALT 15542 Alternate proof of ~ demoi...
eirrlem 15545 Lemma for ~ eirr . (Contr...
eirr 15546 ` _e ` is irrational. (Co...
egt2lt3 15547 Euler's constant ` _e ` = ...
epos 15548 Euler's constant ` _e ` is...
epr 15549 Euler's constant ` _e ` is...
ene0 15550 ` _e ` is not 0. (Contrib...
ene1 15551 ` _e ` is not 1. (Contrib...
xpnnen 15552 The Cartesian product of t...
znnen 15553 The set of integers and th...
qnnen 15554 The rational numbers are c...
rpnnen2lem1 15555 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem2 15556 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem3 15557 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem4 15558 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem5 15559 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem6 15560 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem7 15561 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem8 15562 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem9 15563 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem10 15564 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem11 15565 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem12 15566 Lemma for ~ rpnnen2 . (Co...
rpnnen2 15567 The other half of ~ rpnnen...
rpnnen 15568 The cardinality of the con...
rexpen 15569 The real numbers are equin...
cpnnen 15570 The complex numbers are eq...
rucALT 15571 Alternate proof of ~ ruc ....
ruclem1 15572 Lemma for ~ ruc (the reals...
ruclem2 15573 Lemma for ~ ruc . Orderin...
ruclem3 15574 Lemma for ~ ruc . The con...
ruclem4 15575 Lemma for ~ ruc . Initial...
ruclem6 15576 Lemma for ~ ruc . Domain ...
ruclem7 15577 Lemma for ~ ruc . Success...
ruclem8 15578 Lemma for ~ ruc . The int...
ruclem9 15579 Lemma for ~ ruc . The fir...
ruclem10 15580 Lemma for ~ ruc . Every f...
ruclem11 15581 Lemma for ~ ruc . Closure...
ruclem12 15582 Lemma for ~ ruc . The sup...
ruclem13 15583 Lemma for ~ ruc . There i...
ruc 15584 The set of positive intege...
resdomq 15585 The set of rationals is st...
aleph1re 15586 There are at least aleph-o...
aleph1irr 15587 There are at least aleph-o...
cnso 15588 The complex numbers can be...
sqrt2irrlem 15589 Lemma for ~ sqrt2irr . Th...
sqrt2irr 15590 The square root of 2 is ir...
sqrt2re 15591 The square root of 2 exist...
sqrt2irr0 15592 The square root of 2 is an...
nthruc 15593 The sequence ` NN ` , ` ZZ...
nthruz 15594 The sequence ` NN ` , ` NN...
divides 15597 Define the divides relatio...
dvdsval2 15598 One nonzero integer divide...
dvdsval3 15599 One nonzero integer divide...
dvdszrcl 15600 Reverse closure for the di...
dvdsmod0 15601 If a positive integer divi...
p1modz1 15602 If a number greater than 1...
dvdsmodexp 15603 If a positive integer divi...
nndivdvds 15604 Strong form of ~ dvdsval2 ...
nndivides 15605 Definition of the divides ...
moddvds 15606 Two ways to say ` A == B `...
modm1div 15607 A number greater than 1 di...
dvds0lem 15608 A lemma to assist theorems...
dvds1lem 15609 A lemma to assist theorems...
dvds2lem 15610 A lemma to assist theorems...
iddvds 15611 An integer divides itself....
1dvds 15612 1 divides any integer. Th...
dvds0 15613 Any integer divides 0. Th...
negdvdsb 15614 An integer divides another...
dvdsnegb 15615 An integer divides another...
absdvdsb 15616 An integer divides another...
dvdsabsb 15617 An integer divides another...
0dvds 15618 Only 0 is divisible by 0. ...
dvdsmul1 15619 An integer divides a multi...
dvdsmul2 15620 An integer divides a multi...
iddvdsexp 15621 An integer divides a posit...
muldvds1 15622 If a product divides an in...
muldvds2 15623 If a product divides an in...
dvdscmul 15624 Multiplication by a consta...
dvdsmulc 15625 Multiplication by a consta...
dvdscmulr 15626 Cancellation law for the d...
dvdsmulcr 15627 Cancellation law for the d...
summodnegmod 15628 The sum of two integers mo...
modmulconst 15629 Constant multiplication in...
dvds2ln 15630 If an integer divides each...
dvds2add 15631 If an integer divides each...
dvds2sub 15632 If an integer divides each...
dvds2subd 15633 Natural deduction form of ...
dvdstr 15634 The divides relation is tr...
dvdsmultr1 15635 If an integer divides anot...
dvdsmultr1d 15636 Natural deduction form of ...
dvdsmultr2 15637 If an integer divides anot...
ordvdsmul 15638 If an integer divides eith...
dvdssub2 15639 If an integer divides a di...
dvdsadd 15640 An integer divides another...
dvdsaddr 15641 An integer divides another...
dvdssub 15642 An integer divides another...
dvdssubr 15643 An integer divides another...
dvdsadd2b 15644 Adding a multiple of the b...
dvdsaddre2b 15645 Adding a multiple of the b...
fsumdvds 15646 If every term in a sum is ...
dvdslelem 15647 Lemma for ~ dvdsle . (Con...
dvdsle 15648 The divisors of a positive...
dvdsleabs 15649 The divisors of a nonzero ...
dvdsleabs2 15650 Transfer divisibility to a...
dvdsabseq 15651 If two integers divide eac...
dvdseq 15652 If two nonnegative integer...
divconjdvds 15653 If a nonzero integer ` M `...
dvdsdivcl 15654 The complement of a diviso...
dvdsflip 15655 An involution of the divis...
dvdsssfz1 15656 The set of divisors of a n...
dvds1 15657 The only nonnegative integ...
alzdvds 15658 Only 0 is divisible by all...
dvdsext 15659 Poset extensionality for d...
fzm1ndvds 15660 No number between ` 1 ` an...
fzo0dvdseq 15661 Zero is the only one of th...
fzocongeq 15662 Two different elements of ...
addmodlteqALT 15663 Two nonnegative integers l...
dvdsfac 15664 A positive integer divides...
dvdsexp 15665 A power divides a power wi...
dvdsmod 15666 Any number ` K ` whose mod...
mulmoddvds 15667 If an integer is divisible...
3dvds 15668 A rule for divisibility by...
3dvdsdec 15669 A decimal number is divisi...
3dvds2dec 15670 A decimal number is divisi...
fprodfvdvdsd 15671 A finite product of intege...
fproddvdsd 15672 A finite product of intege...
evenelz 15673 An even number is an integ...
zeo3 15674 An integer is even or odd....
zeo4 15675 An integer is even or odd ...
zeneo 15676 No even integer equals an ...
odd2np1lem 15677 Lemma for ~ odd2np1 . (Co...
odd2np1 15678 An integer is odd iff it i...
even2n 15679 An integer is even iff it ...
oddm1even 15680 An integer is odd iff its ...
oddp1even 15681 An integer is odd iff its ...
oexpneg 15682 The exponential of the neg...
mod2eq0even 15683 An integer is 0 modulo 2 i...
mod2eq1n2dvds 15684 An integer is 1 modulo 2 i...
oddnn02np1 15685 A nonnegative integer is o...
oddge22np1 15686 An integer greater than on...
evennn02n 15687 A nonnegative integer is e...
evennn2n 15688 A positive integer is even...
2tp1odd 15689 A number which is twice an...
mulsucdiv2z 15690 An integer multiplied with...
sqoddm1div8z 15691 A squared odd number minus...
2teven 15692 A number which is twice an...
zeo5 15693 An integer is either even ...
evend2 15694 An integer is even iff its...
oddp1d2 15695 An integer is odd iff its ...
zob 15696 Alternate characterization...
oddm1d2 15697 An integer is odd iff its ...
ltoddhalfle 15698 An integer is less than ha...
halfleoddlt 15699 An integer is greater than...
opoe 15700 The sum of two odds is eve...
omoe 15701 The difference of two odds...
opeo 15702 The sum of an odd and an e...
omeo 15703 The difference of an odd a...
z0even 15704 2 divides 0. That means 0...
n2dvds1 15705 2 does not divide 1. That...
n2dvds1OLD 15706 Obsolete version of ~ n2dv...
n2dvdsm1 15707 2 does not divide -1. Tha...
z2even 15708 2 divides 2. That means 2...
n2dvds3 15709 2 does not divide 3. That...
n2dvds3OLD 15710 Obsolete version of ~ n2dv...
z4even 15711 2 divides 4. That means 4...
4dvdseven 15712 An integer which is divisi...
m1expe 15713 Exponentiation of -1 by an...
m1expo 15714 Exponentiation of -1 by an...
m1exp1 15715 Exponentiation of negative...
nn0enne 15716 A positive integer is an e...
nn0ehalf 15717 The half of an even nonneg...
nnehalf 15718 The half of an even positi...
nn0onn 15719 An odd nonnegative integer...
nn0o1gt2 15720 An odd nonnegative integer...
nno 15721 An alternate characterizat...
nn0o 15722 An alternate characterizat...
nn0ob 15723 Alternate characterization...
nn0oddm1d2 15724 A positive integer is odd ...
nnoddm1d2 15725 A positive integer is odd ...
sumeven 15726 If every term in a sum is ...
sumodd 15727 If every term in a sum is ...
evensumodd 15728 If every term in a sum wit...
oddsumodd 15729 If every term in a sum wit...
pwp1fsum 15730 The n-th power of a number...
oddpwp1fsum 15731 An odd power of a number i...
divalglem0 15732 Lemma for ~ divalg . (Con...
divalglem1 15733 Lemma for ~ divalg . (Con...
divalglem2 15734 Lemma for ~ divalg . (Con...
divalglem4 15735 Lemma for ~ divalg . (Con...
divalglem5 15736 Lemma for ~ divalg . (Con...
divalglem6 15737 Lemma for ~ divalg . (Con...
divalglem7 15738 Lemma for ~ divalg . (Con...
divalglem8 15739 Lemma for ~ divalg . (Con...
divalglem9 15740 Lemma for ~ divalg . (Con...
divalglem10 15741 Lemma for ~ divalg . (Con...
divalg 15742 The division algorithm (th...
divalgb 15743 Express the division algor...
divalg2 15744 The division algorithm (th...
divalgmod 15745 The result of the ` mod ` ...
divalgmodcl 15746 The result of the ` mod ` ...
modremain 15747 The result of the modulo o...
ndvdssub 15748 Corollary of the division ...
ndvdsadd 15749 Corollary of the division ...
ndvdsp1 15750 Special case of ~ ndvdsadd...
ndvdsi 15751 A quick test for non-divis...
flodddiv4 15752 The floor of an odd intege...
fldivndvdslt 15753 The floor of an integer di...
flodddiv4lt 15754 The floor of an odd number...
flodddiv4t2lthalf 15755 The floor of an odd number...
bitsfval 15760 Expand the definition of t...
bitsval 15761 Expand the definition of t...
bitsval2 15762 Expand the definition of t...
bitsss 15763 The set of bits of an inte...
bitsf 15764 The ` bits ` function is a...
bits0 15765 Value of the zeroth bit. ...
bits0e 15766 The zeroth bit of an even ...
bits0o 15767 The zeroth bit of an odd n...
bitsp1 15768 The ` M + 1 ` -th bit of `...
bitsp1e 15769 The ` M + 1 ` -th bit of `...
bitsp1o 15770 The ` M + 1 ` -th bit of `...
bitsfzolem 15771 Lemma for ~ bitsfzo . (Co...
bitsfzo 15772 The bits of a number are a...
bitsmod 15773 Truncating the bit sequenc...
bitsfi 15774 Every number is associated...
bitscmp 15775 The bit complement of ` N ...
0bits 15776 The bits of zero. (Contri...
m1bits 15777 The bits of negative one. ...
bitsinv1lem 15778 Lemma for ~ bitsinv1 . (C...
bitsinv1 15779 There is an explicit inver...
bitsinv2 15780 There is an explicit inver...
bitsf1ocnv 15781 The ` bits ` function rest...
bitsf1o 15782 The ` bits ` function rest...
bitsf1 15783 The ` bits ` function is a...
2ebits 15784 The bits of a power of two...
bitsinv 15785 The inverse of the ` bits ...
bitsinvp1 15786 Recursive definition of th...
sadadd2lem2 15787 The core of the proof of ~...
sadfval 15789 Define the addition of two...
sadcf 15790 The carry sequence is a se...
sadc0 15791 The initial element of the...
sadcp1 15792 The carry sequence (which ...
sadval 15793 The full adder sequence is...
sadcaddlem 15794 Lemma for ~ sadcadd . (Co...
sadcadd 15795 Non-recursive definition o...
sadadd2lem 15796 Lemma for ~ sadadd2 . (Co...
sadadd2 15797 Sum of initial segments of...
sadadd3 15798 Sum of initial segments of...
sadcl 15799 The sum of two sequences i...
sadcom 15800 The adder sequence functio...
saddisjlem 15801 Lemma for ~ sadadd . (Con...
saddisj 15802 The sum of disjoint sequen...
sadaddlem 15803 Lemma for ~ sadadd . (Con...
sadadd 15804 For sequences that corresp...
sadid1 15805 The adder sequence functio...
sadid2 15806 The adder sequence functio...
sadasslem 15807 Lemma for ~ sadass . (Con...
sadass 15808 Sequence addition is assoc...
sadeq 15809 Any element of a sequence ...
bitsres 15810 Restrict the bits of a num...
bitsuz 15811 The bits of a number are a...
bitsshft 15812 Shifting a bit sequence to...
smufval 15814 The multiplication of two ...
smupf 15815 The sequence of partial su...
smup0 15816 The initial element of the...
smupp1 15817 The initial element of the...
smuval 15818 Define the addition of two...
smuval2 15819 The partial sum sequence s...
smupvallem 15820 If ` A ` only has elements...
smucl 15821 The product of two sequenc...
smu01lem 15822 Lemma for ~ smu01 and ~ sm...
smu01 15823 Multiplication of a sequen...
smu02 15824 Multiplication of a sequen...
smupval 15825 Rewrite the elements of th...
smup1 15826 Rewrite ~ smupp1 using onl...
smueqlem 15827 Any element of a sequence ...
smueq 15828 Any element of a sequence ...
smumullem 15829 Lemma for ~ smumul . (Con...
smumul 15830 For sequences that corresp...
gcdval 15833 The value of the ` gcd ` o...
gcd0val 15834 The value, by convention, ...
gcdn0val 15835 The value of the ` gcd ` o...
gcdcllem1 15836 Lemma for ~ gcdn0cl , ~ gc...
gcdcllem2 15837 Lemma for ~ gcdn0cl , ~ gc...
gcdcllem3 15838 Lemma for ~ gcdn0cl , ~ gc...
gcdn0cl 15839 Closure of the ` gcd ` ope...
gcddvds 15840 The gcd of two integers di...
dvdslegcd 15841 An integer which divides b...
nndvdslegcd 15842 A positive integer which d...
gcdcl 15843 Closure of the ` gcd ` ope...
gcdnncl 15844 Closure of the ` gcd ` ope...
gcdcld 15845 Closure of the ` gcd ` ope...
gcd2n0cl 15846 Closure of the ` gcd ` ope...
zeqzmulgcd 15847 An integer is the product ...
divgcdz 15848 An integer divided by the ...
gcdf 15849 Domain and codomain of the...
gcdcom 15850 The ` gcd ` operator is co...
divgcdnn 15851 A positive integer divided...
divgcdnnr 15852 A positive integer divided...
gcdeq0 15853 The gcd of two integers is...
gcdn0gt0 15854 The gcd of two integers is...
gcd0id 15855 The gcd of 0 and an intege...
gcdid0 15856 The gcd of an integer and ...
nn0gcdid0 15857 The gcd of a nonnegative i...
gcdneg 15858 Negating one operand of th...
neggcd 15859 Negating one operand of th...
gcdaddmlem 15860 Lemma for ~ gcdaddm . (Co...
gcdaddm 15861 Adding a multiple of one o...
gcdadd 15862 The GCD of two numbers is ...
gcdid 15863 The gcd of a number and it...
gcd1 15864 The gcd of a number with 1...
gcdabs 15865 The gcd of two integers is...
gcdabs1 15866 ` gcd ` of the absolute va...
gcdabs2 15867 ` gcd ` of the absolute va...
modgcd 15868 The gcd remains unchanged ...
1gcd 15869 The GCD of one and an inte...
gcdmultipled 15870 The greatest common diviso...
gcdmultiplez 15871 The GCD of a multiple of a...
gcdmultiple 15872 The GCD of a multiple of a...
dvdsgcdidd 15873 The greatest common diviso...
6gcd4e2 15874 The greatest common diviso...
bezoutlem1 15875 Lemma for ~ bezout . (Con...
bezoutlem2 15876 Lemma for ~ bezout . (Con...
bezoutlem3 15877 Lemma for ~ bezout . (Con...
bezoutlem4 15878 Lemma for ~ bezout . (Con...
bezout 15879 Bézout's identity: ...
dvdsgcd 15880 An integer which divides e...
dvdsgcdb 15881 Biconditional form of ~ dv...
dfgcd2 15882 Alternate definition of th...
gcdass 15883 Associative law for ` gcd ...
mulgcd 15884 Distribute multiplication ...
absmulgcd 15885 Distribute absolute value ...
mulgcdr 15886 Reverse distribution law f...
gcddiv 15887 Division law for GCD. (Con...
gcdmultipleOLD 15888 Obsolete proof of ~ gcdmul...
gcdmultiplezOLD 15889 Obsolete proof of ~ gcdmul...
gcdzeq 15890 A positive integer ` A ` i...
gcdeq 15891 ` A ` is equal to its gcd ...
dvdssqim 15892 Unidirectional form of ~ d...
dvdsmulgcd 15893 A divisibility equivalent ...
rpmulgcd 15894 If ` K ` and ` M ` are rel...
rplpwr 15895 If ` A ` and ` B ` are rel...
rppwr 15896 If ` A ` and ` B ` are rel...
sqgcd 15897 Square distributes over GC...
dvdssqlem 15898 Lemma for ~ dvdssq . (Con...
dvdssq 15899 Two numbers are divisible ...
bezoutr 15900 Partial converse to ~ bezo...
bezoutr1 15901 Converse of ~ bezout for w...
nn0seqcvgd 15902 A strictly-decreasing nonn...
seq1st 15903 A sequence whose iteration...
algr0 15904 The value of the algorithm...
algrf 15905 An algorithm is a step fun...
algrp1 15906 The value of the algorithm...
alginv 15907 If ` I ` is an invariant o...
algcvg 15908 One way to prove that an a...
algcvgblem 15909 Lemma for ~ algcvgb . (Co...
algcvgb 15910 Two ways of expressing tha...
algcvga 15911 The countdown function ` C...
algfx 15912 If ` F ` reaches a fixed p...
eucalgval2 15913 The value of the step func...
eucalgval 15914 Euclid's Algorithm ~ eucal...
eucalgf 15915 Domain and codomain of the...
eucalginv 15916 The invariant of the step ...
eucalglt 15917 The second member of the s...
eucalgcvga 15918 Once Euclid's Algorithm ha...
eucalg 15919 Euclid's Algorithm compute...
lcmval 15924 Value of the ` lcm ` opera...
lcmcom 15925 The ` lcm ` operator is co...
lcm0val 15926 The value, by convention, ...
lcmn0val 15927 The value of the ` lcm ` o...
lcmcllem 15928 Lemma for ~ lcmn0cl and ~ ...
lcmn0cl 15929 Closure of the ` lcm ` ope...
dvdslcm 15930 The lcm of two integers is...
lcmledvds 15931 A positive integer which b...
lcmeq0 15932 The lcm of two integers is...
lcmcl 15933 Closure of the ` lcm ` ope...
gcddvdslcm 15934 The greatest common diviso...
lcmneg 15935 Negating one operand of th...
neglcm 15936 Negating one operand of th...
lcmabs 15937 The lcm of two integers is...
lcmgcdlem 15938 Lemma for ~ lcmgcd and ~ l...
lcmgcd 15939 The product of two numbers...
lcmdvds 15940 The lcm of two integers di...
lcmid 15941 The lcm of an integer and ...
lcm1 15942 The lcm of an integer and ...
lcmgcdnn 15943 The product of two positiv...
lcmgcdeq 15944 Two integers' absolute val...
lcmdvdsb 15945 Biconditional form of ~ lc...
lcmass 15946 Associative law for ` lcm ...
3lcm2e6woprm 15947 The least common multiple ...
6lcm4e12 15948 The least common multiple ...
absproddvds 15949 The absolute value of the ...
absprodnn 15950 The absolute value of the ...
fissn0dvds 15951 For each finite subset of ...
fissn0dvdsn0 15952 For each finite subset of ...
lcmfval 15953 Value of the ` _lcm ` func...
lcmf0val 15954 The value, by convention, ...
lcmfn0val 15955 The value of the ` _lcm ` ...
lcmfnnval 15956 The value of the ` _lcm ` ...
lcmfcllem 15957 Lemma for ~ lcmfn0cl and ~...
lcmfn0cl 15958 Closure of the ` _lcm ` fu...
lcmfpr 15959 The value of the ` _lcm ` ...
lcmfcl 15960 Closure of the ` _lcm ` fu...
lcmfnncl 15961 Closure of the ` _lcm ` fu...
lcmfeq0b 15962 The least common multiple ...
dvdslcmf 15963 The least common multiple ...
lcmfledvds 15964 A positive integer which i...
lcmf 15965 Characterization of the le...
lcmf0 15966 The least common multiple ...
lcmfsn 15967 The least common multiple ...
lcmftp 15968 The least common multiple ...
lcmfunsnlem1 15969 Lemma for ~ lcmfdvds and ~...
lcmfunsnlem2lem1 15970 Lemma 1 for ~ lcmfunsnlem2...
lcmfunsnlem2lem2 15971 Lemma 2 for ~ lcmfunsnlem2...
lcmfunsnlem2 15972 Lemma for ~ lcmfunsn and ~...
lcmfunsnlem 15973 Lemma for ~ lcmfdvds and ~...
lcmfdvds 15974 The least common multiple ...
lcmfdvdsb 15975 Biconditional form of ~ lc...
lcmfunsn 15976 The ` _lcm ` function for ...
lcmfun 15977 The ` _lcm ` function for ...
lcmfass 15978 Associative law for the ` ...
lcmf2a3a4e12 15979 The least common multiple ...
lcmflefac 15980 The least common multiple ...
coprmgcdb 15981 Two positive integers are ...
ncoprmgcdne1b 15982 Two positive integers are ...
ncoprmgcdgt1b 15983 Two positive integers are ...
coprmdvds1 15984 If two positive integers a...
coprmdvds 15985 Euclid's Lemma (see ProofW...
coprmdvds2 15986 If an integer is divisible...
mulgcddvds 15987 One half of ~ rpmulgcd2 , ...
rpmulgcd2 15988 If ` M ` is relatively pri...
qredeq 15989 Two equal reduced fraction...
qredeu 15990 Every rational number has ...
rpmul 15991 If ` K ` is relatively pri...
rpdvds 15992 If ` K ` is relatively pri...
coprmprod 15993 The product of the element...
coprmproddvdslem 15994 Lemma for ~ coprmproddvds ...
coprmproddvds 15995 If a positive integer is d...
congr 15996 Definition of congruence b...
divgcdcoprm0 15997 Integers divided by gcd ar...
divgcdcoprmex 15998 Integers divided by gcd ar...
cncongr1 15999 One direction of the bicon...
cncongr2 16000 The other direction of the...
cncongr 16001 Cancellability of Congruen...
cncongrcoprm 16002 Corollary 1 of Cancellabil...
isprm 16005 The predicate "is a prime ...
prmnn 16006 A prime number is a positi...
prmz 16007 A prime number is an integ...
prmssnn 16008 The prime numbers are a su...
prmex 16009 The set of prime numbers e...
0nprm 16010 0 is not a prime number. ...
1nprm 16011 1 is not a prime number. ...
1idssfct 16012 The positive divisors of a...
isprm2lem 16013 Lemma for ~ isprm2 . (Con...
isprm2 16014 The predicate "is a prime ...
isprm3 16015 The predicate "is a prime ...
isprm4 16016 The predicate "is a prime ...
prmind2 16017 A variation on ~ prmind as...
prmind 16018 Perform induction over the...
dvdsprime 16019 If ` M ` divides a prime, ...
nprm 16020 A product of two integers ...
nprmi 16021 An inference for composite...
dvdsnprmd 16022 If a number is divisible b...
prm2orodd 16023 A prime number is either 2...
2prm 16024 2 is a prime number. (Con...
2mulprm 16025 A multiple of two is prime...
3prm 16026 3 is a prime number. (Con...
4nprm 16027 4 is not a prime number. ...
prmuz2 16028 A prime number is an integ...
prmgt1 16029 A prime number is an integ...
prmm2nn0 16030 Subtracting 2 from a prime...
oddprmgt2 16031 An odd prime is greater th...
oddprmge3 16032 An odd prime is greater th...
ge2nprmge4 16033 A composite integer greate...
sqnprm 16034 A square is never prime. ...
dvdsprm 16035 An integer greater than or...
exprmfct 16036 Every integer greater than...
prmdvdsfz 16037 Each integer greater than ...
nprmdvds1 16038 No prime number divides 1....
isprm5 16039 One need only check prime ...
isprm7 16040 One need only check prime ...
maxprmfct 16041 The set of prime factors o...
divgcdodd 16042 Either ` A / ( A gcd B ) `...
coprm 16043 A prime number either divi...
prmrp 16044 Unequal prime numbers are ...
euclemma 16045 Euclid's lemma. A prime n...
isprm6 16046 A number is prime iff it s...
prmdvdsexp 16047 A prime divides a positive...
prmdvdsexpb 16048 A prime divides a positive...
prmdvdsexpr 16049 If a prime divides a nonne...
prmexpb 16050 Two positive prime powers ...
prmfac1 16051 The factorial of a number ...
rpexp 16052 If two numbers ` A ` and `...
rpexp1i 16053 Relative primality passes ...
rpexp12i 16054 Relative primality passes ...
prmndvdsfaclt 16055 A prime number does not di...
ncoprmlnprm 16056 If two positive integers a...
cncongrprm 16057 Corollary 2 of Cancellabil...
isevengcd2 16058 The predicate "is an even ...
isoddgcd1 16059 The predicate "is an odd n...
3lcm2e6 16060 The least common multiple ...
qnumval 16065 Value of the canonical num...
qdenval 16066 Value of the canonical den...
qnumdencl 16067 Lemma for ~ qnumcl and ~ q...
qnumcl 16068 The canonical numerator of...
qdencl 16069 The canonical denominator ...
fnum 16070 Canonical numerator define...
fden 16071 Canonical denominator defi...
qnumdenbi 16072 Two numbers are the canoni...
qnumdencoprm 16073 The canonical representati...
qeqnumdivden 16074 Recover a rational number ...
qmuldeneqnum 16075 Multiplying a rational by ...
divnumden 16076 Calculate the reduced form...
divdenle 16077 Reducing a quotient never ...
qnumgt0 16078 A rational is positive iff...
qgt0numnn 16079 A rational is positive iff...
nn0gcdsq 16080 Squaring commutes with GCD...
zgcdsq 16081 ~ nn0gcdsq extended to int...
numdensq 16082 Squaring a rational square...
numsq 16083 Square commutes with canon...
densq 16084 Square commutes with canon...
qden1elz 16085 A rational is an integer i...
zsqrtelqelz 16086 If an integer has a ration...
nonsq 16087 Any integer strictly betwe...
phival 16092 Value of the Euler ` phi `...
phicl2 16093 Bounds and closure for the...
phicl 16094 Closure for the value of t...
phibndlem 16095 Lemma for ~ phibnd . (Con...
phibnd 16096 A slightly tighter bound o...
phicld 16097 Closure for the value of t...
phi1 16098 Value of the Euler ` phi `...
dfphi2 16099 Alternate definition of th...
hashdvds 16100 The number of numbers in a...
phiprmpw 16101 Value of the Euler ` phi `...
phiprm 16102 Value of the Euler ` phi `...
crth 16103 The Chinese Remainder Theo...
phimullem 16104 Lemma for ~ phimul . (Con...
phimul 16105 The Euler ` phi ` function...
eulerthlem1 16106 Lemma for ~ eulerth . (Co...
eulerthlem2 16107 Lemma for ~ eulerth . (Co...
eulerth 16108 Euler's theorem, a general...
fermltl 16109 Fermat's little theorem. ...
prmdiv 16110 Show an explicit expressio...
prmdiveq 16111 The modular inverse of ` A...
prmdivdiv 16112 The (modular) inverse of t...
hashgcdlem 16113 A correspondence between e...
hashgcdeq 16114 Number of initial positive...
phisum 16115 The divisor sum identity o...
odzval 16116 Value of the order functio...
odzcllem 16117 - Lemma for ~ odzcl , show...
odzcl 16118 The order of a group eleme...
odzid 16119 Any element raised to the ...
odzdvds 16120 The only powers of ` A ` t...
odzphi 16121 The order of any group ele...
modprm1div 16122 A prime number divides an ...
m1dvdsndvds 16123 If an integer minus 1 is d...
modprminv 16124 Show an explicit expressio...
modprminveq 16125 The modular inverse of ` A...
vfermltl 16126 Variant of Fermat's little...
vfermltlALT 16127 Alternate proof of ~ vferm...
powm2modprm 16128 If an integer minus 1 is d...
reumodprminv 16129 For any prime number and f...
modprm0 16130 For two positive integers ...
nnnn0modprm0 16131 For a positive integer and...
modprmn0modprm0 16132 For an integer not being 0...
coprimeprodsq 16133 If three numbers are copri...
coprimeprodsq2 16134 If three numbers are copri...
oddprm 16135 A prime not equal to ` 2 `...
nnoddn2prm 16136 A prime not equal to ` 2 `...
oddn2prm 16137 A prime not equal to ` 2 `...
nnoddn2prmb 16138 A number is a prime number...
prm23lt5 16139 A prime less than 5 is eit...
prm23ge5 16140 A prime is either 2 or 3 o...
pythagtriplem1 16141 Lemma for ~ pythagtrip . ...
pythagtriplem2 16142 Lemma for ~ pythagtrip . ...
pythagtriplem3 16143 Lemma for ~ pythagtrip . ...
pythagtriplem4 16144 Lemma for ~ pythagtrip . ...
pythagtriplem10 16145 Lemma for ~ pythagtrip . ...
pythagtriplem6 16146 Lemma for ~ pythagtrip . ...
pythagtriplem7 16147 Lemma for ~ pythagtrip . ...
pythagtriplem8 16148 Lemma for ~ pythagtrip . ...
pythagtriplem9 16149 Lemma for ~ pythagtrip . ...
pythagtriplem11 16150 Lemma for ~ pythagtrip . ...
pythagtriplem12 16151 Lemma for ~ pythagtrip . ...
pythagtriplem13 16152 Lemma for ~ pythagtrip . ...
pythagtriplem14 16153 Lemma for ~ pythagtrip . ...
pythagtriplem15 16154 Lemma for ~ pythagtrip . ...
pythagtriplem16 16155 Lemma for ~ pythagtrip . ...
pythagtriplem17 16156 Lemma for ~ pythagtrip . ...
pythagtriplem18 16157 Lemma for ~ pythagtrip . ...
pythagtriplem19 16158 Lemma for ~ pythagtrip . ...
pythagtrip 16159 Parameterize the Pythagore...
iserodd 16160 Collect the odd terms in a...
pclem 16163 - Lemma for the prime powe...
pcprecl 16164 Closure of the prime power...
pcprendvds 16165 Non-divisibility property ...
pcprendvds2 16166 Non-divisibility property ...
pcpre1 16167 Value of the prime power p...
pcpremul 16168 Multiplicative property of...
pcval 16169 The value of the prime pow...
pceulem 16170 Lemma for ~ pceu . (Contr...
pceu 16171 Uniqueness for the prime p...
pczpre 16172 Connect the prime count pr...
pczcl 16173 Closure of the prime power...
pccl 16174 Closure of the prime power...
pccld 16175 Closure of the prime power...
pcmul 16176 Multiplication property of...
pcdiv 16177 Division property of the p...
pcqmul 16178 Multiplication property of...
pc0 16179 The value of the prime pow...
pc1 16180 Value of the prime count f...
pcqcl 16181 Closure of the general pri...
pcqdiv 16182 Division property of the p...
pcrec 16183 Prime power of a reciproca...
pcexp 16184 Prime power of an exponent...
pcxcl 16185 Extended real closure of t...
pcge0 16186 The prime count of an inte...
pczdvds 16187 Defining property of the p...
pcdvds 16188 Defining property of the p...
pczndvds 16189 Defining property of the p...
pcndvds 16190 Defining property of the p...
pczndvds2 16191 The remainder after dividi...
pcndvds2 16192 The remainder after dividi...
pcdvdsb 16193 ` P ^ A ` divides ` N ` if...
pcelnn 16194 There are a positive numbe...
pceq0 16195 There are zero powers of a...
pcidlem 16196 The prime count of a prime...
pcid 16197 The prime count of a prime...
pcneg 16198 The prime count of a negat...
pcabs 16199 The prime count of an abso...
pcdvdstr 16200 The prime count increases ...
pcgcd1 16201 The prime count of a GCD i...
pcgcd 16202 The prime count of a GCD i...
pc2dvds 16203 A characterization of divi...
pc11 16204 The prime count function, ...
pcz 16205 The prime count function c...
pcprmpw2 16206 Self-referential expressio...
pcprmpw 16207 Self-referential expressio...
dvdsprmpweq 16208 If a positive integer divi...
dvdsprmpweqnn 16209 If an integer greater than...
dvdsprmpweqle 16210 If a positive integer divi...
difsqpwdvds 16211 If the difference of two s...
pcaddlem 16212 Lemma for ~ pcadd . The o...
pcadd 16213 An inequality for the prim...
pcadd2 16214 The inequality of ~ pcadd ...
pcmptcl 16215 Closure for the prime powe...
pcmpt 16216 Construct a function with ...
pcmpt2 16217 Dividing two prime count m...
pcmptdvds 16218 The partial products of th...
pcprod 16219 The product of the primes ...
sumhash 16220 The sum of 1 over a set is...
fldivp1 16221 The difference between the...
pcfaclem 16222 Lemma for ~ pcfac . (Cont...
pcfac 16223 Calculate the prime count ...
pcbc 16224 Calculate the prime count ...
qexpz 16225 If a power of a rational n...
expnprm 16226 A second or higher power o...
oddprmdvds 16227 Every positive integer whi...
prmpwdvds 16228 A relation involving divis...
pockthlem 16229 Lemma for ~ pockthg . (Co...
pockthg 16230 The generalized Pocklingto...
pockthi 16231 Pocklington's theorem, whi...
unbenlem 16232 Lemma for ~ unben . (Cont...
unben 16233 An unbounded set of positi...
infpnlem1 16234 Lemma for ~ infpn . The s...
infpnlem2 16235 Lemma for ~ infpn . For a...
infpn 16236 There exist infinitely man...
infpn2 16237 There exist infinitely man...
prmunb 16238 The primes are unbounded. ...
prminf 16239 There are an infinite numb...
prmreclem1 16240 Lemma for ~ prmrec . Prop...
prmreclem2 16241 Lemma for ~ prmrec . Ther...
prmreclem3 16242 Lemma for ~ prmrec . The ...
prmreclem4 16243 Lemma for ~ prmrec . Show...
prmreclem5 16244 Lemma for ~ prmrec . Here...
prmreclem6 16245 Lemma for ~ prmrec . If t...
prmrec 16246 The sum of the reciprocals...
1arithlem1 16247 Lemma for ~ 1arith . (Con...
1arithlem2 16248 Lemma for ~ 1arith . (Con...
1arithlem3 16249 Lemma for ~ 1arith . (Con...
1arithlem4 16250 Lemma for ~ 1arith . (Con...
1arith 16251 Fundamental theorem of ari...
1arith2 16252 Fundamental theorem of ari...
elgz 16255 Elementhood in the gaussia...
gzcn 16256 A gaussian integer is a co...
zgz 16257 An integer is a gaussian i...
igz 16258 ` _i ` is a gaussian integ...
gznegcl 16259 The gaussian integers are ...
gzcjcl 16260 The gaussian integers are ...
gzaddcl 16261 The gaussian integers are ...
gzmulcl 16262 The gaussian integers are ...
gzreim 16263 Construct a gaussian integ...
gzsubcl 16264 The gaussian integers are ...
gzabssqcl 16265 The squared norm of a gaus...
4sqlem5 16266 Lemma for ~ 4sq . (Contri...
4sqlem6 16267 Lemma for ~ 4sq . (Contri...
4sqlem7 16268 Lemma for ~ 4sq . (Contri...
4sqlem8 16269 Lemma for ~ 4sq . (Contri...
4sqlem9 16270 Lemma for ~ 4sq . (Contri...
4sqlem10 16271 Lemma for ~ 4sq . (Contri...
4sqlem1 16272 Lemma for ~ 4sq . The set...
4sqlem2 16273 Lemma for ~ 4sq . Change ...
4sqlem3 16274 Lemma for ~ 4sq . Suffici...
4sqlem4a 16275 Lemma for ~ 4sqlem4 . (Co...
4sqlem4 16276 Lemma for ~ 4sq . We can ...
mul4sqlem 16277 Lemma for ~ mul4sq : algeb...
mul4sq 16278 Euler's four-square identi...
4sqlem11 16279 Lemma for ~ 4sq . Use the...
4sqlem12 16280 Lemma for ~ 4sq . For any...
4sqlem13 16281 Lemma for ~ 4sq . (Contri...
4sqlem14 16282 Lemma for ~ 4sq . (Contri...
4sqlem15 16283 Lemma for ~ 4sq . (Contri...
4sqlem16 16284 Lemma for ~ 4sq . (Contri...
4sqlem17 16285 Lemma for ~ 4sq . (Contri...
4sqlem18 16286 Lemma for ~ 4sq . Inducti...
4sqlem19 16287 Lemma for ~ 4sq . The pro...
4sq 16288 Lagrange's four-square the...
vdwapfval 16295 Define the arithmetic prog...
vdwapf 16296 The arithmetic progression...
vdwapval 16297 Value of the arithmetic pr...
vdwapun 16298 Remove the first element o...
vdwapid1 16299 The first element of an ar...
vdwap0 16300 Value of a length-1 arithm...
vdwap1 16301 Value of a length-1 arithm...
vdwmc 16302 The predicate " The ` <. R...
vdwmc2 16303 Expand out the definition ...
vdwpc 16304 The predicate " The colori...
vdwlem1 16305 Lemma for ~ vdw . (Contri...
vdwlem2 16306 Lemma for ~ vdw . (Contri...
vdwlem3 16307 Lemma for ~ vdw . (Contri...
vdwlem4 16308 Lemma for ~ vdw . (Contri...
vdwlem5 16309 Lemma for ~ vdw . (Contri...
vdwlem6 16310 Lemma for ~ vdw . (Contri...
vdwlem7 16311 Lemma for ~ vdw . (Contri...
vdwlem8 16312 Lemma for ~ vdw . (Contri...
vdwlem9 16313 Lemma for ~ vdw . (Contri...
vdwlem10 16314 Lemma for ~ vdw . Set up ...
vdwlem11 16315 Lemma for ~ vdw . (Contri...
vdwlem12 16316 Lemma for ~ vdw . ` K = 2 ...
vdwlem13 16317 Lemma for ~ vdw . Main in...
vdw 16318 Van der Waerden's theorem....
vdwnnlem1 16319 Corollary of ~ vdw , and l...
vdwnnlem2 16320 Lemma for ~ vdwnn . The s...
vdwnnlem3 16321 Lemma for ~ vdwnn . (Cont...
vdwnn 16322 Van der Waerden's theorem,...
ramtlecl 16324 The set ` T ` of numbers w...
hashbcval 16326 Value of the "binomial set...
hashbccl 16327 The binomial set is a fini...
hashbcss 16328 Subset relation for the bi...
hashbc0 16329 The set of subsets of size...
hashbc2 16330 The size of the binomial s...
0hashbc 16331 There are no subsets of th...
ramval 16332 The value of the Ramsey nu...
ramcl2lem 16333 Lemma for extended real cl...
ramtcl 16334 The Ramsey number has the ...
ramtcl2 16335 The Ramsey number is an in...
ramtub 16336 The Ramsey number is a low...
ramub 16337 The Ramsey number is a low...
ramub2 16338 It is sufficient to check ...
rami 16339 The defining property of a...
ramcl2 16340 The Ramsey number is eithe...
ramxrcl 16341 The Ramsey number is an ex...
ramubcl 16342 If the Ramsey number is up...
ramlb 16343 Establish a lower bound on...
0ram 16344 The Ramsey number when ` M...
0ram2 16345 The Ramsey number when ` M...
ram0 16346 The Ramsey number when ` R...
0ramcl 16347 Lemma for ~ ramcl : Exist...
ramz2 16348 The Ramsey number when ` F...
ramz 16349 The Ramsey number when ` F...
ramub1lem1 16350 Lemma for ~ ramub1 . (Con...
ramub1lem2 16351 Lemma for ~ ramub1 . (Con...
ramub1 16352 Inductive step for Ramsey'...
ramcl 16353 Ramsey's theorem: the Rams...
ramsey 16354 Ramsey's theorem with the ...
prmoval 16357 Value of the primorial fun...
prmocl 16358 Closure of the primorial f...
prmone0 16359 The primorial function is ...
prmo0 16360 The primorial of 0. (Cont...
prmo1 16361 The primorial of 1. (Cont...
prmop1 16362 The primorial of a success...
prmonn2 16363 Value of the primorial fun...
prmo2 16364 The primorial of 2. (Cont...
prmo3 16365 The primorial of 3. (Cont...
prmdvdsprmo 16366 The primorial of a number ...
prmdvdsprmop 16367 The primorial of a number ...
fvprmselelfz 16368 The value of the prime sel...
fvprmselgcd1 16369 The greatest common diviso...
prmolefac 16370 The primorial of a positiv...
prmodvdslcmf 16371 The primorial of a nonnega...
prmolelcmf 16372 The primorial of a positiv...
prmgaplem1 16373 Lemma for ~ prmgap : The ...
prmgaplem2 16374 Lemma for ~ prmgap : The ...
prmgaplcmlem1 16375 Lemma for ~ prmgaplcm : T...
prmgaplcmlem2 16376 Lemma for ~ prmgaplcm : T...
prmgaplem3 16377 Lemma for ~ prmgap . (Con...
prmgaplem4 16378 Lemma for ~ prmgap . (Con...
prmgaplem5 16379 Lemma for ~ prmgap : for e...
prmgaplem6 16380 Lemma for ~ prmgap : for e...
prmgaplem7 16381 Lemma for ~ prmgap . (Con...
prmgaplem8 16382 Lemma for ~ prmgap . (Con...
prmgap 16383 The prime gap theorem: for...
prmgaplcm 16384 Alternate proof of ~ prmga...
prmgapprmolem 16385 Lemma for ~ prmgapprmo : ...
prmgapprmo 16386 Alternate proof of ~ prmga...
dec2dvds 16387 Divisibility by two is obv...
dec5dvds 16388 Divisibility by five is ob...
dec5dvds2 16389 Divisibility by five is ob...
dec5nprm 16390 Divisibility by five is ob...
dec2nprm 16391 Divisibility by two is obv...
modxai 16392 Add exponents in a power m...
mod2xi 16393 Double exponents in a powe...
modxp1i 16394 Add one to an exponent in ...
mod2xnegi 16395 Version of ~ mod2xi with a...
modsubi 16396 Subtract from within a mod...
gcdi 16397 Calculate a GCD via Euclid...
gcdmodi 16398 Calculate a GCD via Euclid...
decexp2 16399 Calculate a power of two. ...
numexp0 16400 Calculate an integer power...
numexp1 16401 Calculate an integer power...
numexpp1 16402 Calculate an integer power...
numexp2x 16403 Double an integer power. ...
decsplit0b 16404 Split a decimal number int...
decsplit0 16405 Split a decimal number int...
decsplit1 16406 Split a decimal number int...
decsplit 16407 Split a decimal number int...
karatsuba 16408 The Karatsuba multiplicati...
2exp4 16409 Two to the fourth power is...
2exp6 16410 Two to the sixth power is ...
2exp8 16411 Two to the eighth power is...
2exp16 16412 Two to the sixteenth power...
3exp3 16413 Three to the third power i...
2expltfac 16414 The factorial grows faster...
cshwsidrepsw 16415 If cyclically shifting a w...
cshwsidrepswmod0 16416 If cyclically shifting a w...
cshwshashlem1 16417 If cyclically shifting a w...
cshwshashlem2 16418 If cyclically shifting a w...
cshwshashlem3 16419 If cyclically shifting a w...
cshwsdisj 16420 The singletons resulting b...
cshwsiun 16421 The set of (different!) wo...
cshwsex 16422 The class of (different!) ...
cshws0 16423 The size of the set of (di...
cshwrepswhash1 16424 The size of the set of (di...
cshwshashnsame 16425 If a word (not consisting ...
cshwshash 16426 If a word has a length bei...
prmlem0 16427 Lemma for ~ prmlem1 and ~ ...
prmlem1a 16428 A quick proof skeleton to ...
prmlem1 16429 A quick proof skeleton to ...
5prm 16430 5 is a prime number. (Con...
6nprm 16431 6 is not a prime number. ...
7prm 16432 7 is a prime number. (Con...
8nprm 16433 8 is not a prime number. ...
9nprm 16434 9 is not a prime number. ...
10nprm 16435 10 is not a prime number. ...
11prm 16436 11 is a prime number. (Co...
13prm 16437 13 is a prime number. (Co...
17prm 16438 17 is a prime number. (Co...
19prm 16439 19 is a prime number. (Co...
23prm 16440 23 is a prime number. (Co...
prmlem2 16441 Our last proving session g...
37prm 16442 37 is a prime number. (Co...
43prm 16443 43 is a prime number. (Co...
83prm 16444 83 is a prime number. (Co...
139prm 16445 139 is a prime number. (C...
163prm 16446 163 is a prime number. (C...
317prm 16447 317 is a prime number. (C...
631prm 16448 631 is a prime number. (C...
prmo4 16449 The primorial of 4. (Cont...
prmo5 16450 The primorial of 5. (Cont...
prmo6 16451 The primorial of 6. (Cont...
1259lem1 16452 Lemma for ~ 1259prm . Cal...
1259lem2 16453 Lemma for ~ 1259prm . Cal...
1259lem3 16454 Lemma for ~ 1259prm . Cal...
1259lem4 16455 Lemma for ~ 1259prm . Cal...
1259lem5 16456 Lemma for ~ 1259prm . Cal...
1259prm 16457 1259 is a prime number. (...
2503lem1 16458 Lemma for ~ 2503prm . Cal...
2503lem2 16459 Lemma for ~ 2503prm . Cal...
2503lem3 16460 Lemma for ~ 2503prm . Cal...
2503prm 16461 2503 is a prime number. (...
4001lem1 16462 Lemma for ~ 4001prm . Cal...
4001lem2 16463 Lemma for ~ 4001prm . Cal...
4001lem3 16464 Lemma for ~ 4001prm . Cal...
4001lem4 16465 Lemma for ~ 4001prm . Cal...
4001prm 16466 4001 is a prime number. (...
sloteq 16476 Equality theorem for the `...
brstruct 16480 The structure relation is ...
isstruct2 16481 The property of being a st...
structex 16482 A structure is a set. (Co...
structn0fun 16483 A structure without the em...
isstruct 16484 The property of being a st...
structcnvcnv 16485 Two ways to express the re...
structfung 16486 The converse of the conver...
structfun 16487 Convert between two kinds ...
structfn 16488 Convert between two kinds ...
slotfn 16489 A slot is a function on se...
strfvnd 16490 Deduction version of ~ str...
basfn 16491 The base set extractor is ...
wunndx 16492 Closure of the index extra...
strfvn 16493 Value of a structure compo...
strfvss 16494 A structure component extr...
wunstr 16495 Closure of a structure ind...
ndxarg 16496 Get the numeric argument f...
ndxid 16497 A structure component extr...
strndxid 16498 The value of a structure c...
reldmsets 16499 The structure override ope...
setsvalg 16500 Value of the structure rep...
setsval 16501 Value of the structure rep...
setsidvald 16502 Value of the structure rep...
fvsetsid 16503 The value of the structure...
fsets 16504 The structure replacement ...
setsdm 16505 The domain of a structure ...
setsfun 16506 A structure with replaceme...
setsfun0 16507 A structure with replaceme...
setsn0fun 16508 The value of the structure...
setsstruct2 16509 An extensible structure wi...
setsexstruct2 16510 An extensible structure wi...
setsstruct 16511 An extensible structure wi...
wunsets 16512 Closure of structure repla...
setsres 16513 The structure replacement ...
setsabs 16514 Replacing the same compone...
setscom 16515 Component-setting is commu...
strfvd 16516 Deduction version of ~ str...
strfv2d 16517 Deduction version of ~ str...
strfv2 16518 A variation on ~ strfv to ...
strfv 16519 Extract a structure compon...
strfv3 16520 Variant on ~ strfv for lar...
strssd 16521 Deduction version of ~ str...
strss 16522 Propagate component extrac...
str0 16523 All components of the empt...
base0 16524 The base set of the empty ...
strfvi 16525 Structure slot extractors ...
setsid 16526 Value of the structure rep...
setsnid 16527 Value of the structure rep...
sbcie2s 16528 A special version of class...
sbcie3s 16529 A special version of class...
baseval 16530 Value of the base set extr...
baseid 16531 Utility theorem: index-ind...
elbasfv 16532 Utility theorem: reverse c...
elbasov 16533 Utility theorem: reverse c...
strov2rcl 16534 Partial reverse closure fo...
basendx 16535 Index value of the base se...
basendxnn 16536 The index value of the bas...
basprssdmsets 16537 The pair of the base index...
reldmress 16538 The structure restriction ...
ressval 16539 Value of structure restric...
ressid2 16540 General behavior of trivia...
ressval2 16541 Value of nontrivial struct...
ressbas 16542 Base set of a structure re...
ressbas2 16543 Base set of a structure re...
ressbasss 16544 The base set of a restrict...
resslem 16545 Other elements of a struct...
ress0 16546 All restrictions of the nu...
ressid 16547 Behavior of trivial restri...
ressinbas 16548 Restriction only cares abo...
ressval3d 16549 Value of structure restric...
ressress 16550 Restriction composition la...
ressabs 16551 Restriction absorption law...
wunress 16552 Closure of structure restr...
strleun 16579 Combine two structures int...
strle1 16580 Make a structure from a si...
strle2 16581 Make a structure from a pa...
strle3 16582 Make a structure from a tr...
plusgndx 16583 Index value of the ~ df-pl...
plusgid 16584 Utility theorem: index-ind...
opelstrbas 16585 The base set of a structur...
1strstr 16586 A constructed one-slot str...
1strbas 16587 The base set of a construc...
1strwunbndx 16588 A constructed one-slot str...
1strwun 16589 A constructed one-slot str...
2strstr 16590 A constructed two-slot str...
2strbas 16591 The base set of a construc...
2strop 16592 The other slot of a constr...
2strstr1 16593 A constructed two-slot str...
2strbas1 16594 The base set of a construc...
2strop1 16595 The other slot of a constr...
basendxnplusgndx 16596 The slot for the base set ...
grpstr 16597 A constructed group is a s...
grpbase 16598 The base set of a construc...
grpplusg 16599 The operation of a constru...
ressplusg 16600 ` +g ` is unaffected by re...
grpbasex 16601 The base of an explicitly ...
grpplusgx 16602 The operation of an explic...
mulrndx 16603 Index value of the ~ df-mu...
mulrid 16604 Utility theorem: index-ind...
plusgndxnmulrndx 16605 The slot for the group (ad...
basendxnmulrndx 16606 The slot for the base set ...
rngstr 16607 A constructed ring is a st...
rngbase 16608 The base set of a construc...
rngplusg 16609 The additive operation of ...
rngmulr 16610 The multiplicative operati...
starvndx 16611 Index value of the ~ df-st...
starvid 16612 Utility theorem: index-ind...
ressmulr 16613 ` .r ` is unaffected by re...
ressstarv 16614 ` *r ` is unaffected by re...
srngstr 16615 A constructed star ring is...
srngbase 16616 The base set of a construc...
srngplusg 16617 The addition operation of ...
srngmulr 16618 The multiplication operati...
srnginvl 16619 The involution function of...
scandx 16620 Index value of the ~ df-sc...
scaid 16621 Utility theorem: index-ind...
vscandx 16622 Index value of the ~ df-vs...
vscaid 16623 Utility theorem: index-ind...
lmodstr 16624 A constructed left module ...
lmodbase 16625 The base set of a construc...
lmodplusg 16626 The additive operation of ...
lmodsca 16627 The set of scalars of a co...
lmodvsca 16628 The scalar product operati...
ipndx 16629 Index value of the ~ df-ip...
ipid 16630 Utility theorem: index-ind...
ipsstr 16631 Lemma to shorten proofs of...
ipsbase 16632 The base set of a construc...
ipsaddg 16633 The additive operation of ...
ipsmulr 16634 The multiplicative operati...
ipssca 16635 The set of scalars of a co...
ipsvsca 16636 The scalar product operati...
ipsip 16637 The multiplicative operati...
resssca 16638 ` Scalar ` is unaffected b...
ressvsca 16639 ` .s ` is unaffected by re...
ressip 16640 The inner product is unaff...
phlstr 16641 A constructed pre-Hilbert ...
phlbase 16642 The base set of a construc...
phlplusg 16643 The additive operation of ...
phlsca 16644 The ring of scalars of a c...
phlvsca 16645 The scalar product operati...
phlip 16646 The inner product (Hermiti...
tsetndx 16647 Index value of the ~ df-ts...
tsetid 16648 Utility theorem: index-ind...
topgrpstr 16649 A constructed topological ...
topgrpbas 16650 The base set of a construc...
topgrpplusg 16651 The additive operation of ...
topgrptset 16652 The topology of a construc...
resstset 16653 ` TopSet ` is unaffected b...
plendx 16654 Index value of the ~ df-pl...
pleid 16655 Utility theorem: self-refe...
otpsstr 16656 Functionality of a topolog...
otpsbas 16657 The base set of a topologi...
otpstset 16658 The open sets of a topolog...
otpsle 16659 The order of a topological...
ressle 16660 ` le ` is unaffected by re...
ocndx 16661 Index value of the ~ df-oc...
ocid 16662 Utility theorem: index-ind...
dsndx 16663 Index value of the ~ df-ds...
dsid 16664 Utility theorem: index-ind...
unifndx 16665 Index value of the ~ df-un...
unifid 16666 Utility theorem: index-ind...
odrngstr 16667 Functionality of an ordere...
odrngbas 16668 The base set of an ordered...
odrngplusg 16669 The addition operation of ...
odrngmulr 16670 The multiplication operati...
odrngtset 16671 The open sets of an ordere...
odrngle 16672 The order of an ordered me...
odrngds 16673 The metric of an ordered m...
ressds 16674 ` dist ` is unaffected by ...
homndx 16675 Index value of the ~ df-ho...
homid 16676 Utility theorem: index-ind...
ccondx 16677 Index value of the ~ df-cc...
ccoid 16678 Utility theorem: index-ind...
resshom 16679 ` Hom ` is unaffected by r...
ressco 16680 ` comp ` is unaffected by ...
slotsbhcdif 16681 The slots ` Base ` , ` Hom...
restfn 16686 The subspace topology oper...
topnfn 16687 The topology extractor fun...
restval 16688 The subspace topology indu...
elrest 16689 The predicate "is an open ...
elrestr 16690 Sufficient condition for b...
0rest 16691 Value of the structure res...
restid2 16692 The subspace topology over...
restsspw 16693 The subspace topology is a...
firest 16694 The finite intersections o...
restid 16695 The subspace topology of t...
topnval 16696 Value of the topology extr...
topnid 16697 Value of the topology extr...
topnpropd 16698 The topology extractor fun...
reldmprds 16710 The structure product is a...
prdsbasex 16712 Lemma for structure produc...
imasvalstr 16713 Structure product value is...
prdsvalstr 16714 Structure product value is...
prdsvallem 16715 Lemma for ~ prdsbas and si...
prdsval 16716 Value of the structure pro...
prdssca 16717 Scalar ring of a structure...
prdsbas 16718 Base set of a structure pr...
prdsplusg 16719 Addition in a structure pr...
prdsmulr 16720 Multiplication in a struct...
prdsvsca 16721 Scalar multiplication in a...
prdsip 16722 Inner product in a structu...
prdsle 16723 Structure product weak ord...
prdsless 16724 Closure of the order relat...
prdsds 16725 Structure product distance...
prdsdsfn 16726 Structure product distance...
prdstset 16727 Structure product topology...
prdshom 16728 Structure product hom-sets...
prdsco 16729 Structure product composit...
prdsbas2 16730 The base set of a structur...
prdsbasmpt 16731 A constructed tuple is a p...
prdsbasfn 16732 Points in the structure pr...
prdsbasprj 16733 Each point in a structure ...
prdsplusgval 16734 Value of a componentwise s...
prdsplusgfval 16735 Value of a structure produ...
prdsmulrval 16736 Value of a componentwise r...
prdsmulrfval 16737 Value of a structure produ...
prdsleval 16738 Value of the product order...
prdsdsval 16739 Value of the metric in a s...
prdsvscaval 16740 Scalar multiplication in a...
prdsvscafval 16741 Scalar multiplication of a...
prdsbas3 16742 The base set of an indexed...
prdsbasmpt2 16743 A constructed tuple is a p...
prdsbascl 16744 An element of the base has...
prdsdsval2 16745 Value of the metric in a s...
prdsdsval3 16746 Value of the metric in a s...
pwsval 16747 Value of a structure power...
pwsbas 16748 Base set of a structure po...
pwselbasb 16749 Membership in the base set...
pwselbas 16750 An element of a structure ...
pwsplusgval 16751 Value of addition in a str...
pwsmulrval 16752 Value of multiplication in...
pwsle 16753 Ordering in a structure po...
pwsleval 16754 Ordering in a structure po...
pwsvscafval 16755 Scalar multiplication in a...
pwsvscaval 16756 Scalar multiplication of a...
pwssca 16757 The ring of scalars of a s...
pwsdiagel 16758 Membership of diagonal ele...
pwssnf1o 16759 Triviality of singleton po...
imasval 16772 Value of an image structur...
imasbas 16773 The base set of an image s...
imasds 16774 The distance function of a...
imasdsfn 16775 The distance function is a...
imasdsval 16776 The distance function of a...
imasdsval2 16777 The distance function of a...
imasplusg 16778 The group operation in an ...
imasmulr 16779 The ring multiplication in...
imassca 16780 The scalar field of an ima...
imasvsca 16781 The scalar multiplication ...
imasip 16782 The inner product of an im...
imastset 16783 The topology of an image s...
imasle 16784 The ordering of an image s...
f1ocpbllem 16785 Lemma for ~ f1ocpbl . (Co...
f1ocpbl 16786 An injection is compatible...
f1ovscpbl 16787 An injection is compatible...
f1olecpbl 16788 An injection is compatible...
imasaddfnlem 16789 The image structure operat...
imasaddvallem 16790 The operation of an image ...
imasaddflem 16791 The image set operations a...
imasaddfn 16792 The image structure's grou...
imasaddval 16793 The value of an image stru...
imasaddf 16794 The image structure's grou...
imasmulfn 16795 The image structure's ring...
imasmulval 16796 The value of an image stru...
imasmulf 16797 The image structure's ring...
imasvscafn 16798 The image structure's scal...
imasvscaval 16799 The value of an image stru...
imasvscaf 16800 The image structure's scal...
imasless 16801 The order relation defined...
imasleval 16802 The value of the image str...
qusval 16803 Value of a quotient struct...
quslem 16804 The function in ~ qusval i...
qusin 16805 Restrict the equivalence r...
qusbas 16806 Base set of a quotient str...
quss 16807 The scalar field of a quot...
divsfval 16808 Value of the function in ~...
ercpbllem 16809 Lemma for ~ ercpbl . (Con...
ercpbl 16810 Translate the function com...
erlecpbl 16811 Translate the relation com...
qusaddvallem 16812 Value of an operation defi...
qusaddflem 16813 The operation of a quotien...
qusaddval 16814 The base set of an image s...
qusaddf 16815 The base set of an image s...
qusmulval 16816 The base set of an image s...
qusmulf 16817 The base set of an image s...
fnpr2o 16818 Function with a domain of ...
fnpr2ob 16819 Biconditional version of ~...
fvpr0o 16820 The value of a function wi...
fvpr1o 16821 The value of a function wi...
fvprif 16822 The value of the pair func...
xpsfrnel 16823 Elementhood in the target ...
xpsfeq 16824 A function on ` 2o ` is de...
xpsfrnel2 16825 Elementhood in the target ...
xpscf 16826 Equivalent condition for t...
xpsfval 16827 The value of the function ...
xpsff1o 16828 The function appearing in ...
xpsfrn 16829 A short expression for the...
xpsff1o2 16830 The function appearing in ...
xpsval 16831 Value of the binary struct...
xpsrnbas 16832 The indexed structure prod...
xpsbas 16833 The base set of the binary...
xpsaddlem 16834 Lemma for ~ xpsadd and ~ x...
xpsadd 16835 Value of the addition oper...
xpsmul 16836 Value of the multiplicatio...
xpssca 16837 Value of the scalar field ...
xpsvsca 16838 Value of the scalar multip...
xpsless 16839 Closure of the ordering in...
xpsle 16840 Value of the ordering in a...
ismre 16849 Property of being a Moore ...
fnmre 16850 The Moore collection gener...
mresspw 16851 A Moore collection is a su...
mress 16852 A Moore-closed subset is a...
mre1cl 16853 In any Moore collection th...
mreintcl 16854 A nonempty collection of c...
mreiincl 16855 A nonempty indexed interse...
mrerintcl 16856 The relative intersection ...
mreriincl 16857 The relative intersection ...
mreincl 16858 Two closed sets have a clo...
mreuni 16859 Since the entire base set ...
mreunirn 16860 Two ways to express the no...
ismred 16861 Properties that determine ...
ismred2 16862 Properties that determine ...
mremre 16863 The Moore collections of s...
submre 16864 The subcollection of a clo...
mrcflem 16865 The domain and range of th...
fnmrc 16866 Moore-closure is a well-be...
mrcfval 16867 Value of the function expr...
mrcf 16868 The Moore closure is a fun...
mrcval 16869 Evaluation of the Moore cl...
mrccl 16870 The Moore closure of a set...
mrcsncl 16871 The Moore closure of a sin...
mrcid 16872 The closure of a closed se...
mrcssv 16873 The closure of a set is a ...
mrcidb 16874 A set is closed iff it is ...
mrcss 16875 Closure preserves subset o...
mrcssid 16876 The closure of a set is a ...
mrcidb2 16877 A set is closed iff it con...
mrcidm 16878 The closure operation is i...
mrcsscl 16879 The closure is the minimal...
mrcuni 16880 Idempotence of closure und...
mrcun 16881 Idempotence of closure und...
mrcssvd 16882 The Moore closure of a set...
mrcssd 16883 Moore closure preserves su...
mrcssidd 16884 A set is contained in its ...
mrcidmd 16885 Moore closure is idempoten...
mressmrcd 16886 In a Moore system, if a se...
submrc 16887 In a closure system which ...
mrieqvlemd 16888 In a Moore system, if ` Y ...
mrisval 16889 Value of the set of indepe...
ismri 16890 Criterion for a set to be ...
ismri2 16891 Criterion for a subset of ...
ismri2d 16892 Criterion for a subset of ...
ismri2dd 16893 Definition of independence...
mriss 16894 An independent set of a Mo...
mrissd 16895 An independent set of a Mo...
ismri2dad 16896 Consequence of a set in a ...
mrieqvd 16897 In a Moore system, a set i...
mrieqv2d 16898 In a Moore system, a set i...
mrissmrcd 16899 In a Moore system, if an i...
mrissmrid 16900 In a Moore system, subsets...
mreexd 16901 In a Moore system, the clo...
mreexmrid 16902 In a Moore system whose cl...
mreexexlemd 16903 This lemma is used to gene...
mreexexlem2d 16904 Used in ~ mreexexlem4d to ...
mreexexlem3d 16905 Base case of the induction...
mreexexlem4d 16906 Induction step of the indu...
mreexexd 16907 Exchange-type theorem. In...
mreexdomd 16908 In a Moore system whose cl...
mreexfidimd 16909 In a Moore system whose cl...
isacs 16910 A set is an algebraic clos...
acsmre 16911 Algebraic closure systems ...
isacs2 16912 In the definition of an al...
acsfiel 16913 A set is closed in an alge...
acsfiel2 16914 A set is closed in an alge...
acsmred 16915 An algebraic closure syste...
isacs1i 16916 A closure system determine...
mreacs 16917 Algebraicity is a composab...
acsfn 16918 Algebraicity of a conditio...
acsfn0 16919 Algebraicity of a point cl...
acsfn1 16920 Algebraicity of a one-argu...
acsfn1c 16921 Algebraicity of a one-argu...
acsfn2 16922 Algebraicity of a two-argu...
iscat 16931 The predicate "is a catego...
iscatd 16932 Properties that determine ...
catidex 16933 Each object in a category ...
catideu 16934 Each object in a category ...
cidfval 16935 Each object in a category ...
cidval 16936 Each object in a category ...
cidffn 16937 The identity arrow constru...
cidfn 16938 The identity arrow operato...
catidd 16939 Deduce the identity arrow ...
iscatd2 16940 Version of ~ iscatd with a...
catidcl 16941 Each object in a category ...
catlid 16942 Left identity property of ...
catrid 16943 Right identity property of...
catcocl 16944 Closure of a composition a...
catass 16945 Associativity of compositi...
0catg 16946 Any structure with an empt...
0cat 16947 The empty set is a categor...
homffval 16948 Value of the functionalize...
fnhomeqhomf 16949 If the Hom-set operation i...
homfval 16950 Value of the functionalize...
homffn 16951 The functionalized Hom-set...
homfeq 16952 Condition for two categori...
homfeqd 16953 If two structures have the...
homfeqbas 16954 Deduce equality of base se...
homfeqval 16955 Value of the functionalize...
comfffval 16956 Value of the functionalize...
comffval 16957 Value of the functionalize...
comfval 16958 Value of the functionalize...
comfffval2 16959 Value of the functionalize...
comffval2 16960 Value of the functionalize...
comfval2 16961 Value of the functionalize...
comfffn 16962 The functionalized composi...
comffn 16963 The functionalized composi...
comfeq 16964 Condition for two categori...
comfeqd 16965 Condition for two categori...
comfeqval 16966 Equality of two compositio...
catpropd 16967 Two structures with the sa...
cidpropd 16968 Two structures with the sa...
oppcval 16971 Value of the opposite cate...
oppchomfval 16972 Hom-sets of the opposite c...
oppchom 16973 Hom-sets of the opposite c...
oppccofval 16974 Composition in the opposit...
oppcco 16975 Composition in the opposit...
oppcbas 16976 Base set of an opposite ca...
oppccatid 16977 Lemma for ~ oppccat . (Co...
oppchomf 16978 Hom-sets of the opposite c...
oppcid 16979 Identity function of an op...
oppccat 16980 An opposite category is a ...
2oppcbas 16981 The double opposite catego...
2oppchomf 16982 The double opposite catego...
2oppccomf 16983 The double opposite catego...
oppchomfpropd 16984 If two categories have the...
oppccomfpropd 16985 If two categories have the...
monfval 16990 Definition of a monomorphi...
ismon 16991 Definition of a monomorphi...
ismon2 16992 Write out the monomorphism...
monhom 16993 A monomorphism is a morphi...
moni 16994 Property of a monomorphism...
monpropd 16995 If two categories have the...
oppcmon 16996 A monomorphism in the oppo...
oppcepi 16997 An epimorphism in the oppo...
isepi 16998 Definition of an epimorphi...
isepi2 16999 Write out the epimorphism ...
epihom 17000 An epimorphism is a morphi...
epii 17001 Property of an epimorphism...
sectffval 17008 Value of the section opera...
sectfval 17009 Value of the section relat...
sectss 17010 The section relation is a ...
issect 17011 The property " ` F ` is a ...
issect2 17012 Property of being a sectio...
sectcan 17013 If ` G ` is a section of `...
sectco 17014 Composition of two section...
isofval 17015 Function value of the func...
invffval 17016 Value of the inverse relat...
invfval 17017 Value of the inverse relat...
isinv 17018 Value of the inverse relat...
invss 17019 The inverse relation is a ...
invsym 17020 The inverse relation is sy...
invsym2 17021 The inverse relation is sy...
invfun 17022 The inverse relation is a ...
isoval 17023 The isomorphisms are the d...
inviso1 17024 If ` G ` is an inverse to ...
inviso2 17025 If ` G ` is an inverse to ...
invf 17026 The inverse relation is a ...
invf1o 17027 The inverse relation is a ...
invinv 17028 The inverse of the inverse...
invco 17029 The composition of two iso...
dfiso2 17030 Alternate definition of an...
dfiso3 17031 Alternate definition of an...
inveq 17032 If there are two inverses ...
isofn 17033 The function value of the ...
isohom 17034 An isomorphism is a homomo...
isoco 17035 The composition of two iso...
oppcsect 17036 A section in the opposite ...
oppcsect2 17037 A section in the opposite ...
oppcinv 17038 An inverse in the opposite...
oppciso 17039 An isomorphism in the oppo...
sectmon 17040 If ` F ` is a section of `...
monsect 17041 If ` F ` is a monomorphism...
sectepi 17042 If ` F ` is a section of `...
episect 17043 If ` F ` is an epimorphism...
sectid 17044 The identity is a section ...
invid 17045 The inverse of the identit...
idiso 17046 The identity is an isomorp...
idinv 17047 The inverse of the identit...
invisoinvl 17048 The inverse of an isomorph...
invisoinvr 17049 The inverse of an isomorph...
invcoisoid 17050 The inverse of an isomorph...
isocoinvid 17051 The inverse of an isomorph...
rcaninv 17052 Right cancellation of an i...
cicfval 17055 The set of isomorphic obje...
brcic 17056 The relation "is isomorphi...
cic 17057 Objects ` X ` and ` Y ` in...
brcici 17058 Prove that two objects are...
cicref 17059 Isomorphism is reflexive. ...
ciclcl 17060 Isomorphism implies the le...
cicrcl 17061 Isomorphism implies the ri...
cicsym 17062 Isomorphism is symmetric. ...
cictr 17063 Isomorphism is transitive....
cicer 17064 Isomorphism is an equivale...
sscrel 17071 The subcategory subset rel...
brssc 17072 The subcategory subset rel...
sscpwex 17073 An analogue of ~ pwex for ...
subcrcl 17074 Reverse closure for the su...
sscfn1 17075 The subcategory subset rel...
sscfn2 17076 The subcategory subset rel...
ssclem 17077 Lemma for ~ ssc1 and simil...
isssc 17078 Value of the subcategory s...
ssc1 17079 Infer subset relation on o...
ssc2 17080 Infer subset relation on m...
sscres 17081 Any function restricted to...
sscid 17082 The subcategory subset rel...
ssctr 17083 The subcategory subset rel...
ssceq 17084 The subcategory subset rel...
rescval 17085 Value of the category rest...
rescval2 17086 Value of the category rest...
rescbas 17087 Base set of the category r...
reschom 17088 Hom-sets of the category r...
reschomf 17089 Hom-sets of the category r...
rescco 17090 Composition in the categor...
rescabs 17091 Restriction absorption law...
rescabs2 17092 Restriction absorption law...
issubc 17093 Elementhood in the set of ...
issubc2 17094 Elementhood in the set of ...
0ssc 17095 For any category ` C ` , t...
0subcat 17096 For any category ` C ` , t...
catsubcat 17097 For any category ` C ` , `...
subcssc 17098 An element in the set of s...
subcfn 17099 An element in the set of s...
subcss1 17100 The objects of a subcatego...
subcss2 17101 The morphisms of a subcate...
subcidcl 17102 The identity of the origin...
subccocl 17103 A subcategory is closed un...
subccatid 17104 A subcategory is a categor...
subcid 17105 The identity in a subcateg...
subccat 17106 A subcategory is a categor...
issubc3 17107 Alternate definition of a ...
fullsubc 17108 The full subcategory gener...
fullresc 17109 The category formed by str...
resscat 17110 A category restricted to a...
subsubc 17111 A subcategory of a subcate...
relfunc 17120 The set of functors is a r...
funcrcl 17121 Reverse closure for a func...
isfunc 17122 Value of the set of functo...
isfuncd 17123 Deduce that an operation i...
funcf1 17124 The object part of a funct...
funcixp 17125 The morphism part of a fun...
funcf2 17126 The morphism part of a fun...
funcfn2 17127 The morphism part of a fun...
funcid 17128 A functor maps each identi...
funcco 17129 A functor maps composition...
funcsect 17130 The image of a section und...
funcinv 17131 The image of an inverse un...
funciso 17132 The image of an isomorphis...
funcoppc 17133 A functor on categories yi...
idfuval 17134 Value of the identity func...
idfu2nd 17135 Value of the morphism part...
idfu2 17136 Value of the morphism part...
idfu1st 17137 Value of the object part o...
idfu1 17138 Value of the object part o...
idfucl 17139 The identity functor is a ...
cofuval 17140 Value of the composition o...
cofu1st 17141 Value of the object part o...
cofu1 17142 Value of the object part o...
cofu2nd 17143 Value of the morphism part...
cofu2 17144 Value of the morphism part...
cofuval2 17145 Value of the composition o...
cofucl 17146 The composition of two fun...
cofuass 17147 Functor composition is ass...
cofulid 17148 The identity functor is a ...
cofurid 17149 The identity functor is a ...
resfval 17150 Value of the functor restr...
resfval2 17151 Value of the functor restr...
resf1st 17152 Value of the functor restr...
resf2nd 17153 Value of the functor restr...
funcres 17154 A functor restricted to a ...
funcres2b 17155 Condition for a functor to...
funcres2 17156 A functor into a restricte...
wunfunc 17157 A weak universe is closed ...
funcpropd 17158 If two categories have the...
funcres2c 17159 Condition for a functor to...
fullfunc 17164 A full functor is a functo...
fthfunc 17165 A faithful functor is a fu...
relfull 17166 The set of full functors i...
relfth 17167 The set of faithful functo...
isfull 17168 Value of the set of full f...
isfull2 17169 Equivalent condition for a...
fullfo 17170 The morphism map of a full...
fulli 17171 The morphism map of a full...
isfth 17172 Value of the set of faithf...
isfth2 17173 Equivalent condition for a...
isffth2 17174 A fully faithful functor i...
fthf1 17175 The morphism map of a fait...
fthi 17176 The morphism map of a fait...
ffthf1o 17177 The morphism map of a full...
fullpropd 17178 If two categories have the...
fthpropd 17179 If two categories have the...
fulloppc 17180 The opposite functor of a ...
fthoppc 17181 The opposite functor of a ...
ffthoppc 17182 The opposite functor of a ...
fthsect 17183 A faithful functor reflect...
fthinv 17184 A faithful functor reflect...
fthmon 17185 A faithful functor reflect...
fthepi 17186 A faithful functor reflect...
ffthiso 17187 A fully faithful functor r...
fthres2b 17188 Condition for a faithful f...
fthres2c 17189 Condition for a faithful f...
fthres2 17190 A faithful functor into a ...
idffth 17191 The identity functor is a ...
cofull 17192 The composition of two ful...
cofth 17193 The composition of two fai...
coffth 17194 The composition of two ful...
rescfth 17195 The inclusion functor from...
ressffth 17196 The inclusion functor from...
fullres2c 17197 Condition for a full funct...
ffthres2c 17198 Condition for a fully fait...
fnfuc 17203 The ` FuncCat ` operation ...
natfval 17204 Value of the function givi...
isnat 17205 Property of being a natura...
isnat2 17206 Property of being a natura...
natffn 17207 The natural transformation...
natrcl 17208 Reverse closure for a natu...
nat1st2nd 17209 Rewrite the natural transf...
natixp 17210 A natural transformation i...
natcl 17211 A component of a natural t...
natfn 17212 A natural transformation i...
nati 17213 Naturality property of a n...
wunnat 17214 A weak universe is closed ...
catstr 17215 A category structure is a ...
fucval 17216 Value of the functor categ...
fuccofval 17217 Value of the functor categ...
fucbas 17218 The objects of the functor...
fuchom 17219 The morphisms in the funct...
fucco 17220 Value of the composition o...
fuccoval 17221 Value of the functor categ...
fuccocl 17222 The composition of two nat...
fucidcl 17223 The identity natural trans...
fuclid 17224 Left identity of natural t...
fucrid 17225 Right identity of natural ...
fucass 17226 Associativity of natural t...
fuccatid 17227 The functor category is a ...
fuccat 17228 The functor category is a ...
fucid 17229 The identity morphism in t...
fucsect 17230 Two natural transformation...
fucinv 17231 Two natural transformation...
invfuc 17232 If ` V ( x ) ` is an inver...
fuciso 17233 A natural transformation i...
natpropd 17234 If two categories have the...
fucpropd 17235 If two categories have the...
initorcl 17242 Reverse closure for an ini...
termorcl 17243 Reverse closure for a term...
zeroorcl 17244 Reverse closure for a zero...
initoval 17245 The value of the initial o...
termoval 17246 The value of the terminal ...
zerooval 17247 The value of the zero obje...
isinito 17248 The predicate "is an initi...
istermo 17249 The predicate "is a termin...
iszeroo 17250 The predicate "is a zero o...
isinitoi 17251 Implication of a class bei...
istermoi 17252 Implication of a class bei...
initoid 17253 For an initial object, the...
termoid 17254 For a terminal object, the...
initoo 17255 An initial object is an ob...
termoo 17256 A terminal object is an ob...
iszeroi 17257 Implication of a class bei...
2initoinv 17258 Morphisms between two init...
initoeu1 17259 Initial objects are essent...
initoeu1w 17260 Initial objects are essent...
initoeu2lem0 17261 Lemma 0 for ~ initoeu2 . ...
initoeu2lem1 17262 Lemma 1 for ~ initoeu2 . ...
initoeu2lem2 17263 Lemma 2 for ~ initoeu2 . ...
initoeu2 17264 Initial objects are essent...
2termoinv 17265 Morphisms between two term...
termoeu1 17266 Terminal objects are essen...
termoeu1w 17267 Terminal objects are essen...
homarcl 17276 Reverse closure for an arr...
homafval 17277 Value of the disjointified...
homaf 17278 Functionality of the disjo...
homaval 17279 Value of the disjointified...
elhoma 17280 Value of the disjointified...
elhomai 17281 Produce an arrow from a mo...
elhomai2 17282 Produce an arrow from a mo...
homarcl2 17283 Reverse closure for the do...
homarel 17284 An arrow is an ordered pai...
homa1 17285 The first component of an ...
homahom2 17286 The second component of an...
homahom 17287 The second component of an...
homadm 17288 The domain of an arrow wit...
homacd 17289 The codomain of an arrow w...
homadmcd 17290 Decompose an arrow into do...
arwval 17291 The set of arrows is the u...
arwrcl 17292 The first component of an ...
arwhoma 17293 An arrow is contained in t...
homarw 17294 A hom-set is a subset of t...
arwdm 17295 The domain of an arrow is ...
arwcd 17296 The codomain of an arrow i...
dmaf 17297 The domain function is a f...
cdaf 17298 The codomain function is a...
arwhom 17299 The second component of an...
arwdmcd 17300 Decompose an arrow into do...
idafval 17305 Value of the identity arro...
idaval 17306 Value of the identity arro...
ida2 17307 Morphism part of the ident...
idahom 17308 Domain and codomain of the...
idadm 17309 Domain of the identity arr...
idacd 17310 Codomain of the identity a...
idaf 17311 The identity arrow functio...
coafval 17312 The value of the compositi...
eldmcoa 17313 A pair ` <. G , F >. ` is ...
dmcoass 17314 The domain of composition ...
homdmcoa 17315 If ` F : X --> Y ` and ` G...
coaval 17316 Value of composition for c...
coa2 17317 The morphism part of arrow...
coahom 17318 The composition of two com...
coapm 17319 Composition of arrows is a...
arwlid 17320 Left identity of a categor...
arwrid 17321 Right identity of a catego...
arwass 17322 Associativity of compositi...
setcval 17325 Value of the category of s...
setcbas 17326 Set of objects of the cate...
setchomfval 17327 Set of arrows of the categ...
setchom 17328 Set of arrows of the categ...
elsetchom 17329 A morphism of sets is a fu...
setccofval 17330 Composition in the categor...
setcco 17331 Composition in the categor...
setccatid 17332 Lemma for ~ setccat . (Co...
setccat 17333 The category of sets is a ...
setcid 17334 The identity arrow in the ...
setcmon 17335 A monomorphism of sets is ...
setcepi 17336 An epimorphism of sets is ...
setcsect 17337 A section in the category ...
setcinv 17338 An inverse in the category...
setciso 17339 An isomorphism in the cate...
resssetc 17340 The restriction of the cat...
funcsetcres2 17341 A functor into a smaller c...
catcval 17344 Value of the category of c...
catcbas 17345 Set of objects of the cate...
catchomfval 17346 Set of arrows of the categ...
catchom 17347 Set of arrows of the categ...
catccofval 17348 Composition in the categor...
catcco 17349 Composition in the categor...
catccatid 17350 Lemma for ~ catccat . (Co...
catcid 17351 The identity arrow in the ...
catccat 17352 The category of categories...
resscatc 17353 The restriction of the cat...
catcisolem 17354 Lemma for ~ catciso . (Co...
catciso 17355 A functor is an isomorphis...
catcoppccl 17356 The category of categories...
catcfuccl 17357 The category of categories...
fncnvimaeqv 17358 The inverse images of the ...
bascnvimaeqv 17359 The inverse image of the u...
estrcval 17362 Value of the category of e...
estrcbas 17363 Set of objects of the cate...
estrchomfval 17364 Set of morphisms ("arrows"...
estrchom 17365 The morphisms between exte...
elestrchom 17366 A morphism between extensi...
estrccofval 17367 Composition in the categor...
estrcco 17368 Composition in the categor...
estrcbasbas 17369 An element of the base set...
estrccatid 17370 Lemma for ~ estrccat . (C...
estrccat 17371 The category of extensible...
estrcid 17372 The identity arrow in the ...
estrchomfn 17373 The Hom-set operation in t...
estrchomfeqhom 17374 The functionalized Hom-set...
estrreslem1 17375 Lemma 1 for ~ estrres . (...
estrreslem2 17376 Lemma 2 for ~ estrres . (...
estrres 17377 Any restriction of a categ...
funcestrcsetclem1 17378 Lemma 1 for ~ funcestrcset...
funcestrcsetclem2 17379 Lemma 2 for ~ funcestrcset...
funcestrcsetclem3 17380 Lemma 3 for ~ funcestrcset...
funcestrcsetclem4 17381 Lemma 4 for ~ funcestrcset...
funcestrcsetclem5 17382 Lemma 5 for ~ funcestrcset...
funcestrcsetclem6 17383 Lemma 6 for ~ funcestrcset...
funcestrcsetclem7 17384 Lemma 7 for ~ funcestrcset...
funcestrcsetclem8 17385 Lemma 8 for ~ funcestrcset...
funcestrcsetclem9 17386 Lemma 9 for ~ funcestrcset...
funcestrcsetc 17387 The "natural forgetful fun...
fthestrcsetc 17388 The "natural forgetful fun...
fullestrcsetc 17389 The "natural forgetful fun...
equivestrcsetc 17390 The "natural forgetful fun...
setc1strwun 17391 A constructed one-slot str...
funcsetcestrclem1 17392 Lemma 1 for ~ funcsetcestr...
funcsetcestrclem2 17393 Lemma 2 for ~ funcsetcestr...
funcsetcestrclem3 17394 Lemma 3 for ~ funcsetcestr...
embedsetcestrclem 17395 Lemma for ~ embedsetcestrc...
funcsetcestrclem4 17396 Lemma 4 for ~ funcsetcestr...
funcsetcestrclem5 17397 Lemma 5 for ~ funcsetcestr...
funcsetcestrclem6 17398 Lemma 6 for ~ funcsetcestr...
funcsetcestrclem7 17399 Lemma 7 for ~ funcsetcestr...
funcsetcestrclem8 17400 Lemma 8 for ~ funcsetcestr...
funcsetcestrclem9 17401 Lemma 9 for ~ funcsetcestr...
funcsetcestrc 17402 The "embedding functor" fr...
fthsetcestrc 17403 The "embedding functor" fr...
fullsetcestrc 17404 The "embedding functor" fr...
embedsetcestrc 17405 The "embedding functor" fr...
fnxpc 17414 The binary product of cate...
xpcval 17415 Value of the binary produc...
xpcbas 17416 Set of objects of the bina...
xpchomfval 17417 Set of morphisms of the bi...
xpchom 17418 Set of morphisms of the bi...
relxpchom 17419 A hom-set in the binary pr...
xpccofval 17420 Value of composition in th...
xpcco 17421 Value of composition in th...
xpcco1st 17422 Value of composition in th...
xpcco2nd 17423 Value of composition in th...
xpchom2 17424 Value of the set of morphi...
xpcco2 17425 Value of composition in th...
xpccatid 17426 The product of two categor...
xpcid 17427 The identity morphism in t...
xpccat 17428 The product of two categor...
1stfval 17429 Value of the first project...
1stf1 17430 Value of the first project...
1stf2 17431 Value of the first project...
2ndfval 17432 Value of the first project...
2ndf1 17433 Value of the first project...
2ndf2 17434 Value of the first project...
1stfcl 17435 The first projection funct...
2ndfcl 17436 The second projection func...
prfval 17437 Value of the pairing funct...
prf1 17438 Value of the pairing funct...
prf2fval 17439 Value of the pairing funct...
prf2 17440 Value of the pairing funct...
prfcl 17441 The pairing of functors ` ...
prf1st 17442 Cancellation of pairing wi...
prf2nd 17443 Cancellation of pairing wi...
1st2ndprf 17444 Break a functor into a pro...
catcxpccl 17445 The category of categories...
xpcpropd 17446 If two categories have the...
evlfval 17455 Value of the evaluation fu...
evlf2 17456 Value of the evaluation fu...
evlf2val 17457 Value of the evaluation na...
evlf1 17458 Value of the evaluation fu...
evlfcllem 17459 Lemma for ~ evlfcl . (Con...
evlfcl 17460 The evaluation functor is ...
curfval 17461 Value of the curry functor...
curf1fval 17462 Value of the object part o...
curf1 17463 Value of the object part o...
curf11 17464 Value of the double evalua...
curf12 17465 The partially evaluated cu...
curf1cl 17466 The partially evaluated cu...
curf2 17467 Value of the curry functor...
curf2val 17468 Value of a component of th...
curf2cl 17469 The curry functor at a mor...
curfcl 17470 The curry functor of a fun...
curfpropd 17471 If two categories have the...
uncfval 17472 Value of the uncurry funct...
uncfcl 17473 The uncurry operation take...
uncf1 17474 Value of the uncurry funct...
uncf2 17475 Value of the uncurry funct...
curfuncf 17476 Cancellation of curry with...
uncfcurf 17477 Cancellation of uncurry wi...
diagval 17478 Define the diagonal functo...
diagcl 17479 The diagonal functor is a ...
diag1cl 17480 The constant functor of ` ...
diag11 17481 Value of the constant func...
diag12 17482 Value of the constant func...
diag2 17483 Value of the diagonal func...
diag2cl 17484 The diagonal functor at a ...
curf2ndf 17485 As shown in ~ diagval , th...
hofval 17490 Value of the Hom functor, ...
hof1fval 17491 The object part of the Hom...
hof1 17492 The object part of the Hom...
hof2fval 17493 The morphism part of the H...
hof2val 17494 The morphism part of the H...
hof2 17495 The morphism part of the H...
hofcllem 17496 Lemma for ~ hofcl . (Cont...
hofcl 17497 Closure of the Hom functor...
oppchofcl 17498 Closure of the opposite Ho...
yonval 17499 Value of the Yoneda embedd...
yoncl 17500 The Yoneda embedding is a ...
yon1cl 17501 The Yoneda embedding at an...
yon11 17502 Value of the Yoneda embedd...
yon12 17503 Value of the Yoneda embedd...
yon2 17504 Value of the Yoneda embedd...
hofpropd 17505 If two categories have the...
yonpropd 17506 If two categories have the...
oppcyon 17507 Value of the opposite Yone...
oyoncl 17508 The opposite Yoneda embedd...
oyon1cl 17509 The opposite Yoneda embedd...
yonedalem1 17510 Lemma for ~ yoneda . (Con...
yonedalem21 17511 Lemma for ~ yoneda . (Con...
yonedalem3a 17512 Lemma for ~ yoneda . (Con...
yonedalem4a 17513 Lemma for ~ yoneda . (Con...
yonedalem4b 17514 Lemma for ~ yoneda . (Con...
yonedalem4c 17515 Lemma for ~ yoneda . (Con...
yonedalem22 17516 Lemma for ~ yoneda . (Con...
yonedalem3b 17517 Lemma for ~ yoneda . (Con...
yonedalem3 17518 Lemma for ~ yoneda . (Con...
yonedainv 17519 The Yoneda Lemma with expl...
yonffthlem 17520 Lemma for ~ yonffth . (Co...
yoneda 17521 The Yoneda Lemma. There i...
yonffth 17522 The Yoneda Lemma. The Yon...
yoniso 17523 If the codomain is recover...
isprs 17528 Property of being a preord...
prslem 17529 Lemma for ~ prsref and ~ p...
prsref 17530 "Less than or equal to" is...
prstr 17531 "Less than or equal to" is...
isdrs 17532 Property of being a direct...
drsdir 17533 Direction of a directed se...
drsprs 17534 A directed set is a proset...
drsbn0 17535 The base of a directed set...
drsdirfi 17536 Any _finite_ number of ele...
isdrs2 17537 Directed sets may be defin...
ispos 17545 The predicate "is a poset....
ispos2 17546 A poset is an antisymmetri...
posprs 17547 A poset is a proset. (Con...
posi 17548 Lemma for poset properties...
posref 17549 A poset ordering is reflex...
posasymb 17550 A poset ordering is asymme...
postr 17551 A poset ordering is transi...
0pos 17552 Technical lemma to simplif...
isposd 17553 Properties that determine ...
isposi 17554 Properties that determine ...
isposix 17555 Properties that determine ...
pltfval 17557 Value of the less-than rel...
pltval 17558 Less-than relation. ( ~ d...
pltle 17559 "Less than" implies "less ...
pltne 17560 The "less than" relation i...
pltirr 17561 The "less than" relation i...
pleval2i 17562 One direction of ~ pleval2...
pleval2 17563 "Less than or equal to" in...
pltnle 17564 "Less than" implies not co...
pltval3 17565 Alternate expression for t...
pltnlt 17566 The less-than relation imp...
pltn2lp 17567 The less-than relation has...
plttr 17568 The less-than relation is ...
pltletr 17569 Transitive law for chained...
plelttr 17570 Transitive law for chained...
pospo 17571 Write a poset structure in...
lubfval 17576 Value of the least upper b...
lubdm 17577 Domain of the least upper ...
lubfun 17578 The LUB is a function. (C...
lubeldm 17579 Member of the domain of th...
lubelss 17580 A member of the domain of ...
lubeu 17581 Unique existence proper of...
lubval 17582 Value of the least upper b...
lubcl 17583 The least upper bound func...
lubprop 17584 Properties of greatest low...
luble 17585 The greatest lower bound i...
lublecllem 17586 Lemma for ~ lublecl and ~ ...
lublecl 17587 The set of all elements le...
lubid 17588 The LUB of elements less t...
glbfval 17589 Value of the greatest lowe...
glbdm 17590 Domain of the greatest low...
glbfun 17591 The GLB is a function. (C...
glbeldm 17592 Member of the domain of th...
glbelss 17593 A member of the domain of ...
glbeu 17594 Unique existence proper of...
glbval 17595 Value of the greatest lowe...
glbcl 17596 The least upper bound func...
glbprop 17597 Properties of greatest low...
glble 17598 The greatest lower bound i...
joinfval 17599 Value of join function for...
joinfval2 17600 Value of join function for...
joindm 17601 Domain of join function fo...
joindef 17602 Two ways to say that a joi...
joinval 17603 Join value. Since both si...
joincl 17604 Closure of join of element...
joindmss 17605 Subset property of domain ...
joinval2lem 17606 Lemma for ~ joinval2 and ~...
joinval2 17607 Value of join for a poset ...
joineu 17608 Uniqueness of join of elem...
joinlem 17609 Lemma for join properties....
lejoin1 17610 A join's first argument is...
lejoin2 17611 A join's second argument i...
joinle 17612 A join is less than or equ...
meetfval 17613 Value of meet function for...
meetfval2 17614 Value of meet function for...
meetdm 17615 Domain of meet function fo...
meetdef 17616 Two ways to say that a mee...
meetval 17617 Meet value. Since both si...
meetcl 17618 Closure of meet of element...
meetdmss 17619 Subset property of domain ...
meetval2lem 17620 Lemma for ~ meetval2 and ~...
meetval2 17621 Value of meet for a poset ...
meeteu 17622 Uniqueness of meet of elem...
meetlem 17623 Lemma for meet properties....
lemeet1 17624 A meet's first argument is...
lemeet2 17625 A meet's second argument i...
meetle 17626 A meet is less than or equ...
joincomALT 17627 The join of a poset commut...
joincom 17628 The join of a poset commut...
meetcomALT 17629 The meet of a poset commut...
meetcom 17630 The meet of a poset commut...
istos 17633 The predicate "is a toset....
tosso 17634 Write the totally ordered ...
p0val 17639 Value of poset zero. (Con...
p1val 17640 Value of poset zero. (Con...
p0le 17641 Any element is less than o...
ple1 17642 Any element is less than o...
islat 17645 The predicate "is a lattic...
latcl2 17646 The join and meet of any t...
latlem 17647 Lemma for lattice properti...
latpos 17648 A lattice is a poset. (Co...
latjcl 17649 Closure of join operation ...
latmcl 17650 Closure of meet operation ...
latref 17651 A lattice ordering is refl...
latasymb 17652 A lattice ordering is asym...
latasym 17653 A lattice ordering is asym...
lattr 17654 A lattice ordering is tran...
latasymd 17655 Deduce equality from latti...
lattrd 17656 A lattice ordering is tran...
latjcom 17657 The join of a lattice comm...
latlej1 17658 A join's first argument is...
latlej2 17659 A join's second argument i...
latjle12 17660 A join is less than or equ...
latleeqj1 17661 "Less than or equal to" in...
latleeqj2 17662 "Less than or equal to" in...
latjlej1 17663 Add join to both sides of ...
latjlej2 17664 Add join to both sides of ...
latjlej12 17665 Add join to both sides of ...
latnlej 17666 An idiom to express that a...
latnlej1l 17667 An idiom to express that a...
latnlej1r 17668 An idiom to express that a...
latnlej2 17669 An idiom to express that a...
latnlej2l 17670 An idiom to express that a...
latnlej2r 17671 An idiom to express that a...
latjidm 17672 Lattice join is idempotent...
latmcom 17673 The join of a lattice comm...
latmle1 17674 A meet is less than or equ...
latmle2 17675 A meet is less than or equ...
latlem12 17676 An element is less than or...
latleeqm1 17677 "Less than or equal to" in...
latleeqm2 17678 "Less than or equal to" in...
latmlem1 17679 Add meet to both sides of ...
latmlem2 17680 Add meet to both sides of ...
latmlem12 17681 Add join to both sides of ...
latnlemlt 17682 Negation of "less than or ...
latnle 17683 Equivalent expressions for...
latmidm 17684 Lattice join is idempotent...
latabs1 17685 Lattice absorption law. F...
latabs2 17686 Lattice absorption law. F...
latledi 17687 An ortholattice is distrib...
latmlej11 17688 Ordering of a meet and joi...
latmlej12 17689 Ordering of a meet and joi...
latmlej21 17690 Ordering of a meet and joi...
latmlej22 17691 Ordering of a meet and joi...
lubsn 17692 The least upper bound of a...
latjass 17693 Lattice join is associativ...
latj12 17694 Swap 1st and 2nd members o...
latj32 17695 Swap 2nd and 3rd members o...
latj13 17696 Swap 1st and 3rd members o...
latj31 17697 Swap 2nd and 3rd members o...
latjrot 17698 Rotate lattice join of 3 c...
latj4 17699 Rearrangement of lattice j...
latj4rot 17700 Rotate lattice join of 4 c...
latjjdi 17701 Lattice join distributes o...
latjjdir 17702 Lattice join distributes o...
mod1ile 17703 The weak direction of the ...
mod2ile 17704 The weak direction of the ...
isclat 17707 The predicate "is a comple...
clatpos 17708 A complete lattice is a po...
clatlem 17709 Lemma for properties of a ...
clatlubcl 17710 Any subset of the base set...
clatlubcl2 17711 Any subset of the base set...
clatglbcl 17712 Any subset of the base set...
clatglbcl2 17713 Any subset of the base set...
clatl 17714 A complete lattice is a la...
isglbd 17715 Properties that determine ...
lublem 17716 Lemma for the least upper ...
lubub 17717 The LUB of a complete latt...
lubl 17718 The LUB of a complete latt...
lubss 17719 Subset law for least upper...
lubel 17720 An element of a set is les...
lubun 17721 The LUB of a union. (Cont...
clatglb 17722 Properties of greatest low...
clatglble 17723 The greatest lower bound i...
clatleglb 17724 Two ways of expressing "le...
clatglbss 17725 Subset law for greatest lo...
oduval 17728 Value of an order dual str...
oduleval 17729 Value of the less-equal re...
oduleg 17730 Truth of the less-equal re...
odubas 17731 Base set of an order dual ...
pospropd 17732 Posethood is determined on...
odupos 17733 Being a poset is a self-du...
oduposb 17734 Being a poset is a self-du...
meet0 17735 Lemma for ~ odujoin . (Co...
join0 17736 Lemma for ~ odumeet . (Co...
oduglb 17737 Greatest lower bounds in a...
odumeet 17738 Meets in a dual order are ...
odulub 17739 Least upper bounds in a du...
odujoin 17740 Joins in a dual order are ...
odulatb 17741 Being a lattice is self-du...
oduclatb 17742 Being a complete lattice i...
odulat 17743 Being a lattice is self-du...
poslubmo 17744 Least upper bounds in a po...
posglbmo 17745 Greatest lower bounds in a...
poslubd 17746 Properties which determine...
poslubdg 17747 Properties which determine...
posglbd 17748 Properties which determine...
ipostr 17751 The structure of ~ df-ipo ...
ipoval 17752 Value of the inclusion pos...
ipobas 17753 Base set of the inclusion ...
ipolerval 17754 Relation of the inclusion ...
ipotset 17755 Topology of the inclusion ...
ipole 17756 Weak order condition of th...
ipolt 17757 Strict order condition of ...
ipopos 17758 The inclusion poset on a f...
isipodrs 17759 Condition for a family of ...
ipodrscl 17760 Direction by inclusion as ...
ipodrsfi 17761 Finite upper bound propert...
fpwipodrs 17762 The finite subsets of any ...
ipodrsima 17763 The monotone image of a di...
isacs3lem 17764 An algebraic closure syste...
acsdrsel 17765 An algebraic closure syste...
isacs4lem 17766 In a closure system in whi...
isacs5lem 17767 If closure commutes with d...
acsdrscl 17768 In an algebraic closure sy...
acsficl 17769 A closure in an algebraic ...
isacs5 17770 A closure system is algebr...
isacs4 17771 A closure system is algebr...
isacs3 17772 A closure system is algebr...
acsficld 17773 In an algebraic closure sy...
acsficl2d 17774 In an algebraic closure sy...
acsfiindd 17775 In an algebraic closure sy...
acsmapd 17776 In an algebraic closure sy...
acsmap2d 17777 In an algebraic closure sy...
acsinfd 17778 In an algebraic closure sy...
acsdomd 17779 In an algebraic closure sy...
acsinfdimd 17780 In an algebraic closure sy...
acsexdimd 17781 In an algebraic closure sy...
mrelatglb 17782 Greatest lower bounds in a...
mrelatglb0 17783 The empty intersection in ...
mrelatlub 17784 Least upper bounds in a Mo...
mreclatBAD 17785 A Moore space is a complet...
latmass 17786 Lattice meet is associativ...
latdisdlem 17787 Lemma for ~ latdisd . (Co...
latdisd 17788 In a lattice, joins distri...
isdlat 17791 Property of being a distri...
dlatmjdi 17792 In a distributive lattice,...
dlatl 17793 A distributive lattice is ...
odudlatb 17794 The dual of a distributive...
dlatjmdi 17795 In a distributive lattice,...
isps 17800 The predicate "is a poset"...
psrel 17801 A poset is a relation. (C...
psref2 17802 A poset is antisymmetric a...
pstr2 17803 A poset is transitive. (C...
pslem 17804 Lemma for ~ psref and othe...
psdmrn 17805 The domain and range of a ...
psref 17806 A poset is reflexive. (Co...
psrn 17807 The range of a poset equal...
psasym 17808 A poset is antisymmetric. ...
pstr 17809 A poset is transitive. (C...
cnvps 17810 The converse of a poset is...
cnvpsb 17811 The converse of a poset is...
psss 17812 Any subset of a partially ...
psssdm2 17813 Field of a subposet. (Con...
psssdm 17814 Field of a subposet. (Con...
istsr 17815 The predicate is a toset. ...
istsr2 17816 The predicate is a toset. ...
tsrlin 17817 A toset is a linear order....
tsrlemax 17818 Two ways of saying a numbe...
tsrps 17819 A toset is a poset. (Cont...
cnvtsr 17820 The converse of a toset is...
tsrss 17821 Any subset of a totally or...
ledm 17822 The domain of ` <_ ` is ` ...
lern 17823 The range of ` <_ ` is ` R...
lefld 17824 The field of the 'less or ...
letsr 17825 The "less than or equal to...
isdir 17830 A condition for a relation...
reldir 17831 A direction is a relation....
dirdm 17832 A direction's domain is eq...
dirref 17833 A direction is reflexive. ...
dirtr 17834 A direction is transitive....
dirge 17835 For any two elements of a ...
tsrdir 17836 A totally ordered set is a...
ismgm 17841 The predicate "is a magma"...
ismgmn0 17842 The predicate "is a magma"...
mgmcl 17843 Closure of the operation o...
isnmgm 17844 A condition for a structur...
mgmsscl 17845 If the base set of a magma...
plusffval 17846 The group addition operati...
plusfval 17847 The group addition operati...
plusfeq 17848 If the addition operation ...
plusffn 17849 The group addition operati...
mgmplusf 17850 The group addition functio...
issstrmgm 17851 Characterize a substructur...
intopsn 17852 The internal operation for...
mgmb1mgm1 17853 The only magma with a base...
mgm0 17854 Any set with an empty base...
mgm0b 17855 The structure with an empt...
mgm1 17856 The structure with one ele...
opifismgm 17857 A structure with a group a...
mgmidmo 17858 A two-sided identity eleme...
grpidval 17859 The value of the identity ...
grpidpropd 17860 If two structures have the...
fn0g 17861 The group zero extractor i...
0g0 17862 The identity element funct...
ismgmid 17863 The identity element of a ...
mgmidcl 17864 The identity element of a ...
mgmlrid 17865 The identity element of a ...
ismgmid2 17866 Show that a given element ...
lidrideqd 17867 If there is a left and rig...
lidrididd 17868 If there is a left and rig...
grpidd 17869 Deduce the identity elemen...
mgmidsssn0 17870 Property of the set of ide...
grprinvlem 17871 Lemma for ~ grprinvd . (C...
grprinvd 17872 Deduce right inverse from ...
grpridd 17873 Deduce right identity from...
gsumvalx 17874 Expand out the substitutio...
gsumval 17875 Expand out the substitutio...
gsumpropd 17876 The group sum depends only...
gsumpropd2lem 17877 Lemma for ~ gsumpropd2 . ...
gsumpropd2 17878 A stronger version of ~ gs...
gsummgmpropd 17879 A stronger version of ~ gs...
gsumress 17880 The group sum in a substru...
gsumval1 17881 Value of the group sum ope...
gsum0 17882 Value of the empty group s...
gsumval2a 17883 Value of the group sum ope...
gsumval2 17884 Value of the group sum ope...
gsumsplit1r 17885 Splitting off the rightmos...
gsumprval 17886 Value of the group sum ope...
gsumpr12val 17887 Value of the group sum ope...
issgrp 17890 The predicate "is a semigr...
issgrpv 17891 The predicate "is a semigr...
issgrpn0 17892 The predicate "is a semigr...
isnsgrp 17893 A condition for a structur...
sgrpmgm 17894 A semigroup is a magma. (...
sgrpass 17895 A semigroup operation is a...
sgrp0 17896 Any set with an empty base...
sgrp0b 17897 The structure with an empt...
sgrp1 17898 The structure with one ele...
ismnddef 17901 The predicate "is a monoid...
ismnd 17902 The predicate "is a monoid...
isnmnd 17903 A condition for a structur...
sgrpidmnd 17904 A semigroup with an identi...
mndsgrp 17905 A monoid is a semigroup. ...
mndmgm 17906 A monoid is a magma. (Con...
mndcl 17907 Closure of the operation o...
mndass 17908 A monoid operation is asso...
mndid 17909 A monoid has a two-sided i...
mndideu 17910 The two-sided identity ele...
mnd32g 17911 Commutative/associative la...
mnd12g 17912 Commutative/associative la...
mnd4g 17913 Commutative/associative la...
mndidcl 17914 The identity element of a ...
mndbn0 17915 The base set of a monoid i...
hashfinmndnn 17916 A finite monoid has positi...
mndplusf 17917 The group addition operati...
mndlrid 17918 A monoid's identity elemen...
mndlid 17919 The identity element of a ...
mndrid 17920 The identity element of a ...
ismndd 17921 Deduce a monoid from its p...
mndpfo 17922 The addition operation of ...
mndfo 17923 The addition operation of ...
mndpropd 17924 If two structures have the...
mndprop 17925 If two structures have the...
issubmnd 17926 Characterize a submonoid b...
ress0g 17927 ` 0g ` is unaffected by re...
submnd0 17928 The zero of a submonoid is...
mndinvmod 17929 Uniqueness of an inverse e...
prdsplusgcl 17930 Structure product pointwis...
prdsidlem 17931 Characterization of identi...
prdsmndd 17932 The product of a family of...
prds0g 17933 Zero in a product of monoi...
pwsmnd 17934 The structure power of a m...
pws0g 17935 Zero in a product of monoi...
imasmnd2 17936 The image structure of a m...
imasmnd 17937 The image structure of a m...
imasmndf1 17938 The image of a monoid unde...
xpsmnd 17939 The binary product of mono...
mnd1 17940 The (smallest) structure r...
mnd1id 17941 The singleton element of a...
ismhm 17946 Property of a monoid homom...
mhmrcl1 17947 Reverse closure of a monoi...
mhmrcl2 17948 Reverse closure of a monoi...
mhmf 17949 A monoid homomorphism is a...
mhmpropd 17950 Monoid homomorphism depend...
mhmlin 17951 A monoid homomorphism comm...
mhm0 17952 A monoid homomorphism pres...
idmhm 17953 The identity homomorphism ...
mhmf1o 17954 A monoid homomorphism is b...
submrcl 17955 Reverse closure for submon...
issubm 17956 Expand definition of a sub...
issubm2 17957 Submonoids are subsets tha...
issubmndb 17958 The submonoid predicate. ...
issubmd 17959 Deduction for proving a su...
mndissubm 17960 If the base set of a monoi...
resmndismnd 17961 If the base set of a monoi...
submss 17962 Submonoids are subsets of ...
submid 17963 Every monoid is trivially ...
subm0cl 17964 Submonoids contain zero. ...
submcl 17965 Submonoids are closed unde...
submmnd 17966 Submonoids are themselves ...
submbas 17967 The base set of a submonoi...
subm0 17968 Submonoids have the same i...
subsubm 17969 A submonoid of a submonoid...
0subm 17970 The zero submonoid of an a...
insubm 17971 The intersection of two su...
0mhm 17972 The constant zero linear f...
resmhm 17973 Restriction of a monoid ho...
resmhm2 17974 One direction of ~ resmhm2...
resmhm2b 17975 Restriction of the codomai...
mhmco 17976 The composition of monoid ...
mhmima 17977 The homomorphic image of a...
mhmeql 17978 The equalizer of two monoi...
submacs 17979 Submonoids are an algebrai...
mndind 17980 Induction in a monoid. In...
prdspjmhm 17981 A projection from a produc...
pwspjmhm 17982 A projection from a produc...
pwsdiagmhm 17983 Diagonal monoid homomorphi...
pwsco1mhm 17984 Right composition with a f...
pwsco2mhm 17985 Left composition with a mo...
gsumvallem2 17986 Lemma for properties of th...
gsumsubm 17987 Evaluate a group sum in a ...
gsumz 17988 Value of a group sum over ...
gsumwsubmcl 17989 Closure of the composite i...
gsumws1 17990 A singleton composite reco...
gsumwcl 17991 Closure of the composite o...
gsumsgrpccat 17992 Homomorphic property of no...
gsumccatOLD 17993 Obsolete version of ~ gsum...
gsumccat 17994 Homomorphic property of co...
gsumws2 17995 Valuation of a pair in a m...
gsumccatsn 17996 Homomorphic property of co...
gsumspl 17997 The primary purpose of the...
gsumwmhm 17998 Behavior of homomorphisms ...
gsumwspan 17999 The submonoid generated by...
frmdval 18004 Value of the free monoid c...
frmdbas 18005 The base set of a free mon...
frmdelbas 18006 An element of the base set...
frmdplusg 18007 The monoid operation of a ...
frmdadd 18008 Value of the monoid operat...
vrmdfval 18009 The canonical injection fr...
vrmdval 18010 The value of the generatin...
vrmdf 18011 The mapping from the index...
frmdmnd 18012 A free monoid is a monoid....
frmd0 18013 The identity of the free m...
frmdsssubm 18014 The set of words taking va...
frmdgsum 18015 Any word in a free monoid ...
frmdss2 18016 A subset of generators is ...
frmdup1 18017 Any assignment of the gene...
frmdup2 18018 The evaluation map has the...
frmdup3lem 18019 Lemma for ~ frmdup3 . (Co...
frmdup3 18020 Universal property of the ...
mgm2nsgrplem1 18021 Lemma 1 for ~ mgm2nsgrp : ...
mgm2nsgrplem2 18022 Lemma 2 for ~ mgm2nsgrp . ...
mgm2nsgrplem3 18023 Lemma 3 for ~ mgm2nsgrp . ...
mgm2nsgrplem4 18024 Lemma 4 for ~ mgm2nsgrp : ...
mgm2nsgrp 18025 A small magma (with two el...
sgrp2nmndlem1 18026 Lemma 1 for ~ sgrp2nmnd : ...
sgrp2nmndlem2 18027 Lemma 2 for ~ sgrp2nmnd . ...
sgrp2nmndlem3 18028 Lemma 3 for ~ sgrp2nmnd . ...
sgrp2rid2 18029 A small semigroup (with tw...
sgrp2rid2ex 18030 A small semigroup (with tw...
sgrp2nmndlem4 18031 Lemma 4 for ~ sgrp2nmnd : ...
sgrp2nmndlem5 18032 Lemma 5 for ~ sgrp2nmnd : ...
sgrp2nmnd 18033 A small semigroup (with tw...
mgmnsgrpex 18034 There is a magma which is ...
sgrpnmndex 18035 There is a semigroup which...
sgrpssmgm 18036 The class of all semigroup...
mndsssgrp 18037 The class of all monoids i...
pwmndgplus 18038 The operation of the monoi...
pwmndid 18039 The identity of the monoid...
pwmnd 18040 The power set of a class `...
isgrp 18047 The predicate "is a group....
grpmnd 18048 A group is a monoid. (Con...
grpcl 18049 Closure of the operation o...
grpass 18050 A group operation is assoc...
grpinvex 18051 Every member of a group ha...
grpideu 18052 The two-sided identity ele...
grpplusf 18053 The group addition operati...
grpplusfo 18054 The group addition operati...
resgrpplusfrn 18055 The underlying set of a gr...
grppropd 18056 If two structures have the...
grpprop 18057 If two structures have the...
grppropstr 18058 Generalize a specific 2-el...
grpss 18059 Show that a structure exte...
isgrpd2e 18060 Deduce a group from its pr...
isgrpd2 18061 Deduce a group from its pr...
isgrpde 18062 Deduce a group from its pr...
isgrpd 18063 Deduce a group from its pr...
isgrpi 18064 Properties that determine ...
grpsgrp 18065 A group is a semigroup. (...
dfgrp2 18066 Alternate definition of a ...
dfgrp2e 18067 Alternate definition of a ...
isgrpix 18068 Properties that determine ...
grpidcl 18069 The identity element of a ...
grpbn0 18070 The base set of a group is...
grplid 18071 The identity element of a ...
grprid 18072 The identity element of a ...
grpn0 18073 A group is not empty. (Co...
hashfingrpnn 18074 A finite group has positiv...
grprcan 18075 Right cancellation law for...
grpinveu 18076 The left inverse element o...
grpid 18077 Two ways of saying that an...
isgrpid2 18078 Properties showing that an...
grpidd2 18079 Deduce the identity elemen...
grpinvfval 18080 The inverse function of a ...
grpinvfvalALT 18081 Shorter proof of ~ grpinvf...
grpinvval 18082 The inverse of a group ele...
grpinvfn 18083 Functionality of the group...
grpinvfvi 18084 The group inverse function...
grpsubfval 18085 Group subtraction (divisio...
grpsubfvalALT 18086 Shorter proof of ~ grpsubf...
grpsubval 18087 Group subtraction (divisio...
grpinvf 18088 The group inversion operat...
grpinvcl 18089 A group element's inverse ...
grplinv 18090 The left inverse of a grou...
grprinv 18091 The right inverse of a gro...
grpinvid1 18092 The inverse of a group ele...
grpinvid2 18093 The inverse of a group ele...
isgrpinv 18094 Properties showing that a ...
grplrinv 18095 In a group, every member h...
grpidinv2 18096 A group's properties using...
grpidinv 18097 A group has a left and rig...
grpinvid 18098 The inverse of the identit...
grplcan 18099 Left cancellation law for ...
grpasscan1 18100 An associative cancellatio...
grpasscan2 18101 An associative cancellatio...
grpidrcan 18102 If right adding an element...
grpidlcan 18103 If left adding an element ...
grpinvinv 18104 Double inverse law for gro...
grpinvcnv 18105 The group inverse is its o...
grpinv11 18106 The group inverse is one-t...
grpinvf1o 18107 The group inverse is a one...
grpinvnz 18108 The inverse of a nonzero g...
grpinvnzcl 18109 The inverse of a nonzero g...
grpsubinv 18110 Subtraction of an inverse....
grplmulf1o 18111 Left multiplication by a g...
grpinvpropd 18112 If two structures have the...
grpidssd 18113 If the base set of a group...
grpinvssd 18114 If the base set of a group...
grpinvadd 18115 The inverse of the group o...
grpsubf 18116 Functionality of group sub...
grpsubcl 18117 Closure of group subtracti...
grpsubrcan 18118 Right cancellation law for...
grpinvsub 18119 Inverse of a group subtrac...
grpinvval2 18120 A ~ df-neg -like equation ...
grpsubid 18121 Subtraction of a group ele...
grpsubid1 18122 Subtraction of the identit...
grpsubeq0 18123 If the difference between ...
grpsubadd0sub 18124 Subtraction expressed as a...
grpsubadd 18125 Relationship between group...
grpsubsub 18126 Double group subtraction. ...
grpaddsubass 18127 Associative-type law for g...
grppncan 18128 Cancellation law for subtr...
grpnpcan 18129 Cancellation law for subtr...
grpsubsub4 18130 Double group subtraction (...
grppnpcan2 18131 Cancellation law for mixed...
grpnpncan 18132 Cancellation law for group...
grpnpncan0 18133 Cancellation law for group...
grpnnncan2 18134 Cancellation law for group...
dfgrp3lem 18135 Lemma for ~ dfgrp3 . (Con...
dfgrp3 18136 Alternate definition of a ...
dfgrp3e 18137 Alternate definition of a ...
grplactfval 18138 The left group action of e...
grplactval 18139 The value of the left grou...
grplactcnv 18140 The left group action of e...
grplactf1o 18141 The left group action of e...
grpsubpropd 18142 Weak property deduction fo...
grpsubpropd2 18143 Strong property deduction ...
grp1 18144 The (smallest) structure r...
grp1inv 18145 The inverse function of th...
prdsinvlem 18146 Characterization of invers...
prdsgrpd 18147 The product of a family of...
prdsinvgd 18148 Negation in a product of g...
pwsgrp 18149 The product of a family of...
pwsinvg 18150 Negation in a group power....
pwssub 18151 Subtraction in a group pow...
imasgrp2 18152 The image structure of a g...
imasgrp 18153 The image structure of a g...
imasgrpf1 18154 The image of a group under...
qusgrp2 18155 Prove that a quotient stru...
xpsgrp 18156 The binary product of grou...
mhmlem 18157 Lemma for ~ mhmmnd and ~ g...
mhmid 18158 A surjective monoid morphi...
mhmmnd 18159 The image of a monoid ` G ...
mhmfmhm 18160 The function fulfilling th...
ghmgrp 18161 The image of a group ` G `...
mulgfval 18164 Group multiple (exponentia...
mulgfvalALT 18165 Shorter proof of ~ mulgfva...
mulgval 18166 Value of the group multipl...
mulgfn 18167 Functionality of the group...
mulgfvi 18168 The group multiple operati...
mulg0 18169 Group multiple (exponentia...
mulgnn 18170 Group multiple (exponentia...
mulgnngsum 18171 Group multiple (exponentia...
mulgnn0gsum 18172 Group multiple (exponentia...
mulg1 18173 Group multiple (exponentia...
mulgnnp1 18174 Group multiple (exponentia...
mulg2 18175 Group multiple (exponentia...
mulgnegnn 18176 Group multiple (exponentia...
mulgnn0p1 18177 Group multiple (exponentia...
mulgnnsubcl 18178 Closure of the group multi...
mulgnn0subcl 18179 Closure of the group multi...
mulgsubcl 18180 Closure of the group multi...
mulgnncl 18181 Closure of the group multi...
mulgnn0cl 18182 Closure of the group multi...
mulgcl 18183 Closure of the group multi...
mulgneg 18184 Group multiple (exponentia...
mulgnegneg 18185 The inverse of a negative ...
mulgm1 18186 Group multiple (exponentia...
mulgcld 18187 Deduction associated with ...
mulgaddcomlem 18188 Lemma for ~ mulgaddcom . ...
mulgaddcom 18189 The group multiple operato...
mulginvcom 18190 The group multiple operato...
mulginvinv 18191 The group multiple operato...
mulgnn0z 18192 A group multiple of the id...
mulgz 18193 A group multiple of the id...
mulgnndir 18194 Sum of group multiples, fo...
mulgnn0dir 18195 Sum of group multiples, ge...
mulgdirlem 18196 Lemma for ~ mulgdir . (Co...
mulgdir 18197 Sum of group multiples, ge...
mulgp1 18198 Group multiple (exponentia...
mulgneg2 18199 Group multiple (exponentia...
mulgnnass 18200 Product of group multiples...
mulgnn0ass 18201 Product of group multiples...
mulgass 18202 Product of group multiples...
mulgassr 18203 Reversed product of group ...
mulgmodid 18204 Casting out multiples of t...
mulgsubdir 18205 Subtraction of a group ele...
mhmmulg 18206 A homomorphism of monoids ...
mulgpropd 18207 Two structures with the sa...
submmulgcl 18208 Closure of the group multi...
submmulg 18209 A group multiple is the sa...
pwsmulg 18210 Value of a group multiple ...
issubg 18217 The subgroup predicate. (...
subgss 18218 A subgroup is a subset. (...
subgid 18219 A group is a subgroup of i...
subggrp 18220 A subgroup is a group. (C...
subgbas 18221 The base of the restricted...
subgrcl 18222 Reverse closure for the su...
subg0 18223 A subgroup of a group must...
subginv 18224 The inverse of an element ...
subg0cl 18225 The group identity is an e...
subginvcl 18226 The inverse of an element ...
subgcl 18227 A subgroup is closed under...
subgsubcl 18228 A subgroup is closed under...
subgsub 18229 The subtraction of element...
subgmulgcl 18230 Closure of the group multi...
subgmulg 18231 A group multiple is the sa...
issubg2 18232 Characterize the subgroups...
issubgrpd2 18233 Prove a subgroup by closur...
issubgrpd 18234 Prove a subgroup by closur...
issubg3 18235 A subgroup is a symmetric ...
issubg4 18236 A subgroup is a nonempty s...
grpissubg 18237 If the base set of a group...
resgrpisgrp 18238 If the base set of a group...
subgsubm 18239 A subgroup is a submonoid....
subsubg 18240 A subgroup of a subgroup i...
subgint 18241 The intersection of a none...
0subg 18242 The zero subgroup of an ar...
trivsubgd 18243 The only subgroup of a tri...
trivsubgsnd 18244 The only subgroup of a tri...
isnsg 18245 Property of being a normal...
isnsg2 18246 Weaken the condition of ~ ...
nsgbi 18247 Defining property of a nor...
nsgsubg 18248 A normal subgroup is a sub...
nsgconj 18249 The conjugation of an elem...
isnsg3 18250 A subgroup is normal iff t...
subgacs 18251 Subgroups are an algebraic...
nsgacs 18252 Normal subgroups form an a...
elnmz 18253 Elementhood in the normali...
nmzbi 18254 Defining property of the n...
nmzsubg 18255 The normalizer N_G(S) of a...
ssnmz 18256 A subgroup is a subset of ...
isnsg4 18257 A subgroup is normal iff i...
nmznsg 18258 Any subgroup is a normal s...
0nsg 18259 The zero subgroup is norma...
nsgid 18260 The whole group is a norma...
0idnsgd 18261 The whole group and the ze...
trivnsgd 18262 The only normal subgroup o...
triv1nsgd 18263 A trivial group has exactl...
1nsgtrivd 18264 A group with exactly one n...
releqg 18265 The left coset equivalence...
eqgfval 18266 Value of the subgroup left...
eqgval 18267 Value of the subgroup left...
eqger 18268 The subgroup coset equival...
eqglact 18269 A left coset can be expres...
eqgid 18270 The left coset containing ...
eqgen 18271 Each coset is equipotent t...
eqgcpbl 18272 The subgroup coset equival...
qusgrp 18273 If ` Y ` is a normal subgr...
quseccl 18274 Closure of the quotient ma...
qusadd 18275 Value of the group operati...
qus0 18276 Value of the group identit...
qusinv 18277 Value of the group inverse...
qussub 18278 Value of the group subtrac...
lagsubg2 18279 Lagrange's theorem for fin...
lagsubg 18280 Lagrange's theorem for Gro...
cycsubmel 18281 Characterization of an ele...
cycsubmcl 18282 The set of nonnegative int...
cycsubm 18283 The set of nonnegative int...
cyccom 18284 Condition for an operation...
cycsubmcom 18285 The operation of a monoid ...
cycsubggend 18286 The cyclic subgroup genera...
cycsubgcl 18287 The set of integer powers ...
cycsubgss 18288 The cyclic subgroup genera...
cycsubg 18289 The cyclic group generated...
cycsubgcld 18290 The cyclic subgroup genera...
cycsubg2 18291 The subgroup generated by ...
cycsubg2cl 18292 Any multiple of an element...
reldmghm 18295 Lemma for group homomorphi...
isghm 18296 Property of being a homomo...
isghm3 18297 Property of a group homomo...
ghmgrp1 18298 A group homomorphism is on...
ghmgrp2 18299 A group homomorphism is on...
ghmf 18300 A group homomorphism is a ...
ghmlin 18301 A homomorphism of groups i...
ghmid 18302 A homomorphism of groups p...
ghminv 18303 A homomorphism of groups p...
ghmsub 18304 Linearity of subtraction t...
isghmd 18305 Deduction for a group homo...
ghmmhm 18306 A group homomorphism is a ...
ghmmhmb 18307 Group homomorphisms and mo...
ghmmulg 18308 A homomorphism of monoids ...
ghmrn 18309 The range of a homomorphis...
0ghm 18310 The constant zero linear f...
idghm 18311 The identity homomorphism ...
resghm 18312 Restriction of a homomorph...
resghm2 18313 One direction of ~ resghm2...
resghm2b 18314 Restriction of the codomai...
ghmghmrn 18315 A group homomorphism from ...
ghmco 18316 The composition of group h...
ghmima 18317 The image of a subgroup un...
ghmpreima 18318 The inverse image of a sub...
ghmeql 18319 The equalizer of two group...
ghmnsgima 18320 The image of a normal subg...
ghmnsgpreima 18321 The inverse image of a nor...
ghmker 18322 The kernel of a homomorphi...
ghmeqker 18323 Two source points map to t...
pwsdiagghm 18324 Diagonal homomorphism into...
ghmf1 18325 Two ways of saying a group...
ghmf1o 18326 A bijective group homomorp...
conjghm 18327 Conjugation is an automorp...
conjsubg 18328 A conjugated subgroup is a...
conjsubgen 18329 A conjugated subgroup is e...
conjnmz 18330 A subgroup is unchanged un...
conjnmzb 18331 Alternative condition for ...
conjnsg 18332 A normal subgroup is uncha...
qusghm 18333 If ` Y ` is a normal subgr...
ghmpropd 18334 Group homomorphism depends...
gimfn 18339 The group isomorphism func...
isgim 18340 An isomorphism of groups i...
gimf1o 18341 An isomorphism of groups i...
gimghm 18342 An isomorphism of groups i...
isgim2 18343 A group isomorphism is a h...
subggim 18344 Behavior of subgroups unde...
gimcnv 18345 The converse of a bijectiv...
gimco 18346 The composition of group i...
brgic 18347 The relation "is isomorphi...
brgici 18348 Prove isomorphic by an exp...
gicref 18349 Isomorphism is reflexive. ...
giclcl 18350 Isomorphism implies the le...
gicrcl 18351 Isomorphism implies the ri...
gicsym 18352 Isomorphism is symmetric. ...
gictr 18353 Isomorphism is transitive....
gicer 18354 Isomorphism is an equivale...
gicen 18355 Isomorphic groups have equ...
gicsubgen 18356 A less trivial example of ...
isga 18359 The predicate "is a (left)...
gagrp 18360 The left argument of a gro...
gaset 18361 The right argument of a gr...
gagrpid 18362 The identity of the group ...
gaf 18363 The mapping of the group a...
gafo 18364 A group action is onto its...
gaass 18365 An "associative" property ...
ga0 18366 The action of a group on t...
gaid 18367 The trivial action of a gr...
subgga 18368 A subgroup acts on its par...
gass 18369 A subset of a group action...
gasubg 18370 The restriction of a group...
gaid2 18371 A group operation is a lef...
galcan 18372 The action of a particular...
gacan 18373 Group inverses cancel in a...
gapm 18374 The action of a particular...
gaorb 18375 The orbit equivalence rela...
gaorber 18376 The orbit equivalence rela...
gastacl 18377 The stabilizer subgroup in...
gastacos 18378 Write the coset relation f...
orbstafun 18379 Existence and uniqueness f...
orbstaval 18380 Value of the function at a...
orbsta 18381 The Orbit-Stabilizer theor...
orbsta2 18382 Relation between the size ...
cntrval 18387 Substitute definition of t...
cntzfval 18388 First level substitution f...
cntzval 18389 Definition substitution fo...
elcntz 18390 Elementhood in the central...
cntzel 18391 Membership in a centralize...
cntzsnval 18392 Special substitution for t...
elcntzsn 18393 Value of the centralizer o...
sscntz 18394 A centralizer expression f...
cntzrcl 18395 Reverse closure for elemen...
cntzssv 18396 The centralizer is uncondi...
cntzi 18397 Membership in a centralize...
cntrss 18398 The center is a subset of ...
cntri 18399 Defining property of the c...
resscntz 18400 Centralizer in a substruct...
cntz2ss 18401 Centralizers reverse the s...
cntzrec 18402 Reciprocity relationship f...
cntziinsn 18403 Express any centralizer as...
cntzsubm 18404 Centralizers in a monoid a...
cntzsubg 18405 Centralizers in a group ar...
cntzidss 18406 If the elements of ` S ` c...
cntzmhm 18407 Centralizers in a monoid a...
cntzmhm2 18408 Centralizers in a monoid a...
cntrsubgnsg 18409 A central subgroup is norm...
cntrnsg 18410 The center of a group is a...
oppgval 18413 Value of the opposite grou...
oppgplusfval 18414 Value of the addition oper...
oppgplus 18415 Value of the addition oper...
oppglem 18416 Lemma for ~ oppgbas . (Co...
oppgbas 18417 Base set of an opposite gr...
oppgtset 18418 Topology of an opposite gr...
oppgtopn 18419 Topology of an opposite gr...
oppgmnd 18420 The opposite of a monoid i...
oppgmndb 18421 Bidirectional form of ~ op...
oppgid 18422 Zero in a monoid is a symm...
oppggrp 18423 The opposite of a group is...
oppggrpb 18424 Bidirectional form of ~ op...
oppginv 18425 Inverses in a group are a ...
invoppggim 18426 The inverse is an antiauto...
oppggic 18427 Every group is (naturally)...
oppgsubm 18428 Being a submonoid is a sym...
oppgsubg 18429 Being a subgroup is a symm...
oppgcntz 18430 A centralizer in a group i...
oppgcntr 18431 The center of a group is t...
gsumwrev 18432 A sum in an opposite monoi...
symgval 18435 The value of the symmetric...
symgbas 18436 The base set of the symmet...
elsymgbas2 18437 Two ways of saying a funct...
elsymgbas 18438 Two ways of saying a funct...
symgbasf1o 18439 Elements in the symmetric ...
symgbasf 18440 A permutation (element of ...
symghash 18441 The symmetric group on ` n...
symgbasfi 18442 The symmetric group on a f...
symgfv 18443 The function value of a pe...
symgfvne 18444 The function values of a p...
symgplusg 18445 The group operation of a s...
symgov 18446 The value of the group ope...
symgcl 18447 The group operation of the...
idresperm 18448 The identity function rest...
symgmov1 18449 For a permutation of a set...
symgmov2 18450 For a permutation of a set...
symgbas0 18451 The base set of the symmet...
symg1hash 18452 The symmetric group on a s...
symg1bas 18453 The symmetric group on a s...
symg2hash 18454 The symmetric group on a (...
symg2bas 18455 The symmetric group on a p...
symggrplem 18456 Lemma for ~ symggrp and ~ ...
symgtset 18457 The topology of the symmet...
symggrp 18458 The symmetric group on a s...
symgid 18459 The group identity element...
symginv 18460 The group inverse in the s...
galactghm 18461 The currying of a group ac...
lactghmga 18462 The converse of ~ galactgh...
symgtopn 18463 The topology of the symmet...
symgga 18464 The symmetric group induce...
pgrpsubgsymgbi 18465 Every permutation group is...
pgrpsubgsymg 18466 Every permutation group is...
idressubgsymg 18467 The singleton containing o...
idrespermg 18468 The structure with the sin...
cayleylem1 18469 Lemma for ~ cayley . (Con...
cayleylem2 18470 Lemma for ~ cayley . (Con...
cayley 18471 Cayley's Theorem (construc...
cayleyth 18472 Cayley's Theorem (existenc...
symgfix2 18473 If a permutation does not ...
symgextf 18474 The extension of a permuta...
symgextfv 18475 The function value of the ...
symgextfve 18476 The function value of the ...
symgextf1lem 18477 Lemma for ~ symgextf1 . (...
symgextf1 18478 The extension of a permuta...
symgextfo 18479 The extension of a permuta...
symgextf1o 18480 The extension of a permuta...
symgextsymg 18481 The extension of a permuta...
symgextres 18482 The restriction of the ext...
gsumccatsymgsn 18483 Homomorphic property of co...
gsmsymgrfixlem1 18484 Lemma 1 for ~ gsmsymgrfix ...
gsmsymgrfix 18485 The composition of permuta...
fvcosymgeq 18486 The values of two composit...
gsmsymgreqlem1 18487 Lemma 1 for ~ gsmsymgreq ....
gsmsymgreqlem2 18488 Lemma 2 for ~ gsmsymgreq ....
gsmsymgreq 18489 Two combination of permuta...
symgfixelq 18490 A permutation of a set fix...
symgfixels 18491 The restriction of a permu...
symgfixelsi 18492 The restriction of a permu...
symgfixf 18493 The mapping of a permutati...
symgfixf1 18494 The mapping of a permutati...
symgfixfolem1 18495 Lemma 1 for ~ symgfixfo . ...
symgfixfo 18496 The mapping of a permutati...
symgfixf1o 18497 The mapping of a permutati...
f1omvdmvd 18500 A permutation of any class...
f1omvdcnv 18501 A permutation and its inve...
mvdco 18502 Composing two permutations...
f1omvdconj 18503 Conjugation of a permutati...
f1otrspeq 18504 A transposition is charact...
f1omvdco2 18505 If exactly one of two perm...
f1omvdco3 18506 If a point is moved by exa...
pmtrfval 18507 The function generating tr...
pmtrval 18508 A generated transposition,...
pmtrfv 18509 General value of mapping a...
pmtrprfv 18510 In a transposition of two ...
pmtrprfv3 18511 In a transposition of two ...
pmtrf 18512 Functionality of a transpo...
pmtrmvd 18513 A transposition moves prec...
pmtrrn 18514 Transposing two points giv...
pmtrfrn 18515 A transposition (as a kind...
pmtrffv 18516 Mapping of a point under a...
pmtrrn2 18517 For any transposition ther...
pmtrfinv 18518 A transposition function i...
pmtrfmvdn0 18519 A transposition moves at l...
pmtrff1o 18520 A transposition function i...
pmtrfcnv 18521 A transposition function i...
pmtrfb 18522 An intrinsic characterizat...
pmtrfconj 18523 Any conjugate of a transpo...
symgsssg 18524 The symmetric group has su...
symgfisg 18525 The symmetric group has a ...
symgtrf 18526 Transpositions are element...
symggen 18527 The span of the transposit...
symggen2 18528 A finite permutation group...
symgtrinv 18529 To invert a permutation re...
pmtr3ncomlem1 18530 Lemma 1 for ~ pmtr3ncom . ...
pmtr3ncomlem2 18531 Lemma 2 for ~ pmtr3ncom . ...
pmtr3ncom 18532 Transpositions over sets w...
pmtrdifellem1 18533 Lemma 1 for ~ pmtrdifel . ...
pmtrdifellem2 18534 Lemma 2 for ~ pmtrdifel . ...
pmtrdifellem3 18535 Lemma 3 for ~ pmtrdifel . ...
pmtrdifellem4 18536 Lemma 4 for ~ pmtrdifel . ...
pmtrdifel 18537 A transposition of element...
pmtrdifwrdellem1 18538 Lemma 1 for ~ pmtrdifwrdel...
pmtrdifwrdellem2 18539 Lemma 2 for ~ pmtrdifwrdel...
pmtrdifwrdellem3 18540 Lemma 3 for ~ pmtrdifwrdel...
pmtrdifwrdel2lem1 18541 Lemma 1 for ~ pmtrdifwrdel...
pmtrdifwrdel 18542 A sequence of transpositio...
pmtrdifwrdel2 18543 A sequence of transpositio...
pmtrprfval 18544 The transpositions on a pa...
pmtrprfvalrn 18545 The range of the transposi...
psgnunilem1 18550 Lemma for ~ psgnuni . Giv...
psgnunilem5 18551 Lemma for ~ psgnuni . It ...
psgnunilem2 18552 Lemma for ~ psgnuni . Ind...
psgnunilem3 18553 Lemma for ~ psgnuni . Any...
psgnunilem4 18554 Lemma for ~ psgnuni . An ...
m1expaddsub 18555 Addition and subtraction o...
psgnuni 18556 If the same permutation ca...
psgnfval 18557 Function definition of the...
psgnfn 18558 Functionality and domain o...
psgndmsubg 18559 The finitary permutations ...
psgneldm 18560 Property of being a finita...
psgneldm2 18561 The finitary permutations ...
psgneldm2i 18562 A sequence of transpositio...
psgneu 18563 A finitary permutation has...
psgnval 18564 Value of the permutation s...
psgnvali 18565 A finitary permutation has...
psgnvalii 18566 Any representation of a pe...
psgnpmtr 18567 All transpositions are odd...
psgn0fv0 18568 The permutation sign funct...
sygbasnfpfi 18569 The class of non-fixed poi...
psgnfvalfi 18570 Function definition of the...
psgnvalfi 18571 Value of the permutation s...
psgnran 18572 The range of the permutati...
gsmtrcl 18573 The group sum of transposi...
psgnfitr 18574 A permutation of a finite ...
psgnfieu 18575 A permutation of a finite ...
pmtrsn 18576 The value of the transposi...
psgnsn 18577 The permutation sign funct...
psgnprfval 18578 The permutation sign funct...
psgnprfval1 18579 The permutation sign of th...
psgnprfval2 18580 The permutation sign of th...
odfval 18589 Value of the order functio...
odfvalALT 18590 Shorter proof of ~ odfval ...
odval 18591 Second substitution for th...
odlem1 18592 The group element order is...
odcl 18593 The order of a group eleme...
odf 18594 Functionality of the group...
odid 18595 Any element to the power o...
odlem2 18596 Any positive annihilator o...
odmodnn0 18597 Reduce the argument of a g...
mndodconglem 18598 Lemma for ~ mndodcong . (...
mndodcong 18599 If two multipliers are con...
mndodcongi 18600 If two multipliers are con...
oddvdsnn0 18601 The only multiples of ` A ...
odnncl 18602 If a nonzero multiple of a...
odmod 18603 Reduce the argument of a g...
oddvds 18604 The only multiples of ` A ...
oddvdsi 18605 Any group element is annih...
odcong 18606 If two multipliers are con...
odeq 18607 The ~ oddvds property uniq...
odval2 18608 A non-conditional definiti...
odcld 18609 The order of a group eleme...
odmulgid 18610 A relationship between the...
odmulg2 18611 The order of a multiple di...
odmulg 18612 Relationship between the o...
odmulgeq 18613 A multiple of a point of f...
odbezout 18614 If ` N ` is coprime to the...
od1 18615 The order of the group ide...
odeq1 18616 The group identity is the ...
odinv 18617 The order of the inverse o...
odf1 18618 The multiples of an elemen...
odinf 18619 The multiples of an elemen...
dfod2 18620 An alternative definition ...
odcl2 18621 The order of an element of...
oddvds2 18622 The order of an element of...
submod 18623 The order of an element is...
subgod 18624 The order of an element is...
odsubdvds 18625 The order of an element of...
odf1o1 18626 An element with zero order...
odf1o2 18627 An element with nonzero or...
odhash 18628 An element of zero order g...
odhash2 18629 If an element has nonzero ...
odhash3 18630 An element which generates...
odngen 18631 A cyclic subgroup of size ...
gexval 18632 Value of the exponent of a...
gexlem1 18633 The group element order is...
gexcl 18634 The exponent of a group is...
gexid 18635 Any element to the power o...
gexlem2 18636 Any positive annihilator o...
gexdvdsi 18637 Any group element is annih...
gexdvds 18638 The only ` N ` that annihi...
gexdvds2 18639 An integer divides the gro...
gexod 18640 Any group element is annih...
gexcl3 18641 If the order of every grou...
gexnnod 18642 Every group element has fi...
gexcl2 18643 The exponent of a finite g...
gexdvds3 18644 The exponent of a finite g...
gex1 18645 A group or monoid has expo...
ispgp 18646 A group is a ` P ` -group ...
pgpprm 18647 Reverse closure for the fi...
pgpgrp 18648 Reverse closure for the se...
pgpfi1 18649 A finite group with order ...
pgp0 18650 The identity subgroup is a...
subgpgp 18651 A subgroup of a p-group is...
sylow1lem1 18652 Lemma for ~ sylow1 . The ...
sylow1lem2 18653 Lemma for ~ sylow1 . The ...
sylow1lem3 18654 Lemma for ~ sylow1 . One ...
sylow1lem4 18655 Lemma for ~ sylow1 . The ...
sylow1lem5 18656 Lemma for ~ sylow1 . Usin...
sylow1 18657 Sylow's first theorem. If...
odcau 18658 Cauchy's theorem for the o...
pgpfi 18659 The converse to ~ pgpfi1 ....
pgpfi2 18660 Alternate version of ~ pgp...
pgphash 18661 The order of a p-group. (...
isslw 18662 The property of being a Sy...
slwprm 18663 Reverse closure for the fi...
slwsubg 18664 A Sylow ` P ` -subgroup is...
slwispgp 18665 Defining property of a Syl...
slwpss 18666 A proper superset of a Syl...
slwpgp 18667 A Sylow ` P ` -subgroup is...
pgpssslw 18668 Every ` P ` -subgroup is c...
slwn0 18669 Every finite group contain...
subgslw 18670 A Sylow subgroup that is c...
sylow2alem1 18671 Lemma for ~ sylow2a . An ...
sylow2alem2 18672 Lemma for ~ sylow2a . All...
sylow2a 18673 A named lemma of Sylow's s...
sylow2blem1 18674 Lemma for ~ sylow2b . Eva...
sylow2blem2 18675 Lemma for ~ sylow2b . Lef...
sylow2blem3 18676 Sylow's second theorem. P...
sylow2b 18677 Sylow's second theorem. A...
slwhash 18678 A sylow subgroup has cardi...
fislw 18679 The sylow subgroups of a f...
sylow2 18680 Sylow's second theorem. S...
sylow3lem1 18681 Lemma for ~ sylow3 , first...
sylow3lem2 18682 Lemma for ~ sylow3 , first...
sylow3lem3 18683 Lemma for ~ sylow3 , first...
sylow3lem4 18684 Lemma for ~ sylow3 , first...
sylow3lem5 18685 Lemma for ~ sylow3 , secon...
sylow3lem6 18686 Lemma for ~ sylow3 , secon...
sylow3 18687 Sylow's third theorem. Th...
lsmfval 18692 The subgroup sum function ...
lsmvalx 18693 Subspace sum value (for a ...
lsmelvalx 18694 Subspace sum membership (f...
lsmelvalix 18695 Subspace sum membership (f...
oppglsm 18696 The subspace sum operation...
lsmssv 18697 Subgroup sum is a subset o...
lsmless1x 18698 Subset implies subgroup su...
lsmless2x 18699 Subset implies subgroup su...
lsmub1x 18700 Subgroup sum is an upper b...
lsmub2x 18701 Subgroup sum is an upper b...
lsmval 18702 Subgroup sum value (for a ...
lsmelval 18703 Subgroup sum membership (f...
lsmelvali 18704 Subgroup sum membership (f...
lsmelvalm 18705 Subgroup sum membership an...
lsmelvalmi 18706 Membership of vector subtr...
lsmsubm 18707 The sum of two commuting s...
lsmsubg 18708 The sum of two commuting s...
lsmcom2 18709 Subgroup sum commutes. (C...
smndlsmidm 18710 The direct product is idem...
lsmub1 18711 Subgroup sum is an upper b...
lsmub2 18712 Subgroup sum is an upper b...
lsmunss 18713 Union of subgroups is a su...
lsmless1 18714 Subset implies subgroup su...
lsmless2 18715 Subset implies subgroup su...
lsmless12 18716 Subset implies subgroup su...
lsmidm 18717 Subgroup sum is idempotent...
lsmidmOLD 18718 Obsolete proof of ~ lsmidm...
lsmlub 18719 The least upper bound prop...
lsmss1 18720 Subgroup sum with a subset...
lsmss1b 18721 Subgroup sum with a subset...
lsmss2 18722 Subgroup sum with a subset...
lsmss2b 18723 Subgroup sum with a subset...
lsmass 18724 Subgroup sum is associativ...
mndlsmidm 18725 Subgroup sum is idempotent...
lsm01 18726 Subgroup sum with the zero...
lsm02 18727 Subgroup sum with the zero...
subglsm 18728 The subgroup sum evaluated...
lssnle 18729 Equivalent expressions for...
lsmmod 18730 The modular law holds for ...
lsmmod2 18731 Modular law dual for subgr...
lsmpropd 18732 If two structures have the...
cntzrecd 18733 Commute the "subgroups com...
lsmcntz 18734 The "subgroups commute" pr...
lsmcntzr 18735 The "subgroups commute" pr...
lsmdisj 18736 Disjointness from a subgro...
lsmdisj2 18737 Association of the disjoin...
lsmdisj3 18738 Association of the disjoin...
lsmdisjr 18739 Disjointness from a subgro...
lsmdisj2r 18740 Association of the disjoin...
lsmdisj3r 18741 Association of the disjoin...
lsmdisj2a 18742 Association of the disjoin...
lsmdisj2b 18743 Association of the disjoin...
lsmdisj3a 18744 Association of the disjoin...
lsmdisj3b 18745 Association of the disjoin...
subgdisj1 18746 Vectors belonging to disjo...
subgdisj2 18747 Vectors belonging to disjo...
subgdisjb 18748 Vectors belonging to disjo...
pj1fval 18749 The left projection functi...
pj1val 18750 The left projection functi...
pj1eu 18751 Uniqueness of a left proje...
pj1f 18752 The left projection functi...
pj2f 18753 The right projection funct...
pj1id 18754 Any element of a direct su...
pj1eq 18755 Any element of a direct su...
pj1lid 18756 The left projection functi...
pj1rid 18757 The left projection functi...
pj1ghm 18758 The left projection functi...
pj1ghm2 18759 The left projection functi...
lsmhash 18760 The order of the direct pr...
efgmval 18767 Value of the formal invers...
efgmf 18768 The formal inverse operati...
efgmnvl 18769 The inversion function on ...
efgrcl 18770 Lemma for ~ efgval . (Con...
efglem 18771 Lemma for ~ efgval . (Con...
efgval 18772 Value of the free group co...
efger 18773 Value of the free group co...
efgi 18774 Value of the free group co...
efgi0 18775 Value of the free group co...
efgi1 18776 Value of the free group co...
efgtf 18777 Value of the free group co...
efgtval 18778 Value of the extension fun...
efgval2 18779 Value of the free group co...
efgi2 18780 Value of the free group co...
efgtlen 18781 Value of the free group co...
efginvrel2 18782 The inverse of the reverse...
efginvrel1 18783 The inverse of the reverse...
efgsf 18784 Value of the auxiliary fun...
efgsdm 18785 Elementhood in the domain ...
efgsval 18786 Value of the auxiliary fun...
efgsdmi 18787 Property of the last link ...
efgsval2 18788 Value of the auxiliary fun...
efgsrel 18789 The start and end of any e...
efgs1 18790 A singleton of an irreduci...
efgs1b 18791 Every extension sequence e...
efgsp1 18792 If ` F ` is an extension s...
efgsres 18793 An initial segment of an e...
efgsfo 18794 For any word, there is a s...
efgredlema 18795 The reduced word that form...
efgredlemf 18796 Lemma for ~ efgredleme . ...
efgredlemg 18797 Lemma for ~ efgred . (Con...
efgredleme 18798 Lemma for ~ efgred . (Con...
efgredlemd 18799 The reduced word that form...
efgredlemc 18800 The reduced word that form...
efgredlemb 18801 The reduced word that form...
efgredlem 18802 The reduced word that form...
efgred 18803 The reduced word that form...
efgrelexlema 18804 If two words ` A , B ` are...
efgrelexlemb 18805 If two words ` A , B ` are...
efgrelex 18806 If two words ` A , B ` are...
efgredeu 18807 There is a unique reduced ...
efgred2 18808 Two extension sequences ha...
efgcpbllema 18809 Lemma for ~ efgrelex . De...
efgcpbllemb 18810 Lemma for ~ efgrelex . Sh...
efgcpbl 18811 Two extension sequences ha...
efgcpbl2 18812 Two extension sequences ha...
frgpval 18813 Value of the free group co...
frgpcpbl 18814 Compatibility of the group...
frgp0 18815 The free group is a group....
frgpeccl 18816 Closure of the quotient ma...
frgpgrp 18817 The free group is a group....
frgpadd 18818 Addition in the free group...
frgpinv 18819 The inverse of an element ...
frgpmhm 18820 The "natural map" from wor...
vrgpfval 18821 The canonical injection fr...
vrgpval 18822 The value of the generatin...
vrgpf 18823 The mapping from the index...
vrgpinv 18824 The inverse of a generatin...
frgpuptf 18825 Any assignment of the gene...
frgpuptinv 18826 Any assignment of the gene...
frgpuplem 18827 Any assignment of the gene...
frgpupf 18828 Any assignment of the gene...
frgpupval 18829 Any assignment of the gene...
frgpup1 18830 Any assignment of the gene...
frgpup2 18831 The evaluation map has the...
frgpup3lem 18832 The evaluation map has the...
frgpup3 18833 Universal property of the ...
0frgp 18834 The free group on zero gen...
isabl 18839 The predicate "is an Abeli...
ablgrp 18840 An Abelian group is a grou...
ablgrpd 18841 An Abelian group is a grou...
ablcmn 18842 An Abelian group is a comm...
iscmn 18843 The predicate "is a commut...
isabl2 18844 The predicate "is an Abeli...
cmnpropd 18845 If two structures have the...
ablpropd 18846 If two structures have the...
ablprop 18847 If two structures have the...
iscmnd 18848 Properties that determine ...
isabld 18849 Properties that determine ...
isabli 18850 Properties that determine ...
cmnmnd 18851 A commutative monoid is a ...
cmncom 18852 A commutative monoid is co...
ablcom 18853 An Abelian group operation...
cmn32 18854 Commutative/associative la...
cmn4 18855 Commutative/associative la...
cmn12 18856 Commutative/associative la...
abl32 18857 Commutative/associative la...
rinvmod 18858 Uniqueness of a right inve...
ablinvadd 18859 The inverse of an Abelian ...
ablsub2inv 18860 Abelian group subtraction ...
ablsubadd 18861 Relationship between Abeli...
ablsub4 18862 Commutative/associative su...
abladdsub4 18863 Abelian group addition/sub...
abladdsub 18864 Associative-type law for g...
ablpncan2 18865 Cancellation law for subtr...
ablpncan3 18866 A cancellation law for com...
ablsubsub 18867 Law for double subtraction...
ablsubsub4 18868 Law for double subtraction...
ablpnpcan 18869 Cancellation law for mixed...
ablnncan 18870 Cancellation law for group...
ablsub32 18871 Swap the second and third ...
ablnnncan 18872 Cancellation law for group...
ablnnncan1 18873 Cancellation law for group...
ablsubsub23 18874 Swap subtrahend and result...
mulgnn0di 18875 Group multiple of a sum, f...
mulgdi 18876 Group multiple of a sum. ...
mulgmhm 18877 The map from ` x ` to ` n ...
mulgghm 18878 The map from ` x ` to ` n ...
mulgsubdi 18879 Group multiple of a differ...
ghmfghm 18880 The function fulfilling th...
ghmcmn 18881 The image of a commutative...
ghmabl 18882 The image of an abelian gr...
invghm 18883 The inversion map is a gro...
eqgabl 18884 Value of the subgroup cose...
subgabl 18885 A subgroup of an abelian g...
subcmn 18886 A submonoid of a commutati...
submcmn 18887 A submonoid of a commutati...
submcmn2 18888 A submonoid is commutative...
cntzcmn 18889 The centralizer of any sub...
cntzcmnss 18890 Any subset in a commutativ...
cntrcmnd 18891 The center of a monoid is ...
cntrabl 18892 The center of a group is a...
cntzspan 18893 If the generators commute,...
cntzcmnf 18894 Discharge the centralizer ...
ghmplusg 18895 The pointwise sum of two l...
ablnsg 18896 Every subgroup of an abeli...
odadd1 18897 The order of a product in ...
odadd2 18898 The order of a product in ...
odadd 18899 The order of a product is ...
gex2abl 18900 A group with exponent 2 (o...
gexexlem 18901 Lemma for ~ gexex . (Cont...
gexex 18902 In an abelian group with f...
torsubg 18903 The set of all elements of...
oddvdssubg 18904 The set of all elements wh...
lsmcomx 18905 Subgroup sum commutes (ext...
ablcntzd 18906 All subgroups in an abelia...
lsmcom 18907 Subgroup sum commutes. (C...
lsmsubg2 18908 The sum of two subgroups i...
lsm4 18909 Commutative/associative la...
prdscmnd 18910 The product of a family of...
prdsabld 18911 The product of a family of...
pwscmn 18912 The structure power on a c...
pwsabl 18913 The structure power on an ...
qusabl 18914 If ` Y ` is a subgroup of ...
abl1 18915 The (smallest) structure r...
abln0 18916 Abelian groups (and theref...
cnaddablx 18917 The complex numbers are an...
cnaddabl 18918 The complex numbers are an...
cnaddid 18919 The group identity element...
cnaddinv 18920 Value of the group inverse...
zaddablx 18921 The integers are an Abelia...
frgpnabllem1 18922 Lemma for ~ frgpnabl . (C...
frgpnabllem2 18923 Lemma for ~ frgpnabl . (C...
frgpnabl 18924 The free group on two or m...
iscyg 18927 Definition of a cyclic gro...
iscyggen 18928 The property of being a cy...
iscyggen2 18929 The property of being a cy...
iscyg2 18930 A cyclic group is a group ...
cyggeninv 18931 The inverse of a cyclic ge...
cyggenod 18932 An element is the generato...
cyggenod2 18933 In an infinite cyclic grou...
iscyg3 18934 Definition of a cyclic gro...
iscygd 18935 Definition of a cyclic gro...
iscygodd 18936 Show that a group with an ...
cycsubmcmn 18937 The set of nonnegative int...
cyggrp 18938 A cyclic group is a group....
cygabl 18939 A cyclic group is abelian....
cygablOLD 18940 Obsolete proof of ~ cygabl...
cygctb 18941 A cyclic group is countabl...
0cyg 18942 The trivial group is cycli...
prmcyg 18943 A group with prime order i...
lt6abl 18944 A group with fewer than ` ...
ghmcyg 18945 The image of a cyclic grou...
cyggex2 18946 The exponent of a cyclic g...
cyggex 18947 The exponent of a finite c...
cyggexb 18948 A finite abelian group is ...
giccyg 18949 Cyclicity is a group prope...
cycsubgcyg 18950 The cyclic subgroup genera...
cycsubgcyg2 18951 The cyclic subgroup genera...
gsumval3a 18952 Value of the group sum ope...
gsumval3eu 18953 The group sum as defined i...
gsumval3lem1 18954 Lemma 1 for ~ gsumval3 . ...
gsumval3lem2 18955 Lemma 2 for ~ gsumval3 . ...
gsumval3 18956 Value of the group sum ope...
gsumcllem 18957 Lemma for ~ gsumcl and rel...
gsumzres 18958 Extend a finite group sum ...
gsumzcl2 18959 Closure of a finite group ...
gsumzcl 18960 Closure of a finite group ...
gsumzf1o 18961 Re-index a finite group su...
gsumres 18962 Extend a finite group sum ...
gsumcl2 18963 Closure of a finite group ...
gsumcl 18964 Closure of a finite group ...
gsumf1o 18965 Re-index a finite group su...
gsumreidx 18966 Re-index a finite group su...
gsumzsubmcl 18967 Closure of a group sum in ...
gsumsubmcl 18968 Closure of a group sum in ...
gsumsubgcl 18969 Closure of a group sum in ...
gsumzaddlem 18970 The sum of two group sums....
gsumzadd 18971 The sum of two group sums....
gsumadd 18972 The sum of two group sums....
gsummptfsadd 18973 The sum of two group sums ...
gsummptfidmadd 18974 The sum of two group sums ...
gsummptfidmadd2 18975 The sum of two group sums ...
gsumzsplit 18976 Split a group sum into two...
gsumsplit 18977 Split a group sum into two...
gsumsplit2 18978 Split a group sum into two...
gsummptfidmsplit 18979 Split a group sum expresse...
gsummptfidmsplitres 18980 Split a group sum expresse...
gsummptfzsplit 18981 Split a group sum expresse...
gsummptfzsplitl 18982 Split a group sum expresse...
gsumconst 18983 Sum of a constant series. ...
gsumconstf 18984 Sum of a constant series. ...
gsummptshft 18985 Index shift of a finite gr...
gsumzmhm 18986 Apply a group homomorphism...
gsummhm 18987 Apply a group homomorphism...
gsummhm2 18988 Apply a group homomorphism...
gsummptmhm 18989 Apply a group homomorphism...
gsummulglem 18990 Lemma for ~ gsummulg and ~...
gsummulg 18991 Nonnegative multiple of a ...
gsummulgz 18992 Integer multiple of a grou...
gsumzoppg 18993 The opposite of a group su...
gsumzinv 18994 Inverse of a group sum. (...
gsuminv 18995 Inverse of a group sum. (...
gsummptfidminv 18996 Inverse of a group sum exp...
gsumsub 18997 The difference of two grou...
gsummptfssub 18998 The difference of two grou...
gsummptfidmsub 18999 The difference of two grou...
gsumsnfd 19000 Group sum of a singleton, ...
gsumsnd 19001 Group sum of a singleton, ...
gsumsnf 19002 Group sum of a singleton, ...
gsumsn 19003 Group sum of a singleton. ...
gsumpr 19004 Group sum of a pair. (Con...
gsumzunsnd 19005 Append an element to a fin...
gsumunsnfd 19006 Append an element to a fin...
gsumunsnd 19007 Append an element to a fin...
gsumunsnf 19008 Append an element to a fin...
gsumunsn 19009 Append an element to a fin...
gsumdifsnd 19010 Extract a summand from a f...
gsumpt 19011 Sum of a family that is no...
gsummptf1o 19012 Re-index a finite group su...
gsummptun 19013 Group sum of a disjoint un...
gsummpt1n0 19014 If only one summand in a f...
gsummptif1n0 19015 If only one summand in a f...
gsummptcl 19016 Closure of a finite group ...
gsummptfif1o 19017 Re-index a finite group su...
gsummptfzcl 19018 Closure of a finite group ...
gsum2dlem1 19019 Lemma 1 for ~ gsum2d . (C...
gsum2dlem2 19020 Lemma for ~ gsum2d . (Con...
gsum2d 19021 Write a sum over a two-dim...
gsum2d2lem 19022 Lemma for ~ gsum2d2 : show...
gsum2d2 19023 Write a group sum over a t...
gsumcom2 19024 Two-dimensional commutatio...
gsumxp 19025 Write a group sum over a c...
gsumcom 19026 Commute the arguments of a...
gsumcom3 19027 A commutative law for fini...
gsumcom3fi 19028 A commutative law for fini...
gsumxp2 19029 Write a group sum over a c...
prdsgsum 19030 Finite commutative sums in...
pwsgsum 19031 Finite commutative sums in...
fsfnn0gsumfsffz 19032 Replacing a finitely suppo...
nn0gsumfz 19033 Replacing a finitely suppo...
nn0gsumfz0 19034 Replacing a finitely suppo...
gsummptnn0fz 19035 A final group sum over a f...
gsummptnn0fzfv 19036 A final group sum over a f...
telgsumfzslem 19037 Lemma for ~ telgsumfzs (in...
telgsumfzs 19038 Telescoping group sum rang...
telgsumfz 19039 Telescoping group sum rang...
telgsumfz0s 19040 Telescoping finite group s...
telgsumfz0 19041 Telescoping finite group s...
telgsums 19042 Telescoping finitely suppo...
telgsum 19043 Telescoping finitely suppo...
reldmdprd 19048 The domain of the internal...
dmdprd 19049 The domain of definition o...
dmdprdd 19050 Show that a given family i...
dprddomprc 19051 A family of subgroups inde...
dprddomcld 19052 If a family of subgroups i...
dprdval0prc 19053 The internal direct produc...
dprdval 19054 The value of the internal ...
eldprd 19055 A class ` A ` is an intern...
dprdgrp 19056 Reverse closure for the in...
dprdf 19057 The function ` S ` is a fa...
dprdf2 19058 The function ` S ` is a fa...
dprdcntz 19059 The function ` S ` is a fa...
dprddisj 19060 The function ` S ` is a fa...
dprdw 19061 The property of being a fi...
dprdwd 19062 A mapping being a finitely...
dprdff 19063 A finitely supported funct...
dprdfcl 19064 A finitely supported funct...
dprdffsupp 19065 A finitely supported funct...
dprdfcntz 19066 A function on the elements...
dprdssv 19067 The internal direct produc...
dprdfid 19068 A function mapping all but...
eldprdi 19069 The domain of definition o...
dprdfinv 19070 Take the inverse of a grou...
dprdfadd 19071 Take the sum of group sums...
dprdfsub 19072 Take the difference of gro...
dprdfeq0 19073 The zero function is the o...
dprdf11 19074 Two group sums over a dire...
dprdsubg 19075 The internal direct produc...
dprdub 19076 Each factor is a subset of...
dprdlub 19077 The direct product is smal...
dprdspan 19078 The direct product is the ...
dprdres 19079 Restriction of a direct pr...
dprdss 19080 Create a direct product by...
dprdz 19081 A family consisting entire...
dprd0 19082 The empty family is an int...
dprdf1o 19083 Rearrange the index set of...
dprdf1 19084 Rearrange the index set of...
subgdmdprd 19085 A direct product in a subg...
subgdprd 19086 A direct product in a subg...
dprdsn 19087 A singleton family is an i...
dmdprdsplitlem 19088 Lemma for ~ dmdprdsplit . ...
dprdcntz2 19089 The function ` S ` is a fa...
dprddisj2 19090 The function ` S ` is a fa...
dprd2dlem2 19091 The direct product of a co...
dprd2dlem1 19092 The direct product of a co...
dprd2da 19093 The direct product of a co...
dprd2db 19094 The direct product of a co...
dprd2d2 19095 The direct product of a co...
dmdprdsplit2lem 19096 Lemma for ~ dmdprdsplit . ...
dmdprdsplit2 19097 The direct product splits ...
dmdprdsplit 19098 The direct product splits ...
dprdsplit 19099 The direct product is the ...
dmdprdpr 19100 A singleton family is an i...
dprdpr 19101 A singleton family is an i...
dpjlem 19102 Lemma for theorems about d...
dpjcntz 19103 The two subgroups that app...
dpjdisj 19104 The two subgroups that app...
dpjlsm 19105 The two subgroups that app...
dpjfval 19106 Value of the direct produc...
dpjval 19107 Value of the direct produc...
dpjf 19108 The ` X ` -th index projec...
dpjidcl 19109 The key property of projec...
dpjeq 19110 Decompose a group sum into...
dpjid 19111 The key property of projec...
dpjlid 19112 The ` X ` -th index projec...
dpjrid 19113 The ` Y ` -th index projec...
dpjghm 19114 The direct product is the ...
dpjghm2 19115 The direct product is the ...
ablfacrplem 19116 Lemma for ~ ablfacrp2 . (...
ablfacrp 19117 A finite abelian group who...
ablfacrp2 19118 The factors ` K , L ` of ~...
ablfac1lem 19119 Lemma for ~ ablfac1b . Sa...
ablfac1a 19120 The factors of ~ ablfac1b ...
ablfac1b 19121 Any abelian group is the d...
ablfac1c 19122 The factors of ~ ablfac1b ...
ablfac1eulem 19123 Lemma for ~ ablfac1eu . (...
ablfac1eu 19124 The factorization of ~ abl...
pgpfac1lem1 19125 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem2 19126 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem3a 19127 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem3 19128 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem4 19129 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem5 19130 Lemma for ~ pgpfac1 . (Co...
pgpfac1 19131 Factorization of a finite ...
pgpfaclem1 19132 Lemma for ~ pgpfac . (Con...
pgpfaclem2 19133 Lemma for ~ pgpfac . (Con...
pgpfaclem3 19134 Lemma for ~ pgpfac . (Con...
pgpfac 19135 Full factorization of a fi...
ablfaclem1 19136 Lemma for ~ ablfac . (Con...
ablfaclem2 19137 Lemma for ~ ablfac . (Con...
ablfaclem3 19138 Lemma for ~ ablfac . (Con...
ablfac 19139 The Fundamental Theorem of...
ablfac2 19140 Choose generators for each...
issimpg 19143 The predicate "is a simple...
issimpgd 19144 Deduce a simple group from...
simpggrp 19145 A simple group is a group....
simpggrpd 19146 A simple group is a group....
simpg2nsg 19147 A simple group has two nor...
trivnsimpgd 19148 Trivial groups are not sim...
simpgntrivd 19149 Simple groups are nontrivi...
simpgnideld 19150 A simple group contains a ...
simpgnsgd 19151 The only normal subgroups ...
simpgnsgeqd 19152 A normal subgroup of a sim...
2nsgsimpgd 19153 If any normal subgroup of ...
simpgnsgbid 19154 A nontrivial group is simp...
ablsimpnosubgd 19155 A subgroup of an abelian s...
ablsimpg1gend 19156 An abelian simple group is...
ablsimpgcygd 19157 An abelian simple group is...
ablsimpgfindlem1 19158 Lemma for ~ ablsimpgfind ....
ablsimpgfindlem2 19159 Lemma for ~ ablsimpgfind ....
cycsubggenodd 19160 Relationship between the o...
ablsimpgfind 19161 An abelian simple group is...
fincygsubgd 19162 The subgroup referenced in...
fincygsubgodd 19163 Calculate the order of a s...
fincygsubgodexd 19164 A finite cyclic group has ...
prmgrpsimpgd 19165 A group of prime order is ...
ablsimpgprmd 19166 An abelian simple group ha...
ablsimpgd 19167 An abelian group is simple...
fnmgp 19170 The multiplicative group o...
mgpval 19171 Value of the multiplicatio...
mgpplusg 19172 Value of the group operati...
mgplem 19173 Lemma for ~ mgpbas . (Con...
mgpbas 19174 Base set of the multiplica...
mgpsca 19175 The multiplication monoid ...
mgptset 19176 Topology component of the ...
mgptopn 19177 Topology of the multiplica...
mgpds 19178 Distance function of the m...
mgpress 19179 Subgroup commutes with the...
ringidval 19182 The value of the unity ele...
dfur2 19183 The multiplicative identit...
issrg 19186 The predicate "is a semiri...
srgcmn 19187 A semiring is a commutativ...
srgmnd 19188 A semiring is a monoid. (...
srgmgp 19189 A semiring is a monoid und...
srgi 19190 Properties of a semiring. ...
srgcl 19191 Closure of the multiplicat...
srgass 19192 Associative law for the mu...
srgideu 19193 The unit element of a semi...
srgfcl 19194 Functionality of the multi...
srgdi 19195 Distributive law for the m...
srgdir 19196 Distributive law for the m...
srgidcl 19197 The unit element of a semi...
srg0cl 19198 The zero element of a semi...
srgidmlem 19199 Lemma for ~ srglidm and ~ ...
srglidm 19200 The unit element of a semi...
srgridm 19201 The unit element of a semi...
issrgid 19202 Properties showing that an...
srgacl 19203 Closure of the addition op...
srgcom 19204 Commutativity of the addit...
srgrz 19205 The zero of a semiring is ...
srglz 19206 The zero of a semiring is ...
srgisid 19207 In a semiring, the only le...
srg1zr 19208 The only semiring with a b...
srgen1zr 19209 The only semiring with one...
srgmulgass 19210 An associative property be...
srgpcomp 19211 If two elements of a semir...
srgpcompp 19212 If two elements of a semir...
srgpcomppsc 19213 If two elements of a semir...
srglmhm 19214 Left-multiplication in a s...
srgrmhm 19215 Right-multiplication in a ...
srgsummulcr 19216 A finite semiring sum mult...
sgsummulcl 19217 A finite semiring sum mult...
srg1expzeq1 19218 The exponentiation (by a n...
srgbinomlem1 19219 Lemma 1 for ~ srgbinomlem ...
srgbinomlem2 19220 Lemma 2 for ~ srgbinomlem ...
srgbinomlem3 19221 Lemma 3 for ~ srgbinomlem ...
srgbinomlem4 19222 Lemma 4 for ~ srgbinomlem ...
srgbinomlem 19223 Lemma for ~ srgbinom . In...
srgbinom 19224 The binomial theorem for c...
csrgbinom 19225 The binomial theorem for c...
isring 19230 The predicate "is a (unita...
ringgrp 19231 A ring is a group. (Contr...
ringmgp 19232 A ring is a monoid under m...
iscrng 19233 A commutative ring is a ri...
crngmgp 19234 A commutative ring's multi...
ringmnd 19235 A ring is a monoid under a...
ringmgm 19236 A ring is a magma. (Contr...
crngring 19237 A commutative ring is a ri...
mgpf 19238 Restricted functionality o...
ringi 19239 Properties of a unital rin...
ringcl 19240 Closure of the multiplicat...
crngcom 19241 A commutative ring's multi...
iscrng2 19242 A commutative ring is a ri...
ringass 19243 Associative law for the mu...
ringideu 19244 The unit element of a ring...
ringdi 19245 Distributive law for the m...
ringdir 19246 Distributive law for the m...
ringidcl 19247 The unit element of a ring...
ring0cl 19248 The zero element of a ring...
ringidmlem 19249 Lemma for ~ ringlidm and ~...
ringlidm 19250 The unit element of a ring...
ringridm 19251 The unit element of a ring...
isringid 19252 Properties showing that an...
ringid 19253 The multiplication operati...
ringadd2 19254 A ring element plus itself...
rngo2times 19255 A ring element plus itself...
ringidss 19256 A subset of the multiplica...
ringacl 19257 Closure of the addition op...
ringcom 19258 Commutativity of the addit...
ringabl 19259 A ring is an Abelian group...
ringcmn 19260 A ring is a commutative mo...
ringpropd 19261 If two structures have the...
crngpropd 19262 If two structures have the...
ringprop 19263 If two structures have the...
isringd 19264 Properties that determine ...
iscrngd 19265 Properties that determine ...
ringlz 19266 The zero of a unital ring ...
ringrz 19267 The zero of a unital ring ...
ringsrg 19268 Any ring is also a semirin...
ring1eq0 19269 If one and zero are equal,...
ring1ne0 19270 If a ring has at least two...
ringinvnz1ne0 19271 In a unitary ring, a left ...
ringinvnzdiv 19272 In a unitary ring, a left ...
ringnegl 19273 Negation in a ring is the ...
rngnegr 19274 Negation in a ring is the ...
ringmneg1 19275 Negation of a product in a...
ringmneg2 19276 Negation of a product in a...
ringm2neg 19277 Double negation of a produ...
ringsubdi 19278 Ring multiplication distri...
rngsubdir 19279 Ring multiplication distri...
mulgass2 19280 An associative property be...
ring1 19281 The (smallest) structure r...
ringn0 19282 Rings exist. (Contributed...
ringlghm 19283 Left-multiplication in a r...
ringrghm 19284 Right-multiplication in a ...
gsummulc1 19285 A finite ring sum multipli...
gsummulc2 19286 A finite ring sum multipli...
gsummgp0 19287 If one factor in a finite ...
gsumdixp 19288 Distribute a binary produc...
prdsmgp 19289 The multiplicative monoid ...
prdsmulrcl 19290 A structure product of rin...
prdsringd 19291 A product of rings is a ri...
prdscrngd 19292 A product of commutative r...
prds1 19293 Value of the ring unit in ...
pwsring 19294 A structure power of a rin...
pws1 19295 Value of the ring unit in ...
pwscrng 19296 A structure power of a com...
pwsmgp 19297 The multiplicative group o...
imasring 19298 The image structure of a r...
qusring2 19299 The quotient structure of ...
crngbinom 19300 The binomial theorem for c...
opprval 19303 Value of the opposite ring...
opprmulfval 19304 Value of the multiplicatio...
opprmul 19305 Value of the multiplicatio...
crngoppr 19306 In a commutative ring, the...
opprlem 19307 Lemma for ~ opprbas and ~ ...
opprbas 19308 Base set of an opposite ri...
oppradd 19309 Addition operation of an o...
opprring 19310 An opposite ring is a ring...
opprringb 19311 Bidirectional form of ~ op...
oppr0 19312 Additive identity of an op...
oppr1 19313 Multiplicative identity of...
opprneg 19314 The negative function in a...
opprsubg 19315 Being a subgroup is a symm...
mulgass3 19316 An associative property be...
reldvdsr 19323 The divides relation is a ...
dvdsrval 19324 Value of the divides relat...
dvdsr 19325 Value of the divides relat...
dvdsr2 19326 Value of the divides relat...
dvdsrmul 19327 A left-multiple of ` X ` i...
dvdsrcl 19328 Closure of a dividing elem...
dvdsrcl2 19329 Closure of a dividing elem...
dvdsrid 19330 An element in a (unital) r...
dvdsrtr 19331 Divisibility is transitive...
dvdsrmul1 19332 The divisibility relation ...
dvdsrneg 19333 An element divides its neg...
dvdsr01 19334 In a ring, zero is divisib...
dvdsr02 19335 Only zero is divisible by ...
isunit 19336 Property of being a unit o...
1unit 19337 The multiplicative identit...
unitcl 19338 A unit is an element of th...
unitss 19339 The set of units is contai...
opprunit 19340 Being a unit is a symmetri...
crngunit 19341 Property of being a unit i...
dvdsunit 19342 A divisor of a unit is a u...
unitmulcl 19343 The product of units is a ...
unitmulclb 19344 Reversal of ~ unitmulcl in...
unitgrpbas 19345 The base set of the group ...
unitgrp 19346 The group of units is a gr...
unitabl 19347 The group of units of a co...
unitgrpid 19348 The identity of the multip...
unitsubm 19349 The group of units is a su...
invrfval 19352 Multiplicative inverse fun...
unitinvcl 19353 The inverse of a unit exis...
unitinvinv 19354 The inverse of the inverse...
ringinvcl 19355 The inverse of a unit is a...
unitlinv 19356 A unit times its inverse i...
unitrinv 19357 A unit times its inverse i...
1rinv 19358 The inverse of the identit...
