Metamath Proof Explorer |
This is the Unicode version. Change to GIF version |
Ref | Description |
idi 1 | (_Note_: This inference r... |
a1ii 2 | (_Note_: This inference r... |
mp2 9 | A double modus ponens infe... |
mp2b 10 | A double modus ponens infe... |
a1i 11 | Inference introducing an a... |
2a1i 12 | Inference introducing two ... |
mp1i 13 | Inference detaching an ant... |
a2i 14 | Inference distributing an ... |
mpd 15 | A modus ponens deduction. ... |
imim2i 16 | Inference adding common an... |
syl 17 | An inference version of th... |
3syl 18 | Inference chaining two syl... |
4syl 19 | Inference chaining three s... |
mpi 20 | A nested modus ponens infe... |
mpisyl 21 | A syllogism combined with ... |
id 22 | Principle of identity. Th... |
idALT 23 | Alternate proof of ~ id . ... |
idd 24 | Principle of identity ~ id... |
a1d 25 | Deduction introducing an e... |
2a1d 26 | Deduction introducing two ... |
a1i13 27 | Add two antecedents to a w... |
2a1 28 | A double form of ~ ax-1 . ... |
a2d 29 | Deduction distributing an ... |
sylcom 30 | Syllogism inference with c... |
syl5com 31 | Syllogism inference with c... |
com12 32 | Inference that swaps (comm... |
syl11 33 | A syllogism inference. Co... |
syl5 34 | A syllogism rule of infere... |
syl6 35 | A syllogism rule of infere... |
syl56 36 | Combine ~ syl5 and ~ syl6 ... |
syl6com 37 | Syllogism inference with c... |
mpcom 38 | Modus ponens inference wit... |
syli 39 | Syllogism inference with c... |
syl2im 40 | Replace two antecedents. ... |
syl2imc 41 | A commuted version of ~ sy... |
pm2.27 42 | This theorem, sometimes ca... |
mpdd 43 | A nested modus ponens dedu... |
mpid 44 | A nested modus ponens dedu... |
mpdi 45 | A nested modus ponens dedu... |
mpii 46 | A doubly nested modus pone... |
syld 47 | Syllogism deduction. Dedu... |
syldc 48 | Syllogism deduction. Comm... |
mp2d 49 | A double modus ponens dedu... |
a1dd 50 | Double deduction introduci... |
2a1dd 51 | Double deduction introduci... |
pm2.43i 52 | Inference absorbing redund... |
pm2.43d 53 | Deduction absorbing redund... |
pm2.43a 54 | Inference absorbing redund... |
pm2.43b 55 | Inference absorbing redund... |
pm2.43 56 | Absorption of redundant an... |
imim2d 57 | Deduction adding nested an... |
imim2 58 | A closed form of syllogism... |
embantd 59 | Deduction embedding an ant... |
3syld 60 | Triple syllogism deduction... |
sylsyld 61 | A double syllogism inferen... |
imim12i 62 | Inference joining two impl... |
imim1i 63 | Inference adding common co... |
imim3i 64 | Inference adding three nes... |
sylc 65 | A syllogism inference comb... |
syl3c 66 | A syllogism inference comb... |
syl6mpi 67 | A syllogism inference. (C... |
mpsyl 68 | Modus ponens combined with... |
mpsylsyld 69 | Modus ponens combined with... |
syl6c 70 | Inference combining ~ syl6... |
syl6ci 71 | A syllogism inference comb... |
syldd 72 | Nested syllogism deduction... |
syl5d 73 | A nested syllogism deducti... |
syl7 74 | A syllogism rule of infere... |
syl6d 75 | A nested syllogism deducti... |
syl8 76 | A syllogism rule of infere... |
syl9 77 | A nested syllogism inferen... |
syl9r 78 | A nested syllogism inferen... |
syl10 79 | A nested syllogism inferen... |
a1ddd 80 | Triple deduction introduci... |
imim12d 81 | Deduction combining antece... |
imim1d 82 | Deduction adding nested co... |
imim1 83 | A closed form of syllogism... |
pm2.83 84 | Theorem *2.83 of [Whitehea... |
peirceroll 85 | Over minimal implicational... |
com23 86 | Commutation of antecedents... |
com3r 87 | Commutation of antecedents... |
com13 88 | Commutation of antecedents... |
com3l 89 | Commutation of antecedents... |
pm2.04 90 | Swap antecedents. Theorem... |
com34 91 | Commutation of antecedents... |
com4l 92 | Commutation of antecedents... |
com4t 93 | Commutation of antecedents... |
com4r 94 | Commutation of antecedents... |
com24 95 | Commutation of antecedents... |
com14 96 | Commutation of antecedents... |
com45 97 | Commutation of antecedents... |
com35 98 | Commutation of antecedents... |
com25 99 | Commutation of antecedents... |
com5l 100 | Commutation of antecedents... |
com15 101 | Commutation of antecedents... |
com52l 102 | Commutation of antecedents... |
com52r 103 | Commutation of antecedents... |
com5r 104 | Commutation of antecedents... |
imim12 105 | Closed form of ~ imim12i a... |
jarr 106 | Elimination of a nested an... |
jarri 107 | Inference associated with ... |
pm2.86d 108 | Deduction associated with ... |
pm2.86 109 | Converse of axiom ~ ax-2 .... |
pm2.86i 110 | Inference associated with ... |
loolin 111 | The Linearity Axiom of the... |
loowoz 112 | An alternate for the Linea... |
con4 113 | Alias for ~ ax-3 to be use... |
con4i 114 | Inference associated with ... |
con4d 115 | Deduction associated with ... |
mt4 116 | The rule of modus tollens.... |
mt4d 117 | Modus tollens deduction. ... |
mt4i 118 | Modus tollens inference. ... |
pm2.21i 119 | A contradiction implies an... |
pm2.24ii 120 | A contradiction implies an... |
pm2.21d 121 | A contradiction implies an... |
pm2.21ddALT 122 | Alternate proof of ~ pm2.2... |
pm2.21 123 | From a wff and its negatio... |
pm2.24 124 | Theorem *2.24 of [Whitehea... |
jarl 125 | Elimination of a nested an... |
jarli 126 | Inference associated with ... |
pm2.18d 127 | Deduction form of the Clav... |
pm2.18 128 | Clavius law, or "consequen... |
pm2.18OLD 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... |
Copyright terms: Public domain | W3C validator |