0unit 19359 The additive identity is a...
unitnegcl 19360 The negative of a unit is ...
dvrfval 19363 Division operation in a ri...
dvrval 19364 Division operation in a ri...
dvrcl 19365 Closure of division operat...
unitdvcl 19366 The units are closed under...
dvrid 19367 A cancellation law for div...
dvr1 19368 A cancellation law for div...
dvrass 19369 An associative law for div...
dvrcan1 19370 A cancellation law for div...
dvrcan3 19371 A cancellation law for div...
dvreq1 19372 A cancellation law for div...
ringinvdv 19373 Write the inverse function...
rngidpropd 19374 The ring identity depends ...
dvdsrpropd 19375 The divisibility relation ...
unitpropd 19376 The set of units depends o...
invrpropd 19377 The ring inverse function ...
isirred 19378 An irreducible element of ...
isnirred 19379 The property of being a no...
isirred2 19380 Expand out the class diffe...
opprirred 19381 Irreducibility is symmetri...
irredn0 19382 The additive identity is n...
irredcl 19383 An irreducible element is ...
irrednu 19384 An irreducible element is ...
irredn1 19385 The multiplicative identit...
irredrmul 19386 The product of an irreduci...
irredlmul 19387 The product of a unit and ...
irredmul 19388 If product of two elements...
irredneg 19389 The negative of an irreduc...
irrednegb 19390 An element is irreducible ...
dfrhm2 19398 The property of a ring hom...
rhmrcl1 19400 Reverse closure of a ring ...
rhmrcl2 19401 Reverse closure of a ring ...
isrhm 19402 A function is a ring homom...
rhmmhm 19403 A ring homomorphism is a h...
isrim0 19404 An isomorphism of rings is...
rimrcl 19405 Reverse closure for an iso...
rhmghm 19406 A ring homomorphism is an ...
rhmf 19407 A ring homomorphism is a f...
rhmmul 19408 A homomorphism of rings pr...
isrhm2d 19409 Demonstration of ring homo...
isrhmd 19410 Demonstration of ring homo...
rhm1 19411 Ring homomorphisms are req...
idrhm 19412 The identity homomorphism ...
rhmf1o 19413 A ring homomorphism is bij...
isrim 19414 An isomorphism of rings is...
rimf1o 19415 An isomorphism of rings is...
rimrhm 19416 An isomorphism of rings is...
rimgim 19417 An isomorphism of rings is...
rhmco 19418 The composition of ring ho...
pwsco1rhm 19419 Right composition with a f...
pwsco2rhm 19420 Left composition with a ri...
f1ghm0to0 19421 If a group homomorphism ` ...
f1rhm0to0OLD 19422 Obsolete version of ~ f1gh...
f1rhm0to0ALT 19423 Alternate proof for ~ f1gh...
gim0to0 19424 A group isomorphism maps t...
rim0to0OLD 19425 Obsolete version of ~ gim0...
kerf1ghm 19426 A group homomorphism ` F `...
kerf1hrmOLD 19427 Obsolete version of ~ kerf...
brric 19428 The relation "is isomorphi...
brric2 19429 The relation "is isomorphi...
ricgic 19430 If two rings are (ring) is...
isdrng 19435 The predicate "is a divisi...
drngunit 19436 Elementhood in the set of ...
drngui 19437 The set of units of a divi...
drngring 19438 A division ring is a ring....
drnggrp 19439 A division ring is a group...
isfld 19440 A field is a commutative d...
isdrng2 19441 A division ring can equiva...
drngprop 19442 If two structures have the...
drngmgp 19443 A division ring contains a...
drngmcl 19444 The product of two nonzero...
drngid 19445 A division ring's unit is ...
drngunz 19446 A division ring's unit is ...
drngid2 19447 Properties showing that an...
drnginvrcl 19448 Closure of the multiplicat...
drnginvrn0 19449 The multiplicative inverse...
drnginvrl 19450 Property of the multiplica...
drnginvrr 19451 Property of the multiplica...
drngmul0or 19452 A product is zero iff one ...
drngmulne0 19453 A product is nonzero iff b...
drngmuleq0 19454 An element is zero iff its...
opprdrng 19455 The opposite of a division...
isdrngd 19456 Properties that characteri...
isdrngrd 19457 Properties that characteri...
drngpropd 19458 If two structures have the...
fldpropd 19459 If two structures have the...
issubrg 19464 The subring predicate. (C...
subrgss 19465 A subring is a subset. (C...
subrgid 19466 Every ring is a subring of...
subrgring 19467 A subring is a ring. (Con...
subrgcrng 19468 A subring of a commutative...
subrgrcl 19469 Reverse closure for a subr...
subrgsubg 19470 A subring is a subgroup. ...
subrg0 19471 A subring always has the s...
subrg1cl 19472 A subring contains the mul...
subrgbas 19473 Base set of a subring stru...
subrg1 19474 A subring always has the s...
subrgacl 19475 A subring is closed under ...
subrgmcl 19476 A subgroup is closed under...
subrgsubm 19477 A subring is a submonoid o...
subrgdvds 19478 If an element divides anot...
subrguss 19479 A unit of a subring is a u...
subrginv 19480 A subring always has the s...
subrgdv 19481 A subring always has the s...
subrgunit 19482 An element of a ring is a ...
subrgugrp 19483 The units of a subring for...
issubrg2 19484 Characterize the subrings ...
opprsubrg 19485 Being a subring is a symme...
subrgint 19486 The intersection of a none...
subrgin 19487 The intersection of two su...
subrgmre 19488 The subrings of a ring are...
issubdrg 19489 Characterize the subfields...
subsubrg 19490 A subring of a subring is ...
subsubrg2 19491 The set of subrings of a s...
issubrg3 19492 A subring is an additive s...
resrhm 19493 Restriction of a ring homo...
rhmeql 19494 The equalizer of two ring ...
rhmima 19495 The homomorphic image of a...
rnrhmsubrg 19496 The range of a ring homomo...
cntzsubr 19497 Centralizers in a ring are...
pwsdiagrhm 19498 Diagonal homomorphism into...
subrgpropd 19499 If two structures have the...
rhmpropd 19500 Ring homomorphism depends ...
issdrg 19503 Property of a division sub...
sdrgid 19504 Every division ring is a d...
sdrgss 19505 A division subring is a su...
issdrg2 19506 Property of a division sub...
acsfn1p 19507 Construction of a closure ...
subrgacs 19508 Closure property of subrin...
sdrgacs 19509 Closure property of divisi...
cntzsdrg 19510 Centralizers in division r...
subdrgint 19511 The intersection of a none...
sdrgint 19512 The intersection of a none...
primefld 19513 The smallest sub division ...
primefld0cl 19514 The prime field contains t...
primefld1cl 19515 The prime field contains t...
abvfval 19518 Value of the set of absolu...
isabv 19519 Elementhood in the set of ...
isabvd 19520 Properties that determine ...
abvrcl 19521 Reverse closure for the ab...
abvfge0 19522 An absolute value is a fun...
abvf 19523 An absolute value is a fun...
abvcl 19524 An absolute value is a fun...
abvge0 19525 The absolute value of a nu...
abveq0 19526 The value of an absolute v...
abvne0 19527 The absolute value of a no...
abvgt0 19528 The absolute value of a no...
abvmul 19529 An absolute value distribu...
abvtri 19530 An absolute value satisfie...
abv0 19531 The absolute value of zero...
abv1z 19532 The absolute value of one ...
abv1 19533 The absolute value of one ...
abvneg 19534 The absolute value of a ne...
abvsubtri 19535 An absolute value satisfie...
abvrec 19536 The absolute value distrib...
abvdiv 19537 The absolute value distrib...
abvdom 19538 Any ring with an absolute ...
abvres 19539 The restriction of an abso...
abvtrivd 19540 The trivial absolute value...
abvtriv 19541 The trivial absolute value...
abvpropd 19542 If two structures have the...
staffval 19547 The functionalization of t...
stafval 19548 The functionalization of t...
staffn 19549 The functionalization is e...
issrng 19550 The predicate "is a star r...
srngrhm 19551 The involution function in...
srngring 19552 A star ring is a ring. (C...
srngcnv 19553 The involution function in...
srngf1o 19554 The involution function in...
srngcl 19555 The involution function in...
srngnvl 19556 The involution function in...
srngadd 19557 The involution function in...
srngmul 19558 The involution function in...
srng1 19559 The conjugate of the ring ...
srng0 19560 The conjugate of the ring ...
issrngd 19561 Properties that determine ...
idsrngd 19562 A commutative ring is a st...
islmod 19567 The predicate "is a left m...
lmodlema 19568 Lemma for properties of a ...
islmodd 19569 Properties that determine ...
lmodgrp 19570 A left module is a group. ...
lmodring 19571 The scalar component of a ...
lmodfgrp 19572 The scalar component of a ...
lmodbn0 19573 The base set of a left mod...
lmodacl 19574 Closure of ring addition f...
lmodmcl 19575 Closure of ring multiplica...
lmodsn0 19576 The set of scalars in a le...
lmodvacl 19577 Closure of vector addition...
lmodass 19578 Left module vector sum is ...
lmodlcan 19579 Left cancellation law for ...
lmodvscl 19580 Closure of scalar product ...
scaffval 19581 The scalar multiplication ...
scafval 19582 The scalar multiplication ...
scafeq 19583 If the scalar multiplicati...
scaffn 19584 The scalar multiplication ...
lmodscaf 19585 The scalar multiplication ...
lmodvsdi 19586 Distributive law for scala...
lmodvsdir 19587 Distributive law for scala...
lmodvsass 19588 Associative law for scalar...
lmod0cl 19589 The ring zero in a left mo...
lmod1cl 19590 The ring unit in a left mo...
lmodvs1 19591 Scalar product with ring u...
lmod0vcl 19592 The zero vector is a vecto...
lmod0vlid 19593 Left identity law for the ...
lmod0vrid 19594 Right identity law for the...
lmod0vid 19595 Identity equivalent to the...
lmod0vs 19596 Zero times a vector is the...
lmodvs0 19597 Anything times the zero ve...
lmodvsmmulgdi 19598 Distributive law for a gro...
lmodfopnelem1 19599 Lemma 1 for ~ lmodfopne . ...
lmodfopnelem2 19600 Lemma 2 for ~ lmodfopne . ...
lmodfopne 19601 The (functionalized) opera...
lcomf 19602 A linear-combination sum i...
lcomfsupp 19603 A linear-combination sum i...
lmodvnegcl 19604 Closure of vector negative...
lmodvnegid 19605 Addition of a vector with ...
lmodvneg1 19606 Minus 1 times a vector is ...
lmodvsneg 19607 Multiplication of a vector...
lmodvsubcl 19608 Closure of vector subtract...
lmodcom 19609 Left module vector sum is ...
lmodabl 19610 A left module is an abelia...
lmodcmn 19611 A left module is a commuta...
lmodnegadd 19612 Distribute negation throug...
lmod4 19613 Commutative/associative la...
lmodvsubadd 19614 Relationship between vecto...
lmodvaddsub4 19615 Vector addition/subtractio...
lmodvpncan 19616 Addition/subtraction cance...
lmodvnpcan 19617 Cancellation law for vecto...
lmodvsubval2 19618 Value of vector subtractio...
lmodsubvs 19619 Subtraction of a scalar pr...
lmodsubdi 19620 Scalar multiplication dist...
lmodsubdir 19621 Scalar multiplication dist...
lmodsubeq0 19622 If the difference between ...
lmodsubid 19623 Subtraction of a vector fr...
lmodvsghm 19624 Scalar multiplication of t...
lmodprop2d 19625 If two structures have the...
lmodpropd 19626 If two structures have the...
gsumvsmul 19627 Pull a scalar multiplicati...
mptscmfsupp0 19628 A mapping to a scalar prod...
mptscmfsuppd 19629 A function mapping to a sc...
rmodislmodlem 19630 Lemma for ~ rmodislmod . ...
rmodislmod 19631 The right module ` R ` ind...
lssset 19634 The set of all (not necess...
islss 19635 The predicate "is a subspa...
islssd 19636 Properties that determine ...
lssss 19637 A subspace is a set of vec...
lssel 19638 A subspace member is a vec...
lss1 19639 The set of vectors in a le...
lssuni 19640 The union of all subspaces...
lssn0 19641 A subspace is not empty. ...
00lss 19642 The empty structure has no...
lsscl 19643 Closure property of a subs...
lssvsubcl 19644 Closure of vector subtract...
lssvancl1 19645 Non-closure: if one vector...
lssvancl2 19646 Non-closure: if one vector...
lss0cl 19647 The zero vector belongs to...
lsssn0 19648 The singleton of the zero ...
lss0ss 19649 The zero subspace is inclu...
lssle0 19650 No subspace is smaller tha...
lssne0 19651 A nonzero subspace has a n...
lssvneln0 19652 A vector ` X ` which doesn...
lssneln0 19653 A vector ` X ` which doesn...
lssssr 19654 Conclude subspace ordering...
lssvacl 19655 Closure of vector addition...
lssvscl 19656 Closure of scalar product ...
lssvnegcl 19657 Closure of negative vector...
lsssubg 19658 All subspaces are subgroup...
lsssssubg 19659 All subspaces are subgroup...
islss3 19660 A linear subspace of a mod...
lsslmod 19661 A submodule is a module. ...
lsslss 19662 The subspaces of a subspac...
islss4 19663 A linear subspace is a sub...
lss1d 19664 One-dimensional subspace (...
lssintcl 19665 The intersection of a none...
lssincl 19666 The intersection of two su...
lssmre 19667 The subspaces of a module ...
lssacs 19668 Submodules are an algebrai...
prdsvscacl 19669 Pointwise scalar multiplic...
prdslmodd 19670 The product of a family of...
pwslmod 19671 The product of a family of...
lspfval 19674 The span function for a le...
lspf 19675 The span operator on a lef...
lspval 19676 The span of a set of vecto...
lspcl 19677 The span of a set of vecto...
lspsncl 19678 The span of a singleton is...
lspprcl 19679 The span of a pair is a su...
lsptpcl 19680 The span of an unordered t...
lspsnsubg 19681 The span of a singleton is...
00lsp 19682 ~ fvco4i lemma for linear ...
lspid 19683 The span of a subspace is ...
lspssv 19684 A span is a set of vectors...
lspss 19685 Span preserves subset orde...
lspssid 19686 A set of vectors is a subs...
lspidm 19687 The span of a set of vecto...
lspun 19688 The span of union is the s...
lspssp 19689 If a set of vectors is a s...
mrclsp 19690 Moore closure generalizes ...
lspsnss 19691 The span of the singleton ...
lspsnel3 19692 A member of the span of th...
lspprss 19693 The span of a pair of vect...
lspsnid 19694 A vector belongs to the sp...
lspsnel6 19695 Relationship between a vec...
lspsnel5 19696 Relationship between a vec...
lspsnel5a 19697 Relationship between a vec...
lspprid1 19698 A member of a pair of vect...
lspprid2 19699 A member of a pair of vect...
lspprvacl 19700 The sum of two vectors bel...
lssats2 19701 A way to express atomistic...
lspsneli 19702 A scalar product with a ve...
lspsn 19703 Span of the singleton of a...
lspsnel 19704 Member of span of the sing...
lspsnvsi 19705 Span of a scalar product o...
lspsnss2 19706 Comparable spans of single...
lspsnneg 19707 Negation does not change t...
lspsnsub 19708 Swapping subtraction order...
lspsn0 19709 Span of the singleton of t...
lsp0 19710 Span of the empty set. (C...
lspuni0 19711 Union of the span of the e...
lspun0 19712 The span of a union with t...
lspsneq0 19713 Span of the singleton is t...
lspsneq0b 19714 Equal singleton spans impl...
lmodindp1 19715 Two independent (non-colin...
lsslsp 19716 Spans in submodules corres...
lss0v 19717 The zero vector in a submo...
lsspropd 19718 If two structures have the...
lsppropd 19719 If two structures have the...
reldmlmhm 19726 Lemma for module homomorph...
lmimfn 19727 Lemma for module isomorphi...
islmhm 19728 Property of being a homomo...
islmhm3 19729 Property of a module homom...
lmhmlem 19730 Non-quantified consequence...
lmhmsca 19731 A homomorphism of left mod...
lmghm 19732 A homomorphism of left mod...
lmhmlmod2 19733 A homomorphism of left mod...
lmhmlmod1 19734 A homomorphism of left mod...
lmhmf 19735 A homomorphism of left mod...
lmhmlin 19736 A homomorphism of left mod...
lmodvsinv 19737 Multiplication of a vector...
lmodvsinv2 19738 Multiplying a negated vect...
islmhm2 19739 A one-equation proof of li...
islmhmd 19740 Deduction for a module hom...
0lmhm 19741 The constant zero linear f...
idlmhm 19742 The identity function on a...
invlmhm 19743 The negative function on a...
lmhmco 19744 The composition of two mod...
lmhmplusg 19745 The pointwise sum of two l...
lmhmvsca 19746 The pointwise scalar produ...
lmhmf1o 19747 A bijective module homomor...
lmhmima 19748 The image of a subspace un...
lmhmpreima 19749 The inverse image of a sub...
lmhmlsp 19750 Homomorphisms preserve spa...
lmhmrnlss 19751 The range of a homomorphis...
lmhmkerlss 19752 The kernel of a homomorphi...
reslmhm 19753 Restriction of a homomorph...
reslmhm2 19754 Expansion of the codomain ...
reslmhm2b 19755 Expansion of the codomain ...
lmhmeql 19756 The equalizer of two modul...
lspextmo 19757 A linear function is compl...
pwsdiaglmhm 19758 Diagonal homomorphism into...
pwssplit0 19759 Splitting for structure po...
pwssplit1 19760 Splitting for structure po...
pwssplit2 19761 Splitting for structure po...
pwssplit3 19762 Splitting for structure po...
islmim 19763 An isomorphism of left mod...
lmimf1o 19764 An isomorphism of left mod...
lmimlmhm 19765 An isomorphism of modules ...
lmimgim 19766 An isomorphism of modules ...
islmim2 19767 An isomorphism of left mod...
lmimcnv 19768 The converse of a bijectiv...
brlmic 19769 The relation "is isomorphi...
brlmici 19770 Prove isomorphic by an exp...
lmiclcl 19771 Isomorphism implies the le...
lmicrcl 19772 Isomorphism implies the ri...
lmicsym 19773 Module isomorphism is symm...
lmhmpropd 19774 Module homomorphism depend...
islbs 19777 The predicate " ` B ` is a...
lbsss 19778 A basis is a set of vector...
lbsel 19779 An element of a basis is a...
lbssp 19780 The span of a basis is the...
lbsind 19781 A basis is linearly indepe...
lbsind2 19782 A basis is linearly indepe...
lbspss 19783 No proper subset of a basi...
lsmcl 19784 The sum of two subspaces i...
lsmspsn 19785 Member of subspace sum of ...
lsmelval2 19786 Subspace sum membership in...
lsmsp 19787 Subspace sum in terms of s...
lsmsp2 19788 Subspace sum of spans of s...
lsmssspx 19789 Subspace sum (in its exten...
lsmpr 19790 The span of a pair of vect...
lsppreli 19791 A vector expressed as a su...
lsmelpr 19792 Two ways to say that a vec...
lsppr0 19793 The span of a vector paire...
lsppr 19794 Span of a pair of vectors....
lspprel 19795 Member of the span of a pa...
lspprabs 19796 Absorption of vector sum i...
lspvadd 19797 The span of a vector sum i...
lspsntri 19798 Triangle-type inequality f...
lspsntrim 19799 Triangle-type inequality f...
lbspropd 19800 If two structures have the...
pj1lmhm 19801 The left projection functi...
pj1lmhm2 19802 The left projection functi...
islvec 19805 The predicate "is a left v...
lvecdrng 19806 The set of scalars of a le...
lveclmod 19807 A left vector space is a l...
lsslvec 19808 A vector subspace is a vec...
lvecvs0or 19809 If a scalar product is zer...
lvecvsn0 19810 A scalar product is nonzer...
lssvs0or 19811 If a scalar product belong...
lvecvscan 19812 Cancellation law for scala...
lvecvscan2 19813 Cancellation law for scala...
lvecinv 19814 Invert coefficient of scal...
lspsnvs 19815 A nonzero scalar product d...
lspsneleq 19816 Membership relation that i...
lspsncmp 19817 Comparable spans of nonzer...
lspsnne1 19818 Two ways to express that v...
lspsnne2 19819 Two ways to express that v...
lspsnnecom 19820 Swap two vectors with diff...
lspabs2 19821 Absorption law for span of...
lspabs3 19822 Absorption law for span of...
lspsneq 19823 Equal spans of singletons ...
lspsneu 19824 Nonzero vectors with equal...
lspsnel4 19825 A member of the span of th...
lspdisj 19826 The span of a vector not i...
lspdisjb 19827 A nonzero vector is not in...
lspdisj2 19828 Unequal spans are disjoint...
lspfixed 19829 Show membership in the spa...
lspexch 19830 Exchange property for span...
lspexchn1 19831 Exchange property for span...
lspexchn2 19832 Exchange property for span...
lspindpi 19833 Partial independence prope...
lspindp1 19834 Alternate way to say 3 vec...
lspindp2l 19835 Alternate way to say 3 vec...
lspindp2 19836 Alternate way to say 3 vec...
lspindp3 19837 Independence of 2 vectors ...
lspindp4 19838 (Partial) independence of ...
lvecindp 19839 Compute the ` X ` coeffici...
lvecindp2 19840 Sums of independent vector...
lspsnsubn0 19841 Unequal singleton spans im...
lsmcv 19842 Subspace sum has the cover...
lspsolvlem 19843 Lemma for ~ lspsolv . (Co...
lspsolv 19844 If ` X ` is in the span of...
lssacsex 19845 In a vector space, subspac...
lspsnat 19846 There is no subspace stric...
lspsncv0 19847 The span of a singleton co...
lsppratlem1 19848 Lemma for ~ lspprat . Let...
lsppratlem2 19849 Lemma for ~ lspprat . Sho...
lsppratlem3 19850 Lemma for ~ lspprat . In ...
lsppratlem4 19851 Lemma for ~ lspprat . In ...
lsppratlem5 19852 Lemma for ~ lspprat . Com...
lsppratlem6 19853 Lemma for ~ lspprat . Neg...
lspprat 19854 A proper subspace of the s...
islbs2 19855 An equivalent formulation ...
islbs3 19856 An equivalent formulation ...
lbsacsbs 19857 Being a basis in a vector ...
lvecdim 19858 The dimension theorem for ...
lbsextlem1 19859 Lemma for ~ lbsext . The ...
lbsextlem2 19860 Lemma for ~ lbsext . Sinc...
lbsextlem3 19861 Lemma for ~ lbsext . A ch...
lbsextlem4 19862 Lemma for ~ lbsext . ~ lbs...
lbsextg 19863 For any linearly independe...
lbsext 19864 For any linearly independe...
lbsexg 19865 Every vector space has a b...
lbsex 19866 Every vector space has a b...
lvecprop2d 19867 If two structures have the...
lvecpropd 19868 If two structures have the...
sraval 19877 Lemma for ~ srabase throug...
sralem 19878 Lemma for ~ srabase and si...
srabase 19879 Base set of a subring alge...
sraaddg 19880 Additive operation of a su...
sramulr 19881 Multiplicative operation o...
srasca 19882 The set of scalars of a su...
sravsca 19883 The scalar product operati...
sraip 19884 The inner product operatio...
sratset 19885 Topology component of a su...
sratopn 19886 Topology component of a su...
srads 19887 Distance function of a sub...
sralmod 19888 The subring algebra is a l...
sralmod0 19889 The subring module inherit...
issubrngd2 19890 Prove a subring by closure...
rlmfn 19891 ` ringLMod ` is a function...
rlmval 19892 Value of the ring module. ...
lidlval 19893 Value of the set of ring i...
rspval 19894 Value of the ring span fun...
rlmval2 19895 Value of the ring module e...
rlmbas 19896 Base set of the ring modul...
rlmplusg 19897 Vector addition in the rin...
rlm0 19898 Zero vector in the ring mo...
rlmsub 19899 Subtraction in the ring mo...
rlmmulr 19900 Ring multiplication in the...
rlmsca 19901 Scalars in the ring module...
rlmsca2 19902 Scalars in the ring module...
rlmvsca 19903 Scalar multiplication in t...
rlmtopn 19904 Topology component of the ...
rlmds 19905 Metric component of the ri...
rlmlmod 19906 The ring module is a modul...
rlmlvec 19907 The ring module over a div...
rlmvneg 19908 Vector negation in the rin...
rlmscaf 19909 Functionalized scalar mult...
ixpsnbasval 19910 The value of an infinite C...
lidlss 19911 An ideal is a subset of th...
islidl 19912 Predicate of being a (left...
lidl0cl 19913 An ideal contains 0. (Con...
lidlacl 19914 An ideal is closed under a...
lidlnegcl 19915 An ideal contains negative...
lidlsubg 19916 An ideal is a subgroup of ...
lidlsubcl 19917 An ideal is closed under s...
lidlmcl 19918 An ideal is closed under l...
lidl1el 19919 An ideal contains 1 iff it...
lidl0 19920 Every ring contains a zero...
lidl1 19921 Every ring contains a unit...
lidlacs 19922 The ideal system is an alg...
rspcl 19923 The span of a set of ring ...
rspssid 19924 The span of a set of ring ...
rsp1 19925 The span of the identity e...
rsp0 19926 The span of the zero eleme...
rspssp 19927 The ideal span of a set of...
mrcrsp 19928 Moore closure generalizes ...
lidlnz 19929 A nonzero ideal contains a...
drngnidl 19930 A division ring has only t...
lidlrsppropd 19931 The left ideals and ring s...
2idlval 19934 Definition of a two-sided ...
2idlcpbl 19935 The coset equivalence rela...
qus1 19936 The multiplicative identit...
qusring 19937 If ` S ` is a two-sided id...
qusrhm 19938 If ` S ` is a two-sided id...
crngridl 19939 In a commutative ring, the...
crng2idl 19940 In a commutative ring, a t...
quscrng 19941 The quotient of a commutat...
lpival 19946 Value of the set of princi...
islpidl 19947 Property of being a princi...
lpi0 19948 The zero ideal is always p...
lpi1 19949 The unit ideal is always p...
islpir 19950 Principal ideal rings are ...
lpiss 19951 Principal ideals are a sub...
islpir2 19952 Principal ideal rings are ...
lpirring 19953 Principal ideal rings are ...
drnglpir 19954 Division rings are princip...
rspsn 19955 Membership in principal id...
lidldvgen 19956 An element generates an id...
lpigen 19957 An ideal is principal iff ...
isnzr 19960 Property of a nonzero ring...
nzrnz 19961 One and zero are different...
nzrring 19962 A nonzero ring is a ring. ...
drngnzr 19963 All division rings are non...
isnzr2 19964 Equivalent characterizatio...
isnzr2hash 19965 Equivalent characterizatio...
opprnzr 19966 The opposite of a nonzero ...
ringelnzr 19967 A ring is nonzero if it ha...
nzrunit 19968 A unit is nonzero in any n...
subrgnzr 19969 A subring of a nonzero rin...
0ringnnzr 19970 A ring is a zero ring iff ...
0ring 19971 If a ring has only one ele...
0ring01eq 19972 In a ring with only one el...
01eq0ring 19973 If the zero and the identi...
0ring01eqbi 19974 In a unital ring the zero ...
rng1nnzr 19975 The (smallest) structure r...
ring1zr 19976 The only (unital) ring wit...
rngen1zr 19977 The only (unital) ring wit...
ringen1zr 19978 The only unital ring with ...
rng1nfld 19979 The zero ring is not a fie...
rrgval 19988 Value of the set or left-r...
isrrg 19989 Membership in the set of l...
rrgeq0i 19990 Property of a left-regular...
rrgeq0 19991 Left-multiplication by a l...
rrgsupp 19992 Left multiplication by a l...
rrgss 19993 Left-regular elements are ...
unitrrg 19994 Units are regular elements...
isdomn 19995 Expand definition of a dom...
domnnzr 19996 A domain is a nonzero ring...
domnring 19997 A domain is a ring. (Cont...
domneq0 19998 In a domain, a product is ...
domnmuln0 19999 In a domain, a product of ...
isdomn2 20000 A ring is a domain iff all...
domnrrg 20001 In a domain, any nonzero e...
opprdomn 20002 The opposite of a domain i...
abvn0b 20003 Another characterization o...
drngdomn 20004 A division ring is a domai...
isidom 20005 An integral domain is a co...
fldidom 20006 A field is an integral dom...
fidomndrnglem 20007 Lemma for ~ fidomndrng . ...
fidomndrng 20008 A finite domain is a divis...
fiidomfld 20009 A finite integral domain i...
isassa 20016 The properties of an assoc...
assalem 20017 The properties of an assoc...
assaass 20018 Left-associative property ...
assaassr 20019 Right-associative property...
assalmod 20020 An associative algebra is ...
assaring 20021 An associative algebra is ...
assasca 20022 An associative algebra's s...
assa2ass 20023 Left- and right-associativ...
isassad 20024 Sufficient condition for b...
issubassa3 20025 A subring that is also a s...
issubassa 20026 The subalgebras of an asso...
sraassa 20027 The subring algebra over a...
rlmassa 20028 The ring module over a com...
assapropd 20029 If two structures have the...
aspval 20030 Value of the algebraic clo...
asplss 20031 The algebraic span of a se...
aspid 20032 The algebraic span of a su...
aspsubrg 20033 The algebraic span of a se...
aspss 20034 Span preserves subset orde...
aspssid 20035 A set of vectors is a subs...
asclfval 20036 Function value of the alge...
asclval 20037 Value of a mapped algebra ...
asclfn 20038 Unconditional functionalit...
asclf 20039 The algebra scalars functi...
asclghm 20040 The algebra scalars functi...
ascl0 20041 The scalar 0 embedded into...
asclmul1 20042 Left multiplication by a l...
asclmul2 20043 Right multiplication by a ...
ascldimul 20044 The algebra scalars functi...
ascldimulOLD 20045 The algebra scalars functi...
asclinvg 20046 The group inverse (negatio...
asclrhm 20047 The scalar injection is a ...
rnascl 20048 The set of injected scalar...
issubassa2 20049 A subring of a unital alge...
rnasclsubrg 20050 The scalar multiples of th...
rnasclmulcl 20051 (Vector) multiplication is...
rnasclassa 20052 The scalar multiples of th...
ressascl 20053 The injection of scalars i...
asclpropd 20054 If two structures have the...
aspval2 20055 The algebraic closure is t...
assamulgscmlem1 20056 Lemma 1 for ~ assamulgscm ...
assamulgscmlem2 20057 Lemma for ~ assamulgscm (i...
assamulgscm 20058 Exponentiation of a scalar...
reldmpsr 20069 The multivariate power ser...
psrval 20070 Value of the multivariate ...
psrvalstr 20071 The multivariate power ser...
psrbag 20072 Elementhood in the set of ...
psrbagf 20073 A finite bag is a function...
snifpsrbag 20074 A bag containing one eleme...
fczpsrbag 20075 The constant function equa...
psrbaglesupp 20076 The support of a dominated...
psrbaglecl 20077 The set of finite bags is ...
psrbagaddcl 20078 The sum of two finite bags...
psrbagcon 20079 The analogue of the statem...
psrbaglefi 20080 There are finitely many ba...
psrbagconcl 20081 The complement of a bag is...
psrbagconf1o 20082 Bag complementation is a b...
gsumbagdiaglem 20083 Lemma for ~ gsumbagdiag . ...
gsumbagdiag 20084 Two-dimensional commutatio...
psrass1lem 20085 A group sum commutation us...
psrbas 20086 The base set of the multiv...
psrelbas 20087 An element of the set of p...
psrelbasfun 20088 An element of the set of p...
psrplusg 20089 The addition operation of ...
psradd 20090 The addition operation of ...
psraddcl 20091 Closure of the power serie...
psrmulr 20092 The multiplication operati...
psrmulfval 20093 The multiplication operati...
psrmulval 20094 The multiplication operati...
psrmulcllem 20095 Closure of the power serie...
psrmulcl 20096 Closure of the power serie...
psrsca 20097 The scalar field of the mu...
psrvscafval 20098 The scalar multiplication ...
psrvsca 20099 The scalar multiplication ...
psrvscaval 20100 The scalar multiplication ...
psrvscacl 20101 Closure of the power serie...
psr0cl 20102 The zero element of the ri...
psr0lid 20103 The zero element of the ri...
psrnegcl 20104 The negative function in t...
psrlinv 20105 The negative function in t...
psrgrp 20106 The ring of power series i...
psr0 20107 The zero element of the ri...
psrneg 20108 The negative function of t...
psrlmod 20109 The ring of power series i...
psr1cl 20110 The identity element of th...
psrlidm 20111 The identity element of th...
psrridm 20112 The identity element of th...
psrass1 20113 Associative identity for t...
psrdi 20114 Distributive law for the r...
psrdir 20115 Distributive law for the r...
psrass23l 20116 Associative identity for t...
psrcom 20117 Commutative law for the ri...
psrass23 20118 Associative identities for...
psrring 20119 The ring of power series i...
psr1 20120 The identity element of th...
psrcrng 20121 The ring of power series i...
psrassa 20122 The ring of power series i...
resspsrbas 20123 A restricted power series ...
resspsradd 20124 A restricted power series ...
resspsrmul 20125 A restricted power series ...
resspsrvsca 20126 A restricted power series ...
subrgpsr 20127 A subring of the base ring...
mvrfval 20128 Value of the generating el...
mvrval 20129 Value of the generating el...
mvrval2 20130 Value of the generating el...
mvrid 20131 The ` X i ` -th coefficien...
mvrf 20132 The power series variable ...
mvrf1 20133 The power series variable ...
mvrcl2 20134 A power series variable is...
reldmmpl 20135 The multivariate polynomia...
mplval 20136 Value of the set of multiv...
mplbas 20137 Base set of the set of mul...
mplelbas 20138 Property of being a polyno...
mplval2 20139 Self-referential expressio...
mplbasss 20140 The set of polynomials is ...
mplelf 20141 A polynomial is defined as...
mplsubglem 20142 If ` A ` is an ideal of se...
mpllsslem 20143 If ` A ` is an ideal of su...
mplsubglem2 20144 Lemma for ~ mplsubg and ~ ...
mplsubg 20145 The set of polynomials is ...
mpllss 20146 The set of polynomials is ...
mplsubrglem 20147 Lemma for ~ mplsubrg . (C...
mplsubrg 20148 The set of polynomials is ...
mpl0 20149 The zero polynomial. (Con...
mpladd 20150 The addition operation on ...
mplmul 20151 The multiplication operati...
mpl1 20152 The identity element of th...
mplsca 20153 The scalar field of a mult...
mplvsca2 20154 The scalar multiplication ...
mplvsca 20155 The scalar multiplication ...
mplvscaval 20156 The scalar multiplication ...
mvrcl 20157 A power series variable is...
mplgrp 20158 The polynomial ring is a g...
mpllmod 20159 The polynomial ring is a l...
mplring 20160 The polynomial ring is a r...
mpllvec 20161 The polynomial ring is a v...
mplcrng 20162 The polynomial ring is a c...
mplassa 20163 The polynomial ring is an ...
ressmplbas2 20164 The base set of a restrict...
ressmplbas 20165 A restricted polynomial al...
ressmpladd 20166 A restricted polynomial al...
ressmplmul 20167 A restricted polynomial al...
ressmplvsca 20168 A restricted power series ...
subrgmpl 20169 A subring of the base ring...
subrgmvr 20170 The variables in a subring...
subrgmvrf 20171 The variables in a polynom...
mplmon 20172 A monomial is a polynomial...
mplmonmul 20173 The product of two monomia...
mplcoe1 20174 Decompose a polynomial int...
mplcoe3 20175 Decompose a monomial in on...
mplcoe5lem 20176 Lemma for ~ mplcoe4 . (Co...
mplcoe5 20177 Decompose a monomial into ...
mplcoe2 20178 Decompose a monomial into ...
mplbas2 20179 An alternative expression ...
ltbval 20180 Value of the well-order on...
ltbwe 20181 The finite bag order is a ...
reldmopsr 20182 Lemma for ordered power se...
opsrval 20183 The value of the "ordered ...
opsrle 20184 An alternative expression ...
opsrval2 20185 Self-referential expressio...
opsrbaslem 20186 Get a component of the ord...
opsrbas 20187 The base set of the ordere...
opsrplusg 20188 The addition operation of ...
opsrmulr 20189 The multiplication operati...
opsrvsca 20190 The scalar product operati...
opsrsca 20191 The scalar ring of the ord...
opsrtoslem1 20192 Lemma for ~ opsrtos . (Co...
opsrtoslem2 20193 Lemma for ~ opsrtos . (Co...
opsrtos 20194 The ordered power series s...
opsrso 20195 The ordered power series s...
opsrcrng 20196 The ring of ordered power ...
opsrassa 20197 The ring of ordered power ...
mplrcl 20198 Reverse closure for the po...
mplelsfi 20199 A polynomial treated as a ...
mvrf2 20200 The power series/polynomia...
mplmon2 20201 Express a scaled monomial....
psrbag0 20202 The empty bag is a bag. (...
psrbagsn 20203 A singleton bag is a bag. ...
mplascl 20204 Value of the scalar inject...
mplasclf 20205 The scalar injection is a ...
subrgascl 20206 The scalar injection funct...
subrgasclcl 20207 The scalars in a polynomia...
mplmon2cl 20208 A scaled monomial is a pol...
mplmon2mul 20209 Product of scaled monomial...
mplind 20210 Prove a property of polyno...
mplcoe4 20211 Decompose a polynomial int...
evlslem4 20216 The support of a tensor pr...
psrbagfsupp 20217 Finite bags have finite no...
psrbagev1 20218 A bag of multipliers provi...
psrbagev2 20219 Closure of a sum using a b...
evlslem2 20220 A linear function on the p...
evlslem3 20221 Lemma for ~ evlseu . Poly...
evlslem6 20222 Lemma for ~ evlseu . Fini...
evlslem1 20223 Lemma for ~ evlseu , give ...
evlseu 20224 For a given interpretation...
reldmevls 20225 Well-behaved binary operat...
mpfrcl 20226 Reverse closure for the se...
evlsval 20227 Value of the polynomial ev...
evlsval2 20228 Characterizing properties ...
evlsrhm 20229 Polynomial evaluation is a...
evlssca 20230 Polynomial evaluation maps...
evlsvar 20231 Polynomial evaluation maps...
evlsgsumadd 20232 Polynomial evaluation maps...
evlsgsummul 20233 Polynomial evaluation maps...
evlspw 20234 Polynomial evaluation for ...
evlsvarpw 20235 Polynomial evaluation for ...
evlval 20236 Value of the simple/same r...
evlrhm 20237 The simple evaluation map ...
evlsscasrng 20238 The evaluation of a scalar...
evlsca 20239 Simple polynomial evaluati...
evlsvarsrng 20240 The evaluation of the vari...
evlvar 20241 Simple polynomial evaluati...
mpfconst 20242 Constants are multivariate...
mpfproj 20243 Projections are multivaria...
mpfsubrg 20244 Polynomial functions are a...
mpff 20245 Polynomial functions are f...
mpfaddcl 20246 The sum of multivariate po...
mpfmulcl 20247 The product of multivariat...
mpfind 20248 Prove a property of polyno...
selvffval 20257 Value of the "variable sel...
selvfval 20258 Value of the "variable sel...
selvval 20259 Value of the "variable sel...
mhpfval 20260 Value of the "homogeneous ...
mhpval 20261 Value of the "homogeneous ...
ismhp 20262 Property of being a homoge...
mhpmpl 20263 A homogeneous polynomial i...
mhpdeg 20264 All nonzero terms of a hom...
mhp0cl 20265 The zero polynomial is hom...
mhpaddcl 20266 Homogeneous polynomials ar...
mhpinvcl 20267 Homogeneous polynomials ar...
mhpsubg 20268 Homogeneous polynomials fo...
mhpvscacl 20269 Homogeneous polynomials ar...
mhplss 20270 Homogeneous polynomials fo...
psr1baslem 20281 The set of finite bags on ...
psr1val 20282 Value of the ring of univa...
psr1crng 20283 The ring of univariate pow...
psr1assa 20284 The ring of univariate pow...
psr1tos 20285 The ordered power series s...
psr1bas2 20286 The base set of the ring o...
psr1bas 20287 The base set of the ring o...
vr1val 20288 The value of the generator...
vr1cl2 20289 The variable ` X ` is a me...
ply1val 20290 The value of the set of un...
ply1bas 20291 The value of the base set ...
ply1lss 20292 Univariate polynomials for...
ply1subrg 20293 Univariate polynomials for...
ply1crng 20294 The ring of univariate pol...
ply1assa 20295 The ring of univariate pol...
psr1bascl 20296 A univariate power series ...
psr1basf 20297 Univariate power series ba...
ply1basf 20298 Univariate polynomial base...
ply1bascl 20299 A univariate polynomial is...
ply1bascl2 20300 A univariate polynomial is...
coe1fval 20301 Value of the univariate po...
coe1fv 20302 Value of an evaluated coef...
fvcoe1 20303 Value of a multivariate co...
coe1fval3 20304 Univariate power series co...
coe1f2 20305 Functionality of univariat...
coe1fval2 20306 Univariate polynomial coef...
coe1f 20307 Functionality of univariat...
coe1fvalcl 20308 A coefficient of a univari...
coe1sfi 20309 Finite support of univaria...
coe1fsupp 20310 The coefficient vector of ...
mptcoe1fsupp 20311 A mapping involving coeffi...
coe1ae0 20312 The coefficient vector of ...
vr1cl 20313 The generator of a univari...
opsr0 20314 Zero in the ordered power ...
opsr1 20315 One in the ordered power s...
mplplusg 20316 Value of addition in a pol...
mplmulr 20317 Value of multiplication in...
psr1plusg 20318 Value of addition in a uni...
psr1vsca 20319 Value of scalar multiplica...
psr1mulr 20320 Value of multiplication in...
ply1plusg 20321 Value of addition in a uni...
ply1vsca 20322 Value of scalar multiplica...
ply1mulr 20323 Value of multiplication in...
ressply1bas2 20324 The base set of a restrict...
ressply1bas 20325 A restricted polynomial al...
ressply1add 20326 A restricted polynomial al...
ressply1mul 20327 A restricted polynomial al...
ressply1vsca 20328 A restricted power series ...
subrgply1 20329 A subring of the base ring...
gsumply1subr 20330 Evaluate a group sum in a ...
psrbaspropd 20331 Property deduction for pow...
psrplusgpropd 20332 Property deduction for pow...
mplbaspropd 20333 Property deduction for pol...
psropprmul 20334 Reversing multiplication i...
ply1opprmul 20335 Reversing multiplication i...
00ply1bas 20336 Lemma for ~ ply1basfvi and...
ply1basfvi 20337 Protection compatibility o...
ply1plusgfvi 20338 Protection compatibility o...
ply1baspropd 20339 Property deduction for uni...
ply1plusgpropd 20340 Property deduction for uni...
opsrring 20341 Ordered power series form ...
opsrlmod 20342 Ordered power series form ...
psr1ring 20343 Univariate power series fo...
ply1ring 20344 Univariate polynomials for...
psr1lmod 20345 Univariate power series fo...
psr1sca 20346 Scalars of a univariate po...
psr1sca2 20347 Scalars of a univariate po...
ply1lmod 20348 Univariate polynomials for...
ply1sca 20349 Scalars of a univariate po...
ply1sca2 20350 Scalars of a univariate po...
ply1mpl0 20351 The univariate polynomial ...
ply10s0 20352 Zero times a univariate po...
ply1mpl1 20353 The univariate polynomial ...
ply1ascl 20354 The univariate polynomial ...
subrg1ascl 20355 The scalar injection funct...
subrg1asclcl 20356 The scalars in a polynomia...
subrgvr1 20357 The variables in a subring...
subrgvr1cl 20358 The variables in a polynom...
coe1z 20359 The coefficient vector of ...
coe1add 20360 The coefficient vector of ...
coe1addfv 20361 A particular coefficient o...
coe1subfv 20362 A particular coefficient o...
coe1mul2lem1 20363 An equivalence for ~ coe1m...
coe1mul2lem2 20364 An equivalence for ~ coe1m...
coe1mul2 20365 The coefficient vector of ...
coe1mul 20366 The coefficient vector of ...
ply1moncl 20367 Closure of the expression ...
ply1tmcl 20368 Closure of the expression ...
coe1tm 20369 Coefficient vector of a po...
coe1tmfv1 20370 Nonzero coefficient of a p...
coe1tmfv2 20371 Zero coefficient of a poly...
coe1tmmul2 20372 Coefficient vector of a po...
coe1tmmul 20373 Coefficient vector of a po...
coe1tmmul2fv 20374 Function value of a right-...
coe1pwmul 20375 Coefficient vector of a po...
coe1pwmulfv 20376 Function value of a right-...
ply1scltm 20377 A scalar is a term with ze...
coe1sclmul 20378 Coefficient vector of a po...
coe1sclmulfv 20379 A single coefficient of a ...
coe1sclmul2 20380 Coefficient vector of a po...
ply1sclf 20381 A scalar polynomial is a p...
ply1sclcl 20382 The value of the algebra s...
coe1scl 20383 Coefficient vector of a sc...
ply1sclid 20384 Recover the base scalar fr...
ply1sclf1 20385 The polynomial scalar func...
ply1scl0 20386 The zero scalar is zero. ...
ply1scln0 20387 Nonzero scalars create non...
ply1scl1 20388 The one scalar is the unit...
ply1idvr1 20389 The identity of a polynomi...
cply1mul 20390 The product of two constan...
ply1coefsupp 20391 The decomposition of a uni...
ply1coe 20392 Decompose a univariate pol...
eqcoe1ply1eq 20393 Two polynomials over the s...
ply1coe1eq 20394 Two polynomials over the s...
cply1coe0 20395 All but the first coeffici...
cply1coe0bi 20396 A polynomial is constant (...
coe1fzgsumdlem 20397 Lemma for ~ coe1fzgsumd (i...
coe1fzgsumd 20398 Value of an evaluated coef...
gsumsmonply1 20399 A finite group sum of scal...
gsummoncoe1 20400 A coefficient of the polyn...
gsumply1eq 20401 Two univariate polynomials...
lply1binom 20402 The binomial theorem for l...
lply1binomsc 20403 The binomial theorem for l...
reldmevls1 20408 Well-behaved binary operat...
ply1frcl 20409 Reverse closure for the se...
evls1fval 20410 Value of the univariate po...
evls1val 20411 Value of the univariate po...
evls1rhmlem 20412 Lemma for ~ evl1rhm and ~ ...
evls1rhm 20413 Polynomial evaluation is a...
evls1sca 20414 Univariate polynomial eval...
evls1gsumadd 20415 Univariate polynomial eval...
evls1gsummul 20416 Univariate polynomial eval...
evls1pw 20417 Univariate polynomial eval...
evls1varpw 20418 Univariate polynomial eval...
evl1fval 20419 Value of the simple/same r...
evl1val 20420 Value of the simple/same r...
evl1fval1lem 20421 Lemma for ~ evl1fval1 . (...
evl1fval1 20422 Value of the simple/same r...
evl1rhm 20423 Polynomial evaluation is a...
fveval1fvcl 20424 The function value of the ...
evl1sca 20425 Polynomial evaluation maps...
evl1scad 20426 Polynomial evaluation buil...
evl1var 20427 Polynomial evaluation maps...
evl1vard 20428 Polynomial evaluation buil...
evls1var 20429 Univariate polynomial eval...
evls1scasrng 20430 The evaluation of a scalar...
evls1varsrng 20431 The evaluation of the vari...
evl1addd 20432 Polynomial evaluation buil...
evl1subd 20433 Polynomial evaluation buil...
evl1muld 20434 Polynomial evaluation buil...
evl1vsd 20435 Polynomial evaluation buil...
evl1expd 20436 Polynomial evaluation buil...
pf1const 20437 Constants are polynomial f...
pf1id 20438 The identity is a polynomi...
pf1subrg 20439 Polynomial functions are a...
pf1rcl 20440 Reverse closure for the se...
pf1f 20441 Polynomial functions are f...
mpfpf1 20442 Convert a multivariate pol...
pf1mpf 20443 Convert a univariate polyn...
pf1addcl 20444 The sum of multivariate po...
pf1mulcl 20445 The product of multivariat...
pf1ind 20446 Prove a property of polyno...
evl1gsumdlem 20447 Lemma for ~ evl1gsumd (ind...
evl1gsumd 20448 Polynomial evaluation buil...
evl1gsumadd 20449 Univariate polynomial eval...
evl1gsumaddval 20450 Value of a univariate poly...
evl1gsummul 20451 Univariate polynomial eval...
evl1varpw 20452 Univariate polynomial eval...
evl1varpwval 20453 Value of a univariate poly...
evl1scvarpw 20454 Univariate polynomial eval...
evl1scvarpwval 20455 Value of a univariate poly...
evl1gsummon 20456 Value of a univariate poly...
cnfldstr 20475 The field of complex numbe...
cnfldex 20476 The field of complex numbe...
cnfldbas 20477 The base set of the field ...
cnfldadd 20478 The addition operation of ...
cnfldmul 20479 The multiplication operati...
cnfldcj 20480 The conjugation operation ...
cnfldtset 20481 The topology component of ...
cnfldle 20482 The ordering of the field ...
cnfldds 20483 The metric of the field of...
cnfldunif 20484 The uniform structure comp...
cnfldfun 20485 The field of complex numbe...
cnfldfunALT 20486 Alternate proof of ~ cnfld...
xrsstr 20487 The extended real structur...
xrsex 20488 The extended real structur...
xrsbas 20489 The base set of the extend...
xrsadd 20490 The addition operation of ...
xrsmul 20491 The multiplication operati...
xrstset 20492 The topology component of ...
xrsle 20493 The ordering of the extend...
cncrng 20494 The complex numbers form a...
cnring 20495 The complex numbers form a...
xrsmcmn 20496 The "multiplicative group"...
cnfld0 20497 Zero is the zero element o...
cnfld1 20498 One is the unit element of...
cnfldneg 20499 The additive inverse in th...
cnfldplusf 20500 The functionalized additio...
cnfldsub 20501 The subtraction operator i...
cndrng 20502 The complex numbers form a...
cnflddiv 20503 The division operation in ...
cnfldinv 20504 The multiplicative inverse...
cnfldmulg 20505 The group multiple functio...
cnfldexp 20506 The exponentiation operato...
cnsrng 20507 The complex numbers form a...
xrsmgm 20508 The "additive group" of th...
xrsnsgrp 20509 The "additive group" of th...
xrsmgmdifsgrp 20510 The "additive group" of th...
xrs1mnd 20511 The extended real numbers,...
xrs10 20512 The zero of the extended r...
xrs1cmn 20513 The extended real numbers ...
xrge0subm 20514 The nonnegative extended r...
xrge0cmn 20515 The nonnegative extended r...
xrsds 20516 The metric of the extended...
xrsdsval 20517 The metric of the extended...
xrsdsreval 20518 The metric of the extended...
xrsdsreclblem 20519 Lemma for ~ xrsdsreclb . ...
xrsdsreclb 20520 The metric of the extended...
cnsubmlem 20521 Lemma for ~ nn0subm and fr...
cnsubglem 20522 Lemma for ~ resubdrg and f...
cnsubrglem 20523 Lemma for ~ resubdrg and f...
cnsubdrglem 20524 Lemma for ~ resubdrg and f...
qsubdrg 20525 The rational numbers form ...
zsubrg 20526 The integers form a subrin...
gzsubrg 20527 The gaussian integers form...
nn0subm 20528 The nonnegative integers f...
rege0subm 20529 The nonnegative reals form...
absabv 20530 The regular absolute value...
zsssubrg 20531 The integers are a subset ...
qsssubdrg 20532 The rational numbers are a...
cnsubrg 20533 There are no subrings of t...
cnmgpabl 20534 The unit group of the comp...
cnmgpid 20535 The group identity element...
cnmsubglem 20536 Lemma for ~ rpmsubg and fr...
rpmsubg 20537 The positive reals form a ...
gzrngunitlem 20538 Lemma for ~ gzrngunit . (...
gzrngunit 20539 The units on ` ZZ [ _i ] `...
gsumfsum 20540 Relate a group sum on ` CC...
regsumfsum 20541 Relate a group sum on ` ( ...
expmhm 20542 Exponentiation is a monoid...
nn0srg 20543 The nonnegative integers f...
rge0srg 20544 The nonnegative real numbe...
zringcrng 20547 The ring of integers is a ...
zringring 20548 The ring of integers is a ...
zringabl 20549 The ring of integers is an...
zringgrp 20550 The ring of integers is an...
zringbas 20551 The integers are the base ...
zringplusg 20552 The addition operation of ...
zringmulg 20553 The multiplication (group ...
zringmulr 20554 The multiplication operati...
zring0 20555 The neutral element of the...
zring1 20556 The multiplicative neutral...
zringnzr 20557 The ring of integers is a ...
dvdsrzring 20558 Ring divisibility in the r...
zringlpirlem1 20559 Lemma for ~ zringlpir . A...
zringlpirlem2 20560 Lemma for ~ zringlpir . A...
zringlpirlem3 20561 Lemma for ~ zringlpir . A...
zringinvg 20562 The additive inverse of an...
zringunit 20563 The units of ` ZZ ` are th...
zringlpir 20564 The integers are a princip...
zringndrg 20565 The integers are not a div...
zringcyg 20566 The integers are a cyclic ...
zringmpg 20567 The multiplication group o...
prmirredlem 20568 A positive integer is irre...
dfprm2 20569 The positive irreducible e...
prmirred 20570 The irreducible elements o...
expghm 20571 Exponentiation is a group ...
mulgghm2 20572 The powers of a group elem...
mulgrhm 20573 The powers of the element ...
mulgrhm2 20574 The powers of the element ...
zrhval 20583 Define the unique homomorp...
zrhval2 20584 Alternate value of the ` Z...
zrhmulg 20585 Value of the ` ZRHom ` hom...
zrhrhmb 20586 The ` ZRHom ` homomorphism...
zrhrhm 20587 The ` ZRHom ` homomorphism...
zrh1 20588 Interpretation of 1 in a r...
zrh0 20589 Interpretation of 0 in a r...
zrhpropd 20590 The ` ZZ ` ring homomorphi...
zlmval 20591 Augment an abelian group w...
zlmlem 20592 Lemma for ~ zlmbas and ~ z...
zlmbas 20593 Base set of a ` ZZ ` -modu...
zlmplusg 20594 Group operation of a ` ZZ ...
zlmmulr 20595 Ring operation of a ` ZZ `...
zlmsca 20596 Scalar ring of a ` ZZ ` -m...
zlmvsca 20597 Scalar multiplication oper...
zlmlmod 20598 The ` ZZ ` -module operati...
zlmassa 20599 The ` ZZ ` -module operati...
chrval 20600 Definition substitution of...
chrcl 20601 Closure of the characteris...
chrid 20602 The canonical ` ZZ ` ring ...
chrdvds 20603 The ` ZZ ` ring homomorphi...
chrcong 20604 If two integers are congru...
chrnzr 20605 Nonzero rings are precisel...
chrrhm 20606 The characteristic restric...
domnchr 20607 The characteristic of a do...
znlidl 20608 The set ` n ZZ ` is an ide...
zncrng2 20609 The value of the ` Z/nZ ` ...
znval 20610 The value of the ` Z/nZ ` ...
znle 20611 The value of the ` Z/nZ ` ...
znval2 20612 Self-referential expressio...
znbaslem 20613 Lemma for ~ znbas . (Cont...
znbas2 20614 The base set of ` Z/nZ ` i...
znadd 20615 The additive structure of ...
znmul 20616 The multiplicative structu...
znzrh 20617 The ` ZZ ` ring homomorphi...
znbas 20618 The base set of ` Z/nZ ` s...
zncrng 20619 ` Z/nZ ` is a commutative ...
znzrh2 20620 The ` ZZ ` ring homomorphi...
znzrhval 20621 The ` ZZ ` ring homomorphi...
znzrhfo 20622 The ` ZZ ` ring homomorphi...
zncyg 20623 The group ` ZZ / n ZZ ` is...
zndvds 20624 Express equality of equiva...
zndvds0 20625 Special case of ~ zndvds w...
znf1o 20626 The function ` F ` enumera...
zzngim 20627 The ` ZZ ` ring homomorphi...
znle2 20628 The ordering of the ` Z/nZ...
znleval 20629 The ordering of the ` Z/nZ...
znleval2 20630 The ordering of the ` Z/nZ...
zntoslem 20631 Lemma for ~ zntos . (Cont...
zntos 20632 The ` Z/nZ ` structure is ...
znhash 20633 The ` Z/nZ ` structure has...
znfi 20634 The ` Z/nZ ` structure is ...
znfld 20635 The ` Z/nZ ` structure is ...
znidomb 20636 The ` Z/nZ ` structure is ...
znchr 20637 Cyclic rings are defined b...
znunit 20638 The units of ` Z/nZ ` are ...
znunithash 20639 The size of the unit group...
znrrg 20640 The regular elements of ` ...
cygznlem1 20641 Lemma for ~ cygzn . (Cont...
cygznlem2a 20642 Lemma for ~ cygzn . (Cont...
cygznlem2 20643 Lemma for ~ cygzn . (Cont...
cygznlem3 20644 A cyclic group with ` n ` ...
cygzn 20645 A cyclic group with ` n ` ...
cygth 20646 The "fundamental theorem o...
cyggic 20647 Cyclic groups are isomorph...
frgpcyg 20648 A free group is cyclic iff...
cnmsgnsubg 20649 The signs form a multiplic...
cnmsgnbas 20650 The base set of the sign s...
cnmsgngrp 20651 The group of signs under m...
psgnghm 20652 The sign is a homomorphism...
psgnghm2 20653 The sign is a homomorphism...
psgninv 20654 The sign of a permutation ...
psgnco 20655 Multiplicativity of the pe...
zrhpsgnmhm 20656 Embedding of permutation s...
zrhpsgninv 20657 The embedded sign of a per...
evpmss 20658 Even permutations are perm...
psgnevpmb 20659 A class is an even permuta...
psgnodpm 20660 A permutation which is odd...
psgnevpm 20661 A permutation which is eve...
psgnodpmr 20662 If a permutation has sign ...
zrhpsgnevpm 20663 The sign of an even permut...
zrhpsgnodpm 20664 The sign of an odd permuta...
cofipsgn 20665 Composition of any class `...
zrhpsgnelbas 20666 Embedding of permutation s...
zrhcopsgnelbas 20667 Embedding of permutation s...
evpmodpmf1o 20668 The function for performin...
pmtrodpm 20669 A transposition is an odd ...
psgnfix1 20670 A permutation of a finite ...
psgnfix2 20671 A permutation of a finite ...
psgndiflemB 20672 Lemma 1 for ~ psgndif . (...
psgndiflemA 20673 Lemma 2 for ~ psgndif . (...
psgndif 20674 Embedding of permutation s...
copsgndif 20675 Embedding of permutation s...
rebase 20678 The base of the field of r...
remulg 20679 The multiplication (group ...
resubdrg 20680 The real numbers form a di...
resubgval 20681 Subtraction in the field o...
replusg 20682 The addition operation of ...
remulr 20683 The multiplication operati...
re0g 20684 The neutral element of the...
re1r 20685 The multiplicative neutral...
rele2 20686 The ordering relation of t...
relt 20687 The ordering relation of t...
reds 20688 The distance of the field ...
redvr 20689 The division operation of ...
retos 20690 The real numbers are a tot...
refld 20691 The real numbers form a fi...
refldcj 20692 The conjugation operation ...
recrng 20693 The real numbers form a st...
regsumsupp 20694 The group sum over the rea...
rzgrp 20695 The quotient group ` RR / ...
isphl 20700 The predicate "is a genera...
phllvec 20701 A pre-Hilbert space is a l...
phllmod 20702 A pre-Hilbert space is a l...
phlsrng 20703 The scalar ring of a pre-H...
phllmhm 20704 The inner product of a pre...
ipcl 20705 Closure of the inner produ...
ipcj 20706 Conjugate of an inner prod...
iporthcom 20707 Orthogonality (meaning inn...
ip0l 20708 Inner product with a zero ...
ip0r 20709 Inner product with a zero ...
ipeq0 20710 The inner product of a vec...
ipdir 20711 Distributive law for inner...
ipdi 20712 Distributive law for inner...
ip2di 20713 Distributive law for inner...
ipsubdir 20714 Distributive law for inner...
ipsubdi 20715 Distributive law for inner...
ip2subdi 20716 Distributive law for inner...
ipass 20717 Associative law for inner ...
ipassr 20718 "Associative" law for seco...
ipassr2 20719 "Associative" law for inne...
ipffval 20720 The inner product operatio...
ipfval 20721 The inner product operatio...
ipfeq 20722 If the inner product opera...
ipffn 20723 The inner product operatio...
phlipf 20724 The inner product operatio...
ip2eq 20725 Two vectors are equal iff ...
isphld 20726 Properties that determine ...
phlpropd 20727 If two structures have the...
ssipeq 20728 The inner product on a sub...
phssipval 20729 The inner product on a sub...
phssip 20730 The inner product (as a fu...
phlssphl 20731 A subspace of an inner pro...
ocvfval 20738 The orthocomplement operat...
ocvval 20739 Value of the orthocompleme...
elocv 20740 Elementhood in the orthoco...
ocvi 20741 Property of a member of th...
ocvss 20742 The orthocomplement of a s...
ocvocv 20743 A set is contained in its ...
ocvlss 20744 The orthocomplement of a s...
ocv2ss 20745 Orthocomplements reverse s...
ocvin 20746 An orthocomplement has tri...
ocvsscon 20747 Two ways to say that ` S `...
ocvlsp 20748 The orthocomplement of a l...
ocv0 20749 The orthocomplement of the...
ocvz 20750 The orthocomplement of the...
ocv1 20751 The orthocomplement of the...
unocv 20752 The orthocomplement of a u...
iunocv 20753 The orthocomplement of an ...
cssval 20754 The set of closed subspace...
iscss 20755 The predicate "is a closed...
cssi 20756 Property of a closed subsp...
cssss 20757 A closed subspace is a sub...
iscss2 20758 It is sufficient to prove ...
ocvcss 20759 The orthocomplement of any...
cssincl 20760 The zero subspace is a clo...
css0 20761 The zero subspace is a clo...
css1 20762 The whole space is a close...
csslss 20763 A closed subspace of a pre...
lsmcss 20764 A subset of a pre-Hilbert ...
cssmre 20765 The closed subspaces of a ...
mrccss 20766 The Moore closure correspo...
thlval 20767 Value of the Hilbert latti...
thlbas 20768 Base set of the Hilbert la...
thlle 20769 Ordering on the Hilbert la...
thlleval 20770 Ordering on the Hilbert la...
thloc 20771 Orthocomplement on the Hil...
pjfval 20778 The value of the projectio...
pjdm 20779 A subspace is in the domai...
pjpm 20780 The projection map is a pa...
pjfval2 20781 Value of the projection ma...
pjval 20782 Value of the projection ma...
pjdm2 20783 A subspace is in the domai...
pjff 20784 A projection is a linear o...
pjf 20785 A projection is a function...
pjf2 20786 A projection is a function...
pjfo 20787 A projection is a surjecti...
pjcss 20788 A projection subspace is a...
ocvpj 20789 The orthocomplement of a p...
ishil 20790 The predicate "is a Hilber...
ishil2 20791 The predicate "is a Hilber...
isobs 20792 The predicate "is an ortho...
obsip 20793 The inner product of two e...
obsipid 20794 A basis element has unit l...
obsrcl 20795 Reverse closure for an ort...
obsss 20796 An orthonormal basis is a ...
obsne0 20797 A basis element is nonzero...
obsocv 20798 An orthonormal basis has t...
obs2ocv 20799 The double orthocomplement...
obselocv 20800 A basis element is in the ...
obs2ss 20801 A basis has no proper subs...
obslbs 20802 An orthogonal basis is a l...
reldmdsmm 20805 The direct sum is a well-b...
dsmmval 20806 Value of the module direct...
dsmmbase 20807 Base set of the module dir...
dsmmval2 20808 Self-referential definitio...
dsmmbas2 20809 Base set of the direct sum...
dsmmfi 20810 For finite products, the d...
dsmmelbas 20811 Membership in the finitely...
dsmm0cl 20812 The all-zero vector is con...
dsmmacl 20813 The finite hull is closed ...
prdsinvgd2 20814 Negation of a single coord...
dsmmsubg 20815 The finite hull of a produ...
dsmmlss 20816 The finite hull of a produ...
dsmmlmod 20817 The direct sum of a family...
frlmval 20820 Value of the "free module"...
frlmlmod 20821 The free module is a modul...
frlmpws 20822 The free module as a restr...
frlmlss 20823 The base set of the free m...
frlmpwsfi 20824 The finite free module is ...
frlmsca 20825 The ring of scalars of a f...
frlm0 20826 Zero in a free module (rin...
frlmbas 20827 Base set of the free modul...
frlmelbas 20828 Membership in the base set...
frlmrcl 20829 If a free module is inhabi...
frlmbasfsupp 20830 Elements of the free modul...
frlmbasmap 20831 Elements of the free modul...
frlmbasf 20832 Elements of the free modul...
frlmlvec 20833 The free module over a div...
frlmfibas 20834 The base set of the finite...
elfrlmbasn0 20835 If the dimension of a free...
frlmplusgval 20836 Addition in a free module....
frlmsubgval 20837 Subtraction in a free modu...
frlmvscafval 20838 Scalar multiplication in a...
frlmvplusgvalc 20839 Coordinates of a sum with ...
frlmvscaval 20840 Coordinates of a scalar mu...
frlmplusgvalb 20841 Addition in a free module ...
frlmvscavalb 20842 Scalar multiplication in a...
frlmvplusgscavalb 20843 Addition combined with sca...
frlmgsum 20844 Finite commutative sums in...
frlmsplit2 20845 Restriction is homomorphic...
frlmsslss 20846 A subset of a free module ...
frlmsslss2 20847 A subset of a free module ...
frlmbas3 20848 An element of the base set...
mpofrlmd 20849 Elements of the free modul...
frlmip 20850 The inner product of a fre...
frlmipval 20851 The inner product of a fre...
frlmphllem 20852 Lemma for ~ frlmphl . (Co...
frlmphl 20853 Conditions for a free modu...
uvcfval 20856 Value of the unit-vector g...
uvcval 20857 Value of a single unit vec...
uvcvval 20858 Value of a unit vector coo...
uvcvvcl 20859 A coordinate of a unit vec...
uvcvvcl2 20860 A unit vector coordinate i...
uvcvv1 20861 The unit vector is one at ...
uvcvv0 20862 The unit vector is zero at...
uvcff 20863 Domain and range of the un...
uvcf1 20864 In a nonzero ring, each un...
uvcresum 20865 Any element of a free modu...
frlmssuvc1 20866 A scalar multiple of a uni...
frlmssuvc2 20867 A nonzero scalar multiple ...
frlmsslsp 20868 A subset of a free module ...
frlmlbs 20869 The unit vectors comprise ...
frlmup1 20870 Any assignment of unit vec...
frlmup2 20871 The evaluation map has the...
frlmup3 20872 The range of such an evalu...
frlmup4 20873 Universal property of the ...
ellspd 20874 The elements of the span o...
elfilspd 20875 Simplified version of ~ el...
rellindf 20880 The independent-family pre...
islinds 20881 Property of an independent...
linds1 20882 An independent set of vect...
linds2 20883 An independent set of vect...
islindf 20884 Property of an independent...
islinds2 20885 Expanded property of an in...
islindf2 20886 Property of an independent...
lindff 20887 Functional property of a l...
lindfind 20888 A linearly independent fam...
lindsind 20889 A linearly independent set...
lindfind2 20890 In a linearly independent ...
lindsind2 20891 In a linearly independent ...
lindff1 20892 A linearly independent fam...
lindfrn 20893 The range of an independen...
f1lindf 20894 Rearranging and deleting e...
lindfres 20895 Any restriction of an inde...
lindsss 20896 Any subset of an independe...
f1linds 20897 A family constructed from ...
islindf3 20898 In a nonzero ring, indepen...
lindfmm 20899 Linear independence of a f...
lindsmm 20900 Linear independence of a s...
lindsmm2 20901 The monomorphic image of a...
lsslindf 20902 Linear independence is unc...
lsslinds 20903 Linear independence is unc...
islbs4 20904 A basis is an independent ...
lbslinds 20905 A basis is independent. (...
islinds3 20906 A subset is linearly indep...
islinds4 20907 A set is independent in a ...
lmimlbs 20908 The isomorphic image of a ...
lmiclbs 20909 Having a basis is an isomo...
islindf4 20910 A family is independent if...
islindf5 20911 A family is independent if...
indlcim 20912 An independent, spanning f...
lbslcic 20913 A module with a basis is i...
lmisfree 20914 A module has a basis iff i...
lvecisfrlm 20915 Every vector space is isom...
lmimco 20916 The composition of two iso...
lmictra 20917 Module isomorphism is tran...
uvcf1o 20918 In a nonzero ring, the map...
uvcendim 20919 In a nonzero ring, the num...
frlmisfrlm 20920 A free module is isomorphi...
frlmiscvec 20921 Every free module is isomo...
mamufval 20924 Functional value of the ma...
mamuval 20925 Multiplication of two matr...
mamufv 20926 A cell in the multiplicati...
mamudm 20927 The domain of the matrix m...
mamufacex 20928 Every solution of the equa...
mamures 20929 Rows in a matrix product a...
mndvcl 20930 Tuple-wise additive closur...
mndvass 20931 Tuple-wise associativity i...
mndvlid 20932 Tuple-wise left identity i...
mndvrid 20933 Tuple-wise right identity ...
grpvlinv 20934 Tuple-wise left inverse in...
grpvrinv 20935 Tuple-wise right inverse i...
mhmvlin 20936 Tuple extension of monoid ...
ringvcl 20937 Tuple-wise multiplication ...
mamucl 20938 Operation closure of matri...
mamuass 20939 Matrix multiplication is a...
mamudi 20940 Matrix multiplication dist...
mamudir 20941 Matrix multiplication dist...
mamuvs1 20942 Matrix multiplication dist...
mamuvs2 20943 Matrix multiplication dist...
matbas0pc 20946 There is no matrix with a ...
matbas0 20947 There is no matrix for a n...
matval 20948 Value of the matrix algebr...
matrcl 20949 Reverse closure for the ma...
matbas 20950 The matrix ring has the sa...
matplusg 20951 The matrix ring has the sa...
matsca 20952 The matrix ring has the sa...
matvsca 20953 The matrix ring has the sa...
mat0 20954 The matrix ring has the sa...
matinvg 20955 The matrix ring has the sa...
mat0op 20956 Value of a zero matrix as ...
matsca2 20957 The scalars of the matrix ...
matbas2 20958 The base set of the matrix...
matbas2i 20959 A matrix is a function. (...
matbas2d 20960 The base set of the matrix...
eqmat 20961 Two square matrices of the...
matecl 20962 Each entry (according to W...
matecld 20963 Each entry (according to W...
matplusg2 20964 Addition in the matrix rin...
matvsca2 20965 Scalar multiplication in t...
matlmod 20966 The matrix ring is a linea...
matgrp 20967 The matrix ring is a group...
matvscl 20968 Closure of the scalar mult...
matsubg 20969 The matrix ring has the sa...
matplusgcell 20970 Addition in the matrix rin...
matsubgcell 20971 Subtraction in the matrix ...
matinvgcell 20972 Additive inversion in the ...
matvscacell 20973 Scalar multiplication in t...
matgsum 20974 Finite commutative sums in...
matmulr 20975 Multiplication in the matr...
mamumat1cl 20976 The identity matrix (as op...
mat1comp 20977 The components of the iden...
mamulid 20978 The identity matrix (as op...
mamurid 20979 The identity matrix (as op...
matring 20980 Existence of the matrix ri...
matassa 20981 Existence of the matrix al...
matmulcell 20982 Multiplication in the matr...
mpomatmul 20983 Multiplication of two N x ...
mat1 20984 Value of an identity matri...
mat1ov 20985 Entries of an identity mat...
mat1bas 20986 The identity matrix is a m...
matsc 20987 The identity matrix multip...
ofco2 20988 Distribution law for the f...
oftpos 20989 The transposition of the v...
mattposcl 20990 The transpose of a square ...
mattpostpos 20991 The transpose of the trans...
mattposvs 20992 The transposition of a mat...
mattpos1 20993 The transposition of the i...
tposmap 20994 The transposition of an I ...
mamutpos 20995 Behavior of transposes in ...
mattposm 20996 Multiplying two transposed...
matgsumcl 20997 Closure of a group sum ove...
madetsumid 20998 The identity summand in th...
matepmcl 20999 Each entry of a matrix wit...
matepm2cl 21000 Each entry of a matrix wit...
madetsmelbas 21001 A summand of the determina...
madetsmelbas2 21002 A summand of the determina...
mat0dimbas0 21003 The empty set is the one a...
mat0dim0 21004 The zero of the algebra of...
mat0dimid 21005 The identity of the algebr...
mat0dimscm 21006 The scalar multiplication ...
mat0dimcrng 21007 The algebra of matrices wi...
mat1dimelbas 21008 A matrix with dimension 1 ...
mat1dimbas 21009 A matrix with dimension 1 ...
mat1dim0 21010 The zero of the algebra of...
mat1dimid 21011 The identity of the algebr...
mat1dimscm 21012 The scalar multiplication ...
mat1dimmul 21013 The ring multiplication in...
mat1dimcrng 21014 The algebra of matrices wi...
mat1f1o 21015 There is a 1-1 function fr...
mat1rhmval 21016 The value of the ring homo...
mat1rhmelval 21017 The value of the ring homo...
mat1rhmcl 21018 The value of the ring homo...
mat1f 21019 There is a function from a...
mat1ghm 21020 There is a group homomorph...
mat1mhm 21021 There is a monoid homomorp...
mat1rhm 21022 There is a ring homomorphi...
mat1rngiso 21023 There is a ring isomorphis...
mat1ric 21024 A ring is isomorphic to th...
dmatval 21029 The set of ` N ` x ` N ` d...
dmatel 21030 A ` N ` x ` N ` diagonal m...
dmatmat 21031 An ` N ` x ` N ` diagonal ...
dmatid 21032 The identity matrix is a d...
dmatelnd 21033 An extradiagonal entry of ...
dmatmul 21034 The product of two diagona...
dmatsubcl 21035 The difference of two diag...
dmatsgrp 21036 The set of diagonal matric...
dmatmulcl 21037 The product of two diagona...
dmatsrng 21038 The set of diagonal matric...
dmatcrng 21039 The subring of diagonal ma...
dmatscmcl 21040 The multiplication of a di...
scmatval 21041 The set of ` N ` x ` N ` s...
scmatel 21042 An ` N ` x ` N ` scalar ma...
scmatscmid 21043 A scalar matrix can be exp...
scmatscmide 21044 An entry of a scalar matri...
scmatscmiddistr 21045 Distributive law for scala...
scmatmat 21046 An ` N ` x ` N ` scalar ma...
scmate 21047 An entry of an ` N ` x ` N...
scmatmats 21048 The set of an ` N ` x ` N ...
scmateALT 21049 Alternate proof of ~ scmat...
scmatscm 21050 The multiplication of a ma...
scmatid 21051 The identity matrix is a s...
scmatdmat 21052 A scalar matrix is a diago...
scmataddcl 21053 The sum of two scalar matr...
scmatsubcl 21054 The difference of two scal...
scmatmulcl 21055 The product of two scalar ...
scmatsgrp 21056 The set of scalar matrices...
scmatsrng 21057 The set of scalar matrices...
scmatcrng 21058 The subring of scalar matr...
scmatsgrp1 21059 The set of scalar matrices...
scmatsrng1 21060 The set of scalar matrices...
smatvscl 21061 Closure of the scalar mult...
scmatlss 21062 The set of scalar matrices...
scmatstrbas 21063 The set of scalar matrices...
scmatrhmval 21064 The value of the ring homo...
scmatrhmcl 21065 The value of the ring homo...
scmatf 21066 There is a function from a...
scmatfo 21067 There is a function from a...
scmatf1 21068 There is a 1-1 function fr...
scmatf1o 21069 There is a bijection betwe...
scmatghm 21070 There is a group homomorph...
scmatmhm 21071 There is a monoid homomorp...
scmatrhm 21072 There is a ring homomorphi...
scmatrngiso 21073 There is a ring isomorphis...
scmatric 21074 A ring is isomorphic to ev...
mat0scmat 21075 The empty matrix over a ri...
mat1scmat 21076 A 1-dimensional matrix ove...
mvmulfval 21079 Functional value of the ma...
mvmulval 21080 Multiplication of a vector...
mvmulfv 21081 A cell/element in the vect...
mavmulval 21082 Multiplication of a vector...
mavmulfv 21083 A cell/element in the vect...
mavmulcl 21084 Multiplication of an NxN m...
1mavmul 21085 Multiplication of the iden...
mavmulass 21086 Associativity of the multi...
mavmuldm 21087 The domain of the matrix v...
mavmulsolcl 21088 Every solution of the equa...
mavmul0 21089 Multiplication of a 0-dime...
mavmul0g 21090 The result of the 0-dimens...
mvmumamul1 21091 The multiplication of an M...
mavmumamul1 21092 The multiplication of an N...
marrepfval 21097 First substitution for the...
marrepval0 21098 Second substitution for th...
marrepval 21099 Third substitution for the...
marrepeval 21100 An entry of a matrix with ...
marrepcl 21101 Closure of the row replace...
marepvfval 21102 First substitution for the...
marepvval0 21103 Second substitution for th...
marepvval 21104 Third substitution for the...
marepveval 21105 An entry of a matrix with ...
marepvcl 21106 Closure of the column repl...
ma1repvcl 21107 Closure of the column repl...
ma1repveval 21108 An entry of an identity ma...
mulmarep1el 21109 Element by element multipl...
mulmarep1gsum1 21110 The sum of element by elem...
mulmarep1gsum2 21111 The sum of element by elem...
1marepvmarrepid 21112 Replacing the ith row by 0...
submabas 21115 Any subset of the index se...
submafval 21116 First substitution for a s...
submaval0 21117 Second substitution for a ...
submaval 21118 Third substitution for a s...
submaeval 21119 An entry of a submatrix of...
1marepvsma1 21120 The submatrix of the ident...
mdetfval 21123 First substitution for the...
mdetleib 21124 Full substitution of our d...
mdetleib2 21125 Leibniz' formula can also ...
nfimdetndef 21126 The determinant is not def...
mdetfval1 21127 First substitution of an a...
mdetleib1 21128 Full substitution of an al...
mdet0pr 21129 The determinant for 0-dime...
mdet0f1o 21130 The determinant for 0-dime...
mdet0fv0 21131 The determinant of a 0-dim...
mdetf 21132 Functionality of the deter...
mdetcl 21133 The determinant evaluates ...
m1detdiag 21134 The determinant of a 1-dim...
mdetdiaglem 21135 Lemma for ~ mdetdiag . Pr...
mdetdiag 21136 The determinant of a diago...
mdetdiagid 21137 The determinant of a diago...
mdet1 21138 The determinant of the ide...
mdetrlin 21139 The determinant function i...
mdetrsca 21140 The determinant function i...
mdetrsca2 21141 The determinant function i...
mdetr0 21142 The determinant of a matri...
mdet0 21143 The determinant of the zer...
mdetrlin2 21144 The determinant function i...
mdetralt 21145 The determinant function i...
mdetralt2 21146 The determinant function i...
mdetero 21147 The determinant function i...
mdettpos 21148 Determinant is invariant u...
mdetunilem1 21149 Lemma for ~ mdetuni . (Co...
mdetunilem2 21150 Lemma for ~ mdetuni . (Co...
mdetunilem3 21151 Lemma for ~ mdetuni . (Co...
mdetunilem4 21152 Lemma for ~ mdetuni . (Co...
mdetunilem5 21153 Lemma for ~ mdetuni . (Co...
mdetunilem6 21154 Lemma for ~ mdetuni . (Co...
mdetunilem7 21155 Lemma for ~ mdetuni . (Co...
mdetunilem8 21156 Lemma for ~ mdetuni . (Co...
mdetunilem9 21157 Lemma for ~ mdetuni . (Co...
mdetuni0 21158 Lemma for ~ mdetuni . (Co...
mdetuni 21159 According to the definitio...
mdetmul 21160 Multiplicativity of the de...
m2detleiblem1 21161 Lemma 1 for ~ m2detleib . ...
m2detleiblem5 21162 Lemma 5 for ~ m2detleib . ...
m2detleiblem6 21163 Lemma 6 for ~ m2detleib . ...
m2detleiblem7 21164 Lemma 7 for ~ m2detleib . ...
m2detleiblem2 21165 Lemma 2 for ~ m2detleib . ...
m2detleiblem3 21166 Lemma 3 for ~ m2detleib . ...
m2detleiblem4 21167 Lemma 4 for ~ m2detleib . ...
m2detleib 21168 Leibniz' Formula for 2x2-m...
mndifsplit 21173 Lemma for ~ maducoeval2 . ...
madufval 21174 First substitution for the...
maduval 21175 Second substitution for th...
maducoeval 21176 An entry of the adjunct (c...
maducoeval2 21177 An entry of the adjunct (c...
maduf 21178 Creating the adjunct of ma...
madutpos 21179 The adjuct of a transposed...
madugsum 21180 The determinant of a matri...
madurid 21181 Multiplying a matrix with ...
madulid 21182 Multiplying the adjunct of...
minmar1fval 21183 First substitution for the...
minmar1val0 21184 Second substitution for th...
minmar1val 21185 Third substitution for the...
minmar1eval 21186 An entry of a matrix for a...
minmar1marrep 21187 The minor matrix is a spec...
minmar1cl 21188 Closure of the row replace...
maducoevalmin1 21189 The coefficients of an adj...
symgmatr01lem 21190 Lemma for ~ symgmatr01 . ...
symgmatr01 21191 Applying a permutation tha...
gsummatr01lem1 21192 Lemma A for ~ gsummatr01 ....
gsummatr01lem2 21193 Lemma B for ~ gsummatr01 ....
gsummatr01lem3 21194 Lemma 1 for ~ gsummatr01 ....
gsummatr01lem4 21195 Lemma 2 for ~ gsummatr01 ....
gsummatr01 21196 Lemma 1 for ~ smadiadetlem...
marep01ma 21197 Replacing a row of a squar...
smadiadetlem0 21198 Lemma 0 for ~ smadiadet : ...
smadiadetlem1 21199 Lemma 1 for ~ smadiadet : ...
smadiadetlem1a 21200 Lemma 1a for ~ smadiadet :...
smadiadetlem2 21201 Lemma 2 for ~ smadiadet : ...
smadiadetlem3lem0 21202 Lemma 0 for ~ smadiadetlem...
smadiadetlem3lem1 21203 Lemma 1 for ~ smadiadetlem...
smadiadetlem3lem2 21204 Lemma 2 for ~ smadiadetlem...
smadiadetlem3 21205 Lemma 3 for ~ smadiadet . ...
smadiadetlem4 21206 Lemma 4 for ~ smadiadet . ...
smadiadet 21207 The determinant of a subma...
smadiadetglem1 21208 Lemma 1 for ~ smadiadetg ....
smadiadetglem2 21209 Lemma 2 for ~ smadiadetg ....
smadiadetg 21210 The determinant of a squar...
smadiadetg0 21211 Lemma for ~ smadiadetr : v...
smadiadetr 21212 The determinant of a squar...
invrvald 21213 If a matrix multiplied wit...
matinv 21214 The inverse of a matrix is...
matunit 21215 A matrix is a unit in the ...
slesolvec 21216 Every solution of a system...
slesolinv 21217 The solution of a system o...
slesolinvbi 21218 The solution of a system o...
slesolex 21219 Every system of linear equ...
cramerimplem1 21220 Lemma 1 for ~ cramerimp : ...
cramerimplem2 21221 Lemma 2 for ~ cramerimp : ...
cramerimplem3 21222 Lemma 3 for ~ cramerimp : ...
cramerimp 21223 One direction of Cramer's ...
cramerlem1 21224 Lemma 1 for ~ cramer . (C...
cramerlem2 21225 Lemma 2 for ~ cramer . (C...
cramerlem3 21226 Lemma 3 for ~ cramer . (C...
cramer0 21227 Special case of Cramer's r...
cramer 21228 Cramer's rule. According ...
pmatring 21229 The set of polynomial matr...
pmatlmod 21230 The set of polynomial matr...
pmat0op 21231 The zero polynomial matrix...
pmat1op 21232 The identity polynomial ma...
pmat1ovd 21233 Entries of the identity po...
pmat0opsc 21234 The zero polynomial matrix...
pmat1opsc 21235 The identity polynomial ma...
pmat1ovscd 21236 Entries of the identity po...
pmatcoe1fsupp 21237 For a polynomial matrix th...
1pmatscmul 21238 The scalar product of the ...
cpmat 21245 Value of the constructor o...
cpmatpmat 21246 A constant polynomial matr...
cpmatel 21247 Property of a constant pol...
cpmatelimp 21248 Implication of a set being...
cpmatel2 21249 Another property of a cons...
cpmatelimp2 21250 Another implication of a s...
1elcpmat 21251 The identity of the ring o...
cpmatacl 21252 The set of all constant po...
cpmatinvcl 21253 The set of all constant po...
cpmatmcllem 21254 Lemma for ~ cpmatmcl . (C...
cpmatmcl 21255 The set of all constant po...
cpmatsubgpmat 21256 The set of all constant po...
cpmatsrgpmat 21257 The set of all constant po...
0elcpmat 21258 The zero of the ring of al...
mat2pmatfval 21259 Value of the matrix transf...
mat2pmatval 21260 The result of a matrix tra...
mat2pmatvalel 21261 A (matrix) element of the ...
mat2pmatbas 21262 The result of a matrix tra...
mat2pmatbas0 21263 The result of a matrix tra...
mat2pmatf 21264 The matrix transformation ...
mat2pmatf1 21265 The matrix transformation ...
mat2pmatghm 21266 The transformation of matr...
mat2pmatmul 21267 The transformation of matr...
mat2pmat1 21268 The transformation of the ...
mat2pmatmhm 21269 The transformation of matr...
mat2pmatrhm 21270 The transformation of matr...
mat2pmatlin 21271 The transformation of matr...
0mat2pmat 21272 The transformed zero matri...
idmatidpmat 21273 The transformed identity m...
d0mat2pmat 21274 The transformed empty set ...
d1mat2pmat 21275 The transformation of a ma...
mat2pmatscmxcl 21276 A transformed matrix multi...
m2cpm 21277 The result of a matrix tra...
m2cpmf 21278 The matrix transformation ...
m2cpmf1 21279 The matrix transformation ...
m2cpmghm 21280 The transformation of matr...
m2cpmmhm 21281 The transformation of matr...
m2cpmrhm 21282 The transformation of matr...
m2pmfzmap 21283 The transformed values of ...
m2pmfzgsumcl 21284 Closure of the sum of scal...
cpm2mfval 21285 Value of the inverse matri...
cpm2mval 21286 The result of an inverse m...
cpm2mvalel 21287 A (matrix) element of the ...
cpm2mf 21288 The inverse matrix transfo...
m2cpminvid 21289 The inverse transformation...
m2cpminvid2lem 21290 Lemma for ~ m2cpminvid2 . ...
m2cpminvid2 21291 The transformation applied...
m2cpmfo 21292 The matrix transformation ...
m2cpmf1o 21293 The matrix transformation ...
m2cpmrngiso 21294 The transformation of matr...
matcpmric 21295 The ring of matrices over ...
m2cpminv 21296 The inverse matrix transfo...
m2cpminv0 21297 The inverse matrix transfo...
decpmatval0 21300 The matrix consisting of t...
decpmatval 21301 The matrix consisting of t...
decpmate 21302 An entry of the matrix con...
decpmatcl 21303 Closure of the decompositi...
decpmataa0 21304 The matrix consisting of t...
decpmatfsupp 21305 The mapping to the matrice...
decpmatid 21306 The matrix consisting of t...
decpmatmullem 21307 Lemma for ~ decpmatmul . ...
decpmatmul 21308 The matrix consisting of t...
decpmatmulsumfsupp 21309 Lemma 0 for ~ pm2mpmhm . ...
pmatcollpw1lem1 21310 Lemma 1 for ~ pmatcollpw1 ...
pmatcollpw1lem2 21311 Lemma 2 for ~ pmatcollpw1 ...
pmatcollpw1 21312 Write a polynomial matrix ...
pmatcollpw2lem 21313 Lemma for ~ pmatcollpw2 . ...
pmatcollpw2 21314 Write a polynomial matrix ...
monmatcollpw 21315 The matrix consisting of t...
pmatcollpwlem 21316 Lemma for ~ pmatcollpw . ...
pmatcollpw 21317 Write a polynomial matrix ...
pmatcollpwfi 21318 Write a polynomial matrix ...
pmatcollpw3lem 21319 Lemma for ~ pmatcollpw3 an...
pmatcollpw3 21320 Write a polynomial matrix ...
pmatcollpw3fi 21321 Write a polynomial matrix ...
pmatcollpw3fi1lem1 21322 Lemma 1 for ~ pmatcollpw3f...
pmatcollpw3fi1lem2 21323 Lemma 2 for ~ pmatcollpw3f...
pmatcollpw3fi1 21324 Write a polynomial matrix ...
pmatcollpwscmatlem1 21325 Lemma 1 for ~ pmatcollpwsc...
pmatcollpwscmatlem2 21326 Lemma 2 for ~ pmatcollpwsc...
pmatcollpwscmat 21327 Write a scalar matrix over...
pm2mpf1lem 21330 Lemma for ~ pm2mpf1 . (Co...
pm2mpval 21331 Value of the transformatio...
pm2mpfval 21332 A polynomial matrix transf...
pm2mpcl 21333 The transformation of poly...
pm2mpf 21334 The transformation of poly...
pm2mpf1 21335 The transformation of poly...
pm2mpcoe1 21336 A coefficient of the polyn...
idpm2idmp 21337 The transformation of the ...
mptcoe1matfsupp 21338 The mapping extracting the...
mply1topmatcllem 21339 Lemma for ~ mply1topmatcl ...
mply1topmatval 21340 A polynomial over matrices...
mply1topmatcl 21341 A polynomial over matrices...
mp2pm2mplem1 21342 Lemma 1 for ~ mp2pm2mp . ...
mp2pm2mplem2 21343 Lemma 2 for ~ mp2pm2mp . ...
mp2pm2mplem3 21344 Lemma 3 for ~ mp2pm2mp . ...
mp2pm2mplem4 21345 Lemma 4 for ~ mp2pm2mp . ...
mp2pm2mplem5 21346 Lemma 5 for ~ mp2pm2mp . ...
mp2pm2mp 21347 A polynomial over matrices...
pm2mpghmlem2 21348 Lemma 2 for ~ pm2mpghm . ...
pm2mpghmlem1 21349 Lemma 1 for pm2mpghm . (C...
pm2mpfo 21350 The transformation of poly...
pm2mpf1o 21351 The transformation of poly...
pm2mpghm 21352 The transformation of poly...
pm2mpgrpiso 21353 The transformation of poly...
pm2mpmhmlem1 21354 Lemma 1 for ~ pm2mpmhm . ...
pm2mpmhmlem2 21355 Lemma 2 for ~ pm2mpmhm . ...
pm2mpmhm 21356 The transformation of poly...
pm2mprhm 21357 The transformation of poly...
pm2mprngiso 21358 The transformation of poly...
pmmpric 21359 The ring of polynomial mat...
monmat2matmon 21360 The transformation of a po...
pm2mp 21361 The transformation of a su...
chmatcl 21364 Closure of the characteris...
chmatval 21365 The entries of the charact...
chpmatfval 21366 Value of the characteristi...
chpmatval 21367 The characteristic polynom...
chpmatply1 21368 The characteristic polynom...
chpmatval2 21369 The characteristic polynom...
chpmat0d 21370 The characteristic polynom...
chpmat1dlem 21371 Lemma for ~ chpmat1d . (C...
chpmat1d 21372 The characteristic polynom...
chpdmatlem0 21373 Lemma 0 for ~ chpdmat . (...
chpdmatlem1 21374 Lemma 1 for ~ chpdmat . (...
chpdmatlem2 21375 Lemma 2 for ~ chpdmat . (...
chpdmatlem3 21376 Lemma 3 for ~ chpdmat . (...
chpdmat 21377 The characteristic polynom...
chpscmat 21378 The characteristic polynom...
chpscmat0 21379 The characteristic polynom...
chpscmatgsumbin 21380 The characteristic polynom...
chpscmatgsummon 21381 The characteristic polynom...
chp0mat 21382 The characteristic polynom...
chpidmat 21383 The characteristic polynom...
chmaidscmat 21384 The characteristic polynom...
fvmptnn04if 21385 The function values of a m...
fvmptnn04ifa 21386 The function value of a ma...
fvmptnn04ifb 21387 The function value of a ma...
fvmptnn04ifc 21388 The function value of a ma...
fvmptnn04ifd 21389 The function value of a ma...
chfacfisf 21390 The "characteristic factor...
chfacfisfcpmat 21391 The "characteristic factor...
chfacffsupp 21392 The "characteristic factor...
chfacfscmulcl 21393 Closure of a scaled value ...
chfacfscmul0 21394 A scaled value of the "cha...
chfacfscmulfsupp 21395 A mapping of scaled values...
chfacfscmulgsum 21396 Breaking up a sum of value...
chfacfpmmulcl 21397 Closure of the value of th...
chfacfpmmul0 21398 The value of the "characte...
chfacfpmmulfsupp 21399 A mapping of values of the...
chfacfpmmulgsum 21400 Breaking up a sum of value...
chfacfpmmulgsum2 21401 Breaking up a sum of value...
cayhamlem1 21402 Lemma 1 for ~ cayleyhamilt...
cpmadurid 21403 The right-hand fundamental...
cpmidgsum 21404 Representation of the iden...
cpmidgsumm2pm 21405 Representation of the iden...
cpmidpmatlem1 21406 Lemma 1 for ~ cpmidpmat . ...
cpmidpmatlem2 21407 Lemma 2 for ~ cpmidpmat . ...
cpmidpmatlem3 21408 Lemma 3 for ~ cpmidpmat . ...
cpmidpmat 21409 Representation of the iden...
cpmadugsumlemB 21410 Lemma B for ~ cpmadugsum ....
cpmadugsumlemC 21411 Lemma C for ~ cpmadugsum ....
cpmadugsumlemF 21412 Lemma F for ~ cpmadugsum ....
cpmadugsumfi 21413 The product of the charact...
cpmadugsum 21414 The product of the charact...
cpmidgsum2 21415 Representation of the iden...
cpmidg2sum 21416 Equality of two sums repre...
cpmadumatpolylem1 21417 Lemma 1 for ~ cpmadumatpol...
cpmadumatpolylem2 21418 Lemma 2 for ~ cpmadumatpol...
cpmadumatpoly 21419 The product of the charact...
cayhamlem2 21420 Lemma for ~ cayhamlem3 . ...
chcoeffeqlem 21421 Lemma for ~ chcoeffeq . (...
chcoeffeq 21422 The coefficients of the ch...
cayhamlem3 21423 Lemma for ~ cayhamlem4 . ...
cayhamlem4 21424 Lemma for ~ cayleyhamilton...
cayleyhamilton0 21425 The Cayley-Hamilton theore...
cayleyhamilton 21426 The Cayley-Hamilton theore...
cayleyhamiltonALT 21427 Alternate proof of ~ cayle...
cayleyhamilton1 21428 The Cayley-Hamilton theore...
istopg 21431 Express the predicate " ` ...
istop2g 21432 Express the predicate " ` ...
uniopn 21433 The union of a subset of a...
iunopn 21434 The indexed union of a sub...
inopn 21435 The intersection of two op...
fitop 21436 A topology is closed under...
fiinopn 21437 The intersection of a none...
iinopn 21438 The intersection of a none...
unopn 21439 The union of two open sets...
0opn 21440 The empty set is an open s...
0ntop 21441 The empty set is not a top...
topopn 21442 The underlying set of a to...
eltopss 21443 A member of a topology is ...
riinopn 21444 A finite indexed relative ...
rintopn 21445 A finite relative intersec...
istopon 21448 Property of being a topolo...
topontop 21449 A topology on a given base...
toponuni 21450 The base set of a topology...
topontopi 21451 A topology on a given base...
toponunii 21452 The base set of a topology...
toptopon 21453 Alternative definition of ...
toptopon2 21454 A topology is the same thi...
topontopon 21455 A topology on a set is a t...
funtopon 21456 The class ` TopOn ` is a f...
toponrestid 21457 Given a topology on a set,...
toponsspwpw 21458 The set of topologies on a...
dmtopon 21459 The domain of ` TopOn ` is...
fntopon 21460 The class ` TopOn ` is a f...
toprntopon 21461 A topology is the same thi...
toponmax 21462 The base set of a topology...
toponss 21463 A member of a topology is ...
toponcom 21464 If ` K ` is a topology on ...
toponcomb 21465 Biconditional form of ~ to...
topgele 21466 The topologies over the sa...
topsn 21467 The only topology on a sin...
istps 21470 Express the predicate "is ...
istps2 21471 Express the predicate "is ...
tpsuni 21472 The base set of a topologi...
tpstop 21473 The topology extractor on ...
tpspropd 21474 A topological space depend...
tpsprop2d 21475 A topological space depend...
topontopn 21476 Express the predicate "is ...
tsettps 21477 If the topology component ...
istpsi 21478 Properties that determine ...
eltpsg 21479 Properties that determine ...
eltpsi 21480 Properties that determine ...
isbasisg 21483 Express the predicate "the...
isbasis2g 21484 Express the predicate "the...
isbasis3g 21485 Express the predicate "the...
basis1 21486 Property of a basis. (Con...
basis2 21487 Property of a basis. (Con...
fiinbas 21488 If a set is closed under f...
basdif0 21489 A basis is not affected by...
baspartn 21490 A disjoint system of sets ...
tgval 21491 The topology generated by ...
tgval2 21492 Definition of a topology g...
eltg 21493 Membership in a topology g...
eltg2 21494 Membership in a topology g...
eltg2b 21495 Membership in a topology g...
eltg4i 21496 An open set in a topology ...
eltg3i 21497 The union of a set of basi...
eltg3 21498 Membership in a topology g...
tgval3 21499 Alternate expression for t...
tg1 21500 Property of a member of a ...
tg2 21501 Property of a member of a ...
bastg 21502 A member of a basis is a s...
unitg 21503 The topology generated by ...
tgss 21504 Subset relation for genera...
tgcl 21505 Show that a basis generate...
tgclb 21506 The property ~ tgcl can be...
tgtopon 21507 A basis generates a topolo...
topbas 21508 A topology is its own basi...
tgtop 21509 A topology is its own basi...
eltop 21510 Membership in a topology, ...
eltop2 21511 Membership in a topology. ...
eltop3 21512 Membership in a topology. ...
fibas 21513 A collection of finite int...
tgdom 21514 A space has no more open s...
tgiun 21515 The indexed union of a set...
tgidm 21516 The topology generator fun...
bastop 21517 Two ways to express that a...
tgtop11 21518 The topology generation fu...
0top 21519 The singleton of the empty...
en1top 21520 ` { (/) } ` is the only to...
en2top 21521 If a topology has two elem...
tgss3 21522 A criterion for determinin...
tgss2 21523 A criterion for determinin...
basgen 21524 Given a topology ` J ` , s...
basgen2 21525 Given a topology ` J ` , s...
2basgen 21526 Conditions that determine ...
tgfiss 21527 If a subbase is included i...
tgdif0 21528 A generated topology is no...
bastop1 21529 A subset of a topology is ...
bastop2 21530 A version of ~ bastop1 tha...
distop 21531 The discrete topology on a...
topnex 21532 The class of all topologie...
distopon 21533 The discrete topology on a...
sn0topon 21534 The singleton of the empty...
sn0top 21535 The singleton of the empty...
indislem 21536 A lemma to eliminate some ...
indistopon 21537 The indiscrete topology on...
indistop 21538 The indiscrete topology on...
indisuni 21539 The base set of the indisc...
fctop 21540 The finite complement topo...
fctop2 21541 The finite complement topo...
cctop 21542 The countable complement t...
ppttop 21543 The particular point topol...
pptbas 21544 The particular point topol...
epttop 21545 The excluded point topolog...
indistpsx 21546 The indiscrete topology on...
indistps 21547 The indiscrete topology on...
indistps2 21548 The indiscrete topology on...
indistpsALT 21549 The indiscrete topology on...
indistps2ALT 21550 The indiscrete topology on...
distps 21551 The discrete topology on a...
fncld 21558 The closed-set generator i...
cldval 21559 The set of closed sets of ...
ntrfval 21560 The interior function on t...
clsfval 21561 The closure function on th...
cldrcl 21562 Reverse closure of the clo...
iscld 21563 The predicate "the class `...
iscld2 21564 A subset of the underlying...
cldss 21565 A closed set is a subset o...
cldss2 21566 The set of closed sets is ...
cldopn 21567 The complement of a closed...
isopn2 21568 A subset of the underlying...
opncld 21569 The complement of an open ...
difopn 21570 The difference of a closed...
topcld 21571 The underlying set of a to...
ntrval 21572 The interior of a subset o...
clsval 21573 The closure of a subset of...
0cld 21574 The empty set is closed. ...
iincld 21575 The indexed intersection o...
intcld 21576 The intersection of a set ...
uncld 21577 The union of two closed se...
cldcls 21578 A closed subset equals its...
incld 21579 The intersection of two cl...
riincld 21580 An indexed relative inters...
iuncld 21581 A finite indexed union of ...
unicld 21582 A finite union of closed s...
clscld 21583 The closure of a subset of...
clsf 21584 The closure function is a ...
ntropn 21585 The interior of a subset o...
clsval2 21586 Express closure in terms o...
ntrval2 21587 Interior expressed in term...
ntrdif 21588 An interior of a complemen...
clsdif 21589 A closure of a complement ...
clsss 21590 Subset relationship for cl...
ntrss 21591 Subset relationship for in...
sscls 21592 A subset of a topology's u...
ntrss2 21593 A subset includes its inte...
ssntr 21594 An open subset of a set is...
clsss3 21595 The closure of a subset of...
ntrss3 21596 The interior of a subset o...
ntrin 21597 A pairwise intersection of...
cmclsopn 21598 The complement of a closur...
cmntrcld 21599 The complement of an inter...
iscld3 21600 A subset is closed iff it ...
iscld4 21601 A subset is closed iff it ...
isopn3 21602 A subset is open iff it eq...
clsidm 21603 The closure operation is i...
ntridm 21604 The interior operation is ...
clstop 21605 The closure of a topology'...
ntrtop 21606 The interior of a topology...
0ntr 21607 A subset with an empty int...
clsss2 21608 If a subset is included in...
elcls 21609 Membership in a closure. ...
elcls2 21610 Membership in a closure. ...
clsndisj 21611 Any open set containing a ...
ntrcls0 21612 A subset whose closure has...
ntreq0 21613 Two ways to say that a sub...
cldmre 21614 The closed sets of a topol...
mrccls 21615 Moore closure generalizes ...
cls0 21616 The closure of the empty s...
ntr0 21617 The interior of the empty ...
isopn3i 21618 An open subset equals its ...
elcls3 21619 Membership in a closure in...
opncldf1 21620 A bijection useful for con...
opncldf2 21621 The values of the open-clo...
opncldf3 21622 The values of the converse...
isclo 21623 A set ` A ` is clopen iff ...
isclo2 21624 A set ` A ` is clopen iff ...
discld 21625 The open sets of a discret...
sn0cld 21626 The closed sets of the top...
indiscld 21627 The closed sets of an indi...
mretopd 21628 A Moore collection which i...
toponmre 21629 The topologies over a give...
cldmreon 21630 The closed sets of a topol...
iscldtop 21631 A family is the closed set...
mreclatdemoBAD 21632 The closed subspaces of a ...
neifval 21635 Value of the neighborhood ...
neif 21636 The neighborhood function ...
neiss2 21637 A set with a neighborhood ...
neival 21638 Value of the set of neighb...
isnei 21639 The predicate "the class `...
neiint 21640 An intuitive definition of...
isneip 21641 The predicate "the class `...
neii1 21642 A neighborhood is included...
neisspw 21643 The neighborhoods of any s...
neii2 21644 Property of a neighborhood...
neiss 21645 Any neighborhood of a set ...
ssnei 21646 A set is included in any o...
elnei 21647 A point belongs to any of ...
0nnei 21648 The empty set is not a nei...
neips 21649 A neighborhood of a set is...
opnneissb 21650 An open set is a neighborh...
opnssneib 21651 Any superset of an open se...
ssnei2 21652 Any subset ` M ` of ` X ` ...
neindisj 21653 Any neighborhood of an ele...
opnneiss 21654 An open set is a neighborh...
opnneip 21655 An open set is a neighborh...
opnnei 21656 A set is open iff it is a ...
tpnei 21657 The underlying set of a to...
neiuni 21658 The union of the neighborh...
neindisj2 21659 A point ` P ` belongs to t...
topssnei 21660 A finer topology has more ...
innei 21661 The intersection of two ne...
opnneiid 21662 Only an open set is a neig...
neissex 21663 For any neighborhood ` N `...
0nei 21664 The empty set is a neighbo...
neipeltop 21665 Lemma for ~ neiptopreu . ...
neiptopuni 21666 Lemma for ~ neiptopreu . ...
neiptoptop 21667 Lemma for ~ neiptopreu . ...
neiptopnei 21668 Lemma for ~ neiptopreu . ...
neiptopreu 21669 If, to each element ` P ` ...
lpfval 21674 The limit point function o...
lpval 21675 The set of limit points of...
islp 21676 The predicate "the class `...
lpsscls 21677 The limit points of a subs...
lpss 21678 The limit points of a subs...
lpdifsn 21679 ` P ` is a limit point of ...
lpss3 21680 Subset relationship for li...
islp2 21681 The predicate " ` P ` is a...
islp3 21682 The predicate " ` P ` is a...
maxlp 21683 A point is a limit point o...
clslp 21684 The closure of a subset of...
islpi 21685 A point belonging to a set...
cldlp 21686 A subset of a topological ...
isperf 21687 Definition of a perfect sp...
isperf2 21688 Definition of a perfect sp...
isperf3 21689 A perfect space is a topol...
perflp 21690 The limit points of a perf...
perfi 21691 Property of a perfect spac...
perftop 21692 A perfect space is a topol...
restrcl 21693 Reverse closure for the su...
restbas 21694 A subspace topology basis ...
tgrest 21695 A subspace can be generate...
resttop 21696 A subspace topology is a t...
resttopon 21697 A subspace topology is a t...
restuni 21698 The underlying set of a su...
stoig 21699 The topological space buil...
restco 21700 Composition of subspaces. ...
restabs 21701 Equivalence of being a sub...
restin 21702 When the subspace region i...
restuni2 21703 The underlying set of a su...
resttopon2 21704 The underlying set of a su...
rest0 21705 The subspace topology indu...
restsn 21706 The only subspace topology...
restsn2 21707 The subspace topology indu...
restcld 21708 A closed set of a subspace...
restcldi 21709 A closed set is closed in ...
restcldr 21710 A set which is closed in t...
restopnb 21711 If ` B ` is an open subset...
ssrest 21712 If ` K ` is a finer topolo...
restopn2 21713 If ` A ` is open, then ` B...
restdis 21714 A subspace of a discrete t...
restfpw 21715 The restriction of the set...
neitr 21716 The neighborhood of a trac...
restcls 21717 A closure in a subspace to...
restntr 21718 An interior in a subspace ...
restlp 21719 The limit points of a subs...
restperf 21720 Perfection of a subspace. ...
perfopn 21721 An open subset of a perfec...
resstopn 21722 The topology of a restrict...
resstps 21723 A restricted topological s...
ordtbaslem 21724 Lemma for ~ ordtbas . In ...
ordtval 21725 Value of the order topolog...
ordtuni 21726 Value of the order topolog...
ordtbas2 21727 Lemma for ~ ordtbas . (Co...
ordtbas 21728 In a total order, the fini...
ordttopon 21729 Value of the order topolog...
ordtopn1 21730 An upward ray ` ( P , +oo ...
ordtopn2 21731 A downward ray ` ( -oo , P...
ordtopn3 21732 An open interval ` ( A , B...
ordtcld1 21733 A downward ray ` ( -oo , P...
ordtcld2 21734 An upward ray ` [ P , +oo ...
ordtcld3 21735 A closed interval ` [ A , ...
ordttop 21736 The order topology is a to...
ordtcnv 21737 The order dual generates t...
ordtrest 21738 The subspace topology of a...
ordtrest2lem 21739 Lemma for ~ ordtrest2 . (...
ordtrest2 21740 An interval-closed set ` A...
letopon 21741 The topology of the extend...
letop 21742 The topology of the extend...
letopuni 21743 The topology of the extend...
xrstopn 21744 The topology component of ...
xrstps 21745 The extended real number s...
leordtvallem1 21746 Lemma for ~ leordtval . (...
leordtvallem2 21747 Lemma for ~ leordtval . (...
leordtval2 21748 The topology of the extend...
leordtval 21749 The topology of the extend...
iccordt 21750 A closed interval is close...
iocpnfordt 21751 An unbounded above open in...
icomnfordt 21752 An unbounded above open in...
iooordt 21753 An open interval is open i...
reordt 21754 The real numbers are an op...
lecldbas 21755 The set of closed interval...
pnfnei 21756 A neighborhood of ` +oo ` ...
mnfnei 21757 A neighborhood of ` -oo ` ...
ordtrestixx 21758 The restriction of the les...
ordtresticc 21759 The restriction of the les...
lmrel 21766 The topological space conv...
lmrcl 21767 Reverse closure for the co...
lmfval 21768 The relation "sequence ` f...
cnfval 21769 The set of all continuous ...
cnpfval 21770 The function mapping the p...
iscn 21771 The predicate "the class `...
cnpval 21772 The set of all functions f...
iscnp 21773 The predicate "the class `...
iscn2 21774 The predicate "the class `...
iscnp2 21775 The predicate "the class `...
cntop1 21776 Reverse closure for a cont...
cntop2 21777 Reverse closure for a cont...
cnptop1 21778 Reverse closure for a func...
cnptop2 21779 Reverse closure for a func...
iscnp3 21780 The predicate "the class `...
cnprcl 21781 Reverse closure for a func...
cnf 21782 A continuous function is a...
cnpf 21783 A continuous function at p...
cnpcl 21784 The value of a continuous ...
cnf2 21785 A continuous function is a...
cnpf2 21786 A continuous function at p...
cnprcl2 21787 Reverse closure for a func...
tgcn 21788 The continuity predicate w...
tgcnp 21789 The "continuous at a point...
subbascn 21790 The continuity predicate w...
ssidcn 21791 The identity function is a...
cnpimaex 21792 Property of a function con...
idcn 21793 A restricted identity func...
lmbr 21794 Express the binary relatio...
lmbr2 21795 Express the binary relatio...
lmbrf 21796 Express the binary relatio...
lmconst 21797 A constant sequence conver...
lmcvg 21798 Convergence property of a ...
iscnp4 21799 The predicate "the class `...
cnpnei 21800 A condition for continuity...
cnima 21801 An open subset of the codo...
cnco 21802 The composition of two con...
cnpco 21803 The composition of a funct...
cnclima 21804 A closed subset of the cod...
iscncl 21805 A characterization of a co...
cncls2i 21806 Property of the preimage o...
cnntri 21807 Property of the preimage o...
cnclsi 21808 Property of the image of a...
cncls2 21809 Continuity in terms of clo...
cncls 21810 Continuity in terms of clo...
cnntr 21811 Continuity in terms of int...
cnss1 21812 If the topology ` K ` is f...
cnss2 21813 If the topology ` K ` is f...
cncnpi 21814 A continuous function is c...
cnsscnp 21815 The set of continuous func...
cncnp 21816 A continuous function is c...
cncnp2 21817 A continuous function is c...
cnnei 21818 Continuity in terms of nei...
cnconst2 21819 A constant function is con...
cnconst 21820 A constant function is con...
cnrest 21821 Continuity of a restrictio...
cnrest2 21822 Equivalence of continuity ...
cnrest2r 21823 Equivalence of continuity ...
cnpresti 21824 One direction of ~ cnprest...
cnprest 21825 Equivalence of continuity ...
cnprest2 21826 Equivalence of point-conti...
cndis 21827 Every function is continuo...
cnindis 21828 Every function is continuo...
cnpdis 21829 If ` A ` is an isolated po...
paste 21830 Pasting lemma. If ` A ` a...
lmfpm 21831 If ` F ` converges, then `...
lmfss 21832 Inclusion of a function ha...
lmcl 21833 Closure of a limit. (Cont...
lmss 21834 Limit on a subspace. (Con...
sslm 21835 A finer topology has fewer...
lmres 21836 A function converges iff i...
lmff 21837 If ` F ` converges, there ...
lmcls 21838 Any convergent sequence of...
lmcld 21839 Any convergent sequence of...
lmcnp 21840 The image of a convergent ...
lmcn 21841 The image of a convergent ...
ist0 21856 The predicate "is a T_0 sp...
ist1 21857 The predicate "is a T_1 sp...
ishaus 21858 The predicate "is a Hausdo...
iscnrm 21859 The property of being comp...
t0sep 21860 Any two topologically indi...
t0dist 21861 Any two distinct points in...
t1sncld 21862 In a T_1 space, singletons...
t1ficld 21863 In a T_1 space, finite set...
hausnei 21864 Neighborhood property of a...
t0top 21865 A T_0 space is a topologic...
t1top 21866 A T_1 space is a topologic...
haustop 21867 A Hausdorff space is a top...
isreg 21868 The predicate "is a regula...
regtop 21869 A regular space is a topol...
regsep 21870 In a regular space, every ...
isnrm 21871 The predicate "is a normal...
nrmtop 21872 A normal space is a topolo...
cnrmtop 21873 A completely normal space ...
iscnrm2 21874 The property of being comp...
ispnrm 21875 The property of being perf...
pnrmnrm 21876 A perfectly normal space i...
pnrmtop 21877 A perfectly normal space i...
pnrmcld 21878 A closed set in a perfectl...
pnrmopn 21879 An open set in a perfectly...
ist0-2 21880 The predicate "is a T_0 sp...
ist0-3 21881 The predicate "is a T_0 sp...
cnt0 21882 The preimage of a T_0 topo...
ist1-2 21883 An alternate characterizat...
t1t0 21884 A T_1 space is a T_0 space...
ist1-3 21885 A space is T_1 iff every p...
cnt1 21886 The preimage of a T_1 topo...
ishaus2 21887 Express the predicate " ` ...
haust1 21888 A Hausdorff space is a T_1...
hausnei2 21889 The Hausdorff condition st...
cnhaus 21890 The preimage of a Hausdorf...
nrmsep3 21891 In a normal space, given a...
nrmsep2 21892 In a normal space, any two...
nrmsep 21893 In a normal space, disjoin...
isnrm2 21894 An alternate characterizat...
isnrm3 21895 A topological space is nor...
cnrmi 21896 A subspace of a completely...
cnrmnrm 21897 A completely normal space ...
restcnrm 21898 A subspace of a completely...
resthauslem 21899 Lemma for ~ resthaus and s...
lpcls 21900 The limit points of the cl...
perfcls 21901 A subset of a perfect spac...
restt0 21902 A subspace of a T_0 topolo...
restt1 21903 A subspace of a T_1 topolo...
resthaus 21904 A subspace of a Hausdorff ...
t1sep2 21905 Any two points in a T_1 sp...
t1sep 21906 Any two distinct points in...
sncld 21907 A singleton is closed in a...
sshauslem 21908 Lemma for ~ sshaus and sim...
sst0 21909 A topology finer than a T_...
sst1 21910 A topology finer than a T_...
sshaus 21911 A topology finer than a Ha...
regsep2 21912 In a regular space, a clos...
isreg2 21913 A topological space is reg...
dnsconst 21914 If a continuous mapping to...
ordtt1 21915 The order topology is T_1 ...
lmmo 21916 A sequence in a Hausdorff ...
lmfun 21917 The convergence relation i...
dishaus 21918 A discrete topology is Hau...
ordthauslem 21919 Lemma for ~ ordthaus . (C...
ordthaus 21920 The order topology of a to...
xrhaus 21921 The topology of the extend...
iscmp 21924 The predicate "is a compac...
cmpcov 21925 An open cover of a compact...
cmpcov2 21926 Rewrite ~ cmpcov for the c...
cmpcovf 21927 Combine ~ cmpcov with ~ ac...
cncmp 21928 Compactness is respected b...
fincmp 21929 A finite topology is compa...
0cmp 21930 The singleton of the empty...
cmptop 21931 A compact topology is a to...
rncmp 21932 The image of a compact set...
imacmp 21933 The image of a compact set...
discmp 21934 A discrete topology is com...
cmpsublem 21935 Lemma for ~ cmpsub . (Con...
cmpsub 21936 Two equivalent ways of des...
tgcmp 21937 A topology generated by a ...
cmpcld 21938 A closed subset of a compa...
uncmp 21939 The union of two compact s...
fiuncmp 21940 A finite union of compact ...
sscmp 21941 A subset of a compact topo...
hauscmplem 21942 Lemma for ~ hauscmp . (Co...
hauscmp 21943 A compact subspace of a T2...
cmpfi 21944 If a topology is compact a...
cmpfii 21945 In a compact topology, a s...
bwth 21946 The glorious Bolzano-Weier...
isconn 21949 The predicate ` J ` is a c...
isconn2 21950 The predicate ` J ` is a c...
connclo 21951 The only nonempty clopen s...
conndisj 21952 If a topology is connected...
conntop 21953 A connected topology is a ...
indisconn 21954 The indiscrete topology (o...
dfconn2 21955 An alternate definition of...
connsuba 21956 Connectedness for a subspa...
connsub 21957 Two equivalent ways of say...
cnconn 21958 Connectedness is respected...
nconnsubb 21959 Disconnectedness for a sub...
connsubclo 21960 If a clopen set meets a co...
connima 21961 The image of a connected s...
conncn 21962 A continuous function from...
iunconnlem 21963 Lemma for ~ iunconn . (Co...
iunconn 21964 The indexed union of conne...
unconn 21965 The union of two connected...
clsconn 21966 The closure of a connected...
conncompid 21967 The connected component co...
conncompconn 21968 The connected component co...
conncompss 21969 The connected component co...
conncompcld 21970 The connected component co...
conncompclo 21971 The connected component co...
t1connperf 21972 A connected T_1 space is p...
is1stc 21977 The predicate "is a first-...
is1stc2 21978 An equivalent way of sayin...
1stctop 21979 A first-countable topology...
1stcclb 21980 A property of points in a ...
1stcfb 21981 For any point ` A ` in a f...
is2ndc 21982 The property of being seco...
2ndctop 21983 A second-countable topolog...
2ndci 21984 A countable basis generate...
2ndcsb 21985 Having a countable subbase...
2ndcredom 21986 A second-countable space h...
2ndc1stc 21987 A second-countable space i...
1stcrestlem 21988 Lemma for ~ 1stcrest . (C...
1stcrest 21989 A subspace of a first-coun...
2ndcrest 21990 A subspace of a second-cou...
2ndcctbss 21991 If a topology is second-co...
2ndcdisj 21992 Any disjoint family of ope...
2ndcdisj2 21993 Any disjoint collection of...
2ndcomap 21994 A surjective continuous op...
2ndcsep 21995 A second-countable topolog...
dis2ndc 21996 A discrete space is second...
1stcelcls 21997 A point belongs to the clo...
1stccnp 21998 A mapping is continuous at...
1stccn 21999 A mapping ` X --> Y ` , wh...
islly 22004 The property of being a lo...
isnlly 22005 The property of being an n...
llyeq 22006 Equality theorem for the `...
nllyeq 22007 Equality theorem for the `...
llytop 22008 A locally ` A ` space is a...
nllytop 22009 A locally ` A ` space is a...
llyi 22010 The property of a locally ...
nllyi 22011 The property of an n-local...
nlly2i 22012 Eliminate the neighborhood...
llynlly 22013 A locally ` A ` space is n...
llyssnlly 22014 A locally ` A ` space is n...
llyss 22015 The "locally" predicate re...
nllyss 22016 The "n-locally" predicate ...
subislly 22017 The property of a subspace...
restnlly 22018 If the property ` A ` pass...
restlly 22019 If the property ` A ` pass...
islly2 22020 An alternative expression ...
llyrest 22021 An open subspace of a loca...
nllyrest 22022 An open subspace of an n-l...
loclly 22023 If ` A ` is a local proper...
llyidm 22024 Idempotence of the "locall...
nllyidm 22025 Idempotence of the "n-loca...
toplly 22026 A topology is locally a to...
topnlly 22027 A topology is n-locally a ...
hauslly 22028 A Hausdorff space is local...
hausnlly 22029 A Hausdorff space is n-loc...
hausllycmp 22030 A compact Hausdorff space ...
cldllycmp 22031 A closed subspace of a loc...
lly1stc 22032 First-countability is a lo...
dislly 22033 The discrete space ` ~P X ...
disllycmp 22034 A discrete space is locall...
dis1stc 22035 A discrete space is first-...
hausmapdom 22036 If ` X ` is a first-counta...
hauspwdom 22037 Simplify the cardinal ` A ...
refrel 22044 Refinement is a relation. ...
isref 22045 The property of being a re...
refbas 22046 A refinement covers the sa...
refssex 22047 Every set in a refinement ...
ssref 22048 A subcover is a refinement...
refref 22049 Reflexivity of refinement....
reftr 22050 Refinement is transitive. ...
refun0 22051 Adding the empty set prese...
isptfin 22052 The statement "is a point-...
islocfin 22053 The statement "is a locall...
finptfin 22054 A finite cover is a point-...
ptfinfin 22055 A point covered by a point...
finlocfin 22056 A finite cover of a topolo...
locfintop 22057 A locally finite cover cov...
locfinbas 22058 A locally finite cover mus...
locfinnei 22059 A point covered by a local...
lfinpfin 22060 A locally finite cover is ...
lfinun 22061 Adding a finite set preser...
locfincmp 22062 For a compact space, the l...
unisngl 22063 Taking the union of the se...
dissnref 22064 The set of singletons is a...
dissnlocfin 22065 The set of singletons is l...
locfindis 22066 The locally finite covers ...
locfincf 22067 A locally finite cover in ...
comppfsc 22068 A space where every open c...
kgenval 22071 Value of the compact gener...
elkgen 22072 Value of the compact gener...
kgeni 22073 Property of the open sets ...
kgentopon 22074 The compact generator gene...
kgenuni 22075 The base set of the compac...
kgenftop 22076 The compact generator gene...
kgenf 22077 The compact generator is a...
kgentop 22078 A compactly generated spac...
kgenss 22079 The compact generator gene...
kgenhaus 22080 The compact generator gene...
kgencmp 22081 The compact generator topo...
kgencmp2 22082 The compact generator topo...
kgenidm 22083 The compact generator is i...
iskgen2 22084 A space is compactly gener...
iskgen3 22085 Derive the usual definitio...
llycmpkgen2 22086 A locally compact space is...
cmpkgen 22087 A compact space is compact...
llycmpkgen 22088 A locally compact space is...
1stckgenlem 22089 The one-point compactifica...
1stckgen 22090 A first-countable space is...
kgen2ss 22091 The compact generator pres...
kgencn 22092 A function from a compactl...
kgencn2 22093 A function ` F : J --> K `...
kgencn3 22094 The set of continuous func...
kgen2cn 22095 A continuous function is a...
txval 22100 Value of the binary topolo...
txuni2 22101 The underlying set of the ...
txbasex 22102 The basis for the product ...
txbas 22103 The set of Cartesian produ...
eltx 22104 A set in a product is open...
txtop 22105 The product of two topolog...
ptval 22106 The value of the product t...
ptpjpre1 22107 The preimage of a projecti...
elpt 22108 Elementhood in the bases o...
elptr 22109 A basic open set in the pr...
elptr2 22110 A basic open set in the pr...
ptbasid 22111 The base set of the produc...
ptuni2 22112 The base set for the produ...
ptbasin 22113 The basis for a product to...
ptbasin2 22114 The basis for a product to...
ptbas 22115 The basis for a product to...
ptpjpre2 22116 The basis for a product to...
ptbasfi 22117 The basis for the product ...
pttop 22118 The product topology is a ...
ptopn 22119 A basic open set in the pr...
ptopn2 22120 A sub-basic open set in th...
xkotf 22121 Functionality of function ...
xkobval 22122 Alternative expression for...
xkoval 22123 Value of the compact-open ...
xkotop 22124 The compact-open topology ...
xkoopn 22125 A basic open set of the co...
txtopi 22126 The product of two topolog...
txtopon 22127 The underlying set of the ...
txuni 22128 The underlying set of the ...
txunii 22129 The underlying set of the ...
ptuni 22130 The base set for the produ...
ptunimpt 22131 Base set of a product topo...
pttopon 22132 The base set for the produ...
pttoponconst 22133 The base set for a product...
ptuniconst 22134 The base set for a product...
xkouni 22135 The base set of the compac...
xkotopon 22136 The base set of the compac...
ptval2 22137 The value of the product t...
txopn 22138 The product of two open se...
txcld 22139 The product of two closed ...
txcls 22140 Closure of a rectangle in ...
txss12 22141 Subset property of the top...
txbasval 22142 It is sufficient to consid...
neitx 22143 The Cartesian product of t...
txcnpi 22144 Continuity of a two-argume...
tx1cn 22145 Continuity of the first pr...
tx2cn 22146 Continuity of the second p...
ptpjcn 22147 Continuity of a projection...
ptpjopn 22148 The projection map is an o...
ptcld 22149 A closed box in the produc...
ptcldmpt 22150 A closed box in the produc...
ptclsg 22151 The closure of a box in th...
ptcls 22152 The closure of a box in th...
dfac14lem 22153 Lemma for ~ dfac14 . By e...
dfac14 22154 Theorem ~ ptcls is an equi...
xkoccn 22155 The "constant function" fu...
txcnp 22156 If two functions are conti...
ptcnplem 22157 Lemma for ~ ptcnp . (Cont...
ptcnp 22158 If every projection of a f...
upxp 22159 Universal property of the ...
txcnmpt 22160 A map into the product of ...
uptx 22161 Universal property of the ...
txcn 22162 A map into the product of ...
ptcn 22163 If every projection of a f...
prdstopn 22164 Topology of a structure pr...
prdstps 22165 A structure product of top...
pwstps 22166 A structure product of top...
txrest 22167 The subspace of a topologi...
txdis 22168 The topological product of...
txindislem 22169 Lemma for ~ txindis . (Co...
txindis 22170 The topological product of...
txdis1cn 22171 A function is jointly cont...
txlly 22172 If the property ` A ` is p...
txnlly 22173 If the property ` A ` is p...
pthaus 22174 The product of a collectio...
ptrescn 22175 Restriction is a continuou...
txtube 22176 The "tube lemma". If ` X ...
txcmplem1 22177 Lemma for ~ txcmp . (Cont...
txcmplem2 22178 Lemma for ~ txcmp . (Cont...
txcmp 22179 The topological product of...
txcmpb 22180 The topological product of...
hausdiag 22181 A topology is Hausdorff if...
hauseqlcld 22182 In a Hausdorff topology, t...
txhaus 22183 The topological product of...
txlm 22184 Two sequences converge iff...
lmcn2 22185 The image of a convergent ...
tx1stc 22186 The topological product of...
tx2ndc 22187 The topological product of...
txkgen 22188 The topological product of...
xkohaus 22189 If the codomain space is H...
xkoptsub 22190 The compact-open topology ...
xkopt 22191 The compact-open topology ...
xkopjcn 22192 Continuity of a projection...
xkoco1cn 22193 If ` F ` is a continuous f...
xkoco2cn 22194 If ` F ` is a continuous f...
xkococnlem 22195 Continuity of the composit...
xkococn 22196 Continuity of the composit...
cnmptid 22197 The identity function is c...
cnmptc 22198 A constant function is con...
cnmpt11 22199 The composition of continu...
cnmpt11f 22200 The composition of continu...
cnmpt1t 22201 The composition of continu...
cnmpt12f 22202 The composition of continu...
cnmpt12 22203 The composition of continu...
cnmpt1st 22204 The projection onto the fi...
cnmpt2nd 22205 The projection onto the se...
cnmpt2c 22206 A constant function is con...
cnmpt21 22207 The composition of continu...
cnmpt21f 22208 The composition of continu...
cnmpt2t 22209 The composition of continu...
cnmpt22 22210 The composition of continu...
cnmpt22f 22211 The composition of continu...
cnmpt1res 22212 The restriction of a conti...
cnmpt2res 22213 The restriction of a conti...
cnmptcom 22214 The argument converse of a...
cnmptkc 22215 The curried first projecti...
cnmptkp 22216 The evaluation of the inne...
cnmptk1 22217 The composition of a curri...
cnmpt1k 22218 The composition of a one-a...
cnmptkk 22219 The composition of two cur...
xkofvcn 22220 Joint continuity of the fu...
cnmptk1p 22221 The evaluation of a currie...
cnmptk2 22222 The uncurrying of a currie...
xkoinjcn 22223 Continuity of "injection",...
cnmpt2k 22224 The currying of a two-argu...
txconn 22225 The topological product of...
imasnopn 22226 If a relation graph is ope...
imasncld 22227 If a relation graph is clo...
imasncls 22228 If a relation graph is clo...
qtopval 22231 Value of the quotient topo...
qtopval2 22232 Value of the quotient topo...
elqtop 22233 Value of the quotient topo...
qtopres 22234 The quotient topology is u...
qtoptop2 22235 The quotient topology is a...
qtoptop 22236 The quotient topology is a...
elqtop2 22237 Value of the quotient topo...
qtopuni 22238 The base set of the quotie...
elqtop3 22239 Value of the quotient topo...
qtoptopon 22240 The base set of the quotie...
qtopid 22241 A quotient map is a contin...
idqtop 22242 The quotient topology indu...
qtopcmplem 22243 Lemma for ~ qtopcmp and ~ ...
qtopcmp 22244 A quotient of a compact sp...
qtopconn 22245 A quotient of a connected ...
qtopkgen 22246 A quotient of a compactly ...
basqtop 22247 An injection maps bases to...
tgqtop 22248 An injection maps generate...
qtopcld 22249 The property of being a cl...
qtopcn 22250 Universal property of a qu...
qtopss 22251 A surjective continuous fu...
qtopeu 22252 Universal property of the ...
qtoprest 22253 If ` A ` is a saturated op...
qtopomap 22254 If ` F ` is a surjective c...
qtopcmap 22255 If ` F ` is a surjective c...
imastopn 22256 The topology of an image s...
imastps 22257 The image of a topological...
qustps 22258 A quotient structure is a ...
kqfval 22259 Value of the function appe...
kqfeq 22260 Two points in the Kolmogor...
kqffn 22261 The topological indistingu...
kqval 22262 Value of the quotient topo...
kqtopon 22263 The Kolmogorov quotient is...
kqid 22264 The topological indistingu...
ist0-4 22265 The topological indistingu...
kqfvima 22266 When the image set is open...
kqsat 22267 Any open set is saturated ...
kqdisj 22268 A version of ~ imain for t...
kqcldsat 22269 Any closed set is saturate...
kqopn 22270 The topological indistingu...
kqcld 22271 The topological indistingu...
kqt0lem 22272 Lemma for ~ kqt0 . (Contr...
isr0 22273 The property " ` J ` is an...
r0cld 22274 The analogue of the T_1 ax...
regr1lem 22275 Lemma for ~ regr1 . (Cont...
regr1lem2 22276 A Kolmogorov quotient of a...
kqreglem1 22277 A Kolmogorov quotient of a...
kqreglem2 22278 If the Kolmogorov quotient...
kqnrmlem1 22279 A Kolmogorov quotient of a...
kqnrmlem2 22280 If the Kolmogorov quotient...
kqtop 22281 The Kolmogorov quotient is...
kqt0 22282 The Kolmogorov quotient is...
kqf 22283 The Kolmogorov quotient is...
r0sep 22284 The separation property of...
nrmr0reg 22285 A normal R_0 space is also...
regr1 22286 A regular space is R_1, wh...
kqreg 22287 The Kolmogorov quotient of...
kqnrm 22288 The Kolmogorov quotient of...
hmeofn 22293 The set of homeomorphisms ...
hmeofval 22294 The set of all the homeomo...
ishmeo 22295 The predicate F is a homeo...
hmeocn 22296 A homeomorphism is continu...
hmeocnvcn 22297 The converse of a homeomor...
hmeocnv 22298 The converse of a homeomor...
hmeof1o2 22299 A homeomorphism is a 1-1-o...
hmeof1o 22300 A homeomorphism is a 1-1-o...
hmeoima 22301 The image of an open set b...
hmeoopn 22302 Homeomorphisms preserve op...
hmeocld 22303 Homeomorphisms preserve cl...
hmeocls 22304 Homeomorphisms preserve cl...
hmeontr 22305 Homeomorphisms preserve in...
hmeoimaf1o 22306 The function mapping open ...
hmeores 22307 The restriction of a homeo...
hmeoco 22308 The composite of two homeo...
idhmeo 22309 The identity function is a...
hmeocnvb 22310 The converse of a homeomor...
hmeoqtop 22311 A homeomorphism is a quoti...
hmph 22312 Express the predicate ` J ...
hmphi 22313 If there is a homeomorphis...
hmphtop 22314 Reverse closure for the ho...
hmphtop1 22315 The relation "being homeom...
hmphtop2 22316 The relation "being homeom...
hmphref 22317 "Is homeomorphic to" is re...
hmphsym 22318 "Is homeomorphic to" is sy...
hmphtr 22319 "Is homeomorphic to" is tr...
hmpher 22320 "Is homeomorphic to" is an...
hmphen 22321 Homeomorphisms preserve th...
hmphsymb 22322 "Is homeomorphic to" is sy...
haushmphlem 22323 Lemma for ~ haushmph and s...
cmphmph 22324 Compactness is a topologic...
connhmph 22325 Connectedness is a topolog...
t0hmph 22326 T_0 is a topological prope...
t1hmph 22327 T_1 is a topological prope...
haushmph 22328 Hausdorff-ness is a topolo...
reghmph 22329 Regularity is a topologica...
nrmhmph 22330 Normality is a topological...
hmph0 22331 A topology homeomorphic to...
hmphdis 22332 Homeomorphisms preserve to...
hmphindis 22333 Homeomorphisms preserve to...
indishmph 22334 Equinumerous sets equipped...
hmphen2 22335 Homeomorphisms preserve th...
cmphaushmeo 22336 A continuous bijection fro...
ordthmeolem 22337 Lemma for ~ ordthmeo . (C...
ordthmeo 22338 An order isomorphism is a ...
txhmeo 22339 Lift a pair of homeomorphi...
txswaphmeolem 22340 Show inverse for the "swap...
txswaphmeo 22341 There is a homeomorphism f...
pt1hmeo 22342 The canonical homeomorphis...
ptuncnv 22343 Exhibit the converse funct...
ptunhmeo 22344 Define a homeomorphism fro...
xpstopnlem1 22345 The function ` F ` used in...
xpstps 22346 A binary product of topolo...
xpstopnlem2 22347 Lemma for ~ xpstopn . (Co...
xpstopn 22348 The topology on a binary p...
ptcmpfi 22349 A topological product of f...
xkocnv 22350 The inverse of the "curryi...
xkohmeo 22351 The Exponential Law for to...
qtopf1 22352 If a quotient map is injec...
qtophmeo 22353 If two functions on a base...
t0kq 22354 A topological space is T_0...
kqhmph 22355 A topological space is T_0...
ist1-5lem 22356 Lemma for ~ ist1-5 and sim...
t1r0 22357 A T_1 space is R_0. That ...
ist1-5 22358 A topological space is T_1...
ishaus3 22359 A topological space is Hau...
nrmreg 22360 A normal T_1 space is regu...
reghaus 22361 A regular T_0 space is Hau...
nrmhaus 22362 A T_1 normal space is Haus...
elmptrab 22363 Membership in a one-parame...
elmptrab2 22364 Membership in a one-parame...
isfbas 22365 The predicate " ` F ` is a...
fbasne0 22366 There are no empty filter ...
0nelfb 22367 No filter base contains th...
fbsspw 22368 A filter base on a set is ...
fbelss 22369 An element of the filter b...
fbdmn0 22370 The domain of a filter bas...
isfbas2 22371 The predicate " ` F ` is a...
fbasssin 22372 A filter base contains sub...
fbssfi 22373 A filter base contains sub...
fbssint 22374 A filter base contains sub...
fbncp 22375 A filter base does not con...
fbun 22376 A necessary and sufficient...
fbfinnfr 22377 No filter base containing ...
opnfbas 22378 The collection of open sup...
trfbas2 22379 Conditions for the trace o...
trfbas 22380 Conditions for the trace o...
isfil 22383 The predicate "is a filter...
filfbas 22384 A filter is a filter base....
0nelfil 22385 The empty set doesn't belo...
fileln0 22386 An element of a filter is ...
filsspw 22387 A filter is a subset of th...
filelss 22388 An element of a filter is ...
filss 22389 A filter is closed under t...
filin 22390 A filter is closed under t...
filtop 22391 The underlying set belongs...
isfil2 22392 Derive the standard axioms...
isfildlem 22393 Lemma for ~ isfild . (Con...
isfild 22394 Sufficient condition for a...
filfi 22395 A filter is closed under t...
filinn0 22396 The intersection of two el...
filintn0 22397 A filter has the finite in...
filn0 22398 The empty set is not a fil...
infil 22399 The intersection of two fi...
snfil 22400 A singleton is a filter. ...
fbasweak 22401 A filter base on any set i...
snfbas 22402 Condition for a singleton ...
fsubbas 22403 A condition for a set to g...
fbasfip 22404 A filter base has the fini...
fbunfip 22405 A helpful lemma for showin...
fgval 22406 The filter generating clas...
elfg 22407 A condition for elements o...
ssfg 22408 A filter base is a subset ...
fgss 22409 A bigger base generates a ...
fgss2 22410 A condition for a filter t...
fgfil 22411 A filter generates itself....
elfilss 22412 An element belongs to a fi...
filfinnfr 22413 No filter containing a fin...
fgcl 22414 A generated filter is a fi...
fgabs 22415 Absorption law for filter ...
neifil 22416 The neighborhoods of a non...
filunibas 22417 Recover the base set from ...
filunirn 22418 Two ways to express a filt...
filconn 22419 A filter gives rise to a c...
fbasrn 22420 Given a filter on a domain...
filuni 22421 The union of a nonempty se...
trfil1 22422 Conditions for the trace o...
trfil2 22423 Conditions for the trace o...
trfil3 22424 Conditions for the trace o...
trfilss 22425 If ` A ` is a member of th...
fgtr 22426 If ` A ` is a member of th...
trfg 22427 The trace operation and th...
trnei 22428 The trace, over a set ` A ...
cfinfil 22429 Relative complements of th...
csdfil 22430 The set of all elements wh...
supfil 22431 The supersets of a nonempt...
zfbas 22432 The set of upper sets of i...
uzrest 22433 The restriction of the set...
uzfbas 22434 The set of upper sets of i...
isufil 22439 The property of being an u...
ufilfil 22440 An ultrafilter is a filter...
ufilss 22441 For any subset of the base...
ufilb 22442 The complement is in an ul...
ufilmax 22443 Any filter finer than an u...
isufil2 22444 The maximal property of an...
ufprim 22445 An ultrafilter is a prime ...
trufil 22446 Conditions for the trace o...
filssufilg 22447 A filter is contained in s...
filssufil 22448 A filter is contained in s...
isufl 22449 Define the (strong) ultraf...
ufli 22450 Property of a set that sat...
numufl 22451 Consequence of ~ filssufil...
fiufl 22452 A finite set satisfies the...
acufl 22453 The axiom of choice implie...
ssufl 22454 If ` Y ` is a subset of ` ...
ufileu 22455 If the ultrafilter contain...
filufint 22456 A filter is equal to the i...
uffix 22457 Lemma for ~ fixufil and ~ ...
fixufil 22458 The condition describing a...
uffixfr 22459 An ultrafilter is either f...
uffix2 22460 A classification of fixed ...
uffixsn 22461 The singleton of the gener...
ufildom1 22462 An ultrafilter is generate...
uffinfix 22463 An ultrafilter containing ...
cfinufil 22464 An ultrafilter is free iff...
ufinffr 22465 An infinite subset is cont...
ufilen 22466 Any infinite set has an ul...
ufildr 22467 An ultrafilter gives rise ...
fin1aufil 22468 There are no definable fre...
fmval 22479 Introduce a function that ...
fmfil 22480 A mapping filter is a filt...
fmf 22481 Pushing-forward via a func...
fmss 22482 A finer filter produces a ...
elfm 22483 An element of a mapping fi...
elfm2 22484 An element of a mapping fi...
fmfg 22485 The image filter of a filt...
elfm3 22486 An alternate formulation o...
imaelfm 22487 An image of a filter eleme...
rnelfmlem 22488 Lemma for ~ rnelfm . (Con...
rnelfm 22489 A condition for a filter t...
fmfnfmlem1 22490 Lemma for ~ fmfnfm . (Con...
fmfnfmlem2 22491 Lemma for ~ fmfnfm . (Con...
fmfnfmlem3 22492 Lemma for ~ fmfnfm . (Con...
fmfnfmlem4 22493 Lemma for ~ fmfnfm . (Con...
fmfnfm 22494 A filter finer than an ima...
fmufil 22495 An image filter of an ultr...
fmid 22496 The filter map applied to ...
fmco 22497 Composition of image filte...
ufldom 22498 The ultrafilter lemma prop...
flimval 22499 The set of limit points of...
elflim2 22500 The predicate "is a limit ...
flimtop 22501 Reverse closure for the li...
flimneiss 22502 A filter contains the neig...
flimnei 22503 A filter contains all of t...
flimelbas 22504 A limit point of a filter ...
flimfil 22505 Reverse closure for the li...
flimtopon 22506 Reverse closure for the li...
elflim 22507 The predicate "is a limit ...
flimss2 22508 A limit point of a filter ...
flimss1 22509 A limit point of a filter ...
neiflim 22510 A point is a limit point o...
flimopn 22511 The condition for being a ...
fbflim 22512 A condition for a filter t...
fbflim2 22513 A condition for a filter b...
flimclsi 22514 The convergent points of a...
hausflimlem 22515 If ` A ` and ` B ` are bot...
hausflimi 22516 One direction of ~ hausfli...
hausflim 22517 A condition for a topology...
flimcf 22518 Fineness is properly chara...
flimrest 22519 The set of limit points in...
flimclslem 22520 Lemma for ~ flimcls . (Co...
flimcls 22521 Closure in terms of filter...
flimsncls 22522 If ` A ` is a limit point ...
hauspwpwf1 22523 Lemma for ~ hauspwpwdom . ...
hauspwpwdom 22524 If ` X ` is a Hausdorff sp...
flffval 22525 Given a topology and a fil...
flfval 22526 Given a function from a fi...
flfnei 22527 The property of being a li...
flfneii 22528 A neighborhood of a limit ...
isflf 22529 The property of being a li...
flfelbas 22530 A limit point of a functio...
flffbas 22531 Limit points of a function...
flftg 22532 Limit points of a function...
hausflf 22533 If a function has its valu...
hausflf2 22534 If a convergent function h...
cnpflfi 22535 Forward direction of ~ cnp...
cnpflf2 22536 ` F ` is continuous at poi...
cnpflf 22537 Continuity of a function a...
cnflf 22538 A function is continuous i...
cnflf2 22539 A function is continuous i...
flfcnp 22540 A continuous function pres...
lmflf 22541 The topological limit rela...
txflf 22542 Two sequences converge in ...
flfcnp2 22543 The image of a convergent ...
fclsval 22544 The set of all cluster poi...
isfcls 22545 A cluster point of a filte...
fclsfil 22546 Reverse closure for the cl...
fclstop 22547 Reverse closure for the cl...
fclstopon 22548 Reverse closure for the cl...
isfcls2 22549 A cluster point of a filte...
fclsopn 22550 Write the cluster point co...
fclsopni 22551 An open neighborhood of a ...
fclselbas 22552 A cluster point is in the ...
fclsneii 22553 A neighborhood of a cluste...
fclssscls 22554 The set of cluster points ...
fclsnei 22555 Cluster points in terms of...
supnfcls 22556 The filter of supersets of...
fclsbas 22557 Cluster points in terms of...
fclsss1 22558 A finer topology has fewer...
fclsss2 22559 A finer filter has fewer c...
fclsrest 22560 The set of cluster points ...
fclscf 22561 Characterization of finene...
flimfcls 22562 A limit point is a cluster...
fclsfnflim 22563 A filter clusters at a poi...
flimfnfcls 22564 A filter converges to a po...
fclscmpi 22565 Forward direction of ~ fcl...
fclscmp 22566 A space is compact iff eve...
uffclsflim 22567 The cluster points of an u...
ufilcmp 22568 A space is compact iff eve...
fcfval 22569 The set of cluster points ...
isfcf 22570 The property of being a cl...
fcfnei 22571 The property of being a cl...
fcfelbas 22572 A cluster point of a funct...
fcfneii 22573 A neighborhood of a cluste...
flfssfcf 22574 A limit point of a functio...
uffcfflf 22575 If the domain filter is an...
cnpfcfi 22576 Lemma for ~ cnpfcf . If a...
cnpfcf 22577 A function ` F ` is contin...
cnfcf 22578 Continuity of a function i...
flfcntr 22579 A continuous function's va...
alexsublem 22580 Lemma for ~ alexsub . (Co...
alexsub 22581 The Alexander Subbase Theo...
alexsubb 22582 Biconditional form of the ...
alexsubALTlem1 22583 Lemma for ~ alexsubALT . ...
alexsubALTlem2 22584 Lemma for ~ alexsubALT . ...
alexsubALTlem3 22585 Lemma for ~ alexsubALT . ...
alexsubALTlem4 22586 Lemma for ~ alexsubALT . ...
alexsubALT 22587 The Alexander Subbase Theo...
ptcmplem1 22588 Lemma for ~ ptcmp . (Cont...
ptcmplem2 22589 Lemma for ~ ptcmp . (Cont...
ptcmplem3 22590 Lemma for ~ ptcmp . (Cont...
ptcmplem4 22591 Lemma for ~ ptcmp . (Cont...
ptcmplem5 22592 Lemma for ~ ptcmp . (Cont...
ptcmpg 22593 Tychonoff's theorem: The ...
ptcmp 22594 Tychonoff's theorem: The ...
cnextval 22597 The function applying cont...
cnextfval 22598 The continuous extension o...
cnextrel 22599 In the general case, a con...
cnextfun 22600 If the target space is Hau...
cnextfvval 22601 The value of the continuou...
cnextf 22602 Extension by continuity. ...
cnextcn 22603 Extension by continuity. ...
cnextfres1 22604 ` F ` and its extension by...
cnextfres 22605 ` F ` and its extension by...
istmd 22610 The predicate "is a topolo...
tmdmnd 22611 A topological monoid is a ...
tmdtps 22612 A topological monoid is a ...
istgp 22613 The predicate "is a topolo...
tgpgrp 22614 A topological group is a g...
tgptmd 22615 A topological group is a t...
tgptps 22616 A topological group is a t...
tmdtopon 22617 The topology of a topologi...
tgptopon 22618 The topology of a topologi...
tmdcn 22619 In a topological monoid, t...
tgpcn 22620 In a topological group, th...
tgpinv 22621 In a topological group, th...
grpinvhmeo 22622 The inverse function in a ...
cnmpt1plusg 22623 Continuity of the group su...
cnmpt2plusg 22624 Continuity of the group su...
tmdcn2 22625 Write out the definition o...
tgpsubcn 22626 In a topological group, th...
istgp2 22627 A group with a topology is...
tmdmulg 22628 In a topological monoid, t...
tgpmulg 22629 In a topological group, th...
tgpmulg2 22630 In a topological monoid, t...
tmdgsum 22631 In a topological monoid, t...
tmdgsum2 22632 For any neighborhood ` U `...
oppgtmd 22633 The opposite of a topologi...
oppgtgp 22634 The opposite of a topologi...
distgp 22635 Any group equipped with th...
indistgp 22636 Any group equipped with th...
symgtgp 22637 The symmetric group is a t...
tmdlactcn 22638 The left group action of e...
tgplacthmeo 22639 The left group action of e...
submtmd 22640 A submonoid of a topologic...
subgtgp 22641 A subgroup of a topologica...
subgntr 22642 A subgroup of a topologica...
opnsubg 22643 An open subgroup of a topo...
clssubg 22644 The closure of a subgroup ...
clsnsg 22645 The closure of a normal su...
cldsubg 22646 A subgroup of finite index...
tgpconncompeqg 22647 The connected component co...
tgpconncomp 22648 The identity component, th...
tgpconncompss 22649 The identity component is ...
ghmcnp 22650 A group homomorphism on to...
snclseqg 22651 The coset of the closure o...
tgphaus 22652 A topological group is Hau...
tgpt1 22653 Hausdorff and T1 are equiv...
tgpt0 22654 Hausdorff and T0 are equiv...
qustgpopn 22655 A quotient map in a topolo...
qustgplem 22656 Lemma for ~ qustgp . (Con...
qustgp 22657 The quotient of a topologi...
qustgphaus 22658 The quotient of a topologi...
prdstmdd 22659 The product of a family of...
prdstgpd 22660 The product of a family of...
tsmsfbas 22663 The collection of all sets...
tsmslem1 22664 The finite partial sums of...
tsmsval2 22665 Definition of the topologi...
tsmsval 22666 Definition of the topologi...
tsmspropd 22667 The group sum depends only...
eltsms 22668 The property of being a su...
tsmsi 22669 The property of being a su...
tsmscl 22670 A sum in a topological gro...
haustsms 22671 In a Hausdorff topological...
haustsms2 22672 In a Hausdorff topological...
tsmscls 22673 One half of ~ tgptsmscls ,...
tsmsgsum 22674 The convergent points of a...
tsmsid 22675 If a sum is finite, the us...
haustsmsid 22676 In a Hausdorff topological...
tsms0 22677 The sum of zero is zero. ...
tsmssubm 22678 Evaluate an infinite group...
tsmsres 22679 Extend an infinite group s...
tsmsf1o 22680 Re-index an infinite group...
tsmsmhm 22681 Apply a continuous group h...
tsmsadd 22682 The sum of two infinite gr...
tsmsinv 22683 Inverse of an infinite gro...
tsmssub 22684 The difference of two infi...
tgptsmscls 22685 A sum in a topological gro...
tgptsmscld 22686 The set of limit points to...
tsmssplit 22687 Split a topological group ...
tsmsxplem1 22688 Lemma for ~ tsmsxp . (Con...
tsmsxplem2 22689 Lemma for ~ tsmsxp . (Con...
tsmsxp 22690 Write a sum over a two-dim...
istrg 22699 Express the predicate " ` ...
trgtmd 22700 The multiplicative monoid ...
istdrg 22701 Express the predicate " ` ...
tdrgunit 22702 The unit group of a topolo...
trgtgp 22703 A topological ring is a to...
trgtmd2 22704 A topological ring is a to...
trgtps 22705 A topological ring is a to...
trgring 22706 A topological ring is a ri...
trggrp 22707 A topological ring is a gr...
tdrgtrg 22708 A topological division rin...
tdrgdrng 22709 A topological division rin...
tdrgring 22710 A topological division rin...
tdrgtmd 22711 A topological division rin...
tdrgtps 22712 A topological division rin...
istdrg2 22713 A topological-ring divisio...
mulrcn 22714 The functionalization of t...
invrcn2 22715 The multiplicative inverse...
invrcn 22716 The multiplicative inverse...
cnmpt1mulr 22717 Continuity of ring multipl...
cnmpt2mulr 22718 Continuity of ring multipl...
dvrcn 22719 The division function is c...
istlm 22720 The predicate " ` W ` is a...
vscacn 22721 The scalar multiplication ...
tlmtmd 22722 A topological module is a ...
tlmtps 22723 A topological module is a ...
tlmlmod 22724 A topological module is a ...
tlmtrg 22725 The scalar ring of a topol...
tlmscatps 22726 The scalar ring of a topol...
istvc 22727 A topological vector space...
tvctdrg 22728 The scalar field of a topo...
cnmpt1vsca 22729 Continuity of scalar multi...
cnmpt2vsca 22730 Continuity of scalar multi...
tlmtgp 22731 A topological vector space...
tvctlm 22732 A topological vector space...
tvclmod 22733 A topological vector space...
tvclvec 22734 A topological vector space...
ustfn 22737 The defined uniform struct...
ustval 22738 The class of all uniform s...
isust 22739 The predicate " ` U ` is a...
ustssxp 22740 Entourages are subsets of ...
ustssel 22741 A uniform structure is upw...
ustbasel 22742 The full set is always an ...
ustincl 22743 A uniform structure is clo...
ustdiag 22744 The diagonal set is includ...
ustinvel 22745 If ` V ` is an entourage, ...
ustexhalf 22746 For each entourage ` V ` t...
ustrel 22747 The elements of uniform st...
ustfilxp 22748 A uniform structure on a n...
ustne0 22749 A uniform structure cannot...
ustssco 22750 In an uniform structure, a...
ustexsym 22751 In an uniform structure, f...
ustex2sym 22752 In an uniform structure, f...
ustex3sym 22753 In an uniform structure, f...
ustref 22754 Any element of the base se...
ust0 22755 The unique uniform structu...
ustn0 22756 The empty set is not an un...
ustund 22757 If two intersecting sets `...
ustelimasn 22758 Any point ` A ` is near en...
ustneism 22759 For a point ` A ` in ` X `...
elrnust 22760 First direction for ~ ustb...
ustbas2 22761 Second direction for ~ ust...
ustuni 22762 The set union of a uniform...
ustbas 22763 Recover the base of an uni...
ustimasn 22764 Lemma for ~ ustuqtop . (C...
trust 22765 The trace of a uniform str...
utopval 22768 The topology induced by a ...
elutop 22769 Open sets in the topology ...
utoptop 22770 The topology induced by a ...
utopbas 22771 The base of the topology i...
utoptopon 22772 Topology induced by a unif...
restutop 22773 Restriction of a topology ...
restutopopn 22774 The restriction of the top...
ustuqtoplem 22775 Lemma for ~ ustuqtop . (C...
ustuqtop0 22776 Lemma for ~ ustuqtop . (C...
ustuqtop1 22777 Lemma for ~ ustuqtop , sim...
ustuqtop2 22778 Lemma for ~ ustuqtop . (C...
ustuqtop3 22779 Lemma for ~ ustuqtop , sim...
ustuqtop4 22780 Lemma for ~ ustuqtop . (C...
ustuqtop5 22781 Lemma for ~ ustuqtop . (C...
ustuqtop 22782 For a given uniform struct...
utopsnneiplem 22783 The neighborhoods of a poi...
utopsnneip 22784 The neighborhoods of a poi...
utopsnnei 22785 Images of singletons by en...
utop2nei 22786 For any symmetrical entour...
utop3cls 22787 Relation between a topolog...
utopreg 22788 All Hausdorff uniform spac...
ussval 22795 The uniform structure on u...
ussid 22796 In case the base of the ` ...
isusp 22797 The predicate ` W ` is a u...
ressunif 22798 ` UnifSet ` is unaffected ...
ressuss 22799 Value of the uniform struc...
ressust 22800 The uniform structure of a...
ressusp 22801 The restriction of a unifo...
tusval 22802 The value of the uniform s...
tuslem 22803 Lemma for ~ tusbas , ~ tus...
tusbas 22804 The base set of a construc...
tusunif 22805 The uniform structure of a...
tususs 22806 The uniform structure of a...
tustopn 22807 The topology induced by a ...
tususp 22808 A constructed uniform spac...
tustps 22809 A constructed uniform spac...
uspreg 22810 If a uniform space is Haus...
ucnval 22813 The set of all uniformly c...
isucn 22814 The predicate " ` F ` is a...
isucn2 22815 The predicate " ` F ` is a...
ucnimalem 22816 Reformulate the ` G ` func...
ucnima 22817 An equivalent statement of...
ucnprima 22818 The preimage by a uniforml...
iducn 22819 The identity is uniformly ...
cstucnd 22820 A constant function is uni...
ucncn 22821 Uniform continuity implies...
iscfilu 22824 The predicate " ` F ` is a...
cfilufbas 22825 A Cauchy filter base is a ...
cfiluexsm 22826 For a Cauchy filter base a...
fmucndlem 22827 Lemma for ~ fmucnd . (Con...
fmucnd 22828 The image of a Cauchy filt...
cfilufg 22829 The filter generated by a ...
trcfilu 22830 Condition for the trace of...
cfiluweak 22831 A Cauchy filter base is al...
neipcfilu 22832 In an uniform space, a nei...
iscusp 22835 The predicate " ` W ` is a...
cuspusp 22836 A complete uniform space i...
cuspcvg 22837 In a complete uniform spac...
iscusp2 22838 The predicate " ` W ` is a...
cnextucn 22839 Extension by continuity. ...
ucnextcn 22840 Extension by continuity. ...
ispsmet 22841 Express the predicate " ` ...
psmetdmdm 22842 Recover the base set from ...
psmetf 22843 The distance function of a...
psmetcl 22844 Closure of the distance fu...
psmet0 22845 The distance function of a...
psmettri2 22846 Triangle inequality for th...
psmetsym 22847 The distance function of a...
psmettri 22848 Triangle inequality for th...
psmetge0 22849 The distance function of a...
psmetxrge0 22850 The distance function of a...
psmetres2 22851 Restriction of a pseudomet...
psmetlecl 22852 Real closure of an extende...
distspace 22853 A set ` X ` together with ...
ismet 22860 Express the predicate " ` ...
isxmet 22861 Express the predicate " ` ...
ismeti 22862 Properties that determine ...
isxmetd 22863 Properties that determine ...
isxmet2d 22864 It is safe to only require...
metflem 22865 Lemma for ~ metf and other...
xmetf 22866 Mapping of the distance fu...
metf 22867 Mapping of the distance fu...
xmetcl 22868 Closure of the distance fu...
metcl 22869 Closure of the distance fu...
ismet2 22870 An extended metric is a me...
metxmet 22871 A metric is an extended me...
xmetdmdm 22872 Recover the base set from ...
metdmdm 22873 Recover the base set from ...
xmetunirn 22874 Two ways to express an ext...
xmeteq0 22875 The value of an extended m...
meteq0 22876 The value of a metric is z...
xmettri2 22877 Triangle inequality for th...
mettri2 22878 Triangle inequality for th...
xmet0 22879 The distance function of a...
met0 22880 The distance function of a...
xmetge0 22881 The distance function of a...
metge0 22882 The distance function of a...
xmetlecl 22883 Real closure of an extende...
xmetsym 22884 The distance function of a...
xmetpsmet 22885 An extended metric is a ps...
xmettpos 22886 The distance function of a...
metsym 22887 The distance function of a...
xmettri 22888 Triangle inequality for th...
mettri 22889 Triangle inequality for th...
xmettri3 22890 Triangle inequality for th...
mettri3 22891 Triangle inequality for th...
xmetrtri 22892 One half of the reverse tr...
xmetrtri2 22893 The reverse triangle inequ...
metrtri 22894 Reverse triangle inequalit...
xmetgt0 22895 The distance function of a...
metgt0 22896 The distance function of a...
metn0 22897 A metric space is nonempty...
xmetres2 22898 Restriction of an extended...
metreslem 22899 Lemma for ~ metres . (Con...
metres2 22900 Lemma for ~ metres . (Con...
xmetres 22901 A restriction of an extend...
metres 22902 A restriction of a metric ...
0met 22903 The empty metric. (Contri...
prdsdsf 22904 The product metric is a fu...
prdsxmetlem 22905 The product metric is an e...
prdsxmet 22906 The product metric is an e...
prdsmet 22907 The product metric is a me...
ressprdsds 22908 Restriction of a product m...
resspwsds 22909 Restriction of a product m...
imasdsf1olem 22910 Lemma for ~ imasdsf1o . (...
imasdsf1o 22911 The distance function is t...
imasf1oxmet 22912 The image of an extended m...
imasf1omet 22913 The image of a metric is a...
xpsdsfn 22914 Closure of the metric in a...
xpsdsfn2 22915 Closure of the metric in a...
xpsxmetlem 22916 Lemma for ~ xpsxmet . (Co...
xpsxmet 22917 A product metric of extend...
xpsdsval 22918 Value of the metric in a b...
xpsmet 22919 The direct product of two ...
blfvalps 22920 The value of the ball func...
blfval 22921 The value of the ball func...
blvalps 22922 The ball around a point ` ...
blval 22923 The ball around a point ` ...
elblps 22924 Membership in a ball. (Co...
elbl 22925 Membership in a ball. (Co...
elbl2ps 22926 Membership in a ball. (Co...
elbl2 22927 Membership in a ball. (Co...
elbl3ps 22928 Membership in a ball, with...
elbl3 22929 Membership in a ball, with...
blcomps 22930 Commute the arguments to t...
blcom 22931 Commute the arguments to t...
xblpnfps 22932 The infinity ball in an ex...
xblpnf 22933 The infinity ball in an ex...
blpnf 22934 The infinity ball in a sta...
bldisj 22935 Two balls are disjoint if ...
blgt0 22936 A nonempty ball implies th...
bl2in 22937 Two balls are disjoint if ...
xblss2ps 22938 One ball is contained in a...
xblss2 22939 One ball is contained in a...
blss2ps 22940 One ball is contained in a...
blss2 22941 One ball is contained in a...
blhalf 22942 A ball of radius ` R / 2 `...
blfps 22943 Mapping of a ball. (Contr...
blf 22944 Mapping of a ball. (Contr...
blrnps 22945 Membership in the range of...
blrn 22946 Membership in the range of...
xblcntrps 22947 A ball contains its center...
xblcntr 22948 A ball contains its center...
blcntrps 22949 A ball contains its center...
blcntr 22950 A ball contains its center...
xbln0 22951 A ball is nonempty iff the...
bln0 22952 A ball is not empty. (Con...
blelrnps 22953 A ball belongs to the set ...
blelrn 22954 A ball belongs to the set ...
blssm 22955 A ball is a subset of the ...
unirnblps 22956 The union of the set of ba...
unirnbl 22957 The union of the set of ba...
blin 22958 The intersection of two ba...
ssblps 22959 The size of a ball increas...
ssbl 22960 The size of a ball increas...
blssps 22961 Any point ` P ` in a ball ...
blss 22962 Any point ` P ` in a ball ...
blssexps 22963 Two ways to express the ex...
blssex 22964 Two ways to express the ex...
ssblex 22965 A nested ball exists whose...
blin2 22966 Given any two balls and a ...
blbas 22967 The balls of a metric spac...
blres 22968 A ball in a restricted met...
xmeterval 22969 Value of the "finitely sep...
xmeter 22970 The "finitely separated" r...
xmetec 22971 The equivalence classes un...
blssec 22972 A ball centered at ` P ` i...
blpnfctr 22973 The infinity ball in an ex...
xmetresbl 22974 An extended metric restric...
mopnval 22975 An open set is a subset of...
mopntopon 22976 The set of open sets of a ...
mopntop 22977 The set of open sets of a ...
mopnuni 22978 The union of all open sets...
elmopn 22979 The defining property of a...
mopnfss 22980 The family of open sets of...
mopnm 22981 The base set of a metric s...
elmopn2 22982 A defining property of an ...
mopnss 22983 An open set of a metric sp...
isxms 22984 Express the predicate " ` ...
isxms2 22985 Express the predicate " ` ...
isms 22986 Express the predicate " ` ...
isms2 22987 Express the predicate " ` ...
xmstopn 22988 The topology component of ...
mstopn 22989 The topology component of ...
xmstps 22990 An extended metric space i...
msxms 22991 A metric space is an exten...
mstps 22992 A metric space is a topolo...
xmsxmet 22993 The distance function, sui...
msmet 22994 The distance function, sui...
msf 22995 The distance function of a...
xmsxmet2 22996 The distance function, sui...
msmet2 22997 The distance function, sui...
mscl 22998 Closure of the distance fu...
xmscl 22999 Closure of the distance fu...
xmsge0 23000 The distance function in a...
xmseq0 23001 The distance between two p...
xmssym 23002 The distance function in a...
xmstri2 23003 Triangle inequality for th...
mstri2 23004 Triangle inequality for th...
xmstri 23005 Triangle inequality for th...
mstri 23006 Triangle inequality for th...
xmstri3 23007 Triangle inequality for th...
mstri3 23008 Triangle inequality for th...
msrtri 23009 Reverse triangle inequalit...
xmspropd 23010 Property deduction for an ...
mspropd 23011 Property deduction for a m...
setsmsbas 23012 The base set of a construc...
setsmsds 23013 The distance function of a...
setsmstset 23014 The topology of a construc...
setsmstopn 23015 The topology of a construc...
setsxms 23016 The constructed metric spa...
setsms 23017 The constructed metric spa...
tmsval 23018 For any metric there is an...
tmslem 23019 Lemma for ~ tmsbas , ~ tms...
tmsbas 23020 The base set of a construc...
tmsds 23021 The metric of a constructe...
tmstopn 23022 The topology of a construc...
tmsxms 23023 The constructed metric spa...
tmsms 23024 The constructed metric spa...
imasf1obl 23025 The image of a metric spac...
imasf1oxms 23026 The image of a metric spac...
imasf1oms 23027 The image of a metric spac...
prdsbl 23028 A ball in the product metr...
mopni 23029 An open set of a metric sp...
mopni2 23030 An open set of a metric sp...
mopni3 23031 An open set of a metric sp...
blssopn 23032 The balls of a metric spac...
unimopn 23033 The union of a collection ...
mopnin 23034 The intersection of two op...
mopn0 23035 The empty set is an open s...
rnblopn 23036 A ball of a metric space i...
blopn 23037 A ball of a metric space i...
neibl 23038 The neighborhoods around a...
blnei 23039 A ball around a point is a...
lpbl 23040 Every ball around a limit ...
blsscls2 23041 A smaller closed ball is c...
blcld 23042 A "closed ball" in a metri...
blcls 23043 The closure of an open bal...
blsscls 23044 If two concentric balls ha...
metss 23045 Two ways of saying that me...
metequiv 23046 Two ways of saying that tw...
metequiv2 23047 If there is a sequence of ...
metss2lem 23048 Lemma for ~ metss2 . (Con...
metss2 23049 If the metric ` D ` is "st...
comet 23050 The composition of an exte...
stdbdmetval 23051 Value of the standard boun...
stdbdxmet 23052 The standard bounded metri...
stdbdmet 23053 The standard bounded metri...
stdbdbl 23054 The standard bounded metri...
stdbdmopn 23055 The standard bounded metri...
mopnex 23056 The topology generated by ...
methaus 23057 The topology generated by ...
met1stc 23058 The topology generated by ...
met2ndci 23059 A separable metric space (...
met2ndc 23060 A metric space is second-c...
metrest 23061 Two alternate formulations...
ressxms 23062 The restriction of a metri...
ressms 23063 The restriction of a metri...
prdsmslem1 23064 Lemma for ~ prdsms . The ...
prdsxmslem1 23065 Lemma for ~ prdsms . The ...
prdsxmslem2 23066 Lemma for ~ prdsxms . The...
prdsxms 23067 The indexed product struct...
prdsms 23068 The indexed product struct...
pwsxms 23069 The product of a finite fa...
pwsms 23070 The product of a finite fa...
xpsxms 23071 A binary product of metric...
xpsms 23072 A binary product of metric...
tmsxps 23073 Express the product of two...
tmsxpsmopn 23074 Express the product of two...
tmsxpsval 23075 Value of the product of tw...
tmsxpsval2 23076 Value of the product of tw...
metcnp3 23077 Two ways to express that `...
metcnp 23078 Two ways to say a mapping ...
metcnp2 23079 Two ways to say a mapping ...
metcn 23080 Two ways to say a mapping ...
metcnpi 23081 Epsilon-delta property of ...
metcnpi2 23082 Epsilon-delta property of ...
metcnpi3 23083 Epsilon-delta property of ...
txmetcnp 23084 Continuity of a binary ope...
txmetcn 23085 Continuity of a binary ope...
metuval 23086 Value of the uniform struc...
metustel 23087 Define a filter base ` F `...
metustss 23088 Range of the elements of t...
metustrel 23089 Elements of the filter bas...
metustto 23090 Any two elements of the fi...
metustid 23091 The identity diagonal is i...
metustsym 23092 Elements of the filter bas...
metustexhalf 23093 For any element ` A ` of t...
metustfbas 23094 The filter base generated ...
metust 23095 The uniform structure gene...
cfilucfil 23096 Given a metric ` D ` and a...
metuust 23097 The uniform structure gene...
cfilucfil2 23098 Given a metric ` D ` and a...
blval2 23099 The ball around a point ` ...
elbl4 23100 Membership in a ball, alte...
metuel 23101 Elementhood in the uniform...
metuel2 23102 Elementhood in the uniform...
metustbl 23103 The "section" image of an ...
psmetutop 23104 The topology induced by a ...
xmetutop 23105 The topology induced by a ...
xmsusp 23106 If the uniform set of a me...
restmetu 23107 The uniform structure gene...
metucn 23108 Uniform continuity in metr...
dscmet 23109 The discrete metric on any...
dscopn 23110 The discrete metric genera...
nrmmetd 23111 Show that a group norm gen...
abvmet 23112 An absolute value ` F ` ge...
nmfval 23125 The value of the norm func...
nmval 23126 The value of the norm func...
nmfval2 23127 The value of the norm func...
nmval2 23128 The value of the norm func...
nmf2 23129 The norm is a function fro...
nmpropd 23130 Weak property deduction fo...
nmpropd2 23131 Strong property deduction ...
isngp 23132 The property of being a no...
isngp2 23133 The property of being a no...
isngp3 23134 The property of being a no...
ngpgrp 23135 A normed group is a group....
ngpms 23136 A normed group is a metric...
ngpxms 23137 A normed group is a metric...
ngptps 23138 A normed group is a topolo...
ngpmet 23139 The (induced) metric of a ...
ngpds 23140 Value of the distance func...
ngpdsr 23141 Value of the distance func...
ngpds2 23142 Write the distance between...
ngpds2r 23143 Write the distance between...
ngpds3 23144 Write the distance between...
ngpds3r 23145 Write the distance between...
ngprcan 23146 Cancel right addition insi...
ngplcan 23147 Cancel left addition insid...
isngp4 23148 Express the property of be...
ngpinvds 23149 Two elements are the same ...
ngpsubcan 23150 Cancel right subtraction i...
nmf 23151 The norm on a normed group...
nmcl 23152 The norm of a normed group...
nmge0 23153 The norm of a normed group...
nmeq0 23154 The identity is the only e...
nmne0 23155 The norm of a nonzero elem...
nmrpcl 23156 The norm of a nonzero elem...
nminv 23157 The norm of a negated elem...
nmmtri 23158 The triangle inequality fo...
nmsub 23159 The norm of the difference...
nmrtri 23160 Reverse triangle inequalit...
nm2dif 23161 Inequality for the differe...
nmtri 23162 The triangle inequality fo...
nmtri2 23163 Triangle inequality for th...
ngpi 23164 The properties of a normed...
nm0 23165 Norm of the identity eleme...
nmgt0 23166 The norm of a nonzero elem...
sgrim 23167 The induced metric on a su...
sgrimval 23168 The induced metric on a su...
subgnm 23169 The norm in a subgroup. (...
subgnm2 23170 A substructure assigns the...
subgngp 23171 A normed group restricted ...
ngptgp 23172 A normed abelian group is ...
ngppropd 23173 Property deduction for a n...
reldmtng 23174 The function ` toNrmGrp ` ...
tngval 23175 Value of the function whic...
tnglem 23176 Lemma for ~ tngbas and sim...
tngbas 23177 The base set of a structur...
tngplusg 23178 The group addition of a st...
tng0 23179 The group identity of a st...
tngmulr 23180 The ring multiplication of...
tngsca 23181 The scalar ring of a struc...
tngvsca 23182 The scalar multiplication ...
tngip 23183 The inner product operatio...
tngds 23184 The metric function of a s...
tngtset 23185 The topology generated by ...
tngtopn 23186 The topology generated by ...
tngnm 23187 The topology generated by ...
tngngp2 23188 A norm turns a group into ...
tngngpd 23189 Derive the axioms for a no...
tngngp 23190 Derive the axioms for a no...
tnggrpr 23191 If a structure equipped wi...
tngngp3 23192 Alternate definition of a ...
nrmtngdist 23193 The augmentation of a norm...
nrmtngnrm 23194 The augmentation of a norm...
tngngpim 23195 The induced metric of a no...
isnrg 23196 A normed ring is a ring wi...
nrgabv 23197 The norm of a normed ring ...
nrgngp 23198 A normed ring is a normed ...
nrgring 23199 A normed ring is a ring. ...
nmmul 23200 The norm of a product in a...
nrgdsdi 23201 Distribute a distance calc...
nrgdsdir 23202 Distribute a distance calc...
nm1 23203 The norm of one in a nonze...
unitnmn0 23204 The norm of a unit is nonz...
nminvr 23205 The norm of an inverse in ...
nmdvr 23206 The norm of a division in ...
nrgdomn 23207 A nonzero normed ring is a...
nrgtgp 23208 A normed ring is a topolog...
subrgnrg 23209 A normed ring restricted t...
tngnrg 23210 Given any absolute value o...
isnlm 23211 A normed (left) module is ...
nmvs 23212 Defining property of a nor...
nlmngp 23213 A normed module is a norme...
nlmlmod 23214 A normed module is a left ...
nlmnrg 23215 The scalar component of a ...
nlmngp2 23216 The scalar component of a ...
nlmdsdi 23217 Distribute a distance calc...
nlmdsdir 23218 Distribute a distance calc...
nlmmul0or 23219 If a scalar product is zer...
sranlm 23220 The subring algebra over a...
nlmvscnlem2 23221 Lemma for ~ nlmvscn . Com...
nlmvscnlem1 23222 Lemma for ~ nlmvscn . (Co...
nlmvscn 23223 The scalar multiplication ...
rlmnlm 23224 The ring module over a nor...
rlmnm 23225 The norm function in the r...
nrgtrg 23226 A normed ring is a topolog...
nrginvrcnlem 23227 Lemma for ~ nrginvrcn . C...
nrginvrcn 23228 The ring inverse function ...
nrgtdrg 23229 A normed division ring is ...
nlmtlm 23230 A normed module is a topol...
isnvc 23231 A normed vector space is j...
nvcnlm 23232 A normed vector space is a...
nvclvec 23233 A normed vector space is a...
nvclmod 23234 A normed vector space is a...
isnvc2 23235 A normed vector space is j...
nvctvc 23236 A normed vector space is a...
lssnlm 23237 A subspace of a normed mod...
lssnvc 23238 A subspace of a normed vec...
rlmnvc 23239 The ring module over a nor...
ngpocelbl 23240 Membership of an off-cente...
nmoffn 23247 The function producing ope...
reldmnghm 23248 Lemma for normed group hom...
reldmnmhm 23249 Lemma for module homomorph...
nmofval 23250 Value of the operator norm...
nmoval 23251 Value of the operator norm...
nmogelb 23252 Property of the operator n...
nmolb 23253 Any upper bound on the val...
nmolb2d 23254 Any upper bound on the val...
nmof 23255 The operator norm is a fun...
nmocl 23256 The operator norm of an op...
nmoge0 23257 The operator norm of an op...
nghmfval 23258 A normed group homomorphis...
isnghm 23259 A normed group homomorphis...
isnghm2 23260 A normed group homomorphis...
isnghm3 23261 A normed group homomorphis...
bddnghm 23262 A bounded group homomorphi...
nghmcl 23263 A normed group homomorphis...
nmoi 23264 The operator norm achieves...
nmoix 23265 The operator norm is a bou...
nmoi2 23266 The operator norm is a bou...
nmoleub 23267 The operator norm, defined...
nghmrcl1 23268 Reverse closure for a norm...
nghmrcl2 23269 Reverse closure for a norm...
nghmghm 23270 A normed group homomorphis...
nmo0 23271 The operator norm of the z...
nmoeq0 23272 The operator norm is zero ...
nmoco 23273 An upper bound on the oper...
nghmco 23274 The composition of normed ...
nmotri 23275 Triangle inequality for th...
nghmplusg 23276 The sum of two bounded lin...
0nghm 23277 The zero operator is a nor...
nmoid 23278 The operator norm of the i...
idnghm 23279 The identity operator is a...
nmods 23280 Upper bound for the distan...
nghmcn 23281 A normed group homomorphis...
isnmhm 23282 A normed module homomorphi...
nmhmrcl1 23283 Reverse closure for a norm...
nmhmrcl2 23284 Reverse closure for a norm...
nmhmlmhm 23285 A normed module homomorphi...
nmhmnghm 23286 A normed module homomorphi...
nmhmghm 23287 A normed module homomorphi...
isnmhm2 23288 A normed module homomorphi...
nmhmcl 23289 A normed module homomorphi...
idnmhm 23290 The identity operator is a...
0nmhm 23291 The zero operator is a bou...
nmhmco 23292 The composition of bounded...
nmhmplusg 23293 The sum of two bounded lin...
qtopbaslem 23294 The set of open intervals ...
qtopbas 23295 The set of open intervals ...
retopbas 23296 A basis for the standard t...
retop 23297 The standard topology on t...
uniretop 23298 The underlying set of the ...
retopon 23299 The standard topology on t...
retps 23300 The standard topological s...
iooretop 23301 Open intervals are open se...
icccld 23302 Closed intervals are close...
icopnfcld 23303 Right-unbounded closed int...
iocmnfcld 23304 Left-unbounded closed inte...
qdensere 23305 ` QQ ` is dense in the sta...
cnmetdval 23306 Value of the distance func...
cnmet 23307 The absolute value metric ...
cnxmet 23308 The absolute value metric ...
cnbl0 23309 Two ways to write the open...
cnblcld 23310 Two ways to write the clos...
cnfldms 23311 The complex number field i...
cnfldxms 23312 The complex number field i...
cnfldtps 23313 The complex number field i...
cnfldnm 23314 The norm of the field of c...
cnngp 23315 The complex numbers form a...
cnnrg 23316 The complex numbers form a...
cnfldtopn 23317 The topology of the comple...
cnfldtopon 23318 The topology of the comple...
cnfldtop 23319 The topology of the comple...
cnfldhaus 23320 The topology of the comple...
unicntop 23321 The underlying set of the ...
cnopn 23322 The set of complex numbers...
zringnrg 23323 The ring of integers is a ...
remetdval 23324 Value of the distance func...
remet 23325 The absolute value metric ...
rexmet 23326 The absolute value metric ...
bl2ioo 23327 A ball in terms of an open...
ioo2bl 23328 An open interval of reals ...
ioo2blex 23329 An open interval of reals ...
blssioo 23330 The balls of the standard ...
tgioo 23331 The topology generated by ...
qdensere2 23332 ` QQ ` is dense in ` RR ` ...
blcvx 23333 An open ball in the comple...
rehaus 23334 The standard topology on t...
tgqioo 23335 The topology generated by ...
re2ndc 23336 The standard topology on t...
resubmet 23337 The subspace topology indu...
tgioo2 23338 The standard topology on t...
rerest 23339 The subspace topology indu...
tgioo3 23340 The standard topology on t...
xrtgioo 23341 The topology on the extend...
xrrest 23342 The subspace topology indu...
xrrest2 23343 The subspace topology indu...
xrsxmet 23344 The metric on the extended...
xrsdsre 23345 The metric on the extended...
xrsblre 23346 Any ball of the metric of ...
xrsmopn 23347 The metric on the extended...
zcld 23348 The integers are a closed ...
recld2 23349 The real numbers are a clo...
zcld2 23350 The integers are a closed ...
zdis 23351 The integers are a discret...
sszcld 23352 Every subset of the intege...
reperflem 23353 A subset of the real numbe...
reperf 23354 The real numbers are a per...
cnperf 23355 The complex numbers are a ...
iccntr 23356 The interior of a closed i...
icccmplem1 23357 Lemma for ~ icccmp . (Con...
icccmplem2 23358 Lemma for ~ icccmp . (Con...
icccmplem3 23359 Lemma for ~ icccmp . (Con...
icccmp 23360 A closed interval in ` RR ...
reconnlem1 23361 Lemma for ~ reconn . Conn...
reconnlem2 23362 Lemma for ~ reconn . (Con...
reconn 23363 A subset of the reals is c...
retopconn 23364 Corollary of ~ reconn . T...
iccconn 23365 A closed interval is conne...
opnreen 23366 Every nonempty open set is...
rectbntr0 23367 A countable subset of the ...
xrge0gsumle 23368 A finite sum in the nonneg...
xrge0tsms 23369 Any finite or infinite sum...
xrge0tsms2 23370 Any finite or infinite sum...
metdcnlem 23371 The metric function of a m...
xmetdcn2 23372 The metric function of an ...
xmetdcn 23373 The metric function of an ...
metdcn2 23374 The metric function of a m...
metdcn 23375 The metric function of a m...
msdcn 23376 The metric function of a m...
cnmpt1ds 23377 Continuity of the metric f...
cnmpt2ds 23378 Continuity of the metric f...
nmcn 23379 The norm of a normed group...
ngnmcncn 23380 The norm of a normed group...
abscn 23381 The absolute value functio...
metdsval 23382 Value of the "distance to ...
metdsf 23383 The distance from a point ...
metdsge 23384 The distance from the poin...
metds0 23385 If a point is in a set, it...
metdstri 23386 A generalization of the tr...
metdsle 23387 The distance from a point ...
metdsre 23388 The distance from a point ...
metdseq0 23389 The distance from a point ...
metdscnlem 23390 Lemma for ~ metdscn . (Co...
metdscn 23391 The function ` F ` which g...
metdscn2 23392 The function ` F ` which g...
metnrmlem1a 23393 Lemma for ~ metnrm . (Con...
metnrmlem1 23394 Lemma for ~ metnrm . (Con...
metnrmlem2 23395 Lemma for ~ metnrm . (Con...
metnrmlem3 23396 Lemma for ~ metnrm . (Con...
metnrm 23397 A metric space is normal. ...
metreg 23398 A metric space is regular....
addcnlem 23399 Lemma for ~ addcn , ~ subc...
addcn 23400 Complex number addition is...
subcn 23401 Complex number subtraction...
mulcn 23402 Complex number multiplicat...
divcn 23403 Complex number division is...
cnfldtgp 23404 The complex numbers form a...
fsumcn 23405 A finite sum of functions ...
fsum2cn 23406 Version of ~ fsumcn for tw...
expcn 23407 The power function on comp...
divccn 23408 Division by a nonzero cons...
sqcn 23409 The square function on com...
iitopon 23414 The unit interval is a top...
iitop 23415 The unit interval is a top...
iiuni 23416 The base set of the unit i...
dfii2 23417 Alternate definition of th...
dfii3 23418 Alternate definition of th...
dfii4 23419 Alternate definition of th...
dfii5 23420 The unit interval expresse...
iicmp 23421 The unit interval is compa...
iiconn 23422 The unit interval is conne...
cncfval 23423 The value of the continuou...
elcncf 23424 Membership in the set of c...
elcncf2 23425 Version of ~ elcncf with a...
cncfrss 23426 Reverse closure of the con...
cncfrss2 23427 Reverse closure of the con...
cncff 23428 A continuous complex funct...
cncfi 23429 Defining property of a con...
elcncf1di 23430 Membership in the set of c...
elcncf1ii 23431 Membership in the set of c...
rescncf 23432 A continuous complex funct...
cncffvrn 23433 Change the codomain of a c...
cncfss 23434 The set of continuous func...
climcncf 23435 Image of a limit under a c...
abscncf 23436 Absolute value is continuo...
recncf 23437 Real part is continuous. ...
imcncf 23438 Imaginary part is continuo...
cjcncf 23439 Complex conjugate is conti...
mulc1cncf 23440 Multiplication by a consta...
divccncf 23441 Division by a constant is ...
cncfco 23442 The composition of two con...
cncfmet 23443 Relate complex function co...
cncfcn 23444 Relate complex function co...
cncfcn1 23445 Relate complex function co...
cncfmptc 23446 A constant function is a c...
cncfmptid 23447 The identity function is a...
cncfmpt1f 23448 Composition of continuous ...
cncfmpt2f 23449 Composition of continuous ...
cncfmpt2ss 23450 Composition of continuous ...
addccncf 23451 Adding a constant is a con...
cdivcncf 23452 Division with a constant n...
negcncf 23453 The negative function is c...
negfcncf 23454 The negative of a continuo...
abscncfALT 23455 Absolute value is continuo...
cncfcnvcn 23456 Rewrite ~ cmphaushmeo for ...
expcncf 23457 The power function on comp...
cnmptre 23458 Lemma for ~ iirevcn and re...
cnmpopc 23459 Piecewise definition of a ...
iirev 23460 Reverse the unit interval....
iirevcn 23461 The reversion function is ...
iihalf1 23462 Map the first half of ` II...
iihalf1cn 23463 The first half function is...
iihalf2 23464 Map the second half of ` I...
iihalf2cn 23465 The second half function i...
elii1 23466 Divide the unit interval i...
elii2 23467 Divide the unit interval i...
iimulcl 23468 The unit interval is close...
iimulcn 23469 Multiplication is a contin...
icoopnst 23470 A half-open interval start...
iocopnst 23471 A half-open interval endin...
icchmeo 23472 The natural bijection from...
icopnfcnv 23473 Define a bijection from ` ...
icopnfhmeo 23474 The defined bijection from...
iccpnfcnv 23475 Define a bijection from ` ...
iccpnfhmeo 23476 The defined bijection from...
xrhmeo 23477 The bijection from ` [ -u ...
xrhmph 23478 The extended reals are hom...
xrcmp 23479 The topology of the extend...
xrconn 23480 The topology of the extend...
icccvx 23481 A linear combination of tw...
oprpiece1res1 23482 Restriction to the first p...
oprpiece1res2 23483 Restriction to the second ...
cnrehmeo 23484 The canonical bijection fr...
cnheiborlem 23485 Lemma for ~ cnheibor . (C...
cnheibor 23486 Heine-Borel theorem for co...
cnllycmp 23487 The topology on the comple...
rellycmp 23488 The topology on the reals ...
bndth 23489 The Boundedness Theorem. ...
evth 23490 The Extreme Value Theorem....
evth2 23491 The Extreme Value Theorem,...
lebnumlem1 23492 Lemma for ~ lebnum . The ...
lebnumlem2 23493 Lemma for ~ lebnum . As a...
lebnumlem3 23494 Lemma for ~ lebnum . By t...
lebnum 23495 The Lebesgue number lemma,...
xlebnum 23496 Generalize ~ lebnum to ext...
lebnumii 23497 Specialize the Lebesgue nu...
ishtpy 23503 Membership in the class of...
htpycn 23504 A homotopy is a continuous...
htpyi 23505 A homotopy evaluated at it...
ishtpyd 23506 Deduction for membership i...
htpycom 23507 Given a homotopy from ` F ...
htpyid 23508 A homotopy from a function...
htpyco1 23509 Compose a homotopy with a ...
htpyco2 23510 Compose a homotopy with a ...
htpycc 23511 Concatenate two homotopies...
isphtpy 23512 Membership in the class of...
phtpyhtpy 23513 A path homotopy is a homot...
phtpycn 23514 A path homotopy is a conti...
phtpyi 23515 Membership in the class of...
phtpy01 23516 Two path-homotopic paths h...
isphtpyd 23517 Deduction for membership i...
isphtpy2d 23518 Deduction for membership i...
phtpycom 23519 Given a homotopy from ` F ...
phtpyid 23520 A homotopy from a path to ...
phtpyco2 23521 Compose a path homotopy wi...
phtpycc 23522 Concatenate two path homot...
phtpcrel 23524 The path homotopy relation...
isphtpc 23525 The relation "is path homo...
phtpcer 23526 Path homotopy is an equiva...
phtpc01 23527 Path homotopic paths have ...
reparphti 23528 Lemma for ~ reparpht . (C...
reparpht 23529 Reparametrization lemma. ...
phtpcco2 23530 Compose a path homotopy wi...
pcofval 23541 The value of the path conc...
pcoval 23542 The concatenation of two p...
pcovalg 23543 Evaluate the concatenation...
pcoval1 23544 Evaluate the concatenation...
pco0 23545 The starting point of a pa...
pco1 23546 The ending point of a path...
pcoval2 23547 Evaluate the concatenation...
pcocn 23548 The concatenation of two p...
copco 23549 The composition of a conca...
pcohtpylem 23550 Lemma for ~ pcohtpy . (Co...
pcohtpy 23551 Homotopy invariance of pat...
pcoptcl 23552 A constant function is a p...
pcopt 23553 Concatenation with a point...
pcopt2 23554 Concatenation with a point...
pcoass 23555 Order of concatenation doe...
pcorevcl 23556 Closure for a reversed pat...
pcorevlem 23557 Lemma for ~ pcorev . Prov...
pcorev 23558 Concatenation with the rev...
pcorev2 23559 Concatenation with the rev...
pcophtb 23560 The path homotopy equivale...
om1val 23561 The definition of the loop...
om1bas 23562 The base set of the loop s...
om1elbas 23563 Elementhood in the base se...
om1addcl 23564 Closure of the group opera...
om1plusg 23565 The group operation (which...
om1tset 23566 The topology of the loop s...
om1opn 23567 The topology of the loop s...
pi1val 23568 The definition of the fund...
pi1bas 23569 The base set of the fundam...
pi1blem 23570 Lemma for ~ pi1buni . (Co...
pi1buni 23571 Another way to write the l...
pi1bas2 23572 The base set of the fundam...
pi1eluni 23573 Elementhood in the base se...
pi1bas3 23574 The base set of the fundam...
pi1cpbl 23575 The group operation, loop ...
elpi1 23576 The elements of the fundam...
elpi1i 23577 The elements of the fundam...
pi1addf 23578 The group operation of ` p...
pi1addval 23579 The concatenation of two p...
pi1grplem 23580 Lemma for ~ pi1grp . (Con...
pi1grp 23581 The fundamental group is a...
pi1id 23582 The identity element of th...
pi1inv 23583 An inverse in the fundamen...
pi1xfrf 23584 Functionality of the loop ...
pi1xfrval 23585 The value of the loop tran...
pi1xfr 23586 Given a path ` F ` and its...
pi1xfrcnvlem 23587 Given a path ` F ` between...
pi1xfrcnv 23588 Given a path ` F ` between...
pi1xfrgim 23589 The mapping ` G ` between ...
pi1cof 23590 Functionality of the loop ...
pi1coval 23591 The value of the loop tran...
pi1coghm 23592 The mapping ` G ` between ...
isclm 23595 A subcomplex module is a l...
clmsca 23596 The ring of scalars ` F ` ...
clmsubrg 23597 The base set of the ring o...
clmlmod 23598 A subcomplex module is a l...
clmgrp 23599 A subcomplex module is an ...
clmabl 23600 A subcomplex module is an ...
clmring 23601 The scalar ring of a subco...
clmfgrp 23602 The scalar ring of a subco...
clm0 23603 The zero of the scalar rin...
clm1 23604 The identity of the scalar...
clmadd 23605 The addition of the scalar...
clmmul 23606 The multiplication of the ...
clmcj 23607 The conjugation of the sca...
isclmi 23608 Reverse direction of ~ isc...
clmzss 23609 The scalar ring of a subco...
clmsscn 23610 The scalar ring of a subco...
clmsub 23611 Subtraction in the scalar ...
clmneg 23612 Negation in the scalar rin...
clmneg1 23613 Minus one is in the scalar...
clmabs 23614 Norm in the scalar ring of...
clmacl 23615 Closure of ring addition f...
clmmcl 23616 Closure of ring multiplica...
clmsubcl 23617 Closure of ring subtractio...
lmhmclm 23618 The domain of a linear ope...
clmvscl 23619 Closure of scalar product ...
clmvsass 23620 Associative law for scalar...
clmvscom 23621 Commutative law for the sc...
clmvsdir 23622 Distributive law for scala...
clmvsdi 23623 Distributive law for scala...
clmvs1 23624 Scalar product with ring u...
clmvs2 23625 A vector plus itself is tw...
clm0vs 23626 Zero times a vector is the...
clmopfne 23627 The (functionalized) opera...
isclmp 23628 The predicate "is a subcom...
isclmi0 23629 Properties that determine ...
clmvneg1 23630 Minus 1 times a vector is ...
clmvsneg 23631 Multiplication of a vector...
clmmulg 23632 The group multiple functio...
clmsubdir 23633 Scalar multiplication dist...
clmpm1dir 23634 Subtractive distributive l...
clmnegneg 23635 Double negative of a vecto...
clmnegsubdi2 23636 Distribution of negative o...
clmsub4 23637 Rearrangement of 4 terms i...
clmvsrinv 23638 A vector minus itself. (C...
clmvslinv 23639 Minus a vector plus itself...
clmvsubval 23640 Value of vector subtractio...
clmvsubval2 23641 Value of vector subtractio...
clmvz 23642 Two ways to express the ne...
zlmclm 23643 The ` ZZ ` -module operati...
clmzlmvsca 23644 The scalar product of a su...
nmoleub2lem 23645 Lemma for ~ nmoleub2a and ...
nmoleub2lem3 23646 Lemma for ~ nmoleub2a and ...
nmoleub2lem2 23647 Lemma for ~ nmoleub2a and ...
nmoleub2a 23648 The operator norm is the s...
nmoleub2b 23649 The operator norm is the s...
nmoleub3 23650 The operator norm is the s...
nmhmcn 23651 A linear operator over a n...
cmodscexp 23652 The powers of ` _i ` belon...
cmodscmulexp 23653 The scalar product of a ve...
cvslvec 23656 A subcomplex vector space ...
cvsclm 23657 A subcomplex vector space ...
iscvs 23658 A subcomplex vector space ...
iscvsp 23659 The predicate "is a subcom...
iscvsi 23660 Properties that determine ...
cvsi 23661 The properties of a subcom...
cvsunit 23662 Unit group of the scalar r...
cvsdiv 23663 Division of the scalar rin...
cvsdivcl 23664 The scalar field of a subc...
cvsmuleqdivd 23665 An equality involving rati...
cvsdiveqd 23666 An equality involving rati...
cnlmodlem1 23667 Lemma 1 for ~ cnlmod . (C...
cnlmodlem2 23668 Lemma 2 for ~ cnlmod . (C...
cnlmodlem3 23669 Lemma 3 for ~ cnlmod . (C...
cnlmod4 23670 Lemma 4 for ~ cnlmod . (C...
cnlmod 23671 The set of complex numbers...
cnstrcvs 23672 The set of complex numbers...
cnrbas 23673 The set of complex numbers...
cnrlmod 23674 The complex left module of...
cnrlvec 23675 The complex left module of...
cncvs 23676 The complex left module of...
recvs 23677 The field of the real numb...
qcvs 23678 The field of rational numb...
zclmncvs 23679 The ring of integers as le...
isncvsngp 23680 A normed subcomplex vector...
isncvsngpd 23681 Properties that determine ...
ncvsi 23682 The properties of a normed...
ncvsprp 23683 Proportionality property o...
ncvsge0 23684 The norm of a scalar produ...
ncvsm1 23685 The norm of the opposite o...
ncvsdif 23686 The norm of the difference...
ncvspi 23687 The norm of a vector plus ...
ncvs1 23688 From any nonzero vector of...
cnrnvc 23689 The module of complex numb...
cnncvs 23690 The module of complex numb...
cnnm 23691 The norm of the normed sub...
ncvspds 23692 Value of the distance func...
cnindmet 23693 The metric induced on the ...
cnncvsaddassdemo 23694 Derive the associative law...
cnncvsmulassdemo 23695 Derive the associative law...
cnncvsabsnegdemo 23696 Derive the absolute value ...
iscph 23701 A subcomplex pre-Hilbert s...
cphphl 23702 A subcomplex pre-Hilbert s...
cphnlm 23703 A subcomplex pre-Hilbert s...
cphngp 23704 A subcomplex pre-Hilbert s...
cphlmod 23705 A subcomplex pre-Hilbert s...
cphlvec 23706 A subcomplex pre-Hilbert s...
cphnvc 23707 A subcomplex pre-Hilbert s...
cphsubrglem 23708 Lemma for ~ cphsubrg . (C...
cphreccllem 23709 Lemma for ~ cphreccl . (C...
cphsca 23710 A subcomplex pre-Hilbert s...
cphsubrg 23711 The scalar field of a subc...
cphreccl 23712 The scalar field of a subc...
cphdivcl 23713 The scalar field of a subc...
cphcjcl 23714 The scalar field of a subc...
cphsqrtcl 23715 The scalar field of a subc...
cphabscl 23716 The scalar field of a subc...
cphsqrtcl2 23717 The scalar field of a subc...
cphsqrtcl3 23718 If the scalar field of a s...
cphqss 23719 The scalar field of a subc...
cphclm 23720 A subcomplex pre-Hilbert s...
cphnmvs 23721 Norm of a scalar product. ...
cphipcl 23722 An inner product is a memb...
cphnmfval 23723 The value of the norm in a...
cphnm 23724 The square of the norm is ...
nmsq 23725 The square of the norm is ...
cphnmf 23726 The norm of a vector is a ...
cphnmcl 23727 The norm of a vector is a ...
reipcl 23728 An inner product of an ele...
ipge0 23729 The inner product in a sub...
cphipcj 23730 Conjugate of an inner prod...
cphipipcj 23731 An inner product times its...
cphorthcom 23732 Orthogonality (meaning inn...
cphip0l 23733 Inner product with a zero ...
cphip0r 23734 Inner product with a zero ...
cphipeq0 23735 The inner product of a vec...
cphdir 23736 Distributive law for inner...
cphdi 23737 Distributive law for inner...
cph2di 23738 Distributive law for inner...
cphsubdir 23739 Distributive law for inner...
cphsubdi 23740 Distributive law for inner...
cph2subdi 23741 Distributive law for inner...
cphass 23742 Associative law for inner ...
cphassr 23743 "Associative" law for seco...
cph2ass 23744 Move scalar multiplication...
cphassi 23745 Associative law for the fi...
cphassir 23746 "Associative" law for the ...
tcphex 23747 Lemma for ~ tcphbas and si...
tcphval 23748 Define a function to augme...
tcphbas 23749 The base set of a subcompl...
tchplusg 23750 The addition operation of ...
tcphsub 23751 The subtraction operation ...
tcphmulr 23752 The ring operation of a su...
tcphsca 23753 The scalar field of a subc...
tcphvsca 23754 The scalar multiplication ...
tcphip 23755 The inner product of a sub...
tcphtopn 23756 The topology of a subcompl...
tcphphl 23757 Augmentation of a subcompl...
tchnmfval 23758 The norm of a subcomplex p...
tcphnmval 23759 The norm of a subcomplex p...
cphtcphnm 23760 The norm of a norm-augment...
tcphds 23761 The distance of a pre-Hilb...
phclm 23762 A pre-Hilbert space whose ...
tcphcphlem3 23763 Lemma for ~ tcphcph : real...
ipcau2 23764 The Cauchy-Schwarz inequal...
tcphcphlem1 23765 Lemma for ~ tcphcph : the ...
tcphcphlem2 23766 Lemma for ~ tcphcph : homo...
tcphcph 23767 The standard definition of...
ipcau 23768 The Cauchy-Schwarz inequal...
nmparlem 23769 Lemma for ~ nmpar . (Cont...
nmpar 23770 A subcomplex pre-Hilbert s...
cphipval2 23771 Value of the inner product...
4cphipval2 23772 Four times the inner produ...
cphipval 23773 Value of the inner product...
ipcnlem2 23774 The inner product operatio...
ipcnlem1 23775 The inner product operatio...
ipcn 23776 The inner product operatio...
cnmpt1ip 23777 Continuity of inner produc...
cnmpt2ip 23778 Continuity of inner produc...
csscld 23779 A "closed subspace" in a s...
clsocv 23780 The orthogonal complement ...
cphsscph 23781 A subspace of a subcomplex...
lmmbr 23788 Express the binary relatio...
lmmbr2 23789 Express the binary relatio...
lmmbr3 23790 Express the binary relatio...
lmmcvg 23791 Convergence property of a ...
lmmbrf 23792 Express the binary relatio...
lmnn 23793 A condition that implies c...
cfilfval 23794 The set of Cauchy filters ...
iscfil 23795 The property of being a Ca...
iscfil2 23796 The property of being a Ca...
cfilfil 23797 A Cauchy filter is a filte...
cfili 23798 Property of a Cauchy filte...
cfil3i 23799 A Cauchy filter contains b...
cfilss 23800 A filter finer than a Cauc...
fgcfil 23801 The Cauchy filter conditio...
fmcfil 23802 The Cauchy filter conditio...
iscfil3 23803 A filter is Cauchy iff it ...
cfilfcls 23804 Similar to ultrafilters ( ...
caufval 23805 The set of Cauchy sequence...
iscau 23806 Express the property " ` F...
iscau2 23807 Express the property " ` F...
iscau3 23808 Express the Cauchy sequenc...
iscau4 23809 Express the property " ` F...
iscauf 23810 Express the property " ` F...
caun0 23811 A metric with a Cauchy seq...
caufpm 23812 Inclusion of a Cauchy sequ...
caucfil 23813 A Cauchy sequence predicat...
iscmet 23814 The property " ` D ` is a ...
cmetcvg 23815 The convergence of a Cauch...
cmetmet 23816 A complete metric space is...
cmetmeti 23817 A complete metric space is...
cmetcaulem 23818 Lemma for ~ cmetcau . (Co...
cmetcau 23819 The convergence of a Cauch...
iscmet3lem3 23820 Lemma for ~ iscmet3 . (Co...
iscmet3lem1 23821 Lemma for ~ iscmet3 . (Co...
iscmet3lem2 23822 Lemma for ~ iscmet3 . (Co...
iscmet3 23823 The property " ` D ` is a ...
iscmet2 23824 A metric ` D ` is complete...
cfilresi 23825 A Cauchy filter on a metri...
cfilres 23826 Cauchy filter on a metric ...
caussi 23827 Cauchy sequence on a metri...
causs 23828 Cauchy sequence on a metri...
equivcfil 23829 If the metric ` D ` is "st...
equivcau 23830 If the metric ` D ` is "st...
lmle 23831 If the distance from each ...
nglmle 23832 If the norm of each member...
lmclim 23833 Relate a limit on the metr...
lmclimf 23834 Relate a limit on the metr...
metelcls 23835 A point belongs to the clo...
metcld 23836 A subset of a metric space...
metcld2 23837 A subset of a metric space...
caubl 23838 Sufficient condition to en...
caublcls 23839 The convergent point of a ...
metcnp4 23840 Two ways to say a mapping ...
metcn4 23841 Two ways to say a mapping ...
iscmet3i 23842 Properties that determine ...
lmcau 23843 Every convergent sequence ...
flimcfil 23844 Every convergent filter in...
metsscmetcld 23845 A complete subspace of a m...
cmetss 23846 A subspace of a complete m...
equivcmet 23847 If two metrics are strongl...
relcmpcmet 23848 If ` D ` is a metric space...
cmpcmet 23849 A compact metric space is ...
cfilucfil3 23850 Given a metric ` D ` and a...
cfilucfil4 23851 Given a metric ` D ` and a...
cncmet 23852 The set of complex numbers...
recmet 23853 The real numbers are a com...
bcthlem1 23854 Lemma for ~ bcth . Substi...
bcthlem2 23855 Lemma for ~ bcth . The ba...
bcthlem3 23856 Lemma for ~ bcth . The li...
bcthlem4 23857 Lemma for ~ bcth . Given ...
bcthlem5 23858 Lemma for ~ bcth . The pr...
bcth 23859 Baire's Category Theorem. ...
bcth2 23860 Baire's Category Theorem, ...
bcth3 23861 Baire's Category Theorem, ...
isbn 23868 A Banach space is a normed...
bnsca 23869 The scalar field of a Bana...
bnnvc 23870 A Banach space is a normed...
bnnlm 23871 A Banach space is a normed...
bnngp 23872 A Banach space is a normed...
bnlmod 23873 A Banach space is a left m...
bncms 23874 A Banach space is a comple...
iscms 23875 A complete metric space is...
cmscmet 23876 The induced metric on a co...
bncmet 23877 The induced metric on Bana...
cmsms 23878 A complete metric space is...
cmspropd 23879 Property deduction for a c...
cmssmscld 23880 The restriction of a metri...
cmsss 23881 The restriction of a compl...
lssbn 23882 A subspace of a Banach spa...
cmetcusp1 23883 If the uniform set of a co...
cmetcusp 23884 The uniform space generate...
cncms 23885 The field of complex numbe...
cnflduss 23886 The uniform structure of t...
cnfldcusp 23887 The field of complex numbe...
resscdrg 23888 The real numbers are a sub...
cncdrg 23889 The only complete subfield...
srabn 23890 The subring algebra over a...
rlmbn 23891 The ring module over a com...
ishl 23892 The predicate "is a subcom...
hlbn 23893 Every subcomplex Hilbert s...
hlcph 23894 Every subcomplex Hilbert s...
hlphl 23895 Every subcomplex Hilbert s...
hlcms 23896 Every subcomplex Hilbert s...
hlprlem 23897 Lemma for ~ hlpr . (Contr...
hlress 23898 The scalar field of a subc...
hlpr 23899 The scalar field of a subc...
ishl2 23900 A Hilbert space is a compl...
cphssphl 23901 A Banach subspace of a sub...
cmslssbn 23902 A complete linear subspace...
cmscsscms 23903 A closed subspace of a com...
bncssbn 23904 A closed subspace of a Ban...
cssbn 23905 A complete subspace of a n...
csschl 23906 A complete subspace of a c...
cmslsschl 23907 A complete linear subspace...
chlcsschl 23908 A closed subspace of a sub...
retopn 23909 The topology of the real n...
recms 23910 The real numbers form a co...
reust 23911 The Uniform structure of t...
recusp 23912 The real numbers form a co...
rrxval 23917 Value of the generalized E...
rrxbase 23918 The base of the generalize...
rrxprds 23919 Expand the definition of t...
rrxip 23920 The inner product of the g...
rrxnm 23921 The norm of the generalize...
rrxcph 23922 Generalized Euclidean real...
rrxds 23923 The distance over generali...
rrxvsca 23924 The scalar product over ge...
rrxplusgvscavalb 23925 The result of the addition...
rrxsca 23926 The field of real numbers ...
rrx0 23927 The zero ("origin") in a g...
rrx0el 23928 The zero ("origin") in a g...
csbren 23929 Cauchy-Schwarz-Bunjakovsky...
trirn 23930 Triangle inequality in R^n...
rrxf 23931 Euclidean vectors as funct...
rrxfsupp 23932 Euclidean vectors are of f...
rrxsuppss 23933 Support of Euclidean vecto...
rrxmvallem 23934 Support of the function us...
rrxmval 23935 The value of the Euclidean...
rrxmfval 23936 The value of the Euclidean...
rrxmetlem 23937 Lemma for ~ rrxmet . (Con...
rrxmet 23938 Euclidean space is a metri...
rrxdstprj1 23939 The distance between two p...
rrxbasefi 23940 The base of the generalize...
rrxdsfi 23941 The distance over generali...
rrxmetfi 23942 Euclidean space is a metri...
rrxdsfival 23943 The value of the Euclidean...
ehlval 23944 Value of the Euclidean spa...
ehlbase 23945 The base of the Euclidean ...
ehl0base 23946 The base of the Euclidean ...
ehl0 23947 The Euclidean space of dim...
ehleudis 23948 The Euclidean distance fun...
ehleudisval 23949 The value of the Euclidean...
ehl1eudis 23950 The Euclidean distance fun...
ehl1eudisval 23951 The value of the Euclidean...
ehl2eudis 23952 The Euclidean distance fun...
ehl2eudisval 23953 The value of the Euclidean...
minveclem1 23954 Lemma for ~ minvec . The ...
minveclem4c 23955 Lemma for ~ minvec . The ...
minveclem2 23956 Lemma for ~ minvec . Any ...
minveclem3a 23957 Lemma for ~ minvec . ` D `...
minveclem3b 23958 Lemma for ~ minvec . The ...
minveclem3 23959 Lemma for ~ minvec . The ...
minveclem4a 23960 Lemma for ~ minvec . ` F `...
minveclem4b 23961 Lemma for ~ minvec . The ...
minveclem4 23962 Lemma for ~ minvec . The ...
minveclem5 23963 Lemma for ~ minvec . Disc...
minveclem6 23964 Lemma for ~ minvec . Any ...
minveclem7 23965 Lemma for ~ minvec . Sinc...
minvec 23966 Minimizing vector theorem,...
pjthlem1 23967 Lemma for ~ pjth . (Contr...
pjthlem2 23968 Lemma for ~ pjth . (Contr...
pjth 23969 Projection Theorem: Any H...
pjth2 23970 Projection Theorem with ab...
cldcss 23971 Corollary of the Projectio...
cldcss2 23972 Corollary of the Projectio...
hlhil 23973 Corollary of the Projectio...
mulcncf 23974 The multiplication of two ...
divcncf 23975 The quotient of two contin...
pmltpclem1 23976 Lemma for ~ pmltpc . (Con...
pmltpclem2 23977 Lemma for ~ pmltpc . (Con...
pmltpc 23978 Any function on the reals ...
ivthlem1 23979 Lemma for ~ ivth . The se...
ivthlem2 23980 Lemma for ~ ivth . Show t...
ivthlem3 23981 Lemma for ~ ivth , the int...
ivth 23982 The intermediate value the...
ivth2 23983 The intermediate value the...
ivthle 23984 The intermediate value the...
ivthle2 23985 The intermediate value the...
ivthicc 23986 The interval between any t...
evthicc 23987 Specialization of the Extr...
evthicc2 23988 Combine ~ ivthicc with ~ e...
cniccbdd 23989 A continuous function on a...
ovolfcl 23994 Closure for the interval e...
ovolfioo 23995 Unpack the interval coveri...
ovolficc 23996 Unpack the interval coveri...
ovolficcss 23997 Any (closed) interval cove...
ovolfsval 23998 The value of the interval ...
ovolfsf 23999 Closure for the interval l...
ovolsf 24000 Closure for the partial su...
ovolval 24001 The value of the outer mea...
elovolmlem 24002 Lemma for ~ elovolm and re...
elovolm 24003 Elementhood in the set ` M...
elovolmr 24004 Sufficient condition for e...
ovolmge0 24005 The set ` M ` is composed ...
ovolcl 24006 The volume of a set is an ...
ovollb 24007 The outer volume is a lowe...
ovolgelb 24008 The outer volume is the gr...
ovolge0 24009 The volume of a set is alw...
ovolf 24010 The domain and range of th...
ovollecl 24011 If an outer volume is boun...
ovolsslem 24012 Lemma for ~ ovolss . (Con...
ovolss 24013 The volume of a set is mon...
ovolsscl 24014 If a set is contained in a...
ovolssnul 24015 A subset of a nullset is n...
ovollb2lem 24016 Lemma for ~ ovollb2 . (Co...
ovollb2 24017 It is often more convenien...
ovolctb 24018 The volume of a denumerabl...
ovolq 24019 The rational numbers have ...
ovolctb2 24020 The volume of a countable ...
ovol0 24021 The empty set has 0 outer ...
ovolfi 24022 A finite set has 0 outer L...
ovolsn 24023 A singleton has 0 outer Le...
ovolunlem1a 24024 Lemma for ~ ovolun . (Con...
ovolunlem1 24025 Lemma for ~ ovolun . (Con...
ovolunlem2 24026 Lemma for ~ ovolun . (Con...
ovolun 24027 The Lebesgue outer measure...
ovolunnul 24028 Adding a nullset does not ...
ovolfiniun 24029 The Lebesgue outer measure...
ovoliunlem1 24030 Lemma for ~ ovoliun . (Co...
ovoliunlem2 24031 Lemma for ~ ovoliun . (Co...
ovoliunlem3 24032 Lemma for ~ ovoliun . (Co...
ovoliun 24033 The Lebesgue outer measure...
ovoliun2 24034 The Lebesgue outer measure...
ovoliunnul 24035 A countable union of nulls...
shft2rab 24036 If ` B ` is a shift of ` A...
ovolshftlem1 24037 Lemma for ~ ovolshft . (C...
ovolshftlem2 24038 Lemma for ~ ovolshft . (C...
ovolshft 24039 The Lebesgue outer measure...
sca2rab 24040 If ` B ` is a scale of ` A...
ovolscalem1 24041 Lemma for ~ ovolsca . (Co...
ovolscalem2 24042 Lemma for ~ ovolshft . (C...
ovolsca 24043 The Lebesgue outer measure...
ovolicc1 24044 The measure of a closed in...
ovolicc2lem1 24045 Lemma for ~ ovolicc2 . (C...
ovolicc2lem2 24046 Lemma for ~ ovolicc2 . (C...
ovolicc2lem3 24047 Lemma for ~ ovolicc2 . (C...
ovolicc2lem4 24048 Lemma for ~ ovolicc2 . (C...
ovolicc2lem5 24049 Lemma for ~ ovolicc2 . (C...
ovolicc2 24050 The measure of a closed in...
ovolicc 24051 The measure of a closed in...
ovolicopnf 24052 The measure of a right-unb...
ovolre 24053 The measure of the real nu...
ismbl 24054 The predicate " ` A ` is L...
ismbl2 24055 From ~ ovolun , it suffice...
volres 24056 A self-referencing abbrevi...
volf 24057 The domain and range of th...
mblvol 24058 The volume of a measurable...
mblss 24059 A measurable set is a subs...
mblsplit 24060 The defining property of m...
volss 24061 The Lebesgue measure is mo...
cmmbl 24062 The complement of a measur...
nulmbl 24063 A nullset is measurable. ...
nulmbl2 24064 A set of outer measure zer...
unmbl 24065 A union of measurable sets...
shftmbl 24066 A shift of a measurable se...
0mbl 24067 The empty set is measurabl...
rembl 24068 The set of all real number...
unidmvol 24069 The union of the Lebesgue ...
inmbl 24070 An intersection of measura...
difmbl 24071 A difference of measurable...
finiunmbl 24072 A finite union of measurab...
volun 24073 The Lebesgue measure funct...
volinun 24074 Addition of non-disjoint s...
volfiniun 24075 The volume of a disjoint f...
iundisj 24076 Rewrite a countable union ...
iundisj2 24077 A disjoint union is disjoi...
voliunlem1 24078 Lemma for ~ voliun . (Con...
voliunlem2 24079 Lemma for ~ voliun . (Con...
voliunlem3 24080 Lemma for ~ voliun . (Con...
iunmbl 24081 The measurable sets are cl...
voliun 24082 The Lebesgue measure funct...
volsuplem 24083 Lemma for ~ volsup . (Con...
volsup 24084 The volume of the limit of...
iunmbl2 24085 The measurable sets are cl...
ioombl1lem1 24086 Lemma for ~ ioombl1 . (Co...
ioombl1lem2 24087 Lemma for ~ ioombl1 . (Co...
ioombl1lem3 24088 Lemma for ~ ioombl1 . (Co...
ioombl1lem4 24089 Lemma for ~ ioombl1 . (Co...
ioombl1 24090 An open right-unbounded in...
icombl1 24091 A closed unbounded-above i...
icombl 24092 A closed-below, open-above...
ioombl 24093 An open real interval is m...
iccmbl 24094 A closed real interval is ...
iccvolcl 24095 A closed real interval has...
ovolioo 24096 The measure of an open int...
volioo 24097 The measure of an open int...
ioovolcl 24098 An open real interval has ...
ovolfs2 24099 Alternative expression for...
ioorcl2 24100 An open interval with fini...
ioorf 24101 Define a function from ope...
ioorval 24102 Define a function from ope...
ioorinv2 24103 The function ` F ` is an "...
ioorinv 24104 The function ` F ` is an "...
ioorcl 24105 The function ` F ` does no...
uniiccdif 24106 A union of closed interval...
uniioovol 24107 A disjoint union of open i...
uniiccvol 24108 An almost-disjoint union o...
uniioombllem1 24109 Lemma for ~ uniioombl . (...
uniioombllem2a 24110 Lemma for ~ uniioombl . (...
uniioombllem2 24111 Lemma for ~ uniioombl . (...
uniioombllem3a 24112 Lemma for ~ uniioombl . (...
uniioombllem3 24113 Lemma for ~ uniioombl . (...
uniioombllem4 24114 Lemma for ~ uniioombl . (...
uniioombllem5 24115 Lemma for ~ uniioombl . (...
uniioombllem6 24116 Lemma for ~ uniioombl . (...
uniioombl 24117 A disjoint union of open i...
uniiccmbl 24118 An almost-disjoint union o...
dyadf 24119 The function ` F ` returns...
dyadval 24120 Value of the dyadic ration...
dyadovol 24121 Volume of a dyadic rationa...
dyadss 24122 Two closed dyadic rational...
dyaddisjlem 24123 Lemma for ~ dyaddisj . (C...
dyaddisj 24124 Two closed dyadic rational...
dyadmaxlem 24125 Lemma for ~ dyadmax . (Co...
dyadmax 24126 Any nonempty set of dyadic...
dyadmbllem 24127 Lemma for ~ dyadmbl . (Co...
dyadmbl 24128 Any union of dyadic ration...
opnmbllem 24129 Lemma for ~ opnmbl . (Con...
opnmbl 24130 All open sets are measurab...
opnmblALT 24131 All open sets are measurab...
subopnmbl 24132 Sets which are open in a m...
volsup2 24133 The volume of ` A ` is the...
volcn 24134 The function formed by res...
volivth 24135 The Intermediate Value The...
vitalilem1 24136 Lemma for ~ vitali . (Con...
vitalilem2 24137 Lemma for ~ vitali . (Con...
vitalilem3 24138 Lemma for ~ vitali . (Con...
vitalilem4 24139 Lemma for ~ vitali . (Con...
vitalilem5 24140 Lemma for ~ vitali . (Con...
vitali 24141 If the reals can be well-o...
ismbf1 24152 The predicate " ` F ` is a...
mbff 24153 A measurable function is a...
mbfdm 24154 The domain of a measurable...
mbfconstlem 24155 Lemma for ~ mbfconst and r...
ismbf 24156 The predicate " ` F ` is a...
ismbfcn 24157 A complex function is meas...
mbfima 24158 Definitional property of a...
mbfimaicc 24159 The preimage of any closed...
mbfimasn 24160 The preimage of a point un...
mbfconst 24161 A constant function is mea...
mbf0 24162 The empty function is meas...
mbfid 24163 The identity function is m...
mbfmptcl 24164 Lemma for the ` MblFn ` pr...
mbfdm2 24165 The domain of a measurable...
ismbfcn2 24166 A complex function is meas...
ismbfd 24167 Deduction to prove measura...
ismbf2d 24168 Deduction to prove measura...
mbfeqalem1 24169 Lemma for ~ mbfeqalem2 . ...
mbfeqalem2 24170 Lemma for ~ mbfeqa . (Con...
mbfeqa 24171 If two functions are equal...
mbfres 24172 The restriction of a measu...
mbfres2 24173 Measurability of a piecewi...
mbfss 24174 Change the domain of a mea...
mbfmulc2lem 24175 Multiplication by a consta...
mbfmulc2re 24176 Multiplication by a consta...
mbfmax 24177 The maximum of two functio...
mbfneg 24178 The negative of a measurab...
mbfpos 24179 The positive part of a mea...
mbfposr 24180 Converse to ~ mbfpos . (C...
mbfposb 24181 A function is measurable i...
ismbf3d 24182 Simplified form of ~ ismbf...
mbfimaopnlem 24183 Lemma for ~ mbfimaopn . (...
mbfimaopn 24184 The preimage of any open s...
mbfimaopn2 24185 The preimage of any set op...
cncombf 24186 The composition of a conti...
cnmbf 24187 A continuous function is m...
mbfaddlem 24188 The sum of two measurable ...
mbfadd 24189 The sum of two measurable ...
mbfsub 24190 The difference of two meas...
mbfmulc2 24191 A complex constant times a...
mbfsup 24192 The supremum of a sequence...
mbfinf 24193 The infimum of a sequence ...
mbflimsup 24194 The limit supremum of a se...
mbflimlem 24195 The pointwise limit of a s...
mbflim 24196 The pointwise limit of a s...
0pval 24199 The zero function evaluate...
0plef 24200 Two ways to say that the f...
0pledm 24201 Adjust the domain of the l...
isi1f 24202 The predicate " ` F ` is a...
i1fmbf 24203 Simple functions are measu...
i1ff 24204 A simple function is a fun...
i1frn 24205 A simple function has fini...
i1fima 24206 Any preimage of a simple f...
i1fima2 24207 Any preimage of a simple f...
i1fima2sn 24208 Preimage of a singleton. ...
i1fd 24209 A simplified set of assump...
i1f0rn 24210 Any simple function takes ...
itg1val 24211 The value of the integral ...
itg1val2 24212 The value of the integral ...
itg1cl 24213 Closure of the integral on...
itg1ge0 24214 Closure of the integral on...
i1f0 24215 The zero function is simpl...
itg10 24216 The zero function has zero...
i1f1lem 24217 Lemma for ~ i1f1 and ~ itg...
i1f1 24218 Base case simple functions...
itg11 24219 The integral of an indicat...
itg1addlem1 24220 Decompose a preimage, whic...
i1faddlem 24221 Decompose the preimage of ...
i1fmullem 24222 Decompose the preimage of ...
i1fadd 24223 The sum of two simple func...
i1fmul 24224 The pointwise product of t...
itg1addlem2 24225 Lemma for ~ itg1add . The...
itg1addlem3 24226 Lemma for ~ itg1add . (Co...
itg1addlem4 24227 Lemma for itg1add . (Cont...
itg1addlem5 24228 Lemma for itg1add . (Cont...
itg1add 24229 The integral of a sum of s...
i1fmulclem 24230 Decompose the preimage of ...
i1fmulc 24231 A nonnegative constant tim...
itg1mulc 24232 The integral of a constant...
i1fres 24233 The "restriction" of a sim...
i1fpos 24234 The positive part of a sim...
i1fposd 24235 Deduction form of ~ i1fpos...
i1fsub 24236 The difference of two simp...
itg1sub 24237 The integral of a differen...
itg10a 24238 The integral of a simple f...
itg1ge0a 24239 The integral of an almost ...
itg1lea 24240 Approximate version of ~ i...
itg1le 24241 If one simple function dom...
itg1climres 24242 Restricting the simple fun...
mbfi1fseqlem1 24243 Lemma for ~ mbfi1fseq . (...
mbfi1fseqlem2 24244 Lemma for ~ mbfi1fseq . (...
mbfi1fseqlem3 24245 Lemma for ~ mbfi1fseq . (...
mbfi1fseqlem4 24246 Lemma for ~ mbfi1fseq . T...
mbfi1fseqlem5 24247 Lemma for ~ mbfi1fseq . V...
mbfi1fseqlem6 24248 Lemma for ~ mbfi1fseq . V...
mbfi1fseq 24249 A characterization of meas...
mbfi1flimlem 24250 Lemma for ~ mbfi1flim . (...
mbfi1flim 24251 Any real measurable functi...
mbfmullem2 24252 Lemma for ~ mbfmul . (Con...
mbfmullem 24253 Lemma for ~ mbfmul . (Con...
mbfmul 24254 The product of two measura...
itg2lcl 24255 The set of lower sums is a...
itg2val 24256 Value of the integral on n...
itg2l 24257 Elementhood in the set ` L...
itg2lr 24258 Sufficient condition for e...
xrge0f 24259 A real function is a nonne...
itg2cl 24260 The integral of a nonnegat...
itg2ub 24261 The integral of a nonnegat...
itg2leub 24262 Any upper bound on the int...
itg2ge0 24263 The integral of a nonnegat...
itg2itg1 24264 The integral of a nonnegat...
itg20 24265 The integral of the zero f...
itg2lecl 24266 If an ` S.2 ` integral is ...
itg2le 24267 If one function dominates ...
itg2const 24268 Integral of a constant fun...
itg2const2 24269 When the base set of a con...
itg2seq 24270 Definitional property of t...
itg2uba 24271 Approximate version of ~ i...
itg2lea 24272 Approximate version of ~ i...
itg2eqa 24273 Approximate equality of in...
itg2mulclem 24274 Lemma for ~ itg2mulc . (C...
itg2mulc 24275 The integral of a nonnegat...
itg2splitlem 24276 Lemma for ~ itg2split . (...
itg2split 24277 The ` S.2 ` integral split...
itg2monolem1 24278 Lemma for ~ itg2mono . We...
itg2monolem2 24279 Lemma for ~ itg2mono . (C...
itg2monolem3 24280 Lemma for ~ itg2mono . (C...
itg2mono 24281 The Monotone Convergence T...
itg2i1fseqle 24282 Subject to the conditions ...
itg2i1fseq 24283 Subject to the conditions ...
itg2i1fseq2 24284 In an extension to the res...
itg2i1fseq3 24285 Special case of ~ itg2i1fs...
itg2addlem 24286 Lemma for ~ itg2add . (Co...
itg2add 24287 The ` S.2 ` integral is li...
itg2gt0 24288 If the function ` F ` is s...
itg2cnlem1 24289 Lemma for ~ itgcn . (Cont...
itg2cnlem2 24290 Lemma for ~ itgcn . (Cont...
itg2cn 24291 A sort of absolute continu...
ibllem 24292 Conditioned equality theor...
isibl 24293 The predicate " ` F ` is i...
isibl2 24294 The predicate " ` F ` is i...
iblmbf 24295 An integrable function is ...
iblitg 24296 If a function is integrabl...
dfitg 24297 Evaluate the class substit...
itgex 24298 An integral is a set. (Co...
itgeq1f 24299 Equality theorem for an in...
itgeq1 24300 Equality theorem for an in...
nfitg1 24301 Bound-variable hypothesis ...
nfitg 24302 Bound-variable hypothesis ...
cbvitg 24303 Change bound variable in a...
cbvitgv 24304 Change bound variable in a...
itgeq2 24305 Equality theorem for an in...
itgresr 24306 The domain of an integral ...
itg0 24307 The integral of anything o...
itgz 24308 The integral of zero on an...
itgeq2dv 24309 Equality theorem for an in...
itgmpt 24310 Change bound variable in a...
itgcl 24311 The integral of an integra...
itgvallem 24312 Substitution lemma. (Cont...
itgvallem3 24313 Lemma for ~ itgposval and ...
ibl0 24314 The zero function is integ...
iblcnlem1 24315 Lemma for ~ iblcnlem . (C...
iblcnlem 24316 Expand out the forall in ~...
itgcnlem 24317 Expand out the sum in ~ df...
iblrelem 24318 Integrability of a real fu...
iblposlem 24319 Lemma for ~ iblpos . (Con...
iblpos 24320 Integrability of a nonnega...
iblre 24321 Integrability of a real fu...
itgrevallem1 24322 Lemma for ~ itgposval and ...
itgposval 24323 The integral of a nonnegat...
itgreval 24324 Decompose the integral of ...
itgrecl 24325 Real closure of an integra...
iblcn 24326 Integrability of a complex...
itgcnval 24327 Decompose the integral of ...
itgre 24328 Real part of an integral. ...
itgim 24329 Imaginary part of an integ...
iblneg 24330 The negative of an integra...
itgneg 24331 Negation of an integral. ...
iblss 24332 A subset of an integrable ...
iblss2 24333 Change the domain of an in...
itgitg2 24334 Transfer an integral using...
i1fibl 24335 A simple function is integ...
itgitg1 24336 Transfer an integral using...
itgle 24337 Monotonicity of an integra...
itgge0 24338 The integral of a positive...
itgss 24339 Expand the set of an integ...
itgss2 24340 Expand the set of an integ...
itgeqa 24341 Approximate equality of in...
itgss3 24342 Expand the set of an integ...
itgioo 24343 Equality of integrals on o...
itgless 24344 Expand the integral of a n...
iblconst 24345 A constant function is int...
itgconst 24346 Integral of a constant fun...
ibladdlem 24347 Lemma for ~ ibladd . (Con...
ibladd 24348 Add two integrals over the...
iblsub 24349 Subtract two integrals ove...
itgaddlem1 24350 Lemma for ~ itgadd . (Con...
itgaddlem2 24351 Lemma for ~ itgadd . (Con...
itgadd 24352 Add two integrals over the...
itgsub 24353 Subtract two integrals ove...
itgfsum 24354 Take a finite sum of integ...
iblabslem 24355 Lemma for ~ iblabs . (Con...
iblabs 24356 The absolute value of an i...
iblabsr 24357 A measurable function is i...
iblmulc2 24358 Multiply an integral by a ...
itgmulc2lem1 24359 Lemma for ~ itgmulc2 : pos...
itgmulc2lem2 24360 Lemma for ~ itgmulc2 : rea...
itgmulc2 24361 Multiply an integral by a ...
itgabs 24362 The triangle inequality fo...
itgsplit 24363 The ` S. ` integral splits...
itgspliticc 24364 The ` S. ` integral splits...
itgsplitioo 24365 The ` S. ` integral splits...
bddmulibl 24366 A bounded function times a...
bddibl 24367 A bounded function is inte...
cniccibl 24368 A continuous function on a...
itggt0 24369 The integral of a strictly...
itgcn 24370 Transfer ~ itg2cn to the f...
ditgeq1 24373 Equality theorem for the d...
ditgeq2 24374 Equality theorem for the d...
ditgeq3 24375 Equality theorem for the d...
ditgeq3dv 24376 Equality theorem for the d...
ditgex 24377 A directed integral is a s...
ditg0 24378 Value of the directed inte...
cbvditg 24379 Change bound variable in a...
cbvditgv 24380 Change bound variable in a...
ditgpos 24381 Value of the directed inte...
ditgneg 24382 Value of the directed inte...
ditgcl 24383 Closure of a directed inte...
ditgswap 24384 Reverse a directed integra...
ditgsplitlem 24385 Lemma for ~ ditgsplit . (...
ditgsplit 24386 This theorem is the raison...
reldv 24395 The derivative function is...
limcvallem 24396 Lemma for ~ ellimc . (Con...
limcfval 24397 Value and set bounds on th...
ellimc 24398 Value of the limit predica...
limcrcl 24399 Reverse closure for the li...
limccl 24400 Closure of the limit opera...
limcdif 24401 It suffices to consider fu...
ellimc2 24402 Write the definition of a ...
limcnlp 24403 If ` B ` is not a limit po...
ellimc3 24404 Write the epsilon-delta de...
limcflflem 24405 Lemma for ~ limcflf . (Co...
limcflf 24406 The limit operator can be ...
limcmo 24407 If ` B ` is a limit point ...
limcmpt 24408 Express the limit operator...
limcmpt2 24409 Express the limit operator...
limcresi 24410 Any limit of ` F ` is also...
limcres 24411 If ` B ` is an interior po...
cnplimc 24412 A function is continuous a...
cnlimc 24413 ` F ` is a continuous func...
cnlimci 24414 If ` F ` is a continuous f...
cnmptlimc 24415 If ` F ` is a continuous f...
limccnp 24416 If the limit of ` F ` at `...
limccnp2 24417 The image of a convergent ...
limcco 24418 Composition of two limits....
limciun 24419 A point is a limit of ` F ...
limcun 24420 A point is a limit of ` F ...
dvlem 24421 Closure for a difference q...
dvfval 24422 Value and set bounds on th...
eldv 24423 The differentiable predica...
dvcl 24424 The derivative function ta...
dvbssntr 24425 The set of differentiable ...
dvbss 24426 The set of differentiable ...
dvbsss 24427 The set of differentiable ...
perfdvf 24428 The derivative is a functi...
recnprss 24429 Both ` RR ` and ` CC ` are...
recnperf 24430 Both ` RR ` and ` CC ` are...
dvfg 24431 Explicitly write out the f...
dvf 24432 The derivative is a functi...
dvfcn 24433 The derivative is a functi...
dvreslem 24434 Lemma for ~ dvres . (Cont...
dvres2lem 24435 Lemma for ~ dvres2 . (Con...
dvres 24436 Restriction of a derivativ...
dvres2 24437 Restriction of the base se...
dvres3 24438 Restriction of a complex d...
dvres3a 24439 Restriction of a complex d...
dvidlem 24440 Lemma for ~ dvid and ~ dvc...
dvconst 24441 Derivative of a constant f...
dvid 24442 Derivative of the identity...
dvcnp 24443 The difference quotient is...
dvcnp2 24444 A function is continuous a...
dvcn 24445 A differentiable function ...
dvnfval 24446 Value of the iterated deri...
dvnff 24447 The iterated derivative is...
dvn0 24448 Zero times iterated deriva...
dvnp1 24449 Successor iterated derivat...
dvn1 24450 One times iterated derivat...
dvnf 24451 The N-times derivative is ...
dvnbss 24452 The set of N-times differe...
dvnadd 24453 The ` N ` -th derivative o...
dvn2bss 24454 An N-times differentiable ...
dvnres 24455 Multiple derivative versio...
cpnfval 24456 Condition for n-times cont...
fncpn 24457 The ` C^n ` object is a fu...
elcpn 24458 Condition for n-times cont...
cpnord 24459 ` C^n ` conditions are ord...
cpncn 24460 A ` C^n ` function is cont...
cpnres 24461 The restriction of a ` C^n...
dvaddbr 24462 The sum rule for derivativ...
dvmulbr 24463 The product rule for deriv...
dvadd 24464 The sum rule for derivativ...
dvmul 24465 The product rule for deriv...
dvaddf 24466 The sum rule for everywher...
dvmulf 24467 The product rule for every...
dvcmul 24468 The product rule when one ...
dvcmulf 24469 The product rule when one ...
dvcobr 24470 The chain rule for derivat...
dvco 24471 The chain rule for derivat...
dvcof 24472 The chain rule for everywh...
dvcjbr 24473 The derivative of the conj...
dvcj 24474 The derivative of the conj...
dvfre 24475 The derivative of a real f...
dvnfre 24476 The ` N ` -th derivative o...
dvexp 24477 Derivative of a power func...
dvexp2 24478 Derivative of an exponenti...
dvrec 24479 Derivative of the reciproc...
dvmptres3 24480 Function-builder for deriv...
dvmptid 24481 Function-builder for deriv...
dvmptc 24482 Function-builder for deriv...
dvmptcl 24483 Closure lemma for ~ dvmptc...
dvmptadd 24484 Function-builder for deriv...
dvmptmul 24485 Function-builder for deriv...
dvmptres2 24486 Function-builder for deriv...
dvmptres 24487 Function-builder for deriv...
dvmptcmul 24488 Function-builder for deriv...
dvmptdivc 24489 Function-builder for deriv...
dvmptneg 24490 Function-builder for deriv...
dvmptsub 24491 Function-builder for deriv...
dvmptcj 24492 Function-builder for deriv...
dvmptre 24493 Function-builder for deriv...
dvmptim 24494 Function-builder for deriv...
dvmptntr 24495 Function-builder for deriv...
dvmptco 24496 Function-builder for deriv...
dvrecg 24497 Derivative of the reciproc...
dvmptdiv 24498 Function-builder for deriv...
dvmptfsum 24499 Function-builder for deriv...
dvcnvlem 24500 Lemma for ~ dvcnvre . (Co...
dvcnv 24501 A weak version of ~ dvcnvr...
dvexp3 24502 Derivative of an exponenti...
dveflem 24503 Derivative of the exponent...
dvef 24504 Derivative of the exponent...
dvsincos 24505 Derivative of the sine and...
dvsin 24506 Derivative of the sine fun...
dvcos 24507 Derivative of the cosine f...
dvferm1lem 24508 Lemma for ~ dvferm . (Con...
dvferm1 24509 One-sided version of ~ dvf...
dvferm2lem 24510 Lemma for ~ dvferm . (Con...
dvferm2 24511 One-sided version of ~ dvf...
dvferm 24512 Fermat's theorem on statio...
rollelem 24513 Lemma for ~ rolle . (Cont...
rolle 24514 Rolle's theorem. If ` F `...
cmvth 24515 Cauchy's Mean Value Theore...
mvth 24516 The Mean Value Theorem. I...
dvlip 24517 A function with derivative...
dvlipcn 24518 A complex function with de...
dvlip2 24519 Combine the results of ~ d...
c1liplem1 24520 Lemma for ~ c1lip1 . (Con...
c1lip1 24521 C^1 functions are Lipschit...
c1lip2 24522 C^1 functions are Lipschit...
c1lip3 24523 C^1 functions are Lipschit...
dveq0 24524 If a continuous function h...
dv11cn 24525 Two functions defined on a...
dvgt0lem1 24526 Lemma for ~ dvgt0 and ~ dv...
dvgt0lem2 24527 Lemma for ~ dvgt0 and ~ dv...
dvgt0 24528 A function on a closed int...
dvlt0 24529 A function on a closed int...
dvge0 24530 A function on a closed int...
dvle 24531 If ` A ( x ) , C ( x ) ` a...
dvivthlem1 24532 Lemma for ~ dvivth . (Con...
dvivthlem2 24533 Lemma for ~ dvivth . (Con...
dvivth 24534 Darboux' theorem, or the i...
dvne0 24535 A function on a closed int...
dvne0f1 24536 A function on a closed int...
lhop1lem 24537 Lemma for ~ lhop1 . (Cont...
lhop1 24538 L'Hôpital's Rule for...
lhop2 24539 L'Hôpital's Rule for...
lhop 24540 L'Hôpital's Rule. I...
dvcnvrelem1 24541 Lemma for ~ dvcnvre . (Co...
dvcnvrelem2 24542 Lemma for ~ dvcnvre . (Co...
dvcnvre 24543 The derivative rule for in...
dvcvx 24544 A real function with stric...
dvfsumle 24545 Compare a finite sum to an...
dvfsumge 24546 Compare a finite sum to an...
dvfsumabs 24547 Compare a finite sum to an...
dvmptrecl 24548 Real closure of a derivati...
dvfsumrlimf 24549 Lemma for ~ dvfsumrlim . ...
dvfsumlem1 24550 Lemma for ~ dvfsumrlim . ...
dvfsumlem2 24551 Lemma for ~ dvfsumrlim . ...
dvfsumlem3 24552 Lemma for ~ dvfsumrlim . ...
dvfsumlem4 24553 Lemma for ~ dvfsumrlim . ...
dvfsumrlimge0 24554 Lemma for ~ dvfsumrlim . ...
dvfsumrlim 24555 Compare a finite sum to an...
dvfsumrlim2 24556 Compare a finite sum to an...
dvfsumrlim3 24557 Conjoin the statements of ...
dvfsum2 24558 The reverse of ~ dvfsumrli...
ftc1lem1 24559 Lemma for ~ ftc1a and ~ ft...
ftc1lem2 24560 Lemma for ~ ftc1 . (Contr...
ftc1a 24561 The Fundamental Theorem of...
ftc1lem3 24562 Lemma for ~ ftc1 . (Contr...
ftc1lem4 24563 Lemma for ~ ftc1 . (Contr...
ftc1lem5 24564 Lemma for ~ ftc1 . (Contr...
ftc1lem6 24565 Lemma for ~ ftc1 . (Contr...
ftc1 24566 The Fundamental Theorem of...
ftc1cn 24567 Strengthen the assumptions...
ftc2 24568 The Fundamental Theorem of...
ftc2ditglem 24569 Lemma for ~ ftc2ditg . (C...
ftc2ditg 24570 Directed integral analogue...
itgparts 24571 Integration by parts. If ...
itgsubstlem 24572 Lemma for ~ itgsubst . (C...
itgsubst 24573 Integration by ` u ` -subs...
reldmmdeg 24578 Multivariate degree is a b...
tdeglem1 24579 Functionality of the total...
tdeglem3 24580 Additivity of the total de...
tdeglem4 24581 There is only one multi-in...
tdeglem2 24582 Simplification of total de...
mdegfval 24583 Value of the multivariate ...
mdegval 24584 Value of the multivariate ...
mdegleb 24585 Property of being of limit...
mdeglt 24586 If there is an upper limit...
mdegldg 24587 A nonzero polynomial has s...
mdegxrcl 24588 Closure of polynomial degr...
mdegxrf 24589 Functionality of polynomia...
mdegcl 24590 Sharp closure for multivar...
mdeg0 24591 Degree of the zero polynom...
mdegnn0cl 24592 Degree of a nonzero polyno...
degltlem1 24593 Theorem on arithmetic of e...
degltp1le 24594 Theorem on arithmetic of e...
mdegaddle 24595 The degree of a sum is at ...
mdegvscale 24596 The degree of a scalar mul...
mdegvsca 24597 The degree of a scalar mul...
mdegle0 24598 A polynomial has nonpositi...
mdegmullem 24599 Lemma for ~ mdegmulle2 . ...
mdegmulle2 24600 The multivariate degree of...
deg1fval 24601 Relate univariate polynomi...
deg1xrf 24602 Functionality of univariat...
deg1xrcl 24603 Closure of univariate poly...
deg1cl 24604 Sharp closure of univariat...
mdegpropd 24605 Property deduction for pol...
deg1fvi 24606 Univariate polynomial degr...
deg1propd 24607 Property deduction for pol...
deg1z 24608 Degree of the zero univari...
deg1nn0cl 24609 Degree of a nonzero univar...
deg1n0ima 24610 Degree image of a set of p...
deg1nn0clb 24611 A polynomial is nonzero if...
deg1lt0 24612 A polynomial is zero iff i...
deg1ldg 24613 A nonzero univariate polyn...
deg1ldgn 24614 An index at which a polyno...
deg1ldgdomn 24615 A nonzero univariate polyn...
deg1leb 24616 Property of being of limit...
deg1val 24617 Value of the univariate de...
deg1lt 24618 If the degree of a univari...
deg1ge 24619 Conversely, a nonzero coef...
coe1mul3 24620 The coefficient vector of ...
coe1mul4 24621 Value of the "leading" coe...
deg1addle 24622 The degree of a sum is at ...
deg1addle2 24623 If both factors have degre...
deg1add 24624 Exact degree of a sum of t...
deg1vscale 24625 The degree of a scalar tim...
deg1vsca 24626 The degree of a scalar tim...
deg1invg 24627 The degree of the negated ...
deg1suble 24628 The degree of a difference...
deg1sub 24629 Exact degree of a differen...
deg1mulle2 24630 Produce a bound on the pro...
deg1sublt 24631 Subtraction of two polynom...
deg1le0 24632 A polynomial has nonpositi...
deg1sclle 24633 A scalar polynomial has no...
deg1scl 24634 A nonzero scalar polynomia...
deg1mul2 24635 Degree of multiplication o...
deg1mul3 24636 Degree of multiplication o...
deg1mul3le 24637 Degree of multiplication o...
deg1tmle 24638 Limiting degree of a polyn...
deg1tm 24639 Exact degree of a polynomi...
deg1pwle 24640 Limiting degree of a varia...
deg1pw 24641 Exact degree of a variable...
ply1nz 24642 Univariate polynomials ove...
ply1nzb 24643 Univariate polynomials are...
ply1domn 24644 Corollary of ~ deg1mul2 : ...
ply1idom 24645 The ring of univariate pol...
ply1divmo 24656 Uniqueness of a quotient i...
ply1divex 24657 Lemma for ~ ply1divalg : e...
ply1divalg 24658 The division algorithm for...
ply1divalg2 24659 Reverse the order of multi...
uc1pval 24660 Value of the set of unitic...
isuc1p 24661 Being a unitic polynomial....
mon1pval 24662 Value of the set of monic ...
ismon1p 24663 Being a monic polynomial. ...
uc1pcl 24664 Unitic polynomials are pol...
mon1pcl 24665 Monic polynomials are poly...
uc1pn0 24666 Unitic polynomials are not...
mon1pn0 24667 Monic polynomials are not ...
uc1pdeg 24668 Unitic polynomials have no...
uc1pldg 24669 Unitic polynomials have un...
mon1pldg 24670 Unitic polynomials have on...
mon1puc1p 24671 Monic polynomials are unit...
uc1pmon1p 24672 Make a unitic polynomial m...
deg1submon1p 24673 The difference of two moni...
q1pval 24674 Value of the univariate po...
q1peqb 24675 Characterizing property of...
q1pcl 24676 Closure of the quotient by...
r1pval 24677 Value of the polynomial re...
r1pcl 24678 Closure of remainder follo...
r1pdeglt 24679 The remainder has a degree...
r1pid 24680 Express the original polyn...
dvdsq1p 24681 Divisibility in a polynomi...
dvdsr1p 24682 Divisibility in a polynomi...
ply1remlem 24683 A term of the form ` x - N...
ply1rem 24684 The polynomial remainder t...
facth1 24685 The factor theorem and its...
fta1glem1 24686 Lemma for ~ fta1g . (Cont...
fta1glem2 24687 Lemma for ~ fta1g . (Cont...
fta1g 24688 The one-sided fundamental ...
fta1blem 24689 Lemma for ~ fta1b . (Cont...
fta1b 24690 The assumption that ` R ` ...
drnguc1p 24691 Over a division ring, all ...
ig1peu 24692 There is a unique monic po...
ig1pval 24693 Substitutions for the poly...
ig1pval2 24694 Generator of the zero idea...
ig1pval3 24695 Characterizing properties ...
ig1pcl 24696 The monic generator of an ...
ig1pdvds 24697 The monic generator of an ...
ig1prsp 24698 Any ideal of polynomials o...
ply1lpir 24699 The ring of polynomials ov...
ply1pid 24700 The polynomials over a fie...
plyco0 24709 Two ways to say that a fun...
plyval 24710 Value of the polynomial se...
plybss 24711 Reverse closure of the par...
elply 24712 Definition of a polynomial...
elply2 24713 The coefficient function c...
plyun0 24714 The set of polynomials is ...
plyf 24715 The polynomial is a functi...
plyss 24716 The polynomial set functio...
plyssc 24717 Every polynomial ring is c...
elplyr 24718 Sufficient condition for e...
elplyd 24719 Sufficient condition for e...
ply1termlem 24720 Lemma for ~ ply1term . (C...
ply1term 24721 A one-term polynomial. (C...
plypow 24722 A power is a polynomial. ...
plyconst 24723 A constant function is a p...
ne0p 24724 A test to show that a poly...
ply0 24725 The zero function is a pol...
plyid 24726 The identity function is a...
plyeq0lem 24727 Lemma for ~ plyeq0 . If `...
plyeq0 24728 If a polynomial is zero at...
plypf1 24729 Write the set of complex p...
plyaddlem1 24730 Derive the coefficient fun...
plymullem1 24731 Derive the coefficient fun...
plyaddlem 24732 Lemma for ~ plyadd . (Con...
plymullem 24733 Lemma for ~ plymul . (Con...
plyadd 24734 The sum of two polynomials...
plymul 24735 The product of two polynom...
plysub 24736 The difference of two poly...
plyaddcl 24737 The sum of two polynomials...
plymulcl 24738 The product of two polynom...
plysubcl 24739 The difference of two poly...
coeval 24740 Value of the coefficient f...
coeeulem 24741 Lemma for ~ coeeu . (Cont...
coeeu 24742 Uniqueness of the coeffici...
coelem 24743 Lemma for properties of th...
coeeq 24744 If ` A ` satisfies the pro...
dgrval 24745 Value of the degree functi...
dgrlem 24746 Lemma for ~ dgrcl and simi...
coef 24747 The domain and range of th...
coef2 24748 The domain and range of th...
coef3 24749 The domain and range of th...
dgrcl 24750 The degree of any polynomi...
dgrub 24751 If the ` M ` -th coefficie...
dgrub2 24752 All the coefficients above...
dgrlb 24753 If all the coefficients ab...
coeidlem 24754 Lemma for ~ coeid . (Cont...
coeid 24755 Reconstruct a polynomial a...
coeid2 24756 Reconstruct a polynomial a...
coeid3 24757 Reconstruct a polynomial a...
plyco 24758 The composition of two pol...
coeeq2 24759 Compute the coefficient fu...
dgrle 24760 Given an explicit expressi...
dgreq 24761 If the highest term in a p...
0dgr 24762 A constant function has de...
0dgrb 24763 A function has degree zero...
dgrnznn 24764 A nonzero polynomial with ...
coefv0 24765 The result of evaluating a...
coeaddlem 24766 Lemma for ~ coeadd and ~ d...
coemullem 24767 Lemma for ~ coemul and ~ d...
coeadd 24768 The coefficient function o...
coemul 24769 A coefficient of a product...
coe11 24770 The coefficient function i...
coemulhi 24771 The leading coefficient of...
coemulc 24772 The coefficient function i...
coe0 24773 The coefficients of the ze...
coesub 24774 The coefficient function o...
coe1termlem 24775 The coefficient function o...
coe1term 24776 The coefficient function o...
dgr1term 24777 The degree of a monomial. ...
plycn 24778 A polynomial is a continuo...
dgr0 24779 The degree of the zero pol...
coeidp 24780 The coefficients of the id...
dgrid 24781 The degree of the identity...
dgreq0 24782 The leading coefficient of...
dgrlt 24783 Two ways to say that the d...
dgradd 24784 The degree of a sum of pol...
dgradd2 24785 The degree of a sum of pol...
dgrmul2 24786 The degree of a product of...
dgrmul 24787 The degree of a product of...
dgrmulc 24788 Scalar multiplication by a...
dgrsub 24789 The degree of a difference...
dgrcolem1 24790 The degree of a compositio...
dgrcolem2 24791 Lemma for ~ dgrco . (Cont...
dgrco 24792 The degree of a compositio...
plycjlem 24793 Lemma for ~ plycj and ~ co...
plycj 24794 The double conjugation of ...
coecj 24795 Double conjugation of a po...
plyrecj 24796 A polynomial with real coe...
plymul0or 24797 Polynomial multiplication ...
ofmulrt 24798 The set of roots of a prod...
plyreres 24799 Real-coefficient polynomia...
dvply1 24800 Derivative of a polynomial...
dvply2g 24801 The derivative of a polyno...
dvply2 24802 The derivative of a polyno...
dvnply2 24803 Polynomials have polynomia...
dvnply 24804 Polynomials have polynomia...
plycpn 24805 Polynomials are smooth. (...
quotval 24808 Value of the quotient func...
plydivlem1 24809 Lemma for ~ plydivalg . (...
plydivlem2 24810 Lemma for ~ plydivalg . (...
plydivlem3 24811 Lemma for ~ plydivex . Ba...
plydivlem4 24812 Lemma for ~ plydivex . In...
plydivex 24813 Lemma for ~ plydivalg . (...
plydiveu 24814 Lemma for ~ plydivalg . (...
plydivalg 24815 The division algorithm on ...
quotlem 24816 Lemma for properties of th...
quotcl 24817 The quotient of two polyno...
quotcl2 24818 Closure of the quotient fu...
quotdgr 24819 Remainder property of the ...
plyremlem 24820 Closure of a linear factor...
plyrem 24821 The polynomial remainder t...
facth 24822 The factor theorem. If a ...
fta1lem 24823 Lemma for ~ fta1 . (Contr...
fta1 24824 The easy direction of the ...
quotcan 24825 Exact division with a mult...
vieta1lem1 24826 Lemma for ~ vieta1 . (Con...
vieta1lem2 24827 Lemma for ~ vieta1 : induc...
vieta1 24828 The first-order Vieta's fo...
plyexmo 24829 An infinite set of values ...
elaa 24832 Elementhood in the set of ...
aacn 24833 An algebraic number is a c...
aasscn 24834 The algebraic numbers are ...
elqaalem1 24835 Lemma for ~ elqaa . The f...
elqaalem2 24836 Lemma for ~ elqaa . (Cont...
elqaalem3 24837 Lemma for ~ elqaa . (Cont...
elqaa 24838 The set of numbers generat...
qaa 24839 Every rational number is a...
qssaa 24840 The rational numbers are c...
iaa 24841 The imaginary unit is alge...
aareccl 24842 The reciprocal of an algeb...
aacjcl 24843 The conjugate of an algebr...
aannenlem1 24844 Lemma for ~ aannen . (Con...
aannenlem2 24845 Lemma for ~ aannen . (Con...
aannenlem3 24846 The algebraic numbers are ...
aannen 24847 The algebraic numbers are ...
aalioulem1 24848 Lemma for ~ aaliou . An i...
aalioulem2 24849 Lemma for ~ aaliou . (Con...
aalioulem3 24850 Lemma for ~ aaliou . (Con...
aalioulem4 24851 Lemma for ~ aaliou . (Con...
aalioulem5 24852 Lemma for ~ aaliou . (Con...
aalioulem6 24853 Lemma for ~ aaliou . (Con...
aaliou 24854 Liouville's theorem on dio...
geolim3 24855 Geometric series convergen...
aaliou2 24856 Liouville's approximation ...
aaliou2b 24857 Liouville's approximation ...
aaliou3lem1 24858 Lemma for ~ aaliou3 . (Co...
aaliou3lem2 24859 Lemma for ~ aaliou3 . (Co...
aaliou3lem3 24860 Lemma for ~ aaliou3 . (Co...
aaliou3lem8 24861 Lemma for ~ aaliou3 . (Co...
aaliou3lem4 24862 Lemma for ~ aaliou3 . (Co...
aaliou3lem5 24863 Lemma for ~ aaliou3 . (Co...
aaliou3lem6 24864 Lemma for ~ aaliou3 . (Co...
aaliou3lem7 24865 Lemma for ~ aaliou3 . (Co...
aaliou3lem9 24866 Example of a "Liouville nu...
aaliou3 24867 Example of a "Liouville nu...
taylfvallem1 24872 Lemma for ~ taylfval . (C...
taylfvallem 24873 Lemma for ~ taylfval . (C...
taylfval 24874 Define the Taylor polynomi...
eltayl 24875 Value of the Taylor series...
taylf 24876 The Taylor series defines ...
tayl0 24877 The Taylor series is alway...
taylplem1 24878 Lemma for ~ taylpfval and ...
taylplem2 24879 Lemma for ~ taylpfval and ...
taylpfval 24880 Define the Taylor polynomi...
taylpf 24881 The Taylor polynomial is a...
taylpval 24882 Value of the Taylor polyno...
taylply2 24883 The Taylor polynomial is a...
taylply 24884 The Taylor polynomial is a...
dvtaylp 24885 The derivative of the Tayl...
dvntaylp 24886 The ` M ` -th derivative o...
dvntaylp0 24887 The first ` N ` derivative...
taylthlem1 24888 Lemma for ~ taylth . This...
taylthlem2 24889 Lemma for ~ taylth . (Con...
taylth 24890 Taylor's theorem. The Tay...
ulmrel 24893 The uniform limit relation...
ulmscl 24894 Closure of the base set in...
ulmval 24895 Express the predicate: Th...
ulmcl 24896 Closure of a uniform limit...
ulmf 24897 Closure of a uniform limit...
ulmpm 24898 Closure of a uniform limit...
ulmf2 24899 Closure of a uniform limit...
ulm2 24900 Simplify ~ ulmval when ` F...
ulmi 24901 The uniform limit property...
ulmclm 24902 A uniform limit of functio...
ulmres 24903 A sequence of functions co...
ulmshftlem 24904 Lemma for ~ ulmshft . (Co...
ulmshft 24905 A sequence of functions co...
ulm0 24906 Every function converges u...
ulmuni 24907 A sequence of functions un...
ulmdm 24908 Two ways to express that a...
ulmcaulem 24909 Lemma for ~ ulmcau and ~ u...
ulmcau 24910 A sequence of functions co...
ulmcau2 24911 A sequence of functions co...
ulmss 24912 A uniform limit of functio...
ulmbdd 24913 A uniform limit of bounded...
ulmcn 24914 A uniform limit of continu...
ulmdvlem1 24915 Lemma for ~ ulmdv . (Cont...
ulmdvlem2 24916 Lemma for ~ ulmdv . (Cont...
ulmdvlem3 24917 Lemma for ~ ulmdv . (Cont...
ulmdv 24918 If ` F ` is a sequence of ...
mtest 24919 The Weierstrass M-test. I...
mtestbdd 24920 Given the hypotheses of th...
mbfulm 24921 A uniform limit of measura...
iblulm 24922 A uniform limit of integra...
itgulm 24923 A uniform limit of integra...
itgulm2 24924 A uniform limit of integra...
pserval 24925 Value of the function ` G ...
pserval2 24926 Value of the function ` G ...
psergf 24927 The sequence of terms in t...
radcnvlem1 24928 Lemma for ~ radcnvlt1 , ~ ...
radcnvlem2 24929 Lemma for ~ radcnvlt1 , ~ ...
radcnvlem3 24930 Lemma for ~ radcnvlt1 , ~ ...
radcnv0 24931 Zero is always a convergen...
radcnvcl 24932 The radius of convergence ...
radcnvlt1 24933 If ` X ` is within the ope...
radcnvlt2 24934 If ` X ` is within the ope...
radcnvle 24935 If ` X ` is a convergent p...
dvradcnv 24936 The radius of convergence ...
pserulm 24937 If ` S ` is a region conta...
psercn2 24938 Since by ~ pserulm the ser...
psercnlem2 24939 Lemma for ~ psercn . (Con...
psercnlem1 24940 Lemma for ~ psercn . (Con...
psercn 24941 An infinite series converg...
pserdvlem1 24942 Lemma for ~ pserdv . (Con...
pserdvlem2 24943 Lemma for ~ pserdv . (Con...
pserdv 24944 The derivative of a power ...
pserdv2 24945 The derivative of a power ...
abelthlem1 24946 Lemma for ~ abelth . (Con...
abelthlem2 24947 Lemma for ~ abelth . The ...
abelthlem3 24948 Lemma for ~ abelth . (Con...
abelthlem4 24949 Lemma for ~ abelth . (Con...
abelthlem5 24950 Lemma for ~ abelth . (Con...
abelthlem6 24951 Lemma for ~ abelth . (Con...
abelthlem7a 24952 Lemma for ~ abelth . (Con...
abelthlem7 24953 Lemma for ~ abelth . (Con...
abelthlem8 24954 Lemma for ~ abelth . (Con...
abelthlem9 24955 Lemma for ~ abelth . By a...
abelth 24956 Abel's theorem. If the po...
abelth2 24957 Abel's theorem, restricted...
efcn 24958 The exponential function i...
sincn 24959 Sine is continuous. (Cont...
coscn 24960 Cosine is continuous. (Co...
reeff1olem 24961 Lemma for ~ reeff1o . (Co...
reeff1o 24962 The real exponential funct...
reefiso 24963 The exponential function o...
efcvx 24964 The exponential function o...
reefgim 24965 The exponential function i...
pilem1 24966 Lemma for ~ pire , ~ pigt2...
pilem2 24967 Lemma for ~ pire , ~ pigt2...
pilem3 24968 Lemma for ~ pire , ~ pigt2...
pigt2lt4 24969 ` _pi ` is between 2 and 4...
sinpi 24970 The sine of ` _pi ` is 0. ...
pire 24971 ` _pi ` is a real number. ...
picn 24972 ` _pi ` is a complex numbe...
pipos 24973 ` _pi ` is positive. (Con...
pirp 24974 ` _pi ` is a positive real...
negpicn 24975 ` -u _pi ` is a real numbe...
sinhalfpilem 24976 Lemma for ~ sinhalfpi and ...
halfpire 24977 ` _pi / 2 ` is real. (Con...
neghalfpire 24978 ` -u _pi / 2 ` is real. (...
neghalfpirx 24979 ` -u _pi / 2 ` is an exten...
pidiv2halves 24980 Adding ` _pi / 2 ` to itse...
sinhalfpi 24981 The sine of ` _pi / 2 ` is...
coshalfpi 24982 The cosine of ` _pi / 2 ` ...
cosneghalfpi 24983 The cosine of ` -u _pi / 2...
efhalfpi 24984 The exponential of ` _i _p...
cospi 24985 The cosine of ` _pi ` is `...
efipi 24986 The exponential of ` _i x....
eulerid 24987 Euler's identity. (Contri...
sin2pi 24988 The sine of ` 2 _pi ` is 0...
cos2pi 24989 The cosine of ` 2 _pi ` is...
ef2pi 24990 The exponential of ` 2 _pi...
ef2kpi 24991 If ` K ` is an integer, th...
efper 24992 The exponential function i...
sinperlem 24993 Lemma for ~ sinper and ~ c...
sinper 24994 The sine function is perio...
cosper 24995 The cosine function is per...
sin2kpi 24996 If ` K ` is an integer, th...
cos2kpi 24997 If ` K ` is an integer, th...
sin2pim 24998 Sine of a number subtracte...
cos2pim 24999 Cosine of a number subtrac...
sinmpi 25000 Sine of a number less ` _p...
cosmpi 25001 Cosine of a number less ` ...
sinppi 25002 Sine of a number plus ` _p...
cosppi 25003 Cosine of a number plus ` ...
efimpi 25004 The exponential function a...
sinhalfpip 25005 The sine of ` _pi / 2 ` pl...
sinhalfpim 25006 The sine of ` _pi / 2 ` mi...
coshalfpip 25007 The cosine of ` _pi / 2 ` ...
coshalfpim 25008 The cosine of ` _pi / 2 ` ...
ptolemy 25009 Ptolemy's Theorem. This t...
sincosq1lem 25010 Lemma for ~ sincosq1sgn . ...
sincosq1sgn 25011 The signs of the sine and ...
sincosq2sgn 25012 The signs of the sine and ...
sincosq3sgn 25013 The signs of the sine and ...
sincosq4sgn 25014 The signs of the sine and ...
coseq00topi 25015 Location of the zeroes of ...
coseq0negpitopi 25016 Location of the zeroes of ...
tanrpcl 25017 Positive real closure of t...
tangtx 25018 The tangent function is gr...
tanabsge 25019 The tangent function is gr...
sinq12gt0 25020 The sine of a number stric...
sinq12ge0 25021 The sine of a number betwe...
sinq34lt0t 25022 The sine of a number stric...
cosq14gt0 25023 The cosine of a number str...
cosq14ge0 25024 The cosine of a number bet...
sincosq1eq 25025 Complementarity of the sin...
sincos4thpi 25026 The sine and cosine of ` _...
tan4thpi 25027 The tangent of ` _pi / 4 `...
sincos6thpi 25028 The sine and cosine of ` _...
sincos3rdpi 25029 The sine and cosine of ` _...
pigt3 25030 ` _pi ` is greater than 3....
pige3 25031 ` _pi ` is greater than or...
pige3ALT 25032 Alternate proof of ~ pige3...
abssinper 25033 The absolute value of sine...
sinkpi 25034 The sine of an integer mul...
coskpi 25035 The absolute value of the ...
sineq0 25036 A complex number whose sin...
coseq1 25037 A complex number whose cos...
cos02pilt1 25038 Cosine is less than one be...
cosq34lt1 25039 Cosine is less than one in...
efeq1 25040 A complex number whose exp...
cosne0 25041 The cosine function has no...
cosordlem 25042 Lemma for ~ cosord . (Con...
cosord 25043 Cosine is decreasing over ...
cos11 25044 Cosine is one-to-one over ...
sinord 25045 Sine is increasing over th...
recosf1o 25046 The cosine function is a b...
resinf1o 25047 The sine function is a bij...
tanord1 25048 The tangent function is st...
tanord 25049 The tangent function is st...
tanregt0 25050 The real part of the tange...
negpitopissre 25051 The interval ` ( -u _pi (,...
efgh 25052 The exponential function o...
efif1olem1 25053 Lemma for ~ efif1o . (Con...
efif1olem2 25054 Lemma for ~ efif1o . (Con...
efif1olem3 25055 Lemma for ~ efif1o . (Con...
efif1olem4 25056 The exponential function o...
efif1o 25057 The exponential function o...
efifo 25058 The exponential function o...
eff1olem 25059 The exponential function m...
eff1o 25060 The exponential function m...
efabl 25061 The image of a subgroup of...
efsubm 25062 The image of a subgroup of...
circgrp 25063 The circle group ` T ` is ...
circsubm 25064 The circle group ` T ` is ...
logrn 25069 The range of the natural l...
ellogrn 25070 Write out the property ` A...
dflog2 25071 The natural logarithm func...
relogrn 25072 The range of the natural l...
logrncn 25073 The range of the natural l...
eff1o2 25074 The exponential function r...
logf1o 25075 The natural logarithm func...
dfrelog 25076 The natural logarithm func...
relogf1o 25077 The natural logarithm func...
logrncl 25078 Closure of the natural log...
logcl 25079 Closure of the natural log...
logimcl 25080 Closure of the imaginary p...
logcld 25081 The logarithm of a nonzero...
logimcld 25082 The imaginary part of the ...
logimclad 25083 The imaginary part of the ...
abslogimle 25084 The imaginary part of the ...
logrnaddcl 25085 The range of the natural l...
relogcl 25086 Closure of the natural log...
eflog 25087 Relationship between the n...
logeq0im1 25088 If the logarithm of a numb...
logccne0 25089 The logarithm isn't 0 if i...
logne0 25090 Logarithm of a non-1 posit...
reeflog 25091 Relationship between the n...
logef 25092 Relationship between the n...
relogef 25093 Relationship between the n...
logeftb 25094 Relationship between the n...
relogeftb 25095 Relationship between the n...
log1 25096 The natural logarithm of `...
loge 25097 The natural logarithm of `...
logneg 25098 The natural logarithm of a...
logm1 25099 The natural logarithm of n...
lognegb 25100 If a number has imaginary ...
relogoprlem 25101 Lemma for ~ relogmul and ~...
relogmul 25102 The natural logarithm of t...
relogdiv 25103 The natural logarithm of t...
explog 25104 Exponentiation of a nonzer...
reexplog 25105 Exponentiation of a positi...
relogexp 25106 The natural logarithm of p...
relog 25107 Real part of a logarithm. ...
relogiso 25108 The natural logarithm func...
reloggim 25109 The natural logarithm is a...
logltb 25110 The natural logarithm func...
logfac 25111 The logarithm of a factori...
eflogeq 25112 Solve an equation involvin...
logleb 25113 Natural logarithm preserve...
rplogcl 25114 Closure of the logarithm f...
logge0 25115 The logarithm of a number ...
logcj 25116 The natural logarithm dist...
efiarg 25117 The exponential of the "ar...
cosargd 25118 The cosine of the argument...
cosarg0d 25119 The cosine of the argument...
argregt0 25120 Closure of the argument of...
argrege0 25121 Closure of the argument of...
argimgt0 25122 Closure of the argument of...
argimlt0 25123 Closure of the argument of...
logimul 25124 Multiplying a number by ` ...
logneg2 25125 The logarithm of the negat...
logmul2 25126 Generalization of ~ relogm...
logdiv2 25127 Generalization of ~ relogd...
abslogle 25128 Bound on the magnitude of ...
tanarg 25129 The basic relation between...
logdivlti 25130 The ` log x / x ` function...
logdivlt 25131 The ` log x / x ` function...
logdivle 25132 The ` log x / x ` function...
relogcld 25133 Closure of the natural log...
reeflogd 25134 Relationship between the n...
relogmuld 25135 The natural logarithm of t...
relogdivd 25136 The natural logarithm of t...
logled 25137 Natural logarithm preserve...
relogefd 25138 Relationship between the n...
rplogcld 25139 Closure of the logarithm f...
logge0d 25140 The logarithm of a number ...
logge0b 25141 The logarithm of a number ...
loggt0b 25142 The logarithm of a number ...
logle1b 25143 The logarithm of a number ...
loglt1b 25144 The logarithm of a number ...
divlogrlim 25145 The inverse logarithm func...
logno1 25146 The logarithm function is ...
dvrelog 25147 The derivative of the real...
relogcn 25148 The real logarithm functio...
ellogdm 25149 Elementhood in the "contin...
logdmn0 25150 A number in the continuous...
logdmnrp 25151 A number in the continuous...
logdmss 25152 The continuity domain of `...
logcnlem2 25153 Lemma for ~ logcn . (Cont...
logcnlem3 25154 Lemma for ~ logcn . (Cont...
logcnlem4 25155 Lemma for ~ logcn . (Cont...
logcnlem5 25156 Lemma for ~ logcn . (Cont...
logcn 25157 The logarithm function is ...
dvloglem 25158 Lemma for ~ dvlog . (Cont...
logdmopn 25159 The "continuous domain" of...
logf1o2 25160 The logarithm maps its con...
dvlog 25161 The derivative of the comp...
dvlog2lem 25162 Lemma for ~ dvlog2 . (Con...
dvlog2 25163 The derivative of the comp...
advlog 25164 The antiderivative of the ...
advlogexp 25165 The antiderivative of a po...
efopnlem1 25166 Lemma for ~ efopn . (Cont...
efopnlem2 25167 Lemma for ~ efopn . (Cont...
efopn 25168 The exponential map is an ...
logtayllem 25169 Lemma for ~ logtayl . (Co...
logtayl 25170 The Taylor series for ` -u...
logtaylsum 25171 The Taylor series for ` -u...
logtayl2 25172 Power series expression fo...
logccv 25173 The natural logarithm func...
cxpval 25174 Value of the complex power...
cxpef 25175 Value of the complex power...
0cxp 25176 Value of the complex power...
cxpexpz 25177 Relate the complex power f...
cxpexp 25178 Relate the complex power f...
logcxp 25179 Logarithm of a complex pow...
cxp0 25180 Value of the complex power...
cxp1 25181 Value of the complex power...
1cxp 25182 Value of the complex power...
ecxp 25183 Write the exponential func...
cxpcl 25184 Closure of the complex pow...
recxpcl 25185 Real closure of the comple...
rpcxpcl 25186 Positive real closure of t...
cxpne0 25187 Complex exponentiation is ...
cxpeq0 25188 Complex exponentiation is ...
cxpadd 25189 Sum of exponents law for c...
cxpp1 25190 Value of a nonzero complex...
cxpneg 25191 Value of a complex number ...
cxpsub 25192 Exponent subtraction law f...
cxpge0 25193 Nonnegative exponentiation...
mulcxplem 25194 Lemma for ~ mulcxp . (Con...
mulcxp 25195 Complex exponentiation of ...
cxprec 25196 Complex exponentiation of ...
divcxp 25197 Complex exponentiation of ...
cxpmul 25198 Product of exponents law f...
cxpmul2 25199 Product of exponents law f...
cxproot 25200 The complex power function...
cxpmul2z 25201 Generalize ~ cxpmul2 to ne...
abscxp 25202 Absolute value of a power,...
abscxp2 25203 Absolute value of a power,...
cxplt 25204 Ordering property for comp...
cxple 25205 Ordering property for comp...
cxplea 25206 Ordering property for comp...
cxple2 25207 Ordering property for comp...
cxplt2 25208 Ordering property for comp...
cxple2a 25209 Ordering property for comp...
cxplt3 25210 Ordering property for comp...
cxple3 25211 Ordering property for comp...
cxpsqrtlem 25212 Lemma for ~ cxpsqrt . (Co...
cxpsqrt 25213 The complex exponential fu...
logsqrt 25214 Logarithm of a square root...
cxp0d 25215 Value of the complex power...
cxp1d 25216 Value of the complex power...
1cxpd 25217 Value of the complex power...
cxpcld 25218 Closure of the complex pow...
cxpmul2d 25219 Product of exponents law f...
0cxpd 25220 Value of the complex power...
cxpexpzd 25221 Relate the complex power f...
cxpefd 25222 Value of the complex power...
cxpne0d 25223 Complex exponentiation is ...
cxpp1d 25224 Value of a nonzero complex...
cxpnegd 25225 Value of a complex number ...
cxpmul2zd 25226 Generalize ~ cxpmul2 to ne...
cxpaddd 25227 Sum of exponents law for c...
cxpsubd 25228 Exponent subtraction law f...
cxpltd 25229 Ordering property for comp...
cxpled 25230 Ordering property for comp...
cxplead 25231 Ordering property for comp...
divcxpd 25232 Complex exponentiation of ...
recxpcld 25233 Positive real closure of t...
cxpge0d 25234 Nonnegative exponentiation...
cxple2ad 25235 Ordering property for comp...
cxplt2d 25236 Ordering property for comp...
cxple2d 25237 Ordering property for comp...
mulcxpd 25238 Complex exponentiation of ...
cxpsqrtth 25239 Square root theorem over t...
2irrexpq 25240 There exist irrational num...
cxprecd 25241 Complex exponentiation of ...
rpcxpcld 25242 Positive real closure of t...
logcxpd 25243 Logarithm of a complex pow...
cxplt3d 25244 Ordering property for comp...
cxple3d 25245 Ordering property for comp...
cxpmuld 25246 Product of exponents law f...
cxpcom 25247 Commutative law for real e...
dvcxp1 25248 The derivative of a comple...
dvcxp2 25249 The derivative of a comple...
dvsqrt 25250 The derivative of the real...
dvcncxp1 25251 Derivative of complex powe...
dvcnsqrt 25252 Derivative of square root ...
cxpcn 25253 Domain of continuity of th...
cxpcn2 25254 Continuity of the complex ...
cxpcn3lem 25255 Lemma for ~ cxpcn3 . (Con...
cxpcn3 25256 Extend continuity of the c...
resqrtcn 25257 Continuity of the real squ...
sqrtcn 25258 Continuity of the square r...
cxpaddlelem 25259 Lemma for ~ cxpaddle . (C...
cxpaddle 25260 Ordering property for comp...
abscxpbnd 25261 Bound on the absolute valu...
root1id 25262 Property of an ` N ` -th r...
root1eq1 25263 The only powers of an ` N ...
root1cj 25264 Within the ` N ` -th roots...
cxpeq 25265 Solve an equation involvin...
loglesqrt 25266 An upper bound on the loga...
logreclem 25267 Symmetry of the natural lo...
logrec 25268 Logarithm of a reciprocal ...
logbval 25271 Define the value of the ` ...
logbcl 25272 General logarithm closure....
logbid1 25273 General logarithm is 1 whe...
logb1 25274 The logarithm of ` 1 ` to ...
elogb 25275 The general logarithm of a...
logbchbase 25276 Change of base for logarit...
relogbval 25277 Value of the general logar...
relogbcl 25278 Closure of the general log...
relogbzcl 25279 Closure of the general log...
relogbreexp 25280 Power law for the general ...
relogbzexp 25281 Power law for the general ...
relogbmul 25282 The logarithm of the produ...
relogbmulexp 25283 The logarithm of the produ...
relogbdiv 25284 The logarithm of the quoti...
relogbexp 25285 Identity law for general l...
nnlogbexp 25286 Identity law for general l...
logbrec 25287 Logarithm of a reciprocal ...
logbleb 25288 The general logarithm func...
logblt 25289 The general logarithm func...
relogbcxp 25290 Identity law for the gener...
cxplogb 25291 Identity law for the gener...
relogbcxpb 25292 The logarithm is the inver...
logbmpt 25293 The general logarithm to a...
logbf 25294 The general logarithm to a...
logbfval 25295 The general logarithm of a...
relogbf 25296 The general logarithm to a...
logblog 25297 The general logarithm to t...
logbgt0b 25298 The logarithm of a positiv...
logbgcd1irr 25299 The logarithm of an intege...
2logb9irr 25300 Example for ~ logbgcd1irr ...
logbprmirr 25301 The logarithm of a prime t...
2logb3irr 25302 Example for ~ logbprmirr ....
2logb9irrALT 25303 Alternate proof of ~ 2logb...
sqrt2cxp2logb9e3 25304 The square root of two to ...
2irrexpqALT 25305 Alternate proof of ~ 2irre...
angval 25306 Define the angle function,...
angcan 25307 Cancel a constant multipli...
angneg 25308 Cancel a negative sign in ...
angvald 25309 The (signed) angle between...
angcld 25310 The (signed) angle between...
angrteqvd 25311 Two vectors are at a right...
cosangneg2d 25312 The cosine of the angle be...
angrtmuld 25313 Perpendicularity of two ve...
ang180lem1 25314 Lemma for ~ ang180 . Show...
ang180lem2 25315 Lemma for ~ ang180 . Show...
ang180lem3 25316 Lemma for ~ ang180 . Sinc...
ang180lem4 25317 Lemma for ~ ang180 . Redu...
ang180lem5 25318 Lemma for ~ ang180 : Redu...
ang180 25319 The sum of angles ` m A B ...
lawcoslem1 25320 Lemma for ~ lawcos . Here...
lawcos 25321 Law of cosines (also known...
pythag 25322 Pythagorean theorem. Give...
isosctrlem1 25323 Lemma for ~ isosctr . (Co...
isosctrlem2 25324 Lemma for ~ isosctr . Cor...
isosctrlem3 25325 Lemma for ~ isosctr . Cor...
isosctr 25326 Isosceles triangle theorem...
ssscongptld 25327 If two triangles have equa...
affineequiv 25328 Equivalence between two wa...
affineequiv2 25329 Equivalence between two wa...
affineequiv3 25330 Equivalence between two wa...
affineequiv4 25331 Equivalence between two wa...
affineequivne 25332 Equivalence between two wa...
angpieqvdlem 25333 Equivalence used in the pr...
angpieqvdlem2 25334 Equivalence used in ~ angp...
angpined 25335 If the angle at ABC is ` _...
angpieqvd 25336 The angle ABC is ` _pi ` i...
chordthmlem 25337 If M is the midpoint of AB...
chordthmlem2 25338 If M is the midpoint of AB...
chordthmlem3 25339 If M is the midpoint of AB...
chordthmlem4 25340 If P is on the segment AB ...
chordthmlem5 25341 If P is on the segment AB ...
chordthm 25342 The intersecting chords th...
heron 25343 Heron's formula gives the ...
quad2 25344 The quadratic equation, wi...
quad 25345 The quadratic equation. (...
1cubrlem 25346 The cube roots of unity. ...
1cubr 25347 The cube roots of unity. ...
dcubic1lem 25348 Lemma for ~ dcubic1 and ~ ...
dcubic2 25349 Reverse direction of ~ dcu...
dcubic1 25350 Forward direction of ~ dcu...
dcubic 25351 Solutions to the depressed...
mcubic 25352 Solutions to a monic cubic...
cubic2 25353 The solution to the genera...
cubic 25354 The cubic equation, which ...
binom4 25355 Work out a quartic binomia...
dquartlem1 25356 Lemma for ~ dquart . (Con...
dquartlem2 25357 Lemma for ~ dquart . (Con...
dquart 25358 Solve a depressed quartic ...
quart1cl 25359 Closure lemmas for ~ quart...
quart1lem 25360 Lemma for ~ quart1 . (Con...
quart1 25361 Depress a quartic equation...
quartlem1 25362 Lemma for ~ quart . (Cont...
quartlem2 25363 Closure lemmas for ~ quart...
quartlem3 25364 Closure lemmas for ~ quart...
quartlem4 25365 Closure lemmas for ~ quart...
quart 25366 The quartic equation, writ...
asinlem 25373 The argument to the logari...
asinlem2 25374 The argument to the logari...
asinlem3a 25375 Lemma for ~ asinlem3 . (C...
asinlem3 25376 The argument to the logari...
asinf 25377 Domain and range of the ar...
asincl 25378 Closure for the arcsin fun...
acosf 25379 Domain and range of the ar...
acoscl 25380 Closure for the arccos fun...
atandm 25381 Since the property is a li...
atandm2 25382 This form of ~ atandm is a...
atandm3 25383 A compact form of ~ atandm...
atandm4 25384 A compact form of ~ atandm...
atanf 25385 Domain and range of the ar...
atancl 25386 Closure for the arctan fun...
asinval 25387 Value of the arcsin functi...
acosval 25388 Value of the arccos functi...
atanval 25389 Value of the arctan functi...
atanre 25390 A real number is in the do...
asinneg 25391 The arcsine function is od...
acosneg 25392 The negative symmetry rela...
efiasin 25393 The exponential of the arc...
sinasin 25394 The arcsine function is an...
cosacos 25395 The arccosine function is ...
asinsinlem 25396 Lemma for ~ asinsin . (Co...
asinsin 25397 The arcsine function compo...
acoscos 25398 The arccosine function is ...
asin1 25399 The arcsine of ` 1 ` is ` ...
acos1 25400 The arcsine of ` 1 ` is ` ...
reasinsin 25401 The arcsine function compo...
asinsinb 25402 Relationship between sine ...
acoscosb 25403 Relationship between sine ...
asinbnd 25404 The arcsine function has r...
acosbnd 25405 The arccosine function has...
asinrebnd 25406 Bounds on the arcsine func...
asinrecl 25407 The arcsine function is re...
acosrecl 25408 The arccosine function is ...
cosasin 25409 The cosine of the arcsine ...
sinacos 25410 The sine of the arccosine ...
atandmneg 25411 The domain of the arctange...
atanneg 25412 The arctangent function is...
atan0 25413 The arctangent of zero is ...
atandmcj 25414 The arctangent function di...
atancj 25415 The arctangent function di...
atanrecl 25416 The arctangent function is...
efiatan 25417 Value of the exponential o...
atanlogaddlem 25418 Lemma for ~ atanlogadd . ...
atanlogadd 25419 The rule ` sqrt ( z w ) = ...
atanlogsublem 25420 Lemma for ~ atanlogsub . ...
atanlogsub 25421 A variation on ~ atanlogad...
efiatan2 25422 Value of the exponential o...
2efiatan 25423 Value of the exponential o...
tanatan 25424 The arctangent function is...
atandmtan 25425 The tangent function has r...
cosatan 25426 The cosine of an arctangen...
cosatanne0 25427 The arctangent function ha...
atantan 25428 The arctangent function is...
atantanb 25429 Relationship between tange...
atanbndlem 25430 Lemma for ~ atanbnd . (Co...
atanbnd 25431 The arctangent function is...
atanord 25432 The arctangent function is...
atan1 25433 The arctangent of ` 1 ` is...
bndatandm 25434 A point in the open unit d...
atans 25435 The "domain of continuity"...
atans2 25436 It suffices to show that `...
atansopn 25437 The domain of continuity o...
atansssdm 25438 The domain of continuity o...
ressatans 25439 The real number line is a ...
dvatan 25440 The derivative of the arct...
atancn 25441 The arctangent is a contin...
atantayl 25442 The Taylor series for ` ar...
atantayl2 25443 The Taylor series for ` ar...
atantayl3 25444 The Taylor series for ` ar...
leibpilem1 25445 Lemma for ~ leibpi . (Con...
leibpilem2 25446 The Leibniz formula for ` ...
leibpi 25447 The Leibniz formula for ` ...
leibpisum 25448 The Leibniz formula for ` ...
log2cnv 25449 Using the Taylor series fo...
log2tlbnd 25450 Bound the error term in th...
log2ublem1 25451 Lemma for ~ log2ub . The ...
log2ublem2 25452 Lemma for ~ log2ub . (Con...
log2ublem3 25453 Lemma for ~ log2ub . In d...
log2ub 25454 ` log 2 ` is less than ` 2...
log2le1 25455 ` log 2 ` is less than ` 1...
birthdaylem1 25456 Lemma for ~ birthday . (C...
birthdaylem2 25457 For general ` N ` and ` K ...
birthdaylem3 25458 For general ` N ` and ` K ...
birthday 25459 The Birthday Problem. The...
dmarea 25462 The domain of the area fun...
areambl 25463 The fibers of a measurable...
areass 25464 A measurable region is a s...
dfarea 25465 Rewrite ~ df-area self-ref...
areaf 25466 Area measurement is a func...
areacl 25467 The area of a measurable r...
areage0 25468 The area of a measurable r...
areaval 25469 The area of a measurable r...
rlimcnp 25470 Relate a limit of a real-v...
rlimcnp2 25471 Relate a limit of a real-v...
rlimcnp3 25472 Relate a limit of a real-v...
xrlimcnp 25473 Relate a limit of a real-v...
efrlim 25474 The limit of the sequence ...
dfef2 25475 The limit of the sequence ...
cxplim 25476 A power to a negative expo...
sqrtlim 25477 The inverse square root fu...
rlimcxp 25478 Any power to a positive ex...
o1cxp 25479 An eventually bounded func...
cxp2limlem 25480 A linear factor grows slow...
cxp2lim 25481 Any power grows slower tha...
cxploglim 25482 The logarithm grows slower...
cxploglim2 25483 Every power of the logarit...
divsqrtsumlem 25484 Lemma for ~ divsqrsum and ...
divsqrsumf 25485 The function ` F ` used in...
divsqrsum 25486 The sum ` sum_ n <_ x ( 1 ...
divsqrtsum2 25487 A bound on the distance of...
divsqrtsumo1 25488 The sum ` sum_ n <_ x ( 1 ...
cvxcl 25489 Closure of a 0-1 linear co...
scvxcvx 25490 A strictly convex function...
jensenlem1 25491 Lemma for ~ jensen . (Con...
jensenlem2 25492 Lemma for ~ jensen . (Con...
jensen 25493 Jensen's inequality, a fin...
amgmlem 25494 Lemma for ~ amgm . (Contr...
amgm 25495 Inequality of arithmetic a...
logdifbnd 25498 Bound on the difference of...
logdiflbnd 25499 Lower bound on the differe...
emcllem1 25500 Lemma for ~ emcl . The se...
emcllem2 25501 Lemma for ~ emcl . ` F ` i...
emcllem3 25502 Lemma for ~ emcl . The fu...
emcllem4 25503 Lemma for ~ emcl . The di...
emcllem5 25504 Lemma for ~ emcl . The pa...
emcllem6 25505 Lemma for ~ emcl . By the...
emcllem7 25506 Lemma for ~ emcl and ~ har...
emcl 25507 Closure and bounds for the...
harmonicbnd 25508 A bound on the harmonic se...
harmonicbnd2 25509 A bound on the harmonic se...
emre 25510 The Euler-Mascheroni const...
emgt0 25511 The Euler-Mascheroni const...
harmonicbnd3 25512 A bound on the harmonic se...
harmoniclbnd 25513 A bound on the harmonic se...
harmonicubnd 25514 A bound on the harmonic se...
harmonicbnd4 25515 The asymptotic behavior of...
fsumharmonic 25516 Bound a finite sum based o...
zetacvg 25519 The zeta series is converg...
eldmgm 25526 Elementhood in the set of ...
dmgmaddn0 25527 If ` A ` is not a nonposit...
dmlogdmgm 25528 If ` A ` is in the continu...
rpdmgm 25529 A positive real number is ...
dmgmn0 25530 If ` A ` is not a nonposit...
dmgmaddnn0 25531 If ` A ` is not a nonposit...
dmgmdivn0 25532 Lemma for ~ lgamf . (Cont...
lgamgulmlem1 25533 Lemma for ~ lgamgulm . (C...
lgamgulmlem2 25534 Lemma for ~ lgamgulm . (C...
lgamgulmlem3 25535 Lemma for ~ lgamgulm . (C...
lgamgulmlem4 25536 Lemma for ~ lgamgulm . (C...
lgamgulmlem5 25537 Lemma for ~ lgamgulm . (C...
lgamgulmlem6 25538 The series ` G ` is unifor...
lgamgulm 25539 The series ` G ` is unifor...
lgamgulm2 25540 Rewrite the limit of the s...
lgambdd 25541 The log-Gamma function is ...
lgamucov 25542 The ` U ` regions used in ...
lgamucov2 25543 The ` U ` regions used in ...
lgamcvglem 25544 Lemma for ~ lgamf and ~ lg...
lgamcl 25545 The log-Gamma function is ...
lgamf 25546 The log-Gamma function is ...
gamf 25547 The Gamma function is a co...
gamcl 25548 The exponential of the log...
eflgam 25549 The exponential of the log...
gamne0 25550 The Gamma function is neve...
igamval 25551 Value of the inverse Gamma...
igamz 25552 Value of the inverse Gamma...
igamgam 25553 Value of the inverse Gamma...
igamlgam 25554 Value of the inverse Gamma...
igamf 25555 Closure of the inverse Gam...
igamcl 25556 Closure of the inverse Gam...
gamigam 25557 The Gamma function is the ...
lgamcvg 25558 The series ` G ` converges...
lgamcvg2 25559 The series ` G ` converges...
gamcvg 25560 The pointwise exponential ...
lgamp1 25561 The functional equation of...
gamp1 25562 The functional equation of...
gamcvg2lem 25563 Lemma for ~ gamcvg2 . (Co...
gamcvg2 25564 An infinite product expres...
regamcl 25565 The Gamma function is real...
relgamcl 25566 The log-Gamma function is ...
rpgamcl 25567 The log-Gamma function is ...
lgam1 25568 The log-Gamma function at ...
gam1 25569 The log-Gamma function at ...
facgam 25570 The Gamma function general...
gamfac 25571 The Gamma function general...
wilthlem1 25572 The only elements that are...
wilthlem2 25573 Lemma for ~ wilth : induct...
wilthlem3 25574 Lemma for ~ wilth . Here ...
wilth 25575 Wilson's theorem. A numbe...
wilthimp 25576 The forward implication of...
ftalem1 25577 Lemma for ~ fta : "growth...
ftalem2 25578 Lemma for ~ fta . There e...
ftalem3 25579 Lemma for ~ fta . There e...
ftalem4 25580 Lemma for ~ fta : Closure...
ftalem5 25581 Lemma for ~ fta : Main pr...
ftalem6 25582 Lemma for ~ fta : Dischar...
ftalem7 25583 Lemma for ~ fta . Shift t...
fta 25584 The Fundamental Theorem of...
basellem1 25585 Lemma for ~ basel . Closu...
basellem2 25586 Lemma for ~ basel . Show ...
basellem3 25587 Lemma for ~ basel . Using...
basellem4 25588 Lemma for ~ basel . By ~ ...
basellem5 25589 Lemma for ~ basel . Using...
basellem6 25590 Lemma for ~ basel . The f...
basellem7 25591 Lemma for ~ basel . The f...
basellem8 25592 Lemma for ~ basel . The f...
basellem9 25593 Lemma for ~ basel . Since...
basel 25594 The sum of the inverse squ...
efnnfsumcl 25607 Finite sum closure in the ...
ppisval 25608 The set of primes less tha...
ppisval2 25609 The set of primes less tha...
ppifi 25610 The set of primes less tha...
prmdvdsfi 25611 The set of prime divisors ...
chtf 25612 Domain and range of the Ch...
chtcl 25613 Real closure of the Chebys...
chtval 25614 Value of the Chebyshev fun...
efchtcl 25615 The Chebyshev function is ...
chtge0 25616 The Chebyshev function is ...
vmaval 25617 Value of the von Mangoldt ...
isppw 25618 Two ways to say that ` A `...
isppw2 25619 Two ways to say that ` A `...
vmappw 25620 Value of the von Mangoldt ...
vmaprm 25621 Value of the von Mangoldt ...
vmacl 25622 Closure for the von Mangol...
vmaf 25623 Functionality of the von M...
efvmacl 25624 The von Mangoldt is closed...
vmage0 25625 The von Mangoldt function ...
chpval 25626 Value of the second Chebys...
chpf 25627 Functionality of the secon...
chpcl 25628 Closure for the second Che...
efchpcl 25629 The second Chebyshev funct...
chpge0 25630 The second Chebyshev funct...
ppival 25631 Value of the prime-countin...
ppival2 25632 Value of the prime-countin...
ppival2g 25633 Value of the prime-countin...
ppif 25634 Domain and range of the pr...
ppicl 25635 Real closure of the prime-...
muval 25636 The value of the Möbi...
muval1 25637 The value of the Möbi...
muval2 25638 The value of the Möbi...
isnsqf 25639 Two ways to say that a num...
issqf 25640 Two ways to say that a num...
sqfpc 25641 The prime count of a squar...
dvdssqf 25642 A divisor of a squarefree ...
sqf11 25643 A squarefree number is com...
muf 25644 The Möbius function i...
mucl 25645 Closure of the Möbius...
sgmval 25646 The value of the divisor f...
sgmval2 25647 The value of the divisor f...
0sgm 25648 The value of the sum-of-di...
sgmf 25649 The divisor function is a ...
sgmcl 25650 Closure of the divisor fun...
sgmnncl 25651 Closure of the divisor fun...
mule1 25652 The Möbius function t...
chtfl 25653 The Chebyshev function doe...
chpfl 25654 The second Chebyshev funct...
ppiprm 25655 The prime-counting functio...
ppinprm 25656 The prime-counting functio...
chtprm 25657 The Chebyshev function at ...
chtnprm 25658 The Chebyshev function at ...
chpp1 25659 The second Chebyshev funct...
chtwordi 25660 The Chebyshev function is ...
chpwordi 25661 The second Chebyshev funct...
chtdif 25662 The difference of the Cheb...
efchtdvds 25663 The exponentiated Chebyshe...
ppifl 25664 The prime-counting functio...
ppip1le 25665 The prime-counting functio...
ppiwordi 25666 The prime-counting functio...
ppidif 25667 The difference of the prim...
ppi1 25668 The prime-counting functio...
cht1 25669 The Chebyshev function at ...
vma1 25670 The von Mangoldt function ...
chp1 25671 The second Chebyshev funct...
ppi1i 25672 Inference form of ~ ppiprm...
ppi2i 25673 Inference form of ~ ppinpr...
ppi2 25674 The prime-counting functio...
ppi3 25675 The prime-counting functio...
cht2 25676 The Chebyshev function at ...
cht3 25677 The Chebyshev function at ...
ppinncl 25678 Closure of the prime-count...
chtrpcl 25679 Closure of the Chebyshev f...
ppieq0 25680 The prime-counting functio...
ppiltx 25681 The prime-counting functio...
prmorcht 25682 Relate the primorial (prod...
mumullem1 25683 Lemma for ~ mumul . A mul...
mumullem2 25684 Lemma for ~ mumul . The p...
mumul 25685 The Möbius function i...
sqff1o 25686 There is a bijection from ...
fsumdvdsdiaglem 25687 A "diagonal commutation" o...
fsumdvdsdiag 25688 A "diagonal commutation" o...
fsumdvdscom 25689 A double commutation of di...
dvdsppwf1o 25690 A bijection from the divis...
dvdsflf1o 25691 A bijection from the numbe...
dvdsflsumcom 25692 A sum commutation from ` s...
fsumfldivdiaglem 25693 Lemma for ~ fsumfldivdiag ...
fsumfldivdiag 25694 The right-hand side of ~ d...
musum 25695 The sum of the Möbius...
musumsum 25696 Evaluate a collapsing sum ...
muinv 25697 The Möbius inversion ...
dvdsmulf1o 25698 If ` M ` and ` N ` are two...
fsumdvdsmul 25699 Product of two divisor sum...
sgmppw 25700 The value of the divisor f...
0sgmppw 25701 A prime power ` P ^ K ` ha...
1sgmprm 25702 The sum of divisors for a ...
1sgm2ppw 25703 The sum of the divisors of...
sgmmul 25704 The divisor function for f...
ppiublem1 25705 Lemma for ~ ppiub . (Cont...
ppiublem2 25706 A prime greater than ` 3 `...
ppiub 25707 An upper bound on the prim...
vmalelog 25708 The von Mangoldt function ...
chtlepsi 25709 The first Chebyshev functi...
chprpcl 25710 Closure of the second Cheb...
chpeq0 25711 The second Chebyshev funct...
chteq0 25712 The first Chebyshev functi...
chtleppi 25713 Upper bound on the ` theta...
chtublem 25714 Lemma for ~ chtub . (Cont...
chtub 25715 An upper bound on the Cheb...
fsumvma 25716 Rewrite a sum over the von...
fsumvma2 25717 Apply ~ fsumvma for the co...
pclogsum 25718 The logarithmic analogue o...
vmasum 25719 The sum of the von Mangold...
logfac2 25720 Another expression for the...
chpval2 25721 Express the second Chebysh...
chpchtsum 25722 The second Chebyshev funct...
chpub 25723 An upper bound on the seco...
logfacubnd 25724 A simple upper bound on th...
logfaclbnd 25725 A lower bound on the logar...
logfacbnd3 25726 Show the stronger statemen...
logfacrlim 25727 Combine the estimates ~ lo...
logexprlim 25728 The sum ` sum_ n <_ x , lo...
logfacrlim2 25729 Write out ~ logfacrlim as ...
mersenne 25730 A Mersenne prime is a prim...
perfect1 25731 Euclid's contribution to t...
perfectlem1 25732 Lemma for ~ perfect . (Co...
perfectlem2 25733 Lemma for ~ perfect . (Co...
perfect 25734 The Euclid-Euler theorem, ...
dchrval 25737 Value of the group of Diri...
dchrbas 25738 Base set of the group of D...
dchrelbas 25739 A Dirichlet character is a...
dchrelbas2 25740 A Dirichlet character is a...
dchrelbas3 25741 A Dirichlet character is a...
dchrelbasd 25742 A Dirichlet character is a...
dchrrcl 25743 Reverse closure for a Diri...
dchrmhm 25744 A Dirichlet character is a...
dchrf 25745 A Dirichlet character is a...
dchrelbas4 25746 A Dirichlet character is a...
dchrzrh1 25747 Value of a Dirichlet chara...
dchrzrhcl 25748 A Dirichlet character take...
dchrzrhmul 25749 A Dirichlet character is c...
dchrplusg 25750 Group operation on the gro...
dchrmul 25751 Group operation on the gro...
dchrmulcl 25752 Closure of the group opera...
dchrn0 25753 A Dirichlet character is n...
dchr1cl 25754 Closure of the principal D...
dchrmulid2 25755 Left identity for the prin...
dchrinvcl 25756 Closure of the group inver...
dchrabl 25757 The set of Dirichlet chara...
dchrfi 25758 The group of Dirichlet cha...
dchrghm 25759 A Dirichlet character rest...
dchr1 25760 Value of the principal Dir...
dchreq 25761 A Dirichlet character is d...
dchrresb 25762 A Dirichlet character is d...
dchrabs 25763 A Dirichlet character take...
dchrinv 25764 The inverse of a Dirichlet...
dchrabs2 25765 A Dirichlet character take...
dchr1re 25766 The principal Dirichlet ch...
dchrptlem1 25767 Lemma for ~ dchrpt . (Con...
dchrptlem2 25768 Lemma for ~ dchrpt . (Con...
dchrptlem3 25769 Lemma for ~ dchrpt . (Con...
dchrpt 25770 For any element other than...
dchrsum2 25771 An orthogonality relation ...
dchrsum 25772 An orthogonality relation ...
sumdchr2 25773 Lemma for ~ sumdchr . (Co...
dchrhash 25774 There are exactly ` phi ( ...
sumdchr 25775 An orthogonality relation ...
dchr2sum 25776 An orthogonality relation ...
sum2dchr 25777 An orthogonality relation ...
bcctr 25778 Value of the central binom...
pcbcctr 25779 Prime count of a central b...
bcmono 25780 The binomial coefficient i...
bcmax 25781 The binomial coefficient t...
bcp1ctr 25782 Ratio of two central binom...
bclbnd 25783 A bound on the binomial co...
efexple 25784 Convert a bound on a power...
bpos1lem 25785 Lemma for ~ bpos1 . (Cont...
bpos1 25786 Bertrand's postulate, chec...
bposlem1 25787 An upper bound on the prim...
bposlem2 25788 There are no odd primes in...
bposlem3 25789 Lemma for ~ bpos . Since ...
bposlem4 25790 Lemma for ~ bpos . (Contr...
bposlem5 25791 Lemma for ~ bpos . Bound ...
bposlem6 25792 Lemma for ~ bpos . By usi...
bposlem7 25793 Lemma for ~ bpos . The fu...
bposlem8 25794 Lemma for ~ bpos . Evalua...
bposlem9 25795 Lemma for ~ bpos . Derive...
bpos 25796 Bertrand's postulate: ther...
zabsle1 25799 ` { -u 1 , 0 , 1 } ` is th...
lgslem1 25800 When ` a ` is coprime to t...
lgslem2 25801 The set ` Z ` of all integ...
lgslem3 25802 The set ` Z ` of all integ...
lgslem4 25803 Lemma for ~ lgsfcl2 . (Co...
lgsval 25804 Value of the Legendre symb...
lgsfval 25805 Value of the function ` F ...
lgsfcl2 25806 The function ` F ` is clos...
lgscllem 25807 The Legendre symbol is an ...
lgsfcl 25808 Closure of the function ` ...
lgsfle1 25809 The function ` F ` has mag...
lgsval2lem 25810 Lemma for ~ lgsval2 . (Co...
lgsval4lem 25811 Lemma for ~ lgsval4 . (Co...
lgscl2 25812 The Legendre symbol is an ...
lgs0 25813 The Legendre symbol when t...
lgscl 25814 The Legendre symbol is an ...
lgsle1 25815 The Legendre symbol has ab...
lgsval2 25816 The Legendre symbol at a p...
lgs2 25817 The Legendre symbol at ` 2...
lgsval3 25818 The Legendre symbol at an ...
lgsvalmod 25819 The Legendre symbol is equ...
lgsval4 25820 Restate ~ lgsval for nonze...
lgsfcl3 25821 Closure of the function ` ...
lgsval4a 25822 Same as ~ lgsval4 for posi...
lgscl1 25823 The value of the Legendre ...
lgsneg 25824 The Legendre symbol is eit...
lgsneg1 25825 The Legendre symbol for no...
lgsmod 25826 The Legendre (Jacobi) symb...
lgsdilem 25827 Lemma for ~ lgsdi and ~ lg...
lgsdir2lem1 25828 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem2 25829 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem3 25830 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem4 25831 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem5 25832 Lemma for ~ lgsdir2 . (Co...
lgsdir2 25833 The Legendre symbol is com...
lgsdirprm 25834 The Legendre symbol is com...
lgsdir 25835 The Legendre symbol is com...
lgsdilem2 25836 Lemma for ~ lgsdi . (Cont...
lgsdi 25837 The Legendre symbol is com...
lgsne0 25838 The Legendre symbol is non...
lgsabs1 25839 The Legendre symbol is non...
lgssq 25840 The Legendre symbol at a s...
lgssq2 25841 The Legendre symbol at a s...
lgsprme0 25842 The Legendre symbol at any...
1lgs 25843 The Legendre symbol at ` 1...
lgs1 25844 The Legendre symbol at ` 1...
lgsmodeq 25845 The Legendre (Jacobi) symb...
lgsmulsqcoprm 25846 The Legendre (Jacobi) symb...
lgsdirnn0 25847 Variation on ~ lgsdir vali...
lgsdinn0 25848 Variation on ~ lgsdi valid...
lgsqrlem1 25849 Lemma for ~ lgsqr . (Cont...
lgsqrlem2 25850 Lemma for ~ lgsqr . (Cont...
lgsqrlem3 25851 Lemma for ~ lgsqr . (Cont...
lgsqrlem4 25852 Lemma for ~ lgsqr . (Cont...
lgsqrlem5 25853 Lemma for ~ lgsqr . (Cont...
lgsqr 25854 The Legendre symbol for od...
lgsqrmod 25855 If the Legendre symbol of ...
lgsqrmodndvds 25856 If the Legendre symbol of ...
lgsdchrval 25857 The Legendre symbol functi...
lgsdchr 25858 The Legendre symbol functi...
gausslemma2dlem0a 25859 Auxiliary lemma 1 for ~ ga...
gausslemma2dlem0b 25860 Auxiliary lemma 2 for ~ ga...
gausslemma2dlem0c 25861 Auxiliary lemma 3 for ~ ga...
gausslemma2dlem0d 25862 Auxiliary lemma 4 for ~ ga...
gausslemma2dlem0e 25863 Auxiliary lemma 5 for ~ ga...
gausslemma2dlem0f 25864 Auxiliary lemma 6 for ~ ga...
gausslemma2dlem0g 25865 Auxiliary lemma 7 for ~ ga...
gausslemma2dlem0h 25866 Auxiliary lemma 8 for ~ ga...
gausslemma2dlem0i 25867 Auxiliary lemma 9 for ~ ga...
gausslemma2dlem1a 25868 Lemma for ~ gausslemma2dle...
gausslemma2dlem1 25869 Lemma 1 for ~ gausslemma2d...
gausslemma2dlem2 25870 Lemma 2 for ~ gausslemma2d...
gausslemma2dlem3 25871 Lemma 3 for ~ gausslemma2d...
gausslemma2dlem4 25872 Lemma 4 for ~ gausslemma2d...
gausslemma2dlem5a 25873 Lemma for ~ gausslemma2dle...
gausslemma2dlem5 25874 Lemma 5 for ~ gausslemma2d...
gausslemma2dlem6 25875 Lemma 6 for ~ gausslemma2d...
gausslemma2dlem7 25876 Lemma 7 for ~ gausslemma2d...
gausslemma2d 25877 Gauss' Lemma (see also the...
lgseisenlem1 25878 Lemma for ~ lgseisen . If...
lgseisenlem2 25879 Lemma for ~ lgseisen . Th...
lgseisenlem3 25880 Lemma for ~ lgseisen . (C...
lgseisenlem4 25881 Lemma for ~ lgseisen . Th...
lgseisen 25882 Eisenstein's lemma, an exp...
lgsquadlem1 25883 Lemma for ~ lgsquad . Cou...
lgsquadlem2 25884 Lemma for ~ lgsquad . Cou...
lgsquadlem3 25885 Lemma for ~ lgsquad . (Co...
lgsquad 25886 The Law of Quadratic Recip...
lgsquad2lem1 25887 Lemma for ~ lgsquad2 . (C...
lgsquad2lem2 25888 Lemma for ~ lgsquad2 . (C...
lgsquad2 25889 Extend ~ lgsquad to coprim...
lgsquad3 25890 Extend ~ lgsquad2 to integ...
m1lgs 25891 The first supplement to th...
2lgslem1a1 25892 Lemma 1 for ~ 2lgslem1a . ...
2lgslem1a2 25893 Lemma 2 for ~ 2lgslem1a . ...
2lgslem1a 25894 Lemma 1 for ~ 2lgslem1 . ...
2lgslem1b 25895 Lemma 2 for ~ 2lgslem1 . ...
2lgslem1c 25896 Lemma 3 for ~ 2lgslem1 . ...
2lgslem1 25897 Lemma 1 for ~ 2lgs . (Con...
2lgslem2 25898 Lemma 2 for ~ 2lgs . (Con...
2lgslem3a 25899 Lemma for ~ 2lgslem3a1 . ...
2lgslem3b 25900 Lemma for ~ 2lgslem3b1 . ...
2lgslem3c 25901 Lemma for ~ 2lgslem3c1 . ...
2lgslem3d 25902 Lemma for ~ 2lgslem3d1 . ...
2lgslem3a1 25903 Lemma 1 for ~ 2lgslem3 . ...
2lgslem3b1 25904 Lemma 2 for ~ 2lgslem3 . ...
2lgslem3c1 25905 Lemma 3 for ~ 2lgslem3 . ...
2lgslem3d1 25906 Lemma 4 for ~ 2lgslem3 . ...
2lgslem3 25907 Lemma 3 for ~ 2lgs . (Con...
2lgs2 25908 The Legendre symbol for ` ...
2lgslem4 25909 Lemma 4 for ~ 2lgs : speci...
2lgs 25910 The second supplement to t...
2lgsoddprmlem1 25911 Lemma 1 for ~ 2lgsoddprm ....
2lgsoddprmlem2 25912 Lemma 2 for ~ 2lgsoddprm ....
2lgsoddprmlem3a 25913 Lemma 1 for ~ 2lgsoddprmle...
2lgsoddprmlem3b 25914 Lemma 2 for ~ 2lgsoddprmle...
2lgsoddprmlem3c 25915 Lemma 3 for ~ 2lgsoddprmle...
2lgsoddprmlem3d 25916 Lemma 4 for ~ 2lgsoddprmle...
2lgsoddprmlem3 25917 Lemma 3 for ~ 2lgsoddprm ....
2lgsoddprmlem4 25918 Lemma 4 for ~ 2lgsoddprm ....
2lgsoddprm 25919 The second supplement to t...
2sqlem1 25920 Lemma for ~ 2sq . (Contri...
2sqlem2 25921 Lemma for ~ 2sq . (Contri...
mul2sq 25922 Fibonacci's identity (actu...
2sqlem3 25923 Lemma for ~ 2sqlem5 . (Co...
2sqlem4 25924 Lemma for ~ 2sqlem5 . (Co...
2sqlem5 25925 Lemma for ~ 2sq . If a nu...
2sqlem6 25926 Lemma for ~ 2sq . If a nu...
2sqlem7 25927 Lemma for ~ 2sq . (Contri...
2sqlem8a 25928 Lemma for ~ 2sqlem8 . (Co...
2sqlem8 25929 Lemma for ~ 2sq . (Contri...
2sqlem9 25930 Lemma for ~ 2sq . (Contri...
2sqlem10 25931 Lemma for ~ 2sq . Every f...
2sqlem11 25932 Lemma for ~ 2sq . (Contri...
2sq 25933 All primes of the form ` 4...
2sqblem 25934 Lemma for ~ 2sqb . (Contr...
2sqb 25935 The converse to ~ 2sq . (...
2sq2 25936 ` 2 ` is the sum of square...
2sqn0 25937 If the sum of two squares ...
2sqcoprm 25938 If the sum of two squares ...
2sqmod 25939 Given two decompositions o...
2sqmo 25940 There exists at most one d...
2sqnn0 25941 All primes of the form ` 4...
2sqnn 25942 All primes of the form ` 4...
addsq2reu 25943 For each complex number ` ...
addsqn2reu 25944 For each complex number ` ...
addsqrexnreu 25945 For each complex number, t...
addsqnreup 25946 There is no unique decompo...
addsq2nreurex 25947 For each complex number ` ...
addsqn2reurex2 25948 For each complex number ` ...
2sqreulem1 25949 Lemma 1 for ~ 2sqreu . (C...
2sqreultlem 25950 Lemma for ~ 2sqreult . (C...
2sqreultblem 25951 Lemma for ~ 2sqreultb . (...
2sqreunnlem1 25952 Lemma 1 for ~ 2sqreunn . ...
2sqreunnltlem 25953 Lemma for ~ 2sqreunnlt . ...
2sqreunnltblem 25954 Lemma for ~ 2sqreunnltb . ...
2sqreulem2 25955 Lemma 2 for ~ 2sqreu etc. ...
2sqreulem3 25956 Lemma 3 for ~ 2sqreu etc. ...
2sqreulem4 25957 Lemma 4 for ~ 2sqreu et. ...
2sqreunnlem2 25958 Lemma 2 for ~ 2sqreunn . ...
2sqreu 25959 There exists a unique deco...
2sqreunn 25960 There exists a unique deco...
2sqreult 25961 There exists a unique deco...
2sqreultb 25962 There exists a unique deco...
2sqreunnlt 25963 There exists a unique deco...
2sqreunnltb 25964 There exists a unique deco...
2sqreuop 25965 There exists a unique deco...
2sqreuopnn 25966 There exists a unique deco...
2sqreuoplt 25967 There exists a unique deco...
2sqreuopltb 25968 There exists a unique deco...
2sqreuopnnlt 25969 There exists a unique deco...
2sqreuopnnltb 25970 There exists a unique deco...
2sqreuopb 25971 There exists a unique deco...
chebbnd1lem1 25972 Lemma for ~ chebbnd1 : sho...
chebbnd1lem2 25973 Lemma for ~ chebbnd1 : Sh...
chebbnd1lem3 25974 Lemma for ~ chebbnd1 : get...
chebbnd1 25975 The Chebyshev bound: The ...
chtppilimlem1 25976 Lemma for ~ chtppilim . (...
chtppilimlem2 25977 Lemma for ~ chtppilim . (...
chtppilim 25978 The ` theta ` function is ...
chto1ub 25979 The ` theta ` function is ...
chebbnd2 25980 The Chebyshev bound, part ...
chto1lb 25981 The ` theta ` function is ...
chpchtlim 25982 The ` psi ` and ` theta ` ...
chpo1ub 25983 The ` psi ` function is up...
chpo1ubb 25984 The ` psi ` function is up...
vmadivsum 25985 The sum of the von Mangold...
vmadivsumb 25986 Give a total bound on the ...
rplogsumlem1 25987 Lemma for ~ rplogsum . (C...
rplogsumlem2 25988 Lemma for ~ rplogsum . Eq...
dchrisum0lem1a 25989 Lemma for ~ dchrisum0lem1 ...
rpvmasumlem 25990 Lemma for ~ rpvmasum . Ca...
dchrisumlema 25991 Lemma for ~ dchrisum . Le...
dchrisumlem1 25992 Lemma for ~ dchrisum . Le...
dchrisumlem2 25993 Lemma for ~ dchrisum . Le...
dchrisumlem3 25994 Lemma for ~ dchrisum . Le...
dchrisum 25995 If ` n e. [ M , +oo ) |-> ...
dchrmusumlema 25996 Lemma for ~ dchrmusum and ...
dchrmusum2 25997 The sum of the Möbius...
dchrvmasumlem1 25998 An alternative expression ...
dchrvmasum2lem 25999 Give an expression for ` l...
dchrvmasum2if 26000 Combine the results of ~ d...
dchrvmasumlem2 26001 Lemma for ~ dchrvmasum . ...
dchrvmasumlem3 26002 Lemma for ~ dchrvmasum . ...
dchrvmasumlema 26003 Lemma for ~ dchrvmasum and...
dchrvmasumiflem1 26004 Lemma for ~ dchrvmasumif ....
dchrvmasumiflem2 26005 Lemma for ~ dchrvmasum . ...
dchrvmasumif 26006 An asymptotic approximatio...
dchrvmaeq0 26007 The set ` W ` is the colle...
dchrisum0fval 26008 Value of the function ` F ...
dchrisum0fmul 26009 The function ` F ` , the d...
dchrisum0ff 26010 The function ` F ` is a re...
dchrisum0flblem1 26011 Lemma for ~ dchrisum0flb ....
dchrisum0flblem2 26012 Lemma for ~ dchrisum0flb ....
dchrisum0flb 26013 The divisor sum of a real ...
dchrisum0fno1 26014 The sum ` sum_ k <_ x , F ...
rpvmasum2 26015 A partial result along the...
dchrisum0re 26016 Suppose ` X ` is a non-pri...
dchrisum0lema 26017 Lemma for ~ dchrisum0 . A...
dchrisum0lem1b 26018 Lemma for ~ dchrisum0lem1 ...
dchrisum0lem1 26019 Lemma for ~ dchrisum0 . (...
dchrisum0lem2a 26020 Lemma for ~ dchrisum0 . (...
dchrisum0lem2 26021 Lemma for ~ dchrisum0 . (...
dchrisum0lem3 26022 Lemma for ~ dchrisum0 . (...
dchrisum0 26023 The sum ` sum_ n e. NN , X...
dchrisumn0 26024 The sum ` sum_ n e. NN , X...
dchrmusumlem 26025 The sum of the Möbius...
dchrvmasumlem 26026 The sum of the Möbius...
dchrmusum 26027 The sum of the Möbius...
dchrvmasum 26028 The sum of the von Mangold...
rpvmasum 26029 The sum of the von Mangold...
rplogsum 26030 The sum of ` log p / p ` o...
dirith2 26031 Dirichlet's theorem: there...
dirith 26032 Dirichlet's theorem: there...
mudivsum 26033 Asymptotic formula for ` s...
mulogsumlem 26034 Lemma for ~ mulogsum . (C...
mulogsum 26035 Asymptotic formula for ...
logdivsum 26036 Asymptotic analysis of ...
mulog2sumlem1 26037 Asymptotic formula for ...
mulog2sumlem2 26038 Lemma for ~ mulog2sum . (...
mulog2sumlem3 26039 Lemma for ~ mulog2sum . (...
mulog2sum 26040 Asymptotic formula for ...
vmalogdivsum2 26041 The sum ` sum_ n <_ x , La...
vmalogdivsum 26042 The sum ` sum_ n <_ x , La...
2vmadivsumlem 26043 Lemma for ~ 2vmadivsum . ...
2vmadivsum 26044 The sum ` sum_ m n <_ x , ...
logsqvma 26045 A formula for ` log ^ 2 ( ...
logsqvma2 26046 The Möbius inverse of...
log2sumbnd 26047 Bound on the difference be...
selberglem1 26048 Lemma for ~ selberg . Est...
selberglem2 26049 Lemma for ~ selberg . (Co...
selberglem3 26050 Lemma for ~ selberg . Est...
selberg 26051 Selberg's symmetry formula...
selbergb 26052 Convert eventual boundedne...
selberg2lem 26053 Lemma for ~ selberg2 . Eq...
selberg2 26054 Selberg's symmetry formula...
selberg2b 26055 Convert eventual boundedne...
chpdifbndlem1 26056 Lemma for ~ chpdifbnd . (...
chpdifbndlem2 26057 Lemma for ~ chpdifbnd . (...
chpdifbnd 26058 A bound on the difference ...
logdivbnd 26059 A bound on a sum of logs, ...
selberg3lem1 26060 Introduce a log weighting ...
selberg3lem2 26061 Lemma for ~ selberg3 . Eq...
selberg3 26062 Introduce a log weighting ...
selberg4lem1 26063 Lemma for ~ selberg4 . Eq...
selberg4 26064 The Selberg symmetry formu...
pntrval 26065 Define the residual of the...
pntrf 26066 Functionality of the resid...
pntrmax 26067 There is a bound on the re...
pntrsumo1 26068 A bound on a sum over ` R ...
pntrsumbnd 26069 A bound on a sum over ` R ...
pntrsumbnd2 26070 A bound on a sum over ` R ...
selbergr 26071 Selberg's symmetry formula...
selberg3r 26072 Selberg's symmetry formula...
selberg4r 26073 Selberg's symmetry formula...
selberg34r 26074 The sum of ~ selberg3r and...
pntsval 26075 Define the "Selberg functi...
pntsf 26076 Functionality of the Selbe...
selbergs 26077 Selberg's symmetry formula...
selbergsb 26078 Selberg's symmetry formula...
pntsval2 26079 The Selberg function can b...
pntrlog2bndlem1 26080 The sum of ~ selberg3r and...
pntrlog2bndlem2 26081 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem3 26082 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem4 26083 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem5 26084 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem6a 26085 Lemma for ~ pntrlog2bndlem...
pntrlog2bndlem6 26086 Lemma for ~ pntrlog2bnd . ...
pntrlog2bnd 26087 A bound on ` R ( x ) log ^...
pntpbnd1a 26088 Lemma for ~ pntpbnd . (Co...
pntpbnd1 26089 Lemma for ~ pntpbnd . (Co...
pntpbnd2 26090 Lemma for ~ pntpbnd . (Co...
pntpbnd 26091 Lemma for ~ pnt . Establi...
pntibndlem1 26092 Lemma for ~ pntibnd . (Co...
pntibndlem2a 26093 Lemma for ~ pntibndlem2 . ...
pntibndlem2 26094 Lemma for ~ pntibnd . The...
pntibndlem3 26095 Lemma for ~ pntibnd . Pac...
pntibnd 26096 Lemma for ~ pnt . Establi...
pntlemd 26097 Lemma for ~ pnt . Closure...
pntlemc 26098 Lemma for ~ pnt . Closure...
pntlema 26099 Lemma for ~ pnt . Closure...
pntlemb 26100 Lemma for ~ pnt . Unpack ...
pntlemg 26101 Lemma for ~ pnt . Closure...
pntlemh 26102 Lemma for ~ pnt . Bounds ...
pntlemn 26103 Lemma for ~ pnt . The "na...
pntlemq 26104 Lemma for ~ pntlemj . (Co...
pntlemr 26105 Lemma for ~ pntlemj . (Co...
pntlemj 26106 Lemma for ~ pnt . The ind...
pntlemi 26107 Lemma for ~ pnt . Elimina...
pntlemf 26108 Lemma for ~ pnt . Add up ...
pntlemk 26109 Lemma for ~ pnt . Evaluat...
pntlemo 26110 Lemma for ~ pnt . Combine...
pntleme 26111 Lemma for ~ pnt . Package...
pntlem3 26112 Lemma for ~ pnt . Equatio...
pntlemp 26113 Lemma for ~ pnt . Wrappin...
pntleml 26114 Lemma for ~ pnt . Equatio...
pnt3 26115 The Prime Number Theorem, ...
pnt2 26116 The Prime Number Theorem, ...
pnt 26117 The Prime Number Theorem: ...
abvcxp 26118 Raising an absolute value ...
padicfval 26119 Value of the p-adic absolu...
padicval 26120 Value of the p-adic absolu...
ostth2lem1 26121 Lemma for ~ ostth2 , altho...
qrngbas 26122 The base set of the field ...
qdrng 26123 The rationals form a divis...
qrng0 26124 The zero element of the fi...
qrng1 26125 The unit element of the fi...
qrngneg 26126 The additive inverse in th...
qrngdiv 26127 The division operation in ...
qabvle 26128 By using induction on ` N ...
qabvexp 26129 Induct the product rule ~ ...
ostthlem1 26130 Lemma for ~ ostth . If tw...
ostthlem2 26131 Lemma for ~ ostth . Refin...
qabsabv 26132 The regular absolute value...
padicabv 26133 The p-adic absolute value ...
padicabvf 26134 The p-adic absolute value ...
padicabvcxp 26135 All positive powers of the...
ostth1 26136 - Lemma for ~ ostth : triv...
ostth2lem2 26137 Lemma for ~ ostth2 . (Con...
ostth2lem3 26138 Lemma for ~ ostth2 . (Con...
ostth2lem4 26139 Lemma for ~ ostth2 . (Con...
ostth2 26140 - Lemma for ~ ostth : regu...
ostth3 26141 - Lemma for ~ ostth : p-ad...
ostth 26142 Ostrowski's theorem, which...
itvndx 26153 Index value of the Interva...
lngndx 26154 Index value of the "line" ...
itvid 26155 Utility theorem: index-ind...
lngid 26156 Utility theorem: index-ind...
trkgstr 26157 Functionality of a Tarski ...
trkgbas 26158 The base set of a Tarski g...
trkgdist 26159 The measure of a distance ...
trkgitv 26160 The congruence relation in...
istrkgc 26167 Property of being a Tarski...
istrkgb 26168 Property of being a Tarski...
istrkgcb 26169 Property of being a Tarski...
istrkge 26170 Property of fulfilling Euc...
istrkgl 26171 Building lines from the se...
istrkgld 26172 Property of fulfilling the...
istrkg2ld 26173 Property of fulfilling the...
istrkg3ld 26174 Property of fulfilling the...
axtgcgrrflx 26175 Axiom of reflexivity of co...
axtgcgrid 26176 Axiom of identity of congr...
axtgsegcon 26177 Axiom of segment construct...
axtg5seg 26178 Five segments axiom, Axiom...
axtgbtwnid 26179 Identity of Betweenness. ...
axtgpasch 26180 Axiom of (Inner) Pasch, Ax...
axtgcont1 26181 Axiom of Continuity. Axio...
axtgcont 26182 Axiom of Continuity. Axio...
axtglowdim2 26183 Lower dimension axiom for ...
axtgupdim2 26184 Upper dimension axiom for ...
axtgeucl 26185 Euclid's Axiom. Axiom A10...
tgjustf 26186 Given any function ` F ` ,...
tgjustr 26187 Given any equivalence rela...
tgjustc1 26188 A justification for using ...
tgjustc2 26189 A justification for using ...
tgcgrcomimp 26190 Congruence commutes on the...
tgcgrcomr 26191 Congruence commutes on the...
tgcgrcoml 26192 Congruence commutes on the...
tgcgrcomlr 26193 Congruence commutes on bot...
tgcgreqb 26194 Congruence and equality. ...
tgcgreq 26195 Congruence and equality. ...
tgcgrneq 26196 Congruence and equality. ...
tgcgrtriv 26197 Degenerate segments are co...
tgcgrextend 26198 Link congruence over a pai...
tgsegconeq 26199 Two points that satisfy th...
tgbtwntriv2 26200 Betweenness always holds f...
tgbtwncom 26201 Betweenness commutes. The...
tgbtwncomb 26202 Betweenness commutes, bico...
tgbtwnne 26203 Betweenness and inequality...
tgbtwntriv1 26204 Betweenness always holds f...
tgbtwnswapid 26205 If you can swap the first ...
tgbtwnintr 26206 Inner transitivity law for...
tgbtwnexch3 26207 Exchange the first endpoin...
tgbtwnouttr2 26208 Outer transitivity law for...
tgbtwnexch2 26209 Exchange the outer point o...
tgbtwnouttr 26210 Outer transitivity law for...
tgbtwnexch 26211 Outer transitivity law for...
tgtrisegint 26212 A line segment between two...
tglowdim1 26213 Lower dimension axiom for ...
tglowdim1i 26214 Lower dimension axiom for ...
tgldimor 26215 Excluded-middle like state...
tgldim0eq 26216 In dimension zero, any two...
tgldim0itv 26217 In dimension zero, any two...
tgldim0cgr 26218 In dimension zero, any two...
tgbtwndiff 26219 There is always a ` c ` di...
tgdim01 26220 In geometries of dimension...
tgifscgr 26221 Inner five segment congrue...
tgcgrsub 26222 Removing identical parts f...
iscgrg 26225 The congruence property fo...
iscgrgd 26226 The property for two seque...
iscgrglt 26227 The property for two seque...
trgcgrg 26228 The property for two trian...
trgcgr 26229 Triangle congruence. (Con...
ercgrg 26230 The shape congruence relat...
tgcgrxfr 26231 A line segment can be divi...
cgr3id 26232 Reflexivity law for three-...
cgr3simp1 26233 Deduce segment congruence ...
cgr3simp2 26234 Deduce segment congruence ...
cgr3simp3 26235 Deduce segment congruence ...
cgr3swap12 26236 Permutation law for three-...
cgr3swap23 26237 Permutation law for three-...
cgr3swap13 26238 Permutation law for three-...
cgr3rotr 26239 Permutation law for three-...
cgr3rotl 26240 Permutation law for three-...
trgcgrcom 26241 Commutative law for three-...
cgr3tr 26242 Transitivity law for three...
tgbtwnxfr 26243 A condition for extending ...
tgcgr4 26244 Two quadrilaterals to be c...
isismt 26247 Property of being an isome...
ismot 26248 Property of being an isome...
motcgr 26249 Property of a motion: dist...
idmot 26250 The identity is a motion. ...
motf1o 26251 Motions are bijections. (...
motcl 26252 Closure of motions. (Cont...
motco 26253 The composition of two mot...
cnvmot 26254 The converse of a motion i...
motplusg 26255 The operation for motions ...
motgrp 26256 The motions of a geometry ...
motcgrg 26257 Property of a motion: dist...
motcgr3 26258 Property of a motion: dist...
tglng 26259 Lines of a Tarski Geometry...
tglnfn 26260 Lines as functions. (Cont...
tglnunirn 26261 Lines are sets of points. ...
tglnpt 26262 Lines are sets of points. ...
tglngne 26263 It takes two different poi...
tglngval 26264 The line going through poi...
tglnssp 26265 Lines are subset of the ge...
tgellng 26266 Property of lying on the l...
tgcolg 26267 We choose the notation ` (...
btwncolg1 26268 Betweenness implies coline...
btwncolg2 26269 Betweenness implies coline...
btwncolg3 26270 Betweenness implies coline...
colcom 26271 Swapping the points defini...
colrot1 26272 Rotating the points defini...
colrot2 26273 Rotating the points defini...
ncolcom 26274 Swapping non-colinear poin...
ncolrot1 26275 Rotating non-colinear poin...
ncolrot2 26276 Rotating non-colinear poin...
tgdim01ln 26277 In geometries of dimension...
ncoltgdim2 26278 If there are three non-col...
lnxfr 26279 Transfer law for colineari...
lnext 26280 Extend a line with a missi...
tgfscgr 26281 Congruence law for the gen...
lncgr 26282 Congruence rule for lines....
lnid 26283 Identity law for points on...
tgidinside 26284 Law for finding a point in...
tgbtwnconn1lem1 26285 Lemma for ~ tgbtwnconn1 . ...
tgbtwnconn1lem2 26286 Lemma for ~ tgbtwnconn1 . ...
tgbtwnconn1lem3 26287 Lemma for ~ tgbtwnconn1 . ...
tgbtwnconn1 26288 Connectivity law for betwe...
tgbtwnconn2 26289 Another connectivity law f...
tgbtwnconn3 26290 Inner connectivity law for...
tgbtwnconnln3 26291 Derive colinearity from be...
tgbtwnconn22 26292 Double connectivity law fo...
tgbtwnconnln1 26293 Derive colinearity from be...
tgbtwnconnln2 26294 Derive colinearity from be...
legval 26297 Value of the less-than rel...
legov 26298 Value of the less-than rel...
legov2 26299 An equivalent definition o...
legid 26300 Reflexivity of the less-th...
btwnleg 26301 Betweenness implies less-t...
legtrd 26302 Transitivity of the less-t...
legtri3 26303 Equality from the less-tha...
legtrid 26304 Trichotomy law for the les...
leg0 26305 Degenerated (zero-length) ...
legeq 26306 Deduce equality from "less...
legbtwn 26307 Deduce betweenness from "l...
tgcgrsub2 26308 Removing identical parts f...
ltgseg 26309 The set ` E ` denotes the ...
ltgov 26310 Strict "shorter than" geom...
legov3 26311 An equivalent definition o...
legso 26312 The "shorter than" relatio...
ishlg 26315 Rays : Definition 6.1 of ...
hlcomb 26316 The half-line relation com...
hlcomd 26317 The half-line relation com...
hlne1 26318 The half-line relation imp...
hlne2 26319 The half-line relation imp...
hlln 26320 The half-line relation imp...
hleqnid 26321 The endpoint does not belo...
hlid 26322 The half-line relation is ...
hltr 26323 The half-line relation is ...
hlbtwn 26324 Betweenness is a sufficien...
btwnhl1 26325 Deduce half-line from betw...
btwnhl2 26326 Deduce half-line from betw...
btwnhl 26327 Swap betweenness for a hal...
lnhl 26328 Either a point ` C ` on th...
hlcgrex 26329 Construct a point on a hal...
hlcgreulem 26330 Lemma for ~ hlcgreu . (Co...
hlcgreu 26331 The point constructed in ~...
btwnlng1 26332 Betweenness implies coline...
btwnlng2 26333 Betweenness implies coline...
btwnlng3 26334 Betweenness implies coline...
lncom 26335 Swapping the points defini...
lnrot1 26336 Rotating the points defini...
lnrot2 26337 Rotating the points defini...
ncolne1 26338 Non-colinear points are di...
ncolne2 26339 Non-colinear points are di...
tgisline 26340 The property of being a pr...
tglnne 26341 It takes two different poi...
tglndim0 26342 There are no lines in dime...
tgelrnln 26343 The property of being a pr...
tglineeltr 26344 Transitivity law for lines...
tglineelsb2 26345 If ` S ` lies on PQ , then...
tglinerflx1 26346 Reflexivity law for line m...
tglinerflx2 26347 Reflexivity law for line m...
tglinecom 26348 Commutativity law for line...
tglinethru 26349 If ` A ` is a line contain...
tghilberti1 26350 There is a line through an...
tghilberti2 26351 There is at most one line ...
tglinethrueu 26352 There is a unique line goi...
tglnne0 26353 A line ` A ` has at least ...
tglnpt2 26354 Find a second point on a l...
tglineintmo 26355 Two distinct lines interse...
tglineineq 26356 Two distinct lines interse...
tglineneq 26357 Given three non-colinear p...
tglineinteq 26358 Two distinct lines interse...
ncolncol 26359 Deduce non-colinearity fro...
coltr 26360 A transitivity law for col...
coltr3 26361 A transitivity law for col...
colline 26362 Three points are colinear ...
tglowdim2l 26363 Reformulation of the lower...
tglowdim2ln 26364 There is always one point ...
mirreu3 26367 Existential uniqueness of ...
mirval 26368 Value of the point inversi...
mirfv 26369 Value of the point inversi...
mircgr 26370 Property of the image by t...
mirbtwn 26371 Property of the image by t...
ismir 26372 Property of the image by t...
mirf 26373 Point inversion as functio...
mircl 26374 Closure of the point inver...
mirmir 26375 The point inversion functi...
mircom 26376 Variation on ~ mirmir . (...
mirreu 26377 Any point has a unique ant...
mireq 26378 Equality deduction for poi...
mirinv 26379 The only invariant point o...
mirne 26380 Mirror of non-center point...
mircinv 26381 The center point is invari...
mirf1o 26382 The point inversion functi...
miriso 26383 The point inversion functi...
mirbtwni 26384 Point inversion preserves ...
mirbtwnb 26385 Point inversion preserves ...
mircgrs 26386 Point inversion preserves ...
mirmir2 26387 Point inversion of a point...
mirmot 26388 Point investion is a motio...
mirln 26389 If two points are on the s...
mirln2 26390 If a point and its mirror ...
mirconn 26391 Point inversion of connect...
mirhl 26392 If two points ` X ` and ` ...
mirbtwnhl 26393 If the center of the point...
mirhl2 26394 Deduce half-line relation ...
mircgrextend 26395 Link congruence over a pai...
mirtrcgr 26396 Point inversion of one poi...
mirauto 26397 Point inversion preserves ...
miduniq 26398 Uniqueness of the middle p...
miduniq1 26399 Uniqueness of the middle p...
miduniq2 26400 If two point inversions co...
colmid 26401 Colinearity and equidistan...
symquadlem 26402 Lemma of the symetrial qua...
krippenlem 26403 Lemma for ~ krippen . We ...
krippen 26404 Krippenlemma (German for c...
midexlem 26405 Lemma for the existence of...
israg 26410 Property for 3 points A, B...
ragcom 26411 Commutative rule for right...
ragcol 26412 The right angle property i...
ragmir 26413 Right angle property is pr...
mirrag 26414 Right angle is conserved b...
ragtrivb 26415 Trivial right angle. Theo...
ragflat2 26416 Deduce equality from two r...
ragflat 26417 Deduce equality from two r...
ragtriva 26418 Trivial right angle. Theo...
ragflat3 26419 Right angle and colinearit...
ragcgr 26420 Right angle and colinearit...
motrag 26421 Right angles are preserved...
ragncol 26422 Right angle implies non-co...
perpln1 26423 Derive a line from perpend...
perpln2 26424 Derive a line from perpend...
isperp 26425 Property for 2 lines A, B ...
perpcom 26426 The "perpendicular" relati...
perpneq 26427 Two perpendicular lines ar...
isperp2 26428 Property for 2 lines A, B,...
isperp2d 26429 One direction of ~ isperp2...
ragperp 26430 Deduce that two lines are ...
footexALT 26431 Alternative version of ~ f...
footexlem1 26432 Lemma for ~ footex (Contri...
footexlem2 26433 Lemma for ~ footex (Contri...
footex 26434 From a point ` C ` outside...
foot 26435 From a point ` C ` outside...
footne 26436 Uniqueness of the foot poi...
footeq 26437 Uniqueness of the foot poi...
hlperpnel 26438 A point on a half-line whi...
perprag 26439 Deduce a right angle from ...
perpdragALT 26440 Deduce a right angle from ...
perpdrag 26441 Deduce a right angle from ...
colperp 26442 Deduce a perpendicularity ...
colperpexlem1 26443 Lemma for ~ colperp . Fir...
colperpexlem2 26444 Lemma for ~ colperpex . S...
colperpexlem3 26445 Lemma for ~ colperpex . C...
colperpex 26446 In dimension 2 and above, ...
mideulem2 26447 Lemma for ~ opphllem , whi...
opphllem 26448 Lemma 8.24 of [Schwabhause...
mideulem 26449 Lemma for ~ mideu . We ca...
midex 26450 Existence of the midpoint,...
mideu 26451 Existence and uniqueness o...
islnopp 26452 The property for two point...
islnoppd 26453 Deduce that ` A ` and ` B ...
oppne1 26454 Points lying on opposite s...
oppne2 26455 Points lying on opposite s...
oppne3 26456 Points lying on opposite s...
oppcom 26457 Commutativity rule for "op...
opptgdim2 26458 If two points opposite to ...
oppnid 26459 The "opposite to a line" r...
opphllem1 26460 Lemma for ~ opphl . (Cont...
opphllem2 26461 Lemma for ~ opphl . Lemma...
opphllem3 26462 Lemma for ~ opphl : We as...
opphllem4 26463 Lemma for ~ opphl . (Cont...
opphllem5 26464 Second part of Lemma 9.4 o...
opphllem6 26465 First part of Lemma 9.4 of...
oppperpex 26466 Restating ~ colperpex usin...
opphl 26467 If two points ` A ` and ` ...
outpasch 26468 Axiom of Pasch, outer form...
hlpasch 26469 An application of the axio...
ishpg 26472 Value of the half-plane re...
hpgbr 26473 Half-planes : property for...
hpgne1 26474 Points on the open half pl...
hpgne2 26475 Points on the open half pl...
lnopp2hpgb 26476 Theorem 9.8 of [Schwabhaus...
lnoppnhpg 26477 If two points lie on the o...
hpgerlem 26478 Lemma for the proof that t...
hpgid 26479 The half-plane relation is...
hpgcom 26480 The half-plane relation co...
hpgtr 26481 The half-plane relation is...
colopp 26482 Opposite sides of a line f...
colhp 26483 Half-plane relation for co...
hphl 26484 If two points are on the s...
midf 26489 Midpoint as a function. (...
midcl 26490 Closure of the midpoint. ...
ismidb 26491 Property of the midpoint. ...
midbtwn 26492 Betweenness of midpoint. ...
midcgr 26493 Congruence of midpoint. (...
midid 26494 Midpoint of a null segment...
midcom 26495 Commutativity rule for the...
mirmid 26496 Point inversion preserves ...
lmieu 26497 Uniqueness of the line mir...
lmif 26498 Line mirror as a function....
lmicl 26499 Closure of the line mirror...
islmib 26500 Property of the line mirro...
lmicom 26501 The line mirroring functio...
lmilmi 26502 Line mirroring is an invol...
lmireu 26503 Any point has a unique ant...
lmieq 26504 Equality deduction for lin...
lmiinv 26505 The invariants of the line...
lmicinv 26506 The mirroring line is an i...
lmimid 26507 If we have a right angle, ...
lmif1o 26508 The line mirroring functio...
lmiisolem 26509 Lemma for ~ lmiiso . (Con...
lmiiso 26510 The line mirroring functio...
lmimot 26511 Line mirroring is a motion...
hypcgrlem1 26512 Lemma for ~ hypcgr , case ...
hypcgrlem2 26513 Lemma for ~ hypcgr , case ...
hypcgr 26514 If the catheti of two righ...
lmiopp 26515 Line mirroring produces po...
lnperpex 26516 Existence of a perpendicul...
trgcopy 26517 Triangle construction: a c...
trgcopyeulem 26518 Lemma for ~ trgcopyeu . (...
trgcopyeu 26519 Triangle construction: a c...
iscgra 26522 Property for two angles AB...
iscgra1 26523 A special version of ~ isc...
iscgrad 26524 Sufficient conditions for ...
cgrane1 26525 Angles imply inequality. ...
cgrane2 26526 Angles imply inequality. ...
cgrane3 26527 Angles imply inequality. ...
cgrane4 26528 Angles imply inequality. ...
cgrahl1 26529 Angle congruence is indepe...
cgrahl2 26530 Angle congruence is indepe...
cgracgr 26531 First direction of proposi...
cgraid 26532 Angle congruence is reflex...
cgraswap 26533 Swap rays in a congruence ...
cgrcgra 26534 Triangle congruence implie...
cgracom 26535 Angle congruence commutes....
cgratr 26536 Angle congruence is transi...
flatcgra 26537 Flat angles are congruent....
cgraswaplr 26538 Swap both side of angle co...
cgrabtwn 26539 Angle congruence preserves...
cgrahl 26540 Angle congruence preserves...
cgracol 26541 Angle congruence preserves...
cgrancol 26542 Angle congruence preserves...
dfcgra2 26543 This is the full statement...
sacgr 26544 Supplementary angles of co...
oacgr 26545 Vertical angle theorem. V...
acopy 26546 Angle construction. Theor...
acopyeu 26547 Angle construction. Theor...
isinag 26551 Property for point ` X ` t...
isinagd 26552 Sufficient conditions for ...
inagflat 26553 Any point lies in a flat a...
inagswap 26554 Swap the order of the half...
inagne1 26555 Deduce inequality from the...
inagne2 26556 Deduce inequality from the...
inagne3 26557 Deduce inequality from the...
inaghl 26558 The "point lie in angle" r...
isleag 26560 Geometrical "less than" pr...
isleagd 26561 Sufficient condition for "...
leagne1 26562 Deduce inequality from the...
leagne2 26563 Deduce inequality from the...
leagne3 26564 Deduce inequality from the...
leagne4 26565 Deduce inequality from the...
cgrg3col4 26566 Lemma 11.28 of [Schwabhaus...
tgsas1 26567 First congruence theorem: ...
tgsas 26568 First congruence theorem: ...
tgsas2 26569 First congruence theorem: ...
tgsas3 26570 First congruence theorem: ...
tgasa1 26571 Second congruence theorem:...
tgasa 26572 Second congruence theorem:...
tgsss1 26573 Third congruence theorem: ...
tgsss2 26574 Third congruence theorem: ...
tgsss3 26575 Third congruence theorem: ...
dfcgrg2 26576 Congruence for two triangl...
isoas 26577 Congruence theorem for iso...
iseqlg 26580 Property of a triangle bei...
iseqlgd 26581 Condition for a triangle t...
f1otrgds 26582 Convenient lemma for ~ f1o...
f1otrgitv 26583 Convenient lemma for ~ f1o...
f1otrg 26584 A bijection between bases ...
f1otrge 26585 A bijection between bases ...
ttgval 26588 Define a function to augme...
ttglem 26589 Lemma for ~ ttgbas and ~ t...
ttgbas 26590 The base set of a subcompl...
ttgplusg 26591 The addition operation of ...
ttgsub 26592 The subtraction operation ...
ttgvsca 26593 The scalar product of a su...
ttgds 26594 The metric of a subcomplex...
ttgitvval 26595 Betweenness for a subcompl...
ttgelitv 26596 Betweenness for a subcompl...
ttgbtwnid 26597 Any subcomplex module equi...
ttgcontlem1 26598 Lemma for % ttgcont . (Co...
xmstrkgc 26599 Any metric space fulfills ...
cchhllem 26600 Lemma for chlbas and chlvs...
elee 26607 Membership in a Euclidean ...
mptelee 26608 A condition for a mapping ...
eleenn 26609 If ` A ` is in ` ( EE `` N...
eleei 26610 The forward direction of ~...
eedimeq 26611 A point belongs to at most...
brbtwn 26612 The binary relation form o...
brcgr 26613 The binary relation form o...
fveere 26614 The function value of a po...
fveecn 26615 The function value of a po...
eqeefv 26616 Two points are equal iff t...
eqeelen 26617 Two points are equal iff t...
brbtwn2 26618 Alternate characterization...
colinearalglem1 26619 Lemma for ~ colinearalg . ...
colinearalglem2 26620 Lemma for ~ colinearalg . ...
colinearalglem3 26621 Lemma for ~ colinearalg . ...
colinearalglem4 26622 Lemma for ~ colinearalg . ...
colinearalg 26623 An algebraic characterizat...
eleesub 26624 Membership of a subtractio...
eleesubd 26625 Membership of a subtractio...
axdimuniq 26626 The unique dimension axiom...
axcgrrflx 26627 ` A ` is as far from ` B `...
axcgrtr 26628 Congruence is transitive. ...
axcgrid 26629 If there is no distance be...
axsegconlem1 26630 Lemma for ~ axsegcon . Ha...
axsegconlem2 26631 Lemma for ~ axsegcon . Sh...
axsegconlem3 26632 Lemma for ~ axsegcon . Sh...
axsegconlem4 26633 Lemma for ~ axsegcon . Sh...
axsegconlem5 26634 Lemma for ~ axsegcon . Sh...
axsegconlem6 26635 Lemma for ~ axsegcon . Sh...
axsegconlem7 26636 Lemma for ~ axsegcon . Sh...
axsegconlem8 26637 Lemma for ~ axsegcon . Sh...
axsegconlem9 26638 Lemma for ~ axsegcon . Sh...
axsegconlem10 26639 Lemma for ~ axsegcon . Sh...
axsegcon 26640 Any segment ` A B ` can be...
ax5seglem1 26641 Lemma for ~ ax5seg . Rexp...
ax5seglem2 26642 Lemma for ~ ax5seg . Rexp...
ax5seglem3a 26643 Lemma for ~ ax5seg . (Con...
ax5seglem3 26644 Lemma for ~ ax5seg . Comb...
ax5seglem4 26645 Lemma for ~ ax5seg . Give...
ax5seglem5 26646 Lemma for ~ ax5seg . If `...
ax5seglem6 26647 Lemma for ~ ax5seg . Give...
ax5seglem7 26648 Lemma for ~ ax5seg . An a...
ax5seglem8 26649 Lemma for ~ ax5seg . Use ...
ax5seglem9 26650 Lemma for ~ ax5seg . Take...
ax5seg 26651 The five segment axiom. T...
axbtwnid 26652 Points are indivisible. T...
axpaschlem 26653 Lemma for ~ axpasch . Set...
axpasch 26654 The inner Pasch axiom. Ta...
axlowdimlem1 26655 Lemma for ~ axlowdim . Es...
axlowdimlem2 26656 Lemma for ~ axlowdim . Sh...
axlowdimlem3 26657 Lemma for ~ axlowdim . Se...
axlowdimlem4 26658 Lemma for ~ axlowdim . Se...
axlowdimlem5 26659 Lemma for ~ axlowdim . Sh...
axlowdimlem6 26660 Lemma for ~ axlowdim . Sh...
axlowdimlem7 26661 Lemma for ~ axlowdim . Se...
axlowdimlem8 26662 Lemma for ~ axlowdim . Ca...
axlowdimlem9 26663 Lemma for ~ axlowdim . Ca...
axlowdimlem10 26664 Lemma for ~ axlowdim . Se...
axlowdimlem11 26665 Lemma for ~ axlowdim . Ca...
axlowdimlem12 26666 Lemma for ~ axlowdim . Ca...
axlowdimlem13 26667 Lemma for ~ axlowdim . Es...
axlowdimlem14 26668 Lemma for ~ axlowdim . Ta...
axlowdimlem15 26669 Lemma for ~ axlowdim . Se...
axlowdimlem16 26670 Lemma for ~ axlowdim . Se...
axlowdimlem17 26671 Lemma for ~ axlowdim . Es...
axlowdim1 26672 The lower dimension axiom ...
axlowdim2 26673 The lower two-dimensional ...
axlowdim 26674 The general lower dimensio...
axeuclidlem 26675 Lemma for ~ axeuclid . Ha...
axeuclid 26676 Euclid's axiom. Take an a...
axcontlem1 26677 Lemma for ~ axcont . Chan...
axcontlem2 26678 Lemma for ~ axcont . The ...
axcontlem3 26679 Lemma for ~ axcont . Give...
axcontlem4 26680 Lemma for ~ axcont . Give...
axcontlem5 26681 Lemma for ~ axcont . Comp...
axcontlem6 26682 Lemma for ~ axcont . Stat...
axcontlem7 26683 Lemma for ~ axcont . Give...
axcontlem8 26684 Lemma for ~ axcont . A po...
axcontlem9 26685 Lemma for ~ axcont . Give...
axcontlem10 26686 Lemma for ~ axcont . Give...
axcontlem11 26687 Lemma for ~ axcont . Elim...
axcontlem12 26688 Lemma for ~ axcont . Elim...
axcont 26689 The axiom of continuity. ...
eengv 26692 The value of the Euclidean...
eengstr 26693 The Euclidean geometry as ...
eengbas 26694 The Base of the Euclidean ...
ebtwntg 26695 The betweenness relation u...
ecgrtg 26696 The congruence relation us...
elntg 26697 The line definition in the...
elntg2 26698 The line definition in the...
eengtrkg 26699 The geometry structure for...
eengtrkge 26700 The geometry structure for...
edgfid 26703 Utility theorem: index-ind...
edgfndxnn 26704 The index value of the edg...
edgfndxid 26705 The value of the edge func...
baseltedgf 26706 The index value of the ` B...
slotsbaseefdif 26707 The slots ` Base ` and ` ....
vtxval 26712 The set of vertices of a g...
iedgval 26713 The set of indexed edges o...
1vgrex 26714 A graph with at least one ...
opvtxval 26715 The set of vertices of a g...
opvtxfv 26716 The set of vertices of a g...
opvtxov 26717 The set of vertices of a g...
opiedgval 26718 The set of indexed edges o...
opiedgfv 26719 The set of indexed edges o...
opiedgov 26720 The set of indexed edges o...
opvtxfvi 26721 The set of vertices of a g...
opiedgfvi 26722 The set of indexed edges o...
funvtxdmge2val 26723 The set of vertices of an ...
funiedgdmge2val 26724 The set of indexed edges o...
funvtxdm2val 26725 The set of vertices of an ...
funiedgdm2val 26726 The set of indexed edges o...
funvtxval0 26727 The set of vertices of an ...
basvtxval 26728 The set of vertices of a g...
edgfiedgval 26729 The set of indexed edges o...
funvtxval 26730 The set of vertices of a g...
funiedgval 26731 The set of indexed edges o...
structvtxvallem 26732 Lemma for ~ structvtxval a...
structvtxval 26733 The set of vertices of an ...
structiedg0val 26734 The set of indexed edges o...
structgrssvtxlem 26735 Lemma for ~ structgrssvtx ...
structgrssvtx 26736 The set of vertices of a g...
structgrssiedg 26737 The set of indexed edges o...
struct2grstr 26738 A graph represented as an ...
struct2grvtx 26739 The set of vertices of a g...
struct2griedg 26740 The set of indexed edges o...
graop 26741 Any representation of a gr...
grastruct 26742 Any representation of a gr...
gropd 26743 If any representation of a...
grstructd 26744 If any representation of a...
gropeld 26745 If any representation of a...
grstructeld 26746 If any representation of a...
setsvtx 26747 The vertices of a structur...
setsiedg 26748 The (indexed) edges of a s...
snstrvtxval 26749 The set of vertices of a g...
snstriedgval 26750 The set of indexed edges o...
vtxval0 26751 Degenerated case 1 for ver...
iedgval0 26752 Degenerated case 1 for edg...
vtxvalsnop 26753 Degenerated case 2 for ver...
iedgvalsnop 26754 Degenerated case 2 for edg...
vtxval3sn 26755 Degenerated case 3 for ver...
iedgval3sn 26756 Degenerated case 3 for edg...
vtxvalprc 26757 Degenerated case 4 for ver...
iedgvalprc 26758 Degenerated case 4 for edg...
edgval 26761 The edges of a graph. (Co...
iedgedg 26762 An indexed edge is an edge...
edgopval 26763 The edges of a graph repre...
edgov 26764 The edges of a graph repre...
edgstruct 26765 The edges of a graph repre...
edgiedgb 26766 A set is an edge iff it is...
edg0iedg0 26767 There is no edge in a grap...
isuhgr 26772 The predicate "is an undir...
isushgr 26773 The predicate "is an undir...
uhgrf 26774 The edge function of an un...
ushgrf 26775 The edge function of an un...
uhgrss 26776 An edge is a subset of ver...
uhgreq12g 26777 If two sets have the same ...
uhgrfun 26778 The edge function of an un...
uhgrn0 26779 An edge is a nonempty subs...
lpvtx 26780 The endpoints of a loop (w...
ushgruhgr 26781 An undirected simple hyper...
isuhgrop 26782 The property of being an u...
uhgr0e 26783 The empty graph, with vert...
uhgr0vb 26784 The null graph, with no ve...
uhgr0 26785 The null graph represented...
uhgrun 26786 The union ` U ` of two (un...
uhgrunop 26787 The union of two (undirect...
ushgrun 26788 The union ` U ` of two (un...
ushgrunop 26789 The union of two (undirect...
uhgrstrrepe 26790 Replacing (or adding) the ...
incistruhgr 26791 An _incidence structure_ `...
isupgr 26796 The property of being an u...
wrdupgr 26797 The property of being an u...
upgrf 26798 The edge function of an un...
upgrfn 26799 The edge function of an un...
upgrss 26800 An edge is a subset of ver...
upgrn0 26801 An edge is a nonempty subs...
upgrle 26802 An edge of an undirected p...
upgrfi 26803 An edge is a finite subset...
upgrex 26804 An edge is an unordered pa...
upgrbi 26805 Show that an unordered pai...
upgrop 26806 A pseudograph represented ...
isumgr 26807 The property of being an u...
isumgrs 26808 The simplified property of...
wrdumgr 26809 The property of being an u...
umgrf 26810 The edge function of an un...
umgrfn 26811 The edge function of an un...
umgredg2 26812 An edge of a multigraph ha...
umgrbi 26813 Show that an unordered pai...
upgruhgr 26814 An undirected pseudograph ...
umgrupgr 26815 An undirected multigraph i...
umgruhgr 26816 An undirected multigraph i...
upgrle2 26817 An edge of an undirected p...
umgrnloopv 26818 In a multigraph, there is ...
umgredgprv 26819 In a multigraph, an edge i...
umgrnloop 26820 In a multigraph, there is ...
umgrnloop0 26821 A multigraph has no loops....
umgr0e 26822 The empty graph, with vert...
upgr0e 26823 The empty graph, with vert...
upgr1elem 26824 Lemma for ~ upgr1e and ~ u...
upgr1e 26825 A pseudograph with one edg...
upgr0eop 26826 The empty graph, with vert...
upgr1eop 26827 A pseudograph with one edg...
upgr0eopALT 26828 Alternate proof of ~ upgr0...
upgr1eopALT 26829 Alternate proof of ~ upgr1...
upgrun 26830 The union ` U ` of two pse...
upgrunop 26831 The union of two pseudogra...
umgrun 26832 The union ` U ` of two mul...
umgrunop 26833 The union of two multigrap...
umgrislfupgrlem 26834 Lemma for ~ umgrislfupgr a...
umgrislfupgr 26835 A multigraph is a loop-fre...
lfgredgge2 26836 An edge of a loop-free gra...
lfgrnloop 26837 A loop-free graph has no l...
uhgredgiedgb 26838 In a hypergraph, a set is ...
uhgriedg0edg0 26839 A hypergraph has no edges ...
uhgredgn0 26840 An edge of a hypergraph is...
edguhgr 26841 An edge of a hypergraph is...
uhgredgrnv 26842 An edge of a hypergraph co...
uhgredgss 26843 The set of edges of a hype...
upgredgss 26844 The set of edges of a pseu...
umgredgss 26845 The set of edges of a mult...
edgupgr 26846 Properties of an edge of a...
edgumgr 26847 Properties of an edge of a...
uhgrvtxedgiedgb 26848 In a hypergraph, a vertex ...
upgredg 26849 For each edge in a pseudog...
umgredg 26850 For each edge in a multigr...
upgrpredgv 26851 An edge of a pseudograph a...
umgrpredgv 26852 An edge of a multigraph al...
upgredg2vtx 26853 For a vertex incident to a...
upgredgpr 26854 If a proper pair (of verti...
edglnl 26855 The edges incident with a ...
numedglnl 26856 The number of edges incide...
umgredgne 26857 An edge of a multigraph al...
umgrnloop2 26858 A multigraph has no loops....
umgredgnlp 26859 An edge of a multigraph is...
isuspgr 26864 The property of being a si...
isusgr 26865 The property of being a si...
uspgrf 26866 The edge function of a sim...
usgrf 26867 The edge function of a sim...
isusgrs 26868 The property of being a si...
usgrfs 26869 The edge function of a sim...
usgrfun 26870 The edge function of a sim...
usgredgss 26871 The set of edges of a simp...
edgusgr 26872 An edge of a simple graph ...
isuspgrop 26873 The property of being an u...
isusgrop 26874 The property of being an u...
usgrop 26875 A simple graph represented...
isausgr 26876 The property of an unorder...
ausgrusgrb 26877 The equivalence of the def...
usgrausgri 26878 A simple graph represented...
ausgrumgri 26879 If an alternatively define...
ausgrusgri 26880 The equivalence of the def...
usgrausgrb 26881 The equivalence of the def...
usgredgop 26882 An edge of a simple graph ...
usgrf1o 26883 The edge function of a sim...
usgrf1 26884 The edge function of a sim...
uspgrf1oedg 26885 The edge function of a sim...
usgrss 26886 An edge is a subset of ver...
uspgrushgr 26887 A simple pseudograph is an...
uspgrupgr 26888 A simple pseudograph is an...
uspgrupgrushgr 26889 A graph is a simple pseudo...
usgruspgr 26890 A simple graph is a simple...
usgrumgr 26891 A simple graph is an undir...
usgrumgruspgr 26892 A graph is a simple graph ...
usgruspgrb 26893 A class is a simple graph ...
usgrupgr 26894 A simple graph is an undir...
usgruhgr 26895 A simple graph is an undir...
usgrislfuspgr 26896 A simple graph is a loop-f...
uspgrun 26897 The union ` U ` of two sim...
uspgrunop 26898 The union of two simple ps...
usgrun 26899 The union ` U ` of two sim...
usgrunop 26900 The union of two simple gr...
usgredg2 26901 The value of the "edge fun...
usgredg2ALT 26902 Alternate proof of ~ usgre...
usgredgprv 26903 In a simple graph, an edge...
usgredgprvALT 26904 Alternate proof of ~ usgre...
usgredgppr 26905 An edge of a simple graph ...
usgrpredgv 26906 An edge of a simple graph ...
edgssv2 26907 An edge of a simple graph ...
usgredg 26908 For each edge in a simple ...
usgrnloopv 26909 In a simple graph, there i...
usgrnloopvALT 26910 Alternate proof of ~ usgrn...
usgrnloop 26911 In a simple graph, there i...
usgrnloopALT 26912 Alternate proof of ~ usgrn...
usgrnloop0 26913 A simple graph has no loop...
usgrnloop0ALT 26914 Alternate proof of ~ usgrn...
usgredgne 26915 An edge of a simple graph ...
usgrf1oedg 26916 The edge function of a sim...
uhgr2edg 26917 If a vertex is adjacent to...
umgr2edg 26918 If a vertex is adjacent to...
usgr2edg 26919 If a vertex is adjacent to...
umgr2edg1 26920 If a vertex is adjacent to...
usgr2edg1 26921 If a vertex is adjacent to...
umgrvad2edg 26922 If a vertex is adjacent to...
umgr2edgneu 26923 If a vertex is adjacent to...
usgrsizedg 26924 In a simple graph, the siz...
usgredg3 26925 The value of the "edge fun...
usgredg4 26926 For a vertex incident to a...
usgredgreu 26927 For a vertex incident to a...
usgredg2vtx 26928 For a vertex incident to a...
uspgredg2vtxeu 26929 For a vertex incident to a...
usgredg2vtxeu 26930 For a vertex incident to a...
usgredg2vtxeuALT 26931 Alternate proof of ~ usgre...
uspgredg2vlem 26932 Lemma for ~ uspgredg2v . ...
uspgredg2v 26933 In a simple pseudograph, t...
usgredg2vlem1 26934 Lemma 1 for ~ usgredg2v . ...
usgredg2vlem2 26935 Lemma 2 for ~ usgredg2v . ...
usgredg2v 26936 In a simple graph, the map...
usgriedgleord 26937 Alternate version of ~ usg...
ushgredgedg 26938 In a simple hypergraph the...
usgredgedg 26939 In a simple graph there is...
ushgredgedgloop 26940 In a simple hypergraph the...
uspgredgleord 26941 In a simple pseudograph th...
usgredgleord 26942 In a simple graph the numb...
usgredgleordALT 26943 Alternate proof for ~ usgr...
usgrstrrepe 26944 Replacing (or adding) the ...
usgr0e 26945 The empty graph, with vert...
usgr0vb 26946 The null graph, with no ve...
uhgr0v0e 26947 The null graph, with no ve...
uhgr0vsize0 26948 The size of a hypergraph w...
uhgr0edgfi 26949 A graph of order 0 (i.e. w...
usgr0v 26950 The null graph, with no ve...
uhgr0vusgr 26951 The null graph, with no ve...
usgr0 26952 The null graph represented...
uspgr1e 26953 A simple pseudograph with ...
usgr1e 26954 A simple graph with one ed...
usgr0eop 26955 The empty graph, with vert...
uspgr1eop 26956 A simple pseudograph with ...
uspgr1ewop 26957 A simple pseudograph with ...
uspgr1v1eop 26958 A simple pseudograph with ...
usgr1eop 26959 A simple graph with (at le...
uspgr2v1e2w 26960 A simple pseudograph with ...
usgr2v1e2w 26961 A simple graph with two ve...
edg0usgr 26962 A class without edges is a...
lfuhgr1v0e 26963 A loop-free hypergraph wit...
usgr1vr 26964 A simple graph with one ve...
usgr1v 26965 A class with one (or no) v...
usgr1v0edg 26966 A class with one (or no) v...
usgrexmpldifpr 26967 Lemma for ~ usgrexmpledg :...
usgrexmplef 26968 Lemma for ~ usgrexmpl . (...
usgrexmpllem 26969 Lemma for ~ usgrexmpl . (...
usgrexmplvtx 26970 The vertices ` 0 , 1 , 2 ,...
usgrexmpledg 26971 The edges ` { 0 , 1 } , { ...
usgrexmpl 26972 ` G ` is a simple graph of...
griedg0prc 26973 The class of empty graphs ...
griedg0ssusgr 26974 The class of all simple gr...
usgrprc 26975 The class of simple graphs...
relsubgr 26978 The class of the subgraph ...
subgrv 26979 If a class is a subgraph o...
issubgr 26980 The property of a set to b...
issubgr2 26981 The property of a set to b...
subgrprop 26982 The properties of a subgra...
subgrprop2 26983 The properties of a subgra...
uhgrissubgr 26984 The property of a hypergra...
subgrprop3 26985 The properties of a subgra...
egrsubgr 26986 An empty graph consisting ...
0grsubgr 26987 The null graph (represente...
0uhgrsubgr 26988 The null graph (as hypergr...
uhgrsubgrself 26989 A hypergraph is a subgraph...
subgrfun 26990 The edge function of a sub...
subgruhgrfun 26991 The edge function of a sub...
subgreldmiedg 26992 An element of the domain o...
subgruhgredgd 26993 An edge of a subgraph of a...
subumgredg2 26994 An edge of a subgraph of a...
subuhgr 26995 A subgraph of a hypergraph...
subupgr 26996 A subgraph of a pseudograp...
subumgr 26997 A subgraph of a multigraph...
subusgr 26998 A subgraph of a simple gra...
uhgrspansubgrlem 26999 Lemma for ~ uhgrspansubgr ...
uhgrspansubgr 27000 A spanning subgraph ` S ` ...
uhgrspan 27001 A spanning subgraph ` S ` ...
upgrspan 27002 A spanning subgraph ` S ` ...
umgrspan 27003 A spanning subgraph ` S ` ...
usgrspan 27004 A spanning subgraph ` S ` ...
uhgrspanop 27005 A spanning subgraph of a h...
upgrspanop 27006 A spanning subgraph of a p...
umgrspanop 27007 A spanning subgraph of a m...
usgrspanop 27008 A spanning subgraph of a s...
uhgrspan1lem1 27009 Lemma 1 for ~ uhgrspan1 . ...
uhgrspan1lem2 27010 Lemma 2 for ~ uhgrspan1 . ...
uhgrspan1lem3 27011 Lemma 3 for ~ uhgrspan1 . ...
uhgrspan1 27012 The induced subgraph ` S `...
upgrreslem 27013 Lemma for ~ upgrres . (Co...
umgrreslem 27014 Lemma for ~ umgrres and ~ ...
upgrres 27015 A subgraph obtained by rem...
umgrres 27016 A subgraph obtained by rem...
usgrres 27017 A subgraph obtained by rem...
upgrres1lem1 27018 Lemma 1 for ~ upgrres1 . ...
umgrres1lem 27019 Lemma for ~ umgrres1 . (C...
upgrres1lem2 27020 Lemma 2 for ~ upgrres1 . ...
upgrres1lem3 27021 Lemma 3 for ~ upgrres1 . ...
upgrres1 27022 A pseudograph obtained by ...
umgrres1 27023 A multigraph obtained by r...
usgrres1 27024 Restricting a simple graph...
isfusgr 27027 The property of being a fi...
fusgrvtxfi 27028 A finite simple graph has ...
isfusgrf1 27029 The property of being a fi...
isfusgrcl 27030 The property of being a fi...
fusgrusgr 27031 A finite simple graph is a...
opfusgr 27032 A finite simple graph repr...
usgredgffibi 27033 The number of edges in a s...
fusgredgfi 27034 In a finite simple graph t...
usgr1v0e 27035 The size of a (finite) sim...
usgrfilem 27036 In a finite simple graph, ...
fusgrfisbase 27037 Induction base for ~ fusgr...
fusgrfisstep 27038 Induction step in ~ fusgrf...
fusgrfis 27039 A finite simple graph is o...
fusgrfupgrfs 27040 A finite simple graph is a...
nbgrprc0 27043 The set of neighbors is em...
nbgrcl 27044 If a class ` X ` has at le...
nbgrval 27045 The set of neighbors of a ...
dfnbgr2 27046 Alternate definition of th...
dfnbgr3 27047 Alternate definition of th...
nbgrnvtx0 27048 If a class ` X ` is not a ...
nbgrel 27049 Characterization of a neig...
nbgrisvtx 27050 Every neighbor ` N ` of a ...
nbgrssvtx 27051 The neighbors of a vertex ...
nbuhgr 27052 The set of neighbors of a ...
nbupgr 27053 The set of neighbors of a ...
nbupgrel 27054 A neighbor of a vertex in ...
nbumgrvtx 27055 The set of neighbors of a ...
nbumgr 27056 The set of neighbors of an...
nbusgrvtx 27057 The set of neighbors of a ...
nbusgr 27058 The set of neighbors of an...
nbgr2vtx1edg 27059 If a graph has two vertice...
nbuhgr2vtx1edgblem 27060 Lemma for ~ nbuhgr2vtx1edg...
nbuhgr2vtx1edgb 27061 If a hypergraph has two ve...
nbusgreledg 27062 A class/vertex is a neighb...
uhgrnbgr0nb 27063 A vertex which is not endp...
nbgr0vtxlem 27064 Lemma for ~ nbgr0vtx and ~...
nbgr0vtx 27065 In a null graph (with no v...
nbgr0edg 27066 In an empty graph (with no...
nbgr1vtx 27067 In a graph with one vertex...
nbgrnself 27068 A vertex in a graph is not...
nbgrnself2 27069 A class ` X ` is not a nei...
nbgrssovtx 27070 The neighbors of a vertex ...
nbgrssvwo2 27071 The neighbors of a vertex ...
nbgrsym 27072 In a graph, the neighborho...
nbupgrres 27073 The neighborhood of a vert...
usgrnbcnvfv 27074 Applying the edge function...
nbusgredgeu 27075 For each neighbor of a ver...
edgnbusgreu 27076 For each edge incident to ...
nbusgredgeu0 27077 For each neighbor of a ver...
nbusgrf1o0 27078 The mapping of neighbors o...
nbusgrf1o1 27079 The set of neighbors of a ...
nbusgrf1o 27080 The set of neighbors of a ...
nbedgusgr 27081 The number of neighbors of...
edgusgrnbfin 27082 The number of neighbors of...
nbusgrfi 27083 The class of neighbors of ...
nbfiusgrfi 27084 The class of neighbors of ...
hashnbusgrnn0 27085 The number of neighbors of...
nbfusgrlevtxm1 27086 The number of neighbors of...
nbfusgrlevtxm2 27087 If there is a vertex which...
nbusgrvtxm1 27088 If the number of neighbors...
nb3grprlem1 27089 Lemma 1 for ~ nb3grpr . (...
nb3grprlem2 27090 Lemma 2 for ~ nb3grpr . (...
nb3grpr 27091 The neighbors of a vertex ...
nb3grpr2 27092 The neighbors of a vertex ...
nb3gr2nb 27093 If the neighbors of two ve...
uvtxval 27096 The set of all universal v...
uvtxel 27097 A universal vertex, i.e. a...
uvtxisvtx 27098 A universal vertex is a ve...
uvtxssvtx 27099 The set of the universal v...
vtxnbuvtx 27100 A universal vertex has all...
uvtxnbgrss 27101 A universal vertex has all...
uvtxnbgrvtx 27102 A universal vertex is neig...
uvtx0 27103 There is no universal vert...
isuvtx 27104 The set of all universal v...
uvtxel1 27105 Characterization of a univ...
uvtx01vtx 27106 If a graph/class has no ed...
uvtx2vtx1edg 27107 If a graph has two vertice...
uvtx2vtx1edgb 27108 If a hypergraph has two ve...
uvtxnbgr 27109 A universal vertex has all...
uvtxnbgrb 27110 A vertex is universal iff ...
uvtxusgr 27111 The set of all universal v...
uvtxusgrel 27112 A universal vertex, i.e. a...
uvtxnm1nbgr 27113 A universal vertex has ` n...
nbusgrvtxm1uvtx 27114 If the number of neighbors...
uvtxnbvtxm1 27115 A universal vertex has ` n...
nbupgruvtxres 27116 The neighborhood of a univ...
uvtxupgrres 27117 A universal vertex is univ...
cplgruvtxb 27122 A graph ` G ` is complete ...
prcliscplgr 27123 A proper class (representi...
iscplgr 27124 The property of being a co...
iscplgrnb 27125 A graph is complete iff al...
iscplgredg 27126 A graph ` G ` is complete ...
iscusgr 27127 The property of being a co...
cusgrusgr 27128 A complete simple graph is...
cusgrcplgr 27129 A complete simple graph is...
iscusgrvtx 27130 A simple graph is complete...
cusgruvtxb 27131 A simple graph is complete...
iscusgredg 27132 A simple graph is complete...
cusgredg 27133 In a complete simple graph...
cplgr0 27134 The null graph (with no ve...
cusgr0 27135 The null graph (with no ve...
cplgr0v 27136 A null graph (with no vert...
cusgr0v 27137 A graph with no vertices a...
cplgr1vlem 27138 Lemma for ~ cplgr1v and ~ ...
cplgr1v 27139 A graph with one vertex is...
cusgr1v 27140 A graph with one vertex an...
cplgr2v 27141 An undirected hypergraph w...
cplgr2vpr 27142 An undirected hypergraph w...
nbcplgr 27143 In a complete graph, each ...
cplgr3v 27144 A pseudograph with three (...
cusgr3vnbpr 27145 The neighbors of a vertex ...
cplgrop 27146 A complete graph represent...
cusgrop 27147 A complete simple graph re...
cusgrexilem1 27148 Lemma 1 for ~ cusgrexi . ...
usgrexilem 27149 Lemma for ~ usgrexi . (Co...
usgrexi 27150 An arbitrary set regarded ...
cusgrexilem2 27151 Lemma 2 for ~ cusgrexi . ...
cusgrexi 27152 An arbitrary set ` V ` reg...
cusgrexg 27153 For each set there is a se...
structtousgr 27154 Any (extensible) structure...
structtocusgr 27155 Any (extensible) structure...
cffldtocusgr 27156 The field of complex numbe...
cusgrres 27157 Restricting a complete sim...
cusgrsizeindb0 27158 Base case of the induction...
cusgrsizeindb1 27159 Base case of the induction...
cusgrsizeindslem 27160 Lemma for ~ cusgrsizeinds ...
cusgrsizeinds 27161 Part 1 of induction step i...
cusgrsize2inds 27162 Induction step in ~ cusgrs...
cusgrsize 27163 The size of a finite compl...
cusgrfilem1 27164 Lemma 1 for ~ cusgrfi . (...
cusgrfilem2 27165 Lemma 2 for ~ cusgrfi . (...
cusgrfilem3 27166 Lemma 3 for ~ cusgrfi . (...
cusgrfi 27167 If the size of a complete ...
usgredgsscusgredg 27168 A simple graph is a subgra...
usgrsscusgr 27169 A simple graph is a subgra...
sizusglecusglem1 27170 Lemma 1 for ~ sizusglecusg...
sizusglecusglem2 27171 Lemma 2 for ~ sizusglecusg...
sizusglecusg 27172 The size of a simple graph...
fusgrmaxsize 27173 The maximum size of a fini...
vtxdgfval 27176 The value of the vertex de...
vtxdgval 27177 The degree of a vertex. (...
vtxdgfival 27178 The degree of a vertex for...
vtxdgop 27179 The vertex degree expresse...
vtxdgf 27180 The vertex degree function...
vtxdgelxnn0 27181 The degree of a vertex is ...
vtxdg0v 27182 The degree of a vertex in ...
vtxdg0e 27183 The degree of a vertex in ...
vtxdgfisnn0 27184 The degree of a vertex in ...
vtxdgfisf 27185 The vertex degree function...
vtxdeqd 27186 Equality theorem for the v...
vtxduhgr0e 27187 The degree of a vertex in ...
vtxdlfuhgr1v 27188 The degree of the vertex i...
vdumgr0 27189 A vertex in a multigraph h...
vtxdun 27190 The degree of a vertex in ...
vtxdfiun 27191 The degree of a vertex in ...
vtxduhgrun 27192 The degree of a vertex in ...
vtxduhgrfiun 27193 The degree of a vertex in ...
vtxdlfgrval 27194 The value of the vertex de...
vtxdumgrval 27195 The value of the vertex de...
vtxdusgrval 27196 The value of the vertex de...
vtxd0nedgb 27197 A vertex has degree 0 iff ...
vtxdushgrfvedglem 27198 Lemma for ~ vtxdushgrfvedg...
vtxdushgrfvedg 27199 The value of the vertex de...
vtxdusgrfvedg 27200 The value of the vertex de...
vtxduhgr0nedg 27201 If a vertex in a hypergrap...
vtxdumgr0nedg 27202 If a vertex in a multigrap...
vtxduhgr0edgnel 27203 A vertex in a hypergraph h...
vtxdusgr0edgnel 27204 A vertex in a simple graph...
vtxdusgr0edgnelALT 27205 Alternate proof of ~ vtxdu...
vtxdgfusgrf 27206 The vertex degree function...
vtxdgfusgr 27207 In a finite simple graph, ...
fusgrn0degnn0 27208 In a nonempty, finite grap...
1loopgruspgr 27209 A graph with one edge whic...
1loopgredg 27210 The set of edges in a grap...
1loopgrnb0 27211 In a graph (simple pseudog...
1loopgrvd2 27212 The vertex degree of a one...
1loopgrvd0 27213 The vertex degree of a one...
1hevtxdg0 27214 The vertex degree of verte...
1hevtxdg1 27215 The vertex degree of verte...
1hegrvtxdg1 27216 The vertex degree of a gra...
1hegrvtxdg1r 27217 The vertex degree of a gra...
1egrvtxdg1 27218 The vertex degree of a one...
1egrvtxdg1r 27219 The vertex degree of a one...
1egrvtxdg0 27220 The vertex degree of a one...
p1evtxdeqlem 27221 Lemma for ~ p1evtxdeq and ...
p1evtxdeq 27222 If an edge ` E ` which doe...
p1evtxdp1 27223 If an edge ` E ` (not bein...
uspgrloopvtx 27224 The set of vertices in a g...
uspgrloopvtxel 27225 A vertex in a graph (simpl...
uspgrloopiedg 27226 The set of edges in a grap...
uspgrloopedg 27227 The set of edges in a grap...
uspgrloopnb0 27228 In a graph (simple pseudog...
uspgrloopvd2 27229 The vertex degree of a one...
umgr2v2evtx 27230 The set of vertices in a m...
umgr2v2evtxel 27231 A vertex in a multigraph w...
umgr2v2eiedg 27232 The edge function in a mul...
umgr2v2eedg 27233 The set of edges in a mult...
umgr2v2e 27234 A multigraph with two edge...
umgr2v2enb1 27235 In a multigraph with two e...
umgr2v2evd2 27236 In a multigraph with two e...
hashnbusgrvd 27237 In a simple graph, the num...
usgruvtxvdb 27238 In a finite simple graph w...
vdiscusgrb 27239 A finite simple graph with...
vdiscusgr 27240 In a finite complete simpl...
vtxdusgradjvtx 27241 The degree of a vertex in ...
usgrvd0nedg 27242 If a vertex in a simple gr...
uhgrvd00 27243 If every vertex in a hyper...
usgrvd00 27244 If every vertex in a simpl...
vdegp1ai 27245 The induction step for a v...
vdegp1bi 27246 The induction step for a v...
vdegp1ci 27247 The induction step for a v...
vtxdginducedm1lem1 27248 Lemma 1 for ~ vtxdginduced...
vtxdginducedm1lem2 27249 Lemma 2 for ~ vtxdginduced...
vtxdginducedm1lem3 27250 Lemma 3 for ~ vtxdginduced...
vtxdginducedm1lem4 27251 Lemma 4 for ~ vtxdginduced...
vtxdginducedm1 27252 The degree of a vertex ` v...
vtxdginducedm1fi 27253 The degree of a vertex ` v...
finsumvtxdg2ssteplem1 27254 Lemma for ~ finsumvtxdg2ss...
finsumvtxdg2ssteplem2 27255 Lemma for ~ finsumvtxdg2ss...
finsumvtxdg2ssteplem3 27256 Lemma for ~ finsumvtxdg2ss...
finsumvtxdg2ssteplem4 27257 Lemma for ~ finsumvtxdg2ss...
finsumvtxdg2sstep 27258 Induction step of ~ finsum...
finsumvtxdg2size 27259 The sum of the degrees of ...
fusgr1th 27260 The sum of the degrees of ...
finsumvtxdgeven 27261 The sum of the degrees of ...
vtxdgoddnumeven 27262 The number of vertices of ...
fusgrvtxdgonume 27263 The number of vertices of ...
isrgr 27268 The property of a class be...
rgrprop 27269 The properties of a k-regu...
isrusgr 27270 The property of being a k-...
rusgrprop 27271 The properties of a k-regu...
rusgrrgr 27272 A k-regular simple graph i...
rusgrusgr 27273 A k-regular simple graph i...
finrusgrfusgr 27274 A finite regular simple gr...
isrusgr0 27275 The property of being a k-...
rusgrprop0 27276 The properties of a k-regu...
usgreqdrusgr 27277 If all vertices in a simpl...
fusgrregdegfi 27278 In a nonempty finite simpl...
fusgrn0eqdrusgr 27279 If all vertices in a nonem...
frusgrnn0 27280 In a nonempty finite k-reg...
0edg0rgr 27281 A graph is 0-regular if it...
uhgr0edg0rgr 27282 A hypergraph is 0-regular ...
uhgr0edg0rgrb 27283 A hypergraph is 0-regular ...
usgr0edg0rusgr 27284 A simple graph is 0-regula...
0vtxrgr 27285 A null graph (with no vert...
0vtxrusgr 27286 A graph with no vertices a...
0uhgrrusgr 27287 The null graph as hypergra...
0grrusgr 27288 The null graph represented...
0grrgr 27289 The null graph represented...
cusgrrusgr 27290 A complete simple graph wi...
cusgrm1rusgr 27291 A finite simple graph with...
rusgrpropnb 27292 The properties of a k-regu...
rusgrpropedg 27293 The properties of a k-regu...
rusgrpropadjvtx 27294 The properties of a k-regu...
rusgrnumwrdl2 27295 In a k-regular simple grap...
rusgr1vtxlem 27296 Lemma for ~ rusgr1vtx . (...
rusgr1vtx 27297 If a k-regular simple grap...
rgrusgrprc 27298 The class of 0-regular sim...
rusgrprc 27299 The class of 0-regular sim...
rgrprc 27300 The class of 0-regular gra...
rgrprcx 27301 The class of 0-regular gra...
rgrx0ndm 27302 0 is not in the domain of ...
rgrx0nd 27303 The potentially alternativ...
ewlksfval 27310 The set of s-walks of edge...
isewlk 27311 Conditions for a function ...
ewlkprop 27312 Properties of an s-walk of...
ewlkinedg 27313 The intersection (common v...
ewlkle 27314 An s-walk of edges is also...
upgrewlkle2 27315 In a pseudograph, there is...
wkslem1 27316 Lemma 1 for walks to subst...
wkslem2 27317 Lemma 2 for walks to subst...
wksfval 27318 The set of walks (in an un...
iswlk 27319 Properties of a pair of fu...
wlkprop 27320 Properties of a walk. (Co...
wlkv 27321 The classes involved in a ...
iswlkg 27322 Generalization of ~ iswlk ...
wlkf 27323 The mapping enumerating th...
wlkcl 27324 A walk has length ` # ( F ...
wlkp 27325 The mapping enumerating th...
wlkpwrd 27326 The sequence of vertices o...
wlklenvp1 27327 The number of vertices of ...
wksv 27328 The class of walks is a se...
wlkn0 27329 The sequence of vertices o...
wlklenvm1 27330 The number of edges of a w...
ifpsnprss 27331 Lemma for ~ wlkvtxeledg : ...
wlkvtxeledg 27332 Each pair of adjacent vert...
wlkvtxiedg 27333 The vertices of a walk are...
relwlk 27334 The set ` ( Walks `` G ) `...
wlkvv 27335 If there is at least one w...
wlkop 27336 A walk is an ordered pair....
wlkcpr 27337 A walk as class with two c...
wlk2f 27338 If there is a walk ` W ` t...
wlkcomp 27339 A walk expressed by proper...
wlkcompim 27340 Implications for the prope...
wlkelwrd 27341 The components of a walk a...
wlkeq 27342 Conditions for two walks (...
edginwlk 27343 The value of the edge func...
upgredginwlk 27344 The value of the edge func...
iedginwlk 27345 The value of the edge func...
wlkl1loop 27346 A walk of length 1 from a ...
wlk1walk 27347 A walk is a 1-walk "on the...
wlk1ewlk 27348 A walk is an s-walk "on th...
upgriswlk 27349 Properties of a pair of fu...
upgrwlkedg 27350 The edges of a walk in a p...
upgrwlkcompim 27351 Implications for the prope...
wlkvtxedg 27352 The vertices of a walk are...
upgrwlkvtxedg 27353 The pairs of connected ver...
uspgr2wlkeq 27354 Conditions for two walks w...
uspgr2wlkeq2 27355 Conditions for two walks w...
uspgr2wlkeqi 27356 Conditions for two walks w...
umgrwlknloop 27357 In a multigraph, each walk...
wlkRes 27358 Restrictions of walks (i.e...
wlkv0 27359 If there is a walk in the ...
g0wlk0 27360 There is no walk in a null...
0wlk0 27361 There is no walk for the e...
wlk0prc 27362 There is no walk in a null...
wlklenvclwlk 27363 The number of vertices in ...
wlklenvclwlkOLD 27364 Obsolete version of ~ wlkl...
wlkson 27365 The set of walks between t...
iswlkon 27366 Properties of a pair of fu...
wlkonprop 27367 Properties of a walk betwe...
wlkpvtx 27368 A walk connects vertices. ...
wlkepvtx 27369 The endpoints of a walk ar...
wlkoniswlk 27370 A walk between two vertice...
wlkonwlk 27371 A walk is a walk between i...
wlkonwlk1l 27372 A walk is a walk from its ...
wlksoneq1eq2 27373 Two walks with identical s...
wlkonl1iedg 27374 If there is a walk between...
wlkon2n0 27375 The length of a walk betwe...
2wlklem 27376 Lemma for theorems for wal...
upgr2wlk 27377 Properties of a pair of fu...
wlkreslem 27378 Lemma for ~ wlkres . (Con...
wlkres 27379 The restriction ` <. H , Q...
redwlklem 27380 Lemma for ~ redwlk . (Con...
redwlk 27381 A walk ending at the last ...
wlkp1lem1 27382 Lemma for ~ wlkp1 . (Cont...
wlkp1lem2 27383 Lemma for ~ wlkp1 . (Cont...
wlkp1lem3 27384 Lemma for ~ wlkp1 . (Cont...
wlkp1lem4 27385 Lemma for ~ wlkp1 . (Cont...
wlkp1lem5 27386 Lemma for ~ wlkp1 . (Cont...
wlkp1lem6 27387 Lemma for ~ wlkp1 . (Cont...
wlkp1lem7 27388 Lemma for ~ wlkp1 . (Cont...
wlkp1lem8 27389 Lemma for ~ wlkp1 . (Cont...
wlkp1 27390 Append one path segment (e...
wlkdlem1 27391 Lemma 1 for ~ wlkd . (Con...
wlkdlem2 27392 Lemma 2 for ~ wlkd . (Con...
wlkdlem3 27393 Lemma 3 for ~ wlkd . (Con...
wlkdlem4 27394 Lemma 4 for ~ wlkd . (Con...
wlkd 27395 Two words representing a w...
lfgrwlkprop 27396 Two adjacent vertices in a...
lfgriswlk 27397 Conditions for a pair of f...
lfgrwlknloop 27398 In a loop-free graph, each...
reltrls 27403 The set ` ( Trails `` G ) ...
trlsfval 27404 The set of trails (in an u...
istrl 27405 Conditions for a pair of c...
trliswlk 27406 A trail is a walk. (Contr...
trlf1 27407 The enumeration ` F ` of a...
trlreslem 27408 Lemma for ~ trlres . Form...
trlres 27409 The restriction ` <. H , Q...
upgrtrls 27410 The set of trails in a pse...
upgristrl 27411 Properties of a pair of fu...
upgrf1istrl 27412 Properties of a pair of a ...
wksonproplem 27413 Lemma for theorems for pro...
trlsonfval 27414 The set of trails between ...
istrlson 27415 Properties of a pair of fu...
trlsonprop 27416 Properties of a trail betw...
trlsonistrl 27417 A trail between two vertic...
trlsonwlkon 27418 A trail between two vertic...
trlontrl 27419 A trail is a trail between...
relpths 27428 The set ` ( Paths `` G ) `...
pthsfval 27429 The set of paths (in an un...
spthsfval 27430 The set of simple paths (i...
ispth 27431 Conditions for a pair of c...
isspth 27432 Conditions for a pair of c...
pthistrl 27433 A path is a trail (in an u...
spthispth 27434 A simple path is a path (i...
pthiswlk 27435 A path is a walk (in an un...
spthiswlk 27436 A simple path is a walk (i...
pthdivtx 27437 The inner vertices of a pa...
pthdadjvtx 27438 The adjacent vertices of a...
2pthnloop 27439 A path of length at least ...
upgr2pthnlp 27440 A path of length at least ...
spthdifv 27441 The vertices of a simple p...
spthdep 27442 A simple path (at least of...
pthdepisspth 27443 A path with different star...
upgrwlkdvdelem 27444 Lemma for ~ upgrwlkdvde . ...
upgrwlkdvde 27445 In a pseudograph, all edge...
upgrspthswlk 27446 The set of simple paths in...
upgrwlkdvspth 27447 A walk consisting of diffe...
pthsonfval 27448 The set of paths between t...
spthson 27449 The set of simple paths be...
ispthson 27450 Properties of a pair of fu...
isspthson 27451 Properties of a pair of fu...
pthsonprop 27452 Properties of a path betwe...
spthonprop 27453 Properties of a simple pat...
pthonispth 27454 A path between two vertice...
pthontrlon 27455 A path between two vertice...
pthonpth 27456 A path is a path between i...
isspthonpth 27457 A pair of functions is a s...
spthonisspth 27458 A simple path between to v...
spthonpthon 27459 A simple path between two ...
spthonepeq 27460 The endpoints of a simple ...
uhgrwkspthlem1 27461 Lemma 1 for ~ uhgrwkspth ....
uhgrwkspthlem2 27462 Lemma 2 for ~ uhgrwkspth ....
uhgrwkspth 27463 Any walk of length 1 betwe...
usgr2wlkneq 27464 The vertices and edges are...
usgr2wlkspthlem1 27465 Lemma 1 for ~ usgr2wlkspth...
usgr2wlkspthlem2 27466 Lemma 2 for ~ usgr2wlkspth...
usgr2wlkspth 27467 In a simple graph, any wal...
usgr2trlncl 27468 In a simple graph, any tra...
usgr2trlspth 27469 In a simple graph, any tra...
usgr2pthspth 27470 In a simple graph, any pat...
usgr2pthlem 27471 Lemma for ~ usgr2pth . (C...
usgr2pth 27472 In a simple graph, there i...
usgr2pth0 27473 In a simply graph, there i...
pthdlem1 27474 Lemma 1 for ~ pthd . (Con...
pthdlem2lem 27475 Lemma for ~ pthdlem2 . (C...
pthdlem2 27476 Lemma 2 for ~ pthd . (Con...
pthd 27477 Two words representing a t...
clwlks 27480 The set of closed walks (i...
isclwlk 27481 A pair of functions repres...
clwlkiswlk 27482 A closed walk is a walk (i...
clwlkwlk 27483 Closed walks are walks (in...
clwlkswks 27484 Closed walks are walks (in...
isclwlke 27485 Properties of a pair of fu...
isclwlkupgr 27486 Properties of a pair of fu...
clwlkcomp 27487 A closed walk expressed by...
clwlkcompim 27488 Implications for the prope...
upgrclwlkcompim 27489 Implications for the prope...
clwlkcompbp 27490 Basic properties of the co...
clwlkl1loop 27491 A closed walk of length 1 ...
crcts 27496 The set of circuits (in an...
cycls 27497 The set of cycles (in an u...
iscrct 27498 Sufficient and necessary c...
iscycl 27499 Sufficient and necessary c...
crctprop 27500 The properties of a circui...
cyclprop 27501 The properties of a cycle:...
crctisclwlk 27502 A circuit is a closed walk...
crctistrl 27503 A circuit is a trail. (Co...
crctiswlk 27504 A circuit is a walk. (Con...
cyclispth 27505 A cycle is a path. (Contr...
cycliswlk 27506 A cycle is a walk. (Contr...
cycliscrct 27507 A cycle is a circuit. (Co...
cyclnspth 27508 A (non-trivial) cycle is n...
cyclispthon 27509 A cycle is a path starting...
lfgrn1cycl 27510 In a loop-free graph there...
usgr2trlncrct 27511 In a simple graph, any tra...
umgrn1cycl 27512 In a multigraph graph (wit...
uspgrn2crct 27513 In a simple pseudograph th...
usgrn2cycl 27514 In a simple graph there ar...
crctcshwlkn0lem1 27515 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem2 27516 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem3 27517 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem4 27518 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem5 27519 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem6 27520 Lemma for ~ crctcshwlkn0 ....
crctcshwlkn0lem7 27521 Lemma for ~ crctcshwlkn0 ....
crctcshlem1 27522 Lemma for ~ crctcsh . (Co...
crctcshlem2 27523 Lemma for ~ crctcsh . (Co...
crctcshlem3 27524 Lemma for ~ crctcsh . (Co...
crctcshlem4 27525 Lemma for ~ crctcsh . (Co...
crctcshwlkn0 27526 Cyclically shifting the in...
crctcshwlk 27527 Cyclically shifting the in...
crctcshtrl 27528 Cyclically shifting the in...
crctcsh 27529 Cyclically shifting the in...
wwlks 27540 The set of walks (in an un...
iswwlks 27541 A word over the set of ver...
wwlksn 27542 The set of walks (in an un...
iswwlksn 27543 A word over the set of ver...
wwlksnprcl 27544 Derivation of the length o...
iswwlksnx 27545 Properties of a word to re...
wwlkbp 27546 Basic properties of a walk...
wwlknbp 27547 Basic properties of a walk...
wwlknp 27548 Properties of a set being ...
wwlknbp1 27549 Other basic properties of ...
wwlknvtx 27550 The symbols of a word ` W ...
wwlknllvtx 27551 If a word ` W ` represents...
wwlknlsw 27552 If a word represents a wal...
wspthsn 27553 The set of simple paths of...
iswspthn 27554 An element of the set of s...
wspthnp 27555 Properties of a set being ...
wwlksnon 27556 The set of walks of a fixe...
wspthsnon 27557 The set of simple paths of...
iswwlksnon 27558 The set of walks of a fixe...
wwlksnon0 27559 Sufficient conditions for ...
wwlksonvtx 27560 If a word ` W ` represents...
iswspthsnon 27561 The set of simple paths of...
wwlknon 27562 An element of the set of w...
wspthnon 27563 An element of the set of s...
wspthnonp 27564 Properties of a set being ...
wspthneq1eq2 27565 Two simple paths with iden...
wwlksn0s 27566 The set of all walks as wo...
wwlkssswrd 27567 Walks (represented by word...
wwlksn0 27568 A walk of length 0 is repr...
0enwwlksnge1 27569 In graphs without edges, t...
wwlkswwlksn 27570 A walk of a fixed length a...
wwlkssswwlksn 27571 The walks of a fixed lengt...
wlkiswwlks1 27572 The sequence of vertices i...
wlklnwwlkln1 27573 The sequence of vertices i...
wlkiswwlks2lem1 27574 Lemma 1 for ~ wlkiswwlks2 ...
wlkiswwlks2lem2 27575 Lemma 2 for ~ wlkiswwlks2 ...
wlkiswwlks2lem3 27576 Lemma 3 for ~ wlkiswwlks2 ...
wlkiswwlks2lem4 27577 Lemma 4 for ~ wlkiswwlks2 ...
wlkiswwlks2lem5 27578 Lemma 5 for ~ wlkiswwlks2 ...
wlkiswwlks2lem6 27579 Lemma 6 for ~ wlkiswwlks2 ...
wlkiswwlks2 27580 A walk as word corresponds...
wlkiswwlks 27581 A walk as word corresponds...
wlkiswwlksupgr2 27582 A walk as word corresponds...
wlkiswwlkupgr 27583 A walk as word corresponds...
wlkswwlksf1o 27584 The mapping of (ordinary) ...
wlkswwlksen 27585 The set of walks as words ...
wwlksm1edg 27586 Removing the trailing edge...
wlklnwwlkln2lem 27587 Lemma for ~ wlklnwwlkln2 a...
wlklnwwlkln2 27588 A walk of length ` N ` as ...
wlklnwwlkn 27589 A walk of length ` N ` as ...
wlklnwwlklnupgr2 27590 A walk of length ` N ` as ...
wlklnwwlknupgr 27591 A walk of length ` N ` as ...
wlknewwlksn 27592 If a walk in a pseudograph...
wlknwwlksnbij 27593 The mapping ` ( t e. T |->...
wlknwwlksnen 27594 In a simple pseudograph, t...
wlknwwlksneqs 27595 The set of walks of a fixe...
wwlkseq 27596 Equality of two walks (as ...
wwlksnred 27597 Reduction of a walk (as wo...
wwlksnext 27598 Extension of a walk (as wo...
wwlksnextbi 27599 Extension of a walk (as wo...
wwlksnredwwlkn 27600 For each walk (as word) of...
wwlksnredwwlkn0 27601 For each walk (as word) of...
wwlksnextwrd 27602 Lemma for ~ wwlksnextbij ....
wwlksnextfun 27603 Lemma for ~ wwlksnextbij ....
wwlksnextinj 27604 Lemma for ~ wwlksnextbij ....
wwlksnextsurj 27605 Lemma for ~ wwlksnextbij ....
wwlksnextbij0 27606 Lemma for ~ wwlksnextbij ....
wwlksnextbij 27607 There is a bijection betwe...
wwlksnexthasheq 27608 The number of the extensio...
disjxwwlksn 27609 Sets of walks (as words) e...
wwlksnndef 27610 Conditions for ` WWalksN `...
wwlksnfi 27611 The number of walks repres...
wwlksnfiOLD 27612 Obsolete version of ~ wwlk...
wlksnfi 27613 The number of walks of fix...
wlksnwwlknvbij 27614 There is a bijection betwe...
wwlksnextproplem1 27615 Lemma 1 for ~ wwlksnextpro...
wwlksnextproplem2 27616 Lemma 2 for ~ wwlksnextpro...
wwlksnextproplem3 27617 Lemma 3 for ~ wwlksnextpro...
wwlksnextprop 27618 Adding additional properti...
disjxwwlkn 27619 Sets of walks (as words) e...
hashwwlksnext 27620 Number of walks (as words)...
wwlksnwwlksnon 27621 A walk of fixed length is ...
wspthsnwspthsnon 27622 A simple path of fixed len...
wspthsnonn0vne 27623 If the set of simple paths...
wspthsswwlkn 27624 The set of simple paths of...
wspthnfi 27625 In a finite graph, the set...
wwlksnonfi 27626 In a finite graph, the set...
wspthsswwlknon 27627 The set of simple paths of...
wspthnonfi 27628 In a finite graph, the set...
wspniunwspnon 27629 The set of nonempty simple...
wspn0 27630 If there are no vertices, ...
2wlkdlem1 27631 Lemma 1 for ~ 2wlkd . (Co...
2wlkdlem2 27632 Lemma 2 for ~ 2wlkd . (Co...
2wlkdlem3 27633 Lemma 3 for ~ 2wlkd . (Co...
2wlkdlem4 27634 Lemma 4 for ~ 2wlkd . (Co...
2wlkdlem5 27635 Lemma 5 for ~ 2wlkd . (Co...
2pthdlem1 27636 Lemma 1 for ~ 2pthd . (Co...
2wlkdlem6 27637 Lemma 6 for ~ 2wlkd . (Co...
2wlkdlem7 27638 Lemma 7 for ~ 2wlkd . (Co...
2wlkdlem8 27639 Lemma 8 for ~ 2wlkd . (Co...
2wlkdlem9 27640 Lemma 9 for ~ 2wlkd . (Co...
2wlkdlem10 27641 Lemma 10 for ~ 3wlkd . (C...
2wlkd 27642 Construction of a walk fro...
2wlkond 27643 A walk of length 2 from on...
2trld 27644 Construction of a trail fr...
2trlond 27645 A trail of length 2 from o...
2pthd 27646 A path of length 2 from on...
2spthd 27647 A simple path of length 2 ...
2pthond 27648 A simple path of length 2 ...
2pthon3v 27649 For a vertex adjacent to t...
umgr2adedgwlklem 27650 Lemma for ~ umgr2adedgwlk ...
umgr2adedgwlk 27651 In a multigraph, two adjac...
umgr2adedgwlkon 27652 In a multigraph, two adjac...
umgr2adedgwlkonALT 27653 Alternate proof for ~ umgr...
umgr2adedgspth 27654 In a multigraph, two adjac...
umgr2wlk 27655 In a multigraph, there is ...
umgr2wlkon 27656 For each pair of adjacent ...
elwwlks2s3 27657 A walk of length 2 as word...
midwwlks2s3 27658 There is a vertex between ...
wwlks2onv 27659 If a length 3 string repre...
elwwlks2ons3im 27660 A walk as word of length 2...
elwwlks2ons3 27661 For each walk of length 2 ...
s3wwlks2on 27662 A length 3 string which re...
umgrwwlks2on 27663 A walk of length 2 between...
wwlks2onsym 27664 There is a walk of length ...
elwwlks2on 27665 A walk of length 2 between...
elwspths2on 27666 A simple path of length 2 ...
wpthswwlks2on 27667 For two different vertices...
2wspdisj 27668 All simple paths of length...
2wspiundisj 27669 All simple paths of length...
usgr2wspthons3 27670 A simple path of length 2 ...
usgr2wspthon 27671 A simple path of length 2 ...
elwwlks2 27672 A walk of length 2 between...
elwspths2spth 27673 A simple path of length 2 ...
rusgrnumwwlkl1 27674 In a k-regular graph, ther...
rusgrnumwwlkslem 27675 Lemma for ~ rusgrnumwwlks ...
rusgrnumwwlklem 27676 Lemma for ~ rusgrnumwwlk e...
rusgrnumwwlkb0 27677 Induction base 0 for ~ rus...
rusgrnumwwlkb1 27678 Induction base 1 for ~ rus...
rusgr0edg 27679 Special case for graphs wi...
rusgrnumwwlks 27680 Induction step for ~ rusgr...
rusgrnumwwlk 27681 In a ` K `-regular graph, ...
rusgrnumwwlkg 27682 In a ` K `-regular graph, ...
rusgrnumwlkg 27683 In a k-regular graph, the ...
clwwlknclwwlkdif 27684 The set ` A ` of walks of ...
clwwlknclwwlkdifnum 27685 In a ` K `-regular graph, ...
clwwlk 27688 The set of closed walks (i...
isclwwlk 27689 Properties of a word to re...
clwwlkbp 27690 Basic properties of a clos...
clwwlkgt0 27691 There is no empty closed w...
clwwlksswrd 27692 Closed walks (represented ...
clwwlk1loop 27693 A closed walk of length 1 ...
clwwlkccatlem 27694 Lemma for ~ clwwlkccat : i...
clwwlkccat 27695 The concatenation of two w...
umgrclwwlkge2 27696 A closed walk in a multigr...
clwlkclwwlklem2a1 27697 Lemma 1 for ~ clwlkclwwlkl...
clwlkclwwlklem2a2 27698 Lemma 2 for ~ clwlkclwwlkl...
clwlkclwwlklem2a3 27699 Lemma 3 for ~ clwlkclwwlkl...
clwlkclwwlklem2fv1 27700 Lemma 4a for ~ clwlkclwwlk...
clwlkclwwlklem2fv2 27701 Lemma 4b for ~ clwlkclwwlk...
clwlkclwwlklem2a4 27702 Lemma 4 for ~ clwlkclwwlkl...
clwlkclwwlklem2a 27703 Lemma for ~ clwlkclwwlklem...
clwlkclwwlklem1 27704 Lemma 1 for ~ clwlkclwwlk ...
clwlkclwwlklem2 27705 Lemma 2 for ~ clwlkclwwlk ...
clwlkclwwlklem3 27706 Lemma 3 for ~ clwlkclwwlk ...
clwlkclwwlk 27707 A closed walk as word of l...
clwlkclwwlk2 27708 A closed walk corresponds ...
clwlkclwwlkflem 27709 Lemma for ~ clwlkclwwlkf ....
clwlkclwwlkf1lem2 27710 Lemma 2 for ~ clwlkclwwlkf...
clwlkclwwlkf1lem3 27711 Lemma 3 for ~ clwlkclwwlkf...
clwlkclwwlkfolem 27712 Lemma for ~ clwlkclwwlkfo ...
clwlkclwwlkf 27713 ` F ` is a function from t...
clwlkclwwlkfo 27714 ` F ` is a function from t...
clwlkclwwlkf1 27715 ` F ` is a one-to-one func...
clwlkclwwlkf1o 27716 ` F ` is a bijection betwe...
clwlkclwwlken 27717 The set of the nonempty cl...
clwwisshclwwslemlem 27718 Lemma for ~ clwwisshclwwsl...
clwwisshclwwslem 27719 Lemma for ~ clwwisshclwws ...
clwwisshclwws 27720 Cyclically shifting a clos...
clwwisshclwwsn 27721 Cyclically shifting a clos...
erclwwlkrel 27722 ` .~ ` is a relation. (Co...
erclwwlkeq 27723 Two classes are equivalent...
erclwwlkeqlen 27724 If two classes are equival...
erclwwlkref 27725 ` .~ ` is a reflexive rela...
erclwwlksym 27726 ` .~ ` is a symmetric rela...
erclwwlktr 27727 ` .~ ` is a transitive rel...
erclwwlk 27728 ` .~ ` is an equivalence r...
clwwlkn 27731 The set of closed walks of...
isclwwlkn 27732 A word over the set of ver...
clwwlkn0 27733 There is no closed walk of...
clwwlkneq0 27734 Sufficient conditions for ...
clwwlkclwwlkn 27735 A closed walk of a fixed l...
clwwlksclwwlkn 27736 The closed walks of a fixe...
clwwlknlen 27737 The length of a word repre...
clwwlknnn 27738 The length of a closed wal...
clwwlknwrd 27739 A closed walk of a fixed l...
clwwlknbp 27740 Basic properties of a clos...
isclwwlknx 27741 Characterization of a word...
clwwlknp 27742 Properties of a set being ...
clwwlknwwlksn 27743 A word representing a clos...
clwwlknlbonbgr1 27744 The last but one vertex in...
clwwlkinwwlk 27745 If the initial vertex of a...
clwwlkn1 27746 A closed walk of length 1 ...
loopclwwlkn1b 27747 The singleton word consist...
clwwlkn1loopb 27748 A word represents a closed...
clwwlkn2 27749 A closed walk of length 2 ...
clwwlknfi 27750 If there is only a finite ...
clwwlknfiOLD 27751 Obsolete version of ~ clww...
clwwlkel 27752 Obtaining a closed walk (a...
clwwlkf 27753 Lemma 1 for ~ clwwlkf1o : ...
clwwlkfv 27754 Lemma 2 for ~ clwwlkf1o : ...
clwwlkf1 27755 Lemma 3 for ~ clwwlkf1o : ...
clwwlkfo 27756 Lemma 4 for ~ clwwlkf1o : ...
clwwlkf1o 27757 F is a 1-1 onto function, ...
clwwlken 27758 The set of closed walks of...
clwwlknwwlkncl 27759 Obtaining a closed walk (a...
clwwlkwwlksb 27760 A nonempty word over verti...
clwwlknwwlksnb 27761 A word over vertices repre...
clwwlkext2edg 27762 If a word concatenated wit...
wwlksext2clwwlk 27763 If a word represents a wal...
wwlksubclwwlk 27764 Any prefix of a word repre...
clwwnisshclwwsn 27765 Cyclically shifting a clos...
eleclclwwlknlem1 27766 Lemma 1 for ~ eleclclwwlkn...
eleclclwwlknlem2 27767 Lemma 2 for ~ eleclclwwlkn...
clwwlknscsh 27768 The set of cyclical shifts...
clwwlknccat 27769 The concatenation of two w...
umgr2cwwk2dif 27770 If a word represents a clo...
umgr2cwwkdifex 27771 If a word represents a clo...
erclwwlknrel 27772 ` .~ ` is a relation. (Co...
erclwwlkneq 27773 Two classes are equivalent...
erclwwlkneqlen 27774 If two classes are equival...
erclwwlknref 27775 ` .~ ` is a reflexive rela...
erclwwlknsym 27776 ` .~ ` is a symmetric rela...
erclwwlkntr 27777 ` .~ ` is a transitive rel...
erclwwlkn 27778 ` .~ ` is an equivalence r...
qerclwwlknfi 27779 The quotient set of the se...
hashclwwlkn0 27780 The number of closed walks...
eclclwwlkn1 27781 An equivalence class accor...
eleclclwwlkn 27782 A member of an equivalence...
hashecclwwlkn1 27783 The size of every equivale...
umgrhashecclwwlk 27784 The size of every equivale...
fusgrhashclwwlkn 27785 The size of the set of clo...
clwwlkndivn 27786 The size of the set of clo...
clwlknf1oclwwlknlem1 27787 Lemma 1 for ~ clwlknf1oclw...
clwlknf1oclwwlknlem2 27788 Lemma 2 for ~ clwlknf1oclw...
clwlknf1oclwwlknlem3 27789 Lemma 3 for ~ clwlknf1oclw...
clwlknf1oclwwlkn 27790 There is a one-to-one onto...
clwlkssizeeq 27791 The size of the set of clo...
clwlksndivn 27792 The size of the set of clo...
clwwlknonmpo 27795 ` ( ClWWalksNOn `` G ) ` i...
clwwlknon 27796 The set of closed walks on...
isclwwlknon 27797 A word over the set of ver...
clwwlk0on0 27798 There is no word over the ...
clwwlknon0 27799 Sufficient conditions for ...
clwwlknonfin 27800 In a finite graph ` G ` , ...
clwwlknonel 27801 Characterization of a word...
clwwlknonccat 27802 The concatenation of two w...
clwwlknon1 27803 The set of closed walks on...
clwwlknon1loop 27804 If there is a loop at vert...
clwwlknon1nloop 27805 If there is no loop at ver...
clwwlknon1sn 27806 The set of (closed) walks ...
clwwlknon1le1 27807 There is at most one (clos...
clwwlknon2 27808 The set of closed walks on...
clwwlknon2x 27809 The set of closed walks on...
s2elclwwlknon2 27810 Sufficient conditions of a...
clwwlknon2num 27811 In a ` K `-regular graph `...
clwwlknonwwlknonb 27812 A word over vertices repre...
clwwlknonex2lem1 27813 Lemma 1 for ~ clwwlknonex2...
clwwlknonex2lem2 27814 Lemma 2 for ~ clwwlknonex2...
clwwlknonex2 27815 Extending a closed walk ` ...
clwwlknonex2e 27816 Extending a closed walk ` ...
clwwlknondisj 27817 The sets of closed walks o...
clwwlknun 27818 The set of closed walks of...
clwwlkvbij 27819 There is a bijection betwe...
0ewlk 27820 The empty set (empty seque...
1ewlk 27821 A sequence of 1 edge is an...
0wlk 27822 A pair of an empty set (of...
is0wlk 27823 A pair of an empty set (of...
0wlkonlem1 27824 Lemma 1 for ~ 0wlkon and ~...
0wlkonlem2 27825 Lemma 2 for ~ 0wlkon and ~...
0wlkon 27826 A walk of length 0 from a ...
0wlkons1 27827 A walk of length 0 from a ...
0trl 27828 A pair of an empty set (of...
is0trl 27829 A pair of an empty set (of...
0trlon 27830 A trail of length 0 from a...
0pth 27831 A pair of an empty set (of...
0spth 27832 A pair of an empty set (of...
0pthon 27833 A path of length 0 from a ...
0pthon1 27834 A path of length 0 from a ...
0pthonv 27835 For each vertex there is a...
0clwlk 27836 A pair of an empty set (of...
0clwlkv 27837 Any vertex (more precisely...
0clwlk0 27838 There is no closed walk in...
0crct 27839 A pair of an empty set (of...
0cycl 27840 A pair of an empty set (of...
1pthdlem1 27841 Lemma 1 for ~ 1pthd . (Co...
1pthdlem2 27842 Lemma 2 for ~ 1pthd . (Co...
1wlkdlem1 27843 Lemma 1 for ~ 1wlkd . (Co...
1wlkdlem2 27844 Lemma 2 for ~ 1wlkd . (Co...
1wlkdlem3 27845 Lemma 3 for ~ 1wlkd . (Co...
1wlkdlem4 27846 Lemma 4 for ~ 1wlkd . (Co...
1wlkd 27847 In a graph with two vertic...
1trld 27848 In a graph with two vertic...
1pthd 27849 In a graph with two vertic...
1pthond 27850 In a graph with two vertic...
upgr1wlkdlem1 27851 Lemma 1 for ~ upgr1wlkd . ...
upgr1wlkdlem2 27852 Lemma 2 for ~ upgr1wlkd . ...
upgr1wlkd 27853 In a pseudograph with two ...
upgr1trld 27854 In a pseudograph with two ...
upgr1pthd 27855 In a pseudograph with two ...
upgr1pthond 27856 In a pseudograph with two ...
lppthon 27857 A loop (which is an edge a...
lp1cycl 27858 A loop (which is an edge a...
1pthon2v 27859 For each pair of adjacent ...
1pthon2ve 27860 For each pair of adjacent ...
wlk2v2elem1 27861 Lemma 1 for ~ wlk2v2e : ` ...
wlk2v2elem2 27862 Lemma 2 for ~ wlk2v2e : T...
wlk2v2e 27863 In a graph with two vertic...
ntrl2v2e 27864 A walk which is not a trai...
3wlkdlem1 27865 Lemma 1 for ~ 3wlkd . (Co...
3wlkdlem2 27866 Lemma 2 for ~ 3wlkd . (Co...
3wlkdlem3 27867 Lemma 3 for ~ 3wlkd . (Co...
3wlkdlem4 27868 Lemma 4 for ~ 3wlkd . (Co...
3wlkdlem5 27869 Lemma 5 for ~ 3wlkd . (Co...
3pthdlem1 27870 Lemma 1 for ~ 3pthd . (Co...
3wlkdlem6 27871 Lemma 6 for ~ 3wlkd . (Co...
3wlkdlem7 27872 Lemma 7 for ~ 3wlkd . (Co...
3wlkdlem8 27873 Lemma 8 for ~ 3wlkd . (Co...
3wlkdlem9 27874 Lemma 9 for ~ 3wlkd . (Co...
3wlkdlem10 27875 Lemma 10 for ~ 3wlkd . (C...
3wlkd 27876 Construction of a walk fro...
3wlkond 27877 A walk of length 3 from on...
3trld 27878 Construction of a trail fr...
3trlond 27879 A trail of length 3 from o...
3pthd 27880 A path of length 3 from on...
3pthond 27881 A path of length 3 from on...
3spthd 27882 A simple path of length 3 ...
3spthond 27883 A simple path of length 3 ...
3cycld 27884 Construction of a 3-cycle ...
3cyclpd 27885 Construction of a 3-cycle ...
upgr3v3e3cycl 27886 If there is a cycle of len...
uhgr3cyclexlem 27887 Lemma for ~ uhgr3cyclex . ...
uhgr3cyclex 27888 If there are three differe...
umgr3cyclex 27889 If there are three (differ...
umgr3v3e3cycl 27890 If and only if there is a ...
upgr4cycl4dv4e 27891 If there is a cycle of len...
dfconngr1 27894 Alternative definition of ...
isconngr 27895 The property of being a co...
isconngr1 27896 The property of being a co...
cusconngr 27897 A complete hypergraph is c...
0conngr 27898 A graph without vertices i...
0vconngr 27899 A graph without vertices i...
1conngr 27900 A graph with (at most) one...
conngrv2edg 27901 A vertex in a connected gr...
vdn0conngrumgrv2 27902 A vertex in a connected mu...
releupth 27905 The set ` ( EulerPaths `` ...
eupths 27906 The Eulerian paths on the ...
iseupth 27907 The property " ` <. F , P ...
iseupthf1o 27908 The property " ` <. F , P ...
eupthi 27909 Properties of an Eulerian ...
eupthf1o 27910 The ` F ` function in an E...
eupthfi 27911 Any graph with an Eulerian...
eupthseg 27912 The ` N ` -th edge in an e...
upgriseupth 27913 The property " ` <. F , P ...
upgreupthi 27914 Properties of an Eulerian ...
upgreupthseg 27915 The ` N ` -th edge in an e...
eupthcl 27916 An Eulerian path has lengt...
eupthistrl 27917 An Eulerian path is a trai...
eupthiswlk 27918 An Eulerian path is a walk...
eupthpf 27919 The ` P ` function in an E...
eupth0 27920 There is an Eulerian path ...
eupthres 27921 The restriction ` <. H , Q...
eupthp1 27922 Append one path segment to...
eupth2eucrct 27923 Append one path segment to...
eupth2lem1 27924 Lemma for ~ eupth2 . (Con...
eupth2lem2 27925 Lemma for ~ eupth2 . (Con...
trlsegvdeglem1 27926 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem2 27927 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem3 27928 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem4 27929 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem5 27930 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem6 27931 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem7 27932 Lemma for ~ trlsegvdeg . ...
trlsegvdeg 27933 Formerly part of proof of ...
eupth2lem3lem1 27934 Lemma for ~ eupth2lem3 . ...
eupth2lem3lem2 27935 Lemma for ~ eupth2lem3 . ...
eupth2lem3lem3 27936 Lemma for ~ eupth2lem3 , f...
eupth2lem3lem4 27937 Lemma for ~ eupth2lem3 , f...
eupth2lem3lem5 27938 Lemma for ~ eupth2 . (Con...
eupth2lem3lem6 27939 Formerly part of proof of ...
eupth2lem3lem7 27940 Lemma for ~ eupth2lem3 : ...
eupthvdres 27941 Formerly part of proof of ...
eupth2lem3 27942 Lemma for ~ eupth2 . (Con...
eupth2lemb 27943 Lemma for ~ eupth2 (induct...
eupth2lems 27944 Lemma for ~ eupth2 (induct...
eupth2 27945 The only vertices of odd d...
eulerpathpr 27946 A graph with an Eulerian p...
eulerpath 27947 A pseudograph with an Eule...
eulercrct 27948 A pseudograph with an Eule...
eucrctshift 27949 Cyclically shifting the in...
eucrct2eupth1 27950 Removing one edge ` ( I ``...
eucrct2eupth 27951 Removing one edge ` ( I ``...
konigsbergvtx 27952 The set of vertices of the...
konigsbergiedg 27953 The indexed edges of the K...
konigsbergiedgw 27954 The indexed edges of the K...
konigsbergssiedgwpr 27955 Each subset of the indexed...
konigsbergssiedgw 27956 Each subset of the indexed...
konigsbergumgr 27957 The Königsberg graph ...
konigsberglem1 27958 Lemma 1 for ~ konigsberg :...
konigsberglem2 27959 Lemma 2 for ~ konigsberg :...
konigsberglem3 27960 Lemma 3 for ~ konigsberg :...
konigsberglem4 27961 Lemma 4 for ~ konigsberg :...
konigsberglem5 27962 Lemma 5 for ~ konigsberg :...
konigsberg 27963 The Königsberg Bridge...
isfrgr 27966 The property of being a fr...
frgrusgr 27967 A friendship graph is a si...
frgr0v 27968 Any null graph (set with n...
frgr0vb 27969 Any null graph (without ve...
frgruhgr0v 27970 Any null graph (without ve...
frgr0 27971 The null graph (graph with...
frcond1 27972 The friendship condition: ...
frcond2 27973 The friendship condition: ...
frgreu 27974 Variant of ~ frcond2 : An...
frcond3 27975 The friendship condition, ...
frcond4 27976 The friendship condition, ...
frgr1v 27977 Any graph with (at most) o...
nfrgr2v 27978 Any graph with two (differ...
frgr3vlem1 27979 Lemma 1 for ~ frgr3v . (C...
frgr3vlem2 27980 Lemma 2 for ~ frgr3v . (C...
frgr3v 27981 Any graph with three verti...
1vwmgr 27982 Every graph with one verte...
3vfriswmgrlem 27983 Lemma for ~ 3vfriswmgr . ...
3vfriswmgr 27984 Every friendship graph wit...
1to2vfriswmgr 27985 Every friendship graph wit...
1to3vfriswmgr 27986 Every friendship graph wit...
1to3vfriendship 27987 The friendship theorem for...
2pthfrgrrn 27988 Between any two (different...
2pthfrgrrn2 27989 Between any two (different...
2pthfrgr 27990 Between any two (different...
3cyclfrgrrn1 27991 Every vertex in a friendsh...
3cyclfrgrrn 27992 Every vertex in a friendsh...
3cyclfrgrrn2 27993 Every vertex in a friendsh...
3cyclfrgr 27994 Every vertex in a friendsh...
4cycl2v2nb 27995 In a (maybe degenerate) 4-...
4cycl2vnunb 27996 In a 4-cycle, two distinct...
n4cyclfrgr 27997 There is no 4-cycle in a f...
4cyclusnfrgr 27998 A graph with a 4-cycle is ...
frgrnbnb 27999 If two neighbors ` U ` and...
frgrconngr 28000 A friendship graph is conn...
vdgn0frgrv2 28001 A vertex in a friendship g...
vdgn1frgrv2 28002 Any vertex in a friendship...
vdgn1frgrv3 28003 Any vertex in a friendship...
vdgfrgrgt2 28004 Any vertex in a friendship...
frgrncvvdeqlem1 28005 Lemma 1 for ~ frgrncvvdeq ...
frgrncvvdeqlem2 28006 Lemma 2 for ~ frgrncvvdeq ...
frgrncvvdeqlem3 28007 Lemma 3 for ~ frgrncvvdeq ...
frgrncvvdeqlem4 28008 Lemma 4 for ~ frgrncvvdeq ...
frgrncvvdeqlem5 28009 Lemma 5 for ~ frgrncvvdeq ...
frgrncvvdeqlem6 28010 Lemma 6 for ~ frgrncvvdeq ...
frgrncvvdeqlem7 28011 Lemma 7 for ~ frgrncvvdeq ...
frgrncvvdeqlem8 28012 Lemma 8 for ~ frgrncvvdeq ...
frgrncvvdeqlem9 28013 Lemma 9 for ~ frgrncvvdeq ...
frgrncvvdeqlem10 28014 Lemma 10 for ~ frgrncvvdeq...
frgrncvvdeq 28015 In a friendship graph, two...
frgrwopreglem4a 28016 In a friendship graph any ...
frgrwopreglem5a 28017 If a friendship graph has ...
frgrwopreglem1 28018 Lemma 1 for ~ frgrwopreg :...
frgrwopreglem2 28019 Lemma 2 for ~ frgrwopreg ....
frgrwopreglem3 28020 Lemma 3 for ~ frgrwopreg ....
frgrwopreglem4 28021 Lemma 4 for ~ frgrwopreg ....
frgrwopregasn 28022 According to statement 5 i...
frgrwopregbsn 28023 According to statement 5 i...
frgrwopreg1 28024 According to statement 5 i...
frgrwopreg2 28025 According to statement 5 i...
frgrwopreglem5lem 28026 Lemma for ~ frgrwopreglem5...
frgrwopreglem5 28027 Lemma 5 for ~ frgrwopreg ....
frgrwopreglem5ALT 28028 Alternate direct proof of ...
frgrwopreg 28029 In a friendship graph ther...
frgrregorufr0 28030 In a friendship graph ther...
frgrregorufr 28031 If there is a vertex havin...
frgrregorufrg 28032 If there is a vertex havin...
frgr2wwlkeu 28033 For two different vertices...
frgr2wwlkn0 28034 In a friendship graph, the...
frgr2wwlk1 28035 In a friendship graph, the...
frgr2wsp1 28036 In a friendship graph, the...
frgr2wwlkeqm 28037 If there is a (simple) pat...
frgrhash2wsp 28038 The number of simple paths...
fusgreg2wsplem 28039 Lemma for ~ fusgreg2wsp an...
fusgr2wsp2nb 28040 The set of paths of length...
fusgreghash2wspv 28041 According to statement 7 i...
fusgreg2wsp 28042 In a finite simple graph, ...
2wspmdisj 28043 The sets of paths of lengt...
fusgreghash2wsp 28044 In a finite k-regular grap...
frrusgrord0lem 28045 Lemma for ~ frrusgrord0 . ...
frrusgrord0 28046 If a nonempty finite frien...
frrusgrord 28047 If a nonempty finite frien...
numclwwlk2lem1lem 28048 Lemma for ~ numclwwlk2lem1...
2clwwlklem 28049 Lemma for ~ clwwnonrepclww...
clwwnrepclwwn 28050 If the initial vertex of a...
clwwnonrepclwwnon 28051 If the initial vertex of a...
2clwwlk2clwwlklem 28052 Lemma for ~ 2clwwlk2clwwlk...
2clwwlk 28053 Value of operation ` C ` ,...
2clwwlk2 28054 The set ` ( X C 2 ) ` of d...
2clwwlkel 28055 Characterization of an ele...
2clwwlk2clwwlk 28056 An element of the value of...
numclwwlk1lem2foalem 28057 Lemma for ~ numclwwlk1lem2...
extwwlkfab 28058 The set ` ( X C N ) ` of d...
extwwlkfabel 28059 Characterization of an ele...
numclwwlk1lem2foa 28060 Going forth and back from ...
numclwwlk1lem2f 28061 ` T ` is a function, mappi...
numclwwlk1lem2fv 28062 Value of the function ` T ...
numclwwlk1lem2f1 28063 ` T ` is a 1-1 function. ...
numclwwlk1lem2fo 28064 ` T ` is an onto function....
numclwwlk1lem2f1o 28065 ` T ` is a 1-1 onto functi...
numclwwlk1lem2 28066 The set of double loops of...
numclwwlk1 28067 Statement 9 in [Huneke] p....
clwwlknonclwlknonf1o 28068 ` F ` is a bijection betwe...
clwwlknonclwlknonen 28069 The sets of the two repres...
dlwwlknondlwlknonf1olem1 28070 Lemma 1 for ~ dlwwlknondlw...
dlwwlknondlwlknonf1o 28071 ` F ` is a bijection betwe...
dlwwlknondlwlknonen 28072 The sets of the two repres...
wlkl0 28073 There is exactly one walk ...
clwlknon2num 28074 There are k walks of lengt...
numclwlk1lem1 28075 Lemma 1 for ~ numclwlk1 (S...
numclwlk1lem2 28076 Lemma 2 for ~ numclwlk1 (S...
numclwlk1 28077 Statement 9 in [Huneke] p....
numclwwlkovh0 28078 Value of operation ` H ` ,...
numclwwlkovh 28079 Value of operation ` H ` ,...
numclwwlkovq 28080 Value of operation ` Q ` ,...
numclwwlkqhash 28081 In a ` K `-regular graph, ...
numclwwlk2lem1 28082 In a friendship graph, for...
numclwlk2lem2f 28083 ` R ` is a function mappin...
numclwlk2lem2fv 28084 Value of the function ` R ...
numclwlk2lem2f1o 28085 ` R ` is a 1-1 onto functi...
numclwwlk2lem3 28086 In a friendship graph, the...
numclwwlk2 28087 Statement 10 in [Huneke] p...
numclwwlk3lem1 28088 Lemma 2 for ~ numclwwlk3 ....
numclwwlk3lem2lem 28089 Lemma for ~ numclwwlk3lem2...
numclwwlk3lem2 28090 Lemma 1 for ~ numclwwlk3 :...
numclwwlk3 28091 Statement 12 in [Huneke] p...
numclwwlk4 28092 The total number of closed...
numclwwlk5lem 28093 Lemma for ~ numclwwlk5 . ...
numclwwlk5 28094 Statement 13 in [Huneke] p...
numclwwlk7lem 28095 Lemma for ~ numclwwlk7 , ~...
numclwwlk6 28096 For a prime divisor ` P ` ...
numclwwlk7 28097 Statement 14 in [Huneke] p...
numclwwlk8 28098 The size of the set of clo...
frgrreggt1 28099 If a finite nonempty frien...
frgrreg 28100 If a finite nonempty frien...
frgrregord013 28101 If a finite friendship gra...
frgrregord13 28102 If a nonempty finite frien...
frgrogt3nreg 28103 If a finite friendship gra...
friendshipgt3 28104 The friendship theorem for...
friendship 28105 The friendship theorem: I...
conventions 28106

...

conventions-labels 28107

...

conventions-comments 28108

...

natded 28109 Here are typical n...
ex-natded5.2 28110 Theorem 5.2 of [Clemente] ...
ex-natded5.2-2 28111 A more efficient proof of ...
ex-natded5.2i 28112 The same as ~ ex-natded5.2...
ex-natded5.3 28113 Theorem 5.3 of [Clemente] ...
ex-natded5.3-2 28114 A more efficient proof of ...
ex-natded5.3i 28115 The same as ~ ex-natded5.3...
ex-natded5.5 28116 Theorem 5.5 of [Clemente] ...
ex-natded5.7 28117 Theorem 5.7 of [Clemente] ...
ex-natded5.7-2 28118 A more efficient proof of ...
ex-natded5.8 28119 Theorem 5.8 of [Clemente] ...
ex-natded5.8-2 28120 A more efficient proof of ...
ex-natded5.13 28121 Theorem 5.13 of [Clemente]...
ex-natded5.13-2 28122 A more efficient proof of ...
ex-natded9.20 28123 Theorem 9.20 of [Clemente]...
ex-natded9.20-2 28124 A more efficient proof of ...
ex-natded9.26 28125 Theorem 9.26 of [Clemente]...
ex-natded9.26-2 28126 A more efficient proof of ...
ex-or 28127 Example for ~ df-or . Exa...
ex-an 28128 Example for ~ df-an . Exa...
ex-dif 28129 Example for ~ df-dif . Ex...
ex-un 28130 Example for ~ df-un . Exa...
ex-in 28131 Example for ~ df-in . Exa...
ex-uni 28132 Example for ~ df-uni . Ex...
ex-ss 28133 Example for ~ df-ss . Exa...
ex-pss 28134 Example for ~ df-pss . Ex...
ex-pw 28135 Example for ~ df-pw . Exa...
ex-pr 28136 Example for ~ df-pr . (Co...
ex-br 28137 Example for ~ df-br . Exa...
ex-opab 28138 Example for ~ df-opab . E...
ex-eprel 28139 Example for ~ df-eprel . ...
ex-id 28140 Example for ~ df-id . Exa...
ex-po 28141 Example for ~ df-po . Exa...
ex-xp 28142 Example for ~ df-xp . Exa...
ex-cnv 28143 Example for ~ df-cnv . Ex...
ex-co 28144 Example for ~ df-co . Exa...
ex-dm 28145 Example for ~ df-dm . Exa...
ex-rn 28146 Example for ~ df-rn . Exa...
ex-res 28147 Example for ~ df-res . Ex...
ex-ima 28148 Example for ~ df-ima . Ex...
ex-fv 28149 Example for ~ df-fv . Exa...
ex-1st 28150 Example for ~ df-1st . Ex...
ex-2nd 28151 Example for ~ df-2nd . Ex...
1kp2ke3k 28152 Example for ~ df-dec , 100...
ex-fl 28153 Example for ~ df-fl . Exa...
ex-ceil 28154 Example for ~ df-ceil . (...
ex-mod 28155 Example for ~ df-mod . (C...
ex-exp 28156 Example for ~ df-exp . (C...
ex-fac 28157 Example for ~ df-fac . (C...
ex-bc 28158 Example for ~ df-bc . (Co...
ex-hash 28159 Example for ~ df-hash . (...
ex-sqrt 28160 Example for ~ df-sqrt . (...
ex-abs 28161 Example for ~ df-abs . (C...
ex-dvds 28162 Example for ~ df-dvds : 3 ...
ex-gcd 28163 Example for ~ df-gcd . (C...
ex-lcm 28164 Example for ~ df-lcm . (C...
ex-prmo 28165 Example for ~ df-prmo : ` ...
aevdemo 28166 Proof illustrating the com...
ex-ind-dvds 28167 Example of a proof by indu...
ex-fpar 28168 Formalized example provide...
avril1 28169 Poisson d'Avril's Theorem....
2bornot2b 28170 The law of excluded middle...
helloworld 28171 The classic "Hello world" ...
1p1e2apr1 28172 One plus one equals two. ...
eqid1 28173 Law of identity (reflexivi...
1div0apr 28174 Division by zero is forbid...
topnfbey 28175 Nothing seems to be imposs...
9p10ne21 28176 9 + 10 is not equal to 21....
9p10ne21fool 28177 9 + 10 equals 21. This as...
isplig 28180 The predicate "is a planar...
ispligb 28181 The predicate "is a planar...
tncp 28182 In any planar incidence ge...
l2p 28183 For any line in a planar i...
lpni 28184 For any line in a planar i...
nsnlplig 28185 There is no "one-point lin...
nsnlpligALT 28186 Alternate version of ~ nsn...
n0lplig 28187 There is no "empty line" i...
n0lpligALT 28188 Alternate version of ~ n0l...
eulplig 28189 Through two distinct point...
pliguhgr 28190 Any planar incidence geome...
dummylink 28191 Alias for ~ a1ii that may ...
id1 28192 Alias for ~ idALT that may...
isgrpo 28201 The predicate "is a group ...
isgrpoi 28202 Properties that determine ...
grpofo 28203 A group operation maps ont...
grpocl 28204 Closure law for a group op...
grpolidinv 28205 A group has a left identit...
grpon0 28206 The base set of a group is...
grpoass 28207 A group operation is assoc...
grpoidinvlem1 28208 Lemma for ~ grpoidinv . (...
grpoidinvlem2 28209 Lemma for ~ grpoidinv . (...
grpoidinvlem3 28210 Lemma for ~ grpoidinv . (...
grpoidinvlem4 28211 Lemma for ~ grpoidinv . (...
grpoidinv 28212 A group has a left and rig...
grpoideu 28213 The left identity element ...
grporndm 28214 A group's range in terms o...
0ngrp 28215 The empty set is not a gro...
gidval 28216 The value of the identity ...
grpoidval 28217 Lemma for ~ grpoidcl and o...
grpoidcl 28218 The identity element of a ...
grpoidinv2 28219 A group's properties using...
grpolid 28220 The identity element of a ...
grporid 28221 The identity element of a ...
grporcan 28222 Right cancellation law for...
grpoinveu 28223 The left inverse element o...
grpoid 28224 Two ways of saying that an...
grporn 28225 The range of a group opera...
grpoinvfval 28226 The inverse function of a ...
grpoinvval 28227 The inverse of a group ele...
grpoinvcl 28228 A group element's inverse ...
grpoinv 28229 The properties of a group ...
grpolinv 28230 The left inverse of a grou...
grporinv 28231 The right inverse of a gro...
grpoinvid1 28232 The inverse of a group ele...
grpoinvid2 28233 The inverse of a group ele...
grpolcan 28234 Left cancellation law for ...
grpo2inv 28235 Double inverse law for gro...
grpoinvf 28236 Mapping of the inverse fun...
grpoinvop 28237 The inverse of the group o...
grpodivfval 28238 Group division (or subtrac...
grpodivval 28239 Group division (or subtrac...
grpodivinv 28240 Group division by an inver...
grpoinvdiv 28241 Inverse of a group divisio...
grpodivf 28242 Mapping for group division...
grpodivcl 28243 Closure of group division ...
grpodivdiv 28244 Double group division. (C...
grpomuldivass 28245 Associative-type law for m...
grpodivid 28246 Division of a group member...
grponpcan 28247 Cancellation law for group...
isablo 28250 The predicate "is an Abeli...
ablogrpo 28251 An Abelian group operation...
ablocom 28252 An Abelian group operation...
ablo32 28253 Commutative/associative la...
ablo4 28254 Commutative/associative la...
isabloi 28255 Properties that determine ...
ablomuldiv 28256 Law for group multiplicati...
ablodivdiv 28257 Law for double group divis...
ablodivdiv4 28258 Law for double group divis...
ablodiv32 28259 Swap the second and third ...
ablonncan 28260 Cancellation law for group...
ablonnncan1 28261 Cancellation law for group...
vcrel 28264 The class of all complex v...
vciOLD 28265 Obsolete version of ~ cvsi...
vcsm 28266 Functionality of th scalar...
vccl 28267 Closure of the scalar prod...
vcidOLD 28268 Identity element for the s...
vcdi 28269 Distributive law for the s...
vcdir 28270 Distributive law for the s...
vcass 28271 Associative law for the sc...
vc2OLD 28272 A vector plus itself is tw...
vcablo 28273 Vector addition is an Abel...
vcgrp 28274 Vector addition is a group...
vclcan 28275 Left cancellation law for ...
vczcl 28276 The zero vector is a vecto...
vc0rid 28277 The zero vector is a right...
vc0 28278 Zero times a vector is the...
vcz 28279 Anything times the zero ve...
vcm 28280 Minus 1 times a vector is ...
isvclem 28281 Lemma for ~ isvcOLD . (Co...
vcex 28282 The components of a comple...
isvcOLD 28283 The predicate "is a comple...
isvciOLD 28284 Properties that determine ...
cnaddabloOLD 28285 Obsolete version of ~ cnad...
cnidOLD 28286 Obsolete version of ~ cnad...
cncvcOLD 28287 Obsolete version of ~ cncv...
nvss 28297 Structure of the class of ...
nvvcop 28298 A normed complex vector sp...
nvrel 28306 The class of all normed co...
vafval 28307 Value of the function for ...
bafval 28308 Value of the function for ...
smfval 28309 Value of the function for ...
0vfval 28310 Value of the function for ...
nmcvfval 28311 Value of the norm function...
nvop2 28312 A normed complex vector sp...
nvvop 28313 The vector space component...
isnvlem 28314 Lemma for ~ isnv . (Contr...
nvex 28315 The components of a normed...
isnv 28316 The predicate "is a normed...
isnvi 28317 Properties that determine ...
nvi 28318 The properties of a normed...
nvvc 28319 The vector space component...
nvablo 28320 The vector addition operat...
nvgrp 28321 The vector addition operat...
nvgf 28322 Mapping for the vector add...
nvsf 28323 Mapping for the scalar mul...
nvgcl 28324 Closure law for the vector...
nvcom 28325 The vector addition (group...
nvass 28326 The vector addition (group...
nvadd32 28327 Commutative/associative la...
nvrcan 28328 Right cancellation law for...
nvadd4 28329 Rearrangement of 4 terms i...
nvscl 28330 Closure law for the scalar...
nvsid 28331 Identity element for the s...
nvsass 28332 Associative law for the sc...
nvscom 28333 Commutative law for the sc...
nvdi 28334 Distributive law for the s...
nvdir 28335 Distributive law for the s...
nv2 28336 A vector plus itself is tw...
vsfval 28337 Value of the function for ...
nvzcl 28338 Closure law for the zero v...
nv0rid 28339 The zero vector is a right...
nv0lid 28340 The zero vector is a left ...
nv0 28341 Zero times a vector is the...
nvsz 28342 Anything times the zero ve...
nvinv 28343 Minus 1 times a vector is ...
nvinvfval 28344 Function for the negative ...
nvm 28345 Vector subtraction in term...
nvmval 28346 Value of vector subtractio...
nvmval2 28347 Value of vector subtractio...
nvmfval 28348 Value of the function for ...
nvmf 28349 Mapping for the vector sub...
nvmcl 28350 Closure law for the vector...
nvnnncan1 28351 Cancellation law for vecto...
nvmdi 28352 Distributive law for scala...
nvnegneg 28353 Double negative of a vecto...
nvmul0or 28354 If a scalar product is zer...
nvrinv 28355 A vector minus itself. (C...
nvlinv 28356 Minus a vector plus itself...
nvpncan2 28357 Cancellation law for vecto...
nvpncan 28358 Cancellation law for vecto...
nvaddsub 28359 Commutative/associative la...
nvnpcan 28360 Cancellation law for a nor...
nvaddsub4 28361 Rearrangement of 4 terms i...
nvmeq0 28362 The difference between two...
nvmid 28363 A vector minus itself is t...
nvf 28364 Mapping for the norm funct...
nvcl 28365 The norm of a normed compl...
nvcli 28366 The norm of a normed compl...
nvs 28367 Proportionality property o...
nvsge0 28368 The norm of a scalar produ...
nvm1 28369 The norm of the negative o...
nvdif 28370 The norm of the difference...
nvpi 28371 The norm of a vector plus ...
nvz0 28372 The norm of a zero vector ...
nvz 28373 The norm of a vector is ze...
nvtri 28374 Triangle inequality for th...
nvmtri 28375 Triangle inequality for th...
nvabs 28376 Norm difference property o...
nvge0 28377 The norm of a normed compl...
nvgt0 28378 A nonzero norm is positive...
nv1 28379 From any nonzero vector, c...
nvop 28380 A complex inner product sp...
cnnv 28381 The set of complex numbers...
cnnvg 28382 The vector addition (group...
cnnvba 28383 The base set of the normed...
cnnvs 28384 The scalar product operati...
cnnvnm 28385 The norm operation of the ...
cnnvm 28386 The vector subtraction ope...
elimnv 28387 Hypothesis elimination lem...
elimnvu 28388 Hypothesis elimination lem...
imsval 28389 Value of the induced metri...
imsdval 28390 Value of the induced metri...
imsdval2 28391 Value of the distance func...
nvnd 28392 The norm of a normed compl...
imsdf 28393 Mapping for the induced me...
imsmetlem 28394 Lemma for ~ imsmet . (Con...
imsmet 28395 The induced metric of a no...
imsxmet 28396 The induced metric of a no...
cnims 28397 The metric induced on the ...
vacn 28398 Vector addition is jointly...
nmcvcn 28399 The norm of a normed compl...
nmcnc 28400 The norm of a normed compl...
smcnlem 28401 Lemma for ~ smcn . (Contr...
smcn 28402 Scalar multiplication is j...
vmcn 28403 Vector subtraction is join...
dipfval 28406 The inner product function...
ipval 28407 Value of the inner product...
ipval2lem2 28408 Lemma for ~ ipval3 . (Con...
ipval2lem3 28409 Lemma for ~ ipval3 . (Con...
ipval2lem4 28410 Lemma for ~ ipval3 . (Con...
ipval2 28411 Expansion of the inner pro...
4ipval2 28412 Four times the inner produ...
ipval3 28413 Expansion of the inner pro...
ipidsq 28414 The inner product of a vec...
ipnm 28415 Norm expressed in terms of...
dipcl 28416 An inner product is a comp...
ipf 28417 Mapping for the inner prod...
dipcj 28418 The complex conjugate of a...
ipipcj 28419 An inner product times its...
diporthcom 28420 Orthogonality (meaning inn...
dip0r 28421 Inner product with a zero ...
dip0l 28422 Inner product with a zero ...
ipz 28423 The inner product of a vec...
dipcn 28424 Inner product is jointly c...
sspval 28427 The set of all subspaces o...
isssp 28428 The predicate "is a subspa...
sspid 28429 A normed complex vector sp...
sspnv 28430 A subspace is a normed com...
sspba 28431 The base set of a subspace...
sspg 28432 Vector addition on a subsp...
sspgval 28433 Vector addition on a subsp...
ssps 28434 Scalar multiplication on a...
sspsval 28435 Scalar multiplication on a...
sspmlem 28436 Lemma for ~ sspm and other...
sspmval 28437 Vector addition on a subsp...
sspm 28438 Vector subtraction on a su...
sspz 28439 The zero vector of a subsp...
sspn 28440 The norm on a subspace is ...
sspnval 28441 The norm on a subspace in ...
sspimsval 28442 The induced metric on a su...
sspims 28443 The induced metric on a su...
lnoval 28456 The set of linear operator...
islno 28457 The predicate "is a linear...
lnolin 28458 Basic linearity property o...
lnof 28459 A linear operator is a map...
lno0 28460 The value of a linear oper...
lnocoi 28461 The composition of two lin...
lnoadd 28462 Addition property of a lin...
lnosub 28463 Subtraction property of a ...
lnomul 28464 Scalar multiplication prop...
nvo00 28465 Two ways to express a zero...
nmoofval 28466 The operator norm function...
nmooval 28467 The operator norm function...
nmosetre 28468 The set in the supremum of...
nmosetn0 28469 The set in the supremum of...
nmoxr 28470 The norm of an operator is...
nmooge0 28471 The norm of an operator is...
nmorepnf 28472 The norm of an operator is...
nmoreltpnf 28473 The norm of any operator i...
nmogtmnf 28474 The norm of an operator is...
nmoolb 28475 A lower bound for an opera...
nmoubi 28476 An upper bound for an oper...
nmoub3i 28477 An upper bound for an oper...
nmoub2i 28478 An upper bound for an oper...
nmobndi 28479 Two ways to express that a...
nmounbi 28480 Two ways two express that ...
nmounbseqi 28481 An unbounded operator dete...
nmounbseqiALT 28482 Alternate shorter proof of...
nmobndseqi 28483 A bounded sequence determi...
nmobndseqiALT 28484 Alternate shorter proof of...
bloval 28485 The class of bounded linea...
isblo 28486 The predicate "is a bounde...
isblo2 28487 The predicate "is a bounde...
bloln 28488 A bounded operator is a li...
blof 28489 A bounded operator is an o...
nmblore 28490 The norm of a bounded oper...
0ofval 28491 The zero operator between ...
0oval 28492 Value of the zero operator...
0oo 28493 The zero operator is an op...
0lno 28494 The zero operator is linea...
nmoo0 28495 The operator norm of the z...
0blo 28496 The zero operator is a bou...
nmlno0lem 28497 Lemma for ~ nmlno0i . (Co...
nmlno0i 28498 The norm of a linear opera...
nmlno0 28499 The norm of a linear opera...
nmlnoubi 28500 An upper bound for the ope...
nmlnogt0 28501 The norm of a nonzero line...
lnon0 28502 The domain of a nonzero li...
nmblolbii 28503 A lower bound for the norm...
nmblolbi 28504 A lower bound for the norm...
isblo3i 28505 The predicate "is a bounde...
blo3i 28506 Properties that determine ...
blometi 28507 Upper bound for the distan...
blocnilem 28508 Lemma for ~ blocni and ~ l...
blocni 28509 A linear operator is conti...
lnocni 28510 If a linear operator is co...
blocn 28511 A linear operator is conti...
blocn2 28512 A bounded linear operator ...
ajfval 28513 The adjoint function. (Co...
hmoval 28514 The set of Hermitian (self...
ishmo 28515 The predicate "is a hermit...
phnv 28518 Every complex inner produc...
phrel 28519 The class of all complex i...
phnvi 28520 Every complex inner produc...
isphg 28521 The predicate "is a comple...
phop 28522 A complex inner product sp...
cncph 28523 The set of complex numbers...
elimph 28524 Hypothesis elimination lem...
elimphu 28525 Hypothesis elimination lem...
isph 28526 The predicate "is an inner...
phpar2 28527 The parallelogram law for ...
phpar 28528 The parallelogram law for ...
ip0i 28529 A slight variant of Equati...
ip1ilem 28530 Lemma for ~ ip1i . (Contr...
ip1i 28531 Equation 6.47 of [Ponnusam...
ip2i 28532 Equation 6.48 of [Ponnusam...
ipdirilem 28533 Lemma for ~ ipdiri . (Con...
ipdiri 28534 Distributive law for inner...
ipasslem1 28535 Lemma for ~ ipassi . Show...
ipasslem2 28536 Lemma for ~ ipassi . Show...
ipasslem3 28537 Lemma for ~ ipassi . Show...
ipasslem4 28538 Lemma for ~ ipassi . Show...
ipasslem5 28539 Lemma for ~ ipassi . Show...
ipasslem7 28540 Lemma for ~ ipassi . Show...
ipasslem8 28541 Lemma for ~ ipassi . By ~...
ipasslem9 28542 Lemma for ~ ipassi . Conc...
ipasslem10 28543 Lemma for ~ ipassi . Show...
ipasslem11 28544 Lemma for ~ ipassi . Show...
ipassi 28545 Associative law for inner ...
dipdir 28546 Distributive law for inner...
dipdi 28547 Distributive law for inner...
ip2dii 28548 Inner product of two sums....
dipass 28549 Associative law for inner ...
dipassr 28550 "Associative" law for seco...
dipassr2 28551 "Associative" law for inne...
dipsubdir 28552 Distributive law for inner...
dipsubdi 28553 Distributive law for inner...
pythi 28554 The Pythagorean theorem fo...
siilem1 28555 Lemma for ~ sii . (Contri...
siilem2 28556 Lemma for ~ sii . (Contri...
siii 28557 Inference from ~ sii . (C...
sii 28558 Schwarz inequality. Part ...
ipblnfi 28559 A function ` F ` generated...
ip2eqi 28560 Two vectors are equal iff ...
phoeqi 28561 A condition implying that ...
ajmoi 28562 Every operator has at most...
ajfuni 28563 The adjoint function is a ...
ajfun 28564 The adjoint function is a ...
ajval 28565 Value of the adjoint funct...
iscbn 28568 A complex Banach space is ...
cbncms 28569 The induced metric on comp...
bnnv 28570 Every complex Banach space...
bnrel 28571 The class of all complex B...
bnsscmcl 28572 A subspace of a Banach spa...
cnbn 28573 The set of complex numbers...
ubthlem1 28574 Lemma for ~ ubth . The fu...
ubthlem2 28575 Lemma for ~ ubth . Given ...
ubthlem3 28576 Lemma for ~ ubth . Prove ...
ubth 28577 Uniform Boundedness Theore...
minvecolem1 28578 Lemma for ~ minveco . The...
minvecolem2 28579 Lemma for ~ minveco . Any...
minvecolem3 28580 Lemma for ~ minveco . The...
minvecolem4a 28581 Lemma for ~ minveco . ` F ...
minvecolem4b 28582 Lemma for ~ minveco . The...
minvecolem4c 28583 Lemma for ~ minveco . The...
minvecolem4 28584 Lemma for ~ minveco . The...
minvecolem5 28585 Lemma for ~ minveco . Dis...
minvecolem6 28586 Lemma for ~ minveco . Any...
minvecolem7 28587 Lemma for ~ minveco . Sin...
minveco 28588 Minimizing vector theorem,...
ishlo 28591 The predicate "is a comple...
hlobn 28592 Every complex Hilbert spac...
hlph 28593 Every complex Hilbert spac...
hlrel 28594 The class of all complex H...
hlnv 28595 Every complex Hilbert spac...
hlnvi 28596 Every complex Hilbert spac...
hlvc 28597 Every complex Hilbert spac...
hlcmet 28598 The induced metric on a co...
hlmet 28599 The induced metric on a co...
hlpar2 28600 The parallelogram law sati...
hlpar 28601 The parallelogram law sati...
hlex 28602 The base set of a Hilbert ...
hladdf 28603 Mapping for Hilbert space ...
hlcom 28604 Hilbert space vector addit...
hlass 28605 Hilbert space vector addit...
hl0cl 28606 The Hilbert space zero vec...
hladdid 28607 Hilbert space addition wit...
hlmulf 28608 Mapping for Hilbert space ...
hlmulid 28609 Hilbert space scalar multi...
hlmulass 28610 Hilbert space scalar multi...
hldi 28611 Hilbert space scalar multi...
hldir 28612 Hilbert space scalar multi...
hlmul0 28613 Hilbert space scalar multi...
hlipf 28614 Mapping for Hilbert space ...
hlipcj 28615 Conjugate law for Hilbert ...
hlipdir 28616 Distributive law for Hilbe...
hlipass 28617 Associative law for Hilber...
hlipgt0 28618 The inner product of a Hil...
hlcompl 28619 Completeness of a Hilbert ...
cnchl 28620 The set of complex numbers...
htthlem 28621 Lemma for ~ htth . The co...
htth 28622 Hellinger-Toeplitz Theorem...
The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here
h2hva 28678 The group (addition) opera...
h2hsm 28679 The scalar product operati...
h2hnm 28680 The norm function of Hilbe...
h2hvs 28681 The vector subtraction ope...
h2hmetdval 28682 Value of the distance func...
h2hcau 28683 The Cauchy sequences of Hi...
h2hlm 28684 The limit sequences of Hil...
axhilex-zf 28685 Derive axiom ~ ax-hilex fr...
axhfvadd-zf 28686 Derive axiom ~ ax-hfvadd f...
axhvcom-zf 28687 Derive axiom ~ ax-hvcom fr...
axhvass-zf 28688 Derive axiom ~ ax-hvass fr...
axhv0cl-zf 28689 Derive axiom ~ ax-hv0cl fr...
axhvaddid-zf 28690 Derive axiom ~ ax-hvaddid ...
axhfvmul-zf 28691 Derive axiom ~ ax-hfvmul f...
axhvmulid-zf 28692 Derive axiom ~ ax-hvmulid ...
axhvmulass-zf 28693 Derive axiom ~ ax-hvmulass...
axhvdistr1-zf 28694 Derive axiom ~ ax-hvdistr1...
axhvdistr2-zf 28695 Derive axiom ~ ax-hvdistr2...
axhvmul0-zf 28696 Derive axiom ~ ax-hvmul0 f...
axhfi-zf 28697 Derive axiom ~ ax-hfi from...
axhis1-zf 28698 Derive axiom ~ ax-his1 fro...
axhis2-zf 28699 Derive axiom ~ ax-his2 fro...
axhis3-zf 28700 Derive axiom ~ ax-his3 fro...
axhis4-zf 28701 Derive axiom ~ ax-his4 fro...
axhcompl-zf 28702 Derive axiom ~ ax-hcompl f...
hvmulex 28715 The Hilbert space scalar p...
hvaddcl 28716 Closure of vector addition...
hvmulcl 28717 Closure of scalar multipli...
hvmulcli 28718 Closure inference for scal...
hvsubf 28719 Mapping domain and codomai...
hvsubval 28720 Value of vector subtractio...
hvsubcl 28721 Closure of vector subtract...
hvaddcli 28722 Closure of vector addition...
hvcomi 28723 Commutation of vector addi...
hvsubvali 28724 Value of vector subtractio...
hvsubcli 28725 Closure of vector subtract...
ifhvhv0 28726 Prove ` if ( A e. ~H , A ,...
hvaddid2 28727 Addition with the zero vec...
hvmul0 28728 Scalar multiplication with...
hvmul0or 28729 If a scalar product is zer...
hvsubid 28730 Subtraction of a vector fr...
hvnegid 28731 Addition of negative of a ...
hv2neg 28732 Two ways to express the ne...
hvaddid2i 28733 Addition with the zero vec...
hvnegidi 28734 Addition of negative of a ...
hv2negi 28735 Two ways to express the ne...
hvm1neg 28736 Convert minus one times a ...
hvaddsubval 28737 Value of vector addition i...
hvadd32 28738 Commutative/associative la...
hvadd12 28739 Commutative/associative la...
hvadd4 28740 Hilbert vector space addit...
hvsub4 28741 Hilbert vector space addit...
hvaddsub12 28742 Commutative/associative la...
hvpncan 28743 Addition/subtraction cance...
hvpncan2 28744 Addition/subtraction cance...
hvaddsubass 28745 Associativity of sum and d...
hvpncan3 28746 Subtraction and addition o...
hvmulcom 28747 Scalar multiplication comm...
hvsubass 28748 Hilbert vector space assoc...
hvsub32 28749 Hilbert vector space commu...
hvmulassi 28750 Scalar multiplication asso...
hvmulcomi 28751 Scalar multiplication comm...
hvmul2negi 28752 Double negative in scalar ...
hvsubdistr1 28753 Scalar multiplication dist...
hvsubdistr2 28754 Scalar multiplication dist...
hvdistr1i 28755 Scalar multiplication dist...
hvsubdistr1i 28756 Scalar multiplication dist...
hvassi 28757 Hilbert vector space assoc...
hvadd32i 28758 Hilbert vector space commu...
hvsubassi 28759 Hilbert vector space assoc...
hvsub32i 28760 Hilbert vector space commu...
hvadd12i 28761 Hilbert vector space commu...
hvadd4i 28762 Hilbert vector space addit...
hvsubsub4i 28763 Hilbert vector space addit...
hvsubsub4 28764 Hilbert vector space addit...
hv2times 28765 Two times a vector. (Cont...
hvnegdii 28766 Distribution of negative o...
hvsubeq0i 28767 If the difference between ...
hvsubcan2i 28768 Vector cancellation law. ...
hvaddcani 28769 Cancellation law for vecto...
hvsubaddi 28770 Relationship between vecto...
hvnegdi 28771 Distribution of negative o...
hvsubeq0 28772 If the difference between ...
hvaddeq0 28773 If the sum of two vectors ...
hvaddcan 28774 Cancellation law for vecto...
hvaddcan2 28775 Cancellation law for vecto...
hvmulcan 28776 Cancellation law for scala...
hvmulcan2 28777 Cancellation law for scala...
hvsubcan 28778 Cancellation law for vecto...
hvsubcan2 28779 Cancellation law for vecto...
hvsub0 28780 Subtraction of a zero vect...
hvsubadd 28781 Relationship between vecto...
hvaddsub4 28782 Hilbert vector space addit...
hicl 28784 Closure of inner product. ...
hicli 28785 Closure inference for inne...
his5 28790 Associative law for inner ...
his52 28791 Associative law for inner ...
his35 28792 Move scalar multiplication...
his35i 28793 Move scalar multiplication...
his7 28794 Distributive law for inner...
hiassdi 28795 Distributive/associative l...
his2sub 28796 Distributive law for inner...
his2sub2 28797 Distributive law for inner...
hire 28798 A necessary and sufficient...
hiidrcl 28799 Real closure of inner prod...
hi01 28800 Inner product with the 0 v...
hi02 28801 Inner product with the 0 v...
hiidge0 28802 Inner product with self is...
his6 28803 Zero inner product with se...
his1i 28804 Conjugate law for inner pr...
abshicom 28805 Commuted inner products ha...
hial0 28806 A vector whose inner produ...
hial02 28807 A vector whose inner produ...
hisubcomi 28808 Two vector subtractions si...
hi2eq 28809 Lemma used to prove equali...
hial2eq 28810 Two vectors whose inner pr...
hial2eq2 28811 Two vectors whose inner pr...
orthcom 28812 Orthogonality commutes. (...
normlem0 28813 Lemma used to derive prope...
normlem1 28814 Lemma used to derive prope...
normlem2 28815 Lemma used to derive prope...
normlem3 28816 Lemma used to derive prope...
normlem4 28817 Lemma used to derive prope...
normlem5 28818 Lemma used to derive prope...
normlem6 28819 Lemma used to derive prope...
normlem7 28820 Lemma used to derive prope...
normlem8 28821 Lemma used to derive prope...
normlem9 28822 Lemma used to derive prope...
normlem7tALT 28823 Lemma used to derive prope...
bcseqi 28824 Equality case of Bunjakova...
normlem9at 28825 Lemma used to derive prope...
dfhnorm2 28826 Alternate definition of th...
normf 28827 The norm function maps fro...
normval 28828 The value of the norm of a...
normcl 28829 Real closure of the norm o...
normge0 28830 The norm of a vector is no...
normgt0 28831 The norm of nonzero vector...
norm0 28832 The norm of a zero vector....
norm-i 28833 Theorem 3.3(i) of [Beran] ...
normne0 28834 A norm is nonzero iff its ...
normcli 28835 Real closure of the norm o...
normsqi 28836 The square of a norm. (Co...
norm-i-i 28837 Theorem 3.3(i) of [Beran] ...
normsq 28838 The square of a norm. (Co...
normsub0i 28839 Two vectors are equal iff ...
normsub0 28840 Two vectors are equal iff ...
norm-ii-i 28841 Triangle inequality for no...
norm-ii 28842 Triangle inequality for no...
norm-iii-i 28843 Theorem 3.3(iii) of [Beran...
norm-iii 28844 Theorem 3.3(iii) of [Beran...
normsubi 28845 Negative doesn't change th...
normpythi 28846 Analogy to Pythagorean the...
normsub 28847 Swapping order of subtract...
normneg 28848 The norm of a vector equal...
normpyth 28849 Analogy to Pythagorean the...
normpyc 28850 Corollary to Pythagorean t...
norm3difi 28851 Norm of differences around...
norm3adifii 28852 Norm of differences around...
norm3lem 28853 Lemma involving norm of di...
norm3dif 28854 Norm of differences around...
norm3dif2 28855 Norm of differences around...
norm3lemt 28856 Lemma involving norm of di...
norm3adifi 28857 Norm of differences around...
normpari 28858 Parallelogram law for norm...
normpar 28859 Parallelogram law for norm...
normpar2i 28860 Corollary of parallelogram...
polid2i 28861 Generalized polarization i...
polidi 28862 Polarization identity. Re...
polid 28863 Polarization identity. Re...
hilablo 28864 Hilbert space vector addit...
hilid 28865 The group identity element...
hilvc 28866 Hilbert space is a complex...
hilnormi 28867 Hilbert space norm in term...
hilhhi 28868 Deduce the structure of Hi...
hhnv 28869 Hilbert space is a normed ...
hhva 28870 The group (addition) opera...
hhba 28871 The base set of Hilbert sp...
hh0v 28872 The zero vector of Hilbert...
hhsm 28873 The scalar product operati...
hhvs 28874 The vector subtraction ope...
hhnm 28875 The norm function of Hilbe...
hhims 28876 The induced metric of Hilb...
hhims2 28877 Hilbert space distance met...
hhmet 28878 The induced metric of Hilb...
hhxmet 28879 The induced metric of Hilb...
hhmetdval 28880 Value of the distance func...
hhip 28881 The inner product operatio...
hhph 28882 The Hilbert space of the H...
bcsiALT 28883 Bunjakovaskij-Cauchy-Schwa...
bcsiHIL 28884 Bunjakovaskij-Cauchy-Schwa...
bcs 28885 Bunjakovaskij-Cauchy-Schwa...
bcs2 28886 Corollary of the Bunjakova...
bcs3 28887 Corollary of the Bunjakova...
hcau 28888 Member of the set of Cauch...
hcauseq 28889 A Cauchy sequences on a Hi...
hcaucvg 28890 A Cauchy sequence on a Hil...
seq1hcau 28891 A sequence on a Hilbert sp...
hlimi 28892 Express the predicate: Th...
hlimseqi 28893 A sequence with a limit on...
hlimveci 28894 Closure of the limit of a ...
hlimconvi 28895 Convergence of a sequence ...
hlim2 28896 The limit of a sequence on...
hlimadd 28897 Limit of the sum of two se...
hilmet 28898 The Hilbert space norm det...
hilxmet 28899 The Hilbert space norm det...
hilmetdval 28900 Value of the distance func...
hilims 28901 Hilbert space distance met...
hhcau 28902 The Cauchy sequences of Hi...
hhlm 28903 The limit sequences of Hil...
hhcmpl 28904 Lemma used for derivation ...
hilcompl 28905 Lemma used for derivation ...
hhcms 28907 The Hilbert space induced ...
hhhl 28908 The Hilbert space structur...
hilcms 28909 The Hilbert space norm det...
hilhl 28910 The Hilbert space of the H...
issh 28912 Subspace ` H ` of a Hilber...
issh2 28913 Subspace ` H ` of a Hilber...
shss 28914 A subspace is a subset of ...
shel 28915 A member of a subspace of ...
shex 28916 The set of subspaces of a ...
shssii 28917 A closed subspace of a Hil...
sheli 28918 A member of a subspace of ...
shelii 28919 A member of a subspace of ...
sh0 28920 The zero vector belongs to...
shaddcl 28921 Closure of vector addition...
shmulcl 28922 Closure of vector scalar m...
issh3 28923 Subspace ` H ` of a Hilber...
shsubcl 28924 Closure of vector subtract...
isch 28926 Closed subspace ` H ` of a...
isch2 28927 Closed subspace ` H ` of a...
chsh 28928 A closed subspace is a sub...
chsssh 28929 Closed subspaces are subsp...
chex 28930 The set of closed subspace...
chshii 28931 A closed subspace is a sub...
ch0 28932 The zero vector belongs to...
chss 28933 A closed subspace of a Hil...
chel 28934 A member of a closed subsp...
chssii 28935 A closed subspace of a Hil...
cheli 28936 A member of a closed subsp...
chelii 28937 A member of a closed subsp...
chlimi 28938 The limit property of a cl...
hlim0 28939 The zero sequence in Hilbe...
hlimcaui 28940 If a sequence in Hilbert s...
hlimf 28941 Function-like behavior of ...
hlimuni 28942 A Hilbert space sequence c...
hlimreui 28943 The limit of a Hilbert spa...
hlimeui 28944 The limit of a Hilbert spa...
isch3 28945 A Hilbert subspace is clos...
chcompl 28946 Completeness of a closed s...
helch 28947 The unit Hilbert lattice e...
ifchhv 28948 Prove ` if ( A e. CH , A ,...
helsh 28949 Hilbert space is a subspac...
shsspwh 28950 Subspaces are subsets of H...
chsspwh 28951 Closed subspaces are subse...
hsn0elch 28952 The zero subspace belongs ...
norm1 28953 From any nonzero Hilbert s...
norm1exi 28954 A normalized vector exists...
norm1hex 28955 A normalized vector can ex...
elch0 28958 Membership in zero for clo...
h0elch 28959 The zero subspace is a clo...
h0elsh 28960 The zero subspace is a sub...
hhssva 28961 The vector addition operat...
hhsssm 28962 The scalar multiplication ...
hhssnm 28963 The norm operation on a su...
issubgoilem 28964 Lemma for ~ hhssabloilem ....
hhssabloilem 28965 Lemma for ~ hhssabloi . F...
hhssabloi 28966 Abelian group property of ...
hhssablo 28967 Abelian group property of ...
hhssnv 28968 Normed complex vector spac...
hhssnvt 28969 Normed complex vector spac...
hhsst 28970 A member of ` SH ` is a su...
hhshsslem1 28971 Lemma for ~ hhsssh . (Con...
hhshsslem2 28972 Lemma for ~ hhsssh . (Con...
hhsssh 28973 The predicate " ` H ` is a...
hhsssh2 28974 The predicate " ` H ` is a...
hhssba 28975 The base set of a subspace...
hhssvs 28976 The vector subtraction ope...
hhssvsf 28977 Mapping of the vector subt...
hhssims 28978 Induced metric of a subspa...
hhssims2 28979 Induced metric of a subspa...
hhssmet 28980 Induced metric of a subspa...
hhssmetdval 28981 Value of the distance func...
hhsscms 28982 The induced metric of a cl...
hhssbnOLD 28983 Obsolete version of ~ cssb...
ocval 28984 Value of orthogonal comple...
ocel 28985 Membership in orthogonal c...
shocel 28986 Membership in orthogonal c...
ocsh 28987 The orthogonal complement ...
shocsh 28988 The orthogonal complement ...
ocss 28989 An orthogonal complement i...
shocss 28990 An orthogonal complement i...
occon 28991 Contraposition law for ort...
occon2 28992 Double contraposition for ...
occon2i 28993 Double contraposition for ...
oc0 28994 The zero vector belongs to...
ocorth 28995 Members of a subset and it...
shocorth 28996 Members of a subspace and ...
ococss 28997 Inclusion in complement of...
shococss 28998 Inclusion in complement of...
shorth 28999 Members of orthogonal subs...
ocin 29000 Intersection of a Hilbert ...
occon3 29001 Hilbert lattice contraposi...
ocnel 29002 A nonzero vector in the co...
chocvali 29003 Value of the orthogonal co...
shuni 29004 Two subspaces with trivial...
chocunii 29005 Lemma for uniqueness part ...
pjhthmo 29006 Projection Theorem, unique...
occllem 29007 Lemma for ~ occl . (Contr...
occl 29008 Closure of complement of H...
shoccl 29009 Closure of complement of H...
choccl 29010 Closure of complement of H...
choccli 29011 Closure of ` CH ` orthocom...
shsval 29016 Value of subspace sum of t...
shsss 29017 The subspace sum is a subs...
shsel 29018 Membership in the subspace...
shsel3 29019 Membership in the subspace...
shseli 29020 Membership in subspace sum...
shscli 29021 Closure of subspace sum. ...
shscl 29022 Closure of subspace sum. ...
shscom 29023 Commutative law for subspa...
shsva 29024 Vector sum belongs to subs...
shsel1 29025 A subspace sum contains a ...
shsel2 29026 A subspace sum contains a ...
shsvs 29027 Vector subtraction belongs...
shsub1 29028 Subspace sum is an upper b...
shsub2 29029 Subspace sum is an upper b...
choc0 29030 The orthocomplement of the...
choc1 29031 The orthocomplement of the...
chocnul 29032 Orthogonal complement of t...
shintcli 29033 Closure of intersection of...
shintcl 29034 The intersection of a none...
chintcli 29035 The intersection of a none...
chintcl 29036 The intersection (infimum)...
spanval 29037 Value of the linear span o...
hsupval 29038 Value of supremum of set o...
chsupval 29039 The value of the supremum ...
spancl 29040 The span of a subset of Hi...
elspancl 29041 A member of a span is a ve...
shsupcl 29042 Closure of the subspace su...
hsupcl 29043 Closure of supremum of set...
chsupcl 29044 Closure of supremum of sub...
hsupss 29045 Subset relation for suprem...
chsupss 29046 Subset relation for suprem...
hsupunss 29047 The union of a set of Hilb...
chsupunss 29048 The union of a set of clos...
spanss2 29049 A subset of Hilbert space ...
shsupunss 29050 The union of a set of subs...
spanid 29051 A subspace of Hilbert spac...
spanss 29052 Ordering relationship for ...
spanssoc 29053 The span of a subset of Hi...
sshjval 29054 Value of join for subsets ...
shjval 29055 Value of join in ` SH ` . ...
chjval 29056 Value of join in ` CH ` . ...
chjvali 29057 Value of join in ` CH ` . ...
sshjval3 29058 Value of join for subsets ...
sshjcl 29059 Closure of join for subset...
shjcl 29060 Closure of join in ` SH ` ...
chjcl 29061 Closure of join in ` CH ` ...
shjcom 29062 Commutative law for Hilber...
shless 29063 Subset implies subset of s...
shlej1 29064 Add disjunct to both sides...
shlej2 29065 Add disjunct to both sides...
shincli 29066 Closure of intersection of...
shscomi 29067 Commutative law for subspa...
shsvai 29068 Vector sum belongs to subs...
shsel1i 29069 A subspace sum contains a ...
shsel2i 29070 A subspace sum contains a ...
shsvsi 29071 Vector subtraction belongs...
shunssi 29072 Union is smaller than subs...
shunssji 29073 Union is smaller than Hilb...
shsleji 29074 Subspace sum is smaller th...
shjcomi 29075 Commutative law for join i...
shsub1i 29076 Subspace sum is an upper b...
shsub2i 29077 Subspace sum is an upper b...
shub1i 29078 Hilbert lattice join is an...
shjcli 29079 Closure of ` CH ` join. (...
shjshcli 29080 ` SH ` closure of join. (...
shlessi 29081 Subset implies subset of s...
shlej1i 29082 Add disjunct to both sides...
shlej2i 29083 Add disjunct to both sides...
shslej 29084 Subspace sum is smaller th...
shincl 29085 Closure of intersection of...
shub1 29086 Hilbert lattice join is an...
shub2 29087 A subspace is a subset of ...
shsidmi 29088 Idempotent law for Hilbert...
shslubi 29089 The least upper bound law ...
shlesb1i 29090 Hilbert lattice ordering i...
shsval2i 29091 An alternate way to expres...
shsval3i 29092 An alternate way to expres...
shmodsi 29093 The modular law holds for ...
shmodi 29094 The modular law is implied...
pjhthlem1 29095 Lemma for ~ pjhth . (Cont...
pjhthlem2 29096 Lemma for ~ pjhth . (Cont...
pjhth 29097 Projection Theorem: Any H...
pjhtheu 29098 Projection Theorem: Any H...
pjhfval 29100 The value of the projectio...
pjhval 29101 Value of a projection. (C...
pjpreeq 29102 Equality with a projection...
pjeq 29103 Equality with a projection...
axpjcl 29104 Closure of a projection in...
pjhcl 29105 Closure of a projection in...
omlsilem 29106 Lemma for orthomodular law...
omlsii 29107 Subspace inference form of...
omlsi 29108 Subspace form of orthomodu...
ococi 29109 Complement of complement o...
ococ 29110 Complement of complement o...
dfch2 29111 Alternate definition of th...
ococin 29112 The double complement is t...
hsupval2 29113 Alternate definition of su...
chsupval2 29114 The value of the supremum ...
sshjval2 29115 Value of join in the set o...
chsupid 29116 A subspace is the supremum...
chsupsn 29117 Value of supremum of subse...
shlub 29118 Hilbert lattice join is th...
shlubi 29119 Hilbert lattice join is th...
pjhtheu2 29120 Uniqueness of ` y ` for th...
pjcli 29121 Closure of a projection in...
pjhcli 29122 Closure of a projection in...
pjpjpre 29123 Decomposition of a vector ...
axpjpj 29124 Decomposition of a vector ...
pjclii 29125 Closure of a projection in...
pjhclii 29126 Closure of a projection in...
pjpj0i 29127 Decomposition of a vector ...
pjpji 29128 Decomposition of a vector ...
pjpjhth 29129 Projection Theorem: Any H...
pjpjhthi 29130 Projection Theorem: Any H...
pjop 29131 Orthocomplement projection...
pjpo 29132 Projection in terms of ort...
pjopi 29133 Orthocomplement projection...
pjpoi 29134 Projection in terms of ort...
pjoc1i 29135 Projection of a vector in ...
pjchi 29136 Projection of a vector in ...
pjoccl 29137 The part of a vector that ...
pjoc1 29138 Projection of a vector in ...
pjomli 29139 Subspace form of orthomodu...
pjoml 29140 Subspace form of orthomodu...
pjococi 29141 Proof of orthocomplement t...
pjoc2i 29142 Projection of a vector in ...
pjoc2 29143 Projection of a vector in ...
sh0le 29144 The zero subspace is the s...
ch0le 29145 The zero subspace is the s...
shle0 29146 No subspace is smaller tha...
chle0 29147 No Hilbert lattice element...
chnlen0 29148 A Hilbert lattice element ...
ch0pss 29149 The zero subspace is a pro...
orthin 29150 The intersection of orthog...
ssjo 29151 The lattice join of a subs...
shne0i 29152 A nonzero subspace has a n...
shs0i 29153 Hilbert subspace sum with ...
shs00i 29154 Two subspaces are zero iff...
ch0lei 29155 The closed subspace zero i...
chle0i 29156 No Hilbert closed subspace...
chne0i 29157 A nonzero closed subspace ...
chocini 29158 Intersection of a closed s...
chj0i 29159 Join with lattice zero in ...
chm1i 29160 Meet with lattice one in `...
chjcli 29161 Closure of ` CH ` join. (...
chsleji 29162 Subspace sum is smaller th...
chseli 29163 Membership in subspace sum...
chincli 29164 Closure of Hilbert lattice...
chsscon3i 29165 Hilbert lattice contraposi...
chsscon1i 29166 Hilbert lattice contraposi...
chsscon2i 29167 Hilbert lattice contraposi...
chcon2i 29168 Hilbert lattice contraposi...
chcon1i 29169 Hilbert lattice contraposi...
chcon3i 29170 Hilbert lattice contraposi...
chunssji 29171 Union is smaller than ` CH...
chjcomi 29172 Commutative law for join i...
chub1i 29173 ` CH ` join is an upper bo...
chub2i 29174 ` CH ` join is an upper bo...
chlubi 29175 Hilbert lattice join is th...
chlubii 29176 Hilbert lattice join is th...
chlej1i 29177 Add join to both sides of ...
chlej2i 29178 Add join to both sides of ...
chlej12i 29179 Add join to both sides of ...
chlejb1i 29180 Hilbert lattice ordering i...
chdmm1i 29181 De Morgan's law for meet i...
chdmm2i 29182 De Morgan's law for meet i...
chdmm3i 29183 De Morgan's law for meet i...
chdmm4i 29184 De Morgan's law for meet i...
chdmj1i 29185 De Morgan's law for join i...
chdmj2i 29186 De Morgan's law for join i...
chdmj3i 29187 De Morgan's law for join i...
chdmj4i 29188 De Morgan's law for join i...
chnlei 29189 Equivalent expressions for...
chjassi 29190 Associative law for Hilber...
chj00i 29191 Two Hilbert lattice elemen...
chjoi 29192 The join of a closed subsp...
chj1i 29193 Join with Hilbert lattice ...
chm0i 29194 Meet with Hilbert lattice ...
chm0 29195 Meet with Hilbert lattice ...
shjshsi 29196 Hilbert lattice join equal...
shjshseli 29197 A closed subspace sum equa...
chne0 29198 A nonzero closed subspace ...
chocin 29199 Intersection of a closed s...
chssoc 29200 A closed subspace less tha...
chj0 29201 Join with Hilbert lattice ...
chslej 29202 Subspace sum is smaller th...
chincl 29203 Closure of Hilbert lattice...
chsscon3 29204 Hilbert lattice contraposi...
chsscon1 29205 Hilbert lattice contraposi...
chsscon2 29206 Hilbert lattice contraposi...
chpsscon3 29207 Hilbert lattice contraposi...
chpsscon1 29208 Hilbert lattice contraposi...
chpsscon2 29209 Hilbert lattice contraposi...
chjcom 29210 Commutative law for Hilber...
chub1 29211 Hilbert lattice join is gr...
chub2 29212 Hilbert lattice join is gr...
chlub 29213 Hilbert lattice join is th...
chlej1 29214 Add join to both sides of ...
chlej2 29215 Add join to both sides of ...
chlejb1 29216 Hilbert lattice ordering i...
chlejb2 29217 Hilbert lattice ordering i...
chnle 29218 Equivalent expressions for...
chjo 29219 The join of a closed subsp...
chabs1 29220 Hilbert lattice absorption...
chabs2 29221 Hilbert lattice absorption...
chabs1i 29222 Hilbert lattice absorption...
chabs2i 29223 Hilbert lattice absorption...
chjidm 29224 Idempotent law for Hilbert...
chjidmi 29225 Idempotent law for Hilbert...
chj12i 29226 A rearrangement of Hilbert...
chj4i 29227 Rearrangement of the join ...
chjjdiri 29228 Hilbert lattice join distr...
chdmm1 29229 De Morgan's law for meet i...
chdmm2 29230 De Morgan's law for meet i...
chdmm3 29231 De Morgan's law for meet i...
chdmm4 29232 De Morgan's law for meet i...
chdmj1 29233 De Morgan's law for join i...
chdmj2 29234 De Morgan's law for join i...
chdmj3 29235 De Morgan's law for join i...
chdmj4 29236 De Morgan's law for join i...
chjass 29237 Associative law for Hilber...
chj12 29238 A rearrangement of Hilbert...
chj4 29239 Rearrangement of the join ...
ledii 29240 An ortholattice is distrib...
lediri 29241 An ortholattice is distrib...
lejdii 29242 An ortholattice is distrib...
lejdiri 29243 An ortholattice is distrib...
ledi 29244 An ortholattice is distrib...
spansn0 29245 The span of the singleton ...
span0 29246 The span of the empty set ...
elspani 29247 Membership in the span of ...
spanuni 29248 The span of a union is the...
spanun 29249 The span of a union is the...
sshhococi 29250 The join of two Hilbert sp...
hne0 29251 Hilbert space has a nonzer...
chsup0 29252 The supremum of the empty ...
h1deoi 29253 Membership in orthocomplem...
h1dei 29254 Membership in 1-dimensiona...
h1did 29255 A generating vector belong...
h1dn0 29256 A nonzero vector generates...
h1de2i 29257 Membership in 1-dimensiona...
h1de2bi 29258 Membership in 1-dimensiona...
h1de2ctlem 29259 Lemma for ~ h1de2ci . (Co...
h1de2ci 29260 Membership in 1-dimensiona...
spansni 29261 The span of a singleton in...
elspansni 29262 Membership in the span of ...
spansn 29263 The span of a singleton in...
spansnch 29264 The span of a Hilbert spac...
spansnsh 29265 The span of a Hilbert spac...
spansnchi 29266 The span of a singleton in...
spansnid 29267 A vector belongs to the sp...
spansnmul 29268 A scalar product with a ve...
elspansncl 29269 A member of a span of a si...
elspansn 29270 Membership in the span of ...
elspansn2 29271 Membership in the span of ...
spansncol 29272 The singletons of collinea...
spansneleqi 29273 Membership relation implie...
spansneleq 29274 Membership relation that i...
spansnss 29275 The span of the singleton ...
elspansn3 29276 A member of the span of th...
elspansn4 29277 A span membership conditio...
elspansn5 29278 A vector belonging to both...
spansnss2 29279 The span of the singleton ...
normcan 29280 Cancellation-type law that...
pjspansn 29281 A projection on the span o...
spansnpji 29282 A subset of Hilbert space ...
spanunsni 29283 The span of the union of a...
spanpr 29284 The span of a pair of vect...
h1datomi 29285 A 1-dimensional subspace i...
h1datom 29286 A 1-dimensional subspace i...
cmbr 29288 Binary relation expressing...
pjoml2i 29289 Variation of orthomodular ...
pjoml3i 29290 Variation of orthomodular ...
pjoml4i 29291 Variation of orthomodular ...
pjoml5i 29292 The orthomodular law. Rem...
pjoml6i 29293 An equivalent of the ortho...
cmbri 29294 Binary relation expressing...
cmcmlem 29295 Commutation is symmetric. ...
cmcmi 29296 Commutation is symmetric. ...
cmcm2i 29297 Commutation with orthocomp...
cmcm3i 29298 Commutation with orthocomp...
cmcm4i 29299 Commutation with orthocomp...
cmbr2i 29300 Alternate definition of th...
cmcmii 29301 Commutation is symmetric. ...
cmcm2ii 29302 Commutation with orthocomp...
cmcm3ii 29303 Commutation with orthocomp...
cmbr3i 29304 Alternate definition for t...
cmbr4i 29305 Alternate definition for t...
lecmi 29306 Comparable Hilbert lattice...
lecmii 29307 Comparable Hilbert lattice...
cmj1i 29308 A Hilbert lattice element ...
cmj2i 29309 A Hilbert lattice element ...
cmm1i 29310 A Hilbert lattice element ...
cmm2i 29311 A Hilbert lattice element ...
cmbr3 29312 Alternate definition for t...
cm0 29313 The zero Hilbert lattice e...
cmidi 29314 The commutes relation is r...
pjoml2 29315 Variation of orthomodular ...
pjoml3 29316 Variation of orthomodular ...
pjoml5 29317 The orthomodular law. Rem...
cmcm 29318 Commutation is symmetric. ...
cmcm3 29319 Commutation with orthocomp...
cmcm2 29320 Commutation with orthocomp...
lecm 29321 Comparable Hilbert lattice...
fh1 29322 Foulis-Holland Theorem. I...
fh2 29323 Foulis-Holland Theorem. I...
cm2j 29324 A lattice element that com...
fh1i 29325 Foulis-Holland Theorem. I...
fh2i 29326 Foulis-Holland Theorem. I...
fh3i 29327 Variation of the Foulis-Ho...
fh4i 29328 Variation of the Foulis-Ho...
cm2ji 29329 A lattice element that com...
cm2mi 29330 A lattice element that com...
qlax1i 29331 One of the equations showi...
qlax2i 29332 One of the equations showi...
qlax3i 29333 One of the equations showi...
qlax4i 29334 One of the equations showi...
qlax5i 29335 One of the equations showi...
qlaxr1i 29336 One of the conditions show...
qlaxr2i 29337 One of the conditions show...
qlaxr4i 29338 One of the conditions show...
qlaxr5i 29339 One of the conditions show...
qlaxr3i 29340 A variation of the orthomo...
chscllem1 29341 Lemma for ~ chscl . (Cont...
chscllem2 29342 Lemma for ~ chscl . (Cont...
chscllem3 29343 Lemma for ~ chscl . (Cont...
chscllem4 29344 Lemma for ~ chscl . (Cont...
chscl 29345 The subspace sum of two cl...
osumi 29346 If two closed subspaces of...
osumcori 29347 Corollary of ~ osumi . (C...
osumcor2i 29348 Corollary of ~ osumi , sho...
osum 29349 If two closed subspaces of...
spansnji 29350 The subspace sum of a clos...
spansnj 29351 The subspace sum of a clos...
spansnscl 29352 The subspace sum of a clos...
sumspansn 29353 The sum of two vectors bel...
spansnm0i 29354 The meet of different one-...
nonbooli 29355 A Hilbert lattice with two...
spansncvi 29356 Hilbert space has the cove...
spansncv 29357 Hilbert space has the cove...
5oalem1 29358 Lemma for orthoarguesian l...
5oalem2 29359 Lemma for orthoarguesian l...
5oalem3 29360 Lemma for orthoarguesian l...
5oalem4 29361 Lemma for orthoarguesian l...
5oalem5 29362 Lemma for orthoarguesian l...
5oalem6 29363 Lemma for orthoarguesian l...
5oalem7 29364 Lemma for orthoarguesian l...
5oai 29365 Orthoarguesian law 5OA. Th...
3oalem1 29366 Lemma for 3OA (weak) ortho...
3oalem2 29367 Lemma for 3OA (weak) ortho...
3oalem3 29368 Lemma for 3OA (weak) ortho...
3oalem4 29369 Lemma for 3OA (weak) ortho...
3oalem5 29370 Lemma for 3OA (weak) ortho...
3oalem6 29371 Lemma for 3OA (weak) ortho...
3oai 29372 3OA (weak) orthoarguesian ...
pjorthi 29373 Projection components on o...
pjch1 29374 Property of identity proje...
pjo 29375 The orthogonal projection....
pjcompi 29376 Component of a projection....
pjidmi 29377 A projection is idempotent...
pjadjii 29378 A projection is self-adjoi...
pjaddii 29379 Projection of vector sum i...
pjinormii 29380 The inner product of a pro...
pjmulii 29381 Projection of (scalar) pro...
pjsubii 29382 Projection of vector diffe...
pjsslem 29383 Lemma for subset relations...
pjss2i 29384 Subset relationship for pr...
pjssmii 29385 Projection meet property. ...
pjssge0ii 29386 Theorem 4.5(iv)->(v) of [B...
pjdifnormii 29387 Theorem 4.5(v)<->(vi) of [...
pjcji 29388 The projection on a subspa...
pjadji 29389 A projection is self-adjoi...
pjaddi 29390 Projection of vector sum i...
pjinormi 29391 The inner product of a pro...
pjsubi 29392 Projection of vector diffe...
pjmuli 29393 Projection of scalar produ...
pjige0i 29394 The inner product of a pro...
pjige0 29395 The inner product of a pro...
pjcjt2 29396 The projection on a subspa...
pj0i 29397 The projection of the zero...
pjch 29398 Projection of a vector in ...
pjid 29399 The projection of a vector...
pjvec 29400 The set of vectors belongi...
pjocvec 29401 The set of vectors belongi...
pjocini 29402 Membership of projection i...
pjini 29403 Membership of projection i...
pjjsi 29404 A sufficient condition for...
pjfni 29405 Functionality of a project...
pjrni 29406 The range of a projection....
pjfoi 29407 A projection maps onto its...
pjfi 29408 The mapping of a projectio...
pjvi 29409 The value of a projection ...
pjhfo 29410 A projection maps onto its...
pjrn 29411 The range of a projection....
pjhf 29412 The mapping of a projectio...
pjfn 29413 Functionality of a project...
pjsumi 29414 The projection on a subspa...
pj11i 29415 One-to-one correspondence ...
pjdsi 29416 Vector decomposition into ...
pjds3i 29417 Vector decomposition into ...
pj11 29418 One-to-one correspondence ...
pjmfn 29419 Functionality of the proje...
pjmf1 29420 The projector function map...
pjoi0 29421 The inner product of proje...
pjoi0i 29422 The inner product of proje...
pjopythi 29423 Pythagorean theorem for pr...
pjopyth 29424 Pythagorean theorem for pr...
pjnormi 29425 The norm of the projection...
pjpythi 29426 Pythagorean theorem for pr...
pjneli 29427 If a vector does not belon...
pjnorm 29428 The norm of the projection...
pjpyth 29429 Pythagorean theorem for pr...
pjnel 29430 If a vector does not belon...
pjnorm2 29431 A vector belongs to the su...
mayete3i 29432 Mayet's equation E_3. Par...
mayetes3i 29433 Mayet's equation E^*_3, de...
hosmval 29439 Value of the sum of two Hi...
hommval 29440 Value of the scalar produc...
hodmval 29441 Value of the difference of...
hfsmval 29442 Value of the sum of two Hi...
hfmmval 29443 Value of the scalar produc...
hosval 29444 Value of the sum of two Hi...
homval 29445 Value of the scalar produc...
hodval 29446 Value of the difference of...
hfsval 29447 Value of the sum of two Hi...
hfmval 29448 Value of the scalar produc...
hoscl 29449 Closure of the sum of two ...
homcl 29450 Closure of the scalar prod...
hodcl 29451 Closure of the difference ...
ho0val 29454 Value of the zero Hilbert ...
ho0f 29455 Functionality of the zero ...
df0op2 29456 Alternate definition of Hi...
dfiop2 29457 Alternate definition of Hi...
hoif 29458 Functionality of the Hilbe...
hoival 29459 The value of the Hilbert s...
hoico1 29460 Composition with the Hilbe...
hoico2 29461 Composition with the Hilbe...
hoaddcl 29462 The sum of Hilbert space o...
homulcl 29463 The scalar product of a Hi...
hoeq 29464 Equality of Hilbert space ...
hoeqi 29465 Equality of Hilbert space ...
hoscli 29466 Closure of Hilbert space o...
hodcli 29467 Closure of Hilbert space o...
hocoi 29468 Composition of Hilbert spa...
hococli 29469 Closure of composition of ...
hocofi 29470 Mapping of composition of ...
hocofni 29471 Functionality of compositi...
hoaddcli 29472 Mapping of sum of Hilbert ...
hosubcli 29473 Mapping of difference of H...
hoaddfni 29474 Functionality of sum of Hi...
hosubfni 29475 Functionality of differenc...
hoaddcomi 29476 Commutativity of sum of Hi...
hosubcl 29477 Mapping of difference of H...
hoaddcom 29478 Commutativity of sum of Hi...
hodsi 29479 Relationship between Hilbe...
hoaddassi 29480 Associativity of sum of Hi...
hoadd12i 29481 Commutative/associative la...
hoadd32i 29482 Commutative/associative la...
hocadddiri 29483 Distributive law for Hilbe...
hocsubdiri 29484 Distributive law for Hilbe...
ho2coi 29485 Double composition of Hilb...
hoaddass 29486 Associativity of sum of Hi...
hoadd32 29487 Commutative/associative la...
hoadd4 29488 Rearrangement of 4 terms i...
hocsubdir 29489 Distributive law for Hilbe...
hoaddid1i 29490 Sum of a Hilbert space ope...
hodidi 29491 Difference of a Hilbert sp...
ho0coi 29492 Composition of the zero op...
hoid1i 29493 Composition of Hilbert spa...
hoid1ri 29494 Composition of Hilbert spa...
hoaddid1 29495 Sum of a Hilbert space ope...
hodid 29496 Difference of a Hilbert sp...
hon0 29497 A Hilbert space operator i...
hodseqi 29498 Subtraction and addition o...
ho0subi 29499 Subtraction of Hilbert spa...
honegsubi 29500 Relationship between Hilbe...
ho0sub 29501 Subtraction of Hilbert spa...
hosubid1 29502 The zero operator subtract...
honegsub 29503 Relationship between Hilbe...
homulid2 29504 An operator equals its sca...
homco1 29505 Associative law for scalar...
homulass 29506 Scalar product associative...
hoadddi 29507 Scalar product distributiv...
hoadddir 29508 Scalar product reverse dis...
homul12 29509 Swap first and second fact...
honegneg 29510 Double negative of a Hilbe...
hosubneg 29511 Relationship between opera...
hosubdi 29512 Scalar product distributiv...
honegdi 29513 Distribution of negative o...
honegsubdi 29514 Distribution of negative o...
honegsubdi2 29515 Distribution of negative o...
hosubsub2 29516 Law for double subtraction...
hosub4 29517 Rearrangement of 4 terms i...
hosubadd4 29518 Rearrangement of 4 terms i...
hoaddsubass 29519 Associative-type law for a...
hoaddsub 29520 Law for operator addition ...
hosubsub 29521 Law for double subtraction...
hosubsub4 29522 Law for double subtraction...
ho2times 29523 Two times a Hilbert space ...
hoaddsubassi 29524 Associativity of sum and d...
hoaddsubi 29525 Law for sum and difference...
hosd1i 29526 Hilbert space operator sum...
hosd2i 29527 Hilbert space operator sum...
hopncani 29528 Hilbert space operator can...
honpcani 29529 Hilbert space operator can...
hosubeq0i 29530 If the difference between ...
honpncani 29531 Hilbert space operator can...
ho01i 29532 A condition implying that ...
ho02i 29533 A condition implying that ...
hoeq1 29534 A condition implying that ...
hoeq2 29535 A condition implying that ...
adjmo 29536 Every Hilbert space operat...
adjsym 29537 Symmetry property of an ad...
eigrei 29538 A necessary and sufficient...
eigre 29539 A necessary and sufficient...
eigposi 29540 A sufficient condition (fi...
eigorthi 29541 A necessary and sufficient...
eigorth 29542 A necessary and sufficient...
nmopval 29560 Value of the norm of a Hil...
elcnop 29561 Property defining a contin...
ellnop 29562 Property defining a linear...
lnopf 29563 A linear Hilbert space ope...
elbdop 29564 Property defining a bounde...
bdopln 29565 A bounded linear Hilbert s...
bdopf 29566 A bounded linear Hilbert s...
nmopsetretALT 29567 The set in the supremum of...
nmopsetretHIL 29568 The set in the supremum of...
nmopsetn0 29569 The set in the supremum of...
nmopxr 29570 The norm of a Hilbert spac...
nmoprepnf 29571 The norm of a Hilbert spac...
nmopgtmnf 29572 The norm of a Hilbert spac...
nmopreltpnf 29573 The norm of a Hilbert spac...
nmopre 29574 The norm of a bounded oper...
elbdop2 29575 Property defining a bounde...
elunop 29576 Property defining a unitar...
elhmop 29577 Property defining a Hermit...
hmopf 29578 A Hermitian operator is a ...
hmopex 29579 The class of Hermitian ope...
nmfnval 29580 Value of the norm of a Hil...
nmfnsetre 29581 The set in the supremum of...
nmfnsetn0 29582 The set in the supremum of...
nmfnxr 29583 The norm of any Hilbert sp...
nmfnrepnf 29584 The norm of a Hilbert spac...
nlfnval 29585 Value of the null space of...
elcnfn 29586 Property defining a contin...
ellnfn 29587 Property defining a linear...
lnfnf 29588 A linear Hilbert space fun...
dfadj2 29589 Alternate definition of th...
funadj 29590 Functionality of the adjoi...
dmadjss 29591 The domain of the adjoint ...
dmadjop 29592 A member of the domain of ...
adjeu 29593 Elementhood in the domain ...
adjval 29594 Value of the adjoint funct...
adjval2 29595 Value of the adjoint funct...
cnvadj 29596 The adjoint function equal...
funcnvadj 29597 The converse of the adjoin...
adj1o 29598 The adjoint function maps ...
dmadjrn 29599 The adjoint of an operator...
eigvecval 29600 The set of eigenvectors of...
eigvalfval 29601 The eigenvalues of eigenve...
specval 29602 The value of the spectrum ...
speccl 29603 The spectrum of an operato...
hhlnoi 29604 The linear operators of Hi...
hhnmoi 29605 The norm of an operator in...
hhbloi 29606 A bounded linear operator ...
hh0oi 29607 The zero operator in Hilbe...
hhcno 29608 The continuous operators o...
hhcnf 29609 The continuous functionals...
dmadjrnb 29610 The adjoint of an operator...
nmoplb 29611 A lower bound for an opera...
nmopub 29612 An upper bound for an oper...
nmopub2tALT 29613 An upper bound for an oper...
nmopub2tHIL 29614 An upper bound for an oper...
nmopge0 29615 The norm of any Hilbert sp...
nmopgt0 29616 A linear Hilbert space ope...
cnopc 29617 Basic continuity property ...
lnopl 29618 Basic linearity property o...
unop 29619 Basic inner product proper...
unopf1o 29620 A unitary operator in Hilb...
unopnorm 29621 A unitary operator is idem...
cnvunop 29622 The inverse (converse) of ...
unopadj 29623 The inverse (converse) of ...
unoplin 29624 A unitary operator is line...
counop 29625 The composition of two uni...
hmop 29626 Basic inner product proper...
hmopre 29627 The inner product of the v...
nmfnlb 29628 A lower bound for a functi...
nmfnleub 29629 An upper bound for the nor...
nmfnleub2 29630 An upper bound for the nor...
nmfnge0 29631 The norm of any Hilbert sp...
elnlfn 29632 Membership in the null spa...
elnlfn2 29633 Membership in the null spa...
cnfnc 29634 Basic continuity property ...
lnfnl 29635 Basic linearity property o...
adjcl 29636 Closure of the adjoint of ...
adj1 29637 Property of an adjoint Hil...
adj2 29638 Property of an adjoint Hil...
adjeq 29639 A property that determines...
adjadj 29640 Double adjoint. Theorem 3...
adjvalval 29641 Value of the value of the ...
unopadj2 29642 The adjoint of a unitary o...
hmopadj 29643 A Hermitian operator is se...
hmdmadj 29644 Every Hermitian operator h...
hmopadj2 29645 An operator is Hermitian i...
hmoplin 29646 A Hermitian operator is li...
brafval 29647 The bra of a vector, expre...
braval 29648 A bra-ket juxtaposition, e...
braadd 29649 Linearity property of bra ...
bramul 29650 Linearity property of bra ...
brafn 29651 The bra function is a func...
bralnfn 29652 The Dirac bra function is ...
bracl 29653 Closure of the bra functio...
bra0 29654 The Dirac bra of the zero ...
brafnmul 29655 Anti-linearity property of...
kbfval 29656 The outer product of two v...
kbop 29657 The outer product of two v...
kbval 29658 The value of the operator ...
kbmul 29659 Multiplication property of...
kbpj 29660 If a vector ` A ` has norm...
eleigvec 29661 Membership in the set of e...
eleigvec2 29662 Membership in the set of e...
eleigveccl 29663 Closure of an eigenvector ...
eigvalval 29664 The eigenvalue of an eigen...
eigvalcl 29665 An eigenvalue is a complex...
eigvec1 29666 Property of an eigenvector...
eighmre 29667 The eigenvalues of a Hermi...
eighmorth 29668 Eigenvectors of a Hermitia...
nmopnegi 29669 Value of the norm of the n...
lnop0 29670 The value of a linear Hilb...
lnopmul 29671 Multiplicative property of...
lnopli 29672 Basic scalar product prope...
lnopfi 29673 A linear Hilbert space ope...
lnop0i 29674 The value of a linear Hilb...
lnopaddi 29675 Additive property of a lin...
lnopmuli 29676 Multiplicative property of...
lnopaddmuli 29677 Sum/product property of a ...
lnopsubi 29678 Subtraction property for a...
lnopsubmuli 29679 Subtraction/product proper...
lnopmulsubi 29680 Product/subtraction proper...
homco2 29681 Move a scalar product out ...
idunop 29682 The identity function (res...
0cnop 29683 The identically zero funct...
0cnfn 29684 The identically zero funct...
idcnop 29685 The identity function (res...
idhmop 29686 The Hilbert space identity...
0hmop 29687 The identically zero funct...
0lnop 29688 The identically zero funct...
0lnfn 29689 The identically zero funct...
nmop0 29690 The norm of the zero opera...
nmfn0 29691 The norm of the identicall...
hmopbdoptHIL 29692 A Hermitian operator is a ...
hoddii 29693 Distributive law for Hilbe...
hoddi 29694 Distributive law for Hilbe...
nmop0h 29695 The norm of any operator o...
idlnop 29696 The identity function (res...
0bdop 29697 The identically zero opera...
adj0 29698 Adjoint of the zero operat...
nmlnop0iALT 29699 A linear operator with a z...
nmlnop0iHIL 29700 A linear operator with a z...
nmlnopgt0i 29701 A linear Hilbert space ope...
nmlnop0 29702 A linear operator with a z...
nmlnopne0 29703 A linear operator with a n...
lnopmi 29704 The scalar product of a li...
lnophsi 29705 The sum of two linear oper...
lnophdi 29706 The difference of two line...
lnopcoi 29707 The composition of two lin...
lnopco0i 29708 The composition of a linea...
lnopeq0lem1 29709 Lemma for ~ lnopeq0i . Ap...
lnopeq0lem2 29710 Lemma for ~ lnopeq0i . (C...
lnopeq0i 29711 A condition implying that ...
lnopeqi 29712 Two linear Hilbert space o...
lnopeq 29713 Two linear Hilbert space o...
lnopunilem1 29714 Lemma for ~ lnopunii . (C...
lnopunilem2 29715 Lemma for ~ lnopunii . (C...
lnopunii 29716 If a linear operator (whos...
elunop2 29717 An operator is unitary iff...
nmopun 29718 Norm of a unitary Hilbert ...
unopbd 29719 A unitary operator is a bo...
lnophmlem1 29720 Lemma for ~ lnophmi . (Co...
lnophmlem2 29721 Lemma for ~ lnophmi . (Co...
lnophmi 29722 A linear operator is Hermi...
lnophm 29723 A linear operator is Hermi...
hmops 29724 The sum of two Hermitian o...
hmopm 29725 The scalar product of a He...
hmopd 29726 The difference of two Herm...
hmopco 29727 The composition of two com...
nmbdoplbi 29728 A lower bound for the norm...
nmbdoplb 29729 A lower bound for the norm...
nmcexi 29730 Lemma for ~ nmcopexi and ~...
nmcopexi 29731 The norm of a continuous l...
nmcoplbi 29732 A lower bound for the norm...
nmcopex 29733 The norm of a continuous l...
nmcoplb 29734 A lower bound for the norm...
nmophmi 29735 The norm of the scalar pro...
bdophmi 29736 The scalar product of a bo...
lnconi 29737 Lemma for ~ lnopconi and ~...
lnopconi 29738 A condition equivalent to ...
lnopcon 29739 A condition equivalent to ...
lnopcnbd 29740 A linear operator is conti...
lncnopbd 29741 A continuous linear operat...
lncnbd 29742 A continuous linear operat...
lnopcnre 29743 A linear operator is conti...
lnfnli 29744 Basic property of a linear...
lnfnfi 29745 A linear Hilbert space fun...
lnfn0i 29746 The value of a linear Hilb...
lnfnaddi 29747 Additive property of a lin...
lnfnmuli 29748 Multiplicative property of...
lnfnaddmuli 29749 Sum/product property of a ...
lnfnsubi 29750 Subtraction property for a...
lnfn0 29751 The value of a linear Hilb...
lnfnmul 29752 Multiplicative property of...
nmbdfnlbi 29753 A lower bound for the norm...
nmbdfnlb 29754 A lower bound for the norm...
nmcfnexi 29755 The norm of a continuous l...
nmcfnlbi 29756 A lower bound for the norm...
nmcfnex 29757 The norm of a continuous l...
nmcfnlb 29758 A lower bound of the norm ...
lnfnconi 29759 A condition equivalent to ...
lnfncon 29760 A condition equivalent to ...
lnfncnbd 29761 A linear functional is con...
imaelshi 29762 The image of a subspace un...
rnelshi 29763 The range of a linear oper...
nlelshi 29764 The null space of a linear...
nlelchi 29765 The null space of a contin...
riesz3i 29766 A continuous linear functi...
riesz4i 29767 A continuous linear functi...
riesz4 29768 A continuous linear functi...
riesz1 29769 Part 1 of the Riesz repres...
riesz2 29770 Part 2 of the Riesz repres...
cnlnadjlem1 29771 Lemma for ~ cnlnadji (Theo...
cnlnadjlem2 29772 Lemma for ~ cnlnadji . ` G...
cnlnadjlem3 29773 Lemma for ~ cnlnadji . By...
cnlnadjlem4 29774 Lemma for ~ cnlnadji . Th...
cnlnadjlem5 29775 Lemma for ~ cnlnadji . ` F...
cnlnadjlem6 29776 Lemma for ~ cnlnadji . ` F...
cnlnadjlem7 29777 Lemma for ~ cnlnadji . He...
cnlnadjlem8 29778 Lemma for ~ cnlnadji . ` F...
cnlnadjlem9 29779 Lemma for ~ cnlnadji . ` F...
cnlnadji 29780 Every continuous linear op...
cnlnadjeui 29781 Every continuous linear op...
cnlnadjeu 29782 Every continuous linear op...
cnlnadj 29783 Every continuous linear op...
cnlnssadj 29784 Every continuous linear Hi...
bdopssadj 29785 Every bounded linear Hilbe...
bdopadj 29786 Every bounded linear Hilbe...
adjbdln 29787 The adjoint of a bounded l...
adjbdlnb 29788 An operator is bounded and...
adjbd1o 29789 The mapping of adjoints of...
adjlnop 29790 The adjoint of an operator...
adjsslnop 29791 Every operator with an adj...
nmopadjlei 29792 Property of the norm of an...
nmopadjlem 29793 Lemma for ~ nmopadji . (C...
nmopadji 29794 Property of the norm of an...
adjeq0 29795 An operator is zero iff it...
adjmul 29796 The adjoint of the scalar ...
adjadd 29797 The adjoint of the sum of ...
nmoptrii 29798 Triangle inequality for th...
nmopcoi 29799 Upper bound for the norm o...
bdophsi 29800 The sum of two bounded lin...
bdophdi 29801 The difference between two...
bdopcoi 29802 The composition of two bou...
nmoptri2i 29803 Triangle-type inequality f...
adjcoi 29804 The adjoint of a compositi...
nmopcoadji 29805 The norm of an operator co...
nmopcoadj2i 29806 The norm of an operator co...
nmopcoadj0i 29807 An operator composed with ...
unierri 29808 If we approximate a chain ...
branmfn 29809 The norm of the bra functi...
brabn 29810 The bra of a vector is a b...
rnbra 29811 The set of bras equals the...
bra11 29812 The bra function maps vect...
bracnln 29813 A bra is a continuous line...
cnvbraval 29814 Value of the converse of t...
cnvbracl 29815 Closure of the converse of...
cnvbrabra 29816 The converse bra of the br...
bracnvbra 29817 The bra of the converse br...
bracnlnval 29818 The vector that a continuo...
cnvbramul 29819 Multiplication property of...
kbass1 29820 Dirac bra-ket associative ...
kbass2 29821 Dirac bra-ket associative ...
kbass3 29822 Dirac bra-ket associative ...
kbass4 29823 Dirac bra-ket associative ...
kbass5 29824 Dirac bra-ket associative ...
kbass6 29825 Dirac bra-ket associative ...
leopg 29826 Ordering relation for posi...
leop 29827 Ordering relation for oper...
leop2 29828 Ordering relation for oper...
leop3 29829 Operator ordering in terms...
leoppos 29830 Binary relation defining a...
leoprf2 29831 The ordering relation for ...
leoprf 29832 The ordering relation for ...
leopsq 29833 The square of a Hermitian ...
0leop 29834 The zero operator is a pos...
idleop 29835 The identity operator is a...
leopadd 29836 The sum of two positive op...
leopmuli 29837 The scalar product of a no...
leopmul 29838 The scalar product of a po...
leopmul2i 29839 Scalar product applied to ...
leoptri 29840 The positive operator orde...
leoptr 29841 The positive operator orde...
leopnmid 29842 A bounded Hermitian operat...
nmopleid 29843 A nonzero, bounded Hermiti...
opsqrlem1 29844 Lemma for opsqri . (Contr...
opsqrlem2 29845 Lemma for opsqri . ` F `` ...
opsqrlem3 29846 Lemma for opsqri . (Contr...
opsqrlem4 29847 Lemma for opsqri . (Contr...
opsqrlem5 29848 Lemma for opsqri . (Contr...
opsqrlem6 29849 Lemma for opsqri . (Contr...
pjhmopi 29850 A projector is a Hermitian...
pjlnopi 29851 A projector is a linear op...
pjnmopi 29852 The operator norm of a pro...
pjbdlni 29853 A projector is a bounded l...
pjhmop 29854 A projection is a Hermitia...
hmopidmchi 29855 An idempotent Hermitian op...
hmopidmpji 29856 An idempotent Hermitian op...
hmopidmch 29857 An idempotent Hermitian op...
hmopidmpj 29858 An idempotent Hermitian op...
pjsdii 29859 Distributive law for Hilbe...
pjddii 29860 Distributive law for Hilbe...
pjsdi2i 29861 Chained distributive law f...
pjcoi 29862 Composition of projections...
pjcocli 29863 Closure of composition of ...
pjcohcli 29864 Closure of composition of ...
pjadjcoi 29865 Adjoint of composition of ...
pjcofni 29866 Functionality of compositi...
pjss1coi 29867 Subset relationship for pr...
pjss2coi 29868 Subset relationship for pr...
pjssmi 29869 Projection meet property. ...
pjssge0i 29870 Theorem 4.5(iv)->(v) of [B...
pjdifnormi 29871 Theorem 4.5(v)<->(vi) of [...
pjnormssi 29872 Theorem 4.5(i)<->(vi) of [...
pjorthcoi 29873 Composition of projections...
pjscji 29874 The projection of orthogon...
pjssumi 29875 The projection on a subspa...
pjssposi 29876 Projector ordering can be ...
pjordi 29877 The definition of projecto...
pjssdif2i 29878 The projection subspace of...
pjssdif1i 29879 A necessary and sufficient...
pjimai 29880 The image of a projection....
pjidmcoi 29881 A projection is idempotent...
pjoccoi 29882 Composition of projections...
pjtoi 29883 Subspace sum of projection...
pjoci 29884 Projection of orthocomplem...
pjidmco 29885 A projection operator is i...
dfpjop 29886 Definition of projection o...
pjhmopidm 29887 Two ways to express the se...
elpjidm 29888 A projection operator is i...
elpjhmop 29889 A projection operator is H...
0leopj 29890 A projector is a positive ...
pjadj2 29891 A projector is self-adjoin...
pjadj3 29892 A projector is self-adjoin...
elpjch 29893 Reconstruction of the subs...
elpjrn 29894 Reconstruction of the subs...
pjinvari 29895 A closed subspace ` H ` wi...
pjin1i 29896 Lemma for Theorem 1.22 of ...
pjin2i 29897 Lemma for Theorem 1.22 of ...
pjin3i 29898 Lemma for Theorem 1.22 of ...
pjclem1 29899 Lemma for projection commu...
pjclem2 29900 Lemma for projection commu...
pjclem3 29901 Lemma for projection commu...
pjclem4a 29902 Lemma for projection commu...
pjclem4 29903 Lemma for projection commu...
pjci 29904 Two subspaces commute iff ...
pjcmul1i 29905 A necessary and sufficient...
pjcmul2i 29906 The projection subspace of...
pjcohocli 29907 Closure of composition of ...
pjadj2coi 29908 Adjoint of double composit...
pj2cocli 29909 Closure of double composit...
pj3lem1 29910 Lemma for projection tripl...
pj3si 29911 Stronger projection triple...
pj3i 29912 Projection triplet theorem...
pj3cor1i 29913 Projection triplet corolla...
pjs14i 29914 Theorem S-14 of Watanabe, ...
isst 29917 Property of a state. (Con...
ishst 29918 Property of a complex Hilb...
sticl 29919 ` [ 0 , 1 ] ` closure of t...
stcl 29920 Real closure of the value ...
hstcl 29921 Closure of the value of a ...
hst1a 29922 Unit value of a Hilbert-sp...
hstel2 29923 Properties of a Hilbert-sp...
hstorth 29924 Orthogonality property of ...
hstosum 29925 Orthogonal sum property of...
hstoc 29926 Sum of a Hilbert-space-val...
hstnmoc 29927 Sum of norms of a Hilbert-...
stge0 29928 The value of a state is no...
stle1 29929 The value of a state is le...
hstle1 29930 The norm of the value of a...
hst1h 29931 The norm of a Hilbert-spac...
hst0h 29932 The norm of a Hilbert-spac...
hstpyth 29933 Pythagorean property of a ...
hstle 29934 Ordering property of a Hil...
hstles 29935 Ordering property of a Hil...
hstoh 29936 A Hilbert-space-valued sta...
hst0 29937 A Hilbert-space-valued sta...
sthil 29938 The value of a state at th...
stj 29939 The value of a state on a ...
sto1i 29940 The state of a subspace pl...
sto2i 29941 The state of the orthocomp...
stge1i 29942 If a state is greater than...
stle0i 29943 If a state is less than or...
stlei 29944 Ordering law for states. ...
stlesi 29945 Ordering law for states. ...
stji1i 29946 Join of components of Sasa...
stm1i 29947 State of component of unit...
stm1ri 29948 State of component of unit...
stm1addi 29949 Sum of states whose meet i...
staddi 29950 If the sum of 2 states is ...
stm1add3i 29951 Sum of states whose meet i...
stadd3i 29952 If the sum of 3 states is ...
st0 29953 The state of the zero subs...
strlem1 29954 Lemma for strong state the...
strlem2 29955 Lemma for strong state the...
strlem3a 29956 Lemma for strong state the...
strlem3 29957 Lemma for strong state the...
strlem4 29958 Lemma for strong state the...
strlem5 29959 Lemma for strong state the...
strlem6 29960 Lemma for strong state the...
stri 29961 Strong state theorem. The...
strb 29962 Strong state theorem (bidi...
hstrlem2 29963 Lemma for strong set of CH...
hstrlem3a 29964 Lemma for strong set of CH...
hstrlem3 29965 Lemma for strong set of CH...
hstrlem4 29966 Lemma for strong set of CH...
hstrlem5 29967 Lemma for strong set of CH...
hstrlem6 29968 Lemma for strong set of CH...
hstri 29969 Hilbert space admits a str...
hstrbi 29970 Strong CH-state theorem (b...
largei 29971 A Hilbert lattice admits a...
jplem1 29972 Lemma for Jauch-Piron theo...
jplem2 29973 Lemma for Jauch-Piron theo...
jpi 29974 The function ` S ` , that ...
golem1 29975 Lemma for Godowski's equat...
golem2 29976 Lemma for Godowski's equat...
goeqi 29977 Godowski's equation, shown...
stcltr1i 29978 Property of a strong class...
stcltr2i 29979 Property of a strong class...
stcltrlem1 29980 Lemma for strong classical...
stcltrlem2 29981 Lemma for strong classical...
stcltrthi 29982 Theorem for classically st...
cvbr 29986 Binary relation expressing...
cvbr2 29987 Binary relation expressing...
cvcon3 29988 Contraposition law for the...
cvpss 29989 The covers relation implie...
cvnbtwn 29990 The covers relation implie...
cvnbtwn2 29991 The covers relation implie...
cvnbtwn3 29992 The covers relation implie...
cvnbtwn4 29993 The covers relation implie...
cvnsym 29994 The covers relation is not...
cvnref 29995 The covers relation is not...
cvntr 29996 The covers relation is not...
spansncv2 29997 Hilbert space has the cove...
mdbr 29998 Binary relation expressing...
mdi 29999 Consequence of the modular...
mdbr2 30000 Binary relation expressing...
mdbr3 30001 Binary relation expressing...
mdbr4 30002 Binary relation expressing...
dmdbr 30003 Binary relation expressing...
dmdmd 30004 The dual modular pair prop...
mddmd 30005 The modular pair property ...
dmdi 30006 Consequence of the dual mo...
dmdbr2 30007 Binary relation expressing...
dmdi2 30008 Consequence of the dual mo...
dmdbr3 30009 Binary relation expressing...
dmdbr4 30010 Binary relation expressing...
dmdi4 30011 Consequence of the dual mo...
dmdbr5 30012 Binary relation expressing...
mddmd2 30013 Relationship between modul...
mdsl0 30014 A sublattice condition tha...
ssmd1 30015 Ordering implies the modul...
ssmd2 30016 Ordering implies the modul...
ssdmd1 30017 Ordering implies the dual ...
ssdmd2 30018 Ordering implies the dual ...
dmdsl3 30019 Sublattice mapping for a d...
mdsl3 30020 Sublattice mapping for a m...
mdslle1i 30021 Order preservation of the ...
mdslle2i 30022 Order preservation of the ...
mdslj1i 30023 Join preservation of the o...
mdslj2i 30024 Meet preservation of the r...
mdsl1i 30025 If the modular pair proper...
mdsl2i 30026 If the modular pair proper...
mdsl2bi 30027 If the modular pair proper...
cvmdi 30028 The covering property impl...
mdslmd1lem1 30029 Lemma for ~ mdslmd1i . (C...
mdslmd1lem2 30030 Lemma for ~ mdslmd1i . (C...
mdslmd1lem3 30031 Lemma for ~ mdslmd1i . (C...
mdslmd1lem4 30032 Lemma for ~ mdslmd1i . (C...
mdslmd1i 30033 Preservation of the modula...
mdslmd2i 30034 Preservation of the modula...
mdsldmd1i 30035 Preservation of the dual m...
mdslmd3i 30036 Modular pair conditions th...
mdslmd4i 30037 Modular pair condition tha...
csmdsymi 30038 Cross-symmetry implies M-s...
mdexchi 30039 An exchange lemma for modu...
cvmd 30040 The covering property impl...
cvdmd 30041 The covering property impl...
ela 30043 Atoms in a Hilbert lattice...
elat2 30044 Expanded membership relati...
elatcv0 30045 A Hilbert lattice element ...
atcv0 30046 An atom covers the zero su...
atssch 30047 Atoms are a subset of the ...
atelch 30048 An atom is a Hilbert latti...
atne0 30049 An atom is not the Hilbert...
atss 30050 A lattice element smaller ...
atsseq 30051 Two atoms in a subset rela...
atcveq0 30052 A Hilbert lattice element ...
h1da 30053 A 1-dimensional subspace i...
spansna 30054 The span of the singleton ...
sh1dle 30055 A 1-dimensional subspace i...
ch1dle 30056 A 1-dimensional subspace i...
atom1d 30057 The 1-dimensional subspace...
superpos 30058 Superposition Principle. ...
chcv1 30059 The Hilbert lattice has th...
chcv2 30060 The Hilbert lattice has th...
chjatom 30061 The join of a closed subsp...
shatomici 30062 The lattice of Hilbert sub...
hatomici 30063 The Hilbert lattice is ato...
hatomic 30064 A Hilbert lattice is atomi...
shatomistici 30065 The lattice of Hilbert sub...
hatomistici 30066 ` CH ` is atomistic, i.e. ...
chpssati 30067 Two Hilbert lattice elemen...
chrelati 30068 The Hilbert lattice is rel...
chrelat2i 30069 A consequence of relative ...
cvati 30070 If a Hilbert lattice eleme...
cvbr4i 30071 An alternate way to expres...
cvexchlem 30072 Lemma for ~ cvexchi . (Co...
cvexchi 30073 The Hilbert lattice satisf...
chrelat2 30074 A consequence of relative ...
chrelat3 30075 A consequence of relative ...
chrelat3i 30076 A consequence of the relat...
chrelat4i 30077 A consequence of relative ...
cvexch 30078 The Hilbert lattice satisf...
cvp 30079 The Hilbert lattice satisf...
atnssm0 30080 The meet of a Hilbert latt...
atnemeq0 30081 The meet of distinct atoms...
atssma 30082 The meet with an atom's su...
atcv0eq 30083 Two atoms covering the zer...
atcv1 30084 Two atoms covering the zer...
atexch 30085 The Hilbert lattice satisf...
atomli 30086 An assertion holding in at...
atoml2i 30087 An assertion holding in at...
atordi 30088 An ordering law for a Hilb...
atcvatlem 30089 Lemma for ~ atcvati . (Co...
atcvati 30090 A nonzero Hilbert lattice ...
atcvat2i 30091 A Hilbert lattice element ...
atord 30092 An ordering law for a Hilb...
atcvat2 30093 A Hilbert lattice element ...
chirredlem1 30094 Lemma for ~ chirredi . (C...
chirredlem2 30095 Lemma for ~ chirredi . (C...
chirredlem3 30096 Lemma for ~ chirredi . (C...
chirredlem4 30097 Lemma for ~ chirredi . (C...
chirredi 30098 The Hilbert lattice is irr...
chirred 30099 The Hilbert lattice is irr...
atcvat3i 30100 A condition implying that ...
atcvat4i 30101 A condition implying exist...
atdmd 30102 Two Hilbert lattice elemen...
atmd 30103 Two Hilbert lattice elemen...
atmd2 30104 Two Hilbert lattice elemen...
atabsi 30105 Absorption of an incompara...
atabs2i 30106 Absorption of an incompara...
mdsymlem1 30107 Lemma for ~ mdsymi . (Con...
mdsymlem2 30108 Lemma for ~ mdsymi . (Con...
mdsymlem3 30109 Lemma for ~ mdsymi . (Con...
mdsymlem4 30110 Lemma for ~ mdsymi . This...
mdsymlem5 30111 Lemma for ~ mdsymi . (Con...
mdsymlem6 30112 Lemma for ~ mdsymi . This...
mdsymlem7 30113 Lemma for ~ mdsymi . Lemm...
mdsymlem8 30114 Lemma for ~ mdsymi . Lemm...
mdsymi 30115 M-symmetry of the Hilbert ...
mdsym 30116 M-symmetry of the Hilbert ...
dmdsym 30117 Dual M-symmetry of the Hil...
atdmd2 30118 Two Hilbert lattice elemen...
sumdmdii 30119 If the subspace sum of two...
cmmdi 30120 Commuting subspaces form a...
cmdmdi 30121 Commuting subspaces form a...
sumdmdlem 30122 Lemma for ~ sumdmdi . The...
sumdmdlem2 30123 Lemma for ~ sumdmdi . (Co...
sumdmdi 30124 The subspace sum of two Hi...
dmdbr4ati 30125 Dual modular pair property...
dmdbr5ati 30126 Dual modular pair property...
dmdbr6ati 30127 Dual modular pair property...
dmdbr7ati 30128 Dual modular pair property...
mdoc1i 30129 Orthocomplements form a mo...
mdoc2i 30130 Orthocomplements form a mo...
dmdoc1i 30131 Orthocomplements form a du...
dmdoc2i 30132 Orthocomplements form a du...
mdcompli 30133 A condition equivalent to ...
dmdcompli 30134 A condition equivalent to ...
mddmdin0i 30135 If dual modular implies mo...
cdjreui 30136 A member of the sum of dis...
cdj1i 30137 Two ways to express " ` A ...
cdj3lem1 30138 A property of " ` A ` and ...
cdj3lem2 30139 Lemma for ~ cdj3i . Value...
cdj3lem2a 30140 Lemma for ~ cdj3i . Closu...
cdj3lem2b 30141 Lemma for ~ cdj3i . The f...
cdj3lem3 30142 Lemma for ~ cdj3i . Value...
cdj3lem3a 30143 Lemma for ~ cdj3i . Closu...
cdj3lem3b 30144 Lemma for ~ cdj3i . The s...
cdj3i 30145 Two ways to express " ` A ...
The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here
mathbox 30146 (_This theorem is a dummy ...
sa-abvi 30147 A theorem about the univer...
xfree 30148 A partial converse to ~ 19...
xfree2 30149 A partial converse to ~ 19...
addltmulALT 30150 A proof readability experi...
bian1d 30151 Adding a superfluous conju...
or3di 30152 Distributive law for disju...
or3dir 30153 Distributive law for disju...
3o1cs 30154 Deduction eliminating disj...
3o2cs 30155 Deduction eliminating disj...
3o3cs 30156 Deduction eliminating disj...
sbc2iedf 30157 Conversion of implicit sub...
rspc2daf 30158 Double restricted speciali...
nelbOLD 30159 Obsolete version of ~ nelb...
ralcom4f 30160 Commutation of restricted ...
rexcom4f 30161 Commutation of restricted ...
19.9d2rf 30162 A deduction version of one...
19.9d2r 30163 A deduction version of one...
r19.29ffa 30164 A commonly used pattern ba...
eqtrb 30165 A transposition of equalit...
opsbc2ie 30166 Conversion of implicit sub...
opreu2reuALT 30167 Correspondence between uni...
2reucom 30170 Double restricted existent...
2reu2rex1 30171 Double restricted existent...
2reureurex 30172 Double restricted existent...
2reu2reu2 30173 Double restricted existent...
opreu2reu1 30174 Equivalent definition of t...
sq2reunnltb 30175 There exists a unique deco...
addsqnot2reu 30176 For each complex number ` ...
sbceqbidf 30177 Equality theorem for class...
sbcies 30178 A special version of class...
moel 30179 "At most one" element in a...
mo5f 30180 Alternate definition of "a...
nmo 30181 Negation of "at most one"....
reuxfrdf 30182 Transfer existential uniqu...
rexunirn 30183 Restricted existential qua...
rmoxfrd 30184 Transfer "at most one" res...
rmoun 30185 "At most one" restricted e...
rmounid 30186 Case where an "at most one...
dmrab 30187 Domain of a restricted cla...
difrab2 30188 Difference of two restrict...
rabexgfGS 30189 Separation Scheme in terms...
rabsnel 30190 Truth implied by equality ...
rabeqsnd 30191 Conditions for a restricte...
foresf1o 30192 From a surjective function...
rabfodom 30193 Domination relation for re...
abrexdomjm 30194 An indexed set is dominate...
abrexdom2jm 30195 An indexed set is dominate...
abrexexd 30196 Existence of a class abstr...
elabreximd 30197 Class substitution in an i...
elabreximdv 30198 Class substitution in an i...
abrexss 30199 A necessary condition for ...
elunsn 30200 Elementhood to a union wit...
nelun 30201 Negated membership for a u...
disjdifr 30202 A class and its relative c...
rabss3d 30203 Subclass law for restricte...
inin 30204 Intersection with an inter...
inindif 30205 See ~ inundif . (Contribu...
difininv 30206 Condition for the intersec...
difeq 30207 Rewriting an equation with...
eqdif 30208 If both set differences of...
difxp1ss 30209 Difference law for Cartesi...
difxp2ss 30210 Difference law for Cartesi...
undifr 30211 Union of complementary par...
indifundif 30212 A remarkable equation with...
elpwincl1 30213 Closure of intersection wi...
elpwdifcl 30214 Closure of class differenc...
elpwiuncl 30215 Closure of indexed union w...
eqsnd 30216 Deduce that a set is a sin...
elpreq 30217 Equality wihin a pair. (C...
nelpr 30218 A set ` A ` not in a pair ...
inpr0 30219 Rewrite an empty intersect...
neldifpr1 30220 The first element of a pai...
neldifpr2 30221 The second element of a pa...
unidifsnel 30222 The other element of a pai...
unidifsnne 30223 The other element of a pai...
ifeqeqx 30224 An equality theorem tailor...
elimifd 30225 Elimination of a condition...
elim2if 30226 Elimination of two conditi...
elim2ifim 30227 Elimination of two conditi...
ifeq3da 30228 Given an expression ` C ` ...
uniinn0 30229 Sufficient and necessary c...
uniin1 30230 Union of intersection. Ge...
uniin2 30231 Union of intersection. Ge...
difuncomp 30232 Express a class difference...
pwuniss 30233 Condition for a class unio...
elpwunicl 30234 Closure of a set union wit...
cbviunf 30235 Rule used to change the bo...
iuneq12daf 30236 Equality deduction for ind...
iunin1f 30237 Indexed union of intersect...
ssiun3 30238 Subset equivalence for an ...
ssiun2sf 30239 Subset relationship for an...
iuninc 30240 The union of an increasing...
iundifdifd 30241 The intersection of a set ...
iundifdif 30242 The intersection of a set ...
iunrdx 30243 Re-index an indexed union....
iunpreima 30244 Preimage of an indexed uni...
iunrnmptss 30245 A subset relation for an i...
iunxunsn 30246 Appending a set to an inde...
iunxunpr 30247 Appending two sets to an i...
disjnf 30248 In case ` x ` is not free ...
cbvdisjf 30249 Change bound variables in ...
disjss1f 30250 A subset of a disjoint col...
disjeq1f 30251 Equality theorem for disjo...
disjxun0 30252 Simplify a disjoint union....
disjdifprg 30253 A trivial partition into a...
disjdifprg2 30254 A trivial partition of a s...
disji2f 30255 Property of a disjoint col...
disjif 30256 Property of a disjoint col...
disjorf 30257 Two ways to say that a col...
disjorsf 30258 Two ways to say that a col...
disjif2 30259 Property of a disjoint col...
disjabrex 30260 Rewriting a disjoint colle...
disjabrexf 30261 Rewriting a disjoint colle...
disjpreima 30262 A preimage of a disjoint s...
disjrnmpt 30263 Rewriting a disjoint colle...
disjin 30264 If a collection is disjoin...
disjin2 30265 If a collection is disjoin...
disjxpin 30266 Derive a disjunction over ...
iundisjf 30267 Rewrite a countable union ...
iundisj2f 30268 A disjoint union is disjoi...
disjrdx 30269 Re-index a disjunct collec...
disjex 30270 Two ways to say that two c...
disjexc 30271 A variant of ~ disjex , ap...
disjunsn 30272 Append an element to a dis...
disjun0 30273 Adding the empty element p...
disjiunel 30274 A set of elements B of a d...
disjuniel 30275 A set of elements B of a d...
xpdisjres 30276 Restriction of a constant ...
opeldifid 30277 Ordered pair elementhood o...
difres 30278 Case when class difference...
imadifxp 30279 Image of the difference wi...
relfi 30280 A relation (set) is finite...
reldisjun 30281 Split a relation into two ...
0res 30282 Restriction of the empty f...
funresdm1 30283 Restriction of a disjoint ...
fnunres1 30284 Restriction of a disjoint ...
fcoinver 30285 Build an equivalence relat...
fcoinvbr 30286 Binary relation for the eq...
brabgaf 30287 The law of concretion for ...
brelg 30288 Two things in a binary rel...
br8d 30289 Substitution for an eight-...
opabdm 30290 Domain of an ordered-pair ...
opabrn 30291 Range of an ordered-pair c...
opabssi 30292 Sufficient condition for a...
opabid2ss 30293 One direction of ~ opabid2...
ssrelf 30294 A subclass relationship de...
eqrelrd2 30295 A version of ~ eqrelrdv2 w...
erbr3b 30296 Biconditional for equivale...
iunsnima 30297 Image of a singleton by an...
ac6sf2 30298 Alternate version of ~ ac6...
fnresin 30299 Restriction of a function ...
f1o3d 30300 Describe an implicit one-t...
eldmne0 30301 A function of nonempty dom...
f1rnen 30302 Equinumerosity of the rang...
rinvf1o 30303 Sufficient conditions for ...
fresf1o 30304 Conditions for a restricti...
nfpconfp 30305 The set of fixed points of...
fmptco1f1o 30306 The action of composing (t...
cofmpt2 30307 Express composition of a m...
f1mptrn 30308 Express injection for a ma...
dfimafnf 30309 Alternate definition of th...
funimass4f 30310 Membership relation for th...
elimampt 30311 Membership in the image of...
suppss2f 30312 Show that the support of a...
fovcld 30313 Closure law for an operati...
ofrn 30314 The range of the function ...
ofrn2 30315 The range of the function ...
off2 30316 The function operation pro...
ofresid 30317 Applying an operation rest...
fimarab 30318 Expressing the image of a ...
unipreima 30319 Preimage of a class union....
sspreima 30320 The preimage of a subset i...
opfv 30321 Value of a function produc...
xppreima 30322 The preimage of a Cartesia...
xppreima2 30323 The preimage of a Cartesia...
elunirn2 30324 Condition for the membersh...
abfmpunirn 30325 Membership in a union of a...
rabfmpunirn 30326 Membership in a union of a...
abfmpeld 30327 Membership in an element o...
abfmpel 30328 Membership in an element o...
fmptdF 30329 Domain and codomain of the...
fmptcof2 30330 Composition of two functio...
fcomptf 30331 Express composition of two...
acunirnmpt 30332 Axiom of choice for the un...
acunirnmpt2 30333 Axiom of choice for the un...
acunirnmpt2f 30334 Axiom of choice for the un...
aciunf1lem 30335 Choice in an index union. ...
aciunf1 30336 Choice in an index union. ...
ofoprabco 30337 Function operation as a co...
ofpreima 30338 Express the preimage of a ...
ofpreima2 30339 Express the preimage of a ...
funcnvmpt 30340 Condition for a function i...
funcnv5mpt 30341 Two ways to say that a fun...
funcnv4mpt 30342 Two ways to say that a fun...
preimane 30343 Different elements have di...
fnpreimac 30344 Choose a set ` x ` contain...
fgreu 30345 Exactly one point of a fun...
fcnvgreu 30346 If the converse of a relat...
rnmposs 30347 The range of an operation ...
mptssALT 30348 Deduce subset relation of ...
partfun 30349 Rewrite a function defined...
dfcnv2 30350 Alternative definition of ...
fnimatp 30351 The image of a triplet und...
fnunres2 30352 Restriction of a disjoint ...
mpomptxf 30353 Express a two-argument fun...
suppovss 30354 A bound for the support of...
brsnop 30355 Binary relation for an ord...
cosnopne 30356 Composition of two ordered...
cosnop 30357 Composition of two ordered...
cnvprop 30358 Converse of a pair of orde...
brprop 30359 Binary relation for a pair...
mptprop 30360 Rewrite pairs of ordered p...
coprprop 30361 Composition of two pairs o...
gtiso 30362 Two ways to write a strict...
isoun 30363 Infer an isomorphism from ...
disjdsct 30364 A disjoint collection is d...
df1stres 30365 Definition for a restricti...
df2ndres 30366 Definition for a restricti...
1stpreimas 30367 The preimage of a singleto...
1stpreima 30368 The preimage by ` 1st ` is...
2ndpreima 30369 The preimage by ` 2nd ` is...
curry2ima 30370 The image of a curried fun...
supssd 30371 Inequality deduction for s...
infssd 30372 Inequality deduction for i...
imafi2 30373 The image by a finite set ...
unifi3 30374 If a union is finite, then...
snct 30375 A singleton is countable. ...
prct 30376 An unordered pair is count...
mpocti 30377 An operation is countable ...
abrexct 30378 An image set of a countabl...
mptctf 30379 A countable mapping set is...
abrexctf 30380 An image set of a countabl...
padct 30381 Index a countable set with...
cnvoprabOLD 30382 The converse of a class ab...
f1od2 30383 Sufficient condition for a...
fcobij 30384 Composing functions with a...
fcobijfs 30385 Composing finitely support...
suppss3 30386 Deduce a function's suppor...
fsuppcurry1 30387 Finite support of a currie...
fsuppcurry2 30388 Finite support of a currie...
offinsupp1 30389 Finite support for a funct...
ffs2 30390 Rewrite a function's suppo...
ffsrn 30391 The range of a finitely su...
resf1o 30392 Restriction of functions t...
maprnin 30393 Restricting the range of t...
fpwrelmapffslem 30394 Lemma for ~ fpwrelmapffs ....
fpwrelmap 30395 Define a canonical mapping...
fpwrelmapffs 30396 Define a canonical mapping...
creq0 30397 The real representation of...
1nei 30398 The imaginary unit ` _i ` ...
1neg1t1neg1 30399 An integer unit times itse...
nnmulge 30400 Multiplying by a positive ...
lt2addrd 30401 If the right-hand side of ...
xrlelttric 30402 Trichotomy law for extende...
xaddeq0 30403 Two extended reals which a...
xrinfm 30404 The extended real numbers ...
le2halvesd 30405 A sum is less than the who...
xraddge02 30406 A number is less than or e...
xrge0addge 30407 A number is less than or e...
xlt2addrd 30408 If the right-hand side of ...
xrsupssd 30409 Inequality deduction for s...
xrge0infss 30410 Any subset of nonnegative ...
xrge0infssd 30411 Inequality deduction for i...
xrge0addcld 30412 Nonnegative extended reals...
xrge0subcld 30413 Condition for closure of n...
infxrge0lb 30414 A member of a set of nonne...
infxrge0glb 30415 The infimum of a set of no...
infxrge0gelb 30416 The infimum of a set of no...
dfrp2 30417 Alternate definition of th...
xrofsup 30418 The supremum is preserved ...
supxrnemnf 30419 The supremum of a nonempty...
xnn0gt0 30420 Nonzero extended nonnegati...
xnn01gt 30421 An extended nonnegative in...
nn0xmulclb 30422 Finite multiplication in t...
joiniooico 30423 Disjoint joining an open i...
ubico 30424 A right-open interval does...
xeqlelt 30425 Equality in terms of 'less...
eliccelico 30426 Relate elementhood to a cl...
elicoelioo 30427 Relate elementhood to a cl...
iocinioc2 30428 Intersection between two o...
xrdifh 30429 Class difference of a half...
iocinif 30430 Relate intersection of two...
difioo 30431 The difference between two...
difico 30432 The difference between two...
uzssico 30433 Upper integer sets are a s...
fz2ssnn0 30434 A finite set of sequential...
nndiffz1 30435 Upper set of the positive ...
ssnnssfz 30436 For any finite subset of `...
fzne1 30437 Elementhood in a finite se...
fzm1ne1 30438 Elementhood of an integer ...
fzspl 30439 Split the last element of ...
fzdif2 30440 Split the last element of ...
fzodif2 30441 Split the last element of ...
fzodif1 30442 Set difference of two half...
fzsplit3 30443 Split a finite interval of...
bcm1n 30444 The proportion of one bino...
iundisjfi 30445 Rewrite a countable union ...
iundisj2fi 30446 A disjoint union is disjoi...
iundisjcnt 30447 Rewrite a countable union ...
iundisj2cnt 30448 A countable disjoint union...
fzone1 30449 Elementhood in a half-open...
fzom1ne1 30450 Elementhood in a half-open...
f1ocnt 30451 Given a countable set ` A ...
fz1nnct 30452 NN and integer ranges star...
fz1nntr 30453 NN and integer ranges star...
hashunif 30454 The cardinality of a disjo...
hashxpe 30455 The size of the Cartesian ...
hashgt1 30456 Restate "set contains at l...
dvdszzq 30457 Divisibility for an intege...
prmdvdsbc 30458 Condition for a prime numb...
numdenneg 30459 Numerator and denominator ...
divnumden2 30460 Calculate the reduced form...
nnindf 30461 Principle of Mathematical ...
nnindd 30462 Principle of Mathematical ...
nn0min 30463 Extracting the minimum pos...
subne0nn 30464 A nonnegative difference i...
ltesubnnd 30465 Subtracting an integer num...
fprodeq02 30466 If one of the factors is z...
pr01ssre 30467 The range of the indicator...
fprodex01 30468 A product of factors equal...
prodpr 30469 A product over a pair is t...
prodtp 30470 A product over a triple is...
fsumub 30471 An upper bound for a term ...
fsumiunle 30472 Upper bound for a sum of n...
dfdec100 30473 Split the hundreds from a ...
dp2eq1 30476 Equality theorem for the d...
dp2eq2 30477 Equality theorem for the d...
dp2eq1i 30478 Equality theorem for the d...
dp2eq2i 30479 Equality theorem for the d...
dp2eq12i 30480 Equality theorem for the d...
dp20u 30481 Add a zero in the tenths (...
dp20h 30482 Add a zero in the unit pla...
dp2cl 30483 Closure for the decimal fr...
dp2clq 30484 Closure for a decimal frac...
rpdp2cl 30485 Closure for a decimal frac...
rpdp2cl2 30486 Closure for a decimal frac...
dp2lt10 30487 Decimal fraction builds re...
dp2lt 30488 Comparing two decimal frac...
dp2ltsuc 30489 Comparing a decimal fracti...
dp2ltc 30490 Comparing two decimal expa...
dpval 30493 Define the value of the de...
dpcl 30494 Prove that the closure of ...
dpfrac1 30495 Prove a simple equivalence...
dpval2 30496 Value of the decimal point...
dpval3 30497 Value of the decimal point...
dpmul10 30498 Multiply by 10 a decimal e...
decdiv10 30499 Divide a decimal number by...
dpmul100 30500 Multiply by 100 a decimal ...
dp3mul10 30501 Multiply by 10 a decimal e...
dpmul1000 30502 Multiply by 1000 a decimal...
dpval3rp 30503 Value of the decimal point...
dp0u 30504 Add a zero in the tenths p...
dp0h 30505 Remove a zero in the units...
rpdpcl 30506 Closure of the decimal poi...
dplt 30507 Comparing two decimal expa...
dplti 30508 Comparing a decimal expans...
dpgti 30509 Comparing a decimal expans...
dpltc 30510 Comparing two decimal inte...
dpexpp1 30511 Add one zero to the mantis...
0dp2dp 30512 Multiply by 10 a decimal e...
dpadd2 30513 Addition with one decimal,...
dpadd 30514 Addition with one decimal....
dpadd3 30515 Addition with two decimals...
dpmul 30516 Multiplication with one de...
dpmul4 30517 An upper bound to multipli...
threehalves 30518 Example theorem demonstrat...
1mhdrd 30519 Example theorem demonstrat...
xdivval 30522 Value of division: the (un...
xrecex 30523 Existence of reciprocal of...
xmulcand 30524 Cancellation law for exten...
xreceu 30525 Existential uniqueness of ...
xdivcld 30526 Closure law for the extend...
xdivcl 30527 Closure law for the extend...
xdivmul 30528 Relationship between divis...
rexdiv 30529 The extended real division...
xdivrec 30530 Relationship between divis...
xdivid 30531 A number divided by itself...
xdiv0 30532 Division into zero is zero...
xdiv0rp 30533 Division into zero is zero...
eliccioo 30534 Membership in a closed int...
elxrge02 30535 Elementhood in the set of ...
xdivpnfrp 30536 Plus infinity divided by a...
rpxdivcld 30537 Closure law for extended d...
xrpxdivcld 30538 Closure law for extended d...
wrdfd 30539 A word is a zero-based seq...
wrdres 30540 Condition for the restrict...
wrdsplex 30541 Existence of a split of a ...
pfx1s2 30542 The prefix of length 1 of ...
pfxrn2 30543 The range of a prefix of a...
pfxrn3 30544 Express the range of a pre...
pfxf1 30545 Condition for a prefix to ...
s1f1 30546 Conditions for a length 1 ...
s2rn 30547 Range of a length 2 string...
s2f1 30548 Conditions for a length 2 ...
s3rn 30549 Range of a length 3 string...
s3f1 30550 Conditions for a length 3 ...
s3clhash 30551 Closure of the words of le...
ccatf1 30552 Conditions for a concatena...
pfxlsw2ccat 30553 Reconstruct a word from it...
wrdt2ind 30554 Perform an induction over ...
swrdrn2 30555 The range of a subword is ...
swrdrn3 30556 Express the range of a sub...
swrdf1 30557 Condition for a subword to...
swrdrndisj 30558 Condition for the range of...
splfv3 30559 Symbols to the right of a ...
1cshid 30560 Cyclically shifting a sing...
cshw1s2 30561 Cyclically shifting a leng...
cshwrnid 30562 Cyclically shifting a word...
cshf1o 30563 Condition for the cyclic s...
ressplusf 30564 The group operation functi...
ressnm 30565 The norm in a restricted s...
abvpropd2 30566 Weaker version of ~ abvpro...
oppgle 30567 less-than relation of an o...
oppglt 30568 less-than relation of an o...
ressprs 30569 The restriction of a prose...
oduprs 30570 Being a proset is a self-d...
posrasymb 30571 A poset ordering is asymet...
tospos 30572 A Toset is a Poset. (Cont...
resspos 30573 The restriction of a Poset...
resstos 30574 The restriction of a Toset...
tleile 30575 In a Toset, two elements m...
tltnle 30576 In a Toset, less-than is e...
odutos 30577 Being a toset is a self-du...
tlt2 30578 In a Toset, two elements m...
tlt3 30579 In a Toset, two elements m...
trleile 30580 In a Toset, two elements m...
toslublem 30581 Lemma for ~ toslub and ~ x...
toslub 30582 In a toset, the lowest upp...
tosglblem 30583 Lemma for ~ tosglb and ~ x...
tosglb 30584 Same theorem as ~ toslub ,...
clatp0cl 30585 The poset zero of a comple...
clatp1cl 30586 The poset one of a complet...
xrs0 30589 The zero of the extended r...
xrslt 30590 The "strictly less than" r...
xrsinvgval 30591 The inversion operation in...
xrsmulgzz 30592 The "multiple" function in...
xrstos 30593 The extended real numbers ...
xrsclat 30594 The extended real numbers ...
xrsp0 30595 The poset 0 of the extende...
xrsp1 30596 The poset 1 of the extende...
ressmulgnn 30597 Values for the group multi...
ressmulgnn0 30598 Values for the group multi...
xrge0base 30599 The base of the extended n...
xrge00 30600 The zero of the extended n...
xrge0plusg 30601 The additive law of the ex...
xrge0le 30602 The "less than or equal to...
xrge0mulgnn0 30603 The group multiple functio...
xrge0addass 30604 Associativity of extended ...
xrge0addgt0 30605 The sum of nonnegative and...
xrge0adddir 30606 Right-distributivity of ex...
xrge0adddi 30607 Left-distributivity of ext...
xrge0npcan 30608 Extended nonnegative real ...
fsumrp0cl 30609 Closure of a finite sum of...
abliso 30610 The image of an Abelian gr...
gsumsubg 30611 The group sum in a subgrou...
gsumsra 30612 The group sum in a subring...
gsummpt2co 30613 Split a finite sum into a ...
gsummpt2d 30614 Express a finite sum over ...
lmodvslmhm 30615 Scalar multiplication in a...
gsumvsmul1 30616 Pull a scalar multiplicati...
gsummptres 30617 Extend a finite group sum ...
gsumzresunsn 30618 Append an element to a fin...
xrge0tsmsd 30619 Any finite or infinite sum...
xrge0tsmsbi 30620 Any limit of a finite or i...
xrge0tsmseq 30621 Any limit of a finite or i...
cntzun 30622 The centralizer of a union...
cntzsnid 30623 The centralizer of the ide...
cntrcrng 30624 The center of a ring is a ...
isomnd 30629 A (left) ordered monoid is...
isogrp 30630 A (left-)ordered group is ...
ogrpgrp 30631 A left-ordered group is a ...
omndmnd 30632 A left-ordered monoid is a...
omndtos 30633 A left-ordered monoid is a...
omndadd 30634 In an ordered monoid, the ...
omndaddr 30635 In a right ordered monoid,...
omndadd2d 30636 In a commutative left orde...
omndadd2rd 30637 In a left- and right- orde...
submomnd 30638 A submonoid of an ordered ...
xrge0omnd 30639 The nonnegative extended r...
omndmul2 30640 In an ordered monoid, the ...
omndmul3 30641 In an ordered monoid, the ...
omndmul 30642 In a commutative ordered m...
ogrpinv0le 30643 In an ordered group, the o...
ogrpsub 30644 In an ordered group, the o...
ogrpaddlt 30645 In an ordered group, stric...
ogrpaddltbi 30646 In a right ordered group, ...
ogrpaddltrd 30647 In a right ordered group, ...
ogrpaddltrbid 30648 In a right ordered group, ...
ogrpsublt 30649 In an ordered group, stric...
ogrpinv0lt 30650 In an ordered group, the o...
ogrpinvlt 30651 In an ordered group, the o...
gsumle 30652 A finite sum in an ordered...
symgfcoeu 30653 Uniqueness property of per...
symgcom 30654 Two permutations ` X ` and...
symgcom2 30655 Two permutations ` X ` and...
symgcntz 30656 All elements of a (finite)...
odpmco 30657 The composition of two odd...
symgsubg 30658 The value of the group sub...
pmtrprfv2 30659 In a transposition of two ...
pmtrcnel 30660 Composing a permutation ` ...
pmtrcnel2 30661 Variation on ~ pmtrcnel . ...
pmtrcnelor 30662 Composing a permutation ` ...
pmtridf1o 30663 Transpositions of ` X ` an...
pmtridfv1 30664 Value at X of the transpos...
pmtridfv2 30665 Value at Y of the transpos...
psgnid 30666 Permutation sign of the id...
psgndmfi 30667 For a finite base set, the...
pmtrto1cl 30668 Useful lemma for the follo...
psgnfzto1stlem 30669 Lemma for ~ psgnfzto1st . ...
fzto1stfv1 30670 Value of our permutation `...
fzto1st1 30671 Special case where the per...
fzto1st 30672 The function moving one el...
fzto1stinvn 30673 Value of the inverse of ou...
psgnfzto1st 30674 The permutation sign for m...
tocycval 30677 Value of the cycle builder...
tocycfv 30678 Function value of a permut...
tocycfvres1 30679 A cyclic permutation is a ...
tocycfvres2 30680 A cyclic permutation is th...
cycpmfvlem 30681 Lemma for ~ cycpmfv1 and ~...
cycpmfv1 30682 Value of a cycle function ...
cycpmfv2 30683 Value of a cycle function ...
cycpmfv3 30684 Values outside of the orbi...
cycpmcl 30685 Cyclic permutations are pe...
tocycf 30686 The permutation cycle buil...
tocyc01 30687 Permutation cycles built f...
cycpm2tr 30688 A cyclic permutation of 2 ...
cycpm2cl 30689 Closure for the 2-cycles. ...
cyc2fv1 30690 Function value of a 2-cycl...
cyc2fv2 30691 Function value of a 2-cycl...
trsp2cyc 30692 Exhibit the word a transpo...
cycpmco2f1 30693 The word U used in ~ cycpm...
cycpmco2rn 30694 The orbit of the compositi...
cycpmco2lem1 30695 Lemma for ~ cycpmco2 . (C...
cycpmco2lem2 30696 Lemma for ~ cycpmco2 . (C...
cycpmco2lem3 30697 Lemma for ~ cycpmco2 . (C...
cycpmco2lem4 30698 Lemma for ~ cycpmco2 . (C...
cycpmco2lem5 30699 Lemma for ~ cycpmco2 . (C...
cycpmco2lem6 30700 Lemma for ~ cycpmco2 . (C...
cycpmco2lem7 30701 Lemma for ~ cycpmco2 . (C...
cycpmco2 30702 The composition of a cycli...
cyc2fvx 30703 Function value of a 2-cycl...
cycpm3cl 30704 Closure of the 3-cycles in...
cycpm3cl2 30705 Closure of the 3-cycles in...
cyc3fv1 30706 Function value of a 3-cycl...
cyc3fv2 30707 Function value of a 3-cycl...
cyc3fv3 30708 Function value of a 3-cycl...
cyc3co2 30709 Represent a 3-cycle as a c...
cycpmconjvlem 30710 Lemma for ~ cycpmconjv (Co...
cycpmconjv 30711 A formula for computing co...
cycpmrn 30712 The range of the word used...
tocyccntz 30713 All elements of a (finite)...
evpmval 30714 Value of the set of even p...
cnmsgn0g 30715 The neutral element of the...
evpmsubg 30716 The alternating group is a...
evpmid 30717 The identity is an even pe...
altgnsg 30718 The alternating group ` ( ...
cyc3evpm 30719 3-Cycles are even permutat...
cyc3genpmlem 30720 Lemma for ~ cyc3genpm . (...
cyc3genpm 30721 The alternating group ` A ...
cycpmgcl 30722 Cyclic permutations are pe...
cycpmconjslem1 30723 Lemma for ~ cycpmconjs (Co...
cycpmconjslem2 30724 Lemma for ~ cycpmconjs (Co...
cycpmconjs 30725 All cycles of the same len...
cyc3conja 30726 All 3-cycles are conjugate...
sgnsv 30729 The sign mapping. (Contri...
sgnsval 30730 The sign value. (Contribu...
sgnsf 30731 The sign function. (Contr...
inftmrel 30736 The infinitesimal relation...
isinftm 30737 Express ` x ` is infinites...
isarchi 30738 Express the predicate " ` ...
pnfinf 30739 Plus infinity is an infini...
xrnarchi 30740 The completed real line is...
isarchi2 30741 Alternative way to express...
submarchi 30742 A submonoid is archimedean...
isarchi3 30743 This is the usual definiti...
archirng 30744 Property of Archimedean or...
archirngz 30745 Property of Archimedean le...
archiexdiv 30746 In an Archimedean group, g...
archiabllem1a 30747 Lemma for ~ archiabl : In...
archiabllem1b 30748 Lemma for ~ archiabl . (C...
archiabllem1 30749 Archimedean ordered groups...
archiabllem2a 30750 Lemma for ~ archiabl , whi...
archiabllem2c 30751 Lemma for ~ archiabl . (C...
archiabllem2b 30752 Lemma for ~ archiabl . (C...
archiabllem2 30753 Archimedean ordered groups...
archiabl 30754 Archimedean left- and righ...
isslmd 30757 The predicate "is a semimo...
slmdlema 30758 Lemma for properties of a ...
lmodslmd 30759 Left semimodules generaliz...
slmdcmn 30760 A semimodule is a commutat...
slmdmnd 30761 A semimodule is a monoid. ...
slmdsrg 30762 The scalar component of a ...
slmdbn0 30763 The base set of a semimodu...
slmdacl 30764 Closure of ring addition f...
slmdmcl 30765 Closure of ring multiplica...
slmdsn0 30766 The set of scalars in a se...
slmdvacl 30767 Closure of vector addition...
slmdass 30768 Semiring left module vecto...
slmdvscl 30769 Closure of scalar product ...
slmdvsdi 30770 Distributive law for scala...
slmdvsdir 30771 Distributive law for scala...
slmdvsass 30772 Associative law for scalar...
slmd0cl 30773 The ring zero in a semimod...
slmd1cl 30774 The ring unit in a semirin...
slmdvs1 30775 Scalar product with ring u...
slmd0vcl 30776 The zero vector is a vecto...
slmd0vlid 30777 Left identity law for the ...
slmd0vrid 30778 Right identity law for the...
slmd0vs 30779 Zero times a vector is the...
slmdvs0 30780 Anything times the zero ve...
gsumvsca1 30781 Scalar product of a finite...
gsumvsca2 30782 Scalar product of a finite...
prmsimpcyc 30783 A group of prime order is ...
rngurd 30784 Deduce the unit of a ring ...
dvdschrmulg 30785 In a ring, any multiple of...
freshmansdream 30786 For a prime number ` P ` ,...
ress1r 30787 ` 1r ` is unaffected by re...
dvrdir 30788 Distributive law for the d...
rdivmuldivd 30789 Multiplication of two rati...
ringinvval 30790 The ring inverse expressed...
dvrcan5 30791 Cancellation law for commo...
subrgchr 30792 If ` A ` is a subring of `...
rmfsupp2 30793 A mapping of a multiplicat...
primefldchr 30794 The characteristic of a pr...
isorng 30799 An ordered ring is a ring ...
orngring 30800 An ordered ring is a ring....
orngogrp 30801 An ordered ring is an orde...
isofld 30802 An ordered field is a fiel...
orngmul 30803 In an ordered ring, the or...
orngsqr 30804 In an ordered ring, all sq...
ornglmulle 30805 In an ordered ring, multip...
orngrmulle 30806 In an ordered ring, multip...
ornglmullt 30807 In an ordered ring, multip...
orngrmullt 30808 In an ordered ring, multip...
orngmullt 30809 In an ordered ring, the st...
ofldfld 30810 An ordered field is a fiel...
ofldtos 30811 An ordered field is a tota...
orng0le1 30812 In an ordered ring, the ri...
ofldlt1 30813 In an ordered field, the r...
ofldchr 30814 The characteristic of an o...
suborng 30815 Every subring of an ordere...
subofld 30816 Every subfield of an order...
isarchiofld 30817 Axiom of Archimedes : a ch...
rhmdvdsr 30818 A ring homomorphism preser...
rhmopp 30819 A ring homomorphism is als...
elrhmunit 30820 Ring homomorphisms preserv...
rhmdvd 30821 A ring homomorphism preser...
rhmunitinv 30822 Ring homomorphisms preserv...
kerunit 30823 If a unit element lies in ...
reldmresv 30826 The scalar restriction is ...
resvval 30827 Value of structure restric...
resvid2 30828 General behavior of trivia...
resvval2 30829 Value of nontrivial struct...
resvsca 30830 Base set of a structure re...
resvlem 30831 Other elements of a struct...
resvbas 30832 ` Base ` is unaffected by ...
resvplusg 30833 ` +g ` is unaffected by sc...
resvvsca 30834 ` .s ` is unaffected by sc...
resvmulr 30835 ` .s ` is unaffected by sc...
resv0g 30836 ` 0g ` is unaffected by sc...
resv1r 30837 ` 1r ` is unaffected by sc...
resvcmn 30838 Scalar restriction preserv...
gzcrng 30839 The gaussian integers form...
reofld 30840 The real numbers form an o...
nn0omnd 30841 The nonnegative integers f...
rearchi 30842 The field of the real numb...
nn0archi 30843 The monoid of the nonnegat...
xrge0slmod 30844 The extended nonnegative r...
qusker 30845 The kernel of a quotient m...
eqgvscpbl 30846 The left coset equivalence...
qusvscpbl 30847 The quotient map distribut...
qusscaval 30848 Value of the scalar multip...
imaslmod 30849 The image structure of a l...
quslmod 30850 If ` G ` is a submodule in...
quslmhm 30851 If ` G ` is a submodule of...
ecxpid 30852 The equivalence class of a...
eqg0el 30853 Equivalence class of a quo...
qsxpid 30854 The quotient set of a cart...
qusxpid 30855 The Group quotient equival...
qustriv 30856 The quotient of a group ` ...
qustrivr 30857 Converse of ~ qustriv . (...
fply1 30858 Conditions for a function ...
islinds5 30859 A set is linearly independ...
ellspds 30860 Variation on ~ ellspd . (...
0ellsp 30861 Zero is in all spans. (Co...
0nellinds 30862 The group identity cannot ...
rspsnel 30863 Membership in a principal ...
rspsnid 30864 A principal ideal contains...
lbslsp 30865 Any element of a left modu...
lindssn 30866 Any singleton of a nonzero...
lindflbs 30867 Conditions for an independ...
linds2eq 30868 Deduce equality of element...
lindfpropd 30869 Property deduction for lin...
lindspropd 30870 Property deduction for lin...
prmidlval 30873 The class of prime ideals ...
isprmidl 30874 The predicate "is a prime ...
prmidlnr 30875 A prime ideal is a proper ...
prmidl 30876 The main property of a pri...
prmidl2 30877 A condition that shows an ...
prmidlidl 30878 A prime ideal is an ideal....
lidlnsg 30879 An ideal is a normal subgr...
cringm4 30880 Commutative/associative la...
isprmidlc 30881 The predicate "is prime id...
qsidomlem1 30882 If the quotient ring of a ...
qsidomlem2 30883 A quotient by a prime idea...
qsidom 30884 An ideal ` I ` in the comm...
sra1r 30885 The multiplicative neutral...
sraring 30886 Condition for a subring al...
sradrng 30887 Condition for a subring al...
srasubrg 30888 A subring of the original ...
sralvec 30889 Given a sub division ring ...
srafldlvec 30890 Given a subfield ` F ` of ...
drgext0g 30891 The additive neutral eleme...
drgextvsca 30892 The scalar multiplication ...
drgext0gsca 30893 The additive neutral eleme...
drgextsubrg 30894 The scalar field is a subr...
drgextlsp 30895 The scalar field is a subs...
drgextgsum 30896 Group sum in a division ri...
lvecdimfi 30897 Finite version of ~ lvecdi...
dimval 30900 The dimension of a vector ...
dimvalfi 30901 The dimension of a vector ...
dimcl 30902 Closure of the vector spac...
lvecdim0i 30903 A vector space of dimensio...
lvecdim0 30904 A vector space of dimensio...
lssdimle 30905 The dimension of a linear ...
dimpropd 30906 If two structures have the...
rgmoddim 30907 The left vector space indu...
frlmdim 30908 Dimension of a free left m...
tnglvec 30909 Augmenting a structure wit...
tngdim 30910 Dimension of a left vector...
rrxdim 30911 Dimension of the generaliz...
matdim 30912 Dimension of the space of ...
lbslsat 30913 A nonzero vector ` X ` is ...
lsatdim 30914 A line, spanned by a nonze...
drngdimgt0 30915 The dimension of a vector ...
lmhmlvec2 30916 A homomorphism of left vec...
kerlmhm 30917 The kernel of a vector spa...
imlmhm 30918 The image of a vector spac...
lindsunlem 30919 Lemma for ~ lindsun . (Co...
lindsun 30920 Condition for the union of...
lbsdiflsp0 30921 The linear spans of two di...
dimkerim 30922 Given a linear map ` F ` b...
qusdimsum 30923 Let ` W ` be a vector spac...
fedgmullem1 30924 Lemma for ~ fedgmul (Contr...
fedgmullem2 30925 Lemma for ~ fedgmul (Contr...
fedgmul 30926 The multiplicativity formu...
relfldext 30935 The field extension is a r...
brfldext 30936 The field extension relati...
ccfldextrr 30937 The field of the complex n...
fldextfld1 30938 A field extension is only ...
fldextfld2 30939 A field extension is only ...
fldextsubrg 30940 Field extension implies a ...
fldextress 30941 Field extension implies a ...
brfinext 30942 The finite field extension...
extdgval 30943 Value of the field extensi...
fldextsralvec 30944 The subring algebra associ...
extdgcl 30945 Closure of the field exten...
extdggt0 30946 Degrees of field extension...
fldexttr 30947 Field extension is a trans...
fldextid 30948 The field extension relati...
extdgid 30949 A trivial field extension ...
extdgmul 30950 The multiplicativity formu...
finexttrb 30951 The extension ` E ` of ` K...
extdg1id 30952 If the degree of the exten...
extdg1b 30953 The degree of the extensio...
fldextchr 30954 The characteristic of a su...
ccfldsrarelvec 30955 The subring algebra of the...
ccfldextdgrr 30956 The degree of the field ex...
smatfval 30959 Value of the submatrix. (...
smatrcl 30960 Closure of the rectangular...
smatlem 30961 Lemma for the next theorem...
smattl 30962 Entries of a submatrix, to...
smattr 30963 Entries of a submatrix, to...
smatbl 30964 Entries of a submatrix, bo...
smatbr 30965 Entries of a submatrix, bo...
smatcl 30966 Closure of the square subm...
matmpo 30967 Write a square matrix as a...
1smat1 30968 The submatrix of the ident...
submat1n 30969 One case where the submatr...
submatres 30970 Special case where the sub...
submateqlem1 30971 Lemma for ~ submateq . (C...
submateqlem2 30972 Lemma for ~ submateq . (C...
submateq 30973 Sufficient condition for t...
submatminr1 30974 If we take a submatrix by ...
lmatval 30977 Value of the literal matri...
lmatfval 30978 Entries of a literal matri...
lmatfvlem 30979 Useful lemma to extract li...
lmatcl 30980 Closure of the literal mat...
lmat22lem 30981 Lemma for ~ lmat22e11 and ...
lmat22e11 30982 Entry of a 2x2 literal mat...
lmat22e12 30983 Entry of a 2x2 literal mat...
lmat22e21 30984 Entry of a 2x2 literal mat...
lmat22e22 30985 Entry of a 2x2 literal mat...
lmat22det 30986 The determinant of a liter...
mdetpmtr1 30987 The determinant of a matri...
mdetpmtr2 30988 The determinant of a matri...
mdetpmtr12 30989 The determinant of a matri...
mdetlap1 30990 A Laplace expansion of the...
madjusmdetlem1 30991 Lemma for ~ madjusmdet . ...
madjusmdetlem2 30992 Lemma for ~ madjusmdet . ...
madjusmdetlem3 30993 Lemma for ~ madjusmdet . ...
madjusmdetlem4 30994 Lemma for ~ madjusmdet . ...
madjusmdet 30995 Express the cofactor of th...
mdetlap 30996 Laplace expansion of the d...
txomap 30997 Given two open maps ` F ` ...
qtopt1 30998 If every equivalence class...
qtophaus 30999 If an open map's graph in ...
circtopn 31000 The topology of the unit c...
circcn 31001 The function gluing the re...
reff 31002 For any cover refinement, ...
locfinreflem 31003 A locally finite refinemen...
locfinref 31004 A locally finite refinemen...
iscref 31007 The property that every op...
crefeq 31008 Equality theorem for the "...
creftop 31009 A space where every open c...
crefi 31010 The property that every op...
crefdf 31011 A formulation of ~ crefi e...
crefss 31012 The "every open cover has ...
cmpcref 31013 Equivalent definition of c...
cmpfiref 31014 Every open cover of a Comp...
ldlfcntref 31017 Every open cover of a Lind...
ispcmp 31020 The predicate "is a paraco...
cmppcmp 31021 Every compact space is par...
dispcmp 31022 Every discrete space is pa...
pcmplfin 31023 Given a paracompact topolo...
pcmplfinf 31024 Given a paracompact topolo...
metidval 31029 Value of the metric identi...
metidss 31030 As a relation, the metric ...
metidv 31031 ` A ` and ` B ` identify b...
metideq 31032 Basic property of the metr...
metider 31033 The metric identification ...
pstmval 31034 Value of the metric induce...
pstmfval 31035 Function value of the metr...
pstmxmet 31036 The metric induced by a ps...
hauseqcn 31037 In a Hausdorff topology, t...
unitsscn 31038 The closed unit interval i...
elunitrn 31039 The closed unit interval i...
elunitcn 31040 The closed unit interval i...
elunitge0 31041 An element of the closed u...
unitssxrge0 31042 The closed unit interval i...
unitdivcld 31043 Necessary conditions for a...
iistmd 31044 The closed unit interval f...
unicls 31045 The union of the closed se...
tpr2tp 31046 The usual topology on ` ( ...
tpr2uni 31047 The usual topology on ` ( ...
xpinpreima 31048 Rewrite the cartesian prod...
xpinpreima2 31049 Rewrite the cartesian prod...
sqsscirc1 31050 The complex square of side...
sqsscirc2 31051 The complex square of side...
cnre2csqlem 31052 Lemma for ~ cnre2csqima . ...
cnre2csqima 31053 Image of a centered square...
tpr2rico 31054 For any point of an open s...
cnvordtrestixx 31055 The restriction of the 'gr...
prsdm 31056 Domain of the relation of ...
prsrn 31057 Range of the relation of a...
prsss 31058 Relation of a subproset. ...
prsssdm 31059 Domain of a subproset rela...
ordtprsval 31060 Value of the order topolog...
ordtprsuni 31061 Value of the order topolog...
ordtcnvNEW 31062 The order dual generates t...
ordtrestNEW 31063 The subspace topology of a...
ordtrest2NEWlem 31064 Lemma for ~ ordtrest2NEW ....
ordtrest2NEW 31065 An interval-closed set ` A...
ordtconnlem1 31066 Connectedness in the order...
ordtconn 31067 Connectedness in the order...
mndpluscn 31068 A mapping that is both a h...
mhmhmeotmd 31069 Deduce a Topological Monoi...
rmulccn 31070 Multiplication by a real c...
raddcn 31071 Addition in the real numbe...
xrmulc1cn 31072 The operation multiplying ...
fmcncfil 31073 The image of a Cauchy filt...
xrge0hmph 31074 The extended nonnegative r...
xrge0iifcnv 31075 Define a bijection from ` ...
xrge0iifcv 31076 The defined function's val...
xrge0iifiso 31077 The defined bijection from...
xrge0iifhmeo 31078 Expose a homeomorphism fro...
xrge0iifhom 31079 The defined function from ...
xrge0iif1 31080 Condition for the defined ...
xrge0iifmhm 31081 The defined function from ...
xrge0pluscn 31082 The addition operation of ...
xrge0mulc1cn 31083 The operation multiplying ...
xrge0tps 31084 The extended nonnegative r...
xrge0topn 31085 The topology of the extend...
xrge0haus 31086 The topology of the extend...
xrge0tmd 31087 The extended nonnegative r...
xrge0tmdALT 31088 Alternate proof of ~ xrge0...
lmlim 31089 Relate a limit in a given ...
lmlimxrge0 31090 Relate a limit in the nonn...
rge0scvg 31091 Implication of convergence...
fsumcvg4 31092 A serie with finite suppor...
pnfneige0 31093 A neighborhood of ` +oo ` ...
lmxrge0 31094 Express "sequence ` F ` co...
lmdvg 31095 If a monotonic sequence of...
lmdvglim 31096 If a monotonic real number...
pl1cn 31097 A univariate polynomial is...
zringnm 31100 The norm (function) for a ...
zzsnm 31101 The norm of the ring of th...
zlm0 31102 Zero of a ` ZZ ` -module. ...
zlm1 31103 Unit of a ` ZZ ` -module (...
zlmds 31104 Distance in a ` ZZ ` -modu...
zlmtset 31105 Topology in a ` ZZ ` -modu...
zlmnm 31106 Norm of a ` ZZ ` -module (...
zhmnrg 31107 The ` ZZ ` -module built f...
nmmulg 31108 The norm of a group produc...
zrhnm 31109 The norm of the image by `...
cnzh 31110 The ` ZZ ` -module of ` CC...
rezh 31111 The ` ZZ ` -module of ` RR...
qqhval 31114 Value of the canonical hom...
zrhf1ker 31115 The kernel of the homomorp...
zrhchr 31116 The kernel of the homomorp...
zrhker 31117 The kernel of the homomorp...
zrhunitpreima 31118 The preimage by ` ZRHom ` ...
elzrhunit 31119 Condition for the image by...
elzdif0 31120 Lemma for ~ qqhval2 . (Co...
qqhval2lem 31121 Lemma for ~ qqhval2 . (Co...
qqhval2 31122 Value of the canonical hom...
qqhvval 31123 Value of the canonical hom...
qqh0 31124 The image of ` 0 ` by the ...
qqh1 31125 The image of ` 1 ` by the ...
qqhf 31126 ` QQHom ` as a function. ...
qqhvq 31127 The image of a quotient by...
qqhghm 31128 The ` QQHom ` homomorphism...
qqhrhm 31129 The ` QQHom ` homomorphism...
qqhnm 31130 The norm of the image by `...
qqhcn 31131 The ` QQHom ` homomorphism...
qqhucn 31132 The ` QQHom ` homomorphism...
rrhval 31136 Value of the canonical hom...
rrhcn 31137 If the topology of ` R ` i...
rrhf 31138 If the topology of ` R ` i...
isrrext 31140 Express the property " ` R...
rrextnrg 31141 An extension of ` RR ` is ...
rrextdrg 31142 An extension of ` RR ` is ...
rrextnlm 31143 The norm of an extension o...
rrextchr 31144 The ring characteristic of...
rrextcusp 31145 An extension of ` RR ` is ...
rrexttps 31146 An extension of ` RR ` is ...
rrexthaus 31147 The topology of an extensi...
rrextust 31148 The uniformity of an exten...
rerrext 31149 The field of the real numb...
cnrrext 31150 The field of the complex n...
qqtopn 31151 The topology of the field ...
rrhfe 31152 If ` R ` is an extension o...
rrhcne 31153 If ` R ` is an extension o...
rrhqima 31154 The ` RRHom ` homomorphism...
rrh0 31155 The image of ` 0 ` by the ...
xrhval 31158 The value of the embedding...
zrhre 31159 The ` ZRHom ` homomorphism...
qqhre 31160 The ` QQHom ` homomorphism...
rrhre 31161 The ` RRHom ` homomorphism...
relmntop 31164 Manifold is a relation. (...
ismntoplly 31165 Property of being a manifo...
ismntop 31166 Property of being a manifo...
nexple 31167 A lower bound for an expon...
indv 31170 Value of the indicator fun...
indval 31171 Value of the indicator fun...
indval2 31172 Alternate value of the ind...
indf 31173 An indicator function as a...
indfval 31174 Value of the indicator fun...
ind1 31175 Value of the indicator fun...
ind0 31176 Value of the indicator fun...
ind1a 31177 Value of the indicator fun...
indpi1 31178 Preimage of the singleton ...
indsum 31179 Finite sum of a product wi...
indsumin 31180 Finite sum of a product wi...
prodindf 31181 The product of indicators ...
indf1o 31182 The bijection between a po...
indpreima 31183 A function with range ` { ...
indf1ofs 31184 The bijection between fini...
esumex 31187 An extended sum is a set b...
esumcl 31188 Closure for extended sum i...
esumeq12dvaf 31189 Equality deduction for ext...
esumeq12dva 31190 Equality deduction for ext...
esumeq12d 31191 Equality deduction for ext...
esumeq1 31192 Equality theorem for an ex...
esumeq1d 31193 Equality theorem for an ex...
esumeq2 31194 Equality theorem for exten...
esumeq2d 31195 Equality deduction for ext...
esumeq2dv 31196 Equality deduction for ext...
esumeq2sdv 31197 Equality deduction for ext...
nfesum1 31198 Bound-variable hypothesis ...
nfesum2 31199 Bound-variable hypothesis ...
cbvesum 31200 Change bound variable in a...
cbvesumv 31201 Change bound variable in a...
esumid 31202 Identify the extended sum ...
esumgsum 31203 A finite extended sum is t...
esumval 31204 Develop the value of the e...
esumel 31205 The extended sum is a limi...
esumnul 31206 Extended sum over the empt...
esum0 31207 Extended sum of zero. (Co...
esumf1o 31208 Re-index an extended sum u...
esumc 31209 Convert from the collectio...
esumrnmpt 31210 Rewrite an extended sum in...
esumsplit 31211 Split an extended sum into...
esummono 31212 Extended sum is monotonic....
esumpad 31213 Extend an extended sum by ...
esumpad2 31214 Remove zeroes from an exte...
esumadd 31215 Addition of infinite sums....
esumle 31216 If all of the terms of an ...
gsumesum 31217 Relate a group sum on ` ( ...
esumlub 31218 The extended sum is the lo...
esumaddf 31219 Addition of infinite sums....
esumlef 31220 If all of the terms of an ...
esumcst 31221 The extended sum of a cons...
esumsnf 31222 The extended sum of a sing...
esumsn 31223 The extended sum of a sing...
esumpr 31224 Extended sum over a pair. ...
esumpr2 31225 Extended sum over a pair, ...
esumrnmpt2 31226 Rewrite an extended sum in...
esumfzf 31227 Formulating a partial exte...
esumfsup 31228 Formulating an extended su...
esumfsupre 31229 Formulating an extended su...
esumss 31230 Change the index set to a ...
esumpinfval 31231 The value of the extended ...
esumpfinvallem 31232 Lemma for ~ esumpfinval . ...
esumpfinval 31233 The value of the extended ...
esumpfinvalf 31234 Same as ~ esumpfinval , mi...
esumpinfsum 31235 The value of the extended ...
esumpcvgval 31236 The value of the extended ...
esumpmono 31237 The partial sums in an ext...
esumcocn 31238 Lemma for ~ esummulc2 and ...
esummulc1 31239 An extended sum multiplied...
esummulc2 31240 An extended sum multiplied...
esumdivc 31241 An extended sum divided by...
hashf2 31242 Lemma for ~ hasheuni . (C...
hasheuni 31243 The cardinality of a disjo...
esumcvg 31244 The sequence of partial su...
esumcvg2 31245 Simpler version of ~ esumc...
esumcvgsum 31246 The value of the extended ...
esumsup 31247 Express an extended sum as...
esumgect 31248 "Send ` n ` to ` +oo ` " i...
esumcvgre 31249 All terms of a converging ...
esum2dlem 31250 Lemma for ~ esum2d (finite...
esum2d 31251 Write a double extended su...
esumiun 31252 Sum over a nonnecessarily ...
ofceq 31255 Equality theorem for funct...
ofcfval 31256 Value of an operation appl...
ofcval 31257 Evaluate a function/consta...
ofcfn 31258 The function operation pro...
ofcfeqd2 31259 Equality theorem for funct...
ofcfval3 31260 General value of ` ( F oFC...
ofcf 31261 The function/constant oper...
ofcfval2 31262 The function operation exp...
ofcfval4 31263 The function/constant oper...
ofcc 31264 Left operation by a consta...
ofcof 31265 Relate function operation ...
sigaex 31268 Lemma for ~ issiga and ~ i...
sigaval 31269 The set of sigma-algebra w...
issiga 31270 An alternative definition ...
isrnsiga 31271 The property of being a si...
0elsiga 31272 A sigma-algebra contains t...
baselsiga 31273 A sigma-algebra contains i...
sigasspw 31274 A sigma-algebra is a set o...
sigaclcu 31275 A sigma-algebra is closed ...
sigaclcuni 31276 A sigma-algebra is closed ...
sigaclfu 31277 A sigma-algebra is closed ...
sigaclcu2 31278 A sigma-algebra is closed ...
sigaclfu2 31279 A sigma-algebra is closed ...
sigaclcu3 31280 A sigma-algebra is closed ...
issgon 31281 Property of being a sigma-...
sgon 31282 A sigma-algebra is a sigma...
elsigass 31283 An element of a sigma-alge...
elrnsiga 31284 Dropping the base informat...
isrnsigau 31285 The property of being a si...
unielsiga 31286 A sigma-algebra contains i...
dmvlsiga 31287 Lebesgue-measurable subset...
pwsiga 31288 Any power set forms a sigm...
prsiga 31289 The smallest possible sigm...
sigaclci 31290 A sigma-algebra is closed ...
difelsiga 31291 A sigma-algebra is closed ...
unelsiga 31292 A sigma-algebra is closed ...
inelsiga 31293 A sigma-algebra is closed ...
sigainb 31294 Building a sigma-algebra f...
insiga 31295 The intersection of a coll...
sigagenval 31298 Value of the generated sig...
sigagensiga 31299 A generated sigma-algebra ...
sgsiga 31300 A generated sigma-algebra ...
unisg 31301 The sigma-algebra generate...
dmsigagen 31302 A sigma-algebra can be gen...
sssigagen 31303 A set is a subset of the s...
sssigagen2 31304 A subset of the generating...
elsigagen 31305 Any element of a set is al...
elsigagen2 31306 Any countable union of ele...
sigagenss 31307 The generated sigma-algebr...
sigagenss2 31308 Sufficient condition for i...
sigagenid 31309 The sigma-algebra generate...
ispisys 31310 The property of being a pi...
ispisys2 31311 The property of being a pi...
inelpisys 31312 Pi-systems are closed unde...
sigapisys 31313 All sigma-algebras are pi-...
isldsys 31314 The property of being a la...
pwldsys 31315 The power set of the unive...
unelldsys 31316 Lambda-systems are closed ...
sigaldsys 31317 All sigma-algebras are lam...
ldsysgenld 31318 The intersection of all la...
sigapildsyslem 31319 Lemma for ~ sigapildsys . ...
sigapildsys 31320 Sigma-algebra are exactly ...
ldgenpisyslem1 31321 Lemma for ~ ldgenpisys . ...
ldgenpisyslem2 31322 Lemma for ~ ldgenpisys . ...
ldgenpisyslem3 31323 Lemma for ~ ldgenpisys . ...
ldgenpisys 31324 The lambda system ` E ` ge...
dynkin 31325 Dynkin's lambda-pi theorem...
isros 31326 The property of being a ri...
rossspw 31327 A ring of sets is a collec...
0elros 31328 A ring of sets contains th...
unelros 31329 A ring of sets is closed u...
difelros 31330 A ring of sets is closed u...
inelros 31331 A ring of sets is closed u...
fiunelros 31332 A ring of sets is closed u...
issros 31333 The property of being a se...
srossspw 31334 A semiring of sets is a co...
0elsros 31335 A semiring of sets contain...
inelsros 31336 A semiring of sets is clos...
diffiunisros 31337 In semiring of sets, compl...
rossros 31338 Rings of sets are semiring...
brsiga 31341 The Borel Algebra on real ...
brsigarn 31342 The Borel Algebra is a sig...
brsigasspwrn 31343 The Borel Algebra is a set...
unibrsiga 31344 The union of the Borel Alg...
cldssbrsiga 31345 A Borel Algebra contains a...
sxval 31348 Value of the product sigma...
sxsiga 31349 A product sigma-algebra is...
sxsigon 31350 A product sigma-algebra is...
sxuni 31351 The base set of a product ...
elsx 31352 The cartesian product of t...
measbase 31355 The base set of a measure ...
measval 31356 The value of the ` measure...
ismeas 31357 The property of being a me...
isrnmeas 31358 The property of being a me...
dmmeas 31359 The domain of a measure is...
measbasedom 31360 The base set of a measure ...
measfrge0 31361 A measure is a function ov...
measfn 31362 A measure is a function on...
measvxrge0 31363 The values of a measure ar...
measvnul 31364 The measure of the empty s...
measge0 31365 A measure is nonnegative. ...
measle0 31366 If the measure of a given ...
measvun 31367 The measure of a countable...
measxun2 31368 The measure the union of t...
measun 31369 The measure the union of t...
measvunilem 31370 Lemma for ~ measvuni . (C...
measvunilem0 31371 Lemma for ~ measvuni . (C...
measvuni 31372 The measure of a countable...
measssd 31373 A measure is monotone with...
measunl 31374 A measure is sub-additive ...
measiuns 31375 The measure of the union o...
measiun 31376 A measure is sub-additive....
meascnbl 31377 A measure is continuous fr...
measinblem 31378 Lemma for ~ measinb . (Co...
measinb 31379 Building a measure restric...
measres 31380 Building a measure restric...
measinb2 31381 Building a measure restric...
measdivcst 31382 Division of a measure by a...
measdivcstALTV 31383 Alternate version of ~ mea...
cntmeas 31384 The Counting measure is a ...
pwcntmeas 31385 The counting measure is a ...
cntnevol 31386 Counting and Lebesgue meas...
voliune 31387 The Lebesgue measure funct...
volfiniune 31388 The Lebesgue measure funct...
volmeas 31389 The Lebesgue measure is a ...
ddeval1 31392 Value of the delta measure...
ddeval0 31393 Value of the delta measure...
ddemeas 31394 The Dirac delta measure is...
relae 31398 'almost everywhere' is a r...
brae 31399 'almost everywhere' relati...
braew 31400 'almost everywhere' relati...
truae 31401 A truth holds almost every...
aean 31402 A conjunction holds almost...
faeval 31404 Value of the 'almost every...
relfae 31405 The 'almost everywhere' bu...
brfae 31406 'almost everywhere' relati...
ismbfm 31409 The predicate " ` F ` is a...
elunirnmbfm 31410 The property of being a me...
mbfmfun 31411 A measurable function is a...
mbfmf 31412 A measurable function as a...
isanmbfm 31413 The predicate to be a meas...
mbfmcnvima 31414 The preimage by a measurab...
mbfmbfm 31415 A measurable function to a...
mbfmcst 31416 A constant function is mea...
1stmbfm 31417 The first projection map i...
2ndmbfm 31418 The second projection map ...
imambfm 31419 If the sigma-algebra in th...
cnmbfm 31420 A continuous function is m...
mbfmco 31421 The composition of two mea...
mbfmco2 31422 The pair building of two m...
mbfmvolf 31423 Measurable functions with ...
elmbfmvol2 31424 Measurable functions with ...
mbfmcnt 31425 All functions are measurab...
br2base 31426 The base set for the gener...
dya2ub 31427 An upper bound for a dyadi...
sxbrsigalem0 31428 The closed half-spaces of ...
sxbrsigalem3 31429 The sigma-algebra generate...
dya2iocival 31430 The function ` I ` returns...
dya2iocress 31431 Dyadic intervals are subse...
dya2iocbrsiga 31432 Dyadic intervals are Borel...
dya2icobrsiga 31433 Dyadic intervals are Borel...
dya2icoseg 31434 For any point and any clos...
dya2icoseg2 31435 For any point and any open...
dya2iocrfn 31436 The function returning dya...
dya2iocct 31437 The dyadic rectangle set i...
dya2iocnrect 31438 For any point of an open r...
dya2iocnei 31439 For any point of an open s...
dya2iocuni 31440 Every open set of ` ( RR X...
dya2iocucvr 31441 The dyadic rectangular set...
sxbrsigalem1 31442 The Borel algebra on ` ( R...
sxbrsigalem2 31443 The sigma-algebra generate...
sxbrsigalem4 31444 The Borel algebra on ` ( R...
sxbrsigalem5 31445 First direction for ~ sxbr...
sxbrsigalem6 31446 First direction for ~ sxbr...
sxbrsiga 31447 The product sigma-algebra ...
omsval 31450 Value of the function mapp...
omsfval 31451 Value of the outer measure...
omscl 31452 A closure lemma for the co...
omsf 31453 A constructed outer measur...
oms0 31454 A constructed outer measur...
omsmon 31455 A constructed outer measur...
omssubaddlem 31456 For any small margin ` E `...
omssubadd 31457 A constructed outer measur...
carsgval 31460 Value of the Caratheodory ...
carsgcl 31461 Closure of the Caratheodor...
elcarsg 31462 Property of being a Carath...
baselcarsg 31463 The universe set, ` O ` , ...
0elcarsg 31464 The empty set is Caratheod...
carsguni 31465 The union of all Caratheod...
elcarsgss 31466 Caratheodory measurable se...
difelcarsg 31467 The Caratheodory measurabl...
inelcarsg 31468 The Caratheodory measurabl...
unelcarsg 31469 The Caratheodory-measurabl...
difelcarsg2 31470 The Caratheodory-measurabl...
carsgmon 31471 Utility lemma: Apply mono...
carsgsigalem 31472 Lemma for the following th...
fiunelcarsg 31473 The Caratheodory measurabl...
carsgclctunlem1 31474 Lemma for ~ carsgclctun . ...
carsggect 31475 The outer measure is count...
carsgclctunlem2 31476 Lemma for ~ carsgclctun . ...
carsgclctunlem3 31477 Lemma for ~ carsgclctun . ...
carsgclctun 31478 The Caratheodory measurabl...
carsgsiga 31479 The Caratheodory measurabl...
omsmeas 31480 The restriction of a const...
pmeasmono 31481 This theorem's hypotheses ...
pmeasadd 31482 A premeasure on a ring of ...
itgeq12dv 31483 Equality theorem for an in...
sitgval 31489 Value of the simple functi...
issibf 31490 The predicate " ` F ` is a...
sibf0 31491 The constant zero function...
sibfmbl 31492 A simple function is measu...
sibff 31493 A simple function is a fun...
sibfrn 31494 A simple function has fini...
sibfima 31495 Any preimage of a singleto...
sibfinima 31496 The measure of the interse...
sibfof 31497 Applying function operatio...
sitgfval 31498 Value of the Bochner integ...
sitgclg 31499 Closure of the Bochner int...
sitgclbn 31500 Closure of the Bochner int...
sitgclcn 31501 Closure of the Bochner int...
sitgclre 31502 Closure of the Bochner int...
sitg0 31503 The integral of the consta...
sitgf 31504 The integral for simple fu...
sitgaddlemb 31505 Lemma for * sitgadd . (Co...
sitmval 31506 Value of the simple functi...
sitmfval 31507 Value of the integral dist...
sitmcl 31508 Closure of the integral di...
sitmf 31509 The integral metric as a f...
oddpwdc 31511 Lemma for ~ eulerpart . T...
oddpwdcv 31512 Lemma for ~ eulerpart : va...
eulerpartlemsv1 31513 Lemma for ~ eulerpart . V...
eulerpartlemelr 31514 Lemma for ~ eulerpart . (...
eulerpartlemsv2 31515 Lemma for ~ eulerpart . V...
eulerpartlemsf 31516 Lemma for ~ eulerpart . (...
eulerpartlems 31517 Lemma for ~ eulerpart . (...
eulerpartlemsv3 31518 Lemma for ~ eulerpart . V...
eulerpartlemgc 31519 Lemma for ~ eulerpart . (...
eulerpartleme 31520 Lemma for ~ eulerpart . (...
eulerpartlemv 31521 Lemma for ~ eulerpart . (...
eulerpartlemo 31522 Lemma for ~ eulerpart : ` ...
eulerpartlemd 31523 Lemma for ~ eulerpart : ` ...
eulerpartlem1 31524 Lemma for ~ eulerpart . (...
eulerpartlemb 31525 Lemma for ~ eulerpart . T...
eulerpartlemt0 31526 Lemma for ~ eulerpart . (...
eulerpartlemf 31527 Lemma for ~ eulerpart : O...
eulerpartlemt 31528 Lemma for ~ eulerpart . (...
eulerpartgbij 31529 Lemma for ~ eulerpart : T...
eulerpartlemgv 31530 Lemma for ~ eulerpart : va...
eulerpartlemr 31531 Lemma for ~ eulerpart . (...
eulerpartlemmf 31532 Lemma for ~ eulerpart . (...
eulerpartlemgvv 31533 Lemma for ~ eulerpart : va...
eulerpartlemgu 31534 Lemma for ~ eulerpart : R...
eulerpartlemgh 31535 Lemma for ~ eulerpart : T...
eulerpartlemgf 31536 Lemma for ~ eulerpart : I...
eulerpartlemgs2 31537 Lemma for ~ eulerpart : T...
eulerpartlemn 31538 Lemma for ~ eulerpart . (...
eulerpart 31539 Euler's theorem on partiti...
subiwrd 31542 Lemma for ~ sseqp1 . (Con...
subiwrdlen 31543 Length of a subword of an ...
iwrdsplit 31544 Lemma for ~ sseqp1 . (Con...
sseqval 31545 Value of the strong sequen...
sseqfv1 31546 Value of the strong sequen...
sseqfn 31547 A strong recursive sequenc...
sseqmw 31548 Lemma for ~ sseqf amd ~ ss...
sseqf 31549 A strong recursive sequenc...
sseqfres 31550 The first elements in the ...
sseqfv2 31551 Value of the strong sequen...
sseqp1 31552 Value of the strong sequen...
fiblem 31555 Lemma for ~ fib0 , ~ fib1 ...
fib0 31556 Value of the Fibonacci seq...
fib1 31557 Value of the Fibonacci seq...
fibp1 31558 Value of the Fibonacci seq...
fib2 31559 Value of the Fibonacci seq...
fib3 31560 Value of the Fibonacci seq...
fib4 31561 Value of the Fibonacci seq...
fib5 31562 Value of the Fibonacci seq...
fib6 31563 Value of the Fibonacci seq...
elprob 31566 The property of being a pr...
domprobmeas 31567 A probability measure is a...
domprobsiga 31568 The domain of a probabilit...
probtot 31569 The probability of the uni...
prob01 31570 A probability is an elemen...
probnul 31571 The probability of the emp...
unveldomd 31572 The universe is an element...
unveldom 31573 The universe is an element...
nuleldmp 31574 The empty set is an elemen...
probcun 31575 The probability of the uni...
probun 31576 The probability of the uni...
probdif 31577 The probability of the dif...
probinc 31578 A probability law is incre...
probdsb 31579 The probability of the com...
probmeasd 31580 A probability measure is a...
probvalrnd 31581 The value of a probability...
probtotrnd 31582 The probability of the uni...
totprobd 31583 Law of total probability, ...
totprob 31584 Law of total probability. ...
probfinmeasb 31585 Build a probability measur...
probfinmeasbALTV 31586 Alternate version of ~ pro...
probmeasb 31587 Build a probability from a...
cndprobval 31590 The value of the condition...
cndprobin 31591 An identity linking condit...
cndprob01 31592 The conditional probabilit...
cndprobtot 31593 The conditional probabilit...
cndprobnul 31594 The conditional probabilit...
cndprobprob 31595 The conditional probabilit...
bayesth 31596 Bayes Theorem. (Contribut...
rrvmbfm 31599 A real-valued random varia...
isrrvv 31600 Elementhood to the set of ...
rrvvf 31601 A real-valued random varia...
rrvfn 31602 A real-valued random varia...
rrvdm 31603 The domain of a random var...
rrvrnss 31604 The range of a random vari...
rrvf2 31605 A real-valued random varia...
rrvdmss 31606 The domain of a random var...
rrvfinvima 31607 For a real-value random va...
0rrv 31608 The constant function equa...
rrvadd 31609 The sum of two random vari...
rrvmulc 31610 A random variable multipli...
rrvsum 31611 An indexed sum of random v...
orvcval 31614 Value of the preimage mapp...
orvcval2 31615 Another way to express the...
elorvc 31616 Elementhood of a preimage....
orvcval4 31617 The value of the preimage ...
orvcoel 31618 If the relation produces o...
orvccel 31619 If the relation produces c...
elorrvc 31620 Elementhood of a preimage ...
orrvcval4 31621 The value of the preimage ...
orrvcoel 31622 If the relation produces o...
orrvccel 31623 If the relation produces c...
orvcgteel 31624 Preimage maps produced by ...
orvcelval 31625 Preimage maps produced by ...
orvcelel 31626 Preimage maps produced by ...
dstrvval 31627 The value of the distribut...
dstrvprob 31628 The distribution of a rand...
orvclteel 31629 Preimage maps produced by ...
dstfrvel 31630 Elementhood of preimage ma...
dstfrvunirn 31631 The limit of all preimage ...
orvclteinc 31632 Preimage maps produced by ...
dstfrvinc 31633 A cumulative distribution ...
dstfrvclim1 31634 The limit of the cumulativ...
coinfliplem 31635 Division in the extended r...
coinflipprob 31636 The ` P ` we defined for c...
coinflipspace 31637 The space of our coin-flip...
coinflipuniv 31638 The universe of our coin-f...
coinfliprv 31639 The ` X ` we defined for c...
coinflippv 31640 The probability of heads i...
coinflippvt 31641 The probability of tails i...
ballotlemoex 31642 ` O ` is a set. (Contribu...
ballotlem1 31643 The size of the universe i...
ballotlemelo 31644 Elementhood in ` O ` . (C...
ballotlem2 31645 The probability that the f...
ballotlemfval 31646 The value of F. (Contribut...
ballotlemfelz 31647 ` ( F `` C ) ` has values ...
ballotlemfp1 31648 If the ` J ` th ballot is ...
ballotlemfc0 31649 ` F ` takes value 0 betwee...
ballotlemfcc 31650 ` F ` takes value 0 betwee...
ballotlemfmpn 31651 ` ( F `` C ) ` finishes co...
ballotlemfval0 31652 ` ( F `` C ) ` always star...
ballotleme 31653 Elements of ` E ` . (Cont...
ballotlemodife 31654 Elements of ` ( O \ E ) ` ...
ballotlem4 31655 If the first pick is a vot...
ballotlem5 31656 If A is not ahead througho...
ballotlemi 31657 Value of ` I ` for a given...
ballotlemiex 31658 Properties of ` ( I `` C )...
ballotlemi1 31659 The first tie cannot be re...
ballotlemii 31660 The first tie cannot be re...
ballotlemsup 31661 The set of zeroes of ` F `...
ballotlemimin 31662 ` ( I `` C ) ` is the firs...
ballotlemic 31663 If the first vote is for B...
ballotlem1c 31664 If the first vote is for A...
ballotlemsval 31665 Value of ` S ` . (Contrib...
ballotlemsv 31666 Value of ` S ` evaluated a...
ballotlemsgt1 31667 ` S ` maps values less tha...
ballotlemsdom 31668 Domain of ` S ` for a give...
ballotlemsel1i 31669 The range ` ( 1 ... ( I ``...
ballotlemsf1o 31670 The defined ` S ` is a bij...
ballotlemsi 31671 The image by ` S ` of the ...
ballotlemsima 31672 The image by ` S ` of an i...
ballotlemieq 31673 If two countings share the...
ballotlemrval 31674 Value of ` R ` . (Contrib...
ballotlemscr 31675 The image of ` ( R `` C ) ...
ballotlemrv 31676 Value of ` R ` evaluated a...
ballotlemrv1 31677 Value of ` R ` before the ...
ballotlemrv2 31678 Value of ` R ` after the t...
ballotlemro 31679 Range of ` R ` is included...
ballotlemgval 31680 Expand the value of ` .^ `...
ballotlemgun 31681 A property of the defined ...
ballotlemfg 31682 Express the value of ` ( F...
ballotlemfrc 31683 Express the value of ` ( F...
ballotlemfrci 31684 Reverse counting preserves...
ballotlemfrceq 31685 Value of ` F ` for a rever...
ballotlemfrcn0 31686 Value of ` F ` for a rever...
ballotlemrc 31687 Range of ` R ` . (Contrib...
ballotlemirc 31688 Applying ` R ` does not ch...
ballotlemrinv0 31689 Lemma for ~ ballotlemrinv ...
ballotlemrinv 31690 ` R ` is its own inverse :...
ballotlem1ri 31691 When the vote on the first...
ballotlem7 31692 ` R ` is a bijection betwe...
ballotlem8 31693 There are as many counting...
ballotth 31694 Bertrand's ballot problem ...
sgncl 31695 Closure of the signum. (C...
sgnclre 31696 Closure of the signum. (C...
sgnneg 31697 Negation of the signum. (...
sgn3da 31698 A conditional containing a...
sgnmul 31699 Signum of a product. (Con...
sgnmulrp2 31700 Multiplication by a positi...
sgnsub 31701 Subtraction of a number of...
sgnnbi 31702 Negative signum. (Contrib...
sgnpbi 31703 Positive signum. (Contrib...
sgn0bi 31704 Zero signum. (Contributed...
sgnsgn 31705 Signum is idempotent. (Co...
sgnmulsgn 31706 If two real numbers are of...
sgnmulsgp 31707 If two real numbers are of...
fzssfzo 31708 Condition for an integer i...
gsumncl 31709 Closure of a group sum in ...
gsumnunsn 31710 Closure of a group sum in ...
ccatmulgnn0dir 31711 Concatenation of words fol...
ofcccat 31712 Letterwise operations on w...
ofcs1 31713 Letterwise operations on a...
ofcs2 31714 Letterwise operations on a...
plymul02 31715 Product of a polynomial wi...
plymulx0 31716 Coefficients of a polynomi...
plymulx 31717 Coefficients of a polynomi...
plyrecld 31718 Closure of a polynomial wi...
signsplypnf 31719 The quotient of a polynomi...
signsply0 31720 Lemma for the rule of sign...
signspval 31721 The value of the skipping ...
signsw0glem 31722 Neutral element property o...
signswbase 31723 The base of ` W ` is the t...
signswplusg 31724 The operation of ` W ` . ...
signsw0g 31725 The neutral element of ` W...
signswmnd 31726 ` W ` is a monoid structur...
signswrid 31727 The zero-skipping operatio...
signswlid 31728 The zero-skipping operatio...
signswn0 31729 The zero-skipping operatio...
signswch 31730 The zero-skipping operatio...
signslema 31731 Computational part of sign...
signstfv 31732 Value of the zero-skipping...
signstfval 31733 Value of the zero-skipping...
signstcl 31734 Closure of the zero skippi...
signstf 31735 The zero skipping sign wor...
signstlen 31736 Length of the zero skippin...
signstf0 31737 Sign of a single letter wo...
signstfvn 31738 Zero-skipping sign in a wo...
signsvtn0 31739 If the last letter is nonz...
signstfvp 31740 Zero-skipping sign in a wo...
signstfvneq0 31741 In case the first letter i...
signstfvcl 31742 Closure of the zero skippi...
signstfvc 31743 Zero-skipping sign in a wo...
signstres 31744 Restriction of a zero skip...
signstfveq0a 31745 Lemma for ~ signstfveq0 . ...
signstfveq0 31746 In case the last letter is...
signsvvfval 31747 The value of ` V ` , which...
signsvvf 31748 ` V ` is a function. (Con...
signsvf0 31749 There is no change of sign...
signsvf1 31750 In a single-letter word, w...
signsvfn 31751 Number of changes in a wor...
signsvtp 31752 Adding a letter of the sam...
signsvtn 31753 Adding a letter of a diffe...
signsvfpn 31754 Adding a letter of the sam...
signsvfnn 31755 Adding a letter of a diffe...
signlem0 31756 Adding a zero as the highe...
signshf 31757 ` H ` , corresponding to t...
signshwrd 31758 ` H ` , corresponding to t...
signshlen 31759 Length of ` H ` , correspo...
signshnz 31760 ` H ` is not the empty wor...
efcld 31761 Closure law for the expone...
iblidicc 31762 The identity function is i...
rpsqrtcn 31763 Continuity of the real pos...
divsqrtid 31764 A real number divided by i...
cxpcncf1 31765 The power function on comp...
efmul2picn 31766 Multiplying by ` ( _i x. (...
fct2relem 31767 Lemma for ~ ftc2re . (Con...
ftc2re 31768 The Fundamental Theorem of...
fdvposlt 31769 Functions with a positive ...
fdvneggt 31770 Functions with a negative ...
fdvposle 31771 Functions with a nonnegati...
fdvnegge 31772 Functions with a nonpositi...
prodfzo03 31773 A product of three factors...
actfunsnf1o 31774 The action ` F ` of extend...
actfunsnrndisj 31775 The action ` F ` of extend...
itgexpif 31776 The basis for the circle m...
fsum2dsub 31777 Lemma for ~ breprexp - Re-...
reprval 31780 Value of the representatio...
repr0 31781 There is exactly one repre...
reprf 31782 Members of the representat...
reprsum 31783 Sums of values of the memb...
reprle 31784 Upper bound to the terms i...
reprsuc 31785 Express the representation...
reprfi 31786 Bounded representations ar...
reprss 31787 Representations with terms...
reprinrn 31788 Representations with term ...
reprlt 31789 There are no representatio...
hashreprin 31790 Express a sum of represent...
reprgt 31791 There are no representatio...
reprinfz1 31792 For the representation of ...
reprfi2 31793 Corollary of ~ reprinfz1 ....
reprfz1 31794 Corollary of ~ reprinfz1 ....
hashrepr 31795 Develop the number of repr...
reprpmtf1o 31796 Transposing ` 0 ` and ` X ...
reprdifc 31797 Express the representation...
chpvalz 31798 Value of the second Chebys...
chtvalz 31799 Value of the Chebyshev fun...
breprexplema 31800 Lemma for ~ breprexp (indu...
breprexplemb 31801 Lemma for ~ breprexp (clos...
breprexplemc 31802 Lemma for ~ breprexp (indu...
breprexp 31803 Express the ` S ` th power...
breprexpnat 31804 Express the ` S ` th power...
vtsval 31807 Value of the Vinogradov tr...
vtscl 31808 Closure of the Vinogradov ...
vtsprod 31809 Express the Vinogradov tri...
circlemeth 31810 The Hardy, Littlewood and ...
circlemethnat 31811 The Hardy, Littlewood and ...
circlevma 31812 The Circle Method, where t...
circlemethhgt 31813 The circle method, where t...
hgt750lemc 31817 An upper bound to the summ...
hgt750lemd 31818 An upper bound to the summ...
hgt749d 31819 A deduction version of ~ a...
logdivsqrle 31820 Conditions for ` ( ( log `...
hgt750lem 31821 Lemma for ~ tgoldbachgtd ....
hgt750lem2 31822 Decimal multiplication gal...
hgt750lemf 31823 Lemma for the statement 7....
hgt750lemg 31824 Lemma for the statement 7....
oddprm2 31825 Two ways to write the set ...
hgt750lemb 31826 An upper bound on the cont...
hgt750lema 31827 An upper bound on the cont...
hgt750leme 31828 An upper bound on the cont...
tgoldbachgnn 31829 Lemma for ~ tgoldbachgtd ....
tgoldbachgtde 31830 Lemma for ~ tgoldbachgtd ....
tgoldbachgtda 31831 Lemma for ~ tgoldbachgtd ....
tgoldbachgtd 31832 Odd integers greater than ...
tgoldbachgt 31833 Odd integers greater than ...
istrkg2d 31836 Property of fulfilling dim...
axtglowdim2ALTV 31837 Alternate version of ~ axt...
axtgupdim2ALTV 31838 Alternate version of ~ axt...
afsval 31841 Value of the AFS relation ...
brafs 31842 Binary relation form of th...
tg5segofs 31843 Rephrase ~ axtg5seg using ...
lpadval 31846 Value of the ` leftpad ` f...
lpadlem1 31847 Lemma for the ` leftpad ` ...
lpadlem3 31848 Lemma for ~ lpadlen1 (Cont...
lpadlen1 31849 Length of a left-padded wo...
lpadlem2 31850 Lemma for the ` leftpad ` ...
lpadlen2 31851 Length of a left-padded wo...
lpadmax 31852 Length of a left-padded wo...
lpadleft 31853 The contents of prefix of ...
lpadright 31854 The suffix of a left-padde...
bnj170 31867 ` /\ ` -manipulation. (Co...
bnj240 31868 ` /\ ` -manipulation. (Co...
bnj248 31869 ` /\ ` -manipulation. (Co...
bnj250 31870 ` /\ ` -manipulation. (Co...
bnj251 31871 ` /\ ` -manipulation. (Co...
bnj252 31872 ` /\ ` -manipulation. (Co...
bnj253 31873 ` /\ ` -manipulation. (Co...
bnj255 31874 ` /\ ` -manipulation. (Co...
bnj256 31875 ` /\ ` -manipulation. (Co...
bnj257 31876 ` /\ ` -manipulation. (Co...
bnj258 31877 ` /\ ` -manipulation. (Co...
bnj268 31878 ` /\ ` -manipulation. (Co...
bnj290 31879 ` /\ ` -manipulation. (Co...
bnj291 31880 ` /\ ` -manipulation. (Co...
bnj312 31881 ` /\ ` -manipulation. (Co...
bnj334 31882 ` /\ ` -manipulation. (Co...
bnj345 31883 ` /\ ` -manipulation. (Co...
bnj422 31884 ` /\ ` -manipulation. (Co...
bnj432 31885 ` /\ ` -manipulation. (Co...
bnj446 31886 ` /\ ` -manipulation. (Co...
bnj23 31887 First-order logic and set ...
bnj31 31888 First-order logic and set ...
bnj62 31889 First-order logic and set ...
bnj89 31890 First-order logic and set ...
bnj90 31891 First-order logic and set ...
bnj101 31892 First-order logic and set ...
bnj105 31893 First-order logic and set ...
bnj115 31894 First-order logic and set ...
bnj132 31895 First-order logic and set ...
bnj133 31896 First-order logic and set ...
bnj156 31897 First-order logic and set ...
bnj158 31898 First-order logic and set ...
bnj168 31899 First-order logic and set ...
bnj206 31900 First-order logic and set ...
bnj216 31901 First-order logic and set ...
bnj219 31902 First-order logic and set ...
bnj226 31903 First-order logic and set ...
bnj228 31904 First-order logic and set ...
bnj519 31905 First-order logic and set ...
bnj521 31906 First-order logic and set ...
bnj524 31907 First-order logic and set ...
bnj525 31908 First-order logic and set ...
bnj534 31909 First-order logic and set ...
bnj538 31910 First-order logic and set ...
bnj529 31911 First-order logic and set ...
bnj551 31912 First-order logic and set ...
bnj563 31913 First-order logic and set ...
bnj564 31914 First-order logic and set ...
bnj593 31915 First-order logic and set ...
bnj596 31916 First-order logic and set ...
bnj610 31917 Pass from equality ( ` x =...
bnj642 31918 ` /\ ` -manipulation. (Co...
bnj643 31919 ` /\ ` -manipulation. (Co...
bnj645 31920 ` /\ ` -manipulation. (Co...
bnj658 31921 ` /\ ` -manipulation. (Co...
bnj667 31922 ` /\ ` -manipulation. (Co...
bnj705 31923 ` /\ ` -manipulation. (Co...
bnj706 31924 ` /\ ` -manipulation. (Co...
bnj707 31925 ` /\ ` -manipulation. (Co...
bnj708 31926 ` /\ ` -manipulation. (Co...
bnj721 31927 ` /\ ` -manipulation. (Co...
bnj832 31928 ` /\ ` -manipulation. (Co...
bnj835 31929 ` /\ ` -manipulation. (Co...
bnj836 31930 ` /\ ` -manipulation. (Co...
bnj837 31931 ` /\ ` -manipulation. (Co...
bnj769 31932 ` /\ ` -manipulation. (Co...
bnj770 31933 ` /\ ` -manipulation. (Co...
bnj771 31934 ` /\ ` -manipulation. (Co...
bnj887 31935 ` /\ ` -manipulation. (Co...
bnj918 31936 First-order logic and set ...
bnj919 31937 First-order logic and set ...
bnj923 31938 First-order logic and set ...
bnj927 31939 First-order logic and set ...
bnj930 31940 First-order logic and set ...
bnj931 31941 First-order logic and set ...
bnj937 31942 First-order logic and set ...
bnj941 31943 First-order logic and set ...
bnj945 31944 Technical lemma for ~ bnj6...
bnj946 31945 First-order logic and set ...
bnj951 31946 ` /\ ` -manipulation. (Co...
bnj956 31947 First-order logic and set ...
bnj976 31948 First-order logic and set ...
bnj982 31949 First-order logic and set ...
bnj1019 31950 First-order logic and set ...
bnj1023 31951 First-order logic and set ...
bnj1095 31952 First-order logic and set ...
bnj1096 31953 First-order logic and set ...
bnj1098 31954 First-order logic and set ...
bnj1101 31955 First-order logic and set ...
bnj1113 31956 First-order logic and set ...
bnj1109 31957 First-order logic and set ...
bnj1131 31958 First-order logic and set ...
bnj1138 31959 First-order logic and set ...
bnj1142 31960 First-order logic and set ...
bnj1143 31961 First-order logic and set ...
bnj1146 31962 First-order logic and set ...
bnj1149 31963 First-order logic and set ...
bnj1185 31964 First-order logic and set ...
bnj1196 31965 First-order logic and set ...
bnj1198 31966 First-order logic and set ...
bnj1209 31967 First-order logic and set ...
bnj1211 31968 First-order logic and set ...
bnj1213 31969 First-order logic and set ...
bnj1212 31970 First-order logic and set ...
bnj1219 31971 First-order logic and set ...
bnj1224 31972 First-order logic and set ...
bnj1230 31973 First-order logic and set ...
bnj1232 31974 First-order logic and set ...
bnj1235 31975 First-order logic and set ...
bnj1239 31976 First-order logic and set ...
bnj1238 31977 First-order logic and set ...
bnj1241 31978 First-order logic and set ...
bnj1247 31979 First-order logic and set ...
bnj1254 31980 First-order logic and set ...
bnj1262 31981 First-order logic and set ...
bnj1266 31982 First-order logic and set ...
bnj1265 31983 First-order logic and set ...
bnj1275 31984 First-order logic and set ...
bnj1276 31985 First-order logic and set ...
bnj1292 31986 First-order logic and set ...
bnj1293 31987 First-order logic and set ...
bnj1294 31988 First-order logic and set ...
bnj1299 31989 First-order logic and set ...
bnj1304 31990 First-order logic and set ...
bnj1316 31991 First-order logic and set ...
bnj1317 31992 First-order logic and set ...
bnj1322 31993 First-order logic and set ...
bnj1340 31994 First-order logic and set ...
bnj1345 31995 First-order logic and set ...
bnj1350 31996 First-order logic and set ...
bnj1351 31997 First-order logic and set ...
bnj1352 31998 First-order logic and set ...
bnj1361 31999 First-order logic and set ...
bnj1366 32000 First-order logic and set ...
bnj1379 32001 First-order logic and set ...
bnj1383 32002 First-order logic and set ...
bnj1385 32003 First-order logic and set ...
bnj1386 32004 First-order logic and set ...
bnj1397 32005 First-order logic and set ...
bnj1400 32006 First-order logic and set ...
bnj1405 32007 First-order logic and set ...
bnj1422 32008 First-order logic and set ...
bnj1424 32009 First-order logic and set ...
bnj1436 32010 First-order logic and set ...
bnj1441 32011 First-order logic and set ...
bnj1441g 32012 First-order logic and set ...
bnj1454 32013 First-order logic and set ...
bnj1459 32014 First-order logic and set ...
bnj1464 32015 Conversion of implicit sub...
bnj1465 32016 First-order logic and set ...
bnj1468 32017 Conversion of implicit sub...
bnj1476 32018 First-order logic and set ...
bnj1502 32019 First-order logic and set ...
bnj1503 32020 First-order logic and set ...
bnj1517 32021 First-order logic and set ...
bnj1521 32022 First-order logic and set ...
bnj1533 32023 First-order logic and set ...
bnj1534 32024 First-order logic and set ...
bnj1536 32025 First-order logic and set ...
bnj1538 32026 First-order logic and set ...
bnj1541 32027 First-order logic and set ...
bnj1542 32028 First-order logic and set ...
bnj110 32029 Well-founded induction res...
bnj157 32030 Well-founded induction res...
bnj66 32031 Technical lemma for ~ bnj6...
bnj91 32032 First-order logic and set ...
bnj92 32033 First-order logic and set ...
bnj93 32034 Technical lemma for ~ bnj9...
bnj95 32035 Technical lemma for ~ bnj1...
bnj96 32036 Technical lemma for ~ bnj1...
bnj97 32037 Technical lemma for ~ bnj1...
bnj98 32038 Technical lemma for ~ bnj1...
bnj106 32039 First-order logic and set ...
bnj118 32040 First-order logic and set ...
bnj121 32041 First-order logic and set ...
bnj124 32042 Technical lemma for ~ bnj1...
bnj125 32043 Technical lemma for ~ bnj1...
bnj126 32044 Technical lemma for ~ bnj1...
bnj130 32045 Technical lemma for ~ bnj1...
bnj149 32046 Technical lemma for ~ bnj1...
bnj150 32047 Technical lemma for ~ bnj1...
bnj151 32048 Technical lemma for ~ bnj1...
bnj154 32049 Technical lemma for ~ bnj1...
bnj155 32050 Technical lemma for ~ bnj1...
bnj153 32051 Technical lemma for ~ bnj8...
bnj207 32052 Technical lemma for ~ bnj8...
bnj213 32053 First-order logic and set ...
bnj222 32054 Technical lemma for ~ bnj2...
bnj229 32055 Technical lemma for ~ bnj5...
bnj517 32056 Technical lemma for ~ bnj5...
bnj518 32057 Technical lemma for ~ bnj8...
bnj523 32058 Technical lemma for ~ bnj8...
bnj526 32059 Technical lemma for ~ bnj8...
bnj528 32060 Technical lemma for ~ bnj8...
bnj535 32061 Technical lemma for ~ bnj8...
bnj539 32062 Technical lemma for ~ bnj8...
bnj540 32063 Technical lemma for ~ bnj8...
bnj543 32064 Technical lemma for ~ bnj8...
bnj544 32065 Technical lemma for ~ bnj8...
bnj545 32066 Technical lemma for ~ bnj8...
bnj546 32067 Technical lemma for ~ bnj8...
bnj548 32068 Technical lemma for ~ bnj8...
bnj553 32069 Technical lemma for ~ bnj8...
bnj554 32070 Technical lemma for ~ bnj8...
bnj556 32071 Technical lemma for ~ bnj8...
bnj557 32072 Technical lemma for ~ bnj8...
bnj558 32073 Technical lemma for ~ bnj8...
bnj561 32074 Technical lemma for ~ bnj8...
bnj562 32075 Technical lemma for ~ bnj8...
bnj570 32076 Technical lemma for ~ bnj8...
bnj571 32077 Technical lemma for ~ bnj8...
bnj605 32078 Technical lemma. This lem...
bnj581 32079 Technical lemma for ~ bnj5...
bnj589 32080 Technical lemma for ~ bnj8...
bnj590 32081 Technical lemma for ~ bnj8...
bnj591 32082 Technical lemma for ~ bnj8...
bnj594 32083 Technical lemma for ~ bnj8...
bnj580 32084 Technical lemma for ~ bnj5...
bnj579 32085 Technical lemma for ~ bnj8...
bnj602 32086 Equality theorem for the `...
bnj607 32087 Technical lemma for ~ bnj8...
bnj609 32088 Technical lemma for ~ bnj8...
bnj611 32089 Technical lemma for ~ bnj8...
bnj600 32090 Technical lemma for ~ bnj8...
bnj601 32091 Technical lemma for ~ bnj8...
bnj852 32092 Technical lemma for ~ bnj6...
bnj864 32093 Technical lemma for ~ bnj6...
bnj865 32094 Technical lemma for ~ bnj6...
bnj873 32095 Technical lemma for ~ bnj6...
bnj849 32096 Technical lemma for ~ bnj6...
bnj882 32097 Definition (using hypothes...
bnj18eq1 32098 Equality theorem for trans...
bnj893 32099 Property of ` _trCl ` . U...
bnj900 32100 Technical lemma for ~ bnj6...
bnj906 32101 Property of ` _trCl ` . (...
bnj908 32102 Technical lemma for ~ bnj6...
bnj911 32103 Technical lemma for ~ bnj6...
bnj916 32104 Technical lemma for ~ bnj6...
bnj917 32105 Technical lemma for ~ bnj6...
bnj934 32106 Technical lemma for ~ bnj6...
bnj929 32107 Technical lemma for ~ bnj6...
bnj938 32108 Technical lemma for ~ bnj6...
bnj944 32109 Technical lemma for ~ bnj6...
bnj953 32110 Technical lemma for ~ bnj6...
bnj958 32111 Technical lemma for ~ bnj6...
bnj1000 32112 Technical lemma for ~ bnj8...
bnj965 32113 Technical lemma for ~ bnj8...
bnj964 32114 Technical lemma for ~ bnj6...
bnj966 32115 Technical lemma for ~ bnj6...
bnj967 32116 Technical lemma for ~ bnj6...
bnj969 32117 Technical lemma for ~ bnj6...
bnj970 32118 Technical lemma for ~ bnj6...
bnj910 32119 Technical lemma for ~ bnj6...
bnj978 32120 Technical lemma for ~ bnj6...
bnj981 32121 Technical lemma for ~ bnj6...
bnj983 32122 Technical lemma for ~ bnj6...
bnj984 32123 Technical lemma for ~ bnj6...
bnj985 32124 Technical lemma for ~ bnj6...
bnj986 32125 Technical lemma for ~ bnj6...
bnj996 32126 Technical lemma for ~ bnj6...
bnj998 32127 Technical lemma for ~ bnj6...
bnj999 32128 Technical lemma for ~ bnj6...
bnj1001 32129 Technical lemma for ~ bnj6...
bnj1006 32130 Technical lemma for ~ bnj6...
bnj1014 32131 Technical lemma for ~ bnj6...
bnj1015 32132 Technical lemma for ~ bnj6...
bnj1018 32133 Technical lemma for ~ bnj6...
bnj1020 32134 Technical lemma for ~ bnj6...
bnj1021 32135 Technical lemma for ~ bnj6...
bnj907 32136 Technical lemma for ~ bnj6...
bnj1029 32137 Property of ` _trCl ` . (...
bnj1033 32138 Technical lemma for ~ bnj6...
bnj1034 32139 Technical lemma for ~ bnj6...
bnj1039 32140 Technical lemma for ~ bnj6...
bnj1040 32141 Technical lemma for ~ bnj6...
bnj1047 32142 Technical lemma for ~ bnj6...
bnj1049 32143 Technical lemma for ~ bnj6...
bnj1052 32144 Technical lemma for ~ bnj6...
bnj1053 32145 Technical lemma for ~ bnj6...
bnj1071 32146 Technical lemma for ~ bnj6...
bnj1083 32147 Technical lemma for ~ bnj6...
bnj1090 32148 Technical lemma for ~ bnj6...
bnj1093 32149 Technical lemma for ~ bnj6...
bnj1097 32150 Technical lemma for ~ bnj6...
bnj1110 32151 Technical lemma for ~ bnj6...
bnj1112 32152 Technical lemma for ~ bnj6...
bnj1118 32153 Technical lemma for ~ bnj6...
bnj1121 32154 Technical lemma for ~ bnj6...
bnj1123 32155 Technical lemma for ~ bnj6...
bnj1030 32156 Technical lemma for ~ bnj6...
bnj1124 32157 Property of ` _trCl ` . (...
bnj1133 32158 Technical lemma for ~ bnj6...
bnj1128 32159 Technical lemma for ~ bnj6...
bnj1127 32160 Property of ` _trCl ` . (...
bnj1125 32161 Property of ` _trCl ` . (...
bnj1145 32162 Technical lemma for ~ bnj6...
bnj1147 32163 Property of ` _trCl ` . (...
bnj1137 32164 Property of ` _trCl ` . (...
bnj1148 32165 Property of ` _pred ` . (...
bnj1136 32166 Technical lemma for ~ bnj6...
bnj1152 32167 Technical lemma for ~ bnj6...
bnj1154 32168 Property of ` Fr ` . (Con...
bnj1171 32169 Technical lemma for ~ bnj6...
bnj1172 32170 Technical lemma for ~ bnj6...
bnj1173 32171 Technical lemma for ~ bnj6...
bnj1174 32172 Technical lemma for ~ bnj6...
bnj1175 32173 Technical lemma for ~ bnj6...
bnj1176 32174 Technical lemma for ~ bnj6...
bnj1177 32175 Technical lemma for ~ bnj6...
bnj1186 32176 Technical lemma for ~ bnj6...
bnj1190 32177 Technical lemma for ~ bnj6...
bnj1189 32178 Technical lemma for ~ bnj6...
bnj69 32179 Existence of a minimal ele...
bnj1228 32180 Existence of a minimal ele...
bnj1204 32181 Well-founded induction. T...
bnj1234 32182 Technical lemma for ~ bnj6...
bnj1245 32183 Technical lemma for ~ bnj6...
bnj1256 32184 Technical lemma for ~ bnj6...
bnj1259 32185 Technical lemma for ~ bnj6...
bnj1253 32186 Technical lemma for ~ bnj6...
bnj1279 32187 Technical lemma for ~ bnj6...
bnj1286 32188 Technical lemma for ~ bnj6...
bnj1280 32189 Technical lemma for ~ bnj6...
bnj1296 32190 Technical lemma for ~ bnj6...
bnj1309 32191 Technical lemma for ~ bnj6...
bnj1307 32192 Technical lemma for ~ bnj6...
bnj1311 32193 Technical lemma for ~ bnj6...
bnj1318 32194 Technical lemma for ~ bnj6...
bnj1326 32195 Technical lemma for ~ bnj6...
bnj1321 32196 Technical lemma for ~ bnj6...
bnj1364 32197 Property of ` _FrSe ` . (...
bnj1371 32198 Technical lemma for ~ bnj6...
bnj1373 32199 Technical lemma for ~ bnj6...
bnj1374 32200 Technical lemma for ~ bnj6...
bnj1384 32201 Technical lemma for ~ bnj6...
bnj1388 32202 Technical lemma for ~ bnj6...
bnj1398 32203 Technical lemma for ~ bnj6...
bnj1413 32204 Property of ` _trCl ` . (...
bnj1408 32205 Technical lemma for ~ bnj1...
bnj1414 32206 Property of ` _trCl ` . (...
bnj1415 32207 Technical lemma for ~ bnj6...
bnj1416 32208 Technical lemma for ~ bnj6...
bnj1418 32209 Property of ` _pred ` . (...
bnj1417 32210 Technical lemma for ~ bnj6...
bnj1421 32211 Technical lemma for ~ bnj6...
bnj1444 32212 Technical lemma for ~ bnj6...
bnj1445 32213 Technical lemma for ~ bnj6...
bnj1446 32214 Technical lemma for ~ bnj6...
bnj1447 32215 Technical lemma for ~ bnj6...
bnj1448 32216 Technical lemma for ~ bnj6...
bnj1449 32217 Technical lemma for ~ bnj6...
bnj1442 32218 Technical lemma for ~ bnj6...
bnj1450 32219 Technical lemma for ~ bnj6...
bnj1423 32220 Technical lemma for ~ bnj6...
bnj1452 32221 Technical lemma for ~ bnj6...
bnj1466 32222 Technical lemma for ~ bnj6...
bnj1467 32223 Technical lemma for ~ bnj6...
bnj1463 32224 Technical lemma for ~ bnj6...
bnj1489 32225 Technical lemma for ~ bnj6...
bnj1491 32226 Technical lemma for ~ bnj6...
bnj1312 32227 Technical lemma for ~ bnj6...
bnj1493 32228 Technical lemma for ~ bnj6...
bnj1497 32229 Technical lemma for ~ bnj6...
bnj1498 32230 Technical lemma for ~ bnj6...
bnj60 32231 Well-founded recursion, pa...
bnj1514 32232 Technical lemma for ~ bnj1...
bnj1518 32233 Technical lemma for ~ bnj1...
bnj1519 32234 Technical lemma for ~ bnj1...
bnj1520 32235 Technical lemma for ~ bnj1...
bnj1501 32236 Technical lemma for ~ bnj1...
bnj1500 32237 Well-founded recursion, pa...
bnj1525 32238 Technical lemma for ~ bnj1...
bnj1529 32239 Technical lemma for ~ bnj1...
bnj1523 32240 Technical lemma for ~ bnj1...
bnj1522 32241 Well-founded recursion, pa...
exdifsn 32242 There exists an element in...
srcmpltd 32243 If a statement is true for...
prsrcmpltd 32244 If a statement is true for...
zltp1ne 32245 Integer ordering relation....
nnltp1ne 32246 Positive integer ordering ...
nn0ltp1ne 32247 Nonnegative integer orderi...
0nn0m1nnn0 32248 A number is zero if and on...
fisshasheq 32249 A finite set is equal to i...
dff15 32250 A one-to-one function in t...
hashfundm 32251 The size of a set function...
hashf1dmrn 32252 The size of the domain of ...
hashf1dmcdm 32253 The size of the domain of ...
funen1cnv 32254 If a function is equinumer...
f1resveqaeq 32255 If a function restricted t...
f1resrcmplf1dlem 32256 Lemma for ~ f1resrcmplf1d ...
f1resrcmplf1d 32257 If a function's restrictio...
f1resfz0f1d 32258 If a function with a seque...
revpfxsfxrev 32259 The reverse of a prefix of...
swrdrevpfx 32260 A subword expressed in ter...
lfuhgr 32261 A hypergraph is loop-free ...
lfuhgr2 32262 A hypergraph is loop-free ...
lfuhgr3 32263 A hypergraph is loop-free ...
cplgredgex 32264 Any two (distinct) vertice...
cusgredgex 32265 Any two (distinct) vertice...
cusgredgex2 32266 Any two distinct vertices ...
pfxwlk 32267 A prefix of a walk is a wa...
revwlk 32268 The reverse of a walk is a...
revwlkb 32269 Two words represent a walk...
swrdwlk 32270 Two matching subwords of a...
pthhashvtx 32271 A graph containing a path ...
pthisspthorcycl 32272 A path is either a simple ...
spthcycl 32273 A walk is a trivial path i...
usgrgt2cycl 32274 A non-trivial cycle in a s...
usgrcyclgt2v 32275 A simple graph with a non-...
subgrwlk 32276 If a walk exists in a subg...
subgrtrl 32277 If a trail exists in a sub...
subgrpth 32278 If a path exists in a subg...
subgrcycl 32279 If a cycle exists in a sub...
cusgr3cyclex 32280 Every complete simple grap...
loop1cycl 32281 A hypergraph has a cycle o...
2cycld 32282 Construction of a 2-cycle ...
2cycl2d 32283 Construction of a 2-cycle ...
umgr2cycllem 32284 Lemma for ~ umgr2cycl . (...
umgr2cycl 32285 A multigraph with two dist...
dfacycgr1 32288 An alternate definition of...
isacycgr 32289 The property of being an a...
isacycgr1 32290 The property of being an a...
acycgrcycl 32291 Any cycle in an acyclic gr...
acycgr0v 32292 A null graph (with no vert...
acycgr1v 32293 A multigraph with one vert...
acycgr2v 32294 A simple graph with two ve...
prclisacycgr 32295 A proper class (representi...
acycgrislfgr 32296 An acyclic hypergraph is a...
upgracycumgr 32297 An acyclic pseudograph is ...
umgracycusgr 32298 An acyclic multigraph is a...
upgracycusgr 32299 An acyclic pseudograph is ...
cusgracyclt3v 32300 A complete simple graph is...
pthacycspth 32301 A path in an acyclic graph...
acycgrsubgr 32302 The subgraph of an acyclic...
quartfull 32309 The quartic equation, writ...
deranglem 32310 Lemma for derangements. (...
derangval 32311 Define the derangement fun...
derangf 32312 The derangement number is ...
derang0 32313 The derangement number of ...
derangsn 32314 The derangement number of ...
derangenlem 32315 One half of ~ derangen . ...
derangen 32316 The derangement number is ...
subfacval 32317 The subfactorial is define...
derangen2 32318 Write the derangement numb...
subfacf 32319 The subfactorial is a func...
subfaclefac 32320 The subfactorial is less t...
subfac0 32321 The subfactorial at zero. ...
subfac1 32322 The subfactorial at one. ...
subfacp1lem1 32323 Lemma for ~ subfacp1 . Th...
subfacp1lem2a 32324 Lemma for ~ subfacp1 . Pr...
subfacp1lem2b 32325 Lemma for ~ subfacp1 . Pr...
subfacp1lem3 32326 Lemma for ~ subfacp1 . In...
subfacp1lem4 32327 Lemma for ~ subfacp1 . Th...
subfacp1lem5 32328 Lemma for ~ subfacp1 . In...
subfacp1lem6 32329 Lemma for ~ subfacp1 . By...
subfacp1 32330 A two-term recurrence for ...
subfacval2 32331 A closed-form expression f...
subfaclim 32332 The subfactorial converges...
subfacval3 32333 Another closed form expres...
derangfmla 32334 The derangements formula, ...
erdszelem1 32335 Lemma for ~ erdsze . (Con...
erdszelem2 32336 Lemma for ~ erdsze . (Con...
erdszelem3 32337 Lemma for ~ erdsze . (Con...
erdszelem4 32338 Lemma for ~ erdsze . (Con...
erdszelem5 32339 Lemma for ~ erdsze . (Con...
erdszelem6 32340 Lemma for ~ erdsze . (Con...
erdszelem7 32341 Lemma for ~ erdsze . (Con...
erdszelem8 32342 Lemma for ~ erdsze . (Con...
erdszelem9 32343 Lemma for ~ erdsze . (Con...
erdszelem10 32344 Lemma for ~ erdsze . (Con...
erdszelem11 32345 Lemma for ~ erdsze . (Con...
erdsze 32346 The Erdős-Szekeres th...
erdsze2lem1 32347 Lemma for ~ erdsze2 . (Co...
erdsze2lem2 32348 Lemma for ~ erdsze2 . (Co...
erdsze2 32349 Generalize the statement o...
kur14lem1 32350 Lemma for ~ kur14 . (Cont...
kur14lem2 32351 Lemma for ~ kur14 . Write...
kur14lem3 32352 Lemma for ~ kur14 . A clo...
kur14lem4 32353 Lemma for ~ kur14 . Compl...
kur14lem5 32354 Lemma for ~ kur14 . Closu...
kur14lem6 32355 Lemma for ~ kur14 . If ` ...
kur14lem7 32356 Lemma for ~ kur14 : main p...
kur14lem8 32357 Lemma for ~ kur14 . Show ...
kur14lem9 32358 Lemma for ~ kur14 . Since...
kur14lem10 32359 Lemma for ~ kur14 . Disch...
kur14 32360 Kuratowski's closure-compl...
ispconn 32367 The property of being a pa...
pconncn 32368 The property of being a pa...
pconntop 32369 A simply connected space i...
issconn 32370 The property of being a si...
sconnpconn 32371 A simply connected space i...
sconntop 32372 A simply connected space i...
sconnpht 32373 A closed path in a simply ...
cnpconn 32374 An image of a path-connect...
pconnconn 32375 A path-connected space is ...
txpconn 32376 The topological product of...
ptpconn 32377 The topological product of...
indispconn 32378 The indiscrete topology (o...
connpconn 32379 A connected and locally pa...
qtoppconn 32380 A quotient of a path-conne...
pconnpi1 32381 All fundamental groups in ...
sconnpht2 32382 Any two paths in a simply ...
sconnpi1 32383 A path-connected topologic...
txsconnlem 32384 Lemma for ~ txsconn . (Co...
txsconn 32385 The topological product of...
cvxpconn 32386 A convex subset of the com...
cvxsconn 32387 A convex subset of the com...
blsconn 32388 An open ball in the comple...
cnllysconn 32389 The topology of the comple...
resconn 32390 A subset of ` RR ` is simp...
ioosconn 32391 An open interval is simply...
iccsconn 32392 A closed interval is simpl...
retopsconn 32393 The real numbers are simpl...
iccllysconn 32394 A closed interval is local...
rellysconn 32395 The real numbers are local...
iisconn 32396 The unit interval is simpl...
iillysconn 32397 The unit interval is local...
iinllyconn 32398 The unit interval is local...
fncvm 32401 Lemma for covering maps. ...
cvmscbv 32402 Change bound variables in ...
iscvm 32403 The property of being a co...
cvmtop1 32404 Reverse closure for a cove...
cvmtop2 32405 Reverse closure for a cove...
cvmcn 32406 A covering map is a contin...
cvmcov 32407 Property of a covering map...
cvmsrcl 32408 Reverse closure for an eve...
cvmsi 32409 One direction of ~ cvmsval...
cvmsval 32410 Elementhood in the set ` S...
cvmsss 32411 An even covering is a subs...
cvmsn0 32412 An even covering is nonemp...
cvmsuni 32413 An even covering of ` U ` ...
cvmsdisj 32414 An even covering of ` U ` ...
cvmshmeo 32415 Every element of an even c...
cvmsf1o 32416 ` F ` , localized to an el...
cvmscld 32417 The sets of an even coveri...
cvmsss2 32418 An open subset of an evenl...
cvmcov2 32419 The covering map property ...
cvmseu 32420 Every element in ` U. T ` ...
cvmsiota 32421 Identify the unique elemen...
cvmopnlem 32422 Lemma for ~ cvmopn . (Con...
cvmfolem 32423 Lemma for ~ cvmfo . (Cont...
cvmopn 32424 A covering map is an open ...
cvmliftmolem1 32425 Lemma for ~ cvmliftmo . (...
cvmliftmolem2 32426 Lemma for ~ cvmliftmo . (...
cvmliftmoi 32427 A lift of a continuous fun...
cvmliftmo 32428 A lift of a continuous fun...
cvmliftlem1 32429 Lemma for ~ cvmlift . In ...
cvmliftlem2 32430 Lemma for ~ cvmlift . ` W ...
cvmliftlem3 32431 Lemma for ~ cvmlift . Sin...
cvmliftlem4 32432 Lemma for ~ cvmlift . The...
cvmliftlem5 32433 Lemma for ~ cvmlift . Def...
cvmliftlem6 32434 Lemma for ~ cvmlift . Ind...
cvmliftlem7 32435 Lemma for ~ cvmlift . Pro...
cvmliftlem8 32436 Lemma for ~ cvmlift . The...
cvmliftlem9 32437 Lemma for ~ cvmlift . The...
cvmliftlem10 32438 Lemma for ~ cvmlift . The...
cvmliftlem11 32439 Lemma for ~ cvmlift . (Co...
cvmliftlem13 32440 Lemma for ~ cvmlift . The...
cvmliftlem14 32441 Lemma for ~ cvmlift . Put...
cvmliftlem15 32442 Lemma for ~ cvmlift . Dis...
cvmlift 32443 One of the important prope...
cvmfo 32444 A covering map is an onto ...
cvmliftiota 32445 Write out a function ` H `...
cvmlift2lem1 32446 Lemma for ~ cvmlift2 . (C...
cvmlift2lem9a 32447 Lemma for ~ cvmlift2 and ~...
cvmlift2lem2 32448 Lemma for ~ cvmlift2 . (C...
cvmlift2lem3 32449 Lemma for ~ cvmlift2 . (C...
cvmlift2lem4 32450 Lemma for ~ cvmlift2 . (C...
cvmlift2lem5 32451 Lemma for ~ cvmlift2 . (C...
cvmlift2lem6 32452 Lemma for ~ cvmlift2 . (C...
cvmlift2lem7 32453 Lemma for ~ cvmlift2 . (C...
cvmlift2lem8 32454 Lemma for ~ cvmlift2 . (C...
cvmlift2lem9 32455 Lemma for ~ cvmlift2 . (C...
cvmlift2lem10 32456 Lemma for ~ cvmlift2 . (C...
cvmlift2lem11 32457 Lemma for ~ cvmlift2 . (C...
cvmlift2lem12 32458 Lemma for ~ cvmlift2 . (C...
cvmlift2lem13 32459 Lemma for ~ cvmlift2 . (C...
cvmlift2 32460 A two-dimensional version ...
cvmliftphtlem 32461 Lemma for ~ cvmliftpht . ...
cvmliftpht 32462 If ` G ` and ` H ` are pat...
cvmlift3lem1 32463 Lemma for ~ cvmlift3 . (C...
cvmlift3lem2 32464 Lemma for ~ cvmlift2 . (C...
cvmlift3lem3 32465 Lemma for ~ cvmlift2 . (C...
cvmlift3lem4 32466 Lemma for ~ cvmlift2 . (C...
cvmlift3lem5 32467 Lemma for ~ cvmlift2 . (C...
cvmlift3lem6 32468 Lemma for ~ cvmlift3 . (C...
cvmlift3lem7 32469 Lemma for ~ cvmlift3 . (C...
cvmlift3lem8 32470 Lemma for ~ cvmlift2 . (C...
cvmlift3lem9 32471 Lemma for ~ cvmlift2 . (C...
cvmlift3 32472 A general version of ~ cvm...
snmlff 32473 The function ` F ` from ~ ...
snmlfval 32474 The function ` F ` from ~ ...
snmlval 32475 The property " ` A ` is si...
snmlflim 32476 If ` A ` is simply normal,...
goel 32491 A "Godel-set of membership...
goelel3xp 32492 A "Godel-set of membership...
goeleq12bg 32493 Two "Godel-set of membersh...
gonafv 32494 The "Godel-set for the She...
goaleq12d 32495 Equality of the "Godel-set...
gonanegoal 32496 The Godel-set for the Shef...
satf 32497 The satisfaction predicate...
satfsucom 32498 The satisfaction predicate...
satfn 32499 The satisfaction predicate...
satom 32500 The satisfaction predicate...
satfvsucom 32501 The satisfaction predicate...
satfv0 32502 The value of the satisfact...
satfvsuclem1 32503 Lemma 1 for ~ satfvsuc . ...
satfvsuclem2 32504 Lemma 2 for ~ satfvsuc . ...
satfvsuc 32505 The value of the satisfact...
satfv1lem 32506 Lemma for ~ satfv1 . (Con...
satfv1 32507 The value of the satisfact...
satfsschain 32508 The binary relation of a s...
satfvsucsuc 32509 The satisfaction predicate...
satfbrsuc 32510 The binary relation of a s...
satfrel 32511 The value of the satisfact...
satfdmlem 32512 Lemma for ~ satfdm . (Con...
satfdm 32513 The domain of the satisfac...
satfrnmapom 32514 The range of the satisfact...
satfv0fun 32515 The value of the satisfact...
satf0 32516 The satisfaction predicate...
satf0sucom 32517 The satisfaction predicate...
satf00 32518 The value of the satisfact...
satf0suclem 32519 Lemma for ~ satf0suc , ~ s...
satf0suc 32520 The value of the satisfact...
satf0op 32521 An element of a value of t...
satf0n0 32522 The value of the satisfact...
sat1el2xp 32523 The first component of an ...
fmlafv 32524 The valid Godel formulas o...
fmla 32525 The set of all valid Godel...
fmla0 32526 The valid Godel formulas o...
fmla0xp 32527 The valid Godel formulas o...
fmlasuc0 32528 The valid Godel formulas o...
fmlafvel 32529 A class is a valid Godel f...
fmlasuc 32530 The valid Godel formulas o...
fmla1 32531 The valid Godel formulas o...
isfmlasuc 32532 The characterization of a ...
fmlasssuc 32533 The Godel formulas of heig...
fmlaomn0 32534 The empty set is not a God...
fmlan0 32535 The empty set is not a God...
gonan0 32536 The "Godel-set of NAND" is...
goaln0 32537 The "Godel-set of universa...
gonarlem 32538 Lemma for ~ gonar (inducti...
gonar 32539 If the "Godel-set of NAND"...
goalrlem 32540 Lemma for ~ goalr (inducti...
goalr 32541 If the "Godel-set of unive...
fmla0disjsuc 32542 The set of valid Godel for...
fmlasucdisj 32543 The valid Godel formulas o...
satfdmfmla 32544 The domain of the satisfac...
satffunlem 32545 Lemma for ~ satffunlem1lem...
satffunlem1lem1 32546 Lemma for ~ satffunlem1 . ...
satffunlem1lem2 32547 Lemma 2 for ~ satffunlem1 ...
satffunlem2lem1 32548 Lemma 1 for ~ satffunlem2 ...
dmopab3rexdif 32549 The domain of an ordered p...
satffunlem2lem2 32550 Lemma 2 for ~ satffunlem2 ...
satffunlem1 32551 Lemma 1 for ~ satffun : in...
satffunlem2 32552 Lemma 2 for ~ satffun : in...
satffun 32553 The value of the satisfact...
satff 32554 The satisfaction predicate...
satfun 32555 The satisfaction predicate...
satfvel 32556 An element of the value of...
satfv0fvfmla0 32557 The value of the satisfact...
satefv 32558 The simplified satisfactio...
sate0 32559 The simplified satisfactio...
satef 32560 The simplified satisfactio...
sate0fv0 32561 A simplified satisfaction ...
satefvfmla0 32562 The simplified satisfactio...
sategoelfvb 32563 Characterization of a valu...
sategoelfv 32564 Condition of a valuation `...
ex-sategoelel 32565 Example of a valuation of ...
ex-sategoel 32566 Instance of ~ sategoelfv f...
satfv1fvfmla1 32567 The value of the satisfact...
2goelgoanfmla1 32568 Two Godel-sets of membersh...
satefvfmla1 32569 The simplified satisfactio...
ex-sategoelelomsuc 32570 Example of a valuation of ...
ex-sategoelel12 32571 Example of a valuation of ...
prv 32572 The "proves" relation on a...
elnanelprv 32573 The wff ` ( A e. B -/\ B e...
prv0 32574 Every wff encoded as ` U `...
prv1n 32575 No wff encoded as a Godel-...
mvtval 32644 The set of variable typeco...
mrexval 32645 The set of "raw expression...
mexval 32646 The set of expressions, wh...
mexval2 32647 The set of expressions, wh...
mdvval 32648 The set of disjoint variab...
mvrsval 32649 The set of variables in an...
mvrsfpw 32650 The set of variables in an...
mrsubffval 32651 The substitution of some v...
mrsubfval 32652 The substitution of some v...
mrsubval 32653 The substitution of some v...
mrsubcv 32654 The value of a substituted...
mrsubvr 32655 The value of a substituted...
mrsubff 32656 A substitution is a functi...
mrsubrn 32657 Although it is defined for...
mrsubff1 32658 When restricted to complet...
mrsubff1o 32659 When restricted to complet...
mrsub0 32660 The value of the substitut...
mrsubf 32661 A substitution is a functi...
mrsubccat 32662 Substitution distributes o...
mrsubcn 32663 A substitution does not ch...
elmrsubrn 32664 Characterization of the su...
mrsubco 32665 The composition of two sub...
mrsubvrs 32666 The set of variables in a ...
msubffval 32667 A substitution applied to ...
msubfval 32668 A substitution applied to ...
msubval 32669 A substitution applied to ...
msubrsub 32670 A substitution applied to ...
msubty 32671 The type of a substituted ...
elmsubrn 32672 Characterization of substi...
msubrn 32673 Although it is defined for...
msubff 32674 A substitution is a functi...
msubco 32675 The composition of two sub...
msubf 32676 A substitution is a functi...
mvhfval 32677 Value of the function mapp...
mvhval 32678 Value of the function mapp...
mpstval 32679 A pre-statement is an orde...
elmpst 32680 Property of being a pre-st...
msrfval 32681 Value of the reduct of a p...
msrval 32682 Value of the reduct of a p...
mpstssv 32683 A pre-statement is an orde...
mpst123 32684 Decompose a pre-statement ...
mpstrcl 32685 The elements of a pre-stat...
msrf 32686 The reduct of a pre-statem...
msrrcl 32687 If ` X ` and ` Y ` have th...
mstaval 32688 Value of the set of statem...
msrid 32689 The reduct of a statement ...
msrfo 32690 The reduct of a pre-statem...
mstapst 32691 A statement is a pre-state...
elmsta 32692 Property of being a statem...
ismfs 32693 A formal system is a tuple...
mfsdisj 32694 The constants and variable...
mtyf2 32695 The type function maps var...
mtyf 32696 The type function maps var...
mvtss 32697 The set of variable typeco...
maxsta 32698 An axiom is a statement. ...
mvtinf 32699 Each variable typecode has...
msubff1 32700 When restricted to complet...
msubff1o 32701 When restricted to complet...
mvhf 32702 The function mapping varia...
mvhf1 32703 The function mapping varia...
msubvrs 32704 The set of variables in a ...
mclsrcl 32705 Reverse closure for the cl...
mclsssvlem 32706 Lemma for ~ mclsssv . (Co...
mclsval 32707 The function mapping varia...
mclsssv 32708 The closure of a set of ex...
ssmclslem 32709 Lemma for ~ ssmcls . (Con...
vhmcls 32710 All variable hypotheses ar...
ssmcls 32711 The original expressions a...
ss2mcls 32712 The closure is monotonic u...
mclsax 32713 The closure is closed unde...
mclsind 32714 Induction theorem for clos...
mppspstlem 32715 Lemma for ~ mppspst . (Co...
mppsval 32716 Definition of a provable p...
elmpps 32717 Definition of a provable p...
mppspst 32718 A provable pre-statement i...
mthmval 32719 A theorem is a pre-stateme...
elmthm 32720 A theorem is a pre-stateme...
mthmi 32721 A statement whose reduct i...
mthmsta 32722 A theorem is a pre-stateme...
mppsthm 32723 A provable pre-statement i...
mthmblem 32724 Lemma for ~ mthmb . (Cont...
mthmb 32725 If two statements have the...
mthmpps 32726 Given a theorem, there is ...
mclsppslem 32727 The closure is closed unde...
mclspps 32728 The closure is closed unde...
problem1 32805 Practice problem 1. Clues...
problem2 32806 Practice problem 2. Clues...
problem3 32807 Practice problem 3. Clues...
problem4 32808 Practice problem 4. Clues...
problem5 32809 Practice problem 5. Clues...
quad3 32810 Variant of quadratic equat...
climuzcnv 32811 Utility lemma to convert b...
sinccvglem 32812 ` ( ( sin `` x ) / x ) ~~>...
sinccvg 32813 ` ( ( sin `` x ) / x ) ~~>...
circum 32814 The circumference of a cir...
elfzm12 32815 Membership in a curtailed ...
nn0seqcvg 32816 A strictly-decreasing nonn...
lediv2aALT 32817 Division of both sides of ...
abs2sqlei 32818 The absolute values of two...
abs2sqlti 32819 The absolute values of two...
abs2sqle 32820 The absolute values of two...
abs2sqlt 32821 The absolute values of two...
abs2difi 32822 Difference of absolute val...
abs2difabsi 32823 Absolute value of differen...
axextprim 32824 ~ ax-ext without distinct ...
axrepprim 32825 ~ ax-rep without distinct ...
axunprim 32826 ~ ax-un without distinct v...
axpowprim 32827 ~ ax-pow without distinct ...
axregprim 32828 ~ ax-reg without distinct ...
axinfprim 32829 ~ ax-inf without distinct ...
axacprim 32830 ~ ax-ac without distinct v...
untelirr 32831 We call a class "untanged"...
untuni 32832 The union of a class is un...
untsucf 32833 If a class is untangled, t...
unt0 32834 The null set is untangled....
untint 32835 If there is an untangled e...
efrunt 32836 If ` A ` is well-founded b...
untangtr 32837 A transitive class is unta...
3orel2 32838 Partial elimination of a t...
3orel3 32839 Partial elimination of a t...
3pm3.2ni 32840 Triple negated disjunction...
3jaodd 32841 Double deduction form of ~...
3orit 32842 Closed form of ~ 3ori . (...
biimpexp 32843 A biconditional in the ant...
3orel13 32844 Elimination of two disjunc...
nepss 32845 Two classes are unequal if...
3ccased 32846 Triple disjunction form of...
dfso3 32847 Expansion of the definitio...
brtpid1 32848 A binary relation involvin...
brtpid2 32849 A binary relation involvin...
brtpid3 32850 A binary relation involvin...
ceqsrexv2 32851 Alternate elimitation of a...
iota5f 32852 A method for computing iot...
ceqsralv2 32853 Alternate elimination of a...
dford5 32854 A class is ordinal iff it ...
jath 32855 Closed form of ~ ja . Pro...
sqdivzi 32856 Distribution of square ove...
supfz 32857 The supremum of a finite s...
inffz 32858 The infimum of a finite se...
fz0n 32859 The sequence ` ( 0 ... ( N...
shftvalg 32860 Value of a sequence shifte...
divcnvlin 32861 Limit of the ratio of two ...
climlec3 32862 Comparison of a constant t...
logi 32863 Calculate the logarithm of...
iexpire 32864 ` _i ` raised to itself is...
bcneg1 32865 The binomial coefficent ov...
bcm1nt 32866 The proportion of one bion...
bcprod 32867 A product identity for bin...
bccolsum 32868 A column-sum rule for bino...
iprodefisumlem 32869 Lemma for ~ iprodefisum . ...
iprodefisum 32870 Applying the exponential f...
iprodgam 32871 An infinite product versio...
faclimlem1 32872 Lemma for ~ faclim . Clos...
faclimlem2 32873 Lemma for ~ faclim . Show...
faclimlem3 32874 Lemma for ~ faclim . Alge...
faclim 32875 An infinite product expres...
iprodfac 32876 An infinite product expres...
faclim2 32877 Another factorial limit du...
pdivsq 32878 Condition for a prime divi...
dvdspw 32879 Exponentiation law for div...
gcd32 32880 Swap the second and third ...
gcdabsorb 32881 Absorption law for gcd. (...
brtp 32882 A condition for a binary r...
dftr6 32883 A potential definition of ...
coep 32884 Composition with the membe...
coepr 32885 Composition with the conve...
dffr5 32886 A quantifier free definiti...
dfso2 32887 Quantifier free definition...
dfpo2 32888 Quantifier free definition...
br8 32889 Substitution for an eight-...
br6 32890 Substitution for a six-pla...
br4 32891 Substitution for a four-pl...
cnvco1 32892 Another distributive law o...
cnvco2 32893 Another distributive law o...
eldm3 32894 Quantifier-free definition...
elrn3 32895 Quantifier-free definition...
pocnv 32896 The converse of a partial ...
socnv 32897 The converse of a strict o...
sotrd 32898 Transitivity law for stric...
sotr3 32899 Transitivity law for stric...
sotrine 32900 Trichotomy law for strict ...
eqfunresadj 32901 Law for adjoining an eleme...
eqfunressuc 32902 Law for equality of restri...
funeldmb 32903 If ` (/) ` is not part of ...
elintfv 32904 Membership in an intersect...
funpsstri 32905 A condition for subset tri...
fundmpss 32906 If a class ` F ` is a prop...
fvresval 32907 The value of a function at...
funsseq 32908 Given two functions with e...
fununiq 32909 The uniqueness condition o...
funbreq 32910 An equality condition for ...
br1steq 32911 Uniqueness condition for t...
br2ndeq 32912 Uniqueness condition for t...
dfdm5 32913 Definition of domain in te...
dfrn5 32914 Definition of range in ter...
opelco3 32915 Alternate way of saying th...
elima4 32916 Quantifier-free expression...
fv1stcnv 32917 The value of the converse ...
fv2ndcnv 32918 The value of the converse ...
imaindm 32919 The image is unaffected by...
setinds 32920 Principle of set induction...
setinds2f 32921 ` _E ` induction schema, u...
setinds2 32922 ` _E ` induction schema, u...
elpotr 32923 A class of transitive sets...
dford5reg 32924 Given ~ ax-reg , an ordina...
dfon2lem1 32925 Lemma for ~ dfon2 . (Cont...
dfon2lem2 32926 Lemma for ~ dfon2 . (Cont...
dfon2lem3 32927 Lemma for ~ dfon2 . All s...
dfon2lem4 32928 Lemma for ~ dfon2 . If tw...
dfon2lem5 32929 Lemma for ~ dfon2 . Two s...
dfon2lem6 32930 Lemma for ~ dfon2 . A tra...
dfon2lem7 32931 Lemma for ~ dfon2 . All e...
dfon2lem8 32932 Lemma for ~ dfon2 . The i...
dfon2lem9 32933 Lemma for ~ dfon2 . A cla...
dfon2 32934 ` On ` consists of all set...
rdgprc0 32935 The value of the recursive...
rdgprc 32936 The value of the recursive...
dfrdg2 32937 Alternate definition of th...
dfrdg3 32938 Generalization of ~ dfrdg2...
axextdfeq 32939 A version of ~ ax-ext for ...
ax8dfeq 32940 A version of ~ ax-8 for us...
axextdist 32941 ~ ax-ext with distinctors ...
axextbdist 32942 ~ axextb with distinctors ...
19.12b 32943 Version of ~ 19.12vv with ...
exnel 32944 There is always a set not ...
distel 32945 Distinctors in terms of me...
axextndbi 32946 ~ axextnd as a bicondition...
hbntg 32947 A more general form of ~ h...
hbimtg 32948 A more general and closed ...
hbaltg 32949 A more general and closed ...
hbng 32950 A more general form of ~ h...
hbimg 32951 A more general form of ~ h...
tfisg 32952 A closed form of ~ tfis . ...
dftrpred2 32955 A definition of the transi...
trpredeq1 32956 Equality theorem for trans...
trpredeq2 32957 Equality theorem for trans...
trpredeq3 32958 Equality theorem for trans...
trpredeq1d 32959 Equality deduction for tra...
trpredeq2d 32960 Equality deduction for tra...
trpredeq3d 32961 Equality deduction for tra...
eltrpred 32962 A class is a transitive pr...
trpredlem1 32963 Technical lemma for transi...
trpredpred 32964 Assuming it exists, the pr...
trpredss 32965 The transitive predecessor...
trpredtr 32966 The transitive predecessor...
trpredmintr 32967 The transitive predecessor...
trpredelss 32968 Given a transitive predece...
dftrpred3g 32969 The transitive predecessor...
dftrpred4g 32970 Another recursive expressi...
trpredpo 32971 If ` R ` partially orders ...
trpred0 32972 The class of transitive pr...
trpredex 32973 The transitive predecessor...
trpredrec 32974 If ` Y ` is an ` R ` , ` A...
frpomin 32975 Every (possibly proper) su...
frpomin2 32976 Every (possibly proper) su...
frpoind 32977 The principle of founded i...
frpoinsg 32978 Founded, Partial-Ordering ...
frpoins2fg 32979 Founded Partial Induction ...
frpoins2g 32980 Founded Partial Induction ...
frmin 32981 Every (possibly proper) su...
frind 32982 The principle of founded i...
frindi 32983 The principle of founded i...
frinsg 32984 Founded Induction Schema. ...
frins 32985 Founded Induction Schema. ...
frins2fg 32986 Founded Induction schema, ...
frins2f 32987 Founded Induction schema, ...
frins2g 32988 Founded Induction schema, ...
frins2 32989 Founded Induction schema, ...
frins3 32990 Founded Induction schema, ...
orderseqlem 32991 Lemma for ~ poseq and ~ so...
poseq 32992 A partial ordering of sequ...
soseq 32993 A linear ordering of seque...
wsuceq123 32998 Equality theorem for well-...
wsuceq1 32999 Equality theorem for well-...
wsuceq2 33000 Equality theorem for well-...
wsuceq3 33001 Equality theorem for well-...
nfwsuc 33002 Bound-variable hypothesis ...
wlimeq12 33003 Equality theorem for the l...
wlimeq1 33004 Equality theorem for the l...
wlimeq2 33005 Equality theorem for the l...
nfwlim 33006 Bound-variable hypothesis ...
elwlim 33007 Membership in the limit cl...
wzel 33008 The zero of a well-founded...
wsuclem 33009 Lemma for the supremum pro...
wsucex 33010 Existence theorem for well...
wsuccl 33011 If ` X ` is a set with an ...
wsuclb 33012 A well-founded successor i...
wlimss 33013 The class of limit points ...
frecseq123 33016 Equality theorem for found...
nffrecs 33017 Bound-variable hypothesis ...
frr3g 33018 Functions defined by found...
fpr3g 33019 Functions defined by found...
frrlem1 33020 Lemma for founded recursio...
frrlem2 33021 Lemma for founded recursio...
frrlem3 33022 Lemma for founded recursio...
frrlem4 33023 Lemma for founded recursio...
frrlem5 33024 Lemma for founded recursio...
frrlem6 33025 Lemma for founded recursio...
frrlem7 33026 Lemma for founded recursio...
frrlem8 33027 Lemma for founded recursio...
frrlem9 33028 Lemma for founded recursio...
frrlem10 33029 Lemma for founded recursio...
frrlem11 33030 Lemma for founded recursio...
frrlem12 33031 Lemma for founded recursio...
frrlem13 33032 Lemma for founded recursio...
frrlem14 33033 Lemma for founded recursio...
fprlem1 33034 Lemma for founded partial ...
fprlem2 33035 Lemma for founded partial ...
fpr1 33036 Law of founded partial rec...
fpr2 33037 Law of founded partial rec...
fpr3 33038 Law of founded partial rec...
frrlem15 33039 Lemma for general founded ...
frrlem16 33040 Lemma for general founded ...
frr1 33041 Law of general founded rec...
frr2 33042 Law of general founded rec...
frr3 33043 Law of general founded rec...
elno 33050 Membership in the surreals...
sltval 33051 The value of the surreal l...
bdayval 33052 The value of the birthday ...
nofun 33053 A surreal is a function. ...
nodmon 33054 The domain of a surreal is...
norn 33055 The range of a surreal is ...
nofnbday 33056 A surreal is a function ov...
nodmord 33057 The domain of a surreal ha...
elno2 33058 An alternative condition f...
elno3 33059 Another condition for memb...
sltval2 33060 Alternate expression for s...
nofv 33061 The function value of a su...
nosgnn0 33062 ` (/) ` is not a surreal s...
nosgnn0i 33063 If ` X ` is a surreal sign...
noreson 33064 The restriction of a surre...
sltintdifex 33065 If ` A
sltres 33066 If the restrictions of two...
noxp1o 33067 The Cartesian product of a...
noseponlem 33068 Lemma for ~ nosepon . Con...
nosepon 33069 Given two unequal surreals...
noextend 33070 Extending a surreal by one...
noextendseq 33071 Extend a surreal by a sequ...
noextenddif 33072 Calculate the place where ...
noextendlt 33073 Extending a surreal with a...
noextendgt 33074 Extending a surreal with a...
nolesgn2o 33075 Given ` A ` less than or e...
nolesgn2ores 33076 Given ` A ` less than or e...
sltsolem1 33077 Lemma for ~ sltso . The s...
sltso 33078 Surreal less than totally ...
bdayfo 33079 The birthday function maps...
fvnobday 33080 The value of a surreal at ...
nosepnelem 33081 Lemma for ~ nosepne . (Co...
nosepne 33082 The value of two non-equal...
nosep1o 33083 If the value of a surreal ...
nosepdmlem 33084 Lemma for ~ nosepdm . (Co...
nosepdm 33085 The first place two surrea...
nosepeq 33086 The values of two surreals...
nosepssdm 33087 Given two non-equal surrea...
nodenselem4 33088 Lemma for ~ nodense . Sho...
nodenselem5 33089 Lemma for ~ nodense . If ...
nodenselem6 33090 The restriction of a surre...
nodenselem7 33091 Lemma for ~ nodense . ` A ...
nodenselem8 33092 Lemma for ~ nodense . Giv...
nodense 33093 Given two distinct surreal...
bdayimaon 33094 Lemma for full-eta propert...
nolt02olem 33095 Lemma for ~ nolt02o . If ...
nolt02o 33096 Given ` A ` less than ` B ...
noresle 33097 Restriction law for surrea...
nomaxmo 33098 A class of surreals has at...
noprefixmo 33099 In any class of surreals, ...
nosupno 33100 The next several theorems ...
nosupdm 33101 The domain of the surreal ...
nosupbday 33102 Birthday bounding law for ...
nosupfv 33103 The value of surreal supre...
nosupres 33104 A restriction law for surr...
nosupbnd1lem1 33105 Lemma for ~ nosupbnd1 . E...
nosupbnd1lem2 33106 Lemma for ~ nosupbnd1 . W...
nosupbnd1lem3 33107 Lemma for ~ nosupbnd1 . I...
nosupbnd1lem4 33108 Lemma for ~ nosupbnd1 . I...
nosupbnd1lem5 33109 Lemma for ~ nosupbnd1 . I...
nosupbnd1lem6 33110 Lemma for ~ nosupbnd1 . E...
nosupbnd1 33111 Bounding law from below fo...
nosupbnd2lem1 33112 Bounding law from above wh...
nosupbnd2 33113 Bounding law from above fo...
noetalem1 33114 Lemma for ~ noeta . Estab...
noetalem2 33115 Lemma for ~ noeta . ` Z ` ...
noetalem3 33116 Lemma for ~ noeta . When ...
noetalem4 33117 Lemma for ~ noeta . Bound...
noetalem5 33118 Lemma for ~ noeta . The f...
noeta 33119 The full-eta axiom for the...
sltirr 33122 Surreal less than is irref...
slttr 33123 Surreal less than is trans...
sltasym 33124 Surreal less than is asymm...
sltlin 33125 Surreal less than obeys tr...
slttrieq2 33126 Trichotomy law for surreal...
slttrine 33127 Trichotomy law for surreal...
slenlt 33128 Surreal less than or equal...
sltnle 33129 Surreal less than in terms...
sleloe 33130 Surreal less than or equal...
sletri3 33131 Trichotomy law for surreal...
sltletr 33132 Surreal transitive law. (...
slelttr 33133 Surreal transitive law. (...
sletr 33134 Surreal transitive law. (...
slttrd 33135 Surreal less than is trans...
sltletrd 33136 Surreal less than is trans...
slelttrd 33137 Surreal less than is trans...
sletrd 33138 Surreal less than or equal...
bdayfun 33139 The birthday function is a...
bdayfn 33140 The birthday function is a...
bdaydm 33141 The birthday function's do...
bdayrn 33142 The birthday function's ra...
bdayelon 33143 The value of the birthday ...
nocvxminlem 33144 Lemma for ~ nocvxmin . Gi...
nocvxmin 33145 Given a nonempty convex cl...
noprc 33146 The surreal numbers are a ...
brsslt 33151 Binary relation form of th...
ssltex1 33152 The first argument of surr...
ssltex2 33153 The second argument of sur...
ssltss1 33154 The first argument of surr...
ssltss2 33155 The second argument of sur...
ssltsep 33156 The separation property of...
sssslt1 33157 Relationship between surre...
sssslt2 33158 Relationship between surre...
nulsslt 33159 The empty set is less than...
nulssgt 33160 The empty set is greater t...
conway 33161 Conway's Simplicity Theore...
scutval 33162 The value of the surreal c...
scutcut 33163 Cut properties of the surr...
scutbday 33164 The birthday of the surrea...
sslttr 33165 Transitive law for surreal...
ssltun1 33166 Union law for surreal set ...
ssltun2 33167 Union law for surreal set ...
scutun12 33168 Union law for surreal cuts...
dmscut 33169 The domain of the surreal ...
scutf 33170 Functionhood statement for...
etasslt 33171 A restatement of ~ noeta u...
scutbdaybnd 33172 An upper bound on the birt...
scutbdaylt 33173 If a surreal lies in a gap...
slerec 33174 A comparison law for surre...
sltrec 33175 A comparison law for surre...
madeval 33186 The value of the made by f...
madeval2 33187 Alternative characterizati...
txpss3v 33236 A tail Cartesian product i...
txprel 33237 A tail Cartesian product i...
brtxp 33238 Characterize a ternary rel...
brtxp2 33239 The binary relation over a...
dfpprod2 33240 Expanded definition of par...
pprodcnveq 33241 A converse law for paralle...
pprodss4v 33242 The parallel product is a ...
brpprod 33243 Characterize a quaternary ...
brpprod3a 33244 Condition for parallel pro...
brpprod3b 33245 Condition for parallel pro...
relsset 33246 The subset class is a bina...
brsset 33247 For sets, the ` SSet ` bin...
idsset 33248 ` _I ` is equal to the int...
eltrans 33249 Membership in the class of...
dfon3 33250 A quantifier-free definiti...
dfon4 33251 Another quantifier-free de...
brtxpsd 33252 Expansion of a common form...
brtxpsd2 33253 Another common abbreviatio...
brtxpsd3 33254 A third common abbreviatio...
relbigcup 33255 The ` Bigcup ` relationshi...
brbigcup 33256 Binary relation over ` Big...
dfbigcup2 33257 ` Bigcup ` using maps-to n...
fobigcup 33258 ` Bigcup ` maps the univer...
fnbigcup 33259 ` Bigcup ` is a function o...
fvbigcup 33260 For sets, ` Bigcup ` yield...
elfix 33261 Membership in the fixpoint...
elfix2 33262 Alternative membership in ...
dffix2 33263 The fixpoints of a class i...
fixssdm 33264 The fixpoints of a class a...
fixssrn 33265 The fixpoints of a class a...
fixcnv 33266 The fixpoints of a class a...
fixun 33267 The fixpoint operator dist...
ellimits 33268 Membership in the class of...
limitssson 33269 The class of all limit ord...
dfom5b 33270 A quantifier-free definiti...
sscoid 33271 A condition for subset and...
dffun10 33272 Another potential definiti...
elfuns 33273 Membership in the class of...
elfunsg 33274 Closed form of ~ elfuns . ...
brsingle 33275 The binary relation form o...
elsingles 33276 Membership in the class of...
fnsingle 33277 The singleton relationship...
fvsingle 33278 The value of the singleton...
dfsingles2 33279 Alternate definition of th...
snelsingles 33280 A singleton is a member of...
dfiota3 33281 A definition of iota using...
dffv5 33282 Another quantifier free de...
unisnif 33283 Express union of singleton...
brimage 33284 Binary relation form of th...
brimageg 33285 Closed form of ~ brimage ....
funimage 33286 ` Image A ` is a function....
fnimage 33287 ` Image R ` is a function ...
imageval 33288 The image functor in maps-...
fvimage 33289 Value of the image functor...
brcart 33290 Binary relation form of th...
brdomain 33291 Binary relation form of th...
brrange 33292 Binary relation form of th...
brdomaing 33293 Closed form of ~ brdomain ...
brrangeg 33294 Closed form of ~ brrange ....
brimg 33295 Binary relation form of th...
brapply 33296 Binary relation form of th...
brcup 33297 Binary relation form of th...
brcap 33298 Binary relation form of th...
brsuccf 33299 Binary relation form of th...
funpartlem 33300 Lemma for ~ funpartfun . ...
funpartfun 33301 The functional part of ` F...
funpartss 33302 The functional part of ` F...
funpartfv 33303 The function value of the ...
fullfunfnv 33304 The full functional part o...
fullfunfv 33305 The function value of the ...
brfullfun 33306 A binary relation form con...
brrestrict 33307 Binary relation form of th...
dfrecs2 33308 A quantifier-free definiti...
dfrdg4 33309 A quantifier-free definiti...
dfint3 33310 Quantifier-free definition...
imagesset 33311 The Image functor applied ...
brub 33312 Binary relation form of th...
brlb 33313 Binary relation form of th...
altopex 33318 Alternative ordered pairs ...
altopthsn 33319 Two alternate ordered pair...
altopeq12 33320 Equality for alternate ord...
altopeq1 33321 Equality for alternate ord...
altopeq2 33322 Equality for alternate ord...
altopth1 33323 Equality of the first memb...
altopth2 33324 Equality of the second mem...
altopthg 33325 Alternate ordered pair the...
altopthbg 33326 Alternate ordered pair the...
altopth 33327 The alternate ordered pair...
altopthb 33328 Alternate ordered pair the...
altopthc 33329 Alternate ordered pair the...
altopthd 33330 Alternate ordered pair the...
altxpeq1 33331 Equality for alternate Car...
altxpeq2 33332 Equality for alternate Car...
elaltxp 33333 Membership in alternate Ca...
altopelaltxp 33334 Alternate ordered pair mem...
altxpsspw 33335 An inclusion rule for alte...
altxpexg 33336 The alternate Cartesian pr...
rankaltopb 33337 Compute the rank of an alt...
nfaltop 33338 Bound-variable hypothesis ...
sbcaltop 33339 Distribution of class subs...
cgrrflx2d 33342 Deduction form of ~ axcgrr...
cgrtr4d 33343 Deduction form of ~ axcgrt...
cgrtr4and 33344 Deduction form of ~ axcgrt...
cgrrflx 33345 Reflexivity law for congru...
cgrrflxd 33346 Deduction form of ~ cgrrfl...
cgrcomim 33347 Congruence commutes on the...
cgrcom 33348 Congruence commutes betwee...
cgrcomand 33349 Deduction form of ~ cgrcom...
cgrtr 33350 Transitivity law for congr...
cgrtrand 33351 Deduction form of ~ cgrtr ...
cgrtr3 33352 Transitivity law for congr...
cgrtr3and 33353 Deduction form of ~ cgrtr3...
cgrcoml 33354 Congruence commutes on the...
cgrcomr 33355 Congruence commutes on the...
cgrcomlr 33356 Congruence commutes on bot...
cgrcomland 33357 Deduction form of ~ cgrcom...
cgrcomrand 33358 Deduction form of ~ cgrcom...
cgrcomlrand 33359 Deduction form of ~ cgrcom...
cgrtriv 33360 Degenerate segments are co...
cgrid2 33361 Identity law for congruenc...
cgrdegen 33362 Two congruent segments are...
brofs 33363 Binary relation form of th...
5segofs 33364 Rephrase ~ ax5seg using th...
ofscom 33365 The outer five segment pre...
cgrextend 33366 Link congruence over a pai...
cgrextendand 33367 Deduction form of ~ cgrext...
segconeq 33368 Two points that satisfy th...
segconeu 33369 Existential uniqueness ver...
btwntriv2 33370 Betweenness always holds f...
btwncomim 33371 Betweenness commutes. Imp...
btwncom 33372 Betweenness commutes. (Co...
btwncomand 33373 Deduction form of ~ btwnco...
btwntriv1 33374 Betweenness always holds f...
btwnswapid 33375 If you can swap the first ...
btwnswapid2 33376 If you can swap arguments ...
btwnintr 33377 Inner transitivity law for...
btwnexch3 33378 Exchange the first endpoin...
btwnexch3and 33379 Deduction form of ~ btwnex...
btwnouttr2 33380 Outer transitivity law for...
btwnexch2 33381 Exchange the outer point o...
btwnouttr 33382 Outer transitivity law for...
btwnexch 33383 Outer transitivity law for...
btwnexchand 33384 Deduction form of ~ btwnex...
btwndiff 33385 There is always a ` c ` di...
trisegint 33386 A line segment between two...
funtransport 33389 The ` TransportTo ` relati...
fvtransport 33390 Calculate the value of the...
transportcl 33391 Closure law for segment tr...
transportprops 33392 Calculate the defining pro...
brifs 33401 Binary relation form of th...
ifscgr 33402 Inner five segment congrue...
cgrsub 33403 Removing identical parts f...
brcgr3 33404 Binary relation form of th...
cgr3permute3 33405 Permutation law for three-...
cgr3permute1 33406 Permutation law for three-...
cgr3permute2 33407 Permutation law for three-...
cgr3permute4 33408 Permutation law for three-...
cgr3permute5 33409 Permutation law for three-...
cgr3tr4 33410 Transitivity law for three...
cgr3com 33411 Commutativity law for thre...
cgr3rflx 33412 Identity law for three-pla...
cgrxfr 33413 A line segment can be divi...
btwnxfr 33414 A condition for extending ...
colinrel 33415 Colinearity is a relations...
brcolinear2 33416 Alternate colinearity bina...
brcolinear 33417 The binary relation form o...
colinearex 33418 The colinear predicate exi...
colineardim1 33419 If ` A ` is colinear with ...
colinearperm1 33420 Permutation law for coline...
colinearperm3 33421 Permutation law for coline...
colinearperm2 33422 Permutation law for coline...
colinearperm4 33423 Permutation law for coline...
colinearperm5 33424 Permutation law for coline...
colineartriv1 33425 Trivial case of colinearit...
colineartriv2 33426 Trivial case of colinearit...
btwncolinear1 33427 Betweenness implies coline...
btwncolinear2 33428 Betweenness implies coline...
btwncolinear3 33429 Betweenness implies coline...
btwncolinear4 33430 Betweenness implies coline...
btwncolinear5 33431 Betweenness implies coline...
btwncolinear6 33432 Betweenness implies coline...
colinearxfr 33433 Transfer law for colineari...
lineext 33434 Extend a line with a missi...
brofs2 33435 Change some conditions for...
brifs2 33436 Change some conditions for...
brfs 33437 Binary relation form of th...
fscgr 33438 Congruence law for the gen...
linecgr 33439 Congruence rule for lines....
linecgrand 33440 Deduction form of ~ linecg...
lineid 33441 Identity law for points on...
idinside 33442 Law for finding a point in...
endofsegid 33443 If ` A ` , ` B ` , and ` C...
endofsegidand 33444 Deduction form of ~ endofs...
btwnconn1lem1 33445 Lemma for ~ btwnconn1 . T...
btwnconn1lem2 33446 Lemma for ~ btwnconn1 . N...
btwnconn1lem3 33447 Lemma for ~ btwnconn1 . E...
btwnconn1lem4 33448 Lemma for ~ btwnconn1 . A...
btwnconn1lem5 33449 Lemma for ~ btwnconn1 . N...
btwnconn1lem6 33450 Lemma for ~ btwnconn1 . N...
btwnconn1lem7 33451 Lemma for ~ btwnconn1 . U...
btwnconn1lem8 33452 Lemma for ~ btwnconn1 . N...
btwnconn1lem9 33453 Lemma for ~ btwnconn1 . N...
btwnconn1lem10 33454 Lemma for ~ btwnconn1 . N...
btwnconn1lem11 33455 Lemma for ~ btwnconn1 . N...
btwnconn1lem12 33456 Lemma for ~ btwnconn1 . U...
btwnconn1lem13 33457 Lemma for ~ btwnconn1 . B...
btwnconn1lem14 33458 Lemma for ~ btwnconn1 . F...
btwnconn1 33459 Connectitivy law for betwe...
btwnconn2 33460 Another connectivity law f...
btwnconn3 33461 Inner connectivity law for...
midofsegid 33462 If two points fall in the ...
segcon2 33463 Generalization of ~ axsegc...
brsegle 33466 Binary relation form of th...
brsegle2 33467 Alternate characterization...
seglecgr12im 33468 Substitution law for segme...
seglecgr12 33469 Substitution law for segme...
seglerflx 33470 Segment comparison is refl...
seglemin 33471 Any segment is at least as...
segletr 33472 Segment less than is trans...
segleantisym 33473 Antisymmetry law for segme...
seglelin 33474 Linearity law for segment ...
btwnsegle 33475 If ` B ` falls between ` A...
colinbtwnle 33476 Given three colinear point...
broutsideof 33479 Binary relation form of ` ...
broutsideof2 33480 Alternate form of ` Outsid...
outsidene1 33481 Outsideness implies inequa...
outsidene2 33482 Outsideness implies inequa...
btwnoutside 33483 A principle linking outsid...
broutsideof3 33484 Characterization of outsid...
outsideofrflx 33485 Reflexitivity of outsidene...
outsideofcom 33486 Commutitivity law for outs...
outsideoftr 33487 Transitivity law for outsi...
outsideofeq 33488 Uniqueness law for ` Outsi...
outsideofeu 33489 Given a nondegenerate ray,...
outsidele 33490 Relate ` OutsideOf ` to ` ...
outsideofcol 33491 Outside of implies colinea...
funray 33498 Show that the ` Ray ` rela...
fvray 33499 Calculate the value of the...
funline 33500 Show that the ` Line ` rel...
linedegen 33501 When ` Line ` is applied w...
fvline 33502 Calculate the value of the...
liness 33503 A line is a subset of the ...
fvline2 33504 Alternate definition of a ...
lineunray 33505 A line is composed of a po...
lineelsb2 33506 If ` S ` lies on ` P Q ` ,...
linerflx1 33507 Reflexivity law for line m...
linecom 33508 Commutativity law for line...
linerflx2 33509 Reflexivity law for line m...
ellines 33510 Membership in the set of a...
linethru 33511 If ` A ` is a line contain...
hilbert1.1 33512 There is a line through an...
hilbert1.2 33513 There is at most one line ...
linethrueu 33514 There is a unique line goi...
lineintmo 33515 Two distinct lines interse...
fwddifval 33520 Calculate the value of the...
fwddifnval 33521 The value of the forward d...
fwddifn0 33522 The value of the n-iterate...
fwddifnp1 33523 The value of the n-iterate...
rankung 33524 The rank of the union of t...
ranksng 33525 The rank of a singleton. ...
rankelg 33526 The membership relation is...
rankpwg 33527 The rank of a power set. ...
rank0 33528 The rank of the empty set ...
rankeq1o 33529 The only set with rank ` 1...
elhf 33532 Membership in the heredita...
elhf2 33533 Alternate form of membersh...
elhf2g 33534 Hereditarily finiteness vi...
0hf 33535 The empty set is a heredit...
hfun 33536 The union of two HF sets i...
hfsn 33537 The singleton of an HF set...
hfadj 33538 Adjoining one HF element t...
hfelhf 33539 Any member of an HF set is...
hftr 33540 The class of all hereditar...
hfext 33541 Extensionality for HF sets...
hfuni 33542 The union of an HF set is ...
hfpw 33543 The power class of an HF s...
hfninf 33544 ` _om ` is not hereditaril...
a1i14 33545 Add two antecedents to a w...
a1i24 33546 Add two antecedents to a w...
exp5d 33547 An exportation inference. ...
exp5g 33548 An exportation inference. ...
exp5k 33549 An exportation inference. ...
exp56 33550 An exportation inference. ...
exp58 33551 An exportation inference. ...
exp510 33552 An exportation inference. ...
exp511 33553 An exportation inference. ...
exp512 33554 An exportation inference. ...
3com12d 33555 Commutation in consequent....
imp5p 33556 A triple importation infer...
imp5q 33557 A triple importation infer...
ecase13d 33558 Deduction for elimination ...
subtr 33559 Transitivity of implicit s...
subtr2 33560 Transitivity of implicit s...
trer 33561 A relation intersected wit...
elicc3 33562 An equivalent membership c...
finminlem 33563 A useful lemma about finit...
gtinf 33564 Any number greater than an...
opnrebl 33565 A set is open in the stand...
opnrebl2 33566 A set is open in the stand...
nn0prpwlem 33567 Lemma for ~ nn0prpw . Use...
nn0prpw 33568 Two nonnegative integers a...
topbnd 33569 Two equivalent expressions...
opnbnd 33570 A set is open iff it is di...
cldbnd 33571 A set is closed iff it con...
ntruni 33572 A union of interiors is a ...
clsun 33573 A pairwise union of closur...
clsint2 33574 The closure of an intersec...
opnregcld 33575 A set is regularly closed ...
cldregopn 33576 A set if regularly open if...
neiin 33577 Two neighborhoods intersec...
hmeoclda 33578 Homeomorphisms preserve cl...
hmeocldb 33579 Homeomorphisms preserve cl...
ivthALT 33580 An alternate proof of the ...
fnerel 33583 Fineness is a relation. (...
isfne 33584 The predicate " ` B ` is f...
isfne4 33585 The predicate " ` B ` is f...
isfne4b 33586 A condition for a topology...
isfne2 33587 The predicate " ` B ` is f...
isfne3 33588 The predicate " ` B ` is f...
fnebas 33589 A finer cover covers the s...
fnetg 33590 A finer cover generates a ...
fnessex 33591 If ` B ` is finer than ` A...
fneuni 33592 If ` B ` is finer than ` A...
fneint 33593 If a cover is finer than a...
fness 33594 A cover is finer than its ...
fneref 33595 Reflexivity of the finenes...
fnetr 33596 Transitivity of the finene...
fneval 33597 Two covers are finer than ...
fneer 33598 Fineness intersected with ...
topfne 33599 Fineness for covers corres...
topfneec 33600 A cover is equivalent to a...
topfneec2 33601 A topology is precisely id...
fnessref 33602 A cover is finer iff it ha...
refssfne 33603 A cover is a refinement if...
neibastop1 33604 A collection of neighborho...
neibastop2lem 33605 Lemma for ~ neibastop2 . ...
neibastop2 33606 In the topology generated ...
neibastop3 33607 The topology generated by ...
topmtcl 33608 The meet of a collection o...
topmeet 33609 Two equivalent formulation...
topjoin 33610 Two equivalent formulation...
fnemeet1 33611 The meet of a collection o...
fnemeet2 33612 The meet of equivalence cl...
fnejoin1 33613 Join of equivalence classe...
fnejoin2 33614 Join of equivalence classe...
fgmin 33615 Minimality property of a g...
neifg 33616 The neighborhood filter of...
tailfval 33617 The tail function for a di...
tailval 33618 The tail of an element in ...
eltail 33619 An element of a tail. (Co...
tailf 33620 The tail function of a dir...
tailini 33621 A tail contains its initia...
tailfb 33622 The collection of tails of...
filnetlem1 33623 Lemma for ~ filnet . Chan...
filnetlem2 33624 Lemma for ~ filnet . The ...
filnetlem3 33625 Lemma for ~ filnet . (Con...
filnetlem4 33626 Lemma for ~ filnet . (Con...
filnet 33627 A filter has the same conv...
tb-ax1 33628 The first of three axioms ...
tb-ax2 33629 The second of three axioms...
tb-ax3 33630 The third of three axioms ...
tbsyl 33631 The weak syllogism from Ta...
re1ax2lem 33632 Lemma for ~ re1ax2 . (Con...
re1ax2 33633 ~ ax-2 rederived from the ...
naim1 33634 Constructor theorem for ` ...
naim2 33635 Constructor theorem for ` ...
naim1i 33636 Constructor rule for ` -/\...
naim2i 33637 Constructor rule for ` -/\...
naim12i 33638 Constructor rule for ` -/\...
nabi1i 33639 Constructor rule for ` -/\...
nabi2i 33640 Constructor rule for ` -/\...
nabi12i 33641 Constructor rule for ` -/\...
df3nandALT1 33644 The double nand expressed ...
df3nandALT2 33645 The double nand expressed ...
andnand1 33646 Double and in terms of dou...
imnand2 33647 An ` -> ` nand relation. ...
nalfal 33648 Not all sets hold ` F. ` a...
nexntru 33649 There does not exist a set...
nexfal 33650 There does not exist a set...
neufal 33651 There does not exist exact...
neutru 33652 There does not exist exact...
nmotru 33653 There does not exist at mo...
mofal 33654 There exist at most one se...
nrmo 33655 "At most one" restricted e...
meran1 33656 A single axiom for proposi...
meran2 33657 A single axiom for proposi...
meran3 33658 A single axiom for proposi...
waj-ax 33659 A single axiom for proposi...
lukshef-ax2 33660 A single axiom for proposi...
arg-ax 33661 A single axiom for proposi...
negsym1 33662 In the paper "On Variable ...
imsym1 33663 A symmetry with ` -> ` . ...
bisym1 33664 A symmetry with ` <-> ` . ...
consym1 33665 A symmetry with ` /\ ` . ...
dissym1 33666 A symmetry with ` \/ ` . ...
nandsym1 33667 A symmetry with ` -/\ ` . ...
unisym1 33668 A symmetry with ` A. ` . ...
exisym1 33669 A symmetry with ` E. ` . ...
unqsym1 33670 A symmetry with ` E! ` . ...
amosym1 33671 A symmetry with ` E* ` . ...
subsym1 33672 A symmetry with ` [ x / y ...
ontopbas 33673 An ordinal number is a top...
onsstopbas 33674 The class of ordinal numbe...
onpsstopbas 33675 The class of ordinal numbe...
ontgval 33676 The topology generated fro...
ontgsucval 33677 The topology generated fro...
onsuctop 33678 A successor ordinal number...
onsuctopon 33679 One of the topologies on a...
ordtoplem 33680 Membership of the class of...
ordtop 33681 An ordinal is a topology i...
onsucconni 33682 A successor ordinal number...
onsucconn 33683 A successor ordinal number...
ordtopconn 33684 An ordinal topology is con...
onintopssconn 33685 An ordinal topology is con...
onsuct0 33686 A successor ordinal number...
ordtopt0 33687 An ordinal topology is T_0...
onsucsuccmpi 33688 The successor of a success...
onsucsuccmp 33689 The successor of a success...
limsucncmpi 33690 The successor of a limit o...
limsucncmp 33691 The successor of a limit o...
ordcmp 33692 An ordinal topology is com...
ssoninhaus 33693 The ordinal topologies ` 1...
onint1 33694 The ordinal T_1 spaces are...
oninhaus 33695 The ordinal Hausdorff spac...
fveleq 33696 Please add description her...
findfvcl 33697 Please add description her...
findreccl 33698 Please add description her...
findabrcl 33699 Please add description her...
nnssi2 33700 Convert a theorem for real...
nnssi3 33701 Convert a theorem for real...
nndivsub 33702 Please add description her...
nndivlub 33703 A factor of a positive int...
ee7.2aOLD 33706 Lemma for Euclid's Element...
dnival 33707 Value of the "distance to ...
dnicld1 33708 Closure theorem for the "d...
dnicld2 33709 Closure theorem for the "d...
dnif 33710 The "distance to nearest i...
dnizeq0 33711 The distance to nearest in...
dnizphlfeqhlf 33712 The distance to nearest in...
rddif2 33713 Variant of ~ rddif . (Con...
dnibndlem1 33714 Lemma for ~ dnibnd . (Con...
dnibndlem2 33715 Lemma for ~ dnibnd . (Con...
dnibndlem3 33716 Lemma for ~ dnibnd . (Con...
dnibndlem4 33717 Lemma for ~ dnibnd . (Con...
dnibndlem5 33718 Lemma for ~ dnibnd . (Con...
dnibndlem6 33719 Lemma for ~ dnibnd . (Con...
dnibndlem7 33720 Lemma for ~ dnibnd . (Con...
dnibndlem8 33721 Lemma for ~ dnibnd . (Con...
dnibndlem9 33722 Lemma for ~ dnibnd . (Con...
dnibndlem10 33723 Lemma for ~ dnibnd . (Con...
dnibndlem11 33724 Lemma for ~ dnibnd . (Con...
dnibndlem12 33725 Lemma for ~ dnibnd . (Con...
dnibndlem13 33726 Lemma for ~ dnibnd . (Con...
dnibnd 33727 The "distance to nearest i...
dnicn 33728 The "distance to nearest i...
knoppcnlem1 33729 Lemma for ~ knoppcn . (Co...
knoppcnlem2 33730 Lemma for ~ knoppcn . (Co...
knoppcnlem3 33731 Lemma for ~ knoppcn . (Co...
knoppcnlem4 33732 Lemma for ~ knoppcn . (Co...
knoppcnlem5 33733 Lemma for ~ knoppcn . (Co...
knoppcnlem6 33734 Lemma for ~ knoppcn . (Co...
knoppcnlem7 33735 Lemma for ~ knoppcn . (Co...
knoppcnlem8 33736 Lemma for ~ knoppcn . (Co...
knoppcnlem9 33737 Lemma for ~ knoppcn . (Co...
knoppcnlem10 33738 Lemma for ~ knoppcn . (Co...
knoppcnlem11 33739 Lemma for ~ knoppcn . (Co...
knoppcn 33740 The continuous nowhere dif...
knoppcld 33741 Closure theorem for Knopp'...
unblimceq0lem 33742 Lemma for ~ unblimceq0 . ...
unblimceq0 33743 If ` F ` is unbounded near...
unbdqndv1 33744 If the difference quotient...
unbdqndv2lem1 33745 Lemma for ~ unbdqndv2 . (...
unbdqndv2lem2 33746 Lemma for ~ unbdqndv2 . (...
unbdqndv2 33747 Variant of ~ unbdqndv1 wit...
knoppndvlem1 33748 Lemma for ~ knoppndv . (C...
knoppndvlem2 33749 Lemma for ~ knoppndv . (C...
knoppndvlem3 33750 Lemma for ~ knoppndv . (C...
knoppndvlem4 33751 Lemma for ~ knoppndv . (C...
knoppndvlem5 33752 Lemma for ~ knoppndv . (C...
knoppndvlem6 33753 Lemma for ~ knoppndv . (C...
knoppndvlem7 33754 Lemma for ~ knoppndv . (C...
knoppndvlem8 33755 Lemma for ~ knoppndv . (C...
knoppndvlem9 33756 Lemma for ~ knoppndv . (C...
knoppndvlem10 33757 Lemma for ~ knoppndv . (C...
knoppndvlem11 33758 Lemma for ~ knoppndv . (C...
knoppndvlem12 33759 Lemma for ~ knoppndv . (C...
knoppndvlem13 33760 Lemma for ~ knoppndv . (C...
knoppndvlem14 33761 Lemma for ~ knoppndv . (C...
knoppndvlem15 33762 Lemma for ~ knoppndv . (C...
knoppndvlem16 33763 Lemma for ~ knoppndv . (C...
knoppndvlem17 33764 Lemma for ~ knoppndv . (C...
knoppndvlem18 33765 Lemma for ~ knoppndv . (C...
knoppndvlem19 33766 Lemma for ~ knoppndv . (C...
knoppndvlem20 33767 Lemma for ~ knoppndv . (C...
knoppndvlem21 33768 Lemma for ~ knoppndv . (C...
knoppndvlem22 33769 Lemma for ~ knoppndv . (C...
knoppndv 33770 The continuous nowhere dif...
knoppf 33771 Knopp's function is a func...
knoppcn2 33772 Variant of ~ knoppcn with ...
cnndvlem1 33773 Lemma for ~ cnndv . (Cont...
cnndvlem2 33774 Lemma for ~ cnndv . (Cont...
cnndv 33775 There exists a continuous ...
bj-mp2c 33776 A double modus ponens infe...
bj-mp2d 33777 A double modus ponens infe...
bj-0 33778 A syntactic theorem. See ...
bj-1 33779 In this proof, the use of ...
bj-a1k 33780 Weakening of ~ ax-1 . Thi...
bj-nnclav 33781 When ` F. ` is substituted...
bj-jarrii 33782 Inference associated with ...
bj-imim21 33783 The propositional function...
bj-imim21i 33784 Inference associated with ...
bj-peircestab 33785 Over minimal implicational...
bj-stabpeirce 33786 Over minimal implicational...
bj-syl66ib 33787 A mixed syllogism inferenc...
bj-orim2 33788 Proof of ~ orim2 from the ...
bj-currypeirce 33789 Curry's axiom ~ curryax (a...
bj-peircecurry 33790 Peirce's axiom ~ peirce im...
bj-animbi 33791 Conjunction in terms of im...
bj-currypara 33792 Curry's paradox. Note tha...
bj-con2com 33793 A commuted form of the con...
bj-con2comi 33794 Inference associated with ...
bj-pm2.01i 33795 Inference associated with ...
bj-nimn 33796 If a formula is true, then...
bj-nimni 33797 Inference associated with ...
bj-peircei 33798 Inference associated with ...
bj-looinvi 33799 Inference associated with ...
bj-looinvii 33800 Inference associated with ...
bj-jaoi1 33801 Shortens ~ orfa2 (58>53), ...
bj-jaoi2 33802 Shortens ~ consensus (110>...
bj-dfbi4 33803 Alternate definition of th...
bj-dfbi5 33804 Alternate definition of th...
bj-dfbi6 33805 Alternate definition of th...
bj-bijust0ALT 33806 Alternate proof of ~ bijus...
bj-bijust00 33807 A self-implication does no...
bj-consensus 33808 Version of ~ consensus exp...
bj-consensusALT 33809 Alternate proof of ~ bj-co...
bj-df-ifc 33810 Candidate definition for t...
bj-dfif 33811 Alternate definition of th...
bj-ififc 33812 A biconditional connecting...
bj-imbi12 33813 Uncurried (imported) form ...
bj-biorfi 33814 This should be labeled "bi...
bj-falor 33815 Dual of ~ truan (which has...
bj-falor2 33816 Dual of ~ truan . (Contri...
bj-bibibi 33817 A property of the bicondit...
bj-imn3ani 33818 Duplication of ~ bnj1224 ....
bj-andnotim 33819 Two ways of expressing a c...
bj-bi3ant 33820 This used to be in the mai...
bj-bisym 33821 This used to be in the mai...
bj-bixor 33822 Equivalence of two ternary...
bj-axdd2 33823 This implication, proved u...
bj-axd2d 33824 This implication, proved u...
bj-axtd 33825 This implication, proved f...
bj-gl4 33826 In a normal modal logic, t...
bj-axc4 33827 Over minimal calculus, the...
prvlem1 33832 An elementary property of ...
prvlem2 33833 An elementary property of ...
bj-babygodel 33834 See the section header com...
bj-babylob 33835 See the section header com...
bj-godellob 33836 Proof of Gödel's theo...
bj-genr 33837 Generalization rule on the...
bj-genl 33838 Generalization rule on the...
bj-genan 33839 Generalization rule on a c...
bj-mpgs 33840 From a closed form theorem...
bj-2alim 33841 Closed form of ~ 2alimi . ...
bj-2exim 33842 Closed form of ~ 2eximi . ...
bj-alanim 33843 Closed form of ~ alanimi ....
bj-2albi 33844 Closed form of ~ 2albii . ...
bj-notalbii 33845 Equivalence of universal q...
bj-2exbi 33846 Closed form of ~ 2exbii . ...
bj-3exbi 33847 Closed form of ~ 3exbii . ...
bj-sylgt2 33848 Uncurried (imported) form ...
bj-alrimg 33849 The general form of the *a...
bj-alrimd 33850 A slightly more general ~ ...
bj-sylget 33851 Dual statement of ~ sylgt ...
bj-sylget2 33852 Uncurried (imported) form ...
bj-exlimg 33853 The general form of the *e...
bj-sylge 33854 Dual statement of ~ sylg (...
bj-exlimd 33855 A slightly more general ~ ...
bj-nfimexal 33856 A weak from of nonfreeness...
bj-alexim 33857 Closed form of ~ aleximi ....
bj-nexdh 33858 Closed form of ~ nexdh (ac...
bj-nexdh2 33859 Uncurried (imported) form ...
bj-hbxfrbi 33860 Closed form of ~ hbxfrbi ....
bj-hbyfrbi 33861 Version of ~ bj-hbxfrbi wi...
bj-exalim 33862 Distribute quantifiers ove...
bj-exalimi 33863 An inference for distribut...
bj-exalims 33864 Distributing quantifiers o...
bj-exalimsi 33865 An inference for distribut...
bj-ax12ig 33866 A lemma used to prove a we...
bj-ax12i 33867 A weakening of ~ bj-ax12ig...
bj-nfimt 33868 Closed form of ~ nfim and ...
bj-cbvalimt 33869 A lemma in closed form use...
bj-cbveximt 33870 A lemma in closed form use...
bj-eximALT 33871 Alternate proof of ~ exim ...
bj-aleximiALT 33872 Alternate proof of ~ alexi...
bj-eximcom 33873 A commuted form of ~ exim ...
bj-ax12wlem 33874 A lemma used to prove a we...
bj-cbvalim 33875 A lemma used to prove ~ bj...
bj-cbvexim 33876 A lemma used to prove ~ bj...
bj-cbvalimi 33877 An equality-free general i...
bj-cbveximi 33878 An equality-free general i...
bj-cbval 33879 Changing a bound variable ...
bj-cbvex 33880 Changing a bound variable ...
bj-ssbeq 33883 Substitution in an equalit...
bj-ssblem1 33884 A lemma for the definiens ...
bj-ssblem2 33885 An instance of ~ ax-11 pro...
bj-ax12v 33886 A weaker form of ~ ax-12 a...
bj-ax12 33887 Remove a DV condition from...
bj-ax12ssb 33888 The axiom ~ bj-ax12 expres...
bj-19.41al 33889 Special case of ~ 19.41 pr...
bj-equsexval 33890 Special case of ~ equsexv ...
bj-sb56 33891 Proof of ~ sb56 from Tarsk...
bj-ssbid2 33892 A special case of ~ sbequ2...
bj-ssbid2ALT 33893 Alternate proof of ~ bj-ss...
bj-ssbid1 33894 A special case of ~ sbequ1...
bj-ssbid1ALT 33895 Alternate proof of ~ bj-ss...
bj-ax6elem1 33896 Lemma for ~ bj-ax6e . (Co...
bj-ax6elem2 33897 Lemma for ~ bj-ax6e . (Co...
bj-ax6e 33898 Proof of ~ ax6e (hence ~ a...
bj-spimvwt 33899 Closed form of ~ spimvw . ...
bj-spnfw 33900 Theorem close to a closed ...
bj-cbvexiw 33901 Change bound variable. Th...
bj-cbvexivw 33902 Change bound variable. Th...
bj-modald 33903 A short form of the axiom ...
bj-denot 33904 A weakening of ~ ax-6 and ...
bj-eqs 33905 A lemma for substitutions,...
bj-cbvexw 33906 Change bound variable. Th...
bj-ax12w 33907 The general statement that...
bj-ax89 33908 A theorem which could be u...
bj-elequ12 33909 An identity law for the no...
bj-cleljusti 33910 One direction of ~ cleljus...
bj-alcomexcom 33911 Commutation of universal q...
bj-hbalt 33912 Closed form of ~ hbal . W...
axc11n11 33913 Proof of ~ axc11n from { ~...
axc11n11r 33914 Proof of ~ axc11n from { ~...
bj-axc16g16 33915 Proof of ~ axc16g from { ~...
bj-ax12v3 33916 A weak version of ~ ax-12 ...
bj-ax12v3ALT 33917 Alternate proof of ~ bj-ax...
bj-sb 33918 A weak variant of ~ sbid2 ...
bj-modalbe 33919 The predicate-calculus ver...
bj-spst 33920 Closed form of ~ sps . On...
bj-19.21bit 33921 Closed form of ~ 19.21bi ....
bj-19.23bit 33922 Closed form of ~ 19.23bi ....
bj-nexrt 33923 Closed form of ~ nexr . C...
bj-alrim 33924 Closed form of ~ alrimi . ...
bj-alrim2 33925 Uncurried (imported) form ...
bj-nfdt0 33926 A theorem close to a close...
bj-nfdt 33927 Closed form of ~ nf5d and ...
bj-nexdt 33928 Closed form of ~ nexd . (...
bj-nexdvt 33929 Closed form of ~ nexdv . ...
bj-alexbiex 33930 Adding a second quantifier...
bj-exexbiex 33931 Adding a second quantifier...
bj-alalbial 33932 Adding a second quantifier...
bj-exalbial 33933 Adding a second quantifier...
bj-19.9htbi 33934 Strengthening ~ 19.9ht by ...
bj-hbntbi 33935 Strengthening ~ hbnt by re...
bj-biexal1 33936 A general FOL biconditiona...
bj-biexal2 33937 When ` ph ` is substituted...
bj-biexal3 33938 When ` ph ` is substituted...
bj-bialal 33939 When ` ph ` is substituted...
bj-biexex 33940 When ` ph ` is substituted...
bj-hbext 33941 Closed form of ~ hbex . (...
bj-nfalt 33942 Closed form of ~ nfal . (...
bj-nfext 33943 Closed form of ~ nfex . (...
bj-eeanvw 33944 Version of ~ exdistrv with...
bj-modal4 33945 First-order logic form of ...
bj-modal4e 33946 First-order logic form of ...
bj-modalb 33947 A short form of the axiom ...
bj-wnf1 33948 When ` ph ` is substituted...
bj-wnf2 33949 When ` ph ` is substituted...
bj-wnfanf 33950 When ` ph ` is substituted...
bj-wnfenf 33951 When ` ph ` is substituted...
bj-nnfbi 33954 If two formulas are equiva...
bj-nnfbd 33955 If two formulas are equiva...
bj-nnfbii 33956 If two formulas are equiva...
bj-nnfa 33957 Nonfreeness implies the eq...
bj-nnfad 33958 Nonfreeness implies the eq...
bj-nnfe 33959 Nonfreeness implies the eq...
bj-nnfed 33960 Nonfreeness implies the eq...
bj-nnfea 33961 Nonfreeness implies the eq...
bj-nnfead 33962 Nonfreeness implies the eq...
bj-dfnnf2 33963 Alternate definition of ~ ...
bj-nnfnfTEMP 33964 New nonfreeness implies ol...
bj-wnfnf 33965 When ` ph ` is substituted...
bj-nnfnt 33966 A variable is nonfree in a...
bj-nnftht 33967 A variable is nonfree in a...
bj-nnfth 33968 A variable is nonfree in a...
bj-nnfnth 33969 A variable is nonfree in t...
bj-nnfim1 33970 A consequence of nonfreene...
bj-nnfim2 33971 A consequence of nonfreene...
bj-nnfim 33972 Nonfreeness in the anteced...
bj-nnfimd 33973 Nonfreeness in the anteced...
bj-nnfan 33974 Nonfreeness in both conjun...
bj-nnfand 33975 Nonfreeness in both conjun...
bj-nnfor 33976 Nonfreeness in both disjun...
bj-nnford 33977 Nonfreeness in both disjun...
bj-nnfbit 33978 Nonfreeness in both sides ...
bj-nnfbid 33979 Nonfreeness in both sides ...
bj-nnfv 33980 A non-occurring variable i...
bj-nnf-alrim 33981 Proof of the closed form o...
bj-nnf-exlim 33982 Proof of the closed form o...
bj-dfnnf3 33983 Alternate definition of no...
bj-nfnnfTEMP 33984 New nonfreeness is equival...
bj-nnfa1 33985 See ~ nfa1 . (Contributed...
bj-nnfe1 33986 See ~ nfe1 . (Contributed...
bj-19.12 33987 See ~ 19.12 . Could be la...
bj-nnflemaa 33988 One of four lemmas for non...
bj-nnflemee 33989 One of four lemmas for non...
bj-nnflemae 33990 One of four lemmas for non...
bj-nnflemea 33991 One of four lemmas for non...
bj-nnfalt 33992 See ~ nfal and ~ bj-nfalt ...
bj-nnfext 33993 See ~ nfex and ~ bj-nfext ...
bj-stdpc5t 33994 Alias of ~ bj-nnf-alrim fo...
bj-19.21t 33995 Statement ~ 19.21t proved ...
bj-19.23t 33996 Statement ~ 19.23t proved ...
bj-19.36im 33997 One direction of ~ 19.36 f...
bj-19.37im 33998 One direction of ~ 19.37 f...
bj-19.42t 33999 Closed form of ~ 19.42 fro...
bj-19.41t 34000 Closed form of ~ 19.41 fro...
bj-sbft 34001 Version of ~ sbft using ` ...
bj-axc10 34002 Alternate (shorter) proof ...
bj-alequex 34003 A fol lemma. See ~ aleque...
bj-spimt2 34004 A step in the proof of ~ s...
bj-cbv3ta 34005 Closed form of ~ cbv3 . (...
bj-cbv3tb 34006 Closed form of ~ cbv3 . (...
bj-hbsb3t 34007 A theorem close to a close...
bj-hbsb3 34008 Shorter proof of ~ hbsb3 ....
bj-nfs1t 34009 A theorem close to a close...
bj-nfs1t2 34010 A theorem close to a close...
bj-nfs1 34011 Shorter proof of ~ nfs1 (t...
bj-axc10v 34012 Version of ~ axc10 with a ...
bj-spimtv 34013 Version of ~ spimt with a ...
bj-cbv3hv2 34014 Version of ~ cbv3h with tw...
bj-cbv1hv 34015 Version of ~ cbv1h with a ...
bj-cbv2hv 34016 Version of ~ cbv2h with a ...
bj-cbv2v 34017 Version of ~ cbv2 with a d...
bj-cbvaldv 34018 Version of ~ cbvald with a...
bj-cbvexdv 34019 Version of ~ cbvexd with a...
bj-cbval2vv 34020 Version of ~ cbval2vv with...
bj-cbvex2vv 34021 Version of ~ cbvex2vv with...
bj-cbvaldvav 34022 Version of ~ cbvaldva with...
bj-cbvexdvav 34023 Version of ~ cbvexdva with...
bj-cbvex4vv 34024 Version of ~ cbvex4v with ...
bj-equsalhv 34025 Version of ~ equsalh with ...
bj-axc11nv 34026 Version of ~ axc11n with a...
bj-aecomsv 34027 Version of ~ aecoms with a...
bj-axc11v 34028 Version of ~ axc11 with a ...
bj-drnf2v 34029 Version of ~ drnf2 with a ...
bj-equs45fv 34030 Version of ~ equs45f with ...
bj-hbs1 34031 Version of ~ hbsb2 with a ...
bj-nfs1v 34032 Version of ~ nfsb2 with a ...
bj-hbsb2av 34033 Version of ~ hbsb2a with a...
bj-hbsb3v 34034 Version of ~ hbsb3 with a ...
bj-nfsab1 34035 Remove dependency on ~ ax-...
bj-dtru 34036 Remove dependency on ~ ax-...
bj-dtrucor2v 34037 Version of ~ dtrucor2 with...
bj-hbaeb2 34038 Biconditional version of a...
bj-hbaeb 34039 Biconditional version of ~...
bj-hbnaeb 34040 Biconditional version of ~...
bj-dvv 34041 A special instance of ~ bj...
bj-equsal1t 34042 Duplication of ~ wl-equsal...
bj-equsal1ti 34043 Inference associated with ...
bj-equsal1 34044 One direction of ~ equsal ...
bj-equsal2 34045 One direction of ~ equsal ...
bj-equsal 34046 Shorter proof of ~ equsal ...
stdpc5t 34047 Closed form of ~ stdpc5 . ...
bj-stdpc5 34048 More direct proof of ~ std...
2stdpc5 34049 A double ~ stdpc5 (one dir...
bj-19.21t0 34050 Proof of ~ 19.21t from ~ s...
exlimii 34051 Inference associated with ...
ax11-pm 34052 Proof of ~ ax-11 similar t...
ax6er 34053 Commuted form of ~ ax6e . ...
exlimiieq1 34054 Inferring a theorem when i...
exlimiieq2 34055 Inferring a theorem when i...
ax11-pm2 34056 Proof of ~ ax-11 from the ...
bj-sbsb 34057 Biconditional showing two ...
bj-dfsb2 34058 Alternate (dual) definitio...
bj-sbf3 34059 Substitution has no effect...
bj-sbf4 34060 Substitution has no effect...
bj-sbnf 34061 Move nonfree predicate in ...
bj-eu3f 34062 Version of ~ eu3v where th...
bj-sblem1 34063 Lemma for substitution. (...
bj-sblem2 34064 Lemma for substitution. (...
bj-sblem 34065 Lemma for substitution. (...
bj-sbievw1 34066 Lemma for substitution. (...
bj-sbievw2 34067 Lemma for substitution. (...
bj-sbievw 34068 Lemma for substitution. C...
bj-sbievv 34069 Version of ~ sbie with a s...
bj-moeub 34070 Uniqueness is equivalent t...
bj-sbidmOLD 34071 Obsolete proof of ~ sbidm ...
bj-dvelimdv 34072 Deduction form of ~ dvelim...
bj-dvelimdv1 34073 Curried (exported) form of...
bj-dvelimv 34074 A version of ~ dvelim usin...
bj-nfeel2 34075 Nonfreeness in a membershi...
bj-axc14nf 34076 Proof of a version of ~ ax...
bj-axc14 34077 Alternate proof of ~ axc14...
mobidvALT 34078 Alternate proof of ~ mobid...
eliminable1 34079 A theorem used to prove th...
eliminable2a 34080 A theorem used to prove th...
eliminable2b 34081 A theorem used to prove th...
eliminable2c 34082 A theorem used to prove th...
eliminable3a 34083 A theorem used to prove th...
eliminable3b 34084 A theorem used to prove th...
bj-denotes 34085 This would be the justific...
bj-issetwt 34086 Closed form of ~ bj-issetw...
bj-issetw 34087 The closest one can get to...
bj-elissetv 34088 Version of ~ bj-elisset wi...
bj-elisset 34089 Remove from ~ elisset depe...
bj-issetiv 34090 Version of ~ bj-isseti wit...
bj-isseti 34091 Remove from ~ isseti depen...
bj-ralvw 34092 A weak version of ~ ralv n...
bj-rexvw 34093 A weak version of ~ rexv n...
bj-rababw 34094 A weak version of ~ rabab ...
bj-rexcom4bv 34095 Version of ~ rexcom4b and ...
bj-rexcom4b 34096 Remove from ~ rexcom4b dep...
bj-ceqsalt0 34097 The FOL content of ~ ceqsa...
bj-ceqsalt1 34098 The FOL content of ~ ceqsa...
bj-ceqsalt 34099 Remove from ~ ceqsalt depe...
bj-ceqsaltv 34100 Version of ~ bj-ceqsalt wi...
bj-ceqsalg0 34101 The FOL content of ~ ceqsa...
bj-ceqsalg 34102 Remove from ~ ceqsalg depe...
bj-ceqsalgALT 34103 Alternate proof of ~ bj-ce...
bj-ceqsalgv 34104 Version of ~ bj-ceqsalg wi...
bj-ceqsalgvALT 34105 Alternate proof of ~ bj-ce...
bj-ceqsal 34106 Remove from ~ ceqsal depen...
bj-ceqsalv 34107 Remove from ~ ceqsalv depe...
bj-spcimdv 34108 Remove from ~ spcimdv depe...
bj-spcimdvv 34109 Remove from ~ spcimdv depe...
elelb 34110 Equivalence between two co...
bj-pwvrelb 34111 Characterization of the el...
bj-nfcsym 34112 The nonfreeness quantifier...
bj-ax9 34113 Proof of ~ ax-9 from Tarsk...
bj-sbeqALT 34114 Substitution in an equalit...
bj-sbeq 34115 Distribute proper substitu...
bj-sbceqgALT 34116 Distribute proper substitu...
bj-csbsnlem 34117 Lemma for ~ bj-csbsn (in t...
bj-csbsn 34118 Substitution in a singleto...
bj-sbel1 34119 Version of ~ sbcel1g when ...
bj-abv 34120 The class of sets verifyin...
bj-ab0 34121 The class of sets verifyin...
bj-abf 34122 Shorter proof of ~ abf (wh...
bj-csbprc 34123 More direct proof of ~ csb...
bj-exlimvmpi 34124 A Fol lemma ( ~ exlimiv fo...
bj-exlimmpi 34125 Lemma for ~ bj-vtoclg1f1 (...
bj-exlimmpbi 34126 Lemma for theorems of the ...
bj-exlimmpbir 34127 Lemma for theorems of the ...
bj-vtoclf 34128 Remove dependency on ~ ax-...
bj-vtocl 34129 Remove dependency on ~ ax-...
bj-vtoclg1f1 34130 The FOL content of ~ vtocl...
bj-vtoclg1f 34131 Reprove ~ vtoclg1f from ~ ...
bj-vtoclg1fv 34132 Version of ~ bj-vtoclg1f w...
bj-vtoclg 34133 A version of ~ vtoclg with...
bj-rabbida2 34134 Version of ~ rabbidva2 wit...
bj-rabeqd 34135 Deduction form of ~ rabeq ...
bj-rabeqbid 34136 Version of ~ rabeqbidv wit...
bj-rabeqbida 34137 Version of ~ rabeqbidva wi...
bj-seex 34138 Version of ~ seex with a d...
bj-nfcf 34139 Version of ~ df-nfc with a...
bj-zfauscl 34140 General version of ~ zfaus...
bj-unrab 34141 Generalization of ~ unrab ...
bj-inrab 34142 Generalization of ~ inrab ...
bj-inrab2 34143 Shorter proof of ~ inrab ....
bj-inrab3 34144 Generalization of ~ dfrab3...
bj-rabtr 34145 Restricted class abstracti...
bj-rabtrALT 34146 Alternate proof of ~ bj-ra...
bj-rabtrAUTO 34147 Proof of ~ bj-rabtr found ...
bj-ru0 34150 The FOL part of Russell's ...
bj-ru1 34151 A version of Russell's par...
bj-ru 34152 Remove dependency on ~ ax-...
currysetlem 34153 Lemma for ~ currysetlem , ...
curryset 34154 Curry's paradox in set the...
currysetlem1 34155 Lemma for ~ currysetALT . ...
currysetlem2 34156 Lemma for ~ currysetALT . ...
currysetlem3 34157 Lemma for ~ currysetALT . ...
currysetALT 34158 Alternate proof of ~ curry...
bj-n0i 34159 Inference associated with ...
bj-disjcsn 34160 A class is disjoint from i...
bj-disjsn01 34161 Disjointness of the single...
bj-0nel1 34162 The empty set does not bel...
bj-1nel0 34163 ` 1o ` does not belong to ...
bj-xpimasn 34164 The image of a singleton, ...
bj-xpima1sn 34165 The image of a singleton b...
bj-xpima1snALT 34166 Alternate proof of ~ bj-xp...
bj-xpima2sn 34167 The image of a singleton b...
bj-xpnzex 34168 If the first factor of a p...
bj-xpexg2 34169 Curried (exported) form of...
bj-xpnzexb 34170 If the first factor of a p...
bj-cleq 34171 Substitution property for ...
bj-snsetex 34172 The class of sets "whose s...
bj-clex 34173 Sethood of certain classes...
bj-sngleq 34176 Substitution property for ...
bj-elsngl 34177 Characterization of the el...
bj-snglc 34178 Characterization of the el...
bj-snglss 34179 The singletonization of a ...
bj-0nelsngl 34180 The empty set is not a mem...
bj-snglinv 34181 Inverse of singletonizatio...
bj-snglex 34182 A class is a set if and on...
bj-tageq 34185 Substitution property for ...
bj-eltag 34186 Characterization of the el...
bj-0eltag 34187 The empty set belongs to t...
bj-tagn0 34188 The tagging of a class is ...
bj-tagss 34189 The tagging of a class is ...
bj-snglsstag 34190 The singletonization is in...
bj-sngltagi 34191 The singletonization is in...
bj-sngltag 34192 The singletonization and t...
bj-tagci 34193 Characterization of the el...
bj-tagcg 34194 Characterization of the el...
bj-taginv 34195 Inverse of tagging. (Cont...
bj-tagex 34196 A class is a set if and on...
bj-xtageq 34197 The products of a given cl...
bj-xtagex 34198 The product of a set and t...
bj-projeq 34201 Substitution property for ...
bj-projeq2 34202 Substitution property for ...
bj-projun 34203 The class projection on a ...
bj-projex 34204 Sethood of the class proje...
bj-projval 34205 Value of the class project...
bj-1upleq 34208 Substitution property for ...
bj-pr1eq 34211 Substitution property for ...
bj-pr1un 34212 The first projection prese...
bj-pr1val 34213 Value of the first project...
bj-pr11val 34214 Value of the first project...
bj-pr1ex 34215 Sethood of the first proje...
bj-1uplth 34216 The characteristic propert...
bj-1uplex 34217 A monuple is a set if and ...
bj-1upln0 34218 A monuple is nonempty. (C...
bj-2upleq 34221 Substitution property for ...
bj-pr21val 34222 Value of the first project...
bj-pr2eq 34225 Substitution property for ...
bj-pr2un 34226 The second projection pres...
bj-pr2val 34227 Value of the second projec...
bj-pr22val 34228 Value of the second projec...
bj-pr2ex 34229 Sethood of the second proj...
bj-2uplth 34230 The characteristic propert...
bj-2uplex 34231 A couple is a set if and o...
bj-2upln0 34232 A couple is nonempty. (Co...
bj-2upln1upl 34233 A couple is never equal to...
bj-rcleqf 34234 Relative version of ~ cleq...
bj-rcleq 34235 Relative version of ~ dfcl...
bj-reabeq 34236 Relative form of ~ abeq2 ....
bj-disj2r 34237 Relative version of ~ ssdi...
bj-sscon 34238 Contraposition law for rel...
bj-elpwg 34239 If the intersection of two...
bj-vjust 34240 Justification theorem for ...
bj-df-v 34241 Alternate definition of th...
bj-df-nul 34242 Alternate definition of th...
bj-nul 34243 Two formulations of the ax...
bj-nuliota 34244 Definition of the empty se...
bj-nuliotaALT 34245 Alternate proof of ~ bj-nu...
bj-vtoclgfALT 34246 Alternate proof of ~ vtocl...
bj-elsn12g 34247 Join of ~ elsng and ~ elsn...
bj-elsnb 34248 Biconditional version of ~...
bj-pwcfsdom 34249 Remove hypothesis from ~ p...
bj-grur1 34250 Remove hypothesis from ~ g...
bj-bm1.3ii 34251 The extension of a predica...
bj-0nelopab 34252 The empty set is never an ...
bj-brrelex12ALT 34253 Two classes related by a b...
bj-epelg 34254 The membership relation an...
bj-epelb 34255 Two classes are related by...
bj-nsnid 34256 A set does not contain the...
bj-evaleq 34257 Equality theorem for the `...
bj-evalfun 34258 The evaluation at a class ...
bj-evalfn 34259 The evaluation at a class ...
bj-evalval 34260 Value of the evaluation at...
bj-evalid 34261 The evaluation at a set of...
bj-ndxarg 34262 Proof of ~ ndxarg from ~ b...
bj-evalidval 34263 Closed general form of ~ s...
bj-rest00 34266 An elementwise intersectio...
bj-restsn 34267 An elementwise intersectio...
bj-restsnss 34268 Special case of ~ bj-rests...
bj-restsnss2 34269 Special case of ~ bj-rests...
bj-restsn0 34270 An elementwise intersectio...
bj-restsn10 34271 Special case of ~ bj-rests...
bj-restsnid 34272 The elementwise intersecti...
bj-rest10 34273 An elementwise intersectio...
bj-rest10b 34274 Alternate version of ~ bj-...
bj-restn0 34275 An elementwise intersectio...
bj-restn0b 34276 Alternate version of ~ bj-...
bj-restpw 34277 The elementwise intersecti...
bj-rest0 34278 An elementwise intersectio...
bj-restb 34279 An elementwise intersectio...
bj-restv 34280 An elementwise intersectio...
bj-resta 34281 An elementwise intersectio...
bj-restuni 34282 The union of an elementwis...
bj-restuni2 34283 The union of an elementwis...
bj-restreg 34284 A reformulation of the axi...
bj-intss 34285 A nonempty intersection of...
bj-raldifsn 34286 All elements in a set sati...
bj-0int 34287 If ` A ` is a collection o...
bj-mooreset 34288 A Moore collection is a se...
bj-ismoore 34291 Characterization of Moore ...
bj-ismoored0 34292 Necessary condition to be ...
bj-ismoored 34293 Necessary condition to be ...
bj-ismoored2 34294 Necessary condition to be ...
bj-ismooredr 34295 Sufficient condition to be...
bj-ismooredr2 34296 Sufficient condition to be...
bj-discrmoore 34297 The powerclass ` ~P A ` is...
bj-0nmoore 34298 The empty set is not a Moo...
bj-snmoore 34299 A singleton is a Moore col...
bj-0nelmpt 34300 The empty set is not an el...
bj-mptval 34301 Value of a function given ...
bj-dfmpoa 34302 An equivalent definition o...
bj-mpomptALT 34303 Alternate proof of ~ mpomp...
setsstrset 34320 Relation between ~ df-sets...
bj-nfald 34321 Variant of ~ nfald . (Con...
bj-nfexd 34322 Variant of ~ nfexd . (Con...
copsex2d 34323 Implicit substitution dedu...
copsex2b 34324 Biconditional form of ~ co...
opelopabd 34325 Membership of an ordere pa...
opelopabb 34326 Membership of an ordered p...
opelopabbv 34327 Membership of an ordered p...
bj-opelrelex 34328 The coordinates of an orde...
bj-opelresdm 34329 If an ordered pair is in a...
bj-brresdm 34330 If two classes are related...
brabd0 34331 Expressing that two sets a...
brabd 34332 Expressing that two sets a...
bj-brab2a1 34333 "Unbounded" version of ~ b...
bj-opabssvv 34334 A variant of ~ relopabiv (...
bj-funidres 34335 The restricted identity re...
bj-opelidb 34336 Characterization of the or...
bj-opelidb1 34337 Characterization of the or...
bj-inexeqex 34338 Lemma for ~ bj-opelid (but...
bj-elsn0 34339 If the intersection of two...
bj-opelid 34340 Characterization of the or...
bj-ideqg 34341 Characterization of the cl...
bj-ideqgALT 34342 Alternate proof of ~ bj-id...
bj-ideqb 34343 Characterization of classe...
bj-idres 34344 Alternate expression for t...
bj-opelidres 34345 Characterization of the or...
bj-idreseq 34346 Sufficient condition for t...
bj-idreseqb 34347 Characterization for two c...
bj-ideqg1 34348 For sets, the identity rel...
bj-ideqg1ALT 34349 Alternate proof of bj-ideq...
bj-opelidb1ALT 34350 Characterization of the co...
bj-elid3 34351 Characterization of the co...
bj-elid4 34352 Characterization of the el...
bj-elid5 34353 Characterization of the el...
bj-elid6 34354 Characterization of the el...
bj-elid7 34355 Characterization of the el...
bj-diagval 34358 Value of the funtionalized...
bj-diagval2 34359 Value of the funtionalized...
bj-eldiag 34360 Characterization of the el...
bj-eldiag2 34361 Characterization of the el...
bj-imdirval 34364 Value of the functionalize...
bj-imdirval2 34365 Value of the functionalize...
bj-imdirval3 34366 Value of the functionalize...
bj-imdirid 34367 Functorial property of the...
bj-inftyexpitaufo 34376 The function ` inftyexpita...
bj-inftyexpitaudisj 34379 An element of the circle a...
bj-inftyexpiinv 34382 Utility theorem for the in...
bj-inftyexpiinj 34383 Injectivity of the paramet...
bj-inftyexpidisj 34384 An element of the circle a...
bj-ccinftydisj 34387 The circle at infinity is ...
bj-elccinfty 34388 A lemma for infinite exten...
bj-ccssccbar 34391 Complex numbers are extend...
bj-ccinftyssccbar 34392 Infinite extended complex ...
bj-pinftyccb 34395 The class ` pinfty ` is an...
bj-pinftynrr 34396 The extended complex numbe...
bj-minftyccb 34399 The class ` minfty ` is an...
bj-minftynrr 34400 The extended complex numbe...
bj-pinftynminfty 34401 The extended complex numbe...
bj-rrhatsscchat 34410 The real projective line i...
bj-imafv 34425 If the direct image of a s...
bj-funun 34426 Value of a function expres...
bj-fununsn1 34427 Value of a function expres...
bj-fununsn2 34428 Value of a function expres...
bj-fvsnun1 34429 The value of a function wi...
bj-fvsnun2 34430 The value of a function wi...
bj-fvmptunsn1 34431 Value of a function expres...
bj-fvmptunsn2 34432 Value of a function expres...
bj-iomnnom 34433 The canonical bijection fr...
bj-cmnssmnd 34442 Commutative monoids are mo...
bj-cmnssmndel 34443 Commutative monoids are mo...
bj-grpssmnd 34444 Groups are monoids. (Cont...
bj-grpssmndel 34445 Groups are monoids (elemen...
bj-ablssgrp 34446 Abelian groups are groups....
bj-ablssgrpel 34447 Abelian groups are groups ...
bj-ablsscmn 34448 Abelian groups are commuta...
bj-ablsscmnel 34449 Abelian groups are commuta...
bj-modssabl 34450 (The additive groups of) m...
bj-vecssmod 34451 Vector spaces are modules....
bj-vecssmodel 34452 Vector spaces are modules ...
bj-finsumval0 34455 Value of a finite sum. (C...
bj-fvimacnv0 34456 Variant of ~ fvimacnv wher...
bj-isvec 34457 The predicate "is a vector...
bj-flddrng 34458 Fields are division rings....
bj-rrdrg 34459 The field of real numbers ...
bj-isclm 34460 The predicate "is a subcom...
bj-isrvec 34463 The predicate "is a real v...
bj-rvecmod 34464 Real vector spaces are mod...
bj-rvecssmod 34465 Real vector spaces are mod...
bj-rvecrr 34466 The field of scalars of a ...
bj-isrvecd 34467 The predicate "is a real v...
bj-rvecvec 34468 Real vector spaces are vec...
bj-isrvec2 34469 The predicate "is a real v...
bj-rvecssvec 34470 Real vector spaces are vec...
bj-rveccmod 34471 Real vector spaces are sub...
bj-rvecsscmod 34472 Real vector spaces are sub...
bj-rvecsscvec 34473 Real vector spaces are sub...
bj-rveccvec 34474 Real vector spaces are sub...
bj-rvecssabl 34475 (The additive groups of) r...
bj-rvecabl 34476 (The additive groups of) r...
bj-subcom 34477 A consequence of commutati...
bj-lineqi 34478 Solution of a (scalar) lin...
bj-bary1lem 34479 Lemma for ~ bj-bary1 : exp...
bj-bary1lem1 34480 Lemma for bj-bary1: comput...
bj-bary1 34481 Barycentric coordinates in...
taupilem3 34482 Lemma for tau-related theo...
taupilemrplb 34483 A set of positive reals ha...
taupilem1 34484 Lemma for ~ taupi . A pos...
taupilem2 34485 Lemma for ~ taupi . The s...
taupi 34486 Relationship between ` _ta...
dfgcd3 34487 Alternate definition of th...
csbdif 34488 Distribution of class subs...
csbpredg 34489 Move class substitution in...
csbwrecsg 34490 Move class substitution in...
csbrecsg 34491 Move class substitution in...
csbrdgg 34492 Move class substitution in...
csboprabg 34493 Move class substitution in...
csbmpo123 34494 Move class substitution in...
con1bii2 34495 A contraposition inference...
con2bii2 34496 A contraposition inference...
vtoclefex 34497 Implicit substitution of a...
rnmptsn 34498 The range of a function ma...
f1omptsnlem 34499 This is the core of the pr...
f1omptsn 34500 A function mapping to sing...
mptsnunlem 34501 This is the core of the pr...
mptsnun 34502 A class ` B ` is equal to ...
dissneqlem 34503 This is the core of the pr...
dissneq 34504 Any topology that contains...
exlimim 34505 Closed form of ~ exlimimd ...
exlimimd 34506 Existential elimination ru...
exellim 34507 Closed form of ~ exellimdd...
exellimddv 34508 Eliminate an antecedent wh...
topdifinfindis 34509 Part of Exercise 3 of [Mun...
topdifinffinlem 34510 This is the core of the pr...
topdifinffin 34511 Part of Exercise 3 of [Mun...
topdifinf 34512 Part of Exercise 3 of [Mun...
topdifinfeq 34513 Two different ways of defi...
icorempo 34514 Closed-below, open-above i...
icoreresf 34515 Closed-below, open-above i...
icoreval 34516 Value of the closed-below,...
icoreelrnab 34517 Elementhood in the set of ...
isbasisrelowllem1 34518 Lemma for ~ isbasisrelowl ...
isbasisrelowllem2 34519 Lemma for ~ isbasisrelowl ...
icoreclin 34520 The set of closed-below, o...
isbasisrelowl 34521 The set of all closed-belo...
icoreunrn 34522 The union of all closed-be...
istoprelowl 34523 The set of all closed-belo...
icoreelrn 34524 A class abstraction which ...
iooelexlt 34525 An element of an open inte...
relowlssretop 34526 The lower limit topology o...
relowlpssretop 34527 The lower limit topology o...
sucneqond 34528 Inequality of an ordinal s...
sucneqoni 34529 Inequality of an ordinal s...
onsucuni3 34530 If an ordinal number has a...
1oequni2o 34531 The ordinal number ` 1o ` ...
rdgsucuni 34532 If an ordinal number has a...
rdgeqoa 34533 If a recursive function wi...
elxp8 34534 Membership in a Cartesian ...
cbveud 34535 Deduction used to change b...
cbvreud 34536 Deduction used to change b...
difunieq 34537 The difference of unions i...
inunissunidif 34538 Theorem about subsets of t...
rdgellim 34539 Elementhood in a recursive...
rdglimss 34540 A recursive definition at ...
rdgssun 34541 In a recursive definition ...
exrecfnlem 34542 Lemma for ~ exrecfn . (Co...
exrecfn 34543 Theorem about the existenc...
exrecfnpw 34544 For any base set, a set wh...
finorwe 34545 If the Axiom of Infinity i...
dffinxpf 34548 This theorem is the same a...
finxpeq1 34549 Equality theorem for Carte...
finxpeq2 34550 Equality theorem for Carte...
csbfinxpg 34551 Distribute proper substitu...
finxpreclem1 34552 Lemma for ` ^^ ` recursion...
finxpreclem2 34553 Lemma for ` ^^ ` recursion...
finxp0 34554 The value of Cartesian exp...
finxp1o 34555 The value of Cartesian exp...
finxpreclem3 34556 Lemma for ` ^^ ` recursion...
finxpreclem4 34557 Lemma for ` ^^ ` recursion...
finxpreclem5 34558 Lemma for ` ^^ ` recursion...
finxpreclem6 34559 Lemma for ` ^^ ` recursion...
finxpsuclem 34560 Lemma for ~ finxpsuc . (C...
finxpsuc 34561 The value of Cartesian exp...
finxp2o 34562 The value of Cartesian exp...
finxp3o 34563 The value of Cartesian exp...
finxpnom 34564 Cartesian exponentiation w...
finxp00 34565 Cartesian exponentiation o...
iunctb2 34566 Using the axiom of countab...
domalom 34567 A class which dominates ev...
isinf2 34568 The converse of ~ isinf . ...
ctbssinf 34569 Using the axiom of choice,...
ralssiun 34570 The index set of an indexe...
nlpineqsn 34571 For every point ` p ` of a...
nlpfvineqsn 34572 Given a subset ` A ` of ` ...
fvineqsnf1 34573 A theorem about functions ...
fvineqsneu 34574 A theorem about functions ...
fvineqsneq 34575 A theorem about functions ...
pibp16 34576 Property P000016 of pi-bas...
pibp19 34577 Property P000019 of pi-bas...
pibp21 34578 Property P000021 of pi-bas...
pibt1 34579 Theorem T000001 of pi-base...
pibt2 34580 Theorem T000002 of pi-base...
wl-section-prop 34581 Intuitionistic logic is no...
wl-section-boot 34585 In this section, I provide...
wl-luk-imim1i 34586 Inference adding common co...
wl-luk-syl 34587 An inference version of th...
wl-luk-imtrid 34588 A syllogism rule of infere...
wl-luk-pm2.18d 34589 Deduction based on reducti...
wl-luk-con4i 34590 Inference rule. Copy of ~...
wl-luk-pm2.24i 34591 Inference rule. Copy of ~...
wl-luk-a1i 34592 Inference rule. Copy of ~...
wl-luk-mpi 34593 A nested modus ponens infe...
wl-luk-imim2i 34594 Inference adding common an...
wl-luk-imtrdi 34595 A syllogism rule of infere...
wl-luk-ax3 34596 ~ ax-3 proved from Lukasie...
wl-luk-ax1 34597 ~ ax-1 proved from Lukasie...
wl-luk-pm2.27 34598 This theorem, called "Asse...
wl-luk-com12 34599 Inference that swaps (comm...
wl-luk-pm2.21 34600 From a wff and its negatio...
wl-luk-con1i 34601 A contraposition inference...
wl-luk-ja 34602 Inference joining the ante...
wl-luk-imim2 34603 A closed form of syllogism...
wl-luk-a1d 34604 Deduction introducing an e...
wl-luk-ax2 34605 ~ ax-2 proved from Lukasie...
wl-luk-id 34606 Principle of identity. Th...
wl-luk-notnotr 34607 Converse of double negatio...
wl-luk-pm2.04 34608 Swap antecedents. Theorem...
wl-section-impchain 34609 An implication like ` ( ps...
wl-impchain-mp-x 34610 This series of theorems pr...
wl-impchain-mp-0 34611 This theorem is the start ...
wl-impchain-mp-1 34612 This theorem is in fact a ...
wl-impchain-mp-2 34613 This theorem is in fact a ...
wl-impchain-com-1.x 34614 It is often convenient to ...
wl-impchain-com-1.1 34615 A degenerate form of antec...
wl-impchain-com-1.2 34616 This theorem is in fact a ...
wl-impchain-com-1.3 34617 This theorem is in fact a ...
wl-impchain-com-1.4 34618 This theorem is in fact a ...
wl-impchain-com-n.m 34619 This series of theorems al...
wl-impchain-com-2.3 34620 This theorem is in fact a ...
wl-impchain-com-2.4 34621 This theorem is in fact a ...
wl-impchain-com-3.2.1 34622 This theorem is in fact a ...
wl-impchain-a1-x 34623 If an implication chain is...
wl-impchain-a1-1 34624 Inference rule, a copy of ...
wl-impchain-a1-2 34625 Inference rule, a copy of ...
wl-impchain-a1-3 34626 Inference rule, a copy of ...
wl-ax13lem1 34628 A version of ~ ax-wl-13v w...
wl-mps 34629 Replacing a nested consequ...
wl-syls1 34630 Replacing a nested consequ...
wl-syls2 34631 Replacing a nested anteced...
wl-embant 34632 A true wff can always be a...
wl-orel12 34633 In a conjunctive normal fo...
wl-cases2-dnf 34634 A particular instance of ~...
wl-cbvmotv 34635 Change bound variable. Us...
wl-moteq 34636 Change bound variable. Us...
wl-motae 34637 Change bound variable. Us...
wl-moae 34638 Two ways to express "at mo...
wl-euae 34639 Two ways to express "exact...
wl-nax6im 34640 The following series of th...
wl-hbae1 34641 This specialization of ~ h...
wl-naevhba1v 34642 An instance of ~ hbn1w app...
wl-spae 34643 Prove an instance of ~ sp ...
wl-speqv 34644 Under the assumption ` -. ...
wl-19.8eqv 34645 Under the assumption ` -. ...
wl-19.2reqv 34646 Under the assumption ` -. ...
wl-nfalv 34647 If ` x ` is not present in...
wl-nfimf1 34648 An antecedent is irrelevan...
wl-nfae1 34649 Unlike ~ nfae , this speci...
wl-nfnae1 34650 Unlike ~ nfnae , this spec...
wl-aetr 34651 A transitive law for varia...
wl-dral1d 34652 A version of ~ dral1 with ...
wl-cbvalnaed 34653 ~ wl-cbvalnae with a conte...
wl-cbvalnae 34654 A more general version of ...
wl-exeq 34655 The semantics of ` E. x y ...
wl-aleq 34656 The semantics of ` A. x y ...
wl-nfeqfb 34657 Extend ~ nfeqf to an equiv...
wl-nfs1t 34658 If ` y ` is not free in ` ...
wl-equsalvw 34659 Version of ~ equsalv with ...
wl-equsald 34660 Deduction version of ~ equ...
wl-equsal 34661 A useful equivalence relat...
wl-equsal1t 34662 The expression ` x = y ` i...
wl-equsalcom 34663 This simple equivalence ea...
wl-equsal1i 34664 The antecedent ` x = y ` i...
wl-sb6rft 34665 A specialization of ~ wl-e...
wl-cbvalsbi 34666 Change bounded variables i...
wl-sbrimt 34667 Substitution with a variab...
wl-sblimt 34668 Substitution with a variab...
wl-sb8t 34669 Substitution of variable i...
wl-sb8et 34670 Substitution of variable i...
wl-sbhbt 34671 Closed form of ~ sbhb . C...
wl-sbnf1 34672 Two ways expressing that `...
wl-equsb3 34673 ~ equsb3 with a distinctor...
wl-equsb4 34674 Substitution applied to an...
wl-2sb6d 34675 Version of ~ 2sb6 with a c...
wl-sbcom2d-lem1 34676 Lemma used to prove ~ wl-s...
wl-sbcom2d-lem2 34677 Lemma used to prove ~ wl-s...
wl-sbcom2d 34678 Version of ~ sbcom2 with a...
wl-sbalnae 34679 A theorem used in eliminat...
wl-sbal1 34680 A theorem used in eliminat...
wl-sbal2 34681 Move quantifier in and out...
wl-2spsbbi 34682 ~ spsbbi applied twice. (...
wl-lem-exsb 34683 This theorem provides a ba...
wl-lem-nexmo 34684 This theorem provides a ba...
wl-lem-moexsb 34685 The antecedent ` A. x ( ph...
wl-alanbii 34686 This theorem extends ~ ala...
wl-mo2df 34687 Version of ~ mof with a co...
wl-mo2tf 34688 Closed form of ~ mof with ...
wl-eudf 34689 Version of ~ eu6 with a co...
wl-eutf 34690 Closed form of ~ eu6 with ...
wl-euequf 34691 ~ euequ proved with a dist...
wl-mo2t 34692 Closed form of ~ mof . (C...
wl-mo3t 34693 Closed form of ~ mo3 . (C...
wl-sb8eut 34694 Substitution of variable i...
wl-sb8mot 34695 Substitution of variable i...
wl-axc11rc11 34696 Proving ~ axc11r from ~ ax...
wl-ax11-lem1 34698 A transitive law for varia...
wl-ax11-lem2 34699 Lemma. (Contributed by Wo...
wl-ax11-lem3 34700 Lemma. (Contributed by Wo...
wl-ax11-lem4 34701 Lemma. (Contributed by Wo...
wl-ax11-lem5 34702 Lemma. (Contributed by Wo...
wl-ax11-lem6 34703 Lemma. (Contributed by Wo...
wl-ax11-lem7 34704 Lemma. (Contributed by Wo...
wl-ax11-lem8 34705 Lemma. (Contributed by Wo...
wl-ax11-lem9 34706 The easy part when ` x ` c...
wl-ax11-lem10 34707 We now have prepared every...
wl-clabv 34708 Variant of ~ df-clab , whe...
wl-dfclab 34709 Rederive ~ df-clab from ~ ...
wl-clabtv 34710 Using class abstraction in...
wl-clabt 34711 Using class abstraction in...
wl-dfralsb 34718 An alternate definition of...
wl-dfralflem 34719 Lemma for ~ wl-dfralf and ...
wl-dfralf 34720 Restricted universal quant...
wl-dfralfi 34721 Restricted universal quant...
wl-dfralv 34722 Alternate definition of re...
wl-rgen 34723 Generalization rule for re...
wl-rgenw 34724 Generalization rule for re...
wl-rgen2w 34725 Generalization rule for re...
wl-ralel 34726 All elements of a class ar...
wl-dfrexf 34728 Restricted existential qua...
wl-dfrexfi 34729 Restricted universal quant...
wl-dfrexv 34730 Alternate definition of re...
wl-dfrexex 34731 Alternate definition of th...
wl-dfrexsb 34732 An alternate definition of...
wl-dfrmosb 34734 An alternate definition of...
wl-dfrmov 34735 Alternate definition of re...
wl-dfrmof 34736 Restricted "at most one" (...
wl-dfreusb 34738 An alternate definition of...
wl-dfreuv 34739 Alternate definition of re...
wl-dfreuf 34740 Restricted existential uni...
wl-dfrabsb 34742 Alternate definition of re...
wl-dfrabv 34743 Alternate definition of re...
wl-clelsb3df 34744 Deduction version of ~ cle...
wl-dfrabf 34745 Alternate definition of re...
rabiun 34746 Abstraction restricted to ...
iundif1 34747 Indexed union of class dif...
imadifss 34748 The difference of images i...
cureq 34749 Equality theorem for curry...
unceq 34750 Equality theorem for uncur...
curf 34751 Functional property of cur...
uncf 34752 Functional property of unc...
curfv 34753 Value of currying. (Contr...
uncov 34754 Value of uncurrying. (Con...
curunc 34755 Currying of uncurrying. (...
unccur 34756 Uncurrying of currying. (...
phpreu 34757 Theorem related to pigeonh...
finixpnum 34758 A finite Cartesian product...
fin2solem 34759 Lemma for ~ fin2so . (Con...
fin2so 34760 Any totally ordered Tarski...
ltflcei 34761 Theorem to move the floor ...
leceifl 34762 Theorem to move the floor ...
sin2h 34763 Half-angle rule for sine. ...
cos2h 34764 Half-angle rule for cosine...
tan2h 34765 Half-angle rule for tangen...
lindsadd 34766 In a vector space, the uni...
lindsdom 34767 A linearly independent set...
lindsenlbs 34768 A maximal linearly indepen...
matunitlindflem1 34769 One direction of ~ matunit...
matunitlindflem2 34770 One direction of ~ matunit...
matunitlindf 34771 A matrix over a field is i...
ptrest 34772 Expressing a restriction o...
ptrecube 34773 Any point in an open set o...
poimirlem1 34774 Lemma for ~ poimir - the v...
poimirlem2 34775 Lemma for ~ poimir - conse...
poimirlem3 34776 Lemma for ~ poimir to add ...
poimirlem4 34777 Lemma for ~ poimir connect...
poimirlem5 34778 Lemma for ~ poimir to esta...
poimirlem6 34779 Lemma for ~ poimir establi...
poimirlem7 34780 Lemma for ~ poimir , simil...
poimirlem8 34781 Lemma for ~ poimir , estab...
poimirlem9 34782 Lemma for ~ poimir , estab...
poimirlem10 34783 Lemma for ~ poimir establi...
poimirlem11 34784 Lemma for ~ poimir connect...
poimirlem12 34785 Lemma for ~ poimir connect...
poimirlem13 34786 Lemma for ~ poimir - for a...
poimirlem14 34787 Lemma for ~ poimir - for a...
poimirlem15 34788 Lemma for ~ poimir , that ...
poimirlem16 34789 Lemma for ~ poimir establi...
poimirlem17 34790 Lemma for ~ poimir establi...
poimirlem18 34791 Lemma for ~ poimir stating...
poimirlem19 34792 Lemma for ~ poimir establi...
poimirlem20 34793 Lemma for ~ poimir establi...
poimirlem21 34794 Lemma for ~ poimir stating...
poimirlem22 34795 Lemma for ~ poimir , that ...
poimirlem23 34796 Lemma for ~ poimir , two w...
poimirlem24 34797 Lemma for ~ poimir , two w...
poimirlem25 34798 Lemma for ~ poimir stating...
poimirlem26 34799 Lemma for ~ poimir showing...
poimirlem27 34800 Lemma for ~ poimir showing...
poimirlem28 34801 Lemma for ~ poimir , a var...
poimirlem29 34802 Lemma for ~ poimir connect...
poimirlem30 34803 Lemma for ~ poimir combini...
poimirlem31 34804 Lemma for ~ poimir , assig...
poimirlem32 34805 Lemma for ~ poimir , combi...
poimir 34806 Poincare-Miranda theorem. ...
broucube 34807 Brouwer - or as Kulpa call...
heicant 34808 Heine-Cantor theorem: a co...
opnmbllem0 34809 Lemma for ~ ismblfin ; cou...
mblfinlem1 34810 Lemma for ~ ismblfin , ord...
mblfinlem2 34811 Lemma for ~ ismblfin , eff...
mblfinlem3 34812 The difference between two...
mblfinlem4 34813 Backward direction of ~ is...
ismblfin 34814 Measurability in terms of ...
ovoliunnfl 34815 ~ ovoliun is incompatible ...
ex-ovoliunnfl 34816 Demonstration of ~ ovoliun...
voliunnfl 34817 ~ voliun is incompatible w...
volsupnfl 34818 ~ volsup is incompatible w...
mbfresfi 34819 Measurability of a piecewi...
mbfposadd 34820 If the sum of two measurab...
cnambfre 34821 A real-valued, a.e. contin...
dvtanlem 34822 Lemma for ~ dvtan - the do...
dvtan 34823 Derivative of tangent. (C...
itg2addnclem 34824 An alternate expression fo...
itg2addnclem2 34825 Lemma for ~ itg2addnc . T...
itg2addnclem3 34826 Lemma incomprehensible in ...
itg2addnc 34827 Alternate proof of ~ itg2a...
itg2gt0cn 34828 ~ itg2gt0 holds on functio...
ibladdnclem 34829 Lemma for ~ ibladdnc ; cf ...
ibladdnc 34830 Choice-free analogue of ~ ...
itgaddnclem1 34831 Lemma for ~ itgaddnc ; cf....
itgaddnclem2 34832 Lemma for ~ itgaddnc ; cf....
itgaddnc 34833 Choice-free analogue of ~ ...
iblsubnc 34834 Choice-free analogue of ~ ...
itgsubnc 34835 Choice-free analogue of ~ ...
iblabsnclem 34836 Lemma for ~ iblabsnc ; cf....
iblabsnc 34837 Choice-free analogue of ~ ...
iblmulc2nc 34838 Choice-free analogue of ~ ...
itgmulc2nclem1 34839 Lemma for ~ itgmulc2nc ; c...
itgmulc2nclem2 34840 Lemma for ~ itgmulc2nc ; c...
itgmulc2nc 34841 Choice-free analogue of ~ ...
itgabsnc 34842 Choice-free analogue of ~ ...
bddiblnc 34843 Choice-free proof of ~ bdd...
cnicciblnc 34844 Choice-free proof of ~ cni...
itggt0cn 34845 ~ itggt0 holds for continu...
ftc1cnnclem 34846 Lemma for ~ ftc1cnnc ; cf....
ftc1cnnc 34847 Choice-free proof of ~ ftc...
ftc1anclem1 34848 Lemma for ~ ftc1anc - the ...
ftc1anclem2 34849 Lemma for ~ ftc1anc - rest...
ftc1anclem3 34850 Lemma for ~ ftc1anc - the ...
ftc1anclem4 34851 Lemma for ~ ftc1anc . (Co...
ftc1anclem5 34852 Lemma for ~ ftc1anc , the ...
ftc1anclem6 34853 Lemma for ~ ftc1anc - cons...
ftc1anclem7 34854 Lemma for ~ ftc1anc . (Co...
ftc1anclem8 34855 Lemma for ~ ftc1anc . (Co...
ftc1anc 34856 ~ ftc1a holds for function...
ftc2nc 34857 Choice-free proof of ~ ftc...
asindmre 34858 Real part of domain of dif...
dvasin 34859 Derivative of arcsine. (C...
dvacos 34860 Derivative of arccosine. ...
dvreasin 34861 Real derivative of arcsine...
dvreacos 34862 Real derivative of arccosi...
areacirclem1 34863 Antiderivative of cross-se...
areacirclem2 34864 Endpoint-inclusive continu...
areacirclem3 34865 Integrability of cross-sec...
areacirclem4 34866 Endpoint-inclusive continu...
areacirclem5 34867 Finding the cross-section ...
areacirc 34868 The area of a circle of ra...
unirep 34869 Define a quantity whose de...
cover2 34870 Two ways of expressing the...
cover2g 34871 Two ways of expressing the...
brabg2 34872 Relation by a binary relat...
opelopab3 34873 Ordered pair membership in...
cocanfo 34874 Cancellation of a surjecti...
brresi2 34875 Restriction of a binary re...
fnopabeqd 34876 Equality deduction for fun...
fvopabf4g 34877 Function value of an opera...
eqfnun 34878 Two functions on ` A u. B ...
fnopabco 34879 Composition of a function ...
opropabco 34880 Composition of an operator...
cocnv 34881 Composition with a functio...
f1ocan1fv 34882 Cancel a composition by a ...
f1ocan2fv 34883 Cancel a composition by th...
inixp 34884 Intersection of Cartesian ...
upixp 34885 Universal property of the ...
abrexdom 34886 An indexed set is dominate...
abrexdom2 34887 An indexed set is dominate...
ac6gf 34888 Axiom of Choice. (Contrib...
indexa 34889 If for every element of an...
indexdom 34890 If for every element of an...
frinfm 34891 A subset of a well-founded...
welb 34892 A nonempty subset of a wel...
supex2g 34893 Existence of supremum. (C...
supclt 34894 Closure of supremum. (Con...
supubt 34895 Upper bound property of su...
filbcmb 34896 Combine a finite set of lo...
fzmul 34897 Membership of a product in...
sdclem2 34898 Lemma for ~ sdc . (Contri...
sdclem1 34899 Lemma for ~ sdc . (Contri...
sdc 34900 Strong dependent choice. ...
fdc 34901 Finite version of dependen...
fdc1 34902 Variant of ~ fdc with no s...
seqpo 34903 Two ways to say that a seq...
incsequz 34904 An increasing sequence of ...
incsequz2 34905 An increasing sequence of ...
nnubfi 34906 A bounded above set of pos...
nninfnub 34907 An infinite set of positiv...
subspopn 34908 An open set is open in the...
neificl 34909 Neighborhoods are closed u...
lpss2 34910 Limit points of a subset a...
metf1o 34911 Use a bijection with a met...
blssp 34912 A ball in the subspace met...
mettrifi 34913 Generalized triangle inequ...
lmclim2 34914 A sequence in a metric spa...
geomcau 34915 If the distance between co...
caures 34916 The restriction of a Cauch...
caushft 34917 A shifted Cauchy sequence ...
constcncf 34918 A constant function is a c...
idcncf 34919 The identity function is a...
sub1cncf 34920 Subtracting a constant is ...
sub2cncf 34921 Subtraction from a constan...
cnres2 34922 The restriction of a conti...
cnresima 34923 A continuous function is c...
cncfres 34924 A continuous function on c...
istotbnd 34928 The predicate "is a totall...
istotbnd2 34929 The predicate "is a totall...
istotbnd3 34930 A metric space is totally ...
totbndmet 34931 The predicate "totally bou...
0totbnd 34932 The metric (there is only ...
sstotbnd2 34933 Condition for a subset of ...
sstotbnd 34934 Condition for a subset of ...
sstotbnd3 34935 Use a net that is not nece...
totbndss 34936 A subset of a totally boun...
equivtotbnd 34937 If the metric ` M ` is "st...
isbnd 34939 The predicate "is a bounde...
bndmet 34940 A bounded metric space is ...
isbndx 34941 A "bounded extended metric...
isbnd2 34942 The predicate "is a bounde...
isbnd3 34943 A metric space is bounded ...
isbnd3b 34944 A metric space is bounded ...
bndss 34945 A subset of a bounded metr...
blbnd 34946 A ball is bounded. (Contr...
ssbnd 34947 A subset of a metric space...
totbndbnd 34948 A totally bounded metric s...
equivbnd 34949 If the metric ` M ` is "st...
bnd2lem 34950 Lemma for ~ equivbnd2 and ...
equivbnd2 34951 If balls are totally bound...
prdsbnd 34952 The product metric over fi...
prdstotbnd 34953 The product metric over fi...
prdsbnd2 34954 If balls are totally bound...
cntotbnd 34955 A subset of the complex nu...
cnpwstotbnd 34956 A subset of ` A ^ I ` , wh...
ismtyval 34959 The set of isometries betw...
isismty 34960 The condition "is an isome...
ismtycnv 34961 The inverse of an isometry...
ismtyima 34962 The image of a ball under ...
ismtyhmeolem 34963 Lemma for ~ ismtyhmeo . (...
ismtyhmeo 34964 An isometry is a homeomorp...
ismtybndlem 34965 Lemma for ~ ismtybnd . (C...
ismtybnd 34966 Isometries preserve bounde...
ismtyres 34967 A restriction of an isomet...
heibor1lem 34968 Lemma for ~ heibor1 . A c...
heibor1 34969 One half of ~ heibor , tha...
heiborlem1 34970 Lemma for ~ heibor . We w...
heiborlem2 34971 Lemma for ~ heibor . Subs...
heiborlem3 34972 Lemma for ~ heibor . Usin...
heiborlem4 34973 Lemma for ~ heibor . Usin...
heiborlem5 34974 Lemma for ~ heibor . The ...
heiborlem6 34975 Lemma for ~ heibor . Sinc...
heiborlem7 34976 Lemma for ~ heibor . Sinc...
heiborlem8 34977 Lemma for ~ heibor . The ...
heiborlem9 34978 Lemma for ~ heibor . Disc...
heiborlem10 34979 Lemma for ~ heibor . The ...
heibor 34980 Generalized Heine-Borel Th...
bfplem1 34981 Lemma for ~ bfp . The seq...
bfplem2 34982 Lemma for ~ bfp . Using t...
bfp 34983 Banach fixed point theorem...
rrnval 34986 The n-dimensional Euclidea...
rrnmval 34987 The value of the Euclidean...
rrnmet 34988 Euclidean space is a metri...
rrndstprj1 34989 The distance between two p...
rrndstprj2 34990 Bound on the distance betw...
rrncmslem 34991 Lemma for ~ rrncms . (Con...
rrncms 34992 Euclidean space is complet...
repwsmet 34993 The supremum metric on ` R...
rrnequiv 34994 The supremum metric on ` R...
rrntotbnd 34995 A set in Euclidean space i...
rrnheibor 34996 Heine-Borel theorem for Eu...
ismrer1 34997 An isometry between ` RR `...
reheibor 34998 Heine-Borel theorem for re...
iccbnd 34999 A closed interval in ` RR ...
icccmpALT 35000 A closed interval in ` RR ...
isass 35005 The predicate "is an assoc...
isexid 35006 The predicate ` G ` has a ...
ismgmOLD 35009 Obsolete version of ~ ismg...
clmgmOLD 35010 Obsolete version of ~ mgmc...
opidonOLD 35011 Obsolete version of ~ mndp...
rngopidOLD 35012 Obsolete version of ~ mndp...
opidon2OLD 35013 Obsolete version of ~ mndp...
isexid2 35014 If ` G e. ( Magma i^i ExId...
exidu1 35015 Uniqueness of the left and...
idrval 35016 The value of the identity ...
iorlid 35017 A magma right and left ide...
cmpidelt 35018 A magma right and left ide...
smgrpismgmOLD 35021 Obsolete version of ~ sgrp...
issmgrpOLD 35022 Obsolete version of ~ issg...
smgrpmgm 35023 A semigroup is a magma. (...
smgrpassOLD 35024 Obsolete version of ~ sgrp...
mndoissmgrpOLD 35027 Obsolete version of ~ mnds...
mndoisexid 35028 A monoid has an identity e...
mndoismgmOLD 35029 Obsolete version of ~ mndm...
mndomgmid 35030 A monoid is a magma with a...
ismndo 35031 The predicate "is a monoid...
ismndo1 35032 The predicate "is a monoid...
ismndo2 35033 The predicate "is a monoid...
grpomndo 35034 A group is a monoid. (Con...
exidcl 35035 Closure of the binary oper...
exidreslem 35036 Lemma for ~ exidres and ~ ...
exidres 35037 The restriction of a binar...
exidresid 35038 The restriction of a binar...
ablo4pnp 35039 A commutative/associative ...
grpoeqdivid 35040 Two group elements are equ...
grposnOLD 35041 The group operation for th...
elghomlem1OLD 35044 Obsolete as of 15-Mar-2020...
elghomlem2OLD 35045 Obsolete as of 15-Mar-2020...
elghomOLD 35046 Obsolete version of ~ isgh...
ghomlinOLD 35047 Obsolete version of ~ ghml...
ghomidOLD 35048 Obsolete version of ~ ghmi...
ghomf 35049 Mapping property of a grou...
ghomco 35050 The composition of two gro...
ghomdiv 35051 Group homomorphisms preser...
grpokerinj 35052 A group homomorphism is in...
relrngo 35055 The class of all unital ri...
isrngo 35056 The predicate "is a (unita...
isrngod 35057 Conditions that determine ...
rngoi 35058 The properties of a unital...
rngosm 35059 Functionality of the multi...
rngocl 35060 Closure of the multiplicat...
rngoid 35061 The multiplication operati...
rngoideu 35062 The unit element of a ring...
rngodi 35063 Distributive law for the m...
rngodir 35064 Distributive law for the m...
rngoass 35065 Associative law for the mu...
rngo2 35066 A ring element plus itself...
rngoablo 35067 A ring's addition operatio...
rngoablo2 35068 In a unital ring the addit...
rngogrpo 35069 A ring's addition operatio...
rngone0 35070 The base set of a ring is ...
rngogcl 35071 Closure law for the additi...
rngocom 35072 The addition operation of ...
rngoaass 35073 The addition operation of ...
rngoa32 35074 The addition operation of ...
rngoa4 35075 Rearrangement of 4 terms i...
rngorcan 35076 Right cancellation law for...
rngolcan 35077 Left cancellation law for ...
rngo0cl 35078 A ring has an additive ide...
rngo0rid 35079 The additive identity of a...
rngo0lid 35080 The additive identity of a...
rngolz 35081 The zero of a unital ring ...
rngorz 35082 The zero of a unital ring ...
rngosn3 35083 Obsolete as of 25-Jan-2020...
rngosn4 35084 Obsolete as of 25-Jan-2020...
rngosn6 35085 Obsolete as of 25-Jan-2020...
rngonegcl 35086 A ring is closed under neg...
rngoaddneg1 35087 Adding the negative in a r...
rngoaddneg2 35088 Adding the negative in a r...
rngosub 35089 Subtraction in a ring, in ...
rngmgmbs4 35090 The range of an internal o...
rngodm1dm2 35091 In a unital ring the domai...
rngorn1 35092 In a unital ring the range...
rngorn1eq 35093 In a unital ring the range...
rngomndo 35094 In a unital ring the multi...
rngoidmlem 35095 The unit of a ring is an i...
rngolidm 35096 The unit of a ring is an i...
rngoridm 35097 The unit of a ring is an i...
rngo1cl 35098 The unit of a ring belongs...
rngoueqz 35099 Obsolete as of 23-Jan-2020...
rngonegmn1l 35100 Negation in a ring is the ...
rngonegmn1r 35101 Negation in a ring is the ...
rngoneglmul 35102 Negation of a product in a...
rngonegrmul 35103 Negation of a product in a...
rngosubdi 35104 Ring multiplication distri...
rngosubdir 35105 Ring multiplication distri...
zerdivemp1x 35106 In a unitary ring a left i...
isdivrngo 35109 The predicate "is a divisi...
drngoi 35110 The properties of a divisi...
gidsn 35111 Obsolete as of 23-Jan-2020...
zrdivrng 35112 The zero ring is not a div...
dvrunz 35113 In a division ring the uni...
isgrpda 35114 Properties that determine ...
isdrngo1 35115 The predicate "is a divisi...
divrngcl 35116 The product of two nonzero...
isdrngo2 35117 A division ring is a ring ...
isdrngo3 35118 A division ring is a ring ...
rngohomval 35123 The set of ring homomorphi...
isrngohom 35124 The predicate "is a ring h...
rngohomf 35125 A ring homomorphism is a f...
rngohomcl 35126 Closure law for a ring hom...
rngohom1 35127 A ring homomorphism preser...
rngohomadd 35128 Ring homomorphisms preserv...
rngohommul 35129 Ring homomorphisms preserv...
rngogrphom 35130 A ring homomorphism is a g...
rngohom0 35131 A ring homomorphism preser...
rngohomsub 35132 Ring homomorphisms preserv...
rngohomco 35133 The composition of two rin...
rngokerinj 35134 A ring homomorphism is inj...
rngoisoval 35136 The set of ring isomorphis...
isrngoiso 35137 The predicate "is a ring i...
rngoiso1o 35138 A ring isomorphism is a bi...
rngoisohom 35139 A ring isomorphism is a ri...
rngoisocnv 35140 The inverse of a ring isom...
rngoisoco 35141 The composition of two rin...
isriscg 35143 The ring isomorphism relat...
isrisc 35144 The ring isomorphism relat...
risc 35145 The ring isomorphism relat...
risci 35146 Determine that two rings a...
riscer 35147 Ring isomorphism is an equ...
iscom2 35154 A device to add commutativ...
iscrngo 35155 The predicate "is a commut...
iscrngo2 35156 The predicate "is a commut...
iscringd 35157 Conditions that determine ...
flddivrng 35158 A field is a division ring...
crngorngo 35159 A commutative ring is a ri...
crngocom 35160 The multiplication operati...
crngm23 35161 Commutative/associative la...
crngm4 35162 Commutative/associative la...
fldcrng 35163 A field is a commutative r...
isfld2 35164 The predicate "is a field"...
crngohomfo 35165 The image of a homomorphis...
idlval 35172 The class of ideals of a r...
isidl 35173 The predicate "is an ideal...
isidlc 35174 The predicate "is an ideal...
idlss 35175 An ideal of ` R ` is a sub...
idlcl 35176 An element of an ideal is ...
idl0cl 35177 An ideal contains ` 0 ` . ...
idladdcl 35178 An ideal is closed under a...
idllmulcl 35179 An ideal is closed under m...
idlrmulcl 35180 An ideal is closed under m...
idlnegcl 35181 An ideal is closed under n...
idlsubcl 35182 An ideal is closed under s...
rngoidl 35183 A ring ` R ` is an ` R ` i...
0idl 35184 The set containing only ` ...
1idl 35185 Two ways of expressing the...
0rngo 35186 In a ring, ` 0 = 1 ` iff t...
divrngidl 35187 The only ideals in a divis...
intidl 35188 The intersection of a none...
inidl 35189 The intersection of two id...
unichnidl 35190 The union of a nonempty ch...
keridl 35191 The kernel of a ring homom...
pridlval 35192 The class of prime ideals ...
ispridl 35193 The predicate "is a prime ...
pridlidl 35194 A prime ideal is an ideal....
pridlnr 35195 A prime ideal is a proper ...
pridl 35196 The main property of a pri...
ispridl2 35197 A condition that shows an ...
maxidlval 35198 The set of maximal ideals ...
ismaxidl 35199 The predicate "is a maxima...
maxidlidl 35200 A maximal ideal is an idea...
maxidlnr 35201 A maximal ideal is proper....
maxidlmax 35202 A maximal ideal is a maxim...
maxidln1 35203 One is not contained in an...
maxidln0 35204 A ring with a maximal idea...
isprrngo 35209 The predicate "is a prime ...
prrngorngo 35210 A prime ring is a ring. (...
smprngopr 35211 A simple ring (one whose o...
divrngpr 35212 A division ring is a prime...
isdmn 35213 The predicate "is a domain...
isdmn2 35214 The predicate "is a domain...
dmncrng 35215 A domain is a commutative ...
dmnrngo 35216 A domain is a ring. (Cont...
flddmn 35217 A field is a domain. (Con...
igenval 35220 The ideal generated by a s...
igenss 35221 A set is a subset of the i...
igenidl 35222 The ideal generated by a s...
igenmin 35223 The ideal generated by a s...
igenidl2 35224 The ideal generated by an ...
igenval2 35225 The ideal generated by a s...
prnc 35226 A principal ideal (an idea...
isfldidl 35227 Determine if a ring is a f...
isfldidl2 35228 Determine if a ring is a f...
ispridlc 35229 The predicate "is a prime ...
pridlc 35230 Property of a prime ideal ...
pridlc2 35231 Property of a prime ideal ...
pridlc3 35232 Property of a prime ideal ...
isdmn3 35233 The predicate "is a domain...
dmnnzd 35234 A domain has no zero-divis...
dmncan1 35235 Cancellation law for domai...
dmncan2 35236 Cancellation law for domai...
efald2 35237 A proof by contradiction. ...
notbinot1 35238 Simplification rule of neg...
bicontr 35239 Biimplication of its own n...
impor 35240 An equivalent formula for ...
orfa 35241 The falsum ` F. ` can be r...
notbinot2 35242 Commutation rule between n...
biimpor 35243 A rewriting rule for biimp...
orfa1 35244 Add a contradicting disjun...
orfa2 35245 Remove a contradicting dis...
bifald 35246 Infer the equivalence to a...
orsild 35247 A lemma for not-or-not eli...
orsird 35248 A lemma for not-or-not eli...
cnf1dd 35249 A lemma for Conjunctive No...
cnf2dd 35250 A lemma for Conjunctive No...
cnfn1dd 35251 A lemma for Conjunctive No...
cnfn2dd 35252 A lemma for Conjunctive No...
or32dd 35253 A rearrangement of disjunc...
notornotel1 35254 A lemma for not-or-not eli...
notornotel2 35255 A lemma for not-or-not eli...
contrd 35256 A proof by contradiction, ...
an12i 35257 An inference from commutin...
exmid2 35258 An excluded middle law. (...
selconj 35259 An inference for selecting...
truconj 35260 Add true as a conjunct. (...
orel 35261 An inference for disjuncti...
negel 35262 An inference for negation ...
botel 35263 An inference for bottom el...
tradd 35264 Add top ad a conjunct. (C...
gm-sbtru 35265 Substitution does not chan...
sbfal 35266 Substitution does not chan...
sbcani 35267 Distribution of class subs...
sbcori 35268 Distribution of class subs...
sbcimi 35269 Distribution of class subs...
sbcni 35270 Move class substitution in...
sbali 35271 Discard class substitution...
sbexi 35272 Discard class substitution...
sbcalf 35273 Move universal quantifier ...
sbcexf 35274 Move existential quantifie...
sbcalfi 35275 Move universal quantifier ...
sbcexfi 35276 Move existential quantifie...
spsbcdi 35277 A lemma for eliminating a ...
alrimii 35278 A lemma for introducing a ...
spesbcdi 35279 A lemma for introducing an...
exlimddvf 35280 A lemma for eliminating an...
exlimddvfi 35281 A lemma for eliminating an...
sbceq1ddi 35282 A lemma for eliminating in...
sbccom2lem 35283 Lemma for ~ sbccom2 . (Co...
sbccom2 35284 Commutative law for double...
sbccom2f 35285 Commutative law for double...
sbccom2fi 35286 Commutative law for double...
csbcom2fi 35287 Commutative law for double...
fald 35288 Refutation of falsity, in ...
tsim1 35289 A Tseitin axiom for logica...
tsim2 35290 A Tseitin axiom for logica...
tsim3 35291 A Tseitin axiom for logica...
tsbi1 35292 A Tseitin axiom for logica...
tsbi2 35293 A Tseitin axiom for logica...
tsbi3 35294 A Tseitin axiom for logica...
tsbi4 35295 A Tseitin axiom for logica...
tsxo1 35296 A Tseitin axiom for logica...
tsxo2 35297 A Tseitin axiom for logica...
tsxo3 35298 A Tseitin axiom for logica...
tsxo4 35299 A Tseitin axiom for logica...
tsan1 35300 A Tseitin axiom for logica...
tsan2 35301 A Tseitin axiom for logica...
tsan3 35302 A Tseitin axiom for logica...
tsna1 35303 A Tseitin axiom for logica...
tsna2 35304 A Tseitin axiom for logica...
tsna3 35305 A Tseitin axiom for logica...
tsor1 35306 A Tseitin axiom for logica...
tsor2 35307 A Tseitin axiom for logica...
tsor3 35308 A Tseitin axiom for logica...
ts3an1 35309 A Tseitin axiom for triple...
ts3an2 35310 A Tseitin axiom for triple...
ts3an3 35311 A Tseitin axiom for triple...
ts3or1 35312 A Tseitin axiom for triple...
ts3or2 35313 A Tseitin axiom for triple...
ts3or3 35314 A Tseitin axiom for triple...
iuneq2f 35315 Equality deduction for ind...
rabeq12f 35316 Equality deduction for res...
csbeq12 35317 Equality deduction for sub...
sbeqi 35318 Equality deduction for sub...
ralbi12f 35319 Equality deduction for res...
oprabbi 35320 Equality deduction for cla...
mpobi123f 35321 Equality deduction for map...
iuneq12f 35322 Equality deduction for ind...
iineq12f 35323 Equality deduction for ind...
opabbi 35324 Equality deduction for cla...
mptbi12f 35325 Equality deduction for map...
orcomdd 35326 Commutativity of logic dis...
scottexf 35327 A version of ~ scottex wit...
scott0f 35328 A version of ~ scott0 with...
scottn0f 35329 A version of ~ scott0f wit...
ac6s3f 35330 Generalization of the Axio...
ac6s6 35331 Generalization of the Axio...
ac6s6f 35332 Generalization of the Axio...
el2v1 35371 New way ( ~ elv , and the ...
el3v 35372 New way ( ~ elv , and the ...
el3v1 35373 New way ( ~ elv , and the ...
el3v2 35374 New way ( ~ elv , and the ...
el3v3 35375 New way ( ~ elv , and the ...
el3v12 35376 New way ( ~ elv , and the ...
el3v13 35377 New way ( ~ elv , and the ...
el3v23 35378 New way ( ~ elv , and the ...
an2anr 35379 Double commutation in conj...
anan 35380 Multiple commutations in c...
triantru3 35381 A wff is equivalent to its...
eqeltr 35382 Substitution of equal clas...
eqelb 35383 Substitution of equal clas...
eqeqan2d 35384 Implication of introducing...
ineqcom 35385 Two ways of saying that tw...
ineqcomi 35386 Disjointness inference (wh...
inres2 35387 Two ways of expressing the...
coideq 35388 Equality theorem for compo...
nexmo1 35389 If there is no case where ...
3albii 35390 Inference adding three uni...
3ralbii 35391 Inference adding three res...
ssrabi 35392 Inference of restricted ab...
rabbieq 35393 Equivalent wff's correspon...
rabimbieq 35394 Restricted equivalent wff'...
abeqin 35395 Intersection with class ab...
abeqinbi 35396 Intersection with class ab...
rabeqel 35397 Class element of a restric...
eqrelf 35398 The equality connective be...
releleccnv 35399 Elementhood in a converse ...
releccnveq 35400 Equality of converse ` R `...
opelvvdif 35401 Negated elementhood of ord...
vvdifopab 35402 Ordered-pair class abstrac...
brvdif 35403 Binary relation with unive...
brvdif2 35404 Binary relation with unive...
brvvdif 35405 Binary relation with the c...
brvbrvvdif 35406 Binary relation with the c...
brcnvep 35407 The converse of the binary...
elecALTV 35408 Elementhood in the ` R ` -...
brcnvepres 35409 Restricted converse epsilo...
brres2 35410 Binary relation on a restr...
eldmres 35411 Elementhood in the domain ...
eldm4 35412 Elementhood in a domain. ...
eldmres2 35413 Elementhood in the domain ...
eceq1i 35414 Equality theorem for ` C `...
elecres 35415 Elementhood in the restric...
ecres 35416 Restricted coset of ` B ` ...
ecres2 35417 The restricted coset of ` ...
eccnvepres 35418 Restricted converse epsilo...
eleccnvep 35419 Elementhood in the convers...
eccnvep 35420 The converse epsilon coset...
extep 35421 Property of epsilon relati...
eccnvepres2 35422 The restricted converse ep...
eccnvepres3 35423 Condition for a restricted...
eldmqsres 35424 Elementhood in a restricte...
eldmqsres2 35425 Elementhood in a restricte...
qsss1 35426 Subclass theorem for quoti...
qseq1i 35427 Equality theorem for quoti...
qseq1d 35428 Equality theorem for quoti...
brinxprnres 35429 Binary relation on a restr...
inxprnres 35430 Restriction of a class as ...
dfres4 35431 Alternate definition of th...
exan3 35432 Equivalent expressions wit...
exanres 35433 Equivalent expressions wit...
exanres3 35434 Equivalent expressions wit...
exanres2 35435 Equivalent expressions wit...
cnvepres 35436 Restricted converse epsilo...
ssrel3 35437 Subclass relation in anoth...
eqrel2 35438 Equality of relations. (C...
rncnv 35439 Range of converse is the d...
dfdm6 35440 Alternate definition of do...
dfrn6 35441 Alternate definition of ra...
rncnvepres 35442 The range of the restricte...
dmecd 35443 Equality of the coset of `...
dmec2d 35444 Equality of the coset of `...
brid 35445 Property of the identity b...
ideq2 35446 For sets, the identity bin...
idresssidinxp 35447 Condition for the identity...
idreseqidinxp 35448 Condition for the identity...
extid 35449 Property of identity relat...
inxpss 35450 Two ways to say that an in...
idinxpss 35451 Two ways to say that an in...
inxpss3 35452 Two ways to say that an in...
inxpss2 35453 Two ways to say that inter...
inxpssidinxp 35454 Two ways to say that inter...
idinxpssinxp 35455 Two ways to say that inter...
idinxpssinxp2 35456 Identity intersection with...
idinxpssinxp3 35457 Identity intersection with...
idinxpssinxp4 35458 Identity intersection with...
relcnveq3 35459 Two ways of saying a relat...
relcnveq 35460 Two ways of saying a relat...
relcnveq2 35461 Two ways of saying a relat...
relcnveq4 35462 Two ways of saying a relat...
qsresid 35463 Simplification of a specia...
n0elqs 35464 Two ways of expressing tha...
n0elqs2 35465 Two ways of expressing tha...
ecex2 35466 Condition for a coset to b...
uniqsALTV 35467 The union of a quotient se...
imaexALTV 35468 Existence of an image of a...
ecexALTV 35469 Existence of a coset, like...
rnresequniqs 35470 The range of a restriction...
n0el2 35471 Two ways of expressing tha...
cnvepresex 35472 Sethood condition for the ...
eccnvepex 35473 The converse epsilon coset...
cnvepimaex 35474 The image of converse epsi...
cnvepima 35475 The image of converse epsi...
inex3 35476 Sufficient condition for t...
inxpex 35477 Sufficient condition for a...
eqres 35478 Converting a class constan...
brrabga 35479 The law of concretion for ...
brcnvrabga 35480 The law of concretion for ...
opideq 35481 Equality conditions for or...
iss2 35482 A subclass of the identity...
eldmcnv 35483 Elementhood in a domain of...
dfrel5 35484 Alternate definition of th...
dfrel6 35485 Alternate definition of th...
cnvresrn 35486 Converse restricted to ran...
ecin0 35487 Two ways of saying that th...
ecinn0 35488 Two ways of saying that th...
ineleq 35489 Equivalence of restricted ...
inecmo 35490 Equivalence of a double re...
inecmo2 35491 Equivalence of a double re...
ineccnvmo 35492 Equivalence of a double re...
alrmomorn 35493 Equivalence of an "at most...
alrmomodm 35494 Equivalence of an "at most...
ineccnvmo2 35495 Equivalence of a double un...
inecmo3 35496 Equivalence of a double un...
moantr 35497 Sufficient condition for t...
brabidga 35498 The law of concretion for ...
inxp2 35499 Intersection with a Cartes...
opabf 35500 A class abstraction of a c...
ec0 35501 The empty-coset of a class...
0qs 35502 Quotient set with the empt...
xrnss3v 35504 A range Cartesian product ...
xrnrel 35505 A range Cartesian product ...
brxrn 35506 Characterize a ternary rel...
brxrn2 35507 A characterization of the ...
dfxrn2 35508 Alternate definition of th...
xrneq1 35509 Equality theorem for the r...
xrneq1i 35510 Equality theorem for the r...
xrneq1d 35511 Equality theorem for the r...
xrneq2 35512 Equality theorem for the r...
xrneq2i 35513 Equality theorem for the r...
xrneq2d 35514 Equality theorem for the r...
xrneq12 35515 Equality theorem for the r...
xrneq12i 35516 Equality theorem for the r...
xrneq12d 35517 Equality theorem for the r...
elecxrn 35518 Elementhood in the ` ( R |...
ecxrn 35519 The ` ( R |X. S ) ` -coset...
xrninxp 35520 Intersection of a range Ca...
xrninxp2 35521 Intersection of a range Ca...
xrninxpex 35522 Sufficient condition for t...
inxpxrn 35523 Two ways to express the in...
br1cnvxrn2 35524 The converse of a binary r...
elec1cnvxrn2 35525 Elementhood in the convers...
rnxrn 35526 Range of the range Cartesi...
rnxrnres 35527 Range of a range Cartesian...
rnxrncnvepres 35528 Range of a range Cartesian...
rnxrnidres 35529 Range of a range Cartesian...
xrnres 35530 Two ways to express restri...
xrnres2 35531 Two ways to express restri...
xrnres3 35532 Two ways to express restri...
xrnres4 35533 Two ways to express restri...
xrnresex 35534 Sufficient condition for a...
xrnidresex 35535 Sufficient condition for a...
xrncnvepresex 35536 Sufficient condition for a...
brin2 35537 Binary relation on an inte...
brin3 35538 Binary relation on an inte...
dfcoss2 35541 Alternate definition of th...
dfcoss3 35542 Alternate definition of th...
dfcoss4 35543 Alternate definition of th...
cossex 35544 If ` A ` is a set then the...
cosscnvex 35545 If ` A ` is a set then the...
1cosscnvepresex 35546 Sufficient condition for a...
1cossxrncnvepresex 35547 Sufficient condition for a...
relcoss 35548 Cosets by ` R ` is a relat...
relcoels 35549 Coelements on ` A ` is a r...
cossss 35550 Subclass theorem for the c...
cosseq 35551 Equality theorem for the c...
cosseqi 35552 Equality theorem for the c...
cosseqd 35553 Equality theorem for the c...
1cossres 35554 The class of cosets by a r...
dfcoels 35555 Alternate definition of th...
brcoss 35556 ` A ` and ` B ` are cosets...
brcoss2 35557 Alternate form of the ` A ...
brcoss3 35558 Alternate form of the ` A ...
brcosscnvcoss 35559 For sets, the ` A ` and ` ...
brcoels 35560 ` B ` and ` C ` are coelem...
cocossss 35561 Two ways of saying that co...
cnvcosseq 35562 The converse of cosets by ...
br2coss 35563 Cosets by ` ,~ R ` binary ...
br1cossres 35564 ` B ` and ` C ` are cosets...
br1cossres2 35565 ` B ` and ` C ` are cosets...
relbrcoss 35566 ` A ` and ` B ` are cosets...
br1cossinres 35567 ` B ` and ` C ` are cosets...
br1cossxrnres 35568 ` <. B , C >. ` and ` <. D...
br1cossinidres 35569 ` B ` and ` C ` are cosets...
br1cossincnvepres 35570 ` B ` and ` C ` are cosets...
br1cossxrnidres 35571 ` <. B , C >. ` and ` <. D...
br1cossxrncnvepres 35572 ` <. B , C >. ` and ` <. D...
dmcoss3 35573 The domain of cosets is th...
dmcoss2 35574 The domain of cosets is th...
rncossdmcoss 35575 The range of cosets is the...
dm1cosscnvepres 35576 The domain of cosets of th...
dmcoels 35577 The domain of coelements i...
eldmcoss 35578 Elementhood in the domain ...
eldmcoss2 35579 Elementhood in the domain ...
eldm1cossres 35580 Elementhood in the domain ...
eldm1cossres2 35581 Elementhood in the domain ...
refrelcosslem 35582 Lemma for the left side of...
refrelcoss3 35583 The class of cosets by ` R...
refrelcoss2 35584 The class of cosets by ` R...
symrelcoss3 35585 The class of cosets by ` R...
symrelcoss2 35586 The class of cosets by ` R...
cossssid 35587 Equivalent expressions for...
cossssid2 35588 Equivalent expressions for...
cossssid3 35589 Equivalent expressions for...
cossssid4 35590 Equivalent expressions for...
cossssid5 35591 Equivalent expressions for...
brcosscnv 35592 ` A ` and ` B ` are cosets...
brcosscnv2 35593 ` A ` and ` B ` are cosets...
br1cosscnvxrn 35594 ` A ` and ` B ` are cosets...
1cosscnvxrn 35595 Cosets by the converse ran...
cosscnvssid3 35596 Equivalent expressions for...
cosscnvssid4 35597 Equivalent expressions for...
cosscnvssid5 35598 Equivalent expressions for...
coss0 35599 Cosets by the empty set ar...
cossid 35600 Cosets by the identity rel...
cosscnvid 35601 Cosets by the converse ide...
trcoss 35602 Sufficient condition for t...
eleccossin 35603 Two ways of saying that th...
trcoss2 35604 Equivalent expressions for...
elrels2 35606 The element of the relatio...
elrelsrel 35607 The element of the relatio...
elrelsrelim 35608 The element of the relatio...
elrels5 35609 Equivalent expressions for...
elrels6 35610 Equivalent expressions for...
elrelscnveq3 35611 Two ways of saying a relat...
elrelscnveq 35612 Two ways of saying a relat...
elrelscnveq2 35613 Two ways of saying a relat...
elrelscnveq4 35614 Two ways of saying a relat...
cnvelrels 35615 The converse of a set is a...
cosselrels 35616 Cosets of sets are element...
cosscnvelrels 35617 Cosets of converse sets ar...
dfssr2 35619 Alternate definition of th...
relssr 35620 The subset relation is a r...
brssr 35621 The subset relation and su...
brssrid 35622 Any set is a subset of its...
issetssr 35623 Two ways of expressing set...
brssrres 35624 Restricted subset binary r...
br1cnvssrres 35625 Restricted converse subset...
brcnvssr 35626 The converse of a subset r...
brcnvssrid 35627 Any set is a converse subs...
br1cossxrncnvssrres 35628 ` <. B , C >. ` and ` <. D...
extssr 35629 Property of subset relatio...
dfrefrels2 35633 Alternate definition of th...
dfrefrels3 35634 Alternate definition of th...
dfrefrel2 35635 Alternate definition of th...
dfrefrel3 35636 Alternate definition of th...
elrefrels2 35637 Element of the class of re...
elrefrels3 35638 Element of the class of re...
elrefrelsrel 35639 For sets, being an element...
refreleq 35640 Equality theorem for refle...
refrelid 35641 Identity relation is refle...
refrelcoss 35642 The class of cosets by ` R...
dfcnvrefrels2 35646 Alternate definition of th...
dfcnvrefrels3 35647 Alternate definition of th...
dfcnvrefrel2 35648 Alternate definition of th...
dfcnvrefrel3 35649 Alternate definition of th...
elcnvrefrels2 35650 Element of the class of co...
elcnvrefrels3 35651 Element of the class of co...
elcnvrefrelsrel 35652 For sets, being an element...
cnvrefrelcoss2 35653 Necessary and sufficient c...
cosselcnvrefrels2 35654 Necessary and sufficient c...
cosselcnvrefrels3 35655 Necessary and sufficient c...
cosselcnvrefrels4 35656 Necessary and sufficient c...
cosselcnvrefrels5 35657 Necessary and sufficient c...
dfsymrels2 35661 Alternate definition of th...
dfsymrels3 35662 Alternate definition of th...
dfsymrels4 35663 Alternate definition of th...
dfsymrels5 35664 Alternate definition of th...
dfsymrel2 35665 Alternate definition of th...
dfsymrel3 35666 Alternate definition of th...
dfsymrel4 35667 Alternate definition of th...
dfsymrel5 35668 Alternate definition of th...
elsymrels2 35669 Element of the class of sy...
elsymrels3 35670 Element of the class of sy...
elsymrels4 35671 Element of the class of sy...
elsymrels5 35672 Element of the class of sy...
elsymrelsrel 35673 For sets, being an element...
symreleq 35674 Equality theorem for symme...
symrelim 35675 Symmetric relation implies...
symrelcoss 35676 The class of cosets by ` R...
idsymrel 35677 The identity relation is s...
epnsymrel 35678 The membership (epsilon) r...
symrefref2 35679 Symmetry is a sufficient c...
symrefref3 35680 Symmetry is a sufficient c...
refsymrels2 35681 Elements of the class of r...
refsymrels3 35682 Elements of the class of r...
refsymrel2 35683 A relation which is reflex...
refsymrel3 35684 A relation which is reflex...
elrefsymrels2 35685 Elements of the class of r...
elrefsymrels3 35686 Elements of the class of r...
elrefsymrelsrel 35687 For sets, being an element...
dftrrels2 35691 Alternate definition of th...
dftrrels3 35692 Alternate definition of th...
dftrrel2 35693 Alternate definition of th...
dftrrel3 35694 Alternate definition of th...
eltrrels2 35695 Element of the class of tr...
eltrrels3 35696 Element of the class of tr...
eltrrelsrel 35697 For sets, being an element...
trreleq 35698 Equality theorem for the t...
dfeqvrels2 35703 Alternate definition of th...
dfeqvrels3 35704 Alternate definition of th...
dfeqvrel2 35705 Alternate definition of th...
dfeqvrel3 35706 Alternate definition of th...
eleqvrels2 35707 Element of the class of eq...
eleqvrels3 35708 Element of the class of eq...
eleqvrelsrel 35709 For sets, being an element...
elcoeleqvrels 35710 Elementhood in the coeleme...
elcoeleqvrelsrel 35711 For sets, being an element...
eqvrelrel 35712 An equivalence relation is...
eqvrelrefrel 35713 An equivalence relation is...
eqvrelsymrel 35714 An equivalence relation is...
eqvreltrrel 35715 An equivalence relation is...
eqvrelim 35716 Equivalence relation impli...
eqvreleq 35717 Equality theorem for equiv...
eqvreleqi 35718 Equality theorem for equiv...
eqvreleqd 35719 Equality theorem for equiv...
eqvrelsym 35720 An equivalence relation is...
eqvrelsymb 35721 An equivalence relation is...
eqvreltr 35722 An equivalence relation is...
eqvreltrd 35723 A transitivity relation fo...
eqvreltr4d 35724 A transitivity relation fo...
eqvrelref 35725 An equivalence relation is...
eqvrelth 35726 Basic property of equivale...
eqvrelcl 35727 Elementhood in the field o...
eqvrelthi 35728 Basic property of equivale...
eqvreldisj 35729 Equivalence classes do not...
qsdisjALTV 35730 Elements of a quotient set...
eqvrelqsel 35731 If an element of a quotien...
eqvrelcoss 35732 Two ways to express equiva...
eqvrelcoss3 35733 Two ways to express equiva...
eqvrelcoss2 35734 Two ways to express equiva...
eqvrelcoss4 35735 Two ways to express equiva...
dfcoeleqvrels 35736 Alternate definition of th...
dfcoeleqvrel 35737 Alternate definition of th...
brredunds 35741 Binary relation on the cla...
brredundsredund 35742 For sets, binary relation ...
redundss3 35743 Implication of redundancy ...
redundeq1 35744 Equivalence of redundancy ...
redundpim3 35745 Implication of redundancy ...
redundpbi1 35746 Equivalence of redundancy ...
refrelsredund4 35747 The naive version of the c...
refrelsredund2 35748 The naive version of the c...
refrelsredund3 35749 The naive version of the c...
refrelredund4 35750 The naive version of the d...
refrelredund2 35751 The naive version of the d...
refrelredund3 35752 The naive version of the d...
dmqseq 35755 Equality theorem for domai...
dmqseqi 35756 Equality theorem for domai...
dmqseqd 35757 Equality theorem for domai...
dmqseqeq1 35758 Equality theorem for domai...
dmqseqeq1i 35759 Equality theorem for domai...
dmqseqeq1d 35760 Equality theorem for domai...
brdmqss 35761 The domain quotient binary...
brdmqssqs 35762 If ` A ` and ` R ` are set...
n0eldmqs 35763 The empty set is not an el...
n0eldmqseq 35764 The empty set is not an el...
n0el3 35765 Two ways of expressing tha...
cnvepresdmqss 35766 The domain quotient binary...
cnvepresdmqs 35767 The domain quotient predic...
unidmqs 35768 The range of a relation is...
unidmqseq 35769 The union of the domain qu...
dmqseqim 35770 If the domain quotient of ...
dmqseqim2 35771 Lemma for ~ erim2 . (Cont...
releldmqs 35772 Elementhood in the domain ...
eldmqs1cossres 35773 Elementhood in the domain ...
releldmqscoss 35774 Elementhood in the domain ...
dmqscoelseq 35775 Two ways to express the eq...
dmqs1cosscnvepreseq 35776 Two ways to express the eq...
brers 35781 Binary equivalence relatio...
dferALTV2 35782 Equivalence relation with ...
erALTVeq1 35783 Equality theorem for equiv...
erALTVeq1i 35784 Equality theorem for equiv...
erALTVeq1d 35785 Equality theorem for equiv...
dfmember 35786 Alternate definition of th...
dfmember2 35787 Alternate definition of th...
dfmember3 35788 Alternate definition of th...
eqvreldmqs 35789 Two ways to express member...
brerser 35790 Binary equivalence relatio...
erim2 35791 Equivalence relation on it...
erim 35792 Equivalence relation on it...
dffunsALTV 35796 Alternate definition of th...
dffunsALTV2 35797 Alternate definition of th...
dffunsALTV3 35798 Alternate definition of th...
dffunsALTV4 35799 Alternate definition of th...
dffunsALTV5 35800 Alternate definition of th...
dffunALTV2 35801 Alternate definition of th...
dffunALTV3 35802 Alternate definition of th...
dffunALTV4 35803 Alternate definition of th...
dffunALTV5 35804 Alternate definition of th...
elfunsALTV 35805 Elementhood in the class o...
elfunsALTV2 35806 Elementhood in the class o...
elfunsALTV3 35807 Elementhood in the class o...
elfunsALTV4 35808 Elementhood in the class o...
elfunsALTV5 35809 Elementhood in the class o...
elfunsALTVfunALTV 35810 The element of the class o...
funALTVfun 35811 Our definition of the func...
funALTVss 35812 Subclass theorem for funct...
funALTVeq 35813 Equality theorem for funct...
funALTVeqi 35814 Equality inference for the...
funALTVeqd 35815 Equality deduction for the...
dfdisjs 35821 Alternate definition of th...
dfdisjs2 35822 Alternate definition of th...
dfdisjs3 35823 Alternate definition of th...
dfdisjs4 35824 Alternate definition of th...
dfdisjs5 35825 Alternate definition of th...
dfdisjALTV 35826 Alternate definition of th...
dfdisjALTV2 35827 Alternate definition of th...
dfdisjALTV3 35828 Alternate definition of th...
dfdisjALTV4 35829 Alternate definition of th...
dfdisjALTV5 35830 Alternate definition of th...
dfeldisj2 35831 Alternate definition of th...
dfeldisj3 35832 Alternate definition of th...
dfeldisj4 35833 Alternate definition of th...
dfeldisj5 35834 Alternate definition of th...
eldisjs 35835 Elementhood in the class o...
eldisjs2 35836 Elementhood in the class o...
eldisjs3 35837 Elementhood in the class o...
eldisjs4 35838 Elementhood in the class o...
eldisjs5 35839 Elementhood in the class o...
eldisjsdisj 35840 The element of the class o...
eleldisjs 35841 Elementhood in the disjoin...
eleldisjseldisj 35842 The element of the disjoin...
disjrel 35843 Disjoint relation is a rel...
disjss 35844 Subclass theorem for disjo...
disjssi 35845 Subclass theorem for disjo...
disjssd 35846 Subclass theorem for disjo...
disjeq 35847 Equality theorem for disjo...
disjeqi 35848 Equality theorem for disjo...
disjeqd 35849 Equality theorem for disjo...
disjdmqseqeq1 35850 Lemma for the equality the...
eldisjss 35851 Subclass theorem for disjo...
eldisjssi 35852 Subclass theorem for disjo...
eldisjssd 35853 Subclass theorem for disjo...
eldisjeq 35854 Equality theorem for disjo...
eldisjeqi 35855 Equality theorem for disjo...
eldisjeqd 35856 Equality theorem for disjo...
disjxrn 35857 Two ways of saying that a ...
disjorimxrn 35858 Disjointness condition for...
disjimxrn 35859 Disjointness condition for...
disjimres 35860 Disjointness condition for...
disjimin 35861 Disjointness condition for...
disjiminres 35862 Disjointness condition for...
disjimxrnres 35863 Disjointness condition for...
disjALTV0 35864 The null class is disjoint...
disjALTVid 35865 The class of identity rela...
disjALTVidres 35866 The class of identity rela...
disjALTVinidres 35867 The intersection with rest...
disjALTVxrnidres 35868 The class of range Cartesi...
prtlem60 35869 Lemma for ~ prter3 . (Con...
bicomdd 35870 Commute two sides of a bic...
jca2r 35871 Inference conjoining the c...
jca3 35872 Inference conjoining the c...
prtlem70 35873 Lemma for ~ prter3 : a rea...
ibdr 35874 Reverse of ~ ibd . (Contr...
prtlem100 35875 Lemma for ~ prter3 . (Con...
prtlem5 35876 Lemma for ~ prter1 , ~ prt...
prtlem80 35877 Lemma for ~ prter2 . (Con...
brabsb2 35878 A closed form of ~ brabsb ...
eqbrrdv2 35879 Other version of ~ eqbrrdi...
prtlem9 35880 Lemma for ~ prter3 . (Con...
prtlem10 35881 Lemma for ~ prter3 . (Con...
prtlem11 35882 Lemma for ~ prter2 . (Con...
prtlem12 35883 Lemma for ~ prtex and ~ pr...
prtlem13 35884 Lemma for ~ prter1 , ~ prt...
prtlem16 35885 Lemma for ~ prtex , ~ prte...
prtlem400 35886 Lemma for ~ prter2 and als...
erprt 35889 The quotient set of an equ...
prtlem14 35890 Lemma for ~ prter1 , ~ prt...
prtlem15 35891 Lemma for ~ prter1 and ~ p...
prtlem17 35892 Lemma for ~ prter2 . (Con...
prtlem18 35893 Lemma for ~ prter2 . (Con...
prtlem19 35894 Lemma for ~ prter2 . (Con...
prter1 35895 Every partition generates ...
prtex 35896 The equivalence relation g...
prter2 35897 The quotient set of the eq...
prter3 35898 For every partition there ...
axc5 35909 This theorem repeats ~ sp ...
ax4fromc4 35910 Rederivation of axiom ~ ax...
ax10fromc7 35911 Rederivation of axiom ~ ax...
ax6fromc10 35912 Rederivation of axiom ~ ax...
hba1-o 35913 The setvar ` x ` is not fr...
axc4i-o 35914 Inference version of ~ ax-...
equid1 35915 Proof of ~ equid from our ...
equcomi1 35916 Proof of ~ equcomi from ~ ...
aecom-o 35917 Commutation law for identi...
aecoms-o 35918 A commutation rule for ide...
hbae-o 35919 All variables are effectiv...
dral1-o 35920 Formula-building lemma for...
ax12fromc15 35921 Rederivation of axiom ~ ax...
ax13fromc9 35922 Derive ~ ax-13 from ~ ax-c...
ax5ALT 35923 Axiom to quantify a variab...
sps-o 35924 Generalization of antecede...
hbequid 35925 Bound-variable hypothesis ...
nfequid-o 35926 Bound-variable hypothesis ...
axc5c7 35927 Proof of a single axiom th...
axc5c7toc5 35928 Rederivation of ~ ax-c5 fr...
axc5c7toc7 35929 Rederivation of ~ ax-c7 fr...
axc711 35930 Proof of a single axiom th...
nfa1-o 35931 ` x ` is not free in ` A. ...
axc711toc7 35932 Rederivation of ~ ax-c7 fr...
axc711to11 35933 Rederivation of ~ ax-11 fr...
axc5c711 35934 Proof of a single axiom th...
axc5c711toc5 35935 Rederivation of ~ ax-c5 fr...
axc5c711toc7 35936 Rederivation of ~ ax-c7 fr...
axc5c711to11 35937 Rederivation of ~ ax-11 fr...
equidqe 35938 ~ equid with existential q...
axc5sp1 35939 A special case of ~ ax-c5 ...
equidq 35940 ~ equid with universal qua...
equid1ALT 35941 Alternate proof of ~ equid...
axc11nfromc11 35942 Rederivation of ~ ax-c11n ...
naecoms-o 35943 A commutation rule for dis...
hbnae-o 35944 All variables are effectiv...
dvelimf-o 35945 Proof of ~ dvelimh that us...
dral2-o 35946 Formula-building lemma for...
aev-o 35947 A "distinctor elimination"...
ax5eq 35948 Theorem to add distinct qu...
dveeq2-o 35949 Quantifier introduction wh...
axc16g-o 35950 A generalization of axiom ...
dveeq1-o 35951 Quantifier introduction wh...
dveeq1-o16 35952 Version of ~ dveeq1 using ...
ax5el 35953 Theorem to add distinct qu...
axc11n-16 35954 This theorem shows that, g...
dveel2ALT 35955 Alternate proof of ~ dveel...
ax12f 35956 Basis step for constructin...
ax12eq 35957 Basis step for constructin...
ax12el 35958 Basis step for constructin...
ax12indn 35959 Induction step for constru...
ax12indi 35960 Induction step for constru...
ax12indalem 35961 Lemma for ~ ax12inda2 and ...
ax12inda2ALT 35962 Alternate proof of ~ ax12i...
ax12inda2 35963 Induction step for constru...
ax12inda 35964 Induction step for constru...
ax12v2-o 35965 Rederivation of ~ ax-c15 f...
ax12a2-o 35966 Derive ~ ax-c15 from a hyp...
axc11-o 35967 Show that ~ ax-c11 can be ...
fsumshftd 35968 Index shift of a finite su...
riotaclbgBAD 35970 Closure of restricted iota...
riotaclbBAD 35971 Closure of restricted iota...
riotasvd 35972 Deduction version of ~ rio...
riotasv2d 35973 Value of description binde...
riotasv2s 35974 The value of description b...
riotasv 35975 Value of description binde...
riotasv3d 35976 A property ` ch ` holding ...
elimhyps 35977 A version of ~ elimhyp usi...
dedths 35978 A version of weak deductio...
renegclALT 35979 Closure law for negative o...
elimhyps2 35980 Generalization of ~ elimhy...
dedths2 35981 Generalization of ~ dedths...
nfcxfrdf 35982 A utility lemma to transfe...
nfded 35983 A deduction theorem that c...
nfded2 35984 A deduction theorem that c...
nfunidALT2 35985 Deduction version of ~ nfu...
nfunidALT 35986 Deduction version of ~ nfu...
nfopdALT 35987 Deduction version of bound...
cnaddcom 35988 Recover the commutative la...
toycom 35989 Show the commutative law f...
lshpset 35994 The set of all hyperplanes...
islshp 35995 The predicate "is a hyperp...
islshpsm 35996 Hyperplane properties expr...
lshplss 35997 A hyperplane is a subspace...
lshpne 35998 A hyperplane is not equal ...
lshpnel 35999 A hyperplane's generating ...
lshpnelb 36000 The subspace sum of a hype...
lshpnel2N 36001 Condition that determines ...
lshpne0 36002 The member of the span in ...
lshpdisj 36003 A hyperplane and the span ...
lshpcmp 36004 If two hyperplanes are com...
lshpinN 36005 The intersection of two di...
lsatset 36006 The set of all 1-dim subsp...
islsat 36007 The predicate "is a 1-dim ...
lsatlspsn2 36008 The span of a nonzero sing...
lsatlspsn 36009 The span of a nonzero sing...
islsati 36010 A 1-dim subspace (atom) (o...
lsateln0 36011 A 1-dim subspace (atom) (o...
lsatlss 36012 The set of 1-dim subspaces...
lsatlssel 36013 An atom is a subspace. (C...
lsatssv 36014 An atom is a set of vector...
lsatn0 36015 A 1-dim subspace (atom) of...
lsatspn0 36016 The span of a vector is an...
lsator0sp 36017 The span of a vector is ei...
lsatssn0 36018 A subspace (or any class) ...
lsatcmp 36019 If two atoms are comparabl...
lsatcmp2 36020 If an atom is included in ...
lsatel 36021 A nonzero vector in an ato...
lsatelbN 36022 A nonzero vector in an ato...
lsat2el 36023 Two atoms sharing a nonzer...
lsmsat 36024 Convert comparison of atom...
lsatfixedN 36025 Show equality with the spa...
lsmsatcv 36026 Subspace sum has the cover...
lssatomic 36027 The lattice of subspaces i...
lssats 36028 The lattice of subspaces i...
lpssat 36029 Two subspaces in a proper ...
lrelat 36030 Subspaces are relatively a...
lssatle 36031 The ordering of two subspa...
lssat 36032 Two subspaces in a proper ...
islshpat 36033 Hyperplane properties expr...
lcvfbr 36036 The covers relation for a ...
lcvbr 36037 The covers relation for a ...
lcvbr2 36038 The covers relation for a ...
lcvbr3 36039 The covers relation for a ...
lcvpss 36040 The covers relation implie...
lcvnbtwn 36041 The covers relation implie...
lcvntr 36042 The covers relation is not...
lcvnbtwn2 36043 The covers relation implie...
lcvnbtwn3 36044 The covers relation implie...
lsmcv2 36045 Subspace sum has the cover...
lcvat 36046 If a subspace covers anoth...
lsatcv0 36047 An atom covers the zero su...
lsatcveq0 36048 A subspace covered by an a...
lsat0cv 36049 A subspace is an atom iff ...
lcvexchlem1 36050 Lemma for ~ lcvexch . (Co...
lcvexchlem2 36051 Lemma for ~ lcvexch . (Co...
lcvexchlem3 36052 Lemma for ~ lcvexch . (Co...
lcvexchlem4 36053 Lemma for ~ lcvexch . (Co...
lcvexchlem5 36054 Lemma for ~ lcvexch . (Co...
lcvexch 36055 Subspaces satisfy the exch...
lcvp 36056 Covering property of Defin...
lcv1 36057 Covering property of a sub...
lcv2 36058 Covering property of a sub...
lsatexch 36059 The atom exchange property...
lsatnle 36060 The meet of a subspace and...
lsatnem0 36061 The meet of distinct atoms...
lsatexch1 36062 The atom exch1ange propert...
lsatcv0eq 36063 If the sum of two atoms co...
lsatcv1 36064 Two atoms covering the zer...
lsatcvatlem 36065 Lemma for ~ lsatcvat . (C...
lsatcvat 36066 A nonzero subspace less th...
lsatcvat2 36067 A subspace covered by the ...
lsatcvat3 36068 A condition implying that ...
islshpcv 36069 Hyperplane properties expr...
l1cvpat 36070 A subspace covered by the ...
l1cvat 36071 Create an atom under an el...
lshpat 36072 Create an atom under a hyp...
lflset 36075 The set of linear function...
islfl 36076 The predicate "is a linear...
lfli 36077 Property of a linear funct...
islfld 36078 Properties that determine ...
lflf 36079 A linear functional is a f...
lflcl 36080 A linear functional value ...
lfl0 36081 A linear functional is zer...
lfladd 36082 Property of a linear funct...
lflsub 36083 Property of a linear funct...
lflmul 36084 Property of a linear funct...
lfl0f 36085 The zero function is a fun...
lfl1 36086 A nonzero functional has a...
lfladdcl 36087 Closure of addition of two...
lfladdcom 36088 Commutativity of functiona...
lfladdass 36089 Associativity of functiona...
lfladd0l 36090 Functional addition with t...
lflnegcl 36091 Closure of the negative of...
lflnegl 36092 A functional plus its nega...
lflvscl 36093 Closure of a scalar produc...
lflvsdi1 36094 Distributive law for (righ...
lflvsdi2 36095 Reverse distributive law f...
lflvsdi2a 36096 Reverse distributive law f...
lflvsass 36097 Associative law for (right...
lfl0sc 36098 The (right vector space) s...
lflsc0N 36099 The scalar product with th...
lfl1sc 36100 The (right vector space) s...
lkrfval 36103 The kernel of a functional...
lkrval 36104 Value of the kernel of a f...
ellkr 36105 Membership in the kernel o...
lkrval2 36106 Value of the kernel of a f...
ellkr2 36107 Membership in the kernel o...
lkrcl 36108 A member of the kernel of ...
lkrf0 36109 The value of a functional ...
lkr0f 36110 The kernel of the zero fun...
lkrlss 36111 The kernel of a linear fun...
lkrssv 36112 The kernel of a linear fun...
lkrsc 36113 The kernel of a nonzero sc...
lkrscss 36114 The kernel of a scalar pro...
eqlkr 36115 Two functionals with the s...
eqlkr2 36116 Two functionals with the s...
eqlkr3 36117 Two functionals with the s...
lkrlsp 36118 The subspace sum of a kern...
lkrlsp2 36119 The subspace sum of a kern...
lkrlsp3 36120 The subspace sum of a kern...
lkrshp 36121 The kernel of a nonzero fu...
lkrshp3 36122 The kernels of nonzero fun...
lkrshpor 36123 The kernel of a functional...
lkrshp4 36124 A kernel is a hyperplane i...
lshpsmreu 36125 Lemma for ~ lshpkrex . Sh...
lshpkrlem1 36126 Lemma for ~ lshpkrex . Th...
lshpkrlem2 36127 Lemma for ~ lshpkrex . Th...
lshpkrlem3 36128 Lemma for ~ lshpkrex . De...
lshpkrlem4 36129 Lemma for ~ lshpkrex . Pa...
lshpkrlem5 36130 Lemma for ~ lshpkrex . Pa...
lshpkrlem6 36131 Lemma for ~ lshpkrex . Sh...
lshpkrcl 36132 The set ` G ` defined by h...
lshpkr 36133 The kernel of functional `...
lshpkrex 36134 There exists a functional ...
lshpset2N 36135 The set of all hyperplanes...
islshpkrN 36136 The predicate "is a hyperp...
lfl1dim 36137 Equivalent expressions for...
lfl1dim2N 36138 Equivalent expressions for...
ldualset 36141 Define the (left) dual of ...
ldualvbase 36142 The vectors of a dual spac...
ldualelvbase 36143 Utility theorem for conver...
ldualfvadd 36144 Vector addition in the dua...
ldualvadd 36145 Vector addition in the dua...
ldualvaddcl 36146 The value of vector additi...
ldualvaddval 36147 The value of the value of ...
ldualsca 36148 The ring of scalars of the...
ldualsbase 36149 Base set of scalar ring fo...
ldualsaddN 36150 Scalar addition for the du...
ldualsmul 36151 Scalar multiplication for ...
ldualfvs 36152 Scalar product operation f...
ldualvs 36153 Scalar product operation v...
ldualvsval 36154 Value of scalar product op...
ldualvscl 36155 The scalar product operati...
ldualvaddcom 36156 Commutative law for vector...
ldualvsass 36157 Associative law for scalar...
ldualvsass2 36158 Associative law for scalar...
ldualvsdi1 36159 Distributive law for scala...
ldualvsdi2 36160 Reverse distributive law f...
ldualgrplem 36161 Lemma for ~ ldualgrp . (C...
ldualgrp 36162 The dual of a vector space...
ldual0 36163 The zero scalar of the dua...
ldual1 36164 The unit scalar of the dua...
ldualneg 36165 The negative of a scalar o...
ldual0v 36166 The zero vector of the dua...
ldual0vcl 36167 The dual zero vector is a ...
lduallmodlem 36168 Lemma for ~ lduallmod . (...
lduallmod 36169 The dual of a left module ...
lduallvec 36170 The dual of a left vector ...
ldualvsub 36171 The value of vector subtra...
ldualvsubcl 36172 Closure of vector subtract...
ldualvsubval 36173 The value of the value of ...
ldualssvscl 36174 Closure of scalar product ...
ldualssvsubcl 36175 Closure of vector subtract...
ldual0vs 36176 Scalar zero times a functi...
lkr0f2 36177 The kernel of the zero fun...
lduallkr3 36178 The kernels of nonzero fun...
lkrpssN 36179 Proper subset relation bet...
lkrin 36180 Intersection of the kernel...
eqlkr4 36181 Two functionals with the s...
ldual1dim 36182 Equivalent expressions for...
ldualkrsc 36183 The kernel of a nonzero sc...
lkrss 36184 The kernel of a scalar pro...
lkrss2N 36185 Two functionals with kerne...
lkreqN 36186 Proportional functionals h...
lkrlspeqN 36187 Condition for colinear fun...
isopos 36196 The predicate "is an ortho...
opposet 36197 Every orthoposet is a pose...
oposlem 36198 Lemma for orthoposet prope...
op01dm 36199 Conditions necessary for z...
op0cl 36200 An orthoposet has a zero e...
op1cl 36201 An orthoposet has a unit e...
op0le 36202 Orthoposet zero is less th...
ople0 36203 An element less than or eq...
opnlen0 36204 An element not less than a...
lub0N 36205 The least upper bound of t...
opltn0 36206 A lattice element greater ...
ople1 36207 Any element is less than t...
op1le 36208 If the orthoposet unit is ...
glb0N 36209 The greatest lower bound o...
opoccl 36210 Closure of orthocomplement...
opococ 36211 Double negative law for or...
opcon3b 36212 Contraposition law for ort...
opcon2b 36213 Orthocomplement contraposi...
opcon1b 36214 Orthocomplement contraposi...
oplecon3 36215 Contraposition law for ort...
oplecon3b 36216 Contraposition law for ort...
oplecon1b 36217 Contraposition law for str...
opoc1 36218 Orthocomplement of orthopo...
opoc0 36219 Orthocomplement of orthopo...
opltcon3b 36220 Contraposition law for str...
opltcon1b 36221 Contraposition law for str...
opltcon2b 36222 Contraposition law for str...
opexmid 36223 Law of excluded middle for...
opnoncon 36224 Law of contradiction for o...
riotaocN 36225 The orthocomplement of the...
cmtfvalN 36226 Value of commutes relation...
cmtvalN 36227 Equivalence for commutes r...
isolat 36228 The predicate "is an ortho...
ollat 36229 An ortholattice is a latti...
olop 36230 An ortholattice is an orth...
olposN 36231 An ortholattice is a poset...
isolatiN 36232 Properties that determine ...
oldmm1 36233 De Morgan's law for meet i...
oldmm2 36234 De Morgan's law for meet i...
oldmm3N 36235 De Morgan's law for meet i...
oldmm4 36236 De Morgan's law for meet i...
oldmj1 36237 De Morgan's law for join i...
oldmj2 36238 De Morgan's law for join i...
oldmj3 36239 De Morgan's law for join i...
oldmj4 36240 De Morgan's law for join i...
olj01 36241 An ortholattice element jo...
olj02 36242 An ortholattice element jo...
olm11 36243 The meet of an ortholattic...
olm12 36244 The meet of an ortholattic...
latmassOLD 36245 Ortholattice meet is assoc...
latm12 36246 A rearrangement of lattice...
latm32 36247 A rearrangement of lattice...
latmrot 36248 Rotate lattice meet of 3 c...
latm4 36249 Rearrangement of lattice m...
latmmdiN 36250 Lattice meet distributes o...
latmmdir 36251 Lattice meet distributes o...
olm01 36252 Meet with lattice zero is ...
olm02 36253 Meet with lattice zero is ...
isoml 36254 The predicate "is an ortho...
isomliN 36255 Properties that determine ...
omlol 36256 An orthomodular lattice is...
omlop 36257 An orthomodular lattice is...
omllat 36258 An orthomodular lattice is...
omllaw 36259 The orthomodular law. (Co...
omllaw2N 36260 Variation of orthomodular ...
omllaw3 36261 Orthomodular law equivalen...
omllaw4 36262 Orthomodular law equivalen...
omllaw5N 36263 The orthomodular law. Rem...
cmtcomlemN 36264 Lemma for ~ cmtcomN . ( ~...
cmtcomN 36265 Commutation is symmetric. ...
cmt2N 36266 Commutation with orthocomp...
cmt3N 36267 Commutation with orthocomp...
cmt4N 36268 Commutation with orthocomp...
cmtbr2N 36269 Alternate definition of th...
cmtbr3N 36270 Alternate definition for t...
cmtbr4N 36271 Alternate definition for t...
lecmtN 36272 Ordered elements commute. ...
cmtidN 36273 Any element commutes with ...
omlfh1N 36274 Foulis-Holland Theorem, pa...
omlfh3N 36275 Foulis-Holland Theorem, pa...
omlmod1i2N 36276 Analogue of modular law ~ ...
omlspjN 36277 Contraction of a Sasaki pr...
cvrfval 36284 Value of covers relation "...
cvrval 36285 Binary relation expressing...
cvrlt 36286 The covers relation implie...
cvrnbtwn 36287 There is no element betwee...
ncvr1 36288 No element covers the latt...
cvrletrN 36289 Property of an element abo...
cvrval2 36290 Binary relation expressing...
cvrnbtwn2 36291 The covers relation implie...
cvrnbtwn3 36292 The covers relation implie...
cvrcon3b 36293 Contraposition law for the...
cvrle 36294 The covers relation implie...
cvrnbtwn4 36295 The covers relation implie...
cvrnle 36296 The covers relation implie...
cvrne 36297 The covers relation implie...
cvrnrefN 36298 The covers relation is not...
cvrcmp 36299 If two lattice elements th...
cvrcmp2 36300 If two lattice elements co...
pats 36301 The set of atoms in a pose...
isat 36302 The predicate "is an atom"...
isat2 36303 The predicate "is an atom"...
atcvr0 36304 An atom covers zero. ( ~ ...
atbase 36305 An atom is a member of the...
atssbase 36306 The set of atoms is a subs...
0ltat 36307 An atom is greater than ze...
leatb 36308 A poset element less than ...
leat 36309 A poset element less than ...
leat2 36310 A nonzero poset element le...
leat3 36311 A poset element less than ...
meetat 36312 The meet of any element wi...
meetat2 36313 The meet of any element wi...
isatl 36315 The predicate "is an atomi...
atllat 36316 An atomic lattice is a lat...
atlpos 36317 An atomic lattice is a pos...
atl0dm 36318 Condition necessary for ze...
atl0cl 36319 An atomic lattice has a ze...
atl0le 36320 Orthoposet zero is less th...
atlle0 36321 An element less than or eq...
atlltn0 36322 A lattice element greater ...
isat3 36323 The predicate "is an atom"...
atn0 36324 An atom is not zero. ( ~ ...
atnle0 36325 An atom is not less than o...
atlen0 36326 A lattice element is nonze...
atcmp 36327 If two atoms are comparabl...
atncmp 36328 Frequently-used variation ...
atnlt 36329 Two atoms cannot satisfy t...
atcvreq0 36330 An element covered by an a...
atncvrN 36331 Two atoms cannot satisfy t...
atlex 36332 Every nonzero element of a...
atnle 36333 Two ways of expressing "an...
atnem0 36334 The meet of distinct atoms...
atlatmstc 36335 An atomic, complete, ortho...
atlatle 36336 The ordering of two Hilber...
atlrelat1 36337 An atomistic lattice with ...
iscvlat 36339 The predicate "is an atomi...
iscvlat2N 36340 The predicate "is an atomi...
cvlatl 36341 An atomic lattice with the...
cvllat 36342 An atomic lattice with the...
cvlposN 36343 An atomic lattice with the...
cvlexch1 36344 An atomic covering lattice...
cvlexch2 36345 An atomic covering lattice...
cvlexchb1 36346 An atomic covering lattice...
cvlexchb2 36347 An atomic covering lattice...
cvlexch3 36348 An atomic covering lattice...
cvlexch4N 36349 An atomic covering lattice...
cvlatexchb1 36350 A version of ~ cvlexchb1 f...
cvlatexchb2 36351 A version of ~ cvlexchb2 f...
cvlatexch1 36352 Atom exchange property. (...
cvlatexch2 36353 Atom exchange property. (...
cvlatexch3 36354 Atom exchange property. (...
cvlcvr1 36355 The covering property. Pr...
cvlcvrp 36356 A Hilbert lattice satisfie...
cvlatcvr1 36357 An atom is covered by its ...
cvlatcvr2 36358 An atom is covered by its ...
cvlsupr2 36359 Two equivalent ways of exp...
cvlsupr3 36360 Two equivalent ways of exp...
cvlsupr4 36361 Consequence of superpositi...
cvlsupr5 36362 Consequence of superpositi...
cvlsupr6 36363 Consequence of superpositi...
cvlsupr7 36364 Consequence of superpositi...
cvlsupr8 36365 Consequence of superpositi...
ishlat1 36368 The predicate "is a Hilber...
ishlat2 36369 The predicate "is a Hilber...
ishlat3N 36370 The predicate "is a Hilber...
ishlatiN 36371 Properties that determine ...
hlomcmcv 36372 A Hilbert lattice is ortho...
hloml 36373 A Hilbert lattice is ortho...
hlclat 36374 A Hilbert lattice is compl...
hlcvl 36375 A Hilbert lattice is an at...
hlatl 36376 A Hilbert lattice is atomi...
hlol 36377 A Hilbert lattice is an or...
hlop 36378 A Hilbert lattice is an or...
hllat 36379 A Hilbert lattice is a lat...
hllatd 36380 Deduction form of ~ hllat ...
hlomcmat 36381 A Hilbert lattice is ortho...
hlpos 36382 A Hilbert lattice is a pos...
hlatjcl 36383 Closure of join operation....
hlatjcom 36384 Commutatitivity of join op...
hlatjidm 36385 Idempotence of join operat...
hlatjass 36386 Lattice join is associativ...
hlatj12 36387 Swap 1st and 2nd members o...
hlatj32 36388 Swap 2nd and 3rd members o...
hlatjrot 36389 Rotate lattice join of 3 c...
hlatj4 36390 Rearrangement of lattice j...
hlatlej1 36391 A join's first argument is...
hlatlej2 36392 A join's second argument i...
glbconN 36393 De Morgan's law for GLB an...
glbconxN 36394 De Morgan's law for GLB an...
atnlej1 36395 If an atom is not less tha...
atnlej2 36396 If an atom is not less tha...
hlsuprexch 36397 A Hilbert lattice has the ...
hlexch1 36398 A Hilbert lattice has the ...
hlexch2 36399 A Hilbert lattice has the ...
hlexchb1 36400 A Hilbert lattice has the ...
hlexchb2 36401 A Hilbert lattice has the ...
hlsupr 36402 A Hilbert lattice has the ...
hlsupr2 36403 A Hilbert lattice has the ...
hlhgt4 36404 A Hilbert lattice has a he...
hlhgt2 36405 A Hilbert lattice has a he...
hl0lt1N 36406 Lattice 0 is less than lat...
hlexch3 36407 A Hilbert lattice has the ...
hlexch4N 36408 A Hilbert lattice has the ...
hlatexchb1 36409 A version of ~ hlexchb1 fo...
hlatexchb2 36410 A version of ~ hlexchb2 fo...
hlatexch1 36411 Atom exchange property. (...
hlatexch2 36412 Atom exchange property. (...
hlatmstcOLDN 36413 An atomic, complete, ortho...
hlatle 36414 The ordering of two Hilber...
hlateq 36415 The equality of two Hilber...
hlrelat1 36416 An atomistic lattice with ...
hlrelat5N 36417 An atomistic lattice with ...
hlrelat 36418 A Hilbert lattice is relat...
hlrelat2 36419 A consequence of relative ...
exatleN 36420 A condition for an atom to...
hl2at 36421 A Hilbert lattice has at l...
atex 36422 At least one atom exists. ...
intnatN 36423 If the intersection with a...
2llnne2N 36424 Condition implying that tw...
2llnneN 36425 Condition implying that tw...
cvr1 36426 A Hilbert lattice has the ...
cvr2N 36427 Less-than and covers equiv...
hlrelat3 36428 The Hilbert lattice is rel...
cvrval3 36429 Binary relation expressing...
cvrval4N 36430 Binary relation expressing...
cvrval5 36431 Binary relation expressing...
cvrp 36432 A Hilbert lattice satisfie...
atcvr1 36433 An atom is covered by its ...
atcvr2 36434 An atom is covered by its ...
cvrexchlem 36435 Lemma for ~ cvrexch . ( ~...
cvrexch 36436 A Hilbert lattice satisfie...
cvratlem 36437 Lemma for ~ cvrat . ( ~ a...
cvrat 36438 A nonzero Hilbert lattice ...
ltltncvr 36439 A chained strong ordering ...
ltcvrntr 36440 Non-transitive condition f...
cvrntr 36441 The covers relation is not...
atcvr0eq 36442 The covers relation is not...
lnnat 36443 A line (the join of two di...
atcvrj0 36444 Two atoms covering the zer...
cvrat2 36445 A Hilbert lattice element ...
atcvrneN 36446 Inequality derived from at...
atcvrj1 36447 Condition for an atom to b...
atcvrj2b 36448 Condition for an atom to b...
atcvrj2 36449 Condition for an atom to b...
atleneN 36450 Inequality derived from at...
atltcvr 36451 An equivalence of less-tha...
atle 36452 Any nonzero element has an...
atlt 36453 Two atoms are unequal iff ...
atlelt 36454 Transfer less-than relatio...
2atlt 36455 Given an atom less than an...
atexchcvrN 36456 Atom exchange property. V...
atexchltN 36457 Atom exchange property. V...
cvrat3 36458 A condition implying that ...
cvrat4 36459 A condition implying exist...
cvrat42 36460 Commuted version of ~ cvra...
2atjm 36461 The meet of a line (expres...
atbtwn 36462 Property of a 3rd atom ` R...
atbtwnexOLDN 36463 There exists a 3rd atom ` ...
atbtwnex 36464 Given atoms ` P ` in ` X `...
3noncolr2 36465 Two ways to express 3 non-...
3noncolr1N 36466 Two ways to express 3 non-...
hlatcon3 36467 Atom exchange combined wit...
hlatcon2 36468 Atom exchange combined wit...
4noncolr3 36469 A way to express 4 non-col...
4noncolr2 36470 A way to express 4 non-col...
4noncolr1 36471 A way to express 4 non-col...
athgt 36472 A Hilbert lattice, whose h...
3dim0 36473 There exists a 3-dimension...
3dimlem1 36474 Lemma for ~ 3dim1 . (Cont...
3dimlem2 36475 Lemma for ~ 3dim1 . (Cont...
3dimlem3a 36476 Lemma for ~ 3dim3 . (Cont...
3dimlem3 36477 Lemma for ~ 3dim1 . (Cont...
3dimlem3OLDN 36478 Lemma for ~ 3dim1 . (Cont...
3dimlem4a 36479 Lemma for ~ 3dim3 . (Cont...
3dimlem4 36480 Lemma for ~ 3dim1 . (Cont...
3dimlem4OLDN 36481 Lemma for ~ 3dim1 . (Cont...
3dim1lem5 36482 Lemma for ~ 3dim1 . (Cont...
3dim1 36483 Construct a 3-dimensional ...
3dim2 36484 Construct 2 new layers on ...
3dim3 36485 Construct a new layer on t...
2dim 36486 Generate a height-3 elemen...
1dimN 36487 An atom is covered by a he...
1cvrco 36488 The orthocomplement of an ...
1cvratex 36489 There exists an atom less ...
1cvratlt 36490 An atom less than or equal...
1cvrjat 36491 An element covered by the ...
1cvrat 36492 Create an atom under an el...
ps-1 36493 The join of two atoms ` R ...
ps-2 36494 Lattice analogue for the p...
2atjlej 36495 Two atoms are different if...
hlatexch3N 36496 Rearrange join of atoms in...
hlatexch4 36497 Exchange 2 atoms. (Contri...
ps-2b 36498 Variation of projective ge...
3atlem1 36499 Lemma for ~ 3at . (Contri...
3atlem2 36500 Lemma for ~ 3at . (Contri...
3atlem3 36501 Lemma for ~ 3at . (Contri...
3atlem4 36502 Lemma for ~ 3at . (Contri...
3atlem5 36503 Lemma for ~ 3at . (Contri...
3atlem6 36504 Lemma for ~ 3at . (Contri...
3atlem7 36505 Lemma for ~ 3at . (Contri...
3at 36506 Any three non-colinear ato...
llnset 36521 The set of lattice lines i...
islln 36522 The predicate "is a lattic...
islln4 36523 The predicate "is a lattic...
llni 36524 Condition implying a latti...
llnbase 36525 A lattice line is a lattic...
islln3 36526 The predicate "is a lattic...
islln2 36527 The predicate "is a lattic...
llni2 36528 The join of two different ...
llnnleat 36529 An atom cannot majorize a ...
llnneat 36530 A lattice line is not an a...
2atneat 36531 The join of two distinct a...
llnn0 36532 A lattice line is nonzero....
islln2a 36533 The predicate "is a lattic...
llnle 36534 Any element greater than 0...
atcvrlln2 36535 An atom under a line is co...
atcvrlln 36536 An element covering an ato...
llnexatN 36537 Given an atom on a line, t...
llncmp 36538 If two lattice lines are c...
llnnlt 36539 Two lattice lines cannot s...
2llnmat 36540 Two intersecting lines int...
2at0mat0 36541 Special case of ~ 2atmat0 ...
2atmat0 36542 The meet of two unequal li...
2atm 36543 An atom majorized by two d...
ps-2c 36544 Variation of projective ge...
lplnset 36545 The set of lattice planes ...
islpln 36546 The predicate "is a lattic...
islpln4 36547 The predicate "is a lattic...
lplni 36548 Condition implying a latti...
islpln3 36549 The predicate "is a lattic...
lplnbase 36550 A lattice plane is a latti...
islpln5 36551 The predicate "is a lattic...
islpln2 36552 The predicate "is a lattic...
lplni2 36553 The join of 3 different at...
lvolex3N 36554 There is an atom outside o...
llnmlplnN 36555 The intersection of a line...
lplnle 36556 Any element greater than 0...
lplnnle2at 36557 A lattice line (or atom) c...
lplnnleat 36558 A lattice plane cannot maj...
lplnnlelln 36559 A lattice plane is not les...
2atnelpln 36560 The join of two atoms is n...
lplnneat 36561 No lattice plane is an ato...
lplnnelln 36562 No lattice plane is a latt...
lplnn0N 36563 A lattice plane is nonzero...
islpln2a 36564 The predicate "is a lattic...
islpln2ah 36565 The predicate "is a lattic...
lplnriaN 36566 Property of a lattice plan...
lplnribN 36567 Property of a lattice plan...
lplnric 36568 Property of a lattice plan...
lplnri1 36569 Property of a lattice plan...
lplnri2N 36570 Property of a lattice plan...
lplnri3N 36571 Property of a lattice plan...
lplnllnneN 36572 Two lattice lines defined ...
llncvrlpln2 36573 A lattice line under a lat...
llncvrlpln 36574 An element covering a latt...
2lplnmN 36575 If the join of two lattice...
2llnmj 36576 The meet of two lattice li...
2atmat 36577 The meet of two intersecti...
lplncmp 36578 If two lattice planes are ...
lplnexatN 36579 Given a lattice line on a ...
lplnexllnN 36580 Given an atom on a lattice...
lplnnlt 36581 Two lattice planes cannot ...
2llnjaN 36582 The join of two different ...
2llnjN 36583 The join of two different ...
2llnm2N 36584 The meet of two different ...
2llnm3N 36585 Two lattice lines in a lat...
2llnm4 36586 Two lattice lines that maj...
2llnmeqat 36587 An atom equals the interse...
lvolset 36588 The set of 3-dim lattice v...
islvol 36589 The predicate "is a 3-dim ...
islvol4 36590 The predicate "is a 3-dim ...
lvoli 36591 Condition implying a 3-dim...
islvol3 36592 The predicate "is a 3-dim ...
lvoli3 36593 Condition implying a 3-dim...
lvolbase 36594 A 3-dim lattice volume is ...
islvol5 36595 The predicate "is a 3-dim ...
islvol2 36596 The predicate "is a 3-dim ...
lvoli2 36597 The join of 4 different at...
lvolnle3at 36598 A lattice plane (or lattic...
lvolnleat 36599 An atom cannot majorize a ...
lvolnlelln 36600 A lattice line cannot majo...
lvolnlelpln 36601 A lattice plane cannot maj...
3atnelvolN 36602 The join of 3 atoms is not...
2atnelvolN 36603 The join of two atoms is n...
lvolneatN 36604 No lattice volume is an at...
lvolnelln 36605 No lattice volume is a lat...
lvolnelpln 36606 No lattice volume is a lat...
lvoln0N 36607 A lattice volume is nonzer...
islvol2aN 36608 The predicate "is a lattic...
4atlem0a 36609 Lemma for ~ 4at . (Contri...
4atlem0ae 36610 Lemma for ~ 4at . (Contri...
4atlem0be 36611 Lemma for ~ 4at . (Contri...
4atlem3 36612 Lemma for ~ 4at . Break i...
4atlem3a 36613 Lemma for ~ 4at . Break i...
4atlem3b 36614 Lemma for ~ 4at . Break i...
4atlem4a 36615 Lemma for ~ 4at . Frequen...
4atlem4b 36616 Lemma for ~ 4at . Frequen...
4atlem4c 36617 Lemma for ~ 4at . Frequen...
4atlem4d 36618 Lemma for ~ 4at . Frequen...
4atlem9 36619 Lemma for ~ 4at . Substit...
4atlem10a 36620 Lemma for ~ 4at . Substit...
4atlem10b 36621 Lemma for ~ 4at . Substit...
4atlem10 36622 Lemma for ~ 4at . Combine...
4atlem11a 36623 Lemma for ~ 4at . Substit...
4atlem11b 36624 Lemma for ~ 4at . Substit...
4atlem11 36625 Lemma for ~ 4at . Combine...
4atlem12a 36626 Lemma for ~ 4at . Substit...
4atlem12b 36627 Lemma for ~ 4at . Substit...
4atlem12 36628 Lemma for ~ 4at . Combine...
4at 36629 Four atoms determine a lat...
4at2 36630 Four atoms determine a lat...
lplncvrlvol2 36631 A lattice line under a lat...
lplncvrlvol 36632 An element covering a latt...
lvolcmp 36633 If two lattice planes are ...
lvolnltN 36634 Two lattice volumes cannot...
2lplnja 36635 The join of two different ...
2lplnj 36636 The join of two different ...
2lplnm2N 36637 The meet of two different ...
2lplnmj 36638 The meet of two lattice pl...
dalemkehl 36639 Lemma for ~ dath . Freque...
dalemkelat 36640 Lemma for ~ dath . Freque...
dalemkeop 36641 Lemma for ~ dath . Freque...
dalempea 36642 Lemma for ~ dath . Freque...
dalemqea 36643 Lemma for ~ dath . Freque...
dalemrea 36644 Lemma for ~ dath . Freque...
dalemsea 36645 Lemma for ~ dath . Freque...
dalemtea 36646 Lemma for ~ dath . Freque...
dalemuea 36647 Lemma for ~ dath . Freque...
dalemyeo 36648 Lemma for ~ dath . Freque...
dalemzeo 36649 Lemma for ~ dath . Freque...
dalemclpjs 36650 Lemma for ~ dath . Freque...
dalemclqjt 36651 Lemma for ~ dath . Freque...
dalemclrju 36652 Lemma for ~ dath . Freque...
dalem-clpjq 36653 Lemma for ~ dath . Freque...
dalemceb 36654 Lemma for ~ dath . Freque...
dalempeb 36655 Lemma for ~ dath . Freque...
dalemqeb 36656 Lemma for ~ dath . Freque...
dalemreb 36657 Lemma for ~ dath . Freque...
dalemseb 36658 Lemma for ~ dath . Freque...
dalemteb 36659 Lemma for ~ dath . Freque...
dalemueb 36660 Lemma for ~ dath . Freque...
dalempjqeb 36661 Lemma for ~ dath . Freque...
dalemsjteb 36662 Lemma for ~ dath . Freque...
dalemtjueb 36663 Lemma for ~ dath . Freque...
dalemqrprot 36664 Lemma for ~ dath . Freque...
dalemyeb 36665 Lemma for ~ dath . Freque...
dalemcnes 36666 Lemma for ~ dath . Freque...
dalempnes 36667 Lemma for ~ dath . Freque...
dalemqnet 36668 Lemma for ~ dath . Freque...
dalempjsen 36669 Lemma for ~ dath . Freque...
dalemply 36670 Lemma for ~ dath . Freque...
dalemsly 36671 Lemma for ~ dath . Freque...
dalemswapyz 36672 Lemma for ~ dath . Swap t...
dalemrot 36673 Lemma for ~ dath . Rotate...
dalemrotyz 36674 Lemma for ~ dath . Rotate...
dalem1 36675 Lemma for ~ dath . Show t...
dalemcea 36676 Lemma for ~ dath . Freque...
dalem2 36677 Lemma for ~ dath . Show t...
dalemdea 36678 Lemma for ~ dath . Freque...
dalemeea 36679 Lemma for ~ dath . Freque...
dalem3 36680 Lemma for ~ dalemdnee . (...
dalem4 36681 Lemma for ~ dalemdnee . (...
dalemdnee 36682 Lemma for ~ dath . Axis o...
dalem5 36683 Lemma for ~ dath . Atom `...
dalem6 36684 Lemma for ~ dath . Analog...
dalem7 36685 Lemma for ~ dath . Analog...
dalem8 36686 Lemma for ~ dath . Plane ...
dalem-cly 36687 Lemma for ~ dalem9 . Cent...
dalem9 36688 Lemma for ~ dath . Since ...
dalem10 36689 Lemma for ~ dath . Atom `...
dalem11 36690 Lemma for ~ dath . Analog...
dalem12 36691 Lemma for ~ dath . Analog...
dalem13 36692 Lemma for ~ dalem14 . (Co...
dalem14 36693 Lemma for ~ dath . Planes...
dalem15 36694 Lemma for ~ dath . The ax...
dalem16 36695 Lemma for ~ dath . The at...
dalem17 36696 Lemma for ~ dath . When p...
dalem18 36697 Lemma for ~ dath . Show t...
dalem19 36698 Lemma for ~ dath . Show t...
dalemccea 36699 Lemma for ~ dath . Freque...
dalemddea 36700 Lemma for ~ dath . Freque...
dalem-ccly 36701 Lemma for ~ dath . Freque...
dalem-ddly 36702 Lemma for ~ dath . Freque...
dalemccnedd 36703 Lemma for ~ dath . Freque...
dalemclccjdd 36704 Lemma for ~ dath . Freque...
dalemcceb 36705 Lemma for ~ dath . Freque...
dalemswapyzps 36706 Lemma for ~ dath . Swap t...
dalemrotps 36707 Lemma for ~ dath . Rotate...
dalemcjden 36708 Lemma for ~ dath . Show t...
dalem20 36709 Lemma for ~ dath . Show t...
dalem21 36710 Lemma for ~ dath . Show t...
dalem22 36711 Lemma for ~ dath . Show t...
dalem23 36712 Lemma for ~ dath . Show t...
dalem24 36713 Lemma for ~ dath . Show t...
dalem25 36714 Lemma for ~ dath . Show t...
dalem27 36715 Lemma for ~ dath . Show t...
dalem28 36716 Lemma for ~ dath . Lemma ...
dalem29 36717 Lemma for ~ dath . Analog...
dalem30 36718 Lemma for ~ dath . Analog...
dalem31N 36719 Lemma for ~ dath . Analog...
dalem32 36720 Lemma for ~ dath . Analog...
dalem33 36721 Lemma for ~ dath . Analog...
dalem34 36722 Lemma for ~ dath . Analog...
dalem35 36723 Lemma for ~ dath . Analog...
dalem36 36724 Lemma for ~ dath . Analog...
dalem37 36725 Lemma for ~ dath . Analog...
dalem38 36726 Lemma for ~ dath . Plane ...
dalem39 36727 Lemma for ~ dath . Auxili...
dalem40 36728 Lemma for ~ dath . Analog...
dalem41 36729 Lemma for ~ dath . (Contr...
dalem42 36730 Lemma for ~ dath . Auxili...
dalem43 36731 Lemma for ~ dath . Planes...
dalem44 36732 Lemma for ~ dath . Dummy ...
dalem45 36733 Lemma for ~ dath . Dummy ...
dalem46 36734 Lemma for ~ dath . Analog...
dalem47 36735 Lemma for ~ dath . Analog...
dalem48 36736 Lemma for ~ dath . Analog...
dalem49 36737 Lemma for ~ dath . Analog...
dalem50 36738 Lemma for ~ dath . Analog...
dalem51 36739 Lemma for ~ dath . Constr...
dalem52 36740 Lemma for ~ dath . Lines ...
dalem53 36741 Lemma for ~ dath . The au...
dalem54 36742 Lemma for ~ dath . Line `...
dalem55 36743 Lemma for ~ dath . Lines ...
dalem56 36744 Lemma for ~ dath . Analog...
dalem57 36745 Lemma for ~ dath . Axis o...
dalem58 36746 Lemma for ~ dath . Analog...
dalem59 36747 Lemma for ~ dath . Analog...
dalem60 36748 Lemma for ~ dath . ` B ` i...
dalem61 36749 Lemma for ~ dath . Show t...
dalem62 36750 Lemma for ~ dath . Elimin...
dalem63 36751 Lemma for ~ dath . Combin...
dath 36752 Desargues's theorem of pro...
dath2 36753 Version of Desargues's the...
lineset 36754 The set of lines in a Hilb...
isline 36755 The predicate "is a line"....
islinei 36756 Condition implying "is a l...
pointsetN 36757 The set of points in a Hil...
ispointN 36758 The predicate "is a point"...
atpointN 36759 The singleton of an atom i...
psubspset 36760 The set of projective subs...
ispsubsp 36761 The predicate "is a projec...
ispsubsp2 36762 The predicate "is a projec...
psubspi 36763 Property of a projective s...
psubspi2N 36764 Property of a projective s...
0psubN 36765 The empty set is a project...
snatpsubN 36766 The singleton of an atom i...
pointpsubN 36767 A point (singleton of an a...
linepsubN 36768 A line is a projective sub...
atpsubN 36769 The set of all atoms is a ...
psubssat 36770 A projective subspace cons...
psubatN 36771 A member of a projective s...
pmapfval 36772 The projective map of a Hi...
pmapval 36773 Value of the projective ma...
elpmap 36774 Member of a projective map...
pmapssat 36775 The projective map of a Hi...
pmapssbaN 36776 A weakening of ~ pmapssat ...
pmaple 36777 The projective map of a Hi...
pmap11 36778 The projective map of a Hi...
pmapat 36779 The projective map of an a...
elpmapat 36780 Member of the projective m...
pmap0 36781 Value of the projective ma...
pmapeq0 36782 A projective map value is ...
pmap1N 36783 Value of the projective ma...
pmapsub 36784 The projective map of a Hi...
pmapglbx 36785 The projective map of the ...
pmapglb 36786 The projective map of the ...
pmapglb2N 36787 The projective map of the ...
pmapglb2xN 36788 The projective map of the ...
pmapmeet 36789 The projective map of a me...
isline2 36790 Definition of line in term...
linepmap 36791 A line described with a pr...
isline3 36792 Definition of line in term...
isline4N 36793 Definition of line in term...
lneq2at 36794 A line equals the join of ...
lnatexN 36795 There is an atom in a line...
lnjatN 36796 Given an atom in a line, t...
lncvrelatN 36797 A lattice element covered ...
lncvrat 36798 A line covers the atoms it...
lncmp 36799 If two lines are comparabl...
2lnat 36800 Two intersecting lines int...
2atm2atN 36801 Two joins with a common at...
2llnma1b 36802 Generalization of ~ 2llnma...
2llnma1 36803 Two different intersecting...
2llnma3r 36804 Two different intersecting...
2llnma2 36805 Two different intersecting...
2llnma2rN 36806 Two different intersecting...
cdlema1N 36807 A condition for required f...
cdlema2N 36808 A condition for required f...
cdlemblem 36809 Lemma for ~ cdlemb . (Con...
cdlemb 36810 Given two atoms not less t...
paddfval 36813 Projective subspace sum op...
paddval 36814 Projective subspace sum op...
elpadd 36815 Member of a projective sub...
elpaddn0 36816 Member of projective subsp...
paddvaln0N 36817 Projective subspace sum op...
elpaddri 36818 Condition implying members...
elpaddatriN 36819 Condition implying members...
elpaddat 36820 Membership in a projective...
elpaddatiN 36821 Consequence of membership ...
elpadd2at 36822 Membership in a projective...
elpadd2at2 36823 Membership in a projective...
paddunssN 36824 Projective subspace sum in...
elpadd0 36825 Member of projective subsp...
paddval0 36826 Projective subspace sum wi...
padd01 36827 Projective subspace sum wi...
padd02 36828 Projective subspace sum wi...
paddcom 36829 Projective subspace sum co...
paddssat 36830 A projective subspace sum ...
sspadd1 36831 A projective subspace sum ...
sspadd2 36832 A projective subspace sum ...
paddss1 36833 Subset law for projective ...
paddss2 36834 Subset law for projective ...
paddss12 36835 Subset law for projective ...
paddasslem1 36836 Lemma for ~ paddass . (Co...
paddasslem2 36837 Lemma for ~ paddass . (Co...
paddasslem3 36838 Lemma for ~ paddass . Res...
paddasslem4 36839 Lemma for ~ paddass . Com...
paddasslem5 36840 Lemma for ~ paddass . Sho...
paddasslem6 36841 Lemma for ~ paddass . (Co...
paddasslem7 36842 Lemma for ~ paddass . Com...
paddasslem8 36843 Lemma for ~ paddass . (Co...
paddasslem9 36844 Lemma for ~ paddass . Com...
paddasslem10 36845 Lemma for ~ paddass . Use...
paddasslem11 36846 Lemma for ~ paddass . The...
paddasslem12 36847 Lemma for ~ paddass . The...
paddasslem13 36848 Lemma for ~ paddass . The...
paddasslem14 36849 Lemma for ~ paddass . Rem...
paddasslem15 36850 Lemma for ~ paddass . Use...
paddasslem16 36851 Lemma for ~ paddass . Use...
paddasslem17 36852 Lemma for ~ paddass . The...
paddasslem18 36853 Lemma for ~ paddass . Com...
paddass 36854 Projective subspace sum is...
padd12N 36855 Commutative/associative la...
padd4N 36856 Rearrangement of 4 terms i...
paddidm 36857 Projective subspace sum is...
paddclN 36858 The projective sum of two ...
paddssw1 36859 Subset law for projective ...
paddssw2 36860 Subset law for projective ...
paddss 36861 Subset law for projective ...
pmodlem1 36862 Lemma for ~ pmod1i . (Con...
pmodlem2 36863 Lemma for ~ pmod1i . (Con...
pmod1i 36864 The modular law holds in a...
pmod2iN 36865 Dual of the modular law. ...
pmodN 36866 The modular law for projec...
pmodl42N 36867 Lemma derived from modular...
pmapjoin 36868 The projective map of the ...
pmapjat1 36869 The projective map of the ...
pmapjat2 36870 The projective map of the ...
pmapjlln1 36871 The projective map of the ...
hlmod1i 36872 A version of the modular l...
atmod1i1 36873 Version of modular law ~ p...
atmod1i1m 36874 Version of modular law ~ p...
atmod1i2 36875 Version of modular law ~ p...
llnmod1i2 36876 Version of modular law ~ p...
atmod2i1 36877 Version of modular law ~ p...
atmod2i2 36878 Version of modular law ~ p...
llnmod2i2 36879 Version of modular law ~ p...
atmod3i1 36880 Version of modular law tha...
atmod3i2 36881 Version of modular law tha...
atmod4i1 36882 Version of modular law tha...
atmod4i2 36883 Version of modular law tha...
llnexchb2lem 36884 Lemma for ~ llnexchb2 . (...
llnexchb2 36885 Line exchange property (co...
llnexch2N 36886 Line exchange property (co...
dalawlem1 36887 Lemma for ~ dalaw . Speci...
dalawlem2 36888 Lemma for ~ dalaw . Utili...
dalawlem3 36889 Lemma for ~ dalaw . First...
dalawlem4 36890 Lemma for ~ dalaw . Secon...
dalawlem5 36891 Lemma for ~ dalaw . Speci...
dalawlem6 36892 Lemma for ~ dalaw . First...
dalawlem7 36893 Lemma for ~ dalaw . Secon...
dalawlem8 36894 Lemma for ~ dalaw . Speci...
dalawlem9 36895 Lemma for ~ dalaw . Speci...
dalawlem10 36896 Lemma for ~ dalaw . Combi...
dalawlem11 36897 Lemma for ~ dalaw . First...
dalawlem12 36898 Lemma for ~ dalaw . Secon...
dalawlem13 36899 Lemma for ~ dalaw . Speci...
dalawlem14 36900 Lemma for ~ dalaw . Combi...
dalawlem15 36901 Lemma for ~ dalaw . Swap ...
dalaw 36902 Desargues's law, derived f...
pclfvalN 36905 The projective subspace cl...
pclvalN 36906 Value of the projective su...
pclclN 36907 Closure of the projective ...
elpclN 36908 Membership in the projecti...
elpcliN 36909 Implication of membership ...
pclssN 36910 Ordering is preserved by s...
pclssidN 36911 A set of atoms is included...
pclidN 36912 The projective subspace cl...
pclbtwnN 36913 A projective subspace sand...
pclunN 36914 The projective subspace cl...
pclun2N 36915 The projective subspace cl...
pclfinN 36916 The projective subspace cl...
pclcmpatN 36917 The set of projective subs...
polfvalN 36920 The projective subspace po...
polvalN 36921 Value of the projective su...
polval2N 36922 Alternate expression for v...
polsubN 36923 The polarity of a set of a...
polssatN 36924 The polarity of a set of a...
pol0N 36925 The polarity of the empty ...
pol1N 36926 The polarity of the whole ...
2pol0N 36927 The closed subspace closur...
polpmapN 36928 The polarity of a projecti...
2polpmapN 36929 Double polarity of a proje...
2polvalN 36930 Value of double polarity. ...
2polssN 36931 A set of atoms is a subset...
3polN 36932 Triple polarity cancels to...
polcon3N 36933 Contraposition law for pol...
2polcon4bN 36934 Contraposition law for pol...
polcon2N 36935 Contraposition law for pol...
polcon2bN 36936 Contraposition law for pol...
pclss2polN 36937 The projective subspace cl...
pcl0N 36938 The projective subspace cl...
pcl0bN 36939 The projective subspace cl...
pmaplubN 36940 The LUB of a projective ma...
sspmaplubN 36941 A set of atoms is a subset...
2pmaplubN 36942 Double projective map of a...
paddunN 36943 The closure of the project...
poldmj1N 36944 De Morgan's law for polari...
pmapj2N 36945 The projective map of the ...
pmapocjN 36946 The projective map of the ...
polatN 36947 The polarity of the single...
2polatN 36948 Double polarity of the sin...
pnonsingN 36949 The intersection of a set ...
psubclsetN 36952 The set of closed projecti...
ispsubclN 36953 The predicate "is a closed...
psubcliN 36954 Property of a closed proje...
psubcli2N 36955 Property of a closed proje...
psubclsubN 36956 A closed projective subspa...
psubclssatN 36957 A closed projective subspa...
pmapidclN 36958 Projective map of the LUB ...
0psubclN 36959 The empty set is a closed ...
1psubclN 36960 The set of all atoms is a ...
atpsubclN 36961 A point (singleton of an a...
pmapsubclN 36962 A projective map value is ...
ispsubcl2N 36963 Alternate predicate for "i...
psubclinN 36964 The intersection of two cl...
paddatclN 36965 The projective sum of a cl...
pclfinclN 36966 The projective subspace cl...
linepsubclN 36967 A line is a closed project...
polsubclN 36968 A polarity is a closed pro...
poml4N 36969 Orthomodular law for proje...
poml5N 36970 Orthomodular law for proje...
poml6N 36971 Orthomodular law for proje...
osumcllem1N 36972 Lemma for ~ osumclN . (Co...
osumcllem2N 36973 Lemma for ~ osumclN . (Co...
osumcllem3N 36974 Lemma for ~ osumclN . (Co...
osumcllem4N 36975 Lemma for ~ osumclN . (Co...
osumcllem5N 36976 Lemma for ~ osumclN . (Co...
osumcllem6N 36977 Lemma for ~ osumclN . Use...
osumcllem7N 36978 Lemma for ~ osumclN . (Co...
osumcllem8N 36979 Lemma for ~ osumclN . (Co...
osumcllem9N 36980 Lemma for ~ osumclN . (Co...
osumcllem10N 36981 Lemma for ~ osumclN . Con...
osumcllem11N 36982 Lemma for ~ osumclN . (Co...
osumclN 36983 Closure of orthogonal sum....
pmapojoinN 36984 For orthogonal elements, p...
pexmidN 36985 Excluded middle law for cl...
pexmidlem1N 36986 Lemma for ~ pexmidN . Hol...
pexmidlem2N 36987 Lemma for ~ pexmidN . (Co...
pexmidlem3N 36988 Lemma for ~ pexmidN . Use...
pexmidlem4N 36989 Lemma for ~ pexmidN . (Co...
pexmidlem5N 36990 Lemma for ~ pexmidN . (Co...
pexmidlem6N 36991 Lemma for ~ pexmidN . (Co...
pexmidlem7N 36992 Lemma for ~ pexmidN . Con...
pexmidlem8N 36993 Lemma for ~ pexmidN . The...
pexmidALTN 36994 Excluded middle law for cl...
pl42lem1N 36995 Lemma for ~ pl42N . (Cont...
pl42lem2N 36996 Lemma for ~ pl42N . (Cont...
pl42lem3N 36997 Lemma for ~ pl42N . (Cont...
pl42lem4N 36998 Lemma for ~ pl42N . (Cont...
pl42N 36999 Law holding in a Hilbert l...
watfvalN 37008 The W atoms function. (Co...
watvalN 37009 Value of the W atoms funct...
iswatN 37010 The predicate "is a W atom...
lhpset 37011 The set of co-atoms (latti...
islhp 37012 The predicate "is a co-ato...
islhp2 37013 The predicate "is a co-ato...
lhpbase 37014 A co-atom is a member of t...
lhp1cvr 37015 The lattice unit covers a ...
lhplt 37016 An atom under a co-atom is...
lhp2lt 37017 The join of two atoms unde...
lhpexlt 37018 There exists an atom less ...
lhp0lt 37019 A co-atom is greater than ...
lhpn0 37020 A co-atom is nonzero. TOD...
lhpexle 37021 There exists an atom under...
lhpexnle 37022 There exists an atom not u...
lhpexle1lem 37023 Lemma for ~ lhpexle1 and o...
lhpexle1 37024 There exists an atom under...
lhpexle2lem 37025 Lemma for ~ lhpexle2 . (C...
lhpexle2 37026 There exists atom under a ...
lhpexle3lem 37027 There exists atom under a ...
lhpexle3 37028 There exists atom under a ...
lhpex2leN 37029 There exist at least two d...
lhpoc 37030 The orthocomplement of a c...
lhpoc2N 37031 The orthocomplement of an ...
lhpocnle 37032 The orthocomplement of a c...
lhpocat 37033 The orthocomplement of a c...
lhpocnel 37034 The orthocomplement of a c...
lhpocnel2 37035 The orthocomplement of a c...
lhpjat1 37036 The join of a co-atom (hyp...
lhpjat2 37037 The join of a co-atom (hyp...
lhpj1 37038 The join of a co-atom (hyp...
lhpmcvr 37039 The meet of a lattice hype...
lhpmcvr2 37040 Alternate way to express t...
lhpmcvr3 37041 Specialization of ~ lhpmcv...
lhpmcvr4N 37042 Specialization of ~ lhpmcv...
lhpmcvr5N 37043 Specialization of ~ lhpmcv...
lhpmcvr6N 37044 Specialization of ~ lhpmcv...
lhpm0atN 37045 If the meet of a lattice h...
lhpmat 37046 An element covered by the ...
lhpmatb 37047 An element covered by the ...
lhp2at0 37048 Join and meet with differe...
lhp2atnle 37049 Inequality for 2 different...
lhp2atne 37050 Inequality for joins with ...
lhp2at0nle 37051 Inequality for 2 different...
lhp2at0ne 37052 Inequality for joins with ...
lhpelim 37053 Eliminate an atom not unde...
lhpmod2i2 37054 Modular law for hyperplane...
lhpmod6i1 37055 Modular law for hyperplane...
lhprelat3N 37056 The Hilbert lattice is rel...
cdlemb2 37057 Given two atoms not under ...
lhple 37058 Property of a lattice elem...
lhpat 37059 Create an atom under a co-...
lhpat4N 37060 Property of an atom under ...
lhpat2 37061 Create an atom under a co-...
lhpat3 37062 There is only one atom und...
4atexlemk 37063 Lemma for ~ 4atexlem7 . (...
4atexlemw 37064 Lemma for ~ 4atexlem7 . (...
4atexlempw 37065 Lemma for ~ 4atexlem7 . (...
4atexlemp 37066 Lemma for ~ 4atexlem7 . (...
4atexlemq 37067 Lemma for ~ 4atexlem7 . (...
4atexlems 37068 Lemma for ~ 4atexlem7 . (...
4atexlemt 37069 Lemma for ~ 4atexlem7 . (...
4atexlemutvt 37070 Lemma for ~ 4atexlem7 . (...
4atexlempnq 37071 Lemma for ~ 4atexlem7 . (...
4atexlemnslpq 37072 Lemma for ~ 4atexlem7 . (...
4atexlemkl 37073 Lemma for ~ 4atexlem7 . (...
4atexlemkc 37074 Lemma for ~ 4atexlem7 . (...
4atexlemwb 37075 Lemma for ~ 4atexlem7 . (...
4atexlempsb 37076 Lemma for ~ 4atexlem7 . (...
4atexlemqtb 37077 Lemma for ~ 4atexlem7 . (...
4atexlempns 37078 Lemma for ~ 4atexlem7 . (...
4atexlemswapqr 37079 Lemma for ~ 4atexlem7 . S...
4atexlemu 37080 Lemma for ~ 4atexlem7 . (...
4atexlemv 37081 Lemma for ~ 4atexlem7 . (...
4atexlemunv 37082 Lemma for ~ 4atexlem7 . (...
4atexlemtlw 37083 Lemma for ~ 4atexlem7 . (...
4atexlemntlpq 37084 Lemma for ~ 4atexlem7 . (...
4atexlemc 37085 Lemma for ~ 4atexlem7 . (...
4atexlemnclw 37086 Lemma for ~ 4atexlem7 . (...
4atexlemex2 37087 Lemma for ~ 4atexlem7 . S...
4atexlemcnd 37088 Lemma for ~ 4atexlem7 . (...
4atexlemex4 37089 Lemma for ~ 4atexlem7 . S...
4atexlemex6 37090 Lemma for ~ 4atexlem7 . (...
4atexlem7 37091 Whenever there are at leas...
4atex 37092 Whenever there are at leas...
4atex2 37093 More general version of ~ ...
4atex2-0aOLDN 37094 Same as ~ 4atex2 except th...
4atex2-0bOLDN 37095 Same as ~ 4atex2 except th...
4atex2-0cOLDN 37096 Same as ~ 4atex2 except th...
4atex3 37097 More general version of ~ ...
lautset 37098 The set of lattice automor...
islaut 37099 The predictate "is a latti...
lautle 37100 Less-than or equal propert...
laut1o 37101 A lattice automorphism is ...
laut11 37102 One-to-one property of a l...
lautcl 37103 A lattice automorphism val...
lautcnvclN 37104 Reverse closure of a latti...
lautcnvle 37105 Less-than or equal propert...
lautcnv 37106 The converse of a lattice ...
lautlt 37107 Less-than property of a la...
lautcvr 37108 Covering property of a lat...
lautj 37109 Meet property of a lattice...
lautm 37110 Meet property of a lattice...
lauteq 37111 A lattice automorphism arg...
idlaut 37112 The identity function is a...
lautco 37113 The composition of two lat...
pautsetN 37114 The set of projective auto...
ispautN 37115 The predictate "is a proje...
ldilfset 37124 The mapping from fiducial ...
ldilset 37125 The set of lattice dilatio...
isldil 37126 The predicate "is a lattic...
ldillaut 37127 A lattice dilation is an a...
ldil1o 37128 A lattice dilation is a on...
ldilval 37129 Value of a lattice dilatio...
idldil 37130 The identity function is a...
ldilcnv 37131 The converse of a lattice ...
ldilco 37132 The composition of two lat...
ltrnfset 37133 The set of all lattice tra...
ltrnset 37134 The set of lattice transla...
isltrn 37135 The predicate "is a lattic...
isltrn2N 37136 The predicate "is a lattic...
ltrnu 37137 Uniqueness property of a l...
ltrnldil 37138 A lattice translation is a...
ltrnlaut 37139 A lattice translation is a...
ltrn1o 37140 A lattice translation is a...
ltrncl 37141 Closure of a lattice trans...
ltrn11 37142 One-to-one property of a l...
ltrncnvnid 37143 If a translation is differ...
ltrncoidN 37144 Two translations are equal...
ltrnle 37145 Less-than or equal propert...
ltrncnvleN 37146 Less-than or equal propert...
ltrnm 37147 Lattice translation of a m...
ltrnj 37148 Lattice translation of a m...
ltrncvr 37149 Covering property of a lat...
ltrnval1 37150 Value of a lattice transla...
ltrnid 37151 A lattice translation is t...
ltrnnid 37152 If a lattice translation i...
ltrnatb 37153 The lattice translation of...
ltrncnvatb 37154 The converse of the lattic...
ltrnel 37155 The lattice translation of...
ltrnat 37156 The lattice translation of...
ltrncnvat 37157 The converse of the lattic...
ltrncnvel 37158 The converse of the lattic...
ltrncoelN 37159 Composition of lattice tra...
ltrncoat 37160 Composition of lattice tra...
ltrncoval 37161 Two ways to express value ...
ltrncnv 37162 The converse of a lattice ...
ltrn11at 37163 Frequently used one-to-one...
ltrneq2 37164 The equality of two transl...
ltrneq 37165 The equality of two transl...
idltrn 37166 The identity function is a...
ltrnmw 37167 Property of lattice transl...
dilfsetN 37168 The mapping from fiducial ...
dilsetN 37169 The set of dilations for a...
isdilN 37170 The predicate "is a dilati...
trnfsetN 37171 The mapping from fiducial ...
trnsetN 37172 The set of translations fo...
istrnN 37173 The predicate "is a transl...
trlfset 37176 The set of all traces of l...
trlset 37177 The set of traces of latti...
trlval 37178 The value of the trace of ...
trlval2 37179 The value of the trace of ...
trlcl 37180 Closure of the trace of a ...
trlcnv 37181 The trace of the converse ...
trljat1 37182 The value of a translation...
trljat2 37183 The value of a translation...
trljat3 37184 The value of a translation...
trlat 37185 If an atom differs from it...
trl0 37186 If an atom not under the f...
trlator0 37187 The trace of a lattice tra...
trlatn0 37188 The trace of a lattice tra...
trlnidat 37189 The trace of a lattice tra...
ltrnnidn 37190 If a lattice translation i...
ltrnideq 37191 Property of the identity l...
trlid0 37192 The trace of the identity ...
trlnidatb 37193 A lattice translation is n...
trlid0b 37194 A lattice translation is t...
trlnid 37195 Different translations wit...
ltrn2ateq 37196 Property of the equality o...
ltrnateq 37197 If any atom (under ` W ` )...
ltrnatneq 37198 If any atom (under ` W ` )...
ltrnatlw 37199 If the value of an atom eq...
trlle 37200 The trace of a lattice tra...
trlne 37201 The trace of a lattice tra...
trlnle 37202 The atom not under the fid...
trlval3 37203 The value of the trace of ...
trlval4 37204 The value of the trace of ...
trlval5 37205 The value of the trace of ...
arglem1N 37206 Lemma for Desargues's law....
cdlemc1 37207 Part of proof of Lemma C i...
cdlemc2 37208 Part of proof of Lemma C i...
cdlemc3 37209 Part of proof of Lemma C i...
cdlemc4 37210 Part of proof of Lemma C i...
cdlemc5 37211 Lemma for ~ cdlemc . (Con...
cdlemc6 37212 Lemma for ~ cdlemc . (Con...
cdlemc 37213 Lemma C in [Crawley] p. 11...
cdlemd1 37214 Part of proof of Lemma D i...
cdlemd2 37215 Part of proof of Lemma D i...
cdlemd3 37216 Part of proof of Lemma D i...
cdlemd4 37217 Part of proof of Lemma D i...
cdlemd5 37218 Part of proof of Lemma D i...
cdlemd6 37219 Part of proof of Lemma D i...
cdlemd7 37220 Part of proof of Lemma D i...
cdlemd8 37221 Part of proof of Lemma D i...
cdlemd9 37222 Part of proof of Lemma D i...
cdlemd 37223 If two translations agree ...
ltrneq3 37224 Two translations agree at ...
cdleme00a 37225 Part of proof of Lemma E i...
cdleme0aa 37226 Part of proof of Lemma E i...
cdleme0a 37227 Part of proof of Lemma E i...
cdleme0b 37228 Part of proof of Lemma E i...
cdleme0c 37229 Part of proof of Lemma E i...
cdleme0cp 37230 Part of proof of Lemma E i...
cdleme0cq 37231 Part of proof of Lemma E i...
cdleme0dN 37232 Part of proof of Lemma E i...
cdleme0e 37233 Part of proof of Lemma E i...
cdleme0fN 37234 Part of proof of Lemma E i...
cdleme0gN 37235 Part of proof of Lemma E i...
cdlemeulpq 37236 Part of proof of Lemma E i...
cdleme01N 37237 Part of proof of Lemma E i...
cdleme02N 37238 Part of proof of Lemma E i...
cdleme0ex1N 37239 Part of proof of Lemma E i...
cdleme0ex2N 37240 Part of proof of Lemma E i...
cdleme0moN 37241 Part of proof of Lemma E i...
cdleme1b 37242 Part of proof of Lemma E i...
cdleme1 37243 Part of proof of Lemma E i...
cdleme2 37244 Part of proof of Lemma E i...
cdleme3b 37245 Part of proof of Lemma E i...
cdleme3c 37246 Part of proof of Lemma E i...
cdleme3d 37247 Part of proof of Lemma E i...
cdleme3e 37248 Part of proof of Lemma E i...
cdleme3fN 37249 Part of proof of Lemma E i...
cdleme3g 37250 Part of proof of Lemma E i...
cdleme3h 37251 Part of proof of Lemma E i...
cdleme3fa 37252 Part of proof of Lemma E i...
cdleme3 37253 Part of proof of Lemma E i...
cdleme4 37254 Part of proof of Lemma E i...
cdleme4a 37255 Part of proof of Lemma E i...
cdleme5 37256 Part of proof of Lemma E i...
cdleme6 37257 Part of proof of Lemma E i...
cdleme7aa 37258 Part of proof of Lemma E i...
cdleme7a 37259 Part of proof of Lemma E i...
cdleme7b 37260 Part of proof of Lemma E i...
cdleme7c 37261 Part of proof of Lemma E i...
cdleme7d 37262 Part of proof of Lemma E i...
cdleme7e 37263 Part of proof of Lemma E i...
cdleme7ga 37264 Part of proof of Lemma E i...
cdleme7 37265 Part of proof of Lemma E i...
cdleme8 37266 Part of proof of Lemma E i...
cdleme9a 37267 Part of proof of Lemma E i...
cdleme9b 37268 Utility lemma for Lemma E ...
cdleme9 37269 Part of proof of Lemma E i...
cdleme10 37270 Part of proof of Lemma E i...
cdleme8tN 37271 Part of proof of Lemma E i...
cdleme9taN 37272 Part of proof of Lemma E i...
cdleme9tN 37273 Part of proof of Lemma E i...
cdleme10tN 37274 Part of proof of Lemma E i...
cdleme16aN 37275 Part of proof of Lemma E i...
cdleme11a 37276 Part of proof of Lemma E i...
cdleme11c 37277 Part of proof of Lemma E i...
cdleme11dN 37278 Part of proof of Lemma E i...
cdleme11e 37279 Part of proof of Lemma E i...
cdleme11fN 37280 Part of proof of Lemma E i...
cdleme11g 37281 Part of proof of Lemma E i...
cdleme11h 37282 Part of proof of Lemma E i...
cdleme11j 37283 Part of proof of Lemma E i...
cdleme11k 37284 Part of proof of Lemma E i...
cdleme11l 37285 Part of proof of Lemma E i...
cdleme11 37286 Part of proof of Lemma E i...
cdleme12 37287 Part of proof of Lemma E i...
cdleme13 37288 Part of proof of Lemma E i...
cdleme14 37289 Part of proof of Lemma E i...
cdleme15a 37290 Part of proof of Lemma E i...
cdleme15b 37291 Part of proof of Lemma E i...
cdleme15c 37292 Part of proof of Lemma E i...
cdleme15d 37293 Part of proof of Lemma E i...
cdleme15 37294 Part of proof of Lemma E i...
cdleme16b 37295 Part of proof of Lemma E i...
cdleme16c 37296 Part of proof of Lemma E i...
cdleme16d 37297 Part of proof of Lemma E i...
cdleme16e 37298 Part of proof of Lemma E i...
cdleme16f 37299 Part of proof of Lemma E i...
cdleme16g 37300 Part of proof of Lemma E i...
cdleme16 37301 Part of proof of Lemma E i...
cdleme17a 37302 Part of proof of Lemma E i...
cdleme17b 37303 Lemma leading to ~ cdleme1...
cdleme17c 37304 Part of proof of Lemma E i...
cdleme17d1 37305 Part of proof of Lemma E i...
cdleme0nex 37306 Part of proof of Lemma E i...
cdleme18a 37307 Part of proof of Lemma E i...
cdleme18b 37308 Part of proof of Lemma E i...
cdleme18c 37309 Part of proof of Lemma E i...
cdleme22gb 37310 Utility lemma for Lemma E ...
cdleme18d 37311 Part of proof of Lemma E i...
cdlemesner 37312 Part of proof of Lemma E i...
cdlemedb 37313 Part of proof of Lemma E i...
cdlemeda 37314 Part of proof of Lemma E i...
cdlemednpq 37315 Part of proof of Lemma E i...
cdlemednuN 37316 Part of proof of Lemma E i...
cdleme20zN 37317 Part of proof of Lemma E i...
cdleme20y 37318 Part of proof of Lemma E i...
cdleme19a 37319 Part of proof of Lemma E i...
cdleme19b 37320 Part of proof of Lemma E i...
cdleme19c 37321 Part of proof of Lemma E i...
cdleme19d 37322 Part of proof of Lemma E i...
cdleme19e 37323 Part of proof of Lemma E i...
cdleme19f 37324 Part of proof of Lemma E i...
cdleme20aN 37325 Part of proof of Lemma E i...
cdleme20bN 37326 Part of proof of Lemma E i...
cdleme20c 37327 Part of proof of Lemma E i...
cdleme20d 37328 Part of proof of Lemma E i...
cdleme20e 37329 Part of proof of Lemma E i...
cdleme20f 37330 Part of proof of Lemma E i...
cdleme20g 37331 Part of proof of Lemma E i...
cdleme20h 37332 Part of proof of Lemma E i...
cdleme20i 37333 Part of proof of Lemma E i...
cdleme20j 37334 Part of proof of Lemma E i...
cdleme20k 37335 Part of proof of Lemma E i...
cdleme20l1 37336 Part of proof of Lemma E i...
cdleme20l2 37337 Part of proof of Lemma E i...
cdleme20l 37338 Part of proof of Lemma E i...
cdleme20m 37339 Part of proof of Lemma E i...
cdleme20 37340 Combine ~ cdleme19f and ~ ...
cdleme21a 37341 Part of proof of Lemma E i...
cdleme21b 37342 Part of proof of Lemma E i...
cdleme21c 37343 Part of proof of Lemma E i...
cdleme21at 37344 Part of proof of Lemma E i...
cdleme21ct 37345 Part of proof of Lemma E i...
cdleme21d 37346 Part of proof of Lemma E i...
cdleme21e 37347 Part of proof of Lemma E i...
cdleme21f 37348 Part of proof of Lemma E i...
cdleme21g 37349 Part of proof of Lemma E i...
cdleme21h 37350 Part of proof of Lemma E i...
cdleme21i 37351 Part of proof of Lemma E i...
cdleme21j 37352 Combine ~ cdleme20 and ~ c...
cdleme21 37353 Part of proof of Lemma E i...
cdleme21k 37354 Eliminate ` S =/= T ` cond...
cdleme22aa 37355 Part of proof of Lemma E i...
cdleme22a 37356 Part of proof of Lemma E i...
cdleme22b 37357 Part of proof of Lemma E i...
cdleme22cN 37358 Part of proof of Lemma E i...
cdleme22d 37359 Part of proof of Lemma E i...
cdleme22e 37360 Part of proof of Lemma E i...
cdleme22eALTN 37361 Part of proof of Lemma E i...
cdleme22f 37362 Part of proof of Lemma E i...
cdleme22f2 37363 Part of proof of Lemma E i...
cdleme22g 37364 Part of proof of Lemma E i...
cdleme23a 37365 Part of proof of Lemma E i...
cdleme23b 37366 Part of proof of Lemma E i...
cdleme23c 37367 Part of proof of Lemma E i...
cdleme24 37368 Quantified version of ~ cd...
cdleme25a 37369 Lemma for ~ cdleme25b . (...
cdleme25b 37370 Transform ~ cdleme24 . TO...
cdleme25c 37371 Transform ~ cdleme25b . (...
cdleme25dN 37372 Transform ~ cdleme25c . (...
cdleme25cl 37373 Show closure of the unique...
cdleme25cv 37374 Change bound variables in ...
cdleme26e 37375 Part of proof of Lemma E i...
cdleme26ee 37376 Part of proof of Lemma E i...
cdleme26eALTN 37377 Part of proof of Lemma E i...
cdleme26fALTN 37378 Part of proof of Lemma E i...
cdleme26f 37379 Part of proof of Lemma E i...
cdleme26f2ALTN 37380 Part of proof of Lemma E i...
cdleme26f2 37381 Part of proof of Lemma E i...
cdleme27cl 37382 Part of proof of Lemma E i...
cdleme27a 37383 Part of proof of Lemma E i...
cdleme27b 37384 Lemma for ~ cdleme27N . (...
cdleme27N 37385 Part of proof of Lemma E i...
cdleme28a 37386 Lemma for ~ cdleme25b . T...
cdleme28b 37387 Lemma for ~ cdleme25b . T...
cdleme28c 37388 Part of proof of Lemma E i...
cdleme28 37389 Quantified version of ~ cd...
cdleme29ex 37390 Lemma for ~ cdleme29b . (...
cdleme29b 37391 Transform ~ cdleme28 . (C...
cdleme29c 37392 Transform ~ cdleme28b . (...
cdleme29cl 37393 Show closure of the unique...
cdleme30a 37394 Part of proof of Lemma E i...
cdleme31so 37395 Part of proof of Lemma E i...
cdleme31sn 37396 Part of proof of Lemma E i...
cdleme31sn1 37397 Part of proof of Lemma E i...
cdleme31se 37398 Part of proof of Lemma D i...
cdleme31se2 37399 Part of proof of Lemma D i...
cdleme31sc 37400 Part of proof of Lemma E i...
cdleme31sde 37401 Part of proof of Lemma D i...
cdleme31snd 37402 Part of proof of Lemma D i...
cdleme31sdnN 37403 Part of proof of Lemma E i...
cdleme31sn1c 37404 Part of proof of Lemma E i...
cdleme31sn2 37405 Part of proof of Lemma E i...
cdleme31fv 37406 Part of proof of Lemma E i...
cdleme31fv1 37407 Part of proof of Lemma E i...
cdleme31fv1s 37408 Part of proof of Lemma E i...
cdleme31fv2 37409 Part of proof of Lemma E i...
cdleme31id 37410 Part of proof of Lemma E i...
cdlemefrs29pre00 37411 ***START OF VALUE AT ATOM ...
cdlemefrs29bpre0 37412 TODO fix comment. (Contri...
cdlemefrs29bpre1 37413 TODO: FIX COMMENT. (Contr...
cdlemefrs29cpre1 37414 TODO: FIX COMMENT. (Contr...
cdlemefrs29clN 37415 TODO: NOT USED? Show clo...
cdlemefrs32fva 37416 Part of proof of Lemma E i...
cdlemefrs32fva1 37417 Part of proof of Lemma E i...
cdlemefr29exN 37418 Lemma for ~ cdlemefs29bpre...
cdlemefr27cl 37419 Part of proof of Lemma E i...
cdlemefr32sn2aw 37420 Show that ` [_ R / s ]_ N ...
cdlemefr32snb 37421 Show closure of ` [_ R / s...
cdlemefr29bpre0N 37422 TODO fix comment. (Contri...
cdlemefr29clN 37423 Show closure of the unique...
cdleme43frv1snN 37424 Value of ` [_ R / s ]_ N `...
cdlemefr32fvaN 37425 Part of proof of Lemma E i...
cdlemefr32fva1 37426 Part of proof of Lemma E i...
cdlemefr31fv1 37427 Value of ` ( F `` R ) ` wh...
cdlemefs29pre00N 37428 FIX COMMENT. TODO: see if ...
cdlemefs27cl 37429 Part of proof of Lemma E i...
cdlemefs32sn1aw 37430 Show that ` [_ R / s ]_ N ...
cdlemefs32snb 37431 Show closure of ` [_ R / s...
cdlemefs29bpre0N 37432 TODO: FIX COMMENT. (Contr...
cdlemefs29bpre1N 37433 TODO: FIX COMMENT. (Contr...
cdlemefs29cpre1N 37434 TODO: FIX COMMENT. (Contr...
cdlemefs29clN 37435 Show closure of the unique...
cdleme43fsv1snlem 37436 Value of ` [_ R / s ]_ N `...
cdleme43fsv1sn 37437 Value of ` [_ R / s ]_ N `...
cdlemefs32fvaN 37438 Part of proof of Lemma E i...
cdlemefs32fva1 37439 Part of proof of Lemma E i...
cdlemefs31fv1 37440 Value of ` ( F `` R ) ` wh...
cdlemefr44 37441 Value of f(r) when r is an...
cdlemefs44 37442 Value of f_s(r) when r is ...
cdlemefr45 37443 Value of f(r) when r is an...
cdlemefr45e 37444 Explicit expansion of ~ cd...
cdlemefs45 37445 Value of f_s(r) when r is ...
cdlemefs45ee 37446 Explicit expansion of ~ cd...
cdlemefs45eN 37447 Explicit expansion of ~ cd...
cdleme32sn1awN 37448 Show that ` [_ R / s ]_ N ...
cdleme41sn3a 37449 Show that ` [_ R / s ]_ N ...
cdleme32sn2awN 37450 Show that ` [_ R / s ]_ N ...
cdleme32snaw 37451 Show that ` [_ R / s ]_ N ...
cdleme32snb 37452 Show closure of ` [_ R / s...
cdleme32fva 37453 Part of proof of Lemma D i...
cdleme32fva1 37454 Part of proof of Lemma D i...
cdleme32fvaw 37455 Show that ` ( F `` R ) ` i...
cdleme32fvcl 37456 Part of proof of Lemma D i...
cdleme32a 37457 Part of proof of Lemma D i...
cdleme32b 37458 Part of proof of Lemma D i...
cdleme32c 37459 Part of proof of Lemma D i...
cdleme32d 37460 Part of proof of Lemma D i...
cdleme32e 37461 Part of proof of Lemma D i...
cdleme32f 37462 Part of proof of Lemma D i...
cdleme32le 37463 Part of proof of Lemma D i...
cdleme35a 37464 Part of proof of Lemma E i...
cdleme35fnpq 37465 Part of proof of Lemma E i...
cdleme35b 37466 Part of proof of Lemma E i...
cdleme35c 37467 Part of proof of Lemma E i...
cdleme35d 37468 Part of proof of Lemma E i...
cdleme35e 37469 Part of proof of Lemma E i...
cdleme35f 37470 Part of proof of Lemma E i...
cdleme35g 37471 Part of proof of Lemma E i...
cdleme35h 37472 Part of proof of Lemma E i...
cdleme35h2 37473 Part of proof of Lemma E i...
cdleme35sn2aw 37474 Part of proof of Lemma E i...
cdleme35sn3a 37475 Part of proof of Lemma E i...
cdleme36a 37476 Part of proof of Lemma E i...
cdleme36m 37477 Part of proof of Lemma E i...
cdleme37m 37478 Part of proof of Lemma E i...
cdleme38m 37479 Part of proof of Lemma E i...
cdleme38n 37480 Part of proof of Lemma E i...
cdleme39a 37481 Part of proof of Lemma E i...
cdleme39n 37482 Part of proof of Lemma E i...
cdleme40m 37483 Part of proof of Lemma E i...
cdleme40n 37484 Part of proof of Lemma E i...
cdleme40v 37485 Part of proof of Lemma E i...
cdleme40w 37486 Part of proof of Lemma E i...
cdleme42a 37487 Part of proof of Lemma E i...
cdleme42c 37488 Part of proof of Lemma E i...
cdleme42d 37489 Part of proof of Lemma E i...
cdleme41sn3aw 37490 Part of proof of Lemma E i...
cdleme41sn4aw 37491 Part of proof of Lemma E i...
cdleme41snaw 37492 Part of proof of Lemma E i...
cdleme41fva11 37493 Part of proof of Lemma E i...
cdleme42b 37494 Part of proof of Lemma E i...
cdleme42e 37495 Part of proof of Lemma E i...
cdleme42f 37496 Part of proof of Lemma E i...
cdleme42g 37497 Part of proof of Lemma E i...
cdleme42h 37498 Part of proof of Lemma E i...
cdleme42i 37499 Part of proof of Lemma E i...
cdleme42k 37500 Part of proof of Lemma E i...
cdleme42ke 37501 Part of proof of Lemma E i...
cdleme42keg 37502 Part of proof of Lemma E i...
cdleme42mN 37503 Part of proof of Lemma E i...
cdleme42mgN 37504 Part of proof of Lemma E i...
cdleme43aN 37505 Part of proof of Lemma E i...
cdleme43bN 37506 Lemma for Lemma E in [Craw...
cdleme43cN 37507 Part of proof of Lemma E i...
cdleme43dN 37508 Part of proof of Lemma E i...
cdleme46f2g2 37509 Conversion for ` G ` to re...
cdleme46f2g1 37510 Conversion for ` G ` to re...
cdleme17d2 37511 Part of proof of Lemma E i...
cdleme17d3 37512 TODO: FIX COMMENT. (Contr...
cdleme17d4 37513 TODO: FIX COMMENT. (Contr...
cdleme17d 37514 Part of proof of Lemma E i...
cdleme48fv 37515 Part of proof of Lemma D i...
cdleme48fvg 37516 Remove ` P =/= Q ` conditi...
cdleme46fvaw 37517 Show that ` ( F `` R ) ` i...
cdleme48bw 37518 TODO: fix comment. TODO: ...
cdleme48b 37519 TODO: fix comment. (Contr...
cdleme46frvlpq 37520 Show that ` ( F `` S ) ` i...
cdleme46fsvlpq 37521 Show that ` ( F `` R ) ` i...
cdlemeg46fvcl 37522 TODO: fix comment. (Contr...
cdleme4gfv 37523 Part of proof of Lemma D i...
cdlemeg47b 37524 TODO: FIX COMMENT. (Contr...
cdlemeg47rv 37525 Value of g_s(r) when r is ...
cdlemeg47rv2 37526 Value of g_s(r) when r is ...
cdlemeg49le 37527 Part of proof of Lemma D i...
cdlemeg46bOLDN 37528 TODO FIX COMMENT. (Contrib...
cdlemeg46c 37529 TODO FIX COMMENT. (Contrib...
cdlemeg46rvOLDN 37530 Value of g_s(r) when r is ...
cdlemeg46rv2OLDN 37531 Value of g_s(r) when r is ...
cdlemeg46fvaw 37532 Show that ` ( F `` R ) ` i...
cdlemeg46nlpq 37533 Show that ` ( G `` S ) ` i...
cdlemeg46ngfr 37534 TODO FIX COMMENT g(f(s))=s...
cdlemeg46nfgr 37535 TODO FIX COMMENT f(g(s))=s...
cdlemeg46sfg 37536 TODO FIX COMMENT f(r) ` \/...
cdlemeg46fjgN 37537 NOT NEEDED? TODO FIX COMM...
cdlemeg46rjgN 37538 NOT NEEDED? TODO FIX COMM...
cdlemeg46fjv 37539 TODO FIX COMMENT f(r) ` \/...
cdlemeg46fsfv 37540 TODO FIX COMMENT f(r) ` \/...
cdlemeg46frv 37541 TODO FIX COMMENT. (f(r) ` ...
cdlemeg46v1v2 37542 TODO FIX COMMENT v_1 = v_2...
cdlemeg46vrg 37543 TODO FIX COMMENT v_1 ` <_ ...
cdlemeg46rgv 37544 TODO FIX COMMENT r ` <_ ` ...
cdlemeg46req 37545 TODO FIX COMMENT r = (v_1 ...
cdlemeg46gfv 37546 TODO FIX COMMENT p. 115 pe...
cdlemeg46gfr 37547 TODO FIX COMMENT p. 116 pe...
cdlemeg46gfre 37548 TODO FIX COMMENT p. 116 pe...
cdlemeg46gf 37549 TODO FIX COMMENT Eliminate...
cdlemeg46fgN 37550 TODO FIX COMMENT p. 116 pe...
cdleme48d 37551 TODO: fix comment. (Contr...
cdleme48gfv1 37552 TODO: fix comment. (Contr...
cdleme48gfv 37553 TODO: fix comment. (Contr...
cdleme48fgv 37554 TODO: fix comment. (Contr...
cdlemeg49lebilem 37555 Part of proof of Lemma D i...
cdleme50lebi 37556 Part of proof of Lemma D i...
cdleme50eq 37557 Part of proof of Lemma D i...
cdleme50f 37558 Part of proof of Lemma D i...
cdleme50f1 37559 Part of proof of Lemma D i...
cdleme50rnlem 37560 Part of proof of Lemma D i...
cdleme50rn 37561 Part of proof of Lemma D i...
cdleme50f1o 37562 Part of proof of Lemma D i...
cdleme50laut 37563 Part of proof of Lemma D i...
cdleme50ldil 37564 Part of proof of Lemma D i...
cdleme50trn1 37565 Part of proof that ` F ` i...
cdleme50trn2a 37566 Part of proof that ` F ` i...
cdleme50trn2 37567 Part of proof that ` F ` i...
cdleme50trn12 37568 Part of proof that ` F ` i...
cdleme50trn3 37569 Part of proof that ` F ` i...
cdleme50trn123 37570 Part of proof that ` F ` i...
cdleme51finvfvN 37571 Part of proof of Lemma E i...
cdleme51finvN 37572 Part of proof of Lemma E i...
cdleme50ltrn 37573 Part of proof of Lemma E i...
cdleme51finvtrN 37574 Part of proof of Lemma E i...
cdleme50ex 37575 Part of Lemma E in [Crawle...
cdleme 37576 Lemma E in [Crawley] p. 11...
cdlemf1 37577 Part of Lemma F in [Crawle...
cdlemf2 37578 Part of Lemma F in [Crawle...
cdlemf 37579 Lemma F in [Crawley] p. 11...
cdlemfnid 37580 ~ cdlemf with additional c...
cdlemftr3 37581 Special case of ~ cdlemf s...
cdlemftr2 37582 Special case of ~ cdlemf s...
cdlemftr1 37583 Part of proof of Lemma G o...
cdlemftr0 37584 Special case of ~ cdlemf s...
trlord 37585 The ordering of two Hilber...
cdlemg1a 37586 Shorter expression for ` G...
cdlemg1b2 37587 This theorem can be used t...
cdlemg1idlemN 37588 Lemma for ~ cdlemg1idN . ...
cdlemg1fvawlemN 37589 Lemma for ~ ltrniotafvawN ...
cdlemg1ltrnlem 37590 Lemma for ~ ltrniotacl . ...
cdlemg1finvtrlemN 37591 Lemma for ~ ltrniotacnvN ....
cdlemg1bOLDN 37592 This theorem can be used t...
cdlemg1idN 37593 Version of ~ cdleme31id wi...
ltrniotafvawN 37594 Version of ~ cdleme46fvaw ...
ltrniotacl 37595 Version of ~ cdleme50ltrn ...
ltrniotacnvN 37596 Version of ~ cdleme51finvt...
ltrniotaval 37597 Value of the unique transl...
ltrniotacnvval 37598 Converse value of the uniq...
ltrniotaidvalN 37599 Value of the unique transl...
ltrniotavalbN 37600 Value of the unique transl...
cdlemeiota 37601 A translation is uniquely ...
cdlemg1ci2 37602 Any function of the form o...
cdlemg1cN 37603 Any translation belongs to...
cdlemg1cex 37604 Any translation is one of ...
cdlemg2cN 37605 Any translation belongs to...
cdlemg2dN 37606 This theorem can be used t...
cdlemg2cex 37607 Any translation is one of ...
cdlemg2ce 37608 Utility theorem to elimina...
cdlemg2jlemOLDN 37609 Part of proof of Lemma E i...
cdlemg2fvlem 37610 Lemma for ~ cdlemg2fv . (...
cdlemg2klem 37611 ~ cdleme42keg with simpler...
cdlemg2idN 37612 Version of ~ cdleme31id wi...
cdlemg3a 37613 Part of proof of Lemma G i...
cdlemg2jOLDN 37614 TODO: Replace this with ~...
cdlemg2fv 37615 Value of a translation in ...
cdlemg2fv2 37616 Value of a translation in ...
cdlemg2k 37617 ~ cdleme42keg with simpler...
cdlemg2kq 37618 ~ cdlemg2k with ` P ` and ...
cdlemg2l 37619 TODO: FIX COMMENT. (Contr...
cdlemg2m 37620 TODO: FIX COMMENT. (Contr...
cdlemg5 37621 TODO: Is there a simpler ...
cdlemb3 37622 Given two atoms not under ...
cdlemg7fvbwN 37623 Properties of a translatio...
cdlemg4a 37624 TODO: FIX COMMENT If fg(p...
cdlemg4b1 37625 TODO: FIX COMMENT. (Contr...
cdlemg4b2 37626 TODO: FIX COMMENT. (Contr...
cdlemg4b12 37627 TODO: FIX COMMENT. (Contr...
cdlemg4c 37628 TODO: FIX COMMENT. (Contr...
cdlemg4d 37629 TODO: FIX COMMENT. (Contr...
cdlemg4e 37630 TODO: FIX COMMENT. (Contr...
cdlemg4f 37631 TODO: FIX COMMENT. (Contr...
cdlemg4g 37632 TODO: FIX COMMENT. (Contr...
cdlemg4 37633 TODO: FIX COMMENT. (Contr...
cdlemg6a 37634 TODO: FIX COMMENT. TODO: ...
cdlemg6b 37635 TODO: FIX COMMENT. TODO: ...
cdlemg6c 37636 TODO: FIX COMMENT. (Contr...
cdlemg6d 37637 TODO: FIX COMMENT. (Contr...
cdlemg6e 37638 TODO: FIX COMMENT. (Contr...
cdlemg6 37639 TODO: FIX COMMENT. (Contr...
cdlemg7fvN 37640 Value of a translation com...
cdlemg7aN 37641 TODO: FIX COMMENT. (Contr...
cdlemg7N 37642 TODO: FIX COMMENT. (Contr...
cdlemg8a 37643 TODO: FIX COMMENT. (Contr...
cdlemg8b 37644 TODO: FIX COMMENT. (Contr...
cdlemg8c 37645 TODO: FIX COMMENT. (Contr...
cdlemg8d 37646 TODO: FIX COMMENT. (Contr...
cdlemg8 37647 TODO: FIX COMMENT. (Contr...
cdlemg9a 37648 TODO: FIX COMMENT. (Contr...
cdlemg9b 37649 The triples ` <. P , ( F `...
cdlemg9 37650 The triples ` <. P , ( F `...
cdlemg10b 37651 TODO: FIX COMMENT. TODO: ...
cdlemg10bALTN 37652 TODO: FIX COMMENT. TODO: ...
cdlemg11a 37653 TODO: FIX COMMENT. (Contr...
cdlemg11aq 37654 TODO: FIX COMMENT. TODO: ...
cdlemg10c 37655 TODO: FIX COMMENT. TODO: ...
cdlemg10a 37656 TODO: FIX COMMENT. (Contr...
cdlemg10 37657 TODO: FIX COMMENT. (Contr...
cdlemg11b 37658 TODO: FIX COMMENT. (Contr...
cdlemg12a 37659 TODO: FIX COMMENT. (Contr...
cdlemg12b 37660 The triples ` <. P , ( F `...
cdlemg12c 37661 The triples ` <. P , ( F `...
cdlemg12d 37662 TODO: FIX COMMENT. (Contr...
cdlemg12e 37663 TODO: FIX COMMENT. (Contr...
cdlemg12f 37664 TODO: FIX COMMENT. (Contr...
cdlemg12g 37665 TODO: FIX COMMENT. TODO: ...
cdlemg12 37666 TODO: FIX COMMENT. (Contr...
cdlemg13a 37667 TODO: FIX COMMENT. (Contr...
cdlemg13 37668 TODO: FIX COMMENT. (Contr...
cdlemg14f 37669 TODO: FIX COMMENT. (Contr...
cdlemg14g 37670 TODO: FIX COMMENT. (Contr...
cdlemg15a 37671 Eliminate the ` ( F `` P )...
cdlemg15 37672 Eliminate the ` ( (...
cdlemg16 37673 Part of proof of Lemma G o...
cdlemg16ALTN 37674 This version of ~ cdlemg16...
cdlemg16z 37675 Eliminate ` ( ( F `...
cdlemg16zz 37676 Eliminate ` P =/= Q ` from...
cdlemg17a 37677 TODO: FIX COMMENT. (Contr...
cdlemg17b 37678 Part of proof of Lemma G i...
cdlemg17dN 37679 TODO: fix comment. (Contr...
cdlemg17dALTN 37680 Same as ~ cdlemg17dN with ...
cdlemg17e 37681 TODO: fix comment. (Contr...
cdlemg17f 37682 TODO: fix comment. (Contr...
cdlemg17g 37683 TODO: fix comment. (Contr...
cdlemg17h 37684 TODO: fix comment. (Contr...
cdlemg17i 37685 TODO: fix comment. (Contr...
cdlemg17ir 37686 TODO: fix comment. (Contr...
cdlemg17j 37687 TODO: fix comment. (Contr...
cdlemg17pq 37688 Utility theorem for swappi...
cdlemg17bq 37689 ~ cdlemg17b with ` P ` and...
cdlemg17iqN 37690 ~ cdlemg17i with ` P ` and...
cdlemg17irq 37691 ~ cdlemg17ir with ` P ` an...
cdlemg17jq 37692 ~ cdlemg17j with ` P ` and...
cdlemg17 37693 Part of Lemma G of [Crawle...
cdlemg18a 37694 Show two lines are differe...
cdlemg18b 37695 Lemma for ~ cdlemg18c . T...
cdlemg18c 37696 Show two lines intersect a...
cdlemg18d 37697 Show two lines intersect a...
cdlemg18 37698 Show two lines intersect a...
cdlemg19a 37699 Show two lines intersect a...
cdlemg19 37700 Show two lines intersect a...
cdlemg20 37701 Show two lines intersect a...
cdlemg21 37702 Version of cdlemg19 with `...
cdlemg22 37703 ~ cdlemg21 with ` ( F `` P...
cdlemg24 37704 Combine ~ cdlemg16z and ~ ...
cdlemg37 37705 Use ~ cdlemg8 to eliminate...
cdlemg25zz 37706 ~ cdlemg16zz restated for ...
cdlemg26zz 37707 ~ cdlemg16zz restated for ...
cdlemg27a 37708 For use with case when ` (...
cdlemg28a 37709 Part of proof of Lemma G o...
cdlemg31b0N 37710 TODO: Fix comment. (Cont...
cdlemg31b0a 37711 TODO: Fix comment. (Cont...
cdlemg27b 37712 TODO: Fix comment. (Cont...
cdlemg31a 37713 TODO: fix comment. (Contr...
cdlemg31b 37714 TODO: fix comment. (Contr...
cdlemg31c 37715 Show that when ` N ` is an...
cdlemg31d 37716 Eliminate ` ( F `` P ) =/=...
cdlemg33b0 37717 TODO: Fix comment. (Cont...
cdlemg33c0 37718 TODO: Fix comment. (Cont...
cdlemg28b 37719 Part of proof of Lemma G o...
cdlemg28 37720 Part of proof of Lemma G o...
cdlemg29 37721 Eliminate ` ( F `` P ) =/=...
cdlemg33a 37722 TODO: Fix comment. (Cont...
cdlemg33b 37723 TODO: Fix comment. (Cont...
cdlemg33c 37724 TODO: Fix comment. (Cont...
cdlemg33d 37725 TODO: Fix comment. (Cont...
cdlemg33e 37726 TODO: Fix comment. (Cont...
cdlemg33 37727 Combine ~ cdlemg33b , ~ cd...
cdlemg34 37728 Use cdlemg33 to eliminate ...
cdlemg35 37729 TODO: Fix comment. TODO:...
cdlemg36 37730 Use cdlemg35 to eliminate ...
cdlemg38 37731 Use ~ cdlemg37 to eliminat...
cdlemg39 37732 Eliminate ` =/= ` conditio...
cdlemg40 37733 Eliminate ` P =/= Q ` cond...
cdlemg41 37734 Convert ~ cdlemg40 to func...
ltrnco 37735 The composition of two tra...
trlcocnv 37736 Swap the arguments of the ...
trlcoabs 37737 Absorption into a composit...
trlcoabs2N 37738 Absorption of the trace of...
trlcoat 37739 The trace of a composition...
trlcocnvat 37740 Commonly used special case...
trlconid 37741 The composition of two dif...
trlcolem 37742 Lemma for ~ trlco . (Cont...
trlco 37743 The trace of a composition...
trlcone 37744 If two translations have d...
cdlemg42 37745 Part of proof of Lemma G o...
cdlemg43 37746 Part of proof of Lemma G o...
cdlemg44a 37747 Part of proof of Lemma G o...
cdlemg44b 37748 Eliminate ` ( F `` P ) =/=...
cdlemg44 37749 Part of proof of Lemma G o...
cdlemg47a 37750 TODO: fix comment. TODO: ...
cdlemg46 37751 Part of proof of Lemma G o...
cdlemg47 37752 Part of proof of Lemma G o...
cdlemg48 37753 Eliminate ` h ` from ~ cdl...
ltrncom 37754 Composition is commutative...
ltrnco4 37755 Rearrange a composition of...
trljco 37756 Trace joined with trace of...
trljco2 37757 Trace joined with trace of...
tgrpfset 37760 The translation group maps...
tgrpset 37761 The translation group for ...
tgrpbase 37762 The base set of the transl...
tgrpopr 37763 The group operation of the...
tgrpov 37764 The group operation value ...
tgrpgrplem 37765 Lemma for ~ tgrpgrp . (Co...
tgrpgrp 37766 The translation group is a...
tgrpabl 37767 The translation group is a...
tendofset 37774 The set of all trace-prese...
tendoset 37775 The set of trace-preservin...
istendo 37776 The predicate "is a trace-...
tendotp 37777 Trace-preserving property ...
istendod 37778 Deduce the predicate "is a...
tendof 37779 Functionality of a trace-p...
tendoeq1 37780 Condition determining equa...
tendovalco 37781 Value of composition of tr...
tendocoval 37782 Value of composition of en...
tendocl 37783 Closure of a trace-preserv...
tendoco2 37784 Distribution of compositio...
tendoidcl 37785 The identity is a trace-pr...
tendo1mul 37786 Multiplicative identity mu...
tendo1mulr 37787 Multiplicative identity mu...
tendococl 37788 The composition of two tra...
tendoid 37789 The identity value of a tr...
tendoeq2 37790 Condition determining equa...
tendoplcbv 37791 Define sum operation for t...
tendopl 37792 Value of endomorphism sum ...
tendopl2 37793 Value of result of endomor...
tendoplcl2 37794 Value of result of endomor...
tendoplco2 37795 Value of result of endomor...
tendopltp 37796 Trace-preserving property ...
tendoplcl 37797 Endomorphism sum is a trac...
tendoplcom 37798 The endomorphism sum opera...
tendoplass 37799 The endomorphism sum opera...
tendodi1 37800 Endomorphism composition d...
tendodi2 37801 Endomorphism composition d...
tendo0cbv 37802 Define additive identity f...
tendo02 37803 Value of additive identity...
tendo0co2 37804 The additive identity trac...
tendo0tp 37805 Trace-preserving property ...
tendo0cl 37806 The additive identity is a...
tendo0pl 37807 Property of the additive i...
tendo0plr 37808 Property of the additive i...
tendoicbv 37809 Define inverse function fo...
tendoi 37810 Value of inverse endomorph...
tendoi2 37811 Value of additive inverse ...
tendoicl 37812 Closure of the additive in...
tendoipl 37813 Property of the additive i...
tendoipl2 37814 Property of the additive i...
erngfset 37815 The division rings on trac...
erngset 37816 The division ring on trace...
erngbase 37817 The base set of the divisi...
erngfplus 37818 Ring addition operation. ...
erngplus 37819 Ring addition operation. ...
erngplus2 37820 Ring addition operation. ...
erngfmul 37821 Ring multiplication operat...
erngmul 37822 Ring addition operation. ...
erngfset-rN 37823 The division rings on trac...
erngset-rN 37824 The division ring on trace...
erngbase-rN 37825 The base set of the divisi...
erngfplus-rN 37826 Ring addition operation. ...
erngplus-rN 37827 Ring addition operation. ...
erngplus2-rN 37828 Ring addition operation. ...
erngfmul-rN 37829 Ring multiplication operat...
erngmul-rN 37830 Ring addition operation. ...
cdlemh1 37831 Part of proof of Lemma H o...
cdlemh2 37832 Part of proof of Lemma H o...
cdlemh 37833 Lemma H of [Crawley] p. 11...
cdlemi1 37834 Part of proof of Lemma I o...
cdlemi2 37835 Part of proof of Lemma I o...
cdlemi 37836 Lemma I of [Crawley] p. 11...
cdlemj1 37837 Part of proof of Lemma J o...
cdlemj2 37838 Part of proof of Lemma J o...
cdlemj3 37839 Part of proof of Lemma J o...
tendocan 37840 Cancellation law: if the v...
tendoid0 37841 A trace-preserving endomor...
tendo0mul 37842 Additive identity multipli...
tendo0mulr 37843 Additive identity multipli...
tendo1ne0 37844 The identity (unity) is no...
tendoconid 37845 The composition (product) ...
tendotr 37846 The trace of the value of ...
cdlemk1 37847 Part of proof of Lemma K o...
cdlemk2 37848 Part of proof of Lemma K o...
cdlemk3 37849 Part of proof of Lemma K o...
cdlemk4 37850 Part of proof of Lemma K o...
cdlemk5a 37851 Part of proof of Lemma K o...
cdlemk5 37852 Part of proof of Lemma K o...
cdlemk6 37853 Part of proof of Lemma K o...
cdlemk8 37854 Part of proof of Lemma K o...
cdlemk9 37855 Part of proof of Lemma K o...
cdlemk9bN 37856 Part of proof of Lemma K o...
cdlemki 37857 Part of proof of Lemma K o...
cdlemkvcl 37858 Part of proof of Lemma K o...
cdlemk10 37859 Part of proof of Lemma K o...
cdlemksv 37860 Part of proof of Lemma K o...
cdlemksel 37861 Part of proof of Lemma K o...
cdlemksat 37862 Part of proof of Lemma K o...
cdlemksv2 37863 Part of proof of Lemma K o...
cdlemk7 37864 Part of proof of Lemma K o...
cdlemk11 37865 Part of proof of Lemma K o...
cdlemk12 37866 Part of proof of Lemma K o...
cdlemkoatnle 37867 Utility lemma. (Contribut...
cdlemk13 37868 Part of proof of Lemma K o...
cdlemkole 37869 Utility lemma. (Contribut...
cdlemk14 37870 Part of proof of Lemma K o...
cdlemk15 37871 Part of proof of Lemma K o...
cdlemk16a 37872 Part of proof of Lemma K o...
cdlemk16 37873 Part of proof of Lemma K o...
cdlemk17 37874 Part of proof of Lemma K o...
cdlemk1u 37875 Part of proof of Lemma K o...
cdlemk5auN 37876 Part of proof of Lemma K o...
cdlemk5u 37877 Part of proof of Lemma K o...
cdlemk6u 37878 Part of proof of Lemma K o...
cdlemkj 37879 Part of proof of Lemma K o...
cdlemkuvN 37880 Part of proof of Lemma K o...
cdlemkuel 37881 Part of proof of Lemma K o...
cdlemkuat 37882 Part of proof of Lemma K o...
cdlemkuv2 37883 Part of proof of Lemma K o...
cdlemk18 37884 Part of proof of Lemma K o...
cdlemk19 37885 Part of proof of Lemma K o...
cdlemk7u 37886 Part of proof of Lemma K o...
cdlemk11u 37887 Part of proof of Lemma K o...
cdlemk12u 37888 Part of proof of Lemma K o...
cdlemk21N 37889 Part of proof of Lemma K o...
cdlemk20 37890 Part of proof of Lemma K o...
cdlemkoatnle-2N 37891 Utility lemma. (Contribut...
cdlemk13-2N 37892 Part of proof of Lemma K o...
cdlemkole-2N 37893 Utility lemma. (Contribut...
cdlemk14-2N 37894 Part of proof of Lemma K o...
cdlemk15-2N 37895 Part of proof of Lemma K o...
cdlemk16-2N 37896 Part of proof of Lemma K o...
cdlemk17-2N 37897 Part of proof of Lemma K o...
cdlemkj-2N 37898 Part of proof of Lemma K o...
cdlemkuv-2N 37899 Part of proof of Lemma K o...
cdlemkuel-2N 37900 Part of proof of Lemma K o...
cdlemkuv2-2 37901 Part of proof of Lemma K o...
cdlemk18-2N 37902 Part of proof of Lemma K o...
cdlemk19-2N 37903 Part of proof of Lemma K o...
cdlemk7u-2N 37904 Part of proof of Lemma K o...
cdlemk11u-2N 37905 Part of proof of Lemma K o...
cdlemk12u-2N 37906 Part of proof of Lemma K o...
cdlemk21-2N 37907 Part of proof of Lemma K o...
cdlemk20-2N 37908 Part of proof of Lemma K o...
cdlemk22 37909 Part of proof of Lemma K o...
cdlemk30 37910 Part of proof of Lemma K o...
cdlemkuu 37911 Convert between function a...
cdlemk31 37912 Part of proof of Lemma K o...
cdlemk32 37913 Part of proof of Lemma K o...
cdlemkuel-3 37914 Part of proof of Lemma K o...
cdlemkuv2-3N 37915 Part of proof of Lemma K o...
cdlemk18-3N 37916 Part of proof of Lemma K o...
cdlemk22-3 37917 Part of proof of Lemma K o...
cdlemk23-3 37918 Part of proof of Lemma K o...
cdlemk24-3 37919 Part of proof of Lemma K o...
cdlemk25-3 37920 Part of proof of Lemma K o...
cdlemk26b-3 37921 Part of proof of Lemma K o...
cdlemk26-3 37922 Part of proof of Lemma K o...
cdlemk27-3 37923 Part of proof of Lemma K o...
cdlemk28-3 37924 Part of proof of Lemma K o...
cdlemk33N 37925 Part of proof of Lemma K o...
cdlemk34 37926 Part of proof of Lemma K o...
cdlemk29-3 37927 Part of proof of Lemma K o...
cdlemk35 37928 Part of proof of Lemma K o...
cdlemk36 37929 Part of proof of Lemma K o...
cdlemk37 37930 Part of proof of Lemma K o...
cdlemk38 37931 Part of proof of Lemma K o...
cdlemk39 37932 Part of proof of Lemma K o...
cdlemk40 37933 TODO: fix comment. (Contr...
cdlemk40t 37934 TODO: fix comment. (Contr...
cdlemk40f 37935 TODO: fix comment. (Contr...
cdlemk41 37936 Part of proof of Lemma K o...
cdlemkfid1N 37937 Lemma for ~ cdlemkfid3N . ...
cdlemkid1 37938 Lemma for ~ cdlemkid . (C...
cdlemkfid2N 37939 Lemma for ~ cdlemkfid3N . ...
cdlemkid2 37940 Lemma for ~ cdlemkid . (C...
cdlemkfid3N 37941 TODO: is this useful or sh...
cdlemky 37942 Part of proof of Lemma K o...
cdlemkyu 37943 Convert between function a...
cdlemkyuu 37944 ~ cdlemkyu with some hypot...
cdlemk11ta 37945 Part of proof of Lemma K o...
cdlemk19ylem 37946 Lemma for ~ cdlemk19y . (...
cdlemk11tb 37947 Part of proof of Lemma K o...
cdlemk19y 37948 ~ cdlemk19 with simpler hy...
cdlemkid3N 37949 Lemma for ~ cdlemkid . (C...
cdlemkid4 37950 Lemma for ~ cdlemkid . (C...
cdlemkid5 37951 Lemma for ~ cdlemkid . (C...
cdlemkid 37952 The value of the tau funct...
cdlemk35s 37953 Substitution version of ~ ...
cdlemk35s-id 37954 Substitution version of ~ ...
cdlemk39s 37955 Substitution version of ~ ...
cdlemk39s-id 37956 Substitution version of ~ ...
cdlemk42 37957 Part of proof of Lemma K o...
cdlemk19xlem 37958 Lemma for ~ cdlemk19x . (...
cdlemk19x 37959 ~ cdlemk19 with simpler hy...
cdlemk42yN 37960 Part of proof of Lemma K o...
cdlemk11tc 37961 Part of proof of Lemma K o...
cdlemk11t 37962 Part of proof of Lemma K o...
cdlemk45 37963 Part of proof of Lemma K o...
cdlemk46 37964 Part of proof of Lemma K o...
cdlemk47 37965 Part of proof of Lemma K o...
cdlemk48 37966 Part of proof of Lemma K o...
cdlemk49 37967 Part of proof of Lemma K o...
cdlemk50 37968 Part of proof of Lemma K o...
cdlemk51 37969 Part of proof of Lemma K o...
cdlemk52 37970 Part of proof of Lemma K o...
cdlemk53a 37971 Lemma for ~ cdlemk53 . (C...
cdlemk53b 37972 Lemma for ~ cdlemk53 . (C...
cdlemk53 37973 Part of proof of Lemma K o...
cdlemk54 37974 Part of proof of Lemma K o...
cdlemk55a 37975 Lemma for ~ cdlemk55 . (C...
cdlemk55b 37976 Lemma for ~ cdlemk55 . (C...
cdlemk55 37977 Part of proof of Lemma K o...
cdlemkyyN 37978 Part of proof of Lemma K o...
cdlemk43N 37979 Part of proof of Lemma K o...
cdlemk35u 37980 Substitution version of ~ ...
cdlemk55u1 37981 Lemma for ~ cdlemk55u . (...
cdlemk55u 37982 Part of proof of Lemma K o...
cdlemk39u1 37983 Lemma for ~ cdlemk39u . (...
cdlemk39u 37984 Part of proof of Lemma K o...
cdlemk19u1 37985 ~ cdlemk19 with simpler hy...
cdlemk19u 37986 Part of Lemma K of [Crawle...
cdlemk56 37987 Part of Lemma K of [Crawle...
cdlemk19w 37988 Use a fixed element to eli...
cdlemk56w 37989 Use a fixed element to eli...
cdlemk 37990 Lemma K of [Crawley] p. 11...
tendoex 37991 Generalization of Lemma K ...
cdleml1N 37992 Part of proof of Lemma L o...
cdleml2N 37993 Part of proof of Lemma L o...
cdleml3N 37994 Part of proof of Lemma L o...
cdleml4N 37995 Part of proof of Lemma L o...
cdleml5N 37996 Part of proof of Lemma L o...
cdleml6 37997 Part of proof of Lemma L o...
cdleml7 37998 Part of proof of Lemma L o...
cdleml8 37999 Part of proof of Lemma L o...
cdleml9 38000 Part of proof of Lemma L o...
dva1dim 38001 Two expressions for the 1-...
dvhb1dimN 38002 Two expressions for the 1-...
erng1lem 38003 Value of the endomorphism ...
erngdvlem1 38004 Lemma for ~ eringring . (...
erngdvlem2N 38005 Lemma for ~ eringring . (...
erngdvlem3 38006 Lemma for ~ eringring . (...
erngdvlem4 38007 Lemma for ~ erngdv . (Con...
eringring 38008 An endomorphism ring is a ...
erngdv 38009 An endomorphism ring is a ...
erng0g 38010 The division ring zero of ...
erng1r 38011 The division ring unit of ...
erngdvlem1-rN 38012 Lemma for ~ eringring . (...
erngdvlem2-rN 38013 Lemma for ~ eringring . (...
erngdvlem3-rN 38014 Lemma for ~ eringring . (...
erngdvlem4-rN 38015 Lemma for ~ erngdv . (Con...
erngring-rN 38016 An endomorphism ring is a ...
erngdv-rN 38017 An endomorphism ring is a ...
dvafset 38020 The constructed partial ve...
dvaset 38021 The constructed partial ve...
dvasca 38022 The ring base set of the c...
dvabase 38023 The ring base set of the c...
dvafplusg 38024 Ring addition operation fo...
dvaplusg 38025 Ring addition operation fo...
dvaplusgv 38026 Ring addition operation fo...
dvafmulr 38027 Ring multiplication operat...
dvamulr 38028 Ring multiplication operat...
dvavbase 38029 The vectors (vector base s...
dvafvadd 38030 The vector sum operation f...
dvavadd 38031 Ring addition operation fo...
dvafvsca 38032 Ring addition operation fo...
dvavsca 38033 Ring addition operation fo...
tendospcl 38034 Closure of endomorphism sc...
tendospass 38035 Associative law for endomo...
tendospdi1 38036 Forward distributive law f...
tendocnv 38037 Converse of a trace-preser...
tendospdi2 38038 Reverse distributive law f...
tendospcanN 38039 Cancellation law for trace...
dvaabl 38040 The constructed partial ve...
dvalveclem 38041 Lemma for ~ dvalvec . (Co...
dvalvec 38042 The constructed partial ve...
dva0g 38043 The zero vector of partial...
diaffval 38046 The partial isomorphism A ...
diafval 38047 The partial isomorphism A ...
diaval 38048 The partial isomorphism A ...
diaelval 38049 Member of the partial isom...
diafn 38050 Functionality and domain o...
diadm 38051 Domain of the partial isom...
diaeldm 38052 Member of domain of the pa...
diadmclN 38053 A member of domain of the ...
diadmleN 38054 A member of domain of the ...
dian0 38055 The value of the partial i...
dia0eldmN 38056 The lattice zero belongs t...
dia1eldmN 38057 The fiducial hyperplane (t...
diass 38058 The value of the partial i...
diael 38059 A member of the value of t...
diatrl 38060 Trace of a member of the p...
diaelrnN 38061 Any value of the partial i...
dialss 38062 The value of partial isomo...
diaord 38063 The partial isomorphism A ...
dia11N 38064 The partial isomorphism A ...
diaf11N 38065 The partial isomorphism A ...
diaclN 38066 Closure of partial isomorp...
diacnvclN 38067 Closure of partial isomorp...
dia0 38068 The value of the partial i...
dia1N 38069 The value of the partial i...
dia1elN 38070 The largest subspace in th...
diaglbN 38071 Partial isomorphism A of a...
diameetN 38072 Partial isomorphism A of a...
diainN 38073 Inverse partial isomorphis...
diaintclN 38074 The intersection of partia...
diasslssN 38075 The partial isomorphism A ...
diassdvaN 38076 The partial isomorphism A ...
dia1dim 38077 Two expressions for the 1-...
dia1dim2 38078 Two expressions for a 1-di...
dia1dimid 38079 A vector (translation) bel...
dia2dimlem1 38080 Lemma for ~ dia2dim . Sho...
dia2dimlem2 38081 Lemma for ~ dia2dim . Def...
dia2dimlem3 38082 Lemma for ~ dia2dim . Def...
dia2dimlem4 38083 Lemma for ~ dia2dim . Sho...
dia2dimlem5 38084 Lemma for ~ dia2dim . The...
dia2dimlem6 38085 Lemma for ~ dia2dim . Eli...
dia2dimlem7 38086 Lemma for ~ dia2dim . Eli...
dia2dimlem8 38087 Lemma for ~ dia2dim . Eli...
dia2dimlem9 38088 Lemma for ~ dia2dim . Eli...
dia2dimlem10 38089 Lemma for ~ dia2dim . Con...
dia2dimlem11 38090 Lemma for ~ dia2dim . Con...
dia2dimlem12 38091 Lemma for ~ dia2dim . Obt...
dia2dimlem13 38092 Lemma for ~ dia2dim . Eli...
dia2dim 38093 A two-dimensional subspace...
dvhfset 38096 The constructed full vecto...
dvhset 38097 The constructed full vecto...
dvhsca 38098 The ring of scalars of the...
dvhbase 38099 The ring base set of the c...
dvhfplusr 38100 Ring addition operation fo...
dvhfmulr 38101 Ring multiplication operat...
dvhmulr 38102 Ring multiplication operat...
dvhvbase 38103 The vectors (vector base s...
dvhelvbasei 38104 Vector membership in the c...
dvhvaddcbv 38105 Change bound variables to ...
dvhvaddval 38106 The vector sum operation f...
dvhfvadd 38107 The vector sum operation f...
dvhvadd 38108 The vector sum operation f...
dvhopvadd 38109 The vector sum operation f...
dvhopvadd2 38110 The vector sum operation f...
dvhvaddcl 38111 Closure of the vector sum ...
dvhvaddcomN 38112 Commutativity of vector su...
dvhvaddass 38113 Associativity of vector su...
dvhvscacbv 38114 Change bound variables to ...
dvhvscaval 38115 The scalar product operati...
dvhfvsca 38116 Scalar product operation f...
dvhvsca 38117 Scalar product operation f...
dvhopvsca 38118 Scalar product operation f...
dvhvscacl 38119 Closure of the scalar prod...
tendoinvcl 38120 Closure of multiplicative ...
tendolinv 38121 Left multiplicative invers...
tendorinv 38122 Right multiplicative inver...
dvhgrp 38123 The full vector space ` U ...
dvhlveclem 38124 Lemma for ~ dvhlvec . TOD...
dvhlvec 38125 The full vector space ` U ...
dvhlmod 38126 The full vector space ` U ...
dvh0g 38127 The zero vector of vector ...
dvheveccl 38128 Properties of a unit vecto...
dvhopclN 38129 Closure of a ` DVecH ` vec...
dvhopaddN 38130 Sum of ` DVecH ` vectors e...
dvhopspN 38131 Scalar product of ` DVecH ...
dvhopN 38132 Decompose a ` DVecH ` vect...
dvhopellsm 38133 Ordered pair membership in...
cdlemm10N 38134 The image of the map ` G `...
docaffvalN 38137 Subspace orthocomplement f...
docafvalN 38138 Subspace orthocomplement f...
docavalN 38139 Subspace orthocomplement f...
docaclN 38140 Closure of subspace orthoc...
diaocN 38141 Value of partial isomorphi...
doca2N 38142 Double orthocomplement of ...
doca3N 38143 Double orthocomplement of ...
dvadiaN 38144 Any closed subspace is a m...
diarnN 38145 Partial isomorphism A maps...
diaf1oN 38146 The partial isomorphism A ...
djaffvalN 38149 Subspace join for ` DVecA ...
djafvalN 38150 Subspace join for ` DVecA ...
djavalN 38151 Subspace join for ` DVecA ...
djaclN 38152 Closure of subspace join f...
djajN 38153 Transfer lattice join to `...
dibffval 38156 The partial isomorphism B ...
dibfval 38157 The partial isomorphism B ...
dibval 38158 The partial isomorphism B ...
dibopelvalN 38159 Member of the partial isom...
dibval2 38160 Value of the partial isomo...
dibopelval2 38161 Member of the partial isom...
dibval3N 38162 Value of the partial isomo...
dibelval3 38163 Member of the partial isom...
dibopelval3 38164 Member of the partial isom...
dibelval1st 38165 Membership in value of the...
dibelval1st1 38166 Membership in value of the...
dibelval1st2N 38167 Membership in value of the...
dibelval2nd 38168 Membership in value of the...
dibn0 38169 The value of the partial i...
dibfna 38170 Functionality and domain o...
dibdiadm 38171 Domain of the partial isom...
dibfnN 38172 Functionality and domain o...
dibdmN 38173 Domain of the partial isom...
dibeldmN 38174 Member of domain of the pa...
dibord 38175 The isomorphism B for a la...
dib11N 38176 The isomorphism B for a la...
dibf11N 38177 The partial isomorphism A ...
dibclN 38178 Closure of partial isomorp...
dibvalrel 38179 The value of partial isomo...
dib0 38180 The value of partial isomo...
dib1dim 38181 Two expressions for the 1-...
dibglbN 38182 Partial isomorphism B of a...
dibintclN 38183 The intersection of partia...
dib1dim2 38184 Two expressions for a 1-di...
dibss 38185 The partial isomorphism B ...
diblss 38186 The value of partial isomo...
diblsmopel 38187 Membership in subspace sum...
dicffval 38190 The partial isomorphism C ...
dicfval 38191 The partial isomorphism C ...
dicval 38192 The partial isomorphism C ...
dicopelval 38193 Membership in value of the...
dicelvalN 38194 Membership in value of the...
dicval2 38195 The partial isomorphism C ...
dicelval3 38196 Member of the partial isom...
dicopelval2 38197 Membership in value of the...
dicelval2N 38198 Membership in value of the...
dicfnN 38199 Functionality and domain o...
dicdmN 38200 Domain of the partial isom...
dicvalrelN 38201 The value of partial isomo...
dicssdvh 38202 The partial isomorphism C ...
dicelval1sta 38203 Membership in value of the...
dicelval1stN 38204 Membership in value of the...
dicelval2nd 38205 Membership in value of the...
dicvaddcl 38206 Membership in value of the...
dicvscacl 38207 Membership in value of the...
dicn0 38208 The value of the partial i...
diclss 38209 The value of partial isomo...
diclspsn 38210 The value of isomorphism C...
cdlemn2 38211 Part of proof of Lemma N o...
cdlemn2a 38212 Part of proof of Lemma N o...
cdlemn3 38213 Part of proof of Lemma N o...
cdlemn4 38214 Part of proof of Lemma N o...
cdlemn4a 38215 Part of proof of Lemma N o...
cdlemn5pre 38216 Part of proof of Lemma N o...
cdlemn5 38217 Part of proof of Lemma N o...
cdlemn6 38218 Part of proof of Lemma N o...
cdlemn7 38219 Part of proof of Lemma N o...
cdlemn8 38220 Part of proof of Lemma N o...
cdlemn9 38221 Part of proof of Lemma N o...
cdlemn10 38222 Part of proof of Lemma N o...
cdlemn11a 38223 Part of proof of Lemma N o...
cdlemn11b 38224 Part of proof of Lemma N o...
cdlemn11c 38225 Part of proof of Lemma N o...
cdlemn11pre 38226 Part of proof of Lemma N o...
cdlemn11 38227 Part of proof of Lemma N o...
cdlemn 38228 Lemma N of [Crawley] p. 12...
dihordlem6 38229 Part of proof of Lemma N o...
dihordlem7 38230 Part of proof of Lemma N o...
dihordlem7b 38231 Part of proof of Lemma N o...
dihjustlem 38232 Part of proof after Lemma ...
dihjust 38233 Part of proof after Lemma ...
dihord1 38234 Part of proof after Lemma ...
dihord2a 38235 Part of proof after Lemma ...
dihord2b 38236 Part of proof after Lemma ...
dihord2cN 38237 Part of proof after Lemma ...
dihord11b 38238 Part of proof after Lemma ...
dihord10 38239 Part of proof after Lemma ...
dihord11c 38240 Part of proof after Lemma ...
dihord2pre 38241 Part of proof after Lemma ...
dihord2pre2 38242 Part of proof after Lemma ...
dihord2 38243 Part of proof after Lemma ...
dihffval 38246 The isomorphism H for a la...
dihfval 38247 Isomorphism H for a lattic...
dihval 38248 Value of isomorphism H for...
dihvalc 38249 Value of isomorphism H for...
dihlsscpre 38250 Closure of isomorphism H f...
dihvalcqpre 38251 Value of isomorphism H for...
dihvalcq 38252 Value of isomorphism H for...
dihvalb 38253 Value of isomorphism H for...
dihopelvalbN 38254 Ordered pair member of the...
dihvalcqat 38255 Value of isomorphism H for...
dih1dimb 38256 Two expressions for a 1-di...
dih1dimb2 38257 Isomorphism H at an atom u...
dih1dimc 38258 Isomorphism H at an atom n...
dib2dim 38259 Extend ~ dia2dim to partia...
dih2dimb 38260 Extend ~ dib2dim to isomor...
dih2dimbALTN 38261 Extend ~ dia2dim to isomor...
dihopelvalcqat 38262 Ordered pair member of the...
dihvalcq2 38263 Value of isomorphism H for...
dihopelvalcpre 38264 Member of value of isomorp...
dihopelvalc 38265 Member of value of isomorp...
dihlss 38266 The value of isomorphism H...
dihss 38267 The value of isomorphism H...
dihssxp 38268 An isomorphism H value is ...
dihopcl 38269 Closure of an ordered pair...
xihopellsmN 38270 Ordered pair membership in...
dihopellsm 38271 Ordered pair membership in...
dihord6apre 38272 Part of proof that isomorp...
dihord3 38273 The isomorphism H for a la...
dihord4 38274 The isomorphism H for a la...
dihord5b 38275 Part of proof that isomorp...
dihord6b 38276 Part of proof that isomorp...
dihord6a 38277 Part of proof that isomorp...
dihord5apre 38278 Part of proof that isomorp...
dihord5a 38279 Part of proof that isomorp...
dihord 38280 The isomorphism H is order...
dih11 38281 The isomorphism H is one-t...
dihf11lem 38282 Functionality of the isomo...
dihf11 38283 The isomorphism H for a la...
dihfn 38284 Functionality and domain o...
dihdm 38285 Domain of isomorphism H. (...
dihcl 38286 Closure of isomorphism H. ...
dihcnvcl 38287 Closure of isomorphism H c...
dihcnvid1 38288 The converse isomorphism o...
dihcnvid2 38289 The isomorphism of a conve...
dihcnvord 38290 Ordering property for conv...
dihcnv11 38291 The converse of isomorphis...
dihsslss 38292 The isomorphism H maps to ...
dihrnlss 38293 The isomorphism H maps to ...
dihrnss 38294 The isomorphism H maps to ...
dihvalrel 38295 The value of isomorphism H...
dih0 38296 The value of isomorphism H...
dih0bN 38297 A lattice element is zero ...
dih0vbN 38298 A vector is zero iff its s...
dih0cnv 38299 The isomorphism H converse...
dih0rn 38300 The zero subspace belongs ...
dih0sb 38301 A subspace is zero iff the...
dih1 38302 The value of isomorphism H...
dih1rn 38303 The full vector space belo...
dih1cnv 38304 The isomorphism H converse...
dihwN 38305 Value of isomorphism H at ...
dihmeetlem1N 38306 Isomorphism H of a conjunc...
dihglblem5apreN 38307 A conjunction property of ...
dihglblem5aN 38308 A conjunction property of ...
dihglblem2aN 38309 Lemma for isomorphism H of...
dihglblem2N 38310 The GLB of a set of lattic...
dihglblem3N 38311 Isomorphism H of a lattice...
dihglblem3aN 38312 Isomorphism H of a lattice...
dihglblem4 38313 Isomorphism H of a lattice...
dihglblem5 38314 Isomorphism H of a lattice...
dihmeetlem2N 38315 Isomorphism H of a conjunc...
dihglbcpreN 38316 Isomorphism H of a lattice...
dihglbcN 38317 Isomorphism H of a lattice...
dihmeetcN 38318 Isomorphism H of a lattice...
dihmeetbN 38319 Isomorphism H of a lattice...
dihmeetbclemN 38320 Lemma for isomorphism H of...
dihmeetlem3N 38321 Lemma for isomorphism H of...
dihmeetlem4preN 38322 Lemma for isomorphism H of...
dihmeetlem4N 38323 Lemma for isomorphism H of...
dihmeetlem5 38324 Part of proof that isomorp...
dihmeetlem6 38325 Lemma for isomorphism H of...
dihmeetlem7N 38326 Lemma for isomorphism H of...
dihjatc1 38327 Lemma for isomorphism H of...
dihjatc2N 38328 Isomorphism H of join with...
dihjatc3 38329 Isomorphism H of join with...
dihmeetlem8N 38330 Lemma for isomorphism H of...
dihmeetlem9N 38331 Lemma for isomorphism H of...
dihmeetlem10N 38332 Lemma for isomorphism H of...
dihmeetlem11N 38333 Lemma for isomorphism H of...
dihmeetlem12N 38334 Lemma for isomorphism H of...
dihmeetlem13N 38335 Lemma for isomorphism H of...
dihmeetlem14N 38336 Lemma for isomorphism H of...
dihmeetlem15N 38337 Lemma for isomorphism H of...
dihmeetlem16N 38338 Lemma for isomorphism H of...
dihmeetlem17N 38339 Lemma for isomorphism H of...
dihmeetlem18N 38340 Lemma for isomorphism H of...
dihmeetlem19N 38341 Lemma for isomorphism H of...
dihmeetlem20N 38342 Lemma for isomorphism H of...
dihmeetALTN 38343 Isomorphism H of a lattice...
dih1dimatlem0 38344 Lemma for ~ dih1dimat . (...
dih1dimatlem 38345 Lemma for ~ dih1dimat . (...
dih1dimat 38346 Any 1-dimensional subspace...
dihlsprn 38347 The span of a vector belon...
dihlspsnssN 38348 A subspace included in a 1...
dihlspsnat 38349 The inverse isomorphism H ...
dihatlat 38350 The isomorphism H of an at...
dihat 38351 There exists at least one ...
dihpN 38352 The value of isomorphism H...
dihlatat 38353 The reverse isomorphism H ...
dihatexv 38354 There is a nonzero vector ...
dihatexv2 38355 There is a nonzero vector ...
dihglblem6 38356 Isomorphism H of a lattice...
dihglb 38357 Isomorphism H of a lattice...
dihglb2 38358 Isomorphism H of a lattice...
dihmeet 38359 Isomorphism H of a lattice...
dihintcl 38360 The intersection of closed...
dihmeetcl 38361 Closure of closed subspace...
dihmeet2 38362 Reverse isomorphism H of a...
dochffval 38365 Subspace orthocomplement f...
dochfval 38366 Subspace orthocomplement f...
dochval 38367 Subspace orthocomplement f...
dochval2 38368 Subspace orthocomplement f...
dochcl 38369 Closure of subspace orthoc...
dochlss 38370 A subspace orthocomplement...
dochssv 38371 A subspace orthocomplement...
dochfN 38372 Domain and codomain of the...
dochvalr 38373 Orthocomplement of a close...
doch0 38374 Orthocomplement of the zer...
doch1 38375 Orthocomplement of the uni...
dochoc0 38376 The zero subspace is close...
dochoc1 38377 The unit subspace (all vec...
dochvalr2 38378 Orthocomplement of a close...
dochvalr3 38379 Orthocomplement of a close...
doch2val2 38380 Double orthocomplement for...
dochss 38381 Subset law for orthocomple...
dochocss 38382 Double negative law for or...
dochoc 38383 Double negative law for or...
dochsscl 38384 If a set of vectors is inc...
dochoccl 38385 A set of vectors is closed...
dochord 38386 Ordering law for orthocomp...
dochord2N 38387 Ordering law for orthocomp...
dochord3 38388 Ordering law for orthocomp...
doch11 38389 Orthocomplement is one-to-...
dochsordN 38390 Strict ordering law for or...
dochn0nv 38391 An orthocomplement is nonz...
dihoml4c 38392 Version of ~ dihoml4 with ...
dihoml4 38393 Orthomodular law for const...
dochspss 38394 The span of a set of vecto...
dochocsp 38395 The span of an orthocomple...
dochspocN 38396 The span of an orthocomple...
dochocsn 38397 The double orthocomplement...
dochsncom 38398 Swap vectors in an orthoco...
dochsat 38399 The double orthocomplement...
dochshpncl 38400 If a hyperplane is not clo...
dochlkr 38401 Equivalent conditions for ...
dochkrshp 38402 The closure of a kernel is...
dochkrshp2 38403 Properties of the closure ...
dochkrshp3 38404 Properties of the closure ...
dochkrshp4 38405 Properties of the closure ...
dochdmj1 38406 De Morgan-like law for sub...
dochnoncon 38407 Law of noncontradiction. ...
dochnel2 38408 A nonzero member of a subs...
dochnel 38409 A nonzero vector doesn't b...
djhffval 38412 Subspace join for ` DVecH ...
djhfval 38413 Subspace join for ` DVecH ...
djhval 38414 Subspace join for ` DVecH ...
djhval2 38415 Value of subspace join for...
djhcl 38416 Closure of subspace join f...
djhlj 38417 Transfer lattice join to `...
djhljjN 38418 Lattice join in terms of `...
djhjlj 38419 ` DVecH ` vector space clo...
djhj 38420 ` DVecH ` vector space clo...
djhcom 38421 Subspace join commutes. (...
djhspss 38422 Subspace span of union is ...
djhsumss 38423 Subspace sum is a subset o...
dihsumssj 38424 The subspace sum of two is...
djhunssN 38425 Subspace union is a subset...
dochdmm1 38426 De Morgan-like law for clo...
djhexmid 38427 Excluded middle property o...
djh01 38428 Closed subspace join with ...
djh02 38429 Closed subspace join with ...
djhlsmcl 38430 A closed subspace sum equa...
djhcvat42 38431 A covering property. ( ~ ...
dihjatb 38432 Isomorphism H of lattice j...
dihjatc 38433 Isomorphism H of lattice j...
dihjatcclem1 38434 Lemma for isomorphism H of...
dihjatcclem2 38435 Lemma for isomorphism H of...
dihjatcclem3 38436 Lemma for ~ dihjatcc . (C...
dihjatcclem4 38437 Lemma for isomorphism H of...
dihjatcc 38438 Isomorphism H of lattice j...
dihjat 38439 Isomorphism H of lattice j...
dihprrnlem1N 38440 Lemma for ~ dihprrn , show...
dihprrnlem2 38441 Lemma for ~ dihprrn . (Co...
dihprrn 38442 The span of a vector pair ...
djhlsmat 38443 The sum of two subspace at...
dihjat1lem 38444 Subspace sum of a closed s...
dihjat1 38445 Subspace sum of a closed s...
dihsmsprn 38446 Subspace sum of a closed s...
dihjat2 38447 The subspace sum of a clos...
dihjat3 38448 Isomorphism H of lattice j...
dihjat4 38449 Transfer the subspace sum ...
dihjat6 38450 Transfer the subspace sum ...
dihsmsnrn 38451 The subspace sum of two si...
dihsmatrn 38452 The subspace sum of a clos...
dihjat5N 38453 Transfer lattice join with...
dvh4dimat 38454 There is an atom that is o...
dvh3dimatN 38455 There is an atom that is o...
dvh2dimatN 38456 Given an atom, there exist...
dvh1dimat 38457 There exists an atom. (Co...
dvh1dim 38458 There exists a nonzero vec...
dvh4dimlem 38459 Lemma for ~ dvh4dimN . (C...
dvhdimlem 38460 Lemma for ~ dvh2dim and ~ ...
dvh2dim 38461 There is a vector that is ...
dvh3dim 38462 There is a vector that is ...
dvh4dimN 38463 There is a vector that is ...
dvh3dim2 38464 There is a vector that is ...
dvh3dim3N 38465 There is a vector that is ...
dochsnnz 38466 The orthocomplement of a s...
dochsatshp 38467 The orthocomplement of a s...
dochsatshpb 38468 The orthocomplement of a s...
dochsnshp 38469 The orthocomplement of a n...
dochshpsat 38470 A hyperplane is closed iff...
dochkrsat 38471 The orthocomplement of a k...
dochkrsat2 38472 The orthocomplement of a k...
dochsat0 38473 The orthocomplement of a k...
dochkrsm 38474 The subspace sum of a clos...
dochexmidat 38475 Special case of excluded m...
dochexmidlem1 38476 Lemma for ~ dochexmid . H...
dochexmidlem2 38477 Lemma for ~ dochexmid . (...
dochexmidlem3 38478 Lemma for ~ dochexmid . U...
dochexmidlem4 38479 Lemma for ~ dochexmid . (...
dochexmidlem5 38480 Lemma for ~ dochexmid . (...
dochexmidlem6 38481 Lemma for ~ dochexmid . (...
dochexmidlem7 38482 Lemma for ~ dochexmid . C...
dochexmidlem8 38483 Lemma for ~ dochexmid . T...
dochexmid 38484 Excluded middle law for cl...
dochsnkrlem1 38485 Lemma for ~ dochsnkr . (C...
dochsnkrlem2 38486 Lemma for ~ dochsnkr . (C...
dochsnkrlem3 38487 Lemma for ~ dochsnkr . (C...
dochsnkr 38488 A (closed) kernel expresse...
dochsnkr2 38489 Kernel of the explicit fun...
dochsnkr2cl 38490 The ` X ` determining func...
dochflcl 38491 Closure of the explicit fu...
dochfl1 38492 The value of the explicit ...
dochfln0 38493 The value of a functional ...
dochkr1 38494 A nonzero functional has a...
dochkr1OLDN 38495 A nonzero functional has a...
lpolsetN 38498 The set of polarities of a...
islpolN 38499 The predicate "is a polari...
islpoldN 38500 Properties that determine ...
lpolfN 38501 Functionality of a polarit...
lpolvN 38502 The polarity of the whole ...
lpolconN 38503 Contraposition property of...
lpolsatN 38504 The polarity of an atomic ...
lpolpolsatN 38505 Property of a polarity. (...
dochpolN 38506 The subspace orthocompleme...
lcfl1lem 38507 Property of a functional w...
lcfl1 38508 Property of a functional w...
lcfl2 38509 Property of a functional w...
lcfl3 38510 Property of a functional w...
lcfl4N 38511 Property of a functional w...
lcfl5 38512 Property of a functional w...
lcfl5a 38513 Property of a functional w...
lcfl6lem 38514 Lemma for ~ lcfl6 . A fun...
lcfl7lem 38515 Lemma for ~ lcfl7N . If t...
lcfl6 38516 Property of a functional w...
lcfl7N 38517 Property of a functional w...
lcfl8 38518 Property of a functional w...
lcfl8a 38519 Property of a functional w...
lcfl8b 38520 Property of a nonzero func...
lcfl9a 38521 Property implying that a f...
lclkrlem1 38522 The set of functionals hav...
lclkrlem2a 38523 Lemma for ~ lclkr . Use ~...
lclkrlem2b 38524 Lemma for ~ lclkr . (Cont...
lclkrlem2c 38525 Lemma for ~ lclkr . (Cont...
lclkrlem2d 38526 Lemma for ~ lclkr . (Cont...
lclkrlem2e 38527 Lemma for ~ lclkr . The k...
lclkrlem2f 38528 Lemma for ~ lclkr . Const...
lclkrlem2g 38529 Lemma for ~ lclkr . Compa...
lclkrlem2h 38530 Lemma for ~ lclkr . Elimi...
lclkrlem2i 38531 Lemma for ~ lclkr . Elimi...
lclkrlem2j 38532 Lemma for ~ lclkr . Kerne...
lclkrlem2k 38533 Lemma for ~ lclkr . Kerne...
lclkrlem2l 38534 Lemma for ~ lclkr . Elimi...
lclkrlem2m 38535 Lemma for ~ lclkr . Const...
lclkrlem2n 38536 Lemma for ~ lclkr . (Cont...
lclkrlem2o 38537 Lemma for ~ lclkr . When ...
lclkrlem2p 38538 Lemma for ~ lclkr . When ...
lclkrlem2q 38539 Lemma for ~ lclkr . The s...
lclkrlem2r 38540 Lemma for ~ lclkr . When ...
lclkrlem2s 38541 Lemma for ~ lclkr . Thus,...
lclkrlem2t 38542 Lemma for ~ lclkr . We el...
lclkrlem2u 38543 Lemma for ~ lclkr . ~ lclk...
lclkrlem2v 38544 Lemma for ~ lclkr . When ...
lclkrlem2w 38545 Lemma for ~ lclkr . This ...
lclkrlem2x 38546 Lemma for ~ lclkr . Elimi...
lclkrlem2y 38547 Lemma for ~ lclkr . Resta...
lclkrlem2 38548 The set of functionals hav...
lclkr 38549 The set of functionals wit...
lcfls1lem 38550 Property of a functional w...
lcfls1N 38551 Property of a functional w...
lcfls1c 38552 Property of a functional w...
lclkrslem1 38553 The set of functionals hav...
lclkrslem2 38554 The set of functionals hav...
lclkrs 38555 The set of functionals hav...
lclkrs2 38556 The set of functionals wit...
lcfrvalsnN 38557 Reconstruction from the du...
lcfrlem1 38558 Lemma for ~ lcfr . Note t...
lcfrlem2 38559 Lemma for ~ lcfr . (Contr...
lcfrlem3 38560 Lemma for ~ lcfr . (Contr...
lcfrlem4 38561 Lemma for ~ lcfr . (Contr...
lcfrlem5 38562 Lemma for ~ lcfr . The se...
lcfrlem6 38563 Lemma for ~ lcfr . Closur...
lcfrlem7 38564 Lemma for ~ lcfr . Closur...
lcfrlem8 38565 Lemma for ~ lcf1o and ~ lc...
lcfrlem9 38566 Lemma for ~ lcf1o . (This...
lcf1o 38567 Define a function ` J ` th...
lcfrlem10 38568 Lemma for ~ lcfr . (Contr...
lcfrlem11 38569 Lemma for ~ lcfr . (Contr...
lcfrlem12N 38570 Lemma for ~ lcfr . (Contr...
lcfrlem13 38571 Lemma for ~ lcfr . (Contr...
lcfrlem14 38572 Lemma for ~ lcfr . (Contr...
lcfrlem15 38573 Lemma for ~ lcfr . (Contr...
lcfrlem16 38574 Lemma for ~ lcfr . (Contr...
lcfrlem17 38575 Lemma for ~ lcfr . Condit...
lcfrlem18 38576 Lemma for ~ lcfr . (Contr...
lcfrlem19 38577 Lemma for ~ lcfr . (Contr...
lcfrlem20 38578 Lemma for ~ lcfr . (Contr...
lcfrlem21 38579 Lemma for ~ lcfr . (Contr...
lcfrlem22 38580 Lemma for ~ lcfr . (Contr...
lcfrlem23 38581 Lemma for ~ lcfr . TODO: ...
lcfrlem24 38582 Lemma for ~ lcfr . (Contr...
lcfrlem25 38583 Lemma for ~ lcfr . Specia...
lcfrlem26 38584 Lemma for ~ lcfr . Specia...
lcfrlem27 38585 Lemma for ~ lcfr . Specia...
lcfrlem28 38586 Lemma for ~ lcfr . TODO: ...
lcfrlem29 38587 Lemma for ~ lcfr . (Contr...
lcfrlem30 38588 Lemma for ~ lcfr . (Contr...
lcfrlem31 38589 Lemma for ~ lcfr . (Contr...
lcfrlem32 38590 Lemma for ~ lcfr . (Contr...
lcfrlem33 38591 Lemma for ~ lcfr . (Contr...
lcfrlem34 38592 Lemma for ~ lcfr . (Contr...
lcfrlem35 38593 Lemma for ~ lcfr . (Contr...
lcfrlem36 38594 Lemma for ~ lcfr . (Contr...
lcfrlem37 38595 Lemma for ~ lcfr . (Contr...
lcfrlem38 38596 Lemma for ~ lcfr . Combin...
lcfrlem39 38597 Lemma for ~ lcfr . Elimin...
lcfrlem40 38598 Lemma for ~ lcfr . Elimin...
lcfrlem41 38599 Lemma for ~ lcfr . Elimin...
lcfrlem42 38600 Lemma for ~ lcfr . Elimin...
lcfr 38601 Reconstruction of a subspa...
lcdfval 38604 Dual vector space of funct...
lcdval 38605 Dual vector space of funct...
lcdval2 38606 Dual vector space of funct...
lcdlvec 38607 The dual vector space of f...
lcdlmod 38608 The dual vector space of f...
lcdvbase 38609 Vector base set of a dual ...
lcdvbasess 38610 The vector base set of the...
lcdvbaselfl 38611 A vector in the base set o...
lcdvbasecl 38612 Closure of the value of a ...
lcdvadd 38613 Vector addition for the cl...
lcdvaddval 38614 The value of the value of ...
lcdsca 38615 The ring of scalars of the...
lcdsbase 38616 Base set of scalar ring fo...
lcdsadd 38617 Scalar addition for the cl...
lcdsmul 38618 Scalar multiplication for ...
lcdvs 38619 Scalar product for the clo...
lcdvsval 38620 Value of scalar product op...
lcdvscl 38621 The scalar product operati...
lcdlssvscl 38622 Closure of scalar product ...
lcdvsass 38623 Associative law for scalar...
lcd0 38624 The zero scalar of the clo...
lcd1 38625 The unit scalar of the clo...
lcdneg 38626 The unit scalar of the clo...
lcd0v 38627 The zero functional in the...
lcd0v2 38628 The zero functional in the...
lcd0vvalN 38629 Value of the zero function...
lcd0vcl 38630 Closure of the zero functi...
lcd0vs 38631 A scalar zero times a func...
lcdvs0N 38632 A scalar times the zero fu...
lcdvsub 38633 The value of vector subtra...
lcdvsubval 38634 The value of the value of ...
lcdlss 38635 Subspaces of a dual vector...
lcdlss2N 38636 Subspaces of a dual vector...
lcdlsp 38637 Span in the set of functio...
lcdlkreqN 38638 Colinear functionals have ...
lcdlkreq2N 38639 Colinear functionals have ...
mapdffval 38642 Projectivity from vector s...
mapdfval 38643 Projectivity from vector s...
mapdval 38644 Value of projectivity from...
mapdvalc 38645 Value of projectivity from...
mapdval2N 38646 Value of projectivity from...
mapdval3N 38647 Value of projectivity from...
mapdval4N 38648 Value of projectivity from...
mapdval5N 38649 Value of projectivity from...
mapdordlem1a 38650 Lemma for ~ mapdord . (Co...
mapdordlem1bN 38651 Lemma for ~ mapdord . (Co...
mapdordlem1 38652 Lemma for ~ mapdord . (Co...
mapdordlem2 38653 Lemma for ~ mapdord . Ord...
mapdord 38654 Ordering property of the m...
mapd11 38655 The map defined by ~ df-ma...
mapddlssN 38656 The mapping of a subspace ...
mapdsn 38657 Value of the map defined b...
mapdsn2 38658 Value of the map defined b...
mapdsn3 38659 Value of the map defined b...
mapd1dim2lem1N 38660 Value of the map defined b...
mapdrvallem2 38661 Lemma for ~ mapdrval . TO...
mapdrvallem3 38662 Lemma for ~ mapdrval . (C...
mapdrval 38663 Given a dual subspace ` R ...
mapd1o 38664 The map defined by ~ df-ma...
mapdrn 38665 Range of the map defined b...
mapdunirnN 38666 Union of the range of the ...
mapdrn2 38667 Range of the map defined b...
mapdcnvcl 38668 Closure of the converse of...
mapdcl 38669 Closure the value of the m...
mapdcnvid1N 38670 Converse of the value of t...
mapdsord 38671 Strong ordering property o...
mapdcl2 38672 The mapping of a subspace ...
mapdcnvid2 38673 Value of the converse of t...
mapdcnvordN 38674 Ordering property of the c...
mapdcnv11N 38675 The converse of the map de...
mapdcv 38676 Covering property of the c...
mapdincl 38677 Closure of dual subspace i...
mapdin 38678 Subspace intersection is p...
mapdlsmcl 38679 Closure of dual subspace s...
mapdlsm 38680 Subspace sum is preserved ...
mapd0 38681 Projectivity map of the ze...
mapdcnvatN 38682 Atoms are preserved by the...
mapdat 38683 Atoms are preserved by the...
mapdspex 38684 The map of a span equals t...
mapdn0 38685 Transfer nonzero property ...
mapdncol 38686 Transfer non-colinearity f...
mapdindp 38687 Transfer (part of) vector ...
mapdpglem1 38688 Lemma for ~ mapdpg . Baer...
mapdpglem2 38689 Lemma for ~ mapdpg . Baer...
mapdpglem2a 38690 Lemma for ~ mapdpg . (Con...
mapdpglem3 38691 Lemma for ~ mapdpg . Baer...
mapdpglem4N 38692 Lemma for ~ mapdpg . (Con...
mapdpglem5N 38693 Lemma for ~ mapdpg . (Con...
mapdpglem6 38694 Lemma for ~ mapdpg . Baer...
mapdpglem8 38695 Lemma for ~ mapdpg . Baer...
mapdpglem9 38696 Lemma for ~ mapdpg . Baer...
mapdpglem10 38697 Lemma for ~ mapdpg . Baer...
mapdpglem11 38698 Lemma for ~ mapdpg . (Con...
mapdpglem12 38699 Lemma for ~ mapdpg . TODO...
mapdpglem13 38700 Lemma for ~ mapdpg . (Con...
mapdpglem14 38701 Lemma for ~ mapdpg . (Con...
mapdpglem15 38702 Lemma for ~ mapdpg . (Con...
mapdpglem16 38703 Lemma for ~ mapdpg . Baer...
mapdpglem17N 38704 Lemma for ~ mapdpg . Baer...
mapdpglem18 38705 Lemma for ~ mapdpg . Baer...
mapdpglem19 38706 Lemma for ~ mapdpg . Baer...
mapdpglem20 38707 Lemma for ~ mapdpg . Baer...
mapdpglem21 38708 Lemma for ~ mapdpg . (Con...
mapdpglem22 38709 Lemma for ~ mapdpg . Baer...
mapdpglem23 38710 Lemma for ~ mapdpg . Baer...
mapdpglem30a 38711 Lemma for ~ mapdpg . (Con...
mapdpglem30b 38712 Lemma for ~ mapdpg . (Con...
mapdpglem25 38713 Lemma for ~ mapdpg . Baer...
mapdpglem26 38714 Lemma for ~ mapdpg . Baer...
mapdpglem27 38715 Lemma for ~ mapdpg . Baer...
mapdpglem29 38716 Lemma for ~ mapdpg . Baer...
mapdpglem28 38717 Lemma for ~ mapdpg . Baer...
mapdpglem30 38718 Lemma for ~ mapdpg . Baer...
mapdpglem31 38719 Lemma for ~ mapdpg . Baer...
mapdpglem24 38720 Lemma for ~ mapdpg . Exis...
mapdpglem32 38721 Lemma for ~ mapdpg . Uniq...
mapdpg 38722 Part 1 of proof of the fir...
baerlem3lem1 38723 Lemma for ~ baerlem3 . (C...
baerlem5alem1 38724 Lemma for ~ baerlem5a . (...
baerlem5blem1 38725 Lemma for ~ baerlem5b . (...
baerlem3lem2 38726 Lemma for ~ baerlem3 . (C...
baerlem5alem2 38727 Lemma for ~ baerlem5a . (...
baerlem5blem2 38728 Lemma for ~ baerlem5b . (...
baerlem3 38729 An equality that holds whe...
baerlem5a 38730 An equality that holds whe...
baerlem5b 38731 An equality that holds whe...
baerlem5amN 38732 An equality that holds whe...
baerlem5bmN 38733 An equality that holds whe...
baerlem5abmN 38734 An equality that holds whe...
mapdindp0 38735 Vector independence lemma....
mapdindp1 38736 Vector independence lemma....
mapdindp2 38737 Vector independence lemma....
mapdindp3 38738 Vector independence lemma....
mapdindp4 38739 Vector independence lemma....
mapdhval 38740 Lemmma for ~~? mapdh . (C...
mapdhval0 38741 Lemmma for ~~? mapdh . (C...
mapdhval2 38742 Lemmma for ~~? mapdh . (C...
mapdhcl 38743 Lemmma for ~~? mapdh . (C...
mapdheq 38744 Lemmma for ~~? mapdh . Th...
mapdheq2 38745 Lemmma for ~~? mapdh . On...
mapdheq2biN 38746 Lemmma for ~~? mapdh . Pa...
mapdheq4lem 38747 Lemma for ~ mapdheq4 . Pa...
mapdheq4 38748 Lemma for ~~? mapdh . Par...
mapdh6lem1N 38749 Lemma for ~ mapdh6N . Par...
mapdh6lem2N 38750 Lemma for ~ mapdh6N . Par...
mapdh6aN 38751 Lemma for ~ mapdh6N . Par...
mapdh6b0N 38752 Lemmma for ~ mapdh6N . (C...
mapdh6bN 38753 Lemmma for ~ mapdh6N . (C...
mapdh6cN 38754 Lemmma for ~ mapdh6N . (C...
mapdh6dN 38755 Lemmma for ~ mapdh6N . (C...
mapdh6eN 38756 Lemmma for ~ mapdh6N . Pa...
mapdh6fN 38757 Lemmma for ~ mapdh6N . Pa...
mapdh6gN 38758 Lemmma for ~ mapdh6N . Pa...
mapdh6hN 38759 Lemmma for ~ mapdh6N . Pa...
mapdh6iN 38760 Lemmma for ~ mapdh6N . El...
mapdh6jN 38761 Lemmma for ~ mapdh6N . El...
mapdh6kN 38762 Lemmma for ~ mapdh6N . El...
mapdh6N 38763 Part (6) of [Baer] p. 47 l...
mapdh7eN 38764 Part (7) of [Baer] p. 48 l...
mapdh7cN 38765 Part (7) of [Baer] p. 48 l...
mapdh7dN 38766 Part (7) of [Baer] p. 48 l...
mapdh7fN 38767 Part (7) of [Baer] p. 48 l...
mapdh75e 38768 Part (7) of [Baer] p. 48 l...
mapdh75cN 38769 Part (7) of [Baer] p. 48 l...
mapdh75d 38770 Part (7) of [Baer] p. 48 l...
mapdh75fN 38771 Part (7) of [Baer] p. 48 l...
hvmapffval 38774 Map from nonzero vectors t...
hvmapfval 38775 Map from nonzero vectors t...
hvmapval 38776 Value of map from nonzero ...
hvmapvalvalN 38777 Value of value of map (i.e...
hvmapidN 38778 The value of the vector to...
hvmap1o 38779 The vector to functional m...
hvmapclN 38780 Closure of the vector to f...
hvmap1o2 38781 The vector to functional m...
hvmapcl2 38782 Closure of the vector to f...
hvmaplfl 38783 The vector to functional m...
hvmaplkr 38784 Kernel of the vector to fu...
mapdhvmap 38785 Relationship between ` map...
lspindp5 38786 Obtain an independent vect...
hdmaplem1 38787 Lemma to convert a frequen...
hdmaplem2N 38788 Lemma to convert a frequen...
hdmaplem3 38789 Lemma to convert a frequen...
hdmaplem4 38790 Lemma to convert a frequen...
mapdh8a 38791 Part of Part (8) in [Baer]...
mapdh8aa 38792 Part of Part (8) in [Baer]...
mapdh8ab 38793 Part of Part (8) in [Baer]...
mapdh8ac 38794 Part of Part (8) in [Baer]...
mapdh8ad 38795 Part of Part (8) in [Baer]...
mapdh8b 38796 Part of Part (8) in [Baer]...
mapdh8c 38797 Part of Part (8) in [Baer]...
mapdh8d0N 38798 Part of Part (8) in [Baer]...
mapdh8d 38799 Part of Part (8) in [Baer]...
mapdh8e 38800 Part of Part (8) in [Baer]...
mapdh8g 38801 Part of Part (8) in [Baer]...
mapdh8i 38802 Part of Part (8) in [Baer]...
mapdh8j 38803 Part of Part (8) in [Baer]...
mapdh8 38804 Part (8) in [Baer] p. 48. ...
mapdh9a 38805 Lemma for part (9) in [Bae...
mapdh9aOLDN 38806 Lemma for part (9) in [Bae...
hdmap1ffval 38811 Preliminary map from vecto...
hdmap1fval 38812 Preliminary map from vecto...
hdmap1vallem 38813 Value of preliminary map f...
hdmap1val 38814 Value of preliminary map f...
hdmap1val0 38815 Value of preliminary map f...
hdmap1val2 38816 Value of preliminary map f...
hdmap1eq 38817 The defining equation for ...
hdmap1cbv 38818 Frequently used lemma to c...
hdmap1valc 38819 Connect the value of the p...
hdmap1cl 38820 Convert closure theorem ~ ...
hdmap1eq2 38821 Convert ~ mapdheq2 to use ...
hdmap1eq4N 38822 Convert ~ mapdheq4 to use ...
hdmap1l6lem1 38823 Lemma for ~ hdmap1l6 . Pa...
hdmap1l6lem2 38824 Lemma for ~ hdmap1l6 . Pa...
hdmap1l6a 38825 Lemma for ~ hdmap1l6 . Pa...
hdmap1l6b0N 38826 Lemmma for ~ hdmap1l6 . (...
hdmap1l6b 38827 Lemmma for ~ hdmap1l6 . (...
hdmap1l6c 38828 Lemmma for ~ hdmap1l6 . (...
hdmap1l6d 38829 Lemmma for ~ hdmap1l6 . (...
hdmap1l6e 38830 Lemmma for ~ hdmap1l6 . P...
hdmap1l6f 38831 Lemmma for ~ hdmap1l6 . P...
hdmap1l6g 38832 Lemmma for ~ hdmap1l6 . P...
hdmap1l6h 38833 Lemmma for ~ hdmap1l6 . P...
hdmap1l6i 38834 Lemmma for ~ hdmap1l6 . E...
hdmap1l6j 38835 Lemmma for ~ hdmap1l6 . E...
hdmap1l6k 38836 Lemmma for ~ hdmap1l6 . E...
hdmap1l6 38837 Part (6) of [Baer] p. 47 l...
hdmap1eulem 38838 Lemma for ~ hdmap1eu . TO...
hdmap1eulemOLDN 38839 Lemma for ~ hdmap1euOLDN ....
hdmap1eu 38840 Convert ~ mapdh9a to use t...
hdmap1euOLDN 38841 Convert ~ mapdh9aOLDN to u...
hdmapffval 38842 Map from vectors to functi...
hdmapfval 38843 Map from vectors to functi...
hdmapval 38844 Value of map from vectors ...
hdmapfnN 38845 Functionality of map from ...
hdmapcl 38846 Closure of map from vector...
hdmapval2lem 38847 Lemma for ~ hdmapval2 . (...
hdmapval2 38848 Value of map from vectors ...
hdmapval0 38849 Value of map from vectors ...
hdmapeveclem 38850 Lemma for ~ hdmapevec . T...
hdmapevec 38851 Value of map from vectors ...
hdmapevec2 38852 The inner product of the r...
hdmapval3lemN 38853 Value of map from vectors ...
hdmapval3N 38854 Value of map from vectors ...
hdmap10lem 38855 Lemma for ~ hdmap10 . (Co...
hdmap10 38856 Part 10 in [Baer] p. 48 li...
hdmap11lem1 38857 Lemma for ~ hdmapadd . (C...
hdmap11lem2 38858 Lemma for ~ hdmapadd . (C...
hdmapadd 38859 Part 11 in [Baer] p. 48 li...
hdmapeq0 38860 Part of proof of part 12 i...
hdmapnzcl 38861 Nonzero vector closure of ...
hdmapneg 38862 Part of proof of part 12 i...
hdmapsub 38863 Part of proof of part 12 i...
hdmap11 38864 Part of proof of part 12 i...
hdmaprnlem1N 38865 Part of proof of part 12 i...
hdmaprnlem3N 38866 Part of proof of part 12 i...
hdmaprnlem3uN 38867 Part of proof of part 12 i...
hdmaprnlem4tN 38868 Lemma for ~ hdmaprnN . TO...
hdmaprnlem4N 38869 Part of proof of part 12 i...
hdmaprnlem6N 38870 Part of proof of part 12 i...
hdmaprnlem7N 38871 Part of proof of part 12 i...
hdmaprnlem8N 38872 Part of proof of part 12 i...
hdmaprnlem9N 38873 Part of proof of part 12 i...
hdmaprnlem3eN 38874 Lemma for ~ hdmaprnN . (C...
hdmaprnlem10N 38875 Lemma for ~ hdmaprnN . Sh...
hdmaprnlem11N 38876 Lemma for ~ hdmaprnN . Sh...
hdmaprnlem15N 38877 Lemma for ~ hdmaprnN . El...
hdmaprnlem16N 38878 Lemma for ~ hdmaprnN . El...
hdmaprnlem17N 38879 Lemma for ~ hdmaprnN . In...
hdmaprnN 38880 Part of proof of part 12 i...
hdmapf1oN 38881 Part 12 in [Baer] p. 49. ...
hdmap14lem1a 38882 Prior to part 14 in [Baer]...
hdmap14lem2a 38883 Prior to part 14 in [Baer]...
hdmap14lem1 38884 Prior to part 14 in [Baer]...
hdmap14lem2N 38885 Prior to part 14 in [Baer]...
hdmap14lem3 38886 Prior to part 14 in [Baer]...
hdmap14lem4a 38887 Simplify ` ( A \ { Q } ) `...
hdmap14lem4 38888 Simplify ` ( A \ { Q } ) `...
hdmap14lem6 38889 Case where ` F ` is zero. ...
hdmap14lem7 38890 Combine cases of ` F ` . ...
hdmap14lem8 38891 Part of proof of part 14 i...
hdmap14lem9 38892 Part of proof of part 14 i...
hdmap14lem10 38893 Part of proof of part 14 i...
hdmap14lem11 38894 Part of proof of part 14 i...
hdmap14lem12 38895 Lemma for proof of part 14...
hdmap14lem13 38896 Lemma for proof of part 14...
hdmap14lem14 38897 Part of proof of part 14 i...
hdmap14lem15 38898 Part of proof of part 14 i...
hgmapffval 38901 Map from the scalar divisi...
hgmapfval 38902 Map from the scalar divisi...
hgmapval 38903 Value of map from the scal...
hgmapfnN 38904 Functionality of scalar si...
hgmapcl 38905 Closure of scalar sigma ma...
hgmapdcl 38906 Closure of the vector spac...
hgmapvs 38907 Part 15 of [Baer] p. 50 li...
hgmapval0 38908 Value of the scalar sigma ...
hgmapval1 38909 Value of the scalar sigma ...
hgmapadd 38910 Part 15 of [Baer] p. 50 li...
hgmapmul 38911 Part 15 of [Baer] p. 50 li...
hgmaprnlem1N 38912 Lemma for ~ hgmaprnN . (C...
hgmaprnlem2N 38913 Lemma for ~ hgmaprnN . Pa...
hgmaprnlem3N 38914 Lemma for ~ hgmaprnN . El...
hgmaprnlem4N 38915 Lemma for ~ hgmaprnN . El...
hgmaprnlem5N 38916 Lemma for ~ hgmaprnN . El...
hgmaprnN 38917 Part of proof of part 16 i...
hgmap11 38918 The scalar sigma map is on...
hgmapf1oN 38919 The scalar sigma map is a ...
hgmapeq0 38920 The scalar sigma map is ze...
hdmapipcl 38921 The inner product (Hermiti...
hdmapln1 38922 Linearity property that wi...
hdmaplna1 38923 Additive property of first...
hdmaplns1 38924 Subtraction property of fi...
hdmaplnm1 38925 Multiplicative property of...
hdmaplna2 38926 Additive property of secon...
hdmapglnm2 38927 g-linear property of secon...
hdmapgln2 38928 g-linear property that wil...
hdmaplkr 38929 Kernel of the vector to du...
hdmapellkr 38930 Membership in the kernel (...
hdmapip0 38931 Zero property that will be...
hdmapip1 38932 Construct a proportional v...
hdmapip0com 38933 Commutation property of Ba...
hdmapinvlem1 38934 Line 27 in [Baer] p. 110. ...
hdmapinvlem2 38935 Line 28 in [Baer] p. 110, ...
hdmapinvlem3 38936 Line 30 in [Baer] p. 110, ...
hdmapinvlem4 38937 Part 1.1 of Proposition 1 ...
hdmapglem5 38938 Part 1.2 in [Baer] p. 110 ...
hgmapvvlem1 38939 Involution property of sca...
hgmapvvlem2 38940 Lemma for ~ hgmapvv . Eli...
hgmapvvlem3 38941 Lemma for ~ hgmapvv . Eli...
hgmapvv 38942 Value of a double involuti...
hdmapglem7a 38943 Lemma for ~ hdmapg . (Con...
hdmapglem7b 38944 Lemma for ~ hdmapg . (Con...
hdmapglem7 38945 Lemma for ~ hdmapg . Line...
hdmapg 38946 Apply the scalar sigma fun...
hdmapoc 38947 Express our constructed or...
hlhilset 38950 The final Hilbert space co...
hlhilsca 38951 The scalar of the final co...
hlhilbase 38952 The base set of the final ...
hlhilplus 38953 The vector addition for th...
hlhilslem 38954 Lemma for ~ hlhilsbase2 . ...
hlhilsbase 38955 The scalar base set of the...
hlhilsplus 38956 Scalar addition for the fi...
hlhilsmul 38957 Scalar multiplication for ...
hlhilsbase2 38958 The scalar base set of the...
hlhilsplus2 38959 Scalar addition for the fi...
hlhilsmul2 38960 Scalar multiplication for ...
hlhils0 38961 The scalar ring zero for t...
hlhils1N 38962 The scalar ring unity for ...
hlhilvsca 38963 The scalar product for the...
hlhilip 38964 Inner product operation fo...
hlhilipval 38965 Value of inner product ope...
hlhilnvl 38966 The involution operation o...
hlhillvec 38967 The final constructed Hilb...
hlhildrng 38968 The star division ring for...
hlhilsrnglem 38969 Lemma for ~ hlhilsrng . (...
hlhilsrng 38970 The star division ring for...
hlhil0 38971 The zero vector for the fi...
hlhillsm 38972 The vector sum operation f...
hlhilocv 38973 The orthocomplement for th...
hlhillcs 38974 The closed subspaces of th...
hlhilphllem 38975 Lemma for ~ hlhil . (Cont...
hlhilhillem 38976 Lemma for ~ hlhil . (Cont...
hlathil 38977 Construction of a Hilbert ...
ioin9i8 38978 Miscellaneous inference cr...
jaodd 38979 Double deduction form of ~...
nsb 38980 Generalization rule for ne...
sbn1 38981 One direction of ~ sbn , u...
sbor2 38982 One direction of ~ sbor , ...
3rspcedvd 38983 Triple application of ~ rs...
rabeqcda 38984 When ` ps ` is always true...
rabdif 38985 Move difference in and out...
sn-axrep5v 38986 A condensed form of ~ axre...
sn-axprlem3 38987 ~ axprlem3 using only Tars...
sn-el 38988 A version of ~ el with an ...
sn-dtru 38989 ~ dtru without ~ ax-8 or ~...
pssexg 38990 The proper subset of a set...
pssn0 38991 A proper superset is nonem...
psspwb 38992 Classes are proper subclas...
xppss12 38993 Proper subset theorem for ...
elpwbi 38994 Membership in a power set,...
opelxpii 38995 Ordered pair membership in...
iunsn 38996 Indexed union of a singlet...
imaopab 38997 The image of a class of or...
fnsnbt 38998 A function's domain is a s...
fnimasnd 38999 The image of a function by...
dfqs2 39000 Alternate definition of qu...
dfqs3 39001 Alternate definition of qu...
qseq12d 39002 Equality theorem for quoti...
qsalrel 39003 The quotient set is equal ...
fzosumm1 39004 Separate out the last term...
ccatcan2d 39005 Cancellation law for conca...
nelsubginvcld 39006 The inverse of a non-subgr...
nelsubgcld 39007 A non-subgroup-member plus...
nelsubgsubcld 39008 A non-subgroup-member minu...
rnasclg 39009 The set of injected scalar...
selvval2lem1 39010 ` T ` is an associative al...
selvval2lem2 39011 ` D ` is a ring homomorphi...
selvval2lem3 39012 The third argument passed ...
selvval2lemn 39013 A lemma to illustrate the ...
selvval2lem4 39014 The fourth argument passed...
selvval2lem5 39015 The fifth argument passed ...
selvcl 39016 Closure of the "variable s...
frlmfielbas 39017 The vectors of a finite fr...
frlmfzwrd 39018 A vector of a module with ...
frlmfzowrd 39019 A vector of a module with ...
frlmfzolen 39020 The dimension of a vector ...
frlmfzowrdb 39021 The vectors of a module wi...
frlmfzoccat 39022 The concatenation of two v...
frlmvscadiccat 39023 Scalar multiplication dist...
lvecgrp 39024 A left vector is a group. ...
lvecring 39025 The scalar component of a ...
lmhmlvec 39026 The property for modules t...
frlmsnic 39027 Given a free module with a...
uvccl 39028 A unit vector is a vector....
uvcn0 39029 A unit vector is nonzero. ...
c0exALT 39030 Alternate proof of ~ c0ex ...
0cnALT3 39031 Alternate proof of ~ 0cn u...
elre0re 39032 Specialized version of ~ 0...
1t1e1ALT 39033 Alternate proof of ~ 1t1e1...
remulcan2d 39034 ~ mulcan2d for real number...
readdid1addid2d 39035 Given some real number ` B...
sn-1ne2 39036 A proof of ~ 1ne2 without ...
nnn1suc 39037 A positive integer that is...
nnadd1com 39038 Addition with 1 is commuta...
nnaddcom 39039 Addition is commutative fo...
nnaddcomli 39040 Version of ~ addcomli for ...
nnadddir 39041 Right-distributivity for n...
nnmul1com 39042 Multiplication with 1 is c...
nnmulcom 39043 Multiplication is commutat...
addsubeq4com 39044 Relation between sums and ...
sqsumi 39045 A sum squared. (Contribut...
negn0nposznnd 39046 Lemma for ~ dffltz . (Con...
sqmid3api 39047 Value of the square of the...
decaddcom 39048 Commute ones place in addi...
sqn5i 39049 The square of a number end...
sqn5ii 39050 The square of a number end...
decpmulnc 39051 Partial products algorithm...
decpmul 39052 Partial products algorithm...
sqdeccom12 39053 The square of a number in ...
sq3deccom12 39054 Variant of ~ sqdeccom12 wi...
235t711 39055 Calculate a product by lon...
ex-decpmul 39056 Example usage of ~ decpmul...
oexpreposd 39057 Lemma for ~ dffltz . (Con...
cxpgt0d 39058 Exponentiation with a posi...
dvdsexpim 39059 ~ dvdssqim generalized to ...
nn0rppwr 39060 If ` A ` and ` B ` are rel...
expgcd 39061 Exponentiation distributes...
nn0expgcd 39062 Exponentiation distributes...
zexpgcd 39063 Exponentiation distributes...
numdenexp 39064 ~ numdensq extended to non...
numexp 39065 ~ numsq extended to nonneg...
denexp 39066 ~ densq extended to nonneg...
exp11d 39067 ~ sq11d for positive real ...
ltexp1d 39068 ~ ltmul1d for exponentiati...
ltexp1dd 39069 Raising both sides of 'les...
zrtelqelz 39070 ~ zsqrtelqelz generalized ...
zrtdvds 39071 A positive integer root di...
rtprmirr 39072 The root of a prime number...
resubval 39075 Value of real subtraction,...
renegeulemv 39076 Lemma for ~ renegeu and si...
renegeulem 39077 Lemma for ~ renegeu and si...
renegeu 39078 Existential uniqueness of ...
rernegcl 39079 Closure law for negative r...
renegadd 39080 Relationship between real ...
renegid 39081 Addition of a real number ...
reneg0addid2 39082 Negative zero is a left ad...
resubeulem1 39083 Lemma for ~ resubeu . A v...
resubeulem2 39084 Lemma for ~ resubeu . A v...
resubeu 39085 Existential uniqueness of ...
rersubcl 39086 Closure for real subtracti...
resubadd 39087 Relation between real subt...
resubaddd 39088 Relationship between subtr...
resubf 39089 Real subtraction is an ope...
repncan2 39090 Addition and subtraction o...
repncan3 39091 Addition and subtraction o...
readdsub 39092 Law for addition and subtr...
reladdrsub 39093 Move LHS of a sum into RHS...
reltsub1 39094 Subtraction from both side...
reltsubadd2 39095 'Less than' relationship b...
resubcan2 39096 Cancellation law for real ...
resubsub4 39097 Law for double subtraction...
rennncan2 39098 Cancellation law for real ...
renpncan3 39099 Cancellation law for real ...
repnpcan 39100 Cancellation law for addit...
reppncan 39101 Cancellation law for mixed...
resubidaddid1lem 39102 Lemma for ~ resubidaddid1 ...
resubidaddid1 39103 Any real number subtracted...
resubdi 39104 Distribution of multiplica...
re1m1e0m0 39105 Equality of two left-addit...
sn-00idlem1 39106 Lemma for ~ sn-00id . (Co...
sn-00idlem2 39107 Lemma for ~ sn-00id . (Co...
sn-00idlem3 39108 Lemma for ~ sn-00id . (Co...
sn-00id 39109 ~ 00id proven without ~ ax...
re0m0e0 39110 Real number version of ~ 0...
readdid2 39111 Real number version of ~ a...
sn-addid2 39112 ~ addid2 without ~ ax-mulc...
remul02 39113 Real number version of ~ m...
sn-0ne2 39114 ~ 0ne2 without ~ ax-mulcom...
remul01 39115 Real number version of ~ m...
resubid 39116 Subtraction of a real numb...
readdid1 39117 Real number version of ~ a...
resubid1 39118 Real number version of ~ s...
renegneg 39119 A real number is equal to ...
readdcan2 39120 Commuted version of ~ read...
sn-ltaddpos 39121 ~ ltaddpos without ~ ax-mu...
relt0neg1 39122 Comparison of a real and i...
relt0neg2 39123 Comparison of a real and i...
sn-0lt1 39124 ~ 0lt1 without ~ ax-mulcom...
sn-ltp1 39125 ~ ltp1 without ~ ax-mulcom...
remulinvcom 39126 A left multiplicative inve...
remulid2 39127 Commuted version of ~ ax-1...
remulcand 39128 Commuted version of ~ remu...
prjspval 39131 Value of the projective sp...
prjsprel 39132 Utility theorem regarding ...
prjspertr 39133 The relation in ` PrjSp ` ...
prjsperref 39134 The relation in ` PrjSp ` ...
prjspersym 39135 The relation in ` PrjSp ` ...
prjsper 39136 The relation in ` PrjSp ` ...
prjspreln0 39137 Two nonzero vectors are eq...
prjspvs 39138 A nonzero multiple of a ve...
prjsprellsp 39139 Two vectors are equivalent...
prjspeclsp 39140 The vectors equivalent to ...
prjspval2 39141 Alternate definition of pr...
prjspnval 39144 Value of the n-dimensional...
prjspnval2 39145 Value of the n-dimensional...
0prjspnlem 39146 Lemma for ~ 0prjspn . The...
0prjspnrel 39147 In the zero-dimensional pr...
0prjspn 39148 A zero-dimensional project...
dffltz 39149 Fermat's Last Theorem (FLT...
fltne 39150 If a counterexample to FLT...
fltltc 39151 ` ( C ^ N ) ` is the large...
fltnltalem 39152 Lemma for ~ fltnlta . A l...
fltnlta 39153 ` N ` is less than ` A ` ....
binom2d 39154 Deduction form of binom2. ...
cu3addd 39155 Cube of sum of three numbe...
sqnegd 39156 The square of the negative...
negexpidd 39157 The sum of a real number t...
rexlimdv3d 39158 An extended version of ~ r...
3cubeslem1 39159 Lemma for ~ 3cubes . (Con...
3cubeslem2 39160 Lemma for ~ 3cubes . Used...
3cubeslem3l 39161 Lemma for ~ 3cubes . (Con...
3cubeslem3r 39162 Lemma for ~ 3cubes . (Con...
3cubeslem3 39163 Lemma for ~ 3cubes . (Con...
3cubeslem4 39164 Lemma for ~ 3cubes . This...
3cubes 39165 Every rational number is a...
rntrclfvOAI 39166 The range of the transitiv...
moxfr 39167 Transfer at-most-one betwe...
imaiinfv 39168 Indexed intersection of an...
elrfi 39169 Elementhood in a set of re...
elrfirn 39170 Elementhood in a set of re...
elrfirn2 39171 Elementhood in a set of re...
cmpfiiin 39172 In a compact topology, a s...
ismrcd1 39173 Any function from the subs...
ismrcd2 39174 Second half of ~ ismrcd1 ....
istopclsd 39175 A closure function which s...
ismrc 39176 A function is a Moore clos...
isnacs 39179 Expand definition of Noeth...
nacsfg 39180 In a Noetherian-type closu...
isnacs2 39181 Express Noetherian-type cl...
mrefg2 39182 Slight variation on finite...
mrefg3 39183 Slight variation on finite...
nacsacs 39184 A closure system of Noethe...
isnacs3 39185 A choice-free order equiva...
incssnn0 39186 Transitivity induction of ...
nacsfix 39187 An increasing sequence of ...
constmap 39188 A constant (represented wi...
mapco2g 39189 Renaming indices in a tupl...
mapco2 39190 Post-composition (renaming...
mapfzcons 39191 Extending a one-based mapp...
mapfzcons1 39192 Recover prefix mapping fro...
mapfzcons1cl 39193 A nonempty mapping has a p...
mapfzcons2 39194 Recover added element from...
mptfcl 39195 Interpret range of a maps-...
mzpclval 39200 Substitution lemma for ` m...
elmzpcl 39201 Double substitution lemma ...
mzpclall 39202 The set of all functions w...
mzpcln0 39203 Corrolary of ~ mzpclall : ...
mzpcl1 39204 Defining property 1 of a p...
mzpcl2 39205 Defining property 2 of a p...
mzpcl34 39206 Defining properties 3 and ...
mzpval 39207 Value of the ` mzPoly ` fu...
dmmzp 39208 ` mzPoly ` is defined for ...
mzpincl 39209 Polynomial closedness is a...
mzpconst 39210 Constant functions are pol...
mzpf 39211 A polynomial function is a...
mzpproj 39212 A projection function is p...
mzpadd 39213 The pointwise sum of two p...
mzpmul 39214 The pointwise product of t...
mzpconstmpt 39215 A constant function expres...
mzpaddmpt 39216 Sum of polynomial function...
mzpmulmpt 39217 Product of polynomial func...
mzpsubmpt 39218 The difference of two poly...
mzpnegmpt 39219 Negation of a polynomial f...
mzpexpmpt 39220 Raise a polynomial functio...
mzpindd 39221 "Structural" induction to ...
mzpmfp 39222 Relationship between multi...
mzpsubst 39223 Substituting polynomials f...
mzprename 39224 Simplified version of ~ mz...
mzpresrename 39225 A polynomial is a polynomi...
mzpcompact2lem 39226 Lemma for ~ mzpcompact2 . ...
mzpcompact2 39227 Polynomials are finitary o...
coeq0i 39228 ~ coeq0 but without explic...
fzsplit1nn0 39229 Split a finite 1-based set...
eldiophb 39232 Initial expression of Diop...
eldioph 39233 Condition for a set to be ...
diophrw 39234 Renaming and adding unused...
eldioph2lem1 39235 Lemma for ~ eldioph2 . Co...
eldioph2lem2 39236 Lemma for ~ eldioph2 . Co...
eldioph2 39237 Construct a Diophantine se...
eldioph2b 39238 While Diophantine sets wer...
eldiophelnn0 39239 Remove antecedent on ` B `...
eldioph3b 39240 Define Diophantine sets in...
eldioph3 39241 Inference version of ~ eld...
ellz1 39242 Membership in a lower set ...
lzunuz 39243 The union of a lower set o...
fz1eqin 39244 Express a one-based finite...
lzenom 39245 Lower integers are countab...
elmapresaunres2 39246 ~ fresaunres2 transposed t...
diophin 39247 If two sets are Diophantin...
diophun 39248 If two sets are Diophantin...
eldiophss 39249 Diophantine sets are sets ...
diophrex 39250 Projecting a Diophantine s...
eq0rabdioph 39251 This is the first of a num...
eqrabdioph 39252 Diophantine set builder fo...
0dioph 39253 The null set is Diophantin...
vdioph 39254 The "universal" set (as la...
anrabdioph 39255 Diophantine set builder fo...
orrabdioph 39256 Diophantine set builder fo...
3anrabdioph 39257 Diophantine set builder fo...
3orrabdioph 39258 Diophantine set builder fo...
2sbcrex 39259 Exchange an existential qu...
sbcrexgOLD 39260 Interchange class substitu...
2sbcrexOLD 39261 Exchange an existential qu...
sbc2rex 39262 Exchange a substitution wi...
sbc2rexgOLD 39263 Exchange a substitution wi...
sbc4rex 39264 Exchange a substitution wi...
sbc4rexgOLD 39265 Exchange a substitution wi...
sbcrot3 39266 Rotate a sequence of three...
sbcrot5 39267 Rotate a sequence of five ...
sbccomieg 39268 Commute two explicit subst...
rexrabdioph 39269 Diophantine set builder fo...
rexfrabdioph 39270 Diophantine set builder fo...
2rexfrabdioph 39271 Diophantine set builder fo...
3rexfrabdioph 39272 Diophantine set builder fo...
4rexfrabdioph 39273 Diophantine set builder fo...
6rexfrabdioph 39274 Diophantine set builder fo...
7rexfrabdioph 39275 Diophantine set builder fo...
rabdiophlem1 39276 Lemma for arithmetic dioph...
rabdiophlem2 39277 Lemma for arithmetic dioph...
elnn0rabdioph 39278 Diophantine set builder fo...
rexzrexnn0 39279 Rewrite a quantification o...
lerabdioph 39280 Diophantine set builder fo...
eluzrabdioph 39281 Diophantine set builder fo...
elnnrabdioph 39282 Diophantine set builder fo...
ltrabdioph 39283 Diophantine set builder fo...
nerabdioph 39284 Diophantine set builder fo...
dvdsrabdioph 39285 Divisibility is a Diophant...
eldioph4b 39286 Membership in ` Dioph ` ex...
eldioph4i 39287 Forward-only version of ~ ...
diophren 39288 Change variables in a Diop...
rabrenfdioph 39289 Change variable numbers in...
rabren3dioph 39290 Change variable numbers in...
fphpd 39291 Pigeonhole principle expre...
fphpdo 39292 Pigeonhole principle for s...
ctbnfien 39293 An infinite subset of a co...
fiphp3d 39294 Infinite pigeonhole princi...
rencldnfilem 39295 Lemma for ~ rencldnfi . (...
rencldnfi 39296 A set of real numbers whic...
irrapxlem1 39297 Lemma for ~ irrapx1 . Div...
irrapxlem2 39298 Lemma for ~ irrapx1 . Two...
irrapxlem3 39299 Lemma for ~ irrapx1 . By ...
irrapxlem4 39300 Lemma for ~ irrapx1 . Eli...
irrapxlem5 39301 Lemma for ~ irrapx1 . Swi...
irrapxlem6 39302 Lemma for ~ irrapx1 . Exp...
irrapx1 39303 Dirichlet's approximation ...
pellexlem1 39304 Lemma for ~ pellex . Arit...
pellexlem2 39305 Lemma for ~ pellex . Arit...
pellexlem3 39306 Lemma for ~ pellex . To e...
pellexlem4 39307 Lemma for ~ pellex . Invo...
pellexlem5 39308 Lemma for ~ pellex . Invo...
pellexlem6 39309 Lemma for ~ pellex . Doin...
pellex 39310 Every Pell equation has a ...
pell1qrval 39321 Value of the set of first-...
elpell1qr 39322 Membership in a first-quad...
pell14qrval 39323 Value of the set of positi...
elpell14qr 39324 Membership in the set of p...
pell1234qrval 39325 Value of the set of genera...
elpell1234qr 39326 Membership in the set of g...
pell1234qrre 39327 General Pell solutions are...
pell1234qrne0 39328 No solution to a Pell equa...
pell1234qrreccl 39329 General solutions of the P...
pell1234qrmulcl 39330 General solutions of the P...
pell14qrss1234 39331 A positive Pell solution i...
pell14qrre 39332 A positive Pell solution i...
pell14qrne0 39333 A positive Pell solution i...
pell14qrgt0 39334 A positive Pell solution i...
pell14qrrp 39335 A positive Pell solution i...
pell1234qrdich 39336 A general Pell solution is...
elpell14qr2 39337 A number is a positive Pel...
pell14qrmulcl 39338 Positive Pell solutions ar...
pell14qrreccl 39339 Positive Pell solutions ar...
pell14qrdivcl 39340 Positive Pell solutions ar...
pell14qrexpclnn0 39341 Lemma for ~ pell14qrexpcl ...
pell14qrexpcl 39342 Positive Pell solutions ar...
pell1qrss14 39343 First-quadrant Pell soluti...
pell14qrdich 39344 A positive Pell solution i...
pell1qrge1 39345 A Pell solution in the fir...
pell1qr1 39346 1 is a Pell solution and i...
elpell1qr2 39347 The first quadrant solutio...
pell1qrgaplem 39348 Lemma for ~ pell1qrgap . ...
pell1qrgap 39349 First-quadrant Pell soluti...
pell14qrgap 39350 Positive Pell solutions ar...
pell14qrgapw 39351 Positive Pell solutions ar...
pellqrexplicit 39352 Condition for a calculated...
infmrgelbi 39353 Any lower bound of a nonem...
pellqrex 39354 There is a nontrivial solu...
pellfundval 39355 Value of the fundamental s...
pellfundre 39356 The fundamental solution o...
pellfundge 39357 Lower bound on the fundame...
pellfundgt1 39358 Weak lower bound on the Pe...
pellfundlb 39359 A nontrivial first quadran...
pellfundglb 39360 If a real is larger than t...
pellfundex 39361 The fundamental solution a...
pellfund14gap 39362 There are no solutions bet...
pellfundrp 39363 The fundamental Pell solut...
pellfundne1 39364 The fundamental Pell solut...
reglogcl 39365 General logarithm is a rea...
reglogltb 39366 General logarithm preserve...
reglogleb 39367 General logarithm preserve...
reglogmul 39368 Multiplication law for gen...
reglogexp 39369 Power law for general log....
reglogbas 39370 General log of the base is...
reglog1 39371 General log of 1 is 0. (C...
reglogexpbas 39372 General log of a power of ...
pellfund14 39373 Every positive Pell soluti...
pellfund14b 39374 The positive Pell solution...
rmxfval 39379 Value of the X sequence. ...
rmyfval 39380 Value of the Y sequence. ...
rmspecsqrtnq 39381 The discriminant used to d...
rmspecnonsq 39382 The discriminant used to d...
qirropth 39383 This lemma implements the ...
rmspecfund 39384 The base of exponent used ...
rmxyelqirr 39385 The solutions used to cons...
rmxypairf1o 39386 The function used to extra...
rmxyelxp 39387 Lemma for ~ frmx and ~ frm...
frmx 39388 The X sequence is a nonneg...
frmy 39389 The Y sequence is an integ...
rmxyval 39390 Main definition of the X a...
rmspecpos 39391 The discriminant used to d...
rmxycomplete 39392 The X and Y sequences take...
rmxynorm 39393 The X and Y sequences defi...
rmbaserp 39394 The base of exponentiation...
rmxyneg 39395 Negation law for X and Y s...
rmxyadd 39396 Addition formula for X and...
rmxy1 39397 Value of the X and Y seque...
rmxy0 39398 Value of the X and Y seque...
rmxneg 39399 Negation law (even functio...
rmx0 39400 Value of X sequence at 0. ...
rmx1 39401 Value of X sequence at 1. ...
rmxadd 39402 Addition formula for X seq...
rmyneg 39403 Negation formula for Y seq...
rmy0 39404 Value of Y sequence at 0. ...
rmy1 39405 Value of Y sequence at 1. ...
rmyadd 39406 Addition formula for Y seq...
rmxp1 39407 Special addition-of-1 form...
rmyp1 39408 Special addition of 1 form...
rmxm1 39409 Subtraction of 1 formula f...
rmym1 39410 Subtraction of 1 formula f...
rmxluc 39411 The X sequence is a Lucas ...
rmyluc 39412 The Y sequence is a Lucas ...
rmyluc2 39413 Lucas sequence property of...
rmxdbl 39414 "Double-angle formula" for...
rmydbl 39415 "Double-angle formula" for...
monotuz 39416 A function defined on an u...
monotoddzzfi 39417 A function which is odd an...
monotoddzz 39418 A function (given implicit...
oddcomabszz 39419 An odd function which take...
2nn0ind 39420 Induction on nonnegative i...
zindbi 39421 Inductively transfer a pro...
rmxypos 39422 For all nonnegative indice...
ltrmynn0 39423 The Y-sequence is strictly...
ltrmxnn0 39424 The X-sequence is strictly...
lermxnn0 39425 The X-sequence is monotoni...
rmxnn 39426 The X-sequence is defined ...
ltrmy 39427 The Y-sequence is strictly...
rmyeq0 39428 Y is zero only at zero. (...
rmyeq 39429 Y is one-to-one. (Contrib...
lermy 39430 Y is monotonic (non-strict...
rmynn 39431 ` rmY ` is positive for po...
rmynn0 39432 ` rmY ` is nonnegative for...
rmyabs 39433 ` rmY ` commutes with ` ab...
jm2.24nn 39434 X(n) is strictly greater t...
jm2.17a 39435 First half of lemma 2.17 o...
jm2.17b 39436 Weak form of the second ha...
jm2.17c 39437 Second half of lemma 2.17 ...
jm2.24 39438 Lemma 2.24 of [JonesMatija...
rmygeid 39439 Y(n) increases faster than...
congtr 39440 A wff of the form ` A || (...
congadd 39441 If two pairs of numbers ar...
congmul 39442 If two pairs of numbers ar...
congsym 39443 Congruence mod ` A ` is a ...
congneg 39444 If two integers are congru...
congsub 39445 If two pairs of numbers ar...
congid 39446 Every integer is congruent...
mzpcong 39447 Polynomials commute with c...
congrep 39448 Every integer is congruent...
congabseq 39449 If two integers are congru...
acongid 39450 A wff like that in this th...
acongsym 39451 Symmetry of alternating co...
acongneg2 39452 Negate right side of alter...
acongtr 39453 Transitivity of alternatin...
acongeq12d 39454 Substitution deduction for...
acongrep 39455 Every integer is alternati...
fzmaxdif 39456 Bound on the difference be...
fzneg 39457 Reflection of a finite ran...
acongeq 39458 Two numbers in the fundame...
dvdsacongtr 39459 Alternating congruence pas...
coprmdvdsb 39460 Multiplication by a coprim...
modabsdifz 39461 Divisibility in terms of m...
dvdsabsmod0 39462 Divisibility in terms of m...
jm2.18 39463 Theorem 2.18 of [JonesMati...
jm2.19lem1 39464 Lemma for ~ jm2.19 . X an...
jm2.19lem2 39465 Lemma for ~ jm2.19 . (Con...
jm2.19lem3 39466 Lemma for ~ jm2.19 . (Con...
jm2.19lem4 39467 Lemma for ~ jm2.19 . Exte...
jm2.19 39468 Lemma 2.19 of [JonesMatija...
jm2.21 39469 Lemma for ~ jm2.20nn . Ex...
jm2.22 39470 Lemma for ~ jm2.20nn . Ap...
jm2.23 39471 Lemma for ~ jm2.20nn . Tr...
jm2.20nn 39472 Lemma 2.20 of [JonesMatija...
jm2.25lem1 39473 Lemma for ~ jm2.26 . (Con...
jm2.25 39474 Lemma for ~ jm2.26 . Rema...
jm2.26a 39475 Lemma for ~ jm2.26 . Reve...
jm2.26lem3 39476 Lemma for ~ jm2.26 . Use ...
jm2.26 39477 Lemma 2.26 of [JonesMatija...
jm2.15nn0 39478 Lemma 2.15 of [JonesMatija...
jm2.16nn0 39479 Lemma 2.16 of [JonesMatija...
jm2.27a 39480 Lemma for ~ jm2.27 . Reve...
jm2.27b 39481 Lemma for ~ jm2.27 . Expa...
jm2.27c 39482 Lemma for ~ jm2.27 . Forw...
jm2.27 39483 Lemma 2.27 of [JonesMatija...
jm2.27dlem1 39484 Lemma for ~ rmydioph . Su...
jm2.27dlem2 39485 Lemma for ~ rmydioph . Th...
jm2.27dlem3 39486 Lemma for ~ rmydioph . In...
jm2.27dlem4 39487 Lemma for ~ rmydioph . In...
jm2.27dlem5 39488 Lemma for ~ rmydioph . Us...
rmydioph 39489 ~ jm2.27 restated in terms...
rmxdiophlem 39490 X can be expressed in term...
rmxdioph 39491 X is a Diophantine functio...
jm3.1lem1 39492 Lemma for ~ jm3.1 . (Cont...
jm3.1lem2 39493 Lemma for ~ jm3.1 . (Cont...
jm3.1lem3 39494 Lemma for ~ jm3.1 . (Cont...
jm3.1 39495 Diophantine expression for...
expdiophlem1 39496 Lemma for ~ expdioph . Fu...
expdiophlem2 39497 Lemma for ~ expdioph . Ex...
expdioph 39498 The exponential function i...
setindtr 39499 Set induction for sets con...
setindtrs 39500 Set induction scheme witho...
dford3lem1 39501 Lemma for ~ dford3 . (Con...
dford3lem2 39502 Lemma for ~ dford3 . (Con...
dford3 39503 Ordinals are precisely the...
dford4 39504 ~ dford3 expressed in prim...
wopprc 39505 Unrelated: Wiener pairs t...
rpnnen3lem 39506 Lemma for ~ rpnnen3 . (Co...
rpnnen3 39507 Dedekind cut injection of ...
axac10 39508 Characterization of choice...
harinf 39509 The Hartogs number of an i...
wdom2d2 39510 Deduction for weak dominan...
ttac 39511 Tarski's theorem about cho...
pw2f1ocnv 39512 Define a bijection between...
pw2f1o2 39513 Define a bijection between...
pw2f1o2val 39514 Function value of the ~ pw...
pw2f1o2val2 39515 Membership in a mapped set...
soeq12d 39516 Equality deduction for tot...
freq12d 39517 Equality deduction for fou...
weeq12d 39518 Equality deduction for wel...
limsuc2 39519 Limit ordinals in the sens...
wepwsolem 39520 Transfer an ordering on ch...
wepwso 39521 A well-ordering induces a ...
dnnumch1 39522 Define an enumeration of a...
dnnumch2 39523 Define an enumeration (wea...
dnnumch3lem 39524 Value of the ordinal injec...
dnnumch3 39525 Define an injection from a...
dnwech 39526 Define a well-ordering fro...
fnwe2val 39527 Lemma for ~ fnwe2 . Subst...
fnwe2lem1 39528 Lemma for ~ fnwe2 . Subst...
fnwe2lem2 39529 Lemma for ~ fnwe2 . An el...
fnwe2lem3 39530 Lemma for ~ fnwe2 . Trich...
fnwe2 39531 A well-ordering can be con...
aomclem1 39532 Lemma for ~ dfac11 . This...
aomclem2 39533 Lemma for ~ dfac11 . Succ...
aomclem3 39534 Lemma for ~ dfac11 . Succ...
aomclem4 39535 Lemma for ~ dfac11 . Limi...
aomclem5 39536 Lemma for ~ dfac11 . Comb...
aomclem6 39537 Lemma for ~ dfac11 . Tran...
aomclem7 39538 Lemma for ~ dfac11 . ` ( R...
aomclem8 39539 Lemma for ~ dfac11 . Perf...
dfac11 39540 The right-hand side of thi...
kelac1 39541 Kelley's choice, basic for...
kelac2lem 39542 Lemma for ~ kelac2 and ~ d...
kelac2 39543 Kelley's choice, most comm...
dfac21 39544 Tychonoff's theorem is a c...
islmodfg 39547 Property of a finitely gen...
islssfg 39548 Property of a finitely gen...
islssfg2 39549 Property of a finitely gen...
islssfgi 39550 Finitely spanned subspaces...
fglmod 39551 Finitely generated left mo...
lsmfgcl 39552 The sum of two finitely ge...
islnm 39555 Property of being a Noethe...
islnm2 39556 Property of being a Noethe...
lnmlmod 39557 A Noetherian left module i...
lnmlssfg 39558 A submodule of Noetherian ...
lnmlsslnm 39559 All submodules of a Noethe...
lnmfg 39560 A Noetherian left module i...
kercvrlsm 39561 The domain of a linear fun...
lmhmfgima 39562 A homomorphism maps finite...
lnmepi 39563 Epimorphic images of Noeth...
lmhmfgsplit 39564 If the kernel and range of...
lmhmlnmsplit 39565 If the kernel and range of...
lnmlmic 39566 Noetherian is an invariant...
pwssplit4 39567 Splitting for structure po...
filnm 39568 Finite left modules are No...
pwslnmlem0 39569 Zeroeth powers are Noether...
pwslnmlem1 39570 First powers are Noetheria...
pwslnmlem2 39571 A sum of powers is Noether...
pwslnm 39572 Finite powers of Noetheria...
unxpwdom3 39573 Weaker version of ~ unxpwd...
pwfi2f1o 39574 The ~ pw2f1o bijection rel...
pwfi2en 39575 Finitely supported indicat...
frlmpwfi 39576 Formal linear combinations...
gicabl 39577 Being Abelian is a group i...
imasgim 39578 A relabeling of the elemen...
isnumbasgrplem1 39579 A set which is equipollent...
harn0 39580 The Hartogs number of a se...
numinfctb 39581 A numerable infinite set c...
isnumbasgrplem2 39582 If the (to be thought of a...
isnumbasgrplem3 39583 Every nonempty numerable s...
isnumbasabl 39584 A set is numerable iff it ...
isnumbasgrp 39585 A set is numerable iff it ...
dfacbasgrp 39586 A choice equivalent in abs...
islnr 39589 Property of a left-Noether...
lnrring 39590 Left-Noetherian rings are ...
lnrlnm 39591 Left-Noetherian rings have...
islnr2 39592 Property of being a left-N...
islnr3 39593 Relate left-Noetherian rin...
lnr2i 39594 Given an ideal in a left-N...
lpirlnr 39595 Left principal ideal rings...
lnrfrlm 39596 Finite-dimensional free mo...
lnrfg 39597 Finitely-generated modules...
lnrfgtr 39598 A submodule of a finitely ...
hbtlem1 39601 Value of the leading coeff...
hbtlem2 39602 Leading coefficient ideals...
hbtlem7 39603 Functionality of leading c...
hbtlem4 39604 The leading ideal function...
hbtlem3 39605 The leading ideal function...
hbtlem5 39606 The leading ideal function...
hbtlem6 39607 There is a finite set of p...
hbt 39608 The Hilbert Basis Theorem ...
dgrsub2 39613 Subtracting two polynomial...
elmnc 39614 Property of a monic polyno...
mncply 39615 A monic polynomial is a po...
mnccoe 39616 A monic polynomial has lea...
mncn0 39617 A monic polynomial is not ...
dgraaval 39622 Value of the degree functi...
dgraalem 39623 Properties of the degree o...
dgraacl 39624 Closure of the degree func...
dgraaf 39625 Degree function on algebra...
dgraaub 39626 Upper bound on degree of a...
dgraa0p 39627 A rational polynomial of d...
mpaaeu 39628 An algebraic number has ex...
mpaaval 39629 Value of the minimal polyn...
mpaalem 39630 Properties of the minimal ...
mpaacl 39631 Minimal polynomial is a po...
mpaadgr 39632 Minimal polynomial has deg...
mpaaroot 39633 The minimal polynomial of ...
mpaamn 39634 Minimal polynomial is moni...
itgoval 39639 Value of the integral-over...
aaitgo 39640 The standard algebraic num...
itgoss 39641 An integral element is int...
itgocn 39642 All integral elements are ...
cnsrexpcl 39643 Exponentiation is closed i...
fsumcnsrcl 39644 Finite sums are closed in ...
cnsrplycl 39645 Polynomials are closed in ...
rgspnval 39646 Value of the ring-span of ...
rgspncl 39647 The ring-span of a set is ...
rgspnssid 39648 The ring-span of a set con...
rgspnmin 39649 The ring-span is contained...
rgspnid 39650 The span of a subring is i...
rngunsnply 39651 Adjoining one element to a...
flcidc 39652 Finite linear combinations...
algstr 39655 Lemma to shorten proofs of...
algbase 39656 The base set of a construc...
algaddg 39657 The additive operation of ...
algmulr 39658 The multiplicative operati...
algsca 39659 The set of scalars of a co...
algvsca 39660 The scalar product operati...
mendval 39661 Value of the module endomo...
mendbas 39662 Base set of the module end...
mendplusgfval 39663 Addition in the module end...
mendplusg 39664 A specific addition in the...
mendmulrfval 39665 Multiplication in the modu...
mendmulr 39666 A specific multiplication ...
mendsca 39667 The module endomorphism al...
mendvscafval 39668 Scalar multiplication in t...
mendvsca 39669 A specific scalar multipli...
mendring 39670 The module endomorphism al...
mendlmod 39671 The module endomorphism al...
mendassa 39672 The module endomorphism al...
idomrootle 39673 No element of an integral ...
idomodle 39674 Limit on the number of ` N...
fiuneneq 39675 Two finite sets of equal s...
idomsubgmo 39676 The units of an integral d...
proot1mul 39677 Any primitive ` N ` -th ro...
proot1hash 39678 If an integral domain has ...
proot1ex 39679 The complex field has prim...
isdomn3 39682 Nonzero elements form a mu...
mon1pid 39683 Monicity and degree of the...
mon1psubm 39684 Monic polynomials are a mu...
deg1mhm 39685 Homomorphic property of th...
cytpfn 39686 Functionality of the cyclo...
cytpval 39687 Substitutions for the Nth ...
fgraphopab 39688 Express a function as a su...
fgraphxp 39689 Express a function as a su...
hausgraph 39690 The graph of a continuous ...
iocunico 39695 Split an open interval int...
iocinico 39696 The intersection of two se...
iocmbl 39697 An open-below, closed-abov...
cnioobibld 39698 A bounded, continuous func...
itgpowd 39699 The integral of a monomial...
arearect 39700 The area of a rectangle wh...
areaquad 39701 The area of a quadrilatera...
ifpan123g 39702 Conjunction of conditional...
ifpan23 39703 Conjunction of conditional...
ifpdfor2 39704 Define or in terms of cond...
ifporcor 39705 Corollary of commutation o...
ifpdfan2 39706 Define and with conditiona...
ifpancor 39707 Corollary of commutation o...
ifpdfor 39708 Define or in terms of cond...
ifpdfan 39709 Define and with conditiona...
ifpbi2 39710 Equivalence theorem for co...
ifpbi3 39711 Equivalence theorem for co...
ifpim1 39712 Restate implication as con...
ifpnot 39713 Restate negated wff as con...
ifpid2 39714 Restate wff as conditional...
ifpim2 39715 Restate implication as con...
ifpbi23 39716 Equivalence theorem for co...
ifpdfbi 39717 Define biimplication as co...
ifpbiidcor 39718 Restatement of ~ biid . (...
ifpbicor 39719 Corollary of commutation o...
ifpxorcor 39720 Corollary of commutation o...
ifpbi1 39721 Equivalence theorem for co...
ifpnot23 39722 Negation of conditional lo...
ifpnotnotb 39723 Factor conditional logic o...
ifpnorcor 39724 Corollary of commutation o...
ifpnancor 39725 Corollary of commutation o...
ifpnot23b 39726 Negation of conditional lo...
ifpbiidcor2 39727 Restatement of ~ biid . (...
ifpnot23c 39728 Negation of conditional lo...
ifpnot23d 39729 Negation of conditional lo...
ifpdfnan 39730 Define nand as conditional...
ifpdfxor 39731 Define xor as conditional ...
ifpbi12 39732 Equivalence theorem for co...
ifpbi13 39733 Equivalence theorem for co...
ifpbi123 39734 Equivalence theorem for co...
ifpidg 39735 Restate wff as conditional...
ifpid3g 39736 Restate wff as conditional...
ifpid2g 39737 Restate wff as conditional...
ifpid1g 39738 Restate wff as conditional...
ifpim23g 39739 Restate implication as con...
ifpim3 39740 Restate implication as con...
ifpnim1 39741 Restate negated implicatio...
ifpim4 39742 Restate implication as con...
ifpnim2 39743 Restate negated implicatio...
ifpim123g 39744 Implication of conditional...
ifpim1g 39745 Implication of conditional...
ifp1bi 39746 Substitute the first eleme...
ifpbi1b 39747 When the first variable is...
ifpimimb 39748 Factor conditional logic o...
ifpororb 39749 Factor conditional logic o...
ifpananb 39750 Factor conditional logic o...
ifpnannanb 39751 Factor conditional logic o...
ifpor123g 39752 Disjunction of conditional...
ifpimim 39753 Consequnce of implication....
ifpbibib 39754 Factor conditional logic o...
ifpxorxorb 39755 Factor conditional logic o...
rp-fakeimass 39756 A special case where impli...
rp-fakeanorass 39757 A special case where a mix...
rp-fakeoranass 39758 A special case where a mix...
rp-fakeinunass 39759 A special case where a mix...
rp-fakeuninass 39760 A special case where a mix...
rp-isfinite5 39761 A set is said to be finite...
rp-isfinite6 39762 A set is said to be finite...
intabssd 39763 When for each element ` y ...
eu0 39764 There is only one empty se...
epelon2 39765 Over the ordinal numbers, ...
ontric3g 39766 For all ` x , y e. On ` , ...
dfsucon 39767 ` A ` is called a successo...
snen1g 39768 A singleton is equinumerou...
snen1el 39769 A singleton is equinumerou...
sn1dom 39770 A singleton is dominated b...
pr2dom 39771 An unordered pair is domin...
tr3dom 39772 An unordered triple is dom...
ensucne0 39773 A class equinumerous to a ...
ensucne0OLD 39774 A class equinumerous to a ...
nndomog 39775 Cardinal ordering agrees w...
dfom6 39776 Let ` _om ` be defined to ...
infordmin 39777 ` _om ` is the smallest in...
iscard4 39778 Two ways to express the pr...
iscard5 39779 Two ways to express the pr...
elrncard 39780 Let us define a cardinal n...
harsucnn 39781 The next cardinal after a ...
harval3 39782 ` ( har `` A ) ` is the le...
harval3on 39783 For any ordinal number ` A...
en2pr 39784 A class is equinumerous to...
pr2cv 39785 If an unordered pair is eq...
pr2el1 39786 If an unordered pair is eq...
pr2cv1 39787 If an unordered pair is eq...
pr2el2 39788 If an unordered pair is eq...
pr2cv2 39789 If an unordered pair is eq...
pren2 39790 An unordered pair is equin...
pr2eldif1 39791 If an unordered pair is eq...
pr2eldif2 39792 If an unordered pair is eq...
pren2d 39793 A pair of two distinct set...
aleph1min 39794 ` ( aleph `` 1o ) ` is the...
alephiso2 39795 ` aleph ` is a strictly or...
alephiso3 39796 ` aleph ` is a strictly or...
pwelg 39797 The powerclass is an eleme...
pwinfig 39798 The powerclass of an infin...
pwinfi2 39799 The powerclass of an infin...
pwinfi3 39800 The powerclass of an infin...
pwinfi 39801 The powerclass of an infin...
fipjust 39802 A definition of the finite...
cllem0 39803 The class of all sets with...
superficl 39804 The class of all supersets...
superuncl 39805 The class of all supersets...
ssficl 39806 The class of all subsets o...
ssuncl 39807 The class of all subsets o...
ssdifcl 39808 The class of all subsets o...
sssymdifcl 39809 The class of all subsets o...
fiinfi 39810 If two classes have the fi...
rababg 39811 Condition when restricted ...
elintabg 39812 Two ways of saying a set i...
elinintab 39813 Two ways of saying a set i...
elmapintrab 39814 Two ways to say a set is a...
elinintrab 39815 Two ways of saying a set i...
inintabss 39816 Upper bound on intersectio...
inintabd 39817 Value of the intersection ...
xpinintabd 39818 Value of the intersection ...
relintabex 39819 If the intersection of a c...
elcnvcnvintab 39820 Two ways of saying a set i...
relintab 39821 Value of the intersection ...
nonrel 39822 A non-relation is equal to...
elnonrel 39823 Only an ordered pair where...
cnvssb 39824 Subclass theorem for conve...
relnonrel 39825 The non-relation part of a...
cnvnonrel 39826 The converse of the non-re...
brnonrel 39827 A non-relation cannot rela...
dmnonrel 39828 The domain of the non-rela...
rnnonrel 39829 The range of the non-relat...
resnonrel 39830 A restriction of the non-r...
imanonrel 39831 An image under the non-rel...
cononrel1 39832 Composition with the non-r...
cononrel2 39833 Composition with the non-r...
elmapintab 39834 Two ways to say a set is a...
fvnonrel 39835 The function value of any ...
elinlem 39836 Two ways to say a set is a...
elcnvcnvlem 39837 Two ways to say a set is a...
cnvcnvintabd 39838 Value of the relationship ...
elcnvlem 39839 Two ways to say a set is a...
elcnvintab 39840 Two ways of saying a set i...
cnvintabd 39841 Value of the converse of t...
undmrnresiss 39842 Two ways of saying the ide...
reflexg 39843 Two ways of saying a relat...
cnvssco 39844 A condition weaker than re...
refimssco 39845 Reflexive relations are su...
cleq2lem 39846 Equality implies bijection...
cbvcllem 39847 Change of bound variable i...
clublem 39848 If a superset ` Y ` of ` X...
clss2lem 39849 The closure of a property ...
dfid7 39850 Definition of identity rel...
mptrcllem 39851 Show two versions of a clo...
cotrintab 39852 The intersection of a clas...
rclexi 39853 The reflexive closure of a...
rtrclexlem 39854 Existence of relation impl...
rtrclex 39855 The reflexive-transitive c...
trclubgNEW 39856 If a relation exists then ...
trclubNEW 39857 If a relation exists then ...
trclexi 39858 The transitive closure of ...
rtrclexi 39859 The reflexive-transitive c...
clrellem 39860 When the property ` ps ` h...
clcnvlem 39861 When ` A ` , an upper boun...
cnvtrucl0 39862 The converse of the trivia...
cnvrcl0 39863 The converse of the reflex...
cnvtrcl0 39864 The converse of the transi...
dmtrcl 39865 The domain of the transiti...
rntrcl 39866 The range of the transitiv...
dfrtrcl5 39867 Definition of reflexive-tr...
trcleq2lemRP 39868 Equality implies bijection...
al3im 39869 Version of ~ ax-4 for a ne...
intima0 39870 Two ways of expressing the...
elimaint 39871 Element of image of inters...
csbcog 39872 Distribute proper substitu...
cnviun 39873 Converse of indexed union....
imaiun1 39874 The image of an indexed un...
coiun1 39875 Composition with an indexe...
elintima 39876 Element of intersection of...
intimass 39877 The image under the inters...
intimass2 39878 The image under the inters...
intimag 39879 Requirement for the image ...
intimasn 39880 Two ways to express the im...
intimasn2 39881 Two ways to express the im...
ss2iundf 39882 Subclass theorem for index...
ss2iundv 39883 Subclass theorem for index...
cbviuneq12df 39884 Rule used to change the bo...
cbviuneq12dv 39885 Rule used to change the bo...
conrel1d 39886 Deduction about compositio...
conrel2d 39887 Deduction about compositio...
trrelind 39888 The intersection of transi...
xpintrreld 39889 The intersection of a tran...
restrreld 39890 The restriction of a trans...
trrelsuperreldg 39891 Concrete construction of a...
trficl 39892 The class of all transitiv...
cnvtrrel 39893 The converse of a transiti...
trrelsuperrel2dg 39894 Concrete construction of a...
dfrcl2 39897 Reflexive closure of a rel...
dfrcl3 39898 Reflexive closure of a rel...
dfrcl4 39899 Reflexive closure of a rel...
relexp2 39900 A set operated on by the r...
relexpnul 39901 If the domain and range of...
eliunov2 39902 Membership in the indexed ...
eltrclrec 39903 Membership in the indexed ...
elrtrclrec 39904 Membership in the indexed ...
briunov2 39905 Two classes related by the...
brmptiunrelexpd 39906 If two elements are connec...
fvmptiunrelexplb0d 39907 If the indexed union range...
fvmptiunrelexplb0da 39908 If the indexed union range...
fvmptiunrelexplb1d 39909 If the indexed union range...
brfvid 39910 If two elements are connec...
brfvidRP 39911 If two elements are connec...
fvilbd 39912 A set is a subset of its i...
fvilbdRP 39913 A set is a subset of its i...
brfvrcld 39914 If two elements are connec...
brfvrcld2 39915 If two elements are connec...
fvrcllb0d 39916 A restriction of the ident...
fvrcllb0da 39917 A restriction of the ident...
fvrcllb1d 39918 A set is a subset of its i...
brtrclrec 39919 Two classes related by the...
brrtrclrec 39920 Two classes related by the...
briunov2uz 39921 Two classes related by the...
eliunov2uz 39922 Membership in the indexed ...
ov2ssiunov2 39923 Any particular operator va...
relexp0eq 39924 The zeroth power of relati...
iunrelexp0 39925 Simplification of zeroth p...
relexpxpnnidm 39926 Any positive power of a cr...
relexpiidm 39927 Any power of any restricti...
relexpss1d 39928 The relational power of a ...
comptiunov2i 39929 The composition two indexe...
corclrcl 39930 The reflexive closure is i...
iunrelexpmin1 39931 The indexed union of relat...
relexpmulnn 39932 With exponents limited to ...
relexpmulg 39933 With ordered exponents, th...
trclrelexplem 39934 The union of relational po...
iunrelexpmin2 39935 The indexed union of relat...
relexp01min 39936 With exponents limited to ...
relexp1idm 39937 Repeated raising a relatio...
relexp0idm 39938 Repeated raising a relatio...
relexp0a 39939 Absorbtion law for zeroth ...
relexpxpmin 39940 The composition of powers ...
relexpaddss 39941 The composition of two pow...
iunrelexpuztr 39942 The indexed union of relat...
dftrcl3 39943 Transitive closure of a re...
brfvtrcld 39944 If two elements are connec...
fvtrcllb1d 39945 A set is a subset of its i...
trclfvcom 39946 The transitive closure of ...
cnvtrclfv 39947 The converse of the transi...
cotrcltrcl 39948 The transitive closure is ...
trclimalb2 39949 Lower bound for image unde...
brtrclfv2 39950 Two ways to indicate two e...
trclfvdecomr 39951 The transitive closure of ...
trclfvdecoml 39952 The transitive closure of ...
dmtrclfvRP 39953 The domain of the transiti...
rntrclfvRP 39954 The range of the transitiv...
rntrclfv 39955 The range of the transitiv...
dfrtrcl3 39956 Reflexive-transitive closu...
brfvrtrcld 39957 If two elements are connec...
fvrtrcllb0d 39958 A restriction of the ident...
fvrtrcllb0da 39959 A restriction of the ident...
fvrtrcllb1d 39960 A set is a subset of its i...
dfrtrcl4 39961 Reflexive-transitive closu...
corcltrcl 39962 The composition of the ref...
cortrcltrcl 39963 Composition with the refle...
corclrtrcl 39964 Composition with the refle...
cotrclrcl 39965 The composition of the ref...
cortrclrcl 39966 Composition with the refle...
cotrclrtrcl 39967 Composition with the refle...
cortrclrtrcl 39968 The reflexive-transitive c...
frege77d 39969 If the images of both ` { ...
frege81d 39970 If the image of ` U ` is a...
frege83d 39971 If the image of the union ...
frege96d 39972 If ` C ` follows ` A ` in ...
frege87d 39973 If the images of both ` { ...
frege91d 39974 If ` B ` follows ` A ` in ...
frege97d 39975 If ` A ` contains all elem...
frege98d 39976 If ` C ` follows ` A ` and...
frege102d 39977 If either ` A ` and ` C ` ...
frege106d 39978 If ` B ` follows ` A ` in ...
frege108d 39979 If either ` A ` and ` C ` ...
frege109d 39980 If ` A ` contains all elem...
frege114d 39981 If either ` R ` relates ` ...
frege111d 39982 If either ` A ` and ` C ` ...
frege122d 39983 If ` F ` is a function, ` ...
frege124d 39984 If ` F ` is a function, ` ...
frege126d 39985 If ` F ` is a function, ` ...
frege129d 39986 If ` F ` is a function and...
frege131d 39987 If ` F ` is a function and...
frege133d 39988 If ` F ` is a function and...
dfxor4 39989 Express exclusive-or in te...
dfxor5 39990 Express exclusive-or in te...
df3or2 39991 Express triple-or in terms...
df3an2 39992 Express triple-and in term...
nev 39993 Express that not every set...
0pssin 39994 Express that an intersecti...
rp-imass 39995 If the ` R ` -image of a c...
dfhe2 39998 The property of relation `...
dfhe3 39999 The property of relation `...
heeq12 40000 Equality law for relations...
heeq1 40001 Equality law for relations...
heeq2 40002 Equality law for relations...
sbcheg 40003 Distribute proper substitu...
hess 40004 Subclass law for relations...
xphe 40005 Any Cartesian product is h...
0he 40006 The empty relation is here...
0heALT 40007 The empty relation is here...
he0 40008 Any relation is hereditary...
unhe1 40009 The union of two relations...
snhesn 40010 Any singleton is hereditar...
idhe 40011 The identity relation is h...
psshepw 40012 The relation between sets ...
sshepw 40013 The relation between sets ...
rp-simp2-frege 40016 Simplification of triple c...
rp-simp2 40017 Simplification of triple c...
rp-frege3g 40018 Add antecedent to ~ ax-fre...
frege3 40019 Add antecedent to ~ ax-fre...
rp-misc1-frege 40020 Double-use of ~ ax-frege2 ...
rp-frege24 40021 Introducing an embedded an...
rp-frege4g 40022 Deduction related to distr...
frege4 40023 Special case of closed for...
frege5 40024 A closed form of ~ syl . ...
rp-7frege 40025 Distribute antecedent and ...
rp-4frege 40026 Elimination of a nested an...
rp-6frege 40027 Elimination of a nested an...
rp-8frege 40028 Eliminate antecedent when ...
rp-frege25 40029 Closed form for ~ a1dd . ...
frege6 40030 A closed form of ~ imim2d ...
axfrege8 40031 Swap antecedents. Identic...
frege7 40032 A closed form of ~ syl6 . ...
frege26 40034 Identical to ~ idd . Prop...
frege27 40035 We cannot (at the same tim...
frege9 40036 Closed form of ~ syl with ...
frege12 40037 A closed form of ~ com23 ....
frege11 40038 Elimination of a nested an...
frege24 40039 Closed form for ~ a1d . D...
frege16 40040 A closed form of ~ com34 ....
frege25 40041 Closed form for ~ a1dd . ...
frege18 40042 Closed form of a syllogism...
frege22 40043 A closed form of ~ com45 ....
frege10 40044 Result commuting anteceden...
frege17 40045 A closed form of ~ com3l ....
frege13 40046 A closed form of ~ com3r ....
frege14 40047 Closed form of a deduction...
frege19 40048 A closed form of ~ syl6 . ...
frege23 40049 Syllogism followed by rota...
frege15 40050 A closed form of ~ com4r ....
frege21 40051 Replace antecedent in ante...
frege20 40052 A closed form of ~ syl8 . ...
axfrege28 40053 Contraposition. Identical...
frege29 40055 Closed form of ~ con3d . ...
frege30 40056 Commuted, closed form of ~...
axfrege31 40057 Identical to ~ notnotr . ...
frege32 40059 Deduce ~ con1 from ~ con3 ...
frege33 40060 If ` ph ` or ` ps ` takes ...
frege34 40061 If as a conseqence of the ...
frege35 40062 Commuted, closed form of ~...
frege36 40063 The case in which ` ps ` i...
frege37 40064 If ` ch ` is a necessary c...
frege38 40065 Identical to ~ pm2.21 . P...
frege39 40066 Syllogism between ~ pm2.18...
frege40 40067 Anything implies ~ pm2.18 ...
axfrege41 40068 Identical to ~ notnot . A...
frege42 40070 Not not ~ id . Propositio...
frege43 40071 If there is a choice only ...
frege44 40072 Similar to a commuted ~ pm...
frege45 40073 Deduce ~ pm2.6 from ~ con1...
frege46 40074 If ` ps ` holds when ` ph ...
frege47 40075 Deduce consequence follows...
frege48 40076 Closed form of syllogism w...
frege49 40077 Closed form of deduction w...
frege50 40078 Closed form of ~ jaoi . P...
frege51 40079 Compare with ~ jaod . Pro...
axfrege52a 40080 Justification for ~ ax-fre...
frege52aid 40082 The case when the content ...
frege53aid 40083 Specialization of ~ frege5...
frege53a 40084 Lemma for ~ frege55a . Pr...
axfrege54a 40085 Justification for ~ ax-fre...
frege54cor0a 40087 Synonym for logical equiva...
frege54cor1a 40088 Reflexive equality. (Cont...
frege55aid 40089 Lemma for ~ frege57aid . ...
frege55lem1a 40090 Necessary deduction regard...
frege55lem2a 40091 Core proof of Proposition ...
frege55a 40092 Proposition 55 of [Frege18...
frege55cor1a 40093 Proposition 55 of [Frege18...
frege56aid 40094 Lemma for ~ frege57aid . ...
frege56a 40095 Proposition 56 of [Frege18...
frege57aid 40096 This is the all imporant f...
frege57a 40097 Analogue of ~ frege57aid ....
axfrege58a 40098 Identical to ~ anifp . Ju...
frege58acor 40100 Lemma for ~ frege59a . (C...
frege59a 40101 A kind of Aristotelian inf...
frege60a 40102 Swap antecedents of ~ ax-f...
frege61a 40103 Lemma for ~ frege65a . Pr...
frege62a 40104 A kind of Aristotelian inf...
frege63a 40105 Proposition 63 of [Frege18...
frege64a 40106 Lemma for ~ frege65a . Pr...
frege65a 40107 A kind of Aristotelian inf...
frege66a 40108 Swap antecedents of ~ freg...
frege67a 40109 Lemma for ~ frege68a . Pr...
frege68a 40110 Combination of applying a ...
axfrege52c 40111 Justification for ~ ax-fre...
frege52b 40113 The case when the content ...
frege53b 40114 Lemma for frege102 (via ~ ...
axfrege54c 40115 Reflexive equality of clas...
frege54b 40117 Reflexive equality of sets...
frege54cor1b 40118 Reflexive equality. (Cont...
frege55lem1b 40119 Necessary deduction regard...
frege55lem2b 40120 Lemma for ~ frege55b . Co...
frege55b 40121 Lemma for ~ frege57b . Pr...
frege56b 40122 Lemma for ~ frege57b . Pr...
frege57b 40123 Analogue of ~ frege57aid ....
axfrege58b 40124 If ` A. x ph ` is affirmed...
frege58bid 40126 If ` A. x ph ` is affirmed...
frege58bcor 40127 Lemma for ~ frege59b . (C...
frege59b 40128 A kind of Aristotelian inf...
frege60b 40129 Swap antecedents of ~ ax-f...
frege61b 40130 Lemma for ~ frege65b . Pr...
frege62b 40131 A kind of Aristotelian inf...
frege63b 40132 Lemma for ~ frege91 . Pro...
frege64b 40133 Lemma for ~ frege65b . Pr...
frege65b 40134 A kind of Aristotelian inf...
frege66b 40135 Swap antecedents of ~ freg...
frege67b 40136 Lemma for ~ frege68b . Pr...
frege68b 40137 Combination of applying a ...
frege53c 40138 Proposition 53 of [Frege18...
frege54cor1c 40139 Reflexive equality. (Cont...
frege55lem1c 40140 Necessary deduction regard...
frege55lem2c 40141 Core proof of Proposition ...
frege55c 40142 Proposition 55 of [Frege18...
frege56c 40143 Lemma for ~ frege57c . Pr...
frege57c 40144 Swap order of implication ...
frege58c 40145 Principle related to ~ sp ...
frege59c 40146 A kind of Aristotelian inf...
frege60c 40147 Swap antecedents of ~ freg...
frege61c 40148 Lemma for ~ frege65c . Pr...
frege62c 40149 A kind of Aristotelian inf...
frege63c 40150 Analogue of ~ frege63b . ...
frege64c 40151 Lemma for ~ frege65c . Pr...
frege65c 40152 A kind of Aristotelian inf...
frege66c 40153 Swap antecedents of ~ freg...
frege67c 40154 Lemma for ~ frege68c . Pr...
frege68c 40155 Combination of applying a ...
dffrege69 40156 If from the proposition th...
frege70 40157 Lemma for ~ frege72 . Pro...
frege71 40158 Lemma for ~ frege72 . Pro...
frege72 40159 If property ` A ` is hered...
frege73 40160 Lemma for ~ frege87 . Pro...
frege74 40161 If ` X ` has a property ` ...
frege75 40162 If from the proposition th...
dffrege76 40163 If from the two propositio...
frege77 40164 If ` Y ` follows ` X ` in ...
frege78 40165 Commuted form of of ~ freg...
frege79 40166 Distributed form of ~ freg...
frege80 40167 Add additional condition t...
frege81 40168 If ` X ` has a property ` ...
frege82 40169 Closed-form deduction base...
frege83 40170 Apply commuted form of ~ f...
frege84 40171 Commuted form of ~ frege81...
frege85 40172 Commuted form of ~ frege77...
frege86 40173 Conclusion about element o...
frege87 40174 If ` Z ` is a result of an...
frege88 40175 Commuted form of ~ frege87...
frege89 40176 One direction of ~ dffrege...
frege90 40177 Add antecedent to ~ frege8...
frege91 40178 Every result of an applica...
frege92 40179 Inference from ~ frege91 ....
frege93 40180 Necessary condition for tw...
frege94 40181 Looking one past a pair re...
frege95 40182 Looking one past a pair re...
frege96 40183 Every result of an applica...
frege97 40184 The property of following ...
frege98 40185 If ` Y ` follows ` X ` and...
dffrege99 40186 If ` Z ` is identical with...
frege100 40187 One direction of ~ dffrege...
frege101 40188 Lemma for ~ frege102 . Pr...
frege102 40189 If ` Z ` belongs to the ` ...
frege103 40190 Proposition 103 of [Frege1...
frege104 40191 Proposition 104 of [Frege1...
frege105 40192 Proposition 105 of [Frege1...
frege106 40193 Whatever follows ` X ` in ...
frege107 40194 Proposition 107 of [Frege1...
frege108 40195 If ` Y ` belongs to the ` ...
frege109 40196 The property of belonging ...
frege110 40197 Proposition 110 of [Frege1...
frege111 40198 If ` Y ` belongs to the ` ...
frege112 40199 Identity implies belonging...
frege113 40200 Proposition 113 of [Frege1...
frege114 40201 If ` X ` belongs to the ` ...
dffrege115 40202 If from the circumstance t...
frege116 40203 One direction of ~ dffrege...
frege117 40204 Lemma for ~ frege118 . Pr...
frege118 40205 Simplified application of ...
frege119 40206 Lemma for ~ frege120 . Pr...
frege120 40207 Simplified application of ...
frege121 40208 Lemma for ~ frege122 . Pr...
frege122 40209 If ` X ` is a result of an...
frege123 40210 Lemma for ~ frege124 . Pr...
frege124 40211 If ` X ` is a result of an...
frege125 40212 Lemma for ~ frege126 . Pr...
frege126 40213 If ` M ` follows ` Y ` in ...
frege127 40214 Communte antecedents of ~ ...
frege128 40215 Lemma for ~ frege129 . Pr...
frege129 40216 If the procedure ` R ` is ...
frege130 40217 Lemma for ~ frege131 . Pr...
frege131 40218 If the procedure ` R ` is ...
frege132 40219 Lemma for ~ frege133 . Pr...
frege133 40220 If the procedure ` R ` is ...
enrelmap 40221 The set of all possible re...
enrelmapr 40222 The set of all possible re...
enmappw 40223 The set of all mappings fr...
enmappwid 40224 The set of all mappings fr...
rfovd 40225 Value of the operator, ` (...
rfovfvd 40226 Value of the operator, ` (...
rfovfvfvd 40227 Value of the operator, ` (...
rfovcnvf1od 40228 Properties of the operator...
rfovcnvd 40229 Value of the converse of t...
rfovf1od 40230 The value of the operator,...
rfovcnvfvd 40231 Value of the converse of t...
fsovd 40232 Value of the operator, ` (...
fsovrfovd 40233 The operator which gives a...
fsovfvd 40234 Value of the operator, ` (...
fsovfvfvd 40235 Value of the operator, ` (...
fsovfd 40236 The operator, ` ( A O B ) ...
fsovcnvlem 40237 The ` O ` operator, which ...
fsovcnvd 40238 The value of the converse ...
fsovcnvfvd 40239 The value of the converse ...
fsovf1od 40240 The value of ` ( A O B ) `...
dssmapfvd 40241 Value of the duality opera...
dssmapfv2d 40242 Value of the duality opera...
dssmapfv3d 40243 Value of the duality opera...
dssmapnvod 40244 For any base set ` B ` the...
dssmapf1od 40245 For any base set ` B ` the...
dssmap2d 40246 For any base set ` B ` the...
sscon34b 40247 Relative complementation r...
rcompleq 40248 Two subclasses are equal i...
or3or 40249 Decompose disjunction into...
andi3or 40250 Distribute over triple dis...
uneqsn 40251 If a union of classes is e...
df3o2 40252 Ordinal 3 is the triplet c...
df3o3 40253 Ordinal 3 , fully expanded...
brfvimex 40254 If a binary relation holds...
brovmptimex 40255 If a binary relation holds...
brovmptimex1 40256 If a binary relation holds...
brovmptimex2 40257 If a binary relation holds...
brcoffn 40258 Conditions allowing the de...
brcofffn 40259 Conditions allowing the de...
brco2f1o 40260 Conditions allowing the de...
brco3f1o 40261 Conditions allowing the de...
ntrclsbex 40262 If (pseudo-)interior and (...
ntrclsrcomplex 40263 The relative complement of...
neik0imk0p 40264 Kuratowski's K0 axiom impl...
ntrk2imkb 40265 If an interior function is...
ntrkbimka 40266 If the interiors of disjoi...
ntrk0kbimka 40267 If the interiors of disjoi...
clsk3nimkb 40268 If the base set is not emp...
clsk1indlem0 40269 The ansatz closure functio...
clsk1indlem2 40270 The ansatz closure functio...
clsk1indlem3 40271 The ansatz closure functio...
clsk1indlem4 40272 The ansatz closure functio...
clsk1indlem1 40273 The ansatz closure functio...
clsk1independent 40274 For generalized closure fu...
neik0pk1imk0 40275 Kuratowski's K0' and K1 ax...
isotone1 40276 Two different ways to say ...
isotone2 40277 Two different ways to say ...
ntrk1k3eqk13 40278 An interior function is bo...
ntrclsf1o 40279 If (pseudo-)interior and (...
ntrclsnvobr 40280 If (pseudo-)interior and (...
ntrclsiex 40281 If (pseudo-)interior and (...
ntrclskex 40282 If (pseudo-)interior and (...
ntrclsfv1 40283 If (pseudo-)interior and (...
ntrclsfv2 40284 If (pseudo-)interior and (...
ntrclselnel1 40285 If (pseudo-)interior and (...
ntrclselnel2 40286 If (pseudo-)interior and (...
ntrclsfv 40287 The value of the interior ...
ntrclsfveq1 40288 If interior and closure fu...
ntrclsfveq2 40289 If interior and closure fu...
ntrclsfveq 40290 If interior and closure fu...
ntrclsss 40291 If interior and closure fu...
ntrclsneine0lem 40292 If (pseudo-)interior and (...
ntrclsneine0 40293 If (pseudo-)interior and (...
ntrclscls00 40294 If (pseudo-)interior and (...
ntrclsiso 40295 If (pseudo-)interior and (...
ntrclsk2 40296 An interior function is co...
ntrclskb 40297 The interiors of disjoint ...
ntrclsk3 40298 The intersection of interi...
ntrclsk13 40299 The interior of the inters...
ntrclsk4 40300 Idempotence of the interio...
ntrneibex 40301 If (pseudo-)interior and (...
ntrneircomplex 40302 The relative complement of...
ntrneif1o 40303 If (pseudo-)interior and (...
ntrneiiex 40304 If (pseudo-)interior and (...
ntrneinex 40305 If (pseudo-)interior and (...
ntrneicnv 40306 If (pseudo-)interior and (...
ntrneifv1 40307 If (pseudo-)interior and (...
ntrneifv2 40308 If (pseudo-)interior and (...
ntrneiel 40309 If (pseudo-)interior and (...
ntrneifv3 40310 The value of the neighbors...
ntrneineine0lem 40311 If (pseudo-)interior and (...
ntrneineine1lem 40312 If (pseudo-)interior and (...
ntrneifv4 40313 The value of the interior ...
ntrneiel2 40314 Membership in iterated int...
ntrneineine0 40315 If (pseudo-)interior and (...
ntrneineine1 40316 If (pseudo-)interior and (...
ntrneicls00 40317 If (pseudo-)interior and (...
ntrneicls11 40318 If (pseudo-)interior and (...
ntrneiiso 40319 If (pseudo-)interior and (...
ntrneik2 40320 An interior function is co...
ntrneix2 40321 An interior (closure) func...
ntrneikb 40322 The interiors of disjoint ...
ntrneixb 40323 The interiors (closures) o...
ntrneik3 40324 The intersection of interi...
ntrneix3 40325 The closure of the union o...
ntrneik13 40326 The interior of the inters...
ntrneix13 40327 The closure of the union o...
ntrneik4w 40328 Idempotence of the interio...
ntrneik4 40329 Idempotence of the interio...
clsneibex 40330 If (pseudo-)closure and (p...
clsneircomplex 40331 The relative complement of...
clsneif1o 40332 If a (pseudo-)closure func...
clsneicnv 40333 If a (pseudo-)closure func...
clsneikex 40334 If closure and neighborhoo...
clsneinex 40335 If closure and neighborhoo...
clsneiel1 40336 If a (pseudo-)closure func...
clsneiel2 40337 If a (pseudo-)closure func...
clsneifv3 40338 Value of the neighborhoods...
clsneifv4 40339 Value of the closure (inte...
neicvgbex 40340 If (pseudo-)neighborhood a...
neicvgrcomplex 40341 The relative complement of...
neicvgf1o 40342 If neighborhood and conver...
neicvgnvo 40343 If neighborhood and conver...
neicvgnvor 40344 If neighborhood and conver...
neicvgmex 40345 If the neighborhoods and c...
neicvgnex 40346 If the neighborhoods and c...
neicvgel1 40347 A subset being an element ...
neicvgel2 40348 The complement of a subset...
neicvgfv 40349 The value of the neighborh...
ntrrn 40350 The range of the interior ...
ntrf 40351 The interior function of a...
ntrf2 40352 The interior function is a...
ntrelmap 40353 The interior function is a...
clsf2 40354 The closure function is a ...
clselmap 40355 The closure function is a ...
dssmapntrcls 40356 The interior and closure o...
dssmapclsntr 40357 The closure and interior o...
gneispa 40358 Each point ` p ` of the ne...
gneispb 40359 Given a neighborhood ` N `...
gneispace2 40360 The predicate that ` F ` i...
gneispace3 40361 The predicate that ` F ` i...
gneispace 40362 The predicate that ` F ` i...
gneispacef 40363 A generic neighborhood spa...
gneispacef2 40364 A generic neighborhood spa...
gneispacefun 40365 A generic neighborhood spa...
gneispacern 40366 A generic neighborhood spa...
gneispacern2 40367 A generic neighborhood spa...
gneispace0nelrn 40368 A generic neighborhood spa...
gneispace0nelrn2 40369 A generic neighborhood spa...
gneispace0nelrn3 40370 A generic neighborhood spa...
gneispaceel 40371 Every neighborhood of a po...
gneispaceel2 40372 Every neighborhood of a po...
gneispacess 40373 All supersets of a neighbo...
gneispacess2 40374 All supersets of a neighbo...
k0004lem1 40375 Application of ~ ssin to r...
k0004lem2 40376 A mapping with a particula...
k0004lem3 40377 When the value of a mappin...
k0004val 40378 The topological simplex of...
k0004ss1 40379 The topological simplex of...
k0004ss2 40380 The topological simplex of...
k0004ss3 40381 The topological simplex of...
k0004val0 40382 The topological simplex of...
inductionexd 40383 Simple induction example. ...
wwlemuld 40384 Natural deduction form of ...
leeq1d 40385 Specialization of ~ breq1d...
leeq2d 40386 Specialization of ~ breq2d...
absmulrposd 40387 Specialization of absmuld ...
imadisjld 40388 Natural dduction form of o...
imadisjlnd 40389 Natural deduction form of ...
wnefimgd 40390 The image of a mapping fro...
fco2d 40391 Natural deduction form of ...
wfximgfd 40392 The value of a function on...
extoimad 40393 If |f(x)| <= C for all x t...
imo72b2lem0 40394 Lemma for ~ imo72b2 . (Co...
suprleubrd 40395 Natural deduction form of ...
imo72b2lem2 40396 Lemma for ~ imo72b2 . (Co...
syldbl2 40397 Stacked hypotheseis implie...
suprlubrd 40398 Natural deduction form of ...
imo72b2lem1 40399 Lemma for ~ imo72b2 . (Co...
lemuldiv3d 40400 'Less than or equal to' re...
lemuldiv4d 40401 'Less than or equal to' re...
rspcdvinvd 40402 If something is true for a...
imo72b2 40403 IMO 1972 B2. (14th Intern...
int-addcomd 40404 AdditionCommutativity gene...
int-addassocd 40405 AdditionAssociativity gene...
int-addsimpd 40406 AdditionSimplification gen...
int-mulcomd 40407 MultiplicationCommutativit...
int-mulassocd 40408 MultiplicationAssociativit...
int-mulsimpd 40409 MultiplicationSimplificati...
int-leftdistd 40410 AdditionMultiplicationLeft...
int-rightdistd 40411 AdditionMultiplicationRigh...
int-sqdefd 40412 SquareDefinition generator...
int-mul11d 40413 First MultiplicationOne ge...
int-mul12d 40414 Second MultiplicationOne g...
int-add01d 40415 First AdditionZero generat...
int-add02d 40416 Second AdditionZero genera...
int-sqgeq0d 40417 SquareGEQZero generator ru...
int-eqprincd 40418 PrincipleOfEquality genera...
int-eqtransd 40419 EqualityTransitivity gener...
int-eqmvtd 40420 EquMoveTerm generator rule...
int-eqineqd 40421 EquivalenceImpliesDoubleIn...
int-ineqmvtd 40422 IneqMoveTerm generator rul...
int-ineq1stprincd 40423 FirstPrincipleOfInequality...
int-ineq2ndprincd 40424 SecondPrincipleOfInequalit...
int-ineqtransd 40425 InequalityTransitivity gen...
unitadd 40426 Theorem used in conjunctio...
gsumws3 40427 Valuation of a length 3 wo...
gsumws4 40428 Valuation of a length 4 wo...
amgm2d 40429 Arithmetic-geometric mean ...
amgm3d 40430 Arithmetic-geometric mean ...
amgm4d 40431 Arithmetic-geometric mean ...
spALT 40432 ~ sp can be proven from th...
elnelneqd 40433 Two classes are not equal ...
elnelneq2d 40434 Two classes are not equal ...
rr-spce 40435 Prove an existential. (Co...
rexlimdvaacbv 40436 Unpack a restricted existe...
rexlimddvcbv 40437 Unpack a restricted existe...
rr-elrnmpt3d 40438 Elementhood in an image se...
rr-phpd 40439 Equivalent of ~ php withou...
suceqd 40440 Deduction associated with ...
tfindsd 40441 Deduction associated with ...
gru0eld 40442 A nonempty Grothendieck un...
grusucd 40443 Grothendieck universes are...
r1rankcld 40444 Any rank of the cumulative...
grur1cld 40445 Grothendieck universes are...
grurankcld 40446 Grothendieck universes are...
grurankrcld 40447 If a Grothendieck universe...
scotteqd 40450 Equality theorem for the S...
scotteq 40451 Closed form of ~ scotteqd ...
nfscott 40452 Bound-variable hypothesis ...
scottabf 40453 Value of the Scott operati...
scottab 40454 Value of the Scott operati...
scottabes 40455 Value of the Scott operati...
scottss 40456 Scott's trick produces a s...
elscottab 40457 An element of the output o...
scottex2 40458 ~ scottex expressed using ...
scotteld 40459 The Scott operation sends ...
scottelrankd 40460 Property of a Scott's tric...
scottrankd 40461 Rank of a nonempty Scott's...
gruscottcld 40462 If a Grothendieck universe...
dfcoll2 40465 Alternate definition of th...
colleq12d 40466 Equality theorem for the c...
colleq1 40467 Equality theorem for the c...
colleq2 40468 Equality theorem for the c...
nfcoll 40469 Bound-variable hypothesis ...
collexd 40470 The output of the collecti...
cpcolld 40471 Property of the collection...
cpcoll2d 40472 ~ cpcolld with an extra ex...
grucollcld 40473 A Grothendieck universe co...
ismnu 40474 The hypothesis of this the...
mnuop123d 40475 Operations of a minimal un...
mnussd 40476 Minimal universes are clos...
mnuss2d 40477 ~ mnussd with arguments pr...
mnu0eld 40478 A nonempty minimal univers...
mnuop23d 40479 Second and third operation...
mnupwd 40480 Minimal universes are clos...
mnusnd 40481 Minimal universes are clos...
mnuprssd 40482 A minimal universe contain...
mnuprss2d 40483 Special case of ~ mnuprssd...
mnuop3d 40484 Third operation of a minim...
mnuprdlem1 40485 Lemma for ~ mnuprd . (Con...
mnuprdlem2 40486 Lemma for ~ mnuprd . (Con...
mnuprdlem3 40487 Lemma for ~ mnuprd . (Con...
mnuprdlem4 40488 Lemma for ~ mnuprd . Gene...
mnuprd 40489 Minimal universes are clos...
mnuunid 40490 Minimal universes are clos...
mnuund 40491 Minimal universes are clos...
mnutrcld 40492 Minimal universes contain ...
mnutrd 40493 Minimal universes are tran...
mnurndlem1 40494 Lemma for ~ mnurnd . (Con...
mnurndlem2 40495 Lemma for ~ mnurnd . Dedu...
mnurnd 40496 Minimal universes contain ...
mnugrud 40497 Minimal universes are Grot...
grumnudlem 40498 Lemma for ~ grumnud . (Co...
grumnud 40499 Grothendieck universes are...
grumnueq 40500 The class of Grothendieck ...
expandan 40501 Expand conjunction to prim...
expandexn 40502 Expand an existential quan...
expandral 40503 Expand a restricted univer...
expandrexn 40504 Expand a restricted existe...
expandrex 40505 Expand a restricted existe...
expanduniss 40506 Expand ` U. A C_ B ` to pr...
ismnuprim 40507 Express the predicate on `...
rr-grothprimbi 40508 Express "every set is cont...
inagrud 40509 Inaccessible levels of the...
inaex 40510 Assuming the Tarski-Grothe...
gruex 40511 Assuming the Tarski-Grothe...
rr-groth 40512 An equivalent of ~ ax-grot...
rr-grothprim 40513 An equivalent of ~ ax-grot...
nanorxor 40514 'nand' is equivalent to th...
undisjrab 40515 Union of two disjoint rest...
iso0 40516 The empty set is an ` R , ...
ssrecnpr 40517 ` RR ` is a subset of both...
seff 40518 Let set ` S ` be the real ...
sblpnf 40519 The infinity ball in the a...
prmunb2 40520 The primes are unbounded. ...
dvgrat 40521 Ratio test for divergence ...
cvgdvgrat 40522 Ratio test for convergence...
radcnvrat 40523 Let ` L ` be the limit, if...
reldvds 40524 The divides relation is in...
nznngen 40525 All positive integers in t...
nzss 40526 The set of multiples of _m...
nzin 40527 The intersection of the se...
nzprmdif 40528 Subtract one prime's multi...
hashnzfz 40529 Special case of ~ hashdvds...
hashnzfz2 40530 Special case of ~ hashnzfz...
hashnzfzclim 40531 As the upper bound ` K ` o...
caofcan 40532 Transfer a cancellation la...
ofsubid 40533 Function analogue of ~ sub...
ofmul12 40534 Function analogue of ~ mul...
ofdivrec 40535 Function analogue of ~ div...
ofdivcan4 40536 Function analogue of ~ div...
ofdivdiv2 40537 Function analogue of ~ div...
lhe4.4ex1a 40538 Example of the Fundamental...
dvsconst 40539 Derivative of a constant f...
dvsid 40540 Derivative of the identity...
dvsef 40541 Derivative of the exponent...
expgrowthi 40542 Exponential growth and dec...
dvconstbi 40543 The derivative of a functi...
expgrowth 40544 Exponential growth and dec...
bccval 40547 Value of the generalized b...
bcccl 40548 Closure of the generalized...
bcc0 40549 The generalized binomial c...
bccp1k 40550 Generalized binomial coeff...
bccm1k 40551 Generalized binomial coeff...
bccn0 40552 Generalized binomial coeff...
bccn1 40553 Generalized binomial coeff...
bccbc 40554 The binomial coefficient a...
uzmptshftfval 40555 When ` F ` is a maps-to fu...
dvradcnv2 40556 The radius of convergence ...
binomcxplemwb 40557 Lemma for ~ binomcxp . Th...
binomcxplemnn0 40558 Lemma for ~ binomcxp . Wh...
binomcxplemrat 40559 Lemma for ~ binomcxp . As...
binomcxplemfrat 40560 Lemma for ~ binomcxp . ~ b...
binomcxplemradcnv 40561 Lemma for ~ binomcxp . By...
binomcxplemdvbinom 40562 Lemma for ~ binomcxp . By...
binomcxplemcvg 40563 Lemma for ~ binomcxp . Th...
binomcxplemdvsum 40564 Lemma for ~ binomcxp . Th...
binomcxplemnotnn0 40565 Lemma for ~ binomcxp . Wh...
binomcxp 40566 Generalize the binomial th...
pm10.12 40567 Theorem *10.12 in [Whitehe...
pm10.14 40568 Theorem *10.14 in [Whitehe...
pm10.251 40569 Theorem *10.251 in [Whiteh...
pm10.252 40570 Theorem *10.252 in [Whiteh...
pm10.253 40571 Theorem *10.253 in [Whiteh...
albitr 40572 Theorem *10.301 in [Whiteh...
pm10.42 40573 Theorem *10.42 in [Whitehe...
pm10.52 40574 Theorem *10.52 in [Whitehe...
pm10.53 40575 Theorem *10.53 in [Whitehe...
pm10.541 40576 Theorem *10.541 in [Whiteh...
pm10.542 40577 Theorem *10.542 in [Whiteh...
pm10.55 40578 Theorem *10.55 in [Whitehe...
pm10.56 40579 Theorem *10.56 in [Whitehe...
pm10.57 40580 Theorem *10.57 in [Whitehe...
2alanimi 40581 Removes two universal quan...
2al2imi 40582 Removes two universal quan...
pm11.11 40583 Theorem *11.11 in [Whitehe...
pm11.12 40584 Theorem *11.12 in [Whitehe...
19.21vv 40585 Compare Theorem *11.3 in [...
2alim 40586 Theorem *11.32 in [Whitehe...
2albi 40587 Theorem *11.33 in [Whitehe...
2exim 40588 Theorem *11.34 in [Whitehe...
2exbi 40589 Theorem *11.341 in [Whiteh...
spsbce-2 40590 Theorem *11.36 in [Whitehe...
19.33-2 40591 Theorem *11.421 in [Whiteh...
19.36vv 40592 Theorem *11.43 in [Whitehe...
19.31vv 40593 Theorem *11.44 in [Whitehe...
19.37vv 40594 Theorem *11.46 in [Whitehe...
19.28vv 40595 Theorem *11.47 in [Whitehe...
pm11.52 40596 Theorem *11.52 in [Whitehe...
aaanv 40597 Theorem *11.56 in [Whitehe...
pm11.57 40598 Theorem *11.57 in [Whitehe...
pm11.58 40599 Theorem *11.58 in [Whitehe...
pm11.59 40600 Theorem *11.59 in [Whitehe...
pm11.6 40601 Theorem *11.6 in [Whitehea...
pm11.61 40602 Theorem *11.61 in [Whitehe...
pm11.62 40603 Theorem *11.62 in [Whitehe...
pm11.63 40604 Theorem *11.63 in [Whitehe...
pm11.7 40605 Theorem *11.7 in [Whitehea...
pm11.71 40606 Theorem *11.71 in [Whitehe...
sbeqal1 40607 If ` x = y ` always implie...
sbeqal1i 40608 Suppose you know ` x = y `...
sbeqal2i 40609 If ` x = y ` implies ` x =...
axc5c4c711 40610 Proof of a theorem that ca...
axc5c4c711toc5 40611 Rederivation of ~ sp from ...
axc5c4c711toc4 40612 Rederivation of ~ axc4 fro...
axc5c4c711toc7 40613 Rederivation of ~ axc7 fro...
axc5c4c711to11 40614 Rederivation of ~ ax-11 fr...
axc11next 40615 This theorem shows that, g...
pm13.13a 40616 One result of theorem *13....
pm13.13b 40617 Theorem *13.13 in [Whitehe...
pm13.14 40618 Theorem *13.14 in [Whitehe...
pm13.192 40619 Theorem *13.192 in [Whiteh...
pm13.193 40620 Theorem *13.193 in [Whiteh...
pm13.194 40621 Theorem *13.194 in [Whiteh...
pm13.195 40622 Theorem *13.195 in [Whiteh...
pm13.196a 40623 Theorem *13.196 in [Whiteh...
2sbc6g 40624 Theorem *13.21 in [Whitehe...
2sbc5g 40625 Theorem *13.22 in [Whitehe...
iotain 40626 Equivalence between two di...
iotaexeu 40627 The iota class exists. Th...
iotasbc 40628 Definition *14.01 in [Whit...
iotasbc2 40629 Theorem *14.111 in [Whiteh...
pm14.12 40630 Theorem *14.12 in [Whitehe...
pm14.122a 40631 Theorem *14.122 in [Whiteh...
pm14.122b 40632 Theorem *14.122 in [Whiteh...
pm14.122c 40633 Theorem *14.122 in [Whiteh...
pm14.123a 40634 Theorem *14.123 in [Whiteh...
pm14.123b 40635 Theorem *14.123 in [Whiteh...
pm14.123c 40636 Theorem *14.123 in [Whiteh...
pm14.18 40637 Theorem *14.18 in [Whitehe...
iotaequ 40638 Theorem *14.2 in [Whitehea...
iotavalb 40639 Theorem *14.202 in [Whiteh...
iotasbc5 40640 Theorem *14.205 in [Whiteh...
pm14.24 40641 Theorem *14.24 in [Whitehe...
iotavalsb 40642 Theorem *14.242 in [Whiteh...
sbiota1 40643 Theorem *14.25 in [Whitehe...
sbaniota 40644 Theorem *14.26 in [Whitehe...
eubiOLD 40645 Obsolete proof of ~ eubi a...
iotasbcq 40646 Theorem *14.272 in [Whiteh...
elnev 40647 Any set that contains one ...
rusbcALT 40648 A version of Russell's par...
compeq 40649 Equality between two ways ...
compne 40650 The complement of ` A ` is...
compab 40651 Two ways of saying "the co...
conss2 40652 Contrapositive law for sub...
conss1 40653 Contrapositive law for sub...
ralbidar 40654 More general form of ~ ral...
rexbidar 40655 More general form of ~ rex...
dropab1 40656 Theorem to aid use of the ...
dropab2 40657 Theorem to aid use of the ...
ipo0 40658 If the identity relation p...
ifr0 40659 A class that is founded by...
ordpss 40660 ~ ordelpss with an anteced...
fvsb 40661 Explicit substitution of a...
fveqsb 40662 Implicit substitution of a...
xpexb 40663 A Cartesian product exists...
trelpss 40664 An element of a transitive...
addcomgi 40665 Generalization of commutat...
addrval 40675 Value of the operation of ...
subrval 40676 Value of the operation of ...
mulvval 40677 Value of the operation of ...
addrfv 40678 Vector addition at a value...
subrfv 40679 Vector subtraction at a va...
mulvfv 40680 Scalar multiplication at a...
addrfn 40681 Vector addition produces a...
subrfn 40682 Vector subtraction produce...
mulvfn 40683 Scalar multiplication prod...
addrcom 40684 Vector addition is commuta...
idiALT 40688 Placeholder for ~ idi . T...
exbir 40689 Exportation implication al...
3impexpbicom 40690 Version of ~ 3impexp where...
3impexpbicomi 40691 Inference associated with ...
bi1imp 40692 Importation inference simi...
bi2imp 40693 Importation inference simi...
bi3impb 40694 Similar to ~ 3impb with im...
bi3impa 40695 Similar to ~ 3impa with im...
bi23impib 40696 ~ 3impib with the inner im...
bi13impib 40697 ~ 3impib with the outer im...
bi123impib 40698 ~ 3impib with the implicat...
bi13impia 40699 ~ 3impia with the outer im...
bi123impia 40700 ~ 3impia with the implicat...
bi33imp12 40701 ~ 3imp with innermost impl...
bi23imp13 40702 ~ 3imp with middle implica...
bi13imp23 40703 ~ 3imp with outermost impl...
bi13imp2 40704 Similar to ~ 3imp except t...
bi12imp3 40705 Similar to ~ 3imp except a...
bi23imp1 40706 Similar to ~ 3imp except a...
bi123imp0 40707 Similar to ~ 3imp except a...
4animp1 40708 A single hypothesis unific...
4an31 40709 A rearrangement of conjunc...
4an4132 40710 A rearrangement of conjunc...
expcomdg 40711 Biconditional form of ~ ex...
iidn3 40712 ~ idn3 without virtual ded...
ee222 40713 ~ e222 without virtual ded...
ee3bir 40714 Right-biconditional form o...
ee13 40715 ~ e13 without virtual dedu...
ee121 40716 ~ e121 without virtual ded...
ee122 40717 ~ e122 without virtual ded...
ee333 40718 ~ e333 without virtual ded...
ee323 40719 ~ e323 without virtual ded...
3ornot23 40720 If the second and third di...
orbi1r 40721 ~ orbi1 with order of disj...
3orbi123 40722 ~ pm4.39 with a 3-conjunct...
syl5imp 40723 Closed form of ~ syl5 . D...
impexpd 40724 The following User's Proof...
com3rgbi 40725 The following User's Proof...
impexpdcom 40726 The following User's Proof...
ee1111 40727 Non-virtual deduction form...
pm2.43bgbi 40728 Logical equivalence of a 2...
pm2.43cbi 40729 Logical equivalence of a 3...
ee233 40730 Non-virtual deduction form...
imbi13 40731 Join three logical equival...
ee33 40732 Non-virtual deduction form...
con5 40733 Biconditional contrapositi...
con5i 40734 Inference form of ~ con5 ....
exlimexi 40735 Inference similar to Theor...
sb5ALT 40736 Equivalence for substituti...
eexinst01 40737 ~ exinst01 without virtual...
eexinst11 40738 ~ exinst11 without virtual...
vk15.4j 40739 Excercise 4j of Unit 15 of...
notnotrALT 40740 Converse of double negatio...
con3ALT2 40741 Contraposition. Alternate...
ssralv2 40742 Quantification restricted ...
sbc3or 40743 ~ sbcor with a 3-disjuncts...
alrim3con13v 40744 Closed form of ~ alrimi wi...
rspsbc2 40745 ~ rspsbc with two quantify...
sbcoreleleq 40746 Substitution of a setvar v...
tratrb 40747 If a class is transitive a...
ordelordALT 40748 An element of an ordinal c...
sbcim2g 40749 Distribution of class subs...
sbcbi 40750 Implication form of ~ sbcb...
trsbc 40751 Formula-building inference...
truniALT 40752 The union of a class of tr...
onfrALTlem5 40753 Lemma for ~ onfrALT . (Co...
onfrALTlem4 40754 Lemma for ~ onfrALT . (Co...
onfrALTlem3 40755 Lemma for ~ onfrALT . (Co...
ggen31 40756 ~ gen31 without virtual de...
onfrALTlem2 40757 Lemma for ~ onfrALT . (Co...
cbvexsv 40758 A theorem pertaining to th...
onfrALTlem1 40759 Lemma for ~ onfrALT . (Co...
onfrALT 40760 The membership relation is...
19.41rg 40761 Closed form of right-to-le...
opelopab4 40762 Ordered pair membership in...
2pm13.193 40763 ~ pm13.193 for two variabl...
hbntal 40764 A closed form of ~ hbn . ~...
hbimpg 40765 A closed form of ~ hbim . ...
hbalg 40766 Closed form of ~ hbal . D...
hbexg 40767 Closed form of ~ nfex . D...
ax6e2eq 40768 Alternate form of ~ ax6e f...
ax6e2nd 40769 If at least two sets exist...
ax6e2ndeq 40770 "At least two sets exist" ...
2sb5nd 40771 Equivalence for double sub...
2uasbanh 40772 Distribute the unabbreviat...
2uasban 40773 Distribute the unabbreviat...
e2ebind 40774 Absorption of an existenti...
elpwgded 40775 ~ elpwgdedVD in convention...
trelded 40776 Deduction form of ~ trel ....
jaoded 40777 Deduction form of ~ jao . ...
sbtT 40778 A substitution into a theo...
not12an2impnot1 40779 If a double conjunction is...
in1 40782 Inference form of ~ df-vd1...
iin1 40783 ~ in1 without virtual dedu...
dfvd1ir 40784 Inference form of ~ df-vd1...
idn1 40785 Virtual deduction identity...
dfvd1imp 40786 Left-to-right part of defi...
dfvd1impr 40787 Right-to-left part of defi...
dfvd2 40790 Definition of a 2-hypothes...
dfvd2an 40793 Definition of a 2-hypothes...
dfvd2ani 40794 Inference form of ~ dfvd2a...
dfvd2anir 40795 Right-to-left inference fo...
dfvd2i 40796 Inference form of ~ dfvd2 ...
dfvd2ir 40797 Right-to-left inference fo...
dfvd3 40802 Definition of a 3-hypothes...
dfvd3i 40803 Inference form of ~ dfvd3 ...
dfvd3ir 40804 Right-to-left inference fo...
dfvd3an 40805 Definition of a 3-hypothes...
dfvd3ani 40806 Inference form of ~ dfvd3a...
dfvd3anir 40807 Right-to-left inference fo...
vd01 40808 A virtual hypothesis virtu...
vd02 40809 Two virtual hypotheses vir...
vd03 40810 A theorem is virtually inf...
vd12 40811 A virtual deduction with 1...
vd13 40812 A virtual deduction with 1...
vd23 40813 A virtual deduction with 2...
dfvd2imp 40814 The virtual deduction form...
dfvd2impr 40815 A 2-antecedent nested impl...
in2 40816 The virtual deduction intr...
int2 40817 The virtual deduction intr...
iin2 40818 ~ in2 without virtual dedu...
in2an 40819 The virtual deduction intr...
in3 40820 The virtual deduction intr...
iin3 40821 ~ in3 without virtual dedu...
in3an 40822 The virtual deduction intr...
int3 40823 The virtual deduction intr...
idn2 40824 Virtual deduction identity...
iden2 40825 Virtual deduction identity...
idn3 40826 Virtual deduction identity...
gen11 40827 Virtual deduction generali...
gen11nv 40828 Virtual deduction generali...
gen12 40829 Virtual deduction generali...
gen21 40830 Virtual deduction generali...
gen21nv 40831 Virtual deduction form of ...
gen31 40832 Virtual deduction generali...
gen22 40833 Virtual deduction generali...
ggen22 40834 ~ gen22 without virtual de...
exinst 40835 Existential Instantiation....
exinst01 40836 Existential Instantiation....
exinst11 40837 Existential Instantiation....
e1a 40838 A Virtual deduction elimin...
el1 40839 A Virtual deduction elimin...
e1bi 40840 Biconditional form of ~ e1...
e1bir 40841 Right biconditional form o...
e2 40842 A virtual deduction elimin...
e2bi 40843 Biconditional form of ~ e2...
e2bir 40844 Right biconditional form o...
ee223 40845 ~ e223 without virtual ded...
e223 40846 A virtual deduction elimin...
e222 40847 A virtual deduction elimin...
e220 40848 A virtual deduction elimin...
ee220 40849 ~ e220 without virtual ded...
e202 40850 A virtual deduction elimin...
ee202 40851 ~ e202 without virtual ded...
e022 40852 A virtual deduction elimin...
ee022 40853 ~ e022 without virtual ded...
e002 40854 A virtual deduction elimin...
ee002 40855 ~ e002 without virtual ded...
e020 40856 A virtual deduction elimin...
ee020 40857 ~ e020 without virtual ded...
e200 40858 A virtual deduction elimin...
ee200 40859 ~ e200 without virtual ded...
e221 40860 A virtual deduction elimin...
ee221 40861 ~ e221 without virtual ded...
e212 40862 A virtual deduction elimin...
ee212 40863 ~ e212 without virtual ded...
e122 40864 A virtual deduction elimin...
e112 40865 A virtual deduction elimin...
ee112 40866 ~ e112 without virtual ded...
e121 40867 A virtual deduction elimin...
e211 40868 A virtual deduction elimin...
ee211 40869 ~ e211 without virtual ded...
e210 40870 A virtual deduction elimin...
ee210 40871 ~ e210 without virtual ded...
e201 40872 A virtual deduction elimin...
ee201 40873 ~ e201 without virtual ded...
e120 40874 A virtual deduction elimin...
ee120 40875 Virtual deduction rule ~ e...
e021 40876 A virtual deduction elimin...
ee021 40877 ~ e021 without virtual ded...
e012 40878 A virtual deduction elimin...
ee012 40879 ~ e012 without virtual ded...
e102 40880 A virtual deduction elimin...
ee102 40881 ~ e102 without virtual ded...
e22 40882 A virtual deduction elimin...
e22an 40883 Conjunction form of ~ e22 ...
ee22an 40884 ~ e22an without virtual de...
e111 40885 A virtual deduction elimin...
e1111 40886 A virtual deduction elimin...
e110 40887 A virtual deduction elimin...
ee110 40888 ~ e110 without virtual ded...
e101 40889 A virtual deduction elimin...
ee101 40890 ~ e101 without virtual ded...
e011 40891 A virtual deduction elimin...
ee011 40892 ~ e011 without virtual ded...
e100 40893 A virtual deduction elimin...
ee100 40894 ~ e100 without virtual ded...
e010 40895 A virtual deduction elimin...
ee010 40896 ~ e010 without virtual ded...
e001 40897 A virtual deduction elimin...
ee001 40898 ~ e001 without virtual ded...
e11 40899 A virtual deduction elimin...
e11an 40900 Conjunction form of ~ e11 ...
ee11an 40901 ~ e11an without virtual de...
e01 40902 A virtual deduction elimin...
e01an 40903 Conjunction form of ~ e01 ...
ee01an 40904 ~ e01an without virtual de...
e10 40905 A virtual deduction elimin...
e10an 40906 Conjunction form of ~ e10 ...
ee10an 40907 ~ e10an without virtual de...
e02 40908 A virtual deduction elimin...
e02an 40909 Conjunction form of ~ e02 ...
ee02an 40910 ~ e02an without virtual de...
eel021old 40911 ~ el021old without virtual...
el021old 40912 A virtual deduction elimin...
eel132 40913 ~ syl2an with antecedents ...
eel000cT 40914 An elimination deduction. ...
eel0TT 40915 An elimination deduction. ...
eelT00 40916 An elimination deduction. ...
eelTTT 40917 An elimination deduction. ...
eelT11 40918 An elimination deduction. ...
eelT1 40919 Syllogism inference combin...
eelT12 40920 An elimination deduction. ...
eelTT1 40921 An elimination deduction. ...
eelT01 40922 An elimination deduction. ...
eel0T1 40923 An elimination deduction. ...
eel12131 40924 An elimination deduction. ...
eel2131 40925 ~ syl2an with antecedents ...
eel3132 40926 ~ syl2an with antecedents ...
eel0321old 40927 ~ el0321old without virtua...
el0321old 40928 A virtual deduction elimin...
eel2122old 40929 ~ el2122old without virtua...
el2122old 40930 A virtual deduction elimin...
eel0000 40931 Elimination rule similar t...
eel00001 40932 An elimination deduction. ...
eel00000 40933 Elimination rule similar ~...
eel11111 40934 Five-hypothesis eliminatio...
e12 40935 A virtual deduction elimin...
e12an 40936 Conjunction form of ~ e12 ...
el12 40937 Virtual deduction form of ...
e20 40938 A virtual deduction elimin...
e20an 40939 Conjunction form of ~ e20 ...
ee20an 40940 ~ e20an without virtual de...
e21 40941 A virtual deduction elimin...
e21an 40942 Conjunction form of ~ e21 ...
ee21an 40943 ~ e21an without virtual de...
e333 40944 A virtual deduction elimin...
e33 40945 A virtual deduction elimin...
e33an 40946 Conjunction form of ~ e33 ...
ee33an 40947 ~ e33an without virtual de...
e3 40948 Meta-connective form of ~ ...
e3bi 40949 Biconditional form of ~ e3...
e3bir 40950 Right biconditional form o...
e03 40951 A virtual deduction elimin...
ee03 40952 ~ e03 without virtual dedu...
e03an 40953 Conjunction form of ~ e03 ...
ee03an 40954 Conjunction form of ~ ee03...
e30 40955 A virtual deduction elimin...
ee30 40956 ~ e30 without virtual dedu...
e30an 40957 A virtual deduction elimin...
ee30an 40958 Conjunction form of ~ ee30...
e13 40959 A virtual deduction elimin...
e13an 40960 A virtual deduction elimin...
ee13an 40961 ~ e13an without virtual de...
e31 40962 A virtual deduction elimin...
ee31 40963 ~ e31 without virtual dedu...
e31an 40964 A virtual deduction elimin...
ee31an 40965 ~ e31an without virtual de...
e23 40966 A virtual deduction elimin...
e23an 40967 A virtual deduction elimin...
ee23an 40968 ~ e23an without virtual de...
e32 40969 A virtual deduction elimin...
ee32 40970 ~ e32 without virtual dedu...
e32an 40971 A virtual deduction elimin...
ee32an 40972 ~ e33an without virtual de...
e123 40973 A virtual deduction elimin...
ee123 40974 ~ e123 without virtual ded...
el123 40975 A virtual deduction elimin...
e233 40976 A virtual deduction elimin...
e323 40977 A virtual deduction elimin...
e000 40978 A virtual deduction elimin...
e00 40979 Elimination rule identical...
e00an 40980 Elimination rule identical...
eel00cT 40981 An elimination deduction. ...
eelTT 40982 An elimination deduction. ...
e0a 40983 Elimination rule identical...
eelT 40984 An elimination deduction. ...
eel0cT 40985 An elimination deduction. ...
eelT0 40986 An elimination deduction. ...
e0bi 40987 Elimination rule identical...
e0bir 40988 Elimination rule identical...
uun0.1 40989 Convention notation form o...
un0.1 40990 ` T. ` is the constant tru...
uunT1 40991 A deduction unionizing a n...
uunT1p1 40992 A deduction unionizing a n...
uunT21 40993 A deduction unionizing a n...
uun121 40994 A deduction unionizing a n...
uun121p1 40995 A deduction unionizing a n...
uun132 40996 A deduction unionizing a n...
uun132p1 40997 A deduction unionizing a n...
anabss7p1 40998 A deduction unionizing a n...
un10 40999 A unionizing deduction. (...
un01 41000 A unionizing deduction. (...
un2122 41001 A deduction unionizing a n...
uun2131 41002 A deduction unionizing a n...
uun2131p1 41003 A deduction unionizing a n...
uunTT1 41004 A deduction unionizing a n...
uunTT1p1 41005 A deduction unionizing a n...
uunTT1p2 41006 A deduction unionizing a n...
uunT11 41007 A deduction unionizing a n...
uunT11p1 41008 A deduction unionizing a n...
uunT11p2 41009 A deduction unionizing a n...
uunT12 41010 A deduction unionizing a n...
uunT12p1 41011 A deduction unionizing a n...
uunT12p2 41012 A deduction unionizing a n...
uunT12p3 41013 A deduction unionizing a n...
uunT12p4 41014 A deduction unionizing a n...
uunT12p5 41015 A deduction unionizing a n...
uun111 41016 A deduction unionizing a n...
3anidm12p1 41017 A deduction unionizing a n...
3anidm12p2 41018 A deduction unionizing a n...
uun123 41019 A deduction unionizing a n...
uun123p1 41020 A deduction unionizing a n...
uun123p2 41021 A deduction unionizing a n...
uun123p3 41022 A deduction unionizing a n...
uun123p4 41023 A deduction unionizing a n...
uun2221 41024 A deduction unionizing a n...
uun2221p1 41025 A deduction unionizing a n...
uun2221p2 41026 A deduction unionizing a n...
3impdirp1 41027 A deduction unionizing a n...
3impcombi 41028 A 1-hypothesis proposition...
trsspwALT 41029 Virtual deduction proof of...
trsspwALT2 41030 Virtual deduction proof of...
trsspwALT3 41031 Short predicate calculus p...
sspwtr 41032 Virtual deduction proof of...
sspwtrALT 41033 Virtual deduction proof of...
sspwtrALT2 41034 Short predicate calculus p...
pwtrVD 41035 Virtual deduction proof of...
pwtrrVD 41036 Virtual deduction proof of...
suctrALT 41037 The successor of a transit...
snssiALTVD 41038 Virtual deduction proof of...
snssiALT 41039 If a class is an element o...
snsslVD 41040 Virtual deduction proof of...
snssl 41041 If a singleton is a subcla...
snelpwrVD 41042 Virtual deduction proof of...
unipwrVD 41043 Virtual deduction proof of...
unipwr 41044 A class is a subclass of t...
sstrALT2VD 41045 Virtual deduction proof of...
sstrALT2 41046 Virtual deduction proof of...
suctrALT2VD 41047 Virtual deduction proof of...
suctrALT2 41048 Virtual deduction proof of...
elex2VD 41049 Virtual deduction proof of...
elex22VD 41050 Virtual deduction proof of...
eqsbc3rVD 41051 Virtual deduction proof of...
zfregs2VD 41052 Virtual deduction proof of...
tpid3gVD 41053 Virtual deduction proof of...
en3lplem1VD 41054 Virtual deduction proof of...
en3lplem2VD 41055 Virtual deduction proof of...
en3lpVD 41056 Virtual deduction proof of...
simplbi2VD 41057 Virtual deduction proof of...
3ornot23VD 41058 Virtual deduction proof of...
orbi1rVD 41059 Virtual deduction proof of...
bitr3VD 41060 Virtual deduction proof of...
3orbi123VD 41061 Virtual deduction proof of...
sbc3orgVD 41062 Virtual deduction proof of...
19.21a3con13vVD 41063 Virtual deduction proof of...
exbirVD 41064 Virtual deduction proof of...
exbiriVD 41065 Virtual deduction proof of...
rspsbc2VD 41066 Virtual deduction proof of...
3impexpVD 41067 Virtual deduction proof of...
3impexpbicomVD 41068 Virtual deduction proof of...
3impexpbicomiVD 41069 Virtual deduction proof of...
sbcoreleleqVD 41070 Virtual deduction proof of...
hbra2VD 41071 Virtual deduction proof of...
tratrbVD 41072 Virtual deduction proof of...
al2imVD 41073 Virtual deduction proof of...
syl5impVD 41074 Virtual deduction proof of...
idiVD 41075 Virtual deduction proof of...
ancomstVD 41076 Closed form of ~ ancoms . ...
ssralv2VD 41077 Quantification restricted ...
ordelordALTVD 41078 An element of an ordinal c...
equncomVD 41079 If a class equals the unio...
equncomiVD 41080 Inference form of ~ equnco...
sucidALTVD 41081 A set belongs to its succe...
sucidALT 41082 A set belongs to its succe...
sucidVD 41083 A set belongs to its succe...
imbi12VD 41084 Implication form of ~ imbi...
imbi13VD 41085 Join three logical equival...
sbcim2gVD 41086 Distribution of class subs...
sbcbiVD 41087 Implication form of ~ sbcb...
trsbcVD 41088 Formula-building inference...
truniALTVD 41089 The union of a class of tr...
ee33VD 41090 Non-virtual deduction form...
trintALTVD 41091 The intersection of a clas...
trintALT 41092 The intersection of a clas...
undif3VD 41093 The first equality of Exer...
sbcssgVD 41094 Virtual deduction proof of...
csbingVD 41095 Virtual deduction proof of...
onfrALTlem5VD 41096 Virtual deduction proof of...
onfrALTlem4VD 41097 Virtual deduction proof of...
onfrALTlem3VD 41098 Virtual deduction proof of...
simplbi2comtVD 41099 Virtual deduction proof of...
onfrALTlem2VD 41100 Virtual deduction proof of...
onfrALTlem1VD 41101 Virtual deduction proof of...
onfrALTVD 41102 Virtual deduction proof of...
csbeq2gVD 41103 Virtual deduction proof of...
csbsngVD 41104 Virtual deduction proof of...
csbxpgVD 41105 Virtual deduction proof of...
csbresgVD 41106 Virtual deduction proof of...
csbrngVD 41107 Virtual deduction proof of...
csbima12gALTVD 41108 Virtual deduction proof of...
csbunigVD 41109 Virtual deduction proof of...
csbfv12gALTVD 41110 Virtual deduction proof of...
con5VD 41111 Virtual deduction proof of...
relopabVD 41112 Virtual deduction proof of...
19.41rgVD 41113 Virtual deduction proof of...
2pm13.193VD 41114 Virtual deduction proof of...
hbimpgVD 41115 Virtual deduction proof of...
hbalgVD 41116 Virtual deduction proof of...
hbexgVD 41117 Virtual deduction proof of...
ax6e2eqVD 41118 The following User's Proof...
ax6e2ndVD 41119 The following User's Proof...
ax6e2ndeqVD 41120 The following User's Proof...
2sb5ndVD 41121 The following User's Proof...
2uasbanhVD 41122 The following User's Proof...
e2ebindVD 41123 The following User's Proof...
sb5ALTVD 41124 The following User's Proof...
vk15.4jVD 41125 The following User's Proof...
notnotrALTVD 41126 The following User's Proof...
con3ALTVD 41127 The following User's Proof...
elpwgdedVD 41128 Membership in a power clas...
sspwimp 41129 If a class is a subclass o...
sspwimpVD 41130 The following User's Proof...
sspwimpcf 41131 If a class is a subclass o...
sspwimpcfVD 41132 The following User's Proof...
suctrALTcf 41133 The sucessor of a transiti...
suctrALTcfVD 41134 The following User's Proof...
suctrALT3 41135 The successor of a transit...
sspwimpALT 41136 If a class is a subclass o...
unisnALT 41137 A set equals the union of ...
notnotrALT2 41138 Converse of double negatio...
sspwimpALT2 41139 If a class is a subclass o...
e2ebindALT 41140 Absorption of an existenti...
ax6e2ndALT 41141 If at least two sets exist...
ax6e2ndeqALT 41142 "At least two sets exist" ...
2sb5ndALT 41143 Equivalence for double sub...
chordthmALT 41144 The intersecting chords th...
isosctrlem1ALT 41145 Lemma for ~ isosctr . Thi...
iunconnlem2 41146 The indexed union of conne...
iunconnALT 41147 The indexed union of conne...
sineq0ALT 41148 A complex number whose sin...
evth2f 41149 A version of ~ evth2 using...
elunif 41150 A version of ~ eluni using...
rzalf 41151 A version of ~ rzal using ...
fvelrnbf 41152 A version of ~ fvelrnb usi...
rfcnpre1 41153 If F is a continuous funct...
ubelsupr 41154 If U belongs to A and U is...
fsumcnf 41155 A finite sum of functions ...
mulltgt0 41156 The product of a negative ...
rspcegf 41157 A version of ~ rspcev usin...
rabexgf 41158 A version of ~ rabexg usin...
fcnre 41159 A function continuous with...
sumsnd 41160 A sum of a singleton is th...
evthf 41161 A version of ~ evth using ...
cnfex 41162 The class of continuous fu...
fnchoice 41163 For a finite set, a choice...
refsumcn 41164 A finite sum of continuous...
rfcnpre2 41165 If ` F ` is a continuous f...
cncmpmax 41166 When the hypothesis for th...
rfcnpre3 41167 If F is a continuous funct...
rfcnpre4 41168 If F is a continuous funct...
sumpair 41169 Sum of two distinct comple...
rfcnnnub 41170 Given a real continuous fu...
refsum2cnlem1 41171 This is the core Lemma for...
refsum2cn 41172 The sum of two continuus r...
elunnel2 41173 A member of a union that i...
adantlllr 41174 Deduction adding a conjunc...
3adantlr3 41175 Deduction adding a conjunc...
nnxrd 41176 A natural number is an ext...
3adantll2 41177 Deduction adding a conjunc...
3adantll3 41178 Deduction adding a conjunc...
ssnel 41179 If not element of a set, t...
elabrexg 41180 Elementhood in an image se...
sncldre 41181 A singleton is closed w.r....
n0p 41182 A polynomial with a nonzer...
pm2.65ni 41183 Inference rule for proof b...
pwssfi 41184 Every element of the power...
iuneq2df 41185 Equality deduction for ind...
nnfoctb 41186 There exists a mapping fro...
ssinss1d 41187 Intersection preserves sub...
elpwinss 41188 An element of the powerset...
unidmex 41189 If ` F ` is a set, then ` ...
ndisj2 41190 A non-disjointness conditi...
zenom 41191 The set of integer numbers...
uzwo4 41192 Well-ordering principle: a...
unisn0 41193 The union of the singleton...
ssin0 41194 If two classes are disjoin...
inabs3 41195 Absorption law for interse...
pwpwuni 41196 Relationship between power...
disjiun2 41197 In a disjoint collection, ...
0pwfi 41198 The empty set is in any po...
ssinss2d 41199 Intersection preserves sub...
zct 41200 The set of integer numbers...
pwfin0 41201 A finite set always belong...
uzct 41202 An upper integer set is co...
iunxsnf 41203 A singleton index picks ou...
fiiuncl 41204 If a set is closed under t...
iunp1 41205 The addition of the next s...
fiunicl 41206 If a set is closed under t...
ixpeq2d 41207 Equality theorem for infin...
disjxp1 41208 The sets of a cartesian pr...
disjsnxp 41209 The sets in the cartesian ...
eliind 41210 Membership in indexed inte...
rspcef 41211 Restricted existential spe...
inn0f 41212 A nonempty intersection. ...
ixpssmapc 41213 An infinite Cartesian prod...
inn0 41214 A nonempty intersection. ...
elintd 41215 Membership in class inters...
ssdf 41216 A sufficient condition for...
brneqtrd 41217 Substitution of equal clas...
ssnct 41218 A set containing an uncoun...
ssuniint 41219 Sufficient condition for b...
elintdv 41220 Membership in class inters...
ssd 41221 A sufficient condition for...
ralimralim 41222 Introducing any antecedent...
snelmap 41223 Membership of the element ...
xrnmnfpnf 41224 An extended real that is n...
nelrnmpt 41225 Non-membership in the rang...
snn0d 41226 The singleton of a set is ...
iuneq1i 41227 Equality theorem for index...
nssrex 41228 Negation of subclass relat...
iunssf 41229 Subset theorem for an inde...
ssinc 41230 Inclusion relation for a m...
ssdec 41231 Inclusion relation for a m...
elixpconstg 41232 Membership in an infinite ...
iineq1d 41233 Equality theorem for index...
metpsmet 41234 A metric is a pseudometric...
ixpssixp 41235 Subclass theorem for infin...
ballss3 41236 A sufficient condition for...
iunincfi 41237 Given a sequence of increa...
nsstr 41238 If it's not a subclass, it...
rexanuz3 41239 Combine two different uppe...
cbvmpo2 41240 Rule to change the second ...
cbvmpo1 41241 Rule to change the first b...
eliuniin 41242 Indexed union of indexed i...
ssabf 41243 Subclass of a class abstra...
pssnssi 41244 A proper subclass does not...
rabidim2 41245 Membership in a restricted...
eluni2f 41246 Membership in class union....
eliin2f 41247 Membership in indexed inte...
nssd 41248 Negation of subclass relat...
iineq12dv 41249 Equality deduction for ind...
supxrcld 41250 The supremum of an arbitra...
elrestd 41251 A sufficient condition for...
eliuniincex 41252 Counterexample to show tha...
eliincex 41253 Counterexample to show tha...
eliinid 41254 Membership in an indexed i...
abssf 41255 Class abstraction in a sub...
fexd 41256 If the domain of a mapping...
supxrubd 41257 A member of a set of exten...
ssrabf 41258 Subclass of a restricted c...
eliin2 41259 Membership in indexed inte...
ssrab2f 41260 Subclass relation for a re...
restuni3 41261 The underlying set of a su...
rabssf 41262 Restricted class abstracti...
eliuniin2 41263 Indexed union of indexed i...
restuni4 41264 The underlying set of a su...
restuni6 41265 The underlying set of a su...
restuni5 41266 The underlying set of a su...
unirestss 41267 The union of an elementwis...
iniin1 41268 Indexed intersection of in...
iniin2 41269 Indexed intersection of in...
cbvrabv2 41270 A more general version of ...
iinssiin 41271 Subset implication for an ...
eliind2 41272 Membership in indexed inte...
iinssd 41273 Subset implication for an ...
ralrimia 41274 Inference from Theorem 19....
rabbida2 41275 Equivalent wff's yield equ...
iinexd 41276 The existence of an indexe...
rabexf 41277 Separation Scheme in terms...
rabbida3 41278 Equivalent wff's yield equ...
resexd 41279 The restriction of a set i...
r19.36vf 41280 Restricted quantifier vers...
raleqd 41281 Equality deduction for res...
ralimda 41282 Deduction quantifying both...
iinssf 41283 Subset implication for an ...
iinssdf 41284 Subset implication for an ...
resabs2i 41285 Absorption law for restric...
ssdf2 41286 A sufficient condition for...
rabssd 41287 Restricted class abstracti...
rexnegd 41288 Minus a real number. (Con...
rexlimd3 41289 * Inference from Theorem 1...
resabs1i 41290 Absorption law for restric...
nel1nelin 41291 Membership in an intersect...
nel2nelin 41292 Membership in an intersect...
nel1nelini 41293 Membership in an intersect...
nel2nelini 41294 Membership in an intersect...
eliunid 41295 Membership in indexed unio...
reximddv3 41296 Deduction from Theorem 19....
reximdd 41297 Deduction from Theorem 19....
unfid 41298 The union of two finite se...
feq1dd 41299 Equality deduction for fun...
fnresdmss 41300 A function does not change...
fmptsnxp 41301 Maps-to notation and cross...
fvmpt2bd 41302 Value of a function given ...
rnmptfi 41303 The range of a function wi...
fresin2 41304 Restriction of a function ...
ffi 41305 A function with finite dom...
suprnmpt 41306 An explicit bound for the ...
rnffi 41307 The range of a function wi...
mptelpm 41308 A function in maps-to nota...
rnmptpr 41309 Range of a function define...
resmpti 41310 Restriction of the mapping...
founiiun 41311 Union expressed as an inde...
rnresun 41312 Distribution law for range...
f1oeq1d 41313 Equality deduction for one...
dffo3f 41314 An onto mapping expressed ...
rnresss 41315 The range of a restriction...
elrnmptd 41316 The range of a function in...
elrnmptf 41317 The range of a function in...
rnmptssrn 41318 Inclusion relation for two...
disjf1 41319 A 1 to 1 mapping built fro...
rnsnf 41320 The range of a function wh...
wessf1ornlem 41321 Given a function ` F ` on ...
wessf1orn 41322 Given a function ` F ` on ...
foelrnf 41323 Property of a surjective f...
nelrnres 41324 If ` A ` is not in the ran...
disjrnmpt2 41325 Disjointness of the range ...
elrnmpt1sf 41326 Elementhood in an image se...
founiiun0 41327 Union expressed as an inde...
disjf1o 41328 A bijection built from dis...
fompt 41329 Express being onto for a m...
disjinfi 41330 Only a finite number of di...
fvovco 41331 Value of the composition o...
ssnnf1octb 41332 There exists a bijection b...
nnf1oxpnn 41333 There is a bijection betwe...
rnmptssd 41334 The range of an operation ...
projf1o 41335 A biijection from a set to...
fvmap 41336 Function value for a membe...
fvixp2 41337 Projection of a factor of ...
fidmfisupp 41338 A function with a finite d...
choicefi 41339 For a finite set, a choice...
mpct 41340 The exponentiation of a co...
cnmetcoval 41341 Value of the distance func...
fcomptss 41342 Express composition of two...
elmapsnd 41343 Membership in a set expone...
mapss2 41344 Subset inheritance for set...
fsneq 41345 Equality condition for two...
difmap 41346 Difference of two sets exp...
unirnmap 41347 Given a subset of a set ex...
inmap 41348 Intersection of two sets e...
fcoss 41349 Composition of two mapping...
fsneqrn 41350 Equality condition for two...
difmapsn 41351 Difference of two sets exp...
mapssbi 41352 Subset inheritance for set...
unirnmapsn 41353 Equality theorem for a sub...
iunmapss 41354 The indexed union of set e...
ssmapsn 41355 A subset ` C ` of a set ex...
iunmapsn 41356 The indexed union of set e...
absfico 41357 Mapping domain and codomai...
icof 41358 The set of left-closed rig...
rnmpt0 41359 The range of a function in...
rnmptn0 41360 The range of a function in...
elpmrn 41361 The range of a partial fun...
imaexi 41362 The image of a set is a se...
axccdom 41363 Relax the constraint on ax...
dmmptdf 41364 The domain of the mapping ...
elpmi2 41365 The domain of a partial fu...
dmrelrnrel 41366 A relation preserving func...
fco3 41367 Functionality of a composi...
fvcod 41368 Value of a function compos...
freld 41369 A mapping is a relation. ...
elrnmpoid 41370 Membership in the range of...
axccd 41371 An alternative version of ...
axccd2 41372 An alternative version of ...
funimassd 41373 Sufficient condition for t...
fimassd 41374 The image of a class is a ...
feqresmptf 41375 Express a restricted funct...
elrnmpt1d 41376 Elementhood in an image se...
dmresss 41377 The domain of a restrictio...
dmmptssf 41378 The domain of a mapping is...
dmmptdf2 41379 The domain of the mapping ...
dmuz 41380 Domain of the upper intege...
fmptd2f 41381 Domain and codomain of the...
mpteq1df 41382 An equality theorem for th...
mptexf 41383 If the domain of a functio...
fvmpt4 41384 Value of a function given ...
fmptf 41385 Functionality of the mappi...
resimass 41386 The image of a restriction...
mptssid 41387 The mapping operation expr...
mptfnd 41388 The maps-to notation defin...
mpteq12da 41389 An equality inference for ...
rnmptlb 41390 Boundness below of the ran...
rnmptbddlem 41391 Boundness of the range of ...
rnmptbdd 41392 Boundness of the range of ...
mptima2 41393 Image of a function in map...
funimaeq 41394 Membership relation for th...
rnmptssf 41395 The range of an operation ...
rnmptbd2lem 41396 Boundness below of the ran...
rnmptbd2 41397 Boundness below of the ran...
infnsuprnmpt 41398 The indexed infimum of rea...
suprclrnmpt 41399 Closure of the indexed sup...
suprubrnmpt2 41400 A member of a nonempty ind...
suprubrnmpt 41401 A member of a nonempty ind...
rnmptssdf 41402 The range of an operation ...
rnmptbdlem 41403 Boundness above of the ran...
rnmptbd 41404 Boundness above of the ran...
rnmptss2 41405 The range of an operation ...
elmptima 41406 The image of a function in...
ralrnmpt3 41407 A restricted quantifier ov...
fvelima2 41408 Function value in an image...
funresd 41409 A restriction of a functio...
rnmptssbi 41410 The range of an operation ...
fnfvelrnd 41411 A function's value belongs...
imass2d 41412 Subset theorem for image. ...
imassmpt 41413 Membership relation for th...
fpmd 41414 A total function is a part...
fconst7 41415 An alternative way to expr...
sub2times 41416 Subtracting from a number,...
abssubrp 41417 The distance of two distin...
elfzfzo 41418 Relationship between membe...
oddfl 41419 Odd number representation ...
abscosbd 41420 Bound for the absolute val...
mul13d 41421 Commutative/associative la...
negpilt0 41422 Negative ` _pi ` is negati...
dstregt0 41423 A complex number ` A ` tha...
subadd4b 41424 Rearrangement of 4 terms i...
xrlttri5d 41425 Not equal and not larger i...
neglt 41426 The negative of a positive...
zltlesub 41427 If an integer ` N ` is les...
divlt0gt0d 41428 The ratio of a negative nu...
subsub23d 41429 Swap subtrahend and result...
2timesgt 41430 Double of a positive real ...
reopn 41431 The reals are open with re...
elfzop1le2 41432 A member in a half-open in...
sub31 41433 Swap the first and third t...
nnne1ge2 41434 A positive integer which i...
lefldiveq 41435 A closed enough, smaller r...
negsubdi3d 41436 Distribution of negative o...
ltdiv2dd 41437 Division of a positive num...
abssinbd 41438 Bound for the absolute val...
halffl 41439 Floor of ` ( 1 / 2 ) ` . ...
monoords 41440 Ordering relation for a st...
hashssle 41441 The size of a subset of a ...
lttri5d 41442 Not equal and not larger i...
fzisoeu 41443 A finite ordered set has a...
lt3addmuld 41444 If three real numbers are ...
absnpncan2d 41445 Triangular inequality, com...
fperiodmullem 41446 A function with period T i...
fperiodmul 41447 A function with period T i...
upbdrech 41448 Choice of an upper bound f...
lt4addmuld 41449 If four real numbers are l...
absnpncan3d 41450 Triangular inequality, com...
upbdrech2 41451 Choice of an upper bound f...
ssfiunibd 41452 A finite union of bounded ...
fzdifsuc2 41453 Remove a successor from th...
fzsscn 41454 A finite sequence of integ...
divcan8d 41455 A cancellation law for div...
dmmcand 41456 Cancellation law for divis...
fzssre 41457 A finite sequence of integ...
elfzelzd 41458 A member of a finite set o...
bccld 41459 A binomial coefficient, in...
leadd12dd 41460 Addition to both sides of ...
fzssnn0 41461 A finite set of sequential...
xreqle 41462 Equality implies 'less tha...
xaddid2d 41463 ` 0 ` is a left identity f...
xadd0ge 41464 A number is less than or e...
elfzolem1 41465 A member in a half-open in...
xrgtned 41466 'Greater than' implies not...
xrleneltd 41467 'Less than or equal to' an...
xaddcomd 41468 The extended real addition...
supxrre3 41469 The supremum of a nonempty...
uzfissfz 41470 For any finite subset of t...
xleadd2d 41471 Addition of extended reals...
suprltrp 41472 The supremum of a nonempty...
xleadd1d 41473 Addition of extended reals...
xreqled 41474 Equality implies 'less tha...
xrgepnfd 41475 An extended real greater t...
xrge0nemnfd 41476 A nonnegative extended rea...
supxrgere 41477 If a real number can be ap...
iuneqfzuzlem 41478 Lemma for ~ iuneqfzuz : he...
iuneqfzuz 41479 If two unions indexed by u...
xle2addd 41480 Adding both side of two in...
supxrgelem 41481 If an extended real number...
supxrge 41482 If an extended real number...
suplesup 41483 If any element of ` A ` ca...
infxrglb 41484 The infimum of a set of ex...
xadd0ge2 41485 A number is less than or e...
nepnfltpnf 41486 An extended real that is n...
ltadd12dd 41487 Addition to both sides of ...
nemnftgtmnft 41488 An extended real that is n...
xrgtso 41489 'Greater than' is a strict...
rpex 41490 The positive reals form a ...
xrge0ge0 41491 A nonnegative extended rea...
xrssre 41492 A subset of extended reals...
ssuzfz 41493 A finite subset of the upp...
absfun 41494 The absolute value is a fu...
infrpge 41495 The infimum of a nonempty,...
xrlexaddrp 41496 If an extended real number...
supsubc 41497 The supremum function dist...
xralrple2 41498 Show that ` A ` is less th...
nnuzdisj 41499 The first ` N ` elements o...
ltdivgt1 41500 Divsion by a number greate...
xrltned 41501 'Less than' implies not eq...
nnsplit 41502 Express the set of positiv...
divdiv3d 41503 Division into a fraction. ...
abslt2sqd 41504 Comparison of the square o...
qenom 41505 The set of rational number...
qct 41506 The set of rational number...
xrltnled 41507 'Less than' in terms of 'l...
lenlteq 41508 'less than or equal to' bu...
xrred 41509 An extended real that is n...
rr2sscn2 41510 The cartesian square of ` ...
infxr 41511 The infimum of a set of ex...
infxrunb2 41512 The infimum of an unbounde...
infxrbnd2 41513 The infimum of a bounded-b...
infleinflem1 41514 Lemma for ~ infleinf , cas...
infleinflem2 41515 Lemma for ~ infleinf , whe...
infleinf 41516 If any element of ` B ` ca...
xralrple4 41517 Show that ` A ` is less th...
xralrple3 41518 Show that ` A ` is less th...
eluzelzd 41519 A member of an upper set o...
suplesup2 41520 If any element of ` A ` is...
recnnltrp 41521 ` N ` is a natural number ...
fiminre2 41522 A nonempty finite set of r...
nnn0 41523 The set of positive intege...
fzct 41524 A finite set of sequential...
rpgtrecnn 41525 Any positive real number i...
fzossuz 41526 A half-open integer interv...
infrefilb 41527 The infimum of a finite se...
infxrrefi 41528 The real and extended real...
xrralrecnnle 41529 Show that ` A ` is less th...
fzoct 41530 A finite set of sequential...
frexr 41531 A function taking real val...
nnrecrp 41532 The reciprocal of a positi...
qred 41533 A rational number is a rea...
reclt0d 41534 The reciprocal of a negati...
lt0neg1dd 41535 If a number is negative, i...
mnfled 41536 Minus infinity is less tha...
infxrcld 41537 The infimum of an arbitrar...
xrralrecnnge 41538 Show that ` A ` is less th...
reclt0 41539 The reciprocal of a negati...
ltmulneg 41540 Multiplying by a negative ...
allbutfi 41541 For all but finitely many....
ltdiv23neg 41542 Swap denominator with othe...
xreqnltd 41543 A consequence of trichotom...
mnfnre2 41544 Minus infinity is not a re...
uzssre 41545 An upper set of integers i...
zssxr 41546 The integers are a subset ...
fisupclrnmpt 41547 A nonempty finite indexed ...
supxrunb3 41548 The supremum of an unbound...
elfzod 41549 Membership in a half-open ...
fimaxre4 41550 A nonempty finite set of r...
ren0 41551 The set of reals is nonemp...
eluzelz2 41552 A member of an upper set o...
resabs2d 41553 Absorption law for restric...
uzid2 41554 Membership of the least me...
supxrleubrnmpt 41555 The supremum of a nonempty...
uzssre2 41556 An upper set of integers i...
uzssd 41557 Subset relationship for tw...
eluzd 41558 Membership in an upper set...
elfzd 41559 Membership in a finite set...
infxrlbrnmpt2 41560 A member of a nonempty ind...
xrre4 41561 An extended real is real i...
uz0 41562 The upper integers functio...
eluzelz2d 41563 A member of an upper set o...
infleinf2 41564 If any element in ` B ` is...
unb2ltle 41565 "Unbounded below" expresse...
uzidd2 41566 Membership of the least me...
uzssd2 41567 Subset relationship for tw...
rexabslelem 41568 An indexed set of absolute...
rexabsle 41569 An indexed set of absolute...
allbutfiinf 41570 Given a "for all but finit...
supxrrernmpt 41571 The real and extended real...
suprleubrnmpt 41572 The supremum of a nonempty...
infrnmptle 41573 An indexed infimum of exte...
infxrunb3 41574 The infimum of an unbounde...
uzn0d 41575 The upper integers are all...
uzssd3 41576 Subset relationship for tw...
rexabsle2 41577 An indexed set of absolute...
infxrunb3rnmpt 41578 The infimum of an unbounde...
supxrre3rnmpt 41579 The indexed supremum of a ...
uzublem 41580 A set of reals, indexed by...
uzub 41581 A set of reals, indexed by...
ssrexr 41582 A subset of the reals is a...
supxrmnf2 41583 Removing minus infinity fr...
supxrcli 41584 The supremum of an arbitra...
uzid3 41585 Membership of the least me...
infxrlesupxr 41586 The supremum of a nonempty...
xnegeqd 41587 Equality of two extended n...
xnegrecl 41588 The extended real negative...
xnegnegi 41589 Extended real version of ~...
xnegeqi 41590 Equality of two extended n...
nfxnegd 41591 Deduction version of ~ nfx...
xnegnegd 41592 Extended real version of ~...
uzred 41593 An upper integer is a real...
xnegcli 41594 Closure of extended real n...
supminfrnmpt 41595 The indexed supremum of a ...
ceilged 41596 The ceiling of a real numb...
infxrpnf 41597 Adding plus infinity to a ...
infxrrnmptcl 41598 The infimum of an arbitrar...
leneg2d 41599 Negative of one side of 'l...
supxrltinfxr 41600 The supremum of the empty ...
max1d 41601 A number is less than or e...
ceilcld 41602 Closure of the ceiling fun...
supxrleubrnmptf 41603 The supremum of a nonempty...
nleltd 41604 'Not less than or equal to...
zxrd 41605 An integer is an extended ...
infxrgelbrnmpt 41606 The infimum of an indexed ...
rphalfltd 41607 Half of a positive real is...
uzssz2 41608 An upper set of integers i...
leneg3d 41609 Negative of one side of 'l...
max2d 41610 A number is less than or e...
uzn0bi 41611 The upper integers functio...
xnegrecl2 41612 If the extended real negat...
nfxneg 41613 Bound-variable hypothesis ...
uzxrd 41614 An upper integer is an ext...
infxrpnf2 41615 Removing plus infinity fro...
supminfxr 41616 The extended real suprema ...
infrpgernmpt 41617 The infimum of a nonempty,...
xnegre 41618 An extended real is real i...
xnegrecl2d 41619 If the extended real negat...
uzxr 41620 An upper integer is an ext...
supminfxr2 41621 The extended real suprema ...
xnegred 41622 An extended real is real i...
supminfxrrnmpt 41623 The indexed supremum of a ...
min1d 41624 The minimum of two numbers...
min2d 41625 The minimum of two numbers...
pnfged 41626 Plus infinity is an upper ...
xrnpnfmnf 41627 An extended real that is n...
uzsscn 41628 An upper set of integers i...
absimnre 41629 The absolute value of the ...
uzsscn2 41630 An upper set of integers i...
xrtgcntopre 41631 The standard topologies on...
absimlere 41632 The absolute value of the ...
rpssxr 41633 The positive reals are a s...
monoordxrv 41634 Ordering relation for a mo...
monoordxr 41635 Ordering relation for a mo...
monoord2xrv 41636 Ordering relation for a mo...
monoord2xr 41637 Ordering relation for a mo...
xrpnf 41638 An extended real is plus i...
xlenegcon1 41639 Extended real version of ~...
xlenegcon2 41640 Extended real version of ~...
gtnelioc 41641 A real number larger than ...
ioossioc 41642 An open interval is a subs...
ioondisj2 41643 A condition for two open i...
ioondisj1 41644 A condition for two open i...
ioosscn 41645 An open interval is a set ...
ioogtlb 41646 An element of a closed int...
evthiccabs 41647 Extreme Value Theorem on y...
ltnelicc 41648 A real number smaller than...
eliood 41649 Membership in an open real...
iooabslt 41650 An upper bound for the dis...
gtnelicc 41651 A real number greater than...
iooinlbub 41652 An open interval has empty...
iocgtlb 41653 An element of a left-open ...
iocleub 41654 An element of a left-open ...
eliccd 41655 Membership in a closed rea...
iccssred 41656 A closed real interval is ...
eliccre 41657 A member of a closed inter...
eliooshift 41658 Element of an open interva...
eliocd 41659 Membership in a left-open ...
icoltub 41660 An element of a left-close...
eliocre 41661 A member of a left-open ri...
iooltub 41662 An element of an open inte...
ioontr 41663 The interior of an interva...
snunioo1 41664 The closure of one end of ...
lbioc 41665 A left-open right-closed i...
ioomidp 41666 The midpoint is an element...
iccdifioo 41667 If the open inverval is re...
iccdifprioo 41668 An open interval is the cl...
ioossioobi 41669 Biconditional form of ~ io...
iccshift 41670 A closed interval shifted ...
iccsuble 41671 An upper bound to the dist...
iocopn 41672 A left-open right-closed i...
eliccelioc 41673 Membership in a closed int...
iooshift 41674 An open interval shifted b...
iccintsng 41675 Intersection of two adiace...
icoiccdif 41676 Left-closed right-open int...
icoopn 41677 A left-closed right-open i...
icoub 41678 A left-closed, right-open ...
eliccxrd 41679 Membership in a closed rea...
pnfel0pnf 41680 ` +oo ` is a nonnegative e...
eliccnelico 41681 An element of a closed int...
eliccelicod 41682 A member of a closed inter...
ge0xrre 41683 A nonnegative extended rea...
ge0lere 41684 A nonnegative extended Rea...
elicores 41685 Membership in a left-close...
inficc 41686 The infimum of a nonempty ...
qinioo 41687 The rational numbers are d...
lenelioc 41688 A real number smaller than...
ioonct 41689 A nonempty open interval i...
xrgtnelicc 41690 A real number greater than...
iccdificc 41691 The difference of two clos...
iocnct 41692 A nonempty left-open, righ...
iccnct 41693 A closed interval, with mo...
iooiinicc 41694 A closed interval expresse...
iccgelbd 41695 An element of a closed int...
iooltubd 41696 An element of an open inte...
icoltubd 41697 An element of a left-close...
qelioo 41698 The rational numbers are d...
tgqioo2 41699 Every open set of reals is...
iccleubd 41700 An element of a closed int...
elioored 41701 A member of an open interv...
ioogtlbd 41702 An element of a closed int...
ioofun 41703 ` (,) ` is a function. (C...
icomnfinre 41704 A left-closed, right-open,...
sqrlearg 41705 The square compared with i...
ressiocsup 41706 If the supremum belongs to...
ressioosup 41707 If the supremum does not b...
iooiinioc 41708 A left-open, right-closed ...
ressiooinf 41709 If the infimum does not be...
icogelbd 41710 An element of a left-close...
iocleubd 41711 An element of a left-open ...
uzinico 41712 An upper interval of integ...
preimaiocmnf 41713 Preimage of a right-closed...
uzinico2 41714 An upper interval of integ...
uzinico3 41715 An upper interval of integ...
icossico2 41716 Condition for a closed-bel...
dmico 41717 The domain of the closed-b...
ndmico 41718 The closed-below, open-abo...
uzubioo 41719 The upper integers are unb...
uzubico 41720 The upper integers are unb...
uzubioo2 41721 The upper integers are unb...
uzubico2 41722 The upper integers are unb...
iocgtlbd 41723 An element of a left-open ...
xrtgioo2 41724 The topology on the extend...
tgioo4 41725 The standard topology on t...
fsumclf 41726 Closure of a finite sum of...
fsummulc1f 41727 Closure of a finite sum of...
fsumnncl 41728 Closure of a nonempty, fin...
fsumsplit1 41729 Separate out a term in a f...
fsumge0cl 41730 The finite sum of nonnegat...
fsumf1of 41731 Re-index a finite sum usin...
fsumiunss 41732 Sum over a disjoint indexe...
fsumreclf 41733 Closure of a finite sum of...
fsumlessf 41734 A shorter sum of nonnegati...
fsumsupp0 41735 Finite sum of function val...
fsumsermpt 41736 A finite sum expressed in ...
fmul01 41737 Multiplying a finite numbe...
fmulcl 41738 If ' Y ' is closed under t...
fmuldfeqlem1 41739 induction step for the pro...
fmuldfeq 41740 X and Z are two equivalent...
fmul01lt1lem1 41741 Given a finite multiplicat...
fmul01lt1lem2 41742 Given a finite multiplicat...
fmul01lt1 41743 Given a finite multiplicat...
cncfmptss 41744 A continuous complex funct...
rrpsscn 41745 The positive reals are a s...
mulc1cncfg 41746 A version of ~ mulc1cncf u...
infrglb 41747 The infimum of a nonempty ...
expcnfg 41748 If ` F ` is a complex cont...
prodeq2ad 41749 Equality deduction for pro...
fprodsplit1 41750 Separate out a term in a f...
fprodexp 41751 Positive integer exponenti...
fprodabs2 41752 The absolute value of a fi...
fprod0 41753 A finite product with a ze...
mccllem 41754 * Induction step for ~ mcc...
mccl 41755 A multinomial coefficient,...
fprodcnlem 41756 A finite product of functi...
fprodcn 41757 A finite product of functi...
clim1fr1 41758 A class of sequences of fr...
isumneg 41759 Negation of a converging s...
climrec 41760 Limit of the reciprocal of...
climmulf 41761 A version of ~ climmul usi...
climexp 41762 The limit of natural power...
climinf 41763 A bounded monotonic noninc...
climsuselem1 41764 The subsequence index ` I ...
climsuse 41765 A subsequence ` G ` of a c...
climrecf 41766 A version of ~ climrec usi...
climneg 41767 Complex limit of the negat...
climinff 41768 A version of ~ climinf usi...
climdivf 41769 Limit of the ratio of two ...
climreeq 41770 If ` F ` is a real functio...
ellimciota 41771 An explicit value for the ...
climaddf 41772 A version of ~ climadd usi...
mullimc 41773 Limit of the product of tw...
ellimcabssub0 41774 An equivalent condition fo...
limcdm0 41775 If a function has empty do...
islptre 41776 An equivalence condition f...
limccog 41777 Limit of the composition o...
limciccioolb 41778 The limit of a function at...
climf 41779 Express the predicate: Th...
mullimcf 41780 Limit of the multiplicatio...
constlimc 41781 Limit of constant function...
rexlim2d 41782 Inference removing two res...
idlimc 41783 Limit of the identity func...
divcnvg 41784 The sequence of reciprocal...
limcperiod 41785 If ` F ` is a periodic fun...
limcrecl 41786 If ` F ` is a real-valued ...
sumnnodd 41787 A series indexed by ` NN `...
lptioo2 41788 The upper bound of an open...
lptioo1 41789 The lower bound of an open...
elprn1 41790 A member of an unordered p...
elprn2 41791 A member of an unordered p...
limcmptdm 41792 The domain of a maps-to fu...
clim2f 41793 Express the predicate: Th...
limcicciooub 41794 The limit of a function at...
ltmod 41795 A sufficient condition for...
islpcn 41796 A characterization for a l...
lptre2pt 41797 If a set in the real line ...
limsupre 41798 If a sequence is bounded, ...
limcresiooub 41799 The left limit doesn't cha...
limcresioolb 41800 The right limit doesn't ch...
limcleqr 41801 If the left and the right ...
lptioo2cn 41802 The upper bound of an open...
lptioo1cn 41803 The lower bound of an open...
neglimc 41804 Limit of the negative func...
addlimc 41805 Sum of two limits. (Contr...
0ellimcdiv 41806 If the numerator converges...
clim2cf 41807 Express the predicate ` F ...
limclner 41808 For a limit point, both fr...
sublimc 41809 Subtraction of two limits....
reclimc 41810 Limit of the reciprocal of...
clim0cf 41811 Express the predicate ` F ...
limclr 41812 For a limit point, both fr...
divlimc 41813 Limit of the quotient of t...
expfac 41814 Factorial grows faster tha...
climconstmpt 41815 A constant sequence conver...
climresmpt 41816 A function restricted to u...
climsubmpt 41817 Limit of the difference of...
climsubc2mpt 41818 Limit of the difference of...
climsubc1mpt 41819 Limit of the difference of...
fnlimfv 41820 The value of the limit fun...
climreclf 41821 The limit of a convergent ...
climeldmeq 41822 Two functions that are eve...
climf2 41823 Express the predicate: Th...
fnlimcnv 41824 The sequence of function v...
climeldmeqmpt 41825 Two functions that are eve...
climfveq 41826 Two functions that are eve...
clim2f2 41827 Express the predicate: Th...
climfveqmpt 41828 Two functions that are eve...
climd 41829 Express the predicate: Th...
clim2d 41830 The limit of complex numbe...
fnlimfvre 41831 The limit function of real...
allbutfifvre 41832 Given a sequence of real-v...
climleltrp 41833 The limit of complex numbe...
fnlimfvre2 41834 The limit function of real...
fnlimf 41835 The limit function of real...
fnlimabslt 41836 A sequence of function val...
climfveqf 41837 Two functions that are eve...
climmptf 41838 Exhibit a function ` G ` w...
climfveqmpt3 41839 Two functions that are eve...
climeldmeqf 41840 Two functions that are eve...
climreclmpt 41841 The limit of B convergent ...
limsupref 41842 If a sequence is bounded, ...
limsupbnd1f 41843 If a sequence is eventuall...
climbddf 41844 A converging sequence of c...
climeqf 41845 Two functions that are eve...
climeldmeqmpt3 41846 Two functions that are eve...
limsupcld 41847 Closure of the superior li...
climfv 41848 The limit of a convergent ...
limsupval3 41849 The superior limit of an i...
climfveqmpt2 41850 Two functions that are eve...
limsup0 41851 The superior limit of the ...
climeldmeqmpt2 41852 Two functions that are eve...
limsupresre 41853 The supremum limit of a fu...
climeqmpt 41854 Two functions that are eve...
climfvd 41855 The limit of a convergent ...
limsuplesup 41856 An upper bound for the sup...
limsupresico 41857 The superior limit doesn't...
limsuppnfdlem 41858 If the restriction of a fu...
limsuppnfd 41859 If the restriction of a fu...
limsupresuz 41860 If the real part of the do...
limsupub 41861 If the limsup is not ` +oo...
limsupres 41862 The superior limit of a re...
climinf2lem 41863 A convergent, nonincreasin...
climinf2 41864 A convergent, nonincreasin...
limsupvaluz 41865 The superior limit, when t...
limsupresuz2 41866 If the domain of a functio...
limsuppnflem 41867 If the restriction of a fu...
limsuppnf 41868 If the restriction of a fu...
limsupubuzlem 41869 If the limsup is not ` +oo...
limsupubuz 41870 For a real-valued function...
climinf2mpt 41871 A bounded below, monotonic...
climinfmpt 41872 A bounded below, monotonic...
climinf3 41873 A convergent, nonincreasin...
limsupvaluzmpt 41874 The superior limit, when t...
limsupequzmpt2 41875 Two functions that are eve...
limsupubuzmpt 41876 If the limsup is not ` +oo...
limsupmnflem 41877 The superior limit of a fu...
limsupmnf 41878 The superior limit of a fu...
limsupequzlem 41879 Two functions that are eve...
limsupequz 41880 Two functions that are eve...
limsupre2lem 41881 Given a function on the ex...
limsupre2 41882 Given a function on the ex...
limsupmnfuzlem 41883 The superior limit of a fu...
limsupmnfuz 41884 The superior limit of a fu...
limsupequzmptlem 41885 Two functions that are eve...
limsupequzmpt 41886 Two functions that are eve...
limsupre2mpt 41887 Given a function on the ex...
limsupequzmptf 41888 Two functions that are eve...
limsupre3lem 41889 Given a function on the ex...
limsupre3 41890 Given a function on the ex...
limsupre3mpt 41891 Given a function on the ex...
limsupre3uzlem 41892 Given a function on the ex...
limsupre3uz 41893 Given a function on the ex...
limsupreuz 41894 Given a function on the re...
limsupvaluz2 41895 The superior limit, when t...
limsupreuzmpt 41896 Given a function on the re...
supcnvlimsup 41897 If a function on a set of ...
supcnvlimsupmpt 41898 If a function on a set of ...
0cnv 41899 If (/) is a complex number...
climuzlem 41900 Express the predicate: Th...
climuz 41901 Express the predicate: Th...
lmbr3v 41902 Express the binary relatio...
climisp 41903 If a sequence converges to...
lmbr3 41904 Express the binary relatio...
climrescn 41905 A sequence converging w.r....
climxrrelem 41906 If a seqence ranging over ...
climxrre 41907 If a sequence ranging over...
limsuplt2 41910 The defining property of t...
liminfgord 41911 Ordering property of the i...
limsupvald 41912 The superior limit of a se...
limsupresicompt 41913 The superior limit doesn't...
limsupcli 41914 Closure of the superior li...
liminfgf 41915 Closure of the inferior li...
liminfval 41916 The inferior limit of a se...
climlimsup 41917 A sequence of real numbers...
limsupge 41918 The defining property of t...
liminfgval 41919 Value of the inferior limi...
liminfcl 41920 Closure of the inferior li...
liminfvald 41921 The inferior limit of a se...
liminfval5 41922 The inferior limit of an i...
limsupresxr 41923 The superior limit of a fu...
liminfresxr 41924 The inferior limit of a fu...
liminfval2 41925 The superior limit, relati...
climlimsupcex 41926 Counterexample for ~ climl...
liminfcld 41927 Closure of the inferior li...
liminfresico 41928 The inferior limit doesn't...
limsup10exlem 41929 The range of the given fun...
limsup10ex 41930 The superior limit of a fu...
liminf10ex 41931 The inferior limit of a fu...
liminflelimsuplem 41932 The superior limit is grea...
liminflelimsup 41933 The superior limit is grea...
limsupgtlem 41934 For any positive real, the...
limsupgt 41935 Given a sequence of real n...
liminfresre 41936 The inferior limit of a fu...
liminfresicompt 41937 The inferior limit doesn't...
liminfltlimsupex 41938 An example where the ` lim...
liminfgelimsup 41939 The inferior limit is grea...
liminfvalxr 41940 Alternate definition of ` ...
liminfresuz 41941 If the real part of the do...
liminflelimsupuz 41942 The superior limit is grea...
liminfvalxrmpt 41943 Alternate definition of ` ...
liminfresuz2 41944 If the domain of a functio...
liminfgelimsupuz 41945 The inferior limit is grea...
liminfval4 41946 Alternate definition of ` ...
liminfval3 41947 Alternate definition of ` ...
liminfequzmpt2 41948 Two functions that are eve...
liminfvaluz 41949 Alternate definition of ` ...
liminf0 41950 The inferior limit of the ...
limsupval4 41951 Alternate definition of ` ...
liminfvaluz2 41952 Alternate definition of ` ...
liminfvaluz3 41953 Alternate definition of ` ...
liminflelimsupcex 41954 A counterexample for ~ lim...
limsupvaluz3 41955 Alternate definition of ` ...
liminfvaluz4 41956 Alternate definition of ` ...
limsupvaluz4 41957 Alternate definition of ` ...
climliminflimsupd 41958 If a sequence of real numb...
liminfreuzlem 41959 Given a function on the re...
liminfreuz 41960 Given a function on the re...
liminfltlem 41961 Given a sequence of real n...
liminflt 41962 Given a sequence of real n...
climliminf 41963 A sequence of real numbers...
liminflimsupclim 41964 A sequence of real numbers...
climliminflimsup 41965 A sequence of real numbers...
climliminflimsup2 41966 A sequence of real numbers...
climliminflimsup3 41967 A sequence of real numbers...
climliminflimsup4 41968 A sequence of real numbers...
limsupub2 41969 A extended real valued fun...
limsupubuz2 41970 A sequence with values in ...
xlimpnfxnegmnf 41971 A sequence converges to ` ...
liminflbuz2 41972 A sequence with values in ...
liminfpnfuz 41973 The inferior limit of a fu...
liminflimsupxrre 41974 A sequence with values in ...
xlimrel 41977 The limit on extended real...
xlimres 41978 A function converges iff i...
xlimcl 41979 The limit of a sequence of...
rexlimddv2 41980 Restricted existential eli...
xlimclim 41981 Given a sequence of reals,...
xlimconst 41982 A constant sequence conver...
climxlim 41983 A converging sequence in t...
xlimbr 41984 Express the binary relatio...
fuzxrpmcn 41985 A function mapping from an...
cnrefiisplem 41986 Lemma for ~ cnrefiisp (som...
cnrefiisp 41987 A non-real, complex number...
xlimxrre 41988 If a sequence ranging over...
xlimmnfvlem1 41989 Lemma for ~ xlimmnfv : the...
xlimmnfvlem2 41990 Lemma for ~ xlimmnf : the ...
xlimmnfv 41991 A function converges to mi...
xlimconst2 41992 A sequence that eventually...
xlimpnfvlem1 41993 Lemma for ~ xlimpnfv : the...
xlimpnfvlem2 41994 Lemma for ~ xlimpnfv : the...
xlimpnfv 41995 A function converges to pl...
xlimclim2lem 41996 Lemma for ~ xlimclim2 . H...
xlimclim2 41997 Given a sequence of extend...
xlimmnf 41998 A function converges to mi...
xlimpnf 41999 A function converges to pl...
xlimmnfmpt 42000 A function converges to pl...
xlimpnfmpt 42001 A function converges to pl...
climxlim2lem 42002 In this lemma for ~ climxl...
climxlim2 42003 A sequence of extended rea...
dfxlim2v 42004 An alternative definition ...
dfxlim2 42005 An alternative definition ...
climresd 42006 A function restricted to u...
climresdm 42007 A real function converges ...
dmclimxlim 42008 A real valued sequence tha...
xlimmnflimsup2 42009 A sequence of extended rea...
xlimuni 42010 An infinite sequence conve...
xlimclimdm 42011 A sequence of extended rea...
xlimfun 42012 The convergence relation o...
xlimmnflimsup 42013 If a sequence of extended ...
xlimdm 42014 Two ways to express that a...
xlimpnfxnegmnf2 42015 A sequence converges to ` ...
xlimresdm 42016 A function converges in th...
xlimpnfliminf 42017 If a sequence of extended ...
xlimpnfliminf2 42018 A sequence of extended rea...
xlimliminflimsup 42019 A sequence of extended rea...
xlimlimsupleliminf 42020 A sequence of extended rea...
coseq0 42021 A complex number whose cos...
sinmulcos 42022 Multiplication formula for...
coskpi2 42023 The cosine of an integer m...
cosnegpi 42024 The cosine of negative ` _...
sinaover2ne0 42025 If ` A ` in ` ( 0 , 2 _pi ...
cosknegpi 42026 The cosine of an integer m...
mulcncff 42027 The multiplication of two ...
subcncf 42028 The addition of two contin...
cncfmptssg 42029 A continuous complex funct...
constcncfg 42030 A constant function is a c...
idcncfg 42031 The identity function is a...
addcncf 42032 The addition of two contin...
cncfshift 42033 A periodic continuous func...
resincncf 42034 ` sin ` restricted to real...
addccncf2 42035 Adding a constant is a con...
0cnf 42036 The empty set is a continu...
fsumcncf 42037 The finite sum of continuo...
cncfperiod 42038 A periodic continuous func...
subcncff 42039 The subtraction of two con...
negcncfg 42040 The opposite of a continuo...
cnfdmsn 42041 A function with a singleto...
cncfcompt 42042 Composition of continuous ...
addcncff 42043 The sum of two continuous ...
ioccncflimc 42044 Limit at the upper bound o...
cncfuni 42045 A complex function on a su...
icccncfext 42046 A continuous function on a...
cncficcgt0 42047 A the absolute value of a ...
icocncflimc 42048 Limit at the lower bound, ...
cncfdmsn 42049 A complex function with a ...
divcncff 42050 The quotient of two contin...
cncfshiftioo 42051 A periodic continuous func...
cncfiooicclem1 42052 A continuous function ` F ...
cncfiooicc 42053 A continuous function ` F ...
cncfiooiccre 42054 A continuous function ` F ...
cncfioobdlem 42055 ` G ` actually extends ` F...
cncfioobd 42056 A continuous function ` F ...
jumpncnp 42057 Jump discontinuity or disc...
cncfcompt2 42058 Composition of continuous ...
cxpcncf2 42059 The complex power function...
fprodcncf 42060 The finite product of cont...
add1cncf 42061 Addition to a constant is ...
add2cncf 42062 Addition to a constant is ...
sub1cncfd 42063 Subtracting a constant is ...
sub2cncfd 42064 Subtraction from a constan...
fprodsub2cncf 42065 ` F ` is continuous. (Con...
fprodadd2cncf 42066 ` F ` is continuous. (Con...
fprodsubrecnncnvlem 42067 The sequence ` S ` of fini...
fprodsubrecnncnv 42068 The sequence ` S ` of fini...
fprodaddrecnncnvlem 42069 The sequence ` S ` of fini...
fprodaddrecnncnv 42070 The sequence ` S ` of fini...
dvsinexp 42071 The derivative of sin^N . ...
dvcosre 42072 The real derivative of the...
dvsinax 42073 Derivative exercise: the d...
dvsubf 42074 The subtraction rule for e...
dvmptconst 42075 Function-builder for deriv...
dvcnre 42076 From compex differentiatio...
dvmptidg 42077 Function-builder for deriv...
dvresntr 42078 Function-builder for deriv...
fperdvper 42079 The derivative of a period...
dvmptresicc 42080 Derivative of a function r...
dvasinbx 42081 Derivative exercise: the d...
dvresioo 42082 Restriction of a derivativ...
dvdivf 42083 The quotient rule for ever...
dvdivbd 42084 A sufficient condition for...
dvsubcncf 42085 A sufficient condition for...
dvmulcncf 42086 A sufficient condition for...
dvcosax 42087 Derivative exercise: the d...
dvdivcncf 42088 A sufficient condition for...
dvbdfbdioolem1 42089 Given a function with boun...
dvbdfbdioolem2 42090 A function on an open inte...
dvbdfbdioo 42091 A function on an open inte...
ioodvbdlimc1lem1 42092 If ` F ` has bounded deriv...
ioodvbdlimc1lem2 42093 Limit at the lower bound o...
ioodvbdlimc1 42094 A real function with bound...
ioodvbdlimc2lem 42095 Limit at the upper bound o...
ioodvbdlimc2 42096 A real function with bound...
dvdmsscn 42097 ` X ` is a subset of ` CC ...
dvmptmulf 42098 Function-builder for deriv...
dvnmptdivc 42099 Function-builder for itera...
dvdsn1add 42100 If ` K ` divides ` N ` but...
dvxpaek 42101 Derivative of the polynomi...
dvnmptconst 42102 The ` N ` -th derivative o...
dvnxpaek 42103 The ` n ` -th derivative o...
dvnmul 42104 Function-builder for the `...
dvmptfprodlem 42105 Induction step for ~ dvmpt...
dvmptfprod 42106 Function-builder for deriv...
dvnprodlem1 42107 ` D ` is bijective. (Cont...
dvnprodlem2 42108 Induction step for ~ dvnpr...
dvnprodlem3 42109 The multinomial formula fo...
dvnprod 42110 The multinomial formula fo...
itgsin0pilem1 42111 Calculation of the integra...
ibliccsinexp 42112 sin^n on a closed interval...
itgsin0pi 42113 Calculation of the integra...
iblioosinexp 42114 sin^n on an open integral ...
itgsinexplem1 42115 Integration by parts is ap...
itgsinexp 42116 A recursive formula for th...
iblconstmpt 42117 A constant function is int...
itgeq1d 42118 Equality theorem for an in...
mbfres2cn 42119 Measurability of a piecewi...
vol0 42120 The measure of the empty s...
ditgeqiooicc 42121 A function ` F ` on an ope...
volge0 42122 The volume of a set is alw...
cnbdibl 42123 A continuous bounded funct...
snmbl 42124 A singleton is measurable....
ditgeq3d 42125 Equality theorem for the d...
iblempty 42126 The empty function is inte...
iblsplit 42127 The union of two integrabl...
volsn 42128 A singleton has 0 Lebesgue...
itgvol0 42129 If the domani is negligibl...
itgcoscmulx 42130 Exercise: the integral of ...
iblsplitf 42131 A version of ~ iblsplit us...
ibliooicc 42132 If a function is integrabl...
volioc 42133 The measure of a left-open...
iblspltprt 42134 If a function is integrabl...
itgsincmulx 42135 Exercise: the integral of ...
itgsubsticclem 42136 lemma for ~ itgsubsticc . ...
itgsubsticc 42137 Integration by u-substitut...
itgioocnicc 42138 The integral of a piecewis...
iblcncfioo 42139 A continuous function ` F ...
itgspltprt 42140 The ` S. ` integral splits...
itgiccshift 42141 The integral of a function...
itgperiod 42142 The integral of a periodic...
itgsbtaddcnst 42143 Integral substitution, add...
itgeq2d 42144 Equality theorem for an in...
volico 42145 The measure of left-closed...
sublevolico 42146 The Lebesgue measure of a ...
dmvolss 42147 Lebesgue measurable sets a...
ismbl3 42148 The predicate " ` A ` is L...
volioof 42149 The function that assigns ...
ovolsplit 42150 The Lebesgue outer measure...
fvvolioof 42151 The function value of the ...
volioore 42152 The measure of an open int...
fvvolicof 42153 The function value of the ...
voliooico 42154 An open interval and a lef...
ismbl4 42155 The predicate " ` A ` is L...
volioofmpt 42156 ` ( ( vol o. (,) ) o. F ) ...
volicoff 42157 ` ( ( vol o. [,) ) o. F ) ...
voliooicof 42158 The Lebesgue measure of op...
volicofmpt 42159 ` ( ( vol o. [,) ) o. F ) ...
volicc 42160 The Lebesgue measure of a ...
voliccico 42161 A closed interval and a le...
mbfdmssre 42162 The domain of a measurable...
stoweidlem1 42163 Lemma for ~ stoweid . Thi...
stoweidlem2 42164 lemma for ~ stoweid : here...
stoweidlem3 42165 Lemma for ~ stoweid : if `...
stoweidlem4 42166 Lemma for ~ stoweid : a cl...
stoweidlem5 42167 There exists a δ as ...
stoweidlem6 42168 Lemma for ~ stoweid : two ...
stoweidlem7 42169 This lemma is used to prov...
stoweidlem8 42170 Lemma for ~ stoweid : two ...
stoweidlem9 42171 Lemma for ~ stoweid : here...
stoweidlem10 42172 Lemma for ~ stoweid . Thi...
stoweidlem11 42173 This lemma is used to prov...
stoweidlem12 42174 Lemma for ~ stoweid . Thi...
stoweidlem13 42175 Lemma for ~ stoweid . Thi...
stoweidlem14 42176 There exists a ` k ` as in...
stoweidlem15 42177 This lemma is used to prov...
stoweidlem16 42178 Lemma for ~ stoweid . The...
stoweidlem17 42179 This lemma proves that the...
stoweidlem18 42180 This theorem proves Lemma ...
stoweidlem19 42181 If a set of real functions...
stoweidlem20 42182 If a set A of real functio...
stoweidlem21 42183 Once the Stone Weierstrass...
stoweidlem22 42184 If a set of real functions...
stoweidlem23 42185 This lemma is used to prov...
stoweidlem24 42186 This lemma proves that for...
stoweidlem25 42187 This lemma proves that for...
stoweidlem26 42188 This lemma is used to prov...
stoweidlem27 42189 This lemma is used to prov...
stoweidlem28 42190 There exists a δ as ...
stoweidlem29 42191 When the hypothesis for th...
stoweidlem30 42192 This lemma is used to prov...
stoweidlem31 42193 This lemma is used to prov...
stoweidlem32 42194 If a set A of real functio...
stoweidlem33 42195 If a set of real functions...
stoweidlem34 42196 This lemma proves that for...
stoweidlem35 42197 This lemma is used to prov...
stoweidlem36 42198 This lemma is used to prov...
stoweidlem37 42199 This lemma is used to prov...
stoweidlem38 42200 This lemma is used to prov...
stoweidlem39 42201 This lemma is used to prov...
stoweidlem40 42202 This lemma proves that q_n...
stoweidlem41 42203 This lemma is used to prov...
stoweidlem42 42204 This lemma is used to prov...
stoweidlem43 42205 This lemma is used to prov...
stoweidlem44 42206 This lemma is used to prov...
stoweidlem45 42207 This lemma proves that, gi...
stoweidlem46 42208 This lemma proves that set...
stoweidlem47 42209 Subtracting a constant fro...
stoweidlem48 42210 This lemma is used to prov...
stoweidlem49 42211 There exists a function q_...
stoweidlem50 42212 This lemma proves that set...
stoweidlem51 42213 There exists a function x ...
stoweidlem52 42214 There exists a neighborood...
stoweidlem53 42215 This lemma is used to prov...
stoweidlem54 42216 There exists a function ` ...
stoweidlem55 42217 This lemma proves the exis...
stoweidlem56 42218 This theorem proves Lemma ...
stoweidlem57 42219 There exists a function x ...
stoweidlem58 42220 This theorem proves Lemma ...
stoweidlem59 42221 This lemma proves that the...
stoweidlem60 42222 This lemma proves that the...
stoweidlem61 42223 This lemma proves that the...
stoweidlem62 42224 This theorem proves the St...
stoweid 42225 This theorem proves the St...
stowei 42226 This theorem proves the St...
wallispilem1 42227 ` I ` is monotone: increas...
wallispilem2 42228 A first set of properties ...
wallispilem3 42229 I maps to real values. (C...
wallispilem4 42230 ` F ` maps to explicit exp...
wallispilem5 42231 The sequence ` H ` converg...
wallispi 42232 Wallis' formula for π :...
wallispi2lem1 42233 An intermediate step betwe...
wallispi2lem2 42234 Two expressions are proven...
wallispi2 42235 An alternative version of ...
stirlinglem1 42236 A simple limit of fraction...
stirlinglem2 42237 ` A ` maps to positive rea...
stirlinglem3 42238 Long but simple algebraic ...
stirlinglem4 42239 Algebraic manipulation of ...
stirlinglem5 42240 If ` T ` is between ` 0 ` ...
stirlinglem6 42241 A series that converges to...
stirlinglem7 42242 Algebraic manipulation of ...
stirlinglem8 42243 If ` A ` converges to ` C ...
stirlinglem9 42244 ` ( ( B `` N ) - ( B `` ( ...
stirlinglem10 42245 A bound for any B(N)-B(N +...
stirlinglem11 42246 ` B ` is decreasing. (Con...
stirlinglem12 42247 The sequence ` B ` is boun...
stirlinglem13 42248 ` B ` is decreasing and ha...
stirlinglem14 42249 The sequence ` A ` converg...
stirlinglem15 42250 The Stirling's formula is ...
stirling 42251 Stirling's approximation f...
stirlingr 42252 Stirling's approximation f...
dirkerval 42253 The N_th Dirichlet Kernel....
dirker2re 42254 The Dirchlet Kernel value ...
dirkerdenne0 42255 The Dirchlet Kernel denomi...
dirkerval2 42256 The N_th Dirichlet Kernel ...
dirkerre 42257 The Dirichlet Kernel at an...
dirkerper 42258 the Dirichlet Kernel has p...
dirkerf 42259 For any natural number ` N...
dirkertrigeqlem1 42260 Sum of an even number of a...
dirkertrigeqlem2 42261 Trigonomic equality lemma ...
dirkertrigeqlem3 42262 Trigonometric equality lem...
dirkertrigeq 42263 Trigonometric equality for...
dirkeritg 42264 The definite integral of t...
dirkercncflem1 42265 If ` Y ` is a multiple of ...
dirkercncflem2 42266 Lemma used to prove that t...
dirkercncflem3 42267 The Dirichlet Kernel is co...
dirkercncflem4 42268 The Dirichlet Kernel is co...
dirkercncf 42269 For any natural number ` N...
fourierdlem1 42270 A partition interval is a ...
fourierdlem2 42271 Membership in a partition....
fourierdlem3 42272 Membership in a partition....
fourierdlem4 42273 ` E ` is a function that m...
fourierdlem5 42274 ` S ` is a function. (Con...
fourierdlem6 42275 ` X ` is in the periodic p...
fourierdlem7 42276 The difference between the...
fourierdlem8 42277 A partition interval is a ...
fourierdlem9 42278 ` H ` is a complex functio...
fourierdlem10 42279 Condition on the bounds of...
fourierdlem11 42280 If there is a partition, t...
fourierdlem12 42281 A point of a partition is ...
fourierdlem13 42282 Value of ` V ` in terms of...
fourierdlem14 42283 Given the partition ` V ` ...
fourierdlem15 42284 The range of the partition...
fourierdlem16 42285 The coefficients of the fo...
fourierdlem17 42286 The defined ` L ` is actua...
fourierdlem18 42287 The function ` S ` is cont...
fourierdlem19 42288 If two elements of ` D ` h...
fourierdlem20 42289 Every interval in the part...
fourierdlem21 42290 The coefficients of the fo...
fourierdlem22 42291 The coefficients of the fo...
fourierdlem23 42292 If ` F ` is continuous and...
fourierdlem24 42293 A sufficient condition for...
fourierdlem25 42294 If ` C ` is not in the ran...
fourierdlem26 42295 Periodic image of a point ...
fourierdlem27 42296 A partition open interval ...
fourierdlem28 42297 Derivative of ` ( F `` ( X...
fourierdlem29 42298 Explicit function value fo...
fourierdlem30 42299 Sum of three small pieces ...
fourierdlem31 42300 If ` A ` is finite and for...
fourierdlem32 42301 Limit of a continuous func...
fourierdlem33 42302 Limit of a continuous func...
fourierdlem34 42303 A partition is one to one....
fourierdlem35 42304 There is a single point in...
fourierdlem36 42305 ` F ` is an isomorphism. ...
fourierdlem37 42306 ` I ` is a function that m...
fourierdlem38 42307 The function ` F ` is cont...
fourierdlem39 42308 Integration by parts of ...
fourierdlem40 42309 ` H ` is a continuous func...
fourierdlem41 42310 Lemma used to prove that e...
fourierdlem42 42311 The set of points in a mov...
fourierdlem43 42312 ` K ` is a real function. ...
fourierdlem44 42313 A condition for having ` (...
fourierdlem46 42314 The function ` F ` has a l...
fourierdlem47 42315 For ` r ` large enough, th...
fourierdlem48 42316 The given periodic functio...
fourierdlem49 42317 The given periodic functio...
fourierdlem50 42318 Continuity of ` O ` and it...
fourierdlem51 42319 ` X ` is in the periodic p...
fourierdlem52 42320 d16:d17,d18:jca |- ( ph ->...
fourierdlem53 42321 The limit of ` F ( s ) ` a...
fourierdlem54 42322 Given a partition ` Q ` an...
fourierdlem55 42323 ` U ` is a real function. ...
fourierdlem56 42324 Derivative of the ` K ` fu...
fourierdlem57 42325 The derivative of ` O ` . ...
fourierdlem58 42326 The derivative of ` K ` is...
fourierdlem59 42327 The derivative of ` H ` is...
fourierdlem60 42328 Given a differentiable fun...
fourierdlem61 42329 Given a differentiable fun...
fourierdlem62 42330 The function ` K ` is cont...
fourierdlem63 42331 The upper bound of interva...
fourierdlem64 42332 The partition ` V ` is fin...
fourierdlem65 42333 The distance of two adjace...
fourierdlem66 42334 Value of the ` G ` functio...
fourierdlem67 42335 ` G ` is a function. (Con...
fourierdlem68 42336 The derivative of ` O ` is...
fourierdlem69 42337 A piecewise continuous fun...
fourierdlem70 42338 A piecewise continuous fun...
fourierdlem71 42339 A periodic piecewise conti...
fourierdlem72 42340 The derivative of ` O ` is...
fourierdlem73 42341 A version of the Riemann L...
fourierdlem74 42342 Given a piecewise smooth f...
fourierdlem75 42343 Given a piecewise smooth f...
fourierdlem76 42344 Continuity of ` O ` and it...
fourierdlem77 42345 If ` H ` is bounded, then ...
fourierdlem78 42346 ` G ` is continuous when r...
fourierdlem79 42347 ` E ` projects every inter...
fourierdlem80 42348 The derivative of ` O ` is...
fourierdlem81 42349 The integral of a piecewis...
fourierdlem82 42350 Integral by substitution, ...
fourierdlem83 42351 The fourier partial sum fo...
fourierdlem84 42352 If ` F ` is piecewise coni...
fourierdlem85 42353 Limit of the function ` G ...
fourierdlem86 42354 Continuity of ` O ` and it...
fourierdlem87 42355 The integral of ` G ` goes...
fourierdlem88 42356 Given a piecewise continuo...
fourierdlem89 42357 Given a piecewise continuo...
fourierdlem90 42358 Given a piecewise continuo...
fourierdlem91 42359 Given a piecewise continuo...
fourierdlem92 42360 The integral of a piecewis...
fourierdlem93 42361 Integral by substitution (...
fourierdlem94 42362 For a piecewise smooth fun...
fourierdlem95 42363 Algebraic manipulation of ...
fourierdlem96 42364 limit for ` F ` at the low...
fourierdlem97 42365 ` F ` is continuous on the...
fourierdlem98 42366 ` F ` is continuous on the...
fourierdlem99 42367 limit for ` F ` at the upp...
fourierdlem100 42368 A piecewise continuous fun...
fourierdlem101 42369 Integral by substitution f...
fourierdlem102 42370 For a piecewise smooth fun...
fourierdlem103 42371 The half lower part of the...
fourierdlem104 42372 The half upper part of the...
fourierdlem105 42373 A piecewise continuous fun...
fourierdlem106 42374 For a piecewise smooth fun...
fourierdlem107 42375 The integral of a piecewis...
fourierdlem108 42376 The integral of a piecewis...
fourierdlem109 42377 The integral of a piecewis...
fourierdlem110 42378 The integral of a piecewis...
fourierdlem111 42379 The fourier partial sum fo...
fourierdlem112 42380 Here abbreviations (local ...
fourierdlem113 42381 Fourier series convergence...
fourierdlem114 42382 Fourier series convergence...
fourierdlem115 42383 Fourier serier convergence...
fourierd 42384 Fourier series convergence...
fourierclimd 42385 Fourier series convergence...
fourierclim 42386 Fourier series convergence...
fourier 42387 Fourier series convergence...
fouriercnp 42388 If ` F ` is continuous at ...
fourier2 42389 Fourier series convergence...
sqwvfoura 42390 Fourier coefficients for t...
sqwvfourb 42391 Fourier series ` B ` coeff...
fourierswlem 42392 The Fourier series for the...
fouriersw 42393 Fourier series convergence...
fouriercn 42394 If the derivative of ` F `...
elaa2lem 42395 Elementhood in the set of ...
elaa2 42396 Elementhood in the set of ...
etransclem1 42397 ` H ` is a function. (Con...
etransclem2 42398 Derivative of ` G ` . (Co...
etransclem3 42399 The given ` if ` term is a...
etransclem4 42400 ` F ` expressed as a finit...
etransclem5 42401 A change of bound variable...
etransclem6 42402 A change of bound variable...
etransclem7 42403 The given product is an in...
etransclem8 42404 ` F ` is a function. (Con...
etransclem9 42405 If ` K ` divides ` N ` but...
etransclem10 42406 The given ` if ` term is a...
etransclem11 42407 A change of bound variable...
etransclem12 42408 ` C ` applied to ` N ` . ...
etransclem13 42409 ` F ` applied to ` Y ` . ...
etransclem14 42410 Value of the term ` T ` , ...
etransclem15 42411 Value of the term ` T ` , ...
etransclem16 42412 Every element in the range...
etransclem17 42413 The ` N ` -th derivative o...
etransclem18 42414 The given function is inte...
etransclem19 42415 The ` N ` -th derivative o...
etransclem20 42416 ` H ` is smooth. (Contrib...
etransclem21 42417 The ` N ` -th derivative o...
etransclem22 42418 The ` N ` -th derivative o...
etransclem23 42419 This is the claim proof in...
etransclem24 42420 ` P ` divides the I -th de...
etransclem25 42421 ` P ` factorial divides th...
etransclem26 42422 Every term in the sum of t...
etransclem27 42423 The ` N ` -th derivative o...
etransclem28 42424 ` ( P - 1 ) ` factorial di...
etransclem29 42425 The ` N ` -th derivative o...
etransclem30 42426 The ` N ` -th derivative o...
etransclem31 42427 The ` N ` -th derivative o...
etransclem32 42428 This is the proof for the ...
etransclem33 42429 ` F ` is smooth. (Contrib...
etransclem34 42430 The ` N ` -th derivative o...
etransclem35 42431 ` P ` does not divide the ...
etransclem36 42432 The ` N ` -th derivative o...
etransclem37 42433 ` ( P - 1 ) ` factorial di...
etransclem38 42434 ` P ` divides the I -th de...
etransclem39 42435 ` G ` is a function. (Con...
etransclem40 42436 The ` N ` -th derivative o...
etransclem41 42437 ` P ` does not divide the ...
etransclem42 42438 The ` N ` -th derivative o...
etransclem43 42439 ` G ` is a continuous func...
etransclem44 42440 The given finite sum is no...
etransclem45 42441 ` K ` is an integer. (Con...
etransclem46 42442 This is the proof for equa...
etransclem47 42443 ` _e ` is transcendental. ...
etransclem48 42444 ` _e ` is transcendental. ...
etransc 42445 ` _e ` is transcendental. ...
rrxtopn 42446 The topology of the genera...
rrxngp 42447 Generalized Euclidean real...
rrxtps 42448 Generalized Euclidean real...
rrxtopnfi 42449 The topology of the n-dime...
rrxtopon 42450 The topology on generalize...
rrxtop 42451 The topology on generalize...
rrndistlt 42452 Given two points in the sp...
rrxtoponfi 42453 The topology on n-dimensio...
rrxunitopnfi 42454 The base set of the standa...
rrxtopn0 42455 The topology of the zero-d...
qndenserrnbllem 42456 n-dimensional rational num...
qndenserrnbl 42457 n-dimensional rational num...
rrxtopn0b 42458 The topology of the zero-d...
qndenserrnopnlem 42459 n-dimensional rational num...
qndenserrnopn 42460 n-dimensional rational num...
qndenserrn 42461 n-dimensional rational num...
rrxsnicc 42462 A multidimensional singlet...
rrnprjdstle 42463 The distance between two p...
rrndsmet 42464 ` D ` is a metric for the ...
rrndsxmet 42465 ` D ` is an extended metri...
ioorrnopnlem 42466 The a point in an indexed ...
ioorrnopn 42467 The indexed product of ope...
ioorrnopnxrlem 42468 Given a point ` F ` that b...
ioorrnopnxr 42469 The indexed product of ope...
issal 42476 Express the predicate " ` ...
pwsal 42477 The power set of a given s...
salunicl 42478 SAlg sigma-algebra is clos...
saluncl 42479 The union of two sets in a...
prsal 42480 The pair of the empty set ...
saldifcl 42481 The complement of an eleme...
0sal 42482 The empty set belongs to e...
salgenval 42483 The sigma-algebra generate...
saliuncl 42484 SAlg sigma-algebra is clos...
salincl 42485 The intersection of two se...
saluni 42486 A set is an element of any...
saliincl 42487 SAlg sigma-algebra is clos...
saldifcl2 42488 The difference of two elem...
intsaluni 42489 The union of an arbitrary ...
intsal 42490 The arbitrary intersection...
salgenn0 42491 The set used in the defini...
salgencl 42492 ` SalGen ` actually genera...
issald 42493 Sufficient condition to pr...
salexct 42494 An example of nontrivial s...
sssalgen 42495 A set is a subset of the s...
salgenss 42496 The sigma-algebra generate...
salgenuni 42497 The base set of the sigma-...
issalgend 42498 One side of ~ dfsalgen2 . ...
salexct2 42499 An example of a subset tha...
unisalgen 42500 The union of a set belongs...
dfsalgen2 42501 Alternate characterization...
salexct3 42502 An example of a sigma-alge...
salgencntex 42503 This counterexample shows ...
salgensscntex 42504 This counterexample shows ...
issalnnd 42505 Sufficient condition to pr...
dmvolsal 42506 Lebesgue measurable sets f...
saldifcld 42507 The complement of an eleme...
saluncld 42508 The union of two sets in a...
salgencld 42509 ` SalGen ` actually genera...
0sald 42510 The empty set belongs to e...
iooborel 42511 An open interval is a Bore...
salincld 42512 The intersection of two se...
salunid 42513 A set is an element of any...
unisalgen2 42514 The union of a set belongs...
bor1sal 42515 The Borel sigma-algebra on...
iocborel 42516 A left-open, right-closed ...
subsaliuncllem 42517 A subspace sigma-algebra i...
subsaliuncl 42518 A subspace sigma-algebra i...
subsalsal 42519 A subspace sigma-algebra i...
subsaluni 42520 A set belongs to the subsp...
sge0rnre 42523 When ` sum^ ` is applied t...
fge0icoicc 42524 If ` F ` maps to nonnegati...
sge0val 42525 The value of the sum of no...
fge0npnf 42526 If ` F ` maps to nonnegati...
sge0rnn0 42527 The range used in the defi...
sge0vald 42528 The value of the sum of no...
fge0iccico 42529 A range of nonnegative ext...
gsumge0cl 42530 Closure of group sum, for ...
sge0reval 42531 Value of the sum of nonneg...
sge0pnfval 42532 If a term in the sum of no...
fge0iccre 42533 A range of nonnegative ext...
sge0z 42534 Any nonnegative extended s...
sge00 42535 The sum of nonnegative ext...
fsumlesge0 42536 Every finite subsum of non...
sge0revalmpt 42537 Value of the sum of nonneg...
sge0sn 42538 A sum of a nonnegative ext...
sge0tsms 42539 ` sum^ ` applied to a nonn...
sge0cl 42540 The arbitrary sum of nonne...
sge0f1o 42541 Re-index a nonnegative ext...
sge0snmpt 42542 A sum of a nonnegative ext...
sge0ge0 42543 The sum of nonnegative ext...
sge0xrcl 42544 The arbitrary sum of nonne...
sge0repnf 42545 The of nonnegative extende...
sge0fsum 42546 The arbitrary sum of a fin...
sge0rern 42547 If the sum of nonnegative ...
sge0supre 42548 If the arbitrary sum of no...
sge0fsummpt 42549 The arbitrary sum of a fin...
sge0sup 42550 The arbitrary sum of nonne...
sge0less 42551 A shorter sum of nonnegati...
sge0rnbnd 42552 The range used in the defi...
sge0pr 42553 Sum of a pair of nonnegati...
sge0gerp 42554 The arbitrary sum of nonne...
sge0pnffigt 42555 If the sum of nonnegative ...
sge0ssre 42556 If a sum of nonnegative ex...
sge0lefi 42557 A sum of nonnegative exten...
sge0lessmpt 42558 A shorter sum of nonnegati...
sge0ltfirp 42559 If the sum of nonnegative ...
sge0prle 42560 The sum of a pair of nonne...
sge0gerpmpt 42561 The arbitrary sum of nonne...
sge0resrnlem 42562 The sum of nonnegative ext...
sge0resrn 42563 The sum of nonnegative ext...
sge0ssrempt 42564 If a sum of nonnegative ex...
sge0resplit 42565 ` sum^ ` splits into two p...
sge0le 42566 If all of the terms of sum...
sge0ltfirpmpt 42567 If the extended sum of non...
sge0split 42568 Split a sum of nonnegative...
sge0lempt 42569 If all of the terms of sum...
sge0splitmpt 42570 Split a sum of nonnegative...
sge0ss 42571 Change the index set to a ...
sge0iunmptlemfi 42572 Sum of nonnegative extende...
sge0p1 42573 The addition of the next t...
sge0iunmptlemre 42574 Sum of nonnegative extende...
sge0fodjrnlem 42575 Re-index a nonnegative ext...
sge0fodjrn 42576 Re-index a nonnegative ext...
sge0iunmpt 42577 Sum of nonnegative extende...
sge0iun 42578 Sum of nonnegative extende...
sge0nemnf 42579 The generalized sum of non...
sge0rpcpnf 42580 The sum of an infinite num...
sge0rernmpt 42581 If the sum of nonnegative ...
sge0lefimpt 42582 A sum of nonnegative exten...
nn0ssge0 42583 Nonnegative integers are n...
sge0clmpt 42584 The generalized sum of non...
sge0ltfirpmpt2 42585 If the extended sum of non...
sge0isum 42586 If a series of nonnegative...
sge0xrclmpt 42587 The generalized sum of non...
sge0xp 42588 Combine two generalized su...
sge0isummpt 42589 If a series of nonnegative...
sge0ad2en 42590 The value of the infinite ...
sge0isummpt2 42591 If a series of nonnegative...
sge0xaddlem1 42592 The extended addition of t...
sge0xaddlem2 42593 The extended addition of t...
sge0xadd 42594 The extended addition of t...
sge0fsummptf 42595 The generalized sum of a f...
sge0snmptf 42596 A sum of a nonnegative ext...
sge0ge0mpt 42597 The sum of nonnegative ext...
sge0repnfmpt 42598 The of nonnegative extende...
sge0pnffigtmpt 42599 If the generalized sum of ...
sge0splitsn 42600 Separate out a term in a g...
sge0pnffsumgt 42601 If the sum of nonnegative ...
sge0gtfsumgt 42602 If the generalized sum of ...
sge0uzfsumgt 42603 If a real number is smalle...
sge0pnfmpt 42604 If a term in the sum of no...
sge0seq 42605 A series of nonnegative re...
sge0reuz 42606 Value of the generalized s...
sge0reuzb 42607 Value of the generalized s...
ismea 42610 Express the predicate " ` ...
dmmeasal 42611 The domain of a measure is...
meaf 42612 A measure is a function th...
mea0 42613 The measure of the empty s...
nnfoctbdjlem 42614 There exists a mapping fro...
nnfoctbdj 42615 There exists a mapping fro...
meadjuni 42616 The measure of the disjoin...
meacl 42617 The measure of a set is a ...
iundjiunlem 42618 The sets in the sequence `...
iundjiun 42619 Given a sequence ` E ` of ...
meaxrcl 42620 The measure of a set is an...
meadjun 42621 The measure of the union o...
meassle 42622 The measure of a set is gr...
meaunle 42623 The measure of the union o...
meadjiunlem 42624 The sum of nonnegative ext...
meadjiun 42625 The measure of the disjoin...
ismeannd 42626 Sufficient condition to pr...
meaiunlelem 42627 The measure of the union o...
meaiunle 42628 The measure of the union o...
psmeasurelem 42629 ` M ` applied to a disjoin...
psmeasure 42630 Point supported measure, R...
voliunsge0lem 42631 The Lebesgue measure funct...
voliunsge0 42632 The Lebesgue measure funct...
volmea 42633 The Lebeasgue measure on t...
meage0 42634 If the measure of a measur...
meadjunre 42635 The measure of the union o...
meassre 42636 If the measure of a measur...
meale0eq0 42637 A measure that is less tha...
meadif 42638 The measure of the differe...
meaiuninclem 42639 Measures are continuous fr...
meaiuninc 42640 Measures are continuous fr...
meaiuninc2 42641 Measures are continuous fr...
meaiunincf 42642 Measures are continuous fr...
meaiuninc3v 42643 Measures are continuous fr...
meaiuninc3 42644 Measures are continuous fr...
meaiininclem 42645 Measures are continuous fr...
meaiininc 42646 Measures are continuous fr...
meaiininc2 42647 Measures are continuous fr...
caragenval 42652 The sigma-algebra generate...
isome 42653 Express the predicate " ` ...
caragenel 42654 Membership in the Caratheo...
omef 42655 An outer measure is a func...
ome0 42656 The outer measure of the e...
omessle 42657 The outer measure of a set...
omedm 42658 The domain of an outer mea...
caragensplit 42659 If ` E ` is in the set gen...
caragenelss 42660 An element of the Caratheo...
carageneld 42661 Membership in the Caratheo...
omecl 42662 The outer measure of a set...
caragenss 42663 The sigma-algebra generate...
omeunile 42664 The outer measure of the u...
caragen0 42665 The empty set belongs to a...
omexrcl 42666 The outer measure of a set...
caragenunidm 42667 The base set of an outer m...
caragensspw 42668 The sigma-algebra generate...
omessre 42669 If the outer measure of a ...
caragenuni 42670 The base set of the sigma-...
caragenuncllem 42671 The Caratheodory's constru...
caragenuncl 42672 The Caratheodory's constru...
caragendifcl 42673 The Caratheodory's constru...
caragenfiiuncl 42674 The Caratheodory's constru...
omeunle 42675 The outer measure of the u...
omeiunle 42676 The outer measure of the i...
omelesplit 42677 The outer measure of a set...
omeiunltfirp 42678 If the outer measure of a ...
omeiunlempt 42679 The outer measure of the i...
carageniuncllem1 42680 The outer measure of ` A i...
carageniuncllem2 42681 The Caratheodory's constru...
carageniuncl 42682 The Caratheodory's constru...
caragenunicl 42683 The Caratheodory's constru...
caragensal 42684 Caratheodory's method gene...
caratheodorylem1 42685 Lemma used to prove that C...
caratheodorylem2 42686 Caratheodory's constructio...
caratheodory 42687 Caratheodory's constructio...
0ome 42688 The map that assigns 0 to ...
isomenndlem 42689 ` O ` is sub-additive w.r....
isomennd 42690 Sufficient condition to pr...
caragenel2d 42691 Membership in the Caratheo...
omege0 42692 If the outer measure of a ...
omess0 42693 If the outer measure of a ...
caragencmpl 42694 A measure built with the C...
vonval 42699 Value of the Lebesgue meas...
ovnval 42700 Value of the Lebesgue oute...
elhoi 42701 Membership in a multidimen...
icoresmbl 42702 A closed-below, open-above...
hoissre 42703 The projection of a half-o...
ovnval2 42704 Value of the Lebesgue oute...
volicorecl 42705 The Lebesgue measure of a ...
hoiprodcl 42706 The pre-measure of half-op...
hoicvr 42707 ` I ` is a countable set o...
hoissrrn 42708 A half-open interval is a ...
ovn0val 42709 The Lebesgue outer measure...
ovnn0val 42710 The value of a (multidimen...
ovnval2b 42711 Value of the Lebesgue oute...
volicorescl 42712 The Lebesgue measure of a ...
ovnprodcl 42713 The product used in the de...
hoiprodcl2 42714 The pre-measure of half-op...
hoicvrrex 42715 Any subset of the multidim...
ovnsupge0 42716 The set used in the defini...
ovnlecvr 42717 Given a subset of multidim...
ovnpnfelsup 42718 ` +oo ` is an element of t...
ovnsslelem 42719 The (multidimensional, non...
ovnssle 42720 The (multidimensional) Leb...
ovnlerp 42721 The Lebesgue outer measure...
ovnf 42722 The Lebesgue outer measure...
ovncvrrp 42723 The Lebesgue outer measure...
ovn0lem 42724 For any finite dimension, ...
ovn0 42725 For any finite dimension, ...
ovncl 42726 The Lebesgue outer measure...
ovn02 42727 For the zero-dimensional s...
ovnxrcl 42728 The Lebesgue outer measure...
ovnsubaddlem1 42729 The Lebesgue outer measure...
ovnsubaddlem2 42730 ` ( voln* `` X ) ` is suba...
ovnsubadd 42731 ` ( voln* `` X ) ` is suba...
ovnome 42732 ` ( voln* `` X ) ` is an o...
vonmea 42733 ` ( voln `` X ) ` is a mea...
volicon0 42734 The measure of a nonempty ...
hsphoif 42735 ` H ` is a function (that ...
hoidmvval 42736 The dimensional volume of ...
hoissrrn2 42737 A half-open interval is a ...
hsphoival 42738 ` H ` is a function (that ...
hoiprodcl3 42739 The pre-measure of half-op...
volicore 42740 The Lebesgue measure of a ...
hoidmvcl 42741 The dimensional volume of ...
hoidmv0val 42742 The dimensional volume of ...
hoidmvn0val 42743 The dimensional volume of ...
hsphoidmvle2 42744 The dimensional volume of ...
hsphoidmvle 42745 The dimensional volume of ...
hoidmvval0 42746 The dimensional volume of ...
hoiprodp1 42747 The dimensional volume of ...
sge0hsphoire 42748 If the generalized sum of ...
hoidmvval0b 42749 The dimensional volume of ...
hoidmv1lelem1 42750 The supremum of ` U ` belo...
hoidmv1lelem2 42751 This is the contradiction ...
hoidmv1lelem3 42752 The dimensional volume of ...
hoidmv1le 42753 The dimensional volume of ...
hoidmvlelem1 42754 The supremum of ` U ` belo...
hoidmvlelem2 42755 This is the contradiction ...
hoidmvlelem3 42756 This is the contradiction ...
hoidmvlelem4 42757 The dimensional volume of ...
hoidmvlelem5 42758 The dimensional volume of ...
hoidmvle 42759 The dimensional volume of ...
ovnhoilem1 42760 The Lebesgue outer measure...
ovnhoilem2 42761 The Lebesgue outer measure...
ovnhoi 42762 The Lebesgue outer measure...
dmovn 42763 The domain of the Lebesgue...
hoicoto2 42764 The half-open interval exp...
dmvon 42765 Lebesgue measurable n-dime...
hoi2toco 42766 The half-open interval exp...
hoidifhspval 42767 ` D ` is a function that r...
hspval 42768 The value of the half-spac...
ovnlecvr2 42769 Given a subset of multidim...
ovncvr2 42770 ` B ` and ` T ` are the le...
dmovnsal 42771 The domain of the Lebesgue...
unidmovn 42772 Base set of the n-dimensio...
rrnmbl 42773 The set of n-dimensional R...
hoidifhspval2 42774 ` D ` is a function that r...
hspdifhsp 42775 A n-dimensional half-open ...
unidmvon 42776 Base set of the n-dimensio...
hoidifhspf 42777 ` D ` is a function that r...
hoidifhspval3 42778 ` D ` is a function that r...
hoidifhspdmvle 42779 The dimensional volume of ...
voncmpl 42780 The Lebesgue measure is co...
hoiqssbllem1 42781 The center of the n-dimens...
hoiqssbllem2 42782 The center of the n-dimens...
hoiqssbllem3 42783 A n-dimensional ball conta...
hoiqssbl 42784 A n-dimensional ball conta...
hspmbllem1 42785 Any half-space of the n-di...
hspmbllem2 42786 Any half-space of the n-di...
hspmbllem3 42787 Any half-space of the n-di...
hspmbl 42788 Any half-space of the n-di...
hoimbllem 42789 Any n-dimensional half-ope...
hoimbl 42790 Any n-dimensional half-ope...
opnvonmbllem1 42791 The half-open interval exp...
opnvonmbllem2 42792 An open subset of the n-di...
opnvonmbl 42793 An open subset of the n-di...
opnssborel 42794 Open sets of a generalized...
borelmbl 42795 All Borel subsets of the n...
volicorege0 42796 The Lebesgue measure of a ...
isvonmbl 42797 The predicate " ` A ` is m...
mblvon 42798 The n-dimensional Lebesgue...
vonmblss 42799 n-dimensional Lebesgue mea...
volico2 42800 The measure of left-closed...
vonmblss2 42801 n-dimensional Lebesgue mea...
ovolval2lem 42802 The value of the Lebesgue ...
ovolval2 42803 The value of the Lebesgue ...
ovnsubadd2lem 42804 ` ( voln* `` X ) ` is suba...
ovnsubadd2 42805 ` ( voln* `` X ) ` is suba...
ovolval3 42806 The value of the Lebesgue ...
ovnsplit 42807 The n-dimensional Lebesgue...
ovolval4lem1 42808 |- ( ( ph /\ n e. A ) -> ...
ovolval4lem2 42809 The value of the Lebesgue ...
ovolval4 42810 The value of the Lebesgue ...
ovolval5lem1 42811 ` |- ( ph -> ( sum^ `` ( n...
ovolval5lem2 42812 |- ( ( ph /\ n e. NN ) ->...
ovolval5lem3 42813 The value of the Lebesgue ...
ovolval5 42814 The value of the Lebesgue ...
ovnovollem1 42815 if ` F ` is a cover of ` B...
ovnovollem2 42816 if ` I ` is a cover of ` (...
ovnovollem3 42817 The 1-dimensional Lebesgue...
ovnovol 42818 The 1-dimensional Lebesgue...
vonvolmbllem 42819 If a subset ` B ` of real ...
vonvolmbl 42820 A subset of Real numbers i...
vonvol 42821 The 1-dimensional Lebesgue...
vonvolmbl2 42822 A subset ` X ` of the spac...
vonvol2 42823 The 1-dimensional Lebesgue...
hoimbl2 42824 Any n-dimensional half-ope...
voncl 42825 The Lebesgue measure of a ...
vonhoi 42826 The Lebesgue outer measure...
vonxrcl 42827 The Lebesgue measure of a ...
ioosshoi 42828 A n-dimensional open inter...
vonn0hoi 42829 The Lebesgue outer measure...
von0val 42830 The Lebesgue measure (for ...
vonhoire 42831 The Lebesgue measure of a ...
iinhoiicclem 42832 A n-dimensional closed int...
iinhoiicc 42833 A n-dimensional closed int...
iunhoiioolem 42834 A n-dimensional open inter...
iunhoiioo 42835 A n-dimensional open inter...
ioovonmbl 42836 Any n-dimensional open int...
iccvonmbllem 42837 Any n-dimensional closed i...
iccvonmbl 42838 Any n-dimensional closed i...
vonioolem1 42839 The sequence of the measur...
vonioolem2 42840 The n-dimensional Lebesgue...
vonioo 42841 The n-dimensional Lebesgue...
vonicclem1 42842 The sequence of the measur...
vonicclem2 42843 The n-dimensional Lebesgue...
vonicc 42844 The n-dimensional Lebesgue...
snvonmbl 42845 A n-dimensional singleton ...
vonn0ioo 42846 The n-dimensional Lebesgue...
vonn0icc 42847 The n-dimensional Lebesgue...
ctvonmbl 42848 Any n-dimensional countabl...
vonn0ioo2 42849 The n-dimensional Lebesgue...
vonsn 42850 The n-dimensional Lebesgue...
vonn0icc2 42851 The n-dimensional Lebesgue...
vonct 42852 The n-dimensional Lebesgue...
vitali2 42853 There are non-measurable s...
pimltmnf2 42856 Given a real-valued functi...
preimagelt 42857 The preimage of a right-op...
preimalegt 42858 The preimage of a left-ope...
pimconstlt0 42859 Given a constant function,...
pimconstlt1 42860 Given a constant function,...
pimltpnf 42861 Given a real-valued functi...
pimgtpnf2 42862 Given a real-valued functi...
salpreimagelt 42863 If all the preimages of le...
pimrecltpos 42864 The preimage of an unbound...
salpreimalegt 42865 If all the preimages of ri...
pimiooltgt 42866 The preimage of an open in...
preimaicomnf 42867 Preimage of an open interv...
pimltpnf2 42868 Given a real-valued functi...
pimgtmnf2 42869 Given a real-valued functi...
pimdecfgtioc 42870 Given a nonincreasing func...
pimincfltioc 42871 Given a nondecreasing func...
pimdecfgtioo 42872 Given a nondecreasing func...
pimincfltioo 42873 Given a nondecreasing func...
preimaioomnf 42874 Preimage of an open interv...
preimageiingt 42875 A preimage of a left-close...
preimaleiinlt 42876 A preimage of a left-open,...
pimgtmnf 42877 Given a real-valued functi...
pimrecltneg 42878 The preimage of an unbound...
salpreimagtge 42879 If all the preimages of le...
salpreimaltle 42880 If all the preimages of ri...
issmflem 42881 The predicate " ` F ` is a...
issmf 42882 The predicate " ` F ` is a...
salpreimalelt 42883 If all the preimages of ri...
salpreimagtlt 42884 If all the preimages of le...
smfpreimalt 42885 Given a function measurabl...
smff 42886 A function measurable w.r....
smfdmss 42887 The domain of a function m...
issmff 42888 The predicate " ` F ` is a...
issmfd 42889 A sufficient condition for...
smfpreimaltf 42890 Given a function measurabl...
issmfdf 42891 A sufficient condition for...
sssmf 42892 The restriction of a sigma...
mbfresmf 42893 A real-valued measurable f...
cnfsmf 42894 A continuous function is m...
incsmflem 42895 A nondecreasing function i...
incsmf 42896 A real-valued, nondecreasi...
smfsssmf 42897 If a function is measurabl...
issmflelem 42898 The predicate " ` F ` is a...
issmfle 42899 The predicate " ` F ` is a...
smfpimltmpt 42900 Given a function measurabl...
smfpimltxr 42901 Given a function measurabl...
issmfdmpt 42902 A sufficient condition for...
smfconst 42903 Given a sigma-algebra over...
sssmfmpt 42904 The restriction of a sigma...
cnfrrnsmf 42905 A function, continuous fro...
smfid 42906 The identity function is B...
bormflebmf 42907 A Borel measurable functio...
smfpreimale 42908 Given a function measurabl...
issmfgtlem 42909 The predicate " ` F ` is a...
issmfgt 42910 The predicate " ` F ` is a...
issmfled 42911 A sufficient condition for...
smfpimltxrmpt 42912 Given a function measurabl...
smfmbfcex 42913 A constant function, with ...
issmfgtd 42914 A sufficient condition for...
smfpreimagt 42915 Given a function measurabl...
smfaddlem1 42916 Given the sum of two funct...
smfaddlem2 42917 The sum of two sigma-measu...
smfadd 42918 The sum of two sigma-measu...
decsmflem 42919 A nonincreasing function i...
decsmf 42920 A real-valued, nonincreasi...
smfpreimagtf 42921 Given a function measurabl...
issmfgelem 42922 The predicate " ` F ` is a...
issmfge 42923 The predicate " ` F ` is a...
smflimlem1 42924 Lemma for the proof that t...
smflimlem2 42925 Lemma for the proof that t...
smflimlem3 42926 The limit of sigma-measura...
smflimlem4 42927 Lemma for the proof that t...
smflimlem5 42928 Lemma for the proof that t...
smflimlem6 42929 Lemma for the proof that t...
smflim 42930 The limit of sigma-measura...
nsssmfmbflem 42931 The sigma-measurable funct...
nsssmfmbf 42932 The sigma-measurable funct...
smfpimgtxr 42933 Given a function measurabl...
smfpimgtmpt 42934 Given a function measurabl...
smfpreimage 42935 Given a function measurabl...
mbfpsssmf 42936 Real-valued measurable fun...
smfpimgtxrmpt 42937 Given a function measurabl...
smfpimioompt 42938 Given a function measurabl...
smfpimioo 42939 Given a function measurabl...
smfresal 42940 Given a sigma-measurable f...
smfrec 42941 The reciprocal of a sigma-...
smfres 42942 The restriction of sigma-m...
smfmullem1 42943 The multiplication of two ...
smfmullem2 42944 The multiplication of two ...
smfmullem3 42945 The multiplication of two ...
smfmullem4 42946 The multiplication of two ...
smfmul 42947 The multiplication of two ...
smfmulc1 42948 A sigma-measurable functio...
smfdiv 42949 The fraction of two sigma-...
smfpimbor1lem1 42950 Every open set belongs to ...
smfpimbor1lem2 42951 Given a sigma-measurable f...
smfpimbor1 42952 Given a sigma-measurable f...
smf2id 42953 Twice the identity functio...
smfco 42954 The composition of a Borel...
smfneg 42955 The negative of a sigma-me...
smffmpt 42956 A function measurable w.r....
smflim2 42957 The limit of a sequence of...
smfpimcclem 42958 Lemma for ~ smfpimcc given...
smfpimcc 42959 Given a countable set of s...
issmfle2d 42960 A sufficient condition for...
smflimmpt 42961 The limit of a sequence of...
smfsuplem1 42962 The supremum of a countabl...
smfsuplem2 42963 The supremum of a countabl...
smfsuplem3 42964 The supremum of a countabl...
smfsup 42965 The supremum of a countabl...
smfsupmpt 42966 The supremum of a countabl...
smfsupxr 42967 The supremum of a countabl...
smfinflem 42968 The infimum of a countable...
smfinf 42969 The infimum of a countable...
smfinfmpt 42970 The infimum of a countable...
smflimsuplem1 42971 If ` H ` converges, the ` ...
smflimsuplem2 42972 The superior limit of a se...
smflimsuplem3 42973 The limit of the ` ( H `` ...
smflimsuplem4 42974 If ` H ` converges, the ` ...
smflimsuplem5 42975 ` H ` converges to the sup...
smflimsuplem6 42976 The superior limit of a se...
smflimsuplem7 42977 The superior limit of a se...
smflimsuplem8 42978 The superior limit of a se...
smflimsup 42979 The superior limit of a se...
smflimsupmpt 42980 The superior limit of a se...
smfliminflem 42981 The inferior limit of a co...
smfliminf 42982 The inferior limit of a co...
smfliminfmpt 42983 The inferior limit of a co...
sigarval 42984 Define the signed area by ...
sigarim 42985 Signed area takes value in...
sigarac 42986 Signed area is anticommuta...
sigaraf 42987 Signed area is additive by...
sigarmf 42988 Signed area is additive (w...
sigaras 42989 Signed area is additive by...
sigarms 42990 Signed area is additive (w...
sigarls 42991 Signed area is linear by t...
sigarid 42992 Signed area of a flat para...
sigarexp 42993 Expand the signed area for...
sigarperm 42994 Signed area ` ( A - C ) G ...
sigardiv 42995 If signed area between vec...
sigarimcd 42996 Signed area takes value in...
sigariz 42997 If signed area is zero, th...
sigarcol 42998 Given three points ` A ` ,...
sharhght 42999 Let ` A B C ` be a triangl...
sigaradd 43000 Subtracting (double) area ...
cevathlem1 43001 Ceva's theorem first lemma...
cevathlem2 43002 Ceva's theorem second lemm...
cevath 43003 Ceva's theorem. Let ` A B...
simpcntrab 43004 The center of a simple gro...
hirstL-ax3 43005 The third axiom of a syste...
ax3h 43006 Recover ~ ax-3 from ~ hirs...
aibandbiaiffaiffb 43007 A closed form showing (a i...
aibandbiaiaiffb 43008 A closed form showing (a i...
notatnand 43009 Do not use. Use intnanr i...
aistia 43010 Given a is equivalent to `...
aisfina 43011 Given a is equivalent to `...
bothtbothsame 43012 Given both a, b are equiva...
bothfbothsame 43013 Given both a, b are equiva...
aiffbbtat 43014 Given a is equivalent to b...
aisbbisfaisf 43015 Given a is equivalent to b...
axorbtnotaiffb 43016 Given a is exclusive to b,...
aiffnbandciffatnotciffb 43017 Given a is equivalent to (...
axorbciffatcxorb 43018 Given a is equivalent to (...
aibnbna 43019 Given a implies b, (not b)...
aibnbaif 43020 Given a implies b, not b, ...
aiffbtbat 43021 Given a is equivalent to b...
astbstanbst 43022 Given a is equivalent to T...
aistbistaandb 43023 Given a is equivalent to T...
aisbnaxb 43024 Given a is equivalent to b...
atbiffatnnb 43025 If a implies b, then a imp...
bisaiaisb 43026 Application of bicom1 with...
atbiffatnnbalt 43027 If a implies b, then a imp...
abnotbtaxb 43028 Assuming a, not b, there e...
abnotataxb 43029 Assuming not a, b, there e...
conimpf 43030 Assuming a, not b, and a i...
conimpfalt 43031 Assuming a, not b, and a i...
aistbisfiaxb 43032 Given a is equivalent to T...
aisfbistiaxb 43033 Given a is equivalent to F...
aifftbifffaibif 43034 Given a is equivalent to T...
aifftbifffaibifff 43035 Given a is equivalent to T...
atnaiana 43036 Given a, it is not the cas...
ainaiaandna 43037 Given a, a implies it is n...
abcdta 43038 Given (((a and b) and c) a...
abcdtb 43039 Given (((a and b) and c) a...
abcdtc 43040 Given (((a and b) and c) a...
abcdtd 43041 Given (((a and b) and c) a...
abciffcbatnabciffncba 43042 Operands in a biconditiona...
abciffcbatnabciffncbai 43043 Operands in a biconditiona...
nabctnabc 43044 not ( a -> ( b /\ c ) ) we...
jabtaib 43045 For when pm3.4 lacks a pm3...
onenotinotbothi 43046 From one negated implicati...
twonotinotbothi 43047 From these two negated imp...
clifte 43048 show d is the same as an i...
cliftet 43049 show d is the same as an i...
clifteta 43050 show d is the same as an i...
cliftetb 43051 show d is the same as an i...
confun 43052 Given the hypotheses there...
confun2 43053 Confun simplified to two p...
confun3 43054 Confun's more complex form...
confun4 43055 An attempt at derivative. ...
confun5 43056 An attempt at derivative. ...
plcofph 43057 Given, a,b and a "definiti...
pldofph 43058 Given, a,b c, d, "definiti...
plvcofph 43059 Given, a,b,d, and "definit...
plvcofphax 43060 Given, a,b,d, and "definit...
plvofpos 43061 rh is derivable because ON...
mdandyv0 43062 Given the equivalences set...
mdandyv1 43063 Given the equivalences set...
mdandyv2 43064 Given the equivalences set...
mdandyv3 43065 Given the equivalences set...
mdandyv4 43066 Given the equivalences set...
mdandyv5 43067 Given the equivalences set...
mdandyv6 43068 Given the equivalences set...
mdandyv7 43069 Given the equivalences set...
mdandyv8 43070 Given the equivalences set...
mdandyv9 43071 Given the equivalences set...
mdandyv10 43072 Given the equivalences set...
mdandyv11 43073 Given the equivalences set...
mdandyv12 43074 Given the equivalences set...
mdandyv13 43075 Given the equivalences set...
mdandyv14 43076 Given the equivalences set...
mdandyv15 43077 Given the equivalences set...
mdandyvr0 43078 Given the equivalences set...
mdandyvr1 43079 Given the equivalences set...
mdandyvr2 43080 Given the equivalences set...
mdandyvr3 43081 Given the equivalences set...
mdandyvr4 43082 Given the equivalences set...
mdandyvr5 43083 Given the equivalences set...
mdandyvr6 43084 Given the equivalences set...
mdandyvr7 43085 Given the equivalences set...
mdandyvr8 43086 Given the equivalences set...
mdandyvr9 43087 Given the equivalences set...
mdandyvr10 43088 Given the equivalences set...
mdandyvr11 43089 Given the equivalences set...
mdandyvr12 43090 Given the equivalences set...
mdandyvr13 43091 Given the equivalences set...
mdandyvr14 43092 Given the equivalences set...
mdandyvr15 43093 Given the equivalences set...
mdandyvrx0 43094 Given the exclusivities se...
mdandyvrx1 43095 Given the exclusivities se...
mdandyvrx2 43096 Given the exclusivities se...
mdandyvrx3 43097 Given the exclusivities se...
mdandyvrx4 43098 Given the exclusivities se...
mdandyvrx5 43099 Given the exclusivities se...
mdandyvrx6 43100 Given the exclusivities se...
mdandyvrx7 43101 Given the exclusivities se...
mdandyvrx8 43102 Given the exclusivities se...
mdandyvrx9 43103 Given the exclusivities se...
mdandyvrx10 43104 Given the exclusivities se...
mdandyvrx11 43105 Given the exclusivities se...
mdandyvrx12 43106 Given the exclusivities se...
mdandyvrx13 43107 Given the exclusivities se...
mdandyvrx14 43108 Given the exclusivities se...
mdandyvrx15 43109 Given the exclusivities se...
H15NH16TH15IH16 43110 Given 15 hypotheses and a ...
dandysum2p2e4 43111 CONTRADICTION PRO...
mdandysum2p2e4 43112 CONTRADICTION PROVED AT 1 ...
adh-jarrsc 43113 Replacement of a nested an...
adh-minim 43114 A single axiom for minimal...
adh-minim-ax1-ax2-lem1 43115 First lemma for the deriva...
adh-minim-ax1-ax2-lem2 43116 Second lemma for the deriv...
adh-minim-ax1-ax2-lem3 43117 Third lemma for the deriva...
adh-minim-ax1-ax2-lem4 43118 Fourth lemma for the deriv...
adh-minim-ax1 43119 Derivation of ~ ax-1 from ...
adh-minim-ax2-lem5 43120 Fifth lemma for the deriva...
adh-minim-ax2-lem6 43121 Sixth lemma for the deriva...
adh-minim-ax2c 43122 Derivation of a commuted f...
adh-minim-ax2 43123 Derivation of ~ ax-2 from ...
adh-minim-idALT 43124 Derivation of ~ id (reflex...
adh-minim-pm2.43 43125 Derivation of ~ pm2.43 Whi...
adh-minimp 43126 Another single axiom for m...
adh-minimp-jarr-imim1-ax2c-lem1 43127 First lemma for the deriva...
adh-minimp-jarr-lem2 43128 Second lemma for the deriv...
adh-minimp-jarr-ax2c-lem3 43129 Third lemma for the deriva...
adh-minimp-sylsimp 43130 Derivation of ~ jarr (also...
adh-minimp-ax1 43131 Derivation of ~ ax-1 from ...
adh-minimp-imim1 43132 Derivation of ~ imim1 ("le...
adh-minimp-ax2c 43133 Derivation of a commuted f...
adh-minimp-ax2-lem4 43134 Fourth lemma for the deriv...
adh-minimp-ax2 43135 Derivation of ~ ax-2 from ...
adh-minimp-idALT 43136 Derivation of ~ id (reflex...
adh-minimp-pm2.43 43137 Derivation of ~ pm2.43 Whi...
eusnsn 43138 There is a unique element ...
absnsb 43139 If the class abstraction `...
euabsneu 43140 Another way to express exi...
elprneb 43141 An element of a proper uno...
oppr 43142 Equality for ordered pairs...
opprb 43143 Equality for unordered pai...
or2expropbilem1 43144 Lemma 1 for ~ or2expropbi ...
or2expropbilem2 43145 Lemma 2 for ~ or2expropbi ...
or2expropbi 43146 If two classes are strictl...
eubrv 43147 If there is a unique set w...
eubrdm 43148 If there is a unique set w...
eldmressn 43149 Element of the domain of a...
iota0def 43150 Example for a defined iota...
iota0ndef 43151 Example for an undefined i...
fveqvfvv 43152 If a function's value at a...
fnresfnco 43153 Composition of two functio...
funcoressn 43154 A composition restricted t...
funressnfv 43155 A restriction to a singlet...
funressndmfvrn 43156 The value of a function ` ...
funressnvmo 43157 A function restricted to a...
funressnmo 43158 A function restricted to a...
funressneu 43159 There is exactly one value...
aiotajust 43161 Soundness justification th...
dfaiota2 43163 Alternate definition of th...
reuabaiotaiota 43164 The iota and the alternate...
reuaiotaiota 43165 The iota and the alternate...
aiotaexb 43166 The alternate iota over a ...
aiotavb 43167 The alternate iota over a ...
iotan0aiotaex 43168 If the iota over a wff ` p...
aiotaexaiotaiota 43169 The alternate iota over a ...
aiotaval 43170 Theorem 8.19 in [Quine] p....
aiota0def 43171 Example for a defined alte...
aiota0ndef 43172 Example for an undefined a...
r19.32 43173 Theorem 19.32 of [Margaris...
rexsb 43174 An equivalent expression f...
rexrsb 43175 An equivalent expression f...
2rexsb 43176 An equivalent expression f...
2rexrsb 43177 An equivalent expression f...
cbvral2 43178 Change bound variables of ...
cbvrex2 43179 Change bound variables of ...
2ralbiim 43180 Split a biconditional and ...
ralndv1 43181 Example for a theorem abou...
ralndv2 43182 Second example for a theor...
reuf1odnf 43183 There is exactly one eleme...
reuf1od 43184 There is exactly one eleme...
euoreqb 43185 There is a set which is eq...
2reu3 43186 Double restricted existent...
2reu7 43187 Two equivalent expressions...
2reu8 43188 Two equivalent expressions...
2reu8i 43189 Implication of a double re...
2reuimp0 43190 Implication of a double re...
2reuimp 43191 Implication of a double re...
ralbinrald 43198 Elemination of a restricte...
nvelim 43199 If a class is the universa...
alneu 43200 If a statement holds for a...
eu2ndop1stv 43201 If there is a unique secon...
dfateq12d 43202 Equality deduction for "de...
nfdfat 43203 Bound-variable hypothesis ...
dfdfat2 43204 Alternate definition of th...
fundmdfat 43205 A function is defined at a...
dfatprc 43206 A function is not defined ...
dfatelrn 43207 The value of a function ` ...
dfafv2 43208 Alternative definition of ...
afveq12d 43209 Equality deduction for fun...
afveq1 43210 Equality theorem for funct...
afveq2 43211 Equality theorem for funct...
nfafv 43212 Bound-variable hypothesis ...
csbafv12g 43213 Move class substitution in...
afvfundmfveq 43214 If a class is a function r...
afvnfundmuv 43215 If a set is not in the dom...
ndmafv 43216 The value of a class outsi...
afvvdm 43217 If the function value of a...
nfunsnafv 43218 If the restriction of a cl...
afvvfunressn 43219 If the function value of a...
afvprc 43220 A function's value at a pr...
afvvv 43221 If a function's value at a...
afvpcfv0 43222 If the value of the altern...
afvnufveq 43223 The value of the alternati...
afvvfveq 43224 The value of the alternati...
afv0fv0 43225 If the value of the altern...
afvfvn0fveq 43226 If the function's value at...
afv0nbfvbi 43227 The function's value at an...
afvfv0bi 43228 The function's value at an...
afveu 43229 The value of a function at...
fnbrafvb 43230 Equivalence of function va...
fnopafvb 43231 Equivalence of function va...
funbrafvb 43232 Equivalence of function va...
funopafvb 43233 Equivalence of function va...
funbrafv 43234 The second argument of a b...
funbrafv2b 43235 Function value in terms of...
dfafn5a 43236 Representation of a functi...
dfafn5b 43237 Representation of a functi...
fnrnafv 43238 The range of a function ex...
afvelrnb 43239 A member of a function's r...
afvelrnb0 43240 A member of a function's r...
dfaimafn 43241 Alternate definition of th...
dfaimafn2 43242 Alternate definition of th...
afvelima 43243 Function value in an image...
afvelrn 43244 A function's value belongs...
fnafvelrn 43245 A function's value belongs...
fafvelrn 43246 A function's value belongs...
ffnafv 43247 A function maps to a class...
afvres 43248 The value of a restricted ...
tz6.12-afv 43249 Function value. Theorem 6...
tz6.12-1-afv 43250 Function value (Theorem 6....
dmfcoafv 43251 Domains of a function comp...
afvco2 43252 Value of a function compos...
rlimdmafv 43253 Two ways to express that a...
aoveq123d 43254 Equality deduction for ope...
nfaov 43255 Bound-variable hypothesis ...
csbaovg 43256 Move class substitution in...
aovfundmoveq 43257 If a class is a function r...
aovnfundmuv 43258 If an ordered pair is not ...
ndmaov 43259 The value of an operation ...
ndmaovg 43260 The value of an operation ...
aovvdm 43261 If the operation value of ...
nfunsnaov 43262 If the restriction of a cl...
aovvfunressn 43263 If the operation value of ...
aovprc 43264 The value of an operation ...
aovrcl 43265 Reverse closure for an ope...
aovpcov0 43266 If the alternative value o...
aovnuoveq 43267 The alternative value of t...
aovvoveq 43268 The alternative value of t...
aov0ov0 43269 If the alternative value o...
aovovn0oveq 43270 If the operation's value a...
aov0nbovbi 43271 The operation's value on a...
aovov0bi 43272 The operation's value on a...
rspceaov 43273 A frequently used special ...
fnotaovb 43274 Equivalence of operation v...
ffnaov 43275 An operation maps to a cla...
faovcl 43276 Closure law for an operati...
aovmpt4g 43277 Value of a function given ...
aoprssdm 43278 Domain of closure of an op...
ndmaovcl 43279 The "closure" of an operat...
ndmaovrcl 43280 Reverse closure law, in co...
ndmaovcom 43281 Any operation is commutati...
ndmaovass 43282 Any operation is associati...
ndmaovdistr 43283 Any operation is distribut...
dfatafv2iota 43286 If a function is defined a...
ndfatafv2 43287 The alternate function val...
ndfatafv2undef 43288 The alternate function val...
dfatafv2ex 43289 The alternate function val...
afv2ex 43290 The alternate function val...
afv2eq12d 43291 Equality deduction for fun...
afv2eq1 43292 Equality theorem for funct...
afv2eq2 43293 Equality theorem for funct...
nfafv2 43294 Bound-variable hypothesis ...
csbafv212g 43295 Move class substitution in...
fexafv2ex 43296 The alternate function val...
ndfatafv2nrn 43297 The alternate function val...
ndmafv2nrn 43298 The value of a class outsi...
funressndmafv2rn 43299 The alternate function val...
afv2ndefb 43300 Two ways to say that an al...
nfunsnafv2 43301 If the restriction of a cl...
afv2prc 43302 A function's value at a pr...
dfatafv2rnb 43303 The alternate function val...
afv2orxorb 43304 If a set is in the range o...
dmafv2rnb 43305 The alternate function val...
fundmafv2rnb 43306 The alternate function val...
afv2elrn 43307 An alternate function valu...
afv20defat 43308 If the alternate function ...
fnafv2elrn 43309 An alternate function valu...
fafv2elrn 43310 An alternate function valu...
fafv2elrnb 43311 An alternate function valu...
frnvafv2v 43312 If the codomain of a funct...
tz6.12-2-afv2 43313 Function value when ` F ` ...
afv2eu 43314 The value of a function at...
afv2res 43315 The value of a restricted ...
tz6.12-afv2 43316 Function value (Theorem 6....
tz6.12-1-afv2 43317 Function value (Theorem 6....
tz6.12c-afv2 43318 Corollary of Theorem 6.12(...
tz6.12i-afv2 43319 Corollary of Theorem 6.12(...
funressnbrafv2 43320 The second argument of a b...
dfatbrafv2b 43321 Equivalence of function va...
dfatopafv2b 43322 Equivalence of function va...
funbrafv2 43323 The second argument of a b...
fnbrafv2b 43324 Equivalence of function va...
fnopafv2b 43325 Equivalence of function va...
funbrafv22b 43326 Equivalence of function va...
funopafv2b 43327 Equivalence of function va...
dfatsnafv2 43328 Singleton of function valu...
dfafv23 43329 A definition of function v...
dfatdmfcoafv2 43330 Domain of a function compo...
dfatcolem 43331 Lemma for ~ dfatco . (Con...
dfatco 43332 The predicate "defined at"...
afv2co2 43333 Value of a function compos...
rlimdmafv2 43334 Two ways to express that a...
dfafv22 43335 Alternate definition of ` ...
afv2ndeffv0 43336 If the alternate function ...
dfatafv2eqfv 43337 If a function is defined a...
afv2rnfveq 43338 If the alternate function ...
afv20fv0 43339 If the alternate function ...
afv2fvn0fveq 43340 If the function's value at...
afv2fv0 43341 If the function's value at...
afv2fv0b 43342 The function's value at an...
afv2fv0xorb 43343 If a set is in the range o...
an4com24 43344 Rearrangement of 4 conjunc...
3an4ancom24 43345 Commutative law for a conj...
4an21 43346 Rearrangement of 4 conjunc...
dfnelbr2 43349 Alternate definition of th...
nelbr 43350 The binary relation of a s...
nelbrim 43351 If a set is related to ano...
nelbrnel 43352 A set is related to anothe...
nelbrnelim 43353 If a set is related to ano...
ralralimp 43354 Selecting one of two alter...
otiunsndisjX 43355 The union of singletons co...
fvifeq 43356 Equality of function value...
rnfdmpr 43357 The range of a one-to-one ...
imarnf1pr 43358 The image of the range of ...
funop1 43359 A function is an ordered p...
fun2dmnopgexmpl 43360 A function with a domain c...
opabresex0d 43361 A collection of ordered pa...
opabbrfex0d 43362 A collection of ordered pa...
opabresexd 43363 A collection of ordered pa...
opabbrfexd 43364 A collection of ordered pa...
f1oresf1orab 43365 Build a bijection by restr...
f1oresf1o 43366 Build a bijection by restr...
f1oresf1o2 43367 Build a bijection by restr...
fvmptrab 43368 Value of a function mappin...
fvmptrabdm 43369 Value of a function mappin...
leltletr 43370 Transitive law, weaker for...
cnambpcma 43371 ((a-b)+c)-a = c-a holds fo...
cnapbmcpd 43372 ((a+b)-c)+d = ((a+d)+b)-c ...
addsubeq0 43373 The sum of two complex num...
leaddsuble 43374 Addition and subtraction o...
2leaddle2 43375 If two real numbers are le...
ltnltne 43376 Variant of trichotomy law ...
p1lep2 43377 A real number increasd by ...
ltsubsubaddltsub 43378 If the result of subtracti...
zm1nn 43379 An integer minus 1 is posi...
readdcnnred 43380 The sum of a real number a...
resubcnnred 43381 The difference of a real n...
recnmulnred 43382 The product of a real numb...
cndivrenred 43383 The quotient of an imagina...
sqrtnegnre 43384 The square root of a negat...
nn0resubcl 43385 Closure law for subtractio...
zgeltp1eq 43386 If an integer is between a...
1t10e1p1e11 43387 11 is 1 times 10 to the po...
deccarry 43388 Add 1 to a 2 digit number ...
eluzge0nn0 43389 If an integer is greater t...
nltle2tri 43390 Negated extended trichotom...
ssfz12 43391 Subset relationship for fi...
elfz2z 43392 Membership of an integer i...
2elfz3nn0 43393 If there are two elements ...
fz0addcom 43394 The addition of two member...
2elfz2melfz 43395 If the sum of two integers...
fz0addge0 43396 The sum of two integers in...
elfzlble 43397 Membership of an integer i...
elfzelfzlble 43398 Membership of an element o...
fzopred 43399 Join a predecessor to the ...
fzopredsuc 43400 Join a predecessor and a s...
1fzopredsuc 43401 Join 0 and a successor to ...
el1fzopredsuc 43402 An element of an open inte...
subsubelfzo0 43403 Subtracting a difference f...
fzoopth 43404 A half-open integer range ...
2ffzoeq 43405 Two functions over a half-...
m1mod0mod1 43406 An integer decreased by 1 ...
elmod2 43407 An integer modulo 2 is eit...
smonoord 43408 Ordering relation for a st...
fsummsndifre 43409 A finite sum with one of i...
fsumsplitsndif 43410 Separate out a term in a f...
fsummmodsndifre 43411 A finite sum of summands m...
fsummmodsnunz 43412 A finite sum of summands m...
setsidel 43413 The injected slot is an el...
setsnidel 43414 The injected slot is an el...
setsv 43415 The value of the structure...
preimafvsnel 43416 The preimage of a function...
preimafvn0 43417 The preimage of a function...
uniimafveqt 43418 The union of the image of ...
uniimaprimaeqfv 43419 The union of the image of ...
setpreimafvex 43420 The class ` P ` of all pre...
elsetpreimafvb 43421 The characterization of an...
elsetpreimafv 43422 An element of the class ` ...
elsetpreimafvssdm 43423 An element of the class ` ...
fvelsetpreimafv 43424 There is an element in a p...
preimafvelsetpreimafv 43425 The preimage of a function...
preimafvsspwdm 43426 The class ` P ` of all pre...
0nelsetpreimafv 43427 The empty set is not an el...
elsetpreimafvbi 43428 An element of the preimage...
elsetpreimafveqfv 43429 The elements of the preima...
eqfvelsetpreimafv 43430 If an element of the domai...
elsetpreimafvrab 43431 An element of the preimage...
imaelsetpreimafv 43432 The image of an element of...
uniimaelsetpreimafv 43433 The union of the image of ...
elsetpreimafveq 43434 If two preimages of functi...
fundcmpsurinjlem1 43435 Lemma 1 for ~ fundcmpsurin...
fundcmpsurinjlem2 43436 Lemma 2 for ~ fundcmpsurin...
fundcmpsurinjlem3 43437 Lemma 3 for ~ fundcmpsurin...
imasetpreimafvbijlemf 43438 Lemma for ~ imasetpreimafv...
imasetpreimafvbijlemfv 43439 Lemma for ~ imasetpreimafv...
imasetpreimafvbijlemfv1 43440 Lemma for ~ imasetpreimafv...
imasetpreimafvbijlemf1 43441 Lemma for ~ imasetpreimafv...
imasetpreimafvbijlemfo 43442 Lemma for ~ imasetpreimafv...
imasetpreimafvbij 43443 The mapping ` H ` is a bij...
fundcmpsurbijinjpreimafv 43444 Every function ` F : A -->...
fundcmpsurinjpreimafv 43445 Every function ` F : A -->...
fundcmpsurinj 43446 Every function ` F : A -->...
fundcmpsurbijinj 43447 Every function ` F : A -->...
fundcmpsurinjimaid 43448 Every function ` F : A -->...
fundcmpsurinjALT 43449 Alternate proof of ~ fundc...
iccpval 43452 Partition consisting of a ...
iccpart 43453 A special partition. Corr...
iccpartimp 43454 Implications for a class b...
iccpartres 43455 The restriction of a parti...
iccpartxr 43456 If there is a partition, t...
iccpartgtprec 43457 If there is a partition, t...
iccpartipre 43458 If there is a partition, t...
iccpartiltu 43459 If there is a partition, t...
iccpartigtl 43460 If there is a partition, t...
iccpartlt 43461 If there is a partition, t...
iccpartltu 43462 If there is a partition, t...
iccpartgtl 43463 If there is a partition, t...
iccpartgt 43464 If there is a partition, t...
iccpartleu 43465 If there is a partition, t...
iccpartgel 43466 If there is a partition, t...
iccpartrn 43467 If there is a partition, t...
iccpartf 43468 The range of the partition...
iccpartel 43469 If there is a partition, t...
iccelpart 43470 An element of any partitio...
iccpartiun 43471 A half-open interval of ex...
icceuelpartlem 43472 Lemma for ~ icceuelpart . ...
icceuelpart 43473 An element of a partitione...
iccpartdisj 43474 The segments of a partitio...
iccpartnel 43475 A point of a partition is ...
fargshiftfv 43476 If a class is a function, ...
fargshiftf 43477 If a class is a function, ...
fargshiftf1 43478 If a function is 1-1, then...
fargshiftfo 43479 If a function is onto, the...
fargshiftfva 43480 The values of a shifted fu...
lswn0 43481 The last symbol of a not e...
nfich1 43484 The first interchangeable ...
nfich2 43485 The second interchangeable...
ichv 43486 Setvar variables are inter...
ichf 43487 Setvar variables are inter...
ichid 43488 A setvar variable is alway...
ichcircshi 43489 The setvar variables are i...
dfich2 43490 Alternate definition of th...
dfich2ai 43491 Obsolete version of ~ dfic...
dfich2bi 43492 Obsolete version of ~ dfic...
dfich2OLD 43493 Obsolete version of ~ dfic...
ichcom 43494 The interchangeability of ...
ichbi12i 43495 Equivalence for interchang...
icheqid 43496 In an equality for the sam...
icheq 43497 In an equality of setvar v...
ichnfimlem1 43498 Lemma for ~ ichnfimlem3 : ...
ichnfimlem2 43499 Lemma for ~ ichnfimlem3 : ...
ichnfimlem3 43500 Lemma for ~ ichnfim : A s...
ichnfim 43501 If in an interchangeabilit...
ichnfb 43502 If ` x ` and ` y ` are int...
ichn 43503 Negation does not affect i...
ichal 43504 Move a universal quantifie...
ich2al 43505 Two setvar variables are a...
ich2ex 43506 Two setvar variables are a...
ichan 43507 If two setvar variables ar...
ichexmpl1 43508 Example for interchangeabl...
ichexmpl2 43509 Example for interchangeabl...
ich2exprop 43510 If the setvar variables ar...
ichnreuop 43511 If the setvar variables ar...
ichreuopeq 43512 If the setvar variables ar...
sprid 43513 Two identical representati...
elsprel 43514 An unordered pair is an el...
spr0nelg 43515 The empty set is not an el...
sprval 43518 The set of all unordered p...
sprvalpw 43519 The set of all unordered p...
sprssspr 43520 The set of all unordered p...
spr0el 43521 The empty set is not an un...
sprvalpwn0 43522 The set of all unordered p...
sprel 43523 An element of the set of a...
prssspr 43524 An element of a subset of ...
prelspr 43525 An unordered pair of eleme...
prsprel 43526 The elements of a pair fro...
prsssprel 43527 The elements of a pair fro...
sprvalpwle2 43528 The set of all unordered p...
sprsymrelfvlem 43529 Lemma for ~ sprsymrelf and...
sprsymrelf1lem 43530 Lemma for ~ sprsymrelf1 . ...
sprsymrelfolem1 43531 Lemma 1 for ~ sprsymrelfo ...
sprsymrelfolem2 43532 Lemma 2 for ~ sprsymrelfo ...
sprsymrelfv 43533 The value of the function ...
sprsymrelf 43534 The mapping ` F ` is a fun...
sprsymrelf1 43535 The mapping ` F ` is a one...
sprsymrelfo 43536 The mapping ` F ` is a fun...
sprsymrelf1o 43537 The mapping ` F ` is a bij...
sprbisymrel 43538 There is a bijection betwe...
sprsymrelen 43539 The class ` P ` of subsets...
prpair 43540 Characterization of a prop...
prproropf1olem0 43541 Lemma 0 for ~ prproropf1o ...
prproropf1olem1 43542 Lemma 1 for ~ prproropf1o ...
prproropf1olem2 43543 Lemma 2 for ~ prproropf1o ...
prproropf1olem3 43544 Lemma 3 for ~ prproropf1o ...
prproropf1olem4 43545 Lemma 4 for ~ prproropf1o ...
prproropf1o 43546 There is a bijection betwe...
prproropen 43547 The set of proper pairs an...
prproropreud 43548 There is exactly one order...
pairreueq 43549 Two equivalent representat...
paireqne 43550 Two sets are not equal iff...
prprval 43553 The set of all proper unor...
prprvalpw 43554 The set of all proper unor...
prprelb 43555 An element of the set of a...
prprelprb 43556 A set is an element of the...
prprspr2 43557 The set of all proper unor...
prprsprreu 43558 There is a unique proper u...
prprreueq 43559 There is a unique proper u...
sbcpr 43560 The proper substitution of...
reupr 43561 There is a unique unordere...
reuprpr 43562 There is a unique proper u...
poprelb 43563 Equality for unordered pai...
2exopprim 43564 The existence of an ordere...
reuopreuprim 43565 There is a unique unordere...
fmtno 43568 The ` N ` th Fermat number...
fmtnoge3 43569 Each Fermat number is grea...
fmtnonn 43570 Each Fermat number is a po...
fmtnom1nn 43571 A Fermat number minus one ...
fmtnoodd 43572 Each Fermat number is odd....
fmtnorn 43573 A Fermat number is a funct...
fmtnof1 43574 The enumeration of the Fer...
fmtnoinf 43575 The set of Fermat numbers ...
fmtnorec1 43576 The first recurrence relat...
sqrtpwpw2p 43577 The floor of the square ro...
fmtnosqrt 43578 The floor of the square ro...
fmtno0 43579 The ` 0 ` th Fermat number...
fmtno1 43580 The ` 1 ` st Fermat number...
fmtnorec2lem 43581 Lemma for ~ fmtnorec2 (ind...
fmtnorec2 43582 The second recurrence rela...
fmtnodvds 43583 Any Fermat number divides ...
goldbachthlem1 43584 Lemma 1 for ~ goldbachth ....
goldbachthlem2 43585 Lemma 2 for ~ goldbachth ....
goldbachth 43586 Goldbach's theorem: Two d...
fmtnorec3 43587 The third recurrence relat...
fmtnorec4 43588 The fourth recurrence rela...
fmtno2 43589 The ` 2 ` nd Fermat number...
fmtno3 43590 The ` 3 ` rd Fermat number...
fmtno4 43591 The ` 4 ` th Fermat number...
fmtno5lem1 43592 Lemma 1 for ~ fmtno5 . (C...
fmtno5lem2 43593 Lemma 2 for ~ fmtno5 . (C...
fmtno5lem3 43594 Lemma 3 for ~ fmtno5 . (C...
fmtno5lem4 43595 Lemma 4 for ~ fmtno5 . (C...
fmtno5 43596 The ` 5 ` th Fermat number...
fmtno0prm 43597 The ` 0 ` th Fermat number...
fmtno1prm 43598 The ` 1 ` st Fermat number...
fmtno2prm 43599 The ` 2 ` nd Fermat number...
257prm 43600 257 is a prime number (the...
fmtno3prm 43601 The ` 3 ` rd Fermat number...
odz2prm2pw 43602 Any power of two is coprim...
fmtnoprmfac1lem 43603 Lemma for ~ fmtnoprmfac1 :...
fmtnoprmfac1 43604 Divisor of Fermat number (...
fmtnoprmfac2lem1 43605 Lemma for ~ fmtnoprmfac2 ....
fmtnoprmfac2 43606 Divisor of Fermat number (...
fmtnofac2lem 43607 Lemma for ~ fmtnofac2 (Ind...
fmtnofac2 43608 Divisor of Fermat number (...
fmtnofac1 43609 Divisor of Fermat number (...
fmtno4sqrt 43610 The floor of the square ro...
fmtno4prmfac 43611 If P was a (prime) factor ...
fmtno4prmfac193 43612 If P was a (prime) factor ...
fmtno4nprmfac193 43613 193 is not a (prime) facto...
fmtno4prm 43614 The ` 4 `-th Fermat number...
65537prm 43615 65537 is a prime number (t...
fmtnofz04prm 43616 The first five Fermat numb...
fmtnole4prm 43617 The first five Fermat numb...
fmtno5faclem1 43618 Lemma 1 for ~ fmtno5fac . ...
fmtno5faclem2 43619 Lemma 2 for ~ fmtno5fac . ...
fmtno5faclem3 43620 Lemma 3 for ~ fmtno5fac . ...
fmtno5fac 43621 The factorisation of the `...
fmtno5nprm 43622 The ` 5 ` th Fermat number...
prmdvdsfmtnof1lem1 43623 Lemma 1 for ~ prmdvdsfmtno...
prmdvdsfmtnof1lem2 43624 Lemma 2 for ~ prmdvdsfmtno...
prmdvdsfmtnof 43625 The mapping of a Fermat nu...
prmdvdsfmtnof1 43626 The mapping of a Fermat nu...
prminf2 43627 The set of prime numbers i...
2pwp1prm 43628 For every prime number of ...
2pwp1prmfmtno 43629 Every prime number of the ...
m2prm 43630 The second Mersenne number...
m3prm 43631 The third Mersenne number ...
2exp5 43632 Two to the fifth power is ...
flsqrt 43633 A condition equivalent to ...
flsqrt5 43634 The floor of the square ro...
3ndvds4 43635 3 does not divide 4. (Con...
139prmALT 43636 139 is a prime number. In...
31prm 43637 31 is a prime number. In ...
m5prm 43638 The fifth Mersenne number ...
2exp7 43639 Two to the seventh power i...
127prm 43640 127 is a prime number. (C...
m7prm 43641 The seventh Mersenne numbe...
2exp11 43642 Two to the eleventh power ...
m11nprm 43643 The eleventh Mersenne numb...
mod42tp1mod8 43644 If a number is ` 3 ` modul...
sfprmdvdsmersenne 43645 If ` Q ` is a safe prime (...
sgprmdvdsmersenne 43646 If ` P ` is a Sophie Germa...
lighneallem1 43647 Lemma 1 for ~ lighneal . ...
lighneallem2 43648 Lemma 2 for ~ lighneal . ...
lighneallem3 43649 Lemma 3 for ~ lighneal . ...
lighneallem4a 43650 Lemma 1 for ~ lighneallem4...
lighneallem4b 43651 Lemma 2 for ~ lighneallem4...
lighneallem4 43652 Lemma 3 for ~ lighneal . ...
lighneal 43653 If a power of a prime ` P ...
modexp2m1d 43654 The square of an integer w...
proththdlem 43655 Lemma for ~ proththd . (C...
proththd 43656 Proth's theorem (1878). I...
5tcu2e40 43657 5 times the cube of 2 is 4...
3exp4mod41 43658 3 to the fourth power is -...
41prothprmlem1 43659 Lemma 1 for ~ 41prothprm ....
41prothprmlem2 43660 Lemma 2 for ~ 41prothprm ....
41prothprm 43661 41 is a _Proth prime_. (C...
quad1 43662 A condition for a quadrati...
requad01 43663 A condition for a quadrati...
requad1 43664 A condition for a quadrati...
requad2 43665 A condition for a quadrati...
iseven 43670 The predicate "is an even ...
isodd 43671 The predicate "is an odd n...
evenz 43672 An even number is an integ...
oddz 43673 An odd number is an intege...
evendiv2z 43674 The result of dividing an ...
oddp1div2z 43675 The result of dividing an ...
oddm1div2z 43676 The result of dividing an ...
isodd2 43677 The predicate "is an odd n...
dfodd2 43678 Alternate definition for o...
dfodd6 43679 Alternate definition for o...
dfeven4 43680 Alternate definition for e...
evenm1odd 43681 The predecessor of an even...
evenp1odd 43682 The successor of an even n...
oddp1eveni 43683 The successor of an odd nu...
oddm1eveni 43684 The predecessor of an odd ...
evennodd 43685 An even number is not an o...
oddneven 43686 An odd number is not an ev...
enege 43687 The negative of an even nu...
onego 43688 The negative of an odd num...
m1expevenALTV 43689 Exponentiation of -1 by an...
m1expoddALTV 43690 Exponentiation of -1 by an...
dfeven2 43691 Alternate definition for e...
dfodd3 43692 Alternate definition for o...
iseven2 43693 The predicate "is an even ...
isodd3 43694 The predicate "is an odd n...
2dvdseven 43695 2 divides an even number. ...
m2even 43696 A multiple of 2 is an even...
2ndvdsodd 43697 2 does not divide an odd n...
2dvdsoddp1 43698 2 divides an odd number in...
2dvdsoddm1 43699 2 divides an odd number de...
dfeven3 43700 Alternate definition for e...
dfodd4 43701 Alternate definition for o...
dfodd5 43702 Alternate definition for o...
zefldiv2ALTV 43703 The floor of an even numbe...
zofldiv2ALTV 43704 The floor of an odd numer ...
oddflALTV 43705 Odd number representation ...
iseven5 43706 The predicate "is an even ...
isodd7 43707 The predicate "is an odd n...
dfeven5 43708 Alternate definition for e...
dfodd7 43709 Alternate definition for o...
gcd2odd1 43710 The greatest common diviso...
zneoALTV 43711 No even integer equals an ...
zeoALTV 43712 An integer is even or odd....
zeo2ALTV 43713 An integer is even or odd ...
nneoALTV 43714 A positive integer is even...
nneoiALTV 43715 A positive integer is even...
odd2np1ALTV 43716 An integer is odd iff it i...
oddm1evenALTV 43717 An integer is odd iff its ...
oddp1evenALTV 43718 An integer is odd iff its ...
oexpnegALTV 43719 The exponential of the neg...
oexpnegnz 43720 The exponential of the neg...
bits0ALTV 43721 Value of the zeroth bit. ...
bits0eALTV 43722 The zeroth bit of an even ...
bits0oALTV 43723 The zeroth bit of an odd n...
divgcdoddALTV 43724 Either ` A / ( A gcd B ) `...
opoeALTV 43725 The sum of two odds is eve...
opeoALTV 43726 The sum of an odd and an e...
omoeALTV 43727 The difference of two odds...
omeoALTV 43728 The difference of an odd a...
oddprmALTV 43729 A prime not equal to ` 2 `...
0evenALTV 43730 0 is an even number. (Con...
0noddALTV 43731 0 is not an odd number. (...
1oddALTV 43732 1 is an odd number. (Cont...
1nevenALTV 43733 1 is not an even number. ...
2evenALTV 43734 2 is an even number. (Con...
2noddALTV 43735 2 is not an odd number. (...
nn0o1gt2ALTV 43736 An odd nonnegative integer...
nnoALTV 43737 An alternate characterizat...
nn0oALTV 43738 An alternate characterizat...
nn0e 43739 An alternate characterizat...
nneven 43740 An alternate characterizat...
nn0onn0exALTV 43741 For each odd nonnegative i...
nn0enn0exALTV 43742 For each even nonnegative ...
nnennexALTV 43743 For each even positive int...
nnpw2evenALTV 43744 2 to the power of a positi...
epoo 43745 The sum of an even and an ...
emoo 43746 The difference of an even ...
epee 43747 The sum of two even number...
emee 43748 The difference of two even...
evensumeven 43749 If a summand is even, the ...
3odd 43750 3 is an odd number. (Cont...
4even 43751 4 is an even number. (Con...
5odd 43752 5 is an odd number. (Cont...
6even 43753 6 is an even number. (Con...
7odd 43754 7 is an odd number. (Cont...
8even 43755 8 is an even number. (Con...
evenprm2 43756 A prime number is even iff...
oddprmne2 43757 Every prime number not bei...
oddprmuzge3 43758 A prime number which is od...
evenltle 43759 If an even number is great...
odd2prm2 43760 If an odd number is the su...
even3prm2 43761 If an even number is the s...
mogoldbblem 43762 Lemma for ~ mogoldbb . (C...
perfectALTVlem1 43763 Lemma for ~ perfectALTV . ...
perfectALTVlem2 43764 Lemma for ~ perfectALTV . ...
perfectALTV 43765 The Euclid-Euler theorem, ...
fppr 43768 The set of Fermat pseudopr...
fpprmod 43769 The set of Fermat pseudopr...
fpprel 43770 A Fermat pseudoprime to th...
fpprbasnn 43771 The base of a Fermat pseud...
fpprnn 43772 A Fermat pseudoprime to th...
fppr2odd 43773 A Fermat pseudoprime to th...
11t31e341 43774 341 is the product of 11 a...
2exp340mod341 43775 Eight to the eighth power ...
341fppr2 43776 341 is the (smallest) _Pou...
4fppr1 43777 4 is the (smallest) Fermat...
8exp8mod9 43778 Eight to the eighth power ...
9fppr8 43779 9 is the (smallest) Fermat...
dfwppr 43780 Alternate definition of a ...
fpprwppr 43781 A Fermat pseudoprime to th...
fpprwpprb 43782 An integer ` X ` which is ...
fpprel2 43783 An alternate definition fo...
nfermltl8rev 43784 Fermat's little theorem wi...
nfermltl2rev 43785 Fermat's little theorem wi...
nfermltlrev 43786 Fermat's little theorem re...
isgbe 43793 The predicate "is an even ...
isgbow 43794 The predicate "is a weak o...
isgbo 43795 The predicate "is an odd G...
gbeeven 43796 An even Goldbach number is...
gbowodd 43797 A weak odd Goldbach number...
gbogbow 43798 A (strong) odd Goldbach nu...
gboodd 43799 An odd Goldbach number is ...
gbepos 43800 Any even Goldbach number i...
gbowpos 43801 Any weak odd Goldbach numb...
gbopos 43802 Any odd Goldbach number is...
gbegt5 43803 Any even Goldbach number i...
gbowgt5 43804 Any weak odd Goldbach numb...
gbowge7 43805 Any weak odd Goldbach numb...
gboge9 43806 Any odd Goldbach number is...
gbege6 43807 Any even Goldbach number i...
gbpart6 43808 The Goldbach partition of ...
gbpart7 43809 The (weak) Goldbach partit...
gbpart8 43810 The Goldbach partition of ...
gbpart9 43811 The (strong) Goldbach part...
gbpart11 43812 The (strong) Goldbach part...
6gbe 43813 6 is an even Goldbach numb...
7gbow 43814 7 is a weak odd Goldbach n...
8gbe 43815 8 is an even Goldbach numb...
9gbo 43816 9 is an odd Goldbach numbe...
11gbo 43817 11 is an odd Goldbach numb...
stgoldbwt 43818 If the strong ternary Gold...
sbgoldbwt 43819 If the strong binary Goldb...
sbgoldbst 43820 If the strong binary Goldb...
sbgoldbaltlem1 43821 Lemma 1 for ~ sbgoldbalt :...
sbgoldbaltlem2 43822 Lemma 2 for ~ sbgoldbalt :...
sbgoldbalt 43823 An alternate (related to t...
sbgoldbb 43824 If the strong binary Goldb...
sgoldbeven3prm 43825 If the binary Goldbach con...
sbgoldbm 43826 If the strong binary Goldb...
mogoldbb 43827 If the modern version of t...
sbgoldbmb 43828 The strong binary Goldbach...
sbgoldbo 43829 If the strong binary Goldb...
nnsum3primes4 43830 4 is the sum of at most 3 ...
nnsum4primes4 43831 4 is the sum of at most 4 ...
nnsum3primesprm 43832 Every prime is "the sum of...
nnsum4primesprm 43833 Every prime is "the sum of...
nnsum3primesgbe 43834 Any even Goldbach number i...
nnsum4primesgbe 43835 Any even Goldbach number i...
nnsum3primesle9 43836 Every integer greater than...
nnsum4primesle9 43837 Every integer greater than...
nnsum4primesodd 43838 If the (weak) ternary Gold...
nnsum4primesoddALTV 43839 If the (strong) ternary Go...
evengpop3 43840 If the (weak) ternary Gold...
evengpoap3 43841 If the (strong) ternary Go...
nnsum4primeseven 43842 If the (weak) ternary Gold...
nnsum4primesevenALTV 43843 If the (strong) ternary Go...
wtgoldbnnsum4prm 43844 If the (weak) ternary Gold...
stgoldbnnsum4prm 43845 If the (strong) ternary Go...
bgoldbnnsum3prm 43846 If the binary Goldbach con...
bgoldbtbndlem1 43847 Lemma 1 for ~ bgoldbtbnd :...
bgoldbtbndlem2 43848 Lemma 2 for ~ bgoldbtbnd ....
bgoldbtbndlem3 43849 Lemma 3 for ~ bgoldbtbnd ....
bgoldbtbndlem4 43850 Lemma 4 for ~ bgoldbtbnd ....
bgoldbtbnd 43851 If the binary Goldbach con...
tgoldbachgtALTV 43854 Variant of Thierry Arnoux'...
bgoldbachlt 43855 The binary Goldbach conjec...
tgblthelfgott 43857 The ternary Goldbach conje...
tgoldbachlt 43858 The ternary Goldbach conje...
tgoldbach 43859 The ternary Goldbach conje...
isomgrrel 43864 The isomorphy relation for...
isomgr 43865 The isomorphy relation for...
isisomgr 43866 Implications of two graphs...
isomgreqve 43867 A set is isomorphic to a h...
isomushgr 43868 The isomorphy relation for...
isomuspgrlem1 43869 Lemma 1 for ~ isomuspgr . ...
isomuspgrlem2a 43870 Lemma 1 for ~ isomuspgrlem...
isomuspgrlem2b 43871 Lemma 2 for ~ isomuspgrlem...
isomuspgrlem2c 43872 Lemma 3 for ~ isomuspgrlem...
isomuspgrlem2d 43873 Lemma 4 for ~ isomuspgrlem...
isomuspgrlem2e 43874 Lemma 5 for ~ isomuspgrlem...
isomuspgrlem2 43875 Lemma 2 for ~ isomuspgr . ...
isomuspgr 43876 The isomorphy relation for...
isomgrref 43877 The isomorphy relation is ...
isomgrsym 43878 The isomorphy relation is ...
isomgrsymb 43879 The isomorphy relation is ...
isomgrtrlem 43880 Lemma for ~ isomgrtr . (C...
isomgrtr 43881 The isomorphy relation is ...
strisomgrop 43882 A graph represented as an ...
ushrisomgr 43883 A simple hypergraph (with ...
1hegrlfgr 43884 A graph ` G ` with one hyp...
upwlksfval 43887 The set of simple walks (i...
isupwlk 43888 Properties of a pair of fu...
isupwlkg 43889 Generalization of ~ isupwl...
upwlkbprop 43890 Basic properties of a simp...
upwlkwlk 43891 A simple walk is a walk. ...
upgrwlkupwlk 43892 In a pseudograph, a walk i...
upgrwlkupwlkb 43893 In a pseudograph, the defi...
upgrisupwlkALT 43894 Alternate proof of ~ upgri...
upgredgssspr 43895 The set of edges of a pseu...
uspgropssxp 43896 The set ` G ` of "simple p...
uspgrsprfv 43897 The value of the function ...
uspgrsprf 43898 The mapping ` F ` is a fun...
uspgrsprf1 43899 The mapping ` F ` is a one...
uspgrsprfo 43900 The mapping ` F ` is a fun...
uspgrsprf1o 43901 The mapping ` F ` is a bij...
uspgrex 43902 The class ` G ` of all "si...
uspgrbispr 43903 There is a bijection betwe...
uspgrspren 43904 The set ` G ` of the "simp...
uspgrymrelen 43905 The set ` G ` of the "simp...
uspgrbisymrel 43906 There is a bijection betwe...
uspgrbisymrelALT 43907 Alternate proof of ~ uspgr...
ovn0dmfun 43908 If a class operation value...
xpsnopab 43909 A Cartesian product with a...
xpiun 43910 A Cartesian product expres...
ovn0ssdmfun 43911 If a class' operation valu...
fnxpdmdm 43912 The domain of the domain o...
cnfldsrngbas 43913 The base set of a subring ...
cnfldsrngadd 43914 The group addition operati...
cnfldsrngmul 43915 The ring multiplication op...
plusfreseq 43916 If the empty set is not co...
mgmplusfreseq 43917 If the empty set is not co...
0mgm 43918 A set with an empty base s...
mgmpropd 43919 If two structures have the...
ismgmd 43920 Deduce a magma from its pr...
mgmhmrcl 43925 Reverse closure of a magma...
submgmrcl 43926 Reverse closure for submag...
ismgmhm 43927 Property of a magma homomo...
mgmhmf 43928 A magma homomorphism is a ...
mgmhmpropd 43929 Magma homomorphism depends...
mgmhmlin 43930 A magma homomorphism prese...
mgmhmf1o 43931 A magma homomorphism is bi...
idmgmhm 43932 The identity homomorphism ...
issubmgm 43933 Expand definition of a sub...
issubmgm2 43934 Submagmas are subsets that...
rabsubmgmd 43935 Deduction for proving that...
submgmss 43936 Submagmas are subsets of t...
submgmid 43937 Every magma is trivially a...
submgmcl 43938 Submagmas are closed under...
submgmmgm 43939 Submagmas are themselves m...
submgmbas 43940 The base set of a submagma...
subsubmgm 43941 A submagma of a submagma i...
resmgmhm 43942 Restriction of a magma hom...
resmgmhm2 43943 One direction of ~ resmgmh...
resmgmhm2b 43944 Restriction of the codomai...
mgmhmco 43945 The composition of magma h...
mgmhmima 43946 The homomorphic image of a...
mgmhmeql 43947 The equalizer of two magma...
submgmacs 43948 Submagmas are an algebraic...
ismhm0 43949 Property of a monoid homom...
mhmismgmhm 43950 Each monoid homomorphism i...
opmpoismgm 43951 A structure with a group a...
copissgrp 43952 A structure with a constan...
copisnmnd 43953 A structure with a constan...
0nodd 43954 0 is not an odd integer. ...
1odd 43955 1 is an odd integer. (Con...
2nodd 43956 2 is not an odd integer. ...
oddibas 43957 Lemma 1 for ~ oddinmgm : ...
oddiadd 43958 Lemma 2 for ~ oddinmgm : ...
oddinmgm 43959 The structure of all odd i...
nnsgrpmgm 43960 The structure of positive ...
nnsgrp 43961 The structure of positive ...
nnsgrpnmnd 43962 The structure of positive ...
nn0mnd 43963 The set of nonnegative int...
gsumsplit2f 43964 Split a group sum into two...
gsumdifsndf 43965 Extract a summand from a f...
gsumfsupp 43966 A group sum of a family ca...
efmnd 43969 The monoid of endofunction...
efmndbas 43970 The base set of the monoid...
elefmndbas 43971 Two ways of saying a funct...
elefmndbas2 43972 Two ways of saying a funct...
efmndbasf 43973 Elements in the monoid of ...
efmndhash 43974 The monoid of endofunction...
efmndbasfi 43975 The monoid of endofunction...
efmndfv 43976 The function value of an e...
efmndtset 43977 The topology of the monoid...
efmndplusg 43978 The group operation of a m...
efmndov 43979 The value of the group ope...
efmndcl 43980 The group operation of the...
efmndtopn 43981 The topology of the monoid...
efmndmgm 43982 The monoid of endofunction...
efmndsgrp 43983 The monoid of endofunction...
ielefmnd 43984 The identity function rest...
efmndid 43985 The identity function rest...
efmndmnd 43986 The monoid of endofunction...
efmnd0nmnd 43987 Even the monoid of endofun...
efmndbas0 43988 The base set of the monoid...
efmnd1hash 43989 The monoid of endofunction...
efmnd1bas 43990 The monoid of endofunction...
efmnd2hash 43991 The monoid of endofunction...
submefmnd 43992 If the base set of a monoi...
sursubmefmnd 43993 The set of surjective endo...
injsubmefmnd 43994 The set of injective endof...
symgsubmefmnd 43995 The symmetric group on a s...
symgsubmefmndALT 43996 The symmetric group on a s...
efmndtmd 43997 The monoid of endofunction...
idressubmefmnd 43998 The singleton containing o...
idresefmnd 43999 The structure with the sin...
smndex1ibas 44000 The modulo function ` I ` ...
smndex1iidm 44001 The modulo function ` I ` ...
smndex1gbas 44002 The constant functions ` (...
smndex1gid 44003 The composition of a const...
smndex1igid 44004 The composition of the mod...
smndex1basss 44005 The modulo function ` I ` ...
smndex1bas 44006 The base set of the monoid...
smndex1mgm 44007 The monoid of endofunction...
smndex1sgrp 44008 The monoid of endofunction...
smndex1mndlem 44009 Lemma for ~ smndex1mnd and...
smndex1mnd 44010 The monoid of endofunction...
smndex1id 44011 The modulo function ` I ` ...
smndex1n0mnd 44012 The identity of the monoid...
nsmndex1 44013 The base set ` B ` of the ...
smndex2dbas 44014 The doubling function ` D ...
smndex2dnrinv 44015 The doubling function ` D ...
smndex2hbas 44016 The halving functions ` H ...
smndex2dlinvh 44017 The halving functions ` H ...
iscllaw 44024 The predicate "is a closed...
iscomlaw 44025 The predicate "is a commut...
clcllaw 44026 Closure of a closed operat...
isasslaw 44027 The predicate "is an assoc...
asslawass 44028 Associativity of an associ...
mgmplusgiopALT 44029 Slot 2 (group operation) o...
sgrpplusgaopALT 44030 Slot 2 (group operation) o...
intopval 44037 The internal (binary) oper...
intop 44038 An internal (binary) opera...
clintopval 44039 The closed (internal binar...
assintopval 44040 The associative (closed in...
assintopmap 44041 The associative (closed in...
isclintop 44042 The predicate "is a closed...
clintop 44043 A closed (internal binary)...
assintop 44044 An associative (closed int...
isassintop 44045 The predicate "is an assoc...
clintopcllaw 44046 The closure law holds for ...
assintopcllaw 44047 The closure low holds for ...
assintopasslaw 44048 The associative low holds ...
assintopass 44049 An associative (closed int...
ismgmALT 44058 The predicate "is a magma"...
iscmgmALT 44059 The predicate "is a commut...
issgrpALT 44060 The predicate "is a semigr...
iscsgrpALT 44061 The predicate "is a commut...
mgm2mgm 44062 Equivalence of the two def...
sgrp2sgrp 44063 Equivalence of the two def...
idfusubc0 44064 The identity functor for a...
idfusubc 44065 The identity functor for a...
inclfusubc 44066 The "inclusion functor" fr...
lmod0rng 44067 If the scalar ring of a mo...
nzrneg1ne0 44068 The additive inverse of th...
0ringdif 44069 A zero ring is a ring whic...
0ringbas 44070 The base set of a zero rin...
0ring1eq0 44071 In a zero ring, a ring whi...
nrhmzr 44072 There is no ring homomorph...
isrng 44075 The predicate "is a non-un...
rngabl 44076 A non-unital ring is an (a...
rngmgp 44077 A non-unital ring is a sem...
ringrng 44078 A unital ring is a (non-un...
ringssrng 44079 The unital rings are (non-...
isringrng 44080 The predicate "is a unital...
rngdir 44081 Distributive law for the m...
rngcl 44082 Closure of the multiplicat...
rnglz 44083 The zero of a nonunital ri...
rnghmrcl 44088 Reverse closure of a non-u...
rnghmfn 44089 The mapping of two non-uni...
rnghmval 44090 The set of the non-unital ...
isrnghm 44091 A function is a non-unital...
isrnghmmul 44092 A function is a non-unital...
rnghmmgmhm 44093 A non-unital ring homomorp...
rnghmval2 44094 The non-unital ring homomo...
isrngisom 44095 An isomorphism of non-unit...
rngimrcl 44096 Reverse closure for an iso...
rnghmghm 44097 A non-unital ring homomorp...
rnghmf 44098 A ring homomorphism is a f...
rnghmmul 44099 A homomorphism of non-unit...
isrnghm2d 44100 Demonstration of non-unita...
isrnghmd 44101 Demonstration of non-unita...
rnghmf1o 44102 A non-unital ring homomorp...
isrngim 44103 An isomorphism of non-unit...
rngimf1o 44104 An isomorphism of non-unit...
rngimrnghm 44105 An isomorphism of non-unit...
rnghmco 44106 The composition of non-uni...
idrnghm 44107 The identity homomorphism ...
c0mgm 44108 The constant mapping to ze...
c0mhm 44109 The constant mapping to ze...
c0ghm 44110 The constant mapping to ze...
c0rhm 44111 The constant mapping to ze...
c0rnghm 44112 The constant mapping to ze...
c0snmgmhm 44113 The constant mapping to ze...
c0snmhm 44114 The constant mapping to ze...
c0snghm 44115 The constant mapping to ze...
zrrnghm 44116 The constant mapping to ze...
rhmfn 44117 The mapping of two rings t...
rhmval 44118 The ring homomorphisms bet...
rhmisrnghm 44119 Each unital ring homomorph...
lidldomn1 44120 If a (left) ideal (which i...
lidlssbas 44121 The base set of the restri...
lidlbas 44122 A (left) ideal of a ring i...
lidlabl 44123 A (left) ideal of a ring i...
lidlmmgm 44124 The multiplicative group o...
lidlmsgrp 44125 The multiplicative group o...
lidlrng 44126 A (left) ideal of a ring i...
zlidlring 44127 The zero (left) ideal of a...
uzlidlring 44128 Only the zero (left) ideal...
lidldomnnring 44129 A (left) ideal of a domain...
0even 44130 0 is an even integer. (Co...
1neven 44131 1 is not an even integer. ...
2even 44132 2 is an even integer. (Co...
2zlidl 44133 The even integers are a (l...
2zrng 44134 The ring of integers restr...
2zrngbas 44135 The base set of R is the s...
2zrngadd 44136 The group addition operati...
2zrng0 44137 The additive identity of R...
2zrngamgm 44138 R is an (additive) magma. ...
2zrngasgrp 44139 R is an (additive) semigro...
2zrngamnd 44140 R is an (additive) monoid....
2zrngacmnd 44141 R is a commutative (additi...
2zrngagrp 44142 R is an (additive) group. ...
2zrngaabl 44143 R is an (additive) abelian...
2zrngmul 44144 The ring multiplication op...
2zrngmmgm 44145 R is a (multiplicative) ma...
2zrngmsgrp 44146 R is a (multiplicative) se...
2zrngALT 44147 The ring of integers restr...
2zrngnmlid 44148 R has no multiplicative (l...
2zrngnmrid 44149 R has no multiplicative (r...
2zrngnmlid2 44150 R has no multiplicative (l...
2zrngnring 44151 R is not a unital ring. (...
cznrnglem 44152 Lemma for ~ cznrng : The ...
cznabel 44153 The ring constructed from ...
cznrng 44154 The ring constructed from ...
cznnring 44155 The ring constructed from ...
rngcvalALTV 44160 Value of the category of n...
rngcval 44161 Value of the category of n...
rnghmresfn 44162 The class of non-unital ri...
rnghmresel 44163 An element of the non-unit...
rngcbas 44164 Set of objects of the cate...
rngchomfval 44165 Set of arrows of the categ...
rngchom 44166 Set of arrows of the categ...
elrngchom 44167 A morphism of non-unital r...
rngchomfeqhom 44168 The functionalized Hom-set...
rngccofval 44169 Composition in the categor...
rngcco 44170 Composition in the categor...
dfrngc2 44171 Alternate definition of th...
rnghmsscmap2 44172 The non-unital ring homomo...
rnghmsscmap 44173 The non-unital ring homomo...
rnghmsubcsetclem1 44174 Lemma 1 for ~ rnghmsubcset...
rnghmsubcsetclem2 44175 Lemma 2 for ~ rnghmsubcset...
rnghmsubcsetc 44176 The non-unital ring homomo...
rngccat 44177 The category of non-unital...
rngcid 44178 The identity arrow in the ...
rngcsect 44179 A section in the category ...
rngcinv 44180 An inverse in the category...
rngciso 44181 An isomorphism in the cate...
rngcbasALTV 44182 Set of objects of the cate...
rngchomfvalALTV 44183 Set of arrows of the categ...
rngchomALTV 44184 Set of arrows of the categ...
elrngchomALTV 44185 A morphism of non-unital r...
rngccofvalALTV 44186 Composition in the categor...
rngccoALTV 44187 Composition in the categor...
rngccatidALTV 44188 Lemma for ~ rngccatALTV . ...
rngccatALTV 44189 The category of non-unital...
rngcidALTV 44190 The identity arrow in the ...
rngcsectALTV 44191 A section in the category ...
rngcinvALTV 44192 An inverse in the category...
rngcisoALTV 44193 An isomorphism in the cate...
rngchomffvalALTV 44194 The value of the functiona...
rngchomrnghmresALTV 44195 The value of the functiona...
rngcifuestrc 44196 The "inclusion functor" fr...
funcrngcsetc 44197 The "natural forgetful fun...
funcrngcsetcALT 44198 Alternate proof of ~ funcr...
zrinitorngc 44199 The zero ring is an initia...
zrtermorngc 44200 The zero ring is a termina...
zrzeroorngc 44201 The zero ring is a zero ob...
ringcvalALTV 44206 Value of the category of r...
ringcval 44207 Value of the category of u...
rhmresfn 44208 The class of unital ring h...
rhmresel 44209 An element of the unital r...
ringcbas 44210 Set of objects of the cate...
ringchomfval 44211 Set of arrows of the categ...
ringchom 44212 Set of arrows of the categ...
elringchom 44213 A morphism of unital rings...
ringchomfeqhom 44214 The functionalized Hom-set...
ringccofval 44215 Composition in the categor...
ringcco 44216 Composition in the categor...
dfringc2 44217 Alternate definition of th...
rhmsscmap2 44218 The unital ring homomorphi...
rhmsscmap 44219 The unital ring homomorphi...
rhmsubcsetclem1 44220 Lemma 1 for ~ rhmsubcsetc ...
rhmsubcsetclem2 44221 Lemma 2 for ~ rhmsubcsetc ...
rhmsubcsetc 44222 The unital ring homomorphi...
ringccat 44223 The category of unital rin...
ringcid 44224 The identity arrow in the ...
rhmsscrnghm 44225 The unital ring homomorphi...
rhmsubcrngclem1 44226 Lemma 1 for ~ rhmsubcrngc ...
rhmsubcrngclem2 44227 Lemma 2 for ~ rhmsubcrngc ...
rhmsubcrngc 44228 The unital ring homomorphi...
rngcresringcat 44229 The restriction of the cat...
ringcsect 44230 A section in the category ...
ringcinv 44231 An inverse in the category...
ringciso 44232 An isomorphism in the cate...
ringcbasbas 44233 An element of the base set...
funcringcsetc 44234 The "natural forgetful fun...
funcringcsetcALTV2lem1 44235 Lemma 1 for ~ funcringcset...
funcringcsetcALTV2lem2 44236 Lemma 2 for ~ funcringcset...
funcringcsetcALTV2lem3 44237 Lemma 3 for ~ funcringcset...
funcringcsetcALTV2lem4 44238 Lemma 4 for ~ funcringcset...
funcringcsetcALTV2lem5 44239 Lemma 5 for ~ funcringcset...
funcringcsetcALTV2lem6 44240 Lemma 6 for ~ funcringcset...
funcringcsetcALTV2lem7 44241 Lemma 7 for ~ funcringcset...
funcringcsetcALTV2lem8 44242 Lemma 8 for ~ funcringcset...
funcringcsetcALTV2lem9 44243 Lemma 9 for ~ funcringcset...
funcringcsetcALTV2 44244 The "natural forgetful fun...
ringcbasALTV 44245 Set of objects of the cate...
ringchomfvalALTV 44246 Set of arrows of the categ...
ringchomALTV 44247 Set of arrows of the categ...
elringchomALTV 44248 A morphism of rings is a f...
ringccofvalALTV 44249 Composition in the categor...
ringccoALTV 44250 Composition in the categor...
ringccatidALTV 44251 Lemma for ~ ringccatALTV ....
ringccatALTV 44252 The category of rings is a...
ringcidALTV 44253 The identity arrow in the ...
ringcsectALTV 44254 A section in the category ...
ringcinvALTV 44255 An inverse in the category...
ringcisoALTV 44256 An isomorphism in the cate...
ringcbasbasALTV 44257 An element of the base set...
funcringcsetclem1ALTV 44258 Lemma 1 for ~ funcringcset...
funcringcsetclem2ALTV 44259 Lemma 2 for ~ funcringcset...
funcringcsetclem3ALTV 44260 Lemma 3 for ~ funcringcset...
funcringcsetclem4ALTV 44261 Lemma 4 for ~ funcringcset...
funcringcsetclem5ALTV 44262 Lemma 5 for ~ funcringcset...
funcringcsetclem6ALTV 44263 Lemma 6 for ~ funcringcset...
funcringcsetclem7ALTV 44264 Lemma 7 for ~ funcringcset...
funcringcsetclem8ALTV 44265 Lemma 8 for ~ funcringcset...
funcringcsetclem9ALTV 44266 Lemma 9 for ~ funcringcset...
funcringcsetcALTV 44267 The "natural forgetful fun...
irinitoringc 44268 The ring of integers is an...
zrtermoringc 44269 The zero ring is a termina...
zrninitoringc 44270 The zero ring is not an in...
nzerooringczr 44271 There is no zero object in...
srhmsubclem1 44272 Lemma 1 for ~ srhmsubc . ...
srhmsubclem2 44273 Lemma 2 for ~ srhmsubc . ...
srhmsubclem3 44274 Lemma 3 for ~ srhmsubc . ...
srhmsubc 44275 According to ~ df-subc , t...
sringcat 44276 The restriction of the cat...
crhmsubc 44277 According to ~ df-subc , t...
cringcat 44278 The restriction of the cat...
drhmsubc 44279 According to ~ df-subc , t...
drngcat 44280 The restriction of the cat...
fldcat 44281 The restriction of the cat...
fldc 44282 The restriction of the cat...
fldhmsubc 44283 According to ~ df-subc , t...
rngcrescrhm 44284 The category of non-unital...
rhmsubclem1 44285 Lemma 1 for ~ rhmsubc . (...
rhmsubclem2 44286 Lemma 2 for ~ rhmsubc . (...
rhmsubclem3 44287 Lemma 3 for ~ rhmsubc . (...
rhmsubclem4 44288 Lemma 4 for ~ rhmsubc . (...
rhmsubc 44289 According to ~ df-subc , t...
rhmsubccat 44290 The restriction of the cat...
srhmsubcALTVlem1 44291 Lemma 1 for ~ srhmsubcALTV...
srhmsubcALTVlem2 44292 Lemma 2 for ~ srhmsubcALTV...
srhmsubcALTV 44293 According to ~ df-subc , t...
sringcatALTV 44294 The restriction of the cat...
crhmsubcALTV 44295 According to ~ df-subc , t...
cringcatALTV 44296 The restriction of the cat...
drhmsubcALTV 44297 According to ~ df-subc , t...
drngcatALTV 44298 The restriction of the cat...
fldcatALTV 44299 The restriction of the cat...
fldcALTV 44300 The restriction of the cat...
fldhmsubcALTV 44301 According to ~ df-subc , t...
rngcrescrhmALTV 44302 The category of non-unital...
rhmsubcALTVlem1 44303 Lemma 1 for ~ rhmsubcALTV ...
rhmsubcALTVlem2 44304 Lemma 2 for ~ rhmsubcALTV ...
rhmsubcALTVlem3 44305 Lemma 3 for ~ rhmsubcALTV ...
rhmsubcALTVlem4 44306 Lemma 4 for ~ rhmsubcALTV ...
rhmsubcALTV 44307 According to ~ df-subc , t...
rhmsubcALTVcat 44308 The restriction of the cat...
opeliun2xp 44309 Membership of an ordered p...
eliunxp2 44310 Membership in a union of C...
mpomptx2 44311 Express a two-argument fun...
cbvmpox2 44312 Rule to change the bound v...
dmmpossx2 44313 The domain of a mapping is...
mpoexxg2 44314 Existence of an operation ...
ovmpordxf 44315 Value of an operation give...
ovmpordx 44316 Value of an operation give...
ovmpox2 44317 The value of an operation ...
fdmdifeqresdif 44318 The restriction of a condi...
offvalfv 44319 The function operation exp...
ofaddmndmap 44320 The function operation app...
mapsnop 44321 A singleton of an ordered ...
mapprop 44322 An unordered pair containi...
ztprmneprm 44323 A prime is not an integer ...
2t6m3t4e0 44324 2 times 6 minus 3 times 4 ...
ssnn0ssfz 44325 For any finite subset of `...
nn0sumltlt 44326 If the sum of two nonnegat...
bcpascm1 44327 Pascal's rule for the bino...
altgsumbc 44328 The sum of binomial coeffi...
altgsumbcALT 44329 Alternate proof of ~ altgs...
zlmodzxzlmod 44330 The ` ZZ `-module ` ZZ X. ...
zlmodzxzel 44331 An element of the (base se...
zlmodzxz0 44332 The ` 0 ` of the ` ZZ `-mo...
zlmodzxzscm 44333 The scalar multiplication ...
zlmodzxzadd 44334 The addition of the ` ZZ `...
zlmodzxzsubm 44335 The subtraction of the ` Z...
zlmodzxzsub 44336 The subtraction of the ` Z...
mgpsumunsn 44337 Extract a summand/factor f...
mgpsumz 44338 If the group sum for the m...
mgpsumn 44339 If the group sum for the m...
exple2lt6 44340 A nonnegative integer to t...
pgrple2abl 44341 Every symmetric group on a...
pgrpgt2nabl 44342 Every symmetric group on a...
invginvrid 44343 Identity for a multiplicat...
rmsupp0 44344 The support of a mapping o...
domnmsuppn0 44345 The support of a mapping o...
rmsuppss 44346 The support of a mapping o...
mndpsuppss 44347 The support of a mapping o...
scmsuppss 44348 The support of a mapping o...
rmsuppfi 44349 The support of a mapping o...
rmfsupp 44350 A mapping of a multiplicat...
mndpsuppfi 44351 The support of a mapping o...
mndpfsupp 44352 A mapping of a scalar mult...
scmsuppfi 44353 The support of a mapping o...
scmfsupp 44354 A mapping of a scalar mult...
suppmptcfin 44355 The support of a mapping w...
mptcfsupp 44356 A mapping with value 0 exc...
fsuppmptdmf 44357 A mapping with a finite do...
lmodvsmdi 44358 Multiple distributive law ...
gsumlsscl 44359 Closure of a group sum in ...
ascl1 44360 The scalar 1 embedded into...
assaascl0 44361 The scalar 0 embedded into...
assaascl1 44362 The scalar 1 embedded into...
ply1vr1smo 44363 The variable in a polynomi...
ply1ass23l 44364 Associative identity with ...
ply1sclrmsm 44365 The ring multiplication of...
coe1id 44366 Coefficient vector of the ...
coe1sclmulval 44367 The value of the coefficie...
ply1mulgsumlem1 44368 Lemma 1 for ~ ply1mulgsum ...
ply1mulgsumlem2 44369 Lemma 2 for ~ ply1mulgsum ...
ply1mulgsumlem3 44370 Lemma 3 for ~ ply1mulgsum ...
ply1mulgsumlem4 44371 Lemma 4 for ~ ply1mulgsum ...
ply1mulgsum 44372 The product of two polynom...
evl1at0 44373 Polynomial evaluation for ...
evl1at1 44374 Polynomial evaluation for ...
linply1 44375 A term of the form ` x - C...
lineval 44376 A term of the form ` x - C...
zringsubgval 44377 Subtraction in the ring of...
linevalexample 44378 The polynomial ` x - 3 ` o...
dmatALTval 44383 The algebra of ` N ` x ` N...
dmatALTbas 44384 The base set of the algebr...
dmatALTbasel 44385 An element of the base set...
dmatbas 44386 The set of all ` N ` x ` N...
lincop 44391 A linear combination as op...
lincval 44392 The value of a linear comb...
dflinc2 44393 Alternative definition of ...
lcoop 44394 A linear combination as op...
lcoval 44395 The value of a linear comb...
lincfsuppcl 44396 A linear combination of ve...
linccl 44397 A linear combination of ve...
lincval0 44398 The value of an empty line...
lincvalsng 44399 The linear combination ove...
lincvalsn 44400 The linear combination ove...
lincvalpr 44401 The linear combination ove...
lincval1 44402 The linear combination ove...
lcosn0 44403 Properties of a linear com...
lincvalsc0 44404 The linear combination whe...
lcoc0 44405 Properties of a linear com...
linc0scn0 44406 If a set contains the zero...
lincdifsn 44407 A vector is a linear combi...
linc1 44408 A vector is a linear combi...
lincellss 44409 A linear combination of a ...
lco0 44410 The set of empty linear co...
lcoel0 44411 The zero vector is always ...
lincsum 44412 The sum of two linear comb...
lincscm 44413 A linear combinations mult...
lincsumcl 44414 The sum of two linear comb...
lincscmcl 44415 The multiplication of a li...
lincsumscmcl 44416 The sum of a linear combin...
lincolss 44417 According to the statement...
ellcoellss 44418 Every linear combination o...
lcoss 44419 A set of vectors of a modu...
lspsslco 44420 Lemma for ~ lspeqlco . (C...
lcosslsp 44421 Lemma for ~ lspeqlco . (C...
lspeqlco 44422 Equivalence of a _span_ of...
rellininds 44426 The class defining the rel...
linindsv 44428 The classes of the module ...
islininds 44429 The property of being a li...
linindsi 44430 The implications of being ...
linindslinci 44431 The implications of being ...
islinindfis 44432 The property of being a li...
islinindfiss 44433 The property of being a li...
linindscl 44434 A linearly independent set...
lindepsnlininds 44435 A linearly dependent subse...
islindeps 44436 The property of being a li...
lincext1 44437 Property 1 of an extension...
lincext2 44438 Property 2 of an extension...
lincext3 44439 Property 3 of an extension...
lindslinindsimp1 44440 Implication 1 for ~ lindsl...
lindslinindimp2lem1 44441 Lemma 1 for ~ lindslininds...
lindslinindimp2lem2 44442 Lemma 2 for ~ lindslininds...
lindslinindimp2lem3 44443 Lemma 3 for ~ lindslininds...
lindslinindimp2lem4 44444 Lemma 4 for ~ lindslininds...
lindslinindsimp2lem5 44445 Lemma 5 for ~ lindslininds...
lindslinindsimp2 44446 Implication 2 for ~ lindsl...
lindslininds 44447 Equivalence of definitions...
linds0 44448 The empty set is always a ...
el0ldep 44449 A set containing the zero ...
el0ldepsnzr 44450 A set containing the zero ...
lindsrng01 44451 Any subset of a module is ...
lindszr 44452 Any subset of a module ove...
snlindsntorlem 44453 Lemma for ~ snlindsntor . ...
snlindsntor 44454 A singleton is linearly in...
ldepsprlem 44455 Lemma for ~ ldepspr . (Co...
ldepspr 44456 If a vector is a scalar mu...
lincresunit3lem3 44457 Lemma 3 for ~ lincresunit3...
lincresunitlem1 44458 Lemma 1 for properties of ...
lincresunitlem2 44459 Lemma for properties of a ...
lincresunit1 44460 Property 1 of a specially ...
lincresunit2 44461 Property 2 of a specially ...
lincresunit3lem1 44462 Lemma 1 for ~ lincresunit3...
lincresunit3lem2 44463 Lemma 2 for ~ lincresunit3...
lincresunit3 44464 Property 3 of a specially ...
lincreslvec3 44465 Property 3 of a specially ...
islindeps2 44466 Conditions for being a lin...
islininds2 44467 Implication of being a lin...
isldepslvec2 44468 Alternative definition of ...
lindssnlvec 44469 A singleton not containing...
lmod1lem1 44470 Lemma 1 for ~ lmod1 . (Co...
lmod1lem2 44471 Lemma 2 for ~ lmod1 . (Co...
lmod1lem3 44472 Lemma 3 for ~ lmod1 . (Co...
lmod1lem4 44473 Lemma 4 for ~ lmod1 . (Co...
lmod1lem5 44474 Lemma 5 for ~ lmod1 . (Co...
lmod1 44475 The (smallest) structure r...
lmod1zr 44476 The (smallest) structure r...
lmod1zrnlvec 44477 There is a (left) module (...
lmodn0 44478 Left modules exist. (Cont...
zlmodzxzequa 44479 Example of an equation wit...
zlmodzxznm 44480 Example of a linearly depe...
zlmodzxzldeplem 44481 A and B are not equal. (C...
zlmodzxzequap 44482 Example of an equation wit...
zlmodzxzldeplem1 44483 Lemma 1 for ~ zlmodzxzldep...
zlmodzxzldeplem2 44484 Lemma 2 for ~ zlmodzxzldep...
zlmodzxzldeplem3 44485 Lemma 3 for ~ zlmodzxzldep...
zlmodzxzldeplem4 44486 Lemma 4 for ~ zlmodzxzldep...
zlmodzxzldep 44487 { A , B } is a linearly de...
ldepsnlinclem1 44488 Lemma 1 for ~ ldepsnlinc ....
ldepsnlinclem2 44489 Lemma 2 for ~ ldepsnlinc ....
lvecpsslmod 44490 The class of all (left) ve...
ldepsnlinc 44491 The reverse implication of...
ldepslinc 44492 For (left) vector spaces, ...
suppdm 44493 If the range of a function...
eluz2cnn0n1 44494 An integer greater than 1 ...
divge1b 44495 The ratio of a real number...
divgt1b 44496 The ratio of a real number...
ltsubaddb 44497 Equivalence for the "less ...
ltsubsubb 44498 Equivalence for the "less ...
ltsubadd2b 44499 Equivalence for the "less ...
divsub1dir 44500 Distribution of division o...
expnegico01 44501 An integer greater than 1 ...
elfzolborelfzop1 44502 An element of a half-open ...
pw2m1lepw2m1 44503 2 to the power of a positi...
zgtp1leeq 44504 If an integer is between a...
flsubz 44505 An integer can be moved in...
fldivmod 44506 Expressing the floor of a ...
mod0mul 44507 If an integer is 0 modulo ...
modn0mul 44508 If an integer is not 0 mod...
m1modmmod 44509 An integer decreased by 1 ...
difmodm1lt 44510 The difference between an ...
nn0onn0ex 44511 For each odd nonnegative i...
nn0enn0ex 44512 For each even nonnegative ...
nnennex 44513 For each even positive int...
nneop 44514 A positive integer is even...
nneom 44515 A positive integer is even...
nn0eo 44516 A nonnegative integer is e...
nnpw2even 44517 2 to the power of a positi...
zefldiv2 44518 The floor of an even integ...
zofldiv2 44519 The floor of an odd intege...
nn0ofldiv2 44520 The floor of an odd nonneg...
flnn0div2ge 44521 The floor of a positive in...
flnn0ohalf 44522 The floor of the half of a...
logcxp0 44523 Logarithm of a complex pow...
regt1loggt0 44524 The natural logarithm for ...
fdivval 44527 The quotient of two functi...
fdivmpt 44528 The quotient of two functi...
fdivmptf 44529 The quotient of two functi...
refdivmptf 44530 The quotient of two functi...
fdivpm 44531 The quotient of two functi...
refdivpm 44532 The quotient of two functi...
fdivmptfv 44533 The function value of a qu...
refdivmptfv 44534 The function value of a qu...
bigoval 44537 Set of functions of order ...
elbigofrcl 44538 Reverse closure of the "bi...
elbigo 44539 Properties of a function o...
elbigo2 44540 Properties of a function o...
elbigo2r 44541 Sufficient condition for a...
elbigof 44542 A function of order G(x) i...
elbigodm 44543 The domain of a function o...
elbigoimp 44544 The defining property of a...
elbigolo1 44545 A function (into the posit...
rege1logbrege0 44546 The general logarithm, wit...
rege1logbzge0 44547 The general logarithm, wit...
fllogbd 44548 A real number is between t...
relogbmulbexp 44549 The logarithm of the produ...
relogbdivb 44550 The logarithm of the quoti...
logbge0b 44551 The logarithm of a number ...
logblt1b 44552 The logarithm of a number ...
fldivexpfllog2 44553 The floor of a positive re...
nnlog2ge0lt1 44554 A positive integer is 1 if...
logbpw2m1 44555 The floor of the binary lo...
fllog2 44556 The floor of the binary lo...
blenval 44559 The binary length of an in...
blen0 44560 The binary length of 0. (...
blenn0 44561 The binary length of a "nu...
blenre 44562 The binary length of a pos...
blennn 44563 The binary length of a pos...
blennnelnn 44564 The binary length of a pos...
blennn0elnn 44565 The binary length of a non...
blenpw2 44566 The binary length of a pow...
blenpw2m1 44567 The binary length of a pow...
nnpw2blen 44568 A positive integer is betw...
nnpw2blenfzo 44569 A positive integer is betw...
nnpw2blenfzo2 44570 A positive integer is eith...
nnpw2pmod 44571 Every positive integer can...
blen1 44572 The binary length of 1. (...
blen2 44573 The binary length of 2. (...
nnpw2p 44574 Every positive integer can...
nnpw2pb 44575 A number is a positive int...
blen1b 44576 The binary length of a non...
blennnt2 44577 The binary length of a pos...
nnolog2flm1 44578 The floor of the binary lo...
blennn0em1 44579 The binary length of the h...
blennngt2o2 44580 The binary length of an od...
blengt1fldiv2p1 44581 The binary length of an in...
blennn0e2 44582 The binary length of an ev...
digfval 44585 Operation to obtain the ` ...
digval 44586 The ` K ` th digit of a no...
digvalnn0 44587 The ` K ` th digit of a no...
nn0digval 44588 The ` K ` th digit of a no...
dignn0fr 44589 The digits of the fraction...
dignn0ldlem 44590 Lemma for ~ dignnld . (Co...
dignnld 44591 The leading digits of a po...
dig2nn0ld 44592 The leading digits of a po...
dig2nn1st 44593 The first (relevant) digit...
dig0 44594 All digits of 0 are 0. (C...
digexp 44595 The ` K ` th digit of a po...
dig1 44596 All but one digits of 1 ar...
0dig1 44597 The ` 0 ` th digit of 1 is...
0dig2pr01 44598 The integers 0 and 1 corre...
dig2nn0 44599 A digit of a nonnegative i...
0dig2nn0e 44600 The last bit of an even in...
0dig2nn0o 44601 The last bit of an odd int...
dig2bits 44602 The ` K ` th digit of a no...
dignn0flhalflem1 44603 Lemma 1 for ~ dignn0flhalf...
dignn0flhalflem2 44604 Lemma 2 for ~ dignn0flhalf...
dignn0ehalf 44605 The digits of the half of ...
dignn0flhalf 44606 The digits of the rounded ...
nn0sumshdiglemA 44607 Lemma for ~ nn0sumshdig (i...
nn0sumshdiglemB 44608 Lemma for ~ nn0sumshdig (i...
nn0sumshdiglem1 44609 Lemma 1 for ~ nn0sumshdig ...
nn0sumshdiglem2 44610 Lemma 2 for ~ nn0sumshdig ...
nn0sumshdig 44611 A nonnegative integer can ...
nn0mulfsum 44612 Trivial algorithm to calcu...
nn0mullong 44613 Standard algorithm (also k...
fv1prop 44614 The function value of unor...
fv2prop 44615 The function value of unor...
submuladdmuld 44616 Transformation of a sum of...
affinecomb1 44617 Combination of two real af...
affinecomb2 44618 Combination of two real af...
affineid 44619 Identity of an affine comb...
1subrec1sub 44620 Subtract the reciprocal of...
resum2sqcl 44621 The sum of two squares of ...
resum2sqgt0 44622 The sum of the square of a...
resum2sqrp 44623 The sum of the square of a...
resum2sqorgt0 44624 The sum of the square of t...
reorelicc 44625 Membership in and outside ...
rrx2pxel 44626 The x-coordinate of a poin...
rrx2pyel 44627 The y-coordinate of a poin...
prelrrx2 44628 An unordered pair of order...
prelrrx2b 44629 An unordered pair of order...
rrx2pnecoorneor 44630 If two different points ` ...
rrx2pnedifcoorneor 44631 If two different points ` ...
rrx2pnedifcoorneorr 44632 If two different points ` ...
rrx2xpref1o 44633 There is a bijection betwe...
rrx2xpreen 44634 The set of points in the t...
rrx2plord 44635 The lexicographical orderi...
rrx2plord1 44636 The lexicographical orderi...
rrx2plord2 44637 The lexicographical orderi...
rrx2plordisom 44638 The set of points in the t...
rrx2plordso 44639 The lexicographical orderi...
ehl2eudisval0 44640 The Euclidean distance of ...
ehl2eudis0lt 44641 An upper bound of the Eucl...
lines 44646 The lines passing through ...
line 44647 The line passing through t...
rrxlines 44648 Definition of lines passin...
rrxline 44649 The line passing through t...
rrxlinesc 44650 Definition of lines passin...
rrxlinec 44651 The line passing through t...
eenglngeehlnmlem1 44652 Lemma 1 for ~ eenglngeehln...
eenglngeehlnmlem2 44653 Lemma 2 for ~ eenglngeehln...
eenglngeehlnm 44654 The line definition in the...
rrx2line 44655 The line passing through t...
rrx2vlinest 44656 The vertical line passing ...
rrx2linest 44657 The line passing through t...
rrx2linesl 44658 The line passing through t...
rrx2linest2 44659 The line passing through t...
elrrx2linest2 44660 The line passing through t...
spheres 44661 The spheres for given cent...
sphere 44662 A sphere with center ` X `...
rrxsphere 44663 The sphere with center ` M...
2sphere 44664 The sphere with center ` M...
2sphere0 44665 The sphere around the orig...
line2ylem 44666 Lemma for ~ line2y . This...
line2 44667 Example for a line ` G ` p...
line2xlem 44668 Lemma for ~ line2x . This...
line2x 44669 Example for a horizontal l...
line2y 44670 Example for a vertical lin...
itsclc0lem1 44671 Lemma for theorems about i...
itsclc0lem2 44672 Lemma for theorems about i...
itsclc0lem3 44673 Lemma for theorems about i...
itscnhlc0yqe 44674 Lemma for ~ itsclc0 . Qua...
itschlc0yqe 44675 Lemma for ~ itsclc0 . Qua...
itsclc0yqe 44676 Lemma for ~ itsclc0 . Qua...
itsclc0yqsollem1 44677 Lemma 1 for ~ itsclc0yqsol...
itsclc0yqsollem2 44678 Lemma 2 for ~ itsclc0yqsol...
itsclc0yqsol 44679 Lemma for ~ itsclc0 . Sol...
itscnhlc0xyqsol 44680 Lemma for ~ itsclc0 . Sol...
itschlc0xyqsol1 44681 Lemma for ~ itsclc0 . Sol...
itschlc0xyqsol 44682 Lemma for ~ itsclc0 . Sol...
itsclc0xyqsol 44683 Lemma for ~ itsclc0 . Sol...
itsclc0xyqsolr 44684 Lemma for ~ itsclc0 . Sol...
itsclc0xyqsolb 44685 Lemma for ~ itsclc0 . Sol...
itsclc0 44686 The intersection points of...
itsclc0b 44687 The intersection points of...
itsclinecirc0 44688 The intersection points of...
itsclinecirc0b 44689 The intersection points of...
itsclinecirc0in 44690 The intersection points of...
itsclquadb 44691 Quadratic equation for the...
itsclquadeu 44692 Quadratic equation for the...
2itscplem1 44693 Lemma 1 for ~ 2itscp . (C...
2itscplem2 44694 Lemma 2 for ~ 2itscp . (C...
2itscplem3 44695 Lemma D for ~ 2itscp . (C...
2itscp 44696 A condition for a quadrati...
itscnhlinecirc02plem1 44697 Lemma 1 for ~ itscnhlineci...
itscnhlinecirc02plem2 44698 Lemma 2 for ~ itscnhlineci...
itscnhlinecirc02plem3 44699 Lemma 3 for ~ itscnhlineci...
itscnhlinecirc02p 44700 Intersection of a nonhoriz...
inlinecirc02plem 44701 Lemma for ~ inlinecirc02p ...
inlinecirc02p 44702 Intersection of a line wit...
inlinecirc02preu 44703 Intersection of a line wit...
nfintd 44704 Bound-variable hypothesis ...
nfiund 44705 Bound-variable hypothesis ...
nfiundg 44706 Bound-variable hypothesis ...
iunord 44707 The indexed union of a col...
iunordi 44708 The indexed union of a col...
spd 44709 Specialization deduction, ...
spcdvw 44710 A version of ~ spcdv where...
tfis2d 44711 Transfinite Induction Sche...
bnd2d 44712 Deduction form of ~ bnd2 ....
dffun3f 44713 Alternate definition of fu...
setrecseq 44716 Equality theorem for set r...
nfsetrecs 44717 Bound-variable hypothesis ...
setrec1lem1 44718 Lemma for ~ setrec1 . Thi...
setrec1lem2 44719 Lemma for ~ setrec1 . If ...
setrec1lem3 44720 Lemma for ~ setrec1 . If ...
setrec1lem4 44721 Lemma for ~ setrec1 . If ...
setrec1 44722 This is the first of two f...
setrec2fun 44723 This is the second of two ...
setrec2lem1 44724 Lemma for ~ setrec2 . The...
setrec2lem2 44725 Lemma for ~ setrec2 . The...
setrec2 44726 This is the second of two ...
setrec2v 44727 Version of ~ setrec2 with ...
setis 44728 Version of ~ setrec2 expre...
elsetrecslem 44729 Lemma for ~ elsetrecs . A...
elsetrecs 44730 A set ` A ` is an element ...
setrecsss 44731 The ` setrecs ` operator r...
setrecsres 44732 A recursively generated cl...
vsetrec 44733 Construct ` _V ` using set...
0setrec 44734 If a function sends the em...
onsetreclem1 44735 Lemma for ~ onsetrec . (C...
onsetreclem2 44736 Lemma for ~ onsetrec . (C...
onsetreclem3 44737 Lemma for ~ onsetrec . (C...
onsetrec 44738 Construct ` On ` using set...
elpglem1 44741 Lemma for ~ elpg . (Contr...
elpglem2 44742 Lemma for ~ elpg . (Contr...
elpglem3 44743 Lemma for ~ elpg . (Contr...
elpg 44744 Membership in the class of...
sbidd 44745 An identity theorem for su...
sbidd-misc 44746 An identity theorem for su...
gte-lte 44751 Simple relationship betwee...
gt-lt 44752 Simple relationship betwee...
gte-lteh 44753 Relationship between ` <_ ...
gt-lth 44754 Relationship between ` < `...
ex-gt 44755 Simple example of ` > ` , ...
ex-gte 44756 Simple example of ` >_ ` ,...
sinhval-named 44763 Value of the named sinh fu...
coshval-named 44764 Value of the named cosh fu...
tanhval-named 44765 Value of the named tanh fu...
sinh-conventional 44766 Conventional definition of...
sinhpcosh 44767 Prove that ` ( sinh `` A )...
secval 44774 Value of the secant functi...
cscval 44775 Value of the cosecant func...
cotval 44776 Value of the cotangent fun...
seccl 44777 The closure of the secant ...
csccl 44778 The closure of the cosecan...
cotcl 44779 The closure of the cotange...
reseccl 44780 The closure of the secant ...
recsccl 44781 The closure of the cosecan...
recotcl 44782 The closure of the cotange...
recsec 44783 The reciprocal of secant i...
reccsc 44784 The reciprocal of cosecant...
reccot 44785 The reciprocal of cotangen...
rectan 44786 The reciprocal of tangent ...
sec0 44787 The value of the secant fu...
onetansqsecsq 44788 Prove the tangent squared ...
cotsqcscsq 44789 Prove the tangent squared ...
ifnmfalse 44790 If A is not a member of B,...
logb2aval 44791 Define the value of the ` ...
comraddi 44798 Commute RHS addition. See...
mvlraddi 44799 Move LHS right addition to...
mvrladdi 44800 Move RHS left addition to ...
assraddsubi 44801 Associate RHS addition-sub...
joinlmuladdmuli 44802 Join AB+CB into (A+C) on L...
joinlmulsubmuld 44803 Join AB-CB into (A-C) on L...
joinlmulsubmuli 44804 Join AB-CB into (A-C) on L...
mvlrmuld 44805 Move LHS right multiplicat...
mvlrmuli 44806 Move LHS right multiplicat...
i2linesi 44807 Solve for the intersection...
i2linesd 44808 Solve for the intersection...
alimp-surprise 44809 Demonstrate that when usin...
alimp-no-surprise 44810 There is no "surprise" in ...
empty-surprise 44811 Demonstrate that when usin...
empty-surprise2 44812 "Prove" that false is true...
eximp-surprise 44813 Show what implication insi...
eximp-surprise2 44814 Show that "there exists" w...
alsconv 44819 There is an equivalence be...
alsi1d 44820 Deduction rule: Given "al...
alsi2d 44821 Deduction rule: Given "al...
alsc1d 44822 Deduction rule: Given "al...
alsc2d 44823 Deduction rule: Given "al...
alscn0d 44824 Deduction rule: Given "al...
alsi-no-surprise 44825 Demonstrate that there is ...
5m4e1 44826 Prove that 5 - 4 = 1. (Co...
2p2ne5 44827 Prove that ` 2 + 2 =/= 5 `...
resolution 44828 Resolution rule. This is ...
testable 44829 In classical logic all wff...
aacllem 44830 Lemma for other theorems a...
amgmwlem 44831 Weighted version of ~ amgm...
amgmlemALT 44832 Alternate proof of ~ amgml...
amgmw2d 44833 Weighted arithmetic-geomet...
young2d 44834 Young's inequality for ` n...
